facta universitatis series: mechanical engineering vol. 16, no 3, 2018, pp. i ii © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd editorial  foreword to the thematic issue: biomedical engineering the readers who are not so familiar with the latest advances in the field of biomedical engineering might be curious to know the reasons why a journal such as facta universitatis: mechanical engineering is devoting the whole issue to the matters in question. yet this is not so difficult to guess knowing that an increasing amount of research is currently being done in biomedical engineering. even this is in itself worth exploring considering so many aspects involved in the given area but for now we would like to stress only two of the more prominent ones. the first reason for increasing research in the field of biomedical engineering is related to the wish to provide for the needs of a rising number of the elderly. median age of the world population has increased from 24 to 30,9 years in the last 30 years 1 . it is predicted that by the year 2050 the number will rise to 36,8 years. this results in a dramatic increase in life expectancy. according to the world population ageing 2017 report 2 , there were 962 million people aged 60 years or over in the whole world, which is an increase of 152% comparing with 383 million of the same population in 1980. this revolutionary change in life expectancy for only 37 years, caused by better nutrition, quality of life and better medicaments, is not in line with the evolutionary changes in the characteristics of human organs that need millennia to adapt. in order to cope up with the problems that could not be resolved by medical means only, health care industry sought help from engineering. the second reason for an increased interest in the field of biomedical engineering is further advancement of those scientific disciplines and technologies that have proven themselves capable of solving the unresolved. a great number of them can be found in the fields of mechanical or electrical engineering, ict, chemical engineering, biomolecular engineering and the like. of those that are close to mechanical engineering it is worth mentioning reverse engineering, additive technologies, nano materials, biomaterials, finite element method, artificial intelligence, robotics and especially nano-robotics. moreover, it is well known that in the seventies of the 20th century there began convergence of two disciplines, namely, of information and telecommunication technologies which in time resulted in the emergence of a new discipline – ict. a similar process is now taking place in the areas of medicine and engineering. it is increasingly difficult to distinguish where medicine ends and engineering begins, and vice versa. therefore, the term biomedical engineering is increasingly used when referring to solving problems in health care. bearing in mind that many solutions are based on machine technologies, we have decided to prepare a thematic issue dedicated to biomedical engineering. 1 median age of the world population from 1990 to 2015 and forecast until 2100, https://www.statista.com/statistics/ 268766/median-age-of-the-world-population/, accessed on dec 7, 2018. 2 united nations, department of economic and social affairs, population division (2017). world population ageing 2017 (st/esa/ser.a/408) https://www.statista.com/statistics/268766/median-age-of-the-world-population/ https://www.statista.com/statistics/268766/median-age-of-the-world-population/ ii foreword to the thematic issue: biomedical engineering the selected state-of-the-art papers presented in this issue illustrate in the best possible way multidisciplinary nature of biomedical engineering. the papers point up the achieved results in various domains of the given field, including biomaterials, reverse engineering, smart devices, additive technologies, specific modeling techniques, to name but a few. each paper, in its own way, contributes to further development of biomedical engineering – an adventure that has just begun while offering a prosperous future. in this sense, this issue gives a large picture of the current state of development while highlighting some important paths for the future development of this inspiring field of research. miroslav trajanović guest editor osiris canciglieri junior guest editor 3198 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 157 165 https://doi.org/10.22190/fume180220002s © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper influence of the cutting parameters on force, moment and surface roughness in the end milling of aluminum 6082-t6 jelena stanojković, miroslav radovanović faculty of mechanical engineering, university of niš, serbia abstract. in this paper the performances, i.e. cutting force, moment and surface roughness, in the end milling of aluminum 6082-t6 with solid carbide end mill were measured and analyzed for different values of the cutting parameters: number of revolutions, feed rate and depth of cut. the cutting force and moment were measured using a kistler piezoelectric dynamometer. surface roughness was measured using a mahr profilometer. the results were analyzed in the minitab 17 software package, in order to determine the influence of the given factors on the performances and modeling of the milling process. key words: cutting force, moment, surface roughness, aluminum 1. introduction the cutting force, moment and surface roughness are the most important indicators of machinability of materials and are very significant for the theory of cutting processes. by separating the cutting layer from the machining surface, the cutting edge of the cutting tool encounters the force. this force removes a layer of material and separates it from the workpiece in the form of chips. depending on the machining conditions, the magnitude of force can vary widely. the force is the main indicator of wear control, the quality of the machined part, and the shape of chips and vibrations. knowing the cutting force enables one, among other things, to determine the energy balance of the machine tool, do the calculation and dimensioning of the elements of the kinematic system of machine tool, do the calculation and dimensioning of the cutting tool, optimize the machining process and enhance the efficiency of the process based on the calculation of the optimal values of the cutting parameters [1]. the force can be determined by measuring the components by the dynamometer. received february 20, 2018 / accepted november 20, 2018 corresponding author: jelena stanojković faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: jstanojkovic@masfak.ni.ac.rs 158 j. stanojković, m. radovanović the force components in the end milling process can be decomposed into [2]: ▪ fc-cutting (tangential) force, ▪ ff-feed (radial) force, and ▪ fp-thrust (axial) force. the machined surface quality is evaluated by surface roughness of the machined part and it is one of the most significant product quality characteristics [3, 4]. surface roughness depends on the cutting conditions, especially the form of cutting tools, tool wear, deposits, vibration, etc. the basic parameters for monitoring surface roughness are: ▪ ra-arithmetic average of the absolute values, ▪ rz-medium unevenness depth, and ▪ rmax-maximum unevenness depth. milling achieves surface roughness from n5 to n12 with the arithmetic average of absolute values ra of 0.4 to 50 μm, respectively. the experimental measurement of cutting force and surface roughness in the milling process has been investigated by a large number of researchers and so has the application of analytical methods for modeling. one of the most important analytical models of the cutting force was created by kienzle and victor [5] in the 1950s. ganesh babu [6] investigated the effects of the cutting parameters (cutting speed, feed rate and depth of cut) on the cutting force during end milling of alsic metal composite material using the response surface methodology (rsm). the experiment was conducted using a four teeth high-speed steel end milling cutter with 10 mm in diameter on the vertical milling machine. the cutting forces were measured with a kistler piezoelectric dynamometer, type 9257b. the cutting forces increased when the depth of cut increased. tsai [7] investigated the influence of the feed per tooth and the tool diameter on the cutting force in milling aluminum 6060-t6. the experiment was conducted using a carbide end milling cutter with two teeth and with diameters of 12, 16, and 20 mm, the spindle speed of 1000 rev/min and the depth of cut of 1 mm, while the feed rate was varied with values of 200, 260, 300, 360 and 400 mm/min. the cutting forces were measured with a kistler piezoelectric dynamometer, type 9257b. the cutting forces were simulated by the recursive least square (rls) method and compared with the experimental values. thamban [8] investigated the machining parameters (spindle speed, feed rate and depth of cut) during end milling aluminum 6061-t6 with coated tungsten carbide and diamond coated end milling cutter with 10mm in diameter. all the components of the cutting forces were measured with a kistler piezoelectric dynamometer, type 9257b. the cutting force was observed to be increasing with depth and feed rate during the end milling for both the cutting tools (coated and uncoated). turgut [9] investigated the effect of machining parameters (cutting speed, feed rate and depth of cut) on the cutting force and surface roughness in the face milling operation of alsic metal matrix composites. the cutting force and surface roughness were measured at cutting speeds of 300, 350, 400 and 450 m/min, feed per tooth of 0.1, 0.15 and 0.20 mm/tooth and depth of cut of 0.5 and 1 mm. the experiment was conducted using coated and uncoated milling cutters of 32 mm in diameter on the vertical machining center johnford vmc-850 without using coolant. the cutting forces were measured with a kistler piezoelectric dynamometer, type 9257b. increasing the feed per tooth and depth of cut increased the cutting force for all the cutting conditions, but increasing cutting speed decreased the cutting force. the best results of the cutting force were obtained with the cutting speed of 400 m/min and the feed rate of 0.1mm/tooth. jeykumar [10] investigated the influence of spindle speed, feed rate and depth of cut on the cutting force, influence of cutting parameters on force, moment and surface roughness in end milling 159 tool wear and surface roughness in the end milling operation of al6061/sic using the response surface methodology. the experiment was conducted using a milling cutter with indexable inserts made of tungsten carbide on the milling machine hmt-fniu. the cutting forces were measured using a kistler piezoelectric dynamometer, type 5070. the experimental results were compared with the mathematical model developed using the response surface methodology. the objective of this study is to determine the influence of the factors (spindle speed, feed rate and depth of cut) on the performances (cutting force, moment and surface roughness) in the end milling of aluminum alloy 6082-t6. the obtained mathematical model facilitates planning the milling process. 2. experimental study in the experimental measurements of the force, moment and surface roughness in the end milling samples of aluminum alloy 6082-t6 were used, with the following dimensions of the workpiece: 50x30x400 mm. the chemical composition of the aluminum alloy 6082-t6 is given in table 1. table 1 chemical composition of al 6082-t6 chemical elements composition [%] al 95.2-98.3 cr 0.25 cu 0.1 fe 0.5 mg 0.6-1.2 mn 0.4-1.0 si 0.7-1.3 ti 0.1 zn 0.2 others 0.15 the cutting tool that was used in the experiment was solid carbide end mill js413160d2sz3.0, manufactured by seco, fig 1. the geometry of the end milling cutter is given in table 2 [11]. fig. 1 solid carbide end mill seco 160 j. stanojković, m. radovanović table 2 cutting geometry of solid carbide end mill diameter-dc[mm] 16 max depth of cut-ap [mm] 32 diameter of tool shrank-dm [mm] 16 length of cutting tooll2 [mm] 100 number of teethzn 3 helix angle- [] 40 cutting tool edge angle- [] 90 rake angle- [] 20 for design of the experiment the selected factors of the milling process were spindle speed (n), feed rate (vf) and depth of cut (ap). they were the main factors that influence the cutting force, moment and surface roughness. the factors were varied on two levels. the levels of factors are shown in table 3. table 3 levels of factors factors levels -1 0 +1 spindle speed-n [rev/min] 320 405 560 feed rate-vf [mm/min] 62 93 175 depth of cut-ap [mm] 0.4 0.7 1.2 the experimental research was carried out on the “prvomajska” ugh universal milling machine, under the laboratory conditions. the force and moment were measured with a kistler piezoelectric dynamometer, type 9123c. the experimental setup is shown in fig. 2. fig. 2 universal milling machine “prvomajska” ugh surface roughness was measured on a mahr profilometer under the laboratory conditions, fig. 3. influence of cutting parameters on force, moment and surface roughness in end milling 161 fig. 3 mahr profilometer for measuring surface roughness the plan of the experiment and measurement results of force fc, moment m and surface roughness ra is shown in table 4. table 4 the plan of the experiment and measurement results no. n [rev/min] vf [mm/min] ap [mm] n [rev/min] vf [mm/min] ap [mm] fc [n] m [nm] ra [μm] 1 -1 -1 -1 320 62 0.4 37.3 0.30 2.907 2 -1 -1 1 320 175 1.2 125.0 0.9 1.600 3 -1 1 -1 320 62 0.4 98.6 0.76 5.191 4 -1 1 1 320 175 1.2 211.8 2.42 4.267 5 1 -1 -1 560 62 0.4 31.6 0.26 1.673 6 1 -1 1 560 175 1.2 76.7 0.69 1.757 7 1 1 -1 560 62 0.4 74.6 0.57 3.348 8 1 1 1 560 175 1.2 174.9 1.45 3.301 9 0 0 0 405 93 0.7 55.0 0.55 1.945 10 0 0 0 405 93 0.7 50.0 0.50 1.976 11 0 0 0 405 93 0.7 60.0 0.58 1.988 3. analysis of results and discussion the cutting force, moment and surface roughness measurement results were analyzed by using the analysis of variance (anova) in the minitab 17 software package. it is clear from the results of anova that depth of cut (ap) and feed rate (vf) are the dominant factors affecting the cutting force. the factors influencing the force are: spindle speed (n), feed rate (vf), followed by spindle speed (n), while the interaction between feed rate and depth of cut (vfap) and that between spindle speed and depth of cut (nap) are also significant. the interaction between spindle speed and feed rate (nvf) and three ways interaction (nvfap) is not significant based on the p-value because its value is greater than 0.1 [12, 13, 14]. f-value is used to determine whether group means are equal, it is just a matter of including the correct variances in the ratio. the analysis of variance for the cutting force is given in table 5. 162 j. stanojković, m. radovanović table 5 analysis of variance for cutting force factors and interactions sum of square mean square f-value p-value n 1650.3 1650.3 66.01 0.015 vf 10461.8 10461.8 418.47 0.002 ap 14990.5 14990.5 599.62 0.002 nvf 6.0 6.0 0.24 0.674 nap 385.0 385.0 15.40 0.059 vfap 814.1 841.1 32.56 0.029 nvfap 110.3 110.3 4.41 0.171 all factors and interactions have a significant effect on the moment. the analysis of variance for the moment is given in table 6. table 6 analysis of variance for moment factors and interactions sum of square mean square f-value p-value n 0.24851 0.24851 152.15 0.007 vf 1.16281 1.16281 711.93 0.001 ap 1.59311 1.59311 975.37 0.001 nvf 0.10351 0.10351 63.38 0.015 nap 0.11281 0.11281 69.07 0.014 vfap 0.28501 0.28501 174.50 0.006 nvfap 0.04651 0.04651 28.48 0.033 all factors and interactions have a significant effect on surface roughness. the analysis of variance for surface roughness is given in table 7. factors and interactions sum of square mean square f-value p-value n 1.8876 1.88762 3834.04 0.000 vf 8.3436 8.34561 16947.08 0.000 ap 0.6017 0.60170 1222.15 0.001 nvf 0.3750 0.37498 761.63 0.001 nap 0.6430 0.64298 1305.98 0.001 vfap 0.0079 0.0794 16.12 0.057 nvfap 0.0330 0.03302 67.08 0.015 based on the obtained data, the influence of the factors on the cutting force, moment and surface roughness in the end milling of aluminum 6082-t6 can be determined. the greatest effect on the cutting force and moment during the end milling of aluminum 6082-t6 with solid carbide end mill has the depth of cut, followed by the feed rate and the spindle speed. by increasing the depth of cut and the feed rate, the cutting force and the moment increase, while increasing the spindle speed causes decrease of the cutting force and moment, fig. 4 (a), (b). the effect on surface roughness during the end milling of aluminum 6082-t6 has the feed rate. by increasing the feed rate, surface roughness increases, while increasing the spindle speed and depth of cut causes decrease of surface roughness, fig. 4 (c). influence of cutting parameters on force, moment and surface roughness in end milling 163 (a) (b) (c) fig. 4 influence of n, vf and ap on: a) cutting force, b) moment and c) surface roughness to simulate the process in terms of the cutting force, a mathematical model was developed using the multiple regression method. the mathematical model is given in eq. (1): ppfpfc naavavnf 94.609.1029.4316.3636.1481.103 −+++−= (1) the coefficient of determination is r2=99.85%, while the adjusted coefficient of determination is r2(adj)=99.26%. the mathematical model of the moment is given in eq. (2): pfpfpfpf anvavnanvavnm 0763.01888.01187.01137.04462.03812.01763.09188.0 −++−−++−= (2) the coefficient of determination for the moment is r2=99.92%, while the adjusted coefficient of determination is r2(adj)=99.58%. 164 j. stanojković, m. radovanović the mathematical model of surface roughness is given by eq. (3): pfpfpfpf anvavnanvavnra 06425.00315.02835.02165.027425.002125.148575.00055.3 −++−−+−= (3) the coefficient of determination of surface roughness is r2=99.99%, while the adjusted coefficient of determination is r2(adj)=99.97%. based on the 3d surface plots of the cutting force, moment and surface roughness one can study the relations among the influencing factors during the end milling of aluminum alloy 6082-t6, fig. 5. (a) (b) (c) fig. 5 3d surface plots of a) cutting force, b) moment and c) surface roughness 4. conclusions investigating the cutting force, moment and surface roughness is important for the process of milling. the cutting force and moment are the basic criteria for evaluation of machinability, while surface roughness is the basic criteria for the quality of parts. the knowledge of these performances facilitates the effective planning of the machining process. the measurement of the cutting force, moment and surface roughness was carried out for different values of spindle speed (320.405 and 560 rev/min), feed rate (62, 93 and 175 mm/min) and depth of cut (0.4, 0.7 and 1.2 mm) in the laboratory conditions on a universal milling machine during the milling of aluminum alloy 6082-t6 with solid carbide end mill, manufactured by seco, without cooling. the cutting force and moment influence of cutting parameters on force, moment and surface roughness in end milling 165 were measured using a kistler piezoelectric dynamometer, while surface roughness was measured on a mahr profilometer. based on the experimental results of the cutting force, moment and surface roughness, an analysis was performed in the minitab 17 software package. in the end milling of aluminum 6082-t6, the greatest impact on the cutting force and moment is achieved by the depth of cut, followed by the feed rate and the spindle speed. by increasing the cutting depth and the feed rate the main cutting force and moment grow as well, while increasing the spindle speed reduces them. on the other hand, the feed rate has the greatest influence on surface roughness. by increasing the feed rate, surface roughness increases as well, while increasing the speed and depth of cut causes decrease of surface roughness, i.e. a better quality of processing is achieved. acknowledgements: the paper is a part of the research done within the project tr35034. the authors would like to thank to the ministry of education and science, republic of serbia references 1. kovač, p., savković, b., mijić, a., sekulić, m., 2011, analytical and experimental study of cutting force components in face milling, journal of production engineering, 3(1), pp. 15-18. 2. madić, m., radovanović, m., 2011, methodology of developing optimal bp-ann model for the prediction of cutting force in turning using earlz stopping method, facta univesitastis-series mechanical engineering, 9(1), pp. 21-32. 3. ribero, j., lopes, h., queijo, l., figueiredo, d., 2017, optimization of cutting parameters to minimize the surface roughness in the end milling process using the taguchi method, periodica polytechnica, mechanical engineering 61(1), pp. 30-35. 4. maheswara, r., venkatasubbaiah, k., 2016, optimization of surface roughness in cnc turning using taguchi method and anova, international journal of advances scientice and tehnology, 93, pp. 1-14. 5. kienzle, o., victor, h., 1952, determination of forces and productivity of tools used for 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jeyakumar, s., marimuthu, k., ramachandran, t., 2013, prediction of cutting force, tool wear and surface roughness of al 6061/sic composite for end milling operation using rsm, journal of mechanical science and techology, 27(9), pp. 2813-2822. 11. stanojković, j., radovanović, m., 2017, selection of solid carbide end mill for machining aluminum 6082-t6 using mcdm method, u.p.b. sci. bull. series d, 79(1), pp. 175-184. 12. khan, r. m., 2013, problem solving and data analysis using minitab: a clear and easy guide to six sigma methodology, west sussex, wiley, united kingdom. 13. yahya, e., ding, g., qin, s., 2015, optimization of machining parameters based on surface roughness prediction for aa6061 using response surface method, american journal of science and technology, 2(5), pp. 220-231. 14. hamidon, r., adesta, e., muhammad, r., yuhan suprianto, m., 2016. influence of cutting parameters on cutting force and cutting temperature during pocketing operations, arpn journal of engineering and applied science, 11(1), pp. 453-459. 6298 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 167 176 https://doi.org/10.22190/fume201004026r © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper design and calibration of the system supervising belt tension and wear in an industrial feeder tomasz ryba, miroslaw rucki, zbigniew siemiatkowski, damian bzinkowski, michal solecki kazimierz pulaski university of technology and humanities in radom, poland abstract. in the paper, the issue of the supervision of belt tension and wear in industrial feeder is addressed. the designed system is based on strain gauges that are built into the roller and are subject to the belt pressure at each revolution. in order to assess the effectiveness of this system, calibration and uncertainty analysis was performed. as a result, it was demonstrated that the main source of uncertainty was the function of approximation, while the others were orders of magnitude smaller. the final function provided results with accuracy of ca. 10% of actually measured value, which was assumed to be a good result for this particular industrial application. key words: industrial feeder, belt, wear, measurement, calibration 1. introduction in the context of smart factories and “industry 4.0,” preventive maintenance based on the concept of flexible and diverse maintenance levels is widely introduced [1, 2]. it is highly desirable to perform condition-based maintenance capable of identifying fault monitoring actual condition of the system obtained from in-situation, no-invasive tests and measurements [3]. the inspection workload for preventive maintenance of a largescale distribution facility is enormous because it encompasses a large amount of equipment such as conveyors and sorters [4]. implementation of the cyber-physical systems for performance monitoring in production intralogistics requires reliable data about actual state of the conveyors and their elements [5]. however, in the area of industrial belt feeders, no such a system for in-situ tension monitoring was proposed so far. extensive theoretical background for work conditions and calculations of the belt feeders can be found in the literature [6] including 3d models of the tensions [7]. analysis of various internal structures and the type of the material falling onto received october 04, 2020 / accepted february 18, 2021 corresponding author: miroslaw rucki faculty of mechanical engineering, kazimierz pulaski university of technology and humanities in radom, krasickiego str. 54, 26-600 radom, poland e-mail: m.rucki@uthrad.pl 168 t. ryba, m. rucki, z. siemiatkowski, d. bzinkowski, m. solecki a conveyor belt and effects thereof on the incurred damage enabled the damage classification [8]. it was demonstrated that the operating characteristics could be predicted based on experimental measurements, with a specific example focusing on the prediction of the contact force – tension force relationship [9]. there are also propositions concerning diagnostics during exploitation, such as a non-invasive system able to monitor the joints of the monitored belt in order to detect critical elongation [10]. another project involved steel ropes inside the belt material, so that the magnetic field could be measured directly on the feeder [11]. some other solutions propose the belts with built-in tensors, but after the belt is damaged or worn out, the entire tensor system is lost with no possibility of further use. recently, the test equipment for real time belts tension detection during the conveyor work was proposed [12] and patented [13]. it was necessary, however, to prove its capability to detect tension releases caused by wear and damages of the belt. for that purpose, the calibration procedure was performed using a special intermediate device described below. 2. device concept and calibration issue the essence of the novel measurement system supervising belt tension and wear is presented in fig. 1. application of strain gauges directly on the roller made it possible to obtain data concerning the belt conditions during its work. transducers are placed inside the empty roll subjected to the load-dependent on the belt tension. fig. 1 scheme of the measurement system the strain gauges of the type cp 152 ns (ø16) were chosen because of low costs, availability in the market, flexibility in the applications, good dynamic characteristics, and a large enough measuring range. their nominal operating voltage was 1.5 [v] in the temperature range from -40 up to 80 [°c]. initial tests provided promising results since the strain gauges placed along the roller gave the measurement results according to the actual pressure distribution. namely, when the belt was under asymmetrical load, one gauge shows higher tension, while the other detected slight release. it is shown in fig. 2, view from the top, with the load placed closer to gauge t2, but in the actual scale this slight force decrease is not clearly distinguishable. design and calibration of the system supervising belt tension and wear in an industrial feeder 169 fig. 2 load registered by two gauges conditions of the belt tension monitoring through measurement of its pressure on the roller are dynamic. as a result, registered pressure reveals an undesirable peak in the very first moment of contact between the belt and the strain gauge, as shown in fig. 3. even though this peak is quite repeatable, it makes difficult to perform the correct analysis of the obtained measurement signal. the nominal strength of belt kn is calculated from the following equation [6]: 1000 max n e b sr k = k k b    (1) where srmax is the maximal force in the belt during startup [n], b is belt width [m], ke is exploitation safety factor, and kb is the factor of tension concentration in joints. hence, the maximal pressure registered with the measurement system should not be a result of gauge excitation. having noticed this feature generated by the dynamic mechanical contact between the belt and the strain gauge, it was decided to modify the fixation of the gauge. it was found necessary to perform calibration of the strain gauge as part of the system, as it works in real conditions. after modification, however, another issue emerged, namely, of how to ensure steady distribution of the pressure on the calibrated strain gauge surface, with stable and repeatable fixation. fig. 3 signal from the strain gauge obtained during rotation of the roller 170 t. ryba, m. rucki, z. siemiatkowski, d. bzinkowski, m. solecki 3. calibration apparatus and conditions to perform the calibration procedure correctly, novel instrumentation was designed. its aim was to ensure a repeatable contact area between the reference mass standard and the strain gauge surface. stable vertical movement transmitting the mass on the gauge surface was achieved through two precise shafts ø10 fixed in the lower body, with linear bearings denoted lm10uu. fig. 4a presents the designed calibration apparatus and 4b the intersection of its main part. the numbers denote as follows: 1 – calibrated strain gauge, 2 – upper body of calibration instrumentation, 3 – reference mass standard, 4 – shaft with rounded upper end, 5 – round nut, 6 – bolt m5×20, 7 – upper body. upper surface of body 7 was shaped in a special way, enabling steady distribution of the reference weight during calibration. fig. 4 concept of the calibration instrumentation (description in the text) to project and produce the instrumentation, solidworks software was used. the models were exported to *.stl files in order to apply additive manufacturing (am) technology. am is a very useful technology for fabricating complex shape details out of polymers [14] and even for very strong elements [15]. a method known as fdm (fused deposition modeling) was applied, where molten fibers are extruded and deposited to print stacks of 2d crosssections and finally form complex 3d products [16]. 3d printer type 4max was used, with working space 220×220×300 [mm] (width × length × height). the material was pla fiber of diameter 1.75 [mm], deposition was performed at temperature 225 [°c], grid method, printing speed 50 [m/s]. fig. 5, left, presents the solidworks model, and fig. 5, right, photo of printed and assembled instrumentation. fig. 5 calibration instrumentation model and its realization design and calibration of the system supervising belt tension and wear in an industrial feeder 171 the calibration procedure was performed in the laboratory of the radwag company in radom, poland. its uncertainty is affected mainly by the following factors: ▪ uncertainty of weights, ▪ uncertainty of reading resolution and approximation error, ▪ uncertainty of environmental conditions. thus, mass indication mi of the strain gauge and its uncertainty can be expressed with the equation as follows: ( ) ( ) ( ) ( ) ( )i x r r app app s s b bm ± k u m = m ± k u m +δm ±k u δm +δm ±k u δm +δm ±k u δm     , (2) where i denotes the nominal weight actually measured, mr is the reference weight, δmapp is the approximation error, δms is the result of stochastic distribution in repeated measurements, and δmb is the buoyancy effect, k is the coverage factor, and u(x) is the respective standard uncertainty of each measured value. air buoyancy is equal to the weight of the displaced air [17]: b a m f = v ρ g = g ρ    , (3) weights class e2 was used, according to the international recommendation oiml r 111-1 [17]. these weights are generally intended for use in the verification or calibration of weighing instruments of special accuracy class i. reading the resolution of the voltage signal from the strain gauge is 20 digits, which is not necessary due to measurement uncertainty and repeatability. environmental conditions were monitored during each repetition, and respective values of temperature, humidity, and atmospheric pressure registered at the start of measurement and at its end are shown in fig. 6. fig. 6 environmental conditions during calibration 172 t. ryba, m. rucki, z. siemiatkowski, d. bzinkowski, m. solecki due to the very stable conditions and from a practical perspective, the buoyancy effect was found negligibly small. the maximum permissible errors (mpe) of e2 class weights with proper certificates are collected in table 1. table 1 maximum permissible errors of the applied weights nominal weight mpe 0.5 kg ±0.8 mg 1.0 kg ±1.6 mg 2.0 kg ±3.0 mg 5.0 kg ±8.0 mg 10.0 kg ±16.0 mg under the load, the strain gauges changed their electrical conductance, which was indicated in siemens [s = ω−1]. the calibration procedure was repeated 10 times for each of three strain gauges thus enabling the statistical analysis of the obtained results. during each repetition, 100 samples were registered. examples of histograms shown in fig. 7 demonstrate that in each repetition, gaussian statistics can be applied. based on normal distribution, type a uncertainty [18] was calculated for each gauge, and the calibration curves were appointed. results are presented and discussed in the next section. fig. 7 examples of the obtained histograms for two repetitions design and calibration of the system supervising belt tension and wear in an industrial feeder 173 4. results and discussion fig. 8 presents the results of 10 repetitions, each with 100 samples registered, for the strain gauge no. 1 conductance indications under a load of 0.5 kg. it was typical for every repeated procedure that the subsequent samples comprised almost straight lines, while the next repetitions provided similar lines at a different level, with different average, but with a quite similar standard deviation below 0.8, as can be seen in table 2. scattering of the average values from 10 repetitions appeared smaller for larger weights. fig. 8 measurement results for the strain gauge no. 1 under load of 0.5 kg table 2 conductance statistics for 10 repetitions, strain gauge no. 1, nominal load 0.5 kg repetition no. 1 2 3 4 5 6 7 8 9 10 average 0.5m [μs] average 0.5m 66.5 98.5 43.6 99.9 99.3 104.8 101.4 118.3 115.7 100.5 94.9 min [μs] 65.2 96.4 42.7 97.9 97.7 102.9 99.8 116.1 111.2 98.6 43.6 max [μs] 67.5 99.8 44.2 101.5 100.7 106.7 103.0 120.0 117.3 101.8 115.7 range r [μs] 2.2 3.4 1.5 3.6 3.0 3.8 3.2 3.8 6.1 3.1 77.3 std.dev. s [μs] 0.471 0.510 0.332 0.677 0.598 0.624 0.653 0.738 0.794 0.568 it can be seen that the dispersion of the results due to the stochastic distribution in repeatability conditions is several orders of magnitude higher than that of other uncertainty sources specified in eq. (2). thus, the type a uncertainty based on the statistical analysis seems to be the most appropriate methodology. it is noteworthy, however, that 10 repetitions allow for a decrease of uncertainty span, as follows [19]: ( ) ( ) u x u x n = , (4) where n is the number of repetitions, here n = 10. thus, the standard uncertainty can be u( m 0.5) = 0.25 instead of u(m0.5) = 0.794, and expanded uncertainty u0.99 = 0.75 [μs]. 174 t. ryba, m. rucki, z. siemiatkowski, d. bzinkowski, m. solecki coverage factor for level of confidence 99% is assumed k = 3. similarly, uncertainty was estimated for each measurement. table 3 uncertainties for each reference weight mr0.5=0.5 kg mr1=1 kg mr2=2 kg mr5=5 kg mr10=10 kg average xm [μs] 94.85 144.78 207.61 352.23 459.49 min [μs] 42.67 94.86 168.77 292.34 417.17 max[μs] 119.98 182.09 251.59 403.81 505.02 range r [μs] 77.31 87.23 82.82 111.48 87.85 std.dev. smax [μs] 0.79 1.02 2.17 2.18 3.16 mx ± u0.99 [μs] 94.85±0.75 144.78±0.97 207.61±2.05 352.23±2.07 459.49±2.99 approximation of the obtained results led to the following conclusions. linear function, which would be the most desirable, provided linearity error ca. 51 [μs] for the strain gauge conductance output 352 [μs], so that the approximation error was almost 15%. so it was found necessary to approximate the function with a polynomial, as follows: y = -3.7x2+77x+65. (5) this function provided a maximal approximation error of 7.73 [μs] for the strain gauge conductance output 94.85 [μs], so that percentage was ca. 8%. both approximation graphs together with calibration points are shown in fig. 9. fig. 9 approximation functions and calibration points the aforementioned results demonstrated that all the uncertainty components are negligibly small compared to the function approximation error. thus, the latter can be considered the main uncertainty source for each measurement result obtained from the strain gauges during the belt tension measurement. design and calibration of the system supervising belt tension and wear in an industrial feeder 175 for practical reasons, indications in conductance units [s] should be recalculated into respective force values [n]. the formula derived from the experimental data presented above is as follows: y = 136x1.876, (6) where x is the conductance [s], and y is the belt pressure on roller [n]. table 4 presents the results of calibration and approximation. table 4 uncertainties for each reference weight load [kg] load [n] conductance [μs] resistance [ω] load indication [n] approximation error [n] 0.50 4.90 94.85 10542.96 4.60 0.30 1.00 9.81 144.78 6907.03 10.16 -0.35 2.00 19.61 207.61 4816.72 19.95 -0.33 5.00 49.03 352.23 2839.05 53.68 -4.65 10.00 98.07 459.49 2176.33 88.31 9.76 it is seen from table 4 that the approximation error is below 10% of the actually measured value, which is highly satisfactory for this application aiming at the belt tension monitoring in industrial conditions. 5. conclusions the research studies and analysis demonstrated that the main source of uncertainty in the calibration procedure was the function of approximation, while the others were orders of magnitude smaller. registered values revealed distribution fairly close to the expected gaussian one, so that type a uncertainty could be estimated from a number of measurements in repeatability conditions. application of mean value from 10 repetitions made it possible to reduce final uncertainty even more, so that expanded uncertainty of conductance was u0.99 = 0.75 [μs], with coverage factor k = 3 for the level of confidence 99%. the maximal approximation error, however, was 7.73 [μs] for the strain gauge conductance output 94.85 [μs], i.e. ca. 8%. when the conductance was calculated to force values, an approximation error below 10% was obtained. this result was found very good due to the industrial application of the analyzed system. acknowledgements: the authors express their gratitude to the radwag wagi elektroniczne, radom, poland, for the possibility to perform measurements in excellent laboratory conditions. references 1. miyata, h.h., nagano, m.s., gupta, j.n.d., 2019, integrating preventive maintenance activities to the no-wait flow shop scheduling problem with dependent-sequence setup times and makespan minimization, computers & industrial engineering, 135, pp. 79-104. 2. ruiz-sarmiento, j.r., monroy, j., moreno, f.a., galindo, c., bpnelo, j.m., gonzalez-jimenez, j., 2020, a predictive model for the maintenance of industrial machinery in the context of industry 4.0, engineering applications of artificial intelligence, 87, article 103289. 176 t. ryba, m. rucki, z. siemiatkowski, d. bzinkowski, m. solecki 3. lin, d., jin, b., chang, d., 2020, a pso approach for the integrated maintenance model, reliability engineering & system safety, 193, article 106625. 4. kuboki, n., takata, s., 2019, selecting the optimum inspection method for preventive maintenance, procedia cirp, 80, pp. 512-517. 5. mörth, o., emmanouilidis, ch., hafner, n., schadler, m., 2020, cyber-physical systems for performance monitoring in production intralogistics, computers & industrial engineering, 142, article 106333. 6. gładysewicz, l., belt feeders: theory and calculations, wroclaw university of technology, wrocław (in polish). 7. fedorko, g., ivančo, v., 2012, analysis of force ratios in conveyor belt of classic belt conveyor, procedia engineering, 48, pp. 123-128. 8. andrejiova, m., grincova, a., marasova, d., 2019, failure analysis of the rubber-textile conveyor belts using classification models, engineering failure analysis, 101, pp. 407-417. 9. molnár, v., fedorko, g., homolka, l., michalik, p., tučková, z., 2019, utilisation of measurements to predict the relationship between contact forces on the pipe conveyor idler rollers and the tension force of the conveyor belt, measurement, 136, pp. 735-744. 10. mazurkiewicz, d., 2011, study on the chosen aspects of maintenance diagnostics of belt conveyors, lublin university of technology, lublin (in polish). 11. nowak, r., grzyb, k., 2008, monitoring and laboratory research on the diagnostics process of feeding belts with steel rods, proc. 16th international symposium “100 lat w służbie polskiego przemysłu wydobywczego”, zakopane, poland, pp. 39-54. 12. ryba, t., 2019, overview of the rubber belts tension test methods in the close transport conveyors, mechanik, 3, pp. 210-212. 13. ryba, t., 2020, patent application no. p.432900, warsaw, poland. 14. daminabo, s. c., goel, s., grammatikos, s. a., nezhad, h. y., thakur, v. k., 2020, fused deposition modeling-based additive manufacturing (3d printing): techniques for polymer material systems, materials today chemistry, 16, article 100248. 15. tyczynski, p., siemiatkowski, z., rucki, m., analysis of the drill base body fabricated with additive manufacturing technology, proceedings of 18th international euspen conference & exhibition, 4-8 june 2018, venice, italy, pp. 287-288 16. kazmer, d., 2017, applied plastics engineering handbook, second edition, elsevier, amsterdam. 17. international organization of legal metrology, 2004, oiml r 111-1: 2004, international recommendation: weights of classes e1, e2, f1, f2, m1, m1–2, m2, m2–3 and m3, part 1: metrological and technical requirements, grande imprimerie de troyes, france. 18. ea-4/02 m: 2013. evaluation of the uncertainty of measurement in calibration. 19. jcgm 100:2008. evaluation of measurement data — guide to the expression of uncertainty in measurement. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 335 343 10.22190/fume191118019l © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper on the effect of the side flow of 316l stainless steel in the finish turning process under dry conditions kamil leksycki, eugene feldshtein, michał ociepa faculty of mechanical engineering, university of zielona gora, poland abstract. the article presents the results of the research on the plastic flow in the finish turning of 316l (x2crnimo17-12-2) stainless steel under dry cutting conditions. the steel was turned at variable cutting speeds and a constant depth of cut. the investigations were based on the parameter space investigation method (psi) which allowed minimizing the number of test points. it was observed that the phenomenon of slide flow occurred in the range of cutting speeds and feed rates under examination and its intensity depended on the values of these parameters. the phenomenon was more intense in the range of medium and higher cutting speeds and lower feed rates. the side flow results in significant changes between the real and theoretical values of roughness parameter rz, which range from 40% up to even 330%. key words: side flow, stainless steel, finish turning, dry cutting, surface layer 1. introduction due to its favorable mechanical and performance properties, 316l (x2crnimo17-12-2) stainless steel is one of the materials most frequently used in the manufacture of medical products as described by fazel-rezai [1] and ramsden et al. [2]. ristić et al. [3] as well as singh et al. [4] informed that in many cases it is an alternative for titanium alloys, which are widely used in the production of medical devices. supriya et al. [5] revealed that 316l stainless steel is characterized by a high cracking strength and fatigue strength index as well as by a high corrosion resistance. according wegener et al. [6], this material is included among materials difficult to process on account of low quality of the surface obtained, quick wear of the tool, low efficiency and high costs of machining. this is caused by a high temperature in the cutting zone as described by mia et al. [7]. in order to reduce the temperature, maruda et al. [8] performed machining with the use of cutting fluids, which, received november 18 , 2019 / accepted april 02, 2020 corresponding author: kamil leksycki faculty of mechanical engineering, university of zielona gora, 4 prof. z. szafrana street, 65-516 zielona gora, poland e-mail: k.leksycki@ibem.uz.zgora.pl 336 k. leksycki, e. feldshtein, m. ociepa however, have a negative impact on the human health and the environment. it is for this reason, according to bagaber et al. [9], that dry cutting should be considered an optimal solution. however, dry cutting entails threats in the form of disturbances which cause imperfections that unfavorably affect the surface integrity as described by acayaba et al. [10]. suresh et al. [11] found that the side flow is one of such tribological disturbances which occur in the cutting zone. it depends on numerous factors, of which the feed rate, nose radius and tool wear are of major importance [12, 13, 14]. pekelharing, and gieszen [15] were among the first to identify the phenomenon of side flow and described its negative impact on the machined surface quality. according to coelho et al. [16], side flow results in deformations of a part of the undeformed chip in a direction opposite to feed f and is caused by direct compression of the material by the minor flank and the work surface, when thickness h of the undeformed chip is smaller than socalled minimum undeformed chip thickness hmin. on the other hand, the minimum thickness of the undeformed chip depends on the values of feed rate f and cutting speed vc, the hardness and structure of the material being machined and the geometry of the cutting wedge. as stated in grzesik [17], the phenomenon of side flow can also be caused by the outflow of the material plasticized under high temperatures and high pressure values generated in the cutting zone and caused by a worn out cutting wedge. sivaiah and chakradhar [18] investigated the influence of cryogenic cooling on 17-4ph stainless steel turning was compared with dry, wet and mql conditions of machining. the following parameters were applied: vc = 78.5 m/min, f = 0.147 mm/rev and ap = 0.20 – 1 mm. increased depth of cut resulted in an increment of surface roughness under all cooling conditions due to a rise in temperature in the cutting zone which brings about a higher tool wear. side flow was observed both at lower and higher depths of cut. fernández-abia et al. [19] analyzed the issues related to high performance machining of austenitic stainless steels and the phenomenon of side flow. it was determined that machining a material at a higher cutting speed resulted in the formation of side flow. a complete plasticization of the material being machined was reported to be the cause of side flow. as a result of plasticization the material partly flows from the major cutting edge towards the minor one. zou et al. [20] analyzed the mechanism of tool wear in the process of the finish turning of stainless steels, i.e., 17-4 ph martensitic and 321 austenitic steels. for both steels the following parameters of cutting were used: f = 0.10 mm/rev and ap = 0.30 – 0.35 mm, whereas for 17-4 ph stainless steel: vc = 350 – 400 m/min, and for 321 stainless steel: vc = 300 – 350 m/min. for both stainless steels the phenomenon of slide flow was observed, in particular for 321 stainless steel. similarly, fernández-abia et al. [19] determined that the occurrence of side flow was related to the action of high pressure and temperature in the contact zone of the chip with the cutting wedge, which as a result cause a complete plasticization of the material. plasticization generates a partial flow of the material from the major to the minor cutting edge and its adhesive bonding to the freshly machined surface of a workpiece. according to the authors, a partial diffusion of 321 steel to the cutting wedge material takes place simultaneously with the side flow, but this opinion has not been verified by tests. liew et al. [21] investigated wear of tools made of pcbn in the turning process of aisi 420 modified stainless steel at low cutting speeds. it was determined that the phenomenon of side flow occurred at cutting speeds of 44 m/min and 130 m/min. as reported by kishawy and elbestawi [22], the analysis of the impact of process parameters on the effect of side flow of 316l stainless steel in the finish turning process under dry conditions 337 on the phenomenon of side flow of a material being hard turned revealed a strong dependence of side flow on the material being machined on the cutting wedge features. an increase in the nose radius and its wear enhances side flow of the material, whereas the feed rate only slightly affects the intensity of the phenomenon. the aim of the investigations presented herein is to determine the impact of the cutting speed and feed rate on the intensity of side flow in the finish turning process of 316l stainless steel under dry conditions. 2. conditions 316l (en x2crnimo17-12-2) stainless steel of the chemical composition as illustrated in table 1 was machined. table 1 chemical composition of en x2crnimo17-12-2 (en 10088-:2014 standard) element c cr fe mi mo ni p si s % ≤ 0.030 16.0 − 18.0 61.9 − 72.0 ≤ 2.0 2.00 − 3.00 10.00 − 14.00 ≤ 0.045 ≤ 1.0 ≤ 0.030 the tests were performed on the cnc lathe, type ctx 510, manufactured by dmg mori. a cutting tool with coroturn sdjcr 2525m11 holder and coroturn dcmx 11 t3 04-wm 1115 insert made of a cemented carbide type gc1115 with (ti,al)n+(al,cr)2o3 coating deposited by the pvd method. the geometry of the cutting edge was as follows: tool cutting edge angle r = 93°, tool rake angle  = 18°, tool clearance angle  = 7°, nose radius r = 0.4 mm, land width of the face bn = 0.1 mm. the cutting data were as follows: cutting speed in the range of 150 − 500 m/min, the feed rate in the range of 0.05 − 0.4 mm/rev, and constant depth of cut equal to 0.5 mm. these data correspond to the finish turning conditions. feifei et al. [23] claim that the plastic side flow contributes to the increment of the rz (maximum height of the roughness profile) roughness parameter when elasto-plastic materials are machined. it was determined that when a material is turned at a feed rate f with a tool which has nose radius r, the real value of parameter rz can be significantly higher than the rzt value calculated according to the well-known equation [24]: rzt = f 2/(8r) (1) where f is the feed and r is the nose radius. the tests were planned on the basis of the parameter space investigation method (psi). the method allows planning an experiment with minimizing the number of test points, which are located in determined places in a multi-dimensional space, as statnikov and matusov described in [25]. it means that the test points projections on the x1, x2 , etc. axes are located at the same distance from one another (fig. 1). the method has been successfully used for the investigation of cutting processes by maruda et al. [8, 26]. the texture of the machined surface was analyzed with the use of an alicona infinite sl optical measuring system and the results of measurements were obtained using the iflaboratory measurement module software package. 338 k. leksycki, e. feldshtein, m. ociepa fig. 1 location of test points in a multi-dimensional space in accordance with the psi method table 2 coordinates of test points according to the psi method factors test points 1 2 3 4 5 6 7 x1 0.5000 0.2500 0.7500 0.8750 0.3750 0.6250 0.1250 x2 0.5000 0.7500 0.2500 0.6250 0.1250 0.3750 0.8750 3. results a scheme of the side flow formation is presented in fig. 2. when very thin chips are formed, the conditions which ensure a high or complete plasticization of the chip material are created. fig. 2 the scheme of the side flow formation (based on [17]) fig. 3 presents images of machined surfaces obtained for 7 test points in accordance with the psi method. side flow of the material was observed at each of the points. the intensity of the phenomenon was higher within the range of medium and higher cutting speeds and at lower feed rates (red color), whereas side flow was less intense in the on the effect of side flow of 316l stainless steel in the finish turning process under dry conditions 339 whole range of cutting speeds, but at higher feed rates, which were within the range under examination (blue color). fig. 3 2d images of 316l stainless steel machined surfaces fig. 4 presents 3d images of machined surfaces obtained for 7 test points in accordance with the psi method. in the zone of an intense plastic side flow (red color) for higher cutting speeds and lower feed rates an irregular distribution of single burrs and a few clear areas of the 340 k. leksycki, e. feldshtein, m. ociepa flowing material were observed, whereas within the range of medium cutting speeds and lower feed rates side flow was regular on the whole length of the machined surface. in turn, in the zone of a slight intensity of plastic side flow (blue color) within the range of medium and higher cutting speeds and medium feed rates minimal and irregular side flow was observed, whereas within the range of lower cutting speeds and higher feed rates minimal side flow was observed on the whole length of the machined surface. fig. 4 3d images of 316l stainless steel machined surfaces fig. 5 illustrates surface roughness profiles and values of surface roughness parameter rz obtained at individual test points in accordance with the psi method. lower values of surface roughness parameter rz, from 3.58 m to 4.42 m were obtained at points 3,5,6 (blue color), medium, from 6.38 m to 6.69 m, at points 1 and 4 (green color), and maximum values, from 9.20 m to 12.2 m, were obtained at points 2 and 7 (red color). thus, it follows that higher and medium cutting speeds from the range under examination and lower feed rates ensure a decrease of surface roughness parameter rz, whereas higher feed rates cause its increase. on the effect of side flow of 316l stainless steel in the finish turning process under dry conditions 341 fig. 5 surface roughness profiles of 316l stainless steel the real and theoretical values of surface roughness parameter rz were also analyzed. fig. 6 illustrates the percentage differences between them. significant differences between the real and theoretical values of roughness parameter rz were observed. within the range of higher cutting speeds and lower feed rates a decrease of the order of 40% in the real value of roughness parameter rz was obtained. in the other ranges of machining parameters an increase from 40% up to 330% was achieved. 342 k. leksycki, e. feldshtein, m. ociepa fig. 6 percentage differences between real values of surface roughness parameter rz obtained after finish turning of 316l stainless steel compared to theoretical (t) values rzt 4. conclusions the paper presents the results of research on plastic side flow of 316l stainless steel (en x2crnimo17-12-2) in the finish turning process under dry cutting conditions. the effect of the cutting speed and feed rate on the intensity of side flow was analyzed. the following conclusions are made: ▪ plastic side flow of the material occurs within the whole range of cutting speeds and feed rates under examination. ▪ in the cutting speed ranges from 280-420 m/min and feed rates 0.1-0.2 mm/rev a higher intensity of plastic side flow was observed, whereas lower was obtained in the range of feed rates 0.1-0.2 mm/rev at cutting speeds 190-460 m/min. ▪ surface roughness parameter rz depends on the values of cutting speed and feed rate. surface roughness parameter rz decreased within the range of cutting speeds 280-420 m/min and feed rates 0.1-0.2 mm/rev, whereas it increased within the range of cutting speeds 190-460 m/min and feed rates 0.2-0.35 mm/rev. ▪ a significant difference, ranging from 40% up to even 330%, was noted between the real and theoretical values of surface roughness parameter rz. 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influence of active additives on the formation of selected indicators of the condition of the x10crni18-8 stainless steel surface layer in mqcl conditions, international journal of surface science and engineering, 9, pp. 452–465. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 105 113 https://doi.org/10.22190/fume201226017p © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper friction under large-amplitude normal oscillations mikhail popov1,2 1technische universität berlin, germany 2national research tomsk state university, russia abstract. building on a recently proposed contact-mechanical theory of friction control by external vibration, the case of large-amplitude normal oscillation is revisited. it is shown that the coefficient of friction can be expressed in particularly simple form if the waveform of the displacement oscillation is triangular or rectangular, and the contact stiffness is constant. the latter requirement limits the scope of the exact solutions to contacts between a plane and a flat-ended cylinder or a curved shape with a wear flat, but the adopted methodology also enables efficient numerical solution in more general cases. key words: contact mechanics, vibration, control of friction, large amplitudes, sliding friction 1. introduction the ability of externally applied vibration to substantially reduce both static and sliding friction is well known and enjoys many practical applications. the classical examples of wire drawing [1,2] and metal forming [3,4] deserve mention, but a thorough review is outside the scope of this paper. while the effect has attracted a fair amount of research, most of the works are of an experimental, application-oriented nature [5-7], and proposed models are at best semi-empirical [8]. for this reason, no consensus has been established concerning the theoretical underpinnings of the phenomenon. a possible physical model based on macroscopic contact mechanics was recently proposed by the author and colleagues [9]. the mechanism of force reduction in this model is based on the observation that stick-slip can arise in an oscillating contact under suitable conditions, if the compliance of the contact is taken into account. during the stick phases the lateral force is by definition subcritical (i.e. less than what is required to sustain sliding), and therefore lowers the average friction force. multiple extensions of this model have since received december 26, 2020 / accepted february 08, 2021 corresponding author: mikhail popov technische universität berlin, institut für mechanik, fachgebiet kontinuumsmechanik und materialtheorie, sekr. ms2, einsteinufer 5, d-10587 berlin e-mail: mpopov@fastmail.fm 106 m. popov been published and were reviewed in a recent paper [10]. here, the same approach is used to analyze the case of large-amplitude normal oscillation, when the amplitude is larger than the mean indentation and the body starts to “jump” over the plane. 2. model for a complete description of the model the reader is referred to previous publications [9,10], but a short overview is provided here for convenience. first and foremost, it is assumed that the contact is quasistatic and that the contact stiffness is independent of indentation depth. both assumptions are nonessential for the model as such, but are required for analytical calculations. together, they allow us to treat the contact as a single linearly elastic massless spring (fig. 1) with normal and lateral stiffness kz and kx, respectively. if the modeled contact is a flat-ended cylinder with radius a, the stiffness values are given by: 2 2 * 1 2 * 1 2 * 1 2 * 1 2 1 11 2 where 2 21 2 where 4 4 z x k e a e ee k g a g gg     − − = = + − − = = + (1) with ei, gi being the elastic and shear moduli of the contacting bodies and νi their poisson numbers. fig. 1 a single massless spring, which serves as a minimal model of a sliding frictional contact. the sliding velocity is constant, while the vertical coordinate oscillates. amontons friction with the constant coefficient of friction µ0 is assumed in the contact. the spring is pulled with a constant velocity v0 while also being subjected to a normal oscillation that is parametrized as ( ) ( )z z zu t u a w ft= + (2) friction under large-amplitude normal oscillations 107 where zu is the mean indentation, az the amplitude, f the frequency and w a zero-mean, unit-amplitude waveform. the lateral displacement ux is the primary unknown of the system. when the contact point is in a sliding state, its velocity can be shown to be 0( ) ( )z x z x k u t a fw ft k  = (3) the contact transitions from slip to stick when this velocity vanishes. the point of stick onset φ1 = ft1 can therefore be written as: 1 1 ( ) ( )w −= (4) where β is one of the dimensionless variables that parametrize the behavior of the system: 0 0 , , xz z z z k va ft u k a f     = = = (5) the eq. (4) does not necessarily have solutions. for stick-slip to be present, it is necessary that max ( ) cw     = (6) where βc is the maximum positive gradient of the oscillation waveform. if the dimensionless velocity β exceeds this threshold value, stick-slip becomes impossible and the macroscopic coefficient of friction  is the same as the intrinsic coefficient of friction µ0. otherwise it is reduced by some amount that depends on α, β and the shape of w. if condition (6) is satisfied and stick is initiated, the spring continues stretching with the constant velocity v0 and the lateral spring force therefore increases linearly with time: stick 0 1 0 1( ) ( ) ( )z xf t f t k v t t= + − (7) this continues while the stick condition fstick < µ0fz(t) holds. substituting fz = kzuz and rearranging gives the end of the stick phase φ2 in implicit form: 2 1 2 1( ) ( ) ( )w w    − = − (8) the stick-slip process is visualized in fig. 2. the macroscopic friction force xf is computed by integrating fx(t) over both the slip and stick periods: 0 1 ( )d t x xf f t t t =  (9) 108 m. popov fig. 2 stick and slip under the influence of a harmonic oscillation. the dotted line represents the tangential force as it would be in pure slip, fslip = µ0fz(t). the solid line is the actual tangential force in the presence of stick-slip. the stick phases are the straight segments, e.g. between t1 and t2, while slip phases are the sinusoidal segments, e.g. between 2t  and t1, repeating periodically. note that fx ≤ fslip everywhere, which is the origin of friction reduction in our model. since fx only differs from µ0fz during the stick phase, it is actually more convenient to determine the absolute force reduction 0x z xf f f = − : 2 1 0 stick 1 ( ( ) ( ))d t x z t f f t f t t t  = − (10) after expanding and rearranging, it is found that δfx can be expressed as 0 ( )x z z wf k a  =  (11) where ψw is a dimensionless “reduction function” that is specific to the waveform w: ( ) 2 1 1 1( ( ) ( ) ( ))dw w w          = − − − (12) the macroscopic coefficient of friction  can then be recovered through 0 0 0 (1 ( )) z x x w z z z f f f f k u       −  = = − = −  (13) this puts the dependence into a very simple form, with most of the complexity contained in a function of one argument, ψw(β). this function, however, needs to be determined numerically in most cases. this concludes our whirlwind tour of the model framework that will be used in the sequel. a less hurried presentation can be found in [10]. friction under large-amplitude normal oscillations 109 3. large amplitudes in the preceding discussion it was implicitly assumed that the amplitude az is smaller than the mean indentation zu , so that the bodies are permanently in contact and the normal force is non-negative. the purpose of this paper is to extend the analysis to z za u , that is, cases where the bodies lose contact periodically. equivalently, z z za u a−   , where we have excluded the trivial no-contact case. this form also makes evident the need for a small re-parametrization: 1 z z u a   = = (14) which avoids the singularity at 0zu = . the first thing to note about the jumping case is that the static coefficient of friction is always zero, because the contact obviously cannot sustain a lateral force while it is “in the air”, and slow creep will therefore be present at arbitrarily small pulling forces. if measurements of the static coefficient of friction under normal oscillation do not go to zero at suitably large amplitudes, this probably indicates a misalignment in the measurement apparatus. the second thing to note is that, in general, the simplicity of eq. (13) can no longer be maintained. the clean separation between α and β is only possible because the stick-slip process is completely independent of mean indentation, so long as the normal force fz is positive throughout. however, when uz(t) becomes negative in the jumping case, this causes fz to become “clipped” at zero (assuming no adhesion). this destroys the invariance w.r.t. zu , because the waveform w effectively becomes “cut off”, and has to be renormalized to maintain the properties of zero mean and unit amplitude. thus, w(φ) should properly be w(γ,φ) in the jumping case. overall, this leads us to expect the coefficient of friction to be a nonlinear function of two parameters (in addition to the waveform dependence): jmp 0 ( , )wg   = (15) in general, the function g needs to be computed numerically. there are, however, a few cases of some practical importance that can be treated analytically. these include square and triangle waves, for which solutions can be obtained in closed form due to their simplicity; and certain self-similar oscillations, for which asymptotic behavior can be deduced. these cases are considered next. 3.1. special case 1: sawtooth and triangle wave of the possible waveforms with triangular shape, here we consider the left-leaning sawtooth function (stl), the right-leaning sawtooth function (str) and the symmetric triangle wave (tri). the normalized functions w for these waveforms can be defined on the unit interval (with periodic extension understood) as: stl str tri 1 2 2 1 4 1, 1/ 2 3 4 , 1/ 2 w w w       = − = − −  =  −  (16) 110 m. popov from geometrical considerations (which come down to determining the area between the waveform and a straight line with the slope β as in fig. 2), it is easy to show that the corresponding reduction functions ψw(β) in the simple non-jumping case are given by: str stl 2 4 ( ) 1 , ( ) , ( ) 2 2 4 tri        −  = −  =  = + + (17) the triangular waves have the unique property that clipping the waveform does not affect the coefficient of friction. to appreciate this, refer once again to fig. 2. the coefficient of friction is given by the ratio of the area under fx to the area under µ0fz. this ratio changes continuously as the waveform is clipped from below by increasingly large amplitudes. if the waveform is triangular, however, then the only effect from the cutoff is that the ramp of the stick phase starts later and later (in the point of first contact). the area ratio is not affected, which means that the coefficient of friction remains constant, despite the fact that ψw formally depends on γ. this means that, for triangular waves, 0( , 1) ( , 1) (1 ( ))w        = = = − (18) using the reduction functions given in eq. (17), this provides the following simple results for the coefficient of friction under large-amplitude oscillation: str 0 stl 0 0 2 ( ) , ( ) , ( ) 2 2 4 tri               = = = + + (19) 3.2. special case 2: self-similar waveforms, square wave the triangular waves are a special case of what could be termed self-similar waveforms. by this we mean that a cut-off waveform can be rescaled in such a fashion as to be identical to the original waveform. assuming that the waveform is also convex ensures that stick is precipitated in the point of first contact, as in the case of the triangular wave. this means that the stick-slip graph of a cut-off waveform can be rescaled (together with the stick ramp) to have the same area ratio – and therefore the same coefficient of friction – as the same waveform at another cutoff. of course, this rescaling also changes the slope β, which must be adjusted accordingly. usually, it is convenient to choose the coefficient of friction at γ = 1 as a reference point, so that the large-amplitude coefficient of friction of a self-similar waveform can be expressed as: 0( , ) (1 ( ( , )))w      = − (20) the function ξ which provides the remapping of β is specific to the waveform. after the triangle, the next-simplest example of a self-similar waveform is the square wave, which alternates between 1 and -1 in equal intervals. it is easy to show that the remapping function for such an oscillation is given by: sqr 2 ( , ) 1      = + (21) using this mapping and the reduction function ψsqr of the square wave (see eq. (36) in [10]), the coefficient of friction under large-amplitude square wave oscillations can be written as: friction under large-amplitude normal oscillations 111 sqr 0 , 2(1 ) 4(1 ) 1 1 , 2(1 )            + + =  + −  +  (22) this result is shown in fig. 3 for the entire range of γ from 1 (starting to separate) to -1 (barely touching). fig. 3 coefficient of friction under large-amplitude square wave oscillations with 11 different normalized indentations γ covering the entire jumping range from -1 to 1. the concept of self-similar waveforms also applies to the harmonic oscillation, to a limited extent. while the entire sine wave is not self-affine according to our definition, it can be approximated piecewise by a parabola over some of its domain. since the parabola is indeed a self-affine function, we can expect the coefficient of friction under harmonic oscillation to have the described behavior asymptotically, although it will not be valid for values of γ close to 1. the remapping function in this case can be shown to be 0 2 1 ( , ) 1       + = + (23) where γ0 is the value of γ at the point where the self-affine scaling behavior started. more generally, for a waveform that can be asymptotically approximated by a power law φn, the corresponding remapping can be shown to be 1/ 01 ( , ) 1 n n       +  =   +  (24) 112 m. popov 3.3. numerical example: harmonic oscillation as an example of asymptotic scaling, fig. 4 shows numerically determined coefficients of friction under large-amplitude harmonic oscillation. one thing to note is that for γ in the range of approximately -0.2 to 1, the coefficient of friction depends only weakly on γ, with all curves bunching fairly closely together. the dependence on γ is also non-monotonous in this range, leading to lower coefficients of friction at first (from γ = 1 to approx. 0.6), and then increasing again (from γ = 0.6 to -1). the value around γ = -0.3 is the point from which the remaining part of the cropped waveform can be regarded as roughly parabolical, and the subsequent behavior of the coefficient of friction can be described by the scaling given in eq. (23). this is also shown in fig. 4 with black dots. fig. 4 numerically computed coefficient of friction under large-amplitude harmonic oscillations with 11 different normalized indentations γ covering the entire jumping range from -1 to 1. note the non-monotonous dependence on γ: the dark red line corresponds to the critical value γ = 1, which separates the jumping and non-jumping regions. from there, the coefficient of friction is first reduced with diminishing γ (red lines and arrow) and then increases again (blue lines and arrow) starting somewhere around γ = 0.6. black dots indicate the expected scaling behavior according to eq. (23) relative to γ0 = -0.3. 4. conclusions the influence of large-amplitude normal oscillation on sliding friction, which has not previously received much attention in the literature, was analyzed in this work, based on a model proposed by the authors in a previous publication. it was shown that the coefficient of friction in the jumping case depends on the same dimensionless variables as in the lowamplitude case, but in a more complicated fashion. at low amplitudes, the influence of the two main variables, α and β is cleanly separated, while at large amplitudes they become friction under large-amplitude normal oscillations 113 entangled and influence the coefficient of friction in a nontrivial manner. this was demonstrated on the example of the harmonic oscillation, where the amplitude-dependence is non-monotonic and can only be determined numerically. however, some simple cases such as triangular, rectangular and more general self-similar waveforms yield relatively simple results, which allow the coefficient of friction to be expressed either in closed form or as an asymptotic scaling relation. acknowledgement: this work was supported in part by the tomsk state university competitiveness improvement programme, which the author gratefully acknowledges. references 1. siegert, k., ulmer, j., 2001, superimposing ultrasonic waves on the dies in tube and wire drawing, journal of engineering materials and technology, 123(4), pp. 517–523. 2. murakawa, m., jin, m., 2001, the utility of radially and ultrasonically vibrated dies in the wire drawing process, journal of materials processing technology, 113(1-3), pp. 81-86. 3. eaves, a., smith, a., waterhouse, w., sansome, d., 1975, review of the application of ultrasonic vibrations to deforming metals, ultrasonics, 13(4), pp. 162-170. 4. ashida, y., aoyama, h., 2007, press forming using ultrasonic vibration, journal of materials processing technology, 187, pp. 118-122. 5. pohlman, r., lehfeldt, e., 1966, influence of ultrasonic vibration on metallic friction, ultrasonics, 4(4), pp. 178-185. 6. godfrey, d., 1967, vibration reduces metal to metal contact and causes an apparent reduction in friction, asle transactions, 10(2), pp. 183-192. 7. chowdhury, m.a., helali, m., 2008, the effect of amplitude of vibration on the coefficient of friction for different materials, tribology international, 41(4), pp. 307-314. 8. de wit, c.c., olsson, h., satrom, k.j., lischinsky, p., 1995, a new model for control of systems with friction, ieee transactions on automatic control, 40(3), pp. 419-425. 9. popov, m., popov, v.l., popov, n.v., 2017, reduction of friction by normal oscillations. i. influence of contact stiffness, friction, 5(1), pp. 45-55. 10. popov, m., 2020, the influence of vibration on friction: a contact-mechanical perspective, frontiers in mechanical engineering, 6, pp. 69. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 123 131 mixed and boundary lubrication in rolling contact: experiment and simulation  udc 621.7 qiang li, roman pohrt technische universität berlin, germany abstract. a new model of mixed lubrication is proposed in the frame of the method of dimensionality reduction (mdr). in this model the dynamic lubricated rolling contact between rough surfaces is simulated based on the results from elastohydrodynamic lubrication (ehl). in order to account for the break-up of the additional boundary layer on a local micro contact area, a supplemental criterion is imposed. for comparison, a twin-disc test rig is set up to measure the electrical resistance between two lubricated rolling surfaces under different normal forces, rotation speeds and temperatures. we have investigated the probability of boundary layer breakthrough for both experiment and simulation and found good agreement. key words: lubricated contact, ehl, electrical resistance, mixed lubrication, mdr 1. introduction countless examples in mechanical engineering require lubrication between components that are in relative motion. on the one hand, it is known from experience that practically no wear at all occurs when these components operate under conditions, where the surfaces and their roughness features are completely separated by a fluid film. on the other hand, current trends in engineering are at a disadvantage to the creation of a fluid film:  downsizing mechanical components demand for higher pressures  low-viscosity oil increases efficiency but decreases film thickness  start/stop cycles force the system through low-speed relative motion as a consequence, it is common practice for mechanical components such as gears, bearings and cams to operate in a mixed lubrication mode. typically the surface roughness  received november 27, 2013 corresponding author: roman pohrt technische universität berlin, institut für mechanik, straße des 17. juni 135, 10623 berlin, germany e-mail: roman.pohrt@tu-berlin.de acknowledgements: the authors acknowledge many useful discussions with v.l. popov. this material is based upon work supported by the deutsche forschungsgemeinschaft (dfg, grant no. po810/24-1). q. li received support through a scholarship from the china scholarship council (csc). 124 q. li, r. pohrt of contacting bodies is of the same order as the lubricant film thickness, so that the top micro roughness features (asperities) will enter into contact and part of the load and shearing will be carried by these asperity contacts. under this regime, a multitude of wear and damage types can occur. in experiments the contact condition for a lubricated system can be observed by measurement of electrical contact resistance cr which is expressed as [1]. 21      i con c a l r (1) where  is the resistivity of contacting materials and ai is the radius of each single contact spot. lcon is the total resulting contact length and defined as the sum of contact diameters. in the case of full hydrodynamic lubrication, the rough surfaces are completely separated by the lubricant film, so the resistance measured will be very high. in contrast, when the asperity contacts carry a major part of the load, a large number of contact spots are formed, thereby decreasing the electrical resistance dramatically. what makes the mixed lubrication problem difficult is the necessity to handle both hydrodynamic lubrication and asperity contacts. the earliest way of modeling mixed lubrication took into consideration the influence of roughness in hydrodynamic systems where the film thickness was considerably larger than the roughness [2]. in 1970s tallian and johnson considered both asperity contact and hydrodynamic lubrication. tallian studied the cases where asperities deformed elastically and plastically while johnson considered only the elastic deformation based on the greenwood and williamson model [3][4]. later micro-ehl models and combined micro-ehl and asperity contact models included the interaction of surface roughness, film thickness and pressure [5]. a stochastic analysis was developed by zhu and cheng (1988) [6]. it combined patir and cheng’s average flow model (1978) [7] for hydrodynamic lubrication and greenwood and tripp’s load compliance relation (1970) [8] for asperity contacts. with the rapid development of numerical simulation techniques and faster computers, researchers were able to investigate more complicated lubrication problems. therefore, more realistic transient, rough surface, thermal and non-newtonian lubrication problems were studied in the past decade. a deterministic model for mixed lubrication in point contacts was developed by jiang et al. (1999) and the contact between asperities was studied when they moved through the ehl region [9]. wang et al. (2004) [10] developed a thermal model for mixed lubrication in point contact. in this paper we have tried a simple mixed-lubrication model and compared its results with experiment. 2. numerical model we deal with the lubricated rolling contact between rough surfaces of cylinders where a boundary layer is present on the two surfaces. most non-conforming lubricated contacts such as roller bearings, journal bearings, cam and followers or gear teeth can be viewed as such systems. we will impose a new model for the micromechanical contact between asperities including the physically or chemically absorbed boundary layer (fig. 1) and apply it to the conditions found in lubricated rolling contacts. simulation of lubricated rolling contact with a reduced model 125 fig. 1 schematic contact between two cylinders and its view of contact area in micro scale. surfaces may either have a positive gap width when separated by a boundary layer or they can be in intimate contact. the contact conductance only has a considerable value, when there is intimate contact, or the boundary layer has decreased to molecular scale the contact between two elastic cylinders is known for having equivalent in the contact between a rigid plane and an elastic cylinder with equivalent modulus of elasticity 2 2 1 2 * 1 2 1 1 1    e e e   and radius * 1 2 1 1 1   r r r , where e1 and e2 are moduli of elasticity, 1 and 2 are poisson’s ratios, r1 and r2 are radiuses of both cylinders. according to hertzian contact theory, the contact width 2a under load fn is equal to * * 4   nf r a l e (2) where l' is length of cylinder. for elastohydrodynamic lubricated rolling contact the bodies are separated by an oil film and its thickness over whole contact area 2a is almost uniform, except for the trailing edge where a small decrease in the film thickness occurs. a common formula of central film thickness is given by hamrock from numerical studies [13] 056.0*166.0 474.0*692.0 0 470.0 0 )2()'/( )(992.2 elf r h n   (3) where v is mean surface velocity v = (v1 + v2) / 2. the values of 0 (viscosity at atmosphere pressure) and  (pressure-viscosity coefficient) are properties of the lubricating medium and are usually temperature-dependent. thus in a case of a known operation scenario, the film thickness excluding roughness can be calculated. fig. 2 reduced model for lubricated contact. the original 3d problem consists of two rough opposing bodies with a clearance stemming from the lubricant film. surfaces constantly move tangentially, so that new asperity contacts may form. the problem is transformed with the mdr onto two one-dimensional rough lines 126 q. li, r. pohrt the lubricated contact area in three dimensional (fig. 2) consists of two moving rectangles with width 2a and length l that are separated by an oil film with average distance h0 where some asperity contacts may take place. we treat the contact problem in this zone by using the dimensionality reduction method proposed in 2007 by popov and geike [11]. it has some major advantages in computational complexity compared to other methods that can simulate asperity contact. it maps three-dimensional contact problems onto one dimension and eliminates the elastic coupling, so that the computing time is dramatically reduced. in the past few years it has been developed for many contact problems, such as elastic and viscoelastic contact, normal and tangential contact including rough contact [12]. fig. 3 one-dimensional contact between an elastic ‘roller’ and a rigid body. the mean gap width between both is obtained by the ehl theory, the resulting micro contacts are analyzed by means of the mdr coordinates x and z are seen in fig. 3. the average of the rough rigid profile is assumed to be zero and the rough roller is a superposition of a parabolic line and roughness. the roughness has power spectral density c1d  q 2h1 , where q is the wave vector and h is the hurst exponent. in this paper, the lines are generated with 10 6 points, corresponding to the perimeter of the roller used in experiments. the spectral density is defined from qmin = 2 / 2amax to qmax = 2 / 10x, where amax is half the contact width and h = 0.7. from the results of ehl, the macroscopic shape of the ‘parabolic line’ in the interval [a, a] is assumed to be flattened out and the average distance between the elastic ‘roller’ and the rigid profile is equal to the thickness of oil film h0. with applied normal force f and rotation speed v1 and v2 (and also temperature), the value of contact width 2a and film thickness h0 can be calculated according to eq. (2) and eq. (3). therefore, the initial contacting profiles at t = 0 are determined. then, the points on the lines enter and transit through the contact width with different velocities v1 and v2. at each time step we check the contact condition. it can be easily observed that some points are in geometrical contact (fig. 4), but in this paper we consider the boundary layer between two contacting bodies; therefore, based on this geometrical contact, the failure of boundary layer must be calculated. simulation of lubricated rolling contact with a reduced model 127 fig. 4 one-dimensional model for the deformation of an elastic body. the 3d surface topography is transformed to give an equivalent line. the indentation of the rough line must be done with densely packed, independent springs for the break-up of the boundary layer, we consider the model of a perfectly plastic material. it is known [14] that if two plates with radius r are pressed together under normal force fn and separated by a layer of material with low limiting shear stress 0, a film remains with thickness 3 02 3  n r h f  . (4) according to the rules of mdr [12], the elastic body is modeled as a series of parallel springs with the normal stiffness *  c e x , where x is the discrete step (fig. 4). the force on each ‘spring’ is defined as ).(δδ)( * ii xzxexf  (5) here z is the displacement of indentation. in the reduced model eq. (4) is written as 3 0 12   l l l d h f . (6) here dl is the local contact length and equal to x times the number of contacting points and fl is the normal force on this local area and equal to fl = e * xzi from eq. (5). for each "geometrical contact" if value hl calculated according to eq. (6) is smaller than the critical thickness of boundary layer hc, the layer is defined as broken up while asperities are in intimate contact. the boundary layer thickness due to adsorption and chemical reactions is about 1...10 nm [15]. in the simulation we considered hc = 5nm and 0 = 10 6 pa. in a single operation case, the change of total contact length on time is recorded as fig. 5 (a). it is seen that at some moments there is no asperity in contact at all. based on it a general contact condition in this operation case can be obtained from it, which is named the probability of boundary layer breakthrough in the paper and calculated as time percentage when real contact occurs. in a well lubricated condition, the probability is close to zero. 128 q. li, r. pohrt fig. 5 (a) contact length over time, data extracted from mdr simulation; (b) electrical resistance over time from experiment data. parameters: 800 n , 100 r min and 40 c 3. experiment measurement a twin-disc test rig is used (fig. 6) is used for validating the results obtained from simulation. two identical cylinders (radius r = 0.05 m, width l' = 0.01 m, roughness  = 0.2 m) are pressed together and rotated at identical speeds so that pure rolling occurs. a synthetic lubricant is constantly fed into the contact zone; mobilgear shc xmp 320 is used because of its wide usage in highly loaded wind turbine gear boxes. the whole test setup can be heated to give stationary temperature for the rollers and the injected oil. we have measured electrical resistance between the two rollers for a range of operating parameters: the normal force is varied from 100 to 1600 n, rotation speeds from 86 to 200 r/min and temperatures from 40 to 80 o c. fig. 6 experimental setup. the left hand side shows the overall test rig. inside the aluminum block, there are two rollers, driven by external drive shafts. the lower block can be lifted pneumatically to exert a normal force. the right hand side shows a picture of the two rollers in contact without lubricant fig. 5 (b) shows a typical sample of the time-dependent resistance measurement. it can be seen that the contact condition rapidly changes from states of good conductivity to very high resistance. simulation of lubricated rolling contact with a reduced model 129 in order to compare the results quantitatively, we have used the classical approach of contact probability [16]. we calculate the percentage of time, for which the electrical resistance is measured to be below 100 . whenever this is the case, we consider the surfaces to be in contact and the electrical current can flow through the contact spots; otherwise they are separated by a lubricant film. we compare this probability of contact to the simulated one of the boundary layer breakthrough from the 1d model. 4. results there are totally 125 operation cases in both simulation and measurement. fig. 7 (a) shows the simulated breakthrough probability as a function of the temperature. in fig. 7 (b) the experimental contact time probability for the same scenarios are shown. for reason of clarity, not all the cases are included. it can be seen that the contact probability increases fig. 7 comparison of boundary layer breakthrough between (a) simulation and (b) experiment (c) with all data. 130 q. li, r. pohrt with temperature and load but decreases with rotation speed in both investigations. fig. 7 (c) gives a direct comparison for the probability of boundary layer breakthrough between simulation and measurement. good agreement can be found qualitatively and quantitatively in most cases. 5. conclusions the method of dimensionality reduction is used to simulate the process of lubricated rolling contact between rough surfaces. a novel criterion for the breakthrough of the chemical or physical boundary layer is introduced, based on the assumption of perfectly plastic material behavior. using this criterion, the breakthrough probability under different working conditions is predicted and compared to experimental findings. the obtained results show good agreement. references 1. braunovic, m., konchits, v.v., myshkin, n.k., 2007, electrical contacts fundamentals, applications and technology, crc press. 2. spikes, h.a., 1997, mixed lubrication-an overview, lubrication science, 9(3) pp. 221-253. 3. tallian, t.e., 1972, the theory of partial elastohydrodynamic contacts, wear, 21(1) pp. 49-101. 4. johnson, k.l., greenwood, j.a., poon, s.y., 1972, a simple theory of asperity contact in elastohydrodynamic lubrication, wear, 19(1) pp. 91-108. 5. patir, n., cheng, h.s., 1979, application of average flow model to lubrication between rough sliding surfaces, asme journal of lubrication technology, 101(2) pp. 220-230. 6. zhu, d., cheng, h. s., 1988, effect of surface roughness on the point contact ehl, asme journal of tribology, 110(1) pp. 32-37. 7. patir, n., cheng, h.s., 1978, an average flow model for determining effects of three-dimensional roughness on partial hydrodynamic 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journal of tribology, 111(2) pp. 146-251. 14. popov, v.l., 2010, contact mechanics and friction. physical principles and applications, springer-verlag. 15. homola, a., israelavili, j., gee, m., mcguiggan, p., 1984, measurements of and relation between the adhesion and friction of two surfaces separated by molecularly thin liquid films , journal of tribology, 111(4) pp. 675-682. 16. crook, a.w., 1957, simulated gear-tooth contacts: some experiments upon their lubrication and subsurface deformations, proceedings of the institution of mechanical engineers, 171(1) pp. 187-214. simulation of lubricated rolling contact with a reduced model 131 mešovito i granično podmazivanje pri kontaktu valjanja: eksperiment i simulacija predlaže se jedan novi model mešovitog podmazivanja u okviru metode redukcije dimenzionalnosti ili mdr-a. kod njega se dinamički podmazani kontakt valjanja između grubih povšrina simulira na osnovu rezultata elastohidrodinamičkog podmazivanja (ehl-a). da bismo objasnili prekid dodatnog graničnog sloja na lokalnoj mikrokontaktnoj površi uvodimo i dopunski kriterijum. upoređenja radi, postavlja se testirna oprema od dva diska radi merenja električnog otpora između dve podmazane valjane površine pod različitim normalnim silama, rotacionim brzinama i temperaturama. ispitali smo verovatnoću proboja graničnog sloja i za eksperiment i za simulaciju i utvrdili dobro slaganje. ključne reči: podmazani kontakt, ehl, električni otpor, mešovito podmazivanje, mdr facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 199 208 https://doi.org/10.22190/fume201205002h © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper hamiltonian-based frequency-amplitude formulation for nonlinear oscillators ji-huan he1,2,3, wei-fan hou1, na qie1, khaled a. gepreel4,5, ali heidari shirazi6, hamid mohammad-sedighi6,7 1school of science, xi'an university of architecture and technology, xi’an, china 2school of mathematics and information science, henan polytechnic university, jiaozuo, china 3national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china 4math. depart. faculty of science, taif university, saudi arabia 5mathematics department, faculty of science zagazig university egypt 6mechanical engineering department, faculty of engineering, shahid chamran university of ahvaz, ahvaz, iran 7drilling center of excellence and research center, shahid chamran university of ahvaz, ahvaz, iran abstract. complex mechanical systems usually include nonlinear interactions between their components which can be modeled by nonlinear equations that describe the sophisticated motion of the system. in order to interpret the nonlinear dynamics of these systems, it is necessary to compute their nonlinear frequencies more precisely. the nonlinear vibration process of a conservative oscillator always follows the law of energy conservation. a variational formulation is constructed and its hamiltonian invariant is obtained. this paper suggests a hamiltonian-based formulation to quickly determine the frequency property of the nonlinear oscillator. an example is given to explicate the solution process. key words: he’s frequency formulation, ancient chinese mathematics, semi-inverse method, periodic solution received december 05, 2020 / accepted january 06, 2021 corresponding author: ji-huan he a school of mathematics and information science, henan polytechnic university, jiaozuo, china; and national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university,199 ren-ai road, suzhou, china e-mail: hejihuan@suda.edu.cn 200 j.-h. he, w.-f. hou, n. qie, k.a. gepreel, a.h. shirazi, h.m. sedighi 1. introduction small amplitude oscillation of a pendulum or vibration in a long slender beam with low amplitude represent examples of the systems that can be well described using linear vibration theories. however, as the system components shift toward more sophisticated interactions, both nonlinear oscillators and their nonlinear characteristic equations play a vital role in explaining the behavior of complex systems. the unique phenomenon that can be modeled only through nonlinear systems, such as jump phenomenon, chaos, multiple steady-state solutions, etc., are the main significance of using the nonlinear oscillators in the vast majority of fields, especially in engineering structures. nonlinear stiffness and friction in dynamical systems [1], complex beam and piezoelectric plate-based self-sustainable electromechanical models [2,3], nonlinear reinforced nanofibers [4], vibration caused by the interaction between vehicle and bridge [5], large amplitude vibration of beams [6-12] and dynamics of micro/nanoelectromechanical systems [13-18] are a few examples of nonlinear systems in the field of mechanical engineering. from the mathematical point of view, the duffing oscillator, van der pol and mathieu are well-known nonlinear equations. several nonlinear systems can be described by utilizing the duffing equation, from a simple pendulum with harmonic motion to the vibration of arched structures [19]. the duffing equation especially emerges in mechanical systems with the presence of nonlinear stiffness springs. in many cases, stiffness is a function of displacement, which leads to cubic terms in the governing equations. ultimately, this forms a nonlinear relation between the applied force to the spring and the resulting displacement. for instance, fig. 1 shows a truck's rear leaf suspension. the chaotic vibration caused by road excitation in vehicles can be studied by modeling the leaf spring with magnets as a double-potential-well duffing oscillator [20]. fig. 1 leaf spring (left) and quarter car diagram of a nonlinear suspension (right) van der pol is another example of nonlinear self-excited limit cycle oscillators that is widely used to describe various systems in electrical and mechanical engineering, seismology, economics, etc. a classical representation of the van der pol oscillator is in oscillator triode circuits [21]. this equation is also used to describe the cardiac pulse modeling [22]. another well-known nonlinear equation is mathieu's equation. this equation was firstly encountered by émile léonard mathieu when he was studying vibrating elliptical drumheads. mathieu’s equation tends to appear in the systems with hamiltonian-based frequency-amplitude formulation for nonlinear oscillators 201 harmonic motion and is a powerful tool for modeling systems with elliptic boundary conditions. for instance, a wind turbine blade under influence of wind shear force and gravitational cyclic force (fig. 2) can be expressed using the forced mathieu equation [23]. fig. 2 wind turbine (left) and cyclic gravitational force on a blade (right) in this paper, based on the energy conservation, a modification of the frequency formulation is proposed in order to obtain the frequency-amplitude formulation of nonlinear systems. it is demonstrated that the proposed formulation is accurate enough for highly nonlinear differential equations containing large nonlinear terms. several examples are also provided to exhibit the integrity of the introduced formulation. 2. problem statement this paper focuses itself on the following conservative oscillator ( ) 0, (0) 0 (0)w p w w w b + = = = (1) for a periodic solution, it requires p(w) / w > 0. there are many analytical methods available for solving eq. (1), see some review articles in refs. [24-26] . this paper will discuss the frequency-amplitude formulation, which was first proposed in 2006; it was obtained according to an ancient chinese algorithm [27-29]. due to its simplicity and accuracy, the formulation has been widely applied to solving various nonlinear oscillators; various modifications appeared in literature [30-38]. the formulation is to find a suitable solution in the form tbw cos= (2) where  is the frequency to be further determined. b residual equation is obtained by introducing eq. (2) into eq. (1), which results in )cos(cos)( 2 tbptbtr  +−= (3) 202 j.-h. he, w.-f. hou, n. qie, k.a. gepreel, a.h. shirazi, h.m. sedighi the average residual can be calculated as = 4/ 0 cos 4~ t tdtr t r  (4) where  /2=t . the formulation is to choose two trial frequencies, e.g., 1 1 = and 2 2 = , and their residuals are respectively calculated as tdtr t r t 1 4/ 0 1 1 1 cos 4~ 1 = (5) tdtr t r t 2 4/ 0 2 2 2 cos 4~ 2 = (6) the frequency-amplitude formulation is obtained as follows [27-29] 21 2 2 11 2 22 ~~ ~~ rr rr − − =   (7) there are many modifications of eq. (7), see for examples, refs [30-38]. this paper will suggest an effective modification based on the hamiltonian invariant. 3. hamiltonian-based frequency-amplitude formulation the above frequency formulation is derived from a differential equation, here we suggests a modification from an energy form. the kinetic energy and the potential energy are changed during the oscillation process, but the total energy will keep unchanged for a conservative oscillator. in 2002, an energy approach to nonlinear oscillations was suggested [39]. the variational principle of eq. (1) can be constructed by the semi-inverse method [40-43], which is dtwpwwj        −= )( 2 1 )( 2 (8) where p(w) is the potential, satisfying the following relation: )()( wpwp dw d = (9) in the variational formulation given in eq. (8), 2 2 1 w is the kinetic energy, and p(w) is the potential energy. the total energy keeps unchanged during the oscillation: hwpw =+ )( 2 1 2 (10) hamiltonian-based frequency-amplitude formulation for nonlinear oscillators 203 where h is the hamiltonian constant, which can be identified by the initial conditions given in eq. (1). finally we obtain the following first order differential equation, 0)()( 2 1 2 =−+ bpwpw (11) we use eq. (11) instead of eq. (1) to re-build the frequency-amplitude formulations. substituting eq. (2) into eq. (11) results in the following residual equation, )()cos(sin)( 222 bptbptbtr −+=  (12) similarly we define two average residuals tdtr t r t 1 4/ 0 1 1 1 cos 4~ 1 = (13) tdtr t r t 2 4/ 0 2 2 2 cos 4~ 2 = (14) a modification of the frequencyamplitude formulation is given as follows 21 2 2 11 2 22 ~~ ~~ rr rr − − =   (15) 4. example consider the following well-known duffing equation, bwwwww ===++ )0(0)0(,03 (16) eq. (16) can be reduced to the following first-order differential equation, 0 4 1 2 1 4 1 2 1 2 1 42422 =−−++ bbwww  (17) we choose two arbitrary frequencies, e.g., 1 = 1ω and 2 = 2ω , and obtain the following residual equations, respectively. 42442222 1 4 1 2 1 cos 4 1 cos 2 1 sin 2 1 bbtbtbtbr  −−++= (18) 42442222 2 4 1 2 1 2cos 4 1 2cos 2 1 2sin2 bbtbtbtbr  −−++= (19) their average residuals can be easily calculated:   30 7 cos 4~ 4 4/ 0 1 1 1 1 b tdtr t r t − ==  (20)   30 307 2cos 4~ 24 4/ 0 2 2 2 2 bb tdtr t r t +− ==  (21) 204 j.-h. he, w.-f. hou, n. qie, k.a. gepreel, a.h. shirazi, h.m. sedighi according to the modified frequency-amplitude formulation, we obtain 2 21 2 12 2 21 10 7 1~~ ~~ b rr rr    += − − = (22) to show its accuracy given in eq. (22), we consider two extremes when 02 →b and →2b . when 12 b eq. (22) can be approximated as 2 20 7 1 b += (23) while the perturbation solution is [32] 2 8 3 1 b += (24) table 1 shows that both eq. (23) and eq. (24) see good accuracy when 12 b . fig. 3 also shows the good agreement between the approximate and the exact solutions. fig. 3 comparison of the approximate solution, the red continuous line is the exact solution, the black discontinuous line is the approximate solution, and the blue circles are perturbation solution hamiltonian-based frequency-amplitude formulation for nonlinear oscillators 205 table 1. comparison of the approximate frequency of eq. (23) with the exact one and the perturbation solution b2 0 0.001 0.0025 0.003 0.005 0.007 0.009 eq.(23) 1 1.00035 1.000875 1.00105 1.00175 1.00245 1.00315 eq.(24) 1 1.000375 1.0009375 1.001125 1.001875 1.002625 1.003375 exact frequency 1 1.000380 1.0009442 1.00113 1.0018726 1.002613 1.003369 when →2b , its approximate period becomes 2 2 5098.7 10 7 1 2 lim 2 lim 22 b b t b app b       = + == →→ (25) the exact period, when →2b , is 2 2/ 0 2 2 4164.7 sin5.01 4 b x dx b tex   =−=  (26) it is obvious that 987.0lim = → app ex t t  (27) the relative error is 1.317% when →2b . the approximate period by the homotopy perturbation method is 2 2 homotopy 2552.7 4 3 1 2 lim 2 lim 22 b b t bb       = + == →→ (28) 022.1lim homotopy = → t tex  (29) the relative error is 2.153% even when →2b , see fig. 4 and table 2. table 2. comparison of the approximate period of eq. (25) with the exact one b2 100 500 1000 1500 2000 b2 →  exact period 0.73629 0.33118 0.23435 0.19140 0.16577 2/4164.7 b eq.(25) 0.74568 0.33537 0.23731 0.19381 0.16787 2/5098.7 b relative error 1.275% 1.265% 1.263% 1.259% 1.267% 1.317% eq.(28) 0.72073 0.32403 0.22928 0.18725 0.16218 2/2552.7 b relative error 2.113% 2.159% 2.163% 2.168% 2.166% 2.153% 206 j.-h. he, w.-f. hou, n. qie, k.a. gepreel, a.h. shirazi, h.m. sedighi fig. 4 comparison of the approximate solution, the red continuous line is the exact solution, the black discontinuous line is the approximate solution, and the blue circles are perturbation solution 4. conclusion this paper suggests a modification of the frequency formulation based on the energy conservation, the obtained result is globally valid for 0  b2 < . the example shows that our result sees a good agreement with the perturbation solution for the weak nonlinearity. even when b2 → , our approximate frequency has also an extremely high accuracy, better 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thermodynamics to fractal variational principle, thermal science, 24(2 part a), pp. 659–681. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 269 282 design of 3d model of customized anatomically adjusted implants  udc 621.7:617.3 miodrag manić 1 , zoran stamenković 1 , milorad mitković 2 , miloš stojković 1 , duncan e.t. shepherd 3 1 university of niš, faculty of mechanical engineering, serbia 2 university of niš, faculty of medicine, serbia 3 university of birmingham, school of mechanical engineering, uk abstract. design and manufacturing of customized implants is a field that has been rapidly developing in recent years. this paper presents an originally developed method for designing a 3d model of customized anatomically adjusted implants. the method is based upon a ct scan of a bone fracture. a ct scan is used to generate a 3d bone model and a fracture model. using these scans, an indicated location for placing the implant is recognized and the design of a 3d model of customized implants is made. with this method it is possible to design volumetric implants used for replacing a part of the bone or a plate type for fixation of a bone part. the sides of the implants, this one lying on the bone, are fully aligned with the anatomical shape of the bone surface which neighbors the fracture. the given model is designed for implants production utilizing any method, and it is ideal for 3d printing of implants. key words: custom implants, bones fixation, ct scan, 3d bone model 1. introduction in orthopedic surgery there is a range of fixation methods for treating various bone fractures or other traumas. in the case of an internal fracture fixation treatment, it is of particular importance to utilize internal fixation whose geometrical and topological characteristics fully correspond to the shape and size of the patient‟s bone since this ensures improved recovery [1]. this paper focuses on the implants which are not of a standardized shape and size, but are adjusted to the patient‟s specific needs (customized, personalized implants) [2]. the term „customization‟ in medicine mainly refers to the use of the treatments which are adjusted to a specific patient. there are not many examples of received july 8, 2015 / accepted september 10, 2015 corresponding author: miodrag manić university of niš, faculty of mechanical engineering in niš, a. medvedeva 14, 18000 niš, serbia e-mail: miodrag.manic@masfak.ni.ac.rs original scientific paper 270 m. manić, z. stamenković, m. mitković, m. stojković, d.e.t. shephard applying this principle when designing orthopaedic implants. in this paper, we will describe the method for enabling the design of the orthopaedic implants which are adjusted to a specific patient (customized implants) and of both shape volumetric and plate type. the patient‟s specific implants, i.e. custom designed implants for an individual patient, are growing in popularity. the geometry and topology of those implants are adjusted to the anatomy and morphology of the bone and fracture of the specific patient. their application has a positive effect on patients, but on the other hand, it requires more time for preoperative planning and manufacturing. therefore, they are used in the areas where the application of predefined fixators can lead to complications in the surgical interventions or in the process of recovery [3]. internal fixation of medical devices are used as a support to treat damaged or diseaseinfected bones brought about as a consequence of old age, disease or an accident. they are made of different kinds of biocompatible materials [4]. there are two kinds of internal fixations – intramedullary and extramedullary. intramedullary nails are used to treat various long bones (e.g. tibia). the nails are inserted into the bone by using the bone‟s intramedullar canal, after which screws are inserted to fix the nail to the bone [5]. extramedullary devices take the form of plates that are fixed to the bone with screws. (fig. 1). these kinds of implants are placed on the external surface of the fractured part of the bone [4, 6]. in this way, the fractured bone fragments are connected into a whole, the transport capacity of the joint is created and position and direction of the fragments are kept. fig. 1 insertion of a tile for the upper part of tibia there can be found only few described examples of the methods for creating a 3d model of anatomically adjusted internal implants which correspond to the bone contour in the literature. papers [7] and [8] present a method for designing customized implants with a cad system by modifying standard implants, based on a patient‟s bones. the finite element analysis is then used to evaluate and compare the proposed design of a custom femoral component with a conventional design. design of 3d model of customized anatomically adjusted implants 271 papers [9] and [10] present an example of designing a fixation with tile shape, as well as the dynamic screw bolt of a hip. in [11] the description of a method and procedure of designing an internal fixation with tile shape type “medially locking plate” (mlp) is described, which is used for treating fractures of the femur from its lateral side. as a basis for design a 3d femur model is used. the position of a tile related to the femur is defined by creating a datum plane. medial sketch of the mlp is created on the sagittal plane. the sketch is then extruded in both normal directions. the inner sides of the fixator are approximated by polygons, and try to follow the bone surface. in [12] and [13] the method which is used for designing an internal fixation according to mitkovic type tpl (tibia-plato-lateral) is shown. the suggested method is based on the application of the maf method of anatomical features and newly developed techniques for designing fixation supporting surfaces. the result of the application of this method is a parametrical fixation model whose shape and geometry could be changed with a change of parameters. with this approach it is allowed to change the shape of a fixation in order to adjust it to the patient‟s bone shape, in this case tibia, based on parameters values (dimensions) read from a suitable x-ray or ct computer tomography – scan. many methods for creating customized implants try to create a mirror image 3d model of a sound part of the patient‟s bone. after mirroring those methods use engineering tools (spline techniques) to create implants. the case related to reverse engineering of sternum (chest bone) body presented in the paper [14] brings out a method for creating customized implants. the missing part of the sternum, affected by cancer, was redesigned in accordance with the virtual model of the sternum bone that was generated from preceding geometrical analyses of the healthy sternum samples which were geometrically and dimensionally similar to the diseased one. the geometrical analyses of healthy samples were used for generating the characteristic shape pattern of the transversal and sagittal cross-sections of the sternum as well as to identify the specific geometrical and dimensional constraints and relations. 2. design of customized implants an implant is a medical device manufactured to replace a missing biological structure, support a damaged biological structure, or fix an existing biological structure. conventional implant manufacturing methods provide only standard parts in a standard size and shape. this means that the implants do not respond to the patient‟s specific needs, and the post-operative recovery is more difficult. what is of highest importance during the process of implants insertion is to create a minimal direct contact between the fixation and the bone surface, while at the same time ensure that fixation follows the bone contours. the stress created when the device rests on the bone should be avoided since it can damage the periosteum which covers the bone surface and nourishes the bone through blood vessels in it. the implant modeling is performed using a computer-aided product development system which, beside geometry and topology, enables integration of product knowledge and applied technologies restrictions for some forms in a virtual model of a product. 272 m. manić, z. stamenković, m. mitković, m. stojković, d.e.t. shephard methods include use of cad software to design internal implants and these methods are based upon the ideas of orthopedic surgeons and engineers. the basis for creating a 3d geometrical model of an internal fixation is an outline of its contour defined in a suitable position in relation to the bone surface. the general engineering techniques for design, analysis and manufacturing of customized implants, for specific bones, used in this research, include several tasks [15] (fig. 2). 1. creating a 3d parametric model of bones. 2. creating a 3d parametric model of a fracture using a ct scan of the patient‟s fractured bones. 3. selecting the places on the bone where the implant will be placed. 4. adjusting the geometry of the implant according to the requirements of the surgeon. 5. creating a customized 3d model of the implant. 6. simulation of implant placement to bones. 7. analysis and optimization of the shape and dimensions of implants. 8. implant manufacturing. fig. 5 phases for designing and manufacturing of customized implants design of 3d model of customized anatomically adjusted implants 273 this methodology gives the opportunity to create a custom implant design adjusted to the patient‟s bone anatomy. 3d models of implants can then be used for the production of implants, after manufacturability analysis. the creation of 3d models of customized and anatomically adjusted implants is based on the 3d models of the patient bones and 3d parametric model of fracture which is made on the 3d model of the bone. the typical design process of anatomical adjusted customized implants, used in this paper, is shown in fig. 3. fig. 3 typical design process of anatomical adjusted customized implants for this purpose, the first step is to create a 3d model of a selected bone. the creation of geometrically accurate 3d models of human bones utilizes a number of different techniques and presents a unique challenge, because their geometry and form are very complex. these types of shapes can be modeled by using surface patches represented by bezier or b-spline surfaces, or by using nurbs patches, which are commonly used in traditional cad applications, e.g. catia [16]. the output of catia is presented in the stl (stereolithography) format, which allows it to be directly transferred into an rp (rapid prototyping) machine. reverse engineering of human bones implies the use of some kind of medical imaging device for the acquisition of medical data (computer tomography – ct, magnetic resonance imaging – mri), then processing the data in medical or cad software, and, finally, creating a valid geometrical model (surface, volume). 2.1. creating 3d parametric model of bones within the project vihos (virtual human osteoarticular system and its application in preclinical and clinical practice) [17] which is carried out at the faculty of mechanical engineering and the faculty of medicine at the university of niš, the 3d parameterized geometrical bone models are developed. for that purpose the method of anatomical features (maf) is developed. generic parametric model of bone building 3d model of bone import full medical image building 3d model of bone import partial medical image building 3d model with fracture design of anatomical adjusted implant 3d model of implant analysis and optimization acceptable design 274 m. manić, z. stamenković, m. mitković, m. stojković, d.e.t. shephard the goal of this method is to find the best possible solution for the creation of a parametric point model of the human bone in a sense of its best application in medical imaging and preoperative planning in orthopedics. with application of this method it is possible to create patient specific geometrical models (polygonal, surface, and volume) and parametric (predictive) models of the human bones. parametric models enable creation of geometrical models even in the cases when the geometrical data about specific bone is incomplete (e.g. bone fractures or diseases). in these situations geometrical models are created by applying the values of parameters measured on the medical images in the parametric functions. a more detailed description of the maf and its various applications are presented in [18, 19, 20, 21]. the maf consists of several procedures which enable the creation of geometrically precise and anatomically correct geometrical models of the human bones. 1. importing and editing point cloud acquired from medical imaging device, 2. tessellation of point cloud and creation of polygonal model (mesh), 3. anatomical and morphological analyses of a selected bone, 4. identification of rges (referential geometrical entities) which are based on the anatomical and morphological characteristics of a selected bone (points, directions, planes and views), 5. creating and editing the curves on a polygonal model of the selected bone, in accordance with the rges, and, 6. creating and editing the surface model of the selected bone‟s outer surface by sweeping, lofting, blending, and trimming the curves. the developed method can possess multiple benefits (or uses), in medicine and technology. the most important characteristic of the created parametric model is its ability to conform to the individual human dimensions of bone, which brings vast benefits in the sense of: preoperative planning (adequate implant for the analogous bone fracture can be selected, implant(s) position on the bone can be defined with greater accuracy, fixation pin positions can be easily determined), for the creation of solid model for structural analysis by fe, for rp of implant prototypes, for manufacturing presentation models, etc. 2.2. creating 3d parametric model of fracture by using bone a 3d model and spline design techniques in cad system, according to ct scan of fractured bones, it is possible to create a 3d parametric model of fracture. from a ct scan of the patient‟s fracture and the surgeon suggestions, the characteristic points of fracture are measured and transmitted to the 3d bone model. by connecting them, the outside contour of fracture is obtained. the complete model of fracture is created using adequate engineering techniques for free form surfaces and spline modeling. for this purpose ao/ota fracture and dislocation classification [22] is used for creating a udf (user defined feature) for each type of fracture. some examples of created 3d model of tibia‟s fracture are shown in fig. 4. design of 3d model of customized anatomically adjusted implants 275 fig. 4 3d model of a fracture 3. design of anatomically adjusted implants the aim of our research is to develop a design method and technique for creating several types of customized implants (fig. 5). the design method for a scaffold is presented in [23]. fig. 5 different types of customized implants the principle of anatomical adaptability implies that the internal fixation with its contact surface fully corresponds to the surface of the part of a bone where the fracture is located. in this paper, we present the process of designing anatomically adjusted 3d volumetric implants for long bones, and type plate, by using catia v5 software package. for both designs of implant the first step is to create the parametrical 3d geometrical model of bone or a part of bone, made on the basis of the patient‟s ct scan [18, 19, 20, 21]. customized implants 3d volumetric implant 3d plate bone implant scaffolds plates for connection special plates 276 m. manić, z. stamenković, m. mitković, m. stojković, d.e.t. shephard 3.1. designing technique of an anatomically adjusted 3d volumetric implants according to a ct scan of the patient‟s fracture, and the surgeon‟s suggestions, the model of fracture is created, using adequate engineering techniques for free form surfaces and spline modeling (fig. 6). fig. 6 3d model of fracture when the fracture is created and its surfaces satisfy the surgeon‟s needs, the healthy bone fragments that do not belong to the implant are removed (fig. 7). by removing the desired bone fragments a basic 3d model of the implant is created. by using advanced engineering techniques for free form modeling, the initial model of implant can be modified. for instance, bevels, rounding and additional screw holes can be added and everything else that is needed for the implant‟s production and implementation. fig. 7 basic 3d model of volumetric implant design of 3d model of customized anatomically adjusted implants 277 3.2. designing technique of an anatomically adjusted implants type plate designing procedures at the beginning are very similar as previously described ones, for 3d volumetric implants. the first step is creating a model of the fracture (fig. 8). fig. 8 a 3d model of fracture close to the model of the fracture, a datum plane, not far from the lateral surface of bone, is created, which is placed opposite the contour of the fracture (fig. 9). the surgeon suggests and defines the position and orientation of this plane. inside it, the contour of the proximal part of the plate is drawn. fig. 9 creating the plane for contour drawing, and creating the contour of the proximal part of the plate 278 m. manić, z. stamenković, m. mitković, m. stojković, d.e.t. shephard following that, the contour extrusion in the direction of the lateral bone side is performed so as to ensure that the extruded contour surface penetrates the bone surface (fig. 10). fig. 10 extrusion of the contour and its penetration through the bone the intersected closed contour of the plate‟s internal side is created in this way. inside of that contour, the curve drawing of the 3d splines that follow the bone contour is performed. after this step, the moment comes when all the surfaces are removed and the only surface left is the one that actually presents the internal side of the plate that is put directly on the bone (fig. 11). now this surface is extruded to transform it into a full model. with this process completed, we get a full 3d model of a proximal fixation plate that is completely anatomically adjusted to the surface of the proximal part of the bone (fig. 11). fig. 11 intersecting contour, 3d splines inside of the contour, extruded surface design of 3d model of customized anatomically adjusted implants 279 the remaining parts of the internal fixator plate type are made with standard technical elements. after the plate has been shaped, the creation of the screw holes on the proximal side of plate part is performed. according to the orthopedist request, an additional scheme of concentric circles with points for screw holes production is created (fig. 12). the process of screw holes creation is based on projection points and created tangent planes and is performed on the part of the proximal plate surface. fig. 12 creating a full model of plate the final model of the internal customized fixator type plate is shown in fig. 13. fig. 13 final model of customized plate the assembling module of cad system can be used to check whether the implant model plate type is appropriate by creating a set – a bone part, plate and fixation elements (fig. 14). in this way, we can check the model, seating, the number of necessary screws, etc. furthermore, this 3d model of a set can be used for fe analysis and dimension and shape optimization. 280 m. manić, z. stamenković, m. mitković, m. stojković, d.e.t. shephard fig. 14 bone and plate set by combining two previously described techniques a 3d volumetric implant and a tile for its fixation can also be made. moreover, a set bone-volumetric implant-fixation tilebonding elements can be designed. the designed set is used for implementation simulation and all the other analyses (fem etc.). this process is illustrated on fig. 15. fig. 15 set bone-volumetric implant-fixation tile 4. conclusion the presented method is based on designing implants directly on a patient‟s parametric three-dimensional (3d) bone model. when this is finished, further model adjustments to the patient's bone and manufacturing methods are performed. when a 3d model is created design of 3d model of customized anatomically adjusted implants 281 in this way, it can be used for the production of implants or plates. the method presented in this paper can be used for various kinds of internal fixators, which are directly attached to any bone surface. the method presents the designing process of a 3d model of customized anatomically adjusted internal implants, both volumetric and plate type. the internal surfaces lying on the bone are fully aligned with the bone surface and fracture surfaces. in this way, it ideally lies on the bone. this method provides the possibility to create a 3d model of positioning and insertion of the implant and tiles and it can also be used as a simulation model. furthermore, it can be used for a full analysis and shape and dimension optimization of fixation material. the method developed and described in this paper is applicable to various other implants of tile type and for any human bone. the requirement that must be fulfilled is the existence of a 3d bone model with imprinted fracture. this method has significantly improved the technique for production of anatomically adjusted internal fixations. the created 3d model of implants and plates is ideal for 3d printing or their production using a cnc machine. acknowledgements: the paper is part of the projects iii41017 virtual human osteoarticular system and its application in preclinical and clinical practice, sponsored by republic of serbia for the period of 2011-2014, and project bioemis, 530423tempus 1 – 2012 – 1 – uk – tempus – jpcr. references 1. manić 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korunovic n., trajanovic m., milovanovic j., trifunovic m., vitkovic n., 2013, design study of anatomically shaped latticed scaffolds for the bone tissue recovery , iii south-east european conference on computational mechanics-seeccm iii, kos, greece, 12-14 june. https://aotrauma.aofoundation.org/structure/education/self-directed-learning/reference-materials/classifications/pages/ao-ota-classification.aspx https://aotrauma.aofoundation.org/structure/education/self-directed-learning/reference-materials/classifications/pages/ao-ota-classification.aspx plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 107 117 doi: 10.22190/fume160831005r © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper deep drawing technology with wall ironing in mass packaging industry udc 621.7:005.4 saša ranđelović 1 , mladomir milutinović 2 , vladislav blagojević 1 1 university of niš, faculty of mechanical engineering, serbia 2 university of novi sad, faculty of technical sciences, serbia abstract aluminum is a metal that is being increasingly used in the packaging industry in the modern metal forming technology, but it also provides a good opportunity for effective advertising and product promotion. processing technologies for aluminum plastic deformation ensure superior packaging that meets the most rigorous demands in the food, pharmaceutical, chemical, and other industries. it is the case of mass production with very little material loss that offers the possibility of multiple recycling. on the other hand, today's products for general purpose consumers cannot be imagined without aggressive advertising that has a major impact on customers. modern graphics techniques for printing images and different basic surfaces offer great opportunities that manufacturers use widely in the promotion and sale of their products. key words: can, deep drawing, packaging, product management, graphic design 1. introduction large investments in research and primary aluminum processing, especially in the production of finished aluminum products, have led to the aluminum industry being the indicator of the most powerful world economies today. one can increasingly hear the information regarding aluminum consumption per capita, the share of aluminum per automobile, the amount of aluminum in the construction industry, or in everyday use, etc. naturally, all of this is further encouraged by the fact that aluminum is very convenient for getting the finest products which now meet the most rigorous demands of the market, i.e. the customer [1]. received august 31, 2016 / accepted december 20, 2016 corresponding author: saša s. ranđelović faculty of mechanical engineering, department of production engineering, a. medvedeva 14, 18000 niš e-mail: sassa@masfak.ni.ac.rs 108 s. ranđelović, m. milutinović, v. blagojević industrial packaging of food products and fast food delivery to the customer with unchanged characteristics is impossible today without the use of various types of foil, cans, wrappers, and curlers, which are all based on aluminum sheet and its finished products. today, most current industry packaging based on aluminum saves energy in the production and transportation of products [2]. the weight of the beverage packaging in cans, for example 0.33l, is only 5% of the weight of the beverage, while in the case of glass packaging the weight is almost identical. cans are promoted as packing material impervious to light, the fastest to cool, simple to open, while keeping the flavor of drinks well, but also as the only packaging based on metal that is 100% recyclable, which significantly contributes to the preservation of the environment. sheet metal forming is one of the most important manufacturing processes for mass production, especially in the automotive and aerospace industries, yet with application in many other production processes as well. deep drawing and ironing are the most frequently used manufacturing processes to produce thin-walled cans. those who are interested in learning more about metal forming mechanics can refer to certain excellent books [3-5]. many investigations show that there is a thickness reduction rate in which the ironing process becomes unstable [6, 7]. this leads to a variation of thickness along the can in the circumferential direction. these problems are generally solved in the industry by trial and error, with changes in the material geometry [8]. gu et al. [9] presented an optimization method for mass customization products which seeks to maximize the manufacturing efficiency. this model suggests increasing the commonality on different bill of materials levels and thereby maximizing the number of “mass production steps” and minimizing the customization steps during the manufacturing process. while this model would help in improving manufacturing efficiency given a certain set of functional requirements, it does not address balancing the customer demand for customization with the manufacturing efficiency. kumar [10] formulated a number of metrics for customization, mass production and modularity, thereby measuring the number of modules, combinations and theoretical production volume per module. the main metrics are: average number of options per feature, maximum number of configurations, average number of configurations per customer, degree of customization and average demand per option per period. these metrics are useful in relation to describing the variety of a product family and yet they are less useful in relation to assessing whether some options are configured less frequently than others potentially rendering them less profitable. furthermore, these methods do not enable assessment of whether the variety offered is actually the variety demanded by customers. 2. technology of mass production of cans an infinite continuously-rolled strip of aluminum sheet is introduced in a combined tool for blanking a workpiece that is immediately subjected to deep drawing. this process of refining metal requires very tight tolerances of material thickness with a special coating layer so as to minimize the coefficient of friction to  = 0.08. the tin plate strip is unwound, its surface coated with a thin film of lubricant and the strip continuously conveyed to the deep-drawing press. the full technological capacity yields an almost unbelievable productivity of 1700 pieces per minute, or 650 million per deep drawing technology with wall ironing in mass packaging industry 109 year. at first a blank is cut out (d = 164 mm, s=0.25 mm) at each individual tool of the press; the drawing ram then presses this blank through the draw ring to form a cup with the diameter of 100 mm and height of 41 mm (fig. 1). deep drawing is a metal forming process targeted for the production of thin walled cup/can shape objects through a combined compression–tension operation [11-13]. as shown in fig. 1, a blank is forced into a die cavity by a punch and it assumes the shape of the punch while being held by the blank holder. the process normally maintains the thickness of the sheet metal and can be used for shallow or deep parts. the tool is made up of 9 to 10 individual tools (for stamping, deep drawing and ironing) which are arranged next to each other. fig. 1 finished part after deep drawing process critical stress r1max is normal stress which occurs in the radial direction, where the workpiece material suffers an elongation [3, 4]. the maximum stress in the first operation of drawing occurs at the moment of full coverage of the rounded edges of tools upon which plastic deformation takes place only through the radius of the matrix [12]: max 1,1 (1 1,6 ) 2 s d sr sri s m r f s r k n k r r s r s              , (1) where ksr is the mean true von mises stress of the material workpiece, rs is the radius of the workpiece at the moment when the maximal force is identified, r and di represent the radius and the diameter of the deep drawing element at the first and i th operation, respectively,  is the coefficient of friction, fd is the force at the blank holder, rm the radius on the die matrix and rm is the maximal stress extension of the material workpiece. the cup is held by the variable pressure of the blank holder, during deep drawing process, to prevent wrinkles with which the flow of the manufacturing process is not possible [12]. wrinkles are caused by excessive clearance between the punch and the die and also due to an improper value of pressure of the blank holder during deep drawing process and an incorrect value of the punch radius. the proper pressure value is determined based on the following equation: 2 00,25 1 200 i d m i d d p r d s             , (2) where d0 is the diameter of workpiece and di the diameter of deep drawing. 110 s. ranđelović, m. milutinović, v. blagojević in certain applications, parts can be deep-drawn in several steps by redrawing. at each step, the cup becomes longer (deeper) and its diameter is reduced. however, if the wall thickness needs to be reduced as well, an ironing operation is implemented. in this process, as the part is redrawn, it is forced through an ironing ring (like an extra die) placed inside the cavity (fig. 2). ironing is the preferred operation for the fabrication of beverage cans [5, 6]. the cup is conveyed to the wall-ironing tool from the top. the ram first pushes it through the redraw ring to reduce its diameter of 65 mm to the punch diameter whilst retaining the sheet thickness constant at 0.25 mm. there is a gap between the punch and the wall-ironing rings 1 to 4 immediately after the redraw ring where the wall thickness of the can is reduced by "ironing" the thin wall (s = 0.15 mm) and consequently lengthening the can to 170 mm. fig. 2 ironing the can wall for typical ironing stress the stress balance is set in the axial direction [5, 12]. especially considered is the conical part of the tool where there is a change in the thickness of the cylindrical wall and the output section with reduced can wall thickness: 2 2 2 2 2 2 2 1 ( )(( ) ) ( ) 2 cos 2 sin 2 0 sin cos sin z z z dx r dr x dx r r x r r tg dx q dx x x                              (3) where  is the die angle, rz is the axial stress inside the ironing wall, r2 = 0.5(ra + rb) is the mean value of the inner radius of the workpiece, x is variable diameter,  is the tangential contact stresses on the die, while  the tangential contact stresses on the punch and, finally, q is the normal stress as a consequence of continuous load. from the above equation of the balance of forces, it can be seen that the frictional force on the contact surface between the punch and materials process helps deep drawing. the explanation lies in the fact that the focus of deformation of the material is flowing along the punch, in the opposite direction to the movement, but with the direction of the force of friction the same as the direction of the force on the punch. for these reasons, the surface should be punched with greater roughness, if permitted by the required quality of the inner surface of the can, in order to maximize the positive impact of friction on that part of the focus of deformation. substituting tangential contact stresses,  and , which are proportional to the normal stress q: deep drawing technology with wall ironing in mass packaging industry 111 1 2and cos q q         (4) where µ1 is the friction coefficient at plastic deformation and µ2 is the friction coefficient at sliding, the normal stress with the plasticity conditions for the plane strain state reads:  tg rk q zr    11 (5) where kr is the true von-mises stress for plane deformation state. the separated element of volume in the focus of deformation from the previous balance equation (fig. 2) takes the form: 2 2 22 ( ) 2 ( ) 0z z z r zr xdx x r dr b x k r dx        (6) and at the entrance of the conical part of the focus of deformation the normal tensile stress rz has a value of 0 to make its exit receive maximum value rp:                    1 1 1 1 155.1 b i i srp a a b b kr (7) where 1/ cos / sin /b n tg        is the coefficient which depends on the matrix angle and the clearance between the punch and the ironing die, and  is the friction coefficient 8, 9, 12. ai-1 and ai are the ring areas of cross section at the wall of the can before and after the reduction in thickness, respectively. on entering the first ironing die, the material thickness has not yet been reduced, and therefore does not show any strain. as the material passes through the ironing die, there is a rise in the ironing force up to the value indicated in fig. 3, where force is plotted in relation to the reduction in thickness. in subsequent ironings, this force continues to increase up to the value of the maximum thickness reduction. fig. 3 shows the influence of the ironing die angle on the ironing force compared to the deformation of thickness or deformation degree. a small variation in force can be seen for a considerable increase in the ironing die angle, where this variation reaches almost zero at the finished ironing 12, 13. fig. 3 comparison of drawing forces for different values of the ironing die angle 112 s. ranđelović, m. milutinović, v. blagojević as can be seen in fig. 4a, the friction coefficient between the material and the ironing die significantly influences the ironing force, and at every stage of ironing this difference increases further. this shows that the greater force applied to the material, the greater the influence of friction on the process. in fig. 4b, it is shown that for a greater clearance there is a reduction in the ironing deep drawing force and for a smaller clearance there is an increased force, where the greatest influence of the clearance is in the finished ironing. this shows that if there is a misalignment between the punch and the ironing die, there will be a significant imbalance in force and consequent excessive wear of the punch and the ironing die 14. a) b) fig. 4 drawing force comparison for different values of: a) friction coefficient and b) clearance in order for the can to have the necessary strength during transport, process filling and sealing, it is necessary that the bottom of the can gets a much higher stiffness of the wall. this is achieved by forming the bottom with a characteristic profile (fig. 5) by deep drawing technology in future operations. fig. 5 increased stiffness on the bottom of the can deep drawing technology with wall ironing in mass packaging industry 113 at the end of this stroke, the punch with the can comes into contact with the base paneling tool and the can base is formed. when the ram is withdrawn, the can is removed from the punch by a stripper and conveyed out of the machine via an unloader belt. all lubricants from a metal surface in previous forming process must be removed by the process of washing. the wallironing lubricant used in the can forming process is removed prior to coating the can internally and externally. the cans are transported to the washer on a wide belt and conveyed through several washing chambers upside down. in this way the outside of the can is rinsed with tap water supplied through the jets located at the top and the inside of the can by the jets located at the bottom. immediately downstream of the washing unit, the can is dried with dry air at a temperature of approximately 200 °c in the drying oven. 3. graphic design and printing the outside of the can from the above, it is not easy to get a superior aluminum product such as a can, because this is a high quality, high productivity, very cheap and reliable product. metal forming technology meets the basic requirements of customers which have been considered conventional and common for many years now. what remains is the most sensitive part, how to reach the customer as soon as possible and earn his trust in the future on the global market. surely the main role is played by the contents of the can, its quality and price [15]. but one element, which has become increasingly crucial, is the visual effect (fig. 6). market conditions, strong competition, the modern way of life are all elements that affect the finished product. the cans are coated on the outside as protection against atmospheric influence and in order to apply a decorative design [16]. white, gold or transparent coating as well as aluminum-colored coating can be used according to customer specifications gained by various research and analyses of the market. nowadays, the coatings are water-based which is in accordance with modern requirements of environmental protection. fig. 6 sample of modern thermo graphic design and free shape (courtesy: chromatic technologies inc.) cans are spaced by an intake wheel and drawn on to the coating mandrel of the mandrel wheel by means of vacuum. they are then set in rotation around their own axis by the rotation belt. the coating film on the coater cylinder is then transferred to the cans positioned on the rotating coating mandrels (fig. 7). the coating is pumped from a 114 s. ranđelović, m. milutinović, v. blagojević coating container to the engraved cylinder which transfers the appropriate quantity to the rubber-coated coating cylinder where it is transferred to the cans. the coated cans are then blown off the coating mandrels and transported to the drying oven on a magnetic conveyor belt. fig. 7 general principle of painting cans and technical solution of best printing while the market recognizes standard cans found in shops and shopping malls, which are produced in large series, today one can very often find cans that are made in far smaller quantities. these are specially designed and shaped packages, with regard to their volume, shape and color, or to the occasion of a social event, festivals and gatherings, and are prepared in relatively small series with the current messages. complex and distributed innovation processes with a multitude actors call for modern information and communication technologies as supporting factors for virtualization and collaborative innovation management [15]. large manufacturers are now facing major business challenges. they have the opportunity to demonstrate their superiority in the market conditions with harsh global competition, and demonstrate their readiness to respond to the project team and complex requirements (fig. 6). they must offer effective design solutions, which need to be very fast in these conditions and found quickly on the product line, while offering innovative solutions with superior quality at the same time. products are customized both in appearance and excellent print quality, but now this goes a step further, where an effective form or effect (thermo cans which can change color) achieve an even stronger impact on the potential buyer. this is a challenge that requires special technological solutions, but also the superior quality that will leave potential customers breathless at a given moment. multidisciplinarity, flexibility, and the real emergency of practice, primarily business results, are very important here. these requirements are today often met by specialized design teams and agencies that take over the whole business of design and development of such products, in all their aspects, for the global market. 4. production process management for mass customization the described technology with the process parameters aims to illustrate the capabilities of a modern and complex technology that is now widely applied. almost unbelievable data only indicate the kind of level to which this process has been brought without any opportunity for error. design and development are entrusted to the main team with extensive experience, which distributes its proven results and achievements to the lowest level of implementation [17, 18]. the technical support, in cooperation with various external partners, has developed new deep drawing technology with wall ironing in mass packaging industry 115 measuring systems to enhance process quality: sensor systems to carry out machine diagnosis in wall-ironing presses, sensor systems to monitor axial force and compressed air support in the die-necker, etc. [19]. new product and process instrumentation and control equipment are developed at the center laboratory and then installed as standard quality assurance equipment at production plants. various sources of ideas are systematically evaluated. these also include the modified requirements of our customers [20]. the development center team collaborates closely with its customers in order to gain precise knowledge of and understand their needs and requirements (qfd methods). in addition, they want not only to implement innovative solutions but also to keep them exclusively for their customers (fig. 6). a very small number of employees and teamwork come to the fore in modern automated systems. process measuring equipment comprises measuring systems to measure and monitor the individual parameters of the production process and monitor compliance with process tolerances in real time. these include, for example, drawing force, deep drawing acceleration, length of cans, position tool, air pressure, temperature, etc. (fig. 8). in the implementation phase of production, only the given parameters are monitored via statistical process control charts where one can see their current trend and deviation, which indicate timely intervention, correction or the possible replacement of the critical elements. number of measurements number of measurements fig. 8 spc control chart for production parameters, washing temperature with target value of 49.1c and pressure with target value of 0.7 mpa the generated problems, daily reports and data production are carefully analyzed in order to keep the system in the specified control limits. it goes as far as having corrections, replacement of necessary tools and interventions on individual elements all performed at a 116 s. ranđelović, m. milutinović, v. blagojević central workplace, so as to correct, return and assemble the tools in one place, with the aim of reducing losses and empty work strokes. construction, design, and testing of machines, devices, tools and equipment are centralized and they are granted to highly skilled teams of professionals. their results and solutions are closely and strictly connected with the industrial exploitation, and are implemented only after market conditions and competition moves have been assessed. 5. conclusion a production system designed in this way shows great robustness and resilience to disturbances that are always present. its flexibility, on the one hand, and the speed of response to disturbance, on the other, is designed exclusively for mass production. the production process for aluminum cans is already technologically very advanced, and therefore it is useful to have the most information possible on material and tooling in order to optimize it. the tooling force calculations, measurements and analysis show that the material is not being exploited to the highest requirements, and therefore a diagram is constructed (fig. 3, 4a, 4b) which shows that if the production wants to reduce the final can thickness, there is a possibility to explore more of the material without causing defects. it was shown, as expected, that the friction coefficient and the clearance between the punch and the ironing die have great influence on the deep drawing force. the ironing die angle, however, did not prove to be essential to the process, and did not influence the deep drawing force very much. each business team only knows its job, which is a very narrow scope of knowledge and skills that are acquired and grown in a very long time. the fact that in europe tool repair and correction are performed in a single place, or that there are teams which specialize only in quick tool change and assembly speaks volumes. for example, design and development are centralized and located in the united states and germany (bonn) for all production capacity. with the above-mentioned productivity losses, the delay in time must be minimal. orientation towards the market is reflected in following the latest trends and design effects that can be detected on the can. customers appropriate such products and treat them as an integral part of their daily consumer basket. viewed from the perspective of business success, this has been the goal all along – to create a product that will generate large profits in the global market in the long run. acknowledgements: this paper is part of the research funded by ball packaging europe belgrade, republic of serbia. deep drawing technology with wall ironing in mass packaging industry 117 references 1. majstorović, v., 2001, quality management (in serbian), mechanical engineering faculty, beograd, 390 p. 2. eikelenberg, n., kok, i., 2003, tempelman, e., the role of product design in closing material loops, proc. of the 3th international symposium on environmentally conscious design and inverse manufacturing, tokyo, japan, december 8-11, pp. 605-610. 3. altan, t., tekkaya, a.e., 2012, sheet metal forming fundamentals, asm international, 296 p. 4. banabic, d., 2010, sheet metal forming processes, springer, 318 p. 5. marciniak, z., duncan, j. 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ranđelović, s, denić, b, mladenović, s, đorđević, g, 2010, aluminium industry, chance for mass customization and advancement of small enterprises, proc. 4th international conference mcp – ce, novi sad, serbia, pp. 130-134. 19. antonio, k., lau w., 2011, critical success factors in managing modular production design: six company case studies in hong kong, china, and singapore, j. of eng. and technol. manag., 28, pp. 168-183. 20. ross, ph j., 1996, taguchi techniques for quality engineering, mcgraw-hill, 455 p. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 217 227 bone healing in mice: does it follow generic mechano-regulation rules?  edoardo borgiani 1,2 , georg duda 1,2 , bettina willie 1,3 , sara checa 1 1 julius wolff institute, charité universitätsmedizin berlin, germany 2 berlin-brandenburg school for regenerative therapies, berlin, germany 3 research center, shriners hospitals for children-canada, department of pediatric surgery, mcgill university, montreal, canada abstract. mechanical signals are known to influence bone healing progression. previous studies have postulated inter-species differences in the mechanical regulation of the bone healing process. the aim of this study is to investigate whether mechanical “rules” explaining tissue formation patterns during bone healing in rat can be translated to a mouse model of bone regeneration. we have used an established mechano-biological computer model that uses finite element techniques to determine the mechanical conditions within the healing region and an agent-based approach to simulate cellular activity. the computer model is set up to simulate the course of bone healing in a femoral osteotomy model stabilized with an external fixator. computer model predictions are compared to corresponding histological data. generic mechanoregulation “rules” able to explain bone healing progression in the rat are not able to describe tissue formation over the course of healing in the mouse. according to the differentiation theory proposed by prendergast, mechanical stimuli within the healing region immediately post-surgery are determined to be favorable for cartilage and fibrous tissue formation. in contrast, in vivo histological data showed initial intramembraneous bone formation at the periosteal side. these results suggest that in mice, bone does not require as much stability as is required in rat to reach timely healing. this finding emphasizes the need to further investigate the species-specific mechano-biological regulation of bone regeneration. key words: mechano-biology, mouse bone healing, agent-based model, finite element analysis, tissue differentiation received november 1, 2015 / accepted november 20, 2015  corresponding author: sara checa charité universitätsmedizin berlin, julius wolff institute, augustenburger platz 1, 13353 berlin, germany e-mail: sara.checa@charite.de original scientific paper 218 e. borgiani, g. duda, b. willie, s. checa 1. introduction although bone is able to self-repair, in many situations its regeneration capacity is impaired, leading to delayed or non-unions. the healing process is known to be influenced by many factors; among them mechanical signals have been shown to play a fundamental role [1, 2]. it is well known that the course of bone healing is related to mechanical stability, which in turn influences the local mechanical conditions within the callus. mechanical instability has been shown to prolong the endochondral healing phase [3, 4], while a lack of mechanical stimulation may inhibit the bone healing response [5]. elucidating mechanical “rules” driving bone regeneration has been the focus of many studies in the last 30 years, since such knowledge has a great relevance in the design of clinical strategies for the treatment of bone fractures. although local mechanical strains/stresses within the healing region cannot be measured experimentally, they can be determined using computer modeling techniques, such as finite element (fe). using fe models to quantify the distribution of biophysical stimuli in a fracture gap, claes and heigele [6], determined a relationship between the magnitude of these stimuli and the distribution of the tissues present in histological sections. they observed intramembranous bone formation for strains smaller than +/5% and hydrostatic pressures smaller than +/ 0.15 mpa. endochondral ossification was associated with compressive pressures larger than about -0.15 mpa and strains smaller than +/15%. all other conditions were related to the formation of connective tissue or fibrous cartilage. although a globally accepted theory explaining the mechanical regulation of tissue repair does not exist [7, 8], it has been shown that callus tissue volume and shape changes due to mechanical loading are a good indicator of further differentiation processes [9]. over the last several years, these theories have been successfully implemented in computer models to simulate the mechanical regulation of tissue repair and differentiation in fracture healing [10, 11]. the role played by the local mechanical conditions during the healing process has been investigated by simulating the influence of fixation stiffness [12], gap size [13], fracture type [14] and external loading conditions on the healing outcome [15, 16]. so far, the majority of these studies have used the sheep as an animal model to compare computer model predictions to experimental data in order to formulate hypotheses about how mechanical signals drive bone healing responses [17]. this is due to the notion that the process of bone healing in sheep is thought to closely mimic that in humans. however, the rat [18, 19, 20] and mouse [21, 22, 23, 24, 25] have become increasingly popular as animal models in experimental bone healing studies due to ease of handling, low costs, and the availability of molecular biological tools. using a computer modeling approach, checa et al. [26] showed inter-species differences in the mechanical regulation of the bone healing process between sheep and rat, where different levels of mechanical stimuli were determined as favorable for the bone formation response. mice allow an additional advantage of relatively easy genetic modification, thus permitting the study of molecular mechanisms controlling fracture healing [27]. unfortunately, few experimental studies exist which have examined how mechanical factors influence bone healing in mice. holstein et al. [22] compared bone healing under rigid and flexible conditions in a closed fracture model using a conventional or a locking nail with higher stiffness. the initial phase of fracture healing was delayed under flexible conditions. gröngröft et al. [25] showed that a rigid internal plate induced solely intramembranous ossification, whereas a semi-rigid plate led to a mixture of endochondral bone healing in mice: does it follow generic mechano-regulation rules? 219 and intramembranous bone formation. röntgen et al. [24] compared the healing outcome using a rigid versus flexible external fixators to stabilize an osteotomy in the mouse femur. they showed delayed fracture healing with a larger callus formation and prolonged endochondral ossification in the flexible compared to the rigid case. steck et al. [28], characterized the time course of strength recovery and callus development of mouse femoral osteotomies stabilized with internal fixation plates that allowed either low or high flexibility (in bending and torsion). they observed earlier bridging of the mineralized callus under less flexible conditions. few computational models have been developed to investigate bone healing progression in mice. geris et al. [29] developed a mathematical model to investigate a murine tibia fracture semi-stabilized by an intramedullary fixating pin. although they were able to show a qualitative agreement between the experimentally measured and numerically simulated cartilage and bone formation, they did not consider the effect of fracture fixation stability on the healing outcome. isaksson et al. [30] investigated the emergence of a double cortex in the remodeling phase of healing in mice using an established remodeling algorithm. they showed that this peculiarity might be a consequence of different mechanical loading in mice, which may result from differences in skeletal structure or posture during gait. however, they did not investigate the influence of these loading conditions in the earlier healing phases. therefore, it remains largely unknown how the local mechanical strains within the healing region regulate intramembranous and endochondral ossification in mice, particularly during the early phases of healing. whether mechanical “rules” able to explain bone and cartilage formation in other species, such as in rat and sheep, can be translated to mice remain to be determined. therefore, the aim of this study is to investigate the mechanical regulation of bone healing in a mouse femoral osteotomy, stabilized with a rigid external unilateral fixator. using an established mechano-biological computer model, we have determined how local mechanical strains within the healing region relate to tissue formation responses over the course of healing. we hypothesize that, due to anatomical similarities between rat and mouse, bone healing in mice can be explained using the same mechanical rules as those earlier derived in rat. 2. material and methods to investigate the mechanical regulation of bone healing in mice, we used a previously established mechano-biological computer model which uses fe techniques to determine the mechanical conditions within the healing region and an agent-based modeling approach to simulate cellular activity [26]. computer model predictions were compared to histological data of an externally stabilized mouse femoral osteotomy model. 2.1. animal model a 0.5 mm osteotomy was performed on femurs from 26 week old (adult) c57bl/6 female mice under general anaesthesia (75 mg/kg ketamin and 1 mg/kg medetomidin intraperitoneal). the fracture was stabilized with an external fixator that was mounted onto the femur in a cranio-lateral direction by four pins (0.45 mm, risystem, switzerland) (fig. 1). after mounting the fixator in the correct position, a 0.5 mm osteotomy gap was created using a gigli saw (0.44 mm, risystem, switzerland). after 7,14 and 21 days of healing, the mice were sacrificed and the femora were fixated with paraformaldehyde, 220 e. borgiani, g. duda, b. willie, s. checa decalcified in edta for 2 weeks, dehydrated with alcohol and xylol, and embedded in paraffin. sections (4 µm-thick) were cut in a longitudinal direction and stained with movat pentachrome. healing proceeded via a combination of intramebranous ossification and endochondral ossification. intramembranous ossification was evident at all three time points (7, 14, and 21 days post-osteotomy) in the periosteal region and also at day 14 and 21 in the endosteal region. endochondral ossification was visible at 14 and 21 days of healing and was located primarily within the intracortical region of the bones. these islands of cartilage centered within the intracortical region extended into the periosteal and endosteal callus region (fig. 3c). 2.2. finite element model a fe model of the stabilized fracture was developed to determine the mechanical conditions within the healing region (fig. 1). the bone was modeled as a hollow cylinder where the inner region represents the medullary cavity (fig. 1). the osteotomy was simulated, creating a 0.5 mm gap between the bone ends. the model was developed in abaqus 6.12 and meshed with c3d8 elements, with an average element size of 0.15 mm (fig. 1b). fig. 1 a) dimensions of the finite element model obtained from histological images (values are reported in mm). b) finite element model developed to determine the local mechanical conditions within the healing region in a 0.5 mm femoral osteotomy in mouse stabilized with an external fixator [31]. different colors represent different material properties material properties were assigned following checa et al. [26] (table 1). polyetheretherketone (peek) material properties (e= 3800 mpa, ν=0.38) were assigned to the external fixator, while titanium properties (e=17000 mpa, ν=0.33) were used for the four nails. bone healing in mice: does it follow generic mechano-regulation rules? 221 table 1 material mechanical properties granulation tissue fibrous tissue cartilage immature bone mature bone cortical bone marrow young’s modulus (mpa) 0.2 2 10 1000 5000 5000 2 permeability (m 4 /n s * 10 -14 ) 1 1 0.5 10 37 0.001 1 poisson’s ratio (-) 0.167 0.167 0.3 0.3 0.3 0.3 0.167 bulk modulus grain (mpa) 2300 2300 3700 13940 13940 13920 2300 bulk modulus fluid (mpa) 2300 2300 2300 2300 2300 2300 2300 loading conditions in mice are largely unknown. based on anatomical similarities, we assumed that mice experience similar loading conditions than rats. we simulated two loading cases: a compression load and a combined compression and bending load. the compression load was equivalent to six times body weight (bw) (f= 1.5 n) and the bending load was such that it would result in the intact bone in a maximum bending moment of 10.7 bwmm (2.7 nmm) at the femoral mid-shaft, as reported by wehner et al. [32]. over the course of healing, a certain percentage of limb loading was simulated [26]. loads were applied in the proximal bone surface, while the distal end was restrained to move in all directions. 2.3. bone healing simulation to simulate the bone healing process inside the callus, a discrete 3d lattice mechanoregulation model was created following checa et al. [26]. briefly, the callus region was divided into a regular 3d grid, where each position represented a possible location for a cell and its extracellular matrix. the distance between two lattice points was considered 10 μm. the healing response was simulated as an iterative process. initially mesenchymal stem cells (mscs) originated from the periosteum and the marrow cavity (30% of volume fraction [32]) and were allowed to migrate and proliferate at a certain rate (table 2), following a random walk model. after cell maturation, considered to be 6 days, 30% of the mature mscs [30] were allowed to differentiate based on the local mechanical stimulus at their location, following prendergast et al. [8]. differentiated cells table 2 cell activity rates according to checa et al. [26] cell type proliferation rate (day -1 ) apoptosis rate (day -1 ) migration rate (μm/h) msc 0.6 0.05 30 fibroblasts 0.55 0.05 30 chondrocytes 0.2 0.1 osteoblasts 0.3 0.16 222 e. borgiani, g. duda, b. willie, s. checa were then assumed to synthesize a new extracellular matrix, changing the material properties of the tissue within the callus. new material properties were then updated in the finite element model and a new iteration started. 3. results 3.1. mechanical conditions within the healing region immediately post-surgery under both loading cases, strains and fluid velocities were the highest within the fracture gap, while lower magnitudes were predicted to occur at the periosteal side. under compression loading, strains up to 20 % were determined at the osteotomy gap. when combined with bending, strains increased up to 50 %, with the maximum located opposite to the fixator (fig. 2). the external load influenced fluid velocity, where higher velocities were predicted for the combined loading case. as for strains, maximum fluid velocities were found within the gap with values up to 0.005 and 0.01 mm/s in the compression and combined loads, respectively. fig. 2 mechanical environment inside the callus determined using finite element analysis. figure shows the predicted minimal principal strain distribution (a, c) and fluid velocity (b, d) in the situation immediately after surgery. two loading cases are shown: only compression (a, b) and combined compression and bending load (c, d) bone healing in mice: does it follow generic mechano-regulation rules? 223 fig. 3 prediction of fibrous tissue (brown), cartilage (green) and bone (yellow) at 7, 14 and 21 days post-fracture under compression loading (a) and combined compression and bending load (b). histology sections stained with movat pentachrome showing in vivo the formation of bone (yellow), cartilage (green), as well as fibrous connective tissue and bone marrow (reddish brown) over the course of healing (c) both loading conditions (compression and combined loading) led to similar tissue formation patterns during the first three weeks of healing (fig. 3). after 14 days, the computer model predicted fibrous tissue and cartilage formation in the fracture gap and in the periosteal region, respectively. no intramembranous ossification was predicted. after 21 days, bone formed through endochondral ossification was predicted in the periosteal region and the external callus partially joining the cortical ends (fig. 3). after 21 days, high amounts of fibrous tissue were predicted to be still present in the osteotomy gap. loading had an influence on tissue formation patterns at later time points. under compression loading, complete healing was predicted after 14 weeks, while combined compression and bending led to a non-union situation (fig. 4). 224 e. borgiani, g. duda, b. willie, s. checa fig. 4 predicted bone (yellow), cartilage (green) and fibrous tissue (brown) formation between 4 and 14 weeks under compression (a) and combined compression and bending load (b) the percentage of the different tissues formed within the callus area clearly showed that both loading conditions led to similar healing at the early stages, and continued following two different healing paths in the later healing phases (fig. 5). under compression loading, at the later stages of healing, the amount of fibrous tissue and cartilage decreased while the amount of bone increased. in contrast, under combined compression and bending load, the amount of fibrous tissue and cartilage remained relatively constant, indicating a nonunion situation. fig. 5 evolution of the healing response, described as temporal variation of the amount of different tissues predicted within the callus area for both loading cases: only compression (a) and compression in combination with bending (b) bone healing in mice: does it follow generic mechano-regulation rules? 225 4. discussion understanding how mechanical signals influence bone regeneration processes has great relevance in the design of clinical strategies for bone fracture treatment. although different species have been shown to respond to different levels of mechanical stimuli, little is known about how fixation stability and therefore mechanical signals influence callus tissue formation processes over time in mice. mice are a popular animal model due to the availability of a broad spectrum of molecular biological tools and ease of genetic modification. therefore, the aim of this study was to investigate the mechanical regulation of bone healing in a mouse femoral osteotomy model. an established mechano-regulation computer model was used to predict tissue formation patterns over the course of healing, which were compared to experimental histological data. generic mechano-regulation rules, which were able to describe bone healing in rat [26], were not able to explain experimentally the observed tissue formation patterns in a femoral osteotomy model in mouse [31]. computer model predictions showed periosteal cartilage formation at the early healing phases, which were not observed in vivo. finite element analyses showed that mechanical strains within the callus immediately postsurgery were significantly higher than those reported in a rat osteotomy model [26]. the mechanical stimuli created within the callus by the external fixation were in the range of those postulated to promote cartilage and fibrous tissue [8]. experimental studies have suggested that mice bone does not require as much stability for timely healing as the one in humans [25]. this could explain higher mechanical strains within the healing region immediately post-surgery in bone osteotomy models leading to uneventful secondary bone healing in mice [31]. experimentally, bony bridging in the mouse was reached after three weeks. in contrast, computer model predictions showed large amounts of fibrous tissue in the gap and no bony bridging after three weeks of healing. bony bridging was predicted at much later time points, approximately after 12 weeks under compression load. one reason for the difference results between the experimental and computational models is the absence of bone formation at the initial healing phases in the computational model. computer models predicted initial bone formation to occur through endochondral ossification, which requires a longer time period than intramembranous ossification. experimentally a combination of endochondral and intramembranous ossification was observed. additionally, in this study we assumed that cellular activity in the mouse occur at the same rate as in rat. faster cellular activity in the mouse compared to the rat could explain the slower bone healing response predicted by the computational model. however, this needs to be further investigated. we investigated two different loading conditions, compression and combined compression and bending. loading mode had an influence on tissue formation pattern predictions, especially at the later stages of healing. we observed that combined bending and compression loads led to a non-union situation. loading conditions in mice are not well understood. here, we assumed that due to the anatomical similarity, loading conditions in rat and mouse are comparable. our loading conditions were therefore based on values reported for the rat [31], scaled to take into account differences in animal weight. isaksson et al. [30] used a computer model to investigate the development of a second cortex during the remodeling phase of healing in mice and showed that its appearance could be explained when external bending loads were considered. they used 226 e. borgiani, g. duda, b. willie, s. checa a compression load of 0.75 n and a load that resulted in a bending moment at the fracture site of 1.8 nmm. following wehner et al. [32], we applied a compression force of 6 times the mouse body weight, which resulted in 1.5 n and a load which resulted in a maximum bending moment of 2.7 nmm at the femoral midshaft. our loads are approximately twice as those reported by isaksson. they estimated the loads based on a remodeling algorithm, to best describe bone shape. we decided to adapt rat derived loading conditions, from a musculoskeletal model, since they should better take into account anatomical and gait patterns. further studies will further investigate the influence of the loading conditions on mechano-biological predictions of bone healing progression. in summary, we have shown that mechano-regulation “rules” able to explain bone healing in rat are not able to explain tissue formation patterns over the course of healing in a mouse osteotomy model. it appears that bone healing in mice occurs under significantly higher magnitudes of mechanical strain compared to rat. these results are relevant if experimental 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bone healing, j bone miner res., 28. 33. wehner t, wolfram u, henzler t, niemeyer f, claes l, simon u, 2010, internal forces and moments in the femur of the rat during gait, j biomech., 43(13), pp. 2473-2479. 34. fan w, crawford r, xiao y, 2008, structural and cellular differences between metaphyseal and diaphyseal periosteum in different aged rats, bone, 42(1), pp. 81-89. 10915 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume230218071p © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper control of a wire tensioning system with force prediction using artificial neural networks vukašin pavlović1, miša tomić1, sergiu-dan stan2, milan banić1, miloš simonović1, miloš milošević1 1university of niš, faculty of mechanical engineering, niš, serbia 2technical university of cluj-napoca, faculty of automotive, mechatronics and mechanical engineering, cluj-napoca, romania abstract. in addition to the textile industry, the wire winding/unwinding process is used in various fields such as mechanical engineering, electronics, mechatronics and for military purposes. the wire that is wound/unwound has a combination of rotational and translation motion, thus exhibiting a complicated behavior. improper wire tensioning leads to problems such as entangling. one of the most crucial factors that affect the wire winding/unwinding process is the regulation of the wire tension. this paper briefly describes the developed wire tensioning system that can measure and control wire tension during the winding/unwinding process. various data was gathered based on the implemented proportional-integral (pi) control and sensors. this data was then used to build a neural network in order to predict force in the wire during the winding/unwinding process. key words: wire, tension control, winding/unwinding, neural network 1. introduction wire tensioning systems can be found in various mechatronic systems as their main sub-systems. these systems are present in machines in the textile industry [1], cranes [2], papermaking [3], medicine [4], etc. depending on the purpose, wire tensioning systems can be generally divided into two groups: constant tension control and variable tension control [5], where constant tension control is used in fields such as printing and papermaking, while variable tension control can be used in fields where wire can be wound/unwound on a winch or reel. many researchers have dealt with the design and control of such systems, and this paper will provide an overview of some of these investigations. received: december 18, 2023 / accepted april 23, 2024 corresponding author: vukašin pavlović university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, niš, serbia e-mail: vukasin.pavlovic@masfak.ni.ac.rs 2 v. pavlović, m. tomić, s-d. stan, m. banić, m. simonović, m. milošević kevac et al. [6] obtained the general mathematical model of a cable winding/unwinding system for several different constructions, where theoretical and simulation results were confirmed through the experimental analysis of one novel construction of the cable winding/unwinding system. kevac et al. [7] analyzed the phenomenon of non-linear and pulsed nature of the dynamic process of rope winding/unwinding on a winch. kang et al. [8] developed a device for cable unwinding in order to follow the cable unwinding behavior under various unwinding conditions. xu et al. [9] proposed a novel tension control method for a winding machine, which can regulate the fiber tension and transport speed of the winding process by governing the outputs of three different driven rollers in three levels. rodriguez et al. [10] introduced a reinforcement learning approach to optimize the wire profile generated by an automated wire winding machine. hultman et al. [11] presented a production method using industrial robots for automation of cable winding of electric machine stators, where the concept was validated through computer simulations and full-scale winding experiments. mousavi et al. [12] proposed a method to optimize the non-negative wire tensions, through the cables which were constrained based on the workspace conditions, in the redundant cable-driven parallel robots. the effectiveness of the proposed method was verified through an experimental study on the robocab cable robot. mishra et al. [13] proposed an unsupervised neural network algorithm to perform real-time forward geometrico-static analysis of an under-constrained suspended cable-driven parallel robot in a suspended configuration under the action of gravity. li et al. [14] designed, developed and evaluated an innovative miniaturized, low friction, back-drivable reducing mechanism for haptic or surgical robot applications, while francis et al. [15] proposed a cable driven robotic palpation system, where an indirect method based on cable tension observation was used to estimate the contact force. imamura et al. [16] designed a filament winding machine that can measure and control winding tension. for this purpose, two kinds of winding tension control were proposed and implemented using a proportional–integral–derivative pid or i-pd control. sheng-le et al. [17] introduced a closed-loop tension control system with the programmable logic controller (plc). lu et al. [18] developed an iterative learning sliding mode control scheme for wire tension control, while a disturbance observer was employed to estimate the wire tension for the implementation of sensorless wire tension control. abjabi et al. [19] designed a sliding-mode (sm) feedback linearization control system for a multi-motor web-winding system without a tension sensor. knittel et al. [20] presented multivariable h∞ robust control with two degrees of freedom and gain scheduling applied to winding systems, where a global controller, a semidecentralized controller, and a semidecentralized controller with overlapping were considered. wang et al. [21] researched and manufactured a closed-loop tension control system, where a neural network was applied to the system to overcome the shortcomings of the traditional proportional–integral–derivative control method. zhang et al. [22] presented a new control scheme for the winding process of stranded wire helical springs on a computer numerical control (cnc) machine to keep the wire tension uniform using the proportion integral neural network, while zhu et al. [23] proposed a neural network-based cable tension prediction model for tension control of the traction winch cable in order to substitute the traditional control that can make the cable too slack or too tight. control of a wire tensioning system with force prediction using artificial neural networks 3 this paper presents the design and control of a wire tensioning system as the subassembly of the larger self-propelled herding and pasturing system called roboshepherd. the main function of the developed system is to maintain a certain tension in the wire by winding or unwinding the wire, as well as to enable different formations of the roboshepherd system. the presented wire tensioning system consists of a force sensor located on one robotic unit and winding reels located on the other robotic unit. the wire is wound or unwound from the winding reels, whereby the sensor detects and sends information about the force in the wire to the control system. the developed control algorithm monitors the force in the wire and tightens or loosens the wire as needed. during the initial tests, it was determined that the traditional closed loop control used to control the wire tensioning system worked well when the robotic units did not move. however, during the movement of robotic units on terrains of different relief, the control algorithm showed certain shortcomings, i.e., it was not always able to respond in an adequate way in real time, for example, insufficient or excessive tension in the wire might occur. in order to overcome these shortcomings, an artificial neural network was built to predict the force in the wire which occurs during the winding/unwinding process of the wire. the paper’s objectives and main contributions are as follows: 1) the development of a wire tensioning system as a subassembly of the robotic unit, 2) the investigation of the behaviour of the developed wire tensioning system during different movement scenarios of the roboshepherd system, 3) during tests, various data were gathered and later used to train the artificial neural network, 4) the paper provides the results from the test of the traditional control of the wire tensioning system and from the simulation of the proposed artificial neural network for force prediction. the rest of the paper is organized as follows. section 2 presents a brief description of the roboshepherd system, and the design and closed-loop control of the wire tensioning system. the experiment in which the closed-loop control was tested is described in section 3, the design and results of the artificial neural network for predicting the force in the wire are presented in section 4, and the conclusion is given in section 5. 2. design and closed-loop control of the wire tensioning system roboshepherd is a swarm robotic system which acts as a movable polygonal electric fence that surrounds livestock animals in a field or forces them to move along the predefined path. the system consists of a minimum of four robotic units (ru), i.e., four movable pillars interconnected by wires that form the electric fence as shown in fig. 1. a pulsed electric current is sent along the wire from an energizer located on the robotic units. this fence serves to keep the animals away from the fence since when an animal touches the wire fence, the electric circuit closes, creating a short, safe, electric shock leading to the animal movement away from the fence. in the same way, the fence serves to protect the animals from the predators that can approach the other side of the electric fence. 4 v. pavlović, m. tomić, s-d. stan, m. banić, m. simonović, m. milošević fig. 1 roboshepherd system robotic units interconnected by wires each robotic unit consists of a vertical pillar, a movable platform on which the vertical pillar rests, a wire tensioning system and a force sensor. the wire tensioning system (fig. 2a) subassembly consists of a motor-reducer on which the upper plate and the lower plate are connected via a screw connection and on which linear ball bushings are mounted, through which a linear guide, carrying the entire subassembly, passes. the motor-reducer has a double side output shaft. winding reels are mounted on the top and bottom sides of the shaft. rotating the electric motor on which the winding reels are mounted, allows wire winding or unwinding, depending on whether the two adjacent robotic units are at a distance or approaching each other. one end of the wire is attached to one robotic unit via a force sensor (fig. 2b), while the other end of the wire is wound on a winding reel located on the motor shaft of the other robot unit. in this way the units are serially interconnected and form a closed loop. a schematic representation of the wire tensioning system is given in fig. 3. fig. 2 subassembly of the wire tensioning system (a) and force sensor (b) control of a wire tensioning system with force prediction using artificial neural networks 5 fig. 3 schematic representation of the wire tensioning system the tension control system is defined as a closed loop. the force in the wire is measured using a force sensor, and the gathered value is compared with the desired value. the difference between the desired and the current tension force determines an error. this error is sent to a pi controller. the pi controller was selected because it best suited the dynamic behaviour of the wire tensioning system. due to the inertia of the wire tensioning system, there was almost no chance for the appearance of the excessive force during the winding process and therefore it was not necessary to use a derivative component during control. based on the calculated error, the pi controller generated the pulse-width modulation (pwm) signal for controlling the voltage of the motor and therefore controlled the motor velocity. the turning of the motor caused the winding reels to turn, which made the wire to tighten or loosen (fig. 4). in this way the force in the wire was controlled. during the experiments, it was determined that the pi controller was able to bring the system to a setpoint. the setpoint or desired tension force was determined experimentally and it depends on the distance between the robotic units and the acceptable sag which occurs due to the weight of the wire itself. in the movement scenarios described in this paper the desired tension force was set to 80 n. fig. 4 block diagram of the control of the wire tensioning system 3. experiment to test the closed loop control of the wire tensioning system, an experiment was performed near the city of niš, serbia. for this purpose, four robotic units (ru1, ru2, ru3 and ru4) were placed on location and this was the initial position of the robotic units (fig. 5). after that, the robotic units were interconnected with two rows of wires, where each row had one pair of wires (positive and negative wire). in the initial position, the distance between the two front robots and the distance between the two back robots (ru16 v. pavlović, m. tomić, s-d. stan, m. banić, m. simonović, m. milošević ru2 and ru3-ru4) was the same and was approximately 34 m. the distance between the two left side robots (ru2-ru3) was the same as the distance between the two right side robots (ru4-ru1) and was around 28 m. during the experiment different movement scenarios of the robotic units were carried out. some of the most important scenarios were where one or more robotic units moved, with the distance between them changing, such as: all robotic units move with a change in the initial distance between them, three robotic units move while one waits, two robotic units move while two wait, one robotic unit moves while others wait. also tested were the scenarios where all robotic units move in the same direction maintaining the distance or some robots move while others wait. this paper presents the scenario in which all robotic units moved. robotic unit 1 (ru1) started from the initial point rs1 and moved to rs1', robotic unit 2 (ru2) started from the initial point rs2 and moved to rs2', etc. in this scenario the back robotic units (ru3 and ru4) maintained the distance between them, the right side robots (ru4 and ru1) slightly increased the distance, while the distance between the front robotic units (ru1 and ru2) decreased during movement. the distance between the left side robotic units (ru2 and ru3) increased significantly as illustrated in fig. 5. fig. 5 initial position and movement scenario control of a wire tensioning system with force prediction using artificial neural networks 7 in order to minimize the error of position of the robotic units below 1 cm, all robotic units were equipped with real time kinematics (rtk) global positioning system (gps) trackers with long range communication radio (lora). these rtk gps trackers take in the signals from the global navigation satellite systems along with a correction stream via lora and calculate the location of the robotic units within 1 cm accuracy in real time. also, all robotic units were equipped with force sensors to measure the forces in wires and current sensors to measure the current of the motor which winds/unwinds wires. each robotic unit had a personal computer equipped with appropriate hardware (to which sensors were connected) and software developed in the labview environment for control. these personal computers were used for data acquisition, control of the movement of the robotic units and control of the wire tensioning systems during the experiments. data from the gps trackers, current and force sensors were recorded for further investigation. during these field tests of the roboshepherd system several problems with the wire tensioning system occurred. fig. 6 shows the variation in tension force in the pair of wires between ru1 and ru2 during the movement of the robotic units in the selected scenario. these variations happened for several reasons. the wires used for the fence were viscoelastic, therefore, they did not act the same during the winding/unwinding processes. another factor that contributed to the force variations was the long time needed for the information from the force sensor to travel to the controller. this led to the slow response of the control of the wire tensioning system, which further led to the inability to maintain the wires in the desired constant tension, resulting in wire loosening. to try to overcome these problems, an artificial neural network for predicting the force in wires was built using the data gathered from different sensors mounted on the robotic units, as well as some data measured during the control of the wire tensioning system. fig. 6 measured total force in pair of wires 4. artificial neural network for prediction of force in wire in this study, a standard backpropagation artificial neural network (ann) was used. the architecture of an ann typically consists of three layers: an input layer, hidden layer, and output layer [24]. the ann created for this study had four layers: an input layer, two hidden layers and an output layer. the variables used in the input network layer were the 8 v. pavlović, m. tomić, s-d. stan, m. banić, m. simonović, m. milošević pulse width modulation (pwm) signal, which was used to control the motor velocity, the current in the motor measured via a current sensor, and the distance between two robotic units. the distance between the robotic units was calculated based on the position data gathered from gps. each robotic unit was equipped with gps. the distance between two robots was calculated using a haversine algorithm. the haversine algorithm calculates the distance using latitude and longitude. this algorithm aims to find the nearest straight line distance from two given locations [25]. the variable in the network output layer was the total force in a pair of wires measured on the force sensor. to create the proposed ann, some data gathered from the above-mentioned scenario where all robotic units moved was used. four sets, each consisting of 523 pieces of data that were used for the creation of the ann, were related to the two front robotic units ru1 and ru2 during the movement. the initial distance between robots ru1-ru2 was 34 m, while the final distance was 14 m. during the test, robotic unit 1 traveled 20.8 m from position rs1 to rs1', while robotic unit 2 traveled 32.7 m from position rs2 to rs2'. the measured total force in the wires ranged from 11 n to 105 n, while the maximum motor current was 15 a. the ann model was created using the matlab (the math works, inc. usa) software package. the input layer of the ann included 3 input neurons, while the output layer included one output neuron. the two hidden layers had 20 hidden neurons each as shown in fig. 7. the trial-and-error approach was used for determining the number of hidden layers and hidden neurons in these layers. fig. 7 artificial neural network with 3 input variables, 2 hidden layers with 20 neurons each, and 1 output variable control of a wire tensioning system with force prediction using artificial neural networks 9 the ann was trained by using the backpropagation levenberg–marquardt algorithm, due to its high accuracy and fast convergence. during the training, the mean squared error was used for performance measuring. the dataset consisted of 4 variables where each variable had 523 samples. these variables were the pwm signal, the distance between the robotic units, the motor current, and the total force in the wires. in order to use these variables in the ann the dataset was divided into two matrices. one matrix consisted of the pwm signal, the distance and the motor current values, and was called “inputs”. the second matrix had the values of the total force in the wires and was called “targets”. when the matrices were imported, training, validation and testing datasets for neural network were randomly selected. using the “inputs” and “targets” data, the ann calculated “outputs”. 367 pieces of data were used as a training sample for network training. the network was adjusted according to the errors during training. for measuring network generalization, a validation sample of 78 pieces of data was used. when the generalization stopped improving, network training was halted. during the ann training process, although a larger number of iterations was expected, it was observed that fewer iterations were sufficient to train the ann (fig. 8). in the end, to verify the network performance, 78 pieces of data were used as an independent testing sample. fig. 8 mean squared error during the ann training process for force prediction in order to measure network performance, a correlation coefficient r was used and the obtained results are shown in fig. 9. this coefficient shows how well the network was trained by matching the predicted “outputs” with real “targets”. the higher value of r means the better network performance, where r = 1 corresponds to the perfectly matching relationships between “targets” and “outputs”. as can be observed in fig. 9, during the training of the ann the correlation coefficient was 0.98693, which means that the network was trained very well. the trained network was tested on the testing dataset where r was 10 v. pavlović, m. tomić, s-d. stan, m. banić, m. simonović, m. milošević 0.97527, while the total network performance of the trained ann was 0.97981, which was recognized as a very satisfying result. fig. 9 the results of network performance for training, validation and test dataset table 1 provides a comparison between several metrics. mean squared error (mse), root mean squared error (rmse) and mean absolute error (mae) are calculated for the desired tension force and measured force using a force sensor, as well as the desired tension force and the outputs given by the trained ann. as can be observed, the ann gives a slightly smaller error thus giving a better result. table 1 comparison between measured and ann predicted force mse (n2) rmse (n) mae (n) measured force 1472.66 38.375 32.49 ann predicted force 1444.18 38.002 32.37 control of a wire tensioning system with force prediction using artificial neural networks 11 having the obtained results in mind, it was concluded that the ann can be used for the prediction of the force in the wire with high accuracy. therefore, the new control of the wire tensioning system where the force sensor can be substituted with the ann was proposed and is graphically presented in fig. 10. the implementation and assessment of performance of this new control algorithm will be the subject in further research. fig. 10 proposed control of the wire tensioning system 5. conclusion this paper briefly described the design of the wire tensioning system used as a subsystem for the swarm robotic system called roboshepherd. the main task of the developed system was to maintain the optimal tension in the wire by rotating the winding reels that wind or unwind the wire during the utilization of the roboshepherd system. in order to obtain relevant data for the design of this system, tests were carried out at different locations that included environments with different relief and vegetation. the problem of wire tensioning control in the dynamic system when the robotic units move was considered. using classical control methods for wire tension led to certain shortcomings and did not provide satisfactory results due to the viscoelasticity of the wires and the long system response time. the solution to this problem was found in the application of modern methods and algorithms from the domain of artificial intelligence. therefore, the use of artificial neural networks for force prediction was investigated. based on the different data gathered from the test including a movement scenario where all robotic units moved, the ann was created. the created ann showed high correlation between the target data – total force in pairs of wires (measured force) used for network training and the data gathered from the simulation of the artificial neural network (predicted force). mse, rmse and mae were also calculated for a comparison between the measured and the ann predicted force. the obtained results are in favour of using the ann as a possible substitution for the traditional control. based on these results, a new type of control of the wire tensioning system using artificial neural networks is proposed. acknowledgement this paper is a part of the research financially supported by the innovation fund of republic of serbia and coming computer engineering through the project roboshepherd, and the ministry of education, science and technological development of the republic of serbia (contract no. 451-03-9/2021-14/200109) references 1. fazal, m.z., khan, s., abbas, m.a., nawab, y., younis, s., 2021, machine learning approach for prediction of crimp in cotton woven fabrics, technical gazette, 28(1), pp. 88-95. 12 v. pavlović, 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scientific paper study of friction compensation model for mobile robot’s joints yanjie cao, norzalilah mohamad nor, zahurin samad school of mechanical engineering, universiti sains malaysia (usm), penang, malaysia orcid ids: yanjie cao https://orcid.org/0000-0002-7878-5309 norzalilah mohamad nor https://orcid.org/0000-0002-7819-172x zahurin samad https://orcid.org/0000-0001-6400-3284 abstract. frictional forces inside the joints of mobile robots hurt robot operation's stability and positioning accuracy. therefore, establishing a suitable friction force compensation model has been a hot research topic in robotics. to explore the robot joint friction compensation model, three friction compensation models: linear, nonlinear, and neural network models, are developed in this paper. based on the deep learning algorithm for three models at low speed, high speed, acceleration, and uniform speed training test, respectively results have been obtained. the test results show that the best friction compensation effect comes from combining neural network models in acceleration and a consistent speed state way. the friction compensation model trained this way yielded superior results to the other combinations tested. finally, using the method, a friction compensation model trained by adding a neural network to the feedforward control torque was tested on a four-wheeled mobile robot platform. the test results show that the relative error of the torque caused by the friction of each joint is reduced by 15%-75% in 8 groups of tests, which indicates that our friction compensation method has a positive effect on improving the accuracy of the joint torque. key words: deep learning, mobile robot joint, friction compensation, neural network 1. introduction the presence of friction, especially nonlinear friction, can cause tracking errors, hysteretic motion, and limit-loop oscillations in industrial robots during low-speed, highprecision operation, affecting the smoothness of robot motion and reducing control accuracy [1]. therefore, to improve the motion accuracy of mobile robots, it is necessary to study and analyze the frictional forces generated by the joints during industrial robot operation and implement compensation. the research on the analysis and settlement of frictional forces of industrial robot joints has become a hot research subject in the field of the industrial robot [2], which has significant application value for improving the operation precision of industrial robot operation and overall performance. received: june 05, 2023 / accepted october 23, 2023 corresponding author: yanjie cao school of mechanical engineering, universiti sains malaysia (usm), 14300 nibong tebal, penang, malaysia e-mail: caoyanjie@student.usm.my https://orcid.org/0000-0002-7878-5309 https://orcid.org/0000-0002-7819-172x https://orcid.org/0000-0001-6400-3284 722 y. cao, n.m. nor, z. samad a friction phenomenon between the joints of industrial robots is more complex, both sliding friction and rolling friction. establishing a suitable friction model is the mainstream approach in current academia. some empirical friction models have been proposed to compensate for the friction of robot joints. according to whether differential equations can describe the friction phenomena, there are static and dynamic friction models [3]. in the past, the linear and nonlinear viscous friction models appeared and were often used together with the coulomb friction model, giving the coulomb+viscous model [4], on which virgala et al. [5] introduced static friction to obtain the static friction + coulomb + viscous friction model. the common point in the above classical friction models is that the linear relationship between velocity and sliding friction and the interconversion between dynamic and static friction is carried out discretely. richard et al. [6] proposed an exponential model to describe the phenomenon in 2007, which was later compensated by nikfar et al. [7], an improved stribeck friction model, in which the speed dependence of friction force was added. keck et al. [8] studied the elastoplastic friction model parameter identification process, whereby a friction compensator was developed to estimate the state of the elastoplastic friction model. beerens et al. [9] proposed a reset integrator control strategy to achieve robust global asymptotic stability for set points with unknown static friction. marek wojtyra [10] discussed the joint friction of a multi-body system with a closed-loop kinematic chain and solved the issue of uniqueness and high parameter sensitivity of the kinematic equation in a static friction zone. although the static friction force compensation model has the characteristics of a simple mathematical model structure and easy identification of parameters, it has a limited effect on improving the system performance. therefore, researchers have studied the dynamic friction model. lampaert et al. [11] designed a dynamic friction model suitable for control purposes, which predicts the stribeck effect and properties such as friction hysteresis, transition behavior, and disengagement forces. xu et al. [12] proposed a friction compensation strategy based on robust adaptive control (arc), which can guarantee the transient performance and final tracking accuracy of the mechanism in nonlinear dynamic friction scenarios. joanna et al. [13] proposed a method to identify lugre kinetic friction parameters that can significantly reduce trajectory tracking errors. the scholar swevers [14] proposed the leuven model to describe the hysteresis phenomenon more adequately, which is more accurate than the lugre model, but recognizing the parameters of the model is more complicated. in addition, lee et al. [15] proposed a new dynamic friction compensation technique that utilizes a pd control system and observer-based self-adaptive evaluation of friction to enhance the performance of tracking for motion control effectively. maged et al. [16] proposed a physically motivated friction model that can be integrated quickly and smoothly into classic dynamic friction models such as lugre and gms. piasek et al. [17] designed a method to identify lugre's static and dynamic friction parameters and proved by experiments that the compensation of the dynamic friction model dramatically reduces the trajectory tracking error. isaac et al. [18] designed the ida-pbc friction compensation method of the lugre model integrated to enhance the controller performance of the underactuated mechanical system at low speed. these have been meaningful explorations, but the dynamic friction compensation model in practical applications is still complex and challenging to identify parameters. in recent years, some scholars have started using neural networks to study friction compensation. ciliz et al. [19] proposed a friction compensation strategy based on the coulomb and viscosity models, applying a neural network to predict the magnitude of friction. doan et study of friction compensation model for mobile robot’s joints 723 al. [20] disclosed a self-adaptation control system for robot operation using artificial neural networks for error compensation, which can compensate for unmodeled friction parts. grami et al. [21] applied a neural network algorithm model to estimate the friction inside the robot joint and realized the estimated friction in the actual friction compensation. in addition, yen et al. [22] designed a robust self-adaptable control approach with a fuzzy wavelet neural network system, which can reduce friction, external disturbance error and parameter variation and achieve a better control effect. ali et al. [23] designed a control algorithm that combines fit-smc, red and ffnn estimators, making the model-based friction compensation of the multi-degree-of-freedom robot system effective. similarly, wei et al. [24] disclosed an adaptive rbf neural network algorithm, which can reduce the influence of nonlinear and uncertain parts in the system on performance. experimental results show that the jitter problem inherent to the sliding mode controller is also suppressed. because of the beneficial effects of deep learning algorithms in dealing with nonlinear and uncertain friction compensation, we decided to explore and study robot joint friction compensation in this direction. we establish linear, nonlinear, and neural network models and use deep learning algorithms to train them in four states: high speed, low speed, acceleration, and uniform speed. the conclusion shows that the neural network model has the best training effect under acceleration and consistent pace and is more suitable for robot joint friction compensation. the deep learning process we study allows automatic friction compensation to be applied at each mobile robot joint to obtain the best and most personalized friction compensation, which has practical value and improved accuracy. the rest of this paper has the following arrangement: section 2 describes the friction compensation modeling method, and the scheme design, data collection, and data division are included. section 3 presents the experiments and discussions. section 4 shows the conclusions and future works. 2. friction compensation modeling 2.1. program design the size of the friction in the robot joint is related to the structure, speed, load torque, lubrication conditions, and temperature of the robot joint. nevmerzhitskiy et al. [25] disclosed the effect of temperature on friction. bittencourt et al. [26] showed that the load significantly influences the friction model. raviola, andrea et al. [27] designed a mathematical model that can identify the dynamic parameters of the manipulator. experiments show that this model reduces the error of the estimated joint torque of the robot. the nonlinear frictional forces corresponding to this friction phenomenon are generated on the mobile robot’s body, but direct measurement cannot obtain the corresponding frictional moment values. therefore, the first task is to simplify the mobile robot dynamics model to get the importance of the friction moments of the robot’s joints. the mathematical expression of the robot’s dynamics model is shown in fig. 1. the parameters involved are shown in table 1. 724 y. cao, n.m. nor, z. samad fig. 1 wheel of mobile robot motion model table 1 motion model parameters parameters description m1 equivalent mass of the first rod me equivalent mass of the second rod and payload l1 length of first rod (wheel distance) lc1 distance to the center of the first rod equivalent mass (wheel distance) lce distance to the center of the second rod equivalent mass i1 equivalent moment of inertia of the first rod ie equivalent moment of inertia of the second rod q1 angle between the first rod and the horizontal axis q2 angle between the first rod and second rod δe angle between the second rod and the vertical axis the kinetic model relates to the friction compensation term f as shown in eq. (1). ( ) ( , ) ( ) ( )m q q c q q q g q f q + + + = (1) where q∈rn is the desired coordinates of the robot in space, q̇∈rn is the joint angular velocity, q ̣̈ ∈rn is the angular acceleration of the joint, m(q)∈rn⨉n is the robot's positive definite inertia matrix, c(q,q̇)∈rn⨉n is the coriolis force matrix, g(q)∈rn is the gravitational vector, f(q̇)∈rn is the robot joint friction,  ∈rn is the joint input torque, and n represents the nth robot joint. +2 cos( )+2 sin( ) + cos( )+2 sin( ) + cos( )+ sin( ) 2 2 2 2 2 2 ( )   =     q q q q m q q q           (2) study of friction compensation model for mobile robot’s joints 725 (-2 sin( )+2 cos( )) ( )+ cos( )) sin( ) cos( )) 2 2 2 2 2 2 2 2 1 ( sin ( , ) ( 0 −  =   −  q q q q q q c q q q q q       (3) ( )+ sin( )+( + 1) cos( )+ ( ) 1 2 1 2 1 1 2 1 2 2cos 2 2cos( ) ( ) 2 2sin  + +  =   + +  e q q e q q e e q g q e q q e q q       (4) where: 2 2 2 1 1 1 1c e e ce ei m l i m l m l = + + + + , 2= +e e ci m l e , 1 cose ce em i l = , 1 sine ce em i l = , 2 1 1 1 1 1 1cl me m i l i m l= − − , 2 1g /e l= the actual compensation for the various frictional forces in the robot joints during motion is done in the following manner (fig. 2). fig. 2 friction compensation closed-loop control system structure because the friction force includes static friction, sliding friction, viscous friction, etc., the influencing factors are relatively complex and have non-linear characteristics. our study uses three different models to model the frictional forces: 1) a traditional linear mathematical model, 2) a nonlinear mathematical model and 3) a deep learning neural network model. 2.2. building the model three models are investigated separately: 1) linear model, 2) nonlinear model, and 3) neural network model. 2.2.1. traditional friction model: linear model the essence of the linear compensation model is to use the form of setting linear coefficients to realize the friction compensation by using the coefficients co-c4: 0 1 2 3 4· · · ·f c c v c c t c q= + + + + (5) where, f is the overall friction force, v is velocity, τ the torque, t is the temperature, q is the position term and co-c4 are the constant (co takes the value 1.0, c1 the value 1.2, c2 the value 3.6, c3 is 1.0, and c4 takes the value 4.3). 726 y. cao, n.m. nor, z. samad 2.2.2. traditional friction model: nonlinear model the conventional friction model can also be expressed in a nonlinear form as: 2 2 ( ) exp tanhbrk c c st st coul v v v f e f f f fv v v v       = − − + +              (6) where: 2st brkv v= (7) 10 brk coul v v = (8) here, f is the overall friction force, fc the coulomb force, fbrk the maximum static friction, vbrk is the maximum static friction velocity, vst the viscous friction velocity threshold, vcoul the coulomb velocity threshold, v the relative velocity and fv is the viscosity coefficient. 2.2.3. neural network model because the robot joint friction has significant nonlinear characteristics, we use a neural network model to solve the problem of friction compensation. the input feature quantities of the neural network model are torque, velocity, position, temperature, etc. its output is the friction force. the weights in the network are adjusted according to the loss function of its error magnitude to make the model converge. the feedforward neural network has a simple structure and is widely used. it can approximate any continuous function and square integrable function with any precision. moreover, it can accurately implement any finite training sample set and obtain complex nonlinear processing capabilities. we designed a feed-forward neural network in which information is input from the input layer, and neurons in each layer receive inputs from the previous layer and output to the next layer up to the output layer. the constructed neural network model includes several stacked structural modules, and the number of modules can be changed through the programming api. each stacking structure module consists of a batch normalization layer, a fully connected layer, and an activation layer (fig. 3). fig. 3 deep learning algorithm model study of friction compensation model for mobile robot’s joints 727 a neural network consists of an input, implicit, and output layer. the number of hidden layers and neurons per layer is adjustable. it is worth noting that the number of different layers and the number of neurons per layer significantly impact the model performance. thoma et al. [28] showed that feedforward neural networks with two hidden layers have better generalization ability than feedforward neural networks with one hidden layer. too few neurons in the hidden layer can lead to underfitting, and conversely, too many neurons in the hidden layer may lead to overfitting. for the friction compensation model, the robot's friction force is a nonlinear function related to various influencing factors. the five input features of the friction compensation algorithm model are the position, velocity, theoretical torque, actual torque, and temperature of each wheel joint. the output is the friction force of each joint. considering the model's fitting ability and convergence difficulty, the number of implied layers is three, and the number of neurons per layer is determined to be 5 (fig. 4). fig. 4 neural network model 2.3. data preparation 2.3.1. data acquisition the complex physical characteristics and mechanical structure inside the robot joints lead to frictional forces such as coulomb friction, static friction, and viscous friction during the motion, and they have nonlinear characteristics. usually, from the relative rest to the relative sliding between contact surfaces, according to the different main factors that determine the magnitude of friction, there are four stages [29], as shown in fig. 5: ▪ i stage – the contact surface elastic deformation stage; ▪ ii stage – the boundary lubrication stage; ▪ iii stage – the partial liquid lubrication stage; ▪ iv stage – the whole liquid lubrication stage. 728 y. cao, n.m. nor, z. samad fig. 5 diagram of relationship between friction and velocity factors that affect the friction need to be considered in the sampling process: 1) when the robot joint moves, the instantaneous angular velocity change will cause the friction force magnitude to change accordingly. 2) the rotation of the robot joint at different positions causes the variation of the friction force, which is due to the other gaps inside its joints when the rotational axis of the robot joint is at different positions. 3) the motor and reducer of the target joint will cause micro-contact surface expansion, shrinkage and lubrication differences due to temperature changes. these factors make the friction change with temperature. 4) the pressure change caused by different radial loads of the target joint causes the friction force to change with the change of the axial load. 5) different axial loads on the target joint lead to pressure variations, resulting in frictional force variations with axial load. 6) the mathematical relationship of "kinetic model (theoretical torque) + compensation model = measured torque". determination of the data collection scheme implies the following steps: 1) acquisition of torque values at different speeds of wheel joints. 2) to collect the torque values of the wheel joints at different positions. 3) collect torque values at different temperatures of wheel joints. 4) based on the change of axle position, the torque data value is calculated according to the theoretical equation. 5) using sensors to measure the actual torque and calibrate the torque accuracy of the servo system, etc. the specific experimental approach was to program the mobile robot to specify eight different motion paths so that each joint axis of the robot moved at a different position and a different speed. the load, position, velocity, torque, and temperature of each joint of the mobile robot were recorded at sampling intervals of 4-10 ms during the eight sets of experiments. 29 sets of data were collected, including data for joints 1 to 4 of the robot, as shown in table 2. study of friction compensation model for mobile robot’s joints 729 table 2 data collection collection data types number of data collected sampling time 1 position planning value of each axis 4 speed planning values for each axis 4 theoretically calculated torque for each axis 4 actual torque of each axis 4 feedback position of each axis 4 feedback speed per axis 4 feedback torque of each axis 4 2.3.2. data processing before the model training, we perform the necessary processing on the collected data, mainly including data cleaning, classification, equalization, enhancement, etc. [30]. 1) when data collection, as the path data of robot movement, may have the same group and missing data, data cleaning can ensure data integrity, uniqueness, and consistency. 2) data classification is performed in advance according to the four states of acceleration, uniform speed, high speed, and low speed in robot motion, which facilitates the corresponding neural network model. 3) because data acquisition cannot traverse all the motion data of robot conditions, data equalization can make model compensation more effective. 4) to improve the robustness of the model, data enhancements such as positive and negative transformations, translational transformations, and noise addition are utilized on the collected data, thus enhancing the generalization capability of the friction compensation model. as the usage of the three different models implies training and testing of the models, in this work, 80% of the data collected render the training set, and 20% are the test set. in fig. 6, the orange part is the test dataset, while the blue part is the training dataset. fig. 6 data set partitioning 730 y. cao, n.m. nor, z. samad 3. experiments and discussions 3.1. model training this study uses two ways of dividing the data, and two different models are trained for each data division: 1) the first one is based on high or low speed and divided into low-speed and highspeed cases (with joint speed of 200 r/min as the boundary to distinguish between high and low speed). 2) the second is according to the acceleration speed rate and uniform speed case. (because the acceleration and deceleration cases are similar, only the acceleration scenario is taken for analysis). first, the data set used for initializing the neural network is prepared. in this study, the number of neurons in the input layer is 5. the number of neurons in the output layer is based on the solution to the problem, i.e., the friction of the robot joints and the neural network is trained using the five items: position, velocity, theoretical torque, actual torque, and temperature of the robot joints as the input to the neural network. second, the loss function when the neural network is defined during training. this study establishes the loss function as the following mathematical expression. 2loss loss(x ,y ) (x y )i i i i= = − , (9) where, xi is the neural network's practical output value, which is the forecast value, yi is the true value, i denotes the number of training data. therefore, the training goal is to reduce the loss value as much as possible. finally, there is the training of the neural network. the exact procedure of the training is as given in fig. 7. step 1: define the optimizer, select the optimization algorithm as well as the learning rate, pass all parameters of the neural network into the optimizer; define the loss function, which is the equation for calculating the error between the predicted and actual values of the neural network, and use eq. (9) here. step 2: the training data is used as the input of the neural network. the training data, i.e., the position, speed, theoretical torque, actual torque, and temperature values of the robot joints, are used to obtain the corresponding output values through the neural network. step 3: the error between the output value of the neural network and the objective value, which is the value of the friction torque of the robot joint, is found according to the error calculation equation. step 4: clear the residual update parameter values from the previous iteration. step 5: error back propagation and calculation of updated parameter values. step 6: make the parameter update values imposed on the parameters of the neural network. step 7: determine if the present maximum iteration number is reached. if the current step number is less than the maximum iteration number, go to step 2. if the current step number equals the maximum iteration step, then end. study of friction compensation model for mobile robot’s joints 731 fig. 7 neural network training flow chart in this study, the learning rate was set to 0.01, and the maximum number of iteration steps was set to 500 during training. the training tool uses the optimizer in pytorch and the stochastic gradient descent algorithm sgd (stochastic gradient descent) [31]. since the stochastic gradient descent algorithm uses one sample in each iteration to update the parameters, the training is faster, and the specific mathematical expression for parameter update is as follows: 1 θj jθ θ a loss+ = +  (10) where a is the learning rate, 𝜃𝑗 is the weight coefficient under the jth iteration, j is the number of iterations, θloss is the gradient of the loss function. when the number of iterative steps reaches 500, the loss value in the training process tends to zero, and the training ends. the final neural network fitting results (figs. 8-11) 732 y. cao, n.m. nor, z. samad show that the output value of the neural network and the target value are the same for a given velocity of the robot joint, and the fitting effect is good. therefore, it is assumed that the initialized neural network has already learned the information about the correlation between the speed of the robot joints and the friction torque: 1) by low-speed and high-speed phases low-speed phase fig. 8 convergence diagram of low-speed stage high-speed phase fig. 9 convergence diagram of high-speed stage study of friction compensation model for mobile robot’s joints 733 2) by acceleration and uniform phase acceleration phase: fig. 10 convergence diagram of the acceleration phase uniform phase: fig. 11 convergence diagram of uniform velocity stage the conclusions are that the algorithm of the friction compensation model divided by acceleration/deceleration/homogeneous speed converged fastest, and the algorithm converged between 40-150 iterations. the algorithm converged fastest in the homogeneous speed condition and converged after 40 iterations because the friction force in the homogeneous speed condition was most stable. the algorithm converges slowest in the low-speed case, going through two convergence processes and converging after 150 iterations, which is due to the existence of multiple state transition processes for the friction inside the joint in the low-speed case, involving various stages such as elastic deformation stage, boundary lubrication stage, and partial liquid lubrication. 734 y. cao, n.m. nor, z. samad 3.2. training results the following shows the training effects of these three modes in four different scenarios: high speed, low speed, acceleration and uniform speed. the training results of the linear model are given in fig 12. fig. 12 coefficient of determination curves tested using the linear model: a) low-speed state, b) high-speed state, c) accelerated situation, d) uniform speed situation a) b) c) d) study of friction compensation model for mobile robot’s joints 735 the training results of the nonlinear model are presented in fig. 13. fig. 13 coefficient of determination curves tested using the nonlinear model: a) low-speed state, b) high-speed state, c) accelerated situation, d) uniform speed situation a) b) c) d) 736 y. cao, n.m. nor, z. samad finally, the training results of the neural network model are depicted in fig. 14. fig. 14 coefficient of determination curves tested using the neural network model: a) lowspeed state, b) high-speed state, c) accelerated situation, d) uniform speed situation a) b) c) d) study of friction compensation model for mobile robot’s joints 737 the following conclusions are drawn from the results presented above: ▪ the learning results of neural networks are significantly better than other mathematical models. ▪ the model trained by the plus and uniform speed stage performs better in the test set than the model trained by the high and low speeds. therefore, the combination of "neural network+training acceleration or homogeneous model" should be adopted to obtain the best friction compensation results. 3.3. experimental validation the experiment uses a four-wheeled mobile robot platform (fig. 15) with a load capacity of 20 kg. the power unit drives four integrated joint modules by an industrial computer. fig. 15 mobile robot test platform fig. 16 mobile robot’s joint each integrated joint module (fig. 16) comprises a harmonic reducer, frameless torque motor, brake, incremental encoder, absolute encoder, and servo driver, which is compact and easy to install. by comparing the position and speed of the feedback of the two encoders and referring to the drive current and motor torque output, the magnitude of the external force on the joint where the module is located can be determined. the programming scheme is designed to test eight data sets in the practical application of mobile robots. the torque accuracy without friction compensation and with friction compensation is compared. the calculation method of torque relative error is eq. 11. 738 y. cao, n.m. nor, z. samad ( ) δ 100%s f a a t t t t + − =  (11) here, ts is the theoretical torque, tf is the frictional force compensation torque, ta is the actual feedback torque. in the calculation of the relative error, the data that are too small to cause the error explosion part of the original data have been excluded, where 𝑇𝐹 is zero before friction compensation. the average relative error refers to the average of the relative error of each joint under different paths. it is given in fig. 17. fig. 17 robot joint compensation error diagram from the test data given in table 3, it can be shown that the relative error δ of each joint of the mobile robot under eight test groups after friction compensation has been reduced to different degrees compared with that before compensation, and the values are concentrated in the range of 15% -75%. table 3 friction compensation relative error table error (δ) joint 1 joint 2 joint 3 joint 4 a b a b a b a b group 1 28% 18% 37% 15% 45% 16% 47% 24% group 2 57% 21% 57% 20% 58% 19% 56% 28% group 3 62% 31% 75% 24% 56% 34% 60% 32% group 4 60% 35% 65% 40% 56% 28% 68% 38% group 5 75% 45% 48% 20% 53% 20% 56% 27% group 6 64% 37% 46% 38% 60% 41% 58% 45% group 7 73% 35% 59% 24% 72% 45% 65% 35% group 8 67% 24% 69% 23% 75% 45% 64% 19% average error 61% 31% 57% 25.5% 59% 31% 59% 31% *note a is “before friction compensation”; b is “after friction compensation” study of friction compensation model for mobile robot’s joints 739 4. conclusions in this study, to decrease the frictional force of mobile robot joints, we explore a suitable frictional force compensation model to give the robot joints the ability to adaptive adjust and improve accuracy. a friction compensation model based on deep learning is developed. the friction compensation model’s inputs are load, position, speed, torque, and temperature data, and its outputs are friction compensation values. this deep learning-based robot joint friction recognition 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precision reducers, applied sciences, 9(10), 2146. 30. maharana, k., mondal, s., nemade, b., 2022, a review: data pre-processing and data augmentation techniques, global transitions proceedings, 3(1), pp. 91-99. 31. gower, r.m., loizou, n., qian, x., sailanbayev, a., shulgin, e., richtárik, p., 2019, sgd: general analysis and improved rates, in international conference on machine learning, pp. 5200-5209. 12301 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 435 447 https://doi.org/10.22190/fume231202006s © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper modeling osteocyte under shock-wave therapeutic loading alexey smolin, galina eremina institute of strength physics and materials science sb ras, tomsk, russia orcid ids: alexey smolin https://orcid.org/0000-0003-0213-1701 galina eremina https://orcid.org/0000-0003-3346-367x abstract. dental implants have substantial significance in modern dentistry, but their osseointegration stage remains the most crucial and time-consuming step for compensating for a lost tooth. one promising approach to improve osseointegration rates is the use of extracorporeal shock wave therapy, which has shown efficacy in treating fractures, bone defects, and bone tissue regeneration in surgical and arthroplasty procedures. to comprehend the potential of shock wave therapy in accelerating implant osseointegration, it is crucial to investigate the influence of mechanical loading on ossification processes of varying scales objectively. this study aims to study numerically the effects of low-energy shock wave therapy at different intensities on the mechanical response of a single osteocyte, the principal bone cell that corresponds to microscale of bone tissue. the investigation employs the method of movable cellular automata for modeling. the computer simulation results and analysis based on mechanobiological principles indicate that low-intensity shock wave loading creates conditions for intramembranous ossification, whereas highintensity shock wave exposure creates conditions for endochondral ossification. key words: dental implant, osseointegration, bone tissue, osteocyte, shock-wave therapy, mechanobiological principles 1. introduction dental implants are becoming an increasingly common procedure to restore the functionality of the human teeth and oral health. the most time-consuming and important stage of the installation of a dental prosthesis is the osseointegration of its metal implant with the bone tissue. therefore, an important task is to find a way to speed up this process. it should be noted here that osseointegration as well as fracture healing are possible due to the fact that in the bones of a healthy body there is a permanent process of destruction of old tissues by special cells (osteoclasts) and creation of new tissues by other cells (osteoblasts), which is referred to as remodeling. received: december 02, 2023 / accepted february 23, 2024 corresponding author: alexey smolin institute of strength physics and materials science sb ras, pr. akademicheskii 2/4, tomsk, 634055, russia e-mail: asmolin@ispms.ru https://orcid.org/0000-0003-0213-1701 https://orcid.org/0000-0003-3346-367x 436 a. smolin, g. eremina the first in vivo and in situ studies of the processes of osseointegration of implants in animals were conducted in 1952 [1]. later microscopic studies of a microcirculation and intravascular behavior of human blood cells near titanium implants on the arms of healthy volunteers revealed no inflammatory processes. the first dental implants for humans were installed in 1965. as a result of numerous studies conducted since then, scientists have come to the conclusion that osseointegration of a dental implant occurs according to the scenario of differentiation of osteoprogenitor cells (osteogenic cells, mesenchymal stem cells) into osteoblasts with subsequent intramembranous ossification, i.e., without the formation of cartilage tissue [2, 3]. it should be noted that the second type of ossification, endochondral ossification, which occurs inside cartilage germs when progenitor cells differentiate into chondroblasts, is typical of fracture healing. at the same time, many authors note that the main stages of the process of dental implant osseointegration in the jaw tissues take place at the scale of hormones, proteins, and cells (micro-scale) [4, 5]. moreover, the surface properties of the implant and new methods of surface treatment additionally stimulate the osteogenic cell response [6, 7]. another possible mechanism for stimulating cell response is the use of electrical signals, which can be generated, for example, by piezoelectric scaffolds [8] that are specially designed and manufactured using additive technologies [9, 10]. the process of osseointegration during dental implant placement at the microand mesoscale (characterized by the structure of the formed tissues) is similar to primary bone healing [3]. initially, blood and tissue fluid are present between the implant and the bone, and a blood clot is formed. the blood clot is engulfed by phagocytic cells such as polymorphonuclear leukocytes, lymphoid cells, and macrophages. as a result of cellular activity, the implant surface is covered with a protein layer, the structure and composition of which are determined by the type of surface. in this layer, cell adhesion, migration, and differentiation begin, which promote the interaction of cells with the implant surface over several hours or days. at the next stage, connective tissue similar to fibrous tissue is formed, up to 4 mm thick [11]. further, under the action of external mechanical loads (which are always present due to physiological activity) in the connective tissue, there is a differentiation of cells into osteoblasts and fibroblasts, which are the basis for bone synthesis: the main cells of bone tissue—osteocytes, collagen, and extracellular matrix. after that, the phase of bone callus comes, where the number of osteoblasts increases. further, under the influence of mechanical load, woven bone is formed, which possesses reduced mechanical characteristics compared to healthy tissue. at the very last stage, woven bone is transformed into healthy (lamellar) bone tissue [12, 13]. according to mechanobiological principles, mechanical stresses and strains affect bone tissue remodeling as follows [14, 15]. hydrostatic pressure with a magnitude of up to 0.15 mpa and a shear strain value of up to 5% promotes intramembranous ossification (formation of osteoblasts), which leads to the formation of cortical and spongy bone tissue. at values of compressive stresses above 0.15 mpa and strains greater than 5%, endochondral ossification takes place (formation of chondrocytes and cartilage tissue). compressive stresses of the order of 0.7–0.8 mpa are the most favorable for cartilage tissue formation. at tensile stresses higher than 0.15 mpa with a strain of more than 5%, processes of fibrous tissue formation occur. at stresses lower than 0.003 mpa, chondroginesis and osteogenesis do not occur. in addition, it was shown by gardinier et al. [18] that the pore pressure of interstitial fluid in the range of 20 kpa to 2 mpa (68 kpa is the most favorable value) is modeling osteocyte under shock-wave therapeutic loading 437 considered optimal for the development of osteoblasts. at strains of less than 1%, osteoclasts develop, and, as a consequence, bone tissue resorption occurs. to accelerate the process of osseointegration, non-invasive methods are currently being developed, which are divided into drug therapy and external mechanical treatment. currently, there are active studies based on mechanobiological principles to investigate the influence of external mechanical factors on the processes of osseointegration of dental implants at the microand mesoscopic scales [19]. so borzabadi-farahani [20] and amid at al. [20, 21] investigated the influence of external low-intensity laser exposure on the processes of proliferation and differentiation of dental mesenchymal cells, including on the implant surface. to increase the speed and quality of osseointegration, it looks promising to use extracorporeal shockwave therapy, which has proven itself for the treatment of fractures and a number of diseases of the musculoskeletal system. the therapeutic effect of shockwave therapy in dentistry is being actively investigated for the treatment of dental caries and parandanthosis [22, 23] and regeneration of alveolar bone tissue in common degenerative diseases [24, 26]. since 2010, the possibilities of using shockwave therapy before and after implant placement in the dental area have been actively investigated [27, 28]. bone remodeling is known to be regulated by osteocytes. osteocytes are the main cells of bone tissue and are located in lacunae (bone cavities) surrounded by the perilacunar matrix (pcm), embedded, in turn, in the bone matrix (or extracellular matrix, ecm). osteocytes are connected to each other by slender cell processes located within small tubes called the canaliculi. it is believed that osteocytes and canaliculi are peculiar mechanosensors and mechanotransducers that can "sense" mechanical loads and convert them into biochemical signals that regulate bone remodeling [29, 30, 31]. thus, to better understand the possibility of using external loadings to accelerate the osseointegration of dental implants, it is important to know what happens under such influences in the main cells of bone—osteocytes. in turn, it is known that the processes of exchange of nutrients and metabolic products, as well as biochemical signals in osteocytes, depend on the flow of tissue (interstitial) fluid [32]. basically, nutrients are transported both by the flow of this fluid and by diffusion within the bone [33, 34]. since the experimental study of the above-mentioned processes in bone tissues at the microscale is a challenge, an effective alternative method for in vitro biomechanical studies is computer simulation [35, 36]. in this case, numerical models of the mechanical behavior of bone tissues must necessarily take into account the role of interstitial fluid. this study aims to investigate of the effects of shockwave therapy in various ranges on the mechanical behavior of osteocytes in the area of implant placement using computer simulation. 2. model description and validation 2.1. movable cellular automaton method for the fluid saturated materials to describe the mechanical behavior of bone tissue, herein we used the model of a poroelastic body implemented in the method of movable cellular automata (mca) [37]. it has been established that this particle method has proven itself to be very promising for modeling the mechanical loading of different materials at the macroand mesoscale [38, 39, 40]. in the mca method, a specimen is considered an ensemble of discrete elements 438 a. smolin, g. eremina of finite size (cellular automata) that interact with each other according to certain rules, which, within the particle approach and due to many-body interaction forces, describe the deformation behavior of the material as an isotropic elastoplastic body. the motion of the ensemble of elements is governed by the newton-euler equations for their translation and rotation. within the framework of the mca method, the value of the averaged stress tensor in the volume of an automaton is calculated as a superposition of forces that act on the areas of interaction of the automaton with its neighbors [38]. it is assumed that stresses are homogeneously distributed in the automaton volume. knowing the components of the averaged stress tensor allows adapting to mca different models of plasticity and fracture in the classical solid mechanics. automata simulating fluid-saturated material are considered as porous and permeable. the pore space of such an automaton is presented by pores and channels that can be saturated with fluid. the characteristics of the pore space are taken into account implicitly using the effective integral characteristics, namely, porosity ϕ (including narrow channels), permeability k, ratio a = 1 − k/ks of the macroscopic value of bulk modulus k to the bulk modulus of the solid skeleton ks [37]. the mechanical influence of the pore fluid on stresses and strains in the solid skeleton of the automaton is taken into account on the basis of the biot linear model of poroelasticity. within the framework of this model, the mechanical response of a "dry" automaton is assumed to be linearly elastic. the mechanical effect of the pore fluid on the behavior of the automaton is described in terms of the local pore pressure ppore, which only affects the diagonal components of the stress tensor. the interstitial fluid is assumed to be linearly compressible and is described by the following equation of state pore pore 0 0 fl( ) (1 ( ) / )p p p k = + − , where ρ and ppore are the current values of the density and fluid pore pressure in the volume of the automaton; ρ0 and p0 are the equilibrium values of fluid density and pressure under atmospheric conditions and kfl is the bulk modulus of the interstitial fluid. the interstitial fluid filtration in the material is governed by darcy’s law       =       k k t fl , where η is the fluid viscosity and k is the permeability coefficient of the solid skeleton, which depends on porosity as k = ϕdch 2 where dch is the diameter of filtration channel. 2.2. description of the model, its verification and validation to investigate the influence of dynamic loading on mechanosensory effects during bone tissue remodeling, numerical models describing the flow of interstitial fluid in a mesoscopic unit of cortical tissue, the osteon, have recently been actively developed, taking into account the systems of canaliculi and lacunae of the microscopic unit of bone tissue, the osteocyte [41, 42]. wang et al. [43] presented a more detailed poroelastic model of a single osteocyte in the extracellular matrix. within the framework of this study, a three-dimensional model of the main segment of the osteon—the osteocyte—was developed to investigate the influence of shock wave exposure on osseointegration during implant placement. the material parameters of the modeling osteocyte under shock-wave therapeutic loading 439 main parts of the osteocyte and its geometrical characteristics corresponded to the data presented by wang et al. [43]. the study of dynamic loading with different shock wave parameters on the mechanical behavior of a single osteocyte, taking into account its structural features, is proposed. we considered an idealized model that consisted of an oval-shaped osteocyte cell body, canaliculi, and cell processes placed in a cube-shaped ecm matrix (figs. 1, 2). the side length of each direction of the ecm cube corresponded to the literature data for humans and is 43 μm [44]. the canaliculi were tubular structures with an outer layer of pcm with a thickness of ~0.5–1 μm [45] and an inner content of cylindrical channels with a diameter of 0.25 μm, whose physical and mechanical properties were similar to those of the material of the osteocyte body [44]. fig. 1 a half of model osteocyte specimen (section) with indication of materials the poroelastic model of biological materials was characterized by the following parameters: density (ρ), bulk modulus (k), bulk modulus of the solid phase of the material (ks), shear modulus (g), porosity (φ), and permeability (k). the properties of biological materials used in the model are presented in table 1. the biological fluid inside the materials had the properties of salt water with the bulk modulus kf = 2.4 gpa, the density ρf = 1000 kg/m3, and the viscosity ηf = 1 mpa∙s. table 1 properties of the materials material density, kg/m3 bulk modulus, mpa bulk modulus of solid phase, mpa shear modulus, gpa porosity permeability, 10−20 m2 ecm 1850 18300 19000 3.59 0.06 1 pcm 1000 0.0667 0.2668 0.0142 0.75 4 osteocyte 1000 0.00516 0.0258 0.001107 0.80 6 the initial conditions represent the equilibrium configuration of the system of automata (particles) in the absence of deformations and stresses (i.e., all forces between automata are zero). the fluid content in biological tissues corresponds to the pore volume and zero pressure. 440 a. smolin, g. eremina for verification and validation of the osteocyte model, uniaxial compression of the model specimen was simulated (fig. 2, a); for the main studies, shock wave loading with different parameters was simulated (fig. 2, b). the boundary conditions corresponded to the chosen scheme of loading: the lower layer of automata was rigidly fixed (their velocities were set equal to zero, as shown in fig. 2), and the automata of the upper layer moved along the vertical axis of loading. the value of the z-component of velocity of the upper layer was constant and equal to 1 mm/s in the case of uniaxial compression (fig. 2,a) and was changed in time according to the specified energy flux density (efd) of the shock wave (see [39] for the detailed description of the loading velocity computation) in the case of shock-wave loading (fig. 2,b). the lateral boundaries of the specimen are free. temperature effects were not accounted for in the calculations. (a) (b) fig. 2 boundary conditions for modeling uniaxial compression (a) and shock wave loading (b) of the model osteocyte specimen the main purpose of verification is to check the efficiency of the numerical scheme for solving the governing equations of the method. the main means of model verification is the analysis of the convergence of the obtained results with increasing discretization. discretization of the domain is considered optimal when further increases in the model resolution lead to a change in the solution of no more than 5%. in this paper, the convergence analysis of a threedimensional osteocyte model was performed for the system stiffness (integral parameter) at different discretizations of the considered domain, i.e., the size of the discrete element. the number of automata (discrete elements) in the model varied from 1239752 to 4141898 (the size of automata varied from 0.5 to 0.3 μm, respectively). the results on the convergence of the model stiffness showed that the maximum difference in the stiffness value between the minimum and maximum discretization did not exceed 2%, and it was less than 0.1% for the two extreme values of the automaton size (fig. 3). such a small difference indicates a good convergence of the numerical model. the smallest difference in the value of the effective stiffness of the system is observed for 2692237 and 4141898 automata. at the same time, the computational times for these systems are 5 and 12 hours. therefore, the sample with the number of automata equal to 2692237 was chosen as optimal for the further calculations. modeling osteocyte under shock-wave therapeutic loading 441 fig. 3 dependence of stiffness of a model osteocyte specimen on the number of automata validation of the developed osteocyte model was performed by comparing the modeling results with literature data [43]. uniaxial compression of the model specimen with different loading rates was considered, and the distribution of fluid pressure in the pores and its maximum value were analyzed (fig. 4). (a) (b) (c) (d) (e) fig. 4 pore fluid pressure distribution for a model osteocyte specimen at an engineering strain of 1.5×10−4 under uniaxial compression with different loading rates: 100 mm/s (a), 10 mm/s (b), 1 mm/s (c), 0.1 mm/s (d), 0.01 mm/s (e) 220 221 222 223 224 225 1 1.5 2 2.5 3 3.5 4 4.5 s , n /m m n×10 6 442 a. smolin, g. eremina the analysis of the obtained results showed that at a strain of 1.5×10−4, the maximum value of the analyzed parameter is 80 kpa at a velocity of 100 mm/s, 50 kpa at 10 mm/s, 20 kpa at 1 mm/s, 9 kpa at 0.1 mm/s, and 8.7 kpa at 0.01 mm/s. thus, it is shown that the osteocyte model under consideration is highly sensitive to loading rate. moreover, at loading rates of 0.1 and 0.01 mm/s, the maximum value of fluid pressure in the pores corresponds to the values of this parameter presented by wang et al. [43] under similar loading. 3. simulation of shock wave loading of osteocyte most of the works on the numerical study of the mechanical behavior of the structural unit of bone tissue are devoted to static loads [42, 46, 47]. sathishkumar et al. [25] showed that the regenerative effect of shock wave therapy on bone remodeling around the tooth is observed at an exposure with an energy flux density (efd) of 100 kj/mm2. therefore, herein, we studied the low-intensity acoustic effect of the shock wave within the range of efd from 30 up to 170 kj/mm2. it is worth noting that the time of pressure increase for such a load is comparable to the characteristic time of interstitial fluid filtration in the studied system. the hydrostatic pressure fields (fig. 5,a, where compression corresponds to negative values) show that under low-intensity shock-wave loading (energy flux density of 40 kj/mm2), there were areas in the extracellular matrix of the model specimen with compression stresses higher than the minimum values necessary for osteogenesis activation (3 kpa), and the threshold values at which chondrocyte differentiation occurs (less than 0.15 mpa) were not observed. in the area of the osteocyte and its outgrowths, the compression stress was about 50 kpa. such values of hydrostatic pressure contribute to the differentiation of osteoblasts. (a) (b) fig. 5 distribution of hydrostatic pressure in the model specimen under shock wave loading with an energy flux density of 40 kj/mm2 (a) and 168 kj/mm2 (b) when the osteocyte model was loaded with a high-intensity shock wave with an energy flux density of 168 kj/mm2, areas with compressive stresses of more than 0.15 mpa were observed in the ecm region (fig. 5,b), which is a condition for cartilage tissue development. modeling osteocyte under shock-wave therapeutic loading 443 (a) (b) fig. 6 distribution of pore fluid pressure in the model specimen under shock wave loading with an energy flux density of 40 kj/mm2 (a) and 168 kj/mm2 (b) from the analysis of the pressure fields of interstitial fluid (fig. 6), it was found that at low and high-intensity loading in the area of the extracellular matrix, osteocyte canaliculi, and osteocyte shell, the necessary level of tissue fluid pressure is observed to trigger the processes of biological cell transfer (more than 68 kpa). (a) (b) fig. 7 distribution of equivalent strain in the model specimen under shock wave loading with an energy flux density of 40 kj/mm2 (a) and 168 kj/mm2 (b) now let us consider the equivalent strain fields in the osteocyte model under shock wave loading. as can be seen from fig. 7, a, under low-intensity loading, the strain level is about 0.5% in the ecm, about 5% in the osteocyte shell, and about 10% in the canaliculi. the maximum strain values (about 20%) are concentrated in the canaliculi near the loading region. according to mechanobiological principles, such values favor the differentiation of osteoclasts and the formation of gaps in the extracellular matrix. however, at large deformations, the osteocyte signals the surrounding tissues to start the process of differentiation of osteoblasts, which serve as the main cells promoting bone tissue growth. 444 a. smolin, g. eremina 4. discussion a numerical model of the mechanical behavior of the main structural unit of the osteon—an osteocyte with canaliculi surrounded by extracellular matrix—was developed on the basis of the movable cellular automaton method, which was verified and validated. analysis of the modeling results according to the mechanobiological principles allowed us to reveal that the conditions for intramembranous ossification are created at low-intensity shock-wave loading. at high-intensity shock-wave exposure conditions, the conditions for endochondral ossification are created. the endochondral ossification at the early stages of dental implant healing was also described by kung et al. [48] and li et al. [49]. irandoust and müftü [19] noted that a large gap between the implant surface and bone tissue promotes the formation of soft tissue (cartilage), which in turn is a negative phenomenon in the process of osseointegration. babayi et al. [50] showed using modeling that the application of loading immediately after implant placement promotes the growth of fibrous cartilage tissue around the implant in the early stages of osseointegration. while delayed loading (only the neighboring teeth were loaded and the implant itself was not loaded) promotes more active bone growth in the early stages. it follows from the analysis of literature that at the early stages of osseointegration, fibrocartilaginous tissue may appear, which later differentiates into bone tissue, and this is a manifestation of endochondral ossification. such conclusions, based on the results of numerical calculations, including the results of this work, contradict the basic idea that the osseointegration of the dental implant occurs by the intramembranous type of ossification. in vivo studies have shown that fibrous tissues can promote osteoblast adhesion by utilizing type i collagen to form the extracellular matrix [51]. this may even promote the migration of osteogenic cells from the surrounding bone to the implant and thereby help to increase the rate of osseointegration [52, 53, 54]. since fibroblasts may play a dual role in osseointegration, the authors of [51] concluded that restricted fibroblast colonies promote osteogenic cell differentiation, thereby promoting the osseointegration of dental implants. 5. conclusions despite the fact that endochondral ossification is not typical for osseointegration, the data obtained from the calculations performed and the literature review confirm that the presence of small areas of cartilage occurs during the placement of a dental implant and their role in the subsequent establishing of a functional connection between the dental implant and bone should be considered and further studied. a micromodel of the main structural unit of the osteon presented herein as well as a macromodel of the jaw segment with a dental implant presented by the authors [55] allow multiscale numerical study of the use 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implant with bone tissues under shock wave treatment, journal of materials and engineering, 1(2), pp. 92–96. 10193 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 575 589 https://doi.org/10.22190/fume211123010p © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper process optimization by applying the response surface methodology (rsm) to the abrasive suspension water jet cutting of phenolic composites andrzej perec1, aleksandra radomska-zalas1, anna fajdek-bieda1, frank pude2 1jacob of paradies university, faculty of technology, gorzow wlkp., poland 2steinbeis consulting center high-pressure waterjet technology, horgau, germany abstract. the paper introduces the study on the cutting of the industrial composite phenolic resin, based on the thermoset materials reinforced with cotton cloth by the abrasive water suspension jet (awsj). the size reduction of abrasive grains during the formation of the jet and the erosion phenomenon are shown. the results of the machining process's critical factors as nozzle length, nozzle diameter, and abrasive mass flow rate on the maximal cutting depth, are indicated. to build a model of the process, the method of the response surface (rsm) was applied. the second-degree multinomial equation is selected for creating the cutting model. the research indicates the optimal control factors of the process, to achieve the best cutting depth performance. key words: awsj, abrasive water jet cutting, abrasive water suspension jet, process optimization, erosion, composite 1. introduction polymer-based composites assure superior mechanical, physical, and thermal properties; over the last few years they have come to be considered a better option than conventional materials [1]. the traditional treatment of composites induces high temperatures. moreover, cutting forces generate different types of damages, i.e., tool wear, fiber extraction, delamination, and surface failure because of non-homogeneous and anisotropic properties of these materials [2]. machining of composite materials with the use of water jet is a desirable alternative in relation to the traditional machining technologies [3]. received: november 23, 2022 / accepted march 10, 2022 corresponding author: andrzej perec jacob of paradies university, faculty of technology, teatralna 25, 66-400 gorzow wlkp., poland e-mail: aperec@ajp.edu.pl 576 a. perec, a. radomska-zalas, a. fajdek-bieda, f. pude a common defective phenomenon in abrasive water jet (awj) cutting of layered composites materials can be delamination. it is, therefore, essential to predict the depth of penetration to eliminate delamination. wang et al. [4] have introduced such a model in the semiempirical way for predicting the depth of jet cutting in the abrasive water injection jet (awij) cutting of polymer matrix composites. the model feasibility was then assessed by analyzing the predicted tendencies of performance measures and by comparison with the test effects. authors have shown that the model allows proper predictions and can be used for cutting processes planning. for the creation of machining models the design of experiment (doe) method can be used, that is, the one which allows us to minimize the needed numbers of tests and to cut the related process time. the tests can be led with a full factorial design. rsm is a fusion of statistical and mathematical modeling methods. it can be utilized in multi-criteria optimization [5]. in addition, it also ensures a join amid process control parameters and the perceived responses. the multinomial equation for making the regression value [6] follows: y = 𝛽0 + ∑ 𝛽𝑖 𝑘 𝑖=1 𝑥𝑖 + ∑ 𝛽𝑖𝑖 𝑘 𝑖=1 𝑥𝑖 2 ±  (1) where: y is dependent variable (response), xi is values of the i-th control parameter, k is number of control parameters, β0, βi, βii are the coefficients of regressions and ε is the error. the theory of doe allows us to simplify the method of determining process parameters, such as in the case of using it to evaluate quality of cuts after cutting aluminum alloy by awij [7], for optimization of abrasive water cutting process using the topsis method [8],or even multi response optimization of process parameters based on the taguchi-fuzzy model for coal cutting by water jet technology [9] and multi response optimization on the awij machining of stainless steel by the vikor approach coupled with s/n ratio methodology [10]. due to different importance of the conflicting criterions, the multicriteria methods are extremely useful in the selection process of the proper machining type [11]. design of experiments is an interdisciplinary field of science bordering on metrology, mathematics, statistics, and computer science. the use of this method has been used in modeling both conventional machining processes such as turning [12], grinding[13]or advanced manufacturing processes such as analysis on numerical modeling and flow monitoring of micro continuous water jet [14], but also in chemical processes optimization with vikor method [15], optimization of catalytic systems [16], and even in the studies of credit decision based on real set of cash loans by machine learning algorithms [17]. the use of doe gives a lot of important information at relatively low cost and time. doe enables, among others: selection of input variables significantly influencing the controlled process, building a mathematical model of the process, i.e. mathematical relationships between the number of input and output devices, determination of input values serving the most desired process effect (optimization process) and determination of the impact of variability of input values on variability of the entire process. cutting composite materials with methods characterized by low temperature in the cutting zone has recently been the subject of research in various research centers. dhanawade et al. [18] noticed that a carbon composite, treated by high jet pressure and low traverse speed, characterize a low roughness of the cut surface. vigneshwaran et al. [19] tested the impact of slot taper and cutting efficiency on sisal polyester composite. traverse speed was accepted to be the most significant parameter affecting the cutting efficiency, while stand-off distance was approved to process optimization by application of the response surface metodology (rsm)... 577 be the most meaningful coefficient affecting slot taper. azmi et al. [20] tested the slot taper and delamination on hybrid carbon/glass composite. lower traverse speed and stand-off distance were accepted to be proper for optimal slot taper. the preliminary research of piercing the cfrp with abrasive water injection jet (awij), which can reduce delamination published popan et al. [21]. the research folds of adding the abrasive particles to the water jet at the very start of jet generation, thus obtaining a mix of abrasive and water jet at first impact with the composite work piece. in the presented research authors must have designed special device and set up based on the proposed piercing method and it is impossible to use for standard cutting head used in the common awj cutting system. possibility of the titanium (ti6al4v) and cfrp multilayer composite machining was tested by the awij machining process presented by pahuja et al. [22]. the erosion properties, slot width and surface roughness were studied as a function of control parameters. authors observed that the surface roughness and slot width variability was strong at poor jet power level. mathematical regression models were made to predict slot width. an energy grounded semi-analytical model was proposed to predict the slot properties. putz et al. [23] have compared the awij principle and the awsj principle and shown that abrasive water jet cutting is an appropriate alternative to commonly used diamond grinding or laser cutting processes. also, they have tested the machining quality of technical ceramics. the investigation effects illustrate that the awsj technology characterizes higher accuracy than the awij technology in range of slot geometry and roughness properties. additionally, the awsj technology provides achievement a higher cutting efficiency. ramesha et al. [24] presented the comparison of the different control parameters results on slot width and surface roughness while using the awsj method for machining gfrp composite in submerged condition. the test outcomes have validated that the surface roughness and slot width decreased in under water machining in relation to free air condition machining. authors shown that the treatment by awsj used with an optimized set of parameters make better efficiency as compared to machining by abrasive water injection jet (awij). perec and radomska-zalas [25] introduced the impact of important machining parameters by abrasive water suspension jet (awsj): abrasive flow rate, awsj nozzle id, and length on cutting depth of an aluminum marine grade material. the test determined the best dimensions of the awsj nozzle and abrasive flow to reach the biggest cutting depth were gained. perec et al. also published research on the optimization of metamorphic rock marble cutting by awsj [26].the disintegration of abrasive grains phenomenon over the erosion process was shown. to model the erosion process, the method of the response surface (rsm) was exerted and the polynomial equation of the second degree was chosen for developing the regression model. studies have exposed the optimal set of parameters for achieving the maximal depth of the cut. abrasive material use is recognized as one of the major abrasive cutting expenses. abrasive recycling can be an effective way for reducing the cost. in addition, it is also beneficial to environmental protection. awsj is more suitable for abrasive recycling than traditional abrasive water injection jet (awij) because awsj does not use dry abrasives. grounded on the idea of concerning for the recycling process easily and efficiently, guo et al. [27] studied the abrasive recycling in the awsj process and found that the reused abrasives with only big particle impurity being sieved out still have a strong cutting ability. a simplified abrasive recovery scheme of the awsj cutting system has been proved to be feasible. with 578 a. perec, a. radomska-zalas, a. fajdek-bieda, f. pude 30% of recharge in each cycle, the abrasive can be fully utilized, and its cutting performance can remain the same in every reuse cycle of continuously recycling process. guo et al. [28] also presented their investigation of the effects of pressure, traverse speed, and radius upon the cut surface roughness of the circular arc cut by abrasive suspension jet (awsj). an orthogonal matrix with design of experiments was utilized for analyzing the control parameters on cutting surface roughness at various depths. decreasing the traverse speed is the most effective way of lowering surface roughness. multiple linear regressions were used to create the cutting surface roughness model at different depths, which was proved to be reliable by experiments. the conclusions can provide theoretical guidance for improving the awsj cutting efficiency. based on the state of art analysis of the problem, it can be concluded that the use of awsj for processing composite materials is possible and justified due to the a cutting efficiency and the achievement of better surface roughness properties of the cut slot. the most numerous groups of composites that are the subject of research in the field of water abrasive cutting are carbon and glass fiber composites. however, the use of this technology for cutting the phenolic composite is not known in the available literature. therefore, the authors have decided to conduct research on cutting the phenolic composite with awsj. also, the authors have chosen one of the doe methods response surface methods (rsm) for the given modeling. additionally, motivation for this research study was to test the feasibility and cutting performance of a phenolic composite that is sensitive to temperature rise in the cutting zone by conventional machining methods. 2. materials and methods 2.1. processed material in this research study, phenolic composite, known under the commercial name micarta, was used as the cut material. phenolic composite is a laminate plastic created when linen, paper, fiberglass, or other fabrics are impregnated with pf (phenol formaldehyde) resins. this is then cured under pressure and high temperature to create the thermoset plastic laminate. phenolic composite was developed by george westinghouse at least as early as 1910 using phenolic resins invented by leo baekeland [29]. phenolic laminate offers excellent heat, stress, and chemical resistance. it tolerates extreme temperatures, is moisture-resistant and provides excellent electrical insulation making it a popular choice in electric and semiconductor applications. it can also be manufactured in a wide variety of colors and does not become brittle over time. this makes it a popular choice for countertops and tool & knife handles in consumer applications. 2.2. abrasive material for the awsj machining quartz sand was used as abrasive material. as the research concerns the feasibility of using the awsj technology to cut material sensitive to temperature increase during cutting only, the cheapest of the available abrasives quartz sand was used. the price of this abrasive is more than ten times lower than the commonly used garnet, and the cost of the abrasive is more than 60% of the total processing costs. the harmfulness of quartz dust is especially dangerous in dry conditions. in this research process optimization by application of the response surface metodology (rsm)... 579 study, it was as much as 60 dm3·min-1, which significantly reduces or even eliminates the volatility of quartz dust. it can be used for parameter optimization under laboratory conditions with a limited amount of consumption but should not be used under significant business conditions with a high amount of consumption because of health endangering reasons by inbreathing of sand dust particles [30]. the quartz group consists of all sio2 oxides. there are several different ways of organizing sio2. silicon and oxygen are the two most common elements in the earth's crust, so perhaps their diverse modes of organization are not so unexpected. the basic form of sio2 is represented by low quartz with its color varieties: violet, rose quartz, smoke quartz (dark brown), and yellow. for the tests, the abrasive was used with the grain distribution shown in fig. 1, with the predominant fraction of 630 µm at the level of 62%, the fraction of 800 µm (18%), and the fraction of 500 µm, amounting to almost 13%. fig. 1 quartz sand #30 grain distribution 2.3. experimental set-up there are basically two systems for generating abrasive water jets. their main difference is the moment of adding the abrasive leading to specific properties of the stream. in the abrasive water injection jet (awij) method, a stream of water that passes through the mixing chamber is generated and enters into the focusing tube (fig. 2a). this creates a vacuum in the chamber which sucks in the dry abrasive. there, the abrasive is mixed and accelerated by the water jet and concentrated in the focusing tube [31]. the method of generating abrasive water suspension jets (awsj) used in the research consists of mixing the water and the abrasive suspension directly under high pressure before the awsj nozzle (fig. 2b). a part of the flow is led through the bypass branch, passing through a high-pressure vessel that is filled with abrasive slurry. the abrasive slurry is pushed out of the vessel and joins the main flow in the mixing chamber where the cutting slurry is formed. next, it is then transported via a flexible high-pressure hose to the cutting head, where it is finally accelerated inside the nozzle and directed to the workpiece [32]. the hydraulic diagram of the test stand is shown in fig. 3. it consists of two highpressure vessels: z1 and z2 with abrasive cut-off valves (za1, za2) and four independent hydraulic branches. this allows the basic flow parameters to be adjusted. each branch consists of valves: shut-off valve (zo), throttle valve (zd) and check valve (zz), as well as pressure gauge (m). the function of the element protecting against pressure increase is performed by the overflow valve (zp1). the high-pressure abrasive suspension water jet flows out of the device through a flexible hose (w1) and is finally accelerated in a cutting head (g) equipped with a awsj nozzle (d). 580 a. perec, a. radomska-zalas, a. fajdek-bieda, f. pude a) b) fig. 2 high-pressure jet cutting systems: a) abrasive water injection jet (awij), b) abrasive water suspension jet (awsj) the source of high pressure is the p26 type pump (fig. 4) made on the basis of highpressure ceramic plungers and a set of seals by woma. fig. 3 schematic diagram of test stand: zo-shut-off valve, zd-throttle valve zz-check valve, m pressure gauge, zp1-overflow valve fig. 4 source or high pressure p26 pump: 1 high pressure pump, 2 electric motor, 3 pressure regulation system, 4 pressure gauge, 5 controller it allows us to achieve a maximum pressure of 75 mpa at a water flow rate of 60 dm3·min-1. it consists of a plunger pump driven by an 89 kw three-phase electric motor with a nominal speed of 1500 rpm. process optimization by application of the response surface metodology (rsm)... 581 2.3. test methodology the materials were cut by pointing the jet at the material and moving it at a constant speed relative to the material. the cutting sample thickness was selected so that the undermost effective processing parameters do not result in a through-cutting. in this way, potential inaccurate measurements of cutting depth were eliminated. process parameters (table 1, fig. 5) were chosen on the basis of previous works involving the authors of the present study [33], and the studies of other investigators [34,35,36]. table 1 process parameters used in research parameter unit values nozzle length l [mm] 50 75 100 nozzle id dn [mm] 2.00 2.25 2.50 abrasive flow rate (afr)ma [g·s-1] 50 70 90 fig. 5 example details of awsj nozzle the abrasive concentration determines the ratio of the abrasive mass to the water mass in the awsj. the mass of the abrasive is set on the feeder, while the mass of water in the jet arises from the flow rate for a given id of the water nozzle at a given pressure, considering discharge coefficient (cd). the maximum cutting depth was selected as the output parameter. this is a widely used parameter [37, 38] that clearly defines the effectiveness of this process. measurements of cutting depth were made by a digital caliper altimeter. 3. results and discussion 3.1. cutting depth the outcomes of studies on the impact of process control parameters (independent variables) on the cutting depth (dependent variable) are indicated in table 2. the method of analysis of variance (anova) for the 95% level of confidence ( = 0.05) was made (table 3). the model coefficient is statistically significant when it reaches p value <0.05. this is illustrated in fig. 6. to estimate multicollinearity, the variance inflation factor (vif) was calculated. it quantifies the intensity of multicollinearity. vif reveals how much the variance of the evaluated regression factor is inflated as caused by multicollinearity in the model. when vif is 1.0, multicollinearity does not occur. for all tested factors, no multicollinearity was observed because vif =1.000. 582 a. perec, a. radomska-zalas, a. fajdek-bieda, f. pude table 2 values of parameters used in experiments and results of cutting depth test no nozzle length l [mm] nozzle id ·dn [mm] afr ·ma [g·s-1] depth h [mm] 1 50 2.00 50 24.76 2 50 2.00 70 29.33 3 50 2.00 90 25.30 4 50 2.25 50 25.23 5 50 2.25 70 29.44 6 50 2.25 90 26.14 7 50 2.50 50 25.49 8 50 2.50 70 28.03 9 50 2.50 90 24.97 10 75 2.00 50 36.74 11 75 2.00 70 37.82 12 75 2.00 90 36.40 13 75 2.25 50 36.56 14 75 2.25 70 37.37 15 75 2.25 90 36.49 16 75 2.50 50 35.58 17 75 2.50 70 35.32 18 75 2.50 90 33.83 19 100 2.00 50 28.31 20 100 2.00 70 31.34 21 100 2.00 90 32.61 22 100 2.25 50 29.32 23 100 2.25 70 31.87 24 100 2.25 90 34.82 25 100 2.50 50 29.41 26 100 2.50 70 33.67 27 100 2.50 90 31.06 table 3 analysis of variance details source df adj ss adj ms f-value p-value vif model 9 471.008 52.334 29.75 0.000 linear 3 113.525 37.842 21.51 0.000 nozzle length l 1 106.191 106.191 60.36 0.000 0.000 nozzle id dn 1 1.531 1.531 0.87 0.364 0.000 afr ma 1 5.803 5.803 3.30 0.087 0.000 square 3 345.432 115.144 65.44 0.000 nozzle length*nozzle length l2 1 318.379 318.379 180.96 0.000 0.000 nozzle id*nozzle id dn 2 1 3.899 3.899 2.22 0.155 0.000 afr*afr ma 2 1 23.154 23.154 13.16 0.002 0.000 2-way interaction 3 12.051 4.017 2.28 0.116 nozzle length*nozzle id l·dn 1 0.644 0.644 0.37 0.553 0.000 nozzle length*afr l·ma 1 9.223 9.223 5.24 0.035 0.000 nozzle id*afr dn·ma 1 2.185 2.185 1.24 0.281 0.000 error 17 29.910 1.759 total 26 500.918 process optimization by application of the response surface metodology (rsm)... 583 fig. 6 pareto chart of the standardized effect. response is cutting depth h, ( = 0.05) the regression standard error s = 1.3264 and r2 factors (r2, r2 adj) are little differing and take on values over 90%. this confirms that the raw data satisfactory match with the line of regression. ℎ = −121.5 + 1.639 𝑙 − 0.011655 𝑙2 − 0.00491 𝑚𝑎 2 + 0.001753 𝑙 ∙ 𝑚𝑎 (2) where h is depth of cut [mm], l is nozzle length [mm], ma is afr [g·s-1]. figs. 7, 8, and 9 are illustrations of eq. (2). the diameter of the water nozzle change has no significant influence on the cutting depth unlike the nozzle length having a bigger influence. the highest value of the cutting depth can be observed for 80 mm nozzle length in whole afr range. a) b) c) fig. 7 effect of nozzle length and nozzle id with afr: a) 50 g·s-1, b) 70 g·s-1, c) 90 g·s-1 584 a. perec, a. radomska-zalas, a. fajdek-bieda, f. pude a) b) c) fig. 8 effect of nozzle length and afr for id: a) 2.00 mm, b) 2.25 mm, c) 2.50 mm a) b) c) fig. 9 effect of nozzle id and afr for nozzle length a) 50 mm, b) 75 mm, c) 100 mm the scattering of the actual and predicted depth of cut values is shown in fig. 10. all points are localized near a straight line and this confirms that the formulated model is satisfactory. fig. 10 example scattering plot for actual and predicted cutting depth process optimization by application of the response surface metodology (rsm)... 585 based on eq. (2), the optimal values of all three tested control parameters were determined in terms of the depth of cut (fig. 11). optimal nozzle length is near 80 mm, optimal nozzle id is 2.2 mm and optimal flow rate is 74.2g·s-1. fig. 11 variability of control parameters and their optimal values this is also confirmed on the contour charts presented in fig. 12. the biggest values of the cutting depth, shown as deep green zones, are reached in the middle of the control parameters for: nozzle length: 75 90 mm, nozzle id: 2.1 2.3 mm, afr: 70-80 g·s-1. moving the value of each control parameter in any direction beyond the selected deep green areas causes the cut depth value to drop. fig. 12 example contour plot of the range of control parameters at optimal conditions fig. 13 presents a microscopic view of the surface of the material cut under optimal conditions. the arched machining traces (fig. 13a) in this material are much clearer than in the case of metal materials, for example, nickel-based superalloy [39], cooper [40] or steel [41], and are visible on the entire surface, although slightly in its upper part weaker than at the bottom. the surface is not dull. in the right part, there is a triangular material undercut, which is a given characteristic of awj cutting. fig. 13b shows a typical sem image of the surface of the cut material, localized in the mid part of the sample. the chains of fibers can be seen, but there are no visible traces of processing. 586 a. perec, a. radomska-zalas, a. fajdek-bieda, f. pude a) b) fig. 13 example of cut surface, machined at optimal control parameters: a) optical microscope view, b) sem view 3.2. abrasive grain disintegration additionally, abrasive grain fragmentation tests were performed. to catch abrasive grains after their exit from the cutting head, a special collector was used [42]. the collector was customized to catch the abrasive grains and to preclude any extra grains disintegration. the underside pvc collector was shielded by a mild steel target to avert perforation. no wear marks were noticed on the safeguarding target after the termination of tests. the caught abrasive grains were then dried. for the used abrasive grain size distribution tests, the retsch sieving system was used. the fragmented garnet left on the sieves was weighed on the laboratory digital scale. the fragmentation test results for a cutting head equipped with a 2.25 mm id nozzle, 75 mm length and 75 g·s-1 afr are presented in fig. 14. fig. 14 example disintegration of the quartz #30 grain at nozzle 75 mm length, id 2.25mm, and afr 70 g·s-1 process optimization by application of the response surface metodology (rsm)... 587 3.3. discussion cutting test results suggest that the depth of cut of the phenolic composite is most dependent on the traverse speed and it complies with other research studies by perec et al. in aluminum [25] and limestone [33] as well as ramesha et al. [24] on gfrp composite by awsj. a similar phenomenon also occurs in the cutting by awij [26]. the effect of pressure on the depth of cut of the phenolic composite in the case of awsj is as important as in the awij and is directly proportional as in the case of natural fiber composites cutting, published by müller et al. [43]. the amount of abrasive has the smallest influence on the depth of cut; however, this happens only when it oscillates around the theoretical optimum, equal of 18% abrasive by mass in the jet. under these conditions, no delamination and surface burn were observed when cutting the phenolic composite by awsj, unlike the cutting tests of other composite materials by awij. wang et al. [4] confirmed the delamination is a major component defect when machining composites or layered materials and popan et al. [21] observed strong delamination of composites, especially with a small flow rate of abrasive. in the case of investigating abrasive behavior in the awsj machining, an intense disintegration of the most numerous fractions of abrasive grains depending on the working pressure was observed. the influence of pressure on the breakage degree is directly proportional. the higher pressure generates bigger abrasive grains velocity in the awsj nozzle, and the processed material and the disintegration process takes place more intensively. this observation is in line with the research on the disintegration of the abrasive in the awij cutting process [42, 44]. 4. conclusions based on the conducted research related to the modeling of phenolic composite cutting, the following conclusions were obtained: ▪ the processing of the phenolic composite by awsj did not cause any thermal changes in the cutting zone; therefore, it seems advisable to continue the research. ▪ length of nozzle has a significant influence on erosive abilities, measured in the form of cutting depth. ▪ abrasive flow rate (afr) has a poor influence and nozzle id has smallest influence on cutting depth. ▪ r-squared (the percentage of variation in the response that is explained by the model) over 94% shows the model fits very well to experimental data. ▪ adjusted r2 value = 90%, which is r2, adjusted for the number of predictors in the model relative to the number of tests, also confirms a very good model fit. ▪ for regression coefficients of the model was observed no multicollinearity. ▪ in the entire tested range optimal settings of awsj cutting parameters from the maximal cutting depth point of view for the examined area are as follows: nozzle length near 80 mm, nozzle id equal 2.2 mm and for 75 g·s-1 afr. at the above parameters of cutting, the maximal depth of cut of more than 38 mm was attained. ▪ in further tests the almandine garnet should be used, because it is safer for the environment and commonly used 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annual set the environment protection – rocznik ochrona srodowiska, 20(1), pp. 0258–0274. 8241 facta universitatis series:mechanical engineering https://doi.org/10.22190/fume220320037t © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper analytic approximate solution for nonlinear dynamicmodeling of the rotating elastic 2d beam with a single crack arash tavakoli maleki1, milad azimi2, samad moradi3 1k.n. toosi university of technology, tehran, iran 2aerospace research institute (ministry of science, research, and technology), tehran, iran 3islamic azad university, north tehran branch, tehran, iran abstract. in this paper, the 2d lateral vibration analysis of a rotating cracked beam as a rotary structure is investigated through the homotopy perturbation analysis and compared with the numerical newmark-beta (nβ) algorithm. the structure and crack are modeled as the euler-bernoulli (eb) theory and simple torsional spring, respectively. the nonlinear equations of motion are derived using galerkin and the assumed mode method (amm). the system’s stability is analyzed through phase plane and time response for different angular velocities of the base, initial values, external disturbances, crack stiffness, and locations. a comparative study presents simulation results for free (first nonlinear frequency) and forced vibration. it is shown that the proposed semi-analytical approach is beneficial as it provides a benchmark for a more precise analysis and further investigation of cracked rotary structures. key words: assumed mode, crack, homotopy perturbation, newmark-beta, nonlinear vibration, rotating beam 1. introduction dynamic modeling and vibration analysis of rotating structures, including rotor blades, helicopters, flexible spacecraft, flexible link manipulators, wind turbines, etc. (for aviation, space, and power generation industries), have been extensively studied. however, they neglect some concerns or factors using approximations in the modeling procedure. several studies have been concerned with getting simplified solutions for a free and forced vibration analysis of rotating structures [1-3]. these approaches reduce received: march 20, 2022 / accepted september 14, 2022 corresponding author: miladazimi aerospace research institute (ministry of science, research and technology), shahrak-e-gharb,iranzamin ave., mahestan st., tehran, iran. e-mail: azimi.m@ari.ac.ir 2 a. tavakoli maleki, m. azimi, s. moradi computational actions and degrees of freedom, while the accuracy incorporates the nonlinear behavior of systems remains. one of the common structural nonlinearity sources is caused by cracks [4, 5]. the structural members, particularly rotating structures, may experience internal/external disturbances that may result in structural cracks. the stiffness of the cracked structure is reduced locally; as a result, the stability and dynamic behaviors of the system are affected by crack characteristics and location. therefore, exact modeling of the defected structure is of great significance in predicting stability, vibration behavior, and structural health monitoring [6-8]. numerous research studies exist on the dynamic response of structures, including cracks [9-11]. this article focuses on cracked and healthy structures with rotating bases. the rotating eb beam with a crack at its edge considering centrifugal forces as an additional stiffness is analyzed and modeled by yashar et al. [12]. they investigated the natural frequencies and the vibration modes along the flap and chord of the cracked rotary beam using rayleigh-ritz and fem. an analytical method for the free vibration analysis of the rotating cracked functionally graded materials (fgm) structure is investigated by wei et al. [13]. the advantage of the proposed method is that the eigenvalues can be extracted with the desired number of cracks. the classical rayleighritz method is used to investigate the effects of angular velocity, crack depth, and location, on nonlinear bending vibrations of large-amplitude rotating timoshenko beam considering the rotational axial stiffness [14]. the crack is modeled as a torsional spring that divides the beam into two sections. the ritz and the differential quadrature approaches are applied to investigate the effects of crack characteristics and material properties on the linear and nonlinear frequency of the fgm based on the timoshenko beam with different boundary conditions by kitiporancha et al. [15]. afshari et al. investigated the vibration modeling of the eb beam with continuous crack (not as a discontinuous torsional spring) in the presence of piezoelectric (pzt) patches. they analyzed the crack growth by applying the pzt function to the structure [16]. a quadratic b-spline fem and galerkin methods are used to study the free vibrations of the rotating eb beam [17]. latalski et al. studied bending-twisting vibrations of the rotating thin-walled composite structure attached to a hub [18]. in this study, the system’s partial differential equations and mathematical model are derived considering rotational inertia, material anisotropy, and transverse shear and reduced to ordinary differential equations by the galerkin approach. zeng et al. analyzed axial-torsional, flap-wise-chord-wise coupled vibrations of a rotating pre-twisted beam using fem and hamiltonian approaches [19]. gawryluk et al. investigated the dynamical response of a rotating composite beam with a constant velocity caused by harmonic excitation from a macro fiber composite (mfc) motor by fem [20]. in this approach, the pzt materials have been used as an additional excitation source. dibble et al. analyzed the aero-elastic eigenvalues of a rotating blade under a variable angular velocity and compressive loading using the boundary value problem [21]. they investigated the effect of compressive and aerodynamic loading on the reduction of the rotary blade velocity. the rayleigh-ritz approach is utilized to study the effects of angular velocity, hub radius, and other characteristics of a pre-twisted rotating composite blade on vibrational behavior (natural frequencies and mode shapes), considering the effects of coriolis and centrifugal forces [22]. a dynamic analysis of rotating (eb) beam analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 3 using power series solution and numerical simulations is done considering tapering effects [23]. different researchers also investigated the dynamic analysis of rotor blades as coupled rigid-flexible systems [24, 25]. several analytical solutions have been presented for nonlinear vibration analysis of non-rotating structures. using a nonlocal strain gradient theory, nonlinear vibration, bending, and buckling of functionally graded nano beams on an elastic foundation are investigated [26]. sedighi et al. propose a parameter expansion method (pem) for an exact nonlinear vibration analysis of a buckled beam considering dead zone boundary conditions [27]. as can be seen from the literature, much less research on analytical approaches for vibration analysis of cracked rotating structures has been reported compared to the numerous numerical investigations of such systems. on the other hand, it is required to precisely study the effects of significant parameters on the dynamic behavior of nonlinear systems. the problem can be easily handled by driving analytical models and solving approaches, especially for complex multi-body dynamics. therefore, new methods have been proposed to deal with nonlinear problems such as hamiltonian [28], energy balanced [29], multiple scales [30], differential transform method [31], variation [32], and homotopy perturbation approaches [33, 34]. the homotopy perturbation method (hpm) is the one that provides remarkably fast convergence with high accuracy in series solutions for highly nonlinear systems and is not restricted by the assumption of a small number that existed in conventional perturbation approaches [35-37]. essentially, this approach is a hybrid of the traditional perturbation and homotopy approaches, which have been successfully applied to nonlinear oscillations, wave, integral, heat conduction/convection/radiation equations, dispersion equations, etc. there are several approaches with the same characteristics, such as modified hpm [38, 39], global error minimization method [40], book-keeping parameter perturbation method [41], energy balance method [42], and he’s frequency formulation [43]. this study proposes a new methodology for free/forced vibration and stability analysis of a rotating cracked structure (a flexible cantilever beam attached to the rotating hub considering centrifugal stiffening effects) using a high-deformation homotopy perturbation approach (a semi-analytical method).the nonlinear partial equations of the motion of the system applying the galerkin method lead to a nonlinear second-order ordinary differential equation (node). next, a semi-analytical technique is developed to establish a more precise and reliable solution for the system's nonlinear natural frequencies and time response to study different parameter effects on the stability analysis of cracked rotating structures. the main contributions are prepared so that the 2d coupled dynamic equations of the motion of a rotating elastic cracked beam are formulated considering centrifugal stiffness while hpm (with high-order deformation configuration) with second-order approximation is constructed to solve the problem. the proposed approach serves as the foundation for generalizing and implementing the hpm for a broader class of structural dynamics problems referred to as rigid-flexible body problems. this paper is organized as follows: the dynamic equation of the motion of the rotating cracked structure is derived in section 2 and solved by a high-order deformation hpm in section 3. in section 4, numerical and analytical simulations are given and compared for verifications. finally, in section 5, the conclusions are drawn. the predefined styles are to be used. 4 a. tavakoli maleki, m. azimi, s. moradi 2. governing equations of motion the cracked rotating-beam configuration is shown in fig. 1. the structural model is considered an eb isotropic and uniform cantilevered beam with density ρ, length l, cross-section area a, modulus of elasticity e, and a single-axis (about z-direction) rotating hub with an angular velocity . the crack is assumed to be perpendicular to the beam's surface and open at all times. it appears as a discontinuity that affects the structure's local stiffness, the same as a mass-less torsional spring (with stiffness k). fig. 1 rotating cantilevered cracked structure the three-dimensional (3d) displacement field for the eb beam is considered as: , ,x y zu u zw y u u w       (1) considering that large deformations involve nonlinear problems, the displacement strains are expressed as:  , , , , 1 , , , , , 2 ij i j j i i k j ku u u u i j k x y z     (2) considering u=0 in ux displacement field, using the energy approach, the equations of the motion can be obtained by the hamilton principle described by [44]:   2 1 0 t t t u w dt    (3) where t, u, and w are the kinetic, potential, and work done by the external forces, respectively, written as:   2 2 2 0 1 1 = d 2 2 l t a w dx    v   2 2 ij 11 0 2 2 0 1 1 = d 2 2 l ij a l zz yy u y zw w dadx m m w n w dx                                (4) analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 5       2 2 0 0 2 2 2 2 2 0 0 1 1 2 2 1 1 2 l l ext r y z r l l y z w w w f f w dx f w dx x f f w dx a l w dx l                                     with: 11 11 11, ,yy zz a a a m z da m yda n da       (5) where wext is the work done by external forces fy, fz (such as fluid or external excitation), wr is the work done by the rotational forces (affecting the structure’s stiffness). now by proper substitution of t, u, and w into lagrange’s equation, applying the calculus of variation, the system of nonlinear differential equations of motion is obtained as:     2 2 2 2 2 2 2 2 2 2 1 1 ( , ) 2 2 1 1 ( , ) 2 2 zz y yy z ae x a ei w a l f x t l ae x aw ei w w w a l w f x t l                                                             (6) physically, a crack appears as a discontinuity in geometry and introduces considerable local flexibility. in vibration problems, the cracks can be approximated by a torsion spring [45]. suppose that the crack only acts on the z-direction. it can divide the continuous structure into two zones. to connect these zones, considering vertical displacement, shear forces, and bending moments from equalities over both sides of the crack, we have: 1 2 1 2 2 1 1( ) ( ) , ( ) ( ) , ( ) ( ) ( )w a w a w a w a w a w a kw a        (7) where k and a are the crack stiffness and location measured from the roots of the beam, respectively. it is also necessary to add four boundary conditions: 1 1 1 1 1 1 1 (0) (0) ( ) ( ) 0 (0) (0) ( ) ( ) 0 w w w l w l l l                (8) with: 1 1 1 1 1 2 2 2 2 2 3 3 3 3 ( ) sinh( ) cosh( ) sin( ) sin( ) ( ) sinh( ) cosh( ) sin( ) sin( ) ( ) sinh( ) cosh( ) sin( ) sin( ) w x a x b x c x d x w x a x b x c x d x v x a x b x c x d x                            (9) applying b.c for w(x) and v(x), determining unknown coefficients ai and bi in the above equation, one can remove the spatial part of the equations using the orthogonality of the linear modes. by applying the galerkin method, the nonlinear system of pde (nspde) eq. (6) is converted to nonlinear node as: 6 a. tavakoli maleki, m. azimi, s. moradi 3 ( ) (0) , (0) 0 mt ct gt kt f t t a t       (10) with: 2 4 2 0 0 2 2 4 2 1 2 0 0 ( ) ( ) ( ) ( ) ( ) l l a l l a m a w x dx v x d a w x dx w x dx v x dx                               (11)                2 2 2 2 0 0 2 2 2 2 2 2 1 1 1 2 2 2 0 0 2 2 l l a l l a ae g w v w w dx v v w v dx ae w v w w dx w v w w dx v v w v dx                                          (12) 2 2 4 2 4 2 2 2 0 0 0 0 2 4 2 2 4 2 2 2 1 2 1 1 0 0 0 1 1 1 2 1 1 2 l l l l a l l a a x x k ei w dx v dx a l w w dx v v dx l l x ei w dx w dx v dx a l w w dx l w                                                                                          2 2 2 2 0 1 1 l l a x x w dx v v dx l l                                 (13)   1 2 0 0 0 ( ) l a l l z y z z y a f t wf vf dx w f dx w f dx vf dx        (14) and c= αm+βk as a rayleigh damping coefficient with positive constants α and β. it is noteworthy that the integral limits defined in m, g, k, and f, as well as eq. (7), introduce the beam as a two-part structure separated by a crack. 3. homotopy perturbation solution (high order deformation) in this section, we discuss the idea of hpm to solve eq. (10). now, let: ( ) ( ) ,t t u t    (15) substituting eq. (15) into the homogenous un-damped form of eq. (10) and defining /k m , /g m , and /f m yields: 2 3 * 0 (0) , (0) 0 u u u u u u       (16) analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 7 the initial solution for u(t) will be: * 0 ( ) cos( )u t u  (17) constructing homotopy for eq. (16), we have:  1 ˆ( ) ( ) ( ) ( )m m m m nu u h r    u (18) with:    , ,q q   (19)   ( 1) ( 1) 0 , ( ) ( 1)! m m n m q q r m q        u (20)    2 3,q    (21) 0 1 1 1 m m    (22) 0 n n n q    (23)   1 2 0 0 1 2 1 , ( ) ....n n n q u u t q u q u q u         (24) where ℕ, , ĥ , q, and ( ) are nonlinear operator, linear operator, auxiliary constant, perturbation, and auxiliary parameters, respectively. for n=1, we have:  1 1 ˆ( ) ( ) ( )nu h r  u (25) where:   0 2 3 1 1 0 0 0 00 0 , ( ) q q r u u u q          u (26) substituting eq. (24) into eq. (21) yields: 2 3 1 1 0 0 0 0u u u u u    (27) also, with substituting eq. (17) into eq. (27), we have:        3 2 * * * 1 1 0 cos cos cosu u u u u       (28) with some simplification, eq. (28) can be expressed as: 8 a. tavakoli maleki, m. azimi, s. moradi             2 * *3 * 1 1 0 *3 2 * * *3 0 3 1 cos cos cos 3 cos 4 4 3 cos cos 3 4 4 u u u u u u u u u                             (29) the secular term cos(τ) has to be eliminated for the next iterations, so its coefficient must be zero: 2 * * *3 0 3 0 4 u u u    (30) which results in: *2 0 3 4 u   (31) rewriting eq. (29) without secular terms yields:   *3 1 1 cos 3 4 u u u   (32) repeating for n=2 and some simplification, we have: 2 2 *2 2 *2 2 *43 3 21 4 2 32 u u u           (33) 0 1 ...u u u   (34) in order to obtain the time response analysis to external disturbance  0 cos  of system eq. (10), reconstructing eq. (29) as:                 2 * *3 * 1 1 0 0 *3 2 * * *3 0 0 3 1 cos cos cos 3 cos cos 4 4 3 cos cos 3 cos 4 4 u u u u u u u u u                                 (35) with ft  , where ωf is defined as an excitation frequency. eliminating secular terms and rewriting eq. (35) as:     *3 1 1 0 1 1cos 3 cos , (0) 0, (0) 0 4 u u u u u      (36) the solution is given by:   *3 *3 1 0 0cos( ) cos(3 23 ) cos 2 3 u u u f f           (37) analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 9 4. simulation results the simulation results have been investigated to study the system’s performance in the main parameter variation. the effective parameters of the problem are: crack location, corresponding torsional stiffness, angular velocity, external force, and initial values. in each case, time responses and phase diagrams have been obtained. in the simulations, the physical parameters used to describe the system are considered to be: l=12 (m), e=210 (gpa), ρ=7800(kg/m3),a=bh=(0.6×0.04) (m2). the system performance in terms of fundamental nonlinear natural frequency (fnnf) is analyzed for different cases. table 1 shows the effects of three different input base angular velocities  (rad/s) on fnnf. in this case, the equivalent torsion spring stiffness is k=1×106(n.mm/rad), the initial value is u*=0.01, and the crack location is a=6 (m) or c=a/l=0.5. table 1 fnnf for different angular velocities  parameter =0 =5 =10 ω 0.8216 6.661 11.5759 table 2 fnnf for different crack location c parameter c=0.1 c=0.3 c=0.5 c=0.7 c=0.9 ω 4.294 5.433 6.661 7.373 8.169 table 3 fnnf for different crack stiffness k parameter k=∞ k=1e6 k=1e5 ω 7.187 6.661 4.663 moreover, the effects of crack characteristics on fnnf are shown in tables 2 and 3, respectively. clearly, with the increase in the angular velocity crack stiffness, the fnnf of the cracked and the uncracked beam becomes closer to each other. in addition, for cracks near the clamped boundary, the fnnf value is decreased. it can also be found that, for fixed crack properties, the fnnf increases with increasing angular velocity due to the centrifugal stiffness. in the following, a comprehensive parameter analysis was performed to examine how the initial value, hub rotational speed, crack stiffness and crack location affect the free and force vibration responses of the system analytically and numerically. in order to demonstrate the effectiveness of the proposed approach, a comparison is carried out between hpm with high deformation and the runge-kutta (4th order) numerical approach. it can be seen that both methods behave similarly, with a slight deviation observed at higher deformation rates. the system performance in the case of free vibration phase diagrams is illustrated in figs. 2-5. in order to verify the accuracy of the hpm, the results are compared with the nβ algorithm. as shown in fig. 2, the system's stability is preserved for all prescribed initial values. only the increase in amplitude can reduce the stability conditions, which means the radius of the circles (phase portraits) increases. the dimensions of circular patterns are associated with the energy equilibrium of the system. thus, as the system's rigidity increases, each loop's size decreases. 10 a. tavakoli maleki, m. azimi, s. moradi fig. 2 phase portrait for different u* (free vibration) fig. 3 phase portrait for different  (free vibration) from fig. 3, the circular pattern of the non-rotating structures strongly becomes elliptic by increasing the angular velocity of the base. hence with an identical value in the vibration amplitude, the rates are sharply reduced, which indicates an increase in rotational stiffness of the system. this happens for other cases where the crack stiffness increases and the crack moves towards the structure's tip at a lower rate, as shown in figs 4 and 5. moreover, a relatively acceptable correspondence is observed between hpm and nβ. analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 11 fig. 4 phase portrait for different c (free vibration) fig. 5 phase portrait for different k (free vibration) the simulations of forced vibration in time response (tip deflection) and phase diagrams are compared in figs. 6-9. similar to free vibration, the force vibration responses in phase diagrams are compared with nβ, and again, a good agreement is observed. it is worth noting that a general response for systems with force vibration is periodic curves with finite cycles, in which the period corresponds to the excitation type. the 12 a. tavakoli maleki, m. azimi, s. moradi effect of different parameters on the stability and performance of the system is investigated. as for the free vibration analysis, the most prominent parameter which affects the system performance in the presence of external disturbance is the angular velocity and the disturbance amplitude (figs. 6 and 7), so that the non-rotating structure has larger tip velocities of the order of ~3.5 times compared to the rotating cases. on the other hand, by increasing the angular velocity to =5, we face an increase in the amplitude of the oscillations. however, there is a significant reduction in the tip amplitude and its rate as the base angular velocity reaches =10. fig. 6 forced vibration response to f0, a) phase portrait b) tip displacement a) b) analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 13 fig. 7 forced vibration response for different , a) phase portrait b) phase diagram (magnified) c) tip displacement a) b) c) 14 a. tavakoli maleki, m. azimi, s. moradi fig. 8 forced vibration response for different c, a) phase portrait b) tip displacement it should be noted that different studies to investigate the effects of crack location and stiffness in the form of closed curves are shown in phase diagrams with a concentric but different radius. in the presence of external disturbances, as the crack gets closer to the base and as the crack stiffness increases, the radius increases, which provides that the system dynamic becomes more sensitive to parameter variations, external disturbances, and changes in the system stability criteria. one can conclude that the phase diagram of free vibration systems shows periodic steady-state orbits whose amplitude varies with initial conditions. in contrast, for forced vibration analysis, this phase diagram represents periodic orbits characterized by a period proportional to the harmonic excitation force. a) b) analytic approximate solution for nonlinear dynamic analysis of rotating elastic beam 15 fig. 9 forced vibration response for different k, a) phase portrait b) tip displacement 5. conclusions this study analyzes the free and forced nonlinear vibration of a cracked eb beam attached to a rotating base. it is shown that the nonlinear governing ordinary differential equation can be extracted from pde and solved using the high-order deformation form of hpm. comprehensive investigations based on nonlinear natural frequency, time response, and the phase diagram are made to analyze the system's vibration a) b) 16 a. tavakoli maleki, m. azimi, s. moradi characteristics and stability, 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john wiley & sons. 45. gounaris, g., dimarogonas, a., 1988, a finite element of a cracked prismatic beam for structural analysis, computers & structures, 28(3), pp. 309-313. facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. i iii editorial foreword to the thematic issue: science of wear valentin popov technische universität berlin, 10623 berlin germany along with the closely related phenomenon of fatigue, wear is one of the main causes for component damage and subsequent failure of machines and devices. its mitigation by appropriate material choice, coatings, surface design, or lubrication is, therefore, of high economic importance. wear belongs to the most complicated tribological phenomena, but still remains not well understood. microscopic and mesoscopic mechanisms causing the macroscopically observable phenomenon of wear are extremely varied and can include abrasive or adhesive debris formation, their transport in the frictional zone, reintegration of previously removed material, oxidation, chemical or mechanical intermixing of the involved surfaces, mechanically induced diffusion and so on. accordingly, the formulation of a general wear law is quite difficult. the present thematic issue is devoted to the phenomenon of wear considered from different points of view. it is opened with the paper by j. benad "numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections". this paper deals with numerical simulations of contact stresses and deformations in technical systems of complicated shape (in this particular case with fir-tree connections in turbines) – as a prerequisite for any wear analysis. the paper is dedicated to the generalization of the boundary element method to contacts of elastic solids of arbitrary three-dimensional shape, while still taking advantage of the fast fourier transformation used in present implementations of bem. today, the fft-based bem is the most efficient simulation method for contact problems. however, it operates only in the half-space approximation. the three-dimensional version sketched in the paper by j. benad is a good candidate for a future universal contact mechanical tool which will be as effective as the present fftbem but without its restrictions. even under laboratory conditions, it is very difficult to control wear. in practical applications such as wear in contact of tires with the road, the controlling parameters can vary drastically depending on the type of the vehicle, type of tires, type and state of the road, weather and so on. however, technical systems are often operated under such poorly defined conditions and it is important to understand to what extent one can predict the wear under these real conditions. the paper by r. pohrt, "tyre wear particle hot spots – review of influencing factors", presents an overview of experimental data describing the influence of various factors in real operation of tires. ii v.l. popov truly vital is the problem of wear in artificial joints (endoprothesis). wear problems can lead to the necessity of repeated surgical interventions. g. eremina and a. smolin analyze exactly this problem in their paper "multilevel numerical model of hip joint accounting for friction in hip resurfacing endoprothesis". they consider not the complete endoprothesis but "resurfacing endoprothesis" which in itself is a more gentle intervention in the living body. they further analyze the wear process based on the simulation of stress state of the system using the method of movable cellular automata (mca). in simulating wear, very often the archard law is used, stating that the wear intensity is proportional to the normal force and inversely proportional to the hardness of the contacting materials. experiments show that this law is only a very rough approximation; as a matter of fact, it is never valid. both experiments and microscopic simulations show that the dependence of wear intensity on normal load is not linear. but exactly this non-linearity could provide a key for the solution of the riddle of the huge variation in the coefficient of adhesive wear! indeed, is it not paradoxical that archard’s equation, which describes adhesive wear, does not contain any parameter characterizing adhesion? from dimensional analysis, it even follows that the coefficient of wear cannot depend on the specific surface energy, since no dimensionless combination can be constructed from the specific surface energy and other available parameters. the situation changes completely if the wear intensity is not proportional to the normal load. then the specific work of separation can be included in the equation of wear. many empirical studies show that the specific energy of separation definitely belongs to the governing parameters of the wear process, along with the modulus of elasticity, and for plastic bodies, also the yield strength. in the paper "generalized archard law of wear based on rabinowicz criterion of wear particle formation", v. popov analyzes power-law wear equations under conditions of stationary wear. he finds that under the additional assumption of homogeneity of wear in the contact plane, the work of separation does not enter into the wear equation. only deviation from this bound (for example due to transport of wear particles) makes the dependency of wear intensity on the specific work of separation possible. in other words, the specific work of separation can only enter the wear equation if the wear process is characterized by some characteristic length. this can be the characteristic rabinowicz' length or some other structural parameter. in the future, it would be extremely interesting to check the found dependencies both experimentally and using direct numerical simulations. if the normal contact of two bodies is superimposed by small tangential oscillations, partial slip occurs within the contact interface, which causes wear. this phenomenon is called fretting and arises in numerous engineering applications. m. heß analyses this problem forthe contact of a rigid (but wearable) indenter and functionally graded materials. lubrication is often used to reduce or to control wear. especially important are additives determining the properties of boundary layers deposited on the surfaces of contact partners. in their paper "synergistic tribological properties of synthetic magnesium silicate hydroxide combined with amphiphilic molecules" wang et al. report on the synthesis of magnesium silicate hydroxide (msh) nanoparticles and their tribological properties combined with amphiphilic molecules (ams) as additives in base oil. this combination reduces wear losses substantially due to the formation of a double molecular layer on the rubbing surfaces under certain test conditions. e. willert considers in his paper "energy loss and wear in the oblique impact of elastic spheres" the wear processes during impacts of particles between each others or generalized law of adhesive wear iii with a solid. such impacts are a serious source of damage and failure in several technical systems like steam generator tubes, mining machinery and others. according to the archard law of wear, the wear volume is directly proportional to the energy loss during a tribological contact. e. willert utilizes this correlation to analyze impact wear. in the short communication "numerical implementation of fretting wear in the framework of the mdr", q. li et al. describe a numerical implementation of the integral transformations used in the method of dimensionality reduction, which guarantees stability of the numerical process independently of the number of iteration steps. the implementation is illustrated on examples of fretting wear simulation. the thematic issue is closed with the paper by tricarico et al. devoted to adhesion in multilayered systems. the diversity of topics and scales considered in the papers presented in this issue reflects the complexity of the wear process. in developing particular models of wear, it is always prudent to bear in mind this real complexity. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 139 155 https://doi.org/10.22190/fume180420016m © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper1 a cloud-based expert system for synthesis and evolutionary optimization of planar linkages udc 621.7 rosen mitrev 1 , boris tudjarov 2 , todor todorov 3 1 technical university, mechanical engineering faculty, sofia, bulgaria 2 technical university, faculty of computer systems and technologies, sofia, bulgaria 3 technical university, faculty of industrial technology, sofia, bulgaria abstract. the present paper introduces a cloud-based expert system for synthesis and evolutionary optimization of planar linkages. the kinematic structure of the linkage is composed by the modular approach based on assur’s groups. the dyads are represented as functional blocks with input and output variables. the applied approach for obtaining the geometrical relationships between the input and the output variables of the dyads is based on the use of homogeneous transformation matrices. the developed software system allows a dimensional synthesis of planar linkages by using genetic optimization algorithms. one feature is remote creation of the models of genetic algorithms as well as the receiving of the results by means of a user-friendly interface. by exploiting the application, the user can produce and edit the initial information about the synthesized or optimized linkage; thus he can receive the calculation results as a web page and/or as ms excel file. an additional mutation of the best chromosome genes by scanning of every gene within its searching space improves the optimal solution. the analyzed numerical case studies show the applicability of the developed software system for mechanism analysis, synthesis and optimization. because the number of genes is not limited, the linkages with a very big number of design variables can be synthesized by exploiting the developed approach. key words: planar linkage, assur’s groups, genetic algorithms, expert system 1. introduction in the past two decades, along with the classical graphical and analytical techniques [1, 2], there has been an increasing interest in the use of computer technologies for received april 20, 2018 / accepted may 28, 2018 corresponding author: rosen mitrev technical university, mechanical engineering faculty, kliment ohridski 8 blvd., sofia, bulgaria e-mail: rosenm@tu-sofia.bg 140 r. mitrev, b. tudjarov, t.todorov modeling and simulation of machines and mechanisms in education and engineering practice [3,4]. an easy applied and widely used approach is the modeling by way of special or general-purpose mechanical dynamics and kinematics software with different functionality realized on different platforms. some programs are fully interactive, offering an easy-to-use environment [5-7] and possessing modules for preprocessing, numerical analysis and results post-processing. simultaneously with undeniable advantages in its use, this type of software has some significant drawbacks: it is usually high-priced, the obtained results are limited to the software capabilities, equations of motion are embedded in the program and cannot be previewed by the user, and, finally yet importantly it does not allow further development of algorithms by the user. software systems where an active involvement of the user in the mechanism simulation model development is required are becoming increasingly popular. this type of software is based primarily on the algorithmic programming languages. for example, the c/c++ compatible object-oriented software [8] provides for a possibility of realizing independent applications in a web environment and capabilities for performing a kinematic and dynamic analysis of a variety of mechanisms as well as synthesizing mechanisms with predefined properties. other applications [9-11] are entirely web-based and platform independent client-server systems, exploiting the advantages of the network computing. typically, in this case, standard feature rich libraries for mechanism visualization, animation and results plotting are available. in some cases, the software systems are equipped with modules for mechanism type or dimensional synthesis, based on analytical or numerical methods [12]. a widely used approach, considerably facilitating the mechanisms creation, analysis and synthesis, is the modular approach [13, 14], which uses predefined blocks and subroutines for composing mechanisms with arbitrary complexity. during the realization of the modular kinematics it is possible to use different modeling approaches and philosophies. such systems as openmodelica [15] use graphical blocks to compose the mechanism kinematical structure while others use a collection of software subroutines for kinematic simulation, written in general-purpose [16] or computer algebra programming languages [17]. despite the presence of a vast number of software systems, the capabilities of the modular approach combined with optimization for the purpose of mechanism synthesis in the web environment are not used enough. the paper presents an open cloud-based expert system for dimensional synthesis and optimization of planar linkages based on the theory of assur’s groups and genetic algorithms. the modular approach applied to the building of the linkages allows for their fast creation, modification, analysis, synthesis and optimization in a user-friendly cloud-based internet environment, fully exploiting the benefits of the network computing. this paper is organized as follows: in section 1 papers dealing with mechanisms different modeling approaches and philosophies are analyzed. section 2 is devoted to the derivation of the assur’s groups position, velocity and acceleration kinematic equations. section 3 gives the structure of the developed cloud-based expert system for synthesis and optimization of planar linkages. section 4 presents the description and discussion of the synthesis of four-bar and six-bar planar linkages. section 5 represents a short conclusion. cloud-based expert system for synthesis and evolutionary optimization of planar linkages 141 2. assur’s groups kinematic equations the idea of decomposition of the mechanisms into assur’s groups is not new. at the beginning of the 20 th century, the russian scientist leonid assur developed a method of composing planar mechanisms of any complexity by the sequential appending of fundamental kinematic chains, which were later named assur’s groups. the number of links n and the number of the fifth class pairs p5 in the assur’s groups are related by the following equation: 5 3 2 p n because n and p5 must be integer numbers, the first possible solution of the above equation is n = 2 and p5 = 3, i.e. the simplest fundamental kinematical chain consists of two links and three fifth class kinematical pairs. internal kinematical pair of the group connects the two group links to each other and two external pairs connect the group to a driver link, to the other groups or to the ground. this simplest type of group is often called a binary group or dyad. each of the dyads has zero mobility and their appending to the mechanism does not change the dof (degree of freedom) of the whole mechanism. one can distinguish the following types of dyads: rrr, rrt, rtr, trt, rtt, where r denotes rotational one dof pair and t – translational one dof pair. the rrr dyad is called assur’s group of the first type. the rest of the dyads are created by replacement of the rotational with the translational pairs. the vast majority of the industrial linkages can be created by the combination of one or more dyads with the addition of one or more rotational or translational driving links. a substantial advantage of using dyads is the possibility to perform an independent kinematical analysis of each group and then compose a solution for the whole mechanism as a combination of partial solutions for different dyads. the primary goal of the solution is to describe the motion of the dyad according to the referential coordinate frame. let us demonstrate the derivation and analytical solution of the kinematic equations for rrr dyad by using rotation and homogeneous transformation matrices, widely used in robotics. in order to specify the position of the dyad pairs, it is necessary to define their cartesian coordinates in the fixed space reference coordinate system {x0y0}. to each rigid link of the dyad is attached a fixed coordinate frame {xkyk}, k = 1,2. the pose of the link is described by the position of its frame origin and the orientation of its x-axis according to the reference coordinate system. fig.1 shows the geometrical relationships between the global and local representations of the dyad specific points. the orientation of link k is specified by the angle of rotation φk of link xk axis relative to x0 axis of the reference coordinate system. angle φk is considered as positive if the rotation of xk axis according to positive x0 axis is counterclockwise. to point o3 is attached a local coordinate system {xeye} parallel to the reference frame. as the input for the position analysis of the rrr dyad are used coordinates (x1,y1) and (x3,y3) of two external rotational pairs and lengths l1 and l2 of the links. as the output are received coordinates (x2,y2) of the internal pair and angles φ1 and φ2. thus, a dyad can be considered as а functional block which has input and output variables, related by known kinematical relationships. this type of dyad representation allows creating subprograms or modules for each dyad type and utilizing them as building blocks when creating linkages. 142 r. mitrev, b. tudjarov, t.todorov 2.1. formulation of the position equations formulation of the position equations constitutes the most difficult part of the kinematical analysis. over the years, various approaches for formulation and analytical or numerical solution of the position equations are used [18-22]. a method to establish the geometrical relationships between coordinates of the group external pairs o1 and o3 is by using homogeneous transformation matrices. they represent a mapping from one frame to another: 2 2 2 1 3 3 1 2 ( ) ( , ) ( , , ) 1 b b b a a a x y x y              r p т 0 (1) where by ( )b a r is denoted the rotation matrix between two arbitrary coordinate systems a and b:   cos sin sin cos b a             r (2) φ – the angle of rotation between x-axes of a and b coordinate systems. by ( , )b a x yp is denoted the vector that represents coordinates of the origin of frame b according to the origin of frame a. x0 y0 o0 φ1 l1 l2 x1 y1 o1 φ12 φ2 x2y2 o2 o3 x1 y1 x2 y2 x3 y3 φ2 xe ye pk fig.1 schematics of rrr dyad the transformation matrix between {x0y0} and {xeye} coordinate frames is obtained by a sequential multiplication of a number of transformation matrices between the adjacent frames: 1 2 0 0 1 1 1 1 12 1 2 2 2( , , ) ( , ,0) ( , ,0)e ex y l l   t t t t (3) cloud-based expert system for synthesis and evolutionary optimization of planar linkages 143 after the expansion and simplification of eq. (3) we get: 3 1 1 1 2 2 3 1 1 1 2 2 1 0 1 0 cos cos 0 1 0 1 sin sin 0 0 1 0 0 1 x x l l y y l l                            (4) equating the elements (1,3) and (2,3) of the left matrix to the corresponding elements of the right matrix leads to the following position equations: 1 1 2 2 1 3 1 1 2 2 cos cos sin sin l l l l               r r (5) where by 1r and 3r are denoted the position vectors in the global frame of points o1 and o3, 1 1 1[ ]tx yr , 3 3 3[ ]tx yr . in eq. (3) we also had in mind that 12 2 1    (6) eqs. (5) constitutes a nonlinear system of transcendental equations with unknown variables φ1 and φ2. after elaborate algebraic manipulations are obtained equations for the unknown angles in an explicit form:     1 2 acos atan2 , p c d e f        (7) where the following notations are used: 1 1 32 ( )a l x x  , 1 1 32 ( )b l y y  , 2 2 1 2c l l   2 2 1 3 1 3( ) ( )x x y y    , 2 2d a b  , atan2( , )b d a d  , 1 3 1 1 2( ( ) sin ) /e y y l l    , 1 3 1 1 2( ( ) cos ) /f x x l l    . parameter p specifies the assembly mode of the rrr group and accepts values +1 and 1. in addition, the coordinates of inner rotational pair o2 are computed as: 2 1 1 1cosx x l   (8) 2 1 1 1siny y l   (9) 2.2 formulation of the velocity and acceleration equations once the position equations are established, the corresponding velocity and acceleration equations are obtained by a straightforward differentiation with respect to the time. the linear velocities of joints o1 and o3 and the angular velocities of links 1 and 2 are related by jacobian matrix j: v = jω (10) where 1 2[ ]t ω is the vector of the unknown angular velocities of the links and 1 3[ ] v r r is the vector of the difference of the known linear velocities of the external rotational pairs. the jacobian for the considered rrr dyad has the following form: 144 r. mitrev, b. tudjarov, t.todorov 1 1 2 2 1 1 2 2 sin sin cos cos l l l l             j (11) the singular configuration for the dyad is determined from the eq. (12). for l1, l2 ≠ 0 it is easy to find that the determinant (12) vanishes when φ1 = φ2 and singularities exist in this particular configuration. 1 2 1 2det( ) sin( )l l    j (12) the unknown angular velocities are determined from the equation 1 ω = j v (13) where by j -1 is denoted the inverse of the jacobian: 2 1 2 11 1 2 1 21 2 cos sin1 cos sinsin( ) l l l l                j (14) the time differentiation of eq. (10) leads to  a jω jω (15) which provides the relationship between the accelerations of external pairs 1 3[ ] a r r and the angular accelerations of links 1 2[ ]t ω . from eq. (15) we obtain the equations for the angular accelerations of the links: 1( ) ω= j a jω (16) when we use eq. (16), we have in mind that 1 1 1 2 2 2 1 1 1 2 2 2 cos cos sin sin l l l l                j (17) the cartesian coordinates, velocity and acceleration of internal joint o2 are calculated by the eqs. (18), (19) and (20): 2 1 1 1 1[cos sin ]tl   r r (18) 2 1 1 1 1 1[ sin cos ]tl    r r (19) 2 2 2 1 1 1 1 1 1 1 1 1 1( sin cos ) ( cos sin ) t l              r r (20) once the unknown coordinates and angles are obtained, the displacement, velocity and acceleration of every point of the links can be determined. a fixed point pk on body k is located from the origin of local frame {xkyk} by vector p ku and from the origin of global frame {x0y0} by vector , 1, 2p k k r (see fig.1). position p kr , velocity p kr and acceleration p kr of the point are computed by the following relations: cloud-based expert system for synthesis and evolutionary optimization of planar linkages 145 0 ( )p k p k k k k r r r u (21) 0 ( )p k p k k k k k  r r r u (22) 2 0 0( ) ( )p k p k p k k k k k k k k     r r r u r u (23) where 0 sin cos ( ) cos sin k kk k k k               r and 0 cos sin ( ) sin cos k kk k k k             r . in a similar manner, the kinematic equations for the other four types of dyads are derived. in figs.2-5 are shown the schematics and closed-form solutions for the rest of the assur’s groups, derived in a similar manner as for the rrr group. for the assur’s groups, containing slider pairs in their kinematic structure, one must take into account that the line of the sliding pair motion is defined by the coordinates of a point and angle, for example, for rrt dyad (see fig.3) are used coordinates (x3,y3) and angle φ3, measured from horizontal x0-axis. the velocities and accelerations of the dyad output parameters are calculated according to the following kinematical equations: -1 out in q = j q (24) ( )-1 out in outq = j q jq (25) whose quantities are shown in table 1, where the following short notations are used: c1 = cosφ1, s1 = sinφ1, c3 = cosφ3, s3 = sinφ3, cα = cosα, cα1 = cos(α+φ1), sα1 = sin(α+φ1), c4 = cosφ4, s4 = sinφ4, 13 1 3x x x   , 13 1 3y y y   , 13 1 3x x x   , 13 1 3y y y   , 14 1 4x x x   , 14 1 4y y y   , 14 1 4x x x   , 14 1 4y y y   . fig. 2 schematics and equations for trt dyad 146 r. mitrev, b. tudjarov, t.todorov fig. 3 schematics and equations for rrt dyad fig. 4 schematics and equations for rtr dyad fig. 5 schematics and equations for rtt dyad cloud-based expert system for synthesis and evolutionary optimization of planar linkages 147 table 1 velocities and accelerations of the dyads output parameters dyad input/output singularities jacobian rrr 13 13 x y        inq , 13 13 x y        inq 1 2          outq , 1 2          outq 1 2 2 k      1 2 2 k    k  1 1 2 2 1 1 2 2 l s l s l c l c         j , 1 1 1 2 2 2 1 1 1 2 2 2 l c l c l s l s            j 2 2 1 1 1 11 2 2 2 1 sin( ) c s l l c s l l                 -1 j rrt 13 13 x y        inq , 13 13 x y        inq 1 s        outq , 1 s        outq 1 3 2 2 k       1 3 2 2 k       k  1 1 3 1 1 3 l s c l c s         j , 1 1 1 3 3 1 1 1 3 3 l c s l s c            j 3 3 1 1 1 3 1 1 1 cos( ) s c l l c s             -1 j rtr 13 13 x y        inq , 13 13 x y        inq 1 s        outq , 1 s        outq 1 coss l   1 1 1 1 1 1 1 1 1 1 l s ss rc c l c sc rs s                   j 1 11 1 21 1 s c n ns l c             j 1 1 1 1 1n sc rs lc     2 1 1 1 1n ss rc l s     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) ss l c sc rs s sc l s ss rc c                          j trt 13 13 x y        inq , 13 13 x y        inq 1 2 s s        outq , 1 2 s s        outq 3 4 2 k      3 4 2 k    k  4 3 4 3 c c s s        j , 4 4 3 3 4 4 3 3 s s c c             j , 3 31 4 43 4 1 sin( ) s c s c           j rtt 14 14 x y        inq , 14 14 x y        inq 1 2s        outq , 1 2s        outq 2 2 k     2 2 k      k      2 3 3 2 3 3 sin cos c s              j 3 31 2 3 2 32 1 cos( ) sin( )cos s c               j 2 3 3 3 2 3 3 cos( ) sin( ) s c               j 148 r. mitrev, b. tudjarov, t.todorov 3. development of an experimental cloud-based expert system in order to combine the modular approach and the optimization synthesis in a common software environment, an experimental cloud-based expert system for synthesis and evolutionary optimization of planar linkages based on the derived in section 2 assur’s groups equations has been developed and tested. the system consists of the following software modules: 1) a module for linkage synthesis and optimization using evolutionary optimization methods, and, 2) a module with a user interface for linkage visualization, results plotting and animation. among the available evolutionary algorithms [23,24] the genetic optimization algorithms (ga) are chosen and implemented in the developed cloud-based expert system [25]. it allows remote creation of the models of genetic algorithms and the receiving of the results by means of a user-friendly interface. by exploiting the application, the user can produce and edit the initial information about the synthesized or optimized linkage; thus he can receive the calculation results as a web page and/or as ms excel file. the ga framework used technologies and realized functions are shown in fig.6a). the sequence of the work with the experimental application is as follows: a) the user creates a description of the specific problem (model of genetic algorithm) that is transported and saved as xml file fig. 6b), where the explanation of the purpose of the elements is given by italic letters); b) php file reads stored xml and automatically generates a new php file for the considered case and the execution is redirected to the generated file; and c) generated php file performs the calculations and transmits a report back to the user according to the user requirements. a) b) fig. 6 a) the ga framework; b) contents of the xml transport file cloud-based expert system for synthesis and evolutionary optimization of planar linkages 149 the sequence of the steps of genetic algorithms is: a) an initial random population of n chromosomes (solutions) is generated; b) the viability f(x) of each chromosome in the population n (target function called "fitness function") is calculated; c) the chromosomes are sorted according to their viability (calculated values of the fitness function) and priority for the next population is given to the best m of them (m<n); d) a new population is established by repeating the following steps until the new population is completed, namely preservation of the predetermined number m of the best solutions (according to their fitness the fitness function values), selection of two parental chromosomes of m chromosomes, use of crossover to cross the parents to form the next generation (children), use of mutation to mutate the newly created chromosomes and paste the new generation in the new population (adding n-m new chromosomes and filling the population); e) loop, go to step b) or stop and return the report if the final check condition is satisfied. the main novelties, implemented in the developed software system, are: 1) an additional mutation at the beginning of step e) of the best chromosome genes by scanning of every gene within its searching space. scanning is performed by the use of the dichotomy method [23] following the user-defined requirements for the number of the scanned points (respectively steps) and for the number of the scanning repetitions. if a better solution is found the best chromosome is substituted by the mutated one; 2) the models of ga calculator (parameters, fitness function and chromosome contents) for linkages synthesis and optimization are created by the use of the assur’s groups kinematic equations. for the results verification by plotting and animation an additional web-based module is developed with options for editing, design and development of planar linkages. the module is developed by means of free means, namely html5, javascript and canvas. the developed experimental expert system is web-based and has all advantages of the network computing systems: it overcomes space and time disposition, allows multi-user and multitasking regimes, reduces design costs, etc. here we have to point to some special outcomes from our research work: chromosome's genes number is not fixed, fitness function is a separate program written in php and it is possible to examine the results by the module for visualization and animation. by performing experiments with the obtained results the user can change the parameters of the generated linkages and/or initial parameters of the ga calculator. 4. numerical case studies and discussion to test and validate the presented cloud-based expert system two numerical case studies are developed and analyzed. the first case study deals with the dimensional synthesis of a fourbar linkage fig.7a) so coupler point m follows the desired path which is an ellipse with a ratio of minor to major semi-axes 1:2 and arbitrary fixed position in the plane. the four-bar linkage is modeled by an rrr dyad, driven by a driver crank. in the second case a watt i six-bar linkage [26] is considered, composed of two rrr dyads and a driver crank – fig.7b). coupler point m follows the same ellipse. for both cases, the optimal values of the design variables are determined so the maximum distance between the desired path, defined by 20 uniformly situated points pi, (i=1, 2,...,20) and 150 r. mitrev, b. tudjarov, t.todorov the calculated coupler point mi curve is minimal. this method can be described by objective function f as follows: 2min{max },( 1,..., 20)if r i   (26) where 2 2 2( ) ( ) i i i ii p m p mr x x y y     (27) is the squared distance between the points (xpi,ypi) of the ellipse and the corresponding points (xmi,ymi) of the coupler. the design variables for the four-bar linkage are l1, l2, l3, l4 and γ1 – see fig.4a) and for the six-bar linkage are l1, l2, l3, l4, l5, l6, l7, l8, γ1, γ2 and γ3 – see fig.7b). also, for both linkages the coordinates of points o and c must be determined – (xo, yo) and (xc, yc). during the synthesis two types of constraints must be satisfied: 1) the design variables (lengths and angles) are changed in predefined limits (see tables 2 and 3 – min. and max. values); 2) the grashof condition for the oabc and bdfe four-bar linkages must be satisfied. in addition, to obtain a proportionate from an engineering point of view linkage, additional relations between the links lengths could be defined. the linkages dimensional synthesis is performed by using a two-stage optimization process. at the initial stage, the stochastic search of the design variables optimal values is performed in a relatively big design variables space. based on the fitness function change, the design variables values found after 500 generations are considered as optimal for the first stage. o c a m b l1 l2 l3 l4 γ1 o c a d b e f m l1 l2 l3 l4 l5 l7 l6 l8 γ1 γ2 γ3 а) b) fig. 7 schematics of the synthesized linkages: a) four-bar; b) watt-1 six-bar mechanism at the second stage, the first stage optimal values are used as initial guesses and the design variables limits are chosen so that the stochastic search is performed in a narrower range of design variables values. in tables 2 and 3 for the considered linkages the limits of the design variables (dimensions for the lengths are in arbitrary linear units, dimensions for the angles are in degrees) for the two stages are shown. the results for the parameters optimal values and objective function f500 after every stage are shown additionally. cloud-based expert system for synthesis and evolutionary optimization of planar linkages 151 table 2 limits of the design variables and optimal values – four-bar linkage l1 l2 l3 l4 γ1 xo yo xc yc stage 1, 500 generations, f500=2.049 min. values 20 20 20 20 0 -50 0 0 -100 max. values 100 100 100 100 90 50 100 100 0 optimal values 34.79 66 80 52.35 83 19.55 39.84 36.01 -63 stage 2, 500 generations, f500=1.565 min. values 30 60 70 40 60 10 30 30 -70 max. values 50 80 90 60 90 30 50 50 -50 optimal values 37 70.96 87 54 73 14.48 41.89 36.65 -67 table 3 limits of the design variables and optimal values – six-bar linkage l1 l2 l3 l4 l5 l6 l7 l8 γ1 γ2 γ3 xo yo xc yc stage 1, 500 generations, f500=1.724 min. values 20 20 20 20 20 20 20 20 -90 -90 -90 -100 -100 -100 -100 max. values 100 100 100 100 100 100 100 100 90 90 90 100 100 100 100 optimal values 32 72.99 72 78 85.07 71 85 75 89 -27 11 -30.1 -6.1 44.03 -81 stage 2, 500 generations, f500=1.151 min. values 25 68 68 75 80 65 80 70 85 -32 6 -35 -11 40 -85 max. values 35 78 78 85 90 75 90 80 95 -22 16 -25 -1 50 -75 optimal values 30.95 72 73 85 86 72 85 74 91 -24 8 -26.9 -8.1 44 -85 the presented results show that for the four-bar linkage the improvement of the fitness value at the second stage in relation to the first stage is about 23.6%, while for the six-bar linkage the same value is about 33.2%. in addition, one can see that the fitness value for the six-bar linkage is smaller than the fitness value for the four-bar linkage by 26.4%, i.e. the six-bar linkage better tracks the desired path. this fact is explained by the presence of more geometrical parameters in the six-bar linkage that can be adjusted to minimize the fitness function. fig. 8 evolution of the fitness function values for the considered linkages 152 r. mitrev, b. tudjarov, t.todorov in fig. 8 the evolution of the fitness function values for the considered case studies is shown. in figs. 9 and 10 the evolution of the linkages geometrical configurations together with the couplers trajectory and the desired path is shown. stage 1 generation 1, f=11.6161 generation 14, f=5.148866 generation 500, f=2.048843 stage 2 generation 1, f=2.911277 generation 14, f= 2.181182 generation 500, f=1.565115 fig. 9 evolution of the four-bar linkage geometrical configurations and the coupler point path cloud-based expert system for synthesis and evolutionary optimization of planar linkages 153 stage 1 generation 1, f=17.4918 generation 16, f=8.8444 generation 500, f=1.7238 stage 2 generation 1, f=3.03592 generation 10, f=1.54891 generation 500, f=1.151466 fig. 10 evolution of the six-bar linkage geometrical configurations and the coupler point path 5. conclusion the paper presents a cloud-based expert system for synthesis and evolutionary optimization of planar linkages. the software system allows composition of linkage kinematical structure by means of the modular approach based on assur’s groups. the used approach for obtaining the geometrical relationships between the input and the output variables of the dyads is based on the use of homogeneous transformation matrices. the representations of the dyads as functional blocks with input and output variables allows creating subprograms or modules for each dyad type and utilizing them as building blocks when creating linkages with arbitrary complexity. the velocity and acceleration equations are obtained by a straightforward differentiation with respect to the time and the relationships between the input and output velocities and accelerations are represented by the use of the jacobian, its inverse and time derivative. the developed expert system allows a dimensional synthesis of planar linkages by using genetic optimization algorithms. one feature is remote creation of the models of genetic algorithms and the receiving of the results by using a user-friendly interface. by exploiting the application, the user can produce and edit the initial information about the synthesized or optimized linkage; thus he can receive the calculation results as a web page 154 r. mitrev, b. tudjarov, t.todorov and/or as ms excel file. an additional mutation of the best chromosome genes by scanning of every gene within its searching space improves the optimal solution. if a better solution is found the best chromosome is substituted by the mutated one. the analyzed numerical case studies show the applicability of the developed software system for mechanism analysis, synthesis and optimization. because the number of genes is not limited, linkages with a very big number of design variables can be synthesized exploiting the developed approach. references 1. artobolevsky, i., 1988, theory of machines and mechanisms, moscow, science, 640 p. 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(in bulgarian). cloud-based expert system for synthesis and evolutionary optimization of planar linkages 155 24. stoven-dubois, a., botzheim, j., kubota, n., 2016, fuzzy gesture expression model for an interactive and safe robot partner, journal of network intelligence, 1(4), pp. 119–129. 25. tudjarov, b., kubota, n., penchev, v., hristov, v., 2011, web based modeling and calculation of genetic algorithms, international workshop on advanced computational intelligence and intelligent informatics, iwaciii 2011, suzhou, china. 26. mccarthy j., soh g., 2011. geometric design of linkages, springer verlag, 448 p. facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 537 553 https://doi.org/10.22190/fume210416052m © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper an integrated decision-making model for efficiency analysis of the forklifts in warehousing systems eldina mahmutagić1, željko stević1, zdravko nunić1, prasenjit chatterjee2, ilija tanackov3 1university of east sarajevo, faculty of transport and traffic engineering, bosnia and herzegovina 2department of mechanical engineering, mckv institute of engineering, india 3university of novi sad, faculty of technical sciences, serbia abstract. in the logistics world, special attention should be given to warehousing systems, cost rationalization, and improvement of all the factors that affect efficiency and contribute to smooth functioning of logistics subsystems. in real time industrial practice, the issue of evaluating and selecting the most appropriate forklift involves a complex decision-making problem that should be formulated through an efficient analytical model. the forklifts efficiency plays a very important role in the company. the forklifts are being used on a daily basis and no logistical processes could be done without them. therefore, it has been decided to determine their efficiency, which will contribute to the optimization of the process in this logistics subsystem. this study puts forward an integrated forklift selection model using data envelopment analysis (dea), full consistency method (fucom) and measurement alternatives and ranking according to the compromise solution (marcos) methods. five input parameters (regular servicing costs, fuel costs, exceptional servicing costs, total number of all minor accidents and damage caused by forklifts) and one output parameter (number of operating hours) were first identified to assess efficiency of eight forklifts in a warehousing system of the natron-hayat company using the dea model. this step allows sorting of efficient forklifts which are subsequently evaluated and ranked using fucom and marcos methods. a sensitivity analysis is also performed in order to check reliability and accuracy of the results. the findings of this research clearly show that the proposed decision-making model can significantly contribute to all spheres of business applications. key words: dea, fucom, marcos, warehouse, forklifts received april 16, 2021 / accepted june 28, 2021 corresponding author: eldina mahmutagić university of east sarajevo, faculty of transport and traffic engineering, vojvode mišića 52, 74000 doboj, bosnia and herzegovina e-mail: mahmutagiceldina24@gmail.com 538 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov 1. introduction the warehouse, in addition to being one of the basic subsystems of logistics, is an important part in every organization for which it deserves special attention in terms of monitoring and analyzing its associated processes and activities. when planning, building and using any warehouse, the greatest attention is paid to size of the warehouse, layout, and training of employees who manage the warehousing system. on the other hand, enough attention is generally not given to selecting the most appropriate forklift which is one of the most important items that can greatly affect the entire warehousing system, optimization of vehicles’ waiting for loading and unloading, efficiency and overall effectiveness of the entire organization. considering that the natron-hayat company is one of the leading ones in the region of bosnia and herzegovina, optimization of parameters in the warehousing system can bring superior results which finally lead to business success. there are currently eight forklifts in the warehouse of the considered company and it is necessary to analyze efficiency of these forklifts in order to justify those procurement investments that will contribute to the business success of the company. data envelopment analysis (dea), which is a linear programming-based method, is here adopted in order to analyze efficiency of forklifts which are currently operating at the considered warehouse. dea is primarily used to determine relative efficiencies of decision-making units (dmus) or alternatives. major reasons for a rapid increase in the use of the dea method lie in the fact that this model is interdisciplinary in applications; it is also suitable in the cases where other approaches do not provide satisfactory results due to their complex or unknown nature of the links between multiple inputs and outputs [1]. the main benefit of this method is its competence to quarter a variety of input-output combinations. another advantage of the dea method is that there is no requirement of clearly stipulating mathematical form for the considered functions which enables it to be used for any input-output measurement. based on the collected data for all the forklifts that serve in the warehousing system of the case company, the dea method has identified the forklifts which are not efficient enough, and they are excluded from further considerations. based on the dea-based results, two popular multi-criteria decisionmaking (mcdm) methods, called full consistency method (fucom) and measurement alternatives and ranking according to the compromise solution (marcos) are further applied to derivation of a complete ranking of the efficient forklifts in order to identify the most efficient forklift alternative. the fucom method is used here to obtain criteria weights, whereas the rankings of forklift alternatives are derived using the marcos method. the contribution of this paper is reflected in the fact that based on the deamcdm model applied to the warehousing system and handling equipment; it can significantly improve the decision-making process, reduce costs and increase work efficiency. based on the recommendations of the proposed model, procurement of handling and transportation equipment can be planned efficiently. the contribution of this research in scientific terms can be observed through the original integration of dea and two mcdm methods for handling efficiency estimation problems when solo application of the dea method is unable to provide for a complete ranking order of alternatives in multi-criteria environment. there are different places and ways of determining efficiency in the company. however, due to many parameters, the dea method proved to be a good method, whose results will show efficient as well as less efficient forklifts. the results of the dea method can help managers with the selection and purchasing of forklifts in the future. however, after the dea method, it has been an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 539 decided to use multi-criteria methods to obtain the best and most realistic results. the main reason for the combination and development of the dea-fucom-marcos model is that any company that faces decision-making challenges can get comprehensive and as realistic results as possible through this model. in addition to introductory considerations, the paper is structured through other five sections. section 2 presents an overview of the literature used throughout the research. section 3 presents the methods used to select the most efficient forklift in the warehousing system, namely dea, fucom and marcos methods. section 4 presents application of dea method, while section 5 provides selection of the most efficient forklift using fucom and marcos methods. section 6 refers to sensitivity analysis of the obtained results and section 7 concludes the paper. 2. literature review a large number of modern studies in which they are used prove the application of dea and mcdm methods in logistics and transport. the charnes, cooper and rhodes (ccr) model is considered as the most primary and vastly applied dea model. the ccr model was first conceptualized and formulated by charnes, cooper and rhodes in 1978 [2]. the basic theme of this model actually originated from earlier work on the basic theory of productivity measurement using single output and single input ratio concept. applications of the dea method are now found in many areas, and dea has become a central technique in productivity and efficiency analyses for comparing organizations, enterprises, regions and countries. the dea method has also made its contribution to logistics [3, 4, 5], warehousing [6] and transportation [7] as its subsystems and supply chains [8, 9]. the paper [4] presents an overview of dea models for measuring the efficiency of supply chains. the process of measuring efficiency in production companies differs greatly from the process of measuring efficiency in service companies. it was concluded that in order to successfully measure efficiency in logistics, it is necessary to consider a large number of inputs and outputs that are heterogeneous (financial, technical, environmental, energy, social, etc.) and are expressed in different units of measurement. in this sense, it is possible to measure energy, environmental, cost and other types of efficiency in logistics. in the paper [10], the dea model was applied in order to analyze the efficiency of the bus subsystem of public passenger transport in belgrade, on a sample of five small, three medium, and two large companies. the subject of the research in [11] is the analysis of the efficiency of airlines in the european union in 2012, where the application of the dea model assessed the individual efficiency of each airline; besides, it also identified inefficient elements of business that could be improved. the paper [12] deals with the analysis of the efficiency of intermodal terminals with the aim of identifying terminals that would serve as models for the improvement of existing and development of new terminals. the dea method was used to determine the efficiency of the terminals, and the research was conducted on a sample of 35 real land trimodal terminals in europe. the results of the research showed that for the defined sample, the parameters storage capacity and length of railway tracks had the greatest influence on achieving the efficiency of the terminal. in addition to assessing efficiency, this method is also used to define degree of safety, i.e. to evaluate risk in transportation and traffic systems [13, 14]. the dea method has also found application in military organization [15, 16, 17]. this is shown in ref. [15] 540 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov presenting the application of the dea and sfa methods for evaluating the efficiency of the work of the selected ten military transport units and 173 military motor vehicles used in cargo transportation tasks for the needs of supply and special needs of the army, individually and within six defined classes. we also see the application of the dea method and the determination of the efficiency of the liaoning port logistics [16]. this paper used dea to analyze the logistics efficiency of four major ports in the liaoning province. then based on the results of dea analysis, the relationship between logistics efficiency and its influencing factors is studied. the results show that the overall pure technical efficiency of port logistics in the liaoning province has been maintained at a high level and the state is stable. also, there is an increasing number of papers related to solving logistics problems using mcdm methods. this research [17] created the hybrid bwm-copras model for the assessment of off-road vehicles. 3. method in this section, fig. 1 presents the methodology which includes application of the dea method for assessing relative efficiencies of the alternative of forklifts and the fucom and marcos methods application to identifying the most efficient forklift. fig. 1 research methodology the research methodology in this paper consists of four phases. the first phase refers to the analysis and collection of necessary data on the operation of forklifts in the warehousing system. in the second phase, the dea method is applied which is presented through three steps, namely: defining input and output parameters, applying ccr model and results of dea method. after obtaining results of the dea ccr model, the third an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 541 phase starts in which the fucom and marcos methods are applied to selecting the most efficient forklift. the last step of the third phase is a sensitivity analysis of the obtained results. finally, at the end of the paper, the results of all the applied steps aim at determining the most efficient forklift. 3.1. dea method this section presents two dea ccr models [2] that have been applied to obtaining the values of alternatives, i.e. dmus according to an input-oriented model (min) and an output-oriented model (max). the dea ccr input oriented model (min) is as follows: 𝐷𝐸𝐴𝑖𝑛𝑝𝑢𝑡 = min ∑ 𝑤𝑖 𝑚 𝑖=1 𝑥𝑖−𝑖𝑛𝑝𝑢𝑡 𝑠𝑡: ∑ 𝑤𝑖 𝑚 𝑖=1 𝑥𝑖𝑗 − ∑ 𝑤𝑖 𝑚+𝑠 𝑖=𝑚+1 𝑦𝑖𝑗 ≥ 0, 𝑗 = 1, … , 𝑛 ∑ 𝑤𝑖 𝑚+𝑠 𝑖=𝑚+1 𝑦𝑖−𝑜𝑢𝑡𝑝𝑢𝑡 = 1 𝑤𝑖 ≥ 0, 𝑖 = 1, . . . , 𝑚 + 𝑠 (1) dmus consist of m input parameters for each alternative xij, while s represents the output parameters for each alternative yij, taking into account weights of the parameters denoted by wi,n represents total number of dmus. the dea ccr output oriented model (max) is as follows: 𝐷𝐸𝐴𝑜𝑢𝑡𝑝𝑢𝑡 = max ∑ 𝑤𝑖 𝑚+𝑠 𝑖=𝑚+1 𝑦𝑖−𝑜𝑢𝑡𝑝𝑢𝑡 𝑠𝑡: −(∑ 𝑤𝑖 𝑚 𝑖=1 𝑥𝑖𝑗) + ∑ 𝑤𝑖 𝑚+𝑠 𝑖=𝑚+1 𝑦𝑖𝑗 ≤ 0, 𝑗 = 1, . . . , 𝑛 ∑ 𝑤𝑖 𝑚 𝑖=1 𝑥𝑖−𝑖𝑛𝑝𝑢𝑡 = 1 𝑤𝑖 ≥ 0, 𝑖 = 1, . . . , 𝑚 + 𝑠 (2) 3.2. fucom method fucom method was developed by pamučar, stević and sremac [18] to determine the criteria weights in mutually conflicting mcdm environment. fucom provides the possibility to perform model validation by calculating error size for the obtained weight vectors by determining degree of consistency [18-21]. fig. 2 presents fucom algorithm including steps of this method. 3.3. marcos method the marcos method (fig. 3) is based on defining relations between an alternative and reference values (ideal and anti-ideal alternatives). based on these defined relations, utility functions of the alternatives are determined and a compromise ranking is made in relation to ideal and anti-ideal solutions. decision preferences are defined based on utility functions. utility functions represent position of an alternative in relation to an ideal and anti-ideal solution. the best alternative is the one which is closest to an ideal as well as furthest from an anti-ideal reference point. the marcos method is implemented through the following simple steps [22-24] presented in fig. 3. 542 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov fig. 2 overview of fucom method and its steps fig. 3 algorithm of the marcos method algorithm: fucom input: expert pairwise comparison of criteria output: optimal values of the weight coefficients of criteria/sub-criteria step 1: expert ranking of criteria/sub-criteria. step 2: determining the vectors of the comparative significance of evaluation criteria. step 3: defining the restrictions of a non-linear optimization model. restriction 1: the ratio of the weight coefficients of criteria is equal to the comparative significance among the observed criteria, i.e. 1 /( 1)k k k kw w + += . restriction 2: the values of weight coefficients should satisfy the condition of mathematical transitivity, i.e. / ( 1) ( 1)/ ( 2) / ( 2) k k k k k k  + + + + = . step 4: defining a model for determining the final values of the weight coefficients of evaluation criteria: ( ) / ( 1) ( 1) ( ) / ( 1) ( 1)/ ( 2) ( 2) 1 min . . , , 1 0, j k k k j k j k k k k k j k n j j j s t w j w w j w w w j       + + + + + + = −   −    =    step 5: calculating the final values of evaluation criteria/sub-criteria ( )1 2, ,..., t nw w w . steps of marcos model ii creating of an extended initial matrix i creating an initial decision-making matrix iv computation of the weighted matrix vi calculation of the utility degree of alternatives iii creating a normalized matrix v calculation of matrix vii calculation of matrix ti viii . determination of utility functions in relation to the ideal and anti-ideal ix determination of the utility function of alternatives x ranking the alternatives an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 543 4. application of the dea method for evaluation of forklift efficiency natron-hayat ltd. is a highly recognized and reputable european company in the field of manufacturing paper and paper packaging applications. in this company, the method of storage is decentralized, and includes four warehouses for storage of finished products (warehouse of paper machine pm4, warehouse of paper machines pm3-pm1, warehouse of production plant paper products and customs warehouse). it is clear that the main handling equipment in the warehouses is a forklift, so it is the reason for conducting this research in order to determine how efficiency of a forklift affects the entire warehousing process. the collected data from the analysis of forklift efficiency and application of the integrated dea-mcdm model cover one calendar year (2019). most of the data were recorded in previous years; however, special attention was paid to some parameters for getting precise and comprehensive data for the considered case study. the data that form the basis for the dea method in this paper are presented in fig. 4. fig. 4 input and output parameters required to determine efficiency of forklifts in order to determine efficiencies of the considered forklifts, data were collected for all forklifts, currently in operation. table 1 shows input and output parameters for all eight forklifts. input parameters for each of the eight forklifts are difficult to be determined, but the most important input parameters on the basis of which the efficiency can be determined using the dea method are: regular servicing costs, fuel costs, exceptional servicing costs, and the total number of minor accidents and damage caused by the forklift. output parameter, i.e. the result of forklift operation is the number of operating hours for each forklift individually. 544 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov table 1 input and output parameters for all forklifts parameters regular servicing costs (bam) fuel costs (bam) exceptional servicing costs (bam) the total number of all minor accidents and damage caused by the forklift the number of operating hours (h) forklift 1 870 483 562.5 12 864 forklift 2 1820 5622 562.5 36 4320 forklift 3 2534 14806 2,222.11 36 5184 forklift 4 3503 13706 5,094.27 36 5184 forklift 5 1500 6097 1,363.85 36 2400 forklift 6 1890 7593 2,887.89 60 3240 forklift 7 1800 4046 562.5 60 2640 forklift 8 2700 8856 562.5 36 3840 table 1 shows all input as well as output parameters for forklifts currently in operation. each of these forklifts caused certain costs as stated, and also a certain number of minor accidents and damage. the number of operating hours is an output parameter that is very important for determining efficiency, and from this table, we can see that forklifts 3 and 4 have the highest number of operating hours, which does not mean that they are the most efficient as there are other forklifts as well with lower costs, fewer accidents and less damage. the input and output dea models are given below by eqs. (1) and (2). by solving the dea models for all the considered forklifts, the obtained values are shown in table 2. this table also presents the final dea values, obtained by comparing the two input and output oriented dea models. the dea final values are being derived from: dea-finaln = dea-outputn / dea-inputn. table 2 results of dea method dea-input dea-output dea-final forklift 1 1.000 1.000 1.000 forklift 2 1.000 1.000 1.000 forklift 3 1.000 1.000 1.000 forklift 4 1.000 1.000 1.000 forklift 5 1.483 0.674 0.454 forklift 6 1.384 0.722 0.521 forklift 7 1.234 0.809 0.655 forklift 8 1.125 0.888 0.789 from table 2, it is clear that forklifts 5, 6, 7 and 8 have values less than 1. they are not considered further into the model since they are neither efficient enough nor do they contribute to the natron-hayat company like other forklifts. after obtaining the results of the dea method, it is observed that the first four forklifts have efficiency values of 1, indicating them as efficient alternatives; the most efficient of these four forklifts will now be selected in the next phase using fucom-marcos methods. an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 545 5. ranking efficient forklifts by applying the fucom and marcos methods the dea model results indicate that the first four forklifts are efficient alternatives, fulfilling their tasks satisfactorily. this section of the paper presents results of mcdm methods as applied for these four forklifts. criteria weights are determined by the fucom method, while the marcos method is used to determine the most efficient forklift, i.e. ranking of these four forklifts is derived according to their efficiency. calculation of criteria weights by the fucom method is performed as follows. the first step is an activity in which the criteria are ranked as follows: c2>c5>c1>c4>c3 after ranking the criteria, they are compared, so the values of comparative priorities are defined according to the second step, which can be seen in table 3. table 3 overview of comparative priorities for comparing criteria criterion name (in accordance with ranking) c2 c5 c1 c4 c3 comparison of criteria 1 1.15 1.3 1.6 2.1 it is then necessary to apply the third, fourth and fifth steps of the fucom method in order to obtain final values of criteria weights. the final results of the fucom method, i.e. significance of the criteria on the basis of which final evaluation of the forklift efficiency is performed, are shown in fig. 5. fig. 5 criteria weights as given by fucom method according to the results of the fucom method, as shown in fig. 5, we can see that out of five criteria, the criterion related to fuel costs is the most significant (c2). it is then subsequently followed by criterion c5 (number of operating hours) and c1 (regular servicing costs). the last two and the least significant criteria are those relating to the total number of all minor accidents and damage caused by the forklift (c4) and exceptional servicing costs (c3). 546 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov after determining the criteria weights, the marcos method is applied in order to derive a complete ranking order of the four forklifts. table 4 presents an extended initial matrix formed according to the second step of the marcos method. essence of forming this matrix is to consider the initial decision matrix while taking into account orientation of the criteria themselves, i.e. the need to minimize or maximize the criteria. it should be emphasized that the first four criteria: regular servicing costs, fuel costs, exceptional servicing costs and the total number of all minor accidents and damage caused by the forklift are minimized, and the total number of operating hours needs to be maximized. accordingly, for the first four criteria, the ideal solution that enters the extended initial decision matrix is the minimum value, while for the fifth criterion, the highest value is the ideal solution. when forming an anti-ideal solution, which is also an integral part of the aforementioned matrix, opposite values are taken. table 4 extended initial matrix c1 c2 c3 c4 c5 aii 3503 14806 5094.3 36.0 864 a1 870 483 562.5 12 864 a2 1820 5622 562.5 36 4320 a3 2534 14806 2222.11 36 5184 a4 3503 13706 5094.27 36 5184 ai 870 483 562.50 12.00 5184 max/min min min min min max normalization of the extended initial matrix is performed according to step 3, and values of the normalized matrix can be seen in table 5: 𝑛𝑖𝑗 = 𝑥𝑎𝑖 𝑥𝑖𝑗 ; 𝑥21 = 870 1820 = 0.478 𝑛𝑖𝑗 = 𝑥𝑖𝑗 𝑥𝑎𝑖 ; 𝑛15 = 864 5184 = 0.167 table 5 normalized matrix c1 c2 c3 c4 c5 aii 0.248 0.033 0.110 0.333 0.167 a1 1.000 1.000 1.000 1.000 0.167 a2 0.478 0.086 1.000 0.333 0.833 a3 0.343 0.033 0.253 0.333 1.000 a4 0.248 0.035 0.110 0.333 1.000 ai 1.000 1.000 1.000 1.000 1.000 weighted normalized matrix is then obtained according to the fourth step of the marcos method by multiplying the values from the normalized matrix by weight coefficients of the criteria, which were previously obtained using the fucom method, as shown in table 6. an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 547 𝑣𝑖𝑗 = 𝑛𝑖𝑗 × 𝑤𝑗; 𝑣11 = 1.000 × 0.206 = 0.206 table 6 weighted normalized matrix c1 c2 c3 c4 c5 aii 0.051 0.009 0.014 0.056 0.039 a1 0.206 0.267 0.127 0.167 0.039 a2 0.098 0.023 0.127 0.056 0.194 a3 0.071 0.009 0.032 0.056 0.233 a4 0.051 0.009 0.014 0.056 0.233 ai 0.206 0.267 0.127 0.167 0.233 calculation of utility degree of ki alternative: according to step 5, utility degrees of the alternatives in relation to an anti-ideal and ideal solution are calculated as follows. 𝐾𝐼 − = 𝑆𝑖 𝑆𝑎𝑎𝑖 = 0.806 0.168 = 4.790, 𝐾𝐼 + = 𝑆𝑖 𝑆𝑎𝑖 = 0.806 1 = 0.806 where expression si(i=1,2,…,m) represents the sum of the weighted matrix elements. 𝑆𝑖 = ∑ 𝑣𝑖𝑗 𝑛 𝑖=1 = 𝑆1 = 0.206 + 0.267 + 0.127 + 0.167 + 0.039 = 0.806 also, we have that 𝑆𝑎𝑎𝑖 = 0.051 + 0.009 + 0.014 + 0.056 + 0.039 = 0.168 determining utility function of alternative f(ki). the utility function represents a compromise of the observed alternative in relation to an ideal and anti-ideal solution. utility functions of alternatives are defined according to the sixth step: 𝑓(𝐾𝑖) = 𝐾𝑖 + + 𝐾𝑖 − 1 + 1−𝑓(𝐾𝑖 +) 𝑓(𝐾𝑖 +) + 1−𝑓(𝐾𝑖 −) 𝑓(𝐾𝐼 −) = 0.806 + 4.790 1 + 1−0.856 0.856 + 1−0.144 0.144 = 0.787 where f (ki -) represents a utility function in relation to an anti-ideal solution, while f (ki +) represents a utility function in relation to an ideal solution. the utility functions in relation to an ideal and anti-ideal solution are determined by applying step 8 as follows 𝑓(𝐾𝐼 −) = 𝐾𝑖 + 𝐾𝑖 + + 𝐾𝑖 − = 0.806 0.806 + 4.790 = 0.144 𝑓(𝐾𝐼 +) = 𝐾𝑖 − 𝐾𝑖 + + 𝐾𝑖 − = 4.790 0.806 + 4.790 = 0.856 the ninth and tenth steps represent ranking of alternatives on the basis of utility functions. it is always preferable when an alternative has the highest possible value of utility function. 548 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov table 7 results of marcos method si kiki+ fkfk+ ki rank a1 0.806 4.790 0.806 0.144 0.856 0.787 1 a2 0.498 2.959 0.498 0.144 0.856 0.486 2 a3 0.400 2.375 0.400 0.144 0.856 0.390 3 a4 0.363 2.155 0.363 0.144 0.856 0.354 4 results of the marcos method of table 7 show that the most efficient forklift is a1, i.e. alternative 1. from table 7, it is observed that utility function of forklift a1 is significantly higher than the obtained values of other forklifts. forklift a2 is less efficient as compared to the forklift a1, and the next position in terms of efficiency is occupied by forklift a3. the least efficient among these four forklifts is forklift a4, i.e. alternative 4 due to its lowest utility function value. 6. sensitivity analysis in order to test accuracy of the obtained results, a sensitivity analysis has been performed. in this paper, the sensitivity analysis has been done in two parts. the first part involves changing criteria weights to determine how the criteria weights affect the results. the second part is a comparison of the obtained results with those of seven other well established mcdm methods. 6.1. changes in weight values of criteria in this part of the sensitivity analysis, impact of changes in criteria weights is analyzed. criteria weights are changed in a range of 15-90%, starting from the most significant criterion c2, followed by criteria c5, c1, c4 to criterion c3. by applying eq. (3), a total of 30 scenarios are formed. 𝑊𝑛𝛽 = (1 − 𝑊𝑛𝛼) 𝑊𝛽 (1−𝑊𝑛) (3) in scenarios s1-s6, weight of the most significant criterion c2 was changed, while in scenarios s7-s12, weight of criterion c5 was changed, followed by subsequent weight changes in criterion c1 for scenarios s13-s18, criterion c4 for scenarios s19-s24 and criterion c3 for scenarios s25-s30, respectively.wnβ represents a new value of criteria c1, c3, c4, c5,wnα represents a reduced value of criterion c2, wβ is an original value of the observed criterion and wn represents an original value of the criterion whose value is reduced – c2 (for the first group of scenarios,s1-s6). wnβ represents a new value of criteria c1, c2, c3, c4, wnα represents a reduced value of criterion c5, wβ is an original value of the observed criterion and wn represents an original value of the criterion whose value is reduced –c5 (for the second group of scenarios,s7-s12).wnβ represents a new value of criteria c2, c3, c4, c5, wnα represents a reduced value of criterion c1, wβ is an original value of the observed criterion and wn represents an original value of the criterion whose value is reduced –c1 (for the third group of scenarios,s13-s18). wnβ represents a new value of criteriac1, c2, c3, c5, wnα represents a reduced value of criterion c4, wβ is an original value of the observed criterion and wn represents an original value of the criterion whose an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 549 value is reduced c4 (for the fourth group of scenarios, s19-s24). wnβ represents a new value of criteria c1, c2, c4, c5, wnα represents a reduced value of criterion c3, wβ is an original value of the observed criterion and wn represents an original value of the criterion whose value is reduced c3 (for the fifth group of scenarios, s25-s30). all simulated values of the criteria through the newly formed 30 scenarios are presented in table 8. table 8 simulated values of criteria through newly formed 30 scenarios w1 w2 w3 w4 w5 w1 w2 w3 w4 w5 s1 0.217 0.227 0.134 0.176 0.245 s16 0.082 0.309 0.147 0.193 0.269 s2 0.228 0.187 0.141 0.185 0.258 s17 0.051 0.319 0.152 0.200 0.278 s3 0.239 0.147 0.148 0.195 0.271 s18 0.021 0.330 0.157 0.206 0.287 s4 0.251 0.107 0.155 0.204 0.283 s19 0.212 0.275 0.131 0.142 0.240 s5 0.262 0.067 0.162 0.213 0.296 s20 0.218 0.283 0.135 0.117 0.247 s6 0.273 0.027 0.169 0.222 0.309 s21 0.224 0.292 0.139 0.092 0.253 s7 0.215 0.280 0.133 0.175 0.198 s22 0.230 0.300 0.143 0.067 0.260 s8 0.224 0.292 0.139 0.182 0.163 s23 0.237 0.308 0.146 0.042 0.267 s9 0.234 0.304 0.145 0.190 0.128 s24 0.243 0.316 0.150 0.017 0.274 s10 0.243 0.316 0.150 0.197 0.093 s25 0.210 0.273 0.108 0.171 0.238 s11 0.252 0.328 0.156 0.205 0.058 s26 0.215 0.279 0.089 0.174 0.243 s12 0.262 0.340 0.162 0.213 0.023 s27 0.219 0.285 0.070 0.178 0.248 s13 0.175 0.278 0.132 0.174 0.242 s28 0.224 0.291 0.051 0.182 0.253 s14 0.144 0.288 0.137 0.180 0.251 s29 0.228 0.297 0.032 0.185 0.258 s15 0.113 0.299 0.142 0.187 0.260 s30 0.233 0.302 0.013 0.189 0.263 fig. 6 results of the sensitivity analysis for new criterion values fig. 6 clearly shows that the initially obtained results do not change with changes in criteria weights which is a clear indicator of invariability of the obtained results. forklift a1 remains the most efficient alternative, followed by forklifts a2, a3 and a4. 550 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov 6.2. comparative analysis in this section, a comparative analysis is performed with seven other mcdm methods, namely aras additive ratio assessment [25], mabac multi-attributive border approximation area comparison [26, 27], saw simple additive weighting method [28], waspas weighted aggregated sum product assessment [29], edas evaluation based on distance from average solution [30], cocoso combined compromise solution [31] and topsis technique for order of preference by similarity to ideal solution [32]. fig. 7 results of comparative analysis with seven other mcdm methods based on the results of comparative analysis with seven other mcdm methods, we can conclude that no changes are observed in terms of forklift ranking. fig. 7 clearly shows that in all the considered mcdm methods, forklift a1 retains its first position, i.e. it is the most efficient forklift; with respect to efficiency, it is followed by forklifts a2, a3 and a4. based on the performed comparative analysis and the applied integrated dea-fucom-marcos model, it can be concluded that the proposed methodology is quite reliable, and any changes in parameters do not affect stability of alternative rankings. 6.3. changing the number of inputs and the creation of the pca-dea model in this part of the sensitivity analysis, the number of inputs was changed, forming four scenarios (s) in which one input was eliminated, starting from the first. subsequently, the principal component analysis pca-dea [33] was applied with 85% of the information retained. the results of this part of the sensitivity analysis are shown in fig. 8. the results shown in fig. 8 show that if we eliminate the first input (regular servicing costs) or third input (exceptional servicing costs), the results do not change, i. e. v1-v4 forklifts are efficient, while the others are not. in the second scenario, when we eliminate the second input (fuel costs), the efficiency of forklifts v2-v4 does not change (1.000). in contrast, the efficiency of the first forklift v1 changes drastically because it gets inefficient and the lowest value. this means that the efficiency of the first forklift is strictly related to the second input. in the fourth scenario, when the total number of all minor accidents and an integrated decision-making model for efficiency analysis of forklifts in warehousing systems 551 damage caused by the forklift is eliminated, the v1 and v2 forklifts are efficient, which is the final rank by applying the integrated dea-fucom-marcos model. finally, by applying the pca-dea model, the results tend to the second scenario. fig. 8 results of the sensitivity analysis with pca-dea and a reduced number of inputs 7. conclusions this paper presents the way of dealing with determining the efficiency of handling and transportation equipment in each company. based on the efficiency analysis (dea), it has been determined which forklifts currently operating in this warehousing system are efficient and which of them do not contribute to work to the extent they should. based on the data (regular servicing costs, fuel costs, exceptional servicing costs, total number of all minor accidents and damage caused by the forklift, number of operating hours) for all eight forklifts operating in the warehousing system of the natron-hayat company, it has been determined that four out of eight forklifts are not efficient enough and do not contribute to work in this company (a5-a8) like other forklifts. after that, using multi-criteria decisionmaking methods, the ranking and selection of the most efficient forklift out of the remaining four ones are carried out. after determining the weight coefficients of the criteria using the fucom method, the marcos method is presented, step by step, and its final results. based on the results of the applied marcos method, it has been determined that forklift a1 is currently the most efficient forklift serving in this system. also, a sensitivity analysis has been performed in order to determine the stability of the final results. after changing the weight values of the criteria, it has been determined that these changes do not affect the final result. then a comparative analysis is carried out, i.e. testing the stability of the results with seven other mcdm methods. after applying all these methods, we have come to the conclusion that there are no changes in terms of determining the efficiency of forklifts, forklift a1 is still the most efficient forklift, followed by forklifts a2, a3 and a4. the contribution of this paper is evident in forming an original integrated model for determining efficiency, which can be applied to other fields as well. specifically in this 552 e. mahmutagić, ž. stević, z. nunić, p. chatterjee, i. tanackov paper, the significance of the applied model is that the warehousing system managers are provided with a quantified analysis based on which they can make further decisions in order to increase the overall efficiency of warehousing systems of the company that is the object of the research. by applying this model, it is possible to easily determine the efficiency of both forklifts and other equipment, and pay more attention to identifying and monitoring input and output parameters. regarding the obtained results, it is necessary that the managers in warehousing systems perform adequate monitoring of all activities done by forklifts, and to rationalize all unnecessary movements and pointless operation of forklifts. the issues with previous works in determining the efficiency are that only 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presented. the framework utilizes distributed anomaly detection, with anomaly detectors based on the recently introduced growing structure multiple model systems (gsmms) models. physics based models of the drive system and its pertinent subsystems were derived and were calibrated using data collected over several actual trips on a high speed train. simulation results demonstrate the ability to localize faults within various parts of the drive train system without the need for models of the underlying faults. in addition, traditional model based diagnosis was utilized for positive identification of faults, with signals emitted by the systems in the presence of those faults being available for modeling and subsequent recognition of faulty behavior. key words: immunity inspired diagnostics, high speed trains, growing structure multiple model systems 1. introduction a growing concern with the environmental impact of air traffic has contributed to the success and growth of high speed rail as a more sustainable transport medium. consequently, in recent years the european and japanese markets have seen a significant transition of traffic from airplanes to high speed rail, especially for journeys up to a few hundred miles long [1]. studies have also been carried out that underline the benefits of high speed rail as a transport system [2]. the growing popularity of high speed rail has inevitably led to investment in the development of the resources required to ensure reliability of the train systems [3], which is critical to the ability of high speed rail to compete with alternative modes of transport. as such, there is a need to develop systems for condition monitoring that would enable received january 26, 2015 / accepted march 30, 2015  corresponding author: dragan đurđanović the university of texas at austin, austin, tx 78712, usa e-mail: dragand@me.utexas.edu original scientific paper 68 a. ul haq, d. đurđanović detection of faults and localization of their root causes within the system 1 . the increasing complexity of trains running at higher speeds has led to greater challenges in the tasks of detecting faults that cascade through the system, and finding their sources. the first and foremost factor driving the need for such reliable and efficient monitoring systems derives from the safety requirements for high speed trains. in addition to this, there is a two-fold financial significance. first, ensuring reliability is critical as it prevents delays, which becomes a factor in retaining passengers. second, accurate fault localization contributes to the reduction of wastage of resources on ineffective maintenance. unfortunately, monitoring systems based on the classical framework are restricted in their diagnostic abilities, due to their reliance on fault models for these tasks. namely, the classical diagnostic paradigm requires models of the relevant faults in order to detect their occurrence. furthermore, these models need to be adequate throughout the operating space which the system experiences. therefore, a monitoring system under the classical framework is unable to deal with faults that have not been foreseen or for systems in operating regimes for which diagnostic models were not trained. this is of particular significance for highly complex system operating under highly variable operating regimes, such as the drive train of a high speed train. for such systems, it becomes unfeasible to build models of all possible faults, under all operating conditions. this strongly implies the need for a precedent-free fault detection and isolation approach. in this paper, the method for precedent-free fault detection and localization introduced in [4], and further developed in [5], is employed to facilitate monitoring of high speed train drive systems. the methodology presented in [4] spans the tasks of fault detection, localization and identification in complex systems of interacting dynamic subsystems. anomalous behavior of the system is detected as a statistically significant departure of its behavior from the normal one. the detection of a fault triggers the distribution of anomaly detectors (ads) across the subsystems of the faulty system, spreading into increasingly granular levels of subsystems that exhibit deviation from their own models of normal behavior. this process of ad proliferation continues as each ad that detects a fault is replaced by multiple ads monitoring the constituent subsystems of the faulty system. thus, the source(s) of the fault(s) is (are) localized, as part(s) of the system surrounded by alarming ads, in a hierarchical manner once the highest possible level of granularity of subsystems is reached. this distributed anomaly detection based on an a priori known structure of the monitored system 2 has been shown to enable precedent-free fault root cause localization [6]. the entire process is based solely on models of normal behavior, thereby bypassing the need for fault models, which, as previously mentioned, is a major constraining factor in the applicability of traditional diagnostic methods. after faulty subsystem(s) is (are) located, the natural next step is fault diagnosis, which involves identification of corresponding fault models so that such behavior may be recognized in the future, and possibly remedied via fault-tolerant controller adaptation or maintenance intervention. this obviously amounts to the traditional diagnostic paradigm of recognizing known faults based on their models, or building new fault models when the currently observed behavior of the monitored system does not match any existing fault model. such fault models can be built based on knowledge about system physics, as well as historical experience and observations of system operation. 1 reliably and efficiently determining which part of the system is at fault 2 knowing what subsystems constitute it and what their respective inputs and outputs are precedent-free fault localization and diagnosis for high speed train drive systems 69 the novel diagnostic framework briefly described above has previously been successfully implemented for fault detection, localization and diagnosis in the electronic throttle and crankshaft systems of an automotive internal combustion (ic) engine [4] [7], exhaust gas recirculation system of an automotive diesel engine [5] and most recently, distributed thermo-fluidic systems [8]. in this paper, this approach is employed for monitoring the drive system of a high speed train. a drive system in a high speed train is a complex system which incorporates linear and non-linear subsystems with continuous as well as discrete inputs and outputs. these factors contribute to a tremendously increased complexity for the monitoring task at hand 3 . the remainder of this paper is organized as follows. section 2 describes the growing structure multiple model systems (gsmms) modeling approach, which is used as the foundation of the ads in this work. it also describes the gsmms-based anomaly detection and isolation procedures. section 3 describes the physics based and data driven modeling of the system and section 4 goes on to describe the implementation of the framework to the system in question and the results thereof. finally, the conclusions and suggested future work are presented in section 5. 2. diagnostic framework the diagnostic framework described in the previous section does not require fault models for localization of the sources of abnormal behavior of the monitored system. instead, it only requires models of normal behavior for all the relevant subsystems, which form the basis of the ads distributed across the system. the recently introduced gsmms approach for modeling nonlinear dynamic systems [4] is exploited to create the aforementioned models and this section will briefly look into the motivation for the use of this modeling paradigm, as well as summarize methodological traits of the gsmms model. further, this section will also discuss how the distributed anomaly detection framework can be used to facilitate precedent-free localization of culprit subsystems causing anomalous behavior of the monitored system. 2.1 modeling of dynamic behavior traditional anomaly detection methods, based on global models of system behavior, focus on characterizing probability distributions of behavioral features and detecting anomalies as changes in those distributions. for systems that do not involve interactions between various constituent subsystems, such anomaly detection approaches are appropriate. however, interactions with other subsystems mean that shifts in the dynamic behavior of a constituent subsystem may not occur solely due to changes in the system dynamics (i.e. real faults), but also due to changes in the operating regime (which should not be seen as anomalies). namely, changes in the upstream subsystems, whose behavior affects the monitored system, cause shifts in the operating regime of the monitored system, potentially leading to changes in the behavior of the modeling residuals of the relevant anomaly detector and, consequently, false alarms. 3 requiring the use of more distributed ads and a higher level ad hierarchy than the cases reported in the literature on precedent-free diagnostics so far 70 a. ul haq, d. đurđanović such a situation necessitates the use of modeling and anomaly detection approaches that have the potential to separate abnormalities caused by unusual operating conditions (which are not truly anomalies) and true anomalies due to changes in the internal dynamics of the monitored system. to that end, one can utilize "divide and conquer" approaches, pursued in e.g. [4, 5, 7, 9, 10], where the operating space of the monitored system is indexed using features from other systems affecting it. divide and conquer models decompose the operating space into regimes of similar dynamic behavior, permitting the diagnostic framework to deal with regime-switching induced behavioral shifts. by postulating relatively tractable models in each operating regime, a set of region-specific anomaly detectors can be utilized. the behavior can then be considered independently in each operating regime and the presence of a fault can be detected as unusual behavior of modeling residuals within any of those operating regimes (i.e. corresponding to any of the local models within the divide and conquer modeling framework). within the gsmms framework, the regionalization of operating regimes of a system is conducted via unsupervised clustering of its inputs 4 and initial conditions using a kohonen self-organizing map (som) [11]. the use of such an unsupervised approach for partitioning the operating space overcomes the drawbacks associated with ad-hoc or variable-by-variable approaches [12-14]. in addition, growing mechanisms, such as those reported in [15-17], enable the determination of the number of local models required to approximate the underlying nonlinear dynamics, with a desired accuracy. the growing structure multiple model system can be seen as a collection of local models, with a local model capturing the dynamic behavior in each operating regime. the simple and tractable linear arx type models were used for the work presented in this paper, allowing easy parameter estimation and interpretation of local models. essentially such a gsmms formulation casts the problem of representing the system dynamics into the framework of interconnected, analytically tractable linear dynamic models. even more simply stated, it approximates a curved surface (non-linear) with a set of appropriately shaped and sized flat tiles (linear models), where the number, shape, size and location of the tiles is determined via a growing som. this structure enables the modeling of complex systems, such as the drive system of a high speed train, while maintaining analytical tractability and an operating regime decomposition that enables regionalized anomaly detection. the gsmms approach has been used successfully for modeling an electronic throttle system in a gasoline engine [9], automotive crankshaft dynamics [7], diesel engine exhaust gas recirculation (egr) system and its subsystems [5], electrical portion of an alternating current generator [10] and a distributed thermo-fluidic system [8]. further details, including the mathematical details and graphical representations, of this modeling approach can be found in [18]. 2.2 method for detection, isolation and diagnosis of an anomaly anomalous behavior can be seen as a statistically significant departure of the current dynamics of the target subsystem away from the normal one. once a gsmms model of normal behavior is built for each system to be monitored, anomaly detection can be 4 these inputs are often outputs of other systems affecting the behavior of the monitored system. precedent-free fault localization and diagnosis for high speed train drive systems 71 accomplished through comparison of the statistical characteristics of its residuals 5 displayed during normal behavior with characteristics of the most recent modeling residuals. since the operating space is decomposed into regions within which a linear model describes the system dynamics, each gsmms region can be equipped with its own decision making scheme that quantifies how close the current residual pattern is to the normal pattern. following [9], the performance within each operating region will be described in this paper using the concept of regional confidence values (cvs), defined as the area of overlap of the probability density function (pdf) of the modeling residuals displayed during normal behavior and the pdf of the residuals corresponding to the current behavior, in that region. based on their universal approximation ability, gaussian mixture models (gmms) were used to approximate the pdfs [19], which allows efficient recursive updating of the pdfs during operation to obtain the most recent distributions [20], as well as analytical and thus, fast calculation of the distribution overlaps (cvs). with the above definition, one can see that the cv will be close to 1 when there has been no significant change in the local dynamics of the monitored system, while any notable shift in the local system dynamics will result in lower cvs, with 0 being the lower bound. following [9] the global cv for the monitored system is then quantified as the geometric mean of the local cvs. this choice of global cv prevents the masking of a fault that is apparent only in certain operating regimes. namely, a low cv in any given operating regime will force a low global cv for the system, even if the performance is not affected in other operating regimes. isolation of the anomaly source can be conducted by proliferating anomaly detectors (ads) to monitor subsystems of the anomalous system, all of which utilize only models of normal behavior of the system they monitor. effectively, once an anomaly is detected, the proliferation of the ads monitoring the pertinent subsystems of that target system is initiated, enabling monitoring and anomaly detection in subsystems of ever finer granularity. such distributed anomaly detection leads to progressively finer localization of the fault through the hierarchy of the overall monitored system, until the finest feasible granularity is reached 6 . once the fault is localized to a subsystem, the next step is to recognize the underlying fault (if the model of that fault exists) or recognize that the underlying fault is unknown. a diagnoser for a specific fault can be constructed following essentially the same approach pursued for the purpose of anomaly detection. signatures emitted in the presence of the fault that the diagnoser needs to recognize can be utilized to estimate the pdfs of the modeling residuals of that diagnoser in the presence of that fault 7 (residuals of the gsmms corresponding that fault). proximity of the most recent system behavior to that fault can then be evaluated via the overlap between the pdf characterizing the most recent residuals of the fault model and that corresponding to the residuals of the fault model observed in the presence of the fault it is supposed to recognize. whenever this 5 the modeling residuals are differences between the system output and the output of the gsmms describing the normal system behavior [28] 6 the level of granularity is effectively determined by the availability of signals from the monitored system and its subsystems. generally, the ideal situation is to have access to all relevant inputs and outputs from all field replaceable units (frus) in the system, which would enable localization of all anomalies to the level of components that can be directly replaced during maintenance. 7 these pdfs serve as the equivalent of the pdfs representing normal behavior in the anomaly detection task. 72 a. ul haq, d. đurđanović cv-like value for a specific fault model is close to 1, it can be concluded that the corresponding fault is present and a value of this cv-like index close to 0 would imply the absence of that fault. if for none of the existing diagnosers this cv-like overlap happens to be close to 1, the presence of an unknown fault can be inferred and a new fault model must be developed to enable recognition of this fault in the future. 3. modeling the high speed train drive system in order to implement the distributed anomaly detection to the drive system of a high speed train, gsmms models for the system and its pertinent subsystems must be developed. to this end, a physics based model was first built based on a combination of expert advice and available literature. the model was built in simulink® and simulated using velocity profiles collected from actual tgv train journeys between paris and metz, in france. the simulations generated data for the inputs and outputs of each of the subsystems of interest, which was then used to develop the relevant gsmms based ads. the overall system receives a reference velocity as the input, while the actual velocity generated by the drive system is the output. it is composed of a controller, electrical supply, drive motor and mechanical transmission, as illustrated in figs. 1 and 2. these figures show the major components of the simulink® model utilized. in addition, it was assumed that sensors were available to collect the input and output data pertaining to each of these subsystems, as well as their component subsystems. fig. 1 high speed drive train system fig. 2 components of controller and mechanical transmission systems the drive motor was taken to be a permanent magnet synchronous motor (pmsm), as per [21]. pmsm modeling has been tackled in the literature in various ways, commonly using a transition of the electrical component from the physical 3-phase structure to an equivalent 2-phase right-angled structure, enabled by the clark transformation [22]. in this paper, we used model of pmsm dynamics developed in [23]. following [24], the controller was taken to be a simple proportional-integral (pi) controller, with a pulse width modulating inverter. finally, the mechanical transmission subsystem consists of two gears and a wheel and axle combination [21], each of which was modeled simply as a precedent-free fault localization and diagnosis for high speed train drive systems 73 proportional gain. the wheel size required for the gain of the wheel and axle was set as per the information available in [25]. in order to make the data generated as representative of the real world conditions as possible, the reference velocity profiles were collected during actual high speed train journeys in europe. four such profiles were collected, one of which was used for training and the other three for testing of the proposed diagnostic approach. these profiles were gathered using a mobile phone based android application called ’my tracks’ [26], which tracks position, velocity and height using gps signals. the measurement of interest here is the velocity profile, an example of which is provided in fig. 3 as a screenshot from the mobile phone. fig. 3 example reference velocity trajectory once the physics based model was built, data collected from the simulations were used to build the required gsmms based ads for all the relevant subsystems of the drive train. the orders of the local arx models within the gsmms, were set ad hoc, although techniques for the automated selection of these parameters can be found in [27]. 4. simulation of distributed anomaly detection with the gsmms based ads available to monitor each subsystem, the distributed anomaly detection approach was put to the test. the hierarchy of the ad distribution is shown in figs. 4 and 5, displaying the ad associated with each monitored system and subsystem. the fault localization process commences at ad1 which monitors the overall drive train system. once a fault is detected by ad1, ad2  ad4 are activated and they begin to monitor the controller, pmsm and mechanical transmission systems respectively. the fault is then localized to one of these subsystems and, depending on which system is faulty, either ad5 and ad6 or ad7, ad8 and ad9 are activated. ad5 monitors the pi controller within the controller; hence it and ad6 would be activated if the fault had been 74 a. ul haq, d. đurđanović signaled by ad2. if the fault had been signaled by ad4, ad7  ad9 would be activated respectively monitoring gear 1, gear 2 and the wheel and axle combination. the faults considered in this paper were limited to the controller and mechanical transmission systems, and were introduced approximately 8 minutes into the journey. fig. 2 levels 1 and 2 of the anomaly detector distribution hierarchy fig. 3 level 3 of the anomaly detector distribution hierarchy 4.1 localization of a fault in the controller the fault in the controller was introduced in the form of a delay in its output, with delays of 0.7 seconds and 1.4 seconds being inserted in 2 different simulations. the results are presented in the form of the cvs associated with each ad and shown in figs. 6, 7 and 8. the fault is detected by ad1 as is highlighted in fig. 6 by the drop in the associated cv. from fig. 7 one can see that, of the 3 ads monitoring the first level of subsystems, only ad2 exhibits a drop in cv. hence, the fault can at this stage be localized to the controller subsystem. finally, it is observed in fig. 8 that the cv associated with ad5 drops significantly, while that associated with ad6 remains high. these results show the fault being tracked through the levels as being local first to the overall system, then the controller subsystem and finally the pi controller. the approach has hence been able to localize the fault without having any signatures or models associated with the fault in question. in addition, it is noted that no fault was signaled at any of the subsystems that were not faulty, including those interacting with the faulty subsystem. with the anomaly detectors having been set up beforehand, the distributed anomaly detection framework is able to detect the fault online. precedent-free fault localization and diagnosis for high speed train drive systems 75 fig. 6 controller fault detection response of the ad monitoring the overall system fig. 7 controller fault localization at first level of subsystems fig. 8 fault localization within the controller system 4.2 localization of a fault in the mechanical transmission a fault in the mechanical transmission was modeled in the form of added noise to the output from gear 2 to simulate a chattering type fault. once again, the fault was introduced about 8 minutes into the journey and 2 simulations were conducted with added noise at 76 a. ul haq, d. đurđanović 6% and 10% of the signal, respectively. the resulting cvs for the relevant ads are shown in figs. 9  11. fig. 9 gear fault detection response of the ad monitoring the overall system fig. 10 gear fault localization at the first level of subsystems fig. 11 gear fault localization within the mechanical transmission system the cvs shown in fig. 9 provide clear indication of the presence of a fault in the overall system. per the proliferation of anomaly detectors described in 2.2, this activates precedent-free fault localization and diagnosis for high speed train drive systems 77 ad2 – ad4 whose cvs are shown in fig. 10 localize the fault to the mechanical transmission, implicated by the falling cvs in ad4. the continued proliferation leads to the activation of ad7 – ad9, in fig. 11 the cvs of these ads isolate gear 2 as the source of the faulty behavior. whenever the next maintenance opportunity arrives the maintenance team will know exactly which component requires their attention, thereby saving time on inspection and offline fault localization. 4.3 fault diagnosis utilizing the methodology described in section 2.2., diagnosers were trained for the 2 faults introduced into the gear (6% and 10% added noise to the output). the diagnosers were connected as shown in fig 12, where diagnoser 1 (d1) refers to the diagnoser trained using signals received in the presence of 6% added noise, and diagnoser 2 (d2) was trained using the signals gathered in the presence of 10% added noise. the diagnosers were tested by introducing each of the faults that they were trained to recognize at the start of a journey and allowing them to persist for the full duration of that journey. during the first trip, the train was in its in normal operating mode, thereafter the fault corresponding to d1 was introduced for the next journey and finally the last journey was completed in presence of the fault corresponding to d2. the results of the diagnosis are presented in fig. 13, where we can see that, during each stage the cv associated with the appropriate diagnoser (or normal operation monitor) is the highest. fig. 12 configuration of the diagnosers within the mechanical transmission fig. 13 diagnosis results for chatter type fault in gear 78 a. ul haq, d. đurđanović however, it is also observed that the crossover of cvs is not instantaneous, rather it takes some time for the diagnosers to raise or lower their cvs in response to the change. this is an inevitable result of the recursive updating of the pdfs used to calculate the cvs. hence, in order for a fault to be diagnosed it must persist for some time. another important observation is that the fault recognition did not perform well if the subsystem remained in steady state operation. however, given that the train is not expected to continue in steady state operation until the maintenance opportunity, this does not present a major difficulty. hence, such diagnosers can be used to recognize a fault that has previously been experienced, or for which a fault model is available a priori. 3. conclusions and future work a recently introduced distributed anomaly detection framework is utilized for precedent-free fault localization in the drive system of a high speed train. the framework uses growing structure multiple model system (gsmms) models of the monitored system to describe its dynamics and a statistical measure of departure away from normal behavior for fault detection. gsmms-based anomaly monitors distributed across the system were then used to localize the sources of anomalous behavior without the need for signatures or models of the underlying faults. the plant was simulated using a physics based model, which was tuned using data collected from several actual tgv journeys. simulations of that model were used to generate the data needed to build gsmms based anomaly detectors for the drive train system and its subsystems. the results of the fault detection and localization accomplished using these ads show that distributed anomaly detection successfully localizes the faulty subsystems, without any prior information regarding the underlying fault. further, data 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[accessed june 2013] 27. jiang l., latronico e., ni j., 2008, a novel method for input selection for the modeling of nonlinear dynamic systems, asme dynamic systems and control conference, ann arbor, mi. 28. isermann r., 2006, fault-diagnosis systems, springer science & business media. 11420 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 689 710 https://doi.org/10.22190/fume221217008f © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper investigation of heat-affected zones of thermite rail weldings szabolcs fischer1, dóra harangozó2, dalma németh1, bence kocsis2, mykola sysyn3, dmytro kurhan4, andrás brautigam5 1széchenyi istván university, department of transport infrastructure and water resources engineering, hungary 2széchenyi istván university, department of materials science and technology, hungary 3technical university dresden, department of planning and design of railway infrastructure, germany 4ukrainian state university of science and technologies, department of transport infrastructure, ukraine 5budapest transport privately held corporation (bkv), hungary orcid ids: szabolcs fischer https://orcid.org/0000-0001-7298-9960 dóra harangozó https://orcid.org/0000-0001-9624-0487 dalma németh https://orcid.org/0009-0009-9666-121x bence kocsis https://orcid.org/0000-0003-4751-5071 mykola sysyn https://orcid.org/0000-0001-6893-0018 dmytro kurhan https://orcid.org/0000-0002-9448-5269 andrás brautigam https://orcid.org/0009-0000-3840-595x abstract. the paper investigates the heat-affected zone (haz) of several rail joints executed by thermite rail welding (tw). the examined rail profile was 54e1 (uic54). the rail steel categories were different: r260 and r400ht. the welding portions of the tws fitted r350ht and r260 rail categories with normal welding gaps. the rail pieces were brand new, i.e., without any usage in the railway track. the authors executed vickers-hardness tests (hv10) and material texture tests on the running surface of the rail head, as well as on slices cut from the rail head. the cutting was performed by the water jet method, five longitudinal direction slices with vertical cutting lines. the considered specimen lengths were 2×70 mm (i.e., 70 mm from the mid-point of the rail joint), however, the depths were 20 mm from the running surface. therefore, the measuring spaces were 5 mm lengthwise and 2 mm in depth. the variation of the hardness values was determined considering the microstructures of the base steel material and the tw. for comparison, previously measured elektrothermit sow5 and earlier own research were taken into consideration. key words: thermite rail welding, heat-affected zone, haz, vickers hardness, brinell hardness, microstructure received: december 17, 2022 / accepted february 09, 2023 corresponding author: dóra harangozó széchenyi istván university, audi hungaria faculty of automotive engineering, department of materials science and technology, h-9026 győr, egyetem tér 1., hungary e-mail: harangozo.dora@sze.hu 690 s. fischer, d. harangozó, d. németh, et al 1. introduction long-distance rail transport became one of the most important means of land transport in the 20th and 21st centuries. it is convenient, safe, fast, and generally has high punctuality, which of course, can vary considerably depending on the country and the railway company. long-distance rail travel is generally worthwhile for distances above 500-1000 km, taking into account travel time, and is less competitive than air travel above this distance [1]. of course, when travel costs are considered (in particular ticket and/or season ticket prices), it is at a significant disadvantage compared to so-called low-cost airlines. on the other hand, it is also worth considering that railway stations and passenger stations in large cities may be located in the center of the city, in the central core (even in the historic city center), thanks to the railway construction of the 18th and 19th centuries. in contrast, airports are usually located far from the cities. it can mean distances of up to 10-50 km. these relatively long distances can only be covered by additional public or private transport, which is a disadvantage for tourists and tourism. in the years 2021-2023, the significant increase in electricity prices and fuel prices (e.g., diesel, kerosene, etc.) will create serious problems and difficulties for both countries and public and private (large) public transport companies (of course, the same can be said for private transport, where petrol, diesel, electricity, natural gas, etc. are the most relevant). the reasons, of course, are many: part of the explanations and justifications is seen in the current problematic political situation worldwide, another part in the prolonged covid pandemic, and in the cyclical world economic stock market boom and bust, etc. however, the most plausible explanation may be a combination of the preceding, to a greater or lesser extent, i.e., no one factor can be neglected entirely. because of the above, it should be noted that transportation itself [2,3], among other things, the rail transport and railway lines are both critical and mandatory areas of major importance for the national economy. in this article, the authors focus on one of the most critical elements of railway tracks: the rail track. in the following paragraphs, the authors have prepared a relevant literature review on railway tracks to present and summarize the essential literature findings on the subject. railway tracks consist of superstructure and substructure, but of course, other related structural elements can also be mentioned (e.g., catenary support columns, safety equipment elements, and components, etc.). if the discussion is restricted to railway tracks, the superstructure consists of the track and the ballast, and special elements of the track, e.g., siding, in the case of ballastless tracks. the substructure includes the additional layer, usually granular material, and the earthwork itself, usually soil material. the tracks' parts are the rails, rail fasteners, and sleepers. the vehicle load must be considered as dynamic, however, in some approximated calculations, only the static values can be considered [4]. it is proved that the higher the geometrical faults in the railway track, the more critical and higher the dynamic evolved effect due to them [4-10]. of course, the vibration and the noise will increase, however, they are not only related to the geometrical faults [11] but the structure of the railway track [8-10], the rail dampers can be noted as one of the most relevant parts [12-13]. of course, safety, as well as risk and economic analysis of railway projects, is an integral part of railway lines' design, construction, and operation [14]. this study focuses on rail connections (or, in other words, rail joints). the rails can be connected with the help of fishplates and bolts, as well as rail welding. investigation of heat-affected zones of thermite rail weldings 691 modern rail production technologies and delivery options limit the length of rails up to a maximum of 120 meters (e.g., see the technology of voest alpine [15]). therefore, rail joints are essential parts of railway systems to secure the continuity of the rails without vertical or horizontal drops or changing directions. rail joints must bear dynamic loads and secure movements caused by dilatation without structural damage. in order to be feasible easily and fast, the joints should contain the less and most simple parts, which are also easy to maintain. generally, rail joints are probably the weakest parts of rail tracks. their malfunctions lead to dangerous situations and a high risk of accidents. global and corporational (i.e., industrial) railway safety requirements control the applicability of the different types [16]. the original method of joining rails is using bolted rail joints (or, in other words: fishplated rail joints). it is a simple, fast, and cost-effective technology with the disadvantage of resulting in a high rate of failures since the bolts cause dynamic shocks in rail wheels, reducing their lifetime and producing noises [17]. to eliminate the high demand for the maintenance of bolted joints, engineers introduced welding techniques as new methods of rail joints in the early 1900s [18]. against bolted joints, the dynamic behavior of the vehicle and the track is more beneficial and less expensive to maintain. flash-butt welding (fbw) and aluminothermic (thermite, tw) welding are the most commonly used welding technologies. fbw is based on electric resistance. the rail ends are heated by electricity (low voltage, approx. 1525 v, and extremely high amperage, approx. 100,000-150,000 a) and hydraulically forged in a stationary plant. then the sections of approx. 400 m can be transported to their final location [19]. however, another solution must be mentioned: the so-called welding machines on trucks or locomotives, which can weld the rails by fbw at the construction site. therefore, the transportation of some hundred-meter-long rails can be avoided [16]. the tw technology is the most common method of joining and repairing rails on-site [20]. the technology was invented by a german professor of chemistry, hans goldschmidt, in 1895. he studied the reaction of metal oxides and aluminum for decades. the rail ends are melted by the heat (approx. 2200-2400 °c) emerging from the reaction of aluminum and iron-oxide [21,22]. generally, the thermite welding material consists of 1820% feo and 79-85% fe2o3 with an average grain size of 0.1-5 mm. additional elements such as carbon, manganese, chromium, nickel, vanadium, and tic may also be added to the mixture to produce similar characteristics to the rail to be welded. it is a fusion welding process since the melted iron flows into the gaps in the mold where the rail ends are welded together. during the exotherm reaction, metal oxides and aluminum become iron and aluminum oxide. by strictly controlling the parameters of the process (such as equipment, composition of the thermite material, preparation, and pre-heating of the rails), a weld free of macroscopic defects can be achieved with mechanical properties comparable to the "parent rail". this way, tw is an effective methodology of on-site joining rails without using electric power [16,23-24]. however, the microstructure of the rails and, thus, the mechanical properties are inevitably affected by the enormous heat arising during the welding process [23,25,26]. it is the so-called heat-affected zone (haz). close to the centerline of the welding, the material reaches the recrystallization temperature. the austenitization of steel depends on the temperature and time of heating [27]. therefore, zones close to the centerline are completely recrystallized, while only partial austenitization could happen at a certain distance. porcaro et al. [25] studied the microstructure of fbw. they discussed haz, which consists of three zones: 692 s. fischer, d. harangozó, d. németh, et al i. close to the central line of the welding, a coarser grain size ferritic-pearlitic microstructure can be observed, which is a sign of decarburization. ii. grain growth region: the temperature was high enough to allow grain growth of austenite crystals. iii. grain refined region: completely recrystallized zone, but the temperature was not so high to coarsen the grain size. regarding the microstructural map of the welding, these zones are followed by the partially austenitized zone and the base metal. similarly to fbw, in the case of tw, the pearlitic main structure of the parent rail with a grain size of 5-6 may transform into a ferritic-pearlitic microstructure [28]. retained austenite may also appear. zones with coarser or finer grain sizes than the original may appear at different distances from the center line of the weld. the cementite of pearlite becomes spherical in the haz, reducing the rail's hardness and strength [29]. a zone with a higher hardness also exists in the haz, which can give rise to fatigue fracture [17,21,22]. in accordance with the microstructural changes, the hardness of the rail also alters. in general, a smaller grain size indicates higher hardness and thus reduces the joint's stiffness, which can lead to fatigue failures. corser grain structure causes lower hardness and wear resistance, resulting in increased impact loading and premature welding failure [30]. detailed studies of the hardness profile of fbw are published in [25,26]. 2 mm below the running surface, the hardness of the centerline of the welding is approx. 300 hv, which is decreased by 50 hv at a distance of 20 mm. the rolling technology of rails indicates a higher hardness on the rail web than on the head and the foot [26]. compared to fbw, the coarse casted structure of tw joints means lower hardness making them less resistant to wear, but it can be improved by heat treatment [16]. the most spectacular advantage of heat treatment of tw joints is the improvement of tensile properties and the change of the fracture mechanism from brittle to ductile. examination of the vehicle-track contact dynamics and failure modes of rails is a topic of contemporary research [31,32]. financial operations with efficient energy use are also popular research topics [33-35] regarding the problems of the 21st century. the most modern digital image correlation (dic) measurement systems' applicability in mechanical engineering, materials science, and electronics has already been proven [36-40]. recent research applies dic to investigate a civil engineering problem [41]. as mentioned earlier, the rail joints are the weak points of rail tracks as they can be potential initiation points of failure. besides traditional microstructure studies, current research focuses on the fracture mechanism of welded joints. [18-20,30,40-43] published detailed numerical studies and performed finite element analysis on the fatigue behavior of tw joints. these methods can be well adapted to predict the fatigue crack initiation, thus, the life cycle of rail welding. nowadays, newly developed types of rails are installed in rail tracks, claiming the need to join different rails without transitional parts. welding of two rail ends with very different hardness may also be required. test welds with transition rails have been prepared and analyzed in a material testing laboratory to examine the possibility of eliminating too low or too high hardness layers. investigation of heat-affected zones of thermite rail weldings 693 2. materials and methods during the research, laboratory hardness tests were executed on aluminothermic rail weldings prepared under non-laboratory conditions but with precise and accurate adherence to the technology. for the tests, rails of different base hardnesses (r260 and r400ht, [46]) have been welded together. the rail profiles for both rail sections and weldings were 54e1 [46]. two weldings have been prepared: i. sample #1: r260-r400ht with a normal non-heat treated welding portion for r260 rails and ii. sample #2: r260-r400ht with a heat-treated welding portion for r350ht rails (see (fig 2.1.1). both weldings have normal welding gaps, i.e., 27-30 mm. in both cases, the rails were unloaded, i.e., brand new. 2.1. preparatory work in the preparation phase of non-track welding, the rail ends should be cleaned of dirt, and the welding gap set according to the technology (in terms of height and width, i.e., vertically and horizontally, respectively) (see fig. 1). fig. 1 setting the rail ends as a function of the welding gap 2.2. preparation of the weldings the execution of the rail welding procedure starts with preparing and unpacking the socalled welding set. then the sand tray, the universal clamping device, and the molding elements (with mold clamping plates) are placed on the previously cleaned and adjusted rail ends. in all cases, the gaps must be sealed with sealing sand. after the slag tray has been installed, the rail head protection plate is also installed. the prepared pots are then placed on top, and the thermite can be loaded into them. in the case of low external temperatures, the rail ends must be pre-heated: usually, over 1-1 m from the rail ends. in 694 s. fischer, d. harangozó, d. németh, et al this study, this pre-heating was only applied for the rail ends, and the temperature was approx. 900 °c. depending on external conditions and rail qualities, it may be necessary to partially preheat the rails over a longer section [47]. the procedure is as follows: fitting the burner head, ignition, pre-heating itself and then fitting the sealing element. pre-heating is achieved with a propane-oxygen gas mixture (the pressure can vary between 1.50 and 4.0 bars. 1.50 bar is related to propane, and 4.0 bar is to oxygen). after inserting the igniter into the thermite, the pot must be covered to allow the chemical reaction. next, the thermite steel (the approx. temperature is 2000 °c) flows into a mold while the liquid slag flows into the tray. when the slag solidifies, the tray and the pot can be removed. finally, the mold clamping plates can be removed. 2.3. post-production of the weldings after the molds are unset, the excess material is removed with a press to half the height of the rail web, and after cooling, the residual material follows. the subsequent operation is rough grinding, followed by cleaning the welding. the final step is to carry out the finishing operation. 2.4. sampling the marking of the samples (based on section 2) can be seen in fig. 2. slices have been taken under laboratory conditions from different parts of the rail head by water jet cutting (machine type: flow) in the laboratory. the samples were first cut into five equal slices (see fig. 3) by waterjet. these slices were cut in half at the welding axis for further testing and then further cut into 7 cm long pieces (see fig. 4) by a metallographic cutting machine. the sampling has not affected the measured results, as appropriate cooling of the metal has been ensured during the process. fig. 2 marking of samples (in the front sample #2, in the background sample #1) investigation of heat-affected zones of thermite rail weldings 695 fig. 3 the samples' rail heads are divided into five slices (the thickness is approximately 22 mm). the identifications will be in the following: from slice #1 to slice #5. fig. 4 the final specimen of 2×70 mm previous research [48,49] has demonstrated that critical points in the heat-affected zone can be found within 70 mm of the welding's axis (for normal-gap thermite welding, i.e., a welding gap of about 27-30 mm), and taking into account that the grinding and polishing equipment used to prepare the samples limited the length that could be tested in a single pass. it was for this reason that samples of 2×70 mm were created. 2.5. metallographic preparation process to be able to perform the hardness measurement and the microstructural analysis, the slices from the water jet cutting must be metal-cleaned so that machining grooves do not affect the accuracy of the measurements. for this reason, the surface must be "mirrorfinished" for low-load tests. conventional steps of the metallographic preparation process: 1. wet grinding with silicon carbide sandpaper in multiple grades: p80, p180, p500, and finally, p800, 2. polishing of slices in multiple steps with diamond suspension: 9, 6, 3, and finally, 1 µm, 3. etching with nital (3% nitric acid, 97% ethyl alcohol) to reveal the microstructure. 7 cm 7 cm 696 s. fischer, d. harangozó, d. németh, et al the result of this preparation process is presented in fig. 5. the structure of the weld and the haz are separated well in the macroscopic image. fig. 5 the outlined microstructure (sample #1, slice #3) 2.6. hardness tests vickers hardness values have been determined by applying f=98.07 n force according to the specifications of msz en iso 6507-1:2018 [50]. these values have been converted to hb (i.e., brinell hardness). furthermore, hb hardness can be measured for pre-grinding and post-grinding measurements on the running surface (the instrument converts from leeb hardness values). however, it has to be noted that this study does not consider surface hardness measurements. a calibrated modern hardness testing machine (type kb30) with a programmable work stage has been used for the experiments. thus, the position of the hardness measurement points can be accurately selected. the hardness measurements have been performed from the welding axis side of slices #2, #3, #4, and #5, as shown in fig. 6. the measurement points are 5 mm apart horizontally and 2 mm apart vertically. on average, 15 points were obtained per slice in the "x" direction (horizontally) and 11 points in the "y" direction (vertically). fig. 6 allocation of hardness measurement points expected results are the hardness profiles in the thermal response of the rail weld as a function of the longitudinal distance from the welding center axis for different configurations investigation of heat-affected zones of thermite rail weldings 697 (consideration of different rail hardness, investigation of different welding technologies, joint and repair welds, etc.) 2.7. microstructural analysis microstructural images of the nital etched specimens have been taken by zeiss axio imager m1m optical microscope (fig. 7) using brightfield illumination and objectives with magnifications of 5× and 20×. fig. 7 zeiss axioimager m1m optical microscope 3. results and discussion the hardness values have been plotted and examined in the function of the distance below the running surface (depth) and the distance from the welding axis. due to a large amount of data, different aspects were selected for analysis: i. the hardness profile below the running surface (–0.5 mm) (comparing r260 data with sow-5 [49] and barna et al. [48], calculating surface hardness ratios) was investigated. ii. the hardness profile at limit depth (20.5 mm below the running surface, i.e., –20.5 mm) (comparing r260 data with sow-5 [49] and barna et al. [48], calculating surface hardness ratios) was investigated. the limit depth means that the hardness values do not change in the rail below this value. it was determined based on the measurements. iii. depth hardness ratios were given for the r260 sides of the rails compared to the measurements published in [48]. iv. in selected perpendicular cross sections, the hardness profile has been determined and plotted in the horizontal direction (i.e., laterally). v. 5× magnification macrostructure images have been used to visualize the parts of the haz. vi. 20× magnification microstructure images have been joined with the corresponding hardness values. it has to be mentioned that in the above point (i), the hardness measurements on the running surface have not been executed during the present study. therefore, these data are related to the –0.5 mm depths. it is an approximation, however, it does not have too many errors. 698 s. fischer, d. harangozó, d. németh, et al in the present paper, a detailed analysis of the haz of tw of railway rails was carried out using laboratory tests on samples of welding performed in non-laboratory conditions. the main objective of the tests was to see how the hardness of the rail steel in the heataffected zone varies as a function of depth (i.e., distance from the running surface of the rail head) and distance from the welding centerline. a further objective was to link microstructural images to the hardness values. figs. 8-11 demonstrate the hardness variation of slice #2s of sample #1 (see figs. 8 and 9), as well as sample #2 (see figs. 10 and 11) considering the –0.5 mm and –20.5 mm depth values under the rails' running surface (i.e., figs. 8 and 10 show the –0.5 mm, and figs. 9 and 11 demonstrate the –20.5 mm depths). for type sample #1, it can be concluded that the hardness increases from the welding axis to the heat-affected zone, reaches a maximum there, and then decreases drastically to a minimum at the outer edge. according to the trend described, the maximum value is approx. 40.5...45.5 mm from the weld axis, the minimum is approx. 10 mm further away. for sample #2, the maximum hardness in the heat-affected zone is no longer an extreme value, the values are in a smaller range. the hardness gradually decreases to the minimum at the outer edge of the haz. based on the described behavior, the hardness starts to decrease gradually after approx. 40.5...45.5 mm and reaches its minimum at a distance of approx. 50.5...55.5 mm from the welding axis. fig. 8 hardness variation at a depth of –0.5 mm below the running surface in slice #2 of sample #1 (considering the literature [48,49]) 200 220 240 260 280 300 320 340 360 380 400 -70 -50 -30 -10 10 30 50 70 h a rd n es s [h b ] horizontal distance from the welding's axis [mm] hardness [hb] welding's axis sow-5 – running surface barna et al. – running surface r400ht rail r260 rail investigation of heat-affected zones of thermite rail weldings 699 fig. 9 hardness variation at a depth of –0.5 mm below the running surface in slice #2 of sample #2 (considering the literature [48,49]) fig. 10 hardness variation at a limit depth (–20.5 mm below the running surface) in slice #2 of sample #1 (considering the literature [48,49]) at the limit depth (–20.5 mm) (see figs. 10 and 11), the hardness values start to vary between smaller intervals. an extremely high value is no longer present in the thermal zone but in an upper "plateau". this "plateau" starts at approx. 20.5 mm from the welding axis, 200 220 240 260 280 300 320 340 360 380 400 -70 -50 -30 -10 10 30 50 70 h a rd n es s [h b ] horizontal distance from the welding's axis [mm] hardness [hb] welding's axis sow-5 – running surface barna et al. – running surface r400ht rail r260 rail 200 220 240 260 280 300 320 340 360 380 400 -70 -50 -30 -10 10 30 50 70 h a rd n es s [h b ] horizontal distance from the welding's axis [mm] hardness [hb] welding's axis sow-5 – running surface barna et al. – running surface r400ht rail r260 rail 700 s. fischer, d. harangozó, d. németh, et al and at approx. 40.5 mm, there is a gradual decrease to a minimum hardness. the hardening up to the base material value can be clearly seen in each diagram. this phenomenon is typical for all samples. the above data were compared to the r260 rail's data with those of sow-5 [49] and barna et al. [48]. these graphs are depicted in figs. 12-15. the shown data are calculated based on the averages of all slices' results in the cases in question. analyzing the data related to 0.5 mm under the running surface, the highest surface area ratio is 50.5 mm from the welding's axis for sample #1 (see fig. 12), where the average of the measured data set reaches its maximum, and the sow-5 data set reaches its minimum. the field surface area measurement is between the two values, so the field measurement deviates only 7% from 100%. most of the data are lower than 100% hardness, so our averages are higher than the reference values but lower at 0.5, 40.5, and 60.5 mm from the welding axis. when examining the surface hardness ratio, the majority of the data are lower than 100% hardness, so the calculated averages are higher than the reference values at a depth of –0.5 mm below the running surface at 40.5 mm from the weld axis. it was lower at a depth of –20.5 mm and extended to 60.5 mm for all samples. in figs. 16 and 17, the depth hardness ratio values are given. they are only calculated for the r260 sides of sample #1 and sample #2. based on figs. 16 and 17, it can be concluded that the most critical zone of sample #1 (see fig. 16) is 40-60 mm from the welding's axis, in all depths. there are relatively high steps (jumps) in the hardness values at 50 mm from the welding's axis in a positive direction. it must be noted that the neighbor values are fluctuating at 100%. compared to them, the measured values in fig. 17 are very interesting. they are related to sample #2. between 0.5 and 20.5 mm from the welding's axis, the depth hardness ratio values are approximately 125-130%, there is a downstep to 95-105% (at 40.5 mm from the welding's axis), and from this point, an up step to 125-130% (it is at 50.5 mm from the welding's axis), and again a down step to 90-105%. the reason for this can be the different welding portions and the evolved variation. at defined distances from the welding axis (10.5 mm, 25.5 mm, 35.5 mm, 50.5 mm, 70.5 mm), the horizontal hardness profile was investigated in cross sections perpendicular to the rail axis. this paper presents only the graphs related to +/–10.5 mm (see figs. 18 and 19), however, the relevant results are discussed for all. up to a distance of approx. 35.5 mm from the welding axis, the hardness values of r260 and r400ht rails do not differ significantly, despite the hardness of the base material. the r400ht rail reaches higher hardness values from the edge of the haz. in the case of sample #1, the r400ht rail approached the values of r260. in sample #2, the r260 rail hardness approaches the hardness of the r400ht rail. while the hardness profiles in the specimens of sample #2 do not describe a characteristic behavior in either the welding zone or in the heat-affected zone, the hardness values of the rail r260 in sample #1 increase from the rail axis. the microstructure images at 5× magnification illustrate the haz sections (see figs. 20 and 21). the procedure is similar to the published method and solution in [29]. the 20× magnification microstructure images were paired with the corresponding hardness values. six zones of the microstructure have been identified for sample #1 (welding axis environment, welding edge, fused mixed zone, coarse-grained haz, fine-grained haz, and intercritical haz). sample #2 had five zones (welding axis environment, fused mixed zone, unvarying grain size haz, intercritical haz, and the base material). investigation of heat-affected zones of thermite rail weldings 701 fig. 11 hardness variation at a limit depth (–20.5 mm below the running surface) in slice #2 of sample #2 (considering the literature [48,49]) fig. 12 surface hardness ratio values of sample #1 considering the –0.5 mm below the running surface (considering the literature [48,49]) 200 220 240 260 280 300 320 340 360 380 400 -70 -50 -30 -10 10 30 50 70 h a rd n es s [h b ] horizontal distance from the welding's axis [mm] hardness [hb] welding's axis sow-5 – running surface barna et al. – running surface r400ht rail r260 rail 75% 80% 85% 90% 95% 100% 105% 110% 0 10 20 30 40 50 60 70 h a rd n es s ra ti o [ % ] distance from the welding's axis [mm] compared to sow-5 compared to barna et al.'s running surface measurement 100% 702 s. fischer, d. harangozó, d. németh, et al fig. 13 surface hardness ratio values of sample #1 considering the –20.5 mm below the running surface (considering the literature [48,49]) fig. 14 surface hardness ratio values of sample #2 considering the –0.5 mm below the running surface (considering the literature [48,49]) 75% 80% 85% 90% 95% 100% 105% 110% 0 10 20 30 40 50 60 70 h a rd n es s ra ti o [ % ] distance from the welding's axis [mm] compared to sow-5 compared to barna et al.'s running surface measurement 100% 70% 75% 80% 85% 90% 95% 100% 105% 110% 115% 0 10 20 30 40 50 60 70 h a rd n es s ra ti o [ % ] distance from the welding's axis [mm] compared to sow-5 compared to barna et al.'s running surface measurement 100% investigation of heat-affected zones of thermite rail weldings 703 fig. 15 surface hardness ratio values of sample #2 considering the –20.5 mm below the running surface (considering the literature [48,49]) fig. 16 depth hardness ratio values of sample #1 considering different depths. the comparisons were made according to the surface hardness values published in [48] 75% 80% 85% 90% 95% 100% 105% 110% 115% 0 10 20 30 40 50 60 70 h a rd n es s ra ti o [ % ] distance from the welding's axis [mm] compared to sow-5 compared to barna et al.'s running surface measurement 100% 75% 85% 95% 105% 115% 125% 135% 145% 0 10 20 30 40 50 60 70 h a rd n es s ra ti o [ % ] distance from the welding's axis [mm] –0.5 mm –2.5 mm –6.5 mm –10.5 mm –14.5 mm –20.5 mm 100% 704 s. fischer, d. harangozó, d. németh, et al fig. 17 depth hardness ratio values of sample #2 considering different depths. the comparisons were made according to the surface hardness values published in [48] fig. 18 variation of hardness in +/–10.5 mm depth from the axis of welding for sample #1 in the cross-section 75% 85% 95% 105% 115% 125% 135% 145% 0 10 20 30 40 50 60 70 h a rd n es s ra ti o [ % ] distance from the welding's axis [mm] –0.5 mm –2.5 mm –6.5 mm –10.5 mm –14.5 mm –20.5 mm 100% slice #2 slice #3 slice #4 slice #5 r260n (+10.5 mm) 277 277 290 290 r400ht (-10.5 mm) 290 271 264 271 0 50 100 150 200 250 300 h a rd n es s [h b ] r a il 's a x is investigation of heat-affected zones of thermite rail weldings 705 fig. 19 variation of hardness in +/–10.5 mm depth from the axis of welding for sample #2 in the cross-section after the decarbonization/centreline, grain growth was observed, followed by a finegrained microstructure undergoing recrystallization. partial austenitization between the fine-grained structure and the matrix characterizes the microstructure. fine pearlite grains and large coarse pearlite grains were formed as a function of the cooling rate. the faster the cooling velocity, the finer the grain size is. fine grains crystallize rapidly and form a hard and wear-resistant microstructure. coarse grains are characterized by slow crystallization with lower hardness values a microstructure more sensitive to abrasion. for the r260 rail, the weld zone is characterized by coarse grains and the heat-affected zone by fine grains. for the r400ht rail, the hardness of the base material is the highest, in contrast to the previous case, where the grain size is the smallest. the border of the recrystallization zone is located at approx. 50.5....60.5 mm from the welding axis. it causes an increase in ductility and a decrease in hardness. the phenomenon of softening can occur here. further conclusions can be derived from figs. 8-21, but mainly figs. 20 and 21: ▪ both weldings (i.e., sample #1 and #2) generally have a hardness variation within 100 hb within a zone of 15-60 mm from the weldings' axis. ▪ sample #1's lengthwise profile shows that within 10 mm of the haz (between 50-60 mm at the end of the haz) the hardness values drop 100 hb, which is extremely unfortunate. on the base material, it jumps immediately to 120 hb at 60-70 mm. it is a very critical 20 mm zone. ▪ there can be seen a similar phenomenon in the case of sample #2's lengthwise profile: between 50 and 60 mm from the welding's axis, and there is a drop of 90 hb; then, within 60-70 mm, it jumps 90 hb. this 20 mm zone is more uniform. slice #2 slice #3 slice #4 slice #5 r260n (+10.5 mm) 277 277 290 290 r400ht (-10.5 mm) 290 271 264 271 0 50 100 150 200 250 300 h a rd n es s [h b ] r a il 's a x is 706 s. fischer, d. harangozó, d. németh, et al ▪ the weldings ensured and "brought" the results as expected, a more uniform hardness change can be obtained when applying the r350ht welding dose on the r400ht side (i.e., in the case of sample #2). fig. 20 hardness profile 0.5 mm below the running surface in case of weld sample #1, slice #3 – r400ht rail investigation of heat-affected zones of thermite rail weldings 707 fig. 21 hardness profile 0.5 mm below the running surface in case of weld sample #2, slice #3 – r400ht rail 4. conclusions in this paper, special aluminothermic rail welding was investigated, where there was a two-step difference between the categories of welded rails. r260 and r400ht rails were welded together under non-laboratory but technologically strictly controlled precision conditions. the relevant technological instructions were entirely and precisely followed. 708 s. fischer, d. harangozó, d. németh, et al two types of samples were prepared. for sample #1, we used an r260 welding portion, and for sample #2, a thermite portion for r350ht rails. based on the results of the laboratory hardness, macro-, and microstructural tests performed, it was found that sample #2 gave more favorable results in running the hardness profiles. it means that if it is not possible to incorporate temporary 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(in hungarian) 48. barna, v., brautigam, a., kocsis, b., harangozó, d., fischer, s., 2022, investigation of the effects of thermit welding on the mechanical properties of the rails, acta polytechnica hungarica, 19(3), pp. 37-49. 49. https://www.gt-railservice.com/fileadmin/user_upload/pdf/schienenverbindung/sow-5_de-en-fr.pdf (last access: 17.12.2022) 50. european committee for standardization, 2018, msz en iso 6507-1:2018, metallic materials. vickers hardness test. part 1: test method, 39 p. thermal effect on free vibration and buckling facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 45 62 doi: 10.22190/fume161115007s © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper thermal effect on free vibration and buckling of a double-microbeam system udc 620.179.13+620.174]:534.1 marija stamenković atanasov 1 , danilo karličić 1 , predrag kozić 2 , goran janevski 2 1 mathematical institute of the serbian academy of sciences and arts, serbia 2 university of niš, department of mechanical engineering, serbia abstract. the paper investigates the problem of free vibration and buckling of an eulerbernoulli double-microbeam system (ebdmbs) under the compressive axial loading with a temperature change effect. the system is composed of two identical, parallel simplysupported microbeams which are continuously joined by the pasternak’s elastic layer. analytical expressions for the critical buckling load, critical buckling temperature, natural frequencies and frequencies of transverse vibration of the ebdmbs represented by the ratios are derived and validated by the results found in the literature. also analytical expressions are obtained for various buckling states and vibration-phase of the ebdmbs. the temperature change effect is assumed to have an influence on both the microbeams. the length scale parameter, temperature change effect, critical buckling load, thickness/material parameter, pasternak’s parameter and poisson’s effect are discussed in detail. also, as a clearer display of the thermo-mechanical response of ebdmbs, the paper introduces a critical scale load ratio of the modified and the local critical buckling loads in lowtemperature environs. numerical results show that the critical buckling temperatures for classical theories are always higher than the critical buckling temperature for mcst systems. key words: thermal effect, double-microbeam system, critical buckling load, pasternak’s parameter, poisson’s effect 1. introduction micro and nano structures became an object of interest in modern science and technology just after their invention. they possess important mechanical, electrical and thermal performances that are higher than conventional structural materials. using micro/nano received november 15, 2016 / accepted february 23, 2017 corresponding author: goran janevski university of niš, department of mechanical engineering, a. medvedeva 14, 18000 nis, serbia e-mail: gocky.jane@gmail.com 46 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski structures in a high temperature environment leads to certain changes in stiffness. recently, the vibration and buckling studies of beams with the microstructure effect have been increasingly present in the scientific community. researchers are motivated to develop theories such as the modified couple stress theory (mcst) which contains the material length scale parameter; also, they are able to describe size effects on the micro-scale. on the other hand, the classical continuum mechanics theories neither contain any internal material length scale parameter nor are they able to describe these effects. the structural elements such as beams, plates, and membranes in the micro or nano length scale are frequently used as components in micro/nano electromechanical systems (mems/nems). with the rapid development of technology, functionally graded (fg) beams and plates are often used in mems/nems, such as the components in the shape of memory thin films alloy with a global thickness in the micro/nano scale, atomic force microscopes (afms), and electrically actuated mems devices [1–8]. as opposed to the strain theories which were introduced by mindlin and eshel [9], with five constants besides the lamé constants, lam et al. [1] presented a modified theory consisting of only three non-classical constants. wang et al. [10] used the above theory to analyze the behavior of micro beams considering euler–bernoulli and timoshenko beam theories. analysis of bending and buckling of a thin beam were presented by lazopoulos and lazopoulos [11]. these results imply that the gradient coefficient has a significant effect on the buckling load while the surface effect of the energy is negligible. the modified couple stress theory has been used by many authors just as it has been mostly applied to micro beams. the non-local bernoulli–euler beam model was proposed by peddieson et al. [12], using a constitutive equation after eringen et al. [13] which contains two additional material constants. the non-local theories for the bernoulli–euler, timoshenko, reddy, and levinson beams were developed by reddy [14] in a unified way using the hamilton principle and the non-local constitutive relation of eringen et al. [13]. park and gao [18] used a modified couple stress theory with an euler-bernoulli formulation for the bending analysis of cantilever beams. ma et al. [19] and reddy et al. [17] developed a modified timoshenko beam theory and investigation of the bending and free vibration of simplysupported beams using the navier solution process and finite elements method. the buckling analysis of functionally graded micro beams based on the mcst is presented in nateghi et al. [20]. the free vibration of a single-layered graphene sheet resting on an elastic matrix as a pasternak foundation model explored by using the modified couple stress theory is presented by bekir and civalek [21]. m. simsek and reddy [22] developed a united higher order beam theory for a functionally graded (fg) microbeam embedded in an elastic pasternak medium using the modified couple stress theory. in the paper of hendou and mohammadi [23], an euler–bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection where thermoelastic damping is considered to be the main damping mechanism and displayed as imaginary stiffness into the equation of motion by evaluating the temperature profile as a function of lateral displacement. free vibration and buckling of microbeams with the temperature change effect is presented by ke et al. [24]. the finding that the thermal effect on the fundamental frequency and critical buckling load is very low when the thickness of the microbeam has a similar value to that of the material length scale parameter and that it becomes significant when the thickness of the microbeam becomes larger, was of particular interest, and the present paper considers what may happen with a double microbeam system. scientists are trying to comprehend the vibration behavior of micro/nano structures with the effect of temperature changes. primarily motivated by the last few studies [22, thermal effect on the free vibration and buckling of a double-microbeam system 47 23, 24], this paper analyzes the free vibration and buckling behavior with the temperature change effect of the euler-bernoulli double-microbeam system (ebdmbs). to solve the higher-order main equations of the ebdmbs we use the bernoulli–fourier method. the system is composed of two identical and parallel, simply-supported beams which are continuously joined by the pasternak’s elastic layer. it is assumed that the temperature change effect has an impact on both the microbeams. the length scale parameter, temperature change effect, critical buckling load, thickness/material parameter, pasternak’s parameter and poisson’s effect are discussed in detail. the paper presents the impact of different above-mentioned parameters on the natural frequency, frequency under compressive axial loading, critical buckling load and critical temperature of ebdmbs with thermal effect. also, results for various buckling state and vibration-phase of the ebdmbs are obtained. the vibration phases include out-of-phase and in-phase modes of vibration. in order to verify the present study, a comparison of the thermal effect on the dimensionless natural frequency of the system for the first three modes with the results found in the literature is given in the tabular form. because of the strong coupling between mechanical and electrical phenomenon in electromechanical microdevices, there is a growing need for results with temperature effect since they can give contribution to the making of modern microsensors. the ability of the mems device is precision and sensitivity without the need for any cumbersome electrical components. 2. formulation on the basis of the mcst, we discuss the oscillatory system of two parallel euler– bernoulli microbeams which are continuously joined by the pasternak elastic layer under the influence of axial loading including the temperature change effect (see fig. 1).the pasternak foundation assumes the presence of shear interaction among the spring elements which is achieved by connecting the ends of the springs to a beam that only undergoes transverse shear deformation, see [25]. the load–deflection relationship is obtained by taking into account the vertical equilibrium of a shear layer. the pressure–deflection relationship is given by ,2,1, 2 0 2 0     i x w gkwp i i (1) where g is the modulus of a shear pasternak's layer and k is the stiffness modulus of a winkler elastic layer. beams are continuously connected with the winkler elastic layer which represents an idealized medium formed of system springs. both the microbeams are rectangular and have the same length l, thickness h, width b. fig. 1 double-microbeam system coupled by the pasternak's layer 48 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski the beams are simply supported at the ends and under the effect of the axial compressive load with the temperature change effect. 2.1. introduction of the modified coupled stress theory the mcst was developed from the classical couple stress theory, which was well grounded by mindlin [26], mindlin and tiersten [27], toupin [28] and koiter [29]. this theory, suggested by yang et al.(2002), holds that energy density is a function of strain as well as curvature. according to the modified couple stress theory, yang [30], park and gao [18], ma et al. [19] and reddy [17], strain energy us in an isotropic linear elastic material occupying area ω with a volume element v, and, can be written as 1 ( : : ) , 2 su dv      m (2) where σ and ε are the cauchy stress tensor and strain tensor, respectively, m is the deviatory part of the couple stress tensor, and χ is the symmetric part of the curvature tensor. these tensors correspond to the geometrical equations: 1 1 [ ( ) ] , [ ( ) ] , , 2 2 t t x y z                        u u (3) where  is the nabla operator, u is the displacement vector. the rotation vector and constitutive equations are defined by , 1 curl 2  u (4)   .2, 2   l2tr  mi  (5) where material length scale parameter l has the dimension of length which is mathematically the square of the ratio of the curvature modulus to the shear modulus and is physically regarded as a material property measuring the effect of couple stress, mindlin [26], μ and λ are the lamé constants that are given as: and . (1 )(1 2 ) 2(1 ) e            (6) in order to implement the linear constitutive relations presented in eq. (6) the microbeam material should be made homogeneous, isotropic and linearly elastic. 2.2. mathematical model of the double-microbeam system based on the euler-bernoulli beam theory, axial displacements u(x,z,t) and transverse displacements of any point of the beam, w(x,z,t) are given by reddy [14] as 0 0 ( , ) ( ) , ( ) 0 , ( ) ( , ), w x t u x,z,t z v x,z,t w x,z,t w x t x       (7) thermal effect on the free vibration and buckling of a double-microbeam system 49 where w0(x,t) is the midplane displacement. from vector eqs. (3)-(5) and displacements (7) it follows that 2 0 0 2 ( , ) ( , ) , 0 , , 0,xx yy zz xz yz xy y x z w x t w x t z xx                        (8) 2 0 2 ( , )1 , 0 , 2 xy yy zz xz yz xy w x t x                (9) 2 2 0 2 ( , ) , 0 ,xy xx yy zz xz yz w x t m μl m m m m m x          (10) 2 0 2 ( , )(1 ) , 0 , (1 )(1 2 ) xx yy zz xz yz xy w x te z t x                           (11) where e is young’s modulus, α is the coefficient of thermal expansion, ν is poisson's ratio and δt=tt0 is the temperature change with a respect to reference temperature t0 and assuming no shear strains are created by temperature change. in this study, the equilibrium equations are derived by the principle of total potential energy [15, 16]. from eqs (2) and (8) (11), the variation of strain energy in the doublebeam system can be determined as 2 2 0 0 2 2 0 ( 2 ) , ( 1,2), l i i s xx xx xy xy xi xyi w w u m dv m y dx i x x                       (12) where mxi and yxyi are the stress resultant moments and couple moments for the first and second microbeam, respectively, defined as ., 2 0 2 2 2 0 2 x w ladamy x w ddazm i xzi a xyxyi i xxi a xxxi ii         (13) the stiffness components in eqs. (13) are defined as [22] 2(1 ) { , } {1, } , , ( 1,2). (1 )(1 2 ) i i i xx xx i xz i a a e a d z da a da i            (14) using the displacement field components given in eq. (7), we obtain the variation of kinetic energy in the form ),2,1(,, 0 0 02 0 2 0      iadamdxw t w mk iii a ii l i i ie i  (15) where ρi is the mass density for the first and second microbeam. the first variation of the additional strain energy caused by the elastic medium is written by 01 01 02 02 01 02 01 02 0 ( ) ( ) , l ad w w w w u k w w w w g g dx x x x x                     (16) 50 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski where k and g are the spring constants of the winkler and pasternak elastic medium, respectively. the first variation of the work done by axial forces fxi=fmi+ft, (i=1,2) can be given as 0 0 0 ( ) , ( 1,2), l i i ext mi t w w w f f dx i x x          (17) where ft=axxαδt is the axial force due to the influence of the temperature change and fmi, (i=1,2) is the axial forces due to the mechanical loading for the first and second microbeams. the main equation and the boundary conditions can be derived by the hamilton principles as follows 0 [ ( )] 0. t e s ad extk u u w dt       (18) if we substitute the expressions for δus, δke, δuad and δwext from eqs. (12), (15), (16) and (17) into eq. (18) and after integrating by parts and then collecting the coefficients of δw01 and δw02, the equations of motion of the double microbeam system are obtained in the form ),()(: 02012 01 2 2 01 2 12 1 2 1 2 2 01 2 0101 wwk x w g x w ff x y x m t w mw tm xyx                 (19) ).()(: 02012 02 2 2 02 2 22 2 2 2 2 2 02 2 0202 wwk x w g x w ff x y x m t w mw tm xyx                 (20) the boundary and initial conditions of the double-microbeam system are assumed to be simply supported and considered as 2 2 0 0 0 0 2 2 (0, ) ( , ) 0, (0, ) ( , ) 0, ( 1,2),i i i i w w w t w l t t l t i x x          (21) 0 0 ( ,0) ( ), ( ,0) ( ), ( 1,2).i i i i w w x f x x g x i t      (22) 3. analytical solution procedure for the sake of simplicity, we assume that the two parallel beams of the elastically connected double-beam system have the same bending stiffness ei1=ei2=ei and crosssectional area a1=a2=a. both microbeams have the same length l and same material characteristics ρ1=ρ2=ρ. the equations of motion can be expressed in the terms of displacements w01 and w02. by substituting eqs. (14) into eqs. (19) and (20) the main equations of ebdmbs in terms of the displacements are given below 4 2 2 2 2 01 01 01 01 0 1 01 024 2 2 2 ( ) ( ) ( ) 0,xx xz m t w w w w d a l m f f g k w w x t x x                 (23) thermal effect on the free vibration and buckling of a double-microbeam system 51 4 2 2 2 2 02 02 02 02 0 2 01 024 2 2 2 ( ) ( ) ( ) 0.xx xz m t w w w w d a l m f f g k w w x t x x                 (24) in order to simplify the solving of eqs. (23) and (24), we will introduce the following dimensionless parameters: ).2,1(,,,, ,,,,,, 2 0 0 0 0   i a a a a d d a f f a f f a g k a kl k m a l t l l l l x l w w xx xz xz xx xx xx xx mi mi xx t t xx p xx w xxi i  (25) assuming time harmonic motion and using separation of variables, the solutions of eqs. (23) and (24) with the main boundary conditions (21) can be written in the form 0 1 ( , ) ( ) ( ), ( ) sin( ), , ( 1,2),i n in n n n n w x s x k k n i             (26) where sin(τ) is the unknown time function, and xn(ξ) is the known mode shape function for a simply supported single microbeam. introducing the general solutions (26) into eqs. (23) and (24) we obtain the following equations 2 4 2 2 1 0 1 1 2[( ) ( ) ] 0 ,n xx xz n m t n p n w n w ns d a l k f f k k k k s k s        (27) 2 4 2 2 2 0 2 2 1[( ) ( ) ] 0 ,n xx xz n m t n p n w n w ns d a l k f f k k k k s k s        (28) the solutions of eqs. (27) and (28) are assumed in the following forms ,1,, 21  jedsecs nn j nn j nn  (29) where ωn marks the natural frequency of the double-microbeam system, and cn and dn present the amplitude coefficients of the two microbeams, respectively. by substituting eqs. (29) into eqs. (27) and (28) the determinant can be written from which the nontrivial solutions for constants cn and dn can be obtained only when this determinant of the coefficients vanishes. this gives the following frequency equations 2 4 2 2 2 0 1 2 2 4 2 2 2 4 2 2 0 1 0 2 2 [2( ) ( ) ( ) 2 2 ] [( ) ( ) ][( ) ( ) ] 0. n xx xz n m t n m t n p n w n xx xz n m t n p n w xx xz n m t n p n w w d a l k f f k f f k k k k d a l k f f k k k k d a l k f f k k k k k                        (30) finally, when the bi-axial compression forces due to the mechanical loading 1 2m mf f = 0 are ignored, the natural frequency of the system is written by the formula 0 2 2 4 2 2 , 0( ) ( ) ,ni ii xx xz n t n p n w wd a l k f k k k k k      (31) where 0 ni is the lower natural frequency and 0 nii is the higher natural frequency of the ebdmbs. when the bi-axial load applied on the double-beam system reaches a certain critical 52 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski value, the double-beam system becomes unstable which means that the system begins to buckle. introducing ωn=0 into eq. (30), and substituting mechanical load ratio ,12 mm ff where ffm 1 and ,2 ffm  we obtain the equation for the critical buckling load as follows 2 4 2 2 2 2 4 0 1 14 1 {[( ) ](1 ) 4 } , 2 cr xx xz n t n p n w n n n f d a l k f k k k k k b k c k            (32) 2 4 2 2 2 1 0[( ) ](1 ) ,xx xz n t n p n w nb d a l k f k k k k k      (33) 2 4 2 2 2 2 1 0[( ) ] .xx xz n t n p n w wc d a l k f k k k k k      (34) based on equation (32), the critical buckling temperature of the ebdmbs for the biaxial compression equal to zero 021  mm ff is of the form 2 2 0 1 ( ) [( ) ].cr mcst xx xz n p xx t d a l k k a     (35) the illustrated analytical expressions for the natural frequency equation (31), critical buckling load equation (32) and the critical buckling temperature (35) are common equations for the ebdmbs with thermal influence. 3.1. out-of-phase modes of vibration and buckling state a detailed analysis for different cases of phase modes of vibration and buckling state is shown in the paper of murmu and adhikari [33, 34, 35]. fig. 2 out-of-phase vibration of the double-microbeam system for the ebdmbs we can use a change in variables by considering w0i(x,t) as the relative displacement of the microbeam-1 with respect to the microbeam-2 ,02010 www  (36) then .02001 www  (37) subtracting eq. (23) from eq. (22) and using eqs. (36) and (37) we obtain 4 2 2 2 2 0 0 0 0 0 04 2 2 2 ( ) ( ) 2 0,xx xz m t w w w w d a l m f f g kw x t x x                (38) thermal effect on the free vibration and buckling of a double-microbeam system 53 4 2 2 2 2 02 02 02 02 0 04 2 2 2 ( ) ( ) .xx xz m t w w w w d a l m f f g kw x t x x               (39) in above eqs. (38) and (39) for sake of simplicity we assume that fm1=fm2=fm. if material length scale parameter l is ignored, the above equations become those of the classical euler–bernoulli beam theory. for the present out-of-phase analysis of the ebdmbs, we see simplicity in using eq. (38). the general solution of eq. (38) is written as ,1,)(00  iexww ti (40) where w0(x) is the corresponding deformation shape of the ebdmbs and ω is frequency. for vibration analysis we know that is fm=0. by introducing eq. (40) in eq. (38) we get 4 2 2 20 0 0 04 2 ( ) ( ) ( ) ( ) ( 2 ) ( ) 0,xx xz t w x w x d a l f g m k w x x x            (41) or ,0)( )()( 032 0 2 24 0 4 1       xwa x xw a x xw a (42) where the coefficients are 2 2 1 2 0 3( ) , ( ) , ( 2 ) .xx xz td a l a f g a m k a     (43) the general solution of eq. (42) can be written as ,coshsinhcossin)( 242312110 xcxcxcxcxw  (44) where ck, (k=1,2,3,4) can be determined from the boundary conditions (21) and 2 2 2 2 1 2 2 1 3 2 2 2 1 3 1 1 1 1 ( 4 ), ( 4 ). 2 2 a a a a a a a a a a        (45) further, the solving of frequency for the out-of-phase vibration is presented in this section. the configuration of the ebdmbs with out-of-phase vibration mode (w01  w02  0) is shown in fig. 2. by using the boundary conditions of simply-supported microbeam system from eq. (21) yields c2=0 and c4=0. from that we can write 1 2 1 2 2 2 2 1 1 2 2 3 sin sinh 0 . ( ) sin ( ) sinh 0xx xz xx xz l l c d a l l d a l l c                         (46) for the nontrivial solution of eq. (46) the determinant is zero, it follows 2 2 2 2 1 2 2 1 2sin [( ) sinh ( ) sinh ] 0.xx xz xx xzl d a l l d a l l        (47) from eq. (47) the frequency equation is ,0sin 1  l (48) 54 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski and implies ...2,1,1  nnl (49) using eq. (45) yields .03 2 12 4 11  aaa  (50) using the dimensionless parameters (25) and eq. (43), the natural frequency of the ebdmbs for out-of-phase vibration mode we get 2 4 2 0( ) ( ) 2 .n xx xz n t p n wd a l k f k k k      (51) using dimensionless parameters (25) and eq. (38) we get the expression of buckling load in out-of-phase sequence as 2 4 2 2 ( ) ( ) 2 , xx xz n t p n w n n d a l k f k k k f k      (52) 3.2. in-phase modes of vibration and buckling state the configuration of the ebdmbs with in-phase modes of vibration is shown in fig. 3. the relative displacements between the two microbeams are absent (w01  w02 = 0). for the mentioned ebdmbs vibration we solve the eq. (39). fig. 3 out-of-phase vibration of the double-microbeam system by applying the same procedure from the previous chapter the natural frequencies of the ebdmbs for in-phase vibration mode can be expressed as 2 4 2 0( ) ( ) .n xx xz n t p nd a l k f k k     (53) the microbeams are buckled in the same direction (synchronous), see fig. 3.using dimensionless parameters (25) and eq. (39) we get the expression of buckling load in inphase sequence as 2 2( ) ( ).n xx xz n t pf d a l k f k    (54) it is shown from eqs. (53) and (54) that the in-phase vibration mode and buckling state of the ebdmbs is independent of the stiffness of the connecting springs while it is dependent on pasternak's layer and temperature effect and hence the ebdmbs can be treated as a single microbeam. a similar analysis for nanobeam system is presented in the paper of murmu and adhikari [33, 35]. thermal effect on the free vibration and buckling of a double-microbeam system 55 4. numerical results and discussion in this section, we have illustrated a comparative study of the analytical results written in this paper and the results found in the literature. the microbeams of the system are made of epoxy with the following properties: ν=0.38, ρ=1220kg/m 3 , e=1.44gpa, l=17.6μm, α=54×10 -6 / ◦ c from [24]. the cross-section shape and length are kept the same by letting b/h=2 and l/h=10 respectively. temperature and material length scale parameter effect on the two different cases of phase vibration modes and buckling state will be presented. a detailed parametric study is carried out by investigating the influence of different parameters on the natural frequency, frequency under the compressive axial loading, critical buckling load and critical temperature of the ebdmbs with thermal effect. 4.1. temperature and material length scale parameter effect on the phase vibration modes and phase buckling of the ebdmbs the frequency results of the ebdmbs are presented in terms of the frequency parameters for out-of-phase in eq. (51) and in phase vibration mode in eq. (53).variation in frequency parameter ωn with material length scale parameter l0, for different phase vibration due to temperature change is shown in fig. 4. for the winkler and pasternak parameter we used constant values of kw=10 and kp=0.1, while for the temperature change we used two different values δt=50 ◦ c and δt=100 ◦ c. it can be noticed from fig. 4 that with increasing material length scale parameter l0, frequency parameter ωn also increases for both considered cases of phase vibration. frequency parameter ωn decreases as the temperature effect increases for both considered cases of phase vibration. the buckling state results of the ebdmbs are presented in terms of the buckling parameters for out-of-phase in eq. (52) and in phase buckling in eq. (54). material length scale parameter (l0) on buckling parameter fn for the different phase buckling due to temperature change is shown in fig. 5. also, it can be seen that as material length scale parameter l0 increases, buckling parameter fn also increases for both considered cases of phase buckling. buckling parameter fn decreases as the temperature effect increases for both considered cases of phase buckling. fig. 4 variation in frequency parameter ωn with material length scale parameter (l0) for different phase vibrations due to temperature change 56 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski fig. 5 material length scale parameter (l0) on buckling parameter fn for the different phase buckling due to temperature change 4.2. thermal effect on the natural frequency of the ebdmbs it is commonly known that the lowest natural frequency and buckling load of systems of a larger number of coupled nano/micro structures correspond to the natural frequency and buckling load of one beam or plate, see karliţiš et al. [31]. in this paper, the results for the lowest natural frequency and critical buckling load of the ebdmbs may be compared with those obtained for a microbeam one, presented in ke et al. [24] and ma et al. [19]. in order to confirm the present analytical method, table 1 shows a comparison of thermal effect on the dimensionless natural frequency of the system from eq. (31) for the first three modes with results ke et al [24]. perfect agreement between the present frequencies and those of ke et al. [24] can be observed from table 1. it is shown that the inclusion of the thermal effect decreases the frequencies of the microbeam one. it is seen that the effect of pasternak parameter kp=0.01, for a greater mode, leads to an increase in natural frequencies. table 1 thermal effect on the dimensionless natural frequencies for the three modes of the microbeams with h/l=2 mode ( )t c n=1 n=2 n=3 n=1 n=2 n=3 n=1 n=2 n=3 ke et al.[24] present study for 0pk present study for 01.0pk 0 20 40 60 80 100 0.3478 0.3322 0.3159 0.2986 0.2804 0.2608 1.2890 1.2727 1.2562 1.2394 1.2225 1.2053 2.6277 2.6099 2.5920 2.5739 2.5558 2.5374 0.3582 0.3429 0.3271 0.3104 0.2927 0.2739 1.4328 1.4178 1.4027 1.3875 1.3719 1.3563 3.2238 3.2089 3.1939 3.1788 3.1637 3.1485 0.4764 0.4651 0.4535 0.4416 0.4294 0.4168 1.5645 1.5508 1.5370 1.5231 1.5090 1.4948 3.3588 3.3444 3.3300 3.3156 3.3011 3.2866 it can be seen from fig. 6 that the natural frequency with poisson’s ratio (i.e. ν=0.38), suggested by the present euler-bernoulli beam model is always higher than that by poisson’s ratio ν=0. the similar results, merely for a timoshenko beam, are presented by ma et al. [19]. we can conclude that there is perfect agreement between the present frequencies and those of ma et al. [19], when we ignore an effect of the elastic medium, i.e. kp=0 and kw=0. thermal effect on the free vibration and buckling of a double-microbeam system 57 the temperature effect and the pasternak’s parameter on the natural frequency can also be noticed in fig. 6. the natural frequency decreases with temperature effect, in this case 100°c, for a given value of h/l. it is noticed that the inclusion of the constant values of pasternak’s parameter (kp=0.01) increases the natural frequency of the ebdmbs. as a significant result, fig. 6 shows that the increase in bending rigidity is suggested by the present model. also important is that the difference between the natural frequency with the poisson’s ratio and the one without it is important only when the beam thickness is too small. fig. 6 the natural frequency of the ebdmbs varying with microbeam thickness, temperature effect and pasternak’s parameter 4.3. the effect of the compression axial load and temperature effect on the ebdmbs to investigate the influence of the compressive axial loading on the natural frequencies of ebdmbs transverse vibration, we can compare the results of natural frequencies under the compressive axial loading and those without axial loading. , 2 4 2 2 222 , cbb iini     (55) 2 4 2 2 2 2 02( ) ( ) ( ) 2 2 ,xx xz n cr t n cr t n p n wb d a l k f f k f f k k k k         (56) 2 4 2 2 2 0 2 4 2 2 2 0 [( ) ( ) ] [( ) ( ) ] , xx xz n cr t n p n w xx xz n cr t n p n w w c d a l k f f k k k k d a l k f f k k k k k               and .1 cr m f f  (57) if we define .,        nii nii ni ni  (58) with vibration mode number n=1 the impact of the compressive axial loading on the natural frequencies of transverse vibration of the ebdmbs presented by ratios of ψ1 and 58 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski ψ2 are shown in fig. 7. fig. 7 shows that the ratios of frequencies ψ1 and ψ2 decrease with increasing axial compressive load . it can be noticed that the effect of the compressive axial loading on lower natural frequency ωni is practically independent of axial compression ratio ϑ, whereas on higher natural frequency ωnii it is dependent on it. for the winkler and pasternak parameter we used constant values of kw=10 and kp=0.1, while for the temperature change δt=50 ◦ c. it can be noticed from fig. 7(b) that as the axial compression ratio ϑ increases, the ratio of frequency ψ2 decreases. also, it can be seen that the axial compression ratio on the ratio of frequency ψ2 is independent of axial compression ratio ϑ for small axial compressive load , while it is significant for a large axial compressive load. fig. 8 shows the thermal effect and effect of the pasternak parameter on critical buckling load fcr for the ebdmbs as a function of axial load ratio ϑ . with the axial load ratio ϑ increase, the critical buckling load decreases. for a taken value of axial load ratio ϑ, the critical buckling load of the ebdmbs decreases with an increase in temperature change. as can be seen, for the higher value of the pasternak parameter of kp=0.1, the critical buckling load has a noticeably higher value. fig. 7 the thermal effect on the relationships between ratios ψ1 and ψ2 and dimensionless parameter  with increasing axial compressive load ratio ϑ fig. 8 the thermal effect and effect of pasternak parameter on the critical buckling load fcr for the ebdmbs as a function of axial load ratio ϑ thermal effect on the free vibration and buckling of a double-microbeam system 59 the critical scale load ratio of the modified and the local critical buckling loads at a low temperature environs is presented as . lcrclassica crmcst cr       (59) fig. 9 length scale parameter tcr at low temperature environs in order to make a better illustration of the thermo-mechanical response of the ebdmbs, we introduced a scale parameter. fig. 9 shows the influence of length scale parameter tcr at the low temperature environs. the influence of nonlocal parameter at low temperature environs is shown in karliţiš et al. [32]. it can be observed from fig. 9 that this parameter increases for a length scale parameter increase. 5. conclusions the thermal effect on the free vibration and buckling of the euler-bernoulli doublemicrobeam system is examined in this paper based on the modified couple stress theory. the system is composed of two identical, parallel, simply-supported beams which are continuously joined by the pasternak’s elastic layer. the temperature change effect is assumed to have an influence on both microbeams. the higher-order main equations and boundary conditions are derived using the hamilton principle. the separation of variables method (known as the fourier method) is used for the main equations to obtain free vibration frequencies and critical buckling loads of the ebdmbs. the length scale parameter, temperature change effect, critical buckling load, thickness/material parameter, pasternak’s parameter and poisson’s effect are discussed in detail. also, the effect of different mentioned parameters on the natural frequency, frequency under the compressive axial loading, critical buckling load and critical temperature of the ebdmbs with thermal effect are presented. effect of the material length scale parameter and thermal effect on the two different cases of phase modes of vibration and buckling state are discussed. based on the presented analysis we conclude that the in-phase vibration mode and buckling state of the ebdmbs is independent of the stiffness of the connecting springs while it is 60 m. stamenkoviš atanasov, d. karliţiš, p. koziš, g. janevski dependent on pasternak's layer and temperature effect and hence the ebdmbs can be treated as a single microbeam. in order to confirm the present study, we have shown in tabular form a comparison of thermal effect on the dimensionless natural frequency of the system for three modes with the results found in the literature. it is concluded that the presented results are in perfect agreement with the results observed in ke et al. [24]. it is shown that the inclusion of the thermal effect decreases the frequencies of the microbeam one. also, the effect of pasternak parameter kp for a greater mode leads to an increase in natural frequencies, but including the temperature change, the frequency is decreased and leads to the decreased stiffness of the system. the numerical results obtained for the natural frequency with poisson’s effect and suggested by the present euler-bernoulli beam model are always higher than those without poisson’s effect. the thermal effect on the natural frequency is very low for the microbeam one of the ebdmbs and with a small ratio of h/l, while it is significant for the microbeam with a large ratio of h/l. the impact of the compressive axial loading on the natural frequencies of the ebdmbs transverse vibration leads to the following observations:  the temperature change effect has an impact on both microbeams.  the lower and higher natural frequency under the compressive axial loading decrease with increasing axial compressive load and also decrease with a temperature change increase. the reason for that is that the thermal effect leads to the reduction in stiffness and such a behavior leads to the softening of the materials of the ebdmbs.  the effect of the compressive axial loading on the lower natural frequency is almost independent of the axial compression ratio, whereas on the higher natural frequency it depends on it.  for a higher value of the pasternak parameter, the critical buckling load has a higher value which decreases with increasing temperature change.  the critical buckling temperature for the presented systems is always lower than for the classical theories.  the critical scale load ratio of the modified and the local critical buckling loads at the low temperature environs increases with the increasing length scale parameter. all these observations can be useful for modern electromechanical systems. physical views of this paper may be useful for 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instability of double-nanobeam-systems, physics letters a, 375(3), pp. 601-608. 10417 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 21 36 https://doi.org/10.22190/fume220111005p © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a low-cost approach to data-driven fuzzy control of servo systems radu-emil precup1, stefan preitl1, claudia-adina bojan-dragos1, elena-lorena hedrea1, raul-cristian roman1, emil m. petriu2 1politehnica university of timisoara, dept. automation and applied informatics, romania 2university of ottawa, school of electrical engineering and computer science, canada abstract. servo systems become more and more important in control systems applications in various fields as both separate control systems and actuators. ensuring very good control system performance using few information on the servo system model (viewed as a controlled process) is a challenging task. starting with authors’ results on data-driven model-free control, fuzzy control and the indirect model-free tuning of fuzzy controllers, this paper suggests a low-cost approach to the data-driven fuzzy control of servo systems. the data-driven fuzzy control approach consists of six steps: (i) openloop data-driven system identification to produce the process model from input-output data expressed as the system step response, (ii) proportional-integral (pi) controller tuning using the extended symmetrical optimum (eso) method, (iii) pi controller parameters mapping onto parameters of takagi-sugeno pi-fuzzy controller in terms of the modal equivalence principle, (iv) closed-loop data-driven system identification, (v) pi controller tuning using the eso method, (vi) pi controller parameters mapping onto parameters of takagi-sugeno pi-fuzzy controller. the steps (iv), (v) and (vi) are optional. the approach is applied to the position control of a nonlinear servo system. the experimental results obtained on laboratory equipment validate the approach. key words: closed-loop data-driven system identification, data-driven fuzzy control, extended symmetrical optimum method, servo systems 1. introduction as specified in the project [1], in contrast to model-based control, data-driven control avoids the system (process) identification by constructing controllers directly from data. that is the reason why data-driven control is also referred to as model-free control (i.e. no received: january 11, 2022 / accepted february 13, 2022 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania e-mail: radu.precup@aut.upt.ro 22 r.-e. precup, s. preitl, c.-a. bojan-dragos, et al. model in controller tuning), justifying the high interest in nonlinear controllers whose parameters are tuned using process input-output data after conducting few experiments, or, more generally, data-driven model-free control [2]. instead, one or more experiments are conducted in order to use the information in controller tuning, and non-parametric system or process models) can be employed in this regard. a concise discussion on the popular data-driven control techniques is given in precup et al. [3] pointing out the following ones that ensure the iterative experiment-based update of controller parameters: iterative feedback tuning (ift) [4], [5], model-free adaptive control (mfac) [6], [7], simultaneous perturbation stochastic approximation [8], [9], correlation-based tuning [10], [11], frequency domain tuning [12], [13], iterative regression tuning [14], and adaptive online ift [15]. a review on data-driven control [16] offers classifications and highlights the role of observers and estimation in control, also leading to non-iterative data-driven control techniques: model-free control (mfc) [17], [18], virtual reference feedback tuning (vrft) [19], [20], active disturbance rejection control (adrc) [21], [22], data-driven predictive control [23], [24], unfalsified control [25], [26], data-driven inversion based control [27], [28], and the investigation of equivalent conditions on the given data under which different analysis and control problems can be solved [29]. other representative techniques are emphasized in book [2]. it is suggestively stated in [3] and [30] that mfc is an efficient tool for machine learning; moreover, as specified earlier in [25], unfalsified control is also an efficient tool for machine learning. as pointed out in the studies conducted in [1] and [31], fuzzy control is an important subject in the area of nonlinear control as the fuzzy controllers are relatively easily understandable and also offer very good control system performance indices. however, the heuristic approach to design and tune fuzzy controllers is compensated by the systematic design of fuzzy controllers that can employ the stable design of fuzzy control systems, the optimal and robust controller design and tuning. classical and recent applications of fuzzy control deal with popov-type stability analysis [32], embedded fuzzy control system for machining processes [33], tire slip control [34], predictive functional control based on fuzzy models [35], stability and sensitivity analysis of fuzzy control systems [36], stability analysis dedicated to the fuzzy control of nonlinear processes [37], robust evolving cloud-based control [38], power control of series hybrid electric vehicles [39], vehicle navigation by fuzzy cognitive maps [40], fuzzy control for the iron ore sintering process [41], type-2 fuzzy control for line following [42], and singularity-free fixed-time fuzzy control for robotic systems [43]. the model-free tuning of fuzzy controllers is an alternative approach to the modelbased design resulting in data-driven fuzzy control [1] to benefit from the advantages of data-driven control and fuzzy control and to mitigate their drawbacks. the combinations of data-driven model-free and fuzzy control include h∞ fuzzy control [44], fault tolerant fuzzy control [45], parameterized data-driven fuzzy control [46], data-driven interpretable fuzzy control [47], mfc merged with proportional-derivative (pd) takagi-sugeno fuzzy control [48], [49], mfac merged with pd takagi-sugeno fuzzy control [50], [51], adrc mixed with pd takagi-sugeno fuzzy control [52] and tuned by vrft [22] as well, fuzzy logic-based adaptive adrc [53], data-driven arithmetic fuzzy control using the distending function [54], and data-driven mfc developed around continuous-time intelligent proportional-integral (pi) control [31]. the indirect model-free tuning of fuzzy controllers has initially been proposed in authors’ papers [55] and [56], and continued in a low-cost approach to data-driven fuzzy control of servo systems 23 [48] and [50] by controller structures that combine data-driven control and fuzzy control in order to incorporate model-free features in fuzzy control system structures. according to the studies carried out in [57]–[59], several auto-tuning approaches, using a single relay, a sequential array of relays or decentralized relays, are available in the literature. relay identification can achieve fine tuning of controllers including fuzzy controllers. since experiments are conducted with the control system, it is justified to consider auto-tuning as an approach to data-driven fuzzy control. some recent approaches to the auto-tuning of fuzzy controllers include the auto-tuning of pi-fuzzy controllers for variable speed wind turbines [60], the pi-fuzzy logic-based tuning of controllers for hybrid wind & photovoltaic power systems [61], the auto-tuning of proportional-integral-derivative (pid)fuzzy controllers for pitch angle control of wind turbines [62], telescope tracking systems [63], and indoor control for renewable air-conditioning [64]. building upon authors’ results on the indirect model-free tuning on fuzzy controllers [55], [56] and the auto-tuning of pi controllers [65], [66], this paper suggests a low-cost approach to data-driven fuzzy control of servo systems focusing on servo systems that can be modeled by second-order systems with an integral (i) component and a small time constant. these servo systems are controlled by takagi-sugeno pi-fuzzy controllers. the data-driven fuzzy control approach consists of six steps which include open-loop and closed-loop data-driven system identification to produce the process model from inputoutput data expressed as system step responses (i.e. non-parametric models), tuning the linear pi controller using the extended symmetrical optimum (eso) method [67], [68], and mapping the parameters of the pi controller onto the parameters of the takagisugeno pi-fuzzy controller in terms of the modal equivalence principle [69]. the approach is important with respect to the state-of-the-art because it is relatively simple as far as both the theoretical support and the implementation are concerned. nevertheless, only few of the steps must be proceeded, depending on the interests of the control systems designers, who do not need to possess strong knowledge on the control systems and the controlled processes. concluding, this work presents a low-cost approach to data-driven fuzzy control. this approach is novel and feasible in practical applications. the rest of the paper treats the following topics: the tuning approach is presented in the next section. section 3 is dedicated to the validation in the illustrative example of position control of a nonlinear servo system using a takagi-sugeno pi-fuzzy controller. experimental results obtained on laboratory equipment [70] are included. the conclusions are pointed out in section 4. 2. the tuning approach it is assumed that the servo system as a controlled process can be modeled by the transfer function p(s): , )1( )( sts k sp p + = (1) where kp is the process gain and t > 0 is the process small time constant or parasitic time constant. the transfer function in (1) includes the actuator dynamics and the measurement instrumentation dynamics. 24 r.-e. precup, s. preitl, c.-a. bojan-dragos, et al. the presence of the i component in the transfer function (1) increases the difficulty of system identification, namely the computation of the two parameters such that (1) to approximate with an acceptable accuracy the behavior of the real-world servo system. two data-driven identification approaches, (i) and (ii) discussed as follows, are considered to be adequate for the process with the transfer function given in (1). (i) the open-loop approach. a step signal input is applied to the servo system around the operating point of interest on a time horizon of approximately t10 . if the step signal u(t) of magnitude u: )()( tutu =  (2) is applied to the input of the servo system, as shown in fig. 1, then a system response expressed in fig. 1 is illustrated, where y(t) is the servo system (or process) output and also the controlled system output,  (t) is the unit step signal, and the subscript  indicates the steady-state value of a certain variable. the step signal u(t) is also the control signal if the servo system is included in a control system structure, and u, which is used in the controller is assumed to be known, and matches one of the operating regimes that are important for the servo control system. fig. 1 input step signal applied as control signal and servo system response. this figure is adapted from preitl et al. [66] fig. 1 highlights that it is not necessarily impose to the servo system to evolve starting with zero initial conditions. the initial conditions are stated by means of the pair of inputoutput data values (u0, y0), which define the initial operating point. therefore, nonzero initial conditions can be accepted, which is the usual situation in servo control systems operation. the expression of the system response is: ).()]1([)( / 0 tuetttkyty tt p −−+=  −   (3) a low-cost approach to data-driven fuzzy control of servo systems 25 for large values of time, i.e. t >> t, the exponential component is vanishing because 0lim / =− → tt t e , so the steady-state response can be approximated by: .)()( 0 −+ uttkyty p (4) as shown in [65] and [66], the time constant td = t, where the asymptote to the system response cuts the t axis in fig. 1, plays the role of a pure time delay for which: .368.0)( 0 += utkyty pd (5) the data-driven identification approach (i) is carried out in terms of the steps (i1), (i2) and (i2) described as follows: step (i1). a unit step signal defined in (2) is applied as a control signal to the servo system viewed as a controlled output and the system response is recorded. step (i2). considering two time moments t1 and t2, the corresponding output values y(t1) and y(t2) are measured on the basis of fig. 1. the expression of the process small time constant is [65], [66]: . )()( )()( 12 1221 tyty tyttyt tt d − − == (6) step (i3). the expression of the process gain results after the manipulation of (5): . 368.0 )( 0  − = ut yty k d p (7) using (4), another (approximate) way to compute pk is: . )( )( 1 01 − −  utt yty kp (8) (ii) the closed-loop approach. a proportional (p) controller with the transfer function c(s): cksc =)( (9) is included in a control loop that represents the control system for the servo system (fig. 2), where kc is the controller gain. fig. 2 servo system control system structure as a control loop the variables and blocks in fig. 2 are: r – reference input or set-point, r~ – filtered reference input, fr – reference input filter, d – disturbance input, which can be applied to the process input, the process output or, as shown in fig. 2, in a certain (informational) place in the process structure, c – controller (a p controller in the framework of the 26 r.-e. precup, s. preitl, c.-a. bojan-dragos, et al. approach (ii)), p – controlled process, i.e. the servo system, which can be modeled using (1), yre −= ~ – control error. using the control system structure illustrated in fig. 2, the transfer functions of the blocks in (1) and (9), assuming the absence of the block fr, the closed-loop control system transfer function with respect to the reference input (assuming a zero disturbance input) is expressed as: , 2 )( 2 00 2 2 0 ++  = ss sh r (10) with the parameters 0 – natural frequency [65], [66]: ,0  = t kk pc (11) and  – damping factor [65], [66]: . 5.0  = tkk pc (12) for an adequately chosen value of kc the control system can be brought in the situation to have two complex conjugated poles and the system response with respect to a step reference input of magnitude r, which is also supposed to be known as u in relation with (2): ),()( trtr =  (13) to exhibit the oscillatory behavior illustrated in fig. 3. the expression of the control system response is: ).()]arccos1sin( 1 1[)( 2 0 2 0 0 trt e yty t +− − −+=  − (14) the following notations are introduced: 2 0 2 0 1 11 ,1 − =  =−= n nn t (15) for the damped natural frequency n and the period of oscillations nt of the response. fig. 3 highlights, similar to the approach (i), that it is not necessarily imposed to the control system to evolve starting with zero initial conditions. in this context, the initial conditions are stated by means of the pair of input-output data values (r0, y0), which define the initial operating point, and nonzero initial conditions can be accepted as well. the considered situation is normal because this approach is applied only if the control systems designer considers that it is of interest to apply it for possible re-tuning the controller. a low-cost approach to data-driven fuzzy control of servo systems 27 fig. 3 reference input step signal (without fr) and control system response. this figure is adapted from preitl et al. [66] and it assumes that all sub-systems of the control systems are implemented accurately the expressions of the time moments specific to the response and illustrated in fig. 3 are as follows [65], [66]: },2,1{ , 1 , 212 2 0 2 0 21  −  =  = −  = m n t t t (16) and the relationship between the system response (output) values in fig. 3, which also gives the overshoot 1, is [65], [66]: .1 1/1 2 == − −−   e y yym (17) aiming an as accurate as possible data-driven identification, the control system must be brought in the situation 0.25 <  < 0.707 (it is recommended in [65] and [66] to set  = 0.5) by the appropriate modification of kc. this “ideal” value  = 0.5 is convenient in order to measure relatively easily the specific numerical values on both axes in fig. 3. the relationships (16) and (17) are equivalent to the following reversed relationships [65], [66]: . 121 1 , 2 , ) ln (1 1 2 21 2 0 21 2 1 −  = − =   =   + = tt t t n n (18) in the conditions of known kc, measured 1 and tn, and computed  and 0, the expressions of the two parameters in (1) are [65], [66]: 28 r.-e. precup, s. preitl, c.-a. bojan-dragos, et al. . , 4 1 2 0 2 =  − =   c p n k t k t t (19) the data-driven identification approach (ii) is carried out in terms of the steps (ii1), (ii2) and (ii2) described as follows: step (ii1). a unit step signal defined in (13) is applied as a reference input to the control system and the system response is recorded. step (ii2). the controller gain kc is set such that to obtain a system response with 0.25 <  < 0.707. if the system response fulfils this condition, the approach continues with the step (ii3). otherwise, the step (ii1) is repeated. step (ii3). the values of ym1, y and t21 are measured. relationships in (18) are next applied to obtain the values of  and 0. finally, relationships in (19) are applied to compute the values of t and next kp. pi controllers can cope with the process modeled in (1). the transfer function of a pi controller is c(s): , ), 1 1( )1( )( icc i c ic tkk st k s stk sc =+= + = (20) where kc > 0 or kc > 0 are two expressions of the controller gain, with their relation specified in (20), and ti > 0 is the integral time constant. the eso method [67], [68] is successfully applied to tune the pi controller parameters in (20) as it guarantees a tradeoff to the empirical control system performance specifications (expressed as maximum values of percent overshoot, settling time and rise time) of the linear control system making use of a single design parameter  within the largest recommended domain 1 <   20. the pi tuning conditions specific to the eso method are as follows [67], [68]: . , 1 , 1 2   =  =  = tt tk k tk k i p c p c (21) the transfer function of the simplest reference input filter out of the two ones recommended in the papers [67] and [68] is: . 1 1 )( st sfr + = (22) the takagi-sugeno fuzzy controller is designed and tuned in terms of transferring in a fuzzy logic-like interpretation the knowledge from the pi controller structure. the structure and the input membership functions of a the low-cost takagi-sugeno fuzzy controller are presented in fig. 4, where q−1 indicates the backward shift operator, td indicates the discrete time index, tiso-fc is the two inputs-single output fuzzy controller, e(td) is the increment of control error, and u(td) is the increment of control signal. a low-cost approach to data-driven fuzzy control of servo systems 29 a b fig. 4 structure (a) and input membership functions (b) of low-cost takagi-sugeno fuzzy controller. this figure is adapted from precup et al. [71] discretizing the continuous-time pi controller by tustin’s method, the recurrent equation of the incremental discrete-time pi controller is as follows [71]: )],( )([)( ddpd tetektu += (23) where [71]: , 2 2 ), 2 ( si ss icp tt tt tkk − =−= (24) and ts > 0 is the sampling period. the tiso-fc block employs the weighted average method for defuzzification, and the sum and prod operators in the inference engine. the complete rule base of the tiso-fc block is expressed as [71]: )].( )([)( then )p is )( and p is )(( or ) zeis )( and p is )(( or )p is )( and n is )(( or ) zeis )( and n is )(( or ) zeis )(( f )],( )([ )( then )p is )( and p is )(( or )n is )( and n is )(( if ddpd ddd ddd ddd ddpdd ddd tetektu tetete tetete tetete tetektute tetete +=    +=  (25) the role of the additional parameter , with the largest domain 0 <  < 1, is to reduce the overshoot of the control system. therefore, (25) and the fuzzy controller structure make this low-cost fuzzy controller behaves as a bumpless interpolator between two linear pi controllers. the modal equivalence principle [69] applied to this takagi-sugeno pi-fuzzy controller leads to the tuning equation: , ee bb = (26) where the parameter be should be chosen according to the experience of the control systems designer. the parameter  is chosen in a similar way. the optimal tuning can be performed with very good results [71] to get the values of these parameters. summarizing all aspects presented in this section, the low-cost data-driven fuzzy control approach consists of the six steps (dd1) to (dd6): 30 r.-e. precup, s. preitl, c.-a. bojan-dragos, et al. step (dd1). the open-loop data-driven system identification approach (i) is applied to produce the process model in (1) using the input-output data of the controlled process (the servo system) expressed as the servo system step response shown in fig. 1. step (dd2). the linear pi controller is tuned using the eso method such that to meet the performance specifications imposed to the control system. step (dd3). the parameters of the pi controller are mapped onto the parameters of the takagi-sugeno pi-fuzzy controller using (26). step (dd4). this step is optional and conducted only if the control systems designer considers that it is relevant. for example, such situations occur if the control system performance indices are deteriorated in time. the closed-loop data-driven system identification approach (ii) is applied to produce the process model in (1) using the inputoutput data of the control system expressed as the (closed-loop) control system step response shown in fig. 3. step (dd5). this step is also optional in the context of the step (dd4). the linear pi controller is tuned using the eso method such that to meet again the performance specifications imposed to the control system. step (dd6). this step is also optional in the context of the step (dd4) and it is identical to the step (dd3). the parameters of the pi controller are mapped onto the parameters of the takagi-sugeno pi-fuzzy controller using (26). since the step (dd4) is applied in terms of the real-world operation of the control system, a special attention should be paid to the transfer from the takagi-sugeno pifuzzy controller to the p controller and vice-versa. bumpless transfers should be ensured in this regard, meaning that the history of the “old” digital control algorithm requires to be modified in order to avoid big modifications of the control system, which might affect negatively the actuators and finally the control system behavior. a simple solution in the linear case is presented in the references [65] and [66]. 3. experimental results the tuning approach presented in the previous section is validated as follows by applying it to the design and tuning of a takagi-sugeno pi-fuzzy controller to the angular position of a nonlinear servo system laboratory equipment [70]. some details on the steps are given as follows. the state-space model of the servo system is expressed as [71]: ,])()([ ]01[)( ),( 0 )( )( 1 0 10 )( )( ,)( if,1 ,)( if, )( ,|)(| if,0 ,)( if, )( ,)( if,1 )( 21 2 1 2 1 t p b ba ab a ac cb cb c b txtxty tm t k tx tx ttx tx utu utuu uu utu utuu utuu uu utu utu tm =         +              −=                  − − − −− − + −− =    (27) a low-cost approach to data-driven fuzzy control of servo systems 31 where the control signal u(t) applied to the direct current (dc) motor is a pulse width modulated duty cycle, x1(t) = (t) (rad) is the angular position, x2(t) = (t) (rad/s) is the angular speed, and the superscript t indicates matrix transposition. the variable m(t) is the output of the saturation and dead zone static nonlinearity, modeled in the first part in (27), with the parameters ua = 0.15, ub = 0.1 and uc = 0.15. the application of the step (dd1) leads to the values of the servo system (i.e. process) parameters kp = 140 and t = 0.92 s. the first steps (dd1) and (dd2) of the approach presented in the previous section are applied. these two steps are applied simultaneously in terms of the optimal tuning [71] of the takagi-sugeno pi-fuzzy controller parameters such that to ensure a reduced parametric sensitivity with respect to one of the two parameters in (1). one set of linear pi controller and takagi-sugeno pi-fuzzy controller parameter values, which ensures the strongest mitigation of the parametric sensitivity with respect to t is recommended in [71]: β = 16.9763, kc = 0.001884, ti = 15.618 s, be = 20, bδe = 0.01281, and η = 0.287. three optimization algorithms were applied in [71], however other ones could be of interest because of the nonlinearity of the process and the controller as, for example, parameterized genetic algorithms [72], [73], various algorithms adapted from their general formulation for community detection in networks [74], metaheuristic algorithms with information feedback models [75], moea/d [76], slime mould algorithms [77], grey wolf optimizers [78], and algorithms specific to neuro-fuzzy model training [79]. the optimization problems in this context should be defined with great care accounting for various constraints, which may be caused by man-computer symbiosis [80], stochastic demands [81], fault detection and isolation and recovery [82], tradeoff to approximation accuracy and complexity [83], and specific structures of fuzzy systems [84], [85], requiring appropriate handling and the modification of the optimization algorithms. using the above parameter values and implementing the low-cost takagi-sugeno fuzzy controller according to the details given in the previous section, fig. 5 offers a sample of experimental results for the fuzzy control system. fig. 5 illustrates that the fig. 5 real-time experimental results expressed as fuzzy control system responses y and u (pwm indicates pulse width modulation) 32 r.-e. precup, s. preitl, c.-a. bojan-dragos, et al. fuzzy control system exhibits good control system dynamics performance with respect to the 40 rad step modifications of the reference input. fig. 5 also outlines the effects of the nonlinearity in (27). the results considered in this section help the reader to understand the effectiveness and the efficacy of the proposed approach. more effective metrics and performance indices could be exploited to assess the advantages of the developed controllers. 5. conclusions starting with a control structure with auto-tuning proportional-integral controller, which was previously developed by the authors, and two open-loop data-driven system identification approaches, this paper gave a low-cost approach to data-driven fuzzy control of servo systems focusing on takagi-sugeno proportional-integral-fuzzy controllers. using well stated tuning relations, which can ensure good control system performance indices, which are selectable according to the needs / application, the extended symmetrical optimum method is initially used to tune the linear proportional-integral controllers. the modal equivalence principle is next involved in mapping the parameters of the linear controller onto the parameters of the fuzzy one. the paper also presented two identification approaches (i) and (ii) of a certain category of servo systems together with the relations for the computation of the parameters based on dynamic regime measurements, which are relatively easily performed and implemented. the authors helped the reader to understand the novelty issues of the developed scheme. the approach suggested in this paper is advantageous as it can be generalized to processes of integral type and several dynamics and delays. the approach can be implemented automatically by the computer-aided computation of the process parameters in the two identification approaches instead of actually representing the system responses. the data-driven approach presented in the paper proves the potential of auto-tuning approaches in data-driven control. the applications had in view belong to the field of electrical driving systems with fast / slow variable parameters as function of the process operation. section 2 should have addressed more details regarding the considered models and tools; in particular, it does not consider the robustness and reliability issues, due for example to uncertainty and disturbance effects, as well as the model-reality mismatch. this point is fundamental when the reliability and robustness features of the proposed solutions have to be verified and validated with respect to real engineering and safety critical systems. therefore, the effectiveness of the methodology proposed in section 2 is a suggested open problem and future issue that could require further investigations. another direction of open research direction is the combination of this data-driven technique with other data-driven techniques in order to reduce the heuristics in the steps (dd2) and (dd3). the optimal tuning of fuzzy controllers will be carried out accounting for stability constraints but with great care to preserve the data-driven feature of the future novel approaches. all these open problems and future issues will contribute to make data-driven fuzzy control clear and non-questionable. acknowledgement: this work was supported by grants of the romanian ministry of education and research, cncs uefiscdi, project numbers pn-iii-p4-id-pce-2020-0269, pn-iii-p1-1.1te-2019-1117, pn-iii-p1-1.1-pd-2019-0637, within pncdi iii, by the cnfis-fdi-2021-0582 project of the politehnica university of timisoara, romania, and by the nserc of canada. a low-cost approach to data-driven fuzzy control of servo systems 33 references 1. data-driven fuzzy control with experimental validation, national exploratory research grant (pce), financed by the executive agency for higher education, research, development and innovation funding uefiscdi), 2021-2023, project code: pn-iii-p4-id-pce-2020-0269, 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increasingly applied to solving machining optimization problems. regardless of numerous capabilities of the monte carlo method, its application for solving machining optimization problems has been given less attention by researchers and practitioners. the aim of this paper is to investigate the monte carlo method applicability for solving single-objective machining optimization problems and to analyze its efficiency by comparing the optimization solutions to those obtained by the past researchers using meta-heuristic algorithms. for this purpose, five machining optimization case studies taken from the literature are considered and discussed. key words: machining, optimization, monte carlo method 1. introduction in recent years, both high resource efficiency and machining processes optimization are vital for manufacturing companies to gain a competitive advantage and become market winners. the ultimate goal of machining optimization is to select machining factor values so that the overall machining performance is enhanced. determination of optimal machining parameters is a continuous engineering task whose goals comprise production costs reduction as well as achievement of the desired product quality [1]. in general, the selection of optimal machining parameter values for a specific machine tool plays the most important role in manufacturing, as the process control parameters of a machine tool are not always precisely understood. thus, it becomes increasingly difficult to recommend the optimum values with an enormous variety of expensive materials in the market [2]. received february 19, 2014 / accepted march 22, 2014  corresponding author: miloš madić university of niš, faculty of mechanical engineering, department of production engineering, niš, serbia e-mail: madic@masfak.ni.ac.rs original scientific paper 28 m. madić, m. radovanović in the real production environment it is a common practice to select machining factor values based on the experience of the machinist (or production planner), machining handbooks and manufacturers recommendations. as a result, the user attempts to optimize the cutting operations by trial-and-error every time he needs to setup the existing equipment for a new different task [3]. the most adverse effect of such a notvery scientific practice is decreased productivity due to sub-optimal use of machining capability [4]. the advances in the machining technology and the developments in related areas (computing, statistics, artificial intelligence, etc.) have led to development of more sophisticated approaches including data storage and retrieval, expert systems, modelbased approach, modeless approach based on the taguchi method, etc. however, the nonavailability of the required technological performance equation represents a major obstacle to the implementation of optimized cutting conditions in practice [4]. the model-based approach, very popular with researchers, integrates experimental, statistical, mathematical and artificial intelligence tools thus providing a means for better understanding of machining processes. using the experimental data, with the help of regression analysis, artificial neural networks and fuzzy logic, different empirical equations for prediction of machining performance characteristic can be developed. subsequently, the (near) optimal machining parameter values are determined by the application of an optimization algorithm such as gradient based, non-gradient, heuristic or meta-heuristic algorithms. in the field of machining process optimization, the current trend is the application of meta-heuristic algorithms such as genetic algorithm (ga), simulated annealing (sa), particle swarm optimization (pso) algorithm, artificial bee colony algorithm and ant colony optimization algorithm [5]. meta-heuristic algorithms perform an efficient and comprehensive exploration of the optimization search space using random the monte-carlo search guided by governing mechanisms which imitate certain strategies taken from nature, social behavior, physical laws, etc. despite numerous capabilities of the monte carlo method, its application for solving machining optimization problems has been given less attention by researchers and practitioners. the present paper has three objectives: (i) to investigate the monte carlo method applicability for solving single-objective machining optimization problems, (ii) to develop a framework for solving machining optimization problems using the monte carlo method, and (iii) to analyze efficiency of the monte carlo method for solving machining optimization problems by comparing the optimization solutions to those obtained by the past researchers using meta-heuristic algorithms. for this purpose, five machining optimization case studies taken from the literature are considered and discussed. 2. monte carlo method many numerical problems in science, engineering, finance, and statistics are solved nowadays by the monte carlo methods, that is, by means of random experiments on a computer [6]. the monte carlo is in fact a class of methods now widely used in computer simulations [7]. the "classical" monte carlo is used as an uncertainty analysis of the deterministic calculation because it yields distribution describing the probability of alternative possible values about the nominal (designed) point [8]. the idea of the monte carlo calculation is much older than the computer. the name monte carlo is relatively recent, and is connected to famous casinos in monaco. it was coined by nicolas metropolis possibilities of using monte carlo method for solving machining optimization problems 29 in 1949 under the name of "statistical sampling". since the pioneer studies in 1940s and 1950s, especially the work by ulam, von newmann, and metropolis, it has been applied in almost all area of simulations, from the ising model to financial market, from molecular dynamics to engineering, and from the routing of the internet to climate simulations [7]. monte carlo methods have been used for a long time but only in the last few decades, they have gained the status of fully rounded numerical methods. in order to obtain reasonably accurate assessment, it is necessary to calculate a large number of special cases as well as to carry out a respective statistical analysis; that is why an effective application of the monte carlo methods begins with the emergence of fast computers. at the heart of any monte carlo method is a random number generator: a procedure that produces an infinite stream of random variables that are independent and identically distributed according to some probability distribution. when this distribution is a uniform one (i. e. it has equal probability in the interval from 0 to 1), the generator is said to be a uniform random number generator [6]. uniform distribution has a wide-ranging application in various problems in engineering modeling and optimization. the monte carlo is not only used for estimation but also for optimization purposes. the optimization based on monte carlo methods can be useful for solving optimization problems with many local optima and complicated constraints, possibly involving a mix of continuous and discrete variables [6]. in order to enhance the accuracy of the method, the multistage approach may be applied in which the stochastic computations are repeated by diminishing the region of search after identifying a near optimal solution. the basic steps in the monte carlo method implementation, illustrated in fig. 1, are followed for solving machining optimization problems in this paper. 3. case studies to investigate the efficiency of the monte carlo method for solving single-objective machining optimization problems, five conventional machining process research papers are considered. although the monte carlo method has universal applicability, the selection of papers is restricted to only those dealing with the explicitly given mathematical models i.e. mathematical models in terms of polynomial equations, because optimization solutions can be readily checked and compared. 3.1. case study 1 sharma et al. [9] have conducted turning experiments on aluminum 6061 alloy and metal matrix composites of aluminum. for turning of al-sic (5%) the authors develop the following relationship between surface roughness and turning parameters:   dfdvfvdf vdfvr sicala   7.4000229.00015.071.42541 000001.04.1044300122.07.18 22 2 %5 (1) where v is the cutting speed (m/min), f is the feed rate (mm/rev), and d is the depth of cut (mm). 30 m. madić, m. radovanović the single-objective machining optimization problem is formulated as follows: fig. 1 monte carlo method flowchart for solving machining optimization problems (mm) 10.4 (mm/rev) 0.10.05 (m/min) 740228:subject to ),,,( minimize     d f v dfvfra (2) in their attempt to obtain minimum surface roughness and corresponding optimal turning parameter values, the authors have applied pso algorithm. 3.2. case study 2 sanjeev et al. [10] have investigated the turning process of polymeric material (polytetrafluoroethylene – ptfe, teflon). the authors have developed regression model for predicting surface roughness in the following form: dvdffv dfvra   143.0002.0234.0 175.087.0675.0309.0 (3) possibilities of using monte carlo method for solving machining optimization problems 31 the single-objective machining optimization problem is formulated as follows: (mm) 5.20.5 (mm/rev) 0.30.1 (m/min) 275150:subject to ),,,( minimize     d f v dfvfra (4) in their attempt to obtain minimum surface roughness and corresponding optimal turning parameter values, the authors have applied ga. 3.3. case study 3 saravanakumar et al. [11] have investigated turning process of the inconel 718. using the experimental data, the authors have developed regression equation for prediction of material removal rate (mrr) in the following form: vfdfdvd vfdfv   78805373431417 17499149311213629819158mrr (5) the single-objective machining optimization problem is formulated as follows: (mm) 0.250.1 (mm/rev) 0.250.15 (m/min) 8060:subject to ),,,(mrr maximize     d f v dfvf (6) in their attempt to obtain maximal mrr and corresponding optimal turning parameter values, the authors have applied ga. 3.4. case study 4 bhushan et al. [12] have investigated turning of al alloy sic particle composite material using carbide inserts. on the basis of the experimental results, the authors have developed the following regression equation for the prediction of surface roughness: drfddv rdfvra   56484.033125.342419.30000174.0 18753.019915.419694.000324.072412.0 22 (7) where r is the tool nose radius (mm). the single-objective machining optimization problem is formulated as follows: (mm) 0.80.4 (mm) 0.60.2 (mm/rev) 0.250.15 (m/min) 21090:subject to ),,,,( minimize      r d f v rdfvfra (8) 32 m. madić, m. radovanović in order to obtain minimum surface roughness and corresponding optimal turning parameter values, the authors have applied ga. 3.5. case study 5 poornima and sukumar [13] have investigated turning of martensitic stainless steel. on the basis of the experimental results, the authors have developed the following regression equation for the prediction of surface roughness: fddvvfdf vdfvra   142.200575.007857.002333.080272.6 00002.047976.030442.101518.051539.1 22 2 (9) the authors have formulated the following single-objective machining optimization problem: (mm) 0.50.5 (mm/rev) 0.220.15 (m/min) 12080:subject to ),,,( minimize a     d f v dfvfr (10) minimum surface roughness and corresponding optimal turning parameter values are determined by using ga. 4. results and discussion in previous research studies the machining optimization problems are solved by using meta-heuristic algorithms such as the ga and pso. in this section the optimization solutions obtained by the past researchers are compared to those obtained by applying the monte carlo method. all calculations are accomplished by the proposed optimization procedure given in fig. 1 by using excel spreadsheet package. the generation of random numbers is done by using function rand(). in this paper for solving the machining optimization problems formulated in previous section, a two-stage monte carlo approach is applied. in the first stage, random numbers for each independent variable (machining parameter) are generated by considering the interval ranges for each variable. subsequently, 5000 estimations of dependent variable (performance characteristic) are calculated by using the given mathematical model. after ranking all solutions, the best solution with extreme (minimal or maximal) value of dependent variable along with corresponding values of independent variables is identified. in the second stage, on the basis of the analysis of the previously identified best solution, the range for each independent variable is modified. subsequently, the stochastic computations are repeated again for 5000 estimations, and the best solution is recorded. the comparison of obtained optimization solutions for the case studies is summarized in table 1. possibilities of using monte carlo method for solving machining optimization problems 33 table 1 comparison of machining optimization solutions case study method machining parameters objective function v (m/min) f (mm/rev) d (mm) r (mm) 1 ihsa **** 740 0.05 0.4 -- ra (µm) 0.22936 pso 233 0.05 0.4 --1.2883 * 0.8745 ** monte carlo method i stage 609.928 0.051 0.421 --0.639 ii stage 739.367 0.05 0.4 --0.2298 2 ga 158.065 0.164 1.719 -- ra (µm) 61.92 monte carlo method i stage 151.813 0.291 2.494 --38.3504 ii stage 150.016 0.3 2.5 --37.5009 3 sa **** 80 0.25 0.1 -- mrr (mm 3 /min) 2124.275 ga 79.99 0.25 0.1 --2122.23 monte carlo method i stage 79.928 0.25 0.133 --2071.87 ii stage 80 0.25 0.1 --2124.06 4 ga 207.055 0.151 0.201 1.199 ra (µm) 1.039 *** 1.0650 ** monte carlo method i stage 209.293 0.187 0.207 1.108 1.1077 ii stage 209.968 0.15 0.2 1.2 1.0499 5 ga 119.93 0.15 0.5 -- ra (µm) 0.74 monte carlo method i stage 119.051 0.151 0.5 --0.7424 ii stage 119.979 0.15 0.5 --0.7315 * results reported by sharma et al. [9] ** corrected values *** results reported by bhushan et al. [12] **** results reported by madić et al. [14]; ihsa – improved harmony search algorithm the analysis of the machining optimization solutions presented in table 1 indicates that: (i) the optimization solutions obtained by the monte carlo method after first stage are comparable to those obtained by past researchers using meta-heuristic algorithms such as ga and pso, and (ii) solutions obtained by monte carlo method after second stage are better than those obtained by past researchers using meta-heuristic algorithms. the optimization solutions presented in this paper indicate that few thousand monte carlo computation runs are efficient for solving multi-dimensional and complex machining single-objective optimization problems. the efficiency of the monte carlo method is assessed by calculating the percentage improvement of the optimization solution for each case study. the comparisons are graphically illustrated in fig. 2. the entire optimization time when using monte carlo method consists of the time needed to formulate machining optimization problem, the time needed to generate random numbers for each independent variable considering interval ranges, the time for monte carlo computation runs i.e. evaluation of dependent variable values, the time needed for ranking the optimization solutions and the time needed for the identification of the best solution. 34 m. madić, m. radovanović -10 -5 0 5 10 15 20 25 30 35 40 45 1 2 3 4 5 p er ce n ta g e im p ro v em en t case study 0 10 20 30 40 50 60 70 80 1 2 3 4 5 p er ce n ta g e im p ro v em en t case study a) b) fig. 2 percentage improvement of the optimization solutions: a) results of monte carlo method after i stage, and b) results of monte carlo method after ii stage when the entire optimization time is considered, the monte carlo method application for solving single-objective machining optimization problems requires only few minutes. the salient advantage of the monte carlo method application is that it is possible to obtain a majority of optimization solutions, which can be particularly advantageous in machining practice considering different machine/tool constraints. furthermore, the monte carlo based optimization approach requires no expert knowledge, setting of algorithm parameters and/or defining an initial solution as in the case of using classical mathematical and meta-heuristic optimization algorithms. 5. conclusion in this paper, an attempt has been made to investigate the monte carlo method applicability for solving single-objective machining optimization problems. five singleobjective machining optimization case studies are considered. in order to analyze the monte carlo efficiency, the optimization solutions obtained are compared to those determined by past researches using meta-heuristic algorithms. on the basis of the analysis of the obtained results the following conclusions related to the monte carlo capabilities for solving single-objective machining optimization problems can be made:  optimization procedure based on the monte carlo method consists of only few steps and it is very easy to implement without the need to write programming code or use specialized software packages,  monte carlo is a parameter-free universal method in which optimization search based on random numbers is independent of initial conditions,  when the computational time is considered, the monte carlo method provides an efficient determination of solutions,  the application of the monte carlo method is well suited for solving machining optimization problems and the quality of solutions is comparable or even better than those obtained by meta-heuristics. using a multiple-stage procedure one could expect further enhancement of the optimization search,  with increasing computational runs, the monte carlo method can be more efficient by avoiding being trapped in local minima,  monte carlo method enables determination of a majority of solutions, possibilities of using monte carlo method for solving machining optimization problems 35  monte carlo method has the capability for solving multi-objective machining optimization problems thus marking the scope of future research. on the basis of obtained results this study proposes the wider usage of the monte carlo method for solving machining optimization problems because of its simplicity, efficiency and wide-ranging capabilities. acknowledgement: the paper is a part of the research done within the project tr35034. the authors would like to thank to the ministry of education and science, republic of serbia. references 1. kramar, d., sekulić, m., kovač, p., gostimirović, m. kopač, j., 2012, the implementation of taguchi method for quality improvement in high-pressure jet assisted turning process, journal of production engineering, 15(2), pp. 23-26. 2. sivarao, thiru, s., kamaruzaman, j., azizah, s., yusoff, m., jano, z., yaakub, y., hasoalan, hadzley, m., shah, i., izan, n., amran, m., taufik, sapto, w., tan, c.f., sivakumar, d., 2013, modelling of co2 laser materials processing by networked neuro-dimension fuzzy intelligent system, australian journal of basic and applied sciences, 7(3), pp. 35-45. 3. nedić, b., janković, p., radovanović, m., globočki lakić, g., 2013, quality of plasma cutting, proc. of 13th int. conference on tribology "serbiatrib-2013", kragujevac, serbia, pp. 314-319. 4. quazi, t.z., more, p., sonawane, v., 2013, a case study of taguchi method in the optimization of turning parameters, international journal of emerging technologies and advanced engineering, 3(2), pp. 616-626. 5. yusup, n., zain, a.m., hashim, s.z.m., 2012, evolutionary techniques in optimizing machining parameters: review and recent applications (2007-2011), expert system with applications, 39(10), pp. 9909-9927. 6. kroese, d., taimre, t., botev, z., 2011, handbook of monte carlo methods, wiley, new york. 7. yang, x.-s., 2010, engineering optimization an introduction with metaheuristic applications, wiley, new york. 8. pokorádi, l., molnár, b., 2011, monte-carlo simulation of the pipeline system to investigate water temperature’s effects, u.p.b. scientific bulletin series d: mechanical engineering, 73(4), pp. 223-226. 9. sharma, a.v.n.l., sandeep kumar, p., gopichand, a., mohan rao, r., 2012, optimal machining conditions for turning of al/sic mmc using pso and regression analysis, international journal of engineering research and applications, 2(6), pp. 497-500. 10. sanjeev kumar, m., kaviarasan, v., venkatesan, r., 2012, machining parameter optimization of polytetrafluoroethylene (ptfe) using genetic algorithm, international journal of modern engineering research, 2(1), pp. 143-149. 11. saravanakumar, k., pratheesh kumar, m.r., shaik dawood, a.k., 2012, optimization of cnc turning process parameters on inconel 718 using genetic algorithm, iracst engineering science and techology: international journal estij, 2(4), pp. 532-537. 12. bhushan, r.k., kumar, s., das, s., 2012, ga approach for optimization of surface roughness parameters in machining of al alloy sic particle composite, journal of materials engineering and performance, 21(8), pp. 1676-1686. 13. poornima, sukumar, 2012, optimization of machining parameters in cnc turning of martensitic stainless steel using rsm and ga, international journal of modern engineering research, 2(2), pp. 539-542. 14. madić, m., marković, d., radovanović, m., 2013, comparison of meta-heuristic algorithms for solving machining optimization problems, facta universitatis series: mechanical engineering, 11(1), pp. 29-44. 36 m. madić, m. radovanović mogućnosti primene monte carlo metode za rešavanje problema optimizacije parametara obrade u savremenim tržišnim uslovima kompanije su fokusirane na povećanje kvaliteta proizvoda, smanjenje troškova i vremena izrade. za postizanje ovih ciljeva optimizacija parametara obrade je od izuzetnog značaja. pored klasičnih metoda optimizacije, poslednjih godina za rešavanje problema optimizacije se sve češće koriste meta-heuristički algoritmi. uprkos brojnim mogućnostima monte carlo metode, primena ove metode za rešavanje problema optimizacije parametara obtrade nija dovoljno istražena. cilj ovog rada je da se istraži mogućnost primene monte carlo metode za rešavanje jednokriterijumskih problema optimizacije parametara obrade. u ovom radu dobijeni rezultati optimizacije su upoređeni sa rezultatima optimizacije dobijenih primenom raličitih metaheurističkih algoritama. u radu su razmatrane pet studije slučaja jednokriterijumske optimizacije procesa mašinske obrade. ključne reči: mašinska obrada, optimizacija, monte carlo metod facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 261 272 https://doi.org/10.22190/fume170512022k © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a thermal analysis of the threaded spindle bearing assembly in numerically controlled machine tools udc 621.8 vladislav krstić, dragan milčić, miodrag milčić faculty of mechanical engineering, university of niš, serbia abstract. a threaded gear in machine tools is a mechanical actuator that converts rotary motion into linear one of the machine axis using a recirculating ball-nut. it provides positioning accuracy, uniform motion, silent operation, reduced wear and an increased service life. the bearing assembly of the threaded spindles should provide load transfer (cutting forces and friction forces) while maintaining high guiding accuracy. due to a high number of the threaded spindle revolutions and the presence of tension in the bearing and a high axial force originating from the cutting and friction forces, the increased heat load due to friction in the bearings is normally expected. for this reason, this paper presents a thermal analysis of the bearing assembly of the threaded spindle which is realized via an axial ball bearing with angular contact of the zkln type, produced by the german manufacturer schaeffler (ina); in other words, a numerical thermal analysis has been performed. key words: threaded spindles, bearing, thermal analysis, thermal load 1. introduction automation of small-batch and batch manufacturing as the dominant type in the metal processing industry is successfully carried by means of numerically controlled machine tools. they are characterized by increased productivity and accuracy. in machine tools there are a number of local heat sources that increase the thermal gradient inside the machine: an electric motor, friction in the mechanical drive and gears, processing, ambient temperature. heat sources cause local deformations, affecting machine accuracy. therefore, the drive (motor and mechanical gear) should be mounted on the outside of the machine; the temperatures resulting from friction in the bearings and sliding spindle received may 12, 2017 / accepted february 05, 2018 corresponding author: dragan milĉić faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: dragan.milcic@masfak.ni.ac.rs 262 v. krstić, d. milĉić, m. milĉić should be eliminated by adequate lubrication, the temperature generated during processing should be eliminated by adequate cooling and metal shavings removal system, and machine structure should be realized in accordance with the thermally-symmetrical design. mechanical gear may be: threaded gear (threaded spindle and nut), spur (sprocket) gear (pinion and rack) and timing (toothed) belt gear or chain gear. the most commonly used mechanical gear in machine tools is a threaded gear consisting of a threaded spindle and a nut. the main task of the threaded spindle in machine tools is to convert the machine axis motion from rotational into linear one using a recirculating ball-nut. the threaded spindle rotates with high speed. the ball thread and the nut have precision-made helical grooves through which the balls circulate and thus provide a very high degree of guiding accuracy ensuring the final product quality. the bearing assembly of threaded spindles is one of the most challenging tasks in mechanical engineering. the threaded spindles are loaded with a high axial force originating both from processing, i.e. cutting forces, and from frictional forces, and that load must be received by the bearing assembly. fig. 1 (on the right) shows an example of the threaded spindle of the machine tool in the bearing assembly. due to the rolling friction at the joint between the threaded spindle and the nut and in the rolling bearings, heat load is generated in the bearing assembly of the threaded spindle. the generated amount of heat induced by friction between the nut and the spindle, as well as friction in the bearing itself, affect the elongation of the spindle, further leading to errors in the guidance which ultimately leads to poor quality products. fig. 1 threaded spindle of the machine tool 2. overview of the previous work with the expansion of cnc machines the volume of research has increased regarding the problem of heat generation due to friction in the drive part of the machine tools (threaded gear) and the impact on the precision of the machine tools. mahmmod [1] in his work focuses on heat generation due to friction induced forces that occur in the threaded spindle and the associated ball-nut. the paper gives an explanation of the phenomenon of heat accumulation that could affect the occurrence of positioning errors due to the main elongation. the deformations caused by temperature increase are estimated based on the temperature distribution to the threaded spindle and the nut, and a conventional finite difference method is obtained. zahedi and movahhedy in their work [2] contributed to the development of a comprehensive model of high speed spindles that includes sustainable models for mechanical and thermal behavior of its main components, i.e. bearings, shaft/axle and housing. the spindle a thermal analysis of the threaded spindle bearing assembly in numerically controlled machine tools 263 housing and the shaft are modeled as elements of the timoshenko beam model in the form of six degrees of freedom. the bearings are modeled as two-node elements with five positions and a component of thermal load in each node. interaction between thermal and structural behavior of the spindle, housing, bearing and shaft, is described by means of thermal expansion and range of heat transfer. the components are combined in the form of the finite elements model for thermo-mechanical analysis of the spindle-bearing system. in order to obtain thermal characteristics of the integrated spindle-bearing system in cnc machines, xiaolei et al. [3] have defined a mathematical model, using the heat source model. heat characteristics of the spindle-bearing system are identified using the derived formulas and as such were inserted into the model, which was tested by the finite element method. four different cases with different amounts of heat were tested, different coefficients of heat transfer as well as the geometric dimensions of the model and the position of the heat sources. using this model on two real systems in practice, the prediction of the thermal field was performed and later results obtained using the model and through concrete temperature measurement were compared. the maximum relative error for both systems was 0.41% and 8.38%, respectively. takafumi et al. [4] in their work gave an analysis of the deformation of rolling elements i.e. balls, as well as the heat generation that occurs in the axial ball bearing with an angular contact surface intended for the threaded spindle bearing assembly in machine tools. they conducted a three-dimensional measurement of movement of the balls, and proposed construction of the axial ball bearing with angular contact, inner diameter of 70 mm, an outer diameter of 110 mm, for the operating speed of n=30000 min -1 . xiao et al. [5], focus on the study of sources of heat that is generated by the spindle of the cnc machine, which works with a high number of revolutions. the whole research was based on a model of thermo-mechanical coupling. yang and wanhua in their work [6] elaborate methods of compensation of axial error caused by thermal load on the spindle in machine tools. wang et al. provide research [7] of effects of the inner ring displacement due to the effect of centrifugal forces on the dynamic characteristics of ball roller bearings with angular contact, designed for high speeds. yasushi in his work [8] gave a description of the status and trends for the support and improvement of high speeds in machine tools. he also made reference to the new ceramic materials with very good friction properties, which further supports the increase in the number of the bearing revolutions, while reducing friction in the same, thereby reducing the generated heat load. yukio et al. [9] give a presentation of so-called “robust” batch of bearings. this type of bearing is intended to increase productivity with lower power consumption, which is very important in terms of energy efficiency, which has been a popular topic for many years. wu and tan [10] have developed a thermo-mechanical coupling analysis model of the spindle-bearing system based on the hertz’s contact theory and the point contact nonnewtonian thermal elastohydrodynamic lubrication (ehl) theory. kumar and rao [12] have considered previous experimental and analytical work, a static-thermal finite element analysis (fea) of a railroad bearing pressed onto an axle and analyzed using the ansys. the manufacturer of the threaded spindles “heidenhain” from germany in its publication [19] provides a detailed description and explanation of possible problems regarding 264 v. krstić, d. milĉić, m. milĉić the threaded spindle operation. it also presented a description of possible impacts that lead to guidance errors. all these impacts are divided into two main groups, one comprising guidance errors due to mechanical influences while the other comprises those induced by thermal loads. the publication offers concrete solutions for compensation and monitoring of errors during the threaded spindle operation. these solutions range from the simplest options of installing measuring laths, through the structure of the threaded spindle itself to expensive software solutions for continuous monitoring of operation. 3. bearing assembly of threaded spindles and construction of zkln type bearings a bearing assembly of threaded spindles is generally resolved in practice in several ways depending on the particular constellation of mechanical system and the expected load. as shown in fig. 2, there are three ways of bearing assembly of threaded spindle (from the top down), i.e.: free/fixed bearing, fixed/fixed bearing and fixed/fixed and prestressed bearing. the third case of bearing assembly is the most demanding because bearing is fixed on both sides, where the bearing is on the right support (additionally prestressed) for achieving increased rigidity and better compensation of the axial forces. because of the present tension in the axial direction during operation increased friction will occur that generates a larger amount of heat, which in turn affects the thermal dilatation of both the bearing assembly and the threaded spindle, thereby endangering the accuracy of guidance. for this reason it is very important to better define the thermal load of the bearing assembly, as early as in the design stage, taking into account all relevant impacts that will be present during the operation in order to compensate axial errors of the threaded spindle as much as possible. fig. 2 basic types of bearing assemblies of threaded spindles [19] in general case, the type of bearing assembly with free/fixed bearing will be used at shorter threaded spindles when greater axial rigidity of the system is not required and when the critical rotational speed of the threaded spindle is high enough. the type of bearing assembly with fixed/fixed bearing is recommended for medium and longer threaded spindles when the system requires high axial rigidity when a critical number of revolutions of the threaded spindle is high, and when a smaller effect of changes in length due to heating is a thermal analysis of the threaded spindle bearing assembly in numerically controlled machine tools 265 expected on positioning. the type of bearing assembly with fixed/fixed and prestressed bearing is recommended in the case of long threaded spindles, in high dynamic threaded spindles when greater longitudinal deformation of the threaded spindle is expected. for bearing assemblies of threaded spindles, the german manufacturer schaeffler has designed special bearings for this purpose. in fact, these are axial ball bearings with an angular contact of zkln and zklf types. as an example, the zkln2557-2z type of bearing will be analyzed in this paper, produced by the german manufacturer schaeffler (ina). in order to better understand the analysis of this bearing, we must firstly inspect its structure. during the threaded gear operation, the maximum load is generated from the associated recirculation nut, which slides over the spindle. theoretically, the threaded spindle bearing assembly receives radial and axial load, but axial load is much higher (over 90%), and for that reason the structure of this bearing is adjusted to this fact. the specificity of the zkln type bearing is a two-piece inner ring, and a much wider angle of contact between the rolling body-balls and the rolling track (an angle of 60°). in addition, this bearing has a precision nut for prestressing the bearing and also a cover through which the bearing is further secured to the machine housing, and all in order to secure the bearing in the axial direction. from the above-mentioned structure of the bearing, a significant heat load of the bearing assembly can be expected at higher speeds. figure 3 presents the case of installing the zkln type bearing. basically this type of bearing does not provide, in its design, any additional bolted connection used for fixing to the housing of the machine. for this reason, its securing is realized using a precision nut for prestressing and cover, which is further secured to the machine housing through the bolted connection. fig. 3 the case of installation of the bearing of zkln type [13] zkln2557-2z bearing, produced by schaeffler (ina) [13], was used as a representative for thermal analysis. 4. thermal load on the threaded spindle bearing assembly for thermal analysis of the bearing of the zkln type it is necessary to define thermal load of the bearing. thermally safe operating speed n is calculated according to din 732 [14, 15]. the basis for the calculation is the heat balance in the bearing, the equilibrium between the frictional 266 v. krstić, d. milĉić, m. milĉić energy as a function of speed and the heat dissipation as a function of temperature. when conditions are in equilibrium, the bearing temperature is constant. the permissible operating temperature determines thermally safe operating speed n of the bearing. for calculation, it is assumed that normal operating clearance and constant operating conditions are present. in addition to the thermally safe operating speed, limiting speed ng must always be observed. the essential operating conditions are:  reference temperature of the ball bearing on stationery outer ring 70 o c,  reference temperature of the ball bearing environment 20 o c,  reference load for axial ball bearings for threaded spindles equals 2% of static load of the bearing for axial bearings with contact angle 45°< α <90° [18]. the contact angle by zkln type is 60°, p1r = 0,02ˑc0. it should also be noted that the standard din 732, part 1 and part 2, will be only partially applicable for the entire analysis. the reason for this is the specific structure of the aforementioned bearing. specifically, the standard covers only standard bearing structures in which the maximum angle of contact is 40°, and at the same time the bearings in their structure do not include bolted connection, cover, precise nut for prestressing and housing to which the bearing will be further secured. in order to properly perform the analysis, it is necessary to comply with all the theoretical basics related to the mechanisms of heat transfer. as is already known, there are three ways to transfer heat: conduction, convection and radiation. for easier understanding, fig. 4 represents a simplification of the transfer of heat through the observed system of bearing assembly (in case of installation of the bearing of zkln type). as can be seen from fig. 4, the thermal energy that occurs in the bearing assembly is at first induced on the surface between the rolling elements and the rolling track and then spreads in the directions as shown by the arrows in the same fig. 4 [13]. fig. 4 schematic representation of the propagation of heat in the bearing assembly (1 cover for additional securing to the machine housing, qi generated amount of thermal energy, qaamount of thermal energy discharged) [13] the entire thermal process for the present case is divided into several branches. one branch describes the movement of generated thermal energy through the bolt, over the contact surface of the cover for additional bearing/housing securing and out into the environment. on this path the thermal energy passes through “solid material” and the contact surface. the next branch shows the movement of thermal energy through “solid material” of the outer ring of the bearing and bolt. here, it is shown that the heat energy passes through the “solid material” but also a thermal analysis of the threaded spindle bearing assembly in numerically controlled machine tools 267 through the bearing/housing contact surface. the next branch shows the movement of thermal energy from the rolling track and the rolling body through the inner ring, the contact surface (inner ring/ bearing prestressing nut) and further into the atmosphere. the last branch describes the flow of thermal energy from the rolling track and the rolling body over the inner ring and the contact surface (inner ring/threaded spindle) and ends at the threaded spindle. the main internal heat transfer mechanisms are: transfer of heat between the rotational elements of bearings, transfer of heat from the stationary bearing elements and conduction between the contact elements of the bearing (fig. 5). fig. 5 heat transfer mechanisms in the bearing [11] 5. the concept of thermal analysis of the threaded spindle bearing assembly for the reasons of great complexity of the system, the thermal analysis will be conducted through simulation for the bearing of zkln2557-2z type [13]. hereinafter, for this type of bearing the abbreviated notation (zkln) will be used. hereafter follows the algorithm of the concept of performed simulation, see fig. 6. the fig. 6 presents an algorithm of thermal analysis of the threaded spindle bearing assembly. the thermal analysis is done in an iterative process as a combination of the analytical procedure of determination of the power loss due to friction in the bearing, which is converted into heat generated in the bearing and numerical calculation for determining the temperature field of the threaded spindle bearing assembly. as an input for determining power losses due to friction according to the din 732-1 we used geometry of the selected zkln bearing and limiting speed ng=2350 min -1 for that bearing which was given in the manufacturer’s catalog [13]. the power loss due to friction between the rolling elements and the outer and inner rings of the bearing nfr is converted into heat q which is transferred through the outer or inner ring. for the purposes of thermal fea analysis it is necessary to determine the heat flux at the outer and inner ring of the bearing. it is assumed that 75% of the heat is surrendered to the outer rolling path, and the remaining 25% to the inner rolling path. this is empirical data, based on experience of schaeffler, i.e. internal recommendation for thermal distribution by bearing. based on the assessment of the propagation of heat through the bearing and further to the bearing assembly and based on the area of path of the inner and outer rings, the specific amount of heat, i.e. heat flux is determined. 268 v. krstić, d. milĉić, m. milĉić based on the prepared model of the bearing in fea software abaqus 6.9-3 numerical calculations, a thermal analysis is performed. for thermal analysis, a bearing assembly of the threaded spindle with zkln2557-2z bearing was adopted as well as different materials of the bearing housing, made of steel, cast iron en-gjl 250 (cast iron 25) and aluminum. further presentation will present the results for the case of steel housing. fig. 6 algorithm of thermal analysis of the threaded spindle bearing assembly in the fea thermal analysis, the simulation model is prepared in the preprocessing phase. the adopted model for the analysis is 2d, and because of the symmetrical bearing assembly and symmetric impact of loads, we observed only half of the bearing assembly. for the purposes of thermal analysis the values of the thermal conductivity coefficient within the system were calculated as well as the heat transfer coefficient by taking into account mechanisms of heat transfer. fig. 7 provides the values of heat transfer coefficients α with appropriate speed n and temperatures t in certain parts of the system for the case of bearing assembly for the zkln2557-2z bearing. a thermal analysis of the threaded spindle bearing assembly in numerically controlled machine tools 269 fig. 7 values of heat transfer coefficients for the case of bearing assembly of the zkln2557-2z bearing in determining the coefficient of heat transfer by radiation, the emissivity coefficient of dark surfaces ɛ=0.8 was adopted. heat transfer coefficients (by convection) in general are defined in accordance with [17], see equation 1:   d kn fluidu   (1) where kfluid is thermal conductivity in the case of heat transfer between the elements of the bearing in w/mk, nu is nusselt's number (dimensionless value), dλ is diameter from which the heat is released in m, and α is coefficient of heat transfer in w/m 2 k. for the purposes of thermal analysis it is also necessary to define the contact load of the bearing at the points of contact between the inner ring of the bearing threaded spindle, the outer ring of the bearing bearing housing, the inner ring of the bearing bearing prestressing nut and bearing cover bearing housing. figure 8 shows the contact load on the zkln2557-2z bearing with the values of the contact pressure due to installing installation. it also shows the heat transfer coefficients for the existing contact surfaces. they serve as an equivalent for describing thermal resistance. 270 v. krstić, d. milĉić, m. milĉić fig. 8 the values of the load of the bearing on the contact surfaces in the case of bearing assembly of the zkln2557-2z bearing after preparing the simulation model in the preprocessing stage, the solver of the fea software abaqus gives a temperature field as a result. figure 9 shows simulation results for the case of the first iteration for the steel housing. fig. 9 temperature field of the bearing assembly of the threaded spindle in case of steel housing for the purposes of determination of the heat load of the threaded spindle bearing assembly, it is necessary that the temperature of the outer ring is 70°c. for this reason, the second iteration is performed. the used number of revolutions of the bearing (threaded spindle) n2 is greater than the number of revolutions used in the first iteration n2=3500 min 1 > ng=2350 min -1 and the procedure from the algorithm in fig. 6 is repeated. the procedure is repeated until the result of the fea thermal analysis shows the temperature of the outer a thermal analysis of the threaded spindle bearing assembly in numerically controlled machine tools 271 ring of the bearing of 70°c (reference condition). in the example analysis of the zkln25572z bearing this case happened in the third iteration for the estimated number of revolutions n3=3750 min -1 . fig. 10 temperature field of the bearing assembly of the threaded spindle in case of steel housing third iteration (the temperature of the outer ring 70°c) table 1 shows the results of the fea simulation of the temperature of the outer ring of the bearing for different materials of the bearing housing. as shown in table 1, neither the housing material nor the threaded spindle length significantly affects the temperature of the outer ring of the bearing, i.e. the heat distribution through the system (threaded spindlebearinghousingcoverprecision nut for prestressing of the bearing). table 1 temperature on the outer ring of bearing type zkln2557-2z for different variations housing materials power loss by friction nfr in w housing materials temperature on the outer ring in °c 36 steel housing 68.6 36 steel housing (bigger housing) 65.6 36 steel (smaller threaded spindle) 69.7 36 steel housing (increased axial removal of heat on the housing) 65.2 36 aluminum housing 68.5 36 cast iron en-gjl 250 68.6 36 mineral casting 70.1 6. conclusion a threaded gear in machine tools is a mechanical actuator that converts rotary motion into linear one of the machine axis using a recirculation ball-nut. the threaded gears primarily have to ensure positioning accuracy. the heat generated due to the rolling and sliding friction in the bearing and the recirculating nut affects accuracy of machine tools. for this reason, it is important to determine thermal load. this paper presents an algorithm 272 v. krstić, d. milĉić, m. milĉić for determining thermal load of the bearing assembly of the threaded spindle, which is of iterative nature and consists of combination of analytical and numerical fea thermal analysis. this paper shows an example of fea thermal analysis on the zkln2557-2z bearing. the main influential factor is the reference surface through which heat is transferred. that means that by other similar type of bearing (for example zklf) the heat distribution will be different because of the extended reference surface. further, it means that this paper presents a new approach to the thermal analysis of the threaded spindle bearing assembly. references 1. mahmmod , a., m., 2011, the effect of the heat generated by friction in the ballscrew-nut system on the precision of high speed machine, journal al-taqani, 24(6), pp. 112-123. 2. zahedi, a., movahhedy, m.r., 2012, thermo-mechanical modeling of high speed spindles, scientia iranica, 19(2), pp. 282-293. 3. xiaolei, d., jianzhong, f., yuwen, z., 2015, a predictive model for temperature rise of spindle–bearing integrated system, journal of manufacturing science and engineering, 137(2), pp.1-10. 4. yoshida, t., tozaki, y., omokawa, h., hamanaka, k., 2001, tribological technology. threedimensional ball motion in angular contact ball bearing for high-speed machine tool spindle, jglobal, 38(6), pp.304-307. 5. xiao, s., guo, j., zhang, b., 2006, research on the motorized spindle’s thermal properties based on thermo-mechanical coupling analysis, technology and innovation conference, itic, hangzhou, china, pp. 1479-1483. 6. yang, l., wanhua, z., 2012, axial thermal error compensation method for the spindle of a precision horizontal machining, international conference mechatronics and automation, icma, pp. 2319 – 2323. 7. wang, b., mei, x., hu, c., wu, z., 2010, effect of inner ring centrifugal displacement on the dynamic characteristics of high-speed angular contact ball bearing, international conference mechatronics and automation, icma, pp. 951-956. 8. morita, y., 2002, high speed and high precision enhancement technology of the bearing for the main spindle of machine tool, journal science of machine, f0147a, 54(9), pp. 935-940. 9. oura, y., katsuno, y., sugita, s., 1999, robust series high-speed precision angular contact ball bearings for machine tool spindles, journal nsk tech j, s0469a, 668, pp. 20-28. 10. wu, l., tan, q., 2016, thermal characteristic analysis and experimental study of a spindle-bearing system, entropy, 18(7), 271; doi:10.3390/e18070271. 11. živković, a., zeljković, m., tabaković, s., 2013, software solution for the analysis of behavior ball bearings –report (technical solution) (in serbian), faculty of technical sciences, university of novi sad. 12. kumar, p.m., rao, c.j., 2015, structural and thermal analysis on a tapered roller bearing, ijiset international journal of innovative science, engineering & technology, 2(1), pp. 502-511. 13. schaeffler gruppe industrie, 2009, lager für gewindetriebe, katalog tpi 123d-d: schaeffler gruppe. 14. deutsches institut für normung, 1994, din 732 teil 1thermische bezugsdrehzahl. 15. deutsches institut für normung, 1994, din 732 teil 2thermische bezugsdrehzahl. 16. krstić, v., 2013, research limit speed of angular contact ball bearings (in serbian), master thesis, faculty of mechanical engineering, university of niš. 17. vdi-geselschaft verfahrenstechnik und chemieingenieur wesen, 2013, vdi-wärmeatlas, springer. 18. deutsches institut für normung, 2004, din iso 15312wälzlagerthermische bezugsdrezahl – berechnung und beiwerte. 19. heidenhain, 2006, genauigkeit von vorschubachsen, http://www.heidenhain.de/fileadmin/pdb/media / img/349843-10.pdf. facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 43 55 https://doi.org/10.22190/fume190609001m © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  exploring structural design of the francis hydro-turbine blades using composite materials iakovos mastrogiannakis, george-christopher vosniakos national technical university of athens, school of mechanical engineering, section of manufacturing technology, greece abstract. composite materials are increasingly exploited in industry especially replacing metallic structures due to their strength/weight ratio. amongst the notable applications, for which composite materials have not challenged metals yet are hydroturbines, which are overwhelmingly made of steel or copper alloys. replacing blade material by laminate composites can reduce weight and inertia, as well as achieve smaller cross-sectional thicknesses, better fatigue strength, damping, and resistance to cavitation. manufacturing techniques are mature enough to respond to the challenge, provided that the laminate composite blades are properly designed. in the current work, the design of the francis carbon blades was studied by employing finite element analysis. the blades were designed sub-optimally with various stratification patterns and different failure and maximum displacement limitations following a systematic methodology for gradual addition of laminate layers or patches. the methodology is still of a trial and error nature driven by the designer but guesses in the individual steps are much more informed due to model analysis and optimization tools available. key words: composites, francis turbine blade, design, finite element analysis, structural optimization 1. introduction traditionally, hydro turbine runners are made of stainless steel [1]. small runners are also made of manganese brass due to its high strength and abrasion/wear resistance with the addition of fe, sn and mn as alloy elements or small percentage of as or sb for anti-corrosion properties. the blades are often made separately by casting or pressing and assembling with the band and crown afterwards. ni-al bronze is often used to produce large blades. despite a very well established practice of design and manufacture of metallic blades and runners, the use of composite materials is a major challenge for the hydroturbine received june 09, 2019 / accepted january 10, 2020 corresponding author: george-christopher vosniakos national technical university of athens, school of mechanical engineering, section of manufacturing technology, heroon polytechniou 9, 15780 zografou, athens, greece e-mail: vosniak@central.ntua.gr 44 i. mastrogiannakis, g.-c. vosniakos industry. the low specific weight of composite materials reduces inertia; their high strength allows a smaller cross-sectional thickness and their fatigue strength is very good. protecting them from factors such as moisture, impact, and erosion etc. is necessary and can be implemented by special coatings. laminate composites have been used very extensively in airfoil design, especially for wind turbines and propellers, and in hydrofoil design, especially concerning boat propellers and water turbines. in some cases, passively adaptive shapes have been achieved, i.e. foils whose shape changes in a desirable way with load. there is rich experience in structural design of composite laminate foils on which design of hydro-turbine blades can draw, but very rarely has work been reported on hydroturbine blades as such. work has mostly been conducted on marine turbines and propellers as well as on wind turbine blades, specially focusing on exploitation of bend-twist behavior of the blade towards achieving passive adaptivity to external load. an advanced composite pelton wheel was designed and fabricated, and its performance was studied for pico/micro hydro power plant application [2]. the advantages of composite materials, used in marine renewable energy structures, were demonstrated in a 2 m prototype of a c-power underwater turbine [3]. a decrease in thrust and an increase in power capture were achieved by the use of properly designed, passively adaptive bend-twist coupled blades in a horizontal axis tidal turbine [4]. a shape-adaptive composite propeller using bend-twist coupling characteristics of composites was developed [5]. the advantages of flexible composite marine propellers were explored in sub-cavitating and cavitating flows [6]. optimization and experiments of composite marine propellers in changeable pitch were conducted [7]. a systematic design methodology utilized bend-twist coupling effects for performance enhancement of self-twisting composite marine propellers [8]. an efficient theoretical model was developed to obtain a first-order estimation of the static divergence speed of self-twisting composite rotors. the methodology is equally applicable to other structures, such as tidal and wind turbines [9]. a composite marine propeller for a fishing boat was designed and its performance was evaluated [10]. theoretical and experimental exploration of bend-twist coupling and damping properties with relation to the lay-up of composite marine propellers were explored [11]. approaches and evaluation were conducted to predict the performance of wind turbines utilizing passive smart blades [12]. a 10 mw wind turbine blade was designed and analyzed using composite materials [13]. a design methodology of high performance composite bendtwist coupled blades for a horizontal axis tidal turbine was developed [14]. this paper reports on the design of the francis hydroturbine blades using laminate composite materials. based on literature review, composites may bring several advantages to hydrοturbines. the main aim of the paper is to study the material replacement in the francis hydroturbine blades with composites. this replacement has been studied for many turbines but not for the francis type. in addition, adaptable flexible francis blades can be designed to increase the performance of the hydroturbine. section 2 presents the design of a sample francis blade, including loads obtained from cfd analysis, and presentation of candidate materials. section 3 presents the designs of the francis blades under failure and maximum displacement limitations. in each case studied, the respective numerical models are analyzed showing the stacking sequence, the maximum total deformation, the maximum stress, and the failure probability. exploring structural design of the francis hydro-turbine blades using composite materials 45 2. blade analysis analysis was performed in ansystm software. 2.1. fluid flow analysis the blade under consideration was hydrodynamically designed at the laboratory of hydraulic turbomachines the national technical university of athens (ntua) as part of the small francis hydroturbine runner with basic dimensions as follows: diameter 413 mm, height 100 mm, and maximum blade thickness 4 mm. fig. 1 illustrates the francis runner and its components, i.e. the blades, the band, and the crown. fluid flow analysis using the fluenttm solver of the entire hydro turbine runner was carried out under the following flow conditions: rotational velocity w=-157 rad/sec, radial component vr=-6.3 m/sec and local component vu=-32.9 m/sec, corresponding to a flow rate of q=0.022743 m3/sec. fig. 2 shows the distributed pressure on the upper and lower surface of an isolated blade fig. 1 francis runner (left) and its blades (right) fig. 2 distributed fluid pressure (in hbar) on the blade (a) upper surface (b) lower surface 46 i. mastrogiannakis, g.-c. vosniakos 2.2. materials two carbon fiber pre-impregnated epoxy resin systems were used, implementing woven fibers and unidirectional (ud) fibers, respectively (see table 1). their density is 1.42∙10-3 and 1.49∙10-3 gr/mm3 respectively. table 1 230gbτμ epoxy carbon pre-preg material properties elasticity woven ud stress woven ud strain woven ud young’s modulus (mpa) tensile strength (mpa) tensile strain x 61340 1.21e+05 x 805 2231 x 0.012 0.017 y 61340 8600 y 805 29 y 0.012 0.003 z 6900 8600 z 50 29 z 0.012 0.003 poisson’s ratio compressive strength (mpa) compressive strain xy 0.04 0.27 x -509 -1082 x -0.010 -0.011 yz 0.30 0.40 y -509 -100 y -0.010 -0.019 xz 0.30 0.27 z -170 -100 z -0.010 -0.019 shear modulus (mpa) shear strength (mpa) shear strain xy 19500 4700 xy 125 60 xy 0.019 0.012 yz 2700 3100 yz 65 32 yz 0.014 0.011 xz 2700 4700 xz 65 60 xz 0.019 0.012 2.3. geometry and modeling a blade was isolated and its pressure surface selected. in order to create the blade model first a reference surface of zero thickness must be defined and then plies must appropriately be added, thus defining the thickness of the reference surface, hence of the blade. the blade pressure surface edge geometry at the water inlet side was approximated by a bevel shape as shown in fig. 3. for the discretization of the blade model, an element size of 1mm was used. in total, a mesh was formed of 7728 shell elements (10 linear triangular and 7718 linear quadrilateral) and 7929 nodes. fig. 3 blade section at water inlet fig. 4 fixed support of the blade 2.4. boundary conditions fig. 4 shows the fixed support points of the blade edges which are fully constrained. the distributed fluid pressure is added to both surfaces of the blade, see fig. 2. the standard earth gravity and the rotational velocity of the turbine were, obviously, taken into account. exploring structural design of the francis hydro-turbine blades using composite materials 47 2.5. failure criteria inverse reserve factor (irf) refers to the inverse margin to failure as a failure probability measure. load divided by irf yields failure load, i.e. irf>1 means failure. in ansystm irf default threshold is 0.25, for which the failure probability is acceptably low. failure criteria for composite blades refer to laminate plies (e.g. max strain, max stress, tsai-wu, tsai-hill, hoffman, hashin, puck, larc and cuntze). the failure probability caused by the fluid pressure in each ply is calculated based on each criterion because each criterion can cause maximum irf for differing stacking sequence. 2.6. design methodology ansys optimization option through response surface methodology (rsm) was tried, see fig. 5. this proved useful in analyses with single layer as well as with reinforcement layers with few variables, but not in the general case, in which the design methodology illustrated in fig. 6 was followed. note that in the francis blade design, because of loads applied and the curved shape, it was observed that the woven fabrics outperform unidirectional (ud) fabrics. in particular, it appeared that the designs with woven fabrics present better results across the studied range of orientations compared to the ud. the latter exhibited comparable results to the woven fabrics only at the optimum point. consequently, the design with woven fabrics was preferred. fig. 5 rsm project schematic attempted 48 i. mastrogiannakis, g.-c. vosniakos the main stages are analyzed below. fig. 6 design methodology flowchart at the initial ply selection stage the number and orientation of plies are selected according to experience. at the initial ply insertion stage the plies are first placed at random orientations, as follows: (a) in the case of a single layer, this is placed so that the strength design limit is reached; (b) in the case of a single layer with reinforcement patches, the thickness of the single layer is reduced to the next commercially available and at the point or points where the design limit is violated, reinforcement layers are added to reach the strength design limit; (c) in the case of a multi-layer laminate, where the strength design limit is violated, reinforcement layers are added. at the stage of assessment and identification of optimal ply orientations, for each ply separately and all plies collectively, all possible orientations are investigated so that the plies are positioned optimally in relation to the design limit. initially, four alternatives (e.g. 0°, ±45° exploring structural design of the francis hydro-turbine blades using composite materials 49 and 90°) are generally considered; then, the point where the best results are observed is closely investigated. if the initially selected orientation is completely wrong, four opposing orientations are proposed to identify the optimal orientation in a short amount of time. however, in the case of the plies with a similar application area and thickness, a similar optimal orientation is observed. thus, identifying the optimal orientation of one ply can lead to the identification of the optimal orientation of the rest if that orientation is taken as the initial one. at the stage of ply thickness increase, thickness is increased to the next value that is commercially available: (a) in the case of a single layer with reinforcement patches, the thickness of plies is increased, the plies at the design limit violation being given priority. at the same time, minor relocation of patches may be examined, priority being given to a possible reduction in their surface area. a comparison is made between the alternative results and the best is selected. (b) in the case of a multi-layer laminate, the thickness of plies is increased and minor relocation of the reinforcement layers is examined. furthermore, possible reduction of the size of the patches, which initially cover the entire surface of the blade, is investigated. as previously, a comparison is made between the alternative results and the best one is selected. at the ply thickness reduction stage, the thickness of plies is reduced to the next value that is commercially available, following the flow of actions of the previous stage. then, the stage of comparison between the result and the previous result follows and when the best result is reached, an assessment is made to identify the optimal number of plies, this being the final result. 3. results and discussion using the aforesaid design methodology, different analyses were performed in three different cases presented next. 3.1. large allowable displacement for this case, two models were developed. in the first model, the blade was designed with a single layer of woven fabric, as thin as possible (0.65 mm) see fig. 7 (a) for the points with increased failure probability. in the second model, the blade was designed with a single layer of reduced thickness and a reinforcement patch, see fig. 7 (b). for both models, the blades were optimally designed using woven fabrics within the threshold of failure criteria. in the first model, the maximum total deformation is observed in the middle of the water inlet increasing at the center of the blade. the maximum stress is observed at both edges of the water inlet. increased failure probability exists at the water inlet and in the middle of the edge that is bonded to the band. in the second model, the maximum total deformation is observed at the center of the blade, slightly increasing in the middle of the water inlet. the maximum stress is observed at both edges of the water inlet slightly increasing at the center of the blade. an increased failure probability exists on most of the blade’s surface. this shows that the blade is designed at its limits and that there is no excess material. table 2 shows the input and output parameters for both optimized blades. 50 i. mastrogiannakis, g.-c. vosniakos a) b) fig. 7 (a) points with increased failure probability in a 0.65mm thick single layer (b) the position of the reinforcement layer in red table 2 input and output parameters for both optimized blades model 1 2 fabric woven woven single layer b ply thickness [mm] 1.25 0.65 ply orientation [°] 0.7 7.1 reinforcement layer from i along be ply thickness [mm] 0.69 ply orientation [°] 79.2 total deformation max [mm] value 0.150 0.319 position mi mb equiv. stress max [mpa] value 136.25 126.68 position 2ei 2ei irf max value 0.249 0.249 position i, mbe b (i: inlet, mi: middle of inlet, b: entire blade, mb: midpoint of entire blade, be: band edge, mbe: middle of band edge, 2ei: both edges of inlet) 3.2. medium allowable displacement for this analysis, five models were constructed to design the blade with a maximum acceptable displacement of 50 μm within the threshold of failure criteria. fabrics of thickness 0.5 mm, 1 mm, 1.5 mm and 2 mm were used. a displacement constraint serves the purpose of faithful shape retention of the blade for hydrodynamic purposes. in the first and second model, the blades were designed with a single layer using woven and ud fabric, respectively, results being shown in fig. 8. in the third, fourth and fifth models, the blades were designed with a single layer and reinforcement patches, using woven fabrics based on the first model. in particular, in the third model the blade was designed with a single layer of reduced thickness (1.5 mm) and one reinforcement patch, see fig. 9 (a) and (b). in the fourth model, two smaller patches are used, see fig. 9(c). in the fifth model the thickness of the single layer was reduced to 1 mm, fig. 10 (a) illustrating the points of increased displacement and fig. 10 (b) depicting the position of the reinforcement patches in the fifth model. exploring structural design of the francis hydro-turbine blades using composite materials 51 a) b) c) fig. 8 first and second model comparison (a) displacement (b) stress (c) failure risk a) b) c) fig. 9 (a) displacement distribution in a 1.5mm thick single ply (b) reinforcement layers for third model (c) reinforcement layers for fourth model 52 i. mastrogiannakis, g.-c. vosniakos a) b) fig. 10 (a) points with increased displacement in a 1 mm thick single layer (b) the reinforcement layers for the fifth model in red in the first model, the maximum total deformation is observed in the middle of the water inlet and at the center of the blade. the maximum stress is observed at both edges of the water inlet. in the second model, the maximum total deformation is observed at the center of the blade and increased in the middle of the water inlet. the maximum stress is observed at the edge of the water inlet bonded to the band. in the third and fourth models, the maximum total deformation is observed in the middle of the water inlet and at the center of the blade. the maximum stress is observed at the edge of the water inlet bonded to the band and increases at the edge of the water inlet bonded to the crown. in the fifth model, the maximum total deformation extends along the midpoint of almost the entire blade. the maximum stress is observed at the edge of the water inlet bonded to the band and increased at the edge of the water inlet bonded to the crown. all models have a very low failure probability and thus, corresponding positions are not reported. table 3 shows the input and output parameters for all designed blades. table 3 input and output parameters for medium allowable displacement model 1 2 3 4 5 fabric woven ud woven woven woven single layer b pt[mm] 2 2 1.5 1.5 1 po [°] 70 70 65 70 70 reinforcement layer i→ cb pt[mm] 0.5 po [°] 20 i pt[mm] 0.5 1 po [°] 30 0 cb pt[mm] 0.5 1 po [°] 60 40 max deformation [mm] value .047 .048 .049 .049 .048 posit mi,cb cb mi,cb mi,cb cb max equiv stress [mpa] value 59.8 126.0 55.0 68.0 76.0 posit 2ei bei bei bei bei max irf value 0.153 0.185 0.17 0.196 0.18 (pt: ply thickness, po: ply orientation, mi: middle of inlet, b: entire blade, cb: centre of blade, i: inlet, bei: band edge of inlet, 2ei: both edges of inlet) exploring structural design of the francis hydro-turbine blades using composite materials 53 3.3. low allowable displacement considering manufacturing cost a model was constructed to design the blade with a maximum acceptable displacement of 5 μm within the threshold of failure criteria. only fabrics with thickness of 0.5 mm were considered. in addition, the blade has been designed for minimum manufacturing cost. a simplified cost model was used, involving material and labor for n plies, as follows: cost = ac ∗ ai n i=1 + lc ∗ pi 100 + ai 2500 + pi 200 n i=1 (1) fabric costs ac=30.86 €/m2. the labor cost (lc) refers to the time that it takes for the technician to mark and cut the profiles on the fabric sheet and to stack each profile on the mold. after consultation with practitioners, marking and cutting time was calculated by dividing fabric ply perimeter pi (in mm) by 100 while stacking time was obtained by dividing fabric area ai (in mm2) by 2500 and adding it to perimeter pi (in mm) divided by 200. this is performed for n plies. finally, labor cost was assumed at lc=12 € per hour. tooling cost was not taken into account. considering that labor costs depended heavily on perimeter pi of each ply, it was favorable to combine smaller plies in the same layer, which were slightly abstracted from each other, into larger single plies. total cost was calculated at 22.96 € of which 20.44 € is attributed to labor. thus, the blade was designed with multiple plies using woven fabrics. in particular, it consists of 15 layers 0.5mm thick, of which 3 layers extend over the entire surface of the blade (shown in blue color in fig.11), 7 layer patches cover most of the surface of the blade, 2 patches are positioned at the water inlet (shown in yellow color in fig. 11), and 3 patches are positioned at the center of the blade (shown in red color in fig.11, at the middle of the blade and towards the water outlet). a) b) c) fig. 11 (a) points with increased displacement in a laminate of max thickness 6.5mm (b) position of the layers (c) stacking sequence fig. 11 (a) illustrates the points with increased displacement in a 6.5 mm thick laminate. fig. 11(b) illustrates the position of the layers, and fig. 11(c) presents the stacking sequence on the blade. 54 i. mastrogiannakis, g.-c. vosniakos the maximum total deformation extends along the midpoint of almost the entire blade. the maximum stress is observed at both edges of the water inlet. table 4 shows the input and output parameters for the designed blade. table 4 input and output parameters for the blade fabric woven layer eb pt [mm] 0.5 po [°] [703] msb pt [mm] 0.5 po [°] [60/652/602/65/70] i pt [mm] 0.5 po [°] [50/10] cb pt [mm] 0.5 po [°] [70/65/5] max displacement [mm] value 0.0049 position meb max equiv. stress [mpa] value 7.57 position 2ei max irf value 0.025 (pt: ply thickness, po: ply orientation, 2ei: both edges of inlet, i: water inlet, cb: centre of blade, eb: entire blade, msb: most of blade surface, meb: midpoint of entire blade) 4. concluding remarks and further work in the present study, sub-optimal francis carbon fiber turbine blades were designed. the blades were as thin as possible with reduced weight and high strength. a design methodology for laminate hydrofoil optimization was defined. based on this methodology, the design of blades with a single layer as well as multiple layers and reinforcement patches was studied under constraints of failure and maximum displacement. it has been observed that the change in thickness affects the results more than the change of orientation when woven fabrics are used. comparison of a woven with a ud fabric showed that the woven fabric reach better results than ud across the range of orientations, which achieve comparable results only at the optimum orientations. the blade design and optimization methodology was similar either with a failure limitation or a maximum displacement limitation. whenever the thickness of plies was reduced or their orientation changed beyond the optimum point, the values of all three output parameters, i.e. failure probability, equivalent stress, and the maximum displacement increased. at different ply thickness there was a different optimal point of orientation. furthermore, the position selection and the reinforcement layers area were critical for the reduction of the thickness of the single layer and the three output parameters. the design and optimization process was terminated when a further reduction in plies thickness or decrease of their area and any change of their orientation led to inferior results. the findings were similar across all analyses. in particular, the maximum displacement was observed in the middle of the water inlet and at the center of the blade, while the maximum stress was observed at the edges of the water inlet. exploring structural design of the francis hydro-turbine blades using composite materials 55 ultimately, a laminate blade of minimal cost, with woven fabrics 0.5mm thick and stricter maximum allowable displacement was designed, and its manufacturing cost was analyzed. the design and optimization methodology was similar to the previous analyses, as were the effects of the maximum displacement and the maximum stress. it was observed that the main cost is attributed to labor, as opposed to material. thus, it was preferred to increase blade volume in order to reduce total cost. moreover, it was generally observed that blade cost is reduced by using the minimum required number of plies, which however has a negative impact on the accuracy and optimality of the results. as short term future research, fatigue strength should be explored, following an established method [15]. in the long run, an adaptable flexible francis blade with partially fixed support, variable thickness and optimized ply orientation will be explored following the same methodology that was presented in this work. references 1. steele, r.d., 2007, hydraulic turbines, in: e.a. avallone, t. baumeister ams, (eds.), marks’ standard handbook for mechanical engineers. 11th ed. mcgraw-hill, pp. 9154-9166. 2. khabirul islam, a.k.m., bhuyan, s., chowdhury, f., 2013, advanced composite pelton wheel design and study its performance for pico/micro hydro power plant application, int j eng innov technol., 2(11), pp. 126-132. 3. mohan, m., 2008, the advantages of composite material in marine renewable energy structures, marine renewable energy conference (rina), london, pp. 41-57. 4. nicholls-lee, r.f., turnock, sr, boyd, sw., 2013, application of bend-twist coupled blades for horizontal axis tidal turbines, renew energy, 50, pp. 541-550. 5. herath, m.t., gangadhara prusty, b., yeoh, g.h., chowdhury, m., john, n.s., 2013, development of a shape-adaptive composite propeller using bend-twist coupling characteristics of composites, third international symposium on marine propulsors, launceston, tasmania, pp.128-135. 6. young, y.l., 2008, fluid–structure interaction analysis of flexible composite marine propellers, j fluids struct., 24(6), pp.799-818. 7. lin, c.-c., lee, y.-j., hung, c.-s., 2009, optimization and experiment of composite marine propellers, compos struct., 89(2), pp. 206-215. 8. liu, z., young, y.l., 2009, utilization of bend–twist coupling for performance enhancement of composite marine propellers, j fluids struct., 25(6), pp. 1102-1116. 9. liu, z., young, y.l., 2010, static divergence of self-twisting composite rotors, j fluids struct., 26(5), pp. 841-847. 10. hara, y., yamatogi, t., murayama, h., uzawa, k., kageyama, k., 2011, performance evaluation of composite marine propeller for a fishing boat by fluid-structure interaction analysis, 18th international conference on composite materials, jeju island, korea, p. 6. 11. ahmed, a., 2012, theoretical and experimental methods on bend-twist coupling and damping properties with the relationship to lay-up of the composite propeller marine: a review, int j eng sci technol., 4(6), pp. 2907-2917. 12. maheri, a., isikveren, a., 2010, performance prediction of wind turbines utilising passive smart blades: approaches and evaluation, wind energy, 2(3), pp. 255-265. 13. cox, k., echtermeyer, a., 2012, structural design and analysis of a 10 mw wind turbine blade, energy procedia, 24, pp.194-201. 14. nicholls-lee, r., boyd, s., turnock, s., 2009, development of high performance composite bend-twist coupled blades for a horizontal axis tidal turbine, 17th international conference on composite materials, uk, p. 10. 15. ciavarella, m., carbone, g., vinogradov, v., 2018, a critical assessment of kassapoglou’s statistical model for composites fatigue, facta universitatis-series mechanical engineering, 16(2) , pp. 115 – 126. 10396 facta universitatis series: mechanical engineering vol. 22, no 1, 2024, pp. 25 44 https://doi.org/10.22190/fume220106024k © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper influence of crossing wear on rolling contact fatigue damage of frog rail lei kou1, mykola sysyn1, jianxing liu2 1institute of railway systems and public transport, tu-dresden, germany 2school of civil engineering, southwest jiaotong university, chengdu, china orcid ids: lei kou https://orcid.org/0000-0001-5372-6305 mykola sysyn https://orcid.org/0000-0001-6893-0018 jianxing liu https://orcid.org/0000-0002-4779-7761 abstract. the damage of the frog rail significantly affects the wear of the crossing rail and restricts the passing speed of the train. a geometric 3d modeling of the vehicle passing through the crossing center is particularly concerned with the cumulative wheel-rail contact of the traffic volume. the frog rail wear is simulated to obtain the dynamic change of the impact force of the wheel on the frog rail as the rail wears. by summarizing the existing experimental results of other scholars, it is clear that the important factors, that cause the damage of the frog rail, are vehicle load, friction coefficient, slip roll ratio and shear stress. this paper combines the theoretical analysis of mechanics and 3d simulation to obtain the position change of the wheel-rail contact point with the wear of the frog rail, and finally compares it with the actual measurement results. it can more accurately predict the area where the maximum damage occurs after a certain amount of traffic for a certain fixed model, the change of wheel-rail contact point at frog rail is simulated with the wear of each component. through theoretical analysis, the main factors determining frog rail damage were determined. then evaluate the possible damage area of the frog track and control the prediction range to 5-10 cm, which reduces the detection time and cost. the worst state of distraction will be detected in time to facilitate replacement or polishing. through further research in this area, the service life of the frog rail can be predicted. key words: railway crossing, rcf, wheel-rail contact, rail surface defect, frog wear, geometric model received: january 06, 2022 / accepted may 12, 2022 corresponding author: lei kou institute of railway systems and public transport, tu-dresden, 01069, germany e-mail: lei.kou@tu-dresden.de https://orcid.org/0000-0001-5372-6305 https://orcid.org/0000-0001-6893-0018 https://orcid.org/0000-0002-4779-7761 26 l. kou, m. sysyn, j. liu 1. introduction since the birth of the railway, one of the main problems faced by the railway is rail failure. as with all high-speed driving modes, the failure of necessary components can have serious consequences. railway companies around the world have been inspecting their most expensive infrastructure assets. the increase in freight rail traffic and the increase in speed have made railway inspections today more important than ever. railway superstructures need to resist a relatively high load due to the more vehicles running on them [1]. although the focus of the inspection seems to be a well-defined piece of steel, the presence of test variables is important and makes the inspection process challenging. crossing is a general term for the connection and crossing device between railroad rolling stocks that transfer from one track to another. this device is called special track work in railway engineering. the crossing conversion system is an important crossing device for the railway to change the track, which directly affects the safety of the train. the increase failure rate in crossing device will affect operational efficiency. therefore, it is urgent and necessary to monitor the status of crossing device, diagnose and predict failures. the crossing can be divided into three units: crossing (or checkrail), frog (or crossing center) and its connecting part (closure panel). switches and crossings are the essential elements of railway lines, allowing a change in direction, or the transfer of a vehicle from one track to another [2]. in the life of the rail system, the fatigue sensitive positions of the track structure in the crossing area are frog, spring bar and track plate. the frog rail is an important part of the system. its structure is relatively weak, but it has to withstand a huge impact when the train changes direction, and there is a "harmful space" at the frog rail of the crossing, which endangers driving safety. risk management plays a key role in railway projects [3]. therefore, it needs to be inspected and maintained regularly to ensure the safety of train operation. many scholars have conducted research on rail damage detection. traditional rail surface detection uses manual inspection, which is inefficient, and the accuracy of detection results fluctuates widely. in order to improve the detection efficiency and obtain standardized detection results, methods such as geometric measurement [4], ultrasonic testing [5], eddy current testing [6], and rail surface defect detection based on machine vision [7, 8] have been successively developed around the world. so far, the research on frog rail is relatively limited, and the changing cross-sectional shape of the frog rail of the crossing has brought a great challenge to the accuracy of detection. for the detection of the frog rail, it is usually necessary to obtain the data of the entire area. the data can only be completed manually. in actual operation, complete collection is not only dangerous, but also takes a lot of time. determining the representative damage area of the frog rail and reducing it to a certain range can reduce the detection time and cost, and the accuracy of the preliminary data can improve the accuracy of computer vision detection. only by understanding the changes in the mechanical characteristics between worn wheels and rails and determining the location of the largest damaged area, we can accurately optimize the geometry of the crossing, improve the optimization efficiency, and improve the speed and safety of the train passing the crossing. the further research on the damage generation process is of great significance. in the crossing area, because the rolling direction of the entering wheel is different from the guiding direction of the frog rail, the decay speed of the longitudinal wheel-rail force is greater than the decay speed of the vertical wheel-rail force. the geometric incompatibility of the crossing during the collision will aggravate the tangential wheel-rail collision, and influence of crossing wear on rolling contact fatigue damage of frog rail 27 the slip area in the wheel-rail contact surface will be significantly enlarged at this time [9]. the articles [10-14] studied wheel-rail contact through geometric modeling, mechanical model analysis of stress distribution, and actual measurement of wheel-rail vibration, and studied the optimal design of the geometry or loading of the crossing in its core rail. the article [15] collects the profile of wheel tread change during the entire life cycle of the wheel and establishes the dynamic model when the wheel passes through the fixed frog rail, which proves that the wheel wear is beneficial to alleviate the damage of the frog rail. the article [16] carried out a numerical simulation of the contact stress between the core rail and the wheel and determined the most important reason for the rapid wear of the frog rail. mykola sysyn et al.'s research on the crack generation process of the frog rail of the crossing gives a lot of inspiration [17] and the statistical analysis of the three-axis acceleration of the frog rail due to the effect of the rail when the train passes the core rail [18]. the study found that the maximum damage area of the frog rail is not necessarily related to the impact distribution of the frog rail. machine learning shows that the damage of the frog rail is only related to a small part of the high-impact with wheel-rail effects. in this article, a three-dimensional geometric model is first established to simulate the evolution of the frog rail's contact with the wheel as the amount of wear increases, and to determine the important area of wheel-rail contact. afterwards, through analysis and summary, the main factors affecting the wear of frog rail are found, and the most important influencing factors are obtained through theoretical comparison. then we explore the location of important influencing factors, determine the area where the largest damage occurs in the life cycle of the frog rail, and finally compare the results with the actual collected location data of the frog rail damage to obtain the theoretical and actual double inspection results. 2. geometric model 2.1. basis data due to the presence of variable cross-section rails at the frog rail, it is difficult to establish a three-dimensional model. the irregularity of the cross-section shape change of the frog rail is one of the important features of the frog rail model. this feature makes the wheel-rail contact characteristics of the crossing area comparable to the general line. there is a fundamental difference. the crossing modeled here is a crossing with an intersection angle of 1:12. we use the unworn wheel profile first. in the paper [19], l. xin et al. showed a universally applicable cross-section data of frog rail to construct a frog track geometric model. the cross nose in the model is constructed using four main cross sections, which are defined by drawings provided by the manufacturer (fig. 1). the distances between the four cross-sections in the figure are 10a, 10a, and 50a, and the unit here is uniformly millimeters. in addition "a" is equal to the intersection angle of the crossing (the crossing in this article is 1:12, so a = 12). the geometric parameters of the wheel tread in fig. 2 are from the paper [20] and are the external dimensions of the lm tread in china, as shown on the left in figure 2. its characteristics: the width of the wheel flange is 32 mm and the height is 27 mm. the inner side of the rim has a guide angle to guide the wheel through the guardrail smoothly. the rail profile is based on the 60 kg/m rail cross section data in accordance with the chinese standard gb2585-2007. 28 l. kou, m. sysyn, j. liu fig. 1 section drawing from the manufacturer fig. 2 wheel tread and rail section 2.2. model building the main method to deal with the characteristics of variable section of rail is to calculate the whole variable section of rail by interpolation. four control sections of frog track with variable cross section of crossing are given in the standard drawing. according to these standard sections, the profile of any section can be obtained by linear interpolation. the wheel and wing rail only need to extend in their direction. here, because only the outline of the rail head is needed to participate in the calculation, the basic rail section in fig. 2 is selected for the wing rail, and the distance between it and the cross section of the heart rail remains 44 mm with the extension of the direction. matlab interpolation calculation can be obtained along the direction of the train every 0.5 mm interpolation and horizontal every 0.05 mm geometric interpolation. thus, a complete geometric model of the wheel passing through the center rail is obtained (fig. 3). influence of crossing wear on rolling contact fatigue damage of frog rail 29 fig. 3 geometric 3d model among the existing studies, the research direction of the paper [20, 21] is mainly focused on the influence of the wear of the frog rail on the interaction relationship between the wheel and crossing rail. it is concluded that the inner rail of the transition range of wheel load is impacted by wheel load due to the harmful space of the fixed frog, and the frog rail's top width of 20 ~ 40 mm is the area with larger vertical wear. the wheel-load transition area is very short, and the wheel-rail impact square is used to complete the wheel-load transition. as the vertical grinding of frog rail increases, the collision point between wheel and frog rail is far away from the theoretical tip, and the contact force fluctuation increases before and after the wheel load transition, the contact force peak increases, and the wheel load transition becomes unstable. in terms of [15] and [22], the research direction is mainly focused on the influence of wheel wear on the interaction relationship between a wheel and crossing rail. it was determined that rcf had the highest risk in the crown of frog rail and that contour wear reduced the dynamic response. the above studies have determined that rail wear at different levels and wheel wear have a significant impact on the overturning wheel-rail contact state because they affect stress and strain states. in this study, it is necessary to consider the wheel-rail contact state under the combination of frog rail wear, wheel wear and wing rail wear. as shown in fig. 4 (a), contour calculation of three wear stages was carried out for the three components according to the wear state in the. fig. 4 (b) shows how to judge the contact point method. if d1 is greater than d2, the wheel is in contact with the wing rail; otherwise, it is in contact with the frog rail. we can set the threshold for simultaneous contact. the contact point between wheel and rail can be obtained according to the method on the right side of fig. 4(c). since the surface damage of frog rail is studied in this paper, the points that contact with the side of frog rail at the same time are not specifically calculated. 30 l. kou, m. sysyn, j. liu fig. 4 wear simulation of three components and model 3. analysis of frog damage the model significantly shows the contact area when the wheel passes the crossing. taking the 1:12 crossing as an example, the interval of 180 mm 450 mm belongs to the transition area, in which a section of wheels can act on the frog rail and wing rail at the same time. when the load increases and the wheel-rail contact stress exceeds the yield limit of the bifurcated material, the material will undergo plastic deformation. under the action of repeated loading, the plastic deformation will accumulate and increase, forming microscopic cracks on the surface and sub-surface of the material. under the action of larger normal and tangential stresses, the microscopic cracks will expand and form fish-scale cracks distributed on the surface of the rail, that is, the phenomenon of "cracking". if the surface of frog rail cracks is not dealt with in time, the surface cracks will spread to the frog rail body along the direction of movement, and then large pieces of peeling. rail stripping is mainly caused by the action of surface friction. with the increase of friction, the peeling phenomenon of material surface increases and the peeling chip block increases. therefore, this area is also the most prone to damage of frog rail, and the wear coefficient will be affected by load, tangential force, sliding speed and initial surface state. in the process of changing from running-in wear stage to stable wear stage, the wear coefficient of texture surface and grinding polished surface gradually approach. in order to determine the damaged area more accurately, theoretical analysis of the interaction through the frog rail is essential. 3.1. the dynamic interaction in the common crossing the dynamic interaction in the common crossing results from the constructive and wear geometrical properties. wheel tread is not horizontal, but inclined. when the wheel travels through the crossing, the geometric locations of the contact points (between wheel and wing rail or between wheel and crossing nose) form a line [23]. this line is the wheel trajectory during the traveling and called constructive stimulation. the deviation of the wing rail creates a relative displacement between the wheel and the wing rail in the direction from wings rail to frog. then there is a vertical wheel sink △z [24]. because of the surface inclination of the frog tip on fig. 2, the wheel set rises after the touchdown point. the trends of rise and descent are reserved in the stump-travel. the length caused by influence of crossing wear on rolling contact fatigue damage of frog rail 31 the increase and decrease is called the wavelength λ. the depth of the wheel descent is shown as amplitude of stimulation zs [25]. according to fig. 5, it is determined that the geometrically exact stimulation is asymmetrical. the steeper rise with stump travel means that the load on the crossing is higher than with tip travel. since both directions of travel occur, a reduction of the considerations on stump-travel is on the safe side. then we design the wheel trajectory is simplified into a symmetrical shape, as shown in fig. 5. fig. 5 the contact process analysis of the vibration system wheel-rail the stimulation is described by two curves (fig. 6), which show a clear transition at the connection point in the pass-through area. when the wheel passes through this area, one should analyze stimulation models. on the one hand, a wave-shaped stimulation in contact area with the wavelength λ and the wave depth zs (soft wave stimulation) is generated on the fig. 6 contact process analysis 32 l. kou, m. sysyn, j. liu other hand there is an impact by the transition (hard impact stimulation) with an angle α. where the impact stimulation happened is the touchdown point and the angle α is touchdown angle [26]. soft wave and hard impact are the two macroscopic effects of wheel-rail contact on the frog rail area of the crossing, and they are important reasons for the wear of the frog rail. 3.2. factors of rail failure steel under the action of alternating stress, the stress is far lower than the tensile strength of static load, even when the yield strength of static load is lower than the sudden failure, this failure is called fatigue failure. the damage to rail caused by wheel rolling over rail is also fatigue damage. analysis of all rolling contact processes shows that the front of the contact point between the wheel and track is in a state of compressive stress and its rear and secondary surfaces are in a state of tensile stress. the tangential plastic deformation of rail is caused by wheel friction, and the deformation accumulates gradually with the increase of cyclic load. as the deformation of the soft surface increases, the cracks begin to nucleate below the surface, and further cyclic loading promotes the formation of cracks parallel to the surface. when the cracks finally extend to the surface, the thin wear sheets peel off and form flake debris, which eventually forms the depressed damage zone. 3.2.1. factors of previous studies the process of contact fatigue is very complex with many influencing factors, and it is difficult to make a decision by simple analysis. the factors affecting wheel/rail surface fatigue can be summarized as follows: 1. with the increase of normal load, the wear amount of wheel and rail and the friction factors in the rolling friction stability stage both increase, and the massive spalling and cracks on the surface also increase with the increase of normal load [27]. 2. main damage types of rail although their failure mechanisms are different, the main influencing factor is the surface friction between wheel and rail. with the increase of surface friction, all kinds of rail damage will be aggravated. therefore, in the section with high surface friction, such as curve, ramp line, rail damage is aggravated [28]. 3. the crack initiation point is limited to a narrow area, with a typical depth of 0.3mm, which is consistent with the position of the maximum shear stress. the strength criterion of contact fatigue is also expressed as the maximum shear stress criterion [29]. 4. sliding may be an important factor to increase fatigue damage. uneven sliding distribution in the hertz contact zone also contributes to increased stress development, which significantly shortens the expected fatigue life of steel [30]. 5. as the slide-roll ratio increases, the surface damage of wheel-rail materials gradually develops into fatigue wear, accompanied by oxidation wear and abrasive wear. the degree of abrasive wear also increases with the increase of slide-roll ratio (damage behavior of wheel-rail materials under different slide-roll ratios) [31]. so many parameters have an important influence on the surface damage of rail, and they have a complex interaction with each other. in the case that the steel and wheel materials are fixed, but the surface state is always changing, the friction factor is always changing. moreover, the research of article [31] also shows that: the initial value of friction coefficient is about 0.1, 0.2 and 0.3 with the increase of roll ratio under different roll ratio. moreover, the friction factor of rolling friction is much smaller than that of influence of crossing wear on rolling contact fatigue damage of frog rail 33 sliding friction. when wheel-rail relative sliding friction occurs, the high temperature caused by high friction will also change the hardness of materials and crystal structure to change the wear resistance of metal. the hardness of metal usually decreases with the increase of temperature, so sliding friction plays a more decisive role in rail damage than friction. 3.2.2. factors of theoretical analysis according to hertz wheel-track contact theory, contact force is elliptic. within the range of contact spots obtained according to normal clearance, it is assumed that the contact spots are about the face scale formed between axle and main contour line (the influence of shaking head angle on contact spot shape is temporarily ignored) [32]. when the normal force n is known, the normal phase contact stress p0 on the contact spot is: 1 2 0 2 2 2 2 0 2 2 2 20 1 2 1 1 1 1 a b a a a b f f n p y x y x dxdy dxdy b a b a− − = − − + − −    (1) the calculation of wheel-rail stress is complicated, but only the normal stress, shear stress and other factors are analyzed here, and the contact surface is simply approximated as a circular spot instead of an ellipse. so: 0 2 n p a = (2) when the contact area reaches the minimum and the load reaches the maximum, p0 reaches the maximum value pmax. as an important parameter, the shear stress needs further analysis. research in [33, 34] shows that the limit value of stability is greatly influenced by tangential force. therefore, under the same normal load, with the increase of tangential force, the probability of plastic deformation of rail will increase rapidly. therefore, in some sections of the line with high tangential force, such as the braking or starting section and the curve section, the rail's ability to resist plastic deformation will decrease rapidly, and the rail's collapsibility will increase rapidly. the friction resistance under free rolling is small, and the friction coefficient at this time is only about one-tenth of the friction coefficient under controlled rolling when the braking distance is applied. shear stress is the main cause of the permanent fatigue crack. with the increase of friction, the shear stress becomes larger and larger, and the peak shear stress is closer and closer to the surface. at this time, the micro crack expands in the deformation layer and the surface layer, and extends to the surface of the core rail, accompanied by the plastic deformation caused by the pressure exceeding the yield stress of the core rail. under the action of repeated loads, the plastic deformation will accumulate and increase, and the cracks will continue to expand and form fish scale cracks distributed on the surface of the rail. if the rail surface crack is not dealt with in time, the surface crack will expand to the rail body along the direction of movement, and then the large strip. when the surface friction force increases to a certain value, the surface wear type changes, the surface is consistent with the direction of motion, there are obvious scratches, and there are also large flake flakes of wear debris, indicating that in addition to abrasive wear, there are obvious fatigue wear and adhesion wear. rail stripping is mainly caused by the action of surface friction. with the increase of friction, the peeling phenomenon of material surface increases and the peeling chip block increases. 34 l. kou, m. sysyn, j. liu shear stress is calculated according to smith's contact stress theory as follows [35] : 2 2 2 2max 2 1 2( 2 2 ) 2 3xz p z z z a x z xz a a           = − + + + − −      (3) where α is the half width of the contact surface; µ is the friction coefficient of the contact surface; ψ1, ψ2 is the derived coefficient, and its expression is as follows: 2 1 1 2 1 2 1 2 1 1 2 1 1 / / 2 / ( 4 ) / k k k k k k k k k a k   + =   + + − (4) 1 2 2 2 2 1 2 1 2 1 2 2 1 / / 2 / ( 4 ) / k k k k k k k k k a k   + =   + + − (5) 2 2 2 2 1 2( ) ; ( )k a x z k a x z= + + = − + (6) because the contact area size here is also very small, the calculation is simplified, then: x = 0, k1 = k2, ψ2 = 0. by substituting the above formula into (3), the following formula can be obtained. ( )2 2 max 2 2 2 2xz z p a z a a z z        = −  + −  +    (7) according to the above formula, it is obvious that the forward pressure n and the contact area play a decisive role in the forward stress and tangential stress. the shear stress reaches its maximum when it is a certain depth away from the surface layer. article [33] shows that this value is about 0.03 mm. it can be concluded that the positive pressure and the change of contact area play a decisive role in rail damage under fixed rolling conditions. 3.2.3. lateral shift of wheel literature [36-39] comprehensively carried out a series of studies on wheel-rail rolling contact fatigue problems, popularized the results of kalker's latest theory and developed kalker's theory. the studies also show that the larger the transverse amount of rail will produce a larger shear stress and normal stress. because of the wheel set left and right lateral movement of the rail produced when sliding friction contact [38]. sliding contact method to contact force of wheel and rail and wheel under the condition of dynamic load and dynamic load coefficient is greater than the value of the rolling contact condition. under the condition of sliding contact spot near the thermal stress is greater than the value of the rolling contact condition under the condition of sliding contact rail wear volume and the surface plastic strain is greater than the value of the rolling contact condition. contact fatigue crack is the main damage of rail under wheel/rail rolling contact. abrasion is the main damage of rail under sliding contact. the maximum values of wheel-rail normal contact force and wheel dynamic load increase with the increase of axle load and sliding speed. the friction coefficient of rail increases with the increase of axle weight, sliding speed and wheel vibration frequency. research [41] shows that when the wheelset traverse is -8~0 mm, the maximum contact stress increases first and then decreases due to the continuous fitting of the wheel rail tread, but the range of change is not large, and influence of crossing wear on rolling contact fatigue damage of frog rail 35 the contact stress varies between 1500~2500 mpa. however, when the rate of traverse changes from 0 mm to 8 mm, the maximum contact stress firstly decreases and then increases sharply. this is because when the value of traverse changes from 0 mm to 4 mm, the wheel-rail profile is more matched and the contact spot area is larger. when the value of traverse continues to increase, the contact area decreases sharply due to the angular contact between wheel and rail gauge, resulting in a sharp increase in contact stress. although the outline of frog rail is not the same as that of steel rail, it can be predicted that the increase of lateral movement will inevitably lead to an instantaneous increase in stress. because the result of computer simulation is the contact area of wheel-rail decreases instantly when there is a large amount of traverse, the stress of frog rail will increase even if the same force is applied. at this time the frog rail bears all the pressure from the wheel. therefore, in this paper, the main parameters of damage of frog rail surface are soft wave, hard impact α, load, lateral displacement. the load is measured mainly from the vertical pressure, while the lateral displacement is mainly expressed as the change of the lateral wheel-rail force. the increase of lateral movement will obviously lead to the increase of roll ratio. 3.3. major factors of frog rail failure there are two special temporal and spatial nodes in the process of the wheel passing through the frog rail. when the wheel just touches the frog rail, it is prone to more and greater impact because of the direct collision between the wheel and the center rail in the forward direction. the other node directly acts on the frog rail when the wheel leaves the side rail completely, and the frog rail bears the entire load at this time. at the stage, before and after the wheel just contacts the frog rail, the wing rail contacts the wheel at the same time, and both sides jointly bear the load from the vehicle (fig. 7). in the actual situation, due to the change of the wear conditions of the various components in the crossing area and the different lateral relative positions of the wheels entering the crossing, the two situations mentioned above may occur simultaneously within a small length space of the frog rail. fig. 7 the main study area of frog experimental measurements by guo et al. [42] show that the surface hardness of frog rail and the depth of hardened layer are the highest in all seven positions of the crossing railhead segment with the cross-section surface width of 20 mm and 50 mm. this region is about 200 36 l. kou, m. sysyn, j. liu 450 mm from the tip of frog rail, which is consistent with the simulation results in this paper. the wheel-rail contact stress with axle load of 21 t can reach more than 1230 mpa when it runs normally on the smooth track. many of today's trains exceed that. therefore, the stress of wheel-rail contact in the frog rail area must exceed the ultimate strength of the frog rail. as long as the wheel passes through the rail, it will inevitably cause damage to the rail. when the wheel travels from the tip of frog rail to the bottom end with a larger cross-section, the impact and load are obviously greater than the impact when the wheel travels from the bottom end to the tip. therefore, this paper chooses the former as a case in the analysis. generally, the area where the frog track and wheelset interact is transition zone. 3.3.1. impact force in the experiment of damage detection on frog rail, conducted in 2019 [18], the distribution of the impact load borne by the frog rail in the longitudinal area of the frog rail is obtained. the research and analysis showed that the rail surface is subjected to higher frequency impact when the wheel entered the rail tip. as shown in fig. 8, in the 1:12 crossing, 70 % of the shocks are concentrated in the 220 320 mm area, of which the transverse and longitudinal shocks are mainly concentrated in 50 g or less shocks. these are common in normal sections, while the vertical shocks are even as high as 300 g and most of the shocks are concentrated in 130 240 g. the study estimates the influence of one impact with an acceleration of more than 250 g to be equivalent to more than one thousand impacts of 50 g. relevant results have been shown in detail in paper [18]. this region is the main impact region, which bears most of the wheel-rail impact during the whole life process. the wing rails in this area bear part of the vehicle's pressure, so if the first damage occurs in this area, it is due to the high frequency of vertical impact load. therefore, it is necessary to understand the action mechanism of impact load. shi imposes a heavy impact load on hard aluminum alloy, to observe the damage behavior of the hard aluminum alloy. the test results show that the heat generated by the heavy impact passes around, making the bulge deformation. the impact on the part of the subtle grain broken flower, increased with the increase of load and the number of times, increasing the impact will micro cracks appear and expand [43]. in their paper [44], hu et al. conducted an experimental study on the wear characteristics of high-hardness steel dies under shock loading conditions. under the action of impact stress and frictional stress, the grinding surface has micro-roughness, interlocking, and continuous slight sliding friction, which makes the surface layer highly, localized damage. as the number of impacts increases, the damage accumulates and microscopic plastic deformation occurs on the local surface. the stress concentration on the grain boundary or the phase interface leads to the generation of friction cracks, which continue to expand, and eventually lead to friction cracks and damage to the surface of the steel mold. however, in the crossing area, the relative position of wheel and frog rail is normal distribution random position, and the change of speed and vibration will affect the impact position of load on frog rail. therefore, each impact load of wheel on frog rail acts on a certain area but distributes at different drop points. the experiment of impact load on a variety of hard metals can be concluded that impact load will cause the deformation of frog rail surface, and there will be large wear and crack generation under long term impact action. a predictable conclusion can be drawn from the distribution points, experimental results that deformation, and crack are most likely to occur on the surface. instantaneous impact and hard impact alpha are mainly caused by wear and cracks rather than the direct cause of peeling damage. influence of crossing wear on rolling contact fatigue damage of frog rail 37 fig. 8 the distributions of the longitudinal coordinates of the impact and of the corresponding maximal vertical accelerations for the frog rail of a common crossing [18] 3.3.2. load and lateral movement first, as the falling depth zs of the soft wave increases, the load acting on the frog rail will also increase. this is an obvious factor. another position that may cause greater wear is in the section where the wheel leaves the wing rail and fully acts on the frog rail, considering the normal wheel width is 135 mm to 140 mm, sd is 32.5 mm. the wheel has a certain lateral movement range, the maximum is 11.5 mm the maximum probability according to the normal distribution law of the experimental statistics is 4 6 mm. the distance between the wing rail and the frog rail is a discrete point with normal distribution. when the wheel completely leaves the wing rail and falls on the frog rail, the surface of the wing rail has a drop of 1-2mm compared with the frog rail. at this time, the train is still moving forward. after the wheel completely leaves the wing rail and advances a short distance, it falls within the range of about 330 430 mm on the surface of the frog rail (as shown in fig. 9). now before the wheel completely falls on the frog rail, the shape of the outer part of the wheel and the outer part of the wing rail as shown in the figure leads to the inward extrusion of the rail on the wheel, resulting in a large amount of transverse movement of the wheel at this time. when the wheel falls on the frog rail and moves inward, it will hit the guardrail, and the collision with the guardrail will make the wheel move outward. therefore, within that range the wheels are going to be moving sideways a lot. the wheel and the frog track slide in the lateral direction. ren and wan study in [45, 46] have confirmed the above conclusion. the vehicle-rail space model and coupling with vibration constructed by the team analyzes the vibration characteristics of the vehicle through the 38 l. kou, m. sysyn, j. liu crossing system. the results show that the lateral acceleration response of the train will be larger. when the wheels enter the crossing and leave the wing rail, but the latter is especially obvious at 1.83 m/s2. the specific action process is that the wheelset moves toward the inner rail due to lateral extrusion when it breaks away from the side rail, but under the action of the guide curve, it quickly moves toward the outer gauge and reaches the maximum lateral movement. according to the team's simulation data, it can be moved inwards by as much as 6 mm in an instant and outwards by as much as 11 mm again. in order to verify this view, the research team also measured the lateral acceleration of frog rail along the traveling direction, which was consistent with the theoretical results. relevant research has been described in paper [18]. fig. 9 distance of leaving the wing rail area in this area, a portion of the wheel starts to be fixed in a small area where impact and wheel traverse occur. the generation and development of wheel-rail rolling contact fatigue depend on the normal force and tangential (creep) force on the contact spot. the sliding friction and lateral force generated by transverse movement will inevitably lead to the decrease of the wheel-rail contact area and the increase of the normal and tangential force, reaching the maximum value. therefore, this paper considers that the probability of surface stripping in the departure area of the frog rail is far greater than that in other areas, and surface stripping will aggravate the wear of the frog rail and cause more serious damage. therefore, the region where sufficient lateral displacement occurs in the fixed frog rail is the largest damage area. at the same time, it needs to be simulated to check whether the depth of the soft wave in this area will also increase 4. results according to cast high manganese steel frog minor damaged criteria, at the cross section of frog rail width 40 mm, the frog rail wears vertically (excluding the heightened part of wing rail); rails of 50 kg/m and below: exceed than 4 mm on the main line, exceed than 6 mm on the branch line, and exceed than 8 mm on other lines; rails of 60 kg/m and above: exceed 4 mm on the main line with an allowable speed greater than 120 km/h, exceed 6 mm on other main lines, exceed 8 mm on branch lines, and exceed 10 mm on other lines. influence of crossing wear on rolling contact fatigue damage of frog rail 39 in this paper, several models are used to simulate the evolution process of contact points when the vertical wear of the frog rail is from 0 to 5 mm. the position between the touchdown point and the tip of the frog rail when the wheel touches the frog rail for the first time and when the wheel leaves the wing rail is recorded respectively. in order to be able to determine the touchdown point, two further parameters (wave depth zs and touchdown angle α) must be calculated. first, the wheel trajectory should be preserved, because the wheel is not a regular plane, but an area with a slope and curvature. therefore, the size of the wheel trajectory cannot be determined based on the value of the contact points. the line connecting the minimum distance of each section is actually the wheel trajectory on the plane formed by the xand zaxes, but its shape varies, since the wear of the frog changes the geometry during the service life. the stimulation soft wave depth zs and the contact angle α were calculated from the wheel trajectory as fig. 10 presents them. here only the range from x = 0 mm to x = 500 mm is selected for the calculation. the wave depth could be clearly identified, but the angle of contact α can be calculated. fig. 10 change of impact angle and wave depth as the accumulated traffic mass increases, the angle of touchdown becomes smaller, but the depth of the shaft increases. we know that the touch down angle α stands for hard impact and the wave depth zs for soft impact. by changing these two parameters, one can speculate with the increase in the accumulated traffic mass about the changes in the loads on the crossinges. the impact angle and touch down point of the frog appear within a certain range, and the wave depth keeps increasing. for a frog, the xtp of the touch point does not always increase, but only fluctuates within an area. the wheel trajectory is also significantly different due to wear. the reason is that the wear of the wing rail is not considered in advance. if we only use geometry data that does not consider wing rail wear, the frog tip will continue to wear over time. therefore, a geometric model of the control variable is required. 40 l. kou, m. sysyn, j. liu then simulated (in fig. 11) the change of position of touchdown point respectively in abrasion of the wheel at 0.5 mm and 1 mm and also with the frog rail abrasion. fig. 11 (a) shows where the wheel begins to contact the frog rail. fig. 11 (b) shows the position where the wheel is completely disengaged from the wing rail. with the frog rail wearing, the positions of the touchdown point points are significantly away from the frog rail tip. when the vertical wear is reached the slight injury standard, the distance has exceeded 330 mm. in this range, the track gauge wheel leaves the wing rail in advance. a certain number of wheels have had multiple impacts on a small range. with the increase in frog rail wear, this situation will become more serious. the wheel wear also has a significant impact on the movement of the touchdown point. with the increase in wheel wear, the touchdown point also moves back significantly. in order to understand the influence of wing rail wear on contact point change, the wing rail wear of 0.5 mm and 1 mm is simulated (fig. 12). although the wear of the wing rail prevents the backward movement of the touchdown point, the improvement effect is not obvious. fig. 11 changes in contact points after wheel wear fig.12 changes in contact points after wing rail wear the research results obtained in article [47, 48] can prove that when the contact point moves backward, the frog rail is subjected to greater and greater impact load. in the end of frog influence of crossing wear on rolling contact fatigue damage of frog rail 41 rail, the frog rail is worn with the increase of impact, and then the impact point moves backward. when the backward moving area reaches 350 mm, the vertical impact force increases. some wheels have been in the stage of only acting on frog rail in this area, constantly influencing in a fixed area and generating huge load. this situation increases with the backward movement of the contact point, the impact increases, and the transverse movement will increase, causing peeling and other damage to the frog rail surface. because the impact area is relatively fixed, the small spalling expands to the damage of complete damage to the frog rail. when the wheel reaches the area away from the wing rail, even if the wear changes, some wheels with a large distance from the frog rail will not move back. with the increase of wear, the contact point moves back, and the effect of more wheels on the frog rail surface is concentrated in a certain area. as the wave depth increases, the frog rail load increases accordingly. stronger impact and greater lateral movement are concentrated in this area. moreover, it will not move due to the increase of wear. repeated action, even without too high frequency, will quickly lead to the peeling and damage of frog rail surface based on existing wear. in order to verify the correctness of the analysis, jianxing liu collected 25 frog rail surface damage images in different wear periods in china, sorted them into data, and measured and counted the location of maximum surface damage of frog rail, as shown in fig. 13. fig. 13 location of maximum surface damage of frog rail although the maximum damage area of simulation and measurement results is concentrated in the 6cm area of 350 410 mm, the sampling and simulation data are very limited. as the allowable error, we define this range as 10cm, i.e. 330 430 mm. for computer vision inspection, only one sampling is needed to obtain the photos that can see the micro cracks best. the above measured results verify the accuracy of this study. although the 230 330 mm area considered by the traditional view will also produce surface damage in the early stage, it has been proved that the maximum damage position occurs after the wheel leaves the wing rail and moves laterally. with the wear of frog rail and wheel, the location of the maximum damage is concentrated in the range of 350 mm to 420 mm. because the geometric model cannot consider the vehicle body shaking and tilting and the lack of corresponding mechanical feedback data, there must be some errors. however, the results of data feedback from real measurement are completely consistent with the results of theoretical analysis. this study shows that properly reducing the height of wing rail can slow down the backward movement of contact point, reduce the impact of the wheel on the center rail so as to prolong the service life of crossing. 42 l. kou, m. sysyn, j. liu 5. conclusions the surface damage detection of frog rail is always a complicated problem, and the main cause of its damage formation has not been determined yet. in this paper, the contact process of wheel passing frog rail is calculated by establishing geometric model. the changes of contact points with increasing wear of wheel, wing rail and frog rail were simulated and the main contact area was established. through literature research and summary, the main parameters of rail damage are determined. through further theoretical analysis, we find the main areas where the maximum damage may occur with the increase of service life of frog rail surface. the following conclusions can be drawn: 1. the forces acting on the frog are mainly caused by two factors: wave excitation and impact excitation. the resulting wave excitation and impact excitation creates loads when the wheel passes the point facing or vice versa.the train speed and wave depth affect the value of the soft impact. the size of the hard impact is affected by the speed and the angle of touchdown. 2. the larger the area of the touchdown distribution, the longer the service life of the common crossing. the position of the touchdown point will change by changing the position of the wheel set in the track gauge. the elastic vertical deformation of the frog also influences the position of the touchdown point. as can be seen from the trends of the parameters, the soft impact grows with raising of the accumulated traffic mass, on the contrary, the hard impact becomes smaller. 3. vehicle load, friction coefficient, slip roll ratio and shear stress are the main factors affecting the service life of the track. their increase accelerates rail fatigue damage. 4. when the material is fixed, the load and wheel lateral movement are the most important factors causing the damage of frog rail surface. lateral movement mainly reduces the contact area and sharply increases the tangential stress. 5. wheel wear and vertical wear of frog rail will lead to the acceleration of backward movement of contact point, the increase of wheel rail force and the damage of frog rail surface. the vertical wear of wing rail is beneficial to prolong the service life of frog rail. 6. in the area where the wheel geometry cannot act on the frog rail and wing rail at the same time, with the increase of wear, this area will produce concentrated wheel rail contact, causing rapid damage to the frog rail surface. 7. the maximum damage area of frog track is determined at the 1:12 fixed frog crossing 330 mm to 430 mm from the vertex, and the traditional 200 500 mm area is reduced by 60 %. the time loss and cost of testing are greatly reduced. at the same time, the method has universality and can be extended to any type of crossing damage research. of course, this paper also has some limitations, crossing wear simulation and reality are not completely consistent, for the exact force is not fully understood, and these problems will be further completed in the follow-up study. the research results in this paper have good practical application value and lay a foundation for the future research on frog tunnel surface damage identification and damage development prediction. influence of crossing wear on rolling contact fatigue damage of frog rail 43 references 1. szalai, s., eller, b., juhász, e., movahedi, m.r., németh, 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belgrade, serbia 4college of engineering, yuan ze university, taiwan 5faculty of transport and traffic engineering, university of belgrade, belgrade, serbia 6faculty of informatics engineering, idlib university, idlib, syria abstract. the quadratic assignment problem (qap) is an np-hard problem with a wide range of applications in many real-world applications. this study introduces a discrete rat swarm optimizer (drso)algorithm for the first time as a solution to the qap and demonstrates its effectiveness in terms of solution quality and computational efficiency. to address the combinatorial nature of the qap, a mapping strategy is introduced to convert real values into discrete values, and mathematical operators are redefined to make then suitable for combinatorial problems. additionally, a solution quality improvement strategy based on local search heuristics such as 2-opt and 3-opt is proposed. simulations with test instances from the qaplib test library validate the effectiveness of the drso algorithm, and statistical analysis using the wilcoxon parametric test confirms its performance. comparative analysis with other algorithms demonstrates the superior performance of drso in terms of solution quality, convergence speed, and deviation from the best-known values, making it a promising approach for solving the qap. key words: discrete rat swarm optimizer, quadratic assignment problem, combinatorial optimization, swarm intelligence received: june 02, 2023 / accepted august 10, 2023 corresponding author: toufik mzili department of computer science, faculty of science, chouaib doukkali university, avenue jabran khalil jabran, b.p 299-24000, el jadida,morocco e-mail: mzili.t@ucd.ac.ma 530 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi 1. introduction the quadratic assignment problem (qap) is a typical combinatorial optimization problem and also an np-hard problem. since 1957, when koopmans and beckmann [1] first presented the quadratic assignment problem as a combinatorial optimization problem, it has attracted much attention and research from researchers in mathematics, computer science, and many other applications. on the one hand, quadratic assignment problems are widely used in practice, and many real-world problems can be formalized as quadratic assignment problems, such as integrated circuit wiring [2-3], factory location layout [4], typewriter keyboard design, task scheduling [5-6], etc. on the other hand, some classical np-hard combinatorial optimization problems, such as the traveling salesman problem, the triangulation problem, and the max clique problem, can also be transformed into quadratic assignment problems [7-11]. therefore, it is of great theoretical and practical importance to find an efficient algorithm to solve the quadratic assignment problem. traditional methods for solving quadratic assignment problems can be divided into two categories: exact algorithms and approximate algorithms. exact algorithms are able to find the global optimal solution, but the time required increases sharply with the size of the problem and is not suitable for practical applications. approximate algorithms trade accuracy for time, seeking to find a feasible solution as close as possible to the optimal solution in a reasonable amount of computation time. heuristics, as typical approximation algorithms, suffer from poor adaptability and as soon as the problem configuration changes, the original method is no longer superior. the potential of the problem is such that the model has to be redesigned. moreover, for large-scale complex problems, traditional methods can lead to a "combinatorial explosion", as the problem size increases, and the temporal and spatial complexity of the computation grows exponentially. therefore, it is still difficult to design efficient algorithms to solve quadratic assignment problems. in recent years, with the rapid development of computer science, the evolution of artificial intelligence technologies, and the simulation techniques of nature and predators, a number of new methods have emerged to solve combinatorial optimization problems using the modeling of predator’s behaviors in nature, such as hunting, attacking, quarreling, and foraging, as well as their prey, thus providing a new way of thinking to solve quadratic assignment problems. we distinguish several heuristics and metaheuristics inspired by nature and predator behavior [12-16], physics [17-21], humans [22-23], and evolutionary [24-28]. moreover, these nature-inspired heuristics and metaheuristics have shown promising results in tackling complex real-world problems, spanning various domains such as logistics, transportation, network design, and scheduling, by mimicking the efficient and adaptive strategies observed in nature and predator-prey interactions, these methods offer innovative approaches to address combinatorial optimization problems with improved efficiency and effectiveness. the fusion of computer science and natural processes opens up exciting possibilities for the advancement of optimization techniques, paving the way for more sophisticated and intelligent problem-solving paradigms in the future. as researchers continue to explore and refine these nature-inspired approaches, we can anticipate significant advancements in solving quadratic assignment problems and other optimization challenges. a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 531 fig. 1 the best-known metaheuristics in this paper, we propose a new optimizer based on the hunting and attacking behavior of rats in the wild. rats are a species of predator known for their intelligence, they participate in group activities such as hunting and attacking prey. these behaviors will be defined mathematically to create an intelligent and robust optimizer capable of solving more than 38 continuous and nonlinear [16] optimization problems. this optimizer could give excellent results in solving continuous optimization problems, which are better than most of the known metaheuristics in this context and also in solving the famous discrete traveling salesman problem [29]. in this paper, we will introduce another version to solve the discrete combinatorial optimization problem by redefining the mathematical operators of this optimizer with discrete maurice clerc [30] operators, and we will add other strategies to improve the solutions and exploit and explore the discrete search space of the quadratic assignment problem to minimize the total cost of the assignment. the motivation and benefits of choosing this optimization are: ▪ the algorithm has a small number of operators compared to other artificial intelligence-based algorithms. ▪ this ai-based algorithm can access information from the entire search space 532 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi and is simple to implement. ▪ the algorithm has fewer parameters, which reduces its storage requirements and complexity. ▪ the algorithm maintains a good balance between exploration and exploitation during the search process. ▪ the algorithm has been shown to be effective at solving 38 continuous and linear problems. ▪ the algorithm was able to solve the well-known discrete traveling salesperson problem and performed well. the main contributions of this work are presented as follows: ▪ this study introduces the drso algorithm for the first time as a solution to the qap. ▪ the study proposes a mapping strategy to convert real values into discrete values to address the combinatorial nature of the qap. ▪ the study redefines mathematical operators to solve combinatorial and discrete optimization problems, specifically the qap. ▪ the study proposes a solution quality improvement strategy based on local search heuristics such as 2-opt and 3-opt. ▪ the effectiveness of the proposed algorithm is demonstrated through simulations and comparisons of test instances from the qaplib test library. ▪ the study proposes a statistical analysis using the wilcoxon parametric test to validate the performance of the proposed algorithm. the organization of the remaining sections of this paper is as follows: section 2 presents related work. section 3 presents the quadratic assignment problem (qap). section 4 presents the presentation of the rat swarm optimizer algorithm and mathematical behavior modeling. section 5 presents the proposed discrete rso algorithm and its modification for solving the qap. section 6 presents the computer results and analysis, including a wilcoxon validation test. finally, in the final section, the concluding remarks and suggestions for future work are presented. 2. related work in recent years, the solution of combinatorial optimization problems by metaheuristics has undergone a great evolution. metaheuristics have undergone a great evolution, which can be summarized by the speed of development of metaheuristic algorithms thanks to the evolution of the capacities of computing machines. this evolution has allowed, on the one hand, to measure the impact of the search for methods on the problems and, on the other hand, it has had a vision of the result of the methods in a reduced time, especially for the future where the time to find a solution has become more and more requested. however, the development of metaheuristics can be seen in the appearance of recent methods from different sources of inspiration, such as the algorithm of the hunting mechanism of owls [31] and that of the hunting of anteaters [32], the algorithm of symbiotic interaction strategies adopted by organisms to survive and propagate in the ecosystem [33], the algorithm of searching for squirrels [34], and the phenomenon of food storage in crows a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 533 [35]. the appearance of heuristics does not indicate the absence or inefficiency of previous methods, but their interest in appearing is mainly based on a logic of intensification and diversification different from the other methods. each algorithm has its own search characteristics that are different from the others, this produces a large variety of metaheuristics. metaheuristics are usually presented in the continuous or discrete case, and the majority are presented in the continuous case to solve benchmarks of a continuous function form. however, in our research case, which is interested in solving combinatorial problems to measure the quality of methods in the face of real problems, it is obvious to migrate to the aspect of finding a continuous algorithm in a combinatorial computational environment that does not admit real type values. this migration of an algorithm is known by the adaptation of the algorithm for the combinatorial case, which leads to indicating a logical formula to convey the search strategy of intensification and diversification in the combinatorial environment. each new adaptation is applied to the richness of the combinatorial optimization problems in order to be compared to existing heuristic methods. for example, in the case of solving the quadratic assignment problem, we have seen a great evolution of metaheuristics to solve it, for example pesoa [36], which is based on a population of penguins, and with random probability, or dcso [37], discrete cat swarm optimization, which is a metaheuristic method based on the natural behavior of cats. the sso swallow swarm optimization algorithm [38] is a bio-inspired algorithm based on the behavior of swarms of swallows, the harmony search algorithm [39] inspired by the analogy to music. in this study, we will present the rso algorithm as a swarm intelligence algorithm that bases its behavior on attacking and arguing when searching for prey. this approach was originally created to improve continuous functions. this algorithm has been compared to seven continuous algorithms. in fact, the algorithm shows good efficiency when applied to solve various continuous optimization problems. implementing and improving the original algorithm to address various challenges. in order to efficiently optimize the traveling salesman problem, mzili, and riffi added additional heuristics of mechanisms to this method, to improve the local search capability to ensure efficient exploitation of the search space and escape local minima. in what follows, we will redefine this optimizer to solve the quadratic assignment problem by taking up these mathematical operators and search mechanisms. 3. quadratic assignment problem (qap) the quadratic assignment problem (qap) is a mathematical optimization model that was introduced in 1957 by koopmans and beckman [1]. it was originally developed to model a plant location problem, and its objective is to find the optimal location of a set of plants taking into account their interactions and distances with other plants. since its introduction, the qap has become a well-known problem in the literature, as it has been studied in a number of research contexts. this is because the qap presents a generic case that can be applied to other problems. in the qap, the data is presented in the form of matrices, the first flow matrix is f=(fi,j), where (fi,j) is the measure of independence between plant 𝑖 and plant 𝑗, the second matrix 534 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi denotes the address of the premises d=(di,j) to which the factories can be assigned, where di,j represents the distance between premises i and premises j. formally, in a qap problem of size 𝑛, i.e., assigning 𝑛 factories in 𝑛 location, with f=(fi,j) and d=(di,j) are the flow and distance matrices, its solution amounts to optimizing the following function: ▪ sets n={1,2,3,…, n} ▪ sn= ø : n → m represents the set of all permutations. ▪ parameters ▪ f=(fi,j) matrix of flow between facilities i and j ▪ d=(di,j) matrix of the distance between locations i and j. 𝑴𝒊𝒏 ∅ ∈ 𝑺𝒏 ∑ ∑ 𝒇𝒊,𝒋 × 𝒅∅(𝒊)∅(𝒋) 𝒏 𝒋 𝒏 𝒊=𝟏 (1) a permutation, where i is the place to which facility i is assigned, is used to indicate the assignment of facilities to locations. the cost of assigning facility i to location ø(i) and facility j to location ø(j) equals the cost of each individual product fi,j × dø(j) ø(j). the solution is a permutation ø of n elements of the search space; each element ø(i) indicates the location of the proposed assignment for plant 𝑖 in the assignment ø. the interest is to find the permutation ø(i) that minimizes the objective function. for each problem of size 𝑛, the number of possible permutations is 𝑛! e.g., for a problem of size 10 the number of possible solutions is 3628800, as the size of increases, the computational time increases in parallel, which presents complexity in this np-hard classified problem [40]. however, qap has a variety of applications in real life, for example [41] have applied qap in a university campus, the interest is to find the ideal allocation of buildings to decrease the traffic of steps in the campus. another example is [42] to use the same concept to find an assignment of blocks of different departments of a hospital to decrease the patient's route, another use presented by [43] the detection of the right places to install the services of a city such as supermarkets and police stations. qap could be applied not on geometric assignment problems but on any kind of assignment, for example the problem of assigning electronic components on computer panels, the objective is to reduce the total length of cabling used for interconnecting components. example: consider the possibility of an installation location issue with four installations (4 emplacements). the following illustration depicts a potential impact: installation 2 is affected at position 1, installation 1 is affected at position 2, installation 4 i s affected at position 3, and installation 3 is affected at position 4. this impact can be written as the permutation p=2,1,4,3, which denotes that installations 2 and 1 are both affected at position 1, installations 4 and 3 are both affected at position 3, and installations 4 and 3 are both affected at position 4. a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 535 fig. 2 facility location problem with four facilities tables 1 and 2 provide descriptions of the distances between facilities and the necessary flows between facilities. these distance values are essential for computing the assignment cost of the permutation, as they help determine the overall cost associated with different facility assignments for the given problem. table 1 the movements between facilities facility i facility j flow (i,j) 1 2 3 1 4 2 2 4 1 3 4 1 table 2 distances between sites location i location j distance (i,j) 1 3 53 2 1 22 2 3 40 3 4 55 the assignment cost of the permutation may then be calculated using the formula: function_objective=flow(1,2)×distance(2,1)+flow(1,4)×distance(2,3)+flow(2,4)×dis tance(1,3)+flow(3,4)×distance(3,4)= 322+240+153+455. the quadratic assignment problem (qap) can be used to optimize the layout of a manufacturing facility by assigning different machines or processes to different locations in the facility in a way that minimizes the total cost of the assignment. to use the qap for the installation of machines in a manufacturing facility, you would need to define the set of facilities (machines) and the set of locations (available positions in the facility) and specify the cost matrix c that represents the cost of assigning each machine to each location. the cost matrix can include various factors that contribute to the total cost of the assignment, such as the distance between locations, the setup time or changeover time between different products, and the capacity or output of the machines. 536 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi it is important to note that the qap is a combinatorial optimization problem and is known to be np-hard, meaning that it is computationally difficult to solve optimally. therefore, it may be necessary to use approximate solutions or heuristics to find a good, but not necessarily optimal, solution to the problem. 4. rso algorithm the rat swarm optimization algorithm is a nature-inspired metaheuristic that mimics the hunting and attacking behavior of a group of rats. this algorithm is based on the collective intelligence of rats and their ability to adapt to their environment. to model this behavior, the algorithm uses a population of "rats" that move through a search space. each rat has a certain position, which determines its movement in the search space. the rats are also assigned a fitness value, which indicates their probability of finding a solution to the problem at hand. the rats in the swarm are able to communicate with each other and share information about their position and fitness value. this allows them to collaborate and search for the optimal solution to the problem. when a rat finds a good solution, it becomes a "leader" and the other rats follow it, which increases the chances of finding a better solution. rats also exhibit "offensive" behavior, that is, they aggressively search for solutions in areas of the search space that have not yet been explored. this combination of collaborative and aggressive search behavior allows the rat swarm optimization algorithm to efficiently explore the search space and find good solutions to complex optimization problems. rats are social predators that prefer to live in groups and perform their many tasks together, including hunting, attacking, and foraging. the two behaviors that serve as the basis for this bio-inspired algorithm are: hunting behavior: in which rats hunt their prey in packs. to locate the prey, the group members designate a captain each time they think they have located it, and they follow him. however, each time they change captains, they cover the entire area. the behavior of dispute with the prey: in order to hunt their prey, the rats enter into dispute with these last ones, this dispute can cause in several cases the death of certain rats which can translate to the cancellation of a certain solution. 4.1. mathematical and logical modeling of behavior this section explains the chasing and fighting behavior of rats. ▪ prey pursuit: rats typically hunt their prey in packs due to their agonistic social behavior, which makes them sociable. we assume that the finest searcher knows the location of the prey in order to define this behavior quantitatively. the best searcher found so far can be updated by other searchers. the following equations are proposed to model this mechanism: 𝐿𝑜𝑐 = 𝛿 × 𝐿𝑜𝑐𝑖 + 𝛽 × (𝑙𝑜𝑐𝐵𝑒𝑠𝑡 − 𝑙𝑜𝑐𝑖) (2) here, locbest is the best optimal solution and loci specifies the locations of the rats, while the parameters δ and β are determined as follows: 𝛿 = 𝜃 − 𝜌 ( 𝜃 𝑀𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 ) , 1 ≤ 𝜃 ≤ 5, 𝜌 = 1,2,3, … , 𝑀𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (3) a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 537 therefore, the two parameters δ and β are sensitive to good exploration and exploitation throughout the iteration, while δ and β are random values between [1, 5] and [0, 2]. ▪ combating a prey in many cases, the chase ends with the death of some rats. the following equation was presented as a mathematical definition of this process: 𝐿𝑜𝑐𝑖+1 = |𝐿𝑜𝑐𝐵𝑒𝑠𝑡 − 𝐿𝑜𝑐𝑖| (4) where loci+1 specifies the rat's next updated position. the best solution is preserved, and the positions of the other search agents in relation to the best search agent are updated. in general, the execution steps of the standard rso algorithm are presented as follows: algorithm 1: basic discrete rso output: optimal solution input: the initial rat population p, initialize rso parameters: a, c, and r. initialize the rat's population pi where i = 1, 2 now, calculate the fitness value of each search agent. choose the best agent fitness value pbest. while (k < maxiteration) do for each agent search do update the positions of current search agents using equation (4) end for update rso parameters: a, c, and r. check whether any search agent goes beyond the boundary limit of the search space and then amend it. calculate the fitness of each search agent. update pbest if there is a better solution than the previous optimal solution. k ← k + 1. end while return pbest 5. proposed discrete rso algorithm for qap the quadratic assignment problem (qap) is a mathematical optimization problem that involves finding the optimal placement of a set of machines in a manufacturing facility. standard rat swarm optimization (rso) is a continuous optimization method that is used to optimize continuous nonlinear functions, but it cannot be used to directly solve discrete problems. to address this, a modified version of rso, called discrete rat swarm optimization (drso), has been developed to solve discrete combinatorial problems, including the qap. in order to apply drso to the qap, the fundamental equations of rso must be modified to include position representation, position update equations, and rso parameters and operators. additionally, neighborhood search techniques are often used in drso to improve the quality of the solution for combinatorial problems. one such technique is the 2-exchange neighborhood function, which is appropriate for use in qaps. 538 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi this function involves exchanging the positions of two machines in the current solution and evaluating the resulting sum of distances between the facilities and the machines. if the sum of distances is improved, the new solution is accepted as the current solution. by utilizing the 2-exchange neighborhood function, the drso algorithm is able to effectively search for the optimal solution to the qap by making small changes to the current solution and evaluating the resulting improvement in the sum of distances. this allows the algorithm to find a high-quality solution to the qap in a relatively short amount of time. the subsection that follows will go into further information about this function. i. update position the rat swarm optimization algorithm simulates the movement of virtual rats in an ndimensional search space (where n is the size of the problem) according to the location described in the basic rso algorithm. as the rat’s search for the optimal solution to the problem, they tend to move towards the best solution found so far, updating their positions (pi) at each time step (t). during the search process, some rats may be eliminated due to conflicts with other rats. in the context of the quadratic assignment problem (qap), each rat represents a potential solution and locbest represents the best solution found by the ith rat. this solution represents the optimal placement of the machines in the manufacturing facility, minimizing the sum of distances between the facilities and the assigned machines. ii. check the position quality during the search process, the rats may engage in "hunting and fighting" against other rats as they compete for the optimal solution. in some cases, this competition may result in the elimination of weaker rats, or solutions. this process can be modeled as follows: each rat represents a potential solution, and the elimination of a rat corresponds to the abandonment of that solution. this process is analogous to the death of a rat during the search process. iii. operator of discrete rso in continuous optimization problems, logical and mathematical operators are applied to real and natural numbers. however, in discrete optimization problems, these operators cannot be used in the same way because they are designed for continuous scenarios. discrete optimization problems, such as order, sequencing, or permutation optimization, require the use of discrete operators. therefore, it is necessary to modify the operators in order to apply them to discrete optimization problems. the addition operator is used to move the current position by one step. in the discrete case, this operator can be represented as a set of permutations that alter the placement of the facilities. this allows the operator to be applied to discrete optimization problems such as the quadratic assignment problem (qap) the subtraction operator locbest – loci in the rat swarm optimization algorithm is used to calculate the set of permutations needed to transform the current position of a rat 𝑙𝑜𝑐𝑖 into the position of the "best" rat 𝑙𝑜𝑐𝐵𝑒𝑠𝑡. this is done by subtracting the position of the "best" rat from the position of the current rat, resulting in a new position 𝑙𝑜𝑐𝐵𝑒𝑠𝑡 .this operator is used to guide the rats towards the optimal solution by allowing them to follow the movements of the "best" rat. the multiplication operator in the rat swarm optimization algorithm is used to reduce the number of permutations needed to transform the current position of a rat (𝑙𝑜𝑐𝑖).) into the position of the "best" rat locbest. this operator is defined as the multiplication of a real a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 539 number (β) with the set of permutations calculated by the subtraction operator rat (locbest – loci). by multiplying these values, the number of permutations needed to reach the "best" position is reduced, allowing the rats to more efficiently search for the optimal solution. iv. the objective function the objective function is used to evaluate the quality of a given solution. it is defined as the sum of the distances between the locations of the facilities and the machines assigned to them, multiplied by the flow between those locations (as seen in eq. (4)). neighborhood search techniques are often used in combinatorial optimization problems to enhance the quality of the solutions. the two-exchange neighborhood function is a reliable technique for use in the quadratic assignment problem (qap). it involves starting from a random placement of the machines in the facilities and repeatedly exchanging the positions of two machines as long as it results in a more optimized placement. v. the 2-opt the algorithm is a general-purpose optimization algorithm that can be applied to a wide range of problems, including the qap. it works by iteratively improving the current solution by making swaps between pairs of facilities and their assigned machines. by making swaps that reduce the overall cost of the assignment, the algorithm is able to progressively improve the solution and find the optimal placement of the machines in the facilities. here is an example of pseudo-code for the 2-opt algorithm for solving the qap: algorithm 2: 2-opt algorithm for the quadratic assignment problem function 2_opt_qap (solution): solution = initial solution best_solution = solution improved = true while improved do improved = false for i = 0 to n-1 do // n: number of facilities for j = i+1 to n do if cost_of_swapping (solution, i, j) < 0 then solution = swap (solution, i, j) improved = true end if end for end for if cost_of_solution(solution) < cost_of_solution(best_solution) then best_solution = solution end if end while return best_solution 540 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi the final version of the discrete rat algorithm presented as follows: algorithm 3 discrete rat swarm optimization with 2-opt for qap require: distance matrix d, flows matrix f , number of rats n , maximum number of iterations imax initialize rats positions x randomly ·x, representing an installation of facility calculate fitness values f for each rat using distance matrix d and flows matrix f. for t = 1 to imax do for i = 1 to n do update rat position using equation (3): xnew = a·xi+b·(xbest−xi) calculate new fitness value fnew using updated position if fnew < f then if 2-opt(xnew) < xnew then xnew ← 2-opt(xnew) end if f ← fnew xbest ← xnew end if end for end for output: best solution founded xbest 6. experimental results and analysis in this study, the rat swarm optimization (rso) algorithm was implemented in the c++ programming language using a quad-core intel core i5 processor with 4 gb of ram. objects from the qablib library were used for testing the algorithm on various instances of the quadratic assignment problem (qap). the instances used for testing ranged in size from 12 to 100, as indicated by the number in the instance name (e.g. the instance "els19" represents an instance with 19 facilities). to compare the performance of the rso algorithm with other metaheuristics in the domain, we defined a set of parameters and comparison criteria, as shown in the table 3: table 3 parameters of discrete rso parameter value the population of rat size: n 60 δ a random value between [1, 5] β a random value between [0, 1] nb iteration 300 a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 541 tables 4-7 present the results of the discrete rat swarm optimization (drso) algorithm applied to 49 selected benchmark instances from the qaplib dataset, including some of the most difficult instances to solve such as tai20a, tai30a, tai40a, tai80a, and tai100a. the drso algorithm was run 20 times independently for each dataset to obtain the experimental results. the "best", "worst", "average", and "dev (%)" values represent the best, worst, average, and deviation of the solution after running the algorithm 30 times, respectively. the deviation of the solution, represented by dev (%), is calculated using the following formula: dev (%) = average-opt opt ×100 (5) the convergence speed of an optimization algorithm is a measure of how quickly the algorithm converges to a solution. a higher convergence speed means that the algorithm reaches a solution in a shorter amount of time. population diversity, on the other hand, refers to the variation within a population of solutions. a more diverse population is characterized by a greater average distance between individuals, indicating a wider range of possible solutions. to test the performance of our optimizer, we will choose some of the most well-known metaheuristics in solving quadratic assignment problem to make a comprehensive comparison. we provide tests on more than 48 instances, but to make the comparison more meaningful and important, we will compare only the most difficult instances, as well as the instances proposed in the articles of the methods, we are going to compare with them, i.e., the instances of the "tai...." and "sko..." families. the comparison is made with recently developed bio-inspiring metaheuristics more known in the solution of combinatorial optimization problems. dsso: the sso (swallow swarm optimization) [38] method is a swarm intelligence approach that is based on the behavior of swallow swarms. phcso: the parallel hybrid chicken swarm optimization (phcso) [44] method is a nature-inspired optimization approach that combines the behavior of chicken swarms with parallel computing techniques. gbsa: the gbsa (generalized binomial search algorithm) [45] is an optimization algorithm that is based on the principle of binary search. dba: the discrete bat algorithm (dba) [10] is a nature-inspired optimization method that is based on the behavior of bats. to enrich the comparison and make it real and meaningful, it is necessary to use parametric or non-parametric statistical tests. in this study, we will choose the parametric test most commonly used in this type of study, namely the wilcoxon signed ranks test. the wilcoxon signed ranks test a non-parametric statistical technique used to compare two related samples. the wilcoxon signed rank test is often used when data are not normally distributed and can be applied in situations where parametric tests such as t-tests cannot be used. this article discusses the assumptions underlying this method, how it works, its advantages and disadvantages compared to other methods, and some examples of its application. in tables 4-7, the sig column is added to indicate the sign of the average time difference of the drso with each other metaheuristic. 542 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi in figs. 3-10 we present a comparison on of deviation and average time between drso and others methods. table 4 comparison between drso and phcso phcso drso instances best-know best dev time best dev time sig sko42 15812 15812 0,355 31,02 15812 0.04 3.40 = sko49 23386 24124 0.765 78.96 23386 0.01 5.31 + sko81 90998 91113 0.567 215.91 91034 0.01 36.38 + tai12a 224416 224416 0 0 224416 0 0.01 = tai15a 388214 388214 0 0.33 388214 0 0.02 = tai17a 491812 491812 0 0.21 491812 0 0.05 = tai20a 703482 703482 0 0.12 703482 0.12 0.23 = tai25a 1167256 1167256 0 8.09 1167256 0.34 0.16 = tai30a 1818146 1824318 0.673 8.44 1818146 0.49 1.68 + tai35a 2422002 2428322 0.563 16.19 2429278 0.23 9.03 = tai40a 3139370 3139370 0.6275 17.15 3168134 0.02 12.11 = tai40b 637250948 637250948 0.556 9.61 637250948 0 1.27 = tai50a 4938796 5090356 0.176 12.19 4938796 0.78 17.06 + tai50b 458821517 458845260 1.344 37.61 458821517 0.03 10.01 + tai60a 7205962 7351256 0.837 59.87 7231162 0.57 27.13 + tai80a 13499184 13657560 0.863 62.13 13505690 0.77 19.05 + tai100a 21052466 21503812 0.136 86.13 21052466 0.12 69.45 + fig. 3 comparison of average time between drso and phcso http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai35a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50b.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50b.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai60a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai60a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai80a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai100a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai100a.sln a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 543 fig. 4 comparison of deviation between drso and phcso table 5 comparison between drso and dsso dsso. drso instances best-know best dev time best dev time sig sko42 15812 15812 0.02 2.80 15812 0.04 3.40 = sko49 23386 23386 0.09 5.50 23386 0.01 5.31 + sko81 90998 91008 0.08 41.37 91034 0.01 36.38 tai12a 224416 224416 0.00 0 224416 0 0.01 = tai15a 388214 388214 0.05 0.05 388214 0 0.02 = tai17a 491812 491812 0.36 0.06 491812 0 0.05 = tai20a 703482 703482 0.62 0.13 703482 0.12 0.23 = tai25a 1167256 1167256 0.95 0.48 1167256 0.34 0.16 = tai30a 1818146 1825384 0.89 1.29 1818146 0.49 1.68 + tai35a 2422002 2435966 1.05 4.36 2429278 0.23 9.03 + tai40a 3139370 3165320 1.17 9.67 3168134 0.02 12.11 tai40b 637250948 637250948 0.00 0.56 637250948 0 1.27 = tai50a 4938796 4995292 2.00 34.88 4938796 0.78 17.06 + tai100a 21052466 21044752 2.36 545.31 21052466 0.12 69.45 + http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai35a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai100a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai100a.sln 544 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi fig. 5 comparison of average time between drso and dssoso fig. 6 comparison of deviation between drso and dsso table 6 comparison between drso and gbsa gbsa drso instances best-know best dev time best dev time sig sko42 15812 15880 0.99 240 15812 0.04 3.40 + sko49 23386 23582 1.13 240 23386 0.01 5.31 + tai12a 224416 224416 0.00 0 224416 0 0.01 = tai15a 388214 388214 0.00 0 388214 0 0.02 = tai17a 491812 491812 0.00 0.12 491812 0 0.05 = tai20a 703482 703482 0.37 32 703482 0.12 0.23 = tai30a 1818146 1841180 2.22 240 1818146 0.49 1.68 + tai40a 3139370 3215360 3.01 240 3168134 0.02 12.11 + tai50a 4938796 5084020 3.53 240 4938796 0.78 17.06 + http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 545 fig. 7 comparison on of average time between drso and gbsa fig. 8 comparison on of deviation between drso and gbsa table 7 comparison between drso and dba dba drso instance best-know best dev time best dev time sig sko42 15812 15812 0.30 42.87 15812 0.04 3.40 = sko49 23386 23421 0.34 97.71 23386 0.01 5.31 + sko56 34458 34524 0.46 159.8 34458 0.01 17.33 + sko64 48498 48656 0.44 265.63 48498 0.10 27.05 + sko72 66256 66422 0.46 361.59 66259 0.01 38.52 + sko81 90998 91252 0.45 512.73 90998 0.01 36.38 + tai12a 224416 224416 0.00 0.00 224416 0 0.01 = tai15a 388214 388214 0.00 0.50 388214 0 0.02 = tai17a 491812 491812 0.00 0.39 491812 0 0.05 = tai20a 703482 703482 0.85 0.18 703482 0.12 0.23 = tai25a 1167256 1172754 1.51 12.10 1167256 0.34 0.16 + tai30a 1818146 1831272 1.34 20h25 1818146 0.49 1.68 + tai35a 2422002 2438440 1.79 35.43 2429276 0.23 9.03 + tai40a 3139370 3139370 2.02 51.12 3168124 0.02 12.11 tai40b 637250948 637250948 0.00 14.86 637250948 0 1.27 = tai50a 4938796 5042654 2.69 100 4983176 0.78 17.06 + tai50b 458821517 458830119 0.11 126.34 458821517 0.03 10.01 + tai60a 7205962 7387482 2.73 166.23 7231162 0.57 27.13 + tai80a 13499184 13810130 2.67 420.62 13505690 0.77 19.05 + tai100a 21052466 21541326 2.5 1045.27 21052466 0.12 69.45 + http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai30a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai35a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai40a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50b.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai50b.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai60a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai60a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai80a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai100a.sln http://anjos.mgi.polymtl.ca/qaplib/soln.d/tai100a.sln 546 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi fig. 9 comparison on of avg time between drso and dba fig. 10 comparison on of deviation between drso and dba 6. discussion and analysis to compare the performance of our optimization algorithm with the other metaheuristics, we will apply the wilcoxon test [46] with a 95% confidence interval (α=0.05). this test will be conducted twice: first, to compare the difference in dev (%) values between the two algorithms for comparison and ranking; and second, to compare the average execution time and justify the comparison of convergence speed. instances with similar values or that are easy to solve for both algorithms will not be considered. n denotes the number of test cases, and w+ represents the scores of cases where the proposed algorithm performs the best. wrepresents the sum of the scores of the cases where the proposed algorithm performs worse than the comparative algorithm. the p-value is compared to the critical value δ = 0.05 in the wilcoxon signed-rank test. if the p-value ≤ δ, it indicates a significant difference in performance between the two algorithms. however, if the p-value > δ, there is no significant difference in performance between the two algorithms. a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 547 in tables 8 and 9, we conducted the wilcoxon signed rank tests to assess and compare the deviation and average time of different metaheuristics. table 8 wilcoxon signed rank test applied to the deviation of metaheuristics comparison n ww+ p-value significantly drso vs dba 20 -16.00 173,0 <0,001 yes drso vs hpcso 17 0 200 0,044 yes drso vs gbsa 9 0 39 0,031 yes drso vs dsso 14 -3 99 <,001 yes table 9 wilcoxon signed rank test applied to the avg time of metaheuristics comparison n ww+ p-value significantly (p < 0.05)? drso vs dba 20 -3,00 187 <0,001 yes drso vs hpcso 17 -8 145 <0,001 yes drso vs gbsa 9 -3 42 0,020 yes drso vs dsso 14 -54,50 50,50 0,915 no in order to study the quality of drso, we will describe each comparison separately according to the tables above. the comparison of drso with other methods is based on three important factors: the deviation of each algorithm, the average time of convergence to the optimum, and the number of times each algorithm reaches the known optimum of qaplib. each comparison will be related to the curves described above from each table. ▪ drso vs hpcso starting with the comparison with phcso in table 4, we found that the results found by drso are 50% better (9 out of 18 tests) than phcso and are 50% equal (9 out of 18 tests), while the convergence is 100% better (18 out of 18 tests), which means that drso requires less time and iteration to converge to the optimum. on the other hand, the deviation of drso in the 18 tests is less than that of hpcso at 100% which shows that the gap between the results obtained by this method and the best-known value of qablib is very large. concerning the ability of the two algorithms to attract the optimum value of qablib: drso was able to attract the best-know value in 13 instances among the 18 test instances at 72.22% while hpcso found the optimum for 8 instances among 18 i.e. at 44.44% with a deference of 27%. the curves of deviation and time in fig. 3 and 4 show a large difference between the two algorithms, and also show that the dev (%) value of drso is very close to 0 in almost all instances which justifies that the latter was able to find the optimum or almost in most of the tested instances. ▪ drso vs dsso in the comparison with dsso in table 5, we found that the results found by drso are 35.71% better (5 out of 14 tests) than dsso, 50% equal (7 out of 14 tests), and 14.28% (2 548 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi out of 14 tests) weak, while the convergence is 57% better (8 tests out of 14) than dsso while it is weak at 42.85% (6 tests out of 14) which means that the methods converge quite close and require less time and iteration to converge to the optimum. on the other hand, the difference in deviation between the two methods is significant and we can see that the deviation values of drso are better in almost all instances at 92.8% (13 instances out of 14) which shows that the difference between the results obtained by these methods and the best-known value of qablib is very large. regarding the ability of the two algorithms to attract the optimal value of qablib: drso was able to attract the best-known value in 13 instances among the 18 test instances at 72.22% while dsso found the optimum for 8 instances among the 18, i.e. at 44.44% with a deference of 27%. the curves id deviation and time in fig. 5 and 6 show a significant difference between the two algorithms, and also show that the value dev (%) of drso is lower than those of dsso in almost all the instances what justifies that this last one was able to find solutions in the neighborhood or equal to the optimum in the majority of the tested instances, on the other hand shows the curve avg time mounted that the two converge perfectly in a reasonable time and that the two curves are very close in almost all the test. ▪ drso vs gbsa in the comparison with gbsa in table 6, the results found by drso are 55.55% better (5 tests out of 9) than gbsa, 44.44% equal (4 tests out of 9). while the convergence is 77.77% better (7 tests out of 9) than gbsa while it is low at 22.22% (2 tests out of 9) which means that drso converges quickly and requires less time and iterations to converge to the optimum than gbsa. on the other hand, the difference in deviation between the two methods is significant and we can see that the deviation values of drso are better in almost all instances at 100% (9 instances out of 9) which shows that the difference between the results obtained by these methods and the best-known value of qablib is very large and the solutions obtained by drso are all closer to the best-know value of qablib than those of gbsa this can be clear also in fig. 7. regarding the ability of the two algorithms to attract the optimal value of qablib: drso was able to attract the best-known value in 8 instances among the 9 test instances at 88.88% while gbsa found the optimum just for 4 instances among the 9 instances, that is at 44.44% with a deference of 44.44%. the curves of deviation and avg time of fig. 7 and 8 show a very significant difference between the two algorithms, and also show that the value dev (%) of drso is lower than those of gbsa in almost all the instances which justifies that the latter was able to find solutions in the neighborhood or equal to the optimum in almost the majority of the tested instances on the other hand the curve avg time showed that gbsa is so slow in. ▪ drso vs dba finally, we compare our method with dba in table 7, the results found by drso are 65% better (13 tests out of 20) than dba, 30% equal (6 tests out of 20) and 5% (1 test out of 20) weak than dba. while the convergence is 100% better (20 tests out of 20) than dba, which means that drso converges quickly and requires fewer time and iterations to reach the optimum compared to dba. on the other hand, the difference in deviation between the two methods is significant, and we can observe that the deviation values of drso are better in almost all cases at 95% a novel discrete rat swarm optimization algorithm for the quadratic assignment problem 549 confidence level (19 cases out of 20). this indicates that the difference between the results obtained by these methods and the best-known value of qablib is substantial. moreover, the solutions obtained by drso are all closer to the best-known value of qablib compared to those of dba. this clarity is also evident in fig. 10. regarding the ability of the two algorithms to attract the optimal value of qablib: drso was able to attract the best-known value in 15 instances among the 20 test instances at 75% while dba found the optimum only for 7 instances among the 20 instances, i.e. at 35% with a deference of 40%. curves of deviation and time in fig. 9 and 10 show a very significant difference between the two algorithms, and also show that the dev (%) value of drso is lower than those of dba in several instances which justifies that the latter was able to find solutions the neighborhood or equal to the optimum in almost the majority of the tested instances; on the other hand, the avg time curve showed that dba is very slow in the large instances which contain more facility. the superior performance of drso is attributed to the behavior of the rats and their ability to efficiently share and exploit information, as well as rapidly explore the entire search space. these statistical analyzes can be evaluated and confirmed by the results of the wilcoxon signed rank tests carried out and cited in the tables 8 and 9, in these tests, it will be found that the difference in a deviation between drso and the other methods is very significant. at α=0.05 in a 95% confidence interval, which confirms the results of the analyzes carried out below, and justifies that the results and the solutions obtained by drso are much closer or equal to the optimum. on the other hand, the wsr test described in table 9 confirms that the difference in mean times between drso and dba, gbsa, phcso are so significant at α=0.05 in a 95% confidence interval, whereas the difference with dsso and insignificant and almost zero, which confirms the analyses, and approves that the drso converges quickly and requires months of iteration time to find solutions equal to or closer to the optimum, and at the same level as the dsso but just in convergence time. 7. conclusions in conclusion, the discrete rat swarm optimizer (drso) algorithm has demonstrated great potential in solving the discrete quadratic assignment problem (qap). the algorithm has shown effectiveness in finding high-quality solutions and has outperformed existing algorithms in numerous instances. the algorithm's advantages, such as its simplicity, access to the entire search space, and a balanced approach between exploration and exploitation, contribute to its success. additionally, the algorithm has proven its capability by solving various discrete problems, including the well-known discrete traveling salesperson problem. this study's contributions are significant in several aspects. firstly, it introduces the drso algorithm as a novel solution for the qap, addressing its combinatorial nature. the study proposes a mapping strategy for converting real values into discrete values, redefines mathematical operators suitable for solving combinatorial and discrete optimization problems, and incorporates local search heuristics to enhance solution quality. the effectiveness of the algorithm is demonstrated through extensive simulations and 550 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, m. kurdi comparisons using the qaplib test library. furthermore, the study employs statistical analysis, including the wilcoxon parametric test, to validate the algorithm's performance. comparisons with other algorithms, namely phcso, dsso, gbsa, and dba, further highlight the superiority of drso. the results consistently indicate that drso outperforms these algorithms in terms of solution quality, convergence speed, and deviation from the best-known values of qablib. the curves and statistical tests provide strong evidence supporting the superior performance of drso, with its convergence curve consistently outperforming other algorithms. the success of drso can be attributed to the efficient information sharing and exploitation capabilities of the rat swarm. the algorithm effectively explores the entire search space, allowing for rapid convergence to optimal or near-optimal solutions. the statistical analyses, including the wilcoxon signed rank tests, confirm the significant differences in deviation between drso and other methods, reinforcing the 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2019, parallel hybrid chicken swarm optimization for solving the quadratic assignment problem, international journal of electrical and computer engineering, 9(3), 2064. 45. riffi, m.e., sayoti, f., 2019, hybrid algorithm for solving the quadratic assignment problem, international journal of interactive multimedia & artificial intelligence, 5(4), 68. 46. taheri, s. m., hesamian, g., 2012, a generalization of the wilcoxon signed-rank test and its applications, statistical papers, 54(2), pp. 457–470. 13292 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 881 907 https://doi.org/10.22190/fume241130008l © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a rolling bearing fault diagnosis method based on extreme learning machine optimized by improved whale optimization algorithm xin li1, shiliang guo2, dejie sun1, lijun cao1, cong li1, shuyao tian3, peng liu2, yadong qi2 1school of mathematics and information science & technology, hebei normal university of science & technology, qinhuangdao, china 2school of electrical engineering, yanshan university, qinhuangdao, china 3college of electronics and control engineering, north china institute of aerospace engineering, langfang, china orcid ids: xin li https://orcid.org/0009-0009-3792-4083 shiliang guo https://orcid.org/0000-0003-2033-6807 dejie sun https://orcid.org/0009-0006-4882-1053 lijun cao https://orcid.org/0009-0004-5759-9031 cong li https://orcid.org/0009-0005-8217-3785 shuyao tian https://orcid.org/0009-0006-1853-1151 peng liu https://orcid.org/0009-0000-9345-4260 yadong qi https://orcid.org/0009-0003-8268-3369 abstract. rolling bearing is one of the most commonly used components in rotating machinery, and researching fault diagnosis techniques for it has important practical significance. in this paper, a fault diagnosis method based on extreme learning machine optimized by improved whale optimization algorithm (iwoa-elm) is proposed for rolling bearing vibration signals. firstly, variational mode decomposition (vmd) is used to decompose the vibration signal of the bearing, and the energy entropy is calculated to form the eigenvector. secondly, based on the original whale optimization algorithm, a hybrid initialization population strategy is adopted to generate an initial population with a certain quality. selecting convergence factors based on reinforcement learning to improve global search capability, and using adaptive weights and random jumps to update individual positions. in this process, the t-distribution-levy flight variation strategy is introduced to avoid being attracted by local extremum. then, the improved whale optimization algorithm is used to optimize the input weights and hidden layer thresholds of the extreme learning machine (elm). finally, the feature set is input into an improved elm model for training and testing. experiments on fault diagnosis of rolling bearings of different types and degrees have shown that the model proposed in this paper can effectively improve the accuracy of fault classification. key words: fault diagnosis, variational mode decomposition, whale optimization algorithm, reinforcement learning received: november 30, 2024 / accepted february 10, 2025 corresponding author: shiliang guo school of electrical engineering, yanshan university, street address no.438, hebei avenue, haigang district, qinhuangdao 066004, china e-mail: guosl0112@ysu.edu.cn 882 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi 1. introduction rotating machinery plays an important role in modern society, and rolling bearings are critical components that impact its performance. however, the working environment of most rolling bearings is relatively harsh, which makes them prone to failure, resulting in significant economic losses and even catastrophic events. the fault vibration signal of rolling bearings has nonlinear and non-stationary characteristics. how to extract fault information from vibration signals and accurately locate bearing faults are essential for fault diagnosis and normal machinery operation [1-4] the application of machine learning methods in fault diagnosis can minimize manual involvement and greatly improve the accuracy of fault diagnosis, such as support vector machines (svm) [5-7], backpropagation neural networks (bpnn) [8-10], and convolutional neural networks (cnn) [11-13]. however, these models face limitations: svm needs to store large datasets, making it difficult to solve multi classification problems; bpnn and cnn have slow convergence speed and high computational cost, particularly with deeper layers. extreme learning machine (elm) has the characteristics of fast training speed, few learning parameters, strong generalization ability, and is widely used in the field of fault diagnosis [14-16]. wang et al. [17] proposed a bearing fault diagnosis method based on elm for adaptive parameter optimization, and introduced spectral cross-correlation to help select the optimal penalty factor, improving classification accuracy. in order to further enhance the generalization ability of elm, zhang et al. [18] used the multitask beetle antennae swarm algorithm (mbas) to optimize the input weights and biases of elm, which can simultaneously reduce the number of conditions and regression errors. from the above description, it can be seen that optimizing the parameters of elm based on optimization algorithms can further improve the performance of the model. whale optimization algorithm (woa) is a metaheuristic algorithm developed in recent years, which is used to handle various optimization problems in different fields due to its advantage of fewer adjustable parameters [19-21]. however, as the dimensionality of the search space increases, woa faces problems such as easily falling into local optima and slow convergence speed. based on the above analysis, we propose a rolling bearing fault diagnosis method based on extreme learning machine optimized by improved whale optimization algorithm (iwoa-elm), as shown in fig. 1. firstly, the variational mode decomposition (vmd) method is used to extract the fault signal features of rolling bearings, which are then divided into a training set and a testing set. then, based on the woa, a hybrid initialization population strategy, a convergence factor selection based on reinforcement learning, an adaptive weight and random jump update strategy, and a t-distribution-levy flight variation strategy are introduced to solve the problem of the algorithm falling into local optimum. finally, the iwoa was used to optimize the parameter settings of the extreme learning machine, thereby improving the speed and accuracy of bearing fault diagnosis. the iwoaelm algorithm was compared with woa-elm, pso-elm, gwo-elm, and the experimental results verified the effectiveness of the improved rolling bearing fault diagnosis method proposed in this paper. a rolling bearing fault diagnosis method based on extreme learning machine... 883 fig. 1 methodological framework of this study 2. signal decomposition and feature extraction 2.1. vmd theory vmd is an innovative and adaptive signal processing method [22] that can solve the common endpoint effects and mode component mixing problems in empirical mode decomposition (emd). vmd decomposes the initial signal into k discrete sub-signals, ensuring that each decomposition represents a finite bandwidth modal component centered at a specific frequency. at the same time, the sum of estimated bandwidths for each modality is minimized, and the sum of all modes is equal to the original signal as a constraint. the constraint variational expression can be written as:       2 j 2, 1 min ( ( ) j / )* ( ) e s.t. k k k t t k u k kk k t t u t u f    − =     +         =   (1) where, k is the number of decomposed modes. the symbols {uk} and {ωk} refer to the kth mode component and center frequency after decomposing, respectively. (t) is the dirac function where t represents time. * is the convolution operator. furthermore, we introduce the lagrange multiplication operator  and the quadratic penalty factor  to find the optimal solution for the constrained variational problem, and obtain the augmented function: 884 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi    ( )   2 j 2 2 2 , , ( ( ) j / )* ( ) e ( ) ( ) ( ), ( ) ( ) k t k k t k k k k k k l u t t u t f t u t t f t u t      − =  +  + − + −    (2) where the mentioned quadratic penalty factor α is used to reduce the interference of gaussian noise. the alternate direction method of multipliers (admm) iterative algorithm combined with parseval/plancherel and fourier isometry transform is used to optimize each eigenmode component and center frequency, and to search for saddle points of the extended lagrangian function. the expressions for uk, ωk, and λ change after alternating optimization iteration is as follows: 1 2 ˆ ˆˆ ˆ ( ) ( ) ( ) / 2 ( ) 1 2 ( ) in i k k k f u u         +  − +  + −  (3) 2 1 1 0 2 1 0 ˆ ( ) d ) dˆ ( n kn k n k u u        + +  +    (4) 1 1ˆˆ ˆ ˆ( ) ( ) ( ) ( )n n n k k f u      + +   + −     (5) where, γ is the noise margin, 1ˆ ( )n ku + , ˆ ( )iu  , ˆ( )f  and ˆ( )  correspond to the fourier transforms of uk n+1(t), ui(t), f(t) and λ(t) respectively. the main iterative vmd solution process is as follows: (1) initialize 1 1 1ˆˆ , ,k ku   and set the maximum number of iterations n, n←0; (2) using eqs. (3) and (4), update ûk and ωk; (3) update ̂ using eq. (5); (4) accuracy convergence criterion ε>0, if n<n and 2 2 1 2 2 ˆ ˆ ˆ/n n n k k k k u u u + −  are not satisfied, return to the second step. otherwise, the iteration is completed. 2.2. extraction of feature: energy entropy yu [23] proposed the theory of energy entropy, and evaluated the fault information of rolling bearings based on the difference in energy by different frequency bands. the energy for any part of the waveform is defined as: 2 2 1 ) d( g i i ik k e c t t x + − = = = (6) where, g is the number of sample points of the ith imf, and xik is the amplitude. a rolling bearing fault diagnosis method based on extreme learning machine... 885 the original vibration signal is decomposed into n imf vibration components, and the energy of each imf component is ei. the energy of the original vibration wave is total 1 n i i e e = = . introducing pi as the fraction of total signal energy in the i-th imf part: /i i totalp e e= (7) the corresponding energy entropy equation is given as: 1 ( log ) n en i i i h p p = = − (8) 3. iwoa-elm model 3.1. elm theory elm is used to train single hidden layer feedforward neural networks (slfn). fig. 2 illustrates the network structure of elm. fig. 2 the network structure of elm assuming there are f random samples (xi, ti), i = 1, 2,…, f, where xi = [xi1,xi2,…,xin]t  rn and ti = [ti1,ti2,…,tim]t  rm. for a slfn with l hidden layer nodes and an activation function g(x) can be calculated: i i j i i i j i j 1 1 ( ) ( , j 1,2) , , l l i i g x g x b t n   = = = + = =   (9) where, ωi, bi, and βi represent the input weight, deviation, and output weight of the ith node in the hidden layer. we can use a matrix to represent it: 𝐻𝛽 = 𝑇, where h is the output of the hidden layer node, β is the output weight, and t is the output matrix. 886 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi 1 1 1 1 1 1 1 1 1 ( ) ( ) h( , , , , , , , , ) ( ) ( ) l l l l l n l n l n l g x b g x b b b x x g x b g x b        +   +    =   +  +      (10) tt 11 tt 22 tt , t nl n m t t t              = =               (11) thus, the output weight is: h t += (12) where, h+ is the moore-penrose generalized inverse of h. 3.2.woa theory in woa algorithm, each whale represents a set of solutions and can be optimized by its location and speed. when a whale detects its prey, it performs a spiral motion around it and releases a bubble net to trap the prey. the whale individuals can interact with each other in various ways, including "encircling prey," "bubble net attack," and "search for prey," which are described as follows: 3.2.1. encircling prey during hunting, whales will continuously update their position and swim towards their prey after determining its location. in the solution space of the algorithm, assume that the position of the individual whale close to the prey is the optimal position. other individual whales follow the best whale to update their position and gradually approach their prey. the above behavior is expressed as: * ( 1) ( )x t x t a d+ = −   (13) * ( )( )d c tx x t=  −  (14) where, t is the current iteration number, *x  is best position, x  is current position of an individual whale, d stands for distance. a  and c  are coefficient vectors, which are calculated as follows: 12a r =  −  (15) 22c r=   (16) 2 2 t a m = − (17) a rolling bearing fault diagnosis method based on extreme learning machine... 887 where, a is the convergence factor with its value decreasing linearly from 2 to 0 during the iteration, m is the maximum number of iterations. 1r  and 2r  are both random vectors in [0,1]. 3.2.2. bubble-net attacking method (exploitation phase) at this stage, the shrinking encircling mechanism of the woa algorithm is shown in equation (13). in addition, after determining the position of the prey, the humpback whale will move along the spiral towards the prey and update its individual position. the mathematical model is as follows: ' * ( 1 cos(2 )) ( )blx e lt d x t+ = +  (18) ' *( ) ( )d x t x t= −   (19) where, ' d  is the distance between the current whale's position and the best whale's position, b is a logarithmic spiral shape constant, and l is a random value between [-1,1]. the two hunting behaviors of shrinking encircling and spiral movement are performed simultaneously. these two actions are selected by setting the parameter p, where p is a random number between [0,1], and the probability of p<0.5 and p ≥ 0.5 is 50%. the mathematical model is as follows: * ' *      0.5 t ( 1) ( ) cos(2 ) ( ) pbl d p x e t d x t x l x    + =      + −        (20) 3.2.3. search for prey (exploration phase) at this stage, the behavior of whales is controlled by coefficient vectors a  to expand the search range and improve search capabilities. when the value of a is not within the range of [-1,1], individual whales will randomly select other whales as guides to update their position. the mathematical model is as follows: ( 1) randx t x a d+ = −   (21) ( )randd c x x t=  −  (22) where, randx  is the vector of random individual positions. 3.3. improved whale optimization algorithm the woa algorithm was briefly summarized in the previous section. in order to improve the algorithm's ability and efficiency in finding the global optimal solution, this section proposes iwoa. a hybrid initialization population scheme is designed based on the original woa. meanwhile, we select convergence factor based on reinforcement 888 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi learning and design adaptive weight. the random jump update strategy and t-distributionlevy flight variation strategy are implemented to improve the search ability of the algorithm. 3.3.1. hybrid initialization in this section, we use improved tent chaos mapping and opposition-based learning to initialize the population, and design a random exchange strategy for disadvantaged populations. (1) improved tent chaos mapping tent chaotic mapping is a piecewise linear mapping that has more uniform traversal and faster search speed than logistic chaos mapping. the expression of tent chaos mapping is: 1 1         0 2 2 2(1 ) 1            1 2 i i i x x x x x +    =  −   (23) the expression after bernoulli shift transformation is: 1 (2 )mod1i ix x+ = (24) to prevent the tent chaotic sequence from getting stuck in unstable periodic points during the iteration, zhang [24] introduced a random variable to the original tent chaotic mapping equation as follows: 1 1 1 2 (0,1)         0 2 1 1 2(1 ) (0,1) 1 2 i i i x rand x n x x rand x n +  +    =   − +     (25) where n represents the initial population size. the expression after bernoulli shift transformation is: 1 1 (2 ) mod1 (0,1)i ix x rand n + = +  (26) (2) opposition-based learning opposition-based learning mechanism can effectively improve the quality of the population and is applied to the improvement of various optimization algorithms. the expression of obl is as follows: ( )ix lb rand ub lb= + − (27) ' i ix lb ub x= + − (28) where rand is a random value on the interval [0,1], lb and ub are the upper and lower bounds of the search space, respectively. applying the obl idea to the randomly generated n initial solutions  1 2, , ,n nx x x x=  to obtain n reverse solutions  1 2, , ,n nx x x x   =  . in the a rolling bearing fault diagnosis method based on extreme learning machine... 889 process of searching for the optimal solution, the current solution and the reverse solution are simultaneously searched, and the better solution among them is taken as the problem solution, which can expand the search scope and improve the search efficiency of the algorithm. the initial solutions generated by chaotic mapping and opposition-based learning are mixed into a population with 2n initial solutions, and then the n initial solutions with smaller fitness values are selected according to the ranking of fitness values. this move can reduce the loss of good solutions. (3) random exchange of inferior populations in order to further improve the quality of the initial population, two different individuals were randomly selected from the n initial solutions with poor fitness generated in the previous step to form a group (total n/2 groups). the algorithm randomly select k non repeating positions for each individual in each group, and replace the k positions of individual 1 with those of individual 2. similarly, perform the same operation on individual 2 to generate n new initial solution individuals. then the fitness value of each individual is calculated and compared with the previous n smaller initial solutions, and the n initial solutions with smaller fitness values are selected. the random exchange operation for the inferior groups is shown in fig. 3. fig. 3 random exchange 3.3.2. improved hunting patterns (1) adaptive weight in order to improve the ability of traditional woa algorithms to find optimal solutions, we introduce an exponentially changing adaptive weight during the algorithm's prey encirclement stage. the algorithm uses larger weights in the early stage to improve its global search ability, but when approaching the optimal solution, the weight values decrease exponentially to enhance the algorithm's local optimization ability. the weight expression can be written as: ( )10 / e t t  − = (29) taking the above equation into eq. (20), the individual position update model for the encircling prey stage is obtained: * *'                         ( ) cos(2 ) ( ) 0.5 ( 1) 0.5bl x a d p x t xl p t d e t  −   + =     +     (30) where t is the highest number of iterations and t is the current number of iterations. 890 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi (2) random jump update strategy the superiority of woa in the population iteration process depends on the search agent selected from the previous generation of the population. from eqs. (21) and (22), it can be derived that the search agent position update during exploration is based on a randomly selected individual, and the new population convergence speed and accuracy are necessarily affected by the random individual. to address this phenomenon, a random jump update strategy is used to select the search agent more efficiently. the improved formula is as follows: ( )randd xc x t= −   (31) ( ) ( )1x t c d c x t+ = • + •  (32) (3) reinforcement learning based convergence factor selection whales act ‘nonlinearly’ in the real hunting process, and the linear descent strategy used by traditional algorithms cannot fully reflect the true situation of woa. therefore, this paper adopts different nonlinear control strategies to optimize the convergence factor a. ref [25] uses a nonlinear convergence factor based on sinusoidal function to improve whale optimization algorithm, and its mathematical model is: ( )( )sin / 2max max mina a a a t t= − −  . (33) on this basis, several adjustment strategies for control parameter a were proposed in references [26, 27], such as: nonlinear decreasing strategy based on quadratic function: 2( )( / )max max mina a a a t t= − − (34) nonlinear decreasing strategy based on logarithmic form: ( )( )ln 1 (e 1) /max max mina a a a t t= − − + − (35) through numerous experiments in references [26, 27], it can be seen that the convergence factor adjustment strategy has different effects on the utility of test functions. in order to investigate the experimental effect, the various adjustment strategies are considered as actions, and a suitable action is chosen to control the position of the whale at each step of the whale hunt. in addition, the possibility of making the right decision is greater if one can consider a few more steps forward before making a decision. therefore, when whales choose actions, the effectiveness of multiple evolutionary steps can be judged by evaluating the advantages and disadvantages of different actions, which increases the whales' ability to find the best. reinforcement learning can gradually learn the optimal action policy by selecting the action that maximizes the agent’s cumulative return with discount. q-learning is a model independent reinforcement learning algorithm that obtains the optimal action strategy through multi-step learning of the agent to find the action with the maximum discounted return. if the adjustment strategy of the convergence factor is regarded as an action, the individual whale’s choice of the optimal adjustment strategy is transformed into the agent’s choice of the optimal action. based on this, the standard whale optimization algorithm, eq. a rolling bearing fault diagnosis method based on extreme learning machine... 891 (33), eq. (34), and eq. (35) four convergence factor adjustment strategies are selected as the set of actions in this paper. the basic format of q-learning reads ( , ) ( , ) max ( , ) a q s a r s a q s a  = + (36) where, r(s,a) is the immediate reward of the agent after taking action a in state s. s and s represent the two states of the agent, while a and a represent the two actions of the agent. γ(0≤γ<1) is the discount factor, and max q(s,a) denotes the agent the maximum value of the reward that can be obtained by choosing a different action in the next state s. assume that the set of operations available to the agent in any state is a = {a1,a2,…,an},. if the first action taken by the agent is a, then the revenue when the agent chooses the action a after m+1 steps is calculated as: (1) 2 (2) ( )( ) ( ) ( ) ( ) ( )m mq a r a q a q a q a   = + + + + (37) where, a, ( )ia a , and 1 i m  , with the parameter m controlling the number of steps to calculate the q value looking forward. with the above q-learning algorithm, the mapping between woa and q-learning algorithm can be achieved by considering the whale as an agent and various distance control parameter adjustment strategies as a set of agent actions. for any individual whale in the woa, looking forward m+1 steps, the gain q(a) when the whale chooses action a can be calculated by eq. (37). define the individual immediate gain, r(a) = fp(a)fo(a). here, fp(a) is value of fitness function corresponding to parent whale individual and fo(a) is value of fitness function about offspring whale individual generated after selecting action a. if the number of optional actions of whales is n, then nm whales will be generated after m times of evolution. to calculate the optimal evolutionary strategy for the whales, an exponential calculation is required, which is too computationally intensive. for this purpose, the probability of each offspring being retained is calculated using the boltzmann distribution: ( )/ ( )/ 1 ( ) /i i nr a t r a t i i p a e e = =  (38) furthermore, the highest probability of the n whale offspring generated will be retained. after this simplification, only one of the n offspring produced by a whale after one evolution is preserved during the evolution process, which greatly reduces the complexity of the algorithm. the convergence of the evolutionary planning algorithm based on qlearning is proved in [28]. the proof process of woa is similar, and its convergence will not be proved in this paper due to the limitation of space. 3.3.3. t-distribution-levy flight variation strategy (1) t-distribution variation strategy t-distribution has a degree of freedom parameter of n and its probability density function is: 892 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi 1 2 2 1 2 ( ) 2 1 n t n г x p x n n г x n + − +       = + −   +          (39) for each whale, the algorithm generates a random number in the [0,1]. if the variation probability is greater than the generated random number, t-distribution perturbation is applied to it. the specific location update method is as follows: ( )j j new best j bestx x t iteration x+ = (40) where, t(iteration) is the t-distribution with degree of freedom parameter as the number of iterations. j bestx and j newx are the positions of the optimal whale individual in the jth dimension before and after variation perturbation, respectively. if the new solution after tdistribution perturbation has a better fitness value, the old solution is replaced; otherwise, the old solution is retained. (2) levy flight strategy levy flight is a search method that obeys levy distribution [29]. this paper introduces the levy flight strategy into the whale optimization algorithm after t-distribution perturbation. by conducting small-scale exploration around the optimal solution, the search area of the algorithm is further increased to avoid the population falling into local optima. the location update formula is ( )k k new bestx x levy = + (41) where, levy(β) is a method to generate random numbers that obey the levy distribution. in this paper, we take advantage of the normal distribution to resolve random numbers [30]. the steps to generate random numbers submitted to levy distribution are as follows: 1 ~ ( ) u s levy v   = (42) where, u obeys a normal distribution, i.e. u  n(0,2), and v obeys a standard normal distribution, i.e. v  n(0,1). the value of σ is taken as follows: 1 1 2 sin 2 2 1 2       −    (+ )      =  +         (43) generally speaking, the range of values for β is 1≤β≤3, which is taken as 1.5 in this paper. a rolling bearing fault diagnosis method based on extreme learning machine... 893 3.3.4. iwoa algorithm performance test in order to verify the feasibility of the algorithm, 8 benchmark test functions are chosen to evaluate the iwoa algorithm. table 1 displays the test functions. for each benchmark function, iwoa is compared with woa, obcwoa [31], remwoa [32], gwo [33] and goa [34], ssa [35]. the population size is set to 30 and the maximum number of iterations is 500.the parameters of the obcwoa method are configured as follows: a=4, k=1. the relevant parameters of goa algorithm are cmax=1 and cmin=0.00004. the ssa algorithm is set to st=0.8, the proportion of producers is 20% and the proportion of scouts is 10%. table 1 functions for testing fig. 4 illustrates the iterative optimization process of several algorithms. it can be seen that the original woa has a slow convergence rate. although the convergence speed of obcwoa and remwoa has improved, it is easy to fall into local optimum in different test functions. iwoa improves the original whale optimization algorithm by using multiple optimization strategies and improves the optimization capability of woa. the iwoa algorithm can jump out of local optima and find the optimal solution of the function in a timely manner even when encountering local optima, which improves the optimization performance of the algorithm. function dim range fmin peak 2 1 1 ( ) n i i f x x = = 30 [-100,100] 0 single 2 1 1 ( ) nn i i i i f xx x = = = +  30 [-10,10] 0 single 2 3 1 1 ( ) n i j i j f x x = −   =       30 [-100,100] 0 single 4 4 1 ( ) [0,1) n i i f x ix random = = + 30 [-1.28,1.28] 0 single 2 5 1 ( ) 10cos(2 ) 10 n i i i f x x x =  = − +  30 [-5.12,5.12] 0 multiple 2 6 1 1 1 ( ) 20exp 0.2 1 exp cos(2 ) 20 n i i n i i f x x n x e n  = =   = − −        − + +      30 [-32,32] 0 multiple 2 7 1 1 1 ( ) cos 1 400 nn i i i i x f x x i= = = − +  30 [-600,600] 0 multiple 2 211 1 2 8 2 1 3 4 ) ( ) ( i i i i i i x b b x f x a b b x x=  + = −  + +   4 [-5,5] 0.0003 multiple 894 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi (a) (b) (c) (d) a rolling bearing fault diagnosis method based on extreme learning machine... 895 (e) (f) (g) (h) fig. 4 convergence curves of 8 test functions: (a) f1(x); (b) f2(x); (c) f3(x); (d) f4(x); (e) f5(x); (f) f6(x); (g) f7(x); (h) f8(x) 896 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi all algorithms were executed 30 times on the 8 benchmark functions, and the average values were computed. the results of experiment are given in table 2. table 2 optimization performance of 8 benchmark functions table 2 illustrates that the woa is particularly obvious disadvantaged when solving for the optimal value of high dimensional functions. the reason for this result is that there are three ways to update it, but only one way of obtaining the exploration, and the probability of getting it is low. therefore the woa is prone to falling into local optima. compared with woa, the algorithm used in this paper improves the convergence accuracy of the algorithm in later iterations through the choice of convergence factors and the variational strategy. meanwhile, compared with other improved woa, iwoa can also effectively solve the problem of algorithm falling into local extremes. it shows significant advantages over several other algorithms about convergence accuracy and stability. 3.4. iwoa-elm fault diagnosis model in this section, the iwoa algorithm is selected to optimize the input weights and biases of the elm model, and use the training set error rate and validation set error rate of the extremum learning machine as fitness values, as described in equation (44), in order to improve the accuracy and generalization ability of the elm model classification. the feature set is divided into training set, validation set and testing set as the input of elm, and the minimum fitness value fmin and population extreme value gbest and individual extreme value pbest are obtained. the optimal solution, individual extreme value pbest and population extreme value gbest are updated iteratively until the best fitness value is obtained or the maximum number of iterations is reached, and the optimal hidden layer weights and bias are obtained. 2 train verifyfitne accuracyss accuracy −= − (44) where, accuracytrain and accuracyverify represent the correct classification ratio of training set and validation set, respectively. the procedure of iwoa-elm model is shown in fig. 5. function woa remwoa obcwoa iwoa goa ssa gwo f1(x) 5.97×10-72 1.26×10-290 2.06×10-262 0 93.63 3.99×10-60 1.00×10-27 f2(x) 6.11×10−52 1.26×10−147 6.55×10−135 0 77.88 1.79×10−30 7.39×10−17 f3(x) 43722.24 1.27×10−233 6.4×10−225 0 2007.86 2.63×10−24 3.92×10−5 f4(x) 0.003952 8.29×10−5 9.39×10−5 7.25×10-5 0.463665 0.001386 0.002197 f5(x) 0 0 0 0 189.4202 0 2.3886621 f6(x) 4.32×10−15 8.88×10−16 8.88×10−16 8.88×10-16 8.60 8.88×10−16 1.05×10−13 f7(x) 0.014453 0 0 0 2.535341 0 0.002539 f8(x) 0.000584 0.000352 0.000714 0.000397 0.009826 0.000332 0.005038 a rolling bearing fault diagnosis method based on extreme learning machine... 897 fig. 5 fault diagnosis flowchart 4. experimental study 4.1. equipment and data in this paper, we used data from both normal and faulty rolling bearings, which were provided by case western reserve university (cwru). fig. 6 illustrates the experimental simulation platform of cwru. the experimental setup includes a 2 hp motor, a torque transducer, a dynamometer, and the corresponding control electronics. during the experiment, vibration signals were collected using an accelerometer with a sampling frequency of 12 khz. case 1: vibration signals for different fault conditions were diagnosed under varying loads, with the fault diameter in the rolling bearings kept constant. the data that collected from the 6 o'clock position at the 12k drive end serves as the primary source, including fault damage diameter of 0.36mm inner and outer rings, rolling elements, and normal bearing data. the specifics of the bearing data are provided in table 3. case 2: diagnose vibration signals under different fault diameters while keeping the load on the rolling bearing constant. the data source acquisition method remains unchanged, including the normal state of the bearing inner ring and the fault damage diameters of 0.18 mm, 0.36 mm, and 0.53 mm, respectively. the bearing data are presented in table 4. 898 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi fig. 6 baring fault simulation test-bed table 3 overview of the data sets for case 1 table 4 overview of the data sets for case 2 4.2. proposed model evaluation 4.2.1. experiment 1 validation among the collected data, we selected 120,000 vibration signal samples and 10 seconds sampling time. the vibration signals were divided into 400 segments, each containing 300 points. the time-domain waveform of the initial signal is shown in fig. 7. from fig. 7, it can be seen that identifying the fault type from the time-domain waveform is challenging and can be time-consuming even when successful, highlighting the urgent need for the proposed diagnostic method. we use vmd to decompose the vibration signal of rolling bearings, with vmd decomposition k and penalty factor α values of 4 and 1989, respectively [36]. fig. 8 illustrates the modal components along with their corresponding frequency domain diagrams for various fault types. it can be seen from figure 8 that the energy in the low-frequency range is more pronounced for the normal state, while the inner and outer ring failures exhibit the highest energy in their respective frequency bands. the rolling element fault shows significant energy distribution in both low and high-frequency ranges. short-duration low-frequency pulses in the vibration signal indicate shock events due to partial damage in the bearing. additionally, the signal contains resonant components generated by these shock events, which excite highfrequency intrinsic vibrations, modulated by their amplitude. bearing status fault diameter(mm) acronym label normal ball fault inner race fault outer race fault n 1 0.36 bf 2 0.36 irf 3 0.36 orf 4 bearing status fault diameter(mm) acronym label normal inner race fault n 1 0.18 0.36 0.53 irf-0.18 irf-0.36 irf-0.53 2 3 4 a rolling bearing fault diagnosis method based on extreme learning machine... 899 finally, the imfs constituents obtained from the signals of the four fault types are decomposed according to the vmd and the energy entropy is calculated. the characteristic sample sets are obtained by vmd decomposition and calculation of energy entropy, with 300 sets per sample set and a total of 1200 sets. from each group, 250 groups are randomly selected, of which 600 groups are used as the training sample set and 400 groups are used as the test sample set. set the population size of iwoa n=20, the maximum value of iterations m=50, amax=2, amin=0, q-learning discount factor γ=0.5, m=3, the value of nodes of elm hidden layer is set to 100. the parameters of the elm network weights and thresholds are optimized by the iwoa. table 5 shows the results of diagnosing four fault states using the iwoa-elm method. the accuracy of fault diagnosis for the training samples is 98.83%, and for the test samples, it is 97.5%. these findings indicate that the method successfully identify all four fault states of rolling bearings. (a) (b) fig. 7 the time-domain waveforms of rolling bearing fault types: (a) case 1; (b) case 2 900 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi (a) (b) (c) a rolling bearing fault diagnosis method based on extreme learning machine... 901 (d) fig. 8 modal components and frequency distribution diagrams obtained after fault decomposition: (a) normal; (b) ball fault; (c) inner race fault; (d) outer race fault table 5 diagnostic performance results for samples in diverse fault scenarios, divided into training and test groups. 4.2.2. experiment 2 validation in the data collected, we selected 120,000 vibration signal samples from normal and three bearing inner rings with different degrees of failure (sampling time: 10 seconds). the vibration signals were divided into 400 segments, each containing 300 points. fig. 7 shows the time-domain waveform of the raw vibration signal. decompose the vibration signal of rolling bearings using vmd, with the decomposition number k and the penalty factor α value of 4 and 1992, respectively. fig. 9 displays the modal components and their frequency plots for inner ring of the bearing after fault decomposition at damage levels of 0.18 mm and 0.53 mm. since the normal state and the 0.36 mm fault diameter have been discussed, we will not elaborate further here. as shown in fig. 9, the energy associated with the inner ring failure is primarily focused on in the mid-frequency range. furthermore, for the same section of the bearing, the amplitude corresponding to specific frequencies increases with the degree of damage. this comparison shows that as a bearing transitions from a normal state to a failed state, the primary energy in its spectrum is gradually transferred from low frequency band to middle frequency band. subsequently, we calculated the energy entropy of imf based on vmd decomposition of normal state signals and three types of fault vibration signals in the inner ring. bearing status training number/ testing samples n bf irf orf training accuracy testing accuracy n 150/100 149/99 1/1 0/0 0/0 99.33% 99% bf 150/100 1/1 147/97 1/2 1/0 98% 97% irf 150/100 0/0 0/2 148/95 2/3 98.67% 95% orf 150/100 0/1 0/0 1/1 149/98 99.33% 98% 902 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi (a) (b) fig. 9 frequency domain diagrams show inner ring damage with diameters of: (a) 0.18 mm and (b) 0.53 mm finally, 1200 sets of feature samples were obtained through vmd decomposition and energy entropy calculation, including 300 sets for each of the three different fault degrees and normal states. randomly select 250 groups from each group, with 600 groups as training samples and 400 groups as testing samples. set the population size of iwoa to n=20, the maximum iterations number m=50, amax=2, amin=0, q-learning discount factor γ=0.5, m=3, the elm activation function is a sigmoid function, and the number of hidden layer nodes is 100. the weights and thresholds of elm are optimized through iwoa. the results of using iwoa-elm to diagnose normal state and three different degrees of fault samples in the bearing inner race are shown in table 6. the diagnosis accuracy reached 99% for the training samples and 96.75% for the test samples. this discrepancy may be attributed to certain failure feature parameters being less distinct at varying degrees of a rolling bearing fault diagnosis method based on extreme learning machine... 903 failure damage, making it challenging to differentiate between states. however, with the increase of the degree of failure, salient characteristics of the characteristic parameters of the failure state become more obvious. table 6 comparison diagnostic results on training and testing samples across multiple fault severity levels 4.3. comparison of different fault diagnosis models in order to verify the effectiveness of optimization algorithm in improving the accuracy of fault diagnosis, we compared it against several other methods, including svm, elm, psoelm, gwo-elm, and woa-elm. the population sizes for iwoa, woa, and gwo were all set to 30, with a maximum of 100 iterations. the pso algorithm was configured with a population size of 30 and executed for 100 iterations, setting the maximum particle velocity to 0.8 and the minimum particle velocity to -0.8, and the inertia weight to 0.9. for extreme learning machines, g(x) was the sigmoid function and there were 100 hidden layer nodes. in the case of the svm, the penalty coefficient was chosen as c=2 and the kernel width coefficient was set as g=0.2. the diagnostic results of different fault diagnosis methods for case 1 and case 2 are shown in table 7 and table 8, respectively. the convergence curves and comparison between predicted and actual results are shown in fig. 10 and fig. 11, respectively. comparing the results in tables 7 and 8, the iwoa-elm model achieved the highest fault identification accuracy for both different types and levels of bearing fault diagnosis table 7 diagnosis results of different fault diagnosis methods for case 1 model training accuracy rate testing accuracy rate iwoa-elm 98.83% 97.5% woa-elm 98.67% 95.5% gwo-elm 98.33% 95.0% pso-elm 98.17% 95.0% elm 98.17% 90.5% svm 92.00% 91.25% table 8 diagnosis results of different fault diagnosis methods for case 2 model training accuracy rate testing accuracy rate iwoa-elm 99.00% 96.75% woa-elm 98.17% 96.00% gwo-elm 98.00% 94.75% pso-elm 98.17% 94.75% elm 98.00% 92.75% svm 96.17% 94.50% bearing status training number/ testing samples n irf0.18 irf0.36 irf0.53 training accuracy testing accuracy n 150/100 149/99 0/0 1/0 0/1 99.33% 99% irf-0.18 150/100 0/1 145/90 5/9 0/0 96.67% 90% irf-0.36 150/100 0/0 0/0 150/100 0/0 100% 100% irf-0.53 150/100 0/1 0/0 0/1 150/98 100% 98% 904 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi in the training set samples. this suggests that the model effectively leverages the training set, resulting in optimal training outcomes. the test set samples further validate this conclusion, as the iwoa-elm model also achieves the highest diagnostic accuracy. furthermore, comparing the fault diagnosis results with and without optimization shows that the optimized method consistently outperforms the unoptimized approach in terms of accuracy. in addition, while the elm diagnostic model surpasses the svm model on the training set, the svm model performs better on the test set. this discrepancy highlights the differing applicability of these methods to various forms of fault diagnosis, leading to varying diagnostic results. however, this does not affect the diagnostic conclusions. the convergence curves of the different diagnostic models in figs. 10 and 11 show that the iwoa-elm diagnostic model gets better fitness values relative to the optimized elm diagnostic model by other algorithms at the same number of iterations. (a) (b) fig. 10 comparison of different fault diagnosis models under case 1: (a) the convergence curves; (b) comparison of test set prediction results a rolling bearing fault diagnosis method based on extreme learning machine... 905 (a) (b) fig. 11 comparison of different fault diagnosis models under case 2: (a) the convergence curves; (b) comparison of test set prediction results 5. conclusion the setting of parameters has a significant impact on the performance of recognition algorithms for rolling bearing fault diagnosis. in response to this issue, the paper introduces an improved whale optimization algorithm into rolling bearing fault detection and proposes a fault diagnosis model based on the iwoa-elm algorithm. firstly, the vmd method is used to decompose the vibration signals of rolling bearings, and the energy entropy values of imf decomposed for different scenarios are calculated. furthermore, multiple optimization strategies are used to improve the performance of the woa algorithm, and it is combined with elm to optimize the input layer weights and hidden layer thresholds. comparing the improved algorithm with svm, elm, gwo-elm, pso-elm, and woa906 x. li, s. guo, d. sun, l. cao, c. li, s. tian, p. liu, y. qi elm algorithms, the diagnostic results show that the rolling bearing fault diagnosis method based on iwoa-elm can effectively improve the convergence speed and have a higher recognition rate. the idea proposed in this paper of optimizing elm related parameters through improved optimization algorithms to enhance the performance of rolling bearing fault diagnosis can provide some reference for future research on rolling bearing fault diagnosis. acknowledgement: this paper was funded by the science and technology project of hebei education department (grant no.: zd2022087); the hebei innovation center for smart perception and applied technology of agricultural data (grant nos.: adic2023y002, adic2024t001); the central government guides the local 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35. xue, j., shen, b., 2020, a novel swarm intelligence optimization approach: sparrow search algorithm, systems science & control engineering, 8(1), pp. 22-34. 36. he, c., wu, t., gu, r., jin, z., ma, r., qu, h., 2021, rolling bearing fault diagnosis based on composite multiscale permutation entropy and reverse cognitive fruit fly optimization algorithm–extreme learning machine, measurement, 173, 108636. 13385 facta universitatis series: mechanical engineering vol. 23, no 1, 2025, pp. 171 181 https://doi.org/10.22190/fume250114012h © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper surface energy characterization of a single microsphere particle using peakforce quantitative nanomechanical mapping mode of atomic force microscope heebo ha1, sebastian mueller2, svetlana guriyanova2, byungil hwang3 1department of intelligent semiconductor engineering, chung-ang university, republic of korea 2material science, basf se, germany 3school of integrative engineering, chung-ang university, republic of korea orcid ids: heebo ha https://orcid.org/0000-0003-3047-2263 sebastian mueller https://orcid.org/0009-0009-4843-1575 svetlana guriyanova https://orcid.org/0009-0007-2622-6787 byungil hwang https://orcid.org/0000-0001-9270-9014 abstract. the composites are typically created by embedding functional fillers in polymeric matrices. the key indicators of reliable performance include homogeneous filler dispersion and optimal interfacial properties between the fillers and matrix. poor dispersion causes stress concentration, premature failure, and increased filler requirement for achieving functional properties like conductivity. additionally, interfacial characteristics are governed by surface energy differences and significantly influence the mechanical properties like elongation and rupture stress. conventional methods for measuring surface energy, such as contact angle or washburn adsorption, have limitations when applied to microscale powders, including issuers with accurate size handling and bulk material requirements. to overcome these challenges, the peakforce qnm (quantitative nanomechanical mapping) mode of atomic force microscopy (afm) was used to measure the surface energy on the surface of the powder microspheres. this method was used to characterize the surface energies of 2–5 μm carbonyl iron powders with various surface coatings. differences in the surface energies corresponded to variations in the dispersibility and mechanical performances of these powder-based composites. the proposed nanoscale approach provides critical insights into the interfacial mechanisms, enabling enhanced design and optimization of stetchable composites for advanced electronic applications. key words: carbonyl iron powder, composites, surface energy, contact angle, peakforce quantitative nanomechanical mapping, atomic force microscopy received: january 14, 2025 / accepted march 23, 2025 corresponding author: byungil hwang school of integrative engineering, chung-ang university, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr https://orcid.org/0000-0003-3047-2263 https://orcid.org/0009-0009-4843-1575 https://orcid.org/0009-0007-2622-6787 https://orcid.org/0000-0001-9270-9014 172 h. ha, s. mueller, s. guriyanova, b. hwang 1. introduction given the increasing interest in flexible and stretchable electronics, there is increasing demand for compliant composites that are reliable under severe mechanical deformations [1-3]. the main strategy for fabricating such composites involves incorporating functional fillers along with the stretchable polymeric matrix [4-6]. generally, these fillers are one or two-dimensional materials that are dispersed in the polymer matrix via a homogenization process [7-9]. depending on the physical properties of each component, these stretchable composites can be used in various applications [10-12]. in the fabrication of these composites, homogeneous dispersion of the fillers in the matrix is an important step for achieving highly stable stretchable composites [7, 13]. agglomeration of the filler material can cause involuntary stress concentration that can result in failure of the composite at a much lower stress level than that expected for a composite with uniformly dispersed fillers [14, 15]. furthermore, the amount of filler needed to achieve a certain level of functional properties, such as electrical conductivity or magnetic interference, can be much larger for composites with inhomogeneously dispersed fillers. another factor that affects the physical stability of a stretchable composite is the characteristics of the interface between the filler surface and matrix [16, 17]. depending on the status of interface formation, the mechanical properties of a composite such as its elongation or rupture stress is determined; thus, controlling the interfacial characteristics is crucial to obtaining the desired mechanical properties of stretchable composites [18, 19]. the dispersibility of a filler in a matrix as well as the interfacial characteristics between the filler and matrix are closely related to the differences in their surface energies [20, 21]. the higher wettability of a solid filler in a matrix indicates better dispersibility in the matrix [22, 23]. the method of measuring the contact angle using the sessile drop is the most common approach for obtaining the surface energies of each of the components in the composites [24, 25]. however, for microscale powders, the size of the liquid drop is larger than the particle size, which can cause errors in the surface energy calculations [26, 27]. the washburn adsorption method is another option for evaluating the surface energy of a microscale powder [28], where the amount of liquid adsorbed on the powder is converted to the wetting ability of the powder, i.e., its surface energy [29]. however, group of agglomerated particles is required to measure the surface energy using the washburn method; here, the surface energy of the individual particle is ignored even though it is important for understanding the detailed mechanisms of the changes in the mechanical properties of the composites. hence, this method cannot be used for measuring hydrophobic (>90°) powders. to overcome the limitations of the conventional surface energy characterization methods for microscale powders, the peakforce quantitative nanomechanical mapping (pf-qnm) imaging mode of atomic force microscopy (afm) was utilized. in the pf-qnm mode, the adhesion force is measured with nanometer resolution and converted into surface energy using a calibration curve established for well-known silane-coated surfaces [30, 31]. for example, park et al. [32] characterized the surface energy of nanoscale au strips using the afm pf-qnm mode. this work revealed a relationship between adhesion force and contact angle at the nanoscale, representing a novel measurement method for challenging characterization of the nanoscale surface energy. in another work by ha et al. [33], the grain size effect on the surface energy of metal thin film was investigated through pf-qnm method. however, there have been no attempts to characterize the surface energies of microscale powders. herein, the surface energies of various carbonyl iron powders (cips) with diameters surface energy characterization of a single microsphere particle using peakforce… 173 in the range of 2–5 μm and different hydrophobicities were characterized using the pf-qnm mode of afm. different surface coatings on the cips resulted in different surface energy values that could change the dispersibility in the composite states. the surface energies of individual microscale powders will enhance the understanding of the detailed mechanisms regarding the effects of interfacial characteristics on the mechanical properties of the cip-based composites. 2. materials and methods 2.1. materials and sample fabrication in this study, the commercially available carbonyl iron powders (basf se) of different grades (hq, cc, and hq-i) were used as received. additionally, the surface of hq cip was chemically modified by the different functionality. for this, the native oxide layer was needed to be removed, firstly. 100 g of hq grade was added to 250 ml of 0.5 m hydrochloric acid (hcl; 35–37%, daejung chemical co. ltd.) and sonicated for 30 min. a permanent magnet was then used to settle the cips at the bottom of the container. the sedimented cips were later separated from the solvent and sequentially washed thrice with ethanol and deionized (di) water for 15 min each. to synthesize the cip coated by tetraethyl orthosilicate (teos), ethanol and teos were mixed in a 7.6:1 ratio and reacted for a total volume of 200 ml. one drop of hcl was added to the mixture to create an acidic ph environment that acted as the catalyst. the solaging process was conducted by storing the mixture for 24 h to ensure full reaction development. the pretreated hq grade with removed native oxide layer were added to the teos mixture and mechanically stirred at room temperature for 2 h, resulting in the formation of the sio2 layer on the cip surfaces. the teos coated cips were settled at the bottom of the container by using a permanent magnet, and the solvent was removed. the hq@teos particles were sequentially cleaned thrice with ethanol and di water for 15 min each. finally, the hq@teos were dried in a vacuum oven at 120°c for 12 h to evaporate the remaining ethanol and water on their surfaces. all cip grades and their particle size and chemical surface modification are listed in table 1. to compare the conventional sessile drop method for determining the surface energy with the afm pf-qnm technique, a pet substrate was coated with a 30 μm thick acronal dispersion layer (basf se, ludwigshafen, germany) using a doctor blade. afterwards, the cips were sprinkled onto the acronal surface, and the unattached powder particles were removed by using a nitrogen blow gun. to measure the adhesion force on cip surface with nanometer resolution, the afm pfqnm technique needs a calibration curve to convert the adhesion data into a corresponding surface energy as well as contact angle. for this purpose, various self-assembled alkyl silane monolayers (sams) in a wide range of hydrophobicities were prepared on an ultraflat silicon wafer. in a first instance, the silicon wafers were initially ultrasonic cleaned in a 1:1 mixture of acetone:ethanol for 15 min and subsequently dried under a nitrogen gas atmosphere. moreover, the cleaned silicon wafers were treated with oxygen plasma (100 w) for 5 min to activate the surface for the chemical treatment. directly after the plasma treatment, the clean and activated silicon wafers were immersed in a 2% silane/ anhydrous toluene solution for 15 min. subsequently, the wafers were rinsed with toluene to remove any aggregates or contaminants. the sam-treated silicon wafers were dried at 150 °c for 5 min on a hot plate. 174 h. ha, s. mueller, s. guriyanova, b. hwang table 1 carbonyl iron powder (cip) grades and their average particle size and surface coating sample particle size (d50) coating hq 1.7 μm no hq@teos 1.8 μm teos modified cc 4.3 μm silica hq-i 3.6 μm i = insulated 2.2. characterization the surface morphologies of the cips were analyzed using field-emission scanning electron microscopy (fesem; s-3400 n, hitachi). the particle sizes of the cips were determined using a particle size analyzer (psa; microtrac p3000, usa). prior to measuring the particle sizes, calibrations were performed to eliminate the influence of ethanol. the contact angle and surface energy were measured with a drop shape analyzer (dsa100, krüss, germany) by using the sessile drop method. di water, formamide and diiodomethane were used as solvents that represented the polar and nonpolar state, respectively. the surface roughness and adhesion force were measured by afm (dimension icon microscope, bruker, usa) with a standard silicon cantilever (rtespa-300–125; bruker, tip radius: 125 nm, spring constant: 40 n/m). to ensure a consistent tip force, the deflection sensitivity and spring constant were calibrated. 3. results and discussion analyzing particles with different surface properties is a key step in the measurement of single-particle adhesion force at the nanoscale resolution. although most of the particles have similar structures, their surface roughness and coating conditions tend to vary. using pf-qnm, it was possible to analyze individual particles with surface energy across a broad range. the use of cip particles that are commonly used in industries showed that it was possible to investigate diverse sample conditions with different coatings and surface roughness. the cips included hq grade with a native oxide layer as well as silicaor insulate-coated grades, which enable comparisons using the conventional sessile drop method as well as other approaches. additionally, particles with teos-modified surface functionalities and roughness were fabricated in this study. fig. 1 illustrates the sem top-view, afm topography and phase images of the different cip grades with a varying particle size distribution, surface roughness, and coatings. in fig. 1(a), and (b), after removing the native oxide layer of the hq particles, they were coated with silica to produce the teos particles with similar sizes and shapes [34, 35]. consequently, the teos particles coated with thin silica layers exhibited reduced surface roughness. the ra roughness values (calculated from the 1µmx1µm afm topography image) for the hq grade were initially ~2.6 nm but decreased to ~2.2 nm for the teos particles. however, fig. 1(c), and (d) showing silica or insulated coatings presented higher surface roughness; these rough and dotted surfaces resulted in ra roughness values of 4.5, and 5.5 nm, respectively. to measure the contact angles of the cip particles with varying surface structures, the sessile drop method was used as the conventional measurement method. however, significant measurement errors were noted owing to time-dependent variations. the spreading rate of the solvent droplet was influenced by the particle hydrophilicity, which created challenges surface energy characterization of a single microsphere particle using peakforce… 175 in obtaining precise contact angle measurements. furthermore, as the droplet size was larger than the average particle size, there was additional complexity as the effects of both the particles and substrate (acrylic glue used for fixation) could not be disregarded. fig. 1 the sem top-view, afm tapping mode topography and corresponding phase images show the particle distribution, surface structure and roughness of different cip grades: (a) hq, (b) teos-modified, (c) cc, and (d) hq-i fig. 2 shows the changes in the contact angles over time from 0 to 30 s. the hq grade particles showed relatively small reductions from 119° to 102°, while the cc and hq-i grade particles showed steeper reductions from 78° to 58° as well as 43° to 17° over a dwell time of 30 s, respectively. these results represent the strong time dependency of the contact angle measurements, which can decrease the accuracy of the particle surface energy measurements. 176 h. ha, s. mueller, s. guriyanova, b. hwang fig. 2 macro contact angle variation of cip grades according to time measured by using the krüss drop shape analyzer the need for precise nanoscale measurements and particle-specific analysis is increasingly evident. the afm pf-qnm mode offers a high-resolution method for characterizing the mechanical and chemical surface properties at the nanoscale. pf-qnm is an innovative afm mode that uses low frequencies to periodically tap the peak force on the wanted area of the sample, allowing real-time recording of force versus separation curves. the peak force, representing the maximum vertical force applied to the sample surface, is recorded. in pf-qnm mode, the software calculates the force between the afm tip and the sample surface based on feedback that reflects surface morphology and mechanical properties, so the peak force remains constant. fig. 3 presents a schematic illustration of the nanoscopic contact angle (nca) measurement on micro-particles using afm pf-qnm. the macroscopic wettability behavior is correlated with adhesion force and can be evaluated using the pf-qnm mode to calculate the contact angle and surface energy (se) at the nanoscale. a variety of silane-modified silicon wafers are utilized as references to determine the nca and se across a wide hydrophilic-hydrophobic range. on these reference silane layers, the macroscopic contact angle and surface energy are obtained through the sessile drop method, while the adhesion force on silanes was measured using afm to create the calibration curves (which will be explained later in the paper) with linear fit equations: 𝑛𝐶𝐴 = −4.4 ∙ 𝐹𝑎𝑑ℎ + 185.2 (1) 𝑆𝐸 = 2.5 ∙ 𝐹𝑎𝑑ℎ − 23.4 (2) where fadh represents the adhesion force measured from the sample surface. the types of silanes used, along with their corresponding macro contact angle, surface energy, and adhesion force values, are listed in table 2. subsequently, the contact angle and surface energy values of the microscale cip particles are derived by comparing the measured adhesion forces with those from the previously established calibration curve. surface energy characterization of a single microsphere particle using peakforce… 177 fig. 3 schematic illustration of the process to evaluate the nano contact angle or surface energy on micrometer sized cip particle measured by the afm pf-qnm technique table 2 macro contact angles, surface energies, and adhesion forces of different alkyl silane types as reference system for the calibration curve silane type contact angle [°] surface energy [mn/m] adhesion force [nn] octyltrichlorosilane (ots) 105 27 18 (3-aminopropyl)triethoxysilan (aptes) 91 54 21 (3-glycidyloxypropyl)trimethoxysilane (gptms) 77 42 25 methoxy(polyethyleneoxy)propyl]triethoxysilane (pegn-si) 38 61 33 the high-resolution topological afm images of the surface of individual cip particles are shown in fig. 4. variations in adhesion force are influenced by the surface structure, roughness and chemical groups. the particulate surface structure of each cip grade, as shown in fig. 4, was compared with the corresponding fadh presented in fig. 5, and the data were appropriately analyzed for the nca and for the se calculations. fig. 4 afm pf-qnm topography images on the surface of single cip particle of different grades: (a) hq, (b) teos-modified, (c) cc, and (d) hq-i fig. 5 afm pf-qnm adhesion force images on the surface of single cip particle of different grades: (a) hq, (b) teos-modified, (c) cc, and (g) hq-i 178 h. ha, s. mueller, s. guriyanova, b. hwang fig. 6 calibration curves according to adhesion forces from the different alkyl silane monolayers measured by afm pf-qnm versus the (a) macro contact angle and (b) surface energy measured with the krüss drop shape analyzer. (c) adhesion forces measured on cip surface versus the nano contact angle and surface energy calculated from the calibration curve. (d) comparison of the nano and macro contact angle of the cip grades measured with two different measuring systems – krüss drop shape analyzer versus afm pf-qnm using the calibration curve given by eqs. (1) and (2), the adhesion forces of the cip particles with varying surface conditions and roughness from fig. 5 were used to calculate the nca and se values, shown in fig. 6(c) and (d). these results show high correlations between the adhesion forces and conventional macroscopic methods. the hydrophilicity of the silanes used for the calibration curves significantly influenced the fadh, which was correlated with the nca and se (figs. 6 (a) and (b)). the fadh for single cip particles were substituted into the equations to calculate the nca and se values. fig. 6 (c) presents the nca and se values of single cip particles with different surface conditions and roughness; this shows that hydrophobic particles exhibit lower fadh, consistent with their higher nca. conversely, hydrophilic samples show higher fadh, confirming that the fadh are inversely proportional to nca and directly proportional to se. fig. 6 (d) highlights the correlations between the macro contact angle measured at 0.1 s and the nca values, showing their similarity. this indicates the potential applicability of the proposed method to similar types of particles, as it confirms the time intervals suitable for conventional techniques that evolve over time. consequently, the proposed approach is anticipated to be extendable to a wide range of particle types. surface energy characterization of a single microsphere particle using peakforce… 179 the findings of this study demonstrate the feasibility of accurate nanoscale measurements and possibility of in-situ analysis of single particles. this study focused on the surface properties of single particles by assuming stable chemical structures. however, it also shows the possibility of in-situ analysis of chemically unstable particles or surfaces for future research endeavors. 4. conclusions in this study, the feasibility and effectiveness of using pf-qnm mode in afm was demonstrated for characterizing the surface energies of individual microscale particles, specifically microscale cips. by overcoming the limitations of conventional methods such as the sessile drop and washburn adsorption methods, the pf-qnm mode offers a nanoscale resolution approach to surface energy measurements. the results reveal that the surface energies vary significantly depending on the surface modification and roughness of the cips. these variations can directly influence the dispersibility of the particles in composite materials, thereby affecting their mechanical properties and functionalities. the correlations between adhesion force measurements and traditional contact angle methods validate the reliability of the pf-qnm approach. additionally, the high-resolution insights provided by the proposed technique offer new possibilities for in-situ analyses of chemically or structurally unstable particles, extending its applicability to a broader range of materials and industries. the results of this study highlight the critical roles of interfacial characteristics 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yun3, jun young cheong4, sooman lim5, byungil hwang6 1department of intelligent semiconductor engineering, chung-ang university, republic of korea 2material science and engineering, physical science and engineering division, king abdullah university of science and technology (kaust), saudi arabia 3department of molecular science and technology, ajou university, republic of korea 4bavarian center for battery technology (baybatt) and department of chemistry, university of bayreuth, germany 5department of flexible and printable electronics, lanl-jbnu engineering institute, jeonbuk national university, republic of korea 6school of integrative engineering, chung-ang university, republic of korea abstract. this study reviews the potential of flexible strain sensors based on nanomaterials such as carbon nanotubes (cnts), graphene, and metal nanowires (nws). these nanomaterials have excellent flexibility, conductivity, and mechanical properties, which enable them to be integrated into clothing or attached to the skin for the real-time monitoring of various activities. however, the main challenge is balancing high stretchability and sensitivity. this paper explains the basic concept of strain sensors that can convert mechanical deformation into electrical signals. moreover, this paper focuses on simple, flexible, and stretchable resistive and capacitive sensors. it also discusses the important factors in choosing materials and fabrication methods, emphasizing the crucial role of suitable polymers in high-performance strain sensing. this study reviews the fabrication processes, mechanisms, performance, and applications of stretchable strain sensors in detail. it analyzes key aspects, such as sensitivity, stretchability, linearity, response time, and durability. this review provides useful insights into the current status and prospects of stretchable strain sensors in wearable technology and human–machine interfaces. key words: strain sensor, flexible, nanomaterials, mechanical property, deformation received: september 25, 2023 / accepted november 01, 2023 h. ha, n. qaiser, t. yun, j. cheong, s. lim contributed equally to this work. corresponding author: byungil hwang school of integrative engineering, chung-ang university, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr 752 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang 1. introduction flexible and wearable devices that adhere to the skin are garnering interest and are actively developed [1-5]. these devices offer valuable features, such as the ability to interact with and monitor the human body [6-11]. however, conventional strain sensors that rely on metals or semiconductor materials exhibit limited sensitivity and stretchability, making their integration into wearable devices challenging [12-15]. notably, these sensors possess high stretchability to adhere to the skin or body joints effectively. moreover, they must satisfy various requirements, including high elasticity, flexibility, durability, low power consumption, and biocompatibility [16, 17]. fig. 1 state-of-the-art flexible, wearable strain sensors made of nanomaterials and polymer composites. reprinted with permission from [11], copyright 2023 american chemical society. reprinted with permission from [18], copyright 2022 springer nature. reprinted with permission from [19], copyright 2021 wiley. reprinted with permission from [20] under the creative commons attribution 4.0 international (cc by 4.0) license. reprinted with permission from [21], copyright 2020 american chemical society. sensing mechanism and application of mechanical strain sensor: a mini-review 753 hence, carbon nanotubes (cnts) [22-31], graphene [26, 32-36], metal thin films [5, 37], and nanomaterials, such as metal nanowires (nws) [38-45] , have emerged as promising candidates for flexible strain sensors. these materials exhibit remarkable flexibility and conductivity, as well as exceptional mechanical, electrical, optical, and chemical properties. using these nanomaterials, flexible strain sensors can be seamlessly integrated into clothing or adhered directly to the human body for the real-time monitoring of physical activities with impressive efficiency. these materials are often combined with polymers to achieve desired flexibility, durability, and biocompatibility. however, balancing stretchability and sensitivity remains a challenge, hindering the development of high-performance strain sensors [46]. a strain sensor converts mechanical deformations into electrical signals. various types of strain sensors exist, including fiber bragg gratings [47], triboelectric sensors [38], and piezoelectric sensors [48, 49]. however, using strain sensors for wearable applications poses challenges owing to limitations such as the need for sophisticated measurement equipment, low resolution, and dynamic performance issues [50]. some highly sensitive strain sensors can detect small strains and offer excellent electrical responses. however, they tend to lose their conductivity when subjected to relatively large strains, which makes them incapable of detecting subtle changes. for instance [51], while graphene-based strain sensors have a high gauge factor (gf) of 103, indicating substantial changes in electrical resistance owing to deformation, their detectable range is limited to 6%. conversely, highly elastic sensors are better equipped to handle substantial strains but exhibit poor sensitivity. strain sensors based on materials such as finely crushed graphene, graphene nanopaper, and silver nanowires can detect a wide range of strains exceeding 70%. however, within the range of 20% or less strain, they exhibit a low gf value and insufficient sensitivity. wearable sensors are broadly categorized into two types: resistive [52] and capacitive [53]. both require relatively simple systems and offer high flexibility and stretchability. resistive sensors typically consist of conductive sensing films integrated with flexible substrates. as the composite material expanded, the microstructure of the sensing film changed, altering its resistance in response to the applied strain. when the strain was removed, the composite returned to its original state, and the resistance returned to its initial value. however, capacitive sensors rely on electrodes and an insulator. the capacitance changes with the applied strain, leading to alterations in the electric field. an essential consideration in fabricating flexible strain sensors is the selection of the appropriate materials and manufacturing methods. the fabrication process should facilitate low-cost, large-scale production and convenience. to date, stretchable strain sensors have been created by incorporating materials such as carbon black (cb) [54-56], carbon nanotubes (cnts), nws, nanoparticles (nps) [57, 58], silicone-based elastomers (e.g., polydimethylsiloxane (pdms) [11, 26, 59-61], polyurethane (pu) [62-64], or ecoflex [30, 48, 65]), and rubbers (natural rubber and thermoplastic elastomers) [66-69]. the impact of the choice of the polymer type on the strain-sensing performance, although relatively minor, is explored in detail in another section. various methods for fabricating stretchable strain sensors, including filtration [70], printing [27, 37, 71, 72], transfer [73, 74], coating [75, 76], and chemical synthesis [77], have been reported. for example, a highly stretchable and sensitive strain sensor was produced by filtering silver nanowire (agnw)-pdms through a filtration process. this involves coating the agnws onto a silicon substrate, followed by pdms filtration to create a film by curing and peeling. in addition, a flexible 754 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang and sensitive strain sensor was fabricated by spray coating graphene flakes onto a pet substrate [73]. numerous other methods have also been explored, such as strain sensor fabrication through printing and solution mixing to create nanocomposites (fig. 2). fig. 2 fabrication processes of stretchable strain sensors. a) printing technology. reprinted with permission from [71], copyright 2019 american chemical society. b) solution mixing. reprinted with permission from [64], copyright 2021 american chemical society. c) spinning and transfer method. reprinted with permission from [74], copyright 2019 american chemical society. in this paper, we review the fabrication methods, mechanisms, strain-sensing performance, and applications of stretchable strain sensors. we begin by describing the general mechanism of these sensors, followed by an explanation of the responsive behavior mechanisms of both resistive and capacitive sensors. in addition, we discuss critical aspects, such as stretchability, sensitivity, linearity, response time, durability, and other factors that validate the performance of stretchable strain sensors. finally, we provide relevant examples and offer insight into the future of this technology. this review enhances the crucial understanding of the presentation of novel ideas to resolve the limitations related to these sensors. 2. strain-response mechanism 2.1. geometrical effect the response of a strain sensor varies depending on factors such as material, micro/nanostructure, and manufacturing process. the traditional strain resistance response results from the geometry and piezoresistivity of a material [78]. however, unlike sensing mechanism and application of mechanical strain sensor: a mini-review 755 conventional strain gauges, modern stretchable strain sensors utilize mechanisms such as separation between sensing elements, crack propagation in films, and tunneling effects. when a material undergoes stretching, it tends to contract laterally owing to its poisson ratio. in the case of a solid, its resistance (r) can be calculated using the formula: 𝑅 = 𝜑𝐿 𝐴 (1) where φ represents the resistance, l is the length, and a is the cross-sectional area of the solid. as the length increases and the cross-sectional area decreases owing to contraction, the resistance of the solid increases. however, when a capacitive sensor is subjected to stretching, an increase in capacitance arises from changes in the thickness of the dielectric layer and alterations in the capacitor area [79]. the formula for calculating the capacitance (c) is based on the overlap area between the two electrodes (a) and thickness (d) of the dielectric layer. this is expressed as follows: 𝐶 = 𝜀0𝜀𝑟 𝐴 𝑑 (2) where 0 and r are the permittivity of vacuum and relative permittivity of the dielectric, respectively. as the sensor experiences strain, its capacitance of the capacitive sensor increases by preserving the capacitive region while reducing the thickness of the dielectric layer. assuming that the stretchable electrode and the dielectric layer of the stretchable strain sensor share the same poisson ratio, calculating the capacitance is possible during stretching. in this scenario, the capacitance of the capacitive sensors increased linearly with applied strain. however, importantly, if the applied strain becomes excessively large, the linear relationship dictated by poisson's ratio may not hold for polymers. therefore, recognizing that this linear function is valid within a specific range of strains and may not be applicable beyond that range is essential. 2.2. piezoresistive effect piezoresistance refers to the change in the resistance of a material owing to structural deformation [80-83]. this piezoresistance can be expressed as following eq. (3): ∆𝑅 𝑅 = (1 + 2𝑣)𝜏 + ∆𝑝 𝑝 (3) where ν is the poisson’s ratios for the stretchable layer, τ is the applied strain, r is the measured resistance and p is the applied pressure. semiconductor materials such as si and ge can significantly increase the sensor resistance by altering the bandgap between atoms [84]. conversely, some metals and metal alloys exhibit piezoresistive properties that can substantially increase strain gauge resistance. moreover, materials at the nanoscale, such as carbon nanotubes (cnts) and nanowires (nws), can exhibit ultrahigh resistance depending on factors such as chirality and barrier height [85, 86]. this makes semiconductors and nanoscale materials promising for fabricating highly sensitive strain gauges. however, despite their potential, the flexibility required for human attachment and wearable strain sensors surpasses that of conventional strain gauges fabricated using metals and semiconductors. these limitations are attributed to technical challenges, inconsistent 756 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang electrical and mechanical properties, and low elasticity. consequently, swcnts (singlewalled carbon nanotubes) and nw-based sensors have not been widely adopted for strainsensing applications. to address these limitations, novel approaches were introduced to create more flexible and sensitive deformation sensors by incorporating nanomaterials into elastic polymer composites. unlike previous methods, these approaches aimed to enhance the flexibility and sensitivity of strain sensors. however, notably, the contribution of piezoresistive nanomaterials to the overall resistance and strain behavior of elastic strain sensors is expected to be limited. this limitation arises from the weak interfacial bonding strength between the nanomaterials and polymers, as well as the anticipated significant elastic mismatch between the two components [87]. 2.3. disconnection mechanism in a conductive film comprising conductive nanomaterials, electrons traverse the percolation network formed by stacked nanomaterials [16, 88]. as the film underwent stretching, the area of overlap between the connected nanomaterials decreased, increasing the resistance. at the microstructural level, the disconnection of overlapping nanomaterials during stretching occurs because of weak interfacial bonding between the materials and significant elastic mismatch between the nanomaterials and polymer. open circuits within conductive nanomaterials obstruct electrical pathways, resulting in exceptionally high resistance and diminished performance of the strain sensors. as the strain level increases, the disconnection mechanism becomes more pronounced, and the performance of the sensor becomes heavily reliant on this process of disconnection and reconnection, leading to irreversible sensitivity degradation (fig. 3). elastomer composite sensors incorporating nanomaterials, such as graphene flakes [89], agnws [42], and znonws [90], have been reported. similar mechanisms have been observed where disconnections occur at the junctions between nws after expansion, causing increased resistance and decreased performance. stretchable and sensitive strain sensors were developed using thin films of nw networks and graphene flakes, focusing on disconnection mechanisms. for instance [11], in the case of agnws embedded in pdms, where the agnws possess young's modulus of 81–176 gpa, and pdms exhibits a modulus of 2–3.5 mpa, during stretching cycles, the agnws within the pdms slip into the pdms matrix, leading to disconnections at the nw– w junctions. another example involves sensors using agnw, graphene and pdms that exhibit a gf of 9156 under strain of >36% [91]. even with various temperature range, these sensors rely on a disconnection mechanism to detect changes in resistance. in this context, the rigid nature of graphene sheets result in microcrack junctions allowing resistance changes to be observed. similar mechanisms have been identified for sensors that use the rolling resistances of zno nws and graphene flakes. sensing mechanism and application of mechanical strain sensor: a mini-review 757 fig. 3 (a) schematic of the sensing mechanism and sem images of the strain sensors at various stretching states: (b,c) 0%, (d, e) 20%, (f, g) 40%, (h, i) and back to 0%. scale bars in (b), (d), (f) and (h) are 20 μm. scale bars in (c), (e), (g) and (i) are 5 μm. reprinted with permission from [91], copyright 2021 american chemical society. 2.4. crack propagation cracks tend to form in thin and delicate coating layers on the film and can be easily initiated and propagated when subjected to stretching. these cracks typically originate from regions where the stress is concentrated, thereby effectively releasing localized stress. fine cracks developed in the cnts, nws, nps, and graphene films coated on stretched substrates [58, 92, 93]. as the stretching process continued, these cracks expanded within the thin film, significantly increasing the resistance. this mechanism is used to create highly sensitive and flexible strain sensors. moreover, several methods have been explored to enhance the strain sensitivity and gf further. one approach involves peeling off the agnws in conjunction with the pdms and cutting the composite material mechanically [11]. as shown in fig. 4, owing to the structural alterations induced by mechanical cutting, certain areas experienced partial cutting, allowing the strain to reach up to 30% and consequently yielding a gf value of 290.1. notably, the resistance returned to its original state, indicating complete recovery. 758 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang fig. 4 (a) schematic of the sensor and electrical performances of strain sensor under applied strain. (b) optical images of the top surface of the sensor under applied strain showing the crack opening. reprinted with permission from [11]; copyright 2023 american chemical society. 3. performance of stretchable strain sensors 3.1. stretchability the performance of strain sensors depends on various critical factors, including stretchability, sensitivity, gf, linearity, hysteresis, response and recovery times, and durability. these factors are significant for wearable strain sensors that can be exposed to prolonged, repeated strain. the stretchability of a strain sensor exhibits substantial variability, contingent on factors such as the sensor type, utilized materials, structural design, and manufacturing techniques. for instance [94], a resistive sensor composed of an aligned cnt-pdms composite achieved an impressive stretchability of ~280%. this exceptional stretchability was attributed to the consistent propagation of microcracks within the cnt film and interconnection of the aligned cnts. similarly, capacitive sensors demonstrated ~300% stretchability using cnt–silicone elastomer composites owing to the remarkable stretchability of the dielectric material and robust nature of the cnt-based strain electrodes. the integration of 1d nanomaterial-polymer composites plays a pivotal role in significantly enhancing the stretchability of strain sensors [95]. owing to the high aspect ratio of 1d nanomaterials, an effective network can be readily established, ensuring stable electrical and mechanical properties, even under conditions of substantial stretchability. strain sensors employing cb, nps, and graphene films exhibit stretchabilities >70% [9698]. examples include graphene–cellulose composites, fragmentized graphene foam sensing mechanism and application of mechanical strain sensor: a mini-review 759 (fpf)–pdms composites, and graphene–rubber composites. conversely, the limited stretchability observed in cb-, np-, and graphene-based strain sensors can be attributed to the absence of a well-connected network of nanomaterials with lower aspect ratios [99]. 3.2. sensitivity or gauge factor the sensitivity or gf of the strain sensors was determined by the relative change in the electrical signal and slope of the applied strain. the gf can be calculated for both resistive and capacitive sensors using the following equation: 𝐺𝐹 = ∆𝑅 𝑅0𝜌 , 𝐺𝐹 = ∆𝐶 𝐶0𝜌 (3) where c0, r0, and ρ, are the initial capacitance, resistance and the dielectric constant. traditional metal film-based strain gauges typically offer a gf in the range of 2–5, while semiconductor-based strain sensors can provide gf values >100 [16]. by contrast, stretchable strain sensors can exhibit various gf values depending on factors such as the sensing mechanism, nanomaterial used, and structural design of the sensor. among resistive sensors, a broad range of gfs can be achieved by adjusting the density of the agnws. some studies have also aimed to attain high gf values and stretchability by reducing the agnw concentration and employing a disconnection mechanism. however, these approaches have limited reliability. highly sensitive sensors using cracks and disconnection mechanisms with graphene have been reported; however, they tend to have very low stretchability [100]. by contrast, capacitive-type sensors offer high stretchability but typically possess very low gf values. most capacitive sensors have gf values <1. theoretically, a gf of 1 represents the upper limit for capacitive sensors, making achieving a gf higher than this value impossible [101]. 3.3. linearity linearity is a crucial factor affecting the performance of stretchable strain sensors. this is particularly relevant for resistivity-based sensors, where nonlinearity can complicate data processing and adversely affect data collection. nonlinearity often arises when the microstructure of a thin film changes from homogeneous to nonhomogeneous owing to stretching, causing the strain sensor to respond nonlinearly to the applied strain. nonuniform generation and propagation of microcracks during deformation contribute to this nonlinear response in the strain sensor data calculation process. efforts have been made to address these limitations, including enhancing cnts through hybrid structures that incorporate metals or other nanomaterials. however, these efforts mostly resulted in a trade-off between high linearity, sensitivity, and stretchability. conversely, capacitive sensors typically exhibit excellent linearity but lower sensitivity than sensors with a gf of 1 or higher. high-sensitivity strain sensors often exhibit high nonlinearity and low stretchability. consequently, the development of a strain sensor with high stretchability and sensitivity remains a significant challenge. 760 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang 3.4. hysteresis in the context of using strain sensors for body attachment and wearable applications, hysteresis is a critical consideration, particularly when the sensor is subjected to dynamic loading. significant hysteresis can lead to irreversible sensing performance under dynamic loads. capacitive sensors generally exhibit a better hysteresis performance than resistive sensors [53]. the performance of capacitive sensors relies on maintaining a stable overlapping area between the electrodes rather than on the conductivity of the electrodes. however, resistive sensors are more susceptible to hysteresis because they are affected by changes in the conductivity of the thin films, which can contribute to greater hysteresis effects. in addition, hysteresis can stem from interactions between the polymer and nanomaterial fillers as well as the viscoelastic properties of the polymer [94]. strong interfaces between the polymer and filler materials improve strain sensor performance. conversely, weak connections between nanomaterials and the polymers can lead to slipping and increased disconnections during stretching, thereby causing high hysteresis [102]. researchers have endeavored to develop applications that can be attached to the body while significantly reducing hysteresis. for example [103], one approach involved creating a hydrogel using materials like pedot:pss and polyvinyl alcohol (pva), demonstrating a remarkably low hysteresis of ~1.44%, even in a strain range of 300%. however, room for improvement exists in terms of sensitivity, as this particular hydrogel exhibited a relatively low gf value of 4.07. 3.5. response and recovery time the response time is a crucial parameter that indicates how quickly a strain sensor returns to its normal state after experiencing a strain. because of the viscoelastic properties of polymers, all polymer-based strain sensors inherently exhibit some degree of response delay. typically, the standard response time of stretchable sensors is defined using a 90% time constant. for resistivity sensors based on materials such as agnw/pdms [104] and agnw/agnp/pdms [43], response times of 200 and 51 ms, respectively, were reported. by contrast, capacitive sensors exhibit faster response times. capacitive sensors based on mxene/pvp [105] and agnw/pdms [106] show response times of 30 ms. the recovery time is another important performance factor for stretchable strain sensors. capacitive sensors generally have recovery times in milliseconds, whereas resistive sensors require few hundred milliseconds to recover. for instance, capacitive strain sensor exhibit a 15 ms recovery time [105], while resistive strain sensors of aunp/pdms [107] and cnt/pdms [108] composites require 250 and 500 ms to recover. the recovery time of nanocomposite-based strain sensors depends on the interactions between the fillers and polymers and varies based on the type and ratio of the fillers used. capacitive sensors offer advantages such as high linearity, stretchability, and low hysteresis. however, they have a very low gf and can exhibit unpredictable responses owing to unstable overlapping areas and interactions. conversely, resistive sensors provide high gf values but often respond to applied strains with high hysteresis and nonlinearity. sensing mechanism and application of mechanical strain sensor: a mini-review 761 4. applications 4.1. human motion detection and healthcare flexible strain sensors can continuously record a wide spectrum of bodily movements ranging from significant joint actions to subtle deformations associated with physiological signals. consequently, these strain sensors have garnered considerable attention as essential components in healthcare monitoring devices, serving purposes such as human motion detection and diverse healthcare applications. developing highly sensitive sensors capable of generating high-resolution and accurate signals is important to meet the demands of detecting minute variations such as pulses or respiration [20, 25, 43, 44, 64]. fig. 5a illustrates the application of strain sensors employing agnw/pdms composites to monitor pulse rates by attaching them to the brachial and radial arteries. after undergoing structural transformation via partial cutting, the sensors exhibited a strain of up to 30% and an impressive gf value of 290.1. in addition, the sensors demonstrated a rapid increase in resistance through a disconnection mechanism and exhibited exceptional performance stability even after 1000 reliability test cycles. the sensors respond to changes in blood flow pressure resulting from the pulse, enabling the tracking of various pulse characteristics. furthermore, these sensors can be affixed to the back of a person to discern body movements in different directions, thereby facilitating human motion detection and healthcare monitoring, particularly for spinal issues [11]. in addition to the pulse and major body motions, wearable strain sensors can discern alterations in facial expressions and emotional states [64]. capacitive sensors strategically placed around the eyes and mouth enable the detection of actions such as laughter, crying, and frowning with high sensitivity. monitoring diverse emotional expressions is valuable in assessing conditions such as alzheimer's disease. fig. 5b further illustrates the application of these sensors to the neck, capturing changes in resistance corresponding to vocal cord movements, including phonetic sounds like “i,” “a,” and “m.” moreover, wearable sensors can gauge vital parameters, such as respiratory rate, oxygen saturation, and heart rate, and transmit these data wirelessly. these sensors can connect to emergency service centers to dispatch alarm messages during emergencies. this strain sensor was engineered to concurrently measure respiratory volume and rate using polystyrene, au, and pt [20]. these sensors were attached to the rib cage and abdomen to measure volume variations during respiration and to calculate the respiratory rate and volume. the precision was confirmed by comparing the resistance changes due to respiration with spirometer readings, yielding a high-precision level with an r2 of 0.92. unlike conventional wearable sensors, which typically measure only the respiratory rate, this sensor can also estimate the respiratory volume. it can monitor breathing across different activities, is not limited to stationary positions, and is invaluable for identifying individuals with chronic respiratory conditions through everyday wireless bluetooth devices. these versatile sensors, which are suitable for placement on various parts of the human body, hold promise for applications in diverse fields, including exercise, health monitoring, and disease management. 762 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang fig. 5 demonstrations of the strain sensor for personal health monitoring. (a) blood pulse on the radial artery and brachial artery and spine for monitoring lower back strain. reprinted with permission from [11], copyright 2023 american chemical society. (b) detection of human joint movements and the vocal cord vibration. reprinted with permission from [64], copyright 2021 american chemical society. sensing mechanism and application of mechanical strain sensor: a mini-review 763 4.2. soft robotics wearable strain sensors play a pivotal role in soft robotics, a field of technology characterized by robots crafted from pliable materials. these sensors are instrumental in overseeing and directing the shape, movement, and interactions of soft robots in their environment. unlike their traditional counterparts, soft robots exhibit natural fluidity of motion and adaptability to their surroundings. they are typically affixed to soft grippers and detect the force and positioning required for secure object manipulation and lifting. the sensors transmit these data to the robot control system, enabling real-time control and interaction. applications of soft robots span diverse fields, including medicine, structural inspections, environmental exploration, and disaster relief [8, 19, 24, 109]. fig. 6 recognition of finger dexterity, gestures, and temperatures. (a) relationship between the δr/r0 of mwnts/pdms fibers recorded by a tablet oscilloscope and the angle of finger joint bending. metacarpophalangeal point (mp), proximal interphalangeal point (pip), and distal interphalangeal point (dip) are three important joints in the hand. (b–e) photograph of the smart glove for recognition of (b) the bending angles of the finger, (c) numbers, and (d) alphabets. reprinted with permission from [21], copyright 2020 american chemical society. for example, a hydrogel-type strain sensor comprising pedot:pss and pva was applied in soft robotics [103]. this strain sensor was attached to the gripper where alterations in the gripper angle corresponding to the size of the grasped object were observed, changing 764 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang the resistance. this relationship was linearly corrected by analyzing the bending angle and resistance variations. furthermore, this study demonstrates that an industrial robot can be remotely guided by hand manipulation by integrating these sensors. the strain sensors continuously monitor the bending motion of each finger, allowing the controller to direct the movements of the robot based on automatically recognized signals. the experiment underscores that industrial robots can be precisely, reliably, and immediately controlled with minimal hysteresis, thereby ensuring stable performance and high resilience. in addition, other research explored the creation of a smart glove equipped with a strain sensor using mwcnt/pdms fibers [21]. (fig. 6) when worn, this smart glove detects resistance changes when the fingers are bent or flexed. numbers and alphabets can be recognized according to the bending of the finger joints, and the corresponding information can be checked through the tablet on the back of the hand. remarkably, this system exhibited an impressive high linear coefficient r2 of >0.995 within the 0-120% strain range. moreover, it demonstrated excellent durability, enduring 20,000 cycles of stretching and release with a 50% strain applied. nevertheless, it shows the limitation of having very low gf values from 0.45 to 2.08 with different. 4.3. virtual reality another compelling application of wearable strain sensors is as motion sensors for virtual reality (vr) and augmented reality (ar). by affixing these sensors to various parts of the human body, such as gloves, stockings, and clothing, users can engage in immersive gaming experiences that leverage motion detection, pressure sensing, and hand movements, all of which are facilitated by these motion sensors. moreover, wearable strain sensors not only detect the user's movements and touches but also govern interactions within virtual environments, playing a pivotal role in gaming, educational simulations, and other forms of entertainment [110, 111]. as shown in fig. 7a, this research [19] introduced a sensor embedded in a smart glove designed to track the intricate movements of an individual's fingers. these smart gloves, equipped with printed pcb chips, established bluetooth connections with the robot’s hands, enabling them to respond to an array of gestures. the strain sensor using liquid metal and pdms exhibited an impressive stretch capacity of up to 105% strain and a remarkable detection limit as low as 0.05%. by demonstrating remarkable durability over 8000 cycles at 50% strain, these smart gloves proved adept at tracking the movements of various body parts in the virtual reality realm. fig. 7b shows the implementation of a strain sensor based on the mxene/swcnt/pva composite [18]. for human–machine interaction purposes, strain sensors were strategically positioned on the elbow, knee and other joints. these sensors were linked to a data acquisition circuit board that was meticulously designed to interpret finger movements, gather data, and transmit them to a computer via bluetooth. this setup allows users to control vr movements by interacting with the strain sensors. furthermore, these sensors highlight their potential as versatile motion sensors for both real-world movements and immersive realms of virtual and augmented reality. notably, they offer several advantages over traditional camera setups and motion detection systems, including less power consumption, high sensitivity of > 1000, cost-effectiveness, and high accuracy. sensing mechanism and application of mechanical strain sensor: a mini-review 765 fig. 7 (a) demonstration of robotic hand operated by remote system. reprinted with permission from [19], copyright 2021 wiley. (b) photos of the remote control system and snapshots of playing the game with fingers to control all the actions. reprinted with permission from [18], copyright 2022 springer nature. 766 h. ha, n. qaiser, t. g. yun, j. y. cheong, s. lim, b. hwang 5. conclusion and future work table 1 summarizes the strain sensor types, materials, and main features of their applications. this study provides an overview of recent developments in the production methods, mechanisms, strain-sensing capabilities, and practical applications of stretchable and wearable strain sensors. wearable sensors using various materials, including carbonbased materials, 1d and 2d materials, and metals, have exhibited remarkable performance enhancements. this paper also discusses several prominent applications of wearable sensors. however, further performance improvements are vital for real-world applications. within the scope of wearable sensors, capacitive sensors offer advantages such as stretchability, linearity, and low hysteresis but exhibit limited sensitivity. by contrast, resistive sensors provide significantly higher sensitivity and stretchability but suffer from drawbacks such as high hysteresis and nonlinear responses. table 1 summary of the strain sensor types, materials, and main features materials type key performance pros and cons [ref] agnw@pdms resistive gf : 290.1 (30% strain) ▪ high gf and multifunctional ▪ low stretchability and still limited sensing area [11] au/pt@polystyrene resistive high linearity ▪ multifunctional ▪ low gf (2.64) [20] pedot:pss@pva resistive ultralow hysteresis (<1.5%) linearity (r2 = 0.98) ▪ ultralow hysteresis, high linearity (r2 = 0.98) and high stretchability (~300%) ▪ low gf (4.07) and limited sensing area [103] graphene nanosheet@pu fiber resistive gf : 661.59 (0%–75% strain) ▪ high stretchability(~500%), gf and reliability (>10,000 cycles) ▪ high hysteresis and limitations depending on strain area [112] cnt/mxene@tpu resistive gf : 363 light weight ▪ high gf and wide working strain range ▪ nonlinearity depending on strain area [64] lm@pdms capacitive ultralow detection limit (<0.05%) ▪ excellent mechanical robustness (>8,000 cycles) and fast response time (58 ms) ▪ limited sensitivity [19] agnw@pmma capacitive gf : 1 ▪ transparent and low detection limit (<12 mg) ▪ limited sensitivity [113] recent advancements in material science and structural research led to the development of composite materials and functional structures to address these challenges. however, issues such as durability degradation over time, sensitivity concerns, and varying gf values persist as strain sensors exhibit nonlinear responses and variable gf values depending on the strain applied, making them suitable only for specific applications. furthermore, to make these sensors practically viable, they must be evaluated under diverse conditions, sensing mechanism and application of mechanical strain sensor: a mini-review 767 including temperature and humidity variations, which can influence the sensor performance. precise measurements are crucial for elastomers, which are commonly used elastic materials, because their physical properties can change with temperature. in addition, when fabricating wearable sensors from elastomers, they must be designed to ensure breathability and comfort for end users. despite the proposals for fiber and nanowire composites, performance enhancements are still required. furthermore, research is needed into new microstructures and nanostructures to enable multimodal sensing capabilities. addressing these challenges holds the potential for the application of strain sensors in physical stimulation, wearable medical devices, smart surgical equipment, drug delivery monitoring, and artificial organ research. furthermore, the development of multifunctional strain sensors may open doors for various research and applications, including those involving biological components. acknowledgement: this research was supported 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universitatis series: mechanical engineering vol. 21, no 2, 2023, pp. 293 305 https://doi.org/10.22190/fume200828008z © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper novel methodology for real-time structural analysis assistance in custom product design milan zdravković, nikola korunović faculty of mechanical engineering in niš, university of niš, serbia abstract. mass-customization is related to optimizing the balance between flexibility, strongly required by the customer-focused industries and manufacturing efficiency, which is critical for market competitiveness. in the conventional industries, the process of designing, validating and manufacturing a product is long and expensive. some of the common approaches for addressing those issues are parametric product modeling and finite element analysis (fea). however, the costs involved are still relatively high because of the very special expertise needed and the cost of the specialized software. also, the specific design of the product cannot be validated in a real-time, which often leads to making hard compromises between the specific customer requirements and the structural properties of the product in its exploitation. in this paper, we propose the novel methodology for real-time structural analysis assistance for custom product design. we introduce the concept of so-called compiled fea model, a machine learning (ml) model, consisting of dataset of characteristic product parameters and associated physical quantities and properties, selected ml algorithms and the sets of associated hyperparameters. a case study of creating a compiled fea model for the case of internal orthopedic fixator is provided. key words: machine learning, gradient boosting, finite element analysis, parametric modeling 1. introduction the contemporary production has shown significant progress in adopting disrupting technologies such as rapid prototyping, cloud-based storage, enhanced interoperability of diverse enterprise information systems in the value chain and, last but not the least, internet of things. two of the most qualitative effects of such digitalization to the production processes are more efficient mass-customization [1] and streamlined collaboration-based value chain [2]. while the latter unleashed the vast, diverse real-time data about operations, logistics and received august 28, 2020 / accepted january 02, 2021 corresponding author: milan zdravković faculty of mechanical engineering in niš, university of niš, ul. aleksandra medvedeva 14, niš, serbia e-mail: milan.zdravkovic@gmail.com 294 m. zdravković, n. korunović product lifecycle, the former pushed the trend of servitization [3] over the limits. this trend created the opportunities for enhanced collaboration in a product value chain and affordable use of high-end services in the whole production span, starting from structural product analysis to marketing automation and micro-customer segmentation. recently, advances in applied artificial intelligence (ai) have made possible notable acceleration and quality improvement in the product design stage, especially by considering the integrated product lifecycle management [4], extended lifecycle in circular economy [5] while benefiting for the integrated access to vast data about its design and exploitation [6]. in this paper, we explore the practical impact of using the new disrupting technologies (namely, machine learning and cloud-based integration) to resolving the problem of costand time-efficient validation of the design of the custom product, based on product family generic design. such generic design is often represented by the parametric model of the complex product geometry, with other associated relevant features, such as exploitation and environment properties, material properties and others. customized product design problem is often solved by parametric modeling. instead of designing the custom product instance from scratch (or by adapting the existing model to new desired properties), designers can choose the appropriate values of the previously defined, critical features of the product family geometric and structural properties, namely, the product parameters. those choices are made based on the different criteria, including customer requirements, part and material market cost and availability, product pricing policies, exploitation conditions, manufacturability and others. the set of the product’s parameters combined with the other fixed properties is called a parametric product model. the benefits of such a product modeling approach for customization are numerous. the parameters can be used to adapt the product design to the different aspects of its exploitation, as well as manufacturing, such as manufacturability (if the design service is outsourced), cost (including materials and manufacturing complexity), assembly restrictions and customerfocused requirements, such as usability and customer-tailored design (for example, special medical devices that need to be fully adopted to the patient's physiognomy and physiology), among others. in any stage of the custom product design, the single design instance may be validated. such validation can be relatively simple and quick (for example, inspection of the product visual properties by the customer) but sometimes, very troublesome and significant costincurring, such as testing of the product instance physical properties and its integrity in the exploitation conditions. some of the physical quantities and properties that can be of great importance for the design are deformation, stress and product mass. testing in exploitation conditions is often replaced by simulating those conditions by using the finite element analysis (fea) method [7]. fea relies on the finite element method (fem), which is a numerical method highly used in structural, thermal, and various multi-physics analyses to simulate product behavior in exploitation and calculate the fields of various physical quantities. fea could help to calculate the extreme values of physical quantities and properties, such as deformation, stress or strain and compare them to critical ones, before the product is prototyped and tested in realistic environments. unfortunately, despite numerous researchers’ efforts to consider possibility of real-time simulation [8][9], the fea analysis of the customized products is often removed from the design pipeline due to the mass-customization related time/cost pressures (fea software annual subscription rates are as high as tens of thousands of dollars), long duration (complex product fea alone, novel methodology for real-time structural analysis assistance in custom product design 295 even without considering fea model preparation, can last for hours, even days), highlevel expertise requirements and consequently, high service cost. we addressed the above issues by assuming the following scenario (see fig. 1). a manufacturing company maintains a parametric model of the product family design. upon the customer request, the designer needs to create this model’s instance, so this instance meets all the given requirements. instead of launching the fea on the specific instance, the designer is assisted in a real-time by the software which is using the model we call the “compiled” generic fea model. this software is integrated with the cad package used by the designer. fig. 1 concept of using compiled fea models for real-time assistance in product design and validation the compiled fea model is based on the physical quantities and properties (for example, level of mechanical stresses in critical product areas, product mass and the like) of the number of “characteristic” data instances. characteristic data instances dataset is a relatively large collection of product model parameter (lengths, widths, distances, material properties, etc.) values in the selected regions, associated with previously calculated mechanical quantities and properties (such as stresses and product mass). those are calculated once (by using fea software) for each of the parameters' instances, for the whole product family and then used to fit the prediction function, derived by using a machine learning (ml) algorithm [10]. therefore, the compiled fea model is actually a serialized ml model and it involves the dataset with characteristic instances, the selected ml algorithm and the best performing hyper-parameters. from the performance point of view, predicting the physical quantities and properties based on the specific set of the parametric model values is trivial and such service can be executed in a real-time, during the custom product design. more important, no additional cost is incurred. the key hypothesis of the research work behind this paper is that based on the above dataset, ml models can be developed for predicting physical quantities and properties of the custom product which was instantiated by selecting the appropriate design parameters with sufficient accuracy. another hypothesis, which will not be addressed in this paper is that multi-criteria optimization methods [11] can be used to identify all local optimums, 296 m. zdravković, n. korunović namely, to identify the characteristic instances from the dataset that are associated with the best combination of physical quantities and properties. some initial work addressing the optimization problem has been already done [12]. the concept of the solution has been already proposed by the authors [13]. in this paper, the concept is further elaborated and demonstrated by considering realistic design and exploitation aspects (dataset), with improved methodology, analysis of the results and their visualization. the remainder of the paper is structured as follows. first, a novel methodology for facilitating real-time assistance in validating the custom product design is presented. then, the methodology is demonstrated in the case study of validating the design of the internal fixator medical device. finally, the guidelines for the implementation of the methodology and its use in daily practice are provided. 2. methodology the process in which the compiled fea model is built consists of two major activities: design of experiment and training the prediction model. design-of-experiment (doe) feature of the selected fea tool is used to create the dataset of characteristic product instances, based on the selected product family parametric model. doe feature of standard fea tool is usually a part of design exploration functionality and serves as a basis for design surface-based optimization. design surfaces are fitted to the dataset obtained from doe and serve as meta-models predicting the relations between input and output parameters. various experimental plans are usually available when doe is performed. the choice of experimental plan depends on the non-linearity of the relations between design parameters and output parameters (such as deformation or stress). highly nonlinear relations require experimental plans that contain more data points and cover the whole design space, including its extreme values. the ml-based model proposed here requires that a detailed dataset should be created. if this cannot be accomplished using standard detailed experimental plans, a custom experimental plan may be used. after the experimental dataset is created by doe, the ml prediction model is created by fitting the selected ml algorithm with the dataset above, where the design parameters are considered as input and the physical quantities and properties as output features. the prediction model is developed by using the python programming language. its development follows the typical ml pipeline, namely correlation analysis, feature selection, algorithm selection and optimization of the selected algorithm hyper-parameters. correlation analysis aim is to reduce the problem dimensionality. for very complex products, number of design parameters can be measured in hundreds. while creating a compiled model for such a product is one-time job and thus it does not have a significant effect to a process, prediction (including necessary data pre-processing) may come with a computational cost and consequently slower performance which could affect the user’s experience. by selecting the most relevant product geometrical properties, we can address this problem. two-way correlation analysis will be performed. first, correlation of the individual parameters with the physical quantities and properties will be assessed by looking at the pearson coefficients. second, the recursive feature elimination (rfe) [14] method will be used to assess the combined relevance of all n-tuples of input features to each of the individual output features. novel methodology for real-time structural analysis assistance in custom product design 297 different ml algorithms will be tested to choose the one with the least mean absolute error (mae) a key indicator for assessing the accuracy. selected algorithms are linear regression, k-nearest neighbors, support vector machine regressor, decision tree and two ensemble methods, namely random forest and gradient boosting. k-nearest neighbors [15] is a non-parametric method used since the beginning of 1970-ties. it is so-called instance-based method; it stores all available cases/instances and classifies new cases based on a similarity measure (namely, a distance function). support vector machine (svm) belongs to the group of kernel methods [16]. it was initially developed for two-group classification problems. decision tree or in this case so-called regression tree is the method in which observations about some item, represented as branches are used to make decisions about its target values, represented as leaves. random forest [17] belongs to a group of ensemble methods that combine a number of decision trees, and then adopt a mean forecast of the predictions of the individual trees. random forest is today considered as one of the most powerful algorithms in the machine learning without considering artificial neural networks, namely deep learning architectures. gradient boosting [18] adopts the idea of boosting an optimization of a suitable cost function [19], where an ensemble of weak prediction models, namely decision trees are staged one after another. some of the selected algorithms, namely k-nearest neighbors and support vector machine regressor require that before training data is normalized (scaled in (0,1) range). feature scaling is required to reduce the training time and improve the prediction accuracy. those algorithms will be used to develop respective prediction models and test their accuracies. the algorithms with the best performances, as validated by k-fold cross validation method will be selected, trained and produced models will be serialized those models are actually what we call compiled models for real-time structural analysis assistance. standard deviations of the output features will be used as reference values for assessing the accuracies. k-fold cross validation is a method which produces reliable prediction accuracy metrics for a given dataset. instead of a single split between data for training and testing, it does k-1 splits where each of the folds/sections of data is used as a test set. hence, the model will be validated in k test runs, each of which will produce an accuracy measure. mean of those values is then adopted as accuracy of the prediction model. validation is being carried out for predicting each of the output features, namely, product physical quantities and properties. it is expected that the performance of the models based on different estimators will differ for some of the physical quantities and properties data. thus, all models, associated with the set of the optimal settings will be serialized. obviously, those with the best performances for the specific quantities and properties will be used for prediction. each of the estimators used is associated with the set of so-called hyper-parameters, which define its different properties, related to the way how the model is trained and validated. the best prediction performance for the given dataset is achieved only with a unique set of their values. this set, for each of the estimators and each of the output features is typically determined in a process called grid search optimization [20]. grid search calculates accuracy score (per defined scoring function using the specific metric) across defined hyper-parameter space (defined by the value ranges and/or enumerations), most desirably by using the k-fold cross validation method. the search through the 298 m. zdravković, n. korunović combination of the hyper-parameter value from the defined space can be exhaustive (all combinations) or randomized. all steps that involve use of estimators, namely rfe and training models with data, must be carried out with the same conditions. this applies to using not only the same parameters (k) but also the same data in different steps of k-fold cross validation. this is especially important for the models which are trained with a small number of instances ntuples of parameters and physical quantities and properties; in those cases, the results (especially in the optimization step) can be misleading. the exception from this rule may be the optimization process for regression problems, in the case that nmae is selected as a model metrics (which is the case here). then, mean squared error (mse) or r^2 regression score could be used. python implementation of the above methods and functions is scikit-learn [21] library. it is initially released in 2007. it consists of number of classification, regression and clustering algorithms, ensemble methods, data pre-processing tools, metrics, feature engineering tools and others. 3. case study the compiled fea model is developed for the case of orthopedic device – internal fixator, used in subtrochanteric fractures of thigh bone (femur). this is the case of a highly customizable product which needs to be fitted to the different requirements arising from patient physical and physiological properties, some of many different types of fractures, etc. the process in which this fitting is carried out is out of the scope of the research behind this paper. the fixator parametric model has been created by using solidworks cad software. in this case, it is defined by 6 relevant geometry parameters and fixed design. the illustration of the model is provided on fig. 2 below. fig. 2 parametric model of internal fixator, used in subtrochanteric fractures of thighbone (femur) design explorer module of ansys fea software, which was used for calculation, features the design-of-experiment functionality. namely, it is used to generate the set of novel methodology for real-time structural analysis assistance in custom product design 299 values of input parameters that defines the collection of characteristic product instances. these values are then used to create the cad model instances in solidworks and send them back to ansys for calculation of physical quantities. central composite design (ccd)/face centered/enhanced method was applied in planning the experiment (dataset generation). central composite designs are five level fractional factorial designs, which are appropriate for calibrating the quadratic response model. by default, ccd varies the input parameters on three levels each, but still generates less data than a full factorial plan. face centered type of ccd ensured that the extreme values of the input parameters were included in the dataset. „enhanced” option was used to add more data between extreme and middle values of input parameters, resulting in more extensive datasets. created dataset [22] is used as input to the typical ml pipeline. this dataset, with six parameters counts 89 rows. a small number of data instances were used in a case study for the practical reasons (single instance calculation of physical quantities and properties by ansys takes time) as well as because of strong representativeness of data generated by design-of-experiment feature, namely balanced distribution of parameters value over the given range. distribution of the output features data in the dataset is illustrated by using boxplots in the fig. 3 below. fig. 3 distribution of the physical quantities and properties values in the dataset (boxplots) standard deviations for total deformation maximum, equivalent stress and fixator mass are as it follows: std(def) = 1.075536549693965 std(str) = 88.62041183766537 std(mas) = 0.029059066340160897 3.1. correlation analysis analysis of data correlation was carried out by considering pearson linear correlation and recursive feature elimination (rfe) methods. the aim of the analysis is to determine if the dimensionality of the problem can be reduced, namely if it is reasonable to exclude some of the input variables from the training dataset. 300 m. zdravković, n. korunović it was found that there existed a significant linear correlation between: ▪ bar length and total deformation (p=-0.950) ▪ bar length and fixator mass (p=0.899) ▪ bar length and equivalent stress (p=-0.655) ▪ bar end thickness and equivalent stress (p=-0.633) a notable linear correlation is found between: ▪ bar diameter and total deformation maximum (p=-0.250) ▪ bar diameter and fixator mass (p=0.397) other input values did not have a notable linear correlation with output variables, namely deformation, stress and mass. this implies that some features could be removed from the model, namely, radius trochanteric unit, radius bar end and clamp distance. the correlation of the individual geometrical features with the physical quantities and properties is also illustrated with the scatter plots, displayed in fig. 4. fig. 4 correlation of the individual geometrical features with the physical quantities and properties (pearson) the issue of the linear correlation based on the pearson coefficient is that it can help in assessing only the relevance of individual input features for prediction of the output ones. in other words, while one specific input feature may have a very low correlation with the output, it may appear that in combination with the other ones, its changes may significantly affect the output features. thus, to complement the correlation analysis, a recursive feature elimination (rfe) method is applied in order to explore the relevance ranking of the subsets of the input novel methodology for real-time structural analysis assistance in custom product design 301 features. rfe is a method which performs backward feature elimination. the algorithm begins with the set of all features and successively eliminates the feature which induces the smallest effect on the output features. it can be applied by using the selected algorithms, in our case simple linear regression, svm, decision trees, random forest and gradient boosting regressors. knn is excluded because its regressor does not expose the attributes relevant to rfe. rfe method calculates ranks which are the measures of the relevance of individual input features in combination with others for predicting the output features. ranks are calculated in the range (1,5) where less value means better relevance/correlation. all results are then displayed in bar charts to provide an effective illustration of the ranks by different features (fig. 5). the relevance of the first three input features is clearly confirmed by the rfe method and all algorithms. the exception is svm regressor which produced outlier results because data was not normalized (requirement for svr) before rfe estimation. rfe with some of the algorithms suggest some relevance of the input feature #5, namely clamp distance for predicting equivalent stress and deformation mass. the behavior and performance of many ml algorithms are referred to as stochastic because they involve randomness (random state initialization of the models, random selection of data in k-fold cross validation, etc.). for that reason, the indicators that are produced by ml models are typically calculated as a statistical measure (for example, mean) of the population of the specific indicator values produced by the ml models in multiple runs. the minor relevance of clamp distance for predicting some of the output features is not continuously visible in multiple rfe runs. therefore, it should be excluded from the final set of features, together with trochanteric unit radius (feature #3) and bar end radius (feature #4). fig. 5 correlation ranks of the individual geometrical features with the physical properties (recursive feature elimination method) however, it is important to strongly highlight that the decision to reduce the dimensionality of the parameter set in this specific case would not be practical because the complexity of the product is very low, so the possible savings in the computational performance are infinitesimal. still, in the cases of very complex products with hundreds of parameters, this methodological step could help achieving critical benefits. 302 m. zdravković, n. korunović 3.2. compiled models according to the proposed concept, the compiled model is actually a serialized ml model for predicting the physical quantities and properties of the product, based on the parameter values. three characteristic quantities need to be predicted by the compiled model: maximum total deformation, maximum equivalent stress and fixator mass. before the model is compiled, the algorithm with the best performance needs to be selected from the pre-selected list that includes: linear regression, k-nearest neighbors, support vector machine regressor, decision tree regressor, random forest and gradient boosting regressor. in the following step, the selected ml algorithms with default hyper-parameters were fitted with the dataset and the outcomes of the resulting models’ accuracies were compared. k-fold cross validation (k=4) was used for validation and negative mean absolute error (nmae) was used as indicator. same kfold object will be used in all relevant steps in order to get comparable data. object is set not to shuffle data because randomness in selecting data for the folds with such a small dataset would not be beneficial. testing produced the results as shown in table 1. table 1 nmae for different algorithms with default set of hyper-parameters lre knn svr dtr rfr gbr total deformation maximum -0.182 -0.799 -0.209 -0.052 -0.090 -0.034 equivalent stress -39.678 -57.302 -65.971 -20.904 -21.127 -18.423 fixator mass -0.0041 -0.0203 -0.0243 -0.0015 -0.0026 -0.0009 all nmae indicators are well within the standard deviations for the considered output features and certainly within the limits of acceptable error in structural analysis of products of this type. given the high linear correlation, as found by the pearson coefficients, the expectation that the linear regression method will produce good results is confirmed. 3.3. estimator optimization grid search method was used for optimization of hyperparameters. while nmae is used in the table below, for the optimization, another metrics will be used, namely r^2 (coefficient of determination) regression score. best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). grid search optimization is implemented in iterative fashion where the specific set of optimal hyper-parameters is determined for each output feature (physical property) and each estimator. it produced the sets of hyper parameters which significantly improved the performance of models based on k-nearest neighbors and svr. notable improvement was made in predicting equivalent stresses with random forest and gradient boosting estimators. based on the r2 score values, several conclusions can be made. as expected, random forest and gradient boosting have shown the best general performance. both of those ensemble methods are most often used for addressing regression and classification problems by practitioners, despite possible overfitting issues. in our case, with balanced distribution of input features values, overfitting is not a serious concern. with optimized hyper-parameters, gradient boosting estimator produces excellent results in predicting fixator mass and total maximum deformation, with r2>0.99 and in prediction equivalent stresses (r2=0.91). comparable performance was achieved by random forest, in predicting fixator mass novel methodology for real-time structural analysis assistance in custom product design 303 (r2=0.92) and maximum total deformation (r2=0.95) and by svr, in predicting maximum total deformation (r2=0.98). values of nmae indicator after training the estimators by using the optimal sets of hyper-parameters are shown in table 2. table 2 nmae for different algorithms with set of optimal hyper-parameters lre knn svr dtr rfr gbr total deformation maximum -0.182 -0.331 -0.094 -0.133 -0.106 -0.047 equivalent stress -39.678 -37.421 -26.51 -15.726 -14.555 -14.778 fixator mass -0.0041 -0.0094 -0.0243 -0.0044 -0.0041 -0.0015 4. implementation to conclude, the considered research hypotheses have been convincingly confirmed in a case study. the developed ml model can be serialized as a compiled fea model and used in hypothetical cad tool add-on – container for compiled models of selected product families. cad model enriched with this add-on can provide real-time structural analysis assistance of custom product design and thus, significantly reduce its time and cost. fig. 6 depicts the design of the infrastructure for the implementation of the proposed solution for real-time assistance in customized product design. the process starts with the development of parametric model and design of experiment. design of experiment data is used to develop a compiled fea model, as described above. it is then deployed as a web service resource. web infrastructure facilitates: ▪ the deployment of compiled fea models and parametric models, ▪ management (including versioning) of non-geometric model parameters (in the above example, maximal equivalent stress over the product and product mass) ▪ end user authentication and tracking logic and ▪ a business model (subscription based, pay per view, etc.) of choice. fig. 6 concept of the integration of cad system with prediction services based on compiled models 304 m. zdravković, n. korunović it should be exposed by rest api with authentication and key verification functionalities. client is considered as add-on to one of the commonly used cad platforms. add-on facilitates: ▪ user login, ▪ definition and serialization of non-geometric model parameters (e.g. exploitation and environment effects, material properties) ▪ display of user interface with the add-on toolbox and visualization of predicted physical quantities and properties ▪ synchronous rest calls to a web service using associated compiled fea model, where input is current set of parameters (geometric and nongeometric) and output – predicted physical quantities and properties. 5. conclusion mass-customization trend, implying the need for design and manufacturing of custom product designs with efficiency near to mass-production is a new industrial reality. quite obviously, this trend creates new challenges in manufacturing and custom product design domains. most of the challenges are due to the efforts in finding the right balance between flexibility, strongly required by the customer-focused industries and efficiency, which is critical for market competitiveness. in more conventional industries, this balance is often searched for by implementing outsourcing practices, even for critical activities in the manufacturing process. another way is digitalization, which helps to facilitate fast decision-making processes and thus, quick responses to the variety of demand and supply circumstances. today, with the advance of ai methods and tools, it becomes possible to digitalize even knowledge-intensive operations and thus, not only reduce the lead time but also significantly reduce the total cost of product manufacturing. the proposed solution aims to solve the problem of a long and expensive custom product design process, and in specific, the need for a special (expensive) expertise in building fea models, lots of computational resources needed and expensive fea software. each parametric model is defined by the finite set of parameters, mostly related to geometrical features. the values of those parameters, in most of the cases vary within the specific range in order to keep the integrity of the design. the level of correlation of those values with the actual physical quantities and properties of the product define the guidelines important for the ordering process. this process now includes customization sub-process, in which the customer and the designer actually negotiate the design that fits the customer's requirements in the best possible way, in a real time, while still maintaining the integrity of the product in the target exploitation conditions and its manufacturability. the centerpiece of the proposed novel methodology is so-called compiled fea model, offering the best approximations of non-geometric parameters, vital for the exploitation behavior and manufacturability of the custom product design. the use of compiled fea model during geometric parameter tuning facilitates real-time review of the critical nongeometric features and immediate assessment of the designed product physical quantities and properties. moreover, the proposed solution creates opportunities for new collaborative business models, in which the roles of cad and fea specialists are separated across the enterprises and fea can be implemented as online service. novel methodology for real-time structural analysis assistance in custom product design 305 acknowledgements: this research was financially supported by the ministry of education, science and technological development of the republic of serbia. references 1. da silveira, g., borenstein, d., fogliatto, f.s., 2001, mass customization: literature review and research directions, international journal of production economics, 72(1), pp. 1–13. 2. simatupang, t.m., sridharan, r., 2005, the collaboration index: a measure for supply chain collaboration, international journal of physical distribution & logistics management, 35(1), pp. 44–62. 3. vandermerwe, s., rada, j., 1988, servitization of business: adding value by adding services, european management journal, 6(4), pp. 314–324. 4. kwong, c.k., jiang, h., luo, x.g., 2016, ai-based methodology of integrating affective design, engineering, and marketing for defining design specifications of new products, engineering applications of artificial intelligence, 47, pp. 49–60. 5. ghoreishi, m., happonen, a., 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vol. 17, no 2, 2019, pp. 207 215 https://doi.org/10.22190/fume190404026d © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper verification of rabinowicz’ criterion by direct molecular dynamics modeling andrey i. dmitriev 1,2 , anton yu. nikonov 1,2 , werner österle 3 , bai cheng jim 4 1 institute of strength physics and materials science sb ras, tomsk, russia 2 tomsk state university, tomsk, russia 3 federal institute for materials research and testing, berlin, germany 4 institute for composite materials, kaiserslautern, germany abstract. in the paper we use direct molecular dynamics modeling to validate the criterion for formation of wear debris proposed by e. rabinowicz in 1958. a conventional molecular dynamics using a classical tersoff’s potential was applied to simulate the sliding behavior within a thin film corresponding to a tribofilm formed from silica nano-particles in amorphous-like state. the simulation was carried out by varying the initial temperature and the spatial size of the simulated crystallite. the results show the change in sliding behavior of silica-based tribofilm depending on the temperature and the size parameter of the system under consideration. thus increasing the temperature provides smooth sliding while at moderate conditions wear process can occur via debris formation. our estimations show good correlation between predicted critical size of the simulated system and calculated energetic characteristics. key words: molecular dynamics, adhesive wear, wear debris, critical size, surface microasperities, thermal conditions, shear resistance force 1. introduction the problem of investigating relationships between surface roughness of solids and friction mechanisms is of great practical and scientific interest. as mentioned in [1–3], e. rabinowicz was one of the first to explore this issue. in 1958 he proposed an original criterion that determines the contact mode of two friction surfaces by means of estimating contact stresses [4]. as long as the stresses in the vicinity of contacting microasperities are received april 04, 2019 / accepted june 18, 2019 corresponding author: andrey i. dmitriev institute of strength physics and materials science sb ras, 634055, pr. akademicheski 2/4, tomsk, russia. e-mail: dmitr@ispms.ru 208 a.i. dmitirev, a.yu. nikonov, w. österle, b.c. jim less than the yield strength of a material, it is plastically deformed. in the case the stresses exceed the yield strength, cracking of the surface layer of the material and formation of wear debris can occur. thus, according to this criterion, the formation of wear debris and transition to adhesive wear are possible only when the contact area is larger than a critical value. the criterion proposed by e. rabinowicz can be written as follows 2 0 2g w d    , (1) where σ 2 0 – shear strength, g – shear modulus and ∆w – increment of surface energy. although this assumption can be verified numerically, there were some difficulties caused by the dominance of the methods of continual description of simulated medium. the limitation of continual models lies in their inability to explicitly simulate (without loss of accuracy) wear process because of complicated description of the formation of debonded fragments and new contact spots. therefore, the particle method and its various modifications seem to be substantially more effective [5–7]. in terms of this discrete approach the processes of multiple cracking and rebinding of the medium can be explicitly simulated. molecular dynamic simulation is one of the most widespread methods of discrete description. most commonly, atomistic simulation of adhesive wear predicts continuous smoothening of surfaces rather than sliding of contacting bodies with formation of wear debris. first of all, this is due to the limitations on spatial dimensions of the simulated system. in this regard, it is worth to emphasize the studies [8, 9] done by the group of molinari, which have firstly predicted the formation of wear particles induced by the damage of the surface layer of a material during adhesive interaction of surface microasperities within the framework of empirical molecular dynamics. this result was obtained by elaborating simple pair potentials with tuned inelastic properties, which then could be associated with macroscopic behavior of actual materials. the authors have proposed a simple analytic expression that is completely similar to eq. (1) and that predicts the transition to the adhesive wear mechanism in terms of single asperity size: * 2( )j w d g     (2) where σ 2 j like σ 2 0 in eq. (1) characterizes shear strength, while 𝜆 is the coefficient related to the surface roughness geometry. in order to study the relation between surface roughness parameters and wear mechanisms a single contact model elaborated within the framework of empirical molecular dynamics was used in [8, 9], which did not imply a tight link to the scale of a simulated sample. at the same time, worth mentioning is the study of the authors of the present paper [10], where md simulation of a tribofilm consisting of amorphous-like silicon dioxide particles resulted in the prediction of the conditions of wear mode transition of two contacting bodies from a laminar sliding regime to an abrasive wear regime with the formation of wear debris. in contrast to empirical molecular dynamics with phenomenological pair interaction, the tersoff three-particle interatomic potential [11] adapted for calculations of different si-o compounds like α-quartz and sio2 glass [12] was used there. moreover, in [10] it was considered a continuous tribofilm with verification of rabinowicz criterion by direct molecular dynamics modeling 209 periodical boundary conditions rather than single contacting microasperities with various dimensions, and the tribofilm temperature was used as a criterion for transition between the friction regimes. when the kinetic temperature of the sample was above the glasstransition temperature of sio2, the sliding regime was unstable and accompanied by the formation of an agglomerate of silicon and oxygen atoms. when simulating shear deformation of the tribolayer at the temperature corresponding to extreme conditions of a thermal burst in a contact patch, the sliding turns into the laminar regime. considering equations (1) and (2), it can be supposed that the system temperature specified in the simulation determines energetic characteristics of the simulated medium, in particular the ∆w and σ 2 j(σ 2 0) values. however, this assumption requires detailed verification. thus, the objective of this work is to study the sliding behavior of silica-based tribofilm in order to understand the influence of temperature of the atomic system and size parameter of the simulated sample on adhesive wear mechanisms. 2. numerical model the initial structure of amorphous silica sample was created by the following algorithm. at the beginning the initial structure of alpha-quartz (sio2) was formed by bonding si–o tetrahedra together. the bulk sample in a shape of the parallelepiped with the following geometry: 15.0 x 11.6 x 8.3 nm along x (lx), y (ly) and z (lz) directions respectively was used as a basic specimen (see fig. 1a). (a) (b) fig. 1 (a) initial structure of an alpha-quartz sample before heating. (b) resulting structure of the amorphous silica sample and a loading scheme for sliding simulation. hereafter the atoms colored in yellow are used to indicate the features of deformation along the sample. the following zones are indicated: a – atoms under loading (loaded layers), b – crystallite layers, c – melted/amorphous area. purple color denotes o atoms, green – si atoms 210 a.i. dmitirev, a.yu. nikonov, w. österle, b.c. jim to study the influence of size parameter on sliding regime of the simulated sample three other specimens where the width (lx) was equal to 10.0, 22.5 and 30.0 nm were generated as well. thus, the width of the narrowest specimen was corresponded to 0.67 width of the base sample, and the number of atoms in it did not exceed 110 thousand. the width of two other samples corresponded to width of the base sample multiplied by 1.5 and 2.0, which contained 270 and 360 thousand atoms, respectively. each specimen was divided into three zones: a, b and c, as shown in fig. 1. to prepare the amorphous layer the central zone c in each specimen was subjected to heating up to a temperature of 6000 k within the framework of a microcanonical ensemble nve until this fragment was completely melted. height (ly) of zone c in all samples was identical and equal to 10 nm. after 10 ps at 6000 k the melted fragment was quenched to the required temperature during 170 ps. at the end of this procedure, we obtained a block with amorphous silica interlayer as shown in fig. 1b. for generation of positions and velocities the modeled sample was considered as an nve ensemble that maintained the number of particles n, the occupied volume v and the energy of the system e. the integration of nose-hoover style non-hamiltonian equations of motion was utilized with a time step δt of 0.5 fs. to imitate the extension into a tribofilm, periodic boundary conditions were assigned along x and z direction. all simulations were performed using lammps [13], while for the visualization of the simulation results and structure analysis the visualization tool ovito [14] was used. the calculations were performed on the skif cyberia supercomputer resource of tomsk state university. the modeled structure was located between the two loaded layers and subjected to a sliding loading with constant velocities (v) +15 and –15 m/s applied to the upper and lower layers, respectively as shown schematically in fig. 1b. thus the total sliding velocity was 30 m/s. despite the rather large values, these are the characteristic velocities used in md simulation, since the calculation times are usually only a few nanoseconds. simultaneously with shear loading a normal force (f) of 1.3 nn was applied along +yand –y-directions yielding a total contact pressure of about 350 mpa. the temperature of the whole specimen was kept constant within nvt ensemble and was varied in the range from 300k (ambient temperature conditions) to 1100 k (for sliding simulation under flash temperature conditions at the local contact) [15]. the given temperature is achieved by using the velocity rescaling method during the md simulation process from the energy balance described by eq. (3): 2 1 3 2 2 n i i b i m v k t n  (3) where n is the atom number, kb is the boltzmann constant, mi and vi are the i th atom mass and velocity, respectively. since the mechanical properties of amorphous material is closely related to its density, the resulting density was kept constant at 70 at./nm³ (2.25g/cm3) for all tests. the integration of nose-hoover style non-hamiltonian equations of motion was utilized with a time step δt of 0.5 fs. note that due to the action of periodic boundary conditions along the direction of applied load, we have not a spatially independent tribofilm, but a spatially limited fragment that regularly repeats in this direction. however, within this fragment, the implementation of various physical processes and mechanism of deformation are not limited. this means that the variable width of the fragment being modeled can be that critical spatial parameter determining the character of the wear process. verification of rabinowicz criterion by direct molecular dynamics modeling 211 3. results and discussion as mentioned above, the performed earlier sliding simulation using the tersoff potential showed a pronounced effect of the system temperature on the ability of a silica tribofilm to exhibit smooth sliding [9]. the latter was obtained at the high temperature only, whereas at 300 k a stick-slip type of sliding was observed. following the goal of the paper, we simulate the sliding of amorphous silica sample with the width of 15.0 nm at few different temperatures: 300 k, 400 k, 500 k, 600 k, 700 k, and 1100 k. (a) (b) (c) (d) (e) fig. 2 structures of the central fragment of different silica specimens after 1 ns of sliding: (a) lx = 10.0 nm, t = 300 k; (b) lx = 15.0 nm, t = 500 k; (c) lx = 22.5 nm, t = 500 k; (d) lx = 30.0 nm, t = 500 k. (e) trajectories of atoms of the specimen with lx = 15.0 nm after 1.0 ns of sliding at 500 k. arrow marks the direction of atoms rotation according to the simulation results the critical temperature (tc) above which the transition from unstable behavior with debris formation to smooth sliding conditions is taking place is about 500 k. for two other specimens with lx=22.5 nm and lx=30.0 nm, temperature tc is also close to 500 k, while for the narrowest specimen with the width of only 10.0 nm the critical temperature is much lower and is about 300 k. this means that the size of the 212 a.i. dmitirev, a.yu. nikonov, w. österle, b.c. jim fragment being modeled is so small that debris formation in it is possible only under conditions of increasing viscosity (adhesive properties) of the modeled system in comparison with large-sized samples. fig. 2 denotes the resulting structures of all considered specimens at the stage of sliding where wear debris forms in the amorphous layer. according to the trajectories of atoms (see fig. 2e) these wear particles seem to adopt the function of nano-sized rollers. the same samples were also studied at elevated temperatures. fig. 3 denotes the resulting structures of the specimens at the conditions of smooth sliding. in comparison with the data presented in fig. 2, only the temperature was increased by 100 k, while all other parameters remained the same. it is well seen that the velocity accommodation between the upper and lower part of the specimen in that case occurs almost entirely within the amorphous layer. in contrast to the conditions of the critical temperature, no damaged regions are observed and shearing takes place homogeneously within the layer. (a) (b) (c) (d) (e) fig. 3 structures of the central fragment of different silica specimens after 1 ns of sliding: (a) lx = 10.0 nm, t = 400 k; (b) lx = 15.0 nm, t = 600 k; (c) lx = 22.5 nm, t = 600 k; (d) lx = 30.0 nm, t = 600 k. (e) trajectories of atoms of the specimen with lx = 15.0 nm after 2.0 ns of sliding at 700 k arrows show the velocity distribution of atoms instead of the movement of aggregates of linked atoms, now single atoms move as they do in liquid films – in opposite directions along the interface between the upper and lower part and a neutral layer in the middle. this is well indicated by arrows in fig. 3e, where atoms trajectories at steady state motion are shown. comparing figs. 2 and 3 it can be verification of rabinowicz criterion by direct molecular dynamics modeling 213 seen that increase in temperature above the critical value (~500k in our case for most samples) leads to change in the sliding regime from unstable with possible formation of wear debris to smooth sliding. moreover, it was found that the critical temperature decreases rapidly with decreasing sample width of less than 15 nm, which corresponds to the width of the basic sample in our calculations. thus, as we assumed, the temperature of the system acts as a controlled parameter that ensures a change in the adhesive properties of the model material. increasing the temperature leads to a decrease in adhesion forces, and hence to a change in the sliding regime. at the same time, as mentioned early in the introduction, there is a critical length scale of the system, which can change the conditions for transitions of the sliding regime. fig. 4 demonstrates the dependence of the sample width on the tc value. fig. 4 dependence of critical conditions for transitions of sliding regime on sample width and temperature conditions in order to verify eqs. (1) and (2) we calculated the values of the resistance force to tangential motion within the amorphous interlayer acting on the atoms belonging to the loaded layers for the base sample at various temperatures. in addition, the estimation of the corresponding change in specific surface energy was made. the latter is calculated as the difference between the energy of a crystallite with a free surface and the same crystallite with periodic boundary conditions divided by the area of the free surface. calculated results are presented in fig. 5 and summarized in table 1. according to the data, as the temperature rises, both energy characteristics fall. so the peak value for the shear resistance force at 300 k is ~1590 ev/å and at 400 k is about 1500 ev/å. at the same time, the specific surface energy decreases from 3.93 j/m 2 to 3.74 j/m 2 . in this case, the shear modulus, as clearly seen from fig. 5, remains practically unchanged. substituting the obtained values into eqs. (1) or (2), we can find that the ratio of the critical size parameter d * (d) for temperatures 300k and 400k is about 0.93, while the estimation according to data in fig. 4 gives a close value of ~ 0.83. thus, the results of direct md modeling indicate that the criterion proposed by e. rabinowicz can be used to estimate the critical size parameter leading to the formation of wear debris in the model of tribofilm as well. 214 a.i. dmitirev, a.yu. nikonov, w. österle, b.c. jim fig. 5 time dependence of the resistance force to tangential motion within the silica amorphous interlayer in the base sample at various thermal conditions table 1 energetic parameters calculated for basic sample at various thermal conditions temperature, k specific surface energy, j/m 2 resistance force, ev/å 300 3,9349 1590 400 3,7791 1500 500 3,5737 1420 600 3,3939 1300 700 3,3493 1150 1100 3,1029 950 5. conclusion direct molecular dynamics modeling of the shear deformation of a silica sample containing an amorphous interlayer has demonstrated the effect of temperature and size of the system on the character of its sliding while other loading conditions remained constant. it was found that the conditions for changing the regime of sliding for the modeled tribofilm can be qualitatively and quantitatively described by the criterion proposed by e. rabinowicz [4], namely the existence of critical size of a single roughness leading to friction mode switching from smooth sliding to abrasive wear. calculations showed that the ratio of two critical sizes of the system, obtained by direct md simulation and the ratio of two critical length parameters calculated on the basis of eqs. (1) or (2), give similar values. this confirms the validity of the rabinowicz’s criterion not only for the macroscopic systems but also for atomic objects. as limitations of the proposed model of the tribofilm, note that the use of periodic boundary conditions along the loading direction has certain effects on the conditions of wear debris formation and therefore has to be studied additionally in details. verification of rabinowicz criterion by direct molecular dynamics modeling 215 acknowledgements: investigations have been carried out with the financial support from russian science foundation for basic research grant no 18-508-12054 and the fundamental research program of the state academies of sciences for 2013-2020, line of research iii.23.2.4. references 1. popov, v.l., 2018, adhesive wear: generalized rabinowicz’ criteria, facta universitatis-series mechanical engineering, 16(1), pp. 29-39. 2. popov, v.l., pohrt, r., 2018, adhesive wear and particle emission: numerical approach based on asperityfree formulation of rabinowicz criterion, friction, 6(3), pp. 260-273. 3. popov, v.l., popova, e., 2018, 60 year of rabinowicz’ criterion for adhesive wear, friction, 6(3), pp. 341-348. 4. rabinowicz, e., 1958, the effect of size on the looseness of wear fragments, wear, 2, pp. 4–8. 5. cheng, h., shuku, t., thoeni, k., temppone, p., luding, s., magnanimo, v., 2019, an iterative bayesian filtering framework for fast and automated calibration of dem models, computer methods in applied mechanics and engineering, 350, pp. 268-294. 6. österle, w., dmitriev, a.i., kloss h., 2012, does ultra-mild wear play any role for dry friction applications, such as automotive braking?, faraday discussions, 156, pp. 159-171. 7. dmitriev, a.i., nikonov, a.y., österle, w., 2017, molecular dynamics sliding simulations of amorphous ni, ni-p and nanocrystalline ni films, computational material science, 129, pp. 231-238. 8. aghababaei, r., warner, d.h., molinari, j.-f., 2016, critical length scale controls adhesive wear mechanisms, nature communications, 7, pp. 11816/1-11816/8. 9. molinari, j.-f., aghababaei, r., brink, t., frerot, l., milanese, e., 2018, adhesive wear mechanisms uncovered by atomistic simulations, friction, 6(3), pp. 245-259. 10. dmitriev, a.i., nikonov, a.yu., österle, w., 2016, md sliding simulations of amorphous tribofilms consisting of either sio2 or carbon, lubricants, 4(3), pp. 1-24. 11. tersoff, j., 1988, new empirical approach for the structure and energy of covalent systems, physical review b, 37, pp. 6991-7000. 12. munetoh, s., motooka, t., moriguchi, k., shintani, a., 2007, interatomic potential for si–o systems using tersoff parameterization, computational materials science, 39(2), 334-339. 13. plimpton, s., 1995, fast parallel algorithms for short-range molecular dynamics, journal of computational physics, 117(1), pp. 1-19. 14. stukowski, a., 2010, visualization and analysis of atomistic simulation data with ovito–the open visualization tool, modelling and simulation in materials science and engineering, 18, pp. 015012/1015012/7. 15. dmitriev, a.i., österle, w., 2014, modelling the sliding behaviour of tribofilms forming during automotive braking: impact of loading parameters and property range of constituents, tribology letter, 53, pp. 337–351. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 81 90 1application of the method of dimensionality reduction to contacts under normal and torsional loading udc 539.3 emanuel willert 1 , markus hess 1 , valentin l. popov 1,2 1 berlin university of technology, germany 2 national research tomsk state university, russia abstract. recently the method of dimensionality reduction (mdr) has been introduced to solve axisymmetric contact problems easily and exactly. the list of tasks that this method can deal with comprises normal, tangential, adhesive and rolling contacts with simply connected contact areas between elastic or viscoelastic bodies. due to its simplicity and easy applicability the mdr provides the possibility of fast and comprehensive studies of contact problems in technological or biological systems, for example bearings, artificial hip joints, wheel-rail systems or others. within the complicated three-dimensional contact theory those studies, in most cases, cannot be done without a tremendous mathematical or numerical effort. in view of all this, the torsional contact problems have been disregarded until now, although it is known that torsion is a major reason of wear and possible failure of system components. therefore, in the present paper, we extend the mdr to contacts of axisymmetric profiles under superimposed normal and torsional loading. key words: contact mechanics, method of dimensionality reduction, torsion, friction, stick, slip 1. introduction pure torsional contacts or normal and torsional contacts coupled by friction were not given much attention in the past, although torsional loading is known to be a major reason of wear and fatigue. in [1] lubkin gave the shear stress distribution for the torsional contact between two elastic spheres with partial slip. hetényi and mcdonald jr. calculated the stresses and displacements for the full-sliding contact between an elastic sphere and an elastic halfspace using hankel transforms [2]. also based on hankel transforms, i.e. bessel functions, kartal received may 29, 2015 / accepted july 1, 2015 corresponding author: emanuel willert technische universität berlin, institut für mechanik, str. des 17. juni 135, 10623 berlin, germany e-mail: e.willert@tu-berlin.de original scientific paper 82 e. willert, m. hess, v.l. popov et al. analyzed the torsional contact with partial slip between flat-ended elastic cylinders [3]. jäger determined the stress distributions in the form of abel transforms for the torsional contact with partial slip between axisymmetric bodies of arbitrary profile shape [4]. he thereby used a superposition of flat-punch-solutions to solve the contact problem of arbitrarily shaped bodies. in the experimental work [5] trejo et al. investigated the friction between an elastomer and a randomly rough surface using a torsional contact configuration. in a series of recent papers, popov and collaborators have introduced the so-called method of dimensionality reduction (mdr), which allows solving normal contact problems of axisymmetric elastic and viscoelastic bodies as well as tangential contact problems with a constant coefficient of friction for arbitrary loading histories [6, 7]. in the monographs [8] and [9], the mdr has been summarized and many applications have been provided. moreover, an introduction into its usage in the form of a user's handbook can be found in [10]. in the present paper, we are showing that the contact with torsion (rotation around the normal axis to the contact plane) can also be described with the mdr as long as there is no slip in the contact or the thickness of the slip annulus at the edge of the contact is small compared to the contact radius. the paper is organized as follows: in the section 2 we reproduce, for convenience of the reader, the derivation of the mdr equations for the normal contact following [9]. in the section 3 the application of the mdr to contacts with torsion without slip is discussed. in the section 4 the torsional contact with a narrow slip region is considered. section 5 closes the paper. 2. method of dimensionality reduction for the normal contact in this section, the equations of the mdr for the normal contact of axisymmetric profiles f (r) with a compact area of contact are deduced. thereby we follow the idea of jäger [11] to derive the solution of an axisymmetric contact problem by summation of differential flat punch solutions. a flat punch of radius a, indenting an elastic half space with effective elastic modulus e * by indentation depth d, produces displacement uz , 2 arcsin( / ), z d r a u d a r r a         . (1) the resulting pressure distribution will be * 2 2 1 , ( ) 0 , e d r a p r a r r a         (2) and the total normal force *2nf e ad . (3) hence, contact stiffness kz is *d ( ) 2 d n z f k a e a d   . (4) note that this equation is valid for any profile shape, if a is understood as the current contact radius. application of the method of dimensionality reduction to contacts under normal and torsional loading 83 let us assume a contact between a rigid indenter of shape z = f (r) and an elastic half space. the indentation depth due to normal force fn will be d and contact radius a. for any given profile shape any of those three parameters will unambiguously define the other two. especially, the indentation depth is a definite function of the contact radius, which we will denote by ( )d g a . (5) firstly we show that the complete solution of the normal contact problem will be unambiguously determined by function d = g(a). analyzing the complete process of indentation from its very first moment until the final indentation, the current values of the normal force, indentation depth and contact radius are given by nf , d and a . during the indentation process, the indentation depth changes from d = 0 to d = d, the contact radius accordingly from a = 0 to a = a and the normal force from nf = 0 to nf = fn. the final normal force can be written as 0 0 d d d d dd nf a n n n f d f f a ad    . (6) if we take into account that the differential contact stiffness of an area with radius a is given by (4) and the indentation depth by (5), we get * 0 d ( ) 2 d d a n g a f e a a a   , (7) which gives after partial integration  * * 0 0 2 ( ) ( )d 2 ( ) d a a nf e a g a g a a e d g a a                   . (8) let us calculate the pressure distribution within the contact area. an infinitesimal indentation d d of an area with radius a will, due to (2), produce the pressure * 2 2 1 d ( ) d e p r d a r   . (9) the pressure distribution at the end of the indentation process is given by the sum of all infinitesimal pressure components, * * 2 2 2 2 ( ) 1 1 d ( ) ( ) d d d d a d r r e e g a p r d a aa r a r        . (10) hence, function d = g(a) unambiguously defines the pressure distribution and therefore the total normal force as well. that is why the solution of the contact problem is reduced to the determination of this function. this can be done as follows: infinitesimal indentation dd mentioned above will, due to (1), produce surface displacement at r a a  2 d ( ) arcsin dz a u a d a        . (11) again, the total displacement can be understood as a sum of all infinitesimal indentations: 84 e. willert, m. hess, v.l. popov 0 0 2 2 d ( ) ( ) arcsin d arcsin d d d a z a a g a u a d a a a a                 (12) on the other hand, this displacement is given by ( ) ( )zu a d f a  : 0 2 d ( ) ( ) arcsin d d a a g a d f a a a a          , (13) which gives after partial integration 2 2 0 2 ( ) ( ) d a g a f a a a a    . (14) this is an abel integration equation, which can be inverted [12]: 2 2 0 ( ) ( ) d a f a g a a a a a     . (15) the mdr is mainly an interpretation of the equations (5), (8), (10) and (15), which, on the one hand, can be interpreted as just a mnemonic rule. however, in many ways it has a deeper physical meaning. let us assume an elastic foundation of independent equal springs, each at distance x from each other and with stiffness kz = e * x, as shown in fig. 1. also, we define a one-dimensional profile g(x) as a formal transformation of the threedimensional axisymmetric profile z = f (r) according to 2 2 0 ( ) ( ) d x f r g x x r x r     . (16) this transformation is illustrated in fig. 2. fig. 1 equivalent elastic foundation fig. 2 axisymmetric three-dimensional profile and one-dimensional analogue within the framework of the mdr (see eq. (16)) application of the method of dimensionality reduction to contacts under normal and torsional loading 85 the transformed profile is now pressed into the elastic foundation described above. this is shown in fig. 3. the surface displacement in normal direction at any point x will be given by the difference of indentation depth d and profile shape g(x): 1 ( ) ( )d zu x d g x  . (17) fig. 3 mdr-model for the normal contact for contacts without adhesion the displacement vanishes at the edge of the contact: 1 ( ) ( ) 0d zu a d g a   . (18) the normal force in a single spring is given by *( ) ( ( )) ( ( ))n zf x k d g x e d g x x       , (19) from which the total normal force in the equilibrium state can be calculated by summation over all springs. in limiting case 0x  the sum will be the integral * 1 * 0 ( )d 2 ( ( ))d a a d n z a f e u x x e d g x x      . (20) it can be seen easily that the equations (18), (20) and (16) reproduce (5), (8) and (15). hence, transformed profile g(x) is the geometrical interpretation of dependence d = g(a) for the given three-dimensional profile shape. by the equivalence of the equations presented above it is also shown that, instead of analyzing the three-dimensional contact problem, the described equivalent one-dimensional problem can be solved, obtaining the correct and exact results for the original contact. in the next paragraph we are going to show how this can be done, by the same method, for torsional contact as well. 3. description of the torsional contact with the method of dimensionality reduction again, we start with the known solution for the torsional contact of a rigid flat cylinder. if a rigid flat-ended punch is pressed on an elastic half space and rotated around the axis of the cylinder by angle , the produced torsional moment, surface displacement and stress distribution will be given by equations [13] 316 3 zm ga  , (21) 86 e. willert, m. hess, v.l. popov 2 2 ( ) 2 arcsin 1 , , r r a u r a a r a r a r r                       , (22) 2 2 4 ( ) 0 , , g r r a r a r r a         , (23) where g is the shear modulus. in the case of rotational symmetry and of no slip, the torsional contact problem completely decouples from the normal contact problem. we analyze the torsional contact problem that is analogical to the normal contact problem described in the previous section, i.e. we imprint rotational surface displacement u(r) = r(  (r)) into an elastic half space and want to determine the shear stresses due to this displacement. (r) is the deviation of the torsional angle from the pure constant rotation with . this displacement is understood as a sum of infinitesimal torsional loadings of flat punches [4]. in analogy to (5) we introduce the function ( ).a   (24) the complete torsional moment after the process of torque loading can be calculated as 3 00 0 16d d ( ) d ( ) d d d d 3d d a a z z m z m a a gm m a a a a a       . (25) we introduce variable uy(x) in the following differential way: d 0 < d ( ) 0 , , y x x a u x x a      . (26) at the end of the described process of infinitesimal torsional loadings this field will be ( ) d ( ) ( ) d d d a y x x a u x x x a a        . (27) dividing (27) by x and differentiating with respect to x we get ( ) ( )d d ( ) ( ) d d y yu x u ax x a x x x a            . (28) equation (25) can then be written in the following way: 3 0 ( ) ( )16 d ( ) d 3 d a y y z u a u a m g a a a a a a a               , (29) which gives after partial integration 0 ( d16 )y a z gm u aaa  . (30) it is obvious that this equation can also be interpreted within a one-dimensional model. application of the method of dimensionality reduction to contacts under normal and torsional loading 87 let us assume an elastic foundation with tangential stiffness 8yk g x   . (31) the force of a single spring is given by ( )y y yf k u x   (32) and the resulting torsional moment by 8 ( )z ym gxu x x   . (33) the total moment can be calculated again by integration and will be 0 8 ( )d 16 ( )d , a a z y y a m g x u x x g x u x x       (34) which coincides with (30). to complete the solution of the described torsional contact problem, let us calculate function (a) and the stress distribution. according to (23) the stress distribution can – analogically to the previous section – be calculated from function ( )a : 2 2 4 d ( ) ( ) d , d a r g r a r a aa r       (35) or with equation (28) 2 2 ( ) ( )4 d ( ) ( ) d , d a y y r u a u ag r r a a a a a aa r                  (36) which is equivalent to 2 2 2 2 ( ) ( )4 d ( ) d . d a y y r u a u ag r r r a a a aa r a r                (37) the displacement at the edge of the contact, i.e. at r a a  , will be, according to (22), 2 2 0 2 d ( ) ( ) ( ( )) arcsin 1 d . d a a a a u a a a a a a a a a                      (38) in the next section we will analyze the case of slip in the contact area. this will inevitably lead to requirement  (a) = 0. this given, (27) can be written as ( ) ( ( ))yu x x x  (39) and partial integration of (38) will give 2 2 2 2 0 4 ( ) ( ) d . a a a a a a a a       (40) this is again an abel integration equation, which can be inverted [12]: 2 2 2 0 1 d d ( ) ( ( )) . 2 d a r a r r a r a r     (41) 88 e. willert, m. hess, v.l. popov 4. torsional contact with a narrow slip region the boundary conditions at the surface of the half space in the presence of slip can generally be written in the form ( ) 0 , for , ( ) ( ) , for r r c r p r c r a         , (42) with the radius of stick domain c and coefficient of friction . from (41) it is obvious that ( ) 0 , for x x c   . (43) hence, the shear stresses within the contact area will be 2 2 2 2 d ( ) d , for d ( ) d ( ) d , for d 4 4 a c a r g r a r g r a a a r c a r a r a c r a a                   . (44) note, that (44) can always be written in the form  ( ) ( ) , for ( ) ( ) , for a c a p r p r r c r p r c r a           . (45) here pa(r) and pc(r) denote known pressure distribution p(r) with respective contact radii a and c. thus the shear stress distribution in the whole contact area is known. the contact problem will be solved completely, if the integral equation 2 * 22 2 d ( ) 1 d ( ) d ( d 4 ) d d a a a r r a e g ag r a r a p r a a aa r           (46) can be solved for function ( )a . in case of a very small area of the slip domain, i.e. if a c a , this solution is elementary, because r  a and therefore d d , for g c x a    (47) with * 4 e ga    . geometrically, (47) together with (43) describes the following indentation process: first the indenter is pressed into the half space in a pure normal direction until the radius of stick domain c is reached. after that any infinitesimal indentation is a superposition of normal and torque loading, bound by (47). the solution of (47) with the boundary conditions (24) and (5) is given by ( ) ( ( ) ), for x g x d c x a       . (48) application of the method of dimensionality reduction to contacts under normal and torsional loading 89 hence, , for ( ) ( ) , for y z x c u x x u x c x a        . (49) again, as we assumed a c a , it is x a in the slip domain and therefore * , for ( ) ( ) , for 4 y z x x c u x e u x c x a g          (50) if we choose * = 2 as the equivalent coefficient of friction in the mdr-model for torsion, (50) can be written in the form * , for ( ) ( ) , for y z z y x x c u x k u x c x a k          . (51) the radius of the stick domain is given by * * *8 2 ( ( ) ( )) ( ( ) ( ))g c e g a g c e g a g c      , (52) which agrees with the condition of no slip for the springs at the edge of the stick domain in the mdr-model. that is why this torsional contact problem with a finite coefficient of friction can be described by the mdr. we emphasize again that the derivation starting with (47) is only valid for a c a . in the case of general partial slip the solution for ( a ) in the slip domain is given by the inverse abel transform of (46) [12]: 2 2 ( )d ( ) ( ) , for 2 a x p r r a x c x a g r x         (53) however, the resulting necessary spring displacement of the mdr-model 2 2 ( )d ( ) ( ( ) ( )) , for 2 a y x x p r r u x x a x c x a g r x          (54) cannot be interpreted as easily as in the case described above in (47) (51). 5. conclusion in the present paper, we have extended the method of dimensionality reduction to contacts subjected to a superimposed normal and tangential loading. for the case of the simultaneous normal and torsional loading we have shown that the consideration of the original three-dimensional contact problem can be replaced by a contact with a onedimensional elastic foundation with a properly defined coefficient of friction and normal and tangential stiffness if the slip annulus is small compared to the contact radius. 90 e. willert, m. hess, v.l. popov acknowledgements: this work was supported in parts by the german research society and the tomsk state university academic d.i. mendeleev fund program. references 1. lubkin j.l., 1951, the torsion of elastic spheres in contact, journal of applied mechanics, 18, pp. 183187. 2. hetényi m., mcdonald jr. p.h., 1958, contact stresses under combined pressure and twist, journal of applied mechanics, 25, pp. 396-401. 3. kartal m.e., hills d.a., nowell d., barber j.r., 2010, torsional contact between elastically similar flat-ended cylinders, international journal of solids and structures, 47, pp. 1375-1380. 4. jaeger j., 1995, axi-symmetric bodies of equal material in contact under torsion or shift, archive of applied mechanics, 65, pp. 478-487. 5. trejo m., fretigny c., chateauminois a., 2013, friction of viscoelastic elastomers with rough surfaces under torsional contact conditions, physical review, e88, 052401. 6. popov v.l., psakhie s.g., 2007, numerical simulation methods in tribology, tribology international, 40, pp. 916–923. 7. hess m., 2011, über die exakte abbildung ausgewählter dreidimensionaler kontakte auf systeme mit niedrigerer räumlicher dimension, cullier verlag, berlin. 8. popov v.l., hess m., 2015, method of dimensionality reduction in contact mechanics and friction, springer. 9. popov v.l., 2015, kontaktmechanik und reibung, 3. überarbeitete auflage, springer. 10. popov v.l., hess m., 2014, method of dimensionality reduction in contact mechanics and friction: a users handbook. i. axially symmetric contacts, facta universitatis, mechanical engineering, 12(1), pp. 1-14. 11. jaeger j., 2005, new solutions in contact mechanics, wit press. 12. bracewell r., 1965, the fourier transform and its applications, mcgraw-hill book company. 13. johnson k.l., 2001, contact mechanics, cambridge university press. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 285 305 https://doi.org/10.22190/fume200629040t © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a new c0 third-order shear deformation theory for the nonlinear free vibration analysis of stiffened functionally graded plates received june 29, 2020 / accepted october 03, 2020 corresponding author: hoang lan ton-that department of civil engineering, hcmc university of architecture, 196 pasteur, district 3, hcmc, vietnam. e-mail: lan.tonthathoang@uah.edu.vn hoang lan ton-that department of civil engineering, hcmc university of architecture, vietnam abstract. nonlinear free vibration of stiffened functionally graded plates is presented by using the finite element method based on the new c0 third-order shear deformation theory. the material properties are assumed to be graded in the thickness direction by a power-law distribution. based on the von karman theory and the third-order shear deformation theory, the nonlinear governing equations of motion are derived from the hamilton’s principle. an iterative procedure based on the newton-raphson method is employed in computing the natural frequencies and mode shape. the comparison between these solutions and the other available ones suggests that this procedure is characterized by accuracy and efficiency. key words: nonlinear free vibration, functionally graded material, stiffened plate, third-order shear deformation theory 1. introduction the plates with stiffeners are often used in several fields of engineering such as medical, weapon, nuclear reactor construction, aerospace, etc. to improve stiffness of the structures. many different ways are applied to analyzing plate structures in general as well as stiffened plate structures in particular; they are listed as rayleigh-ritz method [1], finite difference method [2, 3], finite element method (fem) [4-15], constraint method [16, 17], mesh-free method [18-20], semi analytical finite defference method [21, 22], finite strip method [23-25], boundary element method [26, 27], integral transform approach [28], etc. the most important issue of this type of structure is the connection between the plate and the stiffeners. for example, peng et al [19] used the first order shear deformation theory as well as the element-free galerkin method to study the compatibility conditions between the plate and the stiffeners when they work together, etc. evidently numerical methods are essential for calculating stiffened structures, in which the fem is the most popular because of 286 h.l. ton-that its efficiency and stability. chattopadhyay et al. as well as holopainen [29, 30] analyzed nonlinear static of composite stiffened plates based on the first order shear deformation theory and the fem method or used a new finite model for a linear free vibration analysis of stiffened plates, which is based on the nine-node quadrilateral element related to mixed interpolation of tensorial components. functionally graded materials with two constituents e.g. ceramic and metal from ceramic surface to metal surface are widely applied. in recent years, many surveys have been carried out in the area of functionally graded plates. the thermoelastic deformations and vibration behaviors with exact solutions were given by vel and batra [31]. stress-driven nonlocal elasticity for nonlinear vibration characteristics of carbon/boron-nitride hetero-nanotube subject to magneto-thermal environment was firstly introduced by sedighi and malikan in [32]. the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity was studied in [33] by ouakad et al. on the other hand, qian et al. [34] also analyzed this kind of structures based on the meshless local petrov-galerkin (mlpg) method. thau and choi showed the bending and free vibration behaviors of functionally graded plates based on the first-order shear deformation theory. furthermore, concerning the mlpg method, the highorder shear and normal deformation plate theory was used to analyze thick functionally graded plates by gilhooly et al. some papers of liew reviewed the meshless methods for composite plate/shell structures. a review of jha involved the listing of studies for this structure. further, reddy proposed a general formulation related to the third-order shear deformation plate theory and the finite element model. an isogeometric analysis and a collocation method employing the shear deformation theory were also applied to the analysis of functionally graded plates by valizadeh et al. or ferreira et al., zhang and zhou, prakash as well as singha who also proposed a formulation to study linear and nonlinear behaviors of the functionally graded plates with respect to the physical neutral surface. an efficient three-node finite shell element for linear and nonlinear analyses of composite structures was also given by marinkovíc et al. [14, 35]. shi's third-order shear deformation theory with its necessary stability was first used for a functionally graded plate structure analysis in thermal environment by bui et al. [7]. besides, the c0 type of this theory was also used in the analysis of functionally graded skew plates by ton [12]. and now, the c0 type of shi’s theory is applied to analyzing stiffened functionally graded plate structures. with the third-order shear deformation theory, we recognize that it is widely used because it does not need shear correction factors while it gives accurate transverse shear stresses. but with low-order finite elements such as four-node quadrilateral element, the need of c1 continuous approximation for the displacement fields in the third-order shear deformation theory causes some impediments. to overcome these shortcomings, the third-order shear deformation theory is a revised form which only requires c0 continuity for displacement fields. in the c0 third-order shear deformation theory, two additional variables are joined, and thence the first derivative of transverse displacements is only required, respectively. the body of this paper is organized into four sections. in sect.2, finite element formulation based on the c0 new third-order shear deformation theory for stiffened functionally graded plates is presented. several examples are subsequently presented in sect.3. the paper ends with some concluding remarks in the last section. a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 287 2. finite element formulation let us consider a stiffened functionally graded plate with geometry as plotted in fig. 1a. the bottom and top faces of plate are to be fully metallic and ceramic, respectively. the mid-plane of the plate is xy-plane, while the z-axis is perpendicular to the xy-plane. (a) (b) fig. 1 (a) the stiffened functionally graded plate and (b) the variation of volume fraction the volume fraction of ceramic (vc) and metal (vm) are formulated in eq. (1) and the variation of volume fraction for several volume fraction coefficients of a functionally graded plate using the power-law distribution is plotted by fig.1b. 288 h.l. ton-that 1 2 n c z v h   = +    1m cv v= − 0n  (1) where z is the thickness coordinate variable with -h / 2 z h / 2  as well as c, m and n represent the ceramic, metal constituents and the non-negative volume fraction gradient index, respectively. all values of e, ,  and  that vary through the thickness of plate are also formulated as below 1 ( ) ( ) 2 n m c m z e z e e e h   = + − +    (2) 1 ( ) ( ) 2 n m c m z z h   = + − +        (3) 1 ( ) ( ) 2 n m c m z z h   = + − +        (4) 1 ( ) ( ) 2 n m c m z z h   = + − +        (5) according to the new theory of shi [36], a three-dimensional displacement field ( , , )u v w was given as below 3 3 0 0,2 2 5 4 1 5 ( , , ) ( , ) ( , ) ( , ) 4 43 3 x xu x y z u x y z z x y z z w x y h h     = + − + −         (6) 3 3 0 0,2 2 5 4 1 5 ( , , ) ( , ) ( , ) ( , ) 4 43 3 y yv x y z v x y z z x y z z w x y h h     = + − + −         (7) 0( , , ) ( , )w x y z w x y= (8) this three-dimensional displacement field can be expressed in terms of the c0 thirdorder shear deformation theory and seven unknown variables as follows 3 3 0 2 2 1 5 5 4 ( , , ) ( , ) ( , ) ( , ) 4 43 3 b s x xu x y z u x y z z x y z z x y h h     = + − + −          (9) 3 3 0 2 2 1 5 5 4 ( , , ) ( , ) ( , ) ( , ) 4 43 3 b s y yv x y z v x y z z x y z z x y h h     = + − + −          (10) 0( , , ) ( , )w x y z w x y= (11) it can be seen that the present theory is composed of seven unknowns including three axial and transverse displacements, and four rotations due to the bending and shear effects. the strain-displacement relations based on the small strain assumptions can be given as follows a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 289 3 0, , , , ,2 3 0, , , , ,2 3 0, 0, , , , , , , ,2 1 5 (5 ) ( ) 4 3 1 5 (5 ) ( ) 4 3 1 5 (5 5 ) ( 4 3 s b s b x x x x x x x x x s b s b x y y y y y y y y y y s s b b s s xy y x x y y x x y y x x y y x x yz xz u z z h v z z h u v z z h −  + + + +    −   + + + +       −    = + + + + + + + +                                 , 2 , 2 2 , 2 ) 5 1 5 ( ) 4 4 5 1 5 ( ) 4 4 b b y y x s b s b y y y y y s b s b x x x x x w z h w z h                +     −    + + + +           −    + + + +                   (12) or matrix form ( ) ( ) ( ) ( ) ( )0 1 3 2 3 20 0 0 0 z z z                  = + + +                                (13) with the membrane strains obtained from ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 l nl u w wx x v w x wy y yu v w w y x y x                        = + = +                      +              (14) the bending strains are given by ( ) , , 1 , , , , , , (5 ) 1 (5 ) 4 (5 5 ) s b x x x x s b y y y y s s b b x y y x x y y x  +   = +   + + +           ( ) , , 3 , ,2 , , , , 5 3 s b x x x x s b y y y y s s b b x y y x x y y x h          + −   = +   + + +   (15) and the shear strains are basically written by ( ) , 0 , 5 1 4 4 5 1 4 4 s b y y y s b x x x w w        + +   =    + +    ( )2 2 5 s b y y s b x xh       +−   =   +   (16) the membrane, bending and shear strains can be then expressed as ( ) 4 1 1 l l i i i= =b q ( ) 4 1 1 1 2 nl nl i i i= = b q ( ) 4 1 2 1 i i i= =b q ( ) 4 3 3 1 i i i= =b q (17) 290 h.l. ton-that ( ) 4 0 4 1 i i i= =b q ( ) 4 2 5 1 i i i= =b q (18) in which , 1 , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i x l i i y i y i x n n n n     =        , , 2 , , , , , , 0 0 0 5 0 0 1 0 0 0 0 5 0 4 0 0 0 5 5 i x i x i i y i y i y i x i y i x n n n n n n n n     =        (19) , , 3 , ,2 , , , , 0 0 0 0 0 5 0 0 0 0 0 3 0 0 0 i x i x i i y i y i y i x i y i x n n n n h n n n n     = −        , 4 , 5 1 0 0 0 0 4 4 5 1 0 0 0 0 4 4 i y i i x n n     =         (20) 0 0 0 0 1 0 15 0 0 0 1 0 1 0h   = −      0 0 1 0 0 , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t nl i i x i y w w x y w w y x n n        =                 (21) where ni, ni,x and ni,y are called the shape function and two derivatives of it in x-direction and y-direction. the global stiffness matrix is computed by l nl g= + +k k k k (22) the element linear stiffness matrix * * , ( ) e t t l e i j i s j d  = + k b d b s d s (23) with 1 2 3( ) ( ) ( )l t t t i i i i  =  b b b b 4 5( ) ( )t t i i i  =  s b b (24) and the element nonlinear stiffness matrix ** ** ** , 1 1 2 2 e t t t nl e li lj nli lj nli nlj d    = + +     k b d b b d b b d b (25) where   t li i i=b b s 1 0 t nl nli i  =  b b * ** * 0 0 s   =     d d d (26) a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 291 on the other hand, the element geometric stiffness matrix , ( ) e t g e i j d  = k g ng (27) with , , 0 0 0 0 0 0 0 0 0 0 0 0 i x i i y n n   =     g x xy xy y n n n n   =     n (28) let us consider that the addition of the stiffener is in the x-direction and by transforming three eqs. (6-8) as follows 3 3 0 2 2 1 5 5 4 ( , , ) ( , ) ( , ) ( , ) 4 43 3 b s st st xst xstu x y z u x y z z x y z z x y h h     = + − + −          (29) ( , , ) 0stv x y z = (30) 0( , , ) ( , )st stw x y z w x y= (31) only the plate elements having an edge coinciding with the stiffener are considered; the establishment of formulation is quite similar. the global stiffness matrix for the stiffener is given by st lst nlst gst= + +k k k k (32) the correlation of the displacements between the plate and the stiffener are presented in matrix form as 00 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 st st ss xxst s y bb xxst b y uu e e v ww                              =                               (33) where ( ) / 2se h h= + is the eccentricity between the plate and the stiffener, respectively. besides, the mass matrix of plate element is shown /2 /2 ( ) ( ) e e h t t t t e v s h z dv z dz ds −   = =       m n l ln n l l n  (34) with 292 h.l. ton-that 3 3 2 2 3 3 2 2 1 5 5 4 1 0 0 4 43 3 1 5 5 4 0 1 0 4 43 3 0 0 1 0 0 z z z z xh h z z z z yh h       − −               = − −                l (35) and exactly the same way for the stiffener element /2 /2 ( ) ( ) st est est st h t l t l est st st st st v s h z dv z dz ds −   = =         m n l l n n l l n  (36) with 3 3 2 2 1 5 5 4 1 0 0 4 43 3 0 0 0 0 0 0 0 1 0 0 z z z z xh h       − −           =        l (37) for vibration of the stiffened functionally graded plate, the equation can be described as ( ) ( )st st+ +m m q + k k q = 0 (38) 2( ) ( )st st  + − + k k m m q = 0 (39) 3. numerical results in this section, the numerical solutions for the nonlinear free vibration analysis of stiffened functionally graded plates are presented. not only the fully simply supported but also the fully clamped boundary conditions are used in this paper. the fully simply supported boundary conditions (ssss) for this procedure 0 0 0s b y yv w  = = = = , at 0,x a= and 0 0 0s b x xu w  = = = = , at 0,y b= (40) and the fully clamped boundary conditions (cccc) 0 0 0 s b s b x x y yu v w    = = = = = = at 0,x a= and 0,y b= (41) a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 293 table 1 material properties of the plate and the stiffener material i e = 3x107 pa, ν = 0.3 and ρ = 2820 kg/m3 material ii e = 211x109 pa, ν = 0.3 and ρ = 7830 kg/m3 material iii em = 322.7 x 109 pa, m = 0.28, m = 2370 kg/m3, ec = 207.79 x 109 pa, c = 0.28, c = 8166 kg/m3 3.1. verification firstly, the fully simply supported rectangular plate with two stiffeners as depicted in fig. 2 is studied in order to verify reliability and validity of the proposed method. the material properties are material i as e = 3x107 pa, ν = 0.3 and ρ = 2820 kg/m3 in table 1 for both the plate and the stiffener. the first five natural frequencies are given in table 2 and compared with the solutions of peng et al. [19] as well as the results from ansys software. it can be seen that the values presented in this paper have a good agreement with the references. fig. 2 the geometric properties of the stiffened rectangular plate with two stiffeners perpendicular to each other next, the other fully simply supported rectangular plate with one stiffener in the middle is considered. the geometric properties of the plate are a = 0.6m, b = 0.41m and h = 0.00633m. 294 h.l. ton-that table 2 the comparison of first five natural frequencies (hz) of the fully simply supported rectangular with two stiffeners results mode 1 2 3 4 5 ansys 0.0812 0.0849 0.1035 0.1090 0.1292 l. x. peng et al. [19] 0.0816 0.0856 0.1000 0.1028 0.1311 present 0.0819 0.0861 0.1055 0.1104 0.1320 table 3 the comparison of first three natural frequencies (hz) of the fully simply supported rectangular with one stiffener results mode 1 2 3 mukherjee et al. [37] 257.05 272.10 524.70 harik et al. [38] 253.59 282.02 513.50 aksu et al. [2] 254.94 269.46 511.64 dayi ou et al. [39] 258.79 273.89 527.29 present 259.47 283.72 525.30 furthermore, the geometric properties of the stiffener along edge b of plate are hs = 0.0222m and bs = 0.001277m. the material properties are material ii with e = 211x109 pa, ν = 0.3 and ρ = 7830 kg/m3 as table 1 for both the plate and the stiffener. the first three natural frequencies based on the proposed method are compared with the others related to mukherjee et al. [37], harik et al. [38], aksu et al. [2] and dayi et al. [39]. from table 3, it is interesting to note that the obtained numerical solutions match very well with the others. the last example in this section is related to the nonlinear free vibration analysis for a fully simply supported functionally graded si3n4/sus304 square plate with a = b = 0.4m and thickness h = 0.005m. the material properties are material iii with em = 322.7 x 109 pa, m = 0.28, m = 2370 kg/m3, ec = 207.79 x 109 pa, c = 0.28, c = 8166 kg/m3. the nonlinear to linear frequency ratios nl/l with n = 2 as given in table 4 are compared with the results of shen [40]. once again, the accuracy and efficiency of the proposed method are proved by the very small errors between the results of two methods. table 4 the nonlinear to linear frequency ratio nl/l of the fully simply supported square functionally graded plate n = 2 wmax / h results 0.0 0.2 0.4 0.6 0.8 1 h. s. shen [40] 1.00 1.021 1.081 1.174 1.293 1.432 present 1.00 1.020 1.079 1.169 1.278 1.414 a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 295 3.2. verification of the nonlinear free vibration of functionally graded plate with one stiffener by adding one stiffener in the middle as illustrated in fig. 3 for the functionally graded si3n4/sus304 square plate as example above with a/h = 10 and 20, the nonlinear to linear frequency ratios nl/l are calculated. fig. 3 the stiffened functionally graded plate with one stiffener in the middle the correlations of geometry between the plate and the stiffener are introduced as bs = a/30, hs = 5h or bs = a/50, hs = 5h. two types of boundary condition and six values of n (0, 0.5, 1, 2, 5 and 10) are also used in this example. the numerical results based on this proposed method are given in tables 5-12 and displayed in figs. 4-7. we have found out that the nonlinear to linear frequency ratios nl/l decrease with increasing the volume fraction coefficient n. this order does not change when we change ratio a/h or the boundary conditions. besides, the first four mode shapes for fully simply supported stiffened functionally graded plate with case a/h = 10, bs = a/30 and n = 2 are also depicted in fig. 8. table 5 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 10) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0211 1.0491 1.0828 1.1213 1.1543 0.5 1.0197 1.0461 1.0777 1.1139 1.1536 1 1.0191 1.0443 1.0748 1.1095 1.1475 2 1.0184 1.0426 1.0717 1.1048 1.1411 5 1.0178 1.0410 1.0688 1.1005 1.1350 10 1.0176 1.0404 1.0678 1.0988 1.1328 296 h.l. ton-that table 6 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 20) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0210 1.0473 1.0772 1.1107 1.1470 0.5 1.0194 1.0432 1.0706 1.1012 1.1344 1 1.0183 1.0407 1.0667 1.0956 1.1269 2 1.0173 1.0384 1.0629 1.0902 1.1195 5 1.0164 1.0364 1.0594 1.0851 1.1127 10 1.0160 1.0354 1.0578 1.0828 1.1048 table 7 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 10) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0032 1.0114 1.0242 1.0415 1.0625 0.5 1.0029 1.0106 1.0227 1.0390 1.0590 1 1.0027 1.0100 1.0217 1.0374 1.0566 2 1.0025 1.0095 1.0206 1.0355 1.0539 5 1.0024 1.0089 1.0195 1.0337 1.0511 10 1.0023 1.0087 1.0190 1.0329 1.0501 table 8 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 20) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0019 1.0066 1.0138 1.0236 1.0358 0.5 1.0013 1.0050 1.0112 1.0197 1.0304 1 1.0009 1.0042 1.0098 1.0176 1.0275 2 1.0007 1.0036 1.0086 1.0157 1.0248 5 1.0005 1.0030 1.0075 1.0140 1.0224 10 1.0003 1.0027 1.0070 1.0132 1.0212 table 9 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 10) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0213 1.0502 1.0875 1.1288 1.1793 0.5 1.0196 1.0471 1.0822 1.1230 1.1683 1 1.0188 1.0455 1.0792 1.1184 1.1619 2 1.0181 1.0438 1.0761 1.1137 1.1551 5 1.0176 1.0424 1.0734 1.1093 1.1489 10 1.0175 1.0420 1.0725 1.1079 1.1468 a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 297 fig. 4 the effects of n on the nonlinear to linear frequency ratio nl/l for case ssss with a/b = 1, a/h = 10 and bs = a/30, hs = 5h. fig. 5 the effects of n on the nonlinear to linear frequency ratio nl/l for case ssss with a/b = 1, a/h = 20 and bs = a/30, hs = 5h. 298 h.l. ton-that fig. 6 the effects of n on the nonlinear to linear frequency ratio nl/l for case cccc with a/b = 1, a/h = 10 and bs = a/30, hs = 5h. fig. 7 the effects of n on the nonlinear to linear frequency ratio nl/l for case cccc with a/b = 1, a/h = 20 and bs = a/30, hs = 5h. a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 299 table 10 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 20) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0211 1.0486 1.0819 1.1198 1.1614 0.5 1.0194 1.0449 1.0756 1.1104 1.1486 1 1.0187 1.0430 1.0722 1.1054 1.1416 2 1.0179 1.0411 1.0690 1.1005 1.1348 5 1.0173 1.0395 1.0660 1.0959 1.1285 10 1.0170 1.0388 1.0647 1.0940 1.1258 mode 1 mode 2 mode 3 mode 4 fig. 8 the first four mode shapes of the stiffened functionally graded plate with one stiffener table 11 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 10) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0035 1.0127 1.0272 1.0467 1.0702 0.5 1.0032 1.0118 1.0255 1.0438 1.0661 1 1.0030 1.0113 1.0244 1.0420 1.0634 2 1.0028 1.0107 1.0232 1.0399 1.0604 5 1.0027 1.0101 1.0219 1.0378 1.0573 10 1.0026 1.0099 1.0215 1.0371 1.0563 300 h.l. ton-that table 12 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 20) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0022 1.0078 1.0166 1.0284 1.0431 0.5 1.0016 1.0063 1.0138 1.0242 1.0371 1 1.0013 1.0056 1.0126 1.0221 1.0341 2 1.0011 1.0049 1.0114 1.0202 1.0313 5 1.0009 1.0044 1.0103 1.0184 1.0287 10 1.0008 1.0041 1.0098 1.0176 1.0276 3.3. nonlinear free vibration of a functionally graded plate with two stiffeners the last example is related to the analysis of functionally graded si3n4/sus304 square plate with two stiffeners perpendicular to each other in the middle as illustrated in fig. 9. fig. 9 the stiffened functionally graded plate with two stiffeners table 13 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 10) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0225 1.0476 1.0765 1.1083 1.1381 0.5 1.0219 1.0459 1.0735 1.1038 1.1362 1 1.0211 1.0450 1.0719 1.1013 1.1328 2 1.0205 1.0440 1.0702 1.0987 1.1291 5 1.0202 1.0432 1.0686 1.0963 1.1257 10 1.0201 1.0429 1.0681 1.0954 1.1245 a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 301 table 14 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 20) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0222 1.0468 1.0736 1.1022 1.1323 0.5 1.0217 1.0456 1.0713 1.0987 1.1276 1 1.0210 1.0447 1.0699 1.0966 1.1246 2 1.0201 1.0438 1.0683 1.0942 1.1214 5 1.0200 1.0429 1.0667 1.0918 1.1181 10 1.0199 1.0425 1.0660 1.0907 1.1166 table 15 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 10) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0024 1.0084 1.0180 1.0308 1.0467 0.5 1.0022 1.0079 1.0170 1.0292 1.0444 1 1.0021 1.0076 1.0164 1.0283 1.0430 2 1.0020 1.0073 1.0158 1.0273 1.0415 5 1.0019 1.0070 1.0152 1.0263 1.0401 10 1.0019 1.0069 1.0150 1.0259 1.0395 table 16 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 20) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0015 1.0047 1.0097 1.0164 1.0246 0.5 1.0012 1.0040 1.0085 1.0145 1.0221 1 1.0010 1.0036 1.0078 1.0135 1.0207 2 1.0009 1.0033 1.0072 1.0125 1.0193 5 1.0008 1.0030 1.0066 1.0116 1.0180 10 1.0007 1.0028 1.0063 1.0112 1.0174 table 17 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 10) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0224 1.0481 1.0793 1.1144 1.1527 0.5 1.0216 1.0462 1.0755 1.1087 1.1448 1 1.0212 1.0459 1.0736 1.1056 1.1405 2 1.0208 1.0448 1.0716 1.1025 1.1361 5 1.0205 1.0439 1.0698 1.0997 1.1320 10 1.0202 1.0436 1.0692 1.0988 1.1306 302 h.l. ton-that table 18 nonlinear to linear frequency ratio nl/l of (ssss) square fgm stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 20) a/h n (ssss) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0220 1.0475 1.0761 1.1074 1.1410 0.5 1.0214 1.0460 1.0734 1.1032 1.1350 1 1.0211 1.0451 1.0718 1.1007 1.1315 2 1.0207 1.0442 1.0701 1.0980 1.1278 5 1.0204 1.0433 1.0684 1.0955 1.1242 10 1.0200 1.0429 1.0678 1.0945 1.1227 table 19 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 10) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0026 1.0094 1.0201 1.0344 1.0521 0.5 1.0024 1.0087 1.0188 1.0323 1.0491 1 1.0023 1.0084 1.0181 1.0312 1.0473 2 1.0022 1.0080 1.0173 1.0299 1.0454 5 1.0021 1.0076 1.0166 1.0286 1.0436 10 1.0020 1.0075 1.0163 1.0282 1.0296 table 20 nonlinear to linear frequency ratio nl/l of (cccc) square fgm stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 20) a/h n (cccc) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0017 1.0055 1.0114 1.0193 1.0291 0.5 1.0013 1.0046 1.0099 1.0170 1.0258 1 1.0012 1.0042 1.0091 1.0158 1.0241 2 1.0010 1.0039 1.0084 1.0147 1.0225 5 1.0009 1.0035 1.0078 1.0136 1.0210 10 1.0008 1.0033 1.0075 1.0131 1.0203 the parameters to be changed are given as exactly the same as in the previous example. once again, the numerical results based on this proposed method are given in tables 13 20. furthermore, the first four mode shapes for a fully simply supported stiffened functionally graded plate with case a/h = 10, bs = a/30 and n = 2 are also depicted in fig. 10. a new c0-tsdt for nonlinear free vibration analysis of stiffened functionally graded plates 303 mode 1 mode 2 mode 3 mode 4 fig. 10 the first four mode shapes of the functionally graded plate with two stiffeners 4. conclusions an efficient numerical method based on the new c0 third-order shear deformation theory with respect to the shi theory is firstly developed for a nonlinear free vibration analysis of stiffened functionally graded plates. the shi's third-order shear deformation theory with its necessary stability is then a revised form which only requires c0 continuity for displacement fields. in this c0 third-order shear deformation theory, two additional variables are joined, and thence the first derivative of transverse displacements is only required, respectively. furthermore, the functionally graded materials with excellent characteristics of ceramic in corrosive resistances combined with the great toughness of metals in absorb energy and plastically deform, lead to outstanding advanced materials that can withstand extreme conditions of reality. this is even more wonderful if they are reinforced by stiffeners. from the above notions, this paper aims to provide mechanical information for this type of structure. in each case of the study with different data, the achieved results are found to agree well with the solutions of other numerical methods. based on this proposed method, the present numerical 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starčević 1,2 1 berlin institute of technology, germany 2 national research tomsk state university, russia abstract. the paper is devoted to an experimental investigation of the sliding friction force between a rapidly oscillating sample and a rotating steel plate. the sliding friction force is studied experimentally as a function of the oscillating amplitude, the sliding velocity and the normal force. the results have proved the hypothesis that the coefficient of friction is a function of dimensionless oscillation amplitude and dimensionless velocity. key words: sliding friction, ultrasonic oscillations, tribospectroscopy 1. introduction ultrasonic vibration is widely used in manufacturing processes for the reduction of friction, such as tube and wire drawing, metal forming and drilling, etc. [1-5]. the effect of ultrasonic oscillation on the sliding friction has been studied for a few decades, including all possible directions of vibration with consideration of various frequencies and oscillation amplitudes [6-10]. the pioneer experimental studies of godfrey [11] and lenkiewicz [12] have found that the sliding friction reduction is due to vibration. later the influence of oscillations on both friction and wear of metals was investigated by weishaupt [13] and goto et al. [14]. in the recent few years the theoretical models have been also developed to explain this reduction phenomenon for different directions of vibration [15, 6-7]. in the recent paper [10], the influence of ultrasonic oscillations for the basic configuration (in-plane-oscillations: oscillations in the contact plane along and perpendicular to the sliding direction; out-of-plane oscillations: oscillations perpendicular to the contact surface) is systematically studied in the relevant interval of the oscillation amplitudes and sliding velocities. the theoretical models are developed to support the experimental data. its results show that the relation between the sliding velocity and the oscillation one is important for received january 12, 2015 / accepted february 28, 2015 corresponding author: natalie milahin technische universität berlin, 10623 berlin, germany e-mail: natalie.milahin@tu-berlin.de original scientific paper 28 n. milahin, q. li, j. starčević the reduction of friction for all these three basic configurations, and this effect can be expected to be significant when the sliding velocity is much smaller than the oscillation one. however, the work is carried out only under constant normal load (fn = 9 n). paper [16] has demonstrated that the normal load conditions also play an important role in the static friction force in the presence of ultrasonic oscillations. this effect is supposed to be observed in the sliding contact. therefore, in this paper a complete experimental study for the out-of-plane configuration will be presented. 2. theoretical analysis we consider the dry contact of a sphere indenter pressed against a substrate moving with constant velocity v . under normal load fn , indentation depth (0) zu is generated. after applying the ultrasonic oscillation, indentation depth (0) zu becomes a periodic function of time t : (0) cosz z zu u u t   , where  is the circular frequency of oscillations. in paper [17], it is argued that that the main governing parameter in most of the contact and frictional problems is indentation depth (0) zu . the only velocity scale which can be produced using this characteristic spatial scale (0) zu is combination (0) zu . with these characteristic space and velocity scales, we can define dimensionless oscillation amplitude (0)ˆ /  z z zu u u (1) and dimensionless velocity (0)ˆ /( )zv v u . (2) according to general ideas of the paper [17], we can expect the resulting coefficient of friction  in the presence of oscillations to be a function of only these two dimensionless variables: ˆ ˆ( , )zf u v   . (3) indentation depth (0) zu is difficult to measure directly but it can be estimated if the normal force is known. it is shown in [18] that for a wide class of surface profiles, the indentation depth is a power function of the applied normal force (0) z nu c f    (4) with c contact stiffness and 0 2 / 3  , while 0  for "very rough" surfaces (white noise) and 2 / 3  corresponds to the hertz solution for a single smooth asperity. for macroscopic profiles with superimposed roughness it is shown that there can be a crossover from a smaller value of  to a larger one when the normal force is increasing [19]. substituting eq. (4) into eq. (1) and (2), we can reformulate the hypothesis (3) in the form ,            z n n u v f cf cf . (5) in this paper this hypothesis will be proved from experiment investigation. influence of the normal force on sliding friction under ultrasonic oscillations 29 3. experiment measurement and results the experimental measurements are carried out on the ultrasonic pin-on-disk tribometer (set-up in fig. 1). to investigate the sliding friction force as a function of the normal force under the ultrasonic oscillations, we have used a specimen made of steel with build-in piezo elements with natural frequency 30 khz  and a small steel bearing ball (100cr6) fixed on the one side. the rotation speed of the disc made of structural steel is controlled by the step motor and the specimen is installed as perpendicular to the disc. during the measurements the specimen is pressed against the rotating disc under different normal forces and the specimen performs oscillations perpendicular to the sliding direction (so called out-of-plane-mode). the variable amplitude of the oscillations is controlled by the ultrasound generator and measured by the high-precision laser vibrometer. the friction and normal forces are measured by the force sensor. fig. 1 experimental set-up, schematic view on the pin-on-disc-tribometer [10]. the sliding friction force is measured for five different excitation states, oscillation amplitude 0 zu , 0.04 , 0.08 , 0.16 and 0.27 m, four different normal loads fn = 4, 9, 18 and 32 n, and also for different sliding velocities v ranging from 0.001 to 0.009 m/s . with these small velocities, the maximum influence of ultrasonic oscillation can be achieved. each measurement (constant sliding velocity, constant normal load and constant amplitude) lasts for 1 minute and the frictional force is averaged by 60 measured values during this time. the coefficient of friction is obtained from the division of averaged frictional force by the applied normal force. 30 n. milahin, q. li, j. starčević fig. 2 the coefficient of sliding friction as a function of oscillation amplitude uz and the sliding velocity for four different normal loads: 4,9,18,32 nnf . fig. 3 the coefficient of sliding friction in the axes (uz / cf  n,  / cf  n). influence of the normal force on sliding friction under ultrasonic oscillations 31 the measurement results are shown in fig. 2. the coefficient of friction  is plotted as a function of oscillation amplitude uz and sliding velocity v, the colors representing different normal loads. the sliding coefficient of friction can become even zero, if the sliding velocity is very small and the oscillation amplitude very high. when the oscillation amplitude is larger, the sliding friction is reduced, but with an increasing normal load this reduction effect becomes less for the same oscillation amplitudes and sliding velocities. now we come to prove the hypothesis (5). we have plotted experimentally measured values of  in the axes (uz / cf  n,  / cf  n) with  = 1/3, as shown in fig. 3. it is found that in these variables, the data indeed are placed on one surface thus defining a function of the form eq. (5) (with exception of the case of very high forces, which can be related to the above mentioned crossover). 4. conclusions a recent paper [17] shows that sliding friction coefficient  in contact with oscillation will be a function of the oscillation amplitude, the sliding velocity and the indentation depth. due to the measurement difficulty of indentation depth, we have proposed an empirical model of frictional contact so that the sliding friction coefficient will be a function of the dimensionless oscillation amplitude and the dimensionless velocity where the normal force is involved because the indentation depth is a power function of the applied normal load for a wide class of surfaces [18]. therefore, an experimental investigation of steel-steel contact is carried out using the pin-on-disc apparatus to measure the coefficient of sliding friction for different oscillating amplitudes, sliding velocities and especially the normal forces. the results show that the effect of friction reduction decreases with the increasing normal force for the same oscillation amplitudes and sliding velocities, and our hypothesis is proved with these experiment data. furthermore, we are looking forward to investigating the influence of contact geometry on sliding friction. also the possibility of a temporary loss of contact which is assumed to appear at large amplitudes, lower normal forces and sliding velocities, will be analyzed in detail. acknowledgements: the authors thank v.l. popov for valuable discussions. this work is supported in part by the deutsche forschungsgemeinschaft and the tomsk state university academic d.i. mendeleev fund program. references 1. siegert, k., ulmer, j., 2001, superimposing ultrasonic waves on the dies in tube and wire drawing, journal of engineering materials and technology, 123(4), pp. 517-523. 2. murakawa, m., 2001, the utility of radially and ultrasonically vibrated dies in the wire drawing process, j. mater. process. technol. 113(1-3), pp. 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and contact geometry on the static force of friction of an oscillating sample, physical mesomechanics, 17(3), pp. 228-231. 17. popov, v.l., psakhie, s., popov, m., 2014, on the role of scales in elastomer friction, international conference on physical mesomechanics of multilevel systems, aip conf. proc. 1623, pp. 507-510. 18. pohrt, r., popov, v.l., 2013, contact stiffness of randomly rough surfaces, scientific reports, 3, 3293. 19. pohrt, r., popov, v.l., 2013, contact mechanics of rough spheres: crossover from fractal to hertzian behavior, advances in tribology, 974178. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 191 205 https://doi.org/10.22190/fume190215025c © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper some simple results on the multiscale viscoelastic friction michele ciavarella, antonio papangelo politecnico di bari, center of excellence in computational mechanics, bari, italy abstract. the coefficient of friction due to bulk viscoelastic losses corresponding to multiscale roughness can be computed with persson's theory. in the search for a more complete understanding of the parametric dependence of the friction coefficient, we show asymptotic results at low or large speed for a generalized maxwell viscoelastic material, or for a material showing power law storage and loss factors at low frequencies. the ascending branch of friction coefficient at low speeds highly depends on the rms slope of the surface roughness (and hence on the large wave vector cutoff), and on the ratio of imaginary and absolute value of the modulus at the corresponding frequency, as noticed earlier by popov. however, the precise multiplicative coefficient in this simplified equation depends in general on the form of the viscoelastic modulus. vice versa, the descending (unstable) branch at high speed mainly on the amplitude of roughness, and this has apparently not been noticed before. hence, for very broad spectrum of roughness, friction would remain high for quite few decades in sliding velocity. unfortunately, friction coefficient does not depend on viscoelastic losses only, and moreover there are great uncertainties in the choice of the large wave vector cutoff, which affect friction coefficient by orders of magnitudes, so at present these theories do not have much predictive capability. key words: roughness, contact mechanics, rubber friction, persson's theories, adhesion 1. introduction in contact mechanics and tribology, roughness plays a fundamental role for adhesion, friction, lubrication, sealing, despite it is very difficult to make any quantitative predictions depending on it. in particular, in elastic contact fractal roughness leads to a "ill-posed" received february 15, 2019 / accepted may 25, 2019 corresponding author: michele ciavarella politecnico di bari, center of excellence in computational mechanics, viale gentile 182, 70126 bari, italy e-mail: mciava@poliba.it 192 m. ciavarella, a. papangelo solution (ciavarella et al., [1], persson [2]), in which the "real area of contact" and many (but not all) other physical quantities, cannot be precisely defined because they strongly depend on the tail of the power spectrum of roughness (persson et al., [3]) and in particular on where this tail is truncated. the classical asperity models like greenwood & williamson [4] initially did not recognize this problem, and anyway solve less accurately the mathematical problem of elastic rough contact, where persson’s theory has today elucidated many aspects (see putignano et al., [5]). the original persson's theory was aimed at a very practical problem with many scientific and technological applications, estimating friction due to viscoelastic losses in sliding of viscoelastic bodies on hard substrates. however, it seems that most often the pure viscoelastic contribution of friction is not sufficient to explain some experimental results, and possible other mechanisms like adhesion are also put forward (lorenz et al., [6]). this perhaps somehow limits the interest in obtaining exact results on the viscoelastic losses, and anyway suggests simple formulations should be preferred given the uncertainties on the considerable number of arbitrary parameters which need to be estimated anyway. the spectrum of roughness is often assumed to be of power law form although is not precisely defined neither at very low wave vectors, nor at high wave vectors, due to limitations in measuring instruments, and the need to use various instruments to obtain a measurement over a very broad range of scales. in particular, the truncation of the spectrum of the surface will critically affect some results. for example, lorenz et al. [6] suggest that (perhaps in typical present tyre-road contacts?) the truncation wave vector should occur where the rms slope reaches 1( ) 1.3rmsh q  , (1) although we have found no data to interpret the motivation and hence substantiate the generality of this recommendation, and no independent researcher has suggested alternative criteria. other authors (carbone & putignano [7]) suggest many factors could be associated to the truncation cutoff, including small dirt particles or rubber wear particles, but do not give precise suggestions. hence, while the mathematical problem is relatively easy to formulate, and the multiscale nature of roughness is postulated to be of critical importance, the actual practical solution of this crucial point is left somehow more obscure. even less discussed is the other truncation in the spectrum, that at low wave vectors, despite being clear that by enlarging the size of the specimen it is very likely that some power content appears – and anyway, one could argue that even shape or undulations of the pavement could occur and may, in principle, play a role for friction, as we shall see more in details where in particular, in the following. the theory by persson [2] is based on exact solution for sliding of a rough rigid random surface in full contact with a viscoelastic medium. the exact result for the friction coefficient would be 1 0 2 3 2 0 0 1 ( cos( )) d ( ) d cos( ) im 2 (1 ) q q e qv q q u q             , (2) where σ0 is the nominal contact stress, u (q) the surface displacements power spectrum (defined as a function of wave vector q), where q0 is the smallest (relevant) roughness wave vector. also, e (ω) is the complex modulus of the viscoelastic material. some simple results on the multiscale viscoelastic friction 193 however, since full contact is a very remote condition in practical applications, the real condition of partial contact is solved by making various clever approximations: 1) the psd (power spectrum density) of displacements, u (q), which would be strictly required to make a correct theory, is approximated with the psd of the roughness c (q) ( ) ( ) or ( ) ( )zu h u q c qx x (3) since power loss occurs mainly in the contact area, and since it is known that the two psds are parallel curves, the error can be fixed a posteriori with corrective factors. 2) the power loss integration is weighted by function p (q) = a (ζ) / a0 which is the relative contact area when the interface is observed at magnification ζ = q / q0, where a0 is the nominal contact area. on the grounds that only the portion of the area which is actually in contact really forces the surface to deform and undergo the viscoelastic deformation 0 1 ( ) ( ) / erf 2 p q a a g          , (4) 0 2 2 3 2 0 0 1 ( cos( )) d ( ) d 8 (1 ) q q e qv g q q c q           , (5) where the argument of the complex modulus is the projection of the wave vector on the direction of sliding. notice that for negative frequencies, the complex modulus becomes the complex conjugate of the modulus at positive frequencies, which permits to solve the integrals such as 2 /2 0 0 4     , and corresponds also to the fact that sliding in negative direction makes the same contribution as sliding in the positive direction. 3) there is a further factor s (q), a correction factor which at large magnifications can be taken as s (q) ≃ 1/2, and otherwise results from some fitting calculations of the stiffness of the contact (which were done for elastic contact), as s(q) = ½ + ½ p 2 (q). all these corrections result in a final calculation which involves four nested integrals 1 0 2 3 2 0 0 1 ( cos( )) d ( ) ( ) ( ) d cos( ) im 2 (1 ) q q e qv q q c q s q p q            , (6) which is not trivial to compute (at least, in our experience), and certainly does not elucidate the parametric dependences of the various branches of the friction coefficient (low, intermediate, and high velocities), as we shall attempt here. persson’s much earlier model [8] suggested that the friction coefficient was more simply 0 0 im ( ) ( ) e c e    , (7) 194 m. ciavarella, a. papangelo where e (ω0) is the complex viscoelastic modulus at the frequency ω0 ∼ v / l of the cyclic deformation at velocity v of an asperity of diameter l. c is a constant of order unity, and this form is much easier to understand as well as to recognize the crucial role of the truncation of the roughness spectrum, here in the choice if asperity diameter l. adding that im e (ω0) / |e (ω0)| is also of the order unity at the frequency where this ratio assumes a maximum, friction itself would be of the order unity as maximum value, as it is correct in terms of order of magnitude. previous fundamental contributions were made by williams, landel & ferry [9] which relate temperature and rate dependence of viscoelastic properties, interpreted by grosch [10] which justify a single "master curve" for the temperature and velocity dependence of friction. grosch himself had established the main results for friction of rubbery materials long ago, finding experimentally that friction could show two maxima, one attributed to adhesion with the track, where ideally should be existing even for a perfectly smooth surface [11], and the other at higher frequency due to viscoelastic losses. the present scientific and technological challenge is on how to make more "quantitative" models, following the hope that, measuring in details the viscoelastic properties of the materials, the roughness of the substrate, one could estimate friction: but there remains a few parameters to estimate, and particularly on the adhesive component, we are left ultimately with a "fitting" exercise, which at engineering level would compete with other possible alternative, including perhaps "artificial intelligence" [12]. popov ([13], eq. (16.12)) suggests an equation similar to eq. (7) of persson [8], which can be essentially attributed to a single "scale" of asperities (same diameter), which however, recognizes in factor c the contribution due to the rms slope of surface h′rms (q1), 0 1 0 im ( ) ( ) ( ) rms e h q e     , (8) and therefore clearly identifies that the friction coefficient strongly depends on the truncation of the psd of roughness. since this implies that μ ≤ h′rms (q1) as at most we can have a peak im e (ω0) / |e (ω0)| close to 1, we recognize that for this theory to be predictive, one needs to postulate a very high slope of the surface, which in turn means a measurement of roughness down to very small scales, to the order of microns in wavelength, and below. we shall see, however, that this simplified formulation is valid only in a very crude sense. we assume persson's theory is the most accurate analytical model presently available, although present comparison with full numerical simulations (see e.g. scaraggi & persson [14]) is necessarily very limited in terms of broadness of the roughness spectrum bandwidth. unfortunately, it involves four nested integrals which in our experience are not easy to compute numerically, and it seems important to see:  under which conditions the "simple" formulations by early persson or popov (8), see also ciavarella [15] are valid  what the order of corrective coefficient k is in a more general equation 1 1 1 im ( ) ( ) ( ) rms e q v kh q e q v   (9)  what the main parametric dependencies are, if the dependence is not that given by this form (9) some simple results on the multiscale viscoelastic friction 195 here, we shall make a more detailed study, based on simple material models, like the generalized maxwell model, or one with power law trends of the loss and storage moduli at low frequencies, to make illustrative examples. 2. generalized maxwell materials viscoelastic materials are typically represented with complex elastic moduli ( ) re ( ) im ( ) ( ) ( )e e i e e ie         , (10) where e′ (ω), the "storage modulus", is the material stiffness while the "loss modulus" e′′(ω) is the irreversible damping: also, the ratio between loss modulus and storage modulus is dissipation factor tan δ(ω), which indicates the degree of viscoelasticity of a material. these factors are often determined using the dynamic mechanical analysis (dma), by a forced oscillation at a constant frequency, and interpolation and extrapolation for a wider frequency range is obtained using the time / temperature shift wlf-equation by william, landel, ferry [9]. very often, a prony series describes the complex modulus (see fig.1), with a model comprising a parallel connection of several maxwell elements 2 2 2 2 1 re ( ) 1 n i i i i e e e            , (11) 2 2 1 im ( ) 1 n i i i i e e        , (12) where e∞ is the final, equilibrium long term modulus of a tensile test (if shear modulus is measured, usually the conversion if done to young’s modulus assuming a frequencyindependent poisson’s ratio), while 1 n t i i e e e    (13) is the instantaneous modulus. the relaxation times are defined as τi = ηi / ei, where ηi is the damping viscosity of the i-th element, and are usually ordered such that τ1 << τ2 … fig. 1 a typical prony series of a generalized maxwell material 196 m. ciavarella, a. papangelo there are various approaches to determine the prony series although this is not a welldefined exercise so that often the relaxation constants are assumed, and the moduli are found by some optimization to reduce the error with experimental data [16, 17]. 2.1. a single relaxation time let us consider for simplicity just one characteristic time 2 2 1 1 2 2 1 re ( ) 1 e e e         , (14) 1 1 2 2 1 im ( ) 1 e e       . (15) we obtain from the general persson’s model (6) 0 3 1 1 d ( ) ( ) ( ) 2 q q q q c q s q p q i  , (16) where 2 / 2 2 1 1 12 2 2 2 0 00 0 1 1 2 2 10 1 ( cos( )) 4 cos d cos( ) im d (1 ) (1 ) 1 ( ) cos 4 1 1 2(1 ) 1 ( ) e qv i e qv qv e qv qv                                   (17) and 0 3 2 1 d ( ) 8 q q g q q c q i  , (18) with 2 2 2 2 2 1 12 2 2 2 2 0 0 0 1 ( cos ) 2 1 d ( 2 ) 1 (1 ) (1 ) 1 ( ) e qv i e e e e qv                             . (19) for a power law psd of roughness between cutoff wave vectors q ϵ [q0, q1] (a choice without loss of generality if the results will turn out to depend only on the tail of the psd which is indeed very often a power law)  2 1 0( ) h c q c q    , (20) for which we can give all integrals except the final one in series form involving hypergeometric functions (mathematica can compute it, but it is here not reproduced for brevity, although some care should be used for very low velocities, where it is better to use the asymptotic expressions). it is interesting, however, to give the behavior at low and high speeds 2 2 2 2 20 low 02 2 2 0 1 ( ) ( ) 18(1 ) h h v c g q e q q h         , (21) some simple results on the multiscale viscoelastic friction 197 2 2 2 2 20 high 1 02 2 2 0 1 ( ) ( ) ( ) 18(1 ) h h v c g q e e q q h          , (22) which is obviously the result due to the complex modulus being essentially real at high and low frequencies, and given by e∞ and e∞ + e1, respectively. further, ow 1 1,l 12 0 4 ( ) 2(1 ) v e i q qv       , (23) 1 1,high 2 10 4 ( ) 2(1 ) v e i q qv      (24) while ow 2 2,l 2 2 0 4 ( ) 2(1 ) v e i q        , (25) 2 2,high 12 2 0 4 ( ) ( ) 2(1 ) vi q e e         . (26) therefore, using 1 1 erf 2 g g       , (27) as can be done even for large area ratios, we obtain   1 0 ow 2 2 1 l 0 1 2 2 2 2 0 2 1 1 0 1 1 1 1 low ( ) 2 (1 ) d 2 (1 ) 2 im ( )2 1 2 ( ) q h v h h q h rms v e q q c h v q e q q e q c h v e h e q vh h q h e q v                     , (28) where we used that rms slopes 1 0 1 1( ) 1 h rms c h q q h    . (29) therefore, notice that corrective coefficient k with respect to eq. (9) depends only on the hurst exponent of the surface and, 2 1 2 h k h    (30) and for example k ≃ 0.133 for h ≃ 0.8. 198 m. ciavarella, a. papangelo this confirms that the friction at low speed highly depends on q1 as suggested originally by persson [8], then by popov [13] and ciavarella [15], but also finds that the corrective coefficient can be significantly lower than one. perhaps this is due to the nature of the single relaxation constant, which is probably a limit case. nothing can be said a priori on how far this approximation goes, however, in terms of speed range. therefore, it is useful to move to the other extreme at high speeds, where we find with the same procedure that 1 0 2 0 1 high 2 2 2 2 1 1 10 2 (1 ) ( ) d ( ) q h rms v h h q c h heq q q h v e e vq q               , (31) 1 high 1 1 high im ( ) ( ) ( ) ( ) v rms v e q v q h q h e q v   , (32) where 1 0 2 0 2 2 2 2 0 2 ( ) (1 ) d 0.85 at 0.8, for example. q h h h h q q h q h h q h q q          (33) eq. (32) is therefore completely different in form with respect to what suggested originally by persson [8], then by popov [13] and ciavarella [15], so there can be no corrective coefficient with respect to eq. (9). in particular, as it is evident in the form (31), it does not depend on truncating large wave vector. as an example, we introduce the case of psd having h = 0.86, c0 = 1.152×10 −3 [m 6−2h ], and q0 = 10 2.7 [1/m], which are realistic values for road surfaces, whereas the high truncating wave vector is initially set, according to lorentz et al. [6], such that h′rms = 1.3. for the material, we take e∞ = 10mpa, e1 = 1000mpa, τ1 = 7 × 10 −4 s and fig. 2 shows the storage and loss moduli in terms of frequency. fig. 2 an example maxwell material with e∞ = 10mpa, e1 = 1000mpa, τ1 = 7 × 10 −4 s. black line indicates the storage, while blue line the loss modulus fig. 3 shows the results for the friction coefficient, showing that the low and high velocity approximations work quite well whereas using the full approximate (9) "simple" equation using for k = 0.098 as in (30) does not estimate the peak friction coefficient with great accuracy, and at high speeds, it is better to use directly the high velocity approximation (31). some simple results on the multiscale viscoelastic friction 199 fig. 3 friction coefficient for the example maxwell material of fig. 2. black solid line is the numerical solution of the friction coefficient with the full persson’s model, red dashed line indicate the asymptotic results (28), (31), and blue dashed line is the approximate "simple" equation (9) using for k = 0.098 as in (30) fig. 4 shows the effect of varying the truncation cutoff, given this choice is arbitrary, results also in dramatic differences. in particular the solid lines indicate the full persson’s solution with h′rms = 0.69, 1.3, 1.8, 2.1 (black, blue, red and green lines, respectively). notice that the high velocity branch of the friction coefficient curve does not change, as we expected from the result (31) which is instead determined by the rms amplitude of roughness, not sensitive to the truncating cutoff. it is obvious that by small changes in the choice of the truncation in h′rms, namely just tripling its value from 0.69 to 2.1, we have in the low branch a change in the friction coefficient which is by orders of magnitude! notice also that for very broad spectra of roughness, one could expect a persistently high friction coefficient for a wide range of velocities. in this sense, the "single scale" approximation (9) is increasingly poor for broad spectra, as can be expected. fig. 4 friction coefficient for the example maxwell material of fig. 2, while varying the upper wave vector truncation q1 such that h′rms = 0.69, 1.3 1.8, 2.1 (black, blue, red and green solid lines, respectively, for the full persson’s solution). dashed lines of the corresponding color are the approximate "simple" eq. (9) using for k = 0.098 as in eq. (30) 200 m. ciavarella, a. papangelo 2.2. multiple relaxation times let us reconsider the more general case of various relaxation times so as to cover more realistic materials. it is clear that repeating the derivation   2 22 1 2 2 2 1 00 22 10 ( ) cos d cos (1 ) 1 ( ) cos 4 1 1 2 1 ( )1 n i i i i n i i i i e qv i qv e qv qv                              , (34) so that i1,lowv remains the same as with a single relaxation time (23), where now we replace the modulus at low or high frequency ow1,l 2 2 low 10 0 4 4 ( ) im ( ) 2 2(1 ) (1 ) n i v i i e i q qv e qv              , (35) 1,high 2 2 high 10 0 4 4 ( ) im ( ) 2 2(1 ) (1 ) n i v i i e i q e qv qv             , (36) since low high 1 1 1 im ( ) , whereas im ( ) n n i i i i i e e e             . (37) similarly for i2, but we can also say ow 2 2,l 2 2 0 2 ( ) (1 ) v e i q        , (38) 2 2,high 2 2 1 0 2 ( ) (1 ) n v i i i q e e                . (39) we obtain from the general persson’s model (6) (and again for a power law psd of roughness), a similar behavior at low and high speeds 2 2 2 2 20 low 02 2 2 0 1 ( ) ( ) 18(1 ) h h v c g q e q q h         , (40) 2 2 2 2 20 high 02 2 2 10 1 ( ) ( ) 18(1 ) n h h v i i c g q e e q q h                  . (41) therefore, using eq. (27), we obtain the same equations (28) and (32) as for the single relaxation time. this seems to suggest that these results should have greater generality. however, in many cases, experimentally we observe that loss and storage modulus have power law form in frequency for many compounds of interest, as we shall discuss in the next paragraph. some simple results on the multiscale viscoelastic friction 201 3. power law form of the loss and storage modulus it appears that in the practical range of interest for rubber compounds used in tires (see lorenz et al. [6]), we observe storage and loss moduli which seem to follow power laws (see fig. 5 for three distinct examples). let us therefore consider this case, which permits also a simple estimate of the relevant equations in closed form at low speeds. (a) (b) (c) fig. 5 real (solid blue) and imaginary (solid black) parts of the viscoelastic modulus in lorenz et al. [6] rubber compound a, b, c (respectively in fig. 5 a,b,c) together with power law approximations (dashed lines) at low frequencies. these are e = 10 1.5 f 0.05 + i10 0.49 f 0.07 , e = 10 1.175 f 0.075 + i10 0.243 f 0.09 , e = 10 1.175 f 0.05 + i10 0.075 f 0.07 , for a, b, c, respectively 202 m. ciavarella, a. papangelo let us consider the following low-frequency end approximation of power law behavior, re ( ) 10 and im ( ) 10 i ir re e        (42) where for example lorenz et al. [6] rubber compound a, b, c suggest that βr ≃ 0.05 − 0.075 whereas βi ≃ 0.07 − 0.09. this seems to suggest that while for the real part, we are not too far from the constant values which in principle we expect at very low frequencies for a (even generalized) maxwell material, the imaginary part is certainly remote from the single relaxation constant material, which would have βi = 1. now, with the same notation as for the maxwell materials, we obtain the integrals 1 2 0 (1 / 2)4 10 ( ) 2 (3/ 2 / 2)(1 ) i i i i i qv            (43) and / 2 2 2 2 2 0 0 2 2 2 2 0 4 d 10 ( cos ) (1 ) (1/ 2 )4 10 ( ) 2 (1 )(1 ) r r r r r r i qv qv                         , (44) where we have considered that at low v the real part of the modulus dominates over the imaginary. also, 2( 1 ) 2( 1 ) 2 2 0 02 2 0 (1/ 2 )1 4 10 8 2 (1 ) 2( 1 )(1 ) r r r r h h r r r q q g v c h                       . (45) therefore, using eq. (27), 1 0 1 2 0 2( 1 ) 2( 1 ) 0 1 1 1 low (1 / 2) 10 (3/ 2 / 2) d (1/ 2 ) (1 ) 2( 1 ) (1 / 2) (3 / 2 / 2) im ( )1 ( ) 1 ( )(1/ 2 ) (1 ) 2( 1 ) i r i r i r r i q h i h h q r r r i i rms i r vr r r v q c q q q h e q vh h q h e q v h                                                           , (46) since 1 1 1 low im ( ) 10 ( ) . ( ) i r i r v e q v q v e q v      (47) eq. (46) is an equation of the form suggested originally by persson [8], then by popov [13] and ciavarella [15], but where the corrective coefficient as in eq. (9) is given by some simple results on the multiscale viscoelastic friction 203 (1 / 2) (3/ 2 / 2)1 1 (1/ 2 ) (1 ) 2( 1 ) i i i r r r r h k h h                        , (48) which in particular we find that for the maxwell material having βr = 0 and βi = 1, we return to eq. (30). notice therefore from eq. (48), that the actual behavior at low speeds changes significantly the coefficients k needed in the simple eq. (9) to estimate the ascending branch of the friction curve. fig. 6 compares some example coefficients k as in eq. (48) as a function of βi in the range from 10 −3 to 1, for a few cases of βr = 0, 0.05, 0.1, 0.15 (black, blue, red, green curves, respectively). it is evident that k can vary significantly both below and above 1, tends to be independent on βi at low βi. however, it is quite close to 1 for the typical materials shown in lorenz et al. [6] rubber compound a,b,c, which explains why there appeared to be even too success with the simple equation in a previous paper (ciavarella [15]). fig. 6 the multiplicative coefficient k as in eq. (48) for using the approximate "simple" eq. (9) at low velocities, for generic power law approximations of the storage and loss moduli as a function of βi in the range from 0 to 1, for a few cases of βr = 0, 0.05, 0.1, 0.15 (black, blue, red, green curves, respectively) 4. discussion recently [18], some actual data from tolpekina and persson [19] have been analyzed using the simplified formulation, and adhesive contribution. it has been remarked that the adhesive contribution seems more important than the viscoelastic one, perhaps more important also than what expected, for example, from earlier studies of grosch. unfortunately, the adhesive contribution fundamentally to date is modeled with empirical fitting equations, which have a "bell-shape" which is quite similar to the viscoelastic friction contribution, in large parts of the velocity spectrum. hence, it results that it is quite difficult to estimate with 204 m. ciavarella, a. papangelo precision the relative contribution since various choices of the truncation wave vector, and other fundamentally arbitrary constants, result in equally satisfactory fitting of the experimental data. additionally, viscoelastic materials nonlinear effects seem to strongly affect the adhesive models (and strictly speaking also the viscoelastic ones, although it is not clear if tolpekina and persson [19] consider this), and their role depends also crucially on the choice of the truncating wavelength of roughness. hence, while we have made a significant progress in the mathematical solution of the viscoelastic contribution in highly idealized conditions, this seems very remote from practical "predictive" capabilities, and the complexities of the full multiscale theories do not help in clarifying the subject for a large audience. what is needed is, instead, that more researchers work actively in this field by comparing theories and results, and ideally, collecting "round robin" results and benchmark cases. our effort here is a small attempt in this direction. 5. conclusions we have derived some simple results for the fully multiscale persson’s theory as applied to quite important materials models, like generalized maxwell, or with power-law storage and loss modulus at low frequencies. this has permitted to elucidate that the friction coefficient at low velocity of sliding, where we expect most contacts operate, show a simple dependency on the ratio of imaginary and absolute value of the modulus and on the rms slope of the profile noticed earlier by persson, popov and also the present author, which also implies a very high sensitivity to the choice of the truncating wave vector of roughness. however, we also find that this "simple equation" has a multiplicative coefficient which depends on the form of the viscoelastic modulus, and we have given approximate but simple closed form results for simple cases. the peak of the friction coefficient, however, cannot be estimated accurately with the simple equation. at high velocities, it is found that the descending (unstable) branch at high speed mainly on the amplitude of roughness, and, therefore, would not depend on the choice of the truncating wave vector. there remain great uncertainties in "predicting" friction coefficient since orders of magnitude variation can be found changing the truncating wave vector. acknowledgements: a.p. is thankful to the dfg (german research foundation) for funding the projects ho 3852/11-1 and pa 3303/1-1. references 1. ciavarella, m., demelio, g., barber, j.r., jang, y.h., 2000, linear elastic contact of the weierstrass profile, proceedings of the royal society of london, series a: mathematical, physical and engineering sciences, 456(1994), pp. 387-405. 2. persson, b.n.j., 2001, theory of rubber friction and contact mechanics, the journal of chemical physics, 115(8), pp. 3840-3861. 3. persson, b.n.j., albohr, o., tartaglino, u., volokitin, a.i., tosatti, e., 2005, on the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion , journal of physics: condensed matter, 17(1), pp. 1–62. 4. greenwood, j.a., williamson, j.b.p., 1966, contact of nominally flat surfaces, proceedings of the royal society of london, series a, mathematical and physical sciences, 295(1442), pp 300-319. some simple results on the multiscale viscoelastic friction 205 5. putignano, c., afferrante, l., carbone, g., demelio, g., 2012, a new efficient numerical method for contact mechanics of rough surfaces, international journal of solids and structures, 49(2), pp. 338–343. 6. lorenz, b., oh, y.r., nam, s.k., jeon, s.h., persson, b.n.j., 2015, rubber friction on road surfaces: experiment and theory for low sliding speeds, the journal of chemical physics, 142(19), p. 194701. 7. carbone, g., putignano, c., 2014, rough viscoelastic sliding contact: theory and experiments, physical review e, 89(3), p. 032408. 8. persson, b.n.j., 1998, on the theory of rubber friction, surface science, 401(3), pp. 445-454. 9. williams, m.l., landel, r.f., ferry, j.d., 1955, the temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, journal of the american chemical society, 77(14), pp. 701-3707. 10. grosch, k.a., 1963, the relation between the friction and visco-elastic properties of rubber, proceedings of the royal society of london, series a, mathematical and physical sciences, 274(1356), pp. 21-39. 11. grosch, k.a., 1996, the rolling resistance, wear and traction properties of tread compounds, rubber chemistry and technology, 69(3), pp. 495-568. 12. khaleghian, s., emami, a., taheri, s., 2017, a technical survey on tire-road friction estimation, friction, 5(2), pp. 123-146. 13. popov, v.l., 2010, contact mechanics and friction: physical principles and applications, springer science & business media, berlin, heidelberg. 14. scaraggi, m., persson, b.n.j., 2015, friction and universal contact area law for randomly rough viscoelastic contacts, journal of physics: condensed matter, 27(10), p. 105102. 15. ciavarella, m., 2018, a simplified version of persson’s multiscale theory for rubber friction due to viscoelastic losses, journal of tribology, 140(1), p. 011403. 16. https://it.mathworks.com/matlabcentral/fileexchange/68710-dma2prony_opt (last access: 10.05.2019) 17. jalocha, d., constantinescu, a., neviere, r., 2015, revisiting the identification of generalized maxwell models from experimental results, international journal of solids and structures, 67, pp. 169-181. 18. genovese, a., carputo, f., ciavarella, m., farroni, f., papangelo, a., sakhnevych, a., 2019, analysis of multiscale theories for viscoelastic rubber friction, proceedings aimeta 2019 xxiv conference the italian association of theoretical and applied mechanics rome, italy, 15-19 september. 19. tolpekina t.v., persson b.n.j., 2019, adhesion and friction for three tire tread compounds, lubricants, 7(3), p. 20. https://it.mathworks.com/matlabcentral/fileexchange/68710-dma2prony_opt 10743 facta universitatis series: mechanical engineering vol. 20, no 2, 2022, pp. 445 455 https://doi.org/10.22190/fume220420023m © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper simplified hamiltonian-based frequency-amplitude formulation for nonlinear vibration systems hongjin ma school of science, xi'an university of architecture and technology, xi’an, china abstract. the hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. this paper gives a systematical analysis of the formulation, and two simplified formulations are suggested. the cubic-quintic duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. it can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory. key words: he’s frequency formulation, duffing equation, hamilton principle, one-step solving 1. introduction nonlinear vibration systems arise everywhere in mechanical engineering, as in thin walled structures [1], rotor vibration [2,3], gyroscope systems [4,5], micro-electro mechanical systems [6,7], packaging systems [8], building vibration [9], heavy machinery [10], to name but a few. the most famous nonlinear oscillators include the pendulum systems [11, 12], van der pol oscillator and duffing oscillator [13]. the spring-pendulum systems have many applications in satellites, submarines, aircraft, and energy harvesting device [14]. the recent research frontier in the nonlinear vibration theory is the fractal vibration; hence the description of a vibrating system in a fractal space [15-18]. generally a nonlinear vibration system is complex and it is difficult to solve it exactly. in the last 200 years, engineers had little mathematical tools. the most available one was the perturbation method. however, a small parameter assumption involved in the perturbation method hinders its widely applications for accurate prediction of the periodic properties of a practical problem; hence it is valid only for weakly nonlinear oscillators. received april 20, 2022 / accepted may 17, 2022 corresponding author: hongjin ma school of science, xi'an university of architecture and technology, xi’an, china e-mail: hjma@xauat.edu.cn 446 h. ma the homotopy perturbation method [19-21] and the variational iteration method [22-24] are much advanced ones for strongly nonlinear oscillators, but both methods require some special mathematics skills for problem-solving; especially the former asks for a suitable homotopy equation, and the latter requires a complex identification of the lagrange multiplier involved in the iteration algorithm. for engineering problems, as claimed by the chinese mathematician ji-huan he, the simpler is better for a complex problem [18]. now the condition is completely changed; the hamiltonian-based frequency formulation [25] sheds a promising light on a fast but reliable insight into the physical properties of a nonlinear oscillator with simple calculation. the hamiltonian-based frequency formulation is mathematically beautiful and physically penetrating since only a simple integration calculation is needed. it gives a vivid picture of the whole solution properties. the 2021 hamiltonian-based frequency formulation [25] is a modification of he’s 2006 frequency formulation [26]. the formulation observes a nonlinear vibration system in an energy view by establishing a hamilton principle, which can elucidate clearly the energy conservation of the vibrating process, that is the total energy of kinetic and potential energy keeps unchanged during oscillation. as the formulation is derived from an energy form of the nonlinear oscillator, its approximate solution is valid for the whole solution domain without limitations such as those in the traditional perturbation method. the frequency formulation begins with two arbitrary guesses of the frequency, and two residual integrals have to be calculated to estimate an accurate frequency. it is amazing that the accurate result does not depend upon the initial choice of two arbitrary frequencies even for strong nonlinearity. due to these unprecedented advantages, the formulation caught immediate attention after publication due to its extremely simplicity and remarkable reliability. it has been successfully applied to kundu-mukherjee-naskar oscillator [27], nonlinear vibration of embedded carbon nanotube [28], fractal vibration systems [29,30], packaging systems [31] and mems oscillators [32]. it has been hailed as an unprecedented success. though the achievement has contributed to the fast development of the nonlinear vibration theory, the solving process is relative complex, so a modification is needed. 2. problem statement consider the following conservative oscillator [25]: ( ) 0, (0) 0 (0)w p w w w b + = = = (1) where p(w) is an odd function of the displacement w and b is the displacement amplitude. its hamilton principle can be easily established by the semi-inverse method [33] dtwpwwj        −= )( 2 1 )( 2 (2) where p(w) is the potential energy defined so that ( ) ( ) d p w p dw w= (3) simplified hamiltonian-based frequency-amplitude formulation for nonlinear vibration systems 447 according to hamilton principle, we have the conversation law 21 ( ) 2 w p hw + = (4) where h is the hamiltonian constant, which depends only upon the initial conditions, so the following first-order differential equation is obtained: 0)()( 2 1 2 =−+ bpwpw (5) the hamiltonian-based frequency formulation is to find an approximate solution in the form tbtw ωcos)( = (6) where ω is the frequency to be solved later. the hamiltonian-based frequency formulation begins with two arbitrary tries ω1 and ω2, and the following residual equations are obtained： 2 2 2 1 1 1 1 1 ( ) sin ( cos ) ( ) 2 r t b t p b t p b= + −ω ω ω (7) 2 2 2 2 2 2 2 1 ( ) sin ( cos ) ( ) 2 r t b t p b t p b= + −ω ω ω (8) the residual integrals are defined as tdtr t r t 1 4/ 0 1 1 1 cos 4~ 1 = (9) tdtr t r t 2 4/ 0 2 2 2 cos 4~ 2 = (10) where t1=2π/ω1, t2=2π/ω2. the frequency formulation is [25] 2 2 2 1 2 2 1 1 2 r r r r = − − ω ω ω (11) we can see that the frequency formulation requires only a simple calculation of two integrals, making it much attractive and promising in the nonlinear vibration theory. 3. modified he’s frequency formulations the present author shows an interest in two arbitrary choices of ω1 and ω2, e.g., ω1=1 and ω2=2, or ω1=100 and ω2=0.1, all cases leading to the same result. this two-trial assumption goes back to an ancient chinese algorithm [26], which was developed to he’s frequency formulation to solve nonlinear oscillators by ji-huan he [26], and chun-hui he’s iteration algorithm for numerical simulation [34,35]. 448 h. ma it is amazing that the hamiltonian-based frequency formulation does not depend upon the choice of ω1 and ω2. here a strict proof is given. from eqs. (7) and (8), we have 2 2 21 1 1 1 2 2 2 2 21 1 1 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos ( ) 2 2 cos b r t t p b t p b b p b b t p b t = + − − = − + ω ω ω ω ω ω ω (12) 2 2 22 2 2 2 2 2 2 2 22 2 2 2 ( ) sin ( cos ) ( ) 2 2 ( ) cos ( ) 2 2 cos b r t t p b t p b b p b b t p b t = + − − = − + ω ω ω ω ω ω ω (13) the residual integrals become 1 2 2 / 4 1 21 1 1 1 0 0 1 6 ( )4 2 cos ( 1 ) 3 t b p b r r tdt p b x dx t   − = = + −  ω ω (14) 2 2 2 / 4 1 22 2 2 0 0 2 6 ( )4 2 cos ( 1 ) 3 t b p b r r tdt p b x dx t   − = = + − 2 ω ω (15) so a modified frequency formulation is obtained, which is 2 2 1 2 2 2 2 1 2 2 1 1 2 2 2 2 2 1 1 2 2 2 21 2 2 1 0 0 2 2 2 2 1 2 1 2 2 2 1 2 0 2 2 2 1 2 2 + 6 ( ) 6 ( )2 2 ( 1 ) ( 1 ) 3 3 6 ( ) 6 ( ) 3 3 6 ( ) 2 ( 1 ) ( ) 3 3 6 ( ) 6 ( 1 ) = = ( ) ( ) r r r r b p b b p b p b x dx p b x dx b p b b p b p b p b x dx b p b p b x d          − − = − − − + − + − − − − − − − − − − =               ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω 1 0 2 x b  (16) this modification seems much simpler than the original one because no calculation of integrals is needed, so it is easily applicable. simplified hamiltonian-based frequency-amplitude formulation for nonlinear vibration systems 449 alternatively, we locate at [26] 1 2 cos cost t k= =ω ω (17) eqs. (7) and (8) become 2 2 2 1 1 1 (1 ) ( ) ( ) 2 r b k p kb p b= − + −ω (18) 2 2 2 2 2 1 (1 ) ( ) ( ) 2 r b k p kb p b= − + −ω (19) another modification of the hamiltonian-based frequency formulation is 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 2 2 2 1 2 1 1 ( (1 ) ( ) ( )) ( (1 ) ( ) ( )) 2 2 1 1 ( (1 ) ( ) ( )) ( (1 ) ( ) ( )) 2 2 ( ) ( ) ( ) ( ) 1 (1 ) ( ) 2 ( ) ( ) 1 (1 2 r r r r b k p kb p b b k p kb p b b k p kb p b b k p kb p b p kb p b k b p b p kb − = − − + − − − + − = − + − − − + − − − − = − − − = − ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω 2 2 )k b (20) we recommend k=1/√2 , the simplified hamiltonian-based frequency formulation is 2 2 4( ( ) ( / 2))p b p b b − =ω (21) it is astonishing that all intermediate steps involved in the original hamiltonian-based frequency formulation have disappeared, making the two modified formulations unprecedentedly simple. 4. examples consider the following well-known duffing equation [25] 3 0, (0) 0 (0)w w w w w b + + = = =ε (22) for this example, we have 2 41 1 ( ) 2 4 p w w w= + ε (23) in order to compare the two modifications with the original one, we illustrate first the solving process of the original hamiltonian-based frequency formulation, which begins with two arbitrary choices ω1 and ω2, and the following residual equations are obtained: 450 h. ma 2 2 2 1 1 1 1 2 2 2 2 2 4 2 41 1 1 1 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos 2 2 4 r t b t p b t p b b p b b b b t t = + − − − = + + ω ω ω ω ω ε ω ω (24) 2 2 2 2 2 2 2 2 2 2 2 2 4 2 42 2 2 2 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos 2 2 4 r t b t p b t p b b p b b b b t t = + − − − = + + ω ω ω ω ω ε ω ω (25) the residual integrals are: 1 2 4 2 / 4 2 1 1 1 1 0 1 4 4 10 30 ( ) cos 3 15 t b b b p b r r tdt t   + − = = + ε ω ω (26) 2 4 4 2 / 4 2 2 2 2 2 0 2 4 4 10 30 ( ) cos 3 15 t b b b p b r r tdt t   + − = = + ε ω ω (27) the hamiltonian-based frequency formulation leads to the result 2 2 2 21 2 2 1 1 2 7 1 10 r r b r r = = − + − ω ω ω ε (28) now the modified formulation of eq. (16) gives a one-step solving process: 1 2 2 0 2 1 2 4 2 2 4 4 2 0 2 2 4 2 4 2 2 4 2 2 6 ( ) 6 ( 1 ) 1 1 1 1 6 6 (1 ) ( 2 1) 2 4 2 4 3 4 3 2 2 5 7 10 7 1 10 p b p b x dx b b b b x b x x dx b b b b b b b b b b − − = + − − + − + = + − + = + = = +                           ω ε ε ε ε ε ε (29) which is the same as eq. (28). simplified hamiltonian-based frequency-amplitude formulation for nonlinear vibration systems 451 the modified formulation of eq. (21) is also simple, which leads to the following result 2 2 2 4 2 4 2 2 4 2 2 4( ( ) ( / 2)) 1 1 1 1 4 2 4 4 16 3 4 3 1 4 p b p b b b b b b b b b b b − =   + − −    = + = = + ω ε ε ε ε (30) this result is the same as those obtained by the variational iteration method or the homotopy perturbation method [26]. consider another example of the cubic-quintic duffing oscillator: 3 5 0, (0) 0 (0)w w aw bw w w b + + + = = = (31) the original hamiltonian-based frequency formulation is to calculate the residuals: 2 2 2 1 1 1 1 2 2 2 2 2 4 6 2 4 61 1 1 1 1 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos cos 2 2 4 6 r t b t p b t p b b p b b b ab bb t t t = + − − − = + + + ω ω ω ω ω ω ω ω (32) 2 2 2 2 2 2 2 2 2 2 2 2 4 6 2 4 62 2 2 2 2 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos cos 2 2 4 6 r t b t p b t p b b p b b b ab bb t t t = + − − − = + + + ω ω ω ω ω ω ω ω (33) and the residual integrals: 1 2 6 4 2 / 4 2 1 1 1 1 0 1 4 16 28 70 210 ( ) cos 3 105 t b bb ab b p b r r tdt t   + + − = = + ω ω (34) 2 2 6 4 2 / 4 2 2 2 2 2 0 2 4 16 28 70 210 ( ) cos 3 105 t b bb ab b p b r r tdt t   + + − = = + ω ω (35) the original hamiltonian-based frequency formulation gives the following result 2 2 2 41 2 2 1 1 2 2 7 19 1 10 35 r r ab bb r r = = − + + − ω ω ω (36) 452 h. ma eq. (16) gives immediately the following results 2 4 61 1 1 ( ) + 2 4 6 p w w aw bw= + (37) and 1 2 2 0 2 1 2 4 6 2 2 4 2 2 6 2 3 0 2 2 4 6 2 4 6 2 2 4 6 2 2 4 6 ( ) 6 ( 1 ) 1 1 1 1 1 1 6 6 (1 ) (1 ) (1 ) 2 4 6 2 4 6 3 4 16 3 2 2 5 35 7 19 10 35 7 19 1 10 35 p b p b x dx b b ab bb b x b x bb x dx b b ab bb b ab bb b b ab bb b ab bb − − = + + − − + − + − = + + − + + = + + = = + +                           ω (38) this is the same with eq. (36). the one-step solving process by eq. (21) leads to the following result 2 2 2 4 6 2 4 6 2 2 4 6 2 2 4 4( ( ) ( / 2)) 1 1 1 1 1 1 4 2 4 6 4 16 48 3 7 4 12 3 7 1 4 12 p b p b b b ab bb b ab bb b b ab bb b ab bb − =   + + − − −    = + + = = + + ω (39) fig. 1 shows the comparison of approximate solution w(t)=bcosωt, where the frequency is determined by eq. (38) or eq. (39), with the exact one when a=b=1 for various values of amplitudes. 4. conclusions this paper simplifies hamiltonian-based frequency formulation for nonlinear oscillators. all the intermediate steps involved in the original formulation become totally unnecessary, making the formulation unprecedentedly simple. both the simplified frequency formulations will play an important role in the nonlinear vibration theory for a fast and reliable insight into the periodic properties of a complex nonlinear oscillator, and it will open up a flood of opportunities for various applications. the one-step solving process given in this paper confirms again ji-huan he’s famous saying: “the simpler is the better for the nonlinear vibration theory”. simplified hamiltonian-based frequency-amplitude formulation for nonlinear vibration systems 453 fig. 1 comparison of the exact solutions with the approximate solutions based on eqs. 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https://doi.org/10.1209/0295-5075/ac3d6b https://doi.org/10.1142/s0218348x22500463 7833 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 647 663 https://doi.org/10.22190/fume210714068b © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper improving corrosion resistance of magnesium nanocomposites by using electroless nickel coatings sudip banerjee1, pujan sarkar2, prasanta sahoo1 1department of mechanical engineering, jadavpur university, kolkata, india 2jaipaiguri polytechnic institute, jalpaiguri, west bengal, india abstract. the present study aims at improving corrosion resistance of magnesium nanocomposites through autocatalytic ni-p coating. electroless ni-p coatings with different concentration of sodium hypophosphite are deposited on 2% wc incorporated magnesium nanocomposites (az31-2wc) and the coated samples are further heattreated. basic characterizations and compositional analyses are done by using scanning electron microscope (sem), energy dispersive x-ray analysis (edax), and x-ray diffraction analysis (xrd). microhardness values of the developed materials are also evaluated. the attempt is made to improve corrosion resistance of az31-2wc by modifying surface roughness. corrosion characteristics of ni-p coated az31-2wc nanocomposites are examined by performing potentiodynamic polarization test and electrochemical impedance spectroscopy (eis). corrosion resistance improves with enhancement of surface quality. corrosion resistance of az31-2wc nanocomposite also improves due to application of ni-p coating. finally, corrosion morphologies are scrutinized by sem micrographs of corroded surface. key words: corrosion resistance, magnesium, nanocomposite, ni-p, coating 1. introduction automobile industries are in search of lightweight materials due to the sky rocketing price of fuels and ever-stricter emission norms. in this regard, magnesium and magnesiumbased composites have got attention of research fraternity, as they possess noticeable properties like high strength/weight ratio, superior hardness, high stiffness and good castability [1]. accordingly, automotive sectors are trying to reinstate different ferrous components (transmission cases, steering shaft, power trains, housings, pistons etc.) with magnesium based lightweight materials [2]. literature reveals that incorporation of received july 14, 2021 / accepted november 14, 2021 corresponding author: prasanta sahoo department of mechanical engineering, jadavpur university, kolkata-700032, india e-mail: psjume@gmail.com; prasanta.sahoo@jadavpuruniversity.in 648 s. banerjee, p. sarkar, p. sahoo ceramic-based reinforcements can provide enhanced mechanical and tribological properties. however, particle size (micro/nano) of reinforcement phases also has noteworthy importance along with types of reinforcements while trying to attain enhanced properties. usually, a higher amount of reinforcement is needed for micron size reinforcement compared to nano-sized reinforcements. researchers reported that nearly two percentage of nano-sized reinforcement is adequate to achieve desired properties [3]. in this context, the researchers have fortified nanoparticles like sic, tic, tib2, wc, zro2, al2o3, bn, zno, graphite in magnesium matrix [3-9]. it was observed that mechanical and tribological properties are enhanced due to incorporation of those nanoparticles having varying weight percentage. consequently, banerjee et al. [9-12] incorporated different wt. % of wc nanoparticles in az31 matrix through ultrasonic vibration associated stir casting method and evaluated mechanical as well as tribological behavior at different experimental conditions. microhardness, nanohardness and elastic modulus of 2% wc incorporated magnesium nanocomposites (az31-2wc) were found to be enhanced by 53%, 122% and 170%, respectively, compared to az31 alloy. even tribological behavior (friction and wear) of az31-2wc nanocomposite possessed significant improvement comparing to az31 alloy in dry sliding, under elevated temperature conditions and also in abrasive conditions for different parametric variations of sliding speed, load, and sliding distance. thus, overall az31-2wc nanocomposite is found to possess excellent mechanical and tribological properties. however, the chemical affinity of magnesium-based materials in aqueous solution makes it difficult to conduct electrochemical treatment. even the existence of other alloying elements possesses electrochemical heterogeneity and increases complicacy [13]. accordingly, corrosion characteristics of az31-2wc nanocomposite get deteriorated compared to az31 alloy [14]. however, improved corrosion resistance is essential for better durability; the same can be achieved through surface modification and treatment. consequently, surface treatments of magnesium-based materials are of immense importance for various applications. in this respect, enhancement of surface quality and employment of coating (organic, inorganic, metallic) on magnesium are widely acknowledged [15]. to enhance surface quality, sample surface should be polished properly. alvarez et al. [16] revealed the importance of surface roughness on corrosion behavior. walter and kannan [17] iterated that corrosion tendency of az91 alloy enhanced with enhancement in surface roughness. among different coating methods, electroless nickel (en) coating on magnesium is increasingly used to enhance corrosion resistance since a uniform layer can be formed on the material surface without having any curtailment on shape [18]. literature shows that en coating can impart a high level of corrosion resistance making it important for electronic industry [19]. microstructure of coatings can be further modified by applying heat treatment that influences corrosion behavior. mahallawy et al. [20] have investigated the effect of ni-p coating on corrosion characteristics of az31b, ae42 and zre1 magnesium alloys and reported significant improvement in hardness and corrosion resistance. elsentriecy and azumi [21] deposited ni-p coating on az91d alloy and examined corrosion characteristics in 3.5% nacl solution. experimental results revealed the highest corrosion resistance since the coated sample and coating layer provided complete coverage to the substrate. wang et al. [22] deposited ni-p coating on az31 alloy using electroless plating technique and annealing was also done on the coated sample. the coated sample possessed higher hardness than the substrate; crystallization further improved hardness value. electrochemical polarization study revealed the best improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 649 corrosion resistance for the coated sample followed by crystallized sample and substrate. abdi-alghanab et al. [23] developed electroless ni-p coating on am60b magnesium alloy considering ldh coating as under-layer and hydrothermal processing was further conducted for different processing time. corrosion resistance was found to enhance linearly with increase in processing time. corrosion current density (icorr) of sample processed for 8 hours was almost 1/7 times of sample processed for 2 hours. buchtik et al. [24] investigated corrosion behavior of electroless ni-p coated ze10 magnesium alloy in 0.1 m nacl solution. presence of ni-p coating resulted in significant improvement in corrosion resistance. thus, extensive review of existing literature reveals that incorporation of nanoparticles improves the strength and tribological characteristics while deposition of electroless ni-p coating may impart higher corrosion resistance. this will essentially yield high strength magnesium-based material having higher corrosion resistance. to the best of authors’ knowledge, such studies have not been reported in literature. the present study tries to fill in that gap in the referential literature. in this study, autocatalytic ni-p coating is deposited on az31-2wc nanocomposite to enhance its corrosion resistance. role of surface roughness, ni-p coating and heat treatment on corrosion resistance is experimentally evaluated. corrosion morphology is further studied using scanning electron microscopy. 2. experimental details 2.1. fabrication of nanocomposites in the present study, az31 magnesium alloy is considered as base matrix and 2 wt% of wc nanoparticles (80 nm) are used as reinforcement to produce az31-2wc nanocomposite. compositional details of az31 alloy and wc particles are presented in table 1 and table 2, respectively. az31-2wc nanocomposite is developed in a specially oriented furnace that has resistance heating facility, powder heating set up, mechanical stirrer, ultrasonic vibrator, mold heater, and bottom pouring arrangement (fig.1). initially, az31 alloy is fed into the crucible of the main furnace and temperature is set at 750°c. simultaneously, the required amount of wc particulates is heated at 300°c before fortifying in the melt. pre-heating typically helps to eliminate moisture content, enhance wettability and protect reinforcements from getting burn out. alloy ingots melt after a definite time. the mechanical stirrer is used inside the melt at a speed of 500 rpm to generate vortex. pre-heated wc particulates are injected inside that vortex constantly. table 1 chemical conformation of az31 alloy element si fe zn al mn mg wt. % 0.10 0.005 1.20 3.20 0.28 balance after completion of particle injection, the stirring speed is escalated to 600 rpm and continued for 5 minutes. afterwards, the mechanical stirrer is replaced by the ultrasonic horn to provide a high frequency ultrasonic vibration in the mixture. ultrasonic vibration is continued for 5 minutes. next, vacuum pouring is done through the bottom pouring hole. finally, the solidified bar is removed from the split die. samples are then machined properly to produce desired samples. fabrication method is minutely discussed elsewhere [9]. 650 s. banerjee, p. sarkar, p. sahoo table 2 features of wc particles features wc nano-powder purity 99.9% elastic modulus (e) 530-700 gpa melting point 2870°c color black particle size 80 nm table 3 specifications for electroless ni-p bath bath parameters details nickel sulfate 20 g/l nickel chloride 20 g/l lead acetate 1 mg/l sodium succinate 12 g/l sodium hypophosphite 10-20 g/l temperature 90 ºc ph 4.5 volume of bath 200 ml coating duration 2 hours 2.2. coating deposition az31-2wc nanocomposite samples with size (20×20×8) mm of roughness grade ra = 0.4 μm are prepared for ni-p coating deposition. coating deposition set up is shown as fig. 2. primarily, the substrates are rinsed properly to get rid of any foreign particle and corrosion products. then the samples are air dried followed by pickling treatment in 18% hcl for 1 min and rinsed in deionised water. initially, a number of trials were performed to finalize composition of bath. bath decomposition after 15-30 minutes was encountered. bath decomposition was prevented by adding a limited amount of lead acetate. details of the bath constituents and the operating parameters are tabulated in table 3. acid-pickled fig. 1 pictorial view of fabrication unit fig. 2 pictorial view of coating deposition set up improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 651 and rinsed sample is dipped in palladium chloride solution for 30 sec to activate it and finally dipped in the heated bath (90°c). palladium chloride helps to achieve good deposition rate from the very beginning. coating process is continued for 2 hours during which ph of solution is maintained at 4.5. after completion of coating process, samples are taken out of the bath and washed in distilled water. two different levels of sodium hypophosphite (10 g/l & 20 g/l) are considered. some coated samples are also heat treated at 400°c for 1 hour and air-cooled. 2.3. characterization basic characterization of base alloy, az31-2wc nanocomposite and coated samples are performed with the help of sem (jeol, japan & zeiss) micrographs. compositional analyses of those samples are carried out using edx (jsm-6360, japan) and xrd analysis (rigaku, ultima iii). for xrd, cu kα source is employed and analysis is performed between scanning ranges of 20°-90° for scanning rate of 2° 2θ/min. characterization of base alloy and nanocomposite is performed after preparing the sample surfaces. samples are initially polished with different grit (1500, 2000 & 2500) emery paper. then those samples are polished on fine clothes using distilled water and polishing agent. final polished samples are etched in acetic-picral solution (4 g picric acid, 10 ml acetic acid, 70 ml ethanol and 10 ml distilled water). corroded samples are finally characterized using sem micrographs. 2.4. microhardness vicker’s microhardness of az31 alloy, az31-2wc nanocomposite, as-coated ni-p sample and heat-treated sample are studied according to astm e384-16 with the help of the uhl microhardness tester (technische mikroskopie). a diamond indenter having apex angle 136° is used for this study. as experimental condition, 50 gf load and 10 s dwell time have been fixed. typically, the average of five readings is taken as microhardness value. 2.5. surface roughness test effect of surface roughness (sr) on corrosion behavior is widely acknowledged by scientific community. accordingly, in this study initial sr of the specimens is assessed with 3d surface profiler (bruker). afterwards, a similar sample is smoothened with abrasive paper (400 grit, sic) and sr is measured with the same instruments. surface profiler provides 2d as well as 3d surface plots of the tested specimens in attached computerized system. 2.6 corrosion study corrosion behavior of the developed samples is investigated by performing potentiodynamic polarization test and electrochemical impedance spectroscopy (eis) using 3.5% nacl solution in a potentiostat (gill ac, uk). contact area of each sample with electrolyte is 1 cm2. composite and coated specimens are cleaned properly using acetone and employed as working electrode. the working electrode is attached to the potentiostat at actual position. saturated calomel electrode is chosen as reference electrode because it comes up with steady potential. here, platinum electrode is considered as auxiliary electrode as it helps alternate current to find an alternative way in electrolyte. these two electrodes are placed properly in the glass cylinder and working electrode (sample) is clamped properly so that 1 cm2 area comes in contact with electrolyte solution. open circuit 652 s. banerjee, p. sarkar, p. sahoo voltage takes 15 minutes to become steady. the potentiostat coupled with a computerized system helps to explore the corrosion results. for this study, the corrosion potentials at the beginning and end of study are -250 mv and 250 mv whilst the scanning rate is 1 mv/s. eis results are scrutinized with the pre-installed software with the best fitting semi-circle technique. corrosion current density (icorr) and potential (ecorr) are examined from the tafel plot by the tafel extrapolation technique. 3. results and discussion 3.1. characterization basic characterization of the developed materials is performed using sem micrographs, edx analysis and xrd analysis. primarily, sem micrographs are examined to check the distribution of reinforcement phase in matrix. sem micrographs of az31, az31-2wc, ascoated sample and heat-treated sample are presented in figs. 3a, 3b, 3c & 3d, respectively. in sem micrographs of az31 and az31-2wc, α-mg and β-mg17al12 are present. sem micrograph of nanocomposite (fig. 3b) discloses clear composite structures having finer grain structures. wc particles are almost uniformly dispersed in matrix without having any significant agglomeration. sem micrograph of as-coated sample surface (fig. 3c) is optically smooth without having any significant porosity and surface damage. many globular particles are observed on the surface of as-coated sample. sem micrograph of heat-treated sample (fig. 3d) possesses some modifications in both microstructure and crystal structure. after heat treatment, crystal growths of microcrystalline deposits enhance. (a) (b) (c) (d) fig. 3 sem micrograph (a) az31 alloy, (b) az31-2wc, (c) ni-p coating and (d) heattreated ni-p coating improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 653 furthermore, compositional details of samples are scrutinized using edax and xrd analysis. edax spectrum of az31-2wc and as-coated ni-p coating is shown in figs. 4a and 4b, respectively. fig. 4a shows spectrums of mg, zn, w, c and o. basic elements of base alloy are present. peak of w confirms inclusion of wc particles. however, the exact percentages of each element are not present as a small area is scanned and particles are of nano-metric scale. edax spectrum of as-coated (fig. 4b) sample shows peak of ni and p. basic composition of as-coated sample is determined with regard to wt.% of ni and p from edax spectrum. weight percentages of ni and p for as-coated and heat-treated samples are tabulated in table 4. amounts of phosphorous in as-coated and heat-treated sample are 9.2% and 11.4%, respectively. hence, coatings recline in high phosphorous scale. typically, crystallinity as well as microstructural integrity highly build upon the amount of phosphorous which consequently decides hardness value. from edax result amorphous nature of coating can be expected. phase structures are further revealed by xrd study. xrd results (fig. 5a) of az31-2wc reveals peak of mg (2θ = 32.16°, 34.58°, 48.14°, 57.17° & 63.54°), wc (36.01° & 48.18°) and w2c (36.01° & 48.18°). xrd result of heat-treated coating sample is illustrated in fig. 5b. fig. 5b possesses peak of ni, ni2p and ni3p. this result is in line with phase transformation characteristics of amorphous coating (ni-p) developed through en method. (a ) (b) fig. 4 edax spectra (a) az31-2wc nanocomposite and (b) ni-p coated az31-2wc sample 654 s. banerjee, p. sarkar, p. sahoo table 4 wt. % of different elements in coatings sample % of nickel % of phosphorous total as-coated 90.8 9.2 100 heat treated (400°c) 88.6 11.4 100 (a) (b) fig. 5 xrd plot (a) mg-2wc nanocomposite and (b) heat-treated coating sample fig. 6 microhardness plot: a comparison between base alloy az31, nanocomposite az312wc, as-deposited ni-p coating and heat treated ni-p coating 3.2. microhardness it is well established in literature that incorporation of reinforcements or deposition of coating enhances microhardness of base alloy. effects of incorporation of wc particles and deposition of ni-p coating on microhardness are shown in fig. 6. it is observed that incorporation of 2wt. % of wc enhances microhardness by 52.45% as compared to az31 alloy. as-coated ni-p (20 g/l) helps to enhance microhardness value significantly. microhardness value enhances by almost seven times than the base alloy for as-coated ni-p sample while the heat-treated coating sample possesses around ten times enhancement. improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 655 3.3. corrosion test potentiodynamic polarization plots of az31, az31-2wc and polished sample (az312wc polished with 400 grit sic paper) are shown in fig. 7. each plot contains anodic section and cathodic section. study of anodic section is necessary to examine corrosion characteristics as anodic section mainly helps to understand transformation technique of material from sample surface [25]. for clarity, surface roughness profile of polished az312wc sample is shown in fig. 8 and surface roughness values of normal sample and polished sample are tabulated in table 5. fig. 7 illustrates that corrosion performance initially decreases due to incorporation of 2wt. % of wc but polished sample possesses better corrosion resistance. however, no noticeable differences in cathodic current values are observed. hence, corrosion characteristics of examined materials are mainly contingent on anodic characteristics. however, clarity can be brought by precisely calculating current density (icorr) and corrosion potential (ecorr). potentiodynamic polarization plots are extrapolated by tafel extrapolation to compute values of ecorr and icorr for each curve. computed values are presented in table 6. literature reported that subordinate value of icorr and superior value of ecorr imply better corrosion characteristics [26]. table 6 yields that polished sample possesses better corrosion resistance than base alloy and az31-2wc nanocomposite as corrosion potential of polished sample is -1.42v whereas that of az31 is -1.43v. this trend is in line with observations reported in literature [16]. fig. 7 potentiodynamic polarization plot for az31, az31-2wc and polished samples table 5 surface roughness of normal sample & polished sample sample roughness (ra), μm normal sample 0.916 polished sample 0.568 656 s. banerjee, p. sarkar, p. sahoo fig. 8 surface roughness profile of polished az31-2wc sample potentiodynamic polarization plot of as-coated ni-p samples with different nah2po2. h2o concentrations (10 g/l & 20 g/l) and corresponding heat-treated samples are presented in fig. 9. tafel extrapolation results of these plots are presented in table 6. it is noticed that icorr value decreases remarkably compared to base alloy and az31-2wc sample due to application of ni-p coating on az31-2wc sample. ecorr value of as-coated sample also swings toward positive direction compared to substrate sample (az312wc). moreover, icorr decreases further and ecorr enhances as concentration of nah2po2. h2o changes from 10 g/l to 20 g/l. these results iterate that corrosion resistance of az31-2wc enhances due to application of ni-p coating and corrosion resistance enhances further with enhancement in concentration of nah2po2.h2o. similar conclusion has been drawn by diegle et al. [27]. potentiodynamic polarization curves of heat-treated coated samples are also presented in fig. 9 and tafel extrapolation results are presented in table 6. for heat-treated samples, no specific trend is available as corrosion resistance deteriorates compared to as-coated sample for 10 g/l heat-treated sample while corrosion resistance enhances for 20 g/l heat treated sample compare to as-coated sample. the highest corrosion resistance is observed for the heat-treated sample having 20g/l sodium hypophosphite. literature reveals that a higher amount of sodium hypophosphite accompanies to incorporate a higher amount of phosphorous which helps to resist pitting corrosion [28]. improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 657 table 6 tafel extrapolation results material ecorr (v) icorr (ma/cm2) az31 -1.43 0.0714 az31-2wc -1.50 0.1162 polished sample -1.42 0.0687 as-coated (10 g/l) -0.485 0.0056 heat-treated (10 g/l) -0.502 0.0101 as-coated (20 g/l) -0.394 0.0028 heat-treated (20 g/l) -0.377 0.0025 fig. 9 potentiodynamic polarization plot of ni-p coated samples and heat-treated samples fig. 10 nyquist plots of az31, az31-2wc and polished sample 658 s. banerjee, p. sarkar, p. sahoo fig. 11 nyquist plot of as-coated ni-p samples heat-treated samples corrosion characteristics are further analyzed with the help of nyquist plot. nyquist plots of az31, az31-2wc and polished sample tested at 3.5% nacl are presented in fig. 10. nyquist plots of each sample contain a high frequency capacitive loop, one medium frequency capacitive loop along with a low frequency inductive loop. typically, rudimentary nature of plots is similar for az31, az31-2wc and polished samples apart from diameter of plots. this nature is similar to the findings of wu et al. [29] who reported that polarization resistance is proportional to the diameter of capacitance loop. on the other hand, diameter of capacitance curve is also proportional to corrosion resistance. thus, it can be concluded from fig. 10 that sample polished with 400 grit paper is the most corrosion resistive one. literature also reveals that high frequency capacitive arc relates electrolytesurface film interface while low frequency capacitive arc signifies diffusion of ion between material and generated film. low frequency inductive arc indicates localized corrosion due to electrolyte diffusion between film and material [25]. as the polished sample possesses the highest capacitive loop (fig. 10) among these three samples (az31, az31-2wc & polished sample), it can be said that there must have more protective layer at filmelectrolyte interface, which helps to protect the sample surface from destructive ions. nyquist plot of as-coated ni-p samples with varying sodium hypophosphite and corresponding heat-treated samples are presented in fig. 11. here nyquist plot of all samples possess high frequency capacitive loop having single semicircular shape with varying diameter within frequency range of 10 khz to 0.01 hz. similar shape with variation in diameter of semicircle signifies similar elementary mechanism of corrosion but for different amount of area. literature reveals that diameter of capacitive loop indicate charge transfer resistance which is directly related to corrosion resistance of that sample. hence, sample having capacitive loop with highest diameter is the most corrosion resistive one within experimental range. among as-coated samples, the sample with sodium hypophosphite concentration 20 g/l shows semicircle with the highest diameter. hence, that sample has the highest corrosion resistance. it is obvious from fig. 11 that samples possess charge transfer controlled behavior in electrolyte solution. however, as-coated sample having sodium hypophosphite concentration 10 g/l possesses proneness of starting another semicircle. probable reason of such nature is piercing by electrolyte and forming local electrochemical cell which deteriorates corrosion resistance. similar observation is noted by zhang et al. [30]. improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 659 obtained curves are analyzed to obtain values of rct and cdl. among tested samples, ascoated sample with sodium hypophosphite concentration 20 g/l shows the highest rct value (34 kohm.cm2) and the lowest cdl value (26 μf). rct value and cdl value of as-coated sample with sodium hypophosphite concentration 10 g/l is 14.7 kohm.cm2 and 41 μf, respectively. literature revealed that corrosion characteristics of ni-p coating mainly rely on amorphous nature of coating and amount of phosphorous. ni-p coating having high phosphorous content results higher corrosion resistance because of more amorphous and homogeneous structure. structural homogeneity and higher phosphorous content reduce porosity of coating. reduced porosity also protects sample surface from infiltration of corrosive parameters. these findings are in line with literature [30]. nyquist plot of heat-treated coated samples are also presented in fig. 11. it is observed that the largest semicircle is present for heat-treated coating sample with 20 g/l sodium hypophosphite. hence, this sample is the most corrosion resistive one among all tested samples. eis results are following same trend of previously discussed tafel plot. among heat-treated samples, sample with sodium hypophosphite concentration 20 g/l shows the highest rct value (59.5 kohm.cm2) and the lowest cdl value (21.9 μf). hence, this is the most corrosion resistive sample among heat-treated sample. literature reveals that ni-p coating become denser as well as less porous after heat treatment. heat treatment also generates secondary phase like niapb which protects the sample surface from corrosion [31]. similar phenomenon may have occurred for the present study. 3.4. corrosion morphology corrosion characteristics of the developed materials are further analyzed using sem images of corroded surfaces. sem micrograph of corroded surface helps to understand typical corrosion mechanism which occurred during experimentation. sem images of corroded surfaces of az31, az31-wc nanocomposite and polished sample are presented in fig. 12. sem micrographs of corroded zones of these three specimens are filled with volcano-like sediments which are mainly insoluble corrosion by-products. cracks of different shapes and sizes are also present in corroded surfaces. similar nature is also mentioned by ascencio et al. [32] in their study. literature also illustrated that cracks are usually formed because of dehydration of deposited layers [33]. destructive electrolytes (cl-) of nacl solution infiltrate through those cracks and touch material surface. as a result, corrosion rate increases [17]. hence, illustration of position, nature and intensity of crack is necessary to predict corrosion characteristics of az31, az31-2wc nanocomposite and polished sample. sem micrographs (fig. 12) clearly show that the corroded surface of the polished sample (fig. 12c) contains lesser amount and intensity cracks than the corroded surfaces az31 (fig. 12a) and az31-2wc (fig. 12b). on the other hand, the corroded surface of az31 possesses lesser intensity cracks than the corroded surface of az31-2wc. hence, the intensity of cracks lies in order of az31-2wc > az31 > polished sample. so, the chances of penetration of clions are greater for az31-2wc followed by az31 and polished sample. thus, corrosion resistance should lie in order of polished sample > az31 > az31-2wc. similar trend is observed in tafel plot and nyquist plot. 660 s. banerjee, p. sarkar, p. sahoo (a) (b) (c) fig. 12 sem micrographs of corroded surface (a) az31, (b) az31-2wc and (c) polished sample sem images of as-coated and heat-treated (ht) samples are presented in fig. 13. sem images of as-coated and ht samples almost possess similar morphology; that is why coating having the highest corrosion resistance (sodium hypophosphite-20 g/l) is discussed here for brevity. very fine pores are present in the corroded surface of both as-coated and heat-treated (a) (b) fig. 13 corroded surface (a) as-coated sample (sodium hypophosphite-20 g/l) and (b) heattreated sample improving corrosion resistance of magnesium nanocomposites by the use of electroless nickel coatings 661 samples. these pores make route for electrolyte solution to reach the sample surface. thus, galvanic cells are formed between ni-p coat and substrate surface. fig. 13 also shows that the pores in heat-treated samples are scanty compared to as-coated sample. that is why the heattreated sample exhibits a much lesser deleterious effect compared to as-coated sample. even the less porous nature of the heat-treated sample intercepts the propagation of harmful effect below the coating and protects the substrate surface. however, sem images of both as-coated and ht samples do not possess cracks like substrates surface. thus, as-coated and ht samples must result in superior corrosion resistance compared to base alloy, nanocomposite sample and polished sample, which is already shown by tafel plot and nyquist plot. similar results are reported by el mahallawy et al. [20]. over the years, en coating has emerged as a widely acknowledged method for surface modification [34]. nevertheless, the conventional coating methods (electro-deposition & electroless plating) suffer from environmental issues due to disposal of toxic elements [35]. as a result, the concepts of cleaner production using chemical vapor deposition (cvd) and physical vapor deposition (pvd) are coming up [36]. however, en coating cannot be overlooked as it possesses some useful advantages like mass production capability, coating ability on any substrate and shape. accordingly, incorporation of the concept cleaner production in electroless coating (making the bath more eco-friendly) is gradually drawing attention of researchers [37]. 4. conclusion in the present study, the result of fortification of wc nanoparticles in az31 alloy, surface roughness, ni-p coating and its heat-treatment on corrosion behavior is examined using 3.5% nacl solution. az31-2wc nanocomposites are produced using ultrasonic vibration supported stir casting technique. electroless ni-p coatings with different concentration of sodium hypophosphite (10 g/l & 20 g/l) are deposited on az31-2wc nanocomposite and the coated sample is further heat-treated. conclusions from the present study are listed below: ▪ sem micrograph of nanocomposite reveals that the composite has finer grain structures. sem micrograph of as-coated sample surface is optically smooth without having any significant porosity and surface damage. many globular particles are observed on the surface of as-coated sample. edax analysis confirms inclusion of wc particles. basic composition of as-coated sample is determined with regard to wt. % of ni and p from edax spectrum. amounts of phosphorous in as-coated and heat-treated sample are 9.2% and 11.4%, respectively. xrd result of heat-treated coating sample illustrated peak of ni, ni2p and ni3p. ▪ incorporation of 2wt. % of wc enhances microhardness by 52.45% compared to base alloy. microhardness value enhances by almost seven times for as-coated ni-p compared to base alloy while heat-treated coating sample possesses around 10 times enhancement. ▪ potentiodynamic tests and eis study are carried out using 3.5% nacl solution for all samples. among az31, az31-2wc and polished sample, polished sample possesses better corrosion resistance than base alloy and az31-2wc nanocomposite. corrosion resistance of az31-2wc enhances due to deposition of ni-p coating; it enhances further with higher concentration of nah2po2. h2o (20 g/l) in electroless bath. heat-treated coating sample with 20 g/l sodium hypophosphite is the most corrosion resistive one among all tested 662 s. banerjee, p. sarkar, p. sahoo samples. very fine pores are present in corroded surface of both as-coated and heat-treated samples. these pores make route for electrolyte solution to reach the sample surface. pores in heat-treated samples are scanty compared to as-coated sample. that is why heat-treated 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springer international publishing, pp. 247-280. 36. navinšek, b., panjan, p., milošev, i., 1999, pvd coatings as an environmentally clean alternative to electroplating and electroless processes, surface and coatings technology, 116, pp. 476-487. 37. bonin, l., vitry, v., delaunois, f., 2020, replacement of lead stabilizer in electroless nickel-boron baths: synthesis and characterization of coatings from bismuth stabilized bath, sustainable materials and technologies, 23, e00130. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 269 283 https://doi.org/10.22190/fume190530030m © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree of freedom dragan marinković, gil rama, manfred zehn department of structural mechanics, berlin institute of technology, germany abstract. integration of classical, passive structures and active elements based on multifunctional materials resulted in a novel structural concept denoted as active structures. the new structural systems are characterized by self-sensing and actuation. coupling the two distinctive features by means of a controller enables a number of exquisite functionalities such as vibration suppression, noise attenuation, shape control, structural health monitoring, etc. reliable, accurate and highly efficient modeling tools are an important ingredient of the active structure design. this paper addresses the abaqus implementation of a recently developed piezoelectric 3-node shell element. the element uses co-rotational formulation to cover geometric nonlinearities. special techniques are used to address the issues originating from low-order interpolation functions. the discrete shear gap is used to resolve the shear locking, while the assumed natural deviatoric strain technique improves the membrane behavior. examples are computed in abaqus upon implementation of the developed element. key words: abaqus, corotational fem, piezoelectric shell element, discrete shear gap, assumed natural deviatoric strain 1. introduction over two decades ago, a novel structural concept denoted as ‘active’, ‘adaptive’, or ‘smart’ structures has seen the light of day [1]. it features integration of classical, passive structures and active elements based on multi-functional materials. in this manner, artificial systems are given the ability of self-sensing and actuation coupled by control capabilities. the concept is obviously the result of mimicking the natural systems that received may 30, 2019 / accepted july 15, 2019 corresponding author: dragan marinkovic department of structural mechanics, tu berlin, str. d. 17. juni 135, 10623 berlin e-mail: dragan.marinkovic@tu-berlin.de 270 d. marinković, g. rama, m. zehn actively react to environmental stimuli in order to protect their integrity and maintain optimal functionality. this provides them with a number of exquisite functionalities such as vibration suppression [2], noise attenuation [3], structural health monitoring [4], shape control [5], etc. in this manner, active systems offer significant benefits over passive ones, including improved safety, ergonomics, robustness, product lifespan, etc. such an enticing research field has attracted many researchers, whereby, generally speaking, two major directions of work can be distinguished. one group of researchers focused their work on resolving some practical problems such as design of active elements, types of materials applied, ways of embedding active elements into passive structures, etc. the other group dedicated their work to development of modeling tools thus enabling reliable simulation and therewith less expensive and successful design. the finite element method (fem), as the most powerful method in the field of structural analysis, is addressed and many authors proposed various types of elements that enable modeling of active structures. in case of thin-walled structures, active elements are typically in the form of small, rather thin patches made of piezoelectric materials that operate based on the e31-effect. this actually means that the patches couple the electric field acting across the thickness to the in-plane strains. the same patches are used as both sensors and actuators with the essential difference only in the boundary conditions. when exposed to mechanical deformation, sensor patches deliver electric signals, i.e. voltages, proportional to the average in-plane strains in the area covered by the piezoelectric patch. oppositely, when supplied with electric voltage, actuator patches produce distributed in-plane forces proportional to the electric voltage and acting perpendicularly to the edges of the patch. the patches are usually placed at the maximum offset distance from the mid-plane in order to produce the maximum bending moment uniformly distributed over the patch edges. they are commonly glued onto the outer shell surfaces. the effect may be amplified by using a pair of such patches collocated with respect to the structure’s midsurface by placing one of them on the upper while the other one on the lower surface, and they are additionally oppositely polarized in order to produce only a bending moment when exposed to the same electric voltage. a large number of shell type of finite elements were proposed for piezoelectric active shell structures. benjeddou [6] gave a thorough survey of the development in the 90’s and the development in the beginning of the new millennium retained the same intensity and innovativeness. various approaches to modeling were investigated. one should notice that the application of piezopatches onto a passive shell structure results in a multilayered structure, even if the passive structure consists of a single layer of material. however, not rarely the passive structure is actually made of a composite laminate. the mainstream developments implemented the equivalent single-layer approach. some of them were based on the classical laminate theory (kirchhoff-love kinematics) [7, 8], while others used the first-order shear deformation theory (mindlin-reissner kinematics) [9, 10]. the latter was more frequently used. one reason for this choice is to be sought in the greater generality as the formulation includes the transverse shear effects, but an equally important reason is the attractiveness of the c0-continuity needed from the element shape functions, whereas the kirchhoff-love elements demand the c1-continuity. to improve the accuracy of shell formulations at the cost of somewhat higher numerical effort, layerwise theories were also addressed. one of the elements based on this approach is the abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree... 271 9-node plate element proposed by cinefra et al. [11] and which implements the technique of mixed interpolation of tensor components and variable through-the-thickness layerwise kinematics. the approach was recently extended by carrera et al. [12, 13] in order to locally increase the accuracy by means of node-dependent kinematics. furthermore, both geometrically [14, 15, 16] and materially [17] nonlinear problems in the behavior of piezoelectric thin-walled structures were also considered in the work of many researchers. as it is of utmost importance to enable users to apply developed numerical tools in modeling and simulation, this paper addresses abaqus implementation of the recently developed piezoelectric 3-node shell element. similar works were already reported in available literature [18, 19]. in what follows, the most important aspects of the element development are briefly presented together with the results of several test cases computed by using the element implemented in abaqus. 2. piezoelectric 3-node shell element with drilling degree of freedom the developed element combines features of already developed and in literature available shell elements. it implements the mechanical field of the element proposed by rama et al. [20] that was also extended to composite laminates by rama et al. [21], and the electric field, including the both way coupling between the two, as described in the work by marinkovic and rama [22]. as the mentioned references provide detailed derivations, the element description given in this section will not go into all the details, but for the sake of completeness, some major aspects of the development will be presented together with the most important matrices. as is the case with most shell elements, the formulation of this one strongly relies on the use of local coordinate frame x, y, z, which is defined so as to have one of its axis, the z-axis, perpendicular to the element, while the other two axes lie in the element’s plain, fig. 1. in the case of an isotropic material, their orientation is mainly of importance for proper evaluation of the results, as stresses and strains are determined and typically given with respect to this coordinate system. on the other hand, in the case of a composite laminate, it is of crucial importance for defining the material elastic properties. the same coordinate system comes also quite handy in the presence of piezoelectric layers, as the mathematical description of the considered e31-piezoelectric effect is straightforward with respect to it. fig. 1 3-node shell element with the local frame and mechanical degrees of freedom 272 d. marinković, g. rama, m. zehn since the 3-node element is a flat element, its mechanical field is effectively a superposition of a plate and a membrane element, fig. 2. fig. 2 3-node shell element as a superposition of plate and membrane elements according to the mindlin-reissner kinematics, the displacement field with respect to the local coordinate frame is given as follows: 3 3 1 12 0 i i yi i i i i x i i i u u h v n v n t w w                                        , (1) where ui, vi, and wi , i=13, are the nodal displacements along the x-, yand z-axis, respectively, xi and yi, are the nodal rotations about the xand y-axis, respectively, ni are the typical nodal shape functions of a linear triangular element, hi are the nodal values of shell thickness and t is the natural coordinate in the thickness direction. the plate behavior of the element is determined by plate stiffness and transverse shear stiffness. the plate stiffness is obtained by deriving the bending strains directly from the displacement field. the resulting bending strain-displacement matrix reads [20]: 23 31 12 32 13 21 23 32 31 13 12 21 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 pb y y y b x x x a y x y x y x                           , (2) where a is the element area, while xij= xixj and yij= yiyj. since shell elements are notorious for shear locking effects, particularly when loworder shape functions are used, special measures are typically needed to mitigate the effect. the developed element implements the discrete shear gap technique originally proposed by bletzinger et al. [23] and improved by the cell strain smoothing technique suggested by ngyen-thoi et al. [24]. the resulting transverse shear strain-displacement matrix for a triangular element is given by [20]: 32 13 1 3 21 2 3 23 31 4 2 12 4 1 01 02 s x a x a a x a a b y a y a a y a aa                  , (3) where: abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree... 273 1 12 13 2 31 21 3 21 13 4 12 31 1 1 1 1 , , , 2 2 2 2 a y x a y x a x x a y y           . (4) in order to avoid the dependence of the strain-displacement matrix on the node numbering and to improve the accuracy, the strain smoothing technique [24] is applied. it implies that the element is divided into three sub-elements by an additional node at the element centroid. equation (3) is then applied to each of the sub-elements and, at the same time, the assumption is used that the displacement at the element centroid is given as the average value of the displacements at the original three nodes. in this manner, the central node is condensed out. finally, the resulting transverse shear strain-displacement matrix is simply given by averaging the matrices of all three sub-elements, [bs i] : 3 1 1 3 i ps s i b b         . (5) since the strain-displacement matrices are constant over the element area, the plate stiffness is obtained in the following manner:     t t p pb ps pb pb ps psk k k a b d b b f b                              , (6) where [d] and [f] are the laminate bending and transverse shear stiffness matrices obtained by integrating the corresponding constants from the hooke’s matrix across the thickness. a particular weakness of low-order 3-node shell elements resides in their membrane behavior. andes formulation by felipa and militello [25] aims to resolve this weakness and to improve the behavior of the element to the level of at least a quadratic element. it represents a combination of the free formulation (ff) proposed by bergan and nygard [26] and a modification of the assumed natural strain (ans) formulation derived by park and stanley [27]. the formulation abandons the discretized displacement field (eq. (1)) and instead, represents the displacements as a superposition of carefully chosen linearly independent modes consisting of rigid-body, constant-strain and linear-strain modes. the first two groups are denoted as basic modes, while the third group as higher-order modes. accordingly, the stiffness is divided into basic and higher-order stiffness, each determining the behavior of the corresponding modes. for a unique transformation between the modal and nodal degrees of freedom, the number of modes is the same as the number of nodal degrees of freedom. the so-called drilling degree of freedom, i.e. the rotation around the local thickness axis (z-axis) plays the crucial role in the definition of the modes. on the other hand, one should notice that this degree of freedom is not a part of the discretized displacement field given by eq. (1). already this important difference speaks in favor of the new quality offered by andes formulation. the basic membrane stiffness is the same as in the ff formulation [26]:       t / ( )mbk l a l ah , (7) where [a] is the membrane stiffness of a laminate, i.e. a part of the abd matrix, while [l] reads: 274 d. marinković, g. rama, m. zehn   23 32 32 23 23 32 12 32 31 12 31 13 12 21 31 13 13 31 31 32 12 13 12 23 12 21 23 32 12 21 21 12 12 32 12 21 0 0 ( ) / 6 ( ) / 6 ( ) / 3 0 0 2 ( ) / 6 ( ) / 6 ( ) / 3 0 0 ( ) / 6 ( y x x y y y y x x x x y x y y x h l x y y y y x x x x y x y y x x y y y y x x                                                  23 31 23 32 31 13) / 6 ( ) / 3x x y x y                                   , (8) with  denoting a non-dimensional parameter. the higher-order modes cover the in-plane bending, as illustrated by the three modes on the left in fig. 3. the linear dependency of those three modes is the reason to introduce the fourth, torsional mode (the same rotation around the z-axis at all three nodes) shown on the right in fig. 3. fig. 3 higher-order modes – three bending modes (left) and a torsional mode (right) without interpretation of single matrices that appear below, the higher-order membrane stiffness is given by [20]: t 0 9 4 mh u uk t k t                 , (9) where 0 is a non-dimensional parameter, and furthermore [20]: 32 32 13 13 21 21 32 32 13 13 21 21 32 32 13 13 21 21 4 0 0 1 0 4 0 4 0 0 4 u x y a x y x y t x y x y a x y a x y x y x y a                            , (10)     t tt 4 4 5 5 6 6 1 3 nat nat natk q a q q a q q a q                                  , (11)   t nat nat nata t a t           , (12) abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree... 275       2 2 2 23 13 21 23 13 21 23 31 32 13 21 2 2 2 31 21 32 31 21 32 31 12 13 21 322 2 2 2 12 32 13 12 32 13 12 23 21 32 13 1 4 nat y y l x x l y x x y l t y y l x x l y x x y l a y y l x x l y x x y l                                   , (13) with lij denoting the length of the element edge between nodes i and j. furthermore:             4 1 2 5 2 3 6 1 3 1 1 1 , , 2 2 2 q q q q q q q q q                    , (14)     3 9 7 9 5 61 2 4 2 2 2 2 2 2 2 2 2 21 21 21 21 21 21 21 21 21 5 6 3 8 94 1 2 1 2 32 2 2 2 2 2 2 32 32 32 32 32 32 32 3 7 8 9 6 54 2 2 2 2 2 2 13 13 13 13 13 13 2 2 2 , , 3 3 3 l l l l l l l l l a a a q q q l l l l l l l l l l l l l l                                                                 7 2 2 2 32 32 1 2 2 2 13 13 13 l l l l                       . (15) there are different possibilities for the choice of parameters  and 0-9. this aspect is elaborated in [21] and the following choice explained: =1/8, 0=2/4, 1=1, 2=2, 3=1, 4=0, 5=1, 6= –1, 7= –1, 8= –1, 9= –2. upon the very compact description of the mechanical field in the element, the attention needs to be turned to the electric field in the piezoelectric layers. the function describing the electric field distribution across the thickness of piezolayers is supposed to be consistent with the maxwell’s equations for dielectrics and with the element kinematics. in the case of mindlin-reissner kinematics, the consistent electric field is linear across the thickness [28]. however, it has also been shown [28] that the classical assumption of constant electric field over the thickness provides sufficient accuracy for typical, rather thin piezopatches. the electric field of the kth piezolayer is then given by: k k k e h   , (16) where k is the difference of electric potentials between the electrodes of the kth piezolayer and hk denotes the thickness of the piezolayer, leading to the typical diagonal form of the electric field – electric potential matrix:   1 k b h               , (17) the dielectric stiffness matrix is then defined by:       t v k b d b dv         , (18) 276 d. marinković, g. rama, m. zehn where [d] is the matrix of dielectric constants at constant strain. finally, the piezoelectric stiffness matrix reads:    tt u u mf v k k b e b dv              , (19) where [bmf] collects the strain-displacement terms that define membrane and flexural strains, because the formulation considers the e31-piezoelectric effects that couples the inplane strains with the thickness-oriented electric field. in eq. (19) [e] is the matrix of piezoelectric constants. the integration in eqs. (18) and (19) needs to cover all the piezoelectric layers across the thickness of the structure. in order to cover geometric nonlinearities, the element implements corotational fe formulation [29]. beside the updated and total lagrangian formulations [30] that both represent standard solutions in major commercially available fe codes, the corotational fe formulation has gained lately in importance in many applications ranging from realtime simulations [31] to engineering solutions [32]. all the details are available in the literature (including above mentioned references), and only the basic idea is briefly outlined here. the general idea is to introduce a new, corotational coordinate frame attached to the element that performs the same rigid-body motion as the element itself and to measure all necessary quantities, such as strains and stresses, with respect to it (fig. 4). with the presented shell element, the local frame is used as corotational. small strains are assumed despite finite displacements and rotations. observing the problem in this way allows for further simplifications, such as to use linear dependency between deformational displacement (i.e. with the rigid-body motion removed from the overall displacements) and strains. with the rigid-body motion extracted from the overall motion, one may further determine all other mechanical and electric quantities in a relatively straightforward manner. for the sake of brevity, the equations are not presented here, but an interested reader may consult the above references for more details. fig. 4 the idea behind the corotational fe formulation abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree... 277 3. numerical examples the element was implemented in abaqus by means of user subroutine (uel) and currently supports up to two electric degrees of freedom per element, i.e. two piezolayers that can be used both as sensors and actuators. the below given examples are of academic nature. as the focus is on numerical results, all the values are given as dimensionless. of course, any set of consistent units corresponding to the quantities (both mechanical and electric quantities and material parameters, including dielectric and piezoelectric parameters) can be associated with them. 3.1. bimorph beam bimorph beam is one of the classical benchmark cases to test the developed numerical tools for piezoelectric active structures. the entire beam structure consists of two piezoelectric layers with opposite polarization. as explained in the introduction, when both layers are exposed to the same electric voltage, bending moment uniformly distributed over the edges of the structure is induced, thus causing bending. the left end of the structure is clamped (fig. 5) and the displacement along the beam length is observed as a representative solution. the length of the beam is l=0.1, the width is w=0.005 and the overall thickness (both layers together) h=0.001. the young’s modulus of the material is 2109, piezoelectric constant of the e31-coupling is 0.046 and the dielectric constant 0.106210-9. the difference of electric voltages supplied to the electrodes placed on the outer surfaces of the beam is 1. fig. 5 bimorph beam the analytical solution for deflection [33] is obtained by assuming beam kinematics (implying poisson’s ratio equal to 0) and it is a quadratic function in x. fig. 6 gives a screen-shot of the deformed configuration with the contour plot of displacements obtained in abaqus. the same case is also considered as a sensor case by exposing the structure to two transverse forces of magnitude 0.1, acting at the two free beam corners. it is assumed that the whole beam acts as a single sensor (each face completely covered by a single electrode) and the resulting electric voltage reflects the average strain in the whole beam. the obtained result for the electric voltage of 165 coincides with the analytical solution. a sensitivity analysis of the electric voltage to mesh distortion is performed. in order to summarize the results of both cases, fig. 7a shows the diagram of deflection normalized with respect to the analytical value of the displacement at the beam tip, while fig. 7b 278 d. marinković, g. rama, m. zehn represents the voltage sensitivity to mesh distortion, by normalizing it in the same manner, i.e. with respect to the analytical value of 165. the distortion is performed by increasing the value of parameter e from 0 to 20 with the increment size of 5 (see fe mesh within the diagram in fig. 7b). fig. 6 deformed bimorph beam – abaqus contour plot for deflection fig. 7 bimorph beam: a) actuator case – normalized deflection; b) sensor case – normalized sensor voltage 3.2. piezolaminated arch a curved structure is considered in this example. it is a semi-cylindrical arch with dimensions and boundary conditions as depicted in fig. 7. the single layer of passive material has thickness of 5.842, the young’s modulus is 68.95103 and the poisson’s ratio 0.3. each of the two outer piezolayers has the thickness of 0.254 and the following material properties: the young’s modulus 63103 and poisson’s ratio 0.3. furthermore, the same piezoelectric constant e31=16.1110-6 is assumed in all in-plane directions, and the dielectric constant is 1.6510-11. a vertical force of 200, acting upwards at the free tip, deforms the structure. the free tip displacements in the xand ydirections and the induced sensor voltage in the inner piezolayer are observed in a geometrically nonlinear analysis performed in abaqus. a) b) abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree... 279 fig. 8 semi-cylindrical arch – dimensions and boundary conditions the obtained results are compared with those reported by zhang [34], who originally proposed the example. fig. 9 depicts the deformed configuration computed using the developed 3-node shell element implemented in abaqus. the contour plot in fig. 9, left, corresponds to the displacement in the global x-direction (u1), while the one in fig. 9, right, to the displacement in the global y-direction (u2). fig. 10 shows the time history of displacements in the xand y-directions with the increasing force, computed by the implemented 3-node shell element, abaqus 3-node shell element and the results by zhang [34]. fig. 9 abaqus contour plots for displacements in x(left) and y-direction (right) fig. 10 displacements in xand y-directions versus force 280 d. marinković, g. rama, m. zehn fig. 11 compares the results obtained for the sensor voltage by the present element with the results reported by zhang [34]. fig. 11 sensor voltage versus force 3.3. modal analysis of a piezolaminated composite plate under various electric boundary conditions the third example considers the change of natural frequency of a piezolaminated composite plate with the change of electric boundary conditions. the modal analysis is performed with short-circuited and open electrodes. short-circuited electrodes imply zero voltage so that the behavior of the structure is purely mechanical (as if there was no piezoelectric effect present). oppositely, with the electrodes left open, electric voltage is induced as a consequence of deformation, which further gives rise to mechanical stresses and, therewith, changes in the natural frequencies of the structure. the considered structure is a composite plate with the sequence of layers [p/0/90/0/p] with respect to the global x-axis (fig. 11), where ‘p’ stands for piezoelectric layer, while the composite layers are made of graphite-epoxy and have the following mechanical properties with respect to the principal material orientations: young’s moduli e11=1.324105 and e22=1.08104, poisson’s ratio 12=0.33 and shear modulus g23=6.6103. the piezoelectric layers have the following mechanical properties: e11=8.13104 and e22=8.13104, poisson’s ratio 12=0.33 and shear modulus g23=2.56104. the piezoelectric constant is e31=14.810-6 and the dielectric constant 1.150510-11. fig. 12 piezolaminated plate with: short-circuited (left) and open electrodes (right) abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree... 281 the span of the plate is a=200. the thickness of each composite layer is 1.068 and of each piezolayer 0.4. the modal analysis is performed for all edges simply supported. to demonstrate the influence of electric boundary conditions onto the resulting natural frequency, only the first eigenmode of the plate is observed. it is depicted in fig. 13. fig. 13 the first eigenmode of the piezolaminated composite plate for the purely mechanical case, the results by abaqus linear shell element (s3) are also provided, while the result obtained using a fine mesh of abaqus quadratic shell elements (s8) is used as a reference solution. this example was originally proposed by saravanos et al. [35]. hence, their results are used as a reference for the open circuit case that is affected by the piezoelectric coupling. all the results are summarized in table 1. table 1 convergence analysis – normalized first natural frequency short-circuited fref = 22915 hz (ref: abaqus s8 2424 mesh) open circuit fref = 24594 hz (ref: saravanos et al. [35]) elements present abaqus s3 [35] present [35] 32 1.190 1.220 1.090 1.107 1.109 128 1.031 1.050 1.034 1.028 1.054 288 1.008 1.024 1.023 1.005 1.044 4. conclusions development of reliable, accurate and highly efficient numerical tools for modeling and simulation of thin-walled piezoelectric active structures is an important prerequisite for successful design of those modern structural systems. the presented shell element combines features of already available elements. since it is a 3-node element, it offers high meshing ability and numerical efficiency. geometric nonlinearities are accounted for based on the corotational fe formulation. the major weaknesses typical for low-order interpolation elements are addressed by means of various existing techniques in order to offer a versatile shell type finite element that also covers the electro-mechanical coupling. in order to bring the element closer to the end user, it was implemented in abaqus. several test cases were considered to 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1993, piezoelectric shells: distributed sensing and control of continua, kluwer academic publishers, amsterdam. 34. zhang, s., 2014, nonlinear fe simulation and active vibration control of piezoelectric laminated thinwalled smart structures, phd thesis, institute of general mechanics, rwth aachen university. 35. saravanos, d.a., heyliger, p.r, hopkins, d.a., 1997, layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates, international journal of solids and structures, 34(3), pp. 359–378. friction behavior of aluminum bronze reinforced by boron carbide particles facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 51 65 https://doi.org/10.22190/fume201226013s © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper friction behavior of aluminum bronze reinforced by boron carbide particles alexey yu. smolin1,2, andrey v. filippov1, evgeny v. shilko1,2 1institute of strength physics and materials science sb ras, tomsk, russia 2national research tomsk state university, tomsk, russia abstract. a promising composite material for tribotechnical applications based on aluminum bronze with reinforcing boron carbide particles fabricated by a special electron beam additive deposition technique was studied experimentally and numerically. tribological experiments showed that reinforcing by carbide particles allowed reducing the coefficient of friction from 0.26 to 0.19 and improving the wear resistance by 2.2 times. computer modeling reveals two main factors playing a significant role in the friction behavior of the studied metal matrix composite: the mechanical effect of reinforcing ceramic inclusions and effective hardening of the metal matrix due to the peculiarities of the 3d electron beam printing. the mechanical effect of hardening inclusions determines a more rounded shape of wear particles, preventing wedging, and thereby increasing the stability of friction. strengthening the metal matrix leads to reducing the number of wear particles. key words: mmc, aluminum bronze, boron carbide, friction, computer simulation 1. introduction aluminum bronze is widely used for producing corrosion and wear-resistant products intended for seawater, water supply, and petroleum chemistry applications. traditional powder metallurgy [1] as well as arc melting technologies [2] are mainly used for this material. the most advanced technologies used for manufacturing products from copper alloys are wire arc additive manufacturing [3, 4], selective laser melting (slm) [5, 6], and electron beam additive manufacturing (ebam) [7]. the latest seems the most promising since it needs vacuum and therefore the risk of interlayer oxidizing is excluded. furthermore, ebam is more energy-efficient as compared with slm since no powerful lasers are needed for forming a stable melted pool. to enhance the mechanical properties of metals they are reinforced by introducing hard inclusions thereby making metal matrix composite (mmc). received december 26, 2020 / accepted february 02, 2021 corresponding author: alexey smolin institute of strength physics and materials science sb ras, pr. akademicheskii 2/4, tomsk, 634055, russia e-mail: asmolin@ispms.ru 52 a. yu. smolin, a. v. filippov, e. v. shilko for fabricating the copper-based mmc, various powder compounds like oxides, carbides, or antifriction disulfides may be used as fillers [8]. carbon, as one of the most frequently used fillers, has many disadvantages in tribotechnical applications; one of them is its low resistance to high-temperature oxidizing in the air, which may cause the loss of solid lubricating in high temperature or/and speed loading conditions [9]. therefore, more stable ceramic materials such as, for example, hexagonal boron nitride or boron carbide are preferable [10]. boron carbide (b4c) proved to be an effective material for reinforcing mmc [11, 12, 13] as well as for improving the wear resistance of aa7075 [14]. recently, a special electron beam additive deposition technique was proposed for fabricating an aluminum bronze based mmc filled with b4c [15]. such a composite possesses a gradient distribution of the reinforcing particles and looks very promising for tribotechnical applications. when developing new materials, it is very useful to have a computational model that could help predict their properties and behavior in the operating conditions. for computer simulation of the mechanical behavior of materials in the friction zone, the numerical methods of computational fluid or/and solid mechanics are usually used [16]. to take into account the elastic-plastic deformation of the main parts of the interacted bodies, as well as stirring of the material in the contact zone, the so-called arbitrary lagrangian-eulerian approach, as well as smoothed particle hydrodynamics, are the most suitable for solving such problems [16, 17, 18]. however, these methods consider the material as a continuum and, therefore, often fail to describe its severe deformation with the accounting of the initial material structure and especially for the formation of new structures in the friction zone. recently, computational techniques based on particle mechanics have been rapidly developed and utilized for such phenomena. among them, we would like to point out the discrete element method and movable cellular automaton method as the most promising ones [19, 20]. thus, the main purpose of this paper is to study the peculiarities of sliding friction behavior of novel aluminum bronze based composites filled with b4c reinforcing particles fabricated by additive electron beam deposition method and to develop a computational particle-based model for numerical studying friction and wear of such heterogeneous materials. 2. experimental study 2.1. materials and methods the samples studied here experimentally were printed using an electron beam additive deposition technique that was described in [21]. a sequential application of aluminum bronze cu-7.5wt%al wire-feed and b4c+al powder-bed electron beam methods were used for preparing gradient heterogeneous cual-b4c samples [15]. two (b4c)xal1−x powder mixtures with different boron carbide/aluminum bronze weight ratios x corresponding to (b4c)0.5al0.5 (sample 1) and (b4c)0.25al0.75 (sample 2), respectively, were prepared and then sequentially deposited on a previously deposited cu-7.5 wt%al alloy layers. to prepare the powder mixtures, a boron carbide powder with an average particle size of ~200 µm and an aluminum powder with an average particle size of ~5 µm were used. friction behavior of aluminum bronze reinforced by boron carbide particles 53 the microhardness of the manufactured samples was measured using a «duramin-5» (stuers аps, denmark) microhardness tester at 50 n load. it was shown that mechanical strength and hardness of the cu-7.5 wt%al matrix were enhanced by incorporating carbide particles as follows from the dependencies of microhardness on the distance below the surface, which correlate with the content of b4c [15]. the maximum >1.25 gpa gain in hardness was observed at a distance of ~1.2 mm below the surface, i.e. down to the middle part of the cual-b4c samples. the microhardness of sample 1 with higher boron carbide content is increased more sharply as compared to that of sample 2. the maximum hardness numbers of samples 1 and 2 are 4.63 gpa and 2.35 gpa, respectively. the bottom parts of both cual-b4c samples possess microhardness negligibly higher than that of cu-7.5 wt%al matrix. samples for tribological experiments were prepared from the additively manufactured cual-b4c samples by cutting them by planes normal to the printed layers. the referenced pure metallic samples were cut off from the deposited cu-7.5 wt%al alloy samples. all the samples were mechanically ground on an emery paper to reach the final roughness of ra = 2.5 µm. a tribometer tribotester (tribotechnic, france) was operated in a reciprocal sliding mode with 3 hz frequency during 105 cycles; the study was performed in accordance with astm g-133 standard. standard 52100 bearing steel 6 mm diameter ball was used as a counterbody. the normal load on the ball was 22 n, which corresponded to a mean contact pressure of about 25 mpa. during the tribological tests, the temperature was 25°c and humidity 45%. we performed three tests per sample. metallographic structures and worn surfaces were examined using a confocal laser scanning microscope lext 4100 (olympus, japan) and scanning electron microscope (sem) tescan mira 3 lmu (tescan, czech republic) attached with the add-on for energy-dispersive x-ray spectroscopy (eds). 2.2. results the highest coefficient of friction (cof) ~0.26 was demonstrated at sliding of a b4c-free cu-7.5 wt%al metal against a steel ball (fig. 1a). reinforcing the matrix metal with boron carbide particles allowed reducing the cof to ~0.23 and ~0.19 for samples 1 and 2, respectively, i.e. higher content of boron carbide and higher microhardness of sample 1 provided sliding with a coefficient of friction higher than that of sample 2. the reason may be more severe abrading the steel ball surface by pulled-out b4c particles. quantification of wear performed by determining the wear groove cross-section area (fig. 1b) allows concluding that the total wear of sample 2 was minimal with the value as low as 2.16×10−3 µm2, i.e. of ~19% lower than that of sample 1 (2.67×10−3 µm2) and of ~2.2 times lower than that of cu-7.5 wt%al. the wear of sample 1 was 1.75 times lower than that of cu-7.5 wt%al. maximum wear groove depths were 39 µm, 34 µm, and 64 µm, for samples 1, 2, and cu-7.5 wt%al correspondingly. 54 a. yu. smolin, a. v. filippov, e. v. shilko fig. 1 time dependencies of the coefficient of friction for deposited metal and composite samples (a) and their wear groove cross-section profiles (b) the worn surfaces of all the samples were represented by wear grooves of different depths and widths formed by the material removal. the wear groove surfaces of all samples were covered by red-brown patch areas identified as oxidized areas, mechanically mixed layers (mml) formed by adhesion transfer, and some amount of black wear debris. sem back scattering electron (bse) images of the worn surfaces demonstrate at least three areas with different bse contrast levels (the images for sample 1 are shown in fig. 2, for example): bright, dark, and black. fig. 2 sem bse image and eds element maps of the worn surface of sample 1 friction behavior of aluminum bronze reinforced by boron carbide particles 55 according to eds element maps shown in fig. 2 bright areas in the sem bse image correspond to less oxidized layers (see maps for o), while the darker ones represent high oxygen areas. black particles or agglomerates correspond to boron carbides (see maps for b and c in fig. 2) and were detected on the worn surfaces of samples 1 and 2. at the same time, the b-enriched particles do not always coincide with the c-enriched ones in fig. 2 what may be evidence in favor of boron carbide decomposition in the melted pool under electron beam irradiation. therefore, at least part of the black particles in the worn surfaces may be free carbon black ones. mechanically mixed layers composed of wear debris and oxides were found on all samples when examining the sample sections by the planes parallel to the sliding direction and perpendicular to the sample surface (fig. 3, this pictures also depict the spatial distribution of reinforcing boron carbide particles in the composite samples). the mml with a thickness of 22 µm was generated on the most ductile cu-7.5 wt%al sample (fig. 3a) while thicknesses of 7 µm and 13 µm were obtained on the hardest cual-b4c sample 1 and less hard sample 2, respectively (fig. 3b and c). these results look reasonable since the higher hardness of sample 1 means less ductility and less penetration of plastic deformation into the sample. therefore, only thin mmls may be generated by subsurface plastic deformation in hard materials. fig. 3 longitudinal cross-sections of wear grooves with mml layers on cu-7.5 wt%al (a), cual-b4c sample 1 (b), and sample 2 (c) 3. computer simulation 3.1. model description the numerical model of sliding friction used herein is based on the method of movable cellular automata (mca), which is a representative of simply deformed discrete element method, i.e. particle mechanics, which looks very promising for the simulation of heterogeneous materials at mesoscale, where the structure has to be considered explicitly [20, 22, 23]. in this method, it is assumed that any material consists of a certain number of discrete elements (called movable cellular automata) of a finite size that can move, rotate, and change their state due to interaction with neighbors thereby simulating real deformation and physical processes. the motion of the elements is governed by newtoneuler equations. the force acting between the automata is written in the many-body form and defined by the local strain parameters and usual moduli of elasticity. in plastic flow, 56 a. yu. smolin, a. v. filippov, e. v. shilko the dependence of the inter-automata force on strain is defined by the local stress-strain curve of the material, which is called the response function of the automaton. actually, the plastic flow theory using the von mises criterion is adopted for modeling plasticity in mca. a pair of automata can be in one of two possible states: bonded and unbonded. thus, in mca, fracture and coupling of fragments (crack healing, cold micro-welding, etc.) are simulated by the corresponding switching of the pair state. switching criteria depend on the physical mechanisms of material behavior. it worth noting, that the mca method has been successfully being used for the simulation of friction for about 20 years. thus, the main model and corresponding software have been validated for modeling the elastoplastic behavior of metallic materials in contact loading and friction (see, for example, [24]). the novelty of the model used herein consists in the new fracture criterion defined by the ultimate value of the strain energy, and the new algorithm and criterion for automata coupling (considered as a process, which takes several time steps) defined by the plastic heat value combined with the condition of compression [23]. besides, in the used mca-model such physical phenomena as thermo-elasticity and heat transfer are also implemented. herewith we model the metal matrix composite based on aluminum bronze with reinforcing boron carbide inclusions. the properties of the components were obtained from the literature, namely, the elastoplastic properties of the bronze matrix were taken from [25, 26], while the elastic-brittle properties for the carbide inclusions from [27]. mechanical and thermophysical properties of aluminum bronze: density 7800 kg/m3, young’s modulus 118 gpa, poisson’s ratio 0.295, yield strength 360 mpa, tensile strength 560 mpa, coefficient of linear thermal expansion 1.78·10−5 k−1, specific thermal conductivity 80 w/(m·k), specific heat 418 j/(kg·k). the plastic heat value used as a criterion for coupling the bronze automata is 141 j/m3 combined with a condition of compressive stress of 360 mpa. properties of b4c: density 2510 kg/m3, young’s modulus 440 gpa, poisson’s ratio 0.17, ultimate compressive strength 5680 mpa, tensile strength 2270 mpa, coefficient of linear thermal expansion 0.4·10−5 k−1, specific thermal conductivity 40 w/(m·k), specific heat 1200 j/(kg·k). fig. 4 view of 2d sample for modeling friction and scheme of its loading two contacting rough surfaces of metallic materials with the same properties were modeled, mimicking a mesoscopic fragment of the friction contact zone of macroscopic bodies (fig. 4). in 2d formulation, the approximation of a plane-stress or plane-strain state was used. the linear dimensions of the model were l = 1000 µm along the x-axis and h = 220 µm along the y-axis. the size of the discrete element was d = 1 µm. the surface of each friction behavior of aluminum bronze reinforced by boron carbide particles 57 of the contacting bodies (hereinafter referred to as blocks) contains more than 40 asperities. the average distance between asperities smi = 23 µm, maximum profile height rmax = 5 µm. the friction of the blocks was simulated by their relative tangential displacement at a constant speed and a given constant value of the contact pressure σn = 36 mpa, which was set by applying a force fn to the upper surface of the upper block (fig. 4). the xcoordinates of the elements of the upper surface of the sample were fixed. the base of the system (the lower surface of the lower block) was fixed in the y direction and moves along the x-axis at a constant velocity vslide = 5 m/s. to reveal the influence of boron carbide inclusions on the friction behavior of the aluminum bronze, we model the pure bronze samples as well as composite samples. fig. 5 shows an example of the structure of a model composite material with a matrix of aluminum bronze and 5 vol. % of dispersed ceramic inclusions. the inclusions are represented by both individual elements (1 μm in size) and agglomerates of these elements up to 6–8 μm in size. fig. 5 2d sample for modeling friction of mmc with 5% of b4c inclusions 3.2. results of computer simulation fig. 6 shows the time dependencies of the friction coefficient change for the samples of pure bronze and composite for the case of the plane-stress state. the similarity of these curves can be noted; the length of the initial stage of the growth of the friction coefficient to values above 1 corresponds approximately to each other. at the same time, the transitional stage in the development of the friction, accompanied by a multiple decreases in the fig. 6 time dependence of the instantaneous cof on the relative tangential displacement of the blocks. the red line depicts averaging the instantaneous values in a window of 100 points 58 a. yu. smolin, a. v. filippov, e. v. shilko coefficient of friction, is almost three times longer for the composite than for pure bronze. it consists of two stages: the stage of a rapid decrease in the coefficient of friction and the stage of a smooth transition of the coefficient of friction to a steady value, which is approximately equal in length. it is also important to note two main points at the stage of the steady-state friction of the composite: i) the average value of cof (~0.48) is significantly lower than that of pure bronze (~0.55); ii) the characteristic amplitude of cof variation is also significantly lower. analysis of the simulation results showed that these differences are associated with the peculiarities of the formation of wear particles in the composite material. indeed, already at the stage of running-in, the process of formation of new contact spots on wear particles proceeds much faster (fig. 7). in particular, the presence of relatively large b4c particles leads to the adhesion of additional surface layers of the material on them. this is due to the fact that hard ceramic particles are stress concentrators. therefore, more intense strain localization occurs in the adjacent matrix layer, which leads to the rapid formation of local interface cracks. ceramic particles detached from the blocks carry with them a matrix layer of the corresponding thickness. or, being on the surface of wear particles, they act on the adjoining surface of the blocks as indenters and abrasives. the result of these effects is, in addition to accelerating the growth of wear particles, their more rounded shape. fig. 7 2d sample for modeling friction of mmc (a) and aluminum bronze (b) at friction path lx=240 µm (correspond to the end of stage ii in fig. 6) at the steady-state stage, the rounded shape of the wear particles (fig. 8) prevents wedging in the contact zone and thereby helps to reduce the amplitude of fluctuations in the coefficient of friction. at the same time, the weak adhesion of the particles to the surfaces (due to the interface zones of ceramic inclusions with bronze, the formation of cracks in which occurs earlier than in the bulk of the bronze matrix), determines a significant decrease in the overall level of the friction coefficient. friction behavior of aluminum bronze reinforced by boron carbide particles 59 fig. 8 2d sample for modeling friction of mmc with 5% of b4c inclusions at friction path lx=650 µm note that the features of the friction model of aluminum bronze described in [23] (the formation of large wedges/prows and a high coefficient of friction) under mechanically constrained conditions of a plane-strain state are also valid for material with carbide inclusions. in this case, the value of the coefficient of friction for such material (~ 1.2– 1.3) is even slightly higher than that for pure aluminum bronze. this allows us to assume that the differences in the values of the friction coefficient of the bronze and the composite in the 3d case will be less than in the case of a mechanically unconstrained system considered here (plane-stress state). 3.3. role of the mechanical characteristics of the matrix in the friction of mmc the presented simulation results allow a better understanding of the mechanisms that are able to determine the effect of reducing the friction coefficient obtained in an experimental study in the material with a gradient structure fabricated by the method of electron-beam deposition (by approximately 0.05 in comparison with aluminum bronze). indeed, the experimentally established change in the friction coefficient in a material with a gradient composite structure (cual-b4c), obtained by electron-beam deposition, can be caused not only by the mechanical effect of inclusions but also by a change in the mechanical characteristics of the bronze matrix [15]. these changes can be determined by the following factors: 1) in the process of cooling the crystallized composite from the melting point to room temperature, “additional” geometrically necessary dislocations (gnds) are formed in the layer of the metal matrix surrounding the surface of the ceramic particles. the density of these dislocations can be an order of magnitude higher than the density of dislocations far from the surface of the inclusions. an increase in the dislocation density provides an increase in the yield stress of the interface layer of the matrix. 2) an additional increase in the density of the gnds in the interface layer of the matrix can also be achieved during deformation of the composite material due to the 60 a. yu. smolin, a. v. filippov, e. v. shilko difference in elastic moduli and plastic properties of the phases and, as a consequence, a high gradient of deformations at the interface. this effect provides an effective increase in the strain-hardening coefficient of the interface zone. 3) carbide particles can partially dissolve in the molten bronze, which can lead to the appearance of impurities of the corresponding chemical elements in the crystal lattice of the matrix and cause an increase in its yield stress and, possibly, the coefficient of strain hardening. to study the influence of the first two factors, a theoretical assessment of the increase in the yield point and the coefficient of deformation hardening of the considered aluminum bronze was carried out. the influence of the first of them is assessed on the basis of the relationships proposed by shibata [28]: thermyy  agb+= , (1) where: p p p cte therm 1 212 f f bd t −  = , (2) here σ’y is the value of the yield stress of the bronze matrix, taking into account the gnds that have arisen during cooling (σy = 360 mpa), a is the dimensionless taylor coefficient, g and b are the shear modulus and the burgers vector, ρtherm is the density of thermally induced gnds, δcte is the difference in the value of the coefficient of thermal expansion of the matrix and inclusion, δt is the difference between the melting point of bronze and room temperature, fp = (r/r)3, where r and r are half the average particle size of inclusions dp and half the average distance between the particles. note that the effect of thermally induced gnds on the loading diagram is taken into account by "raising" the unified hardening curve by δσy = σ’y −σy . the influence of the second factor (an increase in the strain hardening coefficient) was estimated based on the relations of the nonlocal theory of plasticity by h. gao and y. huang [29] 3 p 22 2 pp 2527)()( frbg  += , (3) where σ(εp) is the curve of uniaxial loading of the material in terms of plastic axial deformation εp taking into account the correction δσy from eq. (1) due to thermally induced gnd, f is the volume fraction of inclusions, η is the dimensionless coefficient of the model. the performed estimates show that for the material under consideration, the main influence on the change in the mechanical properties of the composite should be provided by the gnds arising during cooling from the melt. the influence of gnds arising during deformation due to "plastic inconsistency" is a factor of a smaller order of magnitude (black and blue curves in fig. 9). note that the above-described increase in the gnd occurs only in the interface layer of the matrix, the thickness of which is estimated theoretically based on the linear dimensions of inclusions, their concentration, the characteristic distance between them, the mechanical and atomic characteristics of the phases, etc. [28]. due to the current lack of comprehensive quantitative experimental information on the characteristics of the distribution of b4c inclusions, in the calculations presented below, an “upper estimate” is used, which assumes friction behavior of aluminum bronze reinforced by boron carbide particles 61 that the entire volume of the bronze matrix is subjected to dislocation hardening (as the parameter r in eq. (2), half of the average distance between the inclusions). fig. 9 piecewise linear approximation of the experimental diagram of uniaxial tension of a sample of aluminum bronze (black curve) and diagrams of a hardened bronze matrix due to gnd, estimated theoretically (blue curve) and experimental+theoretically (lilac curve) when conducting compression tests of small samples of the experimentally obtained material with a gradient structure (in a layer with a b4c content of ~9 vol.%; this volume concentration approximately corresponds to a two-dimensional concentration of 5-6%, which is considered in our calculations), it was found that its yield point is approximately 1.8 times higher than that of the aluminum bronze samples. this significantly exceeds the theoretically estimated value and may be due to the presence of impurity atoms. at the same time, the experimentally determined strain-hardening coefficient of the obtained material increased insignificantly in comparison with aluminum bronze and correlates well with the estimates according to eq. (3). in fact, the difference between the experimental and theoretically estimated data on the hardening curve of a composite material obtained by electron beam deposition consists in the difference in the value of the yield stress. to assess the effect of this parameter, a modified matrix is also considered (lilac curve in fig. 9), in which the value of the yield stress corresponds to the experimentally determined value, and the strain-hardening coefficient is modified according to eq. (3). for model composites with modified matrices, friction was simulated under the same loading conditions as in the previous cases considered. the new value of yield stress was considered as the minimum contact pressure at which the coupling of the corresponding automata is possible. fig. 10 shows the change in the coefficient of friction for the samples with modified matrices under conditions of a plane-stress state. it can be noted that an increase in the yield point leads to the following changes. 1) the length of the running-in stage increases significantly. in the problems considered, the value of the relative tangential displacement of the blocks was approximately equal to the length of the system (about 1 mm). nevertheless, this turned out to be insufficient for reaching the steady-state regime. 62 a. yu. smolin, a. v. filippov, e. v. shilko 2) the friction coefficient decreases with the increase in the yield point (figs. 6 and 10). in particular, an increase in the yield stress of the matrix by 1.8 times is accompanied by a decrease in the steady-state coefficient of friction by at least 0.1 in comparison with a composite material with the original matrix. fig. 10 the dynamics of the instantaneous friction coefficient in a pair of blocks of the composite material: a) taking into account the strengthening of the matrix due to the gnds estimated theoretically; b) taking into account the strengthening of the matrix due to gnds estimated based on the analysis of experimental data (thermally induced gnd) and theoretically (gnds due to “plastic inconsistency” of the phases) the noted differences are associated with the peculiarities of the formation of wear particles in a composite material with a hardened matrix (fig. 11). so, already at the initial stage of running-in (after cutting and/or smoothing the initial asperities of the surface), the number of new contact spots (wear particles) is several times lower than in the samples with the unmodified matrix. at the same time, the higher the yield point, the lower the number of formed wear particles at the initial stage of friction. so, in fig. 11, corresponding to the hardest matrix, one can see the formation of only two wear particles, which later determine the dynamics of change in the friction coefficient and wear. the reason for this effect is an increase in the “bonding threshold” of materials due to an increase in the yield stress (the contact pressure required for coupling the matrix automata). in this case, only those initial fragments of the “third body” begin to grow, in which the local contact pressure is sufficient for cold welding with the rest materials surface. with an increase in the yield point, such spots become less and less, which determines the formation of a small number of wear particles. thus, in contrast to the original aluminum bronze (or a composite with a matrix with initial mechanical properties), in the case of a hardened matrix, the initially formed wear particles are stable and long-lived formations that determine the further development of friction and wear processes. note also that the result obtained correlates with the discussion in [30, 31] of the features of shaping prows/wedges in a variety of materials. indeed, as shown in this work, the linear dimensions and stability (duration) of the existence of prows/wedges in friction pairs are largely determined by the mechanical parameters of the materials, first of all, by the hardness, which is largely determined by the yield point. the simulation results allowed us to offer one of the possible explanations for this feature. friction behavior of aluminum bronze reinforced by boron carbide particles 63 fig. 11 the structure of the contact zone of the blocks of the composite material at different times of friction: (a) lx = 275 μm; (b) lx = 575 μm; (c) lx = 870 μm; the mechanical properties of the matrix correspond to the lilac curve in fig. 9 4. conclusion friction and wear of the aluminum bronze based composites with reinforcing boron carbide particles fabricated by a combination of wire-feed and powder-bed electron beam deposition methods were studied experimentally. it was shown that reinforcement of the aluminum bronze matrix with b4c particles allowed improving the wear resistance of the composites as compared to that of pure aluminum bronze deposited. coefficient of friction reduced from 0.26 to 0.19. the composite of composition (b4c)0.25al0.75 showed its wear resistance by a factor of 2.2 higher as compared to that of printed cu-7.5 wt%al bronze. using computer modeling, the factors have been identified that have a significant effect on the value of the coefficient of friction of metal matrix composite based on aluminum bronze and reinforced by boron carbide particles. these factors include the mechanical effect of reinforcing ceramic inclusions and effective hardening of the metal matrix due to the peculiarities of the 3d electron beam printing process. it is shown that the mechanical effect of hardening inclusions determines a more rounded shape of wear particles, preventing wedging, and thereby increasing the stability of the friction coefficient. at the same time, the weaker integral adhesion of wear particles to the surfaces contributes to a 64 a. yu. smolin, a. v. filippov, e. v. shilko slight decrease in the coefficient of friction. strengthening the metal matrix leads to reduce the number of wear particles and therefore is a factor in determining the overall level of the coefficient of friction. thus, the results of computer modeling presented herein confirm the supposition derived from the experimental evidence that electron beam additive manufacturing of cual–b4c composite leads to additional reinforcing of the aluminum bronze matrix by boron carbide dissociation and precipitation. so, when contacting the copper melt bead, the boron carbide may dissociate into boron and carbon; boron then may dissolve in copper and form cu–b 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23, pp. 135-146. 13250 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume241203017c © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper simulation and analysis of spray distribution and droplet size of large sprinkler machines loaded with atomizing micro-sprinklers hua cao1,2, yongshen fan1,2, chunya ma1,2, peng li1,2, wei zhan1,2, yinbo cao1,2, fuyi duan1,2 1institute of farmland irrigation, chinese academy of agricultural sciences (caas), xinxiang, china 2key laboratory of water-saving irrigation engineering, ministry of agriculture and rural affairs, xinxiang, china abstract. current agricultural spraying faces issues such as excessive application, pesticide waste, and environmental pollution. this paper analyzes the hydraulic performance of several atomizing micro-sprayers (hollow cone, solid cone, and fanshaped) used in large-scale irrigation machines, providing a theoretical basis for selecting spraying nozzles. three types of micro-sprayers were tested at pressures of 0.2mpa, 0.3mpa, and 0.5mpa, and ground heights of 0.5m, 0.8m, 1.2m, and 1.5m. each test was repeated three times. the results show that: (1) the hollow cone sprayer has a bimodal water distribution, the solid cone is unimodal, and the fan-shaped sprayer is long-strip shaped. as pressure increases, water distribution increases, while height increases reduce water distribution. (2) the droplet size distribution follows a normal distribution. higher pressure increases the number of larger droplets, while lower pressure increases smaller droplets. larger aperture sprayers generate more droplets, with the fan-shaped sprayer producing the most. (3) the particle size of the hollow cone sprayer ranges from 0.312mm to 1.187mm, with speeds below 1.4m/s; the solid cone sprayer ranges from 0.312mm to 6.5mm, with speeds below 2.4m/s; and the fan-shaped sprayer ranges from 0.312mm to 2.75mm, with speeds below 2.5m/s. the experimental results provide a theoretical basis for selecting and using atomizing micro-sprayers in large-scale irrigation, offering guidance for reducing pesticide use and improving agricultural efficiency. key words: pesticide application, large-scale irrigation machine, atomizing microsprayer, particle size, velocity received: december 03, 2024 / accepted april 12, 2025 corresponding author: fuyi duan 1institute of farmland irrigation, chinese academy of agricultural sciences (caas), xinxiang 453002, china 2key laboratory of water-saving irrigation engineering, ministry of agriculture and rural affairs, xinxiang 453002, china e-mail: irrigation2024@126.com 2 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan 1. introduction under the backdrop of global climate change, the situation of pests and diseases is becoming increasingly severe [1-4]. there is an urgent need to establish a technical system for rapidly and effectively eliminating pests and diseases over large areas [5-6], to accelerate the fundamental transformation of pest and disease control models, and to achieve intelligent plant protection and green ecology. the growth of crops is inseparable from efficient and safe plant protection machinery and pesticide application techniques [79]. currently, pesticides can be applied through four methods: manual spraying, ground mechanical spraying, unmanned aerial vehicle (uav) spraying, or through irrigation systems. manual spraying makes it difficult to grasp the timing and precision of pesticide application, which not only leads to environmental pollution but also poses a significant risk to the health of farmers due to the close contact with pesticides [10-12]. ground mechanical spraying is costly and has a low effective utilization rate of pesticides, and the movement of machinery can cause mechanical damage to crops and soil compaction near the wheels [13]. uav spraying is a viable alternative, but compared with ground mechanical spraying, the phenomenon of droplet drift during uav application is more severe [14-17], which can cause environmental pollution due to pesticide drift [18-20]. in actual uav operations, if the wind field is too strong, crops on both sides of the flight path may lodge, and the plant damage caused by lodging essentially provides favorable conditions for the spread of diseases [21]. large sprinkler machines (including linear and center-pivot sprinkler machines) equipped with atomizing micro-sprinklers can be used for chemical irrigation, effectively avoiding mechanical crushing of crops during pesticide application, and reducing the drift loss of chemicals [22-25]. to reduce the drift loss of pesticide droplets, scholars worldwide have conducted numerous studies and experiments on droplet drift. sushilendra et al. [26]investigated the impact of different nozzle types, forward speeds, and spray heights on droplet size of uav sprayers, and the results showed that nozzle type had a significant effect on droplet size, while forward speed and spray height had a smaller impact. fattahi et al. [27] developed various artificial neural network methods to study the factors affecting the drift of pesticide sprays. sensitivity analysis of the neural network model indicated that the impacts of independent variables, including wind speed, spraying pressure, boom height, and nozzle type, were 42%, 27%, 16%, and 15%, respectively. singh et al. [28] explored the impact of pulse width modulation (pwm) systems at different frequencies on spray coverage and droplet size uniformity. park et al. [29] utilized computational fluid dynamics (cfd) simulations to explore the effects of droplet size and wind speed on collection performance. large sprinkler machines equipped with atomizing micro-sprinklers for pesticide application offer the advantage of more precise control over pesticide use, reducing pesticide waste and environmental pollution, while enhancing crop protection effectiveness. however, current research primarily focuses on traditional spraying technologies, and further investigation is needed on the spray distribution, droplet size, and movement characteristics of this new type of equipment. therefore, this paper aims to simulate and analyze the spray distribution and droplet size of large sprinkler machines equipped with atomizing micro-sprinklers, filling the research gap and providing new theoretical foundations and practical directions for water resource conservation and pesticide reduction in agriculture. simulation and analysis of spray distribution and droplet size of large sprinkler machines... 3 2. experimental design and methods the experiment selected three types of micro-sprinklers, namely solid cone, hollow cone, and fan-shaped. the solid cone nozzle models are: m4 and m8, totaling two models. the hollow cone nozzle models are: lnn3w and lnn4w, also totaling two models. one model of the fan-shaped nozzle was chosen for the experiment. the experiment selected three pressures: 0.2 mpa, 0.3 mpa, and 0.5 mpa, and the micro-sprinkler heights above the ground were chosen as 0.5 m, 0.8 m, 1.2 m, and 1.5 m. a rain gauge was used to measure the water distribution of each sprinkler over a period of 1 hour. the size of droplets and the uniformity of droplet deposition have a significant impact on the efficacy of pesticides. a parsivel2 laser raindrop spectrometer was used to measure the droplet size, particle velocity, and deposition distribution of the atomization micro-sprinklers. the structure and dimensions of the lnn series and m series atomization micro-sprinklers, as well as the fan-shaped atomization micro-sprinkler, are shown in fig. 1. a) schematic diagram of the lnn series atomization micro-sprinkler structure: 1 insert, 2 swirl core, 3 prinkler body, 4swirl core seat, 5 filter frame, 6 l-shaped filter mesh, 7 base b) schematic diagram of the m series atomization spray head structure: 1 swirl core seat, 2 swirl core, 3 insert standard angle, 4 main body c) dimensional diagram of the fan-shaped atomization spray head fig. 1 schematic and dimensional diagrams of different types of atomization spray heads 4 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan the basic parameters of the spray nozzles are given in table 1. table 1 basic parameters of spray nozzles micro-sprinkler number type of microsprinkler rated orifice diameter (mm) total length (mm) base (hexagonal) (mm) 1 lnn3w 0.99 47.5 20.6 2 lnn4w 1.5 47.5 20.6 3 m4 1.1 21.5 14.2 4 m8 1.5 21.5 14.2 5 fan-shaped nozzle 0.91 23 14 micro-sprinkler number sprinkler (hexagonal) pressure (mpa) flow rate (l/h) spraying angle (degrees) 1 17.4 0.2 9.7 157 2 17.4 0.2 16.1 155 3 0.2 12.9 72 4 0.2 26 85 5 0.2 30 110 the experiment was conducted in the quality inspection and testing hall of the institute of farmland irrigation, chinese academy of agricultural sciences, china. the experimental setup is shown in fig. 2. fig. 2 nozzle spray test setup diagram: 1 reservoir; 2 water pump; 3 control valve; 4 plastic pe hose; 5 pressure gauge; 6 rain gauge; 7 atomization micro-sprinkler; 8 water pipe; 9 height adjustment stand simulation and analysis of spray distribution and droplet size of large sprinkler machines... 5 the parsivel2 laser raindrop spectrometer uses laser remote sensing technology to derive droplet size distribution, precipitation kinetic energy, particle concentration, and radar reflectivity from the diameter and velocity of precipitation. the measurement area is 54 m2. the size range is 0.062 to 54.5 mm, and the velocity range is 0.05 to 20.8 m/s, both with a resolution of 32 levels, and the measurement accuracy is ±1 level. in this experiment, the parsivel2 laser raindrop spectrometer is used to measure the droplet size and particle velocity of various types and pressure conditions of atomization microsprinklers. during the measurement, the distance between each atomization microsprinkler and the parsivel2 laser raindrop spectrometer is 0.5 m. the parsivel2 laser raindrop spectrometer measures the range of droplet sizes and the range of particle velocities as shown in the following table 2. table 2 correspondence table of particle groups with size and velocity relationships group number average diameter (mm) average velocity (m/s) 1 0.062 0.05 2 0.187 0.15 3 0.312 0.25 4 0.437 0.35 5 0.562 0.45 6 0.687 0.55 7 0.812 0.65 8 0.937 0.75 9 1.062 0.85 10 1.187 0.95 11 1.375 1.1 12 1.625 1.3 13 1.875 1.5 14 2.125 1.7 15 2.375 1.9 16 2.750 2.2 17 3.250 2.6 18 3.750 3 19 4.250 3.4 20 4.750 3.8 21 5.500 4.4 22 6.500 5.2 23 7.500 6 24 8.500 6.8 25 9.500 7.6 26 11.000 8.8 27 13.000 10.4 28 15.000 12 29 17.000 13.6 30 19.000 15.2 31 21.500 17.6 32 24.500 20.8 6 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan 3. experimental results and analysis 3.1 comparative analysis of water distribution isolines for atomization microsprinklers the water distribution of no. 1 atomization micro-sprinkler under different pressures and heights above the ground is plotted as an isoline map, the x-axis represents the lateral spraying range, and the y-axis represents the longitudinal spraying range, as shown in fig. 3. for the no. 1 atomization micro-sprinkler, the experiment selected three pressures and four heights above the ground, adopting a complete test design, resulting in a total of 12 test groups. it can be seen from the figure that the water distribution of the no. 1 atomization micro-sprinkler tends to increase with the increase of pressure, indicating that agricultural spraying can increase the amount of pesticide applied by increasing the pressure of the spraying nozzle, thereby improving the spraying effect. as the height of the no. 1 atomization micro-sprinkler above the ground increases, the water distribution generally tends to decrease, indicating that the height of the spraying nozzle from the crop pest position is a key factor affecting the effectiveness of the medication. with the increase of the height of the atomization micro-sprinkler from the ground, the water distribution tends to decrease, indicating that to improve the effectiveness of the medication, the spraying nozzle needs to be as close as possible to the crop foliage. simulation and analysis of spray distribution and droplet size of large sprinkler machines... 7 fig. 3 isolines of water distribution under different test conditions for no. 1 atomization micro-sprinkler the water distribution of no. 2 atomization micro-sprinkler under different pressures and heights above the ground is plotted as an isoline map, as shown in fig. 4. it can be seen from the figure that under the same conditions, the spraying water volume of no. 2 atomization micro-sprinkler tends to increase with the increase of pressure, and the water distribution shows a bimodal pattern. the reason is that no. 2 nozzle is a hollow cone atomization micro-sprinkler, hence there is a smaller water spraying volume at the center and a larger water spraying volume around the periphery. when the nozzle height above the ground is 0.5 m, 0.8 m, and 1.2 m, the bimodal pattern is more pronounced, and when the nozzle height above the ground is 1.5 m, the spraying water volume basically shows a unimodal pattern. the reason is that when the distance from the ground is higher, the water spraying volume is more affected by gravity and air resistance in the air, and the water spraying state of the hollow cone atomization nozzle approaches that of the solid cone atomization nozzle. 8 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan simulation and analysis of spray distribution and droplet size of large sprinkler machines... 9 fig. 4 isolines of water distribution under different test conditions for no. 2 atomization micro-sprinkler the isolines of water distribution for the no. 3 atomization micro-sprinkler under different test conditions are shown in the following fig. 5. it can be observed from the figure that as the pressure increases, the water volume of the atomization micro-sprinkler tends to increase; as the height above the ground increases, the water distribution of the atomization micro-sprinkler tends to decrease. since the no. 3 atomization micro-sprinkler is a solid cone atomization micro-sprinkler, it can be seen from the figure that the water distribution at each point shows a unimodal pattern, that is, the center point is the largest, and the surrounding areas are smaller. 10 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan fig. 5 isolines of water distribution under different test conditions for no. 3 atomization microsprinkler it can be seen from fig. 6 that as the pressure increases, the no. 4 atomization microsprinkler shows an increasing trend in water spraying volume, and as the height above the ground increases, the water spraying volume shows a decreasing trend. simulation and analysis of spray distribution and droplet size of large sprinkler machines... 11 fig. 6 isolines of water distribution under different test conditions for no. 4 atomization microsprinkler 12 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan it can be observed from fig. 7 that the no. 5 fan-shaped atomization micro-sprinkler exhibits an increasing trend in water spraying volume with the increase of pressure, and a decreasing trend in water spraying volume with the increase of height above the ground. the spraying pattern of the no. 5 fan-shaped atomization micro-sprinkler is approximately elongated, with higher water volume at the center and lower water volume around the periphery. simulation and analysis of spray distribution and droplet size of large sprinkler machines... 13 fig. 7 isolines of water distribution under different test conditions for no. 5 atomization microsprinkler 3.2 droplet size distribution under different test conditions for different types of atomization microsprinklers the degree of pesticide atomization refers to the size dispersion of the droplet group produced during the spraying process, that is, the number of droplets formed after a certain volume of liquid is dispersed through a certain atomization method. the more droplets that can be formed from the same volume of liquid, the finer the droplets are indicated to be, and vice versa, they are coarser. the finer the droplets, the greater the surface area of the target object that can be deposited and covered, which means that the control effect will be improved, and the amount of pesticide used will be reduced. however, fine spraying is not necessary in every situation. for example, in the use of herbicides, fine spraying is more likely to cause droplet drift, which can cause damage to neighboring crops or environmental pollution. the experiment adopts three different pressure conditions of 0.2 mpa, 0.3 mpa, and 0.5 mpa, and uses the parsivel2 laser raindrop spectrometer to test the number of droplets under different spraying droplet size conditions at different pressures, and draw a curve chart as shown in fig. 8. for the lnn series atomizing nozzles (nozzle no. 1 and nozzle no. 2), fig. 8(a) indicates that the number of particle sizes is roughly normally distributed with respect to droplet sizes. as the pressure increases, the peak value of the droplet size quantity shifts to larger droplet sizes, meaning that higher pressure results in more droplets with larger sizes, while lower pressure results in more droplets with smaller sizes. under the same pressure conditions, nozzle no. 2 (lnn4w) produces more droplets than nozzle no. 1 (lnn3w), and the nozzle diameter of nozzle no. 2 is 34% larger than that of nozzle no. 1. this suggests that under the same experimental conditions, an atomizing micro-sprinkler nozzle with a larger orifice can generate a greater number of droplets. 14 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan for the m series atomizing micro-sprinklers (nozzle no. 3 and nozzle no. 4), fig. 8(b) shows that at higher pressures, there are more droplets distributed in the larger size ranges. taking nozzle no. 4 (m8) as an example, when the pressures are 0.2 mpa, 0.3 mpa, and 0.5 mpa, the proportion of droplets with sizes greater than 1.375 mm in the total number of droplets for nozzle no. 4 are 0.58%, 8.58%, and 12.64%, respectively, with a ratio of 1:14.74:21.72. when the pressures are 0.2 mpa, 0.3 mpa, and 0.5 mpa, the proportion of droplets with sizes less than 1.375 mm in the total number of droplets for nozzle no. 4 are 99.42%, 91.42%, and 87.36%, respectively, with a ratio of 1.14:1.05:1. for the fan-shaped series of atomizing micro-sprinklers (nozzle no. 5), fig. 8(c) indicates that the particle size range of the fan-shaped atomizing micro-sprinkler is a) b) c) fig. 8 particle size distribution under different pressure conditions for different series of micro-sprinklers: a) lnn series, b) m series, c) fan-shaped micro-sprinkler simulation and analysis of spray distribution and droplet size of large sprinkler machines... 15 primarily distributed between 0.473 mm and 1.062 mm, accounting for more than 90% of the total proportion. when the pressures are 0.2 mpa, 0.3 mpa, and 0.5 mpa, the proportion of droplets with sizes distributed between 0.473 mm and 1.062 mm to the total number of droplets for the fan-shaped atomizing micro-sprinkler are 99.75%, 98.73%, and 97.89%, respectively. compared to the conical atomizing micro-sprinklers of the lnn series and m series, the fan-shaped atomizing micro-sprinkler has the highest total number of droplets for several reasons: on one hand, the nozzles of fan-shaped spray heads are typically designed to cover a larger area, so they may have a larger orifice area, which allows more water flow to pass through in a unit of time. on the other hand, the water flow distribution pattern produced by the fan-shaped spray head is within a fan-shaped area, rather than being distributed within a conical space like the conical spray heads. this design enables the fan-shaped spray head to produce more droplets over a broader area. at the same working pressure, the fan-shaped spray head may convert pressure into kinetic energy more effectively due to its internal structure design, resulting in the production of more droplets. the spraying angle of the fan-shaped spray head is usually larger, which means it can distribute droplets over a wider fan-shaped area. 3.3 distribution charts of droplet size vs. operating speed for atomization micro-sprinklers the radar charts in fig. 9 below illustrate the relationship between particle size and velocity under different pressure conditions for different models of atomizing microsprinklers. from fig. 9(a), it can be seen that for the hollow cone atomizing microsprinklers of the lnn series, the particle size range is mainly distributed between 0.312 mm and 1.187 mm, and the velocity range is primarily within 1.4 m/s. the characteristic is that the range of droplet sizes is relatively small, all distributed in the range of smaller sizes, and the droplet travel speed is also low. from fig. 9(b), it can be observed that for the solid cone atomizing micro-sprinklers of the m series, the particle size range is primarily distributed between 0.312 mm and 6.5 mm, and the velocity range is mainly within 2.4 m/s. the characteristic is that the droplet size range is relatively large, predominantly distributed in the larger size range, and the operating speed is also comparatively higher. from fig. 9(c), it can be seen that for the fan-shaped atomizing micro-sprinklers, the particle size range is mainly distributed between 0.312 mm and 2.75 mm, and the velocity range is primarily within 2.5 m/s. the characteristic is that the particle size range is also relatively large, mainly distributed in the medium size range, and the operating speed is relatively high. the distribution of droplet size and velocity under different pressure conditions for different series of atomization micro-sprinklers is shown in the following fig. 10. it can be observed from fig. 10 that under the same pressure conditions, the lnn series has a narrower droplet size distribution range, all of which are distributed within a smaller range, while the m series has a larger droplet size distribution range, and the droplet sizes are larger than those of the lnn series. for micro-sprinklers of the same series, the larger the nozzle orifice, the more the sprayed droplet size distribution tends towards the larger droplet size range. for instance, compared to the no. 3 atomization micro-sprinkler, the no. 4 atomization micro-sprinkler produces larger droplet sizes. 16 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan for the same type of atomization micro-sprinkler, under the same pressure conditions, the larger the orifice diameter, the larger the sprayed droplet size, but the droplet velocity decreases. for the same atomization micro-sprinkler, the greater the pressure, the larger the size of the sprayed droplets, but the droplet velocity is lower. a) b) c) fig. 9 radar chart of droplet size vs. velocity of atomizing micro-sprinklers from different series under various pressure conditions a) lnn series, b) m series, c) fanshaped micro-sprinkler simulation and analysis of spray distribution and droplet size of large sprinkler machines... 17 a) b) c) fig. 10 size-velocity distribution under different pressure conditions for different series of atomization micro-sprinklers: a) at 0.2 mpa, b) at 0.3 mpa, c) at 0.5 mpa 4. conclusion this study experimentally investigated the water distribution, droplet size characteristics, and spraying effects of different types of atomizing micro-sprinklers under varying pressure and height conditions. the results indicate that the spraying performance of atomizing micro-sprinklers is significantly affected by pressure, sprinkler type, and spraying height. the main findings are as follows: • water distribution: the water distribution of atomizing micro-sprinklers increases with increasing pressure, while the water distribution generally decreases as the height of the sprinkler above the ground increases. • hollow cone sprinkler water distribution: the water distribution of the hollow cone atomizing sprinkler exhibits a bimodal pattern, with smaller water volume at the center and larger water volume around the edges. as the height from 18 h. cao, y. fan, z. chen, h. sun, p. li, c. ma, p. li, w. zhan, y. cao, f. duan the ground increases, the water distribution is more influenced by gravity and air resistance, and it approaches the behavior of a solid cone atomizing sprinkler. • solid cone sprinkler water distribution: the water distribution of the solid cone atomizing sprinkler follows a unimodal pattern, with the highest water volume at the center and smaller volumes around the edges. • fan-shaped sprinkler water distribution: the water distribution of the fanshaped atomizing sprinkler is elongated in shape, making it suitable for spraying over a larger area. • droplet size distribution and pressure relationship: the droplet size distribution of lnn and m series atomizing micro-sprinklers follows a normal distribution. as pressure increases, the number of larger droplets increases. conversely, when pressure decreases, the number of smaller droplets increases. • nozzle aperture and droplet size relationship: under the same experimental conditions, nozzles with larger apertures generate more droplets. additionally, as the nozzle aperture increases, the droplet size also increases, but the droplet's velocity decreases. • droplet size characteristics of different sprinklers: the lnn series hollow cone atomizing micro-sprinkler has a smaller droplet size range and lower droplet velocity. the m series solid cone atomizing micro-sprinkler has a larger droplet size range and higher droplet velocity. the fan-shaped atomizing sprinkler has a relatively large droplet size range and also higher droplet velocity. • effect of pressure on droplet size and velocity: for the same type of atomizing micro-sprinkler, as the pressure increases, the sprayed droplet size increases, but the droplet velocity decreases. acknowledgment. this paper was supported by central-level basic scientific research business fee special project for research institutes (ifi2024-07; ifi2023-07; firi2022-20), henan province science and technology key project (242102110339), and 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cycles and frequencies, applied engineering in agriculture, 41(2), pp. 119-124. 29. park, j., lee, s. y., choi, l. y., hong, s. w., noh, h., yu, s. h., 2022, airborne-spray-drift collection efficiency of nylon screens: measurement and cfd analysis, agronomy, 12(11), 2865. https://doi.org/10.1007/s10668-023-04452-x facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 665 676 https://doi.org/10.22190/fume200128015m © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical simulation of fluid-structure interaction between fishing wobbler and water miroslav mijajlović1, sonja vidojković2,3, dušan ćirić1, dragan marinković1 1university of niš, faculty of mechanical engineering, serbia 2tu delft, faculty of civil engineering and geosciences, netherlands 3university of belgrade, ihtm, serbia abstract. this paper deals with modeling, discretization, and numerical analysis of the two-way fluid-structure interaction between a fishing wobbler and a water stream. the structural domain is an assembly of several bodies that have multiple mutual structure-tostructure interactions. these interactions are mostly nonlinear contacts that significantly influence the time step used in simulations. as a result of these nonlinearities, the numerical solving of such a model requires significant computer resources and a long computational time. the paper also presents the creation and numerical simplifications of the model. however, the model remains very realistic. it is concluded that solving the structural domain in a model that retains the interaction between solid bodies is more computationally sensitive and more demanding than solving the fluid domain. key words: two way fluid-structure interaction, fishing wobbler, structure modeling, contact modeling, fluid modeling 1. introduction the first artificial bait fishing probably occurred 2000 years ago in china or mesopotamia, when fishers used a silk rope with a weighted bronze (shiny) hook as a fish bait [1]. these hooks were cast into the water and allowed to dive. afterwards, the fishers would cyclically pull the rope, which would cause the hook to jiggle up and down. such movement of the hook would create an illusion of an injured animal struggling to reach the water surface. intrigued by the motion of an object resembling an easy meal, the predatory fish aggressively attacked the hook and became the prey. received january 28, 2020 / accepted march 28, 2020 corresponding author: miroslav mijajlović university of niš, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: miroslav.mijajlovic@masfak.ni.ac.rs 666 m. mijajlović, s. vidojković, d. ćirić, d. marinković artificial baits have evolved in many directions, and now there are hundreds of bait types. all of them behave differently in the water and target different species of fish. one group of artificial baits is well-known for its advantage represented by its complicated deflection in the water, as the preeminent action in luring the fish. the baits from this group are known as wobblers or crankbait [2]. during the initial diving, a wobbler wiggles in an irregular, oscillatory manner. when it reaches a quasi-equilibrium condition, it ceases to dive. afterwards, the wobbler travels through the water predictably and oscillatorily. such behavior is a result of a complex equilibrium of the forces that “surround” the wobbler [3]. this wobbler-water interaction is a typical example of fluid-structure interaction (fsi), where both the solid bodies and the fluids move. govardhan et al. [4] modeled the swimming of a fish as fsi of the elastic body in the water. their results show the disturbances of the water due to the kármán vortex street effect. they used the dual-numeric equation solving methodology for full newton–raphson. recently, luo et al. [5] presented the effectiveness of a caudal peduncle-fin on a robotic fish model. they investigated it numerically using a fully coupled fsi, finite volume (fv) approach. a comprehensive review of analytical and numerical fsi models applicable to the double-moving fsi is present in the works of surana et al. [6], chen and zha [7] and chakrabarti [8]. several of the previously mentioned authors reported the limitations of commercial software as the main reason for the marginal application of numerical fsi. the software limitations were reflected in slow convergence, long calculation times, modeling contacts, mapping and meshing difficulties, and the number of executed iterations. 2. real wobbler and environment the fishing wobbler looks like a predatory fish’s common prey – an insect, frog, small mammal, bird, squid, or fish. therefore, the proper shape of the fishing wobbler is of importance for fishing success. also, the fishing lure is surfaced and colored in such a manner as to get fish even more attracted. usually, it has big eyes, almost always shiny and intense colors for major details, the bottom part of the wobbler is brighter than the top of the lure, etc. wobblers are hollow with embedded noise-making parts inside the body (e.g. small steel balls), to attract the fish even more. fig. 1 typical fishing wobbler: a) real model, b) detailed section view [9] a typical small fish-like wobbler has a hollow body made of two symmetric, plastic or wooden pieces (left and right). the pieces are glued to each other in order to create a body. if necessary, the body is balanced with a certain amount of the contingency mass inserted at numerical simulation of fluid-structure interaction between fishing wobbler and water 667 prescribed locations. a steel wireframe, embedded into the body from the nose to the tail, reinforces the wobbler. the wobbler has a transparent diving lip at the nose of the lure. it is of a prescribed size, shape, position, and inclination concerning the wobbler’s mass and shape. usually, two treble hooks are attached to the frame at the tail and middle-belly of the wobbler. the small steel helical-shaped rings connect the hooks and the wireframe. the fishing line is tied to the swivel that has to prevent torsion of the fishing line. the swivel is, using a steel ring, mounted on the diving lip or wobbler’s body [fig. 1]. afterwards, the prepared fishing line – wobbler fish system is used for fishing in the stagnant (standing) or flowing water (fig. 2; fig. 4, a and b). the goal is to enforce quasioscillatory movement of the wobbler in the water surroundings, in situations where the wobbler is not being pulled (vw=0) and a water stream (vf0) exists. the other two cases are when the wobbler is trolled (vw0) in the standing water (vf=0), or the wobbler is trolled/winded (vw0) in the flowing water (vf0). fig. 2 the wobbler in the river the interaction of the wobbler and the water generates forces that, combined with the gravity, hydrostatic pressure and mechanical force delivered by the fishing line, form an active force couple that moves the wobbler left-right and up-down in the water [3]. this exchange of the mutual influences of the water and the wobbler is fsi. 3. numerical model the first step in numerical modeling is to establish a computer-aided geometrical model (cagm). for the shape of the wobbler, the small fish named “belica” (sunbleak or leucaspius delineates in latin) is adopted. the creating of cagm combines the volumetric imaging methods (vim) on leucaspius delineates and assembling the other, standard, geometrical bodies (fig. 3). as an assembly, they represent a solid (continuum) domain. 668 m. mijajlović, s. vidojković, d. ćirić, d. marinković fig. 3 the geometrical model of the wobbler and body interactions the materials and properties of the bodies from the solid domain are given in table 1. table 1 parts, used materials, mechanical properties of materials part/body…is made of… frame, hooks, rings, etc. fishing line wobbler property / name steel nylon pa 6.6 density (kg/m3) 7850 1130 1150 poisson’s ratio (-) 0.3 0.4 0.4 young’s modulus (mpa) 200000 2000 3500 bulk modulus (gpa) 166.67 3.33 5.83 shear stress (gpa) 76.923 7.143 1.250 yield strength (mpa) 250 70 850 ultimate strength (mpa) 460 the fluid (continuum) domain (water) is modeled as a parallelepipedic environment surrounding the wobbler and accessories. both domains are modeled and discretized (meshed). the solid continuum is firstly simplified: the slim bodies (fishing line, treble hooks, rings, swivel) are modeled as 1d elements, while the massive bodies (wobbler and steel ball) are modeled as 3d elements. the usage of 1d elements in the structure modeling is not a novelty, but there are different opinions about its implementation in fsi. gasche et al. [10] presented the fsi of a suction valve and a comparison of the 1d model to the 3d model regarding the computational time. they also observed a calculation time decrease of 15% in the model using 1d elements. finally, they found the lack of stresses in line bodies (except axial stresses) [10]. 3.1. model discretization the wobbler is modeled/discretized with 3d-high order tet10 elements, while the fishing line and other accessories are modeled with beam elements. the mesh of the structural domain is stationary. the fishing line has a rounded cross-section of 0.25 mm in diameter, and it is 200 mm long. the water domain is dominantly discretized with linear 3d tet4 elements while the water near the domain’s boundary surfaces is discretized using thin prism-based elements (boundary layers, necessary to capture flow separation zones, velocity changes, etc.). numerical simulation of fluid-structure interaction between fishing wobbler and water 669 3.2. structural model and constrains all bodies from the solid domain have some mutual interactions: the steel ball occasionally hits the inner wall of the wobbler, the hooks interact with the rings, and sometimes they hit the outer wall of the wobbler; the fishing line is constrained at the free side while the other side is tied to the swivel, etc. these interactions are modeled as bonds, frictional contacts or joints with a certain degree of freedom restrained. with all interactions involved, the numerical model of the wobbler and accessories is very complex and difficult for solving. it shows extreme nonlinear behavior that delivers convergence issues in simulation. therefore, several simplifications are performed: the steel ball is neglected, and the 1d hook models are connected with the wobbler over one node (a 1d element’s node is merged with a 3d element’s node). the particular emphasis is set on modeling the interaction of the fishing line and the wobbler. the real model joint between the fishing line and the fishing lure is over two metal rings where one ring is fixed to the fishing lure, and the fishing line ties the other ring. the ringto-ring contact (fig. 4, c) allows sliding that the results in the weak restraining of the fishing lure: the fishing line does not enable the wobbler to flow away, but the wobbler can oscillate around its meta-stable position. oscillation is possible around the main axis of the ring tied to the lure (axis 1), around the main axis of the ring tied to the line (axis 3), and only partially possible around the coplanar axis (axis 2). the relative translation of the rings is limited/minor due to the closed geometry of the rings – the fishing lure and the fishing line stay connected. this relative positioning of the rings and the weak restraining of the fishing lure is one of the main reasons why the fishing wobbler can be guided through the water, leaving the ability to move like a wounded fish. such a joining is quite difficult to be numerically modeled: one has to model a fixed joint between the fishing line and the ring, then a frictional contact/joint between (with specially constrained degrees of freedom) the two fig. 4 the wobbler, water and accessories a) flowing water, winded wobbler, b) flowing water, constrained fishing line, c) fishing line to wobbler joint, d) universal joint 670 m. mijajlović, s. vidojković, d. ćirić, d. marinković rings and a fixed joint between the second ring and the fishing lure. such a configuration requires a very fine 3d mesh (or a combination of 1d and 3d), an extremely small time step (needed for convergence), and a powerful computer system. for those reasons, this configuration is simplified as the universal joint (fig. 4, d) – the rings are neglected, the fishing line is modeled as a 1d structure, connected to the fishing lure as a point-to-surface universal joint. the local coordinate system is set to the contact point where the x-axis and the z-axis are tangent to the surface of the fishing lure, and the y-axis is normal. this universal joint restrains deformation/translation in all directions (x, y, z) and rotation in the y-direction, while rotation in the x-direction and the y-direction is free. the universal joint is a quite suitable representation of the real model: (x, y, z) directions are compliant to the (1, 2, 3) directions. the missing constraint is a partial rotation around the real model’s (2) axis, which is fully constrained around the y-axis in the model. as a weakness of the model, it is neglected. the only real constraining is achieved over the pulling point of the fishing line (a, fig. 4, a and b). the point is pulling the fishing line and lure through the water at a certain speed, and the reactive force is creating equilibrium with all other forces that are loading the fishing lure. the real model concerns pulling the wobbler in one direction, while the water is flowing in the opposite direction. the model is simplified – the fishing line is not pulled (fig. 4, b). pulling point a in the numerical model is fixed (no rotation, no translation) but the flow rate of the water is increased for the value of the neglected pulling speed. such behavior results in a smaller water continuum (faster calculation) and a more stable numerical model (the model with one or more fixation points becomes less sensitive to the rigid body motion). 3.3. fluid model and constraints the central part of the water domain follows the shape of the wobbler, and it ends at the boundary surfaces (bottom, riverside, water inlet and outlet, and air-interface). to effectively capture fluid-based phenomena near the solid body, the elements that represent fluid, near the wobbler, have to be small and of good quality. for the fluid solver, the interface with the solid domain is a rough, movable, coupled wall. the bottom and riverside are static, rough walls. all the remaining surfaces are pressure and velocity based boundaries, without surface roughness (no-slip walls) (fig. 2). the water enters the water domain through the inlet surface, and leaves it from the outlet surface, while the free surface is a constant pressure-based wall. this representation of the free surface is an imprecise simplification of the open channel flow (density 998 kg/m3, total pressure 1.01325 mpa, at 0 m altitude and temperature of 20c). the maximal depth/height of the water domain is 250 mm, and the nominal water-flow through the domain is 0.4 m/s. the surface roughness at the walls is modeled using the equivalent sand-grain roughness [11, 12]. to accurately capture the fluid separation near the boundaries, the mesh generator creates prism elements as layers near the boundary walls. the size and the number of layers are selected to provide the y+ wall distance for the predicted high reynolds y+ treatment [13, 14]. the fluid flow numerical model is a transient, pressure-based model, without gravity. the contact stress is defined as constant surface tension stress. the turbulence in the fluid is modeled using the realizable k- model (2 transport equations). the model uses scalable numerical simulation of fluid-structure interaction between fishing wobbler and water 671 wall functions (it benefits from the prism elements) for the near-wall treatment and estimation of fluid velocities in the border layers. the solution is obtained using a pressure-velocity coupling scheme, with a secondorder upwind discretization scheme for solving reynolds average navier-stokes (rans) equations. the coupled “presto!” scheme defines the pressure estimation in the fluid domain. the compressed volume fraction estimation scheme with the transient first-order implicit formulation is active, as well. 4. fsi modeling the solid domain and the fluid domain have a strong, mutual, two-way interaction: the fishing lure (solid domain) influences the water (fluid domain) by pure mechanical deformation of the fluid continuum. the fluid continuum – water changes some of its parameters (e.g. pressure), which results in the appearance of the forces that act on the fishing lure. the forces tend to reach equilibrium, and as a result of this tendency, the fishing lure moves. this equilibrium is the basis of the two-way fluid-structure interaction. numerically, fsi is a set of calculations within the discretized domains in discretized time intervals (steps). initially, a set of equations from one of the domains is calculated (either structural or fluid) for the initial time (t=0 s), the results are transferred to the other set of equations (the ones of the domain that is not calculated), and then the equations are solved (fig. 5). the fsi module checks the convergence, mapping, and precision of the data transfer. afterwards, time is increased for the value of the time step (t+t, t=0.001 s), and the calculation procedure is repeated. this procedure is repeated until the end of the prescribed time (tmax=10 s). selecting the proper time step is a complicated task. one approach is to choose it in a manner of preserving the quality of the generated mesh [15]. in the presented case, the time step is selected intuitively. fig. 5 fsi schematic 672 m. mijajlović, s. vidojković, d. ćirić, d. marinković 5. results and discussion the movement of the wobbler in the water appears to be oscillatory and complex. the wobbler shows a tendency to dive and slightly emerge towards the water surface, while the left-right twisting pushes the wobbler downward. for the understanding of the wobbler’s wiggling movement, eight characteristic points on the wobbler are defined, and each of them is defined with the hodograph (fig. 6). fig. 6 the hodograph of the characteristic points (iso view) at the initial time, the wobbler starts to dive until it “mounts” the equilibrium location. it does not show any oscillatory movement before that. afterwards, it starts to oscillate regularly to the z-direction (fig x), deflecting up to 30 mm (points 7 and 8 in fig. 6). fig. 7 the hodograph of the characteristic points (xz plane view) the majority of the points (points 4 to 8, fig. 6) create almost twisted hodographs (shaped like the number eight), while the rest of the points have different hodograph patterns. numerical simulation of fluid-structure interaction between fishing wobbler and water 673 the wobbler’s body suffers very low stresses (fig. 8). such behavior is expected since the wobbler is not constrained ordinarily. when the forces emerge on the wobbler’s body, it bends/deflects slightly and evades the loading. regardless of this fact, the simulation shows that the extreme values of stresses appear at the roots of the wobbler fins and the universal joint. fig. 8 the von mises stress in the wobbler the water flow shows a typical eddy behavior around the swimming wobbler (figs. 9 and 10). it reaches the maximal flow of 0.539 m/s in a vortex near the wobbler’s left side, at the time moment of 4.8 s. fig. 9 the velocity streamline 674 m. mijajlović, s. vidojković, d. ćirić, d. marinković fig. 10 the velocity isosurface of 0.4 m/s the velocity isosurface (set on the nominal value of 0.4 m/s, fig. 10) shows that the diving lip produces a significant water-flow disturbance effect. in this manner, it is the major component of the wobbler that defines the wobbler’s deflection. it seems that it should be smaller and less edgy than it is in this configuration (fig. 10). fig. 11 the absolute pressure contour plot in plane x=20 mm numerical simulation of fluid-structure interaction between fishing wobbler and water 675 the contour plot of the water’s absolute pressure shows a vortex effect in the lateral direction of the wobbler (fig. 11). there is no excessive increase or decrease of the pressure around the wobbler’s body (max 0.5%). however, this slight change of pressure causes the significant deflection of the wobbler. 6. conclusions the preparations of the numerical model, initial simulations, and final results showed several different contradictory facts about the modeling and simulation of such an fsi case. initially, the numerical modeling of the structure and body interactions in the structure appeared to be a challenging and significant convergence issue. due to them, the time step had to be drastically decreased. it might be useful to neglect many of them or somehow to “linearize” them. modeling and preserving the fish fins in simulation influences the pattern of the wobbler’s deflection in the water. visually, the wobbler has more fish-like topography, but it is questionable how much the preserved fins affect the fishing success of the wobbler. furthermore, as thin bodies, they can easily get broken in real fishing. neglecting gravity and simulating the air pressure as constant (to avoid calculating the time-demanding open-channel flow and volume of the fluid approach), affects the precision of the results. however, if only a global picture of the interaction is essential, they can be neglected. the boundary layer modeling (for capturing the flow-separation effect near the walls) should not be neglected. moreover, it has to be modeled very carefully. references 1. new world encyclopedia, 2019, fishing lure, https://www.newworldencyclopedia.org/entry/fishing_lure (accessed 09 october 2019). 2. fishingnoob, 2019, types of fishing lures, http://fishingnoob.com/68/types-of-fishing-lures/ (accessed 09 october 2019). 3. mijajlović, m., ćirić, d., 2018, two way coupled fluid-structure interaction analysis of the grasshopper fishing lure’s movement in the water stream, proc. the 4th international conference mechanical engineering in xxi century masing 2018, niš, serbia. 4. govardhan, d., ramulu, p.j., prasad, p.v.s.r., anbusagar, n.r.r., 2018. hydrodynamics of a fish using fluid structure interaction, materials today: proceedings, 5, 13, part 3, pp. 27205-27212. 5. luo, y., xiao, q., shi, g., yuan, z., wen, l., 2019. a fluid-structure interaction study on a passively deformed fish fin, proc. asme 2019, 38th international conference on ocean, offshore and arctic engineering, omae 2019, glasgow, united kingdom, 9/06/19 14/06/19. 6. surana, k.s., blackwell, b., powell, m., reddy, j.n., 2014. mathematical models for fluid-solid interaction and their numerical solutions, journal of fluids and structures, 50, pp. 184-216. 7. chen, x.y., zha, g.c., 2005, fully coupled fluid–structural interactions using an efficient high resolution upwind scheme, journal of fluids and structures, 20, 8, pp. 1105-1125. 8. chakrabarti, s.k., 2005, numerical models in fluid-structure interaction, wit press, offshore structure analysis inc., usa. 9. ujka vobleri, 2019, bd4, http://ujkavobleri.com/ujka/model-bd4.html#video, (accessed 09 october 2019). 10. gasche, j.l., dias, a.d.s.l., bueno, d.d., lacerda, j.f., 2016, numerical simulation of a suction valve: comparison between a 3d complete model and a 1d model, proc. 23rd international compressor engineering conference at purdue, pp. 2-8. 11. tauviqirrahman, m., ichsan, b.c., jamari, m., 2019, influence of roughness on the behavior of threedimensional journal bearing based on fluid-structure interaction approach, jour. mech sci technol, 33, pp. 4783. 676 m. mijajlović, s. vidojković, d. ćirić, d. marinković 12. adams, t., grant, c., watson, h., 2012, a simple algorithm to relate measured surface roughness to equivalent sand-grain roughness, international journal of mechanical engineering and mechatronics, 1(1), pp. 66-71. 13. terziev, m., tezdogan, t., oguz, e., gourlay, t., demirel, y.k., incecik, a., 2018, numerical investigation of the behaviour and performance of ships advancing through restricted shallow waters, journal of fluids and structures, 76, pp. 185-215. 14. ansys support, 14.5 release, 2012. lecture 7: turbulence. introduction to ansys fluent, ansys. 15. konyukhov, a., izi, r., 2015, introduction to computational contact mechanics: a geometrical approach, wiley series in computational mechanics, john wiley & sons, chichester, uk. 12692 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 385 397 https://doi.org/10.22190/fume240414028w © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper influence of tangential sliding on the contact area of a macroscopic adhesive contact josefine wilhayn, iakov a. lyashenko, qiang li, valentin l. popov department of system dynamics and friction physics, technische universität berlin, berlin, germany orcid ids: josefine wilhayn https://orcid.org/0009-0007-2635-2302 iakov a. lyashenko https://orcid.org/0000-0001-7511-3163 qiang li https://orcid.org/0000-0001-7458-9450 valentin l. popov https://orcid.org/0000-0003-0506-3804 abstract. influence of tangential force on the adhesive contact, in particular on the adhesive strength and the contact area have been a subject of investigation and discussions for many years. depending on the system under consideration, both increase and decrease of adhesion strength due to shearing of an adhesive contact have been reported. the same is valid for change of the contact area. while it is generally accepted that the contact area decreases when applying tangential traction, also opposite behaviour is possible. in the present paper we study theoretically and experimentally the conditions under which the contact area may increase or decrease due to shear. key words: adhesion, contact area, shear, tangential loading 1. introduction influence of tangential traction and shear on adhesive contacts has been subject of interest both from the viewpoint of fundamental contact mechanics [1][2] and biological applications as gecko feet [3]. one of the problems in the discussion of adhesive contacts under shear is that it is assumed that the term “adhesive contact” is well defined. in reality, the concept of adhesive contact can hide systems of different physical nature. adhesive contact can mean a “glued” contact, which may be destroyed by both normal and tangential movement of contacting bodies. in this case, the combined energy release rate can be defined as it was done in [1] or [4]. the same approach based on the energy release rate is used in the paper [5] for analyzing contact area between a glass plate and pdms bar where an intermediate situation is realized (both sticking and sliding are possible). the adhesive contact described in the classical theory by johnson, kendall and roberts (jkr) [6] is, on the contrary, of completely different nature. one of assumptions of the jkr theory is the absence of friction. according to jkr, shearing of an received: april 14, 2024 / accepted may 17, 2024 corresponding author: valentin l. popov technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: v.popov@tu-berlin.de https://orcid.org/0009-0007-2635-2302 https://orcid.org/0000-0001-7511-3163 https://orcid.org/0000-0001-7458-9450 https://orcid.org/0000-0003-0506-3804 386 j. wilhayn, i.a. lyashenko, q. li, v.l. popov adhesive contact does not lead to appearance of tangential force and does not influence any properties of the adhesive contact. this contradicts observations of real adhesive contacts. however, the jkr theory illustrates that the presence of any kind of friction is by no means a necessary, intrinsic property of adhesive contacts. at least in the model, one can imagine contacts without friction. this is e.g. the case of two absolutely smooth surfaces interacting by van der waals forces as described in the lifshitz’s theory [7]. in the present paper, we will discuss a model very closely related to the (frictionless) jkr model but with consideration of surface roughness or surface profiling. we will show that already this model exhibits all basic properties of adhesive contacts which are observed experimentally. in particular, it shows adhesion hysteresis during all stages of indenting and detaching. the existence of the adhesive hysteresis which was observed experimentally and described theoretically in different context in [8-12], means that the state of adhesive contact is not unique but does depend on the prehistory of loading. we will show that this prehistory influences essentially the interplay of normal and tangential loadings. 2. adhesion hysteresis in frictionless profiled contact the simplest model of an adhesive contact of bodies with rough (or profiled) surfaces was considered by fuller and tabor [13]. we will use this model in an even more simplified form described in [14]. instead of a continuous smooth surface, a set of individual asperities is considered. the main physical difference from the case of smooth surfaces is that the surface profile prevents propagation of the adhesive contact (which is essentially the griffith’s crack [15]). we will be interested in the macroscopic contacts between a flat and a parabolically shaped bodies, so the individual independent asperities will be placed on the same level. for illustration, we consider asperities as springs of equal stiffness (fig. 1), however, the true three-dimensional distribution of stresses is considered below in numerical simulation. the model presented in fig. 1 exhibits pronounced adhesion hysteresis. the starting nocontact position of indenter is shown in fig. 1a. when it comes in touch with the substrate and indents further (fig. 1b), only the springs (summits, asperities) come into contact which correspond to the geometrical intersection of the profiles of contacting bodies. the adhesion does not play any role at the stage of indentation, so the behavior is identical with that of a nonadhesive contact. if we would consider elastic interrelation of asperities, we would become pure hertzian behavior [16]. at the stage of pull-off (fig. 1c), the springs which came into contact during indentation will first follow the indenter until they do not reach some critical elongation. thus, the system behaves as an adhesive contact. if as detachment criterion the heß’s criterion is used, then the resulting force-displacement curve will coincide with that of the jkr theory [17]. our experiments described in [11] exhibit exactly such behavior: practically pure hertzian indentation (with only very minor jump into contact), and strong jkr-adhesion during detaching. note that this hysteresis is not connected with any frictional forces in the interface, as we consider here frictionless adhesive contact, but solely with preventing interfacial crack propagation due to surface profile (or roughness). to retain the model as simple as possible, we will neglect in the following the frictional force in the interface. we thus assume that any individual spring can slide frictionless along the surface of the counter body (as jkr-contact does). influence of tangential sliding on the contact area of a macroscopic adhesive contact 387 fig. 1 illustration of indentation, pull-off and tangential sliding of an indenter on an elastic substrate. (a)-(c) show indentation and pull-off, (d) sliding after indentation, (e) sliding after indentation and lifting, (f) similar to (e) but lifting to a negative depth. we start with a detailed discussion of the normal contact problem. if a rigid indenter is pressed into an elastomer to some depth, and then the movement is reversed, then there will be a phase when the contact area does not change (until the edge springs achieve a critical elongation). this leads to appearance of a linear part of the force-displacement relation. after arriving at this critical value, the force follows the curve for an adhesive contact. let us now consider two cases: i. the indenter is monotonically pressed into the elastic counter body, stopped, and then moved tangentially at the achieved indentation (fig. 1d). note that each individual spring is assumed to have a frictionless contact with the indenter. this means, that the springs freely can slide along the indenter. if we move the indenter by one spacing between springs, the last spring on the leading edge will be pressed down, and the indenter will come into contact with the next spring. this means that the leading edge will move in the direction of the movement of indenter with the same velocity. the spring at the trailing edge, on the contrary, will move upwards (by sliding along the profile of the indenter), and thus will remain at the same place. this means that the contact area will become larger at the leading half of the contact area and remain at place at the training part, thus the contact area will increase, and the contact area becomes elongated in the direction of movement. 388 j. wilhayn, i.a. lyashenko, q. li, v.l. popov as soon as the edge spring at the trailing edge achieves the critical elongation, the trailing boundary starts moving together with the indenter and the contact area remains constant. ii. the indenter is pressed into the elastic counter body, then lifted to the end of the plateau of the contact area, and only then moved tangentially (fig. 1e). lifting to the end of the plateau means that now the edge springs are already in the critical state. if we now move the indenter to the right, the springs at the leading edge will first be pressed down, and the boundary on the leading edge will remain at place. on the trailing edge, on the contrary, the springs were already in the critical state. they will slide up the profile and detach immediately. this means, that the trailing edge will move forward (in the same direction as indenter and with the same velocity). thus, the contact area will decrease and will become elongated in the direction perpendicular to the direction of sliding. iii. the indenter is pressed into the elastic counter body, then lifted to the zero or negative indentation depth, and then moved tangentially (fig. 1f). note that if the indenter is pulled off to the zero or moderate negative indentation depth in the normal direction, the adhesive contact is still not destroyed, and the edge springs are in the critical state, so that the initial state is very similar to that of the case ii. tangential movement of the indenter will lead to pressing down the springs on the leading edge and detaching of springs on the trailing edge. this means that the leading edge will remain at place. however, now no new springs will come into contact because of the negative indentation depth. at the same time, the trailing edge will move forward together with the indenter. the contact area will shrink until it disappears completely. experiments described in [11] confirm qualitatively the existence of the above described three different cases. however, the scenarios described above present only a very rough zero-order “impressionist” picture of how the prehistory of loading influences the reaction of the system to shear. the model used has a number of assumptions which are not valid for real systems and will influence the details of happening. this is in particular the assumption of frictionless contact for each individual asperity. experiments show that much better assumption is existence of a constant shear stress which has to be overcome to move an asperity [11]. we further assumed that the detachment condition of an individual spring is constant. in reality it is a function of the whole contact configuration. finally, non-linear effects may come into play making the behavior very complicated [18]. 3. experiment in our experiments, a spherical indenter with a radius r = 100 mm was pressed into a layer of elastomer with a thickness h = 5 mm to a depth of dmax = 0.3 mm, then either lifted to different indentation depth until complete disappearance of contact, and subsequently shifted in the tangential direction. a photo of the setup is shown in fig. 2. the detailed description of the experimental setup is provided in the previous work [19]. the indenter was moved at a velocity of v =3 μm/s throughout all phases of indentation, pull-off and shearing. at such low velocity, the contact can be considered as quasi-static, and viscosity can be neglected. we conducted experiments with two types of elastomers – a softer tanac crg n0505, and a harder tanac crg n3005. influence of tangential sliding on the contact area of a macroscopic adhesive contact 389 fig. 2 view of the experiment setup: 1 − steel indenter with a radius of r = 100 mm; 2 − a sheet of elastomer placed on a solid glass plate; 3 − an omnidirectional led lighting; 4 − force sensor for measuring all three components of the contact force. 3.1. indentation in softer material tanac crg n0505 fig. 3 illustrates the dependencies of the normal force fn and the contact area a on the indentation depth d during the indentation into the softer material crg n0505. fig. 3 the dependencies of the normal force fn (a) and the contact area a (b) on the indentation depth d during the indentation of an indenter with a radius r = 100 mm into a layer of elastomer tanac crg n0505 with a thickness h = 5 mm. the figure shows the results of three consecutive cycles of indentation/detachment. one can see that there is a small jump-like increase in contact area at the point of initial contact, after which the area a increases linearly with further indentation. when the direction of the indenter motion is changed towards withdrawal, the contact area remains constant for some period of time before it starts decreasing. while the contact area remains 390 j. wilhayn, i.a. lyashenko, q. li, v.l. popov constant, a constant contact stiffness k = ∂fn/∂d is realized, hence immediately after changing the direction of motion, a linear segment of the force dependence fn(d) is observed. fig. 3 demonstrates the results of three consecutive experiments, with curves on both panels of the figure overlapping, indicating the repeatability of the experiments. therefore, in the experiments on tangential contact, only one experiment will be conducted for each loading scenario. the results of a series of experiments on tangential loading of the contact are shown in fig. 4. here, in all experiments, the indenter was first immersed into the elastomer to the maximum depth d = dmax = 0.3 mm, then lifted to a smaller value d = d0, and subsequently shifted tangentially by a distance x = 1 mm at a fixed value of indentation depth d = d0. vertical lines on all panels of the figure denote the moments of the start of tangential displacement of the indenter in all 4 experiments with different values of d0. in the first experiment, the tangential displacement of the indenter was carried out immediately after reaching the depth dmax, i.e., in this experiment d0 = dmax. this corresponds to the case i of section 2. as seen in fig. 4a, the normal force remains close to that corresponding to the maximum indentation depth after the start of tangential displacement, while the tangential force begins to increase. fig. 4b shows the coordinates of the leading and trailing edges of the contact zone. additionally, for clarity, dashed lines close to horizontal are shown on fig. 4b, representing straight lines defined by equations 0.18coordinate b t= + (1) here, the coefficient 0.18 corresponds to the velocity of the indenter motion v = 3 μm/s. if the edges of the contact were moving exactly at the speed of the indenter after the start of its tangential motion, the experimental dependencies shown in fig. 4b would coincide with these dashed lines. however, upon closer examination of the experimental dependencies, it can be observed that immediately after the start of tangential motion, only the trailing edge of the contact moves at the speed of the indenter, which occurs in the time interval t = 1.65 – 3.1 min. the leading edge of the contact begins to move at a speed slightly exceeding the speed of the indenter. this indicates the spreading of the contact at the leading edge after the start of tangential motion, as well as the fact that the contact boundary moves together with the indenter at the trailing edge. however, the situation changes at times t > 3.1 min, when contact failure begins at the trailing edge, while at the leading edge, its propagation continues as before. the basic features of this behavior: a more rapid propagation of the leading edge of the contact and increase of the contact area are the same as predicted by the simple model (section 2, case i). different from prediction, the rear edge does first slowly move together with indenter (which might relate to final interfacial friction contrary to the model). as predicted, after achieving some critical position, it accelerates but moves faster than the indenter, which may be related with the assumption of independent springs (which is not valid in the real system). the main conclusion of theory remains valid: if tangential loading occurs at the position of maximum indentation, then the contact area increases due to shear. influence of tangential sliding on the contact area of a macroscopic adhesive contact 391 fig. 4 the dependencies of the normal fn and tangential fx contact forces (a), coordinates of the leading and trailing contact edges (b), and the contact area a (c) on the experiment time t during indentation into the softer material crg n0505. behavior after some initial lift also qualitatively corresponds to that described in section 2. in particular, at the zero indentation depth the contact finally disappears completely. to provide a more detailed understanding of the contact spreading characteristics after the start of the indenter motion, we additionally present fig. 5. this figure shows contact photographs corresponding to six consecutive states, which are shown on all panels in fig. 392 j. wilhayn, i.a. lyashenko, q. li, v.l. popov 4. these states are separated by equal time intervals. in addition to the contact photographs, the left panel of fig. 5 shows profiles of the contact boundary, which allow easy tracking of the evolution of the contact boundary spreading. the results confirm the conclusions made above. specifically, the first four contact configurations demonstrate the spreading of the contact at its leading edge, while the contact boundary at the trailing edge remains almost stationary. however, the subsequent configurations 5 and 6 show a faster displacement of the contact boundary at the trailing edge, while the spreading characteristics of the contact remain unchanged at the leading edge. fig. 5 right panel – photographs of the contact area corresponding to points 1–6 in fig. 4. left panel – contact boundaries corresponding to the photographs in the right panel. experiment on indentation into the softer material crg n0505 is shown. let’s go back to fig. 4 and look at forces. if indenter is pressed into counter body and then immediately moved tangentially, then, the contact configuration on the leading edge will not change essentially, but the springs on the rear edge will slide up, so that attractive part of the contact force will increase. this means decreasing of the total normal force. this is exactly what is observed in experiment (fig. 4a, red curve corresponding to indentation 0.3 mm). if, however, the indenter is pressed into the counter body and then lifted at last to the end of the plateau of the contact area (blue, brawn and green curves in fig. 4a), then the contact situation on the rear end will not change while at the front edge the springs will glide downwards and so the normal force will increase. this behavior is observed in experiment. 3.2. indentation in hard material tanac crg n3005 we would like to stress that this behavior is universal. a set of experiments conducted with rubber tanac crg n3005 which is 10 times stiffer than the previous one show very similar results with the main difference that the forces are scaled by the factor of 10 (fig. 6). interestingly, this scaling is equal both to normal and tangential forces indicating that the tangential force strongly correlates with the adhesion force, as it should be accordingly to the model described in section 2. influence of tangential sliding on the contact area of a macroscopic adhesive contact 393 fig. 6 dependences of normal fn and tangential fx forces on the time of experiment t during indentation into the softer material crg n3005. the lower curves corresponding to the depth d0 = 0.0 mm are not labelled. 4. numerical simulation numerical simulation was carried out using the boundary element method for jkrtype adhesive contacts [20, 21]. we consider a rough sphere in contact with an elastic rough substrate. the geometry of the rough sphere is modeled by a parabolic profile superposed with a two-dimensional waviness. the surface of elastic substrate is also a two-dimensional waviness with the same amplitude δ and wave length λ, but rotated in plane by an angle π/8. the amplitude and wavelength are much smaller than the sphere radius, δ/λ =0.028, and λ/r = 0.005. the following results are normalized by the jkr-solution for smooth sphere [6] jkr 3 2 f r= , 1/3 2 2 jkr * 3 64 r d e    =     , 1/3 2 jkr * 9 8 r a e   =     . (2) here γ is the work of adhesion per unit area, and e* is the effective elastic modulus. in the following we first show the simulation of a whole indentation process including both indenting and pull-off, then the sliding contact corresponding to cases i and ii in section 2. 4.1. simulation of indentation test fig. 7 shows simulation of an indentation test: indenter is brought into contact (position 1) and pressed to the position 3 (with indentation depth d=6|djkr|), then it is lifted until it is completely detached. due to roughness, the curves for indenting and pull-off do not coincide with the analytical jkr-solution for smooth sphere (gray lines), and the adhesion hysteresis can be clearly observed. note that after indenting to position 3, when the movement direction changes from indenting to pull-off, the contact area remains unchanged up to position 4, well seen in contact area-normal force relation in fig. 7b and from contact areas shown in fig. 7c. this behavior is also described in experiment in fig. 3. the mean contact radius in following figures is defined as a=(acon/π)1/2. 394 j. wilhayn, i.a. lyashenko, q. li, v.l. popov fig. 7 simulation of indenting and pull-off of a rigid rough sphere in contact with an elastic rough substrate. dependence of the normal force on indentation depth (a) and mean contact radius (b). figure (c) shows contact area changing during the whole process. 4.2. simulation of sliding based on the normal contact in fig. 7, we consider two sliding cases: (1) the indenter is pressed into position 3, then it moves tangentially; (2) the indenter is pressed to the position 3, lifted to the position 4 (the contact area remains unchanged during this stage), then the indenter slides tangentially. we focus on the change in contact area during the sliding process. indentation depth keeps constant during sliding. fig. 8a presents the dependence of the mean contact radius on the sliding distance. the contact area increases during the sliding. the first image in fig.8b is the contact area when the indenter is pressed to position 3. the blue color in second and third images represents the increased area in comparison with state 3, and the orange color represents the reduced area. one can clearly see the enlargement of contact region after sliding. it occurs at the rear edge as explained in section 2, case i. fig.8c and fig.8d show the normal force decreasing during the sliding and the dependence of tangential force on the sliding distance. fig. 9 shows the case of sliding after indentation and pull-off to position 4. the first image in fig. 9b shows the contact area when the indenter is pulled off to position 4. from both curve in fig. 9a and images of contact area, it is seen that the contact area is reduced during sliding in this case. these results agree with the prediction in section 2. influence of tangential sliding on the contact area of a macroscopic adhesive contact 395 fig. 8 sliding after indenting to position “3” shown in fig.7. (a) mean contact radius as function of sliding distance. (b) images of contact area. the first image corresponds to the point 3 in fig.7c. images 3-a and 3-b are contact areas when sliding. orange color represents the area lost compared to state “3” and blue color represents the increased area. (c) dependence of normal force and (d) tangential force on sliding distance. fig. 9 sliding after indenting and pull-off to position “4” shown in fig.7. (a) mean contact radius changing during the sliding. (b) images of contact area during sliding. (c) dependence of normal force and (d) tangential force on sliding distance. 396 j. wilhayn, i.a. lyashenko, q. li, v.l. popov 5. conclusions analysis of experimental data on shearing adhesive contacts after pre-loading shows that the reaction of the contact on shear does depend on the preloading history. depending on preloading, shear can lead to increase or decrease of contact area and to decrease or increase of normal contact force. a very rough understanding of all possible situations is provided already by a very simple model based on fuller and tabor model. more detailed understanding can be achieved by considering correct three-dimensional stress distribution (e.g. using bem simulation). numerical simulations show smaller effects of area increase and decrease depending as in experiment. this discrepancy may be caused by relaxation effects. the complete understanding probably needs consideration of material and geometrical non-linearities. acknowledgement: this work has been 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methods for computational contact mechanics, frontiers in mechanical engineering, 6, 61. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 157 170 https://doi.org/10.22190/fume180404015j © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a numerical and experimental analysis of the dynamic stability of hydraulic excavators udc 621 dragoslav janošević, jovan pavlović, vesna jovanović, goran petrović university of niš, faculty of mechanical engineering, serbia abstract. the paper presents the results of a numerical and experimental analysis of the dynamic stability of hydraulic excavators. the analysis has employed the software developed on the basis of a defined general dynamic mathematical model of an excavator grounded in newton-euler equations as well as the measured quantities of the excavator operating state in exploitation conditions. the defined model is used to model the members of the excavator kinematic chain using rigid bodies while the actuators (hydraulic cylinders and hydraulic motors) of the excavator drive mechanisms are modeled with elastically dampened elements. the elastically dampened characteristics of the actuators are defined with regard to the size of the actuator as well as to the compressibility and temperature of the hydraulic oil used in the excavator hydrostatic drive system. to illustrate the analysis, the paper provides the results of the analysis of the dynamic stability of a 16000 kg tracked excavator equipped with a manipulator digging bucket of 0.6 m3 in capacity. key words: hydraulic excavators, dynamic stability, oil temperature, modeling 1. introduction in modern hydraulic excavators one of the most important parameters is the indicator of the excavator’s stable operation. international standards provide the conditions for examination and determination of the static indicators of the excavator stability and hydraulic stability of the excavator drive mechanisms (sae j1097). previous research in the area has been related to the development of mathematical models for the analysis of the static stability of hydraulic excavators [1-4]. however, the studies have shown that hydraulic excavators (and other hydraulic mobile machines) act like dynamic oscillating received april 04, 2018 / accepted may 15, 2018 corresponding author: dragoslav janošević university of niš, faculty of mechanical engineering, department for material handling systems and logistics, a. medvedeva 14, 18000 niš, serbia e-mail: janos@masfak.ni.ac.rs 158 d. janošević, j. pavlović, v. jovanović, g. petrović systems during their cyclic operation [5-7] where actuators (hydraulic motors and hydraulic cylinders) of the excavator drive mechanisms appear as extremely elastically dampened system members in the shape of hydraulic springs. they occur due to the compressibility of oil in the actuator displacement and ducts of the excavator hydrostatic drive system. the dynamic behavior of an excavator is pronounced during the material transfer and unloading operations when disturbances which may cause the dynamic instability of the excavator occur. elastically dampened characteristics of the excavator actuators, therefore, the dynamic stability of the excavator itself, are greatly influenced by the temperature of the hydraulic oil in the excavator drive system. the paper contains an analysis of the influence that the oil temperature in the excavator hydrostatic drive system exerts on the excavator’s dynamic stability. 2. dynamic excavator model when analyzing the dynamic stability of an excavator one observes the physical model of the machine with the kinematic chain of general configuration comprising: support and movement member l1 (fig. 1a), rotating member l2, boom l3, stick l4 and bucket l5. the members of the excavator kinematic chain form the rotating kinematic pairs of the fifth class. the members of the kinematic pairs are joined, directly or indirectly, by hydraulic actuators: hydraulic motor c2 (fig. 1b) of the platform rotation drive and hydraulic cylinders c3, c4, c5 of the manipulator drive which are powered, through the hydraulic distributor 3, by hydraulic pumps 2 driven by the diesel engine 1. with regard to the physical model, a dynamic mathematical model of the excavator (fig. 2) was adopted with the following assumptions: a) the mechanical model of the excavator is a non–conservative system with stationary and ideal links, b) small system oscillations are observed around the stable balance position, c) the excavator support base has elastically dampened properties, d) the members of the excavator kinematic chain are rigid bodies, e) the hydraulic actuators of the drive mechanisms are elastically dampened systems due to the oil compressibility and viscosity, and f) the bulk modulus of the hydraulic oil is constant for a certain oil pressure and temperature. fig. 1 hydraulic excavators: a) physical model, b) hydrostatic drive system a numerical and experimental analysis of the dynamic stability of hydraulic excavators 159 in the mathematical model, a member of excavator kinematic chain li is defined, in its local coordinate system oi xi yi zi, by a set of geometric, kinematic and dynamic parameters (fig. 2): { , , , }i i i i il m s t j (1) where: si – the vector of the centre position of joint oi+1 which links chain member li to next member li–1 (vector magnitude si represents the kinematic length of the member), ti – the vector of the mass centre position of member li, mi – the member mass, ji – the moment of the member inertia. fig. 2 mathematical model of the excavator 160 d. janošević, j. pavlović, v. jovanović, g. petrović in the mathematical model, the parameters of the actuators (hydraulic motors and hydraulic cylinders) of the excavator drive mechanisms are determined by a set of quantities: 1 2{ , , , , , }i i i ip ik ci cic d d c c m n (2) where: di1, di2 – the parameters of actuator sizes (minimal and maximal specific flow of hydraulic motors, i.e. the piston diameter and the piston rod diameter in the hydraulic cylinder), cip – the minimal length of the hydraulic cylinder when the piston rod is fully drawn in, cik – the maximal length of the hydraulic cylinder when the piston rod is fully drawn out, mci – the actuator mass, nci – the number of drive mechanism actuators. the movement of the adopted dynamic mathematical model of the excavator is defined by a set of generalized coordinates (fig. 2):  543210 ,,,,,θ  0 1 2 3 4 5{ , , , , , }       (3) where: θ0 – the vertical movement of the mass centre of the support and movement member, θ1 – the angle of rotation around the main longitudinal central axis of inertia o1y1 of the support and movement member, θ2 – the angle of rotation of the rotating member around the o2z2 axis of the axial bearing, θ3 – the angle of rotation of the manipulator boom around the o3y3 axis of the joint to which the rotating member is connected, θ4 – the angle of rotation of the manipulator stick around the o4y4 axis of the joint at the end of the boom, θ5 – the angle of rotation of the manipulator bucket around the o5y5 axis of the joint at the end of the stick. the generalized coordinates are assumed to be small quantities measured from the position of the system’s stable balance. critical positions of the excavator are analyzed when the manipulator plane is perpendicular to the plane of the tracked support and movement member. the position of the excavator is defined in relation to the immovable (absolute) coordinate system oxyz. coordinate beginning o of the absolute system is in the centre of the support and movement member mass, while the ox axis is directed towards the main transverse central axis of inertia of the same member in the position of the static balance of the entire system. the vector of position rti of the mass centre of member li of the excavator kinematic chain is determined in relation to the absolute coordinate system with the following equation: ioij 1i 1j ojoti ta a     ssr (4) where: s0 = [0 0 zc] t – the vector of deformation of the excavator model support base, i.e. the vector of displacement of the mass centre of the support and movement member in relation to the absolute coordinate system, sj – the vector of the position of joint oj centre in relation to previous joint oj–1, determined in the local coordinate system, ti – the vector of the position of member li mass centre in relation to joint oi centre, aoj – the transitional vector matrix from local coordinate system ojxjyjzj to the absolute coordinate system [7, 8]. a numerical and experimental analysis of the dynamic stability of hydraulic excavators 161 3. differential equations of motion differential equations of motion of the defined mathematical model of the excavator are determined by using lagrange’s equations of the second kind: 0 ee dt d ii p i k              (5) where: ek, ep, φ – the kinetic and potential energy and function of the system dissipation, respectively [9]. 3.1. kinetic energy for the established dynamic model of the excavator, the kinetic energy of the system is expressed using the equation: 2 5y5 2 5t5 2 4y4 2 4t4 2 3y3 2 3t3 2 2z2 2 1y2 2 2t2 2 1y1 2 1t1k jvmjvmjvm jjvmjvme2       (6) where: mi – the mass of an excavator kinematic chain member, (i=1,…,5), vti – the absolute velocities of the mass centers of the excavator kinematic chain members, jiy,jiz – the appropriate main central axial moments of inertia of the masses of the excavator kinematic chain members. differentiating equation (4), in the absolute coordinate system, yields the vector of velocity vti of the mass centers of the excavator kinematic chain members in the following form: ioiioij 1i 1j ojj 1i 1j ojoti aaaa ttsssv        (7) where: ojoiijo a,a,,,  tss – the derivatives of the appropriate vectors of position and transitional matrix systems, where, along the assumptions of small quantities of generalized coordinates θi, the approximate values cosθi ≈1 and sinθi ≈ θi are introduced, thus linearizing the elements of transitional matrices aij. 3.2. potential energy for the established dynamic model of the excavator, the potential energy of the system is determined using the equation: 22 2 2 2 2 1 1 1 1 2 2 2 3 3 32 5 22 2 2 2 3 3 4 4 4 4 4 5 5 5 5 5 1 2 ( ) ( ) ( ) + ( ) ( ) 2 ( ) p o o o o o o o o o o s s s i i ois s s i e k a k k a k k k k k k k k g m z z q q q q q q q q q q q q q q q q q q (8) where: zi, zoi – the current and initial coordinates of the positions of mass centers of the excavator hydraulic chain members in relation to the absolute coordinate system, ko – the stiffness of the elastic support base of the excavator, θo1, θo2 – the static deflections of the 162 d. janošević, j. pavlović, v. jovanović, g. petrović elastic base beneath the support and movement member of the excavator, k3,k4,k5 – the torsional stiffnesses of springs equivalent to the stiffnesses of the elastic hydraulic actuators of drive mechanisms of the manipulator boom, stick and bucket, θ3s,θ4s,θ5s – the angles of static deflections of torsion springs of the manipulator boom, stick and bucket. by neglecting the influence of the generalized coordinate of platform θ2 rotation, as a small quantity, and using the approximate equivalences: 2 sin ; 1 cos sin ; 1 cos 1,2,3 2 i q q q q q q q (9) the elastic action of an actuator with stiffness kci (fig. 2a), in relation to the joint of the drive mechanism which it powers, is substituted by an equivalent action of torsion springs (fig. 2b), whose stiffness ki is determined by the following equation: 2 ck 2, 3, 4, 5i ci ik i i (10) where: ici – the transitional function of the excavator drive mechanism moment [6]. the base (ground) stiffness beneath the support and movement mechanism of the excavator is determined using the equation [9]: 1o rok e a (11) where: ero – the ground reaction modulus, a1 – the footprint surface area of the support and movement member of the excavator. actuator displacements and hydraulic ducts in the drive mechanisms are filled with compressible hydraulic oil of the excavator hydrostatic drive system. due to the hydraulic oil compressibility, the actuators are modeled using springs with an appropriate stiffness kci, determined with the equation [10]: 2 2 1 2 1 1 2 2 2 21 21 22 21 3, 4, 5 -for hydraulic cylinders [ ( ) ] [ ( ) ] = 2 ; -for hydraulic motors i u i u i i i ip i i ik i i ci u a e a e a c c v a c c v k d e v v v q (12) where: ai1, ai2 – the operating surface areas of the hydraulic cylinder on the piston and piston rod end, eu – the elasticity (bulk) modulus of the hydraulic oil, ci,cip,cik – the current, initial and final length of the hydraulic cylinder, d21 – the maximal specific flow (displacement) of the hydraulic motor, vi2 – the displacement of the actuator hydraulic ducts. 3.3. dissipation function for the established dynamic model of the excavator, the potential energy of the system is determined using the equation: 2 2 2 2 2 2 1 1 2 2 3 3 4 4 5 52 ( ) ( )o o o ob a b a b b b bq q q q q q q q (13) a numerical and experimental analysis of the dynamic stability of hydraulic excavators 163 where: bo – the damping coefficient of the excavator support base, bi – the damping coefficient of the hydraulic actuators of drive mechanisms of the platform, boom, stick and bucket, reduced to the joints of the excavator kinematic chain powered by them. the damping coefficient of the excavator support base is determined by the following equation [11]: pooo ek2b  (14) where: ko – the stiffness of the ground, epo – the damping modulus of the ground. the damping coefficient of the hydraulic actuators of drive mechanisms is determined by the following expression: 2 i 2,3,4,5i ci cb i b i (15) where the damping coefficient of hydraulic cylinder bci is defined by the following equation [11]: 2 1 1 2 11 3 3 3,4,5 4 ( )( ) u i i i ci i ii i l d d b i d dd d p h (16) where: ηu – the dynamic viscosity of hydraulic oil, di1 – the piston diameter, li – the piston length, and di – the internal diameter of the hydraulic cylinder. substituting the expression for kinetic (6) and potential (8) energy, and the expression for the dissipation function (13) into lagrange’s equations of the second kind (5), one obtains a system of six homogenous differential equations of the oscillating system: 0  kfm  (17) where: m – the inertial matrix, f – the matrix of resistance forces, k – the stiffness matrix. the equations of motion of the excavator dynamic system are determined using the method of addition of the main forms of oscillation with the next equation uv (18) where: v – modal matrix of eigenvectors, u – vector of normal coordinates [12]. 4. example as an example, the numerical and experimental analysis of dynamic stability was performed on a 16000 kg tracked excavator equipped with a manipulator digging bucket of 0.6 m 3 in capacity. 4.1. numerical analysis using the developed programs, as an example, the analysis was performed on the influence of oil temperature of the drive system on the dynamic stability of a tracked excavator with the mass of 16000 kg, equipped with a manipulator with the bucket capacity 164 d. janošević, j. pavlović, v. jovanović, g. petrović of 0.6 m 3 . the following files are input at the start of the program: the excavator kinematic chain file (with geometric and dynamic parameters of each member), the drive mechanisms file, the support base characteristics, and the characteristics of the oil in the hydrostatic drive system of the excavator. the program output yields the generalized system coordinates in the function of time. the numerical analysis monitored the change in the generalized coordinates (fig. 2):  431oi ,,,   1 3 4, , ,oq q q q (19) the set system conditions and parameters are given in table t1. the position of the excavator was observed during the unloading of the material when, according to the excavator measurements in exploitation conditions, primary movements of the support and movement member and the manipulation boom appeared [11, 13]. table 1 system conditions and parameters initial position coordinates of the support and movement mechanism and the platform θo=0°, θ1=0° ,θ2=0° initial position coordinates of the manipulator kinematic chain θ3o=35°, θ4o =0°, θ5o =-120° bulk modulus of hydraulic oil at 80oc eu=1.4∙109 [n/m2] bulk modulus of hydraulic oil at 90oc eu=1.2∙109 [n/m2] modulus of reaction of the excavator support base ero=5.5∙ 108 [n/m3] damping modulus of the excavator support base epo=0.005 [s] initial boom angle velocity 3 2 oq [rad/s] fig. 3 changes in generalized coordinates θ0 of the support and movement member for the initial position of the manipulator and at different oil temperatures a numerical and experimental analysis of the dynamic stability of hydraulic excavators 165 fig. 4 changes in the generalized coordinates for the initial position of the manipulator and at different oil temperatures: a) θ1 of the support and movement member, b) θ3 of the boom angle, c) θ4 of the stick angle 166 d. janošević, j. pavlović, v. jovanović, g. petrović based on certain matrices and the given initial conditions of movement, and using the developed program, the differential equations were solved and the changes in the generalized coordinates of free oscillations of the excavator were obtained (figs. 3 and 4). diagrams of the vertical θo (fig. 3) and angular θ1 displacement (fig. 4a) of the support and movement member and diagrams of the change in the generalized coordinate of the boom θ3 (fig. 4b) and stick θ4 (fig. 4c) show that the oscillatorydampened movement of the excavator kinematic chain members with different amplitudes and oscillation periods for different temperatures of the hydraulic oil in the excavator drive system. it is noticeable that at the higher oil temperature 90°c, when the dynamic viscosity and modulus of elasticity are smaller, amplitudes and periods of oscillation of the movement mechanism are greater in relation to the lower oil temperature 80°c. furthermore, changes in the generalized coordinates of boom θ3 (fig. 4b) and stick θ4 (fig. 4c) possess a similar oscillatory-dampened character. it is characteristic that the damping time of oscillations of all generalized coordinates is approximately the same and it amounts to 4 s. only the generalized boom coordinate, whose initial movement conditions cause the disturbance in the dynamic system of the excavator, has a longer damping period. 4.2. experimental analysis the experimental analysis of the dynamic stability of the excavator was performed on the basis of the changes in measured quantities in exploitation conditions. during the excavator testing, the following quantities of the excavator exploitation working condition were measured: the vertical displacement of support and movement member c1, the angle of platform rotation c2, the kinematic length of the hydraulic cylinders of boom c3, stick c4 and bucket c5 and the pressure in the actuators of the excavator drive mechanisms. fig. 5 mathematical model of the excavator for the experimental analysis a numerical and experimental analysis of the dynamic stability of hydraulic excavators 167 on the basis of the measured quantities of the excavator operating state, and by using the developed program, the change in the generalized coordinates (θ0,θ1,θ3) of the positions of the excavator kinematic chain members was determined during various manipulation tasks of the excavator in exploitation conditions (fig. 5). table 2 measured quantities in exploitation conditions sensors measured quantities index dim м1 the vertical displacement of the support and movement member c1 m м2 the angle of the platform rotation c2 rad м3 the stroke of the boom hydraulic cylinder c3 m м4 the stroke of the stick hydraulic cylinder c4 m м5 the stroke of the bucket hydraulic cylinder c5 m м6 the pressure in the one duct of the hydraulic motor for platform rotation drive p21 mpa м7 the pressure in the another duct of the hydraulic motor for platform rotation drive p22 mpa м8 the pressure in the boom hydraulic cylinder on the piston end p31 mpa м9 the pressure in the boom hydraulic cylinder on the piston rod end p32 mpa м10 the pressure in the stick hydraulic cylinder on the piston end p41 mpa м11 the pressure in the stick hydraulic cylinder on the piston rod end p42 mpa м12 the pressure in the bucket hydraulic cylinder on the piston end p51 mpa м13 the pressure in the bucket hydraulic cylinder on the piston rod end p52 mpa generalized coordinates θ1 of the support and movement member of tested excavator are determined by the following equation (fig. 5): 1 1 1 1 1 1 1 2 arctg 0 2 2 arctg 0 2 c c l a c c a l q 1 1 1 1 1 1 1 2 arctg 0 2 2 arctg 0 2 c c l a c c a l q (20) where: a1  the coordinate of the sensor position, l  the length of the excavator track footprint. the changes in the following generalized coordinates of the excavator kinematic chain members are selected and presented here: θ0 – the vertical displacement of the centre of mass of the support and movement member (fig. 6), θ1 – the angle of displacement of the support and movement member (fig. 7), θ3 – the angle of the rotation of the manipulator boom (fig. 8). the results yielded by the experimental dynamic analysis of the excavator show that the hydraulic excavators act like very sensitive dynamic oscillatory systems where as elastically dampened elements of the system occur hydrostatic actuators (hydraulic motors and hydraulic cylinders) of excavator drive mechanisms, due to hydraulic oil compressibility. diagrams of the changes in generalized coordinates θ0 (fig. 6), θ1 (fig. 7), θ3 (fig. 8) show that a pronounced oscillatory state of the excavator occurs at the end of the digging operation, when the digging resistance drops rapidly, and it appears as a force impulse 168 d. janošević, j. pavlović, v. jovanović, g. petrović acting on the excavator’s elastically dampened system. the oscillatory displacement also occurs at the beginning of the material moving operation when the boom begins to move. however, primary oscillatory displacement of the member of the support and movement mechanism kinematic chain appear during the material unloading operation at the moment when the bucket is being emptied and when the system dynamic parameters – the mass and the moment of inertia of the scooped material – change rapidly. the results of the experimental dynamic analysis of the excavator stability show the oscillatory-dampened character of the generalized coordinates with the damping time of around 4 s, which approximately corresponds to the time obtained in the numerical analysis. fig. 6 the change in the vertical displacement of the support and movement member θ0 during the manipulation task of the excavator obtained in the experimental analysis fig. 7 the change in the angular displacement of the support and movement member θ1 during the manipulation task of the excavator obtained in the experimental analysis a numerical and experimental analysis of the dynamic stability of hydraulic excavators 169 fig. 8 the change in the vertical displacement of manipulator θ3 during the manipulation task of the excavator obtained in the experimental analysis 5. conclusion on the basis of the obtained results it can be concluded that the hydraulic excavators act like very sensitive dynamic oscillatory systems when performing manipulation tasks. hydrostatic actuators (hydraulic motors and hydraulic cylinders) of drive mechanisms of the excavator kinematic chain appear as extremely elastically dampened system elements in the form of hydraulic springs caused by the hydraulic oil compressibility. oil temperature in the excavator hydrostatic drive system affects the dynamic stability of the excavator. with an increase in oil temperature, amplitudes and periods of dampened oscillation of the excavator kinematic chain members also increase. the comparison of the character of changes and the duration of the calculated movement of the excavator, on the grounds of the defined mathematical model and the developed program, with the results obtained on the basis of the measured quantities of the excavator when operating in exploitation conditions show that the defined mathematical model possesses a sufficient accuracy to analyze the dynamic stability of the excavator. acknowledgements: the paper was prepared within the project tr 35049 financed by the ministry of education and science of the republic of serbia. references 1. nikolaevich, b.e., nyrgaiazovich, b.i, sergeevich z.s., 2015, enhancing the stability of the timber harvesting machine of manipulator type by using an active suspension system, journal of applied engineering science, 13(2), pp. 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interacting mechanical and hydraulic dynamics, master thesis, jomo kenyatta university of agriculture and technology. 13. fox, b., jennings, l.s., zomaya, a.y., 2002, on the modeling of actuator dynamics and the computation of prescribed trajectories, computers & structures, 80(7-8), pp. 605-614. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 315 328 https://doi.org/10.22190/fume181201001f © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper measuring deformations in the telescopic boom under static and dynamic load conditions cristiano fragassa, giangiacomo minak, ana pavlovic department of industrial engineering, university of bologna, italy abstract. the interest in pushing the mechanical structures closer to their limits of usage makes necessary to combine the traditional design with the implementation of specific tests able to definitely confirm and guarantee their safety. exploring the case of a large telescopic boom, the present study analyses the response to intense loads prevenient from static and dynamic conditions. the measure of deformations was oriented to validate several design assumptions, but also to investigate the presence of phenomena of local instability, not easily predictable within theoretical formulations. key words: telescopic arm, experimental mechanics, deformation measurement, metal structures, buckling, mechanical hysteresis 1. introduction the possibility of using mechanical structures in the load situations near the resistance limits of their materials represents, on one hand, a real opportunity for improving their functionality, and, on the other hand, a serious risk that only a perfect design can prevent. this is precisely the case of the telescopic boom object of this study. among the diverse applications for which a telescopic boom can be designed [1-3], this one is intended to be firstly mounted on fire trucks. in this case, the ability to work very close to the material limits, which involves perfect management of sheets thickness [4, 5], contacts and sliding forces, aiming at reducing the risks of instability [6-8], allows the design of ever-slimmer and lighter structures. or, in different terms, keeping an identical weight, the same design requirements make it possible to create structures with ever-increasing outreaches, enlarging their functionality [2, 3]. received december 01, 2018 / accepted january 15, 2019 corresponding author: cristiano fragassa department of industrial engineering, alma mater studiorum university of bologna, viale risorgimento 2, 40136, bologna, italy e-mail: cristiano.fragassa@unibo.it 316 c. fragassa, g. minak, a. pavlovic this general concept is not as simple as it may appear in terms of safety design [9, 10]. mechanical structures are often designed through a 'classic approach' which provides for various simplifications and ways such as the determination of the safety coefficient [1]. however, the same concepts are likely to be overcome when structures are designed for extreme conditions of use, since other unpredicted phenomena can occur [11, 12]. referring, for instance, to the case of the thin metal sheets the telescopic boom is made of, the more their thickness is reduced, the more unstable their behavior becomes [13]. thus, every action dealing with a reduction in thickness needs to be thoroughly analyzed by theories including the study of second order effects, such as buckling of plates and sections [14-17]. an alternative solution is represented by the massive use of numerical calculations by means of tools such as fem algorithms [16-18]. however, also in this case the information available can be non-conclusive, leaving the execution of validation tests as a last resource. in this paper, a complex experiment is defined and detailed with the scope to validate the functionality and the level of safety offered by a large telescopic boom. external loads were applied and deformations measured in such a way as to investigate the overall behavior of the boom in static and dynamic conditions, together with the presence of unattended phenomena of instability. 2. materials and methods 2.1. the case-study this investigation is based on a specific telescopic boom, installed on relatively light vehicles and used for moving aerial platforms on fire trucks (but can be used with other vehicles as well), and which provide:  working height: 17 m  platform height: 15 m  load capacity: 200 kg  outreach: 9 m in particular, the research intends to join experimental evidence with the theoretical and fem considerations already available from a previous study [19] with the aim to improve the overall comprehension of the structure response in respect to working loads. in this sense, it is specifically interesting to validate with measures the limits of utilization of the telescopic boom in terms of its structural resistance and stability. the experiment started with the development of a structural reconstruction of the complex mechanism used for joining the different parts of the telescopic boom and consisting of two rectangular tubes coupled by two pairs of sliding pads (fig. 1a). this connection was mounted inside special equipment specifically designed to provide conditions of forces and constrains representative of the real telescopic boom during its usage (fig. 1b). in particular, the double hinge offers a situation of fix constrain on one side while the other side is pressed by two pistons in contraposition (figs. 1c and 1d), moved by hydraulic pumps and able to provide a vertical force up to 200 bar each. the full equipment made by welding was 5 mm sheets of aisi4130 steel. measuring deformations in the telescopic boom under static and dynamic load conditions 317 fig. 1 experiment design: (a) model of boom showing tubular parts, slides and supports; (b) boom inside the testing equipment; (c) fixing and (d) loading systems the theoretical evaluation of forces and moments was carried out in [19] simplifying the model by the use of a beam, rigidly fixed in one side (fig. 2a). this simplification deals with the evidence that the straight prismatic geometry is limited by two flat perpendicular bases, able to generate side surfaces very close to the points of application of loads. in addition, since mass forces are nearly irrelevant compared to the external loads and since different sections are free from constraints for the largest part of their length, the system can be led back to a lamé’s problem furthermore, in accordance with the de saint venant’s principle, in the presence of elastic materials (as in this case considering the properties of the steel), it is possible to replace a given system of forces with another one, having the same resultant force and the same resultant torque since diverse applications and distribution of these loads do not significantly influence the beam in the sections far from the point of application [19]. then, in brief, this system can be considered as a slender beam, namely a solid having one predominant dimension comparing to the others, undergoing small displacements and with loads and constraints applied at the ends. an additional stage in the simplification is permitted by the reduced thickness of the sheets that allow the adoption of the de saint venant‟s theory for the beam with thinwall. the related shear force and bending moment are displayed in fig. 2b. fig. 2 design of the experiment: (a) scheme of forces and constrains and (b) diagram of shear force (t) and bending moment (mf) b d c a b b b a b b 318 c. fragassa, g. minak, a. pavlovic fig. 3 images of the telescopic boom: (a) the two extensions of the boom in the workshop while waiting to be assembled; (b) the machine during the tests with the supporting frame in the foreground; and (c) the hydraulic system for the application of external forces 2.2. acquisition of deformations in the experiment, 15 strain gauges were used with the scope to investigate the response of the structure to loads. thanks to a variation of electrical resistance of the monoaxial filament, these strain gauges are able to transform the deformations of the structures they are glued over into electric signals. these strain gauges have innumerable advantages including the fact that they can be placed on the surfaces with any spatial orientation and, if protected by silicone, can remain long even in humid and dusty environments, and thus can be used for more tests. they are also simple, economical and precise sensors that lend themselves particularly well to the measurement of static and possibly dynamic deformations on the surface of the loaded bodies. 2.3. placement of strain gauges taking into account the results of the previous simulations [19], the strain gauges were placed in strategic points, particularly suitable for monitoring the behavior of the telescopic boom subject to bending load (fig. 4). entering in details, strain gauges 1, 2, 3 and 4, considered important for analyzing possible areas of stress concentration, were glued to the lower part of the inner arm, in correspondence with the sliding pads. strain gauges 1 and 3 longitudinally and strain gauges 2 and 4 were placed transversely on the two sides of the beam, in non-horizontal areas near the edges. strain gauge 15 was placed on the opposite side with respect to the previous ones, on the same section with control functions. strain gauges 5 and 6 were glued symmetrically on the upper and lower part of the inner beam, respectively. the same was done with strain gauges 11 and 12 for the external beam. in these areas, there was no provision for stress concentrations. strain gauges 7 and 8 were located on the outer beam at the sliding pads as well as strain gauges 9 and 10 otherwise left in reserve. strain gauges 13 and 14 were glued on the side in an area which is supposed to present problems of instability of the elastic equilibrium. after cleaning and smoothing the surfaces, the strain gauges were glued with acrylic glue and then covered with silicone in order to remain protected during the assembly phase; in this way it became possible to use them for any subsequent experimental campaigns. some of the internal strain gauges were fitted before b c a measuring deformations in the telescopic boom under static and dynamic load conditions 319 the boom was assembled, paying attention that the high temperatures reached during welding would not compromise their functionality. fig. 4 position and orientation of the strain gauges 2.4. anticipating the measurements bearing in mind the position of the strain gauges, with simple considerations derived from the lean beam theory and symmetry conditions it was expected that the:  strain gauges 1 and 3 detect similar maximum values, as well as 2 and 4;  strain gauges 5 and 6 show similar maximum values, but with opposite sign;  strain gauges 11 and 12 offer the same situation of strain gauges 5 and 6;  strain gauges 13 and 14 show very low values compared to the others;  strain gauge 15 detects a value quite similar to that of strain gauges 1 and 3. 2.5. calibration of the experimental system a preliminary operation was carried out to check for any variations in the „zero‟ values read by the internal strain gauges, which were placed before the telescopic arm was assembled. in this way, it was verified that the residual stresses due to the welding of the sheets and the deformations related to the assembly did not significantly interfere with the results of the measurements. in particular, a doubt was related to the possibility that the high temperatures reached during welding could damage the internal strain gauges (1, 2, 3, 4 and 5), despite the accurate cooling system developed to prevent this 320 c. fragassa, g. minak, a. pavlovic problem. however, a test carried out with an infrared sensor ensured that the sheets on which the strain gauges were applied did not exceed 50 °c, a fairly low value compared to a maximum permissible temperature of about 90 °c. as the next step, the acquisition system was calibrated starting with the calibration of the pressure sensors that were checked by respective instrument. it was also verified that their response remained proportional to the pressure rise. then, the calibration moved to the strain gauge system, recording the values read in the absence of external load, but with the arm subjected to its own weight. a first load cycle at 10 bar with the discharge after the nominal value of 0 bar was used to make the system settle and with the acquired values the strain gauges were zeroed. 2.6. application of loads tests were carried out by loading the structure, a force hold and a subsequent unloading. in particular, the load sequences, represented in fig. 5, consisted of:  slow and progressive loading up to the 75% of nominal load in accordance with design conditions, equivalent to 75 bar in the pumps pressure, including different stabilization and measuring points;  progressive loading up to nominal design conditions;  fast loading and unloading cycles up to the nominal load;  loading and unloading cycles with maximum load increase, 200 bar, equal to 3 times the nominal load;  the zeroing of the strain gauges was finally carried out and two tests were carried out, increasing the imbalance between the pumps at 61-39 bar and 106-85 bar respectively. fig. 5 loading and unloading cycles carried out during the tests measuring deformations in the telescopic boom under static and dynamic load conditions 321 3. results 3.1. deformations at nominal load figure 6 shows the typical result of an acquisition (in the case of nominal load of 75 bar). first of all, it can be observed that all strain gauges work regularly with no open circuit effects or perceptible noise on the signals, further confirming that the welding and assembly operations did not cause damage to the sensors. strain gauges 9 and 10 were not acquired, but kept in reserve. the pressure trends in the two pumps can also be noted (in black and with values obtained on the right side): it is clear how, before reaching the nominal pressures, oscillations due to the inertia of the fully manual regulation hydraulic system appear. by enlarging the loading phase (fig. 6b), it is possible to see that the load reaches its maximum in a not completely regular manner and the pumps remain, even if slightly, unbalanced. to improve clarity, only a reduced set of strain gauges is visible. observing the measurements, it immediately becomes evident that there are two "classes" of strain gauges that behave differently. strain gauges 1, 3, 5, 6, 15 (axial) follow almost perfectly the load trend, apart from a slight hysteresis linked to the hydraulic circuit and to the gaps present in the arm; in particular, strain gauge 6, which is far from the covering area, also follows the smallest oscillations of the load (fig. 6b, top). on the contrary, strain gauges 2, 4, 7, 8 (transverse) show a marked hysteresis phenomenon, so that the deformation peaks are delayed and the deformation continues to increase for a while even when the load decreases (fig. 6b, right). it is very likely that this phenomenon is linked to a viscous effect caused by the presence of the sliding pads, which delay the transfer of load between the two extensions of the arm and, in the case of a strong peak, they considerably attenuate it. fig. 6 measures for nominal load (75 bar): (a) entire spectrum and (b) loading phase only comparing measures with expectations:  strain gauges 1-3 and 2-4 have different readings showing that the part of the expected symmetry is missing in the real structure or loads (or both);  strain gauges 5 and 6 actually detect similar values with opposite sign, as well as 11 and 12: de saint venant's theory is applicable in those areas;  strain gauge 6, as expected, detects the maximum deformation;  strain gauges 13 and 14 correctly detect lower values (-13%) than strain gauge 6;  strain gauge 15 has a measure similar to strain gauge 1, but not strain gauge 3. b b b a b b 322 c. fragassa, g. minak, a. pavlovic it is possible to conclude that, for the nominal load conditions: 1. the stress state is similar to that of a slender beam only far from the covering area; 2. there is no evidence of instability phenomena; 3. the loading of the structure appears not symmetrical (or, even if less probable, the structure has internal constraints such as to make its geometry not symmetrical): this asymmetry generates an undesired twisting moment on the arm with longitudinal axis which must be taken into account; 4. a deformation concentration effect on the sliding pads appears, which can be evaluated as the average of measures from strain gauges 1-3 and 2-4. 3.2. measures at maximum load during the phase of loading up to the maximum load of 200 bar, already exceeding 140 bar noise emissions were noticed both in the loading and unloading phases. in correspondence with these noises, the strain gauges positioned at the sliding pads (in particular the 2 and 4) recorded jumps (as can be seen in the red circles in fig. 7). fig. 7 measures for maximum load (200 bar): (a) whole spectrum with stick-slip phenomena highlighted; (b) simultaneous jumps on all strain gauges; (c) unloading phase the most plausible explanation for these events is related to the presence of a stick-slip phenomenon (sliding) between the pads and the internal arm, which could most likely be eliminated by means of suitable lubrication. due to the friction of the first detachment, elastic energy is stored which is released and suddenly deforms the arm emitting an acoustic wave at the same time, following a small slipping. since they were static tests, putting sliding pads was not taken into consideration. however, the measurements were not compromised by this effect since the energy accumulated had extremely low values and was immediately released. in fig. 7b it is possible to better appreciate this behavior in strain gauges 1, 3 (very light) and 2, 4 (accentuated) where it can be noted that all the strain gauges reveal the phenomenon b b b c b b a b b measuring deformations in the telescopic boom under static and dynamic load conditions 323 at the same time through vertical lines. instead, in fig. 7c the trend of the readings in the unloading phase of the structure can be observed twice. while the strain gauges positioned in the areas far from the pads faithfully follow the load, those at the covering have a marked hysteresis to be attributed to the viscoelastic behavior of the ertalon and to the stick-slip phenomena. it should be noted, for example, that while strain gauge 6 closely follows the discharge, strain gauge 4 remains charged for a certain period, even if the arm is being unloaded, because the pad is opposed by friction upon returning to the condition of rest. suddenly, the deformation drops with an intermediate step in correspondence of which the acoustic emission is also recorded, and residual deformation remains, which, however, is only slightly higher than that present at the beginning of the loading phase. but it is not only strain gauge 4 that has such a particular behavior: it is seen that strain gauges 2 and 3 even change their sign before returning correctly to zero, highlighting effects of rebound of the structure. from the quantitative point of view, it is observed that the maximum deformation recorded is about 2.910-3 (strain gauge 6), corresponding to 600 mpa in terms of stress, way below the yield stress. finally, strain gauges 13 and 14, which had to control the onset of elastic equilibrium instability, present no deviation from the linearity, remaining at strain values lower than 0.3510-3, also experimentally denying the risk of structural buckling. 3.3. measures for unbalanced loads during the simulations, it was noticed how slight unbalancing loads between pumps 1 and 2 significantly overload the boom with relevant shear stresses. fig. 8 shows the results of a test with progressively rising loads with an imbalance of 25%, until reaching, on the two pumps, 106 and 85 bar, respectively. fig. 8 measures for “unbalanced” loads: (a) entire spectrum; (b) correspondence between load trends and deformations; (c) loading and (d) unloading phases b b b a b b d b b c b b 324 c. fragassa, g. minak, a. pavlovic in particular, in fig. 8b shows different responses of the strain gauges to the initial pressure peak, not detected by strain gauges 2 and 4, as if these sensors mediated what happened to the structure by charging and discharging more slowly, but detected by 1 and 3 (even if in a slightly attenuated form). in fig. 8c it is possible to see the stick-slip phenomenon again: in fact, in the face of a load increase, while strain gauge 6 follows the ramp perfectly and strain-gauges from 1 to 4 are little affected, they have a discontinuity in the deformation. the usual behavior, with steps and inversions of sign, is visible in the unloading (fig. 8d) where, following re-adjustments in the geometric configuration of the arm, continue strong sound emissions that persist for a few seconds even at nominal load virtually nil are present. 3.4. load – deformation curve the load-deformation curve shows the amplitude of the structure's hysteresis and its tendency to accumulate energy. the reported case is the one recorded at 90 bar by the two strain gauges 11 (black) and 12 (red) (fig. 9). it is possible to observe the first phase of linear loading followed by a phase in which, when the pressure increases (between 20 and 90 bar), the deformation remains constant. in this phase, the recovery of the mechanical and „hydraulic‟ gap and the crushing of the sliding pads are present. then, the level of deformation starts to increase. the discharge phase is more regular except for an initial increase in the deformation due to the unbalancing of the pumps during unloading. with the current loading system, which controls only the pressure in the cylinders, it is not possible to carry out direct loaddeformation graphs from which to draw conclusions on local effects because there is a large hysteresis area so as not to allow the perception of the real trend of the forces, and therefore of the bending moment. fig. 9 load-deformation curve with evidence of hysteresis (only two strain gauges) 11 bb 12 bb measuring deformations in the telescopic boom under static and dynamic load conditions 325 4. discussion 4.1. global behavior every time that a stable pressure condition was reached, with a measuring tape able to guarantee an accuracy of the millimeter, the distance between the bottom of the box and the loaded end of the beam was measured obtaining the total inflection of the arm according to the applied pressure (fig. 10a). this curve was compared with the measures from strain gauges of maximum deformations as a function of the bending moment applied (fig. 10bd). both show non-linear initial behavior of the system with large displacements for small load values followed by a fairly linear tendency. the initial trend is certainly linked to the recovery of gaps, the torsion caused by the asymmetry of the system and the viscoelastic behavior of the sliding pads in ertalon. as the load increases, the behavior of the beam as a whole changes and, in the case of strain gauges near the sliding pads, due to the lack of symmetry of the actual contact conditions, the deformation is also qualitatively different between the symmetrically applied strain gauges. in fig. 10b, for instance, it is evident that strain gauge 2, placed transversely in correspondence with one of the lower sliding pads, diverges from linearity with even changing its sign, while strain gauge 4 has values much higher than its symmetrical. fig. 10 parallelism between the (a) overall behavior and (b-d) local deformations by looking at the trends of strain gauges 2 and 4, the hypothesis that the arm is subjected to torque becomes more and more solid, considering how the first tends to discharge while the second is overloaded. even visually, at the maximum loads applied (over 140 bar), the effects of torsion on the loaded end of the arm began to be noticed (fig. 11). b a d c 326 c. fragassa, g. minak, a. pavlovic fig. 11 telescopic arm under test with the first effects of torsion 4.2. in brief as synthesis of experimental measures, it is possible to say that: 1. all strain gauges worked correctly, 2. the steel structure remained in the linear field without presenting instability of the elastic balance even for loads equal to almost 3 times the nominal one, 3. the presence of plastic sliding pads and mechanical gaps has determined a reduction of the overall rigidity and the non-linearity in the first stretch (with viscous sliding phenomena) of the load-displacement curve, 4. most of the measures are compatible with the beam theory (without prejudice to the errors on the measurement of the load) except those of the strain gauges placed close to the sliding pads, 5. the apparent anomalous measures of strain gauges 2 and 4, placed near the sliding pads, are linked to local effects such as load redistribution, slippages and viscous effects due to pads, which create an asymmetry in the local situation. combining the measures from strain gauges 2 and 4, these effects can be almost compensated, 6. the detection of the loads applied to the arm through custom-made load cells, instrumenting both the connecting rods and the supports of the shoes, would allow to circumvent the hysteresis problems of the hydraulic circuit. 4.3. theory vs. experiment there are both correspondences and differences between forecasts from theory or simulation and measurements through which it is possible to move toward a better understanding of what happens inside a telescopic boom, especially in the connecting areas. in particular, from this comparison, it is possible to clarify that: a. in the areas away from contacts (representing the largest part of the metal sheets), the variations between measures and theory are minimal, practically confirming the presence of a membrane state of stress, b. strain gauge 15 shows a tension much higher than expected which shows a strong transversal deformation in the area between the lower pad, able to influence the reading of the strain gauge (which otherwise would have had to provide only the axial strain), measuring deformations in the telescopic boom under static and dynamic load conditions 327 c. the asymmetry of the applied loads, most likely mainly due to geometric imprecisions of the supports and during the assembly, generated a torque which did not allow in some points a direct comparison between the results of the numerical simulations and the direct measure of the directional deformations, but only between their average values, d. by appropriately correcting the simulation model to take into account the real asymmetry that presents the contact area (explained above), the numerical method can lead to extremely accurate results (e.g. close to strain gauges 1 and 3, a -116 mpa was measured, very close to the -120 mpa of the simulation, as well as around 2 and 4 where 31 mpa are detected against the 38 simulated mpa), e. with a peak of 728 mpa estimated in the contact area under nominal load conditions, the telescopic boom is always in the elastic range (stresses lower than 1300 mpa) and can operate without particular problems, f. as the load increases, the soft sliding pads in ertalon soon reach (as expected) the yield stress and, deforming, enlarge the contact area and redistribute the stress in the steel by lowering the localized peaks: in the upper surface of the internal extension (the most critical for the strength of the structure) not even reaching 300 mpa, g. with maximum load of almost 3 times the nominal one, the strains on the external extension provide axial tension values that are about 140% less than those obtained with the simulation. the most plausible hypothesis leads to consider how on the external extension, which has a perfectly fitted end with two sleeves, the (unwanted) torque is most perceived by its sensors. they only record the longitudinal component of a deformation that instead has an inclination of 45 ° with respect to the axis. the internal extension is less affected by this effect due to the lower stiffness of the shoes, h. it is believed that by improving loads on sliding pads, the plasticized part would have affected the fittings with the lower surface and with the side walls extending the section involved in the covering and reducing the stresses, i. there are no instability phenomena up to the maximum load of 200 bar and, in general, it is believed that geometric instability cannot occur because plasticization problems would arise before, j. in accordance with the simulations where a pure bending moment was provided, the boom should collapse under a pure moment above 1000 knm, corresponding to 300 bar (or for a moment combination of 923 knm and share force of 52,570 kn). these load values, 4 times higher than the design limits and far from the potentialities of the present testing equipment, cannot reasonably be achieved during operation conditions by the structure, even assuming abrupt or unconventional overloads. 5. conclusions the theoretical and fem modeling studies, together with the noticeable experience of the designers, allowed to devise an important and complex experiment that, involving over 7000 kg of sheets, aimed to understand the mechanical behavior of the telescopic arms used in the aerial platforms. combining the results from theory and simulations with these measures, it was possible to refine the interpretative model creating new tools able to support the design of safer and more efficient structures. in conclusion, the experimental 328 c. fragassa, g. minak, a. pavlovic tests have validated the numerical simulations, which therefore can be used reliably in the design and verification of structures similar to the one examined. acknowledgements: the authors wish to thank alessandro fantuzzi and marco faedi for their essential contribution in the experiment design and implementation, and vanda roversi and dino balduzzi for their support in signal measuring and elaboration. references 1. en 280:2000 standard, mobile elevating work platforms. design calculations. stability criteria. construction. safety, examinations and tests. 2. włodzimierz, s., 2003, mobile platforms, construction and exploitation, krosno kabe. 3. joyce, n. 1995, design of a mobile elevating work platform, m. phil thesis, brunel university. 4. timoshenko, s. p., gere, j. m., 2009, theory of elastic stability, courier corporation. 5. tvergaad, t., 1999, studies of elastic-plastic instability, journal of applied mechanics, 66, pp. 3-9. 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yanling, l., 2004, finite element analysis for local stability of telescopic boom of truck crane, transactions of the chinese society of agricultural machinery, 35(6), pp. 48-51. 19. pavlovic, a., fragassa, c., minak, g., 2017, buckling analysis of telescopic boom: theoretical and numerical verification of sliding pads, tehnicki vjesnik, 24(3), pp. 729-735. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 79 89 https://doi.org/10.22190/fume201225007d © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical model of a local contact of a polymer nanocomposite and its experimental validation andrey i. dmitriev1,2 1institute of strength physics and materials science sb ras, tomsk, russia 2tomsk state university, tomsk, russia abstract. in the paper a model of a local contact of a polymer-based nanocomposite was developed within the method of a movable cellular automaton. the features of mechanical behavior of nanocomposite at the mesoscale level under dry sliding were studied with explicit account for the microprofile of the counterbody surface and the characteristic sizes of nanofiller. factors that contribute to the conditions for the formation of a stable tribofilm of silica nanoparticles are analyzed. two other parameters like sample geometry and the value of relative sliding velocity are also examined. it is shown that the thickness of tribofilm depends on stress conditions at the contact, and the friction coefficient decreases with increasing sliding velocity similar to one observed experimentally. to ensure the low friction properties of polymer nanocomposite, particles whose sizes are comparable with the characteristic size of the substrate microprofile are preferred. results of numerical simulation are in good correlation with available experimental data. key words: modelling, nanocomposite, silica nanoparticles, carbon fibers, tribofilm, friction coefficient, movable cellular automata method 1. introduction polymer matrix composites are the most significant class of composites finding widespread applications in almost all sectors of society and industry. these composites have increasingly been objects of an important topic of study due to the possibility of significant changes in their mechanical, physical, and other properties through the use of fillers of various types. in recent years, in the polymer nanocomposites (pncs), along with traditional fillers, various nanosized particles added to the composite mixture have been becoming more widespread. the main advantage of these materials is characterized by the use of a low concentration of nanofiller (1–5% vol.) and by their particle size, in received december 25, 2020 / accepted january 28, 2021 corresponding author: andrey i. dmitriev institute of strength physics and materials science sb ras, 634055, pr. akademicheski 2/4 tomsk, russia e-mail: dmitr@ispms.ru 80 a. i. dmitirev contrast with conventional composites [1-4]. the size reduction from microscopic to nanoscopic scale indicates a dramatic increase in the interfacial area as compared with ordinary composites, resulting in an improvement in the material properties. thus in [5], it was established that the wear rate of pnc based on polyphenylene sulfide decreased when tio2 and cuo particles were used as the fillers but increased with zno and sic fillers. a noticeable increase in microhardness and fracture toughness of nanocomposite with nanosized inclusions is described in [6]. the effect of young's modulus values increasing by almost 20% was found in [7] and an increase in the elastic modulus and yield stresses was shown in [8]. in [9] and [10], the effect of a significant change in the frictional performance of pncs with the addition of carbon nanofibers and other nanoscale fillers was shown. thus, the possibility of a targeted effect on the mechanical properties of a polymer-based nanocomposite makes them unique objects with the potential for widespread use in various modern applications. as it was revealed in [11] the monodisperse silica nanoparticles (snp) (diameter ~20 nm), produced via a sol-gel process, greatly improve the tribological behavior of conventional epoxy (ep, bisphenol a) composites when sliding against standard bearing steel (100cr6). the monodisperse nanoparticles and the conventional fillers, e.g. short carbon fibers (scf) and graphite, are beneficial with respect to desired tribological properties. it was shown earlier [10] that the tribological performance of such polymer composites can be significantly enhanced by the formation of a thin, well-distributed and load-carrying transfer film also called a tribofilm. with the addition of the nanoparticles, the friction coefficient drops very fast within the initial 1 hour, owing to the fast building up of a tribofilm. with enhancing the pressure from 1 to 3 mpa, the friction coefficient of the conventional composite becomes much higher. opposite to the previous observation, the increase in pressure reduces the friction coefficient of polymer-based composite with snp and scf. especially at high pv (the pressure multiplied by the sliding velocity) factors, nanoparticles reduce much more the wear than sub-micron particle combinations do (3 vol.% tio2/4 vol.%zns, diameter ~0.3 μm). although many publications have shown the importance of tribofilms during dry friction applications, only a few studies provide information about their structure. this is because these films are usually very thin and nanocrystalline that requires special tedious and expensive sample preparation techniques such as focused ion beam (fib) machining to characterize them with a transmission electron microscope (tem). a systematic study with changing the surface topography within the range of 0.01 to ~ 1 μm was performed by zhang et al in [12]. they revealed that on very rough counterbody surfaces with ra = ~1 μm two types of the transfer film could be identified. a carbon-based transfer film was formed in deeper grooves while the transfer film made from silica particles was identified in shallow grooves. despite the progress made, a number of key questions, however, remain unanswered. the aims of this work are the following. firstly, to develop a numerical model of a local contact of pnc and to study the impact of filler properties and sample geometry on features of tribolayer formation. secondly, to analyze the effect of pv-factor on tribological properties of hybrid pnc and to compare the numerical results with the available experimental data. numerical model of a local contact of a polymer nanocomposite and its experimental validation 81 2. model description a model of a local contact of polymer-based hybrid nanocomposite was developed within the method of the movable cellular automaton (mca) [13-17]. in the mca method the modeled objects are considered as a system of finite size bodies interacting with each other. in contrast to other particle-based approaches the mca method uses the many-particle interaction where the force calculations are based on the spatial configuration of nearest neighbors. within this approximation the system of newtoneuler equations of motion will be written as follow [17]: ( )        = ++        +=+==   = =  = i ii n j ij i i n j ij v ij nv i n j ij pair i v i pi i i m dt d j fffffff dt rd m 1 2 2 11 2 2        . (1) in eq. (1) ir  and i  are the radius-vector and rotation angle of the movable cellular automaton i, mi and iĵ are the particle mass and its moment of inertia, i pf  and i vf  are the potential and viscous constituents of the total force if  acting on the automaton i, ijm  is the momentum of the total force, ni is the number of neighboring particles for the automaton i. the potential force i pf  is written as a sum of the pair-wise component ij pairf  and the volume-dependent contribution if  : i n j ij pair i p fff i  = +=  1 , (2) while the force if  in case of locally isotropic material can be calculated as:  =  −= in j ijijji i nspaf 1  . (3) in eq. (3) pi is the pressure in the volume of the movable cellular automaton i, sij is the contact square between neighboring automata i and j, ijn  is the unit vector directed along the line goes through the centres of i and j, ai is the proportionality factor, which belongs to the material of the automaton i and in common case can be differ from aj. the concept of the mca method also involves the introduction of a new type of state, namely the state of a pair of automata which may be linked or unlinked. in short, the linked state indicates that the pair is part of a solid body and the unlinked state indicates that during sliding the automata may behave like a granular material. by using criteria for linked-unlinked switching in the framework of the mca method it is possible to simulate such events in the contact area like the discontinuity formation and accumulation of damage, as well as processes connected with mass mixing, velocity accommodation and so on. in the present investigations the “fracture” criteria for linked-unlinked switch was defined as critical value of stress intensity in the interacting pair. so, when the calculated stress intensity in a linked pair reaches the fracture strength of the softer material (epoxy matrix in our case) this pair becomes unlinked. after coming into contact with another automaton the unlinked-linked-transition may take place. the latter switching criteria are 82 a. i. dmitirev controlled by a normal flow stress. that means that an adjustable amount of plastic deformation of the softer constituent is taken as prerequisite for the binding of automata together thus forming an aggregate of linked particles. (a) (b) (c) fig. 1 (a) a general view of the simulated setup and a loading scheme. details of the composite structure are shown by the inset. (b) stress-strain diagrams for materials used in the model of a local contact of the hybrid nc. (c) stress-strain curves of uniaxial tension and shear for silica nanoparticles at ambient and elevated temperatures calculated with use of molecular dynamics method. figure 1a shows the loading scheme and the initial structure of the simulated setup. the region under consideration includes the surface of unmovable rigid substrate created by automata with properties of 100cr6 steel, as well as adjacent surface layers of the nanocomposite containing nanoinclusions. two samples with two characteristic sizes of silica nanoinclusions (20 nm and 60 nm) were considered. the inclusions were implemented by movable cellular automata with the properties of silica and the automata of the matrix possessed the properties of epoxy. the diameter of each automaton was set equal to 10 nm, which corresponds to the average diameter of the smallest silica nanoparticles [18]. the geometry of the sample was 12 x 4 μm along the x and y directions, respectively, and the total number of automata in the simulated setup was about 25 thousand. the evolution of an ensemble of movable cellular automata was defined by solution of the numerical model of a local contact of a polymer nanocomposite and its experimental validation 83 system in eq. (1) with the time step δt = 0.25 ps [13]. the latter is not freely adjustable but depends on the size of automata and on their elastic properties like: e dt  , (4) where d is the automata diameter, ρ and e are the density and elastic modulus of each material uses in the setup. the response functions used to describe the interaction between automata representing different materials are shown in fig. 1b. they were built on the basis of existing experimental data [19-21]. since in our preliminary studies [22] it was not possible to obtain the required response of the hybrid pnc with varying the pv-factor in the present model for silica particles, the temperature dependence of the mechanical properties calculated by the molecular dynamics method was introduced [23, 24]. stress-strain diagrams for silica nanoparticles at conditions of ambient and elevated temperatures are shown in figure 1c. in order to study the relationship between the filler properties and microgeometry of the substrate surface, three regions “a”, “b”, and “c” with different characteristic surface profiles were designed in the counterbody as shown in fig 1a. in region “a”, the surface profile of the substrate was characterized by v-shaped grooves with a width of 70 nm at the base of the triangle. region “b” of the counterbody profile contained rectangular recesses with a width of 40 nm, and region “c” included rectangular recesses of a surface profile with a width of 80 nm. the depth of all types of depressions was the same and equal to 70 nm. to analyze the effect of pv-factor on tribological properties of the hybrid polymer nanocomposite the surface profile of the counterbody was changed to flat, i.e. without additional grooves. this choice of the counterbody surface profile was motivated by the intention to speed-up the process in order to achieve the steady-state sliding regime, when the friction coefficient is localized at a certain value, and the position and thickness of the tribolayer are dynamically stabilized [25]. moreover, as it was shown in [14, 25] the initial roughness profiles of the counterbody on the nanometer scale do not impact significantly on the value of the friction coefficient. loading conditions that simulate relative sliding under dry contact were implemented by additional action on the automata of the upper layer of the sample, which were characterized by the properties of carbon fibre. thus the tangential (sliding) velocity (v) was applied to all automata of the upper layer as shown in fig. 1a. its value was varied in the range from 0.05 to 4 m/s in various tasks while maintaining the specified value throughout the calculating time. simultaneously the additional normal forces (f) simulating a compression p = 30 mpa were set. periodic boundary conditions were used to simulate the extension of the simulated fragment along the x axis. it was assumed that the surface layers initially did not contain any damages on the scale under consideration. furthermore, absence of adhesive forces between the sample and the substrate material was assumed. 3. results of sliding simulation 3.1. the influence of nanofillers size figure 2a shows a fragment of the structure of a hybrid pnc filled by silica nanoparticles with the characteristic size of 20 nm at the steady-state sliding regime. 84 a. i. dmitirev rectangles in the figure indicate areas with different types of microgeometry of the counterbody surface. it can be seen that, regardless of the type of microgeometry (“a”, “b” or “c”), in all types of depressions, there are automata that belong to both the matrix and the inclusion material. this is also well-seen in fig. 2b, where the pairs of automata in the “linked” state are indicated by the segments between centers of two neighboring particles. at the steady-state of sliding, the automata with the matrix properties are mostly represented by single particles while in the area with microgeometry of type “c” the agglomerates of the epoxy still exist. such a distribution of automata with properties of the matrix and snp can be explained by the small size of the nanofiller particles and their ability to settle in all types of considered recesses. fig. 2 the fragment of resulting structure (a) and structure of the inter-automata states (b) of a hybrid pnc with snp 20 nm after ~10 μs of sliding. (c) trajectories of automata for the considered fragment of the structure at the steady sliding in the time interval of 10 ÷ 30 μs. relative sliding velocity v = 1 m/s. hereafter automata with the properties of epoxy are marked in green and with the properties of silica – in grey. arrows indicate the non-zeros trajectories of automata in counterbody grooves. thus, in the case of a polymer-based hybrid nanocomposite with small nanofillers, the formation of a stable tribolayer from snp will be hindered by the fact that sio2 particles will be mixed with epoxy, which will eventually be removed from the tribocontact area due to heating and evaporation. however, this factor determines the duration of the grinding-in stage before the system transfer to the regime of stable sliding with a low friction coefficient. the trajectories in fig. 2c also confirm that the automata mobility remains localized in a narrow layer contacting with the substrate at steady-state friction. changes in the automata positions inside the recesses of the counterbody profile are practically not observed. there is only a slight involvement of the particles from large depressions of the “c” region (shown by arrows). numerical model of a local contact of a polymer nanocomposite and its experimental validation 85 figure 3a shows a fragment of the structure of a polymer-based hybrid nanocomposite filled by snp with the characteristic size of 60 nm at the steady-state sliding regime. in the case of the large size of the nanofiller particles, their filling of the recesses of region “b” of the counterbody profile is less frequently observed. this is explained by the proximity of the characteristic particle sizes of inclusions and the width of the recesses. separate automata with the properties of silica, which can be seen inside depressions of the region “b”, are fragments of larger particles that collapsed during sliding. similar conclusions can be made for the region “a” of the counterbody surface profile. however, the patterns described above for this part of the substrate are less pronounced. at the same time, according to the structure of inter-automata states, shown in fig. 3b, almost all depressions of the counterbody are covered with silica nanoparticles. this arrangement of snp should facilitate the formation of a transfer tribofilm. this conclusion is also well confirmed by the automata trajectories at the steady-state sliding regime presented in fig. 3c. the nonzero trajectories of automata are preserved only in the large grooves of the region “c”, but there are fewer of them than in the sample with smaller snp. fig. 3 the fragment of the resulting structure (a) and the structure of the inter-automata states (b) of a hybrid pnc with snp 60nm after ~10 μs of sliding. (c) trajectories of automata for the considered fragment of structure at the steady stage of sliding in the time interval of 10 ÷ 30 μs. relative sliding velocity v = 1 m/s. thus, in the case of a hybrid pnc with a nanofiller particle comparable with the size of the counterbody surface grooves, the formation of a stable tribolayer from snp will occur more intensively. indeed, in this case, large sio2 particles remain localized in a narrow tribolayer, where intense frictional heating of the material occurs and there is a high probability of the transition of silica particles to a state similar to an amorphous one. as was shown in [23, 24], such a transition determines the regime of stable sliding with a low friction coefficient. the latter characterizes hybrid polymer nanocomposites. 86 a. i. dmitirev 3.2. the influence of sample geometry to study the possible relationship between the geometry of the hybrid pnc and the thickness of the tribolayer two samples differing in size along the y-axis (4 and 2 μm) were simulated. all other parameters of the system related to loading conditions, width, and the conterbody surface profile remained unchanged. according to the results, during the relative sliding, most automata remain in a “linked” state with their neighbors, which are depicted by the segments connecting the centers of such particles. however, in the immediate vicinity of the sample-conterbody interface, a layer of “unlinked” particles was formed, which is necessary to accommodate the velocities of the moving sample relative to the substrate. the thickness of the layer (h), in which the active movement of particles occurs, in the case of higher sample is about 10-12 times of the automata diameter while for a sample with a height of about 2 µm this value is limited to 6-7 diameters of the automata. such a difference in values can be explained by the manifestation of different damping properties and stress conditions of both samples. thus, the geometry of the sample affects the thickness of the formed tribofilm, including the layer of mechanical mixing of particles. an increase in the height of the sample leads to an increase in the thickness of the tribofilm (fig. 4). it should be noted that the calculated average value of the friction coefficient for the simulated pnc sample turns out to be close to the experimentally observed value in the range of about 0.2 ÷ 0.3. fig. 4 the structures of inter-automata states after ~4 μs of sliding for the hybrid pnc samples with different size along the y-axis: (a) 4 μm and (b) 2 μm. the shear strain in both cases is about 25%. 3.3. the effect of pv-factor in order to study the influence of pv-factor on the tribological properties of a polymerbased hybrid nanocomposite the pnc sample with the geometry similar to one depicted in fig. 4a was subjected to sliding loading with different conditions. figure 5a shows the calculated values of the instantaneous and average friction coefficient for the system under consideration at different values of sliding velocity while maintaining the compression numerical model of a local contact of a polymer nanocomposite and its experimental validation 87 conditions. the friction coefficient for the simulated system was calculated as the ratio between the normal and tangential forces. averaging was performed over 100 adjacent instantaneous values of friction coefficient. it is well seen that with an increase of sliding velocity, and hence with an increase in the pv-factor, the friction coefficient for the simulated sample decreases. this is in good agreement with the experimental data, the values of which are depicted in fig. 5b. for comparison the simulation results were shown as individual points, which were obtained by averaging friction coefficient values over the stage of steady-state sliding. thus each point in the graph corresponds to individual plot shown in fig. 5a. a similar slope for both the experimental and theoretical linear dependences is observed. the only difference is that the calculated data are shifted towards large values of pv-factor. this is due to the difference in values of applied compression. the high values of normal forces are easily explained by the small sizes of the model of a local contact spot of a hybrid pnc with a steel counterbody. therefore, this difference can be interpreted as the difference between the nominal and actual compressive stresses. otherwise, the presented dependence is in good agreement with the experimental data, which indicates the adequacy of the developed numerical model. (a) (b) fig. 5 (a) the instantaneous (thin lines) and average (bold curves) dependencies of friction coefficient for the hybrid pnc sample filled by snp with the characteristic size of 60 nm at different values of relative sliding velocity. (b) calculated and experimentally measured [11] data of friction coefficient vs. pv-factor, where solid lines are drawn according to the least squares method. 4. conclusion based on the developed numerical model, the features of mechanical behavior of polymerbased hybrid nanocomposite at mesoscale levels under dry frictional contact were studied with explicit allowance for the microprofile of the counterbody surface and the characteristic sizes of nanofiller. factors that contribute to the conditions for the formation of a stable tribofilm from silica nanoparticles are identified. according to the simulation results, the preferred nanofiller to ensure the improved friction properties of hybrid pnc is that, whose sizes are comparable to the characteristic size of the microprofile of the substrate surface. in this case, solid inclusions are localized in the near-surface area of the counterbody depressions 88 a. i. dmitirev and the conditions necessary for the formation of an amorphized tribolayer from silica particles are created. it has been found that the v-shaped grooves of the recesses are preferable over the rectangular recesses of the substrate surface microprofile to ensure stable friction conditions. calculations showed also that with an increase of pv-factor, the friction coefficient for the simulated sample decreases. this tendency is in good agreement with the available experimental data, which indicates the adequacy of the developed numerical model. acknowledgement: the work was performed according to the government research assignment for ispms sb ras, project fwrw-2021-0006. references 1. vaia, r.a., wagner, h.d., 2004, framework for nanocomposites, materials today, 7, pp. 32–37. 2. kotsilkova, r., silvestre, c., cimmino, s., 2007, thermoset nanocomposites for engineering applications, in: r. kotsilkova (ed.), thermoset nanocomposites for engineering applications, rapra technology, shawbury, uk, pp. 1–12. 3. tauqeer ali, h., akrami, r., fotouhi, s., pashmforoush, f., fragassa, c., fotouhi, m., 2020, effect of the stacking sequence on the impact response of carbon-glass/epoxy hybrid composites, facta universitatisseries mechanical engineering, 18(1), pp. 69-77. 4. zare, y., rhee, k.y., 2020, a modeling approach for young’s modulus of interphase layers in polymer nanocomposites, physical mesomechanics, 23(2), pp. 176–181. 5. bahadur, s., sunkara, c., 2005, effect of transfer film structure, composition and bonding on the tribological behavior of polyphenylene sulfide filled with nano particles of tio2, zno, cuo and sic, wear, 258(9), pp. 1411-1421. 6. zhang, h., zhang, z., friedrich, k., et al., 2006, property improvements of in situ epoxy nanocomposites with reduced interparticle distance at high nanosilica content. acta materialia, 54(7), pp. 1833-1842. 7. hernández-vargas, m.l., castillo-pérez, r., flores-cedillo, o., campillo-illanes, b.f., 2021, modification of macromolecular dynamics in polyacrylic hybrid nanocomposite with un-treated sio2 nanoparticles, materials science and engineering b, 265, 114976. 8. ragosta, g., abbate, m., musto, p., et al., 2005, epoxy-silica particulate nanocomposites: chemical interactions, reinforcement and fracture toughness, polymer, 46(23), pp. 10506-10516. 9. zhang, l., zhang, g., chang, l., et al., 2016, distinct tribological mechanisms of silica nanoparticles in epoxy composites reinforced with carbon nanotubes, carbon fibers and glass fibers, tribology international, 104, pp. 225-236. 10. zhang l., qi h., li g., et al., 2017, impact of reinforcing fillers’ properties on transfer film structure and tribological performance of pom-based materials, tribology international, 109, pp. 58-68. 11. zhang, g., sebastian, r., burkhart, t., friedrich, k., 2012, role of monodispersed nanoparticles on the tribological behaviour of conventional epoxy composites filled with carbon fibers and graphite lubricants. wear, 292-293, pp. 176-187. 12. zhang, g., österle, w., jim, b.c., häusler, i., hesse, r., wetzel, b., 2016, the role of surface topography in the evolving microstructure and functionality of tribofilms of an epoxy-based nanocomposite, wear, 364365, pp. 48–56. 13. dmitriev, a.i., österle, w., wetzel, b., 2015, mesoscale modeling of the mechanical and tribological behavior of a polymer matrix composite based on epoxy and 6vol.% silica nanoparticles, computational materials science, 110, pp. 204-214. 14. dmitriev, a.i., österle, 2010, modeling of brake pad-disc interface with emphasis to dynamics and deformation of structures, tribology international, 43(4), pp. 719-727. 15. psakhie, s.g., smolin, a.yu., shilko, e.v., et al., 1997, about the features of transient to steady state deformation of solids, journal of materials science and technology, 13(1), pp. 69-72. 16. czopor, j., aniszewska, d., rybaczuk, m., 2012, the influence of defects on strength of ceramics modeled with movable cellular automata, computational materials science, 51, pp. 151–155. 17. shilko, e.v., psakhie, s.g., schmauder, s., popov, v.l., astafurov, s.v., smolin, a.yu., 2015, overcoming the limitations of distinct element method for multiscale modeling of materials with multimodal internal structure, computational materials science, 102, pp. 267–285. numerical model of a local contact of a polymer nanocomposite and its experimental validation 89 18. dmitriev, a.i., häusler, i., österle, w., 2016, modeling of the stress–strain behavior of an epoxy-based nanocomposite filled with silica nanoparticles, materials and design, 89, pp. 950-956. 19. basu, s., moseson, a., barsoum, m.w., 2006, on the determination of spherical nanoindentation stress– strain curves, journal of materials research, 21, pp. 2628-2637. 20. tinscher, r., bahnsen c., bomas h., et al., 2001, gefüge und mechanische eigenschaften eines durch sprühkompaktieren hergestellten stahles 100cr6, materialwissenschaft und werkstofftechnik, 32, pp. 607-620. 21. goggin, p.r., 1973, the elastic constants of carbon-fibre composites, journal of materials science, 8(2), pp. 233-244. 22. österle, w., dmitriev, a.i., gradt, t., et al., 2015, exploring the beneficial role of tribofilms formed from an epoxy-based hybrid nanocomposite, tribology international, 88, pp. 126-134. 23. dmitriev, a.i., nikonov, a.yu., österle, w., 2016, md sliding simulations of amorphous tribofilms consisting of either sio2 or carbon, lubricants, 4(3), pp. 1-24. 24. dmitriev, a.i., nikonov, a.yu., österle, w., 2018, molecular dynamics modeling of the sliding performance of an amorphous silica nano-layer the impact of chosen interatomic potentials, lubricants, 6(2), pp. 43. 25. dmitriev, a.i., schargott, m., popov, v.l., 2010, direct modelling of surface topography development in a micro-contact with the movable cellular automata method, wear, 268(7-8), pp. 877-885. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. i ii © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd foreword to the thematic issue adhesion and friction: simulation, experiment, applications sergey g. psakhie institute of strength physics and materials science, russian academy of sciences, tomsk, russia editorial this thematic issue contains selected papers related to presentations at the international workshop “adhesion and friction: simulation, experiment, applications”, technische universität berlin, november 13-16, 2017 organized by the technische universität berlin and the institute of strength physics and materials science of the russian academy of sciences with the financial support of the deutsche forschungsgemeinschaft. the papers of the present issue are devoted to the following topics:  numerical simulation of jkr-type of adhesion, in particular of brush-structured flat-ended indenters (q. li and v.l. popov). this topic is interesting both regarding numerical simulations of adhesive contacts with the boundary element method [1] and in the context of much debated influence of the “contact splitting” on the strength of adhesive contacts,  adhesive contact of gradient media with finite range adhesive forces (e. willert). this work is a generalization of the famous mauger theory [2] with the dugdale interaction potential in the case of gradient media. the gradient media contact is of upmost importance for many medical and technical applications,  adhesion between a rigid indenter and a thin layer on a rigid substrate (a. papangelo). this problem is also extremely important for numerous applications. it generalizes the known solution for parabolic body to an arbitrary power-law profile using a very simple and elegant method of reduction of any axissymmetrical adhesive problem to the corresponding non-adhesive one (suggested by v.l. popov, and independently by m. ciavarella). for further generalizations of the method see this issue (v.l. popov, solution of adhesion problem on the basis of the known solution for non-adhesive one),  an important focus of this thematic issue is adhesive wear. recent work by the group of molinari published in nature communications [3] provides convincing verification of a criterion for adhesive wear suggested by e. rabinowicz in 1958 [4]. this criterion can now be considered as a solid basis for better physical understanding of wear. in the paper “adhesive wear: generalized rabinowicz’ criteria”, v.l. popov discusses ii s. g. psakhie possible generalizations of the logic used by rabinowicz to layered or damaged systems,  the paper by dimaki et al. about the “simulation of fracture using a meshdependent fracture criterion in the discrete element method” develops the ideas of the simulation of adhesive contacts using a mesh-dependent detachment criterion [1] applied to the problem of fracture and wear. it goes from the analogy between adhesive contacts and cracks and considers transfer of the ideas developed in [1] and cited therein to fracture mechanics,  the paper by v. pakhaliuk and a. poliakov is devoted to consideration of wear of a hip joint under realistic daily activities such as walking, jumping and a series of “disturbances” which is of utmost importance both for a realistic estimation of the lifetime of the joint and for an optimized design,  an interesting general discussion of problems in the theory of adhesive wear is provided by m. ciavarella and a. papangelo. they use a humorous name of “contact sport” for the youngest activities in the field of contact mechanics of rough surfaces used by r. carpick in his science paper [5] and analyze critically the state of the art, the usefulness of the present theories and some conclusions which could be drawn specifically for the problem of adhesive wear,  the paper by r. balokhonov et al. considers the process of stir welding which is a combination of friction, adhesion and plasticity problem, and,  the issue is completed with two short communications. a note by m. ciavarella is devoted to an attempt of a simple interpretation of recent observations published in the pnas [6] on the interrelation of the contact area in an adhesive contact and tangential load. a short note by v.l. popov documents the reduction of the adhesive contact problem to a non-adhesive one and generalizes it to non-axisymmetric situations using the notion of the “filling factor” suggested in [1]. the organizers are thankful to the deutsche forschungsgemeinschaft for financial support of the workshop and to j. wallendorf and j. starcevic for organizational assistance. references 1. popov, v.l., pohrt, r., li, q., 2017, strength of adhesive contacts: influence of contact geometry and material gradients, friction, 5(3), pp. 308–325. 2. maugis, d., 1992, adhesion of spheres: the jkr-dmt-transition using a dugdale model, journal of colloid and interface science, 150, pp. 243-269. 3. aghababaei, r., warner, d.h., molinari, j.-f., 2016, critical length scale controls adhesive wear mechanisms, nature communications, 7, 11816. 4. rabinowicz, e., 1958, the effect of size on the looseness of wear fragments, wear, 2, pp. 4–8. 5. carpick, r.w., 2018, the contact sport of rough surfaces, science, 359(6371), pp. 38-38. 6. sahli, r., pallares, g., ducottet, c., ben ali, i.e., al akhrass, s., guibert, m., scheibert, j., 2018, evolution of real contact area under shear, proceedings of the national academy of sciences, 115(3), pp. 471-476. facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 69 77 https://doi.org/10.22190/fume191119010a © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  effect of the stacking sequence on the impact response of carbon-glass/epoxy hybrid composites hafiz tauqeer ali 1 , roya akrami 2 , sakineh fotouhi 3 , farzad pashmforoush 4 , cristiano fragassa 5 , mohammad fotouhi 6 1department of mechanical engineering, taif university, kingdom of saudi arabia 2your engineered structures ltd, glasgow, united kingdom 3department of mechanical engineering, university of tabriz, iran 4department of mechanical engineering, university of maragheh, iran 5department of industrial engineering, university of bologna, italy 6school of engineering, university of glasgow, united kingdom abstract. this paper investigates low-velocity impact response of quasi isotropic (qi) hybrid carbon/glass fiber reinforced polymer composites with alternate stacking sequences. cross-ply woven carbon and glass fibers were used as reinforcing materials to fabricate sandwiched and interlayer hybrid composites. for comparison, the laminates containing only-carbon and only-glass fibers were also studied. drop weight test was used to impact the samples. the images captured by a normal camera demonstrated that localized damages (delamination) existed within plies. the hybrid laminates had smaller load drops, smaller maximum deflection, and higher maximum load compared to the single fiber laminates. in addition, carbon outside interlayer hybrid laminate showed the highest maximum load and energy absorption, showing the significant dependence of the impact performance on hybridization and stacking sequence. it was concluded that a hybrid composite would help improve impact performance of laminated composites compared to non-hybrid composites if they are properly designed. key words: damage, glass/carbon hybrids, impact, stacking sequence 1. introduction glass or carbon fiber reinforced polymer composites have been conveniently used in a large selection of industrial products and applications [1], as aerospace and aircrafts [2,3], automotive [4-6], ships and sailing [7, 8], biomedicine and sports [9, 10]. received november 19, 2019 / accepted january 26, 2020 corresponding author: mohamad fotouhi school of engineering, university of glasgow, glasgow g12 8qq, united kingdom e-mail: mohammad.fotouhi@glasgow.ac.uk 70 h.t. ali, r. akrami, s. fotouhi, f. pashmforoush, c. fragassa, m. fotouhi this is especially due to their high strength to weight ratio and non-corrosive properties. however, brittle failure and poor impact performance are two major drawbacks, which have led to the conservative design and large reserve factors in the design of these composite structures [11, 12]. hybridization is one of the successful methods to achieve gradual failure or pseudoductility [13, 14]. in this method, two types of plies with different fibers are co-cured to achieve a desired performance, and to produce a more cost-effective material [15, 16]. most of the work done to-date on hybrid composites has concerned hybrids under tensile [17, 18], bending and compression loadings [19-21]. there are also some studies on hybrid composites when subjected to indentation and impact loadings [22-28]. it was reported that for given materials, stacking sequence is a major factor governing the hybrid’s behavior [29]. for example, carbon outside carbon/glass hybrid composites was reported to have the highest maximum load when subjected to impact [24, 25]. different combinations of glass/carbon hybrids were studied under impact. they have concluded that increasing the percentage of glass fiber has a positive effect on absorbed energy and maximum load [30]. despite the previous studies, the stacking sequence of hybrid structures becomes a key challenge on how to fully utilize the benefits of hybridization. some of the common stacking sequences used in hybrid composites are intermingled continuous fibers [31], intermingled aligned short fibers [32], interlayer or sandwiched layers [33] and intralayer [34]. this project aims at experimentally investigating the low-velocity impact responses of varied interlayer carbon/glass hybrid configurations, due to the simplicity of the hybridization method in manufacturing. it was found that a hybrid composite would help improve impact performance of laminated composites compared to non-hybrid composites if they are designed in proper sequences. 2. materials, manufacturing and testing a schematic of the investigated composites and their stacking sequence are shown in fig. 1. composites consist of axiom 5180 carbon fiber/epoxy and axiom 3180 glass fiber/epoxy, which are cross-ply woven prepregs commonly used for aerospace applications, and labeled in the figure as ‘c’ and ‘g’, respectively. those prepregs were laid up with different stacking sequences to achieve quasiisotropic mechanical properties, as detailed in table 1, where the fiber orientation is specified for each layer (e.g. 0/90 or ±45) and ‘s’ introduces the symmetry of sequences. a cure of 90 minutes at 130 ᵒc was performed for the investigated composite plates. the plates were manufactured according to astm d7136 standards [35]. current plates have dimensions of 300 x 300 mm². these were then cut into 100 mm² test samples pieces. the pieces were cut in such a way that the laminates which have zero angles are oriented along the short edge direction of the test specimens. fig. 2 illustrates the test rig, clamp and the set up. these were designed in accordance with astm standard d7136 [35]. four rubber tipped toggle clamps were used to secure the cut sample of dimensions 150  100 mm² over a 125  75 mm² hole. effect of the stacking sequence on the impact response of carbon-glass/epoxy hybrid composites 71 fig. 1 schematic and names of the investigated samples table 1 stacking sequence for different cases n. name stacking sequence thickness mm 1 c ([(0/90)(±45)]4c)s 4 2 g ([(0/90)(±45)]4g)s 4 3 c/g ([(0/90)(±45)]2c/[(0/90)(±45)]2g)s 4 4 g/c ([(0/90)(±45)]2g/[(0/90)(±45)]2c)s 4 5 c/g/c/g ([(0/90)(±45)]c/[(0/90)(±45)]g/ [(0/90)(±45)]c/[(0/90)(±45)]g)s 4 6 g/c/g/c ([(0/90)(±45)]g/[(0/90)(±45)]c/ [(0/90)(±45)]g/[(0/90)(±45)]c)s 4 the test samples were accurately centered and then clamped before raising the impactor to the desired height level. the gravitational potential energy equation (energy = mgh, where notations m, g and h have their usual meanings of impactor’s mass, acceleration due to gravity and drop height, respectively) was used to calculate the height of the impactor. in order to introduce a reasonable level of damage, a value of 26 j was assumed. the rubber tipped toggle clamps were used in such a way that there is no lateral movement in the test sample pieces. a load cell consisted of strain gauges and was positioned underneath the clamped down samples. strainsmart® software was used to process the signal received from the gauges used. the signal was received at 10 khz. fig. 2-d shows a 12.7 mm ball bearing hemispherical tip impactor. this hemispherical shape of the tip prevents causing any puncturing to the sample and hence avoiding additional damage in the sample. 72 h.t. ali, r. akrami, s. fotouhi, f. pashmforoush, c. fragassa, m. fotouhi fig. 2 a) diagram of a low-velocity test rig, b) low-velocity test rig (impact tower), d) clamped sample and impactor, and c) astm standard clamp and test set up ten measurements for the mass of the impactor were taken and their mean for the total mass found to be 2.82 kg. at each energy level chosen, at least three repeats were done for each configuration reported in the manuscript. the results showed a good repeatability. to calculate the energy losses in the impactor, equation 1 was used. it was noted that the material absorbed most of the energy. it was also assumed that other energy losses are very minimal and negligible. 2 2 1 1 ( ) 2 e m v v  (1) 3. results and discussions 3.1 load-displacement results a typical load-displacement diagram for investigated c samples is shown in fig. 3. initially, there is a linear region in the load-displacement which is related to the elastic region in the early stage of the loading process. when the load level reaches a critical load level, the damage (delamination) initiates. after the initiation of damage, there is an effect of the stacking sequence on the impact response of carbon-glass/epoxy hybrid composites 73 unstable propagation of the delamination, which is distinguishable from the load drops. the specimens are still integral, and the displacement goes to zero after the impact. fig. 3 load-displacement graph for a typical c impacted at 26j fig. 4 shows the load-displacement curves for the impacted samples with different stacking sequences. considering the shape of the curves, all the samples demonstrate a linear behavior in the early stages of the loading process as an elastic response. this is followed by a drop in the contact force owing to the damage caused by the impact. the main quantitative areas for analysis on these graphs are the critical load at damage initiation, the initial load drop, the maximum contact force, the maximum displacement and the absorbed energy. the fig. 4 comparison of the load-displacement graphs for the investigated samples impacted at 26j 74 h.t. ali, r. akrami, s. fotouhi, f. pashmforoush, c. fragassa, m. fotouhi absorbed energy is calculated using equation 1. as can be seen from figs. 5 and 6, the critical load for initiation of delamination was found to be slightly higher for the samples with outer layer carbon fiber. however, the hybrid laminates had smaller load drops, which are indicating a more gradual occurrence of damage. in addition, smaller deflection, and higher maximum load compared to the single fiber laminates. the carbon outside interlayer hybrid laminate (c/g/c/g) showed the highest maximum load, while c laminate had the highest absorbed energy. the results are showing the significant dependence of the impact performance on hybridization and stacking sequence. fig. 5 comparison of the absorbed energy and maximum load for the investigated samples subjected to 26j fig. 6 comparison of the critical load and maximum displacement for the investigated samples subjected to 26j 3.2 damage analysis the differences in the mechanical properties of the investigated composites are attributed to the impact induced damage mechanisms. fig. 7 shows the impacted and back side of the investigated samples subjected to 26j impact energy. for the c, c/g and c/g/c/g laminates, there are visible dents on the impacted side, and some fiber fracture and delamination on the back face; however, due to the opaque nature of the carbon prepregs, the damage is not easily observable by the naked eye. despite this fact, the visible damage on the c sample looks bigger than the c/g and c/g/c/g laminates. for effect of the stacking sequence on the impact response of carbon-glass/epoxy hybrid composites 75 the g, g/c and g/c/g/c due to the translucent nature of the glass the induced damage can be observed easier. the g sample has a similar size and pattern of delamination on both impacted and non-impacted faces; however, delamination damage is much larger in the impacted side of the g/c/g/c sample compared to the non-impacted side. damage and delamination between the dissimilar interfaces, i.e. the carbon/glass interfaces are visible for the g/c sample. overall, the visible damage in the g composite is greater than the other configurations and the c/g/c/g and c/g configurations have the lowest visible damage size. this is in accordance with the better impact response of these laminates compared with the other laminates. fig. 7 impacted surface and back surface (non-impacted side) view of the investigated samples subjected to 26j 4. conclusions this paper studies the effect of stacking sequence on mechanical response and failure patterns of glass/epoxy hybrid composites under low velocity impact. a comprehensive set of drop test impact tests were done on single fiber laminate and 2 different types of interlayer woven glass-carbon/epoxy hybrid laminates. it was found that using carbon fiber as top and bottom layers for composites (c/g and c/g/c/g laminates) helps to increase the maximum load, reduce the damage size and maximum displacement. in addition the hybrid laminates absorbed 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hezam7 1military academy, university of defence, belgrade, serbia 2department of public safety, government of the brčko district of bosnia and herzegovina, brčko, bosnia and herzegovina 3the college of tourism, academy of applied studies belgrade, belgrade, serbia 4department of mathematics, riphah international university, lahore, pakistan 5college of computer science and information technology, wasit university, wasit, iraq 6technical engineering college, al-ayen university, thi-qar, iraq 7department of statistics and operations research, college of sciences, king saud university, riyadh, saudi arabia orcid ids: darko božanić https://orcid.org/0000-0002-9657-0889 adis puška https://orcid.org/0000-0003-3274-0188 duško tešić https://orcid.org/0000-0002-5277-3270 anđelka štilić https://orcid.org/0000-0002-9131-1642 kifayat ullah https://orcid.org/0000-0002-1438-6413 yousif raad muhsen https://orcid.org/0000-0002-4765-4189 ibrahim m. hezam https://orcid.org/0000-0003-2747-6722 abstract. the paper presents a multi-criteria decision-making (mcdm) model designed to rank combined construction machines specifically, backhoe loaders during procurement for military needs. however, the model can also be applied to construction companies. the ranking is based on criteria specifically defined for this research. the study found that most criteria relate to the structural elements of the backhoe loader, which is also significant for manufacturers working on improving these types of machines. the mcdm model is built on two methods: analytic hierarchy process (ahp) and multi-attributive border approximation area comparison (mabac), both adapted using fuzzy numbers. the ahp method was modified with type 2 fuzzy numbers to calculate criteria's weight coefficients. the mabac method, using classic triangular fuzzy numbers, is employed for ranking alternative solutions. validation of the results involved two steps. first, a sensitivity analysis was performed by modifying the weight coefficients of the criteria. second, a comparative analysis with other methods was performed. the validation process confirmed the stability of the obtained results. key words: fuzzy ahp, fuzzy mabac, mcdm, backhoe loader received: august 01, 2025 / accepted september 24, 2025 corresponding author: darko božanić military academy, university of defence in belgrade, veljka lukića kurjaka 33, 11040, belgrade, serbia. e-mail: darko.bozanic@va.mod.gov.rs. 606 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam 1. introduction decision-making is an integral and essential segment of the planning process, both in the implementation of tasks within companies and in certain segments of life [1]. increasingly, the decision-making process requires the application of special methods, which significantly simplify this process [2, 3]. in this sense, numerous methods have been developed in the field of multi-criteria decision-making (mcdm) to smooth this process, and accordingly, many, most often hybrid, models are being established [4]. given that decision-making processes are often accompanied by a series of uncertainties, mcdm methods are most often combined with various mathematical areas that have the ability to treat uncertainty, such as fuzzy, rough, grey, and other numbers [5]. the most common approach to addressing uncertainty in mcdm is the use of fuzzy numbers. mcdm models have also gained a role in the military. a large portion of the papers focuses on different processes for selecting or ranking locations [6], resources [7], and actions [8, 9]. in other words, mcdm is an area that is recognized and highly valued for addressing decision-making challenges in the military. a backhoe loader is a versatile construction machine designed for a wide array of construction and infrastructure tasks. it plays a significant role in both civilian and military sectors. depending on the attachment used, these machines can be employed for soil excavation, material loading, pushing and spreading materials, lifting loads, and transporting trees. in the military, their importance is notable because they are small, agile, compact, quick, and adaptable. the tasks they perform in the army are diverse, including digging trenches, creating or removing obstacles, and building temporary military facilities related to establishing units and command posts. the multifunctionality of these machines makes them highly valuable for setting up units, including constructing trenches, roads, and shelters, as well as building shelters under harsh conditions often encountered during these activities. the specific use of a backhoe loader and the structure of engineering units in the serbian army determine that its primary function is land excavation and material loading. in contrast, the application in construction and other industries tends to be broader and more varied. the selection of assets in public enterprises, including those in the military, is most often conducted through the public procurement system. most public procurement processes focus mainly on the price of the purchased product. this approach can influence the quality of the acquired product. to avoid emphasizing only the product's price, a hybrid mcdm model was developed in this paper. this approach shifts the focus from price to other key parameters that are important when procuring a backhoe loader for the serbian army units. in other words, the practical goal of this study is to improve the process of equipping army units with the combined construction machine the backhoe loader. the motivation for developing such a model stems from the lack of a standard methodology for solving this type of problem. the new model offers a proposal for standardizing this type of procurement. the issue of selecting a backhoe loader for military and other purposes has not been discussed in the literature so far. some papers address specific aspects related to backhoe loaders, mostly focusing on design improvements and analyses [10, 11], and their integration into particular sectors [12, 13]. however, more studies using mcdm techniques can be found when it comes to selecting other construction machines and equipment. for example, study [14] proposes a model for selecting the appropriate excavation machine for a construction site using ahp and promethee methods. in the paper of ghorabaee et al. [15], the authors developed fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 607 an mcdm model based on swara, critic, and edas methods to evaluate construction equipment considering potential environmental impacts. in [16], fuzzy topsis and fuzzy vikor methods were used to select a hydraulic excavator for open-pit mines. hagag et al. [17] offer an overview of the literature on applying mcdm for machine selection in manufacturing and construction. deepak et al. [18] focus on optimizing concrete pump maintenance in the construction industry using mcdm methodologies. additionally, in [19], an anfis model was introduced for loader selection. the use of ahp methods for selecting earthmoving equipment in construction projects was discussed in [20]. the ahp method has been used for solving mcdm problems for a very long time. this method has found applications in contemporary research, both in its basic form and through various modifications. its use can be seen in works across different fields, such as human resources [21], waste management [22], construction projects [23], military science [24], industry [25], economic assessments [26], and others. similar to the ahp method, although much younger, the mabac method is widely applied in many fields, such as mechanical engineering [27], economics [28, 29], management [30], risk assessment [31], and more. the literature review reveals that the problem of selecting a backhoe loader has not been thoroughly studied, creating a research gap that needs addressing. meanwhile, applying mcdm methods in selecting construction machinery has proven to be very effective. therefore, implementing an mcdm model to solve the backhoe loader selection problem is highly beneficial. additionally, although this issue has been addressed for military purposes, future research on the weighting of criteria could facilitate its application in traditional construction companies. this paper offers several original contributions: 1) it defines specific criteria that are important when choosing a backhoe loader; 2) it performs the calculation of weighting criteria for the defined criteria; 3) it demonstrates models practical use in ranking alternatives; 4) it shows the quality of implementing the ahp and mabac methods for addressing modern challenges. 2. description of the methods applied 2.1. model description the main goal of the model is to assist decision makers in selecting the best available alternative. this goal also guides the application of certain mcdm methods [32]. the model presented in this paper primarily consists of two methods: the ahp method enhanced by applying interval, type 2, fuzzy numbers (f2ahp), and the mabac method, which is fuzzified using type 1 fuzzy numbers (fmabac). the f2ahp method is used to establish the weight coefficients of the criteria, whereas the ranking of alternatives is carried out using the fmabac. as evident, different fuzzy approaches are used for fuzzification, reflecting varying degrees of uncertainty in defining the criteria's weight coefficients and ranking the alternatives. the phases of the f2ahp-fmabac model are illustrated in fig. 1. 608 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam phase 1: selection of experts experts competence evaluation phase 2: defining selection criteria survey sensitivity analysis comparative analysis fmabac method f2ahp method phase 5: model validation phase 4: ranking of alternatives phase 3: calculation of weight coefficients of criteria model phases methods fig. 1 f2ahp-fmabac model as presented in fig. 1, the model executed in 5 phases: ▪ phase 1: during this phase, the selection and evaluation of experts' competence are conducted. the competence of experts can be measured using some well-known methods for calculating expert competence [33]. the competence coefficient (k) ranges from 0 to 1, where a coefficient of k=1 indicates a highly competent expert, and lower values indicate less competence. of course, the minimum competence level depends on the available experts. in this research, only experts with a minimum competence coefficient of 0.5 are considered. ▪ phase 2: at this phase, the key criteria are determined through surveys and consultations with experts who influence the selection of the backhoe loader. ▪ phase 3: in the third phase, criteria are compared in pairs, and the criteria weight coefficients are calculated using the ahp method, which is fuzzified by interval fuzzy numbers. ▪ phase 4: in the fourth, the optimal alternative is identified through the application of fuzzy mabac. ▪ phase 5: during this phase, the model was validated through sensitivity analysis and comparative analysis. this paper focuses on presenting the decision-making model itself. in the next section, basic information on fuzzy numbers of types 1 and 2 will be provided, along with a description of the f2ahp and the fmabac method. the phases related to selecting and fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 609 evaluating experts' competencies and identifying key criteria are not specifically explained, as they are part of a standard procedure. 2.2. fuzzy numbers type 1 and type 2 the foundation of fuzzy logic was established by lotfi zadeh [34]. it proved to be a highly effective tool for describing uncertainties, which are very common in decisionmaking processes [35, 36]. essentially, using fuzzy logic allows spoken language and human knowledge to be translated into mathematical terms across various areas, which can then be processed in different ways using fuzzy arithmetic [37, 38]. this paper uses fuzzy numbers type 1 and type 2. type 1 fuzzy numbers represent the beginning of fuzzy logic but are still widely used today [39, 40]. figure 2 shows triangular fuzzy type 1 numbers, which are used during the fuzzification process of the mabac method. t1 t2 t3 1 ( )t x  0 fig. 2 triangular fuzzy number with the further development of fuzzy logic, the question arises if fuzzy numbers, which describe uncertainty, explain why there was no uncertainty in the membership function defining fuzzy numbers. this was followed by the extension of type 1 fuzzy numbers to include interval fuzzy numbers, known as type 2. in this case, the membership function is also presented as uncertain, as shown in fig. 3. 0 0.2 1 0 0.8 1 d eg re e o f m em b er sh ip 1 1 x=0.7 0.6 0.4 0.2 0.2 0.4 0.6 0.8 a) 0.4 0.6 0.8 0 b) 0.80.60.40.20 x=0.7 fig. 3 membership function type -1 (a) and type-2 (b) 610 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam triangular interval fuzzy numbers are used to fuzzify the ahp method, as shown in fig. 4. 1 k uuullllu mu,ml d eg re e o f m em b er sh ip umf lmf f oo tp ri nt o f u nc er ta in ty f ootprint o f u ncertainty fig. 4 triangular fuzzy number type 2 in interval fuzzy numbers, there are two membership functions describing uncertainty, which are not included in fuzzy numbers type 1: upper membership functions (umf) and lower membership functions (lmf). more information about fuzzy numbers type 2 can be found in [41, 42, 43]. 2.3. the ahp method fuzzification the ahp was developed by thomas saaty [44]. the standard for this method is saaty’s scale, used for comparing criteria or alternatives two at a time [45]. although many alternative scales have been created, saaty's scale remains the most widely used [46]. the widespread use of ahp has led to many modifications, one of the most common being its fuzzification. this process usually involves fuzzifying saaty’s scale. currently, there are various methods to fuzzifying saaty’s scale, which can generally be categorized into two types: sharp and soft fuzzification. sharp fuzzification involves establishing the confidence interval of saaty's scale values before conducting pairwise comparisons [47, 48, 49]. some studies utilize the principle of saaty’s scale but with fewer comparison options, such as a six-point scale [50, 51] or a five-point scale [52, 53]. fuzzification is usually performed using a triangular fuzzy number t = (t1, t2, t3) = (x-1, x, x+1), where x represents a standard value from saaty’s scale. in the papers [54, 55, 56], saaty’s scale is fuzzified using the fuzzy number t = (x-, x, x+), where  is taken from the interval 0.5 2  . further analysis shows that some papers also use fuzzifications by applying other types of fuzzy numbers (gauss curve, trapezoidal, and the like), as well as interval fuzzy numbers [57, 58]. however, most papers primarily use triangular fuzzy numbers. the second group of fuzzification is “soft” fuzzification of saaty's scale. here, the confidence interval is defined with fuzzy numbers after the pairwise comparison based on a new parameter the degree of uncertainty (β). one type of such fuzzification is presented in the papers [59, 60]. in the aforementioned papers, the degree of uncertainty related to the accuracy of all comparisons in the table is defined at the level of saaty’s scale. it ranges from 0 to 1, where 1 indicates that the decision-makers are confident in the pairwise comparisons they performed, and vice versa. based on the level of uncertainty, fuzzy numbers are calculated. this method works well for group decision making, but for fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 611 individual decision making, the results compared to the classic ahp method are not significantly different. the second approach of the so-called "soft" fuzzification was developed based on the previous one, with the difference that the decision-makers define the degree of certainty (γ) for each comparison they make [61]. the degree of certainty ranges from 0 to 1, where the value 1 indicates a high degree of certainty, and vice versa. as the previous two fuzzifications are further developed, the question arises whether other factors might influence the final calculation of the weight coefficients of the criteria or alternatives. clearly, the question arises whether all decision-makers, including experts, are equally capable of making decisions. in that regard, a new approach has been developed that uses the level of competence of decision makers (ke), where e(1,2, ... k), and k represents the total number of experts or decision makers. on the other hand, considering that the process of defining the degree of conviction (γ) cannot be completely precise, interval fuzzy numbers are chosen to better handle the uncertainty. the general form of the fuzzy number shown in the fuzzification is (fig. 4): ( , , , , )u l l ut l l m u u= (1) where mu=ml=m. the expressions used for calculating the interval fuzzy number t are presented in table 1 [62]. table 1 fuzzified values of saaty’s scale definition standard values interval fuzzy number the same importance 1 (1,1,1,1,1) weak dominance 3 2 2(3 ,3 ,3,(2 )3,(2 )3)ji ji ji ji   − − strong dominance 5 2 2(5 ,5 ,5,(2 )5,(2 )5)ji ji ji ji   − − very strong dominance 7 2 2(7 ,7 ,7,(2 )7,(2 )7)ji ji ji ji   − − absolute dominance 9 2 2(9 ,9 ,9,(2 )9,(2 )9)ji ji ji ji   − − in between values 2, 4, 6, 8 2 2( , , ,(2 ) ,(2 ) );ji ji ji jix x x x x   − − 2,4,6,8x = fuzzy number ( , , , , )u l l ut l l m u u= ,  1,9x must meet the following conditions:   2 2 2 , 1 , 1,9 1, 1        =    ji ji u ji x x x l x x (2)   , 1 , 1,9 1, 1       =    ji ji l ji x x x l x x (3)  , 1,9=  m x x (4)  (2 ) , 1,9l jiu x x= −   (5)  2(2 ) , 1,9u jiu x x= −   (6) 612 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam based on the provided expressions, the example is shown in fig. 5. the comparison used as an example is "4 the value between weak and strong dominance", with an expert competence k = 0.75, and different levels of certainty γ[1, 0.6, 0.4]. 1 0.75 7.366.41.60.64 441.44 2.4 5.6 6.563 γ=1 γ=0.4 d eg re e o f m em b er sh ip γ=0.6 fig. 5 example of a comparison from saaty’s scale using interval fuzzy numbers the expressions for calculating the inverse interval fuzzy number 1 (1/ ,1/ ,1/ ,1/ ,1/ )u l l ut u u m l l− = are shown in table 2. table 2 inverse fuzzified values of saaty’s scale definition inverse value interval fuzzy number the same importance 1 (1,1,1,1,1) weak dominance 1/3 2 2(1/ (2 )3,1/ (2 )3,1/ 3,1/ 3 ,1/ 3 )ji ji ji ji   − − strong dominance 1/5 2 2(1/ (2 )5,1/ (2 )5,1/ 5,1/ 5 ,1/ 5 )ji ji ji ji   − − very strong dominance 1/7 2 2(1/ (2 )7,1/ (2 )7,1/ 7,1/ 7 ,1/ 7 )ji ji ji ji   − − absolute dominance 1/9 2 2(1/ (2 )9,1/ (2 )9,1/ 9,1/ 9 ,1/ 9 )ji ji ji ji   − − in between values 1/2, 1/4, 1/6, 1/8 2 2(1/ (2 ) ,1/ (2 ) ,1/ ,1/ ,1/ );ji ji ji jix x x x x   − − 2,4,6,8=x inverse fuzzy number 1 (1/ ,1/ ,1/ ,1/ ,1/ )u l l ut u u m l l− = ,  1,9x must satisfy the following conditions:   2 2 2 1/ , 1/ 1/ 1 1/ , 1,9 1, 1/ 1        =    ji ji u ji x x x l x x (7)   1/ , 1/ 1/ 1 1/ , 1,9 1, 1/ 1       =    ji ji l ji x x x l x x (8)  1/ , 1,9=  m x x (9)  1/ 1/ (2 ) , 1,9l jiu x x= −   (10) fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 613  21/ 1/ (2 ) , 1,9u jiu x x= −   (11) in the next section, the standard ahp process is followed to determine the weight vector w. once the weight vectors are obtained, defuzzification of w is carried out using the method described in [58]: ( ) ( ) ( ) ( ) 3 3 2 u u u u l l l l u l u l m l u l m l l k l w − + − − + −  + + +    = (12) when discussing the degree of certainty (), it could be set in two ways: 1) as a percentage or 2) with fuzzy linguistic descriptors. in this paper, fuzzy linguistic descriptors are used, as shown in fig. 6. 0.2 0.4 0.5 0.6 0.8 1 d eg re e o f m em b er sh ip 1 highmediumsmall fig. 6 fuzzy linguistic descriptors for the evaluation of the degree of certainty 2.4. fuzzy mabac the mabac was first presented in 2015 by pamučar and ćirović [63]. this approach is based on calculating the border approximation area and the distance of alternatives from this area. although it is a relatively recent approach, it has been cited in many papers. the fundamental steps of the fmabac are outlined in table 3. table 3 presentation of the fmabac method steps step number step name step 1 forming of the initial decision matrix ( x ). step 2 normalization of the initial matrix elements ( n ) step 3 calculation of the weighted matrix (v ) elements step 4 determination of the approximate border area matrix ( g ). step 5 calculation of the matrix elements of alternatives distance from the border approximate area ( q ) step 6 defuzzification of the obtained values step 7 ranking of alternatives a more detailed presentation of the fmabac method can be found in [64]. 614 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam 3. results 3.1. criteria description and definition of weight coefficients developing criteria for ranking backhoe loaders involves two steps. first, a review of the existing literature was conducted to identify an initial set of criteria. from this analysis, 11 criteria were established. these criteria were then presented to experts, who had the option to add, refine, modify, or remove any criteria they found unnecessary. at the conclusion of this process, the experts identified six criteria that will be used to rank the alternatives. the criteria are outlined as follows: c1 digging depth is a segment that directly depends on the design of the backhoe loader, such as the excavator arm and its capabilities. the potential for greater digging depth broadens the backhoe loader's range of capabilities and increases its suitability for more tasks. the value of the alternatives based on this criterion is given in meters. c2 capacity of the loading tool (loading bucket) is a fundamental parameter for calculating the loader's performance, indicating how many cubic meters of material a backhoe loader can load per unit of time. the values of the alternatives based on this criterion are presented in cubic meters. c3 – capacity of the standard digging tool (excavator bucket) is the key parameter when calculating excavator performance, indicating how many cubic meters backhoe loaders can dig in a given amount of time. the value of alternatives based on this criterion is measured in cubic meters. c4 – constructional features criterion covers various construction segments of the backhoe loader, which experts believe should not be considered separate criteria because their individual impact is minimal. when combined into one criterion, their influence increases. this includes factors such as the comfort provided to the operator, the training time needed for the tool, the variety and replaceability of working tools, engine power, speed, unloading height, and more. c5 backhoe loader price criterion reflects the market value of the machine. the unit of measure for this criterion is €. c6 – maintenance costs. this criterion considers various factors that influence maintenance expenses, such as the length of the warranty period, parts availability, reliability, and the accessibility and speed of service for routine maintenance and repairs. the criterion is especially important because experience so far shows that resources continue to be used extensively after the warranty expires. according to the previously described criteria, four numerical (c1, c2, c3, and c5) and two linguistic (c4 and c6) criteria stand out. the criteria c1, c2, c3, and c4 are benefit-type, while c5 and c6 are cost-type criteria. to describe the linguistic criteria, fuzzy linguistic descriptors were used, as shown in fig. 7. as noted in fig. 7, linguistic criteria are described with five fuzzy descriptors (d1 to d5). the significance of each linguistic descriptor relative to the criterion is given in table 4. as previously mentioned, the f2ahp was used to evaluate the criteria weight coefficients. each expert individually compared the criteria in pairs, applying saaty’s scale, and determined the degree of certainty for each comparison using fuzzy linguistic descriptors, as shown in fig. 6. the initial decision-making matrix for the first expert is presented in table 5 numbers outside the brackets represent the comparison of two criteria, while the values inside the brackets indicate the degree of certainty in the statement. fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 615 0.25 0.5 0.75 1 d eg re e o f m em b er sh ip 1 d5d3d1 d2 d4 0 0 0.5 fig. 7 overview of fuzzy linguistic descriptors table 4 description of fuzzy linguistic descriptors by criteria linguistic descriptor c4 c6 d1 very bad very small d2 bad small d3 average average d4 good high d5 very good very high table 5 initial decision-making matrix for expert 1 for defining the criteria weight coefficients c1 c2 c3 c4 c5 c6 c1 1 2 (h) 6 (m) 7 (s) 5 (m) 1 (m) c2 1/2(h) 1 4 (h) 5 (m) 2(h) 1/2 (vh) c3 1/6 (m) 1/4 (h) 1 3 (h) 1/3 (s) 1/5 (m) c4 1/7 (s) 1/5 (m) 1/3 (h) 1 1/2 (m) 1/5 (m) c5 1/5 (m) 1/2 (h) 3 (s) 2 (m) 1 1/5 (s) c6 1 (m) 2 (vh) 5 (m) 5 (m) 5 (s) 1 (cr=0.06<0.10) furthermore, the quantification of fuzzy linguistic descriptors, as shown in fig. 6, is performed based on the degree of certainty when comparing criteria in pairs. defuzzification of these values is carried out using the following expression [65]: 3 1 2 1 1(( ) ( )) / 3= − + − +a t t t t t (13) using eq. (13), the values given in table 6 are obtained. 616 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam table 6 defuzzified values of the degree of certainty descriptor name value after defuzzification small (s) 0.13 medium (m) 0.5 high (h) 0.87 very high (vh) 1 the next step is to fuzzify the initial decision-making matrix using the expressions from tables 1 and 2. an example calculation is provided for comparing criteria c1 and c2, where the relation between the two criteria is defined as two, the certainty is indicated by the fuzzy linguistic descriptor "high," and the coefficient of competence k equals 0.5. 22*0.87 1.51= =ul 2*0.87 1.74= =ll 2=m (2 0.87)*2 2.26lu = − = 2(2 0.87 )*2 2.49uu = − = in table 7, the fuzzified initial decision-making matrix for expert 1 is provided: table 7 fuzzified initial decision-making matrix for the expert 1 c1 c2 ... c6 c1 (1,1,1,1,1) (1.51, 1.74, 2, 2.26, 2.49) ... (1,1,1,1,1) c2 (0.4, 0.44, 0.5, 0.57, 0.66) (1,1,1,1,1) ... (0.5, 0.5, 0.5, 0.5,0.5) c3 0.1,0.11,0.17,0.33, 0.67) (0.2, 0.22, 0.25, 0.29, 0.33) ... (0.11, 0.13, 0.20, 0.4, 0.8) c4 (0.07, 0.08, 0.14, 1,1) (0.11, 0.13, 0.2, 0.4, 0.8) ... (0.11, 0.13, 0.20, 0.4, 0.8) c5 (0.11, 0.13, 0.2, 0.4, 0.8) (0.4, 0.44, 0.5, 0.57, 0.66) ... (0.1, 0.11, 0.2, 1,1) c6 (1,1,1,1,1) (2,2,2,2,2) ... (1,1,1,1,1) additionally, the classic steps of the ahp method are executed using standard fuzzy arithmetic. finally, fuzzy weight coefficients for each criterion, calculated separately for each expert, are defuzzified using eq. (12). the weight coefficients of the first expert are displayed in table 8. table 8 weight coefficients of the criteria for expert 1 criterion classic ahp method (wi) f2ahp (wi) c1 0.328 0.300 c2 0.177 0.181 c3 0.061 0.076 c4 0.040 0.059 c5 0.088 0.103 c6 0.305 0.282 in table 8, in addition to the f2ahp method, the criteria weight coefficients obtained by applying the classic ahp are also shown. as noted in the table, there are differences fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 617 between the results obtained. the value range using the ahp method extends from 0.040 to 0.328, while in the application of the f2ahp, it is significantly lower, ranging from 0.059 to 0.300. based on this, the expected conclusion is that using the f2ahp, the results specifically, the weight coefficients of the criteria are similar but not identical. the ranking of criteria has been maintained, indicating that pairwise comparison remains the most valued approach. however, by examining the ratio of the weight coefficients, it is clear that they are not the same for both methods. this suggests that in some other applications, the criteria rank may differ when applying the ahp versus the f2ahp. the same conclusions can be drawn from the results provided by other experts. for the final definition of weight coefficients, it is required to convert the existing weight coefficients set into a single value, which is known in the literature as an aggregated weight coefficient. the calculation of this coefficient can be performed in several ways; in this paper, it is done by applying the following synthesis of individual expert decisions using the geometric mean method (gmm) [66]. table 9 shows the final aggregated weight coefficients for the criteria used to select backhoe loaders. table 9 aggregated (final) weight coefficients of the criteria criterion f2ahp (wi) c1 0.295 c2 0.184 c3 0.082 c4 0.069 c5 0.104 c6 0.266 3.2. selection of backhoe loaders there are many manufacturers of backhoe loaders on the market, and almost all of them have developed several different models. six alternatives were identified for selecting backhoe loaders. the assessment of these alternatives based on each criterion (initial decision-making matrix) is presented in table 10. table 10 initial decision-making matrix ( x ) c1 c2 … c4 c5 c6 a1 (4.04,4.24,4.24) (0.96,1.2,1.26) … d5 (89000,91000,94000) d2 a2 (4.14,4.44,4.44) (0.8,1.1.05) … d4 (72000,75000,79000) d3 a3 (4.14,4.44,4.44) (1.04,1.3,1.37) … d3 (76000,79000,88000) d5 a4 (4.7,4.8,5.7) (0.8,1.1.05) … d3 (84000,89000,98000) d4 a5 (4.6,5.6,5.6) (1.01,1.26,1.33) … d3 (98000,102000,107000) d4 a6 (5.2,5.8,5.8) (0.8,1,1.05) … d4 (102000,104000,111000) d5 next, the quantification of linguistic descriptors or the fuzzification of the initial decision-making matrix was performed, as demonstrated in table 11. 618 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam table 11 fuzzified initial decision-making matrix c1 ... c4 c5 c6 a1 (4.04,4.24,4.24) ... (0.75,1,1) (89000,91000,94000) (0,0.25,0.5) a2 (4.14,4.44,4.44) ... (0.5,0.75,1) (72000,75000,79000) (0.25,0.5,0.75) a3 (4.14,4.44,4.44) ... (0.25,0.5,0.75) (76000,79000,88000) (0.75,1,1) a4 (4.7,4.8,5.7) ... (0.25,0.5,0.75) (84000,89000,98000) (0.5,0.75,1) a5 (4.6,5.6,5.6) ... ((0.25,0.5,0.75) (98000,102000,107000) (0.75,1,1) a6 (5.2,5.8,5.8) ... (0.5,0.75,1) (102000,104000,111000) (0.75,1,1) the steps of the fmabac method, shown in [64], are further applied. finally, the criteria functions of the alternatives are calculated, and ranking is performed based on them, as shown in table 12. 4. model validation 4.1. sensitivity analysis sensitivity analysis has become an essential component of models based on mcdm [67, 68]. the most common method involves sensitivity analysis by adjusting the criteria weight coefficients [69, 70], which is also used here. since this study highlights two criteria with higher weight coefficients (c1 and c6), the analysis was conducted for both. for each criterion, nine scenarios of weight coefficient changes were tested, where the weight of criteria c1 and c6 was reduced by 10%, with the remaining weight redistributed among the other criteria. the weight coefficients for each scenario when reducing criterion c1 are shown in fig. 8. fig. 8 weight coefficient in different scenarios when reducing the weight coefficient of criterion c1 table 12 rank of alternatives is . idef s rank a1 (-0.252,0.089,0.385) 0.074 1 a2 (-0.337,0.011,0.33) 0.001 5 a3 (-0.362,-0.042,0.285) -0.039 6 a4 (-0.345,0,0.479) 0.045 2 a5 (-0.356,0.062,0.389) 0.032 4 a6 (-0.3,0.045,0.364) 0.036 3 fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 619 scenarios for lowering the weighting coefficient of criterion c6 are shown in a similar manner (fig. 9). fig. 9 weight coefficient in different scenarios when reducing the weight coefficient of criterion c6 the ranking of alternatives in the sensitivity analysis when decreasing the weight coefficient c1 is shown in fig. 10. it can be observed that the top-ranked alternative remains consistently a1, but there are notable shifts in the ranking of the other criteria. these changes are expected for two reasons. first, criterion c1 has an extremely high weight coefficient, so larger shifts in rank are likely when the weight of c1 changes significantly. second, the criterion function values for alternatives a4, a5, and a6 are very close, so even minor changes in the weight coefficient can cause rank changes. a similar pattern occurs when analyzing the changes in the weight coefficients for criterion c6 (fig. 11). in this case, alternative a6 moves to the first rank after the second scenario and remains there until the end. meanwhile, alternative a1 drops from the first position to finally rank at number 5. this decline for a1 is due to its evaluation based on criterion c6. fig. 10 ranking of alternatives by applying different scenarios when reducing the weight coefficient of criterion c1 620 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam fig. 11 ranking of alternatives based on different scenarios when reducing the weight coefficient of criterion c6 although the results are fairly stable and expected, it is useful to verify and analyze them using a correlation coefficient. in this case, spearman's rank correlation coefficient (srcc) was used. the values of srcc for the analysis when reducing the weight coefficient of criterion c1 are shown in fig. 12. it can be seen that the values of srcc from s1 to s3 are high and approach an ideal correlation. in this part, the weight of criterion c1 is reduced by up to 30%. as we move forward, the weight coefficient of criterion c1 decreases from 40% to 60% (s4 to s6), and srcc drops accordingly, but overall, the results stay satisfactory. finally, with a substantial reduction in criterion c1 (70% to 90%), there is a notable decline in srcc when comparing the initial scenario (s0) and scenario s1 with scenarios s7 to s9. this outcome is expected, considering that the weight coefficient of criterion c1 is 0.295 in scenario s0, while in scenarios s7 to s9, it falls below 0.1. a similar pattern appears when criterion c6 is reduced (fig. 13). fig. 12 values of srcc when reducing the weight coefficient of criterion c1 fuzzy ahp fuzzy mabac model for ranking a combined construction machine backhoe loader 621 fig. 13 values of srcc when reducing the weight coefficient of criterion c6 the sensitivity analysis performed on the two most important criteria showed that the results are stable and fall within expected variations. the analysis also indicated that small errors in defining the weight coefficients do not influence the final alternatives ranking. 4.2. comparative analysis comparative analysis has become an essential part of validating results in the development of mcdm models [71, 72]. in this paper, the results obtained are compared with those from fuzzy topsis [73] and fuzzy mairca [74] methods. figure 14 displays the ranking of alternatives using the mentioned methods. it is evident that alternative a1 ranks first in all cases. it is also apparent that alternatives a2 and a3 are always last or second to last. alternatives a4, a5, and a6 experience rank fluctuations. these changes are due to differences in the mathematical methods used and the close results among these three alternatives. specifically, even with the fmabac method, these three alternatives have very similar values for the criterion functions. overall, it can be stated that the results obtained using the fmabac method are consistent, while the outcomes of other methods are more likely to vary. additionally, srcc supports this conclusion, ranging from 0.77 to 0.94. fig. 14 ranking of alternatives using different mcdm methods 622 d. božanić, a. puška, d. tešić, a. štilić, k. ullah, y. r. muhsen, i. m. hezam 5. conclusion the hybrid model f2ahp-fmabac has been successfully used to determine the criteria weight coefficients for selecting backhoe loaders and for choosing the best option among the alternatives. by applying the f2ahp method, several levels of uncertainty have been effectively addressed, and potential dilemmas faced by decision-makers have been quantified. the common dilemma of decision-makers when comparing alternatives in pairs using saaty's scale is not overlooked; instead, during weight coefficients calculation, it is quantified by the degree of certainty they have in their comparisons. additionally, the potential knowledge and experience of decision-makers are quantified via their competence coefficient. notably, pairwise comparison remains a key element in defining the weight coefficients of criteria. the degrees of certainty and competence lead to smaller variations in the criteria's weight coefficients, which can become quite significant under greater uncertainty or lower competence levels. when decision-makers are fully confident in their knowledge ( = 1) and entirely competent (k = 1), the standard ahp method is used without fuzzification, emphasizing the importance of improving the ahp method itself. the fmabac method has been successfully used to select the best alternative, considering the uncertainties involved in this type of decision-making. this is especially true when there are linguistic criteria, which is also the case in this study. the defined criteria clearly indicate that the structural features of backhoe loaders are key to the selection process. this is demonstrated by the weight coefficients of the first four criteria related to structural solutions, which together have a total weight of 0.63. additionally, the last criterion is related to structural characteristics in some segments, although it mainly addresses economic costs. criterion c6, which pertains to the purchase price of backhoe loaders, has a weight coefficient of 0.104. this suggests that price is not the most important factor when buying this type of construction equipment. model validation demonstrated that the results are consistent. this was first shown through sensitivity analysis, which involved reducing the weight coefficients of the two most significant criteria. the results stayed within expected variations. additionally, the stability of the results was confirmed by 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(eds) proceedings of international conference on data science and applications. lecture notes in networks and systems, vol. 551, pp. 467-480, springer, singapore. 74. mestanza, j.g., bakhat, r., 2021, a fuzzy ahp-mairca model for overtourism assessment: the case of malaga province, sustainability, 13(11), 6394. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 293 300 doi: 10.22190/fume1603293k original scientific paper the boundary element method for viscoelastic material applied to the oblique impact of spheres udc 539.3 stephan kusche department of system dynamics and the physics of friction, tu berlin, germany abstract. the boundary element method (bem) for elastic materials is extended to deal with viscoelastic media. this is obtained by making use of a similar form of the fundamental solution for both the materials. some considerations are attributed to the difference of the normal and the tangential contact problem. both normal and tangential problems are furthermore assumed to be decoupled. then the oblique impact of hard spheres with an incompressible viscoelastic half-space (linear standard-model) is studied. by assuming stick conditions during impact, one obtains the dependence of the two coefficients of restitution as functions of two input parameters. this result is expressed in an elegant and compact form of the fitting function. key words: contact mechanics, boundary element method, linear viscoelastic material, rebound test, oblique impact, tangential problem 1. introduction the history of analytical research of the spheres’ field of the impact started with hertz [1] in 1882. his theory of normal contacts has been extended to tangential contact theories, considering partial slip [2-9]. like in other fields of contact mechanics, a solution in the case of viscoelastic time dependent material behavior is difficult and its closed form has not been found yet. for the pure normal impact of spheres hunter [10], sabin [11] and calvit [12] discovered some analytical solutions. these solutions are given in an implicit form; some kind of numerical treatment is needed in order to obtain results. in the field of elastic contact mechanics, the boundary element method (bem) is well established. this method is based on the fundamental solution for elastic half-space. since the fundamental solution for the viscoelastic half-space is known as well, it should be possible to adapt many procedures from elastic to viscoelastic problems. this can be achieved by using received september 22, 2016 / accepted november 18, 2016 corresponding author: stephan kusche institute of mechanics, berlin institute of technology, str. d. 17. juni 135, 10623 berlin, germany e-mail: s.kusche@tu-berlin.de 294 s. kusche the prony series for modeling the material behavior. in this paper the given approach is used to study the viscoelastic impact problem of spheres. 2. problem description the problem under consideration is the following: a sphere of mass m and radius r has initial velocities vx0 s , vz0 s and an initial angular velocity ω0 s . after rebound from viscoelastic ground the velocities change to vx1 s , vz1 s and ω1 s (see fig. 1). these velocities have to be calculated. to simplify this problem approach it will be assumed, that the normaland the tangential-contact problems are decoupled. this assumption becomes true under two circumstances. firstly, the sphere is a rigid body and the viscoelastic half-space is incompressible. then the normal load will not lead to any tangential displacement and vice versa. secondly, the displacement of the centre of the sphere in tangential direction ux s is assumed to be much smaller than radius r of the sphere. then only tangential contact force fx has to be taken into account for the calculation of the resulting moment acting on the sphere. furthermore, the contact area will conserve a rotational symmetric shape and the action line of resulting normal force fz points toward the centre of the sphere. fig. 1 velocities before and after impact (left hand side). displacement of the center of the sphere and resulting contact forces during impact (right hand side) with these considerations tangential velocity vx s and angular velocity ω s are not independent of each other. by using the principle of impulse and angular momentum one obtains: 22 , 5 s s x x xmu f mr f r    , (1) and this results in: 0 0 2 ( ) 5 s s s s x xv v r     . (2) this means that the after-impact velocities can be expressed by two numbers: 1 1 11 0 0 0 0 , s s ss x kz z xs s s s z x k v r vv e e v v r v           . (3) the boundary element method for viscoelastic material applied to the oblique impact of spheres 295 for the normal contact problem the coefficient of restitution ez is used. in tangential direction the velocity of the contact point is the major variable. the ratio of the contact point’s velocity after and before impact ex is the second number used in this paper. the tangential and angular velocity after the impact can be calculated from this number. the viscoelastic half-space is modeled by the linear standard-model. the maxwell-element and the kelvin-voigt-element are included as special cases. for the linear standard-model the time dependent relaxation modulus to shear has the following form: ( ) exp t g t g g           . (4) it turns out that the numerical solution for ex and ez can be expressed by the following two dimensionless numbers: 1/5 1/5 0 0 1 22 3 2 3 , , s s z zrv rv g m g m g                      . (5) at first glance it seems to be unreasonable that neither the tangential nor the angular velocity appears therein. but, under consideration that normal and tangential contact problems are decoupled, at least the coefficient of restitution ez cannot be dependent on any tangential velocity. then again, the tangential problem is, under the assumption of a full stick regime, only dependent on the contact point’s velocity and the contact area’s shape during the impact. since the system is linear, the magnitude of the contact point’s velocity has no significance. it is finally used for normalization of ex, as this has been done in the definition stated before. both numbers, δ1 and δ2, can be adapted to the special cases of the material model. the maxwell-element is obtained if g∞ tends towards zero, which means that δ2 becomes infinite. the kelvin-voigt-element is obtained if g tends toward infinity but η is held constant. since δ1~ηg -3/5 and δ2~η, the parameter δ1 becomes zero. finally in the elastic case η can be obtained to be either zero or infinity, which correspondents to an infinite soft or an infinite stiff damper. this means, expressed by the parameters, that either δ2 becomes zeros (and δ1 is finite, the elastic constant is g∞) or δ1 tends towards infinite (and δ2 is finite, the elastic constant is then equal to g∞+g). summarized this means: 2 1 1 2 maxwell-element: kelvin-voigt-element: 0 elastic spring: or 0         (6) 3. the boundary element method the used bem is based on the fundamental solution for a point force acting on a viscoelastic, incompressible half-space [13-15]. let x and y be cartesian coordinates on the surface of a half-space. a point force with components fx, fy and fz, is applied at the centre of the mentioned coordinate system and causes the following deflection of the surface: 296 s. kusche 2 2 2 3 3 2 1 4 4, ( ), ,z x x x y z i i x xy u u r r rr j t r xf y r                          . (7) these equations can only be applied, if the acting force is added at time zero and the force is held to be constant. from this solution, the elastic fundamental solution can be obtained if j(t)→1/g is substituted. herein j(t) is the time dependent creep function for shear: / 0( ) ( ) 1 tt j t j j e       . (8) this creep function (8) is related to the time dependent relaxation modulus g(t) given in eq. (4). in the case of the standard-material: 0 0 0 1 , , , g j j j g g g j g            , (9) and in case of the maxwell-element: 0 1 , 0, j j g g     . (10) the fundamental solution for the viscoelastic material looks similar to the one from elastic material. especially the dependency in space is the same. with this conclusion many schemes from elastic contact mechanics can be used. firstly, a standard procedure is to integrate the fundamental solution (boussinesq) over a rectangle, assuming constant pressure [16]. this solution (love) can be superimposed to find the deflection for an arbitrary, but piecewise constant, pressure distribution [17]. to speed this up, it is common practice to interpret this procedure as a convolution, which can be carried out on massive parallel processing units [18-20]. the next, more complicated problem arising in the field of elastic contact mechanic is the opposite task: solve the indentation problem of a rigid indenter pressed in an elastic half-space with an a priori unknown contact area. then conjugate gradient algorithms are used, like the modified form by polonsky and keer [21]. the procedures named above are supposed to be known and are used in the following calculations. the method described above can be applied to the viscoelastic case, if the acting forces are kept constant during one time step. the overall solution will be superimposed by adding the elastic solution multiplied with the time dependent creep function delayed by the time span since the force has been applied. at this point, a sum appears whose numerical costs grow linear in simulation time. by utilizing the special form of the creep function used here, an iterative algorithm can be derived. by applying this algorithm, only the previous time step is needed, and the contacting indenter shape is modified by a viscoelastic term. by using this method, the normal contact can be solved. a description and the application to the tangential sliding of an indenter can be found in [22]. the algorithm is shown below: the boundary element method for viscoelastic material applied to the oblique impact of spheres 297 1 1 1 , 1 1 0 00 0 0 , 1 1 1 0 0 0 1 1 ( ) exp ex 1 )p ( ) ( t z z n n z z n n h n n n i z n n i i i i i i i a b z zt z n n n t n n n n a b n n t tj j f t t f f f j j j hj j f a h f b f f j j j f a u u                                                 1 0 exp z z zt n n d h j b f j                       . (11) herein uz,n is the normal deflection of the surface and fn is the deflection due to a pressure distribution pn each at time tn. furthermore, a and b are variables from the previous time step, initialized with value zero at the beginning of the simulation, and it is j∞=j0+j. as can be seen in the last line of eq. (11), deformation d z has to been taken into account to include viscoelastic behavior (in the elastic case d z is equal to zero). the only unknown term in eq. (11) is fn+1, which means that pressure distribution pn+1 is unknown. this can be calculated with the elastic algorithms mentioned before. it should be noted that this algorithm can handle only materials with a finite modulus for instant deformation (g(0)<∞ and j(0)>0). that excludes the kelvin-element because it does not fit into the scope of this algorithm. the tangential contact is very similar to the normal one. only the calculation of the deflection in tangential direction ux has to be adopted. in all simulations carried out here, the deformation perpendicular to the plane of the motion is neglected. it has been shown for the case of parabolic bodies that this assumption causes only a negligible error [23,24]. at this point, the contact problem itself is solved. for the integration in time both an explicit euler scheme and the velocity verlet algorithm have been used. in comparison, they show no difference in the global error of the velocities at the end of the simulation and in the contact time itself. for an estimation of step size ht the solution of the elastic impact problem has been used: 1 2 5 4 , 2 0 10 2.87 (4 ( )) t c c elastic s z m h t t r g t v          . (12) since the shear modulus is time dependent, one has to start with g(0) and iteratively set t→tc. with the elastic solution also an approximation for contact radius a can be found: , 0 , 2.94 s c elastic z elastic t v a a r    . (13) the geometric discretisation has been chosen in a way that 256256 points are used. a comparison with a finer discretisation shows only a slight improvement of the error. for implementation it has to be considered that the total deflection in normal direction within the contact area is known at every time step since the indentation depth of the sphere is known. contrariwise in tangential direction: the points coming into contact have a pre-deformation through coupling to the points within the contact area from the previous time steps. this can be handled by adding only the current increment of tangential movement at the boundary of the sphere in each time step. 298 s. kusche 4. results the simulation carried out shows that the assumptions, condensed in formula (3) and (5), are correct. this has been proven by choosing random simulation parameters, that all coincide in the same functional relation. this relation has been fitted to the numerical data with two functions, one for each number ex and ez. in the following contour plots (figs. 2 and 3) the obtained data and the fitting are shown. the fitting function and the parameter are given below. the fitting function for the coefficient of restitution in normal direction ez is (parameters are given in table 1):   1 2 2 2 10 1 1 10 2 2 1 1 tanh( ) 2 2 max 0, log ( ) log ( ) max 0, 2 z p p c c x y x y e                           (14) fig. 2 contour lines of the coefficient of restitution ez (left hand side). special case of the kelvin-voigt-element (δ1→0) and the maxwell-element (δ2→∞) (right hand side) table 1 list of the fitting parameters for the coefficient of restitution in normal direction ez corresponding to the fitting function given in (14) parameter δ1 p δ2 p c1 c2 value -1.342 0.9583 2.079 -2.892 the boundary element method for viscoelastic material applied to the oblique impact of spheres 299 the fitting function for the coefficient of restitution in tangential direction ex is (parameters are given in table 2):   2 1 3 4 2 2 5 6 7 8 9 10 1 1 10 2 2 exp ( ) ( ) ( ) , [1 tanh( )] , 2 1 1 tanh( ) 2exp 2 4 1 t ( ) anh( ) , log log el min el max el x x x x x x x x x x p p e e e e e e e e c x e c c y c y e c c y c c c x x y                                                                      (15) fig. 3 contour lines of the coefficient of restitution ex (left hand side). special case of the kelvin-voigt-element (δ1→0) and the maxwell-element (δ2→∞) (right hand side) table 2 list of the fitting parameters for the coefficient of restitution in normal direction ex corresponding to the fitting function given in eq. (15) parameter ex min ex el ex max δ1 p δ2 p value -0.0399 0.1038 0.2028 -0.4 -0.8 parameter c1 c2 c3 c4 c5 value 2.335 0.5406 -2.29 2.253 1.718 parameter c6 c7 c8 c9 value 2.335 0.7999 -2.105 4.000 300 s. kusche 5. conclusions the bem has been extended to the normal and tangential contact problem of viscoelastic media. the approach has different advantages: it implements a relatively simple adaptation to the existing bem procedures developed for elastic media and its applicability to any other transient problem dealing with time dependent viscoelastic behavior modeled by the prony series. exemplary the impact of a sphere with an elastomer, modeled by a maxwell-element and a standard-model, has been investigated. the obtained compact and elegant solution is given in form of a fitting function for further use. references 1. hertz, h., 1882, über die berührung fester elastischer körper, journal für die reine und angewandte mathematik, 92, pp. 156-171. 2. deresiewicz, h., 1968, a note on 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paper a new integrated grey mcdm model: case of warehouse location selection alptekin ulutaş1, figen balo2, lutfu sua3, ezgi demir4, ayşe topal5, vladimir jakovljević6 1sivas cumhuriyet university, international trade and logistics department, turkey 2fırat university, industrial engineering department, turkey 3school of entrepreneurship and business administration, auca, kyrgyzstan 4piri reis university, management information system department, turkey 5niğde ömer halisdemir university, business department, turkey 6university of sidney, school of mathematics and statistics, australia abstract. warehouses link suppliers and customers throughout the entire supply chain. the location of the warehouse has a significant impact on the logistics process. even though all other warehouse activities are successful, if the product dispatched from the warehouse fails to meet the customer needs in time, the company may face with the risk of losing customers. this affects the performance of the whole supply chain therefore the choice of warehouse location is an important decision problem. this problem is a multicriteria decision-making (mcdm) problem since it involves many criteria and alternatives in the selection process. this study proposes an integrated grey mcdm model including grey preference selection index (gpsi) and grey proximity indexed value (gpiv) to determine the most appropriate warehouse location for a supermarket. this study aims to make three contributions to the literature. psi and piv methods combined with grey theory will be introduced for the first time in the literature. in addition, gpsi and gpiv methods will be combined and used to select the best warehouse location. in this study, the performances of five warehouse location alternatives were assessed with twelve criteria. location 4 is found as the best alternative in gpiv. the gpiv results were compared with other grey mcdm methods, and it was found that gpiv method is reliable. it has been determined from the sensitivity analysis that the change in criteria weights causes a change in the ranking of the locations therefore gpiv method was found to be sensitive to the change in criteria weights. key words: grey preference selection index, grey proximity indexed value, multicriteria decision making, warehouse location selection received april 24, 2021 / accepted august 26, 2021 corresponding author: alptekin ulutaş department of international trade and logistics, sivas cumhuriyet university, sivas, turkey e-mail: aulutas@cumhuriyet.edu.tr 516 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević 1. introduction warehouses are critical elements that affect the performance of an entire supply chain. in addition, warehouses are links between upstream suppliers and downstream customers throughout the entire supply chain. warehouses can also be described as places where efficient using of space and equipment are made in. to react more quickly to client demands with reduced costs, efficient warehousing activities considerably decrease the order picking distance and processing time of item motion for order fulfillment within a warehouse [1]. no matter how successful the warehouse activities are, if the product dispatched from the warehouse fails to meet the customer needs in time, the company will risk losing customers. one of the most important factors in the timely delivery of the product is the location of the warehouse therefore enterprises need to develop effective solutions for the warehouse location selection problem which has a significant impact on logistics processes [2]. the problem of warehouse location selection requires a crucial strategic decision plan for the businesses profitably. deciding on distribution warehouse locations is one of the most important issues to be considered in logistics problems. choosing the right warehouse location provides competition and benefits for companies. at the same time, the location of the distribution warehouse is important in issues such as proximity to distribution locations, cost, and labor. since there are usually more than one alternative and more than one criteria to be considered in the problem of selecting a warehouse location, this problem can be solved by using multi-criteria decision-making (mcdm) methods. mcdm methods are frequently used in the solution of assessment and sequencing problems including many conflicting criteria [3]. they are useful in determining the best alternative in a system with multiple alternatives and multiple criteria to be taken into account. many real-world decision-making problems do not contain crisp data. most of the data consist of uncertainty and vagueness. to deal with the uncertainty, different methodologies were developed such as fuzzy set theory, rough theory, d numbers, and grey theory. in the literature, these methodologies have mostly been used in combination with mcdm methods. for example, ecer and pamucar [4] have used fuzzy bwm (best worst method) to find the relative weights and fuzzy cocoso (combined compromise solution) with bonferroni (cocoso’b) to rank alternatives in sustainable supplier selection problem. ecer and pamucar [5] assessed insurance companies according to their service quality in covid19 by using intuitionistic fuzzy marcos (measurement of alternatives and ranking according to compromise solution). shojaei and bolvardizadeh [6] has used rough ahp (analytic hierarchy process) and rough topsis (technique for order performance by similarity to ideal solution) for assessing suppliers in terms of sustainability in construction industry. stević et al. [7] has used rough piprecia (pivot pairwise relative criteria importance assessment) to evaluate tools in sustainable production and fuzzy marcos to rank forest companies. pamucar et al. [8] ranked zero-carbon strategies in london transportation system with fuzzy bwm-d and todim (an acronym in portuguese for interactive and multi criteria decision making)-d. tian et al. [9] have used ahp and grey correlation topsis for material selection problem in construction industry. tadić et al. [10] assessed location alternatives for dry ports by using delphi, ahp, and codas (combinative distance-based assessment) with grey numbers. a new integrated grey mcdm model: case of warehouse location selection 517 grey numbers’ major benefit is its adaptability in dealing with complicated scenarios. in addition, grey theory can be used successfully compared to fuzzy sets in terms of a small amount of data and limited and incomplete data [11-13]. if the upper and lower values of criteria (including uncertain data) are known, these criteria can be expressed in grey numbers. as they are known in this decision problem, a grey mcdm method is used in this study. decision-makers can also make use of rough set theory to deal with uncertainty. however, in the rough set theory, crisp numbers can be used to handle uncertainty. on the contrary, in grey theory, uncertainty is handled by using interval values, which helps to take the data in a larger framework rather than compressing the data into crisp numbers. therefore, in this study, the grey extensions of mcdm (psi and piv) methods are proposed to solve the warehouse selection problem. this study proposes an integrated grey mcdm model including gpsi (grey preference selection index) and gpiv (grey proximity indexed value) to determine the most appropriate warehouse location for a supermarket. while the gpsi method is used to determine the weights of the criteria, the gpiv method is used to evaluate the performance of the alternatives and to rank these alternatives. psi's main benefit is that, unlike other mcdm approaches, it does not need assigning a relative priority between criteria [14]. compared to other mcdm methods, piv has comparatively straightforward and effective with simple computing steps, and minimizes rank reversal issues [15-16]. the flow of methodology is demonstrated in fig. 1. fig. 1 the steps of grey mcdm methodology for location selection criteria weights • creating a grey decision matrix • developing normalized grey decision matrix • finding mean grey normalized values • calculating the grey preference values • determining the grey deviation values • finding the grey weights alternative ranking • forming the grey decision matrix • normalizing the grey decision matrix • multiplying values in normalized matrix with grey weights of criteria • developing grey weighted proximity index • computing crisp overall proximity values • ranking alternatives sensitivity analysis • using four different criteria weigth sets • ranking alternatives with new weights 518 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević this study aims to make three contributions to the literature. first, psi and piv methods combined with grey theory will be introduced for the first time in the literature. there is no study that combined grey theory with psi and piv methods to the best of our knowledge. combining these two methods with grey theory will aid to effectively handle the uncertainties in the problem. second, gpsi and gpiv methods will be combined and used to select the best warehouse location. it has not been seen in the literature that gpsi and gpiv methods are used together in solving any mcdm problems. the study is organized as follows. in section 2, the literature review is made concerning the warehouse location selection, psi and piv methods. in section 3, the methodology of gpsi and gpiv methods is presented. in section 4, the results of the proposed model are indicated. in section 5, the results of gpsi are compared with the results of other grey mcdm, which are grey topsis [17], grey waspas (weighted aggregated sum-product assessment) [18], and grey copras (complex proportional assessment method) [19] and grey weights of criteria are changed using 4 different scenarios and sensitivity analysis is performed. in the last section, a brief conclusion is indicated. 2. literature review in this section, first, the studies that solve the problem of selecting the warehouse location with mcdm methods will be presented, then, studies using psi and piv methods will be demonstrated. 2.1. warehouse location selection problem the problem of warehouse location selection has been discussed many times in the literature. for example, lee [20] discussed cost, hakimi and kuo [21] discussed maximizing profitability, korpela and lehmusvaara [22] developed a customer-oriented approach and used ahp and mixed integer programming model. in another study, korpela et al. [23] used ahp and dea (data envelopment analysis) with distribution time, quantity and quality of distributions, emergency distributions, frequency situations, special requests and capacity criteria. ho and emrouznejad [24] developed an application for users to select a warehouse location. kuo et al. [25], tabari et al. [26], chen [27], kahraman et al. [28], karmaker and saha [29] discussed warehouse location selection problem only with fuzzy methodologies. they examined scenario-based examples of wrong decisions that can be made in warehouse selection in terms of economic losses. stevenson [30] and frazelle [31] explained warehouse centers as a factor of commercial success and competition. demirel et al. [32] discussed the critical success factors in warehouse selection in terms of logistics management and optimization. in this study, it was revealed that decision-making processes are extremely important in case of success and failure in the selection of warehouse location. at the same time, it was concluded that a balance should be struck between cost and effectiveness in potential locations in organizational terms. in this context, the warehouse selection model was first modeled by kuehn and hamburger [33]. in this study, cost minimization has been aimed with mathematical modeling techniques. efroymson and ray [34] and khumawala [35] selected warehouse locations with the classical linear programming model in their studies. the location problem has been created by weber and friedrich [36] and tellier [37] by minimizing the distances to the final distribution locations. owen and daskin [38] dealt with a comprehensive mathematical a new integrated grey mcdm model: case of warehouse location selection 519 problem in their work. in this study, benefit and cost values have been considered as the main criteria. on the other hand, various criteria and methodologies have been used by introducing the mcdm methodology. badri [39] used the ahp method together with linear programming. this study has been focused on 4 main criteria. these criteria are benefits, costs, risks and opportunities. vlachopoulou et al. [40] suggested a geographic decision support system to choose warehouse location with geographic criteria. kabak and keskin [41] proposed geographical information systems (gis) and ahp models for potential warehouse locations. nine criteria have been proposed in this study. yerlikaya et al. [42] suggested an ahp critic (the criteria importance through intercriteria correlation) –vikor (visekriterijumska optimizacija i kompromisno resenje) based approach. in the study, they used cost, speed, safety criteria. mihajlovic et al. [43] studied fruit warehouse location selection based on ahp and waspas. ma et al. [44] handled the choices of warehouse location utilizing an integrated multi-attribute decision making method based on the cumulative prospect theory. it has been determined a strategic ranking of decisionmaking schemes by presenting a cumulative foreground theory. pamučar and božanić [45] selected location from the suggested choices using a singlevalued neutrosophic (svnn) based mabac (multi attributive border approximation area comparison) model. it was aimed to select the ideal logistics center location by applying an optimization routine reducing transport costs and improving the business performance, competitiveness and profitability. tuzkaya et al. [46] used ahp to rank locations to reduce costs and maximize profit. uysal and tosun [47] developed a grey theory-based method to solve warehouse location problem and compared the results of electre (elimination et choix traduisant la realité) and topsis decision-making models. özcan et al. [48] set the criteria for the most optimal warehouse location in a retail sector and applied grey theory, electre, topsis and ahp. in this study, the authors considered five criteria, which are stock holding capacity, unit price, mean distance to shops, mean distance to movement flexibility, and primary suppliers, when evaluating four warehouse location alternatives. in another study, ashrafzadeh et al. [49] used fuzzy topsis to choose the best warehouse location for an iranian company. they considered fifteen criteria in the evaluation of five alternatives. dey et al. [50] proposed an integrated fuzzy mcdm to solve warehouse location selection problem in a supply chain. garcía et al. [51] utilized ahp method to choose the ideal warehouse location for perishable agricultural products. they took into account six criteria, which are costs, distance, needs, security, acceptance and accessibility, in the evaluation of three alternatives. aktepe and ersöz [2] used ahp, moora (multi-objective optimization method by ratio analysis) and vikor methods to address warehouse location selection problem for a company. six criteria were used in the evaluation of eleven alternatives. dey et al. [52] proposed three fuzzy mcdm methods, namely fuzzy topsis, fuzzy moora and fuzzy simple additive weighting to choose a warehouse location. additionally, to evaluate the objective criteria, the classical normalization technique is used. silva et al. [53] used smarter and lexicographic method to rank products and assign them to the locations of warehouse storage. mangalan et al. [54] utilized weighted moora method to optimize warehouse site. the results of the proposed method and topsis method were compared to prove the applicability of the proposed method. temur [55] proposed cloud based design optimization technique tackling high uncertainty. this technique was used in a warehouse location problem in order to indicate the feasibility and performance of this technique. dey et al. [56] developed a novel multi-criteria group decision-making to determine the best warehouse location for an 520 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević indian company. four criteria, which are space availability, transportation facility, cost and availability of markets, were considered in the evaluation process. raut et al. [57] used ahp to determine the best sustainable warehouse location among four alternatives while considering eleven criteria. the most important criterion was determined as governmental policies, and regulations among eleven criteria. emeç and akkaya [3] integrated fuzzy vikor and stochastic ahp methods to address warehouse location problem for a supermarket. seventeen criteria were considered when determining the best location among four alternatives. micale et al. [58] proposed an interval extension of mcdm methods, which are electre tri and topsis, to solve storage location assignment problem for an italian company. canbolat et al. [59] used decision tree and mabac methodologies for warehouse location selection. ehsanifar et al. [60] prioritized and ranked ten criteria using utastar methodology. the most commonly used criteria in the current studies are cost minimization, profitability and geographical factors. 2.2. literature related to psi method some recent studies, carried out on psi method (developed by maniya and bhatt [61]), in the literature are summarized in table 1. table 1 literature related to psi method authors methods application area vahdani et al. [62] interval-valued fuzzy psi application in human resource management attri and grover [63] psi illustrative examples chamoli [64] psi for an experiment, the determination of optimum roughness parameters akyüz and aka [65] psi manufacturing performance measurement in the glass industry petković et al. [66] psi illustrative examples madić et al. [67] psi in determining of laser cutting process conditions tuş and adalı [68] critic, psi, and codas in solving of a personnel selection problem for a textile firm jha et al. [69] psi in determining of optimum composite combination pathak et al. [70] psi and metaheuristic method in determining of optimum value for parameters of scanning process ulutaş [71] fuzzy psi and fuzzy rov in solving of a green supplier selection problem for a textile company 2.3. literature related to piv method many mcdm methods [72-78] have been developed in recent years. piv is one of the newly developed (by mufazzal and muzakkir [15]) mcdm methods and it minimizes the rank reversal problem. there are few studies about this method in the literature. khan et al. [16] used piv method to indicate the efficacy and applicability of this method. to do this, two illustrative examples related to the e-learning websites selection were analyzed and the results of piv method were compared with the results of other mcdm methods a new integrated grey mcdm model: case of warehouse location selection 521 (copras, vikor, ahp, wedba and wdba). in another study, yahya et al. [79] integrated entropy and piv methods to use for multi-response optimization. nine experiments were evaluated while considering two criteria, which are zeta potential and viscosity. as it can be observed from literature review, there are limited studies about psi and piv methods, and they are mostly used for different decision problems other than location selection. for location selection problem, most of the studies in the literature used crisp mcdm methods, such as ahp and topsis. this clearly shows that there is a gap in the application of new mcdm methods on location selection problem. additionally, in this study, unlike most of the studies in the literature; costs will be handled in two types, which are holding costs (hc) and transportation costs (tc). in addition, geographical features of the supplied materials, which are distance to customers (dc), distance to suppliers (ds), distance to producers (dp), delivery time (dt), and distance to the opponents (do), have been handled separately. the criteria list has been enriched by considering different criteria such as capacity of storage (cs), development rate (dr), and transportation diversity (td). besides, environmental conditions, specifically infrastructure (i) and climatic conditions (cc), have been used in the selection of the warehouse location. unlike the most of the studies, 12 different criteria were considered in this study. in the most of the studies, linear programming and cost-benefit optimization, and crisp mcdm methodologies were used to select warehouse location. also, fuzzy sets were also commonly used in the literature. fuzzy sets have been used in problems with uncertainty, and they address the problem with linguistic expressions. however, in the case of the small amount of data and limited and incomplete data, the fuzzy set theory is not sufficient. in this context, it is thought that the study will fill the following gaps in the literature: ▪ few studies in the literature have used grey mcdm methods for warehouse location selection. ▪ there are no grey extensions of the two mcdm methods (psi and piv) that have few computations steps and reach a solution quickly. 3. methodology in this study, a grey model consisting of gpsi and gpiv methods is proposed for the solution of the warehouse location problem. while the gpsi method is used to determine the weights of the criteria, the gpiv method is used to evaluate the performance of the alternatives and to rank these alternatives. 3.1. grey preference selection index step 1: the linguistic values shown in table 2 will be assigned by the experts as the performance values of the alternatives in the criteria. these performance values are converted to grey values with the help of table 2, thus forming a grey decision matrix (⨂𝐹). ⨂𝐹 = [⨂𝑓𝑖𝑗] 𝑚×𝑛 (1) in eq. (1), ⨂𝑓𝑖𝑗 (⨂𝑓𝑖𝑗 = [𝑓𝑖𝑗 𝑙 , 𝑓𝑖𝑗 𝑢 ]) indicates grey performance value of 𝑖th alternative on 𝑗th criterion. 522 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević table 2 linguistic and grey performance values linguistic performance values grey performance values very high [9, 10] high [7, 9] medium [5, 7] low [3, 5] very low [1, 3] step 2: by utilizing eq. (2) (beneficial criteria) and eq. (3) (cost criteria), ⨂𝐹 can be normalized as below. ⨂𝑘𝑖𝑗 = ⨂𝑓𝑖𝑗 𝑚𝑎𝑥(⨂𝑓𝑖𝑗) = [ 𝑓𝑖𝑗 𝑙 𝑚𝑎𝑥(𝑓𝑖𝑗 𝑢) , 𝑓𝑖𝑗 𝑢 𝑚𝑎𝑥(𝑓𝑖𝑗 𝑢) ] (2) ⨂𝑘𝑖𝑗 = 𝑚𝑖𝑛(⨂𝑓𝑖𝑗) ⨂𝑓𝑖𝑗 = [ 𝑚𝑖𝑛(𝑓𝑖𝑗 𝑙 ) 𝑓𝑖𝑗 𝑢 , 𝑚𝑖𝑛(𝑓𝑖𝑗 𝑙 ) 𝑓𝑖𝑗 𝑙 ] (3) in eqs. (2) and (3), ⨂𝑘𝑖𝑗 indicates the normalized version of ⨂𝑓𝑖𝑗 . step 3: by using eq. (4), the mean grey normalized value (⨂�̅�𝑖𝑗) of each criterion is calculated as, ⨂�̅�𝑖𝑗 = ∑ ⨂𝑘𝑖𝑗 𝑚 𝑖=1 𝑚 = [ ∑ 𝑘𝑖𝑗 𝑙𝑚 𝑖=1 𝑚 , ∑ 𝑘𝑖𝑗 𝑢𝑚 𝑖=1 𝑚 ] (4) step 4: for each criterion, the grey preference value (⨂𝛿𝑗 = [𝛿𝑗 𝑙, 𝛿𝑗 𝑢 ]) is computed with eq. (5). ⨂𝛿𝑗 = ∑ (⨂𝑘𝑖𝑗 − ⨂�̅�𝑖𝑗)2 = [∑ (𝑘𝑖𝑗 𝑙 − �̅�𝑖𝑗 𝑙 )2,𝑚 𝑖=1 ∑ (𝑘𝑖𝑗 𝑢 − �̅�𝑖𝑗 𝑢 )2𝑚 𝑖=1 ]𝑚 𝑖=1 (5) step 5: by eq. (6), the grey deviation value (⨂𝛾𝑗) for each criterion is obtained. ⨂𝛾𝑗 = [𝛾𝑗 𝑙, 𝛾𝑗 𝑢 ] = |1 − ⨂𝛿𝑗| = [|1 − 𝛿𝑗 𝑢|, |1 − 𝛿𝑗 𝑙|] (6) step 6: the grey weight (⨂𝑤𝑗 = [𝑤𝑗 𝑙 , 𝑤𝑗 𝑢]) of each criterion is computed with eq. (7). ⨂𝑤𝑗 = ⨂𝛾𝑗 ∑ ⨂𝛾𝑗 𝑛 𝑗=1 = [ 𝛾𝑗 𝑙 ∑ 𝛾𝑗 𝑢𝑛 𝑗=1 , 𝛾𝑗 𝑢 ∑ 𝛾𝑗 𝑙𝑛 𝑗=1 ] (7) after computing the grey weight of each criterion, these grey weights are dispatched into gpiv. 3.2. grey proximity indexed value gpiv method consists of four steps shown as follows. step 1: in eq. (1), the grey decision matrix is formed. the values in this matrix are normalized by using eq. (8). ⨂𝑒𝑖𝑗 = [𝑒𝑖𝑗 𝑙 , 𝑒𝑖𝑗 𝑢 ] = ⨂𝑓𝑖𝑗 √∑ (⨂𝑓𝑖𝑗)𝑚 𝑖=1 2 = [ 𝑓𝑖𝑗 𝑙 √∑ (𝑓𝑖𝑗 𝑢)𝑚 𝑖=1 2 +∑ (𝑓𝑖𝑗 𝑙 )𝑚 𝑖=1 2 , 𝑓𝑖𝑗 𝑢 √∑ (𝑓𝑖𝑗 𝑢)𝑚 𝑖=1 2 +∑ (𝑓𝑖𝑗 𝑙 )𝑚 𝑖=1 2 ] (8) a new integrated grey mcdm model: case of warehouse location selection 523 in eq. (8), ⨂𝑒𝑖𝑗 is the normalized of ⨂𝑓𝑖𝑗. step 2: these normalized values are multiplied by grey weights of criteria (obtained in gpsi) with eq. (9). ⨂𝑡𝑖𝑗 = [𝑡𝑖𝑗 𝑙 , 𝑡𝑖𝑗 𝑢 ] = ⨂𝑤𝑗 × ⨂𝑒𝑖𝑗 = [𝑤𝑗 𝑙 × 𝑒𝑖𝑗 𝑙 , 𝑤𝑗 𝑢 × 𝑒𝑖𝑗 𝑢 ] (9) step 3: grey weighted proximity index (⨂𝑔𝑖𝑗 = [𝑔𝑖𝑗 𝑙 , 𝑔𝑖𝑗 𝑢 ]) is computed for beneficial (eq. (10)) and cost criteria (eq. (11)) as follows. ⨂𝑔𝑖𝑗 = 𝑚𝑎𝑥(⨂𝑡𝑖𝑗) − ⨂𝑡𝑖𝑗 = [𝑚𝑎𝑥(𝑡𝑖𝑗 𝑙 ) − 𝑡𝑖𝑗 𝑢 , 𝑚𝑎𝑥(𝑡𝑖𝑗 𝑢 ) − 𝑡𝑖𝑗 𝑙 ] (10) ⨂𝑔𝑖𝑗 = ⨂𝑡𝑖𝑗 − 𝑚𝑖𝑛(⨂𝑡𝑖𝑗) = [𝑡𝑖𝑗 𝑙 − 𝑚𝑖𝑛(𝑡𝑖𝑗 𝑢 ), 𝑡𝑖𝑗 𝑢 − 𝑚𝑖𝑛(𝑡𝑖𝑗 𝑙 )] (11) step 4: grey (⨂𝑑𝑖 = [𝑑𝑖 𝑙, 𝑑𝑖 𝑢]) and crisp (𝑑𝑖) overall proximity values are computed respectively with eqs. (12) and (13). ⨂𝑑𝑖 = ∑ ⨂𝑔𝑖𝑗 𝑛 𝑗=1 = [∑ 𝑔𝑖𝑗 𝑙 , ∑ 𝑔𝑖𝑗 𝑢 ,𝑛 𝑗=1 𝑛 𝑗=1 ] (12) 𝑑𝑖 = 𝑑𝑖 𝑙+𝑑𝑖 𝑢 2 (13) finally, alternative with the least crisp overall proximity value is designated as the best alternative. 4. application the application of the integrated grey mcdm model is performed in a supermarket, which has over ten years of experience in the sector. during the covid-19 pandemic, there were delays due to restrictions on transportation of the business. in addition, there has been an increase in costs in terms of warehouse and workforce during the pandemic. in order to overcome these problems, the supermarket chain decided to develop a project. at this point, consultancy service was received for logistics and crisis management. this study was carried out for 6 weeks with five experts who are the general manager of the supermarket chain, the director of logistics and transportation department, and 3 people from the consultancy company. the owner of the supermarket chain has 25 years of experience in the field of business graduate. the logistics director of the supermarket chain has a phd in the logistics field and has 20 years of experience. in addition, 3 people in the consultancy company are industrial engineers with over 15 years of experience in the fields of engineering and logistics. criteria were determined by literature review. in the 6-week meetings, the criteria in the literature were discussed with 5 experts, new criteria were added and removed. while determining the criteria, the cost criteria have been expanded to holding cost and transportation cost due to increasing in the costs during pandemic. unlike the literature review, criteria such as infrastructure and climate conditions have been added. in addition, the criteria were developed by examining the distance functions in detail. totally, twelve criteria were identified for utilizing in warehouse location selection. these criteria are holding cost (hc), transportation costs (tc), distance to customers (dc), distance to suppliers (ds), distance to producers (dp), delivery time (dt), distance to opponents (do), capacity of storage (cs), development rate (dr), transportation diversity (td), infrastructure (i) and climatic conditions (cc). the first 524 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević six criteria are assigned as cost criteria and the others are assigned as beneficial criteria. the expert team identified five suitable alternatives for the warehouse location. the grey data of the first criterion were collected from expert team as actual data. the unit of this grey data is us dollars and represents the holding cost per month. the expert team did not give the tc criterion as actual grey data for commercial reasons. for, tc and the other criteria, the grey data were determined together by the expert team and using the linguistic values shown in table 2. the grey decision matrix was formed with all collected data. this matrix is presented in table 3. table 3 the grey decision matrix criteria locations hc tc dc location 1 [340, 380] [3, 5] [7, 9] location 2 [420, 440] [5, 7] [5, 7] location 3 [320, 360] [5, 7] [3, 5] location 4 [430, 460] [5, 7] [3, 5] location 5 [330, 350] [7, 9] [3, 5] criteria locations ds dp dt location 1 [5, 7] [3, 5] [3, 5] location 2 [3, 5] [1, 3] [5, 7] location 3 [5, 7] [1, 3] [5, 7] location 4 [3, 5] [1, 3] [7, 9] location 5 [7, 9] [3, 5] [5, 7] criteria locations do cs dr location 1 [5, 7] [7, 9] [3, 5] location 2 [3, 5] [5, 7] [7, 9] location 3 [5, 7] [5, 7] [3, 5] location 4 [3, 5] [7, 9] [7, 9] location 5 [5, 7] [7, 9] [1, 3] criteria locations td i cc location 1 [5, 7] [3, 5] [1, 3] location 2 [7, 9] [5, 7] [5, 7] location 3 [5, 7] [5, 7] [5, 7] location 4 [7, 9] [5, 7] [5, 7] location 5 [5, 7] [3, 5] [3, 5] by means of eqs. (2) and (3), the grey decision matrix is normalized. the normalized grey decision matrix is presented in table 4. a new integrated grey mcdm model: case of warehouse location selection 525 table 4 the normalized grey decision matrix (for gpsi) criteria locations hc tc dc location 1 [0.842, 0.941] [0.6, 1] [0.333, 0.429] location 2 [0.727, 0.762] [0.429, 0.6] [0.429, 0.6] location 3 [0.889, 1] [0.429, 0.6] [0.6, 1] location 4 [0.696, 0.744] [0.429, 0.6] [0.6, 1] location 5 [0.914, 0.970] [0.333, 0.429] [0.6, 1] criteria locations ds dp dt location 1 [0.429, 0.6] [0.2, 0.333] [0.6, 1] location 2 [0.6, 1] [0.333, 1] [0.429, 0.6] location 3 [0.429, 0.6] [0.333, 1] [0.429, 0.6] location 4 [0.6, 1] [0.333, 1] [0.333, 0.429] location 5 [0.333, 0.429] [0.2, 0.333] [0.429, 0.6] criteria locations do cs dr location 1 [0.714, 1] [0.778, 1] [0.333, 0.556] location 2 [0.429, 0.714] [0.556, 0.778] [0.778, 1] location 3 [0.714, 1] [0.556, 0.778] [0.333, 0.556] location 4 [0.429, 0.714] [0.778, 1] [0.778, 1] location 5 [0.714, 1] [0.778, 1] [0.111, 0.333] criteria locations td i cc location 1 [0.556, 0.778] [0.429, 0.714] [0.143, 0.429] location 2 [0.778, 1] [0.714, 1] [0.714, 1] location 3 [0.556, 0.778] [0.714, 1] [0.714, 1] location 4 [0.778, 1] [0.714, 1] [0.714, 1] location 5 [0.556, 0.778] [0.429, 0.714] [0.429, 0.714] to give an example of the calculation of the values shown in table 4, the hc (eq. 3) grey normalized values of location 1 are found as follows. ⨂𝑘11 = ⨂𝑓11 𝑚𝑎𝑥(⨂𝑓𝑖𝑗) = [ 𝑚𝑖𝑛(𝑓𝑖𝑗 𝑙 ) 𝑓11 𝑢 , 𝑚𝑖𝑛(𝑓𝑖𝑗 𝑙 ) 𝑓11 𝑙 ] = [ 320 380 , 320) 340 ] = [0.842 , 0.941] the grey preference values (⨂𝛿𝑗), grey deviation values (⨂𝛾𝑗) and grey weights (⨂𝑤𝑗) are calculated by using eqs. (5-7), respectively. table 5 presents the results. 526 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević table 5 the gpsi method’s results criteria results hc tc dc ⨂𝛿𝑗 [0.039, 0.059] [0.037, 0.178] [0.063, 0.298] ⨂𝛾𝑗 [0.941, 0.961] [0.822, 0.963] [0.702, 0.937] ⨂𝑤𝑗 [0.087, 0.101] [0.076, 0.101] [0.065, 0.098] criteria results ds dp dt ⨂𝛿𝑗 [0.055, 0.270] [0.021, 0.533] [0.037, 0.178] ⨂𝛾𝑗 [0.730, 0.945] [0.467, 0.979] [0.822, 0.963] ⨂𝑤𝑗 [0.067, 0.099] [0.043, 0.103] [0.076, 0.101] criteria results do cs dr ⨂𝛿𝑗 [0.097, 0.099] [0.060, 0.060] [0.357, 0.357] ⨂𝛾𝑗 [0.901, 0.903] [0.940, 0.940] [0.643, 0.643] ⨂𝑤𝑗 [0.083, 0.095] [0.087, 0.098] [0.059, 0.067] criteria results td i cc ⨂𝛿𝑗 [0.060, 0.060] [0.097, 0.099] [0.260, 0.260] ⨂𝛾𝑗 [0.940, 0.940] [0.901, 0.903] [0.740, 0.740] ⨂𝑤𝑗 [0.087, 0.098] [0.083, 0.095] [0.068, 0.077] to give an example of the calculation of the values shown in table 5, the grey deviation values (⨂𝛾1) and the grey weights (⨂𝑤1) of hc are computed by eqs. (6) and (7) respectively as follows. ⨂𝛾1 = [𝛾1 𝑙 , 𝛾1 𝑢 ] = [|1 − 𝛿1 𝑢|, |1 − 𝛿1 𝑙|] = [|1 − 0.059|, |1 − 0.039|] = [0.941, 0.961] ⨂𝑤1 = ⨂𝛾1 ∑ ⨂𝛾𝑗 𝑛 𝑗=1 = [ 0.941 0.961 + 0.963 + 0.937 … . +0.740 , 0.961 0.941 + 0.822 + 0.702 … . +0.740 ] = [0.087, 0.101] the grey weights of criteria (⨂𝑤𝑗) found in the gpsi method are transferred to the gpiv method. eq. (8) is applied to the grey decision matrix, which is shown in table 3, in order to develop the normalized grey decision matrix for gpiv. this matrix is presented in table 6. a new integrated grey mcdm model: case of warehouse location selection 527 table 6 the normalized grey decision matrix for gpiv criteria locations hc tc dc location 1 [0.279, 0.311] [0.153, 0.254] [0.400, 0.514] location 2 [0.344, 0.360] [0.254, 0.356] [0.286, 0.400] location 3 [0.262, 0.295] [0.254, 0.356] [0.171, 0.286] location 4 [0.352, 0.377] [0.254, 0.356] [0.171, 0.286] location 5 [0.270, 0.287] [0.356, 0.458] [0.171, 0.286] criteria locations ds dp dt location 1 [0.269, 0.376] [0.303, 0.505] [0.153, 0.254] location 2 [0.161, 0.269] [0.101, 0.303] [0.254, 0.356] location 3 [0.269, 0.376] [0.101, 0.303] [0.254, 0.356] location 4 [0.161, 0.269] [0.101, 0.303] [0.356, 0.458] location 5 [0.376, 0.484] [0.303, 0.505] [0.254, 0.356] criteria locations do cs dr location 1 [0.294, 0.411] [0.302, 0.388] [0.163, 0.272] location 2 [0.176, 0.294] [0.216, 0.302] [0.381, 0.490] location 3 [0.294, 0.411] [0.216, 0.302] [0.163, 0.272] location 4 [0.176, 0.294] [0.302, 0.388] [0.381, 0.490] location 5 [0.294, 0.411] [0.302, 0.388] [0.054, 0.163] criteria locations td i cc location 1 [0.228, 0.319] [0.176, 0.294] [0.061, 0.184] location 2 [0.319, 0.410] [0.294, 0.411] [0.307, 0.429] location 3 [0.228, 0.319] [0.294, 0.411] [0.307, 0.429] location 4 [0.319, 0.410] [0.294, 0.411] [0.307, 0.429] location 5 [0.228, 0.319] [0.176, 0.294] [0.184, 0.307] the normalized values are multiplied by the grey weights of criteria with the aid of eq. (9). the grey weighted proximity values are calculated with eqs. (10) and (11). for example, the grey weighted proximity values of location 1’s hc criterion are computed by eq. (11) as follows. ⨂𝑔11 = ⨂𝑡11 − 𝑚𝑖𝑛(⨂𝑡𝑖𝑗) = [𝑡11 𝑙 − 𝑚𝑖𝑛(𝑡𝑖𝑗 𝑢 ), 𝑡11 𝑢 − 𝑚𝑖𝑛(𝑡𝑖𝑗 𝑙 )] = [0.024 − 0.029, 0.031 − 0.023] = [−0.005, 0.008] calculated all grey weighted proximity values are shown in table 7. 528 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević table 7 the grey weighted proximity values criteria locations hc tc dc location 1 [-0.005, 0.008] [-0.014, 0.014] [-0.002, 0.039] location 2 [0.001, 0.013] [-0.007, 0.024] [-0.009, 0.028] location 3 [-0.006, 0.007] [-0.007, 0.024] [-0.017, 0.017] location 4 [0.002, 0.015] [-0.007, 0.024] [-0.017, 0.017] location 5 [-0.006, 0.006] [0.001, 0.034] [-0.017, 0.017] criteria locations ds dp dt location 1 [-0.009, 0.026] [-0.018, 0.048] [-0.014, 0.014] location 2 [-0.016, 0.016] [-0.027, 0.027] [-0.007, 0.024] location 3 [-0.009, 0.026] [-0.027, 0.027] [-0.007, 0.024] location 4 [-0.016, 0.016] [-0.027, 0.027] [0.001, 0.034] location 5 [-0.002, 0.037] [-0.018, 0.048] [-0.007, 0.024] criteria locations do cs dr location 1 [-0.015, 0.015] [-0.012, 0.012] [0.004, 0.023] location 2 [-0.004, 0.024] [-0.004, 0.019] [-0.011, 0.011] location 3 [-0.015, 0.015] [-0.004, 0.019] [0.004, 0.023] location 4 [-0.004, 0.024] [-0.012, 0.012] [-0.011, 0.011] location 5 [-0.015, 0.015] [-0.012, 0.012] [0.011, 0.030] criteria locations td i cc location 1 [-0.003, 0.020] [-0.004, 0.024] [0.007, 0.029] location 2 [-0.012, 0.012] [-0.015, 0.015] [-0.012, 0.012] location 3 [-0.003, 0.020] [-0.015, 0.015] [-0.012, 0.012] location 4 [-0.012, 0.012] [-0.015, 0.015] [-0.012, 0.012] location 5 [-0.003, 0.020] [-0.004, 0.024] [-0.003, 0.020] by using eq. (12), grey overall proximity value (⨂𝑑𝑖) for each location alternative is computed. the crisp overall proximity value (𝑑𝑖) for each location alternative is computed by eq. (13). for example, the grey overall proximity values and crisp overall proximity value for location 1 are computed by eqs. (12) and (13) respectively as follows. ⨂𝑑1 = ∑ ⨂𝑔𝑖𝑗 𝑛 𝑗=1 = [−0.005 + −0.014 + −0.002 … . +0.007 , 0.008 + 0.014 + 0.039 … . +0.029] = [−0.085, 0.272] 𝑑1 = 𝑑1 𝑙 + 𝑑1 𝑢 2 = −0.085 + 0.272 2 = 0.094 the same operations are repeated for other location alternatives. the results and the rankings of location alternatives are indicated in table 8. a new integrated grey mcdm model: case of warehouse location selection 529 table 8 the results of gpiv results locations ⨂𝑑𝑖 𝑑𝑖 rankings location 1 [-0.085, 0.272] 0.094 4 location 2 [-0.123, 0.225] 0.051 2 location 3 [-0.118, 0.229] 0.056 3 location 4 [-0.130, 0.219] 0.045 1 location 5 [-0.075, 0.287] 0.106 5 according to table 8, the warehouse locations are listed as follows; location 4, location 2, location 3, location 1 and location 5. thus, location 4 is designated as the best warehouse location. 5. discussion the gpiv results are compared with the results of other grey mcdm methods, which are grey topsis, grey waspas, and grey copras, to test whether the gpiv results are accurate. the coefficients of spearman’s correlation for all these grey mcdm are indicated in table 9. table 9 spearman correlation coefficients grey mcdm gpiv grey topsis grey waspas grey copras gpiv 1.000 0.700 1.000 1.000 grey topsis 1.000 0.700 0.700 grey waspas 1.000 1.000 grey copras 1.000 according to table 9, gpiv method has reached the same results as grey waspas and grey copras methods. although the correlation coefficient between the grey topsis and gpiv methods is lower than the other correlation coefficients, the first two locations (location 4 and location 2) are obtained as the same ranked according to the results of both methods. as a result, it has been proved that gpiv method has reached correct results when compared with other grey mcdm methods. compared with other grey mcdm methods, it was observed that the gpiv method is easier and have fewer steps. in order to track the change in the rankings of the locations with regard to the change in criteria weights, the sensitivity analysis is performed. four sets of criteria weights are designated for this analysis. table 10 indicates these sets. 530 a. ulutaş, f. balo, l. sua, e. demir, a. topal, v. jakovljević table 10 criteria weights sets sets criteria set 1 set 2 set 3 set 4 hc [0.210, 0.250] [0.350, 0.380] [0.150, 0.180] [0.120, 0.150] tc [0.030, 0.060] [0.030, 0.060] [0.050, 0.060] [0.060, 0.080] dc [0.070, 0.080] [0.070, 0.080] [0.090, 0.100] [0.050, 0.080] ds [0.070, 0.090] [0.070, 0.090] [0.080, 0.090] [0.060, 0.100] dp [0.050, 0.060] [0.050, 0.060] [0.060, 0.080] [0.050, 0.105] dt [0.060, 0.070] [0.060, 0.070] [0.080, 0.100] [0.170, 0.180] do [0.090, 0.095] [0.070, 0.080] [0.085, 0.090] [0.080, 0.090] cs [0.060, 0.080] [0.010, 0.020] [0.080, 0.090] [0.080, 0.095] dr [0.050, 0.080] [0.040, 0.050] [0.060, 0.070] [0.070, 0.090] td [0.060, 0.080] [0.040, 0.050] [0.080, 0.090] [0.080, 0.090] i [0.090, 0.105] [0.020, 0.030] [0.060, 0.065] [0.030, 0.040] cc [0.040, 0.070] [0.100, 0.120] [0.050, 0.060] [0.020, 0.030] these weights of criteria are utilized to perform the sensitivity analysis. the results are indicated in fig. 2. fig. 2 the sensitivity analysis’s results as it can be observed from sensitivity analysis, there is a change in the ranking of all locations. location 3 is designated as the best location in set 1 and set 2, however, location 4 and location 2 are designated as the best locations in set 3 and set 4 respectively. as a result, the change in criteria weights causes a change in the ranking of the locations. thus, gpiv method was found to be sensitive to the change in criteria weights. although the proposed methods have achieved accurate results with easy calculations, the methods have individual limits. the solution efficiency of the psi method decreases as the number of alternatives increases [67]. also, since the psi method does not take into account the correlation between the criteria, it stands weak compared to the critic method in finding 0 1 2 3 4 5 6 set 1 set 2 set 3 set 4 r an ki n gs sets location 1 location 2 location 3 location 4 location 5 a new integrated grey mcdm model: case of warehouse location selection 531 the objective weights of the criteria. the limits mentioned for the psi method are also valid for the gpsi method. both gpsi and gpiv methods work with grey data. in a situation where there is no grey data and the uncertainty is higher, it may be difficult to reach the correct results and both methods may not work. in addition, both methods do not have a membership function as in fuzzy numbers. this will cause the proposed model to deal with uncertainty in less detail compared to comprehensive fuzzy (interval type 2, intuitionistic, spherical, and fermatean) methods. in addition, in the proposed method, only the objective weights of the criteria are considered. subjective weights of criteria can also be obtained using grey mcdm methods (such as grey ahp, grey swara, and grey fucom), and stronger and more consistent results can be obtained by combining objective and subjective weights of criteria. 6. conclusion this study proposes an integrated grey mcdm model including gpsi and gpiv to determine the most appropriate warehouse location for a supermarket. while the gpsi method is used to determine the weights of the criteria, the gpiv method is used to evaluate the performance of the alternatives and to rank these alternatives. this study aims to make three contributions to the literature. psi and piv methods combined with grey theory will be introduced for the first time in the literature. in addition, gpsi and gpiv methods will be combined and used to select the best warehouse location. in this study, the performances of five location alternatives were measured by considering twelve criteria. according to the results of gpiv, location 4 is designated as the best warehouse location. in this study, the results of the gpiv method and other grey mcdm methods were compared. accordingly, it was found that gpiv method reached the correct results. in addition, a sensitivity analysis was performed by changing the weights of the criteria. it has been observed that the change in criteria weights causes a change in the ranking of the locations. thus, gpiv method was found to be sensitive to the change in criteria weights. this study has been carried out for the selection of warehouse location during the covid-19 pandemic. in this context, the criteria considered has been expanded. these criteria can be used in conjunction with other methodologies to compare results. it is expected that the gpsi based gpiv methodology proposed for the first time in this study will be used in future studies therefore will become widespread and cited in the literature. in the proposed method, only the objective weights of the criteria are considered. future studies use grey mcdm methods (such as grey ahp, grey swara, and grey fucom) to obtain subjective weights of 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regenerative braking energy are considered. in addition, the current situation and improvement opportunities to achieve efficient energy use are examined. seven measurement series were performed with scheduled railjet trains between hegyeshalom and győr railway stations in hungary. this railway section is related to the hungarian state railways' no. 1 main railway line (between budapest-kelenföld and hegyeshalom state board), which is a part of the international railway line between budapest and vienna (capitals of hungary and austria, respectively). this double-track, electrified railway line with traditional ballasted superstructures and continuously welded rail tracks is important due to the international passenger and freight transport between germany, austria, and hungary. the value of the regenerative braking energy can be even 20-30% of the total consumed energy. this quite enormous untapped energy can be used for several aims, e.g., for comfort energy demand (air conditioning, heating-cooling, lighting, etc.) or energy-intensive starts. the article also investigates the optimization of regenerative braking energy by seeking the energy-waste locations and the reasons for the significant consumption. the train operator's driving style and habit have been identified as one of the main reasons. furthermore, train driver assistance systems are recommended to save energy, which is planned for future research. key words: railway, electric locomotive, railjet, regenerative braking energy, acceleration-deceleration, energy optimization received: november 04, 2022 / accepted december 12, 2022 corresponding author: szabolcs kocsis szürke széchenyi istván university, audi hungaria faculty of automotive engineering, department of road and rail vehicles, h-9026 győr, egyetem tér 1., hungary e-mail: kocsis.szabolcs@sze.hu 82 s. fischer, s. kocsis szürke 1. introduction in railway transportation, the electricity consumption [1,2] will be critical due to the so-called worldwide energy crisis and the increased price of energy sources [3]. therefore, it is a logical decision, and it would be a good attitude, if the railway companies tried to save money by reducing the overconsumption of electricity. it is also an important task and a relevant factor that the world must be cleaner in the future, as well as the countries must use green or "greener" energies instead of fossil ones [4]. in addition, a new type of urbanization also has a positive effect on the green total factor and the energy efficiency [5]. therefore, people must take care of the planet and try to use as low energy as possible, as well as emit as low green-house and other pollutant gases and materials as they can [6]. for example, minaminosono et al. [6] reported that train emission in japan was measured as 22 g co2/km for 1 km of passenger transport. this value is one-sixth of the carbon dioxide emissions of passenger cars and one-fifth of those of airplanes, suggesting that electric railways have environmental benefits. furthermore, they gave examples of regenerative energy, reaching the 47% of the running energy on the yamanote line. in the case of railway transport, first, the whole consumed energy of a train must be analyzed to reduce it [1,7,8]. the affecting factors regarding energy consumption can be categorized into three main groups: (i) infrastructure effects, (ii) transportation organization effects, and (iii) external environmental impact [9]. the infrastructure effects contain traction and vehicle characteristics and line characteristics. the transportation organization effects are related to speed characteristics, marshaling characteristics, stop plan, utilization ratio of seat capacity, handling characteristics, and signal displays. the external environmental impacts are, e.g., the altitude, climate, barometric pressure, etc. therefore, the energy consumption of a train is the operational energy consumption (i.e., the sum of the traction energy with a positive sign, as well as the regenerative braking energy with a negative sign), as well as auxiliary energy consumption (i.e., on-board service equipment during train operation including auxiliary equipment, e.g., air conditioner, lighting, and ventilation, etc.). to minimize and optimize the total consumed energy, one of the main possibilities is the usage of the so-called recuperated (regenerative) electric energy [10-12]. the other significant solution is the optimized planning of the time-table considering all of the horizontal and vertical geometries of the railway tracks (i.e., geometrical characteristics), as well as the planned (and/or the operation) speed of the given railway lines [10,13-15]. it has to be mentioned that the best solution is when the above two methods are combined. in the case of time-table optimization, the adequate solution is when the optimization procedure is related not only to theoretical calculations but also to real case studies [9,10,16-18]. the most frequently applied methodologies are the several deep learning methods, fully-connected neural networks, recursive neural networks, standard lstm (long short-term memory), lstm with convolutional layers, lag-lstm (lstm with lagged information) [17], lp (linear programming) [12], milp (mixed-integer linear programming) [10,18,19], minlp (non-linear mixed-integer programming model) [20], deep-q-network and markov decision process for the application of reinforcement learning (rl) [21], multiple-particle operation model and new methods based on mutated dichotomy and differential evolutionary algorithm (mddea) [22], cm-pso (competition mechanism-based particle swarm optimization) [23], etc. detection process of energy loss in electric railway vehicles 83 international research provides examples and connecting solutions for optimization, and they were executed for public railways [1], tramways [24], metro lines [25], as well as maglev (i.e., magnetic levitation) lines [26]. when examining regenerative braking energy, road vehicles can also be considered. based on automotive, the quantity of the potentially produced regenerative energy depends on many factors: vehicle, route characteristics, and traffic conditions [11]. energy recovery systems (erss) can be categorized into three main groups: (i) using exhaust gases, (ii) vertical oscillation of the vehicle body, and (iii) application of vehicle inertia. the first group uses thermal energy (e.g., chemical and steam), and the last two items are based on kinetic energy considering chemical and non-chemical types. there are different subtypes, e.g., electric turbo compound (it is below the exhaust gases and kinetic and chemical category), regenerative shock absorber (it is the subtype of vertical oscillation of the vehicle body and kinetic and chemical category), or the flywheel (it is the subtype of vehicle inertia and kinetic and non-chemical category) [11]. the main differences among the possible systems are (a) whether the applied system can provide continuous or non-continuous energy regeneration, (b) the applicability and conformity to hybrid service/operation, (c) the extra weight and space regarding the vehicle body and its dead load, (d) whether it ensures prompt energy storage, (e) contains wearable equipment or not, (f) it is noisy or silent, as well as (g) the additional price compared to the pure, original assembly. this paper partially investigates the train driver's behavior and recommends localizing the energy-wasting places on railway lines. one of the adequate solutions for storing and using the regenerative braking energy seems to be the on-board battery packs [27-29], supercapacitors [28], and hybrid storage system [30,31]. however, direct charging back to the electric network would be the best solution [32]. fischer [2] examined the energy needed for acceleration of locomotives and electric multiple units (emus). based on calculations, he concluded that usage of different correction factors is adequate to determine the consumed energy during acceleration. his defined factors are related not only to the passenger trains, but also to freight trains. international researchers have dealt in a very detailed manner with train driver assistance systems (or in other words, driver-advisory systems, i.e., dass) [24,33-35]. it seems to be one of the most valuable ways to ensure maximal safety, the lowest consumed energy, and the possibility to guarantee the planned time-table for trains. these systems are also available for train drivers and car, truck, bus, trolley, tram, etc. drivers. the dass' type and operation depend on whether the train is "ordinary" (i.e., driven by traditional human drivers) or autonomous [11,16,36]. in the case of train drivers, different driving strategies can be considered: maximum acceleration, cruising (maintaining a certain speed level), coasting (turning off traction), and maximum braking [35]. in the case of the das, also four driving strategies were taken into consideration: (i) mttc (minimum time train control), (ii) mc (maximum coasting), (iii) rms (reduced maximum speed), as well as (iv) eetc (energy-efficient train control). as a result, the lowest energy consumption was related to eetc and the highest one to the mttc. if someone would like to assess the railway system's initial and annual costs, the maintenance costs are the highest part (i.e., considering the whole life-time) [37-39]. on the other hand, the efficiency of the trains and the railway systems are also relevant [40,41]. the life-time costs (lcc) of the railway track depend on a lot of factors, e.g., the condition of the substructure [37,42], the condition of the superstructure [43-47], railway track geometry [48], condition of rolling stock [49], etc. some researchers have dealt with the 84 s. fischer, s. kocsis szürke projects' risk analysis which is quite significant based on the planning and construction phases of the railway projects [50]. as mentioned earlier, current paper is about the energy loss detection process. the authors' main goal was to give a useful algorithm to localize sections where there are significant energy consumptions. the algorithm applies a dividing process (i.e., dividing into subsections) and searches for the reason for these "steps" and possible peaks. furthermore, the authors decided to analyze the worst and the best cases (i.e., the lowest and the highest consumption data series or cases) in a more detailed manner. in this way, these cases can be compared, and the differences can be explained in all the divided subsections. 2. methodology of possibilities for optimizing energy consumption 2.1. methodologies in general the goal of the examination was to improve the energy efficiency of railroad transportation and investigate the factors generating and influencing the energy loss. the examination of the trains takes into account the routes and the consumed energy. during the test series, the reason for the energy losses is analyzed, and the potential energy recovery opportunities (locations) are examined. based on the energy consumption optimization, four main categories were defined [9]: (i) condition of the permanent way and connecting infrastructure, as well as the rolling stock, (ii) recovered energy from regenerative braking, (iii) external factors (i.e., environmental factors, temperature, etc.), finally (iv) the human factor. this article neglects the condition of the rolling stock and the external factors – they are planned research. during the investigation of the condition of the railway permanent way (i), the goal is to identify the locations which cause, or could cause a significant loss of energy and try to explain them. for example, if the railway track geometry is of inadequate quality, a fault can be expected in the railway track (either in superstructure or substructure). of course, an overhead line problem can be a relevant factor. it is also essential to mention the braking systems of the rolling stock, mainly the engines'. when using recuperative braking energy (ii), the first aim is the opportunity of its applicability. in this case, the achieving of greater utilization of regenerative energy, locations should be found where the quantity of regenerative braking energy can be further increased. in the case of external factors (iii), the target is to examine, for example, the temperature dependency of the spent energy and define modification determinants. as mentioned above, as a part of further work, it is planned to investigate how to improve the methodology. in the case of human factor (iv), it can be noticed that the consumed and recuperated energy significantly relies upon the train operator's driving style. it is essential to mention that the operator is not always the main cause of energy loss. the reasons can often be searched in fault, incorrectly determined and located railway signals (unnecessary or wrong red signals, redundant stopping/braking), which lead to high unnecessary energy consumption – maybe with relatively high values compared to the whole routes of trains. 2.2. analysis and energy optimization process 1. data collection: the analysis of electricity consumption data. during the beginning stage, the more extensive the accessible database, the more accurately the problem can be recognized. composition of the data to be tested: acceleration detection process of energy loss in electric railway vehicles 85 and deceleration data set supplemented by consumed and recuperated energy, accurate time data and speed values. current research applies data collection (sampling) at the relevant time points. of course, the best solution would be more frequent data sampling. the higher the data sampling frequency, the better the energy-wasting location(s) can be defined. it has to be noted that more data is required for generalization and, thus, higher computing (computational) power. a possible option is, e.g., to collect more data during larger accelerations and decelerations, as well as constant data sampling during consentaneous progress. the data collection will be more extensive if there is a 5 km/h speed increasing or decreasing (resulting in acceleration or deceleration, respectively). different evaluation methodology is needed which can be difficult. alternatively, data are analyzed only in parts if a "fault" is found at a particular location. all in all, the higher frequency data sampling is efficient. another solution can be connecting a data logger to the telemetry data and analyzing it only when there is a "fault" or only turning it on if there is an assumed or an observed "fault". 2. comparative analysis: based on the balance of the amount of consumed energy in the same section. it is significant to mention that the direction of the route also has to be considered to determine the exact and accurate longitudinal profile (uphill and downhill slopes). another significant element to consider is the deviation of external factors (e.g., weather conditions, temperature values, etc.). the correction factors for this must be taken into consideration. the more data were available on a given railway line, the easier it would be to create normative electric energy consumption. different normative consumption is needed in winter and summer, especially for passenger trains with cooling, heating or combined cooling-heating air conditioners (acs). the difference is presumably not too big for freight trains, but real measurement data are needed. a weighted ordered averaging (owa) of many measurement data could be used. nevertheless, a simple min/max/avr (min – minimum, max – maximum, avr – average) averaging can also be applied. in addition, calculating the skewness and kurtosis parameters and a statistical distribution calculation and entire statistical analysis with sophisticated up-to-date methods would also be helpful. 3. designation of significant deviations: segregating them into shorter subsections and localizing the positions of energy loss. (such a resolution can be divided into parts following the chainage/sectioning.) evaluating the data will show the differences between several tests on the same route. therefore, it is worth analyzing data from the same subsection of different measurements to set up a better consumption strategy (see fig. 1). in fig. 1, the whole section is divided into smaller subsections to identify energy wastage sites. in approach a, the whole distance was first split into two parts, then into two other parts (further splitting is also possible), thus providing a detailed analysis of the whole section. in approach b, in the first case, three different subsections are selected (i.e., #1, #2 and #3, respectively), and only the section with the worst results is split again (they are shown in fig. 1). this method performs a targeted, iterative fault search, intending to locate energy wasters in as few iterations as possible. the first approach gives more detailed results but it is more computationally demanding. after evaluating and using min, max and avr methods, observing the differences between the different sections (or subsections), and defining a transport strategy can be worthwhile. 86 s. fischer, s. kocsis szürke fig. 1 segregating of sections into shorter subsections 4. determination of possible error factors: according to options determined in section 2.1 (i.e., from points (i) to (iv)). the condition of the railway track (i) is one of the critical issues in terms of energy loss, as there is an inadequate assessment of the condition of the track, so there is a speed restriction where it is not needed. vice versa, if a speed restriction is not yet set at the right location where it is needed, it can cause increased degradation of the track geometry and track structure (which is also dangerous and unsafe). however, its determination and evaluation require more complex measurements and analyses, and of course, they cannot be determined by only a few measurements. the so-called worst case scenario for regenerative braking (ii) is the case that no energy can be recovered or stored. in other cases, battery systems are needed to store energy. with this method, the investment can be returned by covering of the comfort energy as savings. (the so-called best case scenario is when the regenerative braking energy is the maximal, however, the total consumed energy should be considered regarding the recovered part.) the role of the weather (iii) in the wheel-rail contact (i.e., in this case, mainly adhesional friction) is the main factor to be considered. in rainy, humid, wet weather, adhesional traction is significantly reduced by decreasing adhesional friction. therefore, it has an effect during acceleration, although not a direct effect. it can be necessary after speed restrictions because the low adhesional friction can increase the length of the reacceleration phase (due to the decreased adhesional traction force), e.g., it may make the re-acceleration more difficult on steep gradients. human error (iv) and possible omittance can be fundamental, as it is an easily avoidable factor with the help of various assistance systems (i.e., with dass). thus, it is recommended to aim for the highest possible degree of autonomy of dass. 5. possible solution: drafting achievable results according to the assessed data. possible solutions should also be considered according to options from (i) to (iv), determined in section 2.1. a repair or renewal strategy should be developed in case of a problem with the condition of railway track (i). it is usually the most expensive solution and may require an accurate cba (cost-benefit analysis). it determines how long it will take to recover the investment, or, more specifically, when the latest intervention is needed to avoid too much degradation. unfortunately, the impact of speed restrictions can be felt immediately in costs, but the detection process of energy loss in electric railway vehicles 87 deterioration of the track does not stop at reduced speeds. on the other hand, there are costs (in terms of person-travel time) associated with slower travel times. regarding regenerative braking (ii), it may be worth examining where the most regenerative energy can be used or stored immediately. also, which braking strategy is better: the more intensive one or the gradually decelerating one. external temperatures (iii) cannot be influenced, but compensation values (i.e., factors) can be defined, which helps in the planning of energy usage and acceleration-deceleration. another interesting aspect to investigate is how the operating temperature of traction motors varies with the external temperature and resulted consumption. in the case of high electricity consumption due to the human factor (iv), it is worth setting up support systems (dass) to help choose the right driving strategy (ideal choice of acceleration and braking.) 6. cost analysis: formation of alternatives according to price/value among the possible opportunities [50]. first, the optimization aims to obtain minimal energy consumption and/or the time accuracy of the trains according to their time-table. additionally, it has to be decided which optimization strategy is better, i.e., the speed of the optimization process or the accuracy of the results. 3. energy efficiency analysis regarding the energy efficiency analysis, specific results were examined related to scheduled railjets between railway stations hegyeshalom (hh) and győr (gy) (see fig.2). during all seven measurement series (ms1 to ms7), the registered energy consumption data are drawn in fig. 3. the mass of the trains was 479 tons, the length was 206 m. fig. 2 map of the measurement's location and the railjet train (edited figure based on internet sources) fig. 3 illustrates the elapsed time during the measurements (i.e., from zero to approx. 1600 s). the consumed energy values are depicted in the vertical axis (i.e., between zero and approx. 900 kwh). the chart in fig. 3 demonstrates the executed seven measurement series in unique colors. according to the outcomes, it can be recognized that the character of electric energy consumption was very similar in all cases, but differences can also be noticed. also, 88 s. fischer, s. kocsis szürke there are more significant differences between the most efficient and the worst cases. therefore, the following examination stage analyzes the electric energy consumption between the stations along the investigated railway section. based on the procedure described in section 2.2 (part 3), the investigated example section was divided into three parts, i.e., section #1, #2 and #3, respectively (see fig. 4). the easiest way to divide it into parts was to consider each station as a breaking point. there are altogether four railway stations where the railjet trains regularly stop: hegyeshalom, mosonmagyaróvár (mm), öttevény (ot), and győr. however, it is relevant to mention that the trains did not stop at mm and ot during ms1 and ms2, i.e., the published results were only approximated (i.e., interpolated) values in these cases. fig. 3 energy consumption during ms1 to ms7 fig. 4 energy consumption between hegyeshalom and győr 0 100 200 300 400 500 600 700 800 900 1000 0 200 400 600 800 1000 1200 1400 1600 1800 e n er g y [ k w h ] time [s] ms1 ms2 ms3 ms4 ms5 ms6 ms7 0 200 400 600 800 1000 1200 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 e n er g y [k w h ] distance [m] ms1 ms2 ms3 ms4 ms5 ms6 ms7 section #i hegyeshalom mosonmagyaróvár section #ii mosonmagyaróvár öttevény section #iii öttevény győr detection process of energy loss in electric railway vehicles 89 it has to be mentioned that the longitudinal slope between hh and gy is approximately between zero and seven per mile (‰, i.e., 1/1000), and there are only horizontal curves with relatively high radii. it is why modification was neglected during this investigation. tables 1-3 summarize unique measurements regarding stops per railway station. table 1 energy efficiency analysis #1 (energy consumption data) railway station hh mm ot gy chainage [km] 188.0 176.4 154.9 141.2 distance [km] 0 11.6 33.1 46.8 ms #1 [kwh] 0 270 690 804 ms #2 [kwh] 0 307 786 914 ms #3 [kwh] 0 248 620 830 ms #4 [kwh] 0 273 610 819 ms #5 [kwh] 0 323 681 912 ms #6 [kwh] 0 230 558 755 ms #7 [kwh] 0 256 652 861 min [kwh] 0 230 558 755 max [kwh] 0 323 786 914 avr [kwh] 0 272 657 842 table 2 energy efficiency analysis #2 (energy consumption data between stations) railway station hh mm ot gy chainage [km] 188.0 176.4 154.9 141.2 distance [km] 0 11.6 33.1 46.8 ms #1 [kwh] 0 270 420 114 ms #2 [kwh] 0 307 479 128 ms #3 [kwh] 0 248 372 210 ms #4 [kwh] 0 273 337 209 ms #5 [kwh] 0 323 358 231 ms #6 [kwh] 0 230 328 197 ms #7 [kwh] 0 256 396 209 min [kwh] 0 230 328 114 max [kwh] 0 323 479 231 avr [kwh] 0 272 384 185 table 3 energy efficiency analysis #3 (consumed energy during measurements) railway station hh mm ot gy chainage [km] 188.0 176.4 154.9 141.2 distance [km] 0 11.6 33.1 46.8 ms #1 [kwh/km] 0 0.41 0.36 0.12 ms #2 [kwh/km] 0 0.43 0.43 0.13 ms #3 [kwh/km] 0 0.36 0.29 0.26 ms #4 [kwh/km] 0 0.39 0.26 0.25 ms #5 [kwh/km] 0 0.46 0.28 0.28 ms #6 [kwh/km] 0 0.33 0.26 0.24 ms #7 [kwh/km] 0 0.37 0.31 0.25 min [kwh/km] 0 0.33 0.26 0.12 max [kwh/km] 0 0.46 0.43 0.28 avr [kwh/km] 0 0.39 0.31 0.22 the meaning of the data in the rows of tables 1-3 is the following: the first rows are related to the several railway stations (hh, mm, ot, and gy), the second rows connect 90 s. fischer, s. kocsis szürke to the chainage (with kilometer sectioning), the third rows are related to the distance in "km" unit from the starting point (i.e., from hh). the following seven rows illustrate the electric energy consumption data. finally, the last three rows give the min, the max, and the avr consumed energy values (see fig. 5). fig. 5 energy consumption between hegyeshalom and győr in table 1, there are data containing only the hauling and acceleration energy values required to provide the hauling between the two stations. moreover, recuperated energy values are neglected in this case (table 1). there is a relevant difference considering the maximal and the minimal cases and it is approximately 159 kwh. accordingly, the places of the highest deviations were defined, and each stop was examined independently in table 2. for ascertainability, the consumed energy data were separated into shorter distances (i.e., stations), and the differences between single sections were monitored. for a more unequivocal explanation in table 3, the consumed electric energy values are shown simultaneously with the traveled length. in this case, a calculated energy need for 1 km can already be observed in accordance with the measurements. referring to them, the so-called energy-wasting places are able to be effortlessly detected and localized. based on this approach, the ms5 and ms6 between stations hh and mm, ms2 and ms6 between stations mm and ot, as well as ms1 and ms5 between stations ot and gy, are taken for further detailed analysis. these are the socalled detailed analyses #1, #2, and #3 (da1, da2, da3), respectively. according to da1, locations with minimal and maximal energy consumption were localized. fig. 6 illustrates the places with the most relevant deviation in ms5 and ms6. fig. 6 presents the section between railway stations hh and mm related to the ms5 (blue lines) and ms6 (red lines). the location is according to the relative "chainage" 0…12 km, based on table 1. the continuous lines in fig. 6 present the consumed energy reduced by recuperation, the dashed lines illustrate the speed data, and the dotted lines represent the regenerative braking energy during the measurement series. the relevant difference can be observed in consumed energy around the relative distance of 2000 m, possibly due to the various speeds. during ms6, the considered railjet accelerated up to 160 km/h and a lot more energy was required compared to ms5. accordingly, higher 0 100 200 300 400 500 600 700 800 900 1000 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 e n er g y [ k w h ] distance [m] min max avr detection process of energy loss in electric railway vehicles 91 recuperated energy can be generated due to the higher speed. however, significantly higher energy is consumed correspondingly. in accordance with it, it can be concluded that unique acceleration and deceleration have a critical influencing factor on the needed energy, and the chauffeur has a significant role in this. nevertheless, it is essential to mention that acceleration-deceleration can happen due to traffic issues, i.e., the consumed energy does not expand in all the cases due to the chauffeur's driving habit. (the recuperated energy reached approx. 30% of the utilized energy. it can be seen in fig. 6, i.e., 54 and 40 kwh values, respectively.) fig. 6 energy consumption during ms5 and ms6 based on da2, examining best and worst case scenarios can be seen in fig. 7. the difference may be due to the train arriving at a higher speed. the braking of ms2 from 160 km/h to 80 km/h generated more regenerative energy of 47 kwh at the location of 26 km, but the subsequent acceleration needed much more energy. as a result, ms2 showed an energy balance (with regenerative subtracted) of approx. 135 kwh, between sections 26.0 km and 33.1 km. during ms1, the energy balance (with the regenerative subtracted) over the same examined subsection is approx. 43 kwh. the difference is, therefore, 92 kwh, so the vast majority of the difference is due to it. the rest is approx. 60 kwh from the initial acceleration, possibly because of the speed restriction. the analysis of the best and worst case scenarios based on the da3 is shown in fig. 8. in da3, the most considerable difference in consumption was due to acceleration in the 33.1 km and 35.9 km sections. the regenerative energy values were 42 kwh in ms1 and 12 kwh in ms5, respectively. it can be concluded that the application of the braking strategy (see ms1) is recommended. results of the best and worst cases are illustrated in table 4. the values in the last two columns are related to measurement series with the minimal and the maximal energy demand, regarding the several subsections between the considered railway stations. there is a 361 kwh energy difference value between the sum values, which is significant. fig. 9 shows the details of the best and the worst cases for the whole examined length. it visualizes the optimal acceleration-deceleration strategy. 0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 250 300 350 400 450 500 0 2000 4000 6000 8000 10000 12000 14000 s p ee d [ k m /h ] e n er g y [ k w h ] distance [m] energy consumption ms5 energy consumption ms6 regenerative energy ms5 regenerative energy ms6 speed ms5 speed ms6 92 s. fischer, s. kocsis szürke the graphs show the relevant energy wasting and energy efficient locations. the relevant differences can be due to the driving style of the chauffeur. it has to be mentioned that some of the differences can be because the speed restrictions were not applied and/or set in the same places during all the measurement series. it can be seen that accelerations and decelerations have a significant impact on energy consumption and determining the optimal train driving speed is a crucial factor. however, it is not always a possibility. fig. 7 energy consumption between mosonmagyaróvár (mm) and öttevény (ot) fig. 8 energy consumption between öttevény (ot) and győr (gy) 60 80 100 120 140 160 180 0 100 200 300 400 500 600 700 800 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 s p ee d [ k m /h ] e n er g y [ k w h ] distance [m] energy consumption ms2 energy consumption ms6 regenerative energy ms2 regenerative energy ms6 speed ms2 speed ms6 0 20 40 60 80 100 120 140 160 180 0 100 200 300 400 500 600 700 800 900 1000 33100 35100 37100 39100 41100 43100 45100 s p ee d [ k m /h ] e n er g y [ k w h ] distance [m] energy consumption ms1 energy consumption ms5 regenerative energy ms1 regenerative energy ms5 speed ms1 speed ms5 detection process of energy loss in electric railway vehicles 93 table 4 results of best and worst cases analysis of energy efficiency railway station chainage [km] distance from the previous railway station [km] distance from the hegyeshalom (hh) railway station [km] best case [kwh] worst case [kwh] hh 188.0 0 0 0 0 mo 176.4 11.6 11.6 230 323 ot 154.9 21.5 33.1 328 479 gy 141.2 13.7 46.8 114 231 sum. – – 46.8 672 1033 fig. 9 energy consumption between öttevény (ot) and győr (gy) fig. 10 illustrates the average energy consumption related to smaller subsections. the dots connect the considered subsections' end points. the positive values refer to energy consumption due to acceleration and constant speed hauling. the negative values are related to regenerative braking. therefore, it can be concluded that the average energy demand is approximately 0.0243 kwh/m. the advantage of this approach is that it shows the required energy for each distance to be traveled, i.e., how much energy is needed to complete a subsection. however, the approach ignores its relation to the following subsection. for example, once a certain speed has been reached, further acceleration may no longer be required, so the energy demand is reduced. it is therefore recommended to apply a significant sample rate and a small subsection length. it must be noted that several measurement series need to be analyzed separately before the statistical investigation of the whole database. furthermore, it can be concluded that it is rather challenging to use the method with rarely/infrequently sampled data. however, short distances and dense data point-sets can greatly help assess a particular section. 0 20 40 60 80 100 120 140 160 180 200 220 240 0 10000 20000 30000 40000 50000 s p ee d [ k m /h ] distance [m] worst case best case section #i hegyeshalom mosonmagyaróvár section #ii mosonmagyaróvár öttevény section #iii öttevény győr 94 s. fischer, s. kocsis szürke fig. 10 average energy consumption data regarding the several subsections (the indicated dots are related to the considered subsections' end points) 4. utilization of energy obtained from regenerative braking the efficient use of the amount of energy recovered is of great importance for increasing efficiency. fig. 11 illustrates the related measurements. fig. 11 shows that braking occured somewhere during ms2 and ms3. the reason was a speed restriction around ot. relevant differences can also be observed in the quantity of generated regenerative braking energy during all the measurement series, probably due to the chauffeur's driving habit and traffic circumstances. the generated regenerative energy is one relevant factor, and the other is its possible utilization. in the case of optimal conditions, it can be charged into the electric network promptly and directly. nevertheless, in the considered cases, it was not available, i.e., a substitute application would be needed. one choice could be the generated regenerative energy to supply comfort devices (e.g., lighting, air conditioning). the comfort energy request during all the measurement series is shown in fig. 12. the consumed electric energy data are presented in fig. 12. these values are only approximations and highly dependent on external factors. the aim is to store the regenerative braking energy instead of losing it and ensure part of the comfort energy requests. according to the results, quite large amounts of energy can be stored using battery packs, and a relevant part of the comfort energy need can be covered (c.f., values in figs. 11 and 12). fig. 12 illustrates different cases in which the comfort energy consumptions (cecs) are 1.0 kwh during 8, 12, 15 and 18 s (cec1-cec4, respectively, see fig. 12). these consumption values are related to real measurement, however, they are only approximated average values. the more accurate comfort energy consumption data are needed to be considered based on, e.g., on-board measurements. further investigations are needed to solve this problem. -0.05 0.00 0.05 0.10 0.15 0.20 0 10000 20000 30000 40000 50000 e n er g y c o n su m p ti o n [ k w h /m ] distance [m] ms1 ms2 ms3 ms4 ms5 ms6 ms7 section #i hegyeshalom mosonmagyaróvár section #ii mosonmagyaróvár öttevény section #iii öttevény győr detection process of energy loss in electric railway vehicles 95 fig. 11 regenerative energy generated between hegyeshalom (hh) and győr (gy) fig. 12 approximated comfort energy consumption between hegyeshalom (hh) and győr (gy) considering four different trends 5. conclusions the paper deals with energy loss and fault location detection. the research is based on seven measurement series (i.e., from ms1 to ms7) performed by railjet trains in hungary. the considered section was between hegyeshalom (hh) and győr (gy) railway stations. during the measurements, specific parameters were registered to define the accurate hauling and regenerative braking energy. 0 20 40 60 80 100 120 140 160 0 200 400 600 800 1000 1200 1400 1600 1800 e n er g y [ k w h ] time [s] ms1 ms2 ms3 ms4 ms5 ms6 ms7 0 50 100 150 200 250 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 e n er g y [ k w h ] distance [m] cec1 cec2 cec3 cec4 96 s. fischer, s. kocsis szürke the article investigates the reasons for the high energy consumption and optimization of regenerative braking energy. generally, two major influencing factors were determined: the speed restrictions (quantities and places) and the chauffeur's driving habit. therefore, the solutions can contain keeping the rolling stock and the related infrastructure in sufficiently acceptable condition or introducing speed restrictions and/or application of dass. these systems aid with optimal acceleration and deceleration (i.e., with optimal energy consumption), considering the real permanent way's parameters and characteristics. this paper also dealt with the energy loss detection process – as a part of energy optimization. the authors' main goal was to provide a useful algorithm to localize places and/or sections where there are significant energy consumptions. the algorithm applies a dividing process (dividing into sections) and searches for the reason for these steps and possible peaks. furthermore, the authors decided to analyze in a more detailed manner the worst and the best cases (i.e., the data series with the lowest and highest energy consumption). in this way, they can be compared, and the differences can be explained in all the divided subsections. the analyses also present an ideal acceleration and deceleration strategy to help decide when major energy losses are observed. of course, these strategies cannot be taken into account in all traffic situations, but it is an excellent approach to designing and implementing an assistance system in the future. the article also presents the consumed energy per traveled distance strategy, which is found to be inefficient to use on small data volumes (i.e., with low/rare data sampling). therefore, the authors propose investigating the initial fault detection using smaller sampling and more measurements. (it would avoid the initial large data set.) furthermore, after locating the critical section, it must be investigated with a larger data sampling. in this way, the exact amount of energy loss for that subsection can be determined very accurately, and the fault can be more easily detected. the value of the regenerative braking energy can be up to 30% of the total consumed energy. this relevant unused energy can be utilized for multiple goals: comfort energy requests (lighting, air conditioning) or energy-intensive starts. the accurate comfort energy consumption must be determined by (field, or on-board) measurements to be able to take into consideration in the future for optimization, as well as design related battery packs. the authors would like to mention that the official price of electric energy for railway hauling is not available nowadays. it is the so-called "secret" between the provider (here, the mvm group) and the consumer (here, the hungarian state railways ltd., i.e., máv ltd.). therefore, the authors did not want to calculate approximated costs without accurate energy price values. however, an approx. electric energy price can be found on the internet (e.g., on the stock markets' webpages), and an elementary multiplication can be executed based on the published energy consumption data in kwh unit. it must be noted that the prices on the stock markets can significantly differ from the prices in mentioned contracts. the authors clearly know that the time-table is one of the most critical aspects of railway transport. however, in the calculations, the optimization of the time-table was not considered. only the energy consumption of the trains was considered. railjet trains have the highest significance on railway mainline no. 1, so the authors think that the minimized energy is the most relevant related to railjets. the time-table can be adjusted to this aim. the time-table optimization is, therefore, a suitable research direction, and it is planned for 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infrastructure projects in the republic of serbia, acta polytechnica hungarica, 19(3), pp. 103-118. https://doi.org/10.2478/cee-2022-0037 facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 155 174 https://doi.org/10.22190/fume201215043f © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper lightening structures by metal replacement: from traditional gym equipment to an advanced fiber-reinforced composite exoskeleton cristiano fragassa department of industrial engineering, alma mater studiorum university of bologna, italy abstract. a redesign procedure used for introducing new functional properties in innovative gym equipment is here reported. it is based on a metal replacement action where a tempered steel was firstly replaced by an aluminum alloy and then by high strength-to-weight fiber-reinforced polymers. the effect of fiber properties (as strength and volume ratio) and plies stacking sequences (as thicknesses and orientation) were investigated. numerical analyses, done by ansys acp, allowed evaluating the stressstrain behavior in realistic boundaries and quasi-static loads, comparing materials and layouts in terms of stiffness. the single-layered shell method with additional integration points was preferred as a technique for discretizing composite laminates. maximum principal stress and maximum distortion energy (tsai-hill) were applied as anisotropic failure criteria. changes in geometry were also considered given their relevant effects on parts and processes. specifically, this paper is focused on a representative component of the main kinematic chain (the ‘forearm’) and details the different redesign phases for that part. the chosen solution consisted of 14 layers of unidirectional and bidirectional carbon fiber-reinforced pre-pregs, offering a 68 % weight reduction with respect to a solid aluminum component with equal stiffness. the part was manufactured by hand lay-up and cured in autoclave. this redesign practice was extended to the rest of the equipment allowing its transformation into an exoskeleton. key words: material design, functional design, exoskeleton, fiber reinforced polymers, strength-to-weight ratio, exercise machine, rehabilitation 1. introduction the replacement of metals by polymer-based materials has become a common practice nowadays in many engineering areas due to a series of benefits, both in terms of the components and processes they are able to provide. received december 15, 2020 / accepted may 28, 2021 corresponding author: cristiano fragassa department of industrial engineering, university of bologna, viale del risorgimento 2, 40136 italy e-mail: cristiano.fragassa@unibo.it 156 c. fragassa these materials offer the possibility of conducting a thorough system optimization based on weight reduction and performance increase based on a tailored-design premise. designers can also count on thermal stability, resistance to corrosion and oxidation, thermal, acoustic, and electrical insulations through shorter manufacturing processes and reduced material waste, thus increasing overall savings [1]. automotive and aircraft industries were the first industrial sectors to benefit from a progressive metal replacement. a gradual decrease of ferrous (as steel alloy, carbon steel, cast iron and wrought iron) and non-ferrous metals (primarily aluminum) presence in the vehicle design over the years is reported [2]. in these cases, metallic alloys are often replaced by high performance polymers and reinforced polymer sandwiches on the way to improving the parts performance. a representative case is stated in [3] in which reinforced plastic replaced steel achieved 23% of weight reduction in the passenger airbag housing. this new component also successfully passed all the strict validation tests. however, each major industrial sector has been involved in replacing, to some extent, metals with plastic polymers [4]. according to their lower material density, a 20/30% weight reduction can be generically estimated in relation to their use, even more in the presence of high strength-to-weight fiber-reinforced polymers (frp) [4, 5]. therefore, the material replacement has become such a common and effective design strategy that some cases emerge in which designers have even replaced composites with composites despite trying to (re)introduce metals. this situation is evident, for instance, in a work concerning railway sleepers [6]. they were traditionally made by hardwood timbers and the use of metal (carbon steel) or concrete was under evaluation. extending the material alternatives to composites as well, the best solution in terms of strengthening and in-service costs reduction responded to the utilization of fiber (glass or carbon) reinforced composites, either as a single constituent material, or as timber reinforcement layers (e.g. applied by wrapping). at a glance, comparing a quite common carbon fiber-reinforced polymer (cfrp) as toray t700, a structural aluminum alloy as 6061-t6 (al) and a conventional tempered steel as iso c40, it is easy to notice the potential benefits offered by cfrp regarding the combination of lightness and stiffness. the cfrp specific weight (2200 kg/m3) is 20% lower than the mentioned lightweight al alloy (2720 kg/m3) and 3.5 times lighter than sst (7800 kg/m3). at the same time, the cfrp mechanical properties are also remarkable [7] with young's modulus (e) up to 112 gpa, and ultimate tensile strength (utm) up to 1100 mpa), making this material a consistent alternative to both al (71 gpa and 310 mpa) or sst (190 gpa, 510 mpa) in many structural applications [7]. moreover, the variability due to anisotropy (e.g. e and utm ranging between 35-112 and 0.25-1.1 gpa, respectively, for t700) permits optimizing part geometries based on local stress distributions [8]. for instance, in [9] a genetic algorithm was used during the process of design optimization (‘layout design’) of 8-layer composite laminates intended to withstand transverse or in-plane loads. aspects such as fiber type (i.e. glass, carbon, and aramid), ply orientation and thickness were modified following a multi-objective approach while finite elements (fes) permitted verifying failure criteria (i.e. by means of the tsai–wu theory). depending on the specific design target, the algorithm permitted to widely range out in terms of weight (±33%), deflection (±76%) and cost (±43%). however, the design concept that material changes can improve product performances is fairly known among designers. less frequent is the case in which a metal replacement leads to significant changes in the product functionality such as it is reported hereby. lightening structures by metal replacement: from a traditional gym equipment to an advanced... 157 this paper proposes the case of advanced gym equipment, innovative with respect to several functional solutions, and that is able to offer a rather exclusive gym training method. the equipment is lightened thanks to a combination of geometric modifications and metal replacements from steel to aluminum and, finally, to reinforced composites – on the way to being transformed into a wearable exoskeleton, thus incorporating new functionalities. composite exoskeletons, using carbon or analogue extra-light fibers as reinforcement, are available nowadays, although not very popular, especially pondering costs. in most cases they are powered by external systems like electric motors, pneumatics, levers or hydraulics allowing movements with increased strength and endurance. applications include but are not limited to military, airspace, advanced industry (in the presence of heavy loads), civilian (firefighters and rescue workers) and medical applications. the first exoskeleton was probably developed in 1972 in serbia as a medical device for rehabilitation of paraplegics [10]. it was active unwearable equipment, made by rather traditional materials and design solutions. then, a long time passed before the next exoskeleton that rose to prominence was developed in the us in 1985 as a powered suit of armor [11]. since then, interest in exoskeletons has been growing year after year and numerous studies describe their design techniques [12-15], including a special attention to an optimal material selection. for instance, the case of a passive/active exoskeleton for index finger rehabilitation is reported in [16]. this article, where much attention is paid to assuring the highest precision of movements, highlights how the correct definition of allowed trajectories (and, therefore, of kinematic joints) represents one among the most challenging aspects in the exoskeletons design. a study of the state-of-the-art in this field identified several exoskeletons that inspired the present development. in particular, in [17] a wearable structure having links and joints corresponding to a human upper body was patented. similar design assumptions were adopted for choosing and positioning connecting elements that enabled controlling the system movements. additional technical specifications, originating from [17], were used here for the design of gym equipment, even if this prototype was conceived as a non-wearable machine at that time. in line with the mentioned literature review, the current study demonstrates how the redesign procedure of a metallic structure into a fiber-reinforced composite one allowed a significant reduction in weight. therefore, the heavy supporting frame turned out to be unnecessary, transforming the equipment into an exoskeleton and adding new functions. to the best knowledge of the author, this is the first exoskeleton designed to ensure: 1) a completely free/unconstrained movement of the upper body, and 2) a constant and 'fluid' resistance, related to the speed of the movement but independent of location and direction. in addition, this exoskeleton was also designed to be active, thanks to the progressive replacement of the traditional frictional elements with magnetorheological torque transmission devices, already investigated in a previous work [18]. 158 c. fragassa 2. geometry and design 2.1. equipment a gym exercise machine typically creates an opposition load to be overcome as 1. gravity force: the load is constant, generated by the weight of an object (as handlebar, barbell, etc.), often driven and/or constrained by a mechanism. 2. elastic force: the load is proportional for intensity to the displacement, generated according to the spring principle (elastic bands, etc.). 3. isokinetic force: the load depends on the speed, increasing along with the relative speed induced on a mechanism (e.g. friction dumpers). 4. pneumatic / hydraulic force: the load is generated using active mechanisms that also define every aspect of the movements (as hydraulic or pneumatic pistons) all these forces are generally constrained according to two main schemes: a. free configuration, where forces are directed according to a single direction as, e.g., the weight force always downwards; the elastic force with direction along the anchor point; the isokinetic one according to the direction of cables and wires. b. structured configuration, where forces are guided by means of articulated mechanisms. with respect to these concepts, the gym machine under investigation was fully designed to be alternative and innovative. it can reproduce the complex kinematics and dynamics of the upper limbs in accordance with the following postulates: 1. ergonomics: the structure must adapt according to the person's build. the functional elements of training must support the user's motor skills. 2. functionality: the work paths, as well as the positions must be chosen as much as possible by the user and not set by the machine. setting too many constraints penalizes the user's abilities in expressing primary and consequently healthy skills. 3. force: the force of opposition to the movement must exclude the effect of gravity (weight force), the variation of intensity according to the arc of action (elastic force) and the dependence on the speed of execution (isokinetic force). the load distribution must be intrinsic to the instrument and be activated in a resistant form only during its use. the result is necessarily an anthropomorphic structure, dressed on the user (fig. 1). in particular, by means of a vest the athlete wears the system and interacts with it through independent articulated segments. thus, the functionality of the human structure has been resumed, supporting it in its mechanical complexity. main elements are: ▪ base: a fixed base and frame, resting on the ground and supporting moving parts; ▪ jacket: a movable system, to be fixed with straps, used for height adjustment and tutoring during the user’s vertical movements; ▪ trapezoids: a movable mechanism, able to replicate the shoulder movements; ▪ segments: a cinematic chain, able to replicate the upper limb movements. each side includes three segments, staying for arm, forearm and hand, integrated by resistive elements. the hand also includes a handle representing the element of interaction with the athlete and integrating the resistance management consoles. these elements are shown in table 1 together with a prediction of parts’ weight done in accordance with geometry and specific weights. these values can be considered as a plausible lightening structures by metal replacement: from a traditional gym equipment to an advanced... 159 estimate of the 'best attainable weight' that the different parts could have if made of metal. an attention was also paid to consider the proper manufacturing processes: sheet metal working (for steel) and tool machining (for steel or aluminum). accessory elements (such as resistors and joints) were also incorporated as weight. fig. 1 the innovative gym equipment (metal version): it allows free movements in space offering a reaction of constant intensity table 1 main parts of the gym equipment and weight estimation base trapezoids jacket segments steel aluminum aluminum aluminum 155 kg 21 kg 10 kg 17 kg the first prototype was developed, assembled, and applied for functional tests. wherever convenient, tempered steel, very common in the case of gym machines, was preferred with the scope to speed up construction and reduce costs. in particular, the large basement was (simplified and) made by metal sheet. many of the remaining parts were made by cnc machining. an aluminum alloy was also used with the scope of lightening. after a six-month testing period in operational environment (a gym), it was possible to confirm the main design choices and functionalities. nevertheless, also evident was the opportunity to move towards further improvements. the current work starts from this condition and, in particular, from the demand of making the entire structure lighter and leaner. such an opportunity goes beyond the usual reasons and deeply affects the machine functionality. 160 c. fragassa the equipment, as mentioned, works by offering a free movement in space coupled with a constant reaction. this objective is achieved by combining two main effects: 1) elements always balanced and in equilibrium (with respect to any spatial configuration of the equipment); 2) constant resistance concentrated in specific elements, placed at the level of the joints. however, during the movements a further factor emerges: 3) the inertial force, opposed to the motion and proportional in intensity to the speed changes. due to the high inertial masses involved, a general complication in using the equipment was noted for the period of tests, especially during the initial movements. 2.2. design changes the redesign process followed two routes: design changes in shape and in materials. however, the article is limited on those geometric changes strictly related to a proper exploitation of materials (ignoring when they simply aimed at reducing volumes). in addition, in order to simplify the description, it limits the narrative to a single component of the kinematic chain (the 'forearm'), but reports results with respect to the entire machine. specifically, the design work started with a tempered steel component (the 'forearm') and then went through its evolution in aluminum alloy and ended up with the final version in reinforced composite. table 2 reports this design progress showing changes in shape and in weight. weights were estimated considering geometry and specific weight. manufacturing technology are also listed (i.e. tool machining for steel, die casting for aluminum and hand lay-up with autoclave molding for composites). table 3 specifies the main material properties related to the chosen materials [19, 20]. in the case of the composite, properties and weight are here indicative before an optimization based on the definition of proper stratifications. shape changes were introduced to favor the lightening, but also to meet some aspects regarding the different manufacturing processes. specifically, it can be noted that the steel version (a) was made by cnc machine tool in two solid pieces. it represented the proper solution for prototyping the first machine to be functionally tested in gym. table 2 design evolution steel aluminum composite tool machining die casting autoclave 4.4 kg 1.0 kg 0.5 kg lightening structures by metal replacement: from a traditional gym equipment to an advanced... 161 table 3 comparing different versions [19, 20] version a b c material steel aluminum composite type c40 6061-t6 t1000 young modulus (e) gpa 193 71 121 ultimate strength (uts) mpa 586 460 2231 yield strength (ys) mpa 207 250 poisson ratio 0.31 0.33 0.27 density kg/m3 7850 2770 1490 the aluminum version (b) was designed to make the whole machine and its parts lighter. it was planned to be manufactured in die-casting. it was made as a 1:5 scale model using rapid prototyping techniques, but never manufactured in true scale, because it was overtaken by the composite version (c). however, aluminum die-casting offered a broader range of shapes in comparison to steel machining. it allowed the ‘forearm’ to be shaped in the way to reduce it to a single part (instead of two). non-essential areas in terms of function or resistance were also removed thanks to the proper definition of shapes. in addition, an excellent aluminum was selected, the 6061-t6, a lightweight and highresistant structural material that can be further strengthened through alloying and heat treatments. among others, its advantages are low density, high strength-to-weight ratio, good corrosion resistance, ease of fabrication and diversity of form. this first material change achieved a preliminary weight reduction (from 4.4 to 1.0 kg) to be decreased over by using fiber-reinforced plastics. starting with this update (b), several changes in geometry were introduced as well with the scope of considering the composite manufacturing technology. specifically, as said, autoclave molding was preferred: thanks to high temperatures and pressures, it allows the production of composite parts characterized by high concentration of reinforcing fibers. therefore, several changes had to be introduced with respect to this geometry, as shown in fig. 2. a) b) c) fig. 2 shape changes in the case of composite version by eliminating the: a) undercut; b) shim; c) lightening notches (version b, in aluminum, is shown) the aim of the changes is to simplify mold productions and parts extraction, such as: a. elimination of the upper undercut. the undercut is in the form of the fins, arranged in the top of the segment. niches are also present, acting as containment tank for any glue overflow during parts’ assembly. 162 c. fragassa b. elimination of the shim positioned on the terminal. its thickness has the purpose of optimizing the ergonomic appearance. in fact, this area hosts the handle, the element of interaction with the hand of the user. c. elimination of the lightening notches. cut-outs are provided for lightening the part and for driving forces in conditions of flexural-torsional loads. all these design measures, used to reinforce the aluminum component in specific areas, can be replaced by an appropriate layering in the case of composites. 3. materials and methods 3.1. fiber reinforced thermoplastics two high strength-to-weight ratio composites were used, reinforced by t800 and t1000 carbon fibers. manufactured by toray composites materials inc, they are among the strongest fibers (230 gpa) available on the market. the real difference between them is therefore mainly linked to the fact that a unidirectional (ud) weave was chosen for the t1000 while the t800 was bidirectional (bd). these fibers were supplied in the form of pre-impregnated fabrics (prepreg) on epoxy. epoxy produces a strong and stiff structure with good adhesion to the carbon fibers. the fabrics, manufactured in rolls, already contain an optimized balance of epoxy resin and carbon fiber. the low resin-to-carbon ratio accentuates the ability of the carbon fibers on the way to minimizing weight and maximizing properties. in the specific case, t1000 and t800 were able to guarantee the maximum optimization of the fiber content which usually achieves 60% (after curing). cfrp are shown in table 4 in which a marked anisotropy is also evident. table 4 main material properties for epoxy prepreg, carbon (230 gpa) [20] material unit t1000 t800 type ud bd fiber carbon (230gpa) carbon (230gpa) matrix epoxy epoxy form prepreg prepreg thickness mm 4x0.15 0.30 properties density kg/m3 1490 1420 young’s modulus (e) * mpa 121000, 8600, 8600 61340, 61340, 6900 shear modulus (g) ** mpa 4700, 3100, 4700 19500, 2700, 2700 poisson’s ratio (v) ** 0.27, 0.40, 0.27 0.04, 0.30, 0.30 stress limits tensile (uts)* mpa 2231, 29, 29 805, 805, 50 shear** mpa 60, 32, 60 125,65,65 strain limits tensile strength* % 1.67, 0.32, 0.32 1.26, 1.26, 0.80 shear stress limit** % 1.2, 1.1, 1.2 2.2, 1.9, 1.9 along *(x, y, z) **(xy, yz, xz) lightening structures by metal replacement: from a traditional gym equipment to an advanced... 163 ud and bd prepregs were provided with 0.15 and 0.30 mm of thicknesses, respectively. however, very thin ud fabrics were always used by overlapping 4 layers (therefore, in the following section, a udx4 layer means a 0.6 mm stratification of t1000, made by overlapping 4 layers of 0.15mm keeping the fibers along the same directions). ansys workbench 19.3 platform was used for performing the numerical analysis. as material models, the ansys internal library was exploited, selecting “epoxy carbon ud (230 gpa) prepreg” and “epoxy carbon woven (230 gpa) prepreg” for t1000 and t800 respectively where default properties were changed in accordance with table 4. 3.2. stacking sequences ansys acp toolkit was also permitted to build exact layouts. specifically, the composite layout was recreated by defining the series of overlapping fabrics and their orientation, both reciprocal and with respect to the component. acp allows us to do this, by identifying the specific stratification, layer by layer, orientation by orientation, with respect to each zone. ten different layouts (namely l1, l2, ..., l10) were defined and applied. they are detailed in table 5 where material and orientation are reported per each layer. in the case of the l6-l10, a symmetrical doubling of the stratification has to be considered. table 5 composite layouts l1 l2 l3 l4 l5 1 bd 0/90 bd 0/90 bd 0/90 bd ±45 bd ±45 2 bd ±45 udx4 ±45 udx4 0 udx4 0 udx4 0 3 bd ±45 bd 0/90 bd 0/90 bd ±45 udx4 0 4 bd 0/90 bd ±45 layers 4 6 6 6 10 thickness (mm) 1.2 1.2 1.2 1.8 1.2 weight (g/m2) 1.70 1.75 1.75 1.75 2.64 l6 l7 l8 l9 l10 1 bd 0/90 bd 0/90 bd 0/90 bd 0/90 bd 0/90 2 udx4 0 udx4 0 udx4 0 bd ±45 udx4 0 3 bd 0/90 bd ±45 bd ±45 udx4 0 bd ±45 4  symmetry  bd ±45 bd 0/90 bd ±45 udx4 0 5  symmetry   symmetry  bd 0/90 bd ±45 6  symmetry   symmetry  layers 12 14 14 16 22 thickness (mm) 2.4 3.0 3.0 3.6 4.2 weight (g/m2) 3.49 4.34 4.34 5.20 6.32 the laminates were made by overlapping up to 22 layers of ud fabrics, at an orientation of 0° (occasionally 45° and -45°), and bd fabrics, at 0°/90° or ±45°. between these configurations, only the first two (l1 and l2) were quasi-isotropic (i.e. with in-plane isotropic properties). all the others are anisotropic, better responding to the particular physical configuration (geometry, loads, constrains) under examination. thickness varies from 1.2 (l1) to 4.2 mm (l10) with a specific weight slightly changing between 1420 and 1467 kg/m3. however, layouts can be better compared using the weight per unit area, usually expressed in g/m2: it varies between 1700 and 6320 g/m2 164 c. fragassa for a thickness of, respectively, 1.2 (l1) and 4.2 mm (l10). as reference, a 4.2 mm aluminum foil in 6061-t6 has a weight of 11.6 kg/m2. 3.3. loads and constrains the comparison was made through numerical analysis in static load configuration. the load value was considered equal to 750n: the real value of the applied force depends on various parameters (e.g. resistances, inertia masses), but a maximum effort of 750 n on each of the handles was considered adequate to represent a proper load for the proposed use (i.e. exercises in gyms). according to the so-called human engineering, the study of people in their working environment in order to maximize safety or efficiency, this value represents the maximum horizontal push forces exertable intermittently and for short periods of time in the case of male personnel [21]. actually, this load is considered as representative when applied on the back, if braced against a vertical wall and feet anchored on a perfectly nonslip ground. in the case of push and pull, this load is reduced to 500 n when applied on both hands (or one shoulder) and 250 n on one hand. the applied constraints are similar to those expected in the real kinematic chain. in particular, there is a clear difference between version a, with a double constraint (double fixed support), and versions b and c with a single constraint (single fixed support), as in fig. 3. this makes the new configurations more critical in terms of stresses and strains. a) b) c) fig. 3 loads and constrains applied to: a) steel; b) aluminum and c) composite parts 3.4. numerical discretization meshing was done using 3d ten-noded tetrahedral elements (tet10) in the case of steel (version a) and aluminum (version b), and 2d three-noded triangular elements (tri3) in the case of cfrp [22]. finite element (maximum) dimension was kept equal to 2.5mm. details are available in table 6. a control on the deformation energy of fes is allowed to verify the absence of critical situations in the mesh (e.g. ‘hourglass effect’). the so-called single-layer shell method was used for discretizing the composite layout. this simplification, quite common in the presence of geometrically complex structures, permits us to investigate the phenomena at the level of macroscale. with a proper conversion in properties, it reduces a multilayered laminate to an equivalent single layer laminate with shell elements all along the surface and integration points (ip) throughout the thickness [22, 23]. lightening structures by metal replacement: from a traditional gym equipment to an advanced... 165 table 6 numerical discretization by finite elements (fes) finite elements denomination tet10 tet10 tri3 type solid solid shell description 10-noded tetrahedral 10-noded tetrahedral 3-noded triangle dimension 2.5 mm 2.5 mm 2.5 mm elements (tot) 678 235 217 601 166 756 nodes (tot) 970 550 324 752 83 372 specifically, an ip was set per each layer (i.e. without additional ip for investigating the interfaces between layers). in such a way it was possible to limit the number of fe speeding up the simulation focusing the attention on in-plane phenomena [22, 23]. however, this methodological limit has no practical effect on the present study. 4. results and discussion 4.1. aluminum vs. steel fig. 4 shows the results of the simulations performed on the part, in steel (version a) and aluminum (version b) expressed as (equivalent von-mises) stresses and displacements. the values are quite different in the two cases, due to differences in: a) geometry; b) material; c) constraint. the maximum stress reached in the components is very far from the critical conditions of failure (uts) or yield (ys). in particular, maximum stress values of 58.9 and 99.7 mpa are evident for steel and aluminum, much lower than 586 and 460 mpa as uts (or 207 and 250 mpa as ys). thus, it is possible to say both these components work with safety coefficients higher than 2.5 making convenient to prefer the aluminum solution (version b). this version, in fact, besides being 4 times lighter, also present other advantages such as: a) reduced overall dimensions; b) a single fixing point; c) better exploitation of material properties with stress fields more widely distributed; d) use of a production technology suitable to produce large numbers; e) aesthetics and ergonomics. on the other hand, the higher displacement (3.8 vs. 1.3 mm) of aluminum does not generate dysfunctions. 166 c. fragassa a) b) fig. 4 comparing steel (up) and aluminum (down) parts in terms of: a) displacements (in mm); b) equivalent von mises stress (in mpa) 4.2. composite vs. aluminum as the next step, different composite layouts were modeled by numerical analysis and compared with aluminum in terms of stiffness (tab. 5). the first objective was to identify an initial stratification, able to ensure a maximum displacement for the composite part (version c) similar to the aluminum one (version b) despite of a lower weight in the case of composite. at this point, preliminary verified was also the presence of appropriate conditions on the stress distribution, far from the material limit values. for this purpose, the maximum principal stress (mps) in the most stressed layer was compared with the t800 (bd) tensile stress limit (uts), equal to 805 mpa. a better check on failure criteria was postponed to the final configuration. in particular, 10 layouts were considered, as displayed in fig. 5, where a progressive reduction of displacement is evident as the laminate complexity increases. the graph also clarifies the advantage offered by the composites with respect to metals in terms of weight. in each case, compliance with the resistance conditions is also assured. lightening structures by metal replacement: from a traditional gym equipment to an advanced... 167 fig. 5 effect of stacking sequences on mass and displacement the l8 layout, with 14 layers (8 ud) was finally preferred (tab. 5), able to assure the same stiffness of aluminum (fig. 6). beyond the apparent complexity, this layout has thickness of 3.0 mm and a weight of 4.34 g/cm2, permitting a 3.85 mm displacement with a 0.32 kg of weight. it also exhibits a maximum principal stress in the most stressed layer (2nd layer, in ud) far beyond the tensile stress limit. ud fabrics were aligned with unidirectional fibers in the same direction, equivalent to the longitudinal direction of the cantilever beam, which also corresponds to the prevailing direction of the lines of forces. besides, bd fabrics were alternated (0/90 and ±45) to withstand non-prevailing directions of forces. additional measures such as placing: a) ud close to the surface, away from the neutral plane and b) bd at 0/90 as external protective layers allowed us to improve the results. a) b) fig. 6 mechanic behavior of a fiber-reinforced composite component (version c) in terms of: a) displacements; b) maximum principal stress (2nd layer) 168 c. fragassa 4.3. layout design during the layout design, different design criteria were introduced noting that: ▪ ud fabric: a) provides high bending strength against forces aligned with the fibers (0° orientation); b) bends with respect to forces perpendicular to the fibers (90°). in either orientation, it exhibits low torsional strength. ▪ bd fabric: a) provides medium bending strength and medium torsional strength in a 0° or a 90° orientation; b) is more flexible with high torsional strength at 45°. as a practical consequence, the layout design followed these practical considerations: ▪ ud fabrics were used in the presence of and aligned with unidirectional forces. ▪ bd fabrics were used in the absence of forces along prevailing directions. ▪ bd fabrics were overlapped, when possible, considering a 45° rotation. moreover, composite manufacturing techniques also suggested: ▪ layouts were symmetric to avoid differential dilations during cure. ▪ bd was used for outermost layers. ▪ number of layers/cohesive zones between different fabrics was minimized. ▪ proper local reinforcements were considered with respect to the most stressed areas. table 7 reports the effect of the different stacking sequences on the component in terms of maximum displacement and weight. their values permit us to better understand the succession of layouts (l1 l10) defined in table 5 and the redesign process followed. table 7 comparing stiffness and weight of different composite layouts layout l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 layers 4 6 6 6 10 12 14 14 16 22 displacement (mm) 14.9 14.2 10.4 10.4 5.7 5.0 4.0 3.8 3.2 2.4 part weight (kg) 0.12 0.13 0.13 0.13 0.19 0.26 0.32 0.32 0.38 0.45 even if the initial layouts (l1 l5) were not adequate to assure a proper target value of displacement (i.e. 3.8 mm), they still permitted to verify, by numerical methods, the effect of: a) using ud instead of bd fabrics; b) fibers orientation. useful information emerges, in fact, when layouts were compared by means of displacements: ▪ l1 vs. l2: displacements are equivalent (14.96 vs. 14.23) even if layouts are different (for equivalent thickness and weight). it depends on the fact that the two layers of used ud, placed orthogonally, are equivalent to a layer of used bd. ▪ l3 vs. l2: when the fiber orientation is not aligned with the lines of force (l2), the displacement increases by 40%. it highlights the positive effect of a proper ud orientation. ▪ l3 vs. l5: doubling ud layers, the displacement is reduced by a further 45%. it confirms the relevance of a proper ud alignment as criterion for design. ▪ l3 vs. l4: an insignificant difference emerges with changing the bd orientation. it means the bd direction can be (almost) freely defined. in this regard, it should be noted that the bd layouts have to also consider the risk of unexpected forces. although none of these layouts satisfy the target, it is also evident that 8 layers of ud (l5) allow to get quite close (5.7 vs. 3.8 mm). the next layouts (l6-10), with more layers, get the target. they were defined also considering the opportunity of: a) not exceeding 4 overlapping layers of the same kind; b) alternating the orientation of the bd layers. lightening structures by metal replacement: from a traditional gym equipment to an advanced... 169 in particular: ▪ l6 represents a specular doubling of l3 layout, moving from 6 to 12 layers. the effect is to halve the displacement (5.0 mm). ▪ l7 and l8 add two bd layers to l6, on the neutral line, reducing displacement by further 20%. between the two, the best is l8 which alternates directions of bds: with 14 layers, l8 achieves the target value for displacement (3.8 mm) ▪ l9 and l10 are layouts in which attempts are made to lower displacement (3.20 and 2.47 mm) by adding, respectively, bd or ud to l8. 4.4. failure criteria every failure criterion has to consider both the material anisotropy and the specific stacking sequence. in these terms, fig. 7 shows the maximum principal stress for the most representative layers. in particular, visible is the 1st layer, in bd (0/90), and the 2nd, in ud. the additional ud layers (3rd – 5th) are not reported since their behavior largely reflects the 2nd with lower stress values. the following bd layers, the 6th at ±45 and 7th at 0/90, are shown. the layers in the lower part are not reported since their behavior symmetrically reflects the upper layers with lower values of stress. a) b) c) d) fig. 7 maximum principal stress for the most relevant layers: a) 1st, bd 0/90; b) 2nd ud 0; c) 6th bd ±45; d) 7th bd 0/90 specifically, the failures were verified by the application of maximum principal stress and maximum distortion energy (tsai-hill) failure criteria in terms of inverse reverse factor (irf), wherever <<1. the irf is one parameter commonly used in investigating the composite failure. the failure load can be defined as the load value divided by irf. thus when irf>1 the composite fails and if irf<1 it is safe. this is immediately evident in fig. 8 170 c. fragassa where, layer by layer, a visual reference with respect to the existing gap between the maximum principal stress and material strength (i.e. uts) is offered. fig. 8 layer-by-layer stress and material strength comparison in terms of inverse reverse factor (irf) and maximum principal stress to avoid unnecessary bulky structure irf is normally maintained between 0.5 and 0.75. as a result, this layout may appear oversized. this consideration is correct as long as the present load configuration is taken into account. during the normal use, dissimilar loads can emerge (e.g., misaligned or dynamic) moving away from an ideal condition. in fig. 9 failure results with respect to the tsai-hill criterion were also reported in terms of margin of safety (equal to the inverse of irf) and considering a load intensification factor of 300%. not even in this extreme case (i.e. a 3-times higher applied load), any criticality emerges except close to the joint where ud layers (especially the 2nd) seem ready to fail (fig. 9b). this situation is related to the fact that in this zone ud fibers are not always aligned with the lines of force. thus, ud is found to react along directions where it exhibits a very low stiffness (e = 8600 mpa) and strength (uts = 29 mpa). a) b) fig. 9 margin of safety with respect to the tsai-hill criterion, also reporting the most critical layers lightening structures by metal replacement: from a traditional gym equipment to an advanced... 171 this design challenge, unknown in advance, was solved by: ▪ strengthening the whole connecting area by substituting ud with bd. specifically, replacing the same number of layers (i.e. 8) three benefits were locally achieved, increasing thickness (from 3.8 to 5 mm), stiffness (from 8600 to 61340 mpa) and strength (from 29 to 805 mpa) with marginal changes in weight (from 0.32 to 0.33 kg). ▪ inserting a bronze bushing, necessary for the part’s correct coupling with the rest of the equipment. 4.5. prototyping the component (the forearm) was manufactured in two (asymmetrical) parts, by hand lay-up of prepregs on open molds. two aluminum molds were made by cnc machining (fig. 10a). vacuum bags, with breather assemblies, were placed over the layup and attached to the tool; then a vacuum is pulled prior to initiation of cure. this vacuum bagging process consolidated the plies and reduced voids due to the off-gassing that occurred as the matrix progressed through its chemical curing stages. the cure was performed in autoclave where pressure (6 bar) and temperature (135 °c) facilitated a high fiber volume fraction and low void content for maximum structural efficiency. additional details on the manufacturing are also available in [25] where a similar process was used. then, the parts were extracted from the molds (fig. 10b), finished (fig. 10c) and glued. special inserts, also in cfrp, allowed us to improve the connection. a) b) c) d) fig. 10 production stages: a) molds; b) after curing; c) two sections; d) joining inserts 172 c. fragassa 4.6. the exoskeleton the design study was extended to the other parts of the upper limb. with respect to the two additional segments, the essential elements of the analysis remain the same. however, as far as possible, the different conditions expected by the statics of the human body were taken into account. for example, the masses of the different body parts were included; the effect of not orthogonal elements (i.e. arm and forearm are free to rotate between 0 and 142 degree) were considered. this material change permitted to reduce the segments weight from an initial 17 kg (in the case of aluminum) to the current 8.5 kg (with composites). these segments now include arm, forearm, hand, and handle, for both sides (right and left), and also integrate the resistive elements and the console box. with respect to the initial design, the jacket (allowing wearing the exoskeleton) and the trapezoids (simulating the shoulders’ movement) must be completely redesigned for functional scopes. it is evident that many of their original parts lose functionality in the presence of an exoskeleton. the new target weight is 18kg. according to [21], where human strength and handling capacity are also classified, in fact, this value is maximum individual weight that can be associated with a “portable” item. however, the same standard, established for military use, also recognizes a weight of 18.5 kg for trained soldiers in operational conditions and 27.7 kilograms when marching. the expected weight reduction will certainly permit going below this higher threshold, making the base not strictly necessary during gym workout. a support structure, as an option, is still useful to make this new equipment also accessible to the beginner athletes or for different applications (e.g. physiotherapy). 5. conclusions a redesign procedure, based on a metal replacement action, is here detailed with the scope to transform advanced gym equipment into a wearable exoskeleton. a first prototype, was designed, produced, and used as gym workout machine. in terms of training functionality, it was fully validated, permitting new and more complex exercises. however, made in steel sheets and aluminum castings, the weight of its moving parts (>48 kg), acting as inertial masses against free athletes’ movements, made its utilization rather challenging except for professional trainers. furthermore, the excessive weight required a bulky support structure, making the whole machine impractical. the redesign involved the entire upper limb involving changes in shapes, materials, and production processes. regarding the materials, carbon fiber-reinforced composites were chosen with high-strength fibers (230 gpa) used through epoxy unidirectional and bidirectional fabrics. the effect of fiber properties (as strength and volume ratio) and plies stacking sequences (as thicknesses and orientation) were numerically investigated and a specific layout design was carried out taking into account the direction of forces. external loads higher than expected were considered with an analysis limited to static load configurations. the narrative is limited to one element of the limb, the forearm, but then extends the results to the entire kinematic chain. for the forearm, it was possible to estimate a weight reduction from 4.4 kg for steel and 1.0 kg for aluminum, to the definitive 0.3 kg in the case of reinforced composites. this result was obtained thanks to a stratification of 14 plies which, however, had a thickness of (only) 3.0 mm and a weight of 4.34 g/m2 (about half lightening structures by metal replacement: from a traditional gym equipment to an advanced... 173 of an aluminum sheet with similar thickness). the structural resistance was verified with the failure criteria provided for anisotropic materials (maximum principal stress, tsai-hill) highlighting a margin of safety never lower than 25-30%. this margin would allow a further reduction in the stratification. however, the design procedure was addressed to optimize the weight inside the condition to guarantee the same stiffness (compared to solid aluminum). this result was achieved with a 2.85 mm of displacement for a beam of 400 mm subject to a weight of 70 kg. moreover, additional design steps can lead to the full exploitation of the properties of these high performing composites. in this regard, in fact, before part manufacturing, further design measures were introduced such as: the use of bidirectional composites wherever no prevailing directions of forces emerged; a layout reduction where stress levels were extraordinarily low (i.e. side surfaces); addition of reinforcement elements and connection inserts. the part was manufactured by hand lay-up and autoclave curing, confirming the estimated weight reduction of 68% (with respect to aluminum). this redesign practice was extended to the rest of the entire upper limb, halving the section weight, eliminating the need for support frame, and moving in the direction to transform the original equipment into an active exoskeleton. acknowledgements: this paper was developed as part of the 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(eds.), advances in application of industrial biomaterials, springer international publishing, cham, ch, doi: 10.1007/978-3-319-62767-0_2, pp. 21-47. https://www.toraycma.com/page.php?id=661 an advanced coarse-fine search approach for digital image correlation applications facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 63 73 original scientific paper an advanced coarse-fine search approach for digital image correlation applications  udc 004.93 samo simončič1, melita kompolšek2, primož podržaj1 1faculty of mechanical engineering, university of ljubljana, slovenia 2upper-secondary school of electrical and computer engineering and technical gymnasium, ljubljana, slovenia abstract. the paper presents a newly developed fine search algorithm used in the application of digital correlation. in order to evaluate its performance a special purpose application was developed using c# programming language. the algorithm was then tested on a pre-prepared set of the computer generated speckled images. it turned out to be much faster than the conventional fine search algorithm. consequently, it is a major step forward in a never ending quest for a fast digital correlation execution with sub pixel accuracy. key words: digital image correlation, motion detection, deformation, sub-pixel registration, coarse-fine search scheme 1. introduction in the last two decades, the research field of digital image correlation (dic) [1, 2] has been growing at an ever increasing pace. this can be attributed to its distinct advantages such as a simple experimental set-up, low requirements regarding experimental setup and a wide range of applicability. in addition, the digital image correlation has been widely used for deformation and shape measurement, mechanical parameter characterization and numerical-experimental cross validations [3-5]. the basic idea behind the standard and most widely used subset-based dic method [6] is to track the subsets (or sub-images) specified in a reference image through the sequence of deformed images. the result of this procedure gives us the information about the full-field motion and deformation. it is evident that the proposed approach is rather received august 20, 2015 / accepted november 8, 2015 corresponding author: samo simončič faculty of mechanical engineering, university of ljubljana, aškerčeva cesta 6, 1000 ljubljana, slovenia e-mail: samo.simoncic@fs.uni-lj.si mailto:samo.simoncic@fs.uni-lj.si 64 s. simončič, m. kompolšek, p. podrţaj simple and intuitive, but sub-pixel registration accuracy and computational efficiency are two important aspects which need to be considered if the approach is to be applied. in the first case (accuracy of dic measurements) the obtained accuracy depends on various factors, such as speckle pattern [7], subset size [8], correlation criterion [9], shape function [10], subpixel interpolation scheme [11] and sub-pixel registration algorithm [12] where the latter one has the major influence on the registration accuracy of dic. for this purpose different types of sub-pixel registration methods have been developed. currently, an iterative spatial domain cross-correlation approach (for example a newton-raphson [13] (nr) approach) is one of the most widely used sub-pixel registration algorithms. in the last decade, original nr approach [14] has been improved significantly [13, 15] by reducing its computation complexity [16], improving its robustness [17] and extending its applicability. in this field, it is considered as a golden standard for accurate and robust sub-pixel motion detection in digital image. its ability to take into the account the relative deformation and rotation of the target subset is one of the main reasons for its wide applicability in various fields of science. it is also capable of providing the best sub-pixel registration accuracy compared to the other methods. the nr approach, however, needs the correlation function, which by nature is nonlinear with respect to the desired mapping parameters vector. this of course implies nonlinear numerical optimization. in such a case the initial value of mapping parameters vector is required, which presents initial guess in the correlation procedure. it should be noted that the initial guess must be defined as accurately as possible because only in this way the convergence of the nr approach is guaranteed [15]. as already mentioned, the nr approach is very appropriate in the cases where the deformation and/or rotation of the target subset are present. on the other hand, if only rigid body translation of the target subset is considered, then it is possible to use the sub-pixel registration approaches which are not so demanding from the mathematical point of view. in many situations, this fact provides the possibility to use the methods which are very simple from the theoretical point of view and very straightforward for implementation which also means that they are computationally efficient. in our opinion, the coarse-fine search algorithm [18] is well suited for such a task. because of that we have decided to implement it together with some improvements which will be presented in detail. the basic idea behind the coarse-fine search strategy can be described by the following steps. in the first step, it calculates the predefined correlation coefficient for all points of interest in the searching area with a 1 pixel step. in order to improve its accuracy it is logical to reduce the searching step, for example to 0.1 pixel or 0.01 pixel. from practical point of view, the value of the searching step depends on accuracy, which is needed in actual application. in many cases, the coarse-fine searching strategy is able to handle only rigid body motion and consequently the shape changes of the deformed subset cannot be evaluated as in the case of the nr approach. if the searching step is less than 1 pixel, the gray level at sub-pixel locations must be reconstructed and for this purpose a certain interpolation scheme is needed. from the execution time point of view this is the most demanding part of the coarse-fine searching approach. consequently, it should be treated with some care (see reference [19] for example). the conventional fine search methods usually search for the best matching point by a given searching step in the x and y direction, respectively. thus the computation complexity of such searching method is proportional to the given searching step (x_step × y_step) where x_step and y_step present number of steps in the x and y direction, respectively. in this an advanced coarse-fine search approach for digital image correlation applications 65 manner, the fine search calculation is executed (x_step -1 + 1) x (y_step -1 + 1) times for each sample point. to be more precise, if the searching step is 0.01 pixel in both directions, the algorithm has to be executed 101 x 101 (10,201) times for each sample pixel. it is obvious that the presented scheme is very time consuming since for each sample pixel a great number of the correlation coefficients must be calculated. in similar manner, the interpolation coefficients must be calculated at each execution step as well. based on this fact, it is evident that such searching method has a high computational complexity. some work has been done to reduce the computation cost without decreasing the accuracy. the scheme presented in [20] needs to be executed n x 11 x 11 (121n) times for a searching step of 10 -n pixel in both directions. for instance, if the searching step is 0.01 pixel in both directions the fine search approach only needs to be calculated 242 times for the actual sample pixel. this improvement was our inspiration to develop a novel course-fine searching strategy in which the computation cost is significantly reduced compared to the other existing methods in the literature. the proposed approach needs to be executed n x 2 x 2 (4n) times for each sample point with a searching step of 2 -n in both directions. if a searching step is assumed to be 2 -7 (= 0.007812), then the proposed scheme needs to be executed only 28 times for each pixel. from this point of view, it is evident that the proposed approach outperforms the conventional one by a great margin (265 times) in the case when the searching steps are 0.01 and 0.0078 pixels for the conventional and the proposed scheme, respectively. the performance of a novel coarse-fine searching approach is tested and evaluated by a sequence of computer-generated speckled images, where each of them was translated compared to the previous one for some known value. to evaluate the measurement accuracy and effectiveness of the proposed method, we developed a windows application which was developed specially for this purpose in the visual studio development environment using c# programming language. the results obtained by the conventional and the novel approach are also compared and evaluated from the computational complexity point of view. 2. principles of digital image correlation after digital images of the object surface before and after deformation are obtained, the dic method can be used to calculate the movement of each image pixel. if full-field deformation is required, a roi in the reference image must be specified within which image pixels are evenly spaced by virtual grids. the basic idea of the dic method is to match a reference mask (subset) in the original image with a deformed mask (subset) in the image after deformation (deformed image) as illustrated in fig. 1. we have assumed that the reference subset is a square with a reference pixel in its center. once the location of the target subset in the deformed image is found, the displacement components of the reference and target subset centers can be determined. 66 s. simončič, m. kompolšek, p. podrţaj fig. 1 schematic representation of the subset before and after deformation in order to estimate the degree of similarity between the reference and the deformed subsets, a certain correlation criterion should be defined first. as already mentioned, the evaluation of the proposed approach is tested on the set of computer-generated speckled images where the illumination is controlled. due to this fact, we have only used a normalized sum of squared differences (nssd) correlation criterion and not a more robust zeronormalized sum of squared differences (znssd) one. the main drawback of the znssd, however, lies in the fact that it is computationally expensive comparing to the applied nssd correlation criterion. it should be noted that the used nssd correlation criterion is insensitive to the linearly varying illumination intensity but sensitive to the offset in the illumination intensity. the latter one is not true for znssd correlation criterion. however, the nssd correlation criterion is defined by the following equation [1], 2 ),(),(            m mi m mj iiii nssd g yxg f yxf c (1) where f and g are mean intensities of reference and deformed subsets, respectively. it is well known that the digital image is represented by a finite number of pixels. consequently, one might think that accuracy is limited to one pixel, but, as already mentioned, it is possible to find registration methods with accuracy better than one pixel. even in this case, however, the first step is the application of an algorithm with one pixel accuracy. this (one pixel accuracy) can be made by a simple searching scheme within the deformed image or with a more advanced approach [21]. in ref. [21], the authors developed a fast recursive scheme to mathematically reduce the computational complexity of the traditional dic technique with one pixel accuracy. in general, the sub-pixel methods are only used afterwards to get a more accurate displacement evaluation and this step was done by a novel searching scheme which is presented in more detail hereafter. the proposed approach can be used directly (without simple searching scheme or coarse scheme) if the actual displacement at measured image point between two consecutive images is not larger than one pixel. an advanced coarse-fine search approach for digital image correlation applications 67 3. a novel fine searching strategy the starting point for a fine searching method is a square subset of 1 × 1 pixel, where its location in the deformed image is determined by a coarse search. this step is the same as in the other fine searching methods. the obtained square subset is used for searching the best matching point by a given searching step in x and y directions, which is presented by x_step and y_step, respectively. due to the fact that the correlation coefficient for each sub-pixel location in the square subset must be calculated, the computation of these fine search methods is quite time-consuming. more precisely, if the searching step is assumed to be 0.01 pixel in both directions, then the fine searching scheme will be executed 101 x 101 times for each sample point. this fact was the main reason to develop a novel fine search method in which computational complexity would be significantly reduced. for simplicity, we suppose that the searching steps are of the form x_step = y_step = 2 -n (n = 1, 2,…), where parameter n presents its accuracy. if parameter n is for example equal to 5, the obtained motion error would not be greater than 2 -5 (= 0.03125) pixel. based on the known searching step, a novel searching scheme can be defined as follows. in the first step, searching for the matching point at searching steps )5.0(2 1  is performed. the square subset of 1 × 1 pixels centered at the location given by the coarse search is divided in four subsets centered at four sub-pixel locations. for each of them the nssd correlation coefficient is calculated and the location of the point with the best match is denoted as ),( 11 yx . in the second step, the searching step is reduced from 12 to 22 pixel and the searching procedure is performed in the same manner as in the first step, but in this case a square subset of 0.5 × 0.5 pixels centered at ),( 11 yx is used for searching area. thus, the location of the best match when searching with 22 pixel step is denoted as ),( 22 yx . this procedure is repeated until the value of the searching step is sufficiently small compared to some predefined threshold. if the searching step is assumed to be n2 pixels for both directions, then one of the four sub-pixel locations in a square subset of 11 22   nn pixels centered at ),( 11  nn yx presents the best match which is denoted as ),( nn yx . to be clear, the points from ),( 11 yx to ),( nn yx present sub-pixel locations where their gray levels are calculated by predefined sub-pixel interpolation algorithm. the idea behind a novel fine searching scheme is clearly presented in fig. 2. it is evident that the proposed scheme needs to be executed 22n times, if the searching steps are assumed to be n2 pixels in both directions. based on the presented theoretical facts, the numbers of iterations which need to be calculated for each sample point at different searching steps are presented in fig. 3. they are denoted by red and blue bars for the proposed and the conventional approach, respectively. it can clearly be seen that the obtained number of iterations is increasing much slowly than in the case of the conventional approach. for example, if the searching step is 0.0156 pixel in both directions, then the conventional approach needs to be executed 4225 times for each sample point. this means 4201 iterations more than in the case of the novel fine searching scheme. this fact confirms that the proposed approach drastically reduces computational cost which results in a much wider range of applicability. 68 s. simončič, m. kompolšek, p. podrţaj fig. 2 schematic diagram of a novel fine search scheme fig. 3 comparison of the number of iterations needed for calculation of the best match at different searching steps between the conventional and the proposed fine searching scheme for each sample point as shown, the number of iterations is obviously reduced for each sample pixel, but the execution time for each iteration has not been modified in the sense of an improved performance at that point. as mentioned, at each iteration the interpolation coefficients must be calculated. this part of the algorithm has a major impact on execution time. because of that, the calculation of sub-pixel interpolation coefficients must be done carefully. the gray level at sub-pixel locations must be reconstructed and for this propose some kind of sub-pixel interpolation algorithm must be used. the sub-pixel interpolation calculation of a pixel point of a certain reference subset is not only performed in each iteration, but it also needs to be carried out for the same pixel point that appeared in adjacent reference subsets. the repeated interpolation calculation performed at sub-pixel locations consumes a lot of execution time. to eliminate such unnecessary computation, the authors in [19] present an an advanced coarse-fine search approach for digital image correlation applications 69 elegant solution, which provides an efficient calculation of the interpolation coefficients. a more detailed explanation of this idea and its experimental verification are given in [19]. in our experiments, the bicubic interpolation is used to estimate gray level and firstorder gray gradient at sub-pixel locations. they are defined by the following equations:     3 0 3 0 ),( m n nm mn yxyxg      3 1 3 0 1),( m n nm mnx ymxyxg  (2)     3 0 3 1 1),( m n nm mny ynxyxg  where ),( yxg  is gray level at sub-pixel location ),( yx  , ),( yxg x  and ),( yxg y  are first-order gray gradient at the same sub-pixel location ),( yx  in x and y direction, respectively. in bicubic interpolation, the unknown 16 coefficients, i.e., 330100 ,...,, aaa can be calculated by gray intensity of the neighboring 44 pixels centered at the sub-pixel location. it should be noted, that the 16 interpolation coefficients have the same values for each interpolation block, irrespective of the sub-pixel locations within it. in our application, the idea described in [19] was, however, used for efficient calculation of the interpolation coefficients. the basic idea behind it is presented in the following section. 4. pre-computed interpolation coefficient for efficient bicubic interpolation as presented in the previous section, the proposed fine searching approach needs to be implemented by sub-pixel interpolation method which calculates intensity ),( yxg  at each sub-pixel location when that is necessary. this procedure is the same for conventional approach as well and it is extremely time-consuming, because it takes most of the computational time at each iteration step of the fine searching approach. more precisely, if the searching step is assumed to be n2 pixels for both directions, then the proposed approach will be executed n4 times at each sample point. in this case, each pixel within the considered subset will be interpolated repeatedly n4 times. therefore, it is evident that this procedure has a big overlap with the adjacent subsets because the pixel point within one reference subset may also appear in its neighboring reference subset as shown in fig.4. this means that repetitive interpolations should be performed on the same pixel point. considering a reference subset of )12()12(  mm pixels dimension and a grid step of l pixels, each pixel in this subset is then also used in the adjacent 2[floor((2 1) / ) 1]m l   reference subsets. if the considered pixel point is displaced to a sub-pixel location, this pixel point will be interpolated approximately 24 [floor((2 1) / ) 1]n m l     times using the conventional method (parameter n presents the desired accuracy of the proposed approach). the study of other 70 s. simončič, m. kompolšek, p. podrţaj references has shown that the 16 interpolation coefficients which need to be determined, if equation 2 is to be applied, are repeatedly calculated. as already noted in the case of bicubic interpolation, the 16 coefficients are not modified in each interpolation block regardless of the sub-pixel location. due to this fact it is possible to form a global look-up table for each interpolation block in the deformed image before a novel fine search approach is executed. consequently it is possible to determine all the 16 coefficients in advance. as soon as a certain pixel falls in a certain block these coefficients are used to directly reconstruct the intensity at sub-pixel locations. as a consequence, the construction of interpolation coefficients must be performed only once and not 24 [floor((2 1) / ) 1]n m l     times for each interpolation block. for example, if we take a subset of 21×21 pixels and the grid step is set to 5 pixels, the number of interpolations is reduced from 75 to just 1. obviously, a larger subset size and or a smaller grid step will further reduce the computation time of the newly presented approach. fig. 4 schematic representation of the redundant interpolation in the newly presented fine search approach. the displaced pixel point (denoted with a blue dot) is repeatedly appearing in the adjacent subsets (denoted with dashed lines) as the reference subsets defined in reference image are overlapping each other. this means that there are redundant interpolations during the optimization of subsets an advanced coarse-fine search approach for digital image correlation applications 71 5. experimental verification to evaluate the measurement accuracy and effectiveness of the proposed method computer-generated speckled images were generated. each image was translated compared to the previous one for some known value of the displacement vector. as already mentioned the calculation is performed by the conventional and the proposed fine search approaches together with a normalized sum of squared differences (nssd) correlation criterion and zero-order displacement mapping function. the intensity of reference i1(x,y) and deformed ),(2 yxi images were calculated by the following expressions 2 2 0 1 2 1 ( ) ( ) ( , ) exp s k k k k x x y y i x y i d           (3) and 2 2 0 0 0 2 2 1 ( ) ( ) ( , ) exp s k k k k x x u y y v i x y i d             , (4) where s is the total number of speckle granules, r is the size of the speckle granule, (xk, yk) are the positions of each speckle granule with random distribution and i 0 k is the peak intensity of each speckle granule. all computations are executed within an application which has been developed especially for this purpose and is written in c# programming language on a personal computer with an intel pentium i7, 2.30ghz processor and 8 gb memory. the graphical user interface of the developed program is presented in fig. 5. the validation of the proposed and the conventional fine searching schemes is performed using the developed application. fig. 5 graphical user interface of the developed program for measurements displacements by the proposed fine searching scheme 72 s. simončič, m. kompolšek, p. podrţaj fig. 6 compares the computation time of the conventional and the proposed fine searching scheme at different searching steps, ranging from 0.25 to 0.0156 pixels for a fixed subset of 41 x 41 pixel size. as shown, the computation time of the conventional approach begins to increase rapidly as the searching step decreases. fig. 6 comparison of the computation time needed for calculation the best match at different searching steps between the conventional and the proposed fine searching scheme for each sample point for example, if the value of the searching step is assumed to be 0.0156 pixels, the conventional approach needs more than 16 seconds for each sample point to find the best match. on the other hand, the proposed approach only needs few milliseconds at the same searching step. from this fact, it is evident that the computation time is extremely reduced. it should be noted that the results presented in fig. 6 have similar distribution with respect to the searching step as to the number of iterations for each sample point presented in fig. 3. this is mainly due to the fact that the computation time is directly related to the number of iterations. 6. concluding remarks in order to test our idea of a newly developed fine searching method an application was made in visual 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lasers in engineering, 49(7), pp. 841-847. 20. zhang, z.-f., kang, y.-l., wang, h.-w., qin, q.-h., qiu, y., li, x.-q., 2006, a novel coarse-fine search scheme for digital image correlation method, measurement, 9(8), pp. 710-718. 21. jianyong, h., tao, z., xiaochang, p., lei, q., xiaoling, p., chunyang, x., jing, f., 2010, a highefficiency digital image correlation method based on a fast recursive scheme, meas. sci. technol., 21, pp. 1-12. 13266 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 757 785 https://doi.org/10.22190/fume250307026j © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper prediction of contact distribution on rough surfaces using deep learning algorithms ilkwang jang, yong hoon jang yonsei university, school of mechanical engineering, seoul, republic of korea orcid ids: ilkwang jang https://orcid.org/0000-0002-4388-979x yong hoon jang https://orcid.org/0000-0002-5436-5465 abstract. the contact characteristics of rough surfaces play a crucial role in determining friction, wear, thermal resistance, and electrical conductivity. this study proposes a deep learning approach to efficiently predict the contact distribution of rough surfaces based on surface image information alone, and evaluates its effectiveness against numerical methods. a u-net architecture was employed for predicting contact areas under varying scales and load conditions, using a dataset of 100,000 fractal surfaces generated via the random midpoint displacement (rmd) method. the results indicate that the deep learning model achieved performance comparable to conventional numerical methods in predicting both contact areas and electrical contact resistance, with minimal error observed in electrical contact resistance prediction. the model approached the contact prediction as an image segmentation task, enabling faster and more efficient computations than traditional numerical approaches. high performance across metrics such as dice coefficient, jaccard index, bradford factor (bf) score, and pixel accuracy highlighted its ability to maintain prediction accuracy while significantly enhancing computational efficiency. additionally, by leveraging two-dimensional (2d) fast fourier transform (fft) techniques, the model effectively captured both lowand high-frequency characteristics, accurately predicting large-scale and fine-scale features of contact areas, while reducing computation time by more than 95% compared to numerical models. these findings demonstrate that the deep learning algorithms can effectively address multiscale contact problems, offering reliable data for various engineering design applications, including friction, wear, and thermal/electrical resistance, as well as enabling real-time analysis and large-scale simulations. key words: contact distribution, fractal rough surfaces, deep learning, u-net, scale variations, force variations received: march 07, 2025 / accepted july 17, 2025 corresponding author: yong hoon jang yonsei university, 50 yonsei-ro, seodaemun-gu, seoul, 03722, republic of korea e-mail: jyh@yonsei.ac.kr https://orcid.org/0000-0002-4388-979x https://orcid.org/0000-0002-5436-5465 758 i. jang, y.h. jang 1. introduction the interaction between asperities governs the contact behavior of rough surfaces, which typically occurs at the micrometer or submicrometer scales, resulting in the formation of distributed micro-contact regions. these micro-contact regions contribute significantly to frictional resistance by inducing disparities between the normal and tangential forces, thereby impeding relative motion. in addition, they play a crucial role in defining the electrical and thermal contact resistances, thus inhibiting conduction in both domains. given the critical importance of friction and electrical/thermal contact resistance in engineering disciplines such as tribology and heat transfer, the accurate prediction of micro-contact regions is of substantial significance [1-5]. however, obtaining microcontact distributions experimentally poses significant challenges because of the need for sophisticated instrumentation capable of detecting minute surface deformations [6,7]. direct measurements of the contact pressure distributions are even more demanding, rendering numerical analysis practically the only feasible approach for acquiring such detailed contact information. over the years, many numerical models have been developed to facilitate the analysis of contact on rough surfaces, with several models demonstrating considerable efficacy in predicting contact areas [8-10]. nevertheless, numerical methods generally require a preprocessing step in which the surface is discretized into finite elements. the resolution of this discretization is intrinsically linked to the accuracy of the contact analysis. achieving precise results requires a fine discretization scale, which in turn significantly increases the computational costs. multiscale approaches have demonstrated that the surface structures vary with scale, leading to the subdivision of contact areas into smaller clusters. despite these advances, a comprehensive theoretical foundation for the convergence between the applied load and contact area in multiscale contact problems remains underdeveloped [11]. consequently, solving contact problems involving rough surfaces necessitates the incorporation of fine surface details at smaller scales, which drastically increases the number of elements and variables in numerical models, ultimately resulting in exponential growth in computational complexity. as an alternative approach, deep learning techniques, particularly artificial neural networks (anns), have garnered significant interest in recent years. in particular, convolutional neural networks (cnns) have driven remarkable progress in image processing and analysis by excelling in learning local features and structures from images. they have demonstrated superior performance in tasks such as image classification and object detection, while also offering significant computational speed and accuracy. despite these advancements, the application of deep learning in the field of tribology remains nascent [12-16], and leveraging deep learning to address contact mechanics problems remains relatively unexplored. several studies have employed ann-based methods in contact mechanics. however, they primarily focus on predicting scalar outputs such as the total real contact area or other mechanical parameters using pre-calculated scalar surface topography parameters and load conditions as inputs through standard or deep ann architectures. for example, kalliorinne et al. developed an ann model that takes 14 areal roughness parameters, hardness, and load as input to predict various contact mechanical responses, including the real area of contact [17]. similarly, suman and prajapati proposed an optimized deep ann model employing parameters such as root mean square roughness (sq), skewness (ssk), kurtosis (sku), pattern ratio, and applied load to predict the total real contact area [18]. in contrast to these parameter-based prediction of contact distribution on rough surfaces using deep learning algorithms 759 regression approaches, this study employs a distinct deep learning strategy leveraging cnns to predict the contact area distribution directly from surface height images. this study employs deep learning techniques to predict the contact area distributions on rough surfaces. by using the surface height information as input data and conceptualizing the contact state as a classification problem for each pixel, the task can be cast as an image segmentation task. cnn-based segmentation models can then be employed to predict the contact areas efficiently and accurately. specifically, this study proposes a model based on the u-net architecture, which has demonstrated exceptional performance in medical image analysis [19,20], to predict the distribution of contact areas on rough surfaces. this approach is designed to overcome the inherent limitations of the conventional numerical methods by providing a computationally efficient and accurate means of predicting contact distributions. the deep learning algorithm developed in this study offers ease of application because the structures of existing datasets related to rough surfaces and contact area distributions align well with the proposed model. however, contact area distributions are inherently dependent on the spatial resolution, necessitating the acquisition of datasets across multiple scales. this issue parallels the super-resolution problem [21], a well-known challenge in deep learning research. for simplicity, we did not incorporate cross-scale correlations. given the rapid advancements in deep learning technologies and their potential applications in tribology, it is imperative to determine whether deep learning can effectively learn the complex nonlinear relationships inherent in contact mechanics. should such learning prove feasible, identifying the critical factors that facilitate this predictive capability and understanding the limitations of the approach will be paramount. therefore, this study investigated the feasibility of employing deep learning models to replace the preprocessing and computational phases of numerical analysis for prediction of contact phenomena. furthermore, we propose a deep learning architecture capable of capturing the geometric and contact characteristics of interacting surfaces with the ultimate goal of developing a model that can predict contact area distributions under diverse conditions and environments. ultimately, the objective is to address the inherent complexity and convergence issues associated with numerical analysis, while offering a faster and more accurate predictive tool for contact area distributions. 2. deep learning procedures the objective of this study is to derive the contact area distribution from images of rough surfaces. to accomplish this, information regarding the contact pressure distribution and contact area distribution on rough surfaces is essential, along with a detailed methodology for constructing the dataset required to train the deep learning model. this dataset will be used for the design and implementation of the deep learning architecture to predict the contact area distribution under given contact conditions. 2.1. rough surface generation a sufficient number of rough surface contact pressure datasets are required to train the deep learning architecture. because it takes considerable time to measure actual rough surfaces and calculate the contact pressure to construct such a dataset, rough surface and 760 i. jang, y.h. jang contact pressure datasets were constructed by implementing a surface based on fractal theory and applying a method that allows quick computation to efficiently obtain the dataset. actual surfaces consist of rough surfaces that exhibit multiscale properties that continuously change according to the resolution of the measurement devices as surface roughness measuring instruments advance. these rough surfaces exhibit hierarchical scales that persist until a resolution limit is reached [22-24]. in this study, the rough surface was assumed to be a self-affine fractal and generated using the random midpoint displacement (rmd) method [25]. rough surfaces were generated as 1 × 1 mm square sections, which were sufficient to capture the representative roughness characteristics. to ensure the similarity between the fractal surface and the actual surface, the fractal dimension d was limited to 2.5, and the rough surface height (h) was constrained within a specific range of 0 to 50 μm. statistical analysis of the generated surfaces revealed an arithmetic average roughness (ra) of approximately 7 μm, which falls within typical engineering surface roughness values. in the rmd method, the height at the midpoint between nodes defined at the previous scale is determined by a mean of zero, and the variance is influenced by the fractal dimension. according to this definition, the surface becomes rougher as the surface roughness and variance increase. in this study, a comprehensive dataset consisting of 100,000 rough surface geometries and their corresponding contact pressure distributions was constructed to train and test the deep learning model. the generated surfaces exhibited consistent statistical properties while maintaining the specified height range. rough surfaces were obtained according to the given scale 𝐿/2𝑛, where each mesh element had a size of 𝐿/2𝑛 × 𝐿/2𝑛 (l = 1 mm). the multiscale surfaces produced results at scales ranging from n=5 to n=9. 2.2. contact analysis by direct numerical simulations a numerical analysis was required to determine the contact area distribution on rough surfaces generated using the rmd algorithm. the contact problem addressed in this study was limited to elastic bodies with frictionless contact. the findings of this study can also serve as fundamental information for extending the problem to include friction. the contact conditions for the frictionless case are as follows [26]. 𝑢(𝑥, 𝑦) = δ − 𝑔0(𝑥, 𝑦) (𝑥, 𝑦) ∈ 𝐴 (1) 𝑢(𝑥, 𝑦) > δ − 𝑔0(𝑥, 𝑦) (𝑥, 𝑦) ∉ 𝐴 (2) 𝑝(𝑥, 𝑦) > 0 (𝑥, 𝑦) ∈ 𝐴 (3) 𝑝(𝑥, 𝑦) = 0 (𝑥, 𝑦) ∉ 𝐴 (4) where 𝑢(𝑥, 𝑦) = 1 𝜋 𝐸∗ ∬ 𝑝( 𝜉,𝜂)𝑑𝜉𝑑𝜂 √( 𝑥−𝜉)2+( 𝑦−𝜂)2𝐴 , (5) where 𝑔0(𝑥, 𝑦) is the initial gap, δ is the rigid body approach, and 𝑝(𝑥, 𝑦) is the contact pressure. eqs. (1)-(5) indicate that at the rough surface, the gap that includes the surface displacement is zero, and the contact pressure is positive. in the separation region, the gap was greater than zero, and the contact pressure was always zero. additionally, the term 1/𝐸∗ represents (1 − 𝜈1 2)/𝐸1 + (1 − 𝜈2 2)/𝐸2, where 𝐸1, 𝜈1 and 𝐸2, 𝜈2 are the elastic moduli and poisson’s ratios of the two contacting bodies, respectively. the sum of the contact pressures obtained from these contact conditions was in equilibrium with the applied load f. prediction of contact distribution on rough surfaces using deep learning algorithms 761 ∬ 𝑝(𝑥, 𝑦)𝑑𝑥𝑑𝑦 𝐴 = 𝐹. (6) deep learning improves the accuracy through repeated training based on large datasets, making it essential to secure as many datasets as possible for effective learning. however, traditional numerical methods such as finite element analysis require a significant amount of time to derive the contact area distribution on rough surfaces, making it extremely difficult to acquire sufficient datasets. to overcome this issue, this study uses numerical analysis based on the fast fourier transform (fft) method and multi-level multisummation (mlms) technique [8,26], allowing for the rapid derivation of contact area distributions and the collection of a large number of datasets. it is important to note that the dataset used for training our deep learning model relies on the results generated by this specific fft-based mlms numerical solver, chosen primarily for its computational efficiency and accuracy balance, particularly for concentrated contact problems. this efficiency enabled the generation of the large-scale dataset (100,000 samples) essential for deep learning within a practical timeframe. while advantageous for speed and handling non-periodic issues better than standard fft without extensive zero-padding [26], this method, like any numerical technique, operates under specific physical assumptions (e.g., linear elasticity) and is subject to discretization effects. consequently, the training data inherently reflects these characteristics. we acknowledge the existence of various other numerical methods (e.g., fem, bem, other dfft variants) which may possess different strengths, weaknesses, and applicability limits under various conditions, as reviewed by müser et al. [27]. the following paragraph briefly summarizes the numerical analysis method. for a discrete system, in which the 𝑥 and 𝑦 planar spaces are divided into (𝑁𝑥 , 𝑁𝑦) uniform rectangular grids along each axis, eq. (5) yields the following displacement: 𝑢𝑖𝑗 = − ∑ ∑ 𝐾𝑖−𝑘,𝑗−𝑚𝑝𝑘𝑚 𝑁𝑦 𝑚=1 𝑁𝑥 𝑘=1 , (7) where 𝐾𝑖𝑗 is the influence coefficient reflecting eq. (5). the coarse and fine grids are represented by (𝐼, 𝐽) and (𝑖, 𝑗), respectively. the contact pressure and displacement are described as follows: 𝑝𝐼,𝐽 = �̂�2𝑠𝑝𝑖𝑗 , (8) 𝑢𝑖𝑗 = 𝐼2𝑠𝑢𝐼𝐽 + ∑ ∑ 𝐶𝑘𝑚 2𝑠 𝑝𝑖−𝑘,𝑗−𝑚𝑚𝑘 , (9) where 𝐼2𝑠 is the interpolation operator of order 2𝑠 , and 𝐶𝑘𝑚 2𝑠 represents the correction coefficients. using this relationship, the transformation between the coarse and fine grids is performed sequentially. meanwhile, applying the fft technique to the continuous functions defined in the numerical analysis and their discretization yields the following result for eq. (9): �̂�ℎ𝑙 = 𝐾ℎ𝑙�̂�ℎ𝑙 (10) where �̂�ℎ𝑙 , 𝐾ℎ𝑙 , and �̂�ℎ𝑙 are fourier coefficients corresponding to 𝑢𝑖𝑗 , 𝐾𝑖𝑗 , and 𝑝𝑖𝑗 . the surface nodal deflection 𝑢𝑖𝑗 can be obtained from 𝑢ℎ𝑙 using the inverse fft. from the above contact analysis, the contact pressure and micro-contact area distribution were obtained. the rough surface was in contact with a flat opposing rough surface, and the load range was 𝐹=0-50 n. fig.1 shows the rough surface, contact pressure, and contact spot distributions for scales 𝐿/27 and 𝐿/28. the rough surfaces presented in 762 i. jang, y.h. jang fig. 1 are the same as those reconstructed according to each scale. as the scale decreased, the contact area formed at larger scales is divided into smaller contact spots. in addition, the contact spots tended to form clusters. the contact pressure at these spots increased as the scale decreased. thus far, the contact area distribution under elastic contact conditions has shown a continuous reduction as the scale decreased, demonstrating non-convergent results. to address these non-convergent results, attempts have been made to account for new physical phenomena associated with scale changes, such as variations in material properties and adhesion effects [28-31]. however, in this analysis algorithm, no new physical phenomena other than mechanical behavior changes with scale were considered. only the contact area distribution was used in the deep learning process. 2.3. deep learning model: u-net the contact area distribution prediction problem addressed in this study is closely related to image segmentation tasks. this specifically pertains to the classification of contact and noncontact regions in the field of computer vision. the goal is to classify each pixel as either a contact or noncontact region based on the shape of the surface. we adopted the u-net architecture to perform this task [32]. this model demonstrated excellent performance in medical image segmentation. owing to its structural characteristics, u-net can effectively capture the multiscale nature of contact problems, making it highly suitable for the objectives of this study. the u-net architecture used in this study was designed to take a rough surface geometry as the input and output the corresponding contact area distribution, as shown in fig. 2. specifically, the network was configured to process a 2d input image of size 𝑁𝑥 × 𝑁𝑦, where 𝑁𝑥 and 𝑁𝑦 represent the number of grid points of the rough surface in the horizontal and vertical directions, respectively. the input image size was defined to range from 25 × 25 to 29 × 29 pixels, depending on the scale. the u-net structure is divided into two main parts: contracting and expanding paths. these two components align well with the characteristics of the contact problems. the contracting path progressively reduces the spatial dimensions of the input image through a series of convolution operations and max pooling. this reduction process allows the network to extract high-level features [33,34]. this process is crucial for capturing the macroscopic geometry of the rough surface and overall contact pattern. specifically, the contracting path comprises repeated applications of the following layers. prediction of contact distribution on rough surfaces using deep learning algorithms 763 fig. 1 contact analysis simulation results for deep learning network training based on scale variation. rows a) and b) show the results for scales 𝐿/2𝑛 with 𝑛=7 and 𝑛=8, respectively. for each scale, the columns represent: a) rough surface information, b) contact pressure distribution, and c) distribution of contact spots. fig. 2 u-net architecture and network configuration for contact area prediction the convolutional layer applies filters (or kernels) to the input data to extract features. in this study, 3 × 3 filters were used and the number of feature maps doubled after each 764 i. jang, y.h. jang convolution operation. for example, 64 feature maps were generated in the first stage, 128 in the second stage, and 256 in the next stage. this process enabled the network to gradually learn more complex and abstract features. after each convolution operation, a rectified linear unit (relu) activation function was applied to introduce nonlinearity. the relu function is defined as 𝑓(𝑥) = max(0, 𝑥), which converts negative inputs to zero and allows positive inputs to pass. this function is widely used in deep neural networks because it is computationally simple and helps mitigate the vanishing gradient problem. after applying the convolution and activation functions, a 2 × 2 max pooling operation was performed. this operation reduced the spatial dimensions of the feature map by half. this operation selects the largest value in each 2 × 2 region to create a new feature map. max pooling reduces computation and increases the spatial invariance of the features. this contraction process is repeated four times to reduce the input image size to 1/16 of its original dimensions. this process allows the network to learn the macroscopic features of the rough surface and the overall contact pattern. however, the expanding path progressively restores the spatial dimensions of the feature maps through up-sampling and convolution. it then combines the low-level features obtained from the contracting path to accurately predict the fine contact areas. the expanding path consists of the following layers. in this study, up-sampling doubled the spatial dimensions of the feature map using the transposed convolution method. this process can be considered the reverse of max pooling, which converts low-resolution features into high-resolution features. one of the core features of u-net is the skip connection, which is particularly important for contact problems. skip connections help preserve important spatial information that may be lost during the down-sampling process, which is critical for accurately identifying contact areas. by linking features from the contracting path to the expanding path, skip connections ensure that both low-level details and high-level context are available for precise contact predictions. this connection links feature maps obtained at each stage of the contracting path to the corresponding layers of the expanding path. this enables the network to leverage information about the fine surface structure (low-level features) and the overall contact pattern (high-level features) simultaneously. this is crucial for accurately identifying the boundaries of the contact areas and is essential for the precise prediction of the contact area and distribution. a 3 × 3 convolution and relu activation function were applied again to the connected feature maps. during this process, the number of feature maps was gradually reduced, ultimately reaching the desired number of output channels (two in this study, representing the contact and noncontact areas). in the final layer of the expanding path, a 1 × 1 convolution is applied to generate the final output. this output takes the form of a probability map, in which each pixel represents the probability of belonging to a contact area. each convolutional layer in u-net can be expressed as follows: 𝒚𝑐𝑜𝑛𝑣 = 𝜎(𝐵𝑁(𝒘𝑐𝑜𝑛𝑣 ∗ 𝒙𝑐𝑜𝑛𝑣)), (11) where 𝒙𝑐𝑜𝑛𝑣 is the input feature map, 𝒘𝑐𝑜𝑛𝑣 is the convolution kernel, bn is batch normalization, 𝜎 is the activation function, and 𝒚𝑐𝑜𝑛𝑣 is the output feature map. the relu was used as the activation function and is defined as follows: 𝜎(𝒙) = max(0, 𝒙). (12) prediction of contact distribution on rough surfaces using deep learning algorithms 765 the use of the relu activation function allows the network to effectively model the nonlinearity of contact problems. additionally, the application of batch normalization helps stabilize the large variability in rough surface heights and accelerates the model training process. each stage of the u-net addresses specific aspects of the contact problem. the initial layers capture the fine surface structure and local features, which are essential for identifying small-scale contact points. the intermediate layers recognize mid-scale patterns and structures on the rough surface, contributing to the understanding of contact cluster formation. the deeper layers capture the overall surface geometry and large-scale contact patterns, aiding in the prediction of the overall contact area distribution. finally, the up-sampling layers integrate these multiscale features to generate a high-resolution contact area map. the u-net model used in this study consists of 23 convolutional layers. this number of layers was selected to provide sufficient depth for capturing both the fine and coarse details of the rough surface, thus enhancing the ability of the model to learn the complex relationships necessary for accurate contact area predictions. this deep structure allows the network to effectively model the complex nonlinear relationship between the rough surface geometry and contact area distribution. these characteristics of u-net make it particularly suitable for contact area distribution prediction problems. it effectively handles the multiscale nature of rough surfaces, accurately identifies the boundaries of the contact areas, and correctly predicts the overall contact pattern. 2.4. loss function the choice of the loss function in the training process of a deep learning model has a significant impact on its performance and training efficiency [35]. in this study, considering the characteristics of the contact area prediction problem, a combined loss function was used that combines the binary cross-entropy (bce) and dice losses. the bce loss is one of the most widely used loss functions for binary classification problems and is defined as follows: bce = −[𝑦 log(𝑝) + (1 − 𝑦) log(1 − 𝑝)], (13) where 𝑦 represents the true label (0 or 1) and 𝑝 represents the predicted probability. bce loss allows predictions to be interpreted as probabilities and is statistically well defined, because it is based on information theory and maximum likelihood estimation. it also has the advantage of providing meaningful gradients, even in extreme predictions, which stabilizes the learning process. in contact problems, the bce loss independently calculates the loss for each pixel, enabling fine-grained predictions at the pixel level. however, the bce loss is sensitive to class imbalance. when the contact area occupies only a small portion of the total area, the model can predict all pixels as noncontact and achieve a low loss value. this could result in the model failing to properly predict small but important contact areas in contact problems. to compensate for this limitation of bce loss, dice loss is also considered. dice loss is based on the dice coefficient, a metric that directly measures the similarity between the predicted and actual contact areas. to understand this metric, we introduce the concepts of true positive (tp), true negative (tn), false positive (fp), and false negative (fn). these are the 766 i. jang, y.h. jang basic classification outcomes defined by comparing the model predictions with the actual values. in the context of contact problems, these outcomes can be understood as follows: 1. tp: the number of pixels predicted as contact areas that are indeed contact areas. this indicates that the model correctly identified the contact areas. 2. tn: the number of pixels predicted as noncontact areas. this indicates that the model correctly identified the noncontact areas. 3. fp: the number of pixels predicted as contact areas that are actually noncontact areas. this represents a case in which the model incorrectly judged noncontact areas as contact areas, indicating an overestimation of the contact area. 4. fn: the number of pixels predicted as noncontact areas that are actually contact areas. this represents a case in which the model incorrectly judged contact areas as noncontact areas, indicating an underestimation of the contact area. these classification outcomes are crucial for contact problems. for example, a high fp indicates that the model overestimates the contact area, predicting that contact occurs over a larger area than that in reality. conversely, a high fn indicates that the model underestimates the contact area, predicting contact over a smaller area than that in reality. the dice coefficient measures the overlap between the predicted and actual areas, and is calculated as follows: dice coefficient ≡ 2tp 2tp + fp + fn . (14) the dice loss is then defined as: dice loss = 1 − dice coefficient. (15) the dice loss is less sensitive to class imbalance, making it effective in predicting small contact areas. additionally, it is particularly useful for accurately predicting the boundaries of the contact regions, which is critical for accurately determining the contact area and distribution. dice loss is also differentiable, making it easy to apply in the backpropagation process of deep learning models, and ranges between 0 and 1, offering an intuitive interpretation. moreover, it automatically assigns larger weights to difficult samples (e.g., those with small contact areas), which helps address class imbalance. to leverage the strengths of the bce and dice losses, this study uses a combined loss function. the combined loss is defined as follows: combined loss = 𝛼 ⋅ bce + (1 − 𝛼) ⋅ dice loss, (16) where 𝛼 is a hyperparameter that controls the balance between the two losses. the combined loss enables more balanced learning by combining the pixel-level evaluation of the bce with the region-level evaluation of the dice loss. the bce captures local characteristics, whereas the dice loss captures global characteristics, allowing optimization across multiple scales. additionally, by adjusting the value of 𝛼, the relative importance of the two losses can be adjusted depending on the problem characteristics, which provides flexibility. the combined use of loss-of-contact problems is particularly beneficial. given the nature of contact problems, where both accurate boundary predictions of the contact area and an overall understanding of the contact area are essential, combining the local precision of the bce with the global similarity measurement of the dice loss can be highly effective. prediction of contact distribution on rough surfaces using deep learning algorithms 767 2.5. evaluation metrics and implementation to evaluate the performance of the contact area prediction using a deep learning cnn model, the selection of appropriate evaluation metrics is crucial. in this study, the pixel accuracy, jaccard index (intersection over union (iou)), dice coefficient, and bradford factor (bf) score were used as the main evaluation metrics [36,37]. these metrics allow the performance of the model to be assessed from various perspectives, enabling an interpretation that considers the characteristics of the contact problem. pixel accuracy is the most basic evaluation metric. it is calculated based on classification results, represents the proportion of correctly classified pixels out of all pixels, and is defined as follows: pixel accuracy ≡ tp + tn tp + tn + fp + fn . (17) in contact problems, the pixel accuracy indicates the proportion of rough surfaces where contact/noncontact areas are correctly predicted, showing how well the model captures the overall contact pattern. however, this metric is vulnerable to class imbalance. for example, if the contact area occupies a very small portion of the total surface, the model can achieve high pixel accuracy even by predicting all pixels as noncontact. therefore, it can be difficult to evaluate model performance accurately using only pixel accuracy, and this should be considered alongside other metrics. the jaccard index, or iou, is calculated as the ratio of the intersection to the union of the predicted and actual areas: jaccard index ≡ tp tp + fp + fn . (18) this metric ranges from 0 to 1, with values closer to 1 indicating more accurate predictions. in contact problems, the jaccard index intuitively reflects the overlap between predicted and actual contact areas, making it particularly useful for evaluating the accuracy of contact boundaries. the dice coefficient measures the overlap between the predicted and actual areas as defined in eq. (14). similar to the jaccard index, it ranges from 0 to 1, with values closer to 1 indicating more accurate predictions. in contact problems, the dice coefficient can be more sensitive than the jaccard index in predicting small contact areas, making it useful for evaluating finer contact patterns. the bf score evaluates the accuracy of the boundary region and is calculated as the harmonic mean of precision and recall: bf score ≡ 2∗precision∗recall precision+recall , (19) where precision and recall are defined as follows: precision = tp tp + fp . (20) this value represents the proportion of pixels predicted as contact areas that are actually in contact. additionally, recall = tp tp + fn . (21) 768 i. jang, y.h. jang this value represents the proportion of actual contact area pixels that the model correctly predicted as contact areas. the physical meaning of the bf score indicates the accuracy with which the model predicts the contact area boundary. this is particularly useful in problems in which the shape, such as the contact area, is important. a higher bf score indicates that the model accurately predicted the boundary of the contact area. these evaluation metrics can be used as loss functions during the model training process or to assess the performance of the model during the validation and testing stages. because the importance of each metric may vary depending on the nature of the problem, it is important to select appropriate metrics or consider multiple metrics comprehensively, considering the specific characteristics of the problem. 2.6. training the neural network model in this study, a visualization process suitable for contact mechanics was conducted to utilize the fractal surfaces generated by the rmd method and the contact area data calculated by the mlms method to train the deep learning model. the complex geometry of the rough surfaces and the resulting contact area distribution possess inherent statistical characteristics, and to effectively train the model, it is necessary to visually represent these characteristics in the data. the process of visualizing rough surface height information and contact area distribution data was designed to preserve the precision of the numerical data while reflecting the characteristics of contact mechanics. initially, the data were stored in a tagged image file format (tiff); however, to reduce video memory usage and image file size when representing the fine structure of the fractal surfaces, the portable network graphics (png) format was ultimately adopted. png provides lossless compression and supports a 16-bit channel depth, allowing precise representation of the fine surface geometry without loss of detail. the following steps were performed to convert the surface height information into images. first, the fractal surface data were normalized to the range [0, 1] based on the minimum and maximum heights. this normalization ensures that the statistical characteristics of height variation, which are crucial in contact mechanics research, are maintained while enabling a consistent comparison of all data points. the normalized values were then converted to the 16bit integer range [0, 65535]. the reason for selecting 16-bit depth was to accurately capture the complex fine structure of the fractal surface and the corresponding contact mechanism. the converted data were generated as 16-bit grayscale images and stored in png format. the lossless compression of the png format was used to optimize the balance between file size and precision. the contact area distribution data were converted into a binary mask based on the results calculated using the mlms method. the locations where the contact pressure was greater than zero were defined as contact regions, whereas the other locations were defined as noncontact regions. this method precisely represented the discrete contact distribution observed in actual contact scenarios. the binary mask was saved in the png format, where a single-bit depth was sufficient to convey meaningful information. this visualization process offered several advantages. first, using a 16-bit depth, it is possible to maintain high-resolution details of the fine surface geometry. in contact mechanics, fine surface variations significantly affect contact strength, contact area, and contact pressure distribution, making precise representation crucial. the use of the png format allows effective data compression, saves storage space, and enables fast data prediction of contact distribution on rough surfaces using deep learning algorithms 769 loading in most deep learning frameworks. this is an important factor in the training process for large-scale surface datasets. fig. 3 example of a preprocessed dataset for deep learning network training. rows a) and b) show the results for scales 𝐿/2𝑛 with 𝑛=7 and 𝑛=8, respectively. in each scale, column a) shows the height information of the rough surface converted to a 16-bit grayscale png, column b) shows the contact area distribution represented as a binary mask, and column c) shows a zoomed-in section of the contact area distribution. this dataset includes fractal surfaces generated by the rmd method and contact area information calculated using the mlms technique. moreover, visualized data are easy to analyze. converting the surface height map and contact area distribution into images allows researchers to intuitively understand the changes in the contact conditions. this is advantageous for clearly analyzing the effects of the surface roughness or geometric changes on the contact properties. in addition, imagebased data facilitate the application of data augmentation techniques for training deep learning models. thus, the proposed visualization process can be considered an effective data preprocessing method that reflects the characteristics of contact mechanics. the dataset generated through this preprocessing was directly used to train the deep learning model. a height map image was used as the model input, and the binary mask of the contact area was used as the label for the direct numerical simulation. examples of the visualized rough surface height map and the binary mask of the contact area used to build the training dataset are shown in fig. 3. in this study, data augmentation techniques were applied to help the deep learning model learn the complex characteristics of contact mechanics more effectively. the shape 770 i. jang, y.h. jang of the rough surface and distribution of the contact areas exhibited nonlinear and complex patterns, making data augmentation necessary to enable the model to capture this complexity. data augmentation prevents the model from overfitting to specific data patterns and improves its generalization performance in predicting contact areas [38]. an important characteristic of the data used in contact mechanics is that they are independent of orientation. the roughness or shape of a rough surface is statistically isotropic; thus, the distribution of the contact areas also exhibits orientation-independent characteristics. accordingly, flipping (vertical and horizontal) and rotation transformations were applied to the images. these transformations increase the diversity of the data while preserving the physical contact properties. vertical and horizontal flipping was applied with a probability of 50%, and image rotations were performed at 90, 180, and 270 degrees. these transformations generate a variety of training samples without altering the statistical characteristics of the contact area distribution, contact stiffness, or contact pressure, thereby allowing the model to learn a wider range of contact scenarios. for example, eight training samples, including the original image, were generated by applying vertical flipping, horizontal flipping, and three rotations to a rough surface image. this effectively increases the dataset size by eight times, which is particularly useful when there is a limited amount of training data. the data augmentation technique in contact mechanics helps the deep learning model effectively learn the nonlinear behavior that occurs in various contact situations. because the irregular distribution of contact areas and the complex surface geometry pose challenges to the generalization ability of the model, data augmentation plays a key role in mitigating these issues. in real-world scenarios, various surface conditions and orientations exist, making it essential for the model to learn how to handle such variability robustly. consequently, through a data augmentation technique that reflects the complex characteristics of contact mechanics, the proposed model is equipped with more reliable and practical contact area prediction capabilities. a total of 100,000 datasets containing rough surface geometries and contact area distributions were used to train the u-net model. of these, 80% were used as training data and 20% were used as testing data. the batch size was set to 32 during training and the model was trained for 100 epochs. during the training process, the adam optimizer was used to optimize the model parameters [39]. the initial learning rate was set to 0.001 and a strategy was applied to reduce the learning rate by 20% if the validation loss did not improve for 10 consecutive epochs. additionally, an early stopping strategy was employed to prevent overfitting and determine the optimal parameters. specifically, training was halted if the validation loss did not improve for 20 consecutive epochs, or if the learning rate dropped below 0.0001. at the end of each epoch, the model weights were saved, and the performance metrics (dice coefficient, jaccard index, bf score, pixel accuracy) for both training and validation were recorded. all experiments in this study were conducted in a python 3.11 environment and implemented using the pytorch library (version 2.1.2). the hardware was a workstation equipped with an amd epyc 7502 cpu, 256 gb of ram, and an nvidia geforce rtx 3090 gpu. prediction of contact distribution on rough surfaces using deep learning algorithms 771 3. results 3.1. evaluation of the performance of the deep learning model the results of the contact area distribution predictions based on the changes in load and scale are presented. to evaluate the performance of the model, the accuracy of the results was analyzed using various previously introduced metrics. first, the accuracy of the predictions made by the deep learning model trained at different scales was verified. fig. 4 shows an example comparing the direct numerical model results and predictions made by the deep learning model for the contact area distribution at different scales in the test field. in this case, the applied force was 30 n. numerical analysis results indicated that as the scale became finer, the contact area was subdivided into smaller contact regions, which subsequently formed clusters. additionally, it shows that new contact spots emerged at coarser scales, which were not previously present. to quantify the accuracy of the contact area predictions made by the deep learning model in the test field, evaluation metrics, such as the dice coefficient, jaccard index, bf score, and pixel accuracy, are presented in table 1. these evaluation metrics represent the average values calculated from the predictions on 4,000 test datasets, which consisted of samples at various scales under a fixed applied force of 30 n. the predicted results for each scale showed a dice coefficient of approximately 0.9156, jaccard index of 0.8464, bf score of 0.9521, and pixel accuracy of 0.9587. these values indicate that the deep learning prediction model demonstrates excellent overall performance in segmentation and prediction. the high dice coefficient and jaccard index values indicate that the model accurately identified and segmented the contact area distribution. additionally, the high bf score suggests that the model predicts the boundaries of the contact area distribution with precision, whereas the pixel accuracy indicates that there is minimal difference between the predicted and actual contact areas, because the pixels between the distribution of the deep learning model and direct numerical simulation are classified correctly. second, an example comparing the numerical analysis results and predictions made by the deep learning model for the contact area distribution at different loads is shown in fig. 5. the results demonstrated that as the load increased, the contact area expanded and a larger number of new contact spots appeared across the entire contact region under higher loads than under lower loads. the metrics used to evaluate the performance of the deep learning model in predicting the contact area distribution across different loads are presented in table 2. these evaluation metrics represent the average values calculated from predictions on 4,000 test datasets, which consisted of samples at a fixed scale of l/2⁷ and varying loads. the predicted results for each load exhibited a dice coefficient of approximately 0.9289, a jaccard index of 0.8679, a bf score of 0.9864, and a pixel accuracy of 0.9948. compared with the results from table 1, which analyzed scale variations, the overall results were similar, indicating the excellent performance of the prediction model. notably, the dice coefficient tended to be relatively high when the load was 𝐹 = 30 n. 772 i. jang, y.h. jang fig. 4 comparison of contact area distribution according to scale variation 𝐿/2𝑛, (𝑛=5,9). column (a) shows the rough surface image as a 16-bit grayscale png, (b) the contact area distribution from the direct numerical analysis, (c) the contact area distribution from the deep learning model, and (d) a comparison of radial average magnitudes by frequency (the blue line represents the numerical analysis results, and the red dashed line represents the predicted results). table 1 evaluation metrics such as dice coefficient, jaccard index, bf score, and pixel accuracy for the predictions by the trained deep learning model at different scales scale dice coefficient jaccard index bf score pixel accuracy 𝐿/25 0.9169 0.8478 0.9019 0.9647 𝐿/26 0.9129 0.8419 0.8849 0.9520 𝐿/27 0.9188 0.8517 0.9844 0.9515 𝐿/28 0.9138 0.8441 0.9917 0.9605 𝐿/29 0.9154 0.8467 0.9977 0.9647 prediction of contact distribution on rough surfaces using deep learning algorithms 773 a two-dimensional fast fourier transform (2d fft) technique was used to analyze the structural similarity between the predicted and actual contact areas [40]. for a discrete 2d image 𝑓(𝑥, 𝑦) of size 𝑀 × 𝑁, the 2d fft 𝐹(𝑢, 𝑣) is defined as follows: 𝐹(𝑢, 𝑣) = ∑ ∑ 𝑓(𝑥, 𝑦)𝑒−𝑗2𝜋( 𝑢𝑥 𝑀 + 𝑣𝑦 𝑁 )𝑁−1 𝑦=0 𝑀−1 𝑥=0 , (22) where 𝑥 = 0, 1, … , 𝑀 − 1 and 𝑦 = 0, 1, … , 𝑁 − 1 represent the spatial coordinates of the image, indicating the pixel positions and sampled values of the original image, respectively. in contrast, 𝑢 = 0, 1, … , 𝑀 − 1 and 𝑣 = 0, 1, … , 𝑁 − 1 are the frequency coordinates in the horizontal and vertical directions, respectively, representing the magnitude and phase of each frequency component in the transformed domain. both sets of coordinates must be integers because this is inherent in the nature of discrete signals and the discrete fourier transform. the periodic patterns and structural characteristics of the images were effectively analyzed by transforming the spatial domain images into the frequency domain using a 2d fft. the power spectrum 𝑃(𝑢, 𝑣) is computed as follows, 𝑃(𝑢, 𝑣) = |𝐹(𝑢, 𝑣)|2. (23) to facilitate interpretation and visualization, a frequency shift operation was performed on the transformed frequency spectrum by moving the low-frequency components to the center of the spectrum. this shift can be mathematically expressed as 𝐹𝑐(𝑢, 𝑣) = 𝐹 ((𝑢 + 𝑀 2 ) mod 𝑀, (𝑣 + 𝑁 2 ) mod 𝑁) , (24) where the mod function is the modulus operator, which returns the remainder or signed remainder of a division after dividing one number by another. the radial average 𝑅(𝑟) is then calculated by averaging the power spectrum over the concentric circles, as follows: 𝑅(𝑟) = 1 𝑁𝑟 ∑ 𝑃(𝑢, 𝑣) 𝑢,𝑣: √𝑢2+𝑣2 =𝑟 , (25) where 𝑁𝑟 is the number of frequency components at radius 𝑟 , and 𝑟 = √𝑢2 + 𝑣2 represents the radial frequency. radial averaging reduces the two-dimensional (2d) frequency spectrum to a one-dimensional (1d) representation, allowing for a more straightforward analysis of the frequency characteristics. frequency analysis revealed distinct characteristics in different frequency ranges. in the low-frequency domain, the spectrum primarily reflected the overall shape and size of the contact clusters. the magnitudes of these low-frequency components are correlated with the macroscopic contact distribution. conversely, in the high-frequency domain, the spectrum captured the fine features and boundaries of individual contact areas. as shown in figs. 4 and 5, in the cases with high prediction performance, the frequency spectra of the predicted and actual contact areas exhibited highly similar patterns across the entire frequency range. 774 i. jang, y.h. jang fig. 5 contact area distribution according to load variation. column a) numerical analysis results, b) prediction results by the deep learning model, and c) comparison of radial average magnitudes by frequency (the blue and red dashed lines represent the numerical analysis and the predicted results, respectively). table 2. evaluation metrics such as dice coefficient, jaccard index, bf score, and pixel accuracy for the predictions by the trained deep learning model at different load variations force (n) dice coefficient jaccard index bf score pixel accuracy 10 0.9002 0.8186 0.9868 0.9976 20 0.9384 0.8840 0.9868 0.9971 30 0.9535 0.9112 0.9890 0.9964 40 0.9406 0.8880 0.9887 0.9938 50 0.9117 0.8377 0.9809 0.9890 prediction of contact distribution on rough surfaces using deep learning algorithms 775 the similarity in the low-frequency range indicated that the overall shape and size of the contact area were accurately predicted, whereas the match in the high-frequency range suggested that the detailed features and boundaries were precisely reproduced. this indicates that the model accurately captured both the large-scale structure and the fine details of the contact area. this similarity aligns with the dice coefficient values in the range of 0.95-0.96 for contact area distribution across all scales, and 0.91-0.95 range for contact area distribution across all loads, as presented in figs. 4 and 5, respectively. to quantitatively evaluate the prediction performance of the contact area across the entire test set of 20,000 samples, we compared the actual contact areas calculated through numerical analysis with those predicted by the deep-learning model, as shown in fig. 6. this comparison demonstrated the overall prediction accuracy and characteristics of the model. the average error rate across the complete dataset was 12.60%, with a notable trend of decreasing relative error as the contact area increased. a significant observation is that the data points are predominantly distributed below the ideal prediction line (dashed line). this indicates that the model tends to underestimate the actual contact area. the causes and implications of this systematic underprediction are discussed in detail in the following section. the dense distribution of data points around the ideal prediction line indicates consistent model performance across various contact conditions. in particular, in regions where the contact area exceeds 0.3 𝑚𝑚2 , the scatter between the predicted and actual values decreases, demonstrating enhanced model reliability in this range. fig. 6 comparison of contact area between numerical analysis and deep learning predictions for the entire test dataset (20,000 samples). the dashed line represents perfect prediction. 776 i. jang, y.h. jang 3.2. evaluation of the accuracy of the deep learning model from a contact mechanics perspective the elastic contact problem involving rough surfaces under normal loads typically shows that the contact area decreases, following a power law relationship, as the scale becomes finer. as shown in fig. 7(a), the contact area decreased in accordance with the power law relationship as the scale decreased. for the rough surfaces presented in fig. 4, the contact area predicted by the deep learning model shows a similar trend compared to the numerical analysis results, but it tends to underpredict by an average of approximately 10%. the relationship between the applied load and contact area was also investigated. an analysis of fig. 8(a) reveals that the deep learning model captures the fundamental characteristic of elastic contact mechanics: the linear proportionality between the normal load and the true contact area. when examining the quantitative accuracy across different load conditions (10 n-50 n), the model demonstrated consistent prediction capability while maintaining an average deviation of 11.96% from the numerical analysis results. this systematic underprediction was most pronounced at lower loads (18.14% at 10 n) and showed gradual improvement as the load increased (10.01% at 50 n), suggesting that the performance of the model is more reliable under higher load conditions where larger contact areas are formed. this underprediction of the contact area can be primarily attributed to two factors. first, it indicates that the deep learning model misses actual contact areas more often than it mistakenly classifies noncontact areas. second, the inherent class imbalance problem in the dataset may have led to the insufficient learning of smaller contact areas. a detailed analysis of the underprediction of the contact area is presented in the next section and in the discussion section. fig. 7 comparison between numerical analysis results and deep learning predictions of (a) variations in contact area and (b) changes in electrical contact resistance resulting from scale variations. solid square and circle shapes represent the deep learning prediction and the numerical analysis results, respectively. prediction of contact distribution on rough surfaces using deep learning algorithms 777 fig. 8 comparison between numerical analysis results and deep learning predictions of (a) variations in contact area and (b) changes in electrical contact resistance resulting from load variations. solid square and circle shapes represent the deep learning prediction and the numerical analysis results, respectively. currently, the predictive performance of deep-learning algorithms is based solely on the model data generated at each scale. in other words, it does not consider causal relationships where data from a previous (coarse) scale affect data at a subsequent (fine) scale. nonetheless, it is noteworthy that the power law relationship was maintained even in the absence of learned correlations between scale changes. this suggests that the deep learning model can implicitly learn the fundamental physical characteristics of the contact mechanism. the electrical contact resistance (ecr) was calculated at each scale using the predicted contact area distribution. while various theoretical formulations for ecr exist, such as those developed by paggi and barber [41], the method presented by boyer was selected for this study due to its suitability for our pixel-based contact analysis. specifically, the analytical model from section ii of boyer [42], which addresses clean metallic contact without interface films. in particular, we adopted the formulation derived for multiple square-shaped microcontacts, where the total contact resistance r is given by: 𝑅 ≈ 𝑆(𝑁)𝜌 𝑛𝐿 + 𝜌 𝜋𝑛2 ∑ ∑ 1 𝑑𝑖𝑗 𝑖≠𝑗 (26) where 𝑅 represents the total electrical contact resistance, and ρ denotes the electrical resistivity of the contacting material. the variable n is the number of contacting pixels, each of which is treated as a square microcontact, and 𝐿 is the side length of each contact pixel. the term 𝑆(𝑁) is the shape factor for square contacts, for which we used the value 0.868, following boyer’s suggestion. the term 𝑑𝑖𝑗 denotes the center-to-center distance between the 𝑖-th and 𝑗-th contact pixels, and the summation is carried out over all distinct contact pairs. this formulation reflects both the parallel resistance of uniformly sized microcontacts (first term) and the additional resistance arising from spatial interactions among them 778 i. jang, y.h. jang (second term). all required values—such as the number of contacts 𝑛, side length 𝐿, and pairwise distances 𝑑𝑖𝑗—were extracted directly from the predicted binary contact masks. as shown in fig. 7(b), the computed ecr increases and converges as the scale decreases. despite an average deviation of approximately 10% in the predicted contact area relative to the numerical reference, the resulting ecr exhibited a discrepancy of less than 0.5%. this small difference is attributed to the fact that ecr is predominantly governed by long-wavelength surface features, which are well captured by the model, as further supported by the spectral analysis in section 3.3. similarly, the electrical contact resistance was analyzed for different load conditions. fig. 8(b) shows that the electrical contact resistance decreased monotonically with increasing load, following an inverse relationship characteristic of elastic contact mechanics. the deep learning model accurately captured this trend, maintaining a mean error of just 2.61% across all load conditions. interestingly, the model performance showed varying accuracy levels at different loads, with the lowest error of 0.82% at 20 n and a maximum error of 6.42% at 50 n. this pattern differs notably from the contact area predictions, where higher loads generally yield better accuracy. this phenomenon aligns with the fact that the electrical contact resistance is determined by the long wavelength roughness [43]. the current analysis of deep learning models indicates that the prediction of contact points owing to short wavelength asperities is inaccurate, leading to an underestimation of the contact area. however, the distribution of the contact surfaces influenced by long wavelength roughness was captured with a high degree of accuracy. despite the relatively large error observed in the contact area prediction, the model consistently showed low prediction errors for the electrical contact resistance across various scales and load conditions. this suggests that the skip connection feature of the u-net architecture effectively contributes to learning the spatial characteristics of contact regions. finally, the computation times of the direct numerical model and deep learning algorithm were compared. given that the deep learning algorithm requires a very short computation time for a single surface image, the comparison was performed using 100 datasets at a scale of 𝐿/29, including the time required to store the contact distribution results. the total computation time for the direct numerical model was approximately 11,835 s, whereas the deep learning algorithm required only approximately 297 s. on average, the computation time per dataset was approximately 118 s for the direct numerical model and 3 s for the deep learning algorithm. this demonstrates the significantly higher computational efficiency of the deep learning algorithm. 3.3. presentation of poorly matching cases the contact area distribution predicted by the deep learning model often closely matched the results of the numerical analysis; however, in some cases, the prediction led to relatively large errors in localized regions. these mispredictions tended to increase, particularly when the contact area was relatively small. because different evaluation metrics highlight various aspects of performance, it is challenging to assess a model using a single metric. however, this explanation focuses on dice coefficients. figs. 9(a)(a), 9(b)(a), and 9(c)(a) show the contact area distribution obtained through the numerical analysis, whereas figs. 9(a)(b), 9(b)(b), and 9(c)(b) display the contact area distribution predicted using the deep learning model. by comparing and analyzing these prediction of contact distribution on rough surfaces using deep learning algorithms 779 three sets of images, we aimed to evaluate the performance and limitations of the contact area prediction model. fig. 9 three types of contact area distributions showing poor predictions (a, b, and c). (a) results of the direct numerical model, (b) predictions by the deep learning model, and (c) a comparison of radial average magnitudes by frequency (the blue line represents the numerical analysis results, and the red dashed line represents the predicted results). in fig. 9(a), the numerical analysis results show a sparse distribution of contact points, whereas in the predicted image, some regions exhibit either an underestimation or overestimation of contact points. upon analyzing the actual data for this case, the tp was 0.0054, fn was 0.0031, fp was 0.0009, and tn was 0.9907. this indicates that approximately 0.63% of the total pixels were predicted as contact areas (tp + fp), which is slightly lower than the actual contact area (tp + fn ≈ 0.85%). the fact that the fn value 780 i. jang, y.h. jang is approximately 3.4 times higher than the fp value suggests that the model tends to miss the actual contact areas. this demonstrates the limitations of the model in accurately capturing fine contact points when the contact area is relatively small. the dice coefficient for this distribution was 0.7303, indicating a relatively low prediction accuracy. in the comparison of the radial average profile, the predicted image shows a behavior similar to that of the original image in the low-frequency range, but it fails to fully capture the fine characteristics of the actual contact area in the high-frequency range. the actual contact area is 0.0085 𝑚𝑚2 , whereas the predicted contact area is 0.0062 𝑚𝑚2 , indicating a significant error of 26.62%. this substantial error quantitatively demonstrates the significantly degraded prediction performance of the model for very small contact areas. in particular, the fact that the fn value was approximately 3.4 times higher than the fp value suggests that the model struggles to capture fine contact spots. in fig. 9(b), the analysis results show that the tp was 0.0213, fn was 0.0087, fp was 0.0068, and tn was 0.9632. in this case, about 2.81% of the total pixels were predicted as contact areas (tp + fp), which is relatively similar to the actual contact area (tp + fn ≈ 3%). however, both fn and fp values were considerably high, indicating that the model struggled to accurately identify the contact areas. notably, the fn value was approximately 1.28 times higher than the fp value, indicating that the model tends to miss the actual contact areas. this suggests that the model was inconsistent in predicting the varying sizes and densities of the contact points observed in the numerical analysis results. nonetheless, the dice coefficient for this distribution was 0.7332, indicating a slightly higher prediction accuracy than in the first case. this suggests that, as the overall size of the contact area increases, the prediction performance of the model improves to some extent. in the radial average profile, the predicted values match the original to some degree in the lowfrequency range; however, as the frequency increases, the discrepancy increases, and the model fails to sufficiently capture the boundaries and fine details of the predicted contact points. in this case, the actual contact area was 0.0300 𝑚𝑚2, and the predicted contact area was 0.0281 𝑚𝑚2 , showing a relatively small error of 6.50%. this indicates improved model prediction accuracy as the contact area increases. however, the high fn and fp values indicate that the model continues to struggle to precisely identify contact locations. in fig. 9(c), the analysis results show that the tp was 0.0344, fn was 0.0137, fp was 0.0078, and tn was 0.9440. in this case, about 4.22% of the total pixels were predicted as contact areas (tp + fp), which is slightly lower than the actual contact area (tp + fn ≈ 4.81%). the fn value was approximately 1.76 times higher than the fp value, indicating that the model tended to underestimate the actual contact areas. this suggests that the model fails to accurately capture the dense cluster-like contact areas observed in the numerical analyses. however, the dice coefficient for this distribution was 0.7616, showing a relatively higher prediction accuracy compared to the previous two cases. in the radial average profile, discrepancies with the original are observed in both the lowand high-frequency ranges; however, overall, this case shows an improved performance compared with the previous two. in this case, the actual contact area was 0.0482 𝑚𝑚2, whereas the predicted contact area was 0.0422 𝑚𝑚2, with an error of 12.29%. despite having the largest contact area among the three cases, the considerable error suggests that the prediction performance of the model may degrade in contact regions with complex cluster structures. a common issue observed across all three cases was that the fn rate was higher than the fp rate, and the frequency analysis results showed a lower prediction accuracy in the high-frequency range. this suggests that the model tends to underestimate the actual prediction of contact distribution on rough surfaces using deep learning algorithms 781 contact areas and has difficulty accurately capturing fine contact points or complex cluster structures. the frequency-domain analysis clearly demonstrates that although the model captures the overall contact pattern (low-frequency components) to some extent, it struggles to reproduce the fine structures and boundaries (high-frequency components) accurately. in particular, the decreased performance of the model in predicting low-density contact distributions or complex cluster structures raises the possibility that these extreme cases have not been sufficiently learned. the frequency analysis further highlighted these limitations. to identify the root causes of these issues, it is necessary to go beyond merely considering the model structure or quantity of training data to analyze the inherent characteristics of contact mechanics and the essential nature of the data involved. 3.4. discussion the deep learning model developed in this study has shown significant success in predicting the contact area distribution in contact mechanics problems. however, it also revealed several important limitations and areas for improvement. notably, issues stemming from the characteristics of the training dataset were found to have a substantial impact on model performance. the most prominent issue is the severe class imbalance inherent in the contact data. in most cases, the number of pixels corresponding to the noncontact areas far exceeded those in the contact areas. this "intrinsic imbalance" can cause the model to become excessively biased toward the majority class (noncontact areas), resulting in a significant reduction in the prediction accuracy for the minority class (contact areas). according to buda et al. [44], class imbalance in deep neural networks negatively affects the model performance, and this effect becomes more severe as the imbalance ratio increases. this class imbalance is further compounded by the characteristics of learning algorithms. the commonly used cross-entropy loss function assumes a balanced dataset and may not perform optimally under severe imbalance [45]. moreover, in batch-based learning, if the minority class is not sufficiently represented in the batch, the model may fail to effectively learn the characteristics of that class. to address these issues, various approaches can be considered [46]. however, the unique nature of contact mechanics problems must be taken into account. common techniques used in classification problems, such as oversampling or undersampling, are not easily applied to contact area data owing to their specific characteristics. simply increasing or decreasing the contact area data can distort the physical meaning. instead, algorithmlevel approaches may be more appropriate. for example, a weighted loss function that assigns greater weight to the minority class (contact areas) can be considered. this would guide the model to pay more attention to the contact areas, which would help alleviate the class imbalance issue. to address this challenge, we employ a combined loss function. by integrating two different loss functions and adjusting the weighting parameters, the model focuses on mitigating the impact of class imbalance. optimizing these weighting parameters plays a crucial role in improving the prediction accuracy of the model, particularly in scenarios with low contact loads. however, despite these efforts, the class imbalance issue has not been fully resolved and performance degradation is still observed in cases of extreme imbalance. therefore, further development of more effective loss functions and additional training strategies are necessary. 782 i. jang, y.h. jang in addition, strategies that consider class balance when constructing batches can be modified and applied. for example, constructing batches by sampling image patches with smaller contact areas more frequently could help strengthen the information about minor contact areas during the learning process while preserving the physical meaning of the data. a two-stage learning approach could also be considered. in the first stage, the model would learn to extract features from the entire image. in the second stage, it would focus on predicting the contact areas based on the extracted features. this method would allow the model to learn both the overall surface geometry and the localized contact information in a balanced manner. moreover, adding physical constraints from contact mechanics to the loss function could be another potential improvement [47]. for instance, using the relationship between the predicted contact area and actual load as a constraint could lead to physically valid predictions. this hybrid approach, which combines data-driven learning with physics-based modeling, offers a method for mitigating class imbalance while preserving physical relevance. future research should empirically evaluate the effectiveness of these methods and develop extended deep learning algorithms tailored to contact mechanics problems. in particular, expanding the model to predict contact area distribution (a binary classification problem) and contact pressure distribution (a regression problem with continuous values) will be essential. overcoming the intrinsic differences between these two problems requires the development of a new architecture. one promising approach is the application of generative adversarial networks (gans). through competitive learning between a generator and a discriminator network, gans can generate high-quality images [48]. by applying gans to contact-pressure distribution prediction, it is expected that a physically valid and detailed pressure distribution can be generated. this could provide a more realistic and accurate pressure distribution than simple regression models. additionally, the approach developed in this study could be extended to predict surface wear. wear is a result of the complex interaction between contact-pressure distribution and relative motion, which leads to changes in the surface geometry over time. it would be possible to use a deep learning model to predict wear patterns over time by taking the initial surface geometry and motion conditions as input. this would be a significant application in the field of tribology, contributing to the prediction of machine part lifespan and optimal design. furthermore, enhancing the generalization capability of the model developed in this study is another important task. although the current model was trained under specific load conditions and surface characteristics, it should be able to operate under a wider range of conditions for practical engineering applications. to achieve this, transfer-learning techniques can be employed to develop a generalized model for various contact conditions. moreover, concerning the model's generalization capabilities and extensibility, a significant advantage of the proposed image-based deep learning framework lies in its potential independence from the specific numerical simulation method used for data generation. as the model fundamentally learns the mapping from a surface height image (input) to a contact area distribution mask (output), datasets generated using alternative well-established computational methods—such as fem (e.g., using commercial software like abaqus or ansys), bem, or other advanced simulation techniques including various fft-based approaches and green's function molecular dynamics—could theoretically be utilized for training, provided the data is formatted into the consistent image/mask structure. future work could explicitly investigate training and evaluating the model on datasets derived from these diverse numerical sources. integrating data from multiple methods might even enhance the model's robustness and generalizability, potentially making it less sensitive to the inherent assumptions or artifacts of prediction of contact distribution on rough surfaces using deep learning algorithms 783 any single simulation technique and thus increasing its practical value across varied engineering applications. finally, improving the interpretability of deep learning models is a critical area for future research. current deep learning models are often regarded as "black boxes," making it difficult to understand how they make decisions. in contact mechanics, it is important to understand the decision-making process of the model, and applying explainable artificial intelligence (xai) techniques would allow for the interpretation and visualization of the decision processes of the model [49]. these research directions will expand the scope of deep learning applications in contact mechanics and contribute to the development of more accurate and reliable prediction models. this, in turn, will enable more efficient solutions to complex contact problems and significantly improve the engineering design and optimization processes. 4. conclusions this paper presents a deep learning algorithm for predicting the contact area distribution of rough surfaces using only surface image data. the main findings of this study are as follows: 1. the performance of the deep learning model was evaluated under various scale variations and loading conditions to predict the contact area distribution of rough surfaces. the results demonstrated that the deep learning model could accurately predict the changes in the contact area owing to variations in scale and load, and showed results similar to those of the numerical analysis for predicting electrical contact resistance. this indicates that the deep learning model effectively learned the spatial distribution of the contact points and accurately predicted the contact characteristics. 2. using a u-net-based deep learning model, the contact area of rough surfaces was approached as an image segmentation problem, which overcame the high computational complexity of traditional numerical methods and enabled rapid and efficient prediction. the model exhibited high performance in various evaluation metrics such as the dice coefficient and jaccard index. these results confirm that the deep learning model can achieve a prediction accuracy comparable to that of numerical methods while significantly enhancing computational efficiency. 3. the deep learning model demonstrated results similar to the actual data in both the lowand high-frequency domains using the 2d fft technique, indicating that the model effectively learned the geometric characteristics of the contact phenomena. the model showed the ability to accurately predict not only the large-scale structure of the contact area but also its fine features. these findings imply that the deep learning model can adequately reflect the contact characteristics across different frequency domains. 4. the deep learning model significantly improved computational efficiency compared to conventional numerical models, reducing computation time by more than 95% for finescale datasets. the deep learning algorithm developed in this study demonstrated the potential to efficiently predict complex and fine contact areas when applied to contact mechanics problems and significantly reduced computational complexity compared to conventional numerical methods. this algorithm is expected to provide reliable data for various engineering design issues, including friction, wear, and thermal/electrical contact resistance. moreover, by reducing the computation time and cost, the proposed method has potential for application in real-time analysis and large-scale simulations across diverse engineering fields. 784 i. jang, y.h. jang acknowledgment: the authors are pleased to acknowledge support from the national 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secondly, the formulation is extended to the damped case. numerical verification shows that the frequency formulation is mathematically simple and physically insightful and practically applicable. this paper paves a novel way for engineers to use the formulation to study nonlinear vibration system with ease and reliability. key words: duffing equation, he’s frequency formulation, ancient chinese mathematics, residual equation, nonlinear vibrations 1. introduction nonlinear systems are widely present in the field of mechanical engineering and hold significant importance in unraveling the dynamical characteristics of nonlinear vibration systems. in the realm of mechanical engineering, nonlinear vibration systems are omnipresent, encompassing nano-scale vibrations [1], atomic-level lattice vibrations [2], and microelectromechanical systems [3,4], among others. the most renowned nonlinear oscillators in these systems include pendulum oscillators [5,6], van der pol oscillators, and duffing oscillators [7]. with the rapid development of nonlinear science, more and more scholars have shown great interest in nonlinear problems. solving large-scale linear equation systems is no longer a challenge, however, for nonlinear equation systems, traditional perturbation methods [8] have been widely applied in the nonlinear analysis of engineering problems. however, like other nonlinear asymptotic techniques, perturbation methods are also affected by limitations imposed by small parameters, and this point should received: september 09, 2023 / accepted november 30, 2023 corresponding author: jian-gang zhang school of mathematics and physics, lanzhou jiaotong university, lanzhou, p. r. china e-mail: zhangjg7715776@126.com 702 j.g.zhang, q.r.song, j.q.zhang, f.wang also be taken into consideration. as is well known, most nonlinear problems do not involve small parameters. even if appropriate small parameters exist, the approximate solutions obtained through perturbation methods are only effective for weakly nonlinear cases [9]. furthermore, the approximate solutions obtained through perturbation methods often lose some crucial information, thereby failing to accurately reflect the important relationship between amplitude and frequency. the duffing equation serves as an example of this case. therefore, to solve this problem, chinese mathematician, ji-huan he, inspired by ancient chinese algorithms [10-12], proposed a simple method for conserving nonlinear vibrators, and a frequency-amplitude formulation was proposed, which is called he’s frequency formulation in literature [13-15]. the ancient chinese mathematics was also further extended to solve nonlinear differential equations numerically [16], and in literature the method is called as chun-hui he’s iteration method [17,18]. he’s frequency formulation [10-12] not only can quickly unveil the relationship between the frequency and amplitude of various vibrators but also ensures high accuracy. elías-zúñiga, et al. [19] extended the formulation by coupling jacobi elliptic function, ma [20] suggested a hamiltonian-based modification, alyousef, et al. [21] used the formulation to solve a complex nonlinear system with great success. he and liu [22] gave a mathematical proof of the formulation and suggested a powerful modification. all of the above studies considered simple initial conditions, and there is much space to extend the formulation to generalized initial condition. this paper considers a damped duffing oscillator with generalized initial conditions to show the effectiveness and simplicity of he’s frequency formulation. this paper is divided into three sections. the first section considers an undamped duffing oscillator with generalized initial conditions and applies he's frequency formulation to solve the approximate frequency, which is then compared with the rk4 numerical solution. in the second part, he's frequency formulation is applied to a linear damped oscillator with generalized initial condition. in the third part, a nonlinear duffing oscillator with damping is considered to show reliability of he’s frequency formulation. 2. undamped duffing equation we consider a duffing equation 3 0u u u + + = (1) with the generalized initial conditions (0) , (0)u a u b = = (2) where a, b and ε are constants. there is much literature to study the above problem in case of small ε or a=0/b=0 by various methods, for examples, the homotopy perturbation method [23], the energy conservation principle [24] and the variational iteration method [25]. in this section, we will use he's frequency formulation to find the solution to eq. (1). assuming that the solution of eq. (1) has the following form: cos( )u a t= +  (3) where ɑ and σ are constants, ω is the frequency to be determined. the initial conditions lead to the following relations: application of he’s frequency formula to nonlinear oscillators with generalized initial condition 703 (0) sin (0) cos u a a u a b  = − = = =    (4) substituting eq. (3) into eq. (1) gives the residual equation as 2 3 3( ) cos( ) cos( ) cos ( )r t a t a t a t= − + + + + +        (5) therefore, the average residual equation is defined as 2 0 2 cos( )dt t r r t t  = + (6) where t=2π/ω. by choosing two arbitrary frequencies ω1 and ω2, we obtain two residual equations, which are 2 3 3 1 1 1 1 1 2 3 3 2 2 2 2 2 ( ) cos( ) cos( ) cos ( ) ( ) cos( ) cos( ) cos ( ) r t a t a t a t r t a t a t a t = − + + + + + = − + + + + +                 (7) therefore, the average residual equations as 2 32 1 1 1 10 2 32 2 2 2 20 2 3 cos( )dt 2 2 8 2 3 cos( )dt 2 2 8 t t a a r r t a t a a r r t a t = + = − + + = + = − + +           (8) we choose the location point as cos(ω1t+σ)=cos(ω2t+σ)=√3/2, eq. (8) becomes 2 3 1 1 2 3 2 2 3 3 2 2 3 3 2 2 r a a a r a a a = − + + = − + +     (9) according to he’s frequency-amplitude formulation, we have 3 3 2 2 2 2 2 1 2 22 1 1 2 1 2 2 2 1 2 3 3 ( ( ) )( ) 32 2 1 43 ( ) 2 a a r r a r r a + − − = = = + − − −          (10) the result depends upon the location point, to overcome the shortcoming, we use the following formulation 2 2 2 2 1 1 2 1 2 3 2 2 2 1 2 2 2 1 2 3 ( )( ) 32 8 1 4 ( ) 2 r r r r a a a a − = − + − = = + − −          (11) 704 j.g.zhang, q.r.song, j.q.zhang, f.wang from eqs. (4) and (11), ɑ, σ and ω can be determined for given a and b. when a=0 or b=0, our result is the same as those obtained by the homotopy perturbation method or the variational iteration method. we compare the approximate solution based on he’s frequency formulation with the exact solution of the runge-kutta method, as shown in fig. 1. from the figure, we can see that the two curves fit well, reflecting the consistency between the exact solution and the approximate solution. (a, b)=(0.01,0.01) (a, b)=(0.1,0.1) (a, b)=(1.6,1) (a, b)=(5.5,2) fig. 1 the comparison between the exact solution and the approximate one based on he’s frequency formulation for different values of (a, b) 3. linear oscillator with damping in the previous section, we discussed the frequency formulation for a classical duffing oscillator with general initial conditions, but in reality, perfect simple harmonic motion is almost non-existent. in reality, the amplitude of the duffing oscillator described above will decrease due to air damping and other effects, eventually reaching an equilibrium state. there are many types of damping in engineering, and the most commonly used damping is viscous damping. therefore, in this section, the linear oscillator with viscous damping is considered first. application of he’s frequency formula to nonlinear oscillators with generalized initial condition 705 introducing viscous damping into a linear oscillator, the equation of vibration is: 2 0, (0) , (0)u u u u b u a+ + = = = (12) where 2η is the damping coefficient. the magnitude of damping is proportional to the vibration rate, and the direction is opposite to the speed direction. based on he's frequency formula, the approximate solution is found in the following form: ( ) cos( )tu t ae t− = +   (13) where ω′ is the frequency to be solved later. based on he's frequency formulation, an attempt is made to start from two arbitrary values of ω1′and ω2′, to obtain the following residual equations: 2 2 1 1 1 1 1 2 1 1 1 1 2 1 1 ( ) ( ) cos( ) 2 sin( ) 2 cos( ) sin( ) cos( ) (1 )cos( ) t t t t t t r t ae t a e t a e t a e t ae t ae t − − − − − −    = − + + +    − + − + + +  = − − +                          (14) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) cos( ) 2 sin( ) 2 cos( ) sin( ) cos( ) (1 )cos( ) t t t t t t r t ae t a e t a e t a e t ae t ae t − − − − − −    = − + + +    − + − + + +  = − − +                          (15) the residual integrals are given below 2 1 1 10 2 22 1 10 2 2 1 2 ( )cos( )d 2 (1 )cos ( )d 6 4 (1 ) 5 t t t r r t t t t = ae t t t e a          −  − = +  − − + − = − −    (16) 2 2 2 20 2 22 2 20 2 2 2 2 ( )cos( )d 2 (1 )cos ( )d 6 4 (1 ) 5 t t t r r t t t t = ae t t t e a          −  − = +  − − + − = − −    (17) where t=2π/η, then the frequency equation is 706 j.g.zhang, q.r.song, j.q.zhang, f.wang 2 2 1 1 2 1 2 2 2 2 2 2 1 2 2 2 2 1 2 6 4 ( )(1 ) 5 6 4 ( ) 5 1 r r r r e a e a  −  −  −  = − −  − − = −  −  = −          (18) 4. nonlinear duffing oscillator with damping viscous damping is introduced in a nonlinear duffing oscillator, its vibration equation is: 32 0, (0) , (0)u u u u u b u a+ + + = = =  (19) in the first section, the frequency formula of the nonlinear oscillator ü+u+εu3=0 is given by 2 23 1 4 a= +  (20) therefore, the damped nonlinear oscillator can be equivalent to 22 0, (0) , (0)u u u u b u a+ + = = =  (21) similarly, there are approximate solutions of the following form ( ) cos( )tu t ae t− = +   (22) where ω′ is the frequency to be solved later. we try to get the following residual equations starting from two arbitrary ω1′and ω2′: 2 2 1 1 1 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 2 2 ( ) ( ) cos( ) 2 sin( ) 3 2 cos( ) sin( ) (1 )cos( ) 4 3 (1 )cos( ) 4 ( ) ( ) cos( ) 2 sin( t t t t t t t t r t ae t a e t a e t a e t ae a t ae a t r t ae t a e − − − − − − − −    = − + + +    − + − + + + +  = + − − +    = − + +                                   2 2 2 2 2 2 2 2 2 2 2 ) 3 2 cos( ) sin( ) (1 )cos( ) 4 3 (1 )cos( ) 4 t t t t t a e t a e t ae a t ae a t − − − − +    − + − + + + +  = + − − +                    (23) the residual integrals are application of he’s frequency formula to nonlinear oscillators with generalized initial condition 707 2 1 1 10 2 2 22 1 10 2 2 2 1 2 ( )cos( )d 2 3 (1 )cos ( )d 4 3 6 4 (1 ) 4 5 t t t r r t t t t = ae a t t t e a a −  − = +  + − − + − = + − −               (24) 2 2 2 20 2 2 22 2 20 2 2 2 2 2 ( )cos( )d 2 3 (1 )cos ( )d 4 3 6 4 (1 ) 4 5 t t t r r t t t t = ae a t t t e a a −  − = +  + − − + − = + − −               (25) so a frequency formulation is obtained, which is 2 2 2 2 2 2 1 2 2 22 1 1 2 1 2 2 2 2 2 1 3 6 4 ( )(1 ) 34 5 1 4 6 4 ( ) 5 e a a r r a r r e a  −  − −  − + −  −  = = = + − − −  −             (26) in addition, the frequency equation of eq. (20) can be found by ( ) cos( )tu t ae t− = +   (27) (0) cos sinu a a a    = − − = (28) (0) cosu a b= = (29) combining eq. (28) and eq. (29) yields 2 2 2 2 ( cos )a a a b +  = −    (30) in the following, we will use numerical simulation to analyze the impact of the variation of the damping term on nonlinear oscillators with generalized initial conditions. in order to better align with the actual operating conditions of the machinery, we make the values of a and b vary from small to large. fig. 2-fig. 4 show the fitting results of the exact and approximate solutions when the damping η is varied, respectively. 708 j.g.zhang, q.r.song, j.q.zhang, f.wang in fig. 2, we choose η=0.01 and let (a, b) vary from 0.2 to 1000, where the blue curve denotes the exact solution and the red denotes the approximate solution based on he’s frequency formulation. the two curves are highly fitted as can be seen in the figure. (a, b)=(0.2,0.2) (a, b)=(10,10) (a, b)=(99,100) (a, b)=(1000,1000) fig. 2 when η=0.01, the comparison between the exact solution and the approximate one based on he’s frequency formulation for taking different values of (a, b) when η=0.1, considering that the nonlinear vibrations in the actual working conditions are all large parameters, as we take (a, b) from 1 to 1000, respectively. the resulting figure is shown in fig. 3. from the figure, we can not only see that the increase of damping makes the system amplitude gradually decrease, but also find that the exact and approximate solutions are in good agreement. in addition, compared to fig. 2, the magnitude of the system amplitude reduction is more pronounced when the damping is increased from 0.01 to 0.1. application of he’s frequency formula to nonlinear oscillators with generalized initial condition 709 (a, b)=(1,1) (a, b)=(10,10) (a, b)=(99,100) (a, b)=(1000,1000) fig. 3 when η=0.1, the comparison between the exact solution and the approximate one based on he’s frequency formulation for taking different values of (a, b) to further verify the conclusion, we increase the damping so that η=0.15 and (a, b) is varied from 25 to 2500. the image is displayed in fig. 4. the numerical simulation shows that even if we increase the damping and the initial values, the approximate solution based on he’s frequency formulation still agrees with the exact solution from the runge-kutta method. in addition, combined with fig. 2, fig. 3 and fig. 4, we can observe that a small change in the damping term can make the amplitude oscillation of the oscillator solution curve more significant, and the solution curve will gradually approach equilibrium over time. 710 j.g.zhang, q.r.song, j.q.zhang, f.wang (a, b)=(25,30) (a, b)=(100,120) (a, b)=(1100,1300) (a, b)=(2500,3000) fig. 4 when η=0.15, the comparison between the exact solution and the approximate one based on he’s frequency formulation for taking different values of (a, b) 5. conclusion this article mainly utilizes the he’s frequency formula to solve the solution of the nonlinear duffing oscillator with generalized initial conditions, and compares it with the exact solution of the runge-kutta method. we summarize this article in the following two aspects: (1) for the undamped nonlinear duffing oscillator, this paper first utilizes he’s frequency formula to calculate the approximate frequency and derive the corresponding approximate solution. subsequently, we compare the approximate solution with the runge-kutta exact solution through numerical simulation and find consistent results between them. it is worth noting that this study focuses on generalized initial conditions, and when we set the initial condition a to 0 or b to 0, our results are in complete agreement with those obtained by the homotopy perturbation method or the variational iteration method. (2) in this paper, we investigate the duffing oscillator with viscous damping. we divided the duffing oscillator into linear and nonlinear types, and used he's frequency formula to find approximate solutions in both cases, followed by numerical simulations. application of he’s frequency formula to nonlinear oscillators with generalized initial condition 711 the experimental results show that, even when we increase the damping and the initial conditions a and b in the nonlinear duffing oscillator, the approximate solution obtained using he's frequency formula still matches the exact solution obtained by the runge-kutta method very closely. this fully demonstrates that he's frequency formula is equally applicable to nonlinear damped oscillators with general initial conditions. the he's frequency formula method is both simple and fast, and we also observed a gradual decrease in the amplitude of the duffing oscillator's solution, eventually reaching equilibrium as the damping increased. these results are of great significance for a deeper understanding and application of the dynamical behavior of duffing oscillators. in conclusion, this frequency formula possesses simple yet profound mathematical properties and holds significant practical value. it provides engineers with an efficient and reliable approach to studying nonlinear vibration systems. in future research, we aim to further explore the application of this frequency formula to fractional-order nonlinear systems with generalized initial conditions, and thoroughly examine its effectiveness. by applying this formula in broader domains, we can enhance our understanding and analysis of complex vibration systems, offering more reliable and accurate solutions for engineering practices and scientific investigations. acknowledgement: the paper was supported by the national natural science foundation of china (61863022) and the key project of gansu natural science foundation (23jrra882). references 1. yan, w., noel, 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mechanical engineering vol. 16, no 3, 2018, pp. 307 319 https://doi.org/10.22190/fume170913031r © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper surface-treated and fibrin-coated electrospun polyacrylonitrile fiber for endothelial cell growth and proliferation  udc 678.754.3:678.026 nur syazana rashidi 1 , irza sukmana 2 , agung mataram 3 , noor jasmawati 1 , mohd ramdan mohd rofi 1 , mohammed rafiq abdul kadir 1 1 medical devices and technology group (mediteg), faculty of bioscience and medical engineering, universiti teknologi, malaysia 2 faculty of engineering, universitas lampung, indonesia 3 faculty of engineering, university of sriwijaya, indonesia abstract. polyacrylonitrile (pan) is a synthetic biocompatible polymer used as a filtering membrane in hemodialysis and for enzyme immobilization purposes. however, its potential usage in other medical applications is limited due to its poor hydrophilicity. to overcome this problem, two groups of electrospun pan fibers were treated with sodium carbonate (na2co3) and sodium hydroxide (naoh), respectively at 100oc for 5 minutes. fibrin gel was then coated onto the treated samples before seeding human umbilical vein endothelial cells (huvecs) for 1 and 3 days. x-ray diffraction results showed increased crystallinity of the pan fibers when treated with na2co3 and naoh. the contact angle measurements showed that the hydrophilicity of na2co3 treated and naoh treated samples improved from 115° to 88° and 64°, respectively. the fourier transform infrared spectroscopy confirmed that their hydrophilicity was due to the existence of carboxyl and hydroxyl groups. however, tensile strength of the pan fibers was reduced by 34% when treated with na2co3 and 42% when treated with naoh. cytotoxicity tests showed increased absorbance in day 3 for both treated samples. however, the absorbance value for naoh treated pan fibers and na2co3 treated pan fibers showed nearly the same absorbance on day 7. in vitro tests showed increased cell adhesion and proliferation after 3 days of culture. the pan treated fibers coated with fibrin are, therefore, proven to attract huvec cells and promote endothelialization. key words: polyacrylonitrile fibre, surface treatment, scaffold, endothelial cells received september 13, 2017 / accepted october 25, 2018 corresponding author: irza sukmana department of mechanical engineering, faculty of engineering, universitas lampung, jl. prof. soemantri brojonegoro no. 1, bandar lampung 35145, indonesia e-mail: irza.sukmana@eng.unila.ac.id 308 n.s. rashidi, i. sukmana, a. mataram, n. jasmawati, m.r.m. rofi, m.r.a. kadir 1. introduction due to its high biomechanical strength, thermal stability, tolerance to solvents as well as its resistance to bacteria and photo irradiation effects, polyacrylonitrile (pan) has been used extensively to produce various manufactured materials [1]. these include products such as commercial carbon fibers, filtration [2], adsorption [3], as well as weaved products such as blankets, carpets and clothes [4]. pan is also proven as a biocompatible material suitable for biomedical purposes such as tissue-engineered bioartificial skin [5], hemodialysis [6], and biocatalysts immobilization [7]. however, due to its poor hydrophilicity, the use of pan without surface treatment appears limited. therefore, the surface treatments used for overcoming this problem have been described in the referential literature. they include methods such as plasma treatment [8-10], plasma initiated graft polymerization [11,12], and photoinduced grafting [13]. yet, these methods have inherent limitations. compounded by the high cost involved in the treatment process, many of them have been deemed impractical for large scale manufacturing purposes [14]. it has been suggested that the use of na2co3 and naoh solutions in millipore-purified water can be used to provide similar effects without involving exorbitant cost [15, 16]. the promotion of endothelial adhesion and vascularization to prevent thrombosis as well as intimal hyperlasia of a small diameter vascular graft has been tested to the pan fibers. previous studies have shown that without the fibrin coating process, the cell differentiation onto the material surfaces would be non-existent thus rendering the material less useful for vascularization of an engineered-tissue substitute [17]. the role of fibrin actively directs cellular responses through specific receptor-mediated interactions with endothelial cells of the vessel wall [18]. therefore, the use of fibrin as a coating active matrix onto a bioactive scaffold to support endothelialization is of interest. however, no study has been done yet to show that the electrospun pan fibers with na2co3 and naoh surface treatment followed by fibrin coating can form endothelial cells monolayer for blood vessel. the surface treatment using naoh is nevertheless already established for pan materials by authors mostly in ultrafitration and hemodylisis applications [19, 20]. however, the effects of na2co3 treatment on the pan mechanical and physical properties have not been intensively studied compared with naoh surface treatment. research on na2co3 surface treatment onto pan only covered a hydrolysis part while biocompatibility has not been addressed before [15, 20]. the present study aims at analyzing hydrophilicity of the treated pan fibers with na2co3 and naoh. the pan treated surfaces are then coated with fibrin gel to provide covalently attached bioactive compound. therefore, the present study needs more cell attachment for the future tissue engineering; also, it has managed to develop a new technique utilizing fibrin gel coating on the treated pan fiber. this study also determines a change in biophysical surface appearance, surface chemistry and biomechanical properties, as well as bioactivity and biocompatibility of the pan fibers before and after undergoing surface treatment followed by fibrin coating. surface treated and fibrin coated electrospun polyacrylonitrile fiber for endothelial cell growth... 309 2. materials and methods 2.1. materials polyacrylonitrile (pan; mw=150,000; 53.06 g/mol), n,n-dimethylformamide (dmf; 73.10 g/mol) and acrylamide (am; 71.08g/mol) were bought from aldrich chemical and were used without further purification. other materials are sodium hydroxide (naoh; sigma aldrich, usa), sodium carbonate (na2co3; sigma aldrich; usa), fibrinogen and thrombin (sigma aldrich; usa). human umbilical vein endothelial cells (huvec) were purchased from promo-cell (c-12200, heidelberg, germany). 2.2. pan fiber preparation the random pan fibers were developed through electrospinning under optimum conditions. briefly, 14% (w/v) of pan was dissolved in a dmf solution. this solution was loaded into a 5 ml glass syringe and the flow rate of 2 ml.h −1 was controlled by the syringe pump. a high voltage (21 kv) was applied using a high voltage regulated dc power supply (model es 30p-5w, gamma high voltage research, ormond beach, fl). a piece of aluminum foil was placed towards the tip at the distance of 12 cm as grounded collector. afterwards, the electrospun fibers were peeled off from the aluminum foil and the residual solvent was released in an oven at 40 o c. the dried electrospun fibers were then stored in the desiccator prior to further processes. the electrospun pan fibers mat was cut by 2 cm x 2 cm. one percent w/v aqueous solution of na2co3 was prepared. the pan fiber was dropped into boiling na2co3 at temperature 100 o c for 5 minutes and then it was rinsed with plenty of water until ph~7. the substrate then was dried at ambient temperature (t~25 o c) for 24 hours. this step was repeated for naoh surface treatment. the pan fiber was sterilized by immersing it in 10% penicillin-streptomycin under ultraviolet light (uv) for 15 minutes. next, the substrate was dried under uv for 1 hour. in 24 well plates, fibrin gels were prepared using 500 μl/well of fibrinogen solution (2.0 mg.ml -1 ) made in phosphate buffered saline (pbs). this solution was directly mixed with 1 mg / 500 μl (100 u.ml -1 in pbs) for polymerization process of fibrinogen into fibrin. the pan fiber after na2co3 and naoh treatment was transferred and deposited in the prepared fibrin gel. the fibrin-coated pan fiber was then stored in an incubator at 4 o c overnight. afterwards, the substrate was dried on the filter paper at room temperature in sterile conditions. 2.3. surface characterization the morphology of the fabricated scaffolds before and after the surface treatment and the coated pan fibers were observed by a high-resolution field emission scanning electron microscope (fesem; zeiss supra 35vp), operated at an acceleration voltage of 10 kv. the fibers diameters were measured by the sem (jeol, jsm-6390lv) micrographs measured randomly within an average of 50 fibers. the pan fiber composition before and after the surface treatment and coating was determined by the energy dispersive x-ray spectroscopy (eds). the fourier transformed infrared (ftir) spectra of the fibers were collected on the perkin–elmer spectrum 2000 in order to analyze the chemical structure of the electrospun pan fiber over a range of 400-4000 cm -1 with typically 32 scans at a resolution of 4 cm -1 . 310 n.s. rashidi, i. sukmana, a. mataram, n. jasmawati, m.r.m. rofi, m.r.a. kadir also, the crystallinity of the sample was characterized by x-ray diffraction (xrd) (d5000, siemens, germany) in a thin film mode at 40 kv and 40 ma. the data were recorded in 2θ range using cukα radiation of 1.5406 å. the water contact angle of the pan fibers before and after the surface treatment were measured using vca optima (ast product, inc.) mounted with a ccd camera. the fiber was placed on the sample stage and a drop of ultrapure water (milli-q, 2 μl) was dropped to the surface for contact angle measurement. five points per fiber were measured to determine the mean value of the water contact angle (n = 3). 2.4 mechanical properties the samples were cut using a punch die (3 mm wide and 3 cm length) and tested using a tensile machine (instron 8845) with a load cell capacity of 10kn. the appropriate specimen gripping as shown in fig.1 (i) is required to prevent the fibers from breaking or slipping at the grips. according to astm d882, the samples were mounted and had a 25 mm gage length. the samples were prepared for testing at a crosshead speed of 5 mm/min at ambient conditions. the initial modulus, ultimate strength and elongation at ultimate strength were measured. 2.5 endothelial cell responses human umbilical vein endothelial cells (huvecs) were cultured in endothelial cell growth (promo cell) supplemented by 1% antibiotic penicillin-streptomycin (gibco, usa). the cells were replaced twice a week and cultures were maintained at 37 o c in the incubator containing 5% co2. huvec between passages 3 and 5 were used in all experiments. after coating process, the sterile fibrin-coated pan fibers treated with na2co3 and naoh were seeded with huvecs at a density of 3x10 4 cells/cm 2 in 24 well-plates and observed for day 1 and day 3. cell morphology was studied using the axio vert a1 inverted microscope and the sem (carl zeiss). the substrates were fixed in 2% glutaraldehyde (0.1 m phosphate buffered) for 2 h and then washed with ethanol series before being observed under the sem. for staining procedure, huvec-seeded-on-fibers were gently washed with pbs (3 times) and fixed with formaldehyde solution (3.75% wt/v) in pbs for 20 min. the sample was then washed three times with pbs and then permeabilized with a triton x-100 solution (0.5% v/v in pbs) for 15 minutes. the sample was finally rinsed three times in pbs and stained with hoechst 33258 (1:10,000 dilution, catalog #b2883; sigma aldrich) for 1 hour at room temperature and in dark. to assess the intracellular of huvecs, the sample then was washed with pbs three times before being stained with alexa fluor 488, life technologies for 1 hour. finally, after washing with pbs, the sample was conserved with 3 ml per well of pbs and observed under microscope. the mtt assay was used as a measure of relative cell viability. huvecs were harvested and seeded onto the samples at 1x10 4 cells/cm 2 for a specified time of 1, 3, 5 and 7 days in 96-well plate with endothelial cell specific medium changes every 2 days. the cell viability was evaluated using the mtt assay (mtt; sigma), in which 100 ml of mtt (5mg/ml) was added to each well and incubated at 37 o c for 4 h. at the end of the assay, the blue formazan reaction product was dissolved by adding dmso. the absorbance was measured at 570 nm using a microplate reader (thermo scientific; us). surface treated and fibrin coated electrospun polyacrylonitrile fiber for endothelial cell growth... 311 3. results 3.1. morphology of untreated and treated pan fiber fesem images of the electrospun pan fibers are to investigate the effect of the surface treatment upon the fiber morphology structure presented in fig. 1. fig. 1 fesem micrograph and distribution of untreated pan fiber (a, b); fiber treated with na2co3 (c, d) as well as with naoh (e, f). na2co3 (g) and naoh (h) treated fiber then coated with fibrin, (i) mounting tab for tensile test (j) stress strain curve pan control table 1 mechanical properties of pan before surface treatment and pan after surface treatment with na2co3 and naoh mechanical properties pan before surface treatment pan surface treatment with na2co3 pan surface treatment with naoh tensile strength (mpa) 3.77±0.68 2.46±0.73 2.17±0.73 tensile modulus (mpa) 1.35±0.32 0.90±0.11 0.41±0.37 failure strain 35.32±16.63 31.91±13.91 23.60±9.03 fig. 1(a) shows that the micrographs of the untreated pan fibers obtained by electrospinning have bead-free and homogenous morphology. the pan fiber surfaces also seem featureless and smooth as shown in greater detail in the figure. the morphology of the pan after the surface treatment with na2co3 shown in fig. 1(c) has an uneven surface after treatment. as indicated by the arrows, the pan fiber after naoh as shown in fig. 1(e) surface treatment reveals ripples and it is rougher compared with na2co3 surface treatment. the surface treatment processes were proved to give pan fibers with different diameter range before the surface treatment (afd (average fiber diameter): 2.06 ±0.54; maxfd (maximum fiber diameter): 4.15 and minfd (minimum fiber diameter):1.66). next, the pan fiber after the surface treatment with na2co3 becomes (afd (average fiber 312 n.s. rashidi, i. sukmana, a. mataram, n. jasmawati, m.r.m. rofi, m.r.a. kadir diameter): 2.44 ± 0.79; maxfd (maximum fiber diameter): 4.21 and minfd (minimum fiber diameter):1.05) while the pan fiber after the surface treatment with naoh becomes afd (average fiber diameter): 1.75 ± 0.48; maxfd (maximum fiber diameter): 3.32 and minfd (minimum fiber diameter):0.83). the measurements show that the electrospun pan fiber diameter increases for 15% once the surface is treated with na2co3 while it decreases for 18% after being treated with naoh. fig. 1(g-h) shows that the pan fiber after being treated by na2co3 and naoh is fully coated with fibrin gel. the eds results report the existence of fibrin with significant changes of nitrogen and oxygen element of the pan fiber after the coating process. increasing in the carbon composition confirms the existence of fibrin. the coating is fully covering the cellular fibers due to the structure of fibrinogen transformed into fibrin gel, deposited on the surface and forming a mesh of fibrils obviously in-between the junction fibers. 3.2. mechanical properties of untreated and treated pan fiber table 1 summarizes the tensile properties of the electrospun pan fibers before and after reaction with na2co3 and naoh. it is noted that there are significant differences between the pan fibers before and after the surface treatment with na2co3 and naoh for tensile strength and tensile modulus. after the surface treatment, there were in total decreases in tensile strength, tensile modulus and strain to failure due to the loss of chain orientation. tensile strength of the pan before and after being treated with na2co3 and naoh is 3.77mpa, 2.46mpa and 2.17mpa. the decrease of tensile strength after na2co3 and naoh surface treatment of the pan fiber is as much as 34% and 42%, respectively. tensile modulus of the substrate decreases from 1.35mpa to 0.9mpa and 0.41mpa while strain decreases for 9% and 4% after being treated with na2co3 and naoh, respectively; while the value of the pan fiber before and after being treated with na2co3 and naoh is 35%, 31% and 23%. in theory, the tensile modulus decreases after the surface treatment because of the random pan fiber would become gradually aligned during uniaxial tensile test. this also leads to decreasing pan fiber diameter after the surface treatment, causing decrease in tensile strength; thus, the fiber will easily fracture. 3.3. surface characterization of untreated and treated pan fiber surface characterization of the treated and untreated pan fiber is presented in fig. 2. ftir spectra of the pan fibers, before and after treatment are included in fig. 2(a). the control pan molecule consists of functional groups such as methyl (ch3) and nitrile (c≡n). it is found that the surface treatment in the presence of na2co3 and naoh includes the stage of carboxyl, amide and aldehyde. the pan fiber shows a characteristic peak at 2260-2210 cm -1 before and after treatment with naoh and na2co3 indicating the likely and expected presence of acrylonitrile due to nitrile stretch. the alkali reaction of na2co3 and naoh will attack the chain of nitrile to become carboxyl group. the interface between these reactions involves amide formation, as can be seen in the spectra of 3400-3250 cm -1 for 1 o and 2 o amide and then proceeds through a carbonyl group which can be proven by conversion of -cn to -coo groups. atomic ratio (%) c=59.40 n=36.70 o=3.90 surface treated and fibrin coated electrospun polyacrylonitrile fiber for endothelial cell growth... 313 fig. 2 (a) ftir, (b) xrd, and (c) contact angle results of pan fiber, pan treated with na2co3 and pan treated with naoh briefly, the peak at 1645 cm -1 is assigned to carbonyl stretching, but the peak broadened after being treated into 1652 cm -1 for naoh and 1649 cm -1 for na2co3 surface treatment. in conjunction with the peaks at 1320 cm -1 and 1000 cm -1 range (due in part to c-o stretch), 1760-1665 cm -1 (due to c-o-c bend), and broad absorption around 3300 cm -1 and 2500 cm -1 (due to -oh stretch) the carbonyl peak seems to confirm the presence of acrylate (h c=c(ch)co r) as a co-monomer for naoh surface treatment. this carboxylic acid formation will lead to the formation of secondary amines in which the hydroxyl group has been replaced by amine. however, in the presence of na2co3 surface treatment, it neither includes the stage of amide formation for second degree nor results in the complete exhaustion of nitrile groups in a pan considering that na2co3 is not a very strong and reducing agent like naoh; hence it will neither fully attack nor harm the substrate. the obtained xrd patterns are shown in fig. 2(b). a number of crystalline peaks are observed for pure pan, which can be attributed to the semicrystalline structure of this polymer. the polymer exhibits a diffraction peak at 16.9°, which is the typical peak for a polyacrylonitrile polymer. two equatorial peaks are shown with one at 2θ = 29.5å corresponding to a spacing of d ≈3.03 å from the (1 1 0) reflection and the other at 2θ = 17.0å corresponding to a spacing of d ≈5.3 å from the (1 0 0) reflection. these peaks are common to the fiber diffraction pattern of pan with hexagonal crystal system [21]. the weak peak of the pan control is shown from the diffraction pattern with the value of 2θ at 17.0 o . this proves that the fibers fabricated using electrospinning give limited crystallinity. this low crystallinity leads to stretched pan chains solidified rapidly after elongation, preventing crystal formation in the electrospun pan fibers. 314 n.s. rashidi, i. sukmana, a. mataram, n. jasmawati, m.r.m. rofi, m.r.a. kadir in contrast, the pan with na2co3 and naoh surface treatment as increased in crystallinity even after a short-timing treatment shows two diffraction peaks indexed with values of 2θ of 17.0 o and 29.5 o . the crystallinities of the naoh-treated fiber are higher than the pan control and the pan after being treated with na2co3. the main factor depends on the alkali concentrations selected for the treatment and the type of fiber studied [22]. the crystallinity factor will lead to the structural changes of the pan fibers after surface treatments; thus, it will influence the course of their entire degradation process. the contact angle of the pan-modified surfaces treated via na2co3 and naoh as well as the control are shown in figure 2(c). it indicates that the pans treated by na2co3 and naoh are more hydrophilic (a significant lower contact angle) than found in an untreated pan. the contact angle of the pan surface of the pan is reduced from 115° to 88° and 64° after being treated with na2co3 and naoh, respectively. these results clearly indicate that the untreated pan fibers have the least attraction toward deionized water. however, this is improved upon by the surface treatment. the water contact angle of the pan surface gradually decreases after being treated with na2co3 and naoh. it is noteworthy that the receding contact angle decreases for all the pan fibers after the surface treatment. 3.4. endothelial cell adhesion and proliferation once huvecs are seeded onto both untreated and surface-treated and coated pan fibrous scaffolds in presence of fibrin, the cell attachment is evaluated at day 1 and day 3 and presented in fig. 3. after day 1, huvecs cultured on pan fibers are rounded in phenotype for all fibers (fig. 3 (a,d,g)). this is due to the short timing of culture. huvecs seem to be securely attached and spread on the surface in a flatten formation, regardless of the surface treatments and surface coating after day 3. it is evident that the cells are more actively spreading across the modified surfaces (fig. 3 (e) and (h)) while the cells remain separately on pan control even for day 3. this suggests that the huvecs adhered to pan treated with na2co3 and naoh and then coated with fibrin surface are higher than those found on the pan control at the time points of 1 and 3 days. this is proven by staining after day 3 culture (fig. 3 (f,i)) which confirms that not only the confluent huvecs layer is formed, but also that the cells are oriented completely along the aligned fiber direction and that they exhibit an elongated morphology for both surface treated fibers coated with fibrin. the metabolic behavior and cell proliferation are characterized by mtt assay. fig. 3(i) depicts the proliferation of huvecs on the treated pan fiber scaffold in comparison with that on the gold standard, tissue culture treated polystyrene (tcps) and the pan untreated. absorbance of the pan treated with na2co3 and naoh coated with fibrin increases by increasing time as shown in fig. 3(i). however, the pan control increases absorbance slowly because of the reduced toxic of fluorine coming from dmf which is a pan solution during fabrication of fibers. the mtt absorbance of pan after being treated with na2co3-fibrin coated shows increases throughout the testing period while the pan treated with naoh-fibrin coated slowly increases from day 5 to day 7 with absorbance value 0.38± 0.03 and 0.39±0.02, respectively. the pan treated with naohfibrin coated preceding the pan treated with na2co3-fibrin coated shows as much as 4% on day 3 while for day 5 and day 7 absorbance shows 0.37±0.02 and 0.38±0.02, respectively, for the pan treated with na2co3. surface treated and fibrin coated electrospun polyacrylonitrile fiber for endothelial cell growth... 315 fig. 3 sem and images of huvecs cultured on an untreated fiber as control (a,b,c); na2co3 treated and fibrin coated (d,e,f); and naoh treated and fibrin coated (g,h,i) after 1 and 3 days of study. letters “a” indicate huvec cells. mtt test on cytotoxicity of pan before and after surface treated to huvecs also presented (i). “tcps” indicate tissue culture treated polystyrene 316 n.s. rashidi, i. sukmana, a. mataram, n. jasmawati, m.r.m. rofi, m.r.a. kadir 4. discussions the pan fiber has been used for membrane applications [7]. in this study, the pan fiber is prepared to promote endothelial cell adhesion and proliferation. the surface treatment and bioactive coating of polymer fiber may lead to endothelial cell proliferation thus encouraging endothelial network or endothelialization, as presented elsewhere [23]. the objective of the study is to examine the effect of bioactive coating on the pan fiber to the huvec responses, which complements the previous study by groth et al. that used the pan fiber for fibroblast cell for wound healing application [5]. it has been previously shown that the use of na2co3 and naoh removes the natural surface wax thereby exposing the underlying texture of substrates [24]. this improves the wetting and wicking properties of the material which in turn increases the material surfaces for better surfacing coating [25]. in this experiment, it is observed that the pan fiber shows increased hydrophilicity after being treated with na2co3 and naoh thus improving coating of fibrin gel on pan fibers. the bioactive protein, in this case fibrin, has become a positive surface charged at neutral ph; this will attract negative charge on the treated pan sample [19]. the fibrin coating is able to attract huvecs on the pan fiber since fibrin has directs cellular responses through specific receptor [26]. many studies have already discussed naoh surface treatment on various substrates [14, 24]. afra et al. showed porous surface and rough morphology on pet substrate after surface treatment of 2 hours with naoh [13]. a higher concentration of naoh and longer period of surface treatment will cause degradation of surface fiber [19] or grooved serrated surfaces. it has been reported that the substrate with very high surface roughness decreases cell adhesion onto its surface [27]. therefore, in this study the lower concentration for both treatments are used. there are no significant differences in morphology on the pan surface after being treated with both naoh and na2co3 since the concentration for both treatments is as low as 0.1 and 0.2 molar of na2co3and naoh, respectively. even though the morphology of the pan fiber after the surface treatment does not give much difference, the diameter of fibers is affected. this is due to the transformation of nitrile to carboxylic group of the pan surface which may result in a decrease of fiber diameter. however, incomplete exhaustion of nitrile group for na2co3 surface treatment gives an advantage to na2co3 compared to naoh surface treatment in terms of mechanical properties since the fiber diameter is strongly related to strength of the fiber [28]. tensile strength of the pan fiber after na2co3 treatment is higher than naoh surface treatment but still lower than pan before the surface treatment. this is because the fiber’s exposure to the heated surface treatment makes the fiber give an emission and auto exhaustion to the environment [29]. alteration in physical characteristics of fibers will influence tensile properties due to increasing shrinkage and density of fibers [14]. the decreasing result for tensile modulus of pan after treatment with na2co3 and naoh shows that the pan surface is relatively ductile due to exhaustions of nitrile group. this has also led to the changes of crystallinity because acrylonitrile is a reactive chemical that polymerizes spontaneously when heated in the presence of a strong alkali. result of mechanical properties depends on the material used. instead of synthetic fiber, previous study used natural fiber for base material [29]. alteration in ductility of specific material after removal of impurities has modified mechanical result [28]. however, tensile strength in this experiment before and after the surface treatment is still in the range of coronary artery which is from 1.40 mpa to 11.14 mpa [23]. surface treated and fibrin coated electrospun polyacrylonitrile fiber for endothelial cell growth... 317 cytotoxicity test on pan fibers after the surface treatment with na2co3 and naoh shows these fibers are suitable for growth of huvecs, attaining increasing in cell viability until day 7 of culture. these observations suggest that the fibrin assisted increases proliferation of huvecs and allows the proliferation of the cells into the interior of the pan fiber and the uniform distribution of fibrin coated on pan fiber after the surface treatment with na2co3 and naoh has enhanced proliferation of infiltrated cells throughout the fiber volume. the sem micrographs of huvecs on pan control and pan treated with na2co3fibrin coated and naoh scaffolds obtained on day 1 of culture show a normal morphology of cell growth on the fibers. huvec cells disconnects to individual cells, round phenotype for all fibers. this is due to the shortened time of culture. this cell behavior is the same as in the previous report [8], where fibers have low surface density and large inter fiber which does not encourage adhesion of the cells across the nearest fibers in short time seeding. huvecs attach and spread more on pan substrate after surface treatments than pan control by day 3. also, huvecs are numerous and well-spread, forming monolayer cell on the pan treated with naoh scaffolds while reaching sub confluence. some cells aggregate along the fibers on pan treated with na2co3 as confirmed by double-staining with alexa fluor 488 for the cell cytoskeleton (i.e., acting filaments) and with hoechst 33258 for nuclei. these observations point to the significance of the huvecs adhesion increase due to better hydrophilicity caused by the pan fiber surface treatment process subsequently coated with fibrin. it also shows that fibrin was immobilized on the pan, which is important for strong cell adhesion [30]. fibrin contains many integration-binding sites and this reason makes it easier for cells to adhere to coated pan fiber [31]. formation of focal adhesions only occurs if the ligands can withstand cell contractile forces as acting stresses fibers while focal adhesions are critical for cell survival. this indicates that culturing cells on native substrates proves that cell attachment, spreading and growth enhanced, depends on the substrate, which has been reported elsewhere [32]. sem and staining micrographic observations support the trend of huvecs proliferation quantified by the mtt assays. 5. conclusions this study has attempted to attach huvecs on pan via two different surface treatments with na2co3 and naoh, then coated with fibrin gel. the attachment of huvecs on pan after na2co3 surface treatment can be another option for wet chemical treatment rather than common naoh since na2co3 surface treatment gives improvement in wettability and mechanical properties. the presence of carboxyl functions on pan fiber after surface treatment can be an advantage since fibrin can be covalently coupled via a simple wet chemistry. the fibrin-coated pan after the surface treatment enhanced endothelialization, as observed in spreading cell morphology and increasing cell proliferation. this is another successful finding of the pan fiber on cells after another research which has proved it for fibroblast cells. acknowledgements: i. sukmana is supported by universitas lampung and kemenristekdikti indonesia grants. the rest of the authors are supported by the malaysian ministry of higher education (mohe) 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three-dimensional environment, biomacromolecules, 8, pp. 864-873. 32. sukmana, i., kadir, m.r.a., djuansjah, j.r.p., 2013, in vitro angiogenesis assay for the guidance of microvessel containing multi-cellular lumen formation, advanced science letters, 19, pp. 3547-3550. 8346 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 671 684 https://doi.org/10.22190/fume211228019h © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined in a permeable channel jafar hasnain1, hina ghias satti1, mariam sheikh2, zaheer abbas3 1bahria university islamabad campus, pakistan 2university of sialkot, pakistan 3the islamia university of bahawalpur, pakistan abstract. the effects of slips and porosity on the channel walls in the flow of a heatabsorbing/generating dusty ferrofluid streaming through a porous medium are investigated in this article. the channel is upright and subjected to a transverse magnetic flux along with thermal radiation. kerosene with magnetite is used as the base fluid. the basic equations of the channel flow, which seem dimensional, are redesigned in a dimensionless manner utilizing non-dimensional variables. the variable separable method approach is used to solve the obtained equations analytically. the graphs demonstrate the behavior of these parameters on the flow fields, skin friction, and heat transmission rate, and are explained briefly. results reveal that the flow velocity for heat-generating fluids is greater than the heat-absorbing liquids. the fluid velocity upsurges with the improved values of the velocity slip parameter. the heat-generating dusty liquid has a higher heat transmission rate as compared to heatabsorbing dusty liquid. key words: ferro-fluid, dust particles, oscillatory flow, suction/injection, heat absorption/generation 1. introduction many researchers have concentrated in recent decades on integrating nano or micrometersized artefacts in the base liquids because of the increased applications in areas such as lubrication, vibration damping, pressure seals for compressors and blowers, oscillation damping, seals and cooling of loudspeakers, chemical reactors, and several others. the combination of milli or micrometer-sized particles (dust particles) in the base liquids creates dusty fluid. at the same time, nanofluid is formed by combining nanometer-sized particles into the base fluid. magnetic nanofluid or ferrofluid is a subcategory of nanofluids built on tiny received: december 28, 2021 / accepted april 12, 2022 corresponding author: jafar hasnain bahria university, shangrilla road, e-8, islamabad, pakistan e-mail: jafar.buic@bahria.edu.pk 672 j. hasnain, h. g. satti, m. sheikh, z. abbas magnetic particles with only one magnetic domain. when there is no magnetic field in the proximity of these particles, they act similarly to ordinary metallic particles. sunil et al. [1] reported that for very high values of the magnetic parameter, ferrofluid reacts like a usual fluid while examining the influence of dust particles on ferrofluid in the presence of the magnetic field. sekar and raju [2] analyzed the stability of soret effects on dusty ferrofluid. they reported that the system was considered unstable due to the absorption of the porous media and dust particle parameters. in the existence of heat and magnetic field, a numerical result of dusty nanofluid flow was found by sulochana and sundeep [3] along with a stretching/ shrinking cylinder. they noticed that the rise in dust particles' volume fraction deteriorates the heat transfer rate. majeed et al. [4] discussed the flow, including dust and ferro particles with porous media over an extending sheet with heat. they concluded that if the values of the ferromagnetic parameter and eckert number increase, so do the temperature profiles for both the ferrofluid and the dust phase. gireesha et al. [5] considered hall's effects, including magnetic field and heat generation/absorption on two-phase dusty-nanofluid with the observation that on the thermal distribution, hall and magnetic parameters have opposite effects. raizah [6] analyzed a hybrid nanofluid having dust particles moving through an enclosure containing two heated fins. he claimed that at the center of the top sheet, there are the lowest values for the local nusselt number. hatami and jing [7] used an asymmetric wavy channel to describe the nanofluid flow. they concluded that the brownian motion parameter affects the concentration of nanomaterials. azam et al. [8] studied the axisymmetric flow of a non-newtonian nanofluid in the presence of activation energy and reported that as the activation energy parameter was increased, the concentration of nanoparticles increased. kaneez et al. [9] reported the numerical result of the flow of dustconcentrated micropolar fluid having hybrid nano-particles and presented the result that the parameter of the dust fleck association performs a crucial part in the fluid temperature field. the impact of activation energy on cross nanofluid in axisymmetric flow was explored numerically by azam et al. [10]. one of the significant findings was that the concentration of nanoparticles increases as the activation energy increases. nanjundappa et al. [11] theoretically investigated the flow of dusty ferrofluid flow with the perception that in comparison to very small magnetic intensity and ferromagnetic surfaces, strong paramagnetic surfaces have a greater steadying impact on ferro-thermalconvection. mousavi el al. [12] found the dual solution for the flow of a hybrid nanofluid over a thin needle with thermal radiation, noting that the first solution was always thinner than the second. in a cross nanofluid with non-linear radiation, azam et al. [13] considered gyrotatic microorganisms. they concluded that an increase in lewis's number lowered the microorganism field. the fluid field boundary conditions play an essential role in measuring fluid flows in various industrial and medical systems. the dearth of the no-slip state is very obvious in polymer melts that often experience microscopic wall slippage. slip state, either velocity or thermal, is essential in cleaning interval cavities and artificial heart valves, as well as shear skin, hysteresis effect, and spurt. hayat et al. [14] highlighted the influence of mhd and slip in a channel on nanofluid flow and obtained an analytical solution. kamel et al. [15] worked on the two-phase stream of dusty liquid in a planar channel with slip boundary condition. the findings suggested that the condition of slip has a significant impact under certain concentration ranges. guria [16] applied the periodic suction on the vertical channel with radiation and slip boundary condition and found that the slip effect has been noticed to minimize the velocity. panaseti and georgiou [17] discussed viscoplastic fluid flow in the channel in the slip regime with the observation that there has huge spike in the mild slip study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined … 673 region. slip parameter reduces the radial velocity was reported by kashyap et al. [18] while studying the flow of ucm fluid past a channel having a chemical reaction and velocity slip boundary condition. saleem et al. [19] investigated the peristaltic flow of mhd jeffery fluid over a tapered channel with velocity second slip and concluded that a rise in slip parameters decreases the size of the trapping bolus. the heat generation/absorption investigation is carried out under a variety of conditions. fluids go through chemical reactions that are either exothermic or endothermic in these situations. when heat energy is obtained from radioactive waste and water is heated in a solar collector, heat generation/absorption procedures occur. it is also used in packed-bed reactors, radiative cooling of molten steel, semi-conductor wafers, food storage, and underground disposal of nuclear waste, as well as electronic devices. the generation or absorption of heat determines both the fluid flow and the heat transfer rate in the moving fluids. malik et al. [20] discussed the impact of thermal conductivity and heat generation/absorption on williamson fluid flow over a stretching cylinder. they concluded that the heat source increases the temperature profile, whereas the heat sink shows the opposite behavior. pandey and kumar [21] reported that the temperature profiles are expanded according to each heat generation/ absorption parameter value while analyzing mhd nanofluid flow through convergent/ divergent channels under heat generation/absorption. with heat generation/absorption, jha et al. [22] investigated fluid flow through the rotating channel. they found out that fluid’s temperature decreases with the rise in heat absorption while increasing with heat generation. mishra et al. [23] unveiled the effects of heat generation/absorption through stretching/ shrinking channels on mhd nanofluid flow. the highest heat generation/absorption parameter values in the shrinking convergent channel caused a spike in velocity and thermal profiles. prakesh et al. [24] performed an analysis of the mhd flow of viscous fluid within the vertical channel along with heat generation/absorption and variable properties. they reported that the heat transmission rate drops in the left plate with the incorporation of heat generation/ absorption and increases in the right plate. sobamowo [25] studied the temperature of a moving porous fin as a function of internal heat generation and found that the temperature of the fin decreases as internal heat generation increases. azam et al. [26] numerically investigated the flow of cross nanofluid under the influence of heat source and heat sink with the observation that heat source increases the temperature of the fluid. the utility of more than one phase streams analysis, in which solid particles are dispersed in a base liquid, is immense. strong fuel rocket nozzles, crude oil cleaning, nuclear reactor cooling, hydro cyclones, electrical gadgets, cryopreservation, and cancer treatments are descriptions of such flows. suction/injection of fluid through borders is critical in many areas and functional aspects such as aerodynamics, space sciences, film cooling, and boundary layer monitoring. taking these applications into consideration, this study aims to evaluate the effect of dusty-ferrofluid oscillatory flow in a channel. porous media, heat generation/absorption, the velocity slip boundary state, and thermal jump are also taken into account. 2. problem development the laminar flow is considered under the impact of radiative heat flux (rhf) in a permeable media, saturated with incompressible dusty fluid. the medium is drawn along the x-axis and the y-axis is kept perpendicular to the walls of the channel (fig. 1). the 674 j. hasnain, h. g. satti, m. sheikh, z. abbas magnetic field strength b0 is parallel to the y-axis. the fluid temperature on both walls is different. tf is the initial fluid temperature. t0 is the left wall temperature whereas tw is the temperature at the right wall. reynolds number re is considered very small and hence the induced magnetic field becomes insignificant. the fluid is drawn with velocity v0 from the left wall and injected from the right wall. with these assumptions and using boussinesq approximation, the flow equations are ([24], [27], [28]) fig. 1 geometry of the channel (1) (2) (3) with the initial and surface constraints (4) where tw= t0+( tft0) ei ω t. in the above expressions u, up are velocity components for fluid and dust particles in the x-direction, p is pressure, k is porous media permeability, k0 ( ) ( )20 0 0 0 0 1 , n fnf t y x nf yy p nf nf nf n f n k u v u p u u u u b u g t t k − = − + − + − − + −       ( ) ( )0 ,p pt u k u u= − ( ) ( ) ( ) ( ) * 0 1 , nf t o y yy y p p pnf nf nf k q t v t t q t t c c c   − = − + − ( ) ( ) ( ) ( ) ( ) ( ) * * 2 2 * * 1 0 1 0, at , ,0 , , 0, at , , , , 0, at , 0, , p f y p w y y p y u u t t y t y u u u t t t y t a t u u u t t t y t t = = = = =− = = − = = = = + =     study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined … 675 (=6πρυd) is stokes resistance, d is the average radius of a dust particle, g is gravitational constant, γ1 *, γ2 * are slip constants for velocity, α1 *, α2 * are slip constants for temperature, q* is a heat source/sink and a is the width of the channel. the subscripts x, y, t represent the derivatives with respect to x, y, and t respectively. assuming fluid to be thin optically and have low density relatively, the expression for rhf q is given as [29] (5) where ξ is the coefficient of mean radiative absorption. the expressions for effective values viscosity μnf, density ρnf, thermal diffusivity αnf, thermal expansion βnf, specific heat capacitance (ρc)nf, thermal conductivity knf and electrical conductivity σnf for ferrofluid are expressed as [9] (6) here ϕ is the nanomaterial volume fraction. the subscripts nf, f, s stands for nanofluid, fluid and nanoparticles respectively. the thermophysical properties for nanomaterials and base fluid are given in table 1. table 1 thermo-physical properties of nfs and nanoparticles ([30], [31]) liquids and nanoparticles  (kg/m3) cp (jkg−1/k−1) k (wm−1k−1) *10−5 (k−1)  (s/m) kerosene 783 2090 0.145 99 6*10-10 magnetite 5180 670 9.7 0.5 25000 copper 8933 385 401 1.67 59.6*106 silver 10500 235 429 1.89 6.3*107 the forthcoming non-dimensional quantities are used. 2 04 ( ).yq t t= − ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 2.5 * * * * , 1 , , 1 2 2 , 1 , 2 3 1 1 , 1 , . 2 1 nf f nf nf f s nf p nf s f f snf p pnf s f s f f s nf s nf f s f f k c k k k kk c c k k k k k                           = = − + = − + − − = = − + + − − = − + = + = + − − 0 0 2 0 0 2 2 0 , , , , , , , , , re , pr , p p f f f f p f f f f ut tx y u tu x y u t u a a u t t a u cn k ak ua da m l ka k a        − = = = = = = − = = = = = 676 j. hasnain, h. g. satti, m. sheikh, z. abbas (7) the flow equations along with the surface constraints in non-dimensional form can be written as (8) (9) (10) with surface constraints as (11) the bars are discarded for simplicity. the parameters h, gr, l, m, n, p*, pr, q, s, s, α1, α2, γ1, γ2 and λ are hartman number, garshof number, particle concentration, particle mass, radiation, dimensionless pressure, prandtl number, heat source parameter, shape factor for porous zone, a parameter of suction, slip parameters for velocity and temperature and oscillation amplitude for pressure respectively. 3. method of solution the solution of the eqs. (8) (10) is obtained for pure oscillatory flow by letting (12) substituting the values from eq. (12) into eqs. (8-11) we obtain: (13) (14) 22 2 * 2 02 * 2 2 * * * * 020 1 2 1 2 1 2 1 2 (t t )4 , , gr , , , , , , , . f f f f f f f f f f f g aa q a a p n q p k k u u a bv a s h a a a a                 − = = = = − = = = = = = 2 2 * 4 3 1 1 2 1 1 2 1 1 1 1 re ,t x y yy p h q s l l u p s u u u u gr q q q q q q q q q   = − + + − + + + +     re ( ) ,p t pm u u u= − 2 5 5 6repr pr ,t y yyq s q q n q− = + +     ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 1 0, 1 at , ,0 , , 0, at , 1, , , 0, at , 0, . p i t y p y y p y u u y t y u u u e y t t u u u y t t = = = = = − = = − = = = = =           ( ) ( ) ( ) ( ) ( ) ( ) * 0 0 0 , , , , , , . i t i t x i t i t p p p e u y t u y e u y t u y e y t y e − = = = =        ( ) ( ) 2 0 0 8 0 0 3 7 1 1 , yy y u s u q u gr q q q + − = − −   ( ) 0 0 , 1 re p u u i m = +  study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined … 677 (15) with (16) on solving equation (15) with constraints (16), the expression for fluid temperature is obtained as (17) the expressions for fluid and dusty particles velocities are obtained after solving eqs. (13) and (14) respectively along with the constraints (16) and given as (18) (19) the quantities of interest from the engineering point of view are surface friction cf and nusselt number nu, which are expressed as (20) where τf = ρnf υf uy at y=1 represents the shear stress of fluid at the right wall of the channel. and (25) where qw=-knf ty at y=1 represents the heat flux at the right wall of the channel. the constants that appear in the above expressions are given below: ( ) ( ) 2 6 0 5 0 9 0pr 0, yy y q s q q+ + =   ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 2 0 0 2 0 0 1 0 0 1 0 0, 1 at , ,0 , , 0, at , 1, , , 0, at , 0, . p i t y p y y p y u u y t y u u u e y t t u u u y t t = = = = = − = = − = = = = =           ( ) ( )11 12 1 2, , q y q yi ty t e c e c e= + ( ) ( )17 18 11 12 3 4 19 20 21, , q y q y q y q yi tu y t e c e c e q q e q e= + + + + ( ) ( )17 18 11 12 3 4 19 20 21 1 , . 1 re q y q y q y q yi t pu y t e c e c e q q e q e i m = + + + + +   ( )17 18 11 12 3 17 4 18 20 11 21 121 , f q y q y q y q yi t f y y nf nf a c u e c q e c q e q q e q q e u = = = = + + +    ( ) ( )11 12 1 11 2 120,1 0 , q y q yi tw y y nf f aq nu e c q e c q e k t t = = = − = − + −  2.5 1 2 3 4 5 6 22 24 7 1 2 8 9 1 2 1 2 2 2 2 10 6 9 10 6 9 10 5 11 12 13 6 6 ( ) 1 , (1 ) , 1 , , 1 , , ( ) re , re , pr re, 1 re q 4 q 4 q pr , q , q , 2 2 nf p s nfs s f f f p f f c k q q q q q q c k h q l i m q q q q i q n q i q q q q i m q q q q q q q q q q                    = − + = − = − + = = − + =   = = + + + = + −  +  − − − − + − = = = 11 12 1 11 14 14 1 12 15 2 11 16 2 12 1 14 15 13 16 2 2 2 2 2 2 7 7 8 7 7 7 8 713 1 17 18 14 15 13 16 1 , q 1 , (1 ), (1 ), , q q q q 4q q 4q qq , , , q q q q 2 2 q q q q q q e q q e q c q s q s q s q s c q q     = − = − = − = − = − − − + − + +− = = = − 678 j. hasnain, h. g. satti, m. sheikh, z. abbas 4. results and discussion the expressions for the velocity of both fluid and dust particles and temperature of fluid are obtained by solving flow equations with the surface constraints analytically using the method of separating variables. graphs are plotted to express the effects of the physical parameters on the velocity and temperature fields. figure 2 is plotted to present the behavior of fluid velocity u and particle velocity up with the variation in s for both heat absorbing and generating liquids. the figure shows that both u and up are increasing with a rise in suction parameter s (>0) however u decreases with a rise in injection parameter s (<0). the physical fact is that cold fluid particles are injected into the porous channel through the channel's hot wall, and these heated fluid particles are then removed. as the temperature in the channel drops, the convection circulation weakens. as a result, the velocity decreases. the flow is parabolic for higher values of the suction parameter while it shifts towards the right wall for increasing injection. the flow field for heat-generating liquid is higher than the heat-absorbing liquid. fig. 2 variation in velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for s fig. 3 displays the variation in u and up with γ1, γ2 for two values of q. this figure suggests that both the slip parameters have an increasing impact on both flow velocities. however, the fluid and dust particle velocities for a heat-absorbing case are less than the heat-generating case. when velocities of fluid and dusty particles are graphed through fig. 4 to analyze the impact of ϕ, it is found that the ϕ has a decreasing effect on the velocities for heatgenerating liquid. while it has an increasing effect on the case of heat-absorbing liquid. 17 18 11 12 1 3 7 2 3 7 19 20 21 22 1 172 2 2 2 2 1 8 11 7 8 7 11 12 7 8 7 12 23 1 18 24 19 20 1 11 21 1 12 25 2 17 26 2 18 27 19 20 2 11 21 , , , 1 , 1 , q (1 ) (1 q ), q (1 q ), q (1 q ), q (1 q ) q ( q q q q c grq q c grq q q q q q q q q q q q q q s q q q q q s q q q q q q e e q q e e         − − = = = = − − + − + = − = + − + − = + = + = + + + 2 12 24 26 23 27 22 27 24 25 3 4 23 25 22 26 23 25 22 26 1 q ), . q q q q q q q q c c q q q q q q q q + − − = = − − study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined … 679 fig. 3 variation in velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for γ1, γ2 fig. 4 variation in velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for ϕ a comparative analysis is shown in fig. 5(a, b) for the velocities of fluid and dust particles by dispersing different kinds of nanoparticles namely magnetite (magnetic particles) and copper and silver (non-magnetic particles). these figures show that the velocities of liquid and dust particles are lower when magnetic particles are immersed in the fluid as compared to non-magnetic particles. however, the velocities are higher when silver nanoparticles are used in the liquid. the reason for the lower value of velocities for the magnetic nanoparticles is the lorentz force which arises due to the magnetic field affecting the magnetic particles as compared to non-magnetic particles. fig. 5 comparison of velocities of (a) liquid u(y,t) (b) dust particle up(y,t) for different nanoparticles 680 j. hasnain, h. g. satti, m. sheikh, z. abbas fig. 6 is plotted to show the impacts of ϕ, α1, α2 and s on fluid temperature θ. the decreasing impact of ϕ for both heat absorbing and heat-generating liquids can be noticed in fig. 6(a) with the higher temperature for the heat-generating liquid. the effects of slip parameters α1, α2 on the temperature of the dusty liquid can see in fig. 6(b) which shows an increasing behavior with the rise in α1, α2 for the heat-generating liquid. the variation is reversed near the right wall for heat-absorbing liquid. q rises, the fluid becomes more heated, and so the velocity and temperature rise. the temperature of both heat generating/absorbing dusty liquids increase when suction is increased while it reduces with a raise in injection (see fig. 6(c)). in all of the figures above, the magnitude of the fluid's velocity and temperature in the case of heat generation is greater than that of heat absorption. this finding is consistent with expectations: as q rises, the fluid becomes more heated, and so the velocity and temperature rise. fig. 7 exhibits the variation in cf at the right wall for both heat-absorbing and generating fluids. surface friction versus s and γ2 are presented in fig. 7(a) and 7(b) respectively. it can be observed that an increase in ϕ raises the magnitude of surface friction for the heatabsorbing case while it decreases for the heat-generating liquid. the surface friction increases with suction at the wall whilst it reduces with a rise in injection. (fig. 7(a)). the magnitude of surface friction shows a decreasing trend for the heat-generating liquid with the rise in ϕ and γ2. fig. 6 change in fluid temperature θ(y,t) for (a) ϕ (b) α1, α2 (c) s study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined … 681 fig. 7 surface friction cf at right wall for ϕ (a) versus s (b) versus γ2 heat transmission rates versus and γ2 at the right wall for different values of ϕ are presented in fig. 8. an increase in ϕ and s (<0) reduce the heat transmission rate while s (>0) has an enhancing impact on the rate of heat transfer for both heat-absorbing and generating liquids (fig. 8(a)). fig. 8(b) shows a slight increase in the transmission rate for heat-absorbing liquid with α2. however, a significant improvement in the heat transmission rate is noticed for heat-generating liquid. fig. 8 heat transfer rate nu at right wall for ϕ (a) versus s (b) versus α2 fig. 9 is plotted to present the oscillatory behavior for the flow fields for the case of heat-generating fluids with variation in suction/injection parameter. it can be seen the oscillation is higher for the case of suction for all flow fields. fig. 10 presents the agreeable results of the present study with the results of already published literature for limiting values of parameters s=ϕ=γ1=γ2 =α1=α2=q=0. s 682 j. hasnain, h. g. satti, m. sheikh, z. abbas fig. 9 oscillatory behavior of (a) velocities of liquid u(y,t) (b) dust particle up(y,t) (c) fluid temperature θ(y,t) versus t for s fig. 10 comparison of present results with existing literature for (a) velocities of liquid u(y,t) (b) dust particle up(y,t) (c) fluid temperature θ(y,t) study of double slip boundary condition on the oscillatory flow of dusty ferrofluid confined … 683 5. concluding remarks the velocity and thermal slip effects on the oscillatory flow of a ferrofluid in the presence of dusty particles are studied. the heat source/sink is taken into account when investigating heat transfer features. for the flow query, an exact solution is sought. the numerical values are graphed to demonstrate the effect of the physical parameters concerned. the results are summarized below: ▪ when the suction parameter is increased, both the fluid and particle velocities increase; but, when the injection parameter is increased, the fluid velocity decreases. ▪ the fluid velocity of the heat-generating liquid is greater than that of the heatabsorbing liquid. ▪ velocity slip parameters have a growing influence on both flow velocities. ▪ the temperature of the dusty fluid rises as the thermal slip parameters of the heatgenerating fluid increase. ▪ the temperature of both heat generating/absorbing dusty liquids rises as suction rises, while it falls as injection rises. ▪ for heat-generating liquid, there is a significant increase in heat transmission rate. ▪ oscillation in the flow fields is higher for the case of suction. ▪ velocities of liquid and dust particles are higher 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in mhd mixed convection flow of a ferrofluid along a vertical channel, plos one, 10(11), e0141213. 29. cogley, a.c., vincenti, w.g., gilles, s.e., 1968, differential approximation for radiative transfer in a nongrey gas near equilibrium,” aiaa j., 6(3), pp. 551–553. 30. kandelousi, m.s., 2014, effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition, eur. phys. j. plus, 129, 248. 31. rashad, a.m., 2017, impact of thermal radiation on mhd slip flow of a ferrofluid over a non-isothermal wedge, j. magn. magn. mater., 422, pp. 25–31. 10848 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 377 385 https://doi.org/10.22190/fume230428025f © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a circular sector vibration system in a porous medium: a fractal-fractional model and he’s frequency formulation guangqing feng school of mathematics and information science, henan polytechnic university, jiaozuo, china orcid id: guangqing feng https://orcid.org/0000-0001-7851-1458 abstract. a circular sector is commonly used in a linkage mechanism, and its frequency property plays an important role in optimization of the linkage mechanism. fast insight into its vibration property with simple calculation is very meaningful in scientific research. this paper studies the vibration of the circular sector in a porous medium (e.g. water), and a fractal-fractional oscillator is established using the two-scale fractal derivative. he’s frequency formula and ma’s modification are used to elucidate the circular sector’s periodic property in a porous medium, the results show that the fractal dimension of the porous medium plays an important role in vibration attenuation. key words: he’s frequency formula, ma’s modification, circular sector oscillator, fractal variational theory, two-scale transform 1. introduction a circular sector is commonly used in a linkage mechanism, it might vibrate in water or oil, and its vibration property has caught much attention in mechanical engineering [1]. there are many engineering applications in literature, such as the base of structures, the wipers of cars, smoothing filters and many other vibration systems [2-4]. the traditional vibration theory cannot effectively elucidate the effect of the porous medium’s geometry on the vibration property, now the condition is changed, because of the new born fractal vibration theory [5]. the fractal vibration theory can make up for the deficiency of the traditional vibration theory. let's take the vibration of a sector in the air as an example. in all previous vibration theories, the influence of air on vibration characteristics is considered as an air drag; however, the performance of thin air in a microgravity environment is different from that of near surface atmosphere. the vibration received: april 28, 2023 / accepted august 10, 2023 corresponding author: guangqing feng school of mathematics and information science, henan polytechnic university, shanyang district, 454011 jiaozuo, henan, china e-mail: fgqing@hpu.edu.cn https://orcid.org/0000-0001-7851-1458 378 g. feng model constructed by the fractal vibration theory can consider the influence of molecules’ size and distribution on vibration motion, the acceleration in the fractal space is a combination of the damping force and the inertia force in the traditional vibration system [6-10], so the vibration system in a porous medium(e.g. air, water ) can be modelled by a fractal-fractional model without considering the damping effect. in general, the fractal vibration follows a fractal variational theory, that is, the vibration system in a porous medium obeys the fractal energy conservation law, the fractal kinetic energy and the potential energy are changed during the vibrating process, but their total energy keeps unchanged. fractal nonlinear systems truly describe the dynamic problems of engineering science, and its academic research has greatly expanded the field of human cognition. many phenomena can be explained by the fractal theory, which makes us realize that the world is totally discontinuous and nonlinear. the fractal nonlinear vibration can be closer to practical problems in both depth and breadth. it is very important to study the analytical or approximate solutions of nonlinear vibration equations that can provide deep insight into the essence of general properties of a fractal vibrating system. so, there are many analytical and numerical methods to find an approximate solution of a differential equation involving fractal-fractional derivatives, such as the variational iteration method [11-13], the homotopy perturbation [14,15], the hamiltonian approach [16], and the taylor series method [17]. the most important property of a nonlinear system is the frequency-amplitude relationship, so how to estimate quickly its relationship is an urgent problem in practical applications. many researchers devoted their efforts to studying fractal calculus that provides a powerful tool to characterizing the periodic behavior of a nonlinear oscillator [18-20]. chun-hui he suggested a fractal nano/microelectromechanical (n/mems) system [21], ji-huan he gave a tutorial review on fractal space and fractional calculus [22,23]. he, et al. studied the fractal duffing oscillator with arbitrary conditions [24]. ji-huan he suggested a frequency formula for a conservation nonlinear oscillator [25] and it was further improved to an extremely simple formula in refs. [26,27], and it has been widely used to solve nonlinear oscillator problems, for examples, the n/mems oscillator [28], the attachment oscillator [29], the tangent oscillator [30], the fractal undamped duffing equation [31], the large amplitude vibration system [32] and the vibration system on a porous foundation [33]. many modifications were appeared in literature to deal with more complex vibration systems [34-37]. the hamiltonian-based frequency formula is a modification of he’s frequency formula [38]. the frequency formula starts with two arbitrary guesses for the frequency, and two residual integrals must be computed to estimate a more accurate frequency. this paper applies he’s frequency formula and ma’s modification to study the fractal vibration systems and uses the fractal circular sector oscillator as an example. the results show that the two methods are very effective for fractal nonlinear oscillators. 2. fractal circular sector oscillator as the fractal variational theory is helpful in establishing a governing equation in a fractal space, it has become a significantly hot topic in both mathematics and mechanical engineering. many fractal variational principles were appeared for the internal temperature response of a porous concrete [39], the fractal benney-lin equation [40], the fractal shallow water wave [41], the fractal bogoyavlenskii system [42], the fractal schrodinger system [43], the fractal solitary wave [44] and the fractal economics [45]. a circular sector vibration system in a porous medium: a fractal-fractional model... 379 the variational formula of an oscillator in a fractal space is in the form     duf d du uj  −= )}()( 2 1 {)( 2 (1) where (du/d)2/2 is the fractal kinetic energy and f(u) is the potential energy. when f=-cosu, it is the well-known pendulum oscillator [46,47]; when f=u2/2+au4/4, it is the well-known duffing oscillator [48]. when f=gln(3r2/2-2rkcosu)/(2r), its variational formula is:     durkr r g d du uj  −−= )}cos2 2 3 ln( 2 )( 2 1 {)( 22 (2) it meets the following energy balance equation hurkr r g d du =−+ )cos2 2 3 ln( 2 )( 2 1 22  (3) where h is the hamiltonian constant. the fractal nonlinear oscillator can be obtained as follows 0 cos2 2 3 sin )( 2 = − + urkr ugk d du d d   (4) it is the fractal circular sector oscillator [49], where u is the angular displacement, r is the semicircular radius, g is the gravity acceleration, r̅ is the height of mass center, r̅=2rsin/(3),  indicates the semicircular angle, k is the dimensionless geometrical parameter, k= r̅/r=2sin/(3), as illustrated in fig. 1. fig. 1 circular sector oscillator in water by the two-scale transform [50], t=, the fractal circular sector oscillator of eq. (4) can be converted to an ordinary form as follows 0)0(,)0(,0 cos2 2 3 sin 2 === − + uau urkr ugk u (5) where the derivative of the function u with respect to t. 380 g. feng 3. he’s frequency formula he’s frequency formula [26,27] is an effective tool to learn the frequency-amplitude relationship of fractal oscillators in a timely and efficient way. consider the following nonlinear oscillator 0)0(,)0(,0)( ===+ uauuhu (6) where h(u) is a nonlinear function, and it satisfies h(u)/u>0. he’s frequency formula is： audu udh 5.0 2 )( = = (7) or na nah )(2 = (8) where n is a constant, 0<n≤1. n is recommended as√3/2 for nonlinear oscillators in [26, 27]. convert eq. (5) to the following approximate form 02 =+ uu  (9) where the square of the frequency can be calculated by eq.(8): )] 2 3 cos(2 2 3 [ 2 3 ) 2 3 sin( 2 2 arkra agk − = (10) the approximate solution of eq. (4) can be formed as follows )cos()(   au = (11) 4. ma’s modification ma obtained a simplified form of hamiltonian-based frequency formula as follows [38]: 22 2 )1(5.0 )()( al lahah − − = (12) where dh(u)/du=h(u), and l is recommend as 2-0.5 [38], the simplified hamiltonian-based frequency formula converts to )] 2 ()([ 4 2 2 a hah a −= (13) according to eq.(5), we have )cos2 2 3 ln( 2 )( 2 urkr r g uh −= (14) a circular sector vibration system in a porous medium: a fractal-fractional model... 381 ma’s modification leads to the following result )2/cos(2 2 3 )cos(2 2 3 ln 2 2 2 akr akr ra g − − = (15) the approximate solution of eq. (4) can be formed as following )cos()(   = au (16) the obtained approximate analytical solution must be compared with the numerical ones to demonstrate the validity of the above approaches. therefore, figs. 2-4 demonstrate the analytical solution from the simplified hamiltonian-based frequency method (shf), together with the numerical solution (ns) and the solution from he’s frequency method (hf) for different values of a, r and α. the approximate periodic solutions agree well with exact numerical results. the value of α changes from /6 to /3, the change amount is =/6, and the period increment is t=4.5 in fig. 2. the value of α changes from /2 to 2/3, the change amount is =/6, and the period increment is t=14 in fig. 3. the larger the value of α, the more obvious the period change. it concludes that by increasing the semicircular angle α with constant semicircular radius r, the period of the oscillation will increase considerably. in fig. 4, the value of the amplitude a determines the maximum value of u. by comparing figs. 3 and 4, it is found that by reducing the semicircular radius r, the period of the oscillation will decrease. fig. 2 comparisons of the exact solutions (ns) with the approximate solutions based on eqs. (11) (hf) and (16) (shf) for α ranging from /6 to /3 0 10 20 30 40 50 60 70 80 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t u r = 15 , a =  / 8 ,  =  / 6 n s h f s h f 0 10 20 30 40 50 60 70 80 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t u r = 15 , a =  / 8 ,  =  / 3 n s h f s h f 382 g. feng fig. 3 comparisons of the exact solutions (ns) with the approximate solutions based on eqs. (11) (hf) and (16) (shf) for α ranging from /2 to 2/3 fig. 4 comparisons of the exact solutions (ns) with the approximate solutions based on eqs. (11) (hf) and (16) (shf) for amplitude a ranging from /8 to /6 based on eqs. (10) and (15), results are extracted from the analytical solution and are shown in fig. 5. it visually shows the relationship between a, α and the frequency when the value of r is fixed. an increase in the semicircle radius r or semicircle α results in a decrease in frequency, which implies an increase in the oscillation period. five sequences of the fractal parameter θ are considered for eq. (4) with the fixed parameters r=10, a=π/6, α=π/4 and shown together in fig. 6. numerical simulations indicate that the oscillation frequency becomes faster and the vibration attenuation occurs greater for increasing the values of the fractal exponent θ. 0 10 20 30 40 50 60 70 80 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t u r = 15 , a =  / 8 ,  =  / 2 n s h f s h f 0 10 20 30 40 50 60 70 80 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t u r= 15 , a =  / 8 ,  = 2  / 3 n s h f s h f 0 10 20 30 40 50 60 70 80 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t u r = 10 , a =  / 8 ,  =  / 4 r=10, a=/6, =/4r=10, a=/6, =/4 n s h f s h f 0 10 20 30 40 50 60 70 80 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 t u r = 10 , a =  / 6,  =  / 4 n s h f s h f a circular sector vibration system in a porous medium: a fractal-fractional model... 383 fig. 5 sensitivity analysis of frequency (0<a<π/4, 0<α<π/2) with r = 10 (left) and r = 15 (right) fig. 6 eq. (4) at different  values for r=10, a=π/6, α=π/4 5. conclusion the most important property of a fractal nonlinear system is the relationship between frequency and amplitude, which plays a crucial role in construction engineering. this paper presents he’s frequency formula and ma’s modification of hamiltonian-based frequency formula and discusses how to quickly calculate the approximate frequency of the fractal circular sector in a porous medium. the result shows that two methods are accurate tools to quickly calculating the periodic properties of fractal nonlinear oscillators, and that the fractal dimension of a porous medium plays an important role in vibration attenuation. both the frequency formulations will play an important role in the fractal nonlinear vibration theory and open up numerous opportunities to elucidate the effect of the porous medium’s geometry on the vibration property, and also probably pave a new avenue for optimizing the circular sector in a nanofluid [51] where the nanoparticles’ size and distribution can be effectively modelled by the fractal-fractional model. 0 0.2 0.4 0.6 0.8 0 0.4 0.8 1.2 1.6 0.16 0.17 0.18 0.19 0.2 0.21 0.22 aα fr e q u e n cy 0.17 0.18 0.19 0.2 0.21 0 0.2 0.4 0.6 0 0.4 0.8 1.2 1.6 0.1 0.11 0.12 0.13 0.14 0.15 aα fre qu en cy 0.11 0.115 0.12 0.125 0.13 0.135 0 5 10 15 20 25 30 35 40 -0.4 -0.2 0 0.2 0.4 0.6  u θ=0.7 θ=0.9 θ=1 θ=1.1 θ=1.3 384 g. feng references 1. ahad, f.e., shi, d., aftab, s.m., waqas, h.m., 2019, an improved fourier series method for vibration analysis of moderately thick annular and circular sector plates subjected to elastic boundary conditions, journal of vibroengineering, 18(6), pp.3841-3857. 2. aboutalebi, r., eshaghi, m., taghvaeipour, a., 2021, nonlinear vibration analysis of circular/annular/sector sandwich panels incorporating magnetorheological fluid operating in the post-yield region, journal of intelligent material systems and structures, 32(7), pp. 781-796. 3. hiraoka, t., 2020, generation 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quasi-molecular simulations to confirm the criterion for formation of debris, proposed in 1958 by rabinowicz [4]. the work of aghababaei, warner and molinari improves our understanding of adhesive wear but at the same time puts many new questions. the present paper is devoted to the discussion of possible generalizations of the rabinowicz-molinari criterion and its application to a variety of systems differing by the interactions in the interface and by the material properties (elastic and elastoplastic) and structure (homogeneous and layered systems). a generalization of the rabinowicz-molinari criterion for systems with arbitrary complex contact configuration is suggested which does not use the notion of "asperity". key words: plasticity, adhesion, critical length, adhesive wear, layered systems, functionally gradient materials, rabinowicz’ criterion 1. introduction among the basic tribological phenomena of contact, adhesion, friction, lubrication and wear, wear remains the least scientifically understood. this may be due to the complexity and diversity of the processes leading to wear. in particular, wear is not a purely contact mechanical phenomenon, but necessarily also includes fracture phenomena within the material and material transport with the resulting very broad problem of the "third body". at the same time, wear remains one of the most important tribological phenomena in practice, affecting all aspects of our lives and current technologies. wear significantly determines the life time of mechanical systems; it is a key factor in matters of technical safety. wear not only affects machines and mechanical constructions, it is e.g. also an unsatisfactorily solved problem in medicine: many implants, especially artificial joints, have to be replaced after approx. 10 years. wear is also an important issue in terms of received december 26, 2017 / accepted january 29, 2018 corresponding author: valentin l. popov technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: v.popov@tu-berlin.de 30 v. popov emission of wear particles into the environment (such as brakes and tires). in all of these areas, very great efforts are undertaken to get wear under control. up to now, this has largely been done in a purely empirical way. the most common basis for wear calculations is formed by a law which was formulated in 1953 by archard [1] and bears his name. according to archard's law, the wear volume is proportional to the sliding length, the normal force, and inversely proportional to the hardness of the material. the coefficient of proportionality is called wear coefficient. but the devil is in precisely this "coefficient", because empirically measured values of the adhesive coefficient of wear can differ by 7 decimal orders of magnitude [2], which invalidates the influence of hardness. accordingly, general recommendations made on the basis of the archard’s law have a very limited scope of applicability. for example, there is a widely spread opinion that the higher the hardness, the lower the wear, since the hardness stands in the denominator of the archard’s equation. however, kragelsky [3] formulated an almost exactly opposite principle for minimizing wear – the principle of a positive hardness gradient, which states that the surface layers must be softer than the lower layers, otherwise catastrophic wear occurs. these two statements, which seemingly exclude each other, both have empirical confirmation, which only emphasizes that the physics of the wear process has so far been poorly understood at its core. in the last few decades, however, some ideas have been collected which shed new light on the physics of wear. thus, molinari with collaborators picked up and confirmed an old idea by rabinowicz (1958), [4], about the physical mechanism that determines the size of the wear particles and also controls the transition from mild to catastrophic wear. in the criterion of rabinowicz and the theory of molinari et al. based on that [5], it is the interplay of plasticity and adhesion, which leads to the appearance of a characteristic length: if a micro contact is smaller than the characteristic length, it is plastically deformed; if it is larger than the characteristic length, wear particles form. the existence of these two scenarios has been independently confirmed by popov and dimaki using the method of movable cellular automata and has been observed in molecular dynamics simulations [6]. combined with advanced numerical simulation methods of contact between rough surfaces [7], this new understanding advances the old idea of rabinowicz to a new paradigm [8]. the rabinowicz-molinari criterion applies to homogeneous systems. however, the surface region of contacts in most low-wear technical systems is not homogeneous. through the work of gerve et al. [9] and in the last decade, especially by m. scherge and co-workers [10], the role of very thin chemically modified surface layers has been demonstrated for systems with "minimal wear" (including, for example, combustion engines). further, the rabinowicz-molinari criterion uses the notion of “asperity”. however, the development of the contact mechanics of rough surfaces has shown that this notion is poorly defined. thus, it is important to search for formulations of the same physical principles without using the notion of asperity. in the present paper, the basic physical idea underlying the criterion of rabinowicz-molinari will be applied to heterogeneous media. further, “asperity-free” concepts will be discussed. adhesive wear: generalized rabinowicz' criteria 31 2. rabinowicz' criterion for formation of wear debris 2.1. original rabinowicz' criterion for homogeneous media we start with the reproduction of the well-known derivation of the rabinowicz criterion [2, 4, 11] for a homogeneous medium. if two micro heterogeneities collide and form a welded bridge, as suggested by bowden and tabor [12], (see fig. 1) they are plastically deformed, and the maximum stress that can be achieved is of the order of the hardness of the material. in this state, the stored elastic energy is proportional to the third power of contact size: 3 2 0 2 σ d g uel  . (1) this energy can relax by creating a wear particle. the process of detaching a wear particle can, however, only occur if the stored elastic energy exceeds the energy 2 adhu w d   (2) which is needed to create new free surfaces (where w is the work of adhesion per unit area). it follows that only particles larger than some critical size can be detached: 2 0 2g w d    . (3) fig. 1 welded joint of size d created due to contact and shear of two asperities. note that the eq. (3) predicts only the existence of a lower bound of the size of wear particles. thus, there also should be some mechanism suppressing the appearance of too large particles. rabinowicz did not make any suggestions for such a mechanism. however, a possible mechanism could be very simple and follow from the same eq. (3). indeed, as noticed in [13], if some particle has size much larger than (3), it is energetically favorable for it to disintegrate into two smaller ones. this process can only continue until the critical size (3) is reached. thus, the wear particles should all have a size of the same order of magnitude as the critical length given by eq. (3). 2.2. modified rabinowicz' criterion for frictional interaction in the interface consider the same asperity contact as shown in fig. 1, but assume now that the tangential stress  needed to induce macroscopic relative sliding of asperities is determined by the force of friction with the coefficient of friction : =p, where p is the pressure acting in the considered micro contact. the elastic energy stored in the contact immediately before gross sliding can be estimated as 32 v. popov 3 22 2 μ d g p uel  (4) and the criterion (3) is modified as follows: 2 2 2g w d p     . (5) the main difference of this criterion from the classical rabinowicz' criterion is that the critical asperity size depends not only on material parameters but also on the pressure in that considered asperity. however, even in this case there exists some characteristic asperity size which can be estimated by substituting in (5) the average pressure in microcontacts, which in good approximation is given by [14] *1 2 p e z  . (6) the criterion for formation of particles thus takes the form 2 2 w d g z     , (7) with the additional constraint of (3), since the pressure cannot exceed 0. this criterion does not necessarily assume plastic behaviour and can also be applied to purely elastic media. 2.3. modified rabinowicz' criterion for contacts with friction and adhesion in the paper [15], a contact of two bodies has been considered, which interact by adhesion and friction forces at the same time. in the limit of very strong but short ranged adhesive interactions, it has been shown that the critical force at complete sliding is given by the simple equation 2 2 , ,( ) ( ( ) )x slip n jkr c cf a f a a a       , (8) where fn,jkr(a) is the normal force according to the jkr-theory for the corresponding profile [16, 17]. this can be done if 2 ,/ ( ) / ( ) 1c n jkr ca f a a eh     which is the case for typical material parameters of metals and junctions larger than about 1 m. in this case, in the first approximation we can assume that the surfaces are pressed against each other with a constant and high adhesive pressure c. we thus have a contact with the "flow stress" =c, and eq. (3) is directly applicable with the only substitution of 0=c. 3. wear in systems with a soft surface layer rabinowicz has formulated his criterion for homogeneous media. however, many tribological systems have a pronounced layered structure – either artificially designed or developed during tribological loading. in the present paper we repeat the arguments of rabinowicz for such layered systems and find the conditions for plastic smoothing and particle detachment in this case. we follow the presentation of preprint [18]. adhesive wear: generalized rabinowicz' criteria 33 consider an elastic medium with elastic shear modulus g0 covered with a soft elastoplastic layer of thickness h having shear modulus gc and the tangential yield stress c. this layer can be deposited to the surface artificially or it can appear naturally through mechanically induced chemical reactions of the base material with surrounding substances (lubricant, counter-body, air and so on) [9, 10]. assume that due to normal loading and tangential sliding a junction with the diameter d is formed, and that the diameter d is much larger than the thickness of the layer (the opposite case corresponds to a homogeneous medium and is covered by eq.(3)). the components of the stress tensor in the near surroundings of the junction will be of the order of magnitude of c. if the elastic energy stored in the system is not enough for creating new surfaces with an area of the order of d 2 than the only possible process will be plastic smoothing as illustrated in detail in the paper [13]. in the opposite case, the elastic energy can be relaxed by detaching of a wear particle. in the case of debris formation, two limiting cases are possible: 3.1. detachment in the base material in this case, we basically can repeat the line of argument of rabinowicz. the stored elastic energy has the order of (c 2 /2g0)d 3 and the surface energy needed for formation of a wear particle is of the order of wd 2 where w is the work of separation of the base material. the formation of wear particle is possible if (c 2 /2g0)d 3 > wd 2 or 0 2 2 c g w d    (9) which coincides with the rabinowicz’ criterion (3). the only difference from the classical criterion is that the elastic modulus and energy of separation are those of the base material while the critical flow stress is that of the surface layer. 3.2. detachment inside the surface layer in this case, the elastic energy which is released due to particle detachment is on the order of (c 2 /2gc)d 2 h and the energy needed for detachment cd 2 , where c is the work of adhesion inside the soft layer. the formation of particles is thus possible if (c 2 /2gc)d 2 h > cd 2 or 2 2 c c c c g h h     . (10) in this case, the fulfilment of criterion does not depend on the diameter of junction but depends solely on the thickness of the layer. if the thickness of the layer is smaller than the critical one, hc, then formation of particles is not possible, independently of the size of junctions. note that in this case only the properties of the softer surface layer do play a role. 3.3. criteria for formation of “flat” and “spherical” wear particles let us consider in detail the transition between the cases 1 and 2 discussed in the previous section. again consider a junction with some particular diameter d. the following cases are possible: 34 v. popov 1. 0 0 2 2 c g d    but 2 2 c c c g h    . in this case, formation of the in-layer particles is not possible but formation of the basematerial particles is possible. this is the classical “rabinowicz case”. 2. 0 0 2 2 c g d    but 2 2 c c c g h    . (this case is possible if the elastic modulus of the surface layer is sufficiently smaller than that of the base material). in this case, the formation of “bulk” particles is not possible but surface-layer flat wear particles can be formed. 3. in the general case, one could suggest the following generalized estimation. assume that we have a junction of diameter d and detached is a particle of diameter d and thickness h. then, the elastic energy stored in the system is on the order of 2 2 0 2 2 , for 2 , for 2 c c el c c d h h h h h g g u d h h h g                 (11) the energy needed for formation of the above particle is on the order of 2 0 2 , for , for surf c d h h u d h h       (12) the formation of particles is possible if 2 2 2 0 0 2 2 2 , for 2 , for 2 c c c c c d h h h d h h g g d h d h h g                    (13) or 0 0 0 2 2 2 1 , for 2 , for cc c c c g g h h h h g g h h h                   (14) let us display these relations graphically on the plane (h, h), adhesive wear: generalized rabinowicz' criteria 35 fig. 2 schematic representation of conditions given by eq. (14). a completely “wear-less” sliding will occur if the following two conditions are fulfilled: 2 2 c c c g h    (15) and 0 0 02 2 [ ]c c c c d g g g       . (16) due to softness of the surface layer, the critical junction size can be made large enough so that the only condition which has to be observed would be that given by eq. (15). this explains the principle of “positive hardness gradient” as condition for wear-less sliding, which was formulated by kragelsky [3]. of course, even the process of plastic smoothing will lead to effective “wear” due to “squeezing out” of the surface layer. however, it was shown in [9] (see also [11], §17.5) that in this case the effective wear rate is proportional to the square of the ratio of the layer thickness to the linear size l of the frictional contact zone. the wear coefficient will thus be on the order of 2 adh h k l        (17) and can assume extremely small values. note that the smaller is the thickness of the surface layer the smaller is the wear coefficient. 36 v. popov 4. asperity free concepts for adhesive wear rabinowicz' criterion is based on the notion of "asperity". an important parameter for application of this criterion is the knowledge of the "asperity size". however, it is widely recognized that the notion of asperity is a poorly defined notion for real surfaces which often have roughness on many length scales. however, without properly defining the size of an asperity, the rabinowicz' criterion cannot be applied. looking at a contact configuration of bodies with fractal rough surfaces (fig. 3), we see more or less continuous clusters of contact areas instead of separated asperities. we would like to suggest a "modified rabinowicz’ criterion" which is largely independent of the definition of an asperity and is based on the ideas first proposed in [19]. a) b) c) d) fig. 3 numerical simulation of tangential contact between a rough surface and an elastic half-space: a) the surface topography; b) contact area at a given indentation depth. black color shows the stick regions and gray color the slip regions; c) tangential stress distribution in the contact; d) tangential displacement of elastic half-space. black areas show a rigid-body translation and gray areas the slip regions. adhesive wear: generalized rabinowicz' criteria 37 consider as illustration the contact of rough surfaces shown in fig. 3. a rough sphere with roughness having the hurst exponent 0.7 was generated according to the rules described in [20]. the indenter was pressed into the elastic half-space and then moved tangentially by a ux (0) smaller than the displacement corresponding to complete sliding [21]. both the normal indentation and tangential loading were simulated using the boundary element method as described in [7]. unlike in [7], we assumed that any two points of bodies are in the stick state as long as the local stress is smaller than a fixed critical value, c. after beginning of slip, the stress remained constant and equal to c, thus mimicking elastic-ideally-plastic behavior in the contact interface. the tangential contact comes into plastic state by overcoming the critical value c which in this context plays the role of the "yield stress" used by rabinowicz in eq. (1). now consider a circular region centered at an arbitrary point with an arbitrary diameter d. in fig. 3b and 3d, several examples of such regions with different positions and different diameters are shown with red circles. the macroscopic tangential stress in the selected circular area is equal to c, where  is the "filling factor" defined as the ratio of the real contact area in this circle to the area of the circle ~d 2 . assume further, that configuration of the contact in this area corresponds to the plateau of the stiffness, as described in [22]. then the contact acts as a complete contact. the elastic energy which would be released if a wear particle with the characteristic volume ~d 3 would detach, has then the order of magnitude ((c) 2 /2g)d 3 . if it is not enough for creating the free surface of the order of d 2 , the detaching cannot happen. thus the criterion for the possibility of detaching a wear particle with the size d is ((c) 2 /2g)d 3 > d 2 w or 2 2 ( )c g w d    . (18) in the most general case, elastic energy that will be relaxed by detaching the considered particle, can be estimated as 2 4( ) 4 ( ) c el h d u ga d   (19) where ah is the holm-radius of the considered contact configuration [23]. the condition for particle detachment can be written as 2 2 2 2 ( ) ( )h c d g w a d    . (20) generally, the dependence of the holm-radius on the diameter of the circle can only be determined numerically. thus, this equation has to be evaluated using numerical simulations of contact and local stiffness of various areas. note that for using eqs. (19) and (20) there is no need to define what an asperity is. by "probing" various positions and diameters, one can identify the material regions which "can potentially produce wear particles". however, in this concept the real contact area will play an essential role so that further ideas may be needed for determination of a robust criterion which does not depend on fine details of the power density of the surface roughness. a very interesting discussion of these aspects can be found in [24]. 38 v. popov 5. conclusions in the present paper, we applied the rabinowicz-molinari criterion for formation of wear particles for a variety of systems differing by the interactions in the interface and by the material properties (elastic and elastoplastic) and structure (homogeneous and layered systems). of special interest is the result that in the system with a soft layer, no critical size of contact does exist. instead, there appears some critical thickness. we further discuss a generalization of the rabinowicz-molinari criterion for systems with arbitrary complex contact configuration. this formulation does not use the notion of "asperity" and automatically includes "multi-contact" situations. acknowledgement: the author thanks m. popov for reading the draft paper and for useful comments, j.-f. molinari and r. pohrt for interesting discussions related to the present paper, and qiang li for help with preparation of fig. 3. the author acknowledges financial support of the deutsche forschungsgemeinschaft (dfg po 810-55-1). references 1. archard, j. f., 1953, contact and rubbing of flat surfaces, journal of applied physics, 24, pp. 981-988. 2. rabinowicz, e., 1995, friction and wear of materials. second edition, john wiley & sons, inc., 3. kragelski, i.v., 1965, friction and wear, butter worth. 4. rabinowicz, e., 1958, the effect of size on the looseness of wear fragments. wear, 2, pp. 4–8. 5. aghababaei, r., warner, d.h., molinari, j.-f., 2016, critical length scale controls adhesive wear mechanisms. nature communications. 7, 11816. 6. dmitriev, a.i., nikonov, a.y., österle, w., 2016, md sliding simulations of amorphous tribofilms consisting of either sio2 or carbon, lubricants 4 (3), pp. 1-24. 7. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17, pp. 334–340. 8. aghababaei, r., warner, d.h., molinari, j.-f., 2017, on the debris-level origins of adhesive wear, proceedings of the national academy of sciences, 114(30), pp. 7935–7940. 9. popov, v.l., smolin, i.yu., gervé, a., kehrwald, b., 2000, simulation of wear in combustion engines, computational materials science, 19, pp. 285-291. 10. scherge, m., shakhvorostov, d., pöhlmann, k., 2003, fundamental wear mechanism of metals, wear, 255, pp. 395-400. 11. popov, v.l., 2017, contact mechanics and friction. physical principles and applications. springer, berlin. 12. bowden, f.p., tabor, d., 2001, the friction and lubrication of solids, clarendon press. 13. popov, v.l., 2017, generalized rabinowicz’ criterion for adhesive wear for elliptic micro contacts, aip conference proceedings 1909, 020178. 14. hyun, s., pei, l., molinari, j.-f., robbins, m. o., 2004, finite-element analysis of contact between elastic self-affine surfaces, phys. rev. e 70, 026117. 15. popov, v.l., dimaki, a.v., 2017, friction in an adhesive tangential contact in the coulomb-dugdale approximation, the journal of adhesion 93 (14), pp. 1131-1145. 16. johnson, k.l., kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids. proceedings of the royal society of london, series a, 324, pp. 301-313. 17. popov, v.l., heß, m., willert, e., 2017, handbuch der kontaktmechanik. exakte lösungen axialsymmetrischer kontaktprobleme, springer, berlin. 18. popov, v.l., on the rabinowicz like criterion of formation of wear particles in a system with a soft surface layer, arxiv preprint arxiv:1712.06122, 2017 19. li, q., popov, v.l., 2017, on the possibility of frictional damping with reduced wear: a note on the applicability of archard s law of adhesive wear under conditions of fretting, physical mesomechanics, 20(5), pp. 91-95. adhesive wear: generalized rabinowicz' criteria 39 20. pohrt, r, popov, v.l., 2013, contact mechanics of rough spheres: crossover from fractal to hertzian behavior, 2013, 9741782013. 21. grzemba, b., pohrt, r., teidelt, e., popov, v.l., 2014, maximum micro-slip in tangential contact of randomly rough self-affine surfaces, wear, 309, pp. 256-258. 22. pohrt, r., v.l. popov, 2012, normal contact stiffness of elastic solids with fractal rough surfaces, physical review letters, 108(10), 104301. 23. holm r, holm e., 1958, electric contacts handbook. springer, berlin. 24. ciavarella, m., papangelo, a., 2017, discussion of “measuring and understanding contact area at the nanoscale: a review”(jacobs, tdb, and ashlie martini, a., 2017, asme appl. mech. rev., 69 (6), p. 060802), applied mechanics reviews, 69(6), 065502. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 245 256 doi: 10.22190/fume170505012n © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper decoupling control of the tito system supported by the dominant pole placement method udc 62-5:681.5 novak n. nedić 1 , saša lj. prodanović 2 , ljubiša m. dubonjić 1 1 university of kragujevac, faculty of mechanical and civil engineering in kraljevo, serbia 2 university of east sarajevo, faculty of mechanical engineering, bosnia and herzegovina abstract. appropriate approach to the nature of systems is a significant precondition for its successful control. an always actual issue of its mutual coupling is considered in this paper. a multivariable system with two-inputs and two-outputs (tito) is in the focus here. the dominant pole placement method is used in trying to tune the pid controllers that should support the decoupling control. the aim is to determine parameters of the pid controllers which, in combination with decoupler, can obtain a good dynamical behavior of the system. therefore, this kind of the centralized analytically obtained controller is used for object control. another goal is to simplify the tuning procedure of pid controllers and enlarge the possibility for introducing the given approach into practice. but the research results indicate that the proposed procedure leads to the usage of p controllers because they enable the best performances for the considered object. also, it is noticed that some differences from the usual rules in selection of the dominant poles gives better results. the research is supported by simulations and, therefore, the proposed method effectiveness, regarding the system behavior quality, is presented on several examples. key words: decoupling control, pid control, tito process, dominant pole placement method 1. introduction multivariable systems have been in focus of many research studies in recent decades. their decoupling has been studied intensively in [1-5]. neither type of decoupler is universal; hence which of them will provide for appropriate compensation of the mutual coupling depends on the object nature. in the present paper the static inverted decoupler is used for the investigated received may 05, 2017 / accepted july 03, 2017 corresponding author: saša lj. prodanović university of east sarajevo, faculty of mechanical engineering, vuka karadžića 30, 71123 east sarajevo, bosnia and herzegovina e-mail: sasa.prodanovic@ues.rs.ba 246 n. n. nedić, s. lj. prodanović, lj. m. dubonjić object, like in [6]. cantilever beam as an object of control is taken into consideration. its mathematical model is determined in [7]. here the electrohydraulic servosystem designed for structural testing is considered as a system with two inputs and two outputs (tito). the decoupling control enables taking this kind of system as a finite number (in this case two) of siso (single-input single-output) systems. having in mind this fact, a wider spectrum of methods can be used for the tuning of controller parameters. therefore, the dominant pole placement method has also significant place as one of the tuning rules. das et al. [8] tune pid controllers by using the guaranteed dominant pole placement method. investigation of this method for the time delay systems was performed in [9-12]. madady and reza-alikhani considered approaches for the first-order controller design using dominant pole placement, too [13]. besides many other procedures for pid controller tuning, åström and hägglund in [14] presented the dominant pole placement method for several kind of objects. filipović and nedić in [15] showed procedures for pi and pid controller design based on this way. q.-g. wang et al. [16] dealt with the fourth-order object but without zeros. nicolau [17] researched possibilities for pid controller design based on the pole placement technique in the combination with symmetrical optimum criterion. consideration of tracking performance for a continuous-time pid controller (tuned using this method) with anti-windup compensator was described in [18]. the decoupling control, that contains controller designed according to the dominant pole placement, was described and tested in [19]. a further step in the method implementation was made by rasouli et al. [20]. they made fractional order pole placement controller. extension of the original dominant pole placement method for controller design to the multivariable systems is presented in [21-24]. in contrast to the aforementioned research studies, the present paper deals with controller design for the tito object, whose decoupled loops are of the third-order with two left half plane zeros. 2. decoupling of object general transfer function matrix of the considered object is given by eq. (1): 11 12 21 22 g ( ) g ( ) g( ) g ( ) g ( ) s s s s s        (1) where gij(s) are elements of the transfer function matrix. the decoupling control strategy containing inverted decoupler in the combination with pid controllers is shown in fig. 1, where laplace operator s was omitted to make it simpler. taking into account [5,6], decoupler d(s) is calculated as a static decoupler using eq. (2), in order to avoid introducing of additional dynamic into the system: 12 1112 21 21 22 0 (0) 1 (0)1 ( ) ( ) ( ) 1 (0) 1 (0) s g gd s d s d s g g |                   (2) where dij(s) are off-diagonal elements of the decoupler transfer matrix. decoupling control of tito system supported by dominant pole placement method 247 apparent system of equations (3), that should be obtained after decoupling, enables considering of the tito system as a finite number of siso systems (in this case two siso systems q1(s) and q2(s)). fig. 1 inverted decoupling control for the tito object [1] actually, the inverted decoupling is applied because of its utilization of advantages of ideal (simple apparent system q(s)) and simplified decoupling (simple decoupler). hence, 1 2 ( ) 0 ( ) ( ) ( ) 0 ( ) q s q s g s d s q s          (3) controllers will be designed based on diagonal elements of eq. (3). 3. controller design general expression for the decentralized pid controller for the tito process is given by eq. (4) and its elements (two single loop controllers k1(s) and k2(s)) are presented with eq. (5): 1 2 ( ) ( ) ( ) 0 0 s s s k k k         (4) 1 1 1 1 2 2 2 2 ( ) ( ) i p d i p d k k s k k s s k k s k k s s         (5) where kp, ki and kd are proportional, integral and derivative controller gains, respectively. in the inverted decoupling, controllers are designed for the diagonal elements of q(s) and hence: q1(s)=g11(s) and q2(s)=g22(s). therefore, as previously stated, the pid controller design using the dominance pole placement method will be researched for the third-order transfer function with two left half plane zeros (eq. (6)): 2 2 1 0 3 2 2 1 0 ( ) a a a ii b s b s b g s s s s       (6) 248 n. n. nedić, s. lj. prodanović, lj. m. dubonjić where ai and bi are coefficients of the denominator and numerator, respectively. according to that, the characteristic equation of the single loop is expressed by eq. (7-9): 1 ( ) ( ) 0ii ig s k s   (7) 0      s ksksk sss bsbsb ipd 11 2 1 01 2 2 3 01 2 2 aaa 1 (8) 3 2 2 2 2 1 0 2 1 0 1 1 1( a a a ) ( ) ( ) 0d p is s s s b s b s b k s k s k             (9) equation (10) is a general form of the fourth-order characteristic equation. so, there are four poles: two conjugate complex eq. (11) and two real. since the pid controller has three parameters, three dominant poles should be determined. 2 2( ) ( ) ( ) 0n n n ns αω s βω s 2ξ s ω       (10) 2 1 ξjωξωs nn  (11) here ωn is natural frequency and ξ is damping coefficient, while  and  are parameters which serve for pole placement. equalization of the eq. (9) and eq. (10) and large mathematical transformations lead to expressions for the pid controller gains eq. (12): 2 2 (2 ) n p ξ α β ω a k b     0 4 b αβω k n i  (12) 2 2 3 2 1 2 1 1 2 2 0 2 0 1 2 2 2 0 2 2 0 2 1 2 1 0 1 2 (1 2 2 ) ( 2 ) (2 ) (2 ) n n d n n α β αβ b b ω a b b α β α ξ b ω a b k b b b ξ α β b b ω a b b ξ α β b ω a b b b b                    4. examples the proposed procedure is illustrated through three examples that have been examined to check its sensitivity to the model uncertainties and at the same time to start investigation of its applicability to the different objects. 4.1. example 1 electrohydraulic servosystem for structural testing is shown in fig. 2. its mathematical model was obtained by means of an appropriate identification procedure and given by eq. (13) [7]. the control system serves to enable defined load to the cantilever beam. intensity and character of the forces on the piston rods are characteristics that should be controlled by flow rates through the servovalves. forces f1r and f2r are reference values. values f1 and f2 from their transducers are object outputs (controlled variables). decoupling control of tito system supported by dominant pole placement method 249 fig. 2 double actuator electrohydraulic servosystem for structural testing (scheme) [7] 11 12 21 22 ( ) ( )1 ( ) ( ) ( )( ) g s g s g s g s g sδ s        2 4 4 3 7 2 7 9 11( ) 2.926 10 1.9152 10 1.2667 10 5.5825 10 4.7959 10g s s s s s          4 3 7 2 7 9 12 ( ) 3.8382 10 1.7068 10 8.3584 10 6.4967 10g s s s s         (13) 3 3 6 2 7 9 21( ) 4.4533 10 3.2461 10 1.4362 10 1.2403 10g s s s s         2 4 4 3 6 2 6 9 22 ( ) 2.506 10 1.6229 10 6.6134 10 3.0476 10 2.4813 10g s s s s s          5 2 4 4 3 6 2 8 6( ) 1.2308 10 6.993 10 1.5098 10 3.5504 10 8.2333 10δ s s s s s s            static inverted decoupler for the particular system eq. (13) calculated according to eq. (2) is: 0 1 1.35 ( ) 0.5 1s d s |         (14) fifth-order elements of the transfer matrix g11(s)/(s) and g22(s)/(s) were reduced to the third-order using matlab toolbox. effectiveness of the reducing procedure has been checked by comparison of step and sine responses of these elements. these graphics are shown in fig. 3. based on them, it is obvious that the reduced elements well represent identified transfer matrix. this is due to the appropriate matching of the step responses and excellent matching of the sine responses. 250 n. n. nedić, s. lj. prodanović, lj. m. dubonjić a) b) c) d) fig. 3 comparison of step (a, b) and sine (c, d) responses of identified [7] and reduced elements of the transfer matrix presented by eq. (13) laplace operator s was omitted in fig. 3 in order to improve its clarity. decoupling control of tito system supported by dominant pole placement method 251 reduced elements are given by: 2 .1 11 3 2 2 .1 22 3 2 191 593 75911 ( ) 14.3 5620.5 102 1041 39243 ( ) 16.8 5614.8 ex ex s s g s s s s s s g s s s s                   (15) appropriate choice of parameters α, β and ξ defines the position of the poles in the complex plane. the other coefficients are known from eq. (6). in the all three examples the following values of the parameters are taken α=12, β=1 and ξ=1. in this one, according to eq. (15) natural frequency is ωn=7.15 rad/s (for g11 ex.1 ) and ωn=8.4 rad/s (for g22 ex.1 ). controller parameters calculated from eq. (12) are: kp1=0.4866 ; ki1=0.4131 ; kp2=1.0706 ; ki2=1.5224 values for derivative gains are too high; knowing that they cause system instability, they are not taken into consideration for this system. this is a potential drawback of this procedure. 4.2. example 2 in this example, polynomial coefficients in the eq. (15) are increased for 20 % to obtain eq. (16). considering poles, two of three poles have been moved to the left in comparison with example 1. their movement has been carried out because of their wellknown influence to the system behavior. this and following example are used to examine the possibility for appropriate tuning of the controller when the mathematical model of the object is not completely accurate. this case is very often in practice due to changeable functioning conditions and also during process of identification. thus these two examples actually represent variants of example 1. 2 .2 11 3 2 2 .2 22 3 2 229.2 711.6 91093.2 ( ) 17.16 6744.6 122.4 1249.2 47091.6 ( ) 20.16 6737.76 ex ex s s g s s s s s s g s s s s                   (16) according to eq. (16), natural frequency is ωn= 8.58 rad/s (for g11 ex.2 ) and ωn= 10.08 rad/s (for g22 ex.2 ). afterwards, the controller gains from eq. (12) are: kp1=0.4866 ; ki1= 0.7139 ; kp2=1.0706 ; ki2= 2.6308 derivative gains are also too high for this system; that is the reason why they are omitted. 4.3. example 3 coefficients in the eq. (15) are decreased for 20 % in this case. here two of three poles have been moved to the right in comparison with example 1. now diagonal elements of the eq. (1), i.e. eq. (3) are given by eq. (17): 252 n. n. nedić, s. lj. prodanović, lj. m. dubonjić 2 .3 11 3 2 2 .3 22 3 2 171.9 533.7 68319.9 ( ) 12.87 5058.45 91.8 936.9 35318.7 ( ) 15.12 5053.32 ex ex s s g s s s s s s g s s s s                   (17) here, natural frequency is ωn= 6.435 rad/s (for g11 ex.3 ) and ωn= 7.56 rad/s (for g22 ex.3 ). from eq. (12) it follows: kp1=0.4866 ; ki1= 0.3012 ; kp2=1.0706 ; ki2= 1.1098 derivative gains have been avoided like in previous examples. 5. simulation results based on configuration in fig.1, the proposed decoupling control is simulated using matlab/simulink. simulations are carried out for the two cases regarding reference functions (signals) r1 and r2. in the first case (fig. 4) r1 is unit sine function and r2 is unit step function, and vice versa in the second case. fig. 4 block diagram of the control algorithm for electrohydraulic servosystem system responses and their enlarged views are shown in figs. 5 and 6. decoupling control of tito system supported by dominant pole placement method 253 a) b) fig. 5 a) forces on the cylinders (r1 unit sine function for f1, r2 unit step function for f2) b) enlarged view of characteristic response range 254 n. n. nedić, s. lj. prodanović, lj. m. dubonjić a) b) fig. 6 a) forces on the cylinders (r1 unit step function for f1, r2 unit sine function for f2) b) enlarged view of characteristic response range decoupling control of tito system supported by dominant pole placement method 255 these figures show responses for the four types of controllers in the combination with the static inverted decoupler and one response without decoupler that was controlled in [7]. it is noticeable that p controllers give the best reference tracking. this is confirmed by a very small deviation between reference signals r1 and r2 compared to the appropriate responses of the system with applied p controller. the described slight deviation is, in fact, an expected delay of the output in relation to the input signal. this fact cancels the aforementioned possible drawback of the proposed procedure because it is important that at least one type of controller can satisfy the defined requirements for the system dynamic behavior. moreover, it leaves the possibility of its application to other objects. the most appropriate value for proportional gain kp is obtained when non-dominant pole has 12 times higher absolute value of the real part than the three dominant poles. pi controllers give a lower quality of responses. observing the values of kp in the examined three examples, it is also noticeable that the p controller is the least sensitive to the model perturbations, i.e. model uncertainties. in comparison with [7] (the case without decoupling), there is an obvious improvement in the compensation of interaction between loops. 6. conclusions the proposed procedure for the pid controller design is extension of the dominant pole placement method to the third-order objects with two left half plane zeros. after calculating controller gains, the most suitable controller type can be chosen. it is proved that, in some control algorithms, the ratio between the absolute values of the real part of non-dominant and dominant poles should be greater than four, which is the value usually suggested in the literature. suitable reduction of the previously known (identified) transfer matrix, i.e. its diagonal elements, makes easier controller tuning. effectiveness of the applied reduction is proved as good because the designed controllers were tested on the identified (initial) model of the system and they enabled appropriate system behavior. the controllers tuned on the basis of the presented approach are compatible with the previously decoupled objects. this is confirmed on the tito electrohydraulic system for structural testing, where the p controller in the combination with static inverted decoupler enables good system performances, especially regarding reference tracking as well as cancellation of mutual coupling and reducing sensitivity to the mathematical model uncertainties. omitting of the derivative terms due to their high values for the considered cantilever beam does not necessarily be a rule for other systems. this should be the subject of future research studies. acknowledgements: the authors wish to express their gratitude to the serbian mpntr for partly financing of this paper through the project tr33026. references 1. nordfeldt, p., hägglund, t., 2006, decoupler and pid controller design of tito systems, j. process control, l (16), pp. 923-936. 2. vázquez, f., morilla, f., 2002, tuning decentralized pid controllers for mimo systems with decouplers, proceedings of the 15th ifac world congress, barcelona, spain. 256 n. n. nedić, s. lj. prodanović, lj. m. dubonjić 3. morilla f., vázquez, f., garrido, j., 2008, centralized pid control by decoupling for tito processes, proceedings of the 13th ieee international conference on emerging technologies and factory automation. hamburg, germany, pp. 1318-1325. 4. garrido, j., vázquez, f., morilla, f., hägglund, t., 2011, practical advantages of inverted decoupling, proceedings of the institution of mechanical engineers, part i: j. systems and control engineering, 225, pp. 977-992. 5. morilla, f., garrido, j., vázquez, f., 2013, control multivariable por desacoplo, revista iberoamericana de automática e informática industrial, 10, pp. 3-17, (in spanish). 6. prodanović, s., nedić, n., 2016, control improvement of a double actuator electrohydraulic servosystem for structural testing, in proceedings of the 15th youth symposium on experimental solid mechanics ysesm 2016, rimini, italia. 7. singer, g., meashio, y., 1995, analysis of a double actuator electrohydraulic system for structural testing, iee, savoy place, london. 8. das, s., halder, k., pan, i., ghosh, s., gupta, a., 2012, inverse optimal control formulation for guaranteed dominant pole placement with pi/pid controllers, international conference on computer communication and informatics (iccci -2012), coimbatore, india. 9. zítek, p., fišer, j., vyhlídal, t., 2013, dimensional analysis approach to dominant three-pole placement in delayed pid control loops, j. process control, 23, pp. 1063-1074. 10. branlea, i., petrovic, i. and peric, n., 2002, toolkit for pid dominant pole design, 9th international conference on electronics, circuits and systems, 3, pp. 1247-1250. 11. zítek, p., fišer, j., vyhlídal, t., 2012, ultimate-frequency based three-pole dominant placement in delayed pid control loop, proceedings of the 10-th ifac workshop on time delay systems, the international federation of automatic control northeastern university, boston, usa, pp.150-155. 12. li, y., sheng, a., qi, q., 2011, further results on guaranteed dominant pole placement with pid controllers, proceedings of the 30th chinese control conference, yantai, china, pp. 3756-3760. 13. madady, a., reza-alikhani, h.r., 2011, first-order controllers design employing dominant pole placement, 19th mediterranean conference on control and automation aquis corfu holiday palace, corfu, greece, pp.1498-1503. 14. åström, k.j., hägglund, t., 1995, pid controllers: theory, design and tuning, research triangle park, nc: instrumental society of america. 15. filipović, v.ž., nedić, n.n., 2008, pid controllers, university of kragujevac, faculty of mechanical engineering, kraljevo, (in serbian). 16. wang, q-g., zhang, z., åström, k.j., chek, l.s., 2009, guaranteed dominant pole placement with pid controllers, j. process control, 19, pp. 349-352. 17. nicolau, v., 2013, on pid controller design by combining pole placement technique with symmetrical optimum criterion, math. probl. eng., (2013), pp. 1-8. 18. sadalla, t., horla, d., 2015, analysis of simple anti-windup compensation in pole-placement control of a second order oscillatory system, measurement automation monitoring, 61(02), pp. 54-57. 19. nedić, n.n., prodanović, s.lj., dubonjić, lj.m., 2016, some considerations on the decoupling control of tito systems, proc. xiii international saum conference on systems, automatic control and measurements, niš, serbia. 20. rasouli, h., fatehi, a., zamanian, h., 2015, design and implementation of fractional order pole placement controller to control the magnetic flux in damavand tokamak, rev. sci. instrum., 86(033503), pp. 1-11. 21. zhang, y., wang, q-g., åström, k.j., 2000, dominant pole placement for multi-loop control systems, proceedings of the american control conference, chicago, illinois, pp. 1965-1969. 22. lee, j., edgar t.f., 2006, multiloop pi/pid control system improvement via adjusting the dominant pole or the peak amplitude ratio, chem. eng. sci., 61, pp. 1658-1666. 23. mokadam, h.r., patre, b.m., maghade, d.k., 2013, tuning of multivariable pi/pid controllers for tito processes using dominant pole placement approach, int. j. automation and control, 7(1/2), pp. 21-41. 24. maghade, d.k., patre, b.m., 2014, pole placement by pid controllers to achieve time domain specifications for tito systems, trans. inst. meas. control, 36(4), pp. 506-522. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. i ii https://doi.org/10.22190/fume123456001o © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd editorial foreword to the thematic issue: wear particle transport and emission: mechanisms and environmental implications georg-peter ostermeyer1, valentin l. popov2 1technische universität braunschweig, braunschweig, germany 2technische universität berlin, berlin, germany the papers of the present special issue as well as a number of papers of the subsequent issues of facta universitatis – series mechanical engineering represent extended versions of the research works presented at the international workshop "wear particle transport and emission: mechanisms and environmental implications" which was carried out online from 24. to 25. february 2021. the workshop was a part of a series of tribological workshops that prof. popov (tu berlin) and prof. ostermeyer (tu braunschweig) organize annually since 2004, each year devoted to a chosen topical problem. the theme of this year's workshop is closely related to the "third body" problem in tribology [1]. to put it in a somewhat exaggerated way, the problem of the third body is the core problem of tribology and one of its greatest current challenges [2]. the third body is very closely related to all properties of a tribological system and determines the friction, the wear intensity, the chemical composition of the surface layers and the relevant system dynamics. wear changes the surface topography, which in turn influences the frictional force [3]. the material transport from one contact partner to the other [4], [5] as well as the transport in the sliding plane [6] are of central importance for the tribology. this latter process ultimately leads to the emission of wear particles into the environment [7]. especially in our time when people are actively concerned with their health, problems of the emission of abrasive particles, for example from brakes or car tires, are of great importance. the connection and interrelation of friction, wear, and emissions were the core topics of the workshop. the problem of wear, of the third body and emissions from tribological systems are not only topical in scientific and political terms, but also highly complex. only recently, promising approaches for adhesive wear [8], [9] have been proposed as elements of the whole complex process of "boundary layer machine" (ostermeyer et al. [10]). a very important topic is measurement of emission, which is now approached also using new tools such as big data analysis. since processes in tribological contact are difficult to access experimentally, numerical simulation methods play a special role here. development of numerical concepts for fast numerical simulation of systems with geometric and material non-linearity as well as concepts for simulation of the "third body", mixing and surface modification belonged therefore to the central topics of the workshop too. while in recent ii g.p. ostermeyer, v.l. popov years two important tools have been created with the method of dimensionality reduction [11] and the fft-based boundary element method [12], which make it possible for the first time to simulate contact mechanics in their real complexity, there are so far no significant attempts to use these methods to simulate the third body. obviously, there is above all an urgent need to search for new concepts for characterizing and understanding the third body. several adjacent topics related to mechanics of interfacial particles, mechanical and chemical processes in the surface layers, the influence of material structure, as well as basic questions of physics and modeling of relevant processes, including adhesion, have also been included in the discussion. on the other hand, a series of papers focused on engineering and medical applications have also been reported. of course, the outlined field is too big to be dealt within a single workshop. therefore, further meetings on this topic are needed in the future. in the works presented at the workshop, the third body have been considered at very different scales. thus, in the very first presentation by k. de payrebrune, just a single macroscopic particle was considered. it was surprising how complicated and different can be the modes of movement of such a simple system. at the microscale and nanoscale, again single particles were in focus, e.g. in quasi-molecular dynamics simulations. at the mesoscale, the flow of particles was observed. but this flow underwent a self-organization (and selfhealing). as a result, one could see again the rolls. it seems that the formation of rolls is a very common feature independently of material and scale. their movement essentially determines the wear and the friction process. the studies of mechanics, evolution, and related physics of interfacial particles is an important current research topic. references 1. godet, m., 1984, the third-body approach: a mechanical view of wear, wear, 100, pp. 437–452. 2. popov, v.l., 2018, is tribology approaching its golden age? grand challenges in engineering education and tribological research, frontiers in mechanical engineering, 4,16. 3. ostermeyer, g.p., müller, m., 2006, dynamic interaction of friction and surface topography in brake systems, tribology international, 39(5), pp. 370–380. 4. rabinowicz e, tabor d., 1951, metallic transfer between sliding metals: an autoradiographic study, proceedings of the royal society a: mathematical, physical and engineering sciences, 208(1095), pp. 455–475. 5. greenwood, j.a., 2020, metal transfer and wear, frontiers in mechanical engineering, 6, 62. 6. popov, v.l., gervé a., kehrwald, b., smolin, i.y., 2000, simulation of wear in combustion engines, computational materials science, 19(1-4), pp. 285-291. 7. pohrt, r., 2019, tire wear particle hot spots – review of influencing factors, facta universitatis, series: mechanical engineering, 17(1), pp. 17-27. 8. aghababaei, r., warner, d.h., molinari, j.f., 2016, critical length scale controls adhesive wear mechanisms, nature communications, 7, 11816. 9. popov, v.l., pohrt, r., 2018, adhesive wear and particle emission: numerical approach based on asperity-free formulation of rabinowicz criterion, friction, 6, pp. 260–273. 10. ostermeyer, g.p., vietor, t., müller, m., inkermann, d., otto, j., lembeck, h., 2017, the boundary layer machine, proceedings in applied mathematics and mechanics, 17, pp. 159 – 160. 11. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer, 265 p. 12. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17, pp. 334–340. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 545 564 10.22190/fume200510021s © 2020 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper optimal feed rate control strategies for friction drilling roman stryczek, paweł błaszczak faculty of mechanical engineering and computer science, university of bielsko-biala, poland abstract. the presented paper contains the results of research aimed at developing optimal strategies for controlling the feed rate in the friction drilling process. in particular, the use of linear variable feed rate for individual drilling stages and adaptive feed rate control have been tested. the experiments were carried out with the use of a cnc machine tool equipped with an axial force and torque sensor. correlation between thrust force and torque was shown, respectively, in relation to the feed drive load and the drive of machine tool spindle. based on this, a feed rate sensorless control strategy was created to protect against excessive and long-term overload both of the tool and the drives. the following assessment criteria were considered: drilling cycle time, maximum values of thrust and torque, maximum values of feed drive load and drive of machine tool spindle, maximum power and energy effect in the form of work necessary to perform during the drilling process and forming the hole flange. the obtained test results, made for low-carbon steel with a tungsten carbide tool, indicate the advantage of the approach based on the linear variable feed rate and adaptive control over the traditional drilling process based on the step change of the feed rate, according to the recommendations given by the tool manufacturers. key words: friction drilling, feed rate, optimization, adaptive control 1. introduction one of the noticeable trends in modern machine and device constructions is the increasing use of thin-walled components. this results in material saving and a less weight of the designed structures. however, there is a problem of assembling this type of elements with other parts of the product, the solution would be to make drilling holes using the friction drilling technique. friction drilling is an alternative method of holemaking process using heat in sheet metal, pipes and thin-walled profiles made of low received may 10, 2020 / accepted july 05, 2020 corresponding author: roman stryczek faculty of mechanical engineering and computer science, university of bielsko-biala, willowa 2, 43-300, bielsko-biała, poland e-mail: rstryczek@ath.bielsko.pl 546 r. stryczek, p. błaszczak melting point metallic materials. during the friction drilling process, as a result of friction between the drill and the workpiece, there is a rapid increase in temperature, which causes plastic deformation of the workpiece in the area of operation of the drill. the material. displaced in this way forms a bushing, significantly extending the length of the hole and, consequently, also the active length of the formed thread. all material removed from the hole contributes to the formation of the bushing. the properties of the material and its microstructure change during friction drilling due to high temperature and strain. the use of friction drilling as hole-making technique implies an improvement of surface integrity [1]. friction drilling presents a deformed zone which does not appear during conventional drilling. this deformed zone provides an appropriate geometry for next productions steps such as threading or joining processes. in [2, 3], the authors showed that the friction threads lead to much better mechanical, profile and microstructural property. the thread has a significantly larger number of turns, hence the screw connection ensures high strength. stronger threads are produced, also thanks to iso-stress level lines which are parallel to the thread profile. bushing can also be used as a supporting hole for welded and soldered joints. the relatively recently widespread friction drilling technique is quickly gaining new applications, especially in automotive industry, aviation industry, in the production of lighting devices, medical devices, furniture industry, ventilation devices, fitness, etc. its main advantages are: increasing the active length of a hole, improving the strength of the thread connection, wasteless production, productivity, long tool life, up to over 10 000 cycles, possibility to apply to many different construction materials, simple tooling and a clean workplace. numerous publications on friction drilling focus on attempts to create a process model, most often based on the finite element method (fem), to enable better understanding of the complex physical-chemical phenomena associated with this process: material flow, temperature distribution, stress and strain. chow et al. [4] studied the relationship between drill surface temperature, tool wear and axial thrust force in friction drilling aisi 304 stainless steel by tungsten carbide drills with and without coating. li at al. [5] developed an improved theoretical model of the drilling force in friction drilling, which took into account changes in temperature, pressure and friction coefficient. it can be the basis for optimization of tool design. el-bahloul et al. [6, 7] studied combinations of thermal drilling parameters such as tool diameter, tool friction angle, friction contact area ratio (fcar), thickness of the workpiece, feed rate and rotational speed and their effect on thrust, torque, hole diameter error, error roundness and bushing length. to evaluate the results, fuzzy logic elements were used. the paper [8] presents a series of experiments that has been carried out to determine the impact of selected parameters on the quality of the bushing obtained as a result of friction drilling. su et al. [9] showed that the ratio between material thickness and drill diameter has a decisive impact on the busing quality. bustillo et al. [10] proposed a suitable smart manufacturing strategy to the friction-drilling process joining materials with very different mechanical and chemical properties. pereira et al. [1] analyzed the feasibility of friction drilling technique from a technical and environmental point of view. the absence of cutting fluids in machining processes is a key aspect which implies a drastic reduction of environmental footprint. friction drilling is the solution to the problem of joining thin-walled structural elements in a simple, economical, ecological and very effective way. in general, the research confirms the high complexity of the process, resulting from the numerous and diverse set of input parameters which influence the output parameters (fig. optimal feed rate control strategies for friction drilling 547 1). therefore, all previously generated models of friction drilling process are fragmentary in nature, they are mainly limited to three input variables and one to three output parameters. the range of variability of input parameters is very narrow and does not always take into account technically achievable and the most economically justified values. the test stands used, as in [6, 7], for example, have significant technical limitations that make it impossible to test a broader range of process parameters. furthermore, the research most often concerned the operation of new drills, without analysis of the impact of drill wear on the process and output parameters of the drilling cycle. due to the long life of the tool in the friction drilling process, the impact of drill wear on the hole quality can only be tested under the conditions of industrial production. fig. 1 input and output parameters of the friction drilling process experimental research and the development of theoretical models also enable research on parametric optimization of this process. pantawane and ahuja [12] using the statistical analysis method the response surface method (rsm) optimized the drill rotational speed, the feed rate and the tool diameter, due to the hole diameter error, and roughness of the inner surface of the formed bushing. the research demonstrated a noticeable increase in variability of diameter dimensions with an increase in feed rate. in the paper [12], the length of the busing obtained during friction drilling was maximized depending on the rotational speed, thickness of galvanized steel sheet and angle of the conical section of the drill. the established artificial neural network technique model is effectively integrated with simulated annealing algorithm approach to give optimum processing conditions in thermal drilling. jiang et al. [13] used the gray relational analysis to determine the impact of friction angle, fcar, feed rate, and drilling speed on the surface roughness and the bushing length. similar research was performed by ku et al. [14] who stated that surface roughness is mainly influenced by feed rate and rotational speed of the tool, while only fcar has a significant impact on the length of the bushing. in the paper [15] it was found that the surface roughness of the drilled hole was the dominant output characteristic in the thermal drilling process. on the basis of the 548 r. stryczek, p. błaszczak research, it was found that spindle speed and tool angle have a greater impact on the surface roughness of the galvanized steel sheet than the impact of the workpiece thickness. this paper also presents a set of different methods for optimization of friction drilling process. patil and bembrekar [16] analyzed the impact of rotational speed and feed rate on thermal stress, hardness and the bushing length for aluminum and mild steel. at the same time, there is a lack of research on parametric optimization of friction drilling process, in terms of cycle time, tool life, load on machine tool drives and energy expenditure. the potential user is interested in performance and economic aspects of the manufacturing process. therefore, he would be interested in quick selection of machining conditions that guarantee short cycle times and at the same time economic tool life, use of the machine tool production potential, process automation and control of its correct course. manufacturers of friction drills inform the user about the recommended machining parameters, at the same time indicating that these are good starting parameters and require verification along with growing user's experience. so far, in mass production, the friction drilling process has been carried out on specialized devices. in order to extend the scope of application of this technique and at the same time automate the drilling cycle, it would be necessary to adapt universal numerically controlled machine tools (nc), equipping them with simple, programmed process controllers. for nc machine tools, there are no standard machining cycles dedicated to friction drilling yet. the analysis of the scientific publications clearly indicates existence of a research gap in the field of optimization of the friction drilling process from the point of view of user-relevant aspects such as process efficiency, its energy consumption, utilization of the cnc machine tool potential both in terms of the available main drive and feed drive power as well as control functions. in addition to the economic benefits, such approach to optimization generates also ecological progress in the form of reduced energy consumption and less environmental pollution, as the friction drilling process is a clean and waste-free process. the premises determined above have induced the authors of this paper to develop an intelligent, sensorless strategy for control of the friction drilling process. the term intelligence is understood here as the ability to adapt to change. the presented paper describes a new model for control of the friction drilling process, taking into account the intelligent functions adapting the feed rate to the currently performed stage of drilling, the capabilities of the machine tool drives and the condition of the tool. for this purpose, it has been necessary to: ▪ perform a new breakdown of the friction drilling process into stages, the floating limits of which set the maximum load values for the machine tool drives; the proposals for the breakdown of this process into stages known from the literature take into account only the geometrical aspects, which are not useful in the context of adaptive control of the feed rate; ▪ identify the correlations between the thrust and torque and the load of the machine tool drives; ▪ develop an advanced form of the numerical filter in order to limit the impact of the input signal disturbance on the quality of control of the feed rate; ▪ develop an algorithm being capable of correcting of the programmed value of feed rate correction in real time, adjusting it to the capabilities of the machine tool and of the tool, taking into account all restrictions related to the proper course of the process. optimal feed rate control strategies for friction drilling 549 2. test stand test stand (fig. 2) was based on a tug-56 lathe equipped with a 7 kw spindle drive motor and 1.26 kw each feed drive motors. the machine tool is equipped with the sinumerik 810d numerical control system, with an additional external digital input/output panel enabling the control of synchronous actions. the tool is integrated into the machine tool spindle using the er25 collet chuck. the workpiece was square tubing with wall thickness of 2 mm, made of carbon steel s235jrh en 10219. the object was mounted in a tool holder integrated with a dynamometer. each time, before the next drilling cycle, it was again determined to ensure the centricity of the drill axis and the dynamometer axis. piezoelectric dynamometer kistler model 9272a, together with the controller and software for the acquisition, visualization and archiving of measured values, was used to measure axial force and torque during the friction drilling process. other force components occurring between the tool and the workpiece do not have a significant impact on the analysis of this process. the equipment is supplemented with the simatic field pg programmer for real-time recording of: drive loads, programmed feed rate, relative axial position of the tool tip (indications of z-axis of machine tool) and calculation parameters used in the adaptive control algorithm. the latest versions of the cnc controller software already have a built-in tracking and visualization function of the machine tool operating parameters, therefore additional recording devices are unnecessary in such cases. fig. 2 experimental setup: fixture for the tool (1), fixture for the workpiece (2), dynamometer (3), controller (4), programmer (5), cnc control panel (6) a typical friction drilling tool can be divided into five parts: shank, collar, calibration part, conical part and center (fig. 3). in new drill designs, the conical and cylindrical parts have a modified shape (a-a cross-section) to limit the contact area between the tool and the material. fig. 3 shows the key dimensions of the tungsten carbide drill used in the tests. 550 r. stryczek, p. błaszczak fig. 3 regions and key dimensions of the friction drilling tool in numerous publications, the authors divide the friction drilling process into stages on the basis of geometric quantities determined by the mutual positions of the tool and the workpiece. in the presented publication, the boundaries of stages are determined on the basis of observation of the tool load. fig. 4 shows the stages of the friction drilling cycle. the recorded courses of thrust force and torque occurring in the process of friction drilling are typical and coincide with the recorded courses included in numerous works, regardless of whether they concern soft alloys [17] or difficult-to-machine materials [18]. the limits of the basic stages a and b determine the maximum thrust (axial force) and the maximum torque value, respectively. fig. 4 stages displaced in this way forms a bushing, significantly extending the length of the hole and, of friction drilling 3. analysis of thrust force and torque in the friction drilling cycle during the first stage of the process (a), the material is locally heated to a high temperature, depending on the type of material and process parameters, i.e. tool rotation optimal feed rate control strategies for friction drilling 551 speed and axial feed rate. mechanical energy of friction is converted into heat, a deformation of the thermoplastic material occurs, as a result of which an initial, irregular flash is formed. at this stage, a rapid increase in thrust is noticeable, which reaches its maximum value even before the perforation of the hole. when the tool almost penetrates the material, high stress is generated inside the hole due to material compression. torque values gradually increase, but do not reach high values in this phase, which is mainly due to the small diameter of the contact area between the tool and the material. excessive increase in feed rate at this stage results in smaller ductility of the material because there is less time to generate enough heat for plastic deformation. the high thrust causes adhesive and frictional wear of the drill on the conical section, which causes the formation of round grooves on this section of the tool [20, 21]. these grooves also cause greater adhesion of the material to the tool during subsequent drilling cycles, which adversely affects the quality of the worked surface, also reducing tool life. this may cause greater variability in the diameter dimension of the drilled hole [22]. therefore, at this stage, the axial feed rate recommended by the manufacturer should not be significantly exceeded. at the beginning of stage b, high temperature causes decrease of thrust force. the hole is perforated and its inner cylindrical part is formed. as the tool moves deeper into the material, the active tool radius increases, which increases the torque until it reaches its maximum value. increased resistance is caused by friction force at the end of the tapered surface of the drill and deformations of the formed bushing. at stage b, there are significant differences in thrust and torque in subsequent drilling cycles, which indicates that the process is highly unstable at this stage. stage c is a sterile transition of the tool for shaping the external flash and occurs only in cases where the length of the drill is excessive. standard drills are produced in two versions: short and long. rarely the length of the drill is perfectly matched to the size of the bushing produced. during stage c, there is a sharp decrease in both torque and thrust. the implementation of stage c with the same feed rate as stages a and b is then irrational. moreover, we should bear in mind that the external flash formed at stage a quickly lowers the temperature, which is undesirable if it is further formed by the drill flange. stage d consists in shaping and smoothing the upper burr by compressing the flash formed at stage a. there are two options when it comes to this upper burr; one possibility is that it is crushed between the tool ring and the piece, which implies that process is absolutely chipless. the other possibility is that the material is removed from the workpiece by a chip breaker located around the tool shank. during stage d, both axial force and torque increase. in case of difficult-to-machine materials such as: aisi304, ti6al-4v or inconel718 and a low feed rate of tool at stage c, thrust and torque at this stage can reach maximum values, which was confirmed in [18]. in case of soft materials and acceleration of feed rate at stage c, thrust and torque do not reach high levels, which was also demonstrated in this paper. too high values of tool load during this stage mean the necessity to verify the trajectory and parameters of tool feed. at the end of stage d, the axial force and torque decrease to zero, which is associated with deceleration of feed drive. in other publications authors also consider tool retraction phase. this stage is usually carried out with a fast movement, so it has not been considered in this study because it had no effect on the results tested. 552 r. stryczek, p. błaszczak the above analysis shows that both the axial feed rate and the rotational speed of the tool have a significant impact on the level of thrust and torque. friction drilling process requires higher speeds than conventional drilling methods. the required rotational speed of the tool is conditional on the hole diameter, material thickness and type of material. increase in rotational speed causes an increase in temperature, which results in greater ductility of the material, and thus a decrease in thrust and torque, and this entails better working conditions for the tool. we should bear in mind, however, that an increase above the recommended temperature value of the tool 750° c [19], 900° c [23], in turn, causes a rapid decrease in the life of the drill. therefore, it is advisable to use rotational speed recommended by the tool manufacturer. therefore, the user has to choose the axial feed rate as a parameter determining the time of the drilling cycle and the load on the main drive and feed drive. if the machining process is carried out with a worn out tool the friction between the tool and the workpiece increases, and energy consumption increases as well [24]. if the worn out tool is not replaced in due time, it can increase production costs, it can cause downtime or even a machine failure. energy consumption monitoring during subsequent friction drilling cycles may prevent the above-mentioned negative cases. fig. 5 presents a comparison between axial force and torque indications during a cycle for a new and worn out drill after 10 000 operating cycles. fig. 5 comparison of thrust and torque for a new and worn out friction drill machining parameters, i.e. spindle rotation and feed rate during the experiment were kept constant and identical in both tests. significant changes that can be observed are optimal feed rate control strategies for friction drilling 553 mainly due to the tool surface wear in the area located between the conical and cylindrical part of the tool. it results in shifting the maximum main drive load by about 2 mm, which should be included in the process parameters. the boundaries of areas a and b as well as the maximum values of axial force and torque change with the progressive wear of the drill. hence, the optimal tool feed rate should change in adaptive mode adapting to the current condition of the drill. the process parameters recommended by the tool manufacturer are suitable for the new, unused tool. in order to ensure proper operating parameters for the entire tool life, a flexible, automated procedure should be developed for changing its operating parameters with the progressive tool wear. 4. strategies for controlling feed rate in the friction drilling process in a significant part of experimental research, a constant feed rate was adopted for all stages of drilling, which seems irrational from the point of view of the above-said analysis. manufacturers of friction drilling tools provide their customers with recommended feed rates which are constant in subsequent drilling phases. this has obvious benefits, but is still not the optimal solution. therefore, the authors have developed a new method which has not been presented in the technical and scientific literature yet of a linear feed rate change in the individual stages of the drilling cycle, additionally modified with an adaptive strategy for adjustment of the feed rated to the possibility of the assumed load of the machine tool drives, taking into account the restrictions resulting from ensuring of the proper plasticization of the material. the selection of the optimal feed rate is a complex issue and should be considered on a case-by-case basis. in the paper [7], special attention has been paid to the importance of thermal conductivity of the material in this respect. for example, the low thermal conductivity of ti-6al-4v causes a low rate of heat transfer, the workpiece slowly becomes soft and then also it slowly loses heat. long period of time needed to generate sufficient heat and to ensure proper softening of the material causes rapid wear of the drilling tool. on the other hand, it also takes a long time to lower the temperature of the molten material. low thermal conductivity, which causes poor heat transfer in the whole material, is the main cause of severe plastic deformation with surface delamination on the inner periphery of the bushing. fig. 6 shows the recorded thrust force and torque values for three friction drilling cycles that differ in terms of feed rate. the simplest fconst strategy assumes a constant feed rate over the entire drilling process. the strategy, according to the recommendations of the manufacturer of fsec tool provides for a stepped change in the feed rate at individual sections of the drilling process. the flin strategy is based on a linear change of the feed rate in the area of individual process phases, avoiding its abrupt changes. the values of thrust (fig. 7) and torque (fig. 8) in case of fsec and flin strategies are at a similar level, while the cycle time (fig. 9) for the f lin strategy is about 30% shorter than the cycle time for the f sec strategy. in case of the simplest fconst strategy the cycle time is definitely extended, therefore, it should not be taken into account in industrial applications for mass production. 554 r. stryczek, p. błaszczak fig. 6 feed rate for drilling strategies fig. 7 thrust for drilling strategies fig. 8 torque for drilling strategies fig. 9 cycle time for drilling strategies as illustrated by the example, linearly varying feed rates at individual stages of the friction drilling cycle have many significant advantages. however, an unsolved problem remains of how to pre-set the feed rate and how to react to changes in stage boundaries as the tool working surfaces wear out? the solution to the above-said problem may be properly selected strategy of automatic feed rate correction, maximizing, wherever possible, its value and at the same time not allowing to exceed the permissible load on the tool and machine tool drives. therefore, it is necessary to roughly determine the feed rates and limits of their variability and then to optimize their real values in the adaptive mode. 4.1. adaptive control of the friction drilling process because it is very rare that friction drilling machine tools are equipped with a dynamometer to measure thrust and torque, the assumed strategy of adaptive feed rate control uses the load on the feed drives and the spindle as input data. the values of these loads are available in modern numerical control systems of cnc machine tools in the form of system variables, as a percentage of the maximum load of a given drive. this allows the use of such variables as input data in the adaptive control strategy. there is a strong correlation between the quantities measured with a dynamometer, i.e. thrust and torque, and the load on the feed rate, respectively (fig. 10) and spindle drive (fig. 11). optimal feed rate control strategies for friction drilling 555 the initial stroke of spindle drive load observed results from the spindle acceleration to nominal revolutions. fig. 10 thrust and feed drive load fig. 11 torque and spindle drive load the proposed strategy for controlling the feed rate is schematically shown in fig. 12. the actual, currently implemented working feed rate is influenced by the programmed value of linear variable feed rate and programmable, expressed as a percentage of feed rate correction (ovr). an upper limit of ovr has been set, reaching 200%. we can assume that the feed rate actually applied can reach a value in the range of (0, 2∙fpr], where fpr is programmed feed rate value. the ovr value in a given interpolator cycle (ipo) depends on the current main drive load, feed drive load and ovr value in the previous ipo. fig. 12 general scheme of adaptive control of the friction drilling process 556 r. stryczek, p. błaszczak the basic input signals of the control algorithm, i.e. the feed drive load and the spindle drive load are subject to interference resulting from imperfections of their reading and inevitable process instability. signal containing significant interference should therefore be filtered, otherwise the input signal oscillations would also cause oscillations of the ovr output parameter. furthermore, as ovr is also an input signal such adverse oscillations would be amplified. during the research, an advanced filter containing a differential element was selected, allowing, to a large extent, to take into account the expected form of the signal in subsequent interpolator cycles. attempts to apply a signal averaging resulted in a clear delay in response in case of dynamically changing loads. due to a significant role of temperature, in case of friction drilling process there is a much greater dynamics of changes than during e.g. rolling process or conventional drilling. therefore, a digital filter was applied in the form of three components: inertia, current reading and forecasted values. their influence is determined respectively by the weights: wi, wa and wp. a simple filter form was obtained, which is easy to program in synchronous actions 1 1 2( ( )) k k k k d d d d d d f i f a k p f f kl w l w l w l l l − − −= + + + − , (1) where: lfk filtered drive load in k in ipo, d ∊ {c ← spindle drive, z ← feed drive}, wa weight of the current drive load, wi weight of inertia block, wp weight of predicted load value. tests have shown correct functioning of the filter with the weight values respectively: wi=0.4, wa=0.1, wp=0.5, as illustrated in fig. 13. the filter taking into account the predicted values generates a slight delay, about 0.03s, twice smaller than the standard filter, while sufficiently smoothing the signal. in the second step of the method, the lr relative drive load d is calculated as the ratio of the current load value and the lset user's preferred load value as /d f d r f setl l l= . (2) fig. 13 feedback signal of the drive load this allows us to compare the feed drive load and spindle drive, taking into account the preferences and experience of the machine tool user. for further calculations, only higher value of relative load of the spindle drive or feed drive shall be taken into account. optimal feed rate control strategies for friction drilling 557 correction coefficient cf for the feed rate shall be determined. if the relative load value taken into account is higher than 1, then the correction coefficient for the feed rate takes the value between (-1, 0]. if the relative load is less than 1, then the correction coefficient for the feed rate takes the values between [0, 1]. these values are calculated based on the relationship (3). the b factor allows you to control the intensity of the correction. fuzzy functions (4) were used to determine the new ovr’ value. 2 2 1 1( 0.001) (1 ) 1 1 1 1 d r d r b l d d r f b l d r if l c f l e e i  −  − +   − − − → − →    =   +  (3) max max max ( 1) 1 ' 1 d d f r d d df f r c ovr f l ovr ovr covr ovr if l ovr ovr i c → →  +    =    + −     (4) fig. 14 illustrates the functioning of the applied strategy of adaptive feed speed control in friction drilling. the shaded areas indicate the areas affected by exceeding the assumed load values of the feed drives (25%) and the spindle drive (40%) on the shaping of the programmed correction of the feed rate and, consequently, the implemented feed rate. exceeding the permissible lset values resulted in an increase above 1 of relative load value of the lr drive and a decrease below 0 of the value of the cf correction coefficient. negative fc values result in a continuous decrease in the programmed value of ovr feed rate correction, the intensity of the ovr decrease depends on the amount of exceeding the permissible load values of the drives. after overloading the drives, the ovr quickly returns to its maximum value of 200%. the constant feed rate in the first two seconds of the cycle is due to the fact that the adaptive control is turned off during this time to provide time for a sufficient temperature rise and plasticization of the material. the constant feed rate at the first drilling stage is selected in accordance with the tool manufacturer's recommendations. 4.2. selection of nominal feed rate in accordance with the applied adaptive control strategy and based on the tests carried out (fig. 14), the currently implemented feed rate shall be affected by the programmed feed rate in the fpr control program. programmed feed corrector can compensate for the effects of dynamically changing drilling conditions only to some extent. when programming the feed rate in the initial stages of the friction drilling cycle, the user can follow the recommendations of the tool manufacturer and/or his own experience. the proposed adaptive strategy of controlling the feed rate together with the possibility of visualizing the formulation of drive loads and variables ovr and f in the full drilling cycle allows the user in a few steps also to optimally select the fpr feed rate. below the results of three tests have been presented that allow the user to determine whether the programmed feed rates in the next program phases are satisfactory. fig. 15 illustrates the first selected feed rate fpr and the response to such feed rate of the ovr and f variables. 558 r. stryczek, p. błaszczak fig. 14 impact of load on the feed rate fig. 15 test for lz set= 30% and lc set= 40 the analysis of the feed drive load during the first 3mm of drilling clearly indicates that the assumed period of heating and plasticizing of the material is too long and can be limited. after 2 mm of drilling, the feed drive load is clearly reduced. even earlier, after about 1mm, the rapid increase in load of spindle drive stops. therefore, in the next test, it was decided that the first stage should be shortened to 2 mm with a slight increase in the feed rate during the last phase to 250 mm/min. the next observation indicates that at the drilling section between 6mm and 13mm the assumed feed drive load and spindle drive load were exceeded. at this section, the final part of the bushing is formed. therefore, in the next test, a different, trapezoidal feed rate was suggested for this section (fig. 16). lowering the feed in this section is also driven by the need to maintain acceptable quality of the hole. excessive feed rate at this stage has a negative effect on the shape and active length of the hole, causing cracks and petal formation [2]. in case of exceeding 13 mm there was no risk of drive overloading, therefore the maximum feed rate fpr was set there. a slight modification by 0.5 mm was also proposed in the penultimate point of the fpr trajectory. as fig. 16 indicates, as a result of the actions taken, a number of positive effects were achieved: the tool load was smoothed, rapid changes in ovr and f controlled variables were removed, and apparent overloading of the recommended loads on the machine tool drives were avoided. the feed drive load at the critical section oscillates between ± 2% of the value selected by the user. cycle time remained virtually unchanged. optimal feed rate control strategies for friction drilling 559 fig. 16 test lz set= 30% and lc set= 40% fig. 17 test for lz set= 40% and lc set= 50% analysis of the input variable fpr and the output variable f presents the remaining reserves and threats, and thus further optimization of the fpr variable. the most sensitive point of the fpr trajectory remains the point located 7mm from the contact point between the tool and the material. moving it to the left by 0.5 mm should remove the threat. however, extending the trajectory section for which fpr=1000 mm/min applies by 1mm to the left and to the right will not threaten the stability of the process. the control algorithm, however, compensates for this type of "inaccuracy" of the feed rate control trajectory, so it is not necessary to make the above changes. the last test presented concerns another problem: what will be the response of adaptive control to a significant increase in the permissible load of drives. lset values have been increased from 30% to 40% for feed drive and from 40% to 50% for spindle drive, with constant fpr values. the results are shown in fig. 17. during the full cycle, the permissible load values for the drives were not exceeded, despite the fact that the ovr variable reached values over 180%. loads have been smoothed as compared to the previous test, which should be considered as a positive phenomenon. the cycle time was shorter by 18%. therefore, this test indicates a different manner of searching for optimal fpr values. in case of the assumed limit values of drive loads, an ovr similar to that presented in fig. 17 should be obtained. 560 r. stryczek, p. błaszczak 4.3. influence of feed rate control strategy on energy consumption in the friction drilling cycle power and energy analysis in friction drilling process provides basic information concerning machine requirements, such as spindle selection and chuck design. energy consumption during the friction drilling cycle may also be one of the criteria determining the correctness of the adopted process parameters and tool wear. during machining process heat is generated and it is a negative factor in the process. therefore, we try to ensure that most of the heat generated in the process is discharged through the chips and emulsion outside the machining zone. friction drilling is a non-chip drilling method, and we don't use emulsion to cool the tool and workpiece. most of the energy consumed in the friction drilling cycle is converted into heat and transferred to the workpiece and tool. the heat generated is necessary for material ductility, but excessive heat generation limits tool life. the compromise solution is to generate the necessary amount of heat without increasing the mechanical load on the tool and the tool load associated with thermal shock. the test stand is equipped with a dynamometer measuring forces and torque in the friction drilling process, it is possible to calculate the energy expenditure to make the hole and forming the bushing and the flange. the total energy expenditure of the process is of course greater, which results from energy losses associated with the efficiency of the drive and mechanical systems of machine tool. the necessary energy in the friction drilling process is the sum of the ez energy associated with thrust in the direction of the drilling axis and the ec energy associated with overcoming torque in the rotational movement of the tool [25]. both thrust and the torque were measured by a dynamometer. the course of instantaneous energy values related to one ipo cycle (0.01s) was determined according to the equations (5) and (6): ipo ipo z ae f dz=  (5) 20.01 60/ipo c at se =   (6) where: fa average thrust value in the ipo, dzipo the distance covered in the ipo, ta average torque value in the ipo, s spindle rotation. the total energy e necessary to make the hole was expressed as follows: 0 ( ) t ipo ipo z ce e e dt  = + , (7) where δt the total drilling time. the momentary values of energy resulting from thrust (8) and torque (19) are presented below for new and used drill. thrust and torque for this test have been shown earlier in fig. 5. test results indicate a small share of thrust, less than 4% in the total energy needed to make the hole during friction drilling and flange forming. the total energy necessary to make the hole, excluding energy losses resulting from the efficiency of the drives and mechanical systems of machine tool, was 1992 j for a new drill and 2178 j for a used drill. the increase in energy consumption by nearly 10% indicates the possibility of tracking the degree of drill consumption based on the energy consumed in subsequent drilling cycles. optimal feed rate control strategies for friction drilling 561 fig. 18 energy consumption from thrust fig. 19 energy consumption from torque fig. 20 shows the work in a friction drilling cycle concerning the four tested feed rate control strategies. the feed for fconst, fsec and flin was determined according to fig. 6. the feed for fac strategy was determined as in fig. 16. fig. 20 energy consumption for various strategies of feed rate control the maximum energy consumption is in the area where the maximum torque occurs, i.e. at the final section of the hole formation. it should be noted that the maximum energy demand in the drilling cycle with the active function of adaptive control is significantly lower, more than ¼, as compared to the strategy fsec and flin. the conducted research does not confirm the thesis formulated in [25], that "the energy required to drill a hole is independent of the feed rate". 4.4. summary for the purpose of this work, a sensorless method of adaptive feed control was developed during the friction drilling process. friction drilling tests have shown that the variable feed rate during friction drilling has no significant effect on the quality of the drilled hole. the surface quality for all four tested strategies is comparable and enables preparation of the correct thread. the length of the flanged bushing ranged from 7.5 to 8 mm. the height of the petals formed at the end of the bushing did not exceed 1 mm. therefore, the active thread length can be increased more than 3 times. in fig. 21 comparative characteristics of the four feed rate control strategies tested are presented in a graphic form. 562 r. stryczek, p. błaszczak fig. 21 output parameters of four tested feed rate control strategies the maximum thrust for all four tested strategies was at a similar level, which resulted from the assumption made earlier that the tests were comparable. the maximum torque was clearly lower for a constant feed strategy. it remained at a similar level for the other strategies. the maximum power consumed during the cycle was in case of fsec and flin strategies, while in case of fac strategy it was clearly lower, likewise for fconst, which can be seen in fig. 2. the energy consumed during the cycle was diverse for particular strategies. definitely the worst result was obtained for the strategy of constant feed. also the energy consumed for the fsec strategy was clearly higher than in case of the flin and the fac strategies. particularly poor result was achieved in case of fconst strategy in terms of cycle length due to the long "c" stage (fig. 4). this was due to the long cylindrical section of the drill used. but even after choosing the optimal length of the cylindrical part of the drill, this strategy in terms of time will achieve the worst results. cycle times for the fsec and the fac strategies are at a very similar level, hence these solutions should be considered equivalent in terms of maximum thrust, maximum torque, energy consumed and cycle time. the advantage of the fac strategy over the strategy is, in addition to the lower maximum power used in the flin cycle, also the protection of machine tool drives against excessive overload and the ability to automatically change the feed rate adapted to the current state of the tool. it should be emphasized here that synchronous actions necessary for the implementation of the fac strategies do not require the use of advanced and expensive numerical control systems. a properly programmed programmable logic controller allows the implementation of intelligent control strategies, also through synchronous actions. optimal feed rate control strategies for friction drilling 563 5. conclusions due to its high complexity and specific features, the friction drilling process requires an unconventional approach to the issues of parametric optimization. the standard approach, consisting in implementation of the experiment plan for process model building, is not applicable here because of the large number of input and output process variables. therefore, the process models presented so far are fragmentary and for this reason, their practical usefulness is limited. they focus on the selected quality features, ignoring the performance and reliability parameters of the process. a tool for significant improvement of the performed friction drilling processes has been proposed in the presented paper. a new, intelligent approach to parametric optimization of the friction drilling process, based on the sensorless methods of adaptive control through synchronous actions of the feed rate set as linearly variable feed, has been developed. the proposed approach is innovative, because it takes into account the criteria that are important for the users, such as cycle time, load of the machine tool drives and tool condition, not being considered in the scientific papers on this issue so far. thanks to the applied control model, significant improvement of performance, energy and safety 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eng. technol., 8(4), pp. 242-254. 24. ambhore, n., kamble, d., chinchanikar, s., wayal, v., 2015, tool condition monitoring system: a review, mater. today: proc., 2(4-5), pp. 3419-3428. 25. miller, s.f., 2006, experimental analysis and numerical modeling of the friction drilling process, thesis, univ. of michigan, 127 p. 12107 facta universitatis series: mechanical engineering vol. 21, no 3, special issue, 2023, pp. 479 500 https://doi.org/10.22190/fume230901038k © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a hybrid deep learning approach for sentiment analysis in product reviews minghui kuang1, ramin safa2, seyyed ahmad edalatpanah3, robert s. keyser4 1guangzhou city university of technology, guangzhou, guangdong province, china 2department of computer engineering, ayandegan institute of higher education, tonekabon, iran 3department of applied mathematics, ayandegan institute of higher education, tonekabon, iran 4southern polytechnic college of engineering and engineering technology, kennesaw state university, marietta, georgia, usa abstract. product reviews play a crucial role in providing valuable insights to consumers and producers. analyzing the vast amount of data generated around a product, such as posts, comments, and views, can be challenging for business intelligence purposes. sentiment analysis of this content helps both consumers and producers gain a better understanding of the market status, enabling them to make informed decisions. in this study, we propose a novel hybrid approach based on deep neural networks (dnns) for sentiment analysis in product reviews, focusing on the classification of sentiments expressed. our approach utilizes the recursive neural network (rnn) algorithm for sentiment classification. to address the imbalanced distribution of positive and negative samples in social network data, we employ a resampling technique that balances the dataset by increasing samples from the minority class and decreasing samples from the majority class. we evaluate our approach using amazon data, comprising four product categories: clothing, cars, luxury goods, and household appliances. experimental results demonstrate that our proposed approach performs well in sentiment analysis for product reviews, particularly in the context of digital marketing. furthermore, the attentionbased rnn algorithm outperforms the baseline rnn by approximately 5%. notably, the study reveals consumer sentiment variations across different products, particularly in relation to appearance and price aspects. key words: deep learning, recursive neural network (rnn), resampling technique, social media marketing received: september 01, 2023 / accepted october 20, 2023 corresponding authors: minghui kuang, s. a. edalatpanah affiliation, postal address: guangzhou city university of technology, guangzhou, guangdong province, china & department of applied mathematics, ayandegan institute of higher education, tonekabon, iran. e-mail: kuang_minghui@outlook.com & s.a.edalatpanah@aihe.ac.ir mailto:kuang_minghui@outlook.com mailto:s.a.edalatpanah@aihe.ac.ir 480 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser 1. introduction sentiment analysis (sa), also known as opinion mining (om), is an essential application of data mining that aims to collect and extract sentiments and opinions from various documents. with the ever-increasing amount of unstructured data available on the world wide web, processing and interpreting this data pose significant challenges. however, neuro-linguistic programming (nlp) methods can facilitate the analysis [1]. the influence of social media on consumer preferences, behaviors, and attitudes is profound. monitoring social media provides an excellent opportunity to measure customer loyalty and track their sentiments toward different brands or products. however, extracting relevant content from online social networks (osns) is a complex task that requires a comprehensive understanding of dialectal and linguistic semantic rules. sa involves a multi-step process, including feature selection, sentiment data collection, sentiment classification, and sentiment polarity detection. effective feature selection is crucial for understanding the main aspects and enhancing the accuracy of the system. detecting and classifying sentiment has become particularly challenging due to the vast volume, variety, and unstructured nature of web-based social networks. therefore, feature selection plays a critical role in sa feature analysis based on big data. in general, there are three sa levels: (1) document level, (2) sentence level, and (3) aspect level. documentand sentence-level sa assume that only one theme is described in a document or sentence, which is often not the case. therefore, more detailed sa requires further research to identify related entities and aspects and classify sentiment associated with these entities and aspects [2]. instances of entities include products, services, themes, people, organizations, or events, often with dissimilar aspects. this type of sa generally relies on machine learning (ml) techniques. an increased level of abstraction in ml techniques also improves the efficiency of natural language processing (nlp). there are three main tasks for aspect-based sentiment analysis (absa): (1) opinion target extraction, (2) aspect category detection (acd), and (3) sentiment polarity. opinion target extraction involves extracting opinion words, i.e., entity or feature. acd is related to identifying entities and attributes [3]. sentiment polarity is about clarifying the sentiment polarity for aspect words. learning-based sa techniques are generally divided into two categories: lexicon-based and ml-based. lexicon-based techniques require a predefined word source, such as a dictionary containing a list of words used to express emotion and predetermined emotional values. then, based on the sentiment lexicon and the features extracted from the text, the sentences are classified according to their polarity. there is, however, a cost and time involved in creating a valid lexicon for these methods, as they don't need any pre-training. meanwhile, ml-based techniques, such as naive bayesian and support vector machine (svm), have demonstrated significant success in sentence classification and sentiment analysis [4]. ml methods are based on classifying words into sentiment-related labels. the main advantage of ml approaches is their representation of learning ability. ml algorithms require a training dataset that helps automate the classifier, and a test dataset is also used to verify the classifier's performance. therefore, ml-based approaches are better than lexiconbased approaches for sa because of their capacity to deal with large amounts of data. in addition, extensive research has been conducted on sa using ml-based approaches [5]. despite their somewhat acceptable performance, ml-based techniques cannot thoroughly investigate texts' structure and meaning, and disregarding a positive phrase can change the a hybrid deep learning approach for sentiment analysis in product reviews 481 text polarity [6]. alternatively, the extraction of features is a vital part of ml-based techniques, and the extracted features play a crucial role in classifying data. hence, sa and deep learning (dl) techniques are integrated due to the effectiveness of dl in automated machine learning (automl) and the lack of dependence on manual feature extraction. dl includes several networks: convolutional neural network (cnn), deep belief network (dbn), recursive and recurrent neural network (rnn), and many others. albeit dl-based networks outperform other sa algorithms, they still face some limitations [6]. rnn is a sa algorithm. the superiority of rnn over other neural networks (nns) is that it outperforms in predicting structured data for variable input. in the rnn architecture, the input layer includes a word feature vector at time t. the input layer is attached to the hidden layer, containing the information history. in addition to recursive connections to the output layer, the hidden layer has recursive connections to itself. the hidden and output layers also contain neurons for storing values at time t. recurrence permits the network to be deeper than a conventional nn [7]. this study proposes a novel nn-based approach to sa with an emphasis on digital marketing. although simple rnn is considered a very robust tool, it suffers from some fundamental challenges, including vanishing gradient and the inability to learn in deeper layers. therefore, this study seeks to overcome these challenges by proposing a modified rnn approach that uses dl-based techniques. the proposed approach is presented so that it can be compared with the evaluation metrics of other existing approaches. facebook and amazon datasets are also used to evaluate the proposed approach compared to other approaches. although the terms online marketing and digital marketing are prevalent, smm is becoming more popular with the increasing use of social media platforms such as instagram, twitter, facebook, and linkedin [8]. several social media platforms provide companies with data analytics tools for tracking the progress, success, and engagement of ad campaigns. companies can address a wide range of stakeholders through smm. at the strategic level, smm involves handling a marketing campaign and creating a company's intended social media platforms. in data analytics, sa identifies and extracts expressed opinions or feelings. as social networks evolve, research in computer science has become increasingly focused on sa. in particular, many scholars have focused on user behavior toward specific topics based on individuals' interaction and curiosity toward social media platforms [9]. it helps decisionmakers, policy-makers, and managers judge the influence of their products, services, or policies on society. sa is a helpful technique for online marketing, especially on social networks. using sa in commercial products is essential from three perspectives: customer, organization, and ad agencies [10]: 1) customer's perspective: when a person intends to purchase a product, a summary of other people's opinions can be more helpful than extensive information about this product. the customer can also conveniently compare products with summarized reviews or opinions. 2) business organizations and producers' perspective: organizations must improve their products. a product's design and development depend on this information as well as its marketing and evaluation. based on customer feedback and opinions, manufacturing companies may decrease, increase, or replace their products. therefore, you can compare the products and analyze the causes. business organizations and producers can also utilize sentiment summarization. when the number of sentiments increases, it becomes more challenging for producers or consumers to distinguish them. by summarizing sentiments, it becomes easier for a customer to understand other people's feelings about the product, and it becomes easier for producers to know the customer's feelings about the products [11]. 482 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser 3) ad agencies' perspective: feedback and opinions are crucial for advertising agencies as they can elicit market demand ideas. people's general perspective and their favorite product categories can be extracted using opinion mining. hence, the practical implications are: 1) refine marketing campaigns by analyzing sentiment patterns and adjusting messaging, visuals, and overall strategy. 2) identify key customer segments by analyzing sentiment scores, enabling tailored targeting and promotions. 3) improve customer targeting by identifying potential customers with positive sentiments for personalized advertising and partnerships. 4) enhance product development by analyzing sentiment feedback to identify areas of improvement and innovation. 5) monitor brand reputation through real-time sentiment analysis to address negative sentiments and manage pr crises. 6) benchmark competitor performance by comparing sentiment scores and trends to capitalize on market opportunities. a look at the relevant studies shows that dl is one of the new approaches to social network analysis (sna) with different purposes [12]. no need for a domain expert and automated feature detection are essential features that make dl a desirable approach to sna, especially when new classes are constantly emerging, and previous class patterns evolve. another critical feature of dl is that it has a significantly higher learning capacity than traditional ml-based approaches, and thus, it can learn very complicated patterns. by incorporating these two features as an end-to-end (e2e) approach, dl can learn the nonlinear relationship between the input and the corresponding output without splitting the problem into smaller feature selection problems. therefore, the following section reviews related work. 2. related work dl-based approaches have recently become more popularized for sentiment classification. artificial neural networks (anns), initially proposed in 1998, have been used for data recognition to classify short-term sentiment [13]. research demonstrated that a learner's combination of different architectures, including cnn and gated recurrent units (grus), improves sentiment classification performance [14]. due to its significance in business and society, sa is one of nlp's most active research areas, extending beyond computer science, such as management and social sciences. the increasing importance of sa coincides with social media growth. most businesses and social organizations use sa systems since opinions are a fundamental component of human activity and play a critical role in influencing our actions [15]. research in the field of marketing with social media has explored various aspects of social media marketing strategy, including content creation, platform selection, audience targeting, and engagement tactics. these studies have provided valuable insights into effective marketing practices on social media platforms [21] [22]. the book "affective computing and sentiment analysis: emotion, metaphor, and terminology" was published by ahmad [15] in 2011 with 148 pages. this book relies on various research fields, including artificial intelligence (ai), especially reasoning and ml, data mining, linguistics, and psychology. liu's book, "sentiment analysis and opinion mining"[16], was published in 2012. this book includes the results of lectures and extensive studies of data mining, web mining, and text mining. it is a comprehensive introductory book that covers all main topics from the latest relevant developments with over 400 references. a hybrid deep learning approach for sentiment analysis in product reviews 483 2.1. the unbalanced dataset in sentiment analysis the issue of class imbalance is a common challenge in sentiment analysis (sa) and is prevalent in many real-world applications. class imbalance refers to situations where the number of samples in some sentiment classes is significantly higher than in others. this poses a problem for machine learning (ml) algorithms as they tend to prioritize the majority class, leading to a bias and reduced sensitivity in detecting minority-class samples during classification training [17]. consequently, handling classification tasks with imbalanced datasets has become one of the most complex problems in ml. while various algorithms have been proposed to address imbalanced data distribution in structured datasets, the issue of class imbalance has not been extensively explored in the domain of text classification. however, a seminal study by li et al. [18], among the first to investigate imbalanced sentiment classification, demonstrated the adverse effect of imbalanced sample distributions on classifier performance. resampling techniques have been widely adopted to mitigate this effect. these techniques involve manipulating the class distribution by either under-sampling the majority class or oversampling the minority class. previous research has suggested that oversampling approaches tend to outperform under-sampling methods [7]. one popular oversampling technique is the synthetic minority over-sampling technique (smote) [8], which generates synthetic data points based on the similarity between minority samples in the numerical feature space. however, when it comes to text data, oversampling may not yield semantically comparable terms as texts are composed of words or characters. therefore, there is a need to develop an oversampling approach that can generate synthetic texts directly from the contextual space. research conducted in the field of marketing with social media has provided valuable insights into effective marketing practices on social media platforms. leveraging social media platforms allows marketers to establish connections with their target audience, enhance brand awareness, and drive customer engagement [21] [22]. gaining a deep understanding of the dynamics of social media marketing can contribute to the development of more effective sentiment classification models and techniques, enabling marketers to better analyze and interpret customer sentiments and opinions. 3. proposed approach this section proposes a dl-based approach to sa in digital marketing. this approach analyzes users' opinions about different product categories using an appropriate technique to detect user sentiment. their views and ratings also identify trends for the product categories we want to market. figures 1 and 2 illustrate a flowchart for the proposed approach, which comprises two main stages: (a) model generation and (b) product marketing. the first stage generates an rnn-based model using amazon data. this stage consists of four sub-stages: (1) preprocessing, (2) resampling, (3) input vector generation using word2vec, and (4) rnn-based sentiment model generation. the second stage applies facebook data to the generated model and performs product marketing based on different aspects. this stage also includes four sub-stages: (1) preprocessing using stanford corenlp, (2) input vector generation using word2vec, (3) applying facebook data to the model, and (4) marketing the intended product using rating and cosine similarity. 484 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser fig. 1 model generation fig. 2 product marketing 3.1. datasets: amazon data amazon is one of the largest e-commerce websites with countless reviews. in the research analysis section, we use amazon data prepared by amazon web services. since the dataset is unlabeled, we need to label the data to use it in a supervised learning model. we used four json files whose data structure is presented in table 1. table 1 review text features description feature title reviewer identifier reviewerid product identifier asin name of the reviewer reviewername review helpful vote vote text of the review reviewtext rating of the product overall summary of the review summary time of the review (unix time) unixreviewtime time of the review (raw) reviewtime a hybrid deep learning approach for sentiment analysis in product reviews 485 "text of the review" (reviewtext) and the "rating of the product" (overall) are the primary data features used for sentiment classification. in total, four files include about fifteen million reviews. the distribution of amazon data collected for four product categories, including cars, household appliances, jewelry and clothing, and luxury goods, is 1,711,519, 2,070,831, 11,285,464, and 34,278, respectively. as can be seen, the number of samples for jewelry and clothing is much higher than in other categories (75%). on the other hand, the distribution rates for household appliances, cars, and luxury goods are 11.4%, 13.4%, and 0.002, respectively. 3.2. facebook data facebook post reactions dataset is also used for analysis. this dataset comprises facebook posts on the customer service pages of twelve major us and uk chain stores/retail stores, i.e., amazon, tesco, sainsbury, walmart, safeway, aldi uk, target, the home depot, walgreens, best buy, macy's, and publix. most of the posts are released by the customers of these chain stores. besides, we obtained the written text of the posts, the facebook reaction matrix, and posts made by other users with comments attached. posting a reaction on a reply feature was introduced in 2017, and these reactions belong only to the initial post and not to any subsequent replies. therefore, it would lead to generating an incomplete or tiny dataset. reactions encompass "love", "wow", "haha", "sad", and "angry". this dataset contains more than 70,000 posts. of these, 25,969 posts received at least one reaction. the full dataset is available; out of 70,000 posts, 45,386 posts are negative (65%), and 24,614 posts are positive (35%). 3.3. preprocessing preprocessing is an essential step in natural language processing (nlp) tasks, including sentiment analysis and text categorization. it involves transforming raw text data into a format that is suitable for analysis and machine learning algorithms. in the preprocessing phase, six operations are performed to prepare the text data for further analysis: file reading: each product category has a separate file that is read individually to extract the text data. this ensures that the data is organized and processed systematically. text tokenization: tokenization is the process of extracting individual words from the text. it serves as the foundation for subsequent analysis, as it breaks down the text into its constituent units or tokens. each word is treated as a separate token. stop-word removal: stop words are commonly occurring words that do not carry significant meaning, such as articles, prepositions, and conjunctions. in addition to these standard stop words, certain punctuation marks and text separator characters are also considered stop words and removed from the text. this step helps eliminate noise and reduce the dimensionality of the data. selective word removal: this step involves removing specific types of words that are not relevant to the analysis. words like links, stickers, and conjunctions such as "and" and "or" are removed, as they do not contribute to the sentiment or categorization tasks. stemming: stemming is a process that reduces the dimensionality of the input features by converting words to their base or root form. it groups together variations of the same word, making it easier to detect common features and their meanings in the text. the porter stemmer algorithm is used in this case to find word stems. only sentiment words are considered for stemming, and texts without this category of features are removed. for 486 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser example, words like "wait," "waiting," or "waited" are transformed into their root form "wait." text labeling: in the case of the amazon data, text samples are labeled as positive or negative based on their star ratings. texts with ratings of 4 and 5 stars are labeled as positive, indicating favorable sentiment. texts with ratings lower than three stars are labeled as negative, indicating unfavorable sentiment. texts with three-star ratings are considered neutral and are excluded from further analysis. this step ensures the data is labeled for supervised learning tasks. for facebook data, a different preprocessing approach is used. the stanford corenlp parser is applied, which involves the following steps: 1) transforming all words to lowercase letters for uniformity in the data.2) substituting urls with the generic token "url" to remove specific urls and treat them as a single entity. 3) substituting user links with the generic token "at_user" to anonymize specific user information and treat them consistently. 4) eliminating the hash symbol (#) from hashtag references, treating them as regular words. 5) substituting three or more occurrences of the same character in a word with a single occurrence. for example, "looooove" becomes "love" to reduce repeated characters. 6) eliminating sequences containing numbers, as they are not relevant for sentiment analysis. by applying these preprocessing steps, the text data is transformed into a suitable format for sentiment analysis and text categorization tasks, enabling more accurate and efficient analysis of the data 3.4. resampling resampling is a widely used technique to tackle the challenge of imbalanced data distribution in various domains. while oversampling approaches have shown their effectiveness in mitigating the negative impact of imbalanced data, their applicability to imbalanced contextual data remains uncertain. traditional oversampling methods often rely on numerical computations to measure similarities between items. however, when dealing with textual data, which consists of words and characters, numerical approaches may fail to generate semantically consistent new items. therefore, alternative methods are needed to address the oversampling of imbalanced contextual data. researchers have identified limitations in traditional numerical oversampling methods when applied to textual data. for instance, it has been observed that numerical oversampling may struggle to capture the intricate semantic relationships present in texts. the generation of synthetic samples solely based on numerical computations can result in synthetic data that lacks semantic coherence. these limitations underscore the necessity of exploring alternative approaches that can generate synthetic texts instead of relying solely on numerical vectors, thereby enabling effective oversampling of imbalanced contextual data. although oversampling approaches have proven their ability to mitigate the detrimental effect of imbalanced data distribution in multiple domains, their efficacy in dealing with imbalanced contextual data is ambiguous. in conventional sampling methods, similarities between items are measured numerically. however, because texts contain words/characters, numerical computations may not produce semantically consistent new items; hence, we generate synthetic texts instead of numerical vectors to oversample imbalanced contextual data. these studies emphasize the importance of considering the unique characteristics of textual data when designing oversampling methods. by generating synthetic texts instead a hybrid deep learning approach for sentiment analysis in product reviews 487 of relying solely on numerical computations, these approaches aim to create semantically consistent oversampled data. this, in turn, enhances the performance of imbalanced contextual data analysis tasks. in the following section, we will present our novel oversampling method, specifically tailored to address the challenge of imbalanced contextual data. our method focuses on generating synthetic texts that maintain semantic coherence and context, thereby improving the analysis of imbalanced contextual data. the proposed method increases minority-class texts with two strategies: inversion and imitation. during the inversion process, sentimental words in majority-class texts are converted into their opposites with sentiment polarity to generate synthetic texts. despite grammatically incorrect synthetic texts, they must at least be semantically interpretable. during inversion and imitation, we perform the procedures iteratively. to simplify the description, we assume the negative class is the minority class. our first step is to train a classifier based on the original training set. then, a probabilistic prediction is made directly from the training set using the generated classifier. finally, synthetic texts are constructed from positive and negative samples with high and low predicted confidence, respectively, utilizing inversion and imitation methods. we use the label propagation approach to find the counterpart lexicon. we first create a weighted undirected graph for a text with n words, where each node represents a word, and each edge is weighted according to semantic similarity. next, the seed words with robust sentimental polarities are selected, and each word is scored 1. then, each word should be scored between [0,1], where 0 denotes negative words, and 1 represents positive words. the first step constructs a vector for each word (using the technique described in section 3.6). a weighted lexicon graph is formed based on a collection of word vectors, with each node in the graph representing a word. the semantic similarity between two words computes the weight. this study uses cosine similarity. the words wi and wj are connected if their cosine similarity exceeds a threshold value δ. eij = { wi twj if wi twj ‖wi‖‖wj‖ ≥ δ 0 if wi twj ‖wi‖‖wj‖ < δ (1) a set of seed words is selected, and their sentimental polarity is labeled after the lexicon graph is constructed. since the seed words function as labels, they must have a positive or negative sentimental polarity. this study uses the words lexicon proposed in [22] (table 2). table 2 seed words positive seed words negative seed words excellent, good, pleasant, love, delightful, perfect, happy, fortunate, lovely, awesome, amazing, fantastic, nice, successful, best sad, bad, evil, unfortunate, horrible, unhappy, nasty, hate, unpleasant, disgusting, terrible, awful, poor, failure, worst the proposed oversampling approach is identical to boosting the classifier because the reweighting mechanism in promoting the classifier also highlights the significance of misclassified samples. since oversampling increases the training samples and model complexity, we remove the majority-class samples using the inversion technique. samples 488 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser from the majority class for which a counterpart sentence from the minority class was formed are removed from the dataset. in other words, we use oversampling and undersampling techniques simultaneously. 3.5. word embedding vector suppose we have a sentence s = {w1, w2, … , wi, … , wn} consisting of n words and the word aspect wi in sentence s. the aspect-level sentiment classification focuses on the sentiment polarity of the sentence s for the word wi. for example, the sentiment polarity of the sentence "great food, but the service was dreadful!" is positive for "food," while it is negative for "service." when facing a text, we place words in a continuous, lowdimensional, and real-valued vector called word embedding. there is a need for vector representation of words to train dnns aiming to understand the input words deeply. hence, we use the word2vec model [13], a two-layer shallow neural network, in the model's first layer. the word2vec model has two versions: a continuous bag of words (cbow) and a skip-gram (sg). according to previous studies indicating its effectiveness in sa, the proposed approach uses an sg-based model to generate input word vectors. we selected this algorithm in the first layer because of its proven performance in representing semantic words without human involvement. a vector representation of a word is learned based on the context in which it appears. 3.6. rnn-based sentiment classification dl's data representation learning method is based on neural network algorithms, which use multiple layers of modules to analyze and classify inputs, with output from one layer fed into the next. the gradient is computed using an activation function in a process known as backpropagation. the input data (image, audio, or text) generates numeric vectors. since each layer corresponds to a higher level of abstraction, dl implies a hierarchy of more straightforward concepts. a recurrent neural network (rnn) is a popular sa algorithm. in baseline rnns, tokens are fed into recurrent units in a vector of fixed length (e.g., a sentence or document). this method can capture the inherent sequentially of language, i.e., words extend their meaning through their preceding words. rnn models differ from cnn models in that the rnn output is based on prior calculations instead of previously computed data. as a result, it captures text-to-speech dependencies and can model different text lengths. in absa, sentiment classification aims to identify individuals' sentiments according to the words in the text using the interaction among the words and the entire sentence for the model. typical encoder-decoder architectures, such as rnns, may have the problem of encoding irrelevant data, mainly when the input contains much information. the attention mechanism can be used to focus on the part of the original text that is of interest to the model. in this mechanism, attention weights are computed from the lower level, and the weighted vectors are aggregated to represent the higher level. some neural models, such as long short-term memory (lstm), capture contextual data implicitly and cannot explicitly represent primary aspect tokens. only a subset of context words is required to infer sentiment towards an aspect. for instance, in the sentence "great food but the service was dreadful!", "dreadful" is an essential clue to "service," but "great" is not necessary. the standard lstm algorithm operates sequentially and manipulates each context word using the same function, i.e., it cannot explicitly determine the significance a hybrid deep learning approach for sentiment analysis in product reviews 489 of each context word. as part of an optimal solution, context words should be explicitly identified and used in the generation of features after an aspect-based word. the approach proposed by wang et al. [14] addresses the challenge of rnn-based sa. this approach comprises multiple computational layers with standard metrics. layers are contentand location-based models that learn the importance/weight of context words and then compute continuous text representations based on these data. the text representation is viewed as a sentiment classification feature in the last layer. in gradient descent, which uses the cross-entropy error of sentiment classification as a loss function, the entire model can be efficiently trained using differentiable components. the presented approach uses the amazon dataset to generate an sa model and the facebook dataset to test the model and identify the social network users' sentiments for a marketing issue. after developing the model using the amazon dataset, the facebook data preprocessed and prepared to enter the network using the word2vec technique is fed into the generated model. 3.7. product marketing social media is a collective term for communication applications and websites facilitating information interaction, generation, exchange, and dissemination among companies and their customer communities [23, 24]. on the other hand, sajid [25] suggests that social media is an internet-based source for sharing and discussing details among people. finally, delerue et al. [26] define social media as a group of internet-based applications that allow the creation and exchange of user-generated content based on the technological foundations of web 2.0. although social marketing is developing compared to commercial marketing, its role in social development has received intense public attention [24]. most senior corporate managers consider developing, implementing, and revising competitive marketing strategies and policies their primary challenge. therefore, companies' demand for highrank marketing managers is increasing. marketing should not be confused with its classical meaning, i.e., sales. marketing, in a new sense, means meeting customer needs. online reviews of business products and services allow users to share their viewpoints about products purchased or services received, which is of great value to other buyers. this trend has provided new contexts for companies that want to strengthen their presence in the market by investing in digital marketing and gaining a suitable position. due to the increasing attraction in smm, many commercial tools have recently been generated to help organizations analyze the blogosphere on a large scale to extract data related to opinions about their products. however, many existing tools and research efforts are confined to assessing polarity or classifying sentiments based on a minimal set of sentiments [27-29]. marketing aims to provide products based on individuals' attitudes and interests. these attitudes are expressed on social media platforms and selling sites and are an outstanding source for comment marketing. sa and detecting positive and negative opinions about a product help companies and marketers market their target items and sell them through appropriate offerings in the minimum possible time based on people's interests. after sentiment classification and sa processes, the proposed approach determines the number of context sentences in which the desired sentiment is expressed. then, it calculates the average sentiment expressing the desired aspect using eq. (2): 490 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser score(a, ri) = (∑ pos(a,ri) pos(a,ri)+neg(a,ri) sr i=1 ) + idf(a, ri) (2) where ri represents a set of reviews for product i, and sri denotes the number of sentences in context r. the functions pos(a, ri) and neg(a, ri) indicate the positivity and negativity of the aspect in the review dataset. these values are the outputs of the sentiment classification. idf(a, ri) shows the number of occurrences of the aspect in the total review dataset. it is one of the functions of the term frequency-inverse document frequency (tfidf) technique, which is used to retrieve contextual information. its value is determined using eq. (3): idfi = log n ni (3) where n indicates the total number of texts, and ni denotes the number of texts containing the intended word. every product we want to market has certain features, including price, quality, size, and others. these data are contained in a secondary dataset representing a company's products. we intend to identify people's sentiments towards different features and aspects of a product based on the intended product. the most critical issue is the negative sentiments towards the products in the review texts. we aim to market these products based on negative and positive sentiments towards different tools. in this step, we use cosine similarity to match negative review text and product-related text. we compute the cosine similarity between two texts as eq. (4): score (d1, d2) = cos (wd1, wd1) = wd1.wd2 ‖wd1‖2,‖wd2‖2 = ∑ wi,d1.wi,d2 k i=1 √∑ wi,d1 2k i=1 √∑ wi,d1 2k i=1 (4) in summary, cosine similarity is a measure that quantifies the similarity between two vectors, commonly used in text analysis tasks. it is based on the cosine of the angle between the vectors and provides a measure of their directional similarity, regardless of their magnitudes. 4. evaluation results python programming language was used to implement the proposed approach that facilitates the design and implementation of ml and dl algorithms. anaconda was used for the python programming environment. it is a free and open-source distribution of the python and r programming languages for scientific computing (e.g., data science, ml applications, big data processing, predictive analytics, and others) to simplify package management and deployment. more than 15 million users use the anaconda distribution. it includes over 1,500 popular data science packages. this section compares the performance of the proposed approach, which uses the attention-based rnn algorithm, with the baseline rnn algorithm. a hybrid deep learning approach for sentiment analysis in product reviews 491 4.1. evaluation metrics the performance evaluation metrics of the ml algorithm are based on the confusion matrix (table 3). therefore, the supervised ml algorithm uses the confusion matrix to evaluate its performance. regarding classification, the terms "true positive (tp)," "false positive (fp)," "true negative (tn)," and "false negative (fn)" were used to compare the class labels. tp represents the number of samples whose true class is positive, and the classification algorithm recognizes their class as truly positive. fp indicates the number of samples whose true class is negative, and the classification algorithm recognizes their class as falsely positive. fn represents the number of samples whose true class is positive, and the classification algorithm recognizes their class as falsely negative. tp denotes the number of samples whose true class is negative, and the classification algorithm recognizes their class as truly negative. table 3 confusion matrix for performance evaluation actual values negative positive fp tp positive prediction tn fn negative model accuracy is the primary performance evaluation metric of a sentiment classification algorithm in academic research. this metric measures the overall accuracy of the classifier and is widely used for assessing the performance of classification algorithms. it indicates the percentage of correctly classified instances in the test dataset. the classification accuracy is calculated using eq. (5), where tp and tn represent the true positives and true negatives, respectively. p is the number of positives in the experimental samples, and n is the number of negatives in the experimental samples. maximizing tp and tn is crucial in binary or two-class classification problems. accuracy = tp+tn p+n (5) recall and precision are two additional performance evaluation metrics commonly used in sentiment analysis research. recall, calculated as the true positive rate divided by the sum of the true positive rate and false negative rate, reflects the model's ability to identify relevant data points in a dataset correctly. precision, on the other hand, is calculated by dividing true positives by the sum of true positives and false positives. precision indicates the proportion of data points classified as relevant by the model that are indeed relevant in the dataset. precision decreases as recall increases. precision = tp tp+fp (6) in certain scenarios, researchers may prioritize maximizing recall or precision based on their specific objectives. the f1 score or f-measure is often employed as a combined metric to strike a balance between precision and recall. equation (7) represents the combination of recall and precision to calculate the f1 score. 492 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser f1 score = 2 × recall×precision recall+precision (7) in the study, an attention-based rnn algorithm was utilized with five hops. tables 4 and 5 present the weights assigned to different aspects (e.g., quality, price, weight, size, service, design, and appearance) in each hop of the algorithm for the facebook and amazon datasets, respectively. the amazon dataset was divided, with approximately 90% of samples used for training purposes and 10% for testing. table 4 weights of different aspects of the facebook dataset aspect hop 1 hop 2 hop 3 hop 4 hop 5 quality 0.2 0.15 0.14 0.13 0.23 price 0.2 0.1 0.1 0.12 0.13 weight 0.11 0.07 0.08 0.12 0.16 size 0.07 0.08 0.12 0.17 0.078 service 0.03 0.07 0.08 0.12 0.06 design 0.08 0.07 0.08 0.12 0.06 appearance 0.2 0.45 0.45 0.28 0.4 table 5 weights of different aspects of the amazon dataset aspect hop 1 hop 2 hop 3 hop 4 hop 5 quality 0.22 0.15 0.14 0.13 0.23 price 0.21 0.11 0.1 0.11 0.12 weight 0.03 0.11 0.08 0.11 0.06 size 0.1 0.08 0.12 0.13 0.11 service 0.11 0.11 0.08 0.11 0.06 design 0.04 0.11 0.08 0.11 0.06 appearance 0.22 0.32 0.45 0.32 0.43 table 4 shows that "quality," "price," and "appearance" play the same role in the first hop. "appearance" weight increases after the second hop. finally, this model predicts a truly negative polarity for facebook data. it represents the effect of the multi-hop steps. however, as shown in table 5, the model predicts polarity for "appearance" and "price" in the first hop and then "appearance" in the subsequent hops for the amazon data. fig. 3 illustrates the detection accuracy of different aspects of the attention-based rnn model for the facebook dataset. all of these aspects of classification accuracy can be consistently improved by using multiple computational layers. this model works relatively well for all detections. the detection accuracy for "appearance" is higher than other aspects, which varies between 74% for the first and 90% for the ninth hop. on the other hand, the "design" has minimum detection accuracy. a hybrid deep learning approach for sentiment analysis in product reviews 493 fig. 3 the detection accuracy of aspects with different numbers of hops for attention-based rnn algorithm this section evaluates the performance of the proposed approach compared to the baseline rnn-based approach for two-class classification between positives and negatives . fig. 4 illustrates the confusion matrix for the approaches being assessed. figs . 4 (a) and (b) show the confusion matrix for attention-based rnn and baseline rnn approaches, respectively. therefore, the attention-based rnn algorithm can detect samples with an average accuracy of 97.5% and 96% for the baseline rnn. (a) (b) fig. 4 confusion matrix for sa on amazon dataset: (a) attention-based rnn algorithm; (b) baseline rnn algorithm 494 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser fig. 5 illustrates the results obtained using the confusion matrix in fig. 4 according to four metrics: accuracy, recall, precision, f-measure, or f1 score. as shown in fig. 5, the proposed algorithm outperforms the baseline rnn algorithm. this outperformance is noticeable for accuracy because the higher error rate for minority-class samples increases it compared to the rnn algorithm. the classification accuracy of the proposed approach is 97.5% versus 94.4% for the baseline rnn approach. fig. 5 the performance of attention-based rnn and baseline rnn on the amazon dataset the confusion matrix presented in fig. 6 compares the performance of attention-based rnn and baseline rnn algorithms on facebook data. it is important to note that these models were initially trained on amazon data. due to the differences in the data distribution between amazon and facebook, the sentiment classification performance of the models on facebook data is comparatively lower. the facebook dataset used in this evaluation consists of facebook posts pertaining to various products, but it lacks the ability to differentiate between different product categories. (a) (b) fig. 6 confusion matrix for sa on facebook dataset: (a) attention-based rnn algorithm; (b) baseline rnn algorithm a hybrid deep learning approach for sentiment analysis in product reviews 495 despite these challenges, the attention-based rnn achieves an average classification accuracy of approximately 87%, while the baseline rnn achieves around 82.5%. this means that the attention-based rnn performs better than the baseline rnn on the facebook data. the attention-based rnn leverages the mechanism of attention, which allows it to focus on relevant parts of the input during the classification process. this ability to attend to important information contributes to its higher accuracy compared to the baseline rnn. fig. 7 presents additional metrics derived from the confusion matrix shown in fig. 6. the results reveal that the attention-based rnn achieves an average performance of 84%, while the baseline rnn achieves approximately 79%. consequently, the proposed attention-based algorithm outperforms the baseline rnn by a margin of about 5%. these metrics provide further evidence of the superior performance of the attention-based rnn in sentiment classification tasks on facebook data. however, it is worth noting that both algorithms exhibit lower rates in comparison to the other metrics presented. specifically, the attention-based rnn achieves a rate of 80%, and the baseline rnn achieves a rate of 74%. these rates likely correspond to specific subsets or categories within the facebook data that pose challenges for accurate sentiment classification. identifying and addressing the factors contributing to these lower rates could be a potential area for future research and improvement of the models' performance. fig. 7 the performance of attention-based rnn and baseline rnn on the facebook dataset table 6 reports the weight of various aspects with different product categories. the results indicate that the product's "appearance" has the maximum weight for both algorithms. "quality" and "price" get the subsequent ratings. these weights for cars and luxury goods are more than other product categories, indicating people's sentiment and attention to these aspects. in addition, quality ranks lower than appearance and price for luxury goods. another significant aspect is the "service" for household appliances, representing one of the most important aspects for consumers. table 7 presents the rating of different product categories based on various marketing aspects. these values are the average weights calculated for the goods listed for marketing. each product is evaluated using eq. (2). higher ratings indicate the importance of the 496 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser aspect and people's sentiment towards the intended outcome. since the positive sentiments towards the aspect are in the numerator, the higher the number of positive sentiments, the larger the fraction's output. the second part of eq. (2), related to idf, indicates that the more the desired aspect is repeated in more posts, the higher the importance of that aspect and (both positive and negative) sentiments towards it. 5. limitations and ethical considerations while the proposed approach for sentiment analysis in social media marketing demonstrates promising results, there are several limitations that should be acknowledged. firstly, the evaluation of the proposed approach relies primarily on amazon data, encompassing specific product categories. although the results indicate its effectiveness within this context, the generalizability of the approach to other domains and datasets remains to be explored. future research should consider evaluating the approach on diverse datasets to assess its performance in different social media marketing scenarios. secondly, the study addresses the issue of imbalanced data distribution by employing a resampling technique. however, it is important to note that the effectiveness of this approach may vary depending on the characteristics of the dataset. the resampling technique used in this study may not yield optimal results in scenarios with significantly imbalanced classes or when dealing with extremely large datasets. alternative or more sophisticated techniques for handling imbalanced data should be explored in future work. thirdly, the proposed approach utilizes deep neural networks, which often require significant computational resources, particularly for large-scale datasets. in this study, the computational time required for sentiment analysis using the neural network algorithm was manageable. however, it is important to consider the scalability and computational efficiency of the approach when applying it to big data scenarios involving datasets of terabytes or more. future research should explore strategies to optimize the computational requirements of the approach, such as parallel computing or distributed processing. table 6 weights of different aspects for attention-based rnn and baseline rnn on the facebook dataset aspect algorithm cars jewelry and clothing household appliances luxury goods quality attention-based rnn 0.19 0.15 0.14 0.13 rnn 0.27 0.12 0.17 0.06 price attention-based rnn 0.15 0.2 0.1 0.12 rnn 0.13 0.16 0.18 0.16 weight attention-based rnn 0.11 0.07 0.08 0.12 rnn 0.07 0.07 0.11 0.17 size attention-based rnn 0.11 0.1 0.08 0.1 rnn 0.14 0.09 0.06 0.18 service attention-based rnn 0.08 0.07 0.2 0.07 rnn 0.11 0.12 0.16 0.05 design attention-based rnn 0.09 0.07 0.08 0.12 rnn 0.12 0.14 0.06 0.18 appearance attention-based rnn 0.32 0.18 0.18 0.38 rnn 0.24 0.21 0.22 0.27 a hybrid deep learning approach for sentiment analysis in product reviews 497 table 7 average product ratings by different aspects aspect cars jewelry and clothing household appliances luxury goods quality 1.45 0.475 1.54 1.33 price 1.15 1.2 1.47 0.86 weight 0.785 0.688 1.08 0.56 size 0.48 0.63 0.47 0.68 service 1.08 0.78 1.2 1.035 design 1.09 0.68 1.08 1.12 appearance 1.14 1.18 1.04 1.38 furthermore, the study incorporates inversion and imitation techniques to generate synthetic samples for data balancing. while these techniques provide a means to address class imbalance, they may introduce certain limitations in terms of semantic coherence. the inversion of majority-class samples and the replacement of words in minority-class samples may result in synthetic data that lacks complete grammatical or semantic consistency. future research should explore approaches that can generate synthetic samples while ensuring greater semantic coherency. lastly, the evaluation of the proposed approach primarily focuses on sentiment analysis of amazon data. although sentiment analysis on facebook data is briefly mentioned, further exploration and evaluation of the approach on other social media platforms, such as twitter or instagram, would provide a more comprehensive understanding of its effectiveness across different social media contexts. in addition to the mentioned limitations, it is important to address the ethical implications and potential biases associated with sentiment analysis in social media marketing. these considerations should be discussed alongside the limitations to provide a comprehensive understanding of the challenges and ethical concerns in the proposed approach. sentiment analysis in social media marketing can have ethical implications and biases that need to be considered. the analysis of user-generated content on social media platforms raises concerns regarding privacy, consent, and data usage. it is crucial to handle user data responsibly, ensuring compliance with relevant regulations and obtaining appropriate consent for data collection and analysis. transparency in data usage and providing clear explanations of the purpose and potential impact of sentiment analysis can help address ethical concerns and build trust with users. furthermore, biases can emerge in sentiment analysis due to various factors. cultural, demographic, and socioeconomic biases inherent in social media platforms can influence the sentiments expressed by users and introduce skewed results. the representativeness of the data used for sentiment analysis is crucial in minimizing biases. efforts should be made to include diverse data sources that capture a wide range of perspectives. however, it is important to acknowledge that complete representativeness may not always be achievable. steps should be taken to ensure fairness and impartiality in sentiment analysis. the proposed approach can incorporate techniques such as data augmentation and resampling to address imbalances and increase inclusivity. however, it is essential to continuously monitor and evaluate the sentiment analysis model to identify and rectify any biases that may arise. involving diverse and representative annotation teams can also help mitigate potential biases during the data labeling process. additionally, it is important to recognize that biases may still persist in sentiment analysis results due to the inherent biases present in the data itself. interpreting the analysis results with caution and considering the context and limitations of the data sources is essential in avoiding biased conclusions. 498 m. kuang, r. safa, s. a. edalatpanah, r. s. keyser by acknowledging these ethical implications and biases, researchers and practitioners can work towards developing more robust and fair sentiment analysis approaches in social media marketing. 6. conclusions this study developed a social media marketing approach based on neural networks. therefore, an improved deep learning algorithm, called attention-based rnn, was used to classify sentiments into positive and negative categories. we used data from amazon's website, including four product categories: cars, household appliances, luxury goods, and clothing and jewelry. an issue with this data that affects the sentiment classification performance is that the data has an imbalanced distribution between different classes and positive and negative samples. therefore, it influences sentiment classification performance and increases the rate of misclassifications. we addressed this issue using a resampling strategy for data balancing, where minority-class samples increase and majority-class samples decrease. in addition, inversion and imitation techniques were used to generate new samples. the inversion technique inverts some majority-class samples and converts them into minority classes. in contrast, the imitation technique increases minority-class samples by replacing some words. alternative words for inversion or imitation were selected using the weighted lexicon graph technique. the method computes the semantic similarity between words and finds their antonyms or synonyms based on several seed words. the facebook dataset was used for social media-based sentiment analysis after generating a neural network model using the amazon dataset. then, these data were applied to the model generated using amazon data. finally, the products were evaluated and rated based on various aspects such as "price," "quality," "appearance," and others. these ratings express people's sentiments on social media towards different aspects of a product. hence, sellers and marketers can use these analytics for marketing their products based on their intended characteristics. in addition, we performed different experiments to evaluate the proposed model. these experiments modeled the rnn-based algorithm using five hops. first, sentiment classification performance was assessed for both facebook and amazon datasets. the results also showed that sentiment analysis for amazon data outperforms facebook data. the neural network was modeled based on amazon data with different product categories. however, the classifier's performance was acceptable for facebook data (about 85%). the results of applying facebook data to find sentimental aspects for different product 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808525. 28. ibrahim, b., 2022, social media marketing activities and brand loyalty: a meta-analysis examination, journal of promotion management, 28(1), pp. 60-90. solar vacuum tube integrated seawater distillation an experimental study facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 113 120 original scientific paper solar vacuum tube integrated seawater distillation an experimental study  udc 621.1 selcuk selimli, ziyaddin recebli, semra ulker energy systems engineering, university of karabuk, turkey abstract. the subject of this study is the seawater distillation process enhancement through integration of the solar vacuum tube into the system. positive effects on the rate of distillated freshwater achieved by means of the enhanced system have been investigated experimentally. experiments were done in the turkish city of samsun in the black sea region. a distillation pond setup having the volume of 0.015m3 and a water surface area of 0.24m2 was constructed. the distillation pond is covered with a condensation glass and also equipped with a 0.15m solar vacuum tube that is inclined at an angle of 30o to the ground, a feed water tank connected with a ball cock, and distillated fresh water tank. experimental results have shown that the rate of distilled fresh water was enhanced for about 62.5% by integrating the solar vacuum tube and the natural distillation pond. isolation of the condensation glass not only prevents the light transmission to the pond but also heat loss from the pond; hence the rate of the distillated fresh water is increased for about 137.5% due to the natural distillation. key words: solar distillation, solar vacuum tube, solar pond, seawater 1. introduction useable water sources have been depleting rapidly by population growth, increased demands and industrial development. most of the countries, including the united arab emirates, kuwait, libya, japan, qatar, spain, italy, iran, suffer from the lack of fresh water sources; they try to solve this problem by seawater distillation and other alternative methods. a number of studies have been conducted with the similar aim of determining a highly efficient, economical and high rate fresh water distillation process. suarez et al. [1] investigated solar-powered thermal desalination. they found that the heat loss reduction received november 15, 2015 / accepted january 10, 2016 corresponding author: selcuk selimli energys systems engineering, karabuk university, 78050, karabuk, turkey e-mail:selcukselimli@karabuk.edu.tr 114 s. selimli, z. recebli, s. ulker throughout the system would yield higher water fluxes, thus pointing to the need to improve the system efficiency. gonzalez et al. [2] contributed both in the modeling issue, proposing and validating a solar field linear model, and applying an economic optimal control paradigm to solar seawater desalination plants. gray et al. [3] discussed three alternative desalination technologies, namely, membrane distillation, forward osmosis, and capacitive deionization. li et al. [4] reviewed the current solar desalination research activities, discussed the solar-assisted desalination processes and a variety of possible combinations. aybar [5] investigated theoretically an inclined solar water distillation system. khawaji et al. [6] considered the current status, practices, and advances of the seawater desalination technologies. lamei et al. [7] compared the cost of seawater desalination by different solar energy technologies. sampathkumar et al. [8] provided a detailed review of different studies on the active solar distillation system. wu et al. [9] investigated experimentally a new multieffect solar desalination system that was based on the process of humidification/ dehumidification. sebaii and bialy [10] reviewed different designs of solar stills that are double, triple and multi-effect solar stills, vertical stills, tubular type solar stills, finned and corrugated stills, and stepped type solar stills. kalogirou [11] discussed various systems that use renewable energy sources for desalination. aybar et al. [12] studied experimentally a solar water distillation system under the actual environmental conditions. they concluded that the fresh water distillation rate increased about two to three times when wicks were used instead of a bare plate. sampathkumar and senthilkumar [13] studied the effective utilization of the solar water heater for solar still productivity enhancement that operates as a hybrid system. can et al. [14] investigated technically and economically different methods to desalinate seawater in order to obtain, in this way, potable and usable water. aydın and ardalı [15] studied technologies for obtaining fresh water. ketrez et al. [16] discussed the seawater distillation by solar energy. salim et al. [17] compared three units of the solar water distillation setup which are passive solar still, passive solar still coupled with the collector, and solar still coupled with the collector supported by the copper tube to realize the latent heat. they concluded that the solar collector coupled system productivity is for about 42% higher than the passive solar still. rajesh and bharath [18] compared the productivity performance of the seawater distillation by a single basin still and a coupled still with the solar collector. their study showed that the single basin still productivity enhanced for about 40% upon its coupling with the solar collector. shabibi and tahat [19] investigated experimentally the thermal performance of the conventional solar water still with an enhanced solar heating system. it is concluded that the fresh water yielded enhancement of about 50% by the system coupled with the solar preheater. panchal [20] searched a double basin solar still coupled with evacuated tubes. it was found that the distillate output increased for about 56% by adding vacuum tubes and for 65% by adding vacuum tubes and black granite gravel in the double basin solar still. rehim and lasheen [21] presented two modifications for the solar desalination systems. they concluded that the modified solar desalination system efficiency was increased. solar vacuum tube integrated seawaterdistillation, an experimental study 115 2. theoretical background the energy balance of the desalination process is given by the following equation: )()()( vtuegcswgcswsw sw swsw qqqaia dt dt cm        (1) where msw is the mass of seawater, csw is the specific heat of seawater, tsw is the temperature of seawater, i is the radiative flux density, asw is the absorptivity of seawater, asw is the surface area, qc(gsw) is the convective heat transfer from cover glass to seawater, qc(ge) is the convective heat transfer from cover glass to environment and qu(vt) is the useful heat by vacuum tube. convectional heat transfer between condensation glass and seawater qc(gsw) is estimated as follows [22]: )()()( swgswswgswgc tahq   (2) where h(g-sw) represents the convectional heat transfer coefficient [22]: 1 3 3 ( ) ( )0.884( ( )( ) /(268.9 10 ))g sw g sw sw g sw swh t p p t p       (3) with p denoting the partial pressure. convectional heat transfer from the condensation glass to environment (qc(gsw)) is estimated using the following equations [22]: )()()( eggegegc tahq   (4) uh eg 38.2)(  (5) where h(ge) is the convectional heat transfer coefficient, ag is the condensation glass surface area and u is the wind velocity. solar vacuum tube useful energy input to the pond is denoted as qu(vt), and can be given as [23]: ( ) ( )u vt vt vt vt rr e sq a i a e q q    (6) where avt is the absorptivity of vacuum tube, ivt is the vacuum tube insolation, avt is the vacuum tube window area, err is the reradiated energy flux, qe is the convection energy to environment from vacuum tube and, qs is the energy stored by vacuum tube body. in this study, the process of obtaining freshwater by distilling seawater in a solar pond coupled with a solar vacuum tube has been investigated. natural water circulation in the solar vacuum tube is the advantage that enables a higher rate of distilled freshwater without any external circulation energy input. the experimental study has been conducted considering the economic and environmental advantages of this process. 116 s. selimli, z. recebli, s. ulker 3. experimental setup in this experimental study, a pond setup is constructed with 0.24 m 2 water surface area, and 0.015m 3 water capacity. the constructed pond is equipped with a 0.15 m solar vacuum tube. water circulation in the solar vacuum tubes occurs naturally and so it does not require any external circulation energy input into the system. the vacuum tube is inclined at an angle of 30 o for optimal solar light absorption. the solar pond internal surfaces are painted with black dye because of its high absorption coefficient (about 0.96). the solar pond is filled with seawater up to the level of about 0.05m. the feed water tank is attached to the pond with a ball cock to maintain constant water level in the pond. firstly, the water heated by solar energy evaporates; a high temperature steam meets low temperature condensation glass and condenses. condensed water droplets on the condensation glass come together and drip to the fresh water separation channel. distillated water in the channel flows in a pipe line and collects in the fresh water storage tank. water and condensation glass temperature is measured by t-type thermocouple. potential difference between thermocouple ends is read by voltcraft m3850 model multimeter. solar radiation is measured by cem dt-1307 model solarimeter. the experimental setup is illustrated in fig. 1. fig. 1 experimental setup distillation performance is investigated for three different setup configurations. firstly, the solar vacuum tube is isolated and the water is exposed to solar radiation from the condensation glass. secondly, the solar vacuum tube is uncovered and the solar penetration to the water is allowed from the tube and condensation glass. lastly, the condensation glass is isolated and solar radiation allowed only from the solar vacuum tube. except for the aforementioned surfaces, all other surfaces of the setup are isolated to avoid heat losses. each configuration is studied in three consecutive cloudless spring days. measurements were taken every 15 minutes between 10:00 am to 4:30 pm and recorded. solar vacuum tube integrated seawaterdistillation, an experimental study 117 4. results and discussion the obtained measurement data are evaluated and the relation between the thermal parameters and the distilled fresh water rate is assessed in this section. fig. 2 illustrates the temperature variation of the condensation glass for different system configurations and environmental effects with respect to the time. solar radiative density is also given. isolation of condensation glass prevents heat loss from the glass. increase in the water temperature by solar energy from the vacuum tube increases the heat transfer from hot water to glass. the glass temperature fluctuates due to the variation of the flux density and heat loss from the glass surface for the non-isolated glass and vacuum tube configuration. the decrease in the temperature in the period of time between 50-100 min could be related to the low environmental temperature. the high temperature difference between the glass and the environment causes a higher heat loss from the glass than the supplied heat by solar radiation. fig. 2 solar radiative flux density and condensation glass temperature fig. 3 illustrates the distillation pond water temperature and solar radiative heat flux. it shows that the transmitted natural solar radiation and the enforced radiation by the solar vacuum tube tend to increase the overall water temperature. similar results are reported in [20] emphasizing that the integrated vacuum tube increases the water temperature compared to the single basin solar still. however, the low environmental temperature causes a higher heat loss via the glass surface than the heat supplied by solar radiation in the first hours of the experiment (between 50-100 min) and so there is a sudden water temperature decrease. the configuration with the isolated tube, i.e. with the water heating only by the natural solar radiation via the condensation glass, slightly increases the water temperature. the most efficient heating of water is provided by the isolation of the condensation glass so that the heat loss from the system is reduced thus causing a notable increase in the water temperature. 118 s. selimli, z. recebli, s. ulker fig. 3 solar radiative heat flux and solar pond water temperature fig. 4 shows the condensed fresh water quantities obtained by the three different experimental configurations. the quantity of condensed fresh water varies depending on the water temperature, glass temperature and the heat transferred to the water. about 160 ml of fresh water was obtained in the configuration that allowed solar radiation from the condensation glass. low radiation heat transfer and a high heat loss via the condensation glass cause a low temperature difference (about 5 o c) between the water and the condensation glass. low energy transfer means the low vaporization. likewise, a low temperature difference leads to a low condensation rate. the non-isolated condensation glass and the vacuum tube cause a higher temperature difference and a greater heat loss via the glass surface. also, the low temperature difference (about 8 o c) between the water and the condensation glass resulted in 260 ml of distilled fresh water. the highest quantity of fresh water was obtained in the case that combined the isolated condensation glass and the non-isolated solar vacuum tube. the isolated condensation glass means a negligible amount of heat loss via the glass to the environment and the highest amount of the heat gained by the system. additionally, the highest temperature difference (about 12 o c) between the condensation glass and the pond water led to the highest quantity of condensed fresh water of 380 ml. 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 1 2 3 160 260 380 glass isolatednonisolatedtube isolated ml ml ml condensed water fig. 4 condensed fresh water quantity solar vacuum tube integrated seawaterdistillation, an experimental study 119 5. conclusion and outlook in this study, the advantages of seawater distillation conducted on a solar pond with an integrated solar vacuum tube have been investigated experimentally. the process can be evaluated as environment friendly, economical, and efficient. the solar vacuum tube increases the gained heat energy and the quantity of distillated water. the tube has a natural fluid circulation and, hence no external circulation energy input is required. additionally, it has a long service life and a wide range of temperature operability. the investigated distillation process is applicable to the districts that generally suffer from the lack of energy except for solar one. the fresh water distillation rate was increased by 62.5% through integration of the solar vacuum tube into the system. this result was even further enhanced up to 137.5% by the elimination of heat loss from the condensation glass. the problem of an efficient high rate of the fresh water production by alternative methods in order to solve the problem of the people living in the areas with scarce water sources has been investigated by other researchers in the field. the solar pond coupled with a vacuum tube configuration was studied in order to contribute to the studies in the field and to focus on more environment friendly, efficient, and mobile configurations which are independent of exhaustible energy. references 1. suarez, f., ruskowitz, j.a., tyler, s.w., childress, a.e., 2015, renewable water: direct contact membrane distillation coupled with solar ponds, applied energy, 158, pp. 532-539. 2. gonzalez, r., roca, l., rodriguez, f., 2014, economic optimal control applied to a solar seawater desalination plant, computers and chemical engineering, 71, pp. 554-562. 3. gray, s., semiat, r., duke, m., rahardianto, a., cohen, y., 2011, seawater use and desalination technology, earth systems and environmental sciences, 4, pp. 73-109. 4. li, c., yogi, g., elias, s., 2013, solar assisted seawater desalination: a review, renewable and sustainable energy reviews, 19, pp. 136-163. 5. aybar, h.s., 2006, mathematical modelling of and inclined solar water distillation system, desalination, 190, pp. 63-70. 6. khawaji, a.d., kutubkhanah, i.k., wie, j.m., 2008, advances in seawater desalination technologies, desalination, 221, pp. 47-69. 7. lamei, a., zaag, p.v., munch, e.v., 2008, impact of solar energy cost on water production cost of seawater desalination plants in egypt, energy policy, 36, pp. 1748-1756. 8. sampathkumar, k., arjunan, t.v., pitchandi, p., senthilkumar, p., 2010, active solar distillation – a detailed review, renewable and sustainable energy reviews, 14, pp. 1503-1526. 9. wu, g., zheng, h., kang, h., yang, y., cheng, p., chang, z., 2016, experimental investigation of a multi-effect isothermal heat with tandem solar desalination system based on humidification – dehumidification process, desalination, 378, pp. 100-107. 10. sebaii, a.a.e., bialy, e.e., 2015, advanced designs of solar desalination systems: a review, renewable and sustainable energy reviews, 49, pp. 1198-1212. 11. kalogirou, s.a., 2005, seawater desalination using renewable energy sources, progress in energy and combustion science, 31, pp. 242-281. 12. aybar, h.s., egelioglu, f., atikol, u., 2005, an experimental study on an inclined solar water distillation system, desalination, 180, pp. 285-289. 13. sampathkumar, k., senthilkumar, p., 2012, utilization of solar water heater in a single basin solar still – an experimental study, desalination, 297, pp. 8-19. 14. can, m., etemoglu, a.b., avcı, a., 2002, technical and economic analysis of desalination process, uludag university journal of the faculty of engineering and architecture, 7(1), pp. 147-160. 15. aydın, f., ardalı, y., seawater desalination technologies, 2012, sigma journal of engineering and natural sciences, 30, pp. 156-178. 120 s. selimli, z. recebli, s. ulker 16. ketrez, m., usta, h., cay, y., 2012, heat pump assisted solar energy water distillation, journal of sanitary engineering, 127, pp. 5-8. 17. salim, r.d., al-asadi, j.m., hashim, a.y., 2015, design and manufacture three solar distillation units and measuring their productivity, science journal of energy engineering, 3(2), pp. 6-10. 18. rajesh, a.m., bharath, k.n., 2009, solar still coupled with solar collector and storage tank, interjri science and technology, 1(2), pp. 62-72. 19. shabibi, a.m.a., tahat, m., 2015, thermal performance of a single slope solar water still with enhanced solar heating system, renewable energy and power quality journal, 13, pp. 1-3. 20. panchal, h.n., 2015, enhancement of distillate output of double basin solar still with vacuum tubes, journal of king saud university-engineering sciences, 27, pp. 170-175. 21. rehim, z.s.a., lasheen, a., 2005, improving the performance of solar desalination systems, renewable energy, 30, pp. 1955-1971. 22. kumar, d., kumar, p., 2014, mathematical modelling of conventional solar still coupled with solar air heater, international journal of innovative science, engineering & technology, 1(9), pp. 379-383. 23. winter, f.d., 1991, solar collectors, energy storage, and materials, mit press, cambridge 12394 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 473 484 https://doi.org/10.22190/fume231219010k © 2024 by university of niš, serba | creative commons license: cc by-nc-nd original scientific paper longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure khayrulla khudoynazarov samarkand state university, uzbekistan orcid id: khayrulla khudoynazarov https://orcid.org/0000-0001-8994-9738 abstract. the paper considers a cylindrical three-layer structure of arbitrary thickness made of viscoelastic material. it consists of two external bearing layers and a middle layer, the materials of which are generally different. the problem of nonstationary longitudinal-radial vibrations of such a structure is formulated. based on the exact solutions in transformations of the three-dimensional problem of the linear theory of viscoelasticity for a circular cylindrical three-layer body, a mathematical model of its nonstationary longitudinal-radial vibrations is developed. equations are derived that allow, based on the results of solving the vibration equations, to determine the stressstrain state of a cylindrical structure and its layers in arbitrary sections. the results obtained allow for special cases of transition into cylindrical viscoelastic and elastic twolayer structures, as well as into homogeneous single-layer cylindrical structures and round rods. key words: three-layer structure, vibration, stress, torsional displacement, loadbearing layers, non-stationary 1. introduction three-layer structural elements are widely used in aviation and shipbuilding, construction of buildings and structures, the space industry and other industries [1,2]. therefore, the problem of developing effective methods for calculating the stress-strain state of three-layer structural elements, as well as generalizing classical theories using refined models reflecting the dynamic behavior of modern materials, is urgent [3,4]. in this regard, cylindrical structures and round rods are one of the main elements of various engineering structures [5] and studies of their dynamic behavior have important applied values [6,7]. such elements are often under the influence of dynamic loads during operation, which lead to their vibrations [8,9]. recently, special attention has been given to study vibrations of layered structures made of homogeneous and functionally graded materials (fgm) [10]. these include works where vibrations of three-layer plates are considered, considering imperfect [11], slipping received: december 19, 2023 / accepted march 03, 2024 corresponding author: khayrulla khudoynazarov samarkand state university, 100104, 15 university boulevard, samarkand, uzbekistan e-mail: kh.khudoyn@gmail.com https://orcid.org/0000-0001-8994-9738 https://e.mail.ru/compose/?mailto=mailto%3akh.khudoyn@gmail.com 474 kh. khudoynazarov [12] contacts between layers, cylindrical [13] and conical [14] fgm shells. in studies devoted to the dynamic behavior of elements of engineering structures, it is important to develop mathematical foundations for such structures [15,16] for design of new generations of improved lightweight structural materials [17]. one of the main problems in the study of the static and dynamic behavior of shells and rods is the choice of vibration equations, which should be implemented based on the specific physical and mechanical properties of their materials [18]. various methods of derivation of vibration equations are used. one of these methods is the method of using general solutions in transformations of three-dimensional problems of elasticity theory [19,20]. the essence of the method is to study the constructed solutions for various types of external influences [21] and to clarify the conditions under which the displacements or their “main parts” satisfy simple vibration equations, and to find an algorithm that allows calculating approximate values of displacement and stress fields in any cross section for an arbitrary a moment in time. similar studies, but considering more complex physical and mechanical properties, in particular viscoelastic ones, are considered in [22,23]. the analysis of the behavior of elements of engineering structures, taking into account the layering and heterogeneity of structures based on computational models, is relevant for applied problems, as evidenced by publications [24,25]. in addition, a fairly large number of studies of shell dynamics are carried out, which take into account the influence of hyperelastic [26], anisotropic, temperature and other physical and mechanical properties of the material. thus, it can be argued that at present there is a very limited number of works devoted to the practically important task of studying non-stationary longitudinal-radial vibrations of cylindrical three-layer structures of arbitrary thickness. therefore, the problem of creating models for the dynamic calculation of such systems under the influence of dynamic loads, taking into account various physical and mechanical properties of their material, is urgent. in this article, a circular cylindrical three-layer viscoelastic structure of arbitrary thickness with sticking condition between the layers is considered. the task is to study its nonstationary longitudinal-radial vibrations based on the above-mentioned method of exact solutions in transformations. it is envisaged to build a mathematical model of it, including the derivation of general and refined vibration equations and the creation of an algorithm that allows determining the stress-strain state of an arbitrary section of the structure in coordinate and time using the field of desired functions. 2. mathematical model of the problem 2.1. formulation of the problem in the cylindrical coordinate system (r, θ, z), a three-layer circular cylindrical structure made of viscoelastic material is considered. it is assumed, that the structure consists of two layers, hereinafter called load-bearing layers, which are separated by a certain distance using the third layer. the intermediate layer holds the load-bearing layers at a distance. the axis oz of the coordinate system is directed along the axis of symmetry of the structure perpendicular to the cross section and we schematic picture of layers is shown in fig.1. through a and b we denote the inner and outer radii of the cylindrical structure, and through r1 and r2 the inner and outer radii of the middle layer. when deriving the vibration equations, we assume that both the cylindrical structure as a whole and its layers separately longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure 475 strictly obey the mathematical theory of viscoelasticity and are described in an accurate formulation by its three-dimensional equations in a linear formulation. fig. 1 cross section of a three-layer structure with longitudinal radial vibrations of the cylindrical structure, only the components of displacements wm, um, and stresses ( )m rr , ( )m  , ( )m zz , ( )m zr (m=0,1,2), will be different from zero [20]. accordingly, the equations of motion of points of a viscoelastic structure are taken in the form of wave equations with respect to the potentials of longitudinal φm and transverse χm waves in the layers of the structure: ( ) ( ) ( ) 2 0 2 2 0 12 , 2 , , 0,1,2 , , m m m m m m m m m m m r r r r t r m a r r t    = = +      = =              (1) where 2 2 0 2 2 1 r rz r     = + +   , while m r , m r are the integral operators defined by the formula ( ) ( , ), 0 ( ) ( , ) ( ) ( ) ( ) mm t m mr t k t d   = − −                , λm, μm are the (lame)coefficients of layer materials, ( , ) ( ) m k t −   is the kernel of the integral operators. it is assumed that the viscoelastic operators ( , ) ( ) m r    are reversible, and their kernels are arbitrary. here and everywhere else, the index m takes values 0, 1, 2. therefore, in the following, this will not be emphasized every time, implying that this is always the case. it is assumed that the cylindrical structure is at rest prior to loading, and at the moment t=0, external surface’s stresses ( ) ( , )i rf z t , ( ) ( , )i rzf z t (i=1,2) are applied, i.e. it is assumed that the boundary conditions have the form: (1) (1) (1) (1) (2) (2) (2) (2) ( , , ) ( , ), ( , , ) ( , ) , ( , , ) ( , ), ( , , ) ( , ) . rr r rz rz rr r rz rz a z t f z t a z t f z t at r a b z t f z t b z t f z t at r b  = = =  = = =     (2) 476 kh. khudoynazarov in addition, the sticking conditions must be met on the contact surfaces between the layers of the structure, which require equal displacements and stresses, i.e. contact conditions have the following form at r=ri, i=1,2: 0 0 (0) ( ) (0) ( ) ( , , ) ( , , ), ( , , ) ( , , ), ( , , ) ( , , ), ( , , ) ( , , ), ( 1,2). i i i i i i i i rr i rr i rz i rz i w r z t w r z t u r z t u r z t r z t r z t r z t r z t i = =  = = =    (3) the initial conditions of the problem are considered zero, i.e. at t=0 0, 0, 0, 0m m m m t t   = = = =       . (4) 2.2. derivation of vibration equations to solve the formulated problem of torsional vibrations of a three-layer cylindrical viscoelastic shell, the functions ( ) ( , )i rf z t , ( ) ( , )i rzf z t (i=1,2) of external influences under boundary conditions (3) are considered in the class of functions represented as [21]: ( ) ( ) ( ) ( ) 0 ( ) 0 ( ) sin cos ( , ) , ( , ) cos sin i i pt i i pt r r rz rz l l kz kz f dk f k p e dp f dk f k p e dp kz kz    = =  −       . (5) here (l) is an open contour in the plane p adjacent to the right section (–iω0, iω0) of the imaginary axis. furtgermore, the functions ( ) ( , )i rf z t , ( ) ( , )i rzf z t are assumed to be such that the functions ( ) ( , )i rf z t , ( ) ( , )i rzf z t are negligibly small outside the domain {0<k<k0, im|p|<ω0} let’s assume the potential functions φm and χm also in the form of eq. (5). substituting them into the equations of motion (eq. (1)), we obtain ordinary bessel differential equations for the transformed potential functions m , m (m=0,1,2): 2 2 2 2 2 2 1 1 0, 0m m m d d d d r dr r drdr dr         + − = + − =        (6) where 0 2 2 1 2 2 2 1 2 , , , , ( ) ( ) , ( 2 ) 1 ( ) 1 ( ) ( ) 2 ( ) ( ) 2 m m m m m m m pt m m m m m m m m m m m m m m m p r k k p k t e dt r k p r k p k t k t p r k k t  − − −         = + = = + − = − + = + = +                  . (7) the general solution of eq. (6) is (1) (2) (1) (2) 0 0 0 0( ) ( ) ( ), ( ) ( ) ( ),m m m m m m m m m mr a i r a k r r c i r c k r= + = +      (8) where ( ) ( , )i ma k p , ( ) ( , )i mc k p , (m=0,1,2) (i=1,2) are arbitrary, regarding variable r, integration constants. the number of these constants is generally 12. it can be reduced based on the following considerations. the general solutions given by eq. (8) for all three layers have the same structure. they should be finite in the limits r → 0 and r → ∞. at the same time, the boundaries of the inner layer are smooth a and r1, that is a ≤ r ≤ r1. it is bounded from longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure 477 below (from the inside) by the surface r = a, which may tend to zero in the limit, but exceed in no way the values of r1, i.e. it cannot strive for infinity. therefore, in general solutions for the potential functions of the inner layer 1( )r , 1( )r , we can limit ourselves to taking into account their layer in the form of: 1 1 0 1 1 1 0 1( ) ( ), ( ) ( ).r a i r r d i r= =    (9) similarly, the boundaries of the second, outer layer are cylindrical surfaces with radii r = r2 and r = b that is r2 ≤ r ≤b. this layer is bounded externally by the surface of radius r = b, which may tend to infinity, i.e. b → ∞. on the other hand, the inner surface of this layer cannot be tightened to a straight line, because this would lead to a homogeneous rod of circular cross-section of radius r = b. therefore, in general solutions for the potential functions of the outer layer 2 ( )r , 2 ( )r , we can limit ourselves to taking into account its limitations only at r → ∞. based on this, we will take the general solution, eq. (8), for the outer layer as: 2 2 0 2 2 2 0 2 2( )( ) ( ), ( ) ( ), r r br a k r r d k r  = =    . (10) for the middle layer, we will take general solutions. eq. (8), considering that these solutions, in the absence of the two outer layers, should transform into known solutions for a homogeneous cylindrical layer, limited at r → 0 and r → ∞ at the same time: 0 1 0 0 2 0 0 0 1 0 0 2 0 0 1 2( ) ( ) ( ), ( ) ( ) ( ), r r rr b i r b k r r c i r c k r  = + = +      (11) thus, the number of integration constants to be determined from the contact conditions is reduced to eight. let us assume the stress components, both ( ) ( , , )m rr r z t and ( ) ( , , )m rz r z t as ( ) ( ) ( ) ( ) ( ) ( ) 0 0 sin cos , cos sin m m pt m m pt rr rr rz rz l l kz kz dk e dp dk e dp kz kz    = =  −          . (12) substituting eqs. (12) and (5) in the boundary conditions, eq. (2), and expressing them in terms of solutions given by eqs. (9) and (10), we obtain: 1 1 2 2 2 1 (1)1 1 1 0 1 1 1 1 1 0 1 1 1, 1 2 2 1 (1) 1 1 1 1 1 1 1 1 1 2 2 ( ) ( ) ( ) 2 ( ) ( ) ( , ) , 2 ( ) ( ) ( ) ( , ) , r rz k k i a i a a k i a i a d r f k p a a k i a a k i a d r f k p − −      + − − − =           − + =                  (13) 2 2 2 2 2 1 (2)2 2 2 0 2 1 2 2 2 0 2 1 2 2 2 2 1 (2) 2 1 2 2 2 2 1 2 2 2 2 ( ) ( ) ( ) 2 ( ) ( ) ( , ) , 2 ( ) ( ) ( ) ( , ) , r rz k k k b k b a k k b k b d r f k p b b k k b a k k b d r f k p − −      + + − + =           − + = −                  (14) let us express the contact conditions also through general solutions. in this case, the number of unknown constants will be only 8. therefore, due to the fact that we use four boundary conditions, eq. (2), it is sufficient to take four of the eight contact conditions given by eq. (3). in addition, the boundary conditions, eq. (2), are dynamic, therefore, for contact conditions we will use only the kinematic part of the conditions given by eq. (3). let us also assume the displacements in form of eq. (5) and apply together with eq. (12) 478 kh. khudoynazarov to the contact conditions, eq. (3), in the displacements. expressing the obtained contact conditions through general solutions, eqs. (9-11), we find the constants ai, di (i=1,2). substituting the found expressions of the integration constants into eqs. (13) and (14) gives the following system of four equations: 2 1 2 2 ( ) 1 0 2 0 2 2 2 1 2 2 ( ) 3 0 4 0 1 3 3 2 ( , ) ( ) ( , ) ( ) ( ) ( , ) , ( ) 42 ( , ) ( ) ( , ) ( ) ( ) ( , ) , ( ) i i ii i i i i i i i i i r i ii i i i i i i i i rz i r g u r g w r r k f k p r a b r g u r g w r r k f k p r a b − −  + = +   + = +              (15) where 2 2 2 2 2 2 1 2 2 2 2 2 2 2 3 4 1 1 1 1 ( , ) , ( , ) (2 ) 4 4 ( , ) ( 3 ), ( , ) (2 ) i i i i i i i i i i i i i i i i i i g k k g k g k g k k      = + + =  −          = + = − −                 2 2 2 2 , 1, ( ) , 2. a if i a b b if i  = =  = let us highlight the main parts of the longitudinal radial displacement of the middle layer of the shell. for this, the displacements of the points of the median layer ũ0(r,k,p) and 0w (r,k,p) transformed by eq. (5) are expressed in terms of general solutions, eq. (11). we will have         2 0 1 0 0 2 0 0 0 1 0 0 2 0 0 0 0 1 1 0 2 1 0 0 1 1 0 2 1 0 ( , , ) ( ) ( ) ( ) ( ) , ( , , ) ( ) ( ) ( ) ( ) . u r k p k b i r b k r c i r c k r w r k p b i r b k r k c i r c k r = + − + = − − −            (16) we introduce the radius of some intermediate surface of the middle layer of the structure according to the following formula [27]: 1 1 22 r r r   = −      . writing eqs. (16) through the standard decompositions of the bessel functions ii and ki (i=0,1) into series according to the degrees of the radial coordinate r and, limiting them to the first terms at r=ξ, we obtain the indicated main parts, which we denote by 0,0 0,1,u u and 0,0 0,1,w w : 2 2 0,0 10 0 10 0,1 2 0 2 2 2 0,0 0 10 0 10 0,1 2 2 , ( ) / , ( ) / u kb c u kb c w b k c w b kc = − = − = − = −       (17) where: 0 0 10 1 2 10 1 2 1 1 ln (1) , ln (1) 2 2 2 2 b b b c c c     = − − − = − − −              , and ψ(n), n=1,2, … is the logarithmic derivative of the gamma function. writing eqs.(16) for the transformed displacements ũ0(r,k,p) and 0w (r,k,p) through the standard decompositions of the bessel functions ii and ki (i=0,1) into series according to longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure 479 the degrees of the radial coordinate r and substituting in them the values of the constants b10, c10, b2 and c2, found from the system of eqs. (17), we obtain   2 (0) (0) 2 0 0 0,0 1 0,0 0 0,0 0 2 2 2 (0) (0) 0 0,1 0 2 0,1 0,1 3, 2 ( , , ) ( ) ( 2) ( ) ( ) , ( !) n n n n n n n u r k p u q q kw u r u q q kw u r n  = = + − −  − + −         (18)   2 2 (0) (0) 0 0,1 0 0,0 0 1 0,0 0,0 0 2 1 2 2 (0) (0) 2 0 0,1 2 1 0 0,1 0,1 1, ( , , ) ( ) ( 2) ( ) ( ) !( 1)! n n n n n n n w r k p w w q q w ku r r w q q w ku r n n  = + + + = − +  + − −  − + −  +         (19) where: 0 0 (0) 1 (0) 1 1 0 2 0 3, 1 1 1 1 ; 1; ln ; 2 n n k r q r r q r r k − − = = −  =  − = + −    ( ) 2 2 1 2 1(0) 2 (0) (0) (0) 2 20 0 0 0 0 1 2 0 02 2 10 0 ; 0; 1; . n n n n k k n k q q q q − − − = − = = = = = + −          let us introduce the following representations for functions u0,i(r,z,t) and w0,i(r,z,t): 0, 0, 0, 0, 0 ( ) 0 ( ) cos sin , sin cos pt pt i i i i l l kz kz u dk u e dp w dk w e dp kz kz    = =  −      (20) and operators ( ), ( )n n m m    , (m=0,1,2; n=0,1,2,…) 2 2 0 ( ) sin ( ), ( ) ( , ) ( , , ) , cos n n n n pt m m m m l kz dk r k p e dp kz    =   −          (21) similarly to eqs. (18) and (19), we express eq. (15) in terms of the main parts of the transformed longitudinal 0,0 0,1,u u and radial 0,0 0,1,w w displacements. limiting ourselves to the first approximations of (n=0) in infinite series in the obtained equations and applying operators given in eqs. (20) and (21) to them, we obtain: 1 0,0 0,1(0) 1 2 0,0 1 3,0 2 2 2 2 (0) 1 ( ) 2 2 0 0,12 2 2 2 ( ) ln 2 (1 ) ln ( , ) ( ) i i i i i i ii i i i r i u ur g g w g r g q z z r r g q w r f z t r a b z −    − + + − −          − + + = −                  (22) 2 2 0,0 (0) 3 0,0 4 3 3,0 4 2 0,12 2 2 0,1(0) 1 ( ) 4 2 0 2 3 3 2 ( ) ln 42 (1 ) ln ( , ) , ( ) i i i i i i ii i i i rz i w r g u g g r g q u z z wr r g q r f z t zr a b z −    − − + +          + + + = −                   (23) 480 kh. khudoynazarov based on eqs. (7) for the values αm and βm (m=0,1,2), it is easy to show that the operators n m and n m , m=0,1,2; n=0,1,2,… introduced by eqs. (21) during the reverse transition along coordinate z and time t have the following form: 2 2 2 2 1 1 2 2 2 2 ( ) , ( ) 0,1,2; 1,2,3,... m n n n n m m m m mr r t z t z m n − −           = − = −                    = =            (24) the system of eqs. (22) and (23), in accordance with eqs. (21) for operators n m , m=0,1,2; n=0,1,2,… are integro-differential equations. these equations contain the main parts u0,0, u0,1 and w0,0, w0,1, respectively, longitudinal – u0 and radial – w0 displacements of points of some “intermediate” surface of the middle layer of a three-layer cylindrical structure. the specified “intermediate” surface has a radius, the values of which are enclosed in the range r1 ≤ ξ ≤ r2 in accordance with the numerical values of radius ξ, this “intermediate” surface can turn into a median surface at ξ = (r1+r2)/2 and contact surfaces between layers of the structure at ξ = r1 and ξ = r2. therefore, the equations are a system of eqs. (22) and (23) depending on the values of the radius ξ. the equations of vibration of a three-layer cylindrical structure relative to the main parts of the longitudinal and radial displacements of the points of the median or contact surfaces of the median layer can be. these equations, in the absence of external layers, are general equations of longitudinalradial vibrations [19] of a circular cylindrical viscoelastic structure, relative to the main parts of the longitudinal and radial displacements of points on the intermediate surface of the structure. in addition, eqs. (22) and (23) in their right parts correctly considering the forces acting on the outer and inner surfaces of the three-layer structure, reflect (approximately) the relationship and mutual influence of the layers-through the middle layer. it is not difficult to see in these equations the dependence on the viscoelastic operators of m r (m=0,1,2) layers. if specific kernels km and m k are given for viscoelastic layer material operators, then it is not difficult to transform these equations for specific shell layer materials. note that the operators given by eqs. (22) and (23) are derived for the general case of arbitrary operator kernels m r (m=0,1,2). consider the following special case: suppose that the poisson coefficients of the materials of all three layers of the structure are constant in time. then the kernels of the viscoelastic operators rm and m r will be the same, and taking this fact into account, the operators n m and ,n m take the form: 2 2 2 2 1 1 2 2 2 2 2 2 1 1 ( ) , ( ) 0,1,2; 1,2,3,... n n n n m m m m m m m m a t z b t z m n − −           = − = −                    = =         (25) here the operator mm is defined by the formula: 0 ( ) ( ) ( ) ( ) ; t m mm t k t d= − −      (26) longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure 481 am and bm (m=0,1,2) are propagation velocities of longitudinal and transverse elastic waves in layer materials: ( 2 ) ; .m m m m m m mа b= + =     (27) in this case, the operators (0) 1q and (0) 2q are computed in then following way: ( ) ( ) ( )(0) (0) 1 0 0 0 2 0 0 0 0, 2q q= + =− + +       (28) taking into account eqs. (25-28), the equations of longitudinal-radial vibrations of a three-layer cylindrical viscoelastic structure are significantly simplified. 2.3. determination of displacements and stresses to determine the stress-strain state of the structure, it is necessary to determine the displacements and stresses. this procedure must be performed for all three layers. to do this, it will be necessary to express all displacements and stresses through the main parts u0,0, u0,1 and w0,0, w0,1 respectively, longitudinal – u0 and radial – w0 displacements of points of some “intermediate” surface of the middle layer of a three-layer cylindrical structure. first, it is necessary to determine the displacements u0, w0 and the stresses (0) rr , (0) zz , (0)  , (0) zr points of the middle layer. to find u0, w0 it is enough to reverse the expressions in eqs. (18) and (19) for 0 ( , , )u r k p and 0 ( , , )w r k p by p and k. applying transformations given by eqs. (20) and (21) to eqs. (18) and (19) and limiting ourselves to the zeroth order approximation, we obtain: 0 0,0 0,1 0 0,1 0,0( , , ) ; ( , , ) . 2 r u r z t u u w r z t w w r = − = − +   (29) similarly, formulas for stresses are derived, for example 0 2 0,0(0) (0) (0) (0) 1 0,0 1 3, 0 2 2 0 2 0,1(0) (0) 1, 0,1 2 3, 2 1, 0,12 2 (1 ) ( ) 2 ( ) 2 ( ) ( ) rr n n n n n u r q w q r q z z u r w q r q r w zz r  =     = − − + − + −               − − − −                  (30) the displacements and stresses of the other layers are found in a similar way, for example: 2 22 2 2 2 2 2 2 2 2 0 22 2 2 2 2 0 22 2 2 2 ( , , ) ( , , ) 8 ( , , ) ( ) 4 r r u r z t u r z t z z z w r z tr r r z     −   − = − + − +            − + −        (31) solutions of eqs. (22) and (23) for u0,0, u0,1 and w0,0, w0,1, can be used for calculating the displacements u0, w0 of points of an arbitrary cross-section of the median layer for any moment in time using eq. (29) with the desired accuracy along the radial coordinate r, for an arbitrary moment in time. similarly, the stresses of the remaining displacements and stresses at the points of the layers are determined in the case when m=1 and m=2. 482 kh. khudoynazarov 3. results and discussions the obtained equations of longitudinal-radial vibrations of a three-layer circular cylindrical viscoelastic shell, eqs. (22) and (23), are general. in the following, we consider several limiting cases and particular types of vibration equations. 3.1. vibrations of a two-layer viscoelastic structure by a layer, with sticking between the layers at a = r1 the three-layer cylindrical structure transforms into a two-layer structure. the intermediate surface of the structure of radius  passes into the intermediate surface of the inner layer of the structure. in this case, it should be assumed that the external action function (1) (1)( , ), ( , )r rzf z t f z t acts on the surface r = r1 and the operators λ1 = 0, δ1 = 0. then, from the system of eqs. (22) and (23), we obtain a system of equations of a twolayer cylindrical viscoelastic structure. similarly, it is possible to obtain a system of equations of longitudinal-radial vibrations of a two-layer cylindrical viscoelastic structure, where the main unknowns will be the main parts of the longitudinal-radial displacements of the intermediate surface of the outer layer. to do this, it is enough to assume that there is no outer layer and put b = r2 assume that the functions of external action (2) (2)( , ), ( , )r rzf z t f z t act on the surface r = r2, and the operators λ2 = 0, δ2 = 0. if there are no inner and outer layers of the cylindrical structure, i.e. it is homogeneous (single-layer), then a = r1, b = r2 and λi = 0, δi = (i=1,2) should be put into general eqs. (22) and (23). in this case, we will have a system of equations for longitudinal-radial vibrations of a cylindrical homogeneous layer, which exactly coincides with the system of equations derived by khudoynazarov et al. [19]. 3.2. a three-layer cylindrical viscoelastic structure with a thin middle layer and rigid contacts between the layers if r2 = r1(1+ε), where ε > 0 is a small parameter, then the middle layer of the structure is thin (for example, a thin layer of glue, usually applied between layers). in this case, the values of ln(ri/ε) can be assumed to be zero. then eq. (19) for η3,n(ri) is simplified and takes the form: ( )3, 1 1 1 , 0,1,2,..., 1,2. 2 n n i k r n i k= = − = = (32) consequently, eqs. (22) and (23) are also equations of longitudinal-radial vibrations of a three-layer cylindrical structure with a thin middle layer, but with a different value for η3,n(ri), determined by eq. (32). 3.3. three-layer cylindrical elastic structure with sticking between layers if the materials of the layers are elastic, then the expressions for viscoelastic operators will have the equality kμm(t)=0, and therefore we will have ь mr =  . then we get a system of equations that coincides in structure with the system of eqs. (22) and (23), where the integral operators i r ,i=1,2, are replaced by the lame coefficients μ1 and μ2, respectively. longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure 483 in this case, the integral-differential operators ( )n m  and ( )n m  , defined by eqs. (25), pass into the following differential operators: 2 2 2 2 2 2 2 2 2 2 1 1 ( ) , ( ) , 1,2,3,... n n n n m m m m n b t z a t z           = − = − =                            the results obtained coincide with the results by filippov and filippov [20]. in particular cases, from the equations obtained in this way, it is easy to obtain equations of a two-layer elastic structure and a three-layer elastic structure with a thin middle layer, like the limiting cases discussed above. 4. conclusions a new mathematical model of longitudinal-radial vibrations of a circular cylindrical three-layer viscoelastic structure with sticking condition between the layers is proposed. the model includes new vibration equations and an algorithm for calculating the stressstrain state of an arbitrary point of the structure. the vibration equations of the considered structures, including the influence of the moments of inertia and transverse shear deformation, are derived for arbitrary external dynamic loads acting onto the structure surfaces. in the absence of external layers, the results obtained completely coincide with the results of filippov and filippov [20]. a new method has been developed for the dynamic calculation of circular cylindrical three-layer elastic and viscoelastic shells for the action of various external dynamic loads. the method consists in the derivation of vibration equations, both refined ones of the timoshenko type and classical ones of the kirchhoff-love type, and in the development of an algorithm for calculating the stress-strain state system. as special cases of the obtained results, new equations of longitudinal-radial unsteady vibrations of a circular cylindrical three-layer elastic structure are proposed. with restrictions in infinite series, their approximations of different order are followed by kirchhoff-love type vibration equations [21], hermann-mirsky and other refined equations of higher orders. equations 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pp. 118 126. 25. safaei, b., chukwueloka onyibo, e., goren, m., kotrasova, k., yang, z., arman, s., asmael, m., 2023, free vibration investigation on rve of proposed honeycomb sandwich beam and material selection optimization, facta universitatis-series mechanical engineering, 21(1), pp. 31-50. 26. zhao, z., yuan, x., zhang, w., niu, d., zhang, h., 2021, dynamical modeling and analysis of hyperelastic spherical shells under dynamic loads and structural damping, applied mathematical modelling, 95, pp. 468-483. 27. khudoynazarov, kh., 2023, a mathematical model of physically nonlinear torsional vibrations of a circular elastic rod, tomsk state university journal of mathematics and mechanics, 84, pp. 152–166. 12053 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 687 707 https://doi.org/10.22190/fume240215019n © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper machine learning of structure – property relationships: an application to heat generation during plastic deformation filip nikolić1,2, marko čanađija2 1cae department, elaphe propulsion technologies ltd, ljubljana, slovenia 2university of rijeka, faculty of engineering, rijeka, croatia orcid ids: filip nikolić https://orcid.org/0000-0002-4343-5069 marko čanađija https://orcid.org/0000-0001-6550-0258 abstract. in the present work, the heat generation during the plastic deformation of a multiphase material is studied using machine learning (ml) methods. the aim was to predict the temperature increase from the structure-property relationships (spr) of a microstructure considering various taylor–quinney coefficients (tqcs), with the aim of achieving precision and computational efficiency suitable for industry. using automatic microstructure generation to create datasets and finite element analysis (fea) to obtain temperature increase – strain curves, the dataset facilitated the training of an ml model. a 3d convolutional neural network (cnn) was developed using the microstructural configuration and tqc value as input and the temperature increase – strain curve as output. the model demonstrated high prediction accuracy. the results indicated that the hard phase fraction significantly impacts the temperature increase, much more than the tqc values. this underlines the potential of the model for a better understanding of material behavior during deformation and its industrial applicability. key words: deep learning, taylor – quinney coefficient, heat generation, structure – property relationship, finite element analysis 1. introduction it is well-known that the deformation of most materials can be divided into elastic and plastic deformation. both can be either rate dependent or rate independent. while elastic deformation is a reversible process, plastic deformation is irreversible. during plastic deformation, most of the energy is dissipated into heat while during elastic deformation the heat generation is often neglected. for a general overview of the phenomenon, the interested reader is referred to [1] and the references therein. the issue is further received: february 15, 2024 / accepted may 02, 2024 corresponding author: marko čanađija university of rijeka, faculty of engineering, vukovarska 58, rijeka, 51000, croatia. e-mail: marko.canadija@riteh.uniri.hr 688 f. nikolić, m. čanađija complicated by the fact that the response of the material involves the temperature dependence of the properties [2–4]. the fraction of plastic energy that is converted to heat is described by the taylor–quinney coefficient (tqc) and can depend on the type of material, strain rate, amount of plastic strain and many other factors, see [5-7]. it is often assumed to be around 90 % [5] and usually increases with deformation, see for example [2, 5, 7, 8]. moreover, in some special cases this coefficient can be greater than one due to microstructural transformations. at this point, it should only be briefly noted that the microstructural aspects are very important, but the complexity of the phenomenon hinders the possibility of fully addressing all the details using classical techniques. standard multiscale methods can provide some solutions to these issues, but on the other hand they are quite slow [9]. in the last few years, machine learning (ml) has become a very attractive topic in materials science and many different ml algorithms are widely used. shallow algorithms such as the support vector machine can be found in applications such as microstructure classification [10]. azimi et al. [11], on the other hand, used much more complex algorithms such as convolutional neural networks (cnns) for similar tasks. decost et al. [12] applied a cnn model based on segmentation for novel automated applications of microstructure segmentation. applications such as defect detection and determination of distances between secondary dendrite arms using cnns can also be found in [13-15]. optical microstructure images can be related to mechanical properties (yield strength, elongation and ultimate tensile strength) (see [16]). tagimalek et al. [17] used relevance and support vector machine to determine the mechanical properties based on the input process parameters of the friction stir welding process. similarly, milićević et al. [18] predicted the mechanical properties of the 3d printing material pla using neutral networks. zarezadeh et al. [19] investigated the corrosion mechanical and electromechanical behavior of ni coatings using a neural network and an adaptive neuro-fuzzy inference system. li et al. [20] proposed a transfer learning approach for microstructure reconstruction and structure– property relationships (spr). spr is a highly attractive topic in engineering. it can be determined using many methods, but it is most commonly determined using finite element analysis (fea). the most significant issue with fea is the lack of computational efficiency when it comes to simulations over two or more scales. this makes it very difficult for the industry to adopt. for this type of problem, ml methods could be of great advantage. jung et al. [21] for example, have shown that a cnn is tremendously fast in predicting mechanical properties in the spr domain compared to fea simulations. yang et al. [22] employed a cnn to predict the elastic mechanical properties of a high–contrast composite. the input for the cnn was the arrangement of the microstructure and it outperformed all other methods. cecen et al. [23] also developed a cnn to predict mechanical properties based on the microstructure arrangement. they also used principal component analysis (pca) for dimension reduction. dimension reduction techniques could reduce the size of the model input, which in turn could reduce the computational cost of training. however, one should be cautious when using such methods as they may compromise the overall prediction accuracy. wang et al. have presented an ml approach for the prediction of material properties in polymer nanocomposites. latypov and kalidindi [24] performed a variety of fea simulations of digitally generated microstructures to create the dataset for training the ml model. their model successfully predicted the yield strength and strain rate distribution in multiphase materials. to simplify the training and the ml model, they used pca machine learning of structure – property relationships: an application to heat generation... 689 techniques for dimensionality reduction. jung et al. [21] also created an spr ml model that can predict the elongation, ultimate tensile strength and strain localization index based on the arrangement of the microstructure. in addition, they used pca and multidimensional scaling to simplify the ml model. as for the coupled thermoplasticity problems, king et al. [25] developed a physically informed machine learning model that can effectively capture the temperature increase during shear assisted processing and extrusion shape. however, the work does not explicitly model the temperature increases based on the detailed microstructural characteristics or other direct material property relationships. instead, the complex interplay between process parameters and temperature dynamics is captured by the ml model. pantalé et al. [26] used an ml model to accurately replace the analytical formulation of a johnson–cook behavior law in explicit finite element simulations. temperature is included in the formulation, but its influence manifests itself by affecting the flow law. several different network architectures were tested. in the context of predicting stresses and temperature in the thermoelasticity of rubber, the neural network based on the strain invariants shows better performance than the network based on the stress-strain model [27]. however, plastic deformation is not relevant in such cases and was not taken into account. from the above literature review, it can be concluded that although there are some similar research works on spr, none of them deals with the ml prediction of tqc and heat generated during plastic deformation. to this end, the present study aims for the first time to directly predict the temperature increase curve due to plastic deformation by spr using ml methods. the proposed methodology is based on the particular microstructural configuration of the constituents involved and their thermomechanical properties. 2. methods 2.1 dataset generation in order to establish a relationship between the microstructure of the material and the evaluated properties using ml, a large number of different microstructures should be obtained. in this study, representative volume elements (rves) of equiaxed 3d two-phase microstructures consisting of 26x26x26 voxels were generated utilizing the open-source software dream.3d [28]. the microstructure generation in the software relied on a gaussian random distribution, and only the volume fractions of the phases were varied. specifically, the volume fractions of the second (hard) phase ranged from 0% to 15%, while the first (soft) phase ranged from 85% to 100%. to maintain simplicity, all other parameters were kept constant throughout the generation process. the resulting microstructures had hard phase particle sizes ranging from 2 mm to 13 mm, with varying quantities of hard phase particles ranging from 1 to 15. these particles were randomly distributed in the microstructure. it is noteworthy that the generation parameters employed here are consistent with those used in a previous study [29]. for a more comprehensive understanding of the parameters used, interested readers are encouraged to refer to this study. it is emphasized that the present investigation of the mechanical behavior of the material is based on a more realistic 3d scale than the usually considered 2d scale [30–36]. nevertheless, it is important to emphasize that it was not the intention to explicitly replicate real materials, although the methodology is applicable to these as well. instead, the aim was to investigate the impact of different particle sizes and microstructural configurations on the mechanical properties 690 f. nikolić, m. čanađija and behavior of hypothetical composites under different conditions. this approach allows a wide spectrum of particle sizes and microstructural configurations to be explored, some of which could potentially be larger than those commonly found in real composites. finally, 3474 different microstructure arrangements were generated. it should be noted that it is important to obtain a large number of different microstructural configurations since heat generation also depends on the configuration. to determine the plastic behavior of each rve, an fea is performed using abaqus version 2021. for the material properties, youngs moduli of 50 and 500 gpa were used for the first and second phase, respectively, while a poisson’s ratio of 0.33 was chosen for each phase. the true (cauchy) stress true plastic strain curves of each phase are shown in fig. 1. fig. 1 true stress – true plastic strain diagrams of: (a) soft phase and (b) hard phase of the microstructure these curves resemble typical stress-strain curves represented as a series of stress-strain points as determined in a tensile test. no attempt was undertaken to establish a specific equation relating two quantities. the mapping of the properties of each phase to a finite element mesh, the size and element types of the mesh, and the simulation procedure matches our previous study [30]. periodic boundary conditions were enforced, while the machine learning of structure – property relationships: an application to heat generation... 691 load was specified as an enforced displacement, resulting in a strain magnitude of 0.57%. thus, the numerical procedure is equivalent to the macroscopic uniaxial tensile tests. the only difference was that in the present study the influence of temperature on the mechanical properties was neglected for simplicity. the entire dataset took approximately 530 hours to generate using an intel®coret m i7 – 4970 central processing unit (cpu) and a single processing unit. the simulation took between 7 and 12 minutes and on average about 9 minutes, depending on the microstructure configuration. the strain magnitude was kept relatively small due to the computational effort. very high strain magnitudes require a significant number of simulation steps due to convergence issues and drastically increase the simulation time. therefore, a lower strain magnitude was chosen so that the simulations could be performed in a realistic time frame. the output of the analysis was the plastic strain energy, which was given in 20 evenly spaced intervals. thus, the arrangement of the microstructures was the only variable parameter for the fea simulation. the arrangement of the phases with three different hard phase fractions is shown in fig. 2. note that the hard phase fraction varied from 0% to 0.15% in the entire dataset. the hard phase fraction was rather low as the intention was to show the effect of strengthening a weaker material with a stronger material. for each rve in fig. 2, the resulting true von mises stress, the total displacement and the true equivalent plastic strain at the last simulation increment are shown for the corresponding rves (figs. 3, 4, 5, 6). note that the displacement in each figure is scaled by 25. some elements of the rve are intentionally hidden to visualize the material behavior within the rve. it should also be noted that the von mises stress increases as the fraction of the hard phase increases, especially within the elements to which material properties of the hard phase are assigned. for the soft phase, only the non-uniformity of the stress field is higher, while the general level hardly changes, fig. 3. for the highest fraction of the hard phase, the deformation is higher, while there is hardly any difference for the two lower values, fig. 4. both the general level and the nonuniformity of the plastic deformation increase with increasing hard phase fraction, fig. 5. furthermore, it was observed that the level of plastic deformation is significantly lower within the elements to which hard phase material properties are assigned, fig. 6. fig. 2 visualization of the arrangement of the phases for a different rves 692 f. nikolić, m. čanađija fig. 3 von mises stress (mpa) for the rves in fig. 2 fig. 4 total displacement (mm) for the rves in fig. 2 fig. 5 equivalent plastic strain for the rves in fig. 2 machine learning of structure – property relationships: an application to heat generation... 693 fig. 6 equivalent plastic strain for the rves in fig. 2 (only hard phase is shown) knowing the amount of plastic energy dissipated during the deformation of the rve allows the calculation of the average temperature increase in the rve during deformation, as outlined below [1]: ∆𝑇 = 𝑄 · 𝛽 𝜌 · 𝑉 · 𝑐𝑝 (1) where ∆t is the temperature increase in [k], q is the total plastic energy of the rve in [j], β is the tqc, ρ is the mass density in [kg/m3], v is the volume of the rve in [m3] and cp is the specific heat capacity in [j/kg k]. it is important to note that the above equation is only valid for adiabatic conditions, which corresponds to higher strain rates. however, the non – uniformity of the temperature field due to the non-uniform plastic deformation was not taken into account and simplification that the entire rve heats uniformly was considered. the volume of the rve was simply calculated based on its dimensions (26x26x26 mm) and corresponds to v = 1.7576·10−5 m3, while the amount of plastic deformation was obtained from the fea simulation. the mass density and specific heat capacity were both kept constant for the dataset and are: cp = 500 j/kg k and ρ = 2500 kg/m3. for simplicity, the same cp and ρ were assumed for both phases. even though these properties were kept the same for both phases in this study, it should not be a fundamental problem to repeat a similar study with different proportions between cp and ρ. the tqc β is designed to change based on the level of strain experienced by each rve in the dataset: 𝛽 = 𝛽1(18.86𝜀 + 1). (2) the constant β1 is randomly chosen between 0.7 and 0.8 for each rve. for the zero-strain and the maximum tensile strain (0.57%) in the direction of the prescribed displacement, it turns out that  can fall within the range of 0.7 to 0.886. it should be noted that the eq. (2) is not an attempt to physically describe the real evolution of the tqc, but merely serves to promote additional complexity in the heat generation process while still keeping the tqc within reasonable limits for some widely used construction materials. 694 f. nikolić, m. čanađija 2.2 overview of the ml model in general, ml algorithms can be divided into two main groups: shallow learning (e.g., smaller artificial neural networks, vector machine, decision tree, and others) and deep learning (dl) (e.g., recurrent neural networks, deep belief networks, cnn, and others) [37]. many authors have shown that complex dl algorithms typically produce better results [11–15, 20, 38–43]. however, determining the optimal balance between computational resources and desired precision remains an open issue. the ml model used in the present study is a complex 3d-cnn that considers input in the form of a 26x26x26 matrix. the matrix contains information about the arrangement of the hard and soft phases of the microstructure as well as the tqc value. the initial tqc value was used to create the matrix as explained in the previous subsection. the initial matrix consists of zeros (soft phase) and tqc (hard phase). including the tqc value in this manner is a convenient way to pass information about the tqc value to the ml model and simultaneously define the distribution of the hard phase. thus, the tqc value is directly included in the cnn. otherwise, the cnn should be considered as a multiple –input (i.e., information about the phase arrangement and the tqc value), which is more computationally intensive. therefore, the ml model captures any arrangement of phases in the rve and can be used for the same purpose as the macroscopic uniaxial tensile test. since the model was trained using data obtained from the uniaxial tensile test, application to other boundary conditions should be possible, but an initial verification on some test examples is recommended. it should be noted that only the results corresponding to ε greater than 0.00228 were used for training. this threshold corresponds to the last 12 increments, while the first 8 increments were not used. up to ε of 0.00228, the plastic strain is very low and only occurs due to some stress concentration at the interface between the soft and hard phase. therefore, the heat generation is also very low in contrast to the part of the curve that includes ε above 0.00228. as a consequence, the temperature increase within the first 8 points is negligible in contrast to the temperature increase within the following 12 points. thus, excluding the first 8 points of the curve from the learning process, allows the cnn to be much simpler and computationally more efficient, while the loss of accuracy is minimal. additionally, several alternative cnn configurations are examined to compare the performance of the different model structures. while numerous experiments have been conducted with different cnn configurations to determine the best performing one, the focus here is limited to six distinct cnns. the six distinct cnn models were tested on the same test dataset, and their respective performance results are subsequently presented. the performance analysis was concentrated on the maximum global prediction error, mean squared error (mse), and root mean squared error (rmse) metrics, tab. 1. it should be noted that all strain increments were used to evaluate the results. according to the tab. 1, it can be concluded that cnn1 is the best performing model and was selected for the task at hand. a brief comparison of the performances of the different cnns is presented in the results and discussion section. cnn1 was constructed using two blocks built with 3d convolutional layers. each convolution block follows a parametric rectified linear unit (prelu) activation, a batch normalization and a max– pooling layer. for the first block, filter sizes of (8x8x8x32) were used, while filter sizes of (6x6x6x32) were used after the max-pooling layers. for the second block, filter sizes of (5x5x5x32) were used while after the max – pooling layer, filter sizes of (4x4x4x32) were used. the last convolution block is followed by a meta–layer, which is then followed by a fully connected layers of 512, 256, 128, 64 and 20 neurons. each of these layers is followed machine learning of structure – property relationships: an application to heat generation... 695 by a batch normalization and the prelu activation function. the last layer is followed by 12 logistic output neurons that output the 12 uniformly distributed temperature increase intervals. the cnn operates on the feed–forward (sequential) principle. thus, for the microstructure arrangement and the input of the tqc value, the 12 points of the temperature increase vs. true strain curve were output. as already mentioned, dimension reduction techniques such as pca could affect the overall efficiency of the model, so they are avoided in this study. a schematic representation of the model can be found in fig. 7. fig. 7 a schematic representation of the machine learning model the cnn was trained with python version 3.9 and tensorflow version 2.6 using the keras library. convergence was achieved after 300 epochs with a learning rate of 0.00025 and an adam optimizer. the training was completed in approximately one hour using an nvidia quadro k2200 graphics card. the dataset for training was split into the training set, the validation set and the test set in a 70:20:10 ratio. mse was used as the validation loss metric, and the model was validated using the curves from the fea. it is worth emphasizing that the training procedures remained consistent for all different cnns presented in tab. 1, thus, only the cnn configuration itself was changed. unlike some other studies [21, 23, 24, 43], the advantage of the present ml model was that simplifications such as dimension reduction techniques were not necessary, see results and discussion section. in summary, the cnn uses the microstructural configuration of the rve and tqc as input and outputs the temperature increase–strain curves. a schematic representation of the entire learning process is shown in fig. 8. 696 f. nikolić, m. čanađija table 1 cnn models configuration and performance model cnn configuration mse rmse error comp. time cnn1 cl_1 (f-16, (26x26x26x32) – cf(8x8x8x32) 1.94e-5 0.004 6 % 68 min a-prelu), bn, mp_1 ((13x13x13x32) – mpf(6x6x6x32)) cl_2 (f-16, (13x13x13x32) – cf(5x5x5x32) a-prelu)), bn, mp_2 ((8x8x8x32) – mpf(4x4x4x32)) d(512, a-prelu), bn, d(256, a-prelu), bn, d(128, a-prelu), bn, d(64, a-prelu), bn, d(20, a-prelu), bn, output (opt: adam) cnn2 cl_1 (f-16, (26x26x26x32) – cf(5x5x5x32) 2.97e-5 0.005 20 % 54 min a-prelu), bn, mp_1 ((13x13x13x32) – mpf(2x2x2x32)) cl_2 (f-16, (13x13x13x32) – cf(3x3x3x32) a-prelu)), bn, mp_2 ((8x8x8x32) – mpf(2x2x2x32)) d(1024, a-prelu), d(0.5), bn, d(512, a prelu), d(0.5), bn, d(256, a-prelu), d(0.5), bn, d(128, a-prelu), bn, d(64, a-prelu), bn, d(20, a-prelu), bn, output, (opt: adam) cnn3 cl_1 (f-16, (26x26x26x32) – cf(5x5x5x32) 0.01 0.094 400 % 42 min a-relu), bn, mp_1 ((13x13x13x32) – mpf(2x2x2x32)) cl_2 (f-16, (13x13x13x32) – cf(3x3x3x32) a-relu)), bn, mp_2 ((8x8x8x32) – mpf(2x2x2x32)) d(1024, a-relu), d(0.5), bn, d(512, a-relu), d(0.5), bn, d(256, a-relu), d(0.5), bn, d(128, a-relu), bn, d(64, a-relu), bn, d(20, a-relu), bn, output, (opt: sgd) cnn4 cl_1 (f-16, (26x26x26x32) – cf(8x8x8x32) 6.52e-5 0.007 50 % 52 min a-prelu), bn, mp_1 ((13x13x13x32) – mpf(6x6x6x32)) cl_2 (f-16, (13x13x13x32) – cf(5x5x5x32) a-prelu)), bn, mp_2 ((8x8x8x32) – mpf(4x4x4x32)) d(512, a-prelu), bn, d(256, a-prelu), bn, d(128, a-prelu), output, (opt: adam) cnn5 cl_1 (f-16, (26x26x26x32) – cf(5x5x5x32) 0.03 0.17 1400 % 38 min a-relu), bn, mp_1 ((13x13x13x32) – mpf(4x4x4x32)) cl_2 (f-16, (13x13x13x32) – cf(3x3x3x32) a-relu)), bn, mp_2 ((8x8x8x32) – mpf(3x3x3x32)) d(256, a-relu), bn, d(128, a-relu), bn, output, (opt: sgd) cnn6 cl_1 (f-16, (26x26x26x32) – cf(5x5x5x32) 0.07 0.26 2000 % 28 min a-relu), bn, mp_1 ((13x13x13x32) – mpf(4x4x4x32)) d(128, a-relu), bn, output, (opt: sgd) legend: clx – 3d convolutional layer, f – number of filters, cf – number of convolutional filters, a – activation function, mpx – max pooling layer, mpf – number of max pooling filters bn – batch normalization layer, dr – dropout layer, d – dense (fully connected layers), opt optimizer machine learning of structure – property relationships: an application to heat generation... 697 fig. 8 schematic representation of the complete machine learning process 3. results and discussion 3.1. evaluation of different cnn configurations a brief comparison of the predictive capabilities of the six distinct cnns shown in tab 1 can be seen in fig. 9. the cnn5 and cnn6 configurations represent an attempt to address the task at hand using an extremely basic cnn configuration. both models show remarkable similarities and exude simplicity. unfortunately, both performed very poorly in this task. mse and rmse were significantly high compared to the best performing model and the error plots showed an error margin exceeding 1000%. therefore, it can be concluded that such simplified cnn models are not suitable for conducting the spr task at this scale. the cnn4 model introduces a slightly more complex structure and is additionally characterized by the use of an adam optimizer instead of sgd (stochastic gradient decent) and prelu activation functions instead of rectified linear unit (relu). this model demonstrated a significantly improved performance compared to cnn6 and cnn5. however, the results achieved still did not reach the standards required for the task at hand. consequently, it can be concluded that the adam optimizer and the combination of prelu activation functions perform better than the combination of sgd optimizer and the relu activation functions for the current task. moreover, the more complex cnn3 model using an sgd optimizer and relu activation functions, performed worse compared to a simpler cnn4 model, confirming the above hypothesis. in general, it can be concluded that although the performance of the more complex cnn3 model may not be suitable for the current task, it outperforms the simpler cnn6 and cnn5 models that use the same activation functions and optimizers. the cnn2 model shares a similar configuration to the cnn1 model, although it is slightly less complex. it is also worth noting that the comparison between the cnn2 and cnn1 models leads to the conclusion that a small proportion of dropout slightly reduces the performance of the model. the cnn2 model provided commendable performance on the task at hand. however, in terms of metrics and global error, its results were inferior to those of the cnn1 model. in addition, it is also visible that increasing the amount of kernel and max-pooling filters size increased the computation time. for this reason, the cnn1 configuration is 698 f. nikolić, m. čanađija chosen as the primary one in this study. therefore, it can be concluded that the complexity of the model as well as the choice of activation functions and optimizers are crucial for the accurate prediction of the temperature increase-strain curves involving the spr task. it should be emphasized again that the study did not investigate all possible combinations of activation functions, parameters and optimizers. instead, only some of the most popular choices were presented. fig. 9 comparison between the calculated (fea) and the predicted (cnn) value of the temperature increase for all points of the temperature increase curve for: (a) – cnn1 model, (b) – cnn2 model, (c) – cnn3 model, (d) – cnn4 model, (e) – cnn5 model, (f) – cnn6 model in tab. 1 3.2. evaluation of different cnn configurations the performance of the cnn1 model was evaluated on the test dataset using the mse and rmse. rmse and mse of the model were evaluated as 0.004353 °c and 1.9648·10−5 °c2, respectively. it should be noted that none of the microstructural configurations and tqc values used during training were also used in the test set. in addition, the model showed an extremely fast evaluation of the temperature increase, making 348 predictions in about 0.4 seconds. next, the temperature increase during the last twelve calculation steps, which represent the plastic part of the deformation process, is examined. the comparison of the temperature increase curves predicted by the ml model with the curves calculated by the fea for four machine learning of structure – property relationships: an application to heat generation... 699 different hard phase fractions and tqcs is shown in fig. 10. it is evident that the ml model accurately reproduces the temperature increase obtained by the fea. note that the rve is almost entirely in the elastic state during the first eight calculation steps and the heat generated by plastic dissipation is practically nonexistent. it can also be seen that the temperature increase does not change significantly at different tqcs. another factor that affects the amount of heat generated is the arrangement of the soft and hard phases within the microstructure, which is studied in sec 3.3. in the end, it can be noticed that the amount of heat generated depends most strongly on the fraction of the hard phase. fig. 10 comparison of the temperature increase curves as predicted by the ml model with those calculated by the fem (tqc at the end of deformation is shown) the performance of the model was also checked for all strain points and compared with the results of the fea. the comparison of ml and fea is shown in fig. 9 (a). throughout the complete range of temperature increase, the prediction accuracy is very good. it can also be seen that the prediction error for any point in the dataset is less than 6.5%. the errors between the predicted and calculated temperature increase for all strain points were also checked. the maximum error is 0.012 °c, while the average error is 5.87·10−5 °c, while the temperature increase ranges from 0.04 °c to 0.2 °c. the prediction error of the ml model is also analyzed for different tqc values for all strain points, figs. 11, 12. again, it was found that the temperature increase is only weakly related to the tqc value. the prediction error is also not dependent on the tqc value. the error plots for all strain points are shown in figs. 13, 14 for the different hard phase fractions. the general trend of a temperature increase with increasing hard phase fraction could be observed. this means that the stiffer rve configuration increases the amount of 700 f. nikolić, m. čanađija plastic strain and the temperature increase. the effect of stiffness is more influential for the temperature increase than tqc value. nevertheless, a stochastic distribution can be recognized, which is related to the random distribution of the soft and hard phases within the rve. consequently, the highest temperature increase is obtained at the fractions of 0.085, 0.118 and 0.135, as this random configuration generates the largest plastic strain. thus, as already mentioned, the 3d cnn directly captures the phase fractions, the arrangement of the phases and the tqc value. again, the prediction error does not appear to depend on the hard phase fraction, while the hard phase fraction tends to be the most influential factor for the temperature increase. fig. 11 prediction error for the given tqc interval of all 12 points of the temperature increase curve – actual tqc (β) for the last strain point is shown fig. 12 predicted and calculated temperature increase for different temperature values of tqc of all 12 points of the temperature increase curve – actual tqc (β) for the last strain point is shown machine learning of structure – property relationships: an application to heat generation... 701 fig. 13 prediction error for the hard phase fraction interval of all 12 points of the temperature increase curve fig. 14 predicted and calculated temperature increase for different temperature values of the hard phase fraction of all 12 points of the temperature increase curve arrows show the highest temperature increase finally, this approach is significantly more complex compared to classical approaches that do not use machine learning, e.g. [4]. in the latter, the generated heat is estimated by relying on constant tqc, which is of course only an approximation. the former approach, advocated in the present research is certainly more involved and computationally intensive, at least at start. however, once the neural network is trained, it is fast and can account for differences in microstructure and provide more accurate tqc estimates. 3.3. influence of the regularity of phases distribution as shown in fig. 15, the influence of the variations in distances between the centers of gravity (cogs) within the different rves was investigated. the positions of the cogs can serve as an indicator for the regularity of the hard phase distribution. a global cog was calculated for the hard and soft phases of each rve, with the resulting distances between the cogs of the soft and hard phases. it should be noted that each strain point is shown in 702 f. nikolić, m. čanađija the graph. generally, a higher hard phase fractions are associated with a smaller cog distance. it was also observed that some of the rves have significantly higher cog distances than most of the others. this observation can be traced back to scenarios with very low fractions of hard phase, where a limited number of small particles are dispersed away from the central region of the rve. this displacement causes the global cog of the hard phase to deviate significantly from the central point of the rve, which in turn leads to a larger cog distance. recall that the cog of both phases is generally centered, since phase generation in dream.3d is defined via the normal distribution. however, due to the much higher fraction of soft phase in the rve, the cog of the soft phase is generally centered and close to the cog of the rve. conversely, an increase in the hard phase fraction generally leads to a reduction in the cog distance. when fig. 15 (b) is evaluated again, a previously drawn conclusion can be confirmed once more: a larger temperature increase is associated with a larger fraction of the hard phase. furthermore, the prediction error also shows no discernible relationship to the cog distance. again, it can be seen that the main factor for the temperature increase is the hard phase fraction. fig. 15 distance between center of gravity of hard and soft phase of all 12 points of the temperature increase curve: (a) with prediction error and (b) with temperature increase furthermore, a study was carried out to examine the impact of the standard deviation of the distances between all components in the soft and hard phase of the rves from the center, see figs. 16, 17. the main objective is to demonstrate the variability in the dispersion of the data has an influence on the results. from figs. 16, 17 it is clear that there is no significant correlation between the error and the fraction of the hard phase as well as the soft phase, regardless of whether the standard deviation values are higher or lower. it can be observed that most of the values of the standard deviation of the soft phase are mostly concentrated in a narrow range around the value of 3.62. the extreme values of the standard deviations of cog are 3.71 and 3.45, respectively. for the hard phase, it could be observed in fig. 17 that the values of the standard deviation are concentrated in a narrow range around the value of 3.5, while the minimum and maximum are 0.25 and 5.80, respectively. this is in line with the previous conclusion that the cog of the hard phase can vary significantly due to the smaller number of particles and the involved stochastic distribution. machine learning of structure – property relationships: an application to heat generation... 703 fig. 16 standard deviation of center of gravity distances of all elements of an rve from the center of gravity center for all rves of all 12 points of the temperature increase curve – soft phase: (a) with prediction error and (b) with temperature increase fig. 17 standard deviation of center of gravity distances of all elements of an rve from the center of gravity center for all rves of all 12 points of the temperature increase curve– hard phase: (a) with prediction error and (b) with temperature increase fig. 18 illustrates the correlation between the variables through a scatterplot representation that includes aspects such as the cog distances of the hard phase from the center, the standard deviation, the mean distance of the hard and soft phase from the center, the hard phase fractions, and the prediction error. it should be noted that for the calculation of the mean distance, the cogs of the individual elements within the rve are calculated for the individual phase and the mean distance is calculated. upon evaluation of the data, a discernible pattern emerges: the standard deviation of the cog distances and the mean distance of the hard phase are predominantly concentrated in a narrow range around a certain value. only a fraction of the rves deviate from this specific zone, especially those corresponding to a very small amount of hard phase fractions. this pattern confirms the previous assertion of a normal distribution of phases within the rve. the cog distances show a distribution around smaller values, approximately 3 mm, while some rves show higher values up to 15 mm. as already explained, these correlate predominantly with smaller fractions of the hard phase. 704 f. nikolić, m. čanađija it should also be again noted that higher values of the temperature increase are primarily associated with higher fractions of the hard phase. therefore, even though the arrangement of the phases slightly affects the temperature increase and is captured by the ml model, the dominant variable influencing the temperature increase is the hard phase fraction. the error plots are also not correlated with any of the above variables. fig. 18 scatter matrix of all 12 points of the temperature increase curve of all rves – cog distances of the hard phase from the center, standard deviation, mean distance of hard and soft phase from the center, hard phase fractions, and the prediction error 4. conclusion the current study has shown the potential for the use of ml methods in predicting temperature changes resulting from plastic deformation of materials using spr. in particular, more classical approaches based on fea are more efficient when a smaller number of machine learning of structure – property relationships: an application to heat generation... 705 microstructure configurations are to be analyzed. these usually rely on eq. (1). however, if one wants to gain a deeper insight into the problem, the number of configurations increases, which makes the investigation more difficult. an ml model needs to be trained, but once it is trained, it is very efficient. this in turn makes it possible to analyze a larger number of simulations, as the data can be easily processed and the evaluation time is short. this can be useful, for example, when optimizing the microstructure to obtain a specific property since the procedure requires a large number of evaluations. the study examined the impact of various factors, including the fraction of hard phase, the tqc value and the regularity of phase distributions, on the extent of heat generation. the results revealed that within the tqc range investigated, the tqc value had little influence on the outcomes. in contrast, the fraction of the hard phase emerged as the most influential factor in determining the extent of the temperature increase. the effect of the regularity of the phase distribution was also investigated. specifically, the cog distances between the hard and soft phases and the standard deviation of the cogs for the soft and hard phases were examined. however, it was found that none of these values correlated with the prediction error or the magnitude of the temperature increase. on the other hand, the investigation reconfirmed that the fraction of the hard phase and the microstructure configuration remains the most influential factor contributing to temperature increase. moreover, the cnn model developed in the current study exhibits a high computational efficiency, enabling it to generate nearly 350 predictions within 0.4 s. the model performed very well, achieving an rmse of 0.004353 and an mse of 1.9648·10−5. the maximum prediction error of the model is 0.012 °c, while the average error is 5.87·10−5 °c within the temperature range of 0.04 °c to 0.2 °c. the scope of this study is limited to examining the boundary conditions that simulate a 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by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper nonlinear vibration response of piezoelectric nanosensor: influences of surface/interface effects sayyid h. hashemi kachapi department of mechanical engineering, babol noshirvani university of technology, iran abstract. in the present work, surface/interface energy effects for pull-in instability analysis on dimensionless natural frequency (dnf) and nonlinear dynamics response (ndr) of piezoelectric nanosensor (pens) subjected to nonlinear electrostatic excitation and harmonic force are studied using gurtin–murdoch (gm) surface/interface energy approach. to achieve this purpose, the hamilton’s approach, assumed mode and lagrange–euler’s theories and also arc-length continuation and complex averaging methods are used to investigate influences of surface/interface parameters of pens such as lame’s constants, residual stress, piezoelectric constants and mass density for analysis of pull-in instability voltage. key words: pull-in instability, nonlinear dynamic response, piezoelectric nanosensor, surface/interface effects, arc-length continuation, nonlinear electrostatic excitation 1. introduction nanomechanical sensors and resonators especially for piezoelectric nanostructures, due to small size and mass and their unique properties, are widely used in advanced engineering applications and industry [1-3]. in the last decade, there has been a great deal of research into the vibrations and dynamics analysis especially the pull-in instability phenomenon of such nanostructures, some of which will be mentioned in the following. farokhi et al. used fe approach to analyze nonlinear responses of static and dynamic and pull-in behavior for nano resonator with electrostatic excitation [4]. prasanth et al. worked on pull-in behaviour of a micro resonator considering large elastic deflection [5]. surface energy approach is utilized by pourkiaee et al. to study nonlinear vibration of nano beam resonator with piezoelectric layer and electrostatic excitation [6]. karimipour et al. presented nonclassical received june 12, 2021 / accepted october 23, 2021 corresponding author: sayyid h. hashemi kachapi department of mechanical engineering, babol noshirvani university of technology, p.o. box 484, shariati street, babol, mazandaran47148-71167, iran e-mail: sha.hashemi.kachapi@gmail.com mailto:sha.hashemi.kachapi@gmail.com 260 s. h. h. kachapi theories to study pull-in behavior of thin plate activator [7]. in the work by nikpourian et al., nonlinear dynamics analysis in micro beam resonator with piezoelectric layer and electrostatic force investigated using size dependent approach [8]. investigation of pull-in phenomenon in nems and mems is studied by zhang et al. [9]. pull-in instability behavior of microbeams and nano bridge are presented by sedighi et al. using strain gradient approach [10, 11]. pull-in instability behavior of nanostructures such as nano bridges, nems and nanobeams, are presented by tadi beni et al. using non classical theories such as a modified version of coupled stress method [12-14]. dynamic pull-in and damage reconnaissance of micro structures is presented by tajalli et al. [15]. also the modified version of coupled stress method is utilized by farokhi et al. to analyze resonant and pull-in behavior of carbon nanotube based resonators [16]. stability analysis and nonlinear vibrations of structures such as plates and shells in theoretical and experimental cases are also presented by amabili [17]. bornassi et al. utilized nonlocal model to analyze pull-in phenomenon of nems [18]. for consideration of non-classical surface parameters of nanostructures, surface/interface energy approach is presented [19, 20]. recently, using surface effect, buckling and postbuckling analysis and also vibration behavior of nano structures with piezoelectric layers are studied by fang et al. [21, 22]. also, hashemi kachapi et al. presented gurtin–murdoch surface/interface theory to investigate linear and nonlinear vibration analysis of piezoelectric nanostructures [23-28]. in the mentioned papers, the following effects have been studied: the effects of all surface/interface energy parameters; mechanical and structural properties of nanostructures and piezoelectric layers; van der waals force; visco-pasternak effects on the undamped and damped natural frequencies; parametric studies, such as the effects of different boundary conditions; different geometric ratios; structural parameters; electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis. fan utilized surface energy effects to enlarge the capability of piezoelectric/piezomagnetic energy harvester [29]. despite a great number of researches on nanosensors dynamics and vibration, consideration of surface/interface energy effects on pull-in stability and vibration analysis of nanosensor with piezoelectric layers and electrostatic excitation have not been studied yet. in the present study, using arc-length continuation and complex averaging method [30], pull-in instability behavior of piezoelectric nanosensor are studied on dimensionless natural frequency and nonlinear dynamics response considering surface/interface energy effects. 2. mathematical formulation a cylindrical piezoelectric nanosensor subjected to the combined electrostatic force and harmonic excitation is shown in fig. 1. based on the surface/interface effect of gurtin–murdoch theory, the normal stresses 𝜎𝑥𝑥 and 𝜎𝜃𝜃 can be written as [19-26] 𝜎𝑥𝑥(𝑁,𝑝) = 𝐶11(𝑁,𝑝)𝜀𝑥𝑥 + 𝐶12(𝑁,𝑝)𝜀𝜃𝜃 − 𝑒31𝑝�̅�𝑥𝑝 + 𝜐(𝑁,𝑝)𝜎𝑧𝑧 1−𝜐(𝑁,𝑝) (1) 𝜎𝜃𝜃(𝑁,𝑝) = 𝐶21(𝑁,𝑝)𝜀𝑥𝑥 + 𝐶22(𝑁,𝑝)𝜀𝜃𝜃 − 𝑒32𝑝�̅�𝜃𝑝 + 𝜐(𝑁,𝑝)𝜎𝑧𝑧 1−𝜐(𝑁,𝑝) , (2) 𝜎𝑥𝜃(𝑁,𝑝) = 𝐶66(𝑁,𝑝)𝛾𝑥𝜃, (3) where based on nonclassical continuum model, 𝜎𝑧𝑧 is expressed as following https://www.sciencedirect.com/topics/engineering/resonators https://www.sciencedirect.com/topics/engineering/harvesters nonlinear vibration response of piezoelectric nanosensor: influences of surface/interface effects 261 𝜎𝑧𝑧 = 𝑧 ℎ𝑁+ℎ𝑝 ((𝜏0 𝑆 + 𝜏0 𝐼 )( 𝜕2𝑤 𝜕𝑥2 + 1 𝑅2 𝜕2𝑤 𝜕𝜃2 ) − (𝜌𝑆 + 𝜌𝐼) 𝜕2𝑤 𝜕𝑡2 ), (4) fig. 1 piezo-harmoelectrostatic nanosensor subjected to electrostatic harmonic excitation also all coefficients and phrases of eqs. (1-4) such as displacement fields, nonlinear deflection and curvatures, relations of surface/interface theory and etc. can be observed in complete detail in reference [25]. by applying the assumed mode and lagrange-euler approaches, the governing equations and boundary conditions of pens are obtained. for this purpose, the total strain energy of pens considering the surface/interface effect can be presented as: 𝜋 = 1 2 ∫ ∫ { 𝑁𝑥𝑥𝜀𝑥𝑥 0 + 𝑁𝜃𝜃𝜀𝜃𝜃 0 + 𝑁𝑥𝜃𝛾𝑥𝜃 0 + 𝑀𝑥𝑥𝜅𝑥𝑥 +𝑀𝜃𝜃𝜅𝜃𝜃 + 𝑀𝑥𝜃𝜅𝑥𝜃 + 𝜂33�̅�𝑧𝑝 2 ℎ𝑝 } 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥 (5) in eq. (5), the forces (𝑁) and moment (𝑀) resultants are determined in reference hashemi kachapi et al. [25]. the kinetic energy of the pens can be written as: 𝑇 = 1 2 ∬ {𝐼 (( 𝜕𝑢 𝜕𝑡 ) 2 + ( 𝜕𝑣 𝜕𝑡 ) 2 + ( 𝜕𝑤 𝜕𝑡 ) 2 )} 𝑅𝑑𝜃𝑑𝑥 (6) where 𝐼 = ∫ 𝜌𝑁 ℎ𝑁 −ℎ𝑁 𝑑𝑧 + ∫ 𝜌𝑝 −ℎ𝑁 −ℎ𝑁−ℎ𝑝 𝑑𝑧 + ∫ 𝜌𝑝 ℎ𝑁+ℎ𝑝 ℎ𝑁 𝑑𝑧 + 𝜌𝑆,𝐼 = 2𝜌𝑁ℎ𝑁 + 2𝜌𝑝ℎ𝑝 + 2𝜌𝑆 + 2𝜌𝐼 also, the work done on the pens due to the viscoelastic foundation, nonlinear electrostatic excitation and the harmonic force, respectively, can be presented as [25] 262 s. h. h. kachapi 𝑊𝑣𝑚 = − ∫ ∫ ∫ (𝐾𝑤𝑤 − 𝐾𝑝∇2𝑤 + 𝐶𝑤 𝜕𝑤 𝜕𝑡 ) 𝑑𝑤 𝑤 0 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥, (7) 𝑊𝑒 = ∫ ∫ ∫ 𝜋υ𝑉𝐷𝐶 2 √(𝑏−𝑤)(2𝑅+𝑏−𝑤)[𝑐𝑜𝑠ℎ−1(1+ 𝑏−𝑤 𝑅 )] 2 𝑑𝑤 𝑤 0 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥 (8) 𝑊𝑓 = ∫ ∫ ∫ (𝑓𝑐𝑜𝑠𝜔𝑡)𝑑𝑤 𝑤 0 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥, (9) where all coefficients and phrases of eqs. (7-9) can be seen in reference [25]. by applying of hamilton’s principle and substituting eqs. (5-9) into hamilton’s principle and using dimensional parameters [25], the governing equations of motion and boundary conditions for pens are obtained. in following using least-squares polynomial format of electrostatic force and by using the displacement and shear deformation in the assumed mode method and by applying non-dimensional strain and kinetic energies eqs. (5) and (6) and also non-dimensional works eqs. (7-9) and substituting in the lagrange-euler equations, the dimensionless equations of motion is obtained to the following equations: [(𝑀)𝑢 𝑢]{�̈̅�} + [(𝐾)𝑢 𝑢]{�̅�} + [(𝐾)𝑢 𝑣 ]{�̅�} + [(𝐾)𝑢 𝑤]{�̅�} + [(𝑁𝐿)𝑢 𝑤]{�̅�2} = �̅�𝑢𝑝, (10) [(𝑀)𝑣 𝑣]{�̈̅�} + [(𝐾)𝑣 𝑢]{�̅�} + [(𝐾)𝑣 𝑣]{𝑣} + [(𝐾)𝑣 𝑤]{�̅�} + [(𝑁𝐿)𝑣 𝑤]{�̅�2} = �̅�𝑣𝑝, (11) [(𝑀)𝑤 𝑤]{�̈̅�} + [(𝑐)𝑤 𝑤]{�̇̅�} + [(𝐾)𝑤 𝑢 ]{�̅�} + [(𝐾)𝑤 𝑣 ]{�̅�} + [(𝐾)𝑤 𝑤 − �̅�𝑒2(𝐾𝑒)𝑤 𝑤]{�̅�} +[(𝑁𝐿)𝑤 𝑢 ]{�̅��̅�} + [(𝑁𝐿)𝑤 𝑣 ]{�̅��̅�} + [(𝑁𝐿)𝑤2 𝑤 − �̅�𝑒3(𝑁𝐿2𝑒)𝑤 𝑤]{�̅�2} (12) +[(𝑁𝐿)𝑤3 𝑤 − �̅�𝑒4(𝑁𝐿3𝑒)𝑤 𝑤]{�̅�3} = �̅�𝑤𝑒 + �̅�𝑤𝑝 + [�̅�𝑐𝑜𝑠ω̅𝜏], where all the coefficients and phrases of eqs. (10-12) are presented in reference [25]. in order to study the nonlinear dynamics response, arc-length continuation and complex averaging approaches are presented [25-26, 28]. verification, comparison and convergence study is investigated in reference hashemi kachapi et al. with full details for pens [25]. in this section, pull-in instability analysis on dimensionless natural frequency and nonlinear dynamics response are presented. for this purpose, different boundary condition such as clamped edge (cc), simply supported edge (ss), clamped-simply supported edge (cs) and, clamped-free edge (cf) are presented. 3. results and discussions the bulk and surface/interface material properties of aluminum (al) nanoshell and pzt piezoelectric layer are shown in tables 1 and 2, respectively [24]. table 1 1 bulk and surface/interface properties of al 𝐸𝑁(𝐺𝑃𝑎) 𝜐𝑁 𝜌𝑁(𝑘𝑔/ 𝑚3) 𝜆𝐼(𝑁 𝑚⁄ ) 𝜇𝐼(𝑁 𝑚⁄ ) 𝜏0 𝐼 (𝑁 𝑚⁄ ) 𝜌𝐼(𝑘𝑔 𝑚2⁄ ) 70 0.33 2700 3.786 1.95 0.9108 5.46 × 10−7 nonlinear vibration response of piezoelectric nanosensor: influences of surface/interface effects 263 table 2 bulk and surface/interface properties of pzt-4 𝐶11𝑝(𝐺𝑃𝑎) 𝐶22𝑝(𝐺𝑃𝑎) 𝐶12𝑝(𝐺𝑃𝑎) 𝐶21𝑝(𝐺𝑃𝑎) 𝐶66𝑝(𝐺𝑃𝑎) 𝐸𝑝(𝐺𝑃𝑎) 139 139 77.8 77.8 30.5 95 𝜐𝑝 𝜌𝑝(𝑘𝑔 𝑚−3) 𝜂33𝑝(10−8 𝐹 𝑚⁄ ) 𝜆𝑆(𝑁 𝑚⁄ ) 𝜇𝑆(𝑁 𝑚⁄ ) 𝜏0 𝑆(𝑁 𝑚⁄ ) 0.3 7500 8.91 4.488 2.774 0.6048 𝑒31𝑝(𝐶 𝑚2⁄ ) 𝑒32𝑝(𝐶 𝑚2⁄ ) 𝑒31𝑝 𝑆 (𝐶 𝑚⁄ ) 𝑒32𝑝 𝑆 (𝐶 𝑚⁄ ) 𝜌𝑆(𝑘𝑔 𝑚2⁄ ) −5.2 −5.2 −3 × 10−8 −3 × 10−8 5.61 × 10−6 the others geometrical and physical parameters of pens are presented in table 3 [24, 25]. table 3 the geometrical and physical parameters 𝑅(𝑚) 𝐿 𝑅⁄ ℎ𝑁 𝑅⁄ ℎ𝑝 𝑅⁄ 𝑏 𝑅⁄ 𝐶𝑤(𝑁. 𝑆 𝑚⁄ ) 1 × 10−9 10 0.01 0.005 0.1 1 × 10−3 𝐾𝑤(𝑁 𝑚3⁄ ) 𝐾𝑝(𝑁 𝑚⁄ ) 𝑉𝑝(𝑉) 𝑉0 𝑉𝐷𝐶(𝑉) 𝑉𝐴𝐶(𝑉) 8.999503 × 1017 2.071273 1 × 10−2 1 2 1 3.1. surface/interface effects on pull-in instability in current section, the surface/interface parameters effect of piezoelectric nanosensor such as lame’s constants (𝜆𝐼,𝑆, 𝜇𝐼,𝑆), residual stress (𝜏0 𝐼,𝑆), piezoelectric constants (𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 ) and mass density (𝜌𝐼,𝑆) are studied for pull-in instability analysis on dimensionless natural frequency and nonlinear dynamics response in accordance with the data in tables 1-3. first, the effect of direct pull in voltage dc as pull-in instability analysis on the dnf of pens is presented in fig. 2 and for different boundary conditions and two cases of surface/interface densities of table 4. it is obvious from this figure that in higher surface/interface densities of case 1, stiffness is decrease and the natural frequency decrease to without s/i effects. also, with decreasing of surface/interface densities in case 2, stiffness increase and the natural frequency is increased compared to without s/i effects. also, this figure shown that the lowest and highest natural frequency respectively are for case 2 of cc boundary condition and case 1 of cf boundary condition both with s/i effects. in all boundary conditions, case 2 (1) of table 4 considering all s/i effects have maximum (minimum) pull in voltage and natural frequency. table 4 two cases of surface/interface densities case 1 case 2 𝜌𝐼(𝑘𝑔 𝑚2⁄ ) 𝜌𝑆(𝑘𝑔 𝑚2⁄ ) 𝜌𝐼(𝑘𝑔 𝑚2⁄ ) 𝜌𝑆(𝑘𝑔 𝑚2⁄ ) 5.46 × 10−7 5.61 × 10−6 5.46 × 10−8 5.61 × 10−7 264 s. h. h. kachapi fig. 2 the effects of different boundary conditions for pull in voltage on dnf of pens figs. 3 and 4 present the effects of different surface/ interface lame’s constants 𝜆𝑆𝑘 and 𝜆𝐼𝑘 , for pull-in instability analysis respectively on dnf and ndr of pens. fig. 3 the effects of surface/interface lame’s constants 𝜆𝐼,𝑆 for pull in voltage on dnf of ss pens it is clear that increasing both surface/interface lame’s constants 𝜆𝐼,𝑆, due to increasing of pens stiffness, in both analysis of dnf and ndr, pull-in voltage increases. also, it is obvious that for different cases of lame’s constants, the system has stable (for example: 𝐴 to 𝐵1 or 𝐶1 from fig. 4) or unstable (for example: 𝐵1to 𝐵2 or 𝐶1 to 𝐶2 from fig. 4) point as saddle node bifurcation point. nonlinear vibration response of piezoelectric nanosensor: influences of surface/interface effects 265 fig. 4 influence of surface/interface lame’s constants 𝜆𝐼,𝑆 for pull in voltage on ndr of ss pens with �̅�𝑝 = 1 × 10−3; solid line (-): stable, dashed line (--): unstable the effects of different surface and interface lame’s constants 𝜇 𝑆𝑘, and 𝜇𝐼𝑘, for pull-in instability analysis respectively on dnf and ndr of pens are presented in figs. 5 and 6. fig. 5 the effects of surface/interface lame’s constants 𝜇𝐼,𝑆 for pull in voltage on dnf of ss pens similar to 𝜆𝐼,𝑆, it is clear that increasing both surface/interface lame’s constants 𝜇𝐼,𝑆, due to increasing of pens stiffness, in both analysis of dnf and ndr, pull-in voltage increases. 266 s. h. h. kachapi fig. 6 influence of surface/interface lame’s constants 𝜇𝐼,𝑆 for pull in voltage on ndr of ss pens with �̅�𝑝 = 1 × 10−3; solid line (-): stable, dashed line (--): unstable figs. 7 and 8 show the effects of surface and interface residual stress 𝜏0 𝑆𝑘 and 𝜏0 𝐼𝑘, for pull-in instability analysis respectively on dnf and ndr of pens. fig. 7 the effects of surface/interface residual stress 𝜏0 𝐼,𝑆 for pull in voltage on dnf of ss pens as can be show in both analysis of dnf and ndr, increasing both surface/interface residual stress 𝜏0 𝐼,𝑆 lead to increasing of pens stiffness and as a result, pull-in voltage increases. nonlinear vibration response of piezoelectric nanosensor: influences of surface/interface effects 267 fig. 8 influence of surface/interface residual stress 𝜏0 𝐼,𝑆 for pull in voltage on ndr of ss pens with �̅�𝑝 = 1 × 10−3; solid line (-): stable, dashed line (--): unstable the effect of surface piezoelectricity constants 𝑒31𝑝 𝑠𝑘 and 𝑒32𝑝 𝑠𝑘 for pull-in instability analysis respectively on dnf and ndr of pens are presented in figs. 9 and 10. fig. 9 the effects of surface piezoelectricity constants 𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 for pull in voltage on dnf of ss pens it is observed that increasing of negative surface piezoelectricity constants 𝑒31𝑝 𝑠𝑘 and 𝑒32𝑝 𝑠𝑘 leads to increasing of the pens stiffness and as a result, pull-in voltage increases. 268 s. h. h. kachapi fig. 10 influence of surface piezoelectricity constants 𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 for pull in voltage on ndr of ss pens with �̅�𝑝 = 1 × 10−3; solid line (-): stable, dashed line (--): unstable figs. 11 and 12 illustrate the effects of surface and interface mass density 𝝆𝑺𝒌 and 𝝆𝑰𝒌 , for pull-in instability analysis respectively on dnf and ndr of pens. as it can be seen that with increasing surface/interface mass density 𝝆𝑰,𝑺, due to decreasing of the pens stiffness, dnf significantly decrease, but will not have a considerable effect on pull-in voltage and ndr. fig. 11 the effect of surface 𝜌𝑆 and interface 𝜌𝐼 mass density for pull in voltage on dnf of ss pens nonlinear vibration response of piezoelectric nanosensor: influences of surface/interface effects 269 fig. 12 influence of surface 𝜌𝑆 and interface 𝜌𝐼 mass density for pull in voltage on ndr of ss pens with �̅�𝑝 = 1 × 10−3; solid line (-): stable, dashed line(--): unstable in figs. 13 and 14 the effects of surface/interface parameters effects for pull-in instability analysis respectively on dnf and ndr of ss pens are shown. it can be seen that with ignoring the surface/interface density 𝜌𝐼,𝑆, due to increasing of pens stiffness, the system will have a maximum dnf compared to other cases. also considering the surface/interface effects of case 1 in table 4, due to decreasing of nano shell stiffness, has a lower dnf than the case of without s/i effects. according to this figure, regardless the lame’s constants 𝜆𝐼,𝑆 and s/i stress (𝜏0 𝐼,𝑆 ) slightly lead to decreasing of pens stiffness and as a result, the dnf is lower than the case of with all s/i effects. the surface piezoelectric constants (𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 ) do not have much effect on the dnf. fig. 13 the effects of all surface and interface parameters for pull in voltage on dnf of ss pens 270 s. h. h. kachapi fig. 14 influence of surface/interface lame’s constants 𝜆𝐼,𝑆 for pull in voltage on ndr of ss pens with �̅�𝑝 = 1 × 10−3; solid line (-): stable, dashed line (--): unstable also, it can be seen that in dnf analysis, ignoring the surface/interface density 𝜌𝐼,𝑆, surface piezoelectric constants (𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 ) and with all s/i, the ss pens has maximum pull-in voltage, while in ndr analysis of ss pens, in case of without all s/i has maximum pull-in voltage. 4. conclusion in current study, the effect of surface/interface parameters of piezoelectric nanosensor such as lame’s constants (𝜆𝐼,𝑆, 𝜇𝐼,𝑆), residual stress (𝜏0 𝐼,𝑆), piezoelectric constants (𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 ) and mass density (𝜌𝐼,𝑆) are studied for pull-in instability analysis on dimensionless natural frequency and nonlinear dynamics response. the piezoelectric nanosensor is subjected to two piezoelectric layers, nonlinear electrostatic force, harmonic excitations and structural damping. to achieve this purpose, the hamilton’s approach, assumed mode and lagrange–euler’s theories and also arc-length continuation and complex averaging methods are used. some conclusions are obtained from this study: ▪ in the case of lower (higher) surface/interface densities, its stiffness is increase (reduced) that leads to increasing (decreasing) of the natural frequency compared to case of without s/i effects. ▪ the cf (cc) boundary condition with s/i case1 (2) has the lowest (highest) natural frequency. ▪ increasing surface/interface lame’s constants 𝜆𝐼,𝑆 and 𝜇𝐼,𝑆, surface/interface residual stress 𝜏0 𝐼,𝑆 and negative surface piezoelectricity constants 𝑒31𝑝 𝑠𝑘 and 𝑒32𝑝 𝑠𝑘 , due to increasing 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https://www.sciencedirect.com/science/journal/0307904x https://www.sciencedirect.com/science/journal/0307904x https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=2ahukewjuvfurgzrpahwyqhuihysmaeoqfjabegqiebac&url=https%3a%2f%2fonlinelibrary.wiley.com%2fjournal%2f15214001&usg=aovvaw2dqpwnctfmikcxbs1xlo57 https://www.sciencedirect.com/science/article/pii/s0888327018302085#! https://www.sciencedirect.com/science/journal/08883270 facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 47 64 https://doi.org/10.22190/fume190121010h © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space markus heß department of system dynamics and friction physics, berlin university of technology, germany abstract. for the steady wear state of two contact problems involving power-law graded materials, closed-form solutions are derived in terms of pressure distribution and limiting shapes of profile. both gross slip of an initially flat-ended cylindrical punch on a powerlaw graded half-space and the load-controlled fretting wear under partial slip of an initially parabolic indenter are studied. in the case of gross slip at fixed penetration depth there exists a certain exponent of elastic inhomogeneity, for which the effective volume change takes its maximum value. to minimize wear due to fretting under partial slip, an amplitude dependent design of the material gradient is necessary. for large amplitudes of the tangential force a gradient ranged from a soft surface to a hard ground is beneficial, small amplitudes require a reverse gradient characterized by a hard surface and a soft ground. however, the choice of the material gradient also has a decisive influence on the strength of stress singularities at the contact edge and thus the initiation of fretting fatigue cracks, which is why it is discussed in more detail. key words: contact mechanics, gross slip, partial slip, fretting, wear, functionally graded material 1. introduction if the normal contact of two bodies is superimposed by small tangential oscillations, partial slip occurs within the contact interface which causes wear. this phenomenon is called fretting and arises in numerous engineering applications like power steam generator received january 21, 2019 / accepted march 15, 2019 corresponding author: markus heß department of system dynamics and friction physics, berlin university of technology, sekr. c8-4, straße des 17. juni 135, 10623 berlin, germany e-mail: markus.hess@tu-berlin .de 48 m. heß tubes [1], roots of compressor and turbine blades [2], artificial hip joints [3], electrical contacts [4], railway axles [5], etc. extensive investigations have been made in order to tackle the fretting problem, which can be responsible for a shortened life cycle and the failure of machine components. most analytical considerations apply to elastically similar materials and are mainly restricted to the solution of the steady contact state, at which all partial slip is ceased and thus no further wear takes place [6, 7]. particular attention was paid to the calculation of the limiting shape of the profiles, which does not depend on the type of wear criterion [8, 9]. it merely requires knowledge of the solutions of normal contact problems between the rigid cylindrical indenters with appropriately shaped tips and the elastic half space. due to the fact that the (minimum) stick area does not change during the transition from the initial to the steady state, the diameter of the indenter is a priori known [10]. however, the calculation of the stress evolution and profile change during the transient wear process, require an explicit wear law. most numerical wear simulations are based on archard’s well-known wear law [11]. it has been implemented in finite element methods as well as other efficient numerical methods based on an integral equation approach or quadratic programming [12-14]. an extremely efficient and therefore very fast method for simulating fretting wear is the method of dimensionality reduction [15]. for simulations of gross slip wear it was shown that the calculation time using this method is for several orders lower than that of models based on the finite element method [16]. the method was not only used for the numerical study of classic fretting problems [17, 18], but also for the calculation of the limiting shape of the worn profile under multi-mode fretting and the results are in good agreement with experimental ones [19]. all the above mentioned contributions to the study of fretting wear have in common that they only consider elastically homogeneous materials. corresponding studies of elastically inhomogeneous materials are scarce although it is well-known that a controlled gradient of the elastic modulus can lead to a greater resistance to contact and friction damage [20]. for instance, hertzian cone cracks are suppressed due to the reduction of the maximum tensile stresses in the surface [21]. one reason for the lack of research into fretting of so-called functionally graded materials lies in the difficulty of solving such problems. the few works are mostly related to special material gradients and plane contact problems, which are solved numerically [22, 23]. a powerful numerical method for solving three-dimensional fretting contact problems involving arbitrary material gradients goes back to wang et al. [24], however, studies on wear are not included. in this paper we derive analytical solutions for the steady wear state of two frictional contact problems involving power-law graded materials, whose gradient perpendicular to the half-space surface is prescribed by 0 0 ( ) with 1 1 k z e z e k z          . (1) although this gradient is physically unrealistic due to a vanishing or infinitely large elastic modulus at the surface, it reflects qualitative behavior of the real gradient materials quite well. the great advantage of a material gradient according to eq. (1) is that it is probably the only one for which contact problems can be solved in a closed analytic form. whereas complete solutions of frictionless normal contact problems with and without adhesion have been known for some time [25, 26], solutions of tangential contacts were a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 49 only recently developed [27, 28]. taking into account the latter results, in chapt. 2 we study the gross slip of an initially flat-ended cylindrical punch on a power-law graded half-space. chapt. 3 deals with fretting wear under load-controlled conditions in the partial slip regime for the contact between a rigid initially parabolic indenter and a powerlaw graded half-space. for both problems, complete solutions are given in terms of stresses and limiting shapes of the profile. in addition, the influence of the exponent of elastic inhomogeneity as well as the tangential force amplitude on the wear volume is discussed. 2. gross slip wear analysis of a rigid punch on a power-law graded half-space we first consider the contact between a rigid flat-ended cylindrical punch with a circular base and a power-law graded elastic half-space as shown in fig. 1. the rigid indenter is pressed into the half-space by an initial normal force fn,0, giving rise to a prescribed indentation depth d. subsequently, the power-law graded half-space is moved horizontally with constant speed causing wear on the punch. in contrast, wear of the halfspace is assumed to be negligibly small. the indentation depth is kept constant during sliding. from the practical point of view, this is a hardly controllable condition. however, we later explain this choice and discuss the gross slip under a fixed normal force. fig. 1 initial state of a sliding contact between a rigid flat punch and a power-law graded elastic half-space it is supposed that normal and tangential contacts are uncoupled. since the indenter is rigid, uncoupling is, strictly speaking, only valid for a poisson ratio of ν =1/(2+k), known as holl ratio. thus, negative exponents of elastic inhomogeneity would lead to unphysical values. however, more detailed studies show that there is only a very weak coupling if we consider poisson’s ratios of 0.5 instead and all solutions deviate only slightly from the solutions using holl’s ratio [27]. for this reason, we adopt it for negative exponents as well. the solution of the initial contact state of fig. 1 can be taken from booker et al. [25]. the normalized pressure distribution is given by 50 m. heß 1 2 2 2 2 0 0 , 1 ( )( ) 2 2( ) ( ) n k k h k z a r p r a e d        , (2) where hn(k,) is defined in the appendix. the pressure has a singularity at the edge of contact which becomes milder with increasing exponent of elastic inhomogeneity. the displacement of the half-space surface reads   10 2 2 2 1 2 1 for 0 ( ) cos 2 1 1 3 f , ; ; for 1 2 2 2 kk r a w r a k k k a d r a k r r                      , (3) where 2f1(a,b;c;z) represents the gaussian hypergeometric series 2 1 0 ( ) ( ) ( ) ( , ; ; ) with ( ) : ( ) ! ( ) n n n n n n a b z x n f a b c z x c n x        . (4) according to archard’s wear law, the wear rate during sliding will be higher at larger pressures, so that wear takes place primarily at the edge of contact. after a sufficiently long time the pressure distribution becomes uniform and the rigid punch reaches a steady state profile. this final shape coincides with the surface displacement of a circular domain on a power-law graded half-space which is loaded by a constant pressure. the solution of the steady contact state can be taken again from [25]. the displacement of the half-space surface is 2 2 1 2 1 2 2 1 2 1 1 f , ;1; for 0 2 2( ) 1 1 1 f , ;2; for 2 2 2 k k k r r a aw r d k a k k a r a r r                           . (5) according to eqs. (3) and (5), the shape of the indenter and the surface displacements of the power-law graded half-space both in the initial state and the steady wear state are depicted in fig. 2 and 3. the gradient material in fig. 2 is characterized by a negative exponent of elastic inhomogeneity (k = -0.5), which maps a grading from a hard surface to a soft ground. in contrast, the displacements in fig. 3 correspond to a power-law graded half-space with a soft surface and a hard ground (k = 0.5). a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 51 fig. 2 indenter profile and surface displacement of a power-law graded half-space with a negative power-law exponent k=-0.5 in the initial and the steady wear state fig. 3 indenter profile and surface displacement of a power-law graded half-space with a positive power-law exponent k=0.5 in the initial and the steady wear state a comparison of both indicates that the contact depth becomes larger and the surface displacements decrease with increasing the power-law exponent. furthermore, the effective volume changes of the profile shape are highlighted by the shaded areas. note that these volume changes do not represent absolute wear volumes. the effective volume change can be calculated as follows: 0 0 ( ( ) ( ))2 d a v w r w r r r  . (6) inserting eqs. (3) and (5) into eq. (6) yields 2 12 1 1 1 f , ;2;1 2 2 v k k a d           . (7) 52 m. heß fig. 4 illustrates the dependence of the normalized effective volume change on the exponent of elastic inhomogeneity. while it tends to zero at the edges of the exponent's domain, it reaches a maximum of 15.5 % at a defined exponent of k = 0.169. the absence of any change in the profile for k = -1 and k = 1 is a consequence of the vanishing contact stiffness [29]. only in the special case of a linear-inhomogeneous, incompressible halfspace there exists non-zero contact stiffness. this is the famous gibson medium which responds to normal stresses on its surface like a winkler foundation [30]. therefore, once again, the initial shape of the profile does not change. fig. 4 normalized effective volume change of the profile shape as a function of the exponent of the elastic inhomogeneity note that the wear process takes place under constant indentation depth. however, the shape variation leads to a redistribution of the contact pressure and thus a change of the normal force. simple integration of the pressure distribution in the initial and the steady state over the contact area gives ,0 2 0 0 ( , ) 2 (1 )(1 ) k n n k f h k a e da k z      , (8)   , 2 0 02 ( , )(1 ) 2 4(1 )cos k n n kk f h k k a e da z        . (9) fig. 5 demonstrates that the normal force generally decreases during the transition from the initial to the steady state, regardless of the choice of the characteristic depth. one could say that the system is trying to achieve an optimized state. the magnitudes of the normal forces according to eqs. (8) and (9) strongly depend on both the exponent of elastic inhomogeneity k and the characteristic depth z0. the qualitative behavior of the normal force depending on these parameters can be taken from fig. 5 and is compared in table 1 with the behavior of the elastically homogeneous half-space. at small characteristic a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 53 depths coupled with a positive power-law exponent or large characteristic depths coupled with a negative exponent, a larger normal force is required to produce the same indentation depth as for elastically homogeneous materials (exponents closer to -1 or 1 are excepted). in a very rough approximation one can compare the mentioned cases with the indentation of a layered half-space. both a very thin and soft layer on a stiff substrate (es >> e0 >> el) as well as a very thick and stiff layer (el >> e0 >> es) on a soft substrate require a greater normal force for indentation. in a similar way, the tendencies associated with a decrease in the required normal force can be interpreted. fig. 5 normal force in the initial and steady state as a function of the power-law grading exponent for different characteristic depths another interesting result is derived from the ratio of the normal forces given by eqs. (9) and (8):   , ,0 2 (1 )(1 ) 4cos n k n f k k f      . (10) it is independent of the characteristic depth z0 as well as an even function of exponent k and confirms once again the fundamental decrease of the normal force. finally, let us briefly discuss the more practical boundary condition of gross slip under a constant normal force. apart from a scaling factor, the relevant initial and steady state equations are the same as those given here. under fixed normal force the indentation depth generally increases during the transition from the initial to the steady state. the ratio of the indentation depth in the final state to that in the initial state is given by the reverse of eq. (10). analogously, by rearranging of eqs. (8) and (9), we obtain reverse curves to those shown in fig. 5. in other words, very small values of the normal force under fixed penetration depth correspond to very high, and in the limit, even infinitely 54 m. heß large values of the penetration depth under fixed load. this unphysical behavior is due to the choice of a power-law material gradient. hence, we decided to prefer a study under fixed penetration depth. table 1 tendencies of the change of normal force in comparison to an elastic homogeneous material z0 < a z0 > a -1 < k < 0 fn ↓ fn ↑ 0 < k < 1 fn ↑ fn ↓ 3. fretting wear analysis of the contact between a rigid parabolic indenter and a power-law graded half-space we now come to the much more important partial slip contact problem between a rigid initially parabolic shaped indenter and a power-law graded half-space depicted in fig. 6. the rigid indenter is pressed against the half-space by a fixed normal force fn and subsequently subjected by an oscillating tangential force fx whose magnitude does not exceed limiting value μ fn. once again, we suppose uncoupling of the normal and tangential contact as mentioned in the beginning of chapt. 2 and assume that the indenter is rigid although it can wear. by contrast, wear of the power-law graded material should be neglected. at this point, it should be noted that all results are expected to remain valid even for opposite assumptions. due to the change in the surface material properties of the worn gradient material, the redistribution of the pressure during the transient process will be different but the steady state should be unaffected by that. we consider very severe wear which means that wear proceeds until all partial slip is ceased. fig. 6 initial state (left) and steady state (right) of a partial slip contact problem between a rigid (initially) parabolic shaped indenter and a power-law graded half-space in the initial state shown on the left in fig. 6, the solution of the normal contact problem was derived by giannakopoulos and suresh [31]. the penetration depth, normal force, pressure distribution and normal surface displacement outside of the contact area are given by: a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 55 2 0 0 0( ) ( 1) a d a k r   , (11) 3 0 0 0 2 2 0 4 ( , ) ( ) (1 ) ( 1) ( 3) k n n k h k e a f a z k k r        , (12) 1 2 2 0 2 00 ( 3) ( ) 1 2 k nk f r p r aa             , (13)  2 2 22 0 2 0 0 0 2 2 2 0 cos 1 1 3 1 ( ) b ; , b ; , ( 1) 2 2 2 2 ka a ak k r k k w r k r r ra                          , (14) where b( ; , )z x y represents the incomplete beta function according to 1 1 0 b( ; , ) : (1 ) d , z x yz x y t t t x y      . (15) the solution of the tangential contact problem goes back to heß [27]. the (minimum) stick radius c as a function of (fixed) tangential force amplitude fx is 1 3 0 n 1 k xfc a f        , (16) where a0 denotes the initial contact radius [28,32]. herein, it was made use of the usual assumptions: the direction of the tangential stresses coincides with the direction of the applied tangential force. moreover, the (small) slip component perpendicular to the applied tangential force is neglected. in the steady state depicted on the right in fig. 6 an inner part of the contact still remains in the state of stick so that in this region the final indenter profile coincides with the initial one. this stick region of radius b ≤ c is enclosed by an annular domain of outer radius a∞ in which the pressure becomes zero and the surfaces of the two bodies are in glancing contact. thus, the steady contact state can be calculated by using the normal contact solution of a cylindrical punch of radius b with a parabolic tip. the solution can be interpreted as a superposition of two parts: i. the solution for the indentation by a parabolic indenter until contact radius b is reached. corresponding load f1 is given by eq. (12) if we replace a with b. ii. a rigid body translation of the contact area of radius b in vertical direction caused by an applied load fn -f1 which corresponds to the indentation by a flat-ended cylindrical punch whose solution can be adopted from eqs. (2) and (3). 56 m. heß in this way we come to the following solution of the normal contact in the steady state: 1 1 2 22 2 n 11 2 2 ( 1)( ( ))( 3) ( ) ( ) 1 1 2 2 k k k f f bk f b r r p r b bb b                                  , (17)     2 2 2 2 2 2 2 2 2 2 0 n 12 1 2 0 cos 1 1 3 1 ( ) b ; , b ; , ( 1) 2 2 2 2 cos (1 )(1 ) ( ( )) 1 1 b ; , 2 22 ( , ) k kk k n b b k k r b k k w r k r r b r k z f f b b k k h k e b r                                        . (18) the tangential contact in the steady state is characterized by a rigid body (tangential) displacement of the whole stick area. the corresponding distribution of tangential traction due to a tangential force fx is [27] 1 2 2 2 ( 1) ( ) 1 2 k xk f r r bb                 . (19) 3.1. extent of the stick area in the steady state in the following, we will prove that the limiting stick radius b in the steady state coincides with the minimum stick radius c in the initial state. for this purpose, we study the distributions of pressure and tangential tractions at the vicinity of the edge of contact in the steady state. let r = b (1-δ) with δ << 1, then from eqs. (17) and (19) we obtain the asymptotic expressions 1 n 1 2 2 ( 1)( ( )) ( ) (2 ) 2 k k f f b p b         , (20) 1 2 2 ( 1) ( ) (2 ) 2 k xk f b         . (21) both tend towards infinity like δ (k-1)/2 . however, close to the edge of the contact (within the stick area) we expect tangential stresses which are only slightly smaller than the limiting value prescribed by amontons law, i.e. τ∞(δ) = µp∞(δ) ε. using eq. (20) as well as eq. (21) and taking the limit ε→0 yield 1 n n ( ) 1xf f b f f          . (22) remember that the normal force is kept constant during the transient wear process which is why we can substitute the normal force fn on the right side of eq. (22) by the expression given in eq. (12). since the force f1 is also defined by eq. (12) we just have to replace radius a0 by radius b, eq. (22) gives a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 57 1 3 0 n 1 k xfb a f        . (23) a comparison with eq. (16) proves that the radius of the stick area in the steady state is indeed equal to the radius of the minimum stick area in the initial state. 3.2. limiting shape of the profile in the steady state the shape f∞ of the steady state profile in the worn area is determined by the displacement of the free half-space surface due to the indentation by a cylindrical punch of radius c with a parabolic tip. taking into account b = c as well as eq. (12), eq. (18) results in     2 2 2 2 3 2 0 2 2 2 2 cos( ) 2 1 1 1 1 b ; , ( 3) 1 2 2/ cos 3 1 b ; , ( 1) 2 2 k k k cw r c k k k ka r c r r c k k k r                              , (24) where we have introduced = c/a0 and = r/a0. as can be seen in fig. 6 on the right, the steady state wear profile is defined as follows: 0 ( ) for 0 ( ) ( ) for f r r c r a f r d w r c r a               . (25) however, the outer radius of worn region a∞ is still unknown and must be determined from the condition that the initial profile and the steady state profile coincide at this point: 0 0 0( ) ( ) ( ) ( ) ( ) ( )f a f a f a f c w c w a           . (26) normalized by the contact radius in the initial state, fig. 7 shows the limiting radii of the worn annular region in the steady state as a function of the normalized amplitude of the applied tangential force according to eqs. (16) and (26). curves are plotted for both a homogeneous material and two power-law graded materials characterized by exponents k = -0.5 and k = 0.5. it is obvious that the larger the exponent of elastic inhomogeneity the larger the stick radius c and the smaller the radius of the maximum extent of worn region a∞. however, this statement should be treated with caution since the radii have been normalized to contact radius a0 in the initial state and the latter depends strongly on both the exponent of elastic inhomogeneity and the characteristic depth. to be able to compare absolute values, we would have to presuppose power-law graded materials, which exhibit the same contact area in the initial contact state if subjected by the same fixed normal force. such conditions could be of interest in designing an electrical contact, which require both a fixed normal force and a fixed contact area to transmit a certain current (the change in conductivity due to the material gradient is disregarded). 58 m. heß fig. 7 inner radius c and outer radius a∞ of the annular worn region in the steady state versus aspect traction ratio fx /μfn three sample materials that meet these conditions are characterized by the following elastic material gradients: 0 0 0 0 01. ( ) 2. ( ) 0.88 / 3. ( ) 2.06 /e z e e z e a z e z e z a   . (27) that the entire worn area decreases with increasing exponent of elastic inhomogeneity, is also evident in fig. 8. in this figure, the final shapes of the wear profile for 3 different materials are compared: at the top an inhomogeneous material with k = -0.5, which is characterized by a hard surface and a soft ground; in the middle an elastically homogeneous material and at the bottom an inhomogeneous material with k = 0.5, which maps a half-space characterized by a soft surface and a hard ground. in addition to the initial profile of the rigid indenter, its wear profiles are shown for three different amplitudes of the tangential force. it need not be mentioned that in all cases the wear volume grows with increasing amplitude of the tangential force. but fig. 8 also indicates that in comparison to the homogeneous case, the wear volume is greater for negative exponents and smaller for positive ones. in other words, for the three tangential force amplitudes considered, the wear in the case of gradient materials with a soft surface and a hard ground is smaller than that for those with a hard surface and a soft ground. however, that this statement is not universally valid is supported by fig. 9, in which the wear volume is plotted as a function of the amplitude of the tangential force for different exponents of elastic inhomogeneity. the worn volume was determined according to    0 0( ) ( ) 2 d (0) ( ) ( ) 2 d a a c c v f r f r r r w w r f r r r            . (28) a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 59 fig. 8 limiting shapes of the profile for three different tangential force amplitudes and three different exponents of elastic inhomogeneity: k = -0.5 (top), k = 0 (middle) and k = 0.5 (bottom) it is clearly visible that amplitudes less than about 20 % of the limiting value μfn cause more wear if the materials show a gradient ranged from a soft surface to a hard 60 m. heß ground. this is a significant result which can be important for optimized material selection of many technical systems. numerous force-fitted connections are characterized by a very high normal force to realize a preferably large stick region. however, it is wellknown that even the smallest vibrations perpendicular to the normal load result in a small annulus of slip which inevitably causes energy dissipation and wear. according to our results, in the mentioned range of small tangential force amplitudes, the use of a powerlaw gradient material with a negative exponent would reduce wear. thus, a material gradient is needed that ranges from a hard surface to a soft ground. in contrast, an operating area of high tangential force amplitudes requires a power-law graded material with a positive exponent to minimize wear. the tendencies in the behavior of the wear volume depending on the material gradient can be intuited from fig. 8. it reveals that the amount of curvature of the worn area increases significantly with increasing exponent, whereas the worn area itself decreases. for very small slip annuli, the influence of the curvature on the wear volume seems to predominate. fig. 9 normalized representation of the wear volume as a function of the tangential force amplitude for different exponents of elastic inhomogeneity; red curves indicate a negative exponent, blue curves a positive exponent in sec. 3.1, we have already examined the normal and tangential stresses near the edge of the stick area in the fully worn steady state. in fig. 9 the pressure distribution in the fully worn steady state for three different tangential force amplitudes and three different exponents of elastic inhomogeneity are displayed – k = -0.5 at the top, k = 0 in the middle and k = 0.5 at the bottom. once again it is clearly visible that the stick radius and thus the area for transmitting the normal force grow with increasing exponent of elastic inhomogeneity. this is tantamount with a decrease of the mean pressure. furthermore, the stress singularity becomes milder with increasing exponent k. the same applies to the singularity of the tangential stresses according to eq. (21). the stress field resembles the near-tip stress field of a mixed-mode ii/iii crack, whose weighting depends on the surrounding position on the edge of contact. on the x-axis it corresponds to a pure mode a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 61 ii loading, whereas on the y-axis to a pure mode iii loading. the influence of the exponent of elastic inhomogeneity on the stress singularity can be profitably used to suppress plastic yielding and crack nucleation at the edge of contact. fig. 9 pressure distribution in the fully worn steady state for three different tangential force amplitudes and three different exponents of elastic inhomogeneity: k = -0.5 (top), k = 0 (middle) and k = 0.5 (bottom) 62 m. heß 4. conclusions and discussion we have investigated the steady wear state for two types of frictional contact problems between a rigid indenter and a power-law graded material. first, we considered a gross slip between a rigid, cylindrical punch with an initially flat circular base and a power-law graded half-space. under fixed penetration depth and the strong assumption that merely the indenter can wear, the solutions for the initial contact state as well as the steady wear state are given in terms of pressure distribution and normal surface displacements. moreover, the steady wear shape of the profile and the effective volume change were determined. we have found that the effective volume change becomes a maximum of 15.5 % at a defined exponent of k = 0.169. for power-law graded materials characterized by either small characteristic depths coupled with a positive power-law exponent or large characteristic depths coupled with a negative exponent, a larger normal force is required to produce the same indentation depth as for elastically homogeneous material. second, we analyzed fretting wear under load-controlled conditions for the contact between a rigid initially parabolic indenter and a power-law graded half-space. again, complete solutions are given in terms of stresses and normal surface displacements for both the initial and the steady state. following previous studies for homogeneous materials, we prove that the size of the stick zone does not change during the transition from the initial to the steady state. as mentioned above, the power-law graded half-space is assumed to be resistant to wear but the rigid indenter should wear. the dependence of limiting profile shapes f∞ on the exponent of elastic inhomogeneity as well as on the tangential force has been completely determined. one interesting result is that the amount of curvature of the worn area increases significantly with increasing exponent. but probably the most significant result of our analyses is the finding that opposing material gradients for small and large tangential force amplitudes are necessary to minimize the wear. whereas for large amplitudes a gradient ranged from a soft surface to a hard ground is beneficial, small amplitudes require a reverse gradient characterized by a hard surface and a soft ground. although it is frequently stated in the literature that power-law graded materials reflect the qualitative behavior of real gradient materials quite well, appropriate analytical and experimental investigations should be made to ensure this result. the same applies to the influence of the milder stress singularity on fretting fatigue crack initiation. in addition, we would like to emphasize that all our studies are restricted to purely power-law graded elastic material behavior. however, in practice, stress singularities are avoided by plastic deformation of the material and this in turn strongly influence the wear state. by using a finite-element model and considering elastic-plastic material behavior, hu et al. [33] found that the stick-slip boundary can move steadily into the stick region and thus wear continues indefinitely. thus, future investigations on wear of functionally graded materials should consider the elastic-plastic regime. acknowledgement: the author would like to thank dr. r. pohrt and prof. v. l. popov for many valuable discussions. a study on gross slip and fretting wear of contacts involving a power-law graded elastic half-space 63 appendix the function ( , )nh k  in eq. (2) is defined by (see [32])   2(1 )cos 1 2 2 ( , ) , 1 ( , ) ( , )sin 2 2 n k k k h k k k c k k                                     , (29) with 1 3 ( , ) 3 ( , ) 2 2 2 ( , ) (2 ) k k k k k c k k                          (30) and ( , ) (1 ) 1 1 k k k             . 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thouless, m.d., barber, j.r., 2016, effect of plastic deformation on the evolution of wear and local stress fields in fretting, international journal of solids and structures, 82, pp. 1-8. 12096 facta universitatis series: mechanical engineering vol. 21, no 3, special issue, 2023, pp. 359 386 https://doi.org/10.22190/fume230901034m © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper assessment of sustainable wastewater treatment technologies using interval-valued intuitionistic fuzzy distance measure-based mairca method arunodaya raj mishra1, pratibha rani2, fausto cavallaro3, adel fahad alrasheedi4 1department of mathematics, government college raigaon, madhya pradesh, india 2department of engineering mathematics, koneru lakshmaiah education foundation, guntur andhra pradesh, india 3department of economics, university of molise, via de sanctis, campobasso, italy 4statistics and operations research department, college of science, king saud university, riyadh, saudi arabia abstract. effective wastewater treatment has significant effects on saving water and preventing unnecessary water scarcity. an appropriate wastewater treatment technology (wwtt) brings economic benefits through reuse in different sectors and benefits the society and environment. this study aims to develop a decision-making framework for evaluating the sustainable wwtts under interval-valued intuitionistic fuzzy set (ivifs) environment. the proposed mcdm framework is divided into two stages. first, a new hellinger distance measure is developed to determine the degree of difference between ivifss and also discussed its desirable characteristics. second, an interval-valued intuitionistic fuzzy extension of multi-attribute ideal-real comparative analysis (mairca) model is developed using the proposed hellinger distance measurebased weighting tool. further, the proposed model is implemented on an empirical study of sustainable wwtts evaluation problem. sensitivity and comparative studies are made. the results indicate that odor impacts, sludge production, maintenance and operation are the most effective sustainable factors and microbial fuel cell (mfc) technology is the best wwtt followed by natural treatment methods. key words: interval-valued intuitionistic fuzzy sets, distance measure, sustainability, mairca, rank sum model, wastewater treatment received: september 01, 2023 / accepted october 06, 2023 corresponding author: pratibha rani department of engineering mathematics, koneru lakshmaiah education foundation, guntur-522302, andhra pradesh, india e-mail: pratibha138@gmail.com mailto:pratibha138@gmail.com 360 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi 1. introduction most of the water that is not used as an ingredient finally appears in the wastewater stream. wastewater treatment (wwt) is the elimination of impurities from wastewater before it reaches aquifers or natural bodies of water such as lakes, rivers and oceans. it is one of the most imperative necessities of the current scenario that can provide wider environmental and societal benefits including improved ambient water quality, protect wildlife, reduced greenhouse gas emissions and so on [1-3]. treated wastewater also brings economic benefits through reuse in different sectors like agriculture, groundwater recharge, vehicle washing, golf course irrigation, toilet flushes, cooling purposes in thermal power plants and building construction activities. the quality of treated wastewater used in agriculture has a great influence on the operation and performance of the wastewater-soil-plant or aquaculture system. wwt solutions can recover valuable resources from wastewater such as biodiesel, electricity, recycled water and nutrients and serve as the components of fertilizer [4]. thus, it is significant for maintaining the health of human beings and ecosystems. the process of wwt consists of using suitable technology to improve or upgrade the quality of a wastewater. usually, wwt will involve collecting the wastewater in a central, segregated location and subjecting the wastewater to various treatment processes. wwt process needs to be a technology type that will be suitable for a particular development and not necessarily the best available technology. selection of suitable wwt technologies that is highly correlated with sustainable development, presents a challenge to the local, regional, national and global policy makers. in developing countries, there is a need for a decision-making tool to evaluate the wastewater treatment technology (wwtt) selection. many researchers have presented their works on the development new technologies and also reviewed the current systems for wastewater treatment. for instance, choudhury et al. [5] highlighted the working strategies of microbial fuel cell (mfc) technology for wwt and reuse of wastewater for power generation. in a study, tarpani and azapagic [6] assessed the life cycle environmental impacts of advanced wwt methods for removal of pharmaceuticals and personal care products. arroyo and molinos-senante [7] presented a choosing-by-advantages approach to evaluate seven wwt alternatives. based on different aspects of sustainability, they selected the most suitable wwt alternative. a systematic review has presented to explain the advantages and disadvantages of different membrane technologies for water treatment [8]. munoz-cupa et al. [9] highlighted the benefits and technical barriers of current mfc systems for wwt. moreover, they have presented the effects of different reaction conditions on chemical oxygen demand removal and electricity generation from mfcs. zhang et al. [10] stated the current trends of wwt plants in china. their study provided some useful implications to the authorities and policy makers by analyzing the industries’ current status in the wwt process. saravanan et al. [11] reviewed several wwt technologies and presented their remarkable power for toxic pollutants removal from wastewater. in addition, they discussed the difficulties related to commercial development of wwt technologies and suggested the future research directions. saravanan et al. [12] presented sustainable strategy on mfc technology to treat the wastewater for the green energy production. for this purpose, the authors have reviewed diverse mfc technologies and their core performance in the direction of waste management and energy conversion. as per the existing studies, several technologies, assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 361 ranging from conventional to advanced treatment processes, are available to treat the wastewater. finding the most suitable wastewater treatment technology (wwtt) is a complex issue due to involvement of multiple sustainability aspects of criteria. an alternative wwtt is considered “most suitable” to the degree that it is consistent with the economic, environment, social, technical, cultural and political aspects of the society. thus, the multi-criteria decision-making (mcdm) approaches are more appropriate to systematically solve this problem. ullah et al. [13] established a decision-making agenda for the selection of wwtt. for this purpose, they firstly provided the comprehensive review of the state-of-the-art in wwt. srivastava and singh [14] established a decision support model for the selection of most suitable wwtt from multiple criteria perspective in which the criteria weights are determined through full consistency method. salamirad et al. [15] proposed an integrated decision support system using the best worst method and the behavioral technique for order of preference by similarity to ideal solution (topsis). in that study, the authors have evaluated seven wwtts with respect to economic, environmental and social dimensions of sustainability. pennelilini et al. [16] presented a novel utility interval-based evidential reasoning approach to evaluate and prioritize the wwtt alternatives for agricultural reuse. with the use of analytic hierarchy process, ćetković et al. [17] reviewed the current situation and problems related to wwtts and evaluated the optimal variant of wwtts. in the process of multi-criteria wwtt selection, ambiguity arises in some form due to the genuine limits on the human mind and imprecise information. the notion of fuzzy set (fs) has widely been used in practice to handle such types of uncertain decisionmaking problems [18-20]. in order to choose the wwtt, dursun [21] developed a hybrid method by incorporating the decision making trial and evaluation laboratory (dematel) and topsis methods with 2-tuple fuzzy linguistic set and applied to evaluate the wwtt candidates. attri et al. [22] presented the combined use of three mcdm methods such as the multi-objective optimization by ratio analysis (moora), the stepwise weight assessment ratio analysis (swara) and the topsis under fuzzy environment. in this study, fuzzy swara method has applied to evaluate the significance values of the considered criteria, while the fuzzy moora and topsis approaches have used to determine the rank of six wwtts. a hybrid decision support system has developed based on the pivot pairwise relative criteria importance assessment and an interactive mcdm approach with linear diophantine fuzzy information to handle the multi-criteria wwtt selection problem [23]. as the fs only contains a membership grade (mg), therefore, atanassov [24] extended the classical fs and investigated the notion of intuitionistic fuzzy set (ifs), which assigns a mg, a non-membership grade (ng) and an indeterminacy grade (ig) to each element with sum of the mg and ng is bounded to 1. after the pioneering innovation by atanassov [24], various significant results have been achieved based on ifs theory [25-28]. in the intuitionistic fuzzy set theory, the degrees of membership and non-membership are exact numbers, which is hard for the experts to define their exact value in several decisionmaking problems. to conquer this issue, atanassov and gargov [29] gave the idea of interval-valued intuitionistic fuzzy set (ivifs), which deals with uncertainty in practical decision-making problems. its basic feature is that both membership and non-membership functions of an element to a given set are considered and taken as interval values rather than exact numbers. due to the broad range of information coverage, the theory of ivifs is used 362 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi in our proposed work. in the context of ivifs, some relations and basic operations score and accuracy functions have been presented [30]. as an extended version of ifs, the ivifs theory provides a more effective and reasonable way to cope with imprecise and uncertain information. due to its higher flexibility in dealing with uncertain data, the ivifs doctrine has been broadly explored from different perspectives [31-33]. to deal with madm problems, several novel methods have been developed in the recent past [34-39]. the multi-attribute ideal-real comparative analysis (mairca) [40] method is a newly developed mcdm method, which determines the gap between the ideal and empirical ratings during the assessment of alternatives [41]. for each alternative, summation of the gaps for all the criteria determines the total gap and option with the least total distance is considered as the best choice. this ranking technique uses different linear normalization method, which is characterized by easy mathematical computations and solution stability [42]. in a study, the classical mairca method has integrated with the well-known dematel model and applied to evaluate the sites for multimodal logistics centre development from sustainability perspectives. kaya [43] analyzed the consequence of covid-19 pandemic in the countries’ sustainable development level through mairca method. the classical mairca method has been combined with lbwa model and interval rough numbers to develop a hybrid mcdm model [44]. in the context of uncertainty, boral et al. [41] incorporated the standard mairca approach with analytic hierarchy process, failure mode and effects analysis and fuzzy numbers, and developed a hybrid decision support system for solving complicated mcdm problem. further, ecer [45] proposed a hybrid intuitionistic fuzzy mairca method with an application in the assessment of covid-19 vaccines. in the recent past, hezam et al. [46] discussed an approach based on symmetry point of criterion-based mairca method and applied to select the most suitable biomass resources for biofuel production under intuitionistic fuzzy environment. the classical mairca approach has been extended from interval-valued neutrosophic perspective and applied to select the multi-criteria sustainable materials [47]. rani et al. [48] proposed a pythagorean fuzzy information-based mairca model, in which the criteria weights are determined through standard deviation-based method. in addition, they presented the drawbacks of existing studies in the context of pythagorean fuzzy set. based on the literature review, this paper identifies some research gaps in the existing studies, given as ▪ existing distance measures (zhang et al. [49], düğenci [50], baccour and alimi [51], mishra et al. [52]) present some counter-intuitive cases in order to measure the degree of difference between two ivifss. ▪ few authors (kaya [43], božanić et al. [44], boral et al. [42], ecer [45], haq et al. [47], hezam et al. [46], rani et al. [48]) have developed the extensions of classical mairca method from fuzzy, intuitionistic fuzzy, interval-valued neutrosophic, pythagorean fuzzy perspectives, but the these methods cannot deal with ivif information in which the alternatives’ information is represented in terms of intervals rather than the crisp numbers. ▪ in the literature, some authors (ullah et al. [13], srivastava and singh [14], salamirad et al. [15], pennelilini et al. [16], ćetković et al. [17]) have proposed different mcdm methods for solving wwtts assessment problem. as an extension of ifs, the theory of ivifs has the advantage that both membership and non-membership degrees are interval values and can used to characterize the uncertain information more flexibly assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 363 due to its constraint condition. unfortunately, existing decision support models are unable to deal with the interval-valued membership and non-membership degrees. inspired by the limitations of existing studies, this study develops an interval-valued intuitionistic fuzzy madm framework to evaluate and prioritize the wwtt alternatives. the developed methodology does not only evaluate the considered options through ivifdistance measure-mairca model, but also consider the weights of considered criteria and decision makers. the proposed model can assist the decision makers (dmes) to get more confident for ranking the blockchain platforms. the key contributions of this paper are presented as follows: ▪ new hellinger distance measure is proposed to overcome the limitations of existing ivif-distance measures. comparative study is presented to prove the effectiveness of the proposed measure. ▪ a novel extension of mairca method is developed in the context of intervalvalued intuitionistic fuzzy information perspective in which the information about the criteria and decision makers is completely unknown. ▪ in the proposed method, the proposed distance measure and total support degreebased model is presented to compute the criteria weights. ▪ the presented mairca method is implemented on a case study of wwtts assessment, which proves its applicability and powerfulness. the leading question of the wwtt alternative selection problem is “which one is the most appropriate wwtt alternative among a set of alternatives from sustainability perspective?” to solve this problem, the following questions need to be answered: (i) what are the main criteria to evaluate the most appropriate wwtt alternatives with interval-valued intuitionistic fuzzy information? (ii) which is the most significant criterion for wwtts assessment? (iii) which is the most suitable mcdm technique to select and prioritize the wwtt alternatives based on the economic, social, environmental and technical aspects of sustainability? the key objectives of this study are as follows: (i) identify the main criteria to choose the most suitable wwtt alternative through literature survey and de opinions. (ii) find the importance weights of each criterion under the context of uncertain information. introduce a model to compute the weights of considered evaluation criteria. (iii) determine a hybrid mcdm methodology to prioritize the wwtt alternatives under interval-valued intuitionistic fuzzy environment. the rest part of this study is organized as follows: section 2 firstly presents the basic concepts and then proposes a new distance measure for ivifss. section 3 develops a hybrid mairca method for assessing the mcdm problems under ivifs context. section 4 implements the proposed method on a case study of wwtts assessment. section 5 presents the comparative analysis, discussion on the results and implications. section 6 concludes the whole study and recommends for future researches. 2. proposed distance measure for ivifss this section firstly discusses the fundamental notions related to an ivifs and then proposes a new measure to describe the degree of difference between two ivifss. 364 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi 2.1. basic concepts definition 2.1. consider φ = {ϕ1, ϕ2, ..., ϕt} be a finite universal set. in the following way, atanassov and gargov [29] mathematically defined the ivifs q on φ: { , ([ ( ), ( )],[ ( ), ( )]) : },i q i q i q i q i iq          − + − +=   where 0 ( ) ( ) 1,q i q i   − +   0 ( ) ( ) 1q i q i   − +   and 0 ( ) ( ) 1.q i q i   + + +  here, ( ) [ ( ), ( )]q i q i q i     − += and ( ) [ ( ), ( )]q i q i q i     − += define the degrees of interval-valued membership and non-membership of an object ϕi in q, respectively. the function ( ) [ ( ), ( )]q i q i q i     − += represents the indeterminacy degree of ϕi to q, where ( ) 1 ( ) ( )q i q i q i     − + += − − and ( ) 1 ( ) ( ).q i q i q i     + − −= − − for the simplicity, the term ([ ( ), ( )],[ ( ), ( )])q i q i q i q i       − + − + is defined as the “interval-valued intuitionistic fuzzy value/number (ivifv/ivifn)” and denoted by ([ , ],[ , ])       − + − += which fulfills 0 ( ) ( ) 1.  + + +  definition 2.2. xu and gou [30] defined some operational laws on ivifvs 1 1 1 1 1([ , ],[ , ])    − + − += and 2 2 2 2 2([ , ],[ , ]),    − + − += presented as (a) 1 2  if and only if 1 2 1 2 1 2( ) ( ), ( ) ( ), ( ) ( )i i i i i i           − − + + − −   and 1 2( ) ( ), ,i i i    + +   (b) 1 2 = if and only if 1 2  and 1 2 ,  (c) 1 1 1 1 1{( ,[ ( ), ( )],[ ( ), ( )])| },c i i i i i i          − + − +=  definition 2.3. for any ivifn ([ , ],[ , ]),       − + − += xu and gou [30] studied the normalized score function and accuracy function, given by eq. (1) and eq. (2), respectively. ( ) 1 1 ( ) 1 , 2 2        − + − +  = + − − +    (1) 1 ( ) ( ). 2        − + − += + + + (2) definition 2.4. for a set of ivifns ω = {ω1, ω2, ..., ωt}, xu and gou [30] defined the interval-valued intuitionistic fuzzy weighted averaging (ivifwa) and interval-valued intuitionistic fuzzy weighted geometric (ivifwg) operators as 1 1 1 1 1 1 (1 ) , 1 (1 ) , ( ) , ( ) , k k k k t t t tt k k k k k k k k k k k         − + − + = = = = =      =  − − − −              (3) 1 1 1 1 1 ( ) , ( ) , 1 (1 ) , 1 (1 ) . k k k k t t t t k k k k k k k k k k k         − + − + = = = = =       =  − − − −              (4) definition 2.5 [30]. assume that q, r, s ivifss (φ), a real-valued function d: ivifss (φ)×ivifss (φ) → [0,1] is said to be ivif-distance measure if it holds the following axioms: (a1). 0 ≤ d (q, r) ≤ 1, assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 365 (a2). d (q, r) if and only if q = r, (a3). d (q, r) = d (r, q), (a4). if ,q r s  then d (q, r) ≤ d (q, s) and d (r, s) ≤ d (q, s). 2.2. hellinger distance measure on ivifss hellinger [54] originated the idea of hellinger distance to quantify the degree of difference between two discrete probability distributions. based on the concept of hellinger distance, this section proposes a hellinger distance measure on ivifss. moreover, the important characteristics of this measure are discussed in this section. definition 2.6 suppose that q, r ivifss(φ), then the distance measure on ivifss is presented as ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) . 2 ( ) ( ) ( ) ( ) q i r i q i r i t q i r i q i r i i q i r i q i r i d q r t                         − − + + − + + = − − + +  − + −   + − + −=    + − + −    (5) remark 2.1. the bigger value of d (q, r) signifies the larger difference between two ivifss q and r. in a similar way, the lesser value of d (q, r) signifies the smaller difference between two ivifss q and r. the properties of d (q, r) are deduced as follows: property 2.1: for q, r ivifss(φ), 0 ≤ d (q, r) ≤ 1. proof: in ivifs, we have ( ) ( ) ( ) 1,q i q i q i     − − −+ + = ( ) ( ) ( ) 1,q i q i q i     + + ++ + = ( ) ( ) ( ) 1r i r i r i     − − −+ + = and ( ) ( ) ( ) 1.r i r i r i     + + ++ + = thus, we deduce that ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) 1 ( , ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) q i q i r i r i q i q i r i r i q i q i r i r i q i q i r i r i q i q i r i r i q i q i r i d q r t                                                − − − − + + + + − − − − + + + + − − − − + + +  − + +   − + + − = + + − + + − + + − + 1 ( ) t i r i = +    ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) 1 ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) q i q i r i r i q i q i r i r i q i q i r i r i q i q i r i r i q i q i r i r i q i q i r i r i t                                                 − − − − + + + + − − − − + + + − − − − + + + +  − + +   − + + − = + + − + + − + + − + 1 t i=   366 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi ( ) )1 4 2 ( ) ( ) ( ) ( ) ( ) ( ) 1 2 ( ) ( ) ( ) ( ) ( ) ( ) t q i r i q i r i q i r i i q i r i q i r i q i r i t                         − − + + − − + + − − + +=  − + +  = + + +   ( ) )1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1. ( ) ( ) ( ) ( ) ( ) ( ) t q i r i q i r i q i r i i q i r i q i r i q i r i t z                        − − + + − − + + − − + +=  − + + =  + + +   therefore, we can prove that 0 ≤ d (q, r) ≤ 1. property 2.2: d (q, r) = 0, if and only if q = r. proof: for given two ivifss q and r, we have ( ) ( ),q i r i   − −= ( ) ( ),q i r i   + += ( ) ( ),q i r i   − −= ( ) ( ),q i r i   + += ( ) ( )q i r i   − −= and ( ) ( ).q i r i   + += then we find that ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 0 2 ( ) ( ) ( ) ( ) q i r i q i r i t q i r i q i r i i q i r i q i r i d q r t                         − − + + − + + = − − + +  − + −   + − + −= =   + − + −    for any ϕiφ, if d (q, r) = 0, then we have ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 0. 2 ( ) ( ) ( ) ( ) ( ) ( ) q i r i q i r i q i r it i q i r i q i r i q i r i t                         − − + + − − + + − − + +=  − + − + −  =  + − + − + −    it implies ( ) 2 ( ) ( ) 0,q i r i   − −− = ( ) 2 ( ) ( ) 0,q i r i   + +− = ( ) 2 ( ) ( ) 0,q i r i   − −− = ( ) 2 ( ) ( ) 0,q i r i   − − = ( ) 2 ( ) ( ) 0,q i r i   + +− = ( ) 2 ( ) ( ) 0q i r i   − −− = and ( ) 2 ( ) ( ) 0.q i r i   + +− = hence, ( ), 0d q r = if and only if .q r= property 2.3: d (q, r) = d (r, q). proof: for given two ivifss q and r, we have assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 367 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 2 ( ) ( ) ( ) ( ) q i r i q i r i t q i r i q i r i i r i r i q i r i d q r t                         − − + + − − + + = − − + +  − + −   + − + −=    + − + −    ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( , ). 2 ( ) ( ) ( ) ( ) r i q i r i q i t r i q i r i q i i r i q i r i q i d r q t                         − − + + − − + + = − − + +  − + −   + − + −= =   + − + −    hence, d (q, r) = d (r, q). property 2.4: if ,q r s  then d (q, r) ≤ d (q, s) and d (r, s) ≤ d (q, s). proof: for given three ivifss q, r and s, if ,q r s  then ( ) ( ) ( ),q i r i s i     − − −  ( ) ( ) ( ),q i r i s i     + + +  ( ) ( ) ( ),q i r i s i     − − −  ( ) ( ) ( ),q i r i s i     + + +  ( ) ( ) ( )q i r i s i     − − −  and ( ) ( ) ( ).q i r i s i     + + +  ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 2 ( ) ( ) ( ) ( ) q i s i q i s i t q i s i q i s i i q i s i q i s i d q s t                         − − + + − − + + = − − + +  − + −   + − + −=    + − + −    ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( , ). 2 ( ) ( ) ( ) ( ) , q i r i q i r i t q i r i q i r i i q i r i q i r i d q r t                         − − + + − + + = − − + +  − + −   + − + − =   + − + −    similarly, we can prove that when ,q r s  then d (q, r) ≤ d (q, s) and 0 ≤ d (r, s) ≤ d (q, s). 368 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi proposition 2.1: from the properties 2.1-2.4, eq. (5) holds all the necessary conditions of definition 2.5. hence, eq. (5) is valid ivif-distance measure on ivifss (φ). definition 2.7. suppose that q, r, s ivifss(φ), then the weighted hellinger distance measure dα: ivifss(φ)×ivifss(φ) → [0,1] is given by ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )( , ) 2 ( ) ( ) ( ) ( ) , q i r i q i r i t q i r i q i r ii i q i r i q i r i d q r                         − − + + − + + = − − + +  − + −   + − + −=    + − + −    (6) where αi is the weight of ϕi on φ satisfying αi  [0, 1] and 1 1. t ii  = = 2.3. comparative study in this section, we firstly recall some of the previously developed distance measures in the context of ivifs (zhang et al. [49], düğenci [50], baccour and alimi [51], mishra et al. [52]). further, we apply the proposed and existing ivif-distance measures on some common data sets and obtain some useful results. 1 ( ) ( ) ( ) ( )1 ( , ) , 4 ( ) ( ) ( ) ( ) t q i r i q i r i h i q i r i q i r i d q r t                 − − + + − − + + =  − + −  =   + − + −    (7) 2 2 2 2 1 ( ( ) ( )) ( ( ) ( ))1 ( , ) , 4 ( ( ) ( )) ( ( ) ( )) t q i r i q i r i e i q i r i q i r i d q r t                 − − + + − − + + =  − + −  =  + − + −   (8) 1 ( ) ( ) , ( ) ( ) ,1 ( , ) max , 4 ( ) ( ) , ( ) ( ) t q i r i q i r i hh i q i r i q i r i d q r t                 − − + + − − + + =  − −  =   − −    (9) 2 2 2 2 1 ( ( ) ( )) , ( ( ) ( )) ,1 ( , ) max , 4 ( ( ) ( )) , ( ( ) ( )) t q i r i q i r i he i q i r i q i r i d q r t                 − − + + − − + + =  − −  =  − −   (10) 1 ( ) ( ) ( ) ( ) , 1 2 ( , ) max , ( ) ( ) ( ) ( ) 2 q i r i q i r i t z i q i r i q i r i d q r t                 − − + + − − + + =  − + −     =   − + −       (11) assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 369 ( ) 1 ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( ))1 ( , ) 4 1 ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) p p q i r i q i r i t q i r i q i q i d p p p i q i r i q i r i q i r i q i r i d q r t                                      − − − − − − − − + + + += + + + +    − −   +   − − − −   =   +  − −  +    − − − −     1 , p       (12) where  = 2, 3, 4, . . . and p = 1, 2, 3, . . . ( ) ( ) ( ) ( ) 2 2 1 2 2 1 1 ( ) ( ) ( ) ( ) 1 1 ( , ) , 4 4 ( ) ( ) ( ) ( ) t tq i r i q i r i b i i q i r i q i r i d q r t t                 − − + + − + += =     − −       = +     + − + −          (13) 2 2 2 1 1 ( ) ( ) ( ) ( ) 1 1 ( , ) , 8 8 ( ) ( ) ( ) ( ) t tq i r q i r i b i i q i r i q i r i d q r t t                 − − + + − − + + = =     − −    = +       + − + −       (14) 1 1 ( ) ( ) ( ) ( ) 1 1 exp ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) , 1 exp( (1)) , q i r i q i r i t q i r i q i r i i q i r i q i r i m t d q r                                − − + + − − + + = − − + +      − + −          − − + − + −         + − + −        − − =  (15) where γ > 0, γ ≠ 1. table 1 comparisons of diverse ivif-distance measures (bold shows the counter-intuitive cases) qi ([0.25, 0.35], [0.25, 0.35]) [0.25, 0.35], [0.35, 0.45]) ([1, 1], [0, 0]) ([0.5, 0.5], [0.5, 0.5]) ([0.35, 0.45], [0.15, 0.25]) ([0.35, 0.45], [0.15, 0.25]) ri ([0.35, 0.45], [0.35, 0.45]) [0.35, 0.45], [0.25, 0.35]) ([0, 0], [0, 0]) ([0, 0], [0,0]) ([0.45, 0.55], [0.25, 0.35]) ([0.45, 0.55], [0.15, 0.25]) dh (qi, ri) 0.1 0.1 0.5 0.5 0.1 0.05 de (qi, ri) 0.1 0.1 0.707 0.5 0.1 0.071 dhh (qi, ri) 0.025 0.025 0.25 0.125 0.025 0.025 dhe (qi, ri) 0.05 0.05 0.5 0.25 0.05 0.05 dz (qi, ri) 0.100 0.100 1.000 0.500 0.100 0.100 dd (qi, ri) 0.033 0.100 0.500 0.208 0.033 0.050 db1 (qi, ri) 0.007 0.007 0.500 0.375 0.008 0.003 db2 (qi, ri) 0.100 0.100 0.250 0.313 0.100 0.025 dm (qi, ri) 0.252 0.151 1.000 0.802 0.252 0.151 d (qi, ri) 0.165 0.086 1.000 0.804 0.167 0.082 370 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi table 1 presents the computational results obtained by the proposed and previously developed ivif-distance measures (zhang et al. [49], düğenci [50], baccour and alimi [51], mishra et al. [52]). the distance measures dh, de, dhh, dhe, db1, db2, dz, dd and dm present the counter-intuitive results for some examples. for instance, dh (q1, r1) = 0.1 and dh (q2, r2) = 0.1, de (q1, r1) = 0.1 and de (q2, r2) = 0.1, dhh (q1, r1) = 0.025 and dhh (q2, r2) = 0.025, dhe (q1, r1) = 0.05 and dhe (q2, r2) = 0.05, dz (q1, r1) = 0.1 and dz (q2, r2) = 0.1, db1 (q1, r1) = 0.007 and db1 (q2, r2) = 0.007, db2 (q1, r1) = 0.1 and db2 (q2, r2) = 0.1, when q1 = [(0.25, 0.35), (0.25,0.35)], r1 = [(0.35, 0.45), (0.35, 0.45)], q2 = [(0.25, 0.35), (0.35, 0.45)] and r2 = [(0.35, 0.45), (0.25, 0.35)]. also, dhh (q5, r5) = 0.025 and dhh (q6, r6) = 0.025, dhe (q5, r5) = 0.05 and dhe (q6, r6) = 0.05, dz (q5, r5) = 0.1 and dz (q6, r6) = 0.1, when q5 = [(0.35, 0.45), (0.15,0.25)], r5 = [(0.45, 0.55), (0.25, 0.35)], q6 = [(0.35, 0.45), (0.15, 0.25)] and r6 = [(0.45, 0.55), (0.15, 0.25)]. for the above discussed sets, the hellinger distance measure proposed in this study can successfully discriminate the given ivifss and the results are as follows: d (q1, r1) = 0.165 and d (q2, r2) = 0.086, d (q5, r5) = 0.167 and d (q6, r6) = 0.082. for a different pair of ivifss q3 = [(1, 1), (0, 0)], r3 = [(0, 0), (0, 0)] and q4 = [(0.5, 0.5), (0.5, 0.5)], r4 = [(0, 0), (0, 0)], we get dh (q3, r3) = dh (q4, r4) = 0.5, while other measures provide reasonable results. in addition, the counter-intuitive cases arise for q1 = ([0.25, 0.35], [0.25, 0.35]), r1 = ([0.35, 0.45], [0.35, 0.45]), q5 = ([0.35, 0.45], [0.15, 0.25]), r5 = ([0.45, 0.55], [0.25, 0.35]) and r2 = [(0.25, 0.35), (0.35, 0.45)] and r2 = [(0.35, 0.45), (0.25, 0.35)], r6 = [(0.35, 0.45), (0.15, 0.25)] and r6 = [(0.45, 0.55), (0.15, 0.25)], respectively. in all the discussed cases, the present measure is free from all the counter-intuitive cases, see table 1. 3. an integrated ivif-distance measure-based mairca method the current part of the study develops an extended mairca model for solving mcdm problems in which the information about the criteria and dms is completely known. this model combines the proposed ivif-distance measure and the mairca method with ivif information. the proposed model comprises the following procedure (see fig.1): step 1: create the decision matrix. a group of dmes h = {h1, h2, ..., hn} is invited for the assessment of an optimal alternative among a set of options y = {y1, y2, ..., ys} by means of the criteria set w = {w1, w2, ..., wt} the dms present the linguistic assessment rating of each option yi with respect to criteria wj, j = 1, 2, ..., t. let d = (δij (k))s×t be the linguistic assessment matrix (lam), where δij (k) denotes the linguistic variable (lv) of each candidate yi over a criterion wj presented by kth dme. based on the given linguistic scale’s table, the lam is switched into ivif decision matrix (ivifdm). step 2: compute the dmes’ significance values. assume that ([ , ],[ , ]), 1,2,...,k k k k kh k n   − + − += = be the performance of kth dme. then the procedure for estimating the numeric significance value of kth dme is presented in the following steps: step 2a: determine the matrix using score function. each ivifn hk is normalized and computed as mishra et al. [54]: 1 ( )(2 ) , . (( )(2 )) k k k k k n k k k k k h k         − + − + − + − + = + + + =  + + + (16) assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 371 step 2b: determine the rank of dmes’ performances and compute the dme’s significance value n-ρk+1, wherein ρk denotes the priority of kth expert. the normalization process is used to normalize each significance value (zhu et al. [55]): 1 1 , . ( 1) r k k n k k n h k n   = − + =  − + (17) step 2c: compute the weights. in accordance with the combination of eq. (7) and eq. (8), dme’s weighting formula is given as 1 (( ) ( )), , 2 r k k kh h k = +  where 0k  and 1 1. n k k  = = (18) fig. 1 graphical structure of the proposed ivif-distance measure-mairca model 372 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi step 3: aggregate the individual dme’s opinions. as each decision maker has their own opinion regarding the performance of options with respect to the criteria. to make an optimal decision, there is a need to combine the individual decision opinions and create the aggregated ivifdm (a-ivifdm) a = (δij)s×t, where δij denotes the aggregated ivifn, computed based on ivifwa operator. (1) (2) ( )([ , ],[ , ]) ( , ,..., ) .n ij ij ij ij ij ij ij ijivifwa       − + − += = (19) step 4: compute the criteria weights. assume that the significance value of each criterion is different and independent of each other. let x = (x1, x2, ..., xt)t be the weight vector of criteria set, satisfying xj [0, 1] and 1 1. t jj x = = next, we present the procedure to compute the criteria weights as follows: step 4.1: determine the support degree sup (δij, δil) between the considered criteria wj and wl using the proposed ivif-distance measure, which as sup( , ) 1 ( , ), 1,2,..., , , 1,2,..., , ,ij il ij ild i s j l t j l   = − = =  (20) where d (δij, δil) denotes the ivif-distance measure given in eq. (5). step 4.2: compute the total support degree t (δij) for each criterion wj, by means of eq. (21). 1, ( ) sup( , ). t ij ij il l l j t    =  =  (21) step 4.3: compute the rationality degree θj of each criterion wj, given as 1 1 1 ( ), [0,1]. ( 1) s t j ij j i j t s t    = = =  −  (22) step 4.4: determine the comprehensive index (weight of criteria) xj, of jth criterion wj, given as 1 , j j t jj x   = =  (23) where j = 1, 2, …, t. step 5: calculate the positive distance grade psij and the negative distance grade pnij between an element δij in an a-ivifdm a = (δij)s×t and the pis ω+ and the nis ω-, respectively, where ([ , ],[ , ]),ij ij ij ij ij    − + − += ([ , ],[ , ]) j j j j        + + + + + − + − += and ([ , ],[ , ]), j j j j        − − − − − − + − += shown as follows: 2 2 2 2 2 2 1 2 , j j j j j j ij ij ij ij ij ij ij ps                   + + + + + + − − + + − − + + − − + +       − + − + −            =       + − + − + −              (24) assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 373 2 2 2 2 2 2 1 2 . j j j j j j ij ij ij ij ij ij ij pn                   − − − − − − − − + + − − + + − − + +       − + − + −            =       + − + − + −              (25) here, i = 1, 2, …, s and j = 1, 2, …, t. an ivifn has a positive ideal solution ω+ and a negative ideal solution ω-, where ω+ = ([1, 1], [0, 0]) and ω= ([0, 0], [1, 1]) are the ivifns. step 6: with the use of eq. (24) and eq. (25), create the relative closeness decision matrix r = (rcij)s×t, where . ij ij ij ij pn rc pn ps = + (26) step 7: normalize the relative closeness-decision matrix r = (rcij)s×t into the normalized form r = (nrcij)s×t, where min max min max max min ( ) , for is thebenefit criterion, ( ) ( ) ( ) , for is thecost criterion. ( ) ( ) ij ij j ij ij ij ij ij j ij ij rc rc w rc rc nrc rc rc w rc rc −  − =  −  − (27) step 8: make the ivif theoretical matrix t = (εij)s×t, where , iij c jp x = (28) 1 , icp t= (29) where xj signifies the jth criterion’s weight, where j = 1, 2, …, t. step 9: based on the obtained ivif theoretical matrix t = (εij)s×t and the obtained normalized relative closeness decision matrix r = (nrcij)s×t, construct the interval-valued intuitionistic fuzzy real assessment matrix β = (βij)s×t, where . .ij ij ijnrc = (30) step 10: based on the obtained ivif-tm t = (εij)s×t and the obtained real assessment matrix β = (βij)s×t, construct the interval-valued intuitionistic fuzzy gap matrix g = (gij)s×t, where .ij ij ijg  = − (31) step 11: calculate the utility degree ci of alternative yi, shown as follows: 1 , 1,2,..., . t i ij j c g i s = = = (32) 374 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi prioritize the options as per the obtained utility degrees c1, c1, ..., and cs of the alternatives y1, y1, ..., and ys, respectively. the lesser the utility degree of an option yi, the better the ranking order of option yi, where i = 1, 2, …, s. 4. case study: wwtt selection problem environmental challenges related to the chemical and biological pollution of water have become important for the industrial sector, society and public agencies. most of the domestic and industrial activities generate wastewater that contains harmful and undesirable pollutants, thus, it requires a proper management and treatment. the wastewater management and treatment aim to the sustainable development of natural resources together with the protection of environment and public health. in this section, the present mairca model is firstly executed on a case study of the wwtts assessment problem with respect to several factors. further, the sensitivity analysis and comparative study by means of existing methods are discussed under ivifs environment, which shows the stability and robustness of the presented methodology. with the increasing complexity, time boundedness and lack of precise knowledge/ information, it is quite hard to evaluate the candidates with regard to given criteria in realistic situations. in this section, a group of three dms is formed to identify the criteria and evaluate the wwtts based on considered criteria. these des are having more than 15 years of experience in their respective fields. two of them are from the environmental engineering department and the other one is from sustainable planning and management. based on the literature review and online questionnaire, we have considered five wwtt alternatives and nine criteria. the presented case study is for the demonstration purposes to prove the practicality of the proposed method. readers may reduce some criteria or add more criteria as per their requirements. description of the alternatives is presented as follows: ▪ microbial fuel cell (mfc) (y1): mfc is relatively a new promising technology for producing renewable energy while treating wastewater. this technology is a chemical reactor system that generates electricity from the biodegradation of organic materials with the help of suitable microbial substrate. it is used to acquire a higher energy density and pollutants removal. ▪ membrane filtration (y2): it is physiochemical process for the treatment of water from different wastewater streams and makes it possible to reuse. this process of treatment is defined on the size of the material that needs to be separated from the liquid. ▪ automatic variable filtration (avf) (y3): it is a simple water filtration technology used for wastewater treatment where the upward flow of influent is purified or cleaned by the downward flow of filter media. it can effectively remove the bacterial contamination and micro-organisms while treating wastewater. ▪ natural treatment methods (y4): it is a biological treatment method to treat the wastewater naturally by removing contaminants from wastewater. these methods are eco-friendly, cost-effective, and can be jointly driven by public bodies and communities. assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 375 ▪ advanced photo-oxidation process (apop) (y5): apop is a type of chemical treatment that oxidizes organic molecules in wastewater that are hard to manage biologically. table 2 criteria used for wwtt selection extracted from the literature dimensions criteria meanings references economic (ec) maintenance and operation cost (ec1) repair, personnel, chemical and energy costs to manage wwt curiel-esparza et al. [56], molinos-senante et al. [57], saghafi et al. [58], obaideen et al. [59] land requirement (ec2) enough space for wwt plant/future expansion kalbar et al. [60], mahjouri et al. [61], saghafi et al. [58], obaideen et al. [59] environmental (en) energy consumption (en1) energy consumption amount during wwt activities molinos-senante et al. [57], piadeh et al. [62], srdjevic et al. [63], salamirad et al. [15], narayanamoorthy et al. [23] sludge production (en2) sludge generation of the system molinos-senante et al. [57], saghafi et al. [58], salamirad et al. [15] odor impacts (en3) undesired smell potential of the system plakas et al. [64], eseoglu et al. [65], salamirad et al. [15] social (s) public acceptance (s1) public awareness molinos-senante et al. [57], plakas et al. [64], obaideen et al. [59], salamirad et al. [15] aesthetic (s2) acceptability of plant conditions and appearance eseoglu et al. [65], salamirad et al. [15] technical (t) cod removal capacity (t1) the removal capacity of amount of oxygen consumed to oxidize all organic material by chemical oxidants zhang et al. [10], eseoglu et al. [65], srivastava and singh [14] bod removal capacity (t2) the removal capacity of amount of oxygen consumed by microorganisms while decomposing organic matter zhang et al. [10], eseoglu et al. [65], srivastava and singh [14] further, an online survey has been prepared with the purpose of determining the significance of criteria to assess the wwtt alternatives. in addition, the criteria that may have an effect on the wwtt alternatives’ evaluation are assembled through literature survey. table 2 presents the source and type of each considered criterion. fig. 2 presents the hierarchical structure of the considered criteria and alternatives. 376 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi fig. 2 hierarchical structure for sustainable wwtt selection step 1: table 3 presents the linguistic variables and their corresponding ivifns (alrasheedi et al. [32] and mishra et al. [52]). based on the dms’ opinions, the assessment rating of each wwtt alternative with respect to each criterion and form a linguistic assessment matrix in table 4. table 3 lvs for sustainable wwtts assessment lvs ivifns absolutely significant (as) ([0.90,0.95],[0.01,0.05]) very significant (vs) ([0.80,0.90],[0.05,0.10]) significant (s) ([0.70,0.80],[0.10,0.15]) quite significant (qs) ([0.65,0.70],[0.15,0.25]) moderate (m) ([0.55,0.65],[0.20,0.35]) quite insignificant (qi) ([0.40,0.50],[0.40,0.45]) insignificant (i) ([0.25,0.40],[0.45,0.50]) very insignificant (vi) ([0.15,0.20],[0.60,0.75]) absolutely insignificant (ai) ([0.05,0.10],[0.80,0.90]) assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 377 table 4 the lam for sustainable wwtt selection problem criteria y1 y2 y3 y4 y5 w1 (vi,vi,qi) (ai,i,qi) (i,vi,vi) (vi,ai,m) (vi,qi,qi) w2 (s,s,vs) (s,vs,m) (m,s,vs) (qs,qs,s) (m,qs,s) w3 (vi,si,m) (qi,i,m) (qi,i,i) (vi,i,vi) (vi,m,vi) w4 (m,i,vi) (i,vi,ai) (i,vi,m) (i,qi,i) (vi,qi,ai) w5 (vi,i,vi) (qi,i,vi) (vi,vi,m) (i,i,ai) (qi,i,ai) w6 (s,qs,vs) (vs,qs,s) (s,s,qs) (m,vs,s) (vs,vs,s) w7 (vs,qs,qi) (s,qs,qs) (vs,vs,qi) (s,vs,s) (as,m,qs) w8 (qs,as,s) (s,s,qs) (qs,m,s) (vs,s,qi) (vs,m,qi) w9 (m,m,vs) (qs,vs,s) (s,s,s) (m,vs,m) (qi,vs,m) step 2: with the use of linguistic scales of table 3 and eqs. (16-18), the significance values of dms are derived and shown in table 5 for sustainable wwtt selection problem. table 5 dmes’ weights for sustainable wwtt selection dmes h1 h2 h3 lvs h vh eh ivifns ([0.70,0.80],[0.10,0.15]) ([0.80,0.90],[0.05,0.10]) ([0.90,0.95],[0.01,0.05]) n-ρk+1 1 2 3 weights 0.2382 0.3343 0.4274 fig. 3 representation of the criteria weights using ivif-distance measure-based tool 378 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi table 6 aggregated decision matrix for sustainable wwtt selection criteria y1 y2 y3 y4 y5 w1 ([0.268, 0.346], [0.505, 0.603]) ([0.279, 0.389], [0.491, 0.550]) ([0.175, 0.253], [0.560, 0.681]) ([0.328, 0.416], [0.413, 0.576]) ([0.348, 0.441], [0.441, 0.508]) w2 ([0.748, 0.851], [0.074, 0.126]) ([0.688, 0.798], [0.107, 0.188]) ([0.722, 0.830], [0.088, 0.154]) ([0.672, 0.748], [0.126, 0.177]) ([0.652, 0.738], [0.135, 0.202]) w3 ([0.423, 0.520], [0.328, 0.457]) ([0.428, 0.544], [0.309, 0.419]) ([0.289, 0.425], [0.438, 0.488]) ([0.185, 0.273], [0.545, 0.655]) ([0.313, 0.393], [0.416, 0.581]) w4 ([0.299, 0.403], [0.420, 0.546]) ([0.135, 0.214], [0.634, 0.736]) ([0.371, 0.475], [0.350, 0.492]) ([0.304, 0.435], [0.433, 0.483]) ([0.207, 0.281], [0.593, 0.684]) w5 ([0.185, 0.273], [0.545, 0.655]) ([0.250, 0.350], [0.495, 0.580]) ([0.352, 0.438], [0.375, 0.542]) ([0.170, 0.286], [0.575, 0.643]) ([0.229, 0.317], [0.560, 0.627]) w6 ([0.734, 0.830], [0.085, 0.139]) ([0.657, 0.770], [0.135, 0.197]) ([0.680, 0.762], [0.119, 0.170]) ([0.711, 0.819], [0.094, 0.160]) ([0.762, 0.865], [0.067, 0.119]) w7 ([0.614, 0.713], [0.176, 0.240]) ([0.663, 0.728], [0.136, 0.184]) ([0.680, 0.801], [0.122, 0.207]) ([0.738, 0.841], [0.079, 0.131]) ([0.718, 0.794], [0.087, 0.173]) w8 ([0.784, 0.861], [0.051, 0.111]) ([0.596, 0.672], [0.177, 0.276]) ([0.644, 0.734], [0.139, 0.213]) ([0.634, 0.749], [0.153, 0.218]) ([0.580, 0.698], [0.193, 0.289]) w9 ([0.682, 0.795], [0.111, 0.205]) ([0.728, 0.825], [0.087, 0.140]) ([0.700, 0.800], [0.100, 0.150]) ([0.657, 0.770], [0.126, 0.230]) ([0.632, 0.749], [0.148, 0.244]) step 3: by means of table 3, table 4 and eq. (19), the aggregated decision matrix is created to combine the individual opinions of three dmes and presented in table 6. step 4: to determine criteria weights, the total support degree using the proposed ivif-distance measure and rationality degree of the aggregated decision matrix are calculated using eqs. (20-22) and depicted in table 7. based on the rationality degree of each criterion, we determine the weights of criteria for sustainable wwtt selection using eq. (23) and portrayed in table 7. here, fig. 3 presents the significance degrees or weights of considered evaluation criteria in hswpp locations assessment. based on the obtained results, odor impacts (en3) is the most important criterion among a set of twelve criteria for assessing the wwtt. sludge production (en2) is the second most important criterion for wwtt evaluation. maintenance and operation cost (ec1) has third with significance value 0.123, energy consumption (en1) with weight value 0.11 has fourth most important criterion for wwtt evaluation and others are considered crucial sub-criteria for the taken case study. table 7 rationality and comprehensive degree of option for sustainable wwtt selection criteria y1 y2 y3 y4 y5 δj xj w1 2.058 1.789 2.492 1.846 1.592 0.244 0.123 w2 1.723 1.608 1.621 1.672 1.571 0.205 0.103 w3 1.604 1.441 1.769 2.245 1.677 0.218 0.110 w4 1.848 2.660 1.517 1.777 2.170 0.249 0.126 w5 2.443 1.894 1.630 2.275 1.995 0.256 0.129 w6 1.661 1.438 1.473 1.715 2.027 0.208 0.105 w7 1.489 1.525 1.669 1.858 1.720 0.207 0.104 w8 1.896 1.335 1.363 1.513 1.409 0.188 0.095 w9 1.730 1.746 1.533 1.773 1.512 0.207 0.105 assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 379 table 8 normalized relative closeness-decision matrix for sustainable wwtt selection criteria y1 y2 y3 y4 y5 w1 0.509 0.325 1.000 0.120 0.000 w2 1.000 0.403 0.738 0.157 0.000 w3 0.082 0.000 0.460 1.000 0.534 w4 0.248 1.000 0.000 0.173 0.769 w5 0.956 0.541 0.000 1.000 0.816 w6 0.676 0.000 0.128 0.497 1.000 w7 0.000 0.298 0.532 1.000 0.726 w8 1.000 0.017 0.286 0.273 0.000 w9 0.513 1.000 0.729 0.241 0.000 steps 5-6: based on eqs (24-26), we obtain the positive distance grade psij and the negative distance grade pnij and relative closeness matrix, where ω+ = ([1,1], [0, 0]) and ω= ([0,0], [1,1]) are positive and negative ideal solutions, respectively for sustainable wwtt selection as rc11 = 0.431, rc12 = 0.718, rc13 = 0.522, and others. table 9 ivif-theoretical matrix for sustainable wwtt selection criteria y1 y2 y3 y4 y5 w1 0.025 0.025 0.025 0.025 0.025 w2 0.021 0.021 0.021 0.021 0.021 w3 0.022 0.022 0.022 0.022 0.022 w4 0.025 0.025 0.025 0.025 0.025 w5 0.026 0.026 0.026 0.026 0.026 w6 0.021 0.021 0.021 0.021 0.021 w7 0.021 0.021 0.021 0.021 0.021 w8 0.019 0.019 0.019 0.019 0.019 w9 0.021 0.021 0.021 0.021 0.021 step 7: using eq. (27), we obtain the normalized relative closeness decision matrix for sustainable wwtt selection and given in table 8, where nrc11 = 0.509, nrc12 = 1.000, nrc13 = 0.083, and others. step 8: based on eq. (28) and eq. (29), we obtain the interval-valued intuitionistic fuzzy theoretical matrix for sustainable wwtt selection and discussed in table 9, where ε11 = 0.025, ε12 = 0.021, ε13 = 0.022, and others. step 9: based on eq. (30), we obtain the interval-valued intuitionistic fuzzy real assessment matrix for sustainable wwtt selection and presented in table 10, where β11 = 0.013, β12 = 0.021, β13 = 0.002, and others. table 10 ivif-real assessment matrix for sustainable wwtt selection criteria y1 y2 y3 y4 y5 w1 0.013 0.008 0.025 0.003 0.000 w2 0.021 0.008 0.015 0.003 0.000 w3 0.002 0.000 0.010 0.022 0.012 w4 0.006 0.025 0.000 0.004 0.019 w5 0.025 0.014 0.000 0.026 0.021 w6 0.014 0.000 0.003 0.010 0.021 w7 0.000 0.006 0.011 0.021 0.015 w8 0.019 0.000 0.005 0.005 0.000 w9 0.011 0.021 0.015 0.005 0.000 380 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi step 10: based on eq. (31), we obtain the interval-valued intuitionistic fuzzy gap matrix for sustainable wwtt selection and discussed in table 11, where g11 = 0.012, g12 = 0.000, g13 = 0.020, and others. table 11 ivif-gap matrix for sustainable wwtt selection criteria y1 y2 y3 y4 y5 w1 0.012 0.017 0.000 0.022 0.025 w2 0.000 0.012 0.005 0.017 0.021 w3 0.020 0.022 0.012 0.000 0.010 w4 0.019 0.000 0.025 0.021 0.006 w5 0.001 0.012 0.026 0.000 0.005 w6 0.007 0.021 0.018 0.011 0.000 w7 0.021 0.015 0.010 0.000 0.006 w8 0.000 0.019 0.014 0.014 0.019 w9 0.010 0.000 0.006 0.016 0.021 step 11: based on eq. (32), we obtain the utility score ci of alternative yi, where i = 1, 2, 3, 4, 5, c1 = 0.0902, c2 = 0.1171, c3 = 0.1155, c4 = 0.1001 and c5 = 0.1118. because c1 >c4 >c5>c3 >c2, therefore ranking order of the wwtt alternatives y1, y2, y3, y4 and y5 is: y1 y4 y5 y3 y2. thus, the alternative y1 is the best alternative for sustainable wwtt selection. 5. comparative analysis and discussion in this part of the study, we compare the results obtained by the proposed and some of the extant madm methods including ivif-weighted aggregated sum product assessment (ivif-waspas) [66] model, ivif-complex proportional assessment (ivif-copras) [67] model and ivif-combined compromise solution (ivif-cocoso) [32] model and iviftechnique for order of preference by similarity to ideal solution (ivif-topsis) [68] model. here, fig. 4 depicts the obtained ranking results by different mcdm approaches. from fig. 4, we can observe that the ranking orders are different by all the methods but the optimum alternative is same in case of the obtained results by introduced method, ivif-waspas, ivif-copras and ivif-cocoso methods, while ivif-topsis model provides different optimal candidate. the preference orders of wwtt candidate by means of different weighting approaches are shown in fig. 5. to accomplish better insight from the ivif-distance measure-based mairca technique in the assessment of sustainable wwtt, we calculate the utility score of each wwtt over considered sustainable aspects and factors, as given in fig. 5. since mfc (y1) has extensively highest score for all extant and proposed models social, economic environmental aspects of attributes’ weighting (see fig. 5), consequently, it is chosen as the best wwtt option. in accordance with the aforementioned analysis, it can be easily noticed that observing the various weighting frameworks will enhance the utility and effectiveness of the developed ivif-mairca method. table 12 presents the parameters to compare different approaches including proposed and existing mcdm approaches. assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 381 table 12 parameters to compare different approaches standards ivifwaspas ivif-copras ivif-cocoso ivif-topsis proposed method interrelationships between the arguments not considered not considered not considered not considered considered criteria weights computed computed computed assumed computed mcdm procedure group group group group group dmes’ weights not considered computed computed not considered computed does the ranking tool consider type of criteria no no yes no yes preference order y1 y4 y2 y5 y3 y1 y4 y5 y2 y3 y1 y4 y2 y5 y3 y4 y5 y1 y2 y3 y1 y4 y5 y3 y2 optimal option y1 y1 y1 y4 y1 the main advantages of the developed ivif-distance measure-based mairca are listed as ▪ in this method, the weights of the decision makers are computed through score function-based model. thus, the proposed approach gives a more accurate decision as compared to existing ivif mcdm methods. ▪ the distance measure proposed in this study avoids the limitations of several existing ivif-distance measures. thus, the criteria weighting model based on the proposed distance measure provides more efficient result in the assessment of wwtt alternatives. ▪ the mairca method determines the best solution considering the deviation between the defined theoretical and the real results. the key benefits of the mairca approach are presented as follows: (i) it can solve the mcdm problems with mixed qualitative and quantitative assessment criteria; (ii) this method considers the concept of the positive and negative ideal solutions and (iii) the mairca has a distinctive linear normalization algorithm which can obtain highly reliable discrepancies and generate consistent results. fig. 4 comparison of proposed with extant methods for sustainable wwtt selection 382 a. r. mishra, p. rani, f. cavallaro, a. f. alrasheedi 5.1. discussion and implications the weighting outcomes revealed that the odor impacts (en3), sludge production (en2) and maintenance and operation cost (ec1) had become the most significant factors for sustainable wwtt assessment (see fig. 3). as a result, these criteria should be taken sincerely, while energy consumption (en1), public acceptance (s1), bod removal capacity (t2), aesthetic (s2), land requirement (ec2) and cod removal capacity (t1) should be also emphasized with small weights. moreover, assessment outcomes of sustainable aspects of wwtt assessments are prioritized as: environmental (0.365) ≻ economic (0.226) ≻ social (0.209) ≻ technical (0.2), which means environmental dimension have highest impact on prioritization order of sustainable wwtt selection followed by economic, social, technical aspects. the utility scores of wwtt option are 0.0902, 0.1171, 0.1155, 0.1001 and 0.1118, the prioritization order of alternatives y1, y2, y3, y4 and y5 is: 1 4 5 3 2.y y y y y by means of the concept of ivif-distance measure-based mairca methodology, we have combined the weight-determining models based on ivif-distance measure and mairca tool, which reduces information loss during the process of making decision. fig. 5 ranking results obtained by extant methods for wwtts assessment this study suggests the policymakers to understand the performance of wwtt alternatives using different aspects of sustainability from uncertain perspective. the proposed work has the following implications for practitioners and scholars: ▪ the most suitable wwtt alternative performs better with respect to economic, environmental social and technical dimensions of sustainability with minimum cost and higher efficiency in order to remove the effluents from the wastewater. assessment of sustainable wastewater treatment technologies using intervall-valued intuitionistic... 383 ▪ managers and policymakers can use the information presented in this study to support their decision for assessing the wwtt alternative. ▪ the proposed model not only evaluates the significant degrees of considered criteria but also tackles the ambiguity and fuzziness arisen during the process of wwtt alternatives assessment process. 6. conclusions to select the best sustainable wwtt for agricultural purposes, we develop a hybrid mcdm methodology that incorporates the sustainability concept. the methodology presented in this study combines the ivif-distance measure, criteria weighting tool and mairca model, and also considers the uncertainty level of the decision makers. here, we have firstly proposed a new hellinger ivif-distance measure and analyzed the causes of counter-intuitive results of extant distance measures. further, we have proposed a new criteria weight determining model based on the proposed hellinger ivif-distance measure and total support degree-based model. further, we have proposed an integrated mairca approach for dealing with the mcdm problems in which the information about the criteria and decision makers is completely unknown. the proposed mairca method integrates the normalization process and ideal solutions, and then determines the option with the smallest total distance (gap) is the best option, which is the main advantage of the proposed work. comparative assessments have been presented to reveal the outcomes obtained by the hybrid approach. as per the comparative study, it can be observed that the proposed mairca model is very robust and appropriate for the decision support problems under ivifs environment. this study does not consider the geographical and cultural aspects of the criteria, which is one of the main limitations of this research. in addition, we consider only the independent characteristics of the criteria. in future, we can develop a model to evade the drawbacks of present work. in addition, future works should be deliberated towards utilizing a wider number of global dms who will assess the factors affecting the healthcare blockchain platforms evaluation process. 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improve their business performance and productivity, especially in data processing. at the same time, due to the growing number of sensor modules, the amount of data that needs to be processed, in real time, is growing. delays in communication with the cloud environment can lead to poor management decisions or user dissatisfaction. in automation and services, one of the new ict architectures is edge computing in the data processing. edge computing is a networking architecture that brings computing close to the source of data in order to reduce latency and bandwidth use. edge computing brings new power to data processing and the ability to process large amounts of data in real time. this is essential for predicting the behavior of machines, systems, or customers in order to detect errors or provide personalized service as in the case of smart vending machines. in that way, edge computing enables taking steps toward establishing a smart ecosystem in automation and services. key words: iot/iiot, cloud computing, edge computing, ecosystem 1. introduction nowadays the edge between the physical and the digital worlds is starting to disappear. the reason for this is communication and artificial intelligence implemented in machines, and devices, not only in the industrial but also in the non-industrial environment. in that way society can gain added value in every aspect of everyday life, and also, availability to innovate in new ways, using smart ecosystems to create new digital revenue streams. received: may 25, 2022 / accepted july 29, 2022 corresponding author: stevan stankovski 1faculty of technical sciences, university of novi sad, trg dositeja obradovića 6, 21000 novi sad, serbia e-mail: stevan@uns.ac.rs 2 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. one of the key technologies that enable the formation of a smart ecosystem is the internet of things (iot). iot is a network of devices, equipment, and machines capable of interacting with each other. the idea is to obtain information about the environment to understand, control and act according to this information. a specific part of iot is industrial iot (iiot). iiot is implemented in the industrial environment thus providing more accurate and real-time visibility in the flow of materials, parts and products in a production line and warehouse management. also, production systems are investing in the iiot to redesign production workflows, improve tracking of materials, tools, and optimize production costs [1]. iot uses different kinds of sensors and devices to collect data from the targeted environment about status/changes of interest. the changes in the surroundings are detected by the sensor and sent to a device. sensors come in a variety of formats depending on the data of interest, so they can be in form of passive sensors, active sensors, or optical, thermal, mechanical, electrical ones, etc. these sensors forward their data to the devices. a device aggregates data and sends it to the cloud for processing. the device uses a gateway to transmit the data to the network within the cloud. this data is stored in the cloud which in turn makes services and functions accessible [2]. iot system generates numerous data that needs processing, storage, caching, transmission and computation. some applications, especially industrial ones, need responses in a shorter time (real-time). processing a large quantity of data also brings a great burden to the network [3]. this is the main reason why edge computing, cloud computing and fog computing were introduced and are of essential importance in the development of the smart ecosystem. edge computing means computing at the edge of the network, where the edge refers to the resources and equipment along the path between data sources and cloud data centers, especially in the proximity of terminal devices. in edge computing, people utilize open platforms provided with capabilities of computation, storage, communication and application around data sources to supply different services, within the requirements of agile connection, real-time service, data optimization, application intelligence and security protection [4]. an essential part of edge computing is the middle layer, placed between the end device layer and the central cloud layer named edge gateway layer. this layer consists of the edge gateways that are in the vicinity of end devices and have relatively rich resources. in many cases, end devices may not be able to process continuously generated sensor data in real-time. many works aim at offloading computation from end devices to edge gateways to achieve a lower processing latency [5]. cloud computing is a model for enabling ubiquitous, convenient and on-demand network access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service provider interaction [6]. fog computing, on the other hand, represents the link between iot nodes/devices and the cloud by empowering storage, data management, networking, and computing on the network nodes [7]. fog computing reduces the burden of traditional cloud computing data centers and works based on decentralized computing concepts to improve the computational speed [8]. towards a smart ecosystem with automated services 3 2. iot architecture for a better understanding of the steps involved in the design of different iot architectures, we can keep in mind the following and most common ones among many existing definitions of iot:  the interconnection via the internet of computing devices is embedded in everyday objects, enabling them to send and receive data [2],  iot is a recent communication paradigm that envisions a near future, in which the objects of everyday life will be equipped with microcontrollers, transceivers for digital communication, and suitable protocol stacks that will make them able to communicate with one another and with the users, becoming an integral part of the internet [3],  iot is the network of physical objects that contain embedded technology to communicate and sense or interact with their internal states or the external environment [4]. it is obvious that iot is a complex system, and all the elements of the architecture must be synchronized together perfectly in a systematic structure. to organize this complex structure a variety of iot architecture models were suggested. the most widely used iot architecture is the five-layered model [9]. this model consists of the following layers: 1. perception layer the perception layer is the tangible layer that uses devices and sensors for the collection of data from the environment. (sensors, rfid (radio frequency identification) tags, cameras) [10], 2. network layer the network layer connects the devices to other smart devices by the transmission of the data to the cloud servers for processing (wi-fi, zigbee, signal, bluetooth) [10], 3. middleware layer the middleware layer links the database and matches services with equivalent requesters as a software layer (data storage, cloud network) [11], 4. application layer the application layer delivers the operation essential services to the user that depicts large-scale iot applications (smart cities, smart homes, smart health) [10], and, 5. business layer the business layer can manage the overall system and this layer builds assorted business models (business models, monitoring) [11]. when analyzing the complexity of industrial and non-industrial processes, in most cases, there emerges a problem of processing and storing a large quantity of data. in these cases, efficient architecture that can be established is iot-edge-fog-cloud architecture (fig. 1.) and the layers in this architecture are the following [12]: 1. iot things layer the iot things layer mostly manages information sensing and capturing. things implanted with sensors and actuators form a part of this layer (sensors and smart devices) [13], 2. edge layer in the edge layer, the processing is carried out along the network’s edge. this is an intermediate layer between the end-user and the cloud and it provides storage and processing functionalities to an enormous number of iot devices (routers, switches) [13], 3. fog layer in the fog layer, computing is shifted below to the local area network for information to be processed at iot gateways or micro-data centers (fog nodes, satellites) [13], 4 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. 4. cloud layer the cloud layer focuses on big data analytics, industrial-level databases, and processing along with data warehousing. all these calculations are carried out centrally in far-away cloud centers (cloud storage, cloud servers) [13]. fig. 1 iot-edge-fog-cloud architecture based on [12] in some cases, where there is a need for trusted data and security, this architecture needs implementation of blockchain technology. blockchain enables forming of distributed databases, multi-party cooperation, openness and transparency with disabling non-tampering and non-repudiation data access [14]. the services obtained by using self-service automated systems (like vending machines) are just one of the examples showing the complex structure of iot. the growing demand for a fast and personalized service has led to an increase in the use of self-service automated systems in various fields. self-service automated systems are becoming digitalized, enabling them not only to provide a service to customers, but also a marketing service for retailers. with the development and adoption of new ict technologies today, self-service automated systems are becoming smart. they collect their data through iot technologies for processing, in order to improve the service. interconnected systems, which are service-oriented solutions in the cloud environment, enable retailers to provide customers with a better shopping experience. understanding and predicting customer behavior is a very important feature of smart automated systems. the goal is to approach each customer with personalized service and to suggest products that are interesting to him/her based on his/hers habits. real-time data collection and processing with the introduction of business intelligence is necessary to display personalized messages to customers. due to the growing number of sensor modules, the amount of data that needs to be processed in real time is also growing. delays in communication with the cloud environment can lead to wrong management decisions and/or user dissatisfaction. one possible solution to this problem is the application of edge computing, which enables the processing of large amounts of data in real time. this is essential for predicting customer behavior, providing personalized service as well as troubleshooting smart service machines (e.g. vending machines). towards a smart ecosystem with automated services 5 2.1 iot self-service automated systems self-service automated systems are present in almost every aspect of our lives. they provide their services or products to customers 24/7 without the need for employees to be present. this reduces labor costs [15]. one of the best examples of self-service automated systems is a vending machine. according to the european sales association, there are about 4 million vending machines in europe that sell about 80 million items a day. the total turnover of vending machines for 2018 is 15 trillion euros [16]. due to the increasing use of self-service automated systems, various services and products, a constant tendency to improve them also emerges. one way of achieving an improvement is to expand the functionality and implementation of new technologies. the costeffectiveness of any self-service automated system depends on its productivity and the quality of services it provides [17]. however, it is not only the provision of services that is important to users but also the way in which the service is provided, which must be accessible, consistent and reliable. otherwise, users will be dissatisfied and will not use a self-service automated system or will use the service of another manufacturer [18-19]. as more services appear, i.e. new self-service automated systems, the user experience will be the dominant factor for the successful usage of new systems and services [20]. customer satisfaction is becoming one of the most important goals of all the manufacturers who want a long-term relationship with the customer. one of the main aspects that determine customer satisfaction is the perceived quality of service. consumer perceived quality is the ratio between the user's expectations of the service itself and the perception of the service received [21-23]. the quality of service of the self-service automated system is deeply related to customer satisfaction and loyalty (fig.2.) customer loyalty is a consequence of the overall and cumulative experience that customers have with the service provided. customer satisfaction can lead to customer loyalty because customers tend to use services with which they have already had positive experiences. customer loyalty and satisfaction are the most researched areas of marketing today. however, measuring loyalty and satisfaction is not an easy task. what one user perceives as good, another user may consider average or even poor. one of the most commonly used models for measuring the quality of service is servqual, developed in 1988 and designed to analyze customer expectations and perceptions of service in five dimensions believed to represent service quality [24-25], namely:  reliability consistency of service delivery,  tangibility visual experience of customers and quality of equipment,  helpfulness willingness to help customers and provide fast service,  security knowledge and kindness of employees and their ability to inspire trust, and,  empathy care and individual approach to customers. with the concept of the iot, data is collected, analyzed and made available for further use. iot expands the connection between things with minimal need for human intervention. iot means that standard and smart devices and even living beings (plants and animals) are assigned identification numbers and the ability to communicate and exchange data over the internet completely independently. information-communication technologies enable the connection "at any time" and "at any location", while the application of the iot enables the connection of "anything". the concept of the iot is not the result of one new technology, but a combination of several technologies which, when 6 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. combined, enable a connection between the virtual and the physical world. those technologies are the following:  sensors,  data processing,  actuation,  communication and cooperation,  addressability,  identification,  localization, and,  user interface. fig. 2 the relations between service quality, customer satisfaction and loyalty to the manufacturer based on [22] data are collected and analyzed to increase the efficiency and productivity of a selfservice automated system [26-29]. smart systems that are more service-oriented cloud solutions enable retailers to provide customers with a better shopping experience [3031].understanding and predicting customer behavior is a very important feature of smart automated systems [32].the customer experience, as mentioned earlier, has the greatest impact on the customers' intentions toward innovative ways of buying, paying and providing services. the goal is a personalization of customers' service and suggests products that are of interest based on customers' habits. self-service automated system data collection, with the implementation of business intelligence and knowledge management services can enable the displaying of personalized messages to customers to entice them to use a self-service automated system. 3. smart ecosystems with a strong tendency to achieve sustainability, cities are turning to their smartness. smart cities mean city/infrastructures’ ability to adapt to everyday changes by using advanced technologies and resources in an integrated way in order to achieve socially and environmentally viable economic growth. city smartness relies on two pillars [33]: 1. a tangible one, including the physical infrastructures and channels that intertwine the entities who participate in the functioning of the city, and, 2. an intangible one, consisting of the tacit and immaterial factors that underpin the city’s identity and subjectivity. towards a smart ecosystem with automated services 7 in order to establish smart ecosystem implementation, both natural and artificial components must be considered, which are strictly intertwined in the design and implementation of initiatives intended to foster the achievement of sustainability. concerning strategy, three different aspects of a smart ecosystem [34] can be identified, namely: business ecosystem, which centers on an industry and its environment (industry); platform ecosystem, which considers how stakeholders organize around a platform (software providers); and innovation ecosystem (industry, innovation centers, university), which focuses on a particular innovation or new value proposition and the constellation of the actors supporting it. innovation ecosystems require multiple stakeholders to come together and collaborate for a new innovative value proposition to take effect. in [35] authors define a smart ecosystem as a system that is much like a natural biological ecosystem: a distributed, adaptive, open socio-technical system with the properties of self-organization, scalability, and sustainability. usually, in the smart ecosystem devices are connected to a network and allowed to communicate outside their domain (typically by the internet) [36]. an example of devices in a smart ecosystem in [37] is presented hvac (heating, ventilation, and air conditioning) system for controlling the energy supply to an electric motor that powers pumps, fans, compressors, and other equipment (fig. 3). this smart ecosystem can be applied in different environments like water distribution systems [38]. fig. 3 examples of devices in a smart ecosystem based on [37] in [39] authors present a concept and the framework for designing a smart energy network ecosystem for integrated energy services in urban areas. also, they present initial results for the test platform smart otaniemi which is an innovation ecosystem that connects experts, organizations, technologies, and pilots. the platform is based on the framework of the smart energy network ecosystem which is shown in fig. 4. initially, the platform involved 11 companies and three research institutes and the first experiment with the platform began in 2018. using the idea from the framework of the smart energy network ecosystem shown in fig. 4, the university of novi sad and the university of niš, started to develop applications for a smart gas network eco-system in serbia. in the next chapter some of the ideas of the smart gas network ecosystem in serbia will be presented. 8 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. fig. 4 the framework of the smart energy network ecosystem based on [39] towards a smart ecosystem with automated services 9 a typical self-service automated system consists of a lcd touch panel, payment devices (cashless payment device and bill and coin payment device), an embedded industrial pc, a control board, a thermal ticket printer and a communication module (fig. 5.).these components enable a self-service automated system to be part of a smart ecosystem and are present in most automated systems today [40-44]. fig. 5 components of a typical self-service automated system self-service automated systems, in the case of vending machines, are also a typical representative of systems that have two types of users: customer and administrator. their basic roles are shown in fig. 6. in order to develop a new generation of smart ecosystems with self-service automated systems, at the university of novi sad and the university of nis, a model of augmented reality support with self-service automated systems has been developed. the next chapter presents how the model can be applied in a smart shop with a smart vending machine. 10 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. fig. 6 two types of users on the self-service automated system/vending machines 4. smart ecosystems: case studies 4.1 the smart gas network ecosystem for every individual home, buildings, industries, and communities, one of the main challenges concerns the way energy is managed because energy is one of the biggest cost drivers for each of these entities. especially, in the light of global rising energy costs and strict environmental regulations, this challenge is becoming more important. in many buildings and factories, a smart ecosystem to monitor, control, measure, and optimize energy consumption, can be found. usually, these smart ecosystems use available smart energy management systems. in the case of individual homes, another approach can be offered, like using a smart gas network eco-system (to optimize gas consumption). in fig. 7, the framework of the smart gas network ecosystem, based on the framework of the smart energy network ecosystem [39] is represented. fig. 7 the framework of the smart gas network ecosystem towards a smart ecosystem with automated services 11 at the application layer, any customer who has a gas meter can check the current status of the consumed gas on the mobile/web application. one of the images which appear in the application for mobile devices is shown in figs. 8 and 9. the data displayed in the application is obtained by accessing the server located within the data layer. as part of the version i of this application, the customer can have an insight into the current state of his/her gas meter and check the consumption in a certain period. fig. 8 screenshots of the mobile application for displaying gas consumption for 1 day in order to minimize communication requirements, data exchange between the application layer and the data layer is performed once per hour [45]. 12 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. fig. 9 screenshots of the mobile application for displaying gas consumption for 7 days the current read value from the gas meter represents the total gas consumption. in the case where is a problem in receiving data from the gas meter, the value temporarily entered is 0 for the particular hour of reading. the approximate value for the particular hour of reading when the data (m) of gas consumption has not been received, is determined based on the formula m = l + s * (c-l) / h where c represents the current value of gas consumption from the gas meter, l represents the last read value from the gas meter, h represents the difference in the number of hours between the readings of c and l, s represents the difference in the number of hours between the readings of m and l. at the data layer, there are servers and databases. servers allow data to be collected from the physical layer and forwarded to the application layer. databases allow the storage of all collected data. in version ii of the data layer, modules for data analysis will towards a smart ecosystem with automated services 13 be added in order to make better decisions for the management of gas supply, distribution and consumption. at the physical layer, there are smart gas meters, which enable the sending and receiving of data via the lora network. the data sent from the gas meter refers to the gas consumption, the gas pressure in the network, and the received data from the application can be used to open and close the gas control valve. version ii of this application will have the incorporation of consumption prediction based on previous consumption and predicted weather conditions. version iii of this application will have the incorporation of artificial intelligence that will enable the user to set conditions that would allow the application to manage gas consumption. for example, the user sets the desired amount of money (gas consumption) for a certain period, and the application determines how much would be spent each hour during a given period. 4.2 the smart-shop network ecosystem self-service automated systems base their success on the successful realization of the link between service quality, customer satisfaction and customer loyalty (fig. 2).one of the important steps in this realization is the recognition of customers' habits as well as typical customers' behavior. besides the successful implementation of these relations, for the realization of a self-service automated system, the following characteristics should be taken into account (especially in the case of smart vending machines) [46].  cashless payment,  networked location data,  video and touch screen communication,  mobile and facial recognition,  remote experts,  point-of-purchase marketing opportunity, and,  targeted advertising impressions. in addition to these parameters in the implementation of the smart shop, the following parameters were monitored and analyzed in order to determine the typical behavior of users:  customers age,  customers gender,  season of the year, and,  time of day and temperature. a smart shop is a new type of kiosk that can be easily transformed from a standard vendor kiosk to a vendor-less kiosk [47].this vendor-less kiosk is developed based on the proposed architecture in fig.10. an example of this smart shop is shown in fig.11. smart-shop as the iot extends the connectivity between things with minimal need for human intervention [48]. this characteristic is desirable for retailers who seek to ease the shopping experience for the consumers. one of the key preconditions, in order to ease the shopping experience, is to offer iot payment to the customers. iot payment is a process of money transfer based on the user device that is connected to his/her money account. the usage view sets up the technical solution by describing the customers’ journey through all the steps of the use case being implemented. this view would include the key actors, which may be customers and/or machines, and the activities involved. the usage 14 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. view also describes the use case from the point of view of customers’ needs and system capabilities. fig. 10 architecture of the smart-shop network system fig. 11 smart-shop kiosk (image source: http://www.smart-shop.rs) fig. 12 illustrates deloitte’s reference architecture, a typical iot use in a store [49]. applying this scenario, every new user (customer) has much more chance to have good satisfaction using iot payment. more users, and more experience (this means more data), enable the opportunity of using the data analytics process of iot payment as a smart process. the software that is installed in the vending machines at smart shops, based on the concept presented in deloitte’s reference architecture, makes iot payment possible to follow the habits of each customer and, on that basis, to suggest products with a greater towards a smart ecosystem with automated services 15 chance of being firstly selected by the customer. each customer is personalized, which means that all menus on the vending machine are automatically configured according to the customer. in this way, the customer gets the impression that he/she is buying from a salesperson, who knows his/her typical habits. fig. 12 the iot usage view based on [49] the authors of this paper are part of the team that developed this smart shop. for the first time, in serbia, this smart shop will work in conjunction with a salesperson to allow the sale of products that would require specific vending machines. 4.3 discussion in the case studies described, the solutions are developed with the aim to develop smart ecosystems. the achieved results fully justify the approach to the development of such systems. in the study of a smart gas network ecosystem, the customer can monitor gas consumption and, in that way, if possible, correct or reduce gas consumption. also, the customer can immediately estimate gas consumption in terms of the price of gas, bearing in mind that the price of gas consumed is affected by the pressure in the transport system as well as the gas temperature. the next step in the development of this application is that the application will have the incorporation of consumption prediction based on previous consumption and predicted weather conditions. this prediction is important not only for the customers but also for the gas distributor in order to provide for the estimated gas needs in time. the final version of this application will have the incorporation of artificial intelligence that will enable the customer to set conditions that would allow the application to manage gas consumption. also, this function in the application is important not only for the customers but also for the gas distributor. for example, if the customer wants to set the desired amount of money (gas consumption) for a certain period, the gas distributor should allow control valves to be installed at the customer, to accomplish this task. the plan is to implement functions in the application that will determine how much would be spent each hour during a given period. these functions are typically found in scada systems, and, therefore, the gas application will need to be integrated with the existing scada system. in the case of a smart-shop, the system is more demanding because a large number of customers can demand service in different locations at the same time, i.e. to use different 16 i. baranovski, s. stankovski, g. ostojić, v.nikolić, et al. smart-shop. a large number of products are present in every kiosk, which is of special importance when it comes to predicting their consumption not only in an individual kiosk but also in the kiosk network. in addition, each kiosk requires maintenance. one way of solving these problems (tasks) is to introduce an augmented reality support model with self-service automated systems, as shown in fig. 13. fig. 13 smart-shop kiosk architecture based on augmented reality in the model of augmented reality support with self-service automated systems the following units can be defined:  ar model registration, tracking the position and orientation of the customer, as well as the presentation of interactive content,  operational model execution of various functionalities of the self-service automated system,  management model centralized database with monitoring, management and analysis of data from a remote location, and,  model of the self-service machine customers’ interaction and product/service provision. in order to display virtual content in a real environment on a self-service automated system, it is necessary to register the position and orientation of the customer in relation to the reference coordinate system (self-service automated system) via a smartphone. the obtained position and orientation must be updated in real time in order to properly display towards a smart ecosystem with automated services 17 the virtual content. identical self-service automated systems are installed in different locations. using the gps position, it is possible to determine which specific machine the customer is at. the implementation of augmented reality will enable smart-shop customers’ new features. customers will have new opportunities to get information about the product they want to buy while the kiosk administrator will be able to more easily manage and maintain the kiosk itself. 5. conclusion the new ict architecture allows expanding the traditional ways of device management, allowing their users to correct management decisions, considering the economic aspects using of the device. with this approach, we get smart ecosystems. this paper presents some ideas from the concept of the smart gas network ecosystem, which can be part of the energy smart network ecosystem and the smart-shop network ecosystem. the first results from these case studies show justification for the development of smart ecosystems. the development of smart ecosystems is a challenge not only for their producers but also for all participants involved in each phase of the life cycle of a smart ecosystem. certainly, computer technologies such as artificial intelligence, edge computing, and cloud computing will facilitate the 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https://www2.deloitte.com/xe/en/insights/focus/internet-of-things/iot-primer-iot-technologies-applications.html plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 195 207 1a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions udc (531.2+519.6):617.3 hajar razi 1,2 , annette i. birkhold 1,2 , manfred zehn 3 , georg n. duda 1,2 , bettina m. willie 1 , sara checa 1 1 julius wolff institut, charité-universitätsmedizin berlin, germany 2 berlin-brandenburg school for regenerative therapies, berlin, germany 3 technische universität berlin, germany abstract. though bone is known to adapt to its mechanical challenges, the relationship between the local mechanical stimuli and the adaptive tissue response seems so far unclear. a major challenge appears to be a proper characterization of the local mechanical stimuli of the bones (e.g. strains). the finite element modeling is a powerful tool to characterize these mechanical stimuli not only on the bone surface but across the tissue. however, generating a predictive finite element model of biological tissue strains (e.g., physiological-like loading) encounters aspects that are inevitably unclear or vague and thus might significantly influence the predicted findings. we aimed at investigating the influence of variations in bone alignment, joint contact surfaces and displacement constraints on the predicted strains in an in vivo mouse tibial compression experiment. we found that the general strain state within the mouse tibia under compressive loading was not affected by these uncertain factors. however, strain magnitudes at various tibial regions were highly influenced by specific modeling assumptions. the displacement constraints to control the joint contact sites appeared to be the most influential factor on the predicted strains in the mouse tibia. strains could vary up to 150% by modifying the displacement constraints. to a lesser degree, bone misalignment (from 0 to 20°) also resulted in a change of strain (+300 µε = 40%). the definition of joint contact surfaces could lead to up to 6% variation. our findings demonstrate the relevance of the specific boundary conditions in the in vivo mouse tibia loading experiment for the prediction of local mechanical strain values using finite element modeling. key words: finite element model, mouse tibial loading, bone adaptation, bone mechanics received november 01, 2014 / accepted november 30, 2014 corresponding author: sara checa charité universitätsmedizin berlin, julius wolff institut, augustenburger platz 1, 13353 berlin, germany e-mail: sara.checa@charite.de original scientific paper x-apple-data-detectors://3/ x-apple-data-detectors://3/ javascript:void(window.open('/horde/imp/dynamic.php?page=compose&to=sara.checa%40charite.de&popup=1','','width=820,height=610,status=1,scrollbars=yes,resizable=yes')) 196 h. razi, a.i. birkhold, m. zehn, g.n. duda, b.m. willie, s. checa 1. introduction the local mechanical strains play a key role in regulating bone mass and architecture. it is known that changes in the mechanical strains “perceived” at a local position within the bone can result in the deposition or resorption of bone matrix at that location and that this depends, to a large extent, on the strain magnitude [1]. in vivo animal loading experiments have been extensively used to investigate the bone adaptation response to controlled mechanical loading [1-8]. among them, the in vivo mouse tibial compression model has been widely used [2, 4, 5, 7, 8]. in this model, the tibia of the mouse is placed between a concave cup and a platen for the application of a controlled compressive load [3]. finite element techniques are then used to predict the mechanical strains induced in different regions within the bone [3, 8-10]. although these computer models ideally attempt to replicate the experimental set-up, the in vivo experiment includes a certain degree of uncertainty regarding some modeling parameters, e.g. the alignment of the bone within the loading machine or the degrees of freedom at the joint surfaces. a recent finite element study has shown that in order to predict the location of the neutral axis in the mouse tibial loading model, the loading conditions should include both the compressive load being applied during the experiment plus an extra lateral force, which is not included in the experimental protocol [10]. this indicates that the mouse tibia in the loading machine is probably not completely aligned. previous radiographic analyses of the mouse tibia mounted in the loading machine by our group have also confirmed this [3]. in addition, yang and co-workers have shown in situ by removing the soft tissue from the mouse hindlimb that the mouse tibia exhibits a tilt angle in the loading machine [9]. despite these observations, the effect of limb alignment on the predicted strains remains to be investigated. other uncertain parameters in finite element modeling of the mouse tibial compression model relate to the boundary conditions, both in terms of point of application and degrees of freedom. experimental measurement of the contact surfaces at the knee and the ankle through which the load is being transferred is a challenge in vivo. finite element studies of the in vivo mice tibia loading model have assumed different contact surfaces [9-12]. patel and co-workers used micro-computed tomography (µct) to define boundary conditions at the knee and ankle joints of the mouse tibia [10]. they fixed all degrees of freedom in all points within one 2d µct image located at the ankle joint. similarly, the load was uniformly distributed between all points at the most proximal 2d µct image (representing the knee joint) [10]. yang et al., 2014, used two ellipses to approximate the contact surfaces at the knee, while restraining movement in all points on the articular surface of the distal tibia [9]. to what degree the predicted mechanical strains are influenced by the joint contact areas remains unknown. in addition to the contact surfaces, it remains unknown how much the bone can move in the transverse plane perpendicular to the loading axis. the concave cup and platen are designed to hold the tibia in place; however, small movements could be present. yang and co-workers investigated the effect of proximal tibial displacement constraints on the predicted strains [9]. they showed that this parameter introduces the largest index in their sensitivity analysis [9]. the influence of this parameter on the strain state within mouse tibia is still unclear. accurate prediction of strains using finite element modeling relies not only on the geometrical and material properties, but also on correct definition of loading and boundary conditions. in this study, we aimed at investigating the influence of bone alignment and a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions 197 boundary conditions, both in terms of point of application and degrees of freedom, on the mechanical strains predicted using a fe model of the in vivo mouse tibial compression loading experiment. variations in the load direction (bone alignment) and boundary conditions have been investigated separately by isolating one factor at a time (assuming that these parameters are independent). 2. materials and method 2.1. study design this study presents finite element models which have been created to replicate an in vivo strain gauging experiment of the left tibia of an elderly mouse (78 week old female c57bl/6j) as described elsewhere (fig. 1a-b) [7]. briefly, uni-axial strain gauges (one per tibia) were mounted on the anterio-medial surface of the tibia mid-shaft of live animals (fig. 1c). fig. 1 (a) experimental set-up and (b) schematic demonstration of mouse tibia in the loading machine (c) micro-computed tomography image of the mouse tibia demonstrating the position of the strain gauge position in the scan (d) cortical mid-shaft region of interest at the tibial mid-diaphysis and (e) tibial diaphyseal, proximal and distal metaphyseal regions. while the mice remained anesthetized, dynamic compressive loads were applied between the flexed knee and the ankle using a custom-designed in vivo loading device (testbench electroforce lm1, bose, framingham, usa). load and strain were recorded simultaneously. a range of dynamic compressive loads (peak loads ranging from -2 to 12n) were applied to identify the applied load that engendered approximately +1400 µε in vivo at the strain gauge position in mouse tibia. all animal experiments were carried 198 h. razi, a.i. birkhold, m. zehn, g.n. duda, b.m. willie, s. checa out according to the policies and procedures approved by the local legal research animal welfare representative (lageso berlin, g0333/09). 2.2. finite element modeling 2.2.1. geometry and discretization one left tibia from a mouse that underwent the strain gauging experiment was used to create fe models. to acquire accurate bone morphology, ex vivo micro-computed tomography (µct) was performed on the tibia. µct was performed with an isotropic voxel resolution of 9.9 μm (skyscan 1172, kontich, belgium; 100 kvp, 100 µa, 360°, using 0.3° rotation steps, 3 frames averaging). bone geometry was segmented by excluding background and soft tissue voxels from the bone region by applying a global threshold of 0.105 mm -1 (linear attenuation coefficient). the threshold value was determined based on the grey value distribution of the whole bone [13, 14]. following the segmentation, bone surfaces (both cortical and cancellous regions) were defined using a triangular approximation algorithm coupled with best isotropic vertex placement [15] to achieve high triangulation quality. the enclosed bone surfaces were filled with volumetric tetrahedral elements (c3d4 tetrahedrons) resulting in adaptive multiresolution grids using the zibamira software (zuse institute, berlin, germany). the mechanical interaction between the tibia and the fibula at the tibiofibular joint (proximal) was set considering the amount of mineralized tissue connecting them. the influence of mesh density on the predicted strains was investigated using eight models with different number of elements. a range of feasible mesh densities; i.e. minimum density aimed at geometrical fidelity (0.710 6 elements) and maximum density aimed at feasible computational cost (2.210 6 elements), introduced small changes in the strains predicted at the cortical mid-diaphysis and strain gauge site (2% and 7%, respectively). thereafter, fe models were performed using approximately 1.5x10 6 elements in the mouse tibia. linear elastic fe analysis was performed in abaqus 6.12.2 (dassault systemès simulia, ri, usa) to simulate the in vivo tibial loading experiment. to investigate the differences in the induced mechanical strains when varying model parameters, an identical external load (-11 n) was applied to all fems. load was applied through a contact surface selected on the knee side. displacement constraints were applied at ankle side to the talustibialis contact pressure surface. in order to compare the influence of boundary condition variations on the strain state within the bone tissue, bone was assumed isotropic and homogeneous. a linear elastic isotropic young's modulus (e) of 20 gpa and a poisson’s ratio (ν) of 0.3 was assigned to all regions of the bone. 2.2.2. joint contact surfaces the joint contact pressure sites at the knee and ankle joints were modified by changing the surface at which boundary conditions were assigned (fig. 2). in addition, two single points at knee and ankle joints were selected to assign boundary conditions. in these models, the axial compression load was inserted with 0° tilt angle with respect to the tibia longitudinal axis (the description of this axis can be found below; fig. 3) and surface nodes at ankle side were fixed in all degrees of freedom while surface nodes at the knee side were fixed in translation at perpendicular directions to the load (constraint 1). a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions 199 2.2.3. distal and proximal displacement constraints in addition, the displacement constraints at the boundaries were modified in three modes:  constraint 1: surface nodes at ankle side were fixed in all degrees of freedom, surface nodes at the knee side (where load was inserted) were fixed in translation at perpendicular directions to the load.  constraint 2: surface nodes at ankle side were fixed in all translational degrees of freedom and free for rotational moments, surface nodes at the knee side were fixed in translation at perpendicular directions to load.  constraint 3: surface nodes at ankle side were fixed in all degrees of freedom, no displacement constraint was applied at surface nodes at the knee side. in these models, the axial compression load was inserted with 0° tilt angle with respect to the tibia longitudinal axis (fig. 3) and knee/ankle contact surfaces were set as shown in a2b2, fig. 2. fig. 2 contact pressure sites (highlighted in red) at knee (a1 to a4) and ankle (b1 to b4) joints selected for assigning load and displacement constraints in fem. 2.2.4. bone alignment in order to modify the load direction (bone alignment with respect to the loading machine), a longitudinal axis was assigned to the tibia. the proximal-distal (p-d) axis of the bone was defined as the axis passing through the mid-points between the medial and lateral knee tuberosities and the medial and lateral malleouli. load direction was then modified by rotating the p-d axis around the perpendicular axis passing through the lateral and medial knee tuberosities (m-l axis) from 0° to 20° (fig. 3). this a feasible range within which the mouse tibia is visually accommodated by the knee cup and ankle platen. to perform the comparison between different load directions, surface nodes at the ankle side were fixed in all degrees of freedom and surface nodes at the knee side were fixed in translation at perpendicular directions to the load (constraint 1). joint contact pressures were assigned according to a2b2 (fig. 2). 200 h. razi, a.i. birkhold, m. zehn, g.n. duda, b.m. willie, s. checa fig. 3 (a) assignment of the bone longitudinal axis in order to have comparable load trajectory across the models of the in vivo experimental set-up. p-d represents the proximal-distal axis and m-l the medio-lateral axis (shown by the cross). (b) variations in the bone alignment in the in vivo experimental set-up resulting in modified load trajectories in the fe models of the mouse tibia. 2.2.5. data analysis alterations in the induced strains as influenced by the boundary conditions were investigated by comparing the strains in two regions of interest: 1) at the strain gauge position (fig. 1c) and 2) at the cortical mid-diaphysis (5% of bone length) in the fe models (fig. 1d). strain components for the elements at the position of the strain gauge (seen in the ex vivo µct scan of bones) were calculated in the local coordinates of the strain gauge so that εxx is the strain component in the longitudinal strain gauge direction. average and standard deviation for all elements at the strain gauge location are reported. in addition, average and ranges of minimum (compressive) and maximum (tensile) principal strains at the cortical mid-shaft region are reported for different models. 3. results 3.1. joint contact surfaces variation in the joint contact surfaces (fig. 2) resulted in a maximum of 6% and 10% difference in the average compressive and tensile principal strains induced at the tibia mid-diaphysis (table 1). this difference was observed between applying the boundary conditions to the entire surface at both knee and ankle joints (a4b4) and to the single points at joints (fig. 4). this variation was also reflected at the longitudinal strains predicted at the strain gauge position (6% variation in longitudinal strains between models). in addition, up to 70% variation was calculated in the longitudinal strain at the gauge site compared to in vivo measurements (in a4b4). changes in the joint contact surfaces led to variations in the distribution of strains (fig. 5). specifically higher strains were observed at proximal and distal tibia regions when applying the boundary conditions to single nodes compared to larger contact surfaces (fig. 5). however, at the distal diaphyseal region higher strains were predicted when implementing the boundary conditions at the entire knee and ankle surfaces (fig. 5). a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions 201 table 1 mean±sd of principal strains induced at the mid-diaphysis principal strains single nodes a1b1 a2b2 a3b3 a4b4 tensile 350±170 329±147 322±145 321±145 316±144 compressive 760±598 728±588 720±581 718±579 711±574 fig. 4 distribution of minimum (left) and maximum (right) principal strains at the cortical mid-diaphysis of mouse tibia while implementing the boundary conditions at single nodes in ankle and knee surfaces (dashed line) and in a4b4 (filled line) fig. 5 absolute maximum principal strains induced within the mouse tibia while implementing the boundary conditions at single points and four different contact surface conditions at ankle (proximally) and knee (distally) side (a1b1 to a4b4). plots show longitudinal cut through the tibia. circles and arrows point to regions with large differences between models 3.2. distal and proximal displacement constraints modifying the displacement constraints resulted in large differences in the induced strains at the cortical mid-diaphysis of the tibia (fig. 6). the largest differences in maximum and minimum principal strains induced at the cortical midshaft occurred between displacement 202 h. razi, a.i. birkhold, m. zehn, g.n. duda, b.m. willie, s. checa constraints 2 and 3 (20% and 10%, respectively) (fig. 6). in addition, restraining the foot and ankle and leaving the knee free resulted in higher compressive and tensile principal strains at the anterior and posterior side of the distal tibia (respectively) compared to conditions where the knee was partially restrained of movement (constraints 1 and 2) (fig. 7). the influence of the boundary conditions was more evident in the strain predicted at the strain gauge site. longitudinal strains increased by 150% and 120 % between displacement constraints 3 and 2, and between displacement constraints 1 and 2, respectively. compared to the in vivo measurements (+1400 µ), predicted strains at the gauge site were from 28% (constraints 3) to 70% (constraints 1) lower. fig. 6 distribution of minimum (a) and maximum (b) principal strains at the cortical mid-diaphysis of mouse tibia while implementing constraint states 1-3 in ankle and knee surfaces fig. 7 absolute maximum principal strains induced within the mouse tibia while implementing different boundary conditions at a2b2 contact surfaces. plots show longitudinal cut through the tibia. arrows point to regions with large differences between models 3.3. bone alignment increasing the angle between the bone axis and the load direction resulted in a shift towards higher strain values at the cortical mid-shaft of the mouse tibia (fig. 8a). the effect of bone misalignment was also evident in the predicted strain at the strain gauge position. by changing the misalignment angle from 0 to 20°, fem predicted a +300 µε a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions 203 increase in longitudinal strains at the strain gauge position (fig. 8b). compared to the in vivo measurements, strains at the gauge site were 40% to 60% lower. the general strain state within the mouse tibia was not largely affected by introducing the misalignment (fig. 9). however, anterior and posterior proximal diaphyseal regions exhibited high tensile and compressive strain (respectively) by increasing the alignment angles. by increasing the bone misalignment both compression and tension sites (anterior and posterior diaphysis, respectively) were under larger strain magnitudes than the aligned model. fig. 8 (a) distribution of minimum (compressive) and maximum (tensile) principal strains at the cortical mid-diaphysis of mouse tibia while modifying the bone axis with respect to load trajectory from 0° to 20° (b) average and standard deviation of longitudinal strains at strain gauge site in different tibia alignment conditions fig. 9 distribution of absolute maximum principal strains within the mouse tibia (i: longitudinal cross-section of tibia, ii: anteromedial perspective to tibia) while modifying the bone axis with respect to load trajectory from 0° (left) to 10° (middle) and to 20° (right). arrows point to regions with large differences between models 204 h. razi, a.i. birkhold, m. zehn, g.n. duda, b.m. willie, s. checa 4. discussion so far, the direct link between in vivo bone loading, tissue adaptation and local mechanical strain stimulus remains unclear. a major reason is the lack of knowledge on the local mechanical strain and how it depends upon the local in vivo bone loading. finite element modeling is a powerful tool to characterize the local mechanical straining of bone. in vivo mouse tibia loading experiment is a popular method to investigate the mechano-biological tissue adaptation relationships. such controlled animal loading model surpasses the complicated burden of accurately identifying the strain environment during normal physiological daily activities (e.g. walking, running). however, to link local biological reactions to the local mechanical stimuli requires exact modeling of the loading environment in the bones. finite element models to characterize the mechanical environment during tibia loading are subjected to uncertainties regarding the boundary conditions and bone positioning in the experimental set-up. in this study, we investigated the influence of variation in boundary conditions and bone alignment in the loading machine on the predicted strains at the tibia mid-diaphysis and strain gauge position. bone alignment, joint contact surfaces and displacement/rotational constraints at the boundaries were investigated for their potential influence on the predicted strains. we have observed that alterations in joint contact surfaces where the boundary conditions are applied have a small effect on the general strain state within the bone diaphyseal region. however, large variations are observed at closer proximity to the boundary, i.e. proximal and distal metaphyseal regions. previous studies have used different boundary conditions to characterize the mechanical environment in the in vivo tibial compression mouse model [3, 4, 7, 9-12]. patel et al. cropped proximal and distal parts of the bone to apply the boundary conditions to the fem [10] (similar to [12]). other reports have used a 3d reconstructed bone volume and selected the entire articulation surfaces at the knee and ankle joints of murine tibia to apply the boundary conditions [11] (and similarly [4]). yang et al. have assigned the boundary conditions on two elliptical surfaces at the knee side (similar to our a2-b2 mode) [9]. our results are comparable in terms of general strains state within the diaphyseal bone regions to previous reports [4, 9-11]. however, patel et al. have reported up to 3000 µɛ compressive strain at the proximal metaphyseal region [10], which is 2000 µɛ higher than our findings at the same region. these differences can be explained by the fact that in their model cutting the bone boundaries might have eliminated the propagation of deformation into the cancellous bone resulting in the shielding of strain by the outer cortical bone. however, our results at the metaphyseal regions are in close agreement with previous reports where either the entire knee articulation surface [11] or part of this surface [9] is used for application of the boundary conditions. assigning displacement constraints might seem trivial. the general consensus is to fix displacement in all degrees of freedom at one joint and insert the loading conditions in the other joint. in addition, a displacement constraint in the direction perpendicular to the load axis is introduced allowing only axial movement to the corresponding joint [3, 4, 912]. however, it is likely that the bone moves within the cup and platen while the animal is within the loading machine, since there are no restraints to avoid small movements. yang et al. report the highest sensitivity index in their fe model of the mouse tibia compression loading due to the choice of proximal displacement constraints [9]. a large influence is shown by their fe models between allowing and completely restraining the a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions 205 displacement of the knee side in the transverse plane perpendicular to the load. our results are in agreement with this report [9] by showing large differences between predicted strains in these two constraint states (constraint 1 and constraint 3). these results show that proximal constraints largely affect the induced strain magnitudes within the mouse tibia (fig. 6-7). in addition, the predicted strains at the gauge site are largely affected by this variation (up to 150%). another parameter when modeling the in vivo loading experiment is the bone alignment in the experimental set-up. due to the nature of bone being encapsulated in the soft tissue and muscles, and also being in contact with other neighboring bones, accurate estimation of bone alignment within every single experimental set-up is difficult. positions of peak strains have been shown not to be affected by inclusion of misalignment in the rodent tail model [16]. in the mouse axial compression loading experiment, it has been shown that the initial misalignment of the tibia is not affecting the induced strains since it is automatically adjusted during the loading experiment to the axial loading position, likely due to the geometry of the fixtures holding the knee and the foot in place [17]. it remains unclear how the knee and foot are situated within the holders after this initial automatic adjustment. it is shown that in order to match the experimental prediction of the neutral axis in the mouse tibia model, inclusion of a lateral force in the fe model is inevitable [10]. this finding suggests that the mouse tibia in the axial compression loading experiment exhibits a small misalignment. we identified the effect of varied bone misalignments on the induced strains. the general strain state induced at the tibia was not affected by introducing a tilt angle; however, since changing the load direction leads to changes in the bending moment in mouse tibia, strain induced in mouse tibia varied with bone orientation. in the anticipated range of misalignments; from 0° to 20° (higher tilt angles would be visually obvious or intolerable in the machine fixtures) up to +300 µε (40%) difference is introduced in the predicted strains at the strain gauge site. bone material property plays a key role in the determination of the strain environment within bone tissue. in the current study, we have implemented 20 gpa isotropic and homogeneous elastic moduli in order to investigate the effect of various boundary conditions on the predicted strains within the bone. however, it is worth noting that neither of our different configurations of boundary conditions did match the in vivo measured experimental strains at the gauge position. indeed up to 70% difference between the predicted and the measured strains at the gauge position was determined indicating a possible influential role of the material elastic properties. in a recent report, our group has shown how heterogeneous material properties affect the strain magnitudes within the mouse tibia, especially at the sites closer to the proximal and distal regions [14]. characterizing the induced strains within the mouse tibia is a crucial step that helps identifying the mechanical regulation of tibial structural adaptation to external loading regimes. although much effort has been made to characterize the strain environment induced within the mouse tibia in the in vivo compression loading experiment [2-5, 9-12, 17], consistent results are not yet achieved. unclear modeling parameters such as bone alignment, joint contact surfaces and movement/displacement constraints at the proximal and distal bone regions might explain this inconsistency. in this study, we have shown that although the general strain state within the mouse tibia under compressive loading is not affected by these uncertain parameters, the strain magnitudes at various tibial regions are highly influenced by certain modeling assumptions such as displacement constraints. we 206 h. razi, a.i. birkhold, m. zehn, g.n. duda, b.m. willie, s. checa have found that the assignment of displacement constraints has a strong influence on the predicted strains at the mid-shaft region in the mouse tibia. in addition, our results show that the joint contact surfaces mainly influence the strain magnitude and distribution at the proximal metaphyseal region without affecting the diaphyseal region of the tibia. our results clearly demonstrate the importance of appropriate selection of model parameters when developing a finite element models to predict strains induced within the bone. acknowledgements: all funding sources supporting publication of this study: elsbeth bohnhoff foundation and the german research foundation (deutsche forschungsgemeinschaft; wi 3761/1-1, wi 3761/4-1, du298/14-1; ch 1123/4-1) references 1. schulte f.a., ruffoni d., lambers f.m., christen d., webster d.j., kuhn g., muller r., 2013, local mechanical stimuli regulate bone formation and resorption in mice at the tissue level, plos one 8(4): e62172. doi: 10.1371/journal.pone.0062172 2. lynch m.e., main r.p., xu q., walsh d.j., schaffler m.b., wright t.m., van der meulen m.c., 2010, cancellous bone adaptation to tibial compression is not sex dependent in growing mice, journal of applied physiology 109(3), pp. 685-91. 3. willie b.m., birkhold a.i., razi h., thiele t., aido m., kruck b., schill a., checa s., main r.p., duda g.n., 2013, diminished response to in vivo mechanical loading in trabecular and not 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s., willie b.m., 2014, the influence of age on adaptive bone formation and bone resorption, biomaterials 35(34), pp. 9290-301. 8. birkhold a.i., razi h., duda g.n., weinkamer r., checa s., willie b.m., 2014, mineralizing surface is the main target of mechanical stimulation independent of age: 3d dynamic in vivo morphometry, bone 66(), pp. 15-25. 9. yang h., butz k.d., duffy d., niebur g.l., nauman e.a., main r.p.., 2014, characterization of cancellous and cortical bone strain in the in vivo mouse tibial loading model using microct-based finite element analysis, bone, 66, pp. 131-9. 10. patel t.k., brodt m.d., silva m.j., 2014, experimental and finite element analysis of strains induced by axial tibial compression in young-adult and old female c57bl/6 mice, journal of biomechanics 47(2), pp. 451-7. 11. stadelmann v.a., hocke j., verhelle j., forster v., merlini f., terrier a., pioletti d.p., 2009, 3d strain map of axially loaded mouse tibia: a numerical analysis validated by experimental measurements , computer methods in biomechanics and biomedical engineering 12(1), pp. 95-100. 12. connelly j.t., fritton j. c., van der meulen, m. c., 2003, simulation of in vivo loading in the tibia of the c57bl/6 mouse, 49th annual meeting of the orthopaedic research society, poster #0409, new orleans, la. 13. bouxsein m.l., boyd s.k., christiansen b.a., guldberg r.e., jepsen k.j., muller r., 2010, guidelines for assessment of bone microstructure in rodents using micro-computed tomography, journal of bone and mineral research 25(7), pp. 1468-86. a finite element model of in vivo mouse tibial compression loading: influence of boundary conditions 207 14. razi h., birkhold a.i., zaslansky p., weinkamer r., duda g.n., willie b.m., checa s., 2014, skeletal maturity leads to a reduction in the strain magnitudes induced within the bone: a murine tibia study , acta biomaterialia, available online: doi:10.1016/j.actbio.2014.11.021 15. zilske m., lemecker h., zachow s., 2008, adaptive remeshing of non-manifold surfaces. proceedings of eurographics 2008, crete, greece, pp. 207-211. 16. goff m.g., chang k.l., litts e.n., hernandez c.j., 2014, the effects of misalignment during in vivo loading of bone: techniques to detect the proximity of objects in three-dimensional models, journal of biomechanics 47(12), pp. 3156-61. 17. carriero a., abela l., pitsillides a.a., shefelbine s.j., 2014, ex vivo determination of bone tissue strains for an in vivo mouse tibial loading model, journal of biomechanics 47(10), pp. 2490-7. model konačnih elemenata za opterećenje sabijanjem tibije miša in vivo: uticaj graničnih uslova iako je poznata sposobnost kosti da se prilagodi mehaničkim izazovima, odnos između lokalnog mehaničkog stimulansa i reakcije prilagodljivog tkiva za sada izgleda nejasan. glavni izazov izgleda da predstavlja pravilna karakterizacija lokalnog mehaničkog stimulansa kostiju (naprezanja). metod konačnih elemenata je moćan alat za karakterizaciju ovih mehaničkih stimulansa ne samo na površini kostiju već i u tkivu. međutim, razvijanje predvidljivog modela konačnih elemenata za naprezanje biološkog tkiva (na primer, naprezanja poput fiziološkog) nailazi na aspekte koji su neizbežno nejasni ili magloviti što može znatno uticati na predviđene rezultate. naš cilj je bio da ispitamo uticaj varijacija u naleganju kostiju, dodirnih površina zgloba i ograničenja izmeštanja na predviđena naprezanja u eksperimentu sa opterećenjem sabijanjem tibije kod miša in vivo. utvrdili smo da opšte stanje naprezanja u tibiji miša pod opterećenjem sabijanjem nije bilo pod uticajem ovih neizvesnih faktora. međutim, veličine naprezanja na različitim područjima tibije bili su pod velikim uticajem specifičnih pretpostavki modeliranja. ograničenja izmeštanja radi kontrole dodirnih površina zgloba, po svemu sudeći, bili su najznačajniji faktor od uticaja na predviđena naprezanja u tibiji miša. naprezanja su mogla da variraju do 150% izmenom kinematskih ograničenja. u manjoj meri, loše naleganje kostiju (od 0 do 20°) takođe je ishodovalo promenom naprezanja (+300 µε = 40%). izbor dodirnih površina zgloba mogao je da dovede čak do variranja od 6%. naši rezultati pokazuju značaj specifičnih graničnih uslova u eksperimentu sa opterećenjem tibije miša in vivo radi predviđanja vrednosti lokalnih mehaničkih naprezanja pomoću modeliranja konačnim elementima. ključne reči: model konačnih elemenata, opterećenje tibije miša, prilagođavanje kostiju, mehanika kostiju http://dx.doi.org/10.1016/j.actbio.2014.11.021 facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 1 15 https://doi.org/10.22190/fume190103008b © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical methods for the simulation of deformations and stresses in turbine blade fir-tree connections justus benad berlin university of technology, berlin, germany abstract. in this work, different numerical methods for simulating deformations and stresses in turbine blade fir-tree connections are examined. the main focus is on the method of dimensionality reduction (mdr) and the boundary element method (bem). generally, the fir-tree connections require a computationally expensive finite element setup. their complex geometry exceeds the limitations of the faster numerical techniques which are used with great success within the framework of the half-space approximation. ways of extending the application range of the mdr and the bem to the particular problem of highly undulating surfaces of the fir-tree connection are shown and discussed. key words: fir-tree connections, navier equation, boundary element method, fft 1. introduction the rotating components in a gas turbine are a challenge for both design and manufacturing. especially turbine blades lead the way in terms of future technology [1]. improvements of these components may result in lower weight, increased turbine performance, a longer life, and lower operating costs. for aero engines (see fig. 1), such improvements have a positive impact on the entire aircraft [2, 3]. this may lead to lower emissions and a reduced environmental impact. among the most critical parts of the turbine are the fir-tree connections of turbine blade and turbine disk (see fig. 2 and 3). the loads in these connections strongly influence the living of the blade and the disk. indeed, turbine disks are among the components which are most prone to cracking in the entire engine [1]. such a failure may cause severe damage to the aircraft (see fig. 4). the resulting high safety requirements along with the afore mentioned high potential for future development and optimization of turbine blades, disks and their fir-tree connections lead to a demand for highly accurate and rapid simulation tools. received january 03, 2019 / accepted march 11, 2019 corresponding author: justus benad berlin university of technology, strasse des 17. juni 135, 10623 berlin, germany e-mail: mail@jbenad.com 2 j. benad in this work, different numerical methods for simulating deformations and stresses in the turbine blade fir-tree connections are examined. the main focus is on the method of dimensionality reduction (mdr) [4] and the fft-based boundary element method (bem) [5]. both methods have become standard tools in contact mechanics where they are applied with great success within the framework of the half-space approximation [6]. these rapid techniques are well-suited for demanding wear simulations (see for example [7], [8], and [9]). fig. 1 nacelle installation of the rolls-royce trent xwb turbofan engine on the airbus a350-900 aircraft. image: benad fig. 2 fir-tree connections of turbine blades (gold) and disk (silver) on a sectioned rolls-royce turboméca adour turbofan, an engine powering for example the mcdonnell douglas t-45 goshawk aircraft. image: cleynen [10] fir-tree connections generally require a computationally expensive finite element setup. their highly undulating surfaces and other complex geometrical elements in close proximity to the contact region make the application of the half-space theory a challenge. in this study, we closely examine the application range of the mdr and the fft-based bem beyond the numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections 3 half-space approximation. we also discuss ways of extending the methods so as to apply them to the particular problem of the complex geometry of the fir-tree connection. fig. 3 a worn fir-tree connection on the disk of an aircraft turbine. image used with the kind permission of rolls-royce the parts of this work are organized as follows: we first discuss the mdr and show results of its application on a simplified fir-tree model. likewise, the results of a finite element simulation of the same setup are shown. the findings of both methods are compared on a qualitative level. we then turn to the fft-based bem in its well-known and established form for the half-space, propose slight additions to the method and apply it to a fir-tree model. again, the results are compared on a rough qualitative level to finite element results. in the last main section of this paper, we build on two recent studies [11] and [12], where the fft-based bem is performed for completely arbitrary shapes. we present exemplary results of this technique for the two-dimensional navier equation. at the end of this work, the main findings are summarized in a conclusion. a) b) fig. 4 example for the damage caused by an uncontained engine rotor failure. a) general damage to the engine, b) example of internal damage to the left wing. images: australian transport safety bureau, safety report, aviation occurrence investigation: in-flight uncontained engine failure overhead batam island, indonesia, 4. november 2010, airbus a380-842 [13] 4 j. benad 2. method of dimensionality reduction (mdr) the mdr is an efficient tool for calculating surface deformations and stresses of elastic or viscoelastic bodies which are brought into contact. the method was first proposed in 2007 [14] and has since developed to become a standard tool in contact mechanics [4], [15]. the technique is particularly easy to apply for axially-symmetric contacts and is generally used within the framework of the half-space approximation [6]. the mdr maps a three-dimensional contact to an equivalent contact of a transformed profile with a one-dimensional foundation of independent elastic or viscoelastic elements. from a numerical perspective, such a method is very appealing. first, this is due to a very low number of the degrees of freedom, and second, it is because the degrees of freedom are decoupled, which eliminates the need for iterations within the mdr domain. the transformations to the mdr domain and back can also be performed rapidly with an order of computational complexity which does not exceed the order of the number of discretization points of the one-dimensional profile. therefore, the mdr is well suited for demanding wear simulations, which require an underlying highly efficient method to obtain the deformations and stresses. this procedure has been successfully implemented in the past, for example, for high-resolution simulations of gross slip wear [7], fretting-wear [16], and wear analysis of a heterogeneous material [8]. fig. 5 shows a stage within a simple gross slip wear process of a parabolic rigid indenter with a given wear coefficient. this indenter is pressed into and moved across an elastic half-space. also shown is the pressure distribution at the given stage. archard’s law [17] is used as a wear model. in order to obtain a stage in the wear process such as the one displayed in fig. 5, the required computational time is only a fraction of a second on a small personal computer when the mdr is used to calculate the deformations and stresses within the wear iteration. fig. 5 an exemplary stage of a simple gross slip wear process of a parabolic rigid indenter with a given wear coefficient (light red upper shape enclosed with black dotted line) which is pressed into and moved across an elastic half-space (grey area with green dotted line at the surface, the thin horizontal black line marks the undisturbed surface). at the bottom of the image the form of the pressure distribution at the given stage is shown which is zero outside the contact area and takes positive values within the contact area (blue line with dots). the mdr was used to obtain the deformations and stresses while the archard’s law was used to model the wear numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections 5 given the efficiency of the mdr and the success when it comes to wear simulations, it would be desirable to apply the method for calculating deformations and stresses in turbine blade fir-tree connections. fig. 6a shows a schematic view of one tooth of a turbine blade in contact with its counterpart on the disc. a close-up view, which is also stretched in the z-direction, is displayed in fig. 6b. a) b) fig. 6 a) schematic view of one tooth of a turbine blade in contact with its counterpart on the disk, b) close up view of the contacting profiles stretched in z-direction. images: diercks (modified to fit the context). images used with the kind permission of rolls-royce a) b) fig. 7 a) finite element setup for the calculation of deformations and stresses on a simplified fir-tree model with only one tooth in contact, b) exemplary results of the finite element model (blue) and the mdr approximation (red) for the worst principal stress at the disk surface (marked with a thin red line in the left graph a). images: diercks (modified to fit the context). images used with the kind permission of rolls-royce in a rough first approximation, the problem can be regarded as a simple two-dimensional contact of cylindrical lying indenter with a half-space. the transformed profile to be used in the mdr for such a line contact can be obtained directly [6] (see also [18]), or one can make use of fabrikant’s approximation (see [19]) to scale a corresponding rotationally symmetric solution to the real contact area as a rough first estimate. the latter approach was adopted here. with the resulting normal and tangential loads in the contact area, one can then obtain all 6 j. benad components of the stress tensor at the surface using the fundamental solutions of boussinesq and cerruti (see [20]), which makes it possible to display the worst principle stress at the surface. exemplary results for the worst principle stress at the surface obtained in such a fashion are displayed in fig. 7b along with finite element results for comparison. the finite element setup which was used is displayed in fig. 7a. note that for simplicity a blade root with only one tooth was considered in this first model. it transpires that, at least in the contact area, this very crude mdr approach already delivers the results which agree to a certain extent with those obtained with the more appropriate finite element model. the most striking difference is, of course, attained as the bottom of the first lobe is approached (region left of the contact area in fig. 7b). the influence of this notch is not modeled with the half-space approach. in the following section, we will remain within the framework of the half-space theory as we are interested primarily in the properties in the contact area. however, we shall still turn to a less confined model, the boundary element method. the setup will remain the same, with the exception that all four teeth of the fir-tree will be considered. 3. fft-based boundary element method (bem) for the half-space the boundary element method (bem) is used in many engineering applications. for some special cases, such as the simulation of tribological contacts where the technique is applied with the half-space approximation, the tool has set new standards in recent years, becoming the method of choice both in academic and industrial research and development [15]. the integral equations of the bem simplify to convolutions when the boundary can be approximated as a half-space surface. as such, the integrals can easily be obtained with the fast-fourier transformation (fft). the computational complexity to obtain the deformations and stresses at the half-space surface is o(n 2 log n) for a surface with nn discretization points. therein lies the great advantage of using the fft-based half-space bem for when the fft is not used and the entire matrix of the linear system is built, the complexity of the bem is o(n 4 ). therefore, we will for now retain the assumption of a half-space, in order to apply this well-known, established and efficient fft-based half-space bem to the contact problem of the fir-tree connection. the two-dimensional convolution one obtains for the half-space is 0 0 0 0( , ) ( , )a ab b s u x y k x x y y dx dy   (1) where u is the deformation,  is the load, a is the direction of the deformation and b is the direction of the load, (a,b)  {x,y,z} [21]. typically, the half-space approximation is used only when the gradients of the surfaces in the contact region are very small. this is not the case for the fir-tree. however, the error which is made can be kept at bay by careful consideration of the surface constraints such as geometry and friction of the actual curved surfaces. consider a fir-tree connection and coordinate system as displayed in fig. 8. unlike in the previous section where the half-space was aligned with the contacting surfaces of the single lobe, the half-space is now placed along the x-y plane in fig. 8 to run through the entire fir-tree connection. the half-space surface can then be imagined to be perturbed from its original flat state to the highly undulating shape in the z-direction. as a first step, it numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections 7 shall also be assumed that one of the contact partners is elastic and the other is rigid. this is a common approach in many problems in contact mechanics. once the solution is obtained it can generally be transformed back to the original problem with the knowledge of the elastic properties of the contacting bodies [6], [22]. the disk shall be modeled as an elastic half-space, and the blade shall be modeled as a rigid body in the following. a) b) fig. 8 a) geometry of a fir-tree connection of a blade model (light brown) and a disk model (light blue). a rectangular block is cut out of the disk model (sections are displayed yellow) to reveal the global coordinate system and the 2d curves (blue and red). b) extrusion along the y-axis of the 2d curves and contact area (green markers) for an exemplary indentation in x-direction. the disk is modeled as an elastic half-space and the turbine blade as a rigid indenter. the black discretization points of the turbine blade surface are given by a linear interpolation of the disk surface discretization points onto the turbine blade surface along the x-axis. for visualization purposes the extrusion width shown here is small and only a rough discretization is displayed. the beginning and end of the extrusion of the turbine blade profile are slightly rounded of in the direction of lower x to ensure a smooth indentation at these edges. image used with the kind permission of rolls-royce fig. 9 left: components of a shear stress τzx relative to the curved surface. τ * cannot be maintained because (3) is not fulfilled. right: the grid point of the disk (blue) has now slid down along the rigid profile to its equilibrium position where τ * =μp * . this results in a lower τzx and a new component p relative to the even half-space surface. image used with the kind permission of rolls-royce. 8 j. benad consider now a single grid point on the disk profile. for some indentation, that is the displacement of the blade in the x-direction from its position of first contact, the point on the disk is also deflected in the direction of the x-axis by a small distance ux. as the disk profile is modeled as an elastic half-space it can be regarded as even for the determination of the dependencies of surface deformations and forces. under the assumption that the normal and tangential problem are decoupled at the even half-space surface [21] the deflection in the direction of the x-axis is caused by a shear stress τzx at the grid point. however, in order to model the friction conditions, the components of the stress at the grid point relative to the actual curved surface * * cos sin , sin cos . zx zx p p p            (2) need to be considered. this is illustrated in fig. 9. assuming there exists friction according to the coulomb’s law of friction in its simplest form [23], it has to be * *p  (3) so that the equilibrium can be maintained. otherwise, the point will slide. whether the condition (3) is fulfilled depends only on the geometry and the friction coefficient. in the left graph in fig. 9 the condition is not fulfilled. thus, the grid point of the disk profile slides until the relation (3) is fulfilled again (right graph in fig. 9). the corresponding new values for τ * and p * cause a change of the stresses in the reference system of the even half-space which can be determined with the reverse transformation * * * * sin cos , cos sin . zx p p p            (4) fig. 8b displays results which are obtained in such a fashion. the contact area (green dots) is shown for an exemplary indentation. the inner values for the displacements and stresses can be obtained from the conditions at the surface with the fundamental solutions of boussinesq and cerruti (see [20]) as in section 2. fig. 10 displays a qualitative view of the worst principal stress in the disk (upper image). for comparison, the finite element results are displayed below. as in the previous section, it transpires that although there is some rough agreement of the results, there are also quite substantial differences, even on the qualitative level. the most striking difference is once more the notch stress in the lower regions of the lobes. the maximum is clearly shifted towards the contact area for the half-space approach, while it is at a much lower position of the lobes for the finite element results which consider the actual shape more appropriately. while the half-space approach may deliver sufficient results in the contact regions for the analysis of particular problems, the overall results still remain unsatisfactory. therefore, other methods to utilize the efficiency of the convolution for the calculation of deformations and stresses in complex shapes such as the fir-tree must be found. one such approach is described in detail in the next section. numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections 9 fig. 10 a rough qualitative comparison of the results for the worst principal stress as obtained with the fft-based bem for the half-space (upper image) and a finite element simulation (lower image). the maximum of the worst principle stress is attained in the lower notch regions in the fe approach, while for the half-space approach it is shifted upwards to the lower ends of the contact regions. image used with the kind permission of rolls-royce 4. fft-based bem beyond the half-space approximation it is shown in the last two sections that the rapid half-space approach may provide a rough approximation of the results in the contact regions of a fir-tree; yet, the overall results were unsatisfactory. therefore, other methods must be found in order to utilize the efficiency of the convolution for calculating deformations and stresses in complex shapes. various techniques which have been developed in the past may be applied to accomplish this task. in 1991, gao introduced a first-order perturbation method for the half-space approximation to take into account stress concentration effects of slightly undulating surfaces [24]. in later years, various methods were developed to accelerate the classical bem for completely arbitrary shapes which, in part, make use of the low computational complexity of the fft (see for example [25], [26] and [27]), or utilize other techniques such as hierarchical matrices (see for example [28]) to accelerate the calculation. recently, it was illustrated in [11] and [12] that the integral equations of the bem for completely arbitrary shapes (no half-space) can be obtained in a manner very similar to the fft-based half-space approach: for the case of the half-space, the boundary integral (1) is evaluated in the plane of the two coordinates x and y which perfectly aligns with the even half-space surface. this makes it possible to align a regular two-dimensional grid on which the fft is performed with the domain (see fig. 11a). 10 j. benad a) b) fig. 11 a) a uniform grid aligned with the even surface of a half-space, b) an arbitrary shape fully enclosed with a uniform three-dimensional grid for arbitrary shapes, the boundary integral equations represent convolutions over the entire three-dimensional space which encloses the arbitrary shape (see fig. 11b). it is shown in [11] and [12] how one has to appropriately set zeros at grid points which are not in close proximity to the actual surface so as not to distort the results one obtains with this technique. using the fft in such a fashion as to obtain the boundary integrals on arbitrary shapes lowers the computational complexity from o(n 4 ) to o(n 3 log n 1.5 ). while this is still for one dimension higher than the o(n 2 log n) complexity of the half space, the technique opens up opportunities for further reduction in computational complexity through a variable grid (see [12]), and offers the advantage of utilizing highly efficient implementation of the fft, which can be accelerated even further on parallel systems such as the graphics processing unit (gpu). we now build upon the results found in [11] and [12] where the fft is used to accelerate the calculation of the boundary integrals on arbitrary shapes for the laplace equation. the same technique is applied here, but for the two-dimensional navier equation. boundary integrals navier’s equation is , , 1 1 0 1 2 j ji i jj iu u b       , (5) where i,j{1,2,3} and bi is the force density field [20]. for the case of plane displacements, the navier equation remains as in (5), but with i,j {1,2}, [29]. this case can then easily be transformed into the case of plane stress, by replacing  with / (1 )  and leaving the value for μ unchanged [29], see also [30] and [31]. a single unit point force shall act in a point 0x , and in the direction of a unit vector e , so that in eq. (5), it is 0( )i ib x x e  , which yields the equation * * 0, , 1 1 ( ) 1 2 j ji i jj iu u x x e        . (6) the solution ui * of eq. (6) is called the fundamental solution of eq. (5). it can be expressed in terms of a matrix vector product with * * i k kiu e u . (7) numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections 11 with the betti’s theorem [32], the divergence theorem, and the definition of the fundamental solution given in eq. (6), one obtains (see [33]) somigliana’s identity * * 0 0 0( ) ( , ) ( ) ( , ) ( ) ,i ij j ij j s s u x u x x t x ds t x x u x ds   (8) which relates the deformations ( )ju x and stress vector ( ) ( ) ( ) j jk k t x x n x on the boundary s (outward normal vector nk) to the deformation 0( ) j u x at a particular inner point 0x . the term * 0( , )ijt x x which occurs in (8) is briefly explained in the following: note that a certain stress field σij * belongs to the fundamental solution for the deformation field ui * . depending on the position 0x , this sets a corresponding stress vector * * 0 0( , ) ( , ) ( )j jk kt x x x x n x on the given boundary. a matrix tij * is then constructed so as to express this stress vector with * * j i ijt e t . (9) accordingly, and with the material law for a linearly elastic isotropic solid [34], one obtains * * * * , , , 2 ( ) 1 2 ij ik k j ij k k ik j kt u n u n u n        . (10) the fundamental solution for the arbitrary three-dimensional case or for the cases of plane displacements or plane stress can be found in the literature. for simplicity, we will from now on only consider the case of plane displacements. through eq. (7), the two-dimensional fundamental solution of eq. (5) is for this case given with [33] 2 0 0* (3 4 )ln( ) ( )( ) / 8 (1 ) ij i i j j ij r x x x x r u            , (11) where 2 2 0 0( ) ( )r x x y y    . note that we choose to denote xi=1 as x and xi=2 as y. inserting eq. (11) into eq. (10) yields 0 0* 0 2 0 0 ( )( )1 (1 2 ) 2 4 (1 ) (1 2 ) i i j j k k ij ij k j j i i i j x x x x x x t n r r r x x x x n n r r                            (12) providing through eq. (9) the stress vector tj * of the fundamental solution on the boundary for a certain 0x . let us now insert the fundamental solution, that is eqs. (11) and (12), into somigliana’s identity, eq. (8). when we write out the result in detail and choose to denote ui=1 = u and ui=2 = v, we obtain for the first component of the deformations at a particular inner point 0 0 0( , )x x y due to the deformations and stresses at the surface 12 j. benad 2 1 0 2 0 0 0 0 1 2 2 2 0 0 0 1 22 2 2 0 0 14 ( ) ( )( )1 ( , ) (3 4 ) ln( ) 8 (1 ) ( ) ( ) ( )1 1 2 2 4 (1 ) 2( )( ) + ( s s t x x t x x y y u x y r t ds r r x x u x x u y y n n ds r r r x x y y v n r                                            0 2 0 1 0 2 02 ) ( ) 1 2 ( ) ( ) . s x x n y y v n y y n x x ds r              (13) for the second component, we obtain 2 2 0 1 0 0 0 0 2 2 2 2 0 0 0 1 22 2 2 0 0 14 ( ) ( )( )1 ( , ) (3 4 ) ln( ) 8 (1 ) ( ) ( ) ( )1 1 2 2 4 (1 ) 2( )( ) ( s s t y y t x x y y v x y r t ds r r y y v x x v y y n n ds r r r x x y y u n x r                                             0 2 0 2 0 1 02 ) ( ) 1 2 + ( ) ( ) . s x n y y u n x x n y y ds r             (14) exemplary results we now seek to perform an exemplary calculation of the boundary integrals (13) and (14) for an arbitrary shape with the fft making use of the use of technique [12]. in order to test the method, we use a simple analytical solution for the navier equation 0 1 0 2 1 , 2 2 u x c v y c             (15) with the stress distribution 0 , 0 , 0.x y xy      (16) the chosen geometry of the shape and eqs. (15) and (16) then set the analytical solutions for the boundary values for the deformations and the stress vector. both the chosen shape and the stress vector on its boundary are displayed in fig. 12a. the raw fft results for the deformations u and v, as obtained for the chosen numerical values of μ=1, 0.3  and σ0=2, are displayed with colored surfaces in fig. 12b. the corresponding analytical solution for the deformation on the boundary is displayed with the red line in fig. 12b. the numerical results clearly show the linear dependence we expect (see eq. (15)). very close to the boundary there are slight oscillations in the raw results. such small distortions were observed on other problems in previous studies and can generally be eliminated easily without an increase in computational complexity (see [12]). we can conclude from the small example that the fft approach commonly used in contact mechanics to solve a bem numerical methods for simulation of deformations and stresses in turbine blade fir-tree connections 13 problem for a half-space may be adopted in a similar way to solve problems on completely arbitrary shapes. however, we have presented only a first rough example and extensive research is required to develop the method further. a) b) fig. 12 a) an arbitrary two-dimensional shape discretized with boundary elements and fully enclosed with a two-dimensional grid for the application of the fft. the chosen stress vector for the test calculation is displayed with red arrows. the resolution of both the boundary and the fft gird is lower than in the adjacent images merely for purposes of a better visualization. b) raw results of the convolution for the deformations u and v on the boundary of the chosen shape as obtained for numerical values of μ=1, =0.3 and σ0=2. the corresponding analytical solution for the deformation on the boundary is displayed with a red line 5. conclusion in this paper, different numerical methods for simulating deformations and stresses in the turbine blade fir-tree connections were discussed and in part extended for the application on the complex geometry of the fir-tree. it was highlighted that both the method of dimensionality reduction (mdr) and the boundary element method (bem) are rapid simulation techniques for the half-space and are well suited for wear simulations. it was shown that both techniques can provide first insights to better understand the fir-tree. in the region of the contact area of the fir-tree lobes, the results for deformations and stresses obtained with the half-space models are similar to those obtained with finite 14 j. benad element models. generally, however, the complex fir-tree geometry exceeds the limitations of the half-space models. also mentioned are ways of extending the half-space model to the geometry of the fir-tree connection, such as a first-order perturbation method for the half-space approximation to take into account stress concentration effects at slightly undulating surfaces introduced by gao in 1991 [24]. in addition, a recent approach for completely arbitrary shapes (see [11] and [12]) inspired by the rapid fft based half-space bem, was discussed and applied on the navier equation. a small example was presented to indicate that the use of this technique which utilizes the fft to obtain the boundary integrals on completely arbitrary shapes may be a viable method and worthy of further investigation. this is due to the lower computational complexity of the method o(n 3 log n 1.5 ) than the inversion of the standard bem matrix o(n 4 ). other aspects which make the technique appealing are the potential of decreasing this complexity even further through adaptive grids, efficient parallel implementations of the fft, and the similarity of the method to the established and very successful fft-base bem for the half-space. acknowledgements: the author would like to thank v. l. popov, a. golowin, a. lacher, p. duó, and p. diercks for many valuable discussions on the topic. the author is particularly grateful for the kind permission of rolls-royce to use illustrative images throughout this work. references 1. bräunling, w., 2015, flugzeugtriebwerke, 3 ed, springer, berlin. 2. torenbeek, e., 1982, synthesis of subsonic airplane design, kluwer academic publishers, dordrecht. 3. raymer, d., 1999, aircraft design: a conceptual approach, 3 ed, american institute of aeronautics and astronautics, inc., reston. 4. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, 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pp. 2355-2376. 29. irgens, f., 2008, theory of elasticity, continuum mechanics, springer, berlin. 30. galin, l., 2008, plane elasticity theory, contact problems, g. gladwell, editor, springer, dordrecht. 31. kelly, p., 2013, linear elasticity, an introduction to solid mechanics (lecture notes), university of auckland. 32. betti, e., 1872, t , il nuovo cimento, 7(1), pp. 69-97. 33. gaul, l., fiedler, c., 2013, methode der randelemente in statik und dynamik, 2 ed, springer, berlin. 34. gross, d., hauger, w., wriggers, p., 2014, technische mechanik 4, 9 ed, springer, berlin. facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 7 22 https://doi.org/10.22190/fume201221012s © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a discrete element formalism for modelling wear particle formation in contact between sliding metals evgeny v. shilko1,2, aleksandr s. grigoriev1, alexey yu. smolin1,2 1institute of strength physics and materials science sb ras, russia 2national research tomsk state university, russia abstract. the paper describes an advanced discrete-element based mechanical model, which allows modelling contact interaction of ductile materials with taking into account fracture and surface adhesion by the cold welding mechanism. the model describes these competitive processes from a unified standpoint and uses plastic work of deformation as a criterion of both local fracture and chemical bonding of surfaces in contact spots. using this model, we carried out a preliminary study of the formation of wear particles and wedges during the friction of rough metal surfaces and the influence of the type of forming third body (interfacial) elements on the dynamics of the friction coefficient. the qualitative difference of friction dynamics in the areas of the contact zone characterized by different degrees of mechanical confinement is shown. key words: dry friction, cold welding, wear debris, mechanical confinement, computer modeling, discrete element method 1. introduction although adhesive wear during sliding friction of dry surfaces has been intensively studied over the past several decades, it is still a phenomenon extremely difficult to understand and predict [1-5]. the most popular approach to understanding and interpretation of key aspects of friction and wear is the concept of the third body proposed by m. godet [68]. this concept implies the formation and development of material fragments (interfacial elements) between contacting surfaces and the determining contribution of these elements to the dynamics of contact interaction including the value of the friction coefficient and wear rate. in friction pairs formed by ceramic materials (normally characterized by low surface energy), these interfacial elements are usually considered as wear particles [9, 10], while in metallic friction pairs the situation is much more complicated. due to the high values of received december 21, 2020 / accepted february 01, 2021 corresponding author: evgeny v. shilko institute of strength physics and materials science, pr. akademicheskii 2/4, tomsk 634055, russia e-mail: shilko@ispms.tsc.ru 8 e.v. shilko, a.s. grigoriev, a.yu. smolin surface energy and ductility of metallic materials, an important role in the formation and evolution of the third body is played by the adhesion of interfacial elements to contact surfaces and to each other. adhesive strength can be comparable to the cohesive strength of materials themselves. therefore, interfacial elements, being the main load carriers, undergo large plastic deformations and are hardened stronger than the surface layers, and so become strong and stable formations. in the process of friction, these formations can increase in size by accretion of smaller interfacial elements and adherence of material from the surfaces, as well as occasionally crush [11-13]. accordingly, the characteristic dimensions of the main load-carrying interfacial elements may exceed the height of surface asperities (for both initial and current surfaces) by orders of magnitude. therefore, they completely separate the interacting surfaces and form the actual relief of the contact zone (defined as a set of spots of local contact of surfaces with interfacial elements) [5]. the roughness characteristics of the surfaces themselves at the stage of steady-state friction have only an indirect effect on the real contact area and its change, and, consequently, on the dynamics of friction. interfacial elements in metallic friction pairs can have not only significantly different sizes but also geometries. the latter is determined by differences in the conditions and mechanisms of the formation of third body fragments. common names for the largest interfacial elements are prow, wedge, flake (these are generally transfer fragments, but not wear particles), and wear particles/debris themselves [14, 15]. the conditions and mechanisms for the formation of these interfacial elements (and their functional roles) are widely discussed in the literature, but there is still no unambiguous understanding of many key issues. in particular, a recent discussion by j.a. greenwood [5] raises again the question of how wear debris forms in metallic friction pairs and how they relate to wedges/prows. an effective tool for a deeper insight into the physics (in particular, mechanics) of the formation and evolution of prows, wedges and wear particles is computer modelling [16, 17]. macroscopic regularities of friction and wear are traditionally studied using fem or bem with implemented models of contact interaction and wear [18-22]. such studies, however, do not come close to understanding the mechanisms of the formation of wear particles. the most attractive tool for performing such studies is the molecular dynamics method. in recent years, md modelling helped to obtain important information that shed light on the conditions and mechanisms of the formation of nanoscale transfer fragments, which can subsequently contribute to the formation and accretion of wedges, prows, or wear particles [23-28]. however, due to the extreme smallness of the studied spatial and temporal scales (the contact of single nanoasperities is usually considered) and the use of the ideal crystal structure of the initial surface layers, these studies do not allow making unambiguous conclusions about the development of the studied processes at larger spatial scales and times. in recent years, various authors have proposed mesoscale numerical models that allow describing the processes of ploughing, lump/wedge formation, debris particle growth [12, 29-31]. however, these models consider single asperities, while the evolution of the contact zone (including separation of the surfaces and wear) is determined not simply by the independent evolution of single interfacial elements, but also by the mutual influence and competition between interfacial elements even located far from each other in the contact zone. particle-based numerical method of discrete elements (dem) is known as a promising and potentially effective tool for the numerical study of complex processes of formation of transfer fragments and small wear particles, their subsequent coalescence into large wear particles or wedges/prows, as well as competitive growth and occasional fracture of a discrete element formalism for modelling wear particle formation in contact between sliding... 9 such interfacial elements. it was originally developed for the simulation of granular materials, but is now widely used to simulate complex aspects of solids behavior including multiple fractures, contact interaction and friction, mass transfer, and mixing effects. in particular, this method is used to study the mesoscopic mechanisms of the formation of accretionary wedges at convergent plate boundaries in the lithosphere (an example of the contact problem on the geological scale) [32, 33]. the authors of this paper develop the original formulation of dem, namely, the method of homogeneously deformable discrete elements [34]. this formulation makes it possible to implement models of the mechanical behavior of materials with complex rheological properties (including elastic-plastic) [35], as well as to take into account surface adhesion. using these models, the basic modes of deformation and fracture of initial mesoscopic asperities were systematically analyzed for the first time and the influence of surface adhesion and macroscopic material properties was generalized [36, 37]. despite large capabilities, the formalism of the method needs further development to study the evolution of transfer fragments in metallic friction pairs, including the formation of large interfacial elements and the competition of their evolutions. in particular, the model of contact interaction of discrete elements should take into account the key features of cold welding between rough (and potentially contaminated) surfaces [38-40]. the model of element-element bond-breaking should be based on a “universal” (dissipative) criterion, which can be applied both to elastic-plastic material in the initial state and to newly formed (cold-welded) fragments. this paper is devoted to the description of the advanced model of discrete element interaction, which takes into account the features described above. we show that the model allows adequate describing the formation and evolution of large interfacial elements as well as the competition between interfacial elements in the contact zone. 2. discrete element model of contact interaction 2.1. the keystones of the model within the framework of the formalism of dem, the simulated material is represented by an ensemble of interacting elements (particles). a discrete element models a certain finite volume (region or fragment) of a material. this volume is bounded by a welldefined surface and contains a sufficient number of atoms or molecules for an integral description in terms of thermodynamic parameters. mechanical interaction of these finite volumes (discrete elements) leads not only to their translational motion but also to the rotation, which is determined by the moments of interaction forces [41, 42]. the authors use the most widely used implementation of the discrete element method based on the approximation of equivalent disks (2d models) or balls (3d models) [43, 44]. within the framework of this approximation, discrete elements modeling areas of the consolidated material are assumed to have an equiaxial polygonal shape and interact through flat faces. the motion of such elements is described by the classical system of newton-euler equations of motion for inscribed discs or balls having the same values of mass and moment of inertia as describing polyhedra. in this case, the central force of interaction of the two elements leads to their translational motion along the line connecting the centers of the approximating disks or balls, and the tangential force causes motion in the transverse plane and rotation. a pair of interacting elements can be in a 10 e.v. shilko, a.s. grigoriev, a.yu. smolin linked or unlinked state. the first case corresponds to the presence of a chemical bond between the contact surfaces of the elements, while the second case simulates the classical adhesionless contact interaction of bodies. unlike the traditionally used models of rigid discrete elements (they apply only to brittle materials and environments), the authors consider a discrete element as deformable [45, 46]. in this work, the approximation of a uniform distribution of deformations and stresses in the volume of an element is used. this approximation is similar to the widely used approximation of linear interpolation of displacements (first-order polynomial) in the finite element method. within this approximation, the stress-strain state of the discrete element i is characterized by the average stress i   and strain i   tensors (, = x,y,z; xyz is the laboratory coordinate system). the components of the averaged stress tensor are calculated in terms of the forces of interaction of an element (approximated by an equivalent disk) with its neighbors: 0 0 1 [( ) ( ) ]( ) in i i ij ij ij ij ij ij ji r s n t n v     =  =    , (1) where ri is the radius of the equivalent disk/ball, 0 ij s is the initial value of the area of contact of the element i with the neighbor j, vi 0 is the initial volume of the element, ni is the number of neighbors of the element i, ij n  and ij t  are the unit vectors oriented along the normal and in the plane of contact of elements i and j, ij np ijij sf= and ij tp ijij sf= are specific values of central ( np ij f ) and tangential ( tp ij f ) forces of element-element interaction, sij is the current value of the area of contact. the components of the average strain tensor i   are calculated by analogy. the formalism of deformable discrete elements allows modelling the mechanical behavior of materials with complex rheological properties. the specific form of relations for the forces of central and tangential interaction of elements is determined by the form of the constitutive law (elasticity, plasticity, etc.) for the simulated material and usually duplicates it. this paper deals with elastic-plastic metallic materials. the elastic response of element i to mechanical action from element j is written in a form similar to the generalized hooke's law. in hypoelastic form (in increments), the expressions for central and tangential (shear) forces are as follows:      =          −+= ijiij i mean i i ijiij g k g g 2 3 2 12 . (2) here, gi and ki are the shear modulus and bulk modulus of the material of element i, symbol  denotes the increment of the corresponding parameter per integration step, ij and ij are central (normal) strain and shear angle of element i in pair i-j (contributions of the element to the total values of normal and shear strains of the pair), i mean  is mean stress in the element i (the first invariant of the tensor i   ). the mechanical behavior of an element beyond the elastic limit (stress relaxation and accumulation of inelastic strains) is described in the framework of the associated plastic flow law using the von mises yield criterion. within the approximation of a homogeneously deformable element, the second invariant of the deviator of the average stress tensor is used as a yield criterion: a discrete element formalism for modelling wear particle formation in contact between sliding... 11 ( )i in eq y eq =   , (3) where y( in eq ) is the unified hardening curve of the modelled material in terms of true stress, in eq  is the equivalent plastic strain in the element i. the plasticity of the element is numerically implemented using the radial return algorithm, developed by m.l. wilkins. 2.2. the model of cold pressure welding of discrete elements contact interaction and friction of ductile (in particular metallic) materials with high surface energy is typically accompanied by local effects of cold welding between the surfaces [47-52]. it is known that chemical bonding of clean and atomically smooth contacting surfaces of metals can be achieved even in the absence of significant contact pressure. however, complete welding of rough surfaces requires the application of a contact pressure comparable to or exceeding the material yield stress. in the general case of surfaces overlaid by cover layers and/or contaminants, the required contact pressure can be multiples of the yield stress [38-40]. at the same time, numerous experiments indicate that additional plastic shear deformation is capable of providing chemical bonding of surfaces of even dissimilar metallic materials under contact loads comparable to the yield stress (this is observed, in particular, when rolling bimetallic plates or twisting two pressed disks). efficient bonding of contacting surfaces under the condition “compression+shear” is ensured, among other things, due to interpenetration and mixing of the surface layers of counterbodies, as well as dispersion of cover layer and contaminant. thus, the contact pressure can be used as an appropriate criterion for the beginning/ development of the process of cold welding between the surfaces. its value should exceed a certain threshold, sufficient for localization of plastic deformation on the rough surfaces and complete contact of these surfaces, as well as for dispersion of possible cover layers and contaminants. in this case, the condition for achieving complete adhesion of surfaces is the achievement of a certain threshold (minimum) value of accumulated plastic strain in the surface layers. it is clear that the threshold values of the two mentioned parameters should correlate with the material yield stress and the plastic strain before material failure. however, such a relationship is not one-valued due to a large number of other factors (roughness, contaminants, dissimilarity of contacting materials, etc.). therefore, the values of yield stress and plastic strain to fracture (or plastic work of deformation) under tension can be considered as some reference values for characterizing the condition and criterion of cold welding between surfaces [40]. the above considerations, combined with the principles of a well-known phenomenological model by n. bay [38-40], form the basis for a phenomenological numerical model of cold welding of contacting discrete elements, which simulate parts of interacting surfaces or third body fragments. the main theses of the model are as follows: 1) cold welding of a pair of contacting (unlinked) discrete elements i and j proceeds at the current integration step only if the condition (–ij)>p0 weld is satisfied. here, ij is the current specific value of the normal interaction force (it is negative when the pair is compressed), and p0 weld is the contact pressure threshold (input parameter of the model). 2) cold welding between the surfaces of the contacting elements i and j is a progressive process and develops as the plastic straining of these elements. accordingly, the area of the contact of the elements (sij) generally consists of two parts: the area of the 12 e.v. shilko, a.s. grigoriev, a.yu. smolin chemically bonded (welded) part ij linkij ij link kss = and the area of the unbonded (contact) part (1 )ij ij link ij links s k= − . here, 10  ij link k is a fraction of the welded part of the contact surface. we assume linear dependence of the rate of increase in the fraction of chemically bonded contact surface on the rate of plastic straining of the pair: weld ij plastij link w w k  = , (4) where  is the increment in the value of the corresponding variable per time step of numerical integration of the equations of motion, ij plast w is the specific values of plastic work of deformation in a pair (elastic strain energy density dissipated as a result of plastic straining), wweld is the specific value of the plastic work of deformation sufficient for complete cold welding of contacting elements. 3) in the general case, the value of wweld is a decreasing function of the contact pressure pn (pn>p0 weld). the simplest form of dependence is linear: 0 0 1 0 1 0 ( ) ( ) weld weld weld weld n weld n weld weld p p w p w w w p p − = + − − . (5) here, w0 weld=wweld(p0 weld) is the plastic work sufficient for complete welding of elements at the minimum value of contact pressure at which this process is stable, and w1 weld=wweld(p1 weld) is the plastic work sufficient for complete welding at some p1 weld > p0 weld. in the numerical implementation of the model, the value of wweld in eq. (4) is calculated according to eq. (5) at each integration step with the use of |ij| as the contact pressure pn. 4) the increment in the specific work of plastic deformation in a pair i-j ( ij plast w ) is calculated taking into account the fact that the elements in a pair are deformed differently. in other words, the increment of plastic work of deformation of element i in pair i-j ( ( )i j plastw ) is not equal to the increment of plastic work of deformation of element j in this pair ( ( )j i plastw ). therefore, ( ) ( )ij i j j i plast plast plastw w w =  + (plastic straining of both elements in a pair contribute to a tight convergence of the contacting surfaces). the increment in the specific work of plastic deformation of element i in pair i-j ( ( )i j plastw ) is calculated through the increment in its equivalent strain in this pair using a unified hardening curve ( )in y eq  of the material of element i. the increment of the work of plastic deformation of the element j ( ( )j i plastw ) is determined in a similar way. detailed information on the calculation of the element’s equivalent strain in a pair (in the general case, it differs from the equivalent strain calculated using element’s average strain tensor) is given in [34, 46]. the input parameters of the model can be obtained from experimental data or other theoretical constructions. these parameters are (w0 weld, p0 weld) and (w1 weld, p1 weld) for all the pairs of materials and unified hardening curves ( )in y eq  for these materials. 2.3. the model of bond breaking in the pair of linked discrete elements within the framework of the discrete element method, a local fracture is modeled by breaking a chemical bond between elements, that is, by the transition of a pair of elements from a linked to an unlinked state. this transition occurs out when the specified fracture criterion is fulfilled in a pair. the traditional failure criteria in the mechanics of a a discrete element formalism for modelling wear particle formation in contact between sliding... 13 deformable solid are “stress-based” ones. they are formulated in terms of stress tensor invariants (for example, the mises criterion for metals and polymers). to implement such criteria within the framework of the method of deformable discrete elements, we previously developed an approach based on the determination of the local stress tensor on the contact surface of a pair of elements [34, 46, 53]. the invariants of this stress tensor are used to calculate the given “stress-based” criterion of local fracture. the bond between the elements breaks when the criterion exceeds the threshold value (bond strength of the pair). the classical “stress-based” (and similar “strain-based”) fracture criteria make it possible to adequately simulate a single fracture of the material (upon deformation from the initial unloaded state). at the same time, friction of ductile materials with high surface energy can be accompanied by multiple alternating acts of local fracture and cold welding of fragments in contact spots. the use of “stress-based” or “strain-based” fracture criteria (for example, equivalent stress or strain in the pair) for newly formed linked pairs of elements can lead to unphysical premature breakage of chemical bonds in these pairs. the reason for this is the fact that the formation of new linked pairs by the mechanism of cold welding typically occurs under high local stresses. these stresses can be close to or even exceed those required to achieve the threshold value of the fracture criterion. to solve the described fundamental problem, the authors propose to use the dissipative criterion of local fracture (breaking of a chemical bond between discrete elements) instead of “stress-based” or “strain-based”. the dissipative criterion characterizes the specific value of elastic strain energy dissipated during plastic deformation of the material. note that in the case of conventional mechanical tests on samples of ductile material, the dissipative criterion provides the same fracture patterns as the classical mises criterion or an analogous criterion of equivalent strain. in the developed discrete-element model, we use specific works of plastic deformation of elements i and j in the pair i-j ( ( )i j plastw and ( )j i plastw ). the method for calculating these parameters is described in the previous section when describing the numerical implementation of the cold welding model. the formulation of the dissipative fracture criterion is as follows: ( ) ( )max{ , }i j j i plast plast fractw w w= , (6) where wfract is the specified values (model parameter). the values of wfract for the pair of linked elements of the same material is equal to the area under a unified loading curve ( )in y eq  in the range from 0=in eq to =in eq s (s is inelastic equivalent strain corresponding to ultimate stress (strength) and the onset of fracture). the calculation of the parameters ( )i j plastw and ( )j i plastw is carried out in an incremental fashion (the increments of these parameters are calculated at each integration step). note that the initial values of ( )i j plastw and ( )j i plastw are assumed to be zero for both the initially linked pairs of elements and the newly formed (cold-welded) linked pairs. eq. (6) is a condition for the initiation (beginning) of breaking a bond in a pair of elements i and j. the process of local fracture (that is, the formation of two surfaces) is considered as distributed over time. the model of stable crack growth [34] is used as a model of such a process. within the framework of this model, a kinetic equation is specified for the parameter ij link k (the fraction of the chemically bonded part of the contact surface): ( , ,...)ij ij ij link fract linkk f k=  , where f<0 is a specified negative function and ij fract  is a 14 e.v. shilko, a.s. grigoriev, a.yu. smolin value of relative displacement of elements after the start of bond breaking. here, we use the linear function: max,fract ij fract ij link fk −==  , (7) where fract,max is the limiting value of the parameter corresponding to a complete bond breakage between the elements ( 0=ij link k ). the parameter ij fract  increment per integration step ( ij fract  ) is calculated as a combination of the tensile and shear strain increments in the pair i-j. a rough estimate of fract,max is the length of the softening stage (from maximum resistance to zero) in the diagram of uniaxial tension of a sample of ductile material (true stress should be analyzed). 3. results of the illustrative simulation and discussion the adequacy of the developed discrete element model and its advantages in the study of the formation and evolution of large interfacial elements (wear particles and prows/wedges) are illustrated by the example of modeling of the tangential contact of two rough surfaces of metallic materials with the same mechanical properties (fig. 1). the simulated fragment of the contact zone includes more than 40 asperities. the modeling was carried out in a two-dimensional formulation using the approximation of plane stress or plane strain state. the linear dimensions of the model were 1000 µm along the x-axis and 220 µm along the y-axis (the dimension in z-axis is equal to the discrete element size 1 μm). both surfaces have the same roughness parameters: smi23 µm (average step of profile irregularities), rmax5 µm (maximum profile height). fig. 1 structure and loading scheme of the model fragment of the zone of contact interaction of bodies with micro rough surfaces. we considered the model material with mechanical properties close to those of aluminum bronze (copper content 7%) [54, 55]. the following mechanical characteristics of the material were used: density =7800 kg/m3, young's modulus e=118 gpa, poisson's ratio =0.295, yield stress y=360 mpa, tensile strength uts=560 mpa. the material is elastic-plastic with linear hardening (strain hardening modulus h=900 mpa). the value of the parameter wfract of the dissipative fracture criterion was 141 mj/m3 (plastic work of deformation from y to uts). the value of the parameter max,fract  (the length of the softening stage) was assumed to be 5%. the following values of the parameters of cold welding model were used: p0 weld =y = 360 mpa, wweld(pn) =const= a discrete element formalism for modelling wear particle formation in contact between sliding... 15 wfract=141 mj/m3. these values of the parameters of the cold welding model can be considered as an initial ("sighting") approximation for studying the general patterns of formation of interfacial elements (third body fragments) in the contact zone. we analyzed the interaction of the counterbodies during their relative tangential displacement at a constant speed vslide and at a given constant pressing force fn. the force is applied to the upper surface of the upper block, its specific value is 36 mpa, or 10% of the value of the yield stress of the material. the base of the system (the lower surface of the lower block) is fixed in the y direction and moves along the x-axis at a constant speed vslide=5 m/s. periodic boundary conditions are applied along the x-axis. the simulation results have shown that the type of interfacial fragments formed (and the dynamics of the change in the friction coefficient) can qualitatively change when the mechanical confinement change in the direction normal to the sliding direction (normal to the considered plane in fig. 1). in the 2d model, two extreme cases of transverse confinement were considered: no confinement (plane stress approximation) and “rigid” confinement (plane strain approximation, out of plane strains are restricted). in the absence of transverse mechanical confinement, large and approximately equiaxial particles are formed in the contact zone. these particles are weakly bonded to the surfaces. the formation of such particles includes three main stages. at the initial stage of friction (stage i), the surfaces are “matching”. namely, there is cutting or abrasion of the initial surface asperities and the formation of the original (small) fragments of the third body (torn off or cut off fragments of asperities). note that even these small fragments totally separate the surfaces (fig. 2a). some of the fragments undergo strong compression by blocks. as a result, they are strongly hardened (in comparison with the adjacent surface layers of blocks) and adhere to the surfaces by the mechanism of cold welding. in the process of relative tangential displacement of surfaces, such fragments move by rolling. in the course of turning, they tear off layers of material from the adjacent areas of the surfaces and gradually increase in size (figs. 2b and 2c). so, already at the initial stage, the contact spots are determined not by the current roughness of the surfaces, but by the most intensively growing fragments of the third body. fig. 2 an enlarged fragment of the central part of the contact zone (fig. 1) at different values of relative tangential displacement of the blocks: (a) 35 µm; (b) 80 µm. 16 e.v. shilko, a.s. grigoriev, a.yu. smolin the largest (and the most deformed and hardened) fragments of the third body grow faster than the others, since in the process of rolling they sequentially adhere to less hardened areas of both surfaces and tear off layers of material from them. the shape of these particles gradually becomes close to round. as they grow, these particles take on an increasing part of the total contact load and therefore are continuously compacted and hardened. at a certain stage (stage ii), the process of their growth becomes strongly competitive. at this stage, the number of “load carrying” particles gradually decreases (fig. 3). the end of stage ii is associated with a decrease in the number of “load carrying” particles (fig. 4) to a certain minimum level. the growth of the rest (non-bearing) particles stops, and later they coalesce with the largest ones. fig. 3 the structure of the contact zone (stage ii) at different values of relative tangential displacement of blocks: (a) 155 μm; (b) 240 μm. fig. 4 individual “load carrying” particles from the right part of fig. 3b. at stage iii, friction passes into a steady state. here, the “load carrying” particles continue to grow, also due to the attachment of smaller (non-carrying) interfacial fragments (fig. 5). note that, the attached interfacial fragments change the shape of “load carrying” particles to significantly nonequaxial and irregular. rotation of such an irregular “lump” a discrete element formalism for modelling wear particle formation in contact between sliding... 17 constrained by the surfaces is difficult. so, the “lump” stops rotation and acts as a wedge. such wedges become the most loaded interfacial fragments and are destroyed (or lose large fragments). episodically repeating phenomena of wedging and subsequent destruction of the largest and nonequiaxial particles control the maximum characteristic size of these fragments of the third body. fig. 5 the structure of the contact zone (stage iii) at different values of relative tangential displacement of blocks: (a) 425 μm; (b) 650 μm; (c) 870 μm. so, in the absence of transverse mechanical confinement of the contact zone, the sliding friction is actually reduced to adhesive rolling friction caused by the rotational motion of load-carrying particles and accompanied by the accretion of the material layers from surfaces on these particles. the running-in phase (stages i and ii) and transition to the steady-state friction (stage iii) are well manifested in the dependence of the coefficient of friction (cof) on the sliding distance (fig. 6). at the stage of “nucleation” of the initial transfer fragments (stage i), the friction coefficient increases and can exceed 1. at the stage of competitive growth of interfacial fragments (stage ii), the friction coefficient decreases to a level corresponding to the stationary regime. at the stage of transition to the steady-state friction regime (stage iii), the friction coefficient varies around a certain (approximately constant) average value. the most significant deviations are associated with the formation and subsequent destruction of large “wedges” (lumps of fragments). note that cof in fig. 6 characterizes friction of the mesoscale area of the contact zone. it is clear that the amplitude of variations in the macroscopic cof should be many times smaller. 18 e.v. shilko, a.s. grigoriev, a.yu. smolin in the case of “rigid” mechanical confinement of the contact zone (imitated by the approximation of a plane strain state in a 2d formulation), the development of the frictional process differs from that described above not only quantitatively, but qualitatively. after the initial stage of surface matching, the peculiar surface “incrustations” of a lumpy or wedge shape are formed and gradually grow. each of these wedges strongly adheres to one of the surfaces and moves with it. similar to a weakly confined contact zone, the growth of wedges during friction is a competitive process. wedges increase in size due to the attachment of smaller fragments of the “third body” (transfer fragments), as well as due to the shearing of layers of material from the opposite surface (fig. 7). reaching some critical size, these wedges can crush due to a high concentration of shear stresses. therefore, similar to the case of a mechanically unconfined system, the maximum size of the wedges does not exceed a certain characteristic scale. fig. 6 dependence of the instantaneous cof on the relative tangential displacement of the blocks. the red line is the result of averaging the instantaneous values in a window of 100 points. key stages i-iii are designated in roman numerals (stages are delimited by green vertical lines). the horizontal dashed line is the average value of the friction coefficient at stage iii. plane stress state approximation. fig. 7 the structure of the contact zone at different values of relative tangential displacement of blocks: (a) 190 μm; (b) 300 μm. the frame in (b) highlights the largest and the most heavily loaded wedge. plane strain approximation. a discrete element formalism for modelling wear particle formation in contact between sliding... 19 so, under the condition of “rigid” mechanical confinement of the surface layers, the sliding friction proceeds in a classical manner considered in a recent review by j.a. greenwood [5]. note that the friction coefficient in a mechanically confined friction pair reaches 1.2-1.4 at stage i and then decreases only insignificantly (the average value at the stage of steady-state friction is 1.0-1.1). a high value of the local coefficient of friction in mechanically confined areas of the contact zone is defined by the formation of strongly hardened wedges (instead of rolling particles), which act as an abrasive. the simulation results show that the friction can proceed in a significantly different manner in the contact zone regions characterized by different degrees of mechanical confinement of the surface layers. particles of approximately equiaxial (and mostly round) shape are predominantly formed in low confined regions. these particles act as a kind of “rollers” and provide a relatively low value of resistance to tangential movement (low value of local cof). these interfacial fragments are apparently the wear particles that will be carried away from the contact zone due to the relatively weak adhesion to the surfaces. the formation of such particles seems to be the predominant wear mechanism in the regions of the contact zone characterized by low mechanical confinement. wedges, the well-known interfacial elements in tribology, are predominantly formed in highly confined regions of the contact zone. these interfacial fragments have a large contact area and strong adhesion to one of the surfaces, move with it and, generally speaking, are not wear particles. the largest wedges are strongly hardened and therefore can play a role similar to that of an abrasive and produce high characteristic values of the local coefficient of friction. so, wear in highly confined regions of the contact zone is mainly due to the removal of (1) small fragments formed during crushing of the most loaded wedges and (2) small surface irregularities cut by the abrasive wedges. the relative contributions and balance of the described processes in a real threedimensional contact zone (and, consequently, the value of the macroscopic coefficient of friction and the wear rate) should depend on the ratio of squares of low and highly confined regions. 4. conclusion the proposed numerical models of the formation and breakage of chemical bonds in metallic materials are based on the use of a unified energy-based (dissipative) parameter, namely, the plastic work of deformation. the unification ensures the consistency of these models and the unity of the phenomenological description of opposing (competing) processes of fracture and cold welding under the contact interaction of ductile materials. the implementation of these models within the method of homogeneously deformable discrete elements made it possible not only to adequately model the formation and evolution of qualitatively different interfacial (third body) fragments but also to identify conditions favorable for the formation of specific types of interfacial fragments. in particular, we have shown that either equiaxial (rounded) particles weakly adhered to surfaces, or wedges/prows are predominantly formed depending on the degree of mechanical confinement of the contact region. accordingly, the degree of mechanical confinement should largely determine the value of the friction coefficient, as well as the wear rate and the size distribution of wear particles. 20 e.v. shilko, a.s. grigoriev, a.yu. smolin the results of the preliminary study are relevant for a deeper understanding of the conditions for the implementation of various mechanisms of wear during sliding friction, as well as approaches to the control of these mechanisms. note that the shown regularities of friction and wear of ductile materials qualitatively differ from those observed in experiments (and modeling) on the friction of brittle (ceramic) materials. the low surface energy of brittle materials provides the formation of the third body fragments (particles) by means of fracture of asperities or crushing of large third body fragments. this debris not only separates the interacting surfaces and acts as rollers and/or abrasive but fills in the surface irregularities and redistributes (equalizes) the load between them. accordingly, surface profile matters for the tangential contact of brittle materials. this ensures the applicability of analytical and semianalytical models of contact mechanics. as evidenced by the above-shown calculation results, such models are poorly applicable to describe the tangential contact of ductile materials, since the relief of interacting surfaces is of minor importance as it affects only the dynamics of the growth and breakage of “rollers” and wedges. this problem (including the applicability of contact 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behavior and deformation kinetics of gradient structured cu-al alloys with varying stacking fault energy, journal of alloys and compounds, 687, pp. 152-160. 55. huang, c.x., hu, w., yang, g., zhang, z. f., wu, s.d., wang, q.y., gottstein, g., 2012, the effect of stacking fault energy on equilibrium grain size and tensile properties of nanostructured copper and copper–aluminum alloys processed by equal channel angular pressing, materials science and engineering a, 556, pp. 638-647. https://www.sciencedirect.com/science/article/pii/s1365160910002169 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 397 411 https://doi.org/10.22190/fume170505022r original scientific paper casting improvement based on metaheuristic optimization and numerical simulation udc 621.7 radomir radiša 1 , nedeljko dučić 2 , srećko manasijević 1 , nemanja marković 3,4 , žarko ćojbašić 3 1 lola institute belgrade, serbia 2 faculty of technical sciences ĉaĉak, university of kragujevac, serbia 3 faculty of mechanical engineering, university of niš, serbia 4 philip morris operations serbia abstract. this paper presents the use of metaheuristic optimization techniques to support the improvement of casting process. genetic algorithm (ga), ant colony optimization (aco), simulated annealing (sa) and particle swarm optimization (pso) have been considered as optimization tools to define the geometry of the casting part’s feeder. the proposed methodology has been demonstrated in the design of the feeder for casting pelton turbine bucket. the results of the optimization are dimensional characteristics of the feeder, and the best result from all the implemented optimization processes has been adopted. numerical simulation has been used to verify the validity of the presented design methodology and the feeding system optimization in the casting system of the pelton turbine bucket. key words: metaheuristic optimization, sand casting, feeders, numerical simulation 1. introduction the task of the optimization is to find the variables in which the target (criterion) function has extreme (minimum or maximum) value, with the limits, which define the space of potential solutions. optimization is an integral part of natural processes. from the phenomena that take place at the level of micro-scale (e.g. crystallization, in which the molecules occupy the minimum energy position), to the evolutionary process leading to, through the principle of survival of the fittest, the individuals that are better adapted to the received may 05, 2017 / accepted august 04, 2017 corresponding author: nedeljko duĉić faculty of technical sciences ĉaĉak, university of kragujevac, svetog save 65, 32000 ĉaĉak e-mail: nedeljko.ducic@ftn.kg.ac.rs 398 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić conditions in the "environment" – all this serves as an inspiration for several metaheuristic optimization techniques. the implemented metaheuristic optimization methods are based on the idea that, by imitating nature, what should be looked for is the optimum complex function of several variables that represent the mathematical abstraction of a complex engineering problem. the idea developed in this paper is to apply metaheuristic optimization and advanced simulation for improvement of the casting process, which is tested on the problem of design and optimization of the feeder for sand casting of the pelton turbine bucket. when it comes to the implementation of metaheuristic optimization methods in the casting process, the spectrum of published research studies is very wide because of a large number of casting technologies and their respective complexity. gravela et al. [1] presented ant colony optimization (aco) for the solution of an industrial scheduling problem in an aluminum casting center. santos et al. [2] presented the development of a computational algorithm (software) applied to maximizing the quality of steel billets produced by continuous casting. a mathematical model of solidification works integrated with a genetic search algorithm and a knowledge base of operational parameters. surekha et al. [3] presented multi-objective optimization of green sand mould system using evolutionary algorithms, such as genetic algorithm (ga) and particle swarm optimization (pso). slavković et al. [4] presented application of learning machine methods in prediction and optimization of the wear rate of wear resistant casting parts. duĉić et al. [5] presented optimization of chemical composition in the manufacturing process of flotation balls based on intelligent soft sensing. duĉić et al. [6] presented optimization of the gating system for sand casting using genetic algorithm. the implementation of modern cad/cam software systems is frequent in the research projects of the casting process, as well as the combination of modern cad/cam software systems and methods of metaheuristic optimization. dabade et al. [7] used magmasoft software for simulating the casting process and analyzing its various defects, by detecting the cause through simulation dimensionally and positionally different embodiments of the casting and the feeding systems. jie et al. [8] used the pro cast software package to improve the casting process of aluminum alloy, and have concluded that the increase of molten metal temperature and of casting speed solves the problem of porosity. nimbulkara and dalu [9] presented the design of gating and feeding system with the objective of optimizing them by using the auto-cast x1 casting simulation software as well as of preparing the sand mold and casting the part, of comparing the simulated result and experimental results, of reducing the rejection rate and thus enabling the company to again start the production. unlike these studies, in this paper four metaheuristic optimization techniques have been considered, while the obtained results have been tested and verified with the numerical simulation of casting processes using the advanced magma 5 software. 2. optimization aspects of feeding the casting part optimal design of some system is a goal in more or less every engineering discipline. the imperative of optimal design of the feeding system of the cast is to reduce material consumption so that the feeder can successfully compensate shrinkage of the material in the mold cavity. unlike the filling of the mold cavity, feeding is a long, slow process that casting improvement based on metaheuristic optimization and numerical simulation 399 is required during the contraction of the liquid that takes place on freezing. this process takes minutes or hours depending on the size of the casting. during freezing are present three different phases of the contraction volume, i.e. shrinkage: liquid contraction, solidification contraction and solid contraction (fig. 1). fig. 1 schematic illustration of three shrinkage regimes: in the liquid; during freezing; and in the solid [10] volume contraction is manifested in side effects: internal cavities, surface deformation, surface craters. one of the indicators of casting process quality is continuity of molten metal flow in the area of solidification that is fed and compensating deficit caused by solidification. failure in this process will result in deficiencies of the solidification process that is called porosity. in fig. 2 just a generalized classification of porosity is given, as a result of metal shrinkage. open defects, as a result of metal shrinkage, are result of cooling while metal is in liquid state and during solidification. these defects are large-volumed, so they are called macro-shrinkage. closed shrinkage defects manifest themselves as an internal macroporosity and internal microporosity. open defects are exclusively related to the process of metal shrinkage, while closed defects, in addition to process of metal shrinkage, are directly related to nucleation and growth of grains, as the characteristics of crystallization. fig. 2 open and closed defects as a result of metal shrinkage 400 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić the elimination of those side effects can be realized by the proper design of feeders which, after cooling, should be removed from the casting part. by exploring numerous literature references, as a general conclusion, the following sequence of activities in the cast feeding system design [11] is imposed: (a) representation of the casting as a collection of simple, plate-like shapes  locate hot spots, and place a riser on each one  for each plate-like shape, determine edges with and without end effect (b) determination of feeding zones, feeding paths and feeding dimensions. (c) determination of feeding distances (d) determination of riser sizes within this sequence are incorporated rules on valid feeding of casting part, which cambell systematically exposes in his book [10]. numerous literature sources are mainly based on two rules of feeding the cast: (a) the feeder must solidify, at the earliest, at the same time as cast or, of course, later. this rule is called chvorinov's heat-transfer criterion. (b) the feeder must contain sufficient molten metal to compensate to the casting part metal shrinkage, in the extent for which the aforementioned feeder is provided. metaheuristic optimization of a feeder is a certain synthesis of exposed rules and activities in the feeding system design, embedded in standard optimization subjects, such as the fitness function, and appropriate limits. as optimization techniques were used nature-inspired metaheuristic algorithms: genetic algorithm (ga), ant colony optimization (aco), simulated annealing (sa) and particle swarm optimization (pso). 3. metaheuristic optimization techniques two major components of any metaheuristic algorithms are: intensification and diversification, or exploitation and exploration [12]. diversification means to generate diverse solutions so as to explore the search space on a global scale, while intensification means to focus the search in a local region knowing that a current good solution is found in this region. a good balance between intensification and diversification should be found during the selection of the best solutions to improve the rate of algorithm convergence. the selection of the best ensures that solutions will converge to the optimum, while diversification via randomization allows the search to escape from local optima and, at the same time, increases the diversity of solutions. a good combination of these two major components will usually ensure that global optimality is achievable [13]. 3.1. genetic algorithm (ga) genetic algorithms (ga) [13] are probably the most popular and widely used metaheuristic optimization technique. they represent abstraction model of biological natural selection, based on darwin's theory of evolution. application of genetic algorithms assumes the use of concepts from nature such as crossover, mutation, recombination and selection in adaptive and artificial systems. such genetic operators are important elements of the each problemsolving strategy by use of genetic algorithms. genetic algorithms can be described by the following generic representation: casting improvement based on metaheuristic optimization and numerical simulation 401 data: population size n, crossover rate ηc and mutation rate ηm. initialization: create initial population p={pi}, i=1…n, and initialize the best solution best ←void. while {stoppingcriterion not met} evaluate p and update the best solution best. initialize offspring population:r←void. create offsprings: for k=1 to n / 2 do selection stage: select parents q1 and q2 from p, based on fitness. crossover stage: use crossover rate ηc and parents (q1;q2) to create offsprings (s1;s2). mutation stage: use mutation rate ηm to apply stochastic changes to s1 and s2 and create mutated offsprings t1 and t2. add t1 and t2 to offspring population: r ← r u {t1 and t2}. replace current population p with offspring population r: p ←r. elitism: replace the poorest solution in p with the best solution in best. 3.2. ant colony optimization (aco) social ants foraging behavior was the role model for development of ant colony optimization (aco) technique [13]. ants use chemical messenger called pheromone, being social insects that live together in organized colonies and that interact and communicate among themselves. while foraging, ants lay scent chemicals or pheromone and are able to follow the pheromone routes marked by other ants, indicating the trail to food source. the ants follow the route with higher pheromone concentration, and as more and more ants follow the same route, it becomes the favored path with enhanced pheromone which is likely the shortest or more efficient path. evolving, the system converges to a self-organized state. generic representation of ant colony algorithm is: data: population size n, set of components c={c1,…, cn}, evaporation rate evap. initialization: amount of pheromones for each component ph = {ph1, …, phn}; best solution best. while {stopping criterion not met} initialize current population, p=void. create current population of virtual solutions p: for i=1 to n do create feasible solution s. update the best solution, best←void. add solution s to p: p ←p u s apply evaporation: for j=1 to n do phj =phj ·(1–evap) update pheromones for each component: for i=1 to n do for j=1 to n do if component ci is part of solution pj, then update pheromones for this component: phj =phj+fitness(pj) 402 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić 3.3. simulated annealing (sa) simulated annealing (sa) optimization was designed using analogy with metal annealing [14], and is a technique possessing main ability to avoid being trapped in local optima unlike deterministic optimization techniques. it is an optimization method which is alike the process of warming up a solid to melting, then followed by cooling it down until it crystallizes into a perfect lattice. simulated annealing could be considered as markov chain following search [12], which converges under appropriate settings. with each search move, moving trace a piecewise path, acceptance probability is assessed, accepting alterations improving the objective function and also keeping some changes that do not improve the objective [13]. generic representation of simulated annealing is described by: data: initial approximation x0, initial temperature t, number of iteration for a given temperature nt. optimal solution: xbest ← x0. while {stopping criterion not met} n=0; i=0; while (n<nt) do choose a new approximation y accept or reject the new approximation based on the metropolis rule: xi+1 =g(xi,y,t) update optimal solution: if xi+1 is better than xbest, then xbest ← xi+1 next n-iteration (n←n+1) update temperature t next t-iteration (i←i+1) 3.4. particle swarm optimization (pso) particle swarm optimization (pso) is a technique developed on resembling swarm behavior, such as fish and bird schooling [13]. pso has generated a lot of attention in the field of swarm intelligence. system is initialized with a population of random solutions and searches for optima by updating generations. the potential solutions, called particles, fly through the problem space by following the current optimum particles. each particle keeps track of its coordinates which are associated with the best solution (fitness), and its value is also stored. best fitness value is taken as final solution. numerous pso variants exist, as well as hybrid algorithms combining pso with other algorithms. pso is described by the following representation: data: population size n, personal-best weight α, local-best weight β, global-best weight γ, correction factor ε. initialization: create initial population p. while {stopping criterion not met} select the best solution from the current p: best. select global-best solution for all particles: b g . apply swarming: for i=1 to n do select personal-best solution for particle xi:bi p . select local-best solution for particle xi:bi l . casting improvement based on metaheuristic optimization and numerical simulation 403 compute velocity for particle xi: for j=1 to dim do generate correction coefficients: a=α·rand(); b=β·rand(); c=γ·rand(). update velocity of particle xi along dimension j: vij = vij+a·(bij p –xij) + b·(bij l –xij) + c·(bj g –xij) update position of particle xi:xi =xi +ε·vi 4. optimization processes the proposed methodology of combining metaheuristic optimization and advanced casting simulation is applied to the problem of optimization of geometry of a feeder for casting pelton turbine bucket (its cad model is shown in fig. 3). optimization is based on the precise formulation of the objective function and constraints, and abovementioned techniques of metaheuristic optimization were implemented. fig. 3 cad model of buckets pelton turbine 4.1. formulation of the optimization problem the formulation of the optimization problem is actually defining the fitness function. defining the fitness function is one of the major steps of the optimization process. creating the fitness function is based on the rule: the heat-transfer requirement. the heat-transfer requirement for successful feeding can be stated as follows: the freezing time of the feeder must be at least as long as the freezing time of the casting [10]. casting solidification time can be predicted using chvorinov’s rule shown by the equation [11]: 2 s v t c a        , (1) where ts is the total solidification time of the part or feeder, c is a mold constant, v is the volume of metal, and a is the total surface area of the part or feeder. as the feeder serves 404 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić to compensate for the volume shrinkage in the casting part, it needs to be the final link of the solidification chain, i.e. it should be the last to solidify in directional solidification process. the longest solidification time for a given volume is the one where the shape of the part has the minimum surface area. feeder module (mfeeder) is taken to be for a minimum of 20% higher than the module of casting part, i.e. module of part which feeds [10]: _1.2feeder casting partm m  . (2) the modulus of a designed feeder is given by eq. (3), based on the dimensional characteristics (fig. 4): 2 2 2 2 2 2 2 2 ( ) 2 ( ) 3 ( ( ) ( )) 2 2 feeder feeder feeder v h r r r r r l h r r h l m p r r r r h r r r r l h                                . (3) fig. 4 cad model of feeder and dimensional characteristics based on the values of volume and surface of the casting part, obtained by analyzing the cad model (vc=4269327.33 mm 3 and ac=348116.32 mm 2 ), the module of the casting part is mcasting_part=12.26 mm. eq. (4) gives the final expression of fitness function f(r,r,h,l), where all values are entered in [mm], that minimizes: 2 2 2 2 2 2 2 2 ( ) 2 ( ) 12.26 0 3 ( ( ) ( )) 2 2 h r r r r r l h r r h l f r r r r h r r r r l h                                 (4) casting improvement based on metaheuristic optimization and numerical simulation 405 4.2. constraints of optimizations parameter first constraint is based on the rule mass-transfer (volume) requirement. the fulfillment of the conditions given in rule “the heat-transfer requirement”, does not guarantee the fulfillment of the conditions given in rule “mass-transfer (volume) requirement”. although the feeder may have been provided of such a size that it theoretically would contain liquid until after the casting is solid, in fact it may still be too small to deliver the volume of feed liquid that the casting demands. thus, it will be prematurely sucked dry, and the resulting shrinkage cavity will extend into the casting. this is because they are themselves freezing at the same time as the casting, depleting the liquid reserves of the reservoir. effectively, the feeder has to feed both itself and the casting. we can allow for this in the following way. if we denote efficiency ε of the feeder as the ratio (volume of available feed metal) / (volume of feeder, vf) then the volume of feed metal is, of course vf. if the solidification shrinkage of the liquid is α, during freezing, then the feed demand from both the feeder and casting together is α(vf,vc), and hence [10]: ( )f f cv v v   . (5) solving for the feeder volume yields: 2 20.4 ( ) 2 ( ) 0.25c f c c v v v h r r r r r l h r r h l v                     , (6) where α is the solidification shrinkage of the liquid during freezing (α =4, for steels the fcc structure applies above 0.1% carbon where the melt solidifies to austenite) and ε=20% for appropriate feeder. relation (6) is a constraint 1 in the optimization process of geometry feeders. in addition to the abovementioned constraints, other constraints are referred to the mutual relations of values which are optimized and available space on the casting part. table 1 the other constraints constraint 2 2r+l<50 constraint 3 h<155 constraint 4 2r+ l<160 4.3. results of the optimization process the results of the carried out optimization processes are relatively similar and their overview is given in table 2. simultaneous application of several metaheuristic optimization techniques allows for efficient verification of obtained optimization results. table 2 results of optimization process optimization value ga aco sa pso h 153 148 154 150 r 61 58 55 57 r 20 22 23 19 l 26 24 28 27 406 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić based on the obtained results of taken optimizations, the following values of optimized sizes have been adopted: h=155 mm, r=60 mm, r=20 mm and l=25 mm. according to these values, cad model of feeder was created and embedded into assembly of the entire gating system with a casting part. 5. numerical simulation as a verification of the optimization result instead of experimental trial-and-error, the results of mold cavity filling, solidification, critical spots, formation of residual stresses, etc. now optimized in a virtual computer environment with various conditions. using numerical simulations in the development of casting technology not only shortens the time necessary for product development but also reduces expenditures, reduces risks from defects, etc. modern digital tools enable a relatively fast attainment of optimized solutions for casting technology, which results in stable manufacture with lower expenditures [15−19]. digital tools using for optimization of the technical parameters influence the macro and microstructure of a cast. digital tools allow different technological parameters of a casting process to be tested. hence, to perform the simulation, it is necessary to provide a 3d geometric model of the casting and other components (tools for casting, gating system, feeders, etc.) and technological parameters (casting temperature, casting time, alloy composition, etc.). based on the results of experimental investigations, the optimization of relevant technological parameters and their implementation under industrial conditions showed that the properties of a casting, in practice, could be significantly improved. the results show that, under industrial conditions, digital tools provide satisfactory solutions in a short period, starting virtually from scratch. it was shown that the employment of these software packages brings great advantages compared to the conventional manner of adopting new products. for instance, the results obtained in a rapid generation of model elements significantly accelerated the work and improved the productivity of engineers, reduced the time for adopting new products, reduced the scope of traditional prototype testing (creating prototypes and their testing), which is expensive and time consuming, causing standstills in manufacture, and reduced expenditures for adopting new products. the digital tool (in our case used magma 5 ) package itself automatically generates a mesh. the software user can set the fineness of the mesh, i.e., the number of mesh elements by defining the minimum size of the desired elements in all three directions of the coordinate system (fig. 5). fig. 5 mesh quality casting improvement based on metaheuristic optimization and numerical simulation 407 the finer the mesh, i.e., the more elements it has, the more accurate the calculation of the simulation is, but the simulation time is longer. the prepared mesh is used for further calculation. for each element, i.e., part of the mesh, differential equations used to calculate its physical and thermal parameters and the results obtained are boundary conditions for the calculation of the parameters in the neighboring element. thus, the computer calculates element by element in 3d coordinates and finally integrates all the partial results for the overall geometry. the magma 5 software has many criteria for casting process simulation, part filling, but the results of two criteria are present in this paper:  fill_temp, whose results show the distribution of temperatures in the mold cavity from the start to the end of filling; this provides the possibility of monitoring the temperature distribution throughout the mold cavity at any time (fig. 6). fig. 6 fill_temp – filling: a) 25 %, b) 50 %, c) 75 %, and d) 100 %  fill_tracer, which shows which portion of the metal in the casting passed through ingate. the distribution of the melt in the simulation is visible because the respective melt fractions from each ingate are highlighted in different colors. this means that possible changes of the gate positions and gate cross sections as well as changes in the angles at which different sections of the gates are connected could be simulated and the resulting impact on the filling of the mold could be made easily visible, thus substantially facilitating an optimized gating design (fig. 7). 408 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić fig. 7 fill_tracer – filling: a) 25 %, b) 50 %, c) 75 % and d) 100 % in order to define the solidification, i.e., the cooling process of bucket casting, necessary parameters and boundary conditions are entered (the time or temperature at which the solidification simulation stops specified). generally, it is possible to perform simulation without the results of casting solidification. however, if casting simulation is performed before solidification simulation, then the results and temperatures reached at the end of mold filling are taken as the basis for solidification simulation (the fill_temp file) [15]. in the used magma 5 version presented in this paper, this fill_temp file exists in the system, which considerably simplifies the procedure. the solidification results are also shown through several criteria: solid_temp (shows temperature distribution within the selected material in the casting system during solidification), hotspot (determines isolated regions of residual melt at any time during solidification), and porosity (can visualize porosities in the casting). using the solidification criteria results after the simulation is terminated, the program automatically calculates several criteria that can be selected at each point. displaying the criteria results helps find defects in the casting and analyze the solidification behavior. the solidification time is 50 min. this paper presents the results of the following criteria: solid_temp (fig. 8), hotspot and porosity. casting improvement based on metaheuristic optimization and numerical simulation 409 fig. 8 solid_temp – solidification, a) 25 %, b) 50 %, c) 75 % and d) 100 % the results show that the solidification process develops in a directed way and that the last parts that solidify are pour cup and feeder. using the hotspot criterion, isolated regions of residual melt (fig. 9a) are determined at any time during solidification. if the feeding calculation option in the simulation setup is checked, this criterion helps to detect porosities in these residual melt regions. the unit is solidification time in seconds (s, color scale). this can determine the time during solidification at which a particular hot spot develops. it can be clearly seen in fig. 9b that the solidification process develops according to expectations and that the last points of solidification are in feeders. in this case, the solidification time is 47 min. fig. 9 solidification results, a) hotspot and b) porosity 410 r. radiša, n. duĉić, s. manasijević, n. marković, ţ. ćojbašić using the porosity criterion, porosities in the casting can be visualized (fig. 9b). the portion of porosities at the end of solidification simulation is displayed. the unit is percent (%, color scale). the unfilled and thus porous regions are displayed in white. an analysis of porosity problems is the most reliable when using this criterion. 6. conclusion this paper presents a methodology for improvement of casting process, demonstrated on the problem of design and optimization of the feeder’s geometry in the process of casting pelton turbine bucket. four methods of metaheuristic optimization were used: genetic algorithm (ga), ant colony optimization (aco), simulated annealing (sa) and particle swarm optimization (pso), which gave satisfactory and very similar results, mutually confirming validity of the obtained solution. based on the obtained results geometry of feeder was dimensioned and cad model of feeder was created. performed numerical simulations confirm the correctness of the designed optimized feeding system according to all criteria. acknowledgement: the paper is a part of the research done within the projects tr35037, tr35015, tr35016 and tr35005. references 1. gravela, m., priceb, l., w., gagné, c., 2002, scheduling continuous casting of aluminum using a multiple objective ant colony optimization metaheuristic, european journal of operational research, 143 (1), pp. 218-229. 2. santosa, c.a., spimb, j.a., garcia, a., 2003, mathematical modeling and optimization strategies (genetic algorithm and knowledge base) applied to the continuous casting of steel, engineering applications of artificial intelligence, 16 (5-6), pp. 511-527. 3. surekha, b., kaushik, k.l., panduy, k.a., vundavilli, r.p., parappagoudar, b. m., 2012, multiobjective optimization of green sand mould system using evolutionary algorithms, the international journal of advanced manufacturing technology, 58 (1), pp. 9-17. 4. slavković, r., jugović, z., dragićević, s., joviĉić, a., slavković, v., 2013, an application of learning machine methods in prediction of wear rate of wear resistant casting parts, computers and industrial engineering, 64(3), pp. 850-857. 5. duĉić, n., ćojbašić, ţ., slavković, r., jordović, b., purenović, j., 2016, optimization of chemical composition in the manufacturing process of flotation balls based on intelligent soft sensing, hemijska industrija, 70(6), pp. 603-614. 6. duĉić, n., ćojbašić, ţ., manasijević, radiša, r., slavković, r., milićević, i., 2017, optimization of the gating system for sand casting using genetic algorithm, international journal of metalcasting, 11 (2), pp. 222-265. 7. dabade, a. u., bhedasgaonkar, c.r., 2013, casting defect analysis using design of experiments (doe) and computer aided casting simulation technique, procedia cirp, 7, pp. 616-621. 8. jie, z., dongqi, z., pengwei, w., gang, w., feng, l., penglong, d., 2014, numerical simulation research of investment casting for tib2/a356 aluminum base composite, rare metal materials and engineering, 43 (1), pp. 47-51. 9. nimbulkara, l.s., dalu, s.r., 2016, design optimization of gating and feeding system through simulation technique for sand casting of wear plate, perspectives in science, 8, pp. 39-42. 10. campbell, j., 2011, complete casting handbook, 1st edition, elsevier, 661 p. 11. steel founders' society of america, 2001, feeding & risering guidelines for steel castings, steel founders’ society of america, 11 p. casting improvement based on metaheuristic optimization and numerical simulation 411 12. blum, c., roli, a., 2003, metaheuristics in combinatorial optimization: overview and conceptual comparison, acm comput. surv., 35, pp. 268-308. 13. yang, x, -s., 2011, metaheuristic optimization, scholarpedia, 6(8), 11472. 14. kirkpatrick, s., gelatt, c.d., vecchi, m.p., 1983, optimization by simulated annealing, science, 220(4598), pp. 671–680. 15. official website of magmasoft software for casting simulation: http://www.magmasoft.com/en/ (last access 28th april 2017) 16. manasijević, s., radiša, r., petrović, b., 2008, skil and intuition of foundrymen or computer assisted casting, livarstvo-foundry, 47(1), pp. 1–11. 17. radiša, r., marković, s., pristavec, j., kvrgić, v., manasijević, s., 2008, use of cae techniques in virtual design of metalcasting technology-saving in serbian foundries, livarstvo-foundry, 47(1), pp. 12–24. 18. radiša, r., manasijević, s., marković, s., 2009, optimization technology of metal casting using software tool, livarstvo-foundy, 48(3), pp. 30–36. 19. radiša, r., gulišija z., manasijević, s., 2009, optimization of casting process, machine design, 1, pp. 111–114. http://www.scholarpedia.org/article/metaheuristic_optimization http://www.magmasoft.com/en/ plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 199 208 original scientific paper splitting the total exergy destruction into endogenous and exogenous parts of the thermal processes in a real industrial plant udc 621.1:536.7 goran vučković, mića vukić, mirko stojiljković, miloš simonović university of niš, faculty of mechanical engineering, niš, serbia abstract. the total exergy destruction occurring in a component is not only due to the component itself (endogenous exergy destruction) but is also caused by the inefficiencies of the remaining system components (exogenous exergy destruction). hence care must be taken in using the total exergy destruction of a component for making decisions to optimize the overall energy system. in this paper, a complex industrial plant is analyzed by splitting the component’s exergy destruction into its endogenous part (the part resulting totally from the component’s irreversibilities) and its exogenous part (resulting from the irreversibilities of the other components within the system). it is observed that the steam generator has the dominant effect. from the total exergy destruction in the steam generator, 1,097.63 kw or 96.95% come from internal irreversibilities in the component, while the influence of other components on the loss of useful work in the steam generator is only 3.05%. key words: exergy, exergy destruction, endogenous, exogenous, industry 1. introduction with an increasing need to reduce the system waste impact on the environment and an ever increasing global demand for energy, especially in developing countries, it is becoming extremely important to develop even more accurate and systematic approaches for improving the design of energy systems [1]. energy systems, i.e. industrial energy systems transforming some energy forms to others of which at least one is useful have been built since the eighteenth century. the continuously increasing complexity of the industrial systems has led to their construction and operation having a detrimental effect on natural resources and environment. it is, received august 23, 2015 / accepted february 22, 2016 corresponding author: goran vuĉković university of niš, faculty of mechanical engineering niš, a. medvedeva 14, 18000 niš, serbia e-mail: vucko@masfak.ni.ac.rs 200 g. vuĉković, m. vukić, m. stojiljković, m. simonović therefore, necessary to develop and apply methods that will help in design, construction and operation of highly efficient, environmentally friendly and economically viable systems [2]. exergy as a term related to the quality of certain types of energy represents the part of energy that could be converted to any other form of energy, i.e. produce work. exergy can be successfully used as a reference tool for evaluating energy consumption, user and society satisfaction, environmental hazard and economic cost. exergetic analysis is a key element in analyzing and drawing up a plan for sustainable development [3]. conventional exergetic analysis reveals irreversibility within each component of a plant, but it has some limitations [4]. advanced exergetic analyses are needed in order to determine which part of the inefficiencies is caused by component interactions (i.e. the structure of the plant), and which part can be avoided through technological improvements of the plant [5]. the aim of this paper is to explicitly identify the exergy destruction and separate it into endogenous and exogenous exergy destruction, on the example of a real complex industrial plant. 2. technical description of a complex energy system the energy system in a representative factory, fig. 1, consists of four parts: energy supply sector, factory 1, factory 2 (f2) and engineering department (ed). the energy supply sector is a part of the factory complex where chemical and thermal treatment of water is being carried out while superheated steam is produced for the factory’s own use and supply of all other consumers. furthermore, in this section, compressed air and cooling water are prepared for the whole factory complex. the boiler produces superheated steam at the pressure of 10 bars, which is then distributed to factories 1 and 2, and partly reduced at lower pressures in accordance with the needs of consumers. in this paper, it is assumed that the fuel used is pure methane. compressed air at pressure of 7 bars is prepared in the compressed air station where electricity is used to drive the compressors. consumers of compressed air are factory 1, factory 2 and engineering department. cooling water for factories 1 and 2 is prepared in the evaporative cooling tower. hot water from factories 1 and 2 is pumped to the pool of hot water and then transported with pump 3 towards the cooling tower. a spray jet of cooling water, transported from the pool of hot water, spreads over the tube bundle of evaporative heat exchangers and is collected into the reservoir. forced air flow is provided with the centrifugal ventilator, while water drops removal within the air flow is prevented with the droplet eliminator. the cooled water is collected and transported with the pump 4 to consumers in factories and the compressed air station. factory 1 is the largest consumer of energy in the whole complex and it is supplied with energy using the superheated steam at the pressure of 10 bars, compressed air and cooling water. electric energy is provided from a separate transformer station. consumers in the factory use the steam at pressures of 10 and 4 bars. the condensate is collected in the condensate tank 2, and then transported with the pump 6 to the condensate tank 1. the sanitary hot water is prepared in the tank with the capacity of 3000 liters. the primary fluid is steam at 4 bars which, after energy is delivered to the secondary fluid, condenses and goes to the condensate tank 2. the secondary fluid is water with the temperature regime 12/60oc. endogenous and exogenous exergy destruction and exergoeconomics evaluation ... 201 thermal comfort in the manufacturing hall during winter months is maintained with convectors. the steam at 4 bars completely condenses in the convectors and the condensate is transported to the condensate tank 2. fig. 1 flow diagram of the representative industrial plant legend: ca compress air, cg combustion gasses, co – condensate, cp circulation pump, ct condensate tank, cw cold water, ea environmental air, ed engineering department, ep electric power, fw flash water, fp factory products, f2 factory 2, ia indoor air, ha hot air, ho heavy oil, hw hot water, ma moist air, pf primary fuel, rv reduce valve, rw return water, sb steam branch, sd steam distributor, sw supply water, wa waste air, wb water branch, cas compress air station, cop condensate pipeline, cwd cold water distributor, cwt chemical water treatment, dea – deaerator, ect evaporative cooling tower, fwt feed water tank, pfh primary fuel heating, stb steam boiler, stp steam pipeline, shw sanitary hot water, tch thermal comfort hall, tec technologic consumers, tss thermal substation, hwp hot water pool electricity is used for the operation of the fans in the convectors. the total power of the fans is 19.2 kw. in the thermal substation, the energy is transferred from the steam at pressure of 4 bars to hot water, which is used for the radiator heating system in the temperature regime 90/70oc. technology customers produce the final product. technology consumers are using the steam at 10 and 4 bars. for technological reasons, the condensate at 4 bars is discharged into the sewer. the production technology requires the use of compressed air, which is rejected to the environment after use. also, the cooling water in 202 g. vuĉković, m. vukić, m. stojiljković, m. simonović the factory is used for technological needs. during the production process, the temperature of cooling water increases. all consumers of the electricity in the factory, as well as technological consumers, are supplied with the electricity from a separate transformer station. factory 2 is supplied with the steam at pressure of 10 bars. the condensate from factory 2 returns to the condensate tank 1. the engineering department requires steam at 4 bars. the condensate from the engineering department is returned to the condensate tank in the full amount. in this paper, we have discussed in detail factory 1. the effects of factory 2 and the part that deals with the engineering department should be considered on the basis of real values of the superheated steam that they use. the representative system of the considered rubber factory is mathematically modeled with 33 components (k) and 70 streams (j). 3. splitting the total exergy destruction into endogenous and exogenous parts of exergy destruction conventional exergetic analyses pinpoint components and processes with high irreversibility [1, 5]. the total exergy destruction occurring in a component does not result exclusively from the component but is also caused by the inefficiencies of the remaining system components. a part of the exergy destruction within a system component is generally caused by the inefficiencies of the remaining system components (exogenous exergy destruction). if no irreversible processes and exergy losses occur in all remaining system components, the exergy destruction is due exclusively to the component being considered (endogenous exergy destruction). the total exergy destruction within a k-th component ed,k is the sum of these two parts of exergy destruction [6]: , , , en ex d k d k d ke e e  (1) where , en d ke and , ex d ke endogenous and exogenous exergy destruction for k-th component, respectively. the endogenous exergy destruction for k-th component is determined when this component operates under real conditions and all other components of the process are considered to operate without irreversibility (theoretically). the real and theoretical conditions for the most important components in the present industrial plant are shown in table 1 [7]. when the endogenous part of exergy destruction of k-th component is known, the exogenous part of exergy destruction is obtained with eq. (1). a change in the exergy destruction in one subsystem generally affects the exergy destruction in other sub-systems, too [8]. thus, the change in the total exergy input to a system is usually different from the change in the exergy destruction in one system component [9]. in complex thermal systems, it is very difficult and time consuming to separate these two parts of exergy destruction within a system component. to better understand the interactions among components, the exogenous exergy destruction within k-th component is also split. the sum of all the terms is lower than the exogenous exergy destruction within k-th component. the difference is caused by the simultaneous interactions of all (n-1) components. this difference is call mexogenous exergy destruction, but it is not of interest in this paper. exergetic analysis is used to evaluate the performance of energy endogenous and exogenous exergy destruction and exergoeconomics evaluation ... 203 systems from thermodynamic aspect [10]. the essence of the exergetic analysis is that it provides information about the quality of energy [11]. table 1 diffusion coefficient and standard source term component real condition theoretical cond. x b b component real condition theoretical cond. stb 90%; 0lq   100%; 0lq   hwp , ; 0 j j lp t ct q , ; 0 j j lp t ct q pfh min 50 ; 0   l t k q 0 100%; 0l p p q        std  cwd ; . 0    j j l p ct t ct q ; . 0    j j l p ct t ct q dea 20 46 19 23 . ;> > jp ct t t t t ; . 0    j j l p ct t ct q ct (12) ; . 0    j j l p ct t ct q ; . 0    j j l p ct t ct q fwt ; . 0    j j l p ct t ct q ; . 0    j j l p ct t ct q shw min 70 ; 0   l t k q 0 100%; 0l p p q        cp (15) 80%  100%  tch min 70t k  100% cas 90%  26 24 29 26 100%; 0 ; lq t t t t      tss min 60 ; 0   l t k q 0 100%; 0l p p q        4. results and discussion each segment of the system shown in fig. 1 is separately analyzed and mass, energy and exergy balances are defined for each component. the results of the thermodynamic (mass flow rate mj, specific enthalpy hj and specific entropy sj) and exergetic analysis (exergy flow rate ej) for the selected steams j are presented in table 2. input data for the calculation are pressures p and temperatures t in different points of the flows of streams obtained from the existing process of the referent plant. the official data on the lower heating values for the fuel are used. for solving the defined mathematical model, based on the equations in paragraph 2, a specialized software package, called engineering equation solver, is used. the results of exergetic analysis at the component level are presented in table 3. having in mind that the reduce valves typically serve other elements, the reduce valve and the component it serves should be considered together [7]. in the manner of conventional exergetic analysis, exergy of fuel ef,k and product ep,k, exergy destruction ed,k, exergetic efficiency k, maximal exergetic efficiency max k and exergy destruction ratio yd for the selected components are presented in this paper. values of endogenous/exogenous exergy destruction are also presented for the selected components in the advanced exergetic analysis. values of exergy of fuel, exergy of product, total exergy destruction and exergetic efficiency for overall system also presented in table 3. when evaluating any energy system, we mainly focus on its avoidable exergy destruction because it represents the potential for improvement. components pfh, shw, tch and tss have the high values of specific unavoidable exergy destruction of 1.706, 2.512, 2.754 and 1.152, respectively. 204 g. vuĉković, m. vukić, m. stojiljković, m. simonović table 2 calculated variables for selected streams s tr . jm jp jt jh js je s tr . jm jp jt jh js je kg/s bar c kj/kg kj/kgk kw kg/s bar c kj/kg kj/kgk kw 1 0.05 4.01 80.00 -2,255.30 2.207 2,412.10 36 27.79 4.51 21.00 88.44 0.310 82.38 2 0.86 10.29 103.00 432.41 1.340 34.26 37 8.50 4.51 21.00 88.44 0.310 25.20 3 0.86 10.29 182.66 2,782.63 6.584 1,117.55 38 27.79 4.51 28.00 117.71 0.409 80.98 4 1.47 3.51 180.00 -1,477.62 7.038 238.53 39 8.50 4.51 27.00 113.53 0.395 24.47 5 0.05 4.21 45.00 -2,323.15 2.004 2,411.76 40 41.80 1.01 27.41 114.91 0.401 106.20 6 0.002 3.91 156.58 2,767.97 6.975 2.29 41 0.02 1.01 80.00 334.97 1.075 0.43 7 0.002 3.91 142.85 601.51 1.769 0.16 42 0.17 1.01 50.00 209.40 0.704 1.16 8 1.41 1.01 21.00 40.77 5.755 0.04 43 0.83 1.01 12.01 50.52 0.181 3.09 9 0.25 10.29 181.91 2,780.70 6.580 325.78 44 0.48 1.01 12.01 50.52 0.181 1.79 10 0.48 10.29 181.91 2,780.70 6.580 626.11 45 0.82 1.01 80.00 334.97 1.075 17.56 11 0.13 10.29 181.91 2,780.70 6.580 165.09 46 0.82 1.51 80.01 335.03 1.075 17.60 12 0.13 5.51 167.62 2,780.70 6.852 154.81 47 0.83 1.01 12.00 50.46 0.180 3.09 13 0.02 5.51 165.07 2,774.88 6.839 24.40 48 0.83 1.51 12.00 50.52 0.180 3.13 14 0.11 5.51 165.07 2,774.88 6.839 130.17 49 0.48 10.29 181.91 2,780.70 6.580 626.11 15 0.11 3.91 159.74 2,774.88 6.991 125.32 50 0.39 10.29 181.44 2,779.49 6.578 514.57 16 0.06 3.91 156.58 2,767.97 6.975 75.38 51 0.08 10.29 181.44 2,779.49 6.578 111.34 17 0.04 3.91 156.58 2,767.97 6.975 47.42 52 0.08 4.61 164.18 2,779.49 6.929 102.39 18 0.04 1.51 147.87 2,767.97 7.405 42.25 53 0.01 4.61 160.30 2,770.86 6.909 16.40 19 0.01 1.51 147.87 2,767.97 7.405 9.93 54 0.04 4.61 160.30 2,770.86 6.909 45.06 20 0.83 1.51 85.01 356.01 1.134 20.56 55 0.01 4.61 160.30 2,770.86 6.909 16.79 21 0.03 1.51 147.87 2,767.97 7.405 32.31 56 0.02 4.61 160.30 2,770.86 6.909 23.91 22 0.86 1.51 102.88 431.26 1.339 33.43 57 0.12 6.51 12.00 50.99 0.180 0.53 23 0.002 1.51 111.64 2,693.77 7.220 2.52 58 0.12 6.31 60.00 251.67 0.831 1.37 24 0.31 1.01 21.00 294.42 6.847 0.00 59 0.01 4.61 148.86 627.38 1.830 1.22 25 5.51 4.51 21.00 88.44 0.310 16.33 60 3.68 1.01 18.00 291.40 6.836 0.31 26 0.21 7.81 27.00 298.92 6.276 37.38 61 3.68 1.01 45.00 318.59 6.926 2.38 27 0.10 7.81 27.00 298.92 6.276 17.47 62 0.04 4.61 148.86 627.38 1.830 3.34 28 0.00 7.81 27.00 298.92 6.276 0.09 63 0.27 2.51 28.00 117.53 0.409 0.72 29 5.51 4.01 24.50 103.03 0.360 15.44 64 0.27 2.31 55.00 230.42 0.768 2.27 30 33.71 1.01 13.35 25.41 5.701 15.76 65 0.01 4.61 148.86 627.38 1.830 1.25 31 0.35 1.01 12.01 50.52 0.181 1.30 66 0.02 4.61 148.86 627.38 1.830 1.77 32 41.80 2.51 27.42 115.10 0.401 112.50 67 0.01 1.01 100.00 2,675.73 7.354 2.89 33 41.80 1.01 20.97 88.00 0.310 109.30 68 0.08 1.01 100.00 419.07 1.307 2.90 34 34.06 1.01 21.35 60.06 5.821 6.46 69 0.08 1.01 100.00 419.07 1.307 2.90 35 41.80 4.51 21.00 88.44 0.310 123.91 70 0.08 1.01 100.00 419.07 1.307 2.90 essentially, all the components are heat exchangers, whose design allows the use of only the physical exergy. in these components, the superheated steam is used as a primary fluid, and it has a great potential to perform useful work in chemical exergy. endogenous and exogenous exergy destruction and exergoeconomics evaluation ... 205 table 3 results of the exergetic analysis at the component level n u m b er k conventional ea splitting ed ea evaluation ,f ke ,p ke ,d ke , en d ke , ex d ke k max k ,d ky kw kw kw kw kw % % % 1 stb 2,185.50 1,053.29 1,132.21 1,097.6 34.58 48.19 52.69 43.66 2 pfh 2.13 0.34 1.79 1.58 0.21 15.96 36.96 0.07 3 dea 7.24 2.78 4.46 4.33 0.13 38.37 100.00 0.17 4 fwt 46.87 33.43 13.44 4.24 9.20 71.32 100.00 0.52 5 cp1 0.98 0.83 0.15 0.15 0.00 84.42 96.03 0.01 6 cp2 0.05 0.04 0.01 0.01 0.00 75.92 94.70 0.00 7 sd1 1,117.55 1,116.98 0.57 0.58 -0.01 99.95 100.00 0.02 8 sd2 165.09 154.57 10.52 6.46 4.06 93.63 93.79 0.41 9 rv1 10 sd3 130.17 125.09 5.08 2.80 2.28 96.10 96.28 0.20 11 rv2 12 stp 626.11 626.11 0.00 0.00 0.00 100.00 100.00 0.00 13 cas 82.70 54.84 27.86 27.86 0.00 66.31 71.29 1.07 14 cp3 7.87 6.30 1.57 1.54 0.03 80.09 95.02 0.06 15 ect 32.83 4.80 28.03 22.75 5.28 14.62 81.52 1.08 16 hwp 120.89 106.20 14.69 14.49 0.20 87.85 99.96 0.57 17 cp4 18.32 14.61 3.71 3.44 0.27 79.75 94.76 0.14 18 cwd 123.91 123.91 0.00 0.00 0.00 100.00 100.00 0.00 19 ct1 79.61 15.51 64.10 63.55 0.55 19.48 100.00 2.47 20 cwt 3.13 3.13 0.00 0.00 0.00 100.00 100.00 0.00 21 cp5 0.05 0.04 0.01 0.00 0.01 77.01 100.00 0.00 22 sd4 626.11 625.91 0.20 0.00 0.20 99.97 100.00 0.01 23 sd5 111.34 102.16 9.18 8.44 0.74 91.75 100.00 0.35 24 rv4 25 shw 15.33 0.99 14.34 14.16 0.18 6.46 28.47 0.55 26 tch 60.92 2.07 58.85 58.76 0.09 3.40 26.64 2.27 27 tss 15.54 1.55 13.99 13.96 0.03 9.97 46.46 0.54 28 ct2 4.69 2.90 1.79 2.18 -0.39 61.83 99.99 0.07 29 rv3 47.45 42.25 5.20 0.00 5.20 89.04 89.04 0.20 30 cp6 2.90 2.90 0.00 0.00 0.00 100.00 100.00 0.000 31 cop 2.90 2.90 0.00 0.00 0.00 100.00 100.00 0.000 32 sb1 42.25 42.24 0.01 0.00 0.01 99.98 100.00 0.000 33 wb1 3.09 3.09 0.00 0.00 0.00 100.00 100.00 0.000 overall 2,582.59 920.27 1,411.77 35.90 value of exergy loss, el,tot, for overall system is 250.56kw. however, that potential in these devices is not exploited. on the other hand, to obtain the superheated steam, a certain amount of primary fuel (resources) in the steam generator is consumed. 206 g. vuĉković, m. vukić, m. stojiljković, m. simonović fig. 2 presents the values of the current and maximum exergetic efficiency at the component level. it is observed that the most significant limits in the values of the maximum exergetic efficiency are in the heat exchangers: steam-fuel, steam-water and steam-air, or in components shw, tch and tss. these limits are caused by high levels of specific unavoidable exergy destruction in these components. in addition, in the steam generator, from the total exergy destruction of 1,132.21kw, 83.53% is unavoidable, and even 96.94% of destroyed useful work comes from internal irreversibility in the component. therefore, exergetic efficiency higher than 52.69% cannot be achieved in the steam generator. fig. 2 real and maximal exergetic efficiency the exergetic efficiency of the total plant indicates that some potential exists for the improvement of the overall efficiency and reduction of costs, see table 3. exergy losses are mainly associated with the exhaust gases as well as the exergy transfer to the environment. from the total exergy losses, the loss of exhaust gas is 95.19% or 238.53kw. all exergy losses account together for 10.39% or 250.56kw of the fuel exergy supplied to the overall plant. in the plant, the components destroyed 58.53% of the fuel exergy, or 1,411.77kw. the compressed air station has a higher improvement potential, 22.08kw or 79.25%. in the compressed air station, the entire amount of destroyed useful work comes from inside ireversibilities (endogenous exergy destruction 27.86kw). exergy destruction in the deaerator is mainly caused by differences in temperature and pressure of the water streams being mixed. in the deaerator more than 97% or 4.33kw comes from irreversibilities in the component and 2.92% from the other components. fig. 3 presents the total, endogenous and exogenous exergy destruction of the components of the energy system. it is observed that the steam generator has the dominant effect. however, from fig. 3 we can unequivocally conclude that 1,097.63kw or 96.95% from the total exergy destruction of the steam generator comes from the internal irreversibility in components, while the influence of other components of destroyed useful work in the steam generator amounts only to 3.05%, or 34.58kw. significantly, this 34.58kw of destroyed work capacity is the highest value among all the external exergy destruction in the energy system (fig. 3 and table 2). endogenous and exogenous exergy destruction and exergoeconomics evaluation ... 207 fig. 3 shows that the exergy destruction in components ct1, tch, ect and cas is not so small, but the loss of ability originates primarily from the internal irreversibility of the components. fig. 3 endogenous and exogenous exergy destructions 5. conclusions the results of the conventional exergetic analysis are strongly supplemented by the advanced exergetic analysis. irreversibilities identified in the conventional exergetic analysis have been split, according to their origins, in the advanced exergetic analysis. this paper presents the results of the splitting of the total exergy destruction into endogenous and exogenous parts for a real complex industrial plant. the highest exergy destruction is caused by the steam boiler. more than 80% of the total exergy destruction of the overall system comes from the boiler. using the method presented in this paper we know that almost 97% of the total exergy destruction within this component results from the operation of the component itself (endogenous exergy destruction). the advanced exergetic analysis undoubtedly ranks the improvement priority of the steam boiler first, followed by the thermal comfort of the hall and the thermal substation. the combined splitting in advanced exergetic analysis will be investigated in the next step of this study. 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performance evaluation of a typical solar photovoltaic module, thermal science, doi:10.2298/tsci130218147p. 13750 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 183 196 https://doi.org/10.22190/fume250520021r © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper active disturbance rejection control and model-free control tuned via fictitious reference iterative tuning raul-cristian roman1, radu-emil precup1,2, emil m. petriu3 1politehnica university of timisoara, department of automation and applied informatics, timisoara, romania 2romanian academy – timisoara branch, center for fundamental and advanced technical research, timisoara, romania 3university of ottawa, school of electrical engineering and computer science, ottawa, canada orcid ids: raul-cristian roman https://orcid.org/0000-0003-2551-7842 radu-emil precup https://orcid.org/0000-0002-2060-7403 emil m. petriu https://orcid.org/0009-0002-0548-8976 abstract. this paper presents a comparative analysis of two data-driven algorithm combinations: the first-order active disturbance rejection control-fictitious reference iterative tuning (adrc-frit) and the first-order model-free control-fictitious reference iterative tuning (mfc-frit). the objective of both data-driven combinations is to ascertain the tunable parameters through the resolution of an optimization problem and to streamline the heuristic procedures involved. the data-driven algorithms are empirically validated through experimental trials utilizing the 3d laboratory equipment in which the x-, y-, and z-axes are controlled. key words: active disturbance rejection control, model-free control, fictitious reference iterative control, 3d crane 1. introduction in recent years, the utilization of data-driven algorithms [1] in the control domain has seen a gradual increase. within contemporary control systems, data-driven algorithms assume an increasingly pivotal role by using empirical data to design controllers with minimal reliance on explicit mathematical models, which is important in coping with received: may 20, 2025 / accepted july 03, 2025 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania romanian academy – timisoara branch, center for fundamental and advanced technical research, bd. mihai viteazu 24, 300223 timisoara, romania e-mail: radu.precup@upt.ro https://orcid.org/0000-0003-2551-7842 https://orcid.org/0000-0002-2060-7403 https://orcid.org/0009-0002-0548-8976 184 r.-c. roman, r.-e. precup, e. m. petriu complex processing including active those involved in active structures [2]. rather than designing precise system equations, these methodologies derive patterns and dynamics directly from input-output (i/o) data, thereby facilitating adaptation to intricate or uncertain environments. this method is particularly advantageous in contexts where traditional modeling proves impractical due to nonlinearities, noise, or high dimensionality. by seamlessly integrating tools from machine learning and optimization, data-driven control delivers flexible and robust solutions tailored to real-world applications, including robotics, energy systems, and autonomous vehicles. according to the authors, active disturbance rejection control (adrc) [1,3,4], model-free control (mfc) [1,5,6], model-free adaptive control [1,7], virtual reference feedback tuning [1,8], iterative feedback tuning [1,9], fictitious reference iterative tuning (frit) [1,10], or iterative learning control [11,12] are among the most frequently utilized data-driven algorithms. adrc is a feedback control strategy designed to handle uncertainties and external disturbances without relying on an accurate mathematical model of the process. at its core, adrc augments the system’s state with an estimated total disturbance, which encompasses both unknown dynamics and external inputs. this estimate is updated in real time by an observer and then actively canceled in the control law, resulting in improved robustness and disturbance rejection. usually, due to its simplicity, the extended state observer (eso) is preferred. thanks to its model-agnostic design and straightforward tuning rules, adrc has been successfully applied in fields ranging from robotics to process industries, where precise modeling proves challenging. due to its model-independent framework and straightforward tuning guidelines, adrc has been proficiently utilized across various domains. in the realm of tower crane systems, as indicated in [3], two configurations of adrc integrated with the fuzzy logic technique are proposed, ensuring control loop stability as per fuzzy control system stability, with adrc parameters determined through the metaheuristic grey wolf optimizer. in [13], the second-order adrc is combined with the sliding mode technique to enhance control loop performance, proposing adrc stability while optimal controller parameters are ascertained via the metaheuristic slime mould algorithm (sma) with validation on tower crane systems. according to [14], firstorder adrc is mixed with the frit to identify optimal tunable parameters using the metaheuristic african vultures optimization algorithm (avoa), accompanied by a proposed stability mechanism for adrc, also validated on tower crane systems. for 3d crane systems in [15], comparisons are drawn between the first and second orders of adrc algorithms. direct current (dc)-dc buck converters in [16] feature a general error-based adrc with a stability mechanism founded on singular perturbation theory. in permanent magnet synchronous motors, as discussed in [17], a cascaded filter proportional-integralderivative (pid) paradigm is employed for error-based adrc, with system robustness characterized in terms of stability margins. in piezo-actuated beams, as described in [18], adrc is utilized to estimate and compensate for total disturbances, with design based on the governing equation. for gimbal mechanisms, [19] proposes a first-order adrc algorithm alongside a stability mechanism. the application of adrc to rotary dc motors, as stated in [20], involves the use of an anti-windup strategy and hurwitz polynomials. across all these systems, the precise modeling remains a considerable challenge. mfc, often associated with the concept of intelligent pid controllers, presents a viable alternative to traditional model-based control methodologies. instead of relying on a detailed mathematical representation of the system, mfc builds on simplified local models based on real-time data. this approach allows the controller based on mfc to adapt to active disturbance rejection control and model-free control 185 system dynamics and external disturbances in the absence of preliminary identification procedures, making it especially valuable for complex or poorly understood processes. in recent years, mfc has been rigorously improved and validated across various demanding applications, from robotics to process industries, demonstrating its adaptability and efficacy in practical environments. in [5], the discrete-time mfc algorithm is employed, wherein the parameters of ipid controllers are optimally determined utilizing frit via the metaheuristic avoa, subsequently validated through experimental trials on tower crane systems. in [21], the mfc is implemented in advanced processes, including the quanser aero. in [22], the mfc is validated across various scenarios on wendling’s model. in [23], mfc is experimentally applied to cloud and high-performance computing systems. in [24], mfc is utilized to manage processes involving shape memory alloys, specifically controlling a shape memory alloy spring-based actuator initially. in [25], the application of mfc, timedelay estimation mfc, and backstepping-based mfc is demonstrated in coupled mechatronic systems for controlling 2-dof and 3-dof robotic manipulators. in [26], mfc, in combination with flatness-based control, is employed for the management of unmanned surface vehicles. in [27], mfc is applied to the control of a prosthetic hand. in [28], mfc, coupled with a fuzzy technique, is utilized to control a twin rotor aerodynamic system. in all these applications, mfc proves to be highly beneficial due to the complexity and the challenge in determining precise mathematical models for these processes. frit constitutes a data-driven iterative technique devised for the optimization of controllers, obviating the necessity for an explicit process model. the principal concept involves constructing a reference signal, designated as the fictitious reference, which, if pursued by the system, would produce an identical measured output. through the iterative minimization of the error between the actual and desired behaviors, frit facilitates the direct adjustment of controller parameters utilizing solely experimental data, rendering it particularly suitable for systems where modeling is challenging or unreliable. analogous to adrc and mfc, frit has been substantiated across a diverse array of applications, encompassing robotics and process industries, thereby exhibiting its versatility and effectiveness within practical settings. in [29], an enhancement of frit through pseudo-linearization has been implemented on asymmetric bouc-wen systems. according to [30], frit is applied to the control of hydraulic systems and necessitates an offline database for the storage of historical process data. the study in [31] demonstrates the utilization of frit to calibrate the parameters of a fractionaland integer-order pid controller relevant to a benchmark problem encompassing both scenarios with and without time delay, specifically within a flexible transmission model. in [32], frit was employed to manage the cart position within a pendulum cart system. as observed in [33], frit is leveraged for the design of controller parameters in applications concerning a switched reluctance generator, ball screw positioning, two-mass resonance, and switched reluctance motor systems. across these applications, frit effectively manages the feedback loop even when confronted with nonlinear processes. the present paper seeks to build upon and synthesize the authors’ prior work involving the integration of adrc and mfc with frit. initially, as indicated in [5], the discrete-time variant of the first-order mfc, incorporating a pid component, was integrated with frit, with the optimal controller parameters being computed via avoa. subsequently, [34] describes the blending of the continuous-time variant of the first-order mfc with a pi component with frit, with tuning via sma for optimal parameters. furthermore, in [35], a discrete-time, secondorder version of the mfc algorithm with a p component is combined with frit, with optimal parameter settings adjusted using sma. additionally, [14] details the integration of the 186 r.-c. roman, r.-e. precup, e. m. petriu first-order continuous-time adrc with frit, where parameters were optimized utilizing the gauss-newton algorithm. each of these algorithm integrations underwent experimental validation using tower crane systems. accordingly, in the current paper, the authors integrate the continuous-time first-order mfc with a p component and the continuous-time firstorder adrc algorithms with frit, resulting in the adrc-frit and mfc-frit algorithms. both algorithms comprise one user-selected parameter and one tunable parameter whose optimal values will be ascertained through frit using the newton–raphson method. stability mechanisms have been proposed for both adrc and mfc algorithms, and stability is implicitly guaranteed for the adrc-frit and mfc-frit algorithms. the efficacy of these algorithms is demonstrated through experiments performed using the 3d crane laboratory device [36], involving the control of the x-, y-, and z-axes. metaheuristic algorithms such as avoa and sma, employed in solving optimization problems for the algorithms listed below, operate on the same principles as the metaheuristic baso or aso as detailed in [37]. although the exact identification of the 3d crane system constitutes a labor-intensive endeavor, it can be achieved, for instance, through the application of dimension reduction via unsupervised learning [38]. the ensuing sections of this manuscript are systematically arranged as follows: section 2 provides a comprehensive elucidation of the adrc-frit and mfc-frit algorithms. section 3 delivers a succinct description of the 3d crane. section 4 presents a detailed exposition of the experimental results in conjunction with a critical analysis. section 5 underscores the conclusions. 2. the data-driven algorithms mix 2.1. the frit algorithms the integration of continuous-time first-order adrc and mfc with frit is advocated to ascertain the optimal parameters of the adrc and mfc algorithms and to enhance the overall efficacy of the control system architecture. to achieve this objective, it is requisite to conduct an initial closed-loop experiment of the control system incorporating the adrc algorithm. adrc-frit and mfc-frit utilize the i/o data to resolve the optimization problem via the newton–raphson method ,)],(~),([ 1 )(),(minarg 0 2)0( 0 * dttyty t jj ht h  −==   (1) where, in this context, κ* denotes the optimal parameter either of the adrc-frit or the mfc-frit algorithm as it pertains to κ, depending on the practitioner’s choice. the initial parameter of these algorithms is represented by κ(0), with the superscript (0) satisfying its initial value. the cost function associated with the adrc-frit or mfc-frit algorithm is denoted by jκ. notation th is used to specify the duration of the experimental timeframe. furthermore, y0(t, κ(0)) retains the output data obtained post the initial experiment. the fictitious reference model output ỹk(t, k) is determined through offline computation, given the understanding that [10,14,15,34,35] )},,(~)({),(~ 1 = −  srsμlty (2) active disturbance rejection control and model-free control 187 where the notation l–1 is utilized to represent the inverse laplace transform. the reference model transfer function, denoted as m(s), is determined by the designer to yield an output response that meets the performance specifications of the control system, and r(̃t, κ) is the fictitious reference input, also referred to as the virtual set-point )},,(/),({),(),(~ )0( 0 1)0( 0 += − scsultytr (3) r(̃t, κ) constitutes a modified version of the fictitious reference input as presented in [9] for continuous-time systems and corresponds to the improved virtual set-point for continuoustime systems outlined in [14] and [34], and for discrete-time systems in [6] and [35]. the term u0(s, κ(0)) denotes the input data, which is obtained using the laplace transform and gathered from the initial closed-loop experiment, while c(s, κ(0)) represents the transfer function of the proportional component within the adrc or mfc algorithm [10,14,15,34,35] .),( )0( =sc (4) to determine the value of gain κ using the newton-raphson method [39] by computing the first and the second derivative of jκ(κ), i.e., ∂jκ/∂(κ(i)) and ∂2jκ/∂(κ(i))2, and the solution is updated knowing that , )( )( 2 1 2)( 2 )()1( ii ii jj           −=  − + (5) where i is the index of the current iteration, j’ κ and j” κ are the first and the second derivative of jκ(κ). as indicated by [10], the adrc-frit algorithm identifies the i/o data-pair (u0(t, κ(0)), y0(t, κ(0))) as non-trivial, thereby [10,14,15,34,35] .),(~lim)),(~),((lim 0 2 0 2 **  =− →→ hh tt dttedttyty (6) 2.2. the adrc-frit algorithms the adrc algorithm is designed for a continuous-time low-pass filter process ),()()( tutyty =+  (7) where t is the time variable, κ is a member of the real numbers set r, represents the process gain, and τ, also within r, signifies the process’s time constant. the algorithm’s capacity to operate with τ values less than zero enables it to manage unstable processes effectively. here, u(t), a real-valued function, denotes the control input, while y(t), also a real-valued function, indicates the controlled output. the revised process model is derived by incorporating the expression b = b0 + δb, where b0 serves as an estimator of the value of b  r, and δb accounts for the parametric uncertainty or variation in the gain parameter κ of the system [13–15] ).()()()/1()()/1()( 0 tubbttyty +++−=  (8) in (8), the effect of the disturbance δ(t) is evident, as the adrc algorithm mitigates disturbance influences via an eso, specifically a luenberger observer. conversely, in (7), 188 r.-c. roman, r.-e. precup, e. m. petriu this disturbance is excluded to maintain a simplified representation of the nominal system model. the term that gathers all the unknowns of the process is the generalized disturbance term of the adrc algorithm is ),()()/1()()/1()( tubttytf ++−=  (9) where function f(t) within r is contingent upon the known output y(t), in contrast with the unidentified disturbances δ(t) and the modeling error δb, both of which affect the dynamics of the system. within the context of the eso in adrc, y(t) is not directly observable and is instead approximated, thus included in f(t) together with the unknown disturbances δ(t) and the modeling error δb. upon integrating the disturbance term from (8) into the firstorder low-pass filter denoted as (7), the resulting process model transforms into a perturbed integrator [13–15] ).()()( 0 tubtfty += (10) employing the notations z1(t) = y(t) and z2(t) = f(t) within expression (10), the statespace representation of the perturbed integrator model is ),()0 1()( ),( 1 0 )( 0 )( 00 10 )( )( 0 2 1 tty tftu b t tz tz z z =       +      +      =         (11) where z(t) = [z1(t) z2(t)]t, with t representing matrix transposition, the third term in the right-hand side of (11) can be deemed negligible since the eso, as introduced later in the discourse, can estimate and compensate for disturbances. the adrc algorithm does not necessitate explicit disturbance modeling, as the observer performs real-time estimation and mitigation of disturbances. consequently, the system can concentrate on its primary dynamics while the eso addresses disturbances. within the context of the eso is employed to estimate the output of the system rather than the disturbance itself. this arises because, in the framework of adrc, the observer is designed to estimate the system output directly. at the same time, the disturbance is inferred indirectly as a component of the total disturbance. this is achieved by analyzing the discrepancy between the observed output and the estimated output y(t) = ẑ1(t). subsequently, upon substituting ẑ1(t) with ŷ1(t), the model for the eso is adjusted accordingly [13–15]    ),(ˆ )0 1()(ˆ ),( )( 0 )(ˆ 0 1 )(ˆ )(ˆ 2 10 2 1 )(ˆ 2 1 tty ty l l tu b t l l tz tz t z z c lblca z =       +      +      − − =         −     (12) where the eso gain matrices are a  r2x2, b  r2x1, and c  r1x2, while l = [l1 l2]t exerts an influence on the spectrum of matrix (a–lc), and ẑ(t) is the estimate of z(t). the eigenvalues of the matrix (a–lc) are required to reside in the left half-plane because their positions influence the behavior of the control system. within this context, the feasible domain is denoted as dl. here, y(t) represents the controlled output obtained from an initial open-loop experiment, and ẑ1(t) = y(t), which is the primary output produced by the active disturbance rejection control and model-free control 189 observer estimating y(t). the initial open-loop experiment is executed under a dynamic regime that is congruent with the closed-loop control system’s operating conditions. furthermore, the input signal applied to the open-loop experiment must be rich in frequency content. the characteristic polynomial associated with the eso’s dynamic model is [14] .)( 21 2 lslss ++= (13) the stability of the eso is guaranteed by enforcing the characteristic polynomial in (13) to satisfy hurwitz conditions, thereby resulting in the establishment of constraints for determining the elements of l }.0 ,0|] {[ 21 2 21 = lllld t rl (14) the control strategy formulated for the adrc algorithm is ( ),)(ˆ)(ˆ 1 )(ˆ))(ˆ)(( 1 )( 2 0 2 )(ˆ 1 0 tzte b tztztr b tu te −=         −−=  (15) where ê(t) represents the estimated control error, and e(t) = r(t) – z1(t) = r(t) – y(t) denotes the control error, r(t) is the reference input (set-point), and κ is the adrc’s gain. the stability of the control system with the adrc algorithm and eso, denoted as kadrc = [κ l1 l2]t, is ensured by theorem 1 [14] with the proof given in [40]. theorem 1. taking into account the disturbed integrator model (10) alongside the disturbance term (9), in conjunction with the eso dynamic model (12), the adrc control system employing the control law (15) is stable if and only if }.0]/)()( 1 /)( 1 )( 1 [/ ,0/)()( 1 /)( 1 )( 1 ,0/ ,0/)()( 1 ,0 1 ,0 ,0|) {(, 021 2 11 22 10211202 021 2 11 22 102112 00211 12111adrc 3 adrcadrc ++++  ++  ++   ++++  ++  ++  +++    ++= bblllllbblllbbl bblllllbblll bbbblll llllldd t rk (16) in designing the adrc-frit algorithm, theoretical considerations from the current subsection, as well as those of frit in subsection 2.1, are meticulously incorporated. in the context of adrc-frit, for the initial experiment, the practitioner should set a value of κ, and therefore, the domain (16) will be reduced to dl(k) = {[l1 l2]t  r2 | [k l1 l2]t  dadrc}. controller tuning heuristics can be minimized by setting matrix l = [l1 l2]t as the solution l* = [ * 1l * 2l ]t to the optimization problem [14] .)),(( 1 )( ),(minarg 0 2 )( * 1 ==  ht h kd dtte t jj llll ll l l (17) the subsequent implementation of adrc-frit is realized by adhering to the steps outlined below: step 1.1. the designer defines the initial conditions of the open-loop experiment. 190 r.-c. roman, r.-e. precup, e. m. petriu step 1.2. the designer must assign a value other than zero to the estimated coefficient of the control input, denoted as b0. additionally, the designer should establish an appropriate value for κ, which represents the gain of the adrc algorithm (or controller gain), specifically κ(0), as the initial parameter of the adrc-frit algorithm. step 1.3. in the dynamic regime and context tailored to the optimization problem given in (17), constrained by (16), the optimization problem stated in equation (17) is solved using the newton-raphson method to determine the optimal eso parameter vector, l* = [ * 1l * 2l ]t. step 1.4. the adrc algorithm is validated in a closed-loop control experiment. this coincides with the adrc-frit initial experiment. step 1.5. a reference model m(s) is set to ensure the output meets the control system’s performance criteria. step 1.6. to determine the fictitious reference r(̃t, k) accordance with (3), the i/o data pair (u0(t, κ(0)), y0(t, κ(0))) is acquired. next, the fictitious reference model output ỹk(t, k) is derived in accordance with (2). step 1.7. to calculate the optimal parameter k* for the adrc-frit algorithm, the control error from the previous experiment is used as the benchmark input in the gradient experiment. this is carried out with (5), applying the newton-raphson method to solve the optimization challenge outlined in equation (1). step 1.8. an experiment with the control system’s closed-loop structure, employing the adrc algorithm with parameter k* optimally determined via frit, is conducted. the procedures outlined in steps 2.6-2.8 are repeated numerous times to improve the overall efficacy of the control system. the input/output data collected upon the completion of step 2.8 is subsequently employed in step 2.6 for a new iteration. 2.3. the mfc-frit algorithms the mfc algorithms incorporating a p component, commonly identified in contemporary research as the ip controller, are designed based on the first-order local process model similar to (7) [1,5,6,21–23,34,35] ),()()( tutfty += (18) where f(t) represents unmodeled dynamics and disturbances, u(t) and y(t) have the same significance as in the adrc case, while α > 0 for balancing ẏ(t) and αu(t). the mfc algorithm is predicated upon the subsequent specific control law ,/)]()()(ˆ[)( tetrtftu +−−=  (19) where, in the context of the mfc algorithm, κ represents the proportional constant, specifically the gain associated with the p component of the mfc algorithm. )(ˆ tf denotes the estimation of f(t) derived from the i/o data of u(t) and y(t). furthermore, r(t) holds an equivalent significance as observed within the framework of the adrc algorithm, and its first-order derivative is estimated through the utilization of a derivative plus low-pass filter, characterized by [1,5,6,21–23,34,35] ),1/()( stssh ff += (20) active disturbance rejection control and model-free control 191 where the time constant of the first-order derivative filter, denoted as tf, ought to be selected by the practitioner as a small positive value to ensure precise estimation of derivatives. the signals processed through hf(s) in (20), represented as ṙ(t) and ẏ(t), will henceforth be referred to as )(ˆ tr and )(ˆ ty , respectively. additionally, e(t) maintains the same definition as it does within the framework of the adrc algorithm. the derivation of )(ˆ tf is based on the first-order local process model equation given in (18), which is transformed as follows: ).( )(ˆ)(ˆ tutytf −=  (21) the stability of a control system utilizing the mfc algorithm is guaranteed if the characteristic polynomial, obtained by substituting the control law from equation (19) into the local process model delineated in equation (18) ,0)()( =− tete (22) exhibits roots situated within the left half-plane; therefore, κ < 0, and the estimation error denotes )(ˆ)()( tftft −= is assumed to be zero, and ė(t) = ṙ(t) – ẏ(t). the stability of the control system with the mfc algorithm kmfc = κ is ensured by theorem 2, with the proof taken from [41]. theorem 2. considering the first-order local process model as detailed in (18) alongside the control law delineated in (19), as well as the unmodeled dynamics and disturbances specified in (21), assuming δ(t) equals zero, and evaluating the characteristic polynomial of the error dynamics presented in (22), the stability of the control system is ensured if the roots of (22) reside in the left half-plane. therefore, the mfc control system employing the control law (19) is stable if and only if }.0|{,k mfcmfcmfc = dd r (23) the subsequent implementation of mfc-frit is realized by crossing to the outlined steps: step 2.1. the designer defines the initial conditions of the closed-loop experiment. step 2.2. the designer must assign a value other than zero to α. additionally, the designer should establish an appropriate value for κ, which represents the gain of the mfc algorithm (or controller gain), specifically κ(0), as the initial parameter of the mfc-frit algorithm. step 2.3. the designer should establish the derivative plus low-pass filter in (20) by setting a small positive value for tf. step 2.4. the mfc algorithm is validated in a closed-loop control experiment. this coincides with the mfc-frit initial experiment. steps 2.5 to 2.8 are identical to steps 1.5 to 1.8, as is the case with the adrc algorithm, and are applied for the case of the mfc algorithm. therefore, the procedures outlined in steps 2.6-2.8 are repeated numerous times to improve the overall efficacy of the control system. the input/output data collected upon the completion of step 2.8 is subsequently employed in step 2.6 for a new iteration. the block diagrams of the adrc-frit and mfc-frit algorithms are depicted in fig. 1 a) and b), respectively. 192 r.-c. roman, r.-e. precup, e. m. petriu fig. 1 the block diagrams of the adrc-frit in a) and mfc-frit in b). 3. the 3d crane the validation of the continuous-time first-order adrc-frit and mfc-frit is performed with experimental setups involving the 3d crane laboratory equipment, where control is applied to the x-, y-, and z-axes. details of the laboratory configuration and the mathematical model, highlighting the complex dynamics due to nonlinearity, are recorded in [36]. the 3d crane is operated using three control inputs ui  [–1, 1], i  {1, 2, 3}, which modulate the duty cycles of pulse width to control the dc motors, thereby actuating the x-, y-, and z-axes. specifically, x1(m) = y1(m) denotes the cart’s position along the x-axis, x3(m) = y2(m) corresponds to its position along the y-axis, and x9(m) = y3(m) represents the payload’s position along the z-axis. 4. the setup and results to have a fair comparison of the adrc-frit and mfc-frit algorithms, both are validated using experiments considering the same set-up, i.e., 70 s as time horizon, and the reference inputs (set-point) are )(tri obtained as the signals )(ti filtered through }3,2,1{),( 1 ish r : ],70,45(if0],45,30(if05.0],30,15(if1.0],15,0[0)( ],70,40(if0],40,25(if15.0],25,5(if15.0],5,0[0)( ],70,50(if0],50,35(if05.0],35,20(if1.0],20,0[15.0)( 3 2 1 −= −= −= ttttift ttttift ttttift    (24) ).3.01/(1)(),21.01/(1)(),2.01/(1)( 321 sshsshssh rrr +=+=+= (25) the effectiveness of the data-driven algorithms shall be evaluated utilizing the forthcoming performance index:   ++= ht h ue dttetete t kj 0 3 2 32 2 21 2 1, ,)],(),(),([ 1 )( (26) where the time horizon is th = 70 s, the subscript ◊ represents one of the following: adrc, adrc-frit, mfc, mfc-frit, each of which corresponds to the data-driven algorithms active disturbance rejection control and model-free control 193 discussed within this paper. additionally, the subscripts 1, 2, and 3 denote the parameters corresponding to the x-, y-, and z-axes, respectively. according to steps 1.1 to 1.4, the adrc algorithms are designed by considering the open-loop experiment a signal rich in frequency content (step 1.1). next, the practitioner assigns the values of b0 = 1 and κ, which will later be employed as κ(0) in the adrc-frit (step 1.2), obtaining κadrc1 = 1.3432, κadrc2 = 1.3541, and κadrc3 = 1.4135. by solving the optimization problem in (17) in terms of (16) using the newton-raphson method, the optimal parameters of eso are (step 1.3) l1 * = [192.1 9224.1]t, l2 * = [146.8 5388.3]t, l3 * = [172.4 7416.7]t. the validation of the closed-loop adrc algorithm coincides with the initial experiment of the adrc-frit algorithm, considering the reference inputs (set-point) described above for the closed-loop experiments (step 1.4). in the next phase, the designer sets the reference model m(s) chosen to meet the control system performance criteria, and in this case, m(s) is hr(s) as given in (25) (step 1.5). with the i/o data collected in step 1.4, the fictitious reference r(̃t, κ) and fictitious reference model output ỹk(t, k) are computed in terms of (3) and (2) (step 1.6). the optimal parameter κ* of the adrc-frit algorithm is determined after ten iterations by solving (1) using the newton–raphson method and obtaining (steps 1.7 and 1.8) κ* adrc-frit1 = 5.8716, κ* adrc-frit2 = 4.8671, κ* adrc-frit3 = 3.3211. according to steps 2.1 to 2.4, the mfc algorithms are designed by considering the closed-loop experiment, with the same reference inputs as above (step 2.1). next, the practitioner assigns the values of α = 1 and κ, later used as κ(0) in the mfc-frit algorithm (step 2.2), obtaining κmfc1 = –22.9093, κmfc2 = –21.0261, and κmfc3 = –29.0214. the designer establishes that tf = 0.001 for the low-pass filter in (20) (step 2.3). the validation of the closed-loop mfc algorithm coincides with the initial experiment of the mfc-frit algorithm (step 2.4). in the next phase, the designer uses the same reference model m(s) as in the adrc-frit chosen to meet the control system performance (step 2.5). with the i/o data collected in step 2.4, the fictitious reference r(̃t, κ) and fictitious reference model output ỹk(t, k) are computed in terms of (3) and (2) (step 2.6). the optimal parameter κ* of the mfc-frit algorithm is determined after ten iterations by solving (1) using the newton–raphson method and obtaining (steps 2.7 and 2.8) κ* mfc-frit1 = –33.9101, κ* mfcfrit2 = –31.0199, κ* mfc-frit3 = –49.0200. the outcomes of the real-time experiments are illustrated in figs. 1-3 within the supplementary material in [42], as well as in table 2. these outcomes represent the averages of ten distinct sets of real-time experiments conducted to mitigate random disturbances that may arise during an experiment. no additive disturbances were introduced in the real-time experiments. the experimental data distinctly indicate that adrc and mfc exhibit equivalent performance in the initial experiment. subsequent to determining the optimal parameters with frit, the experimental results derived from adrc-frit and mfc-frit surpass those of the initial closed-loop experiments. moreover, similar to the initial closed-loop experiment, adrc-frit and mfc-frit perform equivalently. however, concerning the 3d crane equipment, mfc and mfc-frit demonstrate slightly superior performance. however, these conclusions might be different if combinations of data-driven control with model-based control including fuzzy control [4,12,28,43], [42], and sliding mode control [44,45]. 194 r.-c. roman, r.-e. precup, e. m. petriu table 1 the average and the variance of the performance index in (27) adrc adrc-frit mfc mfc-frit average of je,u 9.8043∙10–4 3.5911∙10–4 5.9425∙10–4 3.3567∙10–4 variance of je,u 1.9881∙10–11 2.0014∙10–11 2.1015∙10–11 1.9988∙10–11 5. conclusions this paper aims to conduct a comparative analysis of two data-driven integrations, specifically the first-order continuous-time adrc-frit and mfc-frit algorithms, as validated through experimental testing on 3d crane laboratory equipment. the innovative aspects of this study include the integration of the first-order continuous-time adrc and mfc with frit, the determination of optimal parameters for adrc and mfc using frit by resolving the optimization problem in (1) via the newton–raphson method, and the validation of the first-order continuous-time adrc, adrc-frit, mfc, and mfc-frit algorithms on the 3d crane system. future research will concentrate on validating the proposed data-driven algorithms presented in this paper on other real-time devices, as well as on the integration and enhancement of data-driven algorithms such as adrc, mfc, or frit. acknowledgement: this work was supported by a grant of the ministry of research, innovation and digitization, cncs/cccdi uefiscdi, project number eranet-enuac-e-mats, within pncdi iv, and the nserc of 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mechanical engineering, 21(2), pp. 307-322. https://uptro29158-my.sharepoint.com/:f:/g/personal/raul-cristian_roman_upt_ro/%0beljhgkk30z1alwd0r0rnnp8btmpi742jm4v8s9mpgdbz2g?e=9roxjd https://uptro29158-my.sharepoint.com/:f:/g/personal/raul-cristian_roman_upt_ro/%0beljhgkk30z1alwd0r0rnnp8btmpi742jm4v8s9mpgdbz2g?e=9roxjd https://uptro29158-my.sharepoint.com/:f:/g/personal/raul-cristian_roman_upt_ro/%0beidrugdukwlgtba5ybq0iqgbqano8m91guxs8t1frqw8la?e=ygqxw4 https://uptro29158-my.sharepoint.com/:f:/g/personal/raul-cristian_roman_upt_ro/%0beidrugdukwlgtba5ybq0iqgbqano8m91guxs8t1frqw8la?e=ygqxw4 12896 facta universitatis series: mechanical engineering vol. 23, no 3, 2025, pp. 555 578 https://doi.org/10.22190/fume240820051r © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology in the logistics industry pratibha rani1, arunodaya raj mishra2, ahmad m. alshamrani3, adel fahad alrasheedi3, predrag atanasković4, vladimir simic5,6,7 1saveetha school of engineering, saveetha institute of medical and technical sciences (simats), chennai, tamil nadu, india 2department of mathematics, government college raigaon, madhya pradesh, india 3department of statistics and operations research, college of science, king saud university, riyadh, saudi arabia 4department of traffic engineering, university of novi sad, novi sad, serbia 5faculty of transport and traffic engineering, university of belgrade, serbia 6yuan ze university, college of engineering, department of industrial engineering and management, taoyuan city, taiwan 7department of computer science and engineering, college of informatics, korea university, seoul, republic of korea orcid ids: pratibha rani https://orcid.org/0000-0002-9186-4167 arunodaya raj mishra https://orcid.org/0000-0001-9949-5813 ahmad m. alshamrani https://orcid.org/0000-0001-7103-8872 adel fahad alrasheedi https://orcid.org/0000-0003-4492-1082 predrag atanasković https://orcid.org/0000-0002-1715-8256 vladimir simic https://orcid.org/0000-0001-5709-3744 abstract. blockchain technology (bt) has appeared as a promising solution for addressing several challenges in the supply chain and logistics. digital transformation in supply chain and logistics can drive significant business growth by updating outdated systems. though, the bt-enabled digital transformation concerning bt for a logistics industry meets the numerous challenges of choosing a suitable blockchain platform for needs of logistics industry. in this paper, we introduce an integrated intuitionistic fuzzy information-based decision support system for selecting and assessing the blockchain platforms in the logistics industry. this method computes the decision experts’ weight and combined weight of criteria with score function-rank sum model and model based on the removal effects of criteria (merec) on intuitionistic fuzzy sets (ifss). in the following, a novel distance measure is proposed for ifss to measure the degree of composite distance of alternatives. next, the developed framework is implemented to a case study of blockchain received: august 20, 2024 / accepted december 18, 2024 corresponding author: pratibha rani saveetha school of engineering, saveetha institute of medical and technical sciences (simats), chennai602105, tamil nadu, india e-mail: pratibha138@gmail.com https://orcid.org/0000-0002-9186-4167 https://orcid.org/0000-0001-9949-5813 https://orcid.org/0000-0001-7103-8872 https://orcid.org/0000-0003-4492-1082 https://orcid.org/0000-0002-1715-8256 https://orcid.org/0000-0001-5709-3744 556 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. platforms assessment problem. sensitivity assessment and comparison with existing methods are discussed for determining the utility and advantages of proposed approach. key words: intuitionistic fuzzy sets, blockchain, logistics, supply chain, distance-based approach, multiple criteria decision-making 1. introduction blockchain is described as a safe, distributed and decentralized database that enables the procedure of recording transactions and tracking resources in a business network. it is a technology which combines several features including distributed notes, smart contracting, storage device and decentralized system, consensus procedure, asymmetric encryption to ensure visibility, network security and transparency [1]. digital signature in blockchain provides a secure and tamper-evident way to safeguard individuals’ online identity and authenticate the transactions [2]. blockchain technology (bt) provides a solution to improve the transparency and immutability in data-driven decision-making which builds trust among stakeholders, as they can independently preserves confidentiality, integrity and availability of all the transactions and data in the blockchain [3]. in the business operations, all latest transaction in blockchain is stored in an immutable block and timestamped to preserve track of the particular product in the end-to-end chain [4-7]. due to its unique features, it has been globally recognized as a powerful system for enabling transparency and trust in data transactions [3]. logistics is the process that comprises the management of arrangements of goods, services, and essential information from production location to the point of consumption [8-9]. it has a significant impact on business, though it faces some challenges that impact the efficacy and effectiveness of supply chain operations. digital transformation (dt) revolutionizes logistics by offering supply chain visibility and powers significant cost savings across logistics operations [10]. bt’s integration into dt facilitates greater efficiency in the business models by enabling the higher levels of transparency and safe exchange of data in logistics supply chain [11-12]. jain et al. [13] gave a study that explores the role of blockchain in logistics industry. moreover, they presented a model for testing the users’ acceptance of bt in logistics and supply chain. based on the comprehensive review, berneis et al. [14] identified the economic advantages of bt implementation in logistics industry. batta et al. [15] examined the actual level of adoption and diffusion of bt in logistics and transportation industries. tan et al. [16] formed a prototype bt-based logistics system with its impact in a real logistics industry. zeng et al. [17] presented a btenabled traceability system for cold-chain pharmaceutical logistics. considering the multiple criteria/factors and decision experts (des), choosing an appropriate blockchain platform for the logistics industry can be defined as a multiple criteria group decisionmaking (mcgdm) problem. as a goal-oriented process, mcgdm approaches allow for a comprehensive evaluation of alternatives based on multiple criteria that may have different weights in the decisionmaking procedure [18-19]. uncertainty is a major issue that may occur during the assessment of alternative in the practical mcgdm problems [20-21]. in general, the classical mcgdm methods are failed to handle the uncertainty of realistic situations. to deal the uncertain situation, zadeh [22] pioneered the notion of fuzzy set (fs), which offer a powerful and flexible tool to handle the ambiguous data [23]. çıkmak et al. [24] used the intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 557 fs-based model to study the effects of blockchain characteristics in the supply chain management (scm). chen et al. [25] presented a fuzzy large-scale dematel model to analyze the influence of bt in humanitarian supply chain. hamidi et al. [26] studied an integrated fuzzy decision support tool for assessing the readiness of bt execution in the maritime logistics industry. hussain and ullah [27] presented a model using sugenoweber operators on spherical fuzzy sets to solve real-life mcgdm problem. sahoo et al. [6] studied comprehensive bibliometric assessment of investigation on material selection problem with various mcgdm models considering the publications from 2010 to 2024. as a refinement of fs, atanassov [28] proposed the idea of intuitionistic fuzzy set (ifs) in which an element is considered by degree of membership (md) (μ) and non-membership (nd) (ν) with a constraint μ + ν < 1. the introduction of ifs theory into mcgdm approaches has increased the practical potential and offered new insights during the procedure of solving decision-making problems [29-33]. wan et al. [34] integrated if-best worst method (bwm) with additive consistency. based on this method, they proposed a non-linear programming model for solving decision-making problems. bajaj & kumar [35] studied a new correlation measure on ifss and implemented to the pattern recognition, medical diagnosis and clustering problems. further, an integrated decision support tool has been to assess a set of options on ifss. majumder et al. [36] gave an intuitionistic fuzzy mcdm model for deriving the significance values of criteria. moreover, they used a unique decision support system to assume a feasibility assessment of solar power plant. şahin et al. [37] used two methods, geographic information measure and ifss-based method to evaluate and rank the locations for solar-wind power plant establishment in netherlands. till now, there is no study that assesses the blockchain platforms in the logistic industry using ifss-based decision-making model. the algorithm of “distance-based approach (dba)” commences with describing the optimum state of total objective, and determines the preferably good ratings of considered criteria during the assessment of alternatives. because of the simple and easy-to-use computational steps, it has been analyzed as a flexible and effective technique for solving mcgdm problems from different perspectives [33,38-39]. for instance, garg et al. [40] set forth an innovative fuzzy dba model for evaluating and ranking the data base management system commercial off-the-shelf components under fuzzy environment. sandhya et al. [41] evaluated the cloud service provider selection problem through classical distance-based approach and ranked the service provider alternatives according to the composite distances. garg & garg [42] used a modified dba model to evaluate, select and rank the robots by means of various aspects and fuzzy information. in addition, their results have validated through comparison with four different approaches. to efficiently select the suitable blockchain platform in the logistics industry, this paper introduces an integrated mcgdm approach based on dba, method using removal effects of criteria (merec) and rank sum (rs) model with intuitionistic fuzzy information (ifi), and named as the “if-merec-rs-dba”. in the proposed framework, the des’ weights are computed through a combined if-score function and rs model-based approach, while weights of criteria are computed via an integrated weighting model considering the objective weight with if-merec tool and subjective weight with if-rs method. combining all these aspects, the proposed dba offers an effective way to evaluate and select the blockchain platforms in the logistics industry. to compute the composite distance matrix in if-merec-rs-dba approach, we propose a new distance measure for ifss that avoids the shortcomings of existing intuitionistic fuzzy distance measures by iqbal & 558 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. rizwan [43], wu et al. [44], ejegwa & agbetayo [45] and li et al. [46]. to the best of authors’ knowledge, no one has combined the dba model with merec and rs model with ifi setting. the combination of objective and subjective weighting methods cannot only compute the criteria weights based on the quantitative data but also consider the des’ opinions during the assessment of criteria weights. to evaluate the blockchain platforms, existing studies have not demonstrated how the combination of objective and subjective weighting approaches impacts the decision results. furthermore, few studies assume the direct weights of attributes without considering the objective weight or subjective weight, which may cause information loss. the novelties and key contributions of the paper are discussed as ▪ this paper proposes a new distance measure for ifss, which quantifies the degree of dissimilarity between ifss. ▪ to solve the mcgdm problems, a hybrid if-merec-rs-dba model is proposed wherein the information about the des and criteria is fully unknown. ▪ in this model, a combined if-score function and if-rs method-based formula is presented to determine weight of des, while an integrated weighting model for criteria is discussed which combines if-merec for objective weight of factors and if-rs approach for subjective weight of factors on ifi. ▪ the developed framework is used on a case study of blockchain platform selection problem with ifi. other sections are prepared as follows. section 2 shows the basic idea and developed if-distance measure. section 3 develops an integrated if-merec-rs-dba model for dealing mcgdm problems. section 4 shows the application of developed model to a case study of blockchain platform selection problem. this section further discusses the sensitivity assessment and comparison with extant mcgdm methods within the context of ifss. section 5 concludes this study and indicates some future research directions. 2. proposed distance measure for ifss this section gives background of the work and develops new if-distance measure to calculate degree of dissimilarity on ifss. 2.1. basic concepts this section shows the fundamental concepts about this work. definition 2.1 [28]. let us consider a finite discourse set ω = {ɩ1, ɩ1, …, ɩp}. atanassov [28] presented the mathematical definition of an ifs e on ω given as ( ) , ( ), ( ) : ,     = k e k e k ke (1) where μe: ω → [0, 1] denotes an md and ve: ω → [0, 1] signifies an nd of an element ɩk to e in ω, satisfying that 0 ≤ μe (ɩk), ve (ɩk) ≤ 1, for each ɩk ∈ ω. for each ɩk ∈ ω, the hesitancy degree is defined as πe (ɩk)= 1μe (ɩk) ve (ɩk). for simplicity, the term “(μe, νe)” is defined as an ‘intuitionistic fuzzy number (ifn)’ [47] and can be denoted as β = (μ, ν). definition 2.2 [48]. for an ifn β = (μ, ν), if-score function and if-accuracy function are defined via eq. (2) and eq. (3), respectively. intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 559 1 ( ) (1 ), 2 o  = −+ (2) where o(β) ∈ [0, 1], ( ) ,  = +a (3) where a∈ (β) [0, 1]. definition 2.3 [47]. to fuse the individual opinions into a group decision, xu [47] introduced the ifwa and ifwg operators for ifns, and given in eq. (4) and eq. (5), respectively. ( )1 2 1 1 1 ( , ,..., ) 1 1 , ,k k p pp w w p k k k k k k k ifwa w      = = =   =  = − −      (4) ( )1 2 1 1 1 ( , ,..., ) , 1 1 ,kk k p pp ww w p k k k k k k ifwg       = = =   =  = − −      (5) where βk = (μk, νk), k = 1, 2, …, p shows ifn and wk is relative weight of ifn βk satisfying wk is non-negative and w1+w2+…+wp =1. definition 2.4 [49]. let e, g, h ifss(ω). a real-valued function ϖ: ifss(ω)×ifss(ω) → [0,1] is said to be a distance measure for ifss if it holds the following requirements: (i). 0 ≤ ϖ (e, g) ≤ 1, (ii). ϖ (e, g) if and only if e = g, (iii). ϖ (e, g) = ϖ (g, e), (iv). if , e g h then ϖ (e, g) ≤ ϖ (e, h) and ϖ (g, h) ≤ ϖ (e, h). 2.2. proposed distance measure on ifss as a mathematical tool, the concept of distance measure is used to compute degree of discrimination between two objects. it has been widely employed in medical diagnosis, pattern recognition and decision-making. in this subsection, we present new distance measure for ifss. theorem 2.1. for e, g ifss(ω), new distance measure on ifss (if-dm) is defined as ( ) ( ) ( ) ( ) 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( )3 . 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 , p e k g k e k g k e k g k e k g k e k g k e k g kk p e g                          =  − − −  + +  + + + + + +   =  (6) proof: (i) for two ifss e and g, we have 0 ≤ μe(ɩk), μg(ɩk) ≤ 1, 0 ≤ ve(ɩk), vg(ɩk) ≤ 1 and 0 ≤ πe(ɩk), πg(ɩk) ≤ 1, for each ɩk ∈ ω. it implies that 0 ≤ μe(ɩk) μg(ɩk) ≤ 1, 0 ≤ ve(ɩk) vg(ɩk) ≤ 1 and 0 ≤ πe(ɩk) πg(ɩk) ≤ 1, for each ɩk ∈ ω. it implies that ( ) ( ) ( ) 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( )3 0 1. 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 p e k g k e k g k e k g k e k g k e k g k e k g kk p                         =  − − −   + +   + + + + + +    hence, 0 ≤ ϖ (e, g) ≤ 1. 560 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. (ii)-(iii) these properties are obvious by the definition; therefore, we have omitted the proof. (iv) since e, g, h ifss(ω) and , e g h thus, 0 ( ) ( ) ( ) 1,        e k g k h k 0 ( ) ( ) ( ) 1        h k g k e k and 0 ( ) ( ) ( ) 1, .           e k g k h k k from eq. (6), we have ( ) ( ) ( ) ( ) 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( )3 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 , p e k g k e k g k e k g k e k g k e k g k e k g kk p e g                          =  − − −  + +  + + + + + +   =  ( ) ( ) ( ) 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( )3 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 p e k h k e k h k e k h k e k h k e k h k e k h kk p                         =  − − −   + +  + + + + + +    ( ), .= e h similarly, we can prove that ϖ (e, h) ≥ ϖ (g, h), as . e g h definition 2.5. suppose that e, g ifss(ω), then the weighted distance measure dw:ifss(ω)×ifss(ω) → [0,1] is given by ( ) ( ) ( ) ( ) 2 2 2 1 ( ) ( ) ( ) ( ) ( ) ( )3 . 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 , p e k g k e k g k e k g k w k e k g k e k g k e k g kk we g                          =  − − −  + +  + + + + + +   =  where wk is the weight of ɩk on ω satisfying wk  [0, 1] and 1 1. = = p kk w further, we apply the proposed and existing if-dms on some data sets and find some useful results. example 2.1. consider four different cases of ifss, which are given as follows: set-i: {e1 = (0.7,0.3), g1 = (0,0), set-ii: {e2 = (0.4,0.6), g2 = (0,0)}, set iii: {e3 = (0.3,0.41), g3 = (0.5,0.344)} and set-iv: {e4 = (0.41,0.2), g4 = (0.22,0.28)}. here, we apply the proposed and existent if-dms (iqbal & rizwan [43], wu et al. [44], ejegwa & agbetayo [45], li et al. [46]) to compute the distance between given pairs of each set. for two different sets (set-i and set-ii), the if-dms by ejegwa & agbetayo [45] and li et al. [46] failed to describe the difference between given pairs in set-i and set-ii, i.e., ϖea (e1, g1) = 1.0 = ϖea (e2, g2) and ϖl (e1, g1) = 1.0 = ϖl (e2, g2), where e1 ≠ e2, g1 = g2. in this case, the proposed if-dm present the reasonable result and given as ϖ (e1, g1) = 0.912 and ϖ (e2, g2) = 0.899. when we compare the pairs of set-iii and set-iv, the if-dms by iqbal & rizwan [43] and wu et al. [44] obtain unreasonable results as ϖir (e3, g3) = 0.013 = ϖir (e4, g4) and ϖw (e3, g3) = 0.139 = ϖw (e4, g4), where e3 ≠ e4, g3 ≠ g4. while the proposed measure successfully compares the given pairs of set-iii and set-iv as ϖ (e3, g3) = 0.187 and ϖ (e4, g4) = 0.175. intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 561 3. proposed if-merec-rs-dba methodology for mcgdm problems this section proposes a hybrid if-merec-rs-dba framework to solve the mcgdm problems under ifss environment. for a ifss-based mcgdm problem, create an expert committee f = {f1, f2, ..., fn} to evaluate the alternatives’ set u = {u1, u2, ..., ur} with respect to the criteria set v = {v1, v2, ..., vt}. each de presents his/her linguistic opinion regarding the performance of each alternative by means of considered evaluation criteria. consequently, we obtain a linguistic assessment matrix (lam) y = (yij (k))r×t , k = 1, 2, …, n, where yij (k) denotes the linguistic variable (lv) of option ui over a criterion vj presented by kth dme and further, we make an if-decision matrix (ifdm). in the following, we present the procedural steps of introduced if-merec-rs-dba framework (see fig. 1): fig. 1 graphical structure of the proposed if-merec-rs-dba model step 1: determination of des’ weights. let fk = (μk, νk), k = 1, 2, …, n be the intuitionistic fuzzy significance rating of kth expert. then, a weighting formula to find the weight of kth expert is given by 562 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. ( ) ( ) ( ) 1 1 2 11 , 2 2 1        = =  − − − +  = +  − − − +     k k k k k n n k k k kk k n r n r (7) where rk signifies the preference of expert, k = 1, 2, …, n. moreover, λk is non-negative and λ1 + λ2 +…+ λn = 1. step 2: creation of an aggregated ifdm (a-ifdm). to get the individual opinion by combining different des’ opinions, we apply the ifwa operator (or ifwg operator) on the ifdm and obtain the a-ifdm z = (zij)r×t, where ( ) ( )(1) (2) ( ) (1) (2) ( )( , ) , y ,..., or , ,..., .n n ij ij ij ij ij ij ij ij ijz yifw y y y ya ifwg = = (8) step 3: computation of criteria weights using if-merec-rs model. assume that w = (w1, w2, ..., wt)t be the weight vector of criteria set, satisfying wj [0, 1] and w1 + w2 +…+ wt = 1. to determine the criteria weights, we discuss an integrated weighting procedure combining the objective and subjective weights of criteria through ifmerec and if-rs models, respectively. case i: finding the objective weight with if-merec. to determine objective weight of criteria, merec (keshavarz-ghorabaee et al. [50]) is applied on ifi setting. here, process of if-merec is given as step 3a: normalization of an a-ifdm. this step involves linear normalization for the purpose of scaling the elements of the a-ifdm z = (zij)r×t and creates the normalized a-ifdm (na-ifdm) n = (ςij)r×t. if vb denotes the set of beneficial types of criteria and vn stands for the non-beneficial criteria set, then eq. (9) is applied for normalization. ( , ), , ( , ) ( ) ( , ), . ij ij ij b ij ij ij c ij ij ij n z j v z j v        =  = =  =  (9) step 3b: create the if-score matrix (if-sm). with the use of score function given by eq. (2), the if-sm ω = (ηij)r×t is created, where ( )0.5 ( ) ( ) 1 . ij ij ij  = − + step 3c: calculation of overall performance of options. the normalized ratings achieved from the step 3b make sure that the smaller ratings of ηij obtain greater performance ratings. here, eq. (10) is applied to compute overall performance of ith option. 1 1 ( ) , .i ij j q ln ln i t     = +         (10) step 3d: finding the performance of option by removing each criterion. let qij stands the overall performance of ith option by the removal of jth criterion. this process is done using eq. (11). , 1 1 ( ) .ij il l l j q ln ln t      = +        (11) intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 563 step 3e: calculate the sum of absolute deviations using eq. (12). .= −j ij i i d q q (12) step 3f: determination of objective weight of jth criterion from eq. (13), where j = 1, 2, …, t. 1 , 1,2,..., . = = =  jo j t j j d w j t d (13) case ii: estimation of subjective weight via if-rs model. step 3g: using ifwa operator, aggregate the ifns provided by the des and obtain an aggregated column matrix b = (bj)1×t. using eq. (2), find if-score value of a-ifn of column matrix b = (bj)1×t. step 3h: on the basis of decreasing score values, determine the preference (mj) of jth criterion and compute weight of criterion using 1= − +s j jw t m , j = 1, 2, 3, ..., t. finally, derive the normalized subjective weight of criteria with eq. (14.) as 1 , = =  s js j t s j j w w w j = 1, 2, …t. (14) case iii: estimation of integrated weight of criteria. we combine the outcomes determined by if-merec and if-rs models. the process of finding combined weight of criteria is given by (1 ) ,o s j j jw w w = + − (15) where satisfying τ [0, 1] is a weight strategy parameter. if τ = 1, then the objective weight by if-merec is fully considered to estimate the criteria weights, while subjective weight of criteria is completely overlooked. conversely, when τ = 0, the subjective weight by ifrs model is fully considered for deriving the criteria weights, while objective weight of criteria is completely neglected. when τ = 0.5, then the combined weight of criteria is estimated as the arithmetic mean of objective and subjective weights of attributes/criteria. step 4: create the weighted na-ifdm. on the basis of obtained criteria weights and eq. (16), the weighted na-ifdm nw = (ξij)r×t. where ˆˆ( , )ij ij ij  = is an aggregated ifn. 1 (1 ) , .j jw w ij j ij ij ijw    =  = − −   (16) step 5: compute the if-ideal solution (if-is) matrix m = (mj)1×t using ( ) ( ) 1 max ( ) , , ( , ) min ( ) , . ij b i j ij ij t ij b i s j v m s j v         = =  (17) 564 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. step 6: analyze the composite distance matrix cd = (cdij)r×t of each alternative from the if-is using eq. (18). 2 2 2ˆˆ ˆ( ) ( ) ( )3 . ˆˆ ˆ2 2 2 2 j ij j ij j ij ij j ij j ij j ij cd              − − −  = + +  + + + + + +   (18) step 7: computation of assessment score (as) li, i = 1, 2, 3, ..., r, of options based on eq. (19) as 1 1 , 1,2,..., . =    = − =      t i ij j l cd i r (19) step 8: prioritize the options based on descending ratings of li, i = 1, 2, 3, ..., r. the option with the maximum assessment score is the most desirable option. 4. results and discussion in this section, we first implement the developed approach on a case study of blockchain platform selection problem. further, we discuss the sensitivity assessment over varying values of criteria weights to test the robustness of introduced method. 4.1. case study: blockchain platform selection in the logistics industry as a promising and revolutionary technology, blockchain helps to ensure the authenticity of information and transparency during upstream transactions in a business network. in a logistics supply chain, blockchain records each step of a product’s journey on an immutable ledger and ensures that each transaction is securely stored and cannot be changed retroactively [51]. it can increase trust, security, transparency among associate organizations by enhancing traceability of data across a business system [52]. new technologies like as internet of things (iot), blockchain, cloud computing (cc) and big data (bd), significantly streamline the logistics process and improve its efficiency [8]. this subsection utilizes the developed framework on a case study of blockchain platform selection with respect to the multiple criteria. to collect the data for assessment, we have organized in-person meetings with the experts; though we have invited ten des, out of which four des, including a chief marketing manager of the company, a chief executive officer, a technology expert and an environmental expert, accepted to cooperate with us for preparing the questionnaires. in the expert committee, the des are having more than 10 years of expertise in the respective disciplines and provided their views in taking an appropriate decision. based on the experts’ opinions, six blockchain platforms are identified as alternatives for this study, which are corda r3 (u1), linux (u2), hydrachain (u3), ibm (u4), chain inc. (u5) and microsoft (u6). considering the literature and des’ views, 17 different factors/criteria are recognized and shown in table 1. intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 565 table 1 details of considered criteria for bt selection factors references type transparency sanka & cheung [53], dabbagh et al. [54], ronaghi [55], siddiqui & haroon [56] benefit speed dabbagh et al. [54], ronaghi [55], chan et al. [57], benefit stability siddiqui & haroon [56], pathak et al. [3] benefit audit gorçün et al. [8], pathak et al. [3] benefit accuracy xu et al. [58], ronaghi [55] benefit productivity siddiqui & haroon [56], gorçün et al. [8], pathak et al. [3] benefit traceability sanka & cheung [53], dabbagh et al. [54], ronaghi [55], siddiqui & haroon [56], gorçün et al. [8], pathak et al. [3] benefit security sanka & cheung [3], dabbagh et al. [54], chan et al. [57], ronaghi [55], siddiqui & haroon [56], pathak et al. [3] benefit cost reduction gorçün et al. [8], pathak et al. [3] benefit flexibility siddiqui & haroon [56], pathak et al. [3] benefit network availability xu et al. [58], dabbagh et al. [54] benefit scope siddiqui & haroon [56], pathak et al. [3] benefit accountability xu et al. [58], ronaghi [55] benefit planning gorçün et al. [8], pathak et al. [3] benefit privacy rana et al. [59], xu et al. [58], jabbar & dani [60], dabbagh et al. [54], chan et al. [57], siddiqui & haroon [56], pathak et al. [3] benefit reliability siddiqui & haroon [56], gorçün et al. [8], pathak et al. [3] benefit fraud prevention lai & liao [61], jabbar & dani [60], pathak et al. [3] benefit the experts’ committee has planned some strategies to evaluate these six blockchain platforms on the basis of considered evaluation criteria. here, table 2 (adopted from [30,31]) shows the linguistic ratings (lrs) and related to ifns. table 3 shows linguistic opinions of four des to assess the bt platforms in the logistics industry over given criteria. in the subsequent steps, we present the computational steps of the developed ifmerec-rs-dba framework in order to prioritize the given six options over considered 17 assessment factors. steps 1-2: by means of table 2 and eq. (7), the weights of four des are calculated and mentioned in table 4. to fuse the individual opinions into a combined opinion of each alternative over diverse criteria, the ifwa operator is applied on intuitionistic fuzzy decision matrix (ifdm) obtained by table 2 and table 3, and therefore, an a-ifdm is created and given in table 5. table 2 lrs and related ifns for criteria and des lrs ifns extremely good (eg) (0.95, 0.05) very very good (vvg) (0.85, 0.10) very good (vg) (0.80, 0.15) good (g) (0.70, 0.20) slightly good (mg) (0.60, 0.30) average (a) (0.50, 0.40) slightly low (ml) (0.40, 0.50) low (l) (0.30,0.60) very very low (vl) (0.20, 0.70) very low (vvl) (0.10, 0.80) extremely low (el) (0.05, 0.95) 566 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. table 3 lam of blockchain platform options over diverse criteria u1 u2 u3 u4 u5 u6 v1 (mg,g,ml,m) (m,g,mg,m) (m,vg,m,g) (m,g,vg,g) (ml,mg,l,m) (mg,g,mg,m) v2 (m,m,ml,vl) (mg,g,m,mg) (m,m,mg,m) (ml,ml,g,ml) (m,g,m,ml) (ml,g,mg,m) v3 (l,g,g,vg) (m,l,ml,g) (ml,m,vl,mg) (mg,l,m,vg) (ml,m,vl,m) (g,ml,m,g) v4 (g,vl,m,l) (l,l,ml,mg) (g,l,ml,m) (g,mg,l,vl) (mg,l,l,g) (vl,g,m,l) v5 (mg,m,g,g) (mg,mg,l,m) (vg,l,m,vg) (m,vvg,g,g) (ml,m,vl,mg) (mg,vg,vl,ml) v6 (ml,mg,vl,m) (g,vg,mg,m) (g,mg,m,mg) (ml,m,m,ml) (ml,m,m,l) (l,vg,mg,mg) v7 (g,ml,ml,l) (ml,l,mg,m) (m,g,m,ml) (m,m,vl,ml) (g,vl,m,ml) (ml,m,l,m) v8 (g,m,vvg,vg) (mg,l,m,mg) (l,m,g,ml) (mg,g,l,vl) (g,g,ml,vg) (vl,g,mg,mg) v9 (ml,m,vg,g) (m,m,ml,g) (mg,l,vg,mg) (m,vvg,g,m) (ml,g,vg,m) (m,vg,g,vg) v10 (ml,ml,mg,g) (g,m,mg,ml) (m,mg,ml,mg) (mg,g,m,vl) (ml,m,l,vl) (mg,vg,m,g) v11 (m,mg,ml,l) (l,g,vg,m) (vg,g,m,mg) (m,mg,g,m) (mg,m,l,vl) (mg,m,vg,g) v12 (m,l,mg,ml) (g,g,vg,m) (l,g,m,mg) (m,mg,m,mg) (g,g,ml,vl) (ml,mg,l,ml) v13 (mg,m,g,g) (m,m,mg,ml) (ml,m,ml,m) (ml,g,m,vl) (vg,g,ml,l) (m,mg,l,ml) v14 (ml,m,g,mg) (g,l,mg,mg) (vl,g,mg,m) (g,vvg,g,m) (ml,mg,vg,vl) (mg,vg,l,l) v15 (m,l,vvg,g) (g,m,ml,vvg) (ml,l,g,mg) (mg,m,m,ml) (mg,g,mg,m) (m,vg,mg,m) v16 (mg,m,ml,vl) (m,mg,g,vg) (ml,mg,g,g) (vg,m,ml,l) (mg,m,ml,mg) (mg,vg,mg,l) v17 (vg,ml,m,ml) (vl,mg,ml,m) (m,ml,m,mg) (mg,l,m,vg) (m,g,mg,l) (m,ml,g,mg) table 4 computational results for weight of des des f1 f2 f3 f4 lrs vg vvg eg g if-score values 0.840 0.8925 0.950 0.770 rk 3 2 1 4 weight 0.2217 0.2793 0.3376 0.1615 table 5 a-ifdm for assessing the blockchain platforms u1 u2 u3 u4 u5 u6 v1 (0.561, 0.333) (0.598, 0.299) (0.644, 0.272) (0.707, 0.212) (0.452, 0.445) (0.617, 0.281) v2 (0.426, 0.472) (0.602, 0.295) (0.536, 0.363) (0.525, 0.367) (0.554, 0.342) (0.581, 0.314) v3 (0.661, 0.244) (0.462, 0.432) (0.411, 0.485) (0.549, 0.359) (0.390, 0.508) (0.567, 0.326) v4 (0.463, 0.428) (0.393, 0.504) (0.478, 0.414) (0.493, 0.397) (0.461, 0.431) (0.492, 0.398) v5 (0.631, 0.266) (0.499, 0.397) (0.613, 0.308) (0.723, 0.192) (0.411, 0.485) (0.555, 0.357) v6 (0.427, 0.468) (0.679, 0.237) (0.595, 0.302) (0.464, 0.436) (0.450, 0.449) (0.627, 0.288) v7 (0.473, 0.420) (0.470, 0.427) (0.554, 0.342) (0.397, 0.501) (0.476, 0.416) (0.417, 0.482) v8 (0.744, 0.183) (0.496, 0.401) (0.533, 0.359) (0.501, 0.388) (0.645, 0.260) (0.570, 0.323) v9 (0.648, 0.27) (0.510, 0.386) (0.630, 0.288) (0.699, 0.215) (0.669, 0.249) (0.719, 0.205) v10 (0.532, 0.363) (0.574, 0.323) (0.518, 0.380) (0.555, 0.338) (0.371, 0.527) (0.661, 0.255) v11 (0.473, 0.425) (0.657, 0.259) (0.659, 0.253) (0.605, 0.292) (0.425, 0.471) (0.678, 0.241) v12 (0.475, 0.421) (0.716, 0.203) (0.549, 0.344) (0.547, 0.352) (0.556, 0.334) (0.436, 0.461) v13 (0.631, 0.266) (0.522, 0.376) (0.446, 0.453) (0.513, 0.379) (0.603, 0.305) (0.458, 0.439) v14 (0.577, 0.317) (0.561, 0.333) (0.554, 0.339) (0.732, 0.184) (0.613, 0.305) (0.564, 0.349) v15 (0.663, 0.251) (0.609, 0.296) (0.536, 0.356) (0.510, 0.389) (0.617, 0.281) (0.641, 0.276) v16 (0.454, 0.443) (0.659, 0.249) (0.621, 0.274) (0.542, 0.37) (0.512, 0.386) (0.639, 0.276) v17 (0.558, 0.355) (0.446, 0.451) (0.493, 0.406) (0.549, 0.359) (0.576, 0.319) (0.573, 0.322) step 3: to derive the objective weight through if-merec, the first step is to normalize the a-ifdm. however, all considered criteria are benefit-type, thus, there is no requirement to normalize a-ifdm and consequently there is no use of eq. (9). next, we find if-score value of each a-ifdm object of table 5. intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 567 fig. 2 variation of criteria weight by if-merec-rs tool for bt platforms assessment on the basis of if-score values, overall performance of each option is derived with eq. (10) and given as q1 = 0.418, q2 = 0.413, q3 = 0.416, q4 = 0.404, q5 = 0.458 and q6 = 0.393. using eq. (11), overall performance of ith option by removing jth criterion is estimated in table 6, where i = 1, 2, 3, ..., 6 and j = 1, 2, 3, ..., 17. further, the sum of the deviations is determined via eq. (12) and given in last second column of table 6. lastly, objective weight of criteria is obtained through eq. (13) and shown in last column of table 6, which as wj o = (0.0474, 0.0571, 0.0640, 0.0710, 0.1116, 0.0580, 0.0715, 0.0502, 0.0403, 0.0583, 0.0505, 0.0564, 0.0593, 0.0468, 0.0474, 0.0520, 0.0584). to find subjective weight of criteria with rs approach [62] on ifss as if-rs, the first step is to obtain the performance rating of each attribute by different expert. further, individual performances are aggregated through ifwa operator and find if-score rating of each a-ifn. table 7 presents computational steps of if-rs model. applying eq. (14), objective weight of criteria is calculated via eq. (13) and shown in last column of table 7, which as wj s = (0.0654, 0.0719, 0.0392, 0.0588, 0.0065, 0.0915, 0.0458, 0.0327, 0.1111, 0.0784, 0.0131, 0.0196, 0.0523, 0.0261, 0.1046, 0.0850, 0.0980). by integrating the objective and subjective weights of factors using eq. (15), combined weight for τ = 0.5 is computed and presented as wj = (0.0564, 0.0645, 0.0516, 0.0649, 0.0590, 0.0747, 0.0586, 0.0414, 0.0757, 0.0684, 0.0318, 0.0380, 0.0558, 0.0365, 0.0760, 0.0685, 0.0782). 568 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. table 6 computational steps of if-merec tool to find objective weight criteria qij ’ values dj wj o u1 u2 u3 u4 u5 u6 v1 0.399 0.396 0.401 0.393 0.432 0.376 0.104 0.0474 v2 0.389 0.397 0.395 0.383 0.439 0.374 0.125 0.0571 v3 0.404 0.387 0.385 0.384 0.427 0.373 0.141 0.0640 v4 0.392 0.381 0.391 0.380 0.433 0.368 0.156 0.0710 v5 0.403 0.390 0.269 0.394 0.429 0.372 0.245 0.1116 v6 0.389 0.401 0.399 0.378 0.432 0.377 0.127 0.0580 v7 0.393 0.388 0.396 0.372 0.434 0.362 0.157 0.0715 v8 0.408 0.390 0.395 0.381 0.444 0.374 0.110 0.0502 v9 0.403 0.391 0.400 0.392 0.445 0.381 0.089 0.0403 v10 0.397 0.395 0.393 0.384 0.425 0.379 0.128 0.0583 v11 0.392 0.399 0.402 0.388 0.430 0.379 0.111 0.0505 v12 0.393 0.402 0.396 0.384 0.439 0.364 0.124 0.0564 v13 0.403 0.391 0.388 0.382 0.442 0.365 0.130 0.0593 v14 0.400 0.394 0.396 0.394 0.442 0.373 0.103 0.0468 v15 0.404 0.397 0.395 0.381 0.443 0.377 0.104 0.0474 v16 0.391 0.400 0.400 0.383 0.436 0.377 0.114 0.0520 v17 0.398 0.386 0.392 0.384 0.440 0.374 0.128 0.0584 fig. 2 presents the pictorial representation of weights of criteria by objective, subjective and integrated weight forms. “fraud prevention (v17)” with weight 0.0782 is the most significant factor during evaluation of blockchain platforms in the logistics industry. “privacy” (v15) with weight 0.0760 is the second significant indicator in blockchain platforms assessment. “cost reduction (v9)” has come out to be third significant indicator with weight 0.0757, “productivity (v6)” with weight value 0.0747 has fourth factor and others are considered crucial factors in the evaluation of blockchain platforms for the given data. table 7 computational steps of if-rs model for finding subjective weight criteria f1 f2 f3 f4 aggregated ifns score values rank wj s v1 m h m ml (0.554, 0.342) 0.606 8 0.0654 v2 h mh m ml (0.568, 0.328) 0.620 7 0.0719 v3 m m mh l (0.510, 0.388) 0.561 12 0.0392 v4 vh mh l ml (0.558, 0.353) 0.602 9 0.0588 v5 mh ml ml l (0.438, 0.46) 0.489 17 0.0065 v6 ml mh h mh (0.603, 0.293) 0.655 4 0.0915 v7 m ml m h (0.516, 0.381) 0.567 11 0.0458 v8 m mh m l (0.504, 0.394) 0.555 13 0.0327 v9 vh m mh mh (0.635, 0.279) 0.678 1 0.1111 v10 ml vh m ml (0.585, 0.331) 0.627 6 0.0784 v11 mh l ml mh (0.464, 0.433) 0.516 16 0.0131 v12 m ml m mh (0.493, 0.406) 0.543 15 0.0196 v13 ml vh ml l (0.547, 0.368) 0.590 10 0.0523 v14 m h l ml (0.500, 0.392) 0.554 14 0.0261 v15 m mh vh l (0.636, 0.283) 0.676 2 0.1046 v16 mh m h m (0.600, 0.297) 0.651 5 0.0850 v17 h ml h mh (0.619, 0.276) 0.671 3 0.0980 intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 569 step 4: in accordance with eq. (16), the weighted normalized a-ifdm is created for assessing the blockchain platforms in the logistics industry and given in table 8. table 8 weighted normalized a-ifdm for assessing the blockchain platforms u1 u2 u3 u4 u5 u6 v1 (0.045, 0.940) (0.050, 0.934) (0.057, 0.929) (0.067, 0.916) (0.033, 0.955) (0.053, 0.931) v2 (0.035, 0.953) (0.058, 0.924) (0.048, 0.937) (0.047, 0.937) (0.051, 0.933) (0.055, 0.928) v3 (0.054, 0.930) (0.031, 0.958) (0.027, 0.963) (0.040, 0.948) (0.025, 0.966) (0.042, 0.944) v4 (0.039, 0.946) (0.032, 0.957) (0.041, 0.944) (0.043, 0.942) (0.039, 0.947) (0.043, 0.942) v5 (0.057, 0.925) (0.040, 0.947) (0.055, 0.933) (0.073, 0.907) (0.031, 0.958) (0.047, 0.941) v6 (0.041, 0.945) (0.082, 0.898) (0.065, 0.914) (0.046, 0.94) (0.044, 0.942) (0.071, 0.911) v7 (0.037, 0.95) (0.037, 0.951) (0.046, 0.939) (0.029, 0.960) (0.037, 0.95) (0.031, 0.958) v8 (0.055, 0.932) (0.028, 0.963) (0.031, 0.958) (0.028, 0.962) (0.042, 0.946) (0.034, 0.954) v9 (0.076, 0.906) (0.053, 0.930) (0.072, 0.91) (0.087, 0.89) (0.08, 0.9) (0.092, 0.887) v10 (0.051, 0.933) (0.057, 0.926) (0.049, 0.936) (0.054, 0.929) (0.031, 0.957) (0.071, 0.911) v11 (0.02, 0.973) (0.033, 0.958) (0.034, 0.957) (0.029, 0.962) (0.017, 0.976) (0.035, 0.956) v12 (0.024, 0.968) (0.047, 0.941) (0.03, 0.96) (0.03, 0.961) (0.03, 0.959) (0.021, 0.971) v13 (0.054, 0.929) (0.04, 0.947) (0.032, 0.957) (0.039, 0.947) (0.05, 0.936) (0.034, 0.955) v14 (0.031, 0.959) (0.03, 0.961) (0.029, 0.961) (0.047, 0.94) (0.034, 0.958) (0.03, 0.962) v15 (0.079, 0.9) (0.069, 0.912) (0.057, 0.924) (0.053, 0.931) (0.07, 0.908) (0.075, 0.907) v16 (0.041, 0.946) (0.071, 0.909) (0.064, 0.915) (0.052, 0.934) (0.048, 0.937) (0.067, 0.916) v17 (0.062, 0.922) (0.045, 0.94) (0.052, 0.932) (0.06, 0.923) (0.065, 0.915) (0.064, 0.915) step 5: applying eq. (17), an if-is is determined as mj = {(0.067, 0.916), (0.058, 0.924), (0.054, 0.93), (0.043, 0.942), (0.073, 0.907), (0.081, 0.898), (0.046, 0.939), (0.055, 0.932), (0.092, 0.887), (0.071, 0.911), (0.035, 0.956), (0.047, 0.941), (0.054, 0.929), (0.047, 0.94), (0.079, 0.9), (0.071, 0.909), (0.065, 0.915)}. step 6: in this step, the composite distance matrix cd = (cdij)6×17 is determined using eq. (18), where cdij is given as cd11 = 0.023, cd12 = 0.026, cd13 = 0, cd14 = 0.004, cd15 = 0.017, cd16 = 0.045, cd17 = 0.011, cd18 = 0, cd19 = 0.018, cd110 = 0.022, cd111 = 0.017, cd112 = 0.025, cd113 = 0, cd114 = 0.018, cd115 = 0, cd116 = 0.035, cd117 = 0.007, cd21 = 0.018, cd22 = 0, cd23 = 0.026, cd24 = 0.014, cd25 = 0.038, cd26 = 0, cd27 = 0.011, cd28 = 0.03, cd29 = 0.043, cd210 = 0.015, cd211 = 0.002, cd212 = 0, cd213 = 0.017, cd214 = 0.02, cd215 = 0.011, cd216 = 0, cd217 = 0.023, cd31 = 0.012, cd32 = 0.011, cd33 = 0.032, cd34 = 0.002, cd35 = 0.023, cd36 = 0.017, cd37 = 0, cd38 = 0.026, cd39 = 0.022, cd310 = 0.025, cd311 = 0.002, cd312 = 0.019, cd313 = 0.026, cd314 = 0.020, cd315 = 0.024, cd316 = 0.007, cd317 = 0.016, cd41 = 0, cd42 = 0.012, cd43 = 0.017, cd44 = 0, c45 = 0, cd46 = 0.040, cd47 = 0.02, cd48 = 0.029, cd49 = 0.005, cd410 = 0.018, cd411 = 0.007, cd412 = 0.019, cd413 = 0.017, cd414 = 0, cd415 = 0.030, cd416 = 0.023, cd417 = 0.007, cd51 = 0.038, cd52 = 0.008, cd53 = 0.034, cd54 = 0.005, cd55 = 0.048, cd56 = 0.042, cd57 = 0.01, cd58 = 0.014, cd59 = 0.013, cd510 = 0.045, cd511 = 0.02, cd512 = 0.018, cd513 = 0.006, cd514 = 0.016, cd515 = 0.009, cd516 = 0.026, cd517 = 0, cd61 = 0.015, cd62 = 0.004, cd63 = 0.013, cd64 = 0, cd65 = 0.031, cd66 = 0.012, cd67 = 0.018, cd68 = 0.022, cd69 = 0, cd610 = 0, cd611 = 0, cd612 = 0.029, cd613 = 0.024, cd614 = 0.02, cd615 = 0.006, cd616 = 0.006 and cd617 = 0.001. 570 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. step 7: using eq. (19), the assessment score of each option is calculated and presented as l1 = (10.27) = 0.73, l2 = (10.2683) = 0.7317, l3 = (10.284) = 0.716, l4 = (10.2447) = 0.7553, l5 = (10.3518) = 0.6482, l6 = (10.2015) = 0.7985. step 8: on the basis of decreasing ratings of assessment scores of options, the ranking order of blockchain platforms is obtained as u6 (0.7985) > u4 (0.7553) > u2 (0.7317) > u1 (0.73) > u3 (0.716) > u5 (0.6482). thus, an option “microsoft (u6)” is the most suitable platform with highest assessment score (0.7985) among a set of six alternatives concerning 17 criteria. 4.2. sensitivity analysis this subsection discusses sensitivity assessment over varying ratings of weight strategy coefficient τ. this analysis can validate the superiority and stability of developed ifmerec-rs-dba framework to assess the blockchain platforms in the logistics industry. in this process, we observed the variations of assessment scores with respect to diverse values of parameter τ and required outcomes are discussed in table 9 and fig. 3. by means of the assessment scores for τ = 0.0 in table 9, the preference of blockchain platforms is u6  u2  u4 u1  u3 u5, whereas prioritization of blockchain platforms for τ = 0.1 to 0.5 is u6  u4 u2  u1 u3  u5, while for τ = 0.6 to 0.9, the ranking order of blockchain alternatives is u6  u4 u1 u2 u3  u5, and at τ = 1.0, the prioritization of blockchain platforms is u6 u4 u1 u2 u3  u5. here, we can easily be observed that the alternative “microsoft (u6)” has always obtained a maximum preference over the others alternatives except at τ = 1.0. thus, the assessment process of alternatives is relied on and sensitive to the parameter τ. fig. 3 sensitivity results of assessment scores based on different values of parameter 𝜏 intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 571 table 9 prioritization outcomes of bt platforms with different τ values τ u1 u2 u3 u4 u5 u6 τ = 0.0 (subjective weight by if-rs) 0.7402 0.7572 0.7345 0.7531 0.6941 0.8497 τ = 0.1 0.7377 0.7515 0.7303 0.7533 0.6842 0.839 τ = 0.2 0.7355 0.7461 0.7264 0.7537 0.6747 0.8285 τ = 0.3 0.7335 0.741 0.7227 0.7541 0.6656 0.8183 τ = 0.4 0.7317 0.7362 0.7192 0.7546 0.6568 0.8083 τ = 0.5 (integrated method by if-merec-rs) 0.73 0.7317 0.716 0.7553 0.6482 0.7985 τ = 0.6 0.7286 0.7274 0.713 0.756 0.6401 0.7888 τ = 0.7 0.7274 0.7235 0.7103 0.7569 0.6322 0.7794 τ = 0.8 0.7264 0.7198 0.7077 0.7578 0.6246 0.7702 τ = 0.9 0.7256 0.7164 0.7054 0.7589 0.6174 0.7612 τ = 1.0 (objective weight by if-merec) 0.725 0.7133 0.7034 0.7601 0.6104 0.7524 4.3. comparative study this subsection compares the proposed if-merec-rs-dba and existing ifinformation based mcgdm methods to check the robustness of introduced framework. to validate the outcomes obtained from the if-merec-rs-dba approach, four well-known mcdm approaches are selected, which are given by qin et al. [29], mishra et al. [30], deb et al. [31] and liu [32]. by applying these methods on the aforesaid case study, we aim to confirm and emphasize the findings determined from developed if-merec-rs-dba method. table 10 optimal performance rating and weighted normalized aggregated ifns c ri te ri a opr u1 u2 u3 u4 u5 u6 v1 (0.067, 0.916) (0.045, 0.94) (0.05, 0.934) (0.057, 0.929) (0.067, 0.916) (0.033, 0.955) (0.053, 0.931) v2 (0.058, 0.924) (0.035, 0.953) (0.058, 0.924) (0.048, 0.937) (0.047, 0.937) (0.051, 0.933) (0.055, 0.928) v3 (0.054, 0.93) (0.054, 0.93) (0.031, 0.958) (0.027, 0.963) (0.04, 0.948) (0.025, 0.966) (0.042, 0.944) v4 (0.043, 0.942) (0.039, 0.946) (0.032, 0.957) (0.041, 0.944) (0.043, 0.942) (0.039, 0.947) (0.043, 0.942) v5 (0.073, 0.907) (0.057, 0.925) (0.04, 0.947) (0.055, 0.933) (0.073, 0.907) (0.031, 0.958) (0.047, 0.941) v6 (0.081, 0.898) (0.041, 0.945) (0.082, 0.898) (0.065, 0.914) (0.046, 0.94) (0.044, 0.942) (0.071, 0.911) v7 (0.046, 0.939) (0.037, 0.95) (0.037, 0.951) (0.046, 0.939) (0.029, 0.96) (0.037, 0.95) (0.031, 0.958) v8 (0.055, 0.932) (0.055, 0.932) (0.028, 0.963) (0.031, 0.958) (0.028, 0.962) (0.042, 0.946) (0.034, 0.954) v9 (0.092, 0.887) (0.076, 0.906) (0.053, 0.93) (0.072, 0.91) (0.087, 0.89) (0.08, 0.9) (0.092, 0.887) v10 (0.071, 0.911) (0.051, 0.933) (0.057, 0.926) (0.049, 0.936) (0.054, 0.929) (0.031, 0.957) (0.071, 0.911) v11 (0.035, 0.956) (0.02, 0.973) (0.033, 0.958) (0.034, 0.957) (0.029, 0.962) (0.017, 0.976) (0.035, 0.956) v12 (0.047, 0.941) (0.024, 0.968) (0.047, 0.941) (0.03, 0.96) (0.03, 0.961) (0.03, 0.959) (0.021, 0.971) v13 (0.054, 0.929) (0.054, 0.929) (0.04, 0.947) (0.032, 0.957) (0.039, 0.947) (0.05, 0.936) (0.034, 0.955) v14 (0.047, 0.94) (0.031, 0.959) (0.03, 0.961) (0.029, 0.961) (0.047, 0.94) (0.034, 0.958) (0.03, 0.962) v15 (0.079, 0.9) (0.079, 0.9) (0.069, 0.912) (0.057, 0.924) (0.053, 0.931) (0.07, 0.908) (0.075, 0.907) v16 (0.071, 0.909) (0.041, 0.946) (0.071, 0.909) (0.064, 0.915) (0.052, 0.934) (0.048, 0.937) (0.067, 0.916) v17 (0.065, 0.915) (0.062, 0.922) (0.045, 0.94) (0.052, 0.932) (0.06, 0.923) (0.065, 0.915) (0.064, 0.915) 572 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. 4.3.1. if-topsis method the if-topsis method proposed by qin et al. [29] is used on aforesaid case study for assessing and prioritizing blockchain platform alternatives by means of the given 17 criteria. using this method, we obtain the ideal solution as {(0.707, 0.212), (0.602, 0.295), (0.661, 0.244), (0.493, 0.397), (0.723, 0.192), (0.679, 0.237), (0.554, 0.342), (0.744, 0.183), (0.719, 0.205), (0.661, 0.255), (0.678, 0.241), (0.716, 0.203), (0.631, 0.266), (0.732, 0.184), (0.663, 0.251), (0.659, 0.249), (0.576, 0.319)} and the anti-ideal solution as {(0.452, 0.445), (0.426, 0.472), (0.390, 0.508), (0.393, 0.504), (0.411, 0.485), (0.427, 0.468), (0.397, 0.501), (0.496, 0.401), (0.510, 0.386), (0.371, 0.527), (0.425, 0.471), (0.436, 0.461), (0.446, 0.453), (0.564, 0.349), (0.510, 0.389), (0.454, 0.443), (0.446, 0.451)}. next, if-distance on each option with if-ideal solution is computed as 0.103, 0.101, 0.103, 0.094, 0.136 and 0.071. similarly, if-distance on each option with if-antiideal solution is computed as 0.110, 0.111, 0.110, 0.120, 0.076 and 0.141. finally, relative closeness coefficient is calculated for each option and given as 0.5162, 0.5223, 0.5160, 0.5610, 0.3588 and 0.6647. on the basis of obtained relative closeness coefficient, the ranking order of blockchain platforms is u6 u4 u2 u1  u3 u5, and the platform “microsoft (u6)” is the most suitable choice among the others. table 11: opd of options for bt platforms assessment criteria score values optimal rating u1 u2 u3 u4 u5 u6 v1 0.075 0.053 0.058 0.064 0.075 0.039 0.061 v2 0.067 0.041 0.067 0.056 0.055 0.059 0.063 v3 0.062 0.062 0.037 0.032 0.046 0.030 0.049 v4 0.051 0.047 0.038 0.048 0.051 0.046 0.050 v5 0.083 0.066 0.047 0.061 0.083 0.036 0.053 v6 0.092 0.048 0.092 0.075 0.053 0.051 0.080 v7 0.054 0.043 0.043 0.054 0.034 0.044 0.037 v8 0.061 0.061 0.033 0.036 0.033 0.048 0.040 v9 0.102 0.085 0.061 0.081 0.098 0.090 0.102 v10 0.080 0.059 0.066 0.056 0.063 0.037 0.080 v11 0.040 0.023 0.038 0.038 0.034 0.021 0.040 v12 0.053 0.028 0.053 0.035 0.034 0.036 0.025 v13 0.063 0.063 0.047 0.038 0.046 0.057 0.039 v14 0.053 0.036 0.034 0.034 0.053 0.038 0.034 v15 0.090 0.090 0.079 0.066 0.061 0.081 0.084 v16 0.081 0.047 0.081 0.074 0.059 0.055 0.076 v17 0.075 0.070 0.053 0.060 0.069 0.075 0.075 opd 1.181 0.923 0.923 0.909 0.947 0.843 0.988 ud 0.7810 0.7816 0.7692 0.8019 0.7138 0.8366 ranking 4 3 5 2 6 1 4.3.2. if-aras method after applying this method on the abovementioned blockchain platforms assessment problem, the optimal performance rating (opr) is determined as {(0.707, 0.212), (0.602, 0.295), (0.661, 0.244), (0.493, 0.397), (0.723, 0.192), (0.679, 0.237), (0.554, 0.342), (0.744, intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 573 0.183), (0.719, 0.205), (0.661, 0.255), (0.678, 0.241), (0.716, 0.203), (0.631, 0.266), (0.732, 0.184), (0.663, 0.251), (0.659, 0.249), (0.576, 0.319)}. next, the weighted normalized aifdm is made and given in table 10. further, the score value of each entry of table 10 is determined and presented in table 11. by adding the corresponding score values of each column in table 11, the optimal performance degree (opd) is calculated as 1.181, 0.923, 0.923, 0.909, 0.947, 0.843 and 0.988. further, the utility degree (ud) of each alternative is derived as 0.7810, 0.7816, 0.7692, 0.8019, 0.7138 and 0.8366, respectively. on account of the decreasing utility degrees, the ranking order of blockchain platforms is u6 u4 u2 u1 u3 u5, and the platform “microsoft (u6)” is the most suitable choice among the others. 4.3.3. if-waspas method based on the execution of if-waspas method on the aforesaid blockchain platforms assessment problem, the additive relative importance of each alternative is determined as (0.561, 0.339), (0.561, 0.339), (0.555, 0.345), (0.571, 0.33), (0.526, 0.372) and (0.590, 0.315), respectively, and their corresponding score values are 0.611, 0.611, 0.605, 0.621, 0.577 and 0.637, respectively. next, the multiplicative relative importance of each alternative is computed as (0.543, 0.357), (0.545, 0.354), (0.546, 0.353), (0.553, 0.348), (0.509, 0.388) and (0.576, 0.328), and their corresponding score values are 0.593, 0.595, 0.596, 0.603, 0.560 and 0.624, respectively. lastly, the total relative importance of each alternative is determined as 0.6020, 0.6032, 0.6006, 0.6118, 0.5688 and 0.6308, respectively. arranging the values of total relative importance, the ranking order of blockchain platforms is u6 u4 u2 u1  u3 u5, and the platform “microsoft (u6)” is the most suitable choice among the others for the logistics industry. 4.3.4. if-cocoso method the if-cocoso method is applied to the aforesaid case study of blockchain platforms evaluation in the logistics industry. the additive and multiplicative importance is same as liu (2024). next, the relative compromise rating of each option is computed as r1 (1) = 0.1664, r2 (1) = 0.1668, r3 (1) = 0.166, r4 (1) = 0.1691, r5 (1) = 0.1608, r6 (1) = 0.1744, r1 (2) = 0.767, r2 (2) = 0.769, r3 (2) = 0.766, r4 (2) = 0.780, r5 (2) = 0.725, r6 (2) = 0.804, r1 (3) = 0.954, r2 (3) = 0.956, r3 (3) = 0.952, r4 (3) = 0.97, r5 (3) = 0.902 and r6 (3) = 1.0. based on the relative compromise rating, the overall compromise rating of each alternative is determined as 0.5626, 0.5638, 0.5613, 0.5717, 0.5339 and 0.5895, respectively. based on decreasing ratings of overall compromise ratings, preference of blockchain platforms is u6 u4 u2 u1  u3 u5, and the platform “microsoft (u6)” is the most suitable choice among the others for the logistics industry. fig. 4 presents the pictorial representation of ranking orders of blockchain platforms using diverse methods. on the basis of obtained results, it can be observed that the proposed and existent approaches attain the same ranking order, which is u6 u4 u2 u1 u3 u5, and the most optimal choice is “microsoft (u6)”. the if-merec-rs-dba framework has the following advantages over the existing models (qin et al. [29], mishra et al. [30], deb et al. [31], liu [32]): ▪ existing methods given by (qin et al. [29], deb et al. [31], and liu [32]) doesn’t consider the des’ weights, while developed if-merec-rs-dba framework estimates weights of des with if-score function and rs model-based weighting 574 p. rani, a. r. mishra, a. m. alshamrani, a. f. alrasheedi, et al. approach. therefore, proposed approach is more suitable for mcgdm problems under ifi setting. ▪ qin et al. [29] and deb et al. [31] assumes weight of criteria without any logical explanation, while liu [32] derives the objective weight of criteria via entropy method. although, developed if-merec-rs-dba framework find weights of criteria through integrated if-merec-rs model consisting of objective weight with if-merec and subjective weight with if-rs approach. thus, the consistency of outcomes is more practical than existing models. ▪ the proposed distance measure computes the composite distance matrix in the ifmerec-rs-dba method, while the existing dba methods by (sandhya et al. [41], garg et al. [40], garg & garg [42]) determine the composite distance matrix through hamming distance formula. fig. 4 assessment degree of blockchain platforms by diverse methods 5. conclusion in the current digital economy era, the dt has offered many opportunities to preserve the competitiveness in the business models. nowadays, the dt in logistics companies is reshaping the businesses by using innovative technologies and improving the productivity with aster, leaner, and more efficient operations. as a decentralized digital ledger, bt improves logistics’ operations by providing transparency, security, and efficiency, and ensures secure and transparent transaction records. to select a suitable blockchain platform in the logistics industry, we have proposed an integrated if-merec-rs-dba framework by combining the weighting model and ranking method within the settings of ifss. in this method, the significance value of each de has been computed through if-score function and if-rs model-based formula. next, weights of criteria have been derived via a combined weighting process consisting of objective weight with if-merec and subjective weight with intuitionistic fuzzy distance measure-based approach for adopting the blockchain technology... 575 if-rs approach. further, an integrated intuitionistic fuzzy distance-based approach has been presented utilizing developed if-distance measure, des’ weighting system and criteria weight-determination model. to this aim, a new distance measure has been introduced with the combination of md, nd and hesitancy degree, and applied to find the composite distance matrix in the proposed if-merec-rs-dba model. moreover, the proposed approach has been applied to a case study of blockchain platform selection problem in the logistics industry, which shows its usefulness. sensitivity and comparative analyses have been performed to validate the stability and robustness of the if-merec-rs-dba framework. in future, some new mcgdm model will be 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2021, a multi-criteria decision-making method based on dnma and critic with linguistic d numbers for blockchain platform evaluation, engineering applications of artificial intelligence, 101, 104200. 62. stillwell, w. g., seaver, d. a., edwards, w., 1981, a comparison of weight approximation techniques in multiattribute utility decision making, organizational behavior and human performance, 28(1), pp. 62-77. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 261 276 1the critical load parameter of a timoshenko beam with one-step change in cross section udc 534.1 goran janevski 1 , marija stamenković 2 , mariana seabra 3 1 university of niš, department of mechanical engineering, serbia 2 mathematical institute of the sasa, belgrade, serbia 3 university of porto, faculty of engineering – feup, portugal abstract. the paper analyzes the transverse vibration of a timoshenko beam with onestep change in cross-section when subjected to an axial force. the axial force is equal in both of the beam portions. three types of beam which occur commonly in engineering application are considered. the frequency equation of the timoshenko beam with onestep change in cross-section is expressed as the fourth order determinant equated to zero. the critical compressive axial force is expressed as a function of the critical load parameter which is tabulated for four classical boundary conditions. apart from the results presented in tables, the paper also provides calculated values of the critical load parameter for other values of system parameters along with the graphic representation of their dependence on the step position parameter. key words: timoshenko beam, frequency equation, critical axial load, critical load parameter, one-step beam 1. introduction mechanical and other engineering structures often contain beams with step changes in cross-section. stepped beams are increasingly used in many fields of engineering and practically always as structural elements. due to the frequent application, they can be very different structures, subjected to various types of loads. for this reason, their dynamic properties have been investigated by many authors. among the first ones are jang and bert [1-2] who have given the first exact results for natural frequencies of a stepped beam. in jang and bert‟s paper [1] the exact solution for fundamental natural frequencies is compared with the results obtained by the use of the finite element method. in the next paper by the same authors [2] higher mode frequencies of a received october 10, 2014 / accepted november 19, 2014 corresponding author: goran janevski university of niš, department of mechanical engineering, serbia e-mail: gocky.jane@gmail.com original scientific paper 262 g. janevski, m. stamenković, m. seabra stepped beam are obtained, expressing the frequency equation as the fourth-order determinant equated to zero. the vibration of euler-bernoulli beam with step changes in cross-section and under axial forces is considered by naguleswaran [2-7]. the author presents the first three frequencies and the first two critical axial forces for beams with several combinations of axial forces in two portions, and for 16 sets of boundary combinations and three types which occur commonly in engineering applications. also, naguleswaran [6] investigates the sensitivity of frequency parameters from the step location factor and the “active“ dimension factor. the sensitivity is presented for the selected system parameters. the vibration of an eulerbernoulli beam with three step changes and elastic end supports is investigated in [7]. the first three frequency parameters are tabulated for the selected sets of system parameters and classical end support and 35 types of elastic end supports. vibration and stability of eulerbernoulli tie-bars are considered by naguleswaran [3]. the first three frequency parameters and the first two buckling axial forces are tabulated for three type arrangements with different number of rings and various end supports. the frequencies, in graphical form, of a uniform euler-bernoulli beam under constant axial compressive and tensile force for the classical boundary conditions are presented by bokaian [8-9]. by means of the adomian decomposition method, mao and pietrzko [10] investigate the free vibrations of a stepped beam. the first four natural frequencies for classical end supports are tabulated. also, the authors obtain the frequencies for the stepped beam with a translational and rotational spring at one and both ends. their results are compared with those obtained in [7]. mao [11] has extended the study in [10] to multiple-stepped beams and compared them with those obtained in [3]. by using the continuous-mass transfer matrix method, wu and chang [12] have investigated free vibration of the axial loaded multi-step timoshenko non-uniform beam carrying any number of concentrated elements. in this paper the authors take into consideration the effects of shear deformation, rotary inertia and their joint action term, and thus determine the exact natural frequencies. zhang at al. [13] have developed an analytic method to study transverse vibrations of double-beam systems, in which two parallel timoshenko beams are connected by discrete springs and coupled with various discontinuities. by dividing the entire structure into a series of distinct components, and then systematically organizing compatibility and boundary conditions with matrix formulations, closed-form expressions for the exact natural frequencies, mode shapes and frequency response functions can be determined. parametric studies are performed for a practical example to illustrate the influences of the parametric variabilities on the dynamic behaviors. in the present paper, the emphasis is on the critical axial force and natural frequencies of a timoshenko stepped beam. the three types of frequent use in engineering applications are considered. the frequency equations for four combinations of the classical boundary conditions are expressed as the second order determinants equated to zero. the critical axial force is presented in the function of the dimensionless parameter. the influence of a flexural rigidity ratio, slenderness ratio and type of beam are considered. the obtained results are displayed graphically for a set of combinations of dimension factors and location parameters. critical load parameter of a timoshenko beam with one-step change in cross section 263 2. problem formulation the basic differential equations of motion for the analysis will be reduced by considering the timoshenko-beam of length l, subjected to axial compressive force f. this will be applied on the basis of the following assumptions:  the behavior of the beam material is linear elastic,  the cross-section is rigid and constant throughout the length of the beam and has one plane of symmetry,  shear deformations of the cross-section of the beam are taken into account while the elastic axial deformations are ignored,  the equations are derived bearing in mind the geometric axial deformations, and,  axial forces f acting on the ends of the beam do not change with time. fig. 1 the coordinate system and notation for the beam: a) timoshenko-beam subjected to an axial compressive force f ; b) deflected differential beam element of length dx a beams element of length dx between two cross-sections taken normal to the deflected axis of the beam is shown in fig. 1b. since the slope of the beam is small, the normal forces acting on the sides of the element can be taken as equal to axial compressive force f. shearing force ft is related to the following relationship:           x v kgaft (1) where v = v(x,t) is the displacement of the cross-section in y-direction, v/x is the global rotation of the cross section,  =  (x,t) is the bending rotation, g is the shear modulus, a is the area of the beam cross-section, and k is the shear correction factor of cross-section. analogously, the relationship between bending moment m and bending angles  is given by: z m ei x     (2) where e is the young‟s modulus and iz is the second moment of the area of the crosssection. finally, forces and moments of inertia are given by: 2 2 2 2 , z w f a j i t t           , (3) 264 g. janevski, m. stamenković, m. seabra respectively, where  is the mass density. the dynamic-forces equilibrium conditions of these forces are given by the following equations: 2 2 2 2 2 2 0 v v v m kga f t x x x                (4) 2 2 2 2 0 v i kga k t x x                 (5) where m = a is the mass per unit length and k = eiz is the flexural rigidity. the equations on motion (4-5), which are coupled together, are reduced by standard procedure, eliminating  , to the following fourth-order partial differential equation: 4 2 4 2 4 4 2 2 2 2 4 1 1 0z z if v v f e v v v k f i m kga x x kga kg x t t kg t                             (6) 3. the frequency equation of the stepped beam in this section of the paper a general theory for the determination of the natural frequencies and the critical buckling load of a beam with step change in cross-section is given. ends a and b are on classical clamped (cl), pinned (pn) and free (fr) supports. each beam portion is made of some material with young‟s modulus e and mass density , and has a cross-section with a uniform cross-section of area ai and moment inertia ii. the flexural rigidity, mass per unit length and the length of the beam portion are ki, mi and li. the axial force in beam portion is fi. the coordinate systems with origin at a and b are in opposite directions. fig. 2 the coordinate system and notation for the stepped beam the dynamics of each beam portion are treated separately. if we apply the abovementioned procedure to a differential element of each beam portion, the following set of coupled differential equations will be obtained: 4 2 4 2 4 1 1 1 1 1 1 1 1 14 2 2 2 42 1 11 1 1 1 1 0 v v v v i vf f e ei f i m kga kga kg t kg tx x x t                            (7) 4 2 4 2 4 2 2 2 2 2 2 2 2 24 2 2 2 42 2 22 2 2 1 1 0 v v v v i vf f e ei f i m kga kga kg t kg tx x x t                            (8) the standard approach to solving eqs. (7, 8) is by separating the variables, and the same procedure may be applied here. thus, assuming that: critical load parameter of a timoshenko beam with one-step change in cross section 265 ( , ) ( ) ( ), 1,2, i i i i v x t y x t t i  (9) where yi(xi), are the known mode shape functions which will be determined on the basis of the type of beam supports. assuming time harmonic motion, the unknown time function can be assumed to have the following form: ( ) , 1 ,j tt t e j   (10) where  denotes the circular natural frequency of the system. to express the set of eqs. (7-8) in the dimensionless form one defines dimensionless abscissas xi, amplitude yi and step position parameter ri as follows: , , , 1,2,i i i i i i x y l x y r i l l l     (11) also, adopt the “reference beam” with the following characteristics: area of the beam cross-section ar, length l, mass per unit mr and flexural rigidity eir. then we can define dimensionless flexural rigidity ratio i, mass per unit ratio i and dimensionless axial force  in the form: 2 , , , 1,2.i i i i r r r ei m fl i ei m ei       (12) introducing the general solutions (9) into eqs. (7-8), considering dimensionless values, one gets the system of dimensionless differential equations: 4 2 4 4 2 4 2 4 ( ) ( ) 1 1 1 1 ( ) 0 , i i i i i r r ri r i i i i i i r r r i i i d y x d y x k dx k dx y x k                                                  (13) where 6.2 g e . in eq. (13) r is the natural frequency parameter and r is the slenderness ratio of the beam: 24 4 2 2 , ,r r r r r r r m l i i ei a l l             (14) where ir is the radius of gyration of the reference beam. the solutions of eq. (13) are assumed to be:   ( ) , 1,2.i iu x i i i y x ae i  (15) for the non-trivial solution, we can determine ui as the solution of the characteristic equation in the form: 2 2 2 4 0 4 4 , 1,2, 2 i i i i i i u i           (16) where: 266 g. janevski, m. stamenković, m. seabra 4 4 2 4 2 4 0 1 , 1 1 , 1 . i i r i r r i r i i i i i r r r i k k k                                                  . (17) with the use of euler‟s formula, solution (15) can be written as: 1, 2, 3, 4, ( ) cosh( ) sinh( ) cosh( ) sinh( ), i i i i i i i i i i i i i i y x c a x c a x c b x c b x    (18) where cj,i (i = 1,2, j = 1,2,3,4) are eight constants to be determined from the initial conditions and: 2 2 2 2 4 0 2 2 4 0 4 4 4 4 , 1,2 2 2 i i i i i i i i i i i i a b i                  . (19) the need for eq. (18) to satisfy the boundary conditions at a and b may be used to eliminate four of the constants. the mode shape of the beam portions may be expressed as: 1 1 2 2 ( ) ( ) ( ), 1,2. i i i i i i i i y x a u x a u x i   (20) where ai1 and ai2 are unknown constants and functions ui1(xi) and ui2(xi) for the classical boundary conditions are: for clamped, 1 2 ( ) cosh( ) cosh( ), ( ) sinh( ) sinh( ). i i i i i i i i i i i i i i u x a x b x a u x a x b x b     for pinned, 1 2 ( ) sinh( ), ( ) sinh( ). i i i i i i i i u x a x u x b x  (21) (21) for free, 2 1 3 2 3 ( ) cosh( ) cosh( ), ( ) sinh( ) sinh( ). i i i i i i i i i i i i i i i i i i i i a u x a x b x b a a u x a x b x b b                 taking into account the opposite direction coordinate axes at a and b, the need to satisfy the continuity of deflection and the slope and compatibility of bending moment and shearing force at c will result in the following equations: 1 1 2 2 1 1 1 2 1 2 ( ) ( ) ( ) ( ), , dy r dy r y r y r dx dx    (22) 2 2 3 3 1 1 2 2 1 1 1 1 2 2 2 2 1 2 1 22 2 3 3 1 2 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) , . d y x d y x d y x dy r d y x dy r dx dx dx dx dx dx                 substituting the solution (20) into eq. (22) and rewriting in the matrix form, one obtains the following homogenous set of four algebraic equations: critical load parameter of a timoshenko beam with one-step change in cross section 267 11 1 12 1 21 22 11 1 12 1 21 22 11 1 1 2 2 12 2 2 2 2 2111 1 12 1 21 22 1 1 2 22 2 2 2 221 1 2 2 11 1 12 1 21 2 2 2 2 2 2 2 222 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u r u r u r u r du r du r du r du r a dx dx dx dx a ad u r d u r d u r d u r adx dx dx dx b r b r b r b r                              0 0 0 0                        , (23) where:   3 3 ( ) ( ) .mn m mn m mn m m m m d u r du r b r dx dx    (24) for the non-trivial solution, the coefficient matrix must be singular, one gets the frequency equation: 11 1 12 1 21 2 22 2 11 1 12 1 21 2 22 2 1 1 2 2 2 2 2 2 11 1 12 1 21 2 22 2 1 1 2 22 2 2 2 1 1 2 2 11 1 12 1 21 2 22 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u r u r u r u r du r du r du r du r dx dx dx dx d u r d u r d u r d u r dx dx dx dx b r b r b r b r          (25) 4. critical axial force, critical load parameter of all the modes of failure, buckling is probably the most common and most catastrophic one. for this reason, the critical buckling load is an important characteristic of a mechanical structure. as it is known, the critical buckling load for uniform beam is: , 2 ,           r ucr k (26) where kr depends on the boundary conditions and one has  kr = 1 for pinned/pinned beam  kr = 1/2 for clamped/clamped beam  kr = 2 for clamped/free beam  kr = 0.69915566 for clamped/pinned beam. in order to verify the proposed method for the analysis of the vibration of the stepped beam, several numerical examples with different boundary conditions, step positions, slenderness ratio and moment of inertia parameter will be discussed in this section. assuming that the stepped beam has a uniform young‟s modulus e and density , the present paper considers three of the types which occur commonly in engineering application. the first two are beams of a rectangular cross-section with step changes in breadth and depth and constant depth and breadth, respectively. the third type of beam has a circular crosssection with the step change in diameter. 268 g. janevski, m. stamenković, m. seabra fig. 3 three types of beam three representative types are shown in fig. 3, where „active‟ dimensions for type 1, 2 and 3 are breadth, depth and diameter, so that: for type 1, 21 , , , , , 12 i r i i i i i r r b h b l             for type 2, 3 21 , , , , , 12 i r i i i i i r r h h b l             (27) for type 3, 2 4 21 , , , , , 1 , 2. 16 i r i i i i i r r d d i d l              without the loss of generality, the beam with length l and the characteristic of the first beam portion, i.e., eir = ei1 and mr = m1, is chosen as the „reference‟ beam. this means that in all the examples 1 = 1, 1 = 1 and 1 = 1. taking the above into account and the fact that r2 = 1  r1, the system dimensionless parameters are 2,  and r1. clearly, the critical load is a function of this parameter. our analysis shows that the critical force can be represented in the form: , 2 , k cr cr u    (28) where k = k (type of beam, 2,  , r1) is a dimensionless critical load parameter (clp). for the selected set of 2,  and r1, to calculate k one writes r = 0 in the frequency equation (25). the roots of the frequency equation (25) are determined by an iterative procedure based on linear interpolation and proposed in [4]. this procedure is used to calculate k for 2 = 0.6, 0.7 & 0.8 r1 = 0.1, 0.2,...,1and  = 1/20 & 1/100. also, for the sake of comparison, we have calculated the clp force values for  = 0 which correspond to the case of eulerbernoulli beam. the clt is tabulated in table 1 for pinned-pinned beam. it can be observed that the values of clp for the uniform euler-bernoulli beam 0 or 1, i.e. the critical force is: 1 2 , 1 , ( 0) , ( 1) cr cr u cr cr u r r        , (29) the previous expression applies to all the types of supports. the clp values for timoshenko-beam are higher which means that the critical force has less value. in order to show the trend of the change in the clp value depending on step position r1, the clt values for other values of r1 with the step of 0.001 are calculated and graphically displayed. critical load parameter of a timoshenko beam with one-step change in cross section 269 the fig. 4 shows the clp dependence on changes r1 for values 2 = 0.8 and  = 1/20 & 1/100 considering the influence of the beam type. it can be noticed that the clp values are maximal for beam type 3, while for beam type 1 are minimal. for larger values of δ (greater influence of shear) in the case when the step position is near the supports, the clp values are maximal for type 1 and minimal for type 3. the clp dependence on changes r1 for types 1 and 3 and value  = 1/20, considering the influence of relations 2, is shown in fig. 5. for smaller values 2, the clp has higher values except in the case when the step position is near the supports; then the clp has higher values for greater values 2. table 1 clp for pn-pn beam table 2 clp for cl-cl beam 1r type  2 1r type  2 0.6 0.7 0.8 0.8 0.7 0.8 0 1 20 1.0125184 1.0179287 1.0286575 0 1 20 1.0496005 1.0710373 1.1135468 100 1.0005022 1.0007193 1.0011497 100 1.0020083 1.0028763 1.0045974 eb 1 1 1 eb 1 1 1 2 20 1.0015053 1.0029331 1.0061206 2 20 1.0060004 1.0116777 1.0243335 100 1.0000603 1.0001175 1.0002453 100 1.0002411 1.0004699 1.000981 eb 1 1 1 eb 1 1 1 3 20 1.0007843 1.0015284 1.0031898 3 20 1.0031296 1.0060937 1.0127052 100 1.0000314 1.0000612 1.0001277 100 1.0001256 1.0002448 1.000511 eb 1 1 1 eb 1 1 1 0.3 1 20 0.9004332 0.8963132 0.8978066 0.3 1 20 0.790734 0.817013 0.8632652 100 0.8881282 0.8788707 0.8701333 100 0.7347316 0.7407416 0.7464902 eb 0.887614 0.8781417 0.8689767 eb 0.7323622 0.7375177 0.7415579 2 20 0.931096 0.9173096 0.9027403 2 20 0.7203619 0.7308869 0.7564571 100 0.9295037 0.9142647 0.8965305 100 0.7112445 0.7154073 0.7280566 eb 0.9294372 0.9141376 0.8962712 eb 0.7108618 0.7147567 0.7268615 3 20 0.942686 0.9280114 0.910252 3 20 0.7310421 0.7198191 0.736254 100 0.9418468 0.9264071 0.9069895 100 0.7257155 0.7107592 0.7202147 eb 0.9418118 0.9263402 0.9068534 eb 0.7254923 0.7103792 0.7195413 0.5 1 20 0.6063826 0.5844888 0.570687 0.5 1 20 0.6444095 0.6400219 0.6578166 100 0.5934776 0.5666266 0.5427583 100 0.5907593 0.5673238 0.5458978 eb 0.5929383 0.5658802 0.541591 eb 0.5884946 0.5642562 0.5411757 2 20 0.7353931 0.6824913 0.6268342 2 20 0.6820122 0.6666843 0.6408012 100 0.7333133 0.6787341 0.6196697 100 0.6724187 0.650626 0.6114626 eb 0.7332265 0.6785773 0.6193705 eb 0.6720159 0.649951 0.6102279 3 20 0.7803152 0.7240699 0.6577481 3 20 0.6807354 0.6787212 0.6520273 100 0.7791806 0.7220238 0.6539021 100 0.6746779 0.6692922 0.635663 eb 0.7791332 0.7219384 0.6537418 eb 0.6744238 0.6688966 0.6349761 0.7 1 20 0.2156296 0.202338 0.198505 0.7 1 20 0.2989858 0.3163953 0.3576915 100 0.2028625 0.1846704 0.1707437 100 0.2562517 0.2540968 0.2552395 eb 0.2023291 0.1839321 0.1695835 eb 0.2544512 0.2514707 0.2509195 2 20 0.3712844 0.2909344 0.2322638 2 20 0.3812923 0.3147333 0.2932055 100 0.3680207 0.2858599 0.22374 100 0.368565 0.2960827 0.2620313 eb 0.3678844 0.2856479 0.2233838 eb 0.3680302 0.2952983 0.2607184 3 20 0.4593584 0.3517904 0.2614351 3 20 0.4648444 0.3611526 0.2934004 100 0.4573784 0.3488207 0.2567818 100 0.4566664 0.3498693 0.2770316 eb 0.4572958 0.3486968 0.2565878 eb 0.4563232 0.3493967 0.276345 1 1 20 0.0125184 0.0179287 0.0286575 1 1 20 0.0496005 0.0710373 0.1135468 100 0.0005022 0.0007193 0.0011497 100 0.0020083 0.0028763 0.0045974 eb 0 0 0 eb 0 0 0 2 20 0.0041728 0.0059762 0.0095525 2 20 0.0165335 0.0236791 0.0378489 100 0.0001674 0.0002398 0.0003832 100 0.0006694 0.0009588 0.0015325 eb 0 0 0 eb 0 0 0 3 20 0.0021755 0.0031157 0.0049801 3 20 0.0086444 0.0123804 0.019789 100 0.000087 0.0001249 0.0001996 100 0.0003487 0.0004994 0.0007983 eb 0 0 0 eb 0 0 0 270 g. janevski, m. stamenković, m. seabra fig 4 variation of clp with r1 for pn-pn beam and 2 0.8  : a) 1 20   , b) 1 100   fig 5 variation of clp with r1 for pn-pn beam and 1 20   : a) type 1, b) type 3 fig. 6 variation of clp with r1 for pn-pn beam and 2 0.8  : a) type 1, b) type 3 the clp dependence on changes r1 for types 1 and 3 and value 2 = 0.8 with influence of change values  are shown in fig. 6. it can be noticed that the clp value is higher for greater values of . the influence of δ is higher for type 1 than for type 3. critical load parameter of a timoshenko beam with one-step change in cross section 271 fig. 7 variation of clp with r1 for cl-cl beam and 2 0.8  : a) 1 20   , b) 1 100   fig. 8 variation of clp with r1 for cl-cl beam and 1 20   : a) type 1, b) type 3 fig. 9 variation of clp with r1 for cl-cl beam and 2 0.8  : a) type 1, b) type 3 the clt is tabulated as shown in table 2 for clamped-clamped beam. in figs.7-9, the clp dependence on change r1 is presented while the beam type influence, dimension ratio 2 and slenderness ratio  are taken into consideration. the influence of the beam type is higher for lower values , as well as that of dimension ratio 2 for beam type 1 in relation to beam type 3. an especially large influence of slenderness ratio  is observed for beam type 1, while for beam type 3 is considerably smaller. 272 g. janevski, m. stamenković, m. seabra table 3 clp for cl-fr beam table 4 clp for cl-pn beam 1r type  2 1r type  2 0.6 0.7 0.8 0.6 0.7 0.8 0 1 20 0.999139839 0.998768088 0.998030899 0 1 20 1.025524562 1.036555975 1.058431446 100 0.999965631 0.999950777 0.999921322 100 1.001027398 1.001471427 1.002351944 eb 1 1 1 eb 1 1 1 2 20 0.999896942 0.999799043 0.999580305 2 20 1.003075743 1.005990663 1.012494595 100 0.999995865 0.999991949 0.999983202 100 1.000123308 1.000240365 1.000501796 eb 1 1 1 eb 1 1 1 3 20 0.999946335 0.99989537 0.999781511 3 20 1.001603102 1.00312319 1.006515908 100 0.999997843 0.999995801 0.999991243 100 1.000064225 1.000125195 1.000261365 eb 1 1 1 eb 1 1 1 0.3 1 20 0.516563665 0.493505096 0.47300968 0.3 1 20 0.746833293 0.762347877 0.788574418 100 0.51772272 0.495023672 0.475235116 100 0.717997588 0.723030128 0.728359103 eb 0.517770844 0.49508674 0.475327561 eb 0.716786788 0.721379896 0.72583265 2 20 0.652268724 0.595615643 0.5401181 2 20 0.719999623 0.716830331 0.728191711 100 0.652459409 0.595956048 0.540735108 100 0.715396893 0.708900445 0.713536209 eb 0.652467341 0.595970201 0.540760754 eb 0.71520442 0.708568591 0.712922479 3 20 0.704485327 0.640117804 0.571758088 3 20 0.737832143 0.71734048 0.717983769 100 0.704588851 0.640308298 0.57210797 100 0.735173101 0.712769566 0.709764639 eb 0.704593159 0.640316223 0.572122522 eb 0.735062002 0.712578473 0.709420883 0.5 1 20 0.231981741 0.214571151 0.200117438 0.5 1 20 0.704941873 0.707837666 0.720463214 100 0.233370034 0.216264853 0.202474946 100 0.676340365 0.6689679 0.660927761 eb 0.233427646 0.216335175 0.202572866 eb 0.675139695 0.667336845 0.658430238 2 20 0.373238269 0.305011992 0.251479256 2 20 0.681825858 0.69320834 0.696071668 100 0.373617013 0.305577286 0.25231405 100 0.676884599 0.685081612 0.681244199 eb 0.373632747 0.305600765 0.252348727 eb 0.676677903 0.684741481 0.680623243 3 20 0.44925751 0.357372974 0.280582333 3 20 0.663081249 0.684467961 0.693158972 100 0.449483015 0.357736179 0.281108551 100 0.659887918 0.679616675 0.684818598 eb 0.449492388 0.357751272 0.281130422 eb 0.659754399 0.679413805 0.684469755 0.7 1 20 0.055186907 0.050063457 0.045640544 0.7 1 20 0.379309729 0.361963181 0.360261413 100 0.056450765 0.051615789 0.047851137 100 0.35135486 0.324284617 0.302297179 eb 0.056503226 0.051680255 0.047942968 eb 0.350182851 0.322704721 0.29986644 2 20 0.108655097 0.079229345 0.061213511 2 20 0.524759271 0.460060425 0.396540649 100 0.109305745 0.079925259 0.062080655 100 0.518851854 0.450425138 0.379786092 eb 0.109332728 0.079954145 0.062116669 eb 0.518604613 0.450021642 0.37908408 3 20 0.156066172 0.101028811 0.07073479 3 20 0.57511257 0.510509096 0.429191083 100 0.156648268 0.1015835 0.071306225 100 0.571486178 0.504829574 0.419799388 eb 0.156672393 0.101606523 0.071329969 eb 0.57133452 0.504592024 0.419406539 1 1 20 -0.00086016 -0.00123191 -0.00196910 1 1 20 0.025524562 0.036555975 0.058431446 100 -0.00003 -0.00005 -0.00008 100 0.001027398 0.001471427 0.002351944 eb 0 0 0 eb 0 0 0 2 20 -0.00028672 -0.00041063 -0.00065636 2 20 0.008508185 0.012185324 0.019477148 100 -0.00001 -0.00002 -0.00003 100 0.000342464 0.000490475 0.000783981 eb 0 0 0 eb 0 0 0 3 20 -0.00014906 -0.00021370 -0.00034159 3 20 0.004440154 0.006359143 0.010164521 100 -0.000006 -0.000009 -0.000014 100 0.000178377 0.000255475 0.000408356 eb 0 0 0 eb 0 0 0 the clt is tabulated as shown in table 3 for clamped-free beam. figs. 10-12 show the clp dependence on changes r1 while the beam type influence, dimension ratio 2 and slenderness ratio  are taken into consideration. here, the influences of parameters on the clp values are totally clear: in the above figs. it can be noticed that the clp values are higher for lower values of dimension ratio 2, maximum for beam type 3, less for beam type 2 and minimum for beam type 1. the influence of slenderness ratio  on the clp values is insignificant, since the influence of shear in this manner is very small. critical load parameter of a timoshenko beam with one-step change in cross section 273 fig. 10 variation of clp with r1 for cl-fr beam and 2 0.8  : a) 1 20   , b) 1 100   fig. 11 variation of clp with r1 for cl-fr beam and 1 20   : a) type 1, b) type 3 fig. 12 variation of clp with r1 for cl-fr beam and 2 0.8  : a) type 1, b) type 3 274 g. janevski, m. stamenković, m. seabra fig. 13 variation of clp with r1 for cl-pn beam and 2 0.8  : a) 1 20   , b) 1 100   fig. 14 variation of clp with r1 for cl-pn beam and 1 20   : a) type 1, b) type 3 fig. 15 variation of clp with r1 for cl-pn beam and 2 0.8  : a) type 1, b) type 3 the clp is tabulated as shown in table 4 for clamped-pinned beam. in figs.13-15, clp dependence on change r1 is presented while the beam type influence, dimension ratio 2 and slenderness ratio  are taken into consideration. the conclusions presented in the previous cases are here confirmed, especially in the part when the step position is near the supports. critical load parameter of a timoshenko beam with one-step change in cross section 275 5. conclusions the frequency equations of one-step timoshenko beams under compressive axial forces and four combinations of classical boundary conditions are expressed as the fourth order determinant equated to zero. the critical axial forces are expressed as a function of the critical load parameter and the critical load for uniform beam. the critical load parameters are tabulated for the beams with the classical boundary conditions. to determine the trend of change and sensibility, the dependence of the critical load parameter is displayed graphically for three types of beam in the function of step position, flexural rigidity ratio and slenderness ratio. less sensitive are the beams with a rectangular cross-section (type 1 and type 2), while the most sensitive is the beam with a circular cross-section (type 3). sensitivity is minor for small values of slenderness ratio due to the small influence of shear. influence of slenderness ratio is almost nonexistent for a clamped-free beam. as far as the type of end support is concerned, there are fields with very small changes of clp in the beams in which one of the support is clamped. regarding a near-supports position, sensitivity is minor for pinned and free end supports and larger for a clamped end support. also, in the beams where one of the supports is clamped there exists a field where sensitivity is very small. the method is applicable for any type of the boundary conditions and other system parameters. references 1. jang, s.k., bert, c.w., 1989, free vibration of stepped beams: exact and numerical solutions, journal of sound and vibration, 130, pp. 342-346. 2. jang, s.k., bert, c.w., 1989, free vibration of stepped beams: higher mode frequencies and effects of steps on frequency, journal of sound and vibration, 132, pp. 164-168. 3. naguleswaran, a., 2006, vibration and stability of ring-stiffened euler-bernoulli tie-bars, applied mathematical modelling, 30, pp. 261-277. 4. naguleswaran, a., 2004, transverse vibration and stability of an euler-bernoulli beam with step change in cross-section and in axial force, journal of sound and vibration, 270, pp.1045-1055. 5. naguleswaran, a., 2003, vibration and stability of an euler-bernoulli beam with up to three-step changes in cross-section and in axial force, international journal of mechanical sciences, 45, pp.1563-157 9. 6. naguleswaran, a., 2002, natural frequencies, sensitivity and mode shape details of an euler-bernoulli beam with one-step change in cross-section and with endes on classical supports, journal of sound and vibration, 252, pp.1045-1055. 7. naguleswaran, a., 2002, vibration of an euler-bernoulli beam on elastic end supports and with up to three step changes in cross-section, international journal of mechanical sciences, 44 pp.2541-2555. 8. bokaian, a., 1990, natural frequncies of beams under tensile axial loads, journal of sound and vibration, 142, pp. 481-498. 9. bokaian, a., 1988, natural frequncies of beams under compressive axial loads, journal of sound and vibration, 126, pp. 49-65. 10. mao, q., 2011, free vibration analysis of ultiple-stepped beams by using adomian decomposition method, mathematical and computer modelling, 54, pp.756-764. 11. mao, q., pietrzko, s., 2010, free vibration analysis of stepped beams by using adomian decomposition method, applied mathematics and computation, 217, pp. 3429-3441. 12. wu, j-s., chang, b-h., 2013, free vibration of axial-loaded multi-step timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method, european journal of mechanics a/solids, 38, pp. 20-37. 13. zhang, z., huang, x., zhang, z., hua, h., 2014, on the transverse vibration of timoshenko double-beam systems coupled with various discontinuities, international journal of mechanical sciences, 89, pp. 222-241. 276 g. janevski, m. stamenković, m. seabra kritiĉna opterećenje timošenkove stepenaste grede sa jednom promenom popreĉnog preseka u radu se analiziraju transferzalne oscilacije timošenkove grede sa jednom promenom poprečnog preseka i pritisnute aksijalnim silama koje su konstantne duž grede. razmatrane su tri tipa grede koje se često koriste u inžinjerskoj praksi. frekventna jednačina timošenkove grede je prikazana u obliku determinante četvrtog reda. kritično opterećenje je izraženo uvodjenjem parametra kritičnog opterećenja čije su vrednost iza gredu sa standardnim graničnim uslovima prikazane u tabelama. na osnovu tih rezultata i drugih vrednosti koje su sračunate za odredjene parametre sistema, zavisnost parametra kritičnog opterećenja je prikazana grafički u funkciji položaja promene poprečnog preseka. ključne reči: timošenkova greda, frekventna jednačina, kritična sila, parametar kritičnog opterećenja, stepenasta greda. original research paper facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 383 395 https://doi.org/10.22190/fume171004023s original research paper method of conversion of highand middle-speed diesel engines into gas diesel engines udc 629.1 mikhail g. shatrov, vladimir v. sinyavski, andrey yu. dunin, ivan g. shishlov, andrey v. vakulenko energo-ecological faculty, moscow automobile and road construction state technical university (madi), moscow, russia abstract. the paper aims at the development of fuel supply and electronic control systems for boosted highand middle-speed transport engines. a detailed analysis of different ways of converting diesel engine to operate on natural gas was carried out. the gas diesel process with minimized ignition portion of diesel fuel injected by the common rail (cr) system was selected. electronic engine control and modular gas feed systems which can be used both on highand middle-speed gas diesel engines were developed. also diesel cr fuel supply systems were developed in cooperation with the industrial partner, namely, those that can be mounted on middle-speed diesel and gas diesel engines. electronic control and gas feed systems were perfected using modeling and engine tests. the high-speed diesel engine was converted into a gas diesel one. after perfection of the gas feed and electronic control systems, bench tests of the high-speed gas diesel engine were carried out showing a high share of diesel fuel substitution with gas, high fuel efficiency and significant decrease of noх and со2 emissions. key words: gas diesel engine, engine control system, gas feed system, diesel fuel supply system, high-speed engine, middle-speed engine 1. introduction at present, natural gas is considered to be one of the most promising types of alternative fuels. the proven resources of gas on the earth are much higher than those of oil. natural gas is cheaper than oil and the engines fueled by it have significantly cleaner exhaust emissions compared with diesel, especially particles. in 2010, the share of natural received: october 04, 2017 / accepted november 16, 2017 corresponding author: vladimir sinyavski moscow automobile and road construction state technical university (madi), energo-ecological faculty, russia 125319 moscow, leningradski pr., 64 e-mail: sinvlad@mail.ru 384 m.g. shatrov, v.v. sinyavski, a.yu. dunin, i.g. shishlov, a.v. vakulenko gas in the world balance of gaseous alternative fuels exceeded 50%. it is forecasted that the share of natural gas will increase to 5.1 billion tons by 2035 and its share in the fuel balance of our planet will increase up to 25% [1, 2]. diesel engine may be converted into spark ignition gas engine operating on a stoichiometric gas-air mixture. the advantages are stable gas combustion and the possibility of using a three-way catalyst similar to that mounted on petrol engines. the location and geometry of the gas supply valves in the intake manifold significantly influence engine operation parameters [3]. using a stoichiometric gas-air mixture does not improve fuel economy compared with diesel engine, but taking into account the fact that natural gas is almost twice cheaper in russia than diesel fuel, expenses for fuel are much lower. the research of fuel consumption by buses having different powertrains running along three different routes show that the buses having the gas engines with stoichiometric gas-air mixture consumed by 10-25% more fuel than those with diesel engine. as natural gas is by 52% cheaper than diesel fuel in serbia, expenses for gas fuel are considerably lower [4]. stoichiometric gas engines have a high temperature of exhaust gases that is dangerous for turbocharger and impedes high boosting. in lean mixture gas engines, the exhaust gases temperature is low and this enables them to have high boosting, high fuel efficiency and low nox emissions which makes it possible to avoid the use of a reduction catalyzer. for middle-speed gas engines operating on a lean gas-air mixture, the prechamber with enriched mixture is used to inflame and control the lean mixture combustion in the main combustion chamber. the fev company developed a prechamber system for gas engine operating on a lean gas-air mixture. to attain the most efficient combustion without knock and low emissions of nox, the number and size of the prechamber holes, their orientation and prechamber size were investigated. a moderate miller cycle was used. calculations were fulfilled using the charge motion design (cmd) package which made it possible to investigate the influence of turbulence on the combustion process. experiments were carried out on a one-cylinder 16 l diesel engine modified for operation on natural gas. after simulation and experimental perfection, the indicated mean effective pressure reached 32 bars, the engine operated without knock and the nox emissions were within the limits of the ta-luft standard [5]. jenbaсher j624 (type 6) gas engines having d/s=190/210 mm used for electric power generation operating at 1500 rpm on a lean gas-air mixture having a gas enriched prechamber attain a high brake mean effective pressure of 2.4 mpa. here, the advanced miller cycle is implemented to avoid knock which also ensures high thermal efficiency up to 48.7% and reduces nox emissions. a two-stage turbocharging system is used to compensate for filling efficiency drop caused by the miller cycle and it also increases the engine effective efficiency by few percent compared with one-stage turbocharging [6, 7]. this is a very good solution for power generation because the engine operates all the time at a high constant speed which enables it to avoid knock. less fuel efficiency compared with gas diesel engines is compensated by a nox much lower price of gas when stationary engines are connected directly to gas pipelines. in russia, gas from the pipeline may be up to 4 times cheaper than that at the gas filling station [3]. the use of this working process on average-speed high boosted transport engines is impeded by knock which originates at low speeds and transfer modes especially for engines having a large cylinder bore. method of conversion of highand middle-speed diesel engines into gas diesel engines 385 gas diesel (or dual-fuel) engines do not have problems of knock. they may have a large size and high boosting. in gas diesel engines using a traditional (mechanical) gas fuel feed system, substitution of diesel fuel by gas is comparatively low because the share of diesel fuel is 20-30% at full loads, it grows with decrease of load and becomes 100% at idle speed [8]. the share of diesel fuel can be lowered if a special hp fuel pump for injecting small portions of diesel fuel is used [9]. substitution of diesel fuel by gas can be increased and most of the engine parameters can be significantly improved in “new generation” gas diesel engines using a minimized portion of diesel fuel injected by the common rail (cr) system for ignition of the gas-air mixture. the analyses conducted in the moscow state technical university named after n.bauman [10] showed that the larger fuel sprays are, the higher is ignition stability of the gas and the lower may be the ignition portion of diesel fuel. if the injector nozzle holes are unchanged, the size of the fuel sprays is determined mainly by the diesel fuel injection pressure (if the pressure increases, the fuel drops are smaller and the fuel spray surface is larger), the speed and direction of gas movement during the compression stroke (if the speed increases, the mixture formation and combustion of the gas-air mixture improve), and the backpressure in the cylinder (if the backpressure increases, the diesel fuel atomization is more efficient). when the engine speed and load decrease, the pressure and speed of the working medium in the cylinder are lower, also the backpressure drops due to lower boost pressure provided by the turbocharger. therefore, a small portion of ignition diesel fuel which can be less than 5% at high load has to be increased at low loads. it is possible to increase the size of the fuel sprays by raising the injection pressure and using a special baffled piston to increase the tangential speed of the gases. in the same paper, a special design of the traditional type fuel system injector is offered which ensures stable injection of small ignition portions of diesel fuel. the injector has a deformable rest. at low injection pressures corresponding to low engine loads, the injector needle valve bumps against a deformable rest and its small lift is stable from cycle to cycle. at higher injection pressures, the rest is deformed and the needle valve lifts to its full height. the combustion process of a gas diesel engine with direct injection of gas into the cylinder (gi engine) was investigated at the kyushu university using a rapid compressionexpansion machine (rcem) [11]. the propagation of flame bodies of diesel and gas fuel during the combustion process was fixed using high-speed cameras via windows in the cylinder head. the intake air turbulence was changed by variation of the location of the intake valve. the egr was imitated by decreasing the content of oxygen in the intake air from 21% to 17.5%. increasing the turbulence by the intake air resulted in the growth of the rate of heat release and a considerable decrease of unburned gas. emissions of nox in the gi engine were by 25% lower than in the diesel engine without egr. the egr enabled to decrease emissions of nox additionally by 75% which was visually confirmed by less brightness of the flame body corresponding to lower combustion temperature. in [12], the problems of combustion in gas diesel engines are addressed: high unburned hc and co emissions due to incomplete combustion in some zones especially at low load operation, cycle to cycle instability at high loads. to cope with this, a number of parameters should be thoroughly controlled: pilot dose of diesel fuel and its phasing, diesel fuel injection timing, quantity of gas, quantity and temperature of air entering the cylinder taking into account engine speed and load. the best way is to use a fast and efficient control algorithm. a 386 m.g. shatrov, v.v. sinyavski, a.yu. dunin, i.g. shishlov, a.v. vakulenko simple 0-dimensional model for simulation of combustion in gas diesel engine was developed to be used for engine control. it is based on vibe formula for diesel fuel combustion; it takes into account heat exchange with the walls and variation of thermodynamic parameters of the working medium. combustion of the natural gas is based on one-step macro reactions of the main components of the mixture. after validation of the model using the results of testing at four-cylinder 2.636 l gas diesel engine, it provides a pretty high agreement of calculated and experimental results. the problem of the gas diesel engines with minimized portion of ignition diesel fuel is probable overheating of the nozzles of standard cr injectors because of their poor cooling by fuel when only 3-5% of diesel fuel is injected. the solution may be to mount the cr fuel system of a smaller engine for which a 3-5% percent ignition portion of diesel fuel amounts almost to full-load fuel supply. in this case, the engine will not be able to operate on diesel fuel only though a reliable gas supply can be ensured for many applications, for example for locomotives and dump trucks that move along fixed routes. on the basis of the analysis carried out, the gas diesel engine using a minimized portion of diesel fuel supplied by the cr system that injects fuel at high pressure at any operation mode was chosen for highand middle-speed engines because it enables high boosting, engine efficiency, ecological parameters; also, it avoids knock. 2. method of development of fuel feed and electronic control systems for highand middle-speed gas diesel engines madi participated in the state programs of development of gas feed and electronic engine control systems for high and middle-speed engines fueled by natural gas and of development of cr diesel fuel supply systems for diesel and gas diesel engines. a highspeed diesel engine was available in madi and a middle-speed engine – not available because its mass production has not yet started. therefore, a method for development of fuel feed and engine control systems for a middle-speed gas diesel engine using a highspeed gas diesel engine as a mockup was proposed which included the following steps:  development of electronic engine control system and modular gas feed system suitable for both the highand middle-speed gas engines;  perfection of both the systems during engine tests on a high-speed gas diesel engine using the diesel fuel supply system of the base diesel engine;  development of fuel supply system for the middle-speed gas diesel engine jointly with the industrial partner. 3. research objects the research was carried out for two in-line 6-cylinder gas diesel engines: high-speed automobile cummins kama engine that was used for experimental perfection of modular gas supply and electronic engine control systems and middle-speed locomotive d200 engine whose mass production has not yet started. the operation parameters of the d200 gas diesel engine were calculated. cr diesel fuel supply system and turbocharging system with the bypass valve at the turbine inlet of the base cummins kama diesel method of conversion of highand middle-speed diesel engines into gas diesel engines 387 engine were used for the gas diesel version. the main parameters of the two base diesel engines are presented in table 1. table 1 main parameters of two engines investigated base diesel engine cylinder diameter (mm) cylinder stroke (mm) rated speed (rpm) rated break mean effective pressure (mpa) compression ratio locomotive d200 200 280 1000 2.0 14.0:1 automobile cummins kama 107 124 2300 1.73 17.3:1 4. engine systems developed for conversion of highand middle speed diesel engines into gas diesel ones, electronic engine control, modular gas feed and cr fuel supply systems were developed. 4.1. electronic engine control system a completely new electronic engine control system for 6-cylinder gas diesel engines was developed which controls supply of gaseous and diesel fuel (fig. 1). the system generates electric control impulses to control actuators; it carries out synchronization and distribution of impulses by the cylinders depending on the engine operation mode on the basis of information received from many sensors. fig. 1 components of the electronic engine control system for gas diesel engines: 1 – information-calculation block; 2, 10 – intake manifold temperature and pressure sensors; 3, 9 – cooling agent temperature sensors; 4 – barometric correction sensor; 5 – crankshaft position sensor; 6 – camshaft position sensor; 7 – crankshaft and camshaft position sensors adapter; 8 – block of thermocouples 388 m.g. shatrov, v.v. sinyavski, a.yu. dunin, i.g. shishlov, a.v. vakulenko 4.2. modular gas feed system the gas feed system has a modular architecture. each module (fig. 2) ensures pressure reduction as well as a supply of natural gas. this enables us to use a different number of modules depending on the engine size. one module is used on the high-speed cummins kama gas diesel engine and three modules – on the middle-speed d200 gas diesel engine. the gas feed system ensures a supply of natural gas under working pressure of 1 mpa for the gas diesel engine with external mixture formation. it has metering valves with electronic control. when three modules are mounted on the d200 engine, three gas supply valves for each cylinder are used: two small valves of the cummins kama engine for injection of small portions of gas at idle and one large valve – at high loads. fig. 2 one module of gas supply system for the gas diesel engine: 1 – main high pressure solenoid valve; 2 – two-stage gas pressure reducer; 3 – pressure and temperature sensors in the reducer; 4 – cooling agent controller; 5 – gas pressure sensor; 6 – gas supply valves; 7 – gas receiver; 8 – gas temperature sensor 4.3. cr fuel supply system for the middle-speed gas diesel engine in partnership with the industrial partner – noginsk factory of fuel systems ojsc, two cr fuel systems were developed which may be used for both middle-speed diesel and gas diesel engines complying with the ecological standards stage iiib:  d200 (6-cylinder, d/s=200/280 mm) manufactured by penzadieselmash ojsc,  m150m (12-cylinder, d/s=150/175 mm) manufactured by zvezda ojsc,  dm185t (6-cylinder, d/s=185/215 mm, 12-cylinder d/s=185/215 mm, 16cylinder d/s=185/215 mm, 20-cylinder d/s=185/215 mm) manufactured by ural diesel engine factory ojsc. to get the highest possible commonality of the model line, the cr fuel systems of the aforementioned engines consist of the maximal possible number of identical components. the layout of the cr fuel system mounted on the 6-cylinder engine with d/s=200/280 mm is shown as an example (fig. 3). the high pressure (hp) fuel pump includes six plunger sections located in radial direction by pairs along the circle over 120 o (fig. 4). method of conversion of highand middle-speed diesel engines into gas diesel engines 389 fig. 3 layout of the cr fuel system on the 6-cylinder diesel engine d/s=200/280 mm: 1 – hp fuel pump; 2 – fuel feed pump; 3 – low pressure fuel line from the fine fuel filter to the hp fuel pump; 4 – low pressure fuel line from the pump 2 to the fine fuel filter; 5 – fine fuel filter; 6 – fuel line from the fuel pump to the first cylinder injector; 7 – cr injector (cri); 8 – fuel lines between the injectors fig. 4 test model of the hp fuel pump: 1, 2 – flanges of the shaft drive and fastening to the cylinder block correspondingly; 3 – head of delivery sections; 4 – nut of high pressure fuel line; 5 – common rail; 6 – solenoid valve; 7, 12 – fittings of the low pressure fuel line and lubricating system correspondingly; 8 – pressure sensor; 9 – fuel feed pump; 10 – bolts fastening fuel pump heads; 11 – fuel pump body 390 m.g. shatrov, v.v. sinyavski, a.yu. dunin, i.g. shishlov, a.v. vakulenko the plungers located in one row operate in the reversed phase – the delivery cycles take place at every 180 o of the crankshaft rotation. in this case, the hp fuel pump cycles of fuel delivery into the high pressure line occur at every 60°. such a sequence of working strokes of the plungers assures the lower (compared with the in-line arrangement of the hp fuel pump) and uniform load on a drive shaft with eccentric cam (compared with direct action fuel systems). power consumption of the hp fuel pump drive decreases. the cr injector (cri) developed is presented in fig. 5. fig. 5 test model of the cri: 1 – nozzle body; 2 – injector nozzle valve; 3 – nozzle nut; 4 – nozzle spring; 5 – floating nozzle bush; 6 – control chamber; 7 – input jet; 8 – output jet; 9 – control valve seat; 10 – control fuel return channels; 11 – channel for fuel input from the common rail of the cri; 12 – control valve; 13 – armature; 14 – electromagnet core; 15 – electromagnet body; 16 – core spring; 17 – magnet spring; 18 – electromagnet power supply wires; 19 – injector body; 20 – sealing rings; 21 – channel for electromagnet power supply wires; 22 – integrated common rail; 23 – clamping element; 24 – feeder nut; 25 – feeder; 26 – holes for mounting fuel tubes; 27 – high pressure fuel lines; 28 – fitting for return of fuel leaks from the feeder; 29 – thrust collar; 30 – channel for fuel supply to the common rail method of conversion of highand middle-speed diesel engines into gas diesel engines 391 the high pressure common rail 22 is integrated into the injector body 19 in order to smooth the pressure oscillations which originate due to a fluctuating fuel supply from the hp pump and to injectors’ operation during the injection process. the fuel rails of the cris are connected to each other by high pressure fuel lines 8 (fig. 3) via feeders 25 (fig. 5). for gas diesel versions of diesel engines d200, m150m and dm185t, a smaller size cr system for supply of the minimized ignition portion of diesel fuel was developed. when developing this experimental cr system, the solutions of design and arrangement of the standard cr systems series on the basis of analysis of the experience of the fuel equipment development for diesel engines was used. the fuel system developed is maximally unified with the cr system of potential consumers of the industrial partner – noginsk factory of fuel systems cjsc. the hp pump, fuel lines and cris inject fuel under pressure of up to 200 mpa which enables us to obtain high parameters of the combustion process [13]. the fuel used for the cris control is drained from every cri to the low pressure line. the fuel pump has a traditional in-line design, location of plungers in a closed block and inserted fuel supply sections. the cams of the camshaft have an eccentric shape. the delivery valve has a traditional design with discharge collar. the plungers have no grooves: the fuel supply is varied by fuel pressure control in the chamber above the plunger. with the aim of unification with the base cr system (fig. 3), dimensions of electrohydraulic valve were preserved and the same floating nozzle bush is used to increase the life time of the cri. 5. computer modeling the parameters of gas diesel engines were calculated by 0-dimensional model according to the method described in [14] using the vibe formula for heat release and the woschni formula for heat losses calculation, empirical formulas for cylinder walls temperature calculation and taking into account composition of the working medium at any instant of the engine cycle. gas exchange was calculated based on the quasistationary approach while the turbocharger maps were used for calculating parameters at the cylinder inlet and outlet. the model was used for the following aims:  calculation of parameters of the highand middle-speed gas diesel engines required for development and adjustment of the gas feed and electronic engine control systems, and,  analysis of experimental parameters obtained during the engine tests of the highspeed gas diesel engine. 6. results and discussions the results of the high-speed gas diesel engine tests by the load and external speed characteristics are shown in figs. 6 and 7. to check the accuracy of the computer model and analyze the experimental results, the calculations of the engine parameters at all operation modes were carried out and 392 m.g. shatrov, v.v. sinyavski, a.yu. dunin, i.g. shishlov, a.v. vakulenko compared with the experimental data. the calculations were conducted with the values of gas consumption gg and diesel fuel consumption gd that were used in the experiments with an optimal ignition advance angle for every operation mode. calculation experiment fig. 6 load characteristic of the high-speed gas diesel engine cummins kama at n=1420 rpm figs. 6 and 7 demonstrate a high effective efficiency ηе=0.43-45 at full engine speed and load. there is also a good agreement of calculated and experimental values of boost pressure рs, gas pressure before turbine pt, air access coefficient α, air consumption ga, boost air temperature тs and effective efficiency ηе at high load and high engine speed. the difference between parameters ga, α and ηе increases at low loads (fig. 6) which may be explained by a not very accurate description of experimental compressor and turbine maps by polynomials at low engine loads. as seen from fig. 7, the values of mean effective pressure pe of gas diesel engine are pretty low at low engine speeds. this may be explained by the decrease of the air quantity in the cylinder and hence – air access coefficient α due to a partial substitution of air with gas fuel compared with the base diesel engine. the measured airflow of the gas diesel engine was approximately by 8% lower than that of the base diesel engine. other reasons may be those described in [10]. while in the diesel engine, all the fuel is located in the combustion chamber while in the gas diesel engine, a part of gas method of conversion of highand middle-speed diesel engines into gas diesel engines 393 penetrates into the gaps between the piston/cylinder head and the piston/liner where it burns incompletely. this effect is especially strong at low engine speed when the air turbulence in the combustion chamber is low. this phenomenon is indirectly confirmed by higher calculated effective efficiency ηе compared with its experimental value at low engine speed (fig. 7) because poor combustion of fuel in the gaps is not taken into account in the computer model used. calculation experiment fig. 7 external speed characteristic of the high-speed gas diesel engine cummins kama some increase of the mean effective pressure at low engine speeds may be achieved by tuning the turbocharger. but to get as high values of pe as in the base diesel engine, the combustion chamber should be redesigned which requires further in-depth investigations. fig. 8 shows the comparison of parameters of the base cummins kama diesel engine and its gas diesel version at three engine speeds and two loads: maximal and partial (30-40%). here the percentage of the ignition portion of diesel fuel to the total amount of fuel is indicated. at full loads, the percentage of diesel fuel is 4.5-6.2%, аt low loads – 8.7-8.9%. the percentage of diesel fuel at idle is 33%. on the average, the effective efficiency of the gas diesel engine is by 2% higher than that of the base diesel engine. co2 emissions decreased for 1.47 and 1.15 times and of nox – 7.4 and 1.52 times at full and partial loads, respectively. 394 m.g. shatrov, v.v. sinyavski, a.yu. dunin, i.g. shishlov, a.v. vakulenko fig. 8 comparison of parameters of the high-speed cummins kama base diesel engine with its gas diesel version 7. conclusion 1. the analysis conducted demonstrated that the gas diesel process using a minimized igniting portion of diesel fuel supplied by the cr system is the most reasonable way of converting high boosted highand middle-speed transport diesel engines to operation on natural gas. 2. the modular gas feed system and that of the engine electronic control were developed to be used both on highand middle-speed gas diesel engines. the base high-speed diesel engine was converted into gas diesel one and used for experimental perfection of these systems. 3. diesel fuel supply systems were developed for middle-speed engines: they are large in size for injecting full portion of diesel fuel for diesel engine or small in size – for injection of ignition portion of diesel fuel for gas diesel engines. method of conversion of highand middle-speed diesel engines into gas diesel engines 395 4. the gas feed and electronic engine control systems were experimentally perfected and engine characteristics were obtained which demonstrated a high degree of diesel fuel substitution by gas: the average diesel fuel portion amounted to 5.6%, 8.8 and 33%, correspondingly, at full load, approximately 35% load and idle. co2 emissions decreased for 1.47 and 1.15 times, of nox – 7.4 and 1.52 times, respectively, at full and partial loads. effective efficiency of the gas diesel was on the average by 2 percent higher than that of the base diesel engine. acknowledgements: applied research and experimental developments of fuel feed systems are carried out with financial support of the state represented by the ministry of education and science of the russian federation under the agreement no 14.580.21.0002 of 27.07.2015, the unique identifier pnier: rfmefi58015x0002. references 1. markov, v.a., gaivoronski, a.i., grehov, l.v., ivaschenko, n.a., 2008, operation of diesel engine on nontraditional fuels, legion-avtodata, moscow, 464 p. 2. shatrov, m.g., khatchijan, a.s., shishlov, i.g., vakulenko, a.v., 2008, analysis of conversion methods of automotive diesel engines to be powered with natural gas, transport na alternativnom toplive/transport on alternative fuel, 4(34), pp. 29-32. 3. luksho, v.a., 2015, a complex method of increasing energy efficiency of gas engines with high compression ratio and shortened intake and exhaust strokes, ph.d. thesis, nami, moscow, 365 p. 4. ivan s. ivković, snežana m. kaplanović, branko m. milovanović, 2017, influence of road and traffic conditions on fuel consumption and fuel cost for different bus technologies. thermal science, 21(1b), pp. 693-706. 5. josé geiger, peter heuser, sven lauer, berthold huchtebrock, harsh sankhla, 2013, combustion system development for a large bore gas engine – efficient combination of simulation and experiment, paper no 80, 27th simac congress, helsinki 6. klausner, j., lang j., trapp, c., 2011, j624 – der weltweit erste gasmotor mit zweistufiger aufladung, mtz – motortechnische zeitschrift ausgabe, 04. 7. grotz, m., böwing, r., lang, j., thalhauser, j., christiner, p., wimmer, a., 2015, efficiency increase of a high performance gas engine for distributed power generation, 6th cimac cascades. dual fuel and gas engines – their impact on application, design and components 8. kudryavtzev, a., lomashov, v., 2010, belaz trucks of xxi century with dm family gas diesel engines, avtogasozapravochniy komplex + alternatvnoye toplivo/autogasfillingcomplex + alternative fuel, 3, pp. 3-6. 9. kapustin, a.a., 2008, fuel feed and control system of a gas diesel engine operating on natural gas. transport na alternativnom toplive/transport on alternative fuel, 4, pp. 46-49. 10. grehov, l.v., ivsachenko, n.a., markov, v.a., 2010, on ways to improve the gas diesel cycle, avtogasozapravochniy komplex + alternatvnoye toplivo/autogasfillingcomplex + alternative fuel, 7(100), pp. 10-14. 11. imhof, d., tsuru, d., tajima, h., takasaki, k, 2013, high-pressure natural gas injection (gi) marine engine research with a rapid compression expansion machine, paper no 12, 26th simac congress, shanghai 12. mikulski, m., wierzbicki, s., 2017, validation of a zero-dimensional and two-phase combustion model for dual-fuel compression ignition engine simulation, thermal science, 21(1b), pp. 387-399. 13. shatrov, m. g., golubkov, l. n., dunin, a. yu., yakovenko, a. l., dushkin, p. v., 2015, influence of high injection pressure on fuel injection perfomances and diesel engine working process, thermal science, 19(6), pp. 2245-2253. 14. khatchijan, a.s., sinyavskiy, v.v., shishlov, i.g., karpov, d.m., 2010, modeling of parameters and characteristics of natural gas powered engines, transport na alternativnom toplive/transport on alternative fuel, 2010, 3(15), pp. 14-19. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 367 381 https://doi.org/10.22190/fume171002020f original scientific paper a new winch construction for the smooth cable winding/unwinding udc 531 mirjana filipović 1 , ljubinko kevac 2 1 mihajlo pupin institute, university of belgrade, serbia 2 school of electrical engineering, university of belgrade, serbia abstract. new constructive solutions of the winches for single-row radial multi-layered cable smooth winding/unwinding are described. two new structural solutions of winches are defined. the nonlinear phenomenon of a cable smooth winding/unwinding process on the winch by using one of the two proposed constructive solutions is defined and analyzed. to facilitate understanding of this concept, the cable winding/unwinding process on only one winch is analyzed. the obtained variables which characterize the kinematics of the cable smooth winding/unwinding process are nonlinear and smooth. this result is important because the systems for the smooth cable winding/unwinding process on the winch could be parts of any cable driven mechanism. these systems can be used in various fields of human activity. for the verification of the presented theoretical contributions, a novel software package named smowind – ow has been developed using matlab. key words: analysis, kinematics, cable winding/unwinding, winch construction, simulation 1. introduction the problem of the cable winding/unwinding on the winch is present in various systems used in different technical areas. these systems can have completely different applications and also different constructions but their characteristic is that they consist of one or more sub-systems for cable winding/unwinding. all these systems require stable control for implementation of defined task. only several systems will be mentioned: measuring mechanism, machines in textile industry, cable logging systems in civil engineering and forestry, cranes, systems in shipping industry, cpr (cable-suspended parallel robot) and other complex cable driven systems. received october 02, 2017 / accepted november 22, 2017 corresponding author: mirjana filipović mihajlo pupin institute, university of belgrade, volgina 15, 11000, belgrade, serbia e-mail: mirjana.filipovic@pupin.rs 368 m. filipović, lj. kevac for many decades, many researchers have dealt with analyzing the cable winding/ unwinding process on the winch. in this paper, only some of these investigations, namely, those that have inspired this research, will be presented. authors of [1] have reviewed application of the cable logging systems. they have defined a user’s manual which helps them to solve several problems such as protection of workers, soil and forests. samset [2] has given a historic review of the systems which use winches for cable winding/ unwinding. the author gives information that these systems are used for more than five millennia which emphasizes their importance. abdel-rahman et al. [3] have analyzed dynamics and control of the cranes having cable winding/unwinding sub-systems as the main part. this is a review paper which shows historic development of cranes. in [4-6], authors present theoretical and experimental contribution to analysis and synthesis of kinematics and dynamics of the winding/unwinding process of thread from a balloon. fluctuating tensions in a perfectly flexible string unwinding from a stationary package is considered by padfield [4]. also, dependence of unwinding tensions on air resistance, unwinding speed, angle of winding on the package etc. is examined. based on results of padfield [4], fraser et al. [5] have considered over-end unwinding of the yarn from a stationary helically wound cylindrical package. an improved theory for the variation of the yarn tension during high speed over-end unwinding from cylindrical yarn packages based upon the theory of the bent and twisted elastic rods is presented by clark et al. [6]. imanishi et al. [7] have presented a dynamic simulation of a wire rope involving both contacts with the winch drum and hydraulic systems using the finite element method. the rapid winch operation often causes a disorderly winding of the wire rope, which is an important quality problem. dynamic simulation is, therefore, required for design of the hydraulic winch system on construction machinery. the wire rope is modeled using truss elements considering a large displacement motion. szczotka et al. [8] have presented the mathematical model of a pipelay spread. in the model, elasto-plastic deflections of the pipe, its large deformations and contact problems are considered. the modification of the rigid finite element method (refm) is used to discretize the pipe. wire-guided control technologies are widely used to increase the targeting accuracy of advanced military weapons through the use of unwinding dispensers to guarantee that unwinding occurs without any problems, such as tangling or cutting. lee et al. [9] have investigated transient behaviors of the cables unwinding from inner-winding cylindrical spool dispensers. the cable is withdrawn from the spool dispenser at a constant velocity through a fixed point located along the axis of the spool dispenser. filipovic et al. [10] have dealt with design, analysis and synthesis of the cable suspended parallel robotic system (cpr system). they have used a well-known winding/ unwinding sub-system presented by von zietzwitz et al. [11]. a wide application of the process of cable winding/unwinding on the winch is important; that is why it deserves a detailed analysis and it is the subject of this paper. in this paper, two new forms of winches for ensuring single-row radial multi-layered cable smooth winding/unwinding have been constructed: a) the first constructive solution is composed of two semicylindrical bodies of different radii: a two-cylinder winch. b) the second constructive solution has a spiral shape: a spiral winch. these two constructive solutions are the main contribution of this paper. the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 369 for analysis and synthesis of the results from this paper a new software package smowind – ow has been synthesized using matlab. kevac et al. [12] have presented a general form of mathematical model of the cable winding/unwinding system. also, it can be noticed that the problem which will be analyzed in this paper is only one special case. after introduction, in section 2 the basic theoretical principles of the kinematics of a smooth single-row radial multi-layered cable winding/unwinding process on the winch are presented. in section 3 the program package smowind – ow is defined. this program package is used to make simulation experiments and these simulations are shown in section 4. finally, the last part of the paper presents the conclusions, remarks, and plans for future research. 2. the basic theoretical principles kinematics of the single-row radial multi-layered smooth cable winding/unwinding process on the winch, hereinafter abbreviated as: the cable smooth winding/unwinding process on the winch, is a very complex and nonlinear process. 2.1. problem definition the solution presented in this paper comes as a result of analyzing the standard winch for single row radial multilayered cable winding (unwinding) from fig. 1. the analysis of the behavior of a circular winch intended for a single-row radial multi-layer cable winding/unwinding system indicates that this system is far from ideal and contains a series of constructive problems. this simple construction of the winch has caused abrupt changes of important variables of this system: radius of cable winding/unwinding: ri, cable length: lwi, and inclination of the cable with respect to yi axis: i. a detailed description of this problem is given by kevac et al. [13]. in this sub-section, the problem of cable winding/unwinding is only presented on standard winch for single row radial multilayered cable winding (unwinding), but it should be emphasized that this problem is present in other forms of cable winding/unwinding systems, for example: a cable winding/unwinding system for a multi-row radial and axial cable winding/unwinding process, see kevac et al. [12]. because of the described effects, it is required to find a solution in the form of a new construction of the winch. constructively, only a winding/unwinding system which does not generate abrupt changes of the variables involved presents a good system for this purpose. the main point of this research is a geometric and mathematic definition of the cable winding/unwinding process on a novel shape of the winch intended for smooth rope (cable) winding/winding, unlike in the work by kevac et al. [12], where the general form of mathematical model of the cable winding/unwinding system is defined. 2.2. problem solution in this paper, a kinematics model of the smooth cable winding/unwinding process on the winch will be developed. the new constructive solution of the winch is presented in fig. 2 in which one can see a new shape of the winch which is adapted to the user’s need that the 370 m. filipović, lj. kevac winding/unwinding process should occur in a smooth and nonlinear fashion. fig. 1 standard winch for the cable winding/unwinding system fig. 2 winch for the smooth cable winding/unwinding process: a) two-cylinder winch, b) spiral winch to achieve a smooth cable winding/unwinding process, two new constructive solutions of the winch are designed: 1) the first constructive solution consists of two semi cylindrical bodies of different radii, as shown in fig. 2a. in view of the characteristics of this winch, it has been named a two-cylinder winch. the two-cylinder winch will be presented in this paper in detail, and 2) the second constructive solution has a spiral shape, as shown in fig. 2b. it has been named a spiral winch. similar effects of the smooth cable winding/unwinding process are achieved for both the spiral and the two-cylinder winch. the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 371 in this paper, the two – cylinder winch will be used for performing the corresponding theoretical analysis. in the continuation of this section, the geometry of the cable smooth winding/unwinding process on one winch will be shown and analyzed in detail. the cable is mounted so that it emerges from the winch at a place where there is a joint of two semi cylindrical bodies. the starting position of the cable during the winding/unwinding process on the winch is shown in fig. 3. the starting position is systematically (by calibration) defined to be at the direction of the negative part of xi axis. selection of the starting position affects further kinematics of the cable winding/unwinding process on the winch. fig. 3 the starting position of the winding/unwinding system a more detailed look at the starting position of the system for cable winding/unwinding is shown in fig. 4a. in order to facilitate understanding of the kinematics of the cable smooth winding/unwinding process on the winch, the geometry of this complex process will be shown. defining the geometry of the cable smooth winding/unwinding process is needed for specifying the kinematic and dynamic models of this process. also, for a good understanding it is sufficient only to analyze the winding process of the cable on the winch; thus the analysis of the process of unwinding will be omitted since the phenomena are the same but only occur in the reverse order with respect to winding. figure 3 shows the starting position of the system for the smooth winding/unwinding of the cable on the winch. this system consists of a newly shaped motorized winch, hereinafter referred to as winch, which is positioned along one axis of the xi yi coordinate system. this winch (winds up)/unwinds a cable of diameter d=0.0008 m. from the other side this cable goes over a smaller pulley (which is not motorized) which has radius r=0.009 m. this pulley is positioned at the center of xi1 yi1 coordinate system. the axis of rotation of the pulley is 372 m. filipović, lj. kevac positioned at the base of this coordinate system, point c. this point in relation to the xi yi coordinate system has coordinates (-a, b). distance a=0.045 m presents a horizontal distance between rotation axes of the winch and the pulley, while distance b=0.524 m presents a vertical distance between these two axes. the new shape of the winch is composed of two semicylindrical bodies. the smaller semicylinder has a basis of radius ri0=0.0136 m with the center at point oi(0, 0). radius of the bigger semicylinder 0ir ~ is geometrically defined by the shape function of this winch in relation to the radius of the smaller semicylinder ri0. the bigger semicylinder has a radius of basis m014.02/drr ~ 0i0i  with the center at point )0,2/d(o ~ i  . the system for smooth winding/unwinding of the cable on the winch is constructed so that when shown in the plane perpendicular to the winding/unwinding axis, it looks as shown in fig. 3. to simplify the terminology, the two semicylinders will further on be referred to as two semicircles. the center of the smaller semicircle is at the origin, oi, of the coordinate system xi yi . the rotation axis of the motorized winch is labeled as oi. unlike point oi, the center of the bigger semicircle, point io ~ , keeps on changing its position constantly during the cable winding/unwinding process on the winch. point io ~ is constantly rotating around point oi at distance d/2. fig. 4 a) the starting position of the winding/unwinding system – close-up; b) position of the winding/unwinding system for 0<i a better view of the winch is shown in fig. 4a. here one can clearly see the place where the cable emerges from the winch. point e belongs to the outer contour of the winch and it is positioned at the intersection of the smaller semicircle and the flat part of the winch. at the starting position, this point belongs to the negative part of xi axis. this point has a stationary position in comparison with the winch during the whole of the cable winding/unwinding process on the winch. it is assumed that the deflection angle between line oie and negative part of xi axis defines the winding/unwinding process of the cable. this deflection is denoted as angle i whose value at the starting position is: 0i  (1) the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 373 unlike point e, point t constantly changes its position on the winch. this point presents the place where the cable touches the winch or the cable wound so far. at the starting position, points e and t overlap, as can be seen in figs. 3 and 4a. the winch and the pulley are positioned so that angle i has a positive value at any moment. angle i presents one of the variables of the system which characterize the cable winding/unwinding process. at the starting position, angle i has the largest value of 0.04116 rad, which is defined by the constructive solution of the mechanical part. this angle is defined in the following fashion: the first leg of the angle is the line running through origin oi in parallel with the tangent drawn at point t to the circle having radius r + d, centered at point c; the second leg is positive part of the yi axis. it is presumed that the cable is wound at constant angular speed, i.e. consti  . this presumption presents an idealized theoretical condition, which has been introduced for an easier and clearer explanation of the smooth cable winding process on the winch. it is assumed that the cable force acts through the central axis of the cable, i.e. along direction of the line sm (see figs. 3 and 4a). usually, it is presumed that the winding/ unwinding radius is calculated at the position where the force acts upon the winch. with this presumption and from figs. 3 and 4a it can be seen that this radius presents distance oiar at the starting position. the value of this radius can be calculated as follows: )cos(aor iiii  (2) at the starting position, point a is positioned at the intersection between line oia and the line which contains point t and is rotated about point t by angle i. the angle between lines oia and oit is labeled as i. this angle has the biggest value at the starting position. the position of point ar is at the intersection between lines sm and or. point b is positioned at the intersection between line sm and the line which contains point c and which is parallel to line or. distance ab labeled as lwi presents a dynamic variable during the cable smooth winding/unwinding process on the winch. this variable has its dynamics of change and thus affects the dynamic response of the system (see fig. 3). figure 4b shows the position of the system when values of angle i meet the following requirement: ii0  (3) for this position of the system, the winding/unwinding radius is calculated as: )cos(aor iiiii  (4) as in the previous case (figs. 3 and 4a), point ar is positioned at the intersection of lines sm and or, while point a is at the intersection of line sm and the line which contains point t and is parallel to line or. in relation to the initial value of angle i, its value is smaller now, while distance lwi is growing in relation to its initial value. the angle between lines oia and oit decreases during the period of winding which is specified by eq. (3). the system’s position which is analyzed next is shown in fig. 5a. this is the position when the following condition is satisfied: ii  (5) at this moment, points ar and a are at the same position and they are lying on line oit. from this moment on, point a is always on line oit during the winding process. at this 374 m. filipović, lj. kevac moment, angle i achieves value 0 and further on it does not affect the cable winding process. in fig. 5a, it can be seen that line oit is lying on x axis (also line oie is lying on this axis) – axis x presents the axis that is deflected by angle i compared to the positive part of xi axis. coordinate system x y is always defined by angle i. at this moment, the cable is tangential to the winch at point t. also, it is the last moment when points e and t overlap. from that moment on, point e maintains its fixed position on the body of the winch, while point t follows winding dynamics of the cable. at this moment, the winding/unwinding radius has the following value: 2/draoaor 0iriii  (6) also, at this moment, distance lwi has the biggest value during the cable winding process on the winch. the cable keeps on winding and angle i takes values from the following range: iii  (7) fig. 5 position of the system for a) i = i and b) i =  the range defined by eq. (7) has been named a con range (constant). throughout this range, radius ri, length lwi, and angle i (important variables of the cable smooth winding/ unwinding process) have constant values which they have acquired at the moment when the condition defined by eq. (5) is satisfied. an example of the position from this range is shown in fig. 5b. in this example line oie overlaps with the positive part of xi axis, i.e. i = . from fig. 5b it can be seen that positions of points e and t are different. an important moment is when the following condition is satisfied: ii  (8) because at that moment the system leaves the con range and a new law of change of all relevant variables: winding/unwinding radius ri length lwi, and angle i starts. it can be seen that at this moment line oie lies on x axis (see fig. 6a), but, in comparison with the positions from fig. 5a, this line is rotated about point oi by angle . after that moment, the cable is starting to wind smoothly on the part of the winch with a bigger radius, i.e. a bigger semicircle. the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 375 from that moment on, angle i takes the values defined by the following equation: iii 2  (9) the range when all the relevant variables are changing their values is named a smvar (smooth variable) and one position of the system within this range is shown in fig. 6b. fig. 6 position of the system for: a) i = i and b)  i <i < i fig. 7 position of the system for: a) i <i < i +, and b)  i +<i < 2 i angle i is changing linearly during the cable winding process, i.e. angular speed of the winch rotation is constant, consti  . the defined smvar range represents the period when cable winding/unwinding radius ri changes its value. upon entering this range, during the cable winding, this radius starts to grow from value ri = ri0 + d/2 to value ri = ri0 + 3(d/2) which is reached at the end of the smvar range. for an easier description of the change of radius ri, two sub-ranges of the smvar range will be considered: 1) the first sub-range is defined by the following change of angle i:  i <i < i +. this period of winding is shown in fig. 7a and in this sub-range of the smvar range, radius ri is calculated by ri = rt + d/2 . radius rt becomes a changing variable in the smvar range, where at the beginning of the range it has value rt = ri0. during the first sub-range, the change of this variable is described by the following eq.: 2 2 t i0 i ir (r d / 2) (d / 2 sin( )) d / 2 cos( )          (10) 376 m. filipović, lj. kevac 2) the second sub-range is defined by the following change of angle i:  i +<i < 2 i. this period is shown in fig. 7b and in this sub-range of the smvar range, radius ri is calculated by ri = rt + d/2, where radius rt is: 2 2 t i0 i ir (r d / 2) (d / 2 sin( )) d / 2 cos( )          (11) fig. 8 a) radius ri and b) the first derivative of radius ri over the smvar range fig. 9 a) length lwi and b) the first derivative of length lwi over the smvar range change of the winding/unwinding radius achieved in this fashion is smooth, as can be seen from fig. 8a, where variation of radius ri over the smvar region is presented. figure 8b shows the first derivative of this radius, i.e. variable ir , where one can see a smooth change of the winding/unwinding radius. it can be seen that during the cable winding process the radius is slowly growing towards the already defined value of ri = ri0 + 3(d/2). upon entering the smvar range, points t and a are slowly changing their positions. because of that distance lwi gradually decreases in its value during this period of winding. this phenomenon can be seen in fig. 9a, where variation of distance lwi over the smvar range is presented. this change has a similar dynamics like radius r, only it is decreasing during the cable winding process. figure 9b shows the change of the first derivative of this variable, iwl  . the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 377 fig. 10 a) angle  and b) the first derivative of angle  over the smvar range fig. 11 a) position of the system for i = and b) angle i: 0 < i < 17 because of the change of positions of points t and a, angle i is decreasing in the smvar range. the change of this angle over the smvar range is shown in fig. 10a, while fig. 10b shows the first derivative of this variable, i . at the moment when angle i reaches the following value: ii 2  (12) the system goes out of the smvar range and enters a new con range. at this moment, winding/unwinding radius ri has the following value: 2/d3rr 0ii  (13) from this moment on, the system for smooth cable winding/unwinding on the winch enters the period when angle i has values defined by: iii 32  (14) winding/unwinding radius ri, length lwi, and angle i have constant values over the new con range and they keep the values achieved when angle i had value of 2 i, see eq. (12). figure 11a shows one example of the system during the new con range defined by eq. (14): position of the system for i=3. 378 m. filipović, lj. kevac based on the process defined in this section, it can be concluded that the smooth cable winding/unwinding process on the winch is accomplished by a cyclical alternation of the con and smvar ranges. 3. the program package smowind – ow based on mathematical eqs. (1–14), a program package smowind – ow (smooth winding for one winch) has been synthesized in matlab. this program package includes definition of the motion dynamics of only one two-cylinder winch for the cable smooth winding/unwinding process. a smooth trajectory of the winch is defined for the overall range of angle i: 0 < i < 17rad. it is presumed that angular speed of this winch is constant. the program is generated so that the motion takes place only in the direction of the cable winding on the winch. it is assumed that the cable unwinding takes place in the same way but in the opposite direction. this program package gives an ability to the user to track the change of dynamics of all the relevant variables, e.g. winding/unwinding radius, length between the winch and hanging point, etc. the program package presumes application of a two-cylinder winch (fig. 2a), but with small modifications this program package can be used for generation of the same results for the spiral winch (fig. 2b), if required. by using this program package, the user can track changes of the first derivatives of all the relevant variables during any part of the winding process within the defined overall range of angle i. the cyclical character of the winding process is shown in detail in the next section where the simulation experiments performed by the program package smowind – ow are presented. 4. cyclicality of the process in section 2, the basic principles of kinematics of the smooth cable winding on one two-cylinder winch, shown in fig. 2a, are presented. the cable winding process has been shown for the values of angle i given by following inequality: 0 < i < i, see fig. 11a. based on this analysis, it is concluded that a smooth cable winding process involves cyclical alterations of the con and smvar ranges. with a constant growth of angle i, i.e. with a continuous cable winding, the previously defined principles of winding should be applied to achieve a smooth growth of radius ri and a smooth decrease of angle i and length lwi. by using the program package smowind – ow, which was outlined in section 3, the simulation experiments for a linear full scale increase of angle i, 0 < i < 17, have been generated. the change of angle i is shown in fig. 11b. from this figure it can be seen that this angle has a linear change, i.e. its first derivative is constant consti  (the angular speed of the winch is constant). figure 12a shows the change of radius ri for this period of winding. during the winding period, a growth of radius ri has a cyclical character. it can be noticed that radius ri has a constant and gradual growth in smvar ranges, while it has a constant value during con ranges. figure 12b shows the first derivative of variable ri. from this figure it can be seen that the first derivative of radius ri has a smooth and cyclical change with maximal value of s/m0025.0r maxi  . the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 379 fig. 12 a) radius ri, and b) the first derivative of radius ri fig. 13 a) angle i and b) the first derivative of angle i angle i is decreasing during the cable winding process on the winch. the change of this variable is similar to the change of the winding/unwinding radius, as can be seen in fig. 13a. also, fig. 13b shows the first derivative of this variable. it can be seen that its maximum absolute value is s/rad0047.0maxi  . length lwi is also decreasing during the cable winding process and the character of its change is almost the same as the character of the change of angle i, as can be seen in fig. 14a. figure 14b shows the change of its first derivative and it can be seen that its maximum absolute value is s/m104.1wl 4 maxi  . the change of the cable movement velocity is much smaller than that achieved with the standard winch form, see kevac et al. [13]. this presents only one advantage of the novel winch in comparison with the standard one. what is essentially important about this new constructive solution of the winch is that abrupt changes of the cable length velocity are avoided. it can be seen that first derivatives of all important variables ir , iwl  , i have small smooth changes and thus their effect on the whole system’s dynamics is much better in terms of the systems’ dynamic response. the figures shown in this section (figs. 11b – 14) correspond to the idealized case of the cable winding process on a two-cylinder winch. also, in these figures only winding of one cable on one winch under ideal conditions, when the angular speed of the winch is constant, consti  , is analyzed. 380 m. filipović, lj. kevac fig. 14 a) length lwi and b) the first derivative of length lwi 5. conclusions the concept of the nonlinear smooth cable winding/unwinding process on the winch has been defined and analyzed. this presents a novelty in the literature published so far. it has been developed due to our intention to define a new form of winches for a single-row radial multi-layered smooth cable winding/unwinding process. two novel mechanical constructions of the winch have been defined: one named a two-cylinder winch and the other named a spiral winch. the two-cylinder winch has been used for further kinematics investigations. to explain the idea, the process of the smooth cable winding/unwinding on one two-cylinder winch has been analyzed and modeled. a detailed synthesis and analysis are performed of the winding process only since it is assumed that the process of unwinding is identical except for its flowing in the reverse direction. idealized conditions during the winding process have been assumed, i.e. it is presumed that the winch is always rotating with a constant angular speed. under these conditions, it is discovered that the smooth cable winding/unwinding process has two ranges: a range smvar when all the relevant variables change their values: winding/unwinding radius ri, angle i, and length lwi and a con range when these variables have constant values. for the verification and validation of the defined theoretical concept, a novel program package smowind – ow has been developed using matlab. by using this program package, the simulation experiments of smooth winding of the cable on one winch have been made and the presented figures show the changes of: radius ri, angle i, length lwi and their first derivatives. from these figures it can be seen that these variables have smooth changes during rotation of the winch. future research in this area would concern the inclusion of the proposed new construction of the winch in a complex cable suspended system, which may consist of n subsystems performing cable winding/unwinding process such in aref et al. [14]. acknowledgements: this research was supported by the ministry of education, science and technological development, government of the republic of serbia through the following two projects: grant tr-35003, "ambientally intelligent service robots of anthropomorphic characteristics", by mihajlo pupin institute, university of belgrade, serbia, and grant oi-174001, "the dynamics of hybrid systems of complex structure", by the sanu institute belgrade and faculty of mechanical engineering university of nis, serbia. we are grateful to prof. dr. katica r. (stevanovic) hedrih from the the basic theoretical principles of the kinematics of smooth cable winding/unwinding on a winch 381 mathematical institute, belgrade for the helpful consultations during this research and we are grateful to our former colleague zivko stikic for his help during this research. references 1. united nations, 1981, cable logging systems, food and agriculture organization of the united nations, roma. 2. samset, i., 1985, winch and cable systems, series forestry sciences, vol. 18, springer netherlands. 3. abdel-rahman, e.m., nayfeh, a.h., masoud, z.n., 2003, dynamics and control of cranes: a review, journal of vibration and control, 9, pp. 863-908. 4. padfield, d.g., 1958, the motion and tension of an unwinding thread. i, proc. r. soc. lond. a, 245(1242), pp. 382–407. 5. fraser, w.b., ghosh, t.k., batra, s.k., 1992, on unwinding yarn from a cylindrical package, proc. r. soc. lond. a, 436(1898), pp. 479-498. 6. clark, j.d., fraser, w.b., stump, d.m., 2001, modelling of tension in yarn package unwinding, journal of engineering mathematics, 40, pp. 59–75. 7. imanishi, e., nanjo, t., kobayashi, t., 2009, dynamic simulation of wire rope with contact, journal of mechanical science and technology, 23, pp. 1083-1088. 8. szczotka, m., wojciech, s., maczynski, a., 2007, mathematical model of a pipelay spread, the archive of mechanical engineering, 54(1), pp. 27-46. 9. lee, j-w., kim, k-w., kim, h-r., yoo, w-s., 2012, prediction of unwinding behaviors and problems of cables from inner-winding spool dispensers, nonlinear dyn., 67, pp.1791–1809. 10. filipovic, m., djuric, a., kevac, lj., 2015, the significance of adopted lagrange principle of virtual work used for modeling aerial robots, applied mathematical modelling, 39(7), pp. 1804-1822. 11. von zietzwitz, j., fehlberg, l., bruckmann, t., vallery, h., 2013, use of passively guided deflection units and energy-storing elements to increase the application range of wire robots, in bruckmann, t., pott, a. (eds.), cable-driven parallel robots, mechanisms and machine science, vol. 12, pp. 167-184. 12. kevac, lj., filipovic, m., 2017, mathematical model of cable winding/unwinding system, journal of mechanics, doi: https://doi.org/10.1017/jmech.2017.59, 13. kevac, lj., filipovic, m., rakic, a., 2017, dynamics of the process of the cable winding (unwinding) on the winch, applied mathematical modelling 48, pp. 821-843. 14. aref, m.m., taghirad, h.d., 2008, geometrical workspace analysis of a cable-driven redundant parallel manipulator: kntu cdrpm. proc. ieee/rsj international conference on intelligent robots and systems (iros), pp. 1958-1963. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 325 337 1 energy analysis of free transverse vibrations of the visco-elastically connected double-membrane system udc 531+534+517. 93(045)=111 julijana simonović 1 , danilo karličić 2 , milan cajić 2 1 faculty of mechanical engineering, university of niš, serbia 2 mathematical institute of the sasa, serbian academy of science and arts, serbia abstract. the presented paper deals with the analysis of energy transfer in the viscoelastically connected circular double-membrane system for free transverse vibration of the membranes. the system motion is described by a set of two coupled non-homogeneous partial differential equations. the solutions are obtained by using the method of separation of variables. once the problem is solved, natural frequencies and mode shape functions are found, and then the form of solution for small transverse deflections of membranes is derived. using the obtained solutions, forms of reduced kinetic, potential and total energies, as functions of dissipation of the whole system and subsystems, are determined. the numerical examples are given as an illustration of the presented theoretical analysis as well as the possibilities to investigate the influence of different parameters and different initial conditions on the energies transfer in the system. key words: double-membrane system, visco-elastic layer, dissipation function, energy transfer, multi-frequency vibration 1. introduction membrane structures and compound systems are widely used in many industrial applications. in addition, in the micro world the system of compound membranes may represent a biological model of a cell membrane. the application involves several disciplines and industrial contexts as, for example, microfiltration systems in biological, medical, food, dairy and beverage products and application in aeronautics, cosmonautics, civil and mechanical engineering (see refs. [1] and [2]). membrane, string, beam, plate or cable [3] structures can perform linear and nonlinear vibrations. despite the fact that the membrane systems are, in general, strongly nonlinear systems, under certain assumptions they can be considered as linear. by assuming that the amplitude of a membrane deflection is received june 1, 2014 / accepted september 10, 2014 corresponding author: juliana simonović university of niš, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: bjulijana@masfak.ni.ac.rs original scientific paper 326 j. simonović, d. karliĉić, m. cajić smaller than its thickness and considering the initial tension on the membrane [4], the dynamic behavior of such structures can be observed within the linear vibration theory. opposite to this, when no or small initial tension is applied and the membrane deflection amplitude is close to the value of the membrane thickness, the system is performing nonlinear vibration [4]. the linear as well as the nonlinear vibration analysis of membrane systems is important from the theoretical and practical point of view. several papers have been published on the vibration analysis of elastically coupled double-membrane or plate-membrane systems, continuously connected by an elastic layer. oniszczuk [5, 6], noga [7, 8] and karliĉić [9] present analytical expressions for the undamped free and forced vibrations of double-membrane and plate-membrane systems connected with the winkler type of layers. knowing the principles of phenomenological mapping and mathematical analogy, [10], it is clear that the composite membrane systems can be studied like coupled plate, beam or belt systems. hedrih and simonović [11-13] have studied transverse vibrations of the rectangular and circular plates connected with elastic or visco-elastic layers for both linear and nonlinear dynamic problems. kelly [14] examines vibration of the elastically connected multiple beams system. energy transfer problem is studied in [15-17] for the vibrations of double-membrane and double-plate systems, connected with elastic or visco-elastic type of layers. in spite of many vibration studies on various types of hybrid systems, where different structures are coupled with different types of layers, according to the best of authors’ knowledge, an energy transfer analysis of free transverse vibration of the visco-elastically connected circular double-membrane system has not been considered in the literature yet. under the assumptions of small transverse vibration of the membranes and constant initial tension for both the membranes, we analyze our system within the linear vibration theory. governing equations of the system are derived and the solution is proposed by using the method of separation of variables. natural frequencies and mode shape functions are determined by solving the boundary and initial value problem. the expressions for time functions are used in order to find analytical forms of reduced kinetic, potential and total energy of the system. 2. formulation of governing equations as a model problem, we consider two circular membranes connected through viscoelastic layer, modeled by continuously distributed elements of the kelvin-voigt type. the scheme of such a mechanical model is depicted in fig. 1. both the membranes are assumed to be thin with mass densities ρi, for i=1,2 corresponding to the lower and upper membrane, respectively, ideally elastic and with neglected thickness. the membranes are stretched and fixed along their entire boundaries in xy plane. the tensions per unit length σi [n/m], caused by stretching, are the same at all points in all directions and do not change during the motion. small transverse deflections of membranes wi(r,φ,t), i=1,2 are considered, where b[ns/m] is denoting the damping coefficient and c [n/m] is the constant stiffness coefficient per surface unit area of the distributed visco-elastic layer between the membranes. using the d`alambert principle, the governing system of non-homogenous coupled partial differential equations for free vibration of the double-membrane system is expressed in the following form: an energy analysis of visco-elastically connected double-membrane system 327 fig. 1 the physical model of circular double-membrane compound system   2 2 21 12 ( , , ) ( , , ) ( , , ) ( , , ) 2 ( , , ) ( , , )i i i i i i i i i w r t w r t w r t c w r t a w r t w r t t tt                        (1) where i = 1,2; ci = (σi /ρi) 1/2 [m/s] are velocities of the transverse wave propagation for both the membranes,  is laplacian operator, and ai 2 = c /ρi, 2δi=b/ρi are notations for reduction coefficients. analytical solutions of the system of coupled partial differential equations are obtained by using the method of separation of variables. for the system of two coupled partial equations eq. (1), for free vibration, we separate variables in wi(r,φ,t), i=1,2 and considering the eigenamplitude functions as wi(nm)(r,φ); n,m=1,2,...,∞ and the time expansion with coefficients in the form of unknown time function as ti(nm)(t) we describe their time evolutions in the form: ( ) ( ) 1 1 ( , ) ( ), 1,2i i nm i nm n m w r w t t i       (2) wherein m =1,2,...,∞ denotes an infinite number of possible vibration modes. here, eigenamplitude functions wi(nm)(r,φ) are the same for both the membranes and written in the following form: w(i)nm(r,) = jn(knmr)cos(n + (i)0n), (3) which are obtained for the decoupled system and for the same boundary conditions of membranes. in eq. (3) jn(knmr) are bessel`s functions of the first kind of n-th order, and knm=xnm/a are characteristic numbers for every m-th root of n-th bessel`s functions over membrane radius a. after introducing eq. (2) into the governing system of eq. (1), we obtain the following system of the homogeneous second order ordinary differential equations with respect to unknown time functions ti(nm)(t) for nm-family mode, in the following form: 2 2 1( ) 1 1( ) 1( ) 1( ) 1 2( ) 1 2( )( ) 2 ( ) ( ) 2 ( ) ( ) 0nm nm nm nm nm nmt t t t t t t t a t t       2 2 2( ) 2 2( ) 2( ) 2( ) 2 1( ) 2 1( )( ) 2 ( ) ( ) 2 ( ) ( ) 0,nm nm nm nm nm nmt t t t t t t t a t t       (4) 328 j. simonović, d. karliĉić, m. cajić where ωi 2 (nm)= ω0i 2 (nm)+ai 2 =ki 2 (nm)ci 2 +ai 2 ; i=1,2; n,m =1,2,...,∞, are natural frequencies for the first and second membrane for the nm-th mode of vibration. to solve the system of two coupled ordinary differential eqs. (4), it is necessary to form the frequent determinant and determine the eigenvalues of the system. we introduce matrices corresponding to the nm-th mode of the observed dynamical system: inertia anm matrix, stiffness coefficients matrix cnm and damping coefficients matrix bnm in the following forms: 2 2 1( ) 11 1 2 2 2 2 2 2( ) 2 21 0 ; ; 2 20 1 nm nm nm nm nm a a                        a b c (5) the coupled system of eqs. (4) for time domain may be treated as a system of equations corresponding to a two-degree-of-freedom discrete system with defined matrices in eq. (5). the characteristic equation of the linearized coupled system is given in the form: 2 2 2 1 1( ) 1 12 2 2 2 2 2 2 2( ) 2 2 0 2 2 nm nm nm nm nm nm nm nm nm nm nm nm nm λ a λ a λ                        a b c . (6) we obtain eigenvalues of the system in nm modes as two pairs of complex conjugate roots in the form: 1,2( )( ) ( )( ) ( )( ) , 1, 2s nm s nm s nmλ i s    (7) where δ (s)(nm) and ω (s)(nm) are real and imaginary parts of the corresponding pair of roots of the characteristic equation. now, the solution of the system of ordinary differential eqs. (4) can be expressed as:  1( )( ) 2( )( ) ( )( ) ( )( ) ( ) s nm s nm st ts i nmi nm is i nm s t c a e a e     . (8) we obtain eigenamplitude numbers a (s) i(nm) and their conjugates ( ) ( ) s i nma from: ( ) ( ) 1( ) 2( ) ( )( ) ( ) 21( ) 22( ) s s nm nm s nms s nm nm a a c k k   or ( ) ( ) 1( ) 2( ) 2 2 2 1 1 ( )( ) ( )( ) 1 ( )( ) 1( )2 2 s s nm nm s s nm s nm s nm nm a a c a            (8a) where k (s) 2i(nm) are cofactors of the determinant in eq. (6) and cs(nm) are known constants determined from the corresponding characteristic equation. considering the time functions in eqs.(8) corresponding to frequencies ω (s)(nm) of the damped coupling and taking into account conjugate complex roots (eq. (7)) yields: ( ) 2 ( ) ( ) ( )( ) ( ) ( )( ) 1 cos( ) sin( )s nm t i nm is nm s nm is nm s nm s t e u t v t            . (9) the constants of integration uis(nm) and vis(nm) are defined as follows: ( ) ( ) ( ) ( ) ( )2 2re( ) im( ) s s is nm s nm s nmi iu a k b k  ,   ( ) ( ) ( ) ( ) 2 2im( ) re( ) s s is nm s nm i is nmv a k b k  . (10) an energy analysis of visco-elastically connected double-membrane system 329 as(nm) and bs(nm) can be obtained through application of the initial conditions. the initial conditions are assumed as known functions: ( , ,0) ( , ),i iw r g r  (11) 0 ( , , ) ( , )i t i w r t g r t      . (12) now, the particular solutions for a non-homogenous system of the coupled partial differential equations, for free vibrations, as membrane deflections, are:     1( ) 2 ( ) 1( ) 1 1( ) 1 1 1 2( ) 2 2( ) 2 ( , , ) ( , ) [ cos( ) sin( )] [ cos( ) sin( )] . nm nm t i i nm i nm i nm n m t i nm i nm w r t w r e u t v t e u t v t                         (13) the solutions from eq. (13) are analytical results of our research on transversal vibrations of the visco-elastically connected double circular membrane system. on the basis of the orthogonality properties of the mode shape functions, unknown constants as(nm) and bs(nm) can be determined from the assumed initial conditions eq. (11) and eq. (12). introducing the known functions of membranes’ point displacements and velocities еq. (11) and eq. (12) into the solutions eq. (2) and applying the classical orthonormality conditions of eigenamplitude functions wi(nm)(r,φ) and wi(sr)(r,φ), in the form: 2 2 ( ) ( ) ( ) 2 ( ) ( ) 0 0 0 0 0 , ( , ) ( , ) [ ( , )] , 1, 2. a a i nmsr i nm i sr i nm i nm sr nm m w r w r rdrd m w r rdrd i sr nm                     , (14) we obtain the values of the initial time functions: 2 ( ) 0 0 0( ) ( ) ( , ) ( , ) a i nm i nm i nm g r w r rdrd t m        and 2 ( ) 0 0 0( ) ( ) ( , ) ( , ) a i nm i nm i nm g r w r rdrd t m        . (15) in order to find the final forms of the transverse vibrations, the initial-value problem has to be solved: 0( ) ( ) ( ) ,nm nm nmt k α (16) where t0(nm)=[t10(nm) t20(nm) t 10(nm) t 20(nm)] is the vector of the known functions eq. (15), α(nm)=[a1(nm) b1(nm) a2(nm) b2(nm)] t is the vector of unknown constants needed for solutions and k(nm) is the functional matrix of the fourth order depending on the system’s properties, given as:                                                    2 222 2 222 2 222 2 222 1 221 1 221 1 221 1 221 2 212 2 212 2 212 2 212 1 211 1 211 1 211 1 211 2 22 2 22 1 22 1 22 2 21 2 21 1 21 1 21 re~im ~ im~re ~ re~im ~ im~re ~ re~im ~ im~re ~ re~im ~ im~re ~ imreimre imreimre kkkkkkkk kkkkkkkk kkkk kkkk nm   k (16a) 330 j. simonović, d. karliĉić, m. cajić from analytical solutions, eq. (13), and corresponding solutions of constant system, eq. (16), we can conclude that in one mode of vibration, two circular frequencies of coupling and a two-frequency time function ti(nm)(t) correspond to one eigenamplitude function. the first time mode is with lower damped frequency ω (1)(nm), and the second one is with higher damped frequency ω (2)(nm). hence, the visco-elastic layer introduces into the system duplication of the number of circular frequencies which correspond to the one eigenamplitude function of the nm-family mode n,m=1,2,...,∞. we can rewrite the solutions for time functions in the form:         i t nminminmi nmiedt  y , (17) where yi(nm) are eigenvectors corresponding to following system matrix: 1 1 .nm nm nm nm nm               0 i q a c a b (18) the constants of integration di(nm) can be calculated by solving a system of simultaneous algebraic equations formulated in the matrix form as: 0( ) ( ) ( ) ,nm i nm i nm i dt y (19) where t0(nm) is the vector of inital condition constants given by eq. (15). the system matrix eq. (18) has two pairs of complex conjugate eigenvalues the same as values eq. (7). therefore, we conclude that the solutions given in eq.(17), eq. (8) and eq. (9) are actually the same. 3. energy analysis we can analyze energy transfer in the double-membrane system by using reduced components of kinetic and potential energy of membranes, potential energy of the light distributed visco-elastic layer and the reduced rayleigh function of dissipation for corresponding nm-family mode. also, we can use the system of two ordinary differential eqs. (4) for corresponding nm-th mode, which is considered as a system with two degrees of freedom. for the system of eqs. (4) it is possible to write forms of kinetic and potential energies by using the matrices eqs. (5). reduced forms of energies corresponding to the nm-th mode are given as: a) reduced kinetic energy ( )k nme ,n,m=1,2,...,∞: ( )1( ) 2 2 ( ) 1( ) 2( ) 1 1( ) 2 2( ) 2( ) ( ) 1 1 ( ) [ ( ) ( ) ] 2 2 k nmnm k nm nm nm nm nm nm i nm et e t t t t t m              a , (20) where ek(nm) and mi(nm) are kinetic energy and orthogonality function eq. (14), respectively. an energy analysis of visco-elastically connected double-membrane system 331 b) reduced potential energy ( )p nme : 1( ) ( ) 1( ) 2( ) 2( ) ( )2 2 2 2 2 2( ) 1( ) 1 01( ) 1( ) 2 02( ) 2( ) ( ) 1 ( ) 2 1 [ ] ( ) ( ) 2 nm p nm nm nm nm p nm nm nm nm nm nm nm i nm t e t t t e c t t t t m                      c , (21) where ep(nm) is potential energy. c) reduced rayleigh function of dissipation ( )nm lay , which is related to the layer, is of the form: ( )lay1( )1 1 1 1 2 ( ) 1( ) 2( ) 1( ) 2( ) 2( )2 2 2 2 ( ) 2 21 1 ( ) [ ] . 2 22 2 p nmnm nm lay nm nm nm nm nm i nm t t t b t t t m                            (22) also, we can separate terms for the kinetic and potential energies which correspond to the first and the second membrane, [8, 14, 15]: a.1) reduced kinetic energy of the membranes: ( )( )2 ( )( ) ( ) ( ) 1 ( ) , 1,2 2 k nm i k nm i i i nm i nm e e t i m    . (23) b.1) reduced potential energy of the membranes and reduced potential energy of a visco-elastic layer for the corresponding membranes: ( )( )2 2 ( )( ) ( ) ( ) ( ) 1 ( ) 2 p nm i p nm i i i nm i nm i nm e e t m    (24) b.2) reduced potential energy of pure interaction between membranes induced by a visco-elastic layer: ( )int2 2 ( ) int 1 1 2 2 2( ) 1( ) ( ) 1 ( ) 2 p nm p nm nm nm i nm e e a a t t m      . (25) b.3) reduced potential energy of the membranes without reduced part of the potential energy of layer for the corresponding membranes: ( )( )2 2 ( )( ) 0 ( ) ( ) ( ) 1 ( ) 2 p nm i b p nm i b i i nm i nm i nm e e t m    . (26) b.4) full reduced potential energy of a visco-elastic layer interaction between the membranes: (1,2)( )2 (1,2)( , ) 2( , ) 1( ) ( ) 1 [ ( ) ( )] . 2 p nm layer p n m layer n m nm i nm e e c t t t t m    (27) from the previous equations, we can see that the reduced potential energy is obtained by considering membranes’ vibration on the elastic foundation of the winkler type. both 332 j. simonović, d. karliĉić, m. cajić the membranes share potential energy of the visco-elastic layer, and only one part is interaction between the membranes depending on layers rigidity and on both time functions of the membranes. in the following, we will introduce: c.1) rayleigh function of dissipation-reduced part of a visco-elastic layer for the corresponding membranes: ( )layer( )2 ( )layer( ) ( ) ( ) ( ) p nm i p nm i i i i nm i nm t m       . (28) c.2) part of the rayleigh dissipation function – pure interaction between the membranes induced by visco-elastic layer   ( )layer(int) ( )layer(int) 1 1 2 2 1( ) 2( ) ( ) p nm p nm nm nm i nm t t m           . (29) if we use the solutions from eq. (9) of free vibrations and their derivatives with respect to time, the reduced total membranes energy for the nm-th mode can be expressed as follows. reduced total energy of the first, i=1, and second, i=2, membranes in the nm-mode are: ( )( ) ( )( )2 2 2 ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) 2 k nm i p nm i nm mi k nm i p nm i i i nm i nm i nm i nm e e e e e t t m            . (30) therefore, the reduced total energy of both membranes is: ( )( 1, 2) ( )(1) ( )(1) ( )(2) ( )(2) ( )( 1) ( )( 2)nm m m k nm p nm k nm p nm nm m nm me e e e e e e      . (31) reduced total energy of the system in the nm-th mode is equal to the sum of the reduced total energy of both the membranes and the reduced potential energy of pure interaction between them (eq. (25)): ( ) ( )( 1, 2) ( )intnm system nm m m p nme e e  . (32) as concluded in [17] for transverse vibrations of the double membrane system with elastic layer, the total energy of the system remains constant and equal to the energy in the initial moment during the whole dynamic process which is indeed a property of conservative systems. here, for a non-conservative system, we have energy dissipation which is equal to: ( ) ( )system ( )2 2 nm system nm nm lay e dt       , (33) where φ(nm)lay is the rayleigh function. this means that the dissipation function is a measure of the degradation of the mechanical energy of system, notwithstanding the choice of system parameters and the initial conditions that is also confirmed in the numerical results section. an energy analysis of visco-elastically connected double-membrane system 333 4. numerical results for the system energy for the numerical calculation and representation of eqs. (32) and (33), the following parameter values are considered: tension per unit length σ1=600[n/m], density ρ1=200[kg/m 3 ] and radius a=1m for the upper membrane and for the lower membrane we consider the same radius but different tension and density. for the discussion of the results their values are presented in table 1. we introduce the following coefficients ξ=ρ2/ρ1 and η=σ2/σ1 with values presented in table 1. all the simulations are performed in the first shape mode nm=01 when k01=2.40483. when we have ξ=η=1, it could be noticed that the damped higher frequency of the second time mode is constant ω 2=4.16528[s -1 ]=k01(σ1 /ρ1) 1/2 =const and that mode is not damped δ 2=0 for different values of layer parameters. the lower frequency aptly corresponds to the changes of stiffness coefficient c and damping coefficient b. lower natural frequency ω 1 increases for an increase of the layers stiffness and it decreases for an increase of the damping coefficient. table 1 the eigenvalues of the system in eq. (7) for proper 01-mode, for different system parameters a=1[m] 50. 1 2 c [n/m] b [ns/m] 50. 1 50. 1 50. 1 100 100 λ1,2(1) λ1,2(2) -0.75 ± 4.28i 0 ± 4.17i -0.53 ± 5.87i -0.22 ± 4.27i -0.27 ± 4.16i -0.23 ± 3.06i -0.5 ± 4.25i 0 ± 4.16i -0.26 ± 4.2i -0.11 ± 2.14i -0.26 ± 4.19i -0.11 ± 3.0i 200 λ1,2(1) λ1,2(2) -1.50 ± 4.07i 0 ± 4.16i -1.10 ± 5.52i -0.39 ± 4.46i -0.5 ± 3.89i -0.5 ± 3.22i -1.0 ± 4.16i 0 ± 4.16i -0.51 ± 4.11i -0.24 ± 2.17i -0.53 ± 4.07i -0.21 ± 3.06i 300 λ1,2(1) λ1,2(2) -2.25 ± 3.71i 0 ± 4.16i -1.9 ± 5.02i -0.35 ± 4.69i -0.3 ± 3.61i -1.2 ± 3.31i -1.5 ± 4.01i 0 ± 4.16i -0.76 ± 3.97i -0.36 ± 2.21i -0.84 ± 3.86i -0.28 ± 3.18i 250 100 λ1,2(1) λ1,2(2) -0.75 ± 4.53i 0± 4.16i -0.57 ± 6.02i -0.18 ± 4.34i -0.32 ± 4.26i -0.18 ± 3.16i -0.5 ± 4.43i 0 ± 4.16i -0.27 ± 4.29i -0.1 ± 2.22i -0.28 ± 4.29i -0.09 ± 3.05i 200 λ1,2(1) λ1,2(2) -1.50 ± 4.34i 0 ± 4.16i -1.18 ± 5.72i -0.32 ± 4.48i -0.64 ± 4.01i -0.35 ± 3.31i -1.0 ± 4.34i 0 ± 4.16i -0.54 ± 4.21i -0.21 ± 2.24i -0.57 ± 4.18i -0.17 ± 3.1i 300 λ1,2(1) λ1,2(2) -2.25 ± 4.00i 0 ± 4.16i -1.94 ± 5.28i -0.31 ± 4.66i -1.21 ± 3.6i -0.29± 3.55i -1.5 ± 4.19i 0 ± 4.16i -0.81 ± 4.07i -0.32 ± 2.28i -0.89 ± 4i -0.23 ± 3.18i 500 100 λ1,2(1) λ1,2(2) -0.75 ± 4.93i 0 ± 4.16i -0.62 ± 6.27i -0.13 ± 4.42i -0.38 ± 4.47i -0.12 ± 3.27i -0.5 ± 4.7i 0 ± 4.16i -0.29 ± 4.45i -0.08 ± 2.32i -0.30 ± 4.46i -0.07 ± 3.11i 200 λ1,2(1) λ1,2(2) -1.50 ± 4.75i 0 ± 4.16i -1.27 ± 6.03i -0.23 ± 4.51i -0.78 ± 4.29i -0.22 ± 3.36i -1.0 ± 4.62i 0 ± 4.16i -0.58 ± 4.38i -0.17 ± 2.34i -0.62 ± 4.37i -0.12 ± 3.15i 300 λ1,2(1) λ1,2(2) -2.25 ± 4.45i 0 ± 4.16i -2 ± 5.66i -0.25 ± 4.63i -1.27 ± 4.0i -0.22 ± 3.48i -1.5 ± 4.48i 0 ± 4.16i -0.87 ± 4.25i -0.25 ± 2.37i -0.96± 4.23i -0.17 ± 3.2i it is interesting to note that the same is true for ξ=η=const, which is marked with gray cells in table 1. this mechanism of changes is also the same for other arbitrary values of σ1 and ρ1 when ξ=η=const, where we have obtained ω 2=k01(σ1 /ρ1 ) 1/2 =const and δ 2=0 for all values of layers parameters. comparing to the other values in table 1, it is obvious that the damped natural frequencies increase for an increase of the layers stiffness, whereas damped lower frequency ω 1 decreases and higher frequency ω 2 increases for an increase of damping coefficient. the final forms of time functions, eqs.(9) rely on the values of vector α(nm). the relations for time functions and their time derivatives determine the values of reduced energies, eqs. (30-32), and dissipative function, eq. (33), of subsystems and the 334 j. simonović, d. karliĉić, m. cajić system itself. it is possible to obtain the diagrams for the system and subsystems energies for every particular value of parameters. for the qualitative analyses several different parameters for membranes and layers are selected for the diagrams of energies and presented in the following figures. fig. 2a shows the reduced total energy of the system, eq. (32), in appropriate 01mode for different values of visco-elastic layer stiffness and the same initial conditions. the parameters used in simulation for this figure are σ1=2σ2=600 [n/m], ρ1=5ρ2= 200 [kg/m 3 ], b=200 [ns/m] for three different values of c ={100, 250 and 500}[n/m], and initial conditions w10=0.0025w01, w20=0.001w01, 10w = 20w =0. as can be seen from fig. 2a, the total energy of the system is larger and more slowly dissipated for higher values of the visco-elastic layer stiffness coefficient than for the lower values. at initial time, total energy depends on the energy given to the system by appropriate initial conditions, as they are the same for those three lines starting from the same appropriate point. fig. 2b shows the double reduced rayleigh function of the dissipation, eq. (29), in the upper part of diagrams, and time exchange of reduced total energy of the system (nm)system/dt in the lower part of diagrams. those diagrams are obtained for the same values of parameters and initial conditions as in fig. 2a. also, as it stems from eq.(33), time exchange of total energy is equal to the negative double value of dissipation function and diagrams from fig. 2b are completely symmetric. figs. 3a and 3b show similar diagrams as figs. 2a and 2b, respectively, with the same initial conditions but for different system parameters: σ1=5σ2=600 [n/m], ρ1=ρ2/2=200 [kg/m 3 ], c =250[n/m] and for varied values of damping coefficient b={100, 200 and 300}[ns/m]. in this case, the total system energy dissipates faster as the damping constant increases. in order to investigate the effect of change of the initial conditions, numerical simulations are performed for the same stiffness and damping coefficients. diagrams in figs. 4a and 4b are similar to those in figs. 2a and 2b, respectively. however, in this case calculations are performed for different system parameters: σ1=5, σ2=600 [n/m], ρ1=ρ2/2=200 [kg/m 3 ], c =250[n/m], b=300 [ns/m]. in addition, three different cases of initial displacements and velocities on the first and the second membranes are employed: i) w10=0.002w01, w20=0, 10w = 20w =0, the lower membrane is at rest in the initial moment; ii) w10=0.002w01, w20=0.002w01, 10w = 20w =0, the membrane points are having the opposite initial positions corresponding to the first amplitude shape function, so-called initially anti-phased membranes, and iii)w10=w20=0.002w01, 10w = 20w =0, the membrane points are having the same initial positions corresponding to the first amplitude shape function, so-called initially phased membranes. the shapes of membranes at the initial moment for the three cases are presented in figs. 5a, 5b and 5c, respectively. the forms of membrane deflection and velocity at the initial moment determine the mechanical energy given to the double membrane system. in fig. 4a one can see different starting points of the total energy function for different initial conditions. the largest values obtained for the case of initially anti-phase membranes, since the system has received the greatest potential energy in the initial moment when the membrane points are having the opposite positions corresponding to the first amplitude shape function. since the system is damped, after some period of time the total energy of the system is dissipated to the zero value. as can be seen from fig. 4 for all the three simulated conditions the period of time needed for dissipation of the total energy does not change and we can say that the system’s damped properties depend only on the system parameters. an energy analysis of visco-elastically connected double-membrane system 335 a) b) fig. 2 a) the reduced total energy of the system and b) the double reduced rayleigh function of dissipation and time exchange of the system’s reduced total energy in appropriate 01-mode for system parameters and different values of stiffness coefficient c a) b) fig. 3 a) the system’s reduced total energy and b) the double reduced rayleigh function of dissipation and time exchange of the system’s reduced total energy in appropriate 01-mode for system parameters and different values of damping coefficient b a) b) fig. 4 a) the system’s reduced total energy and b) the double reduced rayleigh function of dissipation and time exchange of the system’s reduced total energy in appropriate 01-mode for system parameters and different three values of initial displacement and velocity on the first and the second membranes (i, ii, iii) 336 j. simonović, d. karliĉić, m. cajić fig. 5 the shape of membranes at different initial moment: a) the lower membrane is at rest at the initial moment; b) so-called initially anti-phased membranes, and c) socalled initially phased membranes 5. conclusions in this paper, the free vibration and energy analysis of a double-membrane system joined by the kelvin-voigt type layer is performed analytically and numerically. the obtained forms of solutions for time functions eq.(9) in proper mode shape are rather down-to-earth solutions since they are applicable in numerical experiment. furthermore, those forms of solutions could be used in the analysis of forced and nonlinear system oscillations. the analytical analysis showed that the kelvin-voigt type layer is responsible for the appearance of two-frequency regimes, which corresponds to one eigenamplitude function. time functions for different modes of vibrations are uncoupled and energy transfer appears in a single mode. the system is non-conservative and damped by a viscoelastic layer, so the dissipation function is a measure of the degradation of system’s mechanical energy, notwithstanding the choice of system parameters and the change of the initial conditions. the mechanical energy given to the system at the initial time 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plate nonlinear system, international journal of bifurcation and chaos 21(10), pp. 2993-3011. 16. hedrih, k. s., 2009, energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system), archive of applied mechanics 79, pp. 529-540. 17. karliĉić, d., cajić, m., 2012, energy transfer analysis of an elastically connected circular double-membrane compound system, book of abstracts, 8th european solid mechanics conference, graz (2012). energijska analiza slobodnih transverzalnih vibracija sistema visko-elastično spregnutih membrana predstavljeni rad je posvećen analizi prenosa energije kod slobodnih transverzalnih oscilacija sistema visko-elastično spregnute dve membrane. kretanje sistema je opisano sistemom dve spregnute nehomogene parcijalne diferencijalne jednačine. rešenja su dobijena primenom metode razdvajanja promenljivih. rešavajući problem dobijaju se sopstvene kružne frekvencije i osnovni amplitudni oblici oscilovanja sistema, a potom i oblici rešenja za male transverzalne pomeraje membrana. koristeći dobijena rešenja određene su redukovane vrednosti kinetičke, potencijalne energije i funkcije rasipanja kako celog sistema tako i podsistema. primeri numeričkog proračuna su dati kao ilustracija prikazane teorijske analize i kao mogućnost da se prouče uticaji različitih parametara sistema i različitih početnih uslova na prenos energije u sistemu. kljuĉne reĉi: sistem dve membrane, visko-elastični sloj, funkcija rasipanja, prenos energije, višefrekventne oscilacije. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 301 312 doi: 10.22190/fume1603301p original scientific paper modifying and expanding the simulation of wear in the spherical joint with a polymeric component of the total hip prosthesis udc 617.582 vladimir pakhaliuk, aleksandr poliakov, mikhail kalinin, yevgenii pashkov, pavel gadkov sevastopol state university, laboratory of biomechanics, sevastopol, russian federation abstract. the existing model of wear, based on the classical archard equation, in the spherical joint of a total hip prosthesis comprising an acetabular cup of ultra-high molecular weight polyethylene (uhmwpe) in combination with a metal or ceramic femoral head is modified and expanded. with this model, studies are conducted using the finite element analysis in terms of cumulative linear and volumetric wear for the iso 14242-1 demands and additionally for the conditions during walking gait. also they are carried out for the head diameter of 28 mm at the constant and the variable wear factor, where the variable wear factor is adopted from the modified formula for the dependence on the contact pressure. key words: total hip prosthesis, wear, finite element simulation, spherical joint 1. introduction it is widely recognized in medical practice that the minimizing of wear and wear debris is crucial in extending the lifespan of artificial joints [1], where the main factor causing failure of the implant is the loosening of its components due to the impact of wear products and reduced thereby the locking of them in the bone tissues. considering that currently there is a serious trend towards younger patients with indications to be in need of a hip replacement, it is obvious that a lifespan of such prosthesis must be extended up to a maximum of 20 years or more. in this regard, various attempts allowing enhancing received october 03, 2016 / accepted november 16, 2016 corresponding author: vladimir pakhaliuk sevastopol state university, laboratory of biomechanics, universitetskaya 33, 299053 sevastopol, russian federation e-mail: vpakhaliuk@gmail.com 302 v. pakhaliuk, a. poliakov, m. kalinin, y. pashkov, p. gadkov the durability of total hip replacements (thr) are undertaken. for example, new designs are created on the basis of a systematic approach, taking into account the mutual impact of many factors, both structural and operational [2]. to test the engineering solutions and evaluate durability of the existing and newly created thr, the specialized devices to perform a wide range of tests for wear are used, both for the tests regulated by standards and considering the nature of the non-standard behavior of the patient [3]. nevertheless, the problem of accurate assessment of the wear parameters of the implant friction surfaces at the design stage is of current interest. this is especially important for modern total hip replacements, which include the acetabular cup of ultra-high molecular weight polyethylene (uhmwpe) in combination with a metal or ceramic femoral ball head. for preliminary wear evaluation of the sliding surfaces, including modified, the finite element analysis and various wear models (see, e.g., [4, 5] and other sources) are now widely used. in particular, for the thr with a uhmwpe polymeric component the wear simulation is based on the classical archard equation together with all other studies reported in literature referred to, for example, in [6, 7, 8], where the wear factor is taken as a constant value. in [5], the polymer wear in terms of cumulative linear and volume wear when the wear factor is chosen to be a function of contact pressure was first evaluated. herewith, various known wear factor dependencies on the contact pressure were investigated as contained in [9, 10, 11]. since the contact pressure in the above formulae is a power function with a negative degree, these relations give a decrease in wear factor depending on the increase of a contact pressure, which means a reduction of wear values that are not uniquely consistent with the physics of the process. conversely, the zero value of contact pressure can lead to uncertainty in the calculations (division by zero), so this case requires limiting it by some small magnitudes, which can lead to a certain error in the calculations. moreover, study [5] is based only on the demands of iso 14242-1 standard on the loading and angular femoral movement profiles in the joint, as well as a standard design of the implant head with diameter of 32 mm. loading and displacement parameters for wear-testing machines and corresponding environmental conditions for test are regulated in the given standard. at the same time, in [7], at a constant wear factor, there was performed the research for the head of 32 mm and also for standard head diameter of 28 mm. also, it was conducted under conditions during walking gait when the hip implant experiences three-dimensional load and motion patterns at a gait cycle according to the angular positions measured by jonhston and smidt, and load measured by paul [7]. herewith, these patterns differ from those of the iso 14242-1 standard. the objective of this work is to modify the existing wear model and expand the results obtained in [5] by performing simulations using the finite element analysis for the iso 14242-1 demands and additionally for the conditions during walking gait [7]. likewise, it is to be carried out for a head diameter of 28 mm in terms of cumulative linear and volumetric wear at a constant and variable wear factor, where the variable wear factor is adopted from the modified formula for the dependence on the contact pressure. modifying and expanding the simulation of wear in a spherical joint with polymeric component... 303 2. materials and methods the model of the hip joint prosthesis sliding couple contains a solid femoral ball head of cobalt-chromium alloy or ceramics (alumina or zirconia) with widely used standard diameters of 32 mm and 28 mm employed against a soft (uhmwpe) acetabular cup [5]. the radial clearance between them makes up 0.15 mm. the elasticity modulus and poisson's ratio were chosen as 1.4 gpa and 0.46, respectively, for the cup and 210 gpa and 0.3, respectively, for the head [7]. the right hip joint, that corresponds to iso 14242-1 test standard, is defined in anatomically fixed coordinates x ' y ' z ' and shown in fig. 1. in the simulation of wear, it is proposed to use a simplified coordinate system xyz fixed to the cup and placed in its center, see fig. 2. the movable coordinate system used for the euler angles coincides with the center of the cup; it is placed in the center of the head and fixed to the head. the head has three rotational degrees of freedom, known as fe (flexion-extension), aa (abductionadduction), and ior (inward-outward rotation) which correspond to the angular movements profiles in fig. 3. graphs of resultant load vector fres and of its components fx, fz as projections of two coordinate axes x and z are shown in fig. 4. these profiles are related to iso 14242-1 standard. the profile of the resultant load vector is in the anatomical coordinate system within one gait cycle, which corresponds to the period of time of 1s. the profiles during walking gait of angular femoral positions measured by jonhston and smidt are depicted in fig. 5, and patterns of load, measured by paul, in fig. 6 [7]. fig. 2 a simplified coordinate system xyz fig. 1 front view of the right hip joint with the specified directions of rotation (l is a resultant load vector) 304 v. pakhaliuk, a. poliakov, m. kalinin, y. pashkov, p. gadkov fig. 3 graphs of angular movements of the femoral head according to iso 14242-1 fig. 4 graphs of the resultant load vector and its two-dimensional components according to iso 14242-1 the wear simulation was based on the classical archard equation [6, 7]. until now, the total interaction for ideal uniformly loaded isotropic surfaces with a nominal contact pressure in the linear elastic condition is usually adopted for description by the relation: skh wσ , (1) where h is wear depth, kw is a constant empirical wear factor, σ is a contact pressure, s is a sliding distance. in paper [6], relation (1) is proposed in a discrete kind in the parametric form for the evaluation of variables of thr mechanical design and described as following:    n i iiw tstkh 1 ),,θ(),,θ(σ),θ(  , (2) where δh(θ,φ) is an accumulative local linear wear depth at the contact surface in a spherical coordinate system, σ(θ,φ,ti) is a normal contact pressure between the counterface surfaces at the same point of time instant ti of the gait cycle, δs(θ,φ,ti) is an increment of the arc sliding distance between the adjacent measuring points under the same conditions. wear factor kw depends on the material, nature of the surface and, as was found, the nominal contact pressure. for a given material combination, the value of constant wear factor in [6, 7] was adopted as 1.066·10 -6 mm 3 /(nm). but, for example, in [10], the wear factor was proposed in a variable form as an empirical relationship with normal contact pressure: 84,06102  wk , (3) and in [11] as the next relationships: 6 0.57 6 1.44 2.7 10 ( / ) , when / 2.53 6.0 10 ( / ) , when / 2.53 w ref ref w ref ref k p p p p k p p p p           , (4) where p and pref = 1.1mpa are the actual nominal contact pressure and the same for the running-in condition, respectively. modifying and expanding the simulation of wear in a spherical joint with polymeric component... 305 to avoid a power function with a negative degree at using formula (3), eq. (1) can be modified as following: ssskh w 16.06 84.0 6 102 102         , (5) where the value of 2·10 -6 is a proportionality factor in this case. fig. 5 graphs of angular femoral positions measured by jonhston and smidt during walking [7] fig. 6 graphs of three-dimensional patterns of load measured by paul during walking [7] also, eq. (1) can be modified considering eq. (4). resultant formula (5) is already a power function with a positive degree, and in this case, there is an opportunity to avoid the division by zero at calculations by eq. (2). determining sliding track length δs(θ,φ,ti) on the surface of the cup is carried out in [5]. there, marker point k0 is fixed to the head; one set of rotation angles (αi, βi, γi), corresponding to an anatomical sequence of rotations fe → aa → ior, is sequentially rotated with euler rotation angels from the initial position described by vector 0r to a new position ki described by vector ir along the sliding track on the cup surface according to the relationship with rotation matrix ),,( iiixyz r : 0i rrr ),,( iiixyz  then the increment of a sliding track length can be defined with sphere radius r as:             i1i i1i rr rr arccos),,( rts i (6) numerical simulation of wear is carried out using the finite element analysis with ansys and matlab software and presented in details in [5]. this process can be described in short as follows. the model of implant couple is created in ansys software and meshed with hexagonal finite elements (bricks and wedges) that will improve the accuracy of the calculation (fig. 7). then, the contact surface is created, and key points, surfaces and volumes are deleted from the model. it then allows moving the contact surface nodes at the wear calculation. as to the boundary conditions, the outer surface of cup is fully 306 v. pakhaliuk, a. poliakov, m. kalinin, y. pashkov, p. gadkov constrained through its corresponding nodes. during the first simulation, the coordinates of the contact surface nodes are determined in ansys and stored in a text file. the gait cycle is divided into 25 equal time intervals. and in matlab software using the indicated text file, the length of tracks in these areas with angular movements according to fig. 3 or fig. 5 is determined and also stored in a text file. further computational operations are conducted only in the ansys software by custom code for all of 25 time intervals in a loop mode than the computational complexity is reduced in the simulation of wear over a long period of time. at the beginning of each cycle, by solving the contact problem of 3-d surface-to-surface, the values of normal contact pressure σ(θ,φ,ti) at the nodes on the contact surface are defined by the corresponding values of load patterns for this interval according to fig. 4 or fig. 6. fig. 8 illustrates the distribution of contact pressure for the first cycle of third interval of gait corresponding to a maximum resultant load of 3kn according to its pattern in fig. 4. fig. 7 the model of implant couple with hexagonal finite elements fig. 8 the distribution of contact pressure for the first cycle of third interval of gait corresponding to a maximum resultant load of 3kn according to iso 14242-1 thereafter, by eq. (2) considering eq. (5), local wear depth increment δh(θ,φ) is determined in each of the nodes found at the contact surface. by summing the local depth increments of wear, determined at each interval, we can obtain a distribution of the cumulative linear wear of the cup surface at one gait cycle and the volumetric wear, as the volume of part of the material is removed out of the cup. the results obtained are placed in a text file. then the actual geometry of the bearing surface of the cup is adjusted by moving the nodes by the amount of linear wear derived from the specified text file and a new contact surface is generated. after that, the specified computing sequence is repeated in a loop mode to perform the required number of cycles corresponding to a predetermined number of gait cycles. each of the simulations is carried out up to 20 million gait cycles [7], that corresponds to about 20 years of the implant lifespan, with increment of n0 cycles. to increase the accuracy of calculations and reduce the pc operating time, before the moving of contact surface nodes, the linear and volumetric wear, obtained at the end of a gait cycle, are here determined by simply multiplying the wear during one cycle by n0 cycles. modifying and expanding the simulation of wear in a spherical joint with polymeric component... 307 3. results the studies to determine the impact of magnitude of increment n0 on the precision of calculations with a head diameter of 32 mm and according to iso 14242-1 demands were performed. the investigated increments were of 0.2, 0.4 and 1.0 million cycles. it turned out that at a constant wear factor, the linear wear in the case of these increments is almost the same and the volumetric wear differs only at increments of 1.0 million cycles down by about 6% (fig. 9). at a variable wear factor, the linear and volumetric wear differ only at using the increments of 1.0 million cycles down by about 14% and 10%, respectively (fig. 10). this is likely derived from the fact that at the smaller increment value a higher magnitude of error is accumulated at each movement of the contact surface. obviously, this error can be decreased with increasing the mesh grids, but this greatly increases the pc operating time. therefore, knowledge of the magnitude of this error allows accounting it in the evaluation of the simulation results. in order to reflect more accurately the nature of change of the wear parameters obtained as a result of simulation, the present study was carried out in increments of n0 = 0.4 million cycles. fig. 9 graphs on the impact of magnitude of increments of 0.2, 0.4 and 1.0 million cycles on precision of calculations with a head diameter of 32 mm and according to iso 14242-1 demands at a constant wear factor 1.066·10 -6 mm 3 /(nm) in terms of the cumulative linear and volumetric wear fig. 10 graphs on the impact of magnitude of increments of 0.4 and 1.0 million cycles on precision of calculations with a head diameter of 32 mm and according to iso 14242-1 demands at variable wear factor by (5) in terms of the cumulative linear and volumetric wear 308 v. pakhaliuk, a. poliakov, m. kalinin, y. pashkov, p. gadkov the results are obtained in terms of the cumulative linear and volumetric wear of the cup surface at the parameters according to iso 14242-1 test standard and during walking gait. also, it is conducted for the head diameters of 32 and 28 mm at a constant wear factor of 1.066·10 -6 mm 3 /(nm) and of that variable by eq. (5) up to 20 million cycles (see figs. 11, 12, 13 and 14). in the studies at a variable wear factor, relationship (5) is adopted since initial dependence (3) corresponds to the material of the cup taken in this simulation. 4. discussion analysis of the obtained results which are illustrated in figs. 11, 12, 13 and 14 can be expressed as follows. with the head diameter decreasing from 32 mm to 28 mm, the volumetric wear is reduced at a constant and variable wear factor, according to iso 14242-1 demands and during walking gait. moreover, for smaller head diameter of 28 mm at a constant wear factor during walking gait, the linear wear increases by about 36% compared with the diameter of 32 mm, which likely indicates a deeper penetration of the head into the cup, which can lead to more rapid disturbance of the joint biomechanics. on the other hand, the linear wear according to iso 14242-1 requirements for head diameters of 32 mm and 28 mm at a constant wear factor is almost the same (1.78 mm and 1.85 mm, respectively), and for diameter of 32 mm, it practically coincides at the walking gait settings (1.81 mm). that is how it confirms the approximate equivalence of the walking gait conditions and of those according to iso 14242-1 standard in this case. fig. 11 the maximum cumulative linear wear (maximum wear depth) vs. time of the gait cycle according to iso 14242-1 demands for a head diameter of 32 mm at a constant wear factor of 1.066·10 -6 mm 3 /(nm) and of that variable by (5) modifying and expanding the simulation of wear in a spherical joint with polymeric component... 309 fig. 12 the maximum cumulative volume wear vs. time of the gait cycle according to iso 14242-1 demands for a head diameter of 32 mm at a constant wear factor of 1.066·10 -6 mm 3 /(nm) and of that variable by (5) fig. 13 the maximum cumulative linear wear (maximum wear depth) vs. time of the gait cycle according to iso 14242-1 demands for a head diameter of 28 mm at a constant wear factor of 1.066·10 -6 mm 3 /(nm) and of that variable by (5) the parameters of the iso 14242-1 and walking gait have differences, mainly in the magnitude of the load (fig. 4), where the maximum resultant load at iso 14242-1 (3kn) is greater than for about 30% of the maximum load during walking gait (2.23kn) with approximately similar values of the angular parameters of displacement of the femoral head (fig. 5,6). thus, this fact is likely to impact the value of volumetric wear, which, for both head diameters indicated above at a constant and variable wear factor, is less at the walking gait conditions than at those of iso 14242-1. 310 v. pakhaliuk, a. poliakov, m. kalinin, y. pashkov, p. gadkov fig. 14 the maximum cumulative volume wear vs. time of the gait cycle according to iso 14242-1 demands for a head diameter of 28 mm at a constant wear factor of 610066.1  mm 3 /(nm) and of that variable by (5) simulation with a variable wear factor compared to that of the constant showed the following reduction of the amount of wear parameters at 20 million cycles for the head diameter of 32 mm: the linear wear at the iso 14242-1 is approximately of 2 times, for the walking gait in 3 times; the volumetric wear under the iso 14242-1 is approximately of 40%, for the walking gait about of 77%. the same stands for the head diameter of 28 mm: for the linear wear at the iso 14242-1 approximately of 2.6 times, at the walking gait about of 4.3 times; for the volumetric wear at the iso 14242-1 approximately of 46%, for the walking gait about of 53%. such a large scale reduction in the linear wear determines not a very large decrease in the volumetric wear. this likely indicates that the wear surface in this case is redistributed not in depth but in breadth, which positively affects the lifespan of the implant, without greatly disturbing the joint biomechanics. consequently, the simulation at a variable wear factor reflects a more real picture of the contact interaction in the implant couple, which is most clearly illustrated by the magnitude of the volumetric wear. the nature of change of the cumulative linear wear in all the cases studied above is almost a linear relationship. but in the case of the head diameter of 28 mm at a constant wear factor for both iso 14242-1 and for walking gait conditions, a small degree of nonlinearity is observed for the linear wear (fig. 13). it also appears but more weakly at the head diameter of 32 mm in the interval up to 10 million cycles (fig. 11). this could probably be due to the following factors. at the initial interval to 10 million cycles, there is an observed nonlinear increase in the depth of penetration of the head into the cup since the contact surface area has not increased enough to reduce the contact pressure efficiently. moreover, the pressure reduction with the rise of cycle’s number was noted and in [7]. with the deepening of the head, the surface contact area is increased; the contact pressure is reduced and slowed down and so is its penetration process. it has practically no impact on the almost linear nature of the change of volumetric wear since the profile of linear wear is concave-convex with respect to the straight line connecting the point of 0.4 and 20 million cycles, with a contraflexure in the middle of this range (10 modifying and expanding the simulation of wear in a spherical joint with polymeric component... 311 million). it can be assumed that this effect depends on the diameter head value and can be manifested more expressively when using its smaller standard diameter of 22 mm. the results of volumetric wear simulation of 573 mm 3 at a constant wear factor for the walking gait conditions, in the case of the head diameter of 32 mm, are in line up to 10% with the result 633 mm 3 specified in [7] and up to 6% with a score of 541 mm 3 in [6]. this, in turn, confirms the validity of this simulation. 5. conclusion the existing method of numerical wear simulation of the sliding couple in the spherical joint comprising a polymer element in terms of refining the calculation taking into account the parametric dependence of wear factor on the contact pressure according to iso 14242-1 demands and of those during walking gait cycle, is improved. also, studies are conducted at a constant wear factor in all indicated conditions and expanded to the implant head diameter of 28 mm in terms of the cumulative linear and volumetric wear. the use of the archard equation in the form (5) makes it possible to clarify the simulation results only within a few percent, but it makes a more advanced calculation process at a variable wear factor. the developed method is a serious tool for the implementation of the more accurate initial qualifying analysis of design, materials and manufacturing process of the total hip replacement, and thus allows reducing the use of expensive experimental studies using simulators. future studies can be focused on the accounting the daily activity of patients during wear simulation. acknowledgements: this work has been funded by the ministry of education and science of the russian federation in the framework of the base part of state order in the field of scientific activity with the registration no.115041610028. references 1. ingham, e., fisher, j., 2000, biological reactions to wear debris in total joint replacement, proc instn mech engrs, part h: j engineering in medicine, 214(1), pp. 21–37. 2. polyakov, a., pakhaliuk, v., kalinin, m., kramar, v., kolesova, m., kovalenko, o., 2015, system analysis and synthesis of total hip joint endoprosthesis, procedia engineering, 100, pp. 530–538. 3. poliakov, o.m., pakhaliuk, v.i., lazarev, v.b., shtanko, p.k., ivanov, y.m., 2013, stand and control system for wear testing of the spherical joints of vehicle suspension at complex loading conditions, ifac proceedings volumes, 46(25), pp. 106–111. 4. pakhaliuk, v., poliakov, a., kalinin, m., bratan, s., 2016, evaluating the impact and norming the 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commons license: cc by-nc-nd original scientific paper an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal training aircraft for flight training organizations çağlar karamaşa1, darjan karabasevic2, dragisa stanujkic3, alireza rezanezhad kookhdan4, arunodaya raj mishra5, mehmet ertürk6 1faculty of business, department of business administration, anadolu university, eskişehir, turkey 2faculty of applied management, economics and finance, university business academy in novi sad, belgrade, serbia 3technical faculty in bor, university of belgrade, serbia 4 faculty of management, islamic azad university arak branch, arak, iran 5department of mathematics, govt college jaitwara, satna, india 6faculty of aviation and aeronautical sciences, department of civil aviation management, özyeğin university, istanbul, turkey abstract. aircraft’s training is crucial for a flight training organization (fto). therefore, an important decision that these organizations should wisely consider the choice of aircraft to be bought among many alternatives. the criteria for evaluating the optimal training aircraft for ftos are collected based on the survey approach. single valued neutrosophic sets (svns) have the degree of truth, indeterminacy, and falsity membership functions and, as a special case, neutrosophic sets (ns) deal with inconsistent environments. in this regard, this study has extended a single-valued neutrosophic analytic hierarchy process (ahp) based on multi-objective optimization on the basis of ratio analysis plus a full multiplicative form (multimoora) to rank the training aircraft as the alternatives. moreover, a sensitivity analysis is performed to demonstrate the stability of the developed method. finally, a comparison between the results of the developed approach and the existing approaches for validating the developed approach is discussed. this analysis shows that the proposed approach is efficient and with the other methods. key words: training aircraft selection, neutrosophic sets, flight training organization, civil aviation, ahp, multimoora received may 21, 2021 / accepted august 18, 2021 corresponding author: çağlar karamaşa anadolu university, faculty of business, department of business, administration, eskişehir, turkey e-mail: ckaramasa@anadolu.edu.tr; ckaramasa@gmail.com 556 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. 1. introduction aircraft selection offers complex objectives that do not allow the decision-making approach to use only one variable [1-5]. among other reasons, a multi-criteria approach becomes necessary when the number of objectives to be met is greater than or equal to two, and they are contradictory [6]. a good first training aircraft is vital for the success of flight training, so the performance of the first training aircraft is an important factor [7]. the selection of aircraft has historically been of great interest and considerable research effort [8]. for various reasons, most of the published reports have made a choice for military pilot training, although the entire military training is directly applicable to civil aviation [9]. in order to solve the aircraft type selection problem, the anticipated demand for the routes considered and operated by the same aircraft types is taken as input data [10]. the critical problem in this assessment is the selection of an aircraft, as this represents the main part of the required investment [11]. airlines create much more value than other components of the airline industry as a growing service sector and have the lowest return on capital ratio. airline companies are mostly affected by increased competition in the past decades. generally, most aircraft are expected to have a service life of 30 years or more, but several uncertainties could affect the viability and applicability of the aircraft during its service life. for example, the price of fuel is always affected by the economic situation. the load factor changes throughout the year. therefore, these economic uncertainties need to be taken into consideration when deciding to purchase new aircraft for an organization [12]. aircraft selection can be taken as the most important investment decision for airline companies due to having an effect on the type and quality of services presented to customers [13]. aircraft selection is a complex system engineering that includes aircraft parameters, engine performance, communication and navigation technology, flight dynamics, meteorology, art design, financial management, and corporate strategic planning [14]. therefore, all these variables should be taken into consideration when selecting training aircraft. neutrosophic sets (ns) having the degree of truth, indeterminacy, and falsity membership functions as totally independent are introduced by smarandache [15] for explaining decision-makers ambiguity and inconsistent judgments. studies composed of neutrosophic sets based on multi-criteria decision-making methods have increased and gained importance in the last decades. the idea of the proposed method is to extend the analytic hierarchy process (ahp) based multi-objective optimization on the basis of ratio analysis plus a full multiplicative form (multimoora) approach via single-valued neutrosophic sets and show the applicability on aircraft selection as a multiple criteria decision making (mcdm) problem. training plane assessment has always been a big issue for flight training academies. right aircraft selection is very important for both organizations and practitioners. although the selection of the optimal training aircraft for flight training organizations (ftos) is a crucial concept, a few previous studies highlighted this issue in the literature. therefore, a novel approach using ahp-multimoora is developed with svns and applied for aircraft selection problems by considering the decision-makers truth, indeterminacy, and falsity judgments in the real-life environment. sensitivity analysis and comparison with previous methods are executed to show the strength of the approach. for this purpose, svns, as a case of ns is used with an integrated approach. in addition, an aircraft selection problem is handled for showing the approach and presented the an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 557 performance in practical mcdm problems. sensitivity analysis is executed for the visual presentation of the outcome. finally, a comparison is made for showing the strengths of the results. the major outcomes of the paper can be summarized as follows: the rest of this study is organized in the following sections. in the first part information related to flight training organizations and the issue of optimal training aircraft selection is presented. literature review related to training plane selection in terms of multi-criteria decision-making methods is expressed in the second section. a single-valued neutrosophic ahp and neutrosophic multimoora are stated under methodology in the third section. the results of the analysis and findings are presented in the fourth section. finally, conclusions and future suggestions are given in the last section. 2. literature review in past decades, several published papers in aircraft selection have used different decision-making approaches related to classical and fuzzy sets. therefore, a detailed literature review related to the application of conventional and fuzzy decision-making methods in the area of aircraft selection is presented in table 1. ahp [30] and multimoora [31], as mcdm approaches have been widely applied to various realistic problems. the classical ahp and multimoora methods have been handled for overcoming mcdm problems in various environments. the summary of the available study on ahp and multimoora approaches is depicted in table 2. according to tables 1 and 2, there is no study examining the decision problem by using ahp based multimoora method from single-valued neutrosophic sets. that shows the originality and novelty of this method for the application of training aircraft selection. decision-makers can explain their indeterminate and uncertain judgments more flexibly and efficiently than classical, fuzzy, hesitant, and intuitionistic sets. ahp as a subjective weighting method is selected for prioritizing the importance values of criteria due to considering the inconsistency for decision-makers’ judgments and presenting more realistic and practical results for real-world uncertain decision-making problems. multimoora as a robust alternative ranking method is handled because of making a decision based on the dominance of three approaches. fuzzy sets (fss) doctrine originated by zadeh [52], has widely been utilized to cope with the uncertainty that occurred in several fields. in recent times, several extensions of fss such as interval-valued fuzzy sets (ivfss) [53], intuitionistic fuzzy sets (ifss) [54], pythagorean fuzzy sets (pfss) [55], and q-rung orthopair fuzzy sets (q-rofss) [56] have been introduced and implemented in the field of pattern recognition, decision-making, medical diagnosis, etc. [57-59]. however, the notions of fs and its generalizations can only handle incomplete and uncertain information but are unable to tackle the indeterminate and inconsistent information that arises in real-life issues. to handle this concern, smarandache [15] pioneered the idea of neutrosophic set (ns) for describing incomplete, indeterminate and inconsistent information, which is a branch of philosophy as well as a mathematical tool for studying the origin, nature, and scope of neutralities. ns is characterized by the truthmembership, indeterminacy-membership and falsitymembership functions, which are totally independent and lying in ]0-,1+[. for example, assume that 10 voters are taking part in a voting process. suppose four votes are ‘‘yes’’, two votes are ‘‘no’’ and three are undecided. in neutrosophic sense, it can be expressed as (0.4, 0.3, 0.2), which is beyond the 558 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. scope of ifs. according to smarandache [60], it is the generalized concept of fs, ifs, ivifs, pfs, q-rofs, pifs, ternary fuzzy set (tfs), spherical fuzzy set (sfs) and nhyper spherical fuzzy set (n-hsfs). as the ns is a more suitable tool to capture the incomplete, indeterminate and inconsistent information, it has been widely utilized for several purposes [61-63]. table 1 outline of the relevant research studies on aircraft selection author(s) benchmark environment see and lewis [16] multiattribute decision-making methods classical sets wang and chang [7] the technique for order of preference by similarity to ideal solution (topsis) method fuzzy sets yeh and chang [17] a new method for modeling group subjective evaluation with absolute judgments fuzzy sets özdemir et al. [18] analytic network process (anp) classical sets dozic and kalic [8] ahp and sensitivity analysis classical sets gomes et al. [11] novel approach to imprecise assessment and decision environments (naiade) method fuzzy sets bruno et al. [19] ahp fuzzy sets kannan et al. [20] novel framework using topsis fuzzy sets dozic and kalic [10] ahp and even swaps method (esm) classical set lozano et al. [21] ahp and topsis fuzzy sets göleç et al. [22] ahp, simple additive weighting (saw), elimination and choice expressing reality (electre) and topsis classical sets ozdemir and basligil [23] anp and generalized choquet integral fuzzy sets yurdusevimli and ozger[24] ahp and topsis classical sets dozic et al. [1] fuzzy ahp and logarithmic fuzzy preference programming (lfpp) fuzzy sets kiracı and bakir[25] ahp, complex proportional assessment (copras) and multi-objective optimization method by ratio analysis (moora) classical sets ilgın[13] linear physical programming classical sets maywald et al. [4] multi-step heuristic algorithm classical sets durmaz and gencer[26] swara and smaa-2 classical sets kiracı and akan [27] ahp and topsis interval type-2 fuzzy sets sanchez-lozano and rodriguez [6] ahp and reference ideal method fuzzy sets ahmed et al. [28] ahp and efficacy method fuzzy sets hoan and ha [29] aras and fucom classical sets however, without a specific description, it is difficult to implement the nss to scientific and engineering problems as the truth-membership, indeterminacy-membership and falsitymembership functions lie in ]0-,1+[.to overcome this limitation, wang et al. [64] originated the doctrine of single-valued neutrosophic sets (svnss), whose values belong to [0, 1]. svnss can be considered as a subclass of nss. it provides us an additional possibility to represent imprecise, incomplete, uncertain and inconsistent information, which exists in realworld. the notion of svns has been proven as one of the flexible ways for solving group decision-making (gdm) problems. for instance, liu et al. [63] suggested a novel gdm model based on dematel approach with svnss for evaluating the transport service providers. pamučar and božanić [65] evaluated the location for the logistics center by employing single-valued neutrosophic mabac model. rani and mishra [66] designed an integrated decision-making framework based on swara and vikor approaches with an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 559 svnss and applied to deal with eco-industrial thermal power plants. apart from these studies, several aggregation operators [67], information measures [68-69], and decision-making methods [70] have been developed in the context of svnss. however, there is no study in the literature regarding the evaluation of the optimal training aircraft for flight training organizations under svnss environment. table 2 outline of several of the relevant research studies on ahp and multimoora methods author(s) benchmark application(s) brauers and zavadskas [32] multimoora method decide upon a bank loan to buy property balezentis and zeng [33] multimoora extended with type2 fuzzy sets personnel selection aksoy et al. [34] ahp based multimoora and copras evaluating the performance of turkish coal enterprises hafezalkotob et al. [35] interval weighted multimoora materials selection of power gears karabasevic et al. [36] multimoora and step-wise weight assessment ratio analysis (swara) selection of a candidate for the position of the mining engineer for underground mining zavadskas et al. [37] swara and multimoora with single-valued neutrosophic set selection of single-family house elements and materials fattahi and khalilzadeh[38] failure mode and effects analysis (fmea), multimoora, ahp under fuzzy environment risk evaluation of steel industries factory maghsoodi et al. [39] multimoora integrated shannon’s entropy selection of the best performance appraisal methods wang et al. [40] multimoora and choquet integral with triangular fuzzy numbers fine-kinney based risk evaluation of ballast tank maintenance zarch et al. [41] swara based fuzzy multimoora method pharmacological therapy selection of type 2 diabetes liang et al. [42] swara based multimoora with linguistic neutrosophic numbers mining method selection dorfeshan et al. [43] multimoora, moosra, and tpop with interval type-2 fuzzy sets selection of project critical path for aircraft prototype batch and construction project liao et al. [44] unbalanced hesitant fuzzy linguistic multimoora investment selection case related to shared bicycles in china gündoğdu[45] spherical fuzzy multimoora personnel selection lin et al. [46] picture fuzzy multimoora site selection for car sharing stations in beijing asante et al. [47] multimoora-edas exploring and ranking the barriers to renewable energy adoption for ghana rahimi et al. [48] fuzzy bwm, multimoora and gis selecting sustainable landfill site for municipal solid waste in iran tavana et al. [49] fahp, fuzzy multimoora assessing supply chain risk-benefit and supplier selection for a manufacturer of consumer electronic goods in new jersey wu et al. [50] cloud model theory based multimoora determining the ranking order of engineering characteristics for electric vehicle manufacturing organization tanrıverdi and lezki[51] fahp based ftopsis determining the best competition strategy for air cargo carriers 560 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. 3. methodology 3.1. neutrosophic set the fuzzy sets, proposed by zadeh [52], use the membership function for dealing with complex decision-making problems that are associated with inaccuracies and unreliability. some limitations associated with the use of one membership function are eliminated by using two functions: the membership function and the non-membership function, which is considered in intuitionistic and bipolar fuzzy sets. neutrosophic sets (ns) are proposed by smarandache [15] with the degree of truth, indeterminacy, and falsity membership functions that are independent. neutrosophic sets use three membership functions that can be used for expressing accuracy, indeterminacy, and inaccuracy during the evaluation of alternatives in multiple criteria decision-making. using these membership functions, neutrosophic sets provide an efficient and flexible approach for evaluating alternatives even if decision-making problems are related to uncertainty and predictions. a universe of discourse can be symbolized as u and xu. n as an ns can be identified by a truth tn(x) an indeterminacy in(x) and falsity membership functions fn(x), and is shown like n=x:tn(x), in(x),fn(x)xu. also, the functions of tn(x), in(x) and fn(x) are real standard or real nonstandard subsets of ]0-,1+[and can be shown like t,i,f:u→]0-,1+[. the sum of the functions of tn(x), in(x) and fn(x)can be written as 0suptn(x)+supin(x)+supfn(x)3+. the complement of an ns nis represented by nc and described as below: )(1)( xtxt n c n  += , (1) )(1)( xixi n c n  += , (2) )(1)( xfxf n c n  += for all ux . (3) n as ns is contained in other ns p represents, np if and only if inftn(x)inftp(x), suptn(x)suptp(x), infin(x)infip(x), supin(x)supip(x), inffn(x)inffp(x), supfn(x)supfp(x) for all xu [71]. 3.2. single valued neutrosophic sets (svns) svns are proposed by wang et al. [64] for solving real-life problems in an uncertain environment. the interval of [0,1] are considered for real-life applications rather than]0-,1+[. a universe of discourse can be symbolized as u and xu. a svns b in u can be identified by a truth tb(x), an indeterminacy ib(x) and falsity membership functions fb(x). a svns b can be shown as b = ∫x< tb(x), ib(x), fb(x) > / x : xu for continuous values of u. on the other hand, an svns b can be written as b = ∑i n =1∫x< tb(x), ib(x), fb(x) > / xi : xiu for discrete values of u[72]. tb(x), ib(x) and fb(x) functions are real standard subsets of [0,1] that is tb(x):u→0,1, ib(x):u→0,1, and fb(x):u→0,1. additionally, the sum of tb(x), ib(x) and fb(x), are in [0,3] and this can be written as 0 tb(x)+ ib(x)+ fb(x)3 [73]. an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 561 let a single-valued neutrosophic triangular number ã = <(a1, a2, a3); αã, θã, βã> is a special neutrosophic set on r. additionally αã, θã, βã [0,1] and a1,a2,a3 r where a1a2a3.truth, indeterminacy and falsity membership functions of this number can be computed as below [74-75]: t�̃�(𝒙) = { α�̃� ( x−𝑎1 𝑎2−𝑎1 ) α�̃� 𝛼�̃� ( 𝑎3−𝑥 𝑎3−𝑎2 ) 0 (𝑎1 ≤ 𝑥 ≤ 𝑎2) (x = 𝑎2) (𝑎2 < 𝑥 ≤ 𝑎3) otherwise (4) i�̃�(𝒙) = { ( 𝑎2−x+𝜃�̃�(x−𝑎1) 𝑎2−𝑎1 ) θ�̃� ( 𝑥−𝑎2+𝜃�̃�(𝑎3−𝑥) 𝑎3−𝑎2 ) 1 (𝑎1 ≤ 𝑥 ≤ 𝑎2) (x = 𝑎2) (𝑎2 < 𝑥 ≤ 𝑎3) otherwise (5) f�̃�(𝒙) = { ( 𝑎2−x+𝛽�̃�(x−𝑎1) 𝑎2−𝑎1 ) β�̃� ( 𝑥−𝑎2+𝛽�̃�(𝑎3−𝑥) 𝑎3−𝑎2 ) 1 (𝑎1 ≤ 𝑥 ≤ 𝑎2) (x = 𝑎2) (𝑎2 < 𝑥 ≤ 𝑎3) otherwise (6) according to the eqs. (4-6) αã, θã and βã denote maximum truth, minimum indeterminacy and minimum falsity membership degrees, respectively. suppose ã = < (a1, a2, a3); αã, θã, βã > and ñ = < (n1, n2, n3); αñ, θñ, βñ > as two single-valued triangular neutrosophic numbers and 0 as a real number. considering the above-mentioned conditions addition of two single-valued triangular neutrosophic numbers are denoted as follows [74]: �̃� + �̃� = 〈(𝑎1 + 𝑛1, 𝑎2 + 𝑛2, 𝑎3 + 𝑛3); 𝛼�̃� ∧ 𝛼�̃�, 𝜃�̃� ∨ 𝜃�̃�, 𝛽�̃� ∨ 𝛽𝑛〉. (7) subtraction of two single-valued triangular neutrosophic numbers is defined as eq. (8): �̃� − �̃� = 〈(𝑎1 − 𝑛3, 𝑎2 − 𝑛2, 𝑎3 − 𝑛1); 𝛼�̃� ∧ 𝛼�̃�, 𝜃�̃� ∨ 𝜃�̃�, 𝛽𝑎 ∨ 𝛽�̃�〉. (8) the inverse of a single-valued triangular neutrosophic number (ã  0) can be denoted as below: �̃�−1 = 〈( 1 𝑎3 , 1 𝑎2 , 1 𝑎1 ) ; 𝛼�̃�, 𝜃�̃�, 𝛽�̃�〉. (9) multiplication of a single-valued triangular neutrosophic number by a constant value is represented as follows: 𝜆�̃� = { 〈(𝜆𝑎1, 𝜆𝑎2, 𝜆𝑎3); 𝛼�̃�, 𝜃�̃�, 𝛽�̃�〉 𝑖𝑓 (𝜆 > 0) 〈(𝜆𝑎3, 𝜆𝑎2, 𝜆𝑎1); 𝛼�̃�, 𝜃�̃�, 𝛽�̃�〉 𝑖𝑓 (𝜆 < 0) . (10) 562 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. division of a single-valued triangular neutrosophic number by a constant value are denoted as eq. (11): �̃� 𝜆 = { 〈( 𝑎1 𝜆 , 𝑎2 𝜆 , 𝑎3 𝜆 ) ; 𝛼�̃�, 𝜃�̃�, 𝛽�̃�〉 𝑖𝑓 (𝜆 > 0) 〈( 𝑎3 𝜆 , 𝑎2 𝜆 , 𝑎1 𝜆 ) ; 𝛼�̃�, 𝜃�̃�, 𝛽�̃�〉 𝑖𝑓 (𝜆 < 0) . (11) multiplication of two single-valued triangular neutrosophic numbers can be seen as follows: �̃� . �̃� = { 〈(𝑏1𝑐1, 𝑏2𝑐2, 𝑏3𝑐3); 𝛼�̃� ∧ 𝛼�̃�, 𝜃�̃� ∨ 𝜃�̃�, 𝛽�̃� ∨ 𝛽�̃�〉 𝑖𝑓 (𝑏3 > 0, 𝑐3 > 0) 〈(𝑏1𝑐3, 𝑏2𝑐2, 𝑏3𝑐1); 𝛼�̃� ∧ 𝛼�̃�, 𝜃�̃� ∨ 𝜃�̃�, 𝛽�̃� ∨ 𝛽�̃�〉 𝑖𝑓 (𝑏3 < 0, 𝑐3 > 0) 〈(𝑏3𝑐3, 𝑏2𝑐2, 𝑏1𝑐1); 𝛼�̃� ∧ 𝛼�̃�, 𝜃�̃� ∨ 𝜃�̃�, 𝛽�̃� ∨ 𝛽�̃�〉 𝑖𝑓 (𝑏3 < 0, 𝑐3 < 0) . (12) division of two single-valued triangular neutrosophic numbers can be denoted as eq. (13): �̃� �̃� = { 〈( 𝑎1 𝑛3 , 𝑎2 𝑛2 , 𝑎3 𝑛1 ) ; 𝛼�̃� ∧ 𝛼�̃� , 𝜃�̃� ∨ 𝜃�̃�, 𝛽𝑎 ∨ 𝛽�̃�〉 𝑖𝑓 (𝑎3 > 0, 𝑛3 > 0) 〈( 𝑎3 𝑛3 , 𝑎2 𝑛2 , 𝑎1 𝑛1 ) ; 𝛼�̃� ∧ 𝛼�̃� , 𝜃�̃� ∨ 𝜃�̃�, 𝛽�̃� ∨ 𝛽�̃�〉 𝑖𝑓 (𝑎3 < 0, 𝑛3 > 0) 〈( 𝑎3 𝑛1 , 𝑎2 𝑛2 , 𝑎1 𝑛3 ) ; 𝛼�̃� ∧ 𝛼�̃� , 𝜃�̃� ∨ 𝜃�̃�, 𝛽�̃� ∨ 𝛽�̃�〉 𝑖𝑓 (𝑎3 < 0, 𝑛3 < 0) . (13) score function (sa) for a single-valued triangular neutrosophic number a=(a1,a2,a3) can be found as below [76-77]: 𝑠𝑎 = (1 + 𝑎1 − 2 ∗ 𝑎2 − 𝑎3)/2, (14) where sa-1,+1. the maximum distance emax(a,n) between two single-valued triangular neutrosophic numbers such as a =(a1,a2,a3) and n =(n1,n2,n3) can be computed as follows [76]: 𝑒𝑚𝑎𝑥(𝑎, 𝑛) = { |𝑎1 − 𝑛1| 𝑎1, 𝑛1 ∈ ω𝑚𝑎𝑥 |𝑎3 − 𝑛3| 𝑎3, 𝑛3 ∈ ω𝑚𝑖𝑛 . (15) 3.3. neutrosophic ahp steps of neutrosophic ahp can be explained as follows [74-75]: 1. decision problem is constructed as a hierarchical view consisting of goal, criteria, sub-criteria and alternatives, respectively. 2. pairwise comparisons are made to form a neutrosophic evaluation matrix composed of triangular neutrosophic numbers representing decision-makers’ views. neutrosophic pairwise evaluation matrix (õ) is seen as below: �̃� = [ 1̃ �̃�12 ⋯ �̃�1𝑛 ⋮ ⋮ ⋮ ⋮ �̃�𝑛1 �̃�𝑛2 ⋯ 1̃ ]. (16) according to eq. (16), õji = (õij)-1 is valid. 3. neutrosophic pairwise evaluation matrix is constructed by using scale arranged for the neutrosophic environment such as table 3: an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 563 table 3 ahp scale transformed for neutrosophic triangular numbers value explanation neutrosophic triangular scale 1 equally influential 1̃ = 〈(1,1,1); 0.5,0.5,0.5〉 3 slightly influential 3̃ = 〈(2,3,4); 0.3,0.75,0.7〉 5 strongly influential 5̃ = 〈(4,5,6); 0.8,0.15,0.2〉 7 very strongly influential 7̃ = 〈(6,7,8); 0.9,0.1,0.1〉 9 absolutely influential 9̃ = 〈(9,9,9); 1,0,0〉 2 4 6 8 intermediate values between two close scales 2̃ = 〈(1,2,3); 0.4,0.65,0.6〉 4̃ = 〈(3,4,5); 0.6,0.35,0.4〉 6̃ = 〈(5,6,7); 0.7,0.25,0.3〉 8̃ = 〈(7,8,9); 0.85,0.1,0.15〉 4. neutrosophic pairwise evaluation matrix is changed to deterministic pairwise evaluation matrix for obtaining the weights of criterion as follows: let õij = <(d1, e1, f1); αõ, θõ, βõ) be a single-valued neutrosophic number, then the score and accuracy degrees of õij are computed as the following equations: 𝑆(�̃�𝑖𝑗) = 1 16 [𝑑1 + 𝑒1 + 𝑓1]𝑥(2 + 𝛼�̃� − 𝜃�̃� − 𝛽�̃�), (17) 𝐴(�̃�𝑖𝑗) = 1 16 [𝑑1 + 𝑒1 + 𝑓1]𝑥(2 + 𝛼�̃� − 𝜃�̃� + 𝛽�̃�). (18) in order to obtain the score and accuracy degree of õij, the following equations are used. 𝑆(�̃�𝑗𝑖) = 1/𝑆(�̃�𝑖𝑗), (19) 𝐴(�̃�𝑗𝑖) = 1/𝐴(�̃�𝑖𝑗). (20) the deterministic pairwise evaluation matrix is constructed with compensation by score value in the neutrosophic pairwise evaluation matrix. the obtained deterministic matrix can be seen as follows: 𝑂 = [ 1 𝑜12 ⋯ 𝑜1𝑛 ⋮ ⋮ ⋮ ⋮ 𝑜𝑛1 𝑜𝑛2 ⋯ 1 ]. (21) ranking of priorities as eigenvector x is obtained according to the following steps: a) firstly column entries are normalized by dividing each entry to the sum of column b) then row averages are summed. 5. consistency index (ci) and consistency ratio (cr) values are computed for measuring the inconsistency for decision-makers’ judgments in the entire pairwise evaluation matrix. if cr is greater than 0.1, the process should be repeated due to unreliable decisionmakers’ judgments. ci is computed according to the following steps: a) each value in the first column of the pairwise evaluation matrix is multiplied by the priority of the first criterion and this process is applied for all columns. values are summed across the rows to construct the weighted sum vector. b) the elements of the weighted sum vector are divided by corresponding to the priority of each criterion. then the average of values are acquired and represented by max. c) the value of ci is calculated as eq. (22): 564 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. 𝐶𝐼 = 𝜆𝑚𝑎𝑥−𝑛 𝑛−1 . (22) according to eq. (7), the number of elements being compared is denoted by n. after the value of ci is found, cr is computed as follows: 𝐶𝑅 = 𝐶𝐼 𝑅𝐼 (23) where ri denotes the consistency index for randomly generated pairwise evaluation matrix and can be shown as table 4. table 4 ri table used for computing cr value order of random matrix (n) 1 2 3 4 5 6 7 8 9 10 related ri value 0 0 0.58 0.9 1.12 1.24 1.32 1.4 1.45 1.49 6. overall priority values for each alternative are computed and the ranking process is applied. 3.4. multimoora method the multimoora method developed by brauers and zavadskas [31] includes three approaches for ranking alternatives. these approaches are ratio system (rs), reference point (rf), and the full multiplicative form (fmf), respectively. the decision about selecting the best alternative is made according to the theory of dominance for three approaches [78-81]. the overall importance of alternative i in ratio system approach is given as [76]: 𝑧𝑖 = 𝑧𝑖 + − 𝑧𝑖 −, (24) 𝑧𝑖 + = ∑ 𝑤𝑗𝑗∈ω𝑚𝑎𝑥 𝑝𝑖𝑗 (25) 𝑧𝑖 − = ∑ 𝑤𝑗𝑗∈ω𝑚𝑖𝑛 𝑝𝑖𝑗 , (26) 𝑝𝑖𝑗 = 𝑥𝑖𝑗 √∑ 𝑥𝑖𝑗 2𝑛 𝑖=1 . (27) according to eqs. (24)-(27) zi + and zi show overall importance of alternative i acquired for benefit and cost criteria; pij describes the normalized performance of alternative i in terms of criterion j; xij shows the performance of alternative i related with criterion j; sets of benefit and cost criteria are represented by maxandmin; the weight of criterion j is denoted by wj. in terms of ratio system, approach alternatives are ranked in terms of decreasing zi values. according to the reference point approach optimization is shown as eq. (28): 𝑒𝑖 𝑚𝑎𝑥 = max 𝑗 (𝑤𝑗|𝑝𝑗 ∗ − 𝑝𝑖𝑗|). (28) the maximum distance of alternative i to the reference point is shown by emax i and the coordinate j of the reference point is denoted by pj *and obtained as below: 𝑝𝑗 ∗ = { max 𝑖 𝑝𝑖𝑗 𝑗 ∈ ω𝑚𝑎𝑥 min 𝑖 𝑝𝑖𝑗 𝑗 ∈ ω𝑚𝑖𝑛 . (29) an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 565 in terms of reference point, approach alternatives are ranked according to increasing emax i values. the overall utility of the alternative in the full multiplicative form is obtained as: 𝑜𝑢𝑖 = 𝑐𝑖 𝑑𝑖 , (30) 𝑐𝑖 = ∏ 𝑤𝑗𝑝𝑖𝑗𝑗∈ω𝑚𝑎𝑥 , (31) 𝑑𝑖 = ∏ 𝑤𝑗𝑝𝑖𝑗𝑗∈ω𝑚𝑖𝑛 . (32) according to eqs. (30-32) while the product of the weighted performance ratings of benefit criteria for alternative i is represented by ci, di can be used for cost criteria. in terms of full multiplicative form approach alternatives are ranked in terms of decreasing oui values. the ranking list related to three approaches is constructed and the decision is made by considering the theory of dominance [32, 82-85]. 3.5. single valued neutrosophic sets based multimoora method svns based multimoora method can be applied for decision problems consisting of m alternatives and n criteria, and the performances of alternatives are considered by using svns. steps of svns based multimoora method for ratio system approach are expressed as below [76]: 1. by using a single-valued neutrosophic weighted average operator the values of zi +and zi are computed as below: 𝑍𝑖 + = (1 − ∏ (1 − 𝑡𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑎𝑥 , ∏ (𝑖𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑎𝑥 , ∏ (𝑓𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑎𝑥 ), (33) 𝑍𝑖 − = (1 − ∏ (1 − 𝑡𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑖𝑛 , ∏ (𝑖𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑖𝑛 , ∏ (𝑓𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑖𝑛 ). (34) the importance of alternative i acquired for benefit and cost criteria are represented by zi + and zi as single-valued neutrosophic numbers. 2. then the values of zi + and zi are computed via score function as eqs. (35) and (36): 𝑧𝑖 + = 𝑠(𝑍𝑖 +), (35) 𝑧𝑖 − = 𝑠(𝑍𝑖 −). (36) 3. the overall importance for each alternative is obtained as below: 𝑧𝑖 = 𝑧𝑖 + − 𝑧𝑖 −. (37) 4. alternatives are ranked in terms of decreasing zi values. the steps of svns based multimoora method for reference point approach are stated as below [76]: 1. as a single-valued neutrosophic number each coordinate of the reference point is written by p*=p1 *,p2 *,…,pn * and pj *=tj *,ij *,fj * values are acquired as eq. (38): 𝑝𝑗 ∗ = { 〈max 𝑖 𝑡𝑖𝑗 , min 𝑖 𝑖𝑖𝑗 , min 𝑖 𝑓𝑖𝑗〉 𝑗 ∈ ω𝑚𝑎𝑥 〈min 𝑖 𝑡𝑖𝑗 , min 𝑖 𝑖𝑖𝑗 , max 𝑖 𝑓𝑖𝑗〉 𝑗 ∈ ω𝑚𝑖𝑛 . (38) 566 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. coordinate j of the reference point can be written as pj *: 𝑝𝑗 ∗ = { 〈1,0,0〉 𝑗 ∈ ω𝑚𝑎𝑥 〈0,0,1〉 𝑗 ∈ ω𝑚𝑖𝑛 . (39) 2. maximum distance from each alternative to all coordinates of the reference point is calculated by using eq. (40): 𝑒𝑖𝑗 𝑚𝑎𝑥 = 𝑒𝑚𝑎𝑥(𝑝𝑖𝑗 , 𝑝𝑗 ∗)𝑤𝑗 . (40) 3. maximum distance of each alternative is computed via eq. (41): 𝑒𝑖 𝑚𝑎𝑥 = max 𝑗 𝑒𝑖𝑗 𝑚𝑎𝑥. (41) then alternatives are ranked in terms of increasing imax values. the steps of svns based multimoora method for a full multiplicative form approach are explained as follows [76]: 1. the values of ci and di are calculated as follows: 𝐶𝑖 = (∏ (𝑡𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑎𝑥 , 1 − ∏ (1 − 𝑖𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑎𝑥 , 1 − ∏ (1 − 𝑓𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑎𝑥 ), (42) 𝐷𝑖 = (∏ (𝑡𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑖𝑛 , 1 − ∏ (1 − 𝑖𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑖𝑛 , 1 − ∏ (1 − 𝑓𝑗) 𝑤𝑗 𝑗∈ω𝑚𝑖𝑛 ). (43) as single-valued neutrosophic numbers ci and di can be written as ci=tci,ici,fci and di=tdi,idi,fdi, respectively. 2. the values of ci and di are computed via score function as eqs. (44) and (45): 𝑐𝑖 = 𝑠(𝐶𝑖), (44) 𝑑𝑖 = 𝑠(𝐷𝑖). (45) 3. the overall utility for each alternative is obtained as below: 𝑜𝑢𝑖 = 𝑐𝑖 𝑑𝑖 . (46) alternatives are ranked in terms of decreasing oui values and final ranking related to alternatives are made via the theory of dominance in terms of multimoora method. 4. case study and results in this study, nine criteria for choosing the optimal training aircraft for ftos are determined according to literature review [7, 8, 18] and experts’ opinions. additionally, they are weighted via single-valued neutrosophic ahp firstly. for this purpose evaluations of 10 decision-makers related to the selection of training aircraft are considered in terms of the group decision-making (gdm) process. the required criteria are: runway length, capable of ifr/vfr operations, reliability/security, purchasing cost, maintenance cost, operational cost, certified staff number, availability of facilities and ease of finding spare part, respectively. while some of the criteria are common (purchasing cost, maintenance cost, operational cost, reliability/security) , the other criteria are specified and diversified according to the purpose and mission for military, commercial, passenger and cargo aircraft types. an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 567 the neutrosophic evaluation matrix in terms of the criteria considered for the selection of training aircraft is constructed through decision-makers’ linguistic judgments which are seen as table 3. a part of the neutrosophic evaluation matrix for the criteria are shown in appendix 1. after that, the neutrosophic evaluation matrix is transformed into a crisp one by using eq. (17) and taking the geometric means of 10 decision-makers’ judgments. the crisp evaluation matrix for criteria is shown in table 5. table 5 the crisp evaluation matrix for criteria criteria required runway length capable of ifr/vfr operations reliability/ safety purchasing cost maintenance cost operational cost certified staff number availability of facilities ease of finding spare part required runway length 1 0.773 1.0200 0.8695 1.1133 1.2563 0.4357 0.8855 0.591 capable of ifr/vfr operations 1.2922 1 1.1298 2.0885 2.0296 1.2228 1.3450 0.9182 0.715 reliability/safety 0.9803 0.8850 1 1.5795 1.0079 0.8103 0.3959 0.5376 0.663 purchasing cost 1.1500 0.4787 0.6330 1 0.6413 0.9892 0.8545 0.5830 1.247 maintenance cost 0.8981 0.4926 0.9921 1.5592 1 0.9243 0.8374 0.6753 0.793 operational cost 0.7959 0.8177 1.2340 1.0108 1.0818 1 1.0153 0.5583 1.428 certified staff number 2.2947 0.743 2.5254 1.1702 1.194 0.9849 1 0.7840 1.288 availability of facilities 1.1292 1.0890 1.8600 1.7149 1.4807 1.7909 1.2754 1 1.256 ease of finding spare part 1.6901 1.3967 1.5077 0.8015 1.2599 0.7000 0.7759 0.7956 1 the normalized evaluation matrix for criteria is formed as table 6. table 6 the normalized evaluation matrix for criteria criteria required runway length capable of ifr/vfr operations reliability /safety purchasing cost maintenance cost operational cost certified staff number availability of facilities ease of finding spare part required runway length 0.0890 0.1007 0.0857 0.0737 0.1030 0.1298 0.0549 0.1314 0.065 capable of ifr/vfr operations 0.1150 0.1302 0.0949 0.1770 0.1877 0.1263 0.1695 0.1362 0.079 reliability/safety 0.0872 0.1152 0.0840 0.1339 0.093 0.0837 0.0498 0.0797 0.073 purchasing cost 0.1024 0.0623 0.0531 0.0847 0.0593 0.1022 0.1076 0.086 0.138 maintenance cost 0.0799 0.0641 0.0833 0.1322 0.0925 0.0954 0.1055 0.1002 0.088 operational cost 0.070 0.1065 0.1036 0.0857 0.1000 0.1033 0.1279 0.0828 0.158 certified staff number 0.2043 0.0968 0.2121 0.099 0.1104 0.1017 0.1260 0.116 0.143 availability of facilities 0.1005 0.1418 0.1562 0.1454 0.1369 0.1850 0.1607 0.1484 0.139 ease of finding spare part 0.1504 0.1819 0.1266 0.0679 0.1165 0.0723 0.0977 0.1180 0.111 finally, the priorities for the criteria as eigenvector x can be calculated by taking the overall row averages and seen as below: 568 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. 𝑋 = [ 0.092695 0.13521 0.088997 0.088593 0.093533 0.104439 0.134509 0.146125 0.1159 ] according to eigenvector x, while the number of maintenance facilities was found as the most important criterion having a value of 0.146125, the historical cost was obtained as the least important one having a value of 0.088593. then the consistency of decision-makers’ judgments is checked by computing ci and cr values. ci value is found as 0.037 and by using eq. (23) cr value is acquired as 0.025. decision-makers’ evaluations are consistent because of having cr value smaller than 0.1. after obtaining the criteria weights, alternatives have been chosen among the most commonly used aircraft by ftos in turkey. besides, ninety percent of aircraft used for flight training is of a single-engine type [86]. therefore, only six single-engine type aircraft (diamond 40, evektor, the tecnam, pipera, diamond 20 and cessna 172) have been included as alternatives in the research according to the experts’ views in terms of training aircrafts. alternatives are ranked via single-valued neutrosophic sets based on multimoora method. firstly neutrosophic evaluations of six training aircraft obtained by taking the geometric means of 10 decision-makers’ judgments are presented in appendix 2. then in terms of ratio system approach the values of zi +,zi -,zi +,zi and zi are computed and training planes are ranked as seen in table 7. table 7 ranking of training aircraft according to the ratio system approach 𝑍𝑖 + 𝑍𝑖 − 𝑧𝑖 + 𝑧𝑖 − 𝑧𝑖 ranking diamond 40 (0.63,0.35,0.37) (0.33,0.66,0.67) 0.273404 -0.33757 0.610969 3 evektor (0.62,0.37,0.38) (0.32,0.66,0.68) 0.242802 -0.33344 0.576242 5 the tecnam (0.67,0.31,0.33) (0.33,0.65,0.67) 0.360386 -0.31666 0.677044 2 pipera (0.63,0.33,0.37) (0.34,0.64,0.66) 0.298661 -0.30727 0.605931 4 diamond 20 (0.47,0.5,0.53) (0.25,0.72,0.75) -0.02318 -0.47464 0.451462 6 cessna 172 (0.63,0.34,0.37) (0.29,0.68,0.71) 0.298632 -0.38994 0.688570 1 according to table 7 ranking of training aircraft is cessna 172 > the tecnam > diamond 40 > pipera > evektor > diamond 20 in terms of ratio system approach. in addition, the ranking of training aircraft is presented in table 8 in terms of the reference point approach. an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 569 table 8 ranking of training aircraft according to the reference point approach 𝑝1 ∗ 𝑝2 ∗ 𝑝3 ∗ 𝑝4 ∗ 𝑝5 ∗ 𝑝6 ∗ 𝑝7 ∗ 𝑝8 ∗ 𝑝9 ∗ 𝑒𝑖 𝑚𝑎𝑥 ranking diamond 40 0.02 0.033 0.020 0.069 0.074 0.06 0.030 0.052 0.01 0.07 1 evektor 0.02 0.023 0.040 0.067 0.063 0.08 0.053 0.029 0.02 0.08 4 the tecnam 0.02 0.031 0.019 0.068 0.065 0.08 0.020 0.054 0.01 0.08 5 pipera 0.03 0.040 0.017 0.071 0.060 0.08 0.030 0.036 0.02 0.08 6 diamond 20 0.02 0.050 0.036 0.052 0.061 0.06 0.073 0.080 0.03 0.08 3 cessna 172 0.02 0.0360 0.0217 0.0623 0.0645 0.07 0.028 0.040 0.02 0.07 2 according to table 8, the ranking of training aircraft is diamond 40 > cessna 172 > diamond 20 > evektor > the tecnam > pipera in terms of reference point approach. in terms of the full multiplicative form approach the values of ci,di,ci,di and oui are computed and training aircraft are ranked seen as table 9. table 9 ranking of training aircraft according to the full multiplicative form approach 𝐶𝑖 𝐷𝑖 𝑐𝑖 𝑑𝑖 𝑜𝑢𝑖 ranking diamond 40 (0.8,0.18,0.2) (0.92,0.08,0.08) 0.621453 0.836481 0.742937 4 evektor (0.79,0.21,0.21) (0.92,0.07,0.08) 0.580427 0.842982 0.688540 5 the tecnam (0.83,0.16,0.17) (0.92,0.07,0.08) 0.668086 0.851028 0.785034 2 pipera (0.81,0.16,0.19) (0.92,0.07,0.08) 0.650124 0.84683 0.767714 3 diamond 20 (0.67,0.32,0.33) (0.88,0.11,0.12) 0.347624 0.768949 0.452076 6 cessna 172 (0.82,0.16,0.18) (0.9,0.08,0.1) 0.656703 0.819214 0.801625 1 according to table 9, ranking of training aircraft is cessna 172 > the tecnam > pipera > diamond 40 > evektor > diamond 20 in terms of a full multiplicative form approach. the final ranking of training aircraft according to the svns based multimoora method by considering three different approaches is presented in table 10. table 10 the final ranking of training aircraft according to the single-valued neutrosophic sets based multimoora method ranking for ratio system approach ranking for reference point approach ranking for a full multiplicative form approach final ranking based on dominance theory diamond 40 3 1 4 3 evektor 5 4 5 5 the tecnam 2 5 2 2 pipera 4 6 3 4 diamond 20 6 3 6 6 cessna 172 1 2 1 1 according to table 10, three different approaches give different ranking results and the final ranking for dominance theory is cessna 172 > the tecnam > diamond 40 > pipera > evektor > diamond 20 like ratio system approach. 570 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. 5. comparison and sensitivity analysis in this section, the results of the proposed approach are analyzed based on a comparison and a sensitivity analysis. 5.1. comparison with the neutrosophic ahp-topsis method here, a comparison was performed between the results attained from the neutrosophic ahp-multimoora method and those of another approach. to show the efficiency and display the irreplaceable merits of the neutrosophic ahp-multimoora method, the ahp-topsis method [87] is implemented to handle the decision making problem. let pij be the neutrosophic number that describes the normalized performance of alternative i in terms of criterion j; sets of benefit and cost criteria are represented by max and min; weight of criterion j is denoted bywj. in the ahp-topsis approach, the computation of an ideal solution (is) and anti-ideal solution (a-is) of each criterion are important concerns for des. at this time, is and a-is are computed with reference to the neutrosophic fuzzy is (nf-is) and a-is (nfa-is). let pj + and pj denote the nf-is and nfa-is and are calculated using the following formula 𝑝𝑗 + = (𝑡𝑖𝑗 +, 𝑖𝑖𝑗 + , 𝑓𝑖𝑗 +) = { 〈max 𝑖 𝑡𝑖𝑗 , min 𝑖 𝑖𝑖𝑗 , min 𝑖 𝑓𝑖𝑗〉 𝑗 ∈ ω𝑚𝑎𝑥 〈min 𝑖 𝑡𝑖𝑗 , min 𝑖 𝑖𝑖𝑗 , max 𝑖 𝑓𝑖𝑗〉 𝑗 ∈ ω𝑚𝑖𝑛 , (47) 𝑝𝑗 − = (𝑡𝑖𝑗 −, 𝑖𝑖𝑗 − , 𝑓𝑖𝑗 −) = { 〈min 𝑖 𝑡𝑖𝑗 , min 𝑖 𝑖𝑖𝑗 , max 𝑖 𝑓𝑖𝑗〉 𝑗 ∈ ω𝑚𝑎𝑥 〈max 𝑖 𝑡𝑖𝑗 , min 𝑖 𝑖𝑖𝑗 , min 𝑖 𝑓𝑖𝑗〉 𝑗 ∈ ω𝑚𝑖𝑛 . (48) next, calculate the distance measures with the euclidean distance [87]. the separation to the ideal alternative and distance to the anti-ideal alternative are denoted by 𝑑(𝑝𝑖𝑗 , 𝑝𝑗 +) = 1 3 ∑ 𝑤𝑗[|𝑡𝑖𝑗 − 𝑡𝑖𝑗 +| + |𝑖𝑖𝑗 − 𝑖𝑖𝑗 +| + |𝑓𝑖𝑗 − 𝑓𝑖𝑗 +|]𝑛 𝑗=1 , (49) (𝑝𝑖𝑗 , 𝑝𝑗 −) = 1 3 ∑ 𝑤𝑗[|𝑡𝑖𝑗 − 𝑡𝑖𝑗 −| + |𝑖𝑖𝑗 − 𝑖𝑖𝑗 −| + |𝑓𝑖𝑗 − 𝑓𝑖𝑗 −|]𝑛 𝑗=1 . (50) now, compute the relative closeness to the ideal alternative and rank the preference order. the relative closeness of the ith to the ideal alternative concerning the ideal alternative is as follows: 𝐶𝑖 = 𝑑(𝑝𝑖𝑗,𝑝𝑗 −) 𝑑(𝑝𝑖𝑗,𝑝𝑗 −)+𝑑(𝑝𝑖𝑗,𝑝𝑗 +) ; 𝐶𝑖 ∈ [0, 1]. (51) a set of alternatives that can be preference are ranked according to the descending order of ci0,1; then larger means a better alternative. from appendix 2 and eqs. (47)-(48), nf-is and nfa-is are evaluated. now, the whole computational results of neutrosophic ahp-topsis [86] method are depicted in table 11. an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 571 table 11 ranking of training aircraft according to ahp-topsis method 𝑑(𝑝𝑖𝑗 , 𝑝𝑗 +) 𝑑(𝑝𝑖𝑗 , 𝑝𝑗 −) 𝐶𝑖 ranking diamond 40 0.081 0.123 0.602 4 evektor 0.088 0.117 0.571 5 the tecnam 0.070 0.124 0.639 2 pipera 0.073 0.116 0.613 3 diamond 20 0.175 0.112 0.389 6 cessna 172 0.063 0.131 0.673 1 next, to illustrate the advantages of our proposed neutrosophic ahp-multimoora, a comparative analysis is conducted with ahp-topsis [86] model. fig.1 displays the ranking results of the six training aircraft for flight training organizations alternative as yielded using the mentioned methods. from fig. 1, we can observe that given methods suggest cessna 172 as the first choice for the considered training aircraft for flight training organizations. moreover, the ranking orders of the six training aircraft for flight training determined by the proposed method are exactly matched with those derived by the neutrosophic ahp-topsis method. this demonstrates the validity of the proposed decision-making framework. by comparison, the multimoora utilized in our proposed method is more comprehensive in dealing with training aircraft for flight training problems as it utilizes the ratio system, the reference point approach, and the full multiplicative form. therefore, the reliability and veracity of the decision-making results would be improved greatly by using the proposed neutrosophic ahp-multimoora model. compared with the neutrosophic ahp-topsis method in the literature, the ahpmultimoora method developed in this study has the following attractions: ▪ the svnss improve the elicitation of linguistic information when a decisionmaker hesitates among several values to assess a training aircraft for flight training problems. the use of svnss provides a more flexible way to represent decisionmakers’ evaluations. so, an organized method is given to combine expert knowledge and experience for use in selecting the optimal training aircraft for flight training organizations. ▪ the ahp method for importance coefficients of criteria is taken into account in the process of training aircraft for flight training evaluation and selection, which makes the proposed decision-making model more realistic, more practical, and more flexible. ▪ the multimoora method is used for the prioritization of training aircraft for flight training alternatives, which is a robust and powerful mcdm method and is easily implemented relative to other methods such as the topsis and the vikor methods. hence, the proposed method more effectively conducts robust evaluation for a particular manufacturing environment. 572 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. fig. 1 comparison of preference order of training aircraft derived by the different methods 5.2. sensitivity analysis the aim of the sensitivity analysis is to investigate the impact of various settings of criteria weights over different weight sets. in this paper, a sensitivity analysis was done to investigate the proposed approach behavior. nine different criteria weight sets are taken and depicted in table 12. in this table, for every set, one of the criteria has the maximum weight, while the other ones have lower weights. by applying this process, the ample scope of criteria weights was created to investigate the sensitivity of the developed approach to variation of the criteria weights. table 12 different criteria weight sets for ranking of training aircraft required runway length capable of ifr/vfr operations reliability / safety purchasing cost maintenance cost operational cost certified staff number availability of facilities ease of finding spare part set-1 0.0926 0.1352 0.0889 0.0885 0.0935 0.1044 0.1345 0.1461 0.1159 set-2 0.135 0.0889 0.0885 0.0935 0.1044 0.1345 0.1461 0.1159 0.0926 set-3 0.0889 0.0885 0.0935 0.1044 0.1345 0.1461 0.1159 0.0926 0.1352 set-4 0.0885 0.0935 0.1044 0.1345 0.1461 0.1159 0.0926 0.135 0.0889 set-5 0.0935 0.1044 0.1345 0.1461 0.1159 0.0926 0.1352 0.0889 0.0885 set-6 0.1044 0.1345 0.1461 0.1159 0.0926 0.1352 0.0889 0.0885 0.0935 set-7 0.1345 0.1461 0.1159 0.0926 0.1352 0.0889 0.0885 0.0935 0.1044 set-8 0.1461 0.1159 0.0926 0.1352 0.0889 0.0885 0.0935 0.1044 0.1345 set-9 0.1159 0.0926 0.1352 0.0889 0.0885 0.0935 0.1044 0.1345 0.1461 an extended single-valued neutrosophic ahp and multimoora method to evaluate the optimal... 573 table 13 overall utility degree for each training aircraft with different values of weight sets sets diamond 40 evektor the tecnam pipera diamond 20 cessna 172 set-1 0.7427 0.6865 0.7866 0.7661 0.4548 0.8019 set-2 0.7928 0.7133 0.8384 0.8032 0.5357 0.8629 set-3 0.8750 0.7785 0.9147 0.8818 0.6441 0.9294 set-4 0.8483 0.8006 0.8937 0.8907 0.6364 0.9375 set-5 0.8184 0.7102 0.8740 0.8403 0.5794 0.8900 set-6 0.8200 0.7113 0.8489 0.8146 0.5974 0.8701 set-7 0.7711 0.7186 0.8303 0.7973 0.5678 0.8403 set-8 0.7668 0.7165 0.8236 0.7773 0.5714 0.8370 set-9 0.7310 0.6474 0.7812 0.7578 0.4715 0.7899 the sensitivity analysis results described in table 13 and fig. 2 show that the overall utility degree could change over different criteria weight sets and the rank of training aircraft alternative. for example, when decision experts (des) give the different criteria weight sets-1 to 9, the ranking of training aircraft alternative is cessna 172 ≻ the tecnam ≻ diamond 40 ≻ pipera ≻ evektor ≻ diamond 20. and cessna 172 is the optimal alternative. we can observe that in each criterion weight set, cessna 172 is the best option rank but the preference order of training aircraft alternative is also identical, which includes the experiments to manipulate factors and see how outcomes change, sensitivity analysis of a mathematical model reveals how outcomes respond to variate the criteria weight settings. fig. 2 ranking of overall utility for each alternative with different values of weight sets from the above discussion, it is concluded that the training aircraft alternative selection is dependent on and sensitive to these criteria weight sets. therefore, the proposed approach has acceptable stability with different weight sets. 574 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. 5. conclusion in this study, training aircraft are ranked by using a neutrosophic ahp based neutrosophic multimoora approach. for this aim firstly criteria for selecting the training aircraft are determined according to the depth literature review process and weighted via svns based ahp approach. then the six training aircraft as alternatives are ranked by using svns based multimoora method. svns are preferred compared to crisp, fuzzy, interval-valued, and intuitionistic sets due to efficiency, flexibility, and easiness for explaining decision-makers’ indeterminate judgments. furthermore, the selection of training aircraft as a complex realworld decision-making problem can be efficiently solved under neutrosophic sets based environment. for further research studies criteria related to training aircraft selection can be expanded and results can be compared to different multi-criteria decision-making methods. a sensitivity analysis has also been performed with different criteria weights sets to represent the stability of the proposed approach. the analyses of the results show that the proposed approach has good efficiency and stability, and is well consistent with the other methods. besides, alphadiscounting method which was proposed by [59] can be used as an alternative to ahp for prioritizing criteria and obtaining consistent outputs with pairwise comparisons. it works for any number of preferences that can be transformed into a system of homogeneous (and/or non-homogeneous) linear (and/or non-linear) equations (and/or inequalities) 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ahptopsis, neutrosophic sets and systems, 23, pp. 160-171. 87. ye, j., 2014, single-valued neutrosophic minimum spanning tree and its clustering method, journal of intelligent systems, 23(3), pp. 311-324. 578 ç. karamaşa, d. karabasevic, d. stanujkic, a.r. kookhdan, a. r. mishra, et al. appendices appendix 1 neutrosophic evaluation matrix for criteria criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c1 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (2,3,4); 0.3,0.75, 0.7 〉 〈 (4,5,6); 0.8,0.15, 0.2 〉 〈 (2,3,4); 0.3,0.75, 0.7 〉 〈 (2,3,4); 0.3,0.75, 0.7 〉 〈 (6,7,8); 0.9,0.1, 0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (4,5,6); 0.8,0.15, 0.2 〉 〈 (1,2,3); 0.4, 0.65, 0.6 〉 c2 〈 ( 1 4 , 1 3 , 1 2 ) ; 0.3,0.75, 0.7 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (9,9,9); 1,0,0 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (6,7,8); 0.9,0.1, 0.1 〉 c3 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈( 1 9 , 1 9 , 1 9 ) ; 1,0,0 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (2,3,4); 0.3,0.75, 0.7 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (1,2,3); 0.4,0.65, 0.6 〉 〈 (9,9,9); 1,0,0 〉 c4 〈 ( 1 4 , 1 3 , 1 2 ) ; 0.3,0.75, 0.7 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (4,5,6); 0.8,0.15, 0.2 〉 〈 (4,5,6); 0.8,0.15, 0.2 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (2,3,4); 0.3,0.75, 0.7 〉 〈 (2,3,4); 0.3, 0.75,0.7 〉 c5 〈 ( 1 4 , 1 3 , 1 2 ) ; 0.3,0.75 , 0.7 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 ( 1 4 , 1 3 , 1 2 ) ; 0.3,0.75,0.7 〉 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (6,7,8); 0.9,0.1,0.1 〉 〈 (4,5,6); 0.8, 0.15, 0.2 〉 c6 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15,0.2 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (5,6,7); 0.7,0.25, 0.3 〉 〈 (4,5,6); 0.8,0.15, 0.2 〉 〈 (5,6,7); 0.7, 0.25, 0.3 〉 c7 〈 (1,1,1); 0.5,0.5,0.5 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 ( 1 7 , 1 6 , 1 5 ) ; 0.7,0.25, 0.3 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (4,5,6); 0.8,0.15, 0.2 〉 〈 (6,7,8); 0.9,0.1, 0.1 〉 c8 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 ( 1 3 , 1 2 , 1) ; 0.4,0.65, 0.6 〉 〈 ( 1 4 , 1 3 , 1 2 ) ; 0.3,0.75, 0.7 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈 (1,1,1); 0.5,0.5,0.5 〉 〈 (4,5,6); 0.8, 0.15, 0.2 〉 c9 〈 ( 1 3 , 1 2 , 1) ; 0.4,0.65, 0.6 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈( 1 9 , 1 9 , 1 9 ) ; 1,0,0 〉 〈 ( 1 4 , 1 3 , 1 2 ) ; 0.3,0.75, 0.7 〉 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈 ( 1 7 , 1 6 , 1 5 ) ; 0.7,0.25, 0.3 〉 〈( 1 8 , 1 7 , 1 6 ) ; 0.9,0.1,0.1 〉 〈 ( 1 6 , 1 5 , 1 4 ) ; 0.8,0.15, 0.2 〉 〈 (1,1,1); 0.5,0.5, 0.5 〉 appendix 2 neutrosophic evaluation matrix for six training aircraft obtained from 10 decision makers c1 c2 c3 c4 c5 c6 c7 c8 c9 diamond 40 〈0.68,0.3, 0.32 〉 〈 0.75,0.22 , 0.25 〉 〈0.77,0.21, 0.23 〉 〈0.79,0.2, 0.21 〉 〈0.79,0.19, 0.21 〉 〈0.65,0.34, 0.35 〉 〈 0.78, 0.21, 0.22 〉 〈0.64,0.36, 0.36 〉 〈 0.84, 0.15, 0.16 〉 evektor 〈0.75,0.24, 0.25 〉 〈0.82,0.17, 0.18 〉 〈0.54,0.47, 0.46 〉 〈0.77,0.21, 0.23 〉 〈0.67,0.29, 0.33 〉 〈0.78,0.2, 0.22 〉 〈0.6,0.38, 0.4 〉 〈0.8,0.19, 0.2 〉 〈 0.76, 0.22, 0.24 〉 the tecnam 〈0.74,0.21, 0.26 〉 〈0.76,0.23, 0.24 〉 〈0.78,0.2, 0.22 〉 〈0.77,0.21, 0.23 〉 〈0.7,0.28, 0.3 〉 〈0.78,0.19, 0.22 〉 〈0.85,0.14, 0.15 〉 〈0.63,0.35, 0.37 〉 〈0.89,0.1, 0.11 〉 pipera 〈0.66,0.3, 0.34 〉 〈0.7,0.25, 0.3 〉 〈0.8,0.18, 0.2 〉 〈0.81,0.17, 0.19 〉 〈0.64,0.34,0.36〉 〈0.8,0.17, 0.2 〉 〈0.77,0.2, 0.23 〉 〈0.75,0.22, 0.25 〉 〈0.81,0.17, 0.19 〉 diamond 20 〈0.75,0.21, 0.25 〉 〈0.62,0.33, 0.38 〉 〈0.59,0.38, 0.41 〉 〈0.59,0.38, 0.41 〉 〈0.66,0.29, 0.34 〉 〈0.64,0.31, 0.36 〉 〈0.46,0.56, 0.54 〉 〈0.45,0.57, 0.55 〉 〈0.7,0.25, 0.3 〉 cessna 172 〈0.77,0.2, 0.23 〉 〈0.73,0.25, 0.27 〉 〈0.76,0.21, 0.24 〉 〈0.7,0.26, 0.3 〉 〈0.69,0.28, 0.31 〉 〈0.71,0.26, 0.29 〉 〈0.78,0.19, 0.22 〉 〈0.72,0.24, 0.28 〉 〈0.77,0.2, 0.23 〉 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 1 5 https://doi.org/10.22190/fume201211016l © 2021 by university of niš, serbia | creative commons licence: cc by-nc-nd short communication an experimental study on third-body particle transport in sliding contact qiang li1, iakov a. lyshenko1,2, jasminka starcevic1,3 1technische universität berlin, berlin, germany 2sumy state university, sumy, ukraine 3national research tomsk state university, tomsk, russia abstract. an experiment is designed to study the third-body particle transport in a rough contact. to study the influence of particles in a pure form, it is assured that the first bodies have no contact and the sliding is very slow, so that the process can be considered as quasistatic. an example of sliding contact of a 3d printed “rough body” on small spheres artificially located on a rubber layer is presented. the trajectory of particles during the sliding is captured for studying their movement and the correlation to the fluctuation of normal and tangential force. key words: third-body, particle transport, friction, experiment 1. introduction the mechanics and physics of surfaces and interfaces of solid materials determine the tribological properties of contacts. formation of wear debris, chemical reaction, heat transfer, local melting, material transfer and other thermo-mechanical, electro-mechanical coupling phenomena may occur in the interface region [1]. due to these complicated processes, it is meaningful to study mainly only one or two aspects under some specific conditions, for example the orientation of atom layers in structural superlubricity [2], the effect of boundary layer formation in the running-in process [3], or the formation of wear debris due to adhesion [4, 5]. in the present study, we focus on the wear particle transport. it has been found that the wear particle flow is essential for the formation, modification, and transformation of loading structures or films as well as the resulting macroscopic friction and wear rate [6, 7]. to make a step towards understanding of this behavior, we isolate the effect of the presence and dynamics of interfacial particles from others such as wear and consider only the particle transport at the interfacial space and its influence on the friction. received december 11, 2020 / accepted february 04, 2021 corresponding author: qiang, li affiliation: technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: qiang.li@tu-berlin.de 2 q. li, i.a. lyshenko, j. starcevic for this sake, an experiment of a tangential contact between a body with regular waviness and a rubber layer is designed following these concepts: (1) third-body particles are artificially located on the surface and they are much harder than first-bodies; (2) tangential movement of the first-body is very slow; (3) first-bodies have no direct contact. the former two conditions ensure the absence of wear. the latter one simplifies the contact. similar measures were used in a recent study on the dynamics of third-body friction [8]. 2. experiment the experimental setup is illustrated schematically in fig. 1a (see detailed description in [9, 10]). an indenter with rough surface is moved under displacement-control by two linear motors m-403.2dg for the motion in vertical and horizontal directions separately. a 3d force sensor me k3d40 is installed to measure the normal, tangential and lateral forces. the particles are located on a flat transparent rubber sheet tarnac crg n3005, under which a digital camera with resolution 1600x1200 pixels is fixed to observe the movement of particles. in this experiment, we used a 3d printed indenter made of photopolymer (elastic modulus 2 gpa) having two-dimensional waviness generated according to a harmonic function asin(2x/λ)sin(2y/λ) with amplitude a = 0.25 mm and wavelength λ = 1 mm, as shown in fig. 1c. stainless-steel spheres with diameter 1 mm were used as third-body particles. the rubber sheet with thickness 5 mm is very soft with elastic modulus e ≈ 0.324 mpa [9, 10], so it easily sticks due to adhesion to the glass plate below (fig. 1b). a) b) c) fig. 1 (a) scheme of experimental setup for sliding contact between a wavy-shaped indenter and a transparent rubber sheet. metallic spheres with diameter 1 mm are used as third-body particles. (b) photo of the indenter in contact with transparent rubber sheet. (c) photo of the 3d printed sample with wavy surface and some small spheres on it. the subplot below shows the surface waviness of the indenter under a microscope. in the experiment, the indenter was pressed on particles, while the indentation depth was kept constant and equal to 0.5 mm. subsequently, it was moved slowly with a tangential velocity of 5 m/s. we define the zero clearance between the peaks of indenter and sphere as the zero indentation depth, as shown in the left panel of fig. 2. in this case no forces are detected during tangential motion. the contact configuration at the indentation an experimental study on third-body particle transport in sliding contact 3 depth of 0.5 mm is illustrated in fig. 2 right: the particle is trapped between the valley of indenter and the deformed rubber (the indenter and the particle are much harder than the rubber), and there is no contact between indenter and rubber. the tangential force is then induced mainly by the interaction between sliding bodies and particles. the experimental results are presented in the next section. fig. 2 sketch of a three-body contact. the left figure shows the state of zero indentation depth at which the tangential force during sliding is zero. the right figure shows the state with indentation depth 0.5 mm. the comparison of particle size and surface waviness of indenter with the indentation depth can be seen in this figure. 3. results in the experiment 72 spheres with diameter 1 mm were located between indenter and rubber sheet within the camera’s field of view. a snapshot during sliding is shown in fig. 3a. a supplementary video in [11] shows the progress of complete experiment including indenting, sliding forward, sliding backward and pull off. in this analysis we focus only on the part of sliding forward. the indenter is controlled to move tangentially with constant velocity 5 µm/s. the direction of tangential motion was aligned with one direction of the waviness (fig. 3a). no collision between particles was observed during the sliding. with image processing, trajectories of all particles were determined, some of which are shown in fig. 3b. it is seen that some particles travel on a straight line having almost the identical displacement as the indenter (in total 10 mm). these particles stay mainly in the same valleys of waviness and moved together with the indenter during the whole process. the black spots in fig. 3c show these 42 particles. it is noted that some of these particles tried to move over the peaks during the sliding but dropped quickly back to the valleys. the other 30 particles flow from one valley into another one. the red spots and corresponding stars show the particle positions at the beginning and the end of the sliding by taking the indenter as a reference. it is noted that two spheres were squeezed out of contact in the process. it is seen in fig. 3c that the “black” and “red” particles are clustered. 4 q. li, i.a. lyshenko, j. starcevic a) b) c) fig. 3 (a) a snapshot showing all particles between wavy surface and rubber sheet. (b) trajectories of some particles during sliding. (c) positions of particles at the beginning and the end by taking the indenter as a reference. the black spots are particles moving together with the indenter or failed to move over the peaks, and the red ones are having a relative displacement. the red spots are positions of particles trapped on at the beginning and stars are corresponding positions at the end of sliding. the normal and tangential force as well as the coefficient of friction are shown in fig. 4. from the particle trajectories, the instantaneous tangential velocities of particles were determined for analyzing the particle motion (fig. 4b below). here tangential direction is the direction of indenter’s motion (horizontal direction as marked in fig.3a). it is clearly visible that the continuous movement of particles is followed by short jumps or a series of jumps. the jump of motion (velocity) is much larger than the indenter’s velocity. these jumps lead to a "randomly" fluctuating normal and tangential force shown in fig. 4a. a) b) fig. 4 (a) normal and tangential force in sliding contact. (b) coefficient of friction and instantaneous tangential velocities of particles corresponding to trajectories. an experimental study on third-body particle transport in sliding contact 5 4. conclusion an experimental study on particle transport in three-body contact was reported. the experiment was designed to eliminate wear by using the soft rubber sheet and a very small sliding velocity and by avoiding the first-body contact. the normal and tangential forces are then induced mainly by the interaction between artificial small spheres and the surface waviness whose size could be individually designed. the particle trajectory was analyzed to find the movement of spheres which can be transferred from one minimum of the wave into another one. the fluctuating normal and tangential force should be correlated to the motion of particles for example the instantaneous tangential velocities. furthermore, local material property could also play an important role, for example friction, or adhesion such as in this reported case of soft rubber. it is planned to continue this work by systematically studying the contact and varying the system parameters and shape – starting from the simplest case, a single or few particles [12]. the elastic interaction among particle, wavy surface and foundation could be understood by numerical simulation using for example the fast fourier transform assisted boundary element method in which adhesion or friction could be taken into account and the state of particles (sticking, rolling or sliding) could be identified [13, 14]. acknowledgement: this research was partially supported by “the tomsk state university competitiveness improvement program” and by deutsche forschungsgemeinschaft (project dfg po 810-55-1) references 1. vakis, a.i. yastrebov, v.a., schreibert, j., et al., 2018, modeling and simulation in tribology across scales: an overview, tribology international, 125, pp. 169–199. 2. dienwiebel, m., verhoeven, g.s., pradeep, n., frenken, j.w.m., heimberg, j.a., zandbergen, h.w., 2004, superlubricity of graphite, physical review letters, 92, 126101. 3. zhang, y., kovalev a., meng, y., 2018, combined effect of boundary layer formation and surface smoothing on friction and wear rate of lubricated point contacts during normal running-in processes, friction, 6, pp. 274–288. 4. aghababaei, r., warner, d.h., molinari, j.f., 2016, critical length scale controls adhesive wear mechanisms, nature communication, 7, 11816. 5. popov, v.l., pohrt, r., 2018, adhesive wear and particle emission: numerical approach based on asperity-free formulation of rabinowicz criterion, friction, 6, pp. 260-273. 6. ostermeyer, g.p., brumme, s., recke, b., 2017, the wear debris investigator – a new device for studying the formation of particles in the contact area, proc. eurobrake, dresden, eb2017-vdt-019. 7. ostermeyer, g.p., 2003, on the dynamics of the friction coefficient, wear, 254, pp. 852–858. 8. deng, f., tsekenis, g., rubinstein, s.m., 2019, simple law for third-body friction, physical review letters, 122, 135503. 9. lyashenko, i.a., pohrt, r., 2020, adhesion between rigid indenter and soft rubber layer: influence of roughness, frontiers in mechanical engineering. section tribology, 6, 49. 10. lyashenko, i.a., popov, v.l., 2020, the effect of contact duration and indentation depth on adhesion strength: experiment and numerical simulation, technical physics, 65(10), pp. 1695–1707. 11. li, q., lyashenko, i.a., starcevic, j., 2021, an experimental study on third-body particle transport in sliding contact (video of experiment), supplementary video. https://doi.org/10.13140/rg.2.2.31164.67206 12. de payrebrune, k.m., kröger, m., 2015, kinematic analysis of particles in three-body contact, tribology international, 81, pp. 240–247. 13. li, q., 2020, simulation of a single third-body particle in frictional contact, facta universitatis-series mechanical engineering, 18(4), pp. 537–544. 14. rey, v., anciaux, g., molinari, j.f., 2017, normal adhesive contact on rough surfaces: efficient algorithm for fft-based bem resolution, computational mechanics, 60, pp. 69–81. 12432 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 257 273 https://doi.org/10.22190/fume240118033m © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based on rotational motion da-wei man1,2, yu zhang1, li-ping tang1,2, qing-hu xu1,2, dong chen1,2, bang-dong jiang1, ting-ting han1,2, tao xu3, jin-bao li4 1school of civil engineering, anhui jianzhu university, hefei, china 2bim engineering center of anhui province, hefei, china 3china construction first group the fifth construction co., ltd, beijing, china 4jiangsu southeast special engineering & technology co., ltd, nanjing, china orcid ids: da-wei man https://orcid.org/0000-0003-3053-2318 yu zhang https://orcid.org/0009-0006-4473-9183 li-ping tang https://orcid.org/0009-0006-5184-4483 qing-hu xu https://orcid.org/0000-0002-8617-8373 dong chen https://orcid.org/0009-0002-0221-9383 bang-dong jiang https://orcid.org/0009-0003-0063-138x ting-ting han https://orcid.org/0009-0008-2379-214x tao xu https://orcid.org/0009-0002-8384-1608 jin-bao li https://orcid.org/0009-0009-0251-0665 abstract. the paper presents an improved hybrid tri-stable cantilever piezoelectric energy harvester based on rotational motion, thus providing a new perspective for achieving higher efficiency in energy capture for rotational motion. the proposed system comprises a piezoelectric cantilever beam with an innovative dynamic amplifier installed at the edge of a vehicle's wheel hub. through theoretical analysis and numerical simulations, the influence of parameters such as the mass of the tip magnetic on the piezoelectric beam, the wheel hub radius, rotational speed, and the ratio of the dynamic amplifier's spring stiffness are investigated with respect to the system's steady-state dynamic response and time-domain performance. a comparative analysis is also conducted with traditional tri-stable piezoelectric energy harvesters. the results demonstrate that the hybrid tri-stable piezoelectric energy harvester exhibits superior performance in capturing vibrational energy during rotational motion, compared to traditional tri-stable piezoelectric energy harvesters. proper adjustments to the mass of the tip magnetic and the internal spring stiffness of the dynamic amplifier can enhance the system's output voltage and the rotational speed range of inter-well motion. additionally, the rotational speed range of inter-well motion increases with the expansion of the wheel hub radius. however, when the rotational speed is below 100 rpm, the influence of varying the wheel hub radius on the system's output voltage is minimal. key words: rotational motion, piezoelectric energy harvester, tri-stable, dynamic amplifier, steady-state dynamic response received: january 18, 2024 / accepted may 22, 2024 corresponding author: da-wei man school of civil engineering, anhui jianzhu university, hefei 230601, china e-mail: daweiman@hotmail.com https://orcid.org/0000-0003-3053-2318 https://orcid.org/0009-0006-4473-9183 https://orcid.org/0009-0006-5184-4483 https://orcid.org/0000-0002-8617-8373 https://orcid.org/0009-0002-0221-9383 https://orcid.org/0009-0003-0063-138x https://orcid.org/0009-0008-2379-214x https://orcid.org/0009-0002-8384-1608 https://orcid.org/0009-0009-0251-0665 258 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. 1. introduction the widespread application of wireless sensor networks in infrastructure monitoring, ecological environment monitoring, and disaster warning imposes higher requirements on the cleanliness and sustainability of their power sources. traditional chemical batteries are toxic, have limited lifespans, and pose challenges such as time-consuming replacements, high costs, and difficulty in handling waste batteries [1-6]. piezoelectric energy harvesters, capable of capturing vibrational energy from the surrounding environment and converting it into persistent and clean electrical energy, hold significant potential for self-powering wireless sensor nodes due to their simple, compact, and easily integrable structures. this has garnered extensive attention from scholars both domestically and internationally in recent years [7-12]. early linear piezoelectric harvesters were restricted to generating high output power only in resonance, exhibiting low efficiency in wide-frequency or random vibrational environments. to overcome this limitation, researchers introduced nonlinearity to linear systems, resulting in nonlinear piezoelectric harvesters with broader operating frequency bands. this innovation renders them less sensitive to variations in external vibration frequencies, so that they are more suitable for effectively harvesting energy from diverse environmental vibrations [13, 14]. stanton et al. [15, 16] established a magnetic non-linear bi-stable piezoelectric harvester composed of a permanent magnet and a piezoelectric cantilever beam. under low-frequency vibrational excitations, it exhibits oscillations between two potential wells, significantly broadening the operating bandwidth and output power compared to linear piezoelectric harvesters. however, under low intensity vibrational, the bi-stable piezoelectric harvester may struggle to overcome potential energy barriers for substantial inter-well motion [17-20]. consequently, researchers proposed nonlinear multi-stable piezoelectric harvesters, typically utilizing multiple magnets to create multiple potential wells. by adjusting system parameters, the barrier height between wells can be reduced, making it easier for the system to generate high-energy inter-well motion. wang et al. [21] studied the influence mechanism of geometric nonlinearity and gravitational effects on the dynamic response of a tri-stable piezoelectric harvester through theoretical analysis and experimental methods. zhu et al. [22] analyzed the impact of three potential well distribution characteristics on the energy harvesting performance of tri-stable piezoelectric harvesters, showing optimal performance when the barrier heights are equal. yang et al. [23] investigated a hybrid tri-stable vibration energy harvester employing geometric nonlinearity to adjust the resonant frequency and expand the inter-well motion bandwidth. the results indicate that the tri-stable characteristic has the potential to enhance significantly the harvested power. zhou et al. [24] proposed penta-stable cantilever beam piezoelectric harvesters to achieve greater deformation and higher output voltage. most studies on piezoelectric harvesters have focused on harmonic excitation. however, rotational motion is prevalent in both civilian and industrial environments. in recent years, some researchers have focused on using piezoelectric harvesters to achieve self-powering for sensor nodes in rotating machinery such as wheels and wind turbines [25,26]. khameneifar et al. [27] designed a cantilever beam linear piezoelectric harvester that generates continuous vibration and harvests energy through the periodic excitation caused by the tip mass of the beam during rotational motion. zhang et al. [28] proposed a bi-stable piezoelectric harvester mounted on the edge of a hub, capable of converting the rotational and kinetic energy generated by external excitation into electrical energy to power a tire nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 259 pressure monitoring system. mei et al. [29] established a theoretical model of a magnetic tri-stable piezoelectric harvester based on rotational motion, analyzing the impact of different speeds and piezoelectric coupling coefficients on the system's energy harvesting efficiency. currently, effective broadening of the operating frequency band and improvement of the power performance of piezoelectric harvesters under different rotational radius and speeds remain urgent research issues. this paper designs a hybrid tri-stable piezoelectric harvester (htpeh) by attaching a novel dynamic amplifier composed of rotational and vertical springs to the fixed end of a traditional cantilever piezoelectric beam. this amplifier simultaneously increases the rotational amplitude and vertical displacement of the cantilever beam during rotational motion. the influence of rotational radius, magnetic mass at the beam's end, and spring stiffness ratio on the steady-state dynamic response of the htpeh under rotational motion is investigated. 2. modeling of hybrid tri-stable piezoelectric energy harvester figure 1 depicts a htpeh mounted on the edge of a vehicle wheel hub, with r being the radius of the wheel hub. in fig. 2, a schematic illustration of the htpeh during rotational motion is given. it features a piezoelectric cantilever beam, with l and b denoting the length and width of the beam, respectively. the free end of the beam is equipped with a tip magnetic denoted as a and a mass mf is attached to the fixed end through rotational spring kr and vertical spring kf connecting the wheel hub edge and a l-shaped frame. the mass mf combined with springs kr and kf constitutes a novel power amplifier, capable of simultaneously increasing the rotational amplitude and vertical displacement of the cantilever beam during rotational motion. the piezoelectric beam consists of a substrate layer with thickness hs and two piezoelectric layers covering its upper and lower surfaces. the thickness of the substrate layer and piezoelectric layer are hs and tp. the external load resistance rl is connected in series. two external magnets, denoted in the figure by b and c, are arranged on the l-shaped frame, and their appropriate spacing with the tip magnetic tip induces a tri-stable potential energy function for the system. fig. 1 the htpeh installed on a vehicle wheel hub 260 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. z(z ) y xpiezoelectric layer substrate layer fig. 2 the schematic diagram of the htpeh in rotational motion in fig. 2, the x-y-z coordinates represent the fixed coordinate system, and x*-y*-z* is the rotating coordinate system. assuming r0 defines the position of an infinitesimal segment dx on the beam in the rotating coordinate system, w(x,t) and u(x,t) denote the lateral and axial displacements at position x during beam vibration. the angle between the x-axis and the x*-axis is defined by 𝜃 = ∫ 𝜔𝑑𝑡 𝑡 0 , where ω is the angular displacement of the wheel hub. the eccentricities of the tip magnetic and the mass block mf are denoted as et and ef, respectively. the constitutive relationships for the metal base layer and piezoelectric layers of the piezoelectric beam are as follows: s s 1 s 1 p p 1 p 1 31 3 t 3 31 1 33 3 ( ) t y s t y s d e d d t ε e =  = −   = +  (1) here, the subscript/superscript s represents the parameters associated with the substrate layer, while the subscript/superscript p represents those associated with the piezoelectric layer. the x* and y* directions are indicated by subscripts 1 and 3, respectively. t, s, and y represent the stress, strain, and young's modulus of the beam, respectively. d3, d31, and 휀33 𝑇 denote the electric displacement, piezoelectric constants, and dielectric constants, respectively. with v(t) denoting the voltage, the relation 𝐸3 = −𝑉(𝑡)/(2𝑡𝑝) holds, where e3 is the electric field. the relationship between the displacement and strain is expressed as 𝑆1 𝑠 = 𝑆1 𝑝 = −𝑦𝑤", where y represents the distance between the neutral axis of the beam and the point on its cross-section. the lagrangian for the htpeh is expressed as follows: ewutl +−= (2) nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 261 where t represents the kinetic energy of the system, u is the overall potential energy consisting of the strain energy, ue, the gravitational potential energy, ug, the centrifugal potential energy uc, the strain energy, ud, and the magnetic potential energy um. finally, we represents the electrical energy of the system. based on the geometric relationships presented in fig. 1(b), the kinetic energy of the system can be expressed as: ( )  ( )  ( )  2 2 2 0 0 2 2 2 f f 2 2 2 t t 2 2 f t 1 ( , ) ( , ) ( , ) d 2 1 (0, ) (0, ) (0, ) (0, ) 2 1 ( , ) ( , ) ( , ) ( , ) 2 1 1 (0, ) ( , ) 2 2 l t m w x t w x t x r u x t x m w t w t e w t r u t m w l t w l t e w l t l r u l t j w t j w l t        = + + + +   + + − + +   + + + + + +   + +  (3) the strain energy of the system can be expressed as: 2 e p 31 s p 0 1 1 ( , ) ( ) ( ) ( , ) d 2 2 l u yiw x t y bd h t v t w x t x    = − +     (4) where 𝑌𝐼 = 2 3 [𝑌𝑠𝑏ℎ 3 + 𝑌𝑝𝑏(3ℎ 2𝑡𝑝 + 3ℎ𝑡𝑝 2 + 𝑡𝑝 3)] represents the bending stiffness of the beam. the gravitational potential energy of the system is given by:      g 0 f t 0 ( )sin ( , ) cos d g sin (0, )cos g ( )sin ( , ) cos l u m g x r w x t x m r w t m l r w l t     = + + + + + + + (5) the centrifugal potential energy of the system in rotational motion is given by: 2 2 2 2 2 2 2 c 0 0 f t 0 1 1 [ ( ) ( ) ( )] ( , ) d 2 2 l u m r l x m l x m r m l r w x t x    = − + − + + + (6) the strain potential energy of the dynamic amplifier is given by: 2 2 d r f 1 1 (0, ) (0, ) 2 2 u k w t k w t= + (7) the electrical energy of the system is given by: 𝑊𝑒 = 1 4 𝑌𝑝𝑏𝑑31(ℎ𝑠 + 𝑡𝑝)𝑉(𝑡) ∫ 𝑤″(𝑥, 𝑡)𝑑𝑥 𝑙 0 + 𝑏𝑙휀33 𝑠 𝑉(𝑡)2 4𝑡𝑝 (8) where, 휀33 𝑠 = 휀33 𝑇 − 𝑑31 2 𝑌𝑝. utilizing the galerkin method, the latera displacement w(x,t) is expressed as: 1 ( , ) ( ) ( ) n r r r w x t x q t = = (9) here, 𝜙𝑟(𝑥) and qr(t) represent the rth order modal shape function and generalized coordinate of the beam, respectively. the orthogonality condition for the modal shape functions is: 2 0 t t t t t t t t 0 2 f f f f f f f f ( ) ( )d ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) (0) (0) (0) (0) (0)( ) (0) (0) (0) l s r s r s r s r s r s r s r s r s r rs s m s s l m l l m e l l j m e l l m e l m m e j m e m e                       + + + + +    + − + + − =  (10) 262 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. 2 2 2 r f2 20 d ( ) d ( ) d (0) (0) (0) (0) d d l s r m r m r r rs s s yi s k k s s         + + = (11) where, 𝛿𝑟𝑠 is the kronecker delta function. the natural frequency of undamped vibration for the beam is given by 𝜔𝑟 = 𝜆𝑟 2√𝑌𝐼 (𝑚𝑙4⁄ ) , where 𝜆𝑟 is the eigenvalue. the computational method for 𝜆𝑟 and the mode shape functions has been presented by man et al. [30]. considering only the first-order mode, based on the author's previous research, the expression for magnetic potential energy, accounting for the eccentricity of the magnet at the beam tip, is given by: 2 4 6 7 m 0 1 1 2 1 3 1 1 1 1 1 ( ) 2 4 6 u k k q k q k q q= − + + + (12) the expressions for the coefficients in the equation can be found in reference [30]. considering only the first-order mode, substituting eq. (9) into eq. (2) and then into the following lagrangian variational equation: d ( ) dt d ( ) dt l l p t q q l l q t vv     − =           − =    (13) where, 𝑃(𝑡) = −2𝜉1𝜔1�̇�1(𝑡) represents the generalized dissipative force of the system, ω1 is the first-order natural frequency of the system, 𝜉1 is the damping ratio of the system, q(t) represents the generalized output charge, and �̇�(𝑡) = −𝑉(𝑡)/𝑅𝐿, utilizing eq. (13), the electromechanical equation of the htpeh can be derived as follows:  2 2 3 5 1 1 1 1 1 1 c 1 1 1 2 1 3 12 g ( ) cosq q q k q k q k q k q v f t      + + + − + + + − = − + (14) p l 0 v c v q r + + = (15) in eqs. (14) and (15), the expressions for the coefficients kc, 𝜒, 𝜗, 𝛤 and cp can be found in the appendix, while f(t) represents the external excitation arising from the interaction between the vehicle wheel and the road surface. by introducing dimensionless parameters 𝑆(𝜏) = 𝑞1(𝑡)/𝑙, 𝜏 = 𝜔1𝑡, �̄�(𝜏) = 𝑉𝐶𝑝/(𝑙𝜗) into eqs. (14) and (15), we obtain: 2 3 5 1 1 c 2 32 (1 ) ( ) coss s k k s k s k s v g f t    + + − + + + + − =  +  (16) 0v v s+ + = (17) where, 𝐾1 = 𝑘1/𝜔1 2 , 𝜔 = �̇�/𝜔1 , 𝐾2 = 𝑘2𝑙 2/𝜔1 2，𝐾3 = 𝑘3𝑙 4/𝜔1 2 , �̄� = 𝜒/𝑙 , �̄� = 𝛤/ (𝑙𝜔1 2), 𝑓(𝑡) = 𝐹(𝑡)/(𝑙𝜔1 2),𝛩 = 𝜗2/(𝐶𝑝𝜔1 2). this study primarily investigates the rotational motion performance of htpeh under various constant speeds in the absence of external excitations. therefore, let �̈� = 0, 𝑓(𝑡) = 0, nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 263 considering the system's own gravity �̄�𝑔 as an external excitation force f, then eq. (16) can be rewritten as: 2 3 5 1 1 c 2 32 (1 ) coss s k k s k s k s v f  + + − + + + − = (18) 3. theoretical analysis by harmonic balance method the solutions of eqs. (18) and (17) may be assumed in the following form: ( ) ( )sin( ) ( )cos( ) ( ) ( )sin( ) ( )cos( ) s a b v c d           = +  = + (19) substituting eq. (19) into eq. (17) and balancing the coefficients of the sin(ωτ) and cos(ωτ) terms, for steady-state motion, setting all time derivatives to zero, yields: ( ) ( ) 2 2 2 2 c b a d a b            = − +   = − −  + (20) substituting eqs. (19) and (20) into eq. (18), and repeating the above calculations while neglecting higher-order harmonic terms, we obtain: 2 2 2 3 2 5 3 2 432 1 c 2 2 2 2 53 2 (1 ) ( ) ( 2 ) 0 4 8 kk a b k k a a ab a a b ab b a             − − + − + + + + + + − + = + + (21) 2 2 2 2 3 4 2 3 532 1 c 2 2 2 2 53 2 (1 ) ( ) ( 2 ) 4 8 kk b a k k b a b b a b a b b a b f             − + + − + + + + + + + + = + + (22) let the displacement amplitude of the system's steady-state motion be 𝑎 = √𝐴2 + 𝐵2, the voltage amplitude is 𝑢 = √𝐶2 + 𝐷2. squaring both sides of eqs. (21) and (22), and adding them together, we obtain: ( ) 2 22 2 2 2 2 4 2 232 1 c 2 2 2 2 53 1 2 4 8 kk a k k a a a f              − + − + + + + + + =  + +  (23) therefore, the steady-state displacement amplitude of the htpeh in rotational motion can be determined by eq. (23), and the expression for the steady-state output voltage amplitude is 2 2 u a    = + (24) 4. dynamic response analysis in the first part of this section, analytical solutions for the htpeh system are obtained using the harmonic balance method. the study investigates the influence of parameters such as the masses of the two ends of the beam, rotational radius, rotational speed, and the 264 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. stiffness ratio of the springs on the steady-state response of the htpeh system during rotational motion. the correctness of the analytical solutions is verified through frequency sweep simulations. in the second part, the system's energy harvesting performance in the time domain is analyzed using the ode45 solver in matlab. the geometric and material parameters for the htpeh are given in table 1. figures 3-5 depict the variations in steady-state displacement and output voltage amplitude of the htpeh compared to the tpeh with different values of the mt as the rotational speed changes when mt = 25 g, mf = 100 g, r = 0.21 m. the solid lines represent the analytical solutions of the htpeh, the scattered points represent the analytical solutions of the tpeh, and the shaded regions represent the numerical solutions obtained by rotational speed sweeping of the htpeh. from fig. 3, it can be observed that both htpeh and tpeh can generate significant inter-well motion during rotational motion influenced by their respective gravity. compared to tpeh, htpeh exhibits a significantly higher peak value of output voltage and inter-well motion range. as seen from figs. 3 to 5, with the increase of the mt, the displacement, output voltage and inter-well motion range of both piezoelectric harvesters increase significantly, and the energy harvesting advantage of htpeh becomes more significant. when mt = 25 g, the peak voltage and inter-well motion range of htpeh increase by 11.85% and 29.26%, respectively, compared to tpeh. when mt = 27.5 g, these two indicators increase to 11.95% and 29.50%, respectively, and when mt = 30 g, they further increase to 12.34% and 30.42%. the larger mt, the more significant the increase in peak voltage and inter-well motion range. moreover, it can be observed that the numerical simulation results of the steady-state displacement and output voltage amplitude of htpeh under different rotational speeds by rotational speed sweep are in good agreement with the analytical solutions obtained by the harmonic balance method. table 1 the geometry and material parameters of htpeh parameters symbol value piezoelectric film length l 80 mm piezoelectric film width b 20 mm piezoelectric film thickness hs 0.2 mm young's modulus of the metal substrate ys 70 gpa density of the metal substrate s 2700 kg/m3 density of the piezoelectric layer p 7750 kg/m3 magnetization intensity ma, mb, mc 1.22×106 a/m magnetic volume va, vb, vc 1.0×10-6 m-3 damping ratio 1 0.01 young's modulus of the piezoelectric layer yp 60.98 gpa piezoelectric strain constant d31 -1.71×10-10 c/n dielectric constant of the metal substrate s 33 -1.33×10-8 f/m transverse distance of the magnet dh 18 mm vertical distance of the magnet dv 9 mm nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 265 fig. 3 a) dependency of the dimensionless displacement and b) voltage amplitude of the system on the rotational speed for different models and mt = 25 g fig. 4 a) dependency of the dimensionless displacement and b) voltage amplitude of the system on the rotational speed for different models and mt = 27.5 g fig. 5 a) dependency of the dimensionless displacement and b) voltage amplitude of the system on the rotational speed for different models and mt = 30 g 266 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. figure 6 presents the curves of steady-state displacement and voltage amplitude with rotational speed for different values of the spring stiffness ratio �̄� = 𝑘𝑓/𝑘𝑟, when mt = 25 g and r = 0.21 m. it can be observed from fig. 6 that the output voltage-rotational speed response curves for the three kf/kr ratios are essentially overlapping, while the steady-state output displacement, peak voltage and rotational speed range of inter-well motion all increase as kf/kr decreases. fig. 6 a) dependency of the dimensionless displacement and b) voltage amplitude response curves on the rotational speed for different spring stiffness ratios to investigate the time-domain dynamic performance of htpeh and tpeh at different wheel hub rotational speeds, fig. 7 shows the phase diagram and voltage time-history diagram for both piezoelectric harvesters at 300 rpm when mt = 25 g, mf = 100 g, r = 0.21 m. it can be observed from fig. 7 that htpeh quickly transitions from intra-well motion to significant inter-well motion and generates a high output voltage after a short period of intra-well motion. in contrast, tpeh cannot overcome the potential barrier, only exhibiting small oscillations within the intra-wel, resulting in a small output voltage. this indicates that at the same rotational speed, htpeh is more likely to escape the potential well and enter significant inter-well motion. as the rotational speed increases to 350 rpm, it can be seen from fig. 8 that htpeh instantly enters a high-energy orbit to perform substantial inter-well motion, generating a large output voltage. on the other hand, tpeh, after a period of chaotic motion, eventually overcomes the potential barrier and escapes the well, entering significant inter-well motion with a longer time duration and a lower output voltage than htpeh. in summary, at lower rotational speeds, htpeh is more likely to break free from the potential well, generating higher output voltages and capturing more energy during rotational motion, demonstrating the advantages of htpeh in energy harvesting performance. nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 267 fig. 7 (a) three-dimensional phase diagram, (b) two-dimensional phase diagram, and (c) voltage time-history diagram for htpeh and tpeh with the rotational speed of 300 rpm figure 9 presents the curves of the steady-state displacement amplitude and steady-state output voltage amplitude of htpeh at different rotational radii with varying speeds when mt = 25 g, mf = 100 g. it can be observed that when the rotational speed is below 100 rpm, the change in rotational radius has a minimal impact on the steady-state displacement output voltage amplitude of htpeh. however, when the speed exceeds 100 rpm, both the steady-state displacement output voltage amplitude decrease with an increase in the rotational radius. moreover, as the rotational radius becomes larger, the decrease in amplitude becomes more pronounced. taking the peak value of the inter-well motion displacement under different rotational radii as an example, when the rotational radius is between 0.18 m and 0.33 m, the peak voltage of htpeh shows a slow decline with an increase in rotational radius, and the corresponding interwell motion range slowly increases. when the rotational radius is between 0.33 m and 0.37 m, the peak voltage of htpeh sharply decreases with an increase in the rotational radius, and the corresponding inter-well motion range sharply increases. when the rotational radius is greater than 0.33 m, both the displacement and voltage peak values are observed to be at a very low level. this is because at this rotational radius, the htpeh is restricted to intra-well motion. 268 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. fig. 8 (a) three-dimensional phase diagram, (b) two-dimensional phase diagram, and (c) voltage time-history diagram for htpeh and tpeh with the rotational speed of 350 rpm fig. 9 dependency of the dimensionless a) displacement amplitude and (b) the voltage amplitude on rotational speed for htpeh and different rotational radii nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 269 although increasing the rotational radius leads to a decrease in the peak voltage of htpeh, it simultaneously increases the range of inter-well motion. consequently, the system can capture energy from rotational motion over a broader speed range. therefore, an appropriate radius should be set based on practical considerations such as the installation position, the dimensions and the specific application requirements of htpeh. figure 10 depicts the phase portraits and voltage-time history diagram for htpeh at a rotational speed of 300 rpm, with wheel hub radii corresponding to r = 0.18 m and r = 0.21 m. it can be observed that when r = 0.18 m, the system undergoes a period of chaotic motion, subsequently overcoming the potential barrier to perform significant interwell motion and generating a higher output voltage. at r = 0.21 m, the system cannot overcome the potential barrier, remaining in small oscillations within one well and unable to enter inter-well motion, resulting in only a small output voltage. as the rotational increases to 350 rpm, as shown in fig. 11, htpeh under both r = 0.18 m and r = 0.21 m can generate inter-well motion. however, the system with the smaller rotational radius requires less time to enter inter-well motion. when the system with a rotational radius of fig. 10 (a) three-dimensional phase diagram, (b) two-dimensional phase diagram, and (c) voltage-time history diagram for htpeh with different values of wheel hub radius at a rotational speed of 300 rpm 270 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. r = 0.21 m enters stable inter-well motion, its output voltage is significantly lower than that of the system with the rotational radius of r = 0.18 m. figures 9 and 10 illustrate that reducing the rotational radius or increasing the rotational speed can make the htpeh more easily enter substantial inter-well motion, resulting in stable output voltage. fig. 11 (a) three-dimensional phase diagram, (b) two-dimensional phase diagram, and (c) voltage-time history diagram for htpeh with different values of wheel hub radius at a rotational speed of 350 rpm figures 12 and 13 show the output power amplitude curves of the system at various rotational speeds with different load resistances for rotational radii of r = 0.21 m and r = 0.36 m, respectively. it is evident that for htpeh with different rotational radii, substantial inter-well motion can only be generated within a certain rotational speed range. beyond a certain rotational speed, the system transitions into intra-well motion, resulting in a very small output power, and the rotational speed range conducive to inter-well motion generation expands with an increase in the wheel hub radius. nonlinear dynamical characteristics of hybrid tri-stable piezoelectric energy harvester based... 271 fig. 12 a) the variation surface and b) top view of the impact of resistance on the system's output power at the wheel hub radius of 0.21 m fig. 13 a) the variation surface and b) top view of the impact of resistance on the system's output power at the wheel hub radius of 0.36 m 5. conclusion this paper introduces a hybrid tri-stable piezoelectric energy harvester based on rotational motion. the dynamic model of the system is established using lagrangian equations. the influence of various factors such as the mass of the piezoelectric beam tip magnet, wheel hub radius, rotational speed, and the stiffness ratio of the dynamic amplifier on the system's steady-state dynamic response is analyzed. a comparison with traditional tri-stable piezoelectric energy harvesters is conducted. the theoretical analysis is validated through rotational sweep numerical simulations, and the system's time-domain dynamic performance is analyzed using the matlab software. the main conclusions from the theoretical analysis and numerical simulations are as follows: ▪ compared to traditional tri-stable piezoelectric energy harvesters without a dynamic amplifier, the hybrid tri-stable piezoelectric energy harvester exhibits significantly improved output voltage and inter-well motion range during rotational motion. ▪ at lower speeds, changes in wheel hub radius minimally affect the htpeh system's output voltage. while the range of inter-well motion expands with an increase in wheel hub radius, but there is a concurrent decrease in peak output voltage. 272 d.-w. man, y. zhang, l.-p. tang, q.-h. xu, d. chen, et al. ▪ increasing the mass of the tip magnet or reducing the stiffness ratio of the vertical spring to the rotational spring can simultaneously increase the htpeh system's output voltage amplitude and range of inter-well motion. ▪ at higher rotational speeds or smaller wheel hub radii, the htpeh is more likely to enter large inter-well motion, resulting in higher output voltage. acknowledgment: this research was funded by the anhui provincial university provincial 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alexandria engineering journal, 76, pp. 153-165. appendix 2 2 2 2 2 c 0 1 1 1 0 0 0 2 2 2 2 f 1 1 t 1 1 0 0 1 ( ) ( ) d ( ) ( ) d ( ) d 2 ( ) d (0) ( ) ( ) d ( ) l l l l l k m r l x x x l x x x x x m r x x m l r x x l           = − + − −        + − + + −             0 1 f 1 t 1 f f 1 t t 1 0 ( ) ( )d (0) ( ) ( ) (0) ( ) ( ) l m x r x x m r m l r l m e m e l r l      = + + + + − + + p 2 p 31 1 0 ( ) d 2 lt y bd h x x    = +     , 0 1 t 1 f 1 0 ( )d ( ) (0) l m x x m l m   = + + , s 33 p p2 bl c t  = plane thermoelastic waves in infinite half-space caused facta universitatis series:mechanical engineering vol. 14, no 2, 2016, pp. 169 177 original scientific paper brownian heat transfer enhancement in the turbulent regime udc 536.2:532.5 suresh chandrasekhar, vaarin majumdar sharma department of mechanical-mechatronics engineering, lnm institute of information technology, india abstract. the paper presents convection heat transfer of a turbulent flow al2o3/water nanofluid in a circular duct. the duct is a under constant and uniform heat flux. the paper computationally investigates the system’s thermal behavior in a wide range of reynolds number and also volume concentration up to 6%. to obtain the nanofluid thermophysical properties, the hamilton-crosser model along with the brownian motion effect are utilized. then the thermal performance of the system with the nanofluid is compared to the conventional systems which use water as the working fluid. the results indicate that the use of nanofluid of 6% improves the heat transfer rate up to 36.8% with respect to pure water. therefore, using the al2o3/water nanofluid instead of water can be a great choice when better heat transfer is needed. key words: nanofluid, forced convection, heat transfer enhancement, turbulence flow 1. introduction in the past years, many different techniques were utilized to improve the heat transfer rate in order to get higher thermal efficiencies. thus adding solid particles to a base fluid was suggested by maxwell [1, 2]. however, large particles cause many serious problems. choi [3] suggested the use of nano-sized particles. after that, the researcher tried to understand the effects of those nanoparticles on heat transfer efficiency of the system. in recent years, some numerical studies have been done on the forced or free convection of laminar flows of the nanofluids [4-7]. the results reveal that using nanofluid can enhance heat transfer performance of nanofluid compared to pure water. also, some studies have focused on the turbulent flow of nanofluids. heat transfer investigation of nanofluids flow in the turbulent regimes was numerically received april 5, 2016 / accepted june 15, 2016 corresponding author: suresh chandrasekhar department of mechanical-mechatronics engineering, the lnm institute of information technology, india e-mail:sur.chandrasekhar@gmail.com 170 s. chandrasekhar, v. m. sharma performed. manca et al. [8] numerically investigated the turbulent forced convection with nanofluids in 2d channel. they have used a single-phase approach and observed that the heat transfer enhancement increases with the particle volume concentration. besides, roy et al. [9] compared the performance of different nanofluids inside a typical radial flow cooling device. they reported that the heat transfer increases with volume concentration and reynolds number. bianco et al. [10] also numerically studied wateral2o3 nanofluids turbulent convection heat transfer inside a circular tube. they reported that the heat transfer enhancement increases with the particle volume concentration and reynolds number; their results were in good agreement with those of other studies. moreover, the heat transfer enhancement of tio2 and water in a circular tube were investigated by demir et al. [11]. a single-phase model having two-dimensional equations was employed for this simulation and they reported a noticeable jump in the heat transfer rate. in this study, a turbulent flow of al2o3/water nanofluid passes through a tube which is under constant heat flux. to capture the effects of the temperature on the thermophysical properties of the nanofluid, the properties are considered temperature dependent. also, the effect of the brownian motion of nanoparticles is taken into account. to investigate the thermal performance of the system by using this new working fluid, the nusselt number and the convective heat transfer coefficient are presented. 2. mathematical modeling the geometry consists of a tube with diameter (d) of 0.01m and length of l=1m. the al2o3/water nanofluid passes through this tube and a fixed heat flux is subjected to the tube wall. volume concentration of nanoparticles () equals 0% (base fluid), 1%, 2%, 4% and 6%. 2.1. fluid properties in this study, the thermophysical properties of nanofluid are calculated based on temperature dependent models in order to capture the effects of temperature on the properties. the effective properties of the al2o3/water nanofluid are defined as follows: (1 ) nf p p p bf        (1) where p is the volume concentration of nanoparticles, while p, bf, nf are the density of particles, the density of the base fluid that is water and the density of nanofluid, respectively. eq. (1) was originally introduced in [12] for determining the density and then widely employed in literature [4, 6, 10, 13]. for defining the heat capacity, the following equation is used as in many other studies such as [4, 6, 5, 8, 10]: (1 )p p p p bf bf nf nf c c c         (2) in order to get the nanofluid thermal conductivity, many referential studies have used the hamilton and crosser model [14]: ( 1) ( 1) ( ) ( 1) ( ) nf p bf p bf p bf p bf p bf p k k n k n k k k k n k k k             (3) brownian heat transfer enhancement in the turbulent regime 171 in this model, n is an empirical shape factor which accounts for the effect of shape of particles; it can vary from 0.5 to 6.0. for spherical nanoparticles n equals 3. so this case of the hamilton and crosser model (n=3) is: 2 2 ( ) 2 ( ) nf p bf p bf p bf p bf p bf p k k k k k k k k k k          (4) however, this equation only takes into account the nanoparticles shape and volume concentration; it cannot take into account the brownian motion of nanoparticles inside the fluid. however, in this study, we consider the effect of the brownian motion of nanoparticles. for this purpose, we need to modify the hamilton-crosser model by using the formals proposed by koo and kleinstreuer [15]: 45 10 b brownian bf bf p p t k c d      (5) where: 0.08410.0017(100 ) , 1%    (6) besides, the viscosity of nanofluid is calculated based on the einstein model: (1 2.5 )nf bf    (7) 2.2. governing equations to capture the turbulence effect, standard - model is adapted. this - model has two additional equations for turbulent kinetic energy  and the rate of dissipation  in the following form: ( ) ( )t k k v p                   (8) 2 1 2( ) ( )t kv c p c                         (9) where the eddy viscosity is calculated by: 2 t c      (10) 1 21.44, 1.92, 0.09, 1, 1.3kc c c         (11) besides, the governing equations for solving the problem are continuity, momentum and energy equations as follows: .( ) 0v  (12) ' '.( ) .( )vv p v v v        (13) ' '.( ) .( )p pc tv k t c t v      (14) 172 s. chandrasekhar, v. m. sharma 2.3. numerical method in order to discretize the governing equations, a control volume approach is employed by using ansys fluent. also, a collocated grid system is used so that the grid near the wall is smaller in order to capture turbulence effects near the wall. several unstructured grids have been created to assure consistency and accuracy of the numerical results. a grid independency test is done by comparison of the results obtained from employing different grids in terms of the local nusselt number and the relevant errors showed that the 700150 non-uniform grid is good enough to ensure the independency of numerical results from the grid system. this grid consists of 150 nodes in radial (r) and 700 nodes in longitudinal direction (x). the boundary conditions consist of an uniform inlet temperature, tin=293k also a uniform entrance velocity uin which can be calculated from reynolds number based on the following equation, and the non-slip condition is set for the velocities at the walls. re inu d    (15) further, the constant intensity turbulence of 1% is imposed. also, a uniform heat flux is applied at the pipe wall. moreover, both turbulent kinetic energy and dissipation of turbulent kinetic energy are equal to zero. at the channel exit, the fully developed assumption is employed which means that all the axial derivatives are zero. 3. results and discussion in order to validate the numerical method, the current numerical nusselt number of the pure water is compared to some models represented in the following equations presented in [13] versus reynolds numbers in fig. 1. this figure shows that there is a good agreement between the numerical results obtained from the current study and other models and studies. fig.1 comparison of the present numerical results with other works brownian heat transfer enhancement in the turbulent regime 173 gnielinski: 3 1/ 2 2 / 3 (re 10 ) pr 2 2 1 12.7 (pr 1) 2 f f nu f                   (16) dittus-boelter: 0.8 0.40.024re pr heatingnu  (17) petukhov: 2 / 3 re 8 1.07 12.7(pr 1) 8 f pr nu f          (18) fig. 2 shows the local heat transfer coefficient of water and nanofluids 1, 2, 4 and 6% for constant reynolds number of 20000. this figure shows that by replacing the al2o3/water nanofluid by pure water, the heat transfer coefficient increases. also, by increasing the volume concentration of the dispersed nanoparticles in the base fluid, the heat transfer convective coefficient will increase as well. this enhancement in the heat transfer coefficient is because of enhancing the properties of the nanofluid due to adding some solid nano particles in the base fluid. fig. 2 local convective heat transfer coefficient of all working fluids as mentioned before, some studies have already explored the heat transfer performance of al2o3/water nanofluid. therefore, in fig. 3, comparison was made between the present study and the previous studies for the working fluid of al2o3/water nanofluid 1.0%. this comparison shows that the maiga model defined by eq. (19) overestimates the present numerical study results. 0.71 0.350.085re prnu (19) however, our results have a better agreement with the model developed by pak and cho in the following equation: 0.8 0.50.021re prnu  (20) the reason for this difference is because maiga et al. [16] have used a correlation on the experimental data by wang et al. [17] to define the thermophysical properties of 174 s. chandrasekhar, v. m. sharma al2o3-water nanofluid. but those properties are noticeably different from those that pak and cho [12] have used in their work; the reason for this is in different nanoparticles sizes in these two studies. besides, our results agree well with those of bianco et al [10] and takabi and shokouhmand [13] since both the studies use a numerical simulation which is just like the method we use in this work. fig. 3 comparing nu of the present study with other works fig. 4 shows the nusselt number of pure water and nanofluids 1 and 2% as a function of reynolds number. this figure shows that when we use nanofluid instead of pure water as the working fluid, we can get a higher nusselt number. also, when we use higher volume concentrations of nanoparticles in the nanofluid, we will obtain a higher nusselt number. thus we can get higher thermal performance. therefore, from this figure, it can be concluded that we can use nanofluid in a wide range of industrial applications, especially when we need higher thermal performance. for instance, nanofluid is a good choice to use in underwater pipelines [18], or in biomechanics [19], or where we need higher thermal rejection capacity [20,21] since we can improve thermophysical properties such as higher capacity by adding nano-sized solid particles. to be more specific, once nanoparticles are added to the base fluid, the mixture has a higher thermal capacity so that it can absorb shocks and external vibrations. in other words, the nanoparticles in a flow hit the wall and absorb some energy from the wall. so they can decrease extra energy resulting from vibration or thermal shocks on the wall. besides, nanoparticles will fig. 4 nusselt number of pure water, nanofluid 1% and 2% as a function of re brownian heat transfer enhancement in the turbulent regime 175 cause better thermal properties of the mixture. so, as can be seen in this figure, the slope for nanofluid 2% is steeper than for nanofluid 1% and water. therefore, it means the nanofluid 2% has a better thermal performance than water or nanofluid 1%. in order to understand the effect of using nanofluid instead of water quantitatively, we use the average convective heat transfer coefficient ratio as the following equation which has already been used by [13] in a similar work. nf r bf h h h  (21) where bf h and nf h are average convective heat transfer coefficients of the base fluid which is pure water and the nanofluid which is al2o3/water nanofluid. table 1 shows the average convective heat transfer coefficient ratio of al2o3/water nanofluid with different volume concentrations. as can be seen, all the values are greater than 1. it means that using nanofluid will increase the convective heat transfer coefficient that is good for us. also, the concentration for al2o3/water nanofluid 1% is 1.091, while this value for nanofluid of 6% is 1.368. therefore, it shows that by adding five more percent of the volume concentration, 25.4% enhancement in the average convective heat transfer coefficient is achieved. table 1 the average convective heat transfer coefficient ratio of nanofluid nanofluid 1.0%  2.0%  4%  6.0%  r h 1.091 1.166 1.255 1.368 fig. 5 shows wall temperature and bulk temperature of water, nanofluid 1 and 2%. as can be seen, when we use nanofluid, the wall temperature decreases which is favorable. this phenomenon can be used in the laser eye surgery where the laser irradiation is so high that it could damage the cornea tissues. but the use of nanofluid could decrease the temperature on the cornea surface [22]. fig. 5 wall temperature and bulk temperature profile of water and nanofluid and 1.0% 176 s. chandrasekhar, v. m. sharma this behavior is also reported by takabi and shokouhmand [13] and bianco et al [10]. the decrease in the nanofluid wall temperature is caused by the fact that the nanoparticles in the flow hit the wall and absorb the thermal energy of the wall, as discussed earlier. therefore, the wall temperature of nanofluid compared to water decreases. also, the nanoparticles can enhance the fluid’s thermophysical properties. so nanofluid has a higher thermal capacity to absorb thermal energy. furthermore, we can see that after the entrance region of the pipe, the wall temperature and the bulk temperature are getting parallel. it means the flow is fully developed in terms of heat transfer in that region which is in agreement with [23]. 4. conclusions in many industrial applications, we need higher thermal performance for different equipments. thus we can use a better working fluid such as nanofluids. nanofluids are mixtures of some nanoparticles into a base fluid. in this study, the effect of nanofluid with different volume concentration in a circular tube under constant heat flux in a wide range of turbulent flow with the reynolds number between 10000 to 100000 is investigated. that is why a computational 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paper influence of tack welds distribution and welding sequence on the angular distortion of tig welded joint jacek tomków1, konrad sobota2, sławomir krajewski3 1gdańsk university of technology 2damen marine components 2west pomeranian university of technology szczecin abstract. in this paper the influence of tack welds distribution and welding sequence on angular distortion of the tungsten inert gas (tig) welded joint was tested. additionally, the effect of welding current on angular distortion was assessed. for research x2crtinb18 (aisi 441) stainless steel (2.5 mm thick) was chosen. during research specimens were prepared with different distributions of tack welds. then they were welded by different welding sequences with the use of different welding current values. after welding the angular distortion of each specimen was measured by using the coordinate measuring machine. in the next step specimens were cut. cross-sections were polished and the metallographic macroscopic testing was conducted to check the geometry of performed welds. performed experiments allowed determining the optimal tack weld sequence and welding parameters for welding thin stainless steel sheets. key words: angular distortion, tack welds, tig welding, macroscopic testing, weld geometry, welding current 1. introduction tungsten inert gas (tig) welding, named also the gas tungsten arc welding (gta), is one of the widely used joining processes. it allows for the use of lower heat input values in comparison to other welding methods, which can be helpful in joining thin materials. the process is characterized by high accuracy, which leads to a high quality of welded joints. the method is used for welding alloys of various metals, e.g. nickel [1], aluminum [2], magnesium [3], copper [4], and titanium [5]. it is also widely used in performing dissimilar joints [6-9]. the tig is also commonly used for joining stainless steels [10,11]. in the case of steel welding, some typical imperfections could be observed. mahn et al. [12] stated that received may 20, 2020 / accepted october 03, 2020 corresponding author: jacek tomków gdańsk university of technology, gabriela narutowicza 11/12 street, 80-233 gdańsk, poland e-mail: jacek,tomkow@pg.edu.pl 612 j. tomków, k. sobota, s. krajewski tig welded joints can be characterized by the presence of undercut and burn-through. fei et al. [13] in their experiment proved that a high welding heat input provides for undercuts in the joint. it has been demonstrated that the welding parameters should be carefully controlled in order to avoid the formation of defects. the other problem during tig joining is a lack of penetration. the research study of touilem et al. [14] was focused on elimination of this problem. they observed that weld depth can be increased by welding with the flux activated tungsten inert gas (atig) method. wang et al. [15] proposed novel magnetic arc oscillation as a method allowing us to minimize the number of welding imperfections, and to increase penetration during the tig welding. welding deformations are one of the most important factors affecting the proper production of welding structures. regardless of the size of the structure, the materials used and the welding method, deformations could always occur [16]. their causes are the stresses resulting from a welding thermal cycle and a weld shrinkage effect. the thermal cycle depends e.g. on the base metal thickness. metal sheets with lower thickness are characterized by higher susceptibility to distortion as a result of welding process. deng and murakawa [17] presented the thesis that there is almost no temperature gradient through thickness in the thin plate butt joint during welding, which resulted in the formation of distortions in a wide range of welding parameters. shuaibu ahmad et al. [18] proved that after the tig process, welding deformations can be observed. it was also stated that a big deformation induces cracking of the joint. the welding heat input is an important factor in forming weld deformations. it was stated by huang et al. [19], that the relationship between heat input and maximum deflection is not linear. there is a critical heat input beyond which the out-of-plane distortion starts to grow very quickly. qu and xia [20] carried out a research study on welding numerical simulation. they stated that welding deformations depend on the structures in heat-affected zone (haz), which is result of welding heat input. by choosing optimal welding parameters, it is possible to reduce the welding deformations, what was proved by sudhakaran et al. [21]. one type of the welding deformation is angular distortion. the angular distortion consists of the rotation of the structure around the welding line [22]. the effect of the angular distortion was analyzed by huang [23]. the experiment showed that the value of this deformation depends on the shielding gas composition. it was also stated that variations in angular distortion are associated with weld depth. therefore, full penetration welds resulted in lower angular distortion. tseng and chou [24] welded austenitic stainless steel with tig welding process, using a pulsed welding arc. they proved that the angular distortion decreased for materials characterized by higher thermal conductivity. for experiment ferritic stainless x2crtinb18 (aisi 441) steel was chosen. this material is widely used in automotive for parts in the exhaust system, e.g. inlet-/outlet tubes of catalysts and filters [25]. novel trends showed that high-frequency welding could be used as a joining process for these applications [26,27]. however, the tig process is most often used. in comparison to austenitic stainless steel, which is widely used in medicine [28], aisi 441 material is characterized by lower plastic strain hardening and the reduced forming limit [29]. x2crtinb18 steel presents lower thermal expansion, good oxidation resistance and higher resistance to thermal fatigue [30]. the aim of the research was to study influence of tack welds distribution on the angular distortion of tig welded joint. the experiments were performed with the use of different welding sequences. additionally, the influence of different welding current values was influence of tack welds distribution and welding sequence on the angular distortion of tig... 613 tested. the experiments were prepared to check the possibility of reducing angular distortion after tig welding of thin ferritic stainless sheets. 2. materials and methods 2.1. materials as a material for experiment, x2crtinb18 (aisi 441) steel plates with dimensions of 100×50×2.5 mm were chosen. the structure of base metal (bm) consists of fine-grained ferrite with visible grain boundaries. at the grain boundaries also titanium carbonitride precipitates and very fine niobium carbide separation could be found [31]. the microstructure of used steel is presented in fig. 1. fig. 1 the microscopic view of x2crtinb18 (aisi 441) steel the chemical composition of used steel has been tested by emission spectroscopy with a spark excitation method with the use of the glow discharge spectrometer leco gds 500a (glow discharge atomic emission spectrometry). to determine the chemical composition of steel, reference standards for stainless steels were used. the content of elements is presented by mass [%]. as a filler material en iso 14343-a:w 19 12 3 l si rod (1 mm dimension) was selected as a suggested material (by manufacturer) for welding x2crtinb18 stainless steel constructions, which will work in chlorides and chlorinated solutions environment. the chemical compositions of used material are listed in table 1. table 1 chemical composition of used materials, mass % material c n si mn cr nb s p ti other x2crtinb18 (aisi 441) steel 0.023 0.012 0.79 0.95 17.57 0.044 0.002 0.033 0.24 en iso 14343-a:w 19 12 3 l si rod in accordance to manufacturer data* max.0.03 max.0.01 0.80 1.80 19.00 ni – 12.0 mo – 2.8 614 j. tomków, k. sobota, s. krajewski 2.2. methods welding process was carried out by using manually tig welding process. for welding wl15 electrode of 1.6 mm diameter was used. the angle of electrode tip was 30°, which provides good penetration during welding of thin sheets. as a shield gas 99.95% argon was used, and the gas flow rate was the 8 l/mm. the post flow value was 10 s. these parameters were chosen in accordance to preliminary investigations. for experiments 21 specimens were performed. they were welded in seventh groups (three specimens in each) with different distribution of tack welds and different welding sequences. the sequence of welding has a significant role in the properties of welded joints, as it affects the residual stresses and welding deformations [32,33]. the welding plan is presented in table 2. the schematic view of welding procedure is shown in fig. 2. table 2 specimens welding plan group a group b group c group d five tack welds; progressive welding five tack welds; step welding (sequence 1) five tack welds; step welding (sequence 2) four tack welds; step-back welding group e group f group g four tack welds; progressive welding two tack welds; progressive welding four tack welds laid outside welding grove; progressive welding fig. 2 the schematic view of welding schedule; a) group a, b) group b, c) group c, d) group d, e) group e, f) group f, g) group g influence of tack welds distribution and welding sequence on the angular distortion of tig... 615 in each group three specimens have been welded with current (i) values 80 a, 120 a and 160 a. voltage (u), and welding speed (vsp) were chosen to provide similar heat input (ql) values in the range 0.20 – 0.25 kj/mm. the welding speed was calculated after the process, by measuring the welding time and the length of the joint. before welding sheets were joined by tack welds. specimens were named by letters a-g which denote a relevant group. the welding parameters of each specimen are listed in table 3. table 3 welding parameters for prepared specimens specimen no i [a] u [v] vsp [mm/s] ql [kj/mm] specimen no i [a] u [v] vsp [mm/s] ql [kj/m] a1 80 10.8 1.62 0.24 e1 80 10.3 2.27 0.22 a2 120 11.2 3.21 0.25 e2 120 11.5 4.05 0.20 a3 160 12.6 4.52 0.21 e3 160 12.4 5.56 0.21 b1 80 10.5 2.19 0.25 f1 80 10.1 2.27 0.21 b2 120 11.7 3.77 0.23 f2 120 11.5 3.85 0.22 b3 160 12.4 5.59 0.25 f3 160 12.5 5.56 0.22 c1 80 10.4 2.34 0.26 g1 80 10.1 2.42 0.20 c2 120 11.2 4.39 0.23 g2 120 12.1 4.31 0.20 c3 160 12.5 5.48 0.25 g3 160 12.3 5.53 0.21 d1 80 10.3 1.90 0.26 d2 120 11.5 3.13 0.26 d3 160 12.5 4.55 0.26 after welding the angular distortion for each specimen was measured by the use of baty venture coordinate measuring machine. test stand was equipped with measuring table, which allows eliminating measurement interference related to drive and environment vibrations. measurements were taken with the use of 2 mm diameter measurement probe. specimens with the lowest values of angular distortion were selected for macroscopic testing. macroscopic tests were performed in accordance with the en iso 17639:2013 standard. in each specimen two cross-sections were cut, ground, polished and etched by vilella’s etchant. 3. results and discussion 3.1. angular distortion measurements before measurements the measuring table was modified to obtain proper results. the modification consisted in placing stainless steel master plate with threaded holes and specially made blocks, thus enabling quick mounting and facilitating the element's base on the measuring table. each specimen was divided into sections located every 10 mm (8 sections in specimen). in each section three measurements were done. each of these three measurement points were determined by the deflection line. each of the lines was referred to the second sheet plate, which allows accessing the angular distortion of welded joint. the view of measurements results from the baty venture coordinate measuring machine is presented in fig. 3. all of the results are shown in table 4. 616 j. tomków, k. sobota, s. krajewski fig. 3 the view of measurements results by the baty venture coordinate measuring machine table 4 results of angular distortion measurements specimen no avarage angular distortion [°] avarage angular distortion in group [°] specimen no avarage angular distortion [°] avarage angular distortion in group [°] a1 4.12 5.39 e1 2.65 2.52 a2 5.63 e2 2.78 a3 6.42 e3 2.13 b1 4.69 5.23 f1 1.96 1.38 b2 6.48 f2 1.14 b3 4.52 f3 1.03 c1 3.58 3.99 g1 1.61 1.78 c2 3.60 g2 2.23 c3 4.79 g3 1.50 d1 2.48 d2 1.75 1.86 d3 1.34 it was stated in the literature that different welding sequences resulted in different distortion values [34]. this statement was confirmed in the presented study. chen and soares [35] performed experiment of thin plates t-joint welding. they stated that short welding passes in one side of the stiffener provide less deformation than a long pass. in this study it is proved that lower angular distortion is an effect of continuous welding with a longer pass (groups f and g). this may be due to the presence of a higher number of tack welds. the highest values of angular distortion were observed for specimens with five tack welds (groups a-c). in the tack welds areas the high stress concentration could be observed. the stresses generate more welding distortion. similar results were found by influence of tack welds distribution and welding sequence on the angular distortion of tig... 617 fu et al. [36]. they stated that progressive welding must be considered firstly in order to avoid the stress concentration at the weld start/end location. the groups a-c were eliminated from further testing. okano et al. [37] put forward the thesis that weld distortion is not always precisely evaluated by the weld heat input. it was proved that other factors affecting the weld heat input e.g. welding current should be considered separately. experiments presented in this paper focused on the angular distortion of tig filet welded joints. it was proved that, when the welding current increased, the angular distortion decreased. different results were observed by narwadkar and bhosle [38] for mig welding. it is found that angular distortion is bigger when the current increased but this statement was true for butt-joint. the angular distortion values referred to values of welding current for specimens qualified for further experiments (groups d-g) are shown in fig. 4. fig. 4 the angular distortion values referred to values of welding current for specimens qualified to further experiments 3.2. macroscopic testing the aim of metallographic macroscopic testing was to check the depth of penetration in each qualified specimen (groups d-h) and measure weld geometry. the methodology of weld geometry measurements is presented in fig. 5. the welding parameters were chosen to provide similar values of heat input (0.20 – 0.25 kj/mm). preliminary investigations have shown that higher values of heat input affect very high values of angular distortion or provided for melting bm through whole thickness of steel sheet. due to this fact, increasing the welding current affects increasing the welding speed. fig. 5 the methodology of weld geometry measurements 618 j. tomków, k. sobota, s. krajewski based on the obtained results the effects of increasing weld width, decreasing penetration in comparison to increasing welding current were observed. varbai et al. [39] found that a higher welding current provides for increase of the width of weld, which was confirmed in this paper. the effect of welding current on the geometry of performed weld is shown in fig. 6. fig. 6 effect of welding current on the weld geometry – specimens from group e additionally, the lack of penetration was found in the specimens welded with the current equal to 160 a (fig. 6b and fig. 6c), which were welded with a smaller welding speed. one way of increasing the depth of weld is by decreasing the welding speed. however, the welding with smaller welding speed did not allow for forming a proper welded joint. varbai et al. [40] found that the thermal cycle has a significant role in formation structures in haz, which may cause deformations. due to this statement, a lower welding speed may decrease the value of angular distortion. however, during preliminary investigations it was observed that lower welding speed also caused high numbers of undercuts. the results of macroscopic tests are presented in fig. 7. the results of geometry measurements are listed in table 5. fig. 7 results of macroscopic tests; a) specimen d2, b) specimen e3, c) specimen g3, lack of fusion mark by red ellipse influence of tack welds distribution and welding sequence on the angular distortion of tig... 619 table 5 results of geometry measurements specimen no width [mm] a [mm] z [mm] penetration d1 3.49 2.33 1.62 + d2    + d3    − e1    + e2    + e3    − f1    + f2    + f3    − g1    + g2    + g3 4.82 3.57 1.78 5. conclusions in the paper the effects of tack weld distribution and welding sequence on the angular distortion were investigated. additionally, the influence of welding current was assessed. performed experiments showed that all of investigated factors have a significant impact on the angular distortion of tig welded fillet joints. it was proved that an increasing number of tack welds affect increasing angular distortion. also, the step welding technique provides for the increase of the mentioned type of distortion. additionally, it was found that welding with a higher value of welding current leads to the decrease of the angular distortion. however, it also caused lack of penetration. the main conclusions resulting from experiments are: 1. the number of tack welds has a significant impact on the angular distortion of x2crtinb18 (aisi 441) steel filet welded joints. it was proved that an increasing number of tack welds provides for higher values of angular distortion. the lowest values of distortion (1.03° and 1.14°) were observed for specimens from group g, in which two tack welds were laid before test welding. 2. the angular distortion of welded joints made of stainless steel can be changed with different welding techniques. during welding of thick tig welded joints the progressive technique provides for lower values of angular distortion, which was observed for specimens from groups f and g. 3. increasing welding current from 80 to 160 a caused decreasing angular distortion of the performed joints. this effect was observed for each of the tested group. for experiments, specimens welded with different welding current (80 a, 120 a and 160 a) were performed. other parameters were chosen to provide for heat input values in the range 0.20 – 0.25 kj/mm, which allow minimizing the influence of the effect of the heat input on the angular distortion. however, increasing of the welding current caused an increasing welding speed, which resulted in the lack of penetration in specimens welded with 160 a. 4. as referenced parameters for x2crtinb18 (aisi 441) stainless steel filet welded joints, the parameters shown in this paper should be used. it allows avoiding 620 j. tomków, k. sobota, s. krajewski occurring welding imperfections, and minimizing the angular distortion of the performed joints. for welding progressive technique should be used. also, if it is possible, the tack welds should be laid outside the welding groove. if it is not possible, the number of tack welds should be reduced to two. references 1. park, j.h., cheeou, m., cho, s.m., 2020, analysis and characterization of the weld pood and bead geometry of inconel 625 super-tig welds, metals, 10(3), 365. 2. cai, x., lin, s., wang, x., yang, c., fan, c., 2019, characteristics of periodic ultrasonic assisted tig welding for 2219 aluminum alloys, materials, 12(24), 4081. 3. song, g., diao, z., lv, x., liu, l., 2018, tig and laser-tig hybrid filler wire welding of casting and wrought dissimilar 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journal of pressure vessels and piping,176, 103952. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. i iv © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd a special issue in honor of prof. valentin l. popov valentin leonidovich popov is one the leading figures and visionaries in today’s tribology research. it is with great pleasure and honor that we present to the reader this special issue dedicated to his 60th anniversary. to understand mechanics nature, who gives so many useful lectures? who finds the time for all of us, for colleagues, family and science? who shows in friction super class, and wear essence recognizes? he governs friendly like the king the only popov valentin! – ruslan balokhonov – herr popov hat als mein nachfolger in berlin den von mir begonnenen aufbau der reibungsphysik sehr erfolgreich weitergeführt und weltweit sichtbar gemacht. ich schätze unseren wissenschaftlichen austausch sehr und freue mich auf viele treffen, diskussionen und gemeinsame projekte in der zukunft. herzlichen glückwunsch zum geburtstag! – georg-peter ostermeyer – valentin is a person of many accomplishments. being brilliant scientist, well-educated and intellectual, he seems to be deeply engrossed in his research tribology, friction, adhesion and the like. on the other hand, we are happy to know him as a man who is ready to help, as a guy with a keen sense of humor, as a true friend… – varvara romanova – für sie stehen forschung und lehre stets im vordergrund. alles gute zum 60. ! die kolleg*innen aus dem ifm – utz von wagner – russian tribology community sends congratulations and the best wishes to your 60th birthday! we appriciate your input in organization of scientific collaboration and germanrussian conferences and your activities in educating young scientists in mechanics and tribology. – irina goryacheva – please accept my cordial greetings with this great day! i strongly wish you to conserve your young interest to the learning for the decades to come as well as your wonderful ability to see the things which are not visible for others. – sasha filippov – foreword ii m. heß, r. pohrt, j. starcevic as albert einstein said, "if at first, the idea is not absurd, there is no hope for it." all contributing to this special issue were lucky to witness the unfolding of the mdr from prof. popov's bold idea (of mapping three-dimensional contact problems into one dimension) into a quick and reliable guide to hertzian contact mechanics. and perhaps most admirable in this story is the courage and hope that prof. valentin l. popov has shown to pursue his own idea in spite of the critique it has received from fellow scholars. – ivan argatov – valentin's research publications are notable for the breadth of subject matter and for the physical and mathematical insight that he brings to every new problem. also, his excellent books on subjects related to contact mechanics are a great resource both for experts in the field and for researchers and engineers whose primary focus lies elsewhere. i wish him the very best for his 60th birthday. – jim barber – i would like to say that i know prof. popov as a person who combines many unique qualities. he is able to explain in simple language many complex things, which is very important for a scientist, and even more so for a lecturer. i am grateful to him for the fact that with his direct participation, i decided on the choice of my scientific direction, and the solution of tribological problems by numerical methods at various scales became for me the main topic of my research. – andrey dmitriev – your contribution to the development of tribology and contact mechanics is very impressive, but much of this science is not yet known and undiscovered. i wish you siberian health, which will help you to make new discoveries in the physics of friction and will allow you to bring up new outstanding students! – viktor kuznetsov – жизнь измеряется не годами, а трудами. русская пословица – wolfgang müller– even if prejudice is harder than atoms, i believe that one day, the heat of our continuous passion will melt it. – ken nakano – ‘felix qui potuit rerum cognoscere causas’ (vergil) that is, i believe, the impetus driving your research. as a scientist and university professor you have created something lasting and unique. i also thank you in particular for the good long-term cooperation and mutual support. with the most cordially wishes for your 60th birthday. – manfred zehn – per aspera ad astra. a big thank you to professor popov for his immense contribution to the scientific journal facta universitatis series mechanical engineering. – dragan marinkovic – a special issue in honor of prof. valentin l. popov iii i can say that i admire of prof. popov two things: one is the ability to go to the essentials of (tribological) problems, reaching solutions of engineering interest with the cleanest and simplest formulation, not using redundant mathematics to impress the reader (this having also a positive effect in teaching and in educating students at all levels). and the other is to have built a group of collaborators who apparently simply adore him, and this shows he is able to obtain from them the maximum collaboration also without redundant pressure. – michele ciavarella – we fully agree with michele ciavarella’s last statement. indeed, valentin popov has always been and continues to be successful in tying international bonds due to his supportive and generous character. we have reached out to a number of his close collaborators who have in turn contributed to this special edition. their submissions present new advances in the topic of past and current research interests related to him, like wear or the interplay of frictional processes and the behavior of the system. johannes otto and georg-peter ostermeyer employed highly specialized tribotesters to investigate ultrasonic vibrations of brakes that are caused by the boundary layer. while they find a variety of influencing factors, the testing machine used is not one of them. ken nakano and his colleagues investigated a system where the system dynamics indeed play a central role. they examined the stability of a 2dof sliding system by an eigenvalue analysis, considering both the in-plane anisotropy and the in-plane asymmetry. the obtained eigenvalues were organized by using the minimum modal damping ratio as the stability maps, which can be used to reduce the effort for improving frictional properties. colleagues from the chair of utz von wagner present an experimental and theoretical study of a bistable beam equipped with magnets. they show that a discretization of such system should not be based on a single shape function. even in static configuration, mechanical system can interact with frictional contacts in non-trivial ways. sangkyu kim, yong hoon jang and james barber study two-dimensional systems with frictional contact concerning their ability to remain stuck in a wedged state. for every such system, a minimal coefficient of friction can be identified, below which wedging is impossible. irina goryacheva and elena torskaya contribute an analysis of periodic contacts consisting of spherical indenters and a two-layered counterbody. they show how the geometrical and material parameters in such system influence stresses inside the bulk and on the surface. viktor kuznetsov, andrey skorobogatov and vladimir gorgots submitted an in-depth experimental study on nanostructure burnishing. they describe how shear strain depends on burnishing parameters. ruslan balokhonov and varvara romanova investigated a coated steel body under tension. they identify three distinct stages of plastic strain localization. a major contribution of valentin popov is the development of the method of dimensionality reduction which has emerged to a very convenient tool for dealing with a wide class of elastic contact problems. one very active collaborator in this field, ivan argatov traces back the origins of the method and demonstrates why the one-dimensional popov foundation is much more powerful for mapping three-dimensional half-space contacts than the original twodimensional winkler foundation. iv m. heß, r. pohrt, j. starcevic michele ciavarella and antonio papangelo contribute a paper dealing with the multiplicative coefficient for predicting friction in rough contacts due to bulk viscoelastic losses. they refer to an equation given by popov in his textbook and find that the friction coefficient at low speed depends in general on the form of the viscoelastic modulus, while at high speed it mainly depends on the amplitude of roughness. andrey dmitriev and colleagues submitted a paper on a topic that has recently been of high interest to popov’s lab; the formation of wear particles. they used conventional molecular dynamics to study the transition from sliding to abrasive wear. they find the original rabinovicz' criterion to be accurate also for atomic objects! alexander filippov and stanislav gorb considered biological systems and the way they evolute due to an underlying potential. they review ways to generate these potentials such that they account for observable phenomena. the numerical solution of mechanical boundary-value problems with an explicit account for the material microstructure usually requires substantial computational resources. varvara romanova and her colleagues present an explicit dynamic approach that can be successfully used in microstructure-based simulations of quasistatic deformation, considerably reducing the computational costs without losing the information and solution accuracy. in addition to taking into account the aspect of the internal rotational degrees of freedom of a material, micropolar media allow modeling materials with an internal structure. in their study, wolfgang müller and his colleagues lay the foundation for solving a fully coupled flow problem for a micropolar medium undergoing structural change in a funnel-shaped crusher. colleagues from the chair of manfred zehn implemented a new 3-node-shell element in abaqus to model so called active structures. these consist of a combination of passive structures and active elements, for instance piezoelectric sensors and actuators. their potential applications include the suppression of noise and vibration. we welcome the many positive responses to our call for submissions from close colleagues of valentin l. popov. the breadth of submission mirrors his past and current research activity which has touched many topics not only within narrower tribology but also at the level of technical and biological systems as a whole. on behalf of the entire lab “systemdynamik und reibungsphysik” the editors markus heß, roman pohrt, jasminka starcevic 6926 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 73 94 https://doi.org/10.22190/fume201106030a © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd review article recent improvements of the optical and thermal performance of the parabolic trough solar collector systems asaad yasseen al-rabeeah1,3, istván seres2, istván farkas2 1doctoral school of mechanical engineering, szent istván university, hungary 2department of physics and process control, szent istván university, hungary 3department of mechanical engineering, faculty of engineering, university of kufa, iraq abstract. parabolic trough solar collectors (ptscs) are commonly used for applications that reach a temperature of up to 500 °c. recently, improving the efficiency of ptscs has been the focus of research because ptscs have advantages, such as cost and size reduction and improved optical and thermal performance. this study summarizes relevant published research on the preparation, properties and experimental behavior of the optical and thermal properties of ptscs. analyzing of the thermal modeling method presents a steady and transient heat transfer analysis. optical efficiency depends on material properties, such as mirror reflectance, glass cover transmittance, receiver absorption–emission, intercept factor, geometry factor and incidence angle. also analyzed and discussed are the models used in computational fluid dynamics to study the physical properties of ptscs. lastly, studies on ptsc performance and enhancement, including novel designs, enhancement of passive heat transfer and laden flows of nanoparticles inside the absorber tube, are presented and examined separately. nanofluids have illustrated their advantages and ability to increase heat transfer rates. moreover, other works that aimed to enhance the optical and thermal efficiency of ptscs are evaluated. key words: parabolic trough solar collector, optical analysis, heat transfer enhancement, simulation tool analysis, nanofluid received november 06, 2020 / accepted march 11, 2021 corresponding author: asaad y. al-rabeeah mechanical engineering, szent istván university, páter károly utca 1., gödöllő, h-2100, hungary. e-mail: asaady.hussein@uokufa.edu.iq 74 a.y. alrabeeah, i. seres, i. farkas 1. introduction the global energy demand is continuously increasing with the depletion of the conventional energy sources. solar energy is a usable and clean renewable energy source which is used as an alternative for producing energy from fossil fuels. solar radiation is reflected, diffused or absorbed by solid particles, especially by the earth’s surface, depending on many factors, such as climate, weather, agriculture and the earth’s geometry [1, 2]. the parabolic trough solar collector (ptsc) technology is one of the most reliable technologies in the field of solar thermal [3]. it is mainly used for power generation (e.g. generating steam which needs high temperature) and other technological purposes [4, 5]. in the case of ptscs, thermal energy is collected from solar radiation in the focal point of a special geometry to reach a high temperature [6]. the collectors receive direct solar radiation from the sun over a large surface and gather it to the focal point. a fluid flowing inside the tube absorbs the heat energy generated from the focused solar radiation, raising its enthalpy and causing an increase in the temperature of the tube wall [7, 8]. ptsc is an active technology used in the field of solar thermal applications. it consists of a reflecting surface, an absorber tube and the working fluid passing through the tube [9]. the design should be accurate to increase thermal efficiency, and the material of low weight, high mechanical strength and high thermal conductivity is preferable [10, 11]. the thermal conductivity of the absorber tube material affects the performance of ptscs by increasing the heat transfer between the working fluid and the metal [12, 13]. a working fluid is an essential component for enhancing the efficiency of ptscs. the mixing of nanoparticles with the working fluid is an effective method of increasing the collected thermal energy and the nanofluids’ thermophysical properties, such as enthalpy, specific heat capacity, thermal conductivity and density [14, 15]. the thermal efficiency of ptsc depends on the concentration of the volume fraction of nanoparticles in the base fluid [16]. this study primarily aims at summarizing the key advances made in ptscs and identifying the factors to take into consideration in future developments. continuing research and development activities have helped this ptsc technology become the most economically and technologically advanced of all current concentrating solar power technologies. this thorough analysis is conducted in order to study different modeling research and methods used to simulate ptscs. detailed reviews of theoretical studies on thermal and optical performance are performed. moreover, studies on performance improvement techniques and dealing with the alteration of the design of ptscs are evaluated. for increasing heat transfer is the insertion of tabulators in the ptscs design while the use of mono and hybrid nanofluids is to improve efficiency and thus the performance of ptscs. 2. ptsc system 2.1. background in 1883, captain john ericsson used a parabolic trough concentrator (ptc) to work on solar-powered machines for irrigation. however, his experiment on solar engines did not advance to the prototype stage. the invention of the parabolic trough is essential. in 1912, a 45-kw power plant was built in egypt. the system was composed of five solar collectors and was oriented north–south with a system of mechanical tracking [17]. the system generated steam that was used to operate water pumps for irrigation. the recent improvements of optical and thermal performance of parabolic trough solar collector 75 development of the parabolic trough power occurred in the us in the 1970s and in europe in the 1980s [18]. ptscs could produce high temperatures (above 500 °c) to produce industrial process heat. the development was sponsored or conducted by the sandia national laboratories in new mexico. in 1981, the international energy agency developed a small solar power system in tabernas, spain. in 1982, luz international limited (luz) advanced a parabolic trough collector. in 1985, luz built eight power plants of ptscs in california, us. today, according to the database of the national renewable energy laboratory, over 97 plants are at different stages of development of this parabolic trough-based technology. the design of these power plants is to produce electrical power from steam obtained from natural gases or solar fields. the parabolic trough power plants of nevada solar in the us produce 72 mw capacities, and the martin solar plant centre has 75 mw net capacities. the andosol plant is the first parabolic trough power plant in spain. many plants in spain have similar operational characteristics (e.g. andosol with 50 mw and 7.5 h storage energy), some are under construction (e.g. vallesol 50 with 50 mw and 7.5 h storage energy) [19]. the scholars are still trying to improve and increase the parabolic trough power plants efficiency. 2.2. ptsc systems fundamentals a ptsc system is a technology that concentrates solar energy in a focal line to convert it into thermal energy of the high-temperature medium. it can obtain temperatures of up to 500 °c, depending on the application [20]. a mathematical model of the parabola, in terms of the coordinate system, is shown in fig. 1. fig. 1 parabolic concentrator [21] in fig. 1, y=x2/4f represents the parabola equation, a the aperture, f focal distance, hdepth, r the rim radius, ϕthe rim angle, rr the rim radius when the angle ϕ= ϕr. the collector receives the direct solar radiation from the sun over a large surface and focuses on it. the ptsc has a curved reflector or a parabolic mirror for reflecting and 76 a.y. alrabeeah, i. seres, i. farkas concentrating the solar radiation onto specific points or a line. the mirror is manufactured from different materials to reduce absorption losses, such as low iron glass or aluminum. many factors are important in the production of collector mirrors; these factors include solar-weighted reflectivity, durability, abrading properties and cost. the gluing, silvering and protective coating processes are performed after bending the mirror [22]. the heat collection element (hce), also referred to as the receiver, is placed at the focal axis. the heat transfer fluid circulates through the absorber. a fluid flowing inside the tube that absorbs the heat energy generated from the focused solar radiation raises its enthalpy and causes an increase in the temperature of the tube wall. ptcs can be used only in direct solar radiation in the collectors, which are not deviated by dust, fumes or clouds. the absorber tube should be coated by a material of the antireflective layer to minimize the heat losses generated by radiation [23]. the effectiveness of the solar thermal collector is calculated by measuring the fluid temperature difference between the inlet and the outlet and by the flow rate of the working fluid [24]. 2.3. modeling and simulation of ptscs the progress of computing has helped researchers in analyzing the system by modeling and simulation. engineering programs can be used to study the system performance and the effect of several variables with minimum time and low cost [25]. recently, many modeling studies have been performed and have facilitated the development of ptscs; these studies involved thermal and optical analyses through the modeling and simulation of ptscs. by modeling the system, the factors can be analyzed and handled separately (e.g. temperature and the properties of optical materials) [26]. the modeling and simulation of ptscs can be covered as depicted. 3. ptsc-system optical analysis optical efficiency ηo can be obtained by the rate of energy absorbed from radiation in the absorber tube and the amount of the energy incident on the aperture of the collector.  == )0(o , (1) where: ρ the mirror reflectivity, τ the glass envelope transmittance, α the absorptivity of surface coating, γ the mirror interception factor, θ the incidence angle. the efficiency curves are generally calculated at normal incidence; however, the incidence angle for the tracked collector at a single axis changes during the operation. the optical efficiency of ptscs decreases with incidence angle for several reasons, including the increased width of the solar image on the receiver, the decreased transmission of the glazing, the absorption of the absorber and the spillover of the radiation from troughs of finite length. the effect of the angle of incidence must depend on the difference in all optical properties. it can be correlated by a modification called the change in the angle of incidence [27]. a method of reducing the end loss effect in a short trough collector is to recompense the length of the absorber tube [28]. a different way of calculating end loss is presented in cylindrical troughs. the optical design of ptscs is influenced by several factors [29], including apparent changes in the incidence angle effect and the sun’s width, mirror construction and the materials used in recent improvements of optical and thermal performance of parabolic trough solar collector 77 the heat collector element, poor operation, incomplete tracking of the sun’s rays and the manufacturing defects of the ptsc. the next two parts discuss the way in which the analytical and ray tracking approaches for optical errors are perceived and used in the study of ptscs. 3.1. optical analysis of errors optical performance is determined by using an analytical approach to obtain the closed intercept factor. a mathematical expression is determined for the intercept factor by gaussian distribution [30]:   −         = 2 2 2 exp 2 1 )( tottot guss cfd     , (2) 1 2 2 2 2 2 2( )tot sun mirror slop tracking displacement     = + + + + , (3) where: σtot total optical error, σsun beam intensity error, σmirrorsurface mirror error, σslop local slop error, σtracking tracking error, σdisplacement -displacement error, cconcentration ratio. total optical error σtot is obtained from this approach by making all errors in a single term [31]. two groups of optical errors, namely, random and nonrandom, are shown in fig. 2. fig. 2 potential optical error description in ptscs [4] the intercept factor can be obtained from [32] 78 a.y. alrabeeah, i. seres, i. farkas  +                         + +−−+ −         + +−−+ + =     r d d erf d erf r rr r rr r r           0 )cos1( )cos1(2 )cos1()sin1)(cos1(sin )cos1(2 )cos1()sin1)(cos1(sin sin2 cos1 , (4) where: etotal energy, d* the universal nonrandom error parameter due to hce dislocation and mirror profile errors; β* the universal nonrandom error parameter due to angular errors; σ* the universal random error parameter. 3.2. ray tracing the ray-tracing technique is used for analyzing the optical and optical design/optimization performance of ptscs. it benefits the systems that contain many surfaces and newtonian imaging equations and those in which the gaussian is inappropriate. ray tracing supplies a massive amount of detailed information for the optical characteristics of the system [33]. computer technology helps reduce the time for optical analyses. software tools that use the ray tracing technology include optical, asap, tracepro, soltrace and simultrough, using the monte carlo ray tracing (mcrt) method in the optical analysis of a ptscs [34]. 4. heat transfer element for ptsc the heat transfer element in ptscs is a major component and contains an absorber tube; it is an essential part that contributes to the proper performance of the system. solar radiation is focused on the absorber tube, and a heat transfer fluid (e.g. thermal oil, water and nanoparticle-laden fluid) moves through the tube [21]. a schematic of a solar trough parabolic receiver is shown in fig. 3. fig. 3 the schematic figure of losses of the solar trough parabolic receiver [37] recent improvements of optical and thermal performance of parabolic trough solar collector 79 the losses are indicated in the cross section of the tube. an evacuated glass envelope covers the absorber tube to reduce the heat losses. the fluid flow by forced convection in the absorber tube may be in single or two phases. in this case, the flow process in these systems, the heat transfer coefficients and the equation to the hce modeling of heat transfer are much more complex [35, 36]. 5. computational fluid dynamics (cfd) analysis for the numerical modeling of the fluid flow (can be laminar or turbulent flow) inside the tube of ptscs, computational fluid dynamics (cfd) is used to analyze the hce’s overall thermal hydraulic efficiency. the cfd modeling method includes continuity and momentum numerical solutions and energy balance equations. to predict ptsc output correctly during a cfd study, actual boundary conditions must be used. the key to these boundary conditions is the heat flux on the absorber tube of the hce; in the study, this heat flux is typically the leading thermal boundary state [38]. details of studies conducted with cfd are summarized in table 1. table 1 summary of traditional ptsc cfd-analysis ref. type of study findings [39] ansys the difference in the heat flux has a major effect on deciding the overall circumferential hce temperature. [40] fluent with an increase in the nonuniformity of hce distribution, heat loss decreases. when the angle of the incidence decreases, so does heat loss. therefore, the rate of heat loss gradually decreases in accordance with radius ratio (rr) (i.e. relationship between the inner radius and the outer radius of the absorber envelope), which decreases, thus reaching the minimum amount for rr=1,375 if the heat transferred starts after that critical value only through conduction and convection. [41] fluent the critical of rr is less for large-diameter absorber diameters. for a given hce, the critical rr is independent of the hce temperature and outer wind velocity in the weather. in the space of the nonevacuated hce, the contrast of heat transfer losses in individual and variable temperature in a tube in the cases is 1.5%. the rr and wind speed in the evacuated hce have marginal effects on the thermal losses. [42] fluent rising heat transfer at high mass flow rates means the absorber outlet has a high capacity for thermal energy. as the losses in convection rise by wind speeds around the collector, the temperature in the outlet decreases. therefore, the circumferential temperature gradient is nearly even for the absorber tube of copper material compared with one-steel material. [43] fluent /mcrtcode the heat flux distribution becomes gentler as the concentration ratio increases, the angle span of the region decreases, and the absorber’s shadow effect becomes less powerful. increasing the concentration rate can also increase the htf temperature. increasing the angle of the rim reduces as much heat as possible. when the angle of the rim is small, the glass cover reflects many rays; the temperature elevation is much lower. [44] fluent when the htf is steam in different process settings, the thermal stress inside the tube is great. moreover, highly effective solar radiation that focuses on the absorber tube and the high steam temperature contribute to high heat transfer gradients with comparable levels of steam mass flow. 80 a.y. alrabeeah, i. seres, i. farkas [45] ansys fluent /soltrace when the angle of rim increases, the gradient of the circumferential temperature on the surface of the absorber is reduced. the reduction in the peak temperature of the absorber is low as the angle of the rim is greater than 80°. bejan number, a measure in which irreversibility between heat transfer and irreversibility in fluid friction is dominant. it also increases with a reduction of the rim angle and temperature of htf and increase the ratio of concentration. [46] ansys fluent the nusselt number variance is smaller than that of the nonuniform heat transfer flux under uniform heat transfer flux. with the solar elevation angle, the resistance to flow increases. when the number of grashof increases and the number of nusselt increases rapidly with the angle of solar elevation then it starts to decrease slowly at the increase of higher grashof numbers and low on the solar elevation angles. [47] mcrtcode/ansys fluent increased errors in tracking decrease the thermal efficiency. the thermal output drops from 70.64% to 9.41% by raising the error of tracking from 0 mrad to 20 mrad. 6. enhancement of optical efficiency 6.1. selective surface coating on the receiver tube optical efficiency is calculated as the energy ratio of the absorbed energy to the energy incident received on the collector’s aperture [48]. the coating changes have been improving hce performance. the hce output is prone to any difference in the optical properties of the selective coating. many studies have been conducted to enhance absorption and reduce selective surface emission. the microstructure of the material is influenced by extremely high temperature [49]. the coatings should be structurally robust and suitable, safe to handle for extended periods, stable at operating temperatures, environmentally friendly and relatively inexpensive. 6.2. antireflective surface coating on the glass tube in solar applications, borosilicate glass tube should be installed around the absorber and should have high transmissometer properties. selective surface coating on glass increases transmittance from approximately 92% to 96% [50]. 6.3. mirror reflectivity the reflective surfaces are coated by silver, and then followed by layers of copper to increase the quality of the highly polished reflectivity mirror surface 94.5%. the cleaning of mirrors is vital for the efficiency of the solar collector assembly [51]. 6.4. absorber tube intercept factor the intercept factor effects on optical efficiency are determined as part of the ray’s incident angle upon the aperture that reaches the receiver for a given incidence angle. the intercept factor is the parameter that embodies the effect of errors. the local slope and profile errors occur during manufacture. thomas developed a technique to measure the flux distribution around the receiver of ptcs. if the distribution of the flux around the absorber is known, then the intercept factor can be easily calculated [52]. recent improvements of optical and thermal performance of parabolic trough solar collector 81 6.5. incorporating secondary reflectors the essential primary concentrator reflects the solar radiation on the receiver tube either through a mirror or a polished aluminum sheet. the collector that intercepts the radiation flux depends on factors, such as primary focus surface error, rim angle and rigidity of the structure, to withstand wind and self-load and mechanism tracking accuracy. the spillage or dispersion of high-concentration radiation across the source creates a considerable optical and thus thermal efficiency loss [53]. 6.6. dual axis tracking and end losses the geometrical aspect of the collector determines the optical efficiency and performance, decrease of the opening area induced by the irregular effect, blocks, shadows and radiation loss beyond the receiving end. radiation occurring on the concentrator’s edge obverse the solar radiation cannot enter the receiver tube that is called end effect. xu conducted an optical study of the end loss effect and then proposed a mirror design to enhance thermal efficiency. the end loss effect is gradually reduced by increased trough length [54]. 7. performance enhancement techniques many researchers have studied heat-transfer improvement techniques to enhance the thermal performance of ptscs and thus increase their efficiency. ptsc systems can be improved by changing either their heat collector element properties or optical design. various heat-transfer enhancement techniques have been used in ptscs. 7.1. ptsc receiver with glass envelope the materials and dimension of the absorber tube affect the performance of ptscs [55]. the performance of the collector increases with that of the glass cover tube. the glass cover tube reduces the convective heat losses and enhances the performance of the ptsc system by improving the greenhouse effect between the glass and the tube [56]. having a top glass cover increases instant efficiency by 45.56%–62.60% and total efficiency by 10% [57]. kasaeian et al. (2015) designed and manufactured a small prototype model of ptscs to investigate the methods for enhancing the performance of ptcs. the system was compared with different receiver tubes to improve the optical, thermal and heat transfer of the ptscs, with vacuumed steel tube with black paint, black chrome coating copper tube, copper-vacuumed black chrome coating and black chrome coating copper tube with nonevacuated glass cover tube. the test of the different receiver’s tube used mwcnt/oil nanofluids in 0.2% and 0.3% volume fraction. the best results were obtained in the vacuumed receiver, and the efficiency improved by 11% higher than the nonevacuated tube. the maximum optical and thermal efficiency of the vacuum copper receiver system was found to be 61% and 68 %, respectively, due to a high absorption rate of 0.98% [58]. 7.2. novel designs the focus of the novel design focuses on enhancing optical efficiency by increasing the absorbed radiation or decreasing collector heat loss. bader studied the heat transfer 82 a.y. alrabeeah, i. seres, i. farkas analysis of the cylindrical air-based cavity-receiver tube. the receiver efficiency ranged from 45% to 29%. at summer solstice solar noon, the htf inlet temperature was 120 °c, and the htf outlet temperature ranged from 250 °c to 450 °c. the loss of solar radiation on the absorber tube is equal to one third by spillage [59]. the heat loss between two paired horizontal cylinder receivers was studied in conduction and convection in absorber tube from a half-isolated annulus. the application of fibreglass insulation to the half of the annulus away from the parabolic trough increases the reduction of convection heat losses by an average of approximately 25% relative to traditional receivers [60]. demagh studied the possibility of establishing an s-curved/sinusoidal receiver tube in ptcs. the ptsc was replaced with a traditional straight absorber, whose designed scurved/sinusoidal and heat flux density distribution varies on the axial and the azimuthal directions. the heat flux density was distributed on a large surface [61]. xiao designed a tube absorber by a v-cavity on ptscs. the optical efficiency of the absorber improved with reduced aperture distance and increased depth-to-width ratio [62]. 7.3. improving passive heat transfer many researchers studied the collector improvement by passive convective for increasing heat transfer in the absorber tube. various inserts, such as regularly spaced, straight twisted, helically twisted and twisted perforated tapes; protrusion; dimples; wire loops; longitudinal strips; and insert butterfly strings, are used. the thermodynamic, fluid friction and heat transfer performance increase as the width ratio increases and the twist ratio decreases. a significant decrease in the generation of entropy is achieved at a low reynolds number at the twist ratio and decreased the width ratios, while the ideal reynolds number increases. a considerable increase in the heat transfer performance of about 169%, reduction in the absorber tube's circumferential temperature difference is up to 68% while increase in thermal efficiency is up to 10% over a receiver with a plain absorber tube [45]. various porous receiver geometries have been considered for the performance estimate of ptscs. thermal analysis of the receiver tubes was performed for various geometric parameters, such as thickness and ratio of fin aspect and porosity, for varying heat flux conditions. the porous fins inserted into the tubular receiver of the stc enhanced the heat transfer compared with the solid longitudinal fins [63]. porous circular, triangular, square and trapezoidal inserts and the heat losses in all porous inserts were found to be approximately the same [64]. helical fins are utilized in internal tubes for the design of ptscs. many factors, such as thermal loss, pressure loss, thermal fatigue and thermomechanical stress, affect the performance of ptscs [65]. the results show that the parasitic losses associated with the pressure losses in the tube increase with the number of fins and its helix angle. although the thermal losses and temperature gradients are reduced, the energetic and thermal efficiency of the collector increases [66]. on the other hand, several drawbacks in this way, such as increased parasite loads associated with increased pressure loss, noise and additional manufacturing costs, exist. 7.4. nanofluid nanofluid is a term used to describe a fluid in which nanometre-sized particles are suspended with normal scales of 1–100 nm in length [67]. nanoparticles in liquids are suspended to improve thermal conductivity and heat transfer efficiency of basic liquids recent improvements of optical and thermal performance of parabolic trough solar collector 83 [68]. the thermal conductivities of particulate content are typically higher in magnitude, particularly at low volume levels, compared with those of specific fluids, such as water, ethylene glycol and light oils and nanofluids [69]. they can dramatically improve the host fluid thermal efficiency and thermophysical characteristics of ptscs [70, 71]. in the simulations, two major groups emerge: (1) the single-phase modeling that considers the mixture of nanoparticle and base fluid as a single-phase mixture with stable properties and (2) the two-phase modeling that considers the properties and behavior of the nanoparticle separately from that of the base fluid [72]. the pressure drop increases with an increasing volume concentration of nanoparticles in the base fluid. when the reynolds number increases, the pressure drop increases sharply. the pressure drop is a function of the fluid’s thermophysical properties and velocity of inlet fluid in the absorber tube [73]. the force of inter nanoparticles is highly influenced by the concentration of the nanoparticles. the force profiles are influenced by many factors, such as time, size, shape, surfactant concentration and humidity. in greater concentrations nanoparticles increasingly begin to accumulate, swarm, precipitate out of the solution, and adsorb on surfaces [74]. for example, synthetic oils have a temperature of >400 °c, whereas molten salts reach up to 600 °c. by contrast, it is anti-freezing systems due their temperature of solidification about 220 °c [75]. 7.4.1. mono nanofluids a single kind of nanoparticle is suspended with a fluid. in a study, the modeling and simulation of synthesized nanofluids should predict the thermophysical properties to ensure acceptable results. the thermophysical properties for any nanoproduct and fluid become new properties of density, viscosity, specific heat capacity and thermal conductivity [76]. three main parameters involved in calculating the heat transfer rate of the nanofluid are heat capacity, viscosity and thermal conductivity, which may differ from those of the original pure fluid. the density and specific heat of the dispersed liquid are homogenous, and the thermodynamically stable state [77] could be determined from  pfnf +−= )1( , (5) where: ρnf nanofluid density, f fluid density, p –particle density,  the volume fraction of the nanoparticles. furthermore, the specific heat based on the heat capacity concept is as follows [78]: nf pppfpnf nfp cc c   ,, , )1( +− = , (6) where: cp,nf the specific heat capacity of nanofluid, cp,f the specific heat capacity of fluid, cp,p the specific heat capacity of particle. the viscosity of the nanofluid can be estimated with the existing relation )1(  += fnf , (7) where: µnf viscosity of the nanofluid, µf viscosity of the fluid,  intrinsic viscosity is a measure of a solute’s contribution to the viscosity of a solution (brinkman model =2.5) [79]. 84 a.y. alrabeeah, i. seres, i. farkas the equation above is utilized for the calculation of kinematic viscosity, which is applicable to linear viscous fluids with dilution, suspension and spherical particles [80]. a modified einstein’s model for high concentrations of particles up to is as follows:  −−= )1(fnf , (8) krieger and dougherty modified the equation for highly condensed, uniform rigid sphere suspensions. m m fnf     −       −= 1 , (9) where φm represents the maximum factor of particle added to the fluid, ranging from 0.495 to 0.54 under steady-state conditions, and it is approximately 0.605 at a high rate of shear [81]. later, the formula of the dynamic viscosity was modified by [82]                         −         = 3 1 3 1 1 8 9 m m fnf      . (10) a theory incorporating these effects was developed by [83]. the author considered the contribution of the brownian motion to the average stress and obtained the following formula for the effective viscosity, accurate to the second-order in concentration [84]: )2.65.21( 2 ++= fnf . (11) therefore, the frankel model of generalized form includes particle radius and the spacing between particles [85].                                   +      +      ++= 2 12 1 5.45.21 dp h dp h dp h fnf  , (12) where: h nanoparticle diameter , dp the distance between any two nanoparticles. the thermal properties of fluid spherical or cylindrical solid particles were studied; moreover, technically and experimentally outstanding prediction formulations on the efficient thermal conductivity of dispersed substances were proposed. a nanoparticle enhances thermal conductivity in a conventional fluid [86]. the expressions of the conventional models of the effective thermal conductivity of a solid/liquid suspension are as follows [87]: recent improvements of optical and thermal performance of parabolic trough solar collector 85 )(2 )(22 fpfp fpfp fnf kkkk kkkk kk −++ −++ =   , (13) where: knf thermal conductivity of nanofluid, kf thermal conductivity of fluid, kp thermal conductivity of nanoparticle. the solid line defines the relationship expected by the hamilton–crosser prediction equation, in which thermal conductivity is represented as [88] )()1( )()1()1( fpfp fpfp fnf kkknk kknknk kk −+−+ −−+−+ =   , (14)  3 =n , (15) where ψ is the sphericity ratio between a sphere’s surface area and a particle’s surface area with a volume equal to the parts. the thermal conductivity aspect is therefore improved by the irregular motion of the nanoparticles suspended and is shown to be the apparent thermal conductivity of the nanofluid [89]. fc bpp fpfp fpfp fnf r tkc kkkk kkkk kk     32)(2 )(22 + −++ −++ = , (16) where: rc the apparent radius of the clusters, boltzmann constant kb =1.381ˣ10-23 j/k, ttemperature. the heat transfer analysis of the direct absorption receiver system (see fig. 4) under 2d steady state. fig. 4 section view of the direct absorption receiver system [27] 86 a.y. alrabeeah, i. seres, i. farkas the nanofluid’s thermal conductivity depends on the nanofluid’s viscosity and the thermal conductivity of the base liquid and solid particles, as well as the mass, specific heat and volume fraction of the nanoparticles. the heat transfer performance is enhanced by the nanofluid consequent to increasing the properties of the base fluid. the convection heat transfer coefficient is improved due to the increase in volume fraction. the pressure drop increases with the increase in nanofluid density and viscosity [90]. table 2 shows the effects of various nanofluids on the performance of ptscs. table 2 effect of various nanofluids on the performance of ptscs ref. nanofluid nanoparticle/base fluid) volume concentration (%) effect on the performance [91] bh-sio/water tio2/water 3% 3% the thermal efficiency is improved by 0.073%, and the coefficient of heat transfer is 138%. the thermal efficiency is improved by 0.073%, and the coefficient of heat transfer is 128%. [92] al2o3/water sio2/water tio2 /water zno/water al2o3/water au/ water (5, 10, 20) % (1, 5, 25) % (1,10,20,35) % (1, 5, 10) % (0.1, 1, 2) % (0.01) % at low concentrations, only au, tio2, zno and al2o3 nanofluids pose minimal changes compared with water use; however, increasing nanoparticles concentration does not appear to have any benefit with respect to water. at high temperatures, the viscosity decreases, and the thermal conductivity increases [93] cu/therminol vp-1 ag/therminolvp-1 al2o3-thermi-nol vp-1 less than 10% the thermal efficiency for ag-therminolvp-1, cu -therminolvp-1 and al2o3 therminolvp-1 nanofluids improved by 13.9%, 12.5% and 7.2%, respectively. thermal conductivity increased, the efficiency of exergy improved, and performance of heat transfer improved. [94] graphene/therminol vp-1, al/therminol vp-1 0.02% 0.09% graphene has higher solar absorption than nanoparticles in the aluminum particle. dars can transfer heat at 265. [95] al2o3/syltherm 800 cuo /syltherm 800 tio2/ syltherm 800 cu/ syltherm 800 nanofluids boost system efficiency and achieve an increase of up to 1.75% relative to pure thermal oil operations. moreover, al2o3 and cuo must be used at higher concentrations compared with tio2 and cu. [96] cu/water 0.02% adding of cu/water significantly improves its absorption characteristics and optical and thermal efficiency and leads to higher outlet temperatures. [97] al2o3/ synthetic oil 0.02% 0.04% the presence of nanoparticles increases the coefficient of heat transfer of the working fluid in the absorber tube. [98] silica/ ethylene glycol carbon/ ethylene glycol 0.4% the thermal conductivity increases thermal efficiency by adding solid nanoparticles; for mwcnt and nanosilica, the optimal volume fraction is 0.5% and 0.4%, respectively. recent improvements of optical and thermal performance of parabolic trough solar collector 87 [99] al2o3/synthetic oil 0.5% the thermal performance and overall efficiency improved slightly with the use of al2o3– synthetic oil. the essential advantage of using nanofluids is reducing the pumping power. [100] tio2/water ole-tio2/water bh-sio2/water 2% 3% 3% the coefficient of convective heat transfer with tio2/water nanoparticle was increased up to 22.76%, and the maximum efficiency improvement in the ptsc was 8.66% higher than that of the water-based collector. [15] au /water al /water ni/water ag /water tio2 /water 2% by adding different concentrations of nanoparticles, particularly for au–water and al– water nanofluids in a volume concentration of 2%, the measured values are respectively 2.7 and 2.3 times those for pure water; the critical heat flux is significantly improved. [101] cuo/water 0.01% 0.05% 0.1% maximum thermal efficiency improvements are achieved by adding cuo nanoparticles to pure water with 0.01%, 0.05% and 0.1% volume fraction; the results were 3.23, 3.6 and 3.82 times those of pure water. [102] fe3o4 cuo/therminol 66 4% enhancing the reynolds number increases the convective heat transfer coefficient. the results show that fe3o4 nanoparticles have great thermal conductivity from cuo particles under the magnetic field. [103] al2o3/synthetic (1-5)% the addition of 5% of al2o3/synthetic nanoparticles improves the efficiency of relative exergy by about 19%. the exergy efficiencies decrease when the wind speeds increase from 5 m/s to 10 m/s. [104] cuo / water cuo /oil al2o3/ water al2o3/oil 1% 3% 5% at low enthalpy, water performs better than oil as a base fluid. the performance of the base fluid is increased by adding nanoparticle to the oil. as a nanoparticle, cuo has more effect on the energy and energy efficiency of the system than al2o3 because its heat conductivity and density are higher. 7.4.2. hybrid nanofluid a new category of nanofluids have the thermophysical properties; they showed improvement and enhancement of the ptsc. experimental findings allow one to select a suitable model for a given property [105, 106]. the effective properties of the hybrid nanofluids are defined as follows [107, 108]: 2211)1(  pphfhnf ++−= , (17) where; ρhnf hybrid nanofluid density, p1 and p2 – different types of particle density, hthe combined concentration of volume in the hybrid nanofluid of two different types of nanoparticles (1 and 2) as measured. 88 a.y. alrabeeah, i. seres, i. farkas 21  +=h . (18) heat capacity and viscosity of the hybrid nanofluid can be obtained as follows: hnf pppphfp hnfp ccc c   22,11,, , )1( ++− = , (19)  −−= )1( hfhnf . (20) where: cp,hnf the specific heat capacity of hybrid nanofluid, µhnf viscosity of the hybrid nanofluid. the hybrid nanofluid thermal conductivity is described in accordance with maxwell; the following shall be applied:             −+−+ + −+++ + = fhf h pp fhf h pp hnf kkkk kk kkkk kk k       2)(22 ( 2)(22 ( 2211 2211 2211 2211 . (21) 8. conclusion the global energy demand is continuously increasing with conventional energy sources depleting. therefore, fossil fuel resources must be replaced by renewable resources, and the optimal alternative to the traditional energy sources is solar energy due to its inexhaustibility. ptscs are devices that convert solar radiation into heat. the literature reveals that ptscs can enhance the heat transfer distribution inside the collectors when they are well designed. various researchers have studied the ptsc effect and focused their research on modeling, simulation, design and manufacture of the systems in order to determine their performance and the possible improvements that can be made. the goal is to develop the performance further via ptsc modeling and simulation. modeling studies can show the poor side of the design or possible enhancements in the collector. therefore, the parametric analyses of the influence are determined with minimum effort and time and low cost in comparison with experimentation. two main parts which have a considerable effect on the performance of ptsc are the working fluid and the properties of the absorber tube. by using the ansys program optimal working parameters are determined in the analysis of ptscs, which especially heat collector element. further, it applies to the calculation of a flow rate (laminar or turbulent) and the investigation of heat transfer improvements and modifications using nanofluids and absorber tube configurations. to improve the performance of ptscs, different designs have been proposed in the literature. the performance of ptscs can be improved either by modifying their thermal properties or optical design. the receiver tube affects efficiency under vacuum conditions and coating emittance; it is the major criterion for heat loss. therefore, optical efficiency is improved by selecting the coating, glass envelope, material and the reflected surface. heat transfer performance is enhanced by nanofluids consequent to increasing the recent improvements of optical and thermal performance of parabolic trough solar collector 89 properties of the base fluid. a nanofluid improves the thermal and thermodynamic performance of the system. thermal efficiency depends on the volume fraction of nanoparticles and concentration ratio. the performance of a solar parabolic collector is enhanced by increasing the volume fraction of nanoparticles. the improvements in thermal efficiency relate to low concentrations of nanoparticles. hybrid nanofluids, a new advanced nanofluid, contain two types of nanoparticles. hybrid nanofluids enhance the thermal conductivity rate compared with mono nanofluids. economic viability is considered dependent on capital costs. moreover, further emphasis should be placed upon the longevity of the material on the nanofluid in the suspension and on the variable costs in the form of maintenance required to replace damages or relieve blockages. 9. future recommendations this review of ptsc literature offered an in-depth insight into research conducted to enhance optical and thermal performance. few investigations have been performed using nanofluids in compound parabolic-type collectors. the next section highlights the gaps in this area of research for future work to improve the efficiency of ptscs. ▪ coatings for reducing dust particle adhesion and stabilizing temperature operation must be investigated. ▪ studies on different shapes of absorber tubes (e.g. elliptical cross section) and their effects on thermal efficiency and distribution of heat flux are recommended. ▪ future studies must focus on the methods of avoiding reflector corrosion and increasing of mirror reflectivity property. ▪ performance improvements to certain nanoparticles of volume fractions are feasible and can broaden the reach of future research to increase the performance of ptscs at high volume concentrations. ▪ further studies can be conducted with variance in physical geometries to 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140, pp. 1579-1596. 11997 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 287 310 https://doi.org/10.22190/fume231221005b © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical study of a non-newtonian phase change flow in finned rectangular enclosures mohamed boujelbene1, seyed a.m. mehryan2, mikhail sheremet3, mohammad shahabadi4, nasrin b.m. elbashir5, mohammad ghalambaz3 1industrial engineering department, college of engineering, university of ha’il, ha’il, saudi arabia 2young researchers and elite club, yasooj branch, islamic azad university, yasooj, iran 3laboratory on convective heat and mass transfer, tomsk state university, tomsk, russia 4school of aerospace and mechanical engineering, university of oklahoma, norman, ok, united states of america 5department of mathematics, college of science and humanities, prince sattam bin abdulaziz university, al kharj, saudi arabia orcid ids: mohamed boujelbene https://orcid.org/0000-0001-7424-6685 seyed a.m. mehryan https://orcid.org/0000-0001-8812-5905 mikhail sheremet https://orcid.org/0000-0002-4750-0227 mohammad shahabadi https://orcid.org/0000-0003-2753-9519 nasrin b.m. elbashir https://orcid.org/0009-0004-6122-403x mohammad ghalambaz https://orcid.org/0000-0003-0965-2358 abstract. phase change materials (pcms) are widely used for thermal energy storage systems due to their effective thermal properties for energy accumulation. simultaneously, this material has poor thermal conductivity, and for the optimization of such systems, many techniques are used. this study focuses on an analysis of pcms in a vertical cavity with one, two, or three solid fins and differential heating. the finite element procedure has solved governing equations formulated using the power-law approach for the nonnewtonian pcm and enthalpy-porosity method. the developed code has been verified using numerical and experimental data from other authors. effects of fins number and power-law index on flow and thermal structures within the cavity have been studied. it has been found that a rise in the power-law index illustrates a growth of time for the charging level of the storage system, while the addition of a solid fin from one to three allows for reducing the charging time. the extended heat transfer surface can be applied to optimize the thermal energy storage system. key words: non-newtonian phase change material, enthalpy-porosity method, finassisted latent heat energy storage, rectangular enclosures received: december 21, 2023 / accepted february 15, 2024 corresponding authors: mohammad ghalambaz and s.a.m. mehryan laboratory on convective heat and mass transfer, tomsk state university, 634050 tomsk, russia e-mail: m.ghalambaz@gmail.com young researchers and elite club, yasooj branch, islamic azad university, yasooj, iran, e-mail: alal171366244@gmail.com https://orcid.org/0000-0001-7424-6685 https://orcid.org/0000-0001-8812-5905 https://orcid.org/0000-0002-4750-0227 https://orcid.org/0000-0003-2753-9519 https://orcid.org/0009-0004-6122-403x https://orcid.org/0000-0003-0965-2358 288 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. 1. introduction phase change materials (pcms) are widely used for thermal energy storage systems due to their effective thermal properties for energy accumulations. they are important in waste heat recovery systems and sustainable energy [1, 2, 3]. the advantage of pcms as the most common candidate for a thermal energy storage system is that they store and release a huge quantity of energy at a specific temperature [4, 5]. there are several sectors in which the application of pcms is considerable, like heating and cooling of buildings [57], domestic hot water tanks [9, 10], and solar energy systems [11-13]. one important approach in thermal storage systems is to increase the heat transfer rate and consequently improve their performance [4]. one common way to enhance the melting and solidification time of pcms in cavities is to use extended fins. they help enhance heat transfer by thinning and breaking down the thermal boundary layer [15]. in this regard, wu et al. [16] investigated the effects of a fin’s length and location on the pcm’s phase change in a latent heat thermal energy storage (lhtes). they performed transient numerical simulations employing the enthalpy-porosity method. they reported a considerably shortened time of solidification and melting of pcm with embedded fin. two geometric parameters of fin length to cavity length (εl) ratio and fin height to cavity height were considered (εh). for εl = 0.25, 0.5, and 0.75, the melting time was reduced by 10.4%, 20%, and 26.8%, respectively, and for εh = 0.75, 0.5, and 0.25, the melting time was shortened 9.6%, 20%, and 30.8%, respectively. it was seen that increasing the fin length intensively decreases the solidification time, and lowering the fin position exacerbated the non-uniform distribution of temperature during the solidification process. in another study, sinaga et al. [17] inspected the effects of the position and shape of two fins on the melting of pcm in a quadrilateral cavity. they chose lauric acid as the pcm material. they considered three different rectangular, trapezoidal, and triangle shapes for fins and three different heights in the cavity. they discovered that the best melting process happens when the fins are located in the bottom half. in the case of fin shapes, the triangular shape of fins showed the best results and decreased the whole melting time by 1000 seconds. arshad et al. [18] studied the heat transfer and flow-field characteristics of a heat sink, including pcm and fins. the application of their heat sink was in the passive cooling of electronic devices for input heating powers of 4, 5, and 6 w, and fins of 2mm/3mm thickness occupied 9% volume fraction were employed. they observed a uniform heat distribution in the heatsink, and higher power levels led to a higher melting rate of the pcm. also, they found out that heat penetration in a 2 mm thick fin was more than the other one, leading to a shorter melting time. tian et al. [19] examined the influence of fin on the melting of pcm in a rectangular cavity numerically. they evaluated the melting process by six indicators of cost per energy stored, stored energy per mass, mean power, total stored energy, surface averaged nusselt number, and melting time. steel302 stored the highest total energy among other materials, and copper fin had the highest mean power. the results showed that the aluminum fin is preferred, while copper is less beneficial than aluminum. raj et al. [20] investigated the effects of fin thickness, fin shape, the total number of fins, and base thickness on the behavior of a solid-solid pcm thermal management module used in avionics systems for satellites. the study was done for “on” and “off” cycles of 10w at 1200s and at 4800s, respectively. their results showed good heat transfer for straight fin configurations with a square shape; however, triangular ones performed better and showed 25% and 35% lower mass compared to circular and square fins for a fixed volume fraction. numerical study of a non-newtonian phase change flow in finned rectangular enclosures 289 considering different working conditions, a non-linear relation could exist between shear stress and shear rate defined as non-newtonian fluids. there have been investigations on the thermofluidic characteristics of these fluids. ohta et al. [21] numerically investigated the natural convection characteristics of a pseudoplastic fluid in a squared cavity. the cavity was heated from the bottom wall. they reported that the heat transmission rate of pseudoplastic fluid is greater than newtonian fluid due to the easier development of fluid flow, especially close to the walls, by the shear-thinning effects and viscosity reduction. pandey et al. [22] studied the thermofluidic characteristics of a round cylinder within a square cavity containing non-newtonian fluid. their investigations were performed for the prandtl number 10 and rayleigh numbers 1e3 to 1e6. the power-low index range of n = 0.6 to n = 1.6 was considered in their study. at n = 0.6, their results showed a steady to unsteady flow transition due to the shear thinning regime. the highest range of nusselt number was seen for the cylinder case located at the lower part close to the bottom wall. compared to newtonian fluid, the heat transfer rate around the cylinder was improved for the cavity occupied by pseudoplastic fluid and decreased for the dilatant one. in another experimental and numerical study, pandey et al. [23] examined a pseudoplastic fluid’s thermal and flow characteristics within a differentially heated square enclosure in three different rayleigh numbers. they used a new particle image velocimetry approach for the experimental part. they observed good agreement between the numerical and experimental results in the case of magnitude and pattern of velocity at the walls and core of flow. the thermal and hydrodynamic boundary layers were developed close to the side walls quickly. they discovered that there is a directly proportional relationship between the rayleigh number and the mean nusselt number. yang and du [24] conducted a comprehensive review of the convective heat transfer of nan-newtonian fluids heated in a cavity. they introduced two main models of power-law and bingham for formulating non-newtonian fluids. in the case of non-newtonian pcms, ghalambaz et al. [25] investigated the melting process within a cylindrical cavity between two tubes. they used the finite element method for their numerical calculations. they studied the effects of the darcy number, nonnewtonian power-low index, and rayleigh number on the melting of the pcm. they reported a sharp increase in pcm melting by rising rayleigh to 105 as the conductive heat transmission regime changes to convective. the melting process was enhanced by increasing the darcy number as the flow circulation increased in the porous area. mehryan et al. [26] investigated the melting of a power-low non-newtonian pcm containing nanoparticles in the area between two upright cylinders. the temperature of the inner cylinder was higher than that of the outer cylinder. the method used in this study was the galerkin finite element method. they reported that the added nanoparticles in pcm reduced melting owing to the rise of the pcm viscosity and the decrease of the average nusselt number. farahani et al. [27] numerically inspected the phase change process of non-newtonian pcm in a double tube and a finned triple tube. they examined the effects of non-newtonian model parameters, geometrical parameters, and non-uniform magnetic fields on the melting process. they reported an 11-31% increase in the triple-tube configuration compared to the double-tube configuration. besides, long fins led to good melting by about 19-53% and 30-63% improved mean nusselt number compared to non-fin case. the increase in the non-newtonian power low index led to a decrease in the melt fraction and nusselt number. mehryan et al. [28] examined the heat transfer rate of nano-enhanced pcm behaving as a non-newtonian fluid embedded in a porous medium. the solid matrix was made of aluminum foam, pcm was made of n-octadecane, and nanoparticles of mesoporous 290 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. silica (mpsio2) were chosen for this study. they found out that growth in the concentration of the nanoparticles reduced the normalized molten volume and the average nusselt number. also, their results showed that porosity directly affects the phase change phenomena. checking the effects of fins in cavities filled with non-newtonian fluid, shahabadi et al. [29] studied the thermofluidic characteristics of a power-law non-newtonian fluid in a cavity. the cavity of their study contains a flexible fin, insulated top and bottom walls, and hot left and low-temperature right walls. they modeled the deformation of the fin by employing the fluid-structure interaction method. their results showed a significant deflection of a flexible fin of dilatant fluid compared to newtonian and pseudoplastic fluids. the hot wall’s heat transfer decreased as the non-newtonian power low increased. anand et al. [30] studied the fluid-solid interaction in a shallow microchannel filling by a non-newtonian fluid, including a linearly elastic top wall and other rigid walls. they used lubrication theory for the flow problem and two plate theories for the structural problem. they reported an intense reduction of the pressure drop in the microchannel due to the presence of flexible elements. as seen above, there have been different studies on pcm and heat transfer characteristics of storage systems, including pcms and/or non-newtonian fluids. however, there has been no study covering the melting process of non-newtonian pcm inside a cavity, including fins, to see their effects on the melting process. 2. mathematical model the solid-liquid phase change flow of a non-newtonian pcm is investigated in the present study, as shown in schematic fig. 1. as depicted, a high temperature (th) is imposed on the right wall of different configurations of un-finned enclosure and finned rectangular enclosures having 1, 2, and 3 fins. however, the other bounds of the enclosed media are fully insulated. a phase change material with a non-newtonian power law pattern in the liquid phase is poured within the container. it is worth noting that the volume of fins is known as a constraint. volume change of the non-newtonian pcm during the phase transition is insignificant. to capture the effects of the natural convection, the boussinesq linear approximation was employed. considering unsteady, incompressible, laminar, and non-newtonian conditions for the flow, the continuity, energy, and momentum equations, along with the porosity-enthalpy technique, can be presented as the following [1, 25, 31, 32]: * 0u  = , (1) * * * *( ) ( ( ) )pcm b m tr pcm u u u pi u fu t f     +  =   − +  +  + +     , (2) where    ( )1 : , ( ) 2 ( ( ) ) mi r pcm n pcm n t max d d m d u u      −   = =    =  +  (3) numerical study of a non-newtonian phase change flow in finned rectangular enclosures 291 , , , , ) 0 = ( ) ( b x b b x b y b pcm uy fβ g t t f f f i f j f  = + −= , (4) 2 , 3 , , 2 , 3 (1 ) , = (1 ) m x mush m m x m y m y mush f a u e f f i f j f a v e     − = + + − = + , (5) * * *( ) ( ) ( ) mp pcm pcm pc t c u t k t l t t       +  =    −    , (6) * *( ) ( )p sf sf t c k t t   =     , (7) fig. 1 schematic view of the different analyzed configurations the mathematical form of the applied boundary conditions is: on the right wall: , 0ht t u v= = = (8) the other walls: * 0, 0 t n u v  =  = = (9) the computational domain is exposed to the following initial condition: , 0initialt t u v= = = (10) on the edge wall of the solid fin and the pcm: * * , 0sf pcm pcmfin t t k k u v n n         − = − = =   (11) it is worth noting that n * in the eqs. (9) and (11) is the perpendicular direction to the surface. 𝑙𝑒 ℎ 𝑒 𝑥 𝑦 𝑇 ℎ 𝑁 𝑜 𝑛 − 𝑁 𝑒𝑤 𝑡𝑜 𝑛 𝑖𝑎 𝑛 𝑃 𝐶 𝑀 𝑙𝑠𝑓 𝑙𝑒 ℎ 𝑒 ℎ 𝑒 2 ℎ 𝑒 2 𝑡𝑠𝑓 𝑥 𝑦 𝑇 ℎ 𝑙𝑠𝑓 ℎ 𝑒 ℎ 𝑒 2 𝑡𝑠𝑓 2 𝑥 𝑦 ℎ 𝑒 4 ℎ 𝑒 4 𝑡𝑠𝑓 2 𝑇 ℎ 𝑙𝑠𝑓 ℎ 𝑒 𝑡𝑠𝑓 3 𝑥 𝑦 ℎ 𝑒 4 ℎ 𝑒 4 ℎ 𝑒 4 ℎ 𝑒 4 𝑡𝑠𝑓 3 𝑡𝑠𝑓 3 𝑇 ℎ 292 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. 3. the normalized controlling equations before simulating the correlations, the conversion to non-dimensional coordinates is performed: 2 * 2 2 , , , , = , =, fue e e e pcm pcm h e pcm e pcm pc u m f e p p x t tuh vhx y   y   u v , θ h h t t h ρ t fo h h     − = = = = = −  = (12) the equations in non-dimensional form, derived from the aforementioned correlations, can be expressed as: 0u = (13) ( )1( ( ) )n tr b m u u u pi prg u u f f fo − +  =  − +  +  + +  (14) in which ( )( ) ( )( ) [ ] :[ ] , 2 tr min u u g max d d g d  + = = (15) where ming is the minimum shear rate used to avoid numerical divergence at points where velocity gradients are close to zero. , , , ,= 0, b b x b y b x b yf f i f j f f rapr+ = = (16) 2 , 3 , , 2 , 3 (1 ) , = (1 ) m x mush m m x m y m y mush f a u e f f i f j f a v e     − = + + − = + (17) where the rayleigh (ra) and prandtl (pr) numbers were introduced as: 2 2 1 2 2 2 , , n n mush mushn n pcm pcm pcm pcm e e mp pcm pcme p ccm pcm m g t a pr r h m h h a a       − + −  = = = (18) the values of musha and e of the above equations are 55 10 and 310− , respectively. 2 1 u fo ste fo       +  = −   (19) here, the stefan number (ste) is introduced as: , , pcm p pcm h fu c t ste t t t l  =  = − (20) numerical study of a non-newtonian phase change flow in finned rectangular enclosures 293 2 ( ) ( ) p p pcm pcm sf sfc k c fo k      =   (21) it is worth noting that the ratios of ( ) ( )p p pcmsfc c  and csf p mk k are 1.5 and 500, respectively. the thermal and dynamic conditions of the walls in dimensionless format can be expressed as follows, beginning with the condition for the high-temperature boundary: 1, 0 u v = = = (22) the other walls: 0, 0 n u v  =  = = (23) the cavities’ initial conditions for velocity and initial temperature are zero: , 0initial u v = = = (24) at the edge of the fin -pcm: , 0 sf pcm pcmfin k u v k n n          = = =   (25) the key parameters of interest are the nusselt number at the heated surface and the molten fraction. to normalize the melt volume fraction, it is divided by the total volume of the cavity space thus, mvf is defined as: ( ) volume of the melted pcm mvf total volume of the pcm solid and liquid = (26) to evaluate the difference between mvf for different values of n and mvf corresponding to n = 1, the following parameter can be defined: 1 1 n n n n mvf mvf p mvf = = − = (27) the heat transfer rates in non-dimensional form for the fluid adjacent to the hot wall and the fin base are given by: 0 pcm xx nu  =  −  = (28) 0 sf p m f c s x k k x nu  =  −  = (29) additionally, the average nusselt number at the wall is defined as: 1 2 1 2 1 0 + + s s s s total pcm sf pcmnu dy nu dy nu dynu =    (30) 294 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. 3. numerical method the finite element method was applied to solve the equations of mass, momentum, and energy conservation. the initial transformation of the equations into a weak formulation was followed by integration over the physical domain. the solution diagram is shown schematically in fig. 2. backward differential formulation (bdf) was utilized to govern timesteps to fig. 2 flow chart of the numerical steps numerical study of a non-newtonian phase change flow in finned rectangular enclosures 295 maintain a relative computational accuracy of less than 10-4. all equations were completely coupled using the newton technique. [33] provides a full description of the numerical technique. using the finite element technique in conjunction with the bdf and newton approach assures the accuracy and dependability of the governing equations’ solutions. fig. 2’s solution diagram vividly illustrates the method, emphasizing the integration of the weak formulation over the physical domain. the technique outlined in this section is an integral part of the overall strategy for tackling the issue at hand. the reference to [33, 34] further highlights the rigor and depth of the study’s numerical methodology. 4. grid independency examination to discretize the computational domain, triangular elements were chosen. grid indecency examination is conducted for four grids of 8820 (g1), 24700 (g2), 35880 (g3), and 49420 (g4) elements for ra = 1×105, n = 0.6, pr = 1000. considering the dependency of the mvf on the grid size, the grid with the size of 24700 elements is appropriated to continue the computations. fig. 3 shows the impact of mesh size on the melting fraction (mvf) during the melting process. the utilized mesh for all cases is quite fine to prevent solution instabilities while the time step was controlled by the bdf scheme as discussed in the numerical method. as seen, the variation of mesh sizes has a minimal impact on the solution. thus, for a fair trade between the computational accuracy and computational cost, the rest of the results were computed using the g2 case. fig. 3 the effect of grid size on the melted volume fraction (mvf) 5. convergence analysis the developed computational code was validated utilizing numerical data from kebriti and moqtaderi [35] for the natural convection melting of power-law phase change material in a differentially heated cavity. it is worth mentioning that the dimensionless parameters fo m v f 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.2 0.4 0.6 0.8 1 g 1 g 2 g 3 g 4 296 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. in this validation are ra = 1.7×105, pr = 1.0, and ste = 0.5. fig. 4 shows a very good agreement for the melt flow structure and melting volume for different power-law indexes of the phase change material. n = 0.8 n = 1.0 n = 1.6 kebriti and moqtaderi [35] current work fig. 4 comparison between the obtained results for melting fields and data of kebriti and moqtaderi [35] as another validation, the findings of the present model were compared to those of prior research to verify the simulations' precision and accuracy. this research compares its results to those of kamkari et al. [36] and kamkari and amlashi [37], who studied the melting of lauric acid in an enclosure with a vertical hot wall at a temperature of 70 degrees celsius. the container was 50 mm in width and 120 mm in height. fig. 5 displays the findings for two parameters, namely the melting volume fraction (mvf) and the local temperature. the comparison demonstrates that the present study’s calculations are in reasonable agreement with experimental and numerical literature efforts. this comparison is essential for confirming the accuracy and dependability of the model used in the present research. it offers confidence in the robustness and validity of the simulations by proving that the outcomes are consistent with those of prior research. numerical study of a non-newtonian phase change flow in finned rectangular enclosures 297 (a) (b) fig. 5 comparison between the results of the current research and those of kamkari et al. [36] and kamkari and amlashi [37]: (a) local temperature and (b) mvf 6. results and discussion the alterable parameters in this study are non-newtonian index in the range of 0.6 to 1.4, fins number, and dimensionless time (fo). however, the other parameters, i.e., ste, ra, and pr, are considered fixed in 0.1, 105, and 1000, respectively. effects of the abovementioned parameters on flow structures and thermal patterns as well as average nusselt number and melt fraction illustrating the influence of non-newtonian properties have been scrutinized. fig. 6 demonstrates the influence of the power-law index on melting intensity and phase change line location in the case without fins. thus, regardless of the power-law index value, melting occurs from the right heated wall through the upper cavity part due to an influence of the natural convection mode. more intensive melting occurs for the pseudoplastic liquids when the power-law index is less than 1.0. a reduction of the viscosity with a growth of the shear stress can explain such a nature. as a result, the melting flow becomes more effective with high velocity compared to newtonian (n = 1.0) or dilatant liquids (n > 1.0). in the case of dilatant liquids, the liquid viscosity rises with the shear stress. in the case of n = 0.6, the whole melting occurs for fo = 0.5, while for n = 1.0, melting is not so effective, but in this case, a natural convection regime can also be found. for n = 1.4, less obvious distortion of the phase change line can be observed due to attenuation of convective flow and heat transfer with high viscosity with shear stress. time (min) t em p er a tu re (o c ) 0 50 100 150 200 250 300 350 25 30 35 40 45 50 55 60 65 70 t3: present study t12: present study t21: present study t30: present study experimental study [36] time (min) m v f 0 40 80 120 160 200 240 280 0 0.2 0.4 0.6 0.8 1 present study experimental study [36] numerical study [37] 298 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. n = 0.6 n = 1.0 n = 1.4 n = 0.6 n = 1.0 n = 1.4 fo = 0.125 fo = 0.25 fo = 0.375 fo = 0.5 fig. 6 melting contours over time in the cavity without fins for different values of n fig. 7 presents streamlines and velocity contours for different n with time. this figure can be considered an addition to the previous analysis to scrutinize the flow structure. as mentioned previously, for the pseudoplastic liquid (n = 0.6), more intensive melting occurs, where a counter-clockwise convective cell is formed within the melted part of the region. it is interesting to note that the maximum velocity zone location is displaced from the upper part to the bottom one with time, and these values decrease. such nature can be explained by a reduction of the temperature difference with time, and as a result, the influence of the buoyancy force becomes less strong than at the beginning of the process. with time, a displacement of the convective cell core to the bottom wall occurs due to the thermal properties of the melt; namely, the prandtl number of this liquid is high. in the case of newtonian liquid (n = 1.0), the considered time interval (0 < fo ≤ 0.5) is not enough for the whole melting of pcm within the cavity. still, the structure of the flow cell and maximum velocity zone location is similar to the previous case of dilatant liquid. for dilatant liquid (n = 1.4), it is possible to conclude that the melting occurs due to the heat conduction mode where plane-parallel circulation of the melt can be found. numerical study of a non-newtonian phase change flow in finned rectangular enclosures 299 fig. 7 velocity contours and streamlines over time in the cavity without fins for different values of n 300 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. as mentioned above, a rise in the power-law index characterizes a diminution of the velocity circulation within the chamber. a transition between the considered pseudoplastic liquid and newtonian liquid characterizes an essential diminution of the circulation velocity compared to the transition between the newtonian liquid and dilatant liquid. figs. 8 and 9 illustrate the flow structures and melt contours within the chamber with one solid fin. the addition of a solid fin characterizes essential melting compared to the case without fins (see figs. 6 and 7). in the case of pseudoplastic liquid pcm (n = 0.6) for fo = 0.1, one can find a formation of complex flow structure near the right heated wall, while for fo = 0.2, the top part of the cavity is wholly melted, and only the lowest part includes the solid phase of pcm. the whole cavity melting occurs up to fo = 0.4. it should be noted that circulation over the solid fin is weak and for fo > 0.3 one can find obvious circulation only under the solid fin. in the case of newtonian liquid pcm, the melting strength is less compared to that of the pseudoplastic liquid. n = 0.6 n = 1.0 n = 1.4 n = 0.6 n = 1.0 n = 1.4 fo = 0.1 fo = 0.2 fo = 0.3 fo = 0.4 fig. 8 a melting contours over time in the cavity with one fin for different n as shown, the strong circulation can be found only in the upper part of the chamber. due to more intensive melting of the upper part for fo > 0.3, an asymmetrical flow behavior can be found with intensive circulation in the zone over the solid fin. it should be noted that regardless of the whole melting of the upper part the circulation in this part is strong for fo = 0.4 also. in the case of dilatant liquid pcm (n = 1.4) for the considered time interval, symmetrical melting occurs for 0 < fo < 0.3, while for fo ≥ 0.3 more intensive melting occurs in the upper part with strong circulation in this part. numerical study of a non-newtonian phase change flow in finned rectangular enclosures 301 fig. 9 velocity and streamlines contours over time in the cavity with one fin for different values of n 302 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. as mentioned for the case without a fin, the convective mode is weak for dilatant liquids and it is suppressed with the power-law index. it is necessary to highlight that a transition between the considered pseudoplastic liquid and newtonian liquid characterizes more essential diminution of the circulation velocity compared to the transition between the newtonian liquid and dilatant liquid for initial timer level (fo < 0.3), while for fo ≥ 0.3 the opposite nature can be found. an influence of two solid fins on flow structures and melt contours is shown in figs. 10 and 11. an addition of the second fin allows intensifying the melting process and the charging level accelerates. the case of pseudoplastic liquid pcm characterizes strong melting and the time of fo = 0.36 reflects the whole melting within the chamber. it is interesting to note that the pseudoplastic liquid reflects an intensive circulation in the bottom part, while for other cases (n =1.0 and n=1.4) such strong circulation occurs in the middle and upper zones. a huge reduction of flow intensity occurs with a rise of the power-law index for dilatant fluid, where viscosity increases with the shear stress. n = 0.6 n = 1.0 n = 1.4 n = 0.6 n = 1.0 n = 1.4 fo = 0.09 fo = 0.18 fo = 0.27 fo = 0.36 fig. 10 melting contours over time in the cavity with two fins for different values of n numerical study of a non-newtonian phase change flow in finned rectangular enclosures 303 fig. 11 velocity contours and streamlines over time in the cavity with two fins for different values of n 304 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. the appearance of an additional solid fin illustrates a formation of additional recirculation between fins. in the case of n = 0.6 with melting the maximum velocity zone is displaced to the bottom part, while for other cases (n = 1.0 and n = 1.4) these zones are placed in the middle and upper part and fo = 0.36 is not enough for the whole melting of pcm within the energy storage system. taking into account these melting contours, the strong melting occurs in the central part where the extended heat transfer surface allows intensifying the thermal dissipation from the heated wall. figs. 12 and 13 demonstrate melting and velocity contours in the case of three solid fins. these distributions are similar to the previously studied ones. as a result, adding the third fin characterizes more intensive melting of pcm and a formation of additional recirculation between the solid fins. melting is more essential in the central part where solid fins are located, and huge heat dissipation can be found from the heated wall. moreover, comparing these considered cases for different numbers of fins, it is possible to conclude that the growth of the fins' number not only characterizes more essential melting of pcm but also accelerates flow in the convective mode due to flow zone narrowing. it is interesting to emphasize that in the case of pseudoplastic liquid, the velocity circulation decreases with time. in contrast, for other liquids (newtonian and dilatant), one can find an opposite behavior. n = 0.6 n = 1.0 n = 1.4 n = 0.6 n = 1.0 n = 1.4 fo = 0.075 fo = 0.15 fo = 0.225 fo = 0.300 fig. 12 melting contours over time in the cavity with three fins for different values of n numerical study of a non-newtonian phase change flow in finned rectangular enclosures 305 fig. 13 velocity contours and streamlines over time in the cavity with three fins for different values of n 306 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. fig. 14 illustrates the time evolution of the melt volume fraction for different numbers of fins and various power-law indexes. regardless of the number of fins, a rise of the power-law index leads to a growth of time of the whole melting of pcm within the chamber. as mentioned above, more intense melting occurs for the pseudoplastic liquid. an installation of solid fins at the heated wall leads to a decrease in the time of full melting of pcm. regardless of the solid fins number, the heat conduction level when nonnewtonian liquid does not affect the time behavior of the melt volume fraction can be found for the time interval 0 ≤ fo ≤ 0.05. while an addition of solid fins characterizes a growth of mvf during this heat conduction level. in the case of pseudoplastic liquid, a growth of n between 0.6 and 0.8 reflects an essential rise of the time of whole pcm melting, while for the dilatant liquid, a rise of the power-law index between 1.2 and 1.4 has a weak difference in this time of whole melting. moreover, an increment of n illustrates a reduction of time difference for the whole pcm melting between newtonian liquid and considered dilatant liquids. at the same time, the time differences for the pseudoplastic liquids are significant. time (s) m v f 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 1 .0 4 9 0 .5 0 2 0 .7 3 4 0 .9 2 4 1 .1 0 7 fo (a) time m v f 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 0 .7 1 9 0 .5 9 7 0 .7 9 2 0 .4 0 6 0 .8 2 6 1 fo (b) time (s) m v f 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 0 .6 7 6 0 .3 6 6 0 .5 0 3 0 .6 6 6 0 .6 1 1 fo (c) time (s) m v f 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 0 .5 7 4 0 .3 2 9 0 .4 6 1 0 .5 8 3 0 .5 6 5 fo (d) 0.4 fig. 14 normalized melted liquid fraction for the cavity: (a) without fins, (b) with one fin, (c) with two fins, and (d) with three fins different numerical study of a non-newtonian phase change flow in finned rectangular enclosures 307 fig. 15 shows the time evolution of pn with n for different numbers of solid fins. it should be noted that pn illustrates a relative difference in mvf between the current liquid and newtonian fluid. taking into account fig. 15, one can confirm the conclusion mentioned in the previous fig. 12, namely, an essential difference in mvf can be found between pseudoplastic liquid and newtonian liquid. at the same time, an inclusion of solid fins reduces differences between newtonian liquid and other pcm liquids. time p n 0.2 0.4 0.6 0.8 1 0 0.2 0.4 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (a) time p n 0.2 0.4 0.6 0.8 0 0.2 0.4 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (b) time p n 0.2 0.4 0.6 -0.1 0 0.1 0.2 0.3 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (c) time p n 0.2 0.4 0 0.1 0.2 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (d) fig. 15 dependency of pn parameter on the power-law index n for the cavity: (a) without fins, (b) with one fin, (c) with two fins, and (d) with three fins evolution of the average nusselt number at the heated wall with time, power-law index, and solid fins number is demonstrated in fig. 16. a reduction of the average nusselt number with time characterizes a diminution of the temperature gradient close to this wall due to heating of the surrounding fluid. the average nusselt number for dilatant liquids has a weak difference, while this difference is essential for pseudoplastic liquids. moreover, adding solid fins characterizes a rise in the average nusselt number. an achievement of zero value illustrates a whole melting of pcm. 308 m. boujelbene, s.a.m. mehryan, m. sheremet, et al. time n u to ta l 0.2 0.4 0.6 0.8 1 0 5 10 15 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (a) time n u to ta l 0 0.2 0.4 0.6 0.8 100 200 300 400 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (b) time n u to ta l 0.2 0.4 0.6 0 200 400 600 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (c) time n u to ta l 0.2 0.4 0 200 400 600 800 n=0.6 n=0.8 n=1.0 n=1.2 n=1.4 fo (d) fig. 16 dependency of nutotal on n for the cavity: (a) without fins, (b) with one fin, (c) with two fins, and (d) with three fins 7. conclusions numerical analysis of non-newtonian pcm convective melting in a differentiallyheated cavity with solid fins has been performed. governing equations formulated using the power-law approach for non-newtonian liquid, boussinesq approximation for buoyancy force, and enthalpy-porosity approach for description of the melting process have been solved numerically by the finite element technique. the developed computational code has been validated using the literature numerical and experimental outcomes. effects of the power-law index, solid fins number, and time on the flow structures and melting patterns have been scrutinized. it has been found that: ▪ strong melting of pcm occurs for pseudoplastic fluids (n < 1) where the melting time is minimum compared to newtonian and dilatant fluids. ▪ a growth of the power-law index characterizes a rise of the melting time for pcm within the considered thermal energy storage system. numerical study of a non-newtonian phase change flow in finned rectangular enclosures 309 ▪ an increment of n for dilatant liquids does not have essential differences in melting time and average nusselt number. ▪ an addition of solid fins results in a reduction of the melting time and intensification of the phase change process. the melting zone is placed near the thermal dissipation zone, namely, close to the fins' location. ▪ an increment of the power-law index reflects a diminution of the velocity circulation 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convective non-newtonian power-law solid-liquid phase change using the lattice boltzmann method, international journal of thermal sciences, 159, 106574. 36. kamkari, b., shokouhmand, h., bruno, f., 2014, experimental investigation of the effect of inclination angle on convection-driven melting of phase change material in a rectangular enclosure, international journal of heat and mass transfer, 72, pp. 186-200. 37. kamkari, b., amlashi, h.j., 2017, numerical simulation and experimental verification of constrained melting of phase change material in inclined rectangular enclosures, international communications in heat and mass transfer, 88, pp. 211-219. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 1, 2014, pp. 37 50 application of the craig-bampton model order reduction method to a composite structure: mac and xor  udc (531+624.01) humberto peredo fuentes, manfred zehn institute of mechanics, technical university berlin, germany abstract the craig-bampton model order reduction (cbmor) method based on the rayleigh-ritz approach is applied to dynamic behavior simulation of a composite structure in order to verify the method's feasibility and accuracy. the principle of this method is to represent a coupled component model based on the mass, damping and stiffness matrices. the methodology consists of a finite element model based on the classical laminate theory (clt), a design of experiment to improve the modal assurance criteria (mac) and experimental results in order to validate the reduced model based on cbmor method and substructures (super-elements). experimental modal analysis has been performed using a scanner laser doppler vibrometer (sldv) in order to assess the quality of the finite element models. the mac and cross orthogonality mac (xor) values are computed to verify the eigenfrequencies and eigenvectors. this approach demonstrates the feasibility of using cbmor for composite structures. the example is prepared and solved with msc/nastran sol103. the design of experiments (doe) method has been applied in order to identify the critical parameters and thus obtain high mac values. key words: sdtools-matlab, nastran, modal analysis, composites 1. introduction many techniques have been proposed to obtain reduced order finite element models (known as model order reduction (mor) methods) by reducing the order of mass and stiffness matrices of structures made of conventional materials [1-3]. the substitution of conventional materials by composite materials in the aeronautic, space and automotive industry is becoming increasingly important today for the production of industrial highperformance components [11-13]. the state-of-the-art mor techniques are classified in received november 27, 2013 / accepted february 4, 2014  corresponding author: humberto peredo fuentes technical university berlin, institute of mechanics, berlin, germany e-mail: hperedo@mailbox.tu-berlin.de original scientific paper 38 h. peredo fuentes, m. zehn four groups [19]: direct reduction, modal methods, reduction with ritz vectors and the component mode synthesis (cms). according to this classification, the last two groups yield the best results. the ritz vectors improve the accuracy-cost ratio and the cms combines the first three classes of methods. hence the mor method based on the rayleigh-ritz approach is used to improve the accuracy-time ratio in civil and aeronautical engineering applications in many areas of structural dynamics [6, 14, 19, 22, 23]. thus, it is necessary to study the feasibility and efficiency of using the cms with the rayleigh-ritz reduction basis in order to describe the dynamic behavior of a composite structure [14, 19]. the sections 2-4 introduce to mor based on the ritz vectors, classical cms and substructures, respectively. the classical laminate theory (clt) is introduced in section 5. sections 6-8 demonstrate a sensitivity analysis performed by using different tools – design of experiment (doe), finite element method (fem) and modal assurance criteria (mac). 2. model order reduction with ritz vectors it is typical for coupled problems with model sub-structuring [6, 14, 22, 23] to have an accurate second order representation in the form: 2([ ] [ ] [ ]){ } [ ]{ } { } [ ]{ } m s c s k q b u y c q     , (1) where s is the laplace variable, [m], [c], [k] are mass, damping and stiffness matrices, respectively, {q} are generalized degrees of freedom (dofs), [b] and [c] are input and output matrices, respectively, {u} are the inputs describing the time/frequency dependence, and {y} are the physical outputs. in this description, two not very classical and yet important assumptions are made: 1) the decomposition of discretized loads f(s) as the product of the fixed input shape matrix specifying the spatial localization of loads [b] and inputs {u}. 2) the definition of physical outputs {y} is a linear combination of dofs {q}. the ritz/galerkin displacement methods seek approximations of the response within a subspace characterized by matrix [t] associated with generalized dofs {qr}: }]{[}{ rqtq  , (2) where {q} is the original set of dof and {qr} is the reduced set of dof, substituting eq. (2) into eq. (1) leading to an overdetermined set of equations. the ritz approximation assumes that the virtual work of displacements in the dual subspace generated by [t] t is also zero, thus leading to a reduced model:  [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] { ( )} [ ] [ ] { ( )} { ( )} [ ] [ ] { ( )} t 2 t t t r r t m t s t c t s t k t q s t b u s y s c t q s     . (3) application of the craig-bampton model order reductio method to a composite structure: mac and xor 39 3. classical cms bases as approximation of the frequency response the method was first developed by walter hurty in 1964 [1] and later expanded by roy craig and mervyn bampton [2] in 1968. component mode synthesis and model reduction methods provide for the means for building appropriate [t] bases (the subspace spanned rectangular matrix). there are many ways of proving classical bases [22]. their validity is associated with two assumptions: the model needs to be valid over a restricted frequency band and the number of inputs is limited. one needs to translate this hypothesis into the requirement to include mode shapes and static responses into [t] basis. most of the literature on cms implies the fundamental assumption for coupling, which states that the displacement is continuous at the interfaces. considering the response of an elastic structure to applied loads f(s)=[b]{u(s)}, the exact response at a given frequency [h(s)] is given by: 2 1 1[ ( )] [ ]{[ ] [ ]} [ ] [ ][ ( )] [ ]h s c m s k b c z s b    , (4) where [z(s)] is the dynamic stiffness. if there is no external excitation: 1[ ( )] { } {0}j jz    , (5) and the solutions are known as free modes of the structure, where j is j th eigenvalue of the matrix and {j} is j th eigenvector. a reduction model should include these shapes to allow for an accurate representation of the resonances which are associated with the singularities of the dynamic stiffness. a point of particular interest is the static response at s=0. the associated deformation is: 1{ ( 0)} [ (0)] [ ]{ (0)} [ ]{ (0)}sq s z b u t u   . (6) the columns of [ts] are also called attachment modes [22]. for the case of free floating structures (structures with rigid modes), [z(0)] is singular and one defines attachment modes as responses of all modes except for the rigid body modes. the bases combining free modes and attachment modes are valid over a certain frequency range (truncation of the series of free modes) and certain inputs characterized by [b]. one, thus, considers the response of the structure with enforced displacements on a subset of dofs. division of the dofs in two groups – active or interface dofs denoted by i in the subscript, and complementary, denoted by c in the subscript, leads to: [ ( )] [ ( )] { ( )} ( ) , [ ( )] [ ( )] ( ) {0} ii ic i i ci cc c z s z s q s r s z s z s q s                   (7) where <{qi(s)}> and <{0}> denotes a known quantity. the exact solution to this problem is: 1 [ ] { } [ ( )]{ } { } [ ( )] [ ( )] i i cc ci i q t s q q z s z s         . (8) the subspace found here is frequency dependent and can only be used in very restricted applications [23]. a classical approximation is to evaluate the static (s=0) value in this subspace for the active or interface dofs denoted by ci in the subscript, and complementary, cc in the subscript: 40 h. peredo fuentes, m. zehn 1 [ ] [ ] [ ] [ ]cc ci i t k k           . (9) reduction on this basis is known as static or guyan condensation [4]. the columns of [t] are called constraint modes [22]. they correspond to unit displacements of the interface dofs. significant deviations can be expected when [zcc(s)] -1 differs from [zcc(0)] -1 =[kcc] -1 such difference is significant for singularities of [zcc(s)] -1 which are computed by the eigenvalue problem: , [0] [0] 0 0 [0] [ ( )]cc j j cz              . (10) the use of a basis combining constraint, eq. (9), and fixed-interface modes, eq. (10), is proposed in [2]. it yields the craig-bampton method: 1 [ ] [0] [ ] [ ] [ ] [ ]cc ci nm,c i t k k            , (11) where [nm,c] is the interior part of the matrix of kept fixed-interface modes. there are many results reported by balmès et al. [6, 14, 15] obtained by the craig-bampton model order reduction (cbmor) and the rayleigh-ritz vectors approach in order to solve coupled problems related to model sub-structuring (also known as component mode synthesis). one should be aware of the fact that the use of raleigh-ritz vectors leads to dense matrices, as opposed to not reduced fem models characterized by a sparse form of the matrices. 4. substructures or super-elements sub-structuring is a procedure that condenses a group of finite elements into one element. it implies that the whole structure is divided into smaller structures (see figs. 1 and 2), and the resulting elements are referred to as super-elements. in the considered case (fig. 1), the structure is divided into two substructures using 123 nodes at the interface. the model size is reduced from 37,698 dof to 579 dof. a) b) fig. 1 prototype and fem model in nastran and sdtools: a) composite structure – front and back; b) fem model application of the craig-bampton model order reductio method to a composite structure: mac and xor 41 the basic sub-structuring idea is to consider a part of the model separately and extract the degrees of freedom needed to connect this part to the rest of the model. therefore, the result of sub-structuring is a collection of finite elements whose response is defined by the stiffness and mass of the retained degrees of freedom. the categories of modal truncation sub-structuring and static condensation approaches have been widely applied relying on the eigenfrequency information [3, 23]. a) b) fig. 2 sub-structuring: a) substructure 1; b) substructure 2 5. laminate theory the classical laminate theory is applicable to linear and composite elastic materials [21] by means of the discrete kirchhoff theory (dkt) elements [20]. the clt has been used extensively to predict elastic behavior of the traditional fiber-reinforced polymers (frp). frp materials (carbon or glass frp) are widely used in aerospace and construction applications. one important consideration is to have perfectly bonded layers with a uniform thickness (see fig. 3). the mechanical properties measured in ply level experiments are used to populate the stiffness matrix for each ply. the stiffness matrices for the individual plies are combined to form the laminate stiffness matrix – the abc matrix: xx yy xyxy xx yy xyz n n n a b m b c m m                                            . (6) the abc matrix relates forces (ni) and moments (mi) to strains (i) and curvatures (i). the components of the abc matrix are given in eqs. (7-9), where n is the number of plies, qk is the stiffness matrix of each ply, and zk denotes the distance from the laminate's mid-plane to the edges of single plies: 1 1 [ ], in-plane stiffness matrix, n k k k k a q z z      (7) 42 h. peredo fuentes, m. zehn 2 2 1 1 1 [ ], bending-stretching coupling matrix, 2 n k k k k b q z z      (8) 3 3 1 1 1 [ ], bending-stiffness matrix. 3 n k k k k c q z z      (9) fig. 3 configuration of composite layers the prototype and the finite element (fe) model are shown in fig. 1. the composite structure incorporates three parts (properties in table 1). the first component is made of hunts-man ly 564 + hexcel gewebe g0926 (hta-faser) with dimensions of 0.390m  0.810m (fig. 2). the middle shell that connects the two principal parts (fig. 2a) has dimensions of 0.710m  0.030m. finally, there is the c-section hexcel rtm6 + saertex multi-axial-gelege (mag) with a im7-faser with dimensions of 0.710m  0.030m. all the parts have symmetric layer distribution [45/-45/45/-45/]s. 6. design via finite element analysis (full and reduced model) our study is divided into two parts. the first part is a full modal analysis using the same model but with two different solvers for reference purposes. two types of elements have been used: ctria3 shell (from msc/nastran) and pshell (from sdtools). the second part is setting the reduced model by using sdtools for matlab. the reduced model is built up defining two super-elements. super-element 1 (fig. 2a) has 4,753 nodes and 9,219 elements, while super-element 2, (fig. 2b) has 1,615 nodes and 3,026 elements. the defined super-elements share 579 dof distributed in 123 nodes along the common border with different dof per node, according to the cms that has defined an appropriate [t] matrix, used in [3]. we have calculated the same number of modes in each super-element and performed a cross orthogonality mac (xor) evaluation to verify the approximation of the mor used in low (12 mode pairs) and/or high frequency range (29 mode pairs) versus the full model. in order to estimate the main parameters (qualitative and quantitative) that affect our mor based on the number of substructures and nodes, we have performed a doe using first the full model and the experimental analysis. the doe study is performed using the methodology implemented in minitab 16 [7]. fig. 4a shows the main effects of each parameter in the composite structure based on the physical characteristics selected. the application of the craig-bampton model order reductio method to a composite structure: mac and xor 43 main effect is identified through the slope generated due to the eigenfrequency values between the limits defined for each parameter – a bigger slope means a strong parameter effect. due to the number of parameters, it is necessary to perform first a doe-screening with 2 10-5 =32 "runs" and then a full factorial with the identified principal parameters based on the doe-screening. a) b) fig. 4 doe: a) parameters main effects, b) surface response the results shown in fig. 5a (vertical left side) are eigenfrequencies. the mac correlation between the full model and experimental data help us validate the mor results. the young modulus, density, number of nodes and substructure parameters have a strong influence reflected in the slope (fig. 4b) and in the mac values (section 7). once we have selected the main parameters based on the doe-screening, we perform a doe full factorial 2 4 and obtain a surface response (see fig. 4b) that help us find the best model for the parameter limits selected. this process is known in literature as updating. jing [8], barner [9] and xiaoping et al. [10] reported the use of design of experiments in order to quantify and qualify different key parameters in mechanical components (stresses, displacements, low and high cycle fatigue, and frequencies). the doe is a sensitivity analysis tool used to estimate the critical input parameters. a) b) fig. 5 cross orthogonality mac reduced vs. full model: (a) low frequencies (b) higher frequencies (green bars mac, blue bars frequency difference) 44 h. peredo fuentes, m. zehn in fig. 5, we can see the low and high mode pairs selected between the full and reduced model (12 and 29 mode pairs), respectively. the green bars show the eigenvector criteria and the blue bars the eigenfrequency difference between the reduced and the full model. the low frequencies show a larger difference in the 3 rd , 4 th and 12 th mode pair. the largest difference in the frequencies is about 1.2% (low eigenfrequencies) between the full and reduced model. increasing the number of pairs, the eigenfrequency difference increases up to 3% for 29 pairs. however, the mode pairs 3, 4 and 12 have improved suggesting that the accuracy using cbmor method depends on the number of retained constraint modes. most of the pair selections have a correlation above 90%, except for the 12 th mode pair in the low frequency range and the 23 rd and 24 th pair in a high frequency range. table 2 shows the values comparing the full with a reduced model for low frequency. a 3d plot of the xor for high frequency pairs is given in fig. 6. table 1 orthotropic elastic mechanical properties per thickness modulus th1(m) e(gpa) (-) shear g(gpa) ρ(kgm-3) e1 0.035 71.3 0.3 g1 7.0 2600 e2 97.3 0.3 g2 5.0 g3 7.0 modulus th2(m) e(gpa) (-) shear g(gpa) ρ(kgm-3) e1 0.007 71.3 0.2 g1 6.0 1500 e2 68.3 0.2 g2 5.0 g3 6.0 modulus th3(m) e(gpa) (-) shear g(gpa) ρ(kgm-3) e1 0.035 71.3 0.2 g1 6.0 1500 e2 68.3 0.2 g2 5.0 g3 6.0 table 2 mac values: full versus cbrom reduced model  full  reduced df/fa mac 7 57.218 7 57.218 0.0 100 8 106.02 8 106.21 0.2 100 9 167.50 9 167.79 0.2 93 10 168.2 10 168.29 0.1 94 11 234.99 11 235.12 0.1 100 12 236.83 12 236.93 0.0 100 13 315.26 13 315.33 0.0 100 14 323.93 14 326.82 0.9 98 15 401.72 15 401.77 0.0 100 16 408.39 16 408.57 0.0 100 17 432.89 17 433.39 0.1 99 18 494.90 18 501.41 1.3 69 the correlation of nearly double modes 9-10,11-12,13-14 and 15-16 in table 2 suggests the possibility of having bending and torsional modes at close frequencies in the composite structure (mode veering) [24]. thus, a lower mac value is expected in some mode pairs in the experimental validation. there are only three types of structures made of application of the craig-bampton model order reductio method to a composite structure: mac and xor 45 the conventional materials that have been identified to exhibit veering: symmetric or cyclic structures, multi-dimensional structures such as plates having bending and torsion at close frequencies and structures with fully uncoupled substructures. the considered structure corresponds to the second type – multi-dimensional plate structures. 7. modal assurance criterion (mac) there are two general categories for correlation criteria: eigenfrequencies and eigenvectors [18]. the mac is one of the most useful comparison methods that relies on the eigenvector information according to eq. (10). the mac is a known vector correlation between the experimental and the fe model. to approximate the measurements through a polynomial function, (fig. 9), we use the frequency domain identification of structural dynamics applying the pole/residue parameterization [15]. )()()()( )()( )( 11 2 1 kj h l j kjidj h l j idj kj h l j idj cccc cc imac                (10) the mac value of 100 % corresponds to an absolute correlation. the less this value becomes, the worse the eigenvector correlation is (cjid is the j th mode shape at sensors and cjk is the j th analytical mode shape), provided that the observability law for the selection of dofs is not violated. a mac coefficient of a magnitude larger or equal than 90% implies a satisfactory correlation. in fig. 8, we observe some mode shapes of the reduced and full models. figs. 10a, 10b, and 10c, show the mac between the full and the experimental measurements in matlab, nastran, and cbmor model, respectively. the correlation is performed for a low frequency range (up to 400 hz), based on the fitting model generated from the experimental measurements [3, 15]. fig. 6 cross orthogonality mac (xor): higher frequencies  reduced vs. full model 46 h. peredo fuentes, m. zehn fig. 7 frf(blue) and fitting curve (green) of composite model at node 183y 8. experimental modal analysis all the measurements are performed with the scanning laser doppler vibrometer (sldv) psv 840 (fig. 9a). it is a complete and compact system including a sensor head, a pc with dsp boards and windows nt-based application software packages [16]. discrete fourier transform is applied to response x(t) and excitation f(t) to give x(ωi) and f(ωi), respectively [17]. the frequency response function (frf), h(ωi), is defined as the ratio of the transformed excitation [18]: )( )( )( i i i f x h    , (11) where h(i) is the identified (predicted) frf transfer function matrix , h(i) the measured frf transfer function matrix, x(i) the fourier spectrum of response, and f(i) is the fourier spectrum of excitation force. the frf in eq. (11) is the inverse of the dynamic stiffness matrix: 2 1( ) [ [ ] [ ] [ ]]i i ih m c k       . (12) mass [m], damping [c] and stiffness [k] matrices in eq. (12) are dependent on physical parameters such as material's density, young's and shear moduli and poisson ratio. fig. 8 cbmor (in green) vs full model in matlab (in blue) mode 7 at 57.22 hz, mode 7 at 57.22 hz reduced mode 8 at 106 hz, mode 8 at 106.22 hz reduced mode 9 at 167.5 hz, mode 9 at 167.8 hz reduced mode 10 at 168.2 hz, mode 10 at 168.3 hz reduced application of the craig-bampton model order reductio method to a composite structure: mac and xor 47 a) b) c) fig. 9 experiment: a) experimental set-up; b) 153 y-direction sensors; c) 153 sensors in the fem model the sldv employs a laser to sweep over the structure continuously while measuring, capturing the response of the structure from a moving measurement point. various methods have been devised to determine the mode shapes of the structure everywhere along the scan path measurement [16]. a bandwidth of 2% is used in order to localize the eigenfrequencies. the composite structure has rather small internal damping and the experimental modal analysis below 400 hz is performed. the structure is excited by means of a shaker at node 17 (fig. 9a and fig. 9b) that is located in the right bottom corner. the input force is measured using a force transducer type 8200 in combination with a charge to ccld converter type 2646 in order to record the excitation in the transverse direction. the interpolation between the experimental measurements uses frequency response functions (frf) [15], (fig. 7). the frfs allow comparison of the experimental modal parameters (frequency, damping, and mode shape) with those of the fe model. the fast fourier transform (fft) is a fundamental procedure that isolates the inherent dynamic properties of a mechanical structure and in our case with respect to the full and reduced fe model. the mac analysis (fig.10) shows a high correlation between the full model, the reduced model and the experimental measurements. the nearly double correlation in the experimental results identified in table 3 (previously identified applying the cbmor method in table 2), suggests the presence of the veering phenomena (bending and torsional mode at the same frequency) in the considered composite structure. this is reflected in the mac values for the corresponding modes. a) b) c) fig. 10 comparative mac: a) sdtools-exp, b) msc/nastran-exp, c) cbmor-exp 48 h. peredo fuentes, m. zehn fig. 11 experimental mode shapes pierre [25] reported how localization and veering are related to two kinds of "coupling": the physical coupling between structural components, and the modal coupling set up between mode shapes through parameter perturbations. his studies show that, in structures with close eigenvalues, small structural irregularities (could be our case) result in both strong localization of modes and abrupt veering away of the loci of the eigenvalues when these are plotted against a parameter representing the system disorder. the study of the presence of this phenomenon in the composite structure is beyond the scope of this work. table 3 shows the mac values obtained for each case between the full and reduced model versus the experimental results. the mode shapes depicted in fig. 11 are the experimental results. table 3 full and reduced fem model results versus experimental results  experimental  full df/fa mac cbmor df/fa mac 1 49.243 7 57.218 16.2 100 57.218 16.2 100 2 92.265 8 106.02 14.9 97 106.21 15.1 97 3 93.756 8 106.02 13.1 90 106.21 13.3 90 4 145.29 10 168.20 15.8 83 168.29 15.8 84 5 160.05 10 168.20 5.1 86 168.29 5.1 71 6 164.18 9 167.50 2.0 98 167.79 2.2 92 7 226.36 12 236.83 4.6 86 236.93 4.7 85 8 243.40 11 234.99 -3.5 96 235.12 -3.4 97 9 307.33 14 323.93 5.4 81 326.82 5.4 80 10 314.18 14 323.93 3.1 66 326.82 4.0 65 11 324.83 13 315.26 -2.9 74 315.33 -2.9 74 12 329.67 13 315.26 -4.4 90 315.33 -4.3 89 application of the craig-bampton model order reductio method to a composite structure: mac and xor 49 9. conclusions the results have shown a good correlation in dynamic behavior of the composite structure model using the dkt elements with different solvers. the mac values with the full and reduced models have also shown a good agreement with the experimental results. in order to achieve high quality models that can adequately capture the dynamic behavior, the material properties are updated through the doe and are crucial in the mor correlation with the experimental results. the updated mass and stiffness matrices in the full model play an important roll in this procedure. furthermore, the reduced model obtained by means of the craig-bampton mor method (the reduced model couples 2 substructures through 123 nodes and 579 dof) has demonstrated a good agreement with the experimental results. the mac values for the fem models as well with the experimental results suggest a presence of mode veering phenomenon (bending and torsional mode at the same frequency in the considered composite structure). and finally, the experimental results using a sldv as well as the identification of pole/residues used in [15], are suitable to validate the dynamic analysis using modal order reduction. it is improper to draw conclusions from a single example, but the obtained results using two different solvers are coherent. this conducted work obviously leaves much room for further research. other modal assurance criteria need to be performed, such as coordinate modal assurance criteria (comac), enhanced modal assurance criteria (ecomac) and scale coordinate assurance criteria (s-comac) and also other model order reduction and/or mode shape expansion methods should be assessed. references 1. hurty, w. c., 1965, dynamic analysis of structural systems using component modes, aiaa journal, 3(4), pp. 678-685. 2. craig r. j. and bampton m., 1968, coupling of substructures for dynamic analyses, aiaa journal 6(7), pp.1313-1319. 3. sdtools inc. 2011, structural dynamics toolbox and femlink, user's guide, sdtools, ver. 6.4, paris, france. 4. guyan, j. 1965, reduction of stiffness and mass matrices, aiaa journal, 3(380). pp. 5. irons, b. m.., 1965, structural eigenvalue problems elimination of unwanted variables, aiaa journal, 3(5): pp. 961-962. 6. balmès e., 1996, use of generalized interface degrees of freedom in component mode synthesis, international modal analysis conference, pp. 204-210. 7. montgomery, d. c., 2000, design and analysis of experiments, john wiley & sons. 8. fan j., zeng, w., wang r., sherr x., chen z., 2010, research on design and optimization of the turbine blade shroud, 2nd international conference on engineering optimization, lisbon, portugal. 9. barner, n., 2010, isight-abaqus optimization of a ring-stiffened cylinder, simulia customer conference. 10. chen, x., yu, x., and ji b., 2010, study of crankshaft strength based on isight platform and doe methods, international conference on measuring technology and mechatronics automation, pp. 548-551. 11. lauwagie, t., 2005, vibration-based methods for the identification of the elastic properties of layered materials, phd thesis, catholic university of leuven, belgium. 12. reddy, j. n., 2005, mechanics of laminated composite plates and shells theory and analysis, crc, press second edition. 13. berthelot, j. m., 1992, materiaux composites: comportement mecanique et analyse des structures, lavoisier, paris, france 14. balmès e., 1997, efficient sensitivity analysis based on finite element model reduction, international modal analysis conference, imac, pp.1-7. 15. balmès e., 1996, frequency domain identification of structural dynamics using the pole/residue parametrization, international modal analysis conference, pp. 540-546. 50 h. peredo fuentes, m. zehn 16. gade, s., møller, n.b., jacobsen, n.j., and hardonk. b., 2000, modal analysis using a scanning laser doppler vibrometer, sound and vibration measurements, pp. 1015-1019. 17. newland, d.e., 1993, an introduction to random vibration, spectral and wavelet analysis, new york, longman, harlow and john wiley. 18. ewings, d. j., 1995, modal testing: theory and practice, research studies press, letchworth, united kingdom. 19. cunedioğlu, y., muğan, a., akçay, h., 2006, frequency domain analysis of model order reduction techniques, finite elements in analzsis and design, 42, pp. 367-403. 20. batoz, j.l., bathe, k.j., ho, l.w., 1980, a study of three node triangular plate bending elements, international journal for numerical methods in engineering, 15, pp. 1771-1812. 21. batoz, j.l., lardeur, p., 1989, composite plate analysis using a new discrete shear triangular finite element, international journal for numerical methods in engineering, 27, pp. 343-359. 22. craig, r.j., 1987, a review of time-domain and frequency domain component mode synthesis methods. int. j. anal. and exp. modal analysis, 2(2), pp. 59-72. 23. balmès, e., 2000, review and evaluation of shape expansion methods, international modal analysis conference, pp. 555-561. 24. bonisoli e, delprete c., espoito m., mottershead j. e., 2011, structural dynamics with conicident eigenvalues: modeling and testing, modal analysis topics 3, pp 325-337. 25. pierre c., 1988, mode localization and eigenvalue loci of bridges with aeroeslastic effects, journal of engineering mechanics 126(3), pp. 485-502. primena craig-bampton redukcije modela na strukturu od kompozitnog materijala: mac i xor craig-bampton metoda za redukciju modela (cbmor) zasnovana na rayleigh-ritz pristupu je primenjena u simulaciji dinamičkog ponašanja kompozitnih struktura u cilju verifikacije izvodljivosti i tačnosti ove metode. princip ove metode je da predstavi model spregnutih komponenti preko matrica inercije, prigušenja i krutosti. metodologija uključuje model primenom konačnih elemenata (mke) na osnovu klasične teorije laminata (clt), zatim postavku eksperimenta sa ciljem poboljšanja vrednosti koeficijenata poređenja modova (mac), kao i eksperimentalne rezultate sa ciljem validacije redukovanog modela primenom cbmor metode i substruktura (superelemenata). eksperimentalna modalna analiza je sprovedena korišćenjem laserskog doplerovog vibrometra da bi se ocenio kvalitet mke modela. mac vrednosti za pripadajuće i nepripadajuće modove su sračunate da bi se verifikovale sopstvene frekvence i modovi. ovaj postupak pokazuje izvodljivost primene cbmor redukcije modela u slučaju kompozitnih struktura. model je pripremljen i rešen primenom programskog paketa msc/nastran sol103. metodom dizajna eksperimenta identifikovani su kritični parametri, što je kasnije omogućilo dobijanje visokih mac vrednosti. ključne reči: sdtools-matlab, nastran, modalna analiza, kompozitni materijali 11987 facta universitatis series: mechanical engineering vol. 21, no 3, special issue, 2023, pp. 405 432 https://doi.org/10.22190/fume230616020d © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim arithmetic and geometric aggregation operators in single value neutrosophic environment nagarajan deivanayagampillai1, kavikumar jacob2,3, gobinath vellapalayam manohar4, said broumi5 1department of mathematics, rajalakshmi institute of technology, chennai, india 2department of mathematics and statistics, faculty of applied sciences and technology, universiti tun hussein onn malaysia, pagoh campus, malaysia 3department of biosciences, saveetha school of engineering, saveetha institute of medical and technical sciences, chennai, india 4department of mechanical engineering, rajalakshmi institute of technology, chennai, india 5laboratory of information processing, faculty of science ben m’sik, university hassan ii, sidi othman, casablanca, morocco abstract. industry 5.0 acceptance is accelerating, but research is still in its infancy, and existing research covers a small subset of context-specific obstacles. this study aims to enumerate all potential obstacles, quantitatively rank them, and assess interdependencies at the organizational level for industry 5.0 adoption. to achieve this, we thoroughly review the literature, identify obstacles, and investigate causal relationships using a multi-criteria decision-making approach called single value neutrosophic todim. single-valued neutrosophic sets (svns) ensembles are employed in a real-world setting to deal with uncertainty and indeterminacy. the suggested strategy enables the experts to conduct group decision-making by focusing on ranking the smaller collection of criterion values and the comparison with the decision-making trial and evaluation laboratory method (dematel). according to the findings, the most significant hurdles are expenses and the funding system, capacity scalability, upskilling, and reskilling of human labor. as a result, a comfortable atmosphere is produced for decision-making, enabling the experts to handle an acceptable amount of data while still making choices. key words: industry 5.0, risk mitigation, neutrosophic sets, aggregation operators, todim, dematel received: june 16, 2023 / accepted july 29, 2023 corresponding author: nagarajan deivanayagampillai department of mathematics, rajalakshmi institute of technology, chennai, india. e-mail: dnrmsu2002@yahoo.com 406 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi 1. introduction a special case of a neutrosophic set known as a single-valued neutrosophic set (svns) is one in which the membership functions for truth, indeterminacy, and falsehood all fall inside the standard interval [0, 1]. single-valued neutrosophic sets (svns) may effectively and flexibly handle indeterminate and inconformity information in addition to representing imperfect information in the real decision system. they have been proposed recently and are a subclass of neutrosophic sets. using single-valued neutrosophic sets to address challenges involving several factors for making decisions is a crucial application. utilizing human expertise in conjunction with machines to deliver results in the most effective way is the major goal of industry 5.0. this integrates human-machine cooperation, might have an influence on manufacturing companies [1]. they could use the efficiency and development of robots in tandem with human intellect to make well-understood judgements and handle extraordinary situations at the same time. industry 4.0 focuses on digitization and linked automation, whereas industry 5.0 is the upcoming new technology that focuses on the interaction between humans and machines to improve customer engagement [2]. managers also believe that implementing industry 5.0 may lead to expense savings, increased efficiency, mass customization, and improved operational overall effectiveness [2]. many businesses are attempting to implement industry 5.0 to reap the benefits of its advantages. for example, to handle the industrial 5.0 revolution, unity twin inc. is considering implementing the smart industrial software suite. even though the covid19 pandemic has considerably accelerated human-digital tool and technology cooperation and fundamentally altered the nature of employment in the digital age, many sectors of the economy of growing nations should adopt new industrial advancement like industry 5.0 [3]. even though the covid-19 pandemic has considerably accelerated human-digital tool and technology cooperation and fundamentally altered the nature of employment in the digital age, many sectors of the economy of growing nations should adopt new industrial advancement like industry 5.0 [3]. this restraint or absence of a suitable path to address the acquisition hurdles results in delayed growth and progress. we want to create a theoretical structure that will help executives get beyond the obstacles to implementing industry 5.0 while taking this goal into account [4]. several nations' gross domestic products are greatly boosted by the manufacturing and construction sectors (gdp). development projects may be seen as a plan manufacturing company while the goods in the sector are often bigger in size. manufacturing (constructing) ships and aero planes disturbs the distinction between the two industries. the obstacles confronted by the decision-maker when choosing good equipment to work in a manufacturing process or a building project are essentially the same. for many manufacturing and construction companies, choosing the best machine has lately emerged as a crucial issue in decision-making to gain a strategic edge for existence in the international marketplace [5]. poor technology choice choices have a detrimental effect on user satisfaction by raising quality and maintenance expenses and reducing return on investments. decision-makers must have the requisite skills and expertise since technology acquisition is a labor-intensive and challenging procedure. scientists have developed methods to assist decision-makers in response to evaluating the technology selection challenge. the statistical model-based sensible decision sustains approaches offer a comprehensive strategy that makes use of the knowledge already in place and, in addition, provides a glimpse that can aid the decisionmaker in analyzing the decisions that are made consequently [5]. it is difficult to create a investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 407 decision making system for choosing an appropriate technology. it entails selecting from a wide selection of other technologies that are offered on the market. a construction or manufacturing company that is focused on making a profit would also like to reduce the cost of a machine. yet, choosing only one price might lead to poor judgements by ignoring several elements that may be just as significant as price. technical, environmental, logistical, and ergonomic considerations might all be included. due to these factors, the decision support models employed for machine selection adhere to an mcdm framework [5]. the manufacturing techniques and procedures used by smes need to be reviewed, and they must establish an environment created by combining the most recent manufacturing techniques and administrative procedures [6]. the use of new or advanced manufacturing technology (amt) has been seen as a potential means of increasing manufacturing companies' productivity and reducing their expenses. there are problems with the efficient use of these technologies, despite efforts to define and highlight the advantages and contributions of technology for smes. amt design, setup, and challenges may make it difficult for smes to profit from new technology [7]. the performance of businesses adopting amt is not just dependent on whether the technology used is cutting edge or not. while technology itself has not directly affected how effectively amts operate in the workplace, implementation quality is a key consideration [8]. to tackle various decision-making issues and consider the decision maker's (dm) cognitive behavior during the decision-making process in an unstable world, fuzzy todim methods have been employed widely and successfully. the advancement of techniques to improve the management of complex efficiency and decision-making factors involving non-probabilistic insecurity is one way to go about this [9]. this is done with the intention of understanding, developing, and using fuzzy technologies in areas like economics, technology, management, and systemic problems. fuzzy analysis is a technique used in many fields, including engineering, to solve problems involving doubt and ambiguity [9]. it has uses in manufacturing, management, and decision-making issues. the fuzzy set ideas have a great opportunity for implementation to decision-making issues because they include specific modelling properties that are specific modifications of crisp and fuzzy relations, respectively [10]. fuzzy logic can appear in management and decision-making processes with implications for manufacturing, where it is used as a model to assess the effectiveness of modern production systems. industry 5.0 acceptance is accelerating, but research is still in its infancy, especially when it comes to underdeveloped nations. studies already published only look at a small subset of context-specific obstacles. the objective is to outline all potential obstacles to industry 5.0 adoption, rank them empirically, and assess interdependencies at the organizational level. to achieve this, we perform a thorough assessment of the relevant literature, gather opinions from experts, identify obstacles, and explore causal links using a multi-criteria decision-making process called the decision-making trial and evaluation laboratory method (dematel). according to the findings, the most significant hurdles are those related to expenses and the funding system, capacity scalability, upskilling, and reskilling of human labor. finally, taking into consideration expert advice, we offer a thorough explanation of how we used the "green, resilient and inclusive development" (grid) framework to identify appropriate ways to overcome the various constraints we had identified. through the identification and modelling of hurdles, this study advances the study of industry 5.0, and its conclusions can be utilized by policymakers or other key stakeholders to formulate and priorities initiatives to speed up industry 5.0 in organizations. 408 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi 2. literature review we gather and offer the relevant research in this section. this section examines industry 5.0 to know its theoretical definition, breadth, prospects, and acceptance as well as the barriers scientists have identified that might prevent its effective execution in a developing country. while industry 5.0 is still a developing concept that has not reached its pinnacle [11], there are a number of concepts offered by scientists and industry experts, including the following; industry 5.0 intends to combine the intellectual and decision-making strengths of humans with the synergistic advantages of devices; industry 5.0 is expected to be the new industrialization with human involvement; and industry 5.0 is derived on the idea of the 6 r's, which are recognize, reconsider, realize, reduce [12]. industry 5.0 entails the methodical removal of waste, the assurance of high-quality, highly tailored goods, and the integration of human elements with business, technology, and technical features [13]. fig. 1 shows the evaluation of the industry. the foundation of industry 5.0 is efficiency, durability, and human-robot cooperation, but it also encompasses a wide range of cuttingedge innovations which have their origins in industry 4.0 [14]. fig. 1 evolution of industries from 1.0 to 5.0 for a comprehensive list of advanced technologies, see fig. 2. in fig. 2, industry 5.0 represents the convergence of several advanced technologies that are revolutionizing various sectors. robotics, blockchain, 5g/6g, big data, ai, and iot play integral roles in this transformation. security systems ensure the protection of critical assets and data. robots collaborate with humans, leveraging their strengths and automating complex tasks. blockchain technology provides secure and transparent transactions, eliminating intermediaries and enhancing trust. the high-speed connectivity of 5g/6g enables seamless communication and real-time control of industrial processes. big data analytics extract valuable insights from vast amounts of information, facilitating data-driven decision-making. ai empowers machines with cognitive capabilities, enabling intelligent automation and optimization. iot connects devices, sensors, and machines, enabling real-time monitoring, predictive maintenance, and enhanced collaboration. collectively, these technologies revolutionize industries, driving efficiency, productivity, and innovation. industry 5.0 empowers humans to focus investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 409 on creativity, problem-solving, and decision-making, while machines handle repetitive tasks. it fosters a symbiotic relationship between humans and technology, resulting in improved processes, personalized experiences, and sustainable growth. industry 5.0 represents a new era where advanced technologies amplify human potential, reshape industries, and pave the way for a more connected and intelligent future. fig. 2 advanced technologies of industry 5.0 industry 4.0 was first mentioned more than ten years ago in 2011, but studies on industry 5.0 have just recently gathered traction. industry 5.0 depends on the principles and ideas of industry 4.0 in a more comprehensive and futuristic way [11]. over the past five years, academic studies on industry 5.0 have irregularly started in various parts of the world [15]. given that industry 5.0's central tenet is centered on human-technology and manmachine cooperation, this makes sense [16]. the iot, digital and mixed technology, robotic driverless cars, cryptocurrency, and intelligent machines are some major topics under the heading of technical elements [17]. research regarding anticipating the future of employment and learning, particularly their interconnection and the employee development plan for an integrated industry, also included human-centric topics shown in fig. 3 [18]. additionally, we cited research that combined technological and human factors, focusing on issues like future human-robot cooperation capabilities and human engagement in industrial technologies [19]. fig. 3 illustrates the opportunities, limitations and future of industry 5.0. industry 5.0, the next industrial revolution, presents a range of opportunities and limitations as well as a promising future. opportunities lie in the areas of smart infrastructure, clean energy, intelligent transport, and binaural beats. smart infrastructure utilizes technology to enhance efficiency, sustainability, and resilience in areas like energy management and traffic flow. clean energy harnesses renewable sources for sustainable power generation. intelligent transport utilizes technology for safer and more convenient transportation. binaural beats, sound waves played at different frequencies in each ear, offer benefits such as relaxation and improved focus. 410 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi fig. 3 opportunities, limitations and future of industry 5.0 however, industry 5.0 also faces limitations. information security becomes crucial as technology integration increases, necessitating stronger measures against cyber threats. human-machine collaboration requires new approaches and training programs to optimize productivity and cooperation. adaptability becomes essential due to rapid changes in the industry, requiring businesses to embrace new technologies and adapt their models to shifting market conditions. the digital nature of industry 5.0 necessitates digital forensic services to investigate cybercrime and data breaches. additionally, the demand for highskilled workers rises to handle the complexity of industry 5.0, necessitating robust training and development programs. looking ahead, the future of industry 5.0 holds promising developments. hyper-intelligent networking enables real-time data processing for applications like traffic control, self-driving cars, and virtual reality. blockchain, a distributed ledger technology, has the potential to revolutionize industries such as finance and supply chain management. quantum computing, utilizing quantum mechanics, can solve previously unsolvable problems, impacting fields like drug discovery and artificial intelligence. striving for ultimate security, industry 5.0 seeks unbreakable security measures to safeguard data and systems. in summary, industry 5.0 offers significant opportunities in smart infrastructure, clean energy, intelligent transport, and binaural beats. however, limitations in information security, human-machine collaboration, adaptability, digital forensics, and skilled workforce require attention. the future of industry 5.0 looks promising with advancements in hyper-intelligent networking, blockchain, quantum computing, and ultimate security. the implementation of industry 5.0 and the development of several industries, including medicine, hospital, the biosciences, marine, academia, tourism, and entertainment, were the main topics of another batch of studies in the framework of industry 5.0 (fig. 4). the green, resilient and inclusive development (grid) model, put out by the wef and the wb, is consistent with the industry 5.0 pillars of conservation, technical advancement, and social equality [18, 20]. the grid model offers a great lens through which to view the complexities of implementing industry 5.0 [21, 22]. investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 411 fig. 4 applications of industry 5.0 although the technology "industry 5.0" was first used roughly a few years ago, important publications on the topic have just been released. despite important fields like hotels, training, medical services and pharmaceuticals, power generation, and humancentric viewpoints being examined in relation to the implementation and influence of industry 5.0 from a scientific viewpoint, to our understanding none of the researches was concentrated on barriers that industries can inevitably experience while incorporating industry 5.0 [23]. our research is new because, whereas the majority of studies have been conducted from the viewpoint of industrialized nations, none have addressed the views of barriers in industry 5.0 and their mitigation. sachsenmeier [11] on bionics study work is particularly related to industry 5.0 and synthetic biology. the outcome of this study shows the necessities of synthetic biology and bionics in the industrial era. özdemir and hekim [24] made their study on policies that are relevant to industry 5.0 and multiple technologies. the outcome of this study reveals that an industry is a meshwork of complex interconnected networks and various devices. it is important to ensure that the devices that are used in industries must be safe and sustainable. nahavandi [25] introduced the solutions for human centricity for industry 5.0 purposes. these solutions are for the challenges that are faced by manufacturing firms. demir et al. [26] gave the idea of co-working robots with humans. they compared the goals and visions of industry 4.0 and industry 5.0 and crucial barriers are also discussed in this study that considers co-working of humans and robots. abdel basset et al. [19] conducted a study on soft computing and the next generation of iot or the internet of things. when it comes to the development of industry 5.0, soft computing and iot are very important components. fukuda [27] worked on ecosystem models based on sti or science, technology, and innovation. these models review the sti policies and statistical analysis which compare the progress of the ecosystem of japan with germany and the us. sharma et al. [28] worked on the industrial impact on smart cities. their work makes predictions about how smart cities are impacted by industries 5.0 and highlights the modifications that are required for personalization and efficient co-working of humans and 412 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi robots. ciasullo et al. [16] worked on healthcare 4.0 introduction for the awareness of society 5.0. their study shows the importance of healthcare 4.0 and its contribution towards proper care and appropriate treatment for patients suffering from bedsores. bartoloni et al. [29] investigated the application of the design of the quintuple helix. this conceptual model was based on designs that are related to the application method of the quintuple helix. pillai et al. [3] conducted a study focused on covid 19 and redefining operations of hospitality 5.0. this study reflects the principles that are designed in the related of hotel industry 5.0. thakur and sehgal [30] made an architectural proposal that was related to scps or smart cyber-physical system. this architectural structure for scps was proposed for industry 5.0 and used in multiple industries like paper, petroleum, cement, fertilizer, and automobiles. carayannis et al. [31] worked on the energy produced by nuclear fusion and its utilization in future industry 5.0. this work highlights the importance of energy produced in nuclear fusion and its industrial utilization. maddikunta et al. [32] conducted a survey related to industry 5.0 and their industrial application. this work shows different features of technologies and methods of their application along with their challenges. xu et al. [33] conducted a study to clarify different perceptions and for a better understanding of industry 4.0 and industry 5.0. their work elaborates on the differences in technologies, core values, concepts, and challenges between industry 4.0 and industry 5.0. madsen and berg [12] made an analysis exploring new literature related to industry 5.0. their literature survey covers the period between 2015 and 2021. it indicates that industry 5.0 is a novel field. most of the articles were written after 2019 and only 5 belonged to the top used articles. broo et al. [15] worked on the education of engineering for industry 5.0. this work provides a brief history of the learning of engineering and the importance of critical challenges, skills, and plans that are learned and practiced. 3. potential barriers 3.1. security and privacy industry 5.0 is anticipated to introduce a multitude of network topologies, operations, equipment, technologies, systems, and terminals that will help to close the separation between the physical and digital worlds. although the above components constitute a close network, businesses must guarantee that there isn't a security violation without sacrificing service and operational standards [32]. to guarantee the achievement of industry 5.0, businesses must solve security issues relating to verification, accuracy, password protection, and accountability. as industry 5.0 develops, it is also anticipated that data will be openly transmitted over the internet between numerous linked devices, networks, partners, robots, people, and platforms, raising issues about privacy and data protection. data privacy is a major problem for both international companies and governments since data is extremely valuable and, if misused, might have a number of ramifications for the focus business and its associates [34]. a new danger pathway will be created by industry 5.0's usage of ai and enabling technologies. for security reasons, trustworthy implementation is crucial for ai/ml operations. for industry 5.0 implementation to function properly, it is especially important to safeguard the accuracy of the data set in use for ml model training and the al method [35]. for example, industry 5.0 renters must safely exchange scientific data to train ai models or modify models incrementally, as in supervised learning. deploying proactive investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 413 security procedures [36] and mitigating zero-day threats [37] are only a couple of the new security protocols that will result from industry 5.0's increased reliance on ict systems. additionally, industry 5.0 may function in the quantum computing age because of the growth of quantum computing. a quantum computer will significantly simplify protecting old security measures [38]. in such circumstances, industry 5.0 platforms must use postquantum encryption or cryptography that is quantum resilient to guarantee the necessary degree of security [39]. privacy is a crucial need for industry 5.0 services since the whole environment relies on valuable scientific property, cost effective production inputs, and access control [40]. to link robots and people, engineers and other partners, and to transmit surveillance and management data, data is transmitted via the internet in industry 5.0. to maintain the confidence of the cloud-based network, such information cannot be accessible to hostile web users [34]. to prevent a detrimental society effect, particular sociological and moral considerations must be followed while adopting ai. human workers frequently worry that ai will make their occupations obsolete, while industry 5.0 will provide more job prospects. to enable continuous co-production between humans and robots, the moral concerns concerning ai and its effects on people should be resolved [41]. robots must include social decisions, ethical principles, social interactions, and cultural norms [42]. the industrial revolution's authorities must take into consideration the moral dilemmas raised by cooperation between humans and machines. human data protection laws, or the idea that persons have ownership over their data, are one of the privacy concerns. they are entitled to compensation for any data breaches that affect their personal or critical data. in order to protect data privacy, user data must be protected while being used for analytical reasoning and preventative management. 3.2. scalability industry 5.0 is now in its infancy, but as time goes on, it will include a network of thousands of linked equipment, devices, and people that will collectively improve efficiency, performance, and security [43]. to expand up and link to and create such a broad network, businesses must be sensitive to the need and growing activity. therefore, adaptability is the capacity of any technology to adapt to an overall rise in its activity [32]. the systems operating under industry 5.0 must be capable of scaling up to increase production thanks to the many linked devices and software. this is crucial in co-working spaces where people and robots are linked via the web to a variety of gadgets and software. it is a serious reason for worry [28]. using 3d printing e.g., among the most exciting industry 5.0 systems, businesses would be required to take the necessary steps to build barrier ability to be able to expand to meet any growth in market trend, despite the case of customized and unique goods, without lengthening timeframes [44]. once the system's activity varies continuously, scalability may be defined in terms of the system's resilience, adaptability, and reactivity. scalability in industry 5.0 refers to a system's effectiveness in various working situations, regardless of whether there are more or less hyper-connected systems present in the ecosystem. industry 5.0 is designed to link and converse with a variety of systems from other industries as well as a variety of people. scalability is a feature of industry 5.0, which is an improvement of industry 4.0, however creating a companionship between humans and machines or robots by combining their task poses a more serious difficulty [45]. employing service level signals that have been verified in accordance with the service level goals outlined in service level contracts, the scalability may be followed up. this is a 414 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi crucial issue since non-determinism is growing in the rise of data gathering, equipment, manufacturing, and human activity. technology providers in the industry 5.0 ecosystem should be capable of providing services at any capacity, adaptable enough to extend and grow, and give a low-latency solution with no data analysis latencies to assure scalability. the many inquiries received from the adaptable cloud and dynamic edge servers should be processed by ai-based robots with no unforeseen latency. scaling additive manufacturing (am): throughout manufacturing, hundreds of printers may be operating in the same capacity and might be meant for the identical component demand. in these situations, am must grow itself according to the automation infrastructure and increasing expectations. with no traditional limitations, am enables on-demand production to create more personalized items with shorter lead times. they are challenges peculiar to am scaling [44]. the method selection for utilizing am findings and expanding them for manufacturing usage is one of the issues. finding the obstacles to am expanding into manufacturing and developing mitigation strategies is the second problem. the third problem is am's inability to adapt to emerging market dynamics. demand-based counter productiveness can encourage am scaling. using 4d or 5d printing technology along with effective software for designs might improve the challenges with am scaling in manufacturing. for the industry-wide implementation of am, am scaling criteria must also be created and improved. as the amount of data stored on the cloud platform grows, analyzing that data may take a surprisingly long time, delaying responses. micro data centers will now allow the integration of ec with ai at network edge devices. as was previously said, ec will minimize latencies to ensure low latency replies, and the edge server will become a smart edge when ai makes data forecasts there. 3.3. legal and regulatory although most countries have basic rules and guidelines in place relating to the last technological growth, legislation governing advancements in industry 5.0 must be implemented. in addition to human-robot teaming, 6g, digital twins, 3-d printing, drone technology, networked, smart gadgets, and system software, industry 5.0 is anticipated to provide fresh, better methods of working [32]. it is necessary to create new laws and regulations that regulate the general ecosystem layout, as well as rules for humanrobot cooperation and discriminating between different sorts of robots and machines, such as robots and drones, in order to control each of them [26]. in addition, it's important to develop clear regulations for technological advancement and ecosystems that include mobility and adaptability [33]. many rules relevant to both the human and robot should be developed as industry 5.0 seeks to bring again the human labor to collaborate with robotic systems and smart technology [46]. lacking appropriate rules and laws, several problems might develop in this co-production setting [26]. robots should be distinguished from many other robots, such as drones, by laws that are implemented. 3.4. ethical concerns industry 5.0-related ethical problems may be divided into two categories. the first type of ethical quandary of human-robot collaboration may arise, such as putting moral standards into machines, aligning human-robot working morals, and people management and devices collectively in the company to assess their efficiency and conduct [25, 26]. utilizing ai and machine learning (ml), the second type of material difficulties might investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 415 occur. these issues also included deciding whether to use robots in routine, routine tasks instead of people, lowering worker salaries in favor of more affordable machine substitutes, and protecting personal information [42, 47]. 3.5. upskilling, reskilling and talent retention the 10 abilities that production will need in the future have been recognized by the world manufacturing forum [48]. media literacy, ai and big data, innovative problemsolving, an innovative approach, physical and behavioral worker protection, a comprehensive and variability attitude, a confidentiality and security-oriented attitude, the ability to manage growing workload, useful connectivity skills, and an openness to ongoing change are a few of these. only four of the aforementioned abilities are computerized, as can be seen from the list above; the others are more focused on originality, mentality, and innovative thinking [43]. additionally, analysts believe that when industry 5.0 emerges, robots may begin to do routine daily tasks. humans may simultaneously need several and specialized expertise, such as those for managing unique circumstances, innovation, and architecture [48]. this might create a skills deficit and provide possibilities for novel employment that might not otherwise exist. considering that industry 5.0 remains in its very early stages, organizations, community, and the government may not be prepared to provide the necessary instructors and training programs [49]. to tackle any technical, societal, and administrative challenges, standards and legal rules must be implemented since a competent employee in industry 5.0 is required to offer a high-value job in manufacturing. challenges with organizational, personnel, corporate culture, and organizational technology are only a few of the issues involved in providing a trained workforce [14]. insufficient instructors and cost limitations make it difficult for people who work alongside robots to receive the necessary teaching, which is the main skill area difficulty. the need for competent workers and new technologies will increase by the period industry 5.0 is completely implemented, necessitating proper training both for the learners and the future instructors. this may encourage public-private collaborations [50]. reforming the regulatory framework will also be required. the skilled employee can also support operational effectiveness. many sectors would adopt the latest emerging innovation to keep up with digital competency, but administration could not be aware of it. the knowledgeable staff and the environment could take advantage of the company's incompetence. 3.6. acceptance and adaptability modern working methods will be introduced by industry 5.0, such as the fusion of cyber and real workspaces, human-robot collaboration, the employment of modern technology, and the potential automation of human labor to save time and money [33]. because they are afraid of being substituted more by effective machines and robots, the aforementioned might make people resistant, which would prevent them from accepting social variety [26, 49]. humans will once again work alongside robots on the production floor thanks to industry 5.0. although it appears to be a productive technique to create customized goods, several concerns about the interaction between humans and robots should be considered. additionally, when people and robots exchange labor, human job loss anxiety will be reduced. the robot will do the routine tasks, freeing up the human to focus entirely on ideas and creativity. it will be preferable for robots to support humans than the other way around, 416 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi which might lead to organizational instability and disrupt the company's long-standing job competitiveness environment. this could lead to several difficulties, including shifting the responsibility of human resources, robotics in it departments, ethical principles with robots, training people to start competing with robots, legal requirements, mental issues to force people to implement a new approach to working, and job mechanisms assessment for both humans and robots [49]. additionally, robot workers are competent and may have higher expectations than those of a regular work environment. 3.7. cost and funding industry 5.0 would feature excessive technological exploitation, human-robot collaborative work, and mass customization via networked gadgets [27]. industry 5.0 would feature excessive technological exploitation, human-robot collaborative work, and mass customization via networked gadgets. both the main enterprise and the local government would need to make significant expenditures on them. financing becomes a significant barrier since organizations and emerging economies with other objectives may only consider short-term returns and link these investments with high expenses [33]. 3.8. regional disparity an absence of sufficient absorption of industry 5.0 concepts throughout all sectors may result from widening demographic and technical divides between urban and rural areas [27]. consequently, it might pose a serious threat to the effective implementation of industry 5.0. 4. risk mitigation understanding the obstacles to industry 5.0 acceptance is made easier by the understanding of these hurdles. however, merely identifying problems does not guarantee that industry 5.0 will be successfully adopted in the actual world. in order to successfully implement industry 5.0, we need an appropriate mitigation approach. to achieve this, we are working to create a mitigation plan that focuses on the grid architecture [48]. the world economic forum and the world bank have combined their efforts to create this structure [18, 20]. the three main pillars of the grid strategy are sustainability, resilience, and inclusivity. economic equality, universal development, and positive future are the shared goals of each pillar of the grid structure. other theoretical perspectives, such as resourcebased view and dynamic capabilities, recommend that businesses build the capacity to address functional difficulties and achieve a strategic advantage. technology implementation may increase a company's capabilities, but rbv [51] and dcv are mostly mute about how to get through obstacles that a company can have while deploying new technology [2] such as industry 5.0. contrarily, the grid structure distinctly proposes three pillars that can assist businesses in overcoming obstacles. additionally, we introduced the grid structure to professionals, who concurred that it is an appropriate technique for creating a model for the quicker implementation of new technologies, including industry 5.0 [48]. it's noteworthy to note that industry 5.0 technology necessitates tight machine-human-interface interaction. but the majority of the frameworks investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 417 emphasize either men or machinery more. grid systems offer a broad perspective and take both machines and men and women into account. within the grid structure, we offer our mitigation plans for the eight noted barriers [48]. these mitigating measures would speed up the adoption of industry 5.0 by industrial companies. below, we will go into depth about each of these pillars, classifying the eight obstacles that have been found as well as the corresponding suggested mitigation measures. 4.1. the green pillar the green pillar emphasizes development for the future and calls for the transformation of vital industries and activities, such as production, communication, and electricity, in order to build a more environmentally friendly and prosperous future [48]. the most noticeable barrier that connects to all other obstacles is barrier b7 (price and finance). pricing and finance should be the backbone of every technology and growth endeavor, so this is obvious. in addition to 6g, digital twins, 3-d printing, drone technology, networked smart gadgets, and various systems, industry 5.0 is anticipated to provide fresh, better methods of working [32]. modern, appropriate methods must be used to control each of them [26]. speed and adaptability must be taken into account when defining clear regulations for technology and sustainable environment development [33]. in the reason important groups, the barrier b3 (legislative / regulatory) also has a high rating, showing that it has a strong impact on other obstacles. according to rumors, industry 5.0 will entail excessive technological exploitation, human-robot collaboration, and mass customization via linked gadgets. both the main enterprise and the local government would need to make significant expenditures on them. investment may be a key barrier since emerging countries with competing interests and organizations that simply priorities short-term advantages may link these investments with large expenses [27, 33]. the authorities and legislative organizations would be required to offer long-term plans and rewards, like tax refunds and vacations for businesses that use environmental and human-centric activities while still maintaining revenue, to overcome the aforementioned two key impediments [48]. the governmental and legislative organizations must discover new methods of functioning that industry 5.0 intends to offer in order to address the obstacle connected with legal and governmental difficulties. after that, they might review the current laws in order to create new rules that would cover all areas of industry 5.0, including limitations on cutting-edge technology and human-robot collaboration [48]. 4.2. the resilient pillar the resilient pillar focuses on risk mitigation to prevent any risks, whether they be natural or man-made, as well as social and eco-technological disasters. in order to safeguard market segments, equipment, and the general public from crises, this would urge major nations and international organizations to spend in risk mitigation and prevention strategies [48]. this pillar is responsible for barrier b2, which is the second most noticeable barrier and is joined by barrier b1, which is extremely important. systems operating underneath industry 5.0 must be able to expand to meet demand thanks to the many linked devices and programs. this is crucial in office spaces where people and robots are linked via the internet to a network of gadgets and services. it is viewed as a serious source of worry [28,32]. to guarantee the performance of industry 5.0, businesses must solve security issues with relation to verification, reliability, password protection, and accountability [52]. 418 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi additionally, academics anticipate that the emergence of industry 5.0 will result in data being openly transmitted throughout the internet between numerous interconnected devices, networks, partners, robots, people, and platforms, raising worries about data privacy and security [48]. the authorities would be required to step in and impose extra identification and security measures, as well as an independent audit for each action taken across the connection of equipment and systems. businesses would have to shift their attention away from strategic short-term advantages in expansion and towards a longerterm strategy [48]. large businesses may gain from government assistance in the form of tax breaks and substantial interest-free financing. 4.3. the inclusive pillar the comprehensive pillar emphasizes social interaction and human resources, and it works to guarantee that everyone may thrive in a welcoming atmosphere. the reduction of inequities that could exist in resources and results is the pillar's end goal. the greatest obstacle in the temporal group is barrier b6 (acknowledgement and adaptation). a general result of making sure that every person's wants and expectations are taken into account would be less opposition, which would facilitate smoother and faster adoption of novel operating methods through industry 5.0 in terms of technical and social transformation. due to work morals when humans and robots collaborate together, barrier b4 (moral problems) might appear [25] and concerns about robots and technology taking the place of the present employees [42]. additionally, barrier b5, which entails skills training, retraining programs, and talent management, will be addressed concurrently. this means that worries about a skills gap, moral conundrums, and fears related to human-robot collaboration might be eliminated with spending in teaching, learning, and growth of the workers in modern technology and methods of functioning [48]. another significant obstacle, b8 (area inequality), which is brought on by the widening demographic and technical divide between urban and rural regions, particularly in emerging and impoverished countries, is a source of considerable worry [27]. in collaboration with the businesses under discussion, local authorities and organizations should develop area regeneration measures such as refunds, the supply of assets, and technology. the goals of industry 5.0 will be achieved if businesses are encouraged to establish centers and production operations in rural and non-urban locations. with the three aforementioned pillars in place, it is anticipated that all countries and worldwide enterprises in their eagerness to adopt and utilize the benefits of industry 5.0 shall harvest the similar goals of development, conservation, technology progress, and social wealth [48]. in order to achieve this, this study looks at the ranking of and connections between the barriers to industry 5.0 adoption from the managers' perspective who decide which technologies to implement within the company. to the best of the authors' knowledge, this study is the first to explore adoption hurdles for industry 5.0 by delving into their causes. ranking the adoption hurdles of industry 5.0 and understanding how they relate to one another will help determine how the technology will be most successfully adopted and applied in the manufacturing sector, particularly in emerging nations, in the future. 4.4. todim fuzzy sets (fs) are a way to describe and handle data that are not rigid but slightly hazy, with membership values ranging from 0 to 1, according to zadeh [53]. also, he put investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 419 out an original fuzzy set theory that is a useful tool for solving practical issues. the neutrosophic set (ns), according to smarandache [54], deals with truth, falsehood, and indeterminate membership functions, all of which are unrelated to one another. it can be challenging to discuss a decision-making conundrum with ns. as the three degrees of truth, falsehood, and indeterminacy cannot be stated as precise numbers, computing them in real-world contexts is challenging [55]. recently, some researchers developed an environment using interval values to address this problem, where the membership value is established using an interval number rather than a precise integer. although intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets cannot manage indeterminacy, many scholars have widely recognized interval neutrosophic sets (ins) and proposed a similarity measure between inss based on hamming and euclidean distances. in fields like decision-making, information fusion, image processing, and medical diagnostics, the application of ns and ins theories has yielded a number of promising findings [56]. the fuzzy set is incapable of handling all forms of uncertainty, including inconsistent and indeterminate data [57]. there is uncertainty since several experts are unsure of the claim. the fs and ifs, which deal with membership, non-membership, and hesitant functions, do not address these issues. using ns, we can find a solution to this issue. using membership values between [0, 1], aggregation operations in an intuitionistic fuzzy environment can be expanded to ns. consequently, we must focus on aggregation operators on ns in order to aggregate neutrosophic information in decision-making circumstances. the idea that the three membership functions in ns are independent will be very beneficial in the field of information fusion, according to [58]. an original technique for decision support is required in multi-criteria analysis to address uncertainty and human viewpoints. in a multicriteria decision-making process, the decision matrix is made up of criteria and alternatives. we employ fuzzy numbers in real-world circumstances since this matrix is subject to uncertainty. this is a membership grade interval extension. each value in the interval relates to a real number and can be used with an uncertainty declaration. this method was also applied to mcdm by [58]. for determining an exact number as the grade of membership of the decision makers, there are many decision-making techniques, but there are few for calculating an interval-based fuzzy number. researchers gomes et al. [59, 60] are engaged in a project. the todim methodology is an mcdm method based on prospect theory. the decision maker's choice is effectively acted out in risk scenarios, and the global measurement value is established utilizing pt todim. the global multi-attribute value function will combine all loss and gain measures. since each alternative's global value is proportional to its dominance over all other alternatives, todim is comparable to outranking techniques. this approach, which is based on a single parameter, tests particular gains and losses functions. a non-compensatory strategy, as the name implies, does not take into account compromises. to get rid of differences, this technique uses paired comparisons. a technique for iterative mcdm is todim. this approach chooses and reorders the objects in accordance with the advice of the expert. the complete information regarding the attributes will be made available here. insufficient information about attributes is frequently provided by experts, making it challenging to address decisionmaking problems. with incomplete attribute matrices, todim is especially useful since its judgements are more unbiased and widely accepted. mcdm concerns are often divided into two categories: benefit criteria and cost criteria. benefit criterion will typically take precedence over usage convenience. adali et al. [61] claim that measuring the degree to which each alternative dominates the other alternative is the key tenet of the todim 420 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi technique. hence, both qualitative and quantitative criteria can be used using this approach. the applicability of the todim technique for varied fuzzy circumstances was explained by qin et al [62]. one of the mcdm strategies gomes and lima offered is the todim approach, which krohling and souza [63, 64] found employs fuzzy todim and the extended case and is based on prospect theory. gomes et al. [65, 66] used todim and choquet integrals in order to streamline calculations and support interval data. the best option was chosen using a combination of todim and fse by passos et al. in 2014 [67]. wei et al. [68] introduced the hesitant fuzzy linguistic todim technique. to understand heterogeneous data, lourenzutti and krohling [69-71] improved a novel strategy based on the todim method. li et al. [72] suggested intuitionistic fuzzy todim for the distributor problem. fuzziness was used by tosun and akyüz [73] to enhance the shortcomings of a relatively new decision procedure termed todim. ren et al. [74] and zhang et al. [75] introduced the enhanced version of the todim method using the pythagorean fuzzy sets and neutrosophic numbers. this method was repurposed as a decision-making reinforcement method by lin et al. [76]. the interval type-2 fuzzy sets todim technique was developed by sang and liu [77] and utilized to address a challenge in the supplier sector. pramanik et al. [78] introduced the nc-todim method to address an investing issue. qin et al. [62] extended the traditional todim technique for an mcdm problem. sun et al. [79] used extended todim and electre iii techniques for the physician selection challenge. he et al. [80] demonstrated aggregation capabilities for ins utilizing the inwa operator and satisfied decision makers' needs using hmf-todim. the 2tlnns todim methodology, which expands the todim method to the 2-tuple linguistic neutrosophic fuzzy environment, is suggested by wang et al. [9] to provide the criteria values in issues involving multiple attribute group decision-making (magdm) utilizing the enhanced technique of 2-tuple linguistic neutrosophic numbers (2tlnns). a furniture manufacturing company will be required to resolve a real-life case study. in 2019, deng and gao heo [81] expanded the 2tlpfs todim method. thus, the pythagorean fuzzy 2-tuple linguistic environment is added to the original todim technique. davoudabadi et al. [82] presented a novel paradigm in 2020 for dealing with multi-attribute group decision-making issues in fuzzy environments. by merging schweizer-sklar t-conorm and t-norm (sstt) aggregation operators, power average (pa) operators, and todim processes, zindani et al. [83] provide a novel integrated group decision-making framework for decision making in intuitionistic fuzzy settings. while there is no enough study on ins, we used the todim approach to solve an investing dilemma based on the aforementioned literature. fuzzy tri-angular aggregation operators are a brand-new class of fuzzy aggregation operators that ulrich and henri [84] introduced. to analyses ambiguous or erroneous data, another strategy—fuzzy numbers—must be utilized. to do this, we focus on circumstances where it is impossible to quantitatively analyze the available data. liao et al. [85] explained multiple attributes group decision making using the cumulative prospect theory in a probabilistic hesitant fuzzy setting. leoneti et al. [86] explained based on the exponential model of prospect theory, a novel application of the todim method. sun et al., [87] explained using the z-wasserstein distance, an expanded exp-todim approach for multiple attribute decision making. investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 421 4.5. dematel previous studies in this field indicate that mcdm (multi-criteria decision making) techniques are most frequently used to overcome the aforementioned challenges. ahp (analytic hierarchy process), topsis (technique for order of preference by similarity to ideal solution), and anp (analytic network process), to name a few, are just a few of the approaches that fall under the umbrella of the aforementioned mcdm technique. however, the dematel method manages to take into account how various criteria interact and, hence, it is well adapted to tackle real-world issues [88,89]. considering in light of the aforementioned, we approach our research challenge using the dematel (decision making trial and evaluation laboratory) strategy. the dematel methodology was initially created by the battle memorial institute of geneva in the early 1970s [90], and since then it has been widely applied to solve practical problems involving the interrelationship between various factors or criteria [91]. sheng et al. [92] explained the systematic review of dematel. romel et al. [93] explained the analysis of the barrier in waste management. wenping et al. [94] detailed analysis of the variables affecting the supply chain for emergency logistics. 4.6. single valued neutrosophic the phrase "single-valued neutrosophic set" (svns) refers to a particular type of neutrosophic set that is distinguished by having membership functions for truth, indeterminacy, and falsehood that all fall inside the range [0, 1]. riaz et al. [95] explained operators for single-valued neutrosophic reasonable aggregation with multi-criteria judgement. granados [96] discussed the application of quadripartitioned single-valued neutrosophic properties to energy price influencing factors. liu and yang [97] explained college and university physical education teaching quality evaluation using the das method for single-valued neutrosophic number multi attribute group decision-making. d. nagarajan and kavikumar [98] explained neutrosophic hidden markov model with singlevalue and interval values. wang et al. [99] explained that a cost-sensitive, multi-scale, single-valued neutrosophic decision-theoretic rough set is used to determine the best scale. nagarajan et al. [100] explained an innovative method for multi-attribute group decisionmaking based on the neutrosophic bonferroni mean operator of trapezoidal and triangular neutrosophic interval environments. 5. preliminary 5.1. neutrosophic sets in this section, we define some definitions and arithmetic operations for a neutrosophic set used in forthcoming sections. definition 1. [54] let 𝑈 be the universal set, then neutrosophic set is defined as 𝐴 = {(𝑇𝐴(𝑥), 𝐼𝐴(𝑥), 𝐹𝐴(𝑥)), 𝑥 ∈ 𝑈} where 𝑇𝐴(𝑥), 𝐼𝐴(𝑥), 𝐹𝐴(𝑥) ∈]0−, 1+[ and 0− ≤ 𝑇𝐴(𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3+. definition 2. [101] let 𝑈 be the universal set, then svns is defined as �̇� = {(𝑇�̇�(𝑥), 𝐼�̇�(𝑥), 𝐹�̇�(𝑥)), 𝑥 ∈ 𝑈} where 𝑇�̇�(𝑥), 𝐼�̇�(𝑥), 𝐹�̇�(𝑥) ∈ [0,1] and 0 ≤ 𝑇�̇�(𝑥) + 𝐼�̇�(𝑥) + 𝐹�̇�(𝑥) ≤ 3. 422 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi definition 3. [102] the proposed score function for the neutrosophic triangular set is given by, �̇�∗(𝑇𝐴(𝑥), 𝐼𝐴(𝑥), 𝐹𝐴(𝑥)) = 1 2 (1 + 𝑇𝐴(𝑥) − 2 ∗ 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥)). 5.2 single valued neutrosophic sets (svns) let 𝑋 be the universal set and for ∀𝑥 ∈ 𝑋 , svns [103] is defined as 𝑁 = {⟨𝑥, 𝑇𝑁(𝑥), 𝐼𝑁(𝑥), 𝐹𝑁(𝑥)⟩, 𝑥 ∈ 𝑋} where 𝑇𝑁(𝑥) , 𝐼𝑁(𝑥) , 𝐹𝑁(𝑥) → [0−, 1+] are the truth, indeterminacy and falsity-membership function and satisfy the criterion 0− ≤ 𝑇𝑁(𝑥) + 𝐼𝑁(𝑥) + 𝐹𝑁(𝑥) ≤ 3+. the membership function of the embedded sets of an neutrosophic sets satisfy 𝑇𝑁(𝑥) ≤ 𝑇𝑁(𝑥) ≤ 𝑇𝑁(𝑥) , 𝐼𝑁(𝑥) ≤ 𝐼𝑁(𝑥) ≤ 𝐼𝑁(𝑥) and 𝐹𝑁(𝑥) ≤ 𝐹𝑁(𝑥) ≤ 𝐹𝑁(𝑥). definition 4. (single valued neutrosophic weighted operator) [104] let 𝑊𝐴 = ∑ 𝜔𝑖𝐴𝑖 = 1 − ∏ (1 − 𝑇)𝜔𝑖 , ∏ 𝐼𝑤𝑖 ,𝑛 𝑖=1 , ∏ 𝐹𝑤𝑖 ,𝑛 𝑖=1 𝑛 𝑖=1 𝑛 𝑖=1 where 𝜔𝑖 is the weight of 𝐴𝑖 𝑎𝑛𝑑 𝜔𝑖 ∈ [0,1], ∑ 𝜔𝑖 = 1𝑛 𝑖=1 . definition 5. (single valued neutrosophic geometric operator) [104] let 𝑊𝐺 = ∑ 𝜔𝑖𝐺𝑖 = ∏ (1 − 𝑇)𝜔𝑖 , 1 − (1 − ∏ 𝐼𝑤𝑖),𝑛 𝑖=1 1 − (1 − ∏ 𝐹𝑤𝑖),𝑛 𝑖=1 𝑛 𝑖=1 𝑛 𝑖=1 where 𝜔𝑖 is the weight of 𝐴𝑖 𝑎𝑛𝑑 𝜔𝑖 ∈ [0,1], ∑ 𝜔𝑖 = 1𝑛 𝑖=1 . definition 6. (score function) [104] let 𝐴 = (𝑇𝐴, 𝐼𝐴 , 𝐹𝐴) be a single valued neutrosophic set, a score function 𝑆(𝐴) is defined as 𝑆(𝐴) = (2+𝑇𝐴−𝐼𝐴−𝐹𝐴) 3 , 𝑆(𝐴) ∈ [0,1]. the score function for ranking in single value neutrosophic set is defined as 𝑅 = 1+𝑎−2𝑏−𝑐 2 , 𝑅 ∈ [0,1]. the properties of 𝑅 is 𝑅 = 0 if and only if 𝑎 = 2𝑏 + 𝑐 − 1, 𝑅 = 1 if and only if 𝑎 = 2𝑏 + 𝑐 + 1, 𝑅 = −1 if and only if 𝑎 = 2𝑏 + 𝑐 − 3. definition 7. [55] let 𝐴 and 𝐵 be two single-valued neutrosophic sets, the basic operations of single valued neutrosophic sets are: a ⊕ b = (ta + tb − tatb, ia ib, fafb). (1) a ⊗ b = (tatb, ia + ib − ia ib, fa + fb − fafb). (2) λa = 1 − (1 − ta) λ, (ia)λ, (fa)λ, λ > 0. (3) (a)λ = (ta)λ, (ia)λ, 1 − (1 − fa) λ, λ > 0. (4) 6. identification barriers we have assumed that an identification barrier is the best alternative for their mitigation in industry 5.0 [105]. the decision-making board based on four decision-makers who assess the five choices. the five choices are as follows: (a1) regional disparity, scalability and upskilling reskilling and talent retention (𝑎1): growing socioeconomic and technological disparities between urban and rural areas could pose a serious threat to the successful implementation of industry 5.0. any system's resilience to an exponential change in its workload depends on its ability to scale. systems operating under industry 5.0 should be able to scale up to meet demand thanks to the many interconnected devices and applications. this is crucial in co-working settings where humans and robots may be connected over the internet to a landscape of gadgets and applications. humans may need various specialized skill sets, such as handling extraordinary scenarios, investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 423 inventiveness, and designing, whilst robots may take on monotonous day-to-day tasks. this might create newer professions that don't already exist, creating a skills shortage. given that industry 5.0 is still in its infancy, organizations, society, and the government may not yet be prepared to provide the necessary trainers and training programs. (a2) acceptance and adaptability (𝑎2): one major obstacle is the resistance to using robots and other technology at work. concern that machines might supplant people in their current jobs could prevent people from being adaptable. (a3) legal and regulatory (𝑎3): while most countries have general laws and regulations governing the previous industrial revolution, there still has to be enforcement of laws regarding specific advancements in industry 5.0. it is necessary to develop laws and rules that address the broader ecosystem's orientation, directions for human-robot cooperation, and ways to distinguish between different types of robots and other technologies like drones and robots. (a4) cost and funding (𝑎4): according to rumors, industry 5.0 will entail excessive technological exploitation, human-robot collaboration, and mass customization through connected gadgets. both the main enterprise and the local government would need to invest large sums in these. these costs and funding may be seen as a major barrier by developing countries that may have other priorities as well as organizations that may focus on shortterm gains. (a5) ethical concerns (𝑎5): co-working between humans and robots may raise ethical concerns, such as the application of morality to technology. the use of ai and machine learning (ml) may potentially raise ethical concerns, especially if robots begin to take the place of people in routine tasks. 6.1 methods the following steps to find the rank of the attributes using svnwa and svnga step 1: normalizing the data step 2: compute the dominance degree 𝛿(𝐴𝑖 , 𝐴𝑗) = ∑ 𝛾𝑘(𝐴𝑖, 𝐴𝑗), 𝑖, 𝑗 = 1,2,3, … .𝑚𝑛 𝑘=1 step 3: compute the overall value 𝛾𝑘(𝐴𝑖) = ∑ 𝛾𝑘(𝐴𝑖 , 𝐴𝑗) − min (∑ 𝛾𝑘(𝐴𝑖, 𝐴𝑗)) 𝑚 𝑗=1 𝑚 𝑗=1 𝑚𝑎𝑥 ∑ 𝛾𝑘(𝐴𝑖, 𝐴𝑗) − min (∑ 𝛾𝑘(𝐴𝑖 , 𝐴𝑗)) 𝑚 𝑗=1 𝑚 𝑗=1 step 4: ranking all alternatives the primary steps in our research technique are as follows. initially, as stated in sections, we carried out a thorough literature analysis to find any potential obstacles to industry 5.0's successful implementation. the second stage was to present the identified barriers to 14 industry experts so they could conduct a survey to find out how one barrier affected the others. in the third phase, we used the aforementioned data set to rank these obstacles and divide them into cause-and-effect groups using a well-known mcdm (multi-criteria decision making) approach called todim method and compare with dematel (decision making trial and evaluation laboratory). finally, we presented our findings to three business and two academic experts, along with a comprehensive mitigation strategy to address the key obstacles to implementing industry 5.0. given that it takes into account how several criteria interact, it is well adapted to tackle real-world issues. taking into account the aforementioned, we employ the neutrosophic todim technique to solve our study topic. 424 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi 7. numerical problem a straightforward linguistic scale was employed (n = no influence, vl = very low influence, l = low influence, m = medium influence, h = high, and vh = very high impact). if the possible company 𝐴𝑖 (𝑖=1,2,3,4) are determine by the neutrosophic matrix: 𝑅 =̃ [ (0.3,0.6,0.1) (0.4,0.2,0.1) (0.1,0.6,0.1) (0.6,0.2,0.1) (0.8,0.2,0.2) (0.6,0.2,0.3) (0.5,0.6,0.1) (0.7,0.1,0.3) (0.5,0.2,0.1) (0.5,0.7,0.1) (0.6,0.3,0.4) (0.6,0.3,0.1) (0.5,0.3,0.1) (0.3,0.4,0.2) (0.7,0.6,0.1) (0.5,0.7,0.2) (0.8,0.2,0.1) (0.6,0.3,0.1) (0.7,0.6,0.1) (0.5,0.8,0.1)] initially, 𝜔4 = 𝑚𝑎𝑥{𝜔1, 𝜔2, 𝜔3,𝜔4}, and by choosing vector 𝜔 = (0.2,0.1,0.3,0.4)𝑇 and the reference attribute weight 𝜔𝑟 = 0.4, the relative weights are determined as follows: 𝜔1 = 0.75, 𝜔2, = 0.5, 𝜔3, = 0.25, 𝜔4 = 1 and 𝜃 = 1.5 dominance matrix is given as: ∅𝑗 = [[ 0 ⋯ ∅1(𝐴1, 𝐴𝑚) ⋮ ⋱ ⋮ ∅1(𝐴𝑚, 𝐴1) ⋯ 0 ]] ∅1 = [ 0 0.273 −0.146 0.341 0 0.2886 0.182 0.428 0.316 0.230 0.206 0.103 0 0.387 0.258 −0.342 −0.253 0.2581 0.129 −0.309 0 −0.231 −0.253 0.129 0 ] ∅2 = [ 0 0.253 0.279 0.211 0 0.316 −0.334 −0.283 −0.253 0.126 0.2828 0.103 0 0.298 0.235 0.236 0.2113 0.235 0.210 0.3577 0 −0.219 0.283 0.183 0 ] ∅3 = [ 0 −0.593 −0.593 0.247 0 0.438 0.247 0.1666 0.182 0.182 −0.438 −0.473 0 0.506 −0.4 −0.4 −0.432 0.182 0.197 0.211 0 0.197 − 0.166 0.473 0 ] ∅4 = [ 0 −0.297 −0.297 0.494 0 0.219 0.494 0.333 0.365 0.365 −0.219 −0.236 0 0.253 −0.2 −0.2 −0.219 0.365 0.394 0.421 0 0.394 − 0.333 0.236 0 ] based on ∅𝑗, the overall dominance degree is calculated according to the following expression: 𝛿(𝐴𝑖 , 𝐴𝑗) = ∑ ∅𝑧(𝐴𝑖, 𝐴𝑗), 𝑖, 𝑗 = 1,2,3, … . .𝑚𝑛 𝑧=1 , yielding: investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 425 𝛿 = [ 0 −2.264 −0.772 1.752 0 0.027 0.302 0.529 0.541 0.112 −1.867 −1.493 0 0.089 −0.270 −0.871 −0.926 1.459 1.339 −0.429 0 0.119 − −0.152 0.688 0 ] then the values of 𝛿(𝐴𝑖), 𝑖 = 1,2,3,4,5 are as follows: 𝛿(𝐴1) = 0, 𝛿(𝐴2) = 1, 𝛿(𝐴3) = 0.495, 𝛿(𝐴4) = 0.307, 𝛿(𝐴5) = 0.396 we get the order of 𝛿(𝐴𝑖) as: 𝛿(𝐴2), 𝛿(𝐴3), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴1). table 1 gives values of the aggregation operators corresponding attributes. table 1 score value of svnwa and svnga svnwa svnga a1 (0.711,0.188,0.008) (0.0720,0.8926, -0.4437) a2 (0.985,0.005,0.002) (0.3145,0.5157,0.1822) a3 (0.998,0.077,0.002) (0.748,0.834,0.086) a4 (0.721,0.003,0.075) (0.101,0.232,0) a5 (0.968,0.089,0.031) (0.854,0.284,0) table 2 shows the score function of emerging values corresponding to attributes. table 2 score function of the emerging values svnwa svnga a1 0.663354 -0.44372 a2 0.986081 0.050355 a3 0.921302 -0.00262 a4 0.819463 0.31903 a5 0.878935 0.642449 table 3 shows the order of the todim and aggregation operator. table 3 order of the emerging technology enterprises order todim 𝛿(𝐴2), 𝛿(𝐴3), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴1) svnwa 𝛿(𝐴2), 𝛿(𝐴3), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴1) svnmg 𝛿(𝐴2), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴3), 𝛿(𝐴1) 8. comparative analysis table 4 compares the suggested method to other methods already in use [105] according to the value of dematel method. 426 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi table 4 comparison between the proposed and existing methods order todim 𝛿(𝐴2), 𝛿(𝐴3), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴1) svnwa 𝛿(𝐴2), 𝛿(𝐴3), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴1) svnmg 𝛿(𝐴2), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴3), 𝛿(𝐴1) dematel 𝛿(𝐴2), 𝛿(𝐴3), 𝛿(𝐴5), 𝛿(𝐴4), 𝛿(𝐴1) in table 4 the biggest hurdle is the ability to accept and adapt to barriers. experts have confirmed this, saying that while new technology and working methods like co-working with robots are anticipated to become standard in industry 5.0 in the future, people should be open to them. to cut costs and save time, industry 5.0 can opt to replace people with machines. the fear of being replaced by more effective machines and robots may make people resistant to the aforementioned solution. the second in the causative group is the legal and regulatory obstacle. according to experts, while most countries have basic laws and regulations in place surrounding the last industrial revolution, legislation regarding specific advancements in industry 5.0 needs to be implemented. industry 5.0 refers to newer, more intelligent working methods and technologies, such as digital twins, drone technology, and connected, intelligent equipment. these barriers are influenced by the ones we have just talked about. the influencing group is closest to the barrier regional disparity. this means that of all the cause group obstacles, it has the least impact. urban and rural areas are increasingly separated by socio-economic and technological barriers, which may result in uneven adoption of industry 5.0 principles. security and privacy are the next group to take impact. data exchange through third-party interfaces, according to industry analysts, will cause privacy and security problems, especially if it falls into the hands of people who have bad intentions. it is necessary to have authentication, access rights, audit, and authentication controls in place, but doing so cannot jeopardize the ecosystem's overall effectiveness. we have looked at the obstacles so far, but it is also critical to consider the solutions to problems that have been found. fig. 5 depicts the rank selection by using the following methods: todim, svnwa, svnwg and dematel. it reveals that the ranking of results is marginally different because of the legal and regularity cost and funding and ethical concerns. fig. 5 rank selection for barriers through the methods investigation of industry 5.0 hurdles and their mitigation tactics in emerging economies by todim... 427 the neutrosophic todim approach, however, can choose the businesses in a reasonable manner. the proposed method indicates that the best barriers in context with industry 5.0 deal with indeterminacy. this demonstrates the effectiveness and logic of the strategy we suggested. 9. conclusions from the study above, it is clear that the two operators have almost identical barriers, with a few minor differences. however, the svn todim technique can accurately portray how boundaries in industry 5.0 are decided. dematel approaches do not handle ambiguity. neutrosophic todim is, therefore, a realistic and efficient method of choosing the barriers. this work identifies, picks out, ranks, and mitigates the obstacles that stand in the way of effective implementation of industry 5.0. in order to achieve this, we conducted a thorough review of the available literature to identify common hurdles to industry 5.0 and evaluated the findings with fourteen experts. we ultimately identified eight broad-level barriers. we categorise the eight barriers that have been found into three groups: the prominent group, the cause or influencing group, and the effect group through a rigorous procedure that includes the application of the neutrosophic single valued todim technique from the perspective of the multi-criteria decision-making approach. also, we rank these obstacles among the relevant groups and compare dematal techniques. from our analysis, due to their propensity to interact with other barriers the most, the obstacles that belong to the prominent category require immediate attention. the barriers in this category, listed in decreasing order of importance, are a4 (cost and funding) and a5 (scalability) (upskilling, reskilling, and talent retention). similar to how they significantly affect the other obstacles, the cause group's barriers must also be considered. in descending order of importance, the most significant obstacles in this category are a1 (acceptance and adaption), and a2 (legal and regulatory), showing that they were most affected by the other hurdles. we understand the limits of our research. secondly, the eight implementation barriers for industry 5.0 have been narrowed down based on prior research and industry reports involving established countries. although these barriers would be similar to those in developing economies, we acknowledge that emerging countries could need to concentrate more on some particular barriers than others. to overcome the abovementioned constraint, we presented the obstacles to industry and academic professionals who made small suggestions for terminology adjustments to better fit the indian context, somewhat attesting to the barriers' applicability. second, the idea of industry 5.0 is still extremely new and is developing. depending on the market sector and economy, these obstacles may change. thirdly, we employed the well-liked mcdm technique of neutrosophic todim for this study's research methodology. nevertheless, the grey-neutrosophic todiml methodology may be applied in the industry 5.0 environment to eliminate fuzziness and indeterminacy because it is changing and uncertain. we recommend that future scholars examine the correlation between industry 5.0 restrictions in other developed countries and the findings of our study. a longitudinal assessment of the potential barriers may also be conducted to determine whether the potential barriers have changed over the development of industry 5.0. finally, given the notion's immature, constantly changing, and dynamic character, future researchers may employ the neutrosophic markov models' technique to remove the ambiguity, indeterminacy, and fuzziness inherent in studying industry 5.0. further research can evaluate the interrelationships 428 n. deivanayagampillai, k. jacob, g. vellapalayam manohar, s. broumi found in this study using statistical research techniques to assess 5.0 adaptability. we recommend that future work scholars examine the correlation between industry 5.0 restrictions in other developed countries and the findings of our study. a longitudinal assessment of the potential barriers may also be conducted to determine whether the potential barriers have changed over industry 5.0's5.0's development. last but not least, from a methodology standpoint, given the immature, dynamic, and always-growing nature of the notion, future researchers may employ the plithogenic sets and grey-dematel technique to remove uncertainty and fuzziness inherent while investigating industry 5.0. references 1. garrett, g., 2021, council post: how to 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x-h., 2014, interval neutrosophic sets and their application in multicriteria decision making problems, the scientific world journal, 2014, 645953. 105. mukherjee, a.a., raj, a., aggarwal, s., 2023, identification of barriers and their mitigation strategies for industry 5.0 implementation in emerging economies, international journal of production economics, 257, 108770. facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 381 399 https://doi.org/10.22190/fume210130048k © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper study towards the time-based mcda ranking analysis – a supplier selection case study bartłomiej kizielewicz1, jakub więckowski 1, andrii shekhovtsov1, jarosław wątróbski2, radosław depczyński2, wojciech sałabun1 1research team on intelligent decision support systems, department of artificial intelligence methods and applied mathematics, faculty of computer science and information technology, west pomeranian university of technology, szczecin, poland 2institute of management, university of szczecin, szczecin, poland abstract. decision-making processes increasingly use models based on various methods to ensure professional analysis and evaluation of the considered alternatives. however, the abundance of these methods makes it difficult to choose the proper method to solve a given problem. also, it is worth noting whether different results can be obtained using different methods within a single decision problem. in this paper, we used three selected multicriteria decision analysis (mcda) methods called comet, topsis, and spotis in order to examine how the obtained rankings vary. the selection of material suppliers was taken into consideration. the equal weights, entropy and standard deviation methods were used to determine the weights for criteria. final preferences values were then compared with the ws similarity coefficient and weighted spearman correlation coefficient to check the similarity of the received rankings. it was noticed that in the given problem, all of the methods provide highly correlated results, and the obtained positional rankings are not significantly different. however, practical conclusions indicate the need to look for improved solutions in the correct and accurate assessment of suppliers in a given period. key words: mcda, supplier selection, comet, topsis, spotis 1. introduction various methods support the decision-making process with the indicators that favor one solution over another [1, 2]. it allows the experts to choose the optimal alternatives with a greater precision than relying only on their feelings. however, many different methods are used to create such decision-support models, making it a challenge to identify the method which is the right choice for a given decision problem [3, 4]. received january 30, 2021 / accepted may 05, 2021 corresponding author: wojciech sałabun west pomeranian university of technology, szczecin; żołnierska street 49, 71-210 szczecin, poland e-mail: wojciech.salabun@zut.edu.pl 382 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. one of the main approaches to creating support systems is the multi-criteria decision analysis (mcda) methods [5]. they allow evaluating a defined set of alternatives based on the selected criteria describing the quality of alternatives in the considered aspects numerically [6]. however, the number of methods belonging to this group is constantly growing, and the results obtained often give different results within one problem [7, 8, 9]. hence the question arises, how do mcda rankings vary? the multi-criteria decision analysis is of great interest to experts in problems where the choice of an optimal solution is influenced by numerous factors that determine the quality of alternatives. these methods are applied in multiple issues, such as evaluating means of transport [10, 11], the selection of industrial suppliers [12, 13], the assessment of medical patients’ health [14, 15], or the management of resources [16]. the possibilities of a broad application of the mcda methods make them a significant support and are effectively used by a larger group of experts. the calculation of the rankings, in some cases, requires the expert to define the values of the weights for the criteria, thus describing their importance in the evaluation process [17]. it is also possible to determine weights by the methods that do not require expert knowledge, and this is possible by using the methods such as entropy or standard deviation [18, 19]. what is more, in the case of testing the mcda methods’ performance, it is worth determining the extent to which the obtained rankings are similar to each other [20]. for this purpose, similarity coefficients may be used, which use the preferences of alternatives obtained as a result of the mcda methods in order to calculate the correlation value. the selection and evaluation of suppliers play a very important role in an enterprise’s functioning [21]. these activities occupy a key role in implementing the organization’s strategic objectives while being an important determinant of competitive advantage [22]. due to its nature, the problem of correct supplier selection occupies a key role in managing the entire supply chain (supply chain management scm) of the enterprise [23]. in the literature, it is emphasized that the correct selection and assessment of the supplier determines the possibility of focusing the entity on key competencies [24] and the appropriate response of the enterprise to the challenges posed by the market [25]. pro-environmental initiatives and pro-social activities occurring in recent years have been reflected in the current principles of management of economic entities [23, 26]. taking into account environmental imperatives in business activities caused the problem of selection while evaluation of suppliers is considered not only from the technological and economic perspective but also from pro-environmental factors [27]. this is confirmed by numerous literature studies on green supplier selection [22]. additionally, including pro-environmental [26] and pro-social [23] imperatives in the business sphere and an attempt to find solutions satisfying various perspectives [22] resulted in a model issue of sustainable supplier selection [27]. it should be pointed out that, by definition, the problem of selection and evaluation of suppliers requires the consideration of different factors as well as different perspectives in one process [21]. additionally, some of them have a conflicting character, e.g. the quality of services and price. it should be pointed out that these factors may be of both quantitative and qualitative nature, which implies the necessity of applying an appropriate methodological approach [21]. in scientific research, this has shifted the problem of selection and evaluation to the construction of multi-criteria models. the multiplecriteria decision analysis (mcda) methodology is widely used here. the literature on the subject shows a wide spectrum of multi-criteria supplier selection and evaluation study towards the time-based mcda ranking analysis – a supplier selection case study 383 models built using both the methods of the so-called "american school" of multi-criteria decision support [22] and its european counterpart [27]. in the area of the american school, for example, ahp and topsis [28], vikor [29], anp [23] or dematel [24] methods proved their effectiveness in the problems of supplier selection and evaluation. also, the methods based on the superiority relation (european school) are widely used in this issue and, for example, include methods electre i [25], electre ii [30], electre tri [31] or promethee ii [32]. several works have also used rule-based and mixed-mode expansions of mcda methods, and the current state of research in this area is available in ref. [21]. also, fuzzy expansions of the well-known mcda methods have proven their effectiveness in the problem of supplier evaluation and selection [33]. both simple fuzzy developments of the mcda methods based on the triangular or trapezoidal form of membership function (fuzzy ahp [34], bwm and topsis [26, 35]), as well as based on subsequent generalizations, e.g. intuitionist fuzzy sets mcda methods (topsis [36, 37]) proved to be powerful tools for dealing with the uncertainty of measurements and preferences in supplier selection models. the current state of the art in the use of multi-criteria methods in supplier selection and evaluation is contained, for example, in refs. [21, 22, 27]. on the other hand, despite the intensive development of research of the mcda methods development, it should be pointed out that none of them is universal. moreover despite the same input data, the results (supplier rankings) obtained by different mcda methods may differ [38, 39, 40]. confirmation of this fact can be found in literature, where the problem of objectification in the mcda methodology [38, 39, 40] and benchmarking of the mcda methods was analyzed [8, 41, 42, 43, 44]. what is important is that the authors’ conclusions do not contain generalized conclusions. they are fragmentary (limited to a narrow subset of the assessed methods and the specifics of the domain). the authors unambiguously confirm the validity of the recommendations in the paper [3] on the need for a broader analysis and benchmarking of the mcda methods in particular areas (domains) of application. aggregation equal weights expert knowledge based supplier selection problem stucturing set of criteria decision variants set of reference mcda methods topsis method the similarity coefficients comet methodspotis method ws similarity coefficient set of time based supplier evaluation rankings rankings similarity checking weights determining methods analysis and interpretation entropy method standard deviation method performance tables fig. 1 research framework as an answer to the shortcomings of the mcda methods indicated above, in this paper, we used the characteristic objects method (comet), the stable preference ordering towards ideal solution (spotis) and the technique for order preference by similarity to an ideal solution (topsis) methods to solve the problem of the suppliers’ selection. to provide the criteria weights, the methods of determining the weights were used to omit defining this vector by an expert. obtained preferences were then used to 384 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. compare rankings similarity with two selected coefficients, namely the weighted spearman correlation coefficient and ws similarity coefficient. the aim was to check the impact of the used method to received rankings and answer how mcda rankings vary. a detailed framework presenting the study procedure is shown in fig. 1. the rest of the paper is organized as follows. in section 2, preliminaries of three mcda methods are presented, namely comet, spotis and topsis methods. section 3 presents the assumptions of determining the weights for criteria in multi-criteria problems. section 4 presents the correlations coefficients. section 5 includes the study case, in which the theoretical problem of supplier selection was solved. in section 6 the conclusions from the research are presented. 2. preliminaries multi-criteria decision analysis (mcda) methods have been developed to evaluate a considered set of alternatives in order to obtain numerical values indicating the quality of individual options [45, 46]. they are often used as support models in the decision-making process made by the decision-maker [47, 48, 49]. the main assumptions of the selected mcda methods should be presented in order to introduce the principle of their operation. 2.1. the comet method the characteristic objects method (comet) performance is based on the definition of the characteristic objects, which are used in the process of the pairwise comparison made by expert [50, 51]. the preferences for the set of alternatives are then being calculated based on the obtained rule base. the main advantage is that it is completely free of the rank reversal phenomenon [52], meaning that the change in the number of alternatives will not affect the received ranking. the formal notation of the comet method should be shortly recalled [53, 54, 55]. step 1 define the space of the problem – the expert determines the dimensionality of the problem by selecting number r of criteria, c1,c2, ...,cr. then, the set of fuzzy numbers for each criterion ci is selected: 1 2 ,..., }{ , r rn r r rnc c c c= (1) where nr is a number of the fuzzy numbers for criterion r. step 2 generate characteristic objects – the characteristic objects (co) are obtained by using the cartesian product of fuzzy numbers cores for each criterion as follows: 1 2( ) ( ) ( )rco c c c c c c=   (2) step 3 rank the characteristic objects – the expert determines the matrix of expert judgment (mej). it is a result of pairwise comparison of the cos by the problem expert. the mej matrix contains results of comparing characteristic objects by the expert, where αij is the result of comparing coi and coj by the expert. function fexp denotes the mental function of the expert. it depends solely on the knowledge of the expert and can be presented as eq. (3). afterwards, the vertical vector of the summed judgments (sj) is obtained as given in eq. (4). study towards the time-based mcda ranking analysis – a supplier selection case study 385 exp exp exp exp exp exp 0.0, ( ) ( ) 0.5, ( ) ( ) 1.0, ( ) ( ) i j ij i j i j f co f co f co f co f co f co     = =   (3) 1 t i ij j sj  = = (4) finally, the values of preference are approximated for each characteristic object. as a result, vertical vector p is obtained, where i th row contains the approximate value of preference for coi. step 4 the rule base – each characteristic object and value of preference is converted to a fuzzy rule as follows: ( ) ( )1 2 ii iif c c and c c and then p (5) in this way, the complete fuzzy rule base is obtained. step 5 inference and final ranking – each alternative is presented as a set of crisp numbers (e.g., ai = {a1i, a2i, ..., ari}). this set corresponds to criteria c1, c2, ..., cr. mamdani’s fuzzy inference method is used to compute preference of i th alternative. the rule base guarantees that the obtained results are unequivocal. 2.2. the spotis method the stable preference ordering towards ideal solution (spotis) method is a recently developed method [56] and its main assumption is to define the data boundaries so as to determine the ideal solution point (isp). further calculations to obtain the final preferences for alternatives are being made based on the received isp. the method is declared to be fully resistant to the rank reversal phenomenon, similarly to the comet method. for each criterion cj, the data boundaries should be defined. it is required to select the maximum sj max and minimum sj min bound for every cj. the definition of ideal solution point sj * depends on the type of criterion, where for profit type it should meet the condition of sj *=sj max, and for cost type it should be sj *=sj min. the following steps of spotis performance are presented below. step 1 calculation of the normalized distances to ideal solution point: ),( ij j ij i j max min j j s s d a s s s   − = − (6) step 2 calculation of weighted normalized distances d(ai, s*) ∈ [0,1], according to: 1 ,( )( , ) n i j ij i j j d a s w d a s  = = (7) step 3 final ranking should be determined based on d(ai, s*) values. smaller values d(ai, s*) which are preferences of alternatives result in a better position in general ranking. 386 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. 2.3. the topsis method the technique for order preference by similarity to an ideal solution (topsis) method was developed by chen and hwang in 1992 [57, 58]. authors proposed to examine the set of alternatives based on the calculation of the distance to the ideal solution. during this process, the positive ideal solution (pis) and negative ideal solution (nis) are used to calculate alternatives’ final preferences. moreover, topsis requires defining of the weights vector describing the relevance of each criterion [59]. proper application of this method should begin with normalizing the decision matrix. next step is to calculate the weighted normalized decision matrix: , 1, , ; 1, ,ij i ijv w r j j i n=  =  =  (8) positive and negative ideal solutions for a defined decision-making problem should also be identified: 1 1 { , , } {(max | ), (min | )} { , , } {(min | ), (max | )} p c n j ij j ij p c n j ij j ij a v v v i i v i i a v v v i i v i i    − − − =  =   =  =   (9) where ic stands for cost type criteria and ip for profit type. negative and positive distance from an ideal solution should be calculated using the n-dimensional euclidean distance. to apply such calculations, formula presented below should be used: 2 1 2 1 ( ) , 1, , ( ) , 1, , n j ij i i n j ij i i d v v j j d v v j j   = − − = = − =  = − =    (10) the last step is to calculate the relative closeness to the ideal solution: , 1, , ( ) j j j j d c j j d d −   − = =  + (11) 3. weights determining methods the methods for determining the criteria’ weights can be divided into two groups: subjective and objective. in the first one, the weights are chosen based on the expert’s knowledge and feelings, while in the second one, certain features of the data from the decision matrix are used in determining the weights. the calculation of weights in an objective way for the selected three methods is presented below. 3.1. equal weights method this method assigns weights equally to all criteria. the number of criteria equal to n will be the denominator in the calculation of the values of weights, where the nominator is 1 by the necessity of meeting the condition of summing up the values of weights to this study towards the time-based mcda ranking analysis – a supplier selection case study 387 particular number. this method is used in the sensitivity analysis of solutions obtained by different mcda methods. the formula for calculating equal weights is shown below: 1 jw n = (12) 3.2. entropy method the entropy method involves measuring the average amount of information using an appropriate normalization given by eq. (13). based on this, the entropy value for each criterion is calculated according to eq. (14). the weights’ final value is calculated using eq. (15). 1 , 1, , ; 1, , ij ij m ij i x p i m j n x = = =  =   (13) 1 0 ln , 0 , ) 0 ( m i i ij j ij ij j p p p e p =   = =      (14) 1 1 (1 ) j j n i i e w e = − = − (15) 3.3. standard deviation method the standard deviation method in calculating the criteria weights’ values is based on the determination of the standard deviation, where larger values obtained result in assigning a greater weight to the more diverse criteria (eq. (16)). the final values of the criteria weights are determined by means of eq. (17), where normalization is performed using sums of standard deviation values for individual criteria. 2 1 ( ) , 1, , m ij j i j x x j n m  = − = =   (16) 1 / , 1, , n j j j j w j n  = = =  (17) 4. similarity coefficients as the rankings obtained using mcda methods were compared using the weighted spearman correlation coefficient and ws similarity coefficient, their explanations are given below. 388 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. 4.1. weighted spearman’s rank correlation coefficient this coefficient differs from the classic spearman measure because it considers where the alternatives under study were in the rankings under comparison. however, it is a symmetrical measure and does not consider which of the rankings the reference is. this coefficient can be represented by eq. (18). 2 1 3 2 1(6 ( ) ( 1) ( ) 1 ( 1) ) n i i i i i w x y n x n y r n n n n =  − − + + − + = −  + − −  (18) 4.2. rank similarity coefficient the ws ranking similarity coefficient is a new coefficient distinguished by its high sensitivity to significant changes in the ranking [60]. this index is asymmetric and strongly dependent on the difference between the two considered rankings at specific positions. it can be represented by eq. (19). 1 | | 1 2 max{| 1|,| |} i n x i i i i i x y ws x x n − =  − = −   − −   (19) 5. case study the conducted research concerned the evaluation of the quality of suppliers of materials for the manufacturing of metallurgical products. the company produces steel structures using purchased materials, so the price, quality, and security of supply aspects are important elements. using three selected mcda methods, it was decided to solve material suppliers’ problem and evaluate the alternatives considered, the criteria presented in table 1 were taken into account. their relevance determined the selection of criteria in evaluating suppliers and the extent to which they affect their attractiveness. the selection of such a set of criteria resulted from the analysis of reference literature bibliographic studies [61, 62, 63, 64, 65]. on its basis, a set of 53 potential supplier evaluation criteria was identified. subsequently, the company's panel of experts identified criteria relevant from the point of view of the business entity under study. table 1 criteria ci taken into consideration in solving multi-criteria problem ci name c1 price c2 materials quality c3 deliveries timeliness c4 discounts c5 payment condition c6 cooperation assessment c7 supplier communication c8 complaint handling c9 delivery terms study towards the time-based mcda ranking analysis – a supplier selection case study 389 the comet, spotis and topsis methods were used to evaluate the suppliers' quality and materials. this analysis was carried out using data collected over the last four years, taking monthly deliveries. also, in evaluating a set of alternatives, it was decided to use the methods of determining the weights for the criteria to check whether the method used would affect the results obtained in the given problem. the obtained preference results were then subjected to a comparative analysis, which consisted of checking the extent to which the obtained rankings are correlated. for this purpose, the weighted spearman correlation coefficient and ws similarity coefficient were used. the rankings' correlation values are shown in fig. 2 and a1 for the first and second selected coefficients. in individual fig. 2 and a1, the horizontal axis represents the consecutive month number, and the vertical axis represents the value of the obtained correlation coefficient. it should also be noted that the correlation studied concerns rankings obtained by the selected mcda method based on data for two consecutive months. in practice, the resulting broken line represents an analysis of the variability of the rankings of the assessed suppliers. besides, each of figs. (2, 3, 4, and 5, 6, 7) represents results obtained based on different techniques for determining the objectified weights in the model, i.e., entropy, equal and standard. fig. 2 month to month mcda based supplier rankings comparison with weighted spearman correlation coefficient for entropy type weights fig. 3 month to month mcda based supplier rankings comparison with weighted spearman correlation coefficient for equal type weights 390 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. fig. 4 month to month mcda based supplier rankings comparison with weighted spearman correlation coefficient for standard deviation type weights fig. 5 month to month mcda based supplier rankings comparison with weighted spearman correlation coefficient for standard deviation type weights fig. 6 month to month mcda based supplier rankings comparison with ws similarity coefficient for equal type weights study towards the time-based mcda ranking analysis – a supplier selection case study 391 fig. 7 month to month mcda based supplier rankings comparison with ws similarity coefficient for standard deviation type weights a relatively high correlation of rankings is shown in both cases, both for all used methods of determining the relative importance of attributes (entropy, equal and standard deviation methods) and mcda methods (comet, spotis and topsis). as a result of the analysis of figs. 2, 3 and 4, it is easy to show that the highest variability of rankings was observed for the period from month 11 to 14 and for the pairs of months 1 and 2, 23 and 24, 37 and 38 as well as 45 and 46. as a result of the analysis of figs. 5, 6, and 7, it is easy to show that higher sensitivity of ws coefficient concerning weighted spearman's rank correlation coefficient results in the fact that in addition to the previously indicated significant variability of rankings, there was an additional period of significant variability of rankings covering months 18 to 20. in investigating and analyzing the results, periods were sought where the greatest differences in the rankings obtained were recorded, and consequently, where the correlation of the rankings was lowest. tables 2, 3 and 4 present, in turn, the results obtained for the comet, spotis and topsis methods taking into account the lowest similarity suppliers final rankings. for each of the weighing criteria methods, two top lowest correlated rankings are included, along with data describing the year and month in which the suppliers were evaluated. as can be seen, these tables do not differ since, as rw, the ws value was the lowest for these rankings. table 2 lowest correlated suppliers final rankings (comet method compared with ws similarity coefficient) alternative entropy equal std a1 7.0 3.0 2.0 3.0 7.0 3.0 a2 3.0 1.0 3.0 3.0 3.0 1.0 a3 7.0 5.5 5.5 3.0 7.0 5.5 a4 4.0 2.0 5.5 5.0 4.0 2.0 a5 7.0 5.5 1.0 10.0 7.0 5.5 a6 1.0 8.0 8.0 6.0 1.0 8.0 a7 7.0 5.5 11.0 7.5 7.0 5.5 a8 7.0 5.5 7.0 7.5 7.0 5.5 a9 2.0 9.0 4.0 1.0 2.0 9.0 a10 10.0 11.0 10.0 9.0 10.0 11.0 a11 11.0 10.0 9.0 11.0 11.0 10.0 date 2017-11 2017-12 2018-12 2019-01 2017-11 2017-12 392 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. table 3 lowest correlated suppliers final rankings (spotis method compared with ws similarity coefficient) alternative entropy equal std a1 8.0 5.0 8.0 9.0 9.0 4.0 a2 4.0 3.0 8.0 7.0 7.0 5.0 a3 10.0 6.0 2.5 7.0 10.0 8.0 a4 9.0 9.0 8.0 7.0 8.0 9.0 a5 1.0 10.0 10.0 4.0 1.0 10.0 a6 5.0 8.0 5.0 1.5 3.0 7.0 a7 6.5 7.0 5.0 4.0 5.5 6.0 a8 6.5 4.0 5.0 4.0 5.5 3.0 a9 11.0 11.0 11.0 11.0 11.0 11.0 a10 3.0 2.0 2.5 1.5 2.0 2.0 a11 2.0 1.0 1.0 10.0 4.0 1.0 date 2018-02 2018-03 2018-01 2018-02 2018-02 2018-03 table 4 lowest correlated suppliers final rankings (topsis method compared with ws similarity coefficient) alternative entropy equal std a1 7.0 3.0 2.0 3.0 9.0 3.0 a2 2.0 1.0 4.0 3.0 6.0 1.0 a3 7.0 5.5 8.0 3.0 9.0 5.5 a4 4.0 2.0 6.0 5.5 3.0 2.0 a5 7.0 5.5 1.0 5.5 9.0 5.5 a6 1.0 8.0 10.0 11.0 2.0 8.0 a7 7.0 5.5 10.0 8.5 9.0 5.5 a8 7.0 5.5 10.0 8.5 9.0 5.5 a9 3.0 9.0 3.0 1.0 1.0 9.0 a10 10.0 11.0 6.0 10.0 5.0 11.0 a11 11.0 10.0 6.0 7.0 4.0 10.0 date 2017-11 2017-12 2018-12 2019-01 2017-11 2017-12 fig. 8 gives a graphical representation of the correlation between the suppliers month to month rankings obtained for the full 48 month period. the figures below also show detailed correlations for all three mcda methods (topsis, spotis, comet) using ws and rw coefficients. fig. 8 shows the case where equal attribute weights were used in the mcda models. these figures illustrate the density distribution obtained using a month-to-month comparison of the individual similarity coefficients between the different mcda and weighting methods. the most similar results were obtained with the comet and topsis methods. the appendix sets of figs. 9 and 10 were developed for the mcda models using standard deviation and entropy-based weighting methods. study towards the time-based mcda ranking analysis – a supplier selection case study 393 fig. 8 month to month mcda based supplier rankings correlation analysis – equal weights case, 48 month period 394 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. fig. 9 month to month mcda based supplier rankings correlation analysis – standard deviation based weights case, 48 month period study towards the time-based mcda ranking analysis – a supplier selection case study 395 fig. 10 month to month mcda based supplier rankings correlation analysis – entropy based weights case, 48 month period the conducted research clearly shows significant variability of the month-to-monthsuppliers of rankings. objectification of further research leads to the analysis of differences in rankings resulting from different mcda methods (topsis, spotis, comet). one form of such analysis and presentation of detailed quantitative differences between mcda methods in suppliers' rankings is a set of fig. 11. in a nutshell, it can be stated that these are so-called "error charts" showing each time the lack of convergence of the place in the ranking for 2 analyzed mcda methods. a dot plot is expected in perfect correlation, where all dots are located on the main diagonal. it turns out again that the best results were obtained for the comet and topsis methods, whose results are characterized by the most significant similarity. 396 b. kizielewicz, j. więckowski, a. shekhovtsov, et al. fig. 11 month to month mcda based supplier rankings misfits analysis – entropy based weights case, 48 month period 6. conclusions the problem of selecting the best alternative in the considered set, in which many criteria influence the quality assessment, is complex. many support systems are developed to improve the decision-making process, based on different approaches, using mcda methods. however, their number and use of different approaches in solving the problem by these methods often lead to different results. nevertheless, it is worth identifying how obtaining the rankings of mcda methods vary from one another. three methods were selected to solve the problem of evaluating material suppliers in the metallurgical industry: comet, spotis and topsis [2, 5, 66, 67]. it was also decided to use criterion weighting methods to check their performance on the results obtained. the rankings were compared using the ws similarity coefficient and the weighted spearman correlation coefficient to determine the resulting correlations. despite the use of different methods to determine the relevance of the weights and to determine the alternatives’ final quality preferences, similar results were observed, which shows that in the given problem of evaluating material suppliers, the method used did not significantly affect the rankings obtained. this study shows that the significant variability in time-based supplier rankings warrants further research in seeking detailed, reliable mapping models proper reflecting collected data over time. forgetting or recalling functions adopted 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ahp-fuzzy topsis model for supplier evaluation in manufacturing environment, operational research in engineering sciences: theory and applications, 2(1), pp. 72-90. 66. kizielewicz, b., kołodziejczyk, j., 2020, effects of the selection of characteristic values on the accuracy of results in the comet method, procedia computer science, 176, pp. 3581-3590. 67. paradowski, b., więckowski, j., dobryakova, l., 2020, why topsis does not always give correct results?, procedia computer science, 176, pp. 3591-3600. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 335 341 doi: 10.22190/fume1603335p critical velocity of controllability of sliding friction by normal oscillations in viscoelastic contacts udc 539.3 mikhail popov 1,2,3 1 tomsk polytechnic university, tomsk, russia 2 technische universität berlin, berlin, germany 3 tomsk state university, tomsk, russia abstract. sliding friction can be reduced substantially by applying ultrasonic vibration in the sliding plane or in the normal direction. this effect is well known and used in many applications ranging from press forming to ultrasonic actuators. one of the characteristics of the phenomenon is that, at a given frequency and amplitude of oscillation, the observed friction reduction diminishes with increasing sliding velocity. beyond a certain critical sliding velocity, there is no longer any difference between the coefficients of friction with or without vibration. this critical velocity depends on material and kinematic parameters and is a key characteristic that must be accounted for by any theory of influence of vibration on friction. recently, the critical sliding velocity has been interpreted as the transition point from periodic stick-slip to pure sliding and was calculated for purely elastic contacts under uniform sliding with periodic normal loading. here we perform a similar analysis of the critical velocity in viscoelastic contacts using a kelvin material to describe viscoelasticity. a closed-form solution is presented, which contains previously reported results as special cases. this paves the way for more detailed studies of active control of friction in viscoelastic systems, a previously neglected topic with possible applications in elastomer technology and in medicine. key words: active control of friction, ultrasonic vibration, viscoelastic contact, critical velocity received october 22, 2016 / accepted november 30, 2016 corresponding author: mikhail popov technische universität berlin, str. des 17. juni 135, 10623 berlin e-mail:m@popov.name 336 m. popov 1. introduction the reduction of static and sliding friction by ultrasonic oscillation in various directions is a well-known phenomenon with many applications ranging from wire drawing and press forming, stabilization of system dynamics, as in brake squeal suppression, and production of directed motion, as in ultrasonic motors and linear actuators. the effect has been studied for several decades, both experimentally and theoretically. among the proposed explanations, microscopic theories have historically been prevalent. e.g. zaloj et al. [1] suggest that the effect may be due to the dilatation caused by sliding. v. popov et al. point to the possible importance of the microscopic interaction potential [2]. although plausible, microscopic models could never achieve good, quantitative correspondence between theoretical predictions and experimental results, e.g. [3]. opposite to that stand purely macroscopic models, which explain the phenomenon using macroscopic contact mechanics or system dynamics. several system configurations have been considered from that perspective [3, 4, 5] and it was found that the macroscopic models can describe the observed behavior of the systems without fitting parameters. this result is in fact somewhat surprising, considering that these macroscopic theories assume a constant microscopic coefficient of friction and a friction law of the form ff = 0fn. when the average force of friction is determined by integrating the force of friction over time (or integrating stress over time and contact area) and dividing by the integral of the normal force, the direct proportionality of the assumed law of friction will insure that the integrals of normal force will cancel out, with the end result that the average coefficient of friction  must always be equal to 0. this reasoning, however, is subtly flawed, in that it assumes sliding in one direction with a nonzero velocity. it is also possible for the body to temporarily cease motion (e.g. due to increasing normal force or more complicated reasons relating to system dynamics). during such stick phases, the law of friction needs to be written in its static form: ff  0fn. note the less than or equal in this formula, which breaks the proportionality and allows  to be less than 0. to the author’s knowledge, the possibility that the influence of normal oscillations on sliding friction may be explained entirely by the presence of intermittent stick phases has not been made explicit before the publication of the two part-study [6, 7]. in these papers, the stick-induced reduction of friction force was studied in a displacementcontrolled setting with and without in-plane system dynamics. although a closed-form solution for the actual force of friction under the action of normal vibrations does not exist in either case, it has turned out to be possible to calculate the critical velocity vc for a broad class of problems. this critical velocity refers to the maximum sliding velocity, above which vibration no longer has any influence on friction (at a given frequency and amplitude). this is illustrated in fig. 1, which qualitatively describes the behavior of the average coefficient of friction, as it increases from its static value to 0 with increasing sliding velocity. in the theory presented in [6] it was argued that this critical velocity is related to the disappearance of stick in the contact. also in [6], the following expression was obtained for the critical velocity in an entirely displacement-controlled system: 0 * c z * e v u g    , (1) critical velocity of controllability of sliding friction by normal oscillations in viscoelastic contacts 337 where uz is the amplitude of velocity oscillation and e * /g * is the ratio of the normal and tangential stiffness of the contact (the so-called mindlin-ratio). since this ratio is generally of the order of unity, one can roughly say that the critical sliding velocity is equal to the maximum velocity in the normal direction (due to the oscillation) times the microscopic coefficient of friction. this critical velocity also enters into the primary dimensionless parameter characterizing the behavior of the system, which makes accurate analysis of this quantity doubly important. fig. 1 qualitative dependence of the average coefficient of friction (cof) on sliding velocity under action of normal oscillations. of particular interest are the “static cof” at zero velocity, the monotonous increase of the cof with increasing sliding velocity and the critical velocity of controllability, above which the average cof is equal to the microscopic cof, 0, with or without oscillations. if the model is augmented with a system spring and a contact mass, thus enabling inplane system dynamics, the expression for the critical velocity becomes [7]: 2 0 2 z ,c x ,c x c z x,c x k | k k m | v u k | k m |         , (2) where kx,c and kz,c are the tangential and normal stiffness of the contact (in this model, the contact stiffness is assumed to be constant), kx is the tangential stiffness of the surrounding system and m is the mass of the sliding body. the only difference compared to eq. (1) is the additional dependence on the two natural frequencies of the system. indeed, if kx tends to infinity, eq. (2) reduces to the previous result. another notable feature is the presence of two resonant frequencies, in particular xk m where vc becomes infinite. numerical experiments show that in this case, the coefficient of friction reaches a plateau (which is less than 0) at fairly low sliding velocities and does not change thereafter. for the full analysis, the reader is referred to [6, 7]. in the present paper these previous results are extended to also include viscoelastic contacts. active control of friction and system stability seems to be an underexplored topic when viscoelastic contacts are concerned, despite many possible applications in conjunction with the ubiquitous use of elastomers and the rising demands placed on devices in contact with biological tissues in medical technology. with this paper we 338 m. popov would like to begin establishing a quantitative framework for the analysis of viscoelastic friction under oscillation, by proposing that the same methods used in [6, 7] can be applied in viscoelastic contacts in order to calculate the critical velocity in closed form. 2. model and analysis 2.1. formulation of the model the model that will be analyzed in this paper is very similar to the one presented in [7]. it consists of a mass m that is pulled with a constant velocity v0 through a system spring with a constant stiffness kx (see fig. 2). in addition, a displacement-controlled harmonic oscillation is imposed in the direction normal to the plane. the oscillation is defined by: 0z z , zu u u cos t   , (3) where uz is the coordinate of the body in the normal direction, uz,0 the mean indentation depth, uz the oscillation amplitude and  the frequency. the body is connected to the substrate through a contact point, in which amontons’ law of friction with a constant coefficient of friction 0 is assumed. the main difference is that the contact is not elastic but viscoelastic and characterized not only by the constant tangential and normal spring stiffness kx,c and kz,c, but also by the dynamic viscosities γx,c and γz,c. this corresponds to the kelvin material, the simplest model of viscoelasticity. the relevant dynamics of the resulting system is confined to the sliding plane and is characterized by ux, the position of the body and ux,c, the position of the contact point. fig. 2 schematic representation of the considered system, consisting of a mass, a system spring and a viscoelastic contact with the sliding plane. 2.2. analysis of the model 2.2.1. normal force the normal force in the spring-damper combination is given by: 0( )n z,c z z ,c z z ,c z , z z ,c zf k u u k u u cos t u sin t         . (4) critical velocity of controllability of sliding friction by normal oscillations in viscoelastic contacts 339 to ensure that the body is always in contact with the plane, the normal force must always remain positive. this is the case if: 2 2 2 0z ,c z , z z ,c z ,ck u u k     . (5) only this “non-jumping” case is considered in the following. the static force of friction (the force at zero sliding velocity) can be calculated easily by noting that, according to eq. (4), the amplitude of the oscillation of the normal force is equal to: 2 2 2 n z z ,c z ,cf u k      . (6) the static force of friction is equal to the minimal normal force during an oscillation cycle, multiplied with the coefficient of friction: 2 2 2 0 ,0 0 , ,0 , ,( ) ( )s n n z c z z z c z cf f f k u u k         . (7) 2.2.2 tangential movement under the assumption that the immediate contact point is always in the sliding state, the equation of motion of mass m reads: 0 0( )x x x nmu k v t u f   . (8) the equilibrium condition for the “foot point” of the spring-damper combination reads: 0( ) ( )x,c x x,c x,c x x,c nk u u u u f      , (9) where fn is given by eq. (4). equation (8), after inserting eq. (4) on the right hand side, can be easily solved with respect to ux: 0 0 0 0 2 ( ) z ,c z x z , z ,c z ,c x x k u u v t u k cos t sin t k m k             . (10) in our analysis we assume that the material of the contacting elastomer body is isotropic, with a constant (frequency-independent) poisson number. under these conditions, we have: x ,c x ,c z ,c z ,c k k    . (11) equation (9) can also be solved with respect to (ux – ux,c): 0 0( ) z ,c x x,c z. z x ,c k u u u u cos t k      . (12) from eqs. (10) and (12) we can first determine ux,c: 340 m. popov 0 0 0 0 02 1 1 ( ) z ,cz x,c z ,c z , z ,c z ,c z x x,c x,cx ku u v t k u k cos t sin t u cos t k k km k                      (13) and finally x ,cu : 20 0 02 2 20 0 02 2 ( ) ( ) z ,cz x ,c z ,c z ,c z x ,cx z ,c x ,c xz z ,c z x ,cx x ku u v k sin t cos t u sin t km k k k k mu v cos t u sin t km k m k                                    (14) the critical velocity of controllability is given by the condition that the amplitude of the oscillating part of this solution becomes equal to constant sliding velocity vc: 2 2 20 2 ( ) ( ) z ,cz c z ,c x,c x x,cx ku v m k k k| m k |                  . (15) note that the critical velocity depends on the oscillation amplitude but not on the average indentation. in the limit of a very stiff system spring, kx, the critical velocity, eq. (15), is reduced to eq. (1), which thus appears to be valid independently of the viscoelastic properties of the medium. according to the method of dimensionality reduction (mdr) [8], any rotationally symmetric contact can be equivalently represented by a model consisting of a series of independent springs (note that an equivalent one-dimensional model can in fact be constructed for almost arbitrary, e.g. rough, contacts, although there may be no closed-form mapping rule in the general case). as has been argued in [6], the existence an equivalent model with uncoupled spring elements, together with the indentation-independence of eq. (15), implies that the obtained result in eq. (15) is valid not only for the simple considered model with a single spring-damper combination, but also for quite general contacts (so long as the amplitude of oscillation remains small). 3. conclusion while the details of the influence of oscillation on friction may be very complicated at intermediate sliding velocities [7], there are still two simple and nearly universal (except in resonant cases) characteristic points: first, the velocity-dependences all start from the static value at vanishing velocity. second, the coefficient of friction increases monotonically (again, barring exceptional system-dynamical circumstances) until it reaches the microscopic value at some critical velocity. these two points, the static coefficient of friction, and the critical velocity of controllability of friction, are the most important characteristics of any oscillating frictional system. it so happens that both of these points can be determined analytically for very general classes of contacts with and without system dynamics. in the present paper, the critical velocity of controllability was determined for the simplest possible viscoelastic rheology (kelvin body) and the simplest possible contact geometry (contact with constant contact stiffness, e.g. cylindrical punch). eq. (15) provides critical velocity of controllability of sliding friction by normal oscillations in viscoelastic contacts 341 an explicit analytical solution. even under these simple assumptions, the critical velocity depends on almost all system and loading parameters: the local coefficient of friction 0, mass m of the system, the stiffness of the contact and of the system, the frequency of oscillations, the damping coefficient of the contact, and on the amplitude of oscillations. however, it does not depend on absolute indentation, which permits easy generalization to more realistic contact geometry. further, in the case of displacement-controlled horizontal movement (corresponding to an infinitely stiff surrounding system, which eliminates system dynamics in the contact) it was found that the critical velocity is given by eq. (1), without dependence on the rheological properties of the contact: only the ratio of the contact stiffness (mindlin ratio) appears in the expression for this critical velocity. in the future, the critical velocity could also be considered for materials with more general rheology. the method of dimensionality reduction [8] provides a natural theoretical framework for this and for further generalizations to arbitrary contact geometries and loading histories. acknowledgements: the author would like to thank v. l. popov and a. e. filippov for inspiring discussions concerning the present work. references 1. zaloj, v, urbakh, m, klafter, j., 1999, modifying friction by manipulating normal response to lateral motion. physical review letters 82(24), pp. 4823, 1999. 2. popov, vl, starcevic, j, filippov, ae., 2010, influence of ultrasonic in-plane oscillations on static and sliding friction and intrinsic length scale of dry friction processes. tribology letters, 39 (1), 25-30, 2010. 3. teidelt, e, 2015, oscillating contacts: friction induced motion and control of friction, dissertation, tu berlin, germany, 151 p. 4. milahin, n., starcevic, j., 2014, influence of the normal force and contact geometry on the static force of friction of an oscillating sample. physical mesomechanics, 17(3), pp. 84-8. 5. milahin, n., li, q., 2016, friction and wear of a spherical indenter under influence of out-of-plane ultrasonic oscillations. physical mesomechanics, 19(2), pp. 149-153. 6. popov, m., popov, vl, popov, nv, 2016, reduction of friction by normal oscillations. i. influence of contact stiffness, arxiv:1611.07017. 7. mao, x, popov, vl, starcevic, j., popov, m, 2016, reduction of friction by nor-mal oscillations. ii. in-plane system dynamics, arxiv:1611.07018. 8. popov, vl., heß, m. 2015, method of dimensionality reduction in contact mechanics and friction, springer, berlin, heidelberg, 265 p. 10976 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume220829004w © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper an improved bare-bones particle swarm algorithm for multi-objective optimization with application to the engineering structures zhaohua wang1, guobiao yang1, yinxu sun2, yongxin li2, fenghe wu2 1school of mechanical engineering, taiyuan university of science and technology, taiyuan, china 2school of mechanical engineering, yanshan university, qinhuangdao, china abstract. in this paper, an improved bare-bones multi-objective particle swarm algorithm is proposed to solve the multi-objective size optimization problems with non-linearity and constraints in structural design and optimization. firstly, the development of particle individual guide and the randomness of gravity factor are increased by modifying the updated form of particle position. then, the combination of spatial grid density and congestion distance ranking is used to maintain the external archive, which is divided into two parts: feasible solution set and infeasible solution set. next, the global best positions are determined by increasing the probability allocation strategy which varies with time. the algorithmic complexity is given and the performance of solution ability, convergence and constraint processing are analyzed through standard test functions and compared with other algorithms. next, as a case study, a support frame of triangle track wheel is optimized by the bb-mopso and improved bb-mopso. the results show that the improved algorithm improves the cross-region exploration, optimal solution distribution and convergence of the bare-bones particle swarm optimization algorithm, which can effectively solve the multi-objective size optimization problem with non-linearity and constraints. key words: optimization, multi-objective, particle swarm, bb-mopso 1. introduction at present, engineering structures are developing towards high reliability and lightweight design. in the design process, the stiffness, strength and vibration characteristics received: august 29, 2022 / accepted january 16, 2023 corresponding author: fenghe wu school of mechanical engineering, yanshan university, no 438, west of hebei avenue, qinhuangdao, china, 066004. e-mail: risingwu@ysu.edu.cn 2 z. wang, g. yang, y. sun, y. li, f. wu of the structure must be taken into account at the same time [1, 2]. therefore, the multi-objective optimization problem (mop) in the process of structural design is becoming more and more important [3]. in the conceptual design stage, the topology configuration of structure is determined by the topology optimization methods [4]. then, the local size parameters are further optimized to enhance the mechanical performance in the detailed design stage. size optimization needs to consider many mechanical performances of the structure at the same time, including stiffness, strength, fatigue life, etc. local extremum and non-convergence often occur in the calculation process, especially the stress-based optimization, which belongs to a typical nonlinear mop with constraints. with the increasing complexity of multi-objective optimization problems, more and more intelligent algorithms are applied to size optimization. bekdas et al. [5] and assimi et al. [6] improved different intelligent algorithms to optimize the truss size with good results obtained. a new collision box is designed and optimized by using archive-based micro-genetic algorithm and improved, non-dominated sorting genetic algorithm, which improve the energy absorption characteristics and comprehensive crashworthiness [7]. although many achievements have been made in the application of intelligent algorithms to size optimization, there are still problems of local extremum and non-convergence for the solution with multi-constraints. evolutionary algorithm or genetic algorithm [8-9] has a special evaluation mechanism, and has gradually matured after several generations of development, but its local search ability is poor, and the selection of more parameters affects the quality of the solution. simulated annealing method [10-11] draws lessons from the phenomenon of sudden jump of solid properties during annealing heat treatment and becomes a good global optimization method, but it has low efficiency and slow convergence speed. ant colony algorithm [12-13] is a parallel algorithm with fast convergence speed and obvious advantages in dealing with complex combinatorial optimization problems, but it is prone to premature termination. inspired by the predation behavior of birds, particle swarm optimization (pso) [14, 15] is a random iterative parallel algorithm, which is simple and easy to implement, has strong global search ability for nonlinear multi-peak problems, and has obvious advantages in solving multi-objective size optimization problems. however, the algorithm is limited to unconstrained optimization problems, and it is ignored to solve general multi-objective optimization problems with constraints. since the pso does not have the ability to deal with constraint problems, it is often necessary to transform the constraint problems with unconstrained problems. an improved multi-objective particle swarm optimization algorithm (mopso) based on the concept of constraint domination is proposed in [16], which perturbs particles with small probability to improve the diversity of the algorithm. mohamad et al. [17] proposed a mopso with good convergence and constraint processing ability for high-dimensional problems to solve complex engineering problems with many optimization variables. li et al. [18] proposed an adaptive particle swarm optimization algorithm with different learning strategies to solve the path planning problem of mobile robot under different types of constraints in complex environment. xu et al. [19] proposed an improved adaptive weighted pso and applied it to multi-objective optimization design of planetary gears, which solved the problem that pso is not easy to converge or fall into local optimum under complex constraints. it can be seen that many scholars have done extensive research on pso in solving mop and constrained an improved bare-bones particle swarm algorithm for multi-objective optimization... 3 problems [20], but the above algorithms need to set parameters such as learning factor and inertia weight, which affects the computational efficiency of pso. a bare-bones particle swarm optimization algorithm (bb-pso), using a gaussian distribution of personal best positions and global best positions to update particle positions, was first proposed by kennedy [21], which has the advantage of not setting inertia weights, learning factors and other control parameter. zhang et al. [22] first extended the bb-pso to multi-objective optimization problem, and the multi-objective economic/environmental scheduling problems with constraints are solved. however, the bare-bones multi-objective particle swarm optimization algorithm (bb-mopso) focuses more on the globality in particle update and global best positions selection, which makes the algorithm's optimization ability stronger, but the local development ability, boundary search ability and particle diversity are correspondingly reduced. therefore, it needs to be further improved for the problems that there are many local extremum and the optimal solution may be located at the boundary when solving nonlinear multi-objective optimization with constraints. based on the above analysis, this paper proposes an improved bb-mopso to solve the multi-objective size optimization problems with non-linearity and constraints. the mop and bb-mopso are introduced in section 2 and section 3. an improved bb-mopso is proposed in section 4. the algorithmic complexity is given and the performance of solution ability, convergence and constraint processing are analyzed in section 5. as a case study, a support frame of triangle track wheel is optimized in section 6. 2. multi-objective optimization problem a classical multi-objective optimization problem can be defined as finding a decision variable * * * * 1 2[ , , , ]nx x x x that satisfies the following conditions: decision space： min max , 1,2, , ( )i i i ix x x i n x x    . (1) equality constraint： ( ) 0, 1,2, ,jh x j j  . (2) inequality constraint： ( ) 0, 1,2, ,kg k k x . (3) minimized objective function： 1 2( ) [ ( ), ( ), , ( )]mf x f x f x f x . (4) function f(x) is called the objective function. each component value of decision variable x is constrained by two boundary values min ix and max ix . the boundaries of all components of the decision variable constitute the decision space of multi-objective optimization. the output values of m objective functions constitute the objective space of multi-objective optimization. the decision variables that conform to all constraints 4 z. wang, g. yang, y. sun, y. li, f. wu constitute the feasible region of multi-objective optimization, and the solution that conforms to eq. (4) in the feasible region is called the optimal solution. different from the single objective optimization problem, the mop contains many conflicting objective functions, and it is difficult for each objective to achieve the optimal at the same time. therefore, its solution becomes a solution set containing infinite elements. usually, we find the effective solution of this solution set, also known as pareto solution, which is defined as: if there is not x x , make ( ) ( )*f x f x , and then *x is defined as an effective solution to the mop. 3. bb-mopso in the pso, each particle represents a solution of the optimized problem. the position of the i-th particle in the n-dimensional space is represented as xi=[xi,1, xi,2,…, xi,n], and the velocity is represented as vi=[vi,1, vi,2,… , vi,n]. each particle has an adaptation value determined by the optimization function, and it is known that the optimal value of the current particle individual is pi=[pi,1, pi,2,…, pi,n] and the global optimal value of the particle is gi=[g1, g2,…, gn] at the i-th iteration. at the next iteration, the particle determines the next motion state according to its own experience and the experience of particles in the same neighborhood until the end of the iteration. the velocity update equation is, , , 1 1 , , 2 2 ,( 1) ( ) ( ( ) ( )) ( ( ) ( ))i j i j i j i j j i jv t wv t c r p t x t c r g t x t      . (5) where w is the inertia weight, c1 and c2 are learning factors. r1 and r2 are random obeying uniformly distributed u(0,1); j=1, 2,... , n, and i=1, 2,..., n, n is the number of particles. the particle position is, , , ,( 1) ( ) ( 1)i j i j i jx t x t v t    . (6) bb-mopso has the same optimization idea as pso, but it is different from traditional pso. on the one hand, the control parameters such as inertia coefficient and learning factor are deleted to overcome the disadvantage that traditional pso relies too much on parameters. the mathematical model to determine the updating position of particles is realized by a gaussian sampling (normal distribution) of personal best positions and global best positions. the mathematical model of bb-mopso [18] is as follows:  1 , 1 , , , , , [ ( ) (1 ) ( )] / 2, ( ) ( ) (0,1) 0.5 ( 1) ( ) others i j i j i j i j i j i j n r p t r g t p t g t u x t p t            . (7) on the other hand, pso belongs to the single-objective optimization method, which has a definite optimal solution. bb-mopso belongs to the multi-objective optimization method. due to the addition of a non-dominated relationship between each particle, the algorithm may get more than one set of optimal solutions (non-inferior solutions) after iteration. therefore, for each particle, there will be more than one candidate point when updating the global best positions, and these candidate points do not dominate each other. all candidate points are stored in a set different from particle swarm, which is the external archive an improved bare-bones particle swarm algorithm for multi-objective optimization... 5 mentioned above. when the particle position needs to be updated, the global best position of the particle is selected from the external archive, and the external archive element is also submitted to the decision maker as the final result of the algorithm. the flow of the algorithm is shown in fig. 1, and the tmax is the maximum number of iterations. fig. 1 flow chart of bb-mopso 4. improved bb-mopso multi-objective size optimization is a nonlinear multi-objective optimization problem with constraints. there are many local extremum, and the pareto solution may be at the boundary, and the constraint relationship is complex. although bb-mopso has some advantages for solving such problems, but the algorithm focuses more on globality in particle update and global best positions selection, which reduces the algorithm’s boundary and cross-region search ability and particle diversity. therefore, the algorithm is difficult to find the global optimal solution when dealing with multi-objective size optimization problems. in this paper, the bb-mopso is improved from three aspects, including particle update mode, maintenance strategy of external archive and global best positions selection, to increase the algorithm's boundary and cross-region search ability and particle diversity. the calculation process of new algorithm is shown in fig. 2, and the improvement of each part of the algorithm will be introduced in detail in sections 4.1~4.3. 4.1 particle position updates approach the particle position update of bb-mopso (eq. 7) is more inclined to the global best positions. but for the size optimization problems, the optimal solution is likely to appear near the constrained boundary, especially the stress-based optimization. therefore, in the process of particle updating, it is necessary to increase the local development of particles. in this paper, the development of individual particle guides and the randomness of gravitational factors are increased in this paper. the new update mode of particle position is shown in eq. (8). 6 z. wang, g. yang, y. sun, y. li, f. wu  1 , 2 , , , , , [ ( ) ( )] / 2, ( ) ( ) (0,1) 0.5 ( 1) ( ) others i j i j i j i j i j i j n r p t r g t p t g t u x t p t           , (8) where r1 and r2 are random numbers in the range of 0~1. fig. 2 flow chart of improved bb-mopso the new update mode can make the particle have a 50% probability to select the relevant component of the current local optimal particle in each iteration calculation. at the same time, the random numbers r1 and r2 can improve the possibility of particles entering another feasible solution region, which expands the particle search range and theoretically increases the boundary and cross region search ability of the algorithm. the updated particle xi (t+1) needs to be compared with the historical optimal particle pi(t) to select the personal best positions. the principle of selection is as follows: when xi (t+1) dominates pi(t), then pi(t+1)=xi(t+1); when xi(t+1) and pi(t) do not dominate each other, anyone can be selected as pi(t+1), which can increase the probability of "excellent particles" entering the next generation; otherwise, pi(t+1)=pi(t). 4.2 maintenance strategy of external archive the constraint dominance relation is constructed by direct comparison method in bb-mopso, which is easy to fall into local optimum in multi-island problem. in order to reduce the possibility of local optimum, the external archive is divided into two parts: feasible solution set and infeasible solution set. in this way, the development of isolated regions and the global capability of the algorithm can be enhanced [23]. after each update, the feasible solution set is updated first, that is, the particle set with constraint violation degree of 0. a new feasible solution set is composed of the newly obtained particles and the particles in the original feasible solution set. pareto domination an improved bare-bones particle swarm algorithm for multi-objective optimization... 7 relationship is analysed, and the particles that do not dominate each other are left. in order to ensure the diversity of learning particles, the original bb-mopso method uses crowded distances sorting [24] as the external archive maintenance strategy. it needs to calculate the euclidean distance of each particle relative to other particles, which has a large computational complexity. in this paper, the combination of spatial grid and crowded distance sorting are used. the objective space is first divided into m1×m2×…mk grid, k is the objective dimension, mk is the grid number of each dimension. the grid position of each particle is calculated to determine the grid density, which can reduce the computational complexity and maintain the optimal solution distribution in the global range. the grid density can be calculated by the following method. the upper bounds f max i and lower bounds f min i of the objective space are given by analysing the values range of each dimension. the objective width of i-th dimension of each grid is max min( ) /i i i id f f m  . (9) suppose the coordinate of a particle corresponding to the objective space is s=[s1, s2,…,sm], then the particle position in each dimension of the objective space is min( , ) 1i i i il fix s f d   , (10) where fix (a, b) is a rounding function. each grid is numbered and placed into different grids according to the position values of each dimension. the number of particles put into the grids is recorded as n (j, t), which indicates the number of particles in the region where the j-th particle is located at time t. suppose the maximum theoretical capacity of the grid is npi, the density is ( , ) /t i pin j t n  . (11) when the total number of feasible solutions in a certain grid is larger than the grid capacity, the crowding distance of each particle is calculated and npi with larger distance as the new external archive particles are selected, to ensure the distribution of the solution. the crowding distance can be calculated by the following method. the objective values of each dimension are sorted from small to large, and composition of objective value sequence [fi,1, fi,2, …, fi,p, …, fi,sn] , then the crowding distance of a objective value is max min1 2, , 1 , -1 1 1 ( ) / ( ) k k p i p i p i p i i i i dy dy f f f f        , (12) where f max i and f min i are represent the maximum and minimum values of the i-th objective in the current archive. dyi,p is the crowding distance of the p-th particle in the objective value of i-dimension. sn is the size of particle swarm. for the update of the infeasible solution set (the particle set with constraint violation degree greater than 0), the pareto dominance relationship analysis is first performed to obtain the mutually un-dominated particles. and then, the adaptive grid technique is used to divide the particles. for the grid exceeding the capacity, instead of congestion distance sorting, npi particles are randomly selected and re-placed into the grid. although this 8 z. wang, g. yang, y. sun, y. li, f. wu approach may reduce the diversity of infeasible solution set, it can still ensure the ability of the algorithm to explore the unknown feasible region and reduce the computational complexity. 4.3 global best positions selection whether the selection of global best positions is reasonable or not is related to the ability of the algorithm to accurately converge to the pareto front. in the bb-mopso, the global best positions for each particle are selected by the tournament selection method with scale of 2 based on the crowding distance value. the larger the congestion distance value, the more likely it is selected as the global best positions. in this paper, the dynamic probability selection method is used to give the global best positions, that is, the global best positions are selected from the infeasible solution set and the feasible solution set with the probability of psl and 1-psl. 1 2 /sl sl slp p p t t   , (13) where psl1 and psl2 are dynamic probability constants, psl1=0.7 and psl2=0.6. t is the current number of iterations. t is the total number of iterations. in this way, the algorithm has sufficient global search ability. the diversity of learning particles in the early stage is improved, and the convergence in the later stage is accelerated. after determining the global best positions, a probability selection equation (eq. 14) is constructed based on the density values of each grid, so that the probability of selecting a value with small density is high and that of selecting a value with large density is low. the global best positions selected in this way will have good distribution and convergence. 1 1 / ( ) ks t t t i i i i p       , (14) where t ip is the final choice probability. ks is total number of grids. 4.4 analysis of algorithmic complexity the computational cost of the improved bb-mopso mainly concentrates on the update of the reserve set. suppose an optimization problem with m objectives and k constraints, the number of feasible and infeasible solutions in the new particle swarm optimization is np and nq, the feasible reserve capacity is na, and the infeasible reserve capacity is nb. firstly, the renewal process of feasible reserve set is analysed. the size of the new population is np+na, and the number of comparisons is m×(np+na) to judge whether a particle is advantages and disadvantages. considering the worst case (the particles in the population are not dominated by each other), it takes m×(np+na)2 comparisons to select all non-inferior solutions from np+na particles. on the other hand, the computational complexity of using crowded distance to maintain the reserve set is o(m×(np+na)× log(np+na)) and log(np+na)<np+na [25]. as a result, the computational complexity of updating the feasible reserve set is o(m×(np+na)2). next, the update process of infeasible reserve set is analyzed. to judge whether a particle in the new population (new infeasible solution and infeasible reserve set) is dominated by the feasible solution, the number of comparisons is m×(nq+nb)2. because an improved bare-bones particle swarm algorithm for multi-objective optimization... 9 there is no need for crowded distance, the computational complexity of updating infeasible reserve set is o(m×(nq+nb)2). in summary, the computational complexity of the improved bb-mopso is o(m×(np+na)2)+o(m×(nq+nb)2). 5. performance analysis of improved bb-mopso the performance of bb-mopso have been tested by a large number of functions and compared with other intelligent algorithms [22]. therefore, the performance of improved bb-mopso was compared with original bb-mopso, nsga2 and spea2 in this paper. although the improved bb-mopso is proposed for the problem with constraints, it does not change the constraint processing method. so, the classical functions of zdt1, zdt3 and zdt4 [26] are used to test the ability of the algorithm to solve convex functions, the ability to solve discontinuous and multi-connected domain functions, and the ability to solve global optimization. in addition, the function of dtlz3 [27] is used to test the ability to converge to global frontier. in the test, the algorithm runs 30 times independently for each function. zdt1 and zdt3 are taken as 100 dimensions, zdt4 is 30 dimensions, particle number is 100, and iteration number is 300. dtlz3 is taken as 10 dimensions particle number is 500, iteration number is 1000, external archive capacity is 50, and grid capacity is 10. the pareto fronts by each algorithm are shown in figs. 3~6. (a) (b) (c) (d) fig. 3 pareto front of test function zdt1 (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso 10 z. wang, g. yang, y. sun, y. li, f. wu (a) (b) (c) (d) fig. 4 pareto front of test function zdt3 (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso (a) (b) (c) (d) fig. 5 pareto front of test function zdt4 (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso an improved bare-bones particle swarm algorithm for multi-objective optimization... 11 (a) (b) (c) (d) fig. 6 pareto front of test function dtlz3 (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso as can be seen from figs. 3-6, neither nsga2 nor spea2 can find the real frontier in the above test functions. in addition, the solution falls into the local optimum. the improved algorithm is significantly different from the original algorithm in the distribution of the solution. the solution of the improved algorithm converges more uniformly to the frontier. the performance of the four algorithms is quantified by calculating the values of spacing metric (sp) and generational distance metric (gd) [26, 28]. the sp reflects the distribution and diversity of feasible solution set in the objective space, while the gd reflects the distance between feasible solution and pareto frontier, as shown in tables 1-4, where the best results are given in bold. table 1 performance statistics of zdt1 im-bb-mopso bb-mopso nsga2 spea2 gdav 1.53e-4 1.49e-4 4.99e-2 5.09e-2 gdvar 5.35e-5 5.31e-5 5.21e-2 5.10e-3 gdbest 1.53e-4 1.09e-4 1.55e-2 3.97e-2 gdworst 4.38e-4 4.05e-4 2.64e-1 6.04e-2 spav 1.67e-2 2.42e-2 1.64e-2 2.25e-2 spvar 1.70e-3 6.60e-3 3.00e-3 1.08e-2 spbest 1.35e-2 1.30e-2 1.15e-2 1.22e-2 spworst 1.97e-2 4.19e-2 2.46e-2 5.91e-2 12 z. wang, g. yang, y. sun, y. li, f. wu table 2 performance statistics of zdt3 im-bb-mopso bb-mopso nsga2 spea2 gdav 6.53e-4 7.74e-4 7.87e-2 5.31e-2 gdvar 4.57e-5 5.60e-5 4.64e-2 6.50e-3 gdbest 5.87e-4 6.78e-4 2.75e-2 4.38e-2 gdworst 7.10e-4 8.60e-4 2.15e-1 7.50e-2 spav 2.05e-2 2.91e-2 2.07e-2 3.21e-2 spvar 2.50e-3 4.70e-3 4.60e-3 1.78e-2 spbest 1.54e-2 2.13e-2 1.02e-2 1.59e-2 spworst 2.78e-2 4.33e-2 2.97e-2 7.62e-2 table 3 performance statistics of zdt4 im-bb-mopso bb-mopso nsga2 spea2 gdav 2.91e-4 3.38e-4 2.73e+0 6.81e+0 gdvar 4.46e-4 5.70e+4 1.51e+0 3.96e+0 gdbest 1.80e-4 2.80e-4 6.55e-1 2.50e+0 gdworst 4.34e-4 5.80e-4 5.58e+0 1.66e+1 spav 1.80e-2 2.58e-2 1.50e-2 1.41e+0 spvar 1.30e-3 7.80e-3 1.61e-2 5.37e+0 spbest 1.39e-2 1.42e-2 0.00e+0 0.00e+0 spworst 2.04e-2 4.36e-2 4.98e-2 2.29e+1 table 4 performance statistics of dtlz3 im-bb-mopso bb-mopso nsga2 spea2 gdav 3.70e-3 4.50e-3 7.45e+0 1.92e+1 gdvar 5.10e-4 7.95e-4 2.42e+0 6.12e+0 gdbest 2.70e-3 3.50e-3 4.01e+0 7.34e+0 gdworst 3.16e-3 5.50e-3 1.42e+1 2.91e+1 spav 7.73e-2 8.61e-2 4.78e+0 8.50e+1 spvar 4.90e-3 6.40e-3 5.01e+0 3.59e+1 spbest 6.53e-2 7.82e-2 6.17e-1 2.00e+1 spworst 8.94e-2 9.20e-2 1.87e+1 1.73e+2 tables show that the distribution and diversity of feasible solutions of the improved algorithm are much stronger than that of the original algorithm when dealing with zdt1, ztd3, ztd4 and dtlz3. then, the functions of bnh, tnk and dtlz8 [27, 29] are used to test the ability of constraints processing for the improved bb-mopso. bnh and tnk are set as follows: particle number is 100, iteration number is 500, maximum capacity of reserve set is 100, and grid capacity is 10. dtlz8 is set as follows: particle number is 500, an improved bare-bones particle swarm algorithm for multi-objective optimization... 13 iteration number is 1000. similarly, the algorithm runs 30 times independently for each function. the pareto front by each algorithm is shown in figs. 7~9. (a) (b) (c) (d) fig. 7 pareto front of test function bnh (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso (a) (b) (c) (d) fig. 8 pareto front of test function tnk (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso 14 z. wang, g. yang, y. sun, y. li, f. wu (a) (b) (c) (d) fig. 9 pareto front of test function dtlz8 (a) nsga2 (b) spea2 (c) bb-mopso (d) improved bb-mopso in order to express the proportion of the elements that are dominant to each other in the solution set of the improved algorithm and other algorithms, the c measure [30] is used for the test functions of bnh, tnk and dtlz8. the results are shown in tables 5-7 and the best results are marked in bold. table 5 performance statistics of tnk im-bb-mopso bb-mopso nsga2 spea2 c(im-bb) 3.50e-1 2.32e-1 1.63e-1 c(bb-d) 3.72e-1 c(nsga2) 3.25e-1 c(spea2) 2.36e-1 spav 6.70e-3 7.00e-3 5.60e-3 7.20e-3 spvar 8.56e-4 1.00e-3 9.12e-4 1.90e-3 spbest 4.50e-3 5.10e-3 2.40e-3 4.60e-3 spworst 8.40e-3 9.00e-3 7.10e-3 1.47e-2 as can be seen from tables 5-7, the distribution of solutions and the dominant number of solutions of the improved algorithm are better than those of other algorithms when dealing with bnh, tnk and dtlz8. although the distribution of naga2 solutions is better when dealing with tnk, the distribution of solutions of the improved algorithm is an improved bare-bones particle swarm algorithm for multi-objective optimization... 15 more stable. this benefits from external archive maintenance based on grid density and congestion distance sorting, which maintains a certain number of solutions in each region. table 6 performance statistics of bnh im-bb-mopso bb-mopso nsga2 spea2 c(im-bb) 7.40e-2 9.50e-2 2.47e-1 c(bb-d) 9.97e-2 c(nsga2) 2.03e-1 c(spea2) 3.70e-1 spav 8.42e-1 8.77e-1 8.65e-1 9.40e-1 spvar 4.34e-2 1.66e-1 6.75e-2 2.51e-1 spbest 5.07e-1 6.20e-1 5.15e-1 6.13e-1 spworst 6.97e-1 1.39e+0 7.74e-1 1.57e+0 table 7 performance statistics of dtlz8 im-bb-mopso bb-mopso nsga2 spea2 c(im-bb) 1.50e-1 3.55e-2 3.35e-2 c(bb-d) 1.70e-1 c(nsga2) 1.30e-1 c(spea2) 9.40e-2 spav 2.99e-2 3.11e-2 3.60e-2 3.79e-2 spvar 4.44e-3 5.00e-3 9.40e-3 6.90e-3 spbest 1.91e-2 2.16e-2 2.34e-2 2.37e-2 spworst 4.13e-2 4.17e-2 5.80e-2 5.24e-2 in order to further verify the optimization ability of the algorithm, the hybrid composition functions of cf4, cf5 and cf6 provided by liang et al. [31] are used to calculate. the shape of three functions is shown in fig. 10. (a) (b) (c) fig. 10 the shape of three functions (a) cf4 (b) cf5 (c) cf6 16 z. wang, g. yang, y. sun, y. li, f. wu for each test function, each algorithm is run 20 times and the maximum fitness evaluations are set at 50,000 for all algorithms. the results are compared as shown in table 8. the first six groups of data about functions of pso, cpso, clpso, cma-es, g3-pcx, de are from reference [31]. the results show that the improved algorithm has good performance in dealing with complex functions. however, the stability of the solution is low when dealing with cf6 function. table 8 performance statistics of cf4, cf5 and cf6 pso cpso clpso cma-es g3-pcx de im-bbmopso cf4(av) 3.14e+2 5.22e+2 3.22e+2 6.16e+2 4.93e+2 3.25e+2 3.10e+2 cf4(std) 2.01e+1 1.22e+2 2.75e+1 6.72e+2 1.42e+2 1.48e+1 7.55e+0 cf5(av) 8.35e+1 2.56e+2 5.37 e+0 3.59e+2 2.60e+1 1.08e+1 3.22e+0 cf5(std) 1.01e+0 1.76e+2 2.61e+0 1.68e+2 4.16e+1 2.60e+1 7.42e-1 cf6(av) 8.61e+2 8.53e+2 5.01e+2 9.00e+2 7.72e+2 4.91e+2 4.26e+2 cf6(std) 1.26e+2 1.28e+2 7.78e-1 8.32e-2 1.89e+2 3.95e+1 1.07e+1 as can be seen from the above data, the convergence of the improved algorithm is slightly weaker when dealing with zdt1. but the convergence of the improved algorithm is slightly stronger for dealing with ztd3, ztd4 and dtlz3. when dealing with bnh, tnk and dtlz8, the improved algorithm has a higher proportion of dominance. when dealing with cf4, cf5 and cf6 functions, the search results of the improved algorithm are best, this shows that the improved algorithm has a higher accuracy. this is due to the updating of particle location for boundary search and the strategy of selecting local particles, which enhances the ability of developing local particles. based on the above analysis, the new algorithm proposed in this paper has better applicability and accuracy for multi-connected and multi-extreme complex problems, and can be used as a good tool for solving multi-objective size optimization problems with constraints. 6. case analysis as a case study, a support frame of triangle track wheel is optimized by the new algorithm proposed in this paper to verify the applicability and accuracy. the structure is shown in fig. 11, which can be found in reference [32]. in the process of travelling, the triangle track wheel has various working conditions such as climbing, crossing obstacle, starting, braking, turning, crossing the soft road, snow, muddy land, marsh, sand and so on. the support frame has different load-bearing mode under different working-conditions. the three wheel supported ground is the worst working condition, which is considered in this paper. the loads borne by the support frame include: self-gravity 4.5 kn, hydraulic cylinder preload 36 kn, and chassis pressure 78.4 kn. the weight is 0.366 tons. the stiffness and strength of the support frame are more sensitive to the change of mass, and the restriction relationship is more significant. local extremum and non-convergence is easy to occur in the calculation process when the stress is taken as performance index, which is a an improved bare-bones particle swarm algorithm for multi-objective optimization... 17 nonlinear mop with constraints. in this section, multi-objective size optimization of support frame is carried out. guide wheels track drive wheel tensioning device tension wheels support frame load-bearing wheels (a) (b) fig. 11 the structure of support frame (a) triangle track wheel (b) support frame 6.1 mathematical model of multi-objective size optimization the parametric model of the support frame is established as shown in fig. 12. seven important dimensions (plate thickness of local area) are marked including x1~x7, which are design variables. x5 x4 x3 x2 x1 x7 x6 fig. 12 parametric model of the support frame according to the design and manufacturing process [33], each size parameter of the support frame is constrained, and the first-order natural frequency should exceed the excitation frequency transmitted by the engine. the constraints can be expressed as min max 0 , 1,2, 7 . . ( ) i i i i x x x i s t f x f       , (15) where ximin and ximax are the minimum and maximum value of xi. f(xi) is the first-order natural frequency of the support frame. f0 is the excitation frequency of engine, the value is 50 hz. for the support frame, the optimization objective is how to reduce the weight, displacement and stress. so, the objective function can be defined as shown in eq. (16): 18 z. wang, g. yang, y. sun, y. li, f. wu max maxmin [ ( ), ( ), ( )]i i ix d x v x , (16) where σmax(xi) and dmax(xi) are maximum stress and displacement of support frame. v(xi) is the weight. xi is the design parameter. so, the mathematical model of size optimization is established, as shown in eq. (17), which belongs to a typical nonlinear mop with constraints. max max min max 0 find ( 1,2,3,4,5,6,7) min ( ), ( ), ( ) ( ) i i i i i i i i x i x d x v x s.t. x x x f x f          . (17) in the optimization model, the stress, displacement, volume and natural frequency of the support frame lack a functional relationship with the design variables. therefore, the approximate model of structural performance and parameters is established in the next section. 6.2 setting an approximate model the response surface methodology is a method to establish an approximate model. it synthesizes experimental design and mathematical model, and obtains the relationship between design objectives and design variables through limited experimental design of the set of sample points in the designated design space. it is also possible to smooth the response function and reduce the "numerical noise", which is conducive to faster convergence to the global optimum in optimization process. in this paper, a second-order polynomial is used to construct the response surface methodology. the basis function is as follows: 0 11 1 y k k k i i ii i ij i ij ii i i j x x x x             , (18) where β is an unknown coefficient; k is the number of design variables, k=7; y is the predicted response value; β0, βi and βii are deviation term coefficient, linear deviation term coefficient and second-order deviation term coefficient, respectively; βij is the interaction coefficient. four approximate models of response are obtained through experimental design as follows: t 1 1 t 2 2 t 3 3 t 4 4 f = x c f = x c f = x c f = x c     , (19) where f1, f2, f3, f4 represent the approximate model of stress, natural frequency, displacement and volume, respectively. the parameters of approximate model are given in table 9. an improved bare-bones particle swarm algorithm for multi-objective optimization... 19 table 9 parameters of approximate model x c1 c2 c3 c4 constant 2.60e+2 3.36e+1 3.30e-1 2.80e+7 x1 9.84e-1 3.73e-1 -1.10e-2 3.81e+5 x2 -5.42e+0 2.44e-1 -5.00e-3 3.82e+5 x3 2.57e+0 1.19e+0 -4.00e-3 7.96e+4 x4 -1.04e+1 1.79e-1 3.00e-3 4.59e+4 x5 -7.52e+0 3.81e-1 1.00e-2 5.39e+4 x6 -2.11e+0 7.36e-1 -5.00e-3 6.87e+4 x7 -1.82e+0 2.19e-1 2.00e-3 5.16e+4 x1 2 -1.03e-1 -3.00e-3 0 4.29e+2 x2 2 1.10e-1 -2.00e-3 0 4.53e+2 x3 2 -7.80e-2 -2.10e-2 0 6.26e+2 x4 2 2.34e-1 2.00e-3 0 4.89e+2 x5 2 1.42e-1 -5.00e-3 0 5.82e+2 x6 2 4.70e-2 -1.30e-2 0 9.01e+2 x7 2 1.00e-2 1.00e-3 0 5.20e+2 x1·x2 8.40e-2 -6.00e-3 0 -8.96e+1 x1·x3 -3.90e-2 -1.00e-3 0 1.69e+1 x1·x4 2.80e-2 1.00e-3 0 -1.70e+2 x1·x5 2.10e-2 -1.00e-3 0 -1.41e+2 x1·x6 1.00e-3 -1.00e-3 0 -3.64e+2 x1·x7 -2.20e-2 0 0 -1.45e+2 x2·x3 1.40e-2 3.00e-3 0 9.87e+0 x2·x4 2.10e-2 1.00e-3 0 -1.74e+2 x2·x5 4.60e-2 0 0 -1.63e+2 x2·x6 7.00e-3 2.00e-3 0 -3.91e+2 x2·x7 -5.00e-3 -1.00e-3 0 -1.38e+2 x3·x4 -2.70e-2 -2.00e-3 0 -1.19e+1 x3·x5 4.50e-2 -3.00e-3 0 1.44e+0 x3·x6 -3.70e-2 0 0 1.01e+1 x3·x7 -1.70e-2 -1.00e-3 0 3.03e-1 x4·x5 4.40e-2 -1.00e-3 0 -2.86e+1 x4·x6 6.00e-3 -1.00e-3 0 -3.97e+1 x4·x7 5.00e-2 -2.00e-3 0 -8.88e+0 x5·x6 3.00e-3 0 0 -6.02e+0 x5·x7 2.00e-3 0 0 6.68e+0 x6·x7 4.80e-2 3.00e-3 0 1.47e+1 20 z. wang, g. yang, y. sun, y. li, f. wu after establishing the approximate model, it is necessary to evaluate its predictive power. commonly used evaluation index are mean error (ae), maximum error (me), root mean square error (rmse), and correlation coefficient (r2). ae is the average of all errors and can be expressed as ' 1 1 ( ) ae n i i f f n     , (20) where i is the i-th sample point; fi is the finite element analysis result of the i-th sample point; fi’ is the calculated value of the approximate model of the i-th sample point; n is the number of sample points. the parameters of the approximate model are given in table 4. me is the maximum value of all errors, that is max (fi -fi’). rmse also known as standard error and can be expressed as: ' 2 1 ( ) rmse 1 n i i i f f n p       , (21) where p is the number of terms of the polynomial. r2 is an index for evaluating the fitting accuracy of the approximate model to the experimental data. the closer r2 is to 1, the smaller the error. r2 can be described as: 2 ' 2 2 1 1 2 1 ( ) ( ) r ( ) n n i i i i i n i i f f f f f f            , (22) where f is the average of the finite element analysis of all sample points. table 10 shows the error analysis results of approximate model for different performance functions. it shows that the ae of the four responses is less than 0.1, me is less than 0.3, rsme is less than 0.2, r2 is greater than 0.9, which shows that the approximate model has high prediction accuracy. therefore, the approximate model can be used for subsequent optimization design. it is not difficult to see that the accuracy of stress prediction is little poor, because stress is easy to produce numerical errors and has high non-linearity. while the volume prediction accuracy is very high, because volume function of structure is an explicit function of design variables in size optimization. table 10 error analysis variables mass stress displacement mode ae 1.76e-3 5.04e-2 2.04e-2 2.85e-2 me 5.12e-3 2.05e-1 6.58e-2 6.52e-2 rmse 2.17e-3 6.42e-2 2.70e-2 3.39e-2 r2 9.99e-1 9.23e-1 9.86e-1 9.83e-1 an improved bare-bones particle swarm algorithm for multi-objective optimization... 21 6.3 multi-objective size optimization in this section, the mathematical model in eq. (17) are optimized by using the bb-mopso and the improved bb-mopso. the optimization results of design variables are shown in table 11. table 11 the results of size optimization method x1 x2 x3 x4 x5 x6 x7 bb-mopso 20 13.96 10 10 14.68 10 19.69 improved bb-mopso 20 10 11.15 16.49 10 10 20 the model is modified according to the optimization results in table 11. a new model is established and the finite element analysis of the worst working conditions is carried out. the results using the bb-mopso and the improved bb-mopso are listed in table 12 to compare the applicability and accuracy for multi-connected and multi-extreme complex problem. table 12 the comparison of mechanical performance method mass (kg) displacement (mm) stress (mpa) not optimized 376.65 0.217 132.90 bb-mopso 337.28 0.196 73.111 improved bb-mopso 335.70 0.217 68.167 from table 12, the mechanical performance of the support frame is improved by optimization using bb-mopso and improved bb-mopso. however, the optimization model using bb-mopso obtains better results in displacement, but it is worse in mass and stress than the improved bb-mopso. generally, displacement reflects the overall stiffness of the structure, while stress reflects the local strength. the results of table 12 show that the improved bb-mopso has better boundary and cross-region search ability. so, the improved bb-mopso is more suitable for structures with requirements of light weight and high strength. 7. conclusion in this paper, an improved bb-mopso is proposed to solve the multi-objective size optimization problems with non-linearity and constraints in structural design and optimization. the ability of particle searching for boundary and cross-region is enhanced by modifying the updating form of particle location. the optimal solution distribution in the global scope is maintained by combining the spatial grid density with the ranking of crowding distance. the global best positions are determined by increasing the probability allocation strategy changing with time, and the exploration of unknown region and the 22 z. wang, g. yang, y. sun, y. li, f. wu convergence of the algorithm are increased. the performance of the improved algorithm is verified by the test functions. as a case study, a support frame of triangle track wheel is optimized by the bb-mopso and improved bb-mopso. the results show that the improved algorithm can enhance the search ability of boundary and cross-region and diversity, which can effectively solve the multi-objective size optimization problem with non-linearity and constraints. acknowledgement: this research is supported by scientific and technological innovation programs of higher education institutions in shanxi (2021l291), fundamental research program of shanxi province (202103021223290), taiyuan university of science and technology scientific research initial funding (20212040) and national natural science foundation of china (52205278). references 1. wu, f.h., wang, z.h., song, 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creative commons license: cc by-nc-nd original scientific paper towards patient specific plate implants for the human long bones: a distal humerus example udc 617+621.7 mohammed rashid 1 , karim husain 2 , nikola vitković 3 , miodrag manić 3 , slađana petrović 4 1 university of al muthana, iraq 2 university of qadisiya, diwaniya, iraq 3 faculty of mechanical engineering, university of niš, serbia 4 faculty of medicine, university of niš, serbia abstract. plate implants are the most used internal fixators for the surgical treatments of the bone fractures. in clinical cases where there is a requirement to use reconstruction plates, and/or to stabilize the fracture, adaptation of plate shape (e.g. bending) to the patient anatomy is required, and it is usually done during the surgery. in order to eliminate the need for intra-operative bending of plates, precontoured plates can be used. these are patient specific implants whose shape and geometry is adapted to the anatomy and morphology of the specific patient. in order to create a patient specific 3d model of the plate implant, the bone model acquired through medical imaging (e.g. computed tomography ct) is commonly used. by the application of various cad techniques, the volume model of specific plate implant can be created, and used for the production of the plate, by conventional or additive manufacturing technologies. in this paper the authors present a new approach to the creation of a 3d parametric model of the patient specific plate implant for distal humerus. by using such model the surgeon can perform preoperative planning and adapt shape of plate to the specific patient before the surgery, and in this way he can improve pre, intra and post-operative processes. key words: cad, orthopedic, fixator, plate, parametric models, method of anatomical features received june 12, 2017 / accepted may 20, 2018 corresponding author: nikola vitković faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, niš, serbia e-mail: vitko@masfak.ni.ac.rs 348 m. rashid, k. husain, n. vitković, m. manić, s. petrović 1. introduction in the field of orthopedic surgery there is a requirement to provide the best possible medical treatment for the patient with a bone fracture. for the treatment of bone fractures the surgeons apply techniques of internal and external fixation. external fixation is a surgical technique used for stabilization of bone fragments with the fixator positioned outside of the human body (only pins and screws are implanted inside the body) [1]. the alignment of the external fixator can be adjusted externally to provide an optimal position of the bone and bone fragments during the recovery process. internal fixation presumes the use of osteofixation material (screws, pins, plate implants) inside the human body in order to stabilize the bone fracture [2-5]. both internal and external fixation can be used for the healing of the bone fracture, but internal fixation is preferable because it provides better functional recovery of the bone [2]. plate implants are the most used internal fixators for surgical treatments of the bone fractures. they are made in various sizes and shapes in order to be used for different patients [4]. the application of such implants for the treatment of the unique patient bone may initiate a problem because of differences in size and shape of the bone and the plate implant. in such cases it is hard to find proper position of the plate; the patient's treatment may be hampered due to an inadequate transfer of load during the bone healing process, etc. this problem can be reduced by the application of so called patient specific plate implants (pspis). the geometry and shape of pspis are adapted to the anatomy and morphology of the bone belonging to the specific patient [5-8]. application of pspis has a positive effect on patients, but, on the other hand, it requires more time for preoperative planning and their manufacturing. therefore, pspis are used in the cases where the application of predefined implants can lead to both intra-operative and post-operative complications. distal humerus fractures are common fractures of the human arm (elbow). it is of great importance to properly stabilize the elbow while the patient is in the recovery process [9, 10]. for this purpose precontured plates are used. if the quality of the bone is poor (osteoporotic bone), then angular stable plates are used [10]. in the cases when standard plates are used for fixation of the distal humerus fractures, the plate must be adapted to the shape of the patient bone (bending of the plate during the surgery) [10, 11]. in order to improve the pre-, intra-, and post-operative procedures in the treatment of the distal humerus fractures the authors propose application of the pspi created by the new technique presented in this paper. this technique enables the creation of the geometrical model of the pspi whose contact surface with the bone is adapted to the bone’s geometry and morphology. for this purpose, a parametric model of the pspi is created by using one clinical case as an example; hence it should be considered as a prototype model, which will be tested on more samples, and possibly improved in future research. the parametric model is a model whose geometry can be modified by changing the value of parameters (specific dimensions) while its topology remains unchanged. this model is created by the application of the method of anatomical features (maf) which enables the creation of fully geometrically defined anatomical surfaces of the human bones [6-7]. this model can be used as the basis model for the production of the pspi by applying additive or conventional manufacturing technologies. it should be noted that the main intention of this research is not to create a plate parametric model which can be fully applicable to all human bones or part of the bones (or all bones of the towards patient specific plate implants for the human long bones: a distal humerus example 349 same type). that is practically impossible but if the created pspi model can reduce the surgeon’s effort to customize the plate during surgery and to shorten the time of the intervention, that would bring a great benefit to the clinical practice, and, of course, to the patient. this paper is structured as follows. in the second section of the paper, a survey of the internal fixation and fixation components (osteofixation material) is presented. the technique for the creation of the pspis geometrical model for the fixation of distal humerus fractures is presented in the next section of the paper. finally, conclusion is presented together with the prospects for future work. 2. basic principles of internal fixation internal fixation must follow three main principles: it must enable the movement of muscles and joints in the area of fracture; it must provide complete restoration of the bone; and it must enable a direct union of the bone fragments without visible deformation in other areas of tissue (like forming the visible callus) [12]. the main tasks for the internal fixation are to enable stability to the bone and surrounding tissue, to maintain blood supply to the bone, and finally to prevent possible fracture diseases like infection in the area of trauma [12]. in the process of internal fixation the surgeon can invoke two patterns of stability, which will influence the type of bone healing that will occur: absolute stability (results in direct bone healing), and relative stability (results in a secondary or indirect bone union). absolute stability means that there is no movement between bone fragments, and relative stability means that the bone fragments can create motion during their union with the main bone or with each other [13]. in order to enable proper healing of the bone, the surgeons use various mechanical components which provide mechanical and functional stability to the bone during recovery process. the main components which are used for internal fixations are: wires, pins, screws, which are defined in [12-20] and plates [2031]. 2.1 plates in today’s medicine various types of implants are used for the fixation of human bones fractures [21]. the most common ones are plates and their variations. the metal plates have been in use for more than one hundred years. among the first plates were compression plates which use various designs and external devices to enable compression of bone fragments [21]. compression plates with oval holes introduced the dynamic compression plates (dcp) [22]. oval holes were used in order to provide interfragmentary compression during screw tightening. the advantages of the dcp included a low incidence of malunion, stable internal fixation, and no need for external immobilization, thus allowing immediate movement of neighboring joints [21, 22]. to provide adequate stability and to enable functional requirements of the bone, dcps have to be mounted onto the periosteum (the tissue that lines the outer surface of all bones) and should be pressed onto the bone to achieve stability [21-24]. this requirement raises one important issue and that is cortical bone porosis at the site of placement, due to the prohibited blood supply. however, certain doubts about this problem have been reported [24, 25] relating to the usage of plates with a reduced contact area. refracture after plate removal was another problem with dcps. to 350 m. rashid, k. husain, n. vitković, m. manić, s. petrović prevent refracture it was recommended that the plate should not be removed for at least 15– 18 months [21] in order to eliminate a fracture gap between bone fragments. different studies analyzed the reasons for refracture and the conclusion was that refracture was an effect of cortical necrosis [25, 26]. the new plate design was developed in order to reduce the plate’s interference with cortical perfusion and thus decrease cortical necrosis. the design was called the limited contact-dynamic compression plate (lc-dcp) [21]. lc-dcps make less surface to surface contact with the periosteum of the bone in comparison with dcps (about 50%). in this way the necrosis of cortical bone and the osteoporosis under the bone were reduced. also, lc-dcp is constructed with a plate-hole symmetry, which enables dynamic compression from either side of the hole with different intensity [21]. it should be noted that some studies [26] were conducted which shows that lc-dcp does not improve blood flow to the bone, or biomechanical properties of the bone-implant assembly. today, nearly all of the mentioned plate implants are substituted with the plates which are capable for both locking and nonlocking functions, such are locking compression plates (lcp). nevertheless, locked plating cannot completely replace conventional plating [23]. a combination of both plating techniques is possible and should be performed when it is possible [21, 23-25]. lcps provide better fixation and they can withstand more load compared to standard plates (dcp) [27]. in addition to the type of fixation, quality of reduction, soft-tissue handling and the characteristics of the injury, the patient’s general health status also has significant influence on the treatment results. dcp and lcp fixation methods are based on anatomically precontoured plates, reducing or eliminating intraoperative (in-situ) plate modification (usually bending). lcp does not require precise contouring because that is not required when locking screws are used. in such cases plate acts more like a fixator rod. however, greater distance between the plate and the bone can cause a problem [27-29]. it is important to mention that reconstruction plates which are designed with deep notches between the holes can be contoured (bend) in three planes to fit complex surfaces. reconstruction plates are provided in straight and slightly thicker and stiffer precurved lengths. they have oval screw holes, like mentioned compression plates, and they allow potential limited compression [30]. the new objective in plates design and production is to achieve maximum stabilization with minimum damage to the blood supply during fracture healing. also, there is a need for extremely rigid fixation during the healing of fractures, and less rigid fixation during later bone remodeling. in order to demonstrate the complexity of the elbow fractures, in fig. 1 radiographs of 31 year old man with plate fixation are presented [31]. to meet the previously stated requirements, and to properly define plate shape and geometry [32], it is crucial to achieve maximal geometrical accuracy of the bone model. construction of accurate 3d models of human bones is described in study [33], in which different methods for the creation of bone models are described. this is of great importance because with such bone models it is possible to achieve proper stabilization of bone and plate, enable adequate blood supply to the bone, perform pre-bending of plate, etc. towards patient specific plate implants for the human long bones: a distal humerus example 351 fig. 1 radiographs of a 31-year-old male with a right distal humeral fracture (ao type c2) who was treated by perpendicular plating. a: pre-operative anteroposterior radiograph. radiographs showing excellent plate location and fracture union 6 months after the primary procedure (b and c). b: anterior-posterior radiograph. c: lateral radiograph [31]. 3. method of anatomical features (maf) for the cad model creation of the distal part of the human humerus maf was applied. maf is a method which has already been used for the creation of geometrical models of various bones like humerus, femur and tibia. bone models created with the application of maf have good geometrical precision and anatomical correctness. maf presents a new approach to the definition of basic human bone’s geometry based on the anatomical landmarks. maf enables the creation of geometrical models of human bones, by using referential geometrical entities (rges) defined for the each individual bone. rges represent the basis for the construction of bone geometry, and they are defined as the geometrical entities (points, lines, planes, axes, etc.) created in relation to the anatomical landmarks and entities. for the humerus bone, rges are presented in fig. 2 and described in [34]. by using rges, additional 3d models of human bone can be created, like surface, volume and parametric model. in this study 3d surface model of the humerus was created and used for the definition of pspi. the original humerus surface model was already created in previous study [34] conducted by the authors of this one, and geometrical accuracy for the purpose of presentation model was satisfactory. in order to achieve the best possible accuracy of the pspi model, geometrical accuracy of the humerus surface model was improved in this research. this was achieved by using additional curves with more interpolation points (points on the input digitized model acquired from ct). in fig. 3, humerus surface model together with original model acquired from ct scan (toshiba acqulion 64 scanner, slice thickness: 0.5mm, resolution: 512x512px) and previously created surface model are presented. models are created in dassault systems catia v5 r21 software. as it can be seen from fig. 3, a new surface model closely follows the original input model from ct. 352 m. rashid, k. husain, n. vitković, m. manić, s. petrović fig. 2 rges defined on humerus polygonal model [34] fig. 3 surface models and spline curves of the distal humerus (yellow – improved model; purple – previously created model; brown – input model) deviation analysis conducted in catia software, and presented in figs. 4-6, shows that deviation range for most of the surface points of the newly created surface model is around 1mm (fig. 6). it should be noted that, for the analysis, only the points which lie on the periosteum surface of the bone are taken into account because the input polygonal model has lot of points belonging to the inner structure of the bone (e.g. points with deviation above 3.14 mm for a newly created surface model of the distal humerus). fig. 4 deviation analysis between created surfaces towards patient specific plate implants for the human long bones: a distal humerus example 353 fig. 5 deviation analysis between input polygonal model and previously created surface fig. 6 deviation analysis between input polygonal model and newly created surface deviation analysis between newly created surface model and previously created surface model of the distal humerus shows that in 80.34%, distance between points is below 1 mm. in regions of bone with greater curvature, maximum deviation is 6.29 mm, which confirms that additional curves are necessary for the creation of a geometrically precise model of that area. 4. creation of geometrical model of bone-plate contact surface as [10] stated, reconstruction plates are used for the fixation of the distal humerus on the lateral and medial side. on the lateral side, the plate can be placed distally onto the posterior aspect of the capitellum. on the medial side, the plate is usually bent around the epicondyle. the focus of this research was to develop and propose a new technique for the creation of one specific type of medial reconstruction plate model. the proposed technique is developed in order to improve process of plate adaptation to the bone, which is essential for the healing process of bone fracture [10]. maf is used for the construction of plate geometrical model in catia. curves which are used for the creation of a surface model of the distal humerus are used for the construction of the parametric model of the reconstruction plate, as presented in fig. 7. four radiuses are defined and a medial curve was created as an auxiliary curve for surface orientation. radiuses are defined on the spline curves which are used for the creation of the surface model of the distal humerus. each radius defines one arc of adequate length. arc length is a changeable parameter and it defines width of plate (it can be constant). each arc length is defined by four corresponding arc angles. one more parameter is defined, and that is the angle of bending in the lower part of the medial plate. bending angle regulates the distance between plate bottom surface and medial epicondyle surface of the bone. defined radiuses (r1…r4), angles (α1…α4), medial curve and bending angle (bending angle) are presented in fig. 7. 354 m. rashid, k. husain, n. vitković, m. manić, s. petrović fig. 7 defined parameters and surface model of the bone-plate contact surface the values of parameters for this specific patient are presented in table 1. these values of parameters are used for the creation of the surface model of the plate contact surface. that surface is at right distance from the bone surface and the intersection with the surface of the bone is minimal and only at the end of the bended part. table 1 values of parameters measured for the specific patient r1 [mm] r2 [mm] r3 [mm] r4 [mm] bending angle [°] 5.3 3.7 6.1 5.7 132.2° α1 [°] α2 [°] α3 [°] α4 [°] deviation analysis conducted in catia shape module, between surface model of the distal humerus and plate contact surface is presented in fig. 8. it can be concluded that maximum deviation is 0.707 mm, in outer region of the plate surface – closer to edges. deviation range is from 0.177 to 0.707 which is pretty accurate concerning the requirement that plate contact surface should correspond to bone outer surface as maximum as possible [10]. the analysis confirms that nine parameters are enough for the definition of fixator surface shape with respect to the defined requirement. it should be mentioned that during the real surgical intervention, the surgeon can manipulate with the plate, if there is a need for it. the surgeon can rotate, move and perform additional bending (amount of applied bending would be much smaller) in order to adapt the plate to the bone. towards patient specific plate implants for the human long bones: a distal humerus example 355 fig. 8 deviation analysis between plate contact surface and bone outer surface the solid model of the reconstruction plate is created by the application of the thick surface feature in catia (thickness defined as 3mm), and it is presented in fig. 9. fig. 9 example of plate implant solid model 356 m. rashid, k. husain, n. vitković, m. manić, s. petrović 5. conclusion the plate implants are necessary orthopaedic equipment, and their design and ways of production should be constantly improved. this paper presents a new approach for the creation of the patient specific plate implant for distal humerus, which is based on the application of the maf method. more precisely, it represents extensions of the aforementioned method by introducing and defining the corresponding parameters for the purpose of creating a parametric model of the plate. pre-contouring of the plate is achieved by inserting and changing the value of the existing parameters, according to the dimensions values acquired from the 3d humerus model, while topology remains unchanged. the possibility of plate adaptation before surgery, by using presented approach, improves preoperative processes, shortens the time of intervention as well as enables stability of the fracture and satisfies functional properties of the bone and joints. deviation analysis between plate contact surface and bone outer surface in presented clinical case shows that plate shape is adapted to the patient specific bone in accordance with standard recommendations in clinical practise [10, 21-27]. future work will include more bone samples in order to confirm the number and type of parameters which influence the proper construction of the contact surface between the bone and plate. plates defined in this way can be manufactured by means of additive 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http://www.sciencedirect.com/science/article/pii/s0268003306001318 http://www.ncbi.nlm.nih.gov/pubmed/?term=rose%20ps%5bauthor%5d&cauthor=true&cauthor_uid=17097312 http://www.ncbi.nlm.nih.gov/pubmed/?term=adams%20cr%5bauthor%5d&cauthor=true&cauthor_uid=17097312 http://www.ncbi.nlm.nih.gov/pubmed/?term=torchia%20me%5bauthor%5d&cauthor=true&cauthor_uid=17097312 http://www.ncbi.nlm.nih.gov/pubmed/?term=jacofsky%20dj%5bauthor%5d&cauthor=true&cauthor_uid=17097312 http://www.ncbi.nlm.nih.gov/pubmed/?term=haidukewych%20gg%5bauthor%5d&cauthor=true&cauthor_uid=17097312 http://www.ncbi.nlm.nih.gov/pubmed/?term=steinmann%20sp%5bauthor%5d&cauthor=true&cauthor_uid=17097312 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 1, 2014, pp. 85 94 dynamic behavior of hot water boilers during start up  udc 621.181.1 milena todorović, dragoljub živković, marko mančić, dragan milčić faculty of mechanical engineering, university of niš, serbia abstract. the most commonly used central units in the district heating system are most certainly hot water boilers. experience in exploitation as well as their former development have given rise to new demands and possibilities of using hot water boilers for achieving a more efficient utilization of fuel. yet, the current trend is not only for an efficient operation in terms of energy efficiency of primary fuel, but it also comprises many strict requirements when it comes to a cost-effective, reliable and safe operation. a mathematical empirical model of the hot water boiler during the start-up process is presented in this paper. the dynamic behavior of the object will be discussed in the start-up regime because it represents one of the most critical transient regimes during the operation. key words: dynamic behavior, hot water boiler, start-up, transitional operating modes 1. introduction during the hot water boilers' operation, their work is subjected to high temperature and pressure conditions. in addition to their energy efficiency, it is also necessary to take into consideration the system safety and reliability in order to avoid major operation disturbances and downtimes [1]. the basic elements of the hot water boilers are exposed to the effects of high pressure and temperature of the working fluid during the boilers' operation. in transient processes, such as start-up, stopping and load changing, sometimes thermal stresses occur. usually they are caused by thermal expansion of individual elements or assemblies [2, 3]. one of the most important parameters in defining reliability, availability and efficiency of the hot water boiler plant certainly refers to the failures that occur in the boiler piping system [4, 5]. this is confirmed by the results of a research project conducted by the north american electric reliability council, which have shown that with the boiler piping system failures, an average decrease of the plant availability is over 6% [6]. the boiler piping system failures represent the primary cause of delays of thermal recived january 13, 2014 / accepted march 10, 2014  corresponding author: milena todorović university of niš, faculty of mechanical engineering, thermal engineering department, niš, serbia e-mail: milenatod1@yahoo.com original scientific paper 86 m. todorović, d. ţivković, m. manĉić, d. milĉić plants. more than 80% of such failures result in an unplanned downtime, where the average downtime lasts for approximately three days thus leading to high costs [6]. shutting down the hot water boilers and their re-starting and re-integrating into work represent integral parts of every conventional hot water boiler plant as well as thermal one. periodic changes in energy demand also require shutting down of the boilers for inspection and their starting up again. during the process of the boiler load change, as well as in the situations described above, there are significant thermal loads in the boiler elements especially in those with thick walls, which define the permissible intensity of thermal stresses [7]. high thermal stresses, which occur in the thick-walled pressure elements of the boiler during the transient operations, limit both the heating and the cooling rates of temperature changes in the boiler elements [8]. it is considered that one of the most critical modes, where the greatest thermal stress in the boiler's elements occurs, is certainly the process of the boiler plant's start-up. thermal stresses are the results of temperature caused by the differences in the boiler structure. this is due to high time gradients of temperature of combustion products during the process of the boiler start-up. thermal stresses are particularly high in certain parts of the boiler's thick walls, where the greatest temperature difference appears. as far as the hot water boilers are concerned, these thermal stresses are located in the pipe carrying wall of the first deflecting chamber. there is evidence that numerous accidents have occurred on this carrying wall [9]. the precise modeling of the boiler elements' dynamic behavior is also essential for improving the quality of boiler operation monitoring and control. designing a mathematical empirical model describing boiler dynamics is also very important regarding a reliable and safe boiler operation in transient regimes [10]. this paper presents the methods for determining basic operating parameters of the process of boiler start-up, which can be used to analyze the process of starting up not only steam boilers but hot water boilers as well. 2. the process of starting up the boiler when the steam boiler is started, the combustion chamber is partially filled with flame. flame of some burners depends, in size and shape, on the structure and strength of the burner. the burner ignition and fuel used in these cases (oil, gas) should ensure a normal operation of each burner independently of the total load of the combustion chamber. insufficient flame fulfillment of the combustion zone disables all fuel particles that are under the suitable condition for ignition and complete combustion. due to low temperatures in the combustion chamber during the start-up process, insufficient fuel mixing with air forms a large amount of soot; the most massive drops of fuel oil, which do not come into the combustion zone, are cooled in a stream of cold air and flue gases which go out from the combustion chamber unburned. this leads to a decreased efficiency of the combustion chamber and unfavorable working conditions. the terms of ignition and combustion of fuel are also getting worse because of the wall temperature of the combustion chamber [9, 11]. in the start-up process of the hot water boiler, the problem is much simpler. there is a great similarity with the problem that is presented by cwynar, but the hot water boilers have only one burner and regarding the combustion chamber, its role has the flame pipe. problems that may occur in this case are caused by the possibility of tearing the flame and damaging the boiler structure at the end of the flame tube. dynamic behaviour of hot water boilers during start up 87 on the basis of the above mentioned facts it can be concluded that the conditions for the boilers' start-up process taken here into consideration depend on many parameters, which are determined by the burners' structures, the combustion chamber and its technical condition, as well as on the organizational chart of the start-up procedure and of the boiler's thermal state. in the process of the boiler's start-up, it is necessary to determine the changes of the values of parameters related to the operation of the combustion chamber: the amount of heat that is released with combustion and the temperature and the flow rate of those combustion products that are leaving the combustion chamber. for their determination the variables such as combustion chamber efficiency  r c , excess air ratio r and temperature of combustion products  r should be known. 3. combustion chamber efficiency during the boiler's start-up the combustion chamber efficiency during the process of the boiler's start-up is changed as the load changes, depending on the coefficient of excess air and changed conditions of fuel ignition and combustion. load change of the combustion chamber in the process of the boiler's start-up is determined by variable uq. value uq represents the ratio of the current boiler load and the nominal load. r q z q u q  (1) where superscripts r and z refer to the regime in the boiler's start-up and to the nominal one, respectively. therefore, the combustion chamber efficiency during the process of start up ( r c ) in a wide range of the load change, can be described by [11]: 1 /r z c qs u   (2) where s z = 1  z c represents the losses in the combustion chamber during the nominal load. graphical representation of the combustion chamber efficiency change, eq. (2), for some values of s z can be seen in fig. 1. fig. 1 combustion chamber efficiency depending on a wide range of boiler load [11] 88 m. todorović, d. ţivković, m. manĉić, d. milĉić 4. coefficient of excess air ratio during the boiler's start up coefficient of excess air during the boiler's start-up r is defined by the following assumptions [11]:  starting up the boiler is carried out at constant sub-pressure in the combustion chamber, which is equal to the vacuum, which is maintained in the stationary mode, and,  suction of the "waste" air from the atmosphere, which depends on non-hermetic boiler's properties, is considered as a constant value that is determined by the constructive scheme of the fuel supply and technical condition of the boiler. according to these assumptions, mass flow of intake air lf, entering in the combustion chamber from environment during the boiler's start-up is approximately equal to air suction in the combustion chamber at nominal load, z r f f fl l l  . for nominal load it stands: z f zz   1 (3) where  z 1 is the coefficient of excess air in nominal mode and  z f = n1 z 1 , where n1 is the coefficient that characterizes non-hermetic properties of the combustion chamber's lower level (known from the calculations or measurements). the mass balance in the burner zone during the process of the boiler's start-up is given as: 1 1 1 r r r z z z r r r t t p tl b n l b l b    (4) where lt represents a theoretical amount of the air required for complete combustion. from eq. (4) it can be obtained: 1 1 1 ( / )r r z z z r r p t tn l b l b    (5) where b refers to fuel consumption during the process. taking into account the above defined conditions it may be accepted: / /z r z r t t w wl l q q (6) /r r z z q w wu b q b q (7) quantity qw represents the amount of heat that is released by fuel combustion. rearranging eq. (5) with the values from eqs. (6) and (7), the following equation for the coefficient of excess air in the burner zone during the boiler's start-up ( )r p can be obtained: 1 1 1( / )r r z p qn u    (8) or, when 1 1 1 z r    : 1 1(1 / )r r p qn u   (9) graphic dependence of the coefficient of excess air during the process the boiler's start-up r p as a function of qu for different values 11  z and 1n is presented in fig. 2. dynamic behaviour of hot water boilers during start up 89 fig. 2 characteristics of excess air ratio depending on a wide range of boiler load [11] 5. heat received by the combustion chamber walls during the boiler's start-up while changing boiler load, the heat amount that is received by the screens of the combustion chamber is changed also. it can be seen in fig. 3a, where the amount of heat received by the screens of combustion chamber at the stationary mode is q u 0 , compared to the total amount of heat delivered by combustion q u c at the nominal mode, is represented in terms of uq: 0( ) u u q cu q q  (10) with reduction of heat load uq, one part of the heat, which is received by the screens of the combustion chamber, increases. the character of these curves for different boilers is the same. along with reduction uq what is also increased is the heat received by the surface area in the following elements of the boiler, economizers of steam boilers or gas pipes ii and iii pass of hot water boilers (fig. 3b) [11]. a) b) fig. 3 dependence of the relative heating of the combustion chamber walls (a) and flue pipes, or economizer (b) of the boiler load with coal powder combustion op-230, op-650k, bp-1150 and bb-1150 [11] 90 m. todorović, d. ţivković, m. manĉić, d. milĉić 6. cwynar's method for determining the heat during combustion process and temperatures of combustion products in the combustion chamber during the process of starting up the boiler the method is based on the assumption that, on the basis of measurements or calculation, the results are known for the basic parameters, which describe the process of heat transfer in the combustion chamber, for all stationary states of the boiler operation [11, 12]. knowing the function describing the change of specific parameters for different boiler load (i.e., from 60% and more, fig. 4) can be analytically extrapolated to the lowload area. the failure occurring during this procedure depends on the range and accuracy of the known values of the parameters for medium and nominal load. fig. 4 dependence of theoretical combustion temperature (t); temperature of combustion products that leave the combustion chamber (2); and heat that is received by the screens of the combustion chamber (q0) of boiler load with coal powder combustion op-230, op-650k, bp-1150 and bb-1150 [11]. the function describing the change of heat, which is received by screens of combustion chamber during the process of the boiler's start-up can be determined from the equation: 4 4 0 0 [( /100) ( /100) ]r r r r r f g nq n a t t  (11) 2 2 2; 273; 273; 273r r r r r r r g t t t n nt t t t t t t        (12) where  is the parameter, which characterizes the position of burners compared to basic position and af the coefficient of flame emission. in accordance with the assumptions the following parameters are known: dynamic behaviour of hot water boilers during start up 91 0 2, , , ,z z z z z t fq a   (13) or dependence 0 2, , ,u u u u t fq a  of changing these parameters in certain range of loads. coefficient n0, which characterizes the heat transfer in the combustion chamber can be determined by replacing in eq. (11) the values from eq. (13) or the known dependencies 0 2, , ,u u u u t fq a  : 2 2 4 0 0 2( / ){[( 273) /100] [( 273) /100] [( 273) /100] }z z z z z z f t nn q a t      (14) or 2 2 4 0 0 2( / ){[( 273) /100] [( 273) /100] [( 273) /100] }u u u u u u f t nn q a t      (15) knowing dependence n0(uq), eq.(11) for the heat flow that is received by irradiated surfaces of the combustion chamber can be written in the following form [11]: 2 2 4 0 0 2{[( 273) /100] [( 273) /100] [( 273) /100] }r r r r r f t nq n a t       (16) where n0 = n 0 u (uq). function q 0 r (uq) (fig. 4) is determined more precisely in a higher range of loads, for which are known function q0 u (uq),  0 r (uq) and so on, which allows determination of n 0 r (uq). the minimum accuracy will be in the case when only data for nominal load are available, i.e. zn 0 can only be determined. theoretical combustion temperature in the process of starting up the boiler, r t , can be determined from the known dependence [11]: ( ) /r r r r r r r t w c p t l l gp gpq l c t v c    (17) where lc is the average specific heat capacity of air [kj/m 3 k], gpc the average specific heat capacity of combustion products during process of starting up the boiler [kj/m 3 k] and gpv the combustion products volume flow rate during the boiler's start-up [m 3 /s]. temperature of the combustion products that leaves combustion chamber in the process of starting up the boiler, r 2 , can be determined from the heat balance equation: 0 0r r r c gq q q   (18) whereby 2 1( )r z z z z c q q w t lq u q u b q l i   is the heat generated in combustion chamber, rq0 the amount of heat determined with eq. (16) and 2( 273)r r r r r g g gq v c t b  the heat quantity of flue gases that leave the combustion chamber. by substituting the precious values in eq. (18), it can be obtained: 2 2 2( ) 0r rt t     (19) where , ,  are coefficients whose values are: 4 2 010 [( 273) /100]r r r f tn a    r r r g gv c b  (20) 92 m. todorović, d. ţivković, m. manĉić, d. milĉić 4 0 [( 273) /100] 273z r r r r r q f n g gu q n a t v c b     substituting expression 27322  rrt  into eq. (20), the temperature of combustion products, which leave the combustion chamber, can be determined: 2 2 ( 4 ) / 2 273r         (21) 7. gurvich's method for determining the heat during combustion process and temperatures of combustion products in the combustion chamber during the process of starting up the boiler according to the carefully thought-out empirical method of a. m. gurvich for determining heat exchange in the combustion chamber, the temperature of the combustion products that leave the combustion chamber is given with the following equation [13]: 0,6 0,6 0,6 2 273 ( 273)( ) /[ ( ) ( ) ]r r r r r t fbo m a bo     (22) where bo represents boltzmann constant: 8 3(10 / 4.9){ /[ ( 273) ]}r r r r r g g opr tbo b v c h   (23) where  represents the coefficient of cleanliness of heating surfaces. m a bx  (24) in the last equation, it is a = 0.59; b = 0.5 for coal combustion and a = 0.52; b = 0.3 for fuel oil or gas combustion; x = x0 + x, where x0 = h1/h2, h1 is the height of the combustion chamber from the floor up to level of the maximum flame temperature (when the boiler starts to the level of burners position), h2 is the full height of the combustion chamber; x the correctional member, which takes into account the distribution of burners position, their tilting angles, the quality of minced coal, etc. (detailed data for x and the coefficient of flame emission af can be found in [14]). the limited condition for applying the gurvich equation are bo < 10af and 2( 273) /( 273) 0,9r r t    , and it emphasizes the lower boiler load during the process of start-up. knowing the temperature of leaving flue gases r 2 from eq. (22), their flow and heat quantity brought into the combustion chamber z cq r c quq  , the heat quantity rq0 , received by the screens of the combustion chamber (according to eq. (18)) can be determined. if values u g u t u g uu cvbbo ,,,,  are known (fig. 4) for variable boiler loads, the boltzmann constant in the condition for the boiler's start-up can be determined: 3 273 273 r r r u r u g g t u u u r g g t b v c bo bo b v c          (25) dynamic behaviour of hot water boilers during start up 93 8. conclusion the most sensitive elements of the hot water boilers are certainly their pipe systems. they mainly affect the reduction in availability, reliability and time utilization, and even in energy efficiency. stresses in the hot water boiler elements, subjected to a high pressure of the working fluid, are substantially different from those occurring in other machines and objects. the difference is primarily that the stresses in the boiler elements are not only the result of the external forces but internal as well, which depend on the number of structural details, as well as technological and exploitation factors. in addition, many elements of the boiler are exposed to very high temperatures and variable loads, which can cause, especially in the transitional operating mode, extremely high stresses. damages of pipes of heating surfaces of both hot water and steam boilers, especially those with greater capacity, are rare. they usually occur suddenly and can have devastating consequences. they can be prevented only by careful handling and maintenance of the plant and expert monitoring of the processes of the built-in screen pipes exploitation [15, 16]. in the previous studies of the boilers' availability and its models the main focus should be on the ignition and combustion conditions in the combustion chamber, as well as on the operation of the burners and flame pipe in the regimes other than nominal. however, the above described conditions, as well as the heat transfer at low loads (less than 40%) have not been sufficiently explored so far. this paper presents the methods of determining the basic parameters that characterize fuel combustion and heat transfer in the combustion chamber. in the previous studies there are also analyses of the start-up process of the boilers manufactured by "minel kotlogradnja", according to the presented procedure. this analysis has given satisfactory results, which are presented in [9]. for further analysis, it is necessary to measure some of these parameters during the start-up procedure. in this way, more current analyses will show and confirm the above presented procedure for determining significant parameters for the process of the boiler plant's start-up. references 1. katavić, b., jegdić, b., 2007, analysis of damages on water boiler shield pipes, welding and welded structure, 4 pp. 123-130. 2. brkić, lj., ţivanović, t., tucaković, d., 2010, steam boilers, university of belgrade, faculty of mechanical engineering, belgrade. 3. todorović, m., ţivković, d., manĉić, m., ilić, g., 2013, application of energy and exergy analysis to increase efficiency of a hot water gas fired boiler, chem. ind. chem. eng. q., doi:10.2298/ciceq130716033t. 4. djurić, s., milĉić, d., smiljković, m.., 2010, overhaul of screened system in thermal smoke-tube block boiler, imk14 research & development, 36 (3) pp. 97-100. 5. šijaĉki ţeravĉić, v., bakić, g., đukić, m., anċelić, b., 2007, analysis of test results of hot-water boiler as a basis for its integrity assessment, structural integrity and life, 7 (2), pp. 133-140 6. šekeljić, p., bakić, g., 2007, optimization of maintenance measures of piping system of the 60mw boiler in order to raise their availability, energy-economy-environment, 3-4 pp. 45-49. 7. taler, j., michalczyk, k., 2006, thermal and structural stress analysis of boiler drum and central pipe connection in transient conditions, agh university journals mechanics, 25 (1) pp. 41-46. 8. gradziel, s., 2011, determination of temperature and thermal stresses distribution in power boiler elements with use inverse heat conduction method, archives of thermodynamics, 32 (3) pp. 191-200. 9. todorović, m., ţivković, d., manĉić, m.., milĉić, d.., 2013, analysis of dynamic behaviour of hot water boiler during start up, proceedings of the 16th symposium on thermal science and engineering of serbia simterm 2013, sokobanja, srbija, pp. 86-94. 94 m. todorović, d. ţivković, m. manĉić, d. milĉić 10. zima, w., 2011, mathematical modelling of dynamics of boiler surface heated convectively, heat transfer engineering applications, intech, poland. 11. cwinar, l., 1981, poosk parovikh kotlov, moskva: enyergoizdat. 12. cwinar, l., 1971, metoda okreslania warunków pracy przegrzewaczy pary podczas rozruchu kotlów walczakowych, archiwum budowy maszyn, xviii, zeszyt 2, moskva: enyergoizdat. 13. cwinar, l., 1975, programmirovaniye pooska i ostanova kotlov (blokov), byollyetyenʲ postoyannoy komissii po elyektroenyergii 9, moskva: enyergoizdat. 14. orlowski, p., 1972, kotly parowe konstrukcjy i obliczenia, warszawa: wnt. 15. mitovski, m., mitovski, a., 2009, damage of pipes of utilized steam boiler furnace, technique mechanical engineering, 58 (4) pp. 9-15. 16. ţivanović, t., stupar, g., tucaković, d., 2013, causes of damage on hot water boiler mip 6300 gf power of 6,3 mw with repair proposition, proceedings of the 44th international congress & exhibition on heating, refrigeration and air conditioning, belgrade, 4-6. xii 2013, pp. 203-214. dinamičko ponašanje vrelovodnih kotlova za vreme puštanja kotla u rad najzastupljeniji centralni uređaji u sistemima daljinskog grejanja su svakako vrelovodni kotlovi. iskustveni podaci kao i dosadašnji razvoj vrelovodnih kotlova su ukazali na nove zahteve i mogućnosti korišćenja vrelovodnih kotlova kako bi imali efikasno iskorišćenje toplotne moći goriva. ali u današnje vreme se ne zahteva samo efikasan rad u smislu energetske efikasnosti iskorišćenja primarnog goriva, već se nameće niz strogih zahteva u pogledu rentabilnog, pouzdanog i sigurnog rada. matematičko empirijski model vrelovodnog kotla pri puštanju kotla u rad je prezentovan u ovom radu. dinamičko ponašanje objekta je razmatrano u režimu puštanja u rad vrelovodnog kotla, zato što predstavlja jedan od najkritičnijih prelaznih režima rada. kljuĉne reĉi: dinamičko ponašanje, vrelovodni kotlovi, puštanje u rad, prelazni režimi rada plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 101 120 https://doi.org/10.22190/fume210412055k © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator bartłomiej kozakiewicz, tomasz winiarski warsaw university of technology, institute of control and computation engineering, poland abstract. modern robot applications benefit from including variable stiffness actuators (vsa) in the kinematic chain. in this paper, we focus on vsa utilizing a magnetic spring made of two coaxial rings divided into alternately magnetized sections. the torque generated between the rings is opposite to the angular deflection from equilibrium and its value increases as the deflection grows – within a specific range of angles that we call a stable range. beyond the stable range, the spring exhibits negative stiffness what causes problems with prediction and control. in order to avoid it, it is convenient to operate within a narrower range of angles that we call a safe range. the magnetic springs proposed so far utilize few pairs of arc magnets, and their safe ranges are significantly smaller than the stable ones. in order to broaden the safe range, we propose a different design of the magnetic spring, which is composed of flat magnets, as well as a new arrangement of vsa (called attractor) utilizing the proposed spring. correctness and usability of the concept are verified in fem analyses and experiments performed on constructed vsa, which led to formulating models of the magnetic spring. the results show that choosing flat magnets over arc ones enables shaping spring characteristics in a way that broadens the safe range. an additional benefit is lowered cost, and the main disadvantage is a reduced maximal torque that the spring is capable of transmitting. the whole vsa can be perceived as promising construction for further development, miniaturization and possible application in modern robotic mechanisms. key words: variable stiffness actuator, elastic joint, magnetic spring, magnetic clutch, soft robotics 1. introduction construction of most of kinematic chains of robots, which are currently produced, is optimized to maximize their stiffness. low compliance of robot links and joints is especially vital in typical industrial applications like welding, gluing, dispensing, picking and placing received april 12, 2021 / accepted july 16, 2021 corresponding author: tomasz winiarski warsaw university of technology, institute of control and computation engineering, nowowiejska 15/19, 00-665 warsaw, poland e-mail: tomasz.winiarski@pw.edu.pl 102 b. kozakiewicz, t. winiarski because it plays a crucial role in providing high accuracy of the end effector positioning and avoiding undesired oscillations [1]. however, there is a significant downside to this approach – increasing rigidity of kinematic chain results in reducing its potential for safe energy absorption. consequently, due to safety standards, such robots should either have their performance limited or work in safety zones to minimize the risk of collisions. in a context of growing interest in utilizing robots that would be capable of direct cooperation with humans (also in industrial applications) [2], it seems reasonable to intentionally include compliance in the kinematic chain as a means which potentially reduces negative consequences of collision [3–6]. as stated before, the price for that is positioning accuracy. a practical compromise is to make compliance controllable by using variable stiffness actuators (vsa) and to implement a soft-arm tactics [7], which is to perform fast rough movements in safe compliant state and slow precise ones in the more accurate stiff configuration. including compliance in a mechanism – especially when it is variable and controllable – may have some other beneficial consequences [8]. first of all, the capability of energy absorption makes a new class of advanced movements possible, e.g. effective throwing and catching, walking and jumping [9]. some of them like ball kicking and running require adjusting stiffness between consecutive phases of movement [10–12]. moreover, achievable forces [4] and velocities [3, 13] are greater in the case of compliant mechanisms than in the stiff ones. last but not least, the resonance frequency of the elastic kinematic chain can be adjusted to its working cycle resulting in reduced energy consumption [14–16]. all these factors explain why human body actuators, which are antagonistic pairs of muscles, are also capable of in-fly stiffness adjustment [17]. most vsa designs can be assigned to one of three categories [18]: a. stiff constructions controlled in a way that enforces compliant behavior, b. compliant construction controlled in a way that enforces stiffness variability, c. compliant constructions which stiffness is adjusted mechanically. actuators from the first group are the most popular, especially in a field of service and assisting robots – where the safety issues are particularly important [19–21]. this is one of the reasons for the active development of suitable arm control algorithms, which include mechanical interactions with the environment – especially humans. some significant achievements are based on either impedance control [22–24] or force control [25–29]. it is worth noting that there are also some commercially available robots for professional industrial use equipped with vsas belonging to this category [30]. there are not so many constructions belonging to the second group (b) because they are mechanically more complicated than those from the first group, and provide not so many benefits as those from the third group. some notable examples are [31, 32]. it is only possible for the constructions belonging to category c to fully utilize compliant parts capability of energy storage, what makes this specific group peculiarly attractive in the context of the previously mentioned benefits. there are three main methods of achieving stiffness variability by mechanical means [18, 33]: c1. changing pretension of non-linear compliant components (often arranged in an antagonistic way [34]) – e.g. varying air pressure in a pair of pneumatic muscles, c2. changing transmission between the compliant component and output link – e.g. moving pivot point of a lever connecting spring with output arm, c3. changing parameters of compliant component – e.g. adjusting the active length of a leaf spring or its second moment of area. spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 103 in all these categories, the most frequently used compliant components are leaf springs, helix springs and elastic rods [18]. an interesting alternative is to use magnetic parts. the concept of replacing mechanical components with their magnetic equivalents is already present in different branches of engineering. some notable examples are magnetic bearings [35], magnetic clutches and magnetic transmissions [36]. a summary of previous achievements in constructing magnetic springs is presented in section 2. in this paper, we propose a design of magnetic torsional spring composed of flat magnets and also a conception of its application in variable stiffness actuator. the proposed solution increases the scope of magnetic springs applicability in a context of vsas development by increasing ranges of angles for which spring torque-angle characteristics are stable and close to linear. the idea is supported by an analysis of magnetic spring dynamic behavior and its dependency on torque-angle characteristics (section 3). the effect of the proposed solution (section 4) is verified in fem analysis (section 5) and multiple experiments conducted on constructed vsa (section 6). their results were used to develop mathematical models of magnetic spring discussed in section 7. section 8 summarizes the work. 2. related work numerous conceptions of compliant mechanisms based on magnetic components have been developed so far. some of them refer to non-linear springs consisting of magnets in various arrangements which potentially can be used in antagonistic vsas classified in c1 group [37, 38]. at least one conception was successfully implemented in such a setup [39]. a different approach is to use electromagnets and adjust system stiffness by changing current applied to them [40, 41]. such construction belongs to c3 group, but it has one major drawback – high energy consumption also in steady state. fig. 1 variable stiffness torsional spring based on arc magnets – ams another interesting solution belonging to c3 category is shown in fig. 1. it presents a variable stiffness torsional spring, which is made of two coaxial rings consisting of radially magnetized arc magnets aligned in an alternating way. when the components are in equilibrium, rotating one of the rings results in counter-acting torque τsp whose value is approximately proportional to angle φr between the rings if the angle is sufficiently small. stiffness adjustment can be accomplished by translating one of the rings along the axis. moving rings away from the equilibrium by a distance d causes the overlapping part to 104 b. kozakiewicz, t. winiarski become smaller, which reduces magnetic forces between the rings and hence the stiffness of the system decreases. we will refer to this solution as ams – arc magnets spring. such an arrangement has some unique advantages like decoupled position and stiffness control, no need for constant energy supply, inherent maximum torque limit, possible zero stiffness configuration (rings completely decoupled) and an unlimited angular position range. it also has some benefits which are typical for solutions based on magnets: no contact between moving parts, reduced friction and wear, and zero clearances between cooperating parts. to the best of our knowledge, the ams arrangement has been proposed for the first time in [42]. authors of that paper built a magnetic spring using four arc magnets per ring, implemented it in construction of vsa and measured some characteristics like torque-angle relationship for different axial distances between the rings. a continuation of that work was a development of a modified design of a more compact spring [43] with an additional intermediary ring used to increase its maximal torque. the other significant achievement in this field has been made by authors of the paper [44]. they prepared and performed multiple fem analyses for different arrangements of magnets in ams – in particular, they examined an impact of poles number and magnets dimensions on maximum torque, energy and stiffness of magnetic spring. to verify the simulation model, the authors built simple two poles spring and measured its characteristics. 3. magnetic spring characteristics despite of all of the advantages of the arc magnetic springs they have at least one major drawback – they exhibit unstable behavior for specific angles φr. to examine that phenomenon in detail, we shall introduce some additional terms. fig. 2 presents sample torque-angle characteristics of magnetic spring. when external load τld (whose value is lower than maximal torque) is applied to one of the rings, multiple equilibrium points occur. if term dτsp/dφr has a negative sign, the whole system acts like a conventional torsional spring and equilibrium is stable. otherwise, even a small change of resultant torque causes the system to move away from equilibrium, which in this case is unstable, and to accelerate toward the next stable one. the direction of this movement depends heavily on initial disturbance, which makes it nearly impossible to predict and – as a consequence – problematic to control. for this reason, it is better to avoid unstable regions of spring characteristic in typical applications. the critical angle, separating stable and unstable regions, corresponds to maximal torque. to make sure that it is not exceeded, it may be necessary to assume some safety margin and to operate only within the safe range of angles which is narrower than the stable one. it may be beneficial to utilize even a tinier span of angles. control laws could be much simpler if the relationship between torque and deflection were linear (stiffness irrelevant to angle). in fact, in multiple papers concerning modeling and control of robots with compliant joints, the linearity of compliance is one of the main assumptions [45– 47]. the range of angles which provides that real characteristic differs from the linearised one no more than 5% we will call a linearity range. spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 105 fig. 2 terms describing torque-angle characteristics of magnetic spring the part of the stable range of angles that can be utilized as a safe or linear range depends on the shape of torque-angle characteristics (fig. 3). a typical plot for a manypole magnetic spring is close to sinusoid (curve 2). the fewer the poles, the more trapezoidal (curve 1) or even rectangular alike the shape becomes [44] making safe and linear ranges exceptionally narrow. in this context, a triangular (curve 3) or saw (curve 4) alike form of a curve would be more beneficial, however – to the best of our knowledge – it has not been obtained for arc magnets springs so far. in this paper, we propose a solution to broaden both linear and safe ranges of magnetic spring. fig. 3 impact of torque-angle characteristics shape on widths of safe ranges of angles 106 b. kozakiewicz, t. winiarski 4. conception of vsa based on flat permanent magnets the trapezoidal shape of few-poles-ams torque-angle characteristics is related to a non-linear way the magnetic field distribution changes as the rings rotate relative to each other. the idea that we propose is to alter air gap width with respect to the relative angle in a way that compensates these non-linearities and results in a more triangular form of characteristics. a specific configuration of magnets that we suggest is presented in fig. 4. fig. 4 proposed variable stiffness torsional spring based on flat magnets – fms the rings are composed of flat magnets aligned into polygons and magnetized in a direction perpendicular to their walls. we will call this setup fms – flat magnets spring. as will be proven in this paper, it is possible to obtain the effect described above by proper choice of polygon and magnets dimensions. fig. 5 practical setup of magnetic rings while fig. 4 depicts the general conception of magnets arrangement, fig. 5 presents more practical setup visualizing one of the possible ways of supporting cooperating components on shafts and of enclosing the whole assembly to separate it physically and magnetically from its environment. magnets in the outer ring are attached to a ferromagnetic tube with ferromagnetic cups on both ends, while magnets in the inner ring are attached to a ferromagnetic drilled core. both components are mounted to nonferromagnetic shafts. such a choice of materials results in high magnetic flux density in the spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 107 inside of the spring and a negligibly small stream leakage to the outside as it was proved in simulation described in section 5. fig. 6 presents a proposed conceptual design of variable stiffness actuator based on magnetic spring. we will refer to this setup as attractor – variable stiffness magnetic spring actuator. fig. 6 vsa concept – attractor: 1. fixed frame, 2. bearings, 3. output shaft, 4. flange, 5. inner magnetic ring, 6. guiding shafts, 7. linear bushings, 8. rigid carriage, 9. bearings, 10. input shaft, 11. outer magnetic ring, 12. lead screw, 13. bearings, 14. nut, 15,16. clutches, 17,18. dc servomotors, 19,20. absolute optic encoders 5. simulation there were three main goals of fem analyses performed on magnetic spring models: verifying a choice of materials presented in section 4, investigating an impact of magnets number, shape and arrangement on spring torque-angle characteristics and optimizing dimensions of spring used in experiments. multiple models of ams and fms were prepared to perform fem analyses. investigated cases differed from each other with number of poles. all the ams models had the same overall dimensions. in the case of fms, it was impossible to obtain that, so only the inner dimensions (in fig. 7 marked as g, l, w and z) were kept the same. the geometry and dimensions of analyzed models are presented in fig. 7 and specific values are summarized in table 1. material properties were assigned to components according to the description in section 4 and the values presented in table 2. in each case, the spring was surrounded by an air cylinder with neumann boundary conditions on its surface. 108 b. kozakiewicz, t. winiarski table 1 specific values of dimensions of investigated spring models [mm] (parameters which varied across simulations are bold) spring type pole pairs g l u w z g h1 h2 l p1 p2 u w z s t 2 4 40 88° 32 44 4 8 3 84 20 20 88° 63 75 4.0 2 ams 3 4 40 58° 32 44 4 8 3 84 20 20 58° 63 75 4.0 2 4 4 40 43° 32 44 4 8 3 84 20 20 43° 63 75 4.0 2 2 4 40 30 32 44 4 8 3 84 20 20 40 71 84 5.5 2 fms 3 4 40 20 32 44 4 8 3 84 20 20 30 67 74 3.4 2 4 4 40 16 32 44 4 8 3 84 20 20 20 65 70 2.7 2 table 2 material properties material air aluminum struct. steel ndfe35 isotropic relative permeability µr [-] 1.00 1.00 10000 1.10 magnetic coercivity hm [kam-1] 0 0 0 890 fig. 7 shapes and dimensions of investigated spring models (up to scale) an initial analysis was performed to investigate the impact of rings material on magnetic flux density. the results are presented in fig. 8, and they confirm the theses stated in section 4. hence, the material of supporting rings was set as structural steel. in the following analyses, the investigated quantities were torque and axial force between the rings calculated using the virtual work principle [48] for different values of relative angle and distance. fig. 9a presents resultant torque-angle characteristics obtained in a configuration of maximal stiffness (d = 0). examined fmss have lower maximal torque and stiffness than amss of the same number of poles and similar dimensions. the fewer the poles, the difference becomes more significant. fig. 9b visualizes the data normalized – torque is described as a fraction of maximal torque and angle as a fraction of stability limit angle. such representation enables a comparison of characteristics shapes. spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 109 as expected, investigated fmss provide characteristics of more triangular form than amss. again, the fewer the poles, the difference becomes more significant. fig. 8 impact of rings material on magnetic flux density the case arbitrarily selected for further investigation was 6-pole fms. multiple analyses were performed to limit maximal torque of the selected spring below stall torque of the available drive and to maximize the linear and the safe range of angles. the optimization variables were all dimensions listed in table 1. the constraints were a result of space limits and availability of prefabricated components. the resultant characteristics are presented in fig. 9 (labeled as "final"). a) not normalized b) normalized fig. 9 fem analyses results: torque-angle characteristics for d = 0 110 b. kozakiewicz, t. winiarski 6. experiments the conceptions presented in section 4 and dimensions optimized in section 5 were used to design and build magnetic spring (fig. 10a), as well as variable stiffness actuator following attractor concept (fig. 10b and fig. 10c). a) outer and inner magnetic ring b) attractor c) attractor components (numbers correspond to fig. 6) d) experimental setup: 1. attractor, 2. dc servomotors, 3. driver, 4. encoders, 5. controller gathering data from encoders, 6. pc with rtos supervising the experiments, 7. pc used to design experiments and analyze their results, 8. flywheel, 9. removable lever arm, 10. scale, 11. mass hung on a cord wound on the flywheel fig. 10 constructed device spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 111 some additional equipment mountable to the output flange was prepared: flywheel used to increase inertia in order to slow down device dynamics and lever arm used to interact with the output shaft. the whole experimental setup is presented in fig. 10d. the methodology of conducted experiments is described in section 6.1, while their results and analysis are presented in section 6.2. 6.1. methodology the experiments conducted on attractor mechanism can be divided into three categories: one examining the torque-angle relationship of the spring (experiment a), one focused on identifying basic mechanical properties of the system (experiment b) and the last oriented on developing a model of the whole joint – understood as a relationship between position trajectories of output shaft φout(t) and input shaft φin(t) (experiment c). a methodology of experiment a was to apply torque to input shaft using dc motor, to estimate its value τsp basing on indications of rigidly fixed scale pushed with lever arm and to estimate relative angle φr by subtracting readings of absolute encoders mounted on shafts φout-φin. the whole procedure was repeated for different axial distances d between the magnetic rings and all three stable equilibria. due to the instability described in section 3, the range of examined angles was limited (-41° ÷ 41° from stable equilibrium), but it was broader than the stable range (-31.5° ÷ 31.5°) because of static friction. experiment b was performed to identify aggregate inertia j of the output shaft and all components rigidly mounted to it. different masses m were hung on a cord wound on the flywheel and dropped. angular acceleration of output shaft (φöut) was estimated by analyzing encoders readings. the experiment was performed with completely decoupled shafts, so only the parts connected to the output shaft were rotating. experiment c consisted of multiple trials. their goal was to excite system in different ways by providing current profiles i(t) of various shapes (sinusoid, square, step, impulse function) to the motor and to gather information about movements of both shafts φout(t), φin(t) for different distances d. the data were split into two equinumerous sets: teaching and verifying and were used to develop and fit the model of the joint. 6.2. results and analysis data gathered in experiment a was used to find a model of the relationship between spring torque τsp and relative distance d and angle φr between the rings. various general approximating functions were considered: polynomials of different order and number of variables and fourier series terms. the best results were obtained for the model described with eq. (1) and discussed in section 7.1: ( )2 3 4 1 2 3 4 1 2 3( , ) 1 sin sin sin 60 30 15 sp r r r rd a d a d a d a d b b b                = − + + + + + +              (1) where τsp denotes estimated spring torque, d and φr refer to the distance and the angle between the rings, and a…, b… are model parameters. an excerpt of gathered data (marked as dots) and fitted surface described by eq. (1) are presented in fig. 11. spring stiffness for small angles φr can be described with eq. (2) obtained as a partial derivative of eq. (1). 112 b. kozakiewicz, t. winiarski ( )2 3 4 1 2 3 1 2 3 4 0 ( , ) ( ) 1 60 30 15 sp r sp r r d k d b b b a d a d a d a d        =    = − = + + + + + +     (2) eq. (3), which is derived from the euler equation of rotary motion, represents a simple model of a relationship between output shaft acceleration φöut on dropped mass m. 2 fr out mgr mr j   − = + (3) φöut denotes an angular acceleration of output shaft, m – mass attached to the flywheel, g – gravitational acceleration, r – flywheel radius, τfr – friction torque, j – inertia. fig. 11 torque-angle relationship for different distances between the rings inertia j of output shaft was identified by fitting function (3) to the data gathered in experiment b. radius r was known, and friction torque τfr was assumed to be independent of angular position and velocity of output shaft (coulomb's model of solid friction), and was identified as a second parameter of the fitted model. the fitted curve is presented in fig. 12. fig. 12 the curve fitted to the results of experiment b to identify inertia and friction spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 113 if the joint were rigid, the output shaft would always have the same angular position φout as the primary shaft φin. compliance introduces some other kind of relationship between shaft movements. its general form can be predicted theoretically by transforming the euler equation of rotary motion written for the output shaft (eq. 4): ( ) ( ) ( ), ,... ,... , , , ,... ld sp out in fr out out in out out d d j           + − + = (4) where φout denotes an angular position of the output shaft, φin – angular position of the input shaft, τld – external torque applied to the output shaft, τsp – spring torque, d – distance between the rings, τfr – friction torque, j – inertia of output shaft and parts mounted to it. in this model, external load τld was treated as known – its value was zero in the experiments. the model of spring torque τsp(d, φout – φin) has already been found as well as the value of inertia j. multiple models based on eqs. (1), (2), (4) were developed (most interesting cases are listed in table 3 and interpreted in section 7.2) and investigated by fitting their parameters to teaching set (described in section 6.1) and calculating different goodness of fit measures using verifying dataset: mean square error (mse), adjusted coefficient of determination (r2 adj) standard error (se) and bayesian information criterion (bic). also, two partially linearised models were investigated: al and fl. the results are listed in table 4. table 3 models of output shaft acceleration (bold symbols denote fitted parameters) id equation a ( )( )1 ,out sp out inj d   = − b ( ) ( )( )1 , signout sp out in outj d    = − − frτ c ( ) ( )( )1 , tanhout sp out in outj d    = − − frτ v d ( ) ( ) ( )( )1 , tanh sinout sp out in out outj d     = − − + +frτ v w x e ( ) ( ) ( ) ( )( )1 , tanh sinout sp out in out out inj d      = − − + + −frτ v w x y f ( ) ( ) ( ) ( )( )1 , tanh sinout sp out in out out out inj d       = − − + + − −frτ v w x z al ( ) ( )( )( )1 mod 60 ,120 60out sp out inj k d  = − +   −  fl ( ) ( )( ) ( )( ( ) ( )) 1 mod 60 ,120 60 tanh sin out sp out in outj out out in k d       = − +   −  − + + − − frτ v + w x z 114 b. kozakiewicz, t. winiarski table 4 goodness of fit of shaft acceleration models model teaching set verifying set id mse 1r2 adj se bic mse 1r2 adj se bic a 3522 1.895‰ 59.4 164593 4175 3.046‰ 64.6 179356 b 2334 1.255‰ 48.3 156307 3051 2.223‰ 55.2 172617 c 2270 1.221‰ 47.6 155762 2976 2.169‰ 54.6 172092 d 1451 0.780‰ 38.1 146763 2050 1.494‰ 45.3 164094 e 1423 0.765‰ 37.7 146382 2021 1.473‰ 45.0 163795 f 1365 0.734‰ 36.9 145536 2001 1.458‰ 44.7 163590 al 4339 2.335‰ 65.9 168784 148454 92.762‰ 344.2 251313 fl 2122 1.141‰ 46.1 154429 153345 94.811‰ 391.6 256919 fig. 13 long-term output shaft trajectory simulations – not exceeding the critical angle fig. 14 short-term output shaft trajectory simulations – not exceeding the critical angle spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 115 fig. 15 long-term output shaft trajectory simulations – exceeding the critical angle models listed above describe output shaft accelerations. simple approximations (eqs. (5) and (6)) allow deriving differential models of output shaft positions. [ ] [ 1] [ 2] 2 2 ( ) 2 ( ) ( 2 ) 2 ( ) n n ndeff t f t t f t t f f f f t t t − −− − + − − +  = (5) [ ] [ 1]( ) ( ) ( ) n ndeff t f t t f f f t t t −− − −  = (6) for example, the position model obtained in case f is described with eq. (7). ( ) ( ) ( ) ( ) 2 [ ] [ 1] [ 2] [ 1] [ 1] [ 1] [ 1] [ 2] [ 1] [ 1] [ 2] [ 1] [ 2] 2 , tanh sin n n n n n n n n out out out sp out in fr out out n n n n n out out out in in t v d j t z w x t               − − − − − − − − − − − −    = − + − − − +      + + − − − +   (7) position models based on acceleration models were used to simulate output shaft trajectories φout(t) (the model was fed with its output). the simulations were performed using data from a verifying set for two classes of movements: not exceeding (φr ≤ 31.5°) and exceeding (φr > 31.5°) the critical angle. the worst cases are presented in figs. 13, 14 and 15 and discussed in section 7.2 (to maintain readability of the figures only models a, f, al and fl are included; none of the remaining models – b, c, d, e provided better performance than model f). 7. discussion the following sections present a discussion of the obtained results. section 7.1 concerns spring characteristics shape and model, while section 7.2 describes joint models interpretation and usability. 116 b. kozakiewicz, t. winiarski 7.1. spring characteristics the model (eq. (1)) of torque τsp dependency on relative angle φr and distance d, described in section 6.2, has a convenient form of a product of two functions. one is a polynomial dependent on distance d, and the other is composed of the first three terms of sinus series dependant on relative angle φr. the model is well defined, inherently periodical, smooth and easy to analyze. fig. 16 presents a comparison of the surface fitted to experimental data (eq. (1)) and results of the simulation. curves in fig. 16a represent torque-angle relationship for d = 0, while fig. 16b depicts torque-distance relationship for φr = 31.5° (points representing experimental data are plotted for 31° < φr < 32°). the differences in the domain of angles φr are negligible within the stable range of angles (less than 2%); however, beyond this range, they can reach about 20%. in the domain of distances, they are much more distinct and vary from less than 5% in stiff configuration to over 40% in compliant one. the possible reasons for the differences are modeling and numerical errors in fem analysis, manufacturing and mounting inaccuracies, measurement errors, lack of experimental data for angles between 41° and 60° and influence of static friction. although the last factor is unmeasurable in direct ways, its maximum value can be estimated based on the measurements gathered in experiment b – it should be close to kinetic friction identified as about 0.05nm. a) torque-angle relationship b) torque-distance relationship fig. 16 spring characteristics – comparison of experiment and simulation fig. 17 presents the spring characteristics (plotted in the stiff configuration: d = 0) as well as its stable and linear range of angles. it is shaped as intended, and both ranges are relatively wide: their limits are 31.5° (105% of theoretical value 30°) and 19.7° (66%) accordingly. as a reference, in paper [42] where the same number of pole pairs was used these values were about: 26° (87%) and 11° (37%). spring based on flat permanent magnets: design, analysis and use in variable stiffness actuator 117 fig. 17 resultant torque-angle characteristics shape, linear and stable ranges 7.2. joint model all the models examined in section 6.2 and listed in table 3 have one vital feature in common – all their terms have a clear interpretation. it is summarized in table 5. the results presented in table 4 indicate that the including terms corresponding to kinetic friction and wheel unbalance have a significant impact on the goodness of fit, which suggests a profound role of these factors. implementing damping dependant on output angular velocity results in a negligible improvement, so that it can be implied that the influence of viscous friction on the output shaft movement is relatively low. including damping dependant on relative angular velocity provides a noticeable but rather small improvement of the goodness of fit. because there is no contact between rotating rings, the only possible explanation for such damping are magnetic interactions – probably related to inducing eddy currents in conducting components of the spring. the results indicate that – at least for investigated velocities, dimensions and mechanical parameters – this phenomenon has a relatively low impact on output shaft behavior. the results of long-term simulations of output shaft movements (presented in section 6.2 in fig. 13) indicate that – provided the critical angle is not exceeded – the most complex model f can be successfully used as a long-term simulator. the same is true even for its partially linearised form fl. if only short-term predictions are needed (fig. 14), even the simplest models a and al are capable of providing satisfactory results. if the critical angle is exceeded (fig. 15), the simulations based on non-linearised models are not satisfactorily accurate but – contrary to linearised ones – they follow the general trends of real output shaft trajectory for some period of time. this period is the longest in the case of model f (2.6 seconds in presented case). however, it is worth noting that due to the unstable behavior of magnetic spring described in section 3, the predictability of output movement within the unstable range of angles φr may be inherently and inevitably limited. 118 b. kozakiewicz, t. winiarski table 5 interpretation of terms used in examined joint models term interpretation j inertia ( ),sp out ind  − static characteristics of magnetic spring ( )signfr out − model of dry kinetic friction ( )tanhfr outv − differentiable approximation of the model above ( )sin outw x + eccentric mass model of imperfect wheel balance outy− damping dependant on output shaft angular velocity ( )out inz  − − damping dependant on relative angular velocity ( )spk d stiffness in stable equilibrium ( )mod 60 ,120 60out in − +   −  term providing periodicity of 120° 8. conclusions the experimentally verified results of simulations indicate that using flat magnets instead of arc ones for the magnetic spring construction enables alteration of its torqueangle characteristics in a beneficial way by adjusting spring geometry. in the case of a low number of magnet pairs, the choice of optimized fms in the place of ams results in expanding linear and stable ranges of angles, what broadens a space of movements so that the behavior of the spring is well-defined, predictable and easy to control. the additional advantage is lower cost and much better availability of flat magnets as opposed to arc ones. however, the price for all the advantages mentioned above is a reduction of spring maximal torque and stiffness. hence, the best applications for implementing fms are those, in the case of which the wide range of deflections and convenient control is more important than maximization of available torque. since the more pole pairs, the lower the differences between fms and ams are, another field of fms application may be constructions with high pole pairs number. the device constructed accordingly to attractor concept is proven in use and constitutes an example of usability of fms arrangement in vsa development. the movement of the output shaft within the stable range of angles can be well predicted by a simple linearized model and even simulated for a more extended period using a model, which is more complex but still easy to interpret. there is a large potential of device miniaturization what – combined with many important advantages of magnetic mechanisms like reduced wear, zero backlash and possible zero stiffness configuration – justifies considering it as a competitive alternative for conventional mechanical variable stiffness actuators. acknowledgements: this work was funded within the incare aal-2017-059 project ,,integrated solution for innovative elderly care'' by the aal jp and co-funded by the aal jp countries (national centre for research and development, poland under grant aal2/2/incare/2018). the authors would like to thank mateusz baczewski and adam kowalewski for putting their effort in developing hardware and software used to control drives, register data from encoders and conduct experiments. the authors also wish to express appreciation to maciej węgierek for his organisational support and to katarzyna kozakiewicz for her help in gathering experimental data and formatting this paper. spring based on flat permanent magnets: design, analysis and use in 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1987, modeling and control of elastic joint robots, journal of dynamic systems, measurement, and control, 109(4), pp. 310-318. 46. flacco, f., 2012, modeling and control of robots with compliant actuation, phd thesis, sapienza università di roma, italy. 47. albu-schäffer, a., ott, c., hirzinger, g., 2007, a unified passivity-based control framework for position, torque and impedance control of flexible joint robots, the international journal of robotics research, 26(1), pp. 23-39. 48. benhama, a., williamson, a.c., reece, a.b.j., 2000, virtual work approach to the computation of magnetic force distribution from finite element field solutions, iee proceedings – electric power applications, 147(6), pp. 437-442. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 325 336 ontological framework for knowledge management in orthopedic surgery  udc 004.9:617.3 milan zdravković, miroslav trajanović, dragan pavlović faculty of mechanical engineering, university of niš, serbia abstract. efficiency and effectiveness of orthopedic surgery can be achieved by enabling proper decision-making in a shortest period of time, based on complete and updated information on the status, type of fracture and fixators used for a particular fracture. in this way, the risk of possible complications caused by a late intervention can be reduced. in such circumstances, there exist critical needs for an effective and efficient knowledge management approach where different domain models are combined and formally interrelated, so that the decisions are based on the consistent and complete information. in this paper, ontologies are used to propose a framework for implementing such an approach in the domain of orthopedic surgery. the framework combines formal models of the generic products and supply chains for their manufacturing and delivery, anatomical elements, e.g. bones, types of their fractures and fixators – the medical products which are used in the fracture treatments. then the possible uses of this framework for the purpose of knowledge management in orthopedic surgery are discussed in the context of the assumptions of development of next generation enterprise information systems. key words: knowledge management, orthopedic surgery, ontology, systems interoperability, semantic interoperability 1. introduction one of the major challenges of modern health care organizations refers to the possible improvement of the health service quality. to achieve this goal, health care organizations are using standardized clinical protocols in many medical domains [1]. these protocols are now represented in a variety of different formats, languages and formalisms. this variety is considered as a significant obstacle for semantic reconciliation of the models as well as the interoperability of the respective systems that are using those models, thus received september 9, 2015 / accepted october 30, 2015  corresponding author: milan zdravković faculty of mechanical engineering in niš, university of niš, ul. aleksandra medvedeva 14, 18000 niš, serbia e-mail: milan.zdravkovic@gmail.com original scientific paper 326 m. zdravković, m. trajanović, d. pavlović posing a strong need for unification and alignment which will significantly increase the effectiveness of the given systems. one way of resolving this problem, namely, that of achieving the unique representation of the clinical models and protocols, is to use ontologies. according to [2] “an ontology is an explicit specification of a conceptualization.” thus, it represents an approach to formal modeling of a specific reality. ontologies can provide a significant contribution to the design and implementation of the enterprise information systems (eis) in the medical domain. their role in the integration and harmonization of heterogeneous knowledge sources is already considered by many research projects, especially in the field of clinical guidelines and evidence-based medicine [3]. besides the interoperability aspect, the use of ontologies as means for formal representation of the medical concepts will enable the validation of these concepts in terms of consistency and completeness checking and thus it will contribute to more accurate decision-making at the implementation (systems) level. the aim of the research done for this paper is to demonstrate that the ontologies can help in making the decisions regarding conceptually different notions in healthcare, i.e. medical products, management of the supply chain for their manufacturing and delivery and anatomy features. since these notions are handled in different eiss, within or outside the clinical domain, we indirectly aim at demonstrating that these systems can be made interoperable. namely, based on the common, inter-related models, the respective systems that are using these models may exchange the relevant information, that is by default understood by all of these systems. in specific, this paper deals with a set of such decisions being made in the domain of orthopedic surgery. namely, we propose an ontological framework which will facilitate a timely and accurate selection of the fixator – a device that is used in the treatment of the specific types of fractures of so-called long bones. this selection is being done based on the anatomical features of the fractured long bone and correct classification of specific fracture type. finally, the ontological framework enables efficient establishment and management of the supply chain for the manufacturing and delivery of the selected fixator. thus, it will facilitate a prompt response of the medical centre in case of an urgent need for non-standard, customized medical products, typically manufactured in made-toorder fashion. 2. the ontological framework for knowledge management in orthopedic surgery the use of ontologies in medicine is mainly focused on data management, i.e. medical terminologies. data collection (grouping) is becoming one of the most important issues that the researchers in the clinical domain are facing. due to the inconsistency of the formats used for data representation, it is very difficult to develop generic computer algorithms for their interpretation. researchers tend to represent knowledge of their domain in an independent and neutral format so that data can be shared and reused in different platforms. this problem can be solved by using ontologies. ontologies provide a common framework for structured knowledge representation. these ontological frameworks provide common vocabularies for concepts, definitions of concepts, relations and rules, allowing a controlled flow of knowledge into the knowledge base [4]. ontological framework for knowledge management in orthopedic surgery 327 today, ontologies are not only used as modeling assets but also as runtime models, which continuously create and maintain the relationships between the logically related concepts in the different models or systems. thus, they represent not only a tool for the knowledge management in a specific domain, but also act as enablers for the interoperability between the corresponding systems. in this paper, we demonstrate that the ontologies can be used as a facilitator for interoperability among the clinical information systems (cis), which are used to manage the comprehensive patient information, including different aspects of diagnosis and treatment, the decision making systems and even, supply chain management systems. the proposed framework has been developed by using the owl (web ontology language). the owl, adopted by the world wide web consortium (w3c), is a semantic markup language designed for publishing and sharing ontologies on the world wide web. it was developed by extending the resource description framework (rdf) vocabulary and it is based on the experience of developing the daml + oil web ontology language [5]. the framework uses and combines four different ontologies. the ontologies of anatomy and formal representations of the electronic health record (ehr) are used to establish the link with cis, namely to provide a dictionary for pulling information about the specific anatomical features of the patient with the injured bone. the ontology of fractures is used to automatically classify the type of the fracture, based on the information already instantiated in the anatomy and ehr ontologies. the ontology of fixators is used to select the medical device that accurately fits the diagnosis of the injured patient. 2.1. medical ontologies and electronic health records the anatomy domain is that domain of medicine in which, so far, ontologies are most commonly used. in the medical domain, the anatomy is a fundamental discipline that represents the basis for most medical fields [6]. formal anatomical ontologies are an important component of the informatics healthcare infrastructure [7]; also, they are informatics tools used to explore biomedical databases. structural relationship that is primarily used in these ontologies is part of the relationship, because smaller anatomical entities are naturally seen as components of the larger ones [8]. there exist plenty of anatomical ontologies, clinical ontologies or ontologies of other domains in medicine. based on the bioportal statistics, the most frequently used ontology is the foundational model of anatomy (fma) [9]. the fma is a domain ontology that represents a coherent body of explicit declarative knowledge about human anatomy. clinical vocabularies play a strategic role in providing an access to computerized health information because clinicians use a variety of terms for the same concept. when a clinician evaluates a patient, the resulting documentation typically captures free text and unstructured information, such as history and physical findings. the efficiency of payment (reimbursement) processing is probably the key incentive for transforming this free text into more structured data. some of the most commonly used clinical vocabularies today are the logical observation identifiers, names, and codes (loinc) [28] and the systematized nomenclature of medicine clinical terms (snomed) [29]. the loinc for ordering lab tests and the snomed-ct for recording test results provide well-defined meanings for specific terms that can be standardized across applications. the loinc is used to identify individual laboratory results, clinical and diagnostic study observations. it 328 m. zdravković, m. trajanović, d. pavlović is most widely used in laboratory systems. the snomed is designed to be a comprehensive, multi-axial, controlled terminology, created for the indexing of the entire medical record. there are three main organizations that develop and maintain the standards related to ehr messages: the health level seven (hl7), the comité européen de normalization – technical committee (cen tc) 215, and the american society for testing and materials (astm) e31. the hl7 [10] develop the most widely used healthcare-related electronic data exchange standards in north america. the cen tc 215 is the preeminent healthcare it standards developing organization in europe. recently, the research community interest was brought to the openehr [11] open standard specification in health informatics. in contrast to hl7 and cen’s en 13606 standards [12], which are strictly concerned with data exchange between ehr systems, the openehr describes management and storage, retrieval and exchange of health data from ehrs. unfortunately, there is no developed standard rdf/rdfs/owl ontology that could be used to formally describe an ehr yet. this is considered as a major obstacle for semantically interoperable cis, as ehr records often suffer from the vendor-specific realizations of patient record data sets which rarely accommodate to the controlled terminologies [6]. for the proposed framework, we use the openehr owl ontology developed by roman [13]. some preliminary work has been done in integrating the above ontology with snomed [14] and loinc [28] owl representations. 2.2. ontology of fixators for the representation of the fixators, the product ontology [15] is selected, for the reasons of its simplicity vs. the fulfillment of requirements related to modeling fixators and their features. the product ontology is mapped to the unspsc product classification scheme [16], by using the unspsc-skos ontology as a mediator. the skos [17] is a family of formal languages, built upon rdf and rdfs for representation of thesauri, classification schemes, taxonomies or other types of structured controlled vocabulary. the ontology of fixators was developed [18] with the objective to represent the topology of these medical products. it represents a meta-model of their bills of materials (bom). it extends the product ontology by specializing its product, part and feature classes, as illustrated in fig. 1. structural dimensions of the elements fixators depend on certain dimensions, i.e. features, so the feature class contains a subclass named dimension. class dimension contains subclasses of characteristic features that may affect the structural dimensions of fixator elements. note that some of the features above are the features of the bone and not of the fixator itself. however, these features are represented at the level of the product, in order to facilitate the selection of the proper fixators, based on the features of the bone and its fracture. the ontology of fixators is instantiated with two specific fixators: the external fixator “mitković” and hybrid external fixators. hence, the former one consists of the following elements: rod, screw, lateral supporting element, two clamping rings on the lateral supporting element, screw nut, washer, and two clamping ring plates on the clamp ring. each of the part individuals is assigned with the characteristic dimensional features. thus, it becomes possible to select the specific fixator whose features correspond to the geometrical features of the fractured bone, where these features are established by using the x-ray or ct scans of the patient. ontological framework for knowledge management in orthopedic surgery 329 fig. 1 fragment of the ontology of external fixators 2.3. ontology of fractures ontology of fractures formally describes different types of the bone fractures and thus, it makes possible inference of the exact type of fracture, based on its diagnosed features (such as fracture location, number of fragments and their geometry, fracture lines, etc.). this inference is considered as a trivial problem when humans are interpreting the x ray or ct scans. however, the ontology of fractures is intended to be used by the systems, which can automatically interpret the observed specific features of the fracture, e.g. based on image processing and feature recognition. the inferred type can then be used to select the corresponding fixator for the injury treatment. the proposed ontology’s scope is restricted to diaphyseal fractures because these fractures are treated by using the external fixators. the fragment of ontology, related to humerus bone is illustrated in fig. 2. as highlighted above, the ontology of fractures is strictly formal and it uses the relationships of the specific types of diaphyseal fractures with the observations from ct scans to define the necessary conditions for the classification of the specific type. these observations include: number of fracture planes, angle of the fracture planes to a sagittal plane, number of bone fragments and existence of contact between the bone fragments. for example, a simple humerus fracture is characterized by only one fracture and hence, two bone fragments. in the manchester owl syntax, this restriction is represented as follows: 330 m. zdravković, m. trajanović, d. pavlović hasbonefragment exactly 2 bone_fragment the simple humerus fracture can be further decomposed into oblique, transverse and spiral fractures, depending on the angle of the fracture plane to a sagittal plane of the bone and/or its existence. the following restrictions are used to represent the oblique and transverse fractures, respectively: hasfractureplane some (hasfractureangle min 30) hasfractureplane some (hasfractureangle max 29) spiral fractures, caused by torsion, are characterized by the fact that the fracture plane does exist at all, as the fracture line represents the spiral. hence, the condition is: hasfractureplane exactly 0 fracture_plane furthermore, the wedge humerus fracture is characterized by the minimum of 3 bone fragments, where the one is of wedge-shaped. the second condition is that all bone fragments remain in contact with each another. hence, the restrictions are as follows: hasbonefragment min 3 bone_fragment hasbonefragment only (incontactwith some bone_fragment) fig. 2 fragment of the ontology of long bones’ diaphyseal fractures the wedge humerus fracture can be further decomposed into multifragmentary, bending and spiral fractures. each of these subtypes is described by the corresponding conditions. for example, the classification of the bending wedge humerus fractures is enabled by the following restrictions: hasbonefragment exactly 3 bone_fragment hasfractureplane only (oblique_fracture_plane or transverse_fracture_plane) ontological framework for knowledge management in orthopedic surgery 331 in the presented ontological framework, the ontology of fractures is imported by the ontology of fixators, which is then used to execute semantic queries, with the objective to select the specific fixator for a given circumstances of the fracture and anatomical features (e.g. bone length). 2.4. supply chain ontology the response time is one of the critical factors for a successful treatment of the orthopedic disorders. the ontologies of these disorders (i.e., the bone fractures) and the medical products for their treatment (i.e., the fixators), when combined with the atomic observations from the cis (acquired through medical ontologies), provide the framework for accurate and fast decision making and hence, reduction of this response time. however, sometimes, due to the specific anatomy features of the patient or fracture, it is not possible to effectively treat the orthopedic disorder with the medical devices on stock. instead, a custom orthopedic fixator may be needed to facilitate an efficient treatment. needless to say that the process of ordering, manufacturing and delivering such a fixator is extensively time-consuming and may incur the delays that could be critical for a successful treatment. however, the proper knowledge management strategy can significantly reduce this time. with such a strategy implemented, a clinical centre can overtake some of the roles of the medical devices supplier and directly implement the management of the supply chain for their manufacturing and delivery. this approach is facilitated by the supply chain ontologies, namely the scor ontological framework and a semantic web application for supply chain configuration that is using that framework. based on the product’s topology and manufacturing or delivery strategies of each product part (including the services), a sourcing (s) strategy, namely the supply chain configuration is generated by the sc-conf-sys application [19]. the sc-conf-sys is based on the scor reference model for supply chain operations [20], a standard approach for analysis, design and implementation of the core processes in supply chains. the scor ontological framework [21] represents knowledge at three different levels of conceptualization (see fig. 3). first, the implicit scor ontology is used to enable interoperation of the sc-conf-sys with proprietary scor tools. second, the explicit scor-full ontology is an expressive domain ontology which defines the meanings of the implicit scor entities and thus, it facilitates interoperation of the sc-conf-sys with other enterprise applications. fig. 3 supply chain ontology framework 332 m. zdravković, m. trajanović, d. pavlović third, the scor-cfg is application ontology, used to enable the formalization of the competency questions relevant for the framework, namely to enable the representation of the semantic queries. then, the supply chain configuration can be inferred, based on the common rules related to the orderings of the scor source, make and delivery processes in the different cases of the manufacturing strategies: make-to-stock, make-to-order and engineer-to-order; and a capacity of the supplier to deliver the desired part. 4. discussion the digital era in which we live and work today evolves the it infrastructures towards the ubiquitous computing paradigm. the latter assumes an environment in which an increasing number of devices collaboratively collect, process and store an extensive amount of data, information and knowledge. the paradigm of ubiquitous computing gives a boost to development of new storage technologies, such as clouds. however, while the cloud-based systems enable storage and sharing of big data, the common and unified approach which will facilitate management of this data and its subsequent transformation to knowledge has not been developed yet. with the advent of semantic web technologies, such as the rdf/owl, the ontologies are being increasingly used not only as means to represent meta-data structures, but also to serve as runtime models for different applications, i.e. semantic web applications. there exists a visible trend of increasing formalization of the standards, reference models and dictionaries, as well as the developing of transformation tools which enable mapping of less-expressive modeling languages (such as uml) to formal the rdf/owl structures. this trend will have a significant impact on how the information systems of the future are designed and developed. in the recently submitted position paper of the ifac tc5.3 technical committee for enterprise integration and networking [22], the next-generation enterprise information systems (ng eis) are foreseen to be omnipresent, model-driven, open, dynamically reconfigurable, aware and computationally flexible. the list of these properties implies that, in fact, the future eis will be inherently interoperable. in the remainder of this section, we present the scenario of use of the proposed knowledge management framework for the orthopedic surgery domain, by the above-mentioned systems. 4.1. eis infrastructure for knowledge management in orthopedic surgery the omnipresent property of the ng eis means that a computing becomes ubiquitous in the sense that the communication, processing and storage capabilities are not anymore exclusive to computers or smart phones. in fact, a number of devices with these capabilities, connected to internet, rapidly grow, forming so-called internet-of-things (iot). this development has a tremendous impact on the healthcare domain. todays’ medical devices [23] combine sensors for spatio-temporal detection of electrical, thermal, optical, chemical, genetic and other signals with physiological origin, as well as with actuators, e.g. medical dispensers or assisting tools, capable to autonomously and intelligently implement or change the therapy [24]. their operation depends on an extensive amount of information that is continuously being pulled out and pushed back to different medical information systems. ontological framework for knowledge management in orthopedic surgery 333 with an increased number of medical devices with processing capability, the complexity of the overall it infrastructure becomes extremely complex. one of the imminent consequences of this complexity is the rising difficulty to achieve seamless collaboration and exchange of information between all systems within the it environment. this problem is dealt by the ng eis by transferring the data models and business logic based on which these systems operate from their core runtime environment to possibly external, unifying models. hence, the ng eis becomes model-driven. these models are represented in a formal and explicit way – by using ontologies. fig. 4 example architecture of the next generation enterprise information system for knowledge management in orthopedic surgery such ontology-driven approach is illustrated in fig. 4 which describes the basic architecture of the ng eis for knowledge management in orthopedic surgery. it also distinguishes (but also takes into account) the traditional architecture of clinical it infrastructure (right-hand side) from the model-driven one, which also considers new, evolved systems, such as imaging device and so-called smart implant [25] (left-hand side). in the traditional approach, the eiss are typically considered as monolithic applications, driven by static data structures, implemented by database management systems. the integration of eiss is carried out by using separate infrastructures (such as enterprise service buses) whose setup is costly and time-consuming because it involves the negotiation and alignment of different data structures that need to be exchanged between the systems. hence, the scalability of one eis, in terms of its capability to seamlessly work with enterprise service bus facilities is considered as one of the most important properties. in contrast, the ng eis will be driven by dynamic meta-data structures, modeled by far more expressive means than database schemas – the owl ontologies. the increased expressivity implies that these structures will not store only knowledge about data, but also about business logic of eis [26] – thus enabling a truly model-driven approach to running eis. instead of exchanging semi-structured formats, such as xml, the ng eis will communicate by perceiving and reacting to the environment observations, where data about the different stimulus will be transformed into perceptions – sets of logical statements about these 334 m. zdravković, m. trajanović, d. pavlović stimuli [22]. in other words, the ng eiss will become aware of their environment and thus, the need for interoperability mediators, such as esbs, will no longer exist. in such context, all the systems will remain autonomous and the only centralized role will be related to maintaining the consistency of the ontological models used by these systems. hence, in our scenario (see fig. 4), imaging device system and smart implant system will operate autonomously of central cis. however, all these systems will use ehr ontology. imaging device system will use it to validate patient information, based on the computerized physician order entries (cpoe), used to order radiology services; and to automatically assert the atomic perceptions from the scans into ehr. these perceptions will take the form of the logical statements by using the schemas described in ontologies of fractures and anatomy, as presented in the following example: fracture of humerus fracture_plane hasfractureangle 45 fracture hasbonefragment exactly 2 bone_fragment based on the above listed statements, the ontology of fractures can be used to automatically diagnose an oblique simple fracture of humerus. this diagnosis will be asserted to cis, by using the formalisms present in ehr ontology. as argued before, based on the formal description of diagnosis, an automatic selection of the appropriate fixator can be carried out, by exploiting the rules that formalize the correspondences between the geometric features and topology of the fixator and type of fracture, on one hand, and formal description of diagnosis. after the surgery, the healing process of the bone can be tracked by the smart implants, which will host the sensors to measure force, torque, load (e.g. load sharing between the bone and implant), strain, motion (e.g. implant elastic deformations), ph, temperature, and pressure [27]. these measurements will be transformed into perceptions, again, represented by using logical statements and then asserted into cis by using ehr ontology formalisms. sometimes, when it is not possible to select the existing fixator design due to the specific anatomy of the patient, the new, custom design will be done by using the computer aided design (cad) system. then, this new design will be asserted to ontology of fixators. obviously, in this case, it will become necessary to setup and track processes related to the manufacturing of this new fixator design. sc-conf-sys application can be used for this purpose. namely, it will generate a scor-based model of the supply chain and assert this model into the supply chain ontology framework. the sourcing, manufacturing and delivery orders will then become accessible by the suppliers’ enterprise resource planning (erp) systems. 5. conclusions with the variety of models, standards, protocols and other formalisms for representing medical concepts and processes, the healthcare domain is one of the most diversified fields and test-beds for ontologies. many research projects have already demonstrated the number of advantages of using ontologies in healthcare. ontologies can help in building more interoperable information systems in healthcare. they can facilitate transferring, reuse and sharing of patient data. finally, ontologies can support the integration of the ontological framework for knowledge management in orthopedic surgery 335 necessary knowledge and information in healthcare. the work that this paper is based on deals with the latter aspect. in this paper, we presented the knowledge management framework for the representation and use of the knowledge in the domain of orthopedic surgery, by using the ontologies. such a representation combines the anatomical models, ehr data, the types of bone fractures, the models of the medical devices and models of the supply chains that can be swiftly employed for the purpose of manufacturing and delivery of custom fixator. such a framework can enable automated decision-making in the domain, but it will also contribute to achieve the universal and unconditional interoperability between all relevant systems. thus, it will facilitate further development of the paradigms related to so-called next generation enterprise information systems in healthcare domain. acknowledgements: this paper is a result of the work carried out in the project iii41017 “virtual human osteoarticular system and its application in preclinical and clinical practice”, funded by the ministry of education and science of republic of serbia, for the period of 2011-2015. references 1. jiang, g., ogasawara, k., endoh, a., sakurai, t., 2003, context-based ontology building support in clinical domains using formal concept analysis, international journal of medical informatics, 71(1), pp. 71-81. 2. gruber, t., 1993, a translation approach to portable ontology specifications, knowledge acquisition, 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vocabulary. journal of the american medical informatics association, 5(3), pp.276-292. 29. cote, r.a., robboy, s., 1980, progress in medical information management systematized nomenclature of medicine (snomed), the journal of american medical association, 243(8), pp. 756-762. 7527 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 449 457 https://doi.org/10.22190/fume210306038w © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper self-consistency conditions in static three-body elastic tangential contact emanuel willert technische universität berlin, institute of mechanics, germany orcid id: emanuel willert https://orcid.org/0000-0001-7535-7301 abstract. the contact problem for an elastic third-body particle between two elastic half-spaces is considered. the contact is assumed to consist of three hertzian contact spots. the normal and tangential contact problems are analyzed analytically considering partial slip in the contacts and the influence of third-body weight. selfconsistency conditions between global equilibrium and the contact solution are formulated to give criteria, under which circumstances static slip and stationary sliding are possible states for the third-body particle. the sliding case is solved in detail. key words: three-body contact, self-consistent sliding, wear, hertz-mindlin theory 1. introduction the tribological problem of the third body has recently attracted a lot of scientific interest, mostly in connection with the behavior and lifecycle of wear particles, whose understanding plays a critical role in a better description of both the wear process itself as well as the influence of wear and particle transport on other tribological phenomena in mechanical contacts [1]. several aspects of the three-body problem have been analyzed, including the formation of wear debris particles – which was studied both experimentally [2] and numerically [3] – as well as their kinetics [4]. it was also shown that the thirdbody dynamics can have a massive influence on the frictional or other contact mechanical properties of a tribological system [5-7] and that vice versa contact properties like loading forces influence the mode of motion (sliding or rolling) of the wear debris particle [8]. while quite some research has been done on the wear and flow behavior of the debris particles, e.g., based on monte carlo methods [9] or cellular automata [10], there are hitherto very few works on the three-body system as a contact mechanical problem (which it obviously is). li [11] analyzed the elastic three-body contact problem based on the boundary element method, using a starting configuration with only one contact spot on each of the first bodies, which at some threshold will result in rolling of the third body, received march 06, 2021 / accepted april 27, 2021 corresponding author: willert emanuel technische universität berlin, institut für mechanik, sekr. c8-4, straße des 17. juni 135, 10623 berlin e-mail: e.willert@tu-berlin.de https://orcid.org/0000-0001-7535-7301 450 e. willert because of the kinematic indeterminateness of a two-point fixation. an analytic, easy-touse contact theory of the three-body problem, that might be very useful for practitioners or industrial applicants, to the best the author’s knowledge, is lacking completely. whereas a (more or less) spherical particle between two surfaces, that are moved tangentially relative to one another, will usually simply roll, this may not be so easy for a third body of irregular shape. from experience we know that it will make a big difference for the apparent macroscopic friction between the first bodies, whether the third-body particle between them will roll or slide. in this context it is an interesting question whether static slip or stationary sliding are possible states for the particle. in a recent work it was shown analytically that that the same third-body particle can both slide and roll for a given coefficient of friction, depending on the particle’s geometry and orientation [12]. thus, in the present manuscript, the problem of self-consistency for static slip or stationary sliding of the third body will be investigated in analytic fashion, based on a hertz-mindlin formulation of the three-body contact problem. note that the manuscript question is basically whether and how non-rolling configurations are possible for the third-body particle. so, effects of rolling are always excluded from the analysis. 2. global equilibrium conditions let us consider the 2d-model of an elastic third-body particle of some irregular shape between two elastic half-spaces, as shown in fig. 1. for static determinateness let there be three (axisymmetric) hertzian contact spots, where the particle in the vicinity of the contact has radii of curvature ri, i = 1,2,3. in the general 3d case, there should also be contact spots in the lateral direction (outside the plane shown in fig. 1), to ensure lateral stability during tangential motion, but this will be neglected in the following analysis. as the lateral positions of the contact spots do not enter the following equations, the results can be applied directly to the 3d case, if the number of contact spots is adjusted appropriately. the gap height without any loads between the half-spaces is h0. fig. 1 sketch and force diagram for the analyzed three-body contact problem self-consistency conditions in static three-body elastic tangential contact 451 the global equilibrium conditions for the third body are 2 3 1 2 3 1 ( ) 1 1 1 1 2 2 2 2 3 3 3 2 0 , 0 , 0 ( ) ( ) ( ). z x s y f n n n mg f t t t m n x t z n x t z n x t z = = + − − = = + − = = − + − − − − − −    (1) combining the last two equations we obtain 1 1 2 2 3 3 1 1 0 ,n x n x n x t h t h− − =  (2) because the total indentation depth, δh = h0 – h, must be negligible for the hertzian theory to be applicable. 3. normal contact solution for the contact solution in the following sections it shall be generally assumed that the characteristic length of the contact spots (e.g. their radius) is much smaller than the macroscopic dimensions of the third-body particle (so that one can neglect finite size effects for the elastic body) and that the contacting bodies are elastically similar to avoid elastic coupling of the normal and tangential contact problems, i.e. (introducing the shear moduli gi and poisson ratios νi, where the index “3” corresponds to the third-body particle, and “1” and “2” to the upper and lower half-spaces) [13] 31 2 1 2 3 1 21 2 1 2 . g g g   −− − = = (3) with the effective elastic moduli on the upper and lower side [13], 1 1 * * *3 31 2 1 2 3 1 3 2 3 1 11 1 : , : 2 2 2 2 v vv v e e e g g g g − −    − −− − = + = = +        (4) the hertzian normal contact solution reads [13] * 1/2 3/2 1 1 4 , 0, 1,2,3. 3 i i in e r d d i=  = (5) the hertzian solution can obviously only be used if the contact spots do not interact elastically. if the contact spots are very close to each other, one should consider interactions between them [14]. without loss of generality let us assume that the lower half-space is fixed, and the upper half-space is macroscopically displaced by δh. in general, the third-body particle will experience small elastic displacements in all its degrees of freedom (as a rigid body); if we denote the normal and tangential displacement of the center of gravity by ws and us, and the small rotational angle by φ, the indentation depths for the three contact spots are given by 452 e. willert 1 1 2 2 3 3 , , . s s s d h w x d w x d w x    =  − − = + = + (6) for the pure normal contact problem, the tangential forces are absent. the equilibrium conditions for fz and my will then give a nonlinear equation system to determine the two unknown displacements, ws and φ. this equation system will usually be unsolvable in closed analytical form and due to the plenty of influencing parameters a comprehensive solution cannot be shown here. for illustration purposes, however, let us give the solution, if the particle (e.g., due to symmetry) is not rotating. inserting eqs. (6) (with φ = 0) into the first of eqs. (1), we obtain 1/2 1/2* 3/2 3/22 32 1 1* 1/2 * 1/2 1 1 1 1 3 ( ) . 4 r remg d h d e r e r + + =  − (7) introducing dimensionless variables, 1/2 1/2* 3/2 2 31 2 * 1/2 3/2 * 1/2 1 1 1 1 3 : , : , : , 4 r rd emg m h e r h e r   − + = = =   (8) eq. (7) simplifies to 3/2 3/2 3/2(1 ) .m  −+ = − (9) from the derivation above it follows that this equation is independent of the number of contact spots on the upper and lower side – which only changes the value of α – if all contacts on one side have the same indentation depth, i.e., if all “asperities” have the same height. however, considering a height distribution as in classical asperity theories [15] would, of course, be possible without difficulties. the nonlinear eq. (9) cannot be solved in closed form. however, usually the particle weight will be small compared to the contact forces. in this case an asymptotic solution can be found easily. it is given by 0 0 1 0 0 1/2 0 0 ( ) , ( 0) (1 ) , 1d d 2 . d d 3m m m m m m             − = =  + = = = + − = = = − (10) note that δ itself has a weak dependence on the current total indentation δh (via the normalized particle weight). hence, the distribution of the total indentation into the upper and lower side will not be universal during the indentation process, if there are external forces acting on the debris particle. self-consistency conditions in static three-body elastic tangential contact 453 4. tangential contact solution due to the equilibrium condition for the moments of forces, the normal and tangential contact problems are coupled macroscopically and therefore must be solved together. suppose the upper half-space is displaced in the tangential direction by δu, according to some loading history δu = δu(δh). then, the relative tangential displacements in the contact spots are given by 1 1 2 3 2 , . s s u u u z u u u z   =  − + = = − (11) all tangential forces are functions of these displacements, ( ), 1,2,3.i i it t u i= = (12) however, the precise form of those functions depends on the loading history, and therefore, on all system parameters. because of that it is not feasible – although theoretically possible based on the known solution procedures for tangential contact problems with arbitrary loading histories [16] – to give a comprehensive analytic solution for ti. for example, it would be a gross simplification to use the classical cattaneo-mindlin solution, that is valid only for a specific loading history, namely a constant normal force and a subsequently applied increasing tangential force (which clearly contradicts the global equilibrium conditions). nonetheless, a numerical determination for a concrete parameter set is easy, for example within the frameworks of the method of dimensionality reduction [17] or the method of memory diagrams [18]. once the force laws for the tangential forces are known, the global equilibrium conditions (1) provide an equation system for the determination of the three displacements, ws, us and φ. it should be noted that the critical displacement, for which the contacts start to slide globally, is always [13] * , * ,i i c i i i e u d g = (13) with the effective shear moduli 1 1 * * *3 31 2 1 2 3 1 3 2 3 2 22 2 : , : . 4 4 4 4 v vv v g g g g g g g − −    − −− − = + = = +        (14) hence, all contacts on one side will to start to slide at the same time if they have the same indentation depth, independent of their local radii of curvature. when all contacts are sliding, the tangential problem becomes trivial, and all tangential forces are given by the amontons-coulomb law .i i it n= (15) to illustrate the importance of the loading history once again for the tangential contact solution, let us consider a simple (but somewhat academic) non-sliding case, which can be easily solved in exact analytic form, namely the two-contact configuration without external forces. 454 e. willert if we set n3 = t3 = mg = 0, equilibrium of the forces demands that the contact forces on both sides are equal and opposite and the equilibrium condition for the moment of forces reduces to 1 2 : tan . x x t n n h  − = = (16) hence, during loading the tangential contact force always has to be proportional to the normal force and this, of course, also always has to be true for the force increments dn and dt. consider a given equilibrium with the forces n and t and the contact radius a. now, the normal force is increased by dn, which according to the hertzian theory results in a new contact radius [13] 1/3 d d 1 . n a a a n   + = +    (17) irrespective of the previous load history, the entire contact area will initially completely adhere (according to amontons’ law, the slip area is constantly at the limit of possible sticking, a slight increase in pressure leads to complete sticking). by applying an additional incremental force dt, however, local slip can again spread from the edge of the contact. the stick radius c is given by the cattaneo-mindlin theory and equals [13] 1/3 1/3 d d ( d ) 1 1 . ( d ) dt n t c a a a n n n n      = + − = + −    +    (18) if c > a, the contact area increases faster than slip can propagate into the contact, i.e., the contact will always be completely sticking. this leads to the condition tan .  (19) hence, it was shown that – within the hertz-mindlin approximation – there is no partial slip for this loading history; the contact is always completely sticking if µ > tan α (and obviously completely sliding otherwise), because the contact area grows faster than slip can propagate from the contact edge. so, the self-consistency condition for elastic bodies resulting from hertz-mindlin contact mechanics is in this case the same as the one for rigid bodies (i.e., non-rolling configurations are only possible if tan α ≤ µ)! 5. self-consistency conditions for gross sliding in the case of static slip (i.e., macroscopic “sticking”) the tangential forces are bound by the friction law, which imposes a self-consistency condition. for stationary sliding there are two self-consistency conditions (because the tangential forces are given explicitly by the friction law), directly resulting from the global equilibrium, 1 1 0 1 1 2 2 3 3 2 1 1 2 2 3 2 1 1 1 , ( ) ( ) 1 . n h n x n x n x n n n n mg mg n       = − −   = + = +  = −    (20) self-consistency conditions in static three-body elastic tangential contact 455 so, if the upper and lower contacts have the same frictional properties, the weight of the particle will disturb the equilibrium and thus inhibit stationary sliding. that is not restrained to the particle weight or the number of contact spots on each side: in fact, if the coefficients of friction on both sides are the same and all contacts are sliding, obviously any force, which is not aligned with the friction angle, will violate the equilibrium condition. finally, if once again rotation of the particle is absent, using the normal contact solution from eqs. (7) and (8), the first of eqs. (20) can be written in the form 1/2 1/23/2* 32 2 1 0 1 2 3* 1 11 1 ( ) . ( ) re rm h x x x m r re         −  = − +             (21) if the particle weight is negligible compared to the contact forces, this simplifies to 1/2 1/2 2 2 3 3 1 0 1 1/2 1/2 2 3 , r x r x h x r r  + = − + (22) which, interestingly, does not depend on the elastic properties of the system. one possibility for the absence of rotation would be complete symmetry, i.e., x1 = 0, r2 = r3 and x2 = –x3. this results in the self-consistency condition µh0 = 0, so stationary sliding in this case is impossible! 6. example case with equal coefficients of friction to illustrate the above findings let us consider a wear debris particle of some general shape, which in the contact plane forms three contact spots with the first bodies. we want to know whether stationary sliding is a possible state for the third body if the coefficients of friction on the upper and lower surfaces are equal and if we can neglect the particle weight. the contact enumeration shall be as in fig. 1, i.e., spot number “one” is the singular one, “three” is the one on the other surface but on the same side (left/right with respect to the center of gravity) and “two” the one on the other surface and on the other side. all geometrical notations are the same as in fig. 1. as we neglect the particle weight and the friction coefficients are equal on both surfaces, the second of eqs. (20) is always fulfilled and the only remaining condition of self-consistent stationary sliding is the first of eqs. (20). fig. 2 shows the contour line diagram of the normalized position of the third contact spot, ξ3 = x3/x1, as a function of the normalized force distribution, n2 = n2/n1 and the normalized position of the second contact spot, ξ2 = -x2/x1, necessary to enable stationary sliding of the third-body particle. the normalized gap width between the first body surfaces was chosen to be h = µh0/x1 = 1. note that the results for different values of h can be simply obtained by shifting the coordinate ξ2 by δξ2 = δh/n2. 456 e. willert fig. 2 contour line diagram of the normalized position of the third contact spot, ξ3 = x3/x1, as a function of the normalized force distribution, n2 = n2/n1 and the normalized position of the second contact spot, ξ2 = -x2/x1, necessary to enable stationary sliding of the third-body particle if the coefficients of friction on both sides are the same. the normalized gap width between the first body surfaces was chosen to be h = µh0/x1 = 1. geometrical notations as in fig. 1. 7. discussion and conclusions in the considerations above several simplifying assumptions have been made to allow for analytical treatment of the problem, most prominently the amontons-law, linear elasticity and the absence of surface roughness and elastic coupling. however, most findings are a consequence of the principal structure of the three-body contact problem and its resulting static indeterminateness. it was shown how global external forces on the third-body particle – like its weight – can influence the local contact problem and that static slip and stationary sliding are only possible (but they are possible, at least, if one neglects frictional instabilities) for specific system configurations. for example, for stationary sliding, if external forces are absent, the frictional properties of the upper and lower contacts must be the same. that condition is, however, not sufficient, as the equilibrium of the moments of force will impose another restriction for the geometrical “arrangement” of the contact spots. that restriction seems to be independent of the elastic properties but does depend on the local geometry in the vicinity of the contact spots (which strongly influences the respective normal contact solution). references 1. greenwood, j.a., 2020, metal transfer and wear, frontiers in mechanical engineering, 6, 62. 2. ostermeyer, g.p., brumme, s., recke, b., 2017, the wear debris investigator – a new device for studying the formation of particles in the contact area, proc. eurobrake, dresden, eb2017-vdt-019. 3. aghababaei, r., warner, d.h., molinari, j.f., 2016, critical length scale controls adhesive wear mechanisms, nature communication, 7, 11816. self-consistency conditions in static three-body elastic tangential contact 457 4. de payrebrune, k.m., kröger, m., 2015, kinematic analysis of particles in three-body contact, tribology international, 81, pp. 240-247. 5. ostermeyer, g.p., 2003, on the dynamics of the friction coefficient, 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frictional contact, facta universitatis-series mechanical engineering, 18(4), pp. 537-544. 12. bilz, r., de payrebrune, k.m., 2019, analytical investigation of the motion of lapping particles, proceeding in applied mathematics and mechanics, 19, e201900076. 13. popov, v.l., heß, m., willert, e., 2019, handbook of contact mechanics: exact solutions of axisymmetric contact problems, springer-verlag, berlin heidelberg, 347 p. 14. li, l., wang, j., shi, x., ma, s., cai, a., 2021, contact stiffness model of joint surface considering continuous smooth characteristics and asperity interaction, tribology letters, 69, 43. 15. carbone, g., bottiglione, f., 2011, contact mechanics of rough surfaces: a comparison between theories, meccanica, 46, pp. 557-565. 16. jäger, j., 1993, elastic contact of equal spheres under oblique forces, archive of applied mechanics, 63, pp. 402-412. 17. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer-verlag, berlin heidelberg, 265 p. 18. aleshin, v., bou matar, o., van den abeele, k., 2015, method of memory diagrams for mechanical frictional contacts subject to arbitrary 2d loading, international journal of solids and structures, 60-61, pp. 84-95. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 179 197 original scientific paper instability of the rayleigh-benard convection for inclined lower wall with temperature variation udc 536.2 sadoon ayed 1 , gradimir ilić 2 , predrag živković 2 , mića vukić 2 , mladen tomić 3 1university of technology, department of mechanical engineering, baghdad, iraq 2university of niš, faculty of mechanical engineering, niš, serbia 3college of applied technical sciences, niš, serbia abstract. this paper deals with an analysis of a two-dimensional viscous fluid flow between the two parallel plates inclined with respect to the horizontal plane, where the lower plate is heated and the upper one is cooled. the temperature difference between the plates is gradually increased during a certain time period after which it is temporarily constant. the temperature distribution on the lower plate is not constant in x-direction, there is a longitudinal sinusoidal temperature variation imposed on the mean temperature. we have investigated the wave number and amplitude influence of this variation on the subcritical stability and the onset of the rayleigh-bénard convective cells, by direct numerical simulation of 2d navier-stokes and energy equation. key words: dns, rayleigh number, boussinesq approximation, inclined walls, nonlinear stability analysis 1. introduction the rayleigh-bénard flow is a type of natural convection. it is one of the classical problems in fluid mechanics where a fluid is typically bounded by bottom and top walls which are heated and cooled, respectively (fig. (1)). in this research, the fluid is subjected to the spatial temperature modulation at the lower wall and a small inclination of the both walls with respect to the horizontal plane. the reason for this convection is the temperature gradient in the vertical direction, which causes instable density stratification and, consequently, the fluid motion. received september 29, 2015 / accepted april 12, 2016 corresponding author: predrag živković university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, niš, serbia e-mail: pzivkovic@masfak.ni.ac.rs 180 s. ayed, g. ilić, p. živković, m. vukić, m. tomić fig. 1 formation of rayleigh-bénard cells rayleigh gave the first solution of the problem of the conducting state stability when the fluid in a gravitational field is bounded from above and below with constant but unequal temperatures. he obtained the critical value of a dimensionless parameter at which the flow starts. this parameter is named after him: the rayleigh number. another parameter, which measures the relative strength of the non-linearity in the momentum equations, is the prandtl number. they are defined in the following way: 3 1 2( ) a g t t l r a     ; rp a   (1) where g is gravitational acceleration, β is thermal expansion coefficient, t1 is temperature of lower plate and t2 is the temperature of upper plate, l = 2h is the distance between the plates, υ is kinematic viscosity and a is thermal diffusivity. this is a non-dimensional parameter for the measure of the buoyancy/diffusive forces ratio. in the above definition of the rayleigh number, the fluid properties are calculated at mean temperature tm = (t1+t2)/2 as the best reference temperature. in our calculation t1 is a temporal and x-direction variable function as, according to eq. (1), is the ra number. the value of the critical rayleigh number according to the linear stability theory is rac = 1708 at wave number qc = 3.117. beyond this value the fluid starts moving thus forming counter-rotating two-dimensional cells, the cross-section of which is almost square. the cellular flow becomes considerably more complicated as the ra increases. the two-dimensional cells break up in three-dimensional cells, which appear hexagonal in shape when viewed from above. with larger ra numbers, the cells multiply, become oscillatory and, finally, turbulent. the flow becomes turbulent at ra = 104 and pr = 1, for water pr = 7 there is ra = 105, and for higher pr numbers there is ra = 106-107. most of the above is valid for small temperature differences between the plates when the so-called oberbeck-boussinesq approximation is valid. in the oberbeck-boussinesq approximation all the fluid properties are considered constant except density, which is assumed to be a linear function of temperature: 2 2[1 ( )]t t     (2) where ρ2 is fluid density at upper plate, t is fluid temperature. one of the first papers that analyzed the rayleigh-bénard flow with variable viscosity is that of tipelkirch et al. [1]. thereafter, many works have appeared treating the rayleigh-bénard flow with temperature dependant viscosity. some of the authors have treated basic parameters [2,3], variable fluid viscosity [4,5], different enclosures [6,7,8], two-phase flows [9,10], etc. some modern approaches to numerical solution can be found in the recent papers [8,10,11]. instability of rayleigh-bénard convection for inclined lower wall with temperature variation 181 probably the first work where the rayleigh-bénard flow was treated through assuming all the fluid properties as a function of temperature was that of paolucci et al. [12]. the working fluid was air, its dynamic viscosity and thermal conductivity following the sutherland law. the work of fröhlich et all [13] and severin and herwig et al. [14] followed, which is the most recent work on the rayleigh-bénard convection considering all the fluid properties variable and angle-inclined [15-18]. severin and herwig calculated the critical reynolds number using the method of asymptotic expansions and the results were valid only for small heat transfer rates. in this paper we consider a fluid flow for the rayleigh number below the critical value against the wave number very close to the critical one. the numerical simulation of 2d navier-stokes equation in vorticity-stream function form is carried out for temperaturedependant thermophysical properties. in those cases there is spatial temperature modulation at the lower wall with small angle γ, where δm is amplitude and qm is wave number of the temperature modulation around some average value at the lower plate. 2. mathematical model at the rayleigh-bénard convection we can use three different approaches: boussinesq approximation, low mach number approximation, and compressible navier-stokes equations. the incompressible navier-stokes equations for bousinesq approximation, continuity and energy equation read: 2 2 1 ( ) [1 ( )] 0 ( ) v v v p v t t g t v t v t t t                         (3) where kwjviuv   is velocity vector, p pressure, a thermal diffusivity and υ is kinematic viscosity. for the case of the fluid flow between two parallel plates inclined at an angle γ, relative to the horizontal plane, force g  may be expressed as: sin cosg g i g j   at this point in the research, considering the limited computational resources, the problem will be considered only in 2d domain. the boundary conditions are: ( , , ) 0, ( , , ) 0u x y h t u x y h t     (4) ( , , ) 0, ( , , ) 0v x y h t v x y h t     (5) 2 1( , , ) , ( , , )t x y h t t t x y h t t     (6) for the plane flow in the vorticity stream-function formulation we have: 2 ( ) 1 ( )( ) ( ) [1 ( )]( sin cos ) v v v p v t t g i g j t                     (7) 182 s. ayed, g. ilić, p. živković, m. vukić, m. tomić having in mind that per definition p and  υ, and that for plane flow we have kjviuv  0 , and x for any physical value , we obtain the following vorticity equation: 2 2( ) sin ( ) cos ( ) d d v t t t t gk t dy dx                      (8) since 0 0x y z zi j k i j k          in two dimensional fluid flow, our equation after projection onto z direction becomes: 2 2( ) sin ( ) cos ( ) d d v t t t t g t dy dx                      (9) where we have denoted   z. the problem can be made dimensionless by using the following characteristic scales: l = h for length, t = h2/a for time and v = a/h for velocity. after rendering those equations dimensionless, we change the variables introduced in order to transform the physical domain into the computational one which is adopted to the fourier-chebyshev approximation, namely   x*  2 and   y  . the flow variables are reduced by scale factor to obtain the following coordinate transformation: * * * * 2 * * * * * * * * 2 2 * * * * 2 , , 1 1 1 1 , , x lx hx y ly hy a h v vv v t tt t h a l h l h v a a vl l hh                             (10) substituting the above coordinate transformation in the vorticity equation, we have: * * * * * * * 2 2 2* * 1 1 ( ) sin ( ) cos ( ) v v vl vv l l ltt l g d d t t t t l dy dx                                        (11) or in the following forms: * 2 * * * * 2 2* 2 3 * * ( ) sin ( ) cos ( ) v v v g d d v t t t t lt lt l l dy dx                      (12) 2 * 2 * * * * 2 24 * 4 4 * * ( ) sin ( ) cos ( ) a a a g d d v t t t t hh t h h dy dx                      (13) instability of rayleigh-bénard convection for inclined lower wall with temperature variation 183 if we multiply this equation with h4/a2 the above equation yields: 3* * * * * 1 2 2 1 2 * * * 1 2 1 2 ( ) ( ) ( ) ( ) sin cos ( ) ( ) h g t t t t t td d v a a t t t tt dy dx                      (14) 3* * * * * 1 2 2 1 2 * * * 1 2 1 2 ( ) ( ) ( ) ( ) sin cos ( ) ( ) h g t t t t t td d v a a a t t t tt dy dx                       (15) non-dimensional temperature is defined with: * 2 1 2 t t t t     (16) rayleigh number by eq. (1) and prandtl number as ratio of kinematic viscosity to thermal diffusivity: pr a   (17) now, after substitution of eqs. (17) and (16) in (15) we have: * * * * * * * * * ( ) pr pr sin cosv ra t dy dx                       (18) instead of continuity equation the definition of vorticity is used here:    (19) after scalar product with k we obtain: * * * 2 1 0 v v vl l l        (20) and after multiplication with l/v we obtain the above equation in dimensionless form: * * * 0   (21) the energy equation can be transformed in the following way: 2 2 2 ( ) ( )( ) ( ) t t v t t a t t t          (22) and dividing the equation by t1-t2, it yields: 2 2 2 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t v a t t t t t t t            (23) or: * * *( )v a t          (24) 184 s. ayed, g. ilić, p. živković, m. vukić, m. tomić substituting the expressions for coordinate transformations: * * * * * 2 1 1 1 vv a t t l l                (25) or: * * * * * 2 2 1 1a a v a t h hh h                (26) and after dividing by a/h2 we obtain the energy equation in non-dimensional form: * * * * * * * ( )v t          (27) the nondimensional oberbeck-bousinesq system of equations to be solved (18), (21) and (27): * * * * * * * * * * * * * * * * * * * ( ) pr pr 0 ( ) v ra t x v t                              the boundary conditions are the following: * * * * * * * * * * * * * * 0, , 0, , 0, 1 1, , 0, , 0, 1 {( , ) r| 0 2 1 1} at y y at y y d x y x y                             (28) and the initial conditions will be defined later. we have two boundary conditions for stream function and none for vorticity. the problem is solved by influence matrix method [19, 20], so that the first two equations can be solved simultaneously, and the numerical method is described in detail [21]. we describe here only the numerical method used for energy equation. 3. solution procedure the above set of equations should be solved simultaneously in time, so we describe the procedure only for the energy equation. for the solution of the described problem we use the fourier-chebyshev pseudo-spectral method, with the fourier-galerkin method for approximation in homogeneous x-direction and the chebyshev collocation method for non-homogenous y-direction. eq. (16) in developed form is: instability of rayleigh-bénard convection for inclined lower wall with temperature variation 185 * * * 2 * * * * * * * 2* u v s t x y x                (29) where s* is source term, which, in our case, is zero. the fourier approximation in streamwise direction can be expressed by trigonometric polynomials in exponential form: ** * * * * *( , , ) ( , ) k ikx k k k x y t y t e     (30) ** * * * * *( , , ) ( , ) k ikx k k k s x y t s y t e    (31) ** * * * * *( , , ) ( , ) k ikx k k k u x y t u y t e    (32) ** * * * * *( , , ) ( , ) k ikx k k k v x y t v y t e    (33) * * * * * * * * * * * * * 1,* * ( , , ) ( , ) ( , ) k k ikx ikx k k k k kk u x y t u y t e b y t e x x                (34) * * * * * * * * * * * * * 2,* * ( , , ) ( , ) ( , ) k k ikx ikx k k k k kk v x y t v y t e b y t e x y                (35) here * *( , )y t designate fourier coefficients for the exponential form of trigonometric polynomials in eqs. (30)-(35) for any physical value  and 1i . in the following, we will always refer to dimensionless quantities; consequently * are suppressed from superscript, and after substitution of eqs. (30), (31), (34) and (35) in (29), we obtain: * 2 2 2 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) k k k ikx ikx ikx k k k k k k kk k k k k ikx ikx ikx k k k k k k k k k y t e u y t e v y t e t x y y t e y t e s y t e x y                                        (36) after differentiation: * 1, 2, 22 2 2 2 ( , ) ( , ) ( , ) ( ) ( , )( , ) ( , ) ( , ) k k k ikx ikx ikxk k k k k k k k k k k k ikx ilx ikx ikxk k k k k k k k k y t e b y t e b y t e t ik y t e e y t e s y t e y y                          (37) if multiply both the sides with the same weight function as the basis functions, we have: 186 s. ayed, g. ilić, p. živković, m. vukić, m. tomić 1, 2, 2 2 2 ( , )( , ) ( , )( , ) ( , )( , ) ( , )( , ) ( , )( , ) ( , )( , ) ,..., 2 2 k k k ikx ilx ikx ilx ikx ilxk k k k k k k k k k k k ikx ilx ikx ilx ikx ilxk k k k k k k k k x x y t e e b y t e e b y t e e t k y t e e y t e e s y t e e t n n l k                            (38) if we apply the orthogonality condition, so that inner product of exponential functions yields: 1, 2, 2 0 ( , )( , ) ( , )( , ) ( , ) 0 for ( , ) 2 for k k k ikx ilx ikx ilxk k k k k k k k k ikx ilx ikx ilx y t e e b y t e e b y t t k l e e e e k l                    (39) our system of equations is reduced to: 2 2 1, 2, 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,..., 2 2 k k k k k k x x y t b y t b y t k y t y t s y t t t n n l k                (40) for approximation in y-direction we use the chebyshev-collocation method in the following way: 2 (2) 1, 2, , 0 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,..., , cos , 1,..., 1 2 2 yn k j k j k j k j j i j k j i jx x j y y y t b y t b y t k y t d y t s y t t n n k y j n n                  (41) where )2( ,ijd are elements of the chebyshev differentiation matrix. the boundary points j = 0 and j = ny are not included in the above system of equations. if we designate: 1, 2,( , ) ( , ) ( , )k j k j k jb y t b y t b y t  (42) for discretisation in time, the second order finite difference method in generalized form yields: 1 1 1 1 2 1 1 2 1 2 (2) 1 1 , 0 (1 ) ( ) 2 ( ) (1 ) ( ) 2 [ ( ) ( ) (1 ) ( )] [ ( ) (1 ) ( )] [ ( ) (1 ) ( )] ( ) (1 ) ( ) ,..., 2 y n n n k j k j k j n n n n n k j k j k j k j k j n n n n n j i k l k l k l k l i x y y y t b y b y b y k y y d y y s y s y n n k                                                , cos , 1,..., 1, , 0,1,..., 2 jx j y n t y y j n t n t n n n        (43) instability of rayleigh-bénard convection for inclined lower wall with temperature variation 187 with these values the above system of equations becomes: 1 2 1 2 1 22, 0, 0, 1, 0, . . , 1i e                (44) we use the semi-implicit adams-bashworth backward differentiation method for which we have: 1 2 1 (2) 1 , 0 1 1 1 3 ( ) ( ) ( ) 2 4 ( ) ( ) ( ) 2[ ( ) ( )] 2 ,..., , cos , 1,..., 1, , 0,1,..., 2 2 yn n k j n n k j j i k j i n n k j k jn n n k j k j k j jx x j y n t y y k y d y t y y s y b y b y t n n k y j n t n t n n n                                 (45) the values on the right hand side are known, and the values on left hand side are unknown and have to be calculated for each time step. values 1n k  have to be calculated for each wave number k = 0,…,nx/2, and each collocation point yj = cos(j / ny), j = 1,…,ny. the generalized form of a set of eqs. (43) can be solved for γ1=0, γ2=0, in the following way: 2 (2) 1 2 (2) , , 0 0 1 1 1 (1 ) 2 ( ) (1 ) (1 ) ( ) 2 2 (1 ) ( ) ( ) (1 ) ( ) [2 ( ) ( )] 2 ,..., , cos , 1,..., 1, 2 2 y yn n n n j i k j j i k j i i n n n n n k j k j k j k j k j jx x j y n y k d y k d y t t y s y s y y b y t n n k y j n t n                                                        , 0,1,..., tn t n n  (46) if introduce the following matrices: (2) 1 , 4 , 1,..., 1 1,..., 1 [ ] , [ ] 0,..., 0,..., x x i j i j y y i n i n i d d j n j n          (47) and scalars: 2 2 1 0 1 (1 ) 1 , (1 ), 2 2 k k t t t                       (48) the above system of equation is reduced to the following form: 1 1 1 1 4 0 1 4 1 1 1 1 1 1 [ ] [ (1 ) ] (1 ) 2 ( ) ( ) ,..., , , 0,1,..., 2 2 n n n k k k n n n n k k k j k j x x n t i d i d i i s i s b y b y n n k t n t n n                               (49) 188 s. ayed, g. ilić, p. živković, m. vukić, m. tomić where: 0 0 0[ ( ),..., ( )] , [ ( ),..., ( )] , [ ( ),..., ( )]n n n t n n n t n n n t k k k n k k k n k k k ny y s s y s y b b y b y     (50) if we introduce the following notation: 1 1 1 4 0 0 1 4 1 1 1[ ], [ (1 ) ],b i d b i d b i             (51) we finally obtain these matrix equations: 1 1 1 1 1 0 1 1 1(1 ) 2 0,..., 2, , 0,1,..., n n n n n n n k k k k k k k x n t b b b i s i s b b k n t n t n n                    (52) this system of equations should be supplemented by boundary conditions. the horizontal walls are assumed to be isothermal with non-dimensional temperatures θhot = 1 and θcold = 0 at the bottom and top walls: 2 2 2 1 2 1 2 1 1 2 ( ) ( 1) , ( 1) 0 ( ) ( ) ( 1) , ( 1) 1 ( ) cold hot t t t y t y t t t t t y t y t t                   (53) the generalized robin boundary conditions have the form: ( 1) ( 1) ( 1) ( 1) y y g y y y g y                             (54) after the implementation of the chebyshev collocation method we have: (1) 0 , 0 (1) , 0 ( ) ( ) ( ) ( ) cos , 1,..., n n n j n j j n n n n j n j j j y d y g y d y g j y j n n                          (55) in these equations n = ny. boundary conditions on the upper and lower plates can be represented in this way: , , ( ,1, ) ( , ) ( , 1, ) ( , ) k ikx k k k k ikx k k k g x t g y t e g x t g y t e            (56) instability of rayleigh-bénard convection for inclined lower wall with temperature variation 189 if we multiply by eilx and implement orthogonality condition (39) we get: (1) 0 0, , 0 (1) , , 0 ( , ) ( , ) ( ) ( , ) ( , ) ( ) cos , 1,..., n k j k j k j n k n n j k j k j j y t d y t g t y t d y t g t j y j n n                          (57) in our case we have: 1 1(1) (1) (1) (1) 0 0,0 0,1 0,2 0, , (1) (1) (1) (1) ,,0 ,1 ,2 , ( ) ( ) n nk n k kn n n n n k n y d d d d g gd d d d y                                              (58) (1) (1) (1) (1) 0,0 0,1 0,2 0, 6 (1) (1) (1) (1) ,0 ,1 ,2 , 1 ,1 , 1 1 6 , , 0,..., , , 0,1,..., 2 n n n n n n n kn k k n n x k k n t d d d d d d d d d g g g n d g k t n t n n                                              (59) system of equations (52) and (59) can be solved by direct methods: 1 1 1 1 1 1 1 1 1 1 1 6 (1 ) 2 , 0,..., , , 0,1,..., 2 n n n n n n n k k k k k k k n n x k k n t b i s i s b b b h n d g k t n t n n                          (60) if we introduce the following notation:                 1 1 1 6 1 , n k n kn k g h f d b a   (61) the system is reduced to: 1 1 0,..., , 0,1,..., 2 n n k k x t a f n k n n      (62) this system of equations should be solved in a given time step prior the solution of momentum equation in the vorticity-stream function form eqs. (16) and (19), and the calculated values of θn+1 should be used in eq. (16). the system of eqs. (16), (19) and (25) is solved for each time step n = 0,1,…,nt for all wave numbers k = 0,1,…,nx/2. 190 s. ayed, g. ilić, p. živković, m. vukić, m. tomić 4. initial and boundary conditions we consider the initial condition for our simulation for the case of the rayleigh-bénard convection: * ( , , 0) 0, , ( , ,0) 0, , ( , ,0) 0 ( , ) {( , ) r| 0 2 , 1 1} x y t x y x y x y d x y x y                  (63) the temperature at the lower plate is raised gradually according to the following law: ( , 1, ) (1 sin cos )sin 0, 0 / 2 ( , 1, ) (1 sin cos ), / 2 m m m m m m m m x y t q x q x t t x y t q x q x t                          (64) the temperature at the lower wall is not constant in x-direction, it depends on qm wave number modulation, amplitude δm and frequency ω. rayleigh number ra measures average temperature gradient, while the additional spatial modulation is characterized by small amplitude δm and wave number qm. in the absence of forcing (δm = 0), convection cells with wave number qc, bifurcate only for ra above critical rayleigh number rac. the onset of convection is characterized by parabolic linear stability neutral curve in parameter space with ra on abscise and q on ordinate axis. the neutral curve has its minimum at critical wave qc = 3.117 and rayleigh number rac = 1707.8. in contrast, for δm = 0, convection is unavoidable for any finite ra in the simplest case in the form of “forced cells” with wave vector qm. the main goal of the present work is to provide direct numerical simulation of cells and their stability in presence of forcing with small amplitude δm ≈ 0 (0.01) and in ratio qm / qc = 1.2. the stability of the forced cells strongly depends on ratio qm/qc. it is very important to emphasize that our ra number varies temporally and spatially, since temperature at the lower wall is described by the eq. (64), while temperature difference t1-t2 in eq. (1) varies with time t and xcoordinate. our idea is to show the evolution of stream function, vorticity, velocity and temperature field in the transition period 0  t  , for ω = 1. the values we have chosen are subcritical values ra=1000, qm=3.7 according to linear stability analysis, for pr = 7, δt = /300, δm = 0.01, number of fourier modes k = 96, number of nodes nx = 192, number of chebyshev collocation points ny = 192. 5. results of the numerical simulation in a thin layer of the fluid heated from below, convection occurs as a steady pattern of two dimensional cells. the two-dimensional convection cells and stability properties are investigated in detail in [22] and [23]. for the heated layer corresponding to ra > rac the stable roll patterns occur only within a band of wave number centered approximately about critical wave number qc. within the stable band, the cells realized do not necessarily have a preferred length scale. indeed, their wavelengths appear to be dictated by the initial conditions used to select the cells and by the manner that the basic state temperature is prescribed spatially and temporally. the band is bounded on both sides by instabilities that pertain to changing the wavelength of the cells but without changing the form. as the induced cells acquire wave length too large or too small, instability will occur to shift their length scale back to a value close to the critical one. as the value of ra increases, the cells at some point will become unstable and the convection structure will converge to a pattern with a more complex spatial and temporal structure. instability of rayleigh-bénard convection for inclined lower wall with temperature variation 191 in this section we present the results of numerical simulation obtained by our 2d pseudo-spectral method developed in matlab code for the navier-stokes equation in stream function-vorticity formulation and energy equation, with the initial and boundary conditions described above and the walls inclined for 1 angular degree. in fig. (2) we have shown the vorticity distribution for 4 instants of non-dimensional time t = /2, , 3/2 and 2. since we have chosen qm = 3.7, such number of the pair of cells can be seen at t = /2 at the lower wall. this is the moment when temperature on the lower plate reaches its final value with its modulation in x-direction. the distribution in vorticity is not noticeably perturbed. the maximal and minimal values of vorticity are between -11 and 3. in the next instant of time t =  we can see an increase in the vorticity intensity, where it attains the values in the range from -30 to 12, but the distribution is already obviously perturbed in the direction of lifted walls. in the next instant of time t = 3/2 we can see even more vorticity distribution deformation, and the jump in the range of the values from -52 to 33. at t = 2, the convective terms in energy equation start to transport vorticity in upwards left direction and the external vorticity values increase further -75  ω  85, although the temperature at the lower wall is temporarily constant now, but with modulation in x-direction. the results of numerical simulation for stream function are given in fig. (3). we can see that the range of values is increased not only for time period when temperature at the lower wall increases but also when the temperature attains its constant value. the spatial stream function distribution is slightly changed in time period 0  t  /2, but afterwards /2  t   convective terms attain values that significantly change the stream function’s intensity and distribution. the distribution pattern is more perturbed with the passage of time. the results of numerical simulation for velocity in x-direction are given in fig. (4). the u-velocity keeps its form in time period 0  t  /2, but afterwards /2  t   buoyancy effect becomes significant, and convective terms attain values that significantly change the intensity and distribution of u-velocity. in time period /2  t   distribution pattern has been changing and becomes partly disordered. the range of values is increasing between -12 and 14. the fluid heating through the lower wall and fluid cooling through the upper wall are not in balance so that the increase in kinetic and thermal energy in the fluid flow is remarkable. the range of values in t = 3/2 is -22  u 25, and in t = 2 is-47  u  31, and the position of the extremely values is pushed from the lower wall toward the middle of the channel and to the direction of the lifted wall. the lower wall with its temperature modulation is the source of momentum in the initial stages and later the external values are shifted to the middle of the channel. the values near the lower wall are constantly increased. the lower wall serves as a source of momentum in x-direction and feeds the momentum in the middle of channel. in fig. (5) we have shown the v-velocity evolution in period of time 0  t  2. we can see that the values of v-velocity constantly increase and that the upflow and downflow for t  /2 at the center of the channel are disturbed. in this case both buoyancy and shear have a destabilizing effect on the flow pattern and traveling cells are possible flow structures at t  /2. 192 s. ayed, g. ilić, p. živković, m. vukić, m. tomić fig. 2 non-dimensional vorticity evolution in the forced rayleigh-bénard convection instability of rayleigh-bénard convection for inclined lower wall with temperature variation 193 fig. 3 non-dimensional stream function evolution in the forced rayleigh-bénard convection 194 s. ayed, g. ilić, p. živković, m. vukić, m. tomić fig. 4 non-dimensional u-velocity evolution in the forced rayleigh-bénard convection instability of rayleigh-bénard convection for inclined lower wall with temperature variation 195 fig. 5 non-dimensional v-velocity evolution in the forced rayleigh-bénard convection 196 s. ayed, g. ilić, p. živković, m. vukić, m. tomić we can see that the convection is unavoidable for rayleigh number ra = 1000, for δ = 0.01 (forced rbc flow) for each instant of time in the form of forced cells with wave number qm = 3.7 as can be seen in fig. 5. these forced cells develop instability against resonant modes for ra > rac=1708 in wide range of qm/qc; in our case it is qm/qc = 3.7/3.12. only for qm in vicinity of qc the forced cells remain stable up to fairly large ra>rac. in the case of inclined walls this cell has lost its stability at t  /2, in spite of the fact that our qm = 3.7 is in the vicinity of the critical wave number, and rayleigh number ra = 1000 is significantly below critical value rac = 1708. in our simulation the instability is reached at lower values of ra at the value of wave number which is close to the critical one. we have ra < rac but a periodic roll solution exists, since we have δm ≠ 0. for δm = 0 the periodic roll solution can exist only if ra > rac. exploring the stability regime of cells is a demanding task, and even more difficult is the pattern selection, i.e. understating which is spontaneously chosen by system dynamics. the compression of the cells in the interior, which accompanies the enhanced cells curvature, causes the wave number in the cell center to exceed the instability boundary and leads to the formation of the dislocation pairs. the defects then travel toward the wall by climbing in the direction opposite to that of gravity acceleration. the result of this process is reduction of qm to the values less than qc and thus to the restabilization of the pattern. however, the domain walls emit new cells, which gradually re-compress the ones in the center and thus lead to persistent time dependence. 6. conclusions in this paper, dns of the rayleigh-bénard convection (rbc) is performed. in addition to the applied temperature gradient, the parallel walls are inclined and a sinusoidal temperature modulation with amplitude δm and wave number qm is introduced at the lower plate. while in the unforced rbc the heat conduction state becomes unstable at critical rayleigh number rac against convection cells with wave number qc, the forced cell solutions with wave number qm exist at any given ra. various destabilization mechanisms acting on the forced cells depend sensitively on the ratio qm/qc; here the results of simulation when this ratio is 1.2 are presented. additional spatial forcing appears at upper plate and in the horizontal direction as the walls are inclined. this opens up new possibilities, in particular when the system is exposed to the presence of nonequal forcing wave numbers imposed at the two confining plates. it should be also mentioned that even in the absence of an applied uniform temperature gradient, a pure temperature modulation leads to periodic convection patterns. acknowledgements: the authors would like to thank prof. dr. miloš jovanović for his 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serbian soc. of mechnics iconssm, palić, pp. b09-b23. 22. clever, r.m., busse f.h., 1974, transition to time dependant convection, journal of fluid mechanics, 65(4), pp. 625-645. 23. busse, f.h., clever r.m. 1979, instability of convection rolls in a fluid of moderate prandtl number, journal of fluid mechanics, 91(2), pp. 319-335. facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 23 38 https://doi.org/10.22190/fume210108021c © 2021 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper  a fracture-induced adhesive wear criterion and its application to the simulation of wear process of the point contacts under mixed lubrication condition hui cao, yu tian, yonggang meng state key laboratory of tribology, tsinghua university, china abstract. adhesive wear is one of the four major wear mechanisms and very common in almost all macro-, microor nanotribosystems. in an adhesive wear process, tiny material fragments are pulled off from one sliding surface and adhered onto the counterpart. later these fragments form loose particles or transfer between the contact surfaces. because of the topographical and physicochemical property non-uniformity of engineering surfaces, adhesive wear happens heterogeneously on the loaded sliding surfaces, and it is also discontinuous during sliding or rolling motion owing to the damage accumulation and fracture occurred inside the subsurface layers. taking account of these characteristics, a novel fracture-induced adhesive wear criterion has been proposed in this study in order to predict local wear of material in sliding. moreover, the proposed wear criterion is applied to predicting wear particle formation and morphology evolution of mixed lubricated rough surfaces during reciprocating sliding, and the simulation results are compared with the ball-on-disk experimental measurements. key words: adhesive wear, criterion, mixed lubrication, surface energy, wear particles 1. introduction wear of material is ubiquitous in almost all macro-, microor nano systems in motion. usually, it has significant unfavorable influences on operating performances and lives of devices and machinery, such as smearing or fatigue failures frequently occurring in high speed rolling bearings [1]. in some cases, wear could bring about benefits, which are not attainable by other ways, to tribosystems. for instance, an appropriate running-in wear process could lead to a super low coefficient of friction [2] or a stable low wear rate [3]. ultra-flat and ultra-smooth solid surfaces, which are required in manufacturing of modern integrated microelectronic and received january 08, 2021 / accepted february 19, 2021 corresponding author: yonggang, meng affiliation: state key laboratory of tribology, tsinghua university, beijing, 100084, china e-mail: mengyg@tsinghua.edu.cn 24 h. cao, y. tian, y. meng photonic devices, are made with the technology of chemical mechanical polishing process that needs delicate balance between corrosion and mechanical wear [4]. wear is a multiscale and multiphysics phenomenon. there are many intrinsic and extrinsic factors affecting wear behavior of materials. although scientific study on wear can be dated back to holm in 1946 [5], and a tremendous experimental and theoretical effort has been paid since then to the wear problems, it is still impossible to predict formation and evolution of wear debris particles as well as wear life of machine elements in an engineeringly acceptable accuracy. in 1995, meng and ludema [6] reviewed the published work on wear models and predictive equations, and they disappointedly concluded that none of the over 300 equations proposed could ever be found for general and practical use. most of the wear equations are empirical regression relationships between total macroscopic material loss (in terms of mass, volume or depth) and operation parameters including load and speed, such as that proposed by rhee [7], based on experimental data collected in a limited range of test conditions for a specific application, regardless of wear mechanism involved and microscopic debris formation. there are also some mechanism-based wear models; the most famous and widely accepted one of them is the archard law which states that wear volume is proportional to normal load and sliding distance and so is inverse proportion to the hardness of the softer material of contacting bodies [8]. the archard law is derived from the assumptions of adhesive wear occurring at microscopic asperity contacts of rough surfaces and hemispherical shape of wear particles, but it tells nothing about size and number of wear particles. later, a number of variants of archard law are proposed, each presenting a different expression of wear coefficient. in 1961, rabinowicz presented a surface energy criterion for loose of adherent wear particles from solid surfaces [9]. he postulated that loose of adherent wear fragments from surfaces was caused by the release of elastic energy stored in the compressed fragments from a contacted state to a non-contacted state. under the assumption of hemispherical shape of wear fragments, he derived a critical size, dl, which is proportional to the ratio of work of adhesion to hardness. adherent fragments smaller than the critical size could not get loose from surfaces. he indicated that the critical size could predict equilibrium surface roughness after sliding test, asperity junction size, the minimum load effect on wear as well as the minimum size of wear particles [10]. n. p. suh et al. developed a delamination wear model based on contact fatigue mechanism [11]. quinn et al. proposed a mechanism of surface oxidation wear and believed that excessive surface temperature rise in friction process would lead to the oxidation of surface materials. when the oxidation layer reaches a critical thickness, it would automatically loose off from the substrate [12]. in the last two decades, along with the progress in experimental techniques with micro-nanoscale resolutions [13-16], atomic simulations [17-19], discrete dislocation dynamics of crystal materials [20][21] as well as contact, damage and fracture mechanics, understanding of microscopic wear mechanisms and modeling of wear process have developed greatly. distinctions between atomic adhesive wear, plastic wear and fracture-induced adhesive wear have been identified, and the size effect on friction and wear is emphasized more and more. recent coarse grain molecular dynamics (cgmd) simulations [18] have shown that transition from plastic deformation to fractured-induced wear occurs when the junction size exceeds a critical length scale, a re-finding of the critical size concept proposed by rabinowicz based on his surface energy criterion. popov and pohrt [22] extended the rabinowicz criterion to asperity-free case and numerically modeled wear particle emission of dry contacts with the boundary element method. tan et al. proposed that asperity wear is caused by fatigue due to multiple collisions during friction, and deduced formulas for calculating the macroscopic wear a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 25 of a rough surface combined with statistical contact model [23]. however, these wear modeling and simulations neglected the effects of lubrication. undoubtedly, lubrication affects adhesive wear greatly. firstly, a little bit of adsorbed molecular on surfaces, even a small quantity of contaminants, could reduce adhesion strength of contacting surfaces significantly. secondly, most of formulated oils used in industry contain some amount of friction modifiers and antiwear additives which could form boundary lubrication films and reduce wear. thirdly, fluid film lubrication can reduce frictional heating and thus suppress temperature rise of surfaces during sliding, leading to weaker adhesion compared with dry friction. last but not least, hydrodynamic fluid film and boundary film can support a part of or even the whole of the applied load, substantially reducing or even eliminating asperity junctions. to account the effects of fluid film and boundary film on wear, wear modeling should be coupled with the theory and numerical analysis of lubrication properly. on one hand, lubrication alters wear resistance of tribopairs, while, on the other hand, wear brings about nonrecoverable changes in the shape and topography of sliding bodies, which affects lubrication film formation and hydrodynamic pressure. although the theory and numerical modeling of hydrodynamic and elastohydrodynamic lubrication have well developed since the foundation of reynolds equation in 1886, the basic assumption of no wear is still adopted in most of the lubrication simulations, even for the cases of mixed lubrication. in this study, a novel criterion for fracture-induced adhesive wear is proposed, taking into account the effects of work of adhesion and surface energy degradation on adhesive wear. moreover, the stochastic distributions of these physical properties of materials are also considered in the wear criterion. the proposed wear criterion is applied to predict the wear process of a mixed lubricated point contact in sliding motion. morphology evolution of surfaces and wear particles formation during wear process are simulated and compared with experiment measurements. 2. a fracture-induced adhesive wear criterion 2.1. assumptions of fracture-induced adhesive wear process as mentioned above, fracture-induced adhesive wear dominates when the junction size exceeds a critical length scale. from the perspective of material damage mechanism, fracture-induced wear is a process during which the surface and subsurface are gradually damaged and accumulated to a certain extent under repeated external load and frictional force, resulting in the shedding of tiny material from the substrate. generally, it can be divided into brittle damage and ductile damage. however, no matter what type of damage is, chemical bonds between atoms or molecule break, or de-cohesion occurs, and new surfaces are generated if wear of materials happens. therefore, the essence of the fracture-induced wear process can be considered as accumulation of the damages in the surface layer and subsurface layer, and finally a piece of material falls off from the matrix to form new surfaces owing to the input of a part of external mechanical energy into the frictional system. the other parts of external mechanical energy dissipate as thermal energy and other forms of energy. the surface energy degradation here is a comprehensive description of the decrease in the external work needed to generate new surfaces. its essence is that the input frictional work continually leads to nucleation and propagation of subsurface cracks, and hence the energy needed to separate local materials from the substrate decreases. for example, the micro-cracks initially generated in the subsurface layer 26 h. cao, y. tian, y. meng propagate gradually under continuous frictional work, and the slip lines in the subsurface layer gradually tighten and intersect [24], which results in the micro-wear particles naturally falling off from the substrate or are stuck away by the aid of work of adhesion. therefore, wear process is also affected by the adhesive behaviors of contact surfaces, especially for metals. therefore, work of adhesion plays a vital role in adhesive wear [25]. 2.2. a new fracture-induced adhesive wear criterion it is often observed in frictional processes that some asperities are worn off in a relatively short rubbing time while others require a relative longer period. besides the differences in local pressure acting at different asperities, it can be attributed to variations of adhesion strength and material properties over the sliding surface. hence, local wear tends to be discontinuous in space and time. moreover, wear thickness also presents strong randomness. therefore, it can be reasonably speculated that the wear of material happens in the form of removal of fragments with random thickness far greater than the single atomic layer at different time intervals under the action of friction. the shapes and sizes of wear particles display statistical distributions as shown in fig. 1. wear element fig. 1 schematic of wear particles and wear element it is worth noting the distinction between a wear particle and a wear element, a concept introduced in our model, as shown in fig. 1. wear element is defined as the basic minimum unit of wear particles. a cluster of inter-connecting wear elements constitutes a wear particle. the size of a wear element is related to the discretization mesh which should be small enough to depict surface roughness but larger than the critical length scale beyond which fracture-induced wear dominates as mentioned above. the shape of a wear element is assumed as a convex spherical crown with a projection diameter of the mesh size for simplicity. therefore, in simulations based on the wear criterion, when a wear element is formed and removed at a grid point, the surface height changes by a decrement that equals the height of the convex spherical crown at that grid point. unlike the archard law or many previously proposed wear models where wear particle size is assumed to be equal to the junction diameter, here the size of wear elements rather than wear particles is presumed. at a certain moment t during sliding, whether the element located at (x,y) on the sliding surface is worn off or not is judged by the following criterion. a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 27 c 12 c 2 1 w 0 ( , , ) 2 ( , ) ( , , ) d t c aw x y t x y p x y t u u t a    =  − −    (1) if θ≥1, the element is worn off, otherwise no wear element forms but the material degradation progresses. the criterion in eq. (1) expresses the competition between work of adhesion at the interface and the increase in surface energy accompanied with the generation of a wear element. wear occurs at the location (x,y) only when the work of adhesion exceeds or equals the increase in surface energy of the system as a result of falling off the element from the substrate. therefore, the criterion gives the necessary condition for a wear element to generate. however, the criterion is not concerned with any behavior of a wear element after it is generated. in other words, it is assumed that all generated wear elements disappear from the tribosystem after their birth. therefore, the effect of wear particles in the contact region could not be evaluated by using this criterion. in following, the parameters in criterion (1) will be described in detail. the numerator is the product of projection area, ac, and the work of adhesion per unit area w12(x,y,t) at location (x,y) in time t. if there is no adsorption or reaction films at the interface, the work of adhesion, w12, is the sum of the surface energy per unit area of the two surfaces in contact minus the interfacial energy of the two surfaces, as shown in eq. (2), where the surface energy is a material property, and the interfacial energy depends on the compatibility between the contacting bodies. factor cm denotes the magnitude of compatibility and takes a value in the range from 0 to 0.5 [26]. however, for lubricated contacts, work of adhesion changes substantially depending on the local lubrication state. in mixed lubrication, direction contacts without any intervening layer, contacts with a boundary film and contacts separated by a fluid film may co-exist on the sliding surface, and thus work of adhesion probably takes a random value in the range [w12min, w12max] as expressed in equation (3), where u means uniform distribution. ( )12 1 2 12 1 2, ( , ) ( , ) ( , ) ( ( , ) ( , ))mw x y x y x y x y c x y x y    = + − = + (2) 12 12min 12max( , ) ~ ( , )w x y u w w (3) the bracket in the denominator of equation (1) represents the monotonic degradation of surface energy from its initial value of γ(x, y), owing to the accumulation of input frictional work done during the time period [0, t]. when a wear element forms, a convex spherical crown surface and a concave spherical crown surface with the same area of aw are generated at the same time, so the new surface area in the denominator of eq. (1) is 2 times the spherical crown area. for most engineering materials, the surface energy on sliding surface is in general nonhomogeneous. here we assume that the initial surface energy per unit area obeys the normal random distribution as expressed in eq. (4), where n (γm, σγ) means the normal distribution with the mean value of γm and standard deviation σγ. m( , ) ~ ( , )x y n    (4) the integration in the bracket represents the accumulative degradation of surface energy at (x,y) under contact pressure pc, relative sliding speed |u1-u2| and frictional coefficient μc during the time period [0, t]. factor α, referred as conversion coefficient, means how much of the frictional work dissipated as the surface energy degradation. it is a characteristic parameter of 28 h. cao, y. tian, y. meng individual material, and can be estimated experimentally or by atomic simulations. methods for determination of α and calculation of pc will be described in the next section. another key parameter that does not explicitly appear in the criterion (1) but is of importance for prediction of wear volume and morphology evolution is the thickness of wear element. for fracture-induced adhesive wear, the thickness of wear element corresponds to the position within the subsurface layer where crack nucleation and propagation are most probable. considering various intrinsic and extrinsic influence factors, including the microstructures, inclusions and defects distributions in materials produced in manufacturing processes as well as working conditions during service, in this study, the thickness of wear element is assumed to yield a continuous lognormal distribution with a mean value of ∆m and a standard deviation of σ∆, as following. mln ~ (ln , ln )n   (5) in the next sections, the proposed wear criterion is applied to a reciprocating ball-on-plate experiment, and the predicted evolution of surface topography of the plate in wear process is compared with measurements. 3. simulation of wear evolution under mixed lubrication condition and comparison with experiment 3.1. simulation procedure fig. 2 shows the flow chart used to simulate the wear process of a ball-on-plate sliding friction test in mixed lubrication condition, the details of which are to be described below in section 3.4. the upper sample was a bearing steel ball with a diameter of 12.7mm, and the lower sample was a carbon steel plate. the original surface of the steel plate was a ground surface with root mean square roughness rq of 0.21 μm, while the initial surface roughness rq of the steel ball is 0.014 μm, more than 10 times smaller than that of the plate. since the ball was much harder and smoother than the plate, the roughness effect of the ball on lubrication and wear was neglected, and the wear of the ball was not accounted in the simulation. reciprocating friction simulation was carried out across a stroke (about double of hertz contact zone) compatible with the optical field size of the white light interferometer used, for comparison with measurement to be described later. the wear simulation process is a step-by-step repeat of mixed lubrication numerical analysis, updating the 3d surface topography and material properties such as surface energy at every grid point (x,y) according to the situations whether the wear element at that point breaks away the plate surface or not, judged by the wear criterion (1) after an increment of time step for wear updating. the unified reynolds equation established by hu and zhu [26] was used to deal with the mixed lubrication problems of point contacts. to explore the wear particle emission and surface topography evolution in detail, 512×512 finite difference grids, corresponding a mesh size of 1.17μm, were used in numerical simulation, which is denser than conventional mixed lubrication modeling without accounting wear particle generation. considering balance of computational efficiency with accuracy, time step δt for wear updating was set as 50 reciprocating cycles. a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 29 fig. 2 flow chart of wear simulation under mixed lubrication table 1 lists the input parameters for the simulation. the statistical parameters were selected from references [25, 27-28] for the steel materials. tests with two reciprocating frequencies, 2 hz and 0.2 hz, were performed. the lower speed test was done for experimental estimation of the magnitude of the conversion coefficient used in simulations, as described below. 3.2. estimation of conversion coefficient in this paper, an experimental method was used to approximately estimate the magnitude of conversion coefficient α in eq. (1). firstly, a low-speed friction test of the tribopair described in section 3.1 was carried out to ensure the boundary lubrication state. the test conditions were the same as those in section 3.1, except for the reciprocating frequency being set as 0.2 hz instead of 2 hz. table 2 shows the 3d topographies of the plate sample before and after the test, and the variation of friction coefficient recorded during the test. after the test, total wear volume v in a period of friction time tf of 5000 s was measured with a 3d profilometer as 8.18×105 μm3, and frictional work wf was calculated according to the friction coefficient measured during the test as 11.654 j. provided the statistical parameters for surface energy, work of adhesion and thickness of wear element are given as listed in table 1, taking the wear element size as 1.17μm, we can calculate the values of ac and aw as 1.1μm2 and 1.076μm2 30 h. cao, y. tian, y. meng respectively, and volume, vw, of a wear element with the mean thickness of 90nm is 0.0487μm3. the number of equivalent wear elements nw can be obtained as w w v n v = (6) when θ=1, the adhesive wear criterion (1) can be re-written as w m w c 12m et f f [ 0.5 ]n a a w w w w   − = = (7) substituting the values of γm and w12m listed in table 1 and the estimated values of ac, aw, nw and wf shown above, we can get an estimation of conversion coefficient α as 1.66×10-6 for the steel ball-plate tribopair. table 1 parameters for wear simulation parameters symbol value unit normal load lubricant viscosity at 40℃ elastic modulus of disc wl η0 e1 80 0.048 210 n pas gpa poisson's ratio of disc ν1 0.3 mean surface energy of disc γm 1.1 j/m2 standard deviation of surface energy root mean square roughness σγ rq1 0.055 0.21 j/m2 m elastic modulus of ball e2 210 gpa poisson's ratio of ball ν2 0.3 stroke frequency average speed ls fr ue 5 2, 0.2 10 mm hz mm/s maximum of work of adhesion w12max 0.22 j/m2 minimum of work of adhesion w12min 0 j/m2 mean thickness of wear element δm 90 nm standard deviation of thickness of wear element friction coefficient of asperity contact σδ μc 43 0.12 nm table 2 results of friction experiments 3d morphology before wear 3d morphology after wear friction coefficient curve a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 31 3.3. morphology evolution under mixed lubrication due to the reciprocating friction, each contact point within the wear track on the steel plate slides twice in a motion cycle, and the relative sliding distance is one hertz contact width. the wear area on the steel plate is only related to the number of friction cycles, being irrelevant to the actual reciprocating strokes. therefore, in the wear simulation, the friction time is expressed as the number of cycles, and the sliding stroke is selected to be 800μm, matching with the optical field range in the white light interferometery measurement. figs. 3 and 4 show the thickness of primitive wear elements and surface energy distributions over the calculation domain, respectively. fig. 3 thickness distribution of basic wear element fig. 4 surface energy distribution on the surface fig. 5 shows the comparison of the evolution of 3d surface morphologies at different wear stages. it can be seen that with the increase of friction cycles, the portions with higher peaks decreases, and the wear marks become more and more clear. the original surface texture gradually fades, and new surface texture feature appears. 32 h. cao, y. tian, y. meng fig. 5 evolution of 3d morphology during friction process: (a) original surface (b) after 200 cycles (c) after 400 cycles (d) after 600 cycles (e) after 800 cycles fig. 6 shows that the profiles along the sliding direction in the middle section evolve with the sliding cycle. just like in fig. 5, it can be found that the rough peaks are gradually worn off during the wear process and the surface fluctuation presents evident randomness. in addition, it can be observed that at a fixed position, the height changes with great randomness, for the reason that the thickness of wear element was assumed to be stochastic in the modeling, which is in accord with observations in common wear surfaces. fig. 6 2d profile evolution during friction process fig. 7 shows the change of cumulative wear volume during the wear process. it can be seen that with the increase of friction cycles, the slope of cumulative wear volume decreases gradually, indicating that the wear rate decreases gradually. the reason is that the solid bearing ratio keeps decreasing, which leads to a decrease in wear rate. in addition, a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 33 with the decrease of rq, the difference between peak and valley of surface height decreases, resulting in a gradual decrease in pressure peak and frictional work at asperity junctions. therefore, the shedding time of wear element increases and the wear rate decreases. fig. 7 accumulative wear volume with friction cycles fig. 8 shows the change of rq of the surface in the wear track during the wear process. it can be seen that with the increase of friction time, rq decreases from 0.21μm to 0.13 μm, and the rate of decline gradually decreases, approximately reaching a stable state. this is because the peaks are apt to be worn away during the wear process. it should be noted that although the surface roughness approaches a stable value of 0.13 μm, it is still higher than the mean thickness of wear elements presumed in the simulation. this indicates that the set value of mean thickness of wear elements does not affect the roughness of worn surface remarkably. this is displayed more clearly in fig. 9, where three different values, 45nm, 90nm and 150nm, of the mean thickness of wear elements are set in wear simulations. it can be seen from fig. 9(a) that when δm is 45 nm, the surface morphology after 200 cycles does not deviate from the initial one too much. after 800 cycles, however, a new random surface profile forms, which is quite different from that of the virgin surface. fig. 8 evolution of rq in wear track with frictional cycle 34 h. cao, y. tian, y. meng fig. 9 comparison of 2d profile evolution for different δm: (a) δm=45nm; (b) δm=90nm; (c) δm=150nm from figs. 9(b) and 9(c), it can be found that with the increase in the value of δm, the morphology of worn surface rapidly changes apart from the initial topography. the original flat surface becomes more and more concave with the wear time, and the larger the δm, the deeper the wear track for the same period of friction time. fig. 10 shows accumulative wear volume for the different values of δm under the same test condition. it can be seen that wear rate decrease with friction time and wear volume slightly deviates from the linear relationship between the wear volume and the number of cycles. this can be explained by the evolution of surface roughness and severeness of asperity contact under mixed lubrication during the wear process. from fig. 8, we can see that the root mean square roughness becomes smaller with friction cycles, which leads to an increasing film thickness ratio, or, in other words, to a better lubrication condition. in addition, from figs. 6 and 9, we could see that surface height decreases and gradually forms a concave shape of the plate surface with the friction cycles. this means that the initial ball-on-flat nonconformal concentrated contact gradually changes to a more or less conformal contact with a wider and wider contact area, leading to wider but lower contact pressure. from eq. (1), we can find that if pc decreases, it cost a longer t to satisfy the condition of θ>1. a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 35 fig. 10 wear volume vs. frictional cycle for different mean thickness of wear element 3.4. comparison of wear process between experiments and simulations a reciprocating friction and wear tester (rtec instrument, usa) was used to conduct in-situ measurement of evolution of three-dimensional surface morphology during wear under mixed lubrication condition. the test diagram is shown in fig. 11. the upper sample is a bearing steel ball with diameter of 12.7mm, the rms surface roughness is 0.014μm, and the vickers hardness is 810hv. the lower sample is a carbon steel plate with rms surface roughness of 0.16μm and vickers hardness of 395hv. fig. 11 diagram of friction and wear test the experimental procedure was as follows. firstly, the sample table with the lower sample installed was moved to the position under the white light interferometer module, and the initial 3d surface morphology of the sample was recorded. then the sample table was moved along the linear guide rail to the friction module position, and a certain amount of pao lubricating oil was dropped at the sample surface to be tested. after the upper sample touched with the lower sample, it was loaded to the set load of 80 n, and then the reciprocating sliding was started. the sliding friction was suspended every 100 reciprocating cycles. the upper sample was lifted and it gets rid of contact with the lower sample surface. the residual oil on the surface of the lower sample was scrubbed with acetone and dried with compressed nitrogen gas. then, the cleaned sample was shifted to the position under the white light interferometer. after the morphology measurement, the sample table was translated back to the friction test position along the guide rail, 36 h. cao, y. tian, y. meng re-lubricated the sample, applied the normal load, and re-started the friction test at the same reciprocating motion speed. the stroke of the motion was 5 mm, and the frequency set at 2 hz. 10 times magnification objective lens was used for wear morphology measurement. the advantage of the start-stop friction and wear test protocol is to avoid the detachment of the samples for measurement of wear morphology; hence no need was there to re-install the specimen, which may cause misalignment problems between two successive test phases. fig. 12 shows the comparisons of cross-sectional profiles between the experimental and the simulation results for the friction cycles of 300, 600 and 900. it can be seen that the consistence between the simulated wear profile and the experimental measurements is reasonable, and major features of the predicted and measured topographies are in accord with each other. fig. 12 comparison of morphology evolution between experiments and simulations however, there are some discrepancies between the experimental and simulation results, as shown in fig. 12. both the mean lines and fluctuation magnitudes of the predicted profiles do not fit in very well with the experimental ones for a given friction cycle. there are several possible reasons for the discrepancies. among the reasons are inaccurate inputs of the parameters of surface energy, work of adhesion, thickness of wear element and the conversion coefficient used in the simulations. secondly, in the wear model, it is assumed that only adhesive wear happens and the wear particles would escape from the contact area quickly as soon as they formed. however, in a real wear process, some of wear particles may trap at the contact area, resulting in abrasive wear. third, some of adhesive wear elements would attach on the counterpart surface till loosing from the surface later, these transferred materials would change the morphology and work of adhesion, which are not considered in the wear simulation. further discussions on the limitations of the proposed wear criterion are given below. 4. discussions differing from the archard law or other previous wear equations which relate total loss of material in terms of volume, mass or depth with material property and working condition explicitly, the wear criterion proposed in the study provides only a necessary condition for a wear element to generate. the loss of material at certain friction period is implicitly expressed by summation of the volume of all wear elements which satisfy the a fracture-induced adhesive wear criterion and its application to the simulation of wear process... 37 necessary condition during that period, provided that the probability density function of the thickness of wear elements is known. another characteristic of the wear criterion is that it accounts the material degradation owing to frictional work. stochastic feature of local wear observed in practice is also accounted by introducing random distributions of the parameters of surface energy, work of adhesion and the wear element thickness. the wear criterion in eq. (1) can apply to analyzing the wear of both bodies, a and b, in contact. in that case, the surface energy in the denominator in wear criterion (1) takes as the value of the surface energy a or b respectively, while the numerator and the term of frictional work in the denominator for body b are the same as those for body a. both rabinowicz’s wear particle size equation and our local wear criterion include the term of work of adhesion and thus emphasize its vital role in adhesive wear. in this sense, they are similar. but there are several distinctions between them. first, as described in introduction of the manuscript, the rabinowicz equation was derived from the assumption that a loose wear particle forms when the release of elastic deformation energy stored in a particle overcomes the cost of the increase in energy due to the detachment of that particle from the attached surface, which equals to the work of adhesion. in the derivation of our criterion, however, we based on the material degradation mechanism as expressed in the bracket in the denominator of eq. (1), which comes from fatigue and fracture mechanics, rather than the elastic energy restoration mechanism as rabinowicz used. in consequence, it needs a damage accumulation or material surface energy degradation stage [0, t], as expressed in the integration of frictional work over the period [0, t] in the denominator, for a wear element to form, while such a degradation stage is not needed by rabinowicz equation. secondly, rabinowicz assumed that the shape of particles is hemisphere for the sake of simplicity in mathematical derivation. in our model, we did not make any assumptions on the size and shape of wear particles. instead, we assumed the shape of wear elements is spherical crown for the sake of simplicity to estimate ac and aw in equation (1). the size and shape of wear particles are determined by the connectivity of wear elements. thirdly, we introduced probability distribution functions to the material parameters of work of adhesion w12 and surface energy γ, while rabinowicz equation is deterministic. however, it should be mentioned that the wear criterion in eq. (1) does not tell us any information of the thickness of wear elements, which was necessary and assumed in the simulations presented above. a detailed microscopic analysis of damage initiation and propagation is needed to reveal where fracture of material most probably occurs and their statistic distribution for a macro-tribosystem. 5. conclusions based on the understanding of adhesive wear mechanism, a novel fracture-induced adhesive wear criterion has been proposed. the effects of work of adhesion and surface energy degradation on adhesive wear are taken into account in the criterion. moreover, the stochastic distributions of these physical properties of materials are also considered. coupled with the deterministic mixed lubrication theory, the wear criterion has been applied to predicting the wear process of a mixed lubricated point contact in sliding motion. it has been shown that morphology evolution of surfaces and wear particles formation during wear process can be simulated by using the proposed wear criterion and simulation procedure. comparison of simulation results with experiment measurements has been done, and the agreement between them is reasonable. 38 h. cao, y. tian, y. meng acknowledgements: this work was supported by the national natural science foundation of china (nsfc) under grant no. 51635009 and the ministry of industry and information technology, china, with grant number jszl2017213b002. references 1. harris, t.a., kotzalas, m.n., 2007, rolling bearing analysis (5th edition): advanced concepts of bearing technology, taylor & francis. 2. erdemir, a., martin, j.m., luo, j., 2020, pradeep, n., superlubricity (2nd edition), elsevier. 3. zhang, y., kovalev, a., meng, y., 2018, combined effect of boundary layer formation and surface smoothing on friction and wear rate of lubricated point contacts during normal running-in processes, friction, 6, pp. 274–288. 4. zhao, d., lu, x., molinari, j.f., 2013, chemical mechanical polishing: theory and 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et al., 2000, simulation of wear in combustion engines, computational materials science, 19, pp. 285-291. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 181 192 interpreting the meaning of geometric features based on the similarities between associations of semantic network  udc 004.8 milan trifunović, milos stojković, miroslav trajanović, dragan mišić, miodrag manić university of niš, faculty of mechanical engineering, niš, serbia abstract in this paper the method of semantic network analysis which enables semantic network data categorization is presented. the main goal of the network analysis is determination of the similarity of the semantic network associations based on their attribute values. the method allows highly efficient semantic categorization of new concepts, which does not depend on pre-planned inputs and predefined inference rules. also, the method allows different semantic interpretations of the same concept in different semantic contexts. as a specific example for demonstration of the semantic categorization process, a part of mold manufacturing workflow is chosen. key words: artificial intelligence, semantic features, cognitive data processing, active semantic model, manufacturing 1. introduction the semantic categorization as a term from cognitive psychology is used to express the process of assigning meaning to a (new) item [1, 2]. with the semantic data models, this process is usually performed by setting up a semantic relation between the concepts (nodes) in the semantic network. semantic categorization represents one of the biggest challenges faced by modern information technologies. computer networking in the internet has enabled an instant access to huge amounts of data and information. however, it is often rather difficult to find appropriate information, namely the one for which the user  received december 3, 2013 corresponding author: milan trifunović faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: milant@masfak.ni.ac.rs acknowledgements. this paper is part of the project "the research of modern non-conventional technologies application in manufacturing companies with the aim of increase efficiency of use, product quality, reduce of costs and save energy and materials" (project id tr35034), funded by the ministry of education, science and technological development of republic of serbia. 182 m. trifunović, m. stojković, m. trajanović, d. mišić, m. manić searches. the information access quality does not equally follow increases of available information [3]. research in cognitive psychology suggest that the process of searching information contents should include the search of meaning (or according to the meaning) of what is required in order to conduct it in a significantly more precise and even faster way [4, 5]. following these recommendations, the current research trend in the field of information technologies includes two main subdirections which complement each other: a) research and creation of semantic data models and b) research and modeling of cognitive processes. a large piece of knowledge is already embedded into modern computer aided product development (capd) systems. actually, parts of mathematics, physics, chemistry, and other fundamental sciences are turned into algorithms as well as programming code of these systems. on the other side, tacit knowledge [6] with its origin in experience and intuition as well as its liability to permanent updating and changes through a direct touch with the product development process, is often very difficult to turn into strongly defined algorithms. this kind of knowledge has stayed, more or less, out of capd systems. nevertheless, it plays a crucial role in making important assessments, judgments, inferences and decisions. the tacit knowledge products have a huge impact on the strategic product development process features. exploration of the methods for embedding and using tacit knowledge is an important and unavoidable evolutionary step forward in the capd systems development and, certainly, one of the greatest challenges in the field of it application in the product development process. in this paper an approach is described where a new semantic model named active semantic model (asm) is used for modeling and semantic interpretation of geometry elements in a cad model. the asm structure and the method are explained and demonstrated in a given case. the focus is on the method description. 2. active semantic model asm [7] is a new semantic model which has been developed in-house. its primary aim is to capture and interpret semantics of the design features related to manufacturability issues [8]. asm intends to introduce an alternative approach to knowledge representation in comparison with the existing semantic models by moving the focus of data structuring from concepts to semantic relations or associations (the term which is used in asm). this idea of structuring the meaning in associations is chosen to support the thesis stating that the knowledge that people have about items (visual representations, objects, situations, etc.) is contained in inter-concept associations that abstractly represent those items [9]. furthermore, asm has proved itself as a more flexible and productive in capturing and interpreting semantics of data compared to the existing semantic models [7]. here, we explain an algorithm for determining the similarity of associations as a core process in determining semantic inter-concept similarity. 2.1 structure the asm structure consists of: concepts, associations between concepts, concept bodies, and contexts. interpreting the meaning of geometric features based on similarities between associations of semantic network 183 asm concepts are at the same time nodes of the asm semantic network and abstract representations of objects, features, situations, etc. the concept data structure consists of only one parameter – name. concept bodies are their realizations. the asm association structure is characterized by eleven parameters [7]: names (cpti, cptj) of two concepts (or contexts) that are associated (network nodes are part of association); topological parameters: roles (ri, rj) of concepts (i.e. type, subtype), type (t) of associating (i.e. classifying), direction (d) of associating (←, ↔, →) and character (c) of associating (+, -); weight parameters: accuracy (h) of associating for given context (0; 0.25; 0.5; 0.75; 1) and significance (s) of associating for given context (0; 0.25; 0.5; 0.75; 1); and affiliation parameters: context id to which an association belongs, and user id to identify who has created the association. there can be only one concept with given name, but there can be many associations belonging to different contexts associating it with other concepts. type of association states the relationship between the two concepts, i.e. the role of each concept in the association. in this way the values for parameters roles of concepts and type of association are coupled and can be considered as ordered triplets. asm also introduces contexts which are sets of semantically close associations. the context brings abstract meaning of a certain object, situation or event, and, therefore, it is semantically designated. general context is defined and built into asm structure independently of the user. as for others, particular contexts require additional semantic description by the user. all the associations from particular contexts are assigned (usually with different parameters) to the general one. association plexus in asm is, in general, a context subset (mathematical structure) and can be considered without specific abstract meaning. this difference between association plexus and the context has no effect on data processing and their use for the data semantic interpretation in asm. the asm structure is not domain-specific and can be used for knowledge representation in diverse fields. the knowledge from specific domain should be represented through context(s). semantic relations between contexts allow knowledge from one context to be applicable to others. 3. interpreting the semantics of data in asm cognitive data processing (cdp) algorithms represent a set of data processing procedures with which asm attempts to perform data semantic interpretation. they enable asm to acquire new knowledge independently and make meaningful decisions and responses. this group of algorithms consists of: 1. the procedure of determining: a. the class of similarity of associations, and, b. the degree of semantic similarity of concepts 2. the algorithm for determining the similarity of associations (dsa) (the core of this algorithm is the procedure for determining the parameters of association (dpa)) 3. the algorithm for determining the similarity of contexts (dsc) 4. the algorithm or the procedure of contexts upgrading (cup) 5. the algorithm for the creation of heuristics and knowledge "crystallization." 184 m. trifunović, m. stojković, m. trajanović, d. mišić, m. manić 3.1. procedure for determining association parameters determination of semantic similarity of two concepts is usually performed for the concepts which are not directly connected by associations (cptx and cptn), i.e. in the cases when there exists at least one layer of concepts between cptx and cptn known to asm (layer of "connectional" concepts cpti (fig. 1)) through which they are connected. pairs of associations through which concepts cptx and cptn are connected to the same connectional concepts cpti are called common associations (fig. 1). the process of determining semantic similarity of two concepts results in the creation of association(s) between these two concepts. the parameters (type, direction, character, accuracy, and significance) of this (these) association(s) are determined through the procedure of determining association parameters. concepts cptx and cptn which are directly connected with one or more associations (fig. 1) are already semantically categorized. however, determination of semantic similarity for these concepts is possible and needed after every modification of semantic network (after the creation of new associations). fig. 1 schema of semantic relations and concepts with illustration of used terminology the association parameters values, and, therefore, the semantic features of a new concept, generally depend on the set of parameter values of all the association pairs. the associations parameters also depend to a large extent on the contexts containing these associations and semantic elements (concepts and contexts) connected by them. in other words, two concepts can be in completely or partially different semantic relations, depending on the context wherein they are observed. the characteristic of each association in terms of its belonging to a certain context implicitly connects that context with members of this association (concepts or contexts that are connected by this association). this implicit connection between association members and context to which this association belongs is determined by the values of association parameters that are appropriate for given context (fig. 2). therefore, it can be stated that the association parameters values, and, consequently, the new concept’s semantic features in the given context, depend on the set of parameter values of all the association pairs that belong to that context. cptx cpti cptn cpti+3 cpti+2 cpti+1 asm space common associations connectional concepts direct association known concept new concept interpreting the meaning of geometric features based on similarities between associations of semantic network 185 fig. 2 connection between semantic relations (associations) and sets of semantically close associations (contexts) in order to take into account the influence of accuracy and significance of associations, as well as relevance of associations in accordance with context, asm, in all the cases, determines the values of the association parameters between concepts cptx and cptn as follows: 1. asm determines parameters of association for general context (ctx0): a. asm separates a subset of accurate and significant associations of general context. all association pairs that belong to general context whose product of arithmetic mean values of accuracy and significance is smaller than 0.25 are excluded from further process of determining the association parameters values. b. asm performs the procedure of determining the values of parameters of association according to the established algorithm (which will be described in detail later in this section). c. asm creates association between concepts cptx and cptn with the determined values of association parameters. 2. asm determines parameters of association for every particular context (ctxs) which includes associations and concepts connected by them: a. in the first step of refinement for every particular context, asm takes into account the whole set of associations by which concepts cptx and cptn are connected, including associations that are excluded in previous step. asm excludes pairs of associations in which associations belong to different contexts. furthermore, asm excludes a subset of association pairs in which associations belong to the general context, but keeps their variations in particular contexts. at the end of this step, only association pairs in which both associations belong to the same particular context and association pairs in which associations belong only to general context are kept. b. asm groups pairs of associations according to specific context. for every subset, grouped according to specific context, asm selects a subset of accurate and significant association pairs. all association pairs that belong to specific context whose product of arithmetic mean values of accuracy and significance is smaller than 0.25 are excluded from further process of determining the association parameters values. cpt1 cpt2 | r1 | t | c | s | h | d | r2 | ctx1 ctx2 ctx3 { | r1 | t | c | s | h | d | r2 | }ctx3 { | r1 | t | c | s | h | d | r2 | }ctx1 { | r1 | t | c | s | h | d | r2 | }ctx2 186 m. trifunović, m. stojković, m. trajanović, d. mišić, m. manić c. asm performs the procedure of determining the values of parameters of associations according to the established algorithm for every subset of association pairs. d. asm creates associations between concepts cptx and cptn with determined values of association parameters for every specific context separately. 3.1.1 the procedure of determining the type of association there are three characteristic cases that determine what type will be assigned to the association created during the categorization: 1. asm determines complete match of all parameters of all association pairs (fifth class of similarity). in this case asm creates an association of "synonymous" type between these two concepts (cptx is synonym for cptn). 2. asm determines that new concept cptx and known concept cptn have associations of at least the same type toward connectional concepts cpti. in other words, these associations can be of the same or different character, direction, accuracy and significance. therefore, similarity of at least first class is achieved. in this case asm creates association of "similarity" type between these two concepts (cptx is similar to cptn) (fig. 3). fig. 3 determination of the type of association for the case of the second, third or fourth class of similarity 3. asm determines that concepts cptx and cptn share associations of zero class of similarity with concepts cpti. hence, in this case, concepts cptx and cptn have associations of different types with the same (connectional) concepts cpti, i.e. roles of concepts cptx and cptn in these associations are different. the type of association that will connect concepts cptx and cptn will be determined depending on the class of association plexus in which new concept cptx is found. for example, an easily recognizable class of association plexus is the one that decptx cptk cptn end mill milling cutter face mill txk tnk txn = similarity asm space step 1: description – association created by user existing association step 2: categorization – association created by asm [subtype] [similar concept] [type] [subtype] [type] [similar concept] cptj milling [activity] [activity] [argument] [argument] step 3: knowledge widening interpreting the meaning of geometric features based on similarities between associations of semantic network 187 scribes some activity (fig. 4). in this kind of plexus, besides the concept that represents the activity itself, there exists motive for performing that activity, activity subject or subjects (who is performing the activity), activity object or objects (against which the activity is performed), activity argument (with which the activity is performed), product, and result of activity. therefore, it is sufficient to introduce part of association plexus to enable asm to recognize the class of association plexus for the description of activity. in the next step asm starts directed search procedure over the semantic network in order do determine the rest of the associations from the plexus. this procedure is intended for pre-planned types of plexuses (association plexuses that describe activity, assembly, etc.). this procedure is based on the case algorithm which contains explicitly defined causal mechanism (rule) for every pre-planned case. fig. 4 association plexus describing the activity "surface milling" 3.1.2 the procedure of determining accuracy and significance of association the values for accuracy and significance of the association being created between concepts cptx and cptn are determined according to the following equations: 1   1      k xi nii xn s s s k (1) 1   1      k xi nii xn h h h k (2) where k is the number of association pairs formed between concepts cptx and cptn and the same concepts cpti from the semantic network. cptx cptk cptn аsm oval surface surface milling txl tnl txn = subject-activity asm space [concept] [subject] [motive] [result] [object] [activity] cptj milling [argument] [part] [assembly] [product] cptl cptm cptp cptq cptr [attribute] operation op10 surface identifying nc sequence 188 m. trifunović, m. stojković, m. trajanović, d. mišić, m. manić since association accuracy and significance take values from a finite set of standard values (0; 0,25; 0,5; 0,75; 1) for the sake of simplicity, it is necessary (though not required) to adopt values for these parameters that will be different from the calculated values. these values are adopted as follows: table 1 adopting the values for accuracy and significance of association conditional part result sxn ≥ 0.5 sxn = first bigger standard value sxn < 0.5 sxn = first smaller standard value hxn ≥ 0.5 hxn = first bigger standard value sxn < 0.5 hxn = first smaller standard value sxn = 0 or hxn = 0 remove acptx↔cptn 3.1.3 the procedure of determining the character of association determination of the association character, in the case when there is only one association pair between concepts cptx and cptn, is carried out using the following matrix (which is illustrated in fig. 5): table 2 matrix for determining the character of association (only one association pair) cxk cnk cxn + + + +     +  +  fig. 5 typical schema of associations during the determination of association character (only one association pair) in the case when there are multiple association pairs (and, consequently, more connectional concepts) between concepts cptx and cptn, the function for determination of the character of association which should be created between concepts cptx and cptn varies depending on the case: cptx cptk cptn fillet sharp edge оval cxk = + cnk =  cxn =  asm space association created by user existing association association created by asm interpreting the meaning of geometric features based on similarities between associations of semantic network 189 1. when there are multiple association pairs between concepts cptx and cptn which have the same characters. in this case asm creates association of positive character between concepts cptx and cptn. concepts cptx and cptn associate the same connectional concepts cpti in the same way – affirmative or nonaffirmative. 2. when there are multiple association pairs between concepts cptx and cptn which have different characters. in this case asm creates association of negative character between concepts cptx and cptn. if all association pairs have the same types, then asm creates the association of negative character and "antonymous" type. 3.1.4 the procedure of determining the direction of association determination of the association direction in the case when there is only one association pair between concepts cptx and cptn is carried out using the following matrix (which is illustrated in fig. 6): table 3 matrix for determining the direction of association (only one association pair) dxk dnk dxn                            fig. 6 typical schema of association during the determination of association direction (only one association pair) in the case when there are multiple association pairs (and, consequently, more connectional concepts) between concepts cptx and cptn the function for determination of the direction of association which should be created between concepts cptx and cptn is also defined according to the matrix: cptx cptk cptn fillet оval dxk =  dnk =  dxn =  asm space association created by user existing association аssociation created by asm oval surface 190 m. trifunović, m. stojković, m. trajanović, d. mišić, m. manić table 4 matrix for determining the direction of association (multiple association pairs) dx1 d1n dx2 d2n dx3 d3n dxn                 … … … … …   … … … … … 4. case a simplified example of the mold manufacturing workflow will be used for the demonstration of the approach. instead of usual mold cooling canals, the tool constructor has planned conformal cooling canals. the usual mold cooling canals are straight, have a circular cross section, and are manufactured by drilling (deep drilling). the conformal cooling canals extend on a spatial curve (trajectory), and usually have a circular cross section. by following the form of lateral ribs, the conformal cooling canals can approach certain parts of the mold and provide a more complete air outlet. however, the conformal cooling canals, because of their shape, can not be manufactured by drilling. therefore, the regular technological process, which implies manufacturing of cooling canals by drilling, cannot be performed. thus, an exception related to the mold geometry causes disturbance of manufacturing process workflow, preventing the regular technological process. the scenario that shows the asm application in this situation is the following: the asm semantic network already contains a semantic description of the following concepts: opening, hole, canal, cooling canal, straight line, curve, trajectory, circular cross section, drilling, etc. some concepts are part of the "mold" context, while the others belong to the general context. during the introduction of the concept "spatial curve" dpa algorithms will (by using the case algorithm presented in fig. 7) connect concepts "spatial curve" and "straight line" with the "subtype-type" association, whose character will be harmonized with the character of the association between concepts "curve" and "straight line" (negative character). this means that these two concepts are associated nonaffirmative, i.e. one excludes the other. the next association pair that will be processed by dpa algorithms is "spatial curve"↔"straight line"↔"axis". this pair will induce asm to conclude that "spatial curve" and "axis" are similar, but again with negative character. in the last iteration of reasoning asm will connect concept "spatial curve" with concept "drilling" modeled on the association between concepts "axis" and "drilling" with the association of negative character (fig. 8). fig. 7 the schema of decision making for the case with one "similarity" type association cptx cptk cptn asm space [subtype] [subtype] [type] [similar concept] [similar concept] [type] interpreting the meaning of geometric features based on similarities between associations of semantic network 191 fig. 8 part of the asm semantic network representing geometric and technological features of the mold cooling canal. green associations are added based on the analysis of blue associations in the following reasoning iterations, negative character of the association between concepts "spatial curved opening with circular cross section" and "straight opening with circular cross section" will be reflected, in a similar way, on semantic relation between the concepts "spatial curved opening with circular cross section" and "drilling". finally, negative character of the association between concepts "spatial curved opening with circular cross section" and "drilling" will be reflected on semantic relation between the concepts "mold conformal cooling canals" and "drilling" (fig. 8). canal subtype type opening mold cooling canal subtype type h=0.5 hole similar concept similar concept h=0.5 straight opening with circular cross section subtype type h=1.0 circular cross section bottom part assembly axis concept attribute part assembly c =  mold cooling opening subtype type attribute drilling drill product activity argument h=1.0 h=1.0, s=1.0 h=1.0 h=1.0 vent synonym synonym h=1.0 h=0.75 curve spatial curve mold conformal cooling canal spatial curved opening with circular cross section attribute attribute h=1.0 h=1.0 subtype type h=0.75 synonym synonym h=1.0 concept s=1.0 straight line subtype type c =  similar concept similar concept type subtype c =  c =  c =  attribute concept product activity c =  c =  product concept concept similar concept subtype similar concept c =  product activity type subtype similar concept similar concept c =  concept similar concept 192 m. trifunović, m. stojković, m. trajanović, d. mišić, m. manić 5. conclusion active semantic model (and its procedures of cognitive data processing) have succeeded in carrying out the first three steps of semantic interpretation autonomously: 1. introduction – by the initial associating of a new set of data with the data that exist in the semantic network of asm, 2. recognition – by determining of similarity of associations between the new and the known elements of the asm’s semantic network, 3. categorization – by generating associations between the new and the known elements of the asm’s semantic network, the developed functionalities of asm provide for new possibilities for the autonomous generation of relatively precise and useful assessments and predictions in the beforehand unpredicted or insufficiently precise defined input data sets. references 1. deutch, j. a., deutch, d. , 1963, attention: some theoretical considerations, psychological review, 70, 80-90. 2. norman, d. a., 1968, toward a theory of memory and attention, psychological review, 75, 522-536. 3. krill p., 2000, overcoming information overload, infoworld. 4. peterson, l.r., peterson, m.j., miller, g.a., 1961, short term retention and meaningfulness, canadian journal of psychology, 15, 143-147. 5. kostić, a., 2006, kognitivna psihologija, zavod za udžbenike i nastavna sredstva, beograd. 6. nonaka, i., takeuchi, h., 1995, the knowledge creating company: how japanese companies create the dynamics of innovation, oxford university press, new york. 7. stojkovic, m., manic, m., trifunovic, m., misic, d., 2011, semantic categorization of data by determining the similarities of associations of the semantic network, e-society journal research and applications, 2, 3-13. 8. stojkovic, m., 2011, analysis of the manufacturability parameters based on semantic structures of the digital product model, phd thesis, university of nis, faculty of mechanical engineering in nis, nis, serbia. 9. anderson, j.r., bower, g. h., 1973, human associative memory, winston, washington, dc. interpretacija značenja geometrijskih odlika bazirana na sličnosti asocijacija semantičke mreže u radu je predstavljen koncept analize semantičke mreže koji omogućava semantičku kategorizaciju podataka. glavni cilj analize je utvrđivanje sličnosti asocijacija semantičke mreže na osnovu sličnosti vrednosti atributa asocijacija. koncept omogućava efikasnu semantičku kategorizaciju novih pojmova, koja ne zavisi od unapred planiranih ulaza i definisanih pravila zaključivanja. pristup omogućava i različite semantičke interpretacije istog pojma u različitim semantičkim kontekstima. kao primer za demonstraciju procesa semantičke kategorizacije je iskorišćen deo radnog toka izrade kalupa. ključne reči: veštačka inteligencija, semantičke odlike, kognitivna obrada podataka, aktivni semantički model, izrada 12775 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240730046k © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper comprehensive distance-based ranking method for evaluating hydraulic converters in tidal stream turbines utilizing picture fermatean fuzzy set daekook kang1, krishnan suvitha2, samayan narayanamoorthy2 1department of industrial and management engineering, inje university, gyeongsangnam-do, republic of korea 2department of mathematics, bharathiar university, coimbatore, india abstract. in recent years, advancements in hydrokinetic technology and the growing demand for renewable energy have heightened interest in water-based energy extraction. this study proposes a new fuzzy information representation technique, probabilistic picture fermatean fuzzy sets (ppffss), to investigate the selection of hydrokinetic energy harnessing technologies (heht) for various marine and river-based applications. to address the complexity of multiple criteria and alternatives, we developed a new integrated multi-criteria decision-making (mcdm) model that incorporates logarithmic percentage change-driven objective weighting (lopcow) and comprehensive distance-based ranking (cobra) approaches. we define ppffs as the definition of basic functions and the provision of information scoring and accurate operational processes. according to the results, the efficiency factor is the most important criterion, and the seagen tidal stream turbine outperforms all other tidal turbines. the findings are supported by experimental data from the heht and a comparison of how two hydrokinetic energy converters can improve efficiency. as a result, hydrokinetic systems are one of the greatest sustainable energy solutions for distant communities and small-scale applications. key words: multi criteria decision making, hydraulic converter, tidal stream turbine, picture fermatean fuzzy set, comprehensive distance based ranking 1. introduction tidal energy has a global capacity of 120 gw and is capable of generating up to 150 twh annually [1]. unlike other renewable energy sources, wave energy offers long-term received: july 30, 2024 / accepted november 08, 2024 corresponding author: samayan narayanamoorthy department of mathematics, bharathiar university, coimbatore, india e-mail: snmphd@buc.edu.in mailto:snmphd@buc.edu.in 2 d. kang, k. suvitha, s. narayanamoorthy predictability, abundant resources and high energy density, albeit with high maintenance costs. developing an effective and cost-effective wave energy device suitable for various water flow conditions has emerged as a significant challenge [2]. the decision to use hydraulic tidal turbines was a significant step in the development and implementation of tidal energy systems. wave energy, as a sustainable and renewable energy source, has attracted worldwide attention because of its sustainability and potential contribution to the global energy system [3]. however, choosing the right tidal turbine requires careful consideration of various aspects, including cost, efficiency, resilience, environmental impact and maintenance concerns. these criteria are often intertwined and subject to multiple uncertainties [4]. since the construction of a horizontal-axis tidal stream turbine is comparable to that of a horizontal-axis wind turbine, many of its blades resemble wind turbine blades. currently, tidal stream turbines are primarily developed using blade element momentum (pem) theory, which is also used to investigate their hydrodynamic properties [5]. optimization is usually required following the initial design phase to maximize the operating efficiency of tidal stream turbines. this optimization often uses a genetic algorithm, focusing mainly on the distribution of blade chord length, turning angle, and thickness [6]. it involves progressively more sophisticated methods that can efficiently manage ambiguous and imprecise data [7]. fuzzy set theory has emerged in recent years as a useful technique for dealing with uncertainty in mcdm situations. however, due to ambiguity and uncertainty among decision makers (dms), it is challenging and confusing to evaluate the performance of each choice and choose the best one for mcdm difficulties [8]. fermatean fuzzy set (ffs) differs from other extensions of fuzzy set theory by its greater flexibility in describing fuzzy and imprecise information. ffs builds on the concept of intuitionistic fuzzy sets by including a parameter that allows subtle expressions of hesitancy [9]. this paper presents a new method for selecting hydraulic tidal turbines, combining comprehensive distance-based ranking (cobra) and logarithmic percentage change driven objective weighting (lopcow) with probabilistic fermatean fuzzy sets. while the cobra technique best combines multiple criteria and produces a compromise solution that balances opposing objectives, the lopcow approach is widely recognized for its ability to manage quantitative and qualitative criteria [10]. building on the strengths of both methods, the proposed picture probabilistic fermatean fuzzy set-based lopcow-cobra strategy creates a robust and all-encompassing decision framework. the technique effectively integrates the uncertainty and partial truth values associated with various criteria, including probabilistic ffs [11]. the ppffs is a novel extension of the traditional fuzzy set, incorporating both the probabilistic and fermatean fuzzy paradigms. this combination allows ppffs to capture higher levels of uncertainty and ambiguity compared to standard fuzzy sets and other extensions. in particular, ppffs improves the ability to model and analyze situations where information is not only imprecise but also inherently probabilistic. by integrating the probabilistic component with fermatean fuzzy sets, ppffs provides a more fine-grained representation of uncertainty that accommodates both membership degree and non-membership degree with an additional layer of probabilistic uncertainty. the dual nature of ppffs allows for a more robust decision-making framework, especially in data-scarce or highly variable environments. ppffs can be adapted to various application domains, making it a versatile tool for complex decision-making scenarios. comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 3 a comprehensive mcdm technique, the cobra method, combines multiple criteria into a compromise solution. this method balances the requirements to identify the optimal turbine alternative by carefully weighing the trade-offs between several performance parameters [12]. by using the pffs and cobra methods, this work aims to provide a systematic and rigorous approach to hydraulic tidal turbine selection. we provide a robust framework that improves decision-making in the face of uncertainty while improving the validity and reliability of the selection process. a case study illustrates how the suggested technique is applicable and practical in real situations. as stated earlier, these methods are equivalent to ranking alternatives using distance from an established reference point. it is useful to find out which of these methods is better than using euclidean distances to calculate the distance between positive and negative ideals. so far, the literature has not established a methodology for ranking alternatives based on the combined effect of multiple types of distances from different reference sites, which is precisely the research gap that this work attempts to fill. as a result, a new recommended approach was developed that incorporates all the advantages of the distance-based methods discussed earlier in this paper, without the need to consider all distances. the goal of establishing a new mcdm is to develop a more accurate, reliable, consistent, understandable, simple to use, and less cumbersome method. in addition to the previously stated goals, the underlying reason for using the technique proposed in this study is to make dms more complete and reliable. despite progress in fuzzy set theory and mcda techniques, it is still necessary to accurately capture the uncertainty and probabilistic integrity of the wave turbine selection criteria. an innovative approach to overcome these limitations is to combine the pffs and lopcow-cobra methods. the fuzzy cobra deterministic decision framework and improved uncertainty representation of pffs are used in the proposed method. considering the complexity and unpredictability involved, this combination provides a more accurate and reliable way to select hydraulic tidal stream turbines. the study's remaining segment is divided into the following sections: the literature review will be included in section 2. the primary ideas are explained in section 3. a suggested evaluation methodology and framework are provided in section 4. an algorithm for selecting the optimal tidal stream turbines for renewable hydroelectric power is applied to assess the strategies in section 5. discussion of the results, comparison and sensitivity analyses are provided in section 6. in section 7, the results and future work are summarized. 2. literature review hydraulic tidal turbine selection is a complex issue involving trade-offs of many performance, cost, and environmental factors [13]. conventional approaches often fall short of capturing the ambiguity and complexity inherent in this decision-making process [14]. this review of the literature examines current strategies and emphasizes the advantages of ppffs and the cobra technique to overcome these problems. factors in renewable energy systems are complex and variable, including wave energy, requiring sophisticated decision-making techniques [15]. several mcda techniques have been used in studies to solve selection problems for renewable energy. for example, solangi et al [16] evaluated renewable energy solutions using analytic hierarchy process 4 d. kang, k. suvitha, s. narayanamoorthy (ahp) and technique for order preference by similarity to the ideal solution (topsis), emphasizing the value of considering various factors when making decisions. specific research has applied mcda and fuzzy logic to the problem of tidal turbine selection in the context of tidal energy. to select the best tidal turbines, chisale and lee [17] used the fuzzy topsis method, demonstrating the effectiveness of fuzzy approaches in this field. the study found that the arrays had little effect on water levels and maximum water levels, both of which are connected with flood risk. however, the tidal turbine arrays had a greater impact on the tidal currents. the flow velocity field, suspended sediment levels, and faecal bacteria levels rose along the arrays while decreasing upstream and downstream [18]. the model demonstrated the importance of including the interaction of the turbines and the flow in assessing the potential power output by comparing the difference in the estimated power output of the various arrays, both with and without the impact of energy extraction by the turbines in the model [19]. the research elucidates the critical significance of bathymetry in wake recovery and fatigue design, offering vital information for real-world turbine array planning. turbine locations beyond 1.5d upstream of the ridge have a higher rate of wake recovery due to a favourable pressure gradient, whereas places beyond 3d downstream of the ridge have greater turbulence intensity [20]. fuzzy set theory is a way to deal with ambiguity and imprecision in human judgment. it is widely used for decision-making in various fields [21]. fuzzy sets (fss) and their classical extensions such as interval-valued fss [22], hesitant fss [23], neutrosophic fss [24], fermatean fss [25], and spherical fss [26] have received much attention as very convenient operators of evaluation data. fuzzy set extensions, pythagorean fss [27], and intuitionistic fss [28] have improved the ability to express uncertainty. senapati and yagar [29] extended these ideas using ffs, which allows more flexibility and improves uncertainty modeling. several decision-making scenarios use the ffs framework, demonstrating its effectiveness in dealing with inherent uncertainty problems. as a result, many researchers have extended the fs index to solve the problem of uncertainty in various domains such as engineering, medicine, and decision-making [30]. ffss, like pythagorean and intuitionistic fss, can handle a large degree of uncertainty and accurately and effortlessly reflect human error judgments during decision-making [31]. inspired by the dominance of ffss, many studies have used it to solve challenging mcdm problems. for example, extended the application of the weighted aggregated sum product assessment (waspas) approach to specific health care waste disposal facilities in an ffs system [32]. ffs aggregation functions were used in the study by [33] to test the covid-19 capabilities. ffss were used in [34] study to overcome parameter uncertainty in capital budgeting. the work of [35] used ffs parameters and scoring function to solve fuzzy traffic problems. in order to select a bridge, the research of optimized topsis with ffss [36]. using topsis with ffss, a study by [37] determined which purification most effectively reduced covid-19 and suggested a few einstein averaging operators. a study by [38] used elimination et choix traduisant la realite (electre), waspas and preference ranking organization method for enrichment evaluation (promethee-ii) in the ffs system for laboratory selection for covid-19 testing. fuzzy sets handle data unpredictability better when they contain a probabilistic component. developed by [39], probabilistic fuzzy sets combine fuzzy logic with probability theory to provide a robust comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 5 framework for handling uncertainty. this strategy is effectively used in environmental science, technology, and economics. mcdm approaches are important resources when making decisions considering multiple factors. methods like the vikor technique, topsis, and ahp are commonly used [40]. in the field of operations research, mcdm is concerned with the development of mathematical and computational tools that assist decision-makers in their subjective evaluation of predefined criteria [41]. as an objective assessment, there are several mcdm strategies, each with pros and cons. the multiatributive ideal-real comparative analysis (mairca) [42], topsis [43], elimination and choice expressing reality (electre) [44], and vlsekriterijumska optimizacija i kompromisno resenje (vikor) [45] are some of these methods. using a straightforward mathematical tool that exhibits a high degree of consistency concerning changes in the nature and character of the criteria, this approach facilitates decision-making. this method's ability to calculate the probability of each option accounts for its merit. measuring the discrepancy between ideal and actual estimates provides the basic premise of mairca. more recently, krstic et al., [46] presented the cobra approach, which is an extension of the decision-making toolset. it seeks a middle ground where all factors are balanced, which is useful when trade-offs are required in difficult decision-making situations. the literature demonstrates that decision tools have progressed from traditional fuzzy sets to more complex fuzzy and probabilistic extensions and advanced mcdm procedures. however, the inclusion of ppffs in the cobra technique represents a significant advance in the field. this technique seeks to solve the complexity and unpredictability of the problem by providing a more comprehensive and efficient solution for the selection of hydraulic tidal turbines. 3. preliminaries 3.1 fermatean fuzzy set let z be a fixed set, a fermatean fuzzy set (ffs) p on z is defined p = {(z, α(z), β(z)) | z ϵ z} the function α(z), β(z) from z in [0,1] denotes the membership and nonmembership degrees, respectively, which have to satisfy 0 ≤ (α(z))3 + β(z))3 ≤ 1. 3.2 picture fermatean fuzzy set let z be a fixed set, a picture fermatean fuzzy set (pffs) p on z is defined as p = {(z, α(z), β(z), γ(z)) | z ϵ z}, where, α(z) = {ϕ={(σ, ς) | ϕ ϵ α(z)}, β(z) = { χ =(μ, ϖ) | χ ϵ β(z)}, and γ(z)|= { δ=(ξ, ρ) | δ ϵ γ(z)} are three ffss of values in [0,1], indicating the possibility of positive (pmd), neutral (nemd), and negative (nmd) membership degrees. the degrees mentioned above satisfy the requirement of 0 ≤ ϕ+ + χ+ + δ+ ≤ 1. 3.3 picture probabilistic fermatean fuzzy set let z be a fixed set, a picture probabilistic fermatean fuzzy set (ppffs) p on z is defined as, p = {(z, α(z)|u(z), β(z)|v(z), γ(z)|w(z)) | z ϵ z}, where, α(z)|u(z), β(z)|v(z) and γ(z)|w(z) contains a number elements, α(z), β(z), and γ(z) denote the possibility (pmd), (nemd), and (nmd) of z ϵ z 0 ≤ ϕ+ + χ+ + δ+ ≤ 1 requirement to the set p, respectively. 6 d. kang, k. suvitha, s. narayanamoorthy here u(x), v(x), and z(x) are associated with probabilistic data. furthermore, α(z)|u(z), β(z)|v(z), and γ(z)|w(z) satisfy the requirement of 0 ≤ ϕ+ + χ+ + δ+ ≤ 1, and ui, vj, wk ϵ [0,1]. # # # 1 1 1 1, 1, 1i j k i j k u v w            where, ui ϵ u(z), vj ϵ v(z), and wk ϵ w(z). 3.4 basic operations let, p = (α|u, β|v, γ|w), p1 = (α1|u1, β1|v1, γ1|w1) and p2 = (α2|u2, β2|v2, γ2|w2) be three pppffes, κ > 0, then ppffe operations are defined by, , , ( | , | , | ) ({ | },{ | },{ | })cp w v u w v u                      1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 2 2 2 , ,1 2 1 2 1 2 1 2 1 2 , , ({ | },{ | },{ | }p p u u v v w w                                  1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 2 1 2 , ,1 2 1 2 1 2 1 2 1 2 , , 1 2 ({ | },{ | },{ | } p p u u v v w w                                         , , ({1 (1 ) | },{ | },{ | })p w                       , , ({ | },{1 (1 ) | },{1 (1 ) | })p u v w                       3.5 score and accuracy function let p = (α|u, β|v, γ|w) be a ppffe, then the score (s(p)) and accuracy functions (a(p)) are defined as  3 3 3 3 3 31 (( ) ( ) ( ) ( ) ( ) ( ) ) ( ) 2 i i i i i i i i i i i iu u v v w w s p             (1)  3 3 3 3 3 3( ) 1 (( ) ( ) ( ) ( ) ( ) ( ) )i i i i i i i i i i i ia p u u v v w w            (2) let p1 = (α1|u1, β1|v1, γ1|w1) and p2 = (α2|u2, β2|v2, γ2|w2) be two ppffes, then, if s(p1) > s(p2) then (p1) > (p2), if s(p1) < s(p2) then (p1) < (p2), and if s(p1) = s(p2) then (p1) = (p2). if a(p1) > a(p2) then (p1) > (p2), if a(p1) < a(p2) then (p1) < (p2), if a(p1) = a(p2) then (p1) = (p2). 4. proposed methodology 4.1 logarithmic percentage change-driven objective weighting method the following steps were used to apply the lopcow technique. step 1: create a decision matrix using ppff preferences (refer to definition 3.3). this involves comparing the set of alternatives (i=1,2,3…n) against the criteria (j=1,2,3…m) to populate the matrix with appropriate ppff values. comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 7 11 1 1 n ij n nm k k k k k k            (3) step 2: compute the normalized decision matrix. min max min for beneficial criteria ij ij k k k k k    (4) max max min for non-beneficial criteria ij ij k k k k k    (5) step 3: determines the percentage values for each criteria. 2 1 100 m ij i ij k m s in                       (6) where ς and m represent the standard deviation and the number of alternatives. step 4: calculate the variation of preferences in response to each criterion. 1 ij jlopcow m ij i s w s    (7) 4.2 comprehensive distance-based ranking method using the cobra approach, the largest capacity tidal stream turbine is selected as shown in fig. 1. the steps of the cobra method are listed below. step 1: construct the decision matrix [ ]ij m nk k  (8) step 2: compute the normalizes the values max ij ij ij k f k  (9) step 3: determine the weighted normalized the values [ ]ij jlopcowg f w  (10) 8 d. kang, k. suvitha, s. narayanamoorthy fig. 1 the flowchart depicts an optimal performance solving method step 4: negative ideal, positive ideal and mean responses are calculated for each attribute. equations (11) and (13) are used for the benefit attribute, while eqs. (12) and (14) are used for the cost criterion. min ( )j ij jlopcownis f w  (11) max ( )j ij jlopcownis f w  (12) max ( )j ij jlopcowpis f w  (13) comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 9 min ( )j ij jlopcowpis f w  (14) 1 ( ) m ij jlopcow i j f w as m     (15) step 5: the distances between the negative ideal (d (nisj)) and positive ideal (d (pisj)) solutions and from the average solution between negative (d (aqj)) and positive (d (aq+ j)) are calculated as follows. ( ) (e(q )) (e(q )) (t(q ))j j j jd q d d d    (16) equation (17) uses qj to represent pisj, asj, nisj and δ to show the correction coefficient obtained. max ( ( )) min ( ( ))j jd e q d e q   (17) the taxicab and euclidean distances for the obtained positive ideal resolution are denoted by t(qj) and e(qj) in eqs.(18) and (19), respectively. 1 ( ( ) ) m j i j ij jlopcow j d t pis pis f w     (18)   2 1 ( ( ) ) m j i j ij jlopcow j d e pis pis f w     (19) also, the following calculation is performed for these distances for the negative ideal solution. 1 ( ( ) ) m j i j ij jlopcow j d t nis pis f w     (20)   2 1 ( ( ) ) m j i j ij jlopcow j d e nis pis f w     (21) 1( ( ) ) m j j i j ij jlopcow d t aq aq f w      (22)   2 1 ( ( ) ) m j i j ij jlopcow j d e aq aq f w      (23) 1 if 0 if j ij jlopcow j ij jlopcow aq f w aq f w         (24) 1 ( ( ) ) m j i j ij jlopcow j d t aq aq f w      (25) 10 d. kang, k. suvitha, s. narayanamoorthy   2 1 ( ( ) ) m j i j ij jlopcow j d e aq aq f w      (26) 1 if 0 if j ij jlopcow j ij jlopcow aq f w aq f w         (27) step 6: compute the comprehensive distance values d(ci) (pis ) (nis ) (aq ) (aq ) (c ) 4 j i j i j i j i i d d d d d      (28) the optimal solution is determined by selecting the option with the lowest comprehensive distance value. 5. selection of tidal stream turbine for hydrokinetic energy converter the growing demand for sustainable energy sources has fueled much interest in wave energy. similar to underwater wind turbines, tidal turbines convert the kinetic energy of tidal currents into electricity. this case study examines the process of selecting the optimal tidal turbine for a hydrokinetic energy converter project. 5.1 alternatives scortrenewables tidal turbine (p1): the revolutionary floating tidal turbine, called the sr2000, was developed by scotrenewables tidal power, now called orbital marine power. an important development in wave energy technology is the turbine. unlike offshore-mounted systems, this design allows the turbine to be towed into place and anchored, reducing deployment and maintenance costs. compared to conventional fixed turbines, the floating platform design has less impact on the marine environment and coastline [47]. the need for aggressive offshore operations reduces the possibility of towing the turbine to port for repairs. nauticity colmat tidal turbine (p2): the unique design of the colmat tidal turbine aims to efficiently harvest tidal energy. turbines usually use sophisticated hydrodynamic concepts to maximize energy extraction from tidal flow, while precise design details may vary. a colmat tidal turbine produces electricity that can be used for many things, such as providing power maintaining grid stability, and providing green energy for industrial applications [48]. seaflow tidal turbine (p3): the offshore tidal turbine is an important step in the development of marine renewable energy, demonstrating the promise of harnessing tidal currents as a clean and sustainable source of electricity. it aims to collect the kinetic energy of tidal currents and convert it into power [49]. it consists of a horizontal-axis rotor with multiple blades similar to a wind turbine, but specially designed for underwater operation. vertent power tidal turbine (p4): an example of renewable energy technology is the verdent power tidal turbine, which uses tidal current energy to generate electricity [50]. it is a highly adaptable and environmentally friendly alternative to conventional tidal power plants that rely on massive infrastructure such as dams or barrages. comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 11 seagen tidal turbine (p5): seagan is a twin-rotor tidal turbine that harnesses the power of tidal currents. it consists of two horizontal-axis rotors, each with multiple blades, arranged in a cross-beam construction. rotors are positioned in the course of tidal currents, capturing energy as the waves come in and out [51]. seagan has a capacity of about 1.2 mw, making it one of the largest wave energy systems in service at the time of its installation. it is capable of generating large amounts of renewable electricity for use in homes, companies, and other buildings. rer hydro-tidal turbine (p6): hydrokinetic technology harnesses the energy of flowing water without the need for substantial dams or barriers to power the rer hydro-tidal turbine. this makes it possible to use natural water movements, including river currents and waves, to generate energy. turbines can be placed in other aquatic habitats such as rivers, estuaries, and tidal streams [52]. due to its adaptability, hydrokinetic energy can be used in places where other renewable energy sources are not practical. 5.2 attributes availability (c1): the availability of renewable energy is an important indicator for measuring the operational efficiency and reliability of energy systems. this has a direct impact on the efficiency and reliability of renewable energy sources in meeting energy demand. capacity factor (c2): the efficiency factor is an important parameter for evaluating the efficiency of renewable energy systems because it reflects how efficiently a generator converts its capacity into actual energy output over time. conversion efficiency (c3): the conversion efficiency of a device is the ratio of the energy converted into a usable form to the total energy available. economic (c4): the installed capacity of a device is the total power it can generate when it is operating properly and at full power output. traditionally, it refers to the installed capacity of an electrical generator in a device. power-take-off (c5): a system incorporated into a renewable energy device converts energy from the physical motions of the device into usable forms such as electricity. survivability (c6): the ability of a device to remain undamaged and operational in harsh environmental conditions. 6. results and discussion the results of the ppffs based cobra method for selecting hydraulic tidal stream turbines provide a new approach for decision making under uncertainty. this technique combines the advantages of cobra with the flexibility of ppffs to handle imprecise and unpredictable data. the evaluation used seven criteria, and six tidal turbine solutions were examined, each with varying performance levels across parameters. 6.1 results of the criteria weights based on ppffs-lopcow create a decision matrix with dimension m x n, where m is the number of alternative and n is the number of attribute. using definition 3.3 and the linguistic measure of table 1, we construct table 2, which shows the ppffs decision matrix. the ppffs score matrix 12 d. kang, k. suvitha, s. narayanamoorthy is then calculated with eq. (1). the normalized ppffs score matrix was then constructed using eq. (1) and is presented in table 3. table 1 linguistic term and ppffs numbers linguistic variable ffs number very insignificant (0.1, 0.8) not significant (0.2, 0.7) slightly significant (0.3, 0.6) medium significant (0.5, 0.5) significant (0.7, 0.3) very significant (0.8, 0.4) absolutely significant (0.9, 0.1) the equations described above were used to develop specifications for the six key criteria. the most essential factor in selecting techniques is the “power-take-off” value of 0.1835, followed by the “availability” value of 0.1771. however, “conversion efficiency” ranked third with a score of 0.1745. the strategy used the probabilistic properties of fermatean fuzzy sets to account for the uncertainty in the decision process. turbines are rated as most suitable based on their total scores across all criteria as shown in fig. 2. table 2 the ppff decision matrix c1 c2 c3 p1 {(0.5 | 0.4, 0.3 | 0.6), (0.4 | 0.5, 0.2 | 0.5), {(0.7 | 0.4, 0.4 | 0.6), (0.4 | 0.2, 0.2 | 0.8), {(0.2 | 0.7, 0.4 | 0.3), (0.3 | 0.6, 0.9 | 0.4), (0.5 | 0.6, 0.3 | 0.4)} (0.3 | 0.4, 0.2 | 0.6)} (0.4 | 0.4, 0.6 | 0.6)} p2 {(0.7 | 0.5, 0.5 | 0.5), (0.6 | 0.9, 0.4 | 0.1), {(0.8 | 0.7, 0.5 | 0.3), (0.5 | 0.5, 0.3 | 0.5), {(0.6 | 0.2, 0.3 | 0.8), (0.8 | 0.8, 0.4 | 0.2), (0.2 | 0.3, 0.1 | 0.7)} (0.2 | 0.6, 0.5 | 0.3)} (0.2 | 0.3, 0.7 | 0.7)} p3 {(0.4 | 0.7, 0.3 | 0.3), (0.7 | 0.6, 0.2 | 0.4), {(0.9 | 0.9, 0.3|0.1), (0.6 | 0.6, 0.4 | 0.4), {(0.6 | 0.4, 0.4 | 0.6), (0.5 | 0.7, 0.7 | 0.3), (0.5 | 0.5, 0.3 | 0.5)} (0.6 | 0.9, 0.2 | 0.1)} (0.5 | 0.9, 0.8 | 0.1)} p4 {(0.2 | 0.3, 0.1 | 0.7), (0.4 | 0.8, 0.1 | 0.2), {(0.8 | 0.2, 0.6 | 0.8), (0.8 | 0.3, 0.4 | 0.7), {(0.8 | 0.1, 0.5 | 0.9), (0.2 | 0.5, 0.1 | 0.5), (0.6 | 0.3, 0.2 | 0.7)} (0.8 | 0.1, 0.5 | 0.9)} (0.9 | 0.6, 0.5 | 0.4)} p5 {(0.8 | 0.4, 0.5 | 0.6), (0.7 | 0.4, 0.5 | 0.6), {(0.7 | 0.1, 0.5 | 0.9), (0.9 | 0.8, 0.5 | 0.2), {(0.1 | 0.7, 0.7| 0.3), (0.6 | 0.2, 0.3 | 0.8), (0.7 | 0.9, 0.4 | 0.1)} (0.1 | 0.7, 0.7 | 0.3)} (0.2 | 0.3, 0.1 | 0.7)} p6 {(0.3 | 0.2, 0.2 | 0.8), (0.9 | 0.1, 0.4 | 0.9), {(0.6 | 0.7, 0.5 | 0.3), (0.3 | 0.7, 0.2 | 0.3), {(0.8 | 0.4, 0.9 | 0.6), (0.5 | 0.6, 0.7 | 0.4), (0.9 | 0.2, 0.8 | 0.8)} (0.4 | 0.5, 0.1 | 0.5)} (0.3 | 0.8, 0.4 | 0.2)} comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 13 c4 c5 c6 {(0.3 | 0.5, 0.2 | 0.5), (0.4 | 0.4, 0.6 | 0.6), {(0.3 | 0.4, 0.9 | 0.6), (0.8 | 0.6, 0.7 | 0.4), {(0.1 | 0.9, 0.3 | 0.1), (0.6 | 0.1, 0.5 | 0.9), (0.7 | 0.3, 0.8 | 0.7)} (0.2 | 0.5, 0.1 | 0.5)} (0.6 | 0.3, 0.8 | 0.7)} {(0.5 | 0.7, 0.6 | 0.3), (0.7 | 0.7, 0.2 | 0.3), {(0.2 | 0.4, 0.8 | 0.3), (0.9 | 0.5, 0.5 | 0.5), {(0.5 | 0.8, 0.7 | 0.2), (0.8 | 0.2, 0.7 | 0.8), (0.2 | 0.4, 0.4 | 0.6)} (0.4|0.6, 0.5 | 0.4)} (0.9 | 0.4, 0.1 | 0.6)} {(0.7 | 0.9, 0.8 | 0.1), (0.8 | 0.2, 0.6 | 0.8), {(0.1 | 0.9, 0.7 | 0.9), (0.4 | 0.2, 0.2 | 0.8), {(0.9 | 0.7, 0.8 | 0.3), (0.9 | 0.3, 0.3 | 0.7), (0.1 | 0.5, 0.3 | 0.5)} (0.7 | 0.9, 0.4 | 0.1)} (0.2 | 0.5, 0.3 | 0.5)} {(0.1 | 0.8, 0.9 | 0.2), (0.2 | 0.1, 0.1 | 0.9), {(0.6 | 0.4, 0.5 | 0.6), (0.3 | 0.4, 0.1 | 0.6), {(0.6 | 0.6, 0.4 | 0.4), (0.2 | 0.4, 0.1 | 0.6), (0.5 | 0.2, 0.6 | 0.8)} (0.2 | 0.8, 0.3 | 0.2)} (0.4 | 0.8, 0.6 | 0.2)} {(0.7 | 0.4, 0.3|0.6), (0.8 | 0.5, 0.9 | 0.5), {(0.4 | 0.9, 0.2 | 0.1), (0.5 | 0.9, 0.4 | 0.1), {(0.2 | 0.5, 0.3 | 0.5), (0.4 | 0.8, 0.7 | 0.2), (0.9 | 0.9, 0.8 | 0.1)} (0.8 | 0.7, 0.7 | 0.3)} (0.5 | 0.9, 0.3 | 0.1)} {(0.2 | 0.1, 0.5 | 0.9), (0.3 | 0.4, 0.5 | 0.6), {(0.6 | 0.5, 0.8 | 0.5), (0.3 | 0.3, 0.2 | 0.7), {(0.6 | 0.4, 0.8 | 0.6), (0.3 | 0.5, 0.8 | 0.5), (0.2 | 0.8, 0.4 | 0.2)} (0.9 | 0.4, 0.3 | 0.6)} (0.9 | 0.6, 0.1 | 0.4)} table 3 the ppff score normalized the decision matrix c1 c2 c3 c4 c5 c6 p1 1.0000 0.1759 0.7488 0.3166 0.4160 0.4718 p2 0.0000 0.4541 0.1366 0.4227 1.0000 0.3542 p3 0.9085 1.0000 1.0000 0.0000 0.0000 1.0000 p4 0.3062 0.0000 0.0000 0.5315 0.9566 0.6589 p5 0.5982 0.3898 0.0636 1.0000 0.9075 0.5132 p6 0.7114 0.5351 0.3542 0.4224 0.6256 0.0000 fig. 2 criteria weights 14 d. kang, k. suvitha, s. narayanamoorthy 6.2 results of alternative values based on ppff-cobra ppffs was used to depict the uncertainty and probabilistic nature of expert judgments. each criterion for each turbine option was assigned a degree of membership or nonmembership. the cobra technique was used to rank the options according to weighted criteria. table 4 normalized ppff decision matrix c1 c2 c3 c4 c5 c6 p1 1.0000 0.5464 0.8767 0.8053 0.7860 0.8147 p2 0.7246 0.6995 0.5762 0.7401 1.0000 0.7734 p3 0.9748 1.0000 1.0000 1.0000 0.6337 1.0000 p4 0.8089 0.4496 0.5091 0.6732 0.9841 0.8803 p5 0.8893 0.6642 0.4503 0.3853 0.9661 0.8292 p6 0.9205 0.7441 0.6830 0.7402 0.8628 0.6492 equation (9) uses the ppffs score matrix to normalize the decision matrix, as shown in table 4. equation (10) then calculates the weighted normalized result matrix as shown in table 5. equations (11-28) are used to determine the selection of a hydraulic tidal stream turbine. table 6 shows the cobra approach's results and the tidal stream turbines' evaluation. fig. 3 shows that the lowest overall score was the best overall fit for the tidal turbine. the greatest possibilities for high production and established technology are seagen and scotrenewables. table 5 weighted normalized ppff decision matrix c1 c2 c3 c4 c5 c6 p1 0.1771 0.0866 0.1222 0.1343 0.1442 0.1422 p2 0.1284 0.1108 0.0803 0.1234 0.1835 0.1350 p3 0.1727 0.1584 0.1394 0.1667 0.1163 0.1745 p4 0.1433 0.0710 0.0709 0.1122 0.1806 0.1537 p5 0.1575 0.1052 0.0753 0.0642 0.1773 0.1447 p6 0.1631 0.1179 0.0952 0.1234 0.1583 0.1133 table 6 the results of the ppff-cobra method d(nis) d(pis) d(aq+) d(aq-) d(c) rank p1 0.0898 0.1187 0.0349 0.0270 0.0052 2 p2 0.0931 0.1204 0.0237 0.0344 0.0095 3 p3 0.1358 0.2879 0.0882 0.0438 0.0269 6 p4 0.0954 0.1334 0.0375 0.0527 0.0133 5 p5 0.1329 0.0937 0.0172 0.0608 0.0011 1 p6 0.0880 0.1130 0.0116 0.0307 0.0110 4 comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 15 fig. 3 the proposed cobra method ranking of the results 6.3 comparative analysis of different ranking techniques the integrated mcdm model utilized in this study effectively evaluated the different tidal turbine alternates for hydrokinetic energy conversion. the model integrates many mcdm approaches, including waspas, copras, codas, moora, and edas. comparative research revealed that the cobra technique provided rankings that were most consistent with waspas, codas, and moora. the consistency suggests that the ranking process is robust among several mcdm approaches as shown in table 7. application of the proposed method to combine rankings from various mcdm approaches ensures robust and reliable results. the results demonstrate that seagen and scotrenewables is the best substitute, followed by other turbine, with consistent rankings across different techniques as shown in fig. 4. despite the different techniques, the rank orders did not differ significantly, indicating that the best alternatives were consistently and reliably selected. table 7 the comparison results are presented in a ranked order waspas codas copras moora edas proposed method p1 0.4432 -0.0990 0.8834 0.1090 0.3441 0.0052 p2 0.4179 -0.0688 0.8552 0.1043 0.1970 0.0095 p3 0.5181 0.8122 0.9783 0.1235 0.5000 0.0269 p4 0.3937 -0.2569 0.8368 0.1010 0.0901 0.0133 p5 0.3878 -0.3240 1.0000 0.1216 0.5529 0.0011 p6 0.4252 -0.0465 0.8693 0.1069 0.3541 0.0110 16 d. kang, k. suvitha, s. narayanamoorthy fig. 4 comparative ranking of the results 6.4 sensitivity analysis this study investigates the sensitivity of ranking alternatives in selecting tidal turbines for hydrokinetic power generation by changing the weights of the criteria. these weights are subjectively calculated by experts using ppffss-lopcow. sensitivity analysis involves reassigning weights to the original seven criteria and testing two scenarios with multiple decision-makers. case 1: weights of criteria are considered equal. case 2: in this case, the jth beneficial criterion is given equal weight and the others are assigned zero. figure 5 and table 8 of the study demonstrate the resulting rankings for the two cases. weight changes had a significant impact on the ranking of alternatives. the order of preferences for hydroelectric renewable energy is strongly dependent on the weights provided to the criteria. this precision is needed to prioritize sustainability in the selection process for hydroelectric renewable energy sources. the study illustrates the robustness of the integrated mcdm model and its adaptability to various weighting scenarios, emphasizing the need for qualified judgment in criterion weighting. table 8 the psi-dnma method’s findings rank in the order of parameters ranking values ranking order optimal rank case 1 0.0037, 0.0123, 0.0191, 0.0171 p3> p4> p2> p6> p1 0.0051, 0.0110 p5> p1 case 2 -0.0007, 0.0131, 0.122, 0.0233 p4> p2> p3> p6> p1 0.0109, 0.0112 p5> p1 comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 17 fig. 5 the solution is influenced by the parameter the pffs provided a robust framework to handle the inherent uncertainty in expert judgments. it allowed for a more nuanced representation of the experts' confidence levels compared to traditional fuzzy sets. hydrokinetic energy harnessing technologies are pivotal in the sustainable energy landscape, as they offer a renewable and environmentally friendly alternative to traditional energy sources. these technologies capture the kinetic energy of moving water—whether from rivers, tidal currents, or ocean waves—to generate electricity. tidal stream turbines operate similarly to underwater wind turbines. they are installed in tidal streams and ocean currents, capturing the energy from the movement of water. the turbines can be horizontal or vertical axis designs, with blades that rotate as water flows past them. tidal streams offer a predictable and reliable source of energy, as tidal patterns are well-understood and cyclical. mcdm methods, such as the newly proposed picture probabilistic fermatean fuzzy set based lopcow-cobra method, are invaluable in addressing the complex trade-offs involved in turbine selection. these methods allow decision-makers to systematically evaluate and prioritize multiple criteria, incorporating both quantitative and qualitative factors. lopcow is essential for tackling complex problems where traditional methods might fall short. its structured approach helps in breaking down and analyzing intricate issues. by comparing outputs and weights at different levels, lopcow facilitates more informed decision-making. this is particularly important in fields such as economics, engineering, and data science, where results often have significant consequences. the approach helps in refining models and predictions by emphasizing detailed processing and comparison. this leads to more accurate and reliable outcomes. the unique feature of comparing outputs and weights allows for a deeper understanding of relationships and dependencies within the data. lopcow can integrate and process data from various sources, making it adaptable to different kinds of datasets and problems. this flexibility enhances its applicability and effectiveness. despite their low impact, hydrokinetic systems can still affect marine and riverine ecosystems. concerns include the potential for fish and other aquatic organisms to be harmed by turbine blades, changes to sediment transport, and alterations to water flow patterns. the initial cost of hydrokinetic energy systems can be high due to the need for 18 d. kang, k. suvitha, s. narayanamoorthy specialized materials and technologies. additionally, maintenance in underwater environments is complex and costly. however, ongoing advancements and economies of scale are expected to reduce costs over time. sensitivity analysis showed that changes in criteria weights could significantly alter the rankings, highlighting the importance of accurate weight determination. compared to traditional decision-making methods, the pffs-based cobra approach demonstrated superior performance in managing uncertainty and providing reliable rankings. the probabilistic aspect of the fuzzy sets offered a more flexible and realistic modeling of expert opinions. the reliability of results depends on the quality and reliability of expert information. consequently, this work develops an understanding model for factors and objectives before applying a multi-objective optimization method for genetic optimization of power coefficient, a primary indicator of tidal stream turbine hydraulic performance. as tidal power supply is abundant, hydraulic power generation is limited due to turbine conversion efficiency and production costs. 7. conclusion in this study, we introduced a novel hybrid mcdm technique capable of effectively solving practical decision-making (dm) situations. this research aims to present the concept of ppffss, which allows dm experts to provide beneficial and non-beneficial criteria for an alternative set using interval values. we have studied the basic operation rules, scoring, and accuracy functions for ppffns in comparison to ffss and image probabilistic fuzzy sets. next, we developed an extended lopcow-cobra-based approach that uses the proposed functions to handle mcdm problems from a picture probabilistic fermatean fuzzy perspective. finally, to show the model's utility and application, a case study of tidal turbines for hydrokinetic energy converter assessment has been considered on ppffss. for high output and proven technology, the best options are seagen and scotrenewables (orbital marine power). we suggested that basis operators are more useful because they reflect the interconnections of multiple criteria while reducing the detrimental influence of dm excessively high or low evaluation values on final selection decisions. hydrokinetic systems are not weather-dependent and have lower initial capital costs than hydropower, photovoltaics, wave energy converters, and other renewable energy technologies. as a result, hydrokinetic devices are one of the most effective sources of renewable energy for isolated communities and small-scale needs. according to literature works, hydrokinetic technologies can be classified as turbine or non-turbine processes. these properties make our techniques well-suited for real-world mcdm problems. also, a sensitivity analysis was performed to confirm the reliability of the obtained results. finally, to illustrate their applicability and benefits, we compare the ones we developed with some existing models. the significance of this research lies in its ability to provide a holistic evaluation of tidal stream turbine converters, addressing both environmental and technical dimensions of sustainability. a comprehensive distance-based approach provides a valuable tool for policymakers, engineers, and researchers involved in the development and implementation of sustainable wave energy technologies. comprehensive distance-based ranking method for evaluating hydraulic converters in tidal... 19 a high level of technical expertise may be required to properly implement the proposed method. this can be a barrier for organizations or individuals who lack the necessary skills. implementing the lopcow-cobra method can require significant computational resources, which can be challenging for small organizations with limited resources. conduct thorough cost-benefit analyses to justify the initial investment and identify situations where the method's benefits outweigh the costs. establish robust data quality assurance processes to ensure reliable and accurate input data. exploring the integration of tidal stream turbine systems with other renewable energy sources such as wind or solar power may provide new opportunities to improve overall energy sustainability and reliability. to better 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university of niš, serbia 2 institute of physics, photonics center, university of belgrade, serbia 3 university of technology, department of mechanical engineering, baghdad, iraq abstract. the paper presents a physical and mathematical model of the new offset type parabolic concentrator and a numerical procedure for predicting its optical performances. also presented is the process of design and optical ray tracing analysis of a low cost solar concentrator for medium temperature applications. this study develops and applies a new mathematical model for estimating the intercept factor of the solar concentrator based on its geometrical and optical behavior. the solar concentrating system consists of three offset parabolic dish reflectors and a solar thermal absorber at the focus. two types of absorbers are discussed. one is a flat plate circular absorber and the other a spiral smooth pipe absorber. the simulation results could serve as a useful reference for design and optimization of offset parabolic concentrators. key words: optical modeling, parabolic offset reflector, solar energy, ray tracing methodology, geometric factors, characteristics of concentration. 1. introduction solar energy is one of the alternative sources that can meet energy demands. this form of energy can be harnessed at a comparatively low cost. the device which is used to transform solar into heat energy is referred to as a solar thermal collector or stc. depending on the temperatures gained by them, stcs can be divided into low, middle and high temperature systems. mid-temperature systems are applicable for refrigeration systems and industrial processes. among several types of solar concentrators, the multidish concentrator consisting of several discrete small dishes arouses wide interest. they received june 28, 2015 / accepted july 29, 2015 corresponding author: saša pavlović faculty of mechanical engineering, university of nis, aleksandra medvedeva 14, 18000 niš, serbia e-mail:saledoca@gmail.com original scientific paper 170 s. pavlović, d. vasiljević, v. stefanović, z. stamenković, s. ayed are convenient for manufacturing, installation and mending discrete dishes. to avoid design complexity of a specific shaped surface solar concentrator, a common offset parabolic antenna is used to make the solar reflector. these types of antennas are widely used both in developed and underdeveloped countries and they are cheap and easy to obtain. a configuration with several offset parabolic antennas offers many advantages over most common centered-focus solar parabolic concentrators (focal point concentrators), like lower wind resistance and higher stability of the structure because of a lower center of gravity. munir [1] presented an experimental study of transforming solar energy into thermal energy by using a parabolic solar concentrator with a system of continual tracking of the sun. the experiment was carried out on a prototype of concentrator of 1 m in diameter and a copper receiver, 10 cm in diameter and 20 cm length. the receiver was located at the focal plane of the parabola. this model of parabolic solar concentrator led to the levels of temperatures ranging between 200 0 c and 350 0 c. bellel [2] studied the difference in performance of two types of solar cylindrical receivers. bellel dealt particularly with problems of heat transfer in the absorber. in fact, the concentration of solar energy, which the absorber converts into thermal energy, allows one to meet various needs: hot water for different uses, production of electricity through the generation of steam, sterilization of objects (medical devices). bhirud and tandale [3] and delaney [4] developed concentrators that are capable of delivering temperatures in the range of 300 0 c and are technically suitable for medium temperature applications. with a higher concentration ratio, there is an increase in temperature at which heat is delivered. it is observed from the comparison that the two axes tracking paraboloidal dish, which always faces the sun, is the most promising design for the concentrating systems justifying the use of the scheffler concentrator for industrial process heat applications. these concentrators also provide an automatic tracking system. scheffler [5] has shown that about half of the power of sunlight which is collected by the reflector becomes finally available in the cooking vessel. the use of solar energy for the generation of steam is now an economically attractive possibility since the payback period of such a system lies between 1.5 and 2 years. ruelas et al. [6] developed and applied a new mathematical model for estimating the intercept factor of a scheffler-type solar concentrator (stsc) based on the geometric and optical behavior of the concentrator in cartesian coordinates. reinalter et al. [7] investigated thermal and geometric features of the 10 kw receiver cnrs-promes system (parabolic reflector with diameter 3.8m, focal length 4.5m). the validation process of the thermal model was performed by comparing the experimental results reported for the 10 kw cavity receiver cnrs-promes system to the results obtained from the simulation of incoming solar radiation in laboratory conditions by using a solar simulator. munir et al. [8] showed the design principle and calculation of a scheffler fixed focus concentrator with the surface area of 8 m 2 for medium temperature applications. the authors proved mathematically that it is possible to construct a concentrator that can provide a fixed focus for each day in a year. bugarin [9] presented the process of design and construction of a low-cost solar concentrator for cooking. this type of solar cooker produces vertically reflected rays that reach the food pot at the bottom like in conventional stoves. rapp and schwartz [10] constructed and successfully tested a 2m 2 parabolic dish solar concentrator (scheffler concentrator) to focus sunlight onto a stationary target. they described some ideas about the intricate design of the scheffler reflectors and how they are developed. the parabolic scheffler reflectors can provide for a high temperature heat for all types of cooking, steam generation and many optical model and numerical simulation of new offset type parabolic concentrator... 171 other applications. macosko et al. [11] described a solar concentrator system designed to supply both thermal and electrical energy to a hydrogen reduction reactor modeled after the nasa roxygen first generation system. a concept is defined that utilizes an offset parabolic concentrator to collect and focus solar energy directly into the roxygen reactor. the offset parabolic concentrator was rotated to track the sun. a two axis solar tracking system and a concentrator positioning system were utilized to keep the concentrator perfectly aligned with the sun and the focal point centered at the reactor (receiver) aperture. asmelash et al. [12] also designed and manufactured an offset type of solar concentrating system for cooking purposes. the concentrating offset collector had major and minor diameters of 90 by 80 cm respectively. the system was tested for its thermal performance following standard testing procedures of the american society of agricultural engineers (asae) and it was found useful with an encouraging results. maximum temperature of 940 0 c was achieved in the cooking vessel after 25 minutes. patil et al. [13] investigated convection heat loss that inevitably occurs in receivers with high concentrating solar concentrators. they presented an experimental investigation of natural convection heat loss from the hemispherical cavity receiver with diameter 540 mm. the aim of the project was to develop the setup that may be used to identify the total heat loss and convective heat loss from the receiver. cabanillas and kopp [14] developed a solar parabolic dish concentrator with a diameter of 2.44 mm and focal length of 0.92 m. they studied a solar spiral heat exchanger made from carbon steel in order to measure the energy efficiency and net energy gain of the concentration system. saravanan et al. [15] designed solar biomass hybrid dryer for the purpose of drying 40 kg of cashew nut per batch. they have investigated the thermal performance of a solar biomass dryer. their system consists of a solar flat plate collector, a biomass heater, a drying unit, a blower and a chimney. the solar air heating collector system consists of an absorber, a double glass cover, a back plate and insulation. the offset parabolic reflectors are more efficient since objects placed at the focus do not create shadows over the reflecting surface. this paper also describes a prototype of an offset parabolic dish with 2400 mm in diameter. the system consists of three identical offset satellite dishes that are positioned in such a way that they concentrate sun radiation in the integral focal area with the desired area and geometry. this study develops and applies a new mathematical model for estimating the intercept factor of the solar concentrator based on its geometrical and optical behavior. this paper, however, does not devote much attention to experimental measurement and testing of thermal performance of the concentrating solar collectors. as a next step in the research, the experimental verification of simulated results is planned. 2. description of the low-cost parabolic offset concentrating collector for thermal application fig. 1 shows a parabolic offset solar concentrator system installed in the laboratory for thermal engineering at the faculty of mechanical engineering, university of niš. the system consists of three identical offset satellite dishes that are positioned in such a way that they concentrate sun radiation in the integral focal area. the inner surface of the offset parabolic dishes that faces the sun is covered with aluminum foil. the reflectivity of the foil ranges from 58 to 60%. the reflective foil should be cut in such a way that there are no wrinkles when they are glued to the surface of the dish; hence 10 cm wide rectangular 172 s. pavlović, d. vasiljević, v. stefanović, z. stamenković, s. ayed pieces are the optimum cut size for the offset parabolic dish. the concentration of solar radiation in a spiral absorber – receiver (focus surface area) is shown in fig. 2. fig. 1 experimental model of a parabolic offset solar concentrator system fig. 2 concentration of solar radiation at the spiral absorber pipe fig. 3 infrared image of the coil receiver taken at 15:05 pm on september, 30, 2014 fig. 3 shows an infrared image of the coil receiver. the maximum measured surface temperature was 335 c. the day was clear and sunny. the measurement was performed with the thermal imaging camera flir e30. it was expected that the temperature at the coil receiver would be about 400 c in july or august. 2.1. physical model of the offset type parabolic dish concentrator with a thermal spiral tube absorber the offset solar parabolic concentrator is the result of the interception of the open circular area with a small side section of the parabola. it directs solar radiation to a fixed focus point where a solar absorber receiver is installed and secured by a fixed support structure. optical model and numerical simulation of new offset type parabolic concentrator... 173 it was made from low carbon manganese steel. the support structure of the offset solar parabolic dish concentrator was made by welding a large number of metal profiles with a square cross-section (0.03  0.03m). parabolic offset reflectors were placed on the support structure by using adequate trusses which enabled the reflectors to be dismantled from the support structure at any given moment. adjustment of the elevation angle of each of three offset reflectors is possible in the range from 0 to 30 relative to the horizontal plane of the support structure. the mechanical structure should allow the positioning of the three parabolic offset reflectors into the optimal position in which solar rays strike the solar concentrator at angle of 90. the authors performed several numerical simulations of dynamic loads for various wind speeds and angle of attack. the system was designed in the following way. the reflectors track the sun up to the wind speed of vw = 20m/s. at greater speeds, the system stops tracking the sun and gets itself into the most stable position which is the horizontal one. the system was designed with the applied cad technologies and calculated to withstand the maximal wind speed of vwmax = 25m/s (finite element method). figs. 4a and 4b show the solar concentrating system in the working and safety positions. a) b) fig. 4 a) working position of the offset parabolic dish solar concentrator (axonometric view); b) safety position (elevation angle of 900) the main elements of the offset parabolic dish concentrator system in the working position are shown in fig. 4a. one of the essential elements of the solar system is the possibility of a permanent tracking position of the sun (azimuth and elevation). the incident angle of sun rays is parallel to the optical axis of the solar concentrating system. the system remains in the safety position during the night, only to take up the working position after sunrise (tracking mode). 2.2. mathematical modeling of the solar offset parabolic concentrator design parameters of the solar offset parabolic concentrator are illustrated in table 1 [16]. mathematical modeling (calculation) for the whole geometric model of the offset 174 s. pavlović, d. vasiljević, v. stefanović, z. stamenković, s. ayed parabolic concentrator was done in wolfram mathematica v.8.0. the numerical solution of the mathematical model was used to obtain geometric parameters such as the radius of the smaller and larger semi axis of the ellipse, the focal length, and the tangent slope for the three offset reflectors. parameters obtained from mathematical analysis were used as input parameters for 3d modeling in the cad program catia v5r19 dassault systems, usa. the first step was to design a rotational paraboloid with diameter d = 5000 mm and focal length f = 1600 mm. the next step was to design a circular cylinder with diameter d = 1100 mm, which was positioned parallel to the optical axis of the rotational paraboloid at the distance of 100 mm from the optical axis. the three cylinders were positioned parallel to the optical axis and the angle between the axes of the cylinders was 120°. unlike a conventional paraboloid concentrator, the offset parabolic concentrator system fixed focus concentrator is a lateral part of a paraboloid as shown in fig. 5a. fig. 6 shows the structure and geometric model of the new type of solar offset parabolic concentrator. the choice of the optimal position of the solar thermal absorber – receiver was carried out on the basis of monte carlo ray tracing analysis model of the offset type parabolic concentrator dish and receiver. the optimal focal distance is 1155 mm, while the theoretical focal distance is 1600 mm focal hot spot. paraboloid available aperture area lateral section of a paraboloid (offset parabolic concentrator) focus beam solar radiation directrix vertex y x 1100 mm 2500 mm a) b) fig. 5 a) section of the offset parabolic solar dish concentrator from baseline paraboloid; b) process of determining a offset parabolic solar dish concentrator the mathematical representation of the parabolic concentrator is paraboloid which can be represented as a surface obtained by rotating the parabola around the axis. mathematical equations for the parabolic dish solar concentrator in cartesian coordinate systems are defined as: fzyx 422  (1) optical model and numerical simulation of new offset type parabolic concentrator... 175 table 1 geometric parameters of the offset parabolic dish concentrator, pavlovic et al. [16] design parameter numerical value unit major diameter of reflector dref 1.20 [m] minor diameter of reflector dref 1.10 [m] cross-section of opening offset parabolic concentrator aproj 5.31 [m 2 ] sheltered area of concentrator ashadow 0 [m 2 ] effective area of offset concentrator aap,ref 2.85 [m 2 ] receiver diameter 0.20 [m] diameter of baseline paraboloid 5 [m] depth of baseline paraboloid 0.98 [m] depth of offset parabolic concentrator 0.33 [m] focal length 1.6 [m] ψ1 rim angle 43.15 [ 0 ] absorber area 0.0314 [m 2 ] geometric concentration ratio crg = aap/arec 96.01 [-] optical (flux) concentration ratio cr0 = irec/ iap= irec /dni 87,07 [-] surface reflectance ρ 0.60 [-] direct solar radiation id 800 w/m 2 a) b) fig. 6 a) structure of the new type of solar offset parabolic concentrator; b) geometrical model of the new type of solar offset parabolic concentrator usually the paraboloids that are used in solar collectors have rim angles from 10 to 90 degrees. the relationship between the relative aperture and the rim angle is given by: )2/tan(4 1 / rim df   (2) intercept factor φ represents the fraction of the radiation intercepted by the cavity receiver and is calculated by eq. (3) with a change over the term of the fraction of the captured flux, using γ 1.975 instead of γ. the shadow produced by the receiver was considered as a dead area on the projected area of dish reflective surface adish. power consumption for the tracking and controlling systems were neglected. 176 s. pavlović, d. vasiljević, v. stefanović, z. stamenković, s. ayed 1.975 0 8 sin( ) dish if ia            (3) the incident energy received by solar spiral absorber of offset parabolic dish concentrator can be computed from the following equation: ( )s b eq i c   (4) the optical efficiency of that solar concentrator can be computed from: 0 ( )c e    (5) 3. physical models of two types of solar spiral thermal receivers in terms of their configuration, receivers can be flat receivers or cavity receivers. the first ones focal absorber surfaces, whilst the latter ones have the absorber surface surrounded by an insulated cover, with a sufficient opening to capture the radiation from the concentrator. the shape of the receiver depends on its dimension and the concentrator focal length/rim angle. photo thermal conversion in a dish cavity receiver is a key process to efficiently utilize solar energy, mainly because the cavity receiver converts the collected solar radiation into thermal energy by coupling the complicated heat transfer. this paper analyzes two types of solar thermal receivers presented in fig. 7. the first type is the solar thermal receiver with a flat plate. the receiver is a metal disc painted black. the second type is the archimedean spiral smooth pipe directly exposed to reflected radiation from the offset dish concentrator. the receptor portion of the heat exchanger has three primary components, all made of stainless steel. the spiral pipe absorber creates a spiral path for fluid to flow through. a) b) fig. 7 two types of solar thermal receivers: a) flat plate painted black; b) spiral pipe the flow rates in spiral absorber pipe are changeable data. the next step is to create an experimental installation in the mechanical engineering laboratory for testing and thermal hydraulic performance of said heat exchangers in stationary conditions. the optical model and numerical simulation of new offset type parabolic concentrator... 177 installations will be able simulations of real solar modules (offset parabolic concentrator). they will serve as the basis for investigating the work of the spiral absorber at different geometric and spatial configurations, fluid flow, heat input levels, as well as the fluids with significantly different thermo-physical properties (water and a solution of propylene glycol). in order to assess the efficiency of the heat exchanger the measurement of thermal and hydraulic losses will be carried out. the spiral smooth pipe absorber is directly exposed to reflected radiation from the dish concentrator. solar energy systems can improve effectiveness of their receivers by covering them with a selective absorber coating. the spiral pipe receiver is made of stainless steel aisi 304 with a coil of 10 mm in outer diameter and 8 mm in inner diameter rolled into a spiral shape of 200 mm in diameter. the length of spiral pipe absorber is about 2.5 m. usually in this type of receiver the liquid circulates inside the tube, entering from the rim and leaving at the center where the radiation intensity and temperature are higher (fig. 8a and 8b). to avoid pipe deformations, the receiver has a small hole in the center, of about 40 mm in diameter. the incidence angle of the offset parabolic concentrator is zero θi = 0. a) b) fig. 8 a) trace of concentrated rays at the spiral absorber pipe; b) spiral flat coil absorber both types of receivers are painted black on the side facing the concentrator so that they would be able to absorb the maximum amount of thermal energy. the test plate – a metal circular disk, is used to determine where the focal area of the concentrator is. in other words, it is used to determine where the receiver should be placed in relation to the concentrator. 4. results and discussion 4.1. optical analysis of the offset parabolic concentrator the optical analysis of the offset parabolic concentrator was carried out in the commercial software tracepro ver. 7.4.2, lambda research corporation, usa. two types of absorbers were analyzed: the flat plate circular absorber and the spiral smooth pipe absorber. 4.1.1. optical analysis of the offset parabolic concentrator with a flat plate circular type of absorber in tracepro the three offset parabolic reflectors were defined as a surface with reflective coating. the reflectance coefficient was 60%. the receiver was a flat plate circular absorber 178 s. pavlović, d. vasiljević, v. stefanović, z. stamenković, s. ayed with the diameter of 200 mm and 5 mm in thickness. the absorbing surface was defined as the perfect absorber (α =1). the position of the receiver was found by calculating the optimal flux for the given receiver. the optimal position was found at 1155 mm from the vertex of the parabolic dish, after several calculations in tracepro. at a distance of 1155 mm the best value of total flux was obtained solar power and irradiance flux density distribution. after defining the geometry of the offset parabolic solar concentrator, the radiation source was defined. the radiation source was circular with the same diameter as that of the parabolic dish (2400 mm). the radiation source was placed 1500 mm from the vertex of the offset solar parabolic concentrator and had a circular grid pattern for the monte carlo ray tracing. the spatial profile of generated rays was uniform and the angular profile was solar radiation. the input parameter for the optical analysis was the solar irradiance which was set to 800 w/m 2 . experimental values for solar irradiation for the city of niš in serbia are between 750 w/m 2 and 900 w/m 2 . the optical system for the offset parabolic solar concentrator with traced rays is given in fig. 9. since the offset parabolic concentrator continually tracked the sun, the incidence angle of solar rays was always θ = 0 0 . for this type of solar concentrator cosine losses do not exist. x y zz fig. 9 ray tracing model of the offset parabolic solar concentrator with the flat plate circular absorber fig. 10 solar flux distribution on the absorber surface the optical analysis was done by generating 119401 rays and calculating them by the monte carlo ray trace. from all the emitted rays only 76058 rays reached the absorber surface, which was 64% of the emitted rays. the calculated irradiance for the absorbed rays on the receiver was from 3.88∙10 -9 w/m 2 to 3.25∙10 5 w/m 2 . the total calculated flux on the flat plate receiver was 2188 w. fig. 10 shows the total irradiance map for the absorbed flux on the receiver. 4.1.2. optical analysis of the offset parabolic concentrator with a spiral smooth pipe absorber the second optical analysis was performed on the more complicated model of the solar absorber. the solar absorber was the spiral pipe absorber described in the previous section. the position of the receiver was found by calculating the optimal flux for the optical model and numerical simulation of new offset type parabolic concentrator... 179 given receiver. the optimal position of the receiver was 1155 mm from the vertex of the parabolic dish. the optical analysis for the system with the smooth pipe absorber was done by calculating two sets of rays. the first set contained 119401 rays and the second set contained 1078201 rays. all rays were calculated by the monte carlo ray trace. for the first set of rays, from 119401 emitted rays only 49886 rays reached the absorber surface which was 42% of rays. the average irradiance – the local flux density was 57859 w/m 2 . the calculated irradiance for the absorbed rays on the receiver was from 1.625∙10 -15 w/m 2 to 4.124∙10 5 w/m 2 . the total calculated solar flux on the spiral pipe receiver was 1435.4 w. the total irradiance map for the absorbed flux on the receiver for the first set of rays is shown in fig. 11. fig. 11 solar flux distribution on the absorber surface 119 401 simulated rays fig. 12 solar flux distribution on the absorber surface 1078201 simulated rays for the second set of rays, from 1078201 emitted rays only 515418 rays reached the absorber surface which was 48% of rays. the average irradiance – the local flux density was 66195 w/m 2 . the calculated irradiance for the absorbed rays on the receiver was from 1.2207∙10 -15 w/m 2 to 3.8733∙10 5 w/m 2 . the total calculated solar flux on the spiral pipe receiver was 1642.3 w. the total irradiance map for the absorbed flux on the receiver for the second set of rays is shown in fig. 12. from the presented data for the first and second set of rays it can be concluded that it is very important to properly define a set of rays. for a simple solar absorber, such as a plate circular disc, a set of 119401 rays is more than enough, but for a complex type of the solar absorber, such as a spiral smooth pipe absorber, one ought to define much more rays to get valid results. 5. conclusion the authors have developed a completely new model of the offset parabolic dish concentrator. the presented simulation results could serve as a useful reference for the design and optimization of offset parabolic concentrators. as a next step in the research, the experimental verification of simulated results is planned. the authors intend to improve the 180 s. pavlović, d. vasiljević, v. stefanović, z. stamenković, s. ayed design of the absorber by developing a solar receiver with a cavity. this type of receiver has significantly smaller convective and radiation losses and greater conversion efficiency of solar radiation to heat. acknowledgements: the paper is a part of the research done within the projects: iii 42006 research and development of energy and environmentally highly effective polygeneration systems based on renewable energy resources and iii 45016 fabrication and characterization of nanophotonic functional structures in biomedicine and informatics. both projects are financed by the ministry of education, science and technological development of the republic of serbia. the authors would like to acknowledge the help provided by the lambda research corporation by allowing the use of tracepro software for the phd research of saša pavlović. references 1. munir a., 2010, design, development and modeling of a solar distillation system for the processing of medicinal and aromatic plants, phd thesis, universität kassel/witzenhausen, germany 173 p. 2. bellel n., 2011, study of two types of cylindrical absorber of a spherical concentrator, energy procedia, 6, pp. 217–227. 3. bhirud n., tandale s., 2006, field evaluation of a fixed-focus concentrators for industrial oven, in: proceedings of national conference on advances in energy research, december, iit bombay, india, pp. 364–370. 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(vol. i – vol. v), pp. 726-730, http://link.springer.com. 15. saravanan d., wilson v., kumarasamy s., 2014, design and thermal performance of the solar biomass hybrid dryer for cashew drying, facta univesitatis, series: mechanical engineering, 12(3), pp. 277-288. 16. pavlovic s., stefanovic v., mancic m., spasic z., 2015, development of mathematical model of offset type solar parabolic concentrating collector, proceedings of 12th international conference on accomplishments in electrical and mechanical engineering and information technology, faculty of mechanical engineering, banja luka, bosnia and herzegovina. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 121 134 original scientific paper reliable robust controller for half-car active suspension systems based on human-body dynamics udc 681.5:629.3 mohammad gudarzi young researchers and elites club, damavand branch, islamic azad university, iran abstract. the paper investigates a non-fragile robust control strategy for a half-car active suspension system considering human-body dynamics. a 4-dof uncertain vibration model of the driver’s body is combined with the car’s model in order to make the controller design procedure more accurate. the desired controller is obtained by solving a linear matrix inequality formulation. then the performance of the active suspension system with the designed controller is compared to the passive one in both frequency and time domain simulations. finally, the effect of the controller gain variations on the closed-loop system performance is investigated numerically. key words: body acceleration, dynamic modeling, lmis, driver’s biodynamics 1. introduction passive, semi-active and active suspensions are three main types of car suspensions that are used by automotive industries. all suspension systems aim to improve the car performance consisting of ride comfort, handling, road holding, suspension deflection, static deflection, etc. conventional passive suspensions are effective only in a certain frequency range and optimal design performance cannot be achieved when the system and its operating conditions are changed. on the contrary, it has been well recognized that active suspensions have a great potential to meet the tight performance requirements demanded by users. therefore, in recent years more and more attention has been devoted to the development of active suspensions and various approaches have been proposed to solve the crucial problem of designing a suitable control law for these active suspension systems [1-3]. most of the present studies on active suspensions are concerned with vibration reduction of the sprung mass containing vertical acceleration of center of gravity (cog), received may 16, 2016 / accepted june 27, 2016 corresponding author: mohammad gudarzi young researchers and elites club, damavand branch, islamic azad university, iran e-mail: mohammad.gudarzi@gmail.com 122 m. gudarzi pitch acceleration and roll acceleration [4-8]. however, passengers do not sit on the cog of the vehicle and so, vertical, roll and pitch motion of the vehicle affect the acceleration of the passenger’s seats. in addition, around the resonance frequency of the seated human, the passenger’s vibration magnitude due to the seated human dynamics becomes larger than the seated position’s vibration of the sprung mass. moreover, these vibrations cause the driver’s whole body to vibrate. undesirable effects of the driver’s body vibration are experienced when the exposure time is longer than the recommended standard set by iso 2631-1 [9]. so, to take into consideration the driver’s biodynamics in the controller design can improve its performance [10]. in addition to aforementioned aspects, another important goal of the controller design for active suspension systems is to maintain the robustness of the closed-loop systems [11]. in real cars, the total vehicle mass varies due to the changes in passenger load and cargo while characteristics of the actuators change due to aging and nonlinearities. when parameters in the plants change like this, the control performance specified in the design stage tends to deteriorate especially in lqg controllers [12]. however, a constant performance is desired in automotive suspensions, so, this deterioration should be kept as small as possible. therefore, the analysis and synthesis of robust control for active suspension systems have become a research concern in recent years [13]. the application of the standard robust control theory has an assumption that the controller can be realized exactly. however, in practice, many physical limitations such as the effects of finite word length in any digital system, round-off errors in numerical arithmetics, inherent imprecision in analog devices, etc. lead to a loss of precision in controller implementation. accordingly, even though the designed controllers are robust with respect to system uncertainties, they may be very sensitive to their own uncertainties. recently, much attention has been paid to the so-called fragility problems of controllers [14-15]. this problem is basically associated with performance decrease of a closed-loop system due to the inaccuracies in the controller implementation [16]. in recent years, many works have been made to solve the non-fragile controller design problem for linear systems [17-18]. some state feedback non-fragile h∞ controller design method with respect to additive and multiplicative norm-bounded controller gain variations are given in some literature [18-19]. in addition, this kind of controller has been implemented on an active suspension system [20]. the above review indicates that while there exists a notable body of literature on the active car suspension control, a comprehensive investigation on active suspension systems considering accurate human biodynamics based on robust non-fragile controller design seem to be absent. in this study, the problem of robust non-fragile h∞ controller design for a half-car suspension system with norm-bounded parameter uncertainties and controller gain variations is investigated. in order to obtain a better insight of the suspension system performance, a 4-dof uncertain vibration model of the driver’s body is combined with the car’s model. due to no explicit inverse of mass matrix existing in controller design approach, the uncertainties in mass, damping and stiffness can be described more naturally and directly. considered additive controller gain variations in this approach, also, makes this approach more robust and more applicable in engineering practice. finally, the desired controller has been obtained by solving a set of linear matrix inequalities (lmis) using matlab; then the frequency and time domain responses of the active suspension are compared to the passive ones and the effect of controller gain variations on the closed-loop system performance is investigated. reliable robust controller for half-car active suspension systems based on human-body dynamics 123 2. dynamic modeling a half-vehicle model, which is equipped with an active suspension between each unsprung mass and the sprung mass, is shown in fig. 1. the detailed equations of motion for this half-car model, which are bounce, pitch and each unsprung motions are as given in [21]. the schematic model of the driver’s biodynamic consisting of the lumped human linear seat model is illustrated in fig. 2. considered model for the driver is based on a 4dof human body presented in [21]; the related equations could be found there as well. by assuming that the passenger is sitting in the front seat, the obtained dynamic equations by adding structured uncertainties can be rearranged as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),m c k wm z t c z t k z t bu t b w t       (1) fig. 1 half-vehicle model fig. 2 passenger model 124 m. gudarzi where z(t)=[zuf,zur,zcg,θ,zh,zut,zlt,zt,zse] tr9 is the displacement and rotation vector, u(t)=[faf,far] t is the control input vector, and w(t)=[zrf,zrr] t is the external disturbance from the road. mr9×9, cr9×9 and kr9×9 are the mass, damping, and stiffness matrices, respectively; m, c and k are corresponding perturbations, mr9×2 is the input matrix and bwr9×2 is the disturbance matrix. by considering state q(t)=[zt(t),żt(t)]t, eq. (6) can be written as: 0 00 0 ( ) ( ) ( ) ( ). 0 k c wm ii q t q t u t w t k c bm b                            (2) here, w(t) is assumed to be an energy-bounded signal (i.e., w(t)l2[0,)). the objective or measurement signal zo(t) to be controlled is considered as o ( ) ( ) ( )d vz t c z t c z t  (3) where cdr1×9 and cvr1×9. the system can be formulated as: o ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ), wq t q t u t w t z t q t         (4) where:   0 00 00 0 , , , , 00 00 0 , , , k cm w d v w i i m k c c c bb b                                                uncertainty δm is assumed to satisfy: 1 1,m m    (5) where ||.|| refers to the euclidean norm of a vector or matrix. equation (5) implies that ||δee-1||≤δ<1. note that the condition (5) ensures that e+δe is non-singular. in addition: , , k k k k c c c c l f e l f e     (6) where fk≤1, fc≤1, lk, lc, ek, ec are known constant matrices which characterize the influence of uncertain parameters fc, fk in nominal damping matrix c and stiffness matrix k, respectively. the uncertainties in active suspension system (1) satisfying eqs. (5) and (6) are said to be admissible. therefore: [ 0 0 0 0] [ ]k k c c k k k c c c k c f e f e f f l l                  l l (7) with: 0 k kl         l , [ 0],k ke 0 ,c cl         l [0 ].c ce reliable robust controller for half-car active suspension systems based on human-body dynamics 125 the control force, utilizing both displacement and velocity feedback signals, is obtained by: ( ) ( ) ( ) ( ) ( ),d fd v fvu t f z t f z t    (8) where fdr2×9, fvr2×9 are the feedback gain vectors for the displacement and the velocity, respectively, and δfd, δfv their corresponding uncertainties. this can be rearranged as: ( ) ( ) ( ),u t q t  (9) where fr2×18 is the state feedback gain to be calculated, f=[fd,fv], δf=[δfd,δfv] is a norm-bounded feedback gain variation in the form of [19]: ,  f f ff (10) where χf, ef are describing the uncertainty structure and they are known constant matrices of appropriate dimensions, and for the matrix ff, we have ff ≤1. ride comfort quantified by driver’s head vertical acceleration, suspension deflection limitation, road holding and actuator saturation are considered as controlled values. 2 max max 1 3 4 max max ( ) [ ( ) ( )] [ ] ( ) ( ) ( ) ( ) ( ) [1 1] ( ) [ ( ) ( )] [ ] t t o uf ur f r o h f tf tr tr o sf sr t t o af ar f r z t z t z t z z z t z t k z t k z t z t f f z t f t f t f f              (11) the control design objective is to minimize h∞ norm of the closed-loop transfer function from output disturbances w(t) to measurement signal zo(t), tzw(s)∞, so that it remains below a specified bounded value γ>0 for all admissible plant uncertainties and gain variations in f (control). such a designed control law is known as a non-fragile h∞ state feedback controller. 3. controller design in this section, a solution for the problem of robust non-fragile h∞ state feedback control for the active seat suspension system (4) in which both robust closed-loop stability and robust h∞ performance are achieved in spite of parametric uncertainties and controller gain variations is considered. system (4) with the state feedback control gain in eq. (9) becomes: ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ). wq t q t q t w t z t q t           (12) according to the assumption (5), eq. (12) can be written as: 126 m. gudarzi 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ) w w q t q t q t w t q t w t                      (13) where: 1( ) ,( )   1( ) ,  1( ) .w w   according to the bounded real lemma [22-23] for the system (12), following the procedure given in [22] and by applying the schur complement and by some rearrangement of matrix sub-blocks, the result can be expressed as an lmi. therefore, for the uncertain system (4) with given γ>0, η>0, a state feedback control of form (9) can be constructed which could tolerate the system uncertainties δm, δc, δk, and controller gain variations δf so that the resulting closed loop system is robustly stable with disturbance attenuation γ provided that there exist matrices x>0, q>0, y and scalars >0, i>0, i=1,2,3, μi>0, i=1,2,3,4, satisfying the following lmis: 11 12 13 14 22 33 44 11 1 3 3 1 2 2 2 2 2 4 1 2 3 12 32 ( ) ω ω ω ω * ω 0 0 0, * * ω 0 * * * ω ω ( ) , ω [ ], ω diag[ , ( ), t t t t t c c k k k k t t t t t t f f t t t t w k k t k k x q x i y y i i i q x x y i i q                                                     2 4 1 33 2 14 44 3 ( ), ( ), ], ω , * ω [ 0 0 0 0 0] 0 , ω , * [ ] t t k k t t t f t f i x i x i i x i i x i i x i                                (14) and 1 0. ( ) t c t c c c q x x i x         (15) and, a desired robust non-fragile h∞ state feedback control gain matrix is given by f=yx-1. reliable robust controller for half-car active suspension systems based on human-body dynamics 127 4. numerical results and discussion in order to evaluate the effectiveness and robust performance of the controller design method proposed in the above section, an example is introduced in this section. we assume that the vehicle model is a generic sedan car as the parameters given in table 1. table 1 half-car model parameters mass (kg) damping coefficient (ns/m) spring constant (n/m) length (m) moment of inertia (kg.m2) symbol magnitude symbol magnitude symbol magnitude symbol magnitude symbol magnitude mb 950 csf 1000 kt 200000 lf 1.34 ip 1500 mtf 50 csr 1000 ksf 39000 lr 1.46 mtr 50 ksr 37000 px 0.04 hp 0.53 table 2 driver model parameters mass (kg) damping coefficient (ns/m) spring constant (n/m) symbol magnitude symbol magnitude symbol magnitude mh 4.17 ch-ut 310 kh-ut 166990 mut 15 cut-lt 200 kut-lt 10000 mlt 5.5 cut-t 909.1 kut-t 144000 mt 36 clt-t 330 klt-t 20000 mse 35 ct-se 2475 kt-se 49340 cse 150 kse 15000 as mentioned before, in the biodynamic model the seated human body is constructed with four separate mass segments interconnected by five sets of springs and dampers. with a total human mass of 60.67 kg the nominal design parameters for the biodynamic model are listed in table 2. accordingly, the output variables are chosen to be the displacements and velocities of each mass part, therefore, l=i. the uncertainties in the mass, damping, and stiffness matrices are, respectively, modeled as: 1 0.1,m m    (0. ( )1 ) ,k k k k kl f e k f i  (16)  (0.1 ) .c c c c dl f e c f i   assume that the controller gain variation has structure:  1,1,1,1,1,1,1,1,1 , t f fal  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,f (17) where af is an adjustable parameter to describe the level of gain variation (in the range of 102). 128 m. gudarzi the frequency responses of the proposed controller are illustrated in figs. 3 to 6. as it can be seen, the non-fragile controller improves the driver’s head acceleration. the designed controller greatly attenuates road excitation, especially in the frequency range around 10 rad/s, the range that affects ride comfort greatly. in addition, fig. 3 shows that the inaccuracy in the controller gain has no significant effect on the driver’s head acceleration in this frequency range. with respect to suspension deflection from fig. 4, the closed-loop system is not as good as the passive controller, which is an inevitable tradeoff. it is impossible to simultaneously reduce both body acceleration and suspension deflection in the low-frequency range and around the wheel-hop frequency [24]. as for road holding, the proposed controller reduces the value of the first peak in fig. 5. moreover, it may result in handling-loss. one has to note that the variation of the controller gain makes no changes in the frequency responses of suspension deflections and road holdings. fig. 3 frequency response of the head acceleration reliable robust controller for half-car active suspension systems based on human-body dynamics 129 fig. 4 frequency response of suspension deflection fig. 5 frequency response of road holding the aim of active suspension is to reduce acceleration to the greatest extent, which leads to a good ride comfort, and at the same, to keep the suspension deflection and road holding in acceptable ranges. it means that the performance of the suspension deflection and road holding is sacrificed in order to get a good ride comfort, and therefore the frequency response of this case is what is acceptable. 130 m. gudarzi the actuator force frequency responses in fig. 6 are indicators of the energy cost of the two actuators using accurate and inaccurate controllers. this subject is further discussed in the following part of the time responses. fig. 6 frequency response of actuators forces in order to further test the validity of the designed controller, a set of simulations in the time domain are carried out on a current road excitation. for investigation of the active suspension performance, road disturbances can be generally assumed as shocks which are discrete events of relatively short duration and high intensity, caused by, e.g., a pronounced bump or pothole on an otherwise smooth road surface. in this work, this case of road profile is considered first to reveal the transient response characteristic, which is given by: 0 0 0 0 2 1 cos , 0 , 2 0 ( ) , , va l t t t l v z l t v                    (18) and illustrated in fig. 7, where a is the height of the bump, and l is the length of the bump. here we choose a=0.1 m, l=2 m and vehicle speed v0=30 km/h. reliable robust controller for half-car active suspension systems based on human-body dynamics 131 fig. 7 bump input the time responses of the designed controller due to bump disturbance are shown in figs. 8 to 11. it can be seen that the driver’s head acceleration for the designed controller has much lower peaks and the settling time is reduced, as the suspension deflection. the peaks in road holding are a little bigger than for passive suspension. in addition, actuator forces in the time domain are shown in fig. 11, where produced forces are in an acceptable range which can be generated by hydraulic or electrorheological actuators in practice [2526]. as one can see in these figures, the controller variation does not have a sensible effect on the time responses of the closed-loop system. fig. 8 head acceleration due to bump excitation 132 m. gudarzi fig. 9 suspension deflections due to bump excitation fig. 10 road holding due to bump excitation reliable robust controller for half-car active suspension systems based on human-body dynamics 133 fig. 11 actuators forces due to bump excitation it is confirmed that the designed non-fragile robust controller is able to guarantee a better performance under a pronounced bump disturbance and limited actuator control force in spite of the car-driver’s uncertainties and controller variations. 5. conclusions ride performance, suspension deflection, road holding and actuators’ forces of a class of half-car suspension systems considering an uncertain 4-dof driver’s biodynamics using a non-fragile robust h∞ state feedback controller is investigated. the biodynamic model has been added to the system to obtain a good tradeoff between performance and accuracy as well as a better insight of the controller design. a design example demonstrates the effectiveness of the proposed controller 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takabi, b., salehi, s., 2014, augmentation of the heat transfer performance of a sinusoidal corrugated enclosure by employing hybrid nanofluid, advances in mechanical engineering, 6, doi: 10.1155/2014/147059 26. gheitaghy, a.m., takabi, b., alizadeh, m., 2014, modeling of ultrashort pulsed laser irradiation in the cornea based on parabolic and hyperbolic heat equations using electrical analogy, international journal of modern physics c, 25(9), doi: 10.1142/s0129183114500399 7573 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 633 645 https://doi.org/10.22190/fume210317054l © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper thermal oscillation arising in a heat shock of a porous hierarchy and its application fujuan liu1,2, ting zhang3, chun-hui he4, dan tian5 1national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china 2nantong textile & silk industrial technology research institute, nantong, china 3the second affiliated hospital of soochow university, suzhou, china 4school of civil engineering, xi’an university of architecture & technology, xi’an, china 5school of science, xi’an university of architecture & technology, xi’an, china abstract. a building or a bridge might collapse after a heat shock. this paper shows that a porous hierarchy of a coating can effectively prevent a building or a bridge from such damage. a cocoon’s geometrical structure is studied and its resistance to the heat shock is revealed by a thermal oscillator. the theoretical model reveals an extremely low frequency of the thermal oscillator, which is very important for cocoons’ biomechanism, especially in the heat insulation function. at the same time, it shows that the cocoons have the best thickness to protect the pupa from the environment. in addition, surface temperature measurement of hierarchical mulberry leaves is performed. this work provides new insights into biomimetic design of the protective building and coatings. key words: silkworm cocoon, hierarchical structure, heat conduction, microporous capillary, thermal oscillation, freeze-thaw damage 1. introduction a building under a sudden heat shock will suffer a great damage as we can see from the september 11, 2001, attacks on the world trade center, new york, often referred to as 9/11; the high buildings of the world trade center collapsed after the attack. the huge energy was transferred to the metal structure of the building, and the temperature was too high to support the building. it is extremely important to protect a building from the heat shock with a high environment temperature. the general thermal insulation is impossible for heat shock presentation. it seems there is no approach to preventing such damage so far. this paper aims at a new idea for this purpose inspired from a silkworm cocoon, which is received march 17, 2021 / accepted june 03, 2021 corresponding author: fujuan liu college of textile and clothing engineering, soochow university, china e-mail: liufujuan@suda.edu.cn mailto:liufujuan@suda.edu.cn 634 f.j. liu, t. zhang, c.-h. he, d. tian generally considered to protect the pupa from predators and hazards. actually, it can also prevent a sudden heat shock. liu et al. [1] proposed the fractal model for heat transfer in hierarchic cocoons for the first time to explain the fascinating phenomenon of pupa survival under extreme environment. he’s fractional derivative was adopted to study the heat conduction in cocoons which are regarded as fractal medium [2]. moreover, liu et al. [3] defined a new fractional derivative through the variational iteration method and applied it to explain the outstanding thermal protection of insulation clothing with cocoon-like porous structure. because the cocoon is extremely insensitive to environment change and a pupa can survive in a harsh environment, a mathematical explanation to this superior survival ability and an experiment have been carefully carried out to verify the mechanism [4]. these research studies are of great importance for the biomimetic design of functional materials in various fields. a very thin and lightweight cocoon wall has a distinctive hierarchical structure and multiple functions [5-7]; the pupa can suffer from a sudden temperature change in some extreme weather situations, and the cocoon wall plays a critical role in protecting the pupa from energy loss and a sudden heat shock. the hierarchical porous structure of the cocoon wall gives the silkworm pupa unexpected properties, such as energy protection and the heat shock protection. however, a paucity of literature reported about its mechanism of thermal performance, especially a heat shock in the cocoon wall. much literature revealed that a low frequency property of some vibration systems plays an important role in various applications. he, liu and gepreel found that the low frequency property of a porous concrete beam can prevent vibration damage [8]; he, kou, et al. revealed all vibrations in a porous medium have a low frequency property when time tends to infinity [9]; zuo applies the low frequency property of a fractal-like spring system to 3d printing technology [10-12]; he and el-dib studied the frequency property of a fractional kundu–mukherjee–naskar equation [13]. he and his colleagues revealed a long-lost technology to collect water from air by the low frequency theory [14, 15]. the capillary effect is also important in the heat and mass transfer [16, 17]. jin et al. [18] studied the low frequency property of a capillary vibration. lin et al. [19, 20] established a model for a release oscillation in a hollow fiber, and ions release depends upon the low frequency property. in this work, we will study the geometric structure of the silkworm cocoon and analyze its thermal conduction properties and design a porous coating of the building wall which can prevent from a sudden heat shock. 2. geometric analysis of the silkworm cocoon the cocoon wall structure is considered as a porous medium consisting of randomly arranged continuous silk fibers. each layer images of the silkworm cocoon are presented in fig. 1. from the sem micrographs, they reveal that the silkworm cocoons have multilayer and porous structures with double silk fibroin fibers covered by sericin. the morphologies of each layer are remarkably different. from the outer to the inner surface, the density and the amount of bonding of fibers increase. thermal oscillation arising in a heat shock of a porous hierarchy and its application 635 fig. 1 sem micrographs of silkworm cocoon layers from outer (a) to inner (h), scale bar: 30 µm fig. 2 the average diameters of cocoon silks from outer (1) to inner (8) the silk’s diameter of each layer was also measured (fig. 2). it is found that the average diameter of the silk gradually decreases from the outer layer to the inner layer (30.3, 42.3, 34.3, 38, 37.5,33.1, 28.3 and 26.3 µm). in addition, the diameter of the outermost fibers is smaller, which may be due to the component loss caused by the exposure to air. this further illustrates that the cocoon has a unique hierarchical structure. 3. the silkworm cocoon under a sudden heat shock a sudden heat shock will greatly affect a building’s reliability and life. the freeze-thaw damage is the main reason of the failure of a cement-based material [21-29]. if the thermal response to the environment temperature change is slow and the inner temperature will not change much with 12 hours, such freeze-thaw damage can be avoided by covering a porous coating with a structure similar to the cocoon wall. 636 f.j. liu, t. zhang, c.-h. he, d. tian in many cases, a sudden heat shock also refers to a sudden increase or decrease of temperature in harsh environments [30]. in order to cope with this problem, phase change materials (pcm) are widely used in thermal energy systems [31-33]. the main feature of pcm is latent heat storage, which has higher storage density than conventional sensible heat storage due to a high enthalpy change in the phase change process. the latent heat storage based on pcm can be applied in various fields, such as solar heat storage, energy-saving buildings and waste heat recycle, etc [34-37]. however, there always exist diverse defects, such as phase separation, low heat transfer rate, supercooling, leakage in the molten state, instability of performance [38]. especially the buildings on fire, which use phase change materials, will be a very dangerous heat source due to a low heat transfer rate of pcm. this brings great difficulties to firefighters' rescue operations, such as taking longer time to put out the fire, and may even sacrifice more people [39]. hence, thermal conductivity enhancement is one of the main issues for the pcm in the application field of the latent heat storage. in the same way, if the phase change materials are added to the fire suit, the heat cannot be conducted in time, which will directly threaten the lives of firefighters. what we need is a porous hierarchical structure that can not only store heat and keep warm, but also conduct heat in time. the closer it is to the room or the human body, the slower the conduction will be. moreover, people will not feel the discomfort caused by a sudden heat shock. therefore, thermal conductivities of the silkworm cocoon are studied in this paper. the silkworm cocoons are expected to have unique characteristics under a sudden heat shock. thermal conductivities of the silkworm cocoon were carried out using temp.& hum. chamber t/c1000-70. the temperature rise (from the initial conditions of 26 ℃ to 57 ℃) and fall (from 49℃ to 20℃) profiles are obtained and shown in figs. 3 and 4, respectively. fig. 3 temperature profiles for internal of the silkworm cocoon as the ambient temperature increases thermal oscillation arising in a heat shock of a porous hierarchy and its application 637 fig. 4 temperature profiles for internal of the silkworm cocoon as the ambient temperature decreases it can be observed that the temperature inside the silkworm cocoon changes slowly when the ambient temperature change occurs, which indicates that the cocoon has a certain degree of temperature buffering. the internal temperature of the cocoon tends to be close to the surrounding temperature, but the internal temperature will not change immediately when the cocoons encounter a sudden temperature change. therefore, we expect that it may be closely related to the heat flow transfer and the unique porous hierarchical structure of the silkworm cocoon. 4. thermal oscillation and its low frequency property there are many kinds of thermal oscillations. especially, the flow boiling instabilities are undesired phenomena which may cause a premature critical heat flux (chf), high pressure drops, control and operational problems and mechanical vibrations of the system components [40, 41]. chávez et al. studied thermal oscillations during flow boiling of hydrocarbon refrigerants in a microchannels array heat sink [42]. megahed [43] has verified the effect of mass velocity variation on thermal oscillations. recently, kuang et al. [44] found that as the saturation temperature increased, the frequency of the thermal oscillations increased. in general, the frequency and amplitude of the oscillations increase with increasing heat flux. this behavior is due to the intensification of the boiling forces [45]. in addition, kim et al. [46] have investigated the rapid thermal oscillatory flow in an asymmetric micro pulsating heat exchanger (mphe) and demir et al. [47] have studied the dynamics of the bacterial flagellar motor’s angular velocity in response to thermal oscillations while focusing on the effect of frequency. cell growth on nanofibers can be explained by thermal oscillations. fan et al. [48] showed that the capillary-like force which is parallel to the fiber orientation have a good guide for cell orientation. the geometric potential becomes weak when the distance between two adjacent fibers becomes wide. cell orientation can also be guided by the boundary-induced force induced by adjacent nanofibers. besides, lin et al. [19, 20, 49] 638 f.j. liu, t. zhang, c.-h. he, d. tian studied the release oscillation in a hollow fiber and established a fractional model. the results showed that the ions release depends upon the low frequency property. cocoon is a hierarchical porous medium. in this paper, we approximate the porosity in the cocoon wall as a microporous capillary channel for heat transfer. the model schematic is illustrated in fig. 5. heat flow is slowly transferred from the outer layer to the inner layer. fig. 5 schematic of heat transfer in a microporous capillary channel (marked by red lines) for a microporous capillary channel, the newton’s second law for thermal oscillations can be written in the form .. f m x= (1) where x is removal from the equilibrium position, m is the total hot air mass in a microporous capillary channel and f is the force caused by the air pressure difference between the inside and outside the cocoon. force f can be expressed as ( )out inf p p a= − (2) where pout and pin are the hot air pressure outside the cocoon and the air pressure inside the cocoon, respectively. a is a cross-sectional area of the microporous capillary channel. combining eqs. (1) and (2), we have .. ( ) 0out inax x p p a− − = (3) where ρ is hot air density. thermal oscillation arising in a heat shock of a porous hierarchy and its application 639 according to the state equation of gas pv nrt= (4) where p is pressure, v is volume, n is moles of gas, r is the thermodynamic constant and t is temperature. air pressure pin in the cocoon can be written 0 ( ) in in in n rt p v l x a = + − (5) where r is gas constant, v0 is the inner volume of the cocoon (shown in fig. 5), tin is the internal temperature of the cocoon and l is the cocoon thickness. substituting eq. (5) into eq. (3), we get .. 0 0 ( ) in in out n rt ax x p a v l x a − + = + −  (6) after the deformation of eq. (6), it shows .. 0 1 0 ( ) in in outn rt p x ax v l x a x + − = + −  (7) the initial conditions are 0(0)x l= ( ) . 0 0x = (8) the nonlinear equation given in eq. (7) with initial conditions given in eq. (8) can be solved by the homotopy perturbation method [13, 15, 50-52] or by the variational iteration method [53-56], or he’s frequency formulation [57-60]. hereby the taylor series method [61-63] is adopted. after a simple transformation, we obtain the equation: .. 0 0 0 0 0 [ ( ) ] (0) [ ( ) ] out in inp a v l l a n rt x al v l l a + − − = + − (9) the taylor series solution to second order is .. . 2 20 0 0 0 0 0 [ ( ) ](0) ( ) (0) (0) ... 2! 2 [ ( ) ] out in inp a v l l a n rtx x t x x t t l t al v l l a + − − = + + + = + + − (10) according to the oscillation property, we can obtain 2 0 0 0 0 0 0 [ ( ) ] 0 4 2 [ ( ) ] 4 pout in inp tp a v l l a n rtt x l al v l l a  + − −  = + =   + −    (11) where tp is considered the period of the heat flow vibration. from eq. (11), the solution is shown as 640 f.j. liu, t. zhang, c.-h. he, d. tian 0 0 0 0 0 2 [( ) ] 4 [ ( ) ] p out in in a l l a v t l p a v l l a n rt − − = + − −  (12) the frequency of heat flow oscillation is 0 0 0 0 0 2 2 [( ) ] 2 [ ( ) ] p out in in w t a l l a v l p a v l l a n rt = = − − + − −    (13) eq. (13) roughly describes the main factors which affect the frequency of heat flow vibration. the low frequency property is very important for cocoons. it implies the heat cannot be transferred to a long distance, so that the pupa will not be affected by the heat shock outside. in order to ensure that the frequency is small, it can be seen from eq. (13) 0 0 [ ( ) ] 1out in inp a v l l a n rt+ − −  (14) in other words, eq. (14) is equivalent to 0 0 [ ( ) ] 0out in inp a v l l a n rt+ − − = (15) after transformation, we have 0 0 1 in in out n rt l l v a p a   = − −    (16) l0 represents amplitude and l0→0. eq. (16) becomes 0 1 in in out n rt l v a p a   = −    (17) take the derivative of 1/a ( ) 0 1 dl d a = (18) that is 0 2 0in in out n rt v p a − = (19) the optimal pore size can be obtained 0 2 in in opt out n rt a p v = (20) according to the state equation, p rt  = (21) thermal oscillation arising in a heat shock of a porous hierarchy and its application 641 a higher temperature results in a larger pressure (pout), and the porous size should be smaller, so the last cascade of the porous hierarchy of the coating can consist of nanofibers [64]. 5. application in a building exterior wall porous structure with cocoon-like hierarchy has good thermal insulation performance. this kind of structure can play a very good thermal buffer effect when used in building exterior wall coating. the schematic of thermal buffer in hierarchical porous structure is shown in fig. 6. when a sudden heat shock from the external environment reaches the exterior wall coating and passes through the hierarchical porous structure, the temperature changes from fast to slow. the closer to the exterior wall, the extremely slow the temperature change is. finally, the temperature slowly reaches the room temperature. fig. 6 schematic of thermal buffer in hierarchical porous structure leaves have a common hierarchical structure in nature [65-67]. they endure the alternation of temperature difference between day and night. this experiment was to measure the temperature of the upper and lower surfaces of a mulberry leaf every half an hour in the soochow university campus. it was sunny, and the weather favored the measurements. fig. 7 shows the temporal evolution of the leaf surface temperatures from 9:30 a.m. to 3:30 p.m., where the upper and the lower temperatures were marked with the blue and green colors, respectively, and the environment temperature was marked with the red one. the average value of surface temperatures of the mulberry leaf was used in our experiment. it can be seen from fig. 7 that the ambient temperature increases gradually; it reaches the maximum at 2:00 p.m. and then it tends to be stable. due to the sunlight, the temperatures of the upper and 642 f.j. liu, t. zhang, c.-h. he, d. tian lower surfaces of the mulberry leaf change slightly with the rise of the ambient temperature, and both the cases have seen an obvious oscillation in temperature with different frequencies. the upper surface has a higher frequency of the thermal oscillation than the lower one. a higher frequency results in a higher metabolic rate, and it can inspire the specially needed permeability design for cloth and house walls. fig. 7 the temporal evolution of upper and lower surface temperatures of a mulberry leaf 6. conclusions the cocoon wall plays a very good role in protecting silkworm pupae, no matter how the environment changes. the heat flow in the microporous capillary channel moves extremely slowly from the outer layer to the inner layer of the cocoon. it is precisely because of the super slow energy transmission through the cocoon wall that the silkworm pupae will not suffer damage due to either excessively high or low temperatures. the cocoons have a good heat preservation function and the optimal pore size (eq. 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with polygonal cross-sections udc 539.3 qiang li, valentin l. popov department of system dynamics and the physics of friction, tu berlin, germany abstract. using the boundary element method, we numerically study the indentation of prismatic and tapered indenters with polygonal cross-sections. the contact stiffness of punches with flat bases in the form of a triangle and a square as well as a number of higher-order polygons is determined. in particular, the classical results of king (1987) for indenters with triangle and square base shapes are revised and more precise numerical results are provided. for tapered indenters, the equivalent transformed profile used in the method of dimensionality reduction (mdr) is determined. it is shown that the mdr-transformed profile of polygon-based indenters with power function side is given by the power function with the same power; it differs from the 3d profile only by a constant coefficient. these coefficients are listed in the paper for various types of indenters, in particular for pyramidal and paraboloid ones. the determined mdr-transformed profiles can be used for study of other contact problems such as tangential contact, normal contact with elastomers, and, in an approximate way, to adhesive contacts. key words: indentation, contact stiffness, polygonal indenter, boundary element method, mdr transformed profile 1. introduction indentation test is a very common way of probing mechanical properties of materials such as hardness, contact stiffness, elastic modulus and strain-stress relation [1-3]. there is a variety of indenter geometries used in macroand microindentation; the most popular are spherical and pyramidal indenters (e.g. for the vickers hardness test and brinell hardness test) [4]. the contact stiffness of indenters with regular geometries is also important for the foundation design [5]. the analytical solution for contact between a rigid cylindrical flat punch and an elastic half space was given by galin in 1953 (english translation see [6]). his results were later published by sneddon and, in this way, made public to the western world [7]. based on this received september 10, 2016 / accepted november 04, 2016 corresponding author: qiang li institute of mechanics, berlin institute of technology, strasse des 17. juni 135, 10623 berlin, germany e-mail: qiang.li@tu-berlin.de 242 q. li, v.l. popov solution, oliver and pharr proposed an analysis method to determine the hardness and elastic modulus from the load-displacement curves of indentation test [8]. general relations among contact stiffness, contact area, and elastic modulus during indentation have been analytically derived only for axisymmetric indenters. for a non-axisymmetrical geometry, a correction coefficient is needed [9], which can be still found only numerically. in this paper we numerically investigate the indentation of rigid bodies with various geometries: the flat-ended punches in section 2 and tapered indenters in section 3. in both cases we consider different polygonal bases including triangle and square. note that the assumption of a rigid indenter is no restriction as the normal frictionless contact of two elastic bodies with elastic moduli e1 and e2 and poisson numbers ν1 and ν2 can always be reduced to the contact of a rigid indenter and an elastic medium with an effective elastic modulus e * determined as [10] 2 2 1 2 * 1 2 1 11 e ee      . (1) in the present paper, the indentation test is numerically simulated by the high resolution boundary element method (bem), which has recently been generalized to arbitrary contact problems including tangential contact and adhesive contact [11, 12]. 2. indentation of prismatic indenters with polygonal base the normal contact stiffness between a rigid flat cylinder and an elastic half space is given by k=2ae * [7], where a is the radius of the cylinder, and e * is the effective elastic modulus, eq. (1). in the case of a prismatic indenter with an arbitrary base form, the normal contact stiffness is given by [5]: *2 a k e    , (2) where a is the contact area of the base. obviously the value of β is equal to 1 for the flat-ended cylinder. it was proven that eq. (2) is also valid for indenters which have a cross section other than a circle [5]: β=1.034 for triangle and β=1.012 for square. these results were numerically obtained by king in 1987. due to the limitation of computer technology at that time, king used only 200 elements for simulating a triangle indenter, and the fig. 1 prismatic indenters with polygonal bases: m=3 (triangle), m=4 (square), m=5 (pentagon) and m=∞ (circle) indentation of flat-ended and tapered indenters with polygonal cross-sections 243 triangular area looked quite „rugged‟. note that the stiffness of a flat punch and correspondingly factor β are related to the so-called harmonic capacity of the base form of the punch. this analogy was discussed by argatov (2010) [13]. below we repeat the calculations of king using the current high-resolution bem and provide corrected values. using the boundary element method we have numerically carried out the indentation test for different shapes of cross section of indenters: from triangle (m=3), square (m=4), pentagon (m=5) to circle (m=∞) as shown in fig.1. in the simulation, the whole area was divided into 1024x1024 elements where at least 200000 elements were in the contact area. it is at least 1000 times more than in the king‟s simulations; therefore, a much more precise result could be obtained. the values of coefficient β for different m are presented in fig.2 and table1. for the two most popular indenter shapes, the values are: 1.061, for triangle, 1.021, for square,     (3) which is larger than the values reported by king [5]. it can be seen that with the same area of cross section, the stiffness of triangular indenter is for 6% larger than that of a flat cylinder. fig. 2 factor β for different polygonal indenters. the two stars indicate the results obtained numerically by king in 1987 [5] table 1 values of constant β m polygon 3 (triangle) 4 (square) 5 6 7 8 ∞ (cylinder) β 1.061 1.021 1.010 1.005 1.003 1.002 1.000 244 q. li, v.l. popov 3. indentation of tapered indenters with polygonal base and power function side surface now we consider the tapered indenters which have a regular polygonal base, as shown in fig. 3. we begin with the most common type – a pyramid, and then extend it to indenters whose side profile is an arbitrary power function. 3.1. pyramidal indenters for the contact between a rigid cone with profile f (r) = tanθ·r and an elastic half space with effective elastic module e * , the dependence of normal force on indentation depth was analytically found by galin [6] (see also sneddon [7]): * 22 tan n e f d    , (4) where d is indentation depth and θ is defined in fig. 3(c). this solution can be easily reproduced using the method of dimensionality reduction (mdr). in the framework of the mdr [14], any contact problem of an axis-symmetrical profile f(r) with an elastic half-space can be mapped onto a contact of a modified (mdr-transformed) profile g(x): | | 0 2 2 ( ) ( ) | | d x f r g x x r x r     , (5) with properly defined elastic foundation. for a conical profile, f(r) = tanθ·r, the substitution in eq. (5) and integration provides the mdr-transformed profile: ( ) ( / 2) | | tang x x    . (6) a short calculation (see. e.g. [14]) leads to eq. (4). fig. 3 pyramid indenters for n=1 (a)-(c) and parabolic indenters for n=2 (d)-(f) with polygonal base, m=3 (triangle), m=4 (square),and m=∞ (cycle) indentation of flat-ended and tapered indenters with polygonal cross-sections 245 in [15], it was shown that an equivalent mdr-transformed profile does exist not only for axis-symmetrical indenters but also for indenters of arbitrary shape. as shown in [15] and [16], for this sake, quantity l=k/(2e * ) (where k=dfn /dd is the incremental normal stiffness) should be determined numerically as function of indentation depth d. inverse function d(l) is then exactly the unknown mdr transformed profile g(x). let us illustrate this simple procedure on the example of conical indenter. by differentiating eq. (4) with respect of d we get stiffness k=4e * d/(πtanθ) and length l=2d/(πtanθ). inverse relation d=l(π/2)tanθ coincides exactly with the mdr transformed profile (6). this procedure is applicable regardless of whether dependence fn(d) was obtained analytically, numerically or experimentally. in the following, we determine dependence fn(d) numerically and extract from it the mdr-transformed profiles for a number of tapered profiles with polygonal cross-sections (fig. 3). we start with consideration of pyramidal indenters. as shown in fig. 3(a)(b), the bases of the indenter are regular polygons. angle θ is defined as the angle between the ground plane and the 3d indenter side surface as shown in fig.3. in the simulation we calculated the contacts of pyramid indenters with different polygonal bases varying from m=3 to 20, and for each type the angle ranges from =π/64 to 31π/64. all the simulation results show that the one-dimensional profile is still a linear function which can be formulated as: 1d( ) | |g x c x  , (7) with c1d : 1d tanc    , (8) where α is dependent only on polygon order m. for the sake of comparison we can define a fictive rotationally symmetric 3d profile with the same inclination angle: 3d 3d( ) tanf r c r r    . (9) then we can write α =c1d/c3d. the values of α for different shapes of polygons are shown in fig. 4(a) and table 2. for a larger m, the shape of the pyramid indenter is close to a cone, the value of α is almost equal to π/2. fig. 4 coefficient of α for pyramidal indenter n=1 (a) and parabolic indenter n=2 (b) with different polygonal bases 246 q. li, v.l. popov table 2 values of coefficient α m polygon 3 triangle 4 square 5 6 7 10 20 30 ∞ cycle α (n=1) pyramid 1.133 1.356 1.422 1.485 1.510 1.542 1.564 1.568 π/2 α (n=2) paraboloid 1.052 1.493 1.690 1.791 1.848 1.928 1.981 1.993 2 3.2. indenters with arbitrary power function geometry let us now consider the case when the side surface of the indenter is not flat but is given by a power function. an example of parabolic indenter (shape with power 2) is shown in fig. 3(d)-(f). we first remember the corresponding solution for an axisymmetric indenter with an arbitrary power function shape f(r) =cn·r n . according to eq. (5) its one-dimensional mdr-transformed profile is given by: ( ) | |nn ng x c x  , (10) where: ( 2) 2 ( 2 1 2) n n n n        , (11) and γ(n) is gamma function. in particular, for the cone (n=1) κ1= π/2 and for a paraboloid (n=2) κ2=2, corresponding to α =c1d/c3d for m=∞ as shown in fig. 4 and table 2. as in the previous section, we define an axis-symmetrical shape with the same power-law shape as shown in detail in fig. 3. to underline that we have to do with a three-dimensional body which is in contact with a three-dimensional half-space, we denote the corresponding reference shape as 3 3( ) n d df r c r  . (12) this shape coincides with the vertical section of the polygonal indenters (shown by dashed lines in fig. 3). the numerical indentation tests were carried out for different indenters with power function n from 1 to 20 and the polygonal base parameter m from 3 to 30. the results show that the 1d profile for an arbitrary power function is still a power function with the same power. coefficient α =c1d/c3d for the same type of indenter (fixed n and m) is constant (independent of coefficient c3d). an example of parabolic indenter (n=2) is shown in fig. 4 (b), where the values of α for triangle, square and further polygonal based profile are presented. in the limiting case the indenter is a spherical cylinder, and α=2 corresponding to κ2=2 is well-known from the mdr theory [14]. if we use the following parameter instead of α 1 3 d n d c c    , (13) then in the limiting case m=∞, value ξ for any power function n will be equal to 1, ξm=∞=1. some values of ξ, in particular for pyramid and parabolic indenter with triangle and square base are shown in fig. 5 and table 3. indentation of flat-ended and tapered indenters with polygonal cross-sections 247 fig. 5 coefficient of ξ for indenters with power function profile table 3 values of coefficient  n m 3 (triangle) 4 (square) 5 10 20 30 1 (pyramid) 0.723 0.866 0.923 0.986 1.000 1.000 2 (paraboloid) 0.526 0.747 0.845 0.964 0.991 0.997 3 0.384 0.648 0.777 0.947 0.987 0.994 10 0.043 0.241 0.058 0.835 0.957 0.983 20 0.002 0.058 0.190 0.695 0.918 0.964 3.3. consideration of indenters with the same base area in section 2 it is found that the contact stiffnesses of triangular, rectangular indenters and flat cylinder with the same cross-section area are almost the same, and differ at most by 6%. it thus appears to be sensible to try as “reference” indenters the axisymmetrical profiles with the same area of cross-section. this definition is slightly different from the definition in the previous section. for both initial polygonal profile and the reference axisymmetrical profile we carry out the mdr transformation and determine the equivalent 1d-mdr profiles. let us explain the exact procedure on the example of a pyramid indenter (n=1). first, we determine the area of the indenter at different height and construct a cone with exactly the same cross-section areas. then we carry out the three dimensional indentation test of the polygonal indenter by the bem simulation and extract corresponding mdr profile g(x)m-poly and corresponding coefficient c1d,m-poly as described in section 3. for the reference axisymmetrical profile, the corresponding mdr transformed profile and the corresponding coefficient c1d,m=∞ are determined by (5). finally we compare this c1d,m-poly and the coefficient of the axisymmetric conical profile using the ratio 1 , -poly 1 , d m d m c c    . (14) 248 q. li, v.l. popov in an absolute similar way comparisons were also carried out for other power function geometries. the results are shown in fig.6 and table 4. it can be seen that the coefficient c1d of pyramid indenter is close to that of conical indenter: it differs by at most 7% in the case of triangular base (c1d =0.927). it is noted that coefficient c1d cannot directly reflect the contact stiffness. take an example of triangular indenter with power n=20 whose geometry is close to the flat triangular indenter (fig.1a), its ζ is very small ζ =0.295 (m=3, n=20), but the contact stiffness at the large indentation depth is the same to the flat indenter. fig. 6 comparison of coefficient c1d among different indenters with the same base area table 4 coefficient ζ for different power n and polygon m n m 3 (triangle) 4 (square) 5 6 10 20 1 (pyramid) 0.927 0.974 0.988 0.994 1.000 1.000 2 (paraboloid) 0.870 0.951 0.977 0.988 0.997 1.000 3 0.817 0.931 0.966 0.981 0.996 1.000 10 0.536 0.806 0.820 0.893 0.990 1.000 20 0.295 0.651 0.814 0.889 0.973 0.997 4. conclusion indentation of flat-ended and tapered indenters with polygonal base was numerically simulated using the boundary element method. the contact stiffnesses of prismatic punches with the same cross section area are almost same as the cylindrical indenter, where the triangular punch differs at most by 6%. for pyramidal indenter and others with power function side, the one dimensional mdr transformed profile was generated based on the three dimensional simulation of indentation. it is found that the 1d profile is still a power function with the same power and it differs only by a constant factor. the factor was numerically calculated for the indenters with different power function side and different polygonal base. the generated mdr profiles can be used for the further contact problems, such as tangential contact or contact with linear viscoelastic bodies. indentation of flat-ended and tapered indenters with polygonal cross-sections 249 references 1. oliver, w.c., pharr, g.m., 2011, measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology, journal of materials research, 19(1), pp. 3–20. 2. fischer-cripps, a.c., 2000, a review of analysis methods for sub-micron indentation testing, vacuum, 58(4), pp.569-585. 3. hay, j., agee, p., herbert, e., 2010, continuous stiffness measurement during instrumented indentation testing, experimental techniques, 34, pp. 86–94. 4. swadener, j.g., george, e.p., pharr, g.m., 2002, the correlation of the indentation size effect measured with indenters of various shapes, journal of the mechanics and physics of solids, 50(4), pp. 681-694. 5. king, r.b., 1987, elastic analysis of some punch problems for a layered medium, international journal of solids and structures, 23(12), pp. 1657-1664. 6. galin, l.a., 1961, contact problems in the theory of elasticity, north carolina state college, usa 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https://doi.org/10.22190/fume200615026f © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper effect of fiber orientation path on the buckling, free vibration and static analyses of variable angle tow panels nasim fallahi, andrea viglietti, erasmo carrera, alfonso pagani, enrico zappino mul 2 team, department of mechanical and aerospace engineering, politecnico di torino, italy abstract. in this work, the effect of the fiber orientation on the mechanical response of variable angle tow (vat) panels is investigated. a computationally efficient high-order one-dimensional model, derived under the framework of the carrera unified formulation (cuf), is used. in detail, a layerwise approach is adopted to predict the complex phenomena that may appear in vat panels. static, free-vibration and buckling analyses are performed, considering several material properties, geometries, and boundary conditions, and the results are assessed with those obtained using existing approaches. considering the findings of the comparative analysis, several best design practices are suggested to improve the mechanical performances of vat panels. key words: variable angle tow, carrera unified formulation, buckling, free vibration, static analysis 1. introduction advanced tow placement techniques allow fiber to be placed along a curvilinear pattern within each layer. this has led to the emergence of a new class of composite materials named variable angle tow (vat) laminates. compared to the classical composites, the vat laminates provide a more extensive design space and allow engineers to further optimize the final structure in terms of the minimum weight or maximum strength/stiffness [1, 2, 3]. in the vat laminates, the fiber orientation angles vary spatially, owing to which the stiffness received june 15, 2020 / accepted july 18, 2020 corresponding author: nasim fallahi department of mechanical and aerospace engineering, politecnico di torino, corso duca degli abruzzi, 10129, turin, italy. e-mail: nasim.fallahi@polito.it 166 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino properties exhibit local variations. such a stiffness variation can be discrete or continuous with curvilinear fiber paths [4, 5, 6, 7]. it has been reported that vat laminates can improve the structural performance such as the strength and buckling characteristics, and freevibration response of composite structures compared to the corresponding values for classical composites [8, 9]. hyer and lee [10] improved the buckling load in a plate with a circular hole by using variable stiffness composites. several investigations based on numerical models [3, 11, 12] and experimental tests [13] demonstrated the advantages of vat panels in preventing buckling. moreover, the analysis of vat structures has not been limited to simple problems but has dealt with various cases such as composite cylinders [14, 15], thin plates and thinwalled structures [16, 17, 18], for improving first ply failure modes [19], and simply supported rectangular plates under non-uniform uniaxial compression [20]. in addition, the introduction of vat composites can improve the buckling and postbuckling load-carrying capacity under in-plane positive and negative shear loading [21]. furthermore, hao et al. [22] investigated the buckling response of variable-stiffness composite panels by employing the mindlin plate theory in conjunction with the isogeometric analysis and demonstrated that the classical finite element models cannot accurately describe the stiffness variation in vat structures. vat laminates can also be used to modify the dynamic response of composite structures by tailoring the overall structural stiffness. this aspect can be attributed to the effect of the parabolic fiber orientation angles, cutout size, thickness, and boundary conditions on the characteristics of the vat composites [23]. stodieck et al. [24] used a simple mathematical rigid plate model to examine the effect of the vat lay-ups, particularly, the fiber angle variation, on the aeroelastic performance of a wing. furthermore, zhao and kapania [25] used the finite element method to investigate the prestressed free vibration of a simply supported vat laminated plate under uniform end shortening. honda et al. [26] formulated an optimum design methodology to propose novel reinforced composite plates with locally anisotropic structures. in addition, akhavan et al. [27] employed a new p-version finite element method to perform the natural frequency and mode shape analyses of rectangular plates made of variable stiffness composite laminates. labans and bisagni [28] conducted both numerical and experimental investigations pertaining to the buckling and free-vibration response of constant and variable-stiffness cylindrical shells. moreover, samukham et al. [29] used the finite element method to study the dynamic instability of vat composite plates subjected to in-plane loading. the authors employed the first-order shear deformation theory as the displacement field model to derive the governing equations and examined the effect of different parameters including the fiber angle orientation, load parameters, boundary conditions, orthotropy ratio and aspect ratio on the system response. the generalized differential integral quadrature method can be combined with the rayleigh–ritz method to solve the governing differential equation corresponding to the parametric instability problem of vat panels [30]. in a curved panel with vat laminates, the boundary conditions and fiber angles considerably influence the buckling and dynamic instability behavior [31]. furthermore, it has been reported that fiber placement at an angle of 0° and thickness build-up at transversely supported regions of composite panels can help realize a high axial compressive stiffness [32]. viglietti et al. [33]. used refined one-dimensional models based on the carrera unified formulation (cuf) to investigate vat laminates on the dynamic response of complex wing structures. vescovini and dozio [34] proposed an accurate approximation technique effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 167 to perform the vibration and buckling analyses of variable stiffness plates by using the cuf approach in combination with the ritz method. their results pertained to thick variable stiffness laminates such as monolithic and sandwich structures under any combination of boundary conditions. furthermore, a three-dimensional stress field was accurately identified using 1d cuf models with a higher computational efficiency compared to that when using 3d finite element models [35]. the advantages of the layerwise method in the analysis of vat and sandwich beam structure were also demonstrated by patni et al. [36]. specifically, the authors considered the three-dimensional (3d) stress distribution derived using hierarchical serendipity lagrange finite elements and reported on the enhanced structural performance of vat structures compared to that of traditional straight-fiber composites [37]. several studies have reported on the application of the cuf in various cases involving plates, shells, beams, thermoelastics, piezo-electric problems, and aeroelastic applications [38, 39, 40, 41, 42]. in particular, in the 1d cuf beam theory, a polynomial expansion is used to describe the displacement field over the cross-section, allowing the order of the expansion to be considered as a free parameter of the formulation. refined 1d beam models can be modeled based on the equivalent single layer (esl) or layerwise (lw) theories. however, the esl cannot describe the continuity of transverse stresses and zigzag behavior of the displacement along with the thickness of composite layers [43, 44, 45]. in addition, lw theories are defined based on the dependency between the number of unknown parameters and layers [46, 47, 48, 49]. thus, to expand the displacement fields over the cross-section, a taylor expansion with a generic n-order [50, 51] or lagrange polynomial expansion can be used. this discussion indicates that although several reports exist regarding the static and dynamic analyses of vat composite panels performed under the framework of various displacement models as well as different approximation approaches, the effect of the fiber orientation angle on the static, buckling and free-vibration response of vat composite plates under various boundary conditions has not been extensively investigated. therefore, in this work, a high-fidelity one-dimensional beam model is used to study the natural frequencies, critical buckling load, and static response of composite plates made of vat laminates. the results are validated by performing a comparative analysis with several existing approaches. furthermore, a parametric study is performed to examine the influence of various parameters such as the fiber orientation angle, material properties, boundary conditions, and geometry dimensions on the performance of the laminates. in addition, while many works consider the static, free-vibration and bulking analysis separately, in this work all the problems are examined together in order to show that the best panel design is not the one that gives better performances in one of the single analyses, but a better design comes from a trade-off to obtain acceptable performance in all the operational scenarios. 2. vat laminates to represent the variable stiffness properties, the fiber orientation angle can be varied continuously in each ply along either the x or y coordinates or both the coordinates. such design flexibility enables the vat composite to enhance the structural performance. the variation in the vat fiber angles can be mathematically formulated using only a few parameters [52]. thus, in the present work, a vat plate with a linear variation in the fiber 168 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino angle is defined using the following notation to describe the fiber pattern in the vat laminates [8, 53]. 1 0 0 | | ( ) 2( ) y y t t t a     (1) the desired stiffness and strength can be achieved by introducing two different angles, t0 and t1 which denote the lamination angle at the center and edge of the composite laminates, respectively.  y is related to the fiber variation along the y-axis, a is the width of the vat panel, and b is the length of the laminate. fig. 1 illustrates the coordinate system for the vat lamina with a curvilinear fiber design. fig. 1 vat composite model 3. numerical model 3.1. preliminaries the notations and quantities employed in this work are introduced based on continuum mechanics. the vat structures are modeled by using a refined onedimensional model based on the cuf. with no loss of generality, the length of the beam structures is defined along the y axis, and the cross-section is defined on the xz-plane. the displacement vector is denoted as u. superscript t denotes transposition;  and denote the stress and strain vectors, respectively. ( , , ) {     }t x y zx y z u u uu (2) ( , , ) { , , , , , }t xx yy zz xy xz yzx y z       σ (3) ( , , ) { , , , , , }t xx yy zz xy xz yzx y z  (4) where is strain and is defined using a linear differential operator b (6×3 matrix [54]):  bu (5) effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 169 furthermore, based on hooke’s law, the stress vector can be expressed as: σ c (6) where c is the matrix of the elastic coefficients of the material, which can be considered as a variable of the space coordinates in the case of vat panels, as described in detail in the subsequent sections. 3.2. variable kinematics of one-dimensional models the displacement field for the beam structure in the cuf can be defined as: ( , , ) , ( ),         1,2, ,( )x y z f x z y m    u u (7) where f is an arbitrary cross-section expansion over the x,z-plane, u (y) is a generalized displacement vector, and m is the number of expansion terms. the kinematics of the model can be modified according to function f . in this work, 9 and 16 node lagrange elements are used as f expansion polynomials, and they are denoted as l9 and l16, respectively. these expansions are used to formulate cubic and quadratic higher-order kinematics, respectively. the l9 polynomial expansion can be summarized as follows (for more details, please refer to [54, 55]): 1 1 2 2 9 9...x x x xu fu f u f u    1 1 2 2 9 9...y y y yu fu f u f u    (8) 1 1 2 2 9 9...z z z zu fu f u f u    where f1, f2, …., f9 denote the lagrange l9 set over the cross-section, and ux1, uy1, uz1, …., uz9 denote 27 unknown displacement variables that represent the pure displacement components at each node of the l9 element. in the l9 set, the interpolation functions are as follows: 2 2 2 2 2 2 2 2 2 2 1 ( )( )         1,3,5,7 4 1 1 1          2,4,6,8 2 1 1          1 ( )( ) ( )( ) 2 ( ) ) 9( f f f                                 (9) where α and β denote the normalized coordinates and vary over the interval [-1, +1]; for more details, please refer to [56]. the lagrange expansions allow the laminate to be investigated using an lw approach in which an individual kinematics is defined for each layer. therefore, the cross-sections are defined separately in the laminate layers and in each single ply. the use of the lw improves the accuracy of determination of the mechanical behavior compared to that obtained using the classical model based on the esl [57]. in this manner, the actual description of the laminates can be obtained, as shown in fig. 2. 170 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino fig. 2 each layer is modeled independently in the case of layerwise approach where the finite element method is adopted along the y-axis for the discretization of the structure with the generalized displacement vector, which is approximated as: ( ) ( )( , , ) ,          1,2 ,i ix y z f x z n y i k   u q (10) here, index i denotes the number of nodes of the beam element, ni(y) is the shape function, iq denotes the nodal unknowns, and k is the number of nodes on the element. based on the principal virtual displacement, the virtual internal work can be expressed as: int v l dv   t σ (11) where v is the volume of the element, and ‍ is the virtual variation of the strain, which is presented as: ( ,( ) ( ))s j sjf x z n y    b u b q (12) where fs stands as an arbitrary cross-section expansion, nj, is shape function, and qsj is the virtual variation nodal unknown. by combining eqs. (6), (7), (11), and (12), the geometrical relations can be obtained in the linear form. therefore, the virtual variation of the internal work can be defined as: fundamental nucleus ( ) ( ) ( ) ( ), ,int j s iv l n y f x z f x z n y dv    t t sj τiq b cb q t sij sj i   q k q (13) according to cuf, sij k is a (3×3) matrix and is termed as the stiffness fundamental nucleus (fn). in terms of the path function for the vat composites, each layer involves pointby-point continuous angle variations with different values. in the case of vat, the components of the fn are subjected to volume integrals. thus, only two terms of the fn are considered in the following analysis, and the other terms can be obtained by permutations [54]: 22 , , 66 , , 44 , , 23 , , 44 , , ‍‍‍ ‍‍ sij xx x s x i j z s z i j s i y j y v v v sij xy s x i y j x s i j y v v k c f f n n dv c f f n n dv c f f n n dv k c f f n n dv c f f n n dv                (14) in this case, stiffness coefficients c vary within the computational domain; therefore, these coefficients must remain inside the integral of the fn. effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 171 3.3. numerical implementation of the vat concept in general, when using finite elements, integrals can be obtained by using the wellknown gauss–legendre formula. the integral form of the function is evaluated in the (,) domain by considering a natural system (for more details, please refer to carrera et al. [54]). in the vat structure, each fiber path can be defined as an arbitrary function, and the fibers follow a curvilinear pattern. hence, each position corresponds to a different stiffness value. furthermore, in the vat composite, the lamination angle should be accurately defined in the entire domain of the plate, in which c is no longer constant. in this manner, the integral can be introduced in the unique form of the volume, as presented in eq. (14). in this work, the gauss integration technique is used, and the material coefficients in the vat composite can be evaluated in a specific gauss point. therefore, in the cuf framework, the real values of the lamination angle at each gauss point are considered. furthermore, the use of the 1d cuf beam model ensures a smoother approximation of the component stiffness compared with that obtained using the finite element method; for more details, please refer to [58]. fig. 3 illustrates a simplified example of the vat concerning the gauss points for four nodes; in contrast, nine gaussian points were used in the present study. fig. 3 vat definition by gaussian points the cuf material properties can be evaluated in the vat case by defining the correct gauss integration point-to-point in the lamination by calculating the fn [54]. 3.4. linearized buckling equations this section focuses on the linearized buckling problems. the tangent stiffness matrix can be obtained by linearizing the virtual variation of internal strain energy  ( intl ): ( ) ( )t sij t int i sj v l dv         0 q k q σ (15) where (lint) is calculated considering the sum of the linear stiffness and virtual variation work associated with the initial stresses 0 . subsequently, by using the cuf formulation 172 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino in eq. (7) and fem in eqs. (10) and (16), the following formulation can be obtained based on the green–lagrange nonlinear strain and displacement relations (see carrera et al. [55]): ( ) ( )t sij t sij t sij sij int i sj i sj i sjl                   0 0q k q q k q q k k q (16) where k sij is the same as in eq. (14), and a new sij 0k appears in the form of a diagonal matrix, which is investigated as the fn of the geometrical stiffness matrix and is expressed for the buckling case as follows: , , , , , , , , , , , , , , , , , ( ‍‍‍ ‍‍‍ ‍‍‍ sij xx x s x i j yy s i y j x zz z s z i jv v v xy x s i j y xy s x i y j xz x s z i jv v v xz z s x i j yz z s i j y yz s z iv v v f f n n dv f f n n dv f f n n dv f f n n dv f f n n dv f f n n dv f f n n dv f f n n dv f f n                                      0 0 0 0 0 0 0 0 0 0 k , )y jn dv i (17) in eq. (17), the stress tensor is determined by the 9 components corresponding to a 3×3 identity matrix i. moreover, depending on shape function (ni) and function f over the cross-section, any desired beam model can be accessible in the cuf framework. finally, the global matrices are assembled in the classical fem. the critical buckling loads are determined as initial stress states 0 , which render the tangent stiffness matrix singular; i.e., | | 0 0 k k [55]. 3.5. free vibration equations under the cuf framework, the same approach as that based on eqs. (10-13) can be employed for solving free vibration. subsequently, the work done by the inertial forces provides the fundamental nucleus of the mass matrix, as defined in carrera et al. [55]: ‍ t ine v l dv    u u (18) where ρ is the material density, and ü denotes the acceleration vector. from this expression, the fn of the mass matrix can be found straightforwardly and be used to solve usual free vibration problems. 4 results and model validation 4.1. convergence analysis for a single-layer vat panel a convergence analysis is performed considering a single-layer vat plate comparing different kinematic models and meshes with a different refinement level. the results are validated using a classical fe model developed in nastran. the properties of the lamina are as follows: e1=50 gpa, e2= e3=10 gpa, g12= g13= g23=5 gpa, υ12= 0.25, with a thickness of 0.02 m and dimensions a=b=1 m. a linear variation of the fiber orientation is considered and expressed using parameter <t0|t1>, as shown in eq. (1). the boundary conditions are shown in fig. 4. effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 173 fig. 4 boundary conditions of the single-layer vat plate, <t0|t1>= <75°|15°> cubic beam elements, b3, are used along the beam axis (y) with two different polynomial expansions, l9 and l16, to enable the beam kinematics approximation on the cross-section (xzcoordinates). at first, to evaluate the convergence of the models, refinement is performed along the z-direction with nz * = (1, 3, 6, 9) considering the l9 lagrange polynomial expansions on the cross-section. in this case, the number of elements in the x and y directions (10b3) is considered constant (see fig. 5). in the next step, refinement was performed simultaneously along the beam axis with ny * = (5, 7, 10, 15, 20, 30) b3 and along the plate width considering nx * = ny * . this refinement was performed considering both the l9 and l16 expansions on the cross-section, as shown in figs. 6 and 7, respectively. fig. 5 refined elements over the cross-section 174 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino fig. 6 refined elements through the beam with l9 expansion over the cross-section fig. 7 refined elements through the beam with l16 expansion over the cross-section table 1 summarizes the results obtained for the first and second critical buckling loads for each of the models considered. the results indicate the convergence of the proposed cuf approach with a significantly lower number of dof as compared to those used in the nastran model (see figs. 8 and 9). furthermore, the results indicate that the cuf model with l9 polynomial expansions and 10b3 beam elements can converge satisfactorily compared to the nastran model. therefore, this mesh can be employed for further modeling. 4.2. pre-buckling and buckling analyses of a sixteen-layer vat plate a 16-ply balanced symmetric square plate, proposed in [22], is considered in this section. the square plate has a length of a=254 mm. the stacking sequence, in according with eq. (1) is expressed as . the panel is loaded with a pure compression load and is simply supported, as shown in fig. 10. the lamina properties are set as follows: e1=181 gpa, e2=10.270 gpa, g12 = g13= 7.170 gpa, g23 =3.780 gpa, υ12=0.28, and each ply has a thickness of 0.15 mm. three cuf based models with 10b3, 15b3, and 20b3 beam elements have been considered corresponding to 160, 240 and 320 l9 cross-sectional elements, respectively. the results have been compared with a classical fem model presented in [22]. as reported in table 2, the cuf model converged with a significantly lower number of dofs compared to classical fem model. the convergence of the first six critical loads evaluated with the proposed cuf-based models is presented in fig. 11. effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 175 table 1 linear elastic buckling estimates according to the number of elements through the beam and cross-section model dof 1st critical load 2nd critical load nz * nx * = ny * nastran 61206 242406 964806 1.92 n 1.85 n 1.85 n 2.04 n 2.02 n 2.02 n cuf l9 3969 9261 17199 25137 1.91 n 1.90 n 1.90 n 1.90 n 2.04 n 2.03 n 2.03 n 2.03 n 1 3 6 9 10 10 10 10 cuf l9 1089 2025 3969 8649 15129 2.13 n 1.98 n 1.91 n 1.87 n 1.86 n 2.57 n 2.18 n 2.04 n 1.98 n 1.97 n 1 1 1 1 1 5 7 10 15 20 cuf l16 2112 3960 7812 17112 30012 1.97 n 1.92 n 1.88 n 1.86 n 1.85 n 2.48 n 2.14 n 2.02 n 1.97 n 1.96 n 1 1 1 1 1 5 7 10 15 20 nz * = number of elements through the thickness nx * = number of elements along the width ny * = number of elements along the beam axis fig. 8 first critical buckling load vs. dof, based on the refinement of the beam elements 176 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino fig. 9 second critical buckling load vs. dof, based on the refinement of the beam element fig. 10 ssss boundary condition (bc) of the plate table 2 first five critical loads for the sixteen-layer plate [<60°|15°><60°|15°>/<-60°| -15°><60°|15°>] model dof mode 1 (kn) mode 2 (kn) mode 3 (kn) mode 4 (kn) mode 5 (kn) ref. [22] cuf 10b3 cuf 15b3 cuf 20b3 387205 43659 95139 166419 13.62 13.78 13.61 13.67 21.62 22.03 21.69 21.68 35.40 37.67 35.94 35.69 54.46 55.24 54.51 54.60 56.01 60.57 57.65 56.69 effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 177 fig. 11 convergence of the first six critical loads using different cuf-based models pre-buckling and buckling analyses considering various stacking sequences were performed in order to determine non-uniform stress distributions due to the in-plane load and their effects on the critical loads. herein, t0 is a fixed angle, and t1 increases from 0° to 90° in steps of π/15, as reported in table 3. the model with 10b3 elements is considered. table 3 different lay-up considered for the laminate lay-up stacking sequence [<t0°|t1°>/<-t0°|t1°>/<-t0°|-t1°>/<t0°|t1°>]4 t0° t1° 1 60 0 2 60 15 3 60 30 4 60 45 5 60 60 6 60 75 7 60 90 table 4 reports the displacements fields and the normal in-plane stresses of some of the stacking sequences considered. table 5 shows the first five critical buckling modes and the related critical load values for all the lamination schemes considered. the results demonstrate that curvilinear fiber paths can be used to modify the in-plane stress distributions. non-uniform stress fields can be obtained although the panel is subject to a constant uniaxial compression. in the buckling case, as shown in table 5, the variation of fiber orientations, t1, and the resulting change in the stress field lead to significant changes in the critical buckling loads and their modal shapes. 178 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino for instance, increasing t1 from 0° to 45°, the first critical load increases by 47.64% , while, for a lamination angle, t1, higher than 45°, the first mode to appear has two halfwaves instead of one in the direction of the applied load. fig. 12 reports the values of the first six critical loads with the variation of lamination angle t1. these results demonstrate that the proposed cuf model can be used to evaluate the complex stress fields resulting from the use of vat laminas, that is, this approach provides an accurate prediction of the geometric stiffness and critical loads of such complex structures. to validate further the performances of the present model the free vibration of vat laminates is considered in the following section. table 4 pre-buckling displacement and stress distribution for some of the staking sequences considered. the stress fields have been evaluated at the top of the plate. design displacement lay-up 1 lay-up 2 lay-up 4 lay-up 6 lay-up 7 effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 179 table 5 first six critical loads and modal shapes design mode 1 (kn) mode 2 (kn) mode 3 (kn) mode 4 (kn) mode 5(kn) mode 6 (kn) lay-up 1 fcr1=11.44 fcr2= 23.60 fcr3= 42.40 fcr4= 52.08 fcr5= 54.69 fcr6= 68.99 lay-up 2 fcr1= 13.78 fcr2= 22.03 fcr3= 37.67 fcr4= 55.24 fcr5= 60.57 fcr6= 64.98 lay-up 3 fcr1= 16.22 fcr2= 20.62 fcr3= 32.61 fcr4= 50.89 fcr5= 59.69 fcr6 = 67.66 lay-up 4 fcr1= 16.89 fcr2= 18.29 fcr3= 26.46 fcr4= 39.44 fcr5= 57.42 fcr6= 62.02 lay-up 5 fcr1=14.56 fcr2= 14.93 fcr3= 19.50 fcr4= 27.62 fcr5= 39.17 fcr6=54.04 lay-up 6 fcr1= 10.44 fcr2= 11.28 fcr3= 14.05 fcr4=18.90 fcr5=26.63 fcr6=36.73 lay-up 7 fcr1= 7.76 fcr2= 8.42 fcr3=11.43 fcr4= 14.71 fcr5= 21.05 fcr6= 27.68 180 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino fig. 12 first six critical buckling loads for different t1 values 4.3. free vibration analysis of a sixteen-layer vat plate in this section, the free-vibration response of a vat panel has been investigated considering different geometries and boundary conditions, see eq. 19. various stacking sequences see table 3, have been considered. two different geometries have been used: square and rectangle plates with a/t=105.85 and a/t=52.92 (a=0.254, b=0.127 m), as shown in figs. 14 (up) and 14 (down), respectively.                                                             lay up lay up case lay up square case lay up rectangle case lay up case lay up lay up                                    plate geometry bc stackin 1 2 1 3 2 4 3 5 4 6 7         models g sequence 56 (19) effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 181 case 1 case 2 case 3 case 4 fig. 14 four-edge plates subjected to four different bcs, case 1 (ssss), case 2 (ssss-i), case 3 (cccc) and case 4 (cfcf) the first six natural frequencies for the square and rectangle geometries for various bcs are summarized in tables 6 to 9. the results for case 1 (ssss), as described in table 6 and fig. 15, indicate that when the square panel is considered the first natural frequency exhibits a growth of 14.57% with an increase of t1 from 0° to 45°, whereas in lay-ups 5, 6, and 7 (t1 from 60° to 90°) the first frequency values decrease by 16.47%. in contrast, in the rectangular panel, with the increase in the t1 angle from 0° to 90°, the first frequencies increase by 71.91%. the natural frequencies obtained for cases 2 (ssss-i) and 3 (cccc) are listed in tables 7 and 8, respectively. figs.16 and 17 show that the variation of t1 results in similar effects on natural frequencies when cases 1, 2 and 3 are considered. however, in case 4 (cfcf) the results indicate that when the square plate is considered, an increase in the fiber orientation angles (t1) leads to an increase of 130.96% in the first natural frequency values, see tab. 9 and fig. 18. these findings demonstrate that the frequency behaviors are strongly dependent on the geometry, boundary conditions, and stacking sequence designs. the large design space coming from the use of vat composite materials makes it possible to optimize the staking sequence exceeding the performance normally obtained by classic composite materials. table 6 first six natural frequencies for case 1 ssss a/h design mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 =105.85 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 229.50 246.31 260.17 262.96 252.34 230.86 210.77 537.07 590.42 591.19 551.92 495.98 428.17 372.97 659.99 635.41 656.93 688.61 701.96 702.29 635.35 838.68 911.18 948.65 961.98 859.34 728.86 703.71 945.24 984.15 1016.47 983.72 946.74 870.24 769.15 1026.53 1149.90 1131.57 1048.42 1009.32 948.18 893.22 =52.92 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 897.56 988.30 1125.46 1259.12 1379.00 1479.00 1543.10 1332.14 1402.41 1500.44 1576.96 1638.24 1702.01 1758.48 2051.60 2159.78 2338.30 2228.99 2057.49 1923.68 1916.83 2193.36 2303.63 2550.20 3060.88 2645.51 2291.65 2249.48 2323.23 2393.00 2747.62 3136.75 3431.71 2830.93 2754.41 3304.48 3454.18 3392.48 3339.63 3615.06 3593.33 3436.64 182 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino table 7 first six natural frequencies for case 2 ssss-i a/h design mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 =105.85 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 485.38 483.44 470.75 452.97 436.04 424.72 420.64 898.62 935.55 885.60 800.44 716.54 650.46 611.19 1019.77 974.59 984.91 1011.30 1034.19 1004.93 914.40 1389.39 1401.63 1392.64 1321.94 1150.35 1055.29 1076.54 1521.14 1636.83 1533.39 1396.73 1357.23 1326.70 1307.15 1824.45 1697.43 1747.67 1847.50 1748.45 1508.17 1357.56 =52.92 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 897.16 987.92 1125.45 1258.87 1378.90 1479.00 1543.09 1330.80 1401.45 1500.43 1576.27 1637.87 1702.01 1758.35 2050.53 2158.27 2338.29 2227.66 2056.67 1923.67 1916.47 2190.15 2301.56 2550.18 3058.72 2643.98 2291.64 2248.80 2321.18 2390.72 2747.59 3135.22 3429.08 2830.92 2753.17 3298.19 3450.16 3392.46 3337.48 3614.83 3593.32 3434.61 table 8 first six natural frequencies for case 3 cccc a/h design mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 =105.85 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 485.52 483.66 471.05 453.29 436.28 424.89 420.77 899.09 936.15 886.28 801.14 717.03 650.70 611.32 1020.00 975.06 985.77 1012.23 1034.96 1005.31 914.56 1390.14 1402.73 1394.25 1323.38 1151.26 1055.87 1076.99 1522.16 1638.32 1534.78 1398.49 1358.57 1327.43 1307.54 1824.80 1698.27 1749.73 1849.72 1750.10 1508.77 1357.76 =52.92 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 898.47 989.23 1126.72 1260.92 1380.96 1480.54 1544.27 1333.59 1403.79 1502.21 1579.44 1640.75 1703.27 1759.09 2053.71 2161.57 2341.32 2232.79 2060.86 1924.99 1917.67 2195.50 2306.00 2553.10 3066.07 2649.94 2293.35 2250.82 2326.34 2395.29 2751.29 3141.84 3437.33 2833.02 2755.95 3307.40 3456.95 3396.34 3346.17 3620.55 3595.84 3438.23 table 9 first six natural frequencies for case 4 cfcf a/h design mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 =105.85 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 168.47 191.95 226.63 271.46 321.29 364.45 389.11 221.02 251.97 290.17 332.93 378.61 418.01 437.02 438.20 508.91 544.89 560.23 552.72 527.06 495.56 467.69 510.87 618.04 753.28 830.02 738.07 653.91 496.24 574.94 685.38 822.39 897.86 1008.61 929.14 819.45 938.46 928.32 895.22 974.37 1061.33 1064.60 =52.92 lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 715.46 547.99 579.07 846.73 1276.64 1477.97 1500.65 825.44 575.96 597.58 871.56 1331.63 1634.49 1497.18 1217.10 1075.40 1151.93 1375.78 1529.77 1683.48 1554.01 1748.81 1420.49 1440.75 1743.31 1842.07 1791.30 1812.39 1862.21 1440.48 1456.42 2205.91 2269.04 1993.34 2028.78 1990.40 1531.47 1519.91 2211.32 2825.54 2321.45 2381.92 effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 183 fig. 15 sensitivity of the first six frequencies, bc case 1 for different fiber orientations t1 fig. 16 sensitivity of the first six frequencies, bc case 2 for different fiber orientations t1 fig. 17 sensitivity of the first six frequencies, bc case 3 for different fiber orientations t1 184 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino fig. 18 sensitivity of the first six frequency, bc case 4 for different fiber orientations t1 4.4. trade-off analysis identifying the best lamination strategy requires a compromise to fulfill the design requirements in each operational scenario: static, dynamic or buckling. in this section, a preliminary trade-off analysis is performed using the results shown previously. the results of the performed linear buckling, free vibration and static analyses are compared considering the lamination strategies reported in table 3. the fully simply supported boundary condition is used. the results of the buckling analysis are obtained from table 5 corresponding to the boundary condition shown in fig. 10. the results of the natural frequencies for the different vat lay-up schemes were considered based on table 6 where a fully simply supported square plate is investigated. in the case of static analysis, a transverse uniform pressure has been applied to the simply supported panel. to investigate the effect of different lay-up designs of the vat a weighted index has been introduced. the critical buckling loads, natural frequencies and maximum displacement are considered. the normalization in these cases corresponds to the scaling of a variable to have a positive value greater than 0 or a maximum value of 1. consequently, each value includes a weight during the evaluation, as follows: 1 ,max i i w max max x x x i x x         0 1i max x x  (20) where iw is introduced as a weighted index for the various analyses, and xmax denotes the maximum value of xi for each variable in every individual lay-up design. the results for the three analyses are presented in table 10. for both the buckling and free-vibration analyses, the minimum weighted index values correspond to lay-up 7. in table 10, the sum of all the weighted indexes is presented for each lay-up design to evaluate the performance of each fiber orientation angle. in addition, the mean values, and percentages of all the weighted indexes are summarized. considering the buckling modes, lay-up 2 exhibits the highest percentage of 93% among those of all the other lay-up designs. lay-ups 2, 3 and 4 appear as the optimal designs in both the buckling and free-vibration analyses. in terms of the static behavior, the optimal effect of fiber orientation path on the buckling-free vibration, and static analyses of variable angle... 185 models for the vat lay-ups are different. lay-up 1 provides the best compromise among the set-up considered since it can keep a percentage higher than 90% in all the analyses considered. the results indicate that the buckling analysis is more sensitive to the change in the fiber angle orientation compared to the other two analyses. the mean value of the buckling load indexes ranges from 5% to 93% of the maximum value. the displacement obtained in the static analysis ranges from 68% to 100% of the maximum value. finally, the mean value of the natural frequency indexes ranges from 76% to 98%. table 10 weighted indexes for evaluating the lay-up schemes type of analysis iw modes lay-up 1 lay-up 2 lay-up 3 lay-up 4 lay-up 5 lay-up 6 lay-up 7 freevibration analysis mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 0.87 0.91 0.94 0.87 0.93 0.89 0.94 1.00 0.90 0.95 0.97 1.00 0.99 1.00 0.94 0.99 1.00 0.98 1.00 0.93 0.98 1.00 0.97 0.91 0.96 0.84 1.00 0.89 0.93 0.88 0.88 0.72 1.00 0.76 0.86 0.82 0.80 0.63 0.90 0.73 0.76 0.78 sum mean percent (%) 5.42 0.90 90.26 5.76 0.96 95.93 5.90 0.98 98.25 5.79 0.97 96.56 5.50 0.92 91.68 5.04 0.84 84.01 4.60 0.77 76.70 critical buckling load mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 0.68 1.00 1.00 0.94 0.90 1.00 0.82 0.93 0.89 1.00 1.00 0.94 0.96 0.87 0.77 0.92 0.99 0.98 1.00 0.78 0.62 0.71 0.95 0.90 0.86 0.63 0.46 0.50 0.65 0.78 0.62 0.48 0.33 0.34 0.44 0.53 0.46 0.36 0.27 0.27 0.35 0.40 sum mean percent (%) 5.52 0.92 92.05 5.58 0.93 93.00 5.49 0.91 91.50 4.96 0.83 82.67 3.88 0.65 64.74 2.74 0.46 45.69 2.10 0.35 35.01 static maxdisplacement 0.92 0.79 0.71 0.69 0.74 0.86 1.00 structural analysis sum mean percent (%) 0.92 0.92 91.78 0.79 0.79 79.45 0.71 0.71 70.94 0.69 0.69 68.91 0.74 0.74 73.69 0.86 0.86 85.76 1.00 1.00 100.00 5. conclusions in this work, buckling, free-vibration, and static response analyses of vat laminates were performed under the cuf framework. a one-dimensional cuf beam theory was used, and a layer-wise approach was adopted as cross-sectional kinematic. the spatial variation of the fiber orientation has been described in a rigorous manner with a smooth and continuous variation of the panel stiffness. different lay-up was considered to reveal the effects of the use of curvilinear fibers on the static, buckling and free vibration response of a square symmetric vat plate. the results have been compared with those presented in literature and with classical fem models. 186 n. fallahi, a. viglietti, e. carrera, a. pagani, e. zappino a preliminary convergence analysis was conducted to assess the present computational model. the results show the computational efficiency of the current approach that can ensure an accurate prediction of the critical loads with a fraction of the computational cost required by classical fem models. the parametric analysis of a sixteen-layer panel was conducted to study the effects resulting from the variation of the lamination parameters. static, buckling and free-vibration analyses were conducted. the results show that vat lamination schemes can be used to redistribute the in-plane normal stress fields, that is, the critical buckling loads can be modified to fulfill the design requirements. the free vibration analyses have pointed out that different lay-ups led to a variation of the dynamic response of the structure. the effectiveness of the vat depends on the geometry and boundary conditions of the structure. in conclusion, the results pointed out that the use of vat 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kinematic models, composites part b: engineering, 171, pp. 272-283. 11564 facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 1 20 https://doi.org/10.22190/fume230125009b © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper discrete-time model-based sliding mode controllers for tower crane systems anamaria-ioana borlea1, radu-emil precup1,2, raul-cristian roman1 1politehnica university of timisoara, department of automation and applied informatics, timisoara, romania 2romanian academy – timisoara branch, center for fundamental and advanced technical research, timisoara, romania abstract. this paper applies three classical and very popular discrete-time modelbased sliding mode controllers, namely the furuta controller, the gao controller, and the quasi-relay controller due to milosavljević, to the position control of tower crane systems. three single input-single output (siso) control systems are considered, for cart position control, arm angular position control and payload position control, and separate siso controllers are designed in each control system. experimental results are included to support the comparison of the three plus three plus three sliding mode controllers. key words: discrete-time model-based sliding mode controllers, furuta controller, gao controller, tower crane systems 1. introduction sliding mode control (smc) is a particular kind of variable structure system originated in the early 60’s [1]. the main idea of the general variable structure control (vsc) laws is to use a high-speed switching control scheme to drive the process state trajectory onto a specified hyper-surface, which is commonly called the sliding surface or switching surface, and next keep the process state trajectory moving along this surface [2] in order to meet the performance specifications imposed to the control system. the discontinuous nature of the control signal helps to maintain a high performance of smc and vsc by switching between two distinct control structures [3]. because of received: january 25, 2023 / accepted march 05, 2023 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania romanian academy – timisoara branch, center for fundamental and advanced technical research, bd. mihai viteazu 24, 300223 timisoara, romania e-mail: radu.precup@aut.upt.ro 2 a.-i. borlea, r.-e. precup, r.-c. roman this switching behavior, smc can have some dead zones for parameters variations, and is not sensitive to disturbances [4]. this paper is focused on smc of tower crane systems. such control approaches are outlined as follows. in [5], a continuous-time smc law is designed based on a nonlinear model of a tower crane system (tcs). two variants of discrete-time data-driven smc laws for tcss are presented in [6]. many papers on different controllers for cranes have been reported in the past years. the smc problem for overhead crane is the subject of [7], which describes a model based integral smc scheme for discrete-time systems. the authors of [8] offer a modelbased combination of smc. second-order smc for controlling the trolley in the xoy plane is addressed in [9]. an adaptive fuzzy smc is developed by [10] for trolley position and sway control in the xoy plane, where two linear sliding surfaces are defined for the position and sway angle. tower cranes can be found in fewer papers then overhead cranes because of their increased complexity. an adaptive control scheme for underactuated tower cranes is proposed in [11] to achieve simultaneous slew/translation positioning and swing suppression, using this approach is no need for linearize the tower crane dynamical equations around the equilibrium point or to neglect nonlinear terms. controllers which are composed of partial feedback linearization and smc are suggested in [13], guaranteeing the robustness in the case of variations of several system parameters. integral sliding mode control (ismc) for tower cranes is proposed in [13] to ensure precise tracking of the desired position while reducing the oscillations of the payload. the controller in [13] is designed using a high fidelity nonlinear dynamical model, however, the switching gain must be limited to implement the smc on the real tcs, and as a result a steady-state error will be present in the system’s outputs; this shortcoming is overcome using ismc. the purpose of this paper is to apply model-based smc to a representative nonlinear process, namely the tcs. to demonstrate the performance of these approaches, the validation through experiments is presented. three approaches to discrete time sliding mode control (dtsmc) are described based on the mathematical model of the process: the dtsmc law in the approach proposed by k. furuta [14], the dtsmc law in the approach proposed by w.-b. gao et al. [15] and the dtsmc law of quasi-relay type proposed by č. milosavljević [16]. the remainder of this paper is organized as follows. in section 2, the dtsmc problem formulation and the three control laws are described. the validation study is presented in section 3. conclusions are presented in section 4. 2. model-based sliding mode control problem formulation let the dynamical discrete-time system be described by the following single input linear time-invariant state-space mathematical model:    = +=+ , , : 1 kk kkk u p xcy bxax (1) where xk = [xk,1 … xk,n]t  n is the state vector, t indicates matrix transposition, k indicates the discrete time (sampling interval) index, kz, k0, the (scalar) control input discrete-time model-based sliding mode controllers for tower crane systems 3 is uk, yk = [yk,1 … yk,p]t  p is the output vector, and ann, bn1 and cpn are the system matrix, the input matrix, the output matrix, respectively [17]. although the model in (1) is linear, the aim of smc is to control nonlinear processes; however, this linear model is considered as it allows a relatively simple analysis and design of the control systems according to [14], [15] and [16]. using the notation sk for the switching variable, the sliding hyper-surface n kk n k ss === }0{ xkx (2) is determined by choosing the gain matrix of the sliding hyper-surface n 1 k so that the system (1) is stable as long as xk remains on s [14]. 2.1. furuta dtsmc law a type of dtsmc system is proposed by k. furuta in [14], in which the discrete-time control law for the system (1), referred to here as the furuta dtsmc law, is ,d k eq kk uuu += (3) where d ku is discontinuous control input and the equivalent control input is eq ku . imposing sk = 0, the control law to keep the state on (2) is given by ,)()( , 11 n eq keq eq ku − −−= = iakbkf xf (4) with i – the identity matrix of order n. the discontinuous control law is described as .] ... [ , 1 1 nt nd kd d k ff u = = f xf (5) it is proved in [14] that the system (1) controlled with the control law in (3) is stable if the absolute value of the ith element of fd, i.e. fi, i=1…n, satisfies      −  − = ,)(if ,)(if0 ,)(if ,0 , ,0 iikk iikk iikk i xsf xs xsf f    bk bk bk (6) in which xk,i is the ith element of xk and i is defined as ,||)(||5.0 1 , 2 ,0  = = n j jkiki xxf bk (7) with the amplitude of f0 limited by , )( 2 0 1 1 0  =   n j jt f bk (8) 4 a.-i. borlea, r.-e. precup, r.-c. roman where t1 is the first column of the matrix [t1 t2 … tn] in which t1i is the ith element of t1 satisfying kt1=1, kti=1, i=2…n; furthermore t1 and ti, with i=2…n, are linearly independent [14]. the gain matrix k shall be designed so that the system [14] kk xiakbkbax −−= − + )]()([ 1 1 (9) is stable. the performance specifications imposed to the control system are zero stationary control with respect to constant reference inputs and reasonable settling time. the guidelines to design the furuta dtsmc law such that to meet these performance specifications will be given in section 3.1. 2.2. gao dtsmc law a type of smc system is described by w.-b. gao et al. in [15], where the dynamic behavior of the reaching law is as follows: ,)sgn(1 kskskk ststqss −−=−+  (10) where ts is the sampling period, q is a scalar parameter that fulfills qts(0,1) and >0 is a parameter. solving for uk the equation obtained from (1), (2) and (10) leads to the following control law, referred to here as the gao dtsmc law [15]: )}.sgn( ])1([{ )( 1 ksksk ttqu xkxiak bk +−−  −=  (11) imposing the same performance specifications as in section 3.1, the guidelines to design the gao dtsmc law will be presented in section 3.2. 2.3. quasi-relay dtsmc law č. milosavljević describes in detail in [16] the quasi-relay control law for the system (1), referred to as the quasi-relay dtsmc law, with the expression ,1),sgn( 1 , −      −=  = nrsxu k r i ikik  (12) where i>1 are the parameters which must be chosen by the designer. the general condition of existence and achievement of the quasi-sliding mode is [16] .0 ,z ,|||| 1 + kkss kk (13) imposing the same performance specifications as in section 3.1, the guidelines to design the quasi-relay dtsmc law will be specified in section 3.3. discrete-time model-based sliding mode controllers for tower crane systems 5 3. validation case study the tcs is presented in detail in [6] and [18], as representative process to validate various controls algorithms including those discussed in relation with this process. the tcs is a nonlinear electromechanical system with a complex dynamic behavior. the system illustrated in fig. 1 has three controlled outputs, namely the cart position y1(m)=x3(m), the arm angular position y2(rad)=x4(rad) and the payload position y3(m)=x9(m). fig. 1 block diagram of principle of tower crane system [6]. the three actuators shown in fig. 1 are direct current (dc) motors, and pulse width modulation (pwm) is involved. the variable m1[−1, 1] is the output of the saturation and dead zone static nonlinearity specific to the first actuator:          −− − −−−+ −− = ,)( if,1 ,)( if),/())(( ,|)(| if,0 ,)( if),/())(( ,)( if,1 )( 11 1111111 111 1111111 11 1 b baaba ac cbcbc b utu utuuuuutu utuu utuuuuutu utu tm (14) where t is the continuous time argument, tℜ, t≥0, u1(%)[−100, 100] and u1[−1, 1] is the first control input for cart position control, and the values of the parameters in (14) are [6] ua1=0.1925, ub1=1 and uc1=0.2. the variable m2[−1, 1] is the output of the saturation and dead zone static nonlinearity specific to the second actuator:          −− − −−−+ −− = ,)( if,1 ,)( if),/())(( ,|)(| if,0 ,)( if),/())(( ,)( if,1 )( 22 2222222 222 2222222 22 2 b baaba ac cbcbc b utu utuuuuutu utuu utuuuuutu utu tm (15) where u2(%)[−100, 100] and next u2[−1, 1] is the second control input for arm angular position control, the values of the parameters in (15) are [6] ua2=0.18, ub2=1 and uc2=0.1538. the variable m3[−1, 1] is the output of the saturation and dead zone static nonlinearity specific to the third actuator: 6 a.-i. borlea, r.-e. precup, r.-c. roman          −− − −−−+ −− = ,)( if,1 ,)( if),/())(( ,|)(| if,0 ,)( if),/())(( ,)( if,1 )( 33 3333333 333 3333333 33 3 b baaba ac cbcbc b utu utuuuuutu utuu utuuuuutu utu tm (16) where u3(%)[−100, 100] and next u3[−1, 1] is the third control input for payload position control, the values of the parameters in (16) are [6] ua3=0.1, ub3=1 and uc3=0.13. the nonlinear state-space model of the tcs is [18] , , , ),( , , 1 , 1 ),( ),( , , , , 93 42 31 1010 109 2 2 2 8 2 8 1 1 1 7 1 7 66 55 84 73 62 51 xy xy xy fx xx m t k x t x m t k x t x fx fx xx xx xx xx p p = = = = = +−= +−= = = = = = =             (17) in which the expressions of the nonlinear functions f5, f6 and f10 are ),sin2sin2cos2cos2 sin4sinsin2sinsin2sinsin2 cos4cossin2cos2sin2sin coscos44( 2 1 ),,,...,,,,()( 229 2 2 2 2 98 123 2 2 1 2 83 2108211 1 1 21 1 7 21 2 83 187212 2 19 2 819 2 6 21 2 986105 9 2110532155 xmx t k x t xx xmx t k x t xx xxxxxm t k xx t x xxxx xxxxxgxxxxxxx xxxxxxx x mmxxxxxff pp p   +−−+ +−++ −−+− +−== (18) discrete-time model-based sliding mode controllers for tower crane systems 7 ),cossin cossincoscossincoscos coscossin2sinsin2coscos2 cos2( cos 1 ),,,...,,,,()( 2129 2 2 21 2 98 21 1 1 2 1 7 2219 2 8 2 2 83211082196521985 1106 19 2110532166 xxmx t k xx t xx xm t k x t x xxxxx xxxxxxxxgxxxxxxxxx xxx xx mmxxxxxff p p   − ++−+ −−−+− −== (19) ),coscos)(cossin)(cossin)( sinsin2cossin2sincos2 ( coscos1 1 ),,,,...,,,,()( 33 21 2 6 2 5921591269 219652110521106 10 21 3211053211010 l p l l m mk xxxxxxxfxxxfx xxxxxxxxxxxxx gx mxx mmmxxxxxff +++++ −++ +  − + ==  (20) where, as specified in [6], kp1=0.188 m/s and kp2=0.871 rad/s are gains of the first two dc motors, tς1=0.1 s and tς2=0.1 s are time constants for the first two motors in which ml=0.33 kg is the payload mass, μl=1600 kg/s is the viscous coefficient associated with the payload motion, g=9.81 m/s2 is the gravitational acceleration, kp3=200 kg∙m/s2 is process gain of the third dc motor, and zc(m) is the z coordinate of the payload. a part of the model described in eq. (17) is discretized and next extended by adding the discrete-time integral block for zero stationary control error, and introducing a new state of this integral block [19]: ,)/1(,1, kikrkr etxx +=+ (21) where the error is ek=rk−yk, the reference input rk and ti is the integral time constant (in continuous time). this extended model will have the extended state vector: .] [ t r t e xxx = (22) using relations (17) and (22) leads to the state vector for cart position control ,] [ ,,7,3,1 t krkkke xxx=x (23) for arm angular position control ,] [ ,,8,4,2 t krkkke xxx=x (24) and for payload position control .] [ ,,10,9,3 t krkkke xxx=x (25) the gain matrix k is considered to have the following structure, where one subscript will be inserted in order to specify one of the three position control systems: ].[ 21 rkkk=k (26) 8 a.-i. borlea, r.-e. precup, r.-c. roman three separate single input-single output (siso) control systems are considered, namely one for each controlled output y1, y2 and y3. three discrete-time sliding mode controllers discussed in section 2 are design for each siso control system. the unified siso control system structure is illustrated in fig. 2, where dtsmc specifies the sliding mode controller (position), dtsmc  {furuta dtsmc law, gao dtsmc law, quasi-relay dtsmc law}, m  {1, 2, 3} indicates the number of the controlled output, which is also the number of the control system, and dm is the disturbance input, not considered in the model (17). fig. 2 unified siso structure of discrete-time model-based sliding mode control system. the subscript k is omitted for the sake of simplicity. 3.1. furuta dtsmc law design and implementation summarizing the information given in the previous sections, the guidelines to design the furuta dtsmc law consist of the following steps: step f1. set the values of ts to account for the requirements of quasi-continuous digital control and ti, which affects the overshoot and the settling time. step f2. apply eq. (9) to obtain the expression of the system matrix. use eqs. (21) to (25) to express the expression of the extended system matrix (with discrete-time integral block). step f3. choose k so that the system described by the matrix obtained at step f2 is stable. pole placement can be applied in this regard. step f4. apply eq. (8) to get the upper bound of f0, and choose a value which is between limits. these steps are exemplified as follows. for the cart position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.05 s. the expression of the system matrix is obtained using (9) and (23) [20] . 002000 //1/)2000(/ 010 )()( 2212211 1 1 11111           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (27) a set of parameters that guarantee the stability of the system with the matrix in eq. (27) is ].005.02.03[1 −=k (28) for the upper amplitude of f0 described in eq. (8), the matrix [t1 t2 t3] is set to [k–1 0 0]; this gives 0 < f0 < 15.92 and the value is set to f0=1 [20]. discrete-time model-based sliding mode controllers for tower crane systems 9 the cart position reference input trajectory is chosen to be the same as in [6]. the experimental results obtained for the control system with the controller parameters in eq. (28) are illustrated in fig. 3. fig. 3 the results of cart position control system with furuta dtsmc law: (a) u1; (b) r1 (black), y1 (red). for the arm angular position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.012 s. the expression of the system matrix is obtained using eqs. (9) and (24) [20] . 003/25000 //13/)25000(/ 010 )()( 2212212 1 2 22222           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (29) a set of parameters that guarantee the stability of the system with the matrix in (29) is ].0002.02.015.3[2 −=k (30) for the upper amplitude of f0 described in eq. (8), the matrix [t1 t2 t3] is set to [k–1 0 0]; this gives 0 < f0 < 3.63 and the value is set to f0=3 [20]. the arm angular position reference input trajectory is chosen to be the same as in [6]. the experimental results obtained for the control system with the controller parameters in eq. (30) are illustrated in fig. 4. 10 a.-i. borlea, r.-e. precup, r.-c. roman fig. 4 the results of arm angular position control system with furuta dtsmc law: (a) u2; (b) r2 (black), y2 (red). for the payload position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.0001 s. the expression of the system matrix is obtained using eqs. (9) and (25) [20] . 001000000 //1/)1000000(/ 010 )()( 2212213 1 3 33333           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (31) a set of parameters that guarantee the stability of the system with the matrix in (29) is ].0011.00909.59[3 =k (32) for the upper amplitude of f0 described in (8), the matrix [t1 t2 t3] is set to [k–1 0 0]; this gives 0 < f0 < 107.43 and the value is set to f0=106 [20]. the payload position reference input trajectory is chosen to be the same as in [6]. the experimental results obtained for the control system with the controller parameters in eq. (32) are illustrated in fig. 5. discrete-time model-based sliding mode controllers for tower crane systems 11 fig. 5 the results of payload position control system with furuta dtsmc law: (a) u3; (b) r3 (black), y3 (red). 3.2. gao dtsmc law design and implementation summarizing the information given in the previous sections, the guidelines to design the gao dtsmc law consist of the following steps: step g1. this step is identical to step f1. step g2. this step is identical to step f2. step g3. this step is identical to step f3. step g4. set q>0 to fulfill qts (0,1), and >0. these steps are exemplified as follows. for the cart position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.2 s. the expression of the system matrix specific to the furuta dtsmc law results from eqs. (9) and (23) [20] . 00500 //1/)500(/ 010 )()( 2212211 1 1 11111           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (33) a set of parameters that guarantee the stability of the system with the matrix in (33) is ].0011.04.0825.0[1 −=k (34) the other parameters of the control law were set as q=99 such that qts(0,1) and =200 [20]. 12 a.-i. borlea, r.-e. precup, r.-c. roman using the same reference input trajectory as in section 3.1 and [6], the experimental results obtained for the control system with the controller parameters in eq. (34) are given in fig. 6. for the arm angular position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.2 s. the expression of the system matrix specific to the furuta dtsmc law results from eqs. (9) and (24) [20] . 00500 //1/)500(/ 010 )()( 2212212 1 2 22222           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (35) fig. 6 the results of cart position control system with gao dtsmc law: (a) u1; (b) r1 (black), y1 (red). a set of parameters that guarantee the stability of the system with the matrix in eq. (35) is ].0011.04.0825.0[2 −=k (36) the other parameters of the control law were set as q=99 such that qts(0,1) and =500 [20]. using the same reference input trajectory as in section 3.2 and [6], the experimental results obtained for the control system with the controller parameters in eq. (36) are given in fig. 7. discrete-time model-based sliding mode controllers for tower crane systems 13 fig. 7 the results of arm angular position control system with gao dtsmc law: (a) u2; (b) r2 (black), y2 (red). for the payload position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.0025 s. the expression of the system matrix specific to the furuta dtsmc law results from eqs. (9) and (25) [20] . 0079869184599999/1716871947673//141717986918/)5999996871947673(/ 010 )()( 221221 3 1 3 33333           −+= −−= − kkkkkkkk rr eeeex iakbkbaa (37) a set of parameters that guarantee the stability of the system with the matrix in eq. (37) is ].00130.092.61[3 =k (38) the other parameters of the control law were set as q=99 such that qts(0,1) and =300 [20]. using the same reference input trajectory as in section 3.3 and [6], the experimental results obtained for the control system with the controller parameters in eq. (38) are given in fig. 8. 14 a.-i. borlea, r.-e. precup, r.-c. roman fig. 8 the results of payload position control system with gao dtsmc law: (a) u3; (b) r3 (black), y3 (red). 3.3. quasi-relay dtsmc law design and implementation summarizing the information given in the previous sections, the guidelines to design the quasi-relay dtsmc law consist of the following steps: step m1. this step is identical to step f1. step m2. this step is identical to step f2. step m3. this step is identical to step f3. step m4. set the values of the parameters i>1. these steps are exemplified as follows. for the cart position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.25 s. the expression of the system matrix specific to the furuta dtsmc law results from eqs. (9) and (23) [20] . 00500 //1/)500(/ 010 )()( 2212211 1 1 11111           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (39) a set of parameters that guarantee the stability of the system with the matrix in (39) is ].001.04.09.0[1 −=k (40) the other parameters of the control law were set as =[3 1 3] [20]. discrete-time model-based sliding mode controllers for tower crane systems 15 using the same reference input trajectory as in section 3.1 and [6], the experimental results obtained for the control system with the controller parameters in eq. (40) are illustrated in fig. 9. fig. 9 the results of cart position control system with quasi-relay dtsmc law: (a) u1; (b) r1 (black), y1 (red). for the arm angular position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.25 s. the expression of the system matrix specific to the furuta dtsmc law results from eqs. (9) and (24) [20] . 00400 //1/)400(/ 010 )()( 2212212 1 2 22222           − −+=−−= − kkkkkkkk rreeeex iakbkbaa (41) a set of parameters that guarantee the stability of the system with the matrix in eq. (41) is ].0011.03.083.0[2 −=k (42) the other parameters of the control law were set as =[3 1 3] [20]. using the same reference input trajectory as in section 3.2 and [6], the experimental results obtained for the control system with the controller parameters in eq. (42) are illustrated in fig. 10. 16 a.-i. borlea, r.-e. precup, r.-c. roman fig. 10 the results of arm angular position control system with quasi-relay dtsmc law: (a) u2; (b) r2 (black), y2 (red). for the payload position control system, the sampling period and the integral time constant are chosen as ts=0.01 s and ti=0.003 s. the expression of the system matrix specific to the furuta dtsmc law results from eqs. (9) and (25) [20] . 00100000/3//13/)100000(/ 010 )()( 221221 3 1 3 33333           −+= −−= − kkkkkkkk rr eeeex iakbkbaa (43) a set of parameters that guarantee the stability of the system with the matrix in (43) is ].0003.01.07[3 −=k (44) the other parameters of the control law were set as =[50 25 10] [20]. using the same reference input trajectory as in section 3.3 and [6], the experimental results obtained for the control system with the controller parameters in eq. (44) are illustrated in fig. 11. for a fair comparison of the three dtsmc laws, it is accepted that the allowable tolerance zone of the controlled output is of  2 % yst, where yst is the steady-state value of the controlled output. this is reflected in the measurement of the values of the overshoot and the settling time. the performance indices overshoot, settling time and stationary control error (or steady-state value of control error) are considered in the comparison of the control discrete-time model-based sliding mode controllers for tower crane systems 17 systems with the three dtsmc laws designed in this paper. the values of these performance indices are synthesized in table 1, table 2 and table 3, obtained after three test runs for all three control systems with dtsmc laws with extended state vectors defined in eqs. (23), (24) and (25), respectively. the test runs are composed by three step signals, three step signals, and four step signals, respectively. fig. 11 the results of payload position control system with quasi-relay dtsmc law: (a) u3; (b) r3 (black), y3 (red). table 1 overshoot (%) test run step quasi-relay furuta gao cart position 1 2 3 4 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 arm angular position 1 2 3 0 0 0 0 0 0 0 0 0 x 0 0 0 payload position 1 2 3 2.7 0 5 3.5 0 0 − − 0 x 2.6 1.2 − x – average value of the steps which compose the reference signal 18 a.-i. borlea, r.-e. precup, r.-c. roman table 2 settling time (s) test run step quasi-relay furuta gao cart position 1 2 3 4 7.55 6.2 8.2 6 6.5 7 6 8.6 4.2 3.5 4.4 4.5 x 6.987 7.025 4.15 arm angular position 1 2 3 7.4 8.4 6.6 5.5 6 7.4 5.4 5.5 6.5 x 7.467 6.3 5.8 payload position 1 2 3 4.8 2.8 5 9.6 7 9.6 − − 16 x 4.2 8.733 − x – average value of the steps which compose the reference signal table 3 stationary control error (%) cart position step quasi-relay furuta gao arm angular position 1 2 3 4 1 1.2 0.6 1.5 0 0.1 0 0 0 1.2 0.1 0 x 1 0 0.3 payload position 1 2 3 1.6 0 0.2 0 0 0 1.6 0.8 0 x 0.6 0 0.8 1 2 3 0.4 0.3 0 0.1 0 0 − − 5.2 x 0.2 0 − x – average value of the steps which compose the reference signal analyzing the values for the overshoot in table 1 reveals the fact that the smallest average value is exhibited by the furuta dtsmc law. in table 2, the smallest average value of the settling time is obtained by the quasi-relay dtsmc law. table 3 shows that the furuta dtsmc law exhibits the smallest average value as far as the stationary control error is concerned. in conclusion, having in mind all three performance indices discussed here, the best overall performance is ensured by the furuta dtsmc law, the second best one is gao dtsmc low and on the third place is quasi-relay dtscm low. nevertheless, the furuta dtsmc law leads to the smallest number of switching of the control inputs, thus the efforts at the level of actuators (namely, the power electronics part that produces pwm). however, the conclusions of the comparison can be different if other nonlinear processes are subjected to sliding mode control, in various fields, including decision-making [21], man-computer symbiosis [22], 3d printing objects [23], specific structures of fuzzy discrete-time model-based sliding mode controllers for tower crane systems 19 systems [24], [25] including those focused on fuzzy control [26], [27], evolving controllers [28] and fuzzy cognitive maps [29], vanets [30], quantum computing [31], and telesurgical applications [32]. disturbance inputs were not considered (although the role of a part of possible disturbances is highlighted in fig. 2) because the integral block described in [21] ensures the disturbance rejections for certain types of disturbances. it is not guaranteed that the ranking of the sliding mode controllers will be kept after rejecting the disturbances. 4. conclusions this paper proposed the discrete-time model-based sliding mode control of tower crane systems. three popular control discrete-time control laws were considered and compared systematically on the basis of three performance indices obtained after measurements in a specific dynamic regime applied to the laboratory equipment. the furuta law was applied to all three control laws to design the gain matrix k in the conditions of the application of the equivalent control method [2]. however, the control laws are different in terms of different ways to carry out the switching that modifies the control system structures making them belong to the general class of variable structure systems. the parameters involved in switching are also different. this flexibility is an advantage of the control laws designed and implemented in this paper and also confirms the degrees of freedom offered by sliding mode control, also showing certain robustness. one of the main shortcomings of these control laws is the heuristics in the design of the controllers. another shortcoming is the need for certain initial information on the process model. future research will be focused on mitigating these shortcomings. first, the optimal tuning of the free parameters of the sliding mode controllers will be targeted. second, the design of data-driven sliding mode controllers will be carried out starting with [6] and compared to model-based sliding mode controllers. direct relations that involve controller tuning parameters and control system performance indices with respect to both reference and disturbance inputs will be attempted to be derived in all controller designs. acknowledgement: the research reported in this paper was supported by a grant of the romanian ministry of education and research, cncs uefiscdi, project number pn-iii-p4-id-pce2020-0269, within pncdi iii. references 1. emelyanov, s.v., 1967, variable structure control systems. nauka, moscow. 2. utkin, v.i., 1977, variable structure systems with sliding 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of multi-level periodic system of indenters with coated elastic half-space irina g. goryacheva, elena v. torskaya ishlinsky institute for problems in mechanics, russian academy of sciences, russia abstract. the contact of a periodic system of spherical indenters of different heights and radii of curvature with two-layered elastic half-space is considered. numericalanalytical method is developed to determine contact pressure distribution and internal stresses taking into account mutual effect of contact spots. the results for relatively hard and soft coatings are analyzed for different values of input parameters: nominal pressure, contact density, coating thickness. key words: coatings, discrete contact, contact density, internal stresses 1. introduction the widespread use of coatings in different mechanisms actualizes the study of contact and internal stresses and contact fatigue damage accumulation inside the coated bodies. surface roughness of counterbody is one of the parameters which influence contact characteristics (the pressure distribution and the real contact area) and stresses inside the coating and the base. for the case of cycling loading stress concentration near contact spots leads to the different types of the coating failure. roughness in contact problems is often modeled by periodic system of indenters to analyze mutual effect for different model geometry and to study real contact area and the depth of penetration as a function of average load applied to the period. 2-d periodic contact problems were studied mostly by analytical methods; the methods and results are reviewed in different books and papers, for example in [1]. the mutual effect for uncoated elastic solids was studied in [2] both for one-level and multi-level 3-d periodic systems of indenters penetrating into elastic half-space. the most recent studies of periodic contact problems for homogeneous and transversely isotropic elastic half-space [3-5] consider saturation received may 11, 2019 / accepted july 03, 2019 corresponding author: irina g. goryacheva affiliation: ishlinsky institute for problems in mechanics of the russian academy of sciences, 119526 moscow prospect vernadskogo,russia e-mail: goryache@ipmnet.ru 150 i. goryacheva, e. torskaya effect in contact of periodic wavy surface [3] or adhesive contact for a system of indenters of different shape and transversely isotropic half-space [4]. the discrete contact problems for the coated elastic bodies were studied both for normal and sliding contacts in [6-9]. the coating thickness related to geometrical parameters of roughness is specific characteristic, which influences contact and internal stress distributions. two approaches to model the discrete contact taking into account the mutual contact effect were developed almost at the same time. periodic one-level system of spherical indenters penetrating into the two-layered elastic half-space was considered in [6]; 2-d contact problem for a measured profile of counterbody was solved in [7]. the same models of discrete contact were used to take into account roughness in 3d macro contact problem solution [8, 9]; in both studies friction is also considered. for thick coatings some results from [2], which are mostly analytical, can be used for verification of semi analytical and numerical solutions of the similar counter body microgeometry. in this study a multi-level periodic contact problem for a two-layered elastic halfspace is considered. each level has particular geometry of indenters which are uniformly space distributed. the mutual effect of contact spots is taken into account. the model is used to calculate contact and internal stresses for relatively hard and soft coatings and to analyze the effect of microgeometry parameters (space distribution of asperities and their microshapes) on contact characteristics and internal stress distributions. 2. problem formulation and the method of solution 2.1. problem formulation two-layered elastic half-space is considered in contact with a periodic system of rigid indenters (asperities) uniformly distributed at k height levels. in the cylindrical coordinate system related to each indenter, the shape of the indenter is described by the following function: ( ) , 1...m mz f r h m k   (1) here functions fm(r) and hm describe the shape and the height of the asperity of the m-th level respectively, r is the radial coordinate of the polar system coordinates with the center at the point of initial contact of the asperity. in contact interaction the areas of contact regions im are different for the contact spots of various levels (index i indicates the fixed asperity of the m-th level), and the asperities come into contact at definite penetrations of the whole system corresponding to their space location. the example of the contact spots distribution at the surface is presented in fig.1 for the case k=3. in this case the indenters are located at the nodes of the hexagonal lattice, and l is the lattice period. at the first step we introduce the polar system of coordinates at the layer surface with the center at the point of initial contact of the fixed highest asperity (see fig.1). the centers of the other contact spots im have radius-vectors imr . the axis oz is perpendicular to the layer surface and directed inside two-layered half-space. under the assumption that the shear stresses within the contact spots are negligibly small, the boundary conditions at the surface of the layer (z = 0) are the following: contact of multi-level periodic system of indenters with coated elastic half-space 151 (1) (1) (1) ( ) ( ) , , 0, , 1.. , 1,2, , 0, 0, 0 im m im m im z im xz yz w r r f r r h r r m k i r                       (2) where  is the system penetration. fig. 1 scheme of the contact for the system of indenters located at 3 height levels boundary conditions at the layer half-space interface (z=h) correspond to the case of perfect coating-substrate adhesion: (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) , , , , , z z rz rz z z r rw w u u u u                (3) in eqs. (2) and (3) ( ) ( ) ( ) , ,j z j j rz z   are normal and shear stresses, and ( ) ( ) ( ), ,j j j rw u u are normal and tangential displacements (j=1 for the coating, j=2 for the substrate). contact zones im under the asperities are initially unknown. note, that instead of the conditions of perfect adhesion at the interface, the conditions of complete or incomplete sliding can be considered using the approach developed in [10, 11]. using the localization principle [2], we take into account the real pressure distribution only at the fixed number of asperities inside the circle of radius r1 ( 1r r ), which includes a definite number of asperities of all k levels. for known densities cq of the asperities distributions for each level (c=1,2…,k) radius r1 is calculated from the relation (for k=1) 2 1 1 1k c k c c k r q q            (4) here cq is the density of indenters of level n, c is the number of indenters of level c inside the circle. action of indenters outside the circle is replaced by constant average pressure p . inside the circle unknown contact pressure distributions pc(r,) act within the 152 i. goryacheva, e. torskaya contact spots under each indenter (here (r,) are the polar coordinates related to the center of the c-th indenter). to satisfy the equilibrium condition we use the following relation: 1 ( , ) n k n n n p q p r drd       (5) note that due to the mutual effect the asymmetric contact pressure distribution under the axisymmetric indenter occurs, and we take into account this effect in contact problem formulation and solution. the boundary element method together with the iterative procedure is used to solve the contact problem with unknown contact regions. contact pressure is presented as a stepwise function being constant inside each element. we choose the size of a square element so that their number inside a contact region is large enough to provide convergence of the iterative procedure. 2.2. calculation of influence coefficients for the boundary element method to use the boundary element method, the displacement of the two-layered elastic halfspace surface loaded by a pressure q, uniformly distributed inside a square 2a2a, must be calculated from the following boundary conditions at the layer surface (z=0): , | | ,| | 0, | | ,| | 0, 0 z z xz yx q x a y a x a y a              (6) in the case of two-layered elastic half-space stresses and displacements of the layer surface can be calculated by using the method based on double fourier transforms. in particular, normal displacements of the surface are determined by the following relation [11]: /2 1 0 0 '( ', ',0) ( , , , )cos( ' cos )cos( ' sin ) 2 q w x y x y d d g                 (7) here x, y and w are dimensionless coordinates and normal displacement, respectively, g1 is the shear modulus of the layer, =e1(1+2)/e2(1+1) is the relation of the elastic modules of the layer and the substrate, ,  are the internal coordinates in the space of double fourier transforms, =h/a is dimensionless layer thickness. function ( , , , )    represents normal displacements in the space of double fourier transforms. it is obtained from the boundary conditions (3) and (6) using representations of stresses and displacements by a biharmonic function and double fourier transforms of constant pressure q. it makes possible to reduce the problem to linear system of functional equations [11]. the solution of the system provides particularly the analytical representation of ( , , , )    . as the dependence of normal displacements on the value of the constant pressure in eq. (7) is linear, it can be used to find influence coefficients in the boundary element method. contact of multi-level periodic system of indenters with coated elastic half-space 153 2.3. calculation of contact characteristics the boundary element method for a contact problem solution is described in many papers. in [6] it was developed to study the discrete contact problem with mutual effect of contact spots. to solve the contact problem formulated in eqs. (2) and (3), we used the following steps. at the first step, we place the center of the circle, in which we are looking for a real distribution of pressures, in the center of the highest indenter and calculate the surface shape g(x,y) inside the circle with radius r1, eq. (4)), due to constant nominal pressure p outside the circle. the value of p at this step is small enough to provide only the contact of the indenters of the first level. function g(x,y) is used then in formulation of the contact conditions to calculate the pressure distribution under the indenters of the first level and the normal displacements outside the contact spots. then we increase nominal (average) pressure controlling the displacements under indenters of the next level. for pressures 2 1p p p  the indenters of the second level come into contact. then we take the origin of coordinates under the center of indenter of the second level. for calculation of the contact pressure under the indenters of the second level, we take into account the real contact pressure distributions at the contact spots of the first and second levels, and the nominal pressure outside the circle of radius r2, eq. (4), and use the iteration method. using similar procedure for each i-th level we calculate the contact pressure distribution for each level of indenters, normal forces acted at indenters of each level and their redistribution if the nominal pressure increases. additional displacement wa of the periodical system of indenters is also calculated. this value indicates the displacement due to surface microgeometry, and it is calculated from the relationship [2]:     1 ( , ) im i k a i r i w f p r f p     (8) here function im f indicates the surface displacement under the highest indenter due to real contact pressures distribution under asperities located inside the circle of radius r1, irf is related to the displacement caused by average pressure p distributed within the circle of radius r1. the dependence of the additional displacement of the periodical system of indenters on its geometrical characteristics is also analyzed. 3. results and discussion calculations have been completed for the three levels periodic system of spherical indenters with the given heights. the indenters were located at the points of the hexagonal lattice with distance l. the indenter’s shape at the i-th height level was spherical (ri is the radius of curvature of the indenter near the point of initial contact). note that the model under consideration is valid only for small deformations, so the saturation effect cannot be analyzed within the framework of this study. 154 i. goryacheva, e. torskaya the dependence of the contact characteristics on the following model parameters has been analyzed:  coating thickness h;  young modulus ej and poisson ratio j for coating (j=1) and substrate (j=2) materials;  lattice period for the system of indenters, l;  radius of indenters (for each level), ri;  heights of indenters of each level hi. the following dimensionless parameters have been introduced:  dimensionless radius of indenters for i-th level (i=1,2,3), ' /i ir r l (9)  dimensionless contact pressure distribution for i-th level 2'( , ) ( , ) /i ip x y p x y e (10)  dimensionless average pressure 2' /p p e (11) all internal stresses are also related to e2. dimensionless parameter , which characterizes relative compliance of the coating, was introduced in the previous chapter. we do not use special letter symbols for the dimensionless thickness of the coating and the difference in height, but they are related to the lattice period. the contact characteristics calculations for the system of indenters with three height levels contacting with the relatively hard coating bonded to the elastic base have been performed. in step-by-step solution with the increase of the average pressure first we have only one level of indenters in contact with the coating surface. for the chosen space distribution of indenters: (h1-h2)/l=(h2-h3)/l=0.01 mutual effect for the first level was negligible, the pressure under each indenter was axisymmetric. for relatively large average pressure three levels are in contact, the mutual effect becomes stronger. fig.2 illustrates the contact pressure distribution (fig. 2a) and the contact spot configuration (curve 1 in fig. 2b) for this case. we have got not circular contact zone, especially for the indenters of the lowest level. mutual effect can be evaluated by comparison of curves 1 and 2 in fig. 2b; the last one was calculated for the same input parameters, but neglecting the surface deformation due to the penetration of nearby indenters of the first and second levels. maximum value of the contact pressure in fig. 2a is (p3)max=0.07. fig. 3 illustrates the contact pressure distributions calculated for relatively soft coatings characterized by the different poisson ratio. the chosen cross section provides the maximum size of the contact spot. the difference of maximal and minimal distance from the center of the contact zone to the boundary, which characterizes asymmetry due to the mutual effect, is 12 percent for the third level (curve 3); the difference is smaller for the first and the second levels. contact spots for the first level hereinafter are almost axisymmetric. comparison with pressure distributions obtained for isolated indenters loaded by the same forces (dashed lines in fig. 3a,b) leads to conclusion that for the case of relatively soft low-compressive coatings (1=0.45) mutual effect is strong (see also table 1), but for 1=0.2 it is negligible. contact of multi-level periodic system of indenters with coated elastic half-space 155 a) b) fig. 2 contact pressure distribution (a) and contact spot (b) for the 3 rd level of indenters: =2, h/l=0.5, (h1h2)/l=(h2h3)/l=0.01, 3 ' 0.06p  ), 1=0.2, 2=0.3, r1= r2= r3=0.8 a ) b ) fig. 3 pressure distribution under indenters of levels 13 (curves 1-3 respectively) for relatively soft coatings =0.5, h/l=0.5 with 1=0.45 (a), 1=0.2 (b): 2=0.3, (h1h2)/l=(h2h3)/l=0.01, r1= r2= r3=0.8, 3 ' 0.008p  table 1 maximal values of contact pressure related to e2 (results from fig. 3a) 1=0.45 1=0.2 calculation with mutual effect isolated indenters calculation with mutual effect isolated indenters level 1 0.356 0.312 0.320 0.311 level 2 0.298 0.289 0.288 0.287 level 3 0.191 0.232 0.213 0.230 156 i. goryacheva, e. torskaya fig. 4a illustrates the results of calculation of the additional displacement function (8) for relatively hard coating of various thicknesses: h/l=0.12 (curves 1,1’) and h/l=0.24 (curves 2,2’) in the cases of three level model (curves 1 and 2) and one-level model (curves 1’ and 2’). the results indicate that in the case of hard coatings at the softer substrate the additional displacement for three level asperities distribution is higher than for the one level model under the same nominal pressure. decreasing the coating thickness leads to increasing of the additional displacements. it is interesting to note that for the case of relatively thick coatings the difference between curves 2 and 2' (representing different models) is greater than for thinner coatings (curves 1 and 1' respectively). to analyze the mutual effect we calculated the additional displacement function ignoring the surface displacements outside the contact regions (neglecting mutual effect), but taking into account equilibrium conditions. the results of calculations are presented in fig. 4b (solid curves 1 and 2 are the same as in fig. 4a, dashed curves are calculated neglecting mutual effect). there is an essential difference between the curves. the penetration calculated without mutual effect is greater. so neglecting the mutual effect gives the overestimate of additional displacement due to roughness. a) b) fig. 4 dependence of the additional normal displacements on nominal pressures for one-level and three-level models (a) and comparison of the dependences for three level model with ones (curves 1’ and 2’) calculated neglecting mutual effect (b): h/l=0.12 (curves 1, 3), h/l=0.24 (curves 2, 4), (h1h2)/l=(h2h3)/l=0.01 (curves 1, 2), (h1h2)/l=(h2h3)/l=0 (curves 3,4), =02, 1=0.22, 2=0.3, r1= r2= r3=0.8 the results presented in fig. 5 give a possibility to analyze the influence of nominal pressure on contact characteristics (maximal contact pressure and contact size) for two types of microgeometry models: one-level model with period 3l (curves 1), three-level model with the same period 3l for each level (curves 2). the results indicate that the radius of contact spot and maximal contact pressure increases with increasing the nominal pressure. no difference exists between the curves 1 and 2 for small values of average pressure, when the second and third levels are not in contact. contact of multi-level periodic system of indenters with coated elastic half-space 157 curve 3 in fig. 5b illustrates the principal shear stress maximal value. the maximum is localized at the layer-substrate interface for the chosen model parameters. figs. 6 and 7 illustrate the influence of the contact density on the distribution of tensile-compressive and principal shear stresses under an indenter of the first level for the value of the nominal pressure p , which provides three-level contact. all geometrical parameters are related here to the radius of first-level indenter, which allows us to use different value of l associated with the contact density. a) b) fig. 5 dimensionless maximal contact radius am (a), maximal contact pressure (p1)max (b, curves 1-2) and maximal values of the principal shear stress in the coating (1)max (b, curve 3) under first-level indenter as functions of nominal pressure; (h1h2)/l=(h2h3)/l=0.06, r1=1.67, r2=1.25, r3=0.83 (curves 1, 3), one-level model with period 3l and r1=1.67 (curves 2); h/l=0.087,=3, 1=0.22, 2=0.4 we chose these stresses for analysis, because due to their concentration the coating delamination due to brittle fracture (large tension) or contact fatigue (high amplitude values of the principal shear stress) occurs. since the space distribution of indenters influences essentially on the contact spot radius and the maximal contact pressure (see fig. 5), the internal stress distribution depends also on the density and height distribution of asperities. the stress distributions presented in figs. 6 and 7 are typical for relatively hard coatings: tension occurs at the surface near the boundary of contact, and at the coating-substrate interface for thicker coating; principal shear stresses concentrate at the surface-substrate interface, which is often initially damaged. fig. 7 illustrates the results of stress calculation for the same input parameters as used in fig. 6. the values of tensile-compressive stresses (curves 1-3) are very different for the cases of high (b) and low (a) contact densities as at the surface and as at the interface. for the case of high contact density positive tensile stresses occur at the surface under the boundary of the contact zone (curve 1), in coating material at the interface under the center of the contact (curve 2), and only compression realizes in substrate material (curve 3). for the case of low 158 i. goryacheva, e. torskaya density both for the surface and for the interface the curves have the same features, which are compression under the center and tension under the boundary of the contact zone. principal shear stresses for both cases have interface maxima. a) b) fig. 6 tensile-compressive and principal shear stresses under first-level indenter: =3, ' 0.002p  , 1=0.22, h/r1=0.023 2=0.4; r2/r1=0.75, r3/r1=0.5, (h1h2)/r1=(h2h3)/ r1=0.036, l/r1=0.067 (a) l/r1=0.6 (b) a) b) fig. 7 tensile-compressive r (curves 1, 2, 3) and principal shear 1 (curves 1', 2', 3') stresses under first-level indenter: =3, ' 0.002p  , 1=0.22, h/r1=0.023, 2=0.4; , r2/r1=0.75, r3/r1=0.5, (h1h2)/r1=(h2h3)/ r1=0.036, l/r1=0.067 (a) l/r1=0.6 (b) at the surface (curves 1, 1'), at the coating-substrate interface in coating (curves 2, 2') and substrate (curves 3, 3') 4. conclusions the numerical-analytical method of contact problem solution for multi-level periodic system of indenters and two-layered elastic half-space is developed. contact pressure distributions under the indenters of different levels, internal stresses, and additional displacements as a function of average pressure are determined for different values of contact of multi-level periodic system of indenters with coated elastic half-space 159 input parameters: nominal pressure, space distribution of indenters, their contact density, coating thickness and its relative to the base mechanical properties. the mutual effect is analyzed for the case of relatively hard and soft coatings. the results indicate that for the case of thin soft coatings this effect is stronger for materials with relatively high value of poisson ratio. for relatively hard thin coatings due to the coating bending the real contact pressure distribution and the real contact area strongly depend on the contact density and height distribution of indenters, which can be considered as the model of the asperities of the rough surface with regular roughness. it follows from the results that the internal stresses in relatively hard coatings also depend essentially on the contact density and space distribution of asperities. tensile-compressive and principal shear stresses are very different for the cases of low and high density both for the surface and for the interface. for the case of high contact spot density the tension of coating occurs at the layer-substrate interface, but for low density there is the compression. for small contact density, interface tension is realized not under but between indenters. based on the results it may be concluded that the mutual effect is stronger for hard coatings, than for the soft ones. the similar conclusion was made from the problem solution for the one-level asperity model in contact with the two-layered elastic half-space [6]. acknowledgements: this work was carried out under the financial support of the russian foundation for basic research (grants n. 17-58-52030 (russian-taiwan) and 17-01-00352). references 1. goryacheva, i.g., martynyak, r.m., 2014, contact problems for textured surfaces involving frictional effects, proceedings of the institution of mechanical engineers, part j: journal of engineering tribology, 228(7), pp. 707-716. 2. goryacheva, i.g., 1998, contact mechanics in tribology, dordercht: kluwer academic publ, 344 p. 3. yastrebov, v.a., anciaux, g., molinari, j-f., 2014, the contact of elastic regular wavy surfaces revisited, tribology letters, 56, pp. 171-183. 4. argatov, i.i., li, q., popov, v.l., 2019, cluster of the kendall-type adhesive microcontacts as a simple model for load sharing in bioinspired fibrillar adhesives, archive of applied mechanics, 3, pp. 1-26. 5. argatov, i.i., 2012, the contact problem for a periodic cluster of microcontacts, journal of applied mathematics and mechanics, 76, pp. 604–610. 6. goryacheva, i.g.. torskaya, e.v., 2003, stress and fracture analysis in periodic contact problem for coated bodies, fatigue and fracture of engineering materials and structures, 26(4), pp. 343-348. 7. cole, s.j., sayles, r.s., 1991, a numerical model for the contact of layered elastic bodies with real rough surfaces, journal of tribology, 11, pp. 334-340. 8. torskaya, e.v., 2012, modeling of frictional interaction of a rough indenter and a two-layered elastic half-space, physical mesomechanics, 15(3-4), pp. 245-250. 9. nyqvist, j.t., kadiric, a., sayles, r.s., ioannides, e., 2013, three-dimensional analysis of multilayered rough surface contact, proc. 5th world tribology congress, torino, italy. 10. goryacheva, i.g., torskaya, e.v., 2016, modeling the influence of the coating deposition technology on the contact interaction characteristics, mechanics of solids, 51(5), pp. 550-556. 11. nikishin, v.s., shapiro, g.s., 1970, space problems of the elasticity theory for multilayered media, moscow: vts an sssr, 260p. (in russian). 13163 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 741 757 https://doi.org/10.22190/fume241005050r © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper computational study of frequency impact onto the particle parameters and change in acoustic agglomeration time with the number of particles sai manoj rayapureddy1, algirdas maknickas2, jonas matijošius2, darius vainorius3, artūras kilikevičius2 1department of automobile engineering, faculty of transport engineering, gediminas technical university, vilnius, lithuania 2mechanical science institute, vilnius gediminas technical university, vilnius, lithuania 3laboratory of heat equipment research and testing lithuanian energy institute kaunas, lithuania orcid ids: sai manoj rayapureddy https://orcid.org/0009-0006-0766-6050 algirdas maknickas https://orcid.org/0000-0002-8431-2292 jonas matijošius https://orcid.org/0000-0001-6006-9470 darius vainorius https://orcid.org/0000-0002-2927-5142 artūras kilikevičius https://orcid.org/0000-0002-4039-7300 abstract. air pollution, especially from particulate matter (pm), is a significant environmental and health issue worldwide. this research examines the use of acoustic agglomeration (aa) as a novel technique to diminish fine particle emissions, concentrating on the influence of frequency on particle characteristics and agglomeration duration. this study aims to improve the effectiveness of traditional filtration systems by investigating acoustic wave-induced particle agglomeration as an alternative method. the research integrates experimental and computational methodologies. emissions from a 1.9 turbocharged 4-cylinder diesel engine using fat80ip20 fuel were subjected to analysis in a custom-designed acoustic chamber. frequencies of 21.2 khz and 34.6 khz were used, with particle measurements recorded using a fluke 985 particle counter. computational simulations using the discrete element method (dem) were performed to examine particle behaviour at different acoustic frequencies, accounting for first-order factors such orthokinetic interactions and acoustic wake effects. results demonstrate that elevated frequencies (34.6 khz) accelerate particle agglomeration, although the overall quantity of agglomerated particles diminishes. experimental results indicated a 62.78% decrease in 5 μm particles and a 300% rise in 10 μm particles at 34.6 khz. the findings correspond with the computer calculations, which indicated that heightened frequency enhances particle mobility and collisions while diminishing the likelihood of prolonged agglomeration inside the chamber confines. the results confirm the efficacy of acoustic agglomeration for emission control, emphasising a frequency-dependent trade-off between agglomeration speed and the volume of agglomerated particles. this study advances received: october 05, 2024 / accepted november 21, 2024 corresponding author: jonas matijošius mechanical science institute, vilnius gediminas technical university, plytinės str. 25, lt-10105 vilnius, lithuania e-mail: jonas.matijosius@vilniustech.lt https://orcid.org/0000-0001-6006-9470 742 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius the development of sophisticated filtering technologies and more environmentally friendly transportation alternatives. key words: particle emission, acoustic agglomeration, computational simulation, discrete element method, orthokinetic interaction, acoustic wake effect 1. introduction the environment agency of the european union has declared air pollution as europe’s most significant environmental health risk. according to a data published by statista in 2023 [1], the transportation sector accounted for nearly 12% of global greenhouse gas emissions. it is estimated to be the second largest contributor to air pollution [2]. of all emissions, particulate matter or particles are found to have a direct and immediate impact on human health and environment [3,4]. these particles vary in size, with some particles being so small that they can only be seen through a microscope. the size of particles is an important factor in determining their potential impact on human health and the environment. particles less than 10μm in diameter (pm10) can penetrate deep into the lungs and cause respiratory problems, while even smaller particles or fine particles (pm2.5) can penetrate into the bloodstream and have systemic effects on human health [5–7]. although a significant reduction in fine particle matter is observed in the european union by 37% between 2000 and 2020, emissions are now still considered to be at a harmful level [8]. to reduce the impact of particulate matter released from automobiles, countries around the world are taking measures such as monitoring and regulating higher emission standards [9,10] and by focusing on researching new effective solutions such as design of new combustion modes, use of blended fuels and developing advances in particulate filters [11–16]. to increase the efficiency of the conventional filter process and contain the fine particles from releasing into the environment, a particle agglomeration process is implemented [17]. agglomeration is a technique in which smaller particles are excited through various sources and are forced to collide with each other to agglomerate and increase in size as a result of the formation of clusters. this can be achieved through the use of heat, sound, or electric field [18,19]. to achieve better filtration by aggregating the particles using sound waves is termed acoustic agglomeration. researchers consider it an effective and efficient method of agglomerating the particles where the efficiency increases with increasing sound pressure levels [14,20–22]. in a research conducted by dong zhou et al., they observed almost 10% improved efficiency in filtering pm2.5 particles when operated at 1400 hz and 148 db [20,23]. another experimental result shows that with an acoustic field of 21,400 hz frequency, particles of size 10 μm are reduced by >90%, and fine particles (0.3 μm) are reduced by ~45% [24]. acoustic agglomeration (aa) is a process governed by various interactions between the particle and fluid or other particle. the predominant first-order effects that enable the agglomeration of 2 particles in a system are orthokinetic collisions and acoustic wake effect [14,25,26]. orthokinetic interaction is the interactive agglomeration process formed as a result of the direct collision of particles of different velocities [23,27]. particles with different sizes act differently in the system due to their difference in inertia. the primary drawback of this computational study of frequency impact onto the particle parameters and change in acoustic... 743 particular interaction is that it cannot be applied to particles with similar shapes and sizes [28]. the acoustic wake effect works in the hydrodynamic mechanism. it is a particle interaction process in which the acoustic wake causes a reduction in pressure behind the particle moving forward. if another particle follows this wake, it experiences with a drag reduction and moves with an accelerated speed towards the leading particle, causing agglomeration of the 2 particles [29,30]. the acoustic wake effect is a first-order effect that is caused due to the action of hydrodynamic forces [31–33]. at low frequencies particularly for medium-sized particles, orthokinetic interaction forces are found to be more applicable and acoustic wake effect is found to dominate at higher frequencies for particles of different sizes [34]. the simulation of acoustic agglomeration is carried out using the discrete element method (dem). in this computational method, the behaviour of particles under specific circumstances is studied and results are analysed. in dem, each particle is considered as an individual, discrete element [35,36]. the movement of these individual elements, its influence on movement of other elements, and the physical forces between them are considered. by studying individual characteristics and the interaction between the particles, it is possible to estimate the agglomeration effect on a large scale [31]. the use of discrete elements in simulations allows for a more detailed and accurate understanding of particle-particle interaction than any other numerical simulation technique, such as finite element methods or computational fluid dynamics. this is particularly important in applications where the size and shape of individual particles play a significant role in the behaviour of the particulate system [37]. the properties of particles and the system, such as size, density, characteristics of particles, and acoustic medium properties, are predetermined. different forces acting on individual particles and during the interaction with other particles are taken into consideration and studied. the acoustic field in which the particles interact is entered into the system to match the properties of the real world. the result of the simulation consists of the analysis of the acceleration and velocity of the particle in the given medium. the forces acting on the particles, behaviour and change in the size of the particles in the presence of the acoustic field are observed. the formation of agglomeration can be analysed for the given number of particles, and they can also be projected for a larger number of particles which typically consumes huge amounts of data and processing time to simulate the numerical analysis. this study explores the potential of acoustic agglomeration for particulate matter control, but it has limitations. the experiments were conducted in a controlled laboratory environment, which does not fully replicate real-world scenarios. the study focused on two acoustic frequencies, 21.2 khz and 34.6 khz, which may not be applicable to other fuel types or blends with different emission characteristics. the fixed boundaries and design of the chamber could influence particle behavior, potentially overestimating or underestimating agglomeration efficiency. the dem simulations provided valuable insights into particle interactions, but they relied on assumptions that could impact realworld outcomes. the study focused on a specific range of particle sizes and distributions, not including ultrafine particles, which are more challenging to agglomerate but critical for health impacts. the energy consumption and feasibility of the acoustic field were not assessed, and the health and environmental impact of byproducts were not assessed. the experiments were conducted over limited durations, which may not fully capture long-term operational performance or potential wear and tear of the acoustic chamber. 744 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius 2. experimental test the impact of the change in frequency on the particles is tested on an experimental setup. the 1.9 turbocharged 4-cylinder diesel engine is used for testing. engine exhaust emissions are passed through the in-house built acoustic chamber where particle agglomeration is subjected under the application of sound waves. details of the test bench and agglomeration chamber are presented in some of the previous research works [24,38]. a fuel blended volumetrically at 80% fat and 20% isopropanol was tested on 1.9 turbocharged 4-cylinder diesel engine. the particles are subjected to agglomeration under the application of sound waves in an in-house built acoustic chamber. the chamber is maintained at a sound pressure of 140 db and an agglomerator voltage of 100 v. the change in number of particles are measured with an agglomerator frequencies of 21.2 khz and 34.6 khz. the fluke 985 particle counter is used for the measurement of particles. details of the test bench and agglomeration chamber are presented in some of the previous research works [24,38]. at lower frequency, the reduction of particles is found to be proportionate compared to higher frequency as shown in table 1. this can be because with an increase in frequency, particle excitation increases, thereby affecting the probability of agglomeration. although the results show that with increase in frequency, the agglomeration of 10 m in particles of size 10 μm increases but there is an insignificant reduction in small sized particles as presented in fig. 1. table 1 experimental results of fat80ip20 fuel particle size [μm] without agglomeration particles amount, [number] change at frequency 21.2 khz, [%] change at frequency (34.6 khz), [%] 0.3 223,037 -23.07 -5.47 0.5 129,255 -29.83 -16.34 1 71,707 -25.25 -17.39 2 15,168 -13.22 -29.27 5 223 38.12 -62.78 10 3 233.33 300.00 the experimental results clearly show that there is a negative impact on the number of agglomerated particles at higher frequencies. to study the impact of presence of acoustic waves on the particle properties, a theoretical model is developed which is used for a further computational study. in this study, different particle parameters such as size, position of particles, velocity are studied which are comparatively challenging to study on an experimental setup. these studies are done with a reduced number of particles and lower frequencies due to the simulation time involved, which is increased due to the complexions associated with a greater number of particles. to prove this, the change in agglomeration time is studied with an increase in the number of particles. computational study of frequency impact onto the particle parameters and change in acoustic... 745 fig. 1 impact of frequency on particles 3. theoretical model dem can be developed within the framework of the lagrangian approach, various recent research was successful in studying the behaviour of particulate solids using this approach [25,39]. the motion of each particle in dem can be determined using newton’s second law of motion, which states that the mass of the particle and its acceleration is equal to the force acting on the particle. the total forces acting on a single particle can be represented as:        g b flow d dx m f f f dt dt    = + +    (1) where, m is the mass of the particle; 𝑑 𝑑𝑡 ( 𝑑𝑥 𝑑𝑡 ) is the acceleration of the particle with respect to time; fg is gravitational force acting on the particle; fb is the buoyancy force; fflow is the flow of particles in the vibrating medium induced by the sound waves, expressed in drag force. gravitational force is determined by: or gf m g v g=    (2) the buoyancy force equals: b airf v g= −   (3) where, g is 9.8 m/s2; v is volume of the particle; ρ is density of the particle; ρair is the air density. the velocity of the particle that is freely suspended in the medium (gas) which is frequently vibrated due to the presence of sound waves moves in the direction of the gas motion. the equation is generated ignoring complexities associated with fluid-particle interactions [33,37]. 6 flow relf r v =     (4) 746 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius where, μ is dynamic viscosity of the gas; r is radius of the particle; vrel is relative velocity between the particle and gas, also termed as vg vp. considering the orthokinetic interactions and acoustic wake effect as the additional forces acting on the particle. to reflect these forces on the drag force, further modifications are made in terms of velocity. 3.1. orthokinetic interactions for the particles that are interacting in the acoustic filed, previous researchers were successful in determining the sound velocity represented in a sine wave [40,41]. the sound velocity for orthokinetic interaction can be given as follows: ) (g g pv v q sin t x=   − (5) where vg is the gas media velocity amplitude. / ( )g g g airv p c =  (6) where, pg is the sound pressure amplitude; cg is sound velocity; qp is ratio of particle velocity; 𝜔 is angular velocity; f is the frequency. 3.2. acoustic wake effect to understand the effect of acoustic wake induced by a particle on the other particle, we are considering the theory proposed by oseen referred by giona et al. [42]. considering the flowing particle to be a sphere, drag force of the particle can be expressed in simplified terms of oseen regime. 3 6 1 re ( ) 16 flow g pf r v v    =     +  −    (7) where, re is the reynolds number; μ is the dynamic viscosity, vp is particle velocity; vg is the gas velocity. , , g g ac g pvv v v= + (8) where, vg,ac is the vector of gas velocity due to acoustic motion. we consider two particles (k and i) with different radii, rk and ri, respectively, positioned at a distance of r between the centre of two particles making an angle θ with respect to the horizontal x-axis passing through the centre of the particle k. the perturbation gas velocity (vg, pv) in the cartesian coordinate system is determined using the oseen regime by: ( ) ( ) cos 2 2 2 cos 2 cos33 1 1 1 , 2 8 2 33 1 1 sin , 2 8 2 k k k k r k kk r k k r k k k k r e r r r r r r e r                         − − − −    −   =  + −  +                  = −  + −  +         (9) computational study of frequency impact onto the particle parameters and change in acoustic... 747 where the relative velocity (νk) is given as vg,ac vp and: , , , , , , cos sin co sin s g pv x r g pv y r v v v v       = − =    + (10) equations (1-10) are used to analyse the influence of one particle on the trajectory of the other. the theoretical model combining general forces acting on the particle with orthokinetic and awe (acoustic wake effect) presented above is used to compute the particle motion in the acoustic field. the same model is further replicated in the discrete element method (dem) to numerically simulate and analyse the motion and agglomeration effect on particles. 4. simulation results and discussion the generated theoretical model is used as a basis of the computational simulation. 10 particles that are randomly generated with different sizes (table 2). table 2 name and size of the generated particles particle number, [-] particle name, [-] particle radius [m] 1 p1 1.024e-09 2 p2 5.036e-08 3 p3 1.987e-08 4 p4 1.309e-08 5 p5 3.522e-09 6 p6 1.256e-06 7 p7 3.111e-06 8 p8 3.284e-06 9 p9 1.807e-08 10 p10 1.010e-06 each such generated particle is assigned with predetermined properties such as mass, shape, and other physical properties. these properties define the particle interaction and its environment. the sound intensity of the acoustic field is maintained at 140 db throughout the test time. the algorithm of the dem determines which particles are in contact and calculates the forces acting on those particles. besides the gravitational and buoyancy force, orthokinetic interactions and wake effect are considered and applied to the simulation. the same procedure is continued for a desired period using the equation of motion. the position and velocities of the particles are updated accordingly. upon the simulation, the results are further visualized and analysed to understand the behaviour of the particles and their properties such as frequency, size, velocity, and simulation time. 4.1. effect on particle parameters with change in frequency 4.1.1. size all 10 particles are subjected to sound waves at three different frequencies (2 khz, 3 khz, and 5 khz) and the exhibited agglomeration of particles is recorded. 748 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius at 2 khz, the change in size of three of the tested particles is presented. the sudden spike in the size of the particle (y-axis) after certain steps (x-axis) indicates the agglomeration of particles. while 6 of 10 tested particles excited out of the agglomeration testing range, remaining 4 particles agglomerated to form three agglomerated particles at the 11th, 51st, and 61st step (p6, p2, and p10, respectively). it is observed that, at step 11, p7 (3.111e-06 m) agglomerated with p6 (1.256e-06 m) and formed a new particle with a size increase of 153 % compared to p6 and an increase of 2.1 % relative to p7 (3.177e-06 m) as presented in fig. 2. fig. 2 agglomeration of particles at 2 khz at step 51, p2 (5.036e-08 m) and the new p6 (3.177e-06 m) agglomerated and formed a particle with a sharp increase of 6210 % in terms of p2 and almost 0 % with 4.216e-12 m compared to p6. the last two particles left in the test range were agglomerated in the 61st step to form a new particle (3.211e-06 m) with a size increase of 1.05 % compared to newly agglomerated p2 (3.177e-06 m) and a 218 % increase in terms of p10 (1.010e-06 m). the increase in the size of the agglomerated particle de-pends on the impact of collision between the particles. it will vary for each particle depending on its velocity and properties. with an increase in frequency to 3 khz, 5 of 10 tested particles excited out of the agglomeration testing range, p6, which was agglomerated in the 11th step at 2 khz, is now agglomerated in the 9th step with increases in frequency by 1 khz. similarly, with a further increase in frequency to 5 khz, five particles excited off the testing range immediately after the first step of application of frequency, but p6 is found to be agglomerated at the 8th step. of the 10 particles, p6 and p7 are two of the top three particles in terms of size. this proves that the larger particles have a significant chance of agglomerating due to its higher inertia and resonance frequency. the aggregation of p6, which occurred at all three frequencies, is found to occur earlier with an increase in frequency, as shown in fig. 3. as frequency increases, particle agglomeration occurs more rapidly, but the quantity of agglomerated particles decreases. computational study of frequency impact onto the particle parameters and change in acoustic... 749 fig. 3 agglomeration of p6 at different frequencies due to the limited boundary condition of the testing area, some particles are found to be excited out with the presence of frequency within step 1. while three particles agglomerated at 2 khz, only one particle agglomerated at 3 khz and 5 khz. this may be because, in the field of high frequency, the tendency of particles getting more excited and exiting out of the testing area early increases, decreasing the chances of particles colliding with each other and agglomerating. the experimental results of the tested fuel are found to be in line with the computation results. 4.1.2. position of particles (coordinates) the position of a single particle is one of the parameters that is difficult to analyze through experimental tests. the position of the particle within the probability of boundary impacts the collision between particles. particles that are closely placed have the least chance of significant collision, enough to overcome their interparticle forces and agglomerate, unless the velocity of the particle is comparatively higher. the chance of a collision is also impacted by the spatial arrangement. uniformly distributed particles have higher chances compared to random distribution, particularly when the tested particles are limited. the positions of the particles (p6 and p7) that are found to be agglomerating under all 3 frequencies are studied and presented in a 3-cordinate system (x, y, z) in figs. 4-6. interestingly at smaller frequencies, the x-coordinate of the agglomerated particle is found to be greater than that of p6 and p7, except at 5 khz agglomerated particle is at velocity 3.22e-04 which is slightly lesser compared to p7 (velocity is 3.32e-04). in all cases the y-coordinate of the agglomerated particle, is lesser compared to both the particles. in all frequencies, z-coordinate of the agglomerated particle is more inclined towards that of p7. there is no specific tendency observed while calculating the distance between p6 and p7 and its impact on the position of the agglomerated particle. 750 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius distance between two particles is described as: 2 2 2 2 1 2 1 2 1( ) ( ) ( )r x x y y z z− + + −= − (11) additionally, there is no specific tendency observed in the change in the position of the agglomerated particle with increase in the frequency. fig. 4 coordinates of agglomerated particles at 2 khz fig. 5 coordinates of agglomerated particles at 3 khz fig. 6 coordinates of agglomerated particles at 5 khz 4.1.2. velocity of particles in dem simulations, the impact of particle velocity on agglomeration behavior can be significant and is influenced by several factors. particles with higher velocities possess the chance of more significant collisions. it can be said that the velocity of the particles is directly proportional to the number of collisions between the particles. faster-moving particles will experience a higher collision frequency. computational study of frequency impact onto the particle parameters and change in acoustic... 751 the velocity affects the kinetic energy of the particles. higher-velocity particles carry more energy, leading to more collisions that are impactful. oftentimes, the impact of energy of fast-moving particles is sufficient to overcome the inter-particle forces leading to agglomeration. the frequency of sound itself does not affect the speed of individual particles in the material directly. primarily the forces and interactions of the particles in the material or medium determine the velocity of the particles. sound waves composed of pressure variations or compression waves, spread through the medium, but do not change the velocity of the particles directly. conversely, it is observed that velocity, which makes the particles excite and collide with each other to form agglomerates, is found to increase with the increase in frequency. velocities of the particles remaining after their 1st step are presented in tables 3-5. with an increase in frequency from 2k to 3k, the average velocity of particles in the xaxis is found to increase by 3.02 % and in the z-axis by 152.3 %. similarly, with further increase from 3k to 5k, the average velocity of particles is found to increase by 9.26 %, 1.52 % and 20.50 % respectively in the x, y, and z axis. this increase in the velocity of the particles with frequency is the reason for the faster agglomeration of particles as the particles with higher velocity have higher chances of agglomeration. table 3 velocity of particles at frequency of 2 khz particle number, [-] vx, [m/s] vy, [m/s] vz, [m/s] 2 -1.19e-03 -3.53e-04 -3.38e-04 6 -3.10e-01 1.83e-04 7.49e-04 7 -1.72e-01 -1.53e-03 -1.04e-04 10 -2.41e-01 -2.97e-04 -1.66e-05 table 4 velocity of particles at frequency of 3 khz particle number, [-] vx, [m/s] vy, [m/s] vz, [m/s] 2 -1.33e-03 -5.16e-04 -7.29e-04 6 -3.43e-01 1.49e-04 1.61e-03 7 -1.20e-01 -8.76e-04 -5.32e-04 8 -1.08e-01 -2.41e-03 3.38e-04 10 -3.07e-01 -1.40e-04 2.30e-04 table 5 velocity of particles at frequency of 5 khz particle number, [-] vx, [m/s] vy, [m/s] vz, [m/s] 2 -9.72e-04 -5.36e-04 -6.45e-04 6 -3.11e-01 2.22e-04 1.79e-03 7 -7.37e-02 -8.60e-04 -6.00e-04 8 -6.55e-02 -2.41e-03 3.13e-04 10 -3.45e-01 -1.51e-04 2.48e-04 the comparison of the velocity of the p6 and p7 particles that collided and agglomerated across all frequencies is presented in fig. 7. while x coordinate values are relatively small, the velocities of higher frequencies at 3 khz and 5 khz are comparatively higher, which may result in the particles agglomerating faster. 752 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius the resultant velocity of the agglomerated particle for all three frequencies is calculated using the pythagorean theorem and presented below in table 6. ( ) 2 2 2| | x y z = + + (12) (a) (b) fig. 7 velocity of particles at different frequencies: (a) – six particles; (b) – seven particles computational study of frequency impact onto the particle parameters and change in acoustic... 753 the overall velocity of the agglomerated particle is found to decrease with an increase in frequency. there is no direct relation between the reduction in velocity and the change in frequency. the inter-particle force, velocity, or impact of collision of the colliding particles can be determined as some of the reasons for the observed tendency. the impact on velocity of the newly formed agglomerated particle across all three frequencies, presented in fig. 8, is further studied. fig. 8 agglomerated particle velocity comparison the resultant velocity of the agglomerated particle for all three frequencies is calculated using the pythagorean theorem and presented below in table 6. table 6 the resultant velocity of the agglomerated particle frequency [khz] resultant velocity [m/s] 2 1.66e-01 3 1.15e-01 5 6.72e-02 4.2. change of agglomeration time with increase in number of particles inside the confined testing range and frequency, with increase in the number of tested particles, time taken to agglomerate to one particle is presented in table 7 and fig. 9. at 2 khz and 140 db of sound intensity, tests are conducted with 9-15 particles until one particle is left in the agglomeration testing range. reduction of particles can happen either through particle-particle interaction (acoustic agglomeration of particles) or by leaving the particle boundary box. particles leaving the boundary box can be interpreted as agglomeration of particle with the boundary wall. either way, the tests are performed until the volume of the 754 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius particles falls down to 1. the time is measured in steps where each step is accounted for 0.01 seconds (for example, time taken for 20 steps is 0.2 seconds). table 7 steps taken to agglomerate to one particle particles, [number] steps, [number] steps per particle, [-] 9 22 2.44 10 61 6.78 11 11 1.22 12 25 2.78 13 209 23.22 14 65 7.22 15 86 9.56 fig. 9 steps per particle and number of particles it is found that at all seven tested conditions, the steps taken with increasing number of particles is found to increase. although there is a mixed tendency observed in table 7, when plotted on a graph, the tendency is found to be increasing. a linear expression is obtained for the tendency observed from the test results. using this equation, we can estimate the steps ‘y’ (or time) that is expected to take when ‘x’ particles are tested. as presented in fig. 9, it is found that with increase in the number of tested particles, the time taken to agglomerate to one particle is found to increase. this is because with increasing number of particles inside the confined testing range, the combination of interactions increases between the particles and during the long process of combinations, particle gets excited and exits the testing area due to the increased particle-particle interaction shielding effect. computational study of frequency impact onto the particle parameters and change in acoustic... 755 5. conclusion a fuel blended volumetrically at 80 % fat and 20 % isopropanol (fat80ip20) is used on an experimental setup to determine the impact of frequency on acoustic agglomeration of particles. it showed that with increase in frequency from 21.2 khz and 34.6 khz, the number of particles agglomerated are comparatively decreasing. at 21.2 khz, particles of size 5 and 10 μm are found to be increasing at 38.12 % and 233.33 % respectively, while at 34.6 khz, there is a 62.78 % reduction of 5 μm particles and 300 % increase in 10 μm particles. theoretical model of aa which is developed from the law of motion considering the gravitational, buoyancy forces along with orthokinetic interaction and awe acting on the particle. considering the limitations of other numerical simulation techniques in understanding the particle-particle interaction, the discrete element method is chosen over finite element methods or computational fluid dynamics. under specific boundary condition and 140 db of sound intensity of the acoustic field. the results of the computation study of influence of frequency on the particle parameters showed that with increasing frequency, total agglomerated particles decreased (three particles at 2 khz frequency and one particle at 3 and 5 khz). the experimental results of the tested fuel are found to be in line with the computation results. this may be because, in the field of high frequency, the tendency of particles getting more excited and exiting out of the testing area early increases, decreasing the chances of particles colliding with each other and agglomerating inside the boundary condition. the agglomeration of p6, which occurred at all three frequencies, is found to occur earlier with an increase in frequency. at 2 khz, agglomeration of this particle happened at the 11th step, whereas at 3 khz frequency, it is reduced to the 9th step, and, at 5 khz, p6 agglomerated at the 8th step. this leads to an estimated understanding that as frequency increases, particle agglomeration occurs more rapidly, but the quantity of agglomerated particles decreases. it is observed that with an increase in frequency, the velocity of the agglomerated particles is found to be decreasing. the frequency of sound itself does not directly affect the speed of individual particles in the material. the force, velocity, or impact of the collision of the colliding particles can be the reasons for the observed tendency. at 2 khz frequency, when particles are subjected to 140 db of sound intensity, time taken to agglomerate to one particle is found to be increasing with increase in the number of tested particles. this is due to the increase in complexities associated with increase in number of particles. a linear equation is developed using the simulation data which can be used to estimate the steps or time (y) taken to agglomerate ‘x’ number of particles. the study suggests that acoustic agglomeration has significant potential in reducing particulate emissions. however, further research is needed to optimize key acoustic parameters, test with diverse fuels and particle types, integrate with existing filtration technologies, assess the feasibility of acoustic agglomeration in real-world scenarios, and develop particle behavior modeling and simulation techniques. the study also emphasizes the need to investigate the efficiency of acoustic agglomeration in targeting ultrafine particles, which have significant health implications. additionally, a life-cycle analysis of the acoustic agglomeration technology, considering environmental impact, operational costs, and potential energy savings, could help establish its sustainability and market viability compared to existing emission control technologies. these areas of research will help to improve the practical applicability and scientific understanding of acoustic agglomeration. 756 s.m. rayapureddy, a. maknickas, j. matijošius, d. vainorius, a. kilikevičius acknowledgement: the paper is a part of the research done within the project funded from the research council of lithuania (lmtlt), agreement no: s-pd-22-69. this research was supported by the center of excellence project „civil engineering research centre“ (grant no. s-a-uei-23-5). references 1. www.statista.com/statistics/784780/worldwide-co2-emissions-fuel-combustion-by-sector/ (last access: 24.10.2024) 2. zhang, y., zhong, y., wang, j., tan, d., zhang, z., yang, d., 2021, effects of different biodiesel-diesel blend fuel on combustion and emission characteristics of a diesel engine, processes, 9(11), 1984. 3. gabsalikhova, m., makarova, i.v., mukhametdinov, e.m., buyvol, p.a., barinov, a.s., 2019, reasons for forming particulate matter 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operators chiranjib bhowmik1, divya zindani2, prasenjit chatterjee3,4, dragan marinkovic5,6,7, jūratė šliogerienė3 1department of mechanical engineering, techno india university, india 2department of mechanical engineering, sri sivasubramaniya nadar (ssn) college of engineering, india 3faculty of civil engineering, institute of sustainable construction, laboratory of smart building systems, vilnius gediminas technical university, vilnius, lithuania 4college of engineering, yuan ze university, taiwan 5tu berlin, department of structural analysis, berlin, germany 6institute of mechanical science, vilnius gediminas technical university, lithuania 7university college, korea university, 145 anam-ro, seoul, korea orcid ids: chiranjib bhowmik https://orcid.org/0000-0001-9338-7715 divya zindani https://orcid.org/0000-0002-9941-593x prasenjit chatterjee https://orcid.org/0000-0002-7994-4252 dragan marinkovic https://orcid.org/0000-0002-3583-9434 jūratė šliogerienė https://orcid.org/0000-0002-3696-6275 abstract. to address the problem of green energy source selection, this paper proposes a novel decision-making framework using fuzzy-tomada de decisao interativa multicriterio (todim) method in an interval-valued intuitionistic environment. the proposed framework integrates the prospect theory approach with schweizer-sklar and power averaging operators to evaluate the green energy sources including solid waste, solar, tidal, carbon capture storage, hydrogen, marine, hydel, biogas, wind, concentrating solar, geothermal and biomass under the influence of nine conflicting criteria such as annual generation, capacity factor, mitigation potential, useful life, installation period, energy requirement, co2 emission, generating cost and operations and maintenance cost. the vagueness associated with the evaluations as well as biased evaluations is taken care of by schweizersklar and power averaging operators while todim method provides due consideration to the psychological behavior of the decision maker. solar photovoltaic emerges as the best received: july 11, 2024 / accepted september 28, 2024 corresponding authors: prasenjit chatterjee1, dragan marinkovic2 1faculty of civil engineering, institute of sustainable construction, laboratory of smart building systems, vilnius gediminas technical university, vilnius, lithuania 2tu berlin, department of structural analysis, str. d. 17. juni 135, 10623 berlin, germany e-mail: prasenjit.chatterjee@vilniustech.1t, dragan.marinkovic@tu-berlin.de https://orcid.org/0000-0001-9338-7715 https://orcid.org/0000-0002-9941-593x https://orcid.org/0000-0002-7994-4252 https://orcid.org/0000-0002-3583-9434 https://orcid.org/0000-0002-3696-6275 mailto:prasenjit.chatterjee@vilniustech.1t 628 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė energy source. sensitivity analysis has also been performed to assess the robustness of the proposed decision-making framework. key words: green energy source selection, fuzzy logic, schweizer-sklar and power averaging operators, todim, sensitivity analysis 1. introduction green energy refers to energy that is produced from such sources that have minimal impact on the environment. these sources include solar, wind, hydro, geothermal, and biomass energy. green energy is clean and sustainable. green energy sources have emerged as a critical component of the world's energy mix. unlike traditional energy sources such as coal, oil, and gas, green energy sources generate electricity without releasing harmful pollutants into the environment. these energy sources can be harnessed in a sustainable way to meet the growing energy demands of the world population. one of the major challenges in the globe at present is the massive and fast growth in world population, civilization, technology and progress which is correlated with the huge demand and excessive ingesting outline of energy, water, and food resources compared to energy production and limited natural resources such as land, water, minerals and fuel, etc. in addition to that, there are widespread distractions that caused negative effects on the environment which causes various trauma to the common people. the exhaustion and limitations of conventional energy sources are fostering more and more research into green, alternate and renewable energy sources [1]. therefore, the need for a clean environment and energy, and economic security for sustainable development is becoming a real concern for the world communities in the last few years [2]. decrease in ecological discharges, temperament of detrimental wastes, and espousal of green expertise and products are on the precedence list of sustainable planning. one of the primary advantages of green energy sources is their environmental sustainability. unlike fossil fuels, which are finite and contribute to climate change, green energy sources do not emit greenhouse gases or other pollutants. this makes them an ideal solution for reducing our carbon footprint and mitigating the effects of climate change. to minimize dependence on fossil fuels, safeguard the environment, and reduce greenhouse gas emissions, switching from fossil to green energy is seen as an unavoidable trend. green energy is also considered a long-lasting, affordable, and ecofriendly source of energy. multinational enterprises have been at the forefront of advocating for the adoption of clean energy sources in many countries worldwide. the increasing number of businesses committing to the use of green energy and taking tangible measures to accelerate the transition to clean energy is a positive sign for global efforts toward achieving a greener future. as the sustainability concept emerges with increasing awareness about the environment, green energy sources take their positions in future energy planning worldwide for a long time. green energy sources also offer numerous technological benefits such as reduced greenhouse emission, energy security and independence, low operating costs, community empowerment grid stability and resilience. the development of green energy technologies has spurred innovation and advancements in energy storage, grid integration, and other related fields. advances in battery technology have made it possible to store solar and wind energy for use when the sun is not shining or the wind is not blowing. grid integration technologies have made it possible to connect green energy sources to the power grid and manage their output in a more efficient way evaluation of green energy sources: an extended fuzzy-todim approach based on... 629 [3]. the sources of green energy are numerous and diverse. the followings are the most often used for energy production: solid waste, solar, tidal, carbon capture storage, hydrogen, marine, hydel, biogas, wind, concentrating solar, geothermal and biomass. green energy sources are being evaluated to prepare for the future is becoming more and more crucial in recent eras due to the presence of various conflicting criteria. energy managers as well as policy makers are executing several policies and measures to ensure that renewable energy planning is of high environmental standards and conforms to the guidelines. the selection of green energy sources is important for mitigating climate change, improving energy security, protecting the environment, providing economic benefits, and driving technological innovation. the use of green energy sources can help to mitigate the impacts of climate change by reducing greenhouse gas emissions. renewable energy sources do not produce the same level of carbon emissions as fossil fuels, which are a major contributor to climate change. green energy sources can help to improve energy security by reducing reliance on imported fossil fuels. this can help to protect countries from energy supply disruptions, price volatility, and geopolitical risks. the use of green energy sources can provide economic benefits, such as job creation, increased energy independence, and reduced energy costs. as the cost of green energy technologies continues to decline, these benefits are becoming increasingly significant. the development and deployment of green energy sources can drive technological innovation and provide opportunities for new business models and market opportunities. however, the selection of technologies for green energy is a difficult transdisciplinary problem that mainly involves how well the technologies perform in response to several aspects, including environmental, social, technological, and economic ones. multiple criteria decisions making (mcdm) models are thus essential for the selection of green energy sources because it allows decision-makers to take into account multiple criteria when evaluating and selecting energy sources. when selecting green energy sources, decisionmakers need to consider a variety of factors, such as cost, energy efficiency, environmental impact, scalability, reliability, and availability. these criteria may be conflicting, and different stakeholders may have different priorities and preferences. mcdm methods are employed to address such complexities, facilitating the decision-making process in situations where traditional analytical approaches fall short [4]. 2. literature researchers have been investigating in wide array of decision-making problems associated with sustainable technologies using mcdm approaches such as selection of suitable site for charging station [5], selection of suitable site for installing floating photovoltaic system on dam reservoirs [6], healthcare supply chain resilience [7], assessment of management strategies in construction and demolition wastes [8], etc. studies on efficient green energy selection for sustainability planning were also discovered in the literature, focusing on various mcdm methods. some of the highly motivated researchers in the field of green energy source selection are discussed in brief. haralambopoulos and polatidis [9] used preference ranking organization method for enrichment evaluations (promethee ii) method to assess the renewable energy project in greece based on factors like; fuel saved, return on investment, job creation, environmental and risk index. the ranking of renewable energy sources in canada has been investigated 630 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė by nigim et al. [10] considering resource availability, financial viability, social benefit, educational potential, and ecological impact. pohekar and ramachandran [11] applied promethee method for selection of cooking energy substitution using renewable energy generation in india adopting factors like; chulha, improved chulha, kerosene stove, biogas stove, liquefied petroleum gas (lpg) stove, microwave oven, electric oven, solar cooker. patlitzianas et al. [12] utilized ordered weighted averaging (owa) method for the assessment of renewable energy producers in greece using political, social, technological criteria and their sub-factors. as per the study conducted by buchholz et al. [13], multi-criteria analysis (mca) is used for the evaluation of sustainable bioenergy projects in usa taking into account the stakeholder decision-making, application of qualitative data, uncertainty measures, ease of consumption, dynamic revolution, communication of decision process. cavallaro [14] applied promethee method for the evaluation of solar thermal technologies for sustainable energy planning in italy considering investment cost, operation & maintenance cost, mature technology, environmental impact, capacity factor. chung and lee [15] practiced in an integrated manner with elimination and choice expressing reality (electree) ii, evaluation of mixed data (evamix) and regime method for ranking of water management using renewable energy in the republic of korea using technical, environmental, economic criteria. doukas and psarras [16] used technique for order of preference by similarity to ideal solution (topsis) method for accessing renewable energy performance in greece based on reduction in emission of greenhouse gases (ghg), environmental impact, contribution to job creation, local economic development, regional economic development, investment cost and adequacy of energy. kahraman et al. [17] applied the fuzzy analytical hierarchy process (fahp) for the evaluation of renewable energy alternatives in turkey considering technological, environmental, social, economic and energy criteria. ren et al. [18] applied promethee method for the assessment of sustainable residential renewable energy systems in japan based on various factors like: carbon dioxide (co2) emission, cost of operation, investment cost and primary energy consumption. assessment of sustainable energy in greece is carried out by tsoutsos et al. [19] based on investment, operation and maintenance cost, fuel savings, mature technology, safety ratio, energy supply, emission control, local development. their study utilizes promethee i & ii for evaluation purposes. heo et al. [20] applied fahp in republic of korea for the assessment of renewable energy factors considering technological, economic, environmental, market and policy criteria. renewable energy assessment in turkey was put forwarded by kaya and kahraman [21] using vlsekriterijuska optimizacija i komoromisno resenje (vikor) and ahp. the study considers technical, economic, environmental and social criteria. shen et al. [22] applied fahp for the evaluation of renewable energy sources in taiwan based on energy price, security, energy supply, technical maturity and market size. a sustainable energy assessment for canada is done by reza et al. [23] considering factors like; resource depletion, losses in energy, management of waste, climate change, risks for the environment, performance cost, operation & maintenance cost, social acceptability. san-cristóbal [24] applied vikor method for assessing renewable energy projects in romania. their study uses power, investment ratio, period of implementation, operational hours, useful life, cost of operation, costs associated with the maintenance and co2 emissions as evaluating parameters. davoudpour et al. [25] adopted ahp for the selection of renewable energy technology in iran based on market competitiveness, technical factors and environmental factors. an integrated approach namely benefits, opportunities, costs and risks (bocr)-balance scorecard model (bsc)-analytic network process (anp) was proposed by shiue and lin evaluation of green energy sources: an extended fuzzy-todim approach based on... 631 [26] for the assessment of optimal recycling strategy for renewable energy generation in turkey. the study considers various costs, risks and environmental factors. the selection of sustainable renewable energy sources in lithuania is studied by streimikiene et al. [27] using topsis and multi-objective optimization by ratio analysis plus the full multiplicative form (multimoora) based on the economic dimension, environmental dimension and social dimension. balezentiene et al. [28] applied fmultimoora for assessing the sustainable energy in lithuania considering soil carbon sequestration, photosynthesis type, water adaptation, erosion control, input requirement and energy yield. ertay et al. [29] proposed measuring attractiveness by a categorical based evaluation technique (macbeth) and fahp for identifying the accessibility of renewable energy development in turkey based on risk and feasibility, reliability, social acceptance, labor impact, availability of funds and economic value. the selection of a suitable sustainable energy system in the united kingdom is studied by kurka and blackwood [30] using ahp along with delta approximation (deltap) and promithee ii. their study utilizes uncertainty, user-friendliness, flexibility, transparency, communication and multi-stakeholder as an evaluating parameter. yazdanichamzini et al. [31] utilized complex proportional assessment (copras) and ahp for the selection of renewable energy in iran based on power, investment ratio, implementation period, operating hours, useful life, operation & maintenance cost and co2 emission. kabak and dağdeviren [32] applied anp to ranking of renewable energy sources in turkey considering technology, economy, security, global effect and human wellbeing. the ranking of sustainable energy in the usa was investigated by topsis method by kucukvar et al. [33] based on environmental and socio-economic criteria and their sub-factors. troldborg et al. [34] used promithee method for the selection of optimum renewable energy technology in the uk considering power generation, technology maturity, supplier reliability, land requirement, cost and social acceptance as evaluating criteria. evaluation of renewable energy sources for turkey, studied by kuleli pak et al. [35] using topsis and anp based on factors such as: energy use per capita, supply efficiency, dependency on imports, climate change, quality of water, and total area of soil. an integrated ahp-topsis approach is proposed for the ranking of energy sources with potential low-carbon emissions for sustainability in italy by ren and sovacool [36]. their study considers technological maturity, reliability of the resource, acceptability of resources with regard to the social front, support from government and security of energy as an evaluating criterion. şengül et al. [37] applied ftopsis for raking of renewable energy systems in turkey based on technical, economic and social factors. al garni et al. [38] investigated renewable power generation sources for saudi arabia using hybrid ahp method considering technical, socio-political, economic, and environmental criteria. economic, technical, political, social and environmental aspects are taken into consideration by büyüközkan and güleryüz [39] for the selection of most appropriate renewable energy sources in turkey. their study utilizes dematel and anp methods for evaluation purposes. another study was carried out by büyüközkan and güleryüz [40] for the selection of optimum renewable energy sources for turkey using fuzzy preferences with dematel, anp, topsis methods. çolak and kaya [41] applied ahp based type-2 hesitant f-topsis method in an integrated manner for prioritizing renewable energy alternatives in turkey based on technical, economic, social, environmental criteria with their sub-factors. the sustainability behavior of renewable energy sources at algeria is studied by haddah et al. [42] using combined ahp methodology based on social, environmental, economic, and technical concerns. entropy-copras with different normalization tools are employed for the selection of green energy sources in india by bhowmik et al. [43]. their study considers 632 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė power generation, capacity factor, useful life, implementation period, cost factors, and co2 emission. lee and chang [44] applied entropy probability technique for ranking of renewable energy sources in taiwan based on technical, economic, social, environmental criteria with their sub-factors. swara-additive ratio assessment (aras) method is employed for evaluation of sustainability indicators for renewable energy systems in turkey by ghenai et al. [45] considering technical, economic, social, environmental criteria with their subfactors. qazi et al. [46] applied unified theory of acceptance and use of technology (utaut), sentiment analysis and entropy-topsis in an integrated manner for identifying the awareness and intention to accept renewable energy sources in malaysia based on technical efficiency, technological cost, land/price required, job creation, co2 emission, public opinion, awareness, social influence, performance expectancy, effort expectancy, behavior intention. bhowmik et al. [47] proposed an integrated ahp-quality function deployment (qfd) methodology for the selection of renewable energy sources in india based on various environmental, social, economic, and security criteria. evaluation of renewable energy sources for turkey is studied by yürek et al. [48] using integrated pythagorean fuzzy sets (pfss) based ahp-topsis method and considered capacity factor, availability ratio, evaluation of natural reserve potential, economic life cycle, levelized electricity generation cost, avoided emission, impact on ecosystem as an evaluating criterion. salameh et al. [49] applied critic, entropy, and topsis method in an integrated manner for optimal selection and management of renewable energy system in saudi arabia based on various cost factors, capacity, social impact etc. da ponte et al. [50] proposed ahp and topsis, combined with fuzzy logic, for evaluation of renewable energy generation share in brazil based on economic, social, robustness, political and technical criteria. evaluation of renewable energy resources for power generation is studied by sarkodie et al. [51] using critic weighting technique based moora, copras and topsis methodology and considered various cost factors, capacity factor, resource availability, technology maturity, ease of decentralization, efficiency, land use, job creation, socio-political acceptance and levels of co2 emission. swot-based ftopsis method is applied to rank the strategic renewable energy alternatives by akçaba and eminer [52] for northern cyprus based on solar energy potential, geostrategic position in energy transmission, switching to lng fuel use, hydrocarbon potential in eastern mediterranean, interconnected connection in energy transmission, decrease in energy investment costs, increasing awareness of energy efficiency, strong university infrastructure, energy efficiency policies, insufficient installed power, dependency on the use of fuel-oil, renewable energy source, energy management, limited use, energy supply security, geopolitical risks, climate change and environmental obligations, international legal regulations, unprepared entry of electric vehicles into the market. al-barakati et al. [53] used an extended interval-valued pythagorean fwaspas method for evaluation of renewable energy sources based on technical and economic feasibility, economic risks, pollutant emission, land requirement, need of waste disposal, social and political acceptance, water pollutant, investment costs, security of energy supply, source durability as an evaluating criterion. assessment of renewable energy sources for tamil nadu, india is studied by krishankumar et al. [54] using multi-hesitant fuzzy linguistic based choquet integral approach. their study mainly focused on adaptability, job creation, security, durability, pollution control, land use, total production price. apart from the above studies that entail the employability of traditional mcdm tools, some of the researchers have also proposed modern mcdm tools to assess green energy sources. these includes tools such as stable preference ordering towards ideal solution evaluation of green energy sources: an extended fuzzy-todim approach based on... 633 (spotis) [55], characteristic object method (comet) [56], sequential interactive modelling for urban systems (simus) [57], data variability assessment technique for order of preference by similarity to ideal solution (dariatopsi) [58] etc. for the selection of optimal green energy sources, this study proposes an extended ftodim method. according to the existing literature, this study can assert that it is the initial research to apply the proposed methodology in the area of green energy. fuzzy theory is incorporated into the decision-making process as the assessments of decisionmakers regarding criteria and alternatives are typically imprecise [59]. in fuzzy-decision making process, linguistic rapports are casted off to epitomize decision makers’ judgements. this is the main reason of adopting fuzzy theory in green energy sources selection. for illustration purpose, it is quite easier to represent the green energy source selection criteria as good, very good, poor, very poor etc. than in numbers. the experts assign linguistic terminology to the green energy selection parameters and alternatives, which are then summed up using schweizer-sklar and power averaging operators. these operators deal with swayed judgments from decision-makers, which would otherwise have had a negative impact on the final results. todim method ranks the established alternatives when taking into account the decision maker's risk appetite. the option with the highest score rating is the best and is recommended for long-term sustainability, followed by other options. the influence and novelty of the present work can be summarized as: ▪ to propose a novel decision-support system for the evaluation of green energy sources that gives due consideration to the risk appetite of the experts. moreover, the proposed approach also takes care of the swayed judgement during the process which otherwise could adversely affect the decision-making process. ▪ to carry out performance analysis of the green energy sources under the influence of the conflicting factors and hence select the optimum one. ▪ to check the consistency in the decision-making process using sensitivity analysis. 3. methodology this section delivers an outline of the proposed decision-making framework for the selection of optimal green energy sources. the framework models the linguistic evaluation in terms of interval-valued intuitionistic fuzzy numbers [60, 61]. fig. 1 depicts the decisionmaking structure that has been adopted to carry out performance analysis of the considered green energy sources. the proposed framework has several associated advantages: ▪ it pays due respect to the risk appetite of the decision maker while they evaluate the alternatives. ▪ it takes care of the biased evaluations which may be provided by the decision maker during their evaluations. ▪ the decision-making approach provides flexibility to the experts involved in evaluating the alternatives and henceforth in arriving at the final rankingsto check the consistency in the decision-making process using sensitivity analysis. 634 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė fig. 1 proposed framework for green energy selection the following steps are involved in the proposed extended todim approach: step 1: the alternatives {a1, a2, …, an}to be evaluated as well as the criteria {c1, c2, …, cn} under which evaluations are to be made are identified. within this stage, the decision makers are also identified to provide their linguistic evaluations. the evaluations are converted into interval-valued intuitionistic fuzzy (ivif) numbers in accordance with a standard scale. let the decision matrix be represented as; rk=(rij k) m×n (k=1,2,...,q), where rij k=([aij k,bij k],[cij k,dij k]) represents interval-valued intuitionistic fuzzy (ivif) number. step 2: the weight associated with the identified experts is evaluated using eq. (1) as follows: evaluation of green energy sources: an extended fuzzy-todim approach based on... 635 wk ... = [mq +nq { mq mq +nq }]+[mq +nq +{ mq + mq ++nq +}] ∑ [{mq +nq ( mq mq +nq )}+{mq +nq +( mq + mq ++nq +)}] q k=1 ,k=1,2,..., (1) where eq=< [mq-, mq+], [ nq-, nq+] > denotes the rating of the qth decision maker in terms of ivif numbers. step 3: within the scope of this stage of the proposed approach, normalization of the obtained decision matrix is carried out. this process aids in converting the different performance metrics or criteria into similar types. eqs. (2) and (3) are employed to obtain the normalized values for beneficial and non-beneficial criteria, respectively: rij={(aij k ,bij k),(cij k,dij k)} (2) rij={(cij k,dij k),(aij k ,bij k)} (3) step 4: in this step, calculations of support values are used in accordance with eq. (4) sup(rij k,rij t )=1-d(rij k,rij t ) (4) where the support from one ivifn to the other is represented by sup (rij k, rij t) and the distance between two ivifns by d (rij k, rij t). step 5: eq. (5) is followed to arrive at the values of t(rij k): t(rij k)=∑ sup(rij k,rij t ) p &t=1 &t≠k (5) where the summation of all the support values is represented by t(rij k). step 6: the weights (δij k) associated with each ivif evaluation are calculated using eq. (6): δij k= wk{1+t(rij k)} ∑ wk{1+t(rij k)} p k=1 (6) where the weight of an expert is represented by wk. step 7: obtained evaluations in the form of decision matrices are now aggregated into a single decision matrix. this is accomplished with the aid of schweizer-sklar power weighted average (ivifsspwa) operator. the detailed formulation for ivifsspwa operator can be found in [60]. step 8: in the present step of the proposed approach, calculations associated with the criteria weights and hence the relative weights of each of the performance metric are approached. the criteria weights can be obtained using any of the weight determination procedures as per the problem under consideration as for instance suggested by [62]. thus, to uphold the brevity of the piece, the weight determination procedure hasn’t been depicted here. with the aid of derived criteria weights, calculation for the relative weights (wrc) associated with the performance metrics is approached using eq. (7): wrc= wcn wr (7) where wr is the reference weight and wcn is the weight of criteria. 636 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė step 9: at this stage of the proposed approach for appraising the identified alternatives, eq. (8) is used to calculate the dominance degree for each of the identified alternative over the remaining identified alternatives i.e., δ (ri ~, rj ~). δ(r̃i,r̃j)=∑ ϕ j (r̃i,r̃j)∀(i,j)n c=1 (8) where ϕc represents the local dominance score and θ represents attenuation factor. step 10: in the last but one step, the calculations for the overall dominance score, ξ(ai), is processed using eq. (9). step 11: the final rankings of the alternatives are obtained through either of the two approaches: ξ(ai)= [∑ δ{r̃i,r̃j}]mini {∑ δ(r̃i,r̃j) m k=1 m k=1 maxi ∑ δ{r̃i,r̃j}]mini {∑ δ(r̃i,r̃j) m k=1 m k=1 (9) (1) γ levels form the basis of ranking the green energy alternatives. therefore, in this approach, the decision maker is provided with the flexibility to make the final decision in accordance with the different γ levels. (2) in the second approach, the final rankings for the identified alternatives are obtained by the decision-maker through the calculations for the average overall dominance degree which can be obtained by pursuing a comprehensive evaluation approach. eq. (10) depicts the formulation for the average overall dominance degree. this approach is useful when the decision maker doesn’t have understanding on the γ levels. {ξ(ai)}= 1 f ∑ ξf(ai) f f=1 (10) 4. illustrative example this section describes the application of the proposed fuzzy todim approach to evaluate the effectiveness of existing green energy sources in each region to improve its sustainability problem. in the primary stage i.e., in step 1 of the decision-making process, criteria {c1, c2, …, cn}and the alternatives {a1, a2, …, an} are deciphered from various literature and a decision matrix is formed. in the present case, following twelve alternatives have been evaluated: solid waste (a1), solar (a2), tidal (a3), carbon capture and storage (a4), hydrogen (a5), marine (a6), hydel (a7), biogas (a8), wind (a9), concentrating solar (a10), geothermal (a11), and biomass (a12). a brief description of the considered alternatives is as below. solid waste: solid waste can be used as a source of energy through a process known as waste-to-energy (wte) [63]. wte involves converting solid waste into electricity or heat through various technologies, such as incineration, gasification, and pyrolysis [64]. solar: solar energy is a renewable and sustainable source of energy that is generated from the sun's rays. it is considered one of the cleanest and most abundant energy sources available on earth, as it does not produce any harmful emissions or pollutants during its generation or use. there are two main ways to harness solar energy: solar photovoltaics (pv) and solar thermal systems. solar pv systems use solar panels to convert sunlight directly into electricity, while solar thermal systems use sunlight to heat a fluid, which then generates electricity through a turbine or other means. solar energy has several advantages over traditional sources of energy, such as coal and natural gas. it is a clean and renewable energy source, which means it does not contribute to climate change or air pollution. solar evaluation of green energy sources: an extended fuzzy-todim approach based on... 637 energy systems also require very little maintenance, and once installed, they can generate electricity for decades. additionally, solar energy can provide energy independence for households and businesses, as they can generate their own electricity and reduce their dependence on the grid [65]. tidal: tidal energy is a form of renewable energy that is generated from the movement of ocean tides. tidal energy is harnessed using tidal turbines that are installed underwater in areas of high tidal movement, such as bays, estuaries, and channels. these turbines work by capturing the kinetic energy of the moving water and converting it into electricity, much like a wind turbine. tidal energy has several advantages over other forms of renewable energy. tidal currents are predictable and consistent, which means that tidal energy can be generated continuously without the need for large energy storage systems [66]. carbon capture and storage: carbon capture and storage is a technology that is used to capture carbon dioxide (co2) emissions from industrial processes and store them underground or in other secure locations, such as depleted oil and gas or deep saline aquifers. it involves three main steps: capture, transportation, and storage. in the capture stage, co2 is separated from other gases emitted from industrial processes, such as power plants, cement factories, and steel mills. in the transportation stage, the captured co2 is transported via pipelines or other means to the storage location. finally, in the storage stage, the co2 is injected into deep geological formations where it can be securely stored for long periods of time. it has the potential to significantly reduce greenhouse gas emissions from industrial processes, and it can be used to mitigate the impact of climate change [67]. however, technology is still in the early stages of development and deployment, and there are several challenges that need to be addressed. these include the high cost, the energy required to capture and transport co2, and the need for regulatory frameworks to ensure the safe and secure storage of co2. hydrogen: hydrogen is the most abundant element in the universe and can be obtained from a variety of sources, such as water, natural gas, and biomass. hydrogen can be used as a fuel for vehicles, power generation, heating, and other applications. one of the main advantages of hydrogen energy is that it produces no greenhouse gas emissions when used as a fuel. when hydrogen is burned, the only by-product is water vapor. this makes hydrogen a potentially important source of clean energy that could help reduce our dependence on fossil fuels and mitigate climate change [68]. there are also some challenges associated with hydrogen energy. one of the main challenges is the cost of producing, storing, and transporting hydrogen. currently, most hydrogen is produced from natural gas through a process called steam methane reforming, which is energy-intensive and produces carbon dioxide emissions. developing more cost-effective and sustainable methods of producing hydrogen will be essential for its widespread adoption as a source of energy. another challenge is the development of hydrogen infrastructure, including production facilities, storage facilities, and distribution networks. while hydrogen can be transported by pipelines or tanker trucks, these systems are not yet widely available or cost-effective. marine: marine energy refers to the energy that can be derived from the ocean's waves, currents, and thermal gradients. marine energy is a renewable and sustainable source of energy that has the potential to reduce our dependence on fossil fuels and mitigate climate change. there are several types of marine energy technologies, including wave energy converters, tidal energy turbines, ocean current turbines, and ocean thermal energy conversion systems [69]. one of the advantages of marine energy is that it is predictable and consistent. unlike wind and solar energy, which can be intermittent and variable, ocean energy resources are more reliable and can be forecasted with a high degree of accuracy. 638 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė additionally, marine energy has the potential to provide electricity to coastal communities that are not connected to a power grid or to supplement the energy needs of existing coastal infrastructure. hydel: hydel energy, also known as hydroelectric power, is a renewable energy source that is generated by the movement of water. hydel energy is produced by harnessing the energy of falling water to turn turbines, which generate electricity. the process of generating hydel energy starts with the water being collected in a reservoir. when the water is released through the reservoir, it flows through a large pipe that leads to a turbine. the force of the water turns the turbine, which in turn drives a generator to produce electricity. hydel power plants typically use dams to store water in reservoirs, which creates a height difference or head. this head of water is then released through turbines, which turn generators to produce electricity. the amount of energy that can be produced by a hydel power plant depends on the volume of water flow and the height of the head. hydel energy is a renewable and sustainable source of energy, as water is constantly replenished through the water cycle. it also produces no greenhouse gas emissions and has a relatively low operating cost once the dam and hydropower plant are built [70]. biogas: biogas energy refers to the production of energy through the conversion of organic matter, such as agricultural waste, or municipal waste. biogas is primarily composed of methane and co2 and can be used as a fuel for electricity generation or as a replacement for fossil fuels in heating and transportation. it is produced by the breakdown of organic matter in the absence of oxygen, a process known as anaerobic digestion. the process of generating biogas energy starts with the collection and transportation of organic waste to a biogas plant. the waste is then mixed with water and placed in a sealed container called a digester, where microorganisms break down the organic matter and produce biogas. the biogas is then purified to remove impurities such as carbon dioxide and hydrogen sulfide before being used as fuel. however, there are also some challenges associated with biogas energy. the production of biogas can be expensive and requires significant infrastructure, such as biogas plants and transportation systems. in addition, the quality and quantity of biogas can be affected by factors such as the type of organic waste and the efficiency of the digester [71, 72]. wind: wind power is the process of producing electricity by using wind turbines to transform wind energy into usable forms of energy [73]. concentrating solar: concentrating solar power (csp) is a technology that generates electricity by focusing the sun's energy onto a small area to create heat [74]. this heat is then used to generate steam, which drives a turbine and produces electricity. csp systems use mirrors or lenses to concentrate the sunlight onto a receiver, which is typically located at the top of a tower. however, some challenges are associated with csp, such as the high cost of building and maintaining csp systems. in addition, csp systems require a large amount of land, which can have environmental impacts. geothermal: geothermal energy is the energy derived from the natural heat of the earth's interior. the earth's interior is extremely hot, with temperatures reaching up to 6,000 degrees celsius at the core. this heat can be harnessed and converted into electricity using geothermal power plants [75]. biomass: it describes any living or dead biological substance that can be burned as fuel. biomass energy refers to the energy that is derived from organic matter, such as wood, crops, agricultural waste, and other plant and animal materials. one common method of generating biomass energy is through the combustion of biomass materials. biomass materials are burned to produce heat, which is then used to produce steam that drives a turbine to generate electricity. the process is like that used in a coal-fired power plant but with the use of renewable organic materials instead of fossil fuels [76]. evaluation of green energy sources: an extended fuzzy-todim approach based on... 639 table 1 importance weights as linguistic terms [77, 78] linguistic variable ivif number very important (vi) [0.80, 0.95], [0.0, 0.05] important (i) [0.65, 0.75], [0.15, 0.20] modest (m) [0.45, 0.55], [0.30, 0.45] unimportant (u) [0.20, 0.30], [0.55, 0.70] very unimportant (vu) [0.00, 0.10], [0.80, 0.90] table 2 ratings of alternatives using linguistic values [79] linguistic variable ivif number very very poor (vvp)/ very very low (eel) [0.00, 0.10], [0.85, 0.90] very poor (vp)/ very low (el) [0.10, 0.15], [0.70, 0.75] fairly poor (fp)/ moderate low (ol) [0.30, 0.40], [0.45, 0.50] poor (p)/ low (l) [0.15, 0.25], [0.55, 0.60] fair (f)/ moderate (o) [0.40, 0.50], [0.35, 0.40] fairly good (fg)/ moderate high (oh) [0.50, 0.60], [0.25, 0.30] good (g)/ high (h) [0.60, 0.70], [0.20, 0.25] very good (vg)/ very high (eh) [0.70, 0.80], [0.15, 0.20] very very good (vvg)/ very very high (eeh) [0.80, 0.90], [0.05, 0.10] extremely good (xg)/ extremely high (xh)) [0.90, 0.95], [0.0, 0.05] the evaluations have been carried out under the influence of following nine criteria: annual generation (c1), capacity factor (c2), mitigation potential (c3), useful life (c4), installation period (c5), energy requirement (c6), co2 emission (c7), generating cost (c8) and operations and maintenance cost(c9). a group of three experts is subsequently selected to provide their linguistic evaluations. three experts were selected and represented as {d1, d2, …, dn}. one of the experts belonged to the domain of energy engineering and technology and had vast experience of renewable energy technologies, energy storage solutions etc. the second expert had enormous experience in the domain of energy management and planning. the third expert was an environmentalist and had rich experience in environmental science and sustainability those experts were asked to provide the importance of criteria in terms of linguistic information. after this, converted to ivif numbers using the scale represented in table 1. the designated decision makers are also asked to appraise the alternatives with admiration to the considered criteria. thereafter, the evaluations are provided in terms of linguistic information and converted to ivif numbers using the scale depicted in table 2. table 3, table 4 and table 5 depict the provided evaluations respectively. table 3 decision matrix by decision maker 1 dec. maker d1 green energy alternative a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 c1 p p g p g p g vg g f g p c2 g g f p f g f g p p g p c3 p f f p g f f vg g g vg f c4 g f f g f f f vg g f vg f c5 vp p p p f f f vg g g f g c6 p p p f f f f g f f p f c7 vp p p vg vg g g vg vg vg vg g c8 vp p p f g f f vg g f g f c9 vp p vp p f p p f f f g f 640 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė table 4 decision matrix by decision maker 2 dec. maker d2 green energy alternative a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 c1 p g p p f f f g p p g p c2 g vg f p f f f g p p f p c3 p f f p g f g vg g g vg f c4 p f f g f g f vg f f vg f c5 vp p p p f p f vg g g f g c6 p p p f f f p g f f p f c7 p p p vg vg g g vg vg vg vg g c8 vp p p f g f f vg g f g f c9 vg p vp p f p p f f f g p in step 2 of the approach that has been devised for evaluating the identified green energy sources, eq. (1) is applied to arrive at the weights for each of the designated experts. following weight vector was established through the formulation: w= (0.394, 0.274, 0.332)t table 5 decision matrix by decision maker 3 dec. maker d3 green energy alternative a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 c1 g p g p f f f g p p g p c2 g vg f f f f f g p p f p c3 p f g p g f g vg g g vg g c4 p f f g f g f vg f f vg f c5 vp p p p f p f vg g g f g c6 p p p f f f p g f f p f c7 p vp p vg vg g g vg vg vg vg g c8 p p p f g f f vg g f g f c9 g vp vp vp f p p f f p g p in step 3, the decision matrices provided by the three experts are normalized through eqs. (2-3). in the present case, annual generation, capacity factor, mitigation potential and useful life are the beneficial criteria. on the other hand, installation period, energy requirement, co2 emission, generating cost and operations and maintenance cost are the non-beneficial criteria. in step 4, the supports for each of the evaluations are obtained employing eq. (4). in step 5, the values associated with t(rij k) are obtained using eq. (5). in step 6, the calculations for the weights (δij k) associated with each of the evaluations are determined using eq. (6). in the following step (step 7), the decision matrices are aggregated using the equation as given in [59]. the obtained aggregated decision matrix for one alternative has been shown in table 6 due to space limitations. now the relative weights of the performance metrics that have been considered to evaluate the alternatives are calculated using eq. (7). the values derived for the weights have been presented in table 7. evaluation of green energy sources: an extended fuzzy-todim approach based on... 641 table 6 aggregated decision matrix obtained using schweizer–sklar fuzzy todim method for one alternative green energy a1 c1 (0.3, 0.4, 0.506, 0.5) c2 (0.64, 0.743, 0.369, 0.766) c3 (0.3, 0.4, 0.506, 0.5) c4 (0.61, 0.715, 0.362, 0.745) c5 (0.15, 0.25, 0.485, 0.4) c6 (0.3, 0.4, 0.506, 0.5) c7 (0.15, 0.25, 0.485, 0.4) c8 (0.15, 0.25, 0.485, 0.4) c9 (0.15, 0.25, 0.485, 0.4) in step 9, eq. (8) is employed to obtain the results for the dominance degree of each alternative under consideration over the other remaining identified alternatives. the values derived for the dominance degree have been shown in table 7. in step 10, calculations of overall dominance degree are followed. this is accomplished using eq. (9). in step 11, both or either of the following methodologies can be employed to deduce the final ranking results: (1) as per the first approach, the decision-maker considers the value of γ for ranking of the alternatives. therefore, this very approach allows flexibility to the decision-makers as they have the leverage to opt for the ranking results as per their choice of γ values. table 8 depicts the values associated with the overall dominance score at different γ values for the considered alternatives. table 8 overall dominance score {ξ(ai)} of alternatives at different γ and θ values average overall dominance degree green energy gamma level 0 -1 -2 -5 -20 -50 -200 θ = 1 a1 0.9440 0.9640 0.9663 0.9783 0.9796 0.9876 0.9888 a2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 a3 0.9558 0.9748 0.9780 0.9930 0.9951 0.9966 0.9979 a4 0.1343 0.1573 0.1592 0.1752 0.1775 0.1793 0.1804 a5 0.2849 0.3139 0.3170 0.3280 0.3299 0.3315 0.3329 a6 0.2943 0.3163 0.3187 0.3367 0.3388 0.3400 0.3415 a7 0.5115 0.5325 0.5348 0.5558 0.5575 0.5586 0.5603 a8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 a9 0.0326 0.0556 0.0577 0.0697 0.0716 0.0733 0.0745 a10 0.2523 0.2703 0.2720 0.2850 0.2871 0.2883 0.2897 a11 0.3460 0.3680 0.3708 0.3888 0.3914 0.3935 0.3947 a12 0.2125 0.2365 0.2387 0.2577 0.2600 0.2618 0.2637 table 7 weights and relative weights of the different criteria criteria wcn wcr c1 0.1104 0.6671 c2 0.1024 0.6182 c3 0.0939 0.5640 c4 0.0817 0.4937 c5 0.1366 0.8252 c6 0.0874 0.5282 c7 0.1096 0.6622 c8 0.1128 0.6815 c9 0.1656 1.0000 642 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė (2) in this approach, the raking results are derived using the values of the average overall dominance degree {ξ¯(ai)} i.e., the values obtained using eq. (10). the approach is more useful in case the expert doesn’t have any understanding of the γ values. therefore, the experts can arrive at the best alternative solution through an inclusive performance instead of making the final decision based on different γ value. the results have been tabulated in table 9. table 9 average overall dominance degree of alternatives and their ranking results green energy average overall dominance score rank a1 0.9727 3 a2 1.0000 1 a3 0.9844 2 a4 0.1662 10 a5 0.3198 7 a6 0.3266 6 a7 0.5444 4 a8 0.0000 12 a9 0.0621 11 a10 0.2778 8 a11 0.3790 5 a12 0.2473 9 5. sensitivity analysis sensitivity analysis in decision-making is a technique used to determine how different variables or parameters influence the outcome of a decision model. several methods can be used to conduct sensitivity analysis, including the adjustment of criteria weights, proportional variation of weights, and other techniques [80]. the stability of the ranking results is validated in this paper through an analysis of changes in criteria weights. this is achieved by systematically adjusting the weights by 20%, 40%, and 60%, one at a time. the obtained results for the performance appraisal based on the first approach of the proposed methodology have been presented in figs. 2, 3 and 4, respectively. this has been accomplished for θ = 1. the stability of ranking results with respect to varying criteria weights is also illustrated in fig. 5. impact analysis of the changing values of attenuation factor onto the ranking results has also been carried out. different values of attenuation factor i.e., 0.1, 2, 5 and 10 have been considered. the impact of varying attenuation factor values on the ranking results has also been analyzed. attenuation factors of 0.1, 2, 5, and 10 have been considered. the results are shown in fig. 3. while the ranking is noticeably affected at higher attenuation factor values, the overall ranking results remain stable. the analysis provides flexibility to the decision-maker or the policy maker regarding the choice of optimal green energy source. in case the selected optimal green energy source becomes unavailable then the decision maker can adopt other sources of energy with due consideration to the different constraints such as climatic conditions, geographical location etc. in the present study, the outlook of the decision maker has been considered as the key variable entity for the considered criteria weight. the prevailing models on optimal green energy source selection lay emphasis only on one criteria weight while the present study considers a variety of local factors and is therefore inclusive over the prevalent models. the presented study is therefore novel in the domain of green energy source selection for evaluation of green energy sources: an extended fuzzy-todim approach based on... 643 the indian state of tripura. hence, the presented work can be employed by policymakers, decision-makers and energy managers as a benchmark for enhancing reliability in planning towards sustainable development goals. fig. 2 variation of criteria weight by 20% fig. 3 variation of criteria weight by 40% fig. 4 variation of criteria weight by 60% 644 c. bhowmik, d. zindani, p. chatterjee, d. marinkovic, j šliogerienė fig. 5 variation of attenuation factor and its impact on ranking results 6. conclusions green energy within a diverse and mixed energy environment is seen as both a challenge and an opportunity. researchers from various sectors are actively engaged globally in this pursuit, with the aim of generating valuable and practical outcomes. in this complex scenario, uncertainties are managed, and various factors are addressed to achieve optimal energy solutions. as the energy sector evolves, the integration of green energy sources, alongside balancing traditional and renewable alternatives, is considered crucial for advancing sustainability and meeting global energy demands. a pragmatic perspective has been offered in this paper for energy managers, stakeholders, investors, and other key decision-makers, to guide them in making informed choices and analyzing the impact of multiple criteria in green energy selection process. a decision-making framework based on the interval-valued intuitionistic fuzzy todim method is introduced to evaluate various green energy alternatives. these alternatives are rigorously evaluated using both quantitative and qualitative performance indicators, providing a comprehensive analysis of their potential. the analyzed energy alternatives and their operational properties are selected based on a comprehensive literature review. qualitative and quantitative data analyses are performed to identify the optimal green energy alternative. in this study, the proposed decision-making framework integrates prospect theory approach with schweizer-sklar and power-averaging operators to evaluate the energy sources effectively. the vagueness of information associated with all the biasing evaluations is taken care of by the schweizer-sklar and power averaging operators wherein the todim approach takes into consideration the risk appetite of the decision maker while they provide their evaluations. the results obtained through the adopted methodology corroborate with the findings established in the real world-application of the work carried out by earlier researchers. the key conclusions are: ▪ in accordance with the results obtained from the presented work, solar photovoltaic is the best ranked energy source which is then followed with the rest sources of energy. in case the energy from solar photovoltaic gets exhausted, then other sources of energy since their rankings can be adopted for the generation purpose. evaluation of green energy sources: an extended fuzzy-todim approach based on... 645 ▪ out of the considered criteria, the criteria on maintenance cost are weighted the most and, one of the influential factors in opting for the best solution. on the other hand, useful life as one of the criteria has the least impact on the decision-making process as is evident from the derived criteria weights. ▪ the selected green energy source is optimum in terms of annual generation, capacity factor, mitigation potential, useful life, installation period, energy requirement, co2 emission, generating cost and operations and maintenance cost. ▪ the evaluating factors considered in this study ensure sustainable green energy planning for the selected region. ▪ the judgements made by this proposed model ensure that the selected green energy can consistently fulfil the energy demand of the area under consideration. the proposed model shows its application in the conflicting environment of the selected study. the results obtained are based on the performance metrics that have been identified through the exhaustive literature review. there are obvious limitations of the presented work. it is the maturity of the involved experts that has an overbearing 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infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 2, 2014, pp. 149 155 a real time neural network based finite element analysis of shell structure udc 519.95:531.2 žarko ćojbašić, vlastimir nikolić, emina petrović, vukašin pavlović, miša tomić, ivan pavlović, ivan ćirić mechanical engineering faculty, university of niš, serbia abstract. in recent years, the finite element method has been widely used as a powerful tool in the analysis of engineering problems. in the simulation of deformable objects using the finite element method, a complex system of nodes which make a mesh grid is used. the fem model includes material and structural properties, which altogether determine the model’s response to certain loading conditions. a reliable simulation is supposed to provide for an easier, faster and less expensive development of structures. the real-time simulation of shell-type deformable objects using the finite element method for a non-linear analysis is a challenging task because of the need for fast systems that do not demand high computational cost. in this paper, we present an efficient method based on neural networks for simulating the real-time behavior of a thin walled structure modeled by the finite element method in the commercial fe software. using the finite element method, the structures displacements are computed offline, by applying forces in the specified range. in the online application mode, a trained neural network is used for obtaining required results for specified loads. key words: finite element analysis, neural network, simulation, thin walled structure, shell 1. introduction the finite element method (fem) rapidly grew as the most useful numerical analysis tool in the mechanical engineering disciplines (such as aeronautical, biomechanical, and automotive industries). the fem provides for detailed visualization of where structures deform and indicates the distribution of stresses and displacements. continuous development of new structural materials leads to ever increasingly complex structural received march 18, 2014 / accepted july 2, 2014 corresponding author: žarko ćojbašić university of niš, faculty of mechanical engineering, department for mechatronics and control of mef, aleksandra medvedeva 14, 18000 niš, serbia e-mail: zcojba@ni.ac.rs original scientific paper 150 ž. ćojbašić, v. nikolić, e. petrović, v. pavlović, m. tomić, i. pavlović, i. ćirić designs that require a careful analysis [1]. the utilization of numerical methods to solve shell mathematical models of complex structures has become an essential component in the design process [2]. thus, the finite element method has been widely used as the fundamental numerical procedure for the analysis of shells [3]. in order to improve the performance and for optimization purposes, which play an important role in design process and modeling, integration of the finite element method and the neural networks has been considered by some authors. such integration enables a simulating real-time behavior of the structure; it allows the response of a structure in real or nearly real time to external stimuli to be observed. to illustrate the computational methodology javadi et al. [4] have presented three numerical examples of application of the developed intelligent finite element code (named neurofe program) to engineering problems. a neural network is trained using data representing the mechanical response of the material to applied load; the trained network is then used in the finite element analysis to estimate the stress and displacement. an application of the artificial neural networks for definition of the effective constitutive law for a composite is considered in [5]. first, a classical homogenization procedure is directly interpreted with the use of this numerical tool. next, a self-learning finite element code (fe with ann inside) is used in the case when the effective constitutive law is deduced from a numerical experiment (substituting here a purely phenomenological approach). in [6], a combination of the nonlinear contact finite element analysis and the artificial neural network has been applied to explore the possibility of material property execution with automatic ball indentation. this paper addresses the need for a fast system that simulates real-time behavior of shell structures. the main idea in this paper is to combine available tools, namely the finite element method and the neural networks in order to provide for a new quality. in the proposed methodology, the neural network is used for faster and easier obtaining of required results for specified loads. it is trained by using the raw numerical data obtained by the finite element method, representing the displacement of the structure to applied load. 2. modeling of shell structure the recognition of the nature of the global behavior of thin-walled structures allows condensation of the complex 3d-field to the essential ingredients of the structural response described by a 2d approach, and thereby an efficient modeling of such structures is achieved. a general shape implies arbitrarily curved structures. irrespectively of the applied loads, both the membrane and the flexural strains are induced in such a structure. furthermore, the thin-walled structures made of composite laminates require 2d-theories with transverse shear strains and stresses included for adequate modeling. the simplest and most frequently used theory is the first-order shear deformation theory (fsdt), which is based on the mindlin-reissner kinematical assumptions that imply constant transverse shear strains and stresses across the thickness of the structure [7]. within the framework of the fem, the strain field of a shell element based on the aforementioned kinematical assumptions is given in the following general form: a real time neural network based finite element analysis of shell structure 151 { } [ ] { } { } [ ]{ } { } [ ] mf mf e u s s b d b d b ε ε ε ⎧ ⎫ ⎧ ⎫ = = =⎨ ⎬ ⎨ ⎬ ⎩ ⎭ ⎩ ⎭ e , (1) where {εmf} and {εs} are membrane-flexural (in-plane) and transverse shear strains, respectively, [b bmf] and [bsb ] are corresponding strain-displacement matrices and can be summarized into element strain-displacement matrix [bbu], and, finally, {de} are element nodal displacements. for more details on the definition of the strain field, an interested reader is referred to [8]. 3. modeling and analysis of cylindrical arch structure fig. 1 shows a shell that is analyzed in this paper. a simply supported cylindrical arch with radius r=360 mm, central angle 112.5°, and width b=700 mm is considered. the thickness of shell is 30 mm. a uniform mesh is used in this solution. number of elements is 196 (14×14), while type of elements is a quadrilateral shell element with 8 nodes. the type of analysis performed to determine displacements of this structure in the shape of cylindrical arc is non-linear. this type of analysis provides for more accurate results, but it increases computation time, even for this simple shape structure. the considered structure is exposed to rather large loads and, hence, the induced deformations exceed the realm of linear analysis. therefore, the geometrically nonlinear fem analysis is applied to determine the structural response with adequate accuracy. this, furthermore, implies increased computational time because the structural stiffness continuously changes over the course of deformation as a consequence of the change in structural configuration and stress state. the incremental solution procedure based on the iterative newton-raphson method is used to obtain the solution. a) b) fig. 1 a) cylindrical arch with boundary conditions, b) meshed shell the excitation is modeled as a concentrated force with components in xand ydirections. the range of applied forces is 0.3 mn to 4.6 mn in x-direction, and 0.6 mn to 9 mn in y-direction, which results in structural deformation. fig. 4 depicts the shell deformations at different loads obtained in the abaqus. the deformations are given with the scaling factor equal to 1, i.e., the real deformation is depicted. the lower part of the figure shows rather large deformations that definitely require nonlinear computation. 152 ž. ćojbašić, v. nikolić, e. petrović, v. pavlović, m. tomić, i. pavlović, i. ćirić fig. 2 the shell displacements at different loads 4. a real time neural network-based finite element analysis a standard feed-forward, back propagation neural network is used in this study. to provide for an efficient solution, the artificial neural network (ann) [9] has three layers: an input layer, a single hidden layer and an output layer. as network input variables, forces in x and y directions are selected, while selected output network variables are displacements in x, y, and z directions, respectively. using custom script for matlab, the ann model is defined. the model’s input layer consists of two neurons, the hidden one of ten neurons, while the output one has 3 neurons (fig. 3). using a trial and error process, the number of neurons in the hidden layer is determined, while the levenberg–marquardt back propagation algorithm is used for network training [10]. the algorithm uses the approximation 1 1 [ ]t k k tx x j j i jμ − + = ± + e , (2) where j denotes jacobian matrix that contains first derivatives of the network errors with respect to weights and biases, and e is a vector of network errors. as a performance measure during training, the mean squared error is used [11]. the dataset consists of 224 samples. each set consists of 5 variables. these variables are applied forces in x and y directions and measured displacements (in abaqus) in x, y and z directions. to be able to import these variables into the ann, the dataset has to be split into two matrices. one matrix consisting of applied forces in x and y directions is called “input”. the second one consisting of displacements in x, y, and z directions is called “target”. based on “input” and “target” data, the ann searches for a linear regression function and calculates the “output” data. the network performance is measured via “correlation coefficient r (fig. 4)”. a value r = 1 implies that a linear equation describes a real time neural network based finite element analysis of shell structure 153 the relationship between “target” and “output” perfectly, with all data points lying on the line for which “output” increases as “target” increases. when matrices are imported, training, validation and testing sets are randomly generated from the dataset. the network is trained using the training set (comprising 156 of 224 samples), and is adjusted according to its error. to measure network generalization the validation set (34 samples) is used. also, this set is used to halt training when generalization stops improving. finally, the independent test set (34 samples) with no influence on the training is used to measure network performance. this can be observed in fig. 4, which shows the network performance for the training, validation and test set. also, in fig. 4 it can be observed that the total network performance is 0.99999, which is recognized as a fairly satisfying result. fig. 3 artificial neural network with 2 input variables, 1 hidden layer with 10 neurons, and 3 output variables fig. 4 results of network performance for training, validation and test set 154 ž. ćojbašić, v. nikolić, e. petrović, v. pavlović, m. tomić, i. pavlović, i. ćirić 5. conclusion the paper takes into consideration the artificial neural network implementation in the analysis of a cylindrical arch shell structure, that is initially modeled in the abaqus software. the obtained results from the abaqus are used for training the artificial neural network, which is supposed to enable real-time simulation of the shell structure. this property is based on massive parallelism in neural networks which provides for an extremely efficient computational speed by means of suitable hardware and software implementation [12]. in this paper it is proposed that the need for a fast system that simulates real-time behavior of shell structures can be satisfied by using a neural network which is trained by using the raw experimental data obtained by the finite element method, representing the displacements of the structure to the various applied loads. in the second application phase, the trained neural network is capable of fast real-time approximation of shell displacements for the range of loads it is trained for. the differences between the simulated values obtained from the fem analysis and the predicted values from the ann (value obtained by fem / value obtained by ann × 100%) are within the range of 0.0001%, which shows good agreement which is obtained with a relatively modest network structure and, thus, with a modest training effort and moderate dataset. further research aims at providing an analysis of piezoelectric active structures. in the future, active/adaptive structures are expected to become more of a standard solution in many areas of application. this requires reliable and efficient modeling tools [1], which makes the prospective of combining the fem modeling of piezoelectric active structures with artificial neural networks implementation potentially interesting. also, an interesting future research direction could lead to the creation of a real time neural network based finite element analysis of shell structure that provides for the results in a much larger number of points and, possibly, in every node of the mesh. in the latter case, network parallelism could provide for an extremely efficient computation regardless of the possibly large fem models. acknowledgements. the paper is part of the research performed within the bilateral germanserbian project “icoss intelligent control of smart structures”. references 1. marinković, d., marinković, z., 2012, on fem modeling of piezoelectric actuators and sensors for thinwalled structures, smart structures and systems, 9(5), pp. 411-426. 2. noor, a.k. belytschko, t., simo, j.c., 1989, analytical and computational models of shells, the american society of mechanical engineers, new york. 3. bucalem, m.l., bathe, k.j., 1997, finite element analysis of shell structures, archives of computational methods in engineering, 4(1), pp. 3-61. 4. javadi, a.a., tan, t.p., zhang, m., 2003, neural network for constitutive modeling in finite element analysis, computer assisted mechanics and engineering sciences, 10, pp. 523-529. 5. lefik, m., 2004, hybrid, finite element-artificial neural network model for composite materials, journal of theoretical and applied mechanics, 42(3), pp. 539-563. 6. sharma, k., bhasin, v., vaze, k.k., ghosh, a.k., 2011, numerical simulation with finite element and artificial neural network of ball indentation for mechanical property estimation, sadhana, 36(2), pp. 181–192. a real time neural network based finite element analysis of shell structure 155 7. marinkovic, d, zehn, m. marinkovic, z., 2013, the analysis of fem results convergence in modelling piezoelectric active shell structures, transactions of famena, 37(4), pp. 17-28. 8. marinković, d., 2007, a new finite composite shell element for piezoelectric active structures, ph.d. thesis, otto-von-guericke universität magdeburg, fortschritt-berichte vdi, reihe 20: rechnerunterstützte verfahren, nr. 406, vdi verlag, düsseldorf. 9. ćojbašić, ž., nikolić, v., ćirić, i., ćojbašić, lj., 2011, computationally intelligent modelling and control of fluidized bed combustion process, thermal science, 15(2), pp. 321-338. 10. moller, m.f., 1993, a scaled conjugate gradient algorithm for fast supervised learning, neural networks, 6(4), pp. 525–533. 11. ćojbašić, ž., brkić, d., 2013, very accurate explicit approximations for calculation of the colebrook friction factor, international journal of mechanical sciences, 67, pp. 10–13. 12. lukić, s., ćojbašić, ž., jović, n., popović, m., bjelaković, b., dimitrijević, l., bjelaković, lj., 2012, artificial neural networks based prediction of cerebral palsy in infants with central coordination disturbance, early human development, 88(7), pp. 547–553. analiza ljuske u realnom vremenu metodom konačnih elemenata zasnovanoj na neuronskoj mreži u poslednjih nekoliko godina, metoda konačnih elemenata je široko korišćena kao moćno sredstvo u analizi inženjerskih problema. prilikom simulacije deformabilnih objekata metodom konačnih elemenata, koristi se kompleksan sistem čvorova koji formiraju mrežu tačaka. fem model uključuje svojstva materijala i same strukture koja definišu kako će struktura reagovati na određena opterećenja. pouzdana simulacija treba da obezbedi lakši, brži i jeftiniji razvoj struktura. simulacija deformabilnih objekata tipa ljuske u realnom vremenu korišćenjem metode konačnih elemenata je veliki izazov zbog potrebe za brzim sistemima, koji ne zahtevaju velike računarske resurse. u ovom radu smo prikazali efikasan metod zasnovan na neuronskim mrežama za simulaciju u realnom vremenu ponašanja tankozidne ljuske modelirane metodom konačnih elemenata u komercijalnom fe softveru. korišćenjem metode konačnih elemenata, u režimu obučavanja neuro mreže sračunate su deformacije za primenjena opterećenja u određenom opsegu. u režimu aplikacije, korišćenjem obučene neuronske mreže dobijaju se potrebni rezultati za navedena opterećenja u realnom vremenu. ključne reči: metod konačnih elemenata, neuronska mreža, simulacija, tankozidna struktura, ljuska a real time neural network based finite element analysis of shell structure( žarko ćojbašić, vlastimir nikolić, emina petrović, vukašin pavlović, miša tomić, ivan pavlović, ivan ćirić 1. introduction 2. modeling of shell structure 3. modeling and analysis of cylindrical arch structure 4. a real time neural network-based finite element analysis 5. conclusion references 11007 facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 31 50 https://doi.org/10.22190/fume220806042s © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper free vibration investigation on rve of proposed honeycomb sandwich beam and material selection optimization babak safaei1,2, emmanuel chukwueloka onyibo1, mehmet goren1, kamila kotrasova3, zhicheng yang4, samaneh arman5, mohammed asmael1 1department of mechanical engineering, eastern mediterranean university, famagusta, turkey 2department of mechanical engineering science, university of johannesburg, gauteng, south africa 3department of structural mechanics, institute of structural engineering, faculty of civil engineering, technical university of kosice, slovakia 4college of urban and rural construction, zhongkai university of agriculture and engineering, guangzhou, china 5school of science and technology, university of georgia, tbilisi, georgia abstract. in this paper, free vibration, modal and stress state analyses of honeycomb sandwich structures with different boundary conditions was investigated and major factors affecting the sandwich frequencies and stiffness due to material or parameter changes were determined. the representative volume element (rve) method used in this work were analytically and numerically validated by comparing the obtained results to those available in literature. firstly, unit cell method was used to capture the entire effects of different parameters on the free vibration of honeycomb sandwich structure in ansys. this study analyzed the natural frequencies of honeycomb sandwich structures with different core materials combination. the effects of foil thickness, boundary conditions, materials selection, density and presence of crack on sandwich structure were taken into consideration and examined. the proposed core had an inbuilt shaped reinforcement with different materials for effective resonance, fatigue and deformation resistance at much higher frequency. key words: honeycomb sandwich beam; free vibration; representative volume element; mechanical properties received august 06, 2022 / accepted october 25, 2022 corresponding author: babak safaei department of mechanical engineering, eastern mediterranean university, famagusta, north cyprus via mersin 10, turkey department of mechanical engineering science, university of johannesburg, gauteng 2006, south africa e-mail: babak.safaei@emu.edu.tr 32 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... 1. introduction sandwich materials are becoming increasingly used in industry, particularly in the fields of transportation (automotive, aeronautics, shipbuilding, and railroads) and civil engineering [1]. in comparison to other panels, honeycomb panels provide high strengthto-weight ratios and design versatility [2]. sandwich structures are often made up of two thin but strong face sheets and a low-density core [3,4]. expansion method is generally used to create honeycomb for high density honeycomb materials [5], corrugated method is most commonly used as core [6]. in general, using theoretical computation and experimental methods, a number of studies have shown that constituent material, geometrical factors, and core cell topology have significant impacts on the mechanical performance of sandwich structures [7–9]. circular honeycomb structures are supplementing, complementing, and supplanting traditional honeycomb structures. the application of these impact attenuation systems has become very popular among researchers. honeycombs are often made up of hexagonal or square cells [10–12] with the prismatic direction perpendicular to the plane of the sandwich panel's face plates. styrene acrylonitrile (san) foam is a structural closed-cell thermoplastic core material with high toughness and impact resistance. due to their chemical stability and high heat resistance, san foam cores can be used in demanding applications such as the production of cored prepreg laminates that need to withstand the heat and pressure of autoclave processing. several core geometries and materials have been experimented by many researchers [13– 20]. furthermore, lightweight structures continue to advance as base materials. in addition, sandwich structures have fewer lateral deformations, higher buckling resistance, and higher natural frequencies than other structure types. due to the appealing combination of light weight and strong mechanical properties, sandwich structures are commonly used in many manufacturing applications. skins and their distance from the middle surface of composite affect tensile properties and especially flexural rigidity [21]. therefore, in sandwich structures, steel is commonly applied for skins and cores can be made of almost any material [22,23]. the core material of sandwich structures is mostly a homogeneous and isotropic foam, which contributes to the sandwich's intense lightness. the materials commonly used in the core are metallic foams, polymers [24,25] and metals [26]. the behavior of nap-core sandwiches was investigated by ha et al. [27] with special focus on the effect of symmetry. onyibo and safaei [28] applied finite element analysis to honeycomb sandwich structures. mechanical properties and energy absorption capability of aluminum honeycomb structures were found to change with impact velocity [29]. however, honeycomb core is known for its great stiffness. ha et al. [30] investigated the sensitivity of a nap-core sandwich, a unique type of structural composite. chemami et al. [31] added orthogrid to improve the stiffness of soft honeycombs and thereby reduce interfacial mismatch in sandwich structures. faria et al. [32] investigated the dynamic behaviors of sandwich beams with honeycomb cores loaded with magnetorheological (mr) gels and composite material skins under vibration. hence, static deflection, frequency, and transient responses of the multilayer sandwich shell (flat/curved) structures under various forms of mechanical loads were estimated by katariya and panda [33]. fazilati et al. [34] showed that honeycomb structures were commonly used as energy shock absorbers due to their strong crashworthiness characteristics of high energy absorption potential and high strength-to-weight ratio. rama et al. [35,36] presented a linear triangular shell element based on equivalent single-layer technique and an effective free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 33 modeling tool for structures made of fiber-reinforced composite laminates with piezoelectric layers. unwanted structural vibrations can readily impact flexible structures, causing major issues such as structural instability, fatigue, and failure. various research works have been conducted in order to avoid or control these vibratory disturbances in sandwich structures [37,38]. according to the law of vibration, everything (including components and structures) is in constant vibration at a given frequency. zhao et al. [39] investigated the effect of nanotube aspect ratio on the free vibration characteristics of functionally graded nanocomposite cylinders reinforced with wavy single-walled carbon nanotubes (cnts) using a mesh-free method. moreover, a lot of frequency-based vibration on sandwich structures were investigated by many researchers [40–43] and as well as nonlinear bending/instability based vibration [44– 49]. liu et al. [50] examined the nonlinear forced vibrations of functionally graded materials (fgms). li et al. [51] theoretically and experimentally studied the nonlinear vibrations of fiber-reinforced composite cylindrical shells (frccss) with bolted joint boundary conditions. liu et al. [52] developed a novel method for solving nonlinear forced vibrations of functionally graded (fg) piezoelectric shells in multi-physics fields. moradi-dastjerdi and behdinan [53] studied free vibration behaviors of a multifunctional smart sandwich plate (mssp) using an advanced and reliable method. hence, several current research are being carried out on nonlinear forced vibrations [54,55] and nonlinear dynamic responses [56–61] of fgm sandwich structures. in this work, the dynamic behaviors of mechanical structures, particularly composite structures, are well recognized to be crucial to the efficient functioning of a variety of applications, including automotive, civil, and aeronautic sectors. failure may be prevented by understanding practical structural behaviors and unwanted resonance can be minimized or decreased. in general, an overall understanding of modal analysis and natural frequencies for honeycomb sandwich beams with different material properties and boundary conditions are detailed while taking into account their density. moreover, the proposed core orthotropic stiffness matrix is investigated. in general, an elliptical cut on sandwich beam at various positions is analyzed and natural frequency is observed. 2. geometric modelling of honeycomb sandwich beam the software ansys 21.r2 spaceclaim was used to develop the finite element model of the honeycomb sandwich beam. the sandwich structure developed in this research had dimensions of 290 mm x 40 mm x 11 mm. the thicknesses of the honeycomb sandwich beam face-sheet and core were 1 and 9 mm, respectively. the geometries of hexagonal unit cell and proposed unit cell are depicted in figs. 1(a) and (b). moreover, the proposed structure was a honeycomb hexagonal structure with an inbuilt star shaped reinforcement. hence, in order to investigate the natural frequencies of free vibrations with different boundary conditions, inbuilt star shaped cores made of different materials were analyzed. in addition, the considered factors were the mass and stiffness of the overall sandwich beam while considering vibration characteristics, corresponding natural frequencies and mode shapes. in general, rve method in ansys material designer were used to determine effective elastic properties of hexagonal honeycomb and proposed core by capturing their unit cell. the material designer module is used to capture the domain of an entire honeycomb and proposed core structure, as it is expected that the unit cell was repeated 34 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... uniformly over the entire domain of the structure. hence, the result was obtained as an orthotropic stiffness matrix, which could then be used as a material property in larger structures. material properties required in this study are summarized in table 1. in general, carbon fiber, san foam, aluminum alloy and polyvinyl chloride (pvc) foam materials were applied in finite element analysis. san foam has mechanical properties comparable to those of cross-linked pvc but has the toughness and elongation of uncross-linked pvcs. fig. 1 geometry of (a) honeycomb unit cell and (b) honeycomb with star shaped reinforced unit cell table 1 material properties of honeycomb sandwich beams material young’s modulus e (mpa) poisson’s ratio ν shear modulus g (mpa) density ρ (kg/m3) carbon fiber 290000 0.2 9000 1800 san foam 85 0.3 32.69 103 aluminum alloy 71000 0.33 26691.72 2770 pvc foam 102 0.3 39.23 80 2.1. finite element meshing in order to covert the geometry bodies to finite element entities for hexagonal and proposed core unit cell. hence, ‘mesh’ in rve model group was selected with maximum mesh size of 0.25 mm (half the thickness of the unit cell), the periodic meshing and conformal meshing were used for both cores’ unit cell, their number of elements and nodes are depicted in table 2. fig. 2 shows the equivalent models of regular hexagonal honeycomb and proposed reinforced star-shaped core. hexagonal honeycombs are consisted of a ‘unit cell’ repeated many times along one or more spatial directions. this unit cell is usually a fraction of the size of the overall structure under investigation. hence, in ansys workbench 2021, a new feature called “material designer” was used for the homogenization of hexagonal and proposed core of the sandwich beam. moreover, this technique equally reduced the computational time of the analysis. in general, the final result is an orthotropic stiffness matrix, which can subsequently be used as a material property in larger structures and honeycomb structure is replaced with ‘homogenized’ (effective) material properties shown in table 3. material designer gives nine independent material constants including three elastic moduli, three shear constants, and three poisson ratios. a mesh sensitivity analysis was carried out in order to determine the mesh resolution required for accurate results. furthermore, both models were run with various mesh sizes, the resulting stress or displacement values were free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 35 observed. thus, due to the addition of star geometry reinforcement, the proposed core unit cell has more nodes and elements than a normal hexagonal honeycomb. therefore, there was 33.1% increase in the number of elements and 32.2% increase in the number of nodes in the proposed core model when compared with the traditional hexagonal honeycomb core shape. in addition to that, the computation time is almost the same with 3-5 second’s difference in analysis time. table 2 analysis mesh of both unit cell core unit cell number of elements number of nodes hexagonal honeycomb core 2210 4956 proposed core 2942 6553 fig. 2 meshing of (a) hexagonal meshing; (b) proposed core meshing; (c) homogenized model of hexagonal core meshing; and (d) homogenized model of proposed core meshing the rve modeling of hexagonal honeycomb and proposed core were homogenized, geometries were simplified, and material properties of the constituent materials were defined. in general, geometry was meshed for finite element analysis. 36 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... table 3 orthotropic material properties of rve proposed core material hexagonal material star core density (kg/m3) ex = ey (mpa) ez (mpa) ʋxy ʋyz = ʋxz (10-3) gx (mpa) gy = gz (mpa) san foam san foam 11.29 0.1161 9.3191 0.957 3.7 0.01121 1.3762 san foam pvc 10.07 0.1315 10.215 0.951 3.8 0.01142 1.4188 san foam aluminum alloy 151.94 2.4994 3749.12 0.118 0.2 0.01580 1.8990 san foam carbon fiber 100.78 2.5332 1218.78 0.027 0.3 0.01578 1.8960 single hexagonal san foam material 5.86 0.0278 4.84 0.988 1.7 0.00699 0.9548 2.2. modelling boundary conditions the boundary condition of the sandwich beam was chosen as clamped-free and clamped-clamped, as shown in figs. 3(a) and 3(b), respectively, for free vibration analyses. hence, for elliptical cut investigation, clamped-free boundary condition was considered, while elliptical cut position changed along the length of the beam, as shown in fig. 3(c). moreover, elliptical cut depth was throughout the sandwich beam with 30 mm in diameter. table 4 depicts the dimensions of honeycomb sandwich beam. table 4 dimensions of the honeycomb sandwich beam cell size (mm) core thickness (mm) cell thickness (mm) skin (mm) length (mm) width (mm) 10 9 0.5 1 290 40 a) b) c) fig. 3 boundary conditions (a) clamped-free; (b) clamped-clamped; (c) clamped-free with elliptical cut free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 37 3. fem analysis 3.1. analytical models beam theory [62,63] is commonly used to calculate in plane elastic characteristics of honeycombs, for which the radius of curvature in nodes is neglected (fig. 4). the effective young's moduli parallel to ox and oy, effective poisson's ratios and effective shear modulus for commercial honeycomb, i.e. t1 = t and t2 = 2t could be evaluated taking into account bending stresses in cell walls, applicable for when t << ℓ [62,63]: 𝐸1 = 𝐸𝑠 ( 𝑡 ℓ ) 3 𝑐𝑜𝑠 𝜃 ( ℎ ℓ +𝑠𝑖𝑛⁡ 𝜃)𝑠𝑖𝑛2⁡ 𝜃 (1) 𝐸2 = 𝐸𝑠 ( 𝑡 ℓ ) 3 ℎ ℓ +𝑠𝑖𝑛⁡ 𝜃 𝑐𝑜𝑠3 𝜃 (2) 𝑣12 = 𝑐𝑜𝑠2 𝜃 ( ℎ ℓ +𝑠𝑖𝑛⁡ 𝜃)𝑠𝑖𝑛⁡ 𝜃 ; 𝑣21 = 1 𝑣12 (3) 𝐺12 = 𝐸𝑠 ( 𝑡 ℓ ) 3 ℎ ℓ +𝑠𝑖𝑛⁡ 𝜃 ( ℎ ℓ ) 2 (1+ ℎ 4ℓ )𝑐𝑜𝑠⁡ 𝜃 (4) where e1, and e2 are effective young's moduli of honeycomb core in the material system of coordinates, es is young's modulus of solid cell wall material, ν12 and ν21 are effective poisson's ratio of honeycomb core in the material and g12 is effective shear modulus. fig. 4 basic cell used in analytical calculus 3.2. equivalent honeycomb sandwich beam the created equivalent model was used in this research. hence, the time spent for the analysis was reduced and computational time was decreases significantly. homogenization concept is regularly used in the research on the mechanical behaviors of composite materials. this method allows for the avoidance of difficulties associated with heterogeneities. therefore, the conceptualization of this method is basically by considering the genuine constitution of material/geometry arrangement and it is considered to be continuous throughout the entire structure. however, “material designer” analysis system in ansys workbench was used for this technique. 38 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... 4. results and discussions 4.1. free vibration of honeycomb beam in this section, a modal analysis of honeycomb equivalent sandwich beam was presented. the numerically and analytically results obtained in this study were compared with experimental data from the available literature. the honeycomb structures will be parameterized in order to get insight on factors that can affect it natural frequencies by considering the effect of boundary conditions, foil thickness and materials. in general, the proposed reinforced star core was introduced and examined. consequently, density was considered. furthermore, the best material combination for high flexural stiffness, higher frequency and lighter weight was investigated for the proposed core structure. considering a classical honeycomb with an isotropic material having es = 1000 mpa, ρ = 1000 kg/m3 νs = 0.3, t = 1 mm and l = 10 mm, the analytically obtained results (eqs. 1-4) along with those of fem are presented in table 5. however, having validated the regular honeycomb rve with the available literature using ansys material designer analysis system. moreover, the same analysis system was used for the proposed core model unit cell and its orthotropic material properties were equally obtained. table 5 results for the validation of rve models regular honeycomb: l= h =10 mm and θ = 30º model ρ (kg/m3) e1 (mpa) e2 (mpa) g12 (mpa) v12 v21 analytic 115.5 2.18 2.18 0.56 0.96 0.96 present 112.14 2.572 2.572 0.657 0.957 0.957 solid 3d [64] 112.10 2.570 2.570 0.660 0.960 0.960 4.2. effects of materials on free vibration under modal analysis, the natural frequencies of sandwich beams with deferent materials were investigated and their corresponding deformations were noted. howover, considering the effect of free vibration of honeycomb core, fig. 5 shows a hexagonal honeycomb core (hc) with the same dimensions having cantilever beam boundary conditions, simulated with three different materials of aluminum alloy, carbon fiber and san foam in order to investigate their natural frequencies and mode shapes. it was observed that aluminum had much higher natural frequency, followed by carbon fiber and san foam had the lowest natural frequency. as a result of that, aluminum alloy is stiffer than the rest, especially those subjected to double tension and compression cyclic loading. moreover, while considering sandwich structure deformation in line with their corresponding frequencies, fig. 6 depicts the deformation of three sandwich core materials on free vibration analysis. as expected, san foam with the lowest natural frequency tends to vibrate more at any slightest excitation, as result of that it had the highest amount of deformation when compared with the rest materials. it was observed that carbon fiber had the lowest deformation, indicating that carbon fiber has higher stiffness to mass ratio than aluminum alloy. free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 39 fig. 5 effects of materials on natural frequency fig. 6 effects of materials on deformation 4.3. effect of boundary conditions (cf and cc) on the free vibration of sandwich structure considering the effect of boundary conditions on sandwich beam structure. in this section, clamped-free and clamped-clamped boundary condition was applied on sandwich beam while their vibrational response is observed. fig. 7 shows mode 5 directional deformation of the sandwich structure under clamped-free condition with 3 different materials used as core. however, carbon fiber and aluminum core had similar deformation patterns while san foam deviated much with multiple tension and compressive stresses due to lower flexural stiffness. moreover, in order to get insight on sandwich structure and monitor incremental changes, clamp-clamp boundary condition was investigated, as shown in fig. 8. it was observed that 40 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... the peak amplitude of san foam sandwich core was higher, which resulted in larger deformation and underwent 4 tensile and compressive stresses. in general, fixed-fixed boundary conditions increased sandwich beam natural frequency as well as corresponding directional deformation. fig. 7 directional deformation on mode 5 (clamped-free) fig. 8 directional deformation on mode 5 (clamped-clamped) 4.4. effect of hexagonal cell foil thickness on free vibration in this case, the effect of the parameters of hexagonal cell on free vibration were investigated. hence, sandwich structure dimensions including sandwich length and thickness were kept the same and cell dimension were changed. fig. 9 shows the free vibration of foil thickness (ft). it was observed that as the foil thickness was increased, natural frequency was decreased under clamped-free boundary condition. in addition, variations were close and almost negligible, indicating that changing foil thickness did not significantly affect the vibration characteristics of honeycomb sandwich structures and only the weight of the structure was increased, as shown in table 6, such that 10% increase of foil thickness free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 41 increases density by about 9%. furthermore, when foil thickness was increased, young’s modulus was increased such that increase of foil thickness by 10% increased elasticity modulus by 32.7%. further increase of thickness also increased elasticity modulus but in lower rates. therefore, when foil thickness was increased by 7% (from 1.4 to 1.5mm) young’s modulus was increased by 21.7%. table 6 variation of density and foil thickness foil thickness thickness density (kg/m3) ex (mpa) ey (mpa) ʋxy gx (mpa) 1 10 385.39 403.95 403.95 0.92 104.18 1.1 10 422.34 536.20 536.20 0.91 138.88 1.2 10 459.00 689.35 689.35 0.89 180.20 1.3 10 495.38 859.31 859.31 0.88 227.69 1.4 10 531.46 1070.68 1070.68 0.86 284.47 1.5 10 567.26 1303.13 1303.13 0.84 346.21 fig. 9 variation of frequency with respect to foil thickness 4.5. prestressed modal analysis of sandwich structure investigating prestressed modal analysis and how the mode frequencies of the sandwich beams changes with thermal load at different values. hence, in order to generate thermal stresses, the sandwich beam was constrained at both ends and was kept at room temperature. moreover, the total strain developed was equal to the sum of elastic and thermal strains. in this case, since the sandwich was fully constrained along axial direction, total strain was zero in all cases and therefore, the elastic strain developed in the sandwich was equal in magnitude but opposite in sign compared to thermal strain. therefore, when thermal load was higher than reference temperature, thermal strain was positive and this made the elastic strain negative which indicated compressive stresses on sandwich beam. similarly, when thermal load on sandwich beam was less than reference temperature, thermal strain was 42 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... negative. therefore, elastic strain was positive and tensile stresses were developed in the sandwich beam. fig. 10 depicts the changes of mode frequencies with thermal loads. fig. 10 frequency changes with thermal load 4.6. proposed reinforced star core model hexagonal honeycomb was further designed to accommodate star shaped core in the center. however, central core provided excellent torsional resistance while hexagonalshaped one provided extremely lateral bending resistance. hence, a simulation was carried out in ansys workbench, whereby different materials were assigned to the proposed model. in this case, san foam was remained as regular hexagonal material while the star shaped reinforcing materials varied to san foam, pvc foam, aluminum alloy and carbon fiber, as shown in fig. 11. table 7 summarizes the natural frequency of the proposed core under clamped-clamped boundary condition with different material combinations. fig. 12 shows the natural frequency of the proposed core model under clamped-clamped boundary conditions. it was observed that the cores with hexagonal san foam and pvc foam star shaped reinforcements had higher stiffness to mass ratios and natural frequencies than remaining structures. moreover, repeating the same simulation of the proposed core natural frequency under clamped-free condition, pvc and san foams still presented the best material combination for higher frequencies, as shown in fig. 13. free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 43 fig. 11 proposed core material selections fig. 12 natural frequency of the proposed core (clamped-clamped) 44 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... fig. 13 natural frequency of the proposed core (clamped-free) table 7 natural frequency of the free vibration response of the proposed reinforced star core model under clamped-clamped boundary condition mode san foam (sc) san foam (hc) (hz) pvc (sc) san foam (hc) (hz) al(sc) san foam (hc) (hz) cf(sc) san foam (hc) (hz) san foam hc shape only (hz) 1 13.87 15.48 14.08 17.38 11.16 2 29.67 31.95 16.14 19.88 31.1 3 35.82 39.24 16.19 19.9 31.67 4 38.07 42.45 32.43 39.83 33.31 5 63.18 68.37 38.6 47.66 61.92 6 74.43 82.89 41.07 50.67 68.82 density kg/m3 512.88 511.88 627.95 586.1 508.39 4.6.1. comparison of flexural stiffness of the proposed core in order to investigate the deformation resistance of the proposed core, the reinforcement was assigned different materials. the applied force was 100 n which was kept constant for all material combinations. hexagonal shape material was kept constant and only star shaped reinforcement material was changed. the obtained results are summarized in table 8. in line with the obtained results, it was observed that the stresses were almost constant since they were irrespective of rigidity modulus. also, all proposed cores had the same shape, crosssection area and applied load. however, there were significant changes in their deflection, due to material combination stiffness, in which san foam and carbon fiber showed the highest deformation resistance. free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 45 table 8 comparison of material stiffness in the proposed cores san foam (sc) san foam (hc) pvc (sc) san foam (hc) al(sc) san foam (hc) cf(sc) san foam (hc) equivalent stress maximum [mpa] 6.76 6.76 6.72 6.71 total deformation maximum [mm] 113683.82 100417.08 5250.74 5173.66 4.7. elliptical cut on sandwich beam effect of elliptical cut on sandwich beam natural frequency was studied. the sandwich beam has an elliptical cut of 30 mm throughout its layers. hence, fixed-free boundary condition was used for the modal analysis of sandwich beam with elliptical cut. the analysis on sandwich beam was carried out in ansys software. however, it was observed that cut on sandwich might increase or reduce its free vibration characteristics, depending on the size, position, shape and depth of the cut. hence, in this case, elliptical shape cut throughout the thickness of the sandwich was considered. therefore, it was observed that as the elliptical cut moved closer to the fixed support, natural frequency was reduced. in addition, the sandwich with elliptical cut was compared with normal sandwich beam with no cut. however, when the cut was 255 mm away from the fixed support, natural frequency became higher than that of normal sandwich beam without cut. in general, aluminum alloy material was used as core and skin. also, regular hexagonal honeycomb shape was used as core. fig. 14 shows the position of elliptical cut at 6 different distances along the length of the sandwich beam towards the fixed support, illustrating that when the elliptical cut came closer to fixed support, sandwich natural frequency was reduced. the first 3 mode shapes of normal and damaged sandwich beams are depicted in figs. 15 and 16, respectively. fig. 14 positions of elliptical cut vs natural frequency 46 b. safaei, e. chukwueloka onyibo, m. goren, k. kotrasova... fig. 15 first 3 mode shape contours of sandwich beam (a) mode 1 total deformation 203.51 mm, 117.15 hz (bending); (b) mode 2 total deformation 228.62 mm, 353.35 hz (bending); (c) mode 3 total deformation 300.96 mm, 397.83 hz (torsion) fig. 16 first 3 mode shape contours of sandwich beam with elliptical cut (a) mode 1 total deformation 218 mm, 125.19 hz (bending); (b) mode 2 total deformation 248.09 mm, 380.98 hz (bending); (c) mode 3 total deformation 321.27 mm, 400.96 hz (torsion) (a) (c) (b) (a) (b) (c) free vibration investigation on rve of proposed honeycomb sandwich beam and materials ... 47 5. conclusions the results of a detailed honeycomb sandwich beam modal analysis and simulation were shown and plotted, and significant observations were recorded. the free vibration of a honeycomb sandwich beam was examined and validated using available literature and the model was developed using ansys software. however, the validity of the model developed in this study for the computation of in-plane mechanical properties was confirmed by the results of hexagonal rve numerical investigations using ansys material designer. vibrational behavior of honeycomb sandwich beam was affected by many factors as witnessed in parametric analysis results. hence, natural frequency was changed by geometry, material, presence of crack or cut, induced thermal load and boundary conditions. natural frequency was also a function of stiffness to mass ratio. the proposed model was designed to have reinforced inbuilt star shape core which enhanced vibration characteristics, it was further optimized by changing star shaped reinforcement materials. therefore, the material san foam hexagonal and pvc foam star core (sc) reinforcement had higher stiffness to mass ratios and natural frequencies. in addition, increasing foil thickness did not have significant effect on sandwich beam natural frequency but increased the weight and young’s modulus of the structure. in general, presence of a cut in a sandwich beam affects its natural frequency such that the frequency was reduced in under fixed-fixed boundary condition, while under fixed-free boundary condition, it depends on the size, depth and position of the cut. in this experiment, 30 mm elliptical cut throughout the sandwich beam 255 mm away from the fixed support increased natural frequency while for elliptical cut of 55 mm away from the fixed-free boundary support, natural frequency was reduced as compared with 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regulations and increasing consciousness of the customers, the present day manufacturing organizations are continuously striving to engage green suppliers in their supply chain management systems. selection of the most efficient green supplier is now not only dependant on the conventional evaluation criteria but it also includes various other sustainable parameters. this selection process has already been identified as a typical multi-criteria group decision-making task involving subjective judgments of different participating experts. in this paper, a green supplier selection problem for an automobile industry is solved while integrating the cloud model with the technique for order of preference by similarity to an ideal solution (topsis). the adopted method is capable of dealing with both fuzziness and randomness present in the human cognition process while appraising performance of the alternative green suppliers with respect to various evaluation criteria. this model identifies green supplier s4 as the best choice. the derived ranking results using the adopted model closely match with those obtained from other variants of the topsis method. the cloud model can efficiently take into account both fuzziness and randomness in a qualitative attribute, and effectively reconstruct the qualitative attribute into the corresponding quantitative score for effective evaluation and appraisal of the considered green suppliers. comparison of the derived ranking results with other mcdm techniques proves applicability, potentiality and solution accuracy of the cloud topsis model for the green supplier selection. key words: cloud model, topsis, selection, green supplier, rank received march 07, 2020 / accepted may 23, 2020 corresponding author: s. chakraborty department of production engineering, jadavpur university, kolkata e-mail: s_chakraborty00@yahoo.co.in 376 k.r. ramakrishnan, s. chakraborty 1. introduction in today’s enormous competitive environment, the aim of any manufacturing organization must be focused on satisfying its customers with low cost high quality products and prompt services, while keeping in mind their changing demands and perspectives. thus, the production system needs to be so designed as to decrease the related manufacturing cost, increase its flexibility and meet the quality standards. as in any production system, raw materials are usually converted into finished products, any variation in the quality of the input materials may result in deterioration of the final product quality. thus, in the supply chain management, selection of the appropriate suppliers and evaluation of their performance are identified as two of the crucial strategic issues for overall survival of the concerned manufacturing organization. to fulfill the long term objective of the organization and enhance the supply chain efficacy, the selection of the most reliable suppliers for varying input materials has been recognized as of immense importance. in this direction, activities of the purchasing department must be supported and delineated through the deployment of strong mathematical tools and techniques. but, nowadays, manufacturing organizations need to pay more attention to various environmental issues as imposed by the concerned governments. these environmental issues primarily arise from constant decrement in the level of raw materials, increasing pollution and emission of greenhouse gases. those organizations must streamline their manufacturing processes so as to minimally affect the environment. this can only be achieved through the augmentation of a green production system through the involvement of green suppliers in the entire supply chain. it has also been observed that the customers have now become more conscious in procuring more environmentally sensitive products, apart from their primary requirements of low cost high quality products. thus the inclusion of green suppliers in the organizational supply chain has been observed as extremely important with respect to environment friendliness, green service and purchasing, energy conservation, green management, design for environment, carbon footprint and emissions, reverse logistics, water usage and recycling initiatives [1]. besides consciousness about various environmental parameters, the concept of green suppliers should also include green information transfer as well as management and organization practices. governmental regulation, social responsibility, customer pressure and commercial benefits are also responsible for effective green supplier selection. as the selection of the most appropriate green supplier pays more attention to various green factors, it would help the concerned manufacturing organization to supersede its other competitors. it not only helps in influencing the profitability and competitiveness of the organization, but also effectively enhances the performance of the entire supply chain. a wrong green supplier selection decision may adversely affect the health of an organization as well as its goodwill. green supplier selection has now become essential in today’s manufacturing environment considering increasing pollution levels worldwide, which can be attributed to increased consumption as well as innovation and improvement in production techniques and technology. with growing awareness and focus on climate change action, both the manufacturers and suppliers are concerned about the environmental impact of the products produced and consumed. increasing consumer attention on using environment friendly products and manufacturers’ focus on carbon footprint along with the concept of sustainable development have led the manufacturers to rethink and reorient their production strategies, right from the raw material procurement. hence, selection of suppliers who share the common idea of ecoa cloud topsis model for green supplier selection 377 friendliness and being ‘green’ in thinking is very important in today’s competitive manufacturing environment. it not only helps in the fight against pollution, but also helps the organizations in green marketing while improving public perception and trust on their products. for any manufacturing organization, selection of the most apposite green supplier is a complex decision-making task due to the involvement of various experts (decision-makers) from different related departments, like procurement, planning, production, quality control, etc. it has been identified as a multi-criteria decision-making (mcdm) problem where the best green supplier needs to be identified in the presence of a set of conflicting criteria [2]. in the green supplier selection process, a group of experts having dissimilar backgrounds, experiences, expertise and stature usually participate. they usually express their subjective judgments on the relative performance of the candidate green suppliers with respect to several evaluation criteria. the members of the group of experts also have different priority levels and preferences, raising the scope of inclusion of uncertainty, vagueness and hesitancy in the final decision. but, all the experts must aim at identifying a particular green supplier which would be quite similar to the ideal solution. a consensus decision thus must be arrived at after aggregating the individual decisions of all the participating experts. in the process of the green supplier selection and performance appraisal, the individual experts have the difficulty in expressing their judgments with specific numerical values as most of the evaluation criteria are qualitative in nature and the human cognition process is sometimes vague (uncertain). the experts always like to communicate their opinions through linguistic expressions, i.e. imprecise and unquantifiable information. thus, there is an ardent need to deploy an effective mathematical tool to support and transform those linguistic opinions into appropriate quantitative values. the conventional linguistic computational models which have been developed based on different membership functions, ordinal scales and 2-tuple linguistic information can only describe the fuzziness in a group green supplier selection decision-making problem but they are unable to consider the inherent randomness present in that problem. thus, in this paper, a group mcdm method while integrating the cloud model with technique for order of preference by similarity to an ideal solution (topsis) is employed for identifying the most suitable green supplier for an automobile manufacturing unit. this green supplier selection problem consists of five candidate alternatives and 12 evaluation criteria (59 sub-criteria). the cloud model first converts the fuzziness and randomness of linguistic terms present in the group decision-making process into numerical values. the alternative green suppliers are subsequently evaluated and ranked using the topsis method. a comparison of the proposed approach with other fuzzyand intervalbased models ensures its effectiveness in accounting for the inherent randomness and fuzziness present in the green supplier selection process. considering the inherent qualitative evaluation process, it can thus be augmented as an efficient tool in determining the success of supply chain of a manufacturing organization. 2. literature review the present literature is flooded with the applications of various mathematical tools, especially mcdm methods for solving diverse green supplier selection problems for different manufacturing organizations. kuo et al. [3] integrated artificial neural network with data envelopment analysis (dea) and analytic network process (anp) to solve a 378 k.r. ramakrishnan, s. chakraborty green supplier selection problem. using fuzzy decision-making trial and evaluation laboratory (dematel) method, ashlaghi [4] identified the interrelations between different criteria in a green supplier selection problem. the corresponding criteria weights were estimated based on fuzzy anp while a linear physical programming model was later employed to choose the best supplier. dobos and vörösmarty [5] adopted the method composite indicators along with dea approach to identify a suitable weight system for addressing the green factors in a supplier selection problem. yazdani [6] first applied analytic hierarchy process (ahp) to estimate the weights of different green supplier selection criteria while the fuzzy topsis method was subsequently employed to rank the considered suppliers. cao et al. [7] presented an intuitionistic fuzzy mcdm approach for solving green supplier selection problems based on intuitionistic fuzzy criteria values and unknown criteria weights. the topsis method was finally integrated with the proposed model to rank the considered suppliers. for a green supplier selection problem, hashemi et al. [8] first adopted anp to study the interdependencies among different criteria and later applied grey relational analysis (gra) to rank the considered suppliers. chen et al. [9] determined the weights of different criteria using fuzzy ahp and subsequently ranked the candidate green suppliers based on the fuzzy topsis method. doğan et al. [10] applied an mcdm approach in the form of the fuzzy topsis for selecting green suppliers in a manufacturing unit in turkey. ghorabaee et al. [11] proposed the application of weighted aggregated sum product assessment (waspas) method to solve green supplier selection problems with interval type-2 fuzzy sets. a sensitivity analysis was also performed to investigate the effects of criteria weights and model parameters on the ranking results to establish robustness of the novel approach. based on linguistic data, watróbski and sałabun [12] evaluated the performance of 25 green suppliers in a cable bundle manufacturing unit using the fuzzy topsis method. yu and hou [13] applied a modified multiplicative ahp (mmahp) method to deal with a green supplier selection problem in an automobile manufacturing unit. the efficacy of the proposed approach was also validated based on real time data. sahu et al. [14] solved a green supplier selection problem using a fuzzy-based multi-level mcdm approach and compared its performance with respect to the fuzzy topsis method. yazdani et al. [15] incorporated the applications of quality function deployment and house of quality matrix in a green supplier selection and evaluation problem, and finally ranked the candidate suppliers using the waspas method. gavareshki et al. [16] presented an integrated approach for green supplier selection in a brake pad manufacturing unit. at first, interpretive structural modeling and fuzzy micmac (cross-impact matrix multiplication applied to classification) analysis were adopted to highlight the interaction of different categories along with their driving and dependence power. the ahp method was utilized to estimate the criteria weights and vikor (vise kriterijumsko kompromisno rangiranje) method was finally used to rank the candidate suppliers. hashemzahi et al. [17] adopted the topsis method under fuzzy environment to deal with a green supplier selection problem while considering several environmental issues. qin et al. [18] applied todim (tomada de decisao interativa multicriterio), an interactive mcdm tool, to solve a green supplier selection problem based on interval type-2 fuzzy sets. shafique [19] proposed the combined application of dematel, ahp and topsis methods for performance appraisal of green suppliers under fuzzy environment. based on the cloud model and qualitative flexible multiple criteria (qualiflex) method, wang et https://www.worldscientific.com/doi/abs/10.1142/9789813146976_0101 javascript:%20goarcpage('',%20'2133072',%20''); a cloud topsis model for green supplier selection 379 al. [20] evaluated the relative performance of green suppliers in an auto manufacturing unit. badi et al. [21] presented the novel application of combinative distance-based assessment (codas) method to select the most apposite supplier from a pool of six alternatives for a steelmaking company in libya. the proposed approach was based on estimating the euclidean distance and the taxicab distance for evaluating the suitability of a particular supplier. banaeian et al. [22] presented the application of three fuzzy mcdm methods, i.e. topsis, vikor and gra to deal with the selection of green suppliers in an agri-food industry. it was concluded that although all the three methods could provide the same supplier rankings, the fuzzy gra would be the preferred method due to its less computational complexity. under a hesitant fuzzy linguistic environment, zhu and li [23] applied hesitant 2tuple linguistic operator and choquet integral operator to solve a green supplier selection problem. abdullah et al. [24] studied the effects of different preference functions of preference ranking organization method for enrichment evaluation (promethee) in a green supplier selection problem. it was inferred that the best identified green supplier would remain unchanged for all the considered preference functions. alguliyev et al. [25] proposed an mcdm technique for selection of candidates in e-voting based on a set of evaluation criteria. the considered candidates were finally rated using a positional ranking approach. biswas et al. [26] adopted an ensemble approach based on a two-stage framework for effectively resolving portfolio selection problems. for fulfilling the objective, dea, multi-attributive border approximation area comparison (mabac) and entropy methods were integrated. chatterjee and stević [27] integrated fuzzy ahp and fuzzy topsis methods to single out the most appropriate supplier based on a set of quantitative and qualitative criteria to streamline the purchasing process of a manufacturing organization. in order to solve a sustainable supplier selection problem, durmić [28] identified a set of the most significant criteria using full consistency method (fucom) based on the opinions of a group of experts. liu et al. [29] identified green supplier selection as a typical multi-criteria group decision-making problem and presented the application of generalized ordered weighted hesitant fuzzy prioritized average operator to solve the same. lu et al. [30] proposed a novel approach integrating the cloud model and possibility degree for selection of the optimal green supplier in a chinese straw biomass industry. rashidi and cullinane [31] compared the solutions derived from fuzzy topsis and fuzzy dea methods while identifying the best sustainable supplier for logistics service providers in sweden. it was observed that the fuzzy topsis could outperform the other technique with respect to computational complexity and robustness to variations in the number of suppliers. while applying the extended topsis method under interval-valued pythagorean fuzzy environment, yu et al. [32] solved a sustainable supplier selection problem to aid the managers in taking the optimal decision. yucesan et al. [33] combined the best-worst method and interval type-2 fuzzy topsis for identifying the best green supplier in a plastic injection molding unit in turkey. žižović and pamučar [34] developed a new level based weight assessment (lbwa) model for measuring the criteria weights for an mcdm problem. it was proved to be an efficient approach for defining the relations between the considered criteria and providing rational decisionmaking. đalić et al. [35] employed fuzzy pivot pair-wise relative criteria importance assessment (piprecia) and interval rough simple additive weighting (saw) methods to solve a green supplier selection problem. an extensive review of the past research studies clearly reveals that various fuzzy models, mainly intuitionistic model, interval type-2 model, hesitant model, etc. have been 380 k.r. ramakrishnan, s. chakraborty employed to transform the vague qualitative information into numerical values; thereafter, the candidate green suppliers have been subsequently ranked using other mcdm tools, like ahp, anp, topsis, gra, waspas, vikor, saw, piprecia, etc. in this paper, the application of the cloud model is proposed to convert the linguistic information of the participating experts into quantitative data, taking into account both fuzziness and randomness present in the subjective judgments of the experts. the alternative green suppliers for the considered automobile manufacturing unit are later ranked using the topsis method. 3. cloud topsis model 3.1. cloud model let the set u = {x} be the universe of discourse and c be a qualitative attribute corresponding to u. assume µ(x) as a random variable with a probability distribution, having values in [0,1], to represent the membership degree of x in u to considered qualitative attribute c. thus, a membership cloud can be represented as a mapping from the universe of discourse u to the unit interval [0, 1], i.e. µ(x): u  [0, 1] xu, xµ(x) a cloud can be defined as the distribution of x in universe u and cloud drop is the value of every x having membership degree µ(x). the uniqueness of cloud model is that it can efficiently take into account both fuzziness and randomness in a qualitative attribute, and effectively reconstruct the qualitative attribute into the corresponding quantitative numbers using three numerical characteristics, i.e. ex, en and he. ex represents the expected value of the cloud drop in the universe (the most representative qualitative attribute value). on the other hand, en signifies the degree of uncertainty of the considered qualitative attribute (distribution of the attribute), and it combines both fuzziness and randomness of the qualitative attribute. the term he denotes the uncertainty degree of en, which can be measured by the fuzziness and randomness of the entropy. amongst various cloud models, the normal cloud model has become most popular because of its capability to deal with large number of uncertain phenomena in varied decision-making tasks. as there is ‘±3σ’ concept in statistics, the ‘3en’ rule in the cloud model signifies that a cloud drop within the interval [ex − 3en, ex + 3en] can contribute to 99.73% cloud drop in the universe [36]. when there are n clouds xi (exi,eni,hei) (i = 1,2,…,n) in the same universe of discourse and w = (w1,w2,…,wn) ( 1 1   n i iw ) is the weight vector, the weighted average cloud xw can be obtained using the following expression:              n i n i ii n i iiiii n i iw hewenwexwxwx 1 1 2 1 2 1 )(,)(, (1) weighted average cloud xw can be employed to show the complete information of n clouds xi and help in aggregating varying opinions of the decision-makers as involved in a group decision-making task. a cloud topsis model for green supplier selection 381 when there are two clouds xi (exi,eni,hei) and xj (exj,enj,hej) (i ≠ j) in the same universe of discourse, the degree of inconsistency between them can be expressed as the difference of cloud d(xi, xj), which can be denoted as follows: ),(),(),(),( 321 jijijiji hehedenendexexdxxd   (2) )3,3min()3,3max( ),( jjiijjii ji ji enexenexenexenex exex exexd    (3) ),max( ),min( 1),( ji ji ji enen enen enend  (4) ),max( ),min( 1),( ji ji ji hehe hehe hehed  (5) λ1 + λ2 +λ3 = 1 (1 ≥ λ1 ≥ λ2 ≥λ3 ≥ 0) where λ1, λ2 and λ3 are the coefficients of difference, exhibiting the relative degree of importance with respect to the inconsistency of the two clouds in ex, en and he. among the three numerical characteristics of the cloud model, ex has the maximum significance, followed by en and he. coefficients λk (k = 1,2,3) can take different values based on the preferences of the concerned decision-makers. the difference of cloud can be employed to quantitatively determine the difference level between different linguistic variables with respect to the same criterion. in many real time decision-making scenarios, fuzziness and randomness in the accumulated information often appear in the human cognition process. the uncertain information is considered to be quite difficult to convert into quantitative measures. the concept of linguistic variables and interval representation are useful tools for expressing the degree of uncertainty in a decision-making process. there exists an effective way to transform both linguistic variables and interval representation into cloud model [37]. linguistic variables are often considered to express the subjective judgments (opinions) of different decision-makers (experts) and can take various levels, like poor, medium and good. the number of levels usually measures the degree of precision of the linguistic concept. but, in order to consider both precision and accessibility of the linguistic variables, the linguistic concept usually takes five levels, i.e. very poor, poor, medium, good and very good. let the linguistic set considered by the experts be represented as l = {l2= very poor, l1= poor, l0 = medium, l1+ = good, l2+ = very good}. now, each of the elements of the above set can be denoted by a cloud model having an interval [xl, xu], where xu is the upper bound and xl is the lower bound of the interval, respectively. based on the golden section ratio, the numerical characteristics of the five clouds can be derived as follows [36]: ex0 = (x1 + xu)/2 (6) ex2= xl, ex2+ = xu (7) ex1= ex0 – 0.382ex0 = 0.618ex0 (8) 382 k.r. ramakrishnan, s. chakraborty ex1+ = ex0 + 0.382ex0 = 1.382ex0 (9) en1= en1+ = 0.382(xu – xl)/6 (10) en0 = 0.618en1+ (11) en2= en2+ = en1+/0.618 (12) similarly, the value of he0 can be derived as below: he1= he1+ = he0/0.618 (13) he2= he2+ = he1+/0.618 (14) when the values of xl and xu are set as 0 and 1, respectively, the value of he0 becomes 0.01. now, different levels of the linguistic variables can be quantitatively expressed as follows:               5),026.0,103.0,1( 4),016.0,064.0,691.0( 3),01.0,039.0,5.0( 2),016.0,064.0,309.0( 1),026.0,103.0,0( ),,( i i i i i heenexl iiii (15) where li ( i = 1,2,3,4,5) denotes different levels of the linguistic variable, i.e. very poor, poor, medium, good and very good. thus, based on the above-cited mathematical formulations, varying values of the qualitative attributes can be quantitatively expressed through the cloud model. 3.2 integration of the cloud model with topsis the cloud model can be integrated with topsis method using the following procedural steps [38]: step 1: for a group decision-making problem, there are m alternatives ai (i = 1,2,…,m), n evaluation criteria cj (j = 1, 2,…,n) and k decision-makers dk (k = 1,2,…,k). now, based on eqs. (6)-(15), the cloud decision matrix xk= (exkij,enkij,hekij) (k = 1,2,…,k; i = 1,2,…,m; j = 1, 2,…,n) can be formulated. in this matrix, the ratings of i th alternative with respect to j th criterion given by k th decision-maker are provided.              ),,(...),,(),,( ............ ),,(...),,(),,( ),,(...),,(),,( 22221111 22222222222221212121 11111212121211111111 kmnkmnkmnkmnkmkmkmkmkmkmkmkm nknknknkkkkkkkkk nknknknkkkkkkkkk k heenexxheenexxheenexx heenexxheenexxheenexx heenexxheenexxheenexx x (16) step 2: compute the weighted average cloud matrix using eq. (1), the weighted average cloud matrix (xw) can be obtained while multiplying each element of the cloud decision matrix with the corresponding criterion weight. in the weighted average cloud matrix, the overall level of evaluation results while a cloud topsis model for green supplier selection 383 aggregating the opinions of all the decision-makers is provided. the weighted average cloud matrix can be expressed by:              k k k k kij d k k k kij d kkij d k k k k d kw hewenwexwxwx 1 1 2 1 2 1 )(,)(, (17) where d kw (k = 1,2,…,k) shows the relative importance of opinion provided by dk. step 3: determine the criteria weights weight vector w c j (j = 1,2,…,n) depicts the relative importance of n criteria which can be determined by the evaluation process of the participating experts in the group decisionmaking process. step 4: identification of the positive ideal cloud (pic) and the negative ideal cloud (nic) based on topsis methodology, the selected best alternative should have the minimum distance from the pic and the maximum distance from the nic. let )......( 1   nj xxxa and )......( 1   nj xxxa represent the pic and nic, respectively, and can be determined using the following equations:        jjheenex jjheenex x ijijij ijijij j ),min,min,(min ),min,min,(max * (18)        jjheenex jjheenex x ijijij ijijij j ),max,max,(max ),max,max,(min * (19) where j * is the set of beneficial criteria and j  is the set of non-beneficial (cost) criteria. step 5: compute the difference of cloud from the pic (a + ) and the nic (a ) for every alternative ai.      n j jij c jii xxdwaadd 1 2 ),(),( (20)      n j jij c jii xxdwaadd 1 2 ),(),( (21) where w c j is the weight of the criterion cj, and d(xij, xj) is the difference of cloud between cloud xij and cloud xj. step 6: estimate relative closeness degree fi for each alternative ai to the pic using the following expression:     ii i i dd d f (22) step 7: arrange the values of relative closeness degree fi in descending order and rank the considered alternatives. the higher is the value of fi, the better is the alternative ai, because it is closer to the pic. 384 k.r. ramakrishnan, s. chakraborty 4. illustrative example it has already been highlighted that the selection of the most suitable green supplier for any manufacturing organization is extremely crucial for its competitive effectiveness and success of the entire supply chain system. the performance of the alternative suppliers is usually evaluated based on several green criteria, with a scope of inclusion of fuzziness and randomness in the qualitative judgments due to difference in the human cognition process. in this paper, an attempt is put forward to apply the cloud topsis model to identify the best suited green supplier in an automobile manufacturing unit. this illustrative example deals with five potential green suppliers ai (i = 1,2,3,4,5) to be appraised with respect to 12 evaluation criteria cj ( j = 1,2,...,12) by four experts dk (k = 1,2,3,4). these four experts are chosen as one for the procurement, production planning and control, manufacturing and quality control departments of the considered unit. these evaluation criteria are quality, finance, service, delivery, capability of the supplier, environment management, management competency, corporate social responsibility, pollution control, green product, green image and hazardous substance management [39]. it can be clearly noted that the list of the criteria not only includes the conventional evaluation attributes, but also some major green parameters. each of these criteria has also several sub-criteria, as elaborated in tables 1-12. these tables provide the corresponding definitions for each of the sub-criteria as considered in the green supplier selection problem. it is worthwhile to mention here that amongst those sub-criteria, some are beneficial in nature where their higher values are always required, and the remaining are non-beneficial (cost) criteria requiring their lower values. table 1 quality and its different sub-criteria criterion sub-criteria definition quality (c1) quality assurance (c11) desired quality level maintenance certificate issued by third party to ensure green product specification fulfillment rejection rate (c12) percentage of rejection of supplied materials after inspection and testing warranties and claim policies (c13) provision of warranties and claim policies by the supplier or agreements for the faulty products capability of handling abnormal quality (c14) capability to achieve the abnormal customer quality specification without compromising on the existing price of the product quality-related certificates (c15) ensure high quality control of the products and provide the quality concerned certificates, like iso9000, qs9000 etc. table 2 finance and its sub-criteria criterion sub-criteria definition finance (c2) purchasing price (c21) minimize product price without affecting the quality which includes warranty cost, processing cost, cost of greening, etc. price performance value (c22) high level of performance with respect to product value transportation cost (c23) fixed cost of transportation for product supply quantity discount (c24) discount offered by the supplier based on the quantity of purchase a cloud topsis model for green supplier selection 385 table 3 service along with its different sub-criteria criterion sub-criteria definition service (c3) rate of processing order form (c31) satisfactory processing of customer orders capability of delivery on time (c32) ability to deliver product on time according to the customer agreement degree of information modernized (c33) system for tracking of current orders credible delivery (c34) reputation and trust of customer towards the supplier responsiveness (c35) attention given to customer service willingness (c36) concern for the environment and interest to reduce impact on it during production table 4 delivery and its various sub-criteria criterion sub-criteria definition delivery (c4) order fulfillment rate (c41) order delivery at the right time lead time (c42) time between order placement and order arrival order frequency (c43) frequency of orders table 5 capability of supplier and its different sub-criteria criterion sub-criteria definition capability of supplier (c5) supplying capability (c51) ability to fulfill promises to the customer and meet shortcomings level of technique (c52) adoption of novel tools to maintain scheduling and delivery tasks capability of product development (c53) ability to augment innovative designs capability of r & d (c54) proper setup for the related research and development activities technology level (c55) technology development for more efficient production capability of design (c56) competence to design and develop new products to fulfill the end requirements flexibility of supplier (c57) ability of scheduling, modifying and replacing orders on demand supplier stock management (c58) efficient inventory control table 6 sub-criteria for environment management criterion sub-criteria definition environment management (c6) environmental protection policies/plans (c61) efficacy in proposing effective plans used for environment focused management implementation and planning (c62) application of processes for environment management continuous environment improvement (c63) continuous endeavor to use green processes and their improvement to reduce environmental impact energy using product (c64) product design to meet eco-design requirements for energy 386 k.r. ramakrishnan, s. chakraborty table 7 sub-criteria for management competency criterion sub-criteria definition management competency (c7) involvement of partners (c71) motivation of management to use environment friendly and clean production processes exchange of information (c72) willingness to share (or receive) product related information with (from) the customer environment training (c73) training related to obtain a green product table 8 corporate social responsibility and its different sub-criteria criterion sub-criteria definition corporate social responsibility (c8) interests and rights of employees (c81) focus on labor relations, interest of the employees and human rights rights of stakeholder (c82) to meet the interests and rights of the shareholders, customers and communities information disclosure (c83) transparency of information regarding supplier business activities respect for the policy (c84) compliance with local regulations and policies, and avoidance of illegal activities table 9 pollution control with its sub-criteria criterion sub-criteria definition pollution control (c9) use of harmful materials (c91) limit and minimize use of harmful and hazardous materials in production air emission (c92) effective control and treatment of hazardous materials, like so2, nh3, co and hc1 waste water (c93) waste water control and treatment solid waste (c94) capability to treat, use and dispose solid waste energy consumption (c95) energy consumption control table 10 green product and its various sub-criteria criterion sub-criteria definition green product (c10) recycle (c101) ability to convert an already used product into new, reusable product, thereby minimizing damage to environment green packaging (c102) use of green materials in product packaging green certifications (c103) provision of green related certificates by product suppliers green production (c104) use of environment friendly and clean production setup reuse (c105) ability to reutilize previously used products or their components re-manufacture (c106) usage of certain components from waste products for future use disposal (c107) ability to destroy or dispose of the harmful materials in a green way cost of component disposal (c108) cost of treatment and disposal at the end of product life cycle a cloud topsis model for green supplier selection 387 table 11 green image and its five sub-criteria criterion sub-criteria definition green image (c11) materials used in the supplied components to reduce the impact on natural resources (c111) use of materials in the products that reduce impact on the environment and its resources ability to alter process and product for reducing the impact on natural resources (c112) capability of modifying the process as well as product design to trim down the effect on the natural resources green customers’ market share (c113) retention and increase of customers buying green products ratio of green customers to total customers (c114) ratio of customers that buy green products to the total customers of the supplier green innovation (c115) innovative tools focusing on green product development and minimization of impact on environment table 12 sub-criteria for hazardous substance management criterion sub-criteria definition hazardous substance management (c12) management for hazardous substances (c121) proper maintenance and preventive management approaches related to use and disposal of hazardous materials prevention of mixed materials (c122) production procedure standards maintenance for differentiating between green and non-green materials process auditing (c123) effective auditing system to examine process conditions, parameter-setup document management, product change management, disqualified product management, improvement approaches and quality management system for production environment warehouse management (c124) level of warehouse management and space allocation for proper resource storage and maintenance in tables 13-24, where the detailed cloud topsis method-based calculations are exhibited, those beneficial and non-beneficial criteria are distinguished with (+) and (-) symbols, respectively. for simplicity of calculations, all the sub-criteria are assumed to have equal weights and all the four experts (e1, e2, e3 and e4) also have equal importance, i.e. 1 2 3 4 0.25.d d d dw w w w    the values of the coefficients of difference are taken here as λ1 = 1/2, λ2 = 1/3 and λ3 = 1/6. tables 13-24 exhibit the original decision matrices containing judgments by different experts on the considered five alternative green suppliers with respect to all the sub-criteria. in the green supplier performance appraisal and evaluation process by the experts, {very poor, poor, medium, good, very good} = {vp,p,m,g,vg] is the ordered set adopted to describe the human cognition for the beneficial sub-criteria, whereas {very low, low, medium, high, very high}= {vl,l,m,h,vh} is the ordered set employed to highlight the expert’s judgments for the non-beneficial criteria. for example, in table 13, the performance of green supplier s1 with respect to sub-criteria quality assurance (c11) (a beneficial criterion) has been appraised as good (g) by all the participating experts. 388 k.r. ramakrishnan, s. chakraborty table 13 original decision matrix and weighted average cloud matrix for criterion c1 criterion subcriteria green supplier e1 e2 e3 e4 weighted average cloud matrix c1 c11(+) s1 g g g g 0.6910 0.032 0.0080 s2 g f g g 0.6432 0.0294 0.0074 s3 g f g g 0.6432 0.0294 0.0074 s4 g p g g 0.5955 0.032 0.0080 s5 vg vg vg g 0.9227 0.0474 0.0119 c12 (-) s1 l l l m 0.3567 0.0294 0.0074 s2 l l l m 0.3567 0.0294 0.0074 s3 m m l l 0.4045 0.0265 0.0067 s4 l l l m 0.3567 0.0294 0.0074 s5 l m l l 0.3567 0.0294 0.0074 c13 (+) s1 g vg g g 0.7682 0.0378 0.0094 s2 g g g f 0.6432 0.0294 0.0074 s3 g g g g 0.6910 0.032 0.0080 s4 g g g g 0.6910 0.032 0.0080 s5 vg g g vg 0.8455 0.0429 0.0107 c14 (+) s1 g g f g 0.6432 0.0294 0.0074 s2 vg vg g g 0.8455 0.0429 0.0107 s3 g g g g 0.691 0.032 0.0080 s4 g vg f f 0.6727 0.0333 0.0084 s5 vg g g vg 0.8455 0.0429 0.0107 c15 (+) s1 g f g p 0.5477 0.0294 0.0074 s2 g f f f 0.5477 0.0233 0.0059 s3 g g g p 0.5955 0.032 0.0080 s4 g g vg g 0.7682 0.0378 0.0095 s5 vg vg vg g 0.9227 0.0474 0.0119 table 14 original decision matrix and weighted average cloud matrix for criterion c2 criterion subcriteria green supplier e1 e2 e3 e4 weighted average cloud matrix c2 c21 (-) s1 h m h h 0.6432 0.0293 0.0074 s2 l h h h 0.5955 0.032 0.0080 s3 m h m h 0.5955 0.0265 0.0067 s4 l h h h 0.5955 0.032 0.0080 s5 vh vh vh vh 1 0.0515 0.0130 c22 (+) s1 g vg g vg 0.8455 0.0378 0.0108 s2 f g g g 0.6432 0.0356 0.0074 s3 g vg vg g 0.8455 0.0265 0.0108 s4 g vg g vg 0.8455 0.0265 0.0108 s5 vg vg vg vg 1 0.0265 0.0130 c23 (-) s1 h h vl l 0.4227 0.0378 0.0095 s2 l h vh m 0.6250 0.0294 0.0090 s3 m l m h 0.5000 0.0265 0.0067 s4 l h m m 0.5000 0.032 0.0067 s5 h h m m 0.5955 0.0356 0.0067 c24 (+) s1 p vg g g 0.6727 0.032 0.0095 s2 g g p f 0.5477 0.032 0.0074 s3 g f g f 0.5955 0.032 0.0067 s4 g g p g 0.5955 0.0428 0.0080 s5 g vg f g 0.7205 0.0515 0.0089 a cloud topsis model for green supplier selection 389 based on the cloud model and eq. (15), this level of the linguistic expression can be numerically expressed as (0.691, 0.064, 0.016). similarly, all the linguistic decisions as opined by the experts for the five candidate green suppliers with respect to the remaining sub-criteria are converted into the corresponding quantitative values. the last columns of tables 13-24 represent the weighted average cloud matrices for all the considered subcriteria for this green supplier selection problem. the elements of the weighted average cloud matrices are determined based on eq. (17). now, using eqn. (18)-(22), the values of the relative closeness degree are computed for all the green suppliers, as provided in table 25. based on these values, alternative s4 is identified as the best green supplier for the considered automobile manufacturing unit so as to strengthen its supply chain system. amongst the five green suppliers, supplier s1 is the worst preferred choice. table 15 original decision matrix and weighted average cloud matrix for criterion c3 criterion subcriteria green supplier e1 e2 e3 e4 weighted average cloud matrix c3 c31 (+) s1 g g g g 0.6910 0.0429 0.0080 s2 g g g g 0.6910 0.0265 0.0080 s3 g g g g 0.6910 0.0232 0.0080 s4 vg g g vg 0.8455 0.0356 0.0108 s5 vg vg vg vg 1 0.0410 0.0130 c32 (+) s1 vp p p vp 0.1545 0.0356 0.0108 s2 p f p f 0.4045 0.0233 0.0067 s3 f f f g 0.5477 0.0265 0.0059 s4 g vg g f 0.7205 0.0410 0.0090 s5 vg vg g f 0.7977 0.0356 0.0103 c33 (+) s1 f g vg g 0.7205 0.0233 0.0090 s2 g f f f 0.5477 0.0265 0.0059 s3 p f p f 0.4045 0.0410 0.0067 s4 f g vg vg 0.7977 0.0356 0.0103 s5 g f vg g 0.7205 0.0410 0.0089 c34 (+) s1 g vg g f 0.7205 0.0356 0.0090 s2 f f f g 0.5477 0.0233 0.0059 s3 f f p p 0.4045 0.0265 0.0067 s4 vg vg g f 0.7977 0.0410 0.0103 s5 g vg vg f 0.7977 0.0410 0.0103 c35 (+) s1 vg vg g g 0.8455 0.0428 0.0108 s2 vg g f f 0.6727 0.0333 0.0084 s3 g g f f 0.5955 0.0264 0.0067 s4 g g g g 0.6910 0.032 0.0080 s5 vg g f f 0.6727 0.0333 0.0084 c36 (+) s1 f f f f 0.5000 0.0195 0.0050 s2 vg vg g g 0.8455 0.0429 0.0107 s3 vg g f f 0.6727 0.0333 0.0084 s4 g g g g 0.6910 0.032 0.0080 s5 g g f f 0.5955 0.0264 0.0067 390 k.r. ramakrishnan, s. chakraborty table 16 original decision matrix and weighted average cloud matrix for criterion c4 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c4 c41 (+) s1 vg vg vg vg 1 0.0515 0.0130 s2 g vg vg vg 0.9227 0.0474 0.0119 s3 g vg vg g 0.8455 0.0429 0.0108 s4 f g g f 0.5955 0.0265 0.0067 s5 g vg g vg 0.8455 0.0429 0.0108 c42 (+) s1 vp p f f 0.3272 0.0333 0.0084 s2 vp f f f 0.3750 0.0210 0.0078 s3 p f f g 0.5000 0.0265 0.0067 s4 p g g g 0.5955 0.032 0.0080 s5 g g g g 0.6910 0.032 0.0080 c43 (+) s1 g g g g 0.6910 0.032 0.0080 s2 p g f f 0.5000 0.0265 0.0067 s3 f vp f vp 0.2500 0.0389 0.0098 s4 vp f f f 0.3750 0.0310 0.0078 s5 p f f p 0.4045 0.0265 0.0067 table 17 original decision matrix and weighted average cloud matrix for criterion c5 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c5 c51 (+) s1 g vg g g 0.7682 0.0378 0.0095 s2 vg vg vg g 0.9227 0.0474 0.0119 s3 g vg g g 0.7682 0.0378 0.0095 s4 vg g g g 0.7682 0.0378 0.0095 s5 vg vg vg vg 1 0.0515 0.0130 c52 (+) s1 g vg vg vg 0.9227 0.0474 0.0119 s2 vg g vg vg 0.9227 0.0474 0.0119 s3 g g g g 0.6910 0.0320 0.0080 s4 vg g vg vg 0.9227 0.0474 0.0119 s5 g vg vg vg 0.9227 0.0474 0.0119 c53 (+) s1 vg g g g 0.7682 0.0378 0.0095 s2 vg vg g g 0.8455 0.0424 0.0108 s3 vg g g g 0.7682 0.0378 0.0095 s4 vg vg vg g 0.9227 0.0474 0.0119 s5 vg vg vg vg 1 0.0515 0.0130 c54 (+) s1 g g g f 0.6432 0.0294 0.0074 s2 vg g vg vg 0.9227 0.0474 0.0119 s3 g f g g 0.6432 0.0294 0.0074 s4 g f f g 0.5955 0.0265 0.0067 s5 g g g f 0.6432 0.0294 0.0074 c55 (+) s1 p f f f 0.4522 0.0233 0.0059 s2 p f f g 0.5000 0.0265 0.0067 s3 g vg g g 0.7682 0.0378 0.0095 s4 f g g vg 0.7205 0.0356 0.0090 s5 g g vg vg 0.8455 0.0429 0.0107 c56 (+) s1 g g g g 0.6910 0.032 0.0080 s2 g g f f 0.5955 0.0265 0.0067 s3 f f p f 0.4522 0.0233 0.0059 s4 g g g g 0.6910 0.032 0.0080 s5 vg vg vg g 0.9227 0.0474 0.0119 c57 (+) s1 vg vg vg vg 1 0.0515 0.0130 s2 g g g g 0.6910 0.032 0.0080 s3 g g f f 0.5955 0.0265 0.0067 s4 f g f f 0.5477 0.0233 0.0059 s5 g g g f 0.6432 0.0294 0.0074 c58 (+) s1 g g f g 0.6432 0.0294 0.0074 s2 f g g f 0.5955 0.0265 0.0067 s3 g g g g 0.6910 0.032 0.0080 s4 p vp p f 0.2795 0.0356 0.0089 s5 vg g g vg 0.8455 0.0428 0.0108 a cloud topsis model for green supplier selection 391 table 18 original decision matrix and weighted average cloud matrix for criterion c6 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c6 c61 (+) s1 g g g g 0.6910 0.032 0.0080 s2 g g g g 0.6910 0.032 0.0080 s3 vg vg vg g 0.9227 0.0474 0.0119 s4 vg vg vg vg 1 0.0515 0.0130 s5 vg vg vg vg 1 0.0515 0.0130 c62 (+) s1 g g f g 0.6432 0.0294 0.0074 s2 p vp f f 0.3272 0.0333 0.0084 s3 f f f g 0.5477 0.0233 0.0057 s4 g g g g 0.6910 0.032 0.0080 s5 vg vg g g 0.8455 0.0429 0.0107 c63 (+) s1 f f f f 0.5000 0.0195 0.0050 s2 g f f f 0.5477 0.0233 0.0059 s3 g g g vg 0.7682 0.0378 0.0095 s4 vg vg vg vg 1 0.0515 0.0130 s5 p vp f f 0.3272 0.0333 0.0085 c64 (+) s1 g g g g 0.6910 0.032 0.0080 s2 vg g g g 0.7682 0.0378 0.0095 s3 g vg g g 0.7682 0.0378 0.0095 s4 g g vg vg 0.8455 0.0428 0.0108 s5 g g g vg 0.7682 0.0378 0.0095 table 19 original decision matrix and weighted average cloud matrix for criterion c7 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c7 c71 (+) s1 g g g g 0.6910 0.032 0.0080 s2 g g g f 0.6432 0.0294 0.0074 s3 f f g f 0.5477 0.0233 0.0059 s4 g g g vg 0.7682 0.0378 0.0095 s5 vg vg vg g 0.9227 0.0474 0.0119 c72 (+) s1 f f g g 0.5955 0.0265 0.0067 s2 g f g f 0.5955 0.0265 0.0067 s3 g g g g 0.6910 0.032 0.0080 s4 vg g vg vg 0.9227 0.0474 0.0119 s5 g g vg vg 0.8455 0.0429 0.0107 c73 (+) s1 vg vg g g 0.8455 0.0429 0.0107 s2 g vg g vg 0.8455 0.0429 0.0107 s3 g g g g 0.6910 0.0320 0.0080 s4 g g f g 0.6432 0.0294 0.0074 s5 g f g f 0.5955 0.0265 0.0067 392 k.r. ramakrishnan, s. chakraborty table 20 original decision matrix and weighted average cloud matrix for criterion c8 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c8 c81 (+) s1 g vg g g 0.7682 0.0378 0.0095 s2 g f f g 0.5955 0.0265 0.0067 s3 g vg g vg 0.8455 0.0429 0.0107 s4 g vg vg vg 0.9227 0.0474 0.0119 s5 vg vg vg g 0.9227 0.0474 0.0119 c82 (+) s1 f f f p 0.4522 0.0233 0.0059 s2 vg g vg vg 0.9227 0.0474 0.0119 s3 g g g g 0.6910 0.032 0.0080 s4 vg g vg g 0.8455 0.0429 0.0108 s5 vg vg vg g 0.9227 0.0474 0.0119 c83 (+) s1 vg vg vg vg 1 0.0515 0.0130 s2 g vg vg g 0.8455 0.0429 0.0108 s3 vg vg vg vg 1 0.0515 0.0130 s4 vg g g vg 0.8455 0.0429 0.0108 s5 g g f g 0.6432 0.0294 0.0074 c84 (+) s1 vg vg g vg 0.9227 0.0474 0.0119 s2 vg g vg g 0.8455 0.0429 0.0108 s3 g vg vg g 0.8455 0.0429 0.0108 s4 f f f g 0.5477 0.0233 0.0059 s5 vg g g f 0.7205 0.0356 0.0089 table 21 original decision matrix and weighted average cloud matrix for criterion c9 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c9 c91 (-) s1 h h vh h 0.7682 0.0378 0.0095 s2 m h m m 0.5477 0.0233 0.0059 s3 m h h m 0.5955 0.0265 0.0067 s4 l vl m l 0.2795 0.0356 0.0090 s5 l m m l 0.4045 0.0265 0.0067 c92 (-) s1 l l m m 0.4045 0.0265 0.0067 s2 l m m m 0.4522 0.0233 0.0059 s3 vl m m l 0.3272 0.0333 0.0084 s4 l l h m 0.4522 0.0294 0.0074 s5 l l m l 0.3567 0.0294 0.0074 c93 (-) s1 h h h h 0.6910 0.032 0.0080 s2 h h m m 0.5955 0.0265 0.0067 s3 m m m h 0.5477 0.0233 0.0059 s4 m h h m 0.5955 0.0265 0.0067 s5 h h h h 0.6910 0.0320 0.0080 c94 (-) s1 h vh h h 0.7682 0.0378 0.0095 s2 h m m h 0.5955 0.0265 0.0067 s3 h m h m 0.5955 0.0265 0.0067 s4 vl vl l m 0.2022 0.041 0.0103 s5 m l l m 0.4045 0.0265 0.0067 c95 (-) s1 m h m h 0.5955 0.0265 0.0067 s2 l vl l m 0.2795 0.0356 0.0090 s3 m l m m 0.4522 0.0233 0.0059 s4 l vl m l 0.2795 0.0356 0.0090 s5 h h h m 0.6432 0.0294 0.0074 a cloud topsis model for green supplier selection 393 table 22 original decision matrix and weighted average cloud matrix for criterion c10 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c10 c101 (+) s1 f g f f 0.5477 0.0233 0.0059 s2 g g g g 0.6910 0.0320 0.0080 s3 g vg vg vg 0.9227 0.0474 0.0119 s4 g vg vg vg 0.9227 0.0474 0.0119 s5 vg vg vg vg 1 0.0515 0.0130 c102 (+) s1 g f f f 0.5477 0.0233 0.0059 s2 g g vg g 0.7682 0.0378 0.0095 s3 g f f g 0.5955 0.0265 0.0067 s4 vg vg vg vg 1 0.0515 0.0130 s5 g g g g 0.6910 0.032 0.0080 c103 (+) s1 g g vg f 0.7205 0.0356 0.0090 s2 g f g vg 0.7205 0.0356 0.0090 s3 g f g g 0.6432 0.0294 0.0074 s4 vg f g g 0.7205 0.0356 0.0090 s5 vg vg g vg 0.9227 0.0474 0.0119 c104 (+) s1 g g g g 0.6910 0.032 0.0080 s2 g vg g g 0.7682 0.0378 0.0095 s3 g f f vg 0.6727 0.0333 0.0084 s4 g vg vg g 0.8455 0.0428 0.0108 s5 vg vg vg vg 1 0.0515 0.0130 c105 (+) s1 f g f f 0.5477 0.0233 0.0059 s2 g vg g g 0.7682 0.0378 0.0095 s3 g vg vg g 0.8455 0.0429 0.0108 s4 g vg vg vg 0.9227 0.0474 0.0119 s5 vg g vg vg 0.9227 0.0474 0.0119 c106 (+) s1 f g f p 0.5000 0.0265 0.0067 s2 g vg f g 0.7205 0.0356 0.0090 s3 vg vg vg vg 1 0.0515 0.0130 s4 vg vg vg vg 1 0.0515 0.0130 s5 g vg vg vg 0.9227 0.0474 0.0119 c107 (+) s1 f f f f 0.5000 0.0195 0.0050 s2 f f g f 0.5477 0.0233 0.0059 s3 g vg f g 0.7205 0.0356 0.0090 s4 g g g f 0.6432 0.0294 0.0074 s5 vg g vg g 0.8455 0.0429 0.0108 c108 (-) s1 m m h m 0.5477 0.0233 0.0059 s2 m h m h 0.5955 0.0265 0.0067 s3 l m m l 0.4045 0.0265 0.0067 s4 l vl vl l 0.1545 0.0429 0.0108 s5 h vh vh h 0.8455 0.0429 0.0108 394 k.r. ramakrishnan, s. chakraborty table 23 original decision matrix and weighted average cloud matrix for criterion c11 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c11 c111 (+) s1 g g vg vg 0.8455 0.0429 0.0108 s2 f g vg g 0.7205 0.0356 0.0090 s3 g vg vg vg 0.9227 0.0474 0.0119 s4 g g vg vg 0.8455 0.0428 0.0108 s5 vg vg vg g 0.9227 0.0474 0.0119 c112 (+) s1 vg g g g 0.7682 0.0378 0.095 s2 g vg vg vg 0.9227 0.0474 0.0119 s3 vg vg vg vg 1 0.0515 0.0130 s4 vg vg g vg 0.9227 0.0474 0.0119 s5 vg vg vg g 0.9227 0.0474 0.0119 c113 (+) s1 g g f g 0.6432 0.0294 0.0074 s2 g g g f 0.6432 0.0294 0.0074 s3 f g g g 0.6432 0.0294 0.0074 s4 vg vg g g 0.8455 0.0429 0.0108 s5 g g g vg 0.7682 0.0378 0.0095 c114 (+) s1 f f g f 0.5477 0.0233 0.0059 s2 p vp f p 0.2795 0.0356 0.0090 s3 p vp vp p 0.1545 0.0429 0.0108 s4 vg vg g g 0.8455 0.0429 0.0108 s5 vg vg vg vg 1 0.0515 0.0130 c115 (+) s1 g f f f 0.5477 0.0233 0.0059 s2 g g vg g 0.7682 0.0378 0.0095 s3 g f f g 0.5955 0.0265 0.0067 s4 vg vg vg vg 1 0.0515 0.0130 s5 g g g g 0.6910 0.032 0.0080 table 24 original decision matrix and weighted average cloud matrix for criterion c12 criterion sub-criteria green supplier e1 e2 e3 e4 weighted average cloud matrix c12 c121 (+) s1 g g g g 0.6910 0.032 0.0080 s2 g vg g g 0.7682 0.0378 0.0095 s3 g f f vg 0.6727 0.0333 0.0084 s4 g vg vg g 0.8455 0.0428 0.0108 s5 vg vg vg vg 1 0.0515 0.0130 c122 (+) s1 vg vg vg g 0.9227 0.0474 0.0119 s2 vg vg g vg 0.9227 0.0474 0.0119 s3 g g g f 0.6432 0.0294 0.0074 s4 f g f g 0.5955 0.0265 0.0067 s5 vg vg g g 0.8455 0.0429 0.0108 c123 (+) s1 g vg g f 0.7205 0.0356 0.0090 s2 vg g vg g 0.8455 0.0429 0.0108 s3 g vg vg vg 0.9227 0.0474 0.0119 s4 vg vg g g 0.8455 0.0429 0.0108 s5 g g g g 0.6910 0.032 0.0080 c124 (+) s1 vg vg vg vg 1 0.0515 0.0130 s2 g g g g 0.6910 0.0320 0.0080 s3 f f g g 0.5955 0.0265 0.0067 s4 g f g g 0.6432 0.0294 0.0074 s5 g vg g vg 0.8455 0.0429 0.0108 a cloud topsis model for green supplier selection 395 in order to validate the solution accuracy and reliability of the ranking results as derived using the cloud topsis model, the same green supplier selection problem is again solved while employing three other variants of the topsis method, i.e. original topsis, fuzzy topsis and interval topsis. the corresponding rankings of the five green suppliers are provided in table 25. it can be interestingly revealed that for all the four different topsis models, green supplier s4 is the best choice and s1 is the worst choice for the considered automobile manufacturing unit. there are slight variations in the intermediate rankings for the adopted approaches which can only be attributed to the difference in the mathematical complexities involved in these methods. table 25 rankings of the green suppliers green supplier cloud topsis model topsis fuzzy topsis interval topsis d(ai, a +) d(ai, a -) fi rank s1 0.0438 0.0362 0.4529 5 5 5 5 s2 0.0435 0.0402 0.4807 2 4 4 3 s3 0.0435 0.0396 0.4763 4 3 3 4 s4 0.0403 0.0378 0.4842 1 1 1 1 s5 0.0399 0.0368 0.4797 3 2 2 2 6. conclusions in this paper, the cloud topsis model is applied to identify the best performing green supplier in an automobile manufacturing unit. the topsis method has already become popular as an effective mcdm tool due to its various added advantages. but, the topsis method along with its other variants, like fuzzy topsis and interval topsis cannot solve mcdm problems where both fuzziness and randomness are present in the information acquired from different experts while expressing their opinions with respect to the performance of the participating green suppliers in a manufacturing unit. the cloud model is integrated here with the topsis method to deal with this problem arising in a group decision-making environment. the integrated model attempts to quantify the qualitative assessment of the green suppliers by the experts while accounting for the fuzziness and randomness inherent in the decisionmaking procedure. five green supplies are considered in a demonstrative example to be appraised by four experts with respect to 12 evaluation criteria (59 sub-criteria). this model identifies green supplier s4 as the best choice. the derived ranking results using the adopted model closely match with those obtained from the other variants of the topsis method. thus, it can be effectively applied to solving real time group decision-making problems with its better distinction ability. but, it has also few drawbacks like its inability to consider the interactions between different criteria present in the evaluation process, unsuitability to deal with welldefined non-random processes, complexity in the calculations involved, etc. hence, it is advised to develop a software prototype (decision support system) to take care of the varied fuzzy and random opinions of the experts while arriving at the final green supplier selection decision. 396 k.r. ramakrishnan, s. chakraborty references 1. gurel, o., acar, a.z., onden, i., gumus, i., 2015, determinants of the 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supplier selection problem goran petrović 1 , jelena mihajlović 1 , žarko ćojbašić 1 , miloš madić 1 , dragan marinković 2 1university of niš, faculty of mechanical engineering, serbia 2tu berlin, department of structural analysis, berlin, germany abstract. the evaluation and selection of an optimal, efficient and reliable supplier is becoming more and more important for companies in today’s logistics and supply chain management. decision-making in the supplier selection domain, as an essential component of the supply chain management, is a complex process since a wide range of diverse criteria, stakeholders and possible solutions are embedded into this process. this paper shows a fuzzy approach in multi – criteria decision-making (mcdm) process. criteria weights have been determined by fuzzy swara (step-wise weight assessment ratio analysis) method. chosen methods, fuzzy topsis (technique for the order preference by similarity to ideal solution), fuzzy waspas (weighted aggregated sum product assessment) and fuzzy aras (additive ratio assessment) have been used for evaluation and selection of suppliers in the case of procurement of thk linear motion guide components by the group of specialists in the "lagerton" company in serbia. finally, results obtained using different mcdm approaches were compared in order to help managers to identify appropriate method for supplier selection problem solving. key words: supplier selection, fuzzy mcdm methods, linear motion guide, comparative analysis 1. introduction given that supply chains generate a value added of over 80% of the final product [1], nowadays supplier evaluation and selection have been recognized as one of the most important factors which significantly affect company competitiveness, reputation and success in highly competitive markets. the supplier selection process consists of several tasks [2, 3]: problem definition (identification of the needs and specifications), formulation and selection of evaluation criteria, evaluation and pre-qualification of potential suppliers received april 20, 2019 / accepted july 10, 2019 corresponding author: goran s. petrović university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, 18000 niš, serbia e-mail: goran.petrovic@masfak.ni.ac.rs 456 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković with respect to considered criteria and respective significance and evaluation and final selection of supplier. the quality of the final selection largely depends on the quality of all the steps involved in the selection process [4]. among these the formulation and selection of evaluation criteria, attending to cover all important aspects in the selection process as well as the choice of methods for generation of decision (selection) rule play a very important role. starting from the pioneer study of dickson [5], who identified 23 different criteria for evaluation of suppliers, the list of criteria is continuously changing and upgrading, wherein the relative significance of each particular criteria may vary from one case to another. generation of a decision rule, aimed at ranking the considered potential suppliers, relies on efficient processing of information related to attribute values, both quantitative and qualitative which may involve a certain degree of uncertainty and vagueness. given that the supplier selection often involves several decision-makers and requires consideration of a number of conflicting criteria, wherein the entire decision-making process in influenced by uncertainty in practice [6], it can be argued that the supplier selection is a complex task which can be represented as a multi-criteria decision-making (mcdm) problem. the use of mcdm methods has become widely accepted for solving real life supplier selection problems [7]. these methods allow decision-makers to determine compromise solution taking into the account different criteria, type of information (quantitative and qualitative), interest of stakeholders, relative significance of criteria as well as decision-maker preferences [8]. since decision-makers’ judgments are usually imprecise when selecting an alternative with respect to multiple criteria, the fuzzy concept is integrated within the mcdm process [9]. the fuzzy based mcdm methods enable quantification of linguistic attributes and criteria weighting scores which are used by decisionmakers thus enable handling of uncertainty, imprecision and vagueness during decisionmaking process. this section will briefly review the previous research studies focused on the use of fuzzy mcdm methods for supplier evaluation and selection. awasthi et al. [10] used fuzzy technique for ordering preferences by similarity to ideal solution (topsis) methods for solving a supplier selection problem considering the environmental criteria. shaw et al. [11] proposed an integrated fuzzy-ahp and fuzzy multi-objective linear programming for solving a supplier selection problem taking into account greenhouse gas emission, costs, quality, lead time and demand as criteria for evaluating and ranking of suppliers. kumar et al. [12] used fuzzy topsis method to gain more efficient steal manufacturing throughout the supplier selection for raw materials. luthra et al. [13] employed ahp and vikor methods for analyzing and ranking the sustainable suppliers in a supply chain. baneian et al. [14] used fuzzy grey relational analysis (gra), fuzzy topsis and fuzzy vikor in order to evaluate and rank sustainable suppliers in the agricultural industry. senetal [15] employed intuitionistic fuzzy topsis, intuitionistic fuzzy multi-objective optimization by ratio analysis (moora) and intuitionistic gra (if-gra) to facilitate supplier selection in the sustainable supply chain. zeydan et al. [16] combined ahp and fuzzy topsis in a framework which firstly estimated the criteria weights with ahp and then ranked a set of potential suppliers based on fuzzy topsis. büyüközkan and göçer [17] presented the phases of the aras method based on the intuitive phase of the setting at intervals to support the process of selecting digital vendors. in order to cope with vagueness and uncertainty this study employs fuzzy swara (stepwise weight assessment ratio analysis) method for the determination of the considered criteria weights and fuzzy topsis, fuzzy waspas (weighted aggregated sum product assessment) and fuzzy aras (additive ratio assessment) methods for the evaluation and comparison of three fuzzy mcdm methods for solving the supplier selection problem 457 selection of suppliers in the case of procurement of thk linear motion guide components. the data from company management was used so as to determine criteria importance as well as to setup the decision-making matrix. the remainder of this study is structured as follows. in the next section, a theoretical framework of the fuzzy logic, fuzzy decision-making and selected fuzzy mcdm methods are presented. the case study, i.e. thk linear motion guide supplier selection and implementation of the selected fuzzy mcdm methods are then presented in section 3. this section also provides a comparative analysis of the obtained results. concluding remarks and future research directions are given in the last, concluding section 4. 2. the fuzzy logic in mcdm methods today fuzzy logic and fuzzy set theory have numerous applications in artificial intelligence, computer science, medicine, decision theory, expert systems, management science and operations research. fuzzy set theory has been proposed by zadeh [18] with intention to generalize the classical notion of a set. the idea was to accommodate fuzziness as a computational framework for dealing with systems which contain human language, human judgment, their behavior, emotions and decisions. the theory of fuzzy logic provides a mathematical tool to capture the uncertainties associated with linguistic and vague variables such as "not very clear", "probably so", "very likely", etc. a linguistic variable is a variable whose values are sentences in a natural or artificial language. in ordinary set theory, the membership of an element belonging to that set is based upon two valued boolean logic (a member is either in or out of a subset). unlike that fuzzy set theory is based upon multi-valued fuzzy logic which deals with degree of membership. the membership of an element is described in a real unit interval [0,1]. 2.1. fuzzy numbers fuzzy numbers are fuzzy subset of real numbers most often presented in form of triangular fuzzy number (tfn), trapezoidal and gaussian fuzzy numbers [19]. according to numerous definitions[20, 21, 22] tfn is represented as ̃( ) where its membership function ̃ ̃( ) [ ] is given by eq. (1): ̃( ) { (1) values l and u represent lower and upper bounds of the fuzzy number ̃ and b is modal value (see fig. 1). according to [18, 21] the basic algebraic operations with two tfns, ̃ ( ) and ̃ ( ), are put forward:  addition of triangular fuzzy numbers(+): ̃ ( ) ̃ ( )( )( ) ( ) (2) 458 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković  multiplication of fuzzy numbers ( ): ̃ ( ) ̃ ( )( )( ) ( ) (3)  multiplication of a real number k and fuzzy number( ): ( ) ̃ ( )( )( ) ( ) (4)  subtraction of fuzzy numbers ( ): ̃ ( ) ̃ ( )( )( ) ( ) (5)  division of fuzzy numbers (/): ̃ ( ) ̃ ( )( )( ) ( ) (6)  reciprocal of a fuzzy number: ̃ ( ) ( ) (7) fig. 1 the membership function of the triangular fuzzy number 2.2. fuzzy multi criteria decision-making decision-making in solving the supplier selection problem involves the consideration of a number of opposite criteria and possible solutions. a decision-maker has to choose the best alternative among several candidates while considering a set of conflicting criteria. in the case where some ratings of alternatives versus criteria as well as the importance weights of all criteria are assessed in linguistic values represented by fuzzy numbers, such selection can be considered as a fuzzy multi-criteria decision-making (fmcdm) problem. in order to evaluate the overall effectiveness of the candidate alternatives, rank and select the most appropriate (the best) supplier, the primary objective of a fmcdm methodology is to identify the relevant supplier selection problem criteria, assess the alternatives information relating to those criteria and develop methodologies for evaluating the significance of criteria. comparison of three fuzzy mcdm methods for solving the supplier selection problem 459 here, a brief description of the applied fmcdm methods is given. in order to calculate criteria weights, fuzzy swara method is used, while fuzzy topsis, fuzzy waspas and fuzzy aras methods are used for evaluation of alternatives. the first step in all fmcdm methods for evaluation of alternatives (topsis, waspas, aras…) is structuring the fuzzy decision matrix ̃ with fuzzy membership function as shown by eq. (8): ̃ [ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ] [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )] (8) in this expression m is the number of alternative solutions, n is the number of evaluation criteria and ̃ represents aggregated performance of alternative i regarding criteria j. for qualitative criteria boundaries ( ) are aggregated values obtained using singular judgments of decision-makers in form ̃ ( ) ( ). here, values ( ) are assigned to each alternative based on suggestions given in table 1. table 1 the fuzzy scale for the alternative assessment [23] rank triangular fuzzy number attribute grade very low (vl) (0, 0, 0.25) ̃ low (l) (0, 0.25, 0.5) ̃ medium (m) (0.25, 0.5, 0.75) ̃ high (h) (0.5, 0.75, 1.0) ̃ very high (vh) (0.75, 1.0, 1.0) ̃ the aggregated values ( ) for each alternative can be obtained using the minimal, arithmetic mean and maximal value of the corresponding scores (see eq. (9)). ( ) ( ( ) ∑ ( )) (9) if the specific criterion is quantitative, according to [21, 24] two approaches can be applied: a) if no historical (statistical) data are known, triangular fuzzy numbers can be used directly by subjectively expression. for example, if transportation costs are 390 [eur], decision-maker can subjectively estimate lower and upper boundary in triangular fuzzy number as (380, 390, 420). (b) if there are statistical data for some past period, for example, let represent transportation costs of past k periods, the triangular fuzzy number can be obtained using the minimal, geometric mean and maximal value of the corresponding scores: ( ) ( ( ) (∏ ) ( )) (10) 460 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković in this paper the second approach is applied. the further steps of the applied fmcdm methods are different and because of that are briefly described in the following sub-sections 2.2.2 – 2.2.4. 2.2.1. fuzzy swara method the step-wise weight assessment ratio analysis (swara) method was originally introduced by kersuliene et al. [25] in 2010, as a tool for the estimation of criteria weights in mcdm problems considering decision-makers’ preferences. the process of determining the relative weights of criteria using the fuzzy swara method is the same as in the ordinary swara method, such as the following steps [26, 27]: step 1: the criteria should be sorted in descending order based on their expected significances, i.e. the most significant criterion is assigned as rank first, and the least significant criterion is assigned as rank last. step 2: starting from the second criterion, each decision-maker (in total experts) expresses the relative importance of criterion in relation to the previous , for all considered criteria. this ratio is called the comparative importance of average value [25]. the fuzzy comparison scale presented in table 2 should be applied. table 2 the fuzzy comparison scale for the assessment of evaluation criteria [23] linguistic variable response scale equally important (1, 1, 1) moderately less important (2/3, 1, 3/2) less important (2/5, 1/2, 2/3) very less important (2/7, 1/3, 2/5) much less important (2/9, 1/4, 2/7) the aggregated average values of experts’ judgments for evaluation criteria can be obtained, similarly as previously described, using minimal, arithmetic mean and maximal value of the corresponding scores (see eq. (11)). ̃ ( ̃ ̃ ̃ ) ( ( ̃ ) ∑ ̃ ( ̃ )) (11) step 3: obtain coefficient ̃ values, fuzzy weights ̃ and final weights of criteria. coefficient ̃ value is computed as: ̃ { ̃ ̃ ( ) ̃ (12) fuzzy recalculated weights ̃ as: ̃ { ̃ ̃ ̃ (13) final relative weights of criteria ̃ as: ̃ ̃ ∑ ̃ (14) where ̃ ( ̃ ̃ ̃ ) denotes relative importance fuzzy weight of the jth criterion. https://www.emeraldinsight.com/doi/full/10.1108/jmtm-08-2018-0247?fullsc=1& comparison of three fuzzy mcdm methods for solving the supplier selection problem 461 2.2.2. fuzzy topsis method the technique for the order preference by similarity to ideal solution (topsis) method was introduced by hwang and yoon [28] in 1981. the ordinary topsis method is based on the concept that the best alternative should have the shortest euclidian distance from the ideal solution (positive ideal solution – pis) and at the same time the farthest from the anti-ideal solution (negative ideal solution – nis). it is a method of compensatory aggregation that compares a set of alternatives by identifying weights for each criterion. the method was popularized by many researchers from different fields and adapted to deal with fuzzy numbers [22, 29, 30]. in the fuzzy topsis approach an alternative that is nearest to the fuzzy positive ideal solution (fpis) and farthest from the fuzzy negative ideal solution (fnis) is chosen as optimal. an fpis is composed of the best performance values for each alternative whereas the fnis consists of the worst performance values. here, the relevant steps of fuzzy topsis method are given as: step 1: the first step is the same as described in section 2.2 (eq. (8)). step 2: normalizing fuzzy decision matrix ̃ [ ̃ ] : ̃ { ̃ ̃ ̃ ̃ (15) step 3: constructing weighted normalized decision matrix ̃ [ ̃ ] as below: ̃ ̃ ( ) ̃ (16) step 4: determining the fpis ( ̃ ) and fnis( ̃ ) as below: ̃ ( ̃ ̃ ̃ ) ( ̃ ̃ ̃ ) ̃ ( ̃ ̃ ̃ ) ( ̃ ̃ ̃ ) (17) step5: calculating the euclidean distance between each alternative and fpis and fnis as below: ∑ ( ̃ ̃ ) ∑ ( ̃ ̃ ) (18) where, ( ̃ ̃ ) √ [( ) ( ) ( ) ] is the distance measurement between two fuzzy numbers ̃ ̃ . step 6: calculating the relative closeness coefficient as: (19) step 7: ranking the alternatives. the alternative with the smallest value of is considered as the best alternative. 462 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković 2.2.3. fuzzy waspas method the weighted aggregated sum product assessment (waspas) method was proposed by zavadskaset al. [31] in 2012. it consists of two aggregated parts: 1. the weighted sum model (wsm); 2. the weighted product model (wpm). a joint criterion of optimality, upon which final complete ranking of the alternatives is obtained, is derived based on two optimality criteria which are linearly combined using the  coefficient [32]. the popularity of the method has resulted in the development of waspas-g [33], waspas-f [34], and waspas-ivif [35, 36] methods that are intended to work with grey numbers, fuzzy numbers and interval valued intuitionistic fuzzy numbers. also, there are a number of applications of the waspas method, including a number of real cases of solving the supplier selection problems [37]. the wsm determines the overall score of an alternative as a weighted sum of the attribute values, while wpm is developed in order to avoid alternatives with poor attribute values. it determines score of each alternative as a product of the scale rating of each attribute to a power equal to the importance weight of the attribute. the relevant steps of the fuzzy waspas method are as follows [34]: step 1: forming of a fuzzy decision matrix as previously described (see eq. (8)). step 2: the second step (normalization of the fuzzy decision matrix) is the same as in the previous method (see eq. (15)). step 3a: calculating the weighted normalized fuzzy decision matrix for wsm ̃ [ ̃ ]: ̃ ̃ ( ) ̃ (20) step 3b: calculating the weighted normalized fuzzy decision matrix for wpm ̃ [ ̃ ]: ̃ ( ̃ ) ̃ (21) step 4: calculating values of the optimality function: a) according to the wsm for each alternative: ̃ ∑ ̃ (22) b) according to the wpm for each alternative: ̃ ∏ ̃ (23) step 5: calculating crisps values of fuzzy numbers ̃ and ̃ . to derive the crisp value of a fuzzy number few defuzzification methods can be performed (center of gravity, center of area, mean of maxima etc.). here the center-of-area method was used as the simplest approach to apply for defuzzification: ( ̃ ̃ ̃ ) ( ̃ ̃ ̃ ) (24) step 6: calculating the integrated utility function value: a) firstly, based on the assumption that total of all alternatives wsm scores must be equal to the total of wpm scores, coefficient should be calculated: comparison of three fuzzy mcdm methods for solving the supplier selection problem 463 ∑ ∑ ∑ (25) b) finally, the integrated utility function value is: ∑ ( )∑ (26) alternative with maximal value should be chosen as the best alternative. 2.2.4. fuzzy aras method the additive ratio assessment (aras) method was conceptualized and proposed by zavadskas and turskis [38] in 2010. the specificity of this method is that an alternative’s performances are determined with respect to the ideal (optimal) alternative. they argue that the ratio of the sum of normalized and weighted criteria scores, which describe alternative under consideration, to the sum of the values of normalized and weighted criteria, which describes the optimal alternative, is a degree of optimality, which is reached by the alternative under comparison. although it has been developed relatively recently, its application field has been extended by the development of aras-f [39] and aras-g methods [40] for solving mcdm problems involving fuzzy and grey numbers. the relevant steps of fuzzy aras method are as follows [39]: step 1: forming of fuzzy decision matrix as previously described (eq. (8)). step 2: forming of expanded fuzzy decision matrix by adding one row with optimal values of each criterion in the form as: ̃ [ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ] (27) here, the optimal value of criterion is counted as: ̃ ̃ ̃ ̃ ̃ ̃ (28) step 3: normalizing the expanded fuzzy decision matrix ̃ [ ̃ ] : a) the criteria, whose preferable values are maximal, are normalized as follows: ̃ ̃ ∑ ̃ (29) b) the criteria, whose preferable values are minimal, are normalized by applying two-stage procedure: ̃ ̃ ̃ ̃ ∑ ̃ (30) 464 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković step 4: constructing weighted normalized decision matrix ̃ [ ̃ ] as below: ̃ ̃ ( ) ̃ (31) step 5: calculating values of optimality function: ̃ ∑ ̃ (32) where ̃ represents the value of optimality function of i-th alternative. taking into account the calculation process, the optimality function ̃ has a direct and proportional relationship with the values ̃ and weights ̃ of the considered criteria. therefore, the highest value of the optimality function ̃ matches the most effective alternative (in this case it is optimal alternative ). the center-of-area defuzzification method can be applied here too: ( ̃ ̃ ̃ ) (33) step 5: calculating value of the alternative utility degree. the priority orders of considered alternatives ( ) can be determined according to the values (the degree of the alternative utility). the equation used for the calculation of the utility degree of an alternative is given as: (34) where si and s0 are values of optimality function. alternative with maximal value should be chosen as the best alternative. 3. case study – thk linear motion guide supplier selection the proposed fuzzy mcdm methods for supplier evaluation and selection have been implemented in the company "lagerton" (limited liability company) from serbia, which is an authorized distributor of a number of mechanical components. in order to illustrate and validate the applicability of proposed fuzzy mcdm methods a real-life problem, considering evaluation and selection of linear motion guide technologies supplier, is solved here. linear motion guide is a product of thk company from japan. it provides a component that enables linear rolling motion for practical usage in high-precision, high-rigidity, and energy-saving, high-speed machines. for a known buyer the company "lagerton" procures components (fig. 2):  slide blocks srs 12 gm uu;  rail srs 12/570 – 10 – 10. the company acquires components through a selection of the best supplier from european market qualified suppliers. four companies (s1, s2, s3 and s4) have been evaluated and the main criteria for evaluation and selection that were used are: product price (c1), transportation costs (c2), delivery time (c3), company rating (c4) and established cooperation (c5). the first three criteria (quantitative) are minimization criteria where lower attribute values are preferred. the last two criteria (qualitative) are maximization criteria where higher attribute values are preferable. comparison of three fuzzy mcdm methods for solving the supplier selection problem 465 fig. 2 thk linear motion guide components a graphical illustration of decision-making model for thk linear motion guide components supplier selection is shown in fig. 3. fig. 3 a model for supplier selection of thk linear motion guide components in an interview, the management team of the company "lagerton", responsible for evaluation and selection of suppliers, estimated performance ratings of four suppliers and results are shown in table 3. quantitative criteria are evaluated using statistical data for last two years while criteria c4 and c5 (qualitative) are evaluated by five experts of the company management team. table 3 supplier’s performance ratings-decision matrix criteria c1[eur] c2[eur] c3[days] c4[-] c5[-] alternatives min min min max max s1 320 343.22 380 45 51.75 60 10 14.01 20 0.5 0.9 1 0.5 0.9 1.0 s2 380 391.73 420 50 59.73 65 12 18.09 25 0.5 0.85 1 0 0.1 0.5 s3 385 402.83 420 55 62.56 75 10 14.25 18 0.25 0.65 1 0 0.1 0.5 s4 350 372.61 400 50 61.60 70 5 7.02 9 0.25 0.55 1 0 0.05 0.5 466 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković the processing of the obtained data was performed in accordance with the previously defined procedure (section 2.2). in the case of quantitative criteria alternatives are evaluated using eq.10 (data from earlier similar purchases are used) and in the case of qualitative criteria using eq. (9). fuzzy swara method is applied in order to calculate fuzzy criteria weights. according to step 1 the criteria are sorted in descending order based on their expected significances c5 → c1 → c2 → c3 → c4. in second step, the relative importance of each criterion in relation to the previous one is expressed by five experts of the company management team. results are shown in table 4. table 4 comparison of criteria relative importance by "lagerton" experts expert c1 to c5 c2 to c1 c3 to c2 c4 to c3 e1 0.667 1.000 1.500 0.400 0.500 0.667 0.400 0.500 0.667 0.400 0.500 0.667 e2 0.400 0.500 0.667 0.400 0.500 0.667 0.400 0.500 0.667 0.286 0.333 0.400 e3 0.400 0.500 0.667 0.286 0.333 0.400 0.286 0.333 0.400 0.400 0.500 0.667 e4 0.400 0.500 0.667 0.400 0.500 0.667 0.400 0.500 0.667 0.286 0.333 0.400 e5 0.667 1.000 1.500 0.667 1.000 1.500 0.400 0.500 0.667 0.400 0.500 0.667 all further calculations (step 3) and results are shown in table 5. data in column are calculated according to eq. 11. table 5 criteria weights obtained using swara method criteria ̃ ̃ ̃ ̃ c5 1.000 1.000 1.000 1.000 1.000 1.000 0.329 0.417 0.584 c1 0.400 0.700 1.500 1.400 1.700 2.500 0.400 0.588 0.714 0.132 0.245 0.417 c2 0.286 0.567 1.500 1.286 1.567 2.500 0.160 0.375 0.556 0.053 0.157 0.324 c3 0.286 0.467 0.667 1.286 1.467 1.667 0.096 0.256 0.432 0.032 0.107 0.252 c4 0.286 0.433 0.667 1.286 1.433 1.667 0.058 0.179 0.336 0.019 0.074 0.196 in order to evaluate suppliers relative to given criteria three fmcdm methods (fuzzy topsis, fuzzy waspas and fuzzy aras) are used. the application of the proposed fmcdm approaches gives the complete ranking of the suppliers as shown in table 6. table 6 complete rankings of the suppliers according to different fmcdm approaches supplier s1 s2 s3 s4 fuzzy topsis 0.492 (1) 0.343 (3) 0.339 (4) 0.382 (2) fuzzy waspas 0.802 (1) 0.452 (3) 0.446 (4) 0.471 (2) fuzzy aras 0.862 (1) 0.506 (3) 0.503 (4) 0.567 (2) the complete rankings are given according to calculated utility functions for each fuzzy approach. according to this table, the supplier order preference is given below: supplier s1> supplier s4> supplier s2> supplier s3, for all fmcdm methods. the best choice is supplier s1. this order is the result of a strict attitude of "lagerton" company’s management team that comparison of three fuzzy mcdm methods for solving the supplier selection problem 467 the most important criterion is "established cooperation" – c5. thus, they directly favor the supplier s1 with which they have established cooperation through previous purchases of similar components. it is more interesting to study the case in which there is no established cooperation with any supplier. in that new fmcdm problem criterion c5 would have weight ̃. for this hypothetical case calculation of fuzzy criteria weights is given in table 7. table 8 shows the new suppliers order, obtained by application of three fmcdm methods. it can be noticed that the order of alternative suppliers has not changed significantly except that supplier s1 and supplier s4 have switch places. the new order is the follows: supplier s4> supplier s1> supplier s2> supplier s3, for all fmcdm methods. table 7 criteria weights without c5 swara method criteria c1 1.000 1.000 1.000 1.000 1.000 1.000 0.350 0.421 0.561 c2 0.286 0.567 1.500 1.286 1.567 2.500 0.400 0.638 0.778 0.140 0.269 0.436 c3 0.286 0.467 0.667 1.286 1.467 1.667 0.240 0.435 0.605 0.084 0.183 0.339 c4 0.286 0.433 0.667 1.286 1.433 1.667 0.144 0.304 0.471 0.050 0.128 0.264 c5 0 0 0 table 8 complete rankings of the suppliers according to different fmcdm approaches – special case supplier s1 s2 s3 s4 fuzzy topsis 0.441 (2) 0.379 (3) 0.369 (4) 0.452 (1) fuzzy waspas 0.790 (2) 0.699 (3) 0.679 (4) 0.791 (1) fuzzy aras 0.803 (2) 0.715 (3) 0.706 (4) 0.853 (1) 5. conclusions the supplier selection problem is of vital importance for operation of every company because the solution of this problem can directly and substantially affect costs and quality. indeed, for many organizations effective supplier evaluation and purchasing process are critical success factors. this research has demonstrated the applicability of three fuzzy mcdm approaches (fuzzy swara + fuzzy topsis, fuzzy swara + fuzzy waspas and fuzzy swara + fuzzy aras) in the selection of suppliers of mechanical components. in the case of thk linear motion guide components procurement, all considered approaches clearly highlighted supplier s1 as the best. variations in other final ranking scores (supplier s2-4) are insignificant. in the second considered case (no established cooperation with any supplier) alternatives s1 and s4 have switched ranking places. supplier s4 would be suggested to serbian company "lagerton" as the best selection. the developed fuzzy mcdm model can be extended to accompany other relevant criteria which belong to three main criteria groups (quality, environmental and social) so as to achieve continuous supplier monitoring and evaluation. applied fuzzy methods are tools 468 g. petrović, j. mihajlović, ţ. ćojbašić, m. madić, d. marinković that use data in any form like numerical and linguistic, etc. collecting and converting data into the fuzzy model, using considered approaches, reduce subjectivity of each decision maker in group decisioning and also reduce chances of errors caused by units of parameters that can make problem in some mathematical calculations. to overcome those challenges, fuzzy mcdm methods are ideal solutions. the most important future endeavors are directed to the development of an expert and intelligent decision-making system that will be based on fuzzy principles. acknowledgements: the paper is a part of the research done within projects tr-35049 funded by ministry of education, science and technological development of the republic of serbia and "research and development of new generation machine systems in function of the technological development of serbia" funded by the faculty of mechanical 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in order to select the logistic centers location, transport, 25(4), pp. 423-432. 40. turskis, z., zavadskas, e.k., 2010, a novel method for multiple criteria analysis: grey additive ratio assessment (aras-g) method, informatica, 21(4), pp. 597-610. 12378 facta universitatis series: mechanical engineering vol. 22, no 1, 2024, pp. 153 163 https://doi.org/10.22190/fume231205011z © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper experimental study on creep loading of porous coal under different influencing factors jiyu zheng1, longkang wang2 1school of resource and safety engineering, henan university of engineering, zhengzhou, china 2china center for information industry development, beijing, china orcid ids: jiyu zheng https://orcid.org/0000-0001-7697-5400 longkang wang https://orcid.org/0000-0002-3902-5788 abstract. after the gas drainage borehole is formed in the coal seam, its stress deformation process is a creep process without the influence of mining. by arranging boreholes on coal samples and carrying out creep loading test of porous coal through rock mechanics testing machine, the creep process of boreholes in actual coal seams is simulated. the results show that the creep loading test results of drilled coal are quite different from those of conventional loading. creep loading produces large deformation due to rheological and aging characteristics. compared with non-porous coal, the yield of drilled coal occurs earlier and the yield platform is wider during creep loading. also, the larger the borehole diameter, the deeper the borehole depth, and the lower the peak stress. the up dip angle is easier to destroy than the down dip angle. compared with hard coal, soft coal produces greater deformation under smaller peak stress. the research results have guiding significance for drilling gas drainage in the soft coal seam. key words: influencing factors, porous coal, creep, destruction 1. introduction rheological properties and aging characteristics are the inherent mechanical properties of coal and rock. under the action of continuous stress, there is still a strong time effect, resulting in large creep deformation [1, 2]. li [3] has combined natural limestone and artificial samples, and conducted triaxial creep experiments on the combined samples to study the stress-strain curves of rocks under creep and stress relaxation. time has a significant impact on the stress-strain curves. creep and stress relaxation show different stress-strain curves. based on the acoustic emission (ae) theory and riemann liouville type fractional-order calculus operator theory, the creep loading test of salt rock was conducted, and the stress-strain dependence and damage of salt rock samples under received: december 05, 2023 / accepted march 04, 2024 corresponding author: longkang wang china center for information industry development, beijing, 100048, china e-mail: wlkmit@163.com https://orcid.org/0000-0001-7697-5400 https://orcid.org/0000-0002-3902-5788 154 j. zheng, l. wang uniaxial creep loading were studied. the creep curve and acoustic emission characteristic curve of salt rock were obtained. the theoretical curve was used to fit the test curve, and the predicted value could better describe the nonlinear accelerated creep stage [4]. in the process of gas drainage, the coal deformation, crack expansion and collapse around the borehole have a great impact on the gas drainage effect. there are methods based on elastic wave, acoustic emission, electromagnetic wave, microseismic and other methods used for monitoring fracture expansion. these methods are successful in monitoring large roadway deformation and rock burst, but they are not intuitive and cannot observe the specific form and process of fracture expansion [5-7]. however, for the gas drainage boreholes with small borehole diameter, there is often not enough time for prevention due to rapid deformation. a constitutive model of creep deformation of rock materials is proposed by shao et al. [8]. starting from the elastic-plastic model describing short-term behavior, the deformation varying with time is described according to the evolution of microstructure, resulting in the gradual degradation of elastic modulus and failure strength of materials. for soft coal, creep leads to the reduction of the compressive strength of the material, but the deformation is increasing, which is why it is difficult to maintain the drilling in soft coal seam for a long time [9-11]. the defective rock material itself contains initial cracks, which has a great impact on crack propagation [12]. in triaxial compression creep tests, strain history has a certain impact on the initial deformation of damaged coal samples. the initial damage has a great influence on the creep time and peak stress during the creep failure [13]. the soft coal body, also known as the failure coal body, has experienced the compression failure of tectonic movement, which also explains why the soft coal body has rapid deformation and low peak stress [14, 15]. unlike the accelerated creep stage of ordinary rocks, the accelerated creep stage of extremely soft coal rocks exhibits slowly accelerated deformation and instability, with a significant time process [16, 17]. in addition, research on mathematical models for creep deformation, damage, and crack evolution of materials such as rocks can grasp relevant laws [18-21].the creep process under cyclic loading can cause the accumulation of damage to coal and rock, ultimately leading to the failure of coal and rock. after the coal drilling ahead of the mining face undergoes the creep process, it is more likely to cause borehole collapse [22-23]. after the drilling of soft coal seam is completed, the continuous deformation occurs with the extension of time due to the influence of in-situ stress, which conforms to the creep characteristics. therefore, it is necessary to study the creep characteristics of borehole coal. in this study, the creep loading test of drilled coal was carried out by using rmt-301 rock mechanics test system, and the creep loading tests of hard coal and soft coal were carried out respectively. the effects of borehole diameter, borehole depth and borehole inclination on creep were investigated, and the test results were analyzed. 2. test scheme rmt-301 rock mechanics test system is composed of doorframe rigid host, dynamic loading system, control cabinet and computer. the device can realize uniaxial and triaxial compression and mechanical tests under different loading and unloading conditions. its dynamic loading system adopts servo control, with high control accuracy and stability. the creep-loading test of drilled coal is carried out on rmt-301 rock mechanics testing machine, which can control the loading stress to a certain level and keep it unchanged, with good stability. experimental study on creep loading of porous coal under different influencing factors 155 the creep loading test of borehole coal is divided into two categories: hard coal and soft coal. both hard coal and soft coal are compared with a test without a hole as the control. the creep loading test of borehole coal considers the influence of the borehole diameter, borehole depth and borehole inclination on deformation and failure. the test scheme is presented in table 1. the borehole diameter takes values 5 mm and 10 mm. the drilling depth is 25 mm and 50 mm. the dip angles of boreholes are +30° and -30°. the coal sample is of cylindrical shape with a diameter of 50 mm and length of 100 mm. creep refers to the property that the deformation of an object increases as time propagates under the condition of constant stress or small increase of stress. according to the method for determination of creep of coal and rock under unidirectional compression, the initial load stress is about 20% of the uniaxial compressive strength of the specimen. when the deformation difference does not exceed 0.001 in continuous 2h, the next stage of loading can be carried out. table 1 creep loading test scheme of borehole coal classification test name no. drilling depth (mm) borehole inclination (°) loading rate (mpa/s) borehole diameter (mm) hard coal no hole r1 0.5 borehole diameter r2 25 0 0.5 5 r3 25 0 0.5 10 soft coal drilling depth r4 25 0 0.1 5 r5 50 0 0.1 5 borehole inclination r6 25 +30 0.1 5 r7 25 -30 0.1 5 3. analysis of test results firstly, the creep loading test of the non-drilled specimens is carried out. compared with the conventional loading test, the uniaxial creep test of coal shows that the strength decreases during the creep loading, and a strong time effect appears under the action of continuous stress, resulting in large creep deformation. it can be seen from table 2 that with the increase of stress, the initial deformation increases rapidly, then decreases, and then increases rapidly in the later stage. table 2 creep graded loading process of undrilled coal classification time t (min) stress σ (mpa) deformation ε (mm) δε/δσ (mm/mpa) δε/δt (mm/min) 1 37 2.8 0.1560 0.055714 0.004216 2 70.4 5.5 0.2438 0.032519 0.002629 3 275.1 8.1 0.3182 0.028596 0.000363 4 435.7 11.0 0.4182 0.03450 0.000623 5 528.9 13.1 0.4822 0.030476 0.000687 6 601.6 16.2 0.6150 0.042823 0.001826 to facilitate the analysis, the rate of change in deformation of the specimen with respect to the change of stress, δε/δσ, is introduced. the creep rate, δε/δt, is the rate of change of 156 j. zheng, l. wang deformation with respect to time. according to the development of these two quantities, the whole process can be divided into three stages. in the first stage, the deformation per unit stress δε/δσ and the creep rate (δε/δt) show a decreasing trend with time. in the second stage, the deformation per unit stress δε/δσ and the creep rate remain nearly constant, and, hence, the trend of change is represented by nearly a straight line. finally, in the third stage, the deformation per unit stress δε/δσ and the creep rate increase with time. fig. 1 also shows the three stages. the creep loading peak stress of non-porous hard coal is 18.23 mpa and the corresponding deformation is 0.7162 mm, as shown in fig. 1. compared with the conventional loading without holes, the peak stress decreases, but the strain increases. from fig. 2, it can be seen that the time before yielding (plastic deformation, manifested as nonlinearity) accounts for a large proportion of the entire loading process, and an obvious yield platform is generated after yielding. the yield platform prolongs the creep loading strain process and increases the strain. under conventional loading, the failure of coal samples is relatively simple, with obvious tensile shear fracture surface, while under creep loading, the coal samples are broken to a large extent, mostly in debris shape, and the failure form is complex. 0 5 10 15 20 0 0.5 1 deformation/mm s tr es s/ m p a fig. 1 stressdeformation curve (r1) 0 1 2 3 0 100 200 300 400 500 600 700 time/min d ef o rm at io n /m m 0 5 10 15 20 s tr es s/ m p a deformation stress fig. 2 variation of stress and deformation with time (r1) 3.1. effect of borehole diameter on creep loading similar to the creep loading of non-drilled coal, there is no obvious peak stress in the creep process of drilled coal. when there is no hole, the initial yield stress is about 18.2 mpa and the corresponding deformation is about 0.66mm. when the platform drops experimental study on creep loading of porous coal under different influencing factors 157 rapidly, the stress is about 17.58 mpa and the corresponding deformation is 0.75mm. the width of the platform is 0.09 mm. when the borehole diameter is 5 mm, as shown in fig. 3 and fig. 4, the initial yield stress is about 16.31 mpa and the corresponding deformation is about 0.55mm. when the platform drops rapidly, the stress is about 13.98 mpa and the corresponding deformation is 0.74mm. the width of the platform is 0.19 mm. when the borehole diameter is 10 mm, as shown in fig. 5 and fig. 6, the initial yield stress is about 13.29 mpa and the corresponding deformation is about 0.53mm. when the platform drops rapidly, the stress is 11.81 mpa and the corresponding deformation is 0.79mm. the width of the platform is 0.26 mm. hence, the larger the borehole diameter, the lower the yield platform stress, and the larger the borehole diameter, the wider the yield platform. 0 5 10 15 20 0 0.5 1 deformation/mm s tr es s/ m p a fig. 3 stressdeformation curve (r2) 0 1 2 3 0 100 200 300 400 time/min d ef o rm at io n/ m m 0 5 10 15 20 s tr es s/ m p a deformation stess fig. 4 variation of stress and deformation with time (r2) 0 5 10 15 0 0.5 1 deformation/mm s tr es s/ m p a fig. 5 stress-deformation curve (r3) 158 j. zheng, l. wang 0 1 2 3 0 500 1000 time/min d ef o rm at io n /m m 0 5 10 15 s tr es s/ m p a deformation stress fig. 6 variation of stress and deformation with time (r3) in engineering practice, the larger the diameter of the borehole, the larger the exposed area of the coal body, which is more conducive to gas drainage. from the test results, although the creep peak stress decreases with the increase of the diameter of the borehole, the yield platform widens and the peak deformation increases, indicating that the larger the diameter of the borehole, the more deformation time is required. therefore, in the practice of gas drainage engineering, we should increase the borehole diameter as much as possible and expand the exposed area of coal in the borehole, which is conducive to extracting more gas. of course, we should also consider the difficulty and cost of borehole construction and comprehensively consider to select the appropriate borehole diameter. 3.2. effect of drilling depth on creep loading the influence of drilling depth on creep loading of soft coal is shown in figs. 7-10. when the drilling depth is 25 mm, the peak stress is 6.09 mpa and the corresponding deformation is 0.9183 mm. when the drilling depth is 50 mm, the peak stress is 6.09 mpa and the corresponding deformation is 1.2561 mm. the deeper the borehole is, the larger the failure surface of coal is, and the peak creep stress decreases. due to the presence of borehole, the yield is more pronounced and the yield platform is wider compared to then case without the borehole. from the point of view of peak stress and deformation, with the increase of drilling depth, the peak stress decreases, but the peak deformation increases, that is, smaller stress produces greater deformation, which means that the plasticity of coal body increases. in engineering practice, the deeper the drilling depth is, the larger the exposed area of the coal 0 2 4 6 8 0 1 2 deformation/mm s tr es s/ m p a fig. 7 stressdeformation curve (r4) experimental study on creep loading of porous coal under different influencing factors 159 around the drilling hole is, which is again more conducive to gas drainage. therefore, in the practice of gas drainage engineering, the drilling depth should be increased as much as possible to ensure that the drilling depth meets the drainage requirements and avoid blind areas, resulting in excessive local residual gas and large gas emission. 3.3. influence of borehole inclination on creep loading as shown in figs. 11-14, when the borehole inclination is +30°, the peak stress is 6.81 mpa and the peak deformation is 0.75 mm. when the borehole inclination is -30°, the peak stress is 7.32 mpa, while the peak deformation is 0.79 mm. compared with the horizontal drilling, the peak deformation is greatly reduced in both upand down-dip angles. compared with the 0 1 2 3 0 100 200 300 time/min d ef o rm at io n /m m 0 2 4 6 8 s tr es s/ m p a deformation stress fig. 8 variation of stress and deformation with time (r4) 0 2 4 6 8 0 1 2 3 deformation/mm s tr es s/ m p a fig. 9 stressdeformation curve (r5) 0 1 2 3 0 100 200 300 400 500 time/min d ef o rm at io n /m m 0 2 4 6 8 s tr es s/ m p a deformation stress fig. 10 variation of stress and deformation with time (r5) 160 j. zheng, l. wang down-dip angle, the peak stress and deformation of the up-dip angle borehole are smaller when coal is loaded, indicating that the up-dip angle borehole is easier to be damaged. regarding damage, when the drilling angle is +30°, the upper part of the coal sample is broken more severely compared to the lower part, as there are more cracks and those are characterized by a wider crack width. with the drilling angle of -30°, the situation is opposite. the borehole inclination has a great impact on the peak stress and deformation. compared with the horizontal borehole, the peak stress and deformation are greatly reduced. therefore, in the practice of gas drainage engineering, the borehole should be kept horizontal as far as possible. however, in the process of practical application, the borehole construction is generally consistent with the coal seam trend, so it is necessary to consider comprehensively the coal seam inclination and horizontal borehole. drilling along the dip angle of the coal seam makes the borehole longer and the gas extraction effect better. for example, the up dip angle is more likely to reduce the peak stress and deformation of the coal body. for coal seams with a certain inclination angle, drilling from one direction to form a borehole is referred to as upward inclination angle borehole, while drilling in the opposite direction is referred to downward inclination angle borehole. therefore, if conditions permit, the down dip angle drilling should be longer (deeper) and the up dip angle drilling should be shorter (shallower) as much as possible. 0 2 4 6 8 0 0.5 1 1.5 deformation/mm s tr es s/ m p a fig. 11 stressdeformation curve (r6) 0 1 2 3 0 100 200 time/min d ef o rm at io n /m m 0 2 4 6 8 s tr es s/ m p a deformation stress fig. 12 variation of stress and deformation with time (r6) experimental study on creep loading of porous coal under different influencing factors 161 0 2 4 6 8 10 0 1 2 deformation/mm s tr es s/ m p a fig. 13 stressdeformation curve (r7) 0 1 2 3 0 100 200 300 400 time/min d ef o rm at io n /m m 0 2 4 6 8 10 s tr es s/ m p a deformation stress fig. 14 variation of stress and deformation with time (r7) 4. conclusion creep is the process of continuous deformation of materials under constant stress. through the creep loading test of porous coal, the deformation and failure law of porous coal is further understood. the experiment studied the influence of borehole diameter, borehole depth, and borehole inclination angle on creep loading, analyzed the relationship between the stress and strain, and the process of their variation over time. the conclusions are as follows: ▪ the creep loading test results of drilled coal are quite different from those of conventional loading. creep loading produces large deformation due to rheological and aging characteristics. compared with non-porous coal, the yield of drilled coal occurs at lower stress and the yield platform is wider during the creep loading. ▪ the larger the borehole diameter is, the larger the exposed surface of the coal around the borehole is, and the peak stress decreases in the creep process. in terms of strength, it is easier to damage, but the deformation increases and the plasticity range increases with the increase of the borehole diameter. ▪ with the increase of drilling depth, the degree of integrity damage of coal body is greater, thus resulting in the decrease of the peak stress of coal body. this is similar to the effect of drilling diameter on creep loading of coal. although the peak stress decreases, the peak deformation increases. from the perspective of stress and strain, a deeper drilling depth does not imply a weaker coal resistance. 162 j. zheng, l. wang ▪ compared with horizontal boreholes, boreholes with a certain dip angle are easy to be damaged. from the point of view of the peak stress and deformation, the up dip angle is easier 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international journal of geomechanics, 23(4), pp. 565-576. 23. ding, z., feng, x., wang, e., sa, l., wang, d., zhang, q., hu, q., zhao, x., 2023, fracture response and damage evolution features of coal considering the effect of creep damage under dynamic loading, engineering failure analysis, 29(3), pp. 411-423. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 269 284 doi: 10.22190/fume170420006w © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper the influence of viscoelasticity on velocitydependent restitutions in the oblique impact of spheres udc 539.3 emanuel willert 1 , stephan kusche 1 , valentin l. popov 1,2,3 1 berlin university of technology, berlin, germany 2 national research tomsk state university, tomsk, russia 3 national research tomsk polytechnic university, tomsk, russia abstract. we analyse the oblique impact of linear-viscoelastic spheres by numerical models based on the method of dimensionality reduction and the boundary element method. thereby we assume quasi-stationarity, the validity of the half-space hypothesis, short impact times and amontons-coulomb friction with a constant coefficient for both static and kinetic friction. as under these assumptions both methods are equivalent, their results differ only within the margin of a numerical error. the solution of the impact problem written in proper dimensionless variables will only depend on the two parameters necessary to describe the elastic problem and a sufficient set of variables to describe the influence of viscoelastic material behaviour; in the case of a standard solid this corresponds to two additional variables. the full solution of the impact problem is finally determined by comprehensive parameter studies and partly approximated by simple analytic expressions. key words: oblique impacts, friction, viscoelasticity, standard solid model, method of dimensionality reduction, boundary element method 1. introduction collisions of macroscopic particles determine the dynamics of granular gases. as long as the particle density in the granular gas is small enough and hence the impact durations are small compared to the mean free time between two collisions, these will in general be binary. in many cases the difference of the particle velocities before and after the impact received april 20, 2017 / accepted june 21, 2017 corresponding author: willert, emanuel affiliation: berlin university of technology, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: e.willert@tu-berlin.de 270 e. willert, s. kusche, v. l. popov can be described by two coefficients of restitution, one for each the normal and tangential direction of the impact. due to friction, adhesion, viscoelasticity, plasticity or other effects those coefficients of restitution will in general exhibit strong and non-trivial dependencies not only of the geometric or material parameters but of the impact velocities themselves. among the vast literature about granular media only few publication lines account for this velocity-dependence, which is mostly because of two reasons: on the one hand, the various analytical methods of statistical physics applied to deal with granular media are severely complicated by the fact that the restitution coefficients are actually velocity-dependent. on the other hand, the rigorous solution of the single contact-impact problem even in the simplest case of spherical colliding particles is a rather non-trivial task. lun and savage [1] and walton and braun [2] were the first to study the effects of the described velocity-dependence on the granular dynamics using the granular-flow kinetic theory of lun, savage, jeffrey and chepurnity. however, lacking rigorous solutions, they only used an ad-hoc model of a restitution coefficient in normal direction exponentially decreasing with the impact velocity, which can be realistic only in few cases. besides, they did not account for inter-particle friction during the collisions and could hence achieve only rough agreement with their experimental data. only ten years later a research group around brilliantov and pöschel started a series of publications to tackle this problem again. brilliantov et al. [3] gave models for the collisions of spheres accounting for viscoelasticity and friction. however, their material model is equivalent to a kelvinvoigt body, which is only realistic if the time scale of interest is large compared to the relaxation time of the elastomer. as the impact times are short, this might be problematic. moreover, their tribological friction model of broken welds and asperities leads to a stepwise linear dependence of the tangential force on the tangential displacement between the contacting bodies. for spherical profiles this cannot be true due to the profile shape. these collision models have been implemented in granular gas simulations by schwager and pöschel [4], brilliantov and pöschel [5] and dubey et al. [6]. the history of rigorous impact solutions started with hertz [7], who solved the frictionless and non-adhesive normal contact problem of two parabolic surfaces and the associated quasistatic impact problem. hunter [8] studied the influence of the quasi-stationarity and found that the proportion of kinetic energy lost during the impact due to elastic wave propagation is negligible, if the impact velocities are small compared to the speed of sound in the elastic medium. cattaneo [9] and mindlin [10] solved the tangential contact problem of two elastically similar spheres in the case of a constant normal force and an increasing tangential force. the circular contact area will consist of an inner circular stick area and an annular region of local slip. the tangential traction distribution in the contact is a superposition of two hertzian distributions. their work has been extended by mindlin and deresiewicz [11] for various different and by jäger [12] for arbitrary loading protocols. based on the results of mindlin and deresiewicz, maw et al. [13] and barber [14] studied the oblique impact of elastic spheres without adhesion; they found out that the problem written in proper dimensionless variables only depends on two parameters, one describing the elastic and the other (containing a generalized angle of incidence and hence the impact velocities) the frictional properties. moreover, the authors carried out experiments to validate their calculations. the oblique impact problem of elastic spheres with and without adhesion was also studied by thornton and yin [15]. a nice overview of elastic impact problems and several analytical solutions including torsional loading can be found in the paper by jäger [16]. the influence of viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres 271 in a series of publications – see for example [17, 18] and the summarizing book [19] – popov and his co-workers have shown that the generalized hertz-mindlin problem for any convex axisymmetric indenter and arbitrary loading histories can be exactly mapped onto a contact between a properly chosen plain profile and a one-dimensional foundation of independent linear springs in such a way that the solution of the obtained one-dimensional model will exactly coincide with the one of the original three-dimensional problem. due to the enormous simplification and effort reduction of analytical or numerical calculations achieved by this so-called method of dimensionality reduction (mdr) lyashenko and popov [20] were able to give a comprehensive solution for the problem studied earlier by maw and his co-workers in the no-slip regime, i.e. an infinite coefficient of friction. those results have later been generalized by willert and popov [21] for the partial slip regime, i.e. a finite friction coefficient. the viscoelastic contact problem was first addressed by lee and radok [22-24]. from the close relationship between the fundamental equations of elasticity and viscosity the authors deduced a method of functional equations to obtain the solution of a viscoelastic problem if the solution of the associated elastic problem is known and the contact radius is a monotonically increasing function in time. this has been generalized to the case of any number of maxima and minima of the contact radius by graham [25], [26] and ting [27, 28]. an equivalent but somewhat easier formulation of ting’s solution was given by greenwood [29]. however, with every maximum or minimum of the contact radius the analytic calculations get more and more cumbersome. the hertz impact problem for viscoelastic media was treated by pao [30] and hunter [31]. they used arbitrary viscoelastic rheologies to formulate the problem but gave only few concrete solutions. argatov [32] found analytical solutions for the respective flat punch problem in the case of kelvin-voigt-, maxwellor standard solid model. the viscoelastic contact problem in the case of convex axisymmetric indenters and arbitrary loading protocols can also be exactly mapped within the framework of the mdr, which was proven by kürschner and filippov [33] and argatov and popov [34]. hence, the aim of the present paper is to give a comprehensive solution of the viscoelastic oblique impact of spheres with and without slip based on the mdr. very recently kusche [35, 36] presented the no-slip solution of this impact problem using the boundary element method (bem). however, the bem-calculations are numerically much more costly compared with the mdr. as the parameter space for the more general case with slip is larger by one dimension, the comprehensive solution based on bem will be numerically very expensive. nevertheless, the bem-algorithm to solve the impact problem with slip has been implemented and can serve as a validation for the faster mdr-based model. we will use a standard solid for modelling viscoelastic properties because it exhibits all characteristics of general elastomers. as a limiting case the kelvin-voigt solid is also studied at some point. finally, we will focus on the velocity-dependence of the coefficients of restitution as this is the main point of interest for the implementation of the obtained solutions into simulation algorithms for granular media. the paper is organized as follows: in section 2 we will give a formulation of the studied problem. section 3 is devoted to the description of the numerical model based on the mdr, the results of which are given in section 5. section 4 will present a bem-based algorithm to solve the impact problem, which was used to validate the mdr model described before. section 6 will give conclusions. 272 e. willert, s. kusche, v. l. popov 2. problem formulation the present paper is concerned with the oblique impact of two linear-viscoelastic spheres of similar materials. this problem is equivalent to the one of a rigid sphere impacting on a viscoelastic half-space, which is why we will restrict ourselves to the latter one. during contact the frictional interaction between the two surfaces shall be assumed to obey the amontons-coulomb’s law with the static and the kinetic coefficients of friction being constant and equal to each other: μs = μf ≡ μ. the sphere shall have initial velocities vx0 and vz0, z pointing into the half-space, and initial angular velocity ω0. the mass, radius and moment of inertia of the sphere are m, r and j s , respectively. the point on the sphere which first comes into contact shall be denoted as k. the half-space shall possess a constant poisson number ν and a creep function giving the response in shear. actually a viscoelastic material may possess a second creep function for the response to hydrostatic stress, but this shall be neglected. as most elastomers can be considered incompressible (this will also fulfil the condition of elastic similarity) our assumption does not pose a considerable loss of generality. in this case we can introduce time-dependent shear modulus g(t). for the standard solid model g reads: 2 1 2( ) exp . g t g t g g          (1) the kelvin-voigt model can be recovered from this expression via the limit 2 1( ) lim ( ) ( ),kv g g t g t g t     (2) with the dirac δ-distribution. a scheme of the impact with notations is shown in fig. 1. we will make further following assumptions: quasi-stationarity: the impact velocities shall be much smaller than the speed of sound in the viscoelastic material. we therefore neglect all inertia effects like wave propagation. half-space hypothesis: the surface gradients shall be small. for an axisymmetric contact with parabolic indenter shapes in the vicinity of the contact point, this can be written as max max ,d a r (3) with the maximum values of indentation depth d and contact radius a. very short impact: the displacement of the contact point due to the change of position and the rotation of the sphere shall be small compared to the contact radius. this ensures that the contact configuration stays axisymmetric and the contact problem can be treated like a tangential one. rolling will then be accounted for only kinematically. the displacement in vertical direction is of the order of magnitude of the maximum indentation depth. the displacement in tangential direction is of the order of magnitude 0 0 , max 0 .x x k z v r u d v    (4) hence, this assumption will be covered by the half space hypothesis if the ratio of tangential and vertical initial velocity of the contact point is of the order of 1 or smaller. the influence of viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres 273 fig. 1 scheme of the analysed impact problem – a rigid sphere is impacting on a viscoelastic half-space 3. numerical model based on the mdr under the assumptions made, the motion of point k fully determines the motion of the sphere. the normal and tangential displacements of this point shall be uk,z and uk,x. the equations of motion for those displacements are elementary given by 2 k, k, 1 , , x x s z z f mr u m j f u m         (5) where fx and fz are the contact forces while the dots denote the time derivative. to determine these forces and thereby solve the axisymmetric problem described above within the framework of the mdr, two preliminary steps are necessary. first an equivalent plain profile g(x) has to be obtained from axisymmetric indenter profile f(r) via the abel-like integral transform 2 2 0 d d ( ) . d x f r g x x r x r    (6) a spherical indenter in the vicinity of the contact can be described by the parabolic profile 2 ( ) 2 r f r r  (7) and the equivalent profile accordingly is given by the expression 2 ( ) . x g x r  (8) 274 e. willert, s. kusche, v. l. popov fig. 2 single element to model a standard solid fig. 3 single element to model a kelvin-voigt solid as the second step the viscoelastic properties of the half-space must be replaced by a one-dimensional foundation of independent, linear-viscoelastic elements. in case of a linear standard solid with the time-dependent shear modulus given in eq. (1) those elements consist of a spring in series with a dashpot, the pair in parallel with a second spring (see fig. 2). in case of a kelvin-voigt model (see fig. 3) the spring in series with the dashpot is rigid. the elements are at a distance δx of each other. this value is arbitrary if small enough. let us first consider the standard solid and write down the necessary relations of the model and the numerical algorithm. all equations for the kelvin-voigt model can be derived afterwards by the limiting process. the reaction force for a single element at position xi = i δx, with outer and inner displacement vectors iu and iu has the components , 1 , , , , 1 , , , 4 ( ) , 2 2 ( ) . 1 i x i x i x i x i z i z i z i z x f g u u u x f g u u u                 (9) the inner point must fulfil the equilibrium conditions 2 , , , 2 ,z ,z ,z ( ) 0, ( ) 0. i x i x i x i i i g u u u g u u u         (10) for the time integration we will use the least order explicit euler integration scheme with constant time step δt. the current time step number shall be denoted by an upper index j. in the beginning all displacements are set to zero. then, in each time step, first the normal contact problem must be solved. for the elements in contact the normal displacement is enforced by the motion of k, 1 , , k, , for contact.j j j i z i z zu u u t   (11) the elements not in contact are free of forces, i.e. the left side of eqs. (9) is zero, and one obtains the influence of viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres 275 1 1 , , , 1 ( ), for nocontact, ( ) j j j i z i z i zu u u g t         (12) where we introduce relaxation time 2/ g  . an element gets into contact if , j i zu  k, ( )j z iu g x and leaves contact if , 0j i zf  . to solve the tangential contact problem the tangential displacements must be calculated. the elements outside the contact area progress according to 1 1 , , , 1 ( ), for nocontact. ( ) j j j i x i x i xu u u g t         (13) for the elements in contact one has to distinguish between sticking and slipping elements. for all the sticking elements, the displacement is enforced by the movement of k, 1 , , k, , forstickingcontact.j j j i x i x xu u u t   (14) an element in contact is able to stick if the resulting tangential force does not exceed the maximum value given by the amontons-coulomb law, i.e. if , ,z , for stickingcontact.j j i x if f  (15) any element violating this condition will slip. in this case the tangential force is known to be 1 , ,z ,sgn( ), forslippingcontact.j j j i x i i xf f f   (16) after the total contact forces are calculated by summation over all elements, , , ,j j x i x i j j z i z i f f f f       (17) the equations of motion (5) can be solved in each time step. note that it is impossible that contact is re-established by the viscoelastic creep. for the kelvin-voigt model only τ and hence the inner displacements must be set to zero in the equations above. the algorithm was implemented in matlab™. only time steps j and j-1 have to be stored. that is why this algorithm requires only little memory space. also all operations are elementary, which makes the algorithm very fast (this is also why we are able to use a least order explicit integration scheme without stability problems) and enables us to do comprehensive parameter studies on an ordinary desktop pc (the calculation of a single impact took around one or two seconds on a machine with an intel i5 processor). 4. numerical investigation using bem the results acquired with the mdr have been validated using the boundary element method (bem). the bem-solution of the described problem is numerically exact under the assumptions stated before: the half-space approximation, quasi-static conditions and elastic similarity between the contacting surfaces. since the bem does not rely on axissymmetry, this assumption is only made to have results comparable with the mdr. 276 e. willert, s. kusche, v. l. popov the application of the bem consists of two steps. firstly the problem of calculating the deflection field from a given pressure distribution and vice versa must be solved. this can be done by utilizing the fundamental solution for a point load acting on a viscoelastic half-space [37-39]. the material is assumed to be incompressible and components fx, fy, fz of the point load are applied at time zero and are kept constant. the deflection of the surface can then be written as 2 2 2 3 3 , 2 1 ( 4 4 , . ), z x x x y z i i j t r x x xy u u r r rr r f y                        (18) in difference to the mdr, the time-dependent creep function for shear j(t) has been used. it is clear and known that j(t) and g(t) are not independent of each other. the creep function can be written by using the constants introduced in equation (1) in the following form: 2 1 2 1 2 1 1 2 1 ( ) 1 1 exp . ( ) g g g j t t g g g g g                  (19) since the geometric dependences in the viscoelastic and elastic cases are the same, the developed algorithm can be used with only small modifications. in elastic contact mechanics it is a standard procedure to integrate the fundamental solution over a rectangle, assuming constant pressure [40]. this analytic solution is used to find the deflection field for an arbitrary but piecewise constant pressure distribution [41]. this task can be performed very fast and efficiently by using convolution techniques on a parallel computing architecture [4244]. the corresponding inverse problem, namely finding the pressure distribution to a given deflection field can be tackled by using the biconjugate gradient stabilized method [45]. the above described methods have been applied to the viscoelastic problem. since the pressure distribution will change in time, a discretisation is necessary. if, for each time step, the pressure distribution is assumed to be constant, the overall solution in the deflection field can be obtained by adding two solutions in each time step: one to remove the prior load and one to add the current load. based on the fact that the arising sum grows linearly in time, it is crucial to reduce the numerical effort. this can be achieved by applying the special form of the creep function (19) and by observing the following time step. then an iterative algorithm can be developed: 1 1 1 , 1 1 0 00 0 0 , 1 1 1 0 0 0 1 1 1 ( ) ( )exp exp ( ) z z j j z z j j j j j i z j n i i i i i i i a b z z z j j j j j j j a b j j t z t tj j f t t f f f j j j j j f a f b f f j j j f a u t t u                                                    0 exp z zt j j d h j b f j         (20) herein uz,n is the normal deflection of the surface, j∞ = j(t = ∞), j0 = j(t = 0), and fj is the deflection due to a pressure distribution pj – each at the time tj. in the last line of eq. (20) the influence of viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres 277 it can be seen that an additional deformation d z has to been taken into account to include viscoelastic behaviour (in the elastic case d z is equal to zero). the unknown term in the last line of eq. (20) is fj+1, which means that the pressure distribution pj+1 is unknown. this can be calculated with the elastic algorithms mentioned before. it should be noted that this algorithm can handle only materials with a finite modulus of instant deformation, which excludes the kelvin-voigt solid. the tangential contact can be solved very similarly to the normal contact so that the same scheme can be used [46]. only the calculation of the deflection in tangential direction ux, caused by shear stress has to be adopted. if a partial slip is involved, the calculation is modified in the following way: starting with a complete stick area, the deflection is given by the increment of displacement in one time step. if this leads to shear stress that is larger than the value allowed by the coulomb’s law, this part of the contact area will slip. in the slip areas the tangential stress is set to |τ| = μp. then the stress in the remaining stick area is calculated again, under the consideration of the deflection caused by the shear stress in the slip area. this is done until the stick area does not change anymore. in all performed simulations, the deformation perpendicular to the plane of the motion, uy , is neglected. it turns out that this assumption, in the case of parabolic bodies, causes a negligible error [47]. at this point, the contact problem itself is solved. for the integration in time both an explicit euler scheme and the velocity verlet algorithm have been used. in comparison, they show no difference in the global error of the velocities at the end of the simulation and in the contact time itself. for an estimation of the step size δt the mdr solution has been used. for the geometric discretization a matrix of 256256 points has been chosen. the comparison with a finer discretization shows only a slight error reduction. for implementation it has to be considered that the total deflection in normal direction within the contact area is known at every time step since the indentation depth of the sphere is known. contrariwise in tangential direction: the points coming into contact have a pre-deformation through coupling to the points within the contact area from a previous time step. this can be handled by adding only the current increment of tangential movement at the boundary of the sphere in each time step. the systematic investigation of the problem has been done with the mdr. the processing time for the bem is much higher compared to the mdr. therefore, only a few hundred parameter sets spreading over the full range covered by the investigation done with the mdr have been calculated with the bem. it turns out that the relative differences in the coefficients of restitution have always been smaller than 0.5%. therefore, it is reasonable that the mdr can be applied. 5. results of the numerical model: the restitution coefficients as a solution we are interested in the coefficients of restitution in normal and tangential direction , , 0 , , 0 0 ( ) , ( 0) ( ) . ( 0) k z e z z k z z k x e x x k x x u t t v e u t v u t t v r e u t v r                 (21) 278 e. willert, s. kusche, v. l. popov maw et al. [13] have shown that in an ideally elastic case (the coefficient in normal direction being obviously unity) the coefficient in tangential direction only depends on the two dimensionless parameters 2 0 0 0 1 2 2 1 , . 2 2 x s z v rmr vj                  (22) in the case of a sphere impacting on a viscoelastic half space modelled as a linear standard solid, two more dimensionless parameters are of interest, describing the viscoelastic material properties, namely el,1 1 1 1 2 , , (1 ) a g g m g        (23) with the maximum contact radius for the impact with an elastic half space, 1/ 5 2 2 0 el, 15 (1 ) , 1,2 32 z i i mv r a i g        (24) of course, any combinations of those two additional parameters would also be possible to choose as governing variables. for example, in the previous publication on the no-slip impact kusche [35] used the parameters el,1 el,2 3/ 5 1 2 1 1 2 , (1 ) (1 ) a a g m g m              (25) to capture the influence of the material behaviour. however, as we are interested mainly in the velocity-dependence of the coefficients of restitution, it seems convenient to select δ1 and γ, because the latter one is velocity-independent and therefore the velocitydependence due to viscoelasticity can be fully covered by parameter δ1. moreover, the kelvin-voigt model can be recovered as the limiting case γ = 0. also limit γ → ∞ corresponds to the elastic result. to reduce the number of governing parameters, we restrict ourselves mostly to χ = 7/6, which, amongst other cases, corresponds to the case of incompressible, homogenous spheres. to prove that actually 1 1( , ) and ( , , , )z z x xe e e e       (26) we made comprehensive numerical studies, the results of which are shown in the upcoming figures. thereby we first focus on the limiting case of a kelvin-voigt solid and afterwards look at the more general standard solid. in fig. 4 the coefficient of restitution in normal direction is shown for a kelvin-voigt solid as a function of δ1. all free input parameters for the simulations, i.e. velocities, measures of inertia and so on, have been generated randomly. nevertheless, the points create continuous curves and hence our hypothesis is proven for the normal direction. it is easy to interpret the results, as the coefficient of normal restitution shows the often-used quasi-exponentially decreasing behaviour. this, however, only remains true for this material model of a kelvin-voigt solid, which corresponds to an infinitely fast relaxation within the influence of viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres 279 the elastomer. it was already pointed out that this is problematic as the impact times are considered to be small and the relaxation time has to be accounted for in some way. we will see the effects later in the results for the standard solid. fig. 4 coefficient of restitution in normal direction for the impact on a kelvinvoigt solid as a function of δ1 fig. 5 coefficient of restitution in tangential direction as a function of δ1 and ψ with χ = 7/6. online version in colour fig. 5 gives the tangential coefficient of restitution ex as a function of δ1 and ψ for the impact on a kelvin-voigt half-space. the value of χ was fixed at 7/6, all other input parameters for the impact problem have been generated randomly and yet the solutions create continuous, smooth curves. the tangential restitution has a global maximum for an impact without viscosity around ψ = 2. on the right side of the contour plot the behaviour gets quite simple and can be explained the following way: for any material model configurations are possible for which the contact will completely slip during the whole impact. in this case the total tangential force is known due to the coulomb’s law and hence the tangential restitution coefficient for full slip (and any material model) is given by the relation 2 (1 ) 1.fs x ze e      (27) let us now look into the results for the standard solid. we restrict ourselves again to the case χ = 7/6 to spare the generally least important parameter. fig. 6 gives the results for the normal restitution coefficient as a function of δ1. several logarithmically-equally-distributed values for γ have been chosen and all other input parameters for the impact problem, as always, have been generated randomly. nevertheless, the solutions create continuous curves and it is easy to observe the influence of γ on the velocity-dependent restitution: as said before γ → ∞ corresponds to the trivial elastic case and γ = 0 to the monotonically decreasing kelvin-voigt solution. for intermediate values of γ the coefficient of restitution has a global minimum. after that it increases again with increasing δ1, i.e. increasing normal inbound velocities. this distinguishes the general standard solid from its limiting case with infinitely fast relaxation and has, for example, a very interesting consequence for a (driven) granular gas of viscoelastic particles: as on the increasing part of the restitution curve, the coefficient 280 e. willert, s. kusche, v. l. popov of restitution is larger for larger inbound velocities, a region of locally higher internal energy of the granular gas, i.e. higher velocities of the particles, might dissipate less energy than regions of lower energy, which might result in unstable states of the granular gas. fig. 6 coefficient of restitution in normal direction as a function of δ1 (logarithmic) and different values of γ for the impact with a standard solid fig. 7 coefficient of restitution in tangential direction without slip (ψ = 0) as a function of δ1 (logarithmic) and different values of γ for the impact with a standard solid; χ = 7/6 fig. 8 coefficient of restitution in tangential direction as a function of δ1 (logarithmic) and ψ for the impact with a standard solid; χ = 7/6 and γ = 0.0825 fig. 9 coefficient of restitution in tangential direction as a function of δ1 (logarithmic) and ψ for the impact with a standard solid; χ = 7/6 and γ = 1 fig. 7 presents the results for the tangential restitution in the case of no slip, which have been reported by kusche [35] with a slightly different set of governing dimensionless parameters. in fig. 8 and 9 the results are shown for the behaviour with slip. for increasing values of γ a bulb with ex ≈ 0.5 around ψ ≈ 2 is stretching to the left, i.e. the area with less viscosity. the other areas are less affected by the material properties. finally, we come back to the full slip solution and the different regimes for parameter ψ. the influence of viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres 281 in the elastic case maw et al. [13] distinguish three different regimes: for ψ < 1 the impact will start in a completely sticking contact and remain like this during the whole compression phase; for ψ > 4χ – 1 the contact will fully slip during the whole impact; the intermediate values correspond to a mixed regime. now, in the viscoelastic case, the time derivatives of the contact forces in the mdr-model in the very first moment of contact are given by , ,x 2 ( 0) ( 0) ( 0) 1 4 ( 0) ( 0) ( 0) 2 z k z x k x f t g t u t x f t g t u t               (28) hence, for a finite instantaneous stiffness (this excludes the kelvin-voigt body), the impact will begin with sticking contact, if ( 0) ( 0) 1.x zf t f t      (29) in case of the kelvin-voigt body the contact forces in the first moment of contact are nonzero and the no-slip condition will be ( 0) ( 0) 1.x zf t f t      (30) hence, this lower transition value for ψ is unaffected by viscoelasticity. for any standard solid characterised by the two parameters δ1 and γ – and probably any material behaviour – there also exists a value ψc, for which the contact will completely slip during the whole impact if ψ > ψc. for complete slip the tangential coefficient of restitution is given by eq. (27). fig. 10 critical value ψc, for which the contact will completely slip during the whole impact if ψ > ψc for the impact with a standard solid fig. 11 relative error between the numerical result for ψc and the analytic approximation (31) in fig. 10 the value of c is shown for different materials. obviously this transition value strongly correlates with the normal restitution coefficient. the global maximum is elastic case 282 e. willert, s. kusche, v. l. popov ψc = 11/3, and a very good approximation (with a relative error always smaller than 0.2%, see fig. 11) is given by the expression 2 (1 ).c z ze e     (31) 6. conclusions based on the numerical models we investigated the oblique impact of linear-viscoelastic spheres under the assumptions of quasi-stationarity, the validity of the half-space hypothesis, amontons-coulomb friction and short impact times. numerical models based on both the method of dimensionality reduction (mdr) and the boundary element method (bem) have been implemented. as expected both methods in their results only differ within the margin of a numerical error. due to the enormous reduction of mathematical and computational effort achieved by the mdr we were able to perform comprehensive parameter studies for the examined impact problem. it is found that the problem solution, i.e. the coefficients of normal and tangential restitution, written in proper dimensionless variables will depend on exactly four different values, at least two of which contain explicit dependencies on the inbound velocities. by accounting for the finite relaxation time within the elastomer it is possible to increase the normal restitution coefficient with increasing inbound velocities. this is in contrast with most viscoelastic collision models used in the literature about granular media and may have interesting applications in granular chains or gases. as in the elastic case, three different regimes are possible depending on the inbound velocities: the 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universitatis series: mechanical engineering https://doi.org/10.22190/fume231018001s © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper mechanical response of layered structures with internal layers from metamaterials exposed to dynamic loadings vladimir a. skripnyak, maxim chirkov, evgeniya skripnyak, vladimir v. skripnyak national research tomsk state university, russia abstract. the article presents the results of the analysis of mechanical behavior of mechanical layered structures with metamaterial interlayers under dynamic loading. these structures can be used in lightweight structures for damping dynamic loads in transport and aerospace engineering. structural elements with layers of the mechanical metamaterials have a low specific mass density and high specific strength characteristics. these multilayer structures have a high specific ability to absorb and dissipate the energy of external dynamic loads too. the results of numerical simulation of the response of multilayer structures to dynamic impacts obtained in this work indicate high specific energy absorption and dissipative properties, which make it possible to weaken the pulse amplitude after passing through the layered system and attenuate of cyclic impacts amplitudes. the results obtained indicate the possibility of creating effective mechanical damping structures of the type under discussion. key words: layered structures, metamaterial’s interlayers, auxetic metamaterials, pentamode metamaterials, mechanical response, dynamic loading 1. introduction the nonlinear dynamics of constructions containing elements made of metamaterials remains an urgent problem requiring research. the mechanical behavior of such structures under dynamic loading is determined by a combination of nonlinear effects caused by geometric nonlinearities due to large deformation, physical nonlinearities in the constitutive relationships of the component materials, and nonlinearities caused by possible mechanical instabilities of the deformable structural elements. development of construction elements for damping vibration and pulse impacts on the aerospace technics is one of important modern problem [1]. energy-absorbing metamaterials now used in received: october 18, 2023 / accepted january 03, 2024 corresponding author: vladimir a. skripnyak national research tomsk state university, 36, lenin avenue, tomsk, russia, 634050 e-mail: skrp2006@yandex.ru 2 v.a. skripnyak, m. chirkov, e. skripnyak, v.v. skripnyak engineering applications demanding the mitigation of mechanical pulse impact and attenuation of vibration amplitudes [2, 3, 4, 5]. metamaterials and sandwich structures had been studied as dynamic absorbers for vibration reduction in the last decades due to the advances in new manufacturing technologies. yang et al. [6] proposed a structure composed of tuned dynamic absorbers for vibration reduction without the use of viscous vibration damping materials in the broadband frequency range chen et al. [7] analyzed the influence of damping on the wave propagation of the elastic metamaterial. li et al. [8] proposed a novel meta-lattice sandwich structure, and analyzed the speed and stresses in the lower face-sheet under impact and blast load through numerical simulation. metamaterials currently are of interest for a wide variety of applications including damping systems. xue et al. [9] indicated that welding is a unique construction technique for auxetic metamaterials. therefore the perspective investigations of the metamaterials manufacturing by welding are important for reducing of the metamaterials production costs. in this regard, the layered structures and metamaterials made of ti-5al-2.5 sn titanium alloy, which has good weldability and high ductility in a wide range of strain rates were studied in this research. guo et al. [10] showed that mesoscopic deformation of the auxetic structure of a metamaterial goes through three typical stages: convergence, flattening and collapse, ending with the compaction stage. the geometry of the frame structure also has a significant influence on the auxetic characteristics and the resulting stresses [11]. therefore, the adequate describing of a large plastic strain, damage and fracture of material in the elements of layered structures is important for prediction obtaining of the mechanical behavior of these structures under dynamic loads. a number of modifications of models and methods for were proposed in [12, 13, 14] for describing of mechanical behavior alloys patterns under large deformations. to describe the mechanical behavior of materials in 3d printed structures, hybrid machine learning technology can be used [15]. the purpose of this work is to estimate the absorbed and dissipative energy of layered structures with an interlayer of three types of metamaterials subjected to dynamic loading. the model of inelastic deformation and the criterion of plastic fracture of ti-5l-2.5 sn were tested to describe the deformation of three-layer structures with a metamaterial interlayer in a wide range of strain rates equivalent to plastic deformation and stress triaxiality. estimates of energy absorption and dissipation of three-layer titanium structures with metamaterial interlayers under pulse and high-frequency harmonic loads have been obtained. 2. model and computational details 2.1 model of multilayered structures with metamaterials elements the mechanical behavior of a three-layer structure under dynamic loading was studied by numerical simulation of pulse action on a three-layer structure with an internal layer of metamaterial. the loading diagram of the three-layer structure is shown in fig. 1. the outer layers of structural materials in three-layer structures serve to distribute the effective load on the system and form a distributed load on the support. mechanical response of layered structures with internal layers from metamaterials to dynamic... 3 the mechanical responses of three-layer structures made of alpha titanium alloys with an intermediate layer of a pentamode metamaterial and model auxetic metamaterials, the dissipative properties of which were studied in [4, 16], are considered. three-layer structures can be manufactured using selective laser melting technology or frame elements welding [9, 18]. three types of interlayers are considered: pentamode metamaterials and two type auxetic metamaterials, which have significant differences in mechanical properties, including elastic properties. the metamaterials under consideration at the macroscopic level can be considered as isotropic media, for which, at small deformations, the elastic moduli are related by the relation: / ( 1) / [3(0.5 )]b g    (1) where b is the bulk modulus, g is the shear modulus, ν is the poisson’s ratio. for auxetic metamaterials, the poisson's ratio is 0, therefore three-layer systems, with a middle layer of auxetic metamaterials, for which  –1, b/g0, as a result of which the velocities of bulk elastic waves cb = (b/ρ)1/2 are close to zero. therefore, auxetic metamaterials have the ability to be an insulator of external vibration or impulse influences. for pentamode metamaterials, poisson's ratio (0.5), g0, the value of the bulk modulus b depends on the geometric dimensions of the frame elements of the pentamode material. (a) (b) (c) (d) (e) fig. 1 loading scheme of three layer structure (a), three-layer structure with an interlayer of pentamode metamaterial (b), geometry parameters of pentamode frame structural elements (c), structure with an interlayer of auxetic metamaterial (d), and with 3d re-entrant auxetic metamaterial (e) 4 v.a. skripnyak, m. chirkov, e. skripnyak, v.v. skripnyak for a pentamode material with a face-centered cubic frame lattice with the parameter a, and the diameter d in the touching region of frame elements the bulk modulus is determined as b~d/a, and the shear modulus is g ~ (d/a)3 [11, 19]. hedayati et al. [20] developed selective laser melting technology for manufactory of the pentamode metamaterials based on a titanium alloy. the results of [19] show that increasing the smaller diameter 𝑑 and the ratio of larger to smaller diameter can increase the elastic modulus, while the poisson's ratio remains almost independent of these variables and is constant around 0.5 in the considered range. the possibility of additive manufacturing of pentamode materials with a normalized mass density ρ/ρс from ~0.24 to ~4.24 (ρс is the mass density of the material of the frame elements) from the ti-6al-4v alloy powders is shown in [20, 21]. for a pentamode titanium alloy metamaterial, the relationship between effective compressive modulus and effective shear modulus has been shown to be influenced by the presence of stiffening plates, compared to existing theoretical predictions of the ratio of unconfined pentamode lattices [20]. the mechanical behavior of three-layer structure from ti-5al-2.5sn (grade 6) titanium alloys under dynamic loads was studied. the upper and base plates had dimensions in mm: 40×40×2.the thickness of metamaterial interlayer is 40 mm. the elementary cell of the pentamode metamaterial interlayer had dimensions of 10 mm × 4.8 mm × 5 mm, and its frame element dimensions were equal d = 0.4 mm, d = 0.11 mm, h = 1.6 mm. the cubic unit cell of the auxetic metamaterial under consideration (fig. 1(d)) has a frame element width dw = 6 mm, a thickness da = 0.6, and a distance between the upper and lower arcs of the frame elements a = 5.95 mm. the top and base plates had the same dimensions: 275×275×2 mm. the thickness of the metamaterial layer is 215 mm. three-layer structure with 3d re-entrant auxetic metamaterial interlayer (fig. 1(e)) had thickness of upper and base plates 2 mm, and dimension 140 mm × 20 mm. the thickness of frame elements is equal to 1.0 mm. the thickness of the metamaterial interlayer is 87 mm. 2.2 model of the mechanical behavior of a layered alpha titanium structure structural elements are deformed under dynamic loadings on the upper surface of threelayer structures. we investigated loading conditions under which plastic deformations occurred in structural elements, and frame elements of metamaterials changed their initial geometric shape. as a result of plastic deformations, micro damages and cracks can form in frame elements. the mechanical behavior of three-layer structures was described within the framework of the approach of mechanics of damaged elastic-viscoplastic media [12]. the constitutive equation of titanium alloy in frame element of metamaterials and plates was used in form: ( ) 1( )m ij ij ijp f s     (2) where σij is components of effective stress in damaged medium, p(m) is the pressure in condensed phase of frame elements, sij is deviator of effective stress tensor of damaged medium in material particles of frame elements, φ1( f )= 1 – ln (1 – f) is functions linking the effective pressure and its value in condensed phase of medium, superscript (m) denotes pressure in condensed phase of damaged material, f is the damage parameter. mechanical response of layered structures with internal layers from metamaterials to dynamic... 5 in this study, using numerical modeling, we analyzed the mechanical response of threelayer systems with an interlayer of metamaterials created from alpha titanium alloy ti-5al2.5sn with good weldability and resistance to high strain rate plastic deformation under complex stress conditions [12]. a physical and mechanical model for describing alpha titanium alloys at high-strain rates is discussed in [22]. the initial temperature t0 in the material of metamaterial unit cell was assumed equal to 295 k. the temperature increase at high strain rates was calculated in the adiabatic approximation, as proposed in [22]. the flow stress of alpha titanium alloys was described using constitutive equation: ( ) 0 1 2 0 3 4 0exp{ (1 / )} 1 exp{ }exp{ }exp{ ln( / )}m p s s m eq eq eqc t t c k c t c t s s        (3) where seq = [(2/3) έijέij]1/2 , seq0 =γ1exp(–t/γ2)+γ3, έij is the components of strain rate tensor, 0 t p p eq eqdt   is the plastic strain intensity, γ1=2115 s-1, γ2 =38.27 k-1, γ3 = 9.824 10-5 s-1, t is the temperature, t0 is the room temperature, and tm is the melting temperature. the following values of the model coefficients of ti-5al-2.5sn were used in the calculations: σs0 = 0.2 gpa, c1 =385, c2 =0.56 gpa, c3= 0.0016 k-1, c4=0.00009 k-1, k0=85, tm =1875 k [12, 22]. the grid convergence of the calculation results within a few percent was provided by the choice of the discretization step of the grid model. 2.3 damage model the gurson tvergaard damage model was used for the yield stress determination in metamaterial elements [23]: 2 ( ) 2 * ( ) ( ) * 2 1 2 3( / ( ) ) 2 cosh( / 2 ) 1 ( ) 0m m m eq s sq f q p q f       (4) where σs (m) is the yield stress of condensed phase, p(m) = – σkk (m)/3 is the pressure in condensed phase, σeq= [(3/2)σij σij – 0.5 σ2 kk]1/2, σij is the effective stress tensor components, q1 = 1.0, q2 = 1.3, q3 = 1.0, f * is a damage parameter.    1{ (( ) )( ) / ( ) ( ( )}c c c c f c c f cf f h f f + f 1 / q f f f f f h f f )h f f       * (5) where ff = [q1+(q1 2 – q3)1/2]/q3, h(.) is the heaviside function, fc is the material constant. the evolution of damage (growth and nucleation of discontinuities) and fracture of the material of unit cell of pentamode metamaterials was described by the needleman model [23]. 2( / )exp{ 0.5[( ) / ] } (1 ) nucl growth p p nucl eq n n eq n n p growth kk f f f f f s s f f            (6) where εn and sn are the average strains at which damage nucleated and the strain standard deviation, respectively. the generated damages parameter f depends on the value of the parameter fn. the values of model coefficients for alpha titanium alloy determined in [12, 22] were used: fn = 0.04, sn = 0.1, ff = 0.26, fc = 0.117. 6 v.a. skripnyak, m. chirkov, e. skripnyak, v.v. skripnyak 2.4 initial and boundary conditions numerical modeling of the response of three-layer systems to dynamic loads was carried out for initial conditions corresponding to the absence of residual stresses in the material of the elements, a uniform temperature field t0, and the absence of damage (f = 0). options for loading three-layer systems are considered, the diagram of which is presented in fig. 1(a) were given by relations: 1 1 4 41 4 2 3 5 6 1 2 3 1 2 30, 0, ( ), 0, 0, 0, 0z ijs s s ss s s s s s u u u v t u u u           (7) where ui are the components of the velocity vector of material particles, vz (t) is the velocity of material particles on the loading surface, surfaces s1 – s6 are shown in fig.1(a), surfaces s5 and s6 include the surfaces of the metamaterial frame elements. the pulse amplitude ( )amp zv = 100 m/s was specified for pulse loading. during cyclic loading, the speed on the loading surface was specified by a harmonic function of time loading frequencies fcycl at which displacement of the top plate does not lead to complete compaction of the metamaterial layer are considered. the period of harmonic oscillation is equal to tc =1/ fcycl. these frequencies must satisfy the condition: fcycl > 0.25 vz (amp)/[h – m/(ρcf)], where h is the thickness of metamaterial interlayer, m is the mass of metamaterial interlayer, ρc is the mass density of material of metamaterial frame element, f if the square of metamaterial volume projection onto the base plate. loads with frequency 20 khz and amplitudes from 100 m/s and below were considered. numerical simulation of the multilayer structures response to dynamic impacts was performed in the ls dyna environment using the original umat module, which implements the constitutive equations describing the elastic-plastic behavior and damage of titanium alloys in a wide range of strain rates, temperatures, taking into account the influence of complex stress states. 3. results of numerical simulation and discussion the simulation results showed differences in the patterns of deformation development in layers of pentamode and auxetic metamaterials under dynamic loading. the mechanical response of the considered three-layer systems to dynamic influences is determined by the thickness of the metamaterial layer and the patterns of its deformation. in the case of dynamic compression of three-layer systems, the deformation of the metamaterial layer depends on the initial effective mass density. fig. 2(a) shows the changes in average pressures on the lower surface of the top plate versus time (curve 1) and the lower plate (curve 2) for a three-layer system with an intermediate layer of pentamode metamaterial. loading was carried out by a load pulse with amplitude of 100 m/s. pressure oscillations (curve 1) are caused by changes in the compression resistance of the pentamode material in the contact zones of the metamaterial frame elements with the loading plate. pressure relaxation is associated with a decrease in the resistance of the metamaterial layer to deformation at high strain rates as a result of collapse of the frame structure layer and partial fracture. deformation, fracture and intensive compaction of the pentamode metamaterial frame structure in the interlayer is completed at time a (ta = t0+δt ≈ t0+ h/vz(t)). mechanical response of layered structures with internal layers from metamaterials to dynamic... 7 the damping capabilities of the three-layer system after time moment a will be exhausted. the simulation results shown in fig. 2 (d), illustrate the scattering of fragments of the fractured frame structure of the pentamode metamaterial (zone b). fig. 2(b) shows changes in time of the supplied energy (1), kinetic energy (2) and internal energy (3) of considered three-layer structure under the influence of a shock pulse. (a) 0.000 0.005 0.010 0.015 -0.02 -0.01 0.00 0.01 0.02 p re s s u re , g p a time,ms a 1 2 (b) 0.000 0.005 0.010 0.015 0 50 100 150 200 t o ta l, k in e ti c , in te rn a l e n e rg y , j time, ms 3 2 1 b (c) 0.000 0.005 0.010 0.015 0.0 0.2 0.4 0.6 0.8 1.0  e q , g p a time, ms 1 2 3 4 (d) fig. 2 mechanical behavior of three layered titanium structure with pentamode metamaterial interlayer: pressure averaging on surface at upper plate curve 1) and at low plate surface (curve 2) versus time for pentamode interlayer (a), total, kinetic, and internal energy of layered titanium structure versus time (curves 1,2,3) (b), mises equivalent stress in central points and connection zone of frame pentamode elements near top plate surface (curves 1,2 ) and near base plate surface (curves 3,4) (c), deformation and fracture of pentamode metamaterial interlayer in three layer structure under dynamic compression in normal direction to surface of the top plate (d) the results of numerical modeling indicate significant damping properties of a layered structure with a pentamode metamaterial interlayer. the damping properties of such three-layer systems are exhausted at time b, shown in fig. 2(b) when the pentamode metamaterial layer is crushed, destroyed, and compacted. fig. 2 (c) shows the time changes in equivalent stresses in the zone with the maximum diameter of pentamode metamaterial frame element (curves (1), (3)) and in the joint zone of the frame elements (curves (2), (4)). curves (1) and (2) correspond to the frame element connecting with the top plate, and curves (3) and (4) to element in contact with the base plate. oscillations of equivalent stresses in frame elements during compaction of the metamaterial are caused by stress relaxation due to local plastic deformation, accumulation 8 v.a. skripnyak, m. chirkov, e. skripnyak, v.v. skripnyak of damage and fracture. note the plastic deformation and fracture of frame elements in connection zone occurs in the complex stress states, which confirms the importance to take into account stress triaxiality parameter η = – p(m)/ σeq (m) for simulation of the response of pentamode metamaterial to dynamic compression. the results in fig. 3 demonstrate the mechanical response of a three-layer system with an auxetic interlayer (see fig. 1(d)) to impulse impacts with amplitude of 100 m/s. fig. 3(a) shows the changes in pressure versus time on the lower surface of the upper plate (1) and the lower plate (2). the considered auxetic metamaterial has greater compressive rigidity compared to the considered pentamode metamaterial, as a result of which three-layer systems demonstrate higher resistance to pulsed compression. at time moment a, the elastic wave passes through the frame of the auxetic metamaterial and reaches the surface of the base plate. (a) 0.0 0.5 1.0 1.5 2.0 -0.2 0.0 0.2 0.4 0.6 p re s s u re , g p a time, ms 1 2 a (b) 0.0 0.5 1.0 1.5 2.0 0 4000 8000 time, ms t o ta l, k in e ti c , in te rn a l e n e rg y , j 1 2 3 в (c) 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 time, ms 2 1 3 4  e q , g p a (d) fig. 3 mechanical behavior of three layered titanium structure with metamaterial interlayer: pressure at upper plate surface (curve 1) and low plate surface (curve 2) versus time for pentamode interlayer (a), total, kinetic, and internal energy of layered titanium structure versus time (curves 1,2,3) (b), von mises equivalent stress in central points and connection zone of pentamode frame elements near upper surface (curves 1, 2 ) and near lower plate surface (curves 3,4) (c), deformation and fracture of auxetic metamaterial interlayer in three layer structure under dynamic compression (d) pressure oscillations in the loading plate are caused by changes in the compressive resistance of the crushed metamaterial layers shown in fig. 3(d). plastic deformation of frame elements of an auxetic metamaterial is accompanied by dissipation of total mechanical energy. fig. 3(b) shows the changes in mechanical (1), kinetic (2) and internal mechanical response of layered structures with internal layers from metamaterials to dynamic... 9 energy (3) supplied to the volume of a three-layer structure under pulsed loading. the drop of kinetic energy (curve 2) in the time moment b is associated with end of metamaterial compaction. the results indicate the ability of the considered structure to dissipate the supplied mechanical energy under pulsed influences. dissipation of mechanical energy is associated mainly with inelastic deformations of the frame elements of the metamaterial. changes of equivalent stresses in the central zone of frame elements (curves (1), (3)) and in zones of their connections (curves (2), (4)) near the loading plate and base plate are shown in fig. 3(c). oscillations of σeq are caused by the excitation of frame elements vibrations under the influence of a pulse load. the oscillations die out with increasing plastic deformation of the structural elements and compaction of the metamaterial layers. the compaction of the metamaterial layer occurs layer by layer, starting from the zone adjacent to the loaded plate, as shown in fig. 3(d), and ends at time b (fig. 3(b)). fig. 4 presents the results obtained by modeling pulse loading of three-layer systems with an interlayer of a more rigid auxetic metamaterial (fig. 1(e)) [24]. in contrast to the three-layer system with a pentamode interlayer, in the system with an auxetic metamaterial made of alpha titanium alloy, no destruction of frame elements and joint zones was observed. an increase in the thickness of the metamaterial layer for both three-layer systems considered will lead to an increase in their dissipative properties and the duration of deformation until complete compaction. (a) 0.00 0.05 0.10 0.15 0.20 0.25 -0.1 0.0 0.1 0.2 0.3 p re s s u re , g p a time, ms 1 2 (b) 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.02 0.04 0.06  e q p time, ms point 9 point 3 (c) 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4  e q , g p a time, ms point 9 point 3 (d) fig. 4 mechanical behavior of three layered titanium structure with auxetic re-entrant metamaterial (type 2) interlayer: pressure at the upper plate surface versus time (curve 1) and at low plate surfaces (curve 2) (a), equivalent plastic strain versus time in gage points 3, and 9 in frame elements of auxetic re-entrant metamaterial, respectively (b), mises equivalent stress versus time in gage points 3, and 9 (c), mises equivalent stress in three layered structure (d) 10 v.a. skripnyak, m. chirkov, e. skripnyak, v.v. skripnyak pulsed loading of the top plate causes a stress wave to propagate along the metamaterial frame structure, which causes deflections of the frame elements oriented orthogonally to the direction of loading, as well as a decrease in the angles of the connected frame elements. compaction of the frame elements leads to a significant decrease in pressure on the surface of the base plate (2) (fig. 4(a)). a stiff auxetic re-entrant metamaterial under dynamic compression normal to the surface of the loading plate does not experience layer-by-layer collapse, as shown in fig. 4(d).vibrations of deformable frame elements during compression of the stiff auxetic metamaterial layer lead to oscillations of equivalent stresses, which are shown in fig. 4 (c) at gage points 9 and 3, the positions of which are indicated in fig. 4(d). these results suggested that three-layer systems with a stiff auxetic metamaterial interlayer could be used as high-amplitude harmonic dampers. fig. 5(a) shows the damping of the vibration amplitude of frame elements at gage points 10, 5 and 1. (a) 0.0 0.5 1.0 1.5 2.0 -50 -25 0 25 50 u z , m /s time, ms point 10 point 1 point 5 (b) fig. 5 velocity versus time under harmonic loading of layered structure with stiff auxetic re-entrant metamaterial interlayer (a), distribution of local velocity vectors in frame elements (b) the results of modal analysis for the considered three-layer system showed, that natural frequencies are 0.01066, 0.01294, 0.035, 190, 243, 264 hz for the first 6 modes, respectively. the harmonic loading frequency was 20 khz, and amplitude of the velocity oscillation 50 m/s. thus, the response of a three-layer system to harmonic influences was studied far from the induced band gap caused by resonance phenomena. fig. 5(b) shows the amplitudes of the vibration speeds of the elements at the time instant of 1.305 ms. results obtained indicate that three-layer systems with an auxetic intermediate layer can be insulators of high-amplitude vibration impacts. the presence of horizontal frame elements significantly affects the relationship between the effective compression modulus and shear modulus estimates for unconfined pentamode lattices under quasi-static loadings. therefore, the use of effective elastic moduli and theoretical constitutive relations obtained for unconfined pentamode lattices does not provide reliable predictions of the mechanical response of these layered systems under quasi-static and dynamic loading. the results of this research obtained for layered systems with an interlayer of auxetic metamaterials complement the results of [24-28] and confirm the position that topological differences in auxetic metamaterials affect the critical crushing characteristics and energy absorption capacity of systems. note the nonlinear mechanical response of layered structures with internal layers from metamaterials to dynamic... 11 nature of the change in equivalent stresses and pressure in the elements of auxetic metamaterials in fig. 3 and 4 are in qualitative agreement with the results of zang et al. [26] and wang et al. [28] for stresses and strains. the patterns of changes in energy absorption and energy dissipation under dynamic loading of layered systems with an interlayer of auxetic metamaterials are consistent with the results of [26, 27, 28]. the results of layered systems compaction simulation with an interlayer of auxetic materials under dynamic compression and the occurrence of oscillations of compressive stress are consistent with the results obtained by novak et al [24] using a direct experimental researches of dynamic compression of metamaterial samples from titanium alloy, and results obtained by zang et al. [28] for auxetic metamaterials from aluminum alloys. 4. conclusions layered structures with pentamode or auxetic metamaterials interlayer are promising materials for applications in transportation and aerospace systems that require low weight, high strength, and high energy absorption. the results obtained showed that in order to obtain realistic predictions of the mechanical response of these layered structures to dynamic loads, it is necessary to take into account a set of nonlinear geometric effects when describing large deformations of frame elements of metamaterials, nonlinearity of the elastic-viscoplastic constitutive equations and damage of the material at complex stress states. it is shown that the considered layered systems made of alpha titanium alloy have high specific energy absorption and dissipative properties, which make it possible attenuating the amplitude of the stress pulse after passing through the layered system. increasing the rigidity of auxetic metamaterials by 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https://www.researchgate.net/journal/comptes-rendus-physique-1878-1535?_tp=eyjjb250zxh0ijp7imzpcnn0ugfnzsi6inb1ymxpy2f0aw9uiiwicgfnzsi6inb1ymxpy2f0aw9uin19 https://www.sciencedirect.com/journal/theoretical-and-applied-fracture-mechanics https://www.sciencedirect.com/journal/theoretical-and-applied-fracture-mechanics https://www.sciencedirect.com/journal/materials-and-design https://www.sciencedirect.com/journal/composite-structures https://www.sciencedirect.com/journal/composite-structures https://www.sciencedirect.com/journal/composite-structures/vol/288/suppl/c https://www.sciencedirect.com/journal/thin-walled-structures 11506 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 315 328 https://doi.org/10.22190/fume230128016a © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper mechanical characteristics of nano-crystalline material in metallic glass formers hamid al-abboodi1, huiqing fan2, mohammed al-bahrani3, abdelsalam abdelhussien4, barhm mohamad5 1,2state key laboratory of solidification processing, school of materials science and engineering, northwestern polytechnical university, xi’an, china 1kut technical institute, middle technical university, wasit, iraq 3chemical engineering and petroleum industries department, al-mustaqbal university college , babylon, iraq 4school of mechanical engineering, nanjing university of science and technology, nanjing, china 5department of petroleum technology, koya technical institute, erbil polytechnic university, erbil, iraq orcid ids: hamid al-abboodi https://orcid.org/0000-0003-0642-9396 huiqing fan https://orcid.org/0000-0002-1405-9279 mohammed al-bahrani https://orcid.org/0000-0003-0680-0116 abdelsalam abdelhussien https://orcid.org/0000-0003-4676-3292 barhm mohamad https://orcid.org/0000-0001-8107-6127 abstract. in order to evaluate the model metallic glass alloy’s mechanical properties (fe49.7 cr17.1 mn1.9 mo7.4 w1.6 b15.2 c3.8 si2.4) prepared by spark plasma sintering (sps) which have high velocity. we made an apparatus having three-point curve testing. the comparatively bulk sizes of sample in the current study permitted the creation samples for test with a macro scale cross-section (range of mm) consistent test dimensions, and well-controlled sample sizes. cutting using a wire saw produced remarkably sharp notches with a radius that was 3 times smaller than in earlier studies. our three-point bending apparatus allowed us to acquire the 231 gpa and 4.91 mpam1/2 values for notch fracture toughness and young's modulus. additionally, the results of the vickers indentation and flexure tests for young's modulus were reliable. vickers indentation measurements of indentation fracture toughness produced values that were a minimum of 49.9% lower than those obtained flexure using. the method for examine micro scale mechanical properties described in this study and the accompanying scrutinizes are valid to samples with different ones or compositions that are made by further means. key words: metallic glass, three-point bending, fracture toughness, vickers indentation, sps received: january 28, 2023 / accepted may 07, 2023 corresponding author: barhm mohamad department of petroleum technology, koya technical institute, erbil polytechnic university, 44001 erbil, iraq e-mail: (barhm.mohamad@epu.edu.iq https://orcid.org/0000-0003-0642-9396 https://orcid.org/0000-0002-1405-9279 https://orcid.org/0000-0003-0680-0116 https://orcid.org/0000-0003-4676-3292 https://orcid.org/0000-0001-8107-6127 316 h. al-abboodi, h. fan, m. al-bahrani, a. abdelhussien, b. mohamad 1. introduction comparing bulk metallic glasses (bmgs) to crystalline forms of similar compositions, bmgs typically have greater hardness and yield strength values [1, 2]. the metallic glasses are currently being applied include ship manufacturing, boiler fireboxes, the nuclear industry, and the auto industry [3]. however, as their principal weaknesses are frequently low ductility and weak fracture toughness, work is still being done to enhance these qualities [4–9]. this necessitates advancements in material processing as well as the creation of precise methods for calculating fracture toughness. although there are many ways for measuring the mechanical characteristics of the macroscale sample, it is frequently impractical to employ typical testing methodologies for the marginal’s metallic glasses formers because of the processing restrictions on samples size which can be acquired from these types of materials. sps [10–13] is a pressure-assisted pulsed-current process in which the powder samples are loaded in an electrically conducting die and sintered under a uniaxial pressure and could circumvent the samples size problem for the marginals glass former, enabling the measurement of macroscale characteristics [14–16]. indentation has been used to quantify the fractures and toughness of the fe-base metallic glass [17–19], while the notch toughness and test has been done on the sample with the nonstandard shapes [7, 19, 20]. a brittle material's indentation fracture test is unreliable due to the poorly defined cracks in arrested state which makes the analysis more difficult when it is compared with rapid through crack propagations of the standardized test, even though it is most frequently used for samples of small size [21, 22]. as a result, the indentation-based data on mechanical characteristics show significant disparities. consequently, using a standardized method like three-point bending to determine fracture toughness is more accurate. testing metallic glasses for three-point bend, however, may yield results that don't seem reliable. for instance, recent three-point bending tests of the fracture toughness of zrbased metallic glass revealed a significant amount of data scatter [21]. it is acknowledged that thermoplastic forming, such as sps, provides superior controls over the samples heat history is to minimize the processing’s induced dispersion, and that this is type of processing issues rather than testing’s issue [22]. three-point bending has also been used to determine the fractures toughness of the fe-base metallic glass, however, test of specimen and the notch used had irregular geometries which could lead to inaccuracies. metallic glasses have undergone fractures toughness test with the notch machined using the electric discharge machine (edm) [21, 23], the heating brought on by the edm could cause volumes of the materials in heat affected zones to crystallize or even crack, which may result in inaccurate fractures and toughness value. in current study, techniques for evaluating fe based bmgs are developed for increasing precision of the measurements of the fracture’s toughness and the modulus of the marginals glass former. the model marginals glass formers, sam2x5, along with notional compositions of fe49.7 cr17.1 mn1.9 mo7.4 w1.6 b15.2 c3.8 si2.4, was produced in bulk samples using sps. the production of the test’s specimen with the macroscales cross section (in millimetres range) and the well control samples dimension are close to the standardized test. this was made possible in the current investigation due to the comparatively high sample size. the samples' increased sizes also made it possible to examine changes both within and between individual samples. a remarkably sharped notch with radius of (3x smaller than in earlier research) the little samples damage was possible with a wire saw cutting. these samples were tested for fracture toughness and flexural modulus using a mechanical characteristics of nano-crystalline material in metallic glass formers 317 mechanical testing device that was specifically designed for the purpose. there isn’t any data on fracture toughness that can be directly compared because the values presented in this paper were determined using a novel approach on a material that underwent sps processing. as a result, we attempted to compare the numbers obtained from our novel three-point bending technique to those obtained from indentation tests on our material, which are typically used to gauge fracture toughness. to compare the results with those obtained using three-point bend tests, the value for the modulus and the fractures toughness are also determined by using the indentations methods (vickers and nanoindentations). we computed other properties and compared them to known properties using the results of flexure and indentation. the testing of the microscales mechanical property is described in the study. the following analyses are applied to specimens made of other materials or made in different ways. 2. experimental techniques as previously described [13, 15], bulks sam2x5 sample were created using the spark of plasma sintering’s (sps). gas atomization was used to create sam2x5 powders, which were then ground in the fritsch’s p5 planetary and high energy balls mill. after milling, sam2x5 powders particle are about 20µm in the size. these powders were subjected to sintering tests using the sugimoto’s sps 1050 and sps units (sumitomo’s coal mining’s co. ltd., japan), which produced discs with a thickness of around 2 mm using the 19 mm of inner diameter graphite dies and is lined with graphite paper. 9 g of powder, on average, were present in each sample. the applied load was 20 kn, and the sintering temperature was 600 °c. the samples were cylinder-shaped, with a diameter of 19 mm and a thickness of 2 mm. an immersion method was used to gauge the density of the sam2x5 sample. the amorphous structure was evaluated using the x ray diffractions (xrd) pattern from the rigaku ultima iv systems utilizing cuk radiations and the scan rate of the one per second. for the evaluating the impacts of the mechanical grindings and the diamonds lapping on the sample’s crystallization, the xrd of bulk of the samples were carried out as prepared and rough grounded (400 grit) with the diamonds lapped disc. on the thin foils sample that had been dimpling with ion-milled, tem imaging’s with the saed were carried out utilizing the jem 2100 f (jeol) apparatus. for bend testing, the two bulks x ray amorphous such as sam2x5 sample (samples 1 & 2) was chosen. our samples are given the sam2x5 designation by our previous naming policy [13]. because of their slow removal rate and the low heat production when used, surfaces lapping and the wire saw cuttings were used to reduce the effect of the heat production and deformations. the diamond paste was first used to lap the two samples until they were 1860 m thick and their surface was parallel within 1m over 20mm. to conduct experiments on modulus and fracture toughness, the sample was subsequently divided into the small specimen using the wire saw and sic slurry. the bulk sample were divided into the six specimens for the flexural modulus of four specimens for fractures toughness, other samples were divided into the three specimens each for the flexural modulus with fractures toughness. the flexural modulus specimen was rectangular in shape, 1860 mm by 500 mm cross sections in width and thickness, and 10 mm test gauge lengths. although specimens were approximately twice as small as the normal sizes’ and have cross section of the 1mm and 1860 m (width & thickness, respectively), the gauge length 318 h. al-abboodi, h. fan, m. al-bahrani, a. abdelhussien, b. mohamad of the 8 mm, and fractures toughness specimen are adhered to astm e399 prescribed ratios. using a wire saw with the 50 mm diameter wires and the cuttings slurry of the 800 grit sic powders (suspended in the oil) with the medium particles size of the 6.5 mm, fracture toughness specimens were notched. using optical microscopy, the precise measurements of the ensuing notches revealed that there was 900 m in depth (about the half of the specimens thickness), 100 m in width, and has notched root radii of the 40m. to measure modulus and the fractures toughness of millimetre scale specimen, a specially designed and constructed small scales mechanical test apparatus was created. the device is comparable with testing of equipment used by the eberl et al. [24] to evaluate thin films. the microbending machine with sample is shown in fig. 1. fig. 1 microbending machine, close-up of fracture toughness stage and sample the stage's movement is controlled by the actuators, which are managed by matlab software. to track strain, cameras were employed for the digital images correlation (dic) [24]. due to an early fracture, the astm e399-specified fatigue cracks were not obtained. as a result, the reported toughness values are for notch toughness. to contrast the modulus and toughness information received from bending testing, indentation (vickers and nanoindentation) was carried out. the samples were ground with sic paper of 200, 400, 600, 800, & 1200 grits, then polishes with the suspensions of 1 mm diamond particles. the polished samples were subjected to vickers indentations using the leco lm100 indenters. nine vickers indentation was created for each sample using a 300 g load and a 10 s dwell duration. by using the optical measurement of typical crack length originating from indents corner, indentations of toughness were estimated from the averages of nine vicker indents of each sample. anstis et al model's [4, 25], in which half penny of crack shape was assumed, and indentations toughness could be calculated based on it, and is one of most often used methods for the approximation of fracture toughness of bmgs. 3/2 0 c p e k = α hc (1) mechanical characteristics of nano-crystalline material in metallic glass formers 319 where p is indentations load, h is hardness, e is modulus, c0 is half fractures length; kc is toughness estimated from the indentations; and α (0.016 ± 0.004) is calibrations constant with the (empirical constants). a hysitron triboindenter was utilized to create nanoindentations, which were then used to calculate the hardness and young's modulus. for each of the two bulk samples, a total of 36 nanoindenters were created by using the berkovich tips with a 200nm radius, the constant loading rates of the 1000 µn/s, and the maximum loads of 7500 µn. the maximum forces divided by area functions served as the basis for calculating hardness. based on the area function and the stiffness of the unloading curve, reduced modulus as determined by nanoindentation was computed. the lowered modulus was then used to derive young's modulus (under the assumption that v= 0.3). we used our three-point bending equipment to determine the flexural specimens' young's modulus. 5 µm/s was the cross-head stage's movement rate. the movements of piezo actuators served as the basis for calculating displacement values. for each modulus test, a load-displacement curve was created by the subtracting compliance of machines, which was determined independently, from the data from the load cell. the following formula was used to determine young's modulus for the supported prismatic beams with the rectangular cross-section based on the beam bending theory: 3 3 l m e = 4bd (2) where the l is support's length, m is load-displacement curve's slope, b is the specimen's width, and d is its thickness. using a 6061-t6 aluminium specimen that was cast as cast and had a shape identical to the sam2x5 specimens, the apparatus and testing procedures were calibrated. for 6061-t6 al, the youngs modulus value obtained from our testing apparatus were within 3% of standard values of the 69 gpa [26]. it shows sample's top surfaces as well as how it was loaded for the measurements of fractures toughness. for these measurement, cross head stages were moved at the speed of 2µm/s. the displacement of the crack’s mouths opening was measured using the dic. by comparing loads cell data to dic data, the load versus cracks mouth openings displacement curves were developed. fracture toughness was calculated for each specimen using recorded peak loads by astm e399 guidelines. 3. findings and analysis the sam2x5-600 samples utilized in this work had a bulk density of 7.9 g/cm3, which is in good agreement with our earlier findings, which take into account of both theoretical values for sam2x5 powders (7.75 g/cm3) and the structural relaxations which takes place throughout the consolidations processes [13, 15]. the same method of the processing was employed in [13] to create an amorphous alloy. our earlier research supports the fully dense samples and illustrates microstructure of sam2x5. the xrd profiles of the samples are shown in lower insets of the fig. 2. the amorphous natures of sam2x5 sample as seen in bright field tem images is also depicted in fig. 3 320 h. al-abboodi, h. fan, m. al-bahrani, a. abdelhussien, b. mohamad fig. 2 the xrd plots of sam2x5 bmgs indicating amorphous morphology fig. 3 tem image, inset saed pattern the diffusion ring with some of the diffraction’s spots can be seen in the insets of saed patterns of relevant regions. this show that the final product has mixtures of nanograins and amorphous structures and is not entirely amorphous. we assume that the final sample is primarily amorphous even though we do not statistically evaluate volumes percent of the amorphous vs nanograins. the bmg samples were made under various sintering settings (changing time and temperature). the optimal sintering conditions were found using xrd findings for broad diffraction peaks with few crystalline peaks, which are at the 600 °c. the insets of saed patterns in fig. 3 showed some crystallinity. but using sps, this was the best the authors could come up with. even though the presence of crystalline particles can have a negative impact on its toughness, and this is crucial for assessing the material's globally rather than the local toughness because estimates of the parts of life depends on global fractures toughness value. the measured fracture toughness values on bigger specimens, which were bent three times, account for "defect" like crystallinity that would be the inevitable in synthesis of the bmgs, and this guarantees that the fractures toughness measurements obtained through the three points bending may be applied to the life forecasts of the components fabricated from our bmgs. the grindings and lapping’s did not alter the xrd’s profile, indicating that there was no devitrification caused by these specimen preparation techniques. it is plausible to assume that since wire saw cutting generates the less heat than the rough grindings, devitrification is also not a result of this operation. youngs modulus value obtained by three-point bending have an average rate and the first standard deviation of the 227.3 ± 21.3 gpa for six of specimen made from the sample 1 & 232.3 ± 22.2 gpa for the threespecimen made from sample no 2. we use average numbers as reliable approximations of the 230 gpa, or youngs modulus. average and the standard deviation of the youngs modulus value obtained from nanoindentations, in contrast, there are 276.5 gpa & 303.3 gpa, respectively. result from nanoindentations is roughly 20–30% better than those from three-point bending. standard deviations and average vickers hardness values, respectively, were 11.16 ±0.21 gpa and 12.21± 0.21 gpa. youngs modulus of the metallic glass can be calculated using vickers hardness (e ≅ 20 hv) [27]. the projected young's modulus, that are great agreements with the 230 gpa values discovered by our mechanical characteristics of nano-crystalline material in metallic glass formers 321 flexures testing, is between 220 gpa and 240 gpa based on the hardness values discovered by micro indentation. alternately, the weighted modulus of the alloy elements can be used to approximate youngs modulus [27]. this method yields youngs modulus for the sam2x5 of the 300 gpa, that is much in the line with our findings from nanoindentations. according to calculation, vickers hardness is a good way to gauge youngs modulus of bulk material, while the weighted youngs modulus utilizing alloys components yields an accurate representation of the youngs modulus measured by the nanoindentations. the average of hardness value obtained by the nanoindentations was greater than vickers values at 18.0 gpa and 19.1 gpa, respectively. when utilizing the hardness relationship outlined above, hardness numbers don’t replicate youngs modulus result obtained via flexures. fig. 4 shows the sam2x5 nano indenter. along the margins of the indent, the material has accumulated. on our sam2x5 sample, fig. 4 depicts material accumulation along the boundaries of a nano indenter. youngs modulus and the hardness is measured by the nanoindentations can be overestimated by material accumulation. these effective contact areas of nano indenters is altered by significant flexibility, which causes the hardness to be overstated by up to 50% [28]. the assessment of the indent areas is challenging with nanoindentation due to pile-ups, and as a result, the hardness values obtained are not accurate. in light of this, hardness levels determined by the vickers indentations with the higher force of 300g may be much more accurate. by giving the direct correlation between youngs modulus and the hardness and the nanoindentations measured young's modulus is 20–30% higher, we estimate that the hardness is also 20–30% greater. fig. 4 nanoindent of sam2x5. material pile-up can be observed along the edges of the indent after adjusting for the 20–30% discrepancy, the hardness values for samples 1 and 2 are 14–15 gpa and 15–16 gpa, respectively. the young's moduli obtained by applying the e ≅ 20h relationships to the corrected hardness value range from about 280.0 gpa to 300.0 gpa and 290.0 gpa to 320.0 gpa, respectively. this young's modulus is now more consistent with hardness result obtained by the nanoindentations (276.5 with 9.4 gpa and 303.3 with 9.1 gpa). the loads displacement curves for the flexural modulus test is shown in fig. 5. a sudden brittle failure is visible on all flexural modulus curves. none of the modulus test specimens showed any appreciable plastic deformation. young's modulus and the glass transition temperature are both inversely correlated (tg = 2.5e, where tg is in the celsius and the e is in gpa) [27]. the glass transitions temperature of the 575 °c was obtained by 322 h. al-abboodi, h. fan, m. al-bahrani, a. abdelhussien, b. mohamad using ours observed youngs modulus by the flexure or vickers indentations (230 gpa) and the tg = 2.5e relationships, which are very good agreements with previously report glass transitions temperature of the 580 °c for this type of material [29]. by using the direct nanoindentation results, this method would greatly overstate the glass transition temperature, which justifies the adoption of our correction. a typical load vs cracks mouth displacements curve for the fracture toughness tests utilizing three points bending’s is shown in fig. 5. the load against crack mouth displacement curves for all of the fractures toughness test in this study show no changes in the slope before failure, which is consistent with brittle failure at the peak stress. for samples 1 and 2, the average fractures toughness value determined by three points bending are the 5.0 ±0.7 mpam1/2 and 4.7 ±0.8 mpam1/2, respectively. a) b) fig. 5 (a)flexural modulus and (b) fracture toughness load versus displacement curves for sam2x5 samples tested by three-point bending. the difference between the two samples is negligible. therefore, we choose 4.9 mpa.m1/2 as the average values for the notch fracture toughness. both bulk samples had average indentation (vickers) toughness values of 2.5 ±0.1 mpa m1/2 and 2.2 ±0.1 mpa m1/2, respectively. the feb-based bmgs (sam2x5) have been demonstrated to be more brittle and stronger than fec and the fec(b)-based bmgs [30]. as anticipated, sam2x5 (a feb-based bmg) has exhibited vickers indentation toughness values that are slight lower than the those of other fec-based bmgs [16, 18]. the quinn and bradt [30] the point out that it is doubtful that vickers toughness values are true or valid, except from by chance. we discover that these values are 50% less than those we obtain from ours three points bend test. there is a significant degree of inaccuracy in indentation toughness tests [32]. for instance, it can be challenging to get the required half-penny crack geometries, and the computation does not take the irregular and the secondary crack into account (cracks branching). the crack coming from the vickers indents are depicted in fig. 6. half cracks length is optically determined as the separations between indentation's centres and cracks' tips. the assumption made which calculates fractures toughness, is which where the crack that form corners of the indent which are straight with the same lengths. vickers indents caused cracks to emerge in our sam2x5 samples, a) b) mechanical characteristics of nano-crystalline material in metallic glass formers 323 and these irregular cracks added to the computed fracture toughness values' inaccuracies. accurate measurement of the crack lengths is one of the flaws in the vickers indentation toughness determination process. this was the primary impetus for the creation of the micromechanical tester, which more precisely measures toughness levels. this is a problem not only in this study but in other studies that use indentation to measure fracture toughness. based on the research of anstis et al. [25], indentation fracture toughness results are only reliable around ±40% of the time. the "calibration" constant (α) employed in the half crack length, it should be mentioned. vickers indentation data of the fracture toughness of half a crack are shown in fig. 6. fig. 6 fracture toughness measurements of half crack length using vickers indentation. cracks emanating from corners of vickers indents are seldom straight nor symmetric about the center. vickers indents' corner cracks are rarely symmetrical or straight around the centre. it is noted that the “calibration” constant (𝛼) used is taken to be independent of material. in actuality, this value depends on the material and is chosen by comparing it to approved values discovered during bending experiments. vickers indentation assessment of fracture toughness is therefore accurate when it is known, which was not the situation for the many bmgs, where the samples size precludes bending’s experiments. it has been demonstrated that accuracy of indentations findings derived using the common "calibrations" constant of α(0.016) deviates from typical astm kic measurements of sic samples by around 50% [31, 33]. therefore, it has been demonstrated that mechanical property of zr based bmgs are influenced by sample size. for the zr-based bmg wire and foil less than the 1 mm in sizes, conner et al. [34, 35] demonstrated a notable increase in the bends ductility. that is, as the sample size was decreased below the samples thickness of approximately 1mm, the plastic strains to the fractures in bending increased dramatically. according to astm, measurements of fracture toughness acquired by the three points bending of specimen of at least 1mm in length were more equivalent to the measurements made during the large-scale testing, but measurements made by indenting are not indicative of bulk characteristics. upon the impact of the notched versus fatigues pre-cracked specimen on the fractures toughness measures of the zr-based bmgs, varying results have been reported. 324 h. al-abboodi, h. fan, m. al-bahrani, a. abdelhussien, b. mohamad according to reports, the fractures toughness value of the notched sample of different widths are higher (2x) than those of fatigue-pre-cracked samples [36, 37]. the discrepancies in failure mechanisms at the notched roots were one explanation for the disparity in toughness levels. planar crack fronts were present in the fatigue pre-cracked samples in [35], whereas considerable crack bifurcations and shear bands were present in the notched samples. the authors proposed that the variations in observed toughness values could be explained by the energy absorptions from the shear bands and cracked bifurcations in the notched sample. however, additional studies [38] in the zr based and the ti-based bmgs have likewise found no discernible differences in the observed fractures toughness of the notched versus fatigues pre-cracked sample. both the notched and the fatigue pre-cracked samples in those investigations displayed extremely high ductility, displaying pronounced shear banding for the both type of samples. the energy absorptions and the thus assessed toughness level was comparable because the failures mechanisms for the both type of sample was the same. for the extremely brittle bmg like fe based bmg, there hasn't yet been a study that compares the toughness value of the notched vs fatigues pre-cracked specimen (according to the astm e399). the sam2x specimens that were notched were not fatigue pre-cracked by us. although no shear banding or crack bifurcation was seen in any of the notch toughness specimens examined in this study, all of them showed planar crack fronts, suggesting failure processes comparable to those of the fatigue pre-cracked specimens on zr-based bmgs reported in [36]. as a result, even though the current work's toughness specimens lack fatigue cracks, the way cracks spread suggest that the notched toughness value would be consistent with the toughness value for the sam2x5 specimen that have fatigue cracks. nishida et al. [39] examined the impact of the notched root radius of the fracture’s toughness of the fine-grained ceramics with the fracture’s toughness values comparable to the sam2x. the sintered al2o3 sample with the grain sizes of the 1 µm was evaluated utilizing the notch with a range of the root radii (3 µm to 100 µm) using a method identical to ours. the researchers discovered that, below 10 m, values were stable, and fracture toughness declined with decreasing notch root radius. the difference between a root radius of 40 micrometres (as in the current study) and one of 10 micrometres (as in the study) decreased by around 25%, suggesting that the notched toughness values were measured are likely overstated. for more reliable fracture toughness measurements in future, a tiny (sharp) notched root radius (approaching fatigues pre-crack) was preferred. a figure that is 25% lower than what we recorded would result from assuming the sharper roots radius (i.e., less than the 10 µm), which would be simulating fatigue crack. there it appeared to be a lower limit to the relationship between fracture toughness and notch root radius. result from the sintered al2o3 systems indicates that the values measured below the 10µm were identical. according to the theory, fracture toughness values would be comparable as long as notched root radius which is small enough to create same crack "systems" as the fatigues pre-cracked instance. we made sure that the notched root radius was as small as possible because fatigue cracks on the specimens for fracture toughness could not be obtained. our sam2x5 specimens show planar crack fronts without shear banding. we, therefore, think that the results of our three-point bending are fairly close to the results of specimens that had undergone fatigue cracking. three-point bending fracture toughness values may be lower when using a much sharper notch, but we are unsure exactly how much less. however, by employing the three-points bending, we had removed several potential sources of the error in the vickers indentations, including poorly defined as the crack length and the calibration values that could produce value that mechanical characteristics of nano-crystalline material in metallic glass formers 325 differs by either + or -40% of "real" values. we are aware that the notched root radius, which has made an effort to reduce as much as reasonable, maybe the primary cause of the error. since sps sam2x5's absolute fracture toughness cannot currently be measured, we have believed that measuring fractures toughness using our method will result in the more accurate values. we have also provided estimates of the size of the errors and the potential directions of the real values. therefore, it would be very advantageous to apply this method described in the paper for measuring fractures toughness to other brittle bulk metallic glasses. nanoindentation measurements were used to map the hardness and modulus in two dimensions to look at the impact of tungsten on the sam2x5 matrix, as shown in fig. 7 the typical loading curves for the two stages. to examine the mechanical characteristics of the sam2x5 matrix phase, the hardness and modulus maps from a 10 x 10 or 15 x 15 nanoindentation array with a spacing of 4 µm were prepared. nanoindentation has been utilized by many authors to examine bmgs [9,33–35,40]. for instance, a large soft zone and local core of a shear band have been studied using a mapping of nanoscale structural characteristics [2]. recent researches have used the 3-d contour function to assess how heterogeneous structure affects the amorphous matrix phase. fig. 7 typical loading curves in the two phases a weibull curve of the sam2x5-30%w composites' flexural strength or modulus of rupture (mor) is shown in fig. 8. from the weibull plot, the characteristic strength, σo, and weibull modulus, m, were calculated to be 165 and 8.7 mpa, respectively. 326 h. al-abboodi, h. fan, m. al-bahrani, a. abdelhussien, b. mohamad fig. 8 weibull plot for modulus of rupture (mor) calculated for the sam2x5/30%w composites. 4. conclusion for the fe based bulk metallic glasses, the flexural modulus and the fractures (notch) toughness were calculated to be the 230 gpa and the 4.9 mpam1/2, respectively. the flexural modulus measurements were much lower and more accurate representations of bulk properties than those obtained using nanoindentation. the testing procedures and specimens for fracture toughness were as close to astm e399 requirements as possible. with all ratios as recommended, the specimen’s size is roughly half of the recommendation’s values. three-point bending measurements of the notched toughness of the specimen without the fatigue cracks (4.9 mpam1/2) with the relatively sharp notched root radii of (40 µm) are more accurate than vickers indent measurements of fracture toughness (50 percent lower). the observed variations between three-point bending measurements and indentation measurements are anticipated. due to the large specimen’s size, specimens’ geometries such as (rectangular versus the circular), and the specimen’s preparation, bulk of the fracture’s toughness and the modulus measurements of the bmgs utilizing our new testing techniques are much more precise than the value measured by indentations. the inclusions and flaws that are more likely to be present in our larger specimen size make the measured values for fracture toughness more indicative of the entire sample. there are a few sources of measurement of errors for the crack tips in rectangular specimen’s geometries compared to cylindrical specimens because they have constant cross sections. finally, specimens’ preparations utilizing the wire saw cutting reduce the amount of the damage layer by minimizing plasticity and heat damage in comparison to other laboratories' usage of edm. acknowledgments: this work is supported by the national key research and development project (2020yfc1521900 & 2020yfc1521904), the 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tehnički vjesnik, 30(1), pp. 61-67. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 217 229 doi: 10.22190/fume170515010k © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper robot learning of object manipulation task actions from human demonstrations udc (004.896:61):681.5.01 maria kyrarini, muhammad abdul haseeb, danijela ristić-durrant, axel gräser institute of automation, university of bremen, germany abstract. robot learning from demonstration is a method which enables robots to learn in a similar way as humans. in this paper, a framework that enables robots to learn from multiple human demonstrations via kinesthetic teaching is presented. the subject of learning is a high-level sequence of actions, as well as the low-level trajectories necessary to be followed by the robot to perform the object manipulation task. the multiple human demonstrations are recorded and only the most similar demonstrations are selected for robot learning. the high-level learning module identifies the sequence of actions of the demonstrated task. using dynamic time warping (dtw) and gaussian mixture model (gmm), the model of demonstrated trajectories is learned. the learned trajectory is generated by gaussian mixture regression (gmr) from the learned gaussian mixture model. in online working phase, the sequence of actions is identified and experimental results show that the robot performs the learned task successfully. key words: robot learning by demonstration, dynamic time warping, gaussian mixture model, gaussian mixture regression, sequence of actions 1. introduction one of the main research topics in robotics community in the last two decades is development and implementation of methods to teach robots in a “human-like” way to perform particular tasks [1-4]. these methods are generally called “robot learning from demonstration”, “robot programming by demonstration” or “imitation learning”. a human “teacher” shows (demonstrates) his/her knowledge to the robot learner and robot learner uses the demonstrated knowledge to execute particular robotic tasks. received may 15, 2017 / accepted june 29, 2017 corresponding author: maria kyrarini institute of automation, university of bremen, otto-hahn-allee 1, 28359 bremen, germany e-mail: mkyrar@iat.uni-bremen.de 218 m. kyrarini, m. a. haseeb, d. ristic-durrant, a. gräser kinesthetic teaching [5-7] is a popular method for “learning from demonstration”, where the teacher manually guides the robot’s end effector throughout the task while the robot movements are recorded by the robot’s sensors (joints motors’ encoders) thus enabling the robot’s learning of the skills needed for performing the demonstrated task. this method works for light-weighted robots or robots driven by gravity-compensation controllers. however, learning from one human teacher has limitations for, if the teacher makes mistakes during the demonstration, the robot will be vulnerable to those mistakes. a way of overcoming this problem is to enable robot learning from multiple human demonstrations. as the different human demonstrations possibly lead to differently demonstrated tasks, an optimally learned task could be an outcome of a combination of different demonstrations [5]. given a dataset of the task demonstrations that have been acquired using kinesthetic teaching, the robot learner must be able to learn a skill from the acquired data. there are different approaches to abstracting (representing) and reproducing a skill from the datasets of demonstrations. these approaches are grouped, according to [8] and [9], in the following categories: learning a skill at the trajectory level (low-level learning) in this approach, the robot learns particular movements. this approach allows encoding of different types of trajectories that represent different types of gestures but does not allow reproducing of complicated high-level skills such as an assembly task. in [10], the gaussian mixture regression (gmr) is used in order to map the 3d human pose, recorded with a vision system, to the pose of a humanoid robot. multiple humans demonstrate a pose and the different recorded datasets are at first projected in latent spaces of motion by using the principal component analysis (pca) and then aligned temporally using the dynamic time warping (dtw). the aligned signals are encoded in the gaussian mixture model (gmm), which allows an autonomous representation of the gesture. the gmr is used to extract constrains of the gesture and to retrieve such a generalized version of the gesture that the robot can reproduce. symbolic or task learning (high-level learning) in this approach, the task is encoded according to sequences of predefined motion elements which are described symbolically. this approach allows the robot to learn hierarchy, rules and loops, so as to learn high-level tasks [11]. a disadvantage of the symbolic learning is its reliance on a large amount of prior knowledge needed for abstraction of important cues. for abstraction and recognition of highlevel tasks, hidden markov models (hmms) have been widely used. the hmm-based frameworks are used to generalize movements demonstrated to a robot multiple times, as can be seen in [12-14]. the redundancies across all the demonstrations are identified and used for the reproduction of the robot movements. contrary to the above mentioned methods, which are based either on lowor on highlevel learning, in this paper, a framework for robot learning which combines the highlevel learning and low-level learning at the trajectory level is presented. it is based on learning from multiple human demonstrations via kinesthetic teaching. the paper is organized as following: section ii gives an overview of the proposed robot learning framework, section iii presents a detailed analysis of the offline learning phase, section iv explains the online working phase, section v presents the experimental results and section vi concludes the presented work. robot learning of object manipulation task actions from human demonstrations 219 2. overview of the robot learning framework the robot learning framework is separated into two main modules: the offline learning phase and the online working phase, as illustrated in fig. 1. the presented robot learning framework has been developed and implemented onto a two-arm robot manipulator aimed for a collaborative work with human in an industrial assembly scenario. pi4 workerbot 3 [15] is used as a robotic platform. it consists of two ur10 robotic arms [16] and has gravity-compensation controllers, which makes kinesthetic teaching possible. a vacuum gripper is connected as end-effector to each robotic arm. fig. 1 block-diagram of the robot learning and reproduction framework the offline learning phase consists of data acquisition and learning modules. the data acquisition module records and stores into the database the angles and the pose of, respectively, joints and the end-effectors of the robotic arms. also, the gripper actuation status (“on” denoting the activated gripping status and “off” denoting not-activated gripping status) during the human demonstrations of the task via kinesthetic teaching is recorded and stored. 220 m. kyrarini, m. a. haseeb, d. ristic-durrant, a. gräser additionally, the data acquisition module receives and stores the data obtained from the environmental perception module: the pose (position and orientation with respect to world coordinate system) and dimensions of every object in the field of view of the robot vision-based system. in the presented work, a working table with the objects placed on it is in the field of view of the robot vision-based system using the kinect [17] camera. the learning module consists of the following two sub-modules: task or symbolic learning (high-level learning) and learning at the trajectory level (low-level learning). section iii gives details on both sub-modules of the learning module. in the online working phase, the robot has to reproduce the learned task by identifying the objects. a virtual environment for providing situation awareness to the robot has been deployed for visualization of the task actions before they are executed by the robot. section iv provides more details about the online working phase. 3. offline learning phase in the presented work, the robot has to learn the sequence of basic actions needed to perform an object manipulation task. these basic actions are: “grasping of an object”, “moving along an optimal trajectory from grasping to releasing position while carrying the object”, “releasing the object” and “moving away from the working table”. several human teachers are asked to teach the robot the task of assembly of 3 parts (objects). during the task demonstrations, the human teachers had to guide the robotic arm by holding its endeffector (gripper), while the robot arm was in zero-force control mode. there were no other constraints in the teaching of the task. during the demonstrations, the data acquisition module recorded the end-effector’s pose, the gripper status, as well as the pose and dimensions of the objects to be manipulated. the learning module performed learning at two levels: learning at the trajectory level (low-level) and task or symbolic learning (high-level). 3.1. learning at the trajectory level (low-level learning) during the considered multi-human demonstrations of moving the robot’s gripper (end-effector) from one point to another on the working table, the cartesian coordinates (x, y, z) and the orientation (in quaternions) of the gripper’s tip were recorded. an automatic dynamic time warping (dtw)-based algorithm [18] was used to select the most similar demonstrations. further, the dtw was used to align the demonstrations from the selected similar demonstrated trajectories, and the gaussian mixture model (gmm) and the gaussian mixture regression (gmr) methods were used to enable learning of the executed gripper’s trajectory with its constrains [19].  automatic selection of similar demonstrations and alignment of the selected demonstrations the recorded datasets had different number of samples, because every human demonstrator performed the task of guiding the robot arm’s gripper with different speed which caused different lengths of the recorded demonstrations. the dynamic time warping (dtw) [20] is a method for finding an optimal alignment between two given time-series which may vary in speed and time. dtw-based algorithms are currently used for speech recognition [21], gesture recognition [22], robot learning [23], gait analysis [24] and for other sensorbased applications. the fundamental functionality of dtw is to define an optimal robot learning of object manipulation task actions from human demonstrations 221 warping path (alignment) and to calculate the dtw distance (similarity) between two given time-series. the optimal warping path is that with the minimal total cost among all possible ones. the dtw distance is defined as the total cost of the optimal warping path. the algorithm for automatic selection of similar demonstrations [18] is used to select similar demonstrations based on the similarity measurement between the cartesian coordinates (x, y, z) of the end-effector recorded in different demonstrations. however, the method presented in [18] does not take into account the orientation of the end-effector, which is an important parameter for reliable object manipulation. in the approach presented in this paper, the original method [18] is extended to include the recorded orientations of the end-effector in quaternions (qx, qy, qz, qw). the similarity vector is calculated as follows: 7 1 ( ) ( , ), {1,2, , } n d j similarity i dtw i j i n     (1) where n is the total number of the demonstrations and dtw7d(i, j) is the distance matrix in 7 dimensions which is calculated as: ),(),(),(),( ),(),(),(),(7 jidtwjidtwjidtwjidtw jidtwjidtwjidtwjidtw qwqzqyqx zyxd   . (2) matrices dtwx(i, j), dtwy(i, j), dtwz(i, j), dtwqx(i, j), dtwqy(i, j), dtwqz(i, j), dtwqw(i, j) are dtw distances between demonstrations i and j in dimensions x, y, z and quaternions (qx, qy, qz, qw) where i, j{1, 2, …, n}. the smaller the dtw distance is, the more similar the two demonstrations are. for example, if demonstration i is compared with itself, the dtw distance is equal to zero, that is the element (i, i) of the distance matrices is equal to zero. the demonstration that has the smallest value in the vector similarity is the “reference” demonstration and is denoted with r. after deciding on the “reference” demonstration it is needed to find the demonstration which is most similar to the “reference” demonstration. the demonstration which has the minimum 7 ( , ), {1,2, , },ddtw r j j n j r   is selected as the most similar one. the reason that only two demonstrations are selected is because the dtw method is able to align only two time-series at the time. the two selected demonstrations are aligned in time (temporal dimension) by using the dtw for 7 dimensions (x, y, z, qx, qy, qz, qw).  gaussian mixture model and gaussian mixture regression the selected and aligned demonstrations are the input to the learning of trajectories needed to perform the task accurately. the gaussian mixture model (gmm) is used to extract constrains of the aligned trajectories [25] and the gaussian mixture regression (gmr) is used to produce the learned path which can be used to efficiently control robot movement [25]. the pair of selected and previously aligned demonstrations is fed into the learning system that trains the gmm in order to build the probabilistic model of the data [26]. each demonstration consists of data-points l={s, t}, where s  r d–1 , s is spatial variable, t  r, t is temporal variable and d is dimensionality. in the presented work dimensionality d is equal to 8 because each data-point consists of a vector of variables x, y, z, qx, qy ,qz, qw and temporal. in the learning phase, the model is created with a predefined number k of gaussians. each gaussian consists of the following parameters: mean vector, covariance matrix and the prior 222 m. kyrarini, m. a. haseeb, d. ristic-durrant, a. gräser probability. each gaussian has a dimensionality 8 equal to the dimensionality of data-points. the probability density function p(l) for a mixture of k gaussians is calculated according to the following equation [19]: 11 [( ) ( )] 2 1 1 ( ) . (2 ) . t l k k l k k l k d k k p e                 (3) where:  k are prior probabilities,  },{ ,, sktkk   are mean vectors, and,             skstk tsktk k ,, ,, are covariance matrices of the gmm. the parameters (prior, mean and covariance) of the gmm are estimated by the expectation-maximization (em) algorithm [27]. after the gmm parameters are learned for the task, the next step is to generalize the trajectory using gmr algorithm. the gmr retrieves the smooth trajectory through regression and has the advantage that generates a fast and optimal output from the mixture model of gaussians [18]. the trajectory, produced by the gmr, is used directly for efficient control of the robot’s movement. output trajectory ̂ of the gmr, which is stored in the robot task library, is calculated as:          k k skkt a 1 , ˆ,ˆ  (4) where:    k l t t k lp kp 1 ´ )|( )|(    and )()(ˆ , 1 ,,,, tkttkstksksk    , kk ,...,1 . 3.2. task or symbolic learning (high-level learning) the high-level learning module is responsible for the task segmentation into individual actions and learning of sequences of those actions. this module consists of three steps: labeling of objects to be manipulated, mapping of the gripper status onto the learned path and splitting the overall task into individual actions.  object labeling during the demonstration phase, the objects involved in the task are labeled with specific ids which denote the position of the robot’s gripper when the gripper actuation status is on or off and the robotic arm, left or right, which is used for object grasping or releasing. for example, the id “left_pick_1” means that the first object, which was picked up among all identified objects on the working table, was picked up by the left robot-arm and the id “left_place_1” denotes the identified object which was assembled with the object “left_pick_1”. this labeling method also indicates the necessary sequence of actions for the object manipulation task, as the objects to be manipulated are ordered as indicated by the id.  mapping of the gripper status onto the learned trajectory the cartesian pose of the robot’s end-effector (gripper) for the positions when the robot grasped or released an object is compared with the learned trajectory (output of gmr) and the closest point is labeled as an action point which is “grasping” or “releasing” point. robot learning of object manipulation task actions from human demonstrations 223  splitting of the task into individual actions after the mapping of the gripper actuation status onto the learned trajectory, the task is split into actions such as grasping and releasing of an object or moving actions based on the low-level learned trajectories. therefore, in the proposed learning framework, the robot learns the sequence of actions (high-level) to perform the task including the trajectory that needs to be followed (low-level). in the considered example task, the robot learns the following sequence of actions: grasp the object with the id “left_pick_1”, “move the grasped object along the learned path and release it so as to assembly it with the object of the id “left_place_1”. the position, orientation, size and id number of every object involved in the scene, with respect to the world coordinate system, are stored in the task robot library. 4. online working phase after the offline learning phase, the learned task (learned trajectory and learned sequence of actions) is added to the task robot library (trl). during the online phase, the trl is responsible for identifying and retrieving the task to be executed. during the online functioning, the pose and dimensions of every object are provided by the visionbased environmental perception module. trl identifies the objects based on the pose and dimensions and if there is a match with an object in a stored learned task, the trl will retrieve the learned task. in order to illustrate this awareness in an intuitive way to be easily understood by the human collaborator, a virtual environment has been developed using the ros-based tool rviz (ros visualization) [28]. the human collaborator can at first observe the robot performing the task in the virtual environment and subsequently can confirm if he/she is satisfied with the visualized robot’s performing so that the robot can “get a green light” to perform the task in real-world. if the human collaborator is not satisfied, he/she can retrain the robot by providing more demonstrations. 5. experimental results for the evaluation of the proposed learning framework, experimental studies were conducted. five human demonstrators were asked to demonstrate a manipulation task to the pi4 workerbot via kinesthetic teaching. as shown in fig. 2, the object manipulation task consists of the following actions: action 1: pick object a up with the left robot arm action 2: place object a onto the top of object c action 3: move the left robot arm away from the workspace (table) action 4: pick object b up with right robot arm action 5: place object b next to object c action 6: move the right robot arm away from the workspace (table) each human teacher demonstrates the task once. the data acquisition module records the end-effector’s pose for the left and right robot arms. for the sake of simplicity, in this section, only the processing of the cartesian position (x,y,z) for the end-effector of the left robot arm will be shown. fig. 3 shows the data recorded during the 5 demonstrations for the cartesian position of the left robot arm end-effector (gripper). 224 m. kyrarini, m. a. haseeb, d. ristic-durrant, a. gräser fig. 2 overview of the demonstrated task fig. 3 p(x,y,z) of the left robot arm gripper recorded during 5 different human demonstrations of the task robot learning of object manipulation task actions from human demonstrations 225 the first step of the learning module is learning at the trajectory level. the automatic selection of similar trajectories selected the demonstrations 4 and 5 for the left robot-arm. these two selected most similar demonstrations for the left robot-arm are shown in fig. 4, before and after their alignment with dtw. fig. 5-7 show the selected demonstrations after alignment together with the learned gmm models of the selected demonstrations and the trajectories generated by the gmr for each dimension x, y, z. fig. 4 selected demonstrations of the positions (x,y,z) of the left robot-arm gripper before and after alignment using dynamic time warping (dtw) fig. 5 left robot-arm x-dimension: learned gmm (above) and the trajectory generated by gmr (below) 226 m. kyrarini, m. a. haseeb, d. ristic-durrant, a. gräser fig. 6 left robot-arm y-dimension: learned gmm (above) and trajectory generated by gmr (below) fig. 7 left robot-arm z-dimension: learned gmm (above) and trajectory generated by gmr (below) the second step of the learning module is to learn the sequence of actions needed to reproduce the task. firstly, specific ids are assigned to the objects identified on the working table, as shown in fig. 8. it can be seen that object a is labeled as “left_pick_1” and object b is labeled as “right_pick_1”. object c is labeled as “left_place_1” and “right_place_1”, robot learning of object manipulation task actions from human demonstrations 227 since both objects a and b shall be placed next to object c. next, the mapping of the gripper status onto the learned trajectory and the splitting of the learned task (corresponding to the learned trajectory) into sequence of individual actions is completed. an example of mapping of the gripper status onto the dimension z is shown in fig. 9. in the online working phase, the trl recognizes the task based on the objects placed on the working table by comparing the dimensions and pose of the objects with the dimensions and pose of the objects stored in the database during the demonstrations of the task. as shown in fig. 10, the robot performs the learned task successfully. fig. 9 left robot-arm z-dimension: mapping of the gripper status onto the learned trajectory fig. 10 robot execution (reproduction) of learned task fig. 8 labeling of the objects with specific ids 228 m. kyrarini, m. a. haseeb, d. ristic-durrant, a. gräser 6. conclusion in this paper, a framework for the robot learning of the object manipulation tasks via multiple human demonstrations is presented. in the offline learning phase the robot learns the task at the trajectory level by using the algorithms for automatic selection of similar demonstrations, the dynamic time warping (dtw), the gaussian mixture model (gmm) and the gaussian mixture regression (gmr). additionally, with the automatic object labeling and the splitting of the demonstrated task into sequence of actions, the robot is able to learn the actions which are needed to perform the task successfully. the proposed learning framework has been experimentally tested with 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https://www.universal-robots.com/products/ur10-robot/ http://www.xbox.com/en-us/xbox-one/accessories/kinect facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 245 254 https://doi.org/10.22190/fume200129020l © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dynamical model of the asymmetric actuator of directional motion based on power-law graded materials iakov a. lyashenko 1,2 , vadym n. borysiuk 2 , valentin l. popov 1,3,4 1berlin university of technology, germany 2sumy state university, ukraine 3national research tomsk state university, russia 4institute of strength physics and materials science, tomsk, russia abstract. we consider an actuator whose driving bodies are made of power-law graded materials. the directional motion is generated by an asymmetric mechanism producing simultaneously vertical and horizontal oscillations of the indenter. the dynamic contact of gradient materials is described and the equation of motion for the drive is written down and analyzed. it is shown that the exponent of the elastic inhomogeneity significantly affects the average velocity of motion of the cargo, which can be dragged by the drive. key words: piezoelectric actuator, friction, method of dimensionality reduction, normal and tangential contact forces 1. introduction in modern robotics, electro-mechanical devices that convert electrical energy into mechanical energy are widely used to create directional motion. these devices can be realized as servo-drives [1], stepping motors [2], as well as piezoelectric drivers [3-5]. piezoelectric actuators may be based on different principles; yet all of them have piezoelectric elements, which generate superimposed normal and horizontal oscillations. the phase shift between these oscillations defines the modes of produced motion. such drives generate directional movement without an additional transmission mechanism in the system. however, if the actuator includes an asymmetric mechanism it is possible to received january 29 , 2020 / accepted march 30, 2020 corresponding author: iakov a. lyashenko institute of mechanics, berlin institute of technology, strasse des 17. juni 135, 10623 berlin, germany e-mail: i.liashenko@tu-berlin.de 246 i.a. lyashenko, v.n. borysiuk, v.l. popov create directional motion using single source that produces oscillations only in the vertical direction [3, 4, 6]. due to the asymmetry of the structure transmitting force from the piezoelectric element to the indenter, an additional tangential force occurs in the contact, which leads to the appearance of directional motion [3, 4, 6]. to produce repeating stationary directional motion, it is necessary to keep a certain ratio between normal and tangential forces, as this parameter strongly affects the characteristics of produced movement. such actuator of directional motion is a complex dynamic system, and its behavior is strongly affected by the inertial properties (mass of the cargo, moved by the drive) and the friction coefficient between the contacting elements. the elastic properties of the contacting materials and the geometrical shape of the indenter are also very important since these parameters determine the contact properties, and, therefore, the relationship between the normal and tangential forces. there are various methods for theoretical research and modeling of the above-described systems [7–9]. within modern simulation methods the components of the forces (internal parameters), and the trajectory of the indenter, shifted by these forces, can be determined if the geometric sizes and elastic characteristics of the indenter are known. however, if we already know the indenter trajectory, it is possible to use more efficient and faster methods of numerical calculation. thus, varying many different trajectories of the indenter in both directions, one can choose the optimal path which provides the necessary type of movement. in general, the materials with graded elastic properties have found practical application in engineering as cutting tools, engine components, machine parts and other various devices [10]. there are several methods of obtaining of graded materials. one of them is coating as there are always graded properties at the boundary between different materials [10, 11]. the main purpose of using mentioned materials is that in some conditions they exhibit alternative behavior, comparing to homogenous materials, and may somehow improve the properties of certain device of mechanism. in our previous study [12] we considered the behavior of the driving device parts which are made of homogenous materials. thus, knowing that the approach implemented in [12] can be generalized to a similar system with power-law graded materials, it is natural to investigate its behavior and examine its operations mode. another aim of our study is to check if the use of graded materials as contacting parts of the device can lead to any improvement or alternative behavior of the investigated system. in [12] we proposed the dynamic model of a piezoelectric drive, which allowed us to simulate the movement of a cargo (or a slider) from already known indenter trajectories in both directions. in the framework of the proposed approach, we have performed the simulation of the drive experimentally studied in [3]. in the present work, we generalize the approach developed in [12] to power-law graded materials. when the slider of the actuator is made of power-law graded material, it is possible to tune the motion mode more precisely, which opens up new possibilities in robotics. 2. numerical procedure as in the previous work [12], for modeling phenomena we shall use the method of dimensionality reduction (mdr), the framework of which, in the case of power-law graded dynamical model of asymmetric actuator of directional motion based on power-law graded materials 247 materials, is described in detail in [13]. we consider the case where the shear modulus of a power-law graded material e depends on normal coordinate z according to: 0 0 ( ) ( / )ke z e z c , (1) where exponent of elastic inhomogeneity k varies in the range 0 ≤ k < 1. the mdr approach allows finding exact solution for three-dimensional problem of a contact of axially symmetric bodies by mapping it onto an equivalent one-dimensional model. if both contacting bodies are made of the same material (as is the case in our study), the normal and tangential contact problems are decoupled [13, 14] and can be considered independently within the framework of mdr as well. let us consider the contact of an axially symmetric elastic object characterized by three-dimensional profile z = f (r) with elastic half-space. equivalent one-dimensional profile g(x) defined as [13, 15]: | | 2 2 1 0 ( ) ( ) | | d ( ) x k f r g x x r x r . (2) for simplicity let us consider parabolic indenter f (r) = r2/(2r) with radius r, in this case g(x) according to eq. (2) can be written as: 2 ( ) ( 1) x g x k r . when g(x) is known, the next step is to replace the elastic half-space by the winkler elastic foundation: a linear array of non-interacting springs with normal and tangential rigidities: 1 2 2 1 2 1 01 2 02 0 1 1 | | ( ) ( ) ( , ) ( , ) k z n n n x k x c x x x h k e h k e c n n n n , (3) 1 1 01 2 002 1 1 ( , ) ( , | | ( ) ( ) ) t t k x t x k x h k e h x c x x ck en n , (4) where functions hn (k,) and ht (k,) defined in open access literature [13] (see page 267 for exact definition), so we are not writing their exact form here. in eqs. (3) and (4) δx is the step of space discretization (distance between springs), and 1,2 are poisson's ratios of the indenter and half-space materials. elastic parameters e01 and e02 for indenter and half-space are also introduced (see eq. (1)). as mentioned before, we consider the situation where the indenter and the slider are made of the same material i.e. e01 = e02 = e0 and 1 = 2 = . in this case, instead of eqs. (3) and (4) we can use simplified expressions: 0 2 0 ( , ) | | ( ) 2(1 ) k n z h k e x k x x c n n , 0 0 1 | | (( ) 2 , ) t k x x k xh k ex c n , (5) 248 i.a. lyashenko, v.n. borysiuk, v.l. popov in the next step, one-dimensional profile g(x) is indented into the winkler foundation. if the indentation depth (position of the indenter) is d then the displacement of the individual springs in the contact can be obtained as 2 ( ) ( ) ( 1)z x u x d g x d k r . (6) denoting the radius of contact by a, we write uz(a) = 0, and resolve eq. (6) with respect to d: 2 ( 1) a d k r . (7) total normal force fz in the contact area is determined by the sum of the contributions of forces from all the compressed springs (we consider the case without adhesion) in the contact: 3 0 2 2 0 2 ( , ) ( ) ( ) ( )d (1 ) ( 1) ( 3) a k n z n z k a e h k a f a c x u x x c k k r n n . (8) simulation of the dynamic tangential contact within the mdr can be performed as follows. we first assume that all the springs that are in contact with the indenter are displaced along with the indenter during the tangential motion. we denote the tangential displacements of the individual springs as ux(xi). in each iteration the condition ( ) ( ) ( ) n z i t x i c x u x c u xm (9) is checked, where µ is the friction coefficient. it can be re-written as 2 ,( ) ( ) ( , ) ( ) (1 ) z i xt n i u x u h k h xk nm n n . (10) for the springs which fulfill this condition, a new value of displacements should be set: 2 ( , ) ( , ) ( ( ) ) ) 1 ( z t i x i n h k h u k u x x n n n m , (11) where the sign is the same as the sign of ux(xi) prior to the updating of its value. after this, tangential force (friction force) fx is calculated through the equation: 0 0 00 ( , ( ) 2 ( ) ( ) |d ) ( d| ) a a x t x x kt k f a c x u x x u e x x h x c k n . (12) for discretization of the numerical model, integrals (8) and (12) are replaced by the corresponding sums: ( ) ( ) z z z i cont f k x u x , ( ) ( ) x x x i cont f k x u x , (13) where the sum is taken over all the springs in contact. dynamical model of asymmetric actuator of directional motion based on power-law graded materials 249 3. dynamic model of the mechanic driver fig. 1 shows a schematic presentation of the actuator that was experimentally studied in [3]. an analogous system with the slider made of homogeneous material was theoretically studied in [12]. here the slider with mass m is placed on a hard-fixed base. the friction coefficient in the contact between the slider and the base equals µ1. the slider is moved by a parabolic indenter, the friction coefficient in the indenter-slider contact equals µ. the indenter and the slider are made of material with elastic modulus e (1), and poisson ratio . fig. 1 schematic presentation of the driver in fig. 1, horizontal and vertical coordinates of the indenter are denoted x and z, respectively. the indenter movement causes horizontal shift of slider x. the slider located on the fixed base did not move in vertical direction. in this work, we consider the ability of the driver to resist external force fm, which is applied against the direction of movement. therefore, our aim is to find the critical value of constant returning force fm at which the drive is still capable to shift the slider in positive direction ox. we consider the situation with an already known trajectory of an indenter (see fig. 1). in this simplified case, dynamical model for the slider movement consists of a single equation of motion [12]: x base m mx f f f , (14) where fx is tangential friction force between the indenter and the slider, calculated with eq. (13). the friction force in eq. (14) is written with the “+” sign since it is the driving force that causes movement of the slider. as the slider moves, there is a friction force fbase: 1 sgn( ) base z f x fm . (15) in eq. (15), normal force fz calculated with eq. (13). note that to take into account the direction of the acting force in eq. (15) we use standard sign function sgn( )x , with x v , where v is velocity of the slider. one of the most important features of the numerical model is correct determination of the displacement of the springs according to eqs. (9) – (11). here, to check the sliding criterion at each step, the indenter displacement must be added incrementally to the horizontal displacement of springs, and condition (9) must be checked for all springs. however, we consider the case where the slider can move simultaneously with the indenter. therefore, instead of displacement of indenter x, the relative displacement of 250 i.a. lyashenko, v.n. borysiuk, v.l. popov the indenter against slider x should be added to the elongation of the springs at each step. the mentioned displacement of the indenter can be calculated through the equation ( ) ( ) ( ) ( ) i i i i i x x t t x t x t t x t , (16) where ti is incrementally increasing time, δt is the step of numerical integration of eq. (14). to investigate the operating modes of the considered directional drive, we need to set the trajectory of the indenter shown in fig. 1. as such a trajectory, we shall use, analogously to [12], the dependences obtained from the experimental results in [3]. thus, the trajectory of the indenter can be described by a linear dependence as: 1 2 integer t n t t , (17) max 1 2 1 2 1 2 1 1 1 2 max 2 ( ) , ( ) ( ) ( 1)( ) , a t n t t if n t t t n t t t ta n t t t a otherwise t , (18) where amax is an amplitude, t is time, n is an integer and integer(.) is the integer division operation, indicating the period number. usually, in the experiments, the normal oscillation amplitude of the indenter is substantially smaller than the tangential amplitude. thus, we consider the time dependencies of indenter coordinates x(t) and z(t), obtained earlier in [12] that are shown in fig. 2. t, ms x, z ,  m 0 2 4 6 0 10 20 30 fig. 2 time dependencies of the indenter coordinate, shown in fig. 1. dependencies obtained from the eqs. (17) and (18) at parameters amax = 30 µm, t1= 1.9 ms, t2 = 0.1 ms for x(t) and amax =6 µm, t1= 1.7 ms, t2 = 0.3 ms for z(t). we shall use a set of parameters as in our previous work [12]: indenter radius r = 2.5 mm, the friction coefficient between the surfaces of the indenter and the slider µ = 0.3, the friction coefficient between the surfaces of the indenter and the base µ1 = 0.1, slider mass m = 50 g. previously we considered the case of the homogeneous materials with elastic parameters e = 2·1011 pa and ν = 0.3 [12], which correspond to steel. here we dynamical model of asymmetric actuator of directional motion based on power-law graded materials 251 consider the case where the elastic modulus of the indenter and the slider materials e depends on the indentation depth according to eq. (1). in further simulations we will set e0 = 2·1011 pa and ν = 0.3. then, at k = 0 we must arrive at the results, related to homogeneous materials, obtained in [12]. however, in [12] effective parameter for contact of homogeneous materials e* = 0.5/(1 – ν2) was mistakenly determined without coefficient 0.5. in [12] we used constant values of effective elastic modulus e* and effective shear modulus g*. from the relations between e* and g*, it is easy to see that the simulation performed in [12] corresponds to the set of parameters e = 1.9838·1011 pa and ν = 0.74076. such magnitude of poisson ratio is not physically relevant, as for thermodynamically stable materials, poisson ratio ν does not exceed 0.5 [16]. so in proposed work we will consider contact of homogenous materials with k = 0 as an additional specific case because in the case k = 0 we will have the situation e0 = 2·1011 pa and ν = 0.3, how it should have been in [12]. results of simulations for k = 0 shown in fig. 3. 0 4 8 12 16 -100 0 100 200 300 t, ms x ,  m m k=0 fig. 3 time dependencies of the coordinate x of the slider, shown in fig. 1. mass of the cargo m varies from 0 to 1.0 kg with increment 0.2 kg, mass increasing is shown with arrow for numerical solution of the eq. (14) we use the euler method with an integration time step δt = 10-7 s. the discretization step on coordinate (distance between the springs in the mdr method) was set as δx = 10-8 m. fig. 3, as in our previous work [12], presents the movement for varying mass of a cargo moved by the driver shown in fig. 1. the driver performs the work against the force field, where the force, acting against motion defined as gravity force fm = mg. it can be seen that the driver performs directional movement of the load, providing a positive average speed of the slider, provided the mass of the load does not exceed a certain critical value. it also follows from the figure that there are two modes of the driver operation: moving the slider in one direction (upper curves), and stick-slip mode of movement in which the slider velocity periodically changes its sign. in stick-slip mode it is important for the driver to provide a positive average velocity during one period, in this case it will continue to perform the work at moving cargo. next, we determine this average velocity, which depends on the mass of the cargo and the exponent of the elastic inhomogeneity of the material from which the slider and the indenter are made. from eq. (1), it follows that for c0 > 1 m and c0 < 1 m increasing (or decreasing) value of exponent k affects the behavior of the studied system in different ways (due to 252 i.a. lyashenko, v.n. borysiuk, v.l. popov the presence of factor 0 kc ). in this work, we will not consider these features, and will use value c0 = 1 m. results of the performed simulation at different values of the exponent of the elastic inhomogeneity k are shown in fig. 4. left panel of the figure shows the trajectories of the slider at the cargo mass m = 0.1 kg and three values of exponent k. t, ms 0 4 8 12 16 0 100 200 300 x ,  m t, ms 0 2 4 6 8 10 -5 0 5 10 15 f x,  k = 0.0 k = 0.2 k = 0.3 fig. 4 (left panel) time dependence of the slider coordinate x shown in fig. 1 at cargo mass m = 0.1 kg, and different values of the exponent k (shown in figure); (right panel) time dependencies of the tangential forces, acting in contact, related to dependencies in the left panel as can be seen fig.4, increasing of the exponent of the elastic inhomogeneity causes reduction of the efficiency of the device. as parameter k increases, material becomes softer, while normal and tangential forces acting in contact decrease, which leads to mentioned above situation. for instance, the right panel of fig. 4 shows time dependencies of tangential force with decreasing magnitude as the k growth. however, in all studied cases (k = 0, 0.2 and 0.3) the device works as the driver of directional motion, dragging the cargo against gravity force fm = mg. this fact shows that power-law graded materials can be used for manufacturing of such mechanical devices. it is worth mentioning that, according to fig. 4, all considered cases have a transient mode at the beginning of motion, followed by the stationary mode of directional motion after several periods. in [12] we analyzed averaged over period velocity of the slider, depending on the cargo mass m, which define gravity force fm = mg. in the case of power-law graded materials, to perform a similar analysis we need to take into account exponent of elastic inhomogeneity k, as it also affects the efficiency of the driver. we assume that the device reaches the stationary operation mode after first 5 periods (see fig. 4). thus, to estimate average velocity we skipped the first 5 periods, and during the sixth period, the average velocities were calculated. calculated dependencies of average velocity <v> of the slider on cargo mass and exponent k are shown in fig. 5. dynamical model of asymmetric actuator of directional motion based on power-law graded materials 253 m, kg 0 0.2 0.4 0.6 0 4 8 12 16 < v > , m m /s k 0 0.2 0.4 0.6 0.8 1 0 4 8 12 16 < v > , m m /s k = 0.0 k = 0 .1 k = 0 .2 k = 0 .3 m fig. 5 (left panel) dependencies of averaged over period velocity <v> of the slider on cargo mass m at different values of the exponent k; (right panel) dependencies of averaged over period velocity of the slider on exponent k at different values of cargo mass m = 0; 0.01; 0.02; 0.05; 0.1; 0.2; 0.3; 0.4; 0.5; increasing of mass is shown with arrow one can see that with increasing of gradient exponent k and cargo mass m, the driving velocity decreases. therefore, we can conclude that to ensure high drive performance it is best to use homogeneous materials with a constant value of the elastic modulus (with k = 0). however, the dependencies of average velocity on gradient exponent k (right panel of the figure) are not monotonic. so, it is not clear if an increase of exponent k leads to a decrease in the efficiency of the driver of directional motion. we examined the situation in which the indenter and the slider are made of the same power-law graded material. it is possible that in another situation (for example, a rigid indenter and a slider made of power-law graded material) it will be possible to achieve higher driver performance. 4. conclusions we developed a mathematical model of an actuator of directed motion with the driving parts made of power-law graded materials. we show that within the range of gradient exponent 0 ≤ k < 1 the driver can produce the directed motion of the cargo, performing positive work. moreover, although at first glance the situation of using homogeneous materials (k = 0) seems to be optimal, the dependence of the average slider velocity on the gradient parameter, is non-monotonous, which suggests that with different relations between the model parameters, a significant increase in drive efficiency may be achieved. in proposed work, we developed and described a model of the driver made of power-law graded materials; however, to clarify its effectiveness it is necessary to carry out further analysis with respect to additional parameters, which is not the purpose of this work. it is worth noting that another possible solution in the field of producing of the directed motion can be an interaction with a periodic relief created artificially or implemented at the molecular level [17, 18]. 254 i.a. lyashenko, v.n. borysiuk, v.l. popov acknowledgments: this work was partially supported by the fundamental research program of the state academies of sciences for 2013-2020, line of research iii.23 and the tomsk state university competitiveness improvement programme. vnb is grateful to the ministry of education and science of ukraine for financial support (project no. 0117u003923). references 1. ross, r., 2014, investigation into soft-start techniques for driving servos, mechatronics, 24(2), pp. 79-86. 2. wawszczak, w., towarek, z., jagiełło, b., 2004, methodology of the stepper motor rotational motion investigations, mechanics and mechanical engineering, 7(1), pp. 87-95. 3. cheng, t.h., he, m., li, h.y., lu, x.h., zhao, h.w., gao, h.b., 2017, a novel trapezoid-type stick-slip piezoelectric linear actuator using right circular flexure hinge mechanism, ieee transactions on industrial electronics, 64(7), pp. 5545-5552. 4. nguyen, h.x., teidelt, e., popov v.l., fatikow, s., 2016, modeling and waveform optimization of stick–slip micro-drives using the method of dimensionality reduction, archive of applied mechanics, 86, pp. 1771-1785. 5. li, j., zhao, h., shao, m., zhou, x., fan, z., 2015, design and experimental research of an improved stick–slip type piezo-driven linear actuator, advanced in mechanical engineering, 7(9), pp. 1-8. 6. popov, v.l., 2017, oscillation-based methods for actuation and manipulation of nano-objects, aip conference proceedings, 1882, pp. 020056: 1-5. 7. alshamasin, m.s., ionescu, f., al-kasasbeh, r.t., 2012, modelling and simulation of a scara robot using solid dynamics and verification by matlab/simulink, international journal of modelling identification and control, 15(1), pp. 28-38. 8. sui, l., xiong, x., shi, g., 2012, piezoelectric actuator design and application on active vibration control, physics procedia, 25, pp. 1388-1396. 9. kelemen, a., crivii, m., trifa, v., 1987, mathematical modeling and simulation of stepping motor systems, mathematical modelling, 8, pp. 544-549. 10. miyamoto, y., kaysser, w.a., rabin, b.h., kawasaki, a., ford, r.g. 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lifshitz, e.m., 1986, theory of elasticity, new york, pergamon press, 165 pp. 17. popov, v.l., 2004, nanomachines: a general approach to inducing directed motion at the atomic level, international journal of non-linear mechanics, 39(4), pp. 619-633. 18. popov, v.l., 2003, nanomachines: methods to induce a directed motion at nanoscale, physical review e, 68(2), 026608: 1-7. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 467 477 https://doi.org/10.22190/fume160812011s © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper forced convection drying of indian groundnut: an experimental study udc 662.6 ravinder kumar sahdev 1 , mahesh kumar 2 , ashwani kumar dhingra 1 1 mechanical engineering department, maharshi dayanand university, india 2 mechanical engineering department, guru jambheshwar university of sciences & technology, india abstract. in this paper, convective and evaporative heat transfer coefficients of the indian groundnut were computed under indoor forced convection drying (ifcd) mode. the groundnuts were dried as a single thin layer with the help of a laboratory dryer till the optimum safe moisture storage level of 8 – 10%. the experimental data were used to determine the values of experimental constants c and n in the nusselt number expression by a simple linear regression analysis and consequently, the convective heat transfer coefficient (chtc) was determined. the values of chtc were used to calculate the evaporative heat transfer coefficient (ehtc). the average values of chtc and ehtc were found to be 2.48 w/m2 oc and 35.08 w/m2 oc, respectively. the experimental error in terms of percent uncertainty was also estimated. the experimental error in terms of percent uncertainty was found to be 42.55%. the error bars for convective and evaporative heat transfer coefficients are also shown for the groundnut drying under ifcd condition. key words: groundnut/peanut, convective heat transfer coefficient, evaporative heat transfer coefficient, indoor forced convection drying 1. introduction groundnut or peanut (arachis hypogaea) is a perishable oilseed crop grown in tropic and sub-tropic countries [1]. it is rich in proteins (20 – 50%) and edible oil (40 – 50%) which makes it very popular all over the world [2]. it is also known by various names such as monkey nut, wonder nut, earth nut, cashew nut of poor men and so on [3]. it came into existence in india in the 16 th century. the worldwide production of groundnuts has received august 12, 2016 / accepted june 08, 2017 corresponding author: ravinder kumar sahdev maharshi dayanand university, rohtak – 124001, haryana, india e-mail: ravindersahdev1972@gmail.com 468 r. k. sahdev, m. kumar, a. k. dhingra reached about 40 million tons [4]. it is grown on 24 million hectares land throughout the world [4]. china with 16.70 metric million tons is the largest groundnut producing country followed by india with a production of 5 metric million tons [4]. in india about three quarters of groundnut is harvested in the kharif season (june – september) and the remaining quarter in the rabi season (november – march). indian groundnut is famous for its flavor, aroma and crunchiness. export of indian groundnuts has reached the record of 7 lac tons in 2014 – 2015 [5]. drying of agricultural products is the simplest traditional food preservation method which involves the removal of the water present in the product to stop fungus or bacteria growth [6]. groundnuts, just after being dug out from the ground, are required to dry to their safe moisture content of 8 – 10%. in the developing countries, poor farmers dry groundnuts under open sun drying (osd) mode which takes four to five days to dry the groundnuts to their safe moisture level. although the osd is unquestionably the cheapest post-harvest method, it involves many disadvantages such as deterioration of products due to dust, dirt, uncontrolled heating and discoloring of products because of ultra-violet rays, animals, microorganisms and so on. post-harvest losses of the agricultural products are estimated to be about 30 – 40% due to an improper method of drying [7-8]. moreover, farmers are also lacking behind with the better drying facilities. hence, the need is felt to adopt such a method which gives continuous and controlled drying. the convective heat transfer coefficient (chtc) for the drying of groundnut is an important and critical parameter required for the proper design of a dryer. it is governed by the temperature difference between groundnut surface and air, and the physical properties of the humid air which surrounds the groundnut surface. the researchers who have worked on the drying of various commodities under the forced convection drying mode are summarized in table 1. some authors who have also studied the drying of groundnuts under the forced mode are summarized in table 2. table 1 summary of drying of various commodities s. no. researcher ref. commodity mode of drying conclusion/remarks 1 akpinar (2004) [9] apple, strawberry, eggplant, garlic, mulberry, onion, pumpkin, potato indoor forced convection drying (ifcd) the value of chtc was reported to lie within the range of 0.64 – 7.12 w/m2 oc 2 kumar et al. (2011) [10] papad osd and ifcd the values of chtc were found to be 3.54 w/m2 oc and 1.56 w/m2 oc under osd and ifcd modes, respectively 3 anwar and singh (2012) [11] indian gooseberry ifcd the values of chtc were observed to vary from 18.67 to 116.55 w/m2 oc 4 sahdev et al. (2012) [12] vermicelli ifcd the value of chtc was reported to vary from 0.98 to 1.10 w/m2 oc 5 sahdev et al. (2013) [13] corn kernels ifcd the value of chtc was found to vary from 1.02 to 1.04 w/m2 oc. 6 kumar (2014) [14] khoa ifcd the values of chtc and evaporative heat transfer coefficient (ehtc) were observed to vary from 1.93 to 2.51 w/m2 oc and 1.94 to 2.49 w/m2 oc, respectively forced convection drying of indian groundnut: an experimental study 469 table 2 summary of groundnuts drying under forced mode s. no. researcher ref. mode of drying conclusion/remarks 1 ahmed et al. (1967) [15] osd and accelerating drying carried out the comparative study 2 blankenship and chew (1979) [16] trailer drying presented single and double trailer drying 3 troeger and butler (1980, 1980a) [17-18] solar drying dryer used solar heated air, solar heated water and lpg gas 4 nawungkalatusart and tamtawatchai (1989) [19] continuous drying proposed 55 oc suitable for the germination of groundnut seeds 5 premkumar (1990) [20] various dryers proposed forced air oven drying as superior than batch type rotary and conduction dryer 6 gowda et al. (1991) [21] forced drying investigated the drying at different temperatures 7 noomhorn et al. (1992, 1994) [22-23] conduction dryer (cd) proposed the cd for drying groundnuts 8 syarief et al. (1996) [24] convection dryer peanut seeds were investigated in dryer which used coconut as a fuel 9 tumbel et al. (1997) [25] rack dryer presented the rack type dryer for drying groundnuts 10 ertas et al. (1999) [26] trailer dryer proposed semi-trailer dryer for drying peanuts at constant temperature of 35 oc 11 jain et al. (2004) [27] solar dryer proposed forced solar dryer for groundnut drying 12 palacios et al. (2004) [28] batch dryer studied the remoistened peanut in batch type dryer 13 tarigan and tekasakul (2005) [29] indirect solar dryer (isd) isd efficiency was reported to be 23% 14 ezekoya and eneba (2006) [30] modified solar dryer (msd) dryer efficiency was found to be 10% 15 ahmed and mirani (2012) [31] mobile flat-bed dryer moisture was maintained up-to 14% 16 mennouche et al. (2014, 2015) [32-33] direct solar dryer (dsd) and isd yield of oil in dsd and isd dried peanuts were found to be higher the above-listed literature leads to observation that different agricultural products have been dried under ifcd conditions to reduce the drying time and increase the quality. the values of chtc and ehtc for the drying of various products under ifcd mode are reported to vary from 0.16 to 116.55 w/m 2 o c and 1.94 to 2.49 w/m 2 o c, respectively. although groundnuts/peanuts have also been dried by different artificial and mechanical dryer to improve the quality and storage life. studies to evaluate the important parameters such as chtc and ehtc for designing a dryer for groundnut drying have not been found. therefore, the present study has been undertaken to determine the chtc and ehtc of groundnut drying under ifcd mode. this study would be helpful in designing a better dryer for drying groundnut to its safe moisture storage level. 470 r. k. sahdev, m. kumar, a. k. dhingra 2. materials and methods 2.1. experimental set-up the groundnut sample of 180 g was kept in a wire mesh tray (rectangular shape of 0.15×0.25 m 2 size) over the digital weighing balance of 6 kg capacity (least count = 0.1 g). a heat convector (model fh-812t, usha shriram, made in india) was used for blowing hot air over the groundnut surface. the temperature of groundnut surface (tg) was measured by calibrated copper constantan thermocouples connected to a 12–channel digital temperature indicator (least count = 0.1 o c). the thermocouples were calibrated with respect to the zeal thermometer which gives accurate readings. relative humidity (γ) and temperature of exit air (te) was measured by a digital hygrometer (lutron – ht 315, least count: 0.1% rh and 0.1 o c temperature). air velocity (va) over the groundnut surface was measured with a digital anemometer (lutron, am – 4201, taiwan, least count: 0.1 m/s, accuracy ± 2% on the full scale range of 0.2 – 30.0 m/s). the schematic view and photograph of the experimental set up are shown in figs. 1 and 2, respectively. fig. 1 schematic view of the experimental set up under ifcd mode fig. 2 experimental set up under ifcd mode forced convection drying of indian groundnut: an experimental study 471 2.2. sample preparation and experimental procedure fresh groundnuts were purchased directly from the farmer and cleaned to remove immature and broken pods. groundnuts were remoistened by soaking in water for 12 hours. then the samples were conditioned in shed for one hour so that the extra moisture was removed. then the groundnut sample was used for experimentation. the experiment was performed in february, 2016 in the climatic conditions of rohtak, india (28 o 54’0’’n 76 o 34’0’’e). the groundnut sample in a thin layer was kept in a wire mesh tray of size 0.25×0.15 m 2 directly over the digital weighing balance. the difference in weight between two consecutive 30 min. time interval observed readings directly gave the moisture evaporation during the observed time interval. the 30 min. data for the moisture removal, groundnut surface and ambient temperatures, relative humidity and temperature just above the groundnut surface were recorded. the groundnut sample was dried to its optimum safe moisture level of 8 – 10%. 2.3. thermal modeling the chtc under ifcd can be evaluated by using following eq. [1]: ( )nc v h x nu c repr k   (1) where nu is the nusselt number, hc is the convective heat transfer coefficient, x is the characteristic dimension, kv is the thermal conductivity of the humid air, c and n are the experimental constants, re is the reynolds number and pr is the prandtl number. from eq. (1) one can write: ( )nv c k h c repr x  (2) the rate of heat utilized to evaporate moisture, qe, is given by eq. (3) [34]: 0.016 [ ( ) ( )]e c g eq h p t p t  (3) where p(tg) and p(te) are partial vapor pressures at temperatures tg and te, respectively. substituting the value of chtc i.e. hc from eq. (2), eq. (3) becomes 0.016 ( ) [ ( ) ( )]nv e g e k q c repr p t p t x   (4) the moisture evaporated, mev, is determined by dividing eq. (4) by latent heat of vaporization, λ, and multiplying by tray area atray, and time interval, t. 0.016 ( ) [ ( ) ( )] ne v ev tray g e tray q k m ta c repr p t p t ta x       (5) let 0.016 [ ( ) ( )] v g e tray k p t p t ta z x     ( )nevm c repr z   (6) 472 r. k. sahdev, m. kumar, a. k. dhingra taking logarithm on both sides of eq. (6), we get ln ln ln( )evm c n repr z        (7) eq. (7) is the form of a linear equation: cmxy  (8) where        z m lny ev , m = n, x = ln (repr), c = lnc. thus, c = e c the values of m and c in eq. (8) are obtained by using simple linear regression formulae. the ehtc, he, is evaluated as [35] ( ) ( ) 0.016 g e e c g e p t p t h h t t         (9) 2.4. physical properties of the humid air the thermo-physical properties of the humid air, namely, thermal conductivity, kv, dynamic viscosity, μv, density, ρv, specific heat, cv, and partial vapor pressure p(t) were calculated for the mean temperature ti = [(tg+te)/2] by using the following eqs. (10 – 14) [36]: iv t..k 4107673002440  (10) iv t.. 85 106204107181   (11) 3824 1075816101011143402999 iiiv t.t.t..c   (12) 15273 44353 .t . i v   (13) 5144 ( ) exp 25.317 273.15 p t t       (14) 2.5. experimental error and external uncertainty the experimental error was determined in terms of percent uncertainty (internal + external) for the mass of moisture evaporated. eqs. (15) to (17) were used to calculate internal uncertainty [37]: n ... u n* 22 3 2 2 2 1   (15) where σ is the standard deviation and is by eq. (16): ( )i i o x x n     (16) forced convection drying of indian groundnut: an experimental study 473 where xi is the moisture evaporated and ( )i ix x is the deviation of the observations from the mean value, n and no are the number of sets and number of observations in each set respectively. the percent uncertainty was evaluated as: 100 nsobservatioofnumbertotalofaverage u yintuncertaernalint% * (17) the external uncertainty is the least count of all the instruments. 2.6. computation technique the average of groundnut surface temperature (tg) and exit air temperature (te) after the groundnut surface were determined at 30 minutes time interval for corresponding moisture evaporation. the physical properties of the humid air were evaluated for the mean temperature [ti = (tg +te)/2] using eq. (10) to (14). these properties of the humid air and air velocity were used to calculate the prandtl number (pr) and reynolds number (re). the values of experimental constants c and n in eq. (1) were determined by using the linear regression technique analysis, and hence the value of chtc (hc) was evaluated. then, the value of ehtc (he) was calculated by using equation (9). 3. results and discussion the experimental data obtained for groundnut drying under ifcd mode is given in table 3. the data given in table 3 were used to determine the values of the experimental constant ‘c’ and exponent ‘n’ in the nusselt number expression by simple linear regression. then the values of constants c and n in eq. (2) were used to evaluate the chtc (hc). further, the value of ehtc (he) was calculated by substituting the value of hc in eq. (9). the computed values of constants c and n, hc, and he for groundnut drying under ifcd are summarized in table 4. the values of reynolds number (re) and prandtl number (pr) are also given. the product of re and pr were observed to be less than 10 5 , (i.e. repr ≤ 10 5 ). this indicated that the entire groundnut drying under ifcd mode lies within the laminar region [38]. table 3 experimental data for groundnut drying under ifcd condition drying time (min) tg ( o c) te ( o c) mev ×10 -3 (kg) γ (%) re ×10 4 pr 30 29.9 25.83 24.3 38.78 1.87 0.6982 60 44.5 35.33 11.7 22.75 1.74 0.6963 90 45.7 36.52 8.1 19.88 1.73 0.6961 120 46.5 36.98 4.5 19.12 1.72 0.6960 150 46.8 37.42 3.2 18.42 1.72 0.6959 table 4 values of c, n, hc, and he c n hc (w/m 2 o c) hc, avg (w/m 2 o c) he (w/m 2 o c) he, avg (w/m 2 o c) 0.98 0.31 2.45 – 2.49 2.48 28.08 – 38.73 35.08 474 r. k. sahdev, m. kumar, a. k. dhingra from table 3, it is observed that the rate of moisture removal is faster in the initial stage and decreases with the increase in drying time. the values of convective heat transfer coefficient (hc) are observed to vary from 2.45 to 2.49 w/m 2 o c. its average value is found to be 2.48 w/m 2 o c. variation of hc with respect to time is illustrated in fig. 3. it is observed from fig. 3 that the value of hc does not vary much and is almost constant throughout the experiment. the values of evaporative heat transfer coefficient (he) are observed to vary from 28.08 to 38.13 w/m 2 o c. the average value of he for groundnut drying is found to be 35.08 w/m 2 o c. the variability in he is observed to be 37.93% which is more than the variability in hc. the variation of he with time is shown in fig. 4. the photographs of groundnut drying before and after drying are shown in fig. 5. the computed values of experimental error in terms of percent uncertainty (internal + external) are given in table 5. the error in the experimental measurements of chtc and ehtc is shown by the error bar which shows the graphical representation of the variability of data. the variability of chtc and ehtc from its true value is shown by the error bars, with 95% confidence level, in fig. 6 which is drawn with the help of spss software (version 24). fig. 3 variation of hc with time fig. 4 variation of he with time forced convection drying of indian groundnut: an experimental study 475 fig. 5 photograph of groundnut sample before and after drying table 5 experimental percent uncertainties for groundnut drying under ifcd mode internal uncertainty (%) external uncertainty (%) total uncertainty (%) 42.05 0.5 42.55 fig. 6 error bars for chtc and ehtc 4. conclusions the following conclusions are made from the present study in which convective heat transfer coefficient (chtc) and evaporative heat transfer coefficient (ehtc) for groundnut drying under indoor forced convection drying (ifcd) mode are evaluated. 1. the value of chtc for the drying of groundnut drying under ifcd mode is found to vary from 2.45 to 2.49 w/m 2 o c. the average value of the chtc for the drying of groundnut is observed to be 2.48 w/m 2 o c. the chtc is observed to be almost constant throughout the experiment. 2. the value of ehtc for the drying of groundnuts under ifcd mode is found to vary from 28.08 to 38.73 w/m 2 o c. the average value of ehtc for drying of groundnut under ifcd mode is observed to be 35.08 w/m 2 o c. the variability of he is observed to be more than the chtc. 476 r. k. sahdev, m. kumar, a. k. dhingra 3. the experimental error in terms of percent uncertainty (internal + external) for the drying of groundnuts under ifcd mode is computed as 42.55%. 4. this research work will be helpful in designing a better dryer for drying groundnuts to retain their quality during storage so that the farmers in the developing countries can increase the storage life of groundnuts. references 1. woodroof, j. g., 1983, peanut: production, processing, products, westport, ct: avi. 2. sahdev, r. k., kumar, m., 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20, no 3, 2022, pp. 561 587 https://doi.org/10.22190/fume220506029l © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper bending, buckling and free vibration analyses of nanobeam-substrate medium systems suchart limkatanyu1, worathep sae-long2, jaroon rungamornrat3, chinnapat buachart4, piti sukontasukkul5, suraparb keawsawasvong6, prinya chindaprasirt7,8 1department of civil and environmental engineering, faculty of engineering, prince of songkla university, songkhla, thailand 2civil engineering program, school of engineering, university of phayao, phayao, thailand 3applied mechanics and structures research unit, department of civil engineering, faculty of engineering, chulalongkorn university, bangkok, thailand 4department of civil engineering, faculty of engineering, chiang mai university, chiang mai, thailand 5construction and building materials research center, department of civil engineering, king mongkut’s university of technology north bangkok, bangkok, thailand 6department of civil engineering, faculty of engineering, thammasat school of engineering, thammasat university, pathumthani, thailand 7sustainable infrastructure research and development center, department of civil engineering, faculty of engineering, khon kaen university, khon kaen, thailand 8academy of science, royal society of thailand, dusit, bangkok, thailand abstract. this study presents a newly developed size-dependent beam-substrate medium model for bending, buckling, and free-vibration analyses of nanobeams resting on elastic substrate media. the euler-bernoulli beam theory describes the beam-section kinematics and the winkler-foundation model represents interaction between the beam and its underlying substrate medium. the reformulated strain-gradient elasticity theory possessing three non-classical material constants is employed to address the beam-bulk material smallscale effect. the first and second constants is associated with the strain-gradient and couplestress effects, respectively while the third constant is related to the velocity-gradient effect. the gurtin-murdoch surface elasticity theory is adopted to account for the surface-free energy. to obtain the system governing equation as well as corresponding boundary conditions, hamilton’s principle is called for. three numerical simulations are presented to characterize the influences of the material small-scale effect, the surface-energy effect, and the surrounding substrate medium on bending, buckling, and free vibration responses of received: may 06, 2022 / accepted june 11, 2022 corresponding author: worathep sae-long civil engineering program, school of engineering, university of phayao, phayao, 56000, thailand e-mail: worathep.sa@up.ac.th 562 s. limkatanyu, w. sae-long, j. rungamornrat, et al. nanobeam-substrate medium systems. the first simulation focuses on the bending response and shows the ability of the proposed model to eliminate the paradoxical characteristic inherent to nanobeam models proposed in the literature. the second and third simulations perform the sensitivity investigation of the system parameters on the buckling load and the natural frequency, respectively. all analytical results reveal that both material small-scale and surface-energy effects consistently stiffen the system response while the velocity-gradient effect weakens the system response. furthermore, these sized-scale effects are more pronounced when the underlying substrate medium becomes softer. key words: reformulated strain gradient elasticity theory, small-scale effect, surfaceenergy effect, bending analysis, buckling analysis, free vibration analysis 1. introduction with the discovery of carbon nanotubes (cnts) by iijima [1] in 1991, distinct responses of nano-sized structures have drawn high attention from researchers worldwide. as a result, several innovative devices and systems have been possible with superior and unique mechanical characteristics of structures at nanoscale [2-4]. to expedite and support the development of these innovative devices and systems, a thorough understanding of characteristics and behaviours of their nano-sized structural components is of paramount importance. in general, the experimental method is the most straightforward way to characterize mechanical responses of structures. however, there are several limitations on experiments conducted on nano-sized specimens, such as the necessity of a specifically testing apparatus, the requirement of a special testing procedure, and the expensive testing cost, etc. [5-7]. furthermore, available classical structural models (e.g. bar, beam, shell, and plate models) are inadequate to characterize the structural response at nanoscale since the material small-scale effect and the size dependency inherent to nano-sized structures are not considered [5-13]. therefore, a rational mathematical model capable of representing the material smallscale effect and the size dependency is deemed necessary and is the main emphasis of the present study. in the research community, both atomistic-based model [14-17] and enhanced structural mechanics-based model [18-20] have been employed to characterize mechanical responses of nano-sized structures for a wide spectrum of applications [21-24]. although comprehensive details on the mechanical responses of nano-sized structures can be gained with the atomisticbased model [14-17], expensive computational costs and sophisticated modelling efforts are usually required with this numerical model, thus limiting the model access by researchers. consequently, the enhanced structural mechanics-based model [18-20] have gained increasing popularity due to its good compromise between model efficiency and model accuracy [11]. the enhanced structural mechanics-based model is possible with unification of non-classical elasticity theories [25-33] and classical structural models. when the structural dimension is in the order of nanometer, the material small-scale effect induced by long-range interatomic forces becomes pronounced. a family of higher-order elasticity theories have been proposed in the research community to address this material small-scale effect [34-39]. among these higher-order elasticity theories, the nonlocal elasticity theory proposed by eringen [25-26] has been the most popular and was first employed by peddieson et al. [34] to construct the nano-sized beam model. however, the material small-scale effect was not detected by this beam model in the case of the cantilever beam under an end load [35, 38, 40-42]. this peculiar response was bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 563 defined as a “paradox” in the literature [34, 38, 40-43]. romano et al. [43] demonstrated that an ill-posed structural-mechanics problem (bounded domain) arises with adoption of the eringen nonlocal differential model. furthermore, koutsoumaris et al. [44] demonstrated that the eringen nonlocal differential model does not conform to the requirement of the quadratic energy functional form of elasticity. to obtain a more rational small-scale beam model, the strain-gradient elasticity theory has been employed as an alternative [38, 45-49]. the strain-gradient elasticity theory was originally invented by mindlin [27-28] and possesses three forms, namely: form i, form ii, and form iii. the difference among these three rudimentary forms is their expressions of the strain energy density function. form i expresses the strain energy density function in terms of the infinitesimal strain and the second displacement gradient; form ii describes the strain energy density function in terms of the infinitesimal strain and the first strain gradient; and form iii represents the strain energy density function in terms of the infinitesimal strain, the symmetric part of the second displacement gradient; and the rotation gradient (curvature tensor). several researchers [5, 30, 38, 50-54] have formulated their own strain-gradient elasticity models based on these three forms (form i, form ii, and form iii). as the simplest variant of the strain-gradient elasticity theory, the simplified strain-gradient elasticity theory of altan and aifantis [51] is constructed based on form ii but can account only for the strain gradient along the longitudinal direction. to obtain a more refined variant of the strain-gradient elasticity theory, the modified strain-gradient elasticity theory of lam et al. [5] is formulated based on either form i or form iii and contains three material length-scale parameters associated with three strain-gradient measures, namely: the dilatation gradient, the deviatoric stretch gradient, and the symmetric rotation gradient. a thorough study by papargyri-beskou et al. [55] has shown the essence of the velocity gradient on wave dispersion in gradient elastic solids and structures. however, the modified strain-gradient elasticity theory of lam et al. [5] lacks the material length-scale parameter associated with the velocity gradient. therefore, a more versatile strain-gradient elasticity theory is required to construct a mathematical tool for static and dynamic analyses of nano-sized structures. recently, zhang and guo [30] has proposed the reformulated strain-gradient elasticity theory based on form i. strain-gradient measures as well as velocity gradient are both considered in this strain-gradient-type theory. consequently, the reformulated strain-gradient elasticity theory is preferable to the narrowness of the research gap and is adopted in the present work. in addition to the material small-scale effect, the size dependency invoked by the surface-energy effect is crucial when the structural dimension approaches the range of nanometer [36, 38, 56-59]. to incorporate the surface-energy effect into a conventional structural-mechanics model, the surface elasticity theory proposed by gurtin and murdoch [32-33] has been widely employed. in the gurtin-murdoch surface elasticity theory, the surface elastic layer is assumed to be a zero-thickness membrane perfectly bonded to its wrapped bulk material. due to the good compromise between model accuracy and simplicity, several structural-mechanics models have been armed with the ability to account for the size dependency using the gurtin-murdoch surface elasticity theory [11, 36, 38, 57-62]. as a result, the present work adopts the gurtin-murdoch surface elasticity theory to represent the size dependency induced by the surface-energy effect. the concept of a beam on elastic foundation has found a wide spectrum of applications in nanoengineering and nanoscience [21-24]. due to its good compromise between model accuracy and simplicity, the winkler foundation model is the most widely employed foundation model to account for the interactive mechanism between nanobeams and their 564 s. limkatanyu, w. sae-long, j. rungamornrat, et al. underlying substrate media [9, 38, 59, 63]. during the last decade, several nanobeamsubstrate medium models have been proposed in the research community to characterize the nanobeam-substrate medium system [64-68]. for example, azizi et al. [64] assessed the influence of the surface-free energy on the nonlinear vibration characteristics of the simply-supported nanobeam-substrate medium system; niknam and aghdam [65] derived the analytical solution for buckling and vibration analyses of the nonlocal functionally graded (fg) beam on the elastic foundation; demir [66] employed the differential transform method (dtm) to compute the natural frequencies of simply supported and clamped-clamped nanobeams on elastic foundation; ponbunyanon et al. [67] extended the winklerpasternak based beam-foundation of limkatanyu et al. [9] to study the flexural behaviour of the nanobeam-substrate medium system; and jena et al. [68] performed buckling analyses of single-walled carbon nanotubes on elastic substrate media under both low and high temperature environments. to the best of the authors’ knowledge, the reformulated strain-gradient elasticity theory of zhang and guo [30] has not been applied to the problem of nanobeams on substrate media. therefore, there is still room to add a rational nanobeam-substrate model into the research community due to the merit of the reformulated strain-gradient elasticity theory. the objective of the present work is to propose the rational mathematical model for static and free vibration analyses of nanobeams on substrate media. the presentation of the present work is in the following order. first, the kinematics of the euler-bernoulli beam theory is described. then, the reformulated strain-gradient elasticity theory of zhang and guo [30] and the surface elasticity theory of gurtin and murdoch [32-33] are briefly discussed. next, the winkler-based interactive mechanism between the beam and its underlying substrate medium is presented. hamilton’s principle is employed to obtain the system governing equation and classical as well as non-classical boundary conditions for bending, buckling, and free vibration analyses. finally, three numerical simulations are provided to demonstrate behaviours and characteristics of the proposed nanobeamsubstrate medium model. the first simulation is employed to demonstrate the ability of the proposed model to overcome the paradoxical response found in the literature [34, 38, 40-42] and to investigate the material small-scale and surface-energy effects as well as the nanobeam-substrate medium interaction on the bending responses. the second and third simulations examine the variation of the system parameters on the critical buckling load and the natural frequency of the simply-supported nanobeam-substrate medium systems. all symbolic calculations in the present work are carried out using the computer software mathematica [69]. 2. euler-bernoulli beam theory in the present work, the kinematic assumption of the proposed beam model follows the euler-bernoulli beam hypothesis [70] asserting that “plane section remains plane and still normal to the longitudinal axis”. based on this kinematical constraint, the displacements of a generic point along the x-, y-, and zaxes are: 0 0 ( ) ( , ) ; ( , ) ( ); and ( , ) 0x y z v x u x y y u x y v x u x y x  = − = =  (1) bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 565 where ux(x,y), uy(x,y), and uz(x,y) are displacement fields along the x-, y-, and zaxes, respectively; v0(x) is the vertical displacement of the beam section; and y is the vertical distance of the point measured from the reference axis x. 3. reformulated strain-gradient elasticity theory the reformulated strain-gradient elasticity theory of zhang and gao [30] is modified from form i of mindlin’s strain-gradient elasticity theory [27]. the assertion of form i is that the strain energy density function is expressed as a function of the infinitesimal strain tensor and the second gradient of displacement. for a linear isotropic elastic material, form i contains five material constants related to strain gradients in addition to two conventional material constants (lame constants). to simplify form i of mindlin’s strain-gradient elasticity theory, zhang and gao [30] reduced the number of material length-scale parameters from five to two using the symmetry property of the tensor [71]. the first material length-scale parameter addresses the strain-gradient effect while the second material length-scale parameter considers the couple-stress effect. this simplified strain-gradient elasticity theory is named as the “reformulated” strain-gradient elasticity theory and is employed in this study to address the beam-bulk material small-scale effect. based on the reformulated strain-gradient elasticity theory proposed by zhang and gao [30], the stored strain energy u is defined as [54]: 1 ( ) 2 l ij ij ijk ijk ij ij v u m dv     = + + (2) where l ij represents the cauchy stress tensor and is the conjugate-work pair of the strain tensor εij; ijk  represents the symmetric part of the double stress tensor and is the conjugatework pair of the strain gradient tensor ijk  ; mij defines the symmetric part of the couple-stress tensor and is the conjugate-work pair of the curvature tensor χij; and v is the volume of the elastic body. it is worth mentioning that the stored strain energy u of eq. (2) is identical to the one associated with the modified couple-stress theory of yang et al. [29] when the straingradient effect is neglected. in the reformulated strain-gradient elasticity theory, the constitutive relations between stress and strain quantities are given by yin et al. [54] as: 2l ij kk ij ij   = + (3) 22ijk s ijkl   = (4) 22ij m ijm l = (5) where δij is the kronecker delta; μ and λ are lame constants; and ls and lm are the material length-scale parameters associated with the stain-gradient effect and the couple-stress effect, respectively. the strain εij and the strain-gradient measures ijk  , and χij are defined as: , , 1 ( ) 2 ij i j j iu u = + (6) 566 s. limkatanyu, w. sae-long, j. rungamornrat, et al. , , , 1 ( ) 3 ijk i jk j ki k iju u u = + + (7) , , 1 ( ) 2 ij i j j i  = + (8) where ui is the displacement field; and θi is the rotation vector defined as: , 1 2 i ijk k je u = (9) with eijk being the permutation symbol. 4. surface elasticity theory as the structural dimension approaches the range of nanometer, the free energy at the surface of a bulk material can no longer be neglected since it is comparable to the energy stored in the bulk material. this surface-free energy induces the size dependency of nanosized structures as confirmed by both experiments and numerical simulations [36, 57, 72]. to account for this size-dependent characteristic, the proposed nanobeam-substrate medium model employs the surface elasticity theory of gurtin and murdoch [32-33]. based on the assumptions of this surface elasticity theory [32-33], the beam cross section is assumed to be composed of a bulk core and a wrapping outer surface layer shown in fig. 1. the bulk core and its wrapping outer surface layer are assumed to be in a perfect bond condition. the wrapping outer surface layer is taken as a mathematically zero-thickness elastic layer. the constitutive relations of the surface layer are: 0 0 , 0 , , ,[ ( ) ] ( )sur sur sur sur sur sur sur sur sur sur res resu u u u               = + + + + − (10) , sur sur sur n res nu  = (11) where sur  and sur n are, respectively, the in-plane and out-of-plane components of the surface stresses; sur res is the residual surface stress under unconstrained conditions obtained from the atomistic simulation [72]; α and β are the in-plane cartesian coordinates of the lateral surface for the nanobeam; usur is the surface-layer deformation; and 0 sur and 0 sur are the surface elastic constants. d b hz z y y n t nsurface layer t bulk bulk fig. 1 beam cross-sections: beam bulk and surface layer [32-33] bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 567 5. nanobeam-substrate medium interaction in the present work, the interaction between the nanobeam and its underlying substrate medium shown in fig. 2(a) is simulated based on the concept of the winkler foundation model [73]. following this foundation concept, the interactive mechanism of the underlying substrate medium is represented by a set of continuously distributed and non-interconnected springs attached along the nanobeam length shown in fig. 2(b). the relation between the substrate-medium interactive force des(x) and the substrate-medium deformation δes(x) is: ( ) ( )es es esd x k x=  (12) where kes is the elastic modulus of the substrate medium. x l elastic substrate medium ( )0,y v x ( )yq x surface layer beam-bulk 4 4,p u 1 1,p u esk beam ( )0,y v x x 5 5,p u 2 2,p u  1 2 3 4 5 6 t p p p p p p=p  1 2 3 4 5 6 t u u u u u u=u elastic substrate-medium springs 3 3,p u 6 6,p u (a) (b) fig. 2 typical systems: (a) nanobeam resting on elastic substrate medium; and (b) beamsubstrate medium model 6. reformulated strain-gradient nanobeam-substrate medium model 6.1. compatibility conditions and constitutive relations 6.1.1. beam bulk in the case of the planar euler-bernoulli beam problem, the transverse displacement v0(x,t) serves as a primary variable and is a function of the spatial variable x and the temporal variable t since both static and free vibration responses of nanobeam-substrate medium systems are of interest. based on the beam kinematics of eq. (1), non-zero components of the strain εij and the strain-gradient measures ijk  , and χij in eqs. (6) to (9) are expressed in terms of the transverse displacement v0(x,t) as: 2 0 2 ( , ) ( , )xx v x t x t y x   = −  (13) 3 0 3 ( , ) ( , )xxx v x t x t y x    = −  (14) 568 s. limkatanyu, w. sae-long, j. rungamornrat, et al. 2 0 2 ( , )1 ( , ) 3 xxy v x t x t x    =  (15) 2 0 2 ( , )1 ( , ) ( , ) 2 xz zx v x t x t x t x    = =  (16) substituting the compatibility relations of eqs. (13) to (16) into the constitutive relations of eqs. (3) to (5), the corresponding non-zero stress components in terms of the transverse displacement v0(x,t) are: 2 0 2 ( , ) ( , )xx xx v x t x t e y x   = −  (17) 3 2 0 3 ( , ) ( , ) 2xxx s v x t x t l y x    = −  (18) 2 2 0 2 ( , )2 ( , ) 3 xxy s v x t x t l x    =  (19) 2 2 0 2 ( , ) ( , ) ( , )xz zx m v x t m x t m x t l x   = =  (20) where exx defines the elastic modulus of the beam bulk. 6.1.2. outer surface layer for the problem of the planar euler-bernoulli beam, the degenerated forms of eqs. (10) and (11) are given by guo and mahmoud [57] as: ( , ) ( , )sur sur sur sur xx res xx xxx t e x t  − = (21) ( , ) ( , )sur sur sur nx res nxx t x t  = (22) where ( , )sur xx x t denotes the in-plane component of the surface stress and is the conjugatework pair of the in-plane component of the surface strain ( , )sur xx x t ; ( , )sur nx x t denotes the out-of-plane component of the surface stress and is the conjugate-work pair of the out-ofplane component of the surface strain ( , )sur nx x t ; and 0 02sur sur sur xxe  = + is the surface-layer elastic modulus obtained from two surface elastic constants 0 sur and 0 sur [72]. following the beam kinematics of eq. (1) and the full compatibility of the composite nanobeam section of fig. 1, the in-plane ( , )sur xx x t and out-of-plane surface strain ( , )sur nx x t components can be expressed in terms of the transverse displacement v0(x,t) as: 2 0 2 ( , ) ( , )sur xx v x t x t y x   = −  (23) 0 ( , ) ( , )sur nx y v x t x t n x   =  (24) where ny defines the y component of the unit vector n normal to the lateral surface of beam section as shown in fig. 1. bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 569 substituting eqs. (23) and (24) into eqs. (21) and (22), the in-plane ( , )sur xx x t and outof-plane ( , )sur nx x t surface stress components are described in terms of the transverse displacement v0(x,t) as: 2 0 2 ( , ) ( , ) ( , )sur sur sur sur sur xx res xx xx xx v x t x t e x t e y x     − = = −  (25) 0 ( , ) ( , ) ( , )sur sur sur sur nx res nx res y v x t x t x t n x      = =  (26) 6.1.3. substrate medium based on the assumption of the winkler-foundation model [73], the beam and its supporting winkler-spring bed is in a full compatibility condition (δes(x,t) = v0(x,t)). therefore, the substrate-medium interactive force des(x) of eq. (12) can be described in terms of the transverse displacement v0(x,t) as: 0( , ) ( , ) ( , )es es es esd x t k x t k v x t=  = (27) 6.2. equilibrium equation: hamilton’s principle for an isotropic linear elastic material, the stored strain energy u of the nanobeamsubstrate medium system of fig. 2(a) can be defined as: 2 2 2 2 2 20 0 02 2 0 0 0 elastic substrate medium contributionlocal stress contribution couple stress contribution ( , ) ( , )1 1 1 ( ( , )) 2 2 2 1 l l l xx m es v x t v x t u e i dx al dx k v x t dx x x       = + +         +    in-plane component out-of-plane component 2 2 23 2 2 20 0 0 3 2 0 0 0 strain gradient contribution ( , ) ( , ) ( , )1 2 1 2 2 2 3 2 l l l s s v x t v x t v x t il dx al dx p xx x          + −               applied axial load contribution in-plane component out-of 2 22 0 0 2 0 0 surface-induced distributed moment contribution ( , ) ( , )1 1 2 2 l l sur sur xx res dx v x t v x t e i dx s dx xx       + +           -plane component surface-induced shear force contribution (28) where p is the applied axial force; l is the nanobeam length; г is the beam section perimeter; a a da=  is the sectional area; 2 a i y da=  is the second moment of area; 2i y d  =  is the second moment of perimeter; and 2 ys n d  =  . the first variation of the stored strain energy u of eq. (28) during the time interval [0, t] is: 570 s. limkatanyu, w. sae-long, j. rungamornrat, et al. 2 2 2 2 20 0 0 0 2 2 2 2 0 0 0 3 3 2 2 2 20 0 0 0 3 3 2 2 0 0 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )2 2 3 t l l xx m l l s s v x t v x t v x t v x t u e i dx al dx x x x x v x t v x t v x t v x t il dx al x x x x                   = +                          + +                   2 2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ( , ))( ( , )) l l sur sur xx res l l es dx v x t v x t v x t v x t e i dx s dx x xx x v x t v x t k v x t v x t dx p dx dt x x                + +                  + −              (29) to remove all differential operators from the virtual transverse displacement δv0(x,t), the integration by parts is applied to eq. (29), thus leading to the following expression: 6 4 2 0 0 0 0 06 4 2 0 0 5 3 0 0 0 0 5 3 0 0 ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( , ) ( ) ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( ) t l h l sur eff eff es res x l t h l sur eff eff res x v x t v x t v x t u v x t ei ei k v x t s p dxdt x x x v x t v x t v x t v x t ei ei s p dt xx x        =  =     = − + + − −          + − + −      +    4 2 0 0 0 4 2 0 0 2 3 0 0 2 3 0 0 ( , ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( ) x l t h l eff eff x x l t h eff x v x t v x t v x t ei ei dt x x x v x t v x t ei dt x x  = = = =    − +         +        (30) where 2( ) 2h eff sei il= denotes the higher-order flexural rigidity associated with the straingradient contribution; and 2 22 ( ) 3 l sur eff xx s m xxei e i al al e i  = + + + denotes the lower-order flexural rigidity combining the beam-bulk and surface-layer contributions. based on the kinetic energy expression given by papargyri-beskou et al. [55], the first variation of the kinetic energy k with the inclusion of the velocity gradient during the time interval [0, t] is [30]: 2 4 6 4 2 20 0 0 0 0 2 2 2 4 2 2 2 0 0 3 5 2 20 0 0 2 3 2 0 0 ( , ) ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) (2 ) t l v v x l t v v x v x t v x t v x t v x t k v x t a l i l dx dt t x t x t x t v x t v x t v x t al i il x t x t         = =           = − − + −                       + − + +             4 20 0 2 2 0 0 ( , ) ( , ) x l t v x dt v x t v x t il dt x x t   = =     + −          (31) where ρ represents the nanobeam mass density; and lv defines the material length-scale parameter associated with the velocity gradient. it should be noted from the first variation of the kinematic energy in eq. (31) that the beam configurations are prescribed at t = 0 and t = t, resulting in the vanish of the virtual displacements and its derivatives at t = 0 bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 571 and t = t. moreover, the mass density ρ over the time interval [0, t] is constant along the cross section of the nanobeam [30]. the first variation of the external work done w by the transverse distributed load qy(x,t) and end forces p shown in fig. 2(b) during the time interval [0, t] can be defined as: 0 0 0 ( , ) ( , ) t l t yw q x t v x t dx dt     = +      u p (32) where the displacement vector u = [u1 u2 u3 u4 u5 u6]t collects end displacements of the system and the force vector p = [p1 p2 p3 p4 p5 p6]t contains end forces of the system. to establish the governing differential equation of motion and its associated boundary conditions, hamilton’s principal is called for. as a result, eqs. (30), (31), and (32) can be written together as: 0 6 4 2 0 0 0 0 6 4 2 0 0 2 4 6 4 2 20 0 0 0 0 2 2 2 4 2 [ ( )] 0 ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 2 t t l h l sur eff eff res y es v v k u w dt v x t v x t v x t v x t ei ei s p x x x v x t v x t v x t v x t q x t k v x t a l i l t x t x t         − − =        = − + −         + − − − − −            2 2 5 3 0 0 0 0 5 3 0 3 5 2 20 0 2 3 2 0 4 0 0 4 ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( ) ( , ) ( , ) (2 ) ( , ) ( , ) ( ) ( ) t h l sur eff eff res x l v v x h l eff eff dxdt x t v x t v x t v x t v x t ei ei s p xx x v x t v x t al i il dt x t x t v x t v x t ei ei x x        = =            + − + − −     − + +          + −    2 4 20 0 2 2 2 0 0 2 3 0 0 2 3 0 0 ( , ) ( , ) ( , ) ( , ) ( ) x l t v x x l t h t eff x v x t v x t il dt x x t v x t v x t ei dt x x    = = = =   −         + − +       u p (33) considering the arbitrariness of δv0(x,t) during the time interval [0, t], the governing differential equation of motion is unveiled as: 6 4 2 0 0 0 06 4 2 2 4 6 4 2 20 0 0 0 2 2 2 4 2 2 2 ( , ) ( , ) ( , ) ( ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 2 h l sur eff eff res es y v v v x t v x t v x t ei ei s p k v x t q x t x x x v x t v x t v x t v x t a l i l t x t x t x t        − + − − +           = − + −              (34) eq. (34) serves as a fundamental equation for bending, buckling, and free vibration analyses of nanobeam-substrate medium systems. it is worth noting that the governing differential equation of motion for the reformulated strain-gradient nanobeam proposed by zhang and gao [30] can be retrieved from eq. (34) when the surface-energy effect and the underlying substrate medium are neglected. 572 s. limkatanyu, w. sae-long, j. rungamornrat, et al. considering the arbitrariness of δv0(x,t) and δu on boundary terms in eq. (33) provides boundary conditions of the nanobeam-substrate medium system as: boundary conditions: 5 3 0 0 0 1 5 3 3 5 2 20 0 2 3 2 0 4 2 4 20 0 0 2 4 2 2 2 ( , ) ( , ) ( , ) ( ) ( ) ( ) ( , ) ( , ) (2 ) ; ( , ) ( , ) ( , ) ( ) ( ) h l sur eff eff res v v x h l eff eff v v x t v x t v x t p ei ei s p xx x v x t v x t al i il x t x t v x t v x t v x t p ei ei il x x x t     =     = − − + − −      − + +           = − − −     3 0 3 3 0 0 5 3 0 0 0 4 5 3 3 5 2 20 0 2 3 2 4 0 5 4 ( , ) ; ( ) ; ( , ) ( , ) ( , ) ( ) ( ) ( ) ( , ) ( , ) (2 ) ; ( , ) ( ) ( h eff x x h l sur eff eff res v v x l h eff v x t p ei x v x t v x t v x t p ei ei s p xx x v x t v x t al i il x t x t v x t p ei e x   = =  =    = − −           = − + − −      − + +        = −  2 4 3 20 0 0 62 2 2 3 ( , ) ( , ) ( , ) ) ; ( )l h eff v eff x l x l v x t v x t v x t i il p ei x x t x  = =       − = −           (35) 6.3. analytical solution for bending analyses when the applied axial force p and the inertia forces are omitted, the governing equation for bending analyses of nanobeam-substrate medium systems can be obtained from eq. (34) as: 6 4 2 0 0 0 06 4 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0h l sur eff eff res es y v x v x v x ei ei s k v x q x x x x      − + − + − =    (36) the general solution to eq. (36) possesses two parts, namely; the homogeneous solution 0 ( )hv x and the particular solution 0 ( )pv x . 0 0 0( ) ( ) ( )h pv x v x v x= + (37) the homogeneous solution 0 ( )hv x is obtained from eq. (36) when qy(x) = 0 while the particular solution 0 ( )pv x depends on the transversely distributed load qy(x). it is noticed that the governing differential equilibrium equation of the beam on kerr-type foundation and eq. (36) are in the same form. consequently, the following homogeneous solution 0 ( )hv x given by morfidis [74] and avramidis and morfidis [75] can be applied to the proposed nanobeam-substrate medium model. 0 1 1 2 2 3 3 4 4 5 5 6 6( ) ( ) ( ) ( ) ( ) ( ) ( )hv x x c x c x c x c x c x c     = + + + + + (38) where φ1(x), φ2(x), φ3(x), φ4(x), φ5(x) and φ6(x) are the basic displacement functions expressed in appendix a; and c1, c2, c3, c4, c5 and c6 are constants of integration obtained from imposed boundary conditions. bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 573 6.4. analytical solution for buckling analyses to determine the buckling load capacity of nanobeam-substrate medium systems, the transversely distributed load qy(x) and the inertia forces present in eq. (34) are suppressed. therefore, eq. (34) becomes: 6 4 2 0 0 0 06 4 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0h l sur eff eff res es v x v x v x ei ei s p k v x x x x      − + − − + =    (39) for the sake of conciseness, only the simply-supported nanobeam-substrate medium system is considered. the associated classical boundary conditions are defined as: 0 0 ( ) 0 x v x = = and 0 ( ) 0 x l v x = = (40) 4 2 0 0 4 2 0 4 2 0 0 4 2 ( ) ( ) ( ) ( ) ( ) 0 and ( ) ( ) ( ) ( ) ( ) 0 h l eff eff x h l eff eff x l v x v x m x ei ei x x v x v x m x ei ei x x = =    = − + =        = − + =     (41) as suggested by zhang and gao [30], the non-classical boundary conditions associated with the simply-supported nanobeam-substrate medium system are: 2 0 2 0 ( ) ( ) 0 x v x x x  =  = =  and 2 0 2 ( ) ( ) 0 x l v x x x  =  = =  (42) the following analytical displacement solution for the critical (lowest) buckling load is provided by sae-long et al. [38] as: 0 ( ) sin( )v x a x= (43) where a is the generalized coordinate and ψ = π/l. substituting eq. (43) into eq. (39) and subsequently enforcing the non-trivial condition, the critical buckling load proposed crp can be gained as: 6 4 2 2 ( ) ( )h l sur eff eff res esproposed cr ei ei s k p      + + = + (44) it is worth mentioning that eq. (44) can provide the critical buckling load obtained with either the degenerated strain-gradient model or the modified couple-stress model when the associated material length-scale parameter vanishes (lm = 0 or ls = 0). furthermore, the classical euler’s buckling load of the beam-winkler foundation system can be determined from eq. (44) when the material length-scale parameters (ls and lm) and the surface-layer parameters ( sur xxe and sur res ) vanish. 574 s. limkatanyu, w. sae-long, j. rungamornrat, et al. 6.5. analytical solution for free-vibration analyses when the free vibration response of nanobeam-substrate medium systems is of interest, the transversely distributed load qy(x,t) and the applied axial force p present in eq. (34) are suppressed. therefore, the governing differential equation of motion becomes: 6 4 2 0 0 0 06 4 2 2 4 6 4 2 20 0 0 0 2 2 2 4 2 2 2 ( , ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) 2 h l sur eff eff res es v v v x t v x t v x t ei ei s k v x t x x x v x t v x t v x t v x t a l i l t x t x t x t        − + −           = − + −              (45) for the sake of simplicity, only the simply-supported nanobeam-substrate medium system is considered in the present study. classical and non-classical boundary conditions of the simply-supported nanobeam-substrate medium system follow those suggested by zhang and gao [30] as: classical boundary conditions: 0 0 ( , ) 0 x v x t = = and 0 ( , ) 0 x l v x t = = (46) 4 2 4 20 0 0 4 2 2 2 0 4 2 4 20 0 0 4 2 2 2 ( , ) ( , ) ( , ) ( , ) ( ) ( ) 0 and ( , ) ( , ) ( , ) ( , ) ( ) ( ) 0 h l eff eff v x h l eff eff v x l v x t v x t v x t m x t ei ei il x x x t v x t v x t v x t m x t ei ei il x x x t   = =     = − − =           = − − =       (47) non-classical boundary conditions: 2 0 2 0 ( , ) ( , ) 0 x v x t x t x  =  = =  and 2 0 2 ( , ) ( , ) 0 x l v x t x t x  =  = =  (48) to determine the analytical solution to eq. (45), the method of separation of variables is called for. the following transverse displacement v0(x,t) suggested by zhang and gao [30] satisfies all boundary conditions of eqs. (46) to (48). 0 1 ( , ) sin( ) ni t n n n v x t v x e   = = (49) where ϕn = nπ/l; vn is the fourier coefficient; n is the vibration mode; ωn is the natural frequency associated with the nth vibration mode; and i is an imaginary number. substituting eq. (49) into eq. (45), the natural frequency proposed n of the nth vibration mode is obtained by enforcing the non-trivial condition (vn ≠ 0). the resulting explicit form of the nth-vibration mode natural frequency is: 2 4 6 2 2 2 4 2( ) ( ) )( ( ) 1 2 sur l h res n eff n eff n v n s v n n eproposed n s ei ei a l i k l            ++ + = + + + (50) bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 575 the natural frequency proposed n of eq. (50) becomes identical to the natural frequency classical n obtained with the local beam-winkler foundation system when the material length-scale parameters (ls, lm, and lv) and the surface-layer parameters ( sur xxe and sur res ) vanish. furthermore, eq. (50) reduces to the explicit form of the natural frequency obtained with the reformulated strain-gradient beam model of zhang and gao [30] when the underlying substrate medium and surface energy are ignored (kes = sur xxe = sur res = 0). 7. numerical simulations to assess bending, buckling, and free-vibration responses of nanobeam-substrate medium systems, three numerical simulations using the proposed model are presented. the material properties of the lead nanobeam employed in all simulations are obtained from cuenot et al. [76] and wan et al. [77] as summarized in table 1. table 1 material properties of the lead nanobeam [76-77]. material bulk modulus (gpa) poisson’s ratio density (kg/m3) surface elastic modulus (nn/nm) residual surface stress (nn/nm) lead 16 0.4 11,343 8 0.63 7.1. simulation i: bending responses simulation i focuses on bending responses of the proposed model and considers two analysis cases. the first analysis case demonstrates the ability of the proposed model to obtain a rational response of the cantilever nanobeam under the pure-bending state while the second analysis case assesses the influence of system parameters on the bending response of cantilever nanobeam-substrate medium systems. 7.1.1. analysis case i fig. 3 shows a lead cantilever nanobeam of length l = 1,000 nm under an end moment m0 of 3,000 nn-nm, thus inducing the pure-bending state along the whole length. the beam-section shape is square with a dimension h of 100 nm, thus a = 10x103 nm2, i = 8.333x106 nm4, iг = 6.667x105 nm3, and sг = 200 nm. only the material small-scale effect is considered in this analysis case. therefore, the system parameters associated with the underlying substrate medium and the surface energy vanish (kes = sur xxe = sur res = 0). this cantilever nanobeam is used to demonstrate the capability of the proposed nanobeam model in producing the small-scale dependent response under the pure-bending state. it is worth remarking that the eringen nonlocal beam model of limkatanyu et al. [11] and the simplified strain-gradient beam model of sae-long et al. [38] fail to represent such a smallscale dependent response under the pure-bending state. therefore, the cantilever nanobeam of fig. 3 is also analysed by these two beam models to confirm the ability of the proposed model to suppress the paradoxical behaviour. the material length-scale parameters associated with the strain-gradient effect and the couple-stress effect are set to be identical (ls = lm) and are varied through the following relations: 576 s. limkatanyu, w. sae-long, j. rungamornrat, et al. s s h l  = and m m h l  = (51) where ξs and ξm are, respectively, the normalized material length-scale parameters associated with the strain-gradient effect and the couple-stress effect and range from 0 to 2. 1,000l nm= ( )0,y v x x 0 3,000m nn nm= − z y beam section h h 4 3 2 2 ; ; ; 2 12 3 h h a h i i s h = = = = fig. 3 simulation i: bending analysis case i the transverse-displacement responses associated with all beam models are superimposed in fig. 4. obviously, the transverse-displacement response obtained with the local beam model is identical to those obtained with the eringen nonlocal beam model of limkatanyu et al. [11] and the simplified strain-gradient beam model of saelong et al. [38] regardless of their values of material length-scale parameters. this observation confirms the inability of these two nanobeam models to represent the material small-scale dependency under the pure-bending state as observed in limkatanyu et al. [42]. the peculiar transverse-displacement response obtained with the eringen nonlocal beam model of limkatanyu et al. [11] stems from the fact that adoption of the eringen nonlocal differential form leads to an ill-posed structural-mechanics problem [34, 35, 43]. employment of the simplified strain-gradient model leads to a well-posed structural-mechanics problem [38] and is capable of remedying the well-known paradoxical response of a cantilever beam under an end load [42] but fails to represent the small-scale dependent response under the pure-bending state. this failure relies on the fact that only the axial-strain gradient (∂εxx/∂x) along the beam length is accounted for in the simplified strain-gradient beam model of sae-long et al. [38]. nevertheless, the pure-bending state induces the constant axial-strain variation along the beam length, thus vanishing the axial-strain gradient (∂εxx/∂x = 0). in opposition, the proposed nanobeam model is able to represent the material small-scale dependency under pure bending and the obtained transverse-displacement responses become stiffer with enlarging values of material length-scale parameters. this observation is confirmed with both experimental evidence [5, 78] and analytical results [35, 40-43] available in literature. bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 577 -12 -9 -6 -3 0 0 200 400 600 800 1,000 distance along the nanobeam (nm) t ra n sv er se d is p la ce m en t (n m ) proposed model ( )0s m = = proposed model ( )0.5s m = = proposed model ( )1s m = = proposed model ( )2s m = = simplified strain gradient model ( )0for all e a eringen nonlocal model ( )0for all e a local model proposed model ( )0.25s m = = fig. 4 transverse displacement versus distance along nanobeam of simulation i for analysis case i 7.1.2. analysis case ii fig. 5 shows a lead cantilever nanobeam-substrate medium system under an end moment m0 of 30,000 nn-nm. geometric and material properties of this nanobeam follow those employed in analysis case i. to assess the influence of underlying substrate media and surface-free energy on the bending response, the nanobeam-substrate medium system of fig. 5 is analyzed with various values of the substrate-medium stiffness kes and slenderness ratio l/h. three different beam-bulk constitutive models are employed in this analysis case to address the material small-scale effect, namely: (a) the reformulated strain-gradient model; (b) the degenerated strain-gradient model; and (c) the modified couple-stress model. it is worth mentioning that the reformulated strain-gradient model embraces the degenerated strain-gradient model (ls ≠ 0 and lm = 0) as well as the modified couple-stress model (lm ≠ 0 and ls = 0). z y beam section h h 4 3 2 2 ; ; ; 2 12 3 h h a h i i s h = = = = 1,000l nm= ( )0,y v x x 0 30,000m nn nm= − fig. 5 simulation i: bending analysis case ii 578 s. limkatanyu, w. sae-long, j. rungamornrat, et al. fig. 6(a) compares the variation of the normalized end displacement / local v end endv v = with the non-dimensional substrate-medium stiffness esk  = kesl4/(exxi) obtained from all nanobeam models. the non-dimensional substrate-medium stiffness esk  varies from 1 to 100 while the beam depth h is kept constant at 100 nm. this range of the nondimensional substrate-medium stiffness esk  follows that employed by demir et al. [18]. the normalized material-length scale parameters are kept constant at ξs = 1 and ξm = 1. clearly, the material small-scale effect and the surface-energy effect consistently stiffen the system response (ξv < 1) especially for lower values of non-dimensional substratemedium stiffness (softer substrate media). comparison among three nanobeam models indicates that the reformulated strain-gradient beam model leads to the stiffest system response. this superiority of the system stiffness enhancement relies on the fact that both strain-gradient and couple-stress effects are addressed in this beam model and can obviously be observed in eq. (34). furthermore, fig. 6(a) points out that the system stiffness enhancement associated with the couple-stress effect is more pronounced than that associated with the strain-gradient effect. fig. 6(b) shows the variation of the normalized end displacement / local v end endv v = with the slenderness ratio l/h obtained from all nanobeam models. different values of the slenderness ratio l/h are employed to reflect the surface-energy effect as well as the material small-scale effect, ranging from 1 to 100 (very stubby to very slender beams). a specific value of slenderness ratio l/h can be earned by keeping l = 1,000 nm and varying h. as provided by liew et al. [79], a stiffness coefficient of kes = 95x10-3 nn/nm3 is selected to represent the underlying substrate medium as polymer. the material lengthscale parameters are kept constant at ls = 100 nm and lm = 100 nm. fig. 6(b) indicates that with increasing beam slenderness ratio l/h, the system stiffness enhancement induced by the material small-scale effect and the surface-energy effect becomes more pronounced, thus lowering the normalized end displacement / local v end endv v = . it is worth pointing out that the normalized material-length scale parameters (ξs and ξm) become larger with increasing beam slenderness ratio l/h (decreasing beam depth h), thus magnifying the stiffening characteristic induced by the material small-scale effect. furthermore, increasing beam slenderness ratio l/h enlarges the beam surface-area/section-area ratio (as/ab), hence amplifying the stiffness enhancement induced by the surface-energy effect. among three nanobeam models, the reformulated strain-gradient beam model consistently provides the largest system stiffness. however, the beam depth h dictates the system stiffness obtained with the degenerated strain-gradient model and the modified couple-stress model. as shown in the inset of fig. 6(b), the response obtained with the degenerated strain-gradient model is stiffer than that obtained with the modified couplestress model when the beam depth h is larger than 100 nm (l/h < 10). as the beam becomes thinner (h ≤ 100 nm), the system response associated with the modified couplestress model is stiffer than that associated with the degenerated strain-gradient model. this observation relies on the fact that the higher-order flexural rigidity ( )h effei of the degenerated strain-gradient model provides more contribution to the system stiffness when the beam becomes thicker (h > 100 nm). zhang et al. [80] also noticed a similar observation for the reformulated strain-gradient kirchhoff plate model. bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 579 0 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 100 non-dimensional substrate-medium stiffness esk  higher substrate-medium stiffness esk 100h nm= reformulated strain-gradient model ( )1s m = = degenerated strain-gradient model modified couple-stress model ( )1s = ( )1m = vξ 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 slenderness ratio /l h smaller beam depth h 3 395 10 /esk x nn nm−= 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 v /v cl a ss ic a l slenderness ratio proposed model sget model mcst model reformulated strain-gradient model ( )100s ml l nm= = degenerated strain-gradient model modified couple-stress model ( )100sl nm= ( )100ml nm= vξ (a) (b) fig. 6 normalized end displacements with variation of: (a) non-dimensional substratemedium stiffness esk  ; and (b) slenderness ratio l/h 7.2. simulation ii: buckling analysis the second simulation assesses the influence of system parameters on the critical buckling load pcr of the simply-supported lead nanobeam-substrate medium system shown in fig. 7. investigated system parameters include the beam depth h, the strain-gradient material parameter ls, and the substrate-medium stiffness kes. the couple-stress material parameter lm is held constant at 100 nm. the normalized beam-depth parameter h/lm is employed to vary the beam depth h and ranges from 0.2 to 4; the normalized material length-scale parameter ls/lm is employed to examine the strain-gradient effect and varies from 0.2 to 1.2; and the nondimensional substrate-medium stiffness esk = kesl4/ ( )l effei is employed to vary the substratemedium stiffness kes and ranges from 1 to 10. 1,000l nm= ( )0,y v x x esk 4 3 2 2 ; ; ; 2 12 3 h h a h i i s h = = = = z y beam section h hcrp fig. 7 simulation ii: buckling analysis the variation of the normalized buckling load /proposed classical cr cr crp p = with all aforementioned system parameters is presented in fig. 8. the “classical” buckling load classical crp is obtained with the beam-substrate medium model in which the material smallscale and surface-energy effects are all neglected. clearly, decreasing beam depth h and increasing strain-gradient material parameter ls both magnify the normalized buckling load cr , especially for lower values of non-dimensional substrate-medium stiffness esk (softer substrate media). however, different degrees of magnification are observed for 580 s. limkatanyu, w. sae-long, j. rungamornrat, et al. these two parameters. the buckling-load magnification associated with decreasing beam depth h is more pronounced than that associated with increasing strain-gradient material parameter ls. it is worth pointing out that decreasing beam depth h results in a larger beam surface-area/section-area ratio (as/ab) and renders the beam thickness smaller when compared to the couple-stress material parameter lm. the first one amplifies the system stiffening phenomenon induced by the surface-energy effect while the second one magnifies the system stiffening phenomenon associated with the couple-stress effect. 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 0 10 20 30 40 50 60 70 80 0 .2 0 .4 0 .6 0 .8 1 1 .5 2 4 70-80 60-70 50-60 40-50 30-40 20-30 10-20 0-10 1esk = ξ cr 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 0 10 20 30 40 50 60 70 80 0 .2 0 .4 0 .6 0 .8 1 1 .5 2 4 70-80 60-70 50-60 40-50 30-40 20-30 10-20 0-10 4esk = ξ cr (a) (b) 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 0 10 20 30 40 50 60 70 80 0 .2 0 .4 0 .6 0 .8 1 1 .5 2 4 70-80 60-70 50-60 40-50 30-40 20-30 10-20 0-10 7esk = ξ cr 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 0 10 20 30 40 50 60 70 80 0 .2 0 .4 0 .6 0 .8 1 1 .5 2 4 70-80 60-70 50-60 40-50 30-40 20-30 10-20 0-10 10esk = ξ cr (c) (d) fig. 8 normalized buckling load ξcr with variation of normalized parameters ls/lm and h/ lm: (a) esk = 1, (b) esk = 4, (c) esk = 7, and (d) esk = 10 7.3. simulation iii: free vibration analysis the third simulation performs parametric studies on free-vibration responses of the simply-supported lead nanobeam-substrate medium system shown in fig. 9. the material and geometric properties of this nanobeam follow those employed in the first and second simulations. the mass density of the lead nanobeam is 11,343 kg/m3 as provided in table 1. investigated system parameters include the beam depth h, the velocity-gradient material parameter lv, and the substrate-medium stiffness kes. the material length-scale parameters are kept constant at ls = 100 nm and lm = 100 nm. the slenderness ratio l/h is employed to vary the beam depth h and ranges from 5 to 100; the normalized velocity bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 581 gradient material parameter lv/lm is employed to examine the velocity-gradient effect and varies from 1 to 20; and the non-dimensional substrate-medium stiffness esk = kesl4/ ( )l effei is employed to vary the substrate-medium stiffness kes and ranges from 1 to 10. a given value of slenderness ratio l/h can be obtained by keeping l = 1,000 nm and varying h. it is worth remarking that the system size-dependency associated with the material smallscale effect and the surface-energy effect is reflected through the variation of the slenderness ratio l/h. 1,000l nm= ( )0,y v x x esk 4 3 2 2 ; ; ; 2 12 3 h h a h i i s h = = = = z y beam section h h fig. 9 simulation iii: free vibration analysis fig. 10 shows variations of the normalized natural frequency 1 1 1/proposed classical  = for the first vibration mode (n = 1) with normalized velocity-gradient material parameter lv/lm and slenderness ratio l/h for various values of the non-dimensional substratemedium stiffness esk . the “classical” natural frequency 1 classical for the first vibration mode is obtained with the beam-substrate medium model in which material small-scale and surface-energy effects are both neglected. with decreasing beam depth h (increasing slenderness ratio l/h), the normalized natural frequency 1  becomes larger, particularly for lower values of non-dimensional substrate-medium stiffness esk (softer substrate media) as shown in fig.10. it is worth mentioning that decreasing beam depth h leads to a larger beam surface-area/section-area ratio (as/ab) and renders the beam thickness smaller compared to the material length-scale parameters (ls = 100 nm and lm = 100 nm). the first one magnifies the system stiffening phenomenon induced by the surface-energy effect while the second one amplifies the system stiffening phenomenon associated with the material small-scale effect, thus resulting in a higher natural frequency 1 proposed . in opposition, inclusion of the velocity-gradient effect lowers the system stiffness. the normalized natural frequency 1  reduces with increasing normalized velocity-gradient material parameter lv/lm, especially for lower values of non-dimensional substratemedium stiffness esk (softer substrate media). this observation is consistent with that made by zhang and gao [30] and yin et al. [54]. however, the system stiffening phenomenon associated with the material small-scale effect and the surface-energy effect surpasses the system weakening phenomenon associated with the velocity-gradient effect. therefore, the combination of the material small-scale effect, the surface-energy effect, and the velocity-gradient effect results in a higher system natural frequency ( 1  >1.0) for specific values of system parameters investigated in the current study. 582 s. limkatanyu, w. sae-long, j. rungamornrat, et al. 5 2 0 4 0 6 0 8 0 1 0 0 0 1 2 3 4 5 6 7 8 9 10 1 5 1 0 1 5 2 0 9-10 8-9 7-8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 1esk = ωξ 1 5 2 0 4 0 6 0 8 0 1 0 0 0 1 2 3 4 5 6 7 8 9 10 1 5 1 0 1 5 2 0 9-10 8-9 7-8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 4esk = ωξ 1 (a) (b) 5 2 0 4 0 6 0 8 0 1 0 0 0 1 2 3 4 5 6 7 8 9 10 1 5 1 0 1 5 2 0 9-10 8-9 7-8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 7esk = ωξ 1 5 2 0 4 0 6 0 8 0 1 0 0 0 1 2 3 4 5 6 7 8 9 10 1 5 1 0 1 5 2 0 9-10 8-9 7-8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 10esk = ωξ 1 (c) (d) fig. 10 normalized natural frequency 1  with variation of normalized parameter lv/lm and slenderness ratio l/h: (a) esk = 1, (b) esk = 4, (c) esk = 7, and (d) esk = 10 8. summary and conclusions a rational beam-substrate medium model for bending, buckling, and free vibration analyses of nanobeams resting on elastic substrate media is proposed in the present work. the reformulated strain-gradient theory is employed to consider the beam-bulk material smallscale effect. the strain-gradient and couple-stress effects are included in the strain energy density expression while the velocity-gradient effect is included in the kinetic energy expression. the gurtin-murdoch surface elasticity theory is used to address the surfaceenergy effect. the interaction between the nanobeam and its underlying substrate medium is considered through the winkler-foundation model. to obtain the system governing equation and corresponding boundary conditions, hamilton’s principle is called for. three numerical simulations are presented to characterize the influences of the material small-scale effect, the surface-energy effect, and the surrounding substrate medium on bending, buckling, and free vibration responses of nanobeam-substrate medium systems. the first simulation possesses two analysis cases and emphasizes on bending analyses of nanobeam-substrate medium systems. the first one presents the ability of the proposed model to address the size dependency under the pure-bending state and shows the stiffening bending response associated with the material small-scale effect. the second bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 583 one shows that the system-stiffness enhancement associated with the material small-scale and surface-energy effects is more pronounced when the underlying substrate medium becomes softer and the beam becomes more slender. the second simulation assesses the combined impacts of material small-scale, surface energy, and substrate media on the critical buckling load of a simply-supported nanobeamelastic substrate system. it is indicated that the critical buckling load is enhanced by two distinct stiffening phenomena. the first one is associated with the material small-scale effect while the second one is related to the surface-energy effect. however, these two stiffening phenomena are less pronounced when the underlying substrate medium becomes softer. the third simulation investigates the combined effects of material small-scale, surface energy, and substrate media on the fundamental frequency of a simply-supported nanobeam-elastic substrate system. it is found that the material small-scale and surfaceenergy effects enhance the system stiffness while the velocity-gradient effect lowers the system stiffness, particularly for softer substrate media. however, the system stiffening phenomenon associated with the material small-scale effect and the surface-energy effect surpasses the system weakening phenomenon associated with the velocity-gradient effect. therefore, the combination of the material small-scale effect, the surface-energy effect, and the velocity-gradient effect results in a higher system natural frequency ( 1  >1.0) for specific values of system parameters investigated in the current study. the next step in developing the proposed beam model is considering the geometric nonlinearity for analyses of the nanobeam system with large displacement. moreover, the material nonlinearity is included into the model and applying the model to assess the rational micro/nanobeam structures against the failure scenarios in the future. acknowledgement: this study was financially supported by office of the permanent secretary, ministry of higher education, science, research and innovation under research grant for new scholar (rgns 64-134) and by the thailand research fund (trf) under grant (rta6280012). furthermore, special thanks go to a senior lecturer mr. wiwat sutiwipakorn for 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(36) can be expressed as: 6 4 2 0 0 0 1 2 3 06 4 2 ( ) ( for (0, ) ) ( ) ( ) 0 v x v x v x v x x x x x l      + = + +    (a1) where the parameters λ1, λ2, and λ3 are defined as: bending, buckling, and free vibration analyses of nanobeam-substrate medium systems 587 1 2 3; ; ( ) ( ) d ( ) n ( ) a l sur eff res es h h h eff eff eff ei s k ei ei ei     = = =− − (a2) for the sake of conciseness, the parameters λ1, λ2, and λ3 are rewritten in terms of the auxiliary parameters for the expression of the general solution as: 32 3 21 1 2 31 2 (2 / 27) ( / 3)( / 3) ; ; and 3 2          − +− + = =  = + (a3) the general form of homogeneous solution of eq. (a.1) can be expressed as: 0 1 1 2 2 3 3 4 4 5 5 6 6( ) ( ) ( ) ( ) ( ) ( ) ( )hv x x c x c x c x c x c x c     = + + + + + (a4) as suggested by morfidis [74] and avramidis and morfidis [75], there are two possible homogeneous solutions, which depend on the sign of the parameter δ as follows: solution case i: when δ = α3 + β2 > 0 1 1 1 2 3 4 5 6 ( ) ( ) ( i ; ; cos[ ]; s n[ ]; cos[ ]; s ] ) ( ) in( ) ( ) [ r x r x rx rx rx rx e e e qx e x qx e x x x x xqx e qx       − − − = = = = = = (a5) where ( ) ( ) ( ) 2 13 3 1 3 3 1 2 3 2 2 3 ; 2 2 3 ; 2 2 ; 3 2 / 3 r m n m m n m r q m n         + + + − = =          = − +  + − −  − − +  +    = − = − −  + − + − − −  (a6) solution case ii: when δ = α3 + β2 < 0 1 2 1 2 1 2 3 4 5 6( ) ; ( ) ; ( ) ; ( ) ; ( ) ; ( ) r x r x r x r xrx rxx e x e x e x e x e x e     − −−= = = = = = (a7) where 1 1 1 1 31 2 2 2 cos ; 2 cos ; 3 3 3 3 4 2 cos ; cos / 3 3 r r r            − +    = − − = − −        +   = − − = − −      (a8) 10589 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 421 444 https://doi.org/10.22190/fume220203014p © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis valentin l. popov1, aleksandr poliakov2, vladimir pakhaliuk2 1institute of mechanics, technische universität berlin, berlin, germany 2polytechnic institute, sevastopol state university, sevastopol, russian federation abstract. this work is devoted to the study of a one-dimensional phenomenological model of a focal defect regenerative rehabilitation in the articular cartilage. the model is based on six differential equations in partial derivatives of the “diffusion-reaction” type, which was previously used by a number of authors to study cellular processes in various tissues under cell therapy conditions. to take into account the influence of moderate mechanical stimulation of immature tissue, an indirect approach was used, as a result of which some model parameters that directly affect cell proliferation and differentiation were varied considering experimental data. the results of the model study show that moderate stimulation of immature tissue in the early stages of repair the focal articular cartilage defect under conditions of cell therapy leads to an intensification of regenerative processes in the tissue and promotes more rapid formation of the extracellular matrix. key words: synovial joint, articular cartilage, osteoarthritis, cell therapy, tissue engineering, regenerative rehabilitation 1. introduction the articular surfaces of long bones are covered with articular cartilage (ac), which is a type of connective tissue with an almost homogeneous structure at the macro level. however, at the micro level, ac has a complex heterogeneous structure, which is based on a hydrated elastic extracellular matrix (ecm), inside which chondrocytes are located, which are formed from chondroblasts. slightly mobile chondrocytes, due to the ability to divide, provide the synthesis and release of ecm elements. moreover, they are grouped in specific areas lacunae, permeable to low molecular weight metabolites. this allows for interstitial ac growth and its potential for regeneration in the event of damage. received february 03, 2022 / accepted march 21, 2022 corresponding author: valentin l. popov institute of mechanics, technische universität berlin, str. des 17 juni 135, 10623 berlin, germany e-mail: v.popov@tu-berlin.de 422 v. l. popov, a. poliakov, v. pakhaliuk however, in ac, blood, lymphatic vessels and nerves are almost completely absent, and therefore its formation, nutrition and lubrication are carried out only due to the interstitial fluid. the regenerative ac insufficiency determined by this feature in conditions of its progressive destruction leads not only to a variety of tissue damage mechanisms, but also to a variety of morphological phenomena that determine its clinical state. therefore, at present, treatment options for ac in the advanced stages of osteoarthritis are limited to injections of analgesics and total arthroplasty of the joints [1]. at the same time, in the early stages of osteoarthritis and in cases of focal cartilage defects, a greater number of therapies for the disease may be useful, including regenerative medicine-related cell therapy and tissue engineering strategies. when developing new methods and strategies for the treatment of joint diseases, including osteoarthritis, it is necessary to take into account the microstructure of the ac, schematically shown in fig.1. fig. 1 schematic cross-sectional diagram of healthy articular cartilage first of all, this concerns its zonal structuring, which includes three non-mineralized zones: superficial, intermediate, radial (deep) and one mineralized the so-called calcified cartilage [2]. the surface zone includes chondrocytes flat in shape, located close to each other and collagen fibers oriented along tangents to the articular surface. the structure of the intermediate zone differs significantly from the surface one. in it, chondrocytes have an oblique form of organization, and collagen fibers are scattered randomly in space. chondrocytes of the radial zone have a spherical shape and are structured along columns, parallel to which collagen fibers are located, penetrating into the calcified cartilage, thereby ensuring anchoring of the ac on the subchondral bone [3]. chondrocytes function under anaerobic conditions, receiving nutrients by diffusion from the interstitial fluid, and their shape can change not only depending on belonging to a particular ac zone, but also on the physiological state, level of motor activity and age of the patient [4]. to restore small-sized focal ac defects, mosaic chondroblasty technologies are used using native osteochondral grafts. however, their practical applicability is limited by a number of disadvantages, such as the limited size of the defect to be repaired, the difficulty of choosing ac sites with the desired curvature for graft retrieval, the risk of postoperative pain in the joint, etc. elimination of these shortcomings is potentially possible with the use of technologies for implantation the in vitro cultured autologous one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 423 chondrocytes (autologous chondrocytes implantation (aci)), allowing, among other things, to control the process of reparative regeneration of the damaged ac in order to fully restore it. however, chondrocytes undergo dedifferentiation during cultivation and reproduction, as a result of which the expression of chondrocyte markers (collagen ii, aggrecan, transcription factor sox9) decreases, and the expression of fibroblast markers (collagen i, versican) increases [5]. as a result, cells acquire a fibroblast-like phenotype and become unable to produce the components necessary for ac repair. this is a significant challenge in the development of aci technologies. one of the ways to solve it is to select a different type of cells for implantation into the damaged area of the ac, capable of differentiating into articular chondrocytes. it is known that stem cells possess the properties of self-renewal and differentiation into various cell lines. therefore, in modern cell technologies focused on the restoration of not only focal, but also extensive ac defects, mesenchymal stem cells (msc) and embryonic stem cells (esc) (articular stem cell implantation (asi)) are used, which meet the criteria established by the international society for cellular therapy's mesenchymal stromal cell committee (isct msc) [6]. the process of differentiation of progenitor cells into a specific cell line can be controlled by regulatory genes. but the population of cells obtained in this way, as a rule, does not fully correspond to the target tissue in terms of its biochemical and biomechanical properties [7]. this discrepancy can be eliminated by implanting an artificial extracellular matrix (scaffold) into the defect area along with msc / esc. this approach is often used in modern ac recovery technologies. in this case, the scaffold is chosen so that its structure facilitates the creation of an environment in which cells could modulate their environment in vitro and / or in vivo, just as it happens in the target tissue [8]. therefore, in addition to the desired structure, the scaffold must have mechanical properties corresponding to the properties of the target tissue [9]. growth factors that function as signaling molecules that determine intercellular interactions and processes are of great importance for the development of new methods and strategies for ac treating. for example, fibroblast growth factors (fgfs) play a key role in the proliferation and differentiation of a wide range of cells and tissues; bone morphogenetic proteins (bmp), which form a group of multifunctional growth and regeneration factors belonging to the transforming growth factor tgf-β superfamily, are able to regulate various cellular processes in the body. in healthy ac, the growth factors fgf-1 and bmp-2 provide the metabolism and renewal of chondrocytes. extensive information about these molecules, their role in metabolism and regulation of molecular processes of cartilage tissue remodeling is given in numerous literatures [10-18]. it is obvious that innovative strategies for the treatment of ac, including those focused on the complete restoration of damaged tissue areas, should be built taking into account the conditions and the nature of the synovial joint operating as a whole, enduring the action of significant loads. so, for example, the load perceived by a healthy knee joint when a person climbs stairs is 8-10 times greater than his body weight and ac resists it without destruction. the synovial joint elements, touching along the articular surfaces, are subjected to compression, shear and sliding relative to each other in the process of their operating. since ac has a very low permeability, during compression, the hydrostatic pressure of the interstitial fluid present in it increases rapidly, allowing the ac to take on external stress. as the compressive load increases, interstitial fluid begins to be released from the ac, which leads to a redistribution of some of the load on the solid ecm and to its consolidation. upon reaching equilibrium, the flow of interstitial fluid from the ac 424 v. l. popov, a. poliakov, v. pakhaliuk stops, and the ecm takes over the entire load [19]. this effect is known as stress relaxation in ac and characterizes one of its most important characteristics, which determine the compression process and support functions [20]. along with viscoelastic properties, stress relaxation in ac is quite convincingly explained on the basis of its two-phase model, in which it is assumed that ac contains a small number of cells (~ 1% of the tissue mass) and its mechanical performances are mainly determined by ecm, which consists of liquid and solid phases [21]. the liquid phase (~ 80% by weight) is represented by interstitial fluid, and the composite solid phase, as a porous, permeable material bound to fibers, is represented by macromolecules, among which collagen ii and complex proteins proteoglycans, predominate. if we do not take into account the biochemical processes and cellular changes occurring in the cartilaginous tissue during normal operating under the development of pathology or regeneration, then taking into account the above-mentioned features, ac can be considered as a continuous poroelastic medium and according to the theory of effective poroelasticity of m.a. biot [22, 23], to create and investigate various models of cartilage tissue within the framework of continuum mechanics. models considering other characteristic features of ac in synovial joint make it possible to study the interaction of several tissues on each other, cell-cell or cell-ecm interactions, as well as the processes of migration, differentiation and natural cell death. for example, m. blewis and colleagues used the synovial joint (sj) compartment model to predict the dynamic processes occurring in the interstitial fluid [24]. taking into account the mass balance of hyaluronic acid and proteoglycan-4 in synovial fluid, the authors reduced the problem to a system of linear differential equations with three state variables, considering the rate of secretion, decomposition and fluid flow through the synovial membrane. the performances of the model compartments used by the authors can be changed if the characteristics of the synovial fluid change, therefore it is applicable for the analysis of normal and pathological conditions of the joint, as well as for conditions caused by certain therapeutic procedures, for example, hyaluronic acid injections, msc / esc and etc. more complex models are also known, including those applicable to the analysis of rheumatoid arthritis of the joint, depending on changes in the state of inflammatory cells, endothelium, fibroblasts, chondrocytes, cytokines, and growth factors in the sj compartment [25]. that is, the analysis of ac within synovial joint in silico makes it possible to investigate and predict the physiological state, structural rearrangement, cellular and chemical processes occurring in tissues from different points of view. it seems that the results of such studies, as well as a lot of experimental data, contributed to the accumulation of a critical amount of knowledge necessary to create effective technologies for the treatment of various synovial joint diseases. however, up to the present time, such technologies have not yet been created. at the same time, it should be noted that in recent years there has been significant progress in this area of medical science. methods for cultivating and storing cells for therapeutic use, as well as methods for their transplantation, are being improved. tissue engineering is evolving towards the creation of biomimetic and more complex tissue engineering constructs (including those with the desired shape) that could be used to replace larger sections of damaged ac. however, even the use of such modern high-tech methods in many clinical cases does not lead to successful treatment of osteoarthritis, and this fact is the main motive for finding new ways to combat this disease. one of the new areas of medical science, within which an effective solution to the problem of ac recovery can be found, is regenerative rehabilitation, which is based on the combination one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 425 of modern technologies of regenerative and rehabilitation medicine [26-28]. many theoretical and experimental results testify to the synergistic effect that can potentially be achieved as a result of such a combination. from the point of view of regenerative rehabilitation, of interest are studies that have studied the responses of cartilage tissue, cells, and molecules to the action of mechanical stimuli. it has been established that optimal mechanical stimulation promotes collagen formation, cell viability, and fiber organization in tissue engineering constructs [29, 30]. in vitro experiments indicate that both short-term and long-term compressive and/or shear loads change the composition of cartilage tissue [31], its structure [32], and function [33]. during daily physical activity, mechanical signals resulting from the load on the joints and the corresponding deformation of the articular tissues modulate the metabolism of chondrocytes and cause coordinated changes in the biochemical and mechanical properties of ac [34-36]. at the same time, normal cartilage homeostasis is partially associated with cyclic loads that ensure the activation of ecm-associated growth factor tgf-β (bmp-2) [37, 38]. fgf-1 synthesis also presumably depends on the amplitude and frequency of the stimulating force acting on the ac, which increases angiogenesis in the subchondral bone [39] and, consequently, promotes cartilage tissue regeneration. it is generally accepted that mechanical stimulation strongly influences the mechanical properties of the superficial ac zone [40], but is enzymatically regulated in deeper zones [41, 42]. moreover, as shown in [43], it is very likely that the central channel for the transformation of mechanical forces into biological reactions in ac is the mapk (mitogen-activated protein kinase) pathway. the effects described very briefly above, which deserve a detailed systematic study, indicate that regenerative rehabilitation technologies that use ac mechanical stimulation in vivo along with cell therapy and tissue engineering strategies may be effective in the treatment of various synovial joint diseases, including osteoarthritis [44]. for their development, various resources should be involved, including simulation and in silico studies of processes occurring in ac under various conditions, under various tissue conditions and types of stimulation, since there is still no complete understanding of the mechanisms by which mechanical stimulation promotes joint homeostasis, chondroprotection, and the production of pro-inflammatory mediators. such models can be different in form and content, including biological, biophysical, biochemical, biomechanical and combined. in this paper, we investigate one-dimensional biological models of repair the ac focal defects based on the strategies of cell therapy and regenerative rehabilitation in order to compare, first of all, the qualitative results of these strategies implementation. 2. description of the model most of the mathematical models of biological processes in tissues are designed to analyze a relatively small number of related state variables that are most important for the researcher in solving a specific problem. this is explained by the fact that, by their nature, biological tissues are pronounced homeostatic systems, with their inherent unpredictability, instability and striving for balance. in this regard, the processes occurring in them are very variable and depend on many factors, often not completely defined or even random. in addition, even small changes in the internal parameters of a particular tissue can lead to a qualitative restructuring of its homeostasis, i.e. to a bifurcation. therefore, complex 426 v. l. popov, a. poliakov, v. pakhaliuk multifactorial mathematical models of tissue processes can be studied only in stable regions of the space of state variables, the determination of which is a separate rather complicated problem. but important information can also be obtained by studying simplified models of tissue processes, in which some important variables and parameters are taken into account indirectly. as examples, here are a number of models tested in the study of wound healing and bone fractures [45]. in this work, we use a simplified phenomenological biological model built on the basis of the “diffusion-reaction” type system of equations to simulate cellular therapy and regenerative rehabilitation of ac focal defect. previously, such models were used by m. lutianov et al. [46] and later (with some variations) by k. campbell et al. [47, 48] to study various ac regeneration strategies under cell therapy. when substantiating them, the authors used the approach proposed by a. bailón-plaza and m.c. van der meulen to study the effect of growth factors on the healing of bone fractures [49], based on modeling the diffusion migration of cells and the reaction of the environment that promotes their proliferation, differentiation, and death. since the biological processes of cartilage tissue repair and bone tissue fracture healing have a common basis, their biological models can be considered similar. therefore, following [46], the biological model of regenerative rehabilitation of ac focal defect can be represented as a system of partial differential equations with variables: 𝐶𝑆 is the msc density; 𝐶𝐶 is the chondrocyte density; 𝑛 is the nutrient concentration; 𝑚 is the ecm density: 𝜕𝐶𝑆 𝜕𝑡 = ∇[𝐷𝑆∇𝐶𝑆]⏟ + 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑝1𝐶𝑆 𝑛 𝑛+𝑛0 𝐻(𝑛 − 𝑛1)⏟ 𝑝𝑟𝑜𝑙𝑖𝑓𝑒𝑟𝑎𝑡𝑖𝑜𝑛 − 𝑝2𝐶𝑆𝐻(𝐶𝑆 − 𝐶𝑆0)⏟ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛 − 𝑝3𝐶𝑆𝐻(𝑛1 − 𝑛)⏟ 𝑑𝑒𝑎𝑡ℎ , (1a) 𝜕𝐶𝐶 𝜕𝑡 = ∇[𝐷𝐶∇𝐶𝐶]⏟ + 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑝4𝐶𝐶 𝑛 𝑛+𝑛0 𝐻(𝑛 − 𝑛1)⏟ 𝑝𝑟𝑜𝑙𝑖𝑓𝑒𝑟𝑎𝑡𝑖𝑜𝑛 + 𝑝2𝐶𝑆𝐻(𝐶𝑆 − 𝐶𝑆0)⏟ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛 − 𝑝5𝐶𝐶𝐻(𝑛1 − 𝑛)⏟ 𝑑𝑒𝑎𝑡ℎ , (1b) 𝜕𝑛 𝜕𝑡 = ∇[𝐷𝑛∇𝑛]⏟ − 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑛 𝑛+𝑛0 (𝑝6𝐶𝑆 + 𝑝7𝐶𝐶)⏟ 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 , (1c) 𝜕𝑚 𝜕𝑡 = ∇[𝐷𝑚∇𝑚]⏟ + 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑝8 𝑛 𝑛+𝑛0 𝐶𝐶⏟ 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 , (1d) which, if necessary, can be supplemented by equations with variables: 𝑔 is the concentration of growth factors fgf-1 and 𝑏 is the concentration of growth factors bmp-2 [47]: 𝜕𝑔 𝜕𝑡 = ∇[𝐷𝑔∇𝑔]⏟ + 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑝9𝐶𝑆 − 𝑝11𝑔⏟ 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 , (2a) 𝜕𝑏 𝜕𝑡 = ∇[𝐷𝑏∇𝑏]⏟ + 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑝12𝐶𝐶 − 𝑝13𝑔⏟ 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 , (2b) where ∇= 𝜕 𝜕𝑥 𝑖 + 𝜕 𝜕𝑦 𝑗 + 𝜕 𝜕𝑧 �⃗⃗� ; ∇𝜑(𝑥, 𝑦, 𝑧) = 𝜕𝜑 𝜕𝑥 𝑖 + 𝜕𝜑 𝜕𝑦 𝑗 + 𝜕𝜑 𝜕𝑧 �⃗⃗� = 𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗𝜑 ; ∇𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗𝜑 = 𝑑𝑖𝑣𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗𝜑 = 𝜕2𝜑 𝜕𝑥2 + 𝜕2𝜑 𝜕𝑦2 + 𝜕2𝜑 𝜕𝑧2 ; 𝜑(𝑥, 𝑦, 𝑧) is a some scalar function. descriptions of the variables and constant parameters of the model included in eqs (1a – 1d) and (2a – 2b), as well as their possible values in dimensional and dimensionless one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 427 form, are given in appendix a. previously, these values were established experimentally or justified theoretically and used in a number of works on bone fracture healing, implant osseointegration, and ac repair [46-51]. considering the fact that the main goal of this work is to study, first of all, the qualitative results of the various ac recovery strategies implementation, the models of which are characterized by significant uncertainty and depend on many factors, including the physiological state of the intended patient, the nature of the defect, the method of tissue stimulation, etc., these data were used only as estimates of the parameters. in addition, we assumed that mechanical stimulation of ac in the vicinity of the reconstructed region changes the mechanical state of the medium, which also leads to a change in the values of the model parameters. since tissues respond differently to various physiological loads and to stimuli of different types and intensities [52], hypothetically, such changes should lead either to an improvement or deterioration in the processes of damaged tissues repair. in this regard, when modeling, we varied the values of the parameters that change under the influence of mechanical stimuli, taking into account the results of numerous studies available in the literature. as a criterion for the quality of ac repair processes, we used the final density distribution of the synthesized ecm in the region of the repaired ac defect. to evaluate other indicators achieved as a result of the recovery processes implementation, including the phenotype of the resulting tissue, biomechanical models are more suitable, including equations that describe the stress-strain state of the tissue during stimulation [53–55]. to take into account the mechanical stimulation of ac in the process of regenerative rehabilitation, we used the approach proposed by a. andreykiv et al. to model the healing of bone fractures with mechanoregulation of cell differentiation and proliferation [56]. in their model, the authors consider the fact that the diffusion coefficients (𝐷𝑚 , 𝐷𝑓) and proliferation rates (𝑃𝑚, 𝑃𝑓) of msc and fibroblasts, respectively, depend on the volume concentration of the bone (𝑚𝑏) and cartilage (𝑚𝑏 ) fractions of the ecm. at the same time, the initial values of proliferation rates (𝑃𝑚0, 𝑃𝑓0) were calculated in accordance with the mechanoregulation index [53] 𝑆 = 𝛾 𝑎 + 𝜐 𝑏 , where 𝛾 and 𝑣 are the maximum shear stress and maximum velocity of the interstitial fluid, respectively, and 𝑎 and 𝑏 are constants [57]. according to the hypothesis proposed in [53], if 𝑆 is less than a certain threshold value 𝑆𝑚𝑖𝑛, then the mechanical environment of the tissue is favorable for osteoblast differentiation and bone tissue synthesis, and at moderate values (𝑆𝑚𝑖𝑛 < 𝑆 < 𝑆𝑚𝑎𝑥), that do not exceed some maximum value 𝑆𝑚𝑎𝑥, the environment favor the differentiation of chondrocytes and the formation of cartilaginous tissue. and, finally, high values of the index (𝑆 > 𝑆𝑚𝑎𝑥) contribute to the differentiation of fibroblasts and the formation of fibrous tissue in the area of the bone fracture. therefore, in the model [56], the rates of differentiation of fibroblasts, chondrocytes, and osteoblasts (𝑃𝑓 , 𝑃𝑐 , 𝑃𝑏) were also calculated depending on the value of the mechanoregulation index 𝑆. thus, the approach described above makes it possible to indirectly take into account the effect of mechanical stimulation on tissue formation in the process of regenerative rehabilitation. moreover, if we take into account the results of experiments, according to which low-amplitude cyclic tissue deformation leads to an increase in osteoblast proliferation 428 v. l. popov, a. poliakov, v. pakhaliuk by about 1.5 times [58, 59], and also that unstimulated chondrocytes proliferate at a rate similar to osteoblasts [60, 61], a qualitative analysis of the ac regenerative rehabilitation process can be performed by varying the values of cell proliferation and differentiation coefficients without calculating the mechanoregulation index. it should also be noted that during ac repair, bone formation is almost impossible, but there is a possibility of fibrous tissue formation. but with moderate stimulation, according to the hypothesis described above [53], this probability is very small. figure 1 is a schematic representation of the repair process for an ac focal defect using aci/asi cellular technologies and/or regenerative rehabilitation, which shows a schematic cross-section of the defect on an enlarged scale. it is assumed that the defect has small dimensions in width and its properties in planes perpendicular to the defect practically do not change, which allows us to consider this problem as one-dimensional. the origin of the coordinate system is located on the subchondral bone, and the 𝑥 coordinate axis is directed towards the articular surface. the subchondral bone is considered permeable, and msc can diffuse from it into the defect, the flow of which is determined by a certain time function 𝑓(𝑡). the fluxes from the subchondral bone of chondrocytes, growth factors, nutrients, and ecm elements are assumed to be zero, although they can also be specified as specific functions of time if necessary. thus, the boundary conditions at the point 𝑥 = 0 look like: −𝐷𝑆 𝜕𝐶𝑆 𝜕𝑥 = 𝑓(𝑡), 𝐷𝐶 𝜕𝐶𝐶 𝜕𝑥 = 0, 𝐷𝑔 𝜕𝑔 𝜕𝑥 = 0, 𝐷𝑏 𝜕𝑏 𝜕𝑥 = 0, 𝐷𝑛 𝜕𝑛 𝜕𝑥 = 0, 𝐷𝑚 𝜕𝑚 𝜕𝑥 = 0. (3) boundary conditions on the defect surface at 𝑥 = 𝑑 : 𝐷𝑆 𝜕𝐶𝑆 𝜕𝑥 = 0, 𝐷𝐶 𝜕𝐶𝐶 𝜕𝑥 = 0, 𝐷𝑔 𝜕𝑔 𝜕𝑥 = −𝛾𝑔, 𝐷𝑏 𝜕𝑔 𝜕𝑥 = −𝜒𝑏, 𝑛 = 𝑁0, 𝐷𝑚 𝜕𝑚 𝜕𝑥 = 0. (4) that is, it is assumed that the fluxes of mscs, chondrocytes, and ecm elements on the surface of the defect are equal to zero, and nutrients with a constant concentration of 𝑁0 enter the defect from the synovial fluid. the flows of growth factors in the study of the model (1a – 1d, 2a – 2b) calculated in proportion to their concentrations with the coefficients 𝛾 and 𝜒, respectively. the initial conditions of the problem are formulated in accordance with a certain strategy of regenerative rehabilitation. for example, when implementing the asi strategy, it is assumed that mscs are implanted into a defect, and the cells are arranged according to the height of the defect according to a certain law 𝐶𝑆(0) = 𝐶𝑆 (0)ℎ(𝑥). if at the same time a scaffold is implanted into the defect, then the initial ecm density taken equal to (0) = 𝑚3 + 𝑚𝑠 , where 𝑚3 is the initial ecm density and 𝑚𝑠 is the scaffold density. then, taking into account the density of nutrients 𝑛(0) = 𝑁0 and zero values of other state variables at the initial moment of time, the initial conditions have the following form: 𝐶𝑆 = 𝐶𝑆 (0)ℎ(𝑥), 𝑔(0) = 0, 𝑏(0) = 0, 𝐶𝐶 = 0, 𝑛(0) = 𝑁0, 𝑚(0) = 𝑚3 +𝑚𝑠, (5) where 𝐶𝑆 (0) is the initial msc density. the initial conditions corresponding to other ac focal defect restoration strategies are formed in a similar way. the scheme displayed in figure 2 can also be used to describe the process of ac regenerative rehabilitation. in this case, in addition to the conditions of cell therapy and tissue engineering strategies, the effect of stimulating effects on tissue repair should be one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 429 taken into account. at the same time, one should consider the fact that asi/aci technologies involve implantation of msc/esc and/or autologous chondrocytes into the area of the defect. in the case of implementing tissue engineering strategies, cells are preseeded in vitro on an artificial matrix scaffold, in order to increase their viability, proliferation, osteogenic and chondrogenic signal expression. apoptosis differentiation cells in tissue proliferation apoptosis de-differentiation new tissue ecm production stimulated tissue undifferentiated cells mscs chondrocites ecm+scaffold proliferation subchondral bonestimulating load articular cartilage defect area reactive forces fig. 2 scheme of ac focal defect repair process using aci/asi cellular technologies and regenerative rehabilitation the results of a systematic study of ac recovery strategies using asi/aci models, including various combinations of them, are presented in [47, 48]. as a quality criterion for choosing the percentage of implanted cells (msc and autologous chondrocytes), the authors used the average density of the ecm formed new tissue. it has been shown that, in principle, each of the possible strategies (100% asi; 90% asi + 10% aci; …; 10% asi + 90% aci; 100% aci) can be effective. however, from a practical point of view, the effectiveness of ac damage repair technology is also evaluated taking into account the availability of cell types, the overall cost and the effectiveness of the procedure. in this regard, in this study, asi technology (100% asi) was adopted as a standard strategy for cell therapy. that is, we assume that regenerative rehabilitation strategies that involve moderate mechanical stimulation of immature tissue in the damaged area of the ac will contribute to its recovery in all practically significant cases. the study of the ac focal defects restoration processes was carried out using a mathematical model (1a – 1d, 2a – 2b) for various parameter values. based on the hypothesis, confirmed by numerous experimental data, it is possible to select such stimulating effects that 430 v. l. popov, a. poliakov, v. pakhaliuk would enhance the effects of cell therapy and contribute to the creation of the best conditions for the restoration of damaged tissue in vivo. the solution of this problem would contribute to the development of the best strategies for the regenerative rehabilitation of tissues damaged due to injuries or diseases, including osteoarthritis. the scheme (fig. 2) shows that stimulated tissue cells can proliferate, differentiate into chondrocytes, and die. therefore, the main goal of stimulating immature tissue in the area of the ac defect is to provide the best conditions for the viability and differentiation of progenitor cells into chondrocytes, which would promote the production of ecm and the formation of new tissue with the desired phenotype. in this study, we took into account the effect of mechanical stimulation of immature tissue by increasing the coefficients: 𝑝40 is a chondrocyte proliferation constant, 𝑝400 is a chondrocyte proliferation rate (from fgf-1) and 𝑝2 is a msc differentiation rate, because 𝑝40 and 𝑝400 determine the amplitude value of the chondrocyte proliferation rate: 𝑝4 = 𝐵𝑚 (1 − 𝐶𝐶 𝐶𝐶𝑚𝑎𝑥 ) = (𝑝40 𝑚 𝑚2+𝑚2 2 + 𝑝400 𝑔 𝑔+𝑔0 ) (1 − 𝐶𝐶 𝐶𝐶𝑚𝑎𝑥 ) , and 𝑝2 in eq. (1b) determines the modulus of the chondrocyte differentiation rate: 𝑉𝑑𝐶ℎ = 𝑝2𝐶𝑆𝐻(𝐶𝑆 − 𝐶𝑆0), where (𝐶𝑆 − 𝐶𝑆0) = { 0, 𝐶𝑆 ≤ 𝐶𝑆0 1, 𝐶𝑆 > 𝐶𝑆0 is the heaviside function, 𝐶𝑆0 = (𝐶𝑆0𝑚𝑎𝑥 − 𝐶𝑆0𝑚𝑖𝑛)𝑒 −𝛼𝑏 is a msc threshold density, 𝐶𝑆0𝑚𝑎𝑥 is a maximum threshold density, 𝐶𝑆0𝑚𝑖𝑛 is a minimum threshold density, 𝛼 is a threshold msc density reduction factor. following [58-61], when simulation processes of regenerative rehabilitation of ac, the values of these coefficients were taken to be 1.5 times greater than their estimated values given in appendix a. it is clear that such an approach does not allow making any quantitative estimates that could be used in medical practice. but it allows one to obtain and compare qualitative estimates of various ac defect recovery strategies. 3. results and discussion solutions to the equations system (1a – 1d, 2a – 2b) for various model parameters were obtained in a dimensionless form by the finite element method in matlab r2021b (mathworks, usa) using the pdepe function. the dimensionless values of the parameters were calculated taking into account the characteristic values that determine the main features of the model: 𝑑 = 2 𝑚𝑚 is a defect thickness/depth; 𝐶𝑡𝑜𝑡𝑎𝑙,𝑚𝑎𝑥0 = 10 6 𝑁𝑐 𝑚𝑚3 is a maximum total cell density; 𝑚𝑚𝑎𝑥 = 10 −4 𝑔 𝑚𝑚3 is a maximum ecm density; 𝑁0 = 6.5 × 10 −11 𝑁𝑚 𝑚𝑚3 is an initial nutrient concentration; 𝑝80 = 3.75 × 10 −13 𝑔 𝑚𝑚3 ∙ 1 𝑁𝑐 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 is an ecm production constant; 𝑔0 = 10 −10 𝑔 𝑚𝑚3 is a fgf-1 reference concentration; 𝑏0 = 10 −10 𝑔 𝑚𝑚3 is a bmp-2 reference concentration. the dimensionless time was determined by the formula �̃� = 𝑡 𝑝80𝐶𝑡𝑜𝑡𝑎𝑙,𝑚𝑎𝑥0∙ 𝑚𝑚𝑎𝑥 , according to which the time unit corresponds to about 11 days [48]. dimensionless values of other model parameters are presented in appendix a. one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 431 fig. 3 shows the evolution of ecm density of the formed tissue under standard asi conditions, and fig. 4 shows the evolution of ecm density, chondrocites population density, msc population density, bmp-2 concentration and fgf-1 concentration over time with the values of the model parameters given in the appendix a. similar graphs obtained under the conditions of regenerative rehabilitation of the ac injured area under the respective parameter values justified above are shown in figs 5 and 6, respectively. the simulation results indicate that the process of ecm formation occurs in the same way when implementing two ac recovery strategies studied in this work. however, within the framework of the model under study, in the early stages, the maximum values of ecm density are achieved at different points in time (see figs 3 and 5). nevertheless, in all cases, the regenerative rehabilitation strategy provides better conditions for the ac defect repair compared to the asi strategy. so from figs 3 and 5 it follows that ecm density at all stages of evolution is of greater importance in terms of regenerative rehabilitation (table 1). an analysis of the evolution of other model state variables indicates that their maximum values are also reached at different points in time when implementing different strategies (see table 2). in addition, it should be noted that under conditions of regenerative rehabilitation, the maximum values of msc population density and fgf-1 concentration have lower values under asi conditions (see table 3). this is explained by the property of the model (1a – 1d, 2a – 2b), which consists in the fact that during regenerative rehabilitation, part of the msc population differentiates into chondrocytes and this process is accompanied by a decrease in growth factors fgf-1. table 1 ecm density values at various stages of ac recovery strategy time, 𝑡 6 11 16 ecm density ac 0.066150 0.425962 0.470151 regenerative rehabilitation 0.084656 0.469820 0.486250 table 2 time to reach the maximum values of the model state variables (figs 4 and 6) strategy model state variable ecm density chondrocites density msc population density bmp-2 concentration fgf-1 concentration time, 𝑡 (𝑡 = 1 roughly equivalent to 11 days) asi 16 2.21 0.55 3.31 1.66 regenerative rehabilitation 16 2.21 0.55 2.76 1.03 table 3 maximum values of the model state variables for the entire time of evolution (according to figs 4 and 6) strategy model state variable ecm density chondrocites density msc population density bmp-2 concentration fgf-1 concentration asi 0.470151 221.441 486.656 94.851 114.479 regenerative rehabilitation 0.48625 260.617 415.215 114.788 94.446 432 v. l. popov, a. poliakov, v. pakhaliuk and, finally, it should be noted that the constant parameters of the model contribute to the fact that at a certain point in time the ecm is saturated and its density subsequently practically does not change in the positive direction of the x axis (see figs 7 and 8). however, the saturation process is reoriented in the opposite direction, which leads to an increase in density in areas located closer to the subchondral bone. predicting the further process of matrix transformation is quite difficult (if at all possible), since the model we use does not take into account many factors that affect the ac recovery process. fig. 3 the evolution of ecm density m=m(x,t) of the formed tissue under standard asi conditions ( t=1 roughly equivalent to 11 days; x=1 equivalent to d=2 mm) one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 433 fig. 4 time evolution of model state variables under standard asi conditions: a) ecm density; b) chondrocites population density; c) msc population density; d) bmp-2 concentration; e) fgf-1 concentration. t=1 roughly equivalent to 11 days 434 v. l. popov, a. poliakov, v. pakhaliuk fig. 5 the evolution of ecm density m=m(x,t) of the formed tissue under conditions of regenerative rehabilitation ( t=1 roughly equivalent to 11 days; x=1 equivalent to d=2 mm) one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 435 fig. 6 time evolution of model state variables under regenerative rehabilitation conditions: a) ecm density; b) chondrocites population density; c) msc population density; d) bmp-2 concentration; e) fgf-1 concentration. t=1 roughly equivalent to 11 days 436 v. l. popov, a. poliakov, v. pakhaliuk fig. 7 the evolution of ecm density m=m(x,t) of the formed tissue under standard asi conditions for time 𝑡 = 33: a) 3d visualization; b) visualization in the x-z plane; c) visualization in the t-z plane. t=1 roughly equivalent to 11 days; x=1 equivalent to d=2 mm one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 437 fig. 8 the evolution of ecm density m=m(x,t) of the formed tissue under conditions of regenerative rehabilitation for time 𝑡 = 33: a) 3d visualization; b) visualization in the x-z plane; c) visualization in the t-z plane. t=1 roughly equivalent to 11 days; x=1 equivalent to d=2 mm 438 v. l. popov, a. poliakov, v. pakhaliuk 4. conclusion increasing the intensity and quality of injuries and diseases treatment of biological tissues can be achieved by developing new technologies for regenerative rehabilitation, which, according to a large number of well-known experts, contribute to the manifestation of a synergistic effect from the combined use of advanced technologies of regenerative and rehabilitation medicine [62,63]. in this work, we aimed to show that regenerative rehabilitation can be very effective in the treatment of such a common as disease as osteoarthritis. at the same time, it was assumed that various treatment strategies based on the combined use of cell therapy, tissue engineering constructs (scaffolds), and moderate stimulation of immature tissue can be used to treat focal as defects. this assumption was made taking into account the results of a number of experimental in vitro studies published in the scientific literature, indicating the effectiveness of strategies for the regenerative rehabilitation of as compared with conventional cell therapy. however, the results of the implementation of such strategies in experiments in vivo on animal models are very controversial. in this regard, the improvement of ac regenerative rehabilitation strategies that contribute to the achievement of reliable results in the treatment of osteoarthritis is an urgent scientific problem. one approach to its solution is to study various mathematical models of ac regenerative rehabilitation. in this work, we used a simplified phenomenological biological model of regenerative rehabilitation, built on the basis of the “diffusion-reaction” type equations system. the study of the model was carried out by the finite element method, the effectiveness of which was confirmed by the results of many previous works [64]. previously, a similar model was used by a number of authors to study cellular processes in various tissues under conditions of cell therapy. recently, many medical applications tend to use accurate geometric models of the objects under study [65]. however, in this work, a simplified geometric model was used. nevertheless, the results of its study were generally consistent with the results of in vivo experiments, which led to the conclusion about its adequacy. it was shown that a change in the coefficients of the model, which determine the state of the tissue and environmental conditions, have a significant impact on the dynamics of the restored tissue structural elements, such as chondrocytes, ecm, msc, etc. the complication of the model by taking into account a larger number of structural elements leads to a change in the dynamics of their formation, but in general it does not have a significant impact on the final result. to a large extent, its achievement depends on the parameters of the tissue and the environment, which change, including in the process of mechanical stimulation of the cartilage tissue. in this regard, the tasks of more accurate determination of diffusion coefficients, cell proliferation rates, and other parameters, depending on the qualitative and quantitative performances of mechanical stimuli applied to immature tissue, remain topical. at this stage, unfortunately, this is possible only on the basis of various more or less plausible hypotheses formulated based on the results of in vitro studies. therefore, it is premature to say that based on the obtained results it is possible to plan strategies for the practical regenerative rehabilitation of as. first, these results suggest that any strategy will be effective when stimulation of immature tissue will increase msc and chondrocyte proliferation, increase msc differentiation into chondrocytes, and reduce cell death. secondly, the architectonics of the formed ecm, which mainly determines its properties, remains unknown. thirdly, it is not clear at what time period after cell implantation stimulation of the regenerated tissue 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(10−4 ÷ 10−3) 𝑚𝑚2 ℎ𝑜𝑢𝑟 𝐷𝐶 – chondrocyte random motility (diffusion) coefficient 𝐷𝐶 = 𝐷𝐶0 𝑚 𝑚2+𝑚1 2 𝑚𝑚2 ℎ𝑜𝑢𝑟 𝐷𝐶0 – chondrocyte diffusion constant 𝐷𝐶0 = 2𝑚1𝐷𝐶 ∗ 𝑚𝑚2 ℎ𝑜𝑢𝑟 ∙ 𝑔 𝑚𝑚3 0.001 𝐷𝐶 ∗ − maximum chondrocyte diffusion coefficient 𝐷𝐶 ∗ = 3.6 × (10−4 ÷ 10−3) 𝑚𝑚2 ℎ𝑜𝑢𝑟 𝐷𝑛 nutrient diffusion coefficient 𝐷𝑛 = 4.6 𝑚𝑚2 ℎ𝑜𝑢𝑟 100 300 𝐷𝑚 – ecm diffusion coefficient 𝐷𝑚 = 2.5 × 10 −5 𝑚𝑚2 ℎ𝑜𝑢𝑟 0.001 – 0.01 𝐷𝑔 fgf-1 diffusion coefficient 𝐷𝑔 = 2 × 10 −3 𝑚𝑚2 ℎ𝑜𝑢𝑟 1.14 𝐷𝑏 – bmp-2 diffusion coefficient 𝐷𝑏 = 2 × 10 −3 𝑚𝑚2 ℎ𝑜𝑢𝑟 1.14 𝑝1 – msc proliferation rate 𝑝1 = 𝐴𝑚 (1 − 𝐶𝑆 𝐶𝑆𝑚𝑎𝑥 ); 𝑐𝑒𝑙𝑙 ℎ𝑜𝑢𝑟 𝐴𝑚 = 𝑝10 𝑚 𝑚2+𝑚2 2 ; 𝐶𝑆𝑚𝑎𝑥 = 𝐶𝑆𝑚𝑎𝑥0 (1 − 𝑚 𝑚𝑚𝑎𝑥 ) 𝑝10 – msc proliferation constant 𝑝10 = 2𝑚2𝑝1 ∗ = 4 × 10−6 𝑔 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 12 𝑝1 ∗ – maximum msc proliferation rate 𝑝1 ∗ = 0.2 𝑐𝑒𝑙𝑙 ℎ𝑜𝑢𝑟 𝐶𝑆𝑚𝑎𝑥0 – maximum msc density 𝐶𝑆𝑚𝑎𝑥0 = 0 ÷ 10 6 𝑁𝑐 𝑚𝑚3 0.6 𝑚1 – reference ecm density 𝑚1 = 10 −5 (assumed 𝑚𝑚𝑎𝑥 /10 ) 𝑔 𝑚𝑚3 0.1 𝑚2 – reference ecm density 𝑚2 = 10 −5 (assumed 𝑚𝑚𝑎𝑥 /10 ) 𝑔 𝑚𝑚3 0.1 one-dimensional biological model of synovial joints regenerative rehabilitation in osteoarthritis 443 𝑚𝑚𝑎𝑥 – maximum ecm density 𝑚𝑚𝑎𝑥 = 10 −4 𝑔 𝑚𝑚3 𝑝2 – msc differentiation rate 𝑝2 = 3.75 × 10 −3 1 ℎ𝑜𝑢𝑟 1.0 𝑝3 – msc death rate 𝑝3 = 3.75 × 10 −3 1 ℎ𝑜𝑢𝑟 1.0 𝑝4 – chondrocyte proliferation rate 𝑝4 = 𝐵𝑚 (1 − 𝐶𝐶 𝐶𝐶𝑚𝑎𝑥 ); 1 ℎ𝑜𝑢𝑟 𝐵𝑚 = 𝑝40 𝑚 𝑚2+𝑚2 2 + 𝑝400 𝑔 𝑔+𝑔0 ; 𝐶𝐶𝑚𝑎𝑥 = 𝐶𝐶𝑚𝑎𝑥0 (1 − 𝑚 𝑚𝑚𝑎𝑥 ) 𝑔0 – fgf-1 reference concentration 𝑔0 = 10 −10 𝑔 𝑚𝑚3 𝑝40 – chondrocyte proliferation constant 𝑝40 = 2𝑚2𝑝4 ∗ = 4 × 10−9 𝑔 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 0.012 𝑝4 ∗ – maximum chondrocyte proliferation rate 𝑝4 ∗ = 2 × 10−4 1 ℎ𝑜𝑢𝑟 𝑝400 – chondrocyte proliferation rate (from fgf1) 𝑝400 = 2 × 10 −4 1 ℎ𝑜𝑢𝑟 0.012 𝐶с𝑚𝑎𝑥0 – maximum chondrocyte density 𝐶𝐶𝑚𝑎𝑥0 = 0 ÷ 10 6 𝑁𝑐 𝑚𝑚3 0.4 𝐻(𝑛 − 𝑛1) – heaviside function 𝐻(𝑛 − 𝑛1) = { 0, 𝑛 ≤ 𝑛1 1, 𝑛 > 𝑛1 𝑛1 – critical nutrient concentration 𝑛1 = 9.5 × 10 −12 𝑁𝑚 𝑚𝑚3 0.1 𝑛0 – threshold nutrient concentration 𝑛0 = 2.3 × 10 −11 𝑁𝑚 𝑚𝑚3 0.24 – 0.81 𝐻(𝐶𝑆 − 𝐶𝑆0) – heaviside function 𝐻(𝐶𝑆 − 𝐶𝑆0) = { 0, 𝐶𝑆 ≤ 𝐶𝑆0 1, 𝐶𝑆 > 𝐶𝑆0 𝐶𝑆0 – msc threshold density 𝐶𝑆0 = (𝐶𝑆0𝑚𝑎𝑥 − 𝐶𝑆0𝑚𝑖𝑛)𝑒 −𝛼𝑏 + 𝐶𝑆0𝑚𝑖𝑛 𝑁𝑐 𝑚𝑚3 𝐶𝑆0𝑚𝑎𝑥 – maximum threshold msc density 𝐶𝑡𝑜𝑡𝑎𝑙,𝑚𝑎𝑥0 2 𝑁𝑐 𝑚𝑚3 0.35 𝐶𝑆0𝑚𝑖𝑛 – minimum threshold msc density 0.9𝐶𝑆0𝑚𝑎𝑥 𝑁𝑐 𝑚𝑚3 0.315 𝐻(𝑛1 − 𝑛) – heaviside function 𝐻(𝑛1 − 𝑛) = { 0, 𝑛1 ≤ 𝑛 1, 𝑛1 > 𝑛 𝑝5 – chondrocyte death rate 𝑝5 = 3.75 × 10 −3 1 ℎ𝑜𝑢𝑟 1 𝑝6 – nutrient uptake constant by msc 𝑝6 = 1.5 × 10 −14 𝑁𝑚 𝑁𝑐∙ℎ𝑜𝑢𝑟 10000 𝑝7 – nutrient uptake constant by chondrocytes 𝑝6 = 1.5 × 10 −14 𝑁𝑚 𝑁𝑐∙ℎ𝑜𝑢𝑟 10000 444 v. l. popov, a. poliakov, v. pakhaliuk 𝑝8 – ecm synthesis rate 𝑝8 = 𝑝80 − 𝑝81𝑚 𝑔 𝑚𝑚3 ∙ 1 𝑁𝑐/𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 𝑝80 – ecm production constant 𝑝80 = 3.75 × 10 −13 𝑔 𝑚𝑚3 ∙ 1 𝑁𝑐 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 0 1 𝑝800 – fgf-1 ecm deposition rate 𝑝800 = 0 ÷ 1 0 1 𝑝81 – ecm degradation constant 𝑝81 = 3.75 × 10 −9 1 𝑁𝑐 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 1 𝑝9 – fgf-1 production constant 𝑝9 = 10 −17 𝑔 𝑚𝑚3 ∙ 1 𝑁𝑐 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 26.67 𝑝11 – fgf-1 degradation rate 𝑝11 = 5.8 × 10 −2 1 ℎ𝑜𝑢𝑟 15.4 𝑝12 – bmp-2 production constant 𝑝12 = 10 −17 𝑔 𝑚𝑚3 ∙ 1 𝑁𝑐 𝑚𝑚3 ∙ 1 ℎ𝑜𝑢𝑟 26.67 𝑝13 – bmp-2 degradation rate 𝑝13 = 5.8 × 10 −2 1 ℎ𝑜𝑢𝑟 15.4 𝐶𝑡𝑜𝑡𝑎𝑙,𝑚𝑎𝑥0 – maximum total cell density 𝐶𝑡𝑜𝑡𝑎𝑙,𝑚𝑎𝑥0 = (1 − 𝑚 𝑚𝑚𝑎𝑥 ) −1 (𝐶𝑆𝑚𝑎𝑥 + 𝐶𝐶𝑚𝑎𝑥) = 10 6 𝑁𝑐 𝑚𝑚3 𝛼 – threshold stem cell density reduction factor 𝛼 = 1010 1 𝑔/𝑚𝑚3 100 𝛾 – fgf-1 flux coefficient 𝛾 = 10−2 𝑚𝑚 ℎ𝑜𝑢𝑟 0.01 𝜒 – bmp-2 flux coefficient 𝜒 = 10−2 𝑚𝑚 ℎ𝑜𝑢𝑟 1 𝑏0 – bmp-2 reference concentration 𝑏0 = 10 −10 𝑔 𝑚𝑚3 𝐶𝑆 (0) – initial msc density 𝐶𝑆 (0) = 2.5 × 105(based on 106 cells in 20 mm × 20 mm × 10 μm volume) 𝑁𝑐 𝑚𝑚3 0.25 𝐶𝐶 (0) – initial chondrocytes density 𝐶𝑆 (0) = 102 (10−2% of total cell density) 𝑁𝑐 𝑚𝑚3 0.0001 𝑁0 – initial nutrient concentration 𝑁0 = (2.85 ÷ 9.5)10 −11 𝑁𝑚 𝑚𝑚3 𝑚3 – initial ecm density 𝑚3 = 10 −8 (assumed 𝑚𝑚𝑎𝑥 /104) 𝑔 𝑚𝑚3 0.0001 𝑔𝑖𝑛𝑖𝑡 – initial fgf-1 concentration 𝑔𝑖𝑛𝑖𝑡 = 10 −12 𝑔 𝑚𝑚3 0.01 𝑏𝑖𝑛𝑖𝑡 – initial bmp-2 concentration 𝑔𝑖𝑛𝑖𝑡 = 10 −12 𝑔 𝑚𝑚3 0.01 12037 facta universitatis series: mechanical engineering vol. 21, no 3, special issue, 2023, pp. 339 357 https://doi.org/10.22190/fume230805029w © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for resilient green supplier selection peng wang1, baoying zhu1, yang yu1, zeeshan ali2, 3, bandar almohsen4 1school of management science and engineering, shandong university of finance economics, jinan, china 2department of mathematics and statistics, riphah international university islamabad, pakistan 3department of mathematics and statistics, international islamic university islamabad, pakistan 4mathematics department, college of science, king saud university, saudi arabia abstract. one of the main problems faced by resilient supply chain management is how to solve the problem of supplier selection, which is a typical multi-attribute decision-making (madm) problem. given the complexity of the current decision-making environment, the primary influence of this paper is to propose the theory of dombi operational laws based on complex intuitionistic fuzzy (cif) information. moreover, we examined the theory of cif dombi prioritized averaging (cifdpa) and cif weighted dombi prioritized averaging (cifwdpa), where these operators are the modified version of the prioritized aggregation operators and dombi aggregation operators for fuzzy, intuitionistic fuzzy, complex fuzzy and complex intuitionistic fuzzy information. some reliable properties for the above operators are also established. furthermore, to state the art of the proposed operators, an application example in the presence of the invented operators is evaluated for managing resilient green supplier selection problems. finally, through comparative analysis with mainstream technologies, we provide some mechanism explanations for the proposed method to show the supremacy and worth of the invented theory. key words: complex intuitionistic fuzzy sets, prioritized aggregation operators, multi-attribute decision-making problem, dombi t-norm and t-conorm, resilient green supplier selection received: august 05, 2023 / accepted september 16, 2023 corresponding author: peng wang school of management science and engineering, shandong university of finance economics, jinan 250014, china e-mail: pengwang@sdufe.edu.cn 340 p. wang, b. zhu, y. yu, z. ali, b. almohsen 1. introduction supply chain activity refers to planning, controlling material, data, logistics management, and internal and external financial flow among institutions to meet the investor’s needs. valuable and dominant business strategies or policies are needed to cope with internal and external problems and challenges in supply chains and their valuable and essential stages. in the last few years, especially during covid-19, the supply chain has faced many pressures for sustainable business evaluation, including global sourcing, environment, short time to market, and demand ambiguity or uncertainties. the lifetime or long-term theme or goal to survive during the new covid-19 outbreak is to thrive in the “new normal”. moreover, the virus of covid-19 is not limited to control in the supply chain. therefore, coping with all stakeholders in the supply chain is very awkward and challenging. to manage the above problems, the companies must select the right supplier to help reduce costs and remain competitive by meeting stakeholders’ expectations effectively and accurately. various well-known supplier selection techniques have been employed, including the markov chain, bayesian network, and deterministic and stochastic optimization. however, the past existing information rarely evaluates sustainable supplier considerations from a resilience and sustainability point of view. therefore, this information aims to study the best supply chain selection in the presence of some suitable supplier chain management, which is a typical multi-attribute decision-making (madm) problem. decision-making procedures, clustering analysis, and many other valuable techniques are used for evaluating the finest optimal under particular criteria. the madm technique is one of the most valuable and reliable techniques and subparts of the decision-making procedure, which is also used for selecting the optimal alternative from the family of preferences based on some classical set theory. moreover, in the case of classical set theory, we have only two options, such as zero or one; in many real-life cases, we need more options accepted as zero and one. therefore, to accommodating the above problem, the fuzzy set (fs) was proposed by zadeh [1], where the range of fs is more extended than the crisp set because the function defined in fs is defined from universal set to unit interval instead of {0, 1}. many generalizations of the fs and some applications have been proposed by different scholars, such as fuzzy superior mandelbrot set [2], fuzzy n-soft information [3], multi-fuzzy n-soft set theory [4]. the main idea of fs is very famous and reliable because they have received a lot of attention from different scholars. furthermore, the fs has only talked about the grade of membership but has not considered the theory of falsity information, where the grade of non-membership also plays a very critical and valuable role in decision-making analysis. in many situations, we not only face positive information about anyone in real life because many people have given negative information about their enemies. therefore, the theory of intuitionistic fuzzy set (ifs) was invented by atanassov [5]. ifs theory handles two types of information, truth and falsity information, with a valuable and effective option, such as the sum of the duplet, which will be covered in the unit interval. the traditional theory of fs is a special case of the ifs theory because if we remove the falsity information from ifs, then the theory of ifs will be converted to fs theory, where the range of both grades is the same, such as unit interval. many applications and different generalizations of the ifs have been proposed by scholars, such as schweizer-sklar power aggregation operators [6], maclaurin symmetric aggregation operators [7], einstein interactive aggregation operators complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 341 [8], mairca technique [9], twin support vector machine [10], analysis and application of evolutionary procedure [11]. the term “periodic” was critical and essential in complex set theory. in various cases, we failed to cope with the problem without periodic function because of the two-dimensional information in every field of life. in the presence of the fs theory, we can only cope with one-dimensional information at a time. therefore, the concept of complex fuzzy set (cfs) theory was invented by ramot et al. [12] by including the periodic term in the structure of truth grade and given a new shape in the form of the complex number, where the range of the real and unreal parts are unit interval. after the construction of cfs, various applications and generalizations of cfs have been proposed by different scholars, such as distance and entropy measures in the presence of the cfss [13], complex fuzzy n-soft set theory [14], complex multi-fuzzy soft sets [15], and another overview of complex fuzzy logic and their application in decision-making problems [16]. furthermore, the cfs has only talked about the grade of membership but has not considered the theory of falsity information, where the grade of non-membership also plays a very critical and valuable role in decision-making analysis. in many situations, we not only face positive information about anyone in real life because many people have given negative information about their enemies. therefore, the complex intuitionistic fuzzy set (cifs) was invented by alkouri and salleh [17], which covers two types of information, truth and falsity information, with a valuable and effective option such as the sum of the real part (same rule for the imaginary part) of the duplet, which will be covered in the unit interval. the traditional theory of cfs is a special case of the cifs theory because if we remove the falsity of information from cifs, then the theory of cifs will be converted to cfs theory. numerous applications and generalizations of the cifs have been proposed by different scholars, such as prioritized aggregation operators [18], simple aggregation operators [19], novel aggregation operators [20], generalized geometric aggregation operators [21], and robust averaging\geometric aggregation operators [22]. however, most of the existing aggregation operators are based on cifs’s algebraic operation rules, which lack some flexibility in modeling logic operations, such as dealing with information aggregation and decision-making problems in complex situations. the main idea of dombi t-norm and t-conorm was invented by dombi [23] in 1982. dombi t-norm and t-conorm have been used in constructing various aggregation operators. because of parameter , the theory of hamacher (algebraic) t-norm and t-conorm are the special cases of the dombi t-norm and t-conorm. furthermore, evaluating or aggregating information collection into a singleton set is a very challenging task for scholars. hence, the prioritized aggregation operators were established by yager [24] in 2008, which is the modified version of the averaging and geometric aggregation operators. the prioritized aggregation operator is used for aggregating information collection into a singleton set by considering the priority relationship between the aggregated variables. thus, yu and xu [25] presented a novel theory of prioritized aggregation operators for ifss. seikh and mandal [26] exposed the dombi aggregation operators for ifs in the presence of the dombi t-norm and t-conorm. to our knowledge, no scholars have derived the theories of dombi aggregation operators, prioritized aggregation operators, and dombi aggregation operators for cifs theory. however, such operators can effectively improve the flexibility of information integration and the adaptability of decision scenarios. given the unique advantages of cifs in representing decision information and the flexibility brought by the dombi t-norm and t-conorm for aggregation operations, how to propose a set of flexible operation rules around advanced cif information and then 342 p. wang, b. zhu, y. yu, z. ali, b. almohsen develop a class of powerful information aggregation operators to solve decision-making problems in complex reality is a topic worth studying. thus, out of the motives mentioned above, this paper mainly focuses on carrying out the following innovative work: (1) it proposes the theory of dombi operational laws based on cif information and improves further the flexibility of cif information aggregation modeling. (2) it presents the cifdpa and cifwdpa operators, which aggregate information flexibly and consider the priority relationship between integrated parameters. meanwhile, some reliable properties for the above mentioned operators are also established. (3) it constructs an madm technique based on the proposed operators and applies it to the resilient green supplier selection problem. (4) it implements a comparative analysis with various prevailing operators and discusses the operating mechanism and advantages of the proposed method. 2. preliminaries in this section, we collected the basic information about dtn, dtcn, and cifss and their valuable information, such as sv and av. definition 1: [23] for evaluating the theory of dtn and dtcn, we have used any two real numbers ux and uy ( , [0,1]u ux y ), such that 1 1 , , 1 1 1 ( ) 0dtn u u u u u u d x y x y x y =     − −  + +            (1) 1 1 ( ) 0 1 , 1 , 1 1 dtcn u u u u u u d x y x y x y = −       + +     − −       (2) definition 2: [17] assume that u be the family of fixed set with generic elements, then the model of cifs is characterized by: {( , ) : }( ) ( )cif m u n u u ux x x  =  (3) furthermore, we explained the complex form of truthful information and the complex form of false information, such as ( ,( ) ( ) ( ))rp ip m u m u m ux x x =   and ( )n ux = ( , ))( ) (rp ip n u n ux x  with a specific condition: 0 1( ) ( )rp rp m u n ux x +  for real parts and ( ) ( )0 ip ip m u n ux x + 1 for imaginary parts. moreover, the neutral information is stated by: ( )m ux = )( ),( ( )rp ip m u m ux x  = (1 ( ),1 ( ))( ) ( ) ( ) ( )rp rp ip ip m u n u m u n ux x x x−  + −  + and the final and simplest form of cif value (cifv) is stated by: (( , ),( , )), j j j j cif rp ip rp ip j m m n n =     1,2,3, ,j z=  . for example, there exists a cifv whose real part is evaluated as an intuitionistic fuzzy form of (0.3, 0.4) and whose imaginary part is evaluated as an intuitionistic fuzzy form of (0.5, 0.1), then its presentation form ((0.3, 0.5), (0.4, 0.1)). complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 343 definition 3: [19] assume that u be the family of fixed set with generic elements, then in the presence of any two cifvs (( , ),( , )), 1,2 j j j j cif rp ip rp ip j m m n n j =     = , we have: 1 2 1 2 1 2 1 2 1 2 1 21 2 (( , ), ( , ))cif cif rp rp rp rp ip ip ip ip rp rp ip ip m m m m m m m m n n n n  =  + −   + −      (4) 1 2 1 2 1 2 1 2 1 2 1 21 2 (( , ), ( , ))cif cif rp rp ip ip rp rp rp rp ip ip ip ip m m m m n n n n n n n n  =     +  −    +  −   (5) ( )(1 (1 ) ,1 0(1 ) ),(( ) ,( ) ) , j j j j cif rp ip rp ip j m m n n = − − −  −  (6) 0( ) ((( ) ,( ) ),(1 (1 ) ,1 (1 ) )), j j j j cif rp ip rp ip j m m n n =   − − − − (7) definition 4: [19] assume that u be the family of fixed set with generic elements, then in the presence of any cifvs (( , ),( , )), 1,2 j j j j cif rp ip rp ip j m m n n j =     = , we have: 1 (( ) ( )) [ 1,1] 2 ( ) j j j j cif rp ip rp ip j m m n nsv  +  = − +  − (8) 1 (( ) ( )) [0,1] 2 ( ) j j j j cif rp ip rp ip j m m n nav  = + +    + (9) to evaluate any two cifvs, we described some characteristics in the presence of the information in eq. (8) and eq. (9); we have if 1 2( ) ( )cif cifsv sv   , thus 1 2 cif cif ; if 1 2( )( )cif cifs svv    , thus 1 2 cif cif ; if 1 2( )( )cif cifs svv  =  , then if 1( )cifav   2( )cifav  , thus 1 2 cif cif ; if 1 2( ) ( )cif cifav av   , thus 1 2 cif cif . 3. dombi prioritized aggregation operators for cif information before building the dombi prioritized aggregation operators, we must present the basic dombi operational laws in the presence of the cifvs. then, by using the dombi operational laws for cif values, we aim to derive the theory of cifdpa and cifwdpa operators. moreover, we must examine the proposed operators’ basic properties, such as idempotency, monotonicity, and boundedness. definition 5: assume that u be the family of fixed set with generic elements, in the presence of any two cifvs (( , ),( , )), 1,2 j j j j cif rp ip rp ip j m m n n j =     = , we have: 1 1 1 2 1 2 1 2 1 1 1 2 1 2 1 1 1 ,1 , 1 ( ) 1 ( ) 1 1 , 1 ( ) 1 ( ) cif cif a a b b c c d d    − −   + + + +     =        + + + +   (10) 344 p. wang, b. zhu, y. yu, z. ali, b. almohsen 1 1 1 2 1 2 1 2 1 1 1 2 1 2 1 1 , , 1 ( ) 1 ( ) 1 1 1 ,1 1 ( ) 1 ( ) cif cif a a b b c c d d       + + + +     =     − −  + + + +   (11) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , , 1 ( ) 1 ( ) 1 ( ) 1 ( ) cif a b c d            = − −     + + + +     (12) ( )1 1 1 1 1 1 1 1 1 1 1 1 1 , , 1 ,1 1 ( ) 1 ( ) 1 ( ) 1 ( ) cif a b c d            = − −     + + + +     (13) where  and are constants greater than 0; let 1 j j rp m j rp m a    =  −   , 1 j j m ip ip j m b    =  −   , 1 j j rp n j rp n c  −   =     , and 1 j j ip n j ip n d  −   =     , 1,2=j , and these symbols also apply to the following text. definition 6: assume that u be the family of fixed set with generic elements, then in the presence of any two cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , then: 1 2 1 2 1 2 1 1 1 1 1 η η η ( , , , ) η η η η η z cif cif cif cif cif cif z z z z zj j j j j j j z cif j j z j j ifdpa = = = = =     =          =            (14) where, 1η 1= , and ( ) 1 1 η − = =  j j cif k k sv . theorem 1: assume that u is a family of fixed set with generic elements, then in the presence of any two cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , and in the presence of the information in eq. (14), we derive that the below theory, such as: 1 2 1 1 1 1 1 1 1 1 ( , , , ) 1 1 1 1 1 ,1 , , 1 ( ) 1 ( ) 1 ( ) 1 ( ) cif cif cif z z z z z j j j jj j j j cifdpa a b c d = = = =     =          − −     + + + +          (15) complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 345 where 1 η η j j j z j j a a = =  , 1 η η j j j z j j b b = =  , 1 η η j j j z j j c c = =  , and 1 η η j j j z j j d d = =  , and these symbols also apply to the following text. proof: considering the primary procedure of the mathematical induction, we focus on evaluating the data in eq. (15). considering 2=z , we have 1 1 1 1 1 1 2 1 1 1 1 1 η 1 1 1 1 1 ,1 , , 1 1 1 1η cif j j a b c d =       = − −     + + + +     2 2 1 1 1 1 2 2 2 2 2 1 η 1 1 1 1 1 ,1 , , 1 1 1 1η cif j j a b c d =       = − −     + + + +     thus, 1 2 1 2 1 2 2 2 1 1 η η ( , ) η η cif cif cif cif j j j j cifdpa = =   =      1 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , , 1 1 1 1a b c d      = − −     + + + +     1 1 1 1 2 2 2 2 1 1 1 1 1 ,1 , , 1 1 1 1a b c d       − −     + + + +     2 2 2 21 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , , 1 ( ) 1 ( ) 1 ( ) 1 ( )j j j jj j j j a b c d = = = =          = − −     + + + +          for 2=z , we hold our required information; furthermore, we assume that the data in eq. (15) is also held for =n k , such as ( )1 2 1 1 1 1 1 1 1 1 , , , 1 1 1 1 1 ,1 , , 1 ( ) 1 ( ) 1 ( ) 1 ( ) cif cif cif k z z z z j j j jj j j j cifdpa a b c d = = = =     =          − −     + + + +          then, we derive it for 1= +n k , such as 346 p. wang, b. zhu, y. yu, z. ali, b. almohsen 1 2 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 η η η ( , , , ) η η η η η η η η η k cif cif cif cif cif cif k k k k kj j j j j j k j k cif k cif cif k j j k k k kj j j j j j cifdpa + = = = + + + = + + + = = =     =   =          =                1 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , , 1 ( ) 1 ( ) 1 ( ) 1 ( ) k k k k j j j jj j j j a b c d = = = =          = − −     + + + +          1 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , , 1 1 1 1k k k ka b c d+ + + +       − −        + + + +     1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , , 1 ( ) 1 ( ) 1 ( ) 1 ( ) k k k k j j j jj j j j a b c d + + + + = = = =          = − −     + + + +          hence, our final result proved the possible values of z. moreover, we justify the significant properties of the evaluated theory in eq. (15), such as idempotency, monotonicity, and boundedness. property 1: (idempotency) assume that u be the family of fixed set with generic elements, then in the presence of any z cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , if (( , ), ( , ))cif cif rp ip rp ip j m m n n = =     , then 1 2( , , , )cif cif cif cif zcifdpa     = (16) proof: when (( , ), ( , ))cif cif rp ip rp ip j m m n n = =     , then 1 2( , , , )cif cif cif zcifdpa     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ,1 , η η 1 1 η η 1 1 , η η 1 1 η η j j z z m mj jz zj j j j j j z z n nj jz zj j j j a b c d = = = = = = = =               − −             + +               =                      + +                                      complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 347 1 1 1 1 1 1 1 1 1 1 η 1 ,1 , , , 1 1 1 1 1 η jz z jjm m n n j a b c d = =      = − − =        + + + +       1 1 1 1 1 ,1 , , 1 1 1 1 1 1               = − −     − + − +  + −   + −          − −       rp rp ip ip rp rp ip ip m m m m n n n n rp ip rp ip m m n n ((1 (1 ),1 (1 )),( , ))rp ip rp ip m m n n= − − − −   (( , ),( , )) .rp ip rp ip cif m m n n=     = property: (monotonicity) assume that u be the family of fixed set with generic elements, then in the presence of any z cifvs (( , ),( , )) j j j j cif rp ip rp ip j m m n n =     , j = 1, 2, … z, if * cif cif j j , then * * * 1 2 1 2( , , , ) ( , , , )cif cif cif cif cif cif z zcifdpa cifdpa         (17) proof: when * cif cif j j , thus * *,     j j j j rp rp ip ip m m m m and ** **,      j j j j rp rp ip ip n n n n , then * * *1 1       − − j j j j j j rp rp m mrp rp m m rp rp m m * *1 1              − −     j j j j rp rp m m rp rp m m * * 1 1 1 1 η η 1 1η η= = = =              − −         j j j j rp rpj jz z m m rp rp z zj jj jm m j j 1 1 * * 1 1 1 1 η η 1 1 1 1η η= = = =                 +  +       − −               j j j j rp rpj jz z m m rp rp z zj jj jm m j j 1 1 * *1 1 1 1 1 1 η η 1 1 1 1η η = = = =                  + +       − −               j j j j rp rpj j m mz z rp rpj jz zj jm m j j 1 1 * *1 1 1 1 1 1 1 1 η η 1 1 1 1η η = = = =  −  −                + +       − −               j j j j rp rpj j m mz z rp rpj jz zj jm m j j 348 p. wang, b. zhu, y. yu, z. ali, b. almohsen where, by using the same procedure, we get 1 1 * *1 1 1 1 1 1 1 1 η η 1 1 1 1η η = = = =    −  −                + +       − −               j j j j j j ip ip m m ip ipj j m mz z ip i pj jz zj jm m j j furthermore, we can derive the information of falsity data, such as: * *1 1−      − j j j j rp rp rp rp n n n n * * * * 1 1 1 1 1 1 1 1η η η η= = = =        − − − −                                        j j j j j j j j rp rp rp rpj jz z n n n n rp rp rp rp z zj jj jn n n n j j 1 1 * * 1 1 1 1 11η η η η= = = =        − −                                j j j rprpj jz z nj rp rp z zj jj jn n j j 1 1 * * 1 1 1 1 1 1η η 1 1 η η= = = =        −  −         +  +                       j j j j rp rpj jz z n n rp rp z zj jj jn n j j 1 1 * *1 1 1 1 1 1 1 1η η 1 1 η η = = = =          −  −        + +                       j j j j rp rpj j n nz z rp rpj jz zj jn n j j where, by using the same procedure, we get * 1 1 * *1 1 1 1 1 1 1 1η η 1 1 η η = = = =             −  −        + +                       j j j j j j ip ip n n ip ipj j n nz z ip ipj jz zj jn n j j then, in the presence of the data in eq. (8) and eq. (9), we have * * * 1 2 1 2( , , , ) ( , , , )cif cif cif cif cif cif z zcifdpa cifdpa         . property 3: (boundedness) assume that u be the family of fixed set with generic elements, then in the presence of any z cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , if ((min ,min ), (max ,max )) j j j j rp ip rp ip j m m n n j j j j −     = and ((max ,max ), j j rp ip j m m j j +   = (min ,min )) j j rp ip n n j j   , then 1 2( , , , )cif cif cif j z jcifdpa− +       (18) complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 349 proof: select the theory in proposition 1 and proposition 2; we have 1 2 1 2( , , , ) ( , , , )cif cif cif z z jcifdpa cifdpa + + + +         = 1 2 1 2( , , , ) ( , , , )cif cif cif z z jcifdpa cifdpa − − − −         = thus, we have 1 2( , , , ) .cif cif cif j z jcifdpa− +       definition 7: assume that u be the family of fixed set with generic elements, then in the presence of any z cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , then 1 2 1 2 1 2 1 2 1 1 1 1 1 η η ( , , , ) η η ηη η η cif cif cif cif cif z z zj j j jj j jz jcif z cifz z j j z zj j j jj j cifwdpa = = = = =     =           =                   (19) here, 1η 1= and 1 1 η ( ) j j cif k k sv − = =   with weight vector [0,1]j  such that 1 1 = = z j j  . theorem 2: assume that u be the family of fixed set with generic elements, then in the presence of any two cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , then in the presence of the information in eq. (19), we derive that the below theory, such as: 1 2 1 1 1 1 1 1 1 1 1 1 1 ( , , , ) 1 1 1 ,1 , η η 1 ( ) 1 ( ) η η 1 1 , η η 1 ( ) 1 η j j j cif cif cif z j j z zj jrp ip m mj jz zj j j jj j j j z zj jrp nj jz zj jj j cifwdpa = = = = = = = =     =          − −           + +                   +  +                     1 ( ) η j ip n j j                                                         (20) proof: similar to theorem 1, omitted. 350 p. wang, b. zhu, y. yu, z. ali, b. almohsen property 4: (idempotency) assume that u be the family of fixed set with generic elements, then in the presence of any z cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , if (( , ), ( , ))cif cif rp ip rp ip j m m n n = =     , then 1 2( , , , )cif cif cif cif zcifwdpa     = (21) property 5: (monotonicity) assume that u be the family of fixed set with generic elements, then in the presence of any two cifvs (( , ),( , )), j j j j cif rp ip rp ip j m m n n =     1,2, ,j z=  , if * cif cif j j , then * * * 1 2 1 2( , , , ) ( , , , )cif cif cif cif cif cif z zcifwdpa cifwdpa         (22) property 6: (boundedness) assume that u be the family of fixed set with generic elements, then in the presence of any z cifvs (( , ),( , )), 1,2, , j j j j cif rp ip rp ip j m m n n j z =     =  , if ((min ,min ), (max ,max )) j j j j rp ip rp ip j m m n n j j j j −     = and m ),(( ax ,max j j rp ip j m m j j +   = (min ,min )) j j rp ip n n j j   , then 1 2( , , , )cif cif cif j z jcifwdpa− +       (23) 4. an approach for resilient green supplier selection -based proposed theory in this section, we will construct a madm method based on the proposed cifwdpa operator to solve the practical problem of resilient green supplier selection. 4.1 method description to accommodate the above problem, we select the collection of finite values of alternatives, that is 1 2, , ,   al alal g . for every alternative, we consider the finite values of attributes such as 1 2, , ,   at at at z with weight vector [0,1]j  such that 1 1 = = z jj  . in the presence of the above information, such as g-alternatives and z-attributes, we concentrate on computing the decision matrix by using the theory of information of cif values; we explained the complex form of truthful information and the complex form of falsity information such as ( ,( ) ( ) ( ))rp ip m u m u m ux x x =   and ( ,( ) ( ) ( ))rp ip n u n u n ux x x =   with a specific condition: 0 1( ) ( )rp rp m u n ux x +  for the real part and 0 1( ) ( )ip ip m u n ux x +  for imaginary parts. moreover, the neutral information is stated by: ( , ) (1 ( ),1 ( ))( ) ( ) ( ) ( ) ( ) ( ) ( )rp ip rp rp ip ip m u m u m u m u n u m u n ux x x x x x x =   = −  + −  + and the final and simplest form of cifv is stated: (( , ),( , )) hjhj hj hj cif rp ip rp ip hj m m n n =     , 1,2,3, , ;h g=  1,2,3, ,j z=  . therefore, to address some real-life problems, we compute the procedure complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 351 of decision-making for evaluating the finest optimal form of the collection of preferences, such as: step 1: arrange the cifvs as a decision matrix. step 2: normalize the decision matrix if possible (in the case of cost of data) for benfit attribu(( , ), ( , )) (( , ), ( , te for cost attrib)) ute hj hj hj hj hj hj hj hj rp ip rp ip m m n ncif hj rp ip rp ip n n m m       =      step 3: aggregate the data in the decision matrix with the help of the cifwdpa operator as eq. (20), and obtain the final aggregated value )1,2, ,( 3 , = cif h h g for each alternative. step 4: determine the svs 2( )( ) 1, ,3, ,cif hsv h g =  of the aggregated data with the help of data in eq. (8). if necessary, the av is also required. step 5: evaluate or derive the ranking values in the presence of the values of the score data and examine the finest optimal from the collection of preferences. furthermore, we simplify the above procedure of decision-making technique with the help of numerical examples in the environment of the real-life problem and try to evaluate it in the presence of the presented operators. 4.2 application example smart green resilient, where the short form is sgr, refers to an emerging planning technique or approach that themes to enable pragmatic concerns to be evaluated continuously with main environmental problems and to compute long-life resilience in urban planning strategies, sustainability, masterplans, and frameworks for an extensive range of contexts. with the global economic recovery in the post-pandemic era, the selection of resilient green suppliers has become particularly important in the enterprise supply chain management, as it can significantly improve the economic benefits of enterprises. this problem is taken from ref. [27], computed based on sustainable supply chain management. this problem aims to study supplier selection by integrating lean, green agile, and sustainable aspects of the supply chains and resilience. therefore, in this problem, we consider the finest supplier chain for future consideration. for this, we consider five majors from the supplier chain which are used as the alternatives such as: 1 al : sustainable supply chain: the green supply chain is becoming an essential and major part of the strategic thrust in business companies. 2al : lean supply chain: lean supply chain depended on the lean management principles at toyota motor corporation, japan, concentrating on waste lessening and attaining higher earnings. 3al : resilient supply chain: the capability of a supply chain to cope with surprising interruption is described as a resilient supply chain. 4al : green supply chain: it includes efforts to improve environmental performance, cut costs and waste in internal supply chains, external supply chains, and reverse logistics to eliminate negative environmental effects 5al : agile supply chain: in this procedure, flexibility and quick reconfigurability are the main concentrates. 352 p. wang, b. zhu, y. yu, z. ali, b. almohsen to select the best alternative, we give the following features as attribute information: 1 at : response rate, 2at : capability for unexpected disruptions, 3at : capability for reverse logistics, 4at : sustainable product design, and 5at : long-term relationship. therefore, to address the above real-life problem, we use the procedure of decision-making for evaluating the finest optimal form of the collection of preferences, such as: step 1: arrange the cif values as a decision matrix in table 1. step 2: normalize the decision matrix. it’s worth noting that the data in table 1 is not required to be normalized. table 1 mathematical representation of cif decision matrix 1 at 2 at 3 at 4 at 5 at 1 al (0.7,0.7), (0.1,0.2)       (0.71,0.71), (0.11,0.21)       (0.72,0.72), (0.12,0.22)       (0.73,0.73), (0.13,0.23)       (0.74,0.74), (0.14,0.24)       2 al (0.5,0.4), (0.2,0.3)       (0.51,0.41), (0.21,0.31)       (0.52,0.42), (0.22,0.32)       (0.53,0.43), (0.23,0.33)       (0.54,0.44), (0.24,0.34)       3 al (0.5,0.5), (0.2,0.4)       (0.51,0.51), (0.21,0.41)       (0.52,0.52), (0.22,0.42)       (0.53,0.53), (0.23,0.43)       (0.54,0.54), (0.24,0.44)       4 al (0.4,0.6), (0.3,0.1)       (0.41,0.61), (0.31,0.11)       (0.42,0.62), (0.32,0.12)       (0.43,0.63), (0.33,0.13)       (0.44,0.64), (0.34,0.14)       5 al (0.4,0.3), (0.2,0.1)       (0.41,0.31), (0.21,0.11)       (0.42,0.32), (0.22,0.12)       (0.43,0.33), (0.23,0.13)       (0.44,0.34), (0.24,0.14)       step 3: aggregate the data in the decision matrix with the help of the eq. (20) (where 3= ), and obtain the aggregated value 1 ((0.8161,0.8161),(0.0012,0.0431))cif = , 2 ((0.0048,0.0006),(0.3540,0.8769))cif = , 3 ((0.0048,0.0048),(0.3540,0.9840))cif = , 4 ((0.0020,0.1040),(0.6965,0.0045))cif = , 5 ((0.0006,0.0001),(0.3540,0.0139))cif = . step 4: determine the score values of the aggregate data with the help of data in eq. (8), and get 1( ) 0.7940cifsv  = , 2( ) 0.6127cifsv  = − , 3( ) 0.6642cifsv  = − , 4( )cifsv  = 0.2976− , 5( ) 0.1836cifsv  = − . step 5: evaluate or derive the ranking values in the presence of the values of the score data and examine the finest optimal from the collection of preferences. thus, we get the rank result of 1 5 4 2 3    al al al al al , and 1 al is the best decision. 5. parametric analysis and comparative analysis in this section, we compare the proposed operators with some prevailing techniques to show the worth and supremacy of the invented techniques. before making the comparison, complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 353 we check the sensitivity of the parameter by using its different values in the presence of the data in table 1. the experimental results are shown in table 2 and fig. 1. table 2 stability of the parameters for the cifwdpa operator parameters ranking results 1= 1 4 5 2 3 al al al al al       =  2= 1 4 5 2 3 al al al al al        3= 1 5 4 2 3 al al al al al        4= 1 5 4 2 3 al al al al al        5= 1 4 5 2 3 al al al al al        6= 1 4 5 2 3 al al al al al        7= 1 4 5 2 3 al al al al al        8= 1 4 5 2 3 al al al al al        9= 1 4 2 5 3 al al al al al        10= 1 4 2 5 3 al al al al al        fig. 1 score values of alternatives under different values 354 p. wang, b. zhu, y. yu, z. ali, b. almohsen according to table 2, although the ranking results for each parameter are not precisely the same, the optimal results are the same, they are 1 al . meanwhile, as shown in fig.1, as parameter increases, the score value of each alternative increases. from the above parameter analysis, it can be seen that the parameter can reflect the preferences of different decision-makers (dms) so that dms can choose flexibly according to different decision environments and risk preferences. for example, dms who are risk-averse can choose a smaller value. of course, this is also based on the premise that the cifwdpa operator-based madm method can maintain a specific stability of the result. table 3 result representation of the comparative analysis methods ranking results yu and xu’s method [25] −−−−− −−−−−bounded seikh and mandal’s method [26] −−−−− −−−−−bounded senapati’s method [28] −−−−− −−−−−bounded garg and rani’s method [19] (cifwa operator) 1 4 5 2 3 al al al al al     =  =  garg and rani’s method [20] (cifwa operator) 1 4 5 2 3 al al al al al     =  =  garg and rani’s method [21] (cifawg operator) 1 4 5 2 3 al al al al al     =  =  garg and rani’s method [22] (cifwg operator) 1 4 5 2 3 al al al al al     =  =  our method ( 3= ) 1 5 4 2 3 al al al al al        our method ( 5= ) 1 4 5 2 3 al al al al al        our method ( 9= ) 1 4 2 5 3 al al al al al        furthermore, to demonstrate the proposed method’s superiority, we evaluate the comparison between the proposed work and some existing madm methods. for this, we consider the following existing operators: yu and xu [25] presented a novel theory of prioritized aggregation operators for ifss. further, seikh and mandal [26] exposed the major information of dombi aggregation operators for ifs in the presence of the dombi t-norm and t-conorm. senapati et al. [28] derived the aczel-alsina aggregation operators in the presence of the ifs and their application in decision-making. furthermore, aggregation operators for cif values were derived by garg and rani [19], novel aggregation operators for cif information were presented by garg and rani [20], geometric aggregation operators for cifs were exposed by garg and rani [21], and robust averaging and geometric aggregation operators for cifs was presented by garg and rani [22]. based on the data in table 1, the comparative analysis results are listed in table 3. as shown in table 3, we obtained the same ranking results, where the optimal alternatives are all 1 al according to the theory of the cifwdpa operator, which is consistent with the results obtained by methods in [19-22]. this first indicates that the method proposed in this paper is effective. the selection results are not exactly the same, mainly because the characteristics of various methods are different. the advantages of the proposed method and the limitations of the existing work are stated below: the prioritized aggregation operators in [25], the dombi aggregation operators in [26], and the aczel-alsina aggregation operators in [28] are proposed for ifs, and it is clear that complex intuitionistic fuzzy dombi prioritized aggregation operators and their application for... 355 the theory of ifs has the special case of the cifs; therefore, these proposed operators in [25, 26, 28] have failed to evaluate the information in table 1. however, the method proposed in this paper is based on the cifs with stronger information representation ability, which can deal with more complex decision-making and evaluation problems and more conveniently simulate human decision-making cognition. compared to the theory of garg and rani [19-22], the results obtained by their method are significantly different from those obtained by the method proposed in this paper. however, the optimal choice is the same. and the methods in [19-22] cannot effectively distinguish the priority of alternatives 2 al , 3 al , and 5 al . this is because the operators proposed in this paper adopt the dombi operation rules and consider the priority relationship between the aggregated variables. therefore, the constructed aggregation operators are more flexible and have more comprehensive functions. in summary, the characteristics of each method can be summarized in table 4. table 4 comparison of characteristics of different methods characteristics methods stronger information representation ability considering dm’s preference attitude considering attributes’ priority relationship yu and xu’s method [25] no no yes seikh and mandal’s method [26] no yes no senapati’s method [28] no yes no garg and rani’s method [19] yes no no garg and rani’s method [20] yes no no garg and rani’s method [21] yes yes no garg and rani’s method [22] yes no no our method yes yes yes 6. conclusions the prioritized aggregation and dombi norms play a valuable and dominant role in the environment of fuzzy set theory. inspired by the structure of the cifs as well as the idea of dombi prioritized aggregation operators, the significant influence of this paper is listed below: (1) this paper proposed the theory of dombi operational laws based on cif information. (2) this paper examined the theory of the cifdpa and cifwdpa operators, which are the modified versions of the prioritized aggregation operators and dombi aggregation 356 p. wang, b. zhu, y. yu, z. ali, b. almohsen operators for fuzzy, intuitionistic, complex, and complex intuitionistic fuzzy information. some reliable properties for the above operators are also established. (3) this paper derived a madm method based on the proposed operators and applied it to the resilient green supplier selection problem. 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igor ćulum, sonja jozić, dražen bajić, ivana dumanić faculty of electrical engineering, mechanical engineering and naval architecture, university of split, split, croatia abstract. the main goal of this paper is to determine the influence of cooling technique on surface roughness during up and down face milling of aluminum alloy en aw – 2011 t6. along with dry machining, three cooling techniques were observed: cutting fluid (cf), minimum quantity lubricant (mql) and cold compressed air (cca). thirteen experiments were conducted for each technique. following the defined plan of the experiment, cutting speed and feed per tooth were varied. an optical profilometer was used to analyze arithmetic deviation of the profile (ra) and arithmetic mean of the absolute height (sa). down milling produced up to 24 % lower ra and sa values in comparison to up milling. increasing feed per tooth greatly increased surface roughness while increasing cutting speed led to a 12 % to 14 % decrease in surface roughness. using the same cutting parameters, cca produced the lowest, while cf produced the highest ra and sa values. using the test results and regression analysis, mathematical models were generated allowing for precise ra and sa predictions. optimization of the regression models was carried out with the goal of achieving the lowest surface roughness for each milling strategy and applied cooling technique. key words: face milling, minimum quantity lubricant, cold compressed air, dry machining, surface roughness, chip morphology 1. introduction milling is one of the most used machining processes that can precisely produce complex prismatic shapes. to remain competitive in today’s manufacturing industry, it is paramount to reduce production costs. one of the most common cost saving measures involves received: may 23, 2023 / accepted july 29, 2023 corresponding author: sonja jozić faculty of electrical engineering, mechanical engineering and naval architecture, ruđera boškovića 32, 21000, split, croatia e-mail: sonja.jozic@fesb.hr 2 i. ćulum, s. jozić, d. bajić, i. dumanić increasing tool life. the easiest way to increase tool life is to reduce tool wear by introducing cooling, lubricating and flushing into the tool workpiece interface. this, most commonly, involves flooding the tool and workpiece with cutting fluids (cf). although machining using cf provides effective cooling and lubricating, it leads to increased costs associated with cutting fluid filtration, storage, pumping, maintenance and recycling. cutting fluids come in three major variants: oil, gas and water-based fluids. all three variants contain additives to increase machining performance. these additives are not biodegradable and present an environmental hazard so care must be taken when disposing of such fluids [1]. it has been proven that cooling fluids also present a health hazard to the machine tool operator. many skin and lung illnesses have been documented over the years as a result of exposure to cutting fluids. many dermatological illnesses can be traced back to an increased number of bacteria that are found in cutting fluids while inhalation of fluid vapors can lead to conditions such as: asthma, bronchitis and pneumonia [2,3,4]. these issues have led engineers to seek alternative cooling and lubricating techniques. techniques such as: minimum quantity lubrication (mql), cold compressed air cooling (cca) and even dry machining are seen as viable alternatives to cutting fluids. costs associated with cutting fluids sum up from 7 % to 17 % of the entire machining process expenses. the main appeal of alternative cooling techniques is the fact that they require a small amount or even no cutting fluid. reducing or eliminating cutting fluids from the process provides a better surface finish, cleaner chips for recycling, lower maintenance costs, a healthier workpiece environment all the while being more environmentally friendly [5,6,7]. due to low procurement costs, minimal cutting fluid use and favorable cooling and lubricating properties, mql is seen as a leap in the metal machining industry. it involves spraying a fine mist of cutting fluid over the cutting zone using compressed air or gas. the mist forms a fine oil layer on the workpiece surface that provides lubrication while the cold air both cools and blows the hot chips away from cutting zone [6,8]. dry machining has shown to produce no residue and a clean chip that can be easily sold or recycled. the process doesn’t require any additional equipment, fluids or gasses which increase production cost thus ensuring a more competitive product on the market. on another hand, dry machining does not provide lubrication which leads to increased tool wear at the tool workpiece interface. to decrease tool wear, while still avoiding the use of harmful fluids, research has been put into study of cryogenic cooling techniques. these techniques use a liquefied gas such as n2 or co2 to cool the cutting zone. khanna et al. [9] compared surface roughness of dry, cf, mql and cryogenic machined 15-5-ph stainless steel concluding that cf produces the finest surface, followed by cryogenic, mql, and dry machining. khanna et al. [10] also experimented on inconel 718. their study showed that cryogenic cooling produces a lower surface roughness than mql, cf and dry machining. shukla et al. [4] concluded that mql produced a finer surface roughness of al 6061 alloy than both cf and dry machining while an increase in feed rate had the most impact on surface roughness. sreejith et al. [11] noted that an increase in coolant flow rate during mql cooling produced a finer surface. however, machining using cf still produced a finer surface finish than mql. they also showed that an increase in cutting speed negatively influenced surface quality. combining alternative cooling techniques is also possible, as was shown by yıldırım et al. [12]. by combining mql and n2 cryogenic cooling techniques, they achieved a better surface finish than those achievable with either mql or cryogenic cooling. race et al. [13] experimented on sa516 steel concluding that dry machining and mql cooling outperform experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 3 traditional cf techniques in terms of surface roughness. milling strategy plays a crucial role in terms of surface roughness. this was proven by iqbal et al. [14] during testing of ti-6al4v under mql, cryogenic co2 and liquid concluding that the roughness of an up milled surface is higher than a down milled surface regardless of cooling technique. similarly, vakondios et al. [15] experimented on al 7075 – t6 using different milling strategies while varying cutting parameters. they concluded that milling strategies, as well as cutting parameters, greatly influence surface roughness. they also obtained mathematical models used to accurately predict surface roughness based on cutting parameters. joshua et al. [16] also developed mathematical models for mql and dry machining of al 6061 using regression modeling. by varying cutting parameters and measuring surface roughness, they concluded that mql produced a 20 % lower surface roughness than dry machining. pramanik et al. [17] implemented alternative cooling methods for drilling processes by machining al 6061. surface roughness, power consumption, burr formation, diameter and circularity errors were observed under mql, cryogenic ln2 and air cooling. these experiments lead to conclusions that cooling techniques have a negligible influence on power consumption and product accuracy. however, an increase in both power consumption and surface roughness with increased feed rate was observed. decreasing cutting speed led to higher surface roughness values. effects of nano-mos2 particles added in mql during turning of spherical graphite cast iron on machinability was investigated by sertsoz and kacal, [18]. lawal et al. [6] experimented on aisi 9310 steel using a vegetable based mql turning process. their study showed that mql outperformed traditional and alternative cooling techniques by up to 31.6 % in terms of surface finish while reducing cutting fluid related expenses. jozić et al. [19] used box-behneken’s experimental design and regression analysis to determine optimal cutting parameters during turning of 34crnimo6. their findings showed that increasing the feed rate leads to a drastic increase in surface roughness while cutting speed has a far lower influence on the final roughness of the surface. stojanković et al. [20] investigated the influence of cutting parameters on surface roughness of aluminum 6082-t6 alloy during end milling. they concluded that feed rate was the most influential cutting parameter on surface roughness. increasing the feed rate led to an increase in surface roughness while increasing the cutting speed caused a decrease in surface roughness. from this brief literature overview, it is evident that there is space for additional research focusing on testing the influence of cooling technique and cutting parameters on surface roughness and chip morphology. therefore, the goals of this paper are to investigate surface roughness and chip morphology of aluminum alloy en aw – 2011 t6. this aluminum alloy was selected due to its good machinability. along with dry machining, testing will be carried out using three cooling techniques: mql, cf and cca. test results will be used to generate regression models needed for accurate roughness predictions. proving the viability of alternative cooling techniques could greatly reduce health and environmental impacts associated with machining as well as ease recycling of metal chips. 2. experimental work the main idea of this paper is to document the influence of cooling technique on surface roughness and chip morphology on a widely used aluminum en aw – 2011 t6 4 i. ćulum, s. jozić, d. bajić, i. dumanić alloy. experimental research workflow and setup, shown in fig. 1, consisted of up and down face milling, surface scanning and roughness measurement as well as chip morphology analysis. twenty-six plates, measuring 100 mm x 30 mm x 10 mm, were face milled at varying cutting speed and feed per tooth combinations. fig. 1 experimental setup and workflow along with dry machining, three cooling techniques were observed: cf, mql and cca. rhenus tu 30 t coolant was used for cf and mql experiments. coolant properties are shown in table 1. experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 5 table 1 rhenus tu 30 t technical data [21] concentrate emulsion viscosity at 20 °c [mm²/s] content of mineral oil [%] ph – value 5 % corrosion protection (din 51360/2) approx. 160 approx. 18 9.4 4% grade 0 a vortex tube was used for cca testing providing cooling temperatures as low as -34 °c [8]. up and down face milling was performed using a two flute m4132-032-w3202-09 mill manufactured by walter ag. the end mill was equipped with sdht09t304g88 interchangeable carbide inserts. experiments were carried out on a vertical machining centre, spinner vc560. cutting parameters were selected according to tool manufacturer specifications and machine tool limitations. specified cutting parameters, shown in table 2, were input into design expert software. using response surface methodology implemented into the software, a central composite plan of thirteen cutting speed and feed per tooth combinations was generated. axial (ap) and radial (ae) depths of cut were kept at a constant 1.5 mm and 32 mm, respectively. table 2 cutting parameters minimal feed per tooth, ft,min 0.05 mm/tooth maximum feed per tooth, ft,max 0.25 mm/tooth minimum cutting speed, vc,min 500 m/min maximum cutting speed, vc,max 1000 m/min surface roughness of up and down milled specimen was measured using a profilm 3d profilometer. arithmetic deviation of the profile (ra) and arithmetic mean of the absolute height (sa) parameters were measured five times for each specimen and mean values are given in the results and discussion section of this paper. arithemetic deviation of the profile (ra) presents the absolute value of profile deviation from the profile center line within the observed length. arithmetic mean of the absolute height (sa) presents the mean of the average height difference of a measured plane. measurements were carried out in accordance with the international standard iso 8688 – 1 using an 800 μm cut off length. scanning properties were kept constant throughout the testing. a dino-lite am4113zt digital microscope was used to study chip morphology [22]. 3. results and discussion 3.1 influence of cutting parameters and cutting environment on surface roughness 3.1.1 up milled surface finish analysis table 3 shows mean surface roughness values for: dry, cca, cf and mql up milled surfaces. regression models for: dry machining, cca, cf and mql milling are given in table 4 for both ra and sa parameters. 6 i. ćulum, s. jozić, d. bajić, i. dumanić table 3 results of up milled surface roughness measurements run no. vc [m/min] ft [mm/tooth] dry machining cca cf mql ra [μm] sa [μm] ra [μm] sa [μm] ra [μm] sa [μm] ra [μm] sa [μm] 1 750 0.009 0.088 0.099 0.057 0.048 0.077 0.076 0.045 0.050 2 500 0.050 0.331 0.336 0.317 0.321 0.321 0.277 0.304 0.309 3 1000 0.050 0.260 0.247 0.227 0.226 0.316 0.325 0.248 0.255 4 396 0.150 0.824 0.870 0.487 0.490 0.836 0.817 0.828 0.858 5 750 0.150 0.738 0.750 0.433 0.452 0.733 0.749 0.707 0.708 6 750 0.150 0.728 0.762 0.518 0.506 0.755 0.711 0.740 0.733 7 1104 0.150 0.810 0.810 0.706 0.768 0.949 0.911 0.832 0.836 8 750 0.150 0.717 0.743 0.427 0.383 0.886 0.922 0.691 0.733 9 750 0.150 0.764 0.819 0.366 0.374 0.839 0.875 0.761 0.803 10 750 0.150 0.781 0.783 0.369 0.355 0.822 0.872 0.721 0.752 11 1000 0.250 1.027 1.107 0.783 0.849 1.298 1.347 1.027 1.155 12 500 0.250 1.021 1.076 1.258 1.417 1.368 1.702 1.129 1.258 13 750 0.291 1.097 1.148 0.786 0.897 1.615 1.879 1.261 1.353 table 4 regression models of ra and sa surface parameters for up milled surfaces technique regression model r2 equation no. dry machining 4 0.34519 7.44322 *10 5.66744 7 7 2 2 7.66 *10 3.9167 *10 8.79081 ra v f c t v f v f c t c t          0.9935 (1) 4 0.40838 8.28924 *10 5.48256 3 7 2 21.193*10 3.857*10 8.40837 sa v f c t v f v fc t c t          0.9930 (2) cca 3 0.91490 2.96966 *10 5.28624 3 6 2 2 3.845 *10 2.34223 *10 2.53281 ra v f c t v f v f c t c t           0.8277 (3) 31.02164 3.38296*10 5.40680 3 2 24.734*10 2.69594*10 5. 7816 9 2 sa v fc t v f v fc t c t        0.8327 (4) mql 4 0.32138 8.8903 *10 6.04079 4 7 2 2 4.56 *10 5.8747 *10 5.1506 ra v f c t v f v f c t c t           0.9886 (5) 3 0.3916 1.05798 *10 5.865 4 7 2 2 4.83 *10 6.9078 *10 2.95762 sa v f c t v f v f c t c t           0.9949 (6) cf 4 0.2219 6.3058 *10 5.48137 4 7 2 2 6.52 *10 5.1388 *10 0.8767 ra v f c t v f v f c t c t           0.9935 (7) 4 0.1782 1.04715 *10 7.05 4 7 2 2 4.033 *10 2.7538 *10 7.4011 sa v f c t v f v f c t c t            0.9930 (8) experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 7 coefficient of determination (r2) shows how well the model fits measured results. an r2 value closer to 1 is preferred as it indicates that the model covers a high amount of variability of the response data around its regression line. for example: ra model generated for dry machining has an r2 value of 0.9935 indicating that 99.35 % of measured data fits the regression model. using the test results and regression models, surface plots showing the influence of cutting speed and feed per tooth on up milled surface roughness parameters were generated. a comparison of surface plots using dry machining, cca, cf and mql cooling techniques are shown in figs. 2 and 3, for both ra and sa parameters respectively. fig. 2 influence of feed per tooth and cutting speed on surface roughness ra using dry machining a) and cooling techniques: b) cca, c) cf, d) mql b) a) d) c) 8 i. ćulum, s. jozić, d. bajić, i. dumanić fig. 3 influence of feed per tooth and cutting speed on surface roughness sa using dry machining a) and cooling techniques: b) cca, c) cf, d) mql 3.1.2 down milled surface finish analysis table 5 shows mean surface roughness values for: dry, cca, cf and mql down milled surface. regression models for: dry machining, cca, mql, and cf tests were generated using the selected central composite plan and test results. regression models, given in table 6 for both ra and sa parameters, are used for accurate surface roughness predictions. b) a) d) c) experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 9 table 5 results of surface roughness measurements for down milled surface run no. vc [m/min] ft [mm/tooth] dry machining cca cf mql ra [μm] sa [μm] ra [μm] sa [μm] ra [μm] sa [μm] ra [μm] sa [μm] 1 750 0.009 0.083 0.110 0.050 0.059 0.076 0.084 0.050 0.089 2 500 0.050 0.264 0.240 0.228 0.273 0.277 0.251 0.309 0.202 3 1000 0.050 0.251 0.230 0.223 0.212 0.325 0.275 0.255 0.250 4 396 0.150 0.896 0.808 0.447 0.453 0.817 0.877 0.858 0.765 5 750 0.150 0.647 0.754 0.412 0.445 0.749 0.587 0.708 0.580 6 750 0.150 0.713 0.746 0.448 0.466 0.711 0.590 0.733 0.599 7 1104 0.150 0.909 0.759 0.686 0.652 0.911 0.968 0.836 0.806 8 750 0.150 0.838 0.613 0.374 0.356 0.922 0.918 0.733 0.645 9 750 0.150 0.749 0.624 0.366 0.439 0.875 0.749 0.803 0.580 10 750 0.150 0.555 0.596 0.362 0.429 0.872 1.342 0.752 0.515 11 1000 0.250 1.137 1.097 0.644 0.822 1.347 1.342 1.155 0.833 12 500 0.250 1.189 1.010 1.002 1.073 1.702 1.534 1.258 1.012 13 750 0.291 1.310 1.052 0.737 0.827 1.879 1.773 1.353 1.340 table 6 regression models of ra and sa surface parameters for down milled surfaces technique regression model r2 equation no. dry machining 4 0.38891 9.02496 *10 4.94105 4 7 2 2 5.94 *10 4.976 *10 6.41 ra v f c t v f v f c t c t           0.9666 (9) 3 0.53975 1.31824 *10 4.55264 7 2 2 9.72 *10 7.84 *10 5.2 4 315 sa v f c t v f v f c t c t           0.9656 (10) cca 3 0.934 2.74826 *10 3.96665 3 6 2 2 2.217 *10 2.0672 *10 0.7825 ra v f c t v f v f c t c t           0.8187 (11) 3 0.98805 2.7625 *10 3.17097 4 6 2 2 8.84 *10 1.98566 *10 1.60787 sa v f c t v f v f c t c t           0.8548 (12) mql 3 0.47323 1.34772 *10 4.70435 3 6 2 2 1.297 *10 9.96109 *10 0.2005 ra v f c t v f v f c t c t           0.9478 (13) 3 0.32467 1.06477 *10 5.00538 3 6 2 2 2.264 *10 9.121*10 2.15344 sa v f c t v f v f c t c t           0.9329 (14) cf 3 0.691 1.9257 *10 5.3568 4 6 2 2 3.86 *10 1.307 *10 2.117 ra v f c t v f v f c t c t           0.9427 (15) 3 0.7483 2.2195 *10 4.292 3 6 2 2 2.264 *10 1.6817 *10 10.82312 sa v f c t v f v f c t c t           0.9656 (16) 10 i. ćulum, s. jozić, d. bajić, i. dumanić the generated models have a high r2 value which means the models can reliably predict surface roughness parameters. for example: the model generated for dry machining has an r2 value of 0.9666 for ra surface parameter. this means that 96.66 % of measured data fits the generated model. using the test results and regression models, surface plots showing the influence of cutting speed and feed per tooth on up milled surface roughness parameters were generated. a comparison of surface plots using dry machining, cca, cf and mql cooling techniques are shown in figs. 4 and 5 for both ra and sa parameters, respectively. fig. 4 influence of feed per tooth and cutting speed on surface roughness ra using dry machining a) and cooling techniques: b) cca, c) cf, d) mql b) a) d) c) experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 11 fig. 5 influence of feed per tooth and cutting speed on surface roughness sa using dry machining a) and cooling techniques: b) cca, c) cf, d) mql 3.1.3 discussion from the test results, it can be concluded that, generally, up milling provides a higher surface roughness than down milling. observing tests conducted at a cutting speed of 750 m/min and feed per tooth of 0.15 mm/tooth, down milling produced a 24 % lower surface roughness when compared to up milling. similar findings were reported by iqbal et al. [14]. during up milling, chip cross section is at its lowest at the initial contact point between the tool and workpiece. due to the low chip cross section, the tool must be forced into the cut leading to burnishing on the workpiece surface. burnishing generates additional heat which can cause the chips to weld on the rake face changing the tool geometry and leading to higher surface roughness values. during down milling, chip cross section starts at its maximum and later decreases as the tool leaves contact with the workpiece. high chip cross section at the initial contact point of the tool and workpiece makes cutting easier, reduces burnishing and leads to a lower surface roughness value. conducted tests give a good insight on how feed per tooth influences surface roughness. by observing tests with feed per tooth of 0.009 mm/tooth, 0.15 mm/tooth and b) a) d) c) 12 i. ćulum, s. jozić, d. bajić, i. dumanić 0.291 mm/tooth at a constant cutting speed of 750 m/min, an increase in surface roughness was observed with increasing feed per tooth. for example: during dry up milling the surface roughness parameter sa increased from 0.09884 μm at 0.009 mm/tooth to 1.148 μm at 0.291 mm/tooth. similar behavior was noted for all other cooling techniques. by increasing the feed per tooth, the cusp height increases. such effect is shown in fig. 6. the cusp, presented as the hatched surface, increases in both height (h) and length (l) with increasing feed per tooth which negatively influences surface roughness values. this effect will be more closely examined in section 3.3 using an optical profilometer. fig. 6 influence of feed per tooth on the cusp height observing the influence of cutting speed on surface roughness, a 12 % to 14 % decrease in roughness was noted with increasing cutting speed. similar findings were reported by rahman et al. [23], iqbal et al. [14] and others. low cutting speeds lead to increased plastic deformation of the workpiece which cause higher surface roughness. the fact that higher cutting speeds lead to a decrease in surface roughness can be attributed to increased workpiece deformation rate. high deformation rates don’t allow for a high amount of plastic deformation of the chip. speeds higher than 1000 m/min produced a rougher surface than was expected. these are most likely caused by increased adhesion of material on the cutting surfaces such as noted by sreejith et al. [11]. by comparing surface roughness values, shown in tables 3 and 5, it was observed that cca cooling produced a considerably finer surface finish than other techniques. minimum quantity lubricant came in second best followed by dry machining and cf. the effectiveness of cca could be attributed to increased cooling capacity of the technique. high mobility of the chilled compressed air moving in and around the tool workpiece interface provides effective cooling which leads to a better surface finish. minimum quantity lubricant uses a fine mist of cutting fluid which both cools and lubricates the cutting zone. superior surface qualities of cca and mql could additionally be attributed to an increased air flow over the cut surface which could aid in forming a finer passive aluminum oxide layer on the workpiece surface. poor surface roughness achieved by dry milling could be attributed to increased heat generated leading to material adhesion on the tool faces. analyzing generated models, it was observed that feed per tooth has the most impact on surface roughness parameters regardless of cooling technique or milling strategy. experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 13 cutting speed has a much lower impact on ra and sa surface roughness parameters. generated models were used to optimize the machining process. goals of the optimization were to determine optimal cutting parameters while keeping ra and sa values low. optimal cutting parameters and predicted surface roughness parameters are shown in table 7. table 7 optimal cutting parameters cutting technique milling strategy optimal vc [m/min] optimal ft [mm/tooth] predicted ra [μm] predicted sa [μm] dry machining up milling 878 0.05 0.289 0.284 down milling 0.237 0.244 cca up milling 687 0.05 0.119 0.093 down milling 0.146 0.160 mql up milling 721 0.05 0.259 0.256 down milling 0.208 0.205 cf up milling 677 0.05 0.285 0.253 down milling 0.236 0.185 3.2 influence of cutting parameters and cutting environment on chip morphology 3.2.1 influence of cutting speed on chip morphology table 8 shows three different chips produced by dry machining at varied cutting speeds of: 396 m/min, 750 m/min and 1104 m/min and a constant feed rate per tooth of 0.15 mm/tooth. table 8 morphology of dry machined chips with varied cutting speeds at feed per tooth of 0.15 mm/tooth: a) 396 m/min, b) 750 m/min and c) 1104 m/min a) b) c) up milling down milling up milling down milling up milling down milling ra [μm] 0.824 0.896 0.738 0.647 0.810 0.909 sa [μm] 0.870 0.808 0.750 0.754 0.810 0.759 mm mm mm 14 i. ćulum, s. jozić, d. bajić, i. dumanić 3.2.2 influence of feed per tooth on chip morphology table 9 shows three different chips produced dry machining at varied cutting feeds per tooth of: 0.009 mm/tooth, 0.15 mm/tooth and 0.291 mm/tooth. cutting speed was kept at a constant value of 750 m/min. table 9 morphology of dry machined chips with varied feed per tooth at a cutting speed of 750 m/min: a) 0.009 mm/tooth, b) 0.15 mm/tooth and c) 0.291 mm/tooth a) b) c) up milling down milling up milling down milling up milling down milling ra [μm] 0.088 0.083 0.738 0.647 1.097 1.137 sa [μm] 0.099 0.110 0.750 0.754 1.148 1.097 3.2.3 discussion chips between the tool and workpiece cause excess heating, higher friction and serve as a barrier at the point of contact which can have a negative impact on surface roughness. for this reason, smaller chips are preferred as they are easier to evacuate from the cutting zone. low feeds per tooth produced high amounts of thin and short chips. increasing feed per tooth produced a longer and thicker chip with clear signs of increased deformation. similar cases were reported by joshua et al. [16], shukla et al. [4] and others. table 9 a) shows a delaminated chip produced by dry milling at a feed per tooth of 0.009 mm/tooth and cutting speed of 750 m/min. delamination of the chip could be caused by prolonged contact between the tool and workpiece causing excess heating of the chip and its serration at the end of the cut. a similar occurrence was reported by khanna et al. [10] during dry turning of inconel 718. cutting speed has a limited influence on chip morphology as only length and pitch of the chip changed while the thickness remained unchanged. chips produced by alternative cooling techniques showed little to no cutting fluid residues which made them easier to analyze and recycle. mm mm mm experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 15 3.3 specimen surface scanning using a filmetrics profilm 3d optical profilometer it was not only possible to measure various surface roughness parameters but to generate a 3d model of the analyzed surfaces. figs. 7–9 show scans of dry down milled surfaces at a constant cutting speed of 750 m/min with varied feeds per tooth of 0.009 mm/tooth, 0.15 mm/tooth and 0.29 mm/tooth. all surfaces were scanned using the same scan settings. from figs. 8 and 9, it is clear that feed per tooth greatly impacts workpiece surface. as was shown in section 3.1.3, an increase in feed per tooth increases the height and length of the cusp height that leads to an increase in surface roughness measurements. similar observations were made observing the three tested cooling techniques. figs. 10 and 11 show 3d surface scans of down milled dry machined surfaces. the specimen in the figures below were machined using a constant feed per tooth of 0.15 mm/tooth and cutting speed of 396 m/min and 750 m/min. no visible changes in the length of the surface were observed in the figures above. comparing the graph in figs. 10 and 11, a decrease in the height of the profile was noted, indicating a decrease in surface roughness with increased cutting speed. cooling techniques produced comparable results to the ones mentioned above. fig. 7 surface scan of dry down milled specimen at 750 m/min and 0.009 mm/tooth. surface roughness measured for this test were ra = 0.083 μm and sa = 0.110 μm 16 i. ćulum, s. jozić, d. bajić, i. dumanić fig. 8 surface scan of dry down milled specimen at 750 m/min and 0.15 mm/tooth. surface roughness measured for this test were ra = 0.647 μm and sa = 0.754 μm fig. 9 surface scan of dry down milled specimen at 750 m/min and 0.29 mm/tooth. surface roughness measured for this test were ra = 1.310 μm and sa =1.052 μm experimental investigation of face milling aluminum en aw-2011 t6 using various cooling... 17 fig. 10 surface scan of dry down milled specimen at 396 m/min and 0.15 mm/tooth. surface roughness measured for this test were ra = 0.896 μm and sa = 0.808 μm fig. 11 surface scan of dry down milled specimen at 750 m/min and 0.15 mm/tooth. surface roughness measured for this test were ra = 0.647 μm and sa = 0.754 μm 18 i. ćulum, s. jozić, d. bajić, i. dumanić 4. conclusions surface roughness and chip morphology of aluminum alloy en aw – 2011 t6 using four cutting environments were examined in this paper. along with dry machining, three cooling techniques were tested: cold compressed air, minimum quantity lubrication and cutting fluid. from these experiments, the following was concluded:  down milling provides up to 24 % lower surface roughness than up milling. this is a result of a higher chip cross section at the entry point of the tool as compared to a low chip cross section such is produced by up milling. a lower chip cross section at the entry point leads to burnishing of the workpiece surface. burnishing generates excess heat which can cause chips to weld to the rake face of the tool thus negatively influencing surface roughness.  feed per tooth greatly influences surface roughness of the workpiece. increasing feed per tooth produces a rougher surface due to an increased cusp height.  decreasing cutting speed leads to an increase in surface roughness due to increased plastic deformation of the workpiece material.  cold compressed air cooling produced the lowest surface roughness. increased cooling capacity of the chilled air produced by the vortex tube reduces heat generated. surfaces machined using mql cooling performed worse than cca but outperformed dry and cf cooled tests. this could be due to increased air flow over the cut surface which could aid in forming a finer aluminium oxide layer on the workpiece surface. dry machining produces a rougher surface finish than the latter, most likely, owing to increased adhesion of workpiece material on the machined surface.  feed per tooth greatly influence chip morphology. a low feed per tooth produces high amounts of short and thin chips. increasing feed per tooth increases the length and thickness of chips. lower surface roughness parameters could be attributed to smaller chip sizes that do not obstruct the cutting zone and produce less heat at the tool workpiece interface. the influence of tool wear was not observed in this paper due to very short machining times of the test. in future research, multiple passes will be implemented to induce observable tool wear. continuation of research should move in direction of cost analysis. cost 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21. http://www.pingyiao.com/en/rhenus/rh5/pdf/r.rhenus%20tu%2030%20t.pdf, (last access : 28.03.2023.) 22. perec, a., 2022, desirability function analysis (dfa) in multiple responses optimization of abrasive water jet cutting process, reports in mechanical engineering, 3(1), pp. 11-19. 23. rahman, m., senthil kumar, a., manzoor-ui-salem, 2001, evaluation of minimal quantities of lubricant in end milling, the international journal of advanced manufacturing technology, 18, pp. 235-243. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 19 28 https://doi.org/10.22190/fume180102008p © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper ∗ adhesion between a power-law indenter and a thin layer coated on a rigid substrate udc 539.6 antonio papangelo1,2 1politecnico di bari, department of mechanics, mathematics and management, italy 2hamburg university of technology, department of mechanical engineering, germany abstract. in the present paper we investigate indentation of a power-law axisymmetric rigid probe in adhesive contact with a "thin layer" laying on a rigid foundation for both frictionless unbounded and bounded compressible cases. the investigation relies on the "thin layer" assumption proposed by johnson, i.e. the layer thickness being much smaller than the radius of the contact area, and it makes use of the previous solutions proposed by jaffar and barber for the adhesiveless case. we give analytical predictions of the loading curves and provide indentation, load and contact radius at the pull-off. it is shown that the adhesive behavior is strongly affected by the indenter shape; nevertheless below a critical thickness of the layer (typically below 1 µm) the theoretical strength of the material is reached. this is in contrast with the hertzian case, which has been shown to be insensitive to the layer thickness. two cases are investigated, namely, the case of a free layer and the case of a compressible confined layer, the latter being more "efficient", as, due to poisson effects, the same detachment force is reached with a smaller contact area. it is suggested that high sensitive micro-/nanoindentation tests may be performed using probes with different power law profiles for characterization of adhesive and elastic properties of micro-/nanolayers. key words: adhesion, layer, jkr model, adhesion enhancement 1. introduction adhesion is a much debated topic in contact mechanics covering different fields of application, from adhesion of rough surfaces [1-4] to bioinspired adhesive mechanisms [5, 6]. nature has inspired different researchers to try to reproduce the same design strategy adopted by insects such as the "famous" gecko, or to develop an "optimal" profile to reach theoretical adhesive strength on a substrate [6-8]. the progress of technology allows us today to "design" received january 02, 2018 / accepted february 02, 2018 corresponding author: antonio papangelo department of mechanics, mathematics and management, politecnico di bari, viale japigia 182, 70126 bari, italy e-mail: antonio.papangelo@poliba.it 20 a. papangelo surface topography down to the nanoscale. nanopatterned surfaces, with repeating pillars [9] or dimples [10], are nowadays inspiring many researchers to develop pressure sensitive adhesive mechanisms [10-12]. the majority of scientific literature has focused on the case of halfspace geometry; nevertheless, the development of microelectromechanical systems (mems), anti-wear coatings, microelectronics, pressure sensitive adhesive, multilayer coatings calls for a detailed understanding of the contact behavior of the layered surfaces in presence of adhesion. some authors have dealt with axisymmetric contact of an elastic layer supported by a rigid foundation in adhesiveless and adhesive cases, both analytically [13, 14] and numerically [15]. it has been shown that for the hertzian profile the pull-off force does not depend on the elastic properties of the material, similarly to the classical solution of johnson-kendall-roberts (jkr) valid for halfspace geometry [16, 17]. argatov et al. [18] also studied the indentation of an elliptic paraboloid profile in contact with a transversely isotropic layer supported by a rigid foundation in the compressible and incompressible case. to unveil the effect of the indenter profile, in this paper we study the adhesive indentation of an axisymmetric frictionless rigid punch with a power-law profile, which indents a compressible "thin layer" on a rigid foundation in both the bounded and unbounded case. the "thin layer" approximation was first proposed by johnson [19], who assumed that layer thickness b is much smaller than radius of contact a, i.e. b<<a, so that the plane sections remain plane upon deformation. the same hypothesis was used by jaffar [20], who expanded the analysis to the axisymmetric case and later by barber [21] who generalized the formulation for an arbitrary three dimensional problem. following barber [21] we will derive the adhesive solution in the case of short range adhesion (the so-called jkr limit). to this end the energetic derivation of the original jkr model will not be repeated anew. it is in fact known that the adhesive jkr solution can be directly obtained from the adhesiveless solution [22] (see also [23-25]), i.e. indentation δ is 11 2 p/aw ′′′−= δδ (1) where δ1 is the adhesiveless indentation, w is the work of adhesion, a′ is the first derivative of contact area and p1″ the second derivative of the adhesiveless load with respect to δ1. then, adhesive load p is 111 2 p/awppp ′′′′−= (2) where p1′ is the first derivative of the adhesiveless load with respect to indentation δ1. notice that (1, 2) are approximated for a general three-dimensional contact, but they are exact for an axisymmetric or line contact; hence we are not adding any other approximation except the "thin layer" assumption. recently a further generalization of the approach suggested by argatov [22] has been proposed [26], which, in boundary element simulations, introduces a mesh dependent criterion (based on the balance between elastic energy release and adhesion work) to numerically solve adhesive contacts of complex shapes, which in principle can be easily generalized to the case of layered systems. in this paper we will extend the asymptotic solutions for adhesiveless thin layer problems found by johnson [19], jaffar [20], barber [21] to the jkr adhesive case. we shall then discuss implications, particularly regarding the dependence of pull-off and loading curves on the geometry of the profile suggesting the strategies to obtain "optimal" adhesion. the analytical results obtained may be of interest for elastic and adhesive property characterization of thin films, through nano/microindentation test, particularly when a polymeric coating is adopted over metals. adhesion between a power-law indenter and a thin layer 21 2. methods we study indentation by a rigid frictionless punch on a thin layer on a rigid substrate in presence of short range adhesion (jkr type [16]). to this end the adhesiveless relations load-indentation and area-indentation are needed. we follow the line of argument of barber [21], who provided the solution of the adhesiveless problem, for a general three dimensional profile of the punch, for the case of both a frictionless unbounded layer and a bounded compressible layer. the cartesian coordinate system is set as follows: the contact surface of the layer coincides with plane z=0 while two-dimensional spatial coordinates x1, x2 lie in the plane of the layer. we restrict our analysis to the case where layer thickness b is much smaller than contact radius a, i.e., b<<a, so that the deformation through the layer is homogeneous; thus, the plane section remains plane upon compression. this corresponds to the original johnson's approximation, which implies that the in-plane displacements of the layer with components u1, u2 are independent of z. for the case of frictionless foundation barber [21] showed that the contact pressure is ( ) ( 2121 x,x b ex,xp ξ ∗ = ) (3) where indentation ξ is the local interpenetration between the indenter and the layer if it did not deform, e* is the effective elastic modulus e*=e1/(1-ν2), ν the poisson's ratio, b the layer thickness. for the case of bounded compressible layer barber [21] obtained ( ) ( 2121 x,x b zex,xp ξ ∗ = ) (4) where z=(1-ν)2/(1-2ν). the general three dimensional adhesiveless solution could be transformed into the corresponding jkr adhesive solution with the approximation that the "contact shape" is the same in the adhesive and adhesiveless solutions [22-25]. however, here we are dealing with axisymmetric contact; therefore, no approximations are introduced [22-25]; thus, the solution will be exact, provided that the original johnson's approximation of a "thin layer" is fulfilled. 3. axisymmetric contact with power law profile 3.1. frictionless unbounded layer consider the indentation by a rigid axisymmetric frictionless indenter with power law profile rk/(krk-1) on a thin layer supported by a rigid foundation (r is the radial coordinate and r a reference length, see fig. 1). for the case of a frictionless unbounded layer the contact pressure is given by eq. (3), and hence ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= − ∗ 11 k k kr r b erp δ (5) where δ1 is the adhesiveless indentation. 22 a. papangelo fig. 1 schematic representation of an axisymmetric power law profile in contact with a thin elastic layer of thickness b supported by a rigid foundation the total load reads 1 2 1 2 − +∗ + = k k r a b e k p π (6) where a is the contact radius and the subscript "1" stands for the adhesiveless solution. in fact from p(a)=0, the adhesiveless indentation δ1 is 11 1 −= k k r a k δ (7) hence, we have for the repulsive solution ( ) ( ) k/ k kk/k rb e k kp 1 12 2 1 2 1 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = − +∗+ δπ (8) k/ k k/ r ka 2 1 12 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= − δπ (9) computing derivatives a, p1′, p1″ and using eqs. (1) and (2) after some algebra the adhesive solution is obtained ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ∗− +∗ e wb k k rk k b ep k/ k k/k 22 2 1 2 1 1 2 δδπ (10) ∗−= e wb21δδ (11) under force control the pull-off is found imposing p′=0 which gives adhesion between a power-law indenter and a thin layer 23 k/ k popo, e wbra; e wb k 1 1 1 2222 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ == ∗ − ∗δ (12) substituting the first equation from eqs. (12) into eq. (10), the pull-off force is obtained ( ) ( ) ( )k/k k/kk/ po e wbr b e k kp 22 122 2 2 2 + ∗ − ∗ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −= π (13) for a hertzian profile, i.e. imposing k=2, the following results are obtained 41 1 2222 / popopo, e bwra;rwp; e wb ⎟ ⎠ ⎞ ⎜ ⎝ ⎛=−== ∗∗ πδ (14) notice that for the hertzian case, the pull-off does not depend on the elastic properties of the layer, but only on geometry "r" and surface energy "w " similarly to the classical jkr results valid for a sphere pressed against an halfspace, but with an enhanced adhesive strength (in the classical jkr theory the pull-off is equal to -3/2πrw). we define the dimensionless quantities rw pp; b e w π δδ 22 == ∗ (15) dimensionless forms of the adhesive load, eq. (10), and pull-off reads ( ) ( ) ( ) ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = − ∗ − + ke wbr k kp k/ k/k k/k k/k 212 2 2 22 2 2 δδ (16) ( ) ( ) ( )k/k k/kk/ po e wbr k kp 22 22 22 2 − ∗ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −= (17) where eq. (11) is used to express eq. (10) in terms of δ . for a given geometry of the indenter and for given material properties one may think to optimize the thickness of layer b. notice that: for k<2 increasing the layer thickness leads to higher pull-off, for the hertzian case k=2 the pull-off is insensitive to the layer thickness, while for k>2 the layer thickness has to be reduced for increasing the pull-off force. the layer thickness in fact is raised at the power (2-k)/(2k), which is plotted in fig. 2. for high values of exponent k we obtain 21 / po bp −∝ , which suggests that in order to obtain high adhesive strength k>>2 should be adopted together with a very thin layer. bearing in mind that our argumentation is based on the thin layer approximation, in the rest of the paper we will focus on the case k>2. 24 a. papangelo fig. 2 the pull-off force is proportional to .bp k k po 2 2− ∝ here we plot the exponent (2-k)/(2k) versus k to show that for k<2(>2) thicker (thinner) layer increase the pull-off force similarly to gao and yao [7] we assume w=10 mj/m2, e*=1 gpa and for the layer thickness and reference length b=1 µm, r=1 mm. the loading curves are reported in fig. 3 for k=[1.9, 2, 2.1]. two main points arise, i.e., firstly, there is no instability in displacement control (contrary to the classical jkr problem with halfspace), and secondly, the pull-off force is greatly affected by k. this is further confirmed in fig. 4 where the pull-off force is plotted as a function of k: it appears that moving from k=2 to k=3 an enhancement of one order of magnitude is obtained. fig. 3 force-indentation curves for different power law profiles k=[1.9, 2, 2.1]. solid (dashed) curves are stable (unstable) under force control adhesion between a power-law indenter and a thin layer 25 fig. 4 pull-off as a function of the indenter shape, r/b=[102, 103, 104] the strong enhancement obtained increasing k calls for further investigations. first we compute the average tension acting within the contact area at pull-off |σpo| b we k k a p po po po ∗ + == 2 22π σ (18) where we clearly recognize the toughness of material kic=(e*w)1/2 and the dependences on layer thickness b-1/2 and the shape of indenter "k", but independent of the other lengthscale "r" involved in the problem. for k>2, the "optimal" layer thickness, critical thickness bcr below which theoretical strength σth is reached, is easily obtained as po th ic cr ak k kb <<⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = 22 2 2 σ (19) we shall assume reasonable values for σth =10 mpa, w=10 mj/m2, e*=1 gpa as in gao and yao [7] which for k=2 gives bcr=1 µm and is further reduced for k>2. we recall here that the present analysis is valid within the johnson's approximation b<<a. with the same set of parameters we estimate ratio b/apo in the "critical condition", i.e. for b=bcr=1 µm, ( ) 1 2 2 112 21 <<⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = −∗− − k/k th th k/ k/ cr,po cr r /we k k a b σ σ (20) and plot eq. (20) in fig. 5 as a function of k and for r/b=[102, 103, 104]. the curves show that for k>2 the "optimal design" is feasible within the thin layer approximation where the pull-off tension is close to the theoretical strength of the material. also notice that we have used b=bcr= 1 µm, but for any b<bcr→ |σpo |= σth; ; thus, the condition in eq. (20) can be easily fulfilled for our reasonable set of parameter values. the "efficiency" of the adhesive mechanism is easily shown comparing the contact radius at pull-off with that obtained in the classical jkr theory, i.e. ajkr,po=(9/8πr2w/e*)1/3. in fact, ratio apo/ ajkr,po ∝ b1/(2k) and hence, for a sufficiently thin layer, the same detachment force is reached with a much smaller contact area. 26 a. papangelo fig. 5 ratio bcr/ apo,cr plotted for different geometries of the indenter, with r/b=[102, 103, 104]. the thin layer approximation is fulfilled for bcr/apo,cr <<1 3.2. bonded compressible layer barber [21] has shown that if the layer is compressible ν≠0.5 and bounded to the rigid substrate, the contact pressure is given by eq. (4), which differs from the frictionless behavior for a factor z=(1−ν)2/(1−2ν). thus the previous analysis can be repeated to obtain the total adhesiveless load ( ) ( ) k/ k kk/k rb ez k kp 1 12 2 1 2 1 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = − +∗+ δπ (21) computing derivatives a, p1′, p1″ and using eqs. (1) and (2) after some algebra the adhesive solution is obtained ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ∗− +∗ e w z b k k rk kz b ep k/ k k/k 22 2 1 2 1 1 2 δδπ (22) ∗−= e w z b2 1δδ (23) which corresponds to the hertzian case for k=2. to find the minimum we impose p′=0, which gives k/ k popo, e w z bra; e w z b k 1 1 1 2222 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ == ∗ − ∗δ (24) and hence ( ) ( ) ( ) ( ) ( )k/k k/kk/kk/ po e wbzr b e k kp 22 22122 2 2 2 + ∗ −− ∗ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −= π (25) adhesion between a power-law indenter and a thin layer 27 notice that for k>2 pull-off ppo is increased and contact area πa2 po decreased with respect to the frictionless case, thus, due to poisson's effects, the contact is more "efficient" in the bounded configuration. 4. conclusions in this paper we have studied the adhesive indentation by an axisymmetric frictionless rigid indenter with power-law profile on a thin layer supported by a rigid foundation. it has been shown that the detachment force at pull-off is strongly affected by the geometry of the tip. nevertheless we noted that reducing the layer thickness, which also fulfills the "thin layer approximation", leads to reaching the theoretical strength at pull-off. using reasonable data for the elastic and adhesive parameters of the layer, as in gao and yao [7], we have shown that the critical thickness of the layer below which the theoretical strength is reached is of the order of 1 µm. while for the case of hertzian (parabolic) profile the detachment force is independent of the thickness of the layer and on its elastic properties (similarly to the classical results of johnson-kendal-roberts valid for halfspace geometry) when a general power-law is used, this dependence arises. it has been shown that the adhesive mechanism is more efficient when compared to the jkr halfspace solution, particularly for bounded compressible layers, as the same detachment force is obtained with a much smaller contact area. this occurs because the dominant length scale for the stress intensity factor at the contact edge is the layer thickness. the presented analysis is particularly suited for polymeric coating of metallic samples with micro or nanometer thickness. exploiting different probe profiles high sensitivity micro-/nanoindentation test may be performed to determine adhesive and elastic properties (w, e*) of the thin layers coated on "rigid" substrates. possible extension of the present work may consider the adhesive indentation of multilayered systems, which are of interest in tribology, as for the "surface force apparatus", often represented as a three-layer halfspace [27]. references 1. pastewka, l., robbins, m.o., 2014, contact between rough surfaces and a criterion for macroscopic adhesion. proceedings of the national academy of sciences, 111(9), pp. 3298-3303. 2. ciavarella, m., papangelo, a., 2018, a generalized johnson parameter for pull-off decay in the adhesion of rough surfaces, physical mesomechanics, 21(1), pp. 67-75. 3. ciavarella, m., papangelo, a., 2018, a modified form of pastewka--robbins criterion for adhesion, the journal of adhesion, 94(2), pp. 155-165. 4. ciavarella, m., papangelo, a, 2017, a random process asperity model for adhesion between rough surfaces, journal of adhesion science and technology, 31(22), pp. 2445-2467. 5. huber, g, gorb, s, hosoda, n, spolenak, r, arzt, e., 2007, influence of surface roughness on gecko adhesion, acta biomater., 3, pp. 607-610. 6. pugno, n.m., lepore, e., 2008, observation of optimal gecko's adhesion on nanorough surfaces. biosystems, 94, pp. 218-222. 7. gao, h., yao, h., 2004, shape insensitive optimal adhesion of nanoscale fibrillar structures, proceedings of the national academy of sciences of the united states of america, 101(21), pp. 7851-7856. 8. yao, h., gao, h., 2006, optimal shapes for adhesive binding between two elastic bodies, journal of colloid and interface science, 298(2), pp. 564-572. 9. papangelo, a., afferrante, l., ciavarella, m., 2017, 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apparatus, journal of physics d: applied physics, 30(12), pp. 1710–1719. adhesion between a power-law indenter and a thin layer coated on a rigid substrate antonio papangelo1,2 1. introduction 2. methods 3. axisymmetric contact with power law profile 3.1. frictionless unbounded layer 3.2. bonded compressible layer 4. conclusions references plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. i ii foreword to the thematic issue: biomedical engineering one of the greatest engineering challenges miroslav d. trajanović 1 , osiris canciglieri junior 2 1 university of nis, faculty of mechanical engineering, nis, serbia 2 production and system engineering graduate program – ppgeps, pontifical catholic university of parana, pucpr, brazil editorial this thematic issue of facta universitatis, series: mechanical engineering is primarily dedicated to a very wide and important topic, biomedical engineering (bme), while it also tackles the topic of how nature can provide solutions for engineering challenges. the reason is twofold. firstly, the journal receives an increasing number of papers in the field of bme. secondly, it aims to highlight the importance of bme, as one of the increasingly prominent research areas in mechanical engineering. biomedical engineering is one of the youngest branches of engineering which deals with the application of engineering methods to solving problems in medicine. bme is used in all healthcare stages, including diagnostics, monitoring and therapy. today, bme is recognized as a separate engineering discipline although it is based on other engineering disciplines, such as information technologies, mechanical, electrical and chemical engineering. if we take into account that for solving the problems in bme it is necessary to use, in addition to engineering knowledge, that of medicine and biology as well, it can be said that its nature is deeply multidisciplinary. bme has given a great contribution to the analysis of the state and simulation of different processes in the human body, thus creating a better understanding of the physiological processes, which is important for diagnosis and therapy. thanks to bme, the medicine today uses many new materials, tools, devices, systems, artificial tissues and organs, methods and processes, which together broaden and improve medical services and ensure a better quality of people’s life. since the first wood and leather prosthesis dating back from 1069 to 664 b.c, different medical devices have been developed. still, it can be said that the cornerstone of biomedical engineering was set up by wilhelm roentgen who in 1895 made the first x-ray image. the real development of biomedical engineering happened after ww ii. the invention of artificial kidneys (1945), artificial heart valve (1952) and the first external cardiac pacemaker (1958) paved the road towards the creation of new engineering disciplines biomedical engineering. editorial ii m. trajanović, s. o. canciglieri j. as a consequence, the first journal dedicated to biomedical engineering, ieee transactions on biomedical engineering was launched in 1953. the academic community responded quickly, and the john hopkins university in 1962 opened the department of biomedical engineering. today, almost all universities have at least one study program in the field of biomedical engineering and there are many biomedical engineering journals. according to the scimago journal & country rank (http://www.scimagojr.com) there were 181 significant journals in 2014 whose subject category is biomedical engineering. the importance of biomedical engineering is perhaps best illustrated by the data on employment. according to united states department of labor statistics the number of biomedical engineers in usa in 2014 was 20080, with an expected rate of growth of 27% until 2022, much faster than the average for all occupations. in spite of its evident remarkable achievements, biomedical engineering still faces considerable scientific and technical challenges. in order to give floor to researchers to present their achievements, the editorial board of facta universitatis, series: mechanical engineering decided to publish this thematic issue. it contains twelve papers from the field of biomedical engineering. it covers a wide spectrum of topic, from biomaterials, through design and manufacturing of osteofixation devices to simulators. in addition, a couple of papers deal with the application of information technologies in the management of data, knowledge and processes in orthopedics, as well as in the simulation of human organs. 7188 facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 121 135 https://doi.org/10.22190/fume210507063m © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper accuracy analysis of the curved profile measurement with cmm: a case study tomasz mazur1, miroslaw rucki1, yuriy gutsalenko2 1kazimierz pulaski university of technology and humanities in radom, poland 2national technical university “kharkiv polytechnic institute”, kharkiv, ukraine abstract. in the paper, analysis of the curved profile measurement accuracy is described. since there was no cad model or other reference profile for the measured detail, the first step was to generate the reference contour of the cam using the technical drawing and tolerance requirements. the test campaign consisted of three experiments aimed at determining the effect of scanning velocity on the results of form deviation δ measurement, evaluation of deviation δ measurement uncertainty and the measurement repeatability. the scanning time was checked, too. the obtained results demonstrated feasibility of the chosen cmm and measurement strategy. it was found also that the measurement uncertainty did not depend on the scanning sampling step from 0.05 to 0.2 mm, and the true measurement time was for 30-40% longer than that expected from the nominal scanning velocity. key words: curved profile, tolerance, measurement, cmm, uncertainty 1. introduction free-form surfaces and curved profiles are widely used in the design and manufacturing of details with high and strict precision requirements [1,2]. during the machining process, vibrations and other inaccuracies may affect the final state of the curved profile, which leads to the necessity of thorough dimensional and shape inspection to ensure its functionality. thus, the characterization of free-form surfaces is an increasingly important area of metrology [3]. in particular, tolerance of the profile along with the position tolerance is a crucial feature in the design and manufacture of products with curved profiles. the contour of curved profiles can be detected either by sensing or by measuring, and measurement systems are generally based on mobile or stationary coordinate measurement systems with specific software and sensors [4]. the accuracy of the results from coordinate received may 07, 2021 / accepted october 20, 2021 corresponding author: miroslaw rucki kazimierz pulaski university of technology and humanities in radom, malczewskiego 29, 26-600 radom, poland e-mail: m.rucki@uthrad.pl 122 t. mazur, m. rucki, y. gutsalenko measurements depends on the accuracy of the measuring device, workpiece properties, environmental conditions, and especially on the operator and measurement procedures [5]. appropriate planning of a measurement strategy for free-form surfaces is addressed in many publications. for example, it was proposed to establish the local geometric deviation, namely, the difference between each measurement point and the cad model of the measured surface [6]. other papers dealt with two key problems of surfaces and curves profile error measurement: (1) evaluation algorithm of profile error on the basis of minimum zone principle or maximum material condition; (2) computer aided arbitrament for minimum zone principle and maximum material condition [7]. for the measurement of freeform shaped workpieces, the distortions caused by the tip mechanical filtration have impact on measurement accuracy, so that correction is desired in order to restore to the real workpiece surface [8]. menq and chen noted that after measurement with a contact probe, the generated surface differed from the real one because of the radius compensation errors of the probe. minimization of the compensation errors required that the probing directions of the cmm coincide with the normal vectors of the probed points. they emphasized unavailability of the normal vectors of data points when the cad model of a new design did not exist [9]. similarly, in 3d gear measurements, the influence of cmm geometric errors on the results is still unclear because the requirement for a gear measurement standard with ideal geometry cannot be fulfilled [10]. it should be noted that the laser-based measurement methods of curved profiles need compensation, too [11]. a significant share in the overall calculus of errors in scanning measurements performed on coordinate measuring machines (cmm) refers to dynamic errors [12]. since the measurement time and cost increase proportionally as the increase of sampling points, it is essential to study a sampling method [13]. the volumetric probing uncertainty of a cmm is usually determined adding a component of the length measuring uncertainty, considering the distance between 25 points on the calibrating sphere, to get the overall point coordinate uncertainty of the cmm [14]. evaluation of repeatability and reproducibility of the cmm equipment is necessary, too [15]. prior to numerical characterization, filtering is done and it is also essential for extracting information needed to provide process feedback and establish functional correlation [16]. a study of roundness of different artifacts using different algorithm and filters demonstrated impact of filtering on the measurement results [17]. the number of cmm points in the measurement of each feature of a part has to enable achievement of a simulated feature-fit that results in a high-quality representation of the manufactured feature [18]. in order to simplify the calculation and simultaneously retain the accuracy of evaluation, the method was proposed, based on the extraction of key points from scanning data set [19]. other authors emphasized, however, that the simulation of measurement became very complicated when the solution of the measuring task needed construction of elements using measured features [20]. some authors emphasized that there are very few commercially available software systems that offer sweep scan path planning function. moreover, newly proposed methods able to generate a viable sweep scan path automatically require significant user’s knowledge and involvement [21]. from that perspective, it is crucial to keep consistency in the measurement procedure. a dictionary definition of ‘consistency’ is ‘constant adherence to the same principles of thought or action’ [22]. saunders and co-authors noted that this definition is intended to refer to a personal characteristic, but since many personal choices are made during programming, accuracy analysis of the curved profile measurement with cmm: a case study 123 operating, and evaluating of the cmm measured points, the term also works well within the context of measurement [23]. according to the definition of the profile tolerance in iso 1101 [24], the surface profile error can be defined by the minimum diameter covering all measured points of the cluster spheres whose centers lie on the design model. profile tolerance may be related to a basic surface; then its orientation and position are dependent on the definition of bases and base-dependent coordinate system. researchers have proposed various practical approaches towards evaluating form deviations of 2d contour profiles based on coordinate measurement data. for example, a 2d contour was divided into straight and curved parts [25]. there are reports in which extracted points of curved profile deviated from reference data within ±0.1 mm [26]. in the present paper, we focused on the statistical analysis of the results obtained for the curved 2d profile with no available cad model. the measurement results appeared to be dependent on the scanning parameters, so that a balance between accuracy demands and measurement time had to be found. also the problem of basic surface for measurements was challenged in order to keep consistence of the results. 2. materials and methods in the research studies, a cam with complex 2d profile curvature was measured. its dimensions and tolerances are shown in fig. 1. its form tolerance was 0.15 mm , and roughness of curved surface was limited to ra = 1.25 μm and rz = 6.3 μm. using the data from fig. 1, theoretical (reference) profile was generated as a file *.dxf using the program solidworks 2019. fig. 1 technical drawing of the measured cam 124 t. mazur, m. rucki, y. gutsalenko the contour of the cam was generated using the technical drawing, and in form of *.dxf file was input to the cmm control program. in the file, 2829 theoretical (reference) points constituted the profile with coordinates x, y, z in the local coordinate system (lcs) defined according to the technical drawing. the points were located approximately uniformly 1 mm below the base surface of the cam, i.e. z = -1 mm for each point. the length of the contour was lc = 123,49 mm. the measurements were performed with the cmm mitutoyo crysta – apex s 7106. it is a high-accuracy cnc coordinate measuring machine that guarantees a maximum permissible error defined by iso 10360-2:2009 of e0,mpe = (1.7+3l/1000) μm at ambient temperature 20 ±2 °c. l stands for the selected measuring length in mm. the measuring range in three axis is x = 700; y = 1000; z = 600 mm, 3d acceleration a = 2309 mm/s2, and linear velocity in three directions x, y, z is v = 519 mm/s. moreover, the cmm has temperature compensation function in the range of 16÷26 °c, which makes it suitable for working in the industrial conditions. the machine can be equipped with contact scanning probe, non-contact laser probe or vision probe [27]. scanning mode makes it possible to perform measurement with 0.05-1.0 point-point step or distance between points, scanning speeds between 0.5 and 4 mm/s, and permissible probe deflection in the range from 0.15 to 0.4 mm. fixation of the cam with defined lcs is shown in fig 2. fig. 3 presents the initial window of cmm for scanning of an outer closed contour with the set values described below in the text. fig. 2 fixation of the measured cam on the cmm table accuracy analysis of the curved profile measurement with cmm: a case study 125 to perform the experiments, control software mcosmos was used. in this program, ready scripts are available for the measurement of all standardized geometrical tolerances, as well as the measurement of complex shapes in local coordinates. before the measurement, a fixture was made so that the entire profile could be measured in one fixation with one probe. the probe was calibrated with the calibration sphere, and a local coordinate system (lcs) was defined according to the tolerance data provided in the drawing fig 2. the lcs definition covered following elements: ▪ base a as a head surface of the measured cam became the main surface of the lcs, it defined axes x and y from 4 measuring points, ▪ base b defined lcs center in the center of the hole ∅20, using 4 points of the circle, ▪ base c as a line connecting three points, two of them lay in the bisector of the first smaller base hole and the third one was the center of the second smaller base hole; it was moved in parallel up to base b become axis y, ▪ remaining axes x and y are derived from the right-hand coordinate system. fig. 3 window of the mcosmos program for the measurement parameters during the measurement, the cmm program written for that specific purpose is collecting the coordinates of measuring points. according to the definition of the profile deviations, it registers maximal deviation value δmaxand shows its position in the profile. the number of collected points is dependent on the measurement step and only approximately corresponds to the profile length divided by step. apart from deviation value δ, scanning time was registered directly by the cmm program. after the measurement is finished, the program records automatically the measurement report with the value of the largest registered deviation from the reference profile. moreover, the program makes it possible to register all the measuring points from the scanning, in the respective file *.dxf. cnc scanning scanning basic plane --------- cnc parameters step clearance velocity deflection start point --- end point - start direction end direction clean help ok 126 t. mazur, m. rucki, y. gutsalenko 3. test campaign the test campaign consisted of three experiments aimed at determining the effect of scanning velocity on the results of form deviation δ measurement, evaluation of deviation δ measurement uncertainty and the measurement repeatability. during the profile scanning, three parameters could be set: ▪ scanning velocity vs in the range between 0.5 and 4 mm/s, ▪ permissible probe deflection pd in the range between 0.15 and 0.4 mm, and, ▪ scanning step s between 0.05 and 1 mm. the respective values used in each measurement series are presented in the table 1. table 1 parameters of experimental measurements series no. vs [mm/s] pd [mm] s [mm] number of repetitions n 1 1 0.15 0.2 5 2 1 0.2 0.2 5 3 1 0.3 0.2 5 4 1 0.4 0.2 5 5 2 0.15 0.2 5 6 2 0.2 0.2 5 7 2 0.3 0.2 5 8 2 0.4 0.2 5 9 3 0.15 0.2 50 (3×5+1×10+1×25) 10 3 0.2 0.2 5 11 3 0.3 0.2 5 12 3 0.4 0.2 5 13 4 0.15 0.2 5 14 4 0.2 0.2 5 15 4 0.3 0.2 5 16 4 0.4 0.2 5 17 3 0.15 0.2 50 18 3 0.15 0.05 50 the above-mentioned parameters were chosen in order to perform three different experiments, as explained below. 3.1. effect of scanning velocity on results of the form deviation measurements in the first set of experiments, the goal was to determine the effect of different scanning velocities vs on the results of profile form deviation, considering the measurement time. in this set of measurements, the scanning step remained unchanged, s = 0.2 mm. as shown in the table 1 above, the measurements were performed for 16 combinations of vs and pd settings. for each combination, the measurements were repeated 5 times. the number of measuring points differed in a small range from 645 up to 660. 3.2. uncertainty evaluation it can be assumed that the factors having effect on the measurement uncertainty of a cam contour are similar to the ones typical for roundness measurement [28-29]. uncertainty analysis was performed using the type a approach [30] with 50 repetitions made in the accuracy analysis of the curved profile measurement with cmm: a case study 127 repeatability conditions. two series of measurements marked 17 and 18 in table 1 had similar parameters vs = 3 mm/s and pd = 0.15 mm, but different sampling step. this experiment was to demonstrate how the uncertainty of form deviation is dependent on the number of probing points. in the series 17, at sampling step s = 0.2 mm, the number of probing points was ca. 650, while in the series 18 it was 2560-2590, close to the respective number in the reference file *.dxf. 3.3. repeatability in short measurement series considering a relatively long time of a single measurement, which can be as long as 200 s at scanning speed vs = 1 mm/s, it is reasonable to expect that 50 repetitions may not completely conform to the repeatability conditions requirement. to challenge this issue, the third experiment was performed. in the case of the series 9 (table 1, vs = 3 mm/s, pd = 0.15 mm, s = 0.2 mm), the measurement was firstly performed three times with 5 repetitions each time, then with 10 repetitions, and finally with 25 repetitions. this procedure was described in tab. 1 as 3×5+1×10+1×25. thus, the strict repeatability conditions were kept only for each group of repetitions, but not only for the entire sample of 50 similar measurements. as a result, measurement repeatability for a smaller number of repetitions could be compared with the results for a larger number. all these measurements, as well as the ones described in section 3.1, were performed on the same day, with no resetting the cmm. after calibration of the probe, the measured cam was not moved from its fixed position. the coordinate system once established was applied to each measurement due to the specially prepared software program. moreover, to assure a higher level of repeatability, the initial position of the probe before each measurement was identical. the experiments described in section 3.2, however, were performed several months later, and for each of them the cmm was started anew. in this way the obtained results in series 17 and 18 must be treated as two separate experiments, while the others are somewhat interconnected between each other through the same definition of the coordinate system and a relatively short time between the repetitions. 4. results and discussion overall number of the maximal deviation measurement results was 225. in order to determine normality of the results distribution, the kolmogorov-smirnov test was applied to each of tests with 50 repetitions, namely, series no. 9, 17 and 18, as described in the previous section. respective d-values of the statistics were 0.1789, 0.14551 and 0.18754, while p-values were 0.07, 0.22 and 0.05, respectively. hence the measurement results in the series did not differ significantly from gaussian distribution, despite some differences in statistical parameters. fig. 4 presents the respective histograms. 128 t. mazur, m. rucki, y. gutsalenko fig. 4 histograms of the measurement results of 50 repetitions in the series no. 9, 17, and 18 knowing that the distribution of form deviation δ measurement results is normal, it is possible to apply the student-fisher parameters to the smaller series of 5, 10 and 25 repetitions. thus, confidence interval ci can be calculated as follows: ci = tα,n-1 ×sn (1) where: tα,n-1 – student-fisher distribution quantile, n – number of the repetitions in series, sn – standard deviation. assuming confidence level p = 0.99, the respective quantile value for 5 repetitions is tα,n-1 = 4.604, for 10 repetitions tα,n-1 = 3.250, and for 25 repetitions tα,n-1 = 2.797 [31]. 4.1. effect of scanning velocity on form deviation in tables 2-5, there are collected measurement results for form deviation δ obtained at different scanning velocities vs and probe deflection pd, together with the time of measurement t. the series numbers correspond with the ones specified above in table 1. table 4 contains results for probe deflection pd = 0.15 mm only for one series with 5 repetitions. table 2 form deviation δ obtained at vs = 1 mm/s (series 1-4) pd= 0.15 mm pd= 0.2 mm pd= 0.3 mm pd= 0.4 mm repetition no. δ [mm] t [s] δ [mm] t [s] δ [mm] t [s] δ[mm] t [s] 1 0.118 198 0.115 179 0.114 156 0.112 145 2 0.120 203 0.116 181 0.113 155 0.112 145 3 0.116 205 0.115 177 0.112 154 0.111 146 4 0.118 201 0.116 177 0.113 155 0.111 147 5 0.119 199 0.118 178 0.115 154 0.112 146 mean value 0.118 201.2 0.116 178.4 0.113 154.8 0.112 145.8 standard deviation sn 0.001 2.864 0.001 1.673 0.001 0.837 0.001 0.837 confidence interval 0.007 13.18 0.006 7.7 0.005 3.85 0.003 3.85 accuracy analysis of the curved profile measurement with cmm: a case study 129 table 3 form deviation δ obtained at vs = 2 mm/s (series 5-8) pd= 0.15 mm pd= 0.2 mm pd= 0.3 mm pd= 0.4 mm repetition no. δ [mm] t [s] δ [mm] t [s] δ [mm] t [s] δ[mm] t [s] 1 0.119 95 0.115 84 0.114 75 0.112 75 2 0.116 97 0.115 84 0.116 77 0.112 75 3 0.115 95 0.115 84 0.112 77 0.114 73 4 0.116 97 0.115 84 0.113 77 0.112 74 5 0.118 95 0.114 84 0.118 78 0.114 74 mean value 0.117 95.8 0.115 84.0 0.115 76.8 0.113 74.2 standard deviation sn 0.002 1.095 0.000 0.000 0.002 1.095 0.001 0.837 confidence interval 0.008 5.04 0.002 0.000 0.011 5.04 0.005 3.85 table 4 form deviation δ obtained at vs = 3 mm/s (series 9-12) pd= 0.15 mm pd= 0.2 mm pd= 0.3 mm pd= 0.4 mm repetition no. δ [mm] t [s] δ [mm] t [s] δ [mm] t [s] δ[mm] t [s] 1 0.120 61 0.117 56 0.116 47 0.113 47 2 0.117 62 0.116 56 0.116 50 0.110 47 3 0.117 63 0.113 57 0.113 50 0.119 49 4 0.116 62 0.115 57 0.115 51 0.114 50 5 0.116 61 0.113 55 0.116 50 0.112 49 mean value 0.117 61.8 0.115 56.2 0.115 49.6 0.114 48.4 standard deviation sn 0.002 0.837 0.002 0.837 0.001 1.517 0.003 1.342 confidence interval 0.008 3.85 0.008 3.85 0.006 6.98 0.015 6.18 table 5 form deviation δ obtained at vs = 1 mm/s (series 1-4) pd= 0.15 mm pd= 0.2 mm pd= 0.3 mm pd= 0.4 mm repetition no. δ [mm] t [s] δ [mm] t [s] δ [mm] t [s] δ[mm] t [s] 1 0.120 50 0.113 46 0.115 42 0.121 38 2 0.118 49 0.115 46 0.117 40 0.114 38 3 0.117 49 0.117 46 0.115 41 0.114 37 4 0.115 47 0.117 46 0.117 41 0.113 37 5 0.117 48 0.117 46 0.111 41 0.118 38 mean value 0.117 48.6 0.116 46.0 0.115 41.0 0.116 37.6 standard deviation sn 0.002 1.140 0.002 0.000 0.002 0.707 0.003 0.548 confidence interval 0.008 5.25 0.008 0.000 0.011 3.26 0.016 2.52 from the above results, it can be seen that larger permissible probe deflection pd enabled 22-28% shortening of the measurement time at each scanning speed. however, it caused distinguishable 1-6% reduction of the obtained result of form error δ. graph in fig.5 shows how this reduction differs for different scanning speed values vs. from fig. 5 it can be concluded, that at higher scanning speeds, the influence of probe deflection is smaller. for pd≤ 0.2 mm, speed-dependent differences in obtained form deviations δ lay below e0,mpe = (1.7+3l/1000) μm for the used cmm. notably, this range of the deflection values ensured insignificant effect of scanning speed vs on the form deviation results. for each pd = 0.15 and 0.2 mm, differences between obtained δ at various vs were 1 μm. larger probe deflections led to widening of the results span, which indicated 130 t. mazur, m. rucki, y. gutsalenko increased uncertainty of the measurement. due to this observation, we are against application of pd> 0.2 mm for this sort of measurement. fig. 5 form deviation δ obtained at different scanning speeds vs and different probe deflections pd calculations of true scanning velocity vs' revealed substantial differences between them and set values vs. values of vs' were determined from the known length of measured contour lc and automatically registered time of each measurement. table 6 presents the values and percentage differences between them. table 6 true values of scanning speed vs' related to the nominal ones vs for different probe deflections pd (sampling step was 0.2 mm) pd= 0.15 mm pd= 0.2 mm pd= 0.3 mm pd= 0.4 mm vs [mm/s] vs' [mm/s] percen tage vs' [mm/s] percen tage vs' [mm/s] percen tage vs' [mm/s] percen tage 1 0.120 50 0.113 46 0.115 42 0.121 38 2 0.118 49 0.115 46 0.117 40 0.114 38 3 0.117 49 0.117 46 0.115 41 0.114 37 4 0.008 5.25 0.008 0.000 0.011 3.26 0.016 2.52 it should be noted that the true scanning speed never reached its nominal value, and smaller probe deflections reduced its value by almost 40%. considering previous conclusion that the measurement should not be performed with pd> 0.2 mm, this finding becomes extremely important. in the case of 100% inspection of large lots of the cams similar to the one investigated, measurements will take 30-40% longer time than it would be expected from the nominal scanning speed. in the flexible manufacturing systems working in the frames of industry 4.0 concept [32], prolonged inspection time may become an issue. 4.2. results of uncertainty evaluation in the type a uncertainty evaluation, three series of the measurement results were used, 50 repetitions each. it can be assumed that the standard uncertainty is approximately equal to the standard deviation u(x) ≈sn, and the coverage factor for the level of confidence p = 99% can be kp = 2.576 [28]. as described above, series no. 9 followed repeatability accuracy analysis of the curved profile measurement with cmm: a case study 131 conditions, but not as strictly as series no. 17. on the other hand, series no. 18 had a larger number of probing points, close to that of the reference file derived from the technical drawing. table 7 presents values of the calculated standard and expanded uncertainties. examples of the measured profiles with emphasized δmax are shown in figs 6 and 7. table 7 uncertainty estimation based on the series with 50 repetitions series no. 9 17 18  [mm] 0.1151 0.1137 0.1152 sn ≈ u(x) 0.00158 0.00195 0.00142 u0.99 = kp× u(x) 0.004 0.005 0.004 fig. 6 example of the measured profile with the result of δ = 0.113 mm; sampling step s = 0.2 mm, 660 probing points the results presented in table 7 appear a little unexpected. a higher degree of conformity is between series no. 9 and 18 than between no. 17 and any of two others, despite its conditions were “in-between” (same number of probing points as no. 9 and time of experiment closer to no. 18). nevertheless, it should be kept in mind that the differences between both mean values  and expanded uncertainties u0.99 for three series lay below e0,mpe. hence, it may be stated that the influence of sampling is negligibly small when estimating the measurement uncertainty of cmm measurement of the cam profile. 132 t. mazur, m. rucki, y. gutsalenko fig. 7 view of the area with the largest identified form deviation: a) δmax = 0.113 mm after scanning with step s = 0.2 mm, b) δmax = 0.116 mm after scanning with step s = 0.05 mm this conclusion is confirmed by a detailed analysis of the maximal deviation localization on the cam profile. irrespective of what sampling step, scanning velocity or probe deflection was applied ,δmax was identified in the same area. moreover, it should be noted that a large number of probing points effects in increase of the processing time and, hence, the measurement lasts longer. experimental evaluation of the uncertainty demonstrated that it is unnecessary, and the results with similar uncertainty may be obtained at a larger sampling step in a shorter time. 4.3. repeatability in short measurement series table 8 presents the results obtained subsequently for the series no. 9, as described in section 3.3. the first column presents the number of each measurement in the series 9, while the second one is the number of a group, as follows: 5 repetitions in the 1st group, 5 in the 2nd and 3rd, respectively, 10 repetitions in the 4th group and 25 repetitions in the 5th group. for each group, respective standard deviations sn and confidence intervals ci were calculated both for measured deviation δ and for measurement time t. in the last row, overall statistics is added for the entire series no. 9 for t0.01,49=2.6802. fig. 8 shows the histograms of the results with approximated distribution curves for the group no. 5 and for overall statistics. table 8 statistics for form deviation δ and scanning time t obtained at vs = 3 mm/s with sampling step s = 0.2 mm and pd= 0.15 mm in short measurement groups group no.  [mm] sn ci t [s] sn ci 1 0.1172 0.00164 0.0076 61.8 0.837 3.85 2 0.1150 0.00255 0.0117 65.2 0.447 2.06 3 0.1150 0.00173 0.0080 62.6 0.548 2.52 4 0.1153 0.00125 0.0041 60.4 1.35 4.39 5 0.1146 0.00115 0.0032 62.2 1.165 3.26 overall 0.1151 0.00158 0.0042 62.2 1.646 4.41 accuracy analysis of the curved profile measurement with cmm: a case study 133 fig. 8 distribution of the results in short measurement series: a) deviation δ and b) measurement time t interestingly, the first three groups that would be expected to be similar, revealed the following statistics: mean values were similar for groups 2 and 3, with 2.2 μm higher for group 1, but the respective confidence intervals were similar for groups 1 and 3, with ci for the group 2 almost 50% wider. for the groups with 10 and 25 repetitions, confidence intervals reduced substantially, down to 0.0041 and 0.0032, respectively. notably, mean value for group 4 was slightly higher than that for groups 2 and 3, while for group 5 it was slightly lower. the difference was smaller than 0.4 μm, significantly below the maximum permissible error e0,mpe = (1.7+3l/1000) μm for the cmm used in experiments. 5. conclusions the results of the experimental research studies demonstrated that increased probe deflections pd reduced the values of measured form deviation. additionally, pd higher than 0.2 mm increased the results dispersion. it was found also that the measurement uncertainty did not depend on the scanning sampling step from 0.05 to 0.2 mm, but it should be noted that a smaller step increased the measurement time.    134 t. mazur, m. rucki, y. gutsalenko noteworthy, the true measurement time was for 30-40% longer than that declared by nominal scanning velocity. these characteristics must be taken into consideration when projecting the batch inspection procedures, especially when 100% of parts must be measured. moreover, there is no necessity in the increased number of repetitions since even a small number of repetitions gave similar mean results with differences close to the maximum permissible error of the cmm. the most important conclusion is that the highest value of form deviation was identified in the same location irrespective of the applied measurement parameters. expanded uncertainty of form deviation measurement at the level of confidence p = 99% was u0.99 = 0.005 mm, less than 10% of the measured tolerance. this value proved that the chosen cmm as well as the inspection methodology were appropriate for the measurement of the given curved profile. acknowledgements: the paper is a part of the research program conducted at the faculty of mechanical engineering, kazimierz pulaski university of 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dimensional deviation, flank wear, surface roughness and material removal rate in dry turning of c45 steel aleksandar milosevic1, goran simunovic2, zeljko kanovic1, katica simunovic2, zeljko santosi1, mario sokac1, djordje vukelic1 1university of novi sad, faculty of technical sciences, serbia 2university of slavonski brod, mechanical engineering faculty in slavonski brod, croatia orcid ids: aleksandar milosevic https://orcid.org/0000-0002-1596-8159 goran simunovic https://orcid.org/0000-0002-7159-2627 zeljko kanovic https://orcid.org/0000-0003-1456-1135 katica simunovic https://orcid.org/0000-0001-5748-7110 zeljko santosi https://orcid.org/0000-0003-3975-0864 mario sokac https://orcid.org/0000-0002-6245-8629 djordje vukelic https://orcid.org/0000-0003-2420-6778 abstract. the study investigated the turning of c45 steel in a dry environment. the input parameters that were varied were cutting speed, feed, depth of cut, corner radius and insert type. the experimental investigations were carried out according to a custom experimental design using the d-optimality criterion. the measured output parameters were dimensional deviation, flank wear and surface roughness, while the material removal rate was calculated. a detailed analysis and evaluation of the effects of the input parameters on the output parameters was carried out. the model was diagnosed and appropriate regression equations were established. based on the obtained regression equations, multi-objective optimisation was performed using particle swarm optimisation. the objective function was to simultaneously minimise dimensional deviation, flank wear and surface roughness and maximise material removal rate. the optimisation was carried out for different weighting coefficients of each output function for different production requirements. the obtained models and optimal values were verified by additional confirmation experiments. key words: dry turning, dimensional deviation, flank wear, surface roughness received: april 03, 2024 / accepted may 12, 2024 corresponding author: goran simunovic university of slavonski brod, mechanical engineering faculty in slavonski brod, trg ivane brlic mazuranic 2, slavonski brod, croatia e-mail: gsimunovic@unisb.hr https://orcid.org/0000-0002-1596-8159 https://orcid.org/0000-0002-7159-2627 https://orcid.org/0000-0003-1456-1135 https://orcid.org/0000-0001-5748-7110 https://orcid.org/0000-0003-3975-0864 https://orcid.org/0000-0002-6245-8629 https://orcid.org/0000-0003-2420-6778 548 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. 1. introduction turning is a machining operation that can be carried out under dry or wet conditions and using alternative techniques such as minimum coolant and lubricant quantities, cryogenic cooling, etc. [1,2]. all previous methods of cooling and lubrication have their advantages and disadvantages. the main problems with dry turning are short tool life and poor surface quality [3]. dry turning is associated with high temperatures and is therefore not suitable for machining difficult-to-cut materials [4]. the positive aspects of dry turning are that no cooling lubricant is used and it is therefore environmentally friendly. there are also no additional costs for cutting fluid, equipment, disposal, recycling, reuse, etc. therefore, dry turning can be used when the materials are characterized by good machinability but the problems mentioned above need to be solved. this can be achieved through the use of stable machines, reliable locating and clamping, the use of coated inserts and, of course, the obligatory optimisation of all the machining parameters entered [5]. c45 steel is a medium carbon steel with a wide range of industrial applications. it is used for the manufacture of various products such as gears, shafts, axles, equipment, tools, etc. it can be used for the manufacture of complex products that need to be machined with high accuracy and precision. it is characterized by good hardness and wear resistance as well as high tensile strength, which makes it the ideal choice for products that need to withstand strong forces. it is not resistant to moisture and aggressive chemicals, so it has low corrosion resistance. it has good machinability when turned, i.e. it does not belong to the group of difficult-to-machine materials and can therefore be turned economically under certain conditions, even in dry conditions. in the previous period, a large number of researches dealt with the process of turning c45 steel workpieces. grzesik [6] proposed analytical models for estimating the average and maximum temperatures at the tool–chip interface. the models for a three-layer coating agreed well with both the experimental data and the finite element (fe) predictions. piska et al. [7] presented wear morphology. the decohesion between the phases and the different size of the phases led to a weakening of the structural integrity, rapid wear of the coating and a short tool life. stachurski et al. [8] evaluated the influence of cutting parameters on surface roughness. wiper inserts led to a reduction in surface roughness. increasing the feed and corner radius contributed to an increase in surface roughness and cutting force. nieslony et al. [9] presented the influence of constitutive model parameters on the results of fem-based modelling. a sensitivity analysis of the material flow stress in the power constitutive model was performed. sergeto et al. [10] used several digital signals from sensor monitoring to identify and monitor the chip shape. different neural network training algorithms were compared as a decision support system. michal et al. [11] investigated the influence of cutting speed, feed and depth of cut on surface roughness using regression analysis. the results showed that feed and cutting speed had a dominant influence on surface roughness. selvam and senthil [12] studied and optimised the effect of corner radius and turning parameters on surface roughness using the taguchi method and genetic algorithm. corner radius had a greater influence on the surface roughness followed by feed and spindle speed and depth of cut. horvath and lukacs [13] determined prediction models for estimating the cutting force components based on the undeformed chip cross-section. the results showed that the influence of the effective length of the tool cutting edge on specific cutting force components is not negligible. necpal et al. [14] compared experimentally determined cutting forces and those predicted by fe simulation. stress and comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 549 temperature predictions were made for certain values of cutting speed, feed and depth of cut. vereschaka et al. [15] investigated the mechanisms of the formation of longitudinal cracks and delamination in coatings on rake and flank surfaces. the properties of the coatings and the nature of their failures were presented. vereschaka et al. [16] investigated two multilayer nanostructured coatings with the same thickness, elemental composition and basic mechanical properties but different sub-nanolayer thicknesses. a coating with thicker sub-nanolayers was more prone to delamination. a tool with a coating characterized by a thinner sub-nanolayer thickness had the longest tool life. klocke et al. [17] developed mathematical relationships considering strain hardening, dynamic forces and shear strain rate. the external and internal friction during the chip formation process was analysed. dragicevic et al. [18] presented an application of a combined taguchi and fuzzy logic approach for the optimisation and analysis of surface roughness. feed and cutting speed were the main parameters influencing the variation of surface roughness. gjelaj et al. [19] optimised cutting speed, feed, depth of cut as a function of tool path length and cutting force using a genetic algorithm. zmarzły [20] analysed the influence of feed and cutting speed on surface roughness, roundness and cylindricity. increasing the feed causes an increase in the deviation of cylindricity. the cutting speed had a negligible effect on the shape of the cylindrical workpieces. zajac et al. [21] focused on predicting the durability of cutting materials using the taylor model. optimal cutting speeds were determined in order to achieve maximum tool life. usman et al. [22] investigated the effects of cutting speed, feed, depth of cut and ultrasonic vibration amplitude on surface properties using surface function indices. the results showed that the surface function indices obtained in vibration-assisted turning were significantly higher than those obtained in conventional turning. mikołajczyk [23] presented the investigation of the minimal uncut chip thickness. the main cutting edge angle had a significant influence on the process, especially in the range of small uncut chip thickness range. titu et al. [24] simulated the effect of rake and clearance angles on temperature distribution, effective stresses and cutting forces using fe analysis. oleksik et al. [25] analysed vibrations using fourier transform methods. the influence of vibrations on surface roughness was investigated for two design variants of cutting tools. abidi [26] investigated the flank wear of ceramic cutting tools as a function of cutting speed and feed. the results show that the flank wear was more strongly influenced by the cutting speed than by the feed. zaida et al. [27] proposed a method for monitoring flank wear based on the analysis of vibration signals. the method enables estimation of flank wear based on the wavelet transform. ivchenko et al. [28] developed a method for determining the intensity of tool wear by analysing the cutting edge to determine the change in forces. the method was developed based on a simulation of the machining process, taking into account the tribological interaction of the tool and workpieces as well as the corner radius. kovalčík et al. [29] proposed a mathematical cutting force model based on the cutting speed and the coating correction factor as well as the cutting edge geometry. jurko et al. [30] performed optimisation of flank wear and diameter deviation by using grey relation analysis (gra) in combination with the taguchi method. cutting speed was an important factor influencing the multiple responses on the effect of insert wear on average diameter deviation. vukelic et al. [31] modelled surface roughness in finish turning as a function of corner radius, approach angle, rake angle and inclination angle. the surface roughness improved with increasing corner radius, increasing approach angle, increasing rake angle and decreasing inclination angle. niemczewska-wojcik et al. [32] focused on the surface topography of workpieces under 550 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. dry and wet conditions. the mineral oil-based lubricant had a positive influence on tool wear and surface quality. kuruc et al. [33] analysed the plastic deformation and chip compression. plastic deformation decreased with increasing feed and cutting speed. chip compression decreased with increasing cutting speed. moravčíková et al. [34] presented the influence of the heat treatment (soft annealing, normalisation, hardening, quenching and tempering) of c45 steel on roughness, cylindricity and circularity. the heat treatment had a major influence on all output parameters. slusarczyk and franczyk [35] presented a method for determining the feed and tangential components of the cutting force. the method was based on the assumptions that the tool is perfectly sharp, that the chip formation process is continuous, that the chip thickness is constant and that there are no heat losses. in general, the main disadvantage of experimental research is the high time expenditure and the associated costs, therefore the modelling of the turning process and its optimisation is an increasing topic. in previous research, the most common input parameters were cutting speed, feed rate, depth of cut, geometric parameters of the cutting tool, type of coating, etc., while the most common output parameters were surface roughness, wear mechanisms of the cutting tool, cutting forces, tool life, material removal rate, etc. in order to reduce the number of experiments and significantly reduce costs, modelling and optimisation are carried out. the problem with modelling and optimisation is the choice of method, as there are no universal and generally accepted rules. only confirmation of unknown values of input parameters and comparison with measured values can influence the validity of the applied modelling and optimisation method. if the process is modelled and optimised correctly and acceptable errors are achieved, costs and time are reduced. the general validity and independence of the results from the knowledge and experience of the technologists is also greater. in contrast to the previous work, this study shows the influence of a larger number of input parameters on a larger number of output parameters in dry turning of c45 steel. so far, the comprehensive evaluation of the influence of cutting speed, feed, depth of cut, corner radius and insert type on surface roughness, dimensional deviation, flank wear and the material removal rate has not been performed. the influence of the interactions of the input parameters on the output parameters was also investigated. a detailed analysis of the input parameters and an evaluation of their effect on the output parameters of the process were carried out. in addition, the modelling and optimisation of the process and the verification of the solutions obtained were carried out. 2. research methodology the research methodology is shown in fig. 1. the tests were carried out on samples of c45 steel with the following properties: young's modulus 205 gpa, hardness 255, tensile strength 560 mpa and density 7.85 g/cm3. the chemical composition of the steel is as follows: 0.42–0.50 % c, max 0.40 % si, 0.50–0.80 % mn, max 0.40 % ni, max 0.030 % p, max 0.035 % s, max 0.40 % cr and max 0.10 % mo. dry turning was carried out on a cnc lathe. a three-jaw chuck and rotary centre were used for locating and clamping. based on the recommendations of the cutting insert manufacturer and in accordance with the characteristics of the workpiece material, the geometry of the workpiece, the type of sequence, the characteristics of the technological equipment and stability, the machining parameters and the cutting inserts were selected. comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 551 both types of inserts (standard and wiper) with the following characteristics were used for the study: rhombic shape (80°), thickness 4.8 mm, diameter of the inscribed circle 12.7 mm, effective length of the cutting edge 11.7 mm and fixing hole 5.2 mm. the inserts were coated using cvd coating technology (ticn+al2o3+tin coating of 5 μm thickness). the turning parameters were as follows: cutting speed vc = 300–400 (mm/min), feed f = 0.1– 0.25 (mm/rev) and depth of cut ap = 2–3 (mm). fig. 1 research methodology after the experimental investigation, the output parameters were measured and calculated. the measurements of dimensions, surface roughness and flank wear were carried out under controlled microclimatic conditions. the measured output parameters were dimensional deviation (δd), surface roughness (ra) and flank wear (vb) while the material removal rate (mrr) was calculated. the surface roughness was measured using a talysurf measuring device. the measurements were taken in the feed direction with a sampling length of 0.8 mm and an evaluation length of 4 mm using a gaussian filter. the diameter of the workpiece before and after turning was measured using a crysta-apex s 9106 coordinate measuring machine, resolution 0.0001 mm and accuracy of 1.7 + measuring length/1000 μm, with a probe in the form of a ruby ball with a diameter of 5 mm. the large diameter of the probe was chosen to reduce the influence of surface roughness on the relative error of the diameter change. the diameter change represents the difference between the nominal diameter and the measured diameter of the machined surface. the measurement of flank wear was carried out with a leitz orthoplan light optical microscope using the imagej software. 552 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. the fourth output parameter, the material removal rate, was calculated using the equation: 𝑀𝑅𝑅 = 𝑣𝑐 ∙ 𝑎𝑝 ∙ 𝑓 (1) after measurement and calculation, the process was analysed and modelled. the experimental investigations were carried out according to the custom experimental design, which enables the creation of an optimal experimental design with a lower number of trials (minimising costs) and the achievement of reliable and relevant results. the customised experimental design was created using the d-optimal criterion. the experimental design was determined based on the previously defined input and output parameters. in determining the experimental design, an a priori model was defined that includes main effects, their quadratic effects and two-factor interactions as expected influences of the input parameters on the output parameters. the number of experimental points was chosen depending on the complexity of the process and the desire for an accurate estimation of the output parameters of the model. in evaluating the experimental design, it was determined that 72 trials had sufficient statistical power to determine the effects of the input parameters. the experimental design was randomised to eliminate system bias and ensure the reliability of the results. after modelling phase, a multi-objective optimisation was conducted using particle swarm optimisation (pso) algorithm. pso is a global optimisation algorithm suitable for multi-objective optimum search. it is based on behaviour imitation of living beings moving in groups (swarms, flocks, …) and operates with the set of individuals (particles), called swarm. in simple words, particles are moving throughout the search space searching for the point that achieves the best value of objective function. the search is iterative, and is conducted until the defined stopping criterion is fulfilled. each particle in the swarm is defined by its “position“ y, which represents the vector of variables in the optimisation problem, or „coordinates“ in the search space, and by its current “velocity“ v. in other words, particle’s position is the potential solution of the optimisation problem, containing the values of variables, and the velocity determines the way of modification of these values in order to obtain global optimum. in this study, the position of the particle (vector y) is defined by values of input parameters, namely cutting speed vc, depth of cut ap, feed f and corner radius r. also, during the search process, each particle is capable of memorizing its “personal“ best position p, as well as the “global“ best position g achieved by the entire swarm, which is the information “shared“ between all particles. the particle is basically guided by three velocity components: inertia (forces the particle to move in the same direction as in previous iteration), cognitive component (the particle is motivated by its personal experience and guided by its personal best position p), and social component (guides the particle according to global knowledge of the swarm, i.e. towards the global best position g). accordingly, the particle’s position is updated using the following expression: 𝐯(𝑘+1) = 𝑤 ∙ 𝐯(𝑘) + 𝑐𝑝 ∙ 𝑟𝑝(𝑘) ∙ (𝐩(𝑘) − 𝐲(𝑘)) + 𝑐𝑔 ∙ 𝑟𝑔(𝑘) ∙ (𝐠(𝑘) − 𝐲(𝑘)) (2) 𝐲(𝑘+1) = 𝐲(𝑘) + 𝐯(𝑘+1) (3) where k is the number of the current iteration, factors w, cp and cg determine the influence of inertial, cognitive and social components respectively, and factors rp and rg are random numbers uniformly distributed in range [0,1], providing the stochastic nature of the algorithm. in each iteration, personal best position p and global best position g are updated according to the values of objective function. the initial positions of particles are chosen randomly, within the boundaries of the search space. the initial velocities are either set to comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 553 zero or chosen randomly, depending on the variant of the algorithm. the iterative process is repeated until the specified stopping criterion is fulfilled, which is mainly the total number of iterations or, alternatively, the number of successive iterations in which the value of objective function does not change (stalls). finally, the models, regression equations and optimum values of the input and output parameters were validated by additional confirmation tests. 3. results 3.1. experimental research the d-optimality criterion was used for the experimental design, as it ensures an accurate estimation of the effects of the input parameters. the optimum number of experimental points was selected by analysing experimental designs with different numbers of trials. the comparison was carried out using the compare designs function for designs with 32, 48, 64, 72 and 96 experimental points. the minimum number of experimental points required, the performance and characteristics of experimental designs as well as the availability of resources, time and costs of experiment execution were then taken into account using the power analysis. the power analysis makes it possible to determine the ability of the experiment to recognise real effects, i.e. to identify the existence of effects. in the power analysis, the following reference values were defined for evaluating the power of the model: significance level 0.05 and anticipated root mean square error (rmse) 1. the obtained evaluation parameters of different experimental designs (ranging from 0 to 1 and referring to the main effects, their quadratic effects and two-factor interactions) show that the experimental designs for 72 and 96 trials have the statistical power to detect real effects (fig. 2). fig. 2 power analysis 554 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. in view of the small differences in power between these two experimental designs, the design with 72 test points was chosen as it consumes significantly fewer resources. of the total 72 trials, 63 are different trial points and 9 are replicates. the selected levels of the input parameters are listed in table 1. table 1 input parameters and the selected levels input parameter level 1 (minimum) level 2 (middle) level 3 (middle) level 4 (maximum) cutting speed vc (mm/min) 300 350 / 400 feed f (mm/rev) 0.1 0.15 0.2 0.25 depth of cut ap (mm) 2 / / 3 corner radius r (mm) 0.4 0.8 1.2 1.6 insert type standard / / wiper the results of the measurement and calculation of the output parameters is shown in table 2. table 2 results of experiments no. vc (mm/min) ap (mm) f (mm/rev) r (mm) insert type δd (mm) vb (mm) ra (µm) mrr (mm3/min) 1 300 2 0.2 1.2 wiper 0.054 0.102 0.659 120 2 300 3 0.25 1.6 standard 0.111 0.130 2.768 225 3 300 2 0.25 1.6 wiper 0.106 0.097 1.294 150 4 400 3 0.25 0.4 standard 0.289 0.211 7.236 300 5 400 3 0.15 0.8 standard 0.052 0.184 1.303 180 6 400 2 0.25 1.6 wiper 0.111 0.141 1.354 200 7 300 3 0.1 1.6 wiper 0.017 0.090 0.210 90 8 350 2 0.2 0.4 standard 0.178 0.162 4.449 140 9 300 2 0.1 1.6 standard 0.017 0.111 0.435 60 10 300 2 0.15 1.6 standard 0.039 0.115 0.978 90 11 400 3 0.2 1.6 standard 0.074 0.170 1.853 240 12 300 2 0.25 1.6 standard 0.109 0.123 2.717 150 13 300 2 0.1 0.8 standard 0.022 0.129 0.543 60 14 400 3 0.25 0.4 standard 0.291 0.213 7.237 300 15 400 2 0.25 0.4 standard 0.284 0.205 7.109 200 16 400 3 0.25 1.6 wiper 0.112 0.147 1.379 300 17 400 3 0.1 0.4 wiper 0.045 0.172 0.643 120 18 400 3 0.2 0.4 wiper 0.182 0.180 2.573 240 19 300 3 0.1 1.6 standard 0.019 0.118 0.442 90 20 300 2 0.25 0.4 wiper 0.266 0.134 3.773 150 21 400 2 0.1 0.4 standard 0.046 0.192 1.138 80 22 300 3 0.15 1.2 wiper 0.031 0.104 0.377 135 23 400 3 0.25 0.4 wiper 0.284 0.184 4.020 300 24 300 2 0.1 1.6 wiper 0.017 0.084 0.207 60 25 400 3 0.1 1.2 standard 0.015 0.171 0.386 120 26 350 2 0.25 1.2 standard 0.093 0.141 2.317 175 27 300 2 0.25 0.4 standard 0.272 0.160 6.792 150 28 400 2 0.25 0.4 wiper 0.279 0.178 3.950 200 29 300 2 0.1 0.4 standard 0.040 0.148 1.087 60 30 300 3 0.2 0.4 wiper 0.174 0.136 2.460 180 31 350 2 0.1 1.2 standard 0.015 0.129 0.371 70 comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 555 32 350 3 0.25 1.6 wiper 0.111 0.114 1.348 262.5 33 400 2 0.25 1.6 standard 0.114 0.168 2.844 200 34 400 2 0.15 0.4 wiper 0.100 0.170 1.422 120 35 400 2 0.1 1.2 wiper 0.014 0.138 0.172 80 36 300 3 0.25 0.4 standard 0.277 0.167 6.919 225 37 350 3 0.25 0.8 standard 0.142 0.155 3.539 262 38 300 3 0.1 0.4 standard 0.044 0.154 1.107 90 39 300 3 0.1 0.4 standard 0.043 0.153 1.106 90 40 300 3 0.2 1.2 standard 0.059 0.135 1.476 180 41 400 2 0.1 0.4 wiper 0.045 0.165 0.632 80 42 300 3 0.1 0.4 wiper 0.043 0.127 0.615 90 43 400 2 0.2 0.8 standard 0.091 0.182 2.275 160 44 300 3 0.25 1.6 wiper 0.108 0.103 1.318 225 45 350 2 0.25 0.4 wiper 0.273 0.142 3.862 175 46 350 2 0.15 0.8 wiper 0.048 0.117 0.624 105 47 400 3 0.25 1.2 wiper 0.091 0.157 1.096 300 48 300 3 0.25 1.6 standard 0.112 0.131 2.766 225 49 400 2 0.25 1.6 standard 0.115 0.169 2.843 200 50 400 3 0.1 0.8 wiper 0.022 0.153 0.289 120 51 400 2 0.15 1.6 standard 0.041 0.159 1.024 120 52 400 3 0.1 1.6 standard 0.019 0.161 0.463 120 53 350 2 0.2 1.6 wiper 0.070 0.104 0.848 140 54 350 3 0.1 1.6 standard 0.019 0.126 0.453 105 55 400 3 0.25 1.6 standard 0.116 0.174 2.895 300 56 400 3 0.1 1.6 wiper 0.018 0.135 0.221 120 57 400 2 0.1 1.6 wiper 0.018 0.128 0.217 80 58 400 2 0.25 0.8 wiper 0.137 0.160 1.777 200 59 400 3 0.1 0.4 standard 0.044 0.196 1.157 120 60 400 2 0.1 1.6 wiper 0.017 0.126 0.216 80 61 350 3 0.15 0.4 standard 0.102 0.164 2.548 157.5 62 300 2 0.1 0.4 wiper 0.043 0.121 0.604 60 63 300 2 0.1 0.4 wiper 0.042 0.119 0.602 60 64 300 2 0.25 0.4 standard 0.271 0.158 6.790 150 65 350 3 0.1 0.4 wiper 0.044 0.136 0.629 105 66 400 3 0.15 1.6 wiper 0.041 0.139 0.496 180 67 300 2 0.25 1.6 wiper 0.108 0.099 1.292 150 68 400 2 0.1 1.6 standard 0.018 0.155 0.455 80 69 300 3 0.25 0.4 wiper 0.271 0.140 3.844 225 70 400 2 0.1 0.4 standard 0.047 0.193 1.140 80 71 300 3 0.1 1.6 wiper 0.019 0.091 0.212 90 72 300 3 0.25 0.8 wiper 0.133 0.121 1.730 225 3.2. process analysis and modelling the statistical analysis of the experimental results was carried out using sas jmp 14 software. the statistical analysis was performed separately for each output parameter in order to better interpret the results and determine the specific effects of each input parameter. in the statistical analysis, a model was chosen that includes main effects, their quadratic effects and two-factor interactions. the selected regression model consists of 20 terms and defines the variances of δd, vb and ra using the minimum number of required variations. given the irregularities in the distribution of the data and the excessive deviations between the values, a logarithmic transformation (ln) of δd and ra was performed to facilitate the application and interpretation of the statistical method. 556 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. fig. 3 shows the evaluation of the quality of the fitting the regression models to the experimental data (actual) by comparing the dependence of the actual on the predicted output parameters. the resulting plots show a good predictive power of the model as the data points follow a diagonal line, which is an ideal arrangement when the actual output parameters are identical to the predicted output parameters. the proximity of the points to the diagonal indicates a narrow confidence interval, i.e. a high degree of accuracy in the model predictions. fig. 3 actual by predicted plot table 3 shows a summary of the fit, which provides key statistical data on the quality of the selected regression models for each output parameter. the coefficient of determination (rsquare) values for all output parameters are close to 1, indicating a high degree of fit between the model and the data. the coefficients of determination mean that more than 99% of the variability of the responses is explained by the specified input parameters. the difference between the values of rsquare and rsquare adj is small, which means that additional parameters in the model do not contribute to improving the explanation of variability. low values of rmse (0.039579, 0.001286 and 0.028794) indicate that the difference between the actual and predicted values is low. table 3 summary of fit parameter ln δd vb ln ra rsquare 0.998399 0.998371 0.999257 rsquare adj 0.998224 0.998193 0.999188 root mean square error 0.039579 0.001286 0.028794 mean of response -2.71538 0.145361 0.181747 table 4 shows the analysis of variance of the regression models, which provides an assessment of the effectiveness of the selected models in explaining the variability of the data. low values of prob>f (<0.0001) suggest that the models are significant, i.e., there is a real influence of input parameters, that is, they indicate the presence of at least one significant effect in the model. table 5 shows the lack of fit of the model to the data. since the prob>f values for all three output parameters are greater than 0.05, it can be concluded that there is no statistically significant lack of fit. in other words, the models fit the data well and there is no need for further adjustments. comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 557 table 4 analysis of variance output parameter source df sum of squares mean square f ratio prob > f ln δd model 7 62.538877 8.93413 5703.154 <.0001 error 64 0.100258 0.00157 c. total 71 62.639134 vb model 7 0.06487678 0.009268 5604.625 <.0001 error 64 0.00010583 1.65410-6 c. total 71 0.06498261 ln ra model 6 72.433757 12.0723 14561.24 <.0001 error 65 0.053890 0.000829 c. total 71 72.487647 table 5 lack of fit output parameter source df sum of squares mean square f ratio prob > f max rsq ln δd lack of fit 54 0.09138242 0.001692 1.9068 0.1346 0.9999 pure error 10 0.00887509 0.000888 total error 64 0.10025751 vb lack of fit 54 0.00009333 1.7284e-6 1.3827 0.3008 0.9998 pure error 10 0.00001250 1.25e-6 total error 64 0.00010583 ln ra lack of fit 23 0.02598929 0.001130 1.7010 0.0665 0.9996 pure error 42 0.02790028 0.000664 total error 65 0.05388956 table 6 shows the summary of effects of the model sorted by statistical significance for each output parameter. high log worth values indicate a larger effect. regarding the regression model for dimensional deviation, the p-values indicate that the linear effects of all input parameters and the quadratic effects of corner radius and feed are significant, while the interaction terms are not significant. statistically significant terms for flank wear are the linear effects of all input parameters and the quadratic effects of cutting speed and corner radius. for surface roughness, the main effects of feed, corner radius and insert type, the quadratic effects of corner radius and feed and the interaction of the corner radius with insert type are statistically significant. table 6 effect summary ln δd vb ln ra source log worth p value source log worth p value source log worth p value f(0.1,0.25) 86.957 0.00000 vc(300,400) 79.370 0.00000 f(0.1,0.25) 96.952 0.00000 r(0.4,1.6) 67.751 0.00000 r(0.4,1.6) 72.975 0.00000 r(0.4,1.6) 79.872 0.00000 r×r 46.141 0.00000 insert 67.429 0.00000 insert 71.682 0.00000 f×f 23.093 0.00000 f(0.1,0.25) 45.031 0.00000 r×r 55.108 0.00000 vc(300,400) 4.785 0.00002 vc×vc 40.134 0.00000 f×f 32.008 0.00000 insert 2.498 0.00318 ap(2,3) 29.385 0.00000 r×insert 13.908 0.00000 ap(2,3) 1.699 0.02002 r×r 17.689 0.00000 558 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. the least squares estimates of the regression coefficients in regression models for each output parameter are shown in table 7. the p-value (prob>|t|) is a statistical measure that indicates whether the regression coefficient is statistically significant. regression coefficients for which the p-value is less than 0.05 (at a significance level of 0.05) are statistically significant, which means that they have a measurable influence on the output parameters. based on the t-test for each regression coefficient, it can be concluded from table 7 that all regression coefficients make a significant contribution to the model for each output parameter. table 7 parameter estimates output parameter term parameter estimate std. error t ratio prob>|t| ln δd intercept -4.248929915 0.01394 -212.63 <.0001 vc(300,400) 0.0004741868 0.005086 4.66 <.0001 ap(2,3) 0.0222581906 0.004665 2.39 0.0200 f(0.1,0.25) 24.923894213 0.00521 176.07 <.0001 f×f -36.26807853 0.012883 -15.84 <.0001 r(0.4,1.6) -3.61217157 0.005208 -87.96 <.0001 r×r 1.4243724919 0.012883 39.80 <.0001 insert [standard] 0.01430008923 0.004665 3.07 0.0032 insert [wiper] -0.01430008923 0.004665 -3.07 0.0032 vb intercept 0.6515022469 0.000479 268.80 <.0001 vc(300,400) -0.003352888 0.000165 133.91 <.0001 vc×vc 5.421798 10-6 0.000427 31.72 <.0001 ap(2,3) 0.0062934453 0.000152 20.74 <.0001 f(0.1,0.25) 0.0862505098 0.000169 38.18 <.0001 r(0.4,1.6) -0.058217909 0.000169 -106.28 <.0001 r×r 0.0141088 0.000415 12.24 <.0001 insert [standard] 0.0131866107 0.000152 86.93 <.0001 insert [wiper] -0.0131866107 0.000152 -86.93 <.0001 ln ra intercept -1.109854932 0.010118 -5.34 <.0001 f(0.1,0.25) 25.493860539 0.003788 241.16 <.0001 f×f -38.0339955 0.009377 -22.82 <.0001 r(0.4,1.6) -3.651190406 0.003788 -131.51 <.0001 r×r 1.4103920332 0.009365 54.21 <.0001 insert [standard] 0.3334098556 0.003393 98.25 <.0001 insert [wiper] -0.3334098556 0.003393 -98.25 <.0001 r×insert [standard] 0.0376169575 0.003793 9.92 <.0001 r×insert [wiper] -0.0376169575 0.003793 -9.92 <.0001 fig. 4 shows that the residuals are randomly distributed along the horizontal axis (row number, according to table 2) for each output parameter, indicating no trend or systematic irregularities in the distribution of the residuals. fig. 5 shows the normal quantile plots of the residuals. the residual normal quantile plots allow an assessment of the normality of the residuals in the model. the distribution of the points on the plots follows the diagonals and the points are not outside the range, which indicates a normal distribution of the residuals. comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 559 fig. 4 residual by row plot fig. 5 residual normal quantile plot statistical analysis of the data for the selected models yields the following regression equations for dimensional deviation, flank wear and surface roughness, respectively: 𝑙𝑛 ∆𝐷 = −4.248929915 + 0.0222581906 ∙ 𝑎𝑝 + 24.923894213 ∙ 𝑓 − 3.61217157 ∙ 𝑟 + 0.0004741868 ∙ 𝑣𝑐 − 36.26807853 ∙ 𝑓2 + 1.4243724919 ∙ 𝑟2 + 𝑀𝑎𝑡𝑐ℎ (𝐼𝑛𝑠𝑒𝑟𝑡) ( "𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑" ⇒ 0.01430008923 "𝑊𝑖𝑝𝑒𝑟" ⇒ −0.01430008923 𝑒𝑙𝑠𝑒 ⇒ . ) (4) 𝑉𝐵 = 0.6515022469 + 0.0062934453 ∙ 𝑎𝑝 + 0.0862505098 ∙ 𝑓 − 0.058217909 ∙ 𝑟 − 0.003352888 ∙ 𝑣𝑐 + 0.0141088 ∙ 𝑟2 + 5.421798 ∙ 10−6 ∙ 𝑣𝑐 2 + 𝑀𝑎𝑡𝑐ℎ(𝐼𝑛𝑠𝑒𝑟𝑡) ( "𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑" ⇒ 0.0131866107 "𝑊𝑖𝑝𝑒𝑟" ⇒ −0.0131866107 𝑒𝑙𝑠𝑒 ⇒ . ) (5) ln 𝑅𝑎 = −1.109854932 + 25.493860539 ∙ 𝑓 − 3.651190406 ∙ 𝑟 − 1.666666667 ∙ 𝑀𝑎𝑡𝑐ℎ(𝐼𝑛𝑠𝑒𝑟𝑡) ( "𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑" ⇒ 0.0376169575 "𝑊𝑖𝑝𝑒𝑟" ⇒ −0.0376169575 𝑒𝑙𝑠𝑒 ⇒ . ) − 38.0339955 ∙ 𝑓2 + 1.4103920332 ∙ 𝑟2 + 1.666666667 ∙ 𝑟 ∙ 𝑀𝑎𝑡𝑐ℎ(𝐼𝑛𝑠𝑒𝑟𝑡) ( "𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑" ⇒ 0.0376169575 "𝑊𝑖𝑝𝑒𝑟" ⇒ −0.0376169575 𝑒𝑙𝑠𝑒 ⇒ . ) + 𝑀𝑎𝑡𝑐ℎ(𝐼𝑛𝑠𝑒𝑟𝑡) ( "𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑" ⇒ 0.3334098556 "𝑊𝑖𝑝𝑒𝑟" ⇒ −0.3334098556 𝑒𝑙𝑠𝑒 ⇒ . ) (6) 560 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. for the surface roughness, fig. 6 shows a statistically significant interaction between the corner radius and the insert type. the influence of the interaction is most pronounced for the smallest corner radius (r=0.4 mm) and for the standard insert. the influence of the interaction is lowest for the corner radius r=1.2 mm and for the wiper insert. fig. 6 interaction profiles the prediction profilers shown in fig. 7 visually display the optimal values for each output parameter as a function of each input parameter. fig. 7 prediction profiler as shown in fig. 7, the minimum value of the dimensional deviation (δd=0.014 mm) is achieved at the minimum value of the cutting speed (vc=300 mm/min), the minimum value of the depth of cut (ap=2 mm), the minimum value of the feed (f=0.1 mm/rev), the value of corner radius r=1.2 mm and for the wiper insert. the minimum value of the flank wear (vb=0.084 mm) is achieved at the minimum value of the cutting speed (vc=300 mm/min), the minimum value of the depth of cut (ap=2 mm), the minimum value of the feed (f=0.1 mm/rev), the maximum value of the corner radius (r=1.6 mm) and for the wiper insert. the minimum value of the surface roughness (ra=0.172 µm) is achieved at the minimum feed value (f=0.1 mm/rev), the value of corner radius r=1.2 mm and for the wiper cutting insert and does not depend on the cutting speed and the depth of cut. the maximum comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 561 value of the material removal rate (mrr=300 mm3/min) was achieved at the maximum value of the cutting speed (vc=400 mm/min), feed (f=0.25 mm/rev) and depth of cut (ap=3 mm) and does not depend on the corner radius and insert type. given the conflicting requirements of the individual objective functions and in order to find a compromise solution for different production conditions and requirements, it is necessary to perform a multi-objective optimisation. 3.3. process optimisation the objective function is defined as follows: 𝐹𝑐 = 𝑤∆𝐷 ∙ min 𝑥∆𝐷 + 𝑤𝑉𝐵 ∙ min 𝑥𝑉𝐵 + 𝑤𝑅𝑎 ∙ min 𝑥𝑅𝑎 + 𝑤𝑀𝑅𝑅 ∙ max 𝑥𝑀𝑅𝑅 (7) where 𝑤∆𝐷, 𝑤𝑉𝐵, 𝑤𝑅𝑎, 𝑤𝑀𝑅𝑅 are weight coefficients determining the significance of each output parameter in objective function, and 𝑥∆𝐷 , 𝑥𝑉𝐵, 𝑥𝑅𝑎 , 𝑥𝑀𝑅𝑅 are normalized values of output parameters (expression 8). in other words, the problem is defined as multi-objective optimisation seeking for minimum of ∆d, vb and ra, and maximum of mrr. in order to eliminate the influence of the absolute values of the output parameters, their values were normalised using the following expressions: 𝑥𝑅𝑎 = 𝑅𝑎−𝑅𝑎𝑚𝑖𝑛 𝑅𝑎𝑚𝑎𝑥−𝑅𝑎𝑚𝑖𝑛 , 𝑥∆𝐷 = ∆𝐷−∆𝐷𝑚𝑖𝑛 ∆𝐷𝑚𝑎𝑥−∆𝐷𝑚𝑖𝑛 , 𝑥𝑉𝐵 = 𝑉𝐵−𝑉𝐵𝑚𝑖𝑛 𝑉𝐵𝑚𝑎𝑥−𝑉𝐵𝑚𝑖𝑛 , 𝑥𝑀𝑅𝑅 = 𝑀𝑅𝑅−𝑀𝑅𝑅𝑚𝑖𝑛 𝑀𝑅𝑅𝑚𝑎𝑥−𝑀𝑅𝑅𝑚𝑖𝑛 (8) where indices min and max determine the lowest and highest permitted value of each output parameter. as already mentioned, the “coordinates“ of particle’s position vector y in pso algorithm are the input parameters vc, ap, f and r. maximum number of iterations was 1000, and as alternative stopping criterion the number of 100 stall generations was used. the recommendations for adopting these values are not strictly defined in the literature, i.e. they depend on the particular optimisation problem. in practice, these values are determined experimentally, using trial and error procedure, which was also applied in this research. in accordance with experimental research, the following limitations for input parameters were defined: ▪ vcmin ≤ vci≤ vcmax; 300 ≤ vci ≤ 400 ∀i∈[1,…,n]; continuous input parameter, ▪ fmin ≤ fi ≤ fmax; 0.1 ≤ fi ≤ 0.25 ∀i∈[1,…,n]; continuous input parameter, ▪ apmin ≤ api ≤ apmax; 2 ≤ ap ≤ 3 ∀i∈[1,…,n]; continuous input parameter, ▪ rimin ≤ ri ≤ rimax; ri = 0.4 ∨ 0.8 ∨ 1.2 ∨ 1.6; categorical input parameter, ▪ it= standard ∨ wiper; categorical input parameter. the optimisation results for different values of the weighting coefficients are shown in table 8. table 8 optimisation results no. weighting coefficients optimal input parameters wδd wvb wra wmrr vc (mm/min) ap (mm) f (mm/rev) r (mm) insert type 1 1 1 1 1 340 3 0.2 1.2 wiper 2 1 0 1 1 400 3 0.25 1.2 wiper 3 1 0 1 0 328 2.1 0.1 1.2 wiper 562 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. 3.4. confirmation experiments the models obtained were verified by seven additional confirmation experiments. the confirmation experiments were carried out to check the generalisability of the models formed and the validation of the optimal values. the confirmation experiments were carried out both with combinations of input parameters with which the modelling of the turning process was not carried out (experiments 1–4, table 9) and with combinations of input parameters for which optimal output parameters were determined (experiments 5–7, table 9). the results obtained for the seven combinations mentioned above are shown in table 9. the performance evaluation was carried out by calculating the absolute and percentage errors. the graphical interpretation of the obtained minimum, mean and maximum values of the absolute and percentage errors for δd, vb and ra is shown in fig. 8. table 9 confirmation experiments results no. v c ( m m /m in ) a p ( m m ) f (m m /r ev ) r (m m ) in se rt t y p e predicted measure δd (mm) vb (mm) ra (µm) δd (mm) vb (mm) ra (µm) 1 325 2.5 0.125 1.2 standard 0.023 0.125 0.593 0.024 0.128 0.606 2 325 2.5 0.125 1.2 wiper 0.022 0.098 0.297 0.023 0.096 0.303 3 375 2.5 0.175 1.2 standard 0.048 0.151 1.198 0.05 0.154 1.216 4 375 2.5 0.175 1.2 wiper 0.046 0.125 0.601 0.047 0.129 0.609 5 340 3 0.2 1.2 wiper 0.062 0.112 0.795 0.063 0.114 0.807 6 328 2.1 0.1 1.2 wiper 0.015 0.094 0.192 0.016 0.097 0.195 7 400 3 0.25 1.2 wiper 0.098 0.156 1.191 0.099 0.158 1.207 fig. 8 absolute and percentage errors 4. discussion the results of the experiments, statistical analysis, modelling, optimisation and confirmation showed that the different input parameters have a different influence on the output parameters. as the cutting speed increases, the flank wear increases. an identical tendency was found in [26], but some of the studies showed the opposite, i.e. that when the cutting speed is decreased, the wear becomes more intense due to the generation of smaller cutting forces comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 563 [16], but when using multilayer composite nano-structured modified coatings. when the cutting speed increases, the temperature in the machining zone increases, which leads to higher flank wear. the consequence of the higher flank wear is a slight deterioration in the dimensional deviation. it can also be observed that the flank wear has not reached a critical level, so that it does not significantly affect the surface roughness. as the feed increases, surface roughness and flank wear increase. a similar dependence of surface roughness on feed was found in earlier studies [8, 11, 12, 18, 31]. the results differ slightly because different cutting tools/inserts are used, but the trends are very similar. as the feed increases, the cutting forces increase, which contributes to higher flank wear. the intensification of flank wear contributes to an increase in waviness and a deterioration in cylindricity, which in turn leads to an increase in dimensional deviation. a larger feed creates wider valleys on the workpiece, which contributes to an increase in surface roughness. as the corner radius increases, flank wear decreases. as the corner radius increases, the length of the cutting edge in contact with the workpiece increases, dissipating heat from the longer cutting edge and reducing flank wear. several researchers were studying the connection between surface roughness and the corner radius. in previous studies, it was found that the surface roughness can decrease [12, 31] or increase [8] with an increase in the corner radius. in this study, as the corner radius increases, the dimensional deviation and surface roughness initially decrease and then increase slightly. larger corner radii produce smaller peaks on the surface of the workpiece and at the same time less waviness. therefore, a reduction in dimensional deviation and surface roughness up to a certain corner radius value can occur. after reaching this critical corner radius value, small vibrations obviously occur, which have an effect on the deterioration of the dimensional deviation and surface roughness, with the exception of flank wear. the flank wear decreases slightly with increasing depth of cut, similar to [30]. as the depth of cut increases, greater forces are required to break and remove the chips. as a result, the strength of the cutting edge decreases and flank wear increases. higher flank wear contributes to an increase in dimensional deviation. wiper inserts improve surface roughness, which is consistent with the results of an earlier study [8]. wiper inserts are designed with a larger number of wipers and therefore contribute to a significant reduction in surface roughness. in addition, wiper inserts have a longer cutting edge so that they wear more slowly, i.e. the use of wiper inserts reduces flank wear. the optimisation results (table 9) show that it is always advantageous to turn c45 steel with wiper inserts and with a corner radius r = 1.2 mm, regardless of the weights of the output parameters. this is due to the fact that these two input parameters have no influence on the mrr and have a positive effect on δd, vb and ra. in the first case (table 8), where all output parameters are given the same importance (wδd=wvb=wra=wmmr=1), the cutting speed was close to the average level (vc=340 mm/min), the depth of cut was at the maximum level (ap=3 mm) and the feed was close to the average level (f=0.2 mm/rev). increasing the depth of cut had a positive effect on mrr and the least negative effect on δd, ra and vb. increasing the cutting speed and feed had a positive effect on mrr and a negative effect on δd, ra and vb. the negative influence of cutting speed and feed was compensated by the corner radius r=1.2 mm and the wiper insert, which resulted in the best values for δd and ra and almost the best values for vb. 564 a. milosevic, g. simunovic, z. kanovic, k. simunovic, z. santosi, et al. in the second case shown in table 8, where equal importance was given to δd, ra and mrr (wδd=wra=wmmr=1) and no importance was given to vb (wvb=0), the cutting speed was at the maximum level (vc=400 mm/min), the depth of cut was at the maximum level (ap=3 mm) and the feed was at the maximum level (f=0.25 mm/rev). this is due to the fact that the mrr is highest when the cutting speed, depth of cut and feed are at their maximum. in addition, the negative influence of these three parameters on δd and ra was compensated with a corner radius of r=1.2 mm and a wiper insert, and the best values for δd and ra were achieved. in the third case of table 8, where equal importance was given to δd and ra (wδd=wra=1) and no importance was given to vb and mrr (wvb=wmmr=0), the cutting speed was close to the minimum value (vc=328 mm/min), the depth of cut was close to the minimum value (ap=2.1 mm) and the feed was at the minimum value (f=0.1 mm/rev). this is due to the fact that δd and ra decrease with a reduction in cutting speed, depth of cut and feed. 5. conclusion in this study, a scientific approach has been proposed and proven that allows the process of dry turning of c45 steel for defined input parameters to be comprehensively evaluated, modelled and optimised based on the results of experimentation, statistical analysis, modelling, optimisation and confirmation. for different combinations of input parameters, the values of the output parameters vary over wide ranges, namely: ▪ δd in the range of 0.014–0.291 mm and with a ratio of 20.79, ▪ vb in the range of 0.084–0.213 mm and with a ratio of 2.54, ▪ ra in the range of 0.172–7.237 µm and with a ratio of 42.08, ▪ mrr in the range of 60–300 mm3/min and with a ratio of 5. this shows that it is possible to optimise the dry turning of c45 steel for various reasons in order to achieve the required quality characteristics, which is very important with regard to the possibility of practical application of the developed methodology. the most influential input parameters on δd are feed and corner radius, while cutting speed, insert type and depth of cut had a slightly smaller influence. the most influential parameters on vb are cutting speed, corner radius and insert type, while feed and depth of cut have a slightly lower influence. the parameters with the greatest influence on ra are feed, corner radius and insert type. the parameters influencing mrr are cutting speed, feed and depth of cut. the optimal combinations of input parameters vary with the requirements for the defined importance of the output parameters. different combinations of optimal input parameters resulted from the different importance (weighting coefficients) of the output parameters. the results obtained show that dry turning with corner radius of r=1.2 mm and a wiper insert is suitable to be performed independently of other production requirements. the percentage optimisation errors were in the range of 1.01-6.25 % for δd, in the range of 1.27-3.10 % for vb and in the range of 1.31-2.15 % for ra. these percentage errors correspond to absolute errors in the range of 0.001-0.002 mm for δd, in the range of 0.0020.004 for vb and in the range of 0.003-0.018 µm for ra. the values of the percentage comprehensive evaluation of dimensional deviation, flank wear, surface roughness... 565 errors and especially the absolute errors show the possibility of practical implementation of the developed models in a real production environment. the limitations of the applied methodology are reflected in the fact that the models are applicable under the conditions under which the experiments were conducted. therefore, it is planned to consider a larger number of input parameters as well as wider ranges and additional input parameters in future research. acknowledgement: this research has been supported by the ministry of science, technological development and innovation (contract no. 451-03-65/2024-03/200156), the faculty of 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with and without muzzle brake and the muzzle velocity of the projectile were recorded. then the muzzle brake efficiency was calculated. besides, an analytical model based on the orlov method for calculating muzzle brake efficiency was adopted. the obtained results were discussed. the maximum of the muzzle brake force, projectile velocity and muzzle brake efficiency obtained experimentally and analytically were compared. the analytical results were in a good agreement with the experimental ones. key words: muzzle brake, recoil, muzzle brake efficiency, muzzle brake force, orlov method, experiment 1. introduction when a firearm is fired, the burning propellant generates a blast of pressurized gases, which apply an enormous force on the projectile, resulting in accelerating the projectile forward. the same effect is applied to the breech, which induces the backward movement of the recoiling parts. after the projectile exits the barrel, the exhausted gases generate a backward impulse that increases the recoil energy. the firing event has a very short duration but it has a significant impact on the gun structure. according to newton's third law, the recoil momentum is equal to the momentum of the projectile, and the powder gasses ejected from the barrel. in order to reduce the recoil momentum, a muzzle brake can be introduced to reduce the momentum of the exhausted gasses, thus reducing the recoil momentum. received: april 18, 2022 / accepted june 06, 2022 corresponding author: damir jerković university of defence belgrade, military academy belgrade, veljka lukića kurjaka 33, 11042 belgrade e-mail: (damir.jerkovic@va.mod.gov.rs) 2 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci muzzle brakes are the devices attached to the muzzle of the firearm and intended to trap and divert expanding gasses upwards or sideways, [1-6]. there are several designs of the muzzle brake, but they usually are tubular devices that utilize chambers and holes or vents. as the propellant gases impinge on the muzzle brake baffle, a portion of the gasses is deflected from the axial direction, which generates a forward impulse on the muzzle brake, hence decreasing the recoil energy. based on the 2d euler equation, zhang huan-haw et al used a high resolution roe scheme and the structured mesh technique to simulate the flying away of a high speed projectile from the bores through different muzzle brakes, [7]. hong-xia lei et al studied and analyzed the stress of the muzzle brake by using the finite element analysis based on the fluid-solid coupled method to evaluate the muzzle brake’s reliability. first, a 3d numerical simulation of a muzzle brake flow based on cfd technology was conducted, and the muzzle brake force produced by side holes and baffles of each row was analyzed. then, by loading the gas force on the corresponding surface of the muzzle brake using the fluid-solid coupled method, the stress distribution of the muzzle brake was analyzed. this study is helpful to the design and optimization of the muzzle brake, [8]. based on internal ballistics calculation, solid works flow simulation tool and ansys static structural, ekanch chaturvedi et al designed a new perforated muzzle brake and investigated its performance such as velocity distribution, pressure growth and muzzle brake force [9]. in order to reduce the recoil energy of the chain gun, xiao et al studied the modeling and simulation of the low recoil firing of the chain gun by adopting a high efficiency recoil reduction way with new style high efficiency muzzle brake and buffer parameter combined. they studied the variation rules of gun recoil movement and recoil force of muzzle brake on three states: chain gun with the original buffer, chain gun with new high efficiency muzzle brake and original buffer, and chain gun with new high efficiency muzzle brake and new designed buffer combined. the results show that only a high efficiency muzzle brake combined with high efficiency damping buffer matching with reasonable parameters can reduce the recoil significantly [10]. based on the interior ballistics calculations and the numerical simulation of the flow inside the muzzle brake, czyżewska et al. developed a method for predicting the gas-dynamic efficiency coefficient for a perforated muzzle brake. a simple 1d single phase model of interior ballistics was used to determine the parameters of the flow at the inlet of the muzzle brake. then, a 2d axisymmetric simulation of the flow inside the muzzle brake was performed. the mass flux at the inlet and outlet of the muzzle brake were determined and the gas-dynamic efficiency coefficient was calculated, [11]. a test firing of a small caliber gun with a muzzle brake was conducted in order to measure the muzzle brake force based on the strain measurement principle. four strain gauges were pasted symmetrically on the outer surface of the barrel near the muzzle brake. after firing the axial strain of the barrel caused by the muzzle brake was determined and the muzzle brake force was calculated, [12]. kung jiang et al conducted a numerical simulation and experimental test to evaluate the muzzle brake performance with gauge pressure and muzzle brake efficiency. two numerical simulations were carried out to calculate the pressure of the muzzle blast wave and the muzzle brake efficiency. the first was set up with three dimensional unsteady euler equations ignoring the initial flow field and the second was set up with three dimensional unsteady navier-stokes equations with k-ω turbulence model and initial flow field with sliding projectile. for the experimental test, a howitzer was fired to measure the pressure analytical and experimental investigation of the muzzle brake efficiency 3 inside the muzzle brake, recoil displacement, projectile velocity and gauge pressure at the given position. the muzzle brake efficiency was calculated experimentally using the recoil resistance work method. the numerical and the experimental results of the pressure and the muzzle brake efficiency were compared and they were in a good agreement, [13]. an experimental apparatus to measure recoil forces of sporting rifles and shotguns experienced by a typical shooter was developed, [14]. the gun movement during firing was reproduced by the apparatus through both inertial and spring type damping mechanisms. at different shooting conditions and with different ammunition, the horizontal and the vertical components of the recoil force were measured using two load cells. the gun movement was measured using high speed digital camera. the apparatus should prove useful for quantifying differences in the felt recoil as affected by different factors such as recoil reduction devices, ammunition type, gun weight, etc. the force that occurs during the shooting process and depending on the use of the weapon must be adequately absorbed in order to reduce the load and the impact of the wave on the shooter (in the case of sniper rifle). in the case of integration of weapons on the combat platform (light combat vehicle, tank, vessel), the impact of the recoil force is smaller and can be absorbed by a suitable absorber built into the carrier, to reduce platform load and impact on firing conditions. the use of muzzle brakes depends on the recoil force, the type of weapon and the manner of use or carriage of the weapon. the main contribution of the present work is development of an experimental setup in order to measure the recoil force and determine the muzzle brake efficiency. two types of muzzle brake (mb) will be investigated (fig. 1) and the obtained experimental results will be presented. besides, an analytical model for determining the muzzle brake efficiency based on the orlov method is adopted to the case of the investigated muzzle brakes. the experimental results would be used to validate the predictions of the analytical model. fig. 1 muzzle brake types: (left) type 1, (right) type 2 2. mathematical models in this study, the orlov method as an analytical model is used to calculate the muzzle brake efficiency and muzzle brake force [15-18]. the method is based on the geometric parameters of the muzzle brake and the internal ballistic results of the release of gunpowder gases. 4 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci 2.1 muzzle brake efficiency the main characteristic of the method is the muzzle brake efficiency (η), described as: 2 1 · 1 1 · q q                  (1) where ω is the mass of propellant, q is the mass of projectile, β is the after-effect coefficient and α is the structural characteristic of the muzzle brake. after-effect coefficient β is calculated as: 0 1300 v   , if 0 700v m s (2) else,   0 / 1.59  m pck p w s l v       (3) where v0 is the muzzle velocity, pm is the muzzle pressure, l is the travel length of the projectile, wpc the volume of the powder chamber, s is the area of bore cross section and k is the specific heat ratio of gases. structural characteristic α of the muzzle brake is determined by equation:  0 1 0 1 2 1 1 .. 1 · m m i i i i ik k cos              (4) where k0 is the coefficient of the muzzle brake cavity reactivity, σ is the relative amount of unused propellant gas in the muzzle brake chamber, kσ is the coefficient of sidechannel reactivity and ξ is the coefficient that takes into account the effect of gas expansion in the oblique cut. the coefficient of muzzle brake cavity reactivity k0 is expressed as:  0 1 2 31 · 1hk k       (5) where, 1 0.98  (5-a) 2 2 )2 ln(sec ) (tan     (5-b) 3 0.9  (5-c) parameter χ1 is the coefficient that takes into account the dissipative forces, χ2 is the ratio that takes into account the expansion of the gases in the conic entrance of an angle θ analytical and experimental investigation of the muzzle brake efficiency 5 and χ3 is the coefficient that takes into account the non-stationary expiration of gases from the barrel channel. parameter kh is the ideal nozzle reactivity coefficient and it is determined by equation: 1 0 0 2 hk     (6) where λ0 is the ideal dimensionless speed and it depends only on the shape of the entrance to the muzzle brake cavity. this relation is expressed as: 1 1 1 1 2 0 0 2 1 1 1 1 k s k s k s k k                  (7) where ss is the cross-sectional area of the chamber at the beginning of the side channels. the coefficient of side-channel reactivity kσ is calculated in the same way as k0. the relative amount of unused powder gases σ is calculated according to the expression: 0 1 1 s s      (8) where sσ is the small area of the side opening, s0 is the surface of the outlet hole from the muzzle brake chamber and δ is the ratio of specific losses of the propellant gases through the side and central shaft of muzzle brake, and it is calculated as: 0.5 2 2 2 01 01 2 201 01 1 cos cos 1 10.95· · 1 11 1 1 k k k k                         for 2    (9)   0.5 2 3 01 2 1 2 2 012 01 1 1 11 0.95· · 1 cos 11 k k k kk k                         for 2    (10) where ψ is the inclination angle of side openings, and λ01 is the real dimensionless speed in the muzzle brake cavity, which is calculated from the following equation: 2 01 1k k     (11) coefficient ξ is determined by the following equation: 1 tan tankp     (12) 6 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci where, δψ is the additional angle of deviation of the lateral gas flow at the exit of the side opening and ψkp is the angle of the oblique cross-section of the side opening. the relation between δψ, ψ and kσ is expressed in the eq. (13):         1 1 2 2 2 2 2 2 sin sin 1 11 1 1 1 cos cos cos cos cos k k k k k kk k k k k k k k k                                                         (13) 2.2 muzzle brake force during the after-effect period without using muzzle brake, the weapon recoil force is equal to the force of propellant gases pressure, which is calculated with the following equation: 1 2 · · · 1 3 m t q f s p exp b q            (14) parameter t is the time variable during the after-effect period. parameter g is the gravitational acceleration and b is the bravin’s exponent of the after-effect period, and it can be calculated as follows:   00.5 · m v b s p    (15) during the after-effect period when using muzzle brake, weapon recoil force (f) is given by: 1 2 · · · · 1 3 m t q f s p exp b q             (16) parameter χ represents the impulse characteristic of the muzzle brake. it can be expressed, as: 0.5 0.5       (17) muzzle brake force fmb during the after-effect period is given by: analytical and experimental investigation of the muzzle brake efficiency 7   1 2 1 · · · 1 3 mb m t q f s p exp b q              (18) the pressure in the barrel and the projectile velocity were obtained from the internal ballistic calculation, [19]. the structural parameters of the two types of muzzle brakes used during experimental research are shown in table 1. table 1 structural parameters of the muzzle brakes parameter unit value mb type 1 mb type 2 inclination angle of the side opening degrees 120 130 number of chambers 2 3 area of the side opening 10-4 m2 9 first: 7.5 second: 6 third: 4.5 maximum cross-sectional area of chambers 10-4 m2 9 9 surface of the outlet hole from the mb 10-4 m2 1.54 1.54 the parameters of gun and ammunition, used during experimental research are shown in table 2. table 2 specific parameters of gun and ammunition parameter unit value barrel length m 1.007 travel length of the projectile m 0.923 caliber m 0.0127 volume of the powder chamber 10-5 m3 1.8 mass of the projectile 10-3 kg 51.3 mass of the propellant powder 10-3 kg 15.4 based on the above equations, the calculation of the muzzle brake efficiency and muzzle brake force is conducted. based on the internal ballistic calculation, the muzzle velocity of the projectile is calculated, and its value is 818 m/s. table 3 summarizes the muzzle brake efficiency obtained by the analytical model for the two types of muzzle brake. table 3 muzzle brake efficiency parameter mb type 1 mb type 2 muzzle brake efficiency [%] 48.41 48.72 8 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci the muzzle brake force of both muzzle brakes type 1 and type 2 as a function of time are calculated and presented in fig. 2. fig. 2 muzzle brake forces: (left) mb type 1, (right) mb type 2 3. experiment the standard anti-material rifle 12.7 mm was fired without and with two types of muzzle brake type 1 and type 2 (shown in fig. 1) in order to measure recoil force, muzzle brake efficiency and the muzzle velocity of the projectile. the tests were conducted in the test center of zastava arms, kragujevac. measuring system was based on rifle 12.7 mm, force measurement sensor, data acquisition system hbm quantum mx410b (4-channel) and ballistic measuring system for projectile velocity measurement. a schematic diagram of the experimental apparatus is shown in fig. 3, and its picture is shown in fig. 4. the measuring shooting equipment consisted of zastava arms fixed firing rest m2 (mass 400 kg, dimensions 1300 x 420 x 830 mm, elevation angle 3° with drift setting device) and universal receiver (mass 14 kg, dimensions 300 x 215 x 86 mm with manual and electromechanical triggering). fig. 3 schematic of experiment setup the ammunition used in the test is 12.7x108 mm type ball, where the projectile mass and the propellant mass are given in table 2. analytical and experimental investigation of the muzzle brake efficiency 9 fig. 4 recoil measurement apparatus the rifle was fixed on a rigid position to restrain it from moving. the recoil force is measured using a force measurement sensor which consists of four strain gages 350 ω connected in a full wheatstone bridge configuration. the strain gages have to be fixed on the locations corresponding to the biggest stress level. so, a dog bone shaped specimen was created on a cylindrical rod which was fixed at the end of the rifle to allow the concentration of the stress on this area, fig. 5. in order to measure the recoil force which is the axial force, the two pairs of gages were fixed as the following: one pair is in the principal direction of the barrel axis and the other one is in transversal direction used in temperature compensation and bending cancellation [20-23], as shown in fig. 5. fig. 5 force measurement sensor: (left) schematic, (right) picture 10 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci the calibration of the force sensor was conducted in force range of 100 kn, with maximal nonlinearity of 0.3 %. the sampling frequency in the data acquisition channel was set for 96 khz. the initial velocity of the projectile was measured using a ballistic chronograph placed 10 m from the muzzle. the measuring tests of recoil force and projectile velocity were carried out for three different cases: rifle without muzzle brake, rifle with mb type 1, and rifle with mb type 2. the firing test with a series of five shots was conducted to establish the baseline data. to ensure the accuracy of the data, a series of ten shots was rendered for each case, and the recoil forces, as well as projectile velocity were recorded. thereby, the average recoil force and velocity for the three different series were calculated. the coefficient of variation (cv) for the maximal recorded recoil forces (max rf) and velocity was calculated to measure the data dispersion. the results are shown in table 4. table 4 coefficient of variation of the obtained results without mb with mb type 1 with mb type 2 max rf velocity max rf velocity max rf velocity cv [%] 2.3716 0.0652 3.3602 0.1265 2.4881 0.2119 the coefficient of variation does not exceed 4 %, which signifies that the recorded data were consistent and had a low dispersion. fig. 6 experimental averaged recoil forces analytical and experimental investigation of the muzzle brake efficiency 11 the experimental raw data were filtered using a low pass filter in order to eliminate the high frequency noise. fig. 6 represents the average measured recoil forces as a function of time for the three above mentioned cases. it is noticed that without using a muzzle brake, the average maximum of measured recoil force was 17904 n. in the presence of the muzzle brake, the maximum of recoil force is significantly reduced. in the case of using the mb type 1, the average maximum of recoil force is 15729 n, which corresponds to a reduction of 12.9 % compared to the case without muzzle brake. meanwhile, in the case of using the mb type 2, the average maximum of recoil force is equal to 14367 n, with a reduction of 19.8 % compared to the case without muzzle brake. the recoil force described in fig. 6 represents the force in after-effect period. the maximum force recorded during the measurement, while the projectile is in the barrel, is less than the maximum measured force when the projectile leaves the barrel. the average measured muzzle velocity for the three cases is 815.0313 ± 2.23 m/s. 4. results and discussion the experimental muzzle brake efficiency can be calculated as [24]: 2 1 · 1 1 · m q q                  (19) with the use of the muzzle brake, after-effect coefficient βm is given by: 1 · m m i q     (20) where im is the impulse of the recoil force with muzzle brake. without using the muzzle brake, after-effect coefficient β can be calculated as: 1 · i q     (21) where i is the impulse of the recoil force without a muzzle brake. based on the experiment results, the efficiency of the two muzzle brakes is calculated. experimental and analytical velocities, maximum of muzzle brake forces and muzzle brakes efficiency were compared as shown in table 5. the muzzle brake efficiency of mb type 1 is 47.6 %, which corresponds to a relative difference of 1.67 % compared to the one obtained analytically, while the muzzle brake efficiency of mb type 2 is 46.51 % with a relative difference of 4.53 % to the one obtained analytically. for the muzzle brake type 1, the analytical peak value is around 10303 n, while the obtained experimental value is 10507 n, which corresponds to a relative difference of 12 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci 1.94 %. as for the muzzle brake type 2, the observed analytical peak value is 11405 n, with a corresponding experimental value of 11644 n, which amounts to a relative difference of 2.05%. regarding the muzzle velocity, the analytically calculated velocity is 818 m/s while the measured value is 815.03 m/s with a relative difference of 0.36 %. table 5 comparison between experimental and calculated results max. muzzle brake force [n] muzzle brake efficiency [%] muzzle velocity mb type 1 mb type 2 mb type 1 mb type 2 [m/s] experiment 10507 11644 47.60 46.51 815.03±2.23 analytical 10303 11405 48.41 48.72 818 relative difference [%] 1.94 2.05 1.67 4.53 0.36 the relative differences are less than 5 %, which means that the analytical results are in a good agreement with the experimental ones. in this paper, the impulse of the recoil force in the period when the projectile leaves the barrel was used to determine the efficiency of the muzzle brake. if the total period was observed (from firing), the difference in the force impulses acting on the weapon with and without the muzzle brake would be higher due to a longer time, which would lead to a higher value of the muzzle brake efficiency. in the period when the projectile is inside the barrel, the measurement results show a small difference of the impulse of the recoil forces with the muzzle brake in relation to the weapon without the muzzle brake, which is conditioned by a higher total mass of the weapon and muzzle brake, and does not objectively affect the determination of efficiency. the measurement results, for the period in the barrel, very well coincide with the results obtained on the basis of internal ballistic calculation, [19] in terms of maximum force (14 kn) and time (2.2 ms). considering that the aim of the research is to determine the muzzle brake efficiency, that part of the curve of the force, i.e. the impulse of the pressure force of the gunpowder gases was not taken into account, because then it does not act on the muzzle brake. the moment of appearance of a specific shape on the curve of the measured force (saddle shape part of function) in all results of force measurement coincides with the analytically determined duration of the firing process (until the projectile comes out of the barrel). the results of internal ballistics calculations, [19] were used in this paper for analytical determination of muzzle brake efficiency. the difference in the maximum forces of the muzzle brakes type 1 and type 2 according to the analytical and experimental model is about 10 %, that is, the muzzle brake type 2 generates a higher force of the muzzle brake by about 10 %. these differences can be deduced from the recoil forces diagrams shown in fig. 6. however, according to the experiment, the muzzle brake force impulse is 1.1 % higher in type 1, and according to the analytical model, the muzzle brake type 2 has a higher impulse by about 8.9 %. on the basis of correctly measured values of the maximum force and impulses, other devices for the absorption of the recoil force (spring absorber, rubber lining, etc.) can be dimensioned adequately and precisely. analytical and experimental investigation of the muzzle brake efficiency 13 the design of the muzzle brake type 2 is more complex in technological and constructive terms. therefore, the price-quality ratio is on the side of the muzzle brake 1. perforated muzzle brakes are often used for the purpose of absorbing recoil force and represent the simplest muzzle devices, but their efficiency is lower compared to the types of the muzzle brakes presented in this paper. 5. conclusion in the presented study, the muzzle brake efficiency was investigated experimentally and analytically. two muzzle brakes model fitted on a 12.7 mm anti-material rifle were used in this work. an experimental test apparatus to measure the recoil force and the muzzle velocity of the projectile with and without a muzzle brake was developed and applied. then, the muzzle brakes efficiency was calculated. in addition, the orlov method for calculating muzzle brake efficiency combined with the internal ballistics’ calculation was used as an analytical model. the analytical model is validated by comparison with the experiment results. the difference in relation to the maximum of forces is around 2 %. the differences between the analytical and experimental results are less than 5 %, which reveals the feasibility and the accuracy of the analytical model. the results of the measured recoil forces, during three sets of series, were accomplished in the same environmental and experimental conditions. an observed and described very good agreement of the values and the trend of the measured recoil force function in relation to the analytical results, and between three sets of measuring, also can verify the proposed experimental methodology. the reliability of the presented experimental procedure and results can be evaluated, also through the described coefficient of variation, which is small, bellow 4 %, especially according to the impulse nature of the observed force. the direct approach and reliable procedure of the experimental tests, regardless of the developed analytical and numerical investigation, in order to determine the muzzle brake efficiency, is significant. the presented research can be helpful for future studies of the design and optimization of muzzle brakes. acknowledgements: the paper is a part of the research done within the project no. va-tt/2/1618 titled “the research of dynamical behavior of the vehicle under the influence of the firing force of integrated weapon” supported by the university of defence in belgrade. the authors would like to thank to zastava arms, kragujevac, republic of serbia, for providing experimental facilities. references 1. trebinsky, r., leciejewski, z. kijewski, j., gozdzik, d., szupienko, d., 2021, investigations on influence of rifle automatics system action on values of energetic efficiency coefficient of muzzle brakes, defence technology, 2021, doi: 10.1016/j.dt.2021.06.003. 2. carlucci, d.e., jacobson, s.s., 2007, ballistics: theory and design of guns and ammunition, crc press, boca raton, usa, 654 p. 3. petrović, м., 2006, mechanics of automatic weapons, military academy, belgrade, 470 p. 14 n. ziane ahmed, d. jerković, n. hristov, w. boukera abaci 4. ilić, s., kari, a., 2016, introduction to weapon design, media center odbrana, belgrade, 458 p. 5. sherif, m., abdelsalam, o. r., aboul, m., 2018, design optimisation of muzzle brake for sniper rifle, defence science journal, 68(5), pp. 438-444. 6. semenov, i., utkin, p., akhmedyanov, i., menshov, i., pasynkov, p., 2013, numerical investigation of near-muzzle blast levels for perforated muzzle brake using high performance computing , proc. international conference, parallel and distributed computing systems pdcs 2013, kharkiv, ukraine pp. 281-289. 7. huan-hao, z., zhi-hua, c., xiao-hai, j., jun-li h., 2012, investigation of the blast wave structures of a high-speed projectile flying through different muzzle brakes, acta armamentarii, 33(5), pp. 623-629. 8. lei, h.x., wang, z.j., zhao j.l., 2016, stress analysis of muzzle brake by using 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creative commons license: cc by-nc-nd original scientific paper reliability prediction and process parameter optimization of welded joints: artificial neural network and fuzzy logic jaganathan gokulachandran1, mohanavelu thenarasu1, bhadrinath pothkanoori1, madhavarao seshadri narassima2, erfan babaee tirkolaee3,4,5 1department of mechanical engineering, amrita school of engineering, amrita vishwa vidyapeetham, coimbatore, india 2great lakes institute of management, chennai, india 3department of industrial engineering, istinye university, istanbul, turkey 4department of industrial engineering and management, yuan ze university, taoyuan, taiwan 5department of mechanics and mathematics, western caspian university, baku, azerbaijan orcid ids: jaganathan gokulachandran https://orcid.org/0000-0002-7283-1992 mohanavelu thenarasu https://orcid.org/0000-0001-6759-6417 bhadrinath pothkanoori https://orcid.org/0009-0005-8257-9304 madhavarao seshadri narassima https://orcid.org/0000-0002-4113-430x erfan babaee tirkolaee https://orcid.org/0000-0003-1664-9210 abstract. reliability prediction is an upcoming method used in most industries today to correctly estimate and predict each component’s life in a day-to-day application. this field has proven extremely helpful in evolving various methods such as preventive maintenance and non-destructive testing for various machinery and its parts. in this study, mild steel workpieces are welded together according to three parameters: weld current, weld speed, and weld angle. these parameters are varied based on the taguchi l27 orthogonal array design of experiments (doe) to conduct the experiments. the workpieces are then subjected to tensile testing to determine the tensile strength values as well as the failure time. the main objective of this research is to develop a comprehensive, methodical framework to assess the reliability and failure time of welded joints of mild steel material. according to the experimental values, artificial neural network (ann) and fuzzy logic (fl) models are developed to predict reliability percentage error and failure time. based on the findings and in the case of fl implementation, the percentage deviation between the experimental and predicted values received: september 03, 2024 / accepted january 21, 2025 corresponding author: erfan babaee tirkolaee department of industrial engineering, istinye university, 34396 istanbul, turkey e-mail: erfan.babaee@istinye.edu.tr mailto:erfan.babaee@istinye.edu.tr 946 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. is vast, while it is calculated small with the use of ann as a more accurate approach. a sample is also found to have an experimental reliability of 89.5%, the highest among the l27 doe array wherein the optimum weld strength can be achieved by incorporating 100 a weld current, 55º weld angle, and 1.17mm/s of weld speed, respectively. key words: welding, design of experiments, artificial neural network, fuzzy logic, reliability prediction 1. introduction reliability portrays the ability of a product, component, or service to perform its required function without failure for a given period under specific working conditions. reliability engineering deals with predicting, preventing, and managing uncertainty and risk of failure in different systems. reliability prediction is mostly utilized during the product design stage. reliability issues tend to occur during the beginning phases, so vital preventive measures can be taken to save cost and time. the premise of reliability prediction is a measurable examination of a wide array of information accumulated over a period to find the failure rate of a product. this information is then dissected to foster a progression of conditions utilized to demonstrate the comparing failure qualities of the framework. these conditions contain numerous factors that might influence the reliability of the framework, such as stress factors, working environment, temperatures, external loads, etc. welding is a manufacturing or fabrication process used to join materials, usually metals, by coalescence. in this process, the workpieces are melted, and filler material is added to form a pool of molten material, known as a weld pool. this weld pool is further cooled by different processes like air cooling or quenching, forming a strong joint. there are different types of welding processes such as shielded metal arc welding (smaw), gas tungsten arc welding (gtaw), gas metal arc welding (gmaw), flux core arc welding (fcaw), submerged arc welding (saw) and electro slag welding (esw). the welding technique varies depending on the base material type and thickness, application, atmospheric conditions, etc. welding techniques differ with applications, but strength and reliability parameters remain the same. the weld parameters that affect the reliability of the weld are arc voltage, weld current, weld speed, weld angle, wire diameter, extended length, etc. arc voltage is an important input weld parameter that decides the shape and size of the weld. if the required weld width exceeds the arc voltage, it must be increased accordingly. arc voltage is also responsible for weld defects such as undercut, spillage, and slag formation. if the arc voltage is increased, then the area of the heat-affected zone also increases. the current decides the weld’s depth of penetration (dop). a high current is favorable for a strong joint if the base material is thick. however, if high weld currents are used, they can burn the workpiece, or if it is too low, it can cause incomplete penetration. therefore, choosing the optimal weld current is essentially important. welding speed is defined as the speed of the electrode wire for the workpiece. as speed increases, the penetration depth decreases, resulting in weld broadening. if the welding speed is very low, the contrary impact is also seen, as the dop decreases and the energy transfer to a greater depth is impeded by the molten weld pool that already exists at extremely low speeds. understanding and implementing the ideal speed is essential for achieving the required dop and stronger weld. weld angle refers to the angle of the welding electrode relative to the workpiece. it affects the penetration, heat input, and weld bead shape. a steeper angle increases penetration and heat concentration, producing a narrower bead, while a reliability prediction and process parameter optimization of welded joints: artificial neural network... 947 shallower angle spreads heat, resulting in a wider bead. it also influences travel speed and electrode consumption. adjusting the weld angle optimally is crucial for controlling weld quality and attaining stronger weld. the diameter of the wire is responsible for the current density of the electrode. current density is equivalent to the ratio of unity to the square diameter of the electrode. the imbalance in the wire diameter might cause undesired dop, affecting the weld strength. the length of the wire also plays a crucial role in maintaining the weld quality as it directly affects the temperature and inversely affects the weld current. if the length of the wire increases, the weld current decreases, which lowers the dop, creating an improper weld pool and leading to a weak weld. parameter selection is critical where weld reliability also depends on these factors to attain a proper weld. knowing the reliability of random valued parameters is difficult and time-consuming. to avoid this problem, we integrate machine learning (ml) tools to predict the reliability of the weld for varying parameters. these ml tools not only reduce human efforts but also calculate the error existing between the prediction and actual experimental reliability values. ml tools are used to predict various properties and qualities of machine elements in various applications. for example, artificial neural networks (anns) and genetic algorithms (gas) have been used to predict the static strength and fatigue life of resistance spot welding (rsw) joints based on ultrasonic testing results [1]. here, prediction of the ultimate tensile strength (uts) of friction stir welded aluminum alloy joints was carried out by considering spindle speed (n), plunge force (f), and welding speed (v) as input process parameters [2]. ml tools such as ann and fuzzy logic (fl) were deployed and compared to assess the efficiency of the approach. here, it is demonstrated that the fl model provides more accurate results than ann. consequently, the quality levels of defects from gmaw were predicted using imageprocessing neural networks [3]. the integral algorithms of ann, such as the backpropagation (bp) algorithm and differential evolutionary algorithm (dea), were then implemented. the results verified that ann using dea took less time to compute, whereas the prediction provided by ann using bp gave more accurate results. these approaches decrease the number of experimental tests and the cost of experimentation and predict reliability within a limited time. therefore, the research questions to meet the challenges are: how can industries develop and adopt advanced ml techniques for knowing weld reliability analysis in the present manufacturing environments? what are the challenges they need to overcome? after going through different literature reviews, we comprehended the contribution of utilizing artificial intelligence (ai) processes for calculating and assessing the lifecycle and failure event for various parts utilized in separate hardware. the reliability of a system is evaluated by analyzing the conditions and relations between the components. the likelihood of failure is estimated based on the conditions of its parts or components. it is made possible by ml, which consists of computer algorithms that make use of data and recognize patterns in a set of data to conclude the process. ml concentrates on the application of algorithms to statistically estimate complicated functions. ml algorithms are utilized to gather data for a system under study, abstract the process in the form of a model, predict values of the system for the model generated, and detect the way the systems behave under observation. evolutionary/soft computing-based optimization techniques, such as ga, particle swarm optimization (pso), ant colony optimization (aco), teaching learning-based optimization (tlbo), simulated annealing (sa), and random frog (rf) have been also used so far to deal with various manufacturing processes, such as welding process [4]. soft computing-based models, including ann and fl, are 948 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. among the successful ones for modelling and controlling the production processes for this particular experimental investigation and prediction analysis of weld reliability. in this study, ann and fl models were first trained using the noted experimental data points. since ml techniques require a large amount of data to predict the output parameters, only limited input parameters were employed to determine the effectiveness of ml when using lower data sets. the input parameters chosen are weld current, weld speed, and weld angle, which greatly impact the weld strength. the other significance of this work is the utilization of taguchi l27 orthogonal array as the design of experiments (doe) for welding mild steel samples. there are three input parameters set at three different levels; i.e. 60a, 80a, and 100a for the weld current, 1.17mm/s, 1.61mm/s, and 3.25mm/s for the weld speed, 55º, 60º, and 65º for the weld angle. thus, 27 combinations of the input parameters are to be undertaken according to the taguchi l27 orthogonal array. the main advantage of using this l27 doe is that it not only renders the optimum welding parameters and individual parameter contribution towards weld reliability but also provides the significance of combining two or more parameters collectively. the interaction effect of multiple parameters is provided with the help of the taguchi method via linear model analysis of the means versus the input parameters and the signal-to-noise (s/n) ratio [5]. similarly, the ann model was utilized to predict the corrosion behaviour of friction stir welded aa5083 aluminium alloy [6]. three input process parameters were applied for prediction, using a central composite design (ccd) in a feed-forward neural network. the model accuracy was higher as the mean squared error (mse) was close to 0, and the pearson correlation coefficient (r) was equivalent to 1. a study was conducted to determine the efficiency of different algorithms in ann by comparing metrological solar radiation to predicted solar radiation. the results demonstrated that ann models trained by the br method perform better than other algorithm-trained models (shown by the performance score of the corresponding models) with a maximum r of 0.8113 and a minimum root mean squared error (rmse) of 0.2581 [7]. hence, the bp algorithm was implemented in to predict welds’ failure time and reliability due to its higher accuracy and efficiency. there are five subsequent sections which first consist of a literature review in section 2 to identify the gaps and discuss the contributions. section 3 provides a detailed description of the developed methodology. the procedure involved during the experimental investigation is given in section 4. the results obtained in both experimental investigation and prediction modelling are discussed in section 5. finally, section 6 concludes the research and provides an outlook for future studies. 2. literature survey after focusing on failure analysis and reliability prediction and surveying the literature accordingly, it is revealed that the most recent approaches of ml and ai have been offered as methodologies for assessing the dependability of a particular component or system. the outputs of computer numerical control (cnc) milling were estimated by sasindran et al. [8] using two soft computing methodologies, ann and fl, considering the process parameters as inputs. baraya et al. [9] discussed the cost-effectiveness and dependability of ultrasonic welding. both experimental and finite element approaches were taken into account to evaluate joint strength. as a result, the ann technique was used to predict joint strength after the trials were completed. likewise, the fl model was studied by vignesh et al. [10] to comprehend how process parameters affect the tensile strength of a friction reliability prediction and process parameter optimization of welded joints: artificial neural network... 949 stir welded joint. the mamdani fuzzy model was utilized to determine the ideal process parameters. the tests were carried out using a l18 doe array. it was revealed that the fuzzy system performs better than the regression model. to estimate the corrosion rate and potential of the aluminium alloy aa5083 that was put through friction stir processing using potentiodynamic polarization tests. many studies provided a tutorial on constructing and understanding ann models [11-13]. linear regression (lr) and multilayer perceptron (mlp) are two examples of ann. yin et al. [14] utilized k-means approach to gather a significant amount of data the right input settings. they also employed an lr model to yield the most precise findings when predicting the number of failures before they occur. aa6061-t6 was a desirable material in the automotive, aerospace, and marine sectors based on excellent corrosion resistance, great strength, and toughness. the primary issue was the deuteriation of these characteristics in welded joints, which was claimed to be resolved by including synthetic reinforcements. yi and jones [15] developed an ml approach that could predict solder joint reliability in terms of training data, failure variables, and ml approaches. the bp algorithm was applied for feedforward neural network training to calculate the mse for accuracy check. they found that predictions via ml are more precise than those based on the weibull method. ilhe [16] analyzed the strength and mechanical properties of tungsten inert gas (tig) and smaw welding with emphasis on optimization of process parameters to achieve improved productivity and reduce costs and efficient joining of materials by using various filler materials. a fuzzy cognitive map was proposed by huang et al. [17] as a mechanism in which the reliability of dynamic product components is predicted under interaction. this work also examined a case study using a system-in-package when mutual influences affect the life of its components, thus enhancing the accuracy of prediction. hussein et al. [18] discussed how the welding current and time affect material properties such as maximum shear load and nugget diameter in resistance spot welding (rsw) of aisi 304 stainless steel. a fuzzy logic controller (flc) was also implemented to predict the optimal welding parameters in advance and detect the possibility of failure. he et al. [19] developed a probabilistic model for the fatigue life prediction of notched components, combining the weibull distribution with critical distance theory. a comparison of two methods for size effects was done in the study, demonstrating that the highly stressed volume approach fits experimental data better. a tig welding algorithm was offered by kesse et al. [20] based on ai to forecast the bead geometry for tig welding operations. an experimental sample set was simulated using the ai tig welding technique. the results showed 92.59% projected accuracy when compared to the data amassed throughout the trial. soltani et al. [21] employed soft computing and statistical techniques to build up a comparative structure for predicting operational reliability in the automotive manufacturing industry. they demonstrated that the adaptive neuro-fuzzy inference system (anfis) model outperforms statistical models, enhancing reliability and safety. tomaz et al. [22] used a five-factor, five-level ccd matrix for computation for their gtaw trials. utp af ledurit 60 and utp af ledurit 68 were used to create two tubular wires, with an aisi 1020 steel blank as the base. the ann algorithm was applied with a ga to determine the optimal welding parameters and simulate the gtaw process. with a coefficient of determination (r2) of all the data higher than 0.65, the ideal welding parameters of 222a welding current, 25cm/min welding speed, 8mm nozzle deflection distance, 25° travel angle, and 8hz wire feed pulse frequency were attained. pourasl et al. [23] utilized ann and anfis to predict the outputs, revealing the impact of operational parameters on performance measures in applying aisi-d6 steel in die and mould preparation using electrical discharge machining. 950 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. anfis model showed a powerful learning capability and is more reliable than ml techniques as it possessed lower rmse values close to 0, considering the output parameters. abima et al. [24] implemented anfis to predict uts of weld developed via tig-mig hybrid. optimization was done based on taguchi’s approaches using the l9 doe array. metal inert gas (mig) voltage, tig current, and gas flow rate were taken into account as model inputs. the optimum tensile strength among the l27 array was 868.3mpa, and corresponding inputs were found to be 25v, 180a, and 19l/mm, respectively. it was concluded that gas flow makes the highest contribution (42.35%) towards uts, and tig current has the lowest contribution (18.13%). r² values were close to 1, and rmse values for training and testing were 1.8963 and 4.8194, respectively. this meant that anfis yields lower deviations between experimental and predicted values, therefore reducing experimental costs and time consumption. the impact of parameters on the mechanical and microstructural characteristics of friction stir spot welded joints on structural steel 1020 and aa6062 were explored by kumar et al. [25]. they conducted tensile tests on samples to learn the strength of the weld. the parameters employed were tool speed, dwell time, and plunge depth, representing 6.171%, 39.66%, and 35.7% of the contributions. the response surface methodology was employed to predict the parameters and evaluate microstructural parameters such as heat dissipation and grain changes. the prediction yielded results very close to experimental values. an experimental investigation was performed by arivarasu et al. [26] to examine the effect of weld parameters on the mechanical and metallurgical properties of co2 laserwelded nickel alloy 825. the optimal parameters to obtain a 5mm defect-free weld thickness on alloy 825 were 3kw laser power, with a weld speed of 1.5 m min−1. due to the formation of beneficiary tin and al4c3 precipitates in the fusion weld zone, the mechanical strength and hardness increased without influencing ductility. the quality of the produced weldment was indicated by the defect-free 180° root bend test. moganapriya et al. [27] studied the impact of the performance input parameters of coated carbide inserts during the machining of aisi 1015 steel, which were examined by considering output responses such as surface roughness, flank wear, etc. they employed the taguchi design approach integrated with fl and grey theory to explore multi-objective hybrid optimization. the optimized parameters were observed to 500 rpm of speed, 1mm of cutting depth, 0.05mm/rev of feed rate, and rapid cutting fluid flow rate with tialn/wc-c as an ideal coating substrate. ohwoekevwo et al. [28] utilized ann technique to model and forecast the percentage of dilution in aisi 1020 low-carbon steel welds made by tig welding. the determined regression showed an r of 0.9992 as the results of the training test, r of 0.99865 as the progression of the evaluation, and r of 0.85285 as the progression of the training test. finally, it led to an overall r of 0.90007, demonstrating that ann is a useful method for determining the degree of weld dilution. as evidenced by the obtained coefficient of determination (r2 value) of 0.9876, there was a correlation between the experimental and ann findings. feng et al. [29] concluded that fatigue fracture in welded joints causes engineering accidents, wherein the current prediction models could not be used for all service conditions. tan et al. [30] applied a fatigue reliability model for welded structures using the master s/n curve method, addressing issues such as grid sensitivity and joint geometry dependence. the model reduced the computational burden and improved reliability prediction, promoting product innovation and optimal design. similarly, the impact of undercuts and misalignments on the fatigue strength and reliability of load-carrying cruciform welded joints (lcwj) was investigated by song et al. [31] using probabilistic statistics theory and fracture mechanics theory. ml algorithms were explored by gbagba et al. [32] to predict reliability prediction and process parameter optimization of welded joints: artificial neural network... 951 the life span of structures with welding, considering factors such as material type, application, welding method, input parameters, and output parameters. the study highlighted the potential of ml for automation, testing, structural integrity, health monitoring, and damage-tolerant design of welded structures, highlighting its potential for improved efficiency and automation. the fatigue reliability assessment model and design method for welded structures were proposed by zhou et al. [33] based on the structural stress method, which aimed to improve the fatigue reliability of welded joints and structures by analyzing influence variables. prediction and optimization of residual stresses, plastic deformations, and damage induced by laser shock peening (lsp) on a thin ti-6al-4 v used in turbine blades were carried out by ayeb et al. [34] using ann and anfis. they employed a twophase approach including numerical simulation and finite element analysis (fea) to characterize the lsp process and its effects on the material. it was observed that the models made by both ann and anfis learn accurately from the data generated by numerical simulations, allowing them to predict and optimize lsp effects accurately. a comparative study was performed by kiraz et al. [35] in which six ann models were correlated to predict stress concentration factor (scf) for varying training datasets and hidden layer neurons. the models were constructed with undercut depth, reinforcement angle, and deep angle of welding seam as input parameters. among the six ann models, the best prediction model, which had an accuracy of 0.9834, achieved 90% training and five neurons in the hidden layer. it was also found that increasing the number of neurons in the hidden layer will reduce the efficacy of the prediction model. the optimum number of neurons in the hidden layer could be between 5-10. moreover, r was found to be 0.9834, concluding that ann models provide better prediction and save time. a multi-fidelity model for reliability prediction of ball grid array (bga) solder joints was advanced by yu et al. [36], overcoming issues like long simulation time and low accuracy. the model revealed significantly higher prediction accuracy under cost constraints and faster convergence in optimization. the authors demonstrated a research gap in ml and other soft computing techniques for reliability prediction and optimization, and more research must be done in this area. adewuyi et al. [37] explored the impact strength of cr-mo steel bars influenced by welding parameters. pure tungsten with 2% thoriated tig electrodes was used for welding purposes. an ann model was created to predict the steel’s impact strength by taking current, material thickness, number of weld passes, and electrode diameter as input parameters. the sample with 15mm thickness, 90a current, three weld passes, and 2.4mm electrode size represented the highest impact strength upon optimization. material thickness and number of welds passed significantly contributed to the steel’s impact strength. the ann model attained the rmse value close to 4.12%, showing the accuracy of the levenberg-marquardt algorithm (lma) which was thus employed. the above literature review comprises a wide range of studies, including failure analysis, reliability prediction, optimization techniques, and parameter contribution estimation for various welding and manufacturing processes. several methodologies, including ann, fl, and ml techniques, have been studied to assess the reliability and quality estimation of various components used in engineering and science applications. some of the applications mentioned were cnc milling, ultrasonic welding, friction stir welding, rsw, etc. the survey highlights the efficacy of computing techniques such as anfis and ann in predicting outputs such as failure time, reliability, corrosion rate, fatigue life, weld strength, and quality. optimization techniques based on the taguchi method, ga, and ccd have also been employed to determine optimal process parameters for welding operations, machining processes, and material preparation. the findings obtained from ann and anfis revealed accurate results by 952 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. predicting the outcomes close to that of experimental values. the survey emphasizes integrating soft computational modelling techniques to predict welding and other manufacturing processes’ reliability, parameter effect, and efficiency. welding is an important process in the manufacturing industry, allowing for the joining of parts and avoiding failures in crucial components. after welding, it is essential to check the reliability of the weld, as initial welding parameters such as welding current, material thickness, electrode diameter, weld speed, and weld angle. the main contribution of this research is integrating ml tools to learn the weld reliability and effect of weld parameters on an l27 doe array. the ann and anfis techniques are compared to determine the contribution of individual parameters towards weld reliability. further, optimization based on the experimental study is conducted to know the desirable weld parameters. table 1 synthesizes the methodologies and findings from the selected references while identifying gaps or areas needing further exploration. although many research works on ml methods, such as ann and fl, have been studied in different areas of industry, they have not yet been utilized to predict how reliable welded joints will be and when they fail. a significant gap exists in the literature regarding integrating these ml models specifically for weld reliability prediction using real-world parameter variations such as weld current, speed, and angle. this study addresses this gap by implementing ann and fl models trained on data from a taguchi l27 orthogonal array design of tests focused on mild steel welds. the novelty lies in the comparative analysis of these ml models to determine the most accurate approach for predicting reliability and failure time. it meets the need for more efficient, cost-effective, and reliable predictive models in the welding field. this work aids the practical application of weld reliability prediction in industrial settings through a systematic and data-based framework incorporating ml models for accurate reliability assessment. reliability and failure time are the attributes that can be predicted using ann and fl models by considering important welding parameters for improving weld quality and life. these predictive models save significant time, effectively meaning quicker operational decision-making and cost efficiencies by eliminating the need for large-scale physical testing. furthermore, deploying these predictive models in a real-time production environment can lead to preventative maintenance and informed decision-making, enhancing overall weld structure stability for industries such as automotive, aerospace, and construction. 3. methodology design the methodology entails a systematic process for enhancing reliability assessment and failure prediction of welded joints. the methodology involves several steps as follows: 1. identify gaps in reliability assessment and failure prediction of welded joints. 2. select materials and prepare specimens according to standards. 3. conduct experiments under various conditions to gather performance and failure data. 4. analyze data using statistical methods and develop predictive models with ml techniques, particularly in matlab. 5. validate developed models using additional experimental data or real-world datasets. 6. refine weld input parameters based on predictive models to enhance reliability. 7. assess the impact of optimized parameters on reliability through experimentation. reliability prediction and process parameter optimization of welded joints: artificial neural network... 953 table 1 literature summary and gaps identified reference methodology key findings gaps identified amiri et al. [1] ultrasonic testing, ml predicts static & fatigue behaviour of spot-welded joints limited application to other welding techniques dewan et al. [2] anfis, neural network predicts tensile strength of friction stir welds anfis vs. neural network performance comparison karthikeyan et al. [5] design of experiments optimizes tig welding parameters for satellite applications need for broader application beyond satellite components choudhury et al. [4] ann modelling, optimization estimation of weld strength for gtaw of inconel 825 optimization methods for other materials and processes sai et al. [6] ann models predicts corrosion behaviour of friction stir processed aa5083 applicability to other alloys and welding techniques heng et al. [7] ann with different bp algorithms solar radiation prediction limited to meteorological data, needs broader application baraya et al. [9] experimental analysis, predictive modelling enhances smart textile fabrication through ultrasonic welding application to other smart materials sasindran et al. [8] fl, ann optimizes milling parameters for gun metal broader material applicability is needed omoya et al. [12] reliability engineering pipeline design lack of focus on specific welding or manufacturing processes soltanali et al. [21] statistical, soft computing techniques reliability prediction for automotive manufacturing comparison of different techniques across industries ayeb et al. [34] ann, anfis predicts mechanical properties of laser-treated turbine blades generalization to other materials and treatments zhou et al. [33] structural stress method fatigue reliability assessment model for welded structures need for integration with other assessment models gbagba et al. [32] ml techniques fatigue life prediction of welded structures comparison with traditional methods needed feng et al. [29] data-driven methods review of prediction models for fatigue performance of welded joints broader applicability and model integration yu et al. [36] multi-fidelity surrogate model reliability prediction of bga solder joints limited to bga joints, needs a broader focus the choice of ann and fl for the prediction of welding reliability is based on their strong ability to model complex non-linear relationships in welding processes. the deep learning architecture of ann ensures that it captures intricate patterns between such weld parameters as current, speed, and angle, as identified by dewan et al. [2]. it is further supported by fl, which processes uncertainties and variability within the quality of welds. sai et al. [6] also mentioned that it is fl suitable for such type of application, while support vector machines (svms) and decision trees (dts) are feasible models, they have several drawbacks. svms generally involve tedious tuning and have problems related to non-linear and noisy data sets, which are typical in welding scenarios. choudhury et al. [4] pointed 954 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. out that the performance of anns is better than svms for most regression tasks. karthikeyan et al. [5] demonstrated that the dts are prone to overfitting, and their performance may not be as good as those of ann and fl. moreover, fl further enhanced predictive accuracy by handling vagueness in welding processes through human-like reasoning, which is impossible in svms or dts. omoya et al. [12] enlisted the effectiveness of fl in addressing weld parameter selection uncertainties. the ann integrated with fl provided a framework for weld reliability prediction compared to the study that made use of svms and dts. the main aim of this research is to predict the reliability of welded joints, a crucial factor in industries where weld strength impacts product durability. to determine the most effective tool for predicting weld reliability based on the error between predicted and experimental values and to optimize weld input parameters for enhanced reliability, we compare two ml approaches of ann and fl. mild steel, commonly used in welding, was selected as the specimen material, and samples were created to measure 50×30×10 mm to make experiments on universal testing equipment easier. the samples were made using double-butt joints on both sides by arc welding, a widely used method in the industry. experiments were conducted by varying three welding parameters, which are weld angle (55°, 60°, 65°), weld current (60a, 80a, 100a), and weld speed (1.17 mm/s, 1.61 mm/s, 3.25 mm/s). tensile testing established the welded specimens’ tensile strength and failure duration. the collected data was the foundation for developing prediction models using ann and fl in matlab. the levenberg-marquardt training algorithm was employed for the ann model due to its effectiveness in handling non-linear systems. normalizing the data and dividing it into training, validation, and testing sets were two aspects of data preparation. the models were then validated with the help of additional experimental data, and the performance was evaluated using error measures such as mean absolute error (mae), rmse, and r2 values. the ann model is the recommended option for fine-tuning weld input parameters because it outperformed the fl model in terms of accuracy and error reduction. further experiments were carried out to assess the effect of these adjusted parameters on weld strength and dependability. this systematic approach facilitates the identification of optimal welding conditions, enhancing the industrial applicability of the developed models and providing a reliable framework for predicting weld reliability and optimizing welding parameters. both industry standards and previous research informed the selection of weld currents (60a, 80a, 100a). these currents are representative of typical settings used in tig welding, balancing heat input, and weld quality. dewan et al. [2] highlighted the significant impact of weld current on the tensile strength of friction stir welds, supporting the choice of these currents for achieving desirable weld properties. karthikeyan et al. [5] also noted that standard welding practices align with these current ranges, demonstrating their relevance for obtaining optimal weld results. the weld speeds chosen (1.17 mm/s, 1.61 mm/s, 3.25 mm/s) encompass a range that examines both slow and fast welding conditions. preliminary experiments and prior studies guided this range. for example, choudhury et al. [4] used similar speeds in their study of gtaw processes, highlighting their impact on weld quality. moreover, sai et al. [6] found that such speeds effectively influence the mechanical properties and surface quality in friction stir welding, reinforcing the appropriateness of these chosen speeds. the selected weld angles (55º, 60º, 65º) are based on standard industry practices and relevant research. these angles cover a range that is commonly used for achieving optimal weld penetration and bead formation. omoya et al. [12] demonstrated the significant influence of weld angles on mechanical properties and reliability, justifying the choice of reliability prediction and process parameter optimization of welded joints: artificial neural network... 955 these angles for the study. sasindran et al. [8] also supported these angles in their research on milling parameters, indicating their effectiveness in ensuring consistent weld quality. 4. experimental work mild steel material was taken into consideration as it is used widely in various industrial sites and for various purposes. the acquired workpieces are cuboidal in shape, proving difficult to conduce properly during welding. to overcome this, one side of each parent metal is grooved at an angle of 30°, which favours the double-butt joint welding of groove angle of 60° (as planned). before the grooving process, the workpieces were made smooth and precise to the dimensions using the surface grinding process. the final dimensions of the workpieces were 50×30×10 mm. the workpieces were then chamfered using the vertical milling machine using a 30° tool. with the implementation of the doe, it was decided to use the variation of three welding parameters, namely, weld current, weld speed, and weld angle, to ensure the complete diversity in the final weld strength values because of the parameter variations. the weld current varied from 60a to 80a and 100a, which are the industry’s commonly used ranges of weld current. the welded angle was noted to be 60° as a standard usage; hence, to bring in a variation, three weld angles, 55°, 60°, and 65°, were considered and used for welding accordingly. the final parameter was the weld speed. to bring in variation, three weld speeds had to be considered where the welds were made in three specific durations of time, which were calculated with the help of a stopwatch. it was employed to find out the three weld speeds using the distance-speed relation and derived as 1.17mm/s, 1.61mm/s, and 3.25mm/s. the arc welding process was used for the double butt-welding joint for each pair of cuboidal workpieces. tensile strength testing was done on the 27 welded joints after welding according to the l27 doe array. during the process, the time to failure of each sample was taken using a stopwatch. fig. 1(a) shows the 27 mild steel weld specimens that were welded according to the 27 trials generated while fig. 1(b) represents the welded specimens loaded for tensile testing. fig. 1 welding results according to the 27 trials generated 956 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. after acquiring the values of failure time(s), the reliability values of the welded joint samples were computed using the conventional reliability formula so that the values predicted via ann and fl could be compared and trained accordingly. the reliability formula is given by eq. (1): 𝑅(𝑡) = 𝑒(− 𝑡 𝜃 )𝛽 , (1) where 𝜃, 𝛽, and 𝑡 stand for the scale factor, shape factor, and failure time, respectively. a weibull distribution is plotted (fig. 2) using minitab software to obtain the shape and scale factors. the obtained values are used to find the reliability values for each of the 27 welded joints, as shown in table 2. fig. 2 is a distribution analysis graph, showing the relationship between the percentage of a population (y-axis) and a particular variable (xaxis). it is a scatter plot of the data points, a fitted line that represents the overall trend, and a table of statistics that summarizes the data distribution. the scatter plot displays the individual data points. each point stands for a single observation, with its x-coordinate representing the value of the variable and its y-coordinate outlining the percentage of the population corresponding to that value. the fitted line represents the overall trend of the data. it is a straight line drawn through the scatter plot to capture best the general relationship between the variable and the population percentage. it is noteworthy that the table of statistics given in fig. 2 provides numerical summaries of the data distribution; i.e., it includes measures of central tendency (mean and median), dispersion (standard deviation, interquartile range), and shape (weibull distribution parameters). the distribution represents that the shape and scale factors are 2.80588 and 51.7330, respectively. these insights from the weibull distribution are useful for manually calculating experimental reliability. fig. 2 distribution analysis: ml estimates reliability prediction and process parameter optimization of welded joints: artificial neural network... 957 table 2 experimental reliability values experiment no. weld current [a] weld speed [mm/s] weld angle [degrees] tensile strength [n/mm2] failure time [s] experimental reliability 1 60 1.17 55 127.49 32.03 0.770669 2 60 1.17 60 180.81 40.51 0.604405 3 60 1.17 65 149.14 46.12 0.484556 4 60 1.61 55 229.47 40.19 0.611139 5 60 1.61 60 348.28 80.66 0.030898 6 60 1.61 65 315.97 46.58 0.474744 7 60 3.25 55 259.16 28.31 0.831743 8 60 3.25 60 235.68 26.87 0.852886 9 60 3.25 65 279.42 67.84 0.117724 10 80 1.17 55 408.34 81.24 0.028792 11 80 1.17 60 439.26 54.53 0.313734 12 80 1.17 65 441.39 82.04 0.026081 13 80 1.61 55 352.97 60.03 0.219161 14 80 1.61 60 386.06 58.75 0.239579 15 80 1.61 65 407.82 38.71 0.641962 16 80 3.25 55 418.26 47.32 0.459018 17 80 3.25 60 405.44 51.52 0.372138 18 80 3.25 65 433.34 68.75 0.108511 19 100 1.17 55 300.70 23.60 0.895326 20 100 1.17 60 304.57 26.85 0.853169 21 100 1.17 65 272.72 27.62 0.842057 22 100 1.61 55 290.85 34.47 0.726088 23 100 1.61 60 302.02 41.22 0.589391 24 100 1.61 65 267.02 26.64 0.856127 25 100 3.25 55 285.03 32.81 0.756775 26 100 3.25 60 284.76 32.04 0.770493 27 100 3.25 65 307.33 43.06 0.550142 4.1. artificial neural network ann is the first soft computing method implemented in the study. deep learning toolbox of matlab software was utilized to implement it. the toolbox enables the user to select the number of iterations required for training the training algorithm and enables the regression graph viewer. it helps in understanding the training level of the algorithm to extract the lowest error percentage possible. the time to failure of each of the samples, as well as its reliability percentage, is obtained by the ann. the network diagram of the ann, as well as the values of reliability and failure time obtained, are displayed in fig. 3 and table 3. this ann diagram is also called the bp neural network. bp neural networks are multilayer feed-forward networks trained using the bp algorithm. they learn complex input-output relationships without explicit programming, generalize well to unseen data, and apply them to various problems. baraya et al. [9] adopted the bp neural network that predicts the ultrasonic welding parameters for enhanced textile fabrication. the ann architecture was similar to the current study. in fig. 3, the colours represent existing layers in the model. the turquoise colour shows input and output data, respectively, whereas blue represents the layers responsible for computations and carry operations like summations. they require a large 958 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. amount of training data and can overfit the training data. the arrows represent the flow of input data from layer to layer. the plus symbol denotes biases and weights in the equations, which pilots the neural network calculations. the “3” below the input element displays the parameters, i.e., weld current, speed, and angle. the “10” and “1” below the hidden and output layer elements represent the cumulative weights and biases. the “1” below the output layer shows the output parameter, i.e., weld reliability and failure time. however, their internal computations are not easily interpretable. they require a large amount of training data and can overfit the training data. table 3 training conditions of the ann parameter training condition data division random (dividerand) training gradient descent with momentum & adaptive lr (traingdx) performance mse epoch 1000 fig. 3 ann network diagram 4.1.1. structure of the neural network the neural network consists of 3 layers: the input, hidden, and output layers. the input layer gives the input parameters into the network. the hidden layer processes the input values. it relates them with the output values, and the output layer shows the network’s output after the training process in the hidden layer. the ann neurons used in this study are of the ratio 3:10:1 (input: hidden: output). the number of output nodes is taken to be one as the failure time and reliability values are predicted using two different neural networks. the ann structure is given in fig. 4 which is similar to that provided by park et al. [38]. table 4 compares values obtained experimentally with those obtained using ann. fig. 4 proposed structure of the ann reliability prediction and process parameter optimization of welded joints: artificial neural network... 959 table 4 time to failure and reliability values obtained by experimental and ann failure time [experimental] failure time using ann reliability using experimental reliability using ann 32.03 28.3239 0.770669 0.74814 40.51 39.3182 0.604405 0.56773 46.12 47.1409 0.484556 0.45788 40.19 40.6042 0.611139 0.67069 80.66 78.0848 0.030898 0.10239 46.58 47.3944 0.474744 0.46145 28.31 28.0827 0.831743 0.89475 26.87 32.3098 0.852886 0.89471 67.84 66.6089 0.117724 0.11385 81.24 79.7201 0.028792 0.072929 54.53 56.3209 0.313734 0.28509 82.04 78.1896 0.026081 0.13121 60.03 60.1094 0.219161 0.21129 58.75 49.7485 0.239579 0.26222 38.71 39.8371 0.641962 0.57656 47.32 37.6611 0.459018 0.62062 51.52 50.2645 0.372138 0.3551 68.75 71.8491 0.108511 0.097069 23.6 31.1425 0.895326 0.85548 26.85 31.2358 0.853169 0.84185 27.62 30.2877 0.842057 0.83124 34.47 37.8169 0.726088 0.72954 41.22 39.595 0.589391 0.58632 26.64 36.8819 0.856127 0.83045 32.81 35.096 0.756775 0.73678 32.04 30.7815 0.770493 0.71574 43.06 41.6098 0.550142 0.54818 park et al. [38] utilized ann to predict the yield strength of austenitic stainless-steel welds. in fig. 4, the input layer contains three nodes representing the welding parameters: weld current, angle, and speed. these parameters are fed into the network as input data. the hidden layer is a computational layer that processes the input data and extracts meaningful patterns. it consists of multiple interconnected nodes, each performing a non-linear transformation on the input data. the number of nodes in the hidden layer can vary depending on the complexity of the problem. the output layer contains a single node representing the welding process’s predicted reliability or failure time. this output is generated based on the processed information from the hidden layer. figs. 5(a) and 5(b) compare the experimental failure time and reliability with the ann predicted values, respectively, for all the samples of the l27 doe array. 960 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. fig. 5 comparisons of failure time reliability values with their corresponding ann 4.2. fuzzy logic fl is an approach that helps decision-making. it could determine whether a given statement is true or false where the values range from 0 to 1. the input parameters undergo fuzzification, turning a crisp value into a fuzzy one. the input variables are provided with the help of a membership function. a set of rules is provided that relates the input parameter to the output parameter. it goes through a process called defuzzification to obtain the output. this study implemented the fl approach to predict the failure time and reliability output for the varying input conditions, such as weld speed, weld current, and weld angle. the failure time and reliability for various input parameters were determined experimentally for twentyseven samples, which were welded according to the l27 doe array. table 5 displays the training conditions for all the input parameters of the fl model. the comparison of time to failure and reliability values are given in fig. 6 and table 6. reliability prediction and process parameter optimization of welded joints: artificial neural network... 961 table 5 training conditions for the fl model table 6 comparison of the time to failure and reliability values failure time experimental failure time using fl reliability using experimental reliability using fl 32.03 29.9 0.770668644 0.8 40.51 51.9 0.604404957 0.457 46.12 51.9 0.48455637 0.457 40.19 51.9 0.611139182 0.457 80.66 75.1 0.030898086 0.121 46.58 51.9 0.474743642 0.457 28.31 29.9 0.831743344 0.8 26.87 29.9 0.85288615 0.8 67.84 75.1 0.117724492 0.121 81.24 75.1 0.02879167 0.121 54.53 51.9 0.313733557 0.121 82.04 75.1 0.026080583 0.121 60.03 51.9 0.21916138 0.121 58.75 51.9 0.239579293 0.121 38.71 51.9 0.641962295 0.8 47.32 51.9 0.459017936 0.457 51.52 51.9 0.372137975 0.457 68.75 75.1 0.108510609 0.121 23.6 29.9 0.895325963 0.8 26.85 29.9 0.853169446 0.8 27.62 29.9 0.842056643 0.8 34.47 51.9 0.726088324 0.8 41.22 51.9 0.589391195 0.457 26.64 29.9 0.856126709 0.8 32.81 51.9 0.756774669 0.8 32.04 29.9 0.770492753 0.8 43.06 51.9 0.55014172 0.457 parameters conditions range of values weld current low 60 – 76.67 [a] medium 63.33 – 96.67 high 83.27 – 100 weld speed [mm/s] low 1.17 – 1.431 medium 1.34 – 2.402 high 2.191 – 3.25 weld angle [degrees] low 55 – 59.17 medium 55.83 – 64.17 high 60.83 65 failure time [s] low 23.6 – 43.08 medium 39.6 – 63.41 high 60.52 – 82.04 reliability low 2.61 – 21 medium 18.79 – 71.3 high 68.59 – 89.4 member function triangular member function not applicable 962 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. fig. 6 comparisons of failure time reliability values with their corresponding fl predicted values 5. results and discussions based on the reliability percentage values and the failure time found through the experimental procedure, it is possible to train the ann and the fl implementation. the experimental results were used to model and train the network. it helped in finding out the accurately predicted values. they were compared with each other to compute the variation in results and determine the error percentage for each of the two methods, namely, ann and fl implementation. reliability prediction and process parameter optimization of welded joints: artificial neural network... 963 5.1. reliability prediction fig. 7 displays the error percentage between the reliability predicted values of ann and fl on coordinate axes, where the x-axis displays the sample number, and the y-axis represents the error percentage. the reliability values obtained using ann and fl and the corresponding percentages of error prevailing between the experimental reliability values are shown in table 7. from fig. 7, it is inferred that there is only a small error of difference in the average of predicted reliability values from ann compared to the obtained experimental values. meanwhile, the margin of error for fl implementation is relatively higher than that of the ann model. salimiasl et al. [39] conducted a comparative study to assess the best computation ml tool for tool condition monitoring and 𝑅2 for ann was observed to be slightly higher than fl, as 𝑅2 is a measure of the model accuracy, higher 𝑅2 for ann makes it more accurate. however, considering the size of the experiment fl was proved to be more accurate as ann requires large amounts of data sets for computation. table 7 comparison of the reliability values using ann and fl vs. experimental values experimental reliability reliability using ann %error using ann reliability using fl %error using fl 0.770668644 0.74814 2.923259402 0.8 -3.805962091 0.604404957 0.56773 6.067944489 0.457 24.3884428 0.48455637 0.45788 5.505318114 0.457 5.686927531 0.611139182 0.67069 -9.74423175 0.457 25.22161668 0.030898086 0.10239 -231.3797525 0.121 -291.610021 0.474743642 0.46145 2.800172702 0.457 3.737520695 0.831743344 0.89475 -7.575252233 0.8 3.816483055 0.85288615 0.89471 -4.903802187 0.8 6.200845247 0.117724492 0.11385 3.291151672 0.121 -2.78235088 0.02879167 0.072929 -153.2989608 0.121 -320.2604486 0.313733557 0.28509 9.129898965 0.121 61.43224166 0.026080583 0.13121 -403.0945802 0.121 -363.9466825 0.21916138 0.21129 3.591590979 0.121 44.78954285 0.239579293 0.26222 -9.450193467 0.121 49.49480051 0.641962295 0.57656 10.18787164 0.8 -24.61791087 0.459017936 0.62062 -35.20604576 0.457 0.439620199 0.372137975 0.3551 4.578402655 0.457 -22.80391435 0.108510609 0.097069 10.54423101 0.121 -11.50983371 0.895325963 0.85548 4.450442016 0.8 10.64706786 0.853169446 0.84185 1.326752408 0.8 6.23199136 0.842056643 0.83124 1.284550574 0.8 4.994514773 0.726088324 0.72954 -0.475379692 0.8 -10.17943328 0.589391195 0.58632 0.521079294 0.457 22.46236396 0.856126709 0.83045 2.999171564 0.8 6.555888074 0.756774669 0.73678 2.64209013 0.8 -5.711783566 0.770492753 0.71574 7.106199605 0.8 -3.829659256 0.55014172 0.54818 0.356584533 0.457 16.9304957 964 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. fig. 7 error percentage between ann and fl predicted reliability values 5.2. failure time the failure times obtained using ann and fl and their corresponding error percentages prevailing between the experimental reliability values are presented in table 8. the error percentage between the failure time predicted values of ann and fl on coordinate axes are displayed in fig. 8. fig. 8 error percentage between ann and fl predicted failure times it is understood that there is only a small margin of error in the predicted values as compared to the actual values when using ann. in contrast, the margin of variation is comparatively higher when using fl for the failure time prediction. the ann model’s mean absolute deviation (mad) between the predicted and actual data is close to 0.00639. mad obtained for the fl is quite higher than that of ann. şahin et al. [40] observed similar results while studying the efficiency of ml tools such as ann and fl to predict the attendance demand in european football league matches. onyelowe et al. [41] reliability prediction and process parameter optimization of welded joints: artificial neural network... 965 conducted a study to evaluate an accurate computational modelling network between ann and fl using loss function parameters such as mae, rmse, and r-values. biswajeet et al. [42] performed a comparative study on ann and fl to know the prediction ability of l and slide susceptibility mapping in a geographical information system (gis) environment. eren et al. [43] stated that ann parameters such as weights and biases are essential for accurate prediction, and algorithms such as bp help for better optimization and parameter selection. mean absolute percentage deviation (mapd) and mad factors were evaluated to determine the efficient ml tool. mapd values for ann and fl were 0.08 and 0.1, respectively, and mad values for ann and fl were 0.05 and 0.07, respectively, pointing out the ann as the most accurate among the three methods. table 8 comparison of the failure time values using ann and fl vs. experimental values failure time failure time using ann %error using ann failure time using fl %error using fl 32.03 28.3239 11.57071495 29.9 6.65001561 40.51 39.3182 2.941989632 51.9 -28.11651444 46.12 47.1409 -2.213573287 51.9 -12.53252385 40.19 40.6042 -1.030604628 51.9 -29.13660114 80.66 78.0848 3.19266055 75.1 6.893131664 46.58 47.3944 -1.748389867 51.9 -11.42121082 28.31 28.0827 0.802896503 29.9 -5.616389968 26.87 32.3098 -20.24488277 29.9 -11.27651656 67.84 66.6089 1.814711085 75.1 -10.70165094 81.24 79.7201 1.870876416 75.1 7.557853274 54.53 56.3209 -3.284247203 51.9 4.823033193 82.04 78.1896 4.693320332 75.1 8.459288152 60.03 60.1094 -0.1322672 51.9 13.54322839 58.75 49.7485 15.32170213 51.9 11.65957447 38.71 39.8371 -2.911650736 51.9 -34.07388272 47.32 37.6611 20.41187658 51.9 -9.678782756 51.52 50.2645 2.436917702 51.9 -0.73757764 68.75 71.8491 -4.507781818 75.1 -9.236363636 23.6 31.1425 -31.95974576 29.9 -26.69491525 26.85 31.2358 -16.33445065 29.9 -11.3594041 27.62 30.2877 -9.658580739 29.9 -8.254887762 34.47 37.8169 -9.709602553 51.9 -50.56570931 41.22 39.595 3.942261038 51.9 -25.90975255 26.64 36.8819 -38.44557057 29.9 -12.23723724 32.81 35.096 -6.967387991 51.9 -58.18348065 32.04 30.7815 3.927902622 29.9 6.679151061 43.06 41.6098 3.367858802 51.9 -20.52949373 the ann model’s systematic errors can be partly pointed at the limitation of the data, for instance, limited variability within the training data, which may lead to model overfitting and inaccuracies when values beyond the range used in training are forecast. besides, biases related to inappropriate hyper-parameter optimization can affect the model’s accuracy. given that ann is viewed as a “black box,” one cannot indicate the specific source of this error. conversely, the fl model relies on pre-defined rules, which may not fully capture the complex relationships in the data, resulting in larger prediction errors than ann. 966 j. gokulachandran, m. thenarasu, b. pothkanoori, et al. increasing the size and variability of the training dataset would greatly reduce biases and improve the generalization of the data being fed into the system. advanced hyperparameter tuning methods such as grid search and bayesian optimization would also help fine-tune the model’s performance. hybrid approaches that ann has in common with other models will take full advantage of each of the strengths while cancelling the individual weaknesses of each method. interpretability techniques, such as shap or lime, are useful in identifying errors within the ann predictions, thus allowing more lucid insight into the model’s decision-making process. these steps will decrease systematic errors and improve the reliability of ann and fl models throughout the work that follows. 5.3. optimization of process parameters the optimum set of parameters ensures the most reliable weld obtained based on the highest obtained reliability percentage value, 89.5%. it is found that the most optimal weld strength is obtained when the mild steelwork pieces are welded under the following parameters: ▪ weld current is 100a, ▪ weld angle as 55°, ▪ weld speed as 1.17mm/s. the results demonstrate that ann outperforms fl in predicting failure times, with a lower mad of 0.00639 than fl. the optimization of process parameters reveals that the most reliable weld is achieved with a welding current of 100a, weld angle of 55°, and weld speed of 1.17mm/s, resulting in a reliability percentage of 89.5%. trying to explain why ann models outperformed fl in this study, one would point to the fact that the architecture of ann is highly capable of detecting complex non-linear patterns in data, which could easily be due, because of welding processes, to interaction among various parameters such as weld current, speed, and angle. this ability enables ann to learn adaptively and seek the optimal weights and biases via bp algorithm, minimizing the prediction errors observed in our results for lower mae and rmse values. furthermore, the high r-value of ann expresses its strength in mapping the complex relationship between input variables and reliability outcomes. while fl typically relies on rule-based systems and fuzzy sets that can be used in well-structured scenarios or when expert knowledge can easily be codified, in dynamic processes such as welding, the model will not be quite as effective, given its dependence on previous rules, which may not as effectively interpret complex relationships between data points. it was a disadvantage that manifested in our results, where the bigger error margins and prediction variability were consistent with fl. the nature of our experimental data being non-linear and variable further reiterates that ann is superior in this case. ann could henceforth yield more accurate, consistent predictions of both reliability and failure times by capitalizing on its learning capabilities, as opposed to fl, which is inherently more rigid and less amenable to the variability representative of welding processes. the ability of ann to further approximate the continuous functions and handle larger datasets effectively using its network structure containing layered neurons and non-linear activation functions makes it superior. therefore, for those reasons, ann should be superior in performance for the reliability predictions in welding applications on our dataset, which contains complex relationships with high variability. reliability prediction and process parameter optimization of welded joints: artificial neural network... 967 6. conclusion and future work this research focused on developing a comprehensive framework to assess the reliability and failure time of mild steel material welded joints using soft-computing methods. the study involved conducting experiments and developing ann and fl models to predict reliability percentage error and failure time. upon simultaneous evaluation and comparison of predicted values with the experimental reliability and failure times values, it is concluded that the ann showed less error and more accuracy than the fl. the ann model produced average mae, rmse, and r-values of 0.2750, 0.4154, and 0.9983, respectively. in contrast, the fl model produced 0.3737, 0.6654, and 0.9894, respectively, which also proved that the ann model is comparatively more accurate than the fl. the fl implementation proved to be less efficient in the prediction of the two values as the range of values output was wide; hence, the accuracy of each specific value was low, as compared to experimental values. ann was more user-friendly and showed the capability to predict any range of values concerning the given inputs fed into the algorithm, with the least possible error by suitable number of training sessions until the highest regression values are reached for the testing, validation, and training plots. however, the disadvantage associated with the ann is that it is a slow training process and takes time to compute. in contrast, the fl, on the other hand, is less accurate but can process large amounts of data quickly. the results were similar to those of the current study. the ann computation took longer as the ann model requires the conversion of data to ascii format for the matlab package interface and later reconversion to a conventional database. the average failure time errors of the ann and fl were 8.34% and 16.38%, respectively, whereas the average reliability errors of ann and fl were 34.6% and 50.15%, respectively. the variation in results obtained from the ann compared to the fl implementation is on a lesser margin based on error calculation. soft computation tools like ann face large data requirements and computation time challenges. fl’s dependency on human expertise leads to less accurate results. ann’s black-box nature and lack of transparency in computations are drawbacks. standardizing approaches is difficult due to unclear criteria. future studies could explore the ann models to estimate stress states in welded joints, similarly to the work by özden and gökce [44]. genetic programming, bayesian networks, and dts may also be useful for reliability prediction. expanding the study to incorporate alternative welding methods, materials (aluminum, stainless steel), and conditions (temperature, hardness) helps validate the model in multiple scenarios. more parameters, large data, and automation like robotics [45] could improve 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fuzzy logic techniques for l and slide susceptibility mapping, disaster advances, 3(3), pp. 26-34. 43. eren, b., guvenc, m.a., mistikoglu, s., 2021, artificial intelligence applications for friction stir welding: a review, metals and materials international, 27, pp. 193-219. 44. özden, o.b., gökçe, b., 2024, estimation of stress concentration factor using artificial neural networks in t-weld joints forced by bending, tehnički vjesnik, 31(5), pp. 1478-1484. 45. sahoo, s.k., choudhury, b.b., dhal, p.r., 2024, exploring the role of robotics in maritime technology: innovations, challenges, and future prospects, spectrum of mechanical engineering and operational research, 1(1), pp. 159-176. facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 491 511 https://doi.org/10.22190/fume200426030h © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper evaluation of iranian small and medium-sized industries using the dea based on additive ratio model – a review malek hassanpour department of environmental science, ucs, osmania university, telangana state, india abstract. data envelopment analysis (dea) is a prominent procedure in the decisionmaking process with a pivotal role in the sustainable development (sd) assay. project identification is the first step of sustainability assessment in the environmental impact assessment (eia) program for the industrial projects prior to complete establishment. the present review research comprised 405 iranian industries assessment regarding both input and output criteria via the dea integrated with the ratio model of additive ratio assessment (aras) and weighing systems of both kendall and friedman's tests supported by spss software. the findings deployed a classification for iranian industries pertaining to industries' nominal capacity in certain clusters. also, the current review paved the pathway towards executing both energy and materials streams in industries. key words: iranian industries, dea, eia, aras model, assessment 1. introduction the current era, which scholars call the modern world, is characterized by continual change and complexity of structures. in such a situation, only the managers, i.e. those having the right, up-to-date and comprehensive information on how their organization operates, can succeed in making timely and accurate decisions for its continual improvement in line with existing changes [1]. with the expansion of organizations and increasing the supervisory scope of managers, evaluation and control of organizational units become a necessity for managers, which is not possible without evaluating the performance and efficiency of the units under their supervision as well as the evaluation by legal organizations in the eia program [2, 3]. the dea is a good instrument for evaluating the performance of an organization in both the eia program and after [4]. received april 26, 2020 / accepted july 10, 2020 corresponding author: malek hassanpour department of environmental science, ucs, osmania university, hyderabad – 500007, telangana state, india e-mail: malek.hassanpour@yahoo.com 492 m. hassanpour the dea is based on a series of optimizations using linear programming also called nonparametric methods. in the dea, efficient boundary curves are generated from a series of points determined by linear programming [5]. the linear programming method, after a series of optimizations, determines whether the decision-maker unit is inside or outside the performance boundary. in this way, efficient and inefficient units are separated based on inputs and outputs variables. defining inputs will be easy to compute and analyze for singleinput-output units, but in most real-world problems we have units with multiple inputs and outputs; therefore, we require methods that combine inputs and outputs. outputs as a single indicator achieve a good benchmark for measuring efficiency [6, 7]. in the dea model with an input perspective, we seek to obtain technical inefficiencies as a proportion that must be reduced in the inputs so that the output remains unchanged while the unit is at the performance boundary. we need to increase outputs without changing the inputs until the unit reaches the performance limit. in the ccr model of the dea, the values obtained for performance in the two perspectives are equal, unlike in the bcc model where these values are different. the reason for choosing the perspective for a dea model is in the relative evaluation of the performance of units, which in some cases has no control over unit management. the output has no variable and its value is already fixed. on the contrary, in some cases, the input is fixed and the output is variable; in such case the output view is appropriate. inputs and outputs depending on the circumstances of the manager control are determined. scale efficiency represents the link between changes in inputs and outputs of a system. one of the capabilities of the dea method is to apply different patterns corresponding to the returns to different scales as well as to measure the returns to unit scales including fixed and variable scale returns. a) fixed scale returns: that is, every multiple of inputs produces the same multiplier of outputs. the ccr model assumes constant scaling of outputs. b) variable scale returns: that is, any multiple inputs can produce the same multiple outputs, less or more of it in the outputs [8, 9]. in many studies, the criteria used in the research have multiple units (dimensions) that need to be unscaled because it is not possible to use equations for different scaling parameters. one way of finding unscaled data is by using the ratio based systems that include a large number of systems like the aras model. the aras model is recently introduced as a step beyond the simplicity of the simple additive weighting (saw) model. the word aras means evaluating the cumulative ratio. this method is also used to rank research options. the decision matrix of this method is also an optional criterion matrix, the matrix whose columns are the criteria and its rows are the research options. given that the author's study has shown that the dea has had different results in the ratio models, it is, therefore, recommended that similar studies encompassing different scales with impossibility of converting the scale to currency apply a normalization system for ranking alternatives [10]. the non-scaling process with the equations introduced in ranking systems is a way of integrating parameters and options. the difficulty that appeared in shifting units to currency gets back to variations observed in materials stream entered into industrial cycles [11-13]. so by the present review study, we aimed at tabulating whole materials and energy streams in a variety of scales. then we have assigned weighting and ranking models in order to classify and prioritize alternatives, criteria, and options. evaluation of iranian small and medium-sized industries using dea based on additive ratio... 493 2. environmental impact assessment eia of projects is an environmental management practice. the search for environmental management instruments in developing countries and transition economies is itself a search for sd practices. many engineering projects play a major role in the production and dissemination of pollution, which plays a very important role in the destruction of the environment. large-scale impacts are considered in the final environmental threshold assessment, and these effects are likely to be large-scale, global, and to the greatest possible form. this method limits the range of pressures that can disable an ecosystem in its prime and equilibrium conditions. it examines the structure from different aspects, time, quantity, quality, spatial dimensions, and their current and future status. it covers the stages of social and environmental assessment including evaluation, mitigation, impact monitoring, and auditing. comprehensive eia has the following four characteristics such as:  integration with local or national decision-making processes,  integration with environmental management approaches and instruments,  integration with the evaluation process, and  integration with economic, social, health and environmental impacts. in iran, the ultimate goal of eia is to achieve sd in the form of economic programs consistent with the principles of environmental protection as well as to prevent destruction and depletion of renewable and non-prescriptive resources. eia includes the following steps: (1) screening, (2) estimating the scope of work, (3) determining the evaluation report, (4) review, and (5) monitoring. according to the international union of eia views, the process of carrying out an environmental assessment includes the following stages: (1) screening, (2) determining the scope of work, (3) testing options, (4) impact analysis, (5) corrective actions and managing effects, (6) evaluation of index cases, (7) reporting, (8) review, (9) decision-making, and (10) future corrections. screening is a process that determines whether or not there is a need to perform an evaluation and its level. the three general screening criteria are the type of project or development priorities, project size, and project location. the following project executives are also required to carry out a feasibility report and assessment in the eia program [14]. (1) petrochemical plants on any scale (2) the petrochemical refinery on any scale (3) the power plant with a production capacity of more than 100 megawatt (4) steel industries (5) large dams and aquatic structures or a lateral structure larger than 40 hectares or a lake area of more than 400 hectares (6) irrigation projects larger than 5000 hectares (7) tailing dam of any size (8) industrial estates over 100 hectares (9) airport with a runway length greater than 2000 meters (10) agro-industrial units over 5,000 hectares (11) large industrial slaughterhouses (12) landfills for provincial and city centers larger than 200,000 inhabitants and new cities (13) landfill centers (compost factory etc.) (14) oil and gas pipeline designs, oil rigs, oil storage sites, forestry plans, highways, railways, tourism projects (15) related industrial and service plans (16) construction workplaces larger than 10,000 square meters (17) land reclamation (18) accommodation, development of river basins and rural roads. 494 m. hassanpour eia is a process that ensures that all environmental matters are taken into account quite early in the project at planning itself. it takes into consideration not only technical, financial and economic considerations but also traditional aspects like impacts on local people, biodiversity, etc. eia is intended to prevent or minimize potential adverse environmental impacts and enhances the overall quality of a project [15]. the main benefits and advantages of eia are: lower project costs in the long-term, increased project acceptance, improved project design, informed decision-making, environmentally sensitive decision, increased accountability and transparency, reduced environmental damage, improved integration of projects into their environmental and social settings. eia methodology means the structural approaches for doing one or more activities of eia. there are some specific characteristics that an eia methodology should depict. it should be appropriate to the necessary task of the eia process such as impact identification/comparison of alternatives and significantly free from the assessors' bias. it should be economical in terms of costs, as well as of its requirement of data, investigation of time, personnel, equipment, and facilities. impact identification attempts to answer the question of what will happen when a project enters its operational stage. a list of important impacts such as changes in ambient air quality, changes in water and soil qualities, noise levels, wildlife habitats, species diversity, social and cultural systems, employment levels, etc. may be prepared. the important sources of impact like smoke emission, consumption of water and discharge of effluent, etc. are identified. impact evaluation aims to assign relative significance to predicted impacts associated with the projects and to determine the order in which impacts are to be avoided, mitigated or compensated. eia methods encompassed many practices and models such as three-dimension matrix, leopold matrix, patterson matrix, achieve to objectives matrix, time matrix, effective size matrix, weighting matrix, saratoga matrix, favorite matrix, pastakia matrix, monavari method, workshop models, flowdiagrams systems, networks, delphi fuzzy logic, optimal path method, internal method, threshold analysis method, natural threshold method, audit, expert method, checklist method, questionnaires, overlays method, mongol method, ad-hoc method, and battle method, etc. [16,17]. the decision-making process is an integral part of the eia program including lots of multi-criteria decision-making (mcdm) models [11, 18]. recently, the use of mcdm systems containing a variety of variables increased in lots of developing sciences individually or integrated with other models. in the 44 stock market selection used the dea-copras method along with the entropy shannon weighing system for 61 months in india. the return values of each stock have been taken into consideration in currency. by the way, 18 stock markets selected to go through the efficiency assessment steps. the ranking system arranged to assort the companies based on the inputs and outputs criteria in currency [19]. azadi et al. [20] prioritized the 24 green suppliers via the fuzzy double frontier dea model in iran. using the combined vikor and dea models resulted in the estimation of efficiency scores about multiple inputs and outputs. to cope with environmental challenges for the co2 dissipation, scientists used the dea model to estimate the efficiency of emission permissions considering minimum outlays in 29 chinese provinces. the quantities of pollutants reported in co2 (10 thousand tons), gross domestic product (100 million rmb), electricity consumption (100 million kwh), total investment in fixed assets (100 million rmb) [21]. to investigate the footprint of greenhouse gasses, ozone formation and acidification ability caused by european union manufacturing sectors exploited the dea model based on the dissipation rate in currency. the dissipation rate has evaluation of iranian small and medium-sized industries using dea based on additive ratio... 495 been reported as co (t), co2 (t), ch4 (t), n2o (t), nox (t), so2 (t), nh3 (t), volatile organic compounds (t) [2]. the research investigated the chinese industries’ eco-efficiency based on the exploitation of energy and dissipation of environmental pollution via the dea model during 2006-2013. ecological efficiency of provinces was found to be with an average rise [22]. uniting the game theory, the dea model and gray system resulted in developing a model to evaluate the efficiency score of 21 various kinds of cars in iran khodro company. the number of cars entered into pre delivery inspection (pdi), the number of pdi personals, the number of cars exited from pdi, on-time delivery score, customer satisfaction scores were used as the criteria [23]. by research, the dynamic dea applied to appraise the economic development efficiency in seek of co2 dissipation falls out of 28 eu countries in a period from 2009-2013. the input variables devoted to the labor force, real capital stock, energy consumption, fossil fuel consumption and co2 emissions allocated for output variables [24]. with regard to the division done upon the collected data, it was classified as two classes of variables of input and output in the research by bulak and turkyilmaz [25]. input variables included proximity to market, ability to control costs, potential labor force, product quality, prompt advantage, certification, distribution channel, pricing policy and service, capital and machinery equipment track ended up to outputs of market share and net profit margin. the performance efficiency of around 744 small and medium enterprises showed 94 efficient industries in turkey. with regard to some criteria such as store territory, population density, total hours worked by in-store personnel, total hours worked by delivery personnel and weekly expenses, the research considered the use a hybrid approach united with fuzzy analytic hierarchy process (ahp), dea and topsis models to assess the retail efficiency by duman et al [26]. the performance of the supply chain of pharmaceutical companies scrutinized using the dea model supported by factor analysis in the tehran stock exchange. the questionnaires' passed out to 115 experts and they gathered the data for the survey. findings gave a high performance for 12 of the 28 resources investigated [27]. research pursued the performance of china's 31 grid companies via dea and tobit regression at a time interval of 2004 to 2013. the results showed a rise in efficiency scores during the period. anthony et al [28], assessed the performance of 7 indian chemical companies via the traditional dea model along with the weighing system of the friedman test. the criteria reported in currency and composed the frameworks of output and input variables. a range of efficiency scores was found to be between 0 and 1 with high propinquity among companies in terms of the dea score. the thirty-seven regional and national iranian refinement and oil product distribution companies including two inputs and three outputs criteria underwent the dea model and weighted with both entropy and ahp. so, the classification of companies was done in the arranged procedure in iran [29]. azadi et al [30] extended a united dea enhanced russell measure model in fuzzy ambient concerning sd ingredients such as economic, social and environmental to choose the best suppliers among 26 cases in iran. the nineteen facility layout difficulties were evaluated by the dea model in the study of toloo et al [31] via inputs and outputs criteria such as cost, adjacency score and shape ratio, flexibility, quality, and hand-carry utility, respectively. according to the research, the efficiency weights led to the classification of alternatives and removed the problem in iran. the twenty-two indian oil and gas companies scored against efficiency scale analysis using the dea model depend on materials consumed, employee benefit expenses, capital investment, operating revenue and profit after tax as input and output 496 m. hassanpour criteria [32]. the petrochemical industry evaluated for the efficiency score classification based on 5 inputs and 5 outputs criteria via 5 dea models and ahp (weighing system) in saudi arabia [33]. unearthing sustainable technology with the use of the dea model was investigated by kim et al [34] from 1980 to 2010. a study examined the efficiency evaluation using dea model supplemented by ahp about inputs and outputs criteria such as distance from co2 source, minimum flow rate requirement (input), injectivity limit, operating life, product yield, product value, sequestration parameter, reservoir capacity, well security and structural integrity (outputs) [35]. the semiconductor industry evaluated the efficiency steps by the dea model based on 2 inputs values and one output value [36]. to estimate the efficiency of seven bus operators paid attention to fuel cost, labor cost, depreciation expense, other costs, patronage volume, mileage and satisfaction index in nanjing, china. the values reported in currency and then based on mentioned costs applied the dea model to rank the options and ahp to weight the criteria [37]. also nemati and matin [38], saravi et al., [39], ulengin et al., [40], sevinc and eren [41] and woo et al., [42] used the dea model for resource allocation for 60 firms, location selection for the co-firing biomass plants, resources exploitation of 82 automotive companies and operational efficiency of shipping industries regarding lots of criteria and alternatives, respectively. the present review targeted to evaluate 405 iranian industries via dea based on the additive ratio model in the eia of projects with notice to some recent studies carried out in this regard. 3. project identification steps the methodology employed by the present review refers to conjoining the traditional dea model with a newly introduced model of aras in compliance with the saw model. according to the current rules of the iranian environment protection agency and iranian industries organization, the engineering projects should traverse the project identification stage in the eia with encompassing the screening of project properties [43]. the screening of all industrial projects obviously manifests the energy demands, input and output materials flow, the required staff, labors and whole aspects of economic estimation of projects. the weighting systems employed included both friedman and kendall tests due to a full agreement with emerging the highest weights for the highest values and vice versa. zavadskas et al [44] used the kendall test as a weighing system in mcdm models of topsis, copras, and aras to select technological options for installing pilecolumns (driving the rings) in the construction operation. aras model also provided an excellent coherency with the traditional dea model according to our experience during the research. there is no propinquity between the integration of the traditional dea model with other additive ratio models [45]. therefore, to complete the initial screening of the iranian evaluator team for 405 iranian small and medium-sized industries, they have undergone further processing of their ranking and weighing. the findings appeared in the current review published by a few research papers; then we tried to collect main achievements at present review paper. the mentioned steps to complete the review and project identification levels of iranian evaluator team are displayed in fig. 1. evaluation of iranian small and medium-sized industries using dea based on additive ratio... 497 fig 1. the flow-diagram of followed work the collected data are tabulated with regard to inputs materials flow, nominal capacity (nc) of industries and 5 main criteria such as the number of staff, power, water, fuel demands and the land area required to construct industries individually. the nc and input materials flow that is called initial feed (if) included a variety of values in miscellaneous scales such as nc (t), nc (no), nc (pocket), nc (m 2 ), nc (m), nc (skin), nc (ft 2 ), nc (pair), nc (jean), nc (yard), nc (barrel), nc (l), nc (crank), nc (bottles) and if (t), if (no), if (m 2 ), if (m), if (l), if (mm), if (pair), if (m 3 ), if (sheet), if (pieces), if (bundle), if (rolls), if (ft 2 ), if (thread), if (yard) and if (duke). to understand the variables composing the framework of the present review, they depicted in an illustration format. the input and output criteria that design the framework of the dea model are illustrated according to fig. 2. fig. 2 input and output criteria of dea model 498 m. hassanpour the values of weights (w) are calculated by both the friedman and kendall tests using spss software. then the vector of w is introduced into normalized criteria in the aras system to sum the w of alternatives (industries) individually. finally, the weighted average of nc to weighted average of if released the weights for the alternatives; then the ranking system set based on w. in this case, the iranian industrial productivity was investigated to decide in economic growth parallel with political dimensions to expand via dea based on additive ratio models. the ahp sought to determine the values of w. the manpower, remuneration, capital, labor, energy, new investment and value-added were taken into consideration as inputs and outputs in a period [50]. also, azar et al [51] used a novel dea based on the additive ratio model to estimate the ordinary set of w by explaining the procedure via an example. so, iranian industries are classified in certain clusters according to below descriptions based on nc. it needs to explain that all tabulated amounts for energy and materials streams are reported in annual estimation and for the industries before construction. 4. iranian industries by the present review, 9 groups (including 405 industries) of iranian industries passed through the dea-aras model to classify the industries based on materials and energy streams such as iranian wood and cellulose industries (iwci; 16 various types), iranian textile and leather industries (itli; 38 various types), iranian mining and aggregate industries (imai; 26 various types), iranian food industries (ifi; 57 various types), iranian automotive industries (iai; 71 various types), iranian plastic industries (ipi; 21 various types), iranian electronic products manufacturing industries (iepmi; 33 various types), iranian chemical industries (ici; 118 various types) and iranian household appliance industries (ihai; 25 various types). there is no similar study to cover the same implication and concept in this field but in the case of similar studies, they can be related to the cases below. amini and alinezhad [52] asserted that his request to use the dea model to rank 15 iranian industries succeeded to score 8 efficient industries holding a score of 1. a study completed by lu et al., [53] included a method approach to dea to find the performance of many units of industries. the efficiency scores of the 34 chinese life insurers companies evaluated via the dea model included 5 criteria during the period from 2006 to 2010. the dea score ranged from 0.905 to 0.973. iranian regional power companies of azerbaijan, isfahan, bakhtar, tehran, khorasan, khuzestan, zanjan, semnan, sistan, gharb, fars, kerman, gilan, mazandaran, hormozgan, and yazd have undergone the efficiency assessment via a novel dea model. the designed method was also examined with other models to get acceptance for the efficiency analysis as well as its reliability in the mentioned assessment. the power companies got an efficiency score based on input and output stream analysis [54]. a study scrutinized around 23 main manufacturing units via the dea model during the period from 2005-2007. by the way, the industries passed through the efficiency analysis along with the identification of the provinces where engulfed the utmost performance of industries. the minimum and maximum ranges of dea scores placed the industries in an interval of 0 to 1 [55]. the results of using the dea model to assess the efficiency level of manufacturing plants of china and turkey opened the door to demystify the performance of industries between two countries with chinese industries placed in the first rank. it needs to explain that canonical correlation was employed as a weighting system evaluation of iranian small and medium-sized industries using dea based on additive ratio... 499 for both input and output criteria [56]. keramidou et al [57] assessed the efficiency of the greek meat products industry using the dea model in a period from 1994 to 2007. the findings have proved the inefficiency of industries with a rise due to mismanagement and prodigality of the asset. also, 59 iranian manufacturing units underwent efficiency analysis via the dea in terms of the amount of fossil fuel, water, electricity consumption, employee's payment and technical efficiency by rezaee and ghanbarpour [58] and in a time interval from 1995-2009. 5. the values of w for criteria a wide variety of 405 small and medium-sized industries has led to the emergence of different kinds of input and output materials with weight units in iran. a sample from each industry has been studied and reported. the complexity of input materials introduced into industries makes it impossible to convert input and output materials into currency in such studies as well as complexity in identifying commercial name, number of labels and their purity. therefore, the best option for managing the materials is to report them in different units of measurement. the energy consumed and a land area used along with employees number were included as well as the remaining part of input factors in a variety of dimensions. but the output materials of industries which are called industry products were reported as nc different in terms of measurement scales or dimensions. therefore, the tabulated values make the frameworks of criteria to calculate the values of w. table 1 w values for criteria of iwci, imai, and ifi [46, 47] iwci w imai w ifi w nc (t) 12.81 nc (t) 10.92 if (t) 8.58 nc (pocket) 6.06 nc (m2) 3.94 if (no) 10.16 nc (m2) 6.88 employees 7.77 if (l) 3.84 nc (m) 6.13 power (kw) 9.50 if (pieces) 3.70 power (kw) 16.94 water (m3) 7.25 if (m) 4.04 employees 14.66 fuel (gj) 7.40 if (m3) 3.85 fuel (gj) 13 land (m2) 11.31 nc (bottles) 3.88 water (m3) 12.75 if (t) 11.77 nc (no) 5.32 land (m2) 13.88 if (no) 6.19 nc (t) 7.39 if (t) 10.94 if (l) 3.69 employees 49.01 if (m3) 6.63 if (m) 3.98 power (kw) 41.31 if (bundle) 6.03 if (m2) 3.83 water (m3) 0.09 if (no) 11.38 if (m3) 3.44 fuel (gj) 10.79 if (m) 7.03 land (m2) 9.32 if (rolls) 6.63 if (m2) 7.25 if (l) 5.97 if (pieces) 6.06 500 m. hassanpour tables 1 to 3 included the values of w for criteria of iai, ipi, iepmi, ici, ihai, itli, iwci, imai, and ifi. as mentioned above, the values of w for the criteria are estimated by spss software using both friedman and kendall's tests. the friedman and kendall's tests are used to compare a variety of groups to find the ranks among them. introducing non-parametric criteria allows the software to investigate the homogeneity of variances of a variety of data in groups. there are lots of empirical equations to estimate the values of w without applying software. the iranian evaluator teams employed some experimental and empirical equations with a strong base in science and technology in order to estimate the initial values for the inputs and outputs composing the dea framework. therefore, conducting the consistency and reliability tests in the following friedman or kendall tests does not seem common in such studies because the quantities are neither changeable nor have been recognized as expert ideas. but homogeneity of data is a prominent point. kendall's w can support the findings and analysis of the friedman test and can be employed for evaluating agreement among values. so, a high agreement is distinguished among the initial values of data for all groups using kendall's w test [59, 60]. table 2 the w values for criteria of itli, and iai [46, 47] itli w iai w nc (t) 11.29 if (t) 9.87 nc (no) 13.71 if (no) 15.46 nc (m2) 10.75 if (l) 5.24 nc (m) 10.36 if (mm) 5.06 nc (skin) 9.26 if (m2) 5.89 nc (ft2) 9.68 if (m) 5.89 nc (pair) 10.11 if (m3) 4.98 nc (jean) 9.29 if (sheet) 4.99 employees 22.33 nc (m2) 5.05 nc (yard) 9.20 nc (no) 14.98 power (kw) 23.64 nc (t) 4.98 water (m3) 20.29 nc (pair) 5.08 land (m2) 20.87 employees 14.10 if (m2) 10.37 power (kw) 15.46 if (m) 10.30 water (m3) 12.13 if (no) 18.38 fuel (gj) 11.45 if (duke) 9.07 land (m2) 12.39 if (l) 10.03 if (pair) 9.53 if (yard) 9.67 if (thread) 9.11 if (ft2) 10.16 if (t) 18.16 fuel (gj) 20.22 if (sheet) 9.24 evaluation of iranian small and medium-sized industries using dea based on additive ratio... 501 due to the ease of application and high precision and accuracy of spss software to estimate the values of w, the criteria took the values in tables 1 to 3. also, the criteria initially provided the simple use of the decision model of aras-dea to normalize and uniform rates to offer an appropriate decision-making model for industry classification consequently efficiency score [59, 60]. table 3 the w values for criteria of ipi, iepmi, ici, and ihai [47-49, 11, 45] ipi w iepmi w ici w ihai w if (t) 8.02 nc (t) 4.59 nc (t) 9.58 if (t) 3.9 if (no) 7.4 nc (no) 13.17 nc (no) 8.8 if (no) 10 if (m2) 4.05 nc (m2) 4.56 nc (m2) 5.84 if (m2) 2.24 if (m) 5.05 nc (barrel) 4.38 nc (m) 5.79 if (m) 2.82 if (pair) 4.57 nc (crank) 4.61 nc (m3) 5.87 nc (no) 8.24 nc (m2) 4.52 employees 12.89 nc (pair) 5.79 nc (t) 2.4 nc (no) 6.98 power (kw) 14 nc (l) 5.69 employees 8.12 nc (t) 5.93 water (m3) 11.28 employees 15.30 power (kw) 9.7 nc (pair) 4.12 land (m2) 10.88 power (kw) 17.11 water (m3) 6.04 employees 11.69 if (m2) 6.09 water (m3) 13.66 fuel (gj) 5.82 power (kw) 12.90 if (m) 6.2 land (m2) 13.71 land (m2) 6.72 water (m3) 9.74 if (no) 15.66 if (m2) 6.06 fuel (gj) 9.57 if (l) 4.5 if (m) 6.04 land (m2) 10.45 if (pair) 4.97 if (no) 12.54 if (t) 8.45 if (l) 6.85 fuel (kw) 9.77 if (pair) 5.68 if (t) 12.50 fuel (gj) 14.09 the assay accomplished upon the w values for criteria showed the highest correlation around 0.872 between both groups of ifi and ipi industries with no significant differences among the values of w for the criteria of 9 group industries. the sequence diagrams typically do not display a regular graphical functionality of the values of w for criteria. 6. the values of w in the ranking systems the ranking systems applied to the classification of iranian industries produced values for the alternatives considering the criteria. to sum up, the dea-aras model resulted in the below w values. figs. 3 to 5 display the values of w in the ranking system for each cluster industry individually. assigning the aras model to the values of outputs and inputs streams containing different scales resulted in normalizing the values and uniting the quantities to appear as a sum via introducing special vectors of the values of criteria w separately for both inputs and outputs streams in industries. the division of output quantities to input quantities released the values of w in the ranking system. the highest values were devoted to industries of 7, 19 and 12 for the ihai, ipi and imai according to fig. 3, respectively. it conducted the same procedure for the remaining industries groups. 502 m. hassanpour fig. 3 the values of w in the ranking system for ipi, ihai, and imai [11, 46, 49] fig 4 the values of w in the ranking system for iwci and itli [46] according to fig. 4, the highest values of w belong to industries 14 and 31 for the iwci and itli, respectively. also, the highest values of w were devoted to industries 21, 10 and 30 for iai, ifi, and iepmi in fig. 5, respectively. evaluation of iranian small and medium-sized industries using dea based on additive ratio... 503 fig 5 the values of w in the ranking system for iepmi, ifi and iai [47, 48] ici encompassed a variety of industries dealing with the processing of chemical products. this cluster also comprises ipi industries in a certain cluster. but the homogeneity of data and propinquity of project availability in terms of technologies and materials characteristics led to composing a separate cluster for the ipi. the diversity of the values of w encouraged the display of the values in table 4. there is a significant difference around (p-value  0.003) for the ici among the values of w of itli, imai, iwci, ifi, iai, ipi, iepmi and ihai by conducting t-test. also, the highest correlation of about 0.672 was observed between the values of w of iai and iwci via the pearson correlation (sig. 2 tailed) test. 504 m. hassanpour table 4 values of w in the ranking system for ici [11] ici industry ici industry ici industry 0.0920 81 0.0341 41 0.1253 1 0.07768 82 0.0195 42 0.0422 2 0.0476 83 1.2130 43 0.0420 2 0.0186 84 0.0319 44 2.1930 4 0.0016 85 0.0353 45 0.0059 5 6.8818 86 0.0004 46 0.0131 6 0.0414 87 0.0281 47 0.0113 7 0.0293 88 0.0089 48 0.0455 8 5.17e-05 89 0.0504 49 0.0056 9 0.0004 90 0.0809 50 0.1411 10 0.0018 91 0.0653 51 0.0126 11 0.0749 92 0.0152 52 0.3291 12 0.0032 93 0.0156 53 11.526 13 0.0463 94 0.4197 54 9.08e-05 14 0.2009 95 0.1217 55 0.0033 15 0.0705 96 0.0044 56 0.0021 16 0.0552 97 0.07583 57 0.0183 17 0.0661 98 0.0010 58 0.0344 18 0.0115 99 0.0387 59 0.0089 19 0.0084 100 0.1119 60 0.0237 20 0.01194 101 0.0037 61 0.0042 21 0.1188 102 0.0843 62 0.0044 22 0.0221 103 3.6973 63 0.02185 23 0.0012 104 0.1663 64 1.9378 24 3.2977 105 0.0207 65 0.0088 25 0.5322 106 0.1006 66 0.2593 26 0.0164 107 0.5428 67 0.0240 27 0.0054 108 0.1102 68 0.1432 28 0.3924 109 0.1123 69 0.0298 29 0.0090 110 0.0069 70 0.3892 30 13.810 111 0.1267 71 0.0513 31 1.1911 112 0.0648 72 0.0196 32 0.1967 113 0.0426 73 0.01233 33 0.0582 114 0.0358 74 0.0092 34 0.0646 115 0.0312 75 0.0598 35 4.3686 116 4.3707 76 0.0073 36 0.0306 117 0.3729 77 16.449 37 0.0881 118 0.3351 78 0.0008 38 0.0032 79 0.8077 39 0.0133 80 0.2202 40 7. dea rank developed for 9 groups of industries at first industries' information was normalized by the equations of the aras model and then the values of w were released by both friedman and kendall tests assigned to alternatives (industries) in order to collect the final w in the ranking system for the outputs and inputs criteria separately. as discussed above, the weighted average of output divided to the weighted average of inputs generated a numeric value representing the evaluation of iranian small and medium-sized industries using dea based on additive ratio... 505 calculated value for the dea. since the raw data reported by the iranian evaluator team is for all the industries under study, these industries were classified into specific clusters (9 clusters). therefore, the dea rank displays for industries via figs. 6 to 8. fig 6 the dea rank developed for itli, imai, iwci, ipi, and ihai [46, 45, 49] according to fig. 6, the highest dea rank was assigned to industries 31, 12, 4, 19 and 7 for itli, imai, iwci, ipi and, ihai, respectively. the highest dea rank means the industry holding the highest value of w and vice versa. fig 7 the dea rank developed for ici and iai [11, 47] 506 m. hassanpour fig 8 the dea rank developed for iempi and ifi [48, 47] according to figs. 7 and 8, the highest dea rank was allocated to industries 37, 10, 21, and 2 for ici, ifi, iai, and iempi, respectively. clearly, the charts classified 405 iranian industries into 9 groups from the highest to lowest rank according to their annual energy and materials intake. the number of employees required annually is also one of the input factors considered in the calculation by the dea. to prove the findings the dea-aras model conducting a sensitivity analysis based on other mcdm models does not seem a reasonable experience because the results are very different for any model (like aras) in connection or integration with the traditional dea model. but the dea based on other additive ratio models can pursue the objective for determining the efficiency score and classifying industries in similar studies. a variety of reports pointed out that the dea model is an overwhelming procedure for measuring the operational and managerial performance of industries etc. [61]. for instance, the dea model was assigned to find the efficiency score of fifteen insurance companies in a time interval from 2005 to 2012. so, despite identifying the efficient companies, it showed dramatic variations between the values of technical efficiency estimated in the mentioned period [62]. the study by izadikhah et al. [63] determined the efficiency score of 17 suppliers depending on sustainability aspects such as economic, environmental and social factors encompassing 41 criteria. also, mirhedayatian et al. [64] employed a novel dualfactor dea model for supplier selection of 10 companies regarding about 5 main criteria and 26 factors. assessment of the efficiency of wind turbines applied a two-sub process dea model in china. by the way, the models took into consideration wind turbines depend on power generation efficiency [65]. a study managed to estimate the efficiency of around 40 retail stores of cloth companies using the dea model for a time interval from 2010-2013. the technical efficiency employed the regression analysis in the following steps [66]. to select and thrift the appropriate complex product systems of industries suppliers dilated a evaluation of iranian small and medium-sized industries using dea based on additive ratio... 507 dea model. the mentioned model has taken into account the economic, geographic and technical criteria to estimate the productivity in the optimum cost-effective situation. the attempt to examine the criteria sought the selection of the best suppliers among them [67]. to appraise the sustainability of milk packaging systems (prepack, composite box, tetra pak considering the crude oil and natural gas consumption rates, co2, coal, electricity and recycled materials) for commodities about fitness in terms of environmentally friendly aspects the dea model has taken into consideration [68]. the greek meat products investigated the efficiency score via the dea model from 1994 to 2007. so, mismanagement conveyed the industries towards inefficiency levels and mostly the wastage of assets [57]. the study performed by abri and mahmoudzadeh [69], mehdiabadi et al [70], asayesh and raad [71] and azbari et al. [27] utilized the dea, dea-topsis, dea, and dea to appraise the impact of information technology on productivity and efficiency, the rank of efficient units, evaluation of the relative efficiency and supply chain performance in 23 iranian manufacturing industries during 2002–2006, 15 various sectors of industries, gas stations, and one certain iranian pharmaceutical industry, respectively. so, the achievements noticed to it had a positive and statistically significant effect on the productivity of manufacturing industries; 8 efficient units of chemical industries could be considered as the most attracting industries for investment, raking gas station, along with hierarchical classification and latent variable had the highest correlation with supply chain performance and 12 of the 28 supply chains obtained 1 as the highest performance rate and the lowest cases around 0.81, respectively. therefore, the literature proves the vast application of dea techniques in other sciences and techniques [72]. 8. conclusion to the best of our knowledge, this is the first report and research which encompassed whole iranian industries passing through the dea model prior to complete construction. the dea model was found as a strong decision-making model. the present review used the traditional dea model united with the aras model to release the efficiency score due to the presence of various criteria containing different scales. the efficiency score besets the energy consumed and materials entered into industries loops. the prioritization of industries is a way towards classification of existing and upcoming industries pertaining to nc. calculating the values of w via software found the best procedure with high accuracy and precision in this regard. shifting the criteria worth in currency is a suggestion for forthcoming research projects. also, developing any novel and modern technique can be taken into consideration in future studies. the defined input and output criteria can be considered along with intermediate options to deploy new and complex models of the dea and explore efficiency in a more enhanced way. likewise, the industry's information is a useful resource of energy and materials streams and the best media to deploy the relevant database to the comparison of industries at local and global levels depending on the nc and running technologies. accordingly, this review provides a benchmarking for the classification of all industries before construction for future plans in the eia program as well as paving the way towards simplicity in economic and financial studies as well as offering the frameworks of sd for effective expansion. the role of evaluator teams is highlighted by the current study in the 508 m. hassanpour screening plan of the eia program. additionally, the importance of database designing is clear for both the iranian industry's organization and the iranian environment protection agency. the energy and materials stream management 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knowledge enhancement related to am by presenting the details of experimental investigation conducted on fdm of three different materials i.e. polylactic acid (pla), acrylonitrile butadiene styrene (abs) and carbon fiber (epa-cf) and providing the common optimum fabrication parameters for fdm of pla, abs and epa-cf materials. a detailed analysis of the effect of two important fdm parameters, namely infill pattern (ip) and layer thickness (lt), on mechanical properties of the fabricated samples from the materials, has been done. the best mechanical properties of the pla sample were obtained with lt of 0.15 mm and hexagon pattern, of the abs sample were obtained with lt of 0.15 mm and triangle pattern, and of the epacf sample were observed at lt of 0.15 mm and hexagon pattern. it was determined that elongation, yield (ys) and tensile strength (ts) decreased with increasing lt in all materials’ samples. while the fracture surfaces of pla samples were smooth, and the layer lines were evident. whereas the fracture surfaces of abs samples were rough, and the lines were less distinct than pla. on the other side, the fracture surfaces in epa-cf samples had a fibrous structure. the voids formed between layers were seen the most in pla and the least in epa-cf samples. lt of 0.15 mm and hexagon ip. it was observed that the best results in terms of mechanical properties were obtained under lt of 0.15 mm and ip of hexagon conditions. key words: additive manufacturing, fracture, mechanical properties, fdm, pla, rapid prototyping received: august 01, 2024 / accepted september 25, 2024 corresponding author: şenol bayraktar mechanical engineering, faculty of engineering and architecture, recep tayyip erdoğan university, rize, türkiye e-mail: senol.bayraktar@erdogan.edu.tr 2 c. alparslan, ş. bayraktar, k. gupta 1. introduction in recent times, there has been a growing interest in technologies related to am. am technologies are becoming a promising alternative to traditional production methods such as machining, casting, and injection molding, because of their significant benefits such as suitability for mass production, affordability, customization, and quick prototyping [1-3]. the most frequently utilized am techniques are fdm and selective laser sintering (sls). these methods offer remarkable advantages over traditional manufacturing techniques [48]. these advantages include properties such as low initial costs, material diversity, fast manufacture processes and the ability to easily produce complex parts [9-11]. parts manufactured with these methods do not necessarily additional surface engineering processes such as heat treatment, chemical surface modification or mechanical processing. fdm and sls are ideal for applications in sectors like medical, aviation and automotive, by enabling the manufacture of lightweight, high-strength and customized parts [12-15]. additionally, fdm technology allows for the production of parts without geometric limitations and removes the necessity for milling or other post-processing steps commonly required in most metal-based 3d printing technologies. in this way, it makes it possible to reduce or eliminate stress in the produced material. fdm is emerging as a technology used in the production of workpieces with dimensional accuracy, repeatability and high strength. lighter and three-dimensional complex geometries can be easily obtained with the effect of design freedom. for this reason, it is used in the production of prototypes for components used in the aviation, military and automotive sectors before the final product. as a result, faulty product costs are minimized in industrial businesses and the chances of businesses competing in the global market increase. fdm, also referred to as filament freeform fabrication (fff), involves extruding plastic workpiece through heated print equipment. components are built with continuously layering molten thermoplastic workpiece. depending on the geometric structures, additional support structures may be required. initially utilized for creating low-cost prototypes and concept models, this method is now favored for producing high-quality functional prototypes and concept models [16, 17]. the most used materials for fdm 3d printing include abs, polycarbonates (pc), acrylonitrile styrene acrylate (asa), and pla. with the developments in material science, the materials used in the fdm method are constantly changing. the integration of carbon fiber, graphene, and other composite materials importantly develops mechanical properties of parts manufactured with fdm, making them more durable, lightweight, and functional. such developments are shaping the future of am technologies, providing access to wider application areas and industrial sectors [18]. pla turns into harmless lactic acid over time [19]. therefore, it is immunologically inactive [20]. it is preferred for use in the medical implant (screw and pin) sector due to this feature. abs is becoming attractive in the industry to produce demanding consumer products due to its excellent impact and mechanical strength, dimensional stability and electrical insulation properties. therefore, it provides significant advantages in the production of safety helmets, automotive dashboards and other interior components, pipe fittings, home security devices and small household appliances and communication equipment [21]. pa-cf composites are used in the production of equipment such as automotive brackets, inspection gauges, robotic end effectors, and under-hood applications in the automotive industry. in addition, the use of these materials in the industry can contribute to cost reductions and improved operational efficiency [22]. a comparative study on mechanical performance of pla, abs, and cf materials... 3 we have reviewed some previous studies conducted on fabrication of various parts and components using fdm method. yu et al. created specimens using the fdm method by blending pla and pbs (polybutylene succinate) filaments together [23]. they examined the effect of pbs addition on pla filaments with these samples. they concluded that among pbs/pla mixtures, 10% pbs/pla is an ideal mixture for biodegradable raw material. sedlacek and lasova [24] investigated the effect of short carbon fibers on the nylon pa6 polymer used in the fdm process. they examined the mechanical properties of samples printed horizontally and vertically with the fdm method. they found that short cfs significantly affected force (12%) and heat resistance of pa6 components. comparing the longitudinal reinforcement of fibres in pa6 with the transverse reinforcement, they determined a 39% increase in strength and tensile modulus. farashi et al. [25] conducted a meta-analysis study to investigate the effect of the ps and extruder temperature used in the fdm method on the tensile strength of the produced samples. they carried out their search strategies using appropriate search terms and statistical calculation methods. as a result of the study, it was seen that rising extruder temperature had favorable impact on tensile strength (standard mean difference= 2.85, 95% confidence interval [1.71, 3.98]), while increasing printing speed had a negative effect (smd= 2.49, 95% ci [-4.29, -0.69]). bochnia et al. [26] researched mechanical features of fdm parts obtained from pla and pla-cf reinforced materials in terms of thin-walled practices. in their studies, samples with different thicknesses (1, 1.4 and 1.8 mm) and printing orientations (x, y and z) were subjected to tensile testing for two materials. they determined that cf-reinforced pla samples had superior strength than samples made of pure pla. additionally, they observed a 40% lower tensile strength in the z (vertical) direction compared to other sample types. chalgham et al. [27] examined mechanical features of pla parts manufactured by fdm method before and after heat treatment. three-point bending tests were conducted to assess the effect of parameters like layer thickness (lt), printing temperature (pt), and ps on the mechanical properties. they found that the optimum parameters were in the x-z printing direction, spraying temperature of 190 oc, ps of 90 mm/s and lt of 0.3 mm. they stated that the maximum force during bending increased by 2% as a result of subjecting the samples produced with optimum parameters to 75 oc thermal post-treatment. samykano et al. [28] conducted the effect of three different manufacture parameters such as layer height, scanning angle and filling density on the mechanical properties of abs samples in the fdm method. they found that the optimum production parameters were filling rate of 80%, lt of 0.5 mm and scanning angle of 65o. they developed a mathematical equation using surface response methodology (r2: 85.80%) to numerically predict the tensile features of abs workpiece and determine optimum production parameters. khabia et al. [29] studied a comparative study on the mechanical features of components produced with three different fdm printers (accucraft i250, arya uno+, and zortrax m200) using abs filament material combinations (low-cost abs filament and z-abs). they tested tensile parts according to astm d 638 standard on a 3382 electromechanical universal testing system. they determined that the sample exhibiting the best mechanical properties (z-abs filament material) was according to the production parameters found in the ultimaker cura 3.6 slicing software. pazhamannil et al. [30] evaluated the effect of production parameters like filling ratio, infill pattern (ip), extruder temperature, lt, ps and heat treatment processes on mechanical features of pla and cf-pla components prepared by the fdm method. they determined that the maximum mechanical properties of both filaments occurred at ps of 30 mm/s and infill rate of 90% for the gyroid pattern with layer height of 4 c. alparslan, ş. bayraktar, k. gupta 0.1 mm. they found that tensile properties of cf reinforced tensile samples were higher than those of standard polylactic acid. additionally, after an annealing process at 95°c for 60 minutes, they observed a 14.01% increase in the maximum stresses of pla-cf materials. reverte et al. [31] analyzed the mechanical and geometric properties of pla and pla-cf components prepared with fff technique. they carried out tensile, bending and ilss (inter laminar shear strength) tests to obtain mechanical performance of dissimilar specimens. they observed that addition of cfs enhanced the mechanical characteristics of pla-cf composites compared to pure pla. dhinesh et al. [32] investigated different types of raw materials (pla and abs combination) to provide better mechanical features of parts manufactured using am technology. in their study, they produced samples with various compositions using the fdm method in order to present a new filament that could withstand higher tensions than traditional filaments. these different compositions (20% abs and 80% pla; 50% abs and 50% pla; 50% abs and 50% pla; 50% abs and 50% pla; 20% abs and 80% pla; 80% abs and 20% pla) were researched for their resistance to tensile and bending loads. as a result of the study, they found that the abs and pla sandwiching concept exhibited better mechanical properties. am technologies, despite bringing significant innovations to production processes, cannot ignore the challenges and limitations they face. factors such as the properties of materials used in the manufacturing process, printing techniques, and surface quality can directly impact the performance of the final products. anisotropy occurring during the printing process can lead to inconsistencies in the mechanical properties of manufactured parts, which can limit their use in industrial applications. in this context, research and development are focused on advancing am technologies, overcoming these challenges and further improving processes. in the future, with the evolution of am technologies and the exploration of new material combinations, it is expected that this technology will have the capacity to revolutionize industrial manufacture more broadly. in addition, am offers significant advantages in issues such as sustainable production and reduction of environmental impacts. by reducing material waste and increasing energy efficiency, am is poised to play a key role in sustainable manufacturing solutions. with these features, am technologies will continue to have a central position in the manufacturing paradigms of the future. with an aim to contribute to the knowledge base and to develop a deeper understanding of the fdm of different materials, the present research work investigates the effects of different infill patterns (hexagon, triangle and line) and layer thickness (0.15, 0.25 and 0.35) on the mechanical properties of parts made of pla, abs and epacf filaments and fabricated by fdm method. different types of filament materials, as mentioned above, were chosen because they are widely used in industry and can provide the required mechanical properties suitable for a wide range of applications. in this study, we also developed the understanding of critical parameters to optimize the performance of components produced with fdm and to determine common production parameters for three different materials. a comparative study on mechanical performance of pla, abs, and cf materials... 5 2. materials and methods 2.1 material and 3d printing system the experiments were carried out on samples produced using commercially available filaments based on pla, abs, and epa-cf. epa-cf material consists of a nylon 6/66 copolymer with 20% carbon fiber (esun, inc. china). mechanical properties of materials supplied by the manufacturer were given in table 1. table 1 mechanical features of pla, abs and epa-cf [33] properties pla abs epa-cf specific gravity (g/cm3) 1.2 1.04 1.24 tensile strength (mpa) 72 43 140 elongation (%) 11.8 22 10.61 print temperature (°c) 210-260 230-270 250-300 print speed (mm/s) 40-100 40-100 40-100 the materials were supplied as esun [34] brand in the filament wound on a 1.75 mm diameter spool. pla, abs and epa-cf samples were produced using the creator 3 desktop 3d printer manufactured by flashforge [35]. the flashforge creator 3 is equipped with features such as a heated build plate and replaceable nozzles, which enable the use of wide range of workpieces. 3d printing system consists of a heated build plate, an extruder, a temperature control system, a mechanism capable of movement along the xyz axes, and two heated print heads with various nozzle diameters. details of these components of the printer are depicted in fig. 1. fig. 1 schematic representation of built orientation of samples in fdm of abs, pla, and epa-cf 6 c. alparslan, ş. bayraktar, k. gupta table 2 flashforge creator 3 technical specifications parameters value extruder number 2 print resolution ± 0.2 mm layer resolution 0.05-0.4 mm print speed 10-150 mm/s maximum extruder temperature 300 co minimum extruder temperature 120 co table 2 displays technical specifications of the flashforge creator 3 3d printing equipment. samples were modeled with cad software. the modeled files were transferred to the 3d printer as stereolithography files (stl) with good resolution and 0.0171 mm deviation. flashprint 5 software was utilized to import stl files and generate g-code files compatible with the requirements of the flashforge creator 3. this process reveals the 3d printer's ability to work harmoniously with different materials such as pla, abs and epacf, and its efficiency in production. 2.2 fdm machine settings and specimen fabrication the major aim of the study is to analyze the mechanical impact of different lt’es and ip’s on various types of materials. experimental preliminary studies were conducted, taking into account the information provided by the material manufacturer. in this manner, production was repeated with varying parameters, and choosing of fdm printing settings, which influence geometric and mechanical performance of workpieces produced from abs, pla, and epa-cf samples, was determined collaboratively (table 3). tensile test specimens were prepared according to astm d638-14 standard type iv (fig. 2) [36, 37]. a total of 81 tensile samples were fabricated. the samples were prepared according to three different lt (0.15, 0.25 and 0.35) and ip (hexagon, triangle and line). table 3 fdm process parameters considered in the present work parameters value speed of base (mm/s) 60 printing speed (mm/s) 80 height of layer (mm) 0.15, 0.25 and 0.35 printing temperature (oc) 260 platform temperature (oc) 100 nozzle diameter (mm) 0.4 slice profile fine infill pattern hexagon, triangle and line infill density 100% a comparative study on mechanical performance of pla, abs, and cf materials... 7 fig. 2 dimension and shape of specimens utilized for tensile test (dimensions are in mm) irrespective of the 3d printer model in am technology, it is essential to understand how printing thickness and direction affect the mechanical properties of the constructed models. thanks to this, it is possible to correlate the test results presented with different types of printers. fig. 3 shows the variably determined printing parameters. fig. 3 process parameters for fdm printed pla, abs and epa-cf specimens, (a) ip types, (b) lt 8 c. alparslan, ş. bayraktar, k. gupta the tensile samples of the printed examples on the flashforge creator 3 fdm printer are given in fig. 4. upon completion of the printing process, the support material was detached from the printing plate, and any remaining filaments were removed prior to cleaning the models. the geometries of the samples were re-measured after production to ensure compliance with the astm d638-14 type iv standard. this process is crucial to evaluate whether the produced parts provide the specified features and tolerances. this control stage is of great importance in ensuring the accuracy and production quality of 3d printing, as well as ensuring that the final products exhibit the expected performance in their application areas. finally, tensile experiments were conducted on samples. as a result of tensile experiments, the test results were determined with the arithmetic average of ys, ts and elongation. fig. 4 arrangement of parts on the printing plate of the creator 3 print orientation 2.3 procedure for mechanical properties testing tensile tests of pla, abs, and epa-cf specimen were conducted on the utest-profi x6 tensile testing device at constant loading rate of 5 mm/min (fig. 5). test samples were subjected to tensile tests according to each ip and lt. as a result of the tensile tests, ys (n/mm2), ts (n/mm2) and elongation (%) values were obtained. mechanical tests were carried out by following the en iso/iec 17025 standard. in addition, three samples were produced for each parameter to ensure the reliability of the experimental results in the mechanical tests. the final elongation, yield strength, and tensile strength values were determined by the arithmetic average of the results obtained from these samples. fractured surfaces were analyzed using scanning electron microscope (sem). for analysis, all specimens were initially cut to dimensions of 10×5×3 mm and prepared for sem analysis. to make samples suitable for sem analysis, conductive coating is often required. in this process, the surface of the specimen was coated with the sputter coater method (quorum sc7620 machine, 6 minutes, chamber pressure of 200 bar, argon gas and current of 15ma). a comparative study on mechanical performance of pla, abs, and cf materials... 9 this process ensures that the sample becomes conductive under the electron beam, improving image quality and preventing high-energy electrons from damaging the sample. fig. 5 utest profi x6 tensile testing machine 3. results and discussion mechanical features of workpieces produced with fdm technology vary depending on the printing parameters used. these variations are crucial concerning the broad applicability and performance of printed parts. changes in manufacture parameters can directly affect tensile strength, which is a critical mechanical property that expresses the resistance of a material to tensile force. the main fdm parameters that can be adjusted during printing include layer height, infill density, ps and temperature. each of these parameters can influence the material's behavior during printing and the microstructure of the final part. these changes in microstructure are reflected in the macroscopic mechanical features of workpiece. experiments for commonly used fdm filament materials have shown that printing parameters have significant effects on elongation, ys and ts. 3.1 effect of fdm variables on mechanical features it was observed that ts and ys decreased in pla, abs and epa-cf samples with increasing lt (fig. 6). in pla samples, the highest ys (18.95 mpa) and ts (20.45 mpa) were obtained in the hexagon pattern and lt of 0.15 mm, while the lowest ys value (10.17 mpa) was measured in line pattern and lt of 0.35 mm, and the ts value was (8.48 mpa) in the triangle pattern and lt of 0.35 mm (fig 6a). in abs samples, the highest ys (17.11 mpa) was obtained in the hexagon pattern and lt of 0.15 mm, while the highest ts (19.97 mpa) was determined in the triangle pattern and lt of 0.15 mm. the lowest values (ts: 16.90 and ys: 12.52 mpa) were observed in the triangle pattern and lt of 0.35 mm (fig. 10 c. alparslan, ş. bayraktar, k. gupta 6b). in epa-cf samples, the highest ys (22.11 mpa) and ts (25.25 mpa) were obtained in the hexagon pattern and lt of 0.15 mm, while the lowest values (ts: 15.96 and ys: 14.53 mpa) were obtained in the line pattern and lt of 0.35 mm (fig. 6c). it is well-established that carbon fiber reinforced materials exhibit higher strength, attributed to strength of fibers and bonding between the fibers and the matrix [38]. additionally, it is thought that as a result of incorporating carbon fibers into the composite material, the load-carrying capacity of the material increases, thus minimizing deformation under load [39]. moreover, the uniform dispersion (homogeneous) of carbon fibers contributes to an elevated degree of crystallization, effectively preventing localized stress concentrations. therefore, enhances the material's resistance to fracture [40]. upon detailed examination during production, it was observed that the accumulation of individually laid fibers layer by layer results in the creation of a layer through integration of polymer bands within, facilitated by the thermal diffusion process. in this context, it was determined that, in addition to interlayer relationships, layers exist adhesion among the accumulated fibers within a specific layer. changes in the number of structural layers within all materials have been observed to more significantly influence elastic and plastic stress properties as lt increases. this is thought to be due to the combined adhesion force between the intermediate layers. fig. 6 ts and ys depending on lt, (a) pla, (b) abs, (c) epa-cf it was observed that with increasing lt in pla, the elongation decreased by 22.96 (hexagon), 5.05 (triangle) and 11.07% (line) patterns, respectively (fig. 7a), while it a comparative study on mechanical performance of pla, abs, and cf materials... 11 decreased by 23.64 (hexagon), 6.35 (triangle) and 6.96% (line) patterns for abs (fig. 7 b). it was also observed that the epa-cf material decreased by 30.06 (hexagon), 32.33 (triangle) and 33.06% (line), respectively (fig. 7c). the findings revealed that material features were considerably developed by use of certain parameters like lt and ip, with the highest tensile strength values recorded being achieved. however, the effect of lt on elongation, ts and ys cannot be generalized as a linear or one-way relationship. the effect may vary depending on the 3d printing technology used, printing parameters and material properties [41, 42]. when evaluated in general, it was observed that the best results in terms of mechanical properties were obtained under lt of 0.15 mm and ip of hexagon conditions. fig. 7 elongation depending on lt, (a) pla, (b) abs, (c) epa-cf 3.2 influence of fdm parameters on fracture surface characteristics regardless of the ip, it was observed that samples with increased lt exhibit behavior closer to a brittle fracture type. sem images were captured to gather data on impacts of reinforcement elements and structure orientation on morphology of accumulated layers (fig. 8). in pla and abs samples (fig. 8a and b), it was seen that the flat orientation, where the fracture surface was located on a plane, shows brittle behavior. the crosssectional shapes of the layers, their periodicity and formation of intrafilamentary voids between the accumulated strips were determined. this is believed to be due to the lower 12 c. alparslan, ş. bayraktar, k. gupta part of the sample sticking together because of the softened nature of melted filament passing through nozzle during extrusion, while the upper part exhibits a circular, convex structure after cooling [43]. these voids may be the reason for low tensile strength (fig. 8a and b). some air voids between layers appear to be more evident for the pla sample (fig. 8a). fig. 8 fracture surfaces of the examined workpieces, (a) pla, (b) abs, (c) epa-cf when the pla material (fig. 8a), was examined, it was seen that the fracture surface was smooth, and the layer lines were clear. this indicates that the breakage occurs due to interlayer separation caused by insufficient bonding between layers during printing [4446]. the fracture surface of abs samples was observed to be rougher and exhibits distinct layer lines compared to pla samples (fig. 8b). this is evident that fractures in abs materials generally occur at the junctions of the layers. because these regions are weak compared to the general material structure. furthermore, fracture is thought to occur along the direction in which layers of material are stacked. it shows that this situation is due to the anisotropic structure of abs materials [47]. a comparative study on mechanical performance of pla, abs, and cf materials... 13 additionally, whitening occurs in abs materials under strain (fig. 9). this is a sign that the material has undergone plastic deformation before fracture. during tensile testing, stress whitening may occur, especially in areas close to the breaking point. in most scenarios, fracture of thermoplastic components is characterized by stretching of filamentous macromolecules, allowing a high degree of deformation in the material [48]. this situation is associated with the rearrangement and elongation of the mentioned macromolecules. moreover, the fracture surfaces of thermoset polymeric components generally do not show significant plastic deformation due to the high degree of crosslinking between their macromolecules. this is thought to occur as a result of cross-links forming a tight bonding between macromolecular chains and thus limiting the plastic flow of the material [49-51]. for the epa-cf material, carbon fiber reinforcement material is clearly visible on the fractured surfaces (fig. 8c). this, in addition to high strength and hardness, significantly affects the fracture mechanism of the material [52]. additionally, directional fractures were observed on the fracture surface for the epa-cf material. this situation is thought to be due to the material exhibiting anisotropic properties [53-55]. it seems that fiber orientations and lengths are the most effective factors in determining mechanical properties in carbon fiber reinforced materials [56, 57]. in addition, a damage mechanism occurs due to the separation and pulling of fibers from matrix and local deformation occurs in matrix around fibers. when the sem images were examined in detail, no elongation, wear and characteristic voids near some fibers were observed in the fiber-reinforced matrices. it was found that most of cfs were not aligned in the direction of the print layers, as if they were pieces broken off from longer fibers. no printing layer could be observed in the epa-cf material. the layers were roughly identified and determined based on orientation of fibers (fig. 8c). the increase in lt is tough to result in a less homogeneous structure for the epacf material, consequently weakening the connections between layers. this is proven by the decrease in the tensile strength of the part. it is also thought that thicker layers are more pronounced and may increase the roughness of the fracture surfaces. fig. 9 fracture surfaces of the abs specimen 14 c. alparslan, ş. bayraktar, k. gupta more material deposition is required in all samples produced by increasing the lt. therefore, a long printing time is required. longer printing times result in increased thermal effects on the produced samples. this causes thermal stresses in the samples. it is thought that these stresses cause cracks to form on the fracture surfaces and change the fracture behavior of the part. fig. 10 microscopic images of the orientations of hexagon, triangle and line patterns a common defect in fdm manufacturing is the development of internal stresses within the workpiece during printing, which is attributed to the rapid cooling and heating cycles. this situation brings about uneven temperature changes, resulting in the formation of residual stresses. fast cooling ensures rapid solidification of the accumulated layer. when newly extruded filament is deposited onto solidified layer, a local re-melting impact occurs, facilitating bond formation between the filament and solidified layer [58]. however, as the lt increases, the cooling and solidification rate decreases. this causes uneven heating and cooling of the workpiece. due to this uneven temperature change, rise in both previously laid layer and the newly deposited layer. these stresses affect dimensions and shape of final workpieces. materials can exhibit various distortions, including transverse or longitudinal shrinkage, bending, and angular distortion. it is thought that deterioration can be prevented by changing the ips of the samples. because ip structures change the number a a comparative study on mechanical performance of pla, abs, and cf materials... 15 of connections (number of points) and shapes between layers in the samples. it is known that this situation will provide a higher mechanical strength to the material [59]. for this reason, it is thought that internal stresses can be minimized in samples produced by using the optimum ip and lt [60]. microscopic images for hexagon, triangle and line patterns are given in fig. 10. these images show surface conditions according to different infill pattern and height of layer characteristics. orientation lines and small voids between layers were observed on the surfaces of the examined materials. it is thought that these surface conditions occur depending on the production characteristics in line with the independent variables used, as mentioned before. 4. conclusion in this research investigation, mechanical features of pla, abs and epa-cf materials produced with fdm method in different ip and lt were experimentally investigated. it was determined that workpiece with the best mechanical features among the samples produced was the one produced with epa-cf in a lt of 0.15 mm and a hexagon pattern. the best mechanical properties for the pla sample were determined with a lt of 0.15 mm and a hexagon pattern. it was seen that the worst mechanical features were given in lt of 0.35 mm and triangle pattern. it was found that the abs sample exhibited the best mechanical properties with a lt of 0.15mm and a triangle pattern. conversely, the worst mechanical properties were observed with a lt of 0.35 mm and triangle pattern. for the epacf material, the best mechanical properties were observed in samples manufactured with lt of 0.15 mm and a hexagon pattern. the worst mechanical properties were detected in samples with lt of 0.35 mm and line pattern. it was revealed that tensile, ys and elongation values decrease as a result of increasing lt. it was stated that the air voids between the layers occurred the most in the pla sample and the least in the epa-cf sample. the fracture surfaces of samples produced with pla were found to be smooth with distinct layer lines. it was understood that this was due to insufficient bonding between layers. it 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imaging, hybrid techniques and inverse problems, proceedings of the 2017 annual conference on experimental and applied mechanics. springer. 60. zhang, w., wu, a.s., sun, j., quan, z., gu, b., sun, b., chou, t.w., 2017, characterization of residual stress and deformation in additively manufactured abs polymer and composite specimens, composites science and technology, 150, pp. 102-110. facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 473 490 https://doi.org/10.22190/fume200406035s © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper edm process parameter optimization for efficient machining of inconel-718 ankit singh 1 , ranjan kumar ghadai 1 , kanak kalita 2 , prasenjit chatterjee 3 , dragan pamučar 4,5 1 department of mechanical engineering, sikkim manipal institute of technology, india 2 department of mechanical engineering, vel tech rangarajan dr sagunthala r&d institute of science & technology, india 3 department of mechanical engineering, mckv institute of engineering, india 4 informetrics research group, ton duc thang university, ho chi minh city, vietnam 5 faculty of civil engineering, ton duc thang university, ho chi minh city, vietnam abstract. in the present work, multi-response optimization of electro-discharge machining (edm) process is carried out based on an experimental analysis of machining superalloy inconel-718. the study aims at optimizing and determining an optimal set of process variables, namely discharge current ( ), pulse-on duration ( ) and dielectric fluidpressure ( ) for achieving optimal machining performance in edm. nine independent experiments based on l9 orthogonal array are carried out by using tungsten as the electrode. the productivity performance of the edm process is measured in terms of material removal rate (mrr) and its cost parameter is measured in terms of tool wear rate (twr) and electrode wear rate (ewr). the topsis is used in conjunction with five different criterion weight allocation strategies— (namely, mean weight (mw), standard deviation (sdv), entropy, analytic hierarchy process (ahp) and fuzzy). while mw, sdv and entropy are based on the objective evaluation of the decision-maker (dm), the ahp can model the dm’s subjective evaluation. on the other hand, the uncertainty in the dm’s evaluation is analyzed by using the fuzzy weighing approach. key words: edm, parameter optimization, mcdm, topsis, criteria weight received april 06, 2020 / accepted august 03, 2020 corresponding author: prasenjit chatterjee department of mechanical engineering, mckv institute of engineering, india e-mail: prasenjit2007@gmail.com 474 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar 1. introduction over the past two decades, an increasing demand for quality finishing and intricate shapes of high strength materials has led to global interest in non-traditional machining. electrical discharge machining (edm), one of the popular non-traditional machining approaches, is at the forefront and perhaps most robust in modern-day machining technology [1]. it is widely used in industries for cutting intricate shapes, grooving patterns, microholes, boring, craters in alloys and superalloys like inconel, haste alloy, etc. unlike the traditional machining approaches, edm is a non-contact metal removal process that works on the principle of thermo-electrical erosion which is controlled by high-frequency pulses generated in a dielectric medium. it has a unique ability to eliminate mechanical stresses, chatter and vibration issues without being in contact with the working material. in general operation, it develops a significant potential difference between an electrode (tool) and the workpiece, resulting in melting, vaporization and removal of work material debris [2]. edm is advanced machining; hence it is highly preferred for materials difficult to cut like superalloys. though a significant amount of work has been carried out on machining of superalloys with mcdm approaches in edm and wire-edm, optimization analysis related to its machining efficiency with different electrodes and dielectric has not been adequately explored. inconel-718, one of the toughest materials is a ni-based superalloy with a chemical composition of c0.8%, mn0.35%, ni54%, cr20%, ti0.75%, and fe [3, 4]. it is extensively used in making gas turbine blades, jet engine parts, ballistic missiles and automotive applications. its ability to retain the same texture and hardness throughout various temperatures makes it suitable for robust engineering structures. despite being rich in carbides, abrasive contents and high resistance properties, inconel-718 displays sparse heat dissipation and shows poor machinability, resulting in blur formation and rough surface finish while machining in edm. this can be attributed to poor thermal conductivity and rapid work-hardening of inconel. these challenges make it extremely hard to machine even in edm and thus raises concern regarding the usage of a proper electrode, dielectric as well as optimized process parameters. this is where multi-criteria decision-making (mcdm) approaches come handy. several researchers over the past few years have tested different materials for tools and dielectrics for edm operations. while a number of them have made use of metaheuristic approaches like genetic algorithm (ga), particle swarm optimization (pso) etc., others have made use of mcdm approaches like topsis (technique for order performance by similarity to ideal solution), psi (preferential selection index), and wsm (weighted-sum method) in order to improve edm response parameters. mohanty et al. researched inconel 781 using copper, graphite and brass tools in edm. in their study, they observed that the tool material, discharge current and pulse on-time affect machinability of inconel on a wide range. further, they showed experimentally that mrr increases monotonically while using a graphite tool followed by copper and brass [5]. lin et al. investigated the effect of tungsten carbide tool on process parameter optimization of inconel 718 using grey–relation and greytaguchi mcdm technique [6]. kumar et al. studied the effect of brass wire edm on inconel 718 using taguchi l27 with a multi attributed simulated annealing (sa) algorithm. they found that pulse-on time and gap voltage have a more considerable influence on kerf and mrr of material. vikas et al. [7] studied the mcdm methods to optimize process parameters of inconel-718 using the cu-cadmium tool in edm. they applied taguchi l9 to the design variable dataset and used topsis and promethee to compare the responses edm process parameter optimization for efficient machining of inconel-718 475 based on ranking. rahul et al. [8] carried out a research study to determine the appropriate setting of process parameters like gap voltage, discharge current, and flushing pressure for optimal machining of inconel 718 in edm. they used taguchi l25 orthogonal array with pca-topsis to measure performance characteristics, i.e., mrr, ewr, surface roughness, scd and wlt. they found that the peak current plays the most significant role while the flushing parameter is of least interest in machining optimization. huang and liao [9] performed an experimental study in order to optimize machining parameters of wireedm using grey relation and statistical analysis. using taguchi l18 orthogonal array, they found that the feed rate has a significant effect on mrr while the gap width and surface roughness were influenced by pulse on-time. joy et al. [10] performed process parameter optimization using taguchi l9 and promethee approach for inconel 718 in edm. mcdm method was found to be successful in the combined optimization of mrr & twr. singaravel et al. [11] performed the turning process optimization of en25 steel using taguchi and topsis mcdm approach. chakraborty and das [12] used a multivariate quality loss function approach to optimize several non-traditional machining processes. they also used a superiority and inferiority ranking method to identify the best parametric combination of a green edm process [13]. chakraborty et al.[14] used topsis to optimize edm and wedm process of inconel 718 machining. farshid [15] carried out an experimental study on inconel 718 in edm and used an artificial neural network (ann) in conjunction with ga to predict machining conditions and optimize the edm process. he found that mrr is more influenced by the process conditions as compared to surface roughness. both current and pulse on-time are effective for mrr enhancement but the gap voltage is highly influential for mrr. implementation of ann-nsga (artificial neural network-non-sorting genetic algorithm) can efficiently optimize the process conditions. mandeep and hari [16] experimentally studied machining of inconel x-750 using wire-edm. taguchi and grey relation methods were used to optimize the process variables. though the above literature survey shows that a significant amount and a diverse quality of work have been done so far in order to optimize the process parameters for inconel-718 with mcdm, considerable lacuna still remains.  relatively very few studies have been conducted with shannon entropy distribution and improved-ahp with topsis for predicting accurate results.  no comparative study on the application of various subjective and objective weights to experimental data is seen.  very few papers on the comparative assessment of the ranking performance of topsis on the application of fuzzy weights and non-fuzzy weights are seen. therefore, in this paper, an attempt is made to determine an optimal combination of process variables (namely, discharge current, pulse on-time, and dielectric effect) that would effectively increase the mrr while decreasing the twr and ewr. topsis as a widely known and preferred mcdm approach is used in this research as it uniquely converts multiple responses into a single performance criterion at a low computational cost. though the conventional topsis is reliable and easy to implement, it is susceptible to uncertainty and personal influences. entropy and ahp weights are competent in improving the topsis performance through decision/weight matrix via discrete probability distribution and pairwise relative comparison matrix, respectively. these weighted distributions avoid biasing at each level, i.e. experimental/personal and converge the data with consistency and a high degree of accuracy when combined with general topsis steps. this research covers two crucial 476 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar objectives— (1) experimental evaluation of suitable process parameters using an l9 orthogonal array with performance measures being mrr (higher the better), twr (lower the better) and ewr (lower the better), (2) comparison of traditionally used topsis (i.e. mwtopsis) with sdv-topsis, ahp-topsis, entropy-topsis and fuzzy-topsis. 2. materials and methods 2.1. experimental details in this research study, inconel 718 superalloy of square cross-section (100mm×100mm ×4mm) as shown in fig.1a was used as the work material for machining in a die-sinking edm (sparkonix mos, 35a, znc) shown in fig.1b. the mechanical strength and chemical composition of inconel 718 superalloy can be retrieved from [8]. a cylindrical pure tungsten electrode (tool of diameter 2 mm and length 4 cm) as shown in fig.1c and edm oil (dielectric medium) of density 0.764 were used to perform the procedure. the experimental conditions with process parameter levels, i.e., the input responses for controlled operation are depicted in table 1 with a duty cycle of 50%. fig. 1 (a) inconel 718 workpiece (b) tungsten tool (c) machining of inconel with tungsten tool in edm oil table 1 process parameters and their levels process parameter units level 1 level 2 level 3 pulse-on time 50 (a1) 100 (a2) 200 (a3) flushing pressure ⁄ 0.3 (b1) 0.4 (b2) 0.5 (b3) discharge current 18 (c1) 20 (c2) 22 (c3) the experimental design was created using taguchi l9 mixed orthogonal array with equal weight distribution at each level. the factor selection and their levels were developed based on pilot tests and literature surveys. the set of trials were designed and executed on the 3×3 level settings depicted in table 2 on inconel 718 workpiece and tungsten electrode in edm. the working configuration for tool and job piece was initially positioned at the origin, and the machining duration for each trial was set for 14 minutes and 1.5 mm depth. the performance measurement of each run was carried out by measuring mrr ( edm process parameter optimization for efficient machining of inconel-718 477 ) as per eq. (1), twr (g) and ewr (%) as per eq. (2). the weight losses of the tool and material were measured using an electronic weighing balance. table 2 experimental values of mrr, twr and ewr exp. no. pulse-on time ( ) flushing pressure ( ⁄ ) discharge current ( ) parameter combinational mrr ( ) twr ( ) ewr 1 50 0.3 18 a1b1c1 0.07597 0.01192 0.29487 2 100 0.4 18 a2b2c1 0.1225 0.0121015 0.23195 3 200 0.5 18 a3b3c1 0.11189 0.014906 0.28475 4 50 0.4 20 a1b2c2 0.14257 0.01819 0.29984 5 100 0.5 20 a2b3c2 0.19825 0.02149 0.25458 6 200 0.3 20 a3b1c2 0.1988661 0.023233 0.27438 7 50 0.5 22 a1b3c3 0.125326 0.00986166 0.18481 8 100 0.3 22 a2b1c3 0.200095 0.025867 0.30361 9 200 0.4 22 a3b2c3 0.2088095 0.0150057 0.16595 2.2. multi-criteria decision-making by topsis topsis is a robust and widely accepted mcdm technique in operation research and production engineering. in this method, the best alternatives are searched based on the closeness coefficient, i.e. the distances from the ideal best and ideal worst solution. the basic idea behind the method is to evaluate the alternatives on the euclidian distance scale, so that the least span from the ideal best and farthest from the ideal worst solution is achieved. the alternatives are then ordered based on their rank. the steps for the topsis approach are as follows— step 1: decision matrix design and assumption of the weight matrix. let be a decision matrix, where . [ ] (4) weight vector may be expressed as , -, where ∑ ( ) . the strategies regarding the determination of the weight vector are discussed in the subsequent section. step 2: construction of normalized decision matrix of each criterion using eq. (5), √∑ (5) step 3: determination of weighted normalized matrix using eq.(6), for , and , (6) step 4: estimation of the ideal positive (best) and ideal negative (worst) solutions using eqs.(7) and (8), respectively. 478 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar { (7) { (8) where b is a vector of benefit function and c is the vector of the cost function, for ⌈ ⌉ and ⌈ ⌉. step 5: determination of the separation measurement and relative closeness coefficient. in the topsis, the difference of each response from ideal positive (best) solution is given by eq. (9). √∑ ( ) (9) for , and , -. similarly, the difference between each response from the ideal negative (worst) solution is given by eq. (10). √∑ ( ) (10) for , and , corresponding closeness coefficient ( ) of the ith alternative is calculated using: (11) where , , step 6: the final step is to rank the alternatives in decreasing order of closeness coefficient value. 2.3. weight allocation without considering uncertainty 2.3.1. mean weight method this method is based on the principle of equal weight distribution to all output responses. equal weights are assigned in this approach to give equal importance to each parameter while evaluating the influence of each parameter on selecting an optimal set for machining. for the current three criteria (mrr, twr and ewr) edm optimal process parameter selection problem, the weight vector is expressed as , -. henceforth, in this study, for ease of reference, the solutions from these weights applied to topsis are called as mw-topsis. 2.3.2. standard deviation (sdv) method standard deviation constructs an unbiased and unprejudiced assignment of weights. it significantly improves the mcdm approach and lessens the personal assigned weight stress as felt in general-topsis. the sdv weights are calculated through the following equations. first, in order to change various scales of the process parameters and normalize them, eq. (12) is used. edm process parameter optimization for efficient machining of inconel-718 479 ( ) ( ) ( ) (12) √ ∑ ( ) (13) where is the average of the values for the ith measure, where . the weight vector is given by eq. (14) as, ∑ (14) for the current problem, the sdv weight vector is , and henceforth, the solutions from these weights applied to topsis are called sdvtopsis. 2.3.3. entropy method entropy-topsis is a practical and quick approach to predict the best alternative from all responses. topsis combined with entropy weights remove the subjective bias and uses objective decision to derive suitable weight vectors in order to compute the rank of feasible alternatives. the entropy weight function is assumed to be based on the discrete probability distribution. ( ) ∑ ( ) (15) degree of diversity (d) possessed by each criterion is evaluated as, , (16) and the weight objective for each criterion is given by ∑ (17) for the current problem, the entropy weight vector is , and henceforth, in this article, the solutions from these weights applied to topsis are called entropy-topsis. table 3 the preference weight distribution based on saaty nine-point scale scale for influence account for 1 equal influence equally significant 3 weak influence moderately better than other 5 strong influence one has a strong influence over other 7 very strong influence one has a very strong influence over other 9 absolute influence one has absolute influence over other 2.3.4. ahp method analytic hierarchy process (ahp) is known for its outstanding analytical style of solving and predicting convoluted decision-making at different levels of hierarchy or neural system. an improved-ahp can set as many levels of neural/hierarchy and use both 480 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar objective and subjective influence on the neural decision. in this method for predicting weight vector, a pairwise comparison matrix is developed via a scale of the relative importance of attributes and the judgments based on the saaty ahp& ri index given in table 3. the preference weights based on saaty nine-point scale are distributed according to their influence. the ahp weight estimation is based on the following, [ ] [ ] (18) where and { normalized weight is calculated by normalizing the geometric mean of the comparison matrix as, (∏ ) ∑ (∏ ) (19) where , -. the ahp weights for the current problem are computed as , -. the next step is to estimate consistency index ( ), consistency ratio ( ), and eigenvalue. a vector is constructed through the product of pairwise comparison matrix and normalized weighted vector for the significance of alternatives. , , where , (20) for maximum eigen-value ( ) of , ∑ (21) where , and (22) where and . is determined for different size matrixes, and its value is 0.58 for a 3×3 matrix. 2.4. weight allocation under uncertainty by fuzzification in the classical set theory, if a classical set of objects (say ), contains some generic elements (say ), then the belongingness of to is expressed in terms of membership function ( ). ( ) { (23) thus, in the classical set, an element has only two options— either to belong or not to belong to the set. however, not necessarily all real-life situations can be classified in such a bivalent manner. almost all group decision-making tasks are affected by uncertainty; thereby expressing them in terms of classical sets is not always possible. edm process parameter optimization for efficient machining of inconel-718 481 zadeh introduced the fuzzy set theory, where partial membership in sets is possible. the fuzzy set theory uses membership functions to permit the gradual assessment of element membership in a set. thus, a fuzzy set in may be represented as [17], * ( )+ (24) where ( ) , is the membership function of a and ( ) is the degree of membership of in a. ( ) can take any value between , -, capturing partial membership of in fuzzy set . according to madi et al. [17] ―a fuzzy number m is a convex normal fuzzy set m of the real line r such that [18]: there exists exactly one with ( ) ( is called mean value of m) and ( ) is piecewise continuous.‖ among various fuzzy numbers like triangular fuzzy number (tfn), trapezoidal fuzzy number, bell-shaped fuzzy number, etc., tfn is the most commonly used due to its computational simplicity and intuitiveness. tfn is a triplet of three real numbers ( ) (see fig. 2a). fig. 2 (a) a typical triangular fuzzy number (b) linguistic scale selected for the current work table 4 linguistic variables for the importance weight of each output (tfn) importance symbol fuzzy weight extremely low el (0, 0, 0.1) very low vl (0, 0.1, 0.3) low l (0.1, 0.3, 0.5) medium m (0.3, 0.5, 0.7) high h (0.5, 0.7, 0.9) very high vh (0.7, 0.9, 1) extremely high eh (0.9, 1, 1) as seen in fig. 2, and represent the smallest and the largest values; is the mode or core of the tfn. the range expressed by the tfn i.e. ( ) is called support. the membership function of the triangular fuzzy number can be expressed as, 482 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar ( ) { ( ) ( ) ( ) ( ) (25) if say, ̃ ( ) and ̃ ( ) are two triangular fuzzy numbers, then the distance between two fuzzy numbers is calculated using the vertex method [19]. ( ̃ ̃) √ ,( ) ( ) ( ) (26) linguistic variables are adjectives attributed to the parameters or alternatives. the prime advantage of linguistic variables is that they are expressed in natural language. this makes it easier to incorporate the inherent uncertainties associated with decisionmaking processes. in any linguistic scale, linguistic variables are represented by a set of corresponding fuzzy numbers. the linguistic scale selected for the current work is shown in fig. 2b. the fuzzy linguistic terms and their corresponding tfns, as well as tfn supports, are presented in table 4. in this work, uneven supports are used so that more significance is given to moderate attitudes. many researchers argue that such uneven supports stimulate decision-makers to evaluate their decisions more carefully [20]. for any mcdm problem involving criteria and experts (decision-makers), the fuzzy significance coefficients or weights ( ̃ ) are calculated as { } ∑ (27) { } where j = 1, …, n are criteria and k=1, …, k are decision-makers. table 5 ratings given to the three criteria by decision-makers response dm1 dm2 dm3 dm4 dm5 dm6 dm7 mrr vh h eh eh m h h twr h h vh m vh h vh ewr m m h h m m l table 6 aggregated fuzzy weights of the output responses output response aggregated fuzzy weight mrr (0.6143, 0.7857, 0.9143) twr (0.5571, 0.7571, 0.9143) ewr (0.3286, 0.5286, 0.7286) edm process parameter optimization for efficient machining of inconel-718 483 the various steps to be followed during the fuzzy-topsis are as follows, step 1: collect the subjective evaluations of the decision-maker on the importance of weights. the subjective evaluations of the decision-makers for the current work are presented in table 5. step 2: calculate the fuzzy significance coefficients or weights based on the decisionmaker’s subjective evaluations by using table 6 and eq. (27). step 3: form the decision matrix as listed in eq. (4) and normalized decision matrix listed in eq. (5). step 4: form a fuzzy weighted decision matrix by multiplying the normalized decision matrix listed in eq. (5) with corresponding fuzzy weights as per eq. (27). ̃ [ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ] (28) where, ̃ ( ) step 5: the coordinates for fuzzy positive ideal solution are calculated as ̃ { ̃ ̃ (29) the coordinates for fuzzy negative ideal solution are calculated as ̃ { ̃ ̃ (30) where are the index set of beneficial and cost (non-beneficial) criteria, respectively. step 6: the euclidean distance of each alternative from fuzzy positive and negative ideal value is calculated as ∑ ( ̃ ̃ ) (31) ∑ ( ̃ ̃ ) (32) where ( ) represents the distance between two fuzzy numbers calculated by using eq. (26) and depicted in table 7. step 7: closeness coefficient of the alternatives are calculated using eq. (11) and ranked as per descending order in table 8. 484 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar table 7 fuzzy positive and negative ideal value output a+ a mrr (0.2664, 0.3407, 0.3965) (0.097, 0.124, 0.1443) twr (0.1036, 0.1408, 0.17) (0.271, 0.3683, 0.4447) ewr (0.0701, 0.1127, 0.1554) (0.1282, 0.2062, 0.2842) 3. results and discussion 3.1. effect of process parameters on mrr the main effect plot of mrr for various process parameters is shown in fig. 3a. it is seen that the process variables have a significant effect on mrr. the mrr monotonically increases when the current is increased from 18a to 20a, but slightly decreases when the peak current reaches 22a. for pulse-on time, mrr shows a similar trend as seen in the case of the current. it rises when the pulse-on time is increased and then gets saturated depicting no further pulse-on time effect on mrr. in contrast, the flushing pressure is seen as having less influence on mrr compared to discharge current and pulse-on time. the mrr curve remains unaltered when the flushing pressure changes from 0.3 ⁄ to 0.4 ⁄ , but mrr drastically drops when the flushing pressure is increased to 0.5 ⁄ . 3.2. effect of process parameters on twr the main effect plot of twr for various process parameters is shown in fig. 3b. the lower, the better is preferred for twr. it is seen that the process variables have again a non-linear effect on twr as in mrr. the twr increases when the current is increased from 18 to 20a but drastically drops when the peak current reaches about 22a, signifying less tool wear at high current. regarding pulse-on time, twr shows a similar trend as in mrr; it increases rapidly up to 100 but drastically drops at 200 . the flushing pressure, in this case, seems to have a significant influence in comparison to mrr. at 0.3 ⁄ pressure, twr is maximum but it drastically reduces at 0.4 ⁄ whereas, at the 0.5kg/cm2, it shows a slight increment in twr. table 8 separation measures, closeness coefficients and ranking order of alternatives exp. no. triangular membership function didi+ ci rank 1 0.2052 0.3355 0.3795 8 2 0.3225 0.2182 0.5965 3 3 0.2280 0.3127 0.4217 6 4 0.2202 0.3205 0.4073 7 5 0.2958 0.2449 0.5470 4 6 0.2585 0.2822 0.4781 5 7 0.3919 0.1488 0.7248 2 8 0.2013 0.3394 0.3723 9 9 0.4682 0.0725 0.8659 1 edm process parameter optimization for efficient machining of inconel-718 485 fig. 3 effect of the process parameters on (a) mrr, (b) twr, (c) ewr 3.3. effect of process parameters on ewr the main effect plot of ewr for various process parameters is shown in fig. 3c. the lower, the better is also preferred for ewr. it is seen that the process variables have again a non-linear effect on ewr as in mrr and writhe ewr increases slightly when the current 486 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar is increased from 18 to 20a but it sharply drops when the peak current reaches 22a. with pulse-on time, ewr shows the trend opposite to that in the case of mrr and twr. it slightly rises up till 100 but it drastically drops down till 200 . the flushing pressure, in this case, seems to have significant influence and is similar to twr. fig. 4 euclidean distances of each alternative from pis and nis fig. 5 closeness coefficients of the alternatives based on different topsis methods 3.4. optimal process parameter selection in this section, no uncertainty in the weights assigned to the criteria is considered. using the criterion weights derived using the four criteria weight allocation methods (namely mw, sdv, entropy and ahp), their respective weighted normalized matrix is constructed. the weighted-normalized matrix is presented in table 9. using this information, the euclidean distances of each alternative from pis and nis are calculated and presented in fig. 4. it should be noted that for the solution to be effective the euclidean distance of the edm process parameter optimization for efficient machining of inconel-718 487 alternative from the pis should be as low as possible, i.e. in fig. 4 the should be as near the zero line as possible. similarly, the should be as away from the zero line as possible for the solution to be effective. this indicates that the euclidean distance of the alternative from the nis should be as high as possible. the variation of the closeness coefficient for each alterative using different topsis methods is shown in. fig. 5. it is seen that, among all the alternatives, alternative no. 9, i.e. a3b2c3 is the most promising set of process parameters that can be effectively used to simultaneously maximize mrr and minimize twr and ewr. fig. 6 euclidean distances of each alternative from fpis and fnis of fuzzy-topsis table 9 weighted normalized decision matrix mw-topsis sdv-topsis mrr twr ewr mrr twr ewr 0.0526 0.0746 0.1263 0.0540 0.0693 0.1318 0.0848 0.0758 0.0993 0.0872 0.0704 0.1036 0.0775 0.0933 0.1220 0.0796 0.0867 0.1272 0.0987 0.1139 0.1284 0.1014 0.1058 0.1340 0.1373 0.1345 0.1090 0.1410 0.1250 0.1138 0.1377 0.1455 0.1175 0.1415 0.1351 0.1226 0.0868 0.0617 0.0792 0.0892 0.0574 0.0826 0.1385 0.1619 0.1300 0.1424 0.1505 0.1357 0.1446 0.0939 0.0711 0.1486 0.0873 0.0742 entropy-topsis ahp-topsis mrr twr ewr mrr twr ewr 0.0650 0.0945 0.0629 0.0768 0.0975 0.0295 0.1048 0.0959 0.0495 0.1239 0.0989 0.0232 0.0958 0.1181 0.0608 0.1132 0.1219 0.0285 0.1220 0.1441 0.0640 0.1442 0.1487 0.0300 0.1697 0.1703 0.0543 0.2005 0.1757 0.0254 0.1702 0.1841 0.0585 0.2011 0.1899 0.0274 0.1073 0.0781 0.0394 0.1267 0.0806 0.0185 0.1712 0.2050 0.0648 0.2023 0.2115 0.0304 0.1787 0.1189 0.0354 0.2112 0.1227 0.0166 488 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar fig. 7 process performance of edm parameters table 10 ranking of alternatives by various topsis methods alternative mw-topsis sdv-topsis entropy-topsis ahp-topsis fuzzy-topsis 1 7 8 7 9 8 2 3 3 3 3 3 3 6 7 6 8 6 4 8 9 8 6 7 5 4 4 4 4 4 6 5 5 5 5 5 7 2 2 2 2 2 8 9 6 9 7 9 9 1 1 1 1 1 3.5. optimal process parameter selection considering uncertainty in decision to account for the uncertainty in the decision-making process, the current problem is also evaluated by using a fuzzy-topsis approach. the decision regarding the relative importance of mrr, twr and ewr is obtained from 7 decision-makers. the 7-point fuzzy scale-based ratings given by the decision-makers are collected in table 5. next, these are aggregated and the mean fuzzy weights are created, which are reported in table 6. the fuzzy positive ideal value (fpis) and fuzzy negative ideal value (fnis) are reported in table 7. based on these the euclidean distances of each alternative from fpis and fnis are calculated as reported in table 8 and fig. 6. similarly, the closeness coefficients are calculated and reported in fig. 5. it is seen that, in the fuzzy-topsis, the alternative no. 9 is also found to the best compromise solution among all the alternatives. fig. 7 shows the average process performance of the edm process parameters as calculated by different methods. a similar trend is seen for all the methods. the overall optimal set of edm process parameters was found to be a3b3c2. the ranking of the alternatives by various topsis methods is shown in table 10. edm process parameter optimization for efficient machining of inconel-718 489 4. conclusion in the present article, five different weight allocation strategies are used in conjunction with topsis to predict the optimal process parameter combination in edm machining of inconel superalloy. while the methods like mean weight method, standard deviation method, entropy method, and ahp method do not take into account the uncertainty in decision-making process, the fuzzy weight allocation scheme is used to select optimum edm parameters by the collective decision-making process. based on the experimental data and the extensive numerical mcdm analysis it can be concluded that—  all the methods namely, mw-topsis, sdv-topsis, entropy-topsis, ahptopsis and fuzzy-topsis can be effectively used to estimate and optimize the edm process parameters,  the 9th alternative, i.e. a3b2c3, is found to be the optimal setting for achieving high mrr, low twr and low ewr,  from performance characteristics, it is found that both entropy & ahp yielded similar results, and  the optimal set for machining is obtained through proposed mcdm is 22a discharge current, 200µs pulse-on time, and 0.4 ⁄ flushing pressure. it is evident from the experiments that the pure tungsten electrode can be used as a tool in edm for machining inconel 718 at a high discharge current (18a-22a) and high pulseon time (50-200µs) without any rapture. also, pulse-on time and current have a higher influence on mrr and twr as compared to dielectric pressure in edm machining. one limitation of the current study is the use of l9 orthogonal array, which can be effectively used to study the main effects of the process parameters but is unreliable when machine learning based response surface function over the entire domain of the parametric combination is desired. thus, this study can be further improved by using quasi-random low-discrepancy sampling approaches like hammersley, sobol, etc. to design the experiments. further, for multi-response optimization advanced metaheuristic approaches like cuckoo search, grey wolf optimizer, etc. may be used to generate pareto solutions. references 1. pandey, r.k., panda, s.s., 2014, optimization of bone drilling parameters using grey-based fuzzy algorithm, measurement, 47, pp. 386-392. 2. das, b., roy, s., rai, r.n.,saha, s.c., 2016,application of grey fuzzy logic for the optimization of cnc milling parameters for al--4.5% cu--ticmmcs with multi-performance characteristics, engineering science and technology, an international journal, 19, pp. 857-865. 3. tamiloli, n., venkatesan, j., ramnath, b.v., 2016,a grey-fuzzy modeling for evaluating surface roughness and material removal rate of coated end milling insert,measurement, 84, pp. 68-82. 4. singh, r.k., singh, d.k., kumar, v., 2013, parametric optimization of wire-edm by using fuzzy logic, ijbstr research paper, 37, p. 41. 5. mohanty, c.p., mahapatra, s.s., singh, m.r., 2014, an experimental investigation of machinability of inconel 718 in electrical discharge machining, procedia materials science, 6, pp. 605-611. 6. lin, m.y., tsao, c.c., hsu, c.y., chiou, a.h., huang, p.c., lin, y.c., 2013,optimization of micro milling electrical discharge machining of inconel 718 by grey-taguchi method, transactions of nonferrous metals society of china, 23, pp. 661-666. 7. sharma, v., misra, j.p.,singhal, p., 2019,multi-optimization of process parameters for inconel 718 while die-sink edm using multi-criterion decision-making methods, journal of physics: conference series, 012166. 490 a. singh, r.k. ghadai, k. kalita, p. chatterjee, d. pamučar 8. srivastava, a., mishra, d.k., chatterjee, s., datta, s., biswal, b.b., mahapatra, s.s., 2018, multiresponse optimization during electro-discharge machining of super alloy inconel 718: application of pca-topsis, materials today: proceedings, 5, pp. 4269-4276. 9. huang, j.t., liao, y.s., 2003, optimization of machining parameters of wire-edm based on grey relational and statistical analyses, international journal of production research, 41, pp. 1707-1720. 10. sharma, v., misra, j.p.,singhal, p., 2019,optimization of process parameters on combustor material using taguchi & mcdm method in electro-discharge machining (edm), materials today: proceedings, 18, pp. 2672-2678. 11. balasubramaniyan, s., selvaraj, t., 2017, application of integrated taguchi and topsis method for optimization of process parameters for dimensional accuracy in turning of en25 steel, journal of the chinese institute of engineers, 40, pp. 267-274. 12. chakraborty, s., das, p., 2018, a multivariate quality loss function approach for parametric optimization of non-traditional machining processes, management science letters, 8, pp. 873-884. 13. das, p.p., chakraborty, s., 2020, parametric analysis of a green electrical discharge machining process using dematel and sir methods, opsearch, 57, pp. 513-540. 14. chakraborty, s., chatterjee, p., das, p.p., 2019, a doe–topsis method-based meta-model for parametric optimization of non-traditional machining processes, journal of modelling in management, 14, pp. 430-455. 15. jafarian, f., 2020, electro discharge machining of inconel 718 alloy and process optimization, materials and manufacturing processes, 35, pp. 95-103. 16. kumar, m., singh, h., 2016,multi response optimization in wire electrical discharge machining of inconel x-750 using taguchi’s technique and grey relational analysis, cogent engineering, 3, 1266123. 17. madi, e.n., garibaldi, j.m., wagner, c., 2015, a comparison between two types of fuzzy topsis method,in 2015 ieee international conference on systems, pp. 291-297. 18. zimmermann, h.-j., 1992, fuzzy set theory and its applications, kluwer academic publishers. 19. chen, c.-t., 2000, extensions of the topsis for group decision-making under fuzzy environment, fuzzy sets syst, 114, pp. 1-9. 20. stanujkic, d., 2016, an extension of the moora method for solving fuzzy decision-making problems, technological and economic development of economy, 19, pp. 228-255. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 285 308 https://doi.org/10.22190/fume190327035a © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper results and challenges of artificial neural networks used for decision-making and control in medical applications adriana albu, radu-emil precup, teodor-adrian teban politehnica university timisoara, dept. automation and applied informatics, romania abstract. the aim of this paper is to present several approaches by which technology can assist medical decision-making. this is an essential, but also a difficult activity, which implies a large number of medical and technical aspects. but, more important, it involves humans: on the one hand, the patient, who has a medical problem and who requires the best solution; on the other hand, the physician, who should be able to provide, in any circumstances, a decision or a prediction regarding the current and the future medical status of the patient. the technology, in general, and particularly the artificial intelligence (ai) tools could help both of them, and it is assisted by appropriate theory regarding modeling tools. one of the most powerful mechanisms that can be used in this field is the artificial neural networks (anns). this paper presents some of the results obtained by the process control group of the politehnica university timisoara, romania, in the field of anns applied to modeling, prediction and decision-making related to medical systems. an iterative learning control-based approach to batch training a feedforward ann architecture is given. the paper includes authors’ concerns in this domain and emphasizes that these intelligent models, even if they are artificial, are able to make decisions, being useful tools for prevention, early detection and personalized healthcare. key words: artificial neural networks, medical diagnosis, medical prediction, prosthetic hands, recurrent neural networks received march 27, 2019 / accepted july 15, 2019 corresponding author: radu-emil precup politehnica university timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania e-mail: radu.precup@upt.ro 286 a. albu, r.-e. precup, t.-a. teban 1. introduction decision-making and control in medical domain aim to improve the healthcare system and to help physicians by assisting them and offering suggestions or a second opinion. several reasons that justify and encourage the use of these mechanisms should be mentioned here:  there are medical processes that can be modeled by these artificial intelligent tools, with relevant benefits for the patients,  some medical conditions are hardly detected by humans; a suggestion or an alert that is provided by an automated system could be a solid argument for a physician, and,  there is a large number of simple and routine activities that are time consuming and that overload medical staff; these could be easily performed by a machine. artificial intelligence (ai) domain provides a series of techniques, approaches and algorithms that can be used to solve problems which require an intelligent behavior. in order to do this, it is necessary to understand the way humans think and act, and then to build intelligent tools able to reproduce humans’ functionalities. one of the key properties of the human brain activity is learning. although, different kinds of systems having this property have been developed in recent years, the artificial neural networks (anns) are still one of the most popular and efficient forms of learning systems [1]. they follow the structure of the human brain, using a simplified architecture, which is made of basic processing units (artificial neurons), interconnected and working in parallel. anns can be used to solve various problems of classification and generalization. their properties (the ability to execute parallel distributed computation, to tolerate noisy inputs, to learn new associations, new patterns and new functional dependencies, according to [1] and [2]) make them suitable for applications that can be developed to support medical activity. the most common issues in this domain are related to decisionmaking area, where the problems are difficult to be solved using traditional computation. anns are able to supplement the processing power of the digital computers with the ability to make sensible decisions and to learn by ordinary experience, as humans do [2]. the justification of anns is not difficult. ai domain contains more of the other mechanisms that have been used to create automated systems for medical decision-making (e.g. knowledge-based systems and statistical inference, which seem to be the most widespread). the authors of the present paper have been interested in applying them to medical systems [3], [4] and modeling with focus on medical [5], [6] and non-medical process control [7]–[14] including novel training algorithms formulated in the framework of iterative learning control (ilc) [7]–[11] and model predictive control [12]. their conclusions have been similar to those emphasized by the main researchers in ai domain [1], [2]: these conventional mechanisms have remarkable advantages, but they are suitable only for a small area of medical decision-making problems. knowledge-based systems can be easily implemented for simple rules, but it is possible to encounter unexpected results if the problem that has to be solved involves many interconnected premises; therefore, fuzzy logic can be included. in addition to this, the transition from implicit knowledge (of human medical experts) to explicit rules (necessary for symbolic processing) can damage the information content. on the other hand, the most common method of statistical inference, which is bayes’ theorem, assumes that the diseases are mutually exclusive (sometimes this can be true) and, more restrictive, that the considered symptoms of diseases are statistically independent (more often, it is difficult to fulfill this constraint). results and challenges of artificial neural networks for decision-making and control in medical... 287 anns provide an alternative approach by training them to associate diagnoses (or other decisions) to the input data (symptoms, laboratory tests results, images, signals). during the training process, a set of examples consisting in input values, together with the already established decision for them, are repeatedly presented to the network (which has a specific architecture and some features that can be modified in this phase). once the training is completed, the ann should be able to generalize, offering a correct decision for a new set of input data. even if the literature contains an impressive number of papers that describe different ways of using anns, there is enough room for further developments. technology becomes faster and faster. biological neurons are switching at speeds that are million times slower than a computer gate [2]. nevertheless, humans (and even animals) are more efficient in speech recognition and visual information processing than the fastest computer. therefore, the research in anns field is still a challenge, as long as the understanding of human neural system is not completed, yet. this paper describes, in several sections, the research results obtained by the authors in the field of anns developed for medical domain. it continues the work underlined in [15], adding some new results. section 2 presents a literature review, where some current articles are considered, in order to prove that these aspects are still in top of researchers’ concerns. then, the paper continues, in section 3, with the description of three common anns that analyze regular medical data (symptoms and laboratory tests results); one of them is used for detection, suggesting a diagnosis in the field of skin diseases, and the other two were trained to make predictions regarding the evolution of hepatitis b and the risk of stroke. anns are suitable for applications that use medical images in decisional process; section 4 contains a detailed description of such a system. section 5 presents a network of anns that was developed to make predictions regarding several different aspects in the same time. then, section 6 describes results concerning a variable structure recurrent neural network (rnn) and a non-recurrent ann with similar size, which replicate the nonlinear mechanism of a prosthetic hand based on surface myoelectric sensors and produce anns the midcarpal joint angles based on myoelectric signals obtained from surface myoelectric sensors. building upon our results reported in [8], section 7 offers an ilc-based approach to batch training a feedforward ann architecture. conclusions are pointed out in section 8. 2. related work a quick look at recent research papers shows that the influence of ai on human life is continuously increasing. medicine is part of this trend ([16]-[23]) and it could incorporate in the future much more applications based on ai mechanisms in general and, particularly, on machine learning techniques. anns, which are machine learning models inspired from the architecture of the human brain, have been extensively used in various applications developed for medical field. several examples are hereby presented. in [16], the authors are using anns to predict acute rejection in liver transplant patients. the novelty and the advantage of their method is that it uses routine laboratory data only, being non-invasive (while conventional tests need biopsy, which is an invasive procedure). 288 a. albu, r.-e. precup, t.-a. teban the efficiency of anns is also proved in [17], where the authors present an overview regarding the use of anns in lung cancer research. they also underline that even if the literature shows that these tools are suitable for clinical decision support, a strict cooperation between physician and biostatistician is mandatory in order to avoid inaccurate use of anns. another recent example is provided in [18], where anns are used to analyze stomach images, classifying them as normal, benign or malign. the original images are processed in order to reduce their dimension. the accuracy of classification results places this method above others, in the authors’ opinion. medical images are intensively used together with neural networks in decision-making process. for instance, an automated classification of skin lesions is performed by deep convolutional neural networks (cnns) that analyze images in [19]. the results demonstrate that such tools are able to classify skin cancer with accuracy comparable to dermatologists. the use of machine learning in stroke imaging is emphasized in [20]. the authors focus on technical principles, applications and perspectives, stating that these techniques may play an important role in setting the adequate therapeutic method or in predicting the prognosis for stroke patients. as stated in the introduction, ann is just one technique besides many others that are part of ai domain. the comparison between these techniques is debated by researchers for decades. for instance, a study implemented in 2007 on a database of 1069 cases with coronary heart disease, indicated that ann is the best identifier [21]. in 2017, several researchers included anns together with other classifiers in the same project [22], making use of the benefits provided by each of them in order to identify breast cancer. a relevant study is provided in [23], where the authors forecast how medicine (particularly cardiovascular medicine) will incorporate ai in the future. they consider the most common machine learning techniques, including cnns and rnns. even if ai tools have already been used in medicine, the physicians still need to analyze a multitude of variables and relationships between these variables in order to identify the medical status of a patient. meanwhile, the machine learning domain provides a series of algorithms used to represent data structures and to make predictions or classifications. therefore, medicine is still a proper field for automated tools that may support decisionmaking process [15]. 3. artificial neural networks used in diagnosis and prediction anns offer support in decision-making process, so that the physicians can benefit of a faster diagnosis for diseases with various and confusing symptoms [15]. this section contains three relevant examples for this area [24]–[26]. 3.1. skin diseases diagnosis the ann described in [15] and [26] has been created and trained to suggest a diagnosis regarding skin diseases from erythemato-squamous class. it makes the difference between six such diseases: psoriasis, seborrheic dermatitis, lichen planus, pityriasis rosea, chronic dermatitis, and pityriasis rubra pilaris. setting a diagnosis for these diseases is difficult and sometimes inaccurate because patients have multiple and vague symptoms. results and challenges of artificial neural networks for decision-making and control in medical... 289 there are 11 clinical features that are evaluated when a patient is suspected to have an erythemato-squamous disease [26]. these are enumerated in the left column of the graphical user interface provided by fig. 1. some of the erythemato-squamous diseases can be detected using these clinical features only. nevertheless, a biopsy is also required for a correct diagnosis; 22 histopathological features can be evaluated from skin samples [26]. they are listed in the last two columns of fig. 1. fig. 1 the graphical user interface for the skin diseases diagnosis system [15] an analysis of these six diseases from medical point of view shows up that they have similar features. therefore, the diagnosis is difficult for a human. an automated diagnosis system based on ann can be a real help for the dermatologist. anns have the ability to overcome these problems because they learn by examples and then they can generalize and identify patterns. thus, a feedforward ann with back-propagation learning algorithm has been created. since an ann learns by examples, a database of 366 patients with erythemato-squamous diseases has been used [26], [27]. each patient has 33 symptoms and an already established diagnosis (one of the six considered diseases). therefore, the input matrix of ann has 366x33 elements, representing the symptoms of all the patients, and the target matrix has 366x6 elements, representing the diagnosis of all the patients. between the 33 input elements and the 6 output neurons there is a hidden layer with 10 neurons. once the ann is trained on a representative set of data, it can be used to diagnose a new patient. through a user interface (fig. 1 [15]), the physician enters the clinical and the histopathological features of his patient. some of these features are binary values (1 – present, 0 – not present), others have an intensity degree between 0 (not present) and 3 (acute), with discrete intermediate values 1 and 2. 290 a. albu, r.-e. precup, t.-a. teban based on these features, the ann generates the diagnosis. at most one of the six outputs of the network is 1, the rest of them being 0. table 1 contains several examples of inputs and outputs from the dataset [27] that has been used in ann’s training. a relevant difference with respect to [15] is that the current examples emphasize the idea that apparently similar features are defining different diseases, making the diagnosis process more difficult. the superscript t in table 1 indicates matrix transposition. table 1 examples of records from the skin diseases database record input vector diagnosis 1 [2 2 2 1 0 0 0 0 2 2 0 0 0 0 0 0 2 0 2 2 3 2 0 0 0 2 0 0 0 0 0 2 0]t psoriasis 2 [2 2 2 3 2 3 0 2 0 0 0 2 0 0 0 2 2 0 0 0 0 0 0 0 3 0 2 0 2 0 0 2 3]t seborrheic dermatitis 3 [2 2 3 3 2 3 0 1 0 0 0 2 0 0 0 3 2 0 1 0 0 0 0 0 2 0 3 0 3 0 0 2 3]t lichen planus 4 [2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 1 0 0 0 2 0]t pityriasis rosea 5 [2 2 0 2 0 0 1 0 0 0 0 0 0 0 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1]t chronic dermatitis 6 [2 2 1 0 0 0 3 0 2 0 1 0 0 0 0 2 2 1 2 0 0 0 0 0 0 0 0 2 0 3 3 2 0]t pityriasis rubra pilaris the system’s performance is next discussed. its precision (93.7%) reported in [15] and [26] is comparable to that of a human expert; but it is infinitely faster. however, two features of this system should be improved. one refers to the fact that when two diagnoses are plausible, the system identifies just one of them because the outputs are binary values and do not have associated a plausibility score. the other one is connected to uncommon diseases, which are difficult to be identified by the system (because of the small number of examples in the training set). therefore, the research should continue in this field in order to find better solutions. one could be a hybrid expert system that combines the power of ann with the sensitivity of other ai mechanisms. 3.2. hepatitis b prediction the medical status of a patient can be improved if the physician would have some information about a possible future evolution of that patient. the system hereby presented has been developed for liver diseases. an ann has been used to make suggestions regarding the evolution of patients infected with hepatitis b virus [25]. there are six possible severity levels of this infection: easy, medium, serious, prolonged, cholestatic, and comatose. each patient can have a specific evolution, which can be predicted using this mechanism. a function fitting neural network (which is a specialized version of feedforward neural network) has been used. the patient’s current medical status is defined by 22 symptoms, which can be divided into: general features of the patient, subjective signs (that can be seen or easily detected), and objective signs (laboratory test results). an example of these features is given in table 2. ann receives as inputs these values in their initial form. results and challenges of artificial neural networks for decision-making and control in medical... 291 table 2 example of input values for hepatitis b predictions general features of the patient feature gender age living conditions contact transfusion parenteral treatments debut time value male 31 urban no no yes 4 subjective signs feature anorexia nausea queasiness asthenia fever hyperchromic urine value yes yes no yes no yes feature tegument jaundice arthralgia myalgia skin eruption jaundice liver dimension value yes no no no 3 2 objective signs feature biliary retention thymol alanine transaminase (alt) value 4.81 8 840 ann has an output layer, which produces the result. this is a number defining one of the six severity levels of the disease: 1 – easy, 2 – medium, 3 – serious, 4 – prolonged, 5 – cholestatic, 6 – comatose. in the training phase, besides the input values, the network also needs, for each patient, the severity level that was already set by a physician and that is assumed to be correct. between the inputs and the output there is a hidden layer of 10 neurons. this is an empirically established number, which usually is between the number of inputs (22, this case) and the number of outputs (one, for this system). the levenberg-marquardt back-propagation algorithm is used for training. it is the most suitable for this type of problems, and also the fastest. each neuron of ann reacts to its weighted inputs, producing an output. this is internally performed by an activation function. from this point of view, the current system contains two types of neurons: for those belonging to the hidden layer, the activation function is hyperbolic tangent sigmoid, while the neuron from the output layer has a linear activation function. the graphical user interface is divided in several parts, and two of them are presented in fig. 2. the input panel allows the user to enter the features of a new patient. the results are displayed by the output panel, where the accuracy of the system is also provided. the system is using a database that contains 165 patients from the clinical hospital of infectious diseases no. 4 “victor babes”, timisoara. 150 of them are used for training and 15 for testing. 292 a. albu, r.-e. precup, t.-a. teban a) the input panel b) the output panel fig. 2 the graphical user interface for hepatitis b prediction the accuracy of the system is 80%. in order to obtain better results, the database could be improved in two directions: more records and a higher number of fields [25]. on the one hand, any classifier needs as many records as possible, in order to establish the influence of inputs (symptoms) on an output (disease or other prediction). on the other hand, a larger number of variables that describe a patient is desirable, but only if this action significantly increases the accuracy. these features should be relevant for the diseases that are classified. otherwise, the overall performance of the system is depreciated because of the execution time, which is increasing [25]. 3.3. stroke risk prediction as presented in the previous section, anns can be trained to make predictions regarding the future medical status of a patient. another example is the system created for stroke, also called cerebrovascular accident, risk prediction [15], [26]. according to the world health organization [28], stroke is one of the two main frequent causes of death worldwide. it is also responsible for long-term disabilities, and the perspective is getting worse. therefore, the concern of developing automatic tools for predictions of vascular events is fully justified. the risk factors responsible for stroke [26], [29] can be grouped in two distinct categories: controllable (high blood pressure, diabetes, heart diseases, smoking, sedentariness, obesity) and not controllable (age, gender, race, prior stroke, family history). however, it is results and challenges of artificial neural networks for decision-making and control in medical... 293 difficult to present a well-defined influence of these factors on stroke, so that the risk cannot be accurately established. a computational model, able to recognize special aspects, predicting a stroke risk may become crucial in prognostication, identifying the high-risk patients, and may have a future role in treatment selection [30]. anns are able to acquire, to store and to use experiential knowledge [2]. these features make them suitable for the exposed problem. the prediction system [26] has been trained to recognize four levels of stroke risk. the training set contains information about 108 patients, which has been collected from the municipal emergency hospital “clinicile noi” timisoara, romania, between 20152016. there are eleven features for each patient into the dataset and also the risk level, which is a value between 1 and 4 (the worst case is denoted by value 4). the fields of this dataset and some examples of values for several patients with different risk levels are presented in table 3. more than in [15], these examples are indicating that it is difficult to determine the influence that each feature has on the stroke risk level. table 3 the fields of the database for stroke risk prediction (with examples) medical factor 1 age 33 21 37 56 2 gender 1 1 2 2 3 dyslipidemia 100 90 169 149 4 abdominal circumference 77 95 80 90 5 hlv on echocardiogram 90 57 105 88 6 hlv ekg sokolov-lyon 20 24 42 48 7 gfr 30 43 48 18 8 lv wall thickness 0.2 0.7 0.9 0.9 9 postprandial glycaemia 166 250 231 310 10 a jeun glycaemia 102 113 118 150 11 previous condition 0 0 1 3 risk value 1 2 3 4 a feedforward ann has been created for stroke risk prediction [26]. it has 11 inputs (the patient’s medical features involved in decisional process – these are the risk factors responsible for stroke) and one output (the value predicted for stroke risk). it was considered that one hidden layer should be enough (usually, this is sufficient for a large variety of problems). the number of the neurons in the hidden layer is one of the most important considerations when the architecture of the ann is established. nevertheless, there is no conclusive solution appropriate for any task [2]. the current system has a hidden layer of 10 neurons. the back-propagation algorithm has been used for the training process. it is one of the most commonly used algorithms for feedforward anns training. the user interacts with the application through a graphical interface, where he enters the necessary information (the features of the patient that are requested for the decisional process) or he can load this information from an external file (if it is available). then the system provides the stroke risk value and the physician can use it in the decisional process that follows. 294 a. albu, r.-e. precup, t.-a. teban therefore, having this valuable information regarding a potential evolution of a patient, the human expert can choose the actions that are further required to avoid a stroke. this ann is a relevant example of using automated decisional systems to improve patients’ medical condition. further developments can also be performed in this field. for instance, the risk of a second stroke or even the risk of death after a stroke can be predicted using anns [31]. 4. artificial neural networks applied to medical image analysis setting a diagnosis is the first step that should be accomplished when a person has a medical problem. but, as already stated in this paper, this could be a difficult action. sometimes the symptomatic and analytic information presented by the patient are not relevant. for specific diseases, the solution can be provided by medical images, emphasizing internal problems of the human body that cannot be found analyzing the symptoms or the laboratory tests results. the liver represents a great challenge to the radiologists because of the difficulty to appreciate the morphological changes induced by the illness [3]. for this reason, an ann was developed in order to suggest a diagnosis regarding the liver diseases that can be detected analyzing information extracted from medical images obtained by computed tomography (ct). the system is able to detect three hepatic diseases: hepatomegaly, steatosis and tumors. in addition to the previous work described in [15], fig. 3 contains three sequential slides from abdominal tomography of patients with these diseases, together with a healthy liver [3]. in this way the features of each disease can be observed more easily. ann is trained to distinguish between these four types of images. but, as can be seen in fig. 3, a slice of an abdominal tomography contains, besides the liver, some other elements (parts of vertebral column and ribs, a portion of lung, etc.) that are not relevant for the problem that must be solved. for this reason, it is necessary to process a medical image before using it by an ann. the first step is segmentation and it is necessary to locate and to extract an object (the liver, in this case) from the initial image. fig. 4 presents an image before and after segmentation process. then, the processing continues, extracting some features that are relevant for liver’s diseases. one of these is texture and fig. 3 emphasizes the differences of texture between the four states of the liver. the texture features can be extracted using grey level spatial dependence matrices, also called co-occurrence matrices [3], which define the distribution of co-occurring pixel values for a specified offset. the offset is determined by an angle and a distance between pixels. the most used angles are 0°, 45°, 90°, and 135°. the co-occurrence matrices consist of the following elements expressed in a computer programming-like notation (with arguments instead of subscripts): c0,d(i,j)=|{((k,l),(m,n))i: k–m=0, |l-n|=d, i(k,l)=i, i(m,n)=j}|, (1) c45,d(i,j)=|{((k,l),(m,n))i: (k–m=d,l–n=–d) or (k–m=–d,l-n=d), i(k,l)=i, i(m,n)=j}|, (2) c90,d(i,j) = |{((k,l), (m,n))  i: |k–m|=d, l–n=0, i(k,l)=i, i(m,n)=j}|, (3) results and challenges of artificial neural networks for decision-making and control in medical... 295 c135,d(i,j)=|{((k,l),(m,n))i: (k–m=d,l–n=d) or (k–m=–d,l–n=–d), i(k,l)=i, i(m,n)=j}|, (4) where i is the image (in fact the matrix of gray level elements) and d is a distance between pixels. a) healthy liver b) hepatomegaly c) steatosis d) tumors fig. 3 slices of abdominal tomography a) an entire slice b) the liver fig. 4 segmentation 296 a. albu, r.-e. precup, t.-a. teban the co-occurrence matrices have (each one) 256x256 elements, too many to be analyzed by an ann. therefore, the process of analyzing the image continues, extracting a set of six texture features from the co-occurrence matrices: energy, entropy, contrast, maximum element, inverse difference moment, and correlation. these spatial grey tone co-occurrence texture features are usually called haralick features. they were introduced by r. haralick four decades ago, and they are still successfully used in image texture analysis. an image is now described by 24 elements (six features for each of the four matrices). these are the inputs of the ann. it is a feedforward ann, with back-propagation training algorithm. it also has one hidden layer with 10 neurons and one output layer suggesting the diagnosis. this ann has a fixed architecture, therefore, during the training process, the weights of the connections between neurons are modified. this training process has two phases [3]: a preliminary phase, where the parameters receive their initial values, and a main phase, which is iterative, where the parameters are adjusted. the performance of an ann depends not only on the way of modifying its weights, but also on their initial values. for this reason, in order to choose the best neural network, 500 anns with different initial values of the weights have been created and trained. then, the ann with the best accuracy has been used to provide the prediction regarding the healthy level of the liver. the training of this system was performed using abdominal ct images obtained from the “the modeling centre for prosthesis and surgical interventions on the human skeleton” multiple users research base of the politehnica university timisoara. liver diseases can be a real danger for patients’ lives because frequently, they have perceptible symptoms barely in advanced stages. an early detection using an automated diagnosis system that puts together the ability of anns and the accuracy of medical images might save lives. 5. a network of artificial neural networks this section also treats an application in the area of liver diseases. hepatitis c is a serious and frequent one. there is no vaccine against its virus, the treatment is very expensive (in some countries it is supported by national programs), and, more important, this treatment is not always efficient, sometimes causing severe adverse effects. there are three treatments for hepatitis c [3]: simple interferon (ifn), peg interferon α-2a, and peg interferon α-2b. at the beginning of the treatment, for both physician and patient, it would be good to know which of these treatments will have benefits (if there is one), in order to avoid undesired side effects. the treatments for hepatitis c influence four biological indicators (tgp, tgo, ggt, and arn vhc). the system described here (and also mentioned in [3] and [15]) is able to offer, for each of these four biological indicators, predictions regarding the evolution during the next 12 months, indicating its growing tendency, its stabilizing or its decreasing tendency. the physician could use these predictions in order to estimate the evolution of the patient during each possible treatment and to decide which the most suitable treatment for a specific patient is. results and challenges of artificial neural networks for decision-making and control in medical... 297 this system is, in fact, a network of strongly interconnected anns (fig. 5). there are four sections that provide predictions regarding the evolution of the biological indicators after 3, 6, 9, and 12 months of treatment. inside each section there are four anns, one for each biological indicator. all the 16 anns are feedforward neural networks and they are trained using back-propagation algorithm. fig. 5 the network of anns used for hepatitis c prediction each ann has 10 hidden neurons, an output neuron (which predicts the evolution tendency of that biological indicator after 3, 6, 9, or 12 months) and a variable number of inputs. the networks belonging to the first section (the ones that predict the evolution of each biological indicator after the first 3 months of treatment) receive as inputs: the patient’s age, the gender, the location where he/she lives (rural/urban), the treatment that is evaluated, the knodell score, the hepatic fibrosis score and the value of the biological indicator for which the prediction is made, at the initial moment (before the treatment starts). the output of these networks is the value of that biological indicator after 3 months of treatment. the networks in the following sections have a similar structure, but they have as additional inputs the outputs of the networks in the previous sections (referring to the same biological indicator); therefore, the networks in the last section will have 10 inputs (the initial inputs and the values of biological indicators after 3, 6, and 9 months of treatment). the human expert enters, through a graphical user interface (fig. 6), the five features of the patient, the initial values of biological indicators and chooses one of the three treatments. then the system provides the predictions regarding the evolution of the biological indicators during that treatment. in this study, 193 patients have been under observation for 12 months in order to establish the treatment’s influence on the evolution of the biological indicators. the information about these 193 patients (120 women and 73 men, between 14 and 67 years of age) has been collected from the gastroenterology department of the emergency clinical hospital, timisoara, romania. 298 a. albu, r.-e. precup, t.-a. teban fig. 6 the graphical user interface for hepatitis c prediction system [15] regarding the accuracy of this system, each of the 16 anns has its own value. in fact, for each of them have been created and trained 500 neural networks with the same architecture, but different initial parameters. the network with the best accuracy was chosen to be used for predictions. the results are encouraging: most of anns having accuracies around 85% (according to [3] and [15]), as “acc” parameter of each ann from fig. 3 shows. it can be observed that the accuracy of all the networks from the first section is greater than 90% and that in most of the cases the accuracy is decreasing along the chain of networks. one reason could be the fact that not only the prediction is propagating through the network of anns, but also the error. the concern regarding the hepatitis c virus infection is still a major issue in medical domain. even if the treatments for hepatitis c are continuously improved and becoming more and more efficient, the patient evolution during the treatment has to be carefully observed in order to react if something goes wrong. the prediction system presented here provides an overview for 12 months of treatment. this contains information that can be used by the physician in the decisional process of choosing the best treatment for a patient. 6. artificial neural networks applied to modeling finger dynamics this section treats the use of neural networks in modeling the finger dynamics of an amputated hand and the control of a prosthetic hand based on the architecture specified in [6] and [32]. the inputs of the system are 8 myoelectric sensors placed:  four sensors on the flexor digitorum superficialis (fig. 7a),  two sensors on the extensor digitorum and one on extensor digiti minimi (fig. 7b),  one sensor on abductor pollicis longus (fig. 7c). results and challenges of artificial neural networks for decision-making and control in medical... 299 a b c fig. 7 placement of sensors 1 to 4 (a), sensors 5 to 7 (b) and sensor 8 (c) on the hand [6] the main rnn considered in [6] is a long-short term memory (lstm) network [33], [34], which is next extended so as to meet the finger dynamics modeling purpose. the rnn is structured on 4 players as follows: an input layer, an lstm layer, a hidden layer and an output layer. the input is composed of 24 values [6] and is connected to the neural network. each input is sampled at 10 ms and is structured as follows in terms of input vector i: ,] ... ... ... [ 24 ,8,1,8,1,8,1  t nnnnnn ssaazzi (5) where zj.n is the output of myoelectric sensor j, j=1…8, the average of the past ten samples (100 ms) of myoelectric sensor j, j = 1…8, is aj,n: , 10 1 9 0 ,,    i injnj za (6) and the average of the past 100 samples (1 s) of myoelectric sensor j, j=1…8, is sj.n: . 100 1 99 0 ,,    i injnj zs (7) the lstm layer is composed of 300 neurons and connects the input layer to the hidden layer but also memorizes the past 100 time steps (1 second) in order to provide how the signals change in time. 300 a. albu, r.-e. precup, t.-a. teban the hidden layer contains 300 neurons and is connected to the current lstm layer for a new abstractization of the inputs. this layer of abstractization helps the lstm to output values in the first second without having to wait for the first 100 samples to be received the output layer contains five neurons and produces output vector o: ,] [ ,5,4,3,2,1 t nnnnn oooooo (8) which outputs the flex percentage of each finger at time stamp tn. this model contains a total number of 1,112,705 parameters which were trained. they are distributed as follows: 390,000 in the first lstm layer, 721,200 in the hidden lstm layer and 1,505 in the output layer. the rnn structure is described fig. 8 (a multi input-single output system structure) for normal operation. fig. 8 lstm network after the first second [6] the rnn model results are compared to the real hand movements (captured by the flex sensor) using the percent root mean square error (rmse) to express the model accuracy. the number of data samples in the dataset is 18,490 for the validation dataset. as shown in [6], the training of the networks was done on a training data set of 110,374 samples, equivalent to 1,103.74 s. the rmse value for the rnn on the validation data is 8 to 9% depending on the training performance. the other two anns used for comparison are the regular ann (i.e., non-recurrent) with an rmse of 13 to 14% and the same rnn architecture with the validation data used also for training, which generated an rmse of 2%. the regular ann consists of 3 layers, similar in size with the rnn but without the memory, arranged as an input layer of 300 neurons, a hidden layer of 300 neurons and an output layer of 5 neurons. the total number of trained parameters is 99,305 distributed across the layers as follows: input – 7,500, hidden – 90,300 and output – 1,505. the activation function for all the layers is sigmoid and the training was done using the adam optimizer in which a dropout of 0.2 was used for a better generalization of the training. the training of the rnn is the same with the regular ann except that the used optimizer is root mean square propagation, which had a better performance than adam with a 1% increased accuracy. the results are exemplified in fig. 9 for only one finger (index) to simplify the figure. the different lines are a comparison between the expected result (the real finger angle), blue the rnn output (green), the regular ann output (red), and the output of rnn when the validation data was also used for training the rnn (magenta). lstm lstm lstm lstm inn-99 inn-98 inn-1 inn outn hidden results and challenges of artificial neural networks for decision-making and control in medical... 301 fig. 9 results for index finger: expected output (blue), regular ann output (red), rnn output (green) and output of rnn trained with validation data (magenta) the main increase of accuracy for both anns was made by several solutions like changing the number of neurons per layer, the number of layers or the type of network, but the main increase is given by the data preprocessing before the network training. 7. an ilc-based approach to artificial neural network batch training as shown in [8], a feedforward ann architecture is considered, which consists of one hidden layer with a hyperbolic tangent activation function and a single linear neuron. the nonlinear input-output map specific to ann is: )),(),(()()1( kkkky t xvσw (9) where k indicates the discrete time moment, nk ...0 , and also the data sample index, and it is also dropped out as follows if the simplification of notations is intended, y is the output, w is the vector of output layer weights,  is the vector of outputs of hidden layer neurons: ,])(...)(1[ ,]...[ 11 1 10 tt hh t ht hwww xvxvσ w    (10) and hmxxm ...1 ),tanh()(  , are hyperbolic activation tangent activation functions. other activation functions can be considered as well. the input vector is ,]...[ 10 nut nuxxx x (11) 302 a. albu, r.-e. precup, t.-a. teban and the first term in  given in (10) corresponds to the bias of the output neuron. each neuron in the hidden layer neuron is parameterized by the vector of weights v m , m = 1...h, and vector v in (9) is .]...[ ,])(...)()([)( 110 2   nutnu mmm m tthttmtm vvvv vvvv (12) each vector v m includes weight vm 0 of the bias of m th neuron. the number of ann inputs is nu + 1, and the number of neurons in the hidden layer is h. the nonlinear map specific to ann expressed in (9) is next organized as a nonlinear multi input-multi output dynamical system considered in the iteration domain [8]: ,...0)),(,()1( ,...1, , 1 1 nkkk hi i j t jj v j i j i j w jjj i      xvσwy uvv uww (13) where the vector expressions are: ,])(...)0([ ,])1(...)1([ ,]...[ ,]...[ )1)(1( 1 1 10 0         nuntt j t jj nt jjj nutv j v j v j htwh j w j w j n nyy uu uu inuii xxx y u u (14) j is the iteration index, iv j w j uu , are the input vectors, and weight vectors i jj vw , are the state vectors of the dynamical system given in (13). vector xj is a trial-repetitive timeseries disturbance input and also a time-varying parameter vector of (13). vector yj is the output of the nonlinear dynamical system given in (13). using ilc framework, the dynamical system given in (13) is transformed into an input-output static map, which, in fact, ensures an ann. the general goal of ilc is to reduce or minimize the tracking error expressed as the difference between the actual output and a desired output. the desired output vector specific to ann, where the system (or process if the control is targeted) modeling is carried out, is yd: ,)]1( ... )( ... )1([ 1 nt dddd nykyyy (15) where yd(k) is the desired system (process) output. in this regard, the batch training of ann can be considered as a supervised learning approach, where the goal is to minimize the tracking error vector (in terms of ilc), also called training error vector (in terms of ann) ej: .djj yye  (16) however, the input at each iteration can be derived in the framework of norm-optimal ilc as the solution to the optimization problem: results and challenges of artificial neural networks for decision-making and control in medical... 303 , minarg),( 2 2 11 , ** j t jj t j v j w j iv j w j i uquereuu uu   (17) where the stacked vector of inputs is uj: ,])(...)()([ )1(11  nuhhttv j tv j tw jj h uuuu (18) weighting matrices 0t rr  and 0t qq  are diagonal ones (for the sake of simplicity), djj yye   11 is the tracking error at iteration 1j , and  is the euclidean norm of vector  . the second term in the objective function in (17), which weights vector uj, is added in order to prevent over-fitting. the popular nonlinear least-squares approach is applied to obtain the analytical solution to the optimization problem defined in (17). the linearization of the nonlinear input-output map specific to ann of type (9) at the iteration j+1: nkkky i j t jj ...0)),(,()1( 111   xvσw (19) is carried out around i jj vw , for small variations of iv j w j uu , by considering the output as a nonlinear function of the weight vectors as follows: ,...0)),(,,()1( 111 nkkfky i jjj   xvw (20) and input vector )(kx as a parameter vector. the taylor series expansion of (19) in the vicinity of an arbitrary operating point is: .,..)( )( 4 ...)( )( 4 ]))(tanh(...))(tanh(1[))(,()1( 2)()(2)()( 1 1 1 1 11 tohk ee wk ee w kkkky h th j th j t j t j v j t kk h j v j t kk j w j th j t j i j t jj         uxux uxvxvxvσw xvxvxvxv (21) where h.o.t. indicates higher-order terms, which are next neglected. using (19) with j instead of j+1, introducing the notations: ,]))(tanh(...))(tanh(1[)( , )( 4 )( 1 2)()( tth j t jj kk i kkk ee kg ti j ti j xvxvσ xvxv     (22) and neglecting h.o.t. in (21), the result is: .)()(...)()())(()1()1( 1 1 1 1 hv j t h h j v j t j w j t jjj kkgwkkgwkkyky uxuxuxσ  (23) stacking the 1n outputs in (23) over the time argument k leads to [8]: 304 a. albu, r.-e. precup, t.-a. teban . )()(...)()())(( )1()1(...)1()1())1(( )0()0(...)0()0())0(( ,, 1 1 1 1 1 1 ))1(1()1( 1                   nngwnngwn gwgw gwgw t h h j t j t j t h h j t j t j t h h j t j t j j nuhhn jjjjj xxxσ xxxσ xxxσ ψ ψuψyy  (24) but the tracking error vector at iteration j+1 can also be expressed as: ,11 jjjdjjjdjj uψeyuψyyye   (25) which determines the transformation of the optimization problem defined in (17) into the following quadratic one: , 2 minarg 2 2 * j t jjj t jj j ereuzuxuu u  (26) where: . , j t jj t j ψrezqψrψx  (27) using the matrix derivation rules and using the fact that x is symmetric because r and q are symmetric, the analytic solution to the optimization problem defined in (25) is: , ) ()( 11* jjj t jj t j tt j ekerψqψrψzxu   (28) where: . ) ( 1 rψqψrψk t jj t jj  (29) matrix kj can be obtained easily because matrices jj eψ , can be computed at the current iteration. the optimal solution (vector) * ju contains the optimal increments of ann weights. using the following partitioning of kj: ,]...[ 1 t t v j t v j tw jj h kkkk  (30) with )1()1()1()1( ,   nnuv j nhw j i kk , the first two equations in (13) are transformed into: . , 1 1 j v j i j i j j w jjj i ekvv ekww     (31) the two equations in (31) are update laws in the ilc framework, and they depend on the error at current iteration j. concluding, solving the optimization problem defined in (17) at each iteration, results in the expression (15) of ann training equations. this ann training approach is general because the formulation of the objective function in (17) brings a degree of freedom in the learning process. results and challenges of artificial neural networks for decision-making and control in medical... 305 8. conclusions the amount of medical information that describes a patient and that has to be processed in order to generate a decision is continuously increasing. therefore, it is becoming obvious that medicine needs to incorporate some automated tools able to provide valuable support in decision-making process. as shown in the invited paper [15], the current paper continues the presentation of the authors’ results offered in [15] and also shows that anns are suitable for such applications, being reliable tools, which are adapting to the informational environment specific to the problem they are solving. they are providing accuracies that are comparable to those of human experts. more than this, an ann has the advantage that it is faster, more efficient and the experiments it is used for can be easily repeated. the results presented by this paper are encouraging. of course, they can be improved. one of the key factors that influence the accuracy of these tools, increasing their reliability, is the number of available patients used for training. from this point of view, the content of the databases available for these experiments is appropriate (366 patients for skin diseases diagnose, 165 for hepatitis b virus infection, 108 for stroke, 193 for hepatitis c treatment suggestion). however, better results will be obtained if a larger number of records will be used in the training phase. even if medical experts became aware that ai is a necessity and even if they are already using such tools, computer-aided decision-making systems still have enough room for improvements. all these applications lead to a favorized topic nowadays: personalized medicine. the authors intend to continue the research in this direction in order to develop new tools and methods able to improve the healthcare system, in general, and particularly, the patients’ lives. it is still challenging to create a reliable prognostic model able to achieve the requested confidence in order to be used in real clinical circumstances. future research will deal with the combination with fuzzy modeling and the transfer of results to various nonlinear models using results from other modeling and control applications [35]–[40], fuzzy [32], [41]–[44] and other models [45]–[52] applied to the medical field, and the consideration of other nonlinear models that proved to be successful in different fields [53]–[57] including various ann architectures [58]–[61] and optimization techniques [62]–[68]. therefore, anns, which are probably the most common ai techniques could lead to a significant improvement of the medical environment. also, due to the increase number of inputs and the interdependencies between them, deep anns start to surpass the accuracy of the shallow anns and will probably become a necessity in the medical field. the physicians will be able to analyze in a greater depth a much larger quantity of data and the patients will have the chance to benefit from a higher quality healthcare system. acknowledgements: this work was supported by the ceepus network “bg-0722 computer aided design of automated systems for assembling” and by the valuable activity of the team from the faculty of mechanical engineering of the university of niš, serbia. the support of dr. mircearadac in the development of the ilc-based training approach is duly acknowledged. 306 a. albu, r.-e. precup, t.-a. teban references 1. russell, s.j., norvig, p., 2010, artificial intelligence: a modern approach, 3rd ed., pearson, upper saddle river, nj, usa. 2. 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this paper, the mechanical analysis of an advanced body centred cubic (bcc) lattice cell has been performed through a homogenisation procedure to obtain an equivalent set of mechanical properties. the mechanical analyses have been carried out with the use of ansys software and an original ansys parametric design language (apdl) subroutine has been developed for the introduction of the double periodic boundary conditions. the finite element method (fem) is used for the mechanical model, and 3d elements with reduced integration has been employed to guarantee an accurate description of the lattice geometry. different bcc cell configurations have been considered: standard metal bcc cell, metal bcc cell with waved struts, standard metal composite bcc cell. depending on the configuration, the homogenised materials showed isotropic or orthotropic properties. for the evaluation of all the engineering constants, uniaxial traction test and in-plane shear test have been simulated along different loading directions. a parametric study has been conducted varying the struts diameter, the struts waviness and the thickness ratio of the composite struts. finally, the homogenised materials have been tested through the mechanical analysis of sandwich panels with lattice core; a comparison between sandwich panels with homogenised core and sandwich panels with exact lattice cells has been carried out. the parametric study can be useful for the tailoring and optimisation analysis of an advanced component. key words: lattice structure, fem, composite, double periodicity, homogenisation 1. introduction in recent years, the ecological trends are motivating engineers to use fewer amounts of resources and to improve the design of lightweight structures, especially in transportation systems [1, 2]. the spread of additive manufacturing (am) technologies is a clear example response to these trends. the am production processes show a large parameter space, received: january 25, 2024 / accepted march 21, 2024 corresponding author: stefano valvano university of derby, institute of innovation and sustainable engineering, lonsdale house, quaker way de13hd, derby, uk e-mail: s.valvano@derby.ac.uk https://orcid.org/0000-0003-4349-1092 736 s. valvano which highly affects the components mechanical properties without increasing manufacturing costs due to the material usage reduction [3, 4]. recently am technologies are quickly spreading thanks to their promising process flexibility, the implementation of multimaterials and the capability to modify the function, structure shape and material properties, limited production costs with a weight reduction by up to 50% [5]. the parameter space complexity of am production processes permits to derive periodic structures, advantageous for their mechanical properties relative to their weight or volume ratios [4]. in the family of highly periodic structures is possible to collocate the lattice cells and the porous cellular structures; this kind of structures offers a wide applications range and an interesting potential property tailoring [6-8]. the most influential am process parameters should be identified and incorporated into the design and modelling process. in order to use the am full potential, engineers need to develop and select convenient and appropriate design tools [4]. one of the design tools well known in the literature is the representative volume element (rve). the rve can be defined as the smallest volume or cellular element statistically representative of the whole domain with a constitutive response error lower than 5% [9]. the rve should be smaller compared to the whole structural domain, and in the meantime large to contain defects, in order to show a realistic mechanical response [10, 11]. several studies have been proposed to determine the rve size to keep computational costs low, [12, 13]. in the literature there are many important contributions on the topic of cellular materials, among others gibson and ashby [14], masters and evans [15], christensen [16], and wang and mcdowell [17, 18] provided closed-form solutions to obtain the equivalent mechanical properties from the rve definition. the main assumptions lay on the cellular components’ description through the use of euler-bernoulli beams. this approach works well for relatively simple cellular architectures but presents limitations for complex cellular topology. alternative approaches are based on energy equivalence methods [19, 20]. the analysis of the wave propagation in lattice structures and associated frequency characteristics has been investigated in many works [21, 22, 23, 24]. an equivalent continuum representation of periodic solids cellular structures can be investigated through homogenisation techniques. multiscale methods have been developed using asymptotic expansion approaches [25, 26, 27]. differently, maewal [28] studied multiscale method with finite element approach, and in the works of mcdevitt et al. [29, 30] the multiscale strategy has been proposed using the assumed strain method. it is undoubted that fem is the most used tool to analyse continuum micromechanics problems. the periodic homogenisation methods are based on the study of single repeating volume element. the asymptotic homogenisation [31] and the macroscopic degrees of freedom [32] can be used with fem codes. the mechanics of structural genome [33] is a recent finite element based approaches for periodic homogenisation where the structure is analysed and divided in basic repeating elements as genome elements, and it is possible to pass from the macro to micro scale descriptions and vice versa using de-homogenisation and homogenisation techniques. several works have been made on the homogenisation of periodic cellular structures, among others the work of seiler et al. [34] introduced the idea of waviness in 2d periodic grid cellular solids. a simplified homogenisation approach has been used for three-dimensional lattice cells in the work alaimo et al. [35], mantegna et al. [36] and tumino et al. [37]. in this work, the mechanical analysis of an advanced body centred cubic lattice cell has been performed through the derivation of the homogenisation procedure of periodic cellular structures in order to obtain an equivalent set of mechanical properties. an original ansys parametric design language subroutine has been developed for the introduction homogenised properties of lattice metal composite cell 737 of the double periodic boundary conditions. the originality of the work consists in the parametric study of different bcc cell configurations: standard metal bcc cell, metal bcc cell with waved struts, metal composite bcc cell. the approach used for the metal composite cell and the waviness struts present innovative idea for the tailoring of cellular solids, and it can be adapted for the study of multi-layered lattice cells or coated cellular structures. results in terms of the homogenised materials are presented in tabular and figure forms. 2. lattice cell structure sandwich panels are one of the most used light-weight structures [38], and many solutions have been proposed for the material selection and the architecture of sandwich cores [39]. the sandwich core concept, here adopted, is based on lattice structures, in particular the standard bcc cell (with straight struts) has been considered, moreover the author proposed the use of modified bcc cell with waved struts. in order to reduce the computational cost, the sandwich lattice core has been modelled through homogenised properties. in this section, the homogenisation process, based on the use of 3d fem analysis performed in ansys environment, is described. the standard bcc cell structure has been directly drawn in design modeller environment, as shown in fig. 1, a cylindrical strut section has been considered. the waved struts concept is built sweeping the strut circular section on an auxiliary diagonal plane (45°) following a three-point arch path, the single strut is copied mirroring it in the three cartesian directions, see fig. 2. for both the cell structures, the surface boundaries have been obtained through an intersection operation of the struts with a virtual bounding box. the strut material is an isotropic structural steel with the following properties: ρ = 7850 kg/m3, e = 200 gpa, ν = 0.3 and g = 76.9231 gpa. the external bounding box (which defines the cells dimension) is a cube with edge length l = 10 mm. fig. 1 bcc cell geometry definition, straight struts 738 s. valvano fig. 2 bcc cell geometry definition, waved struts based on circumferential arch path 3. straight strut after the bcc cell geometry and the structural material definitions, the 3d meshing has been conducted in ansys using tetrahedral elements with an element dimension around a third of the strut radius, this ratio guarantees an acceptable convergence on the results. in order to apply the boundary conditions, it is important to define the outer boundary surfaces with respect to the direction of its normal. in fig. 3, it is possible to see the four top surfaces (green colour) named z+, the bottom surfaces (purple colour) named z-, the left and the front ones (blue and black colours) called y+ and x-; yand x+ are not reported in the figure for the sake of readability. the boundaries conditions are defined imposing a prescribed displacement on the fem nodes lying on the boundary surfaces. in order to determine the young modulus e, a traction (or compression) has been imposed to the z+ surfaces through a prescribed displacement along the z direction, furthermore the reaction force could be obtained from the lower boundary surfaces zwhere the prescribed displacement is set to zero only for the z component. in order to make the present homogenisation campaign consistent, it is mandatory to ensure the compatibility of displacements along the faces of the representative volume element; in other words, the double periodic boundary conditions are applied to the surfaces with normal different with respect to the loading direction, as shown in fig. 4. fig. 3 bcc cell with straight struts, boundary definition homogenised properties of lattice metal composite cell 739 fig. 4 bcc cell with straight struts, constraint equations applied the periodicity concept implies that the representative volume element has the same deformation mode and there is no separation or overlap between adjacent volume elements [40, 41], the periodicity condition can be introduced as follows: 𝑢𝑖 = 𝜖�̅�𝑘𝑥𝑘 + 𝑢𝑖 ∗ (1) where ϵ̅ik are the average strains and ui * is the unknown periodic part of the displacement components on the boundary surfaces. the displacements on a pair of opposite boundary surfaces can be derived as follows: 𝑢𝑖 𝑗+ = 𝜖�̅�𝑘𝑥𝑘 𝑗+ + 𝑢𝑖 ∗ 𝑢𝑖 𝑗− = 𝜖�̅�𝑘𝑥𝑘 𝑗− + 𝑢𝑖 ∗ (1) where the index j stands for the normal direction of the upper (+) and lower (-) boundary surfaces. the difference between the displacement of the upper and lower boundary surfaces can be defined as: 𝑢𝑖 𝑗+ − 𝑢𝑖 𝑗− = 𝜖�̅�𝑘(𝑥𝑘 𝑗+ − 𝑥𝑘 𝑗− ) = 𝜖�̅�𝑘∆𝑥𝑘 𝑗 (2) for a cubic representative volume element and generally for a parallelepiped volume element, the term δxk j can be considered constant; therefore, a generic periodic boundary condition can be defined as follows: 𝑢𝑖 𝑗+(𝑥, 𝑦, 𝑧) − 𝑢𝑖 𝑗−(𝑥, 𝑦, 𝑧) = 𝑐𝑖 𝑗 (3) where the ci j constants represent the stretch or contraction terms of the representative volume element. it is possible to define the average stresses σ̅ij and strains ϵ̅ij of the representative volume element as: 𝜖�̅�𝑗 = 1 𝑉 ∫ 𝜖𝑖𝑗 𝑑𝑉 𝑉 ; 𝜎𝑖𝑗 = 1 𝑉 ∫ 𝜎𝑖𝑗 𝑑𝑉 𝑉 (4) where ϵij and σij are the local strains and stresses of the volume element v constituents. in order to apply the double periodicity in ansys environment, the author wrote an apdl routine: after checking the same position of two nodes placed on opposite boundary surfaces (one node on the x+ and the other one on the x-), for every pair of outer surface nodes a set of constraint equations is created, as follows: 740 s. valvano 𝑢𝑖 + − 𝑢𝑖 − = ∆𝑢 ; ∆𝑢 = 𝑢𝑟𝑒𝑓 + − 𝑢𝑟𝑒𝑓 − 𝑣𝑖 + − 𝑣𝑖 − = ∆𝑣 ; ∆𝑣 = 𝑣𝑟𝑒𝑓 + − 𝑣𝑟𝑒𝑓 − 𝑤𝑖 + − 𝑤𝑖 − = ∆𝑤 ; ∆𝑤 = 𝑤𝑟𝑒𝑓 + − 𝑤𝑟𝑒𝑓 − (5) in eq. 5 the basic constraint equations are expressed for a single direction, the mechanical displacements for each pair of ith nodes could be linked considering a ∆ term, which represents the stretch/dilatation property of the cell structure, calculated just choosing a reference pair of nodes. applying the double periodicity condition, depending on the structure geometry, some nodes of the mesh could lay on two distinct boundary surfaces with two different normal; those shared nodes must be left free, without any constraint. the loading and boundary conditions applied are different in order to evaluate the young modulus e, the poisson coefficient ν or the shear modulus g, therefore a resume of the used boundaries is listed in table 1. table 1 bcc cell with straight struts, boundary conditions resume outer surfaces x+ xy+ yz+ z e u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=free v=free w=load u=free v=free w=0 ν u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=free v=free w=load u=free v=free w=0 g u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u= load v=free w=0 u=0 v=free w=0 a traction test campaign has been conducted in order to evaluate the young modulus e and the poisson coefficient ν, different strut diameters has been considered, d = 1, 1.5, 2 mm. calling the hooke law: σ = eϵ, it is possible to obtain the young modulus e knowing the stress and the strain. the strain is immediately known because the traction is applied as a prescribed displacement on the top boundaries z+. the equivalent stress can be calculated starting from the reaction force, obtained from the lower boundary surfaces zwhere the prescribed displacement is set to zero only for the z component, furthermore the equivalent stress can be calculated as: 𝜎𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑧 = ∑ 𝐹𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑧 𝐴𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 (6) where aequivalent = l2 and l is the maximum edge dimension of the full cube cell. the poisson coefficient ν is defined as: 𝜈 = − 𝜖𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑒 𝜖𝑙𝑜𝑎𝑑𝑖𝑛𝑔 (7) as mentioned before, ϵloading is known due to the prescribed loading displacement, differently ϵtransverse should be calculated along the x or y direction, simply as: ϵxx=(ux +−ux -)/l or ϵyy=(vy +−vy -)/l. an example of a traction test results is given in fig. 5 for a bcc cell with d = 1 mm strut diameter. in order to evaluate the shear modulus g a shear homogenised properties of lattice metal composite cell 741 test campaign, for various strut diameters d = 1, 1.5, 2 mm, has been carried out. it is possible to obtain the shear modulus g knowing the stress and the strain and using the hooke law: τ = gϵ. as the previous case, the strain is known due to the prescribed displacement on the top boundaries z+ and the equivalent stress is obtained from the reaction force as: 𝜏𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑧𝑥 = ∑ 𝐹𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑧𝑥 𝐴𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 (8) where the loading direction can be x or y direction. shear test results are given in fig. 6 for a bcc cell with d = 2 mm strut diameter. due to the isotropic geometric properties of the bcc cell and the isotropic properties of the considered strut material, it is worth noting that the homogenised properties will be of an isotropic material. finally, in order to completely define the homogenised properties of the standard bcc cell, the equivalent density has been calculated as follows: ρ = ρsteel(vbcc/vcube cell), where vbcc is the volume of the considered bcc cell structure and vcube cell is the volume of the equivalent full cube cell. the homogenised isotropic properties of the standard bcc cell for different strut diameters are reported in table 2. fig. 5 total displacement for bcc cell with thin diameter, traction test fig. 6 total displacement for bcc cell with thick diameter, shear test 742 s. valvano table 2 bcc cell with straight struts, homogenised properties for different strut diameters (d) d [mm] 1 1.5 2 ρ [kg/m3] 386.82 828.41 1396.99 e [mpa] 70.764 408.59 1473.7 ν [-] 0.492 0.479 0.46 g [mpa] 1263.8 2927.7 5384.3 in order to validate the present homogenisation approach, a static analysis of different sandwich panels with lattice core has been conducted. two types of panels have been considered: a sandwich one with the skins made of structural steel and the lattice core based on standard bcc cells; a second panel with the same geometry and the same skins, but with a core layer based on the homogenised properties. the panels have simply-supported boundary conditions on all the lateral surfaces and a constant uniform pressure is applied on the top surface. first of all, a lattice core with 5 × 5 bcc cells has been considered; in fig. 7 it is possible to note the displacement distribution on the top surface, the local effects of the struts are clearly visible. in fig. 8 the displacement distribution on the bottom surface shows a global behaviour. the maximum displacement of the bottom surface is taken for comparison with the other plate model, the maximum bottom displacement for the sandwich with core made of 5 × 5 bcc cells is: −0.014638 mm. fig. 7 total displacement of the sandwich plate with lattice core made of 5×5 pattern bcc cells with straight struts, side and top view fig. 8 total displacement of the sandwich plate with lattice core made of 5×5 pattern bcc cells with straight struts, bottom view homogenised properties of lattice metal composite cell 743 another sandwich configuration is taken into account, the same steel skins and a lattice core with 10 × 10 bcc cells, in fig. 9 the undeformed panel geometry is depicted. the displacement of the loaded structure is represented in fig. 10, it is clear that the top surface displacement shows a more global behaviour, differently with respect to the 5 ×5 pattern core. the maximum displacement of the bottom surface is taken for comparison with the homogenised plate model, the maximum bottom displacement for the sandwich with core made of 10 × 10 bcc cells is: −0.080943 mm. fig. 9 sandwich plate with lattice core made of 10 × 10 pattern bcc cells with straight struts fig. 10 total displacement of the sandwich plate with lattice core made of 10 × 10 pattern bcc cells with straight struts finally, sandwich panels with homogenised core have been analysed, in fig. 11 the 5 × 5 pattern core is represented, it has to be noticed that the local effects, due to the struts of the original core, on the top surface are missing. the maximum deflection of the bottom surface is equal to −0.013473 mm. in fig. 12, the deformed sandwich structure with homogenised core is represented corresponding to 10×10 pattern cells. the displacement distribution of the top surface is similar to the original sandwich panel. the maximum deflection of the bottom surface is equal to −0.076095 mm. 744 s. valvano fig. 11 total displacement of the sandwich plate with homogenised core made of 5 × 5 bcc cells with straight struts fig. 12 total displacement of the sandwich plate with homogenised core made of 10 × 10 bcc cells with straight struts comparing the maximum deflections of the original sandwich panel with 5 ×5 cell pattern core and that of the sandwich one with homogenised core, the homogenised model leads to an error of 7.96 %. moreover, the same comparison for the 10×10 pattern core leads to an error of the homogenised model equal to 6 %. it is clear that, increasing the number of lattice cells in the core pattern, the structural response of the sandwich panel with original lattice core tends to be similar to the behaviour of the homogenised structure (the error decrease), where its core properties have been calculated with double periodicity condition. this ideally means an infinite number of cells in two directions. 4. waved strut the homogenisation campaign continues with a modified bcc geometry, the author proposes here to introduce waviness into the strut path, in particular a three-point-arch path has been used for its definition. different strut diameters, d = 1, 2 mm, and various path curvature, r = 10, 15, 20 mm, have been considered. the maximum cell dimension is kept fixed as the previous standard bcc cell with l = 10 mm. it is clear that the waviness concept introduces a reduction of the geometrical symmetry with respect to the standard bcc cell, consequently the waved bcc cell cannot keep the isotropic homogenised properties of the standard one, and it is well described by orthotropic homogenised homogenised properties of lattice metal composite cell 745 properties. the boundary surfaces are defined as the previous subsection and, for the sake of brevity, the discussion is not reported. in table 3 a resume of the used boundary conditions is given in order to evaluate the orthotropic homogenised properties of the waved bcc cell: ex, ey, ez, νxy, νxz, νyz, gxy, gxz, gyz. table 3 bcc cell with waved struts, boundary conditions resume outer surfaces x+ xy+ yz+ z ex u=load v=free w=free u=0 v=free w=free u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic ez u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=free v=free w=load u=free v=free w=0 νyx u=load v=free w=free u=0 v=free w=free u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic νxz u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=free v=free w=load u=free v=free w=0 gxy u=0 v=load w=free u=0 v=0 w=free u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic gxz u=0 v=free w=load u=0 v=free w=0 u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic gzx u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=periodic v=periodic w=periodic u=load v=free w=0 u=0 v=free w=0 the poisson coefficients νij for orthotropic materials are defined as: 𝜈𝑖𝑗 = − 𝜖𝑗 𝜖𝑖 (9) where ϵj is the strain in the transverse direction and ϵi is the strain in the axial loading direction. considering the first test of the previous subsection, the traction test with the load applied along the z direction, it is possible to obtain the young modulus ez and the poisson coefficients νzx = νzy, in fig. 13 the test is depicted in z direction of a d = 1 mm, r = 10 mm waved bcc cell. the steps for the evaluation of the young modulus and the poisson coefficients are the same of the previous subsection case, for the sake of brevity the discussion is not here reported. moreover, the poisson coefficients rule for orthotropic materials permits to calculate νxz and νyz as follows: 𝜈𝑖𝑗 𝐸𝑖 = 𝜈𝑗𝑖 𝐸𝑗 (10) the validity of this rule, applied to the waved bcc cell, has been verified by the author from the traction tests. due to the geometric asymmetry of the waved bcc cell, another traction test is needed, in fig. 14 the test in x direction of a d = 1 mm, r = 10 mm waved bcc cell is represented. performing this test in x direction it possible to get the young modulus ex and the poisson coefficient νxy. due to the geometric symmetry of the waved 746 s. valvano bcc cell between the x and y directions it is possible directly to assume that ey = ex (for the sake of clarity, this assumption has been verified by the author). fig. 13 directional displacement of the waved cell (d = 1 mm, r = 10 mm) for the traction test, load applied in the z direction fig. 14 directional displacement of the waved cell (d = 1 mm, r = 10 mm) for the traction test, load applied in the x direction in order to complete the homogenised properties definition, the shear moduli have to be defined. in fig. 15 the shear test is represented (d = 2 mm, r = 20 mm), loading in x direction, it is possible to get the shear modulus gzx. differently from the previous standard bcc cell with straight struts, the geometric asymmetry of the waved bcc cell between the x and z directions suggests verifying the shear response of the cell in z direction. in fig. 16 the shear test is depicted (d = 2 mm, r = 20 mm) in z direction, due to the geometric asymmetry the author found that, for the waved bcc cell, gzx ≠ gxz. homogenised properties of lattice metal composite cell 747 fig. 15 directional displacement of the waved cell (d = 2 mm, r = 20 mm) for the shear test, load applied in the x direction fig. 16 directional displacement of the waved cell (d = 2 mm, r = 20 mm) for the shear test, load applied in the z direction considering that for classical elastic material formulation, the shear contribution is taken into account as follows: 𝜏𝑖𝑗 = 𝜏𝑖𝑗 + 𝜏𝑗𝑖 = 𝐺 𝜕𝑢𝑗 𝜕𝑖 + 𝐺 𝜕𝑢𝑖 𝜕𝑗 = 𝐺 ( 𝜕𝑢𝑗 𝜕𝑖 + 𝜕𝑢𝑖 𝜕𝑗 ) (11) rewriting eq. 12 for the present waved bcc cell problem, it follows: 𝜏𝑖𝑗 = 𝜏𝑖𝑗 + 𝜏𝑗𝑖 = 𝐺𝑖𝑗 𝜕𝑢𝑗 𝜕𝑖 + 𝐺𝑗𝑖 𝜕𝑢𝑖 𝜕𝑗 ⇒ ⇒ 𝐺 ( 𝜕𝑢𝑗 𝜕𝑖 + 𝜕𝑢𝑖 𝜕𝑗 ) = 𝐺𝑖𝑗 𝜕𝑢𝑗 𝜕𝑖 + 𝐺𝑗𝑖 𝜕𝑢𝑖 𝜕𝑗 ⇒ ⇒ 𝐺 = 𝐺𝑖𝑗 𝜕𝑢𝑗 𝜕𝑖 + 𝐺𝑗𝑖 𝜕𝑢𝑖 𝜕𝑗 ( 𝜕𝑢𝑗 𝜕𝑖 + 𝜕𝑢𝑖 𝜕𝑗 ) (12) 748 s. valvano the homogenised properties for the shear moduli g̅xz = g̅yz have been obtained as discussed in eq. (13), differently the shear modulus gxy is calculated from a single shear test, due to the symmetry between the x and y directions. the homogenised orthotropic properties of the waved bcc cell for different strut diameters and different curvature radii are reported in table 4. table 4 bcc cell with waved struts, homogenised properties for different strut diameters (d) and waviness radii (r) d [mm] 1 2 r [mm] 10 15 20 10 15 20 ρ [kg/m3] 339.38 309.01 290.66 1201.8 1115.2 1057.2 ex=ey [mpa] 61.032 49.418 43.248 1379.9 1071.1 910.05 ez [mpa] 71.474 39.767 30.325 1845.2 915.93 659.74 νxy [-] 0.5125 0.3564 0.2766 0.5408 0.3687 0.2802 νxz = νyz [-] 0.4355 0.6150 0.7033 0.3432 0.5461 0.6501 gxy [mpa] 710.59 693.50 709.68 4487.0 4018.1 3769.6 gxz = gyz [mpa] 257.45 395.26 492.22 2605.3 3104.6 3260.9 gzx = gzy [mpa] 318.50 450.32 537.33 2751.4 3225.6 3366.1 g̅xz = g̅yz [mpa] 287.97 422.79 514.77 2678.3 3165.1 3313.5 in this subsection, the present 3d homogenisation approach for equivalent orthotropic material is validated considering an infinite sandwich panel using double-periodic boundary conditions. two types of panels have been considered: a sandwich one with the skins made of structural steel and the lattice core based on three waved bcc cells along the thickness direction, see fig. 17; a second panel with the same geometry and the same skins, but with a core layer based on the homogenised properties, see fig. 18. the panels have double-periodic boundary conditions on all the lateral surfaces and a constant uniform displacement is applied on the top surface, the bottom surface has a fixed support along the loading direction z, in other words w(x,y,z=0) = 0. fig. 17 sandwich panel section made of structural steel skins and lattice core based on three waved bcc cells along the thickness direction homogenised properties of lattice metal composite cell 749 fig. 18 sandwich panel section made of structural steel skins and homogenised lattice core based on waved bcc cells the sandwich panel with composite bcc cells shows a global structural response evaluated through the reaction force at the bottom surface, fz = 10488 n. the sandwich panel with the homogenised core has a reaction force of fz = 10635 n, it is clear that the homogenised model leads to an error of 1.5% on the global structural response. considering the stress distribution, it is well known that the lattice cell structures introduce local stress responses. the local stresses cannot be correctly detected by the homogenised models. however, the qualitative stress distribution on the top sandwich panel surface can be well described. the von mises stress in the top surface central point of the sandwich panel with waved bcc cells is 1.341e9 pa, for the homogenised model the von mises stress in the same evaluation point is 1.453e9 pa. the local stress response is described with an error of 8.3% by the homogenised model. this local qualitative description is acceptable because is sufficiently far from the boundary conditions and from the lattice core where the lattice struts introduce stress concentration, see fig. 19. fig. 19 von mises stress distribution along the sandwich panel section with structural steel skins and lattice core based on three waved bcc cells along the thickness direction 750 s. valvano 5. composite strut the bcc geometry with straight strut is here considered. the author proposes to introduce metal matrix composite material with a concentric layup along the strut axis. the concentric multi-layered strut sequence is depicted in fig. 20. fig. 20 composite bcc cell with straight strut with a concentric layup along the strut axis different strut diameters d = 1, 2 mm as well as various thickness layup ratios t̄ = talsic / ttotal = 0.25, 0.5, 0.75, have been considered. the maximum cell dimension is kept fixed as the previous standard bcc cell with l = 10 mm. the composite strut materials are made by an isotropic aluminium and assumed isotropic metal matrix composite alsic with short fibres [42], the properties are resumed in table 5, where vf stands for volume fraction. table 5 elastic properties for the composite strut cells, aluminium and alsic [42] for different volume fractions ρ [kg/m3] e [gpa] g [gpa] ν [-] aluminium 2610 76.8 28.8 0.333 alsic vf 0.41 % 2860 138.9 56 0.239 alsic vf 0.54 % 2940 183.9 74.4 0.235 alsic vf 0.70 % 3020 224.6 91.2 0.231 the concentric material layup along the strut axis does not introduce any kind of asymmetry, the equivalent material has the same symmetry properties of the initial constituent ones, in other words the equivalent material can be assumed as isotropic. the boundary surfaces are defined as the first subsection and, for the sake of brevity, the discussion is not here reported. in table 1 a resume of the used boundary conditions is given. the homogenised isotropic properties for the metal matrix composite bcc cell with concentric strut layup is given in table 6. homogenised properties of lattice metal composite cell 751 table 6 homogenised properties of the metal matrix composite bcc cell for different materials thickness ratios t̄ and different strut diameters d ρ [kg/m3] e [mpa] g [mpa] ν [-] d = 1 aluminium 129.22 27.171 483.77 0.4914 alsic vf 0.41 % 141.59 48.115 879.44 0.4910 alsic vf 0.54 % 145.55 63.649 1164.6 0.4913 alsic vf 0.70 % 149.51 77.671 1422.7 0.4914 t̄ = 0.25 aluminium + alsic vf 0.41% 130.05 27.595 510.05 0.4917 aluminium + alsic vf 0.54% 130.31 27.937 528.45 0.4918 aluminium + alsic vf 0.70% 130.58 28.112 544.61 0.4920 t̄ = 0.5 aluminium + alsic vf 0.41% 132.46 29.567 586.67 0.4921 aluminium + alsic vf 0.54% 133.50 30.954 658.37 0.4926 aluminium + alsic vf 0.70% 134.54 32.116 722.75 0.4931 t̄ = 0.75 aluminium + alsic vf 0.41% 136.35 34.870 709.75 0.4923 aluminium + alsic vf 0.54% 138.63 40.175 870.21 0.4928 aluminium + alsic vf 0.70% 140.92 44.861 1015.0 0.4930 d = 2 aluminium 465.74 565.79 2053.1 0.4611 alsic vf 0.41 % 510.35 983.23 3765.1 0.4603 alsic vf 0.54 % 524.63 1299.7 4988.0 0.4603 alsic vf 0.70 % 538.90 1584.9 6095.7 0.4603 t̄ = 0.25 aluminium + alsic vf 0.41% 468.99 584.50 2171.3 0.4612 aluminium + alsic vf 0.54% 470.03 592.67 2248.2 0.4620 aluminium + alsic vf 0.70% 471.07 598.55 2316.0 0.4626 t̄ = 0.5 aluminium + alsic vf 0.41% 478.12 634.50 2502.5 0.4627 aluminium + alsic vf 0.54% 482.08 671.37 2805.7 0.4646 aluminium + alsic vf 0.70% 486.04 700.88 3075.9 0.4661 t̄ = 0.75 aluminium + alsic vf 0.41% 492.21 741.44 3029.9 0.4633 aluminium + alsic vf 0.54% 500.68 857.57 3710.5 0.4652 aluminium + alsic vf 0.70% 509.15 957.79 4322.8 0.4665 finally, the present 3d homogenisation approach for composite lattice structure is validated considering an infinite sandwich panel using double-periodic boundary conditions. two types of panels have been considered: a sandwich one with the skins made of aluminium and the lattice core based on three composite bcc cells, along the thickness direction, made of aluminium and alsic with thickness layup ratios t̄ = 0.5, see fig. 21; a second panel with the same geometry and the same skins, but with a core layer based on the homogenised properties, 752 s. valvano see fig. 22. the panels have the same boundary conditions of the previous subsection: doubleperiodic boundary conditions on all the lateral surfaces and a constant uniform displacement is applied on the top surface, the bottom surface has a fixed support along the loading direction z, in other words w(x,y,z=0) = 0. fig. 21 sandwich panel section assembled with aluminium skins and lattice core based on three composite bcc cells, along the thickness direction, made of aluminium and alsic with thickness layup ratios t̄ = 0.5 fig. 22 sandwich panel section assembled with aluminium skins and homogenised lattice core based on three composite bcc cells made of aluminium and alsic the sandwich panel with composite bcc cells shows a global structural response evaluated through the reaction force at the bottom surface, fz = 7476.5 n. the sandwich panel with the homogenised core has a reaction force of fz = 7392.3 n, it is clear that the homogenised model leads to an error of 1.1% on the global structural response. as the previous structural example, the local stresses cannot correctly be detected by the homogenised models. however, the qualitative stress distribution on the top sandwich homogenised properties of lattice metal composite cell 753 panel surface can be well described. the von mises stress in the top surface central point of the sandwich panel with waved bcc cells is 8.941e8 pa, for the homogenised model the von mises stress in the same evaluation point is 8.328e8 pa. the local stress response is described with an error of 6.8% by the homogenised model. for the sake of completeness, the von mises stress distribution of the top surface of the sandwich panel with composite bcc cells is depicted in fig. 23, the same stress distribution of the homogenised model is reported in fig. 24. fig. 23 von mises stress distribution on the top surface of the sandwich panel with composite bcc cells fig. 24 von mises stress distribution on the top surface of the sandwich panel with homogenised core based on composite bcc cells 754 s. valvano 6. conclusions in this work, the mechanical analysis of an advanced body centred cubic lattice cell has been performed through the derivation of the homogenisation procedure of periodic cellular structures in order to obtain an equivalent set of mechanical properties. an original ansys parametric design language subroutine has been developed for the introduction of the double periodic boundary conditions. the originality of the work consists in the study of different bcc cell configurations: standard metal bcc cell, metal bcc cell with waved struts, metal composite bcc cell. the approach used for the metal composite cell and the waviness struts present innovative idea for the tailoring of cellular solids, and it can be adapted for the study of multi-layered lattice cells or coated cellular structures. a parametric study has been conducted varying the struts diameter, the struts waviness and the thickness ratio of the composite struts. the following conclusions can be drawn: ▪ the bcc geometry configuration permits to obtain a structure with an equivalent shear modulus g greater with respect to the young modulus e. ▪ increasing the struts diameter, it is possible to increase both the shear and young moduli, and at the same time reducing the poisson ratio ν value. ▪ the strut waviness introduces the concept of orthotropic properties for the description of the equivalent young and shear moduli and the poisson ratios. ▪ the increase of the waviness radius permits to reduce the young moduli, the poisson ratios and the gxy shear modulus; at the same time the shear moduli g̅xz and g̅yz are increasing as the waviness radius. ▪ the composite strut permits to add an input to the tailoring of the lattice cell equivalent 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(eds.), homogenization techniques for composite media. springer-verlag, berlin, pp. 194-275. 41. xia, z., zhang, y., ellyin, f., 2003, a unified periodical boundary conditions for representative volume elements of composites and applications, international journal of solids and structures, 40(8), pp. 1907-1921. 42. kumar, s.s., bai, v.s., rajkumar, k.v., sharma, g.k., jayakumar, t., rajasekharan, t., 2009, elastic modulus of al-si/sic metal matrix composites as a function of volume fraction, journal of physics d: applied physics, 42(17), 175504. facta universitatis series:mechanical engineering vol. 19, no 3, special issue, 2021, pp. 447 471 https://doi.org/10.22190/fume210318047b © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines for enabling mobility darko božanić1, aleksandar milić1, duško tešić1, wojciech sałabun2, dragan pamučar1 1university of defense in belgrade, military academy, belgrade, serbia 2research team on intelligent decision support systems, department of artificial intelligence methods and applied mathematics, faculty of computer science and information technology, west pomeranian university of technology, szczecin, poland abstract. the paper presents a hybrid model for decision-making support based on d numbers, the fucom method and fuzzified rafsi method, used for solving the selection of the group of construction machines for enabling mobility. by applying d numbers, the input parameters for the calculation of the weight coefficients of the criteria were provided. the calculation of the weight coefficients of the criteria was performed using the fucom method. the best alternative was selected using the fuzzified method, which was conditioned by the specificity of the issue so that in this case, the selection of the best alternative was made using the fuzzified rafsi method. key words: d numbers, fucom, fuzzy numbers, rafsi, construction machines 1. introduction the dynamics of living in a modern environment imposes plenty of demands. one of the determining demands is most often expressed through the need for faster transport of goods and services. the way of fulfilling the set of demands is represented by the development of communication transport capacities and possibilities (e.g. quality and branching of roads expressed by meeting certain standards, possibilities of certain means of transport, etc.). the most significant percentage of roads are civil engineering structures roads of high quality and high throughput. enabling mobility on such roads is based on repairing possible damage to certain road sections and reconstructing certain sections to improve received march 18, 2021 / accepted may 30, 2021 corresponding author: darko božanić university of defence in belgrade, military academy, pavla jurišića šturma 33, belgrade, serbia e-mail: dbozanic@yahoo.com 448 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar their features. the construction machines are the main working means in these tasks. therefore, it can be pointed out with certainty that these are the factors on which the quality, scope, and costs of production, respectively, of construction, depend on. having in mind that different construction machines perform numerous works, the requirements are set such as: complete consideration of the scope and type of assignments, precise determination of the required machines and their number (with knowledge of their characteristics and reliability), an appropriate grouping of the machines, consideration of their interdependence and determination of the critical machine. the above-mentioned facts provide the condition for a large share in the total facilities' repair and reconstruction costs. simultaneously with the stated costs, one of the requirements for reducing the operating costs of the vehicle fleet and savings in construction costs stands out [1]. solving resource savings is one of the defining directions of industry and modern economy [2, 3]. one of the cost reduction approaches in the construction sector is presented through: assessment of the performance of different types and subcategories of construction machines in different conditions [4-6], consideration of critical machine performance (engine speed, engine type, operating hours, torque or engine power, weight of machines, type of fuel, service life of equipment) [7-11], the definition of the maximum allowed idling time, the definition of critical machine, change of type of fuel and mixture, use of machines equipped with newer technology and transition to electrical circuit systems [10, 1215]. in addition to the above mentioned, there are other, so-called external parameters affecting the consumption of resources and are related to the performance of construction machinery, such as climatic and soil conditions, driver/operator experience, terrain slope, soil type, density, and volume of sediment being worked on, etc. [12]. many requests for reducing the costs of road repair and reconstruction have resulted in the imposition of different approaches to resolving the set requests. some approaches are based on: precise definition of the set task, clear sizing of the group of machines composition, complete knowledge of the machines' performances (under the stated conditions), the definition of critical machine, or understanding the conditionality of the work process by a machine. all the approaches have their advantages and disadvantages. the engineering units of the serbian army possess construction machines in their service, and these machines are intended for the construction, repair, and reconstruction of temporary military roads. in that regard, engineering units are designed to enable the mobility of other units. it is essential to facilitate mobility during the implementation of combat operations, where possible omissions (or untimely execution of tasks) can have significant consequences. considering that there are many construction machines in the serbian army with the same or similar purpose, the decision-makers are often faced with reaching the optimal composition of the group of devices that will perform a particular task. in this context, a model was developed to select the group of construction machines for enabling mobility, which is primarily based on the structural characteristics of the devices, respectively, the criteria based on these characteristics. other external influences are also combined through the evaluations of the values of alternative solutions by every criterion. selecting the optimal group of machines for earthworks is not a typical research subject in scientific papers. jovanović [16] considered selecting the optimal group of devices for earthworks on a residential and office building by applying compromise programming and multi-criteria ranking of alternative solutions. similar to the presented d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 449 problem, the selection of other different working groups using multi-criteria decisionmaking in the literature has not been, for the most part, considered. karabašević et al. [17] select staff in the company's team, using the swara and aras methods. alencar and de almeida [18] apply the promethee method and group decision-making to select project team members. shipley et al. [19] show the selection of team members during the project using fuzzy logic and the dempster-shafer theory of evidence. zolfani and antucheviciene [20] use the ahp and topsis methods to select team members. bazsova [21] selects members of the project management team using the ahp method. božanić and pamučar [22] select a military unit to remove explosive barriers using a fuzzy logic system. to form an elite security team, dadelo et al [23] use the topsis and saw methods. 2. description of the methods used the specificity of the research issue conditioned the use of methods which take into consideration uncertainty, both for the calculation of the weight coefficients of the criteria, and for the selection of the best alternative. having in mind the simplicity of the mathematical apparatus, on the one hand, as well as the possibilities of the methods on the other hand, the authors decided to use models based on d numbers, the fucom method and fuzzified rafsi method. fig. 1 presents general overview of the model. through the first phase of the model, the criteria influencing the selection are identified, using expert evaluation while the calculation of weight coefficients is made using expert evaluation, d numbers and the fucom method. in the second phase, the identification of alternatives and the selection of the best alternative are performed. in the third phase of model development, the sensitivity analysis is performed by changing the weight coefficients of the criteria. the following text of this unit provides theoretical basis of the applied methods (d numbers, the fucom method and fuzzified rafsi method). 2.1. d numbers the dempster-shafer's theory of evidence is used to process uncertain information [24, 25]. this theory has wide application because it allows direct expression of uncertainty by assigning probability to the elements organized into subsets within a set, rather than to individual objects within a set. although it has been applied in a large number of papers for processing uncertain information, the classic dempster-shafer's theory of evidence has certain limitations as well. one of the well-known problems is the management of contradictions in the case of very conflicting evidence. additionally, the dempster-shafer's theory of evidence implies the exclusivity of elements in discernment, which has greatly limited the practical application of this theory [26, 27]. 450 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar phase 1.1. identification of the criteria by applying expert opinion step 1. ranking all the criteria by significance step 2. comparison of the criteria by applying d numbers step 3. calculation of the final values of the weight coefficients p h a se 1 . id en ti fi ca ti o n o f th e cr it er ia a n d d ef in in g t h e w ei g h t co ef fi ci en ts o f th e cr it er ia phase 1.2. calculation of the weight coefficients of the criteria by applying the fucom method and d numbers p h a se 2 . s el ec ti o n o f th e b es t al te rn at iv e phase 2.1. identification of the alternatives phase 2.2. selection of the best alternative by applying fuzzy rafsi method step 1. forming of fuzzy initial decisionmaking matrix step 2. defining ideal and anti-ideal values step 3. copying the elements of the initial decision-making matrix into the criteria intervals step 4. forming normalized decisionmaking matrix step 5. calculation of fuzzy criteria functions of alternatives and ranking alternatives p h a se 3 . s en si ti v it y a n al y si s b as ed o n ch an g es o f th e w ei g h c o ef fi ci en ts o f th e cr it er ia step 1. making scenarios for the change of the weight coefficients of the criteria step 2. ranking of alternatives by applying different scenarios step 3. calculation of the spearman’s coefficient of rank correlation step 4. adopting final rank of alternatives fig. 1 general overview of the decision-making model including phases and steps due to the mentioned problems, an extension of this theory is performed in order to obtain d numbers, which eliminated certain disadvantages of the dempster-shafer's theory (fig. 2). d numbers can effectively present uncertain information since: 1) the exclusive property of the elements in the frame of discernment is not required, and 2) the completeness constraint is released if necessary (fig. 2b). these improvements provided the use of d numbers in solving numerous practical problems. d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 451 o x medium good very good (a) frame of discernment in dempster-shafer evidence theory o x medium good very good (b) problem domain in d numbers fig. 2 the frame of discernment in the dempster-shafer evidence theory and domain in d numbers 28 the specific application of d numbers can be found in a large number of publications about solving various issues: risk level assessment [29], supplier selection together with the fuzzy ahp method [30], supplier selection in combination with the ahp method [31], determining the quality of logistics services in order to gain adequate insight into the processes of managing service providers with the dematel method and trapezoidal fuzzy numbers [28], evaluation of the green supply chain management practice, where the fuzzy ahp method was used for calculation of weight coefficients [32], in error mode and effect analysis (fmea) in the specific case on the rotor blades for aircraft turbines together with the topsis method [33], selection of an autocannon for integration into combat vehicles in the model with the lbwa and mabac methods [34], selection of suppliers in the tractor production industry with the topsis method [35], etc. basic mathematical formulations of d numbers are presented below. let  be a finite nonempty set, and a d number is a mapping that d: ψ→[0,1], with ( ) 1 ( ) 0 a d a and d    = (1) where ∅ is an empty set and a is any subset of ψ. in the case the condition is met where ∑ 𝐷(a) ≤ 1a⊆ψ the information is considered complete; otherwise, the information is not complete. in discrete set ψ ={b1b2,...bi,bj,...,bn, where bi  r and bi  bj (when i  j), d numbers are presented as 1 1 2 2( ) , ( ) ,..., ( ) , ( ) ,..., ( )i i j j n nd b v d b v d b v d b v d b v= = = = = (2) d numbers presented in expression (2) can be also presented in a simplified way as d={(b1,v1),(b2,v2)...(bi,vi),(bj,vj)...(bn,vn),where the condition is met where vi 0 and ∑ 𝑣𝑖 ≤ 1𝑛 𝑖=1 . 452 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar if two d numbers are provided: d1={(b1,v1),...(bi,vi)...(bn,vn) and d2={(bn,vn),...(bi,vi)... (b1,v1), the combination of d numbers d=d1  d2 is defined as [26] 1 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 1 1 2 2 2 ( ) 0 1 ( ) ( ) ( ), 1 1 ( ) ( ) ( ) ( ) b b bd d b b b b d d b b b b k with k b b q q q b q b d d d d d d  =  =    =   =    − = = =     (3) if d1 and d2 are defined in the frame of discernment and if q1=1 and q2=1, then d number combination rule (3) is transformed into the dempster's rule (4). 1 2 1 2 1 ( ) ( ) ( ) 1 where ( ) ( ) b c a b c m a m b m b k k m b m b  =  = = − =   (4) where a, b and c are three elements of 2ψ, and k is a normalization constant, called a conflict coefficient between two basic probability assignment (bpa) functions. the rule for contamination of d numbers presents a mechanism allowing fusion of uncertain information presented in d numbers: permutation invariability: if there are two d numbers presented as d1={(b1,v1),... (bi,vi)...(bn,vn) and d2={(bn,vn),...(bi,vi)...(b1,v1) than d1  d2, where „“ means „equal to“. integration: for discrete d number d={(b1,v1),(b2,v2)...(bi,vi),(bj,vj)...(bn,vn) the integration operator can be defined as follows: 1 ( ) n i i i i d d v = = (5) where di r+, vi 0 and ∑ 𝑣𝑖 ≤ 1𝑛 𝑖=1 . 2.2. the fucom method the fucom (full consistency method) method is intended for determining the weight coefficients of the evaluation criteria. the method was first presented by pamučar et al. [36]; since then it has been applied in a large number of papers for solving various problems, such as: d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 453 ▪ landfill site selection, together with the codas method [37], ▪ assessment of critical success factors for continuous academic quality assurance and accreditation, in the model with fuzzy ahp method [38], ▪ evaluation of the provisional sizing process in the clothing industry, with the fuzzy piprecia method [39], ▪ selection of the best solution for business balance of the passenger railway operator, as a part of the validation test with the fuzzy ahp method [40], ▪ determination of macro location for railway network, in the model with the fuzzy topsis method [41], ▪ selection of a distribution channel, in combination with the marcos method [42], ▪ solving the case study in the rubber glove industry, used in a hybrid model with the vikor method [43], ▪ for the purpose of assessing human resources, on which the overall efficiency of the enterprise depends, together with the marcos method [44], ▪ mineral potential mapping in greenfields, in the model with the moora and moosra method [45], ▪ selection of vehicles with automatic guidance (agvs), in combination with the rrov (rough range of value) method [46], ▪ improvement of service quality measurement in the hybrid delphi-fucomservqual model [47], ▪ selection of a terrain vehicle for equipping military units, through the validation test of the ahp-dea model, with the bwm method [48], ▪ selection of a sustainable supplier in a construction company, with the copras method, while for the validation of the results the aras, waspas, saw and mabac methods were used in combination with rough numbers [49], ▪ evaluation of the sustainable performance of suppliers, with the mairca method [50], ▪ selection of a location for a textile manufacturing facility, in combination with the gis[51], and, ▪ selection of a fighter aircraft, with the aras method [52]. in addition to the classic fucom method, a fuzzified version of this method was used for solving practical problems, such as: ▪ selection of a system for desalination of renewable energy sources with a perspective of sustainability, with the danp and vector-aided topsis methods [53], ▪ selection and prioritization of appropriate measures for the management of transport requirements in urban mobility system in istanbul, in the fuzzy fucom-dombibonferroni model [54], ▪ in the example of suppliers of electricity from renewable sources [55], ▪ determining sustainability of sewage sludge in terms of energy source with the consideration of hybrid data, together with the fusion approach [56]. the application of the fucom method with rough numbers is discussed in the problem of selecting the location of logistics centers in the spanish autonomous communities with the cocoso method (combined compromise solution) and it is presented in yazdani et al. [57], while the selection of the contractors for solar panel installations is made by applying gray numbers in the gray swara-fucom model [58]. the problem of the group decision-making solved by fucom method is presented in [42, 52,59]. 454 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar the fucom method has a fairly simple mathematical apparatus, providing the results similar or the same as other methods for defining weight coefficients of criteria, such as the ahp and the best-worst methods. the fucom method consists of three steps: step 1 in the first step are ranked all the criteria influencing the decision c={c1,c2,...,cn. the criteria are ranked from the most significant to the least significant criterion, respectively, from the criterion assuming to have the largest weight coefficient to the criterion with the smallest weight coefficient: (1) (2) ( )...j j j kc c c   (6) where k presents the rank of the observed criterion. if there is an opinion of the existence of two or more criteria with the same significance, the sign of equality is placed instead of ">" between these criteria in the expression (6). step 2 in the second step the first-ranked criterion is compared to the other criteria. the comparison of the criteria is performed by experts by applying d numbers. applying expressions (1) to (5), aggregated criteria significance (𝜛𝐶𝑗(𝑘) ) is calculated. in accordance with the calculated comparison, comparative significance of criteria is calculated (φk/(k+1), k=1,2,...,n, where k presents the rank of the criteria). the vector of the comparative priorities of the evaluation criteria are obtained, as in expression (7): 1/2 2/3 /( 1)( , ,..., )k k   + = (7) step 3 in the third step, final values of the weight coefficients of the evaluation criteria (w1,w2,...,wn)t are calculated. final values of the weight coefficients should meet two conditions: / ( 1) 1 k k k k w w  + + = (8) / ( 1) ( 1)/ ( 2) 2 k k k k k k w w  + + + + =  (9) where φk/(k+1) presents the significance (priority) that the criterion of cj(k)rank is compared to the criterion of cj(k+1)rank. the calculation of final values is performed by applying expression (10), and solving the obtained system of equities. ( ) / ( 1) ( 1) ( ) / ( 1) ( 1)/ ( 2) ( 2) 1 min . . , , 1, 0, j k k k j k j k k k k k j k n j j j s t w j w w j w w j w j       + + + + + + = −   −    =     (10) where χ presents maximum consistency, respectively, tends to be χ =0. d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 455 2.3. fuzzy rafsi method ranking of alternatives through functional mapping of criterion sub-intervals into a single interval (rafsi) is a method first presented in the paper by žižović et al. [60]. using the rafsi method, žižović et al. [60] evaluated the researchers who applied for a job in a scientific research center, and the results obtained by their application are compared with those obtained using the topsis, vikor and copras methods. since this method was published in mid-2020, its application in various fields has not been widely represented yet. so far, it has been used in the problem of sustainable health system reorganization in the emergency caused by the covid-19 virus pandemic, along with fuzzy sets and the lbwa and macbeth methods 61. in this paper, the fuzzified rafsi method (frafsi) is used. fuzzification is performed by applying triangular fuzzy numbers t = (t1, t2, t3), as in fig. 3, where t1 presents the left, t3 the right distribution of the confidence interval of fuzzy number t while t2, where the function of fuzzy number membership has a maximum value, one. t1 t2 t3 1 ( )t x  ( ) 21, t x x t = = ( ) 1 1 2 2 1 , t x x t t x t t t  − =   − ( ) 3 2 3 3 2 , t x t x t x t t t  − =   − ( ) 10, t x x t =  ( ) 30, t x x t =   0 αt1 αt2 fig. 3 triangular fuzzy number 62 the steps of the fuzzy rafsi (frafsi) method are presented below 61. step 1 forming fuzzy initial decision-making matrix. this matrix is formed by the evaluation of the defined alternatives from set ai(i=1,2,...,m) in relation to the defined set of criteria cj (j=1,2,...,n). 11 12 1 21 22 2 1 2                =        n n m m mn m n x (11) where 𝜉𝑖𝑗 = (𝜉𝑖𝑗 𝑙 , 𝜉𝑖𝑗 𝑠 , 𝜉𝑖𝑗 𝑢 ) denotes the value of the i-th alternative for the j-th criterion (i=1,2,...,m; j=1,2,...,n). experts can also be engaged in obtaining the elements of the x matrix, where the initial decision-making matrix would be obtained by averaging the elements from 456 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar all expert initial decision-making matrices. considering the specificity of the described problem, a decision will most often be made based on the assessment/calculation of one person. step 2 defining ideal and anti-ideal values. for every criterion cj (j=1,2,...,n) a decisionmaker defines ideal value by criterion cj (𝜉𝐼𝑗 ) and anti-ideal value by criterion cj(𝜉𝑁𝑗 ). defining mentioned values are determined criteria intervals which depend on the character of the criterion: [ , ], [ , ], j j j j n i j i n for benefit criteria c for cost criteria         (12) step 3 copying elements from the decision-making matrix into the criteria intervals. for every alternative from set ai (i=1,2,...,m), function 𝑓𝐴𝑖 (𝐶𝑗) which copies the criteria intervals from the initial decision-making matrix (11) into the criteria interval [n1,nb] is defined, as in expression (13): 11( ) j j i j j j j i n bb a j ij ij i n i n n nn n f c          − − = = + − − (13) where nb and n1 represent the relations showing how better the ideal value is when compared to the anti-ideal value, 𝜉𝐼𝑗 and𝜉𝑁𝑗 respectively, represent ideal and anti-ideal value by criterion cj, while 𝜉𝑖𝑗 denotes the value of the i-th alternative for the j-th criterion from the initial decision-making matrix. the relation of the ideal and anti-ideal value can be different, but it should not be lower than 1:6, respectively, n1=1 and nb=6. applying expression (13) standardized decision-making matrix 𝑇 = [�̃�𝑖𝑗]𝑚𝑥𝑛 (i=1,2,...,m; j=1,2,...,n) is obtained, as in eq. (14). 1 11 12 1 2 21 22 2 1 1 2 2                      =   n n n m m m mn a a t c c a c (14) in matrix t all the elements of the initial decision-making matrix are transferred into interval �̃�𝑖𝑗[𝑛1, 𝑛𝑏]. step 4 forming normalized decision-making matrix 𝑁 = [�̃�𝑖𝑗]𝑚𝑥𝑛(i=1,2,...,m; j=1,2,...,n). 1 11 12 1 2 21 22 2 1 1 2 2                      =   n n n m m m mn a a n c c a c (15) where �̃�𝑖𝑗[0,1] present normalized elements of matrix n. d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 457 the way of normalization of the elements of matrix n depends on the type of criteria. the way of calculation of the normalized values is provided in the expression: , for benefit criteria 2 , for cost criteria 2 ij ij ij a h       =    (16) in expression (16) a represents arithmetic value of elements n1 and nb, which is calculated by applying the expressions: 1 2 bn n a + = (17) value h presents harmonic mean of elements n1 and nb, and it is obtained by applying the expression: 1 2 1 1 b h n n = + (18) step 5 calculation of fuzzy criteria functions of alternatives �̃�(ai) and ranking alternatives. the criteria functions of alternatives �̃�(ai) are calculated by applying the expression: 1 ( ) n i j ij j q a w  = = (19) where wj re represents the weight coefficient of the criteria, and �̃�𝑖𝑗normalized value of the alternative ai (i=1,2,...,m) by the criterion cj ( j=1,2,...,n). the alternatives considered are ranked from the largest (the first-ranked alternative) to the smallest (the last-ranked alternative) value of fuzzy criteria function �̃�(ai). instead of ranking the value of fuzzy criteria function �̃�(ai), defuzzification can be carried out before ranking, thus making the ranking process much simpler. defuzzification can be performed in different ways. one example is provided in expression (20): ( ) ( ( ) 4 ( ) ( ) ) / 6l s u i i i iq a q a q a q a= +  + (20) where q(ai) is the defuzzified value of fuzzy criteria function �̃�(ai), q(ai)l the left distribution of the confidence interval of fuzzy criteria function �̃�(ai), q(ai)u the right distribution of the confidence interval of fuzzy criteria function �̃�(ai), and q(ai)s the value of fuzzy criteria function �̃�(ai) where the membership degree is the highest, receptively, one. 3. application of the d numbers – fucom – frafsi model in this section presents an application of the proposed multi-criteria methodology for the selection of the composition of the group of construction machines for enabling mobility. in the first part, the criteria are determined, on which the selection of the best 458 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar alternative and the calculation of the weight coefficients of the criteria depend. determining the criteria and initial elements for the calculation of the weight coefficients of the criteria was done by engaging seven experts. in the second part of this section the process of selection of the best alternative is presented. 3.1 defining the criteria and their weight coefficients the complexity of the research issue influences the determination of criteria and their weight coefficients to be done in several iterations. at the end of the process, the experts agreed that selecting the best alternative was influenced by six criteria, which are explained below. criterion 1 (c1) performance (m3): expressing the degree of use of construction machines and training of operators is done by work performance [63]. in this specific case, after the calculation, the performance of the key machine is taken as the value according to this criterion. the key machine is the one whose performance is the lowest. it is important to emphasize it because most machines in the group are connected, so that the duration of the key machine's work is also the duration of the whole group work [63]. criterion 2 (c2) operational reliability of the group of construction machines: the reliability of construction machines is usually defined as the probability of performing a specific function without failure under given conditions for a particular time [64]. to evaluate the alternatives according to this criterion, the frequency of failures is estimated (expected number of machine failures in a certain period). the practice has shown that many failures are not expected with the machines of a newer (more recent) production date. simultaneously with the increase in age, the number of expected failures in a certain period of time increases. considering that the group comprises machines with different years of production and made by different producers, special fuzzy linguistic descriptors were made to evaluate this criterion, as presented in fig. 4. the scale shown has six fuzzy linguistic descriptors: very low (vl), low (l), satisfactory (s), medium (m), high (h), and very high (vh). 1 0.8 0.6 0.4 0.2 0 2 3 4 5 6 vl l s h vh 1 m fig. 4 fuzzy linguistic descriptors for the c2 criterion description criterion 3 (c3) possibility of movement outside regulated roads: this criterion presents the possibility of movement of the construction machines to the directions where the terrain is not adjusted to the needs of the machine. since the criterion is evaluated in relation d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 459 to the group, the device's value with the least possibilities of movement outside regulated roads is taken into the calculation. the value of the criterion is expressed in percentage. criterion 4 (c4) the need for a means of transport (tow truck): the movement of the machine on a certain terrain is conditioned by the technical possibilities of the machine itself and the dependency of the machine on the terrain features. during the work engagement of the device, the device needs to be moved from one location to another. in such situations, it is necessary to consider the possibility of self-propelled movement, respectively, the necessity of engaging appropriate means of transport to reduce negative characteristics of the devices for moving construction machines and create necessary conditions for timely arrival at work. the criterion is linguistic, and the values are assigned using fuzzy linguistic descriptors, as in fig. 5. the scale shown has four fuzzy linguistic descriptors: a rarely (r), b occasionally (occ), c often (o), d almost always (aa). criterion 5 (c5) technical capability of fast troubleshooting: it is not possible to engage construction machines without an adequately organized technical support. technical support in combat operations, in addition to ongoing maintenance, is also intended for fast troubleshooting. the speed of troubleshooting depends on several elements: the type of failure, the development of technical support (training of people), the type of machine, the uniformity of devices by types and categories (availability of spare parts), and the like. in this context, a particular linguistic scale is defined to assess this criterion, as in fig. 5. there is a well-developed technical support for older assets, which would monitor the group; however, for the assets in the warranty period, failures are fixed by maintenance companies, which can be a significant problem in combat operations. the scale shown has four fuzzy linguistic descriptors (fig. 5): a very small (vs), b small (s), c medium (m), and d high (h). 1 0.8 0.6 0.4 0.2 0 2 3 4 5 6 a c d 1 7 b fig. 5 fuzzy linguistic descriptors for the c4 criterion description criterion 6 (c6) conveniences of construction features (possibility of setting different types of working tools): its construction features predetermine the purpose of the machine based on its equipment with appropriate tools for realizing its tasks. the possibility of using more tools on one device significantly improves the work process. it can reduce the number of machines in the group or create a better potential for solving problems, which is difficult to predict in the initial phase. the criterion has a numerical 460 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar character, and it is defined through the number of additional work tools, which can be placed on construction machines and thus engage the machine in other tasks. from the previous explanation, it can be concluded that the evaluation of alternatives by criteria is performed numerically (c1, c3 and c6) and linguistically (c2, c4 and c5). in addition to the above mentioned, the set of criteria cj (j=1,2,...,6) can be divided into two subsets: a subset of benefit-type criteria (𝐶𝑗 +where a higher value of the alternative by criteria is more desirable, and which consists of the criteria c1, c2, c3, c5 and c6) and a subset of cost-type criteria (𝐶𝑗 −where a lower value of alternative by criteria is more desirable), which consists of criterion c4. after defining the criteria, the conditions for calculating the weight coefficients of the criteria using d numbers and the fucom method are met. step 1 in the first step, the criteria are ranked from the most important to the least important. the rank of the criteria is reached by the consensus of experts. the experts agreed with the following ranking of criteria: c1 c2 c3 c4c5 c6. step 2 in the second step, every expert compares the first-ranked with the other criteria by applying d numbers, after which their opinions are aggregated into one. the comparison is performed using a scale 𝜛𝐶𝑗(𝑘) [1, 9]. the following are the values of de (where e represents the number of experts e=1,2,...,7) for the comparison of the firstranked (c1) and the second-ranked (c2) criterion: d1={(1,0.2),(1;2,0.2),(2,0.6)} d2={(1,0.5),(1;2,0.3),(2,0.1)} d3={(1,0.1),(2,0.2),(3,0.7)} d4={(1,0.3),(2,0.5),(2;3,0.2)} d5={(2,0.5),(2;3,0.1),(3,0.4)} d6={(2,0.6),(3,0.1),(4,0.1)} d7={(2,0.22),(2;3,0.25),(3,0.5)} after the aggregation, the following values are obtained: d1-2={(1,0.403),(1;2,0.093),(2,0.403)} d3-4={(1,0.097),(2,0.452),(3,0.452)} d5-6={(2,0.702),(3,0.098)} d5-7={(1,0.159),(2,0.741)} d1-4={(2,0.635),(3,0.014)} d1-7={(2,0.698)} based on experts' opinion, the relation of the first-ranked (c1) and the second-ranked (c2) criteria is 𝜛𝐶𝑗(1) = 1.397. the importance of the comparison of the first-ranked (c1) in relation to other criteria is 𝜛𝐶𝑗(𝑘) = (1, 1.397, 1.882, 2.298, 2.601, 4.489). based on the obtained importance values of the criteria, we calculate the comparison importance values of the criteria 𝜑𝐶1/𝐶2 = 1.397/1 = 1.397, 𝜑𝐶2/𝐶3 = 1.882/1.397 = 1.347, 𝜑𝐶3/𝐶4 = 2.298/1.882 = 1.221, 𝜑𝐶4/𝐶5 = 2.601/2.298 = 1.132 and 𝜑𝐶5/𝐶6 = 4.489/2.601 = 1.726. applying expression (10) the final model for determining weight coefficients is defined d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 461 3 51 2 4 2 3 4 5 6 31 2 4 3 4 5 6 6 1 min 1.397 , 1.347 , 1.221 , 1.132 , 1.726 , 1.882 , 1.645 , 1.382 , 1.954 , 1, 0,j j j w ww w w w w w w w ww w w w w w w w w j           = − = − = − = − = − = − = − = − = − = =   by solving the previous expression the weight coefficients of the criteria are obtained, as shown in table 1. table 1 weight coefficients of criteria criteria weight coefficients of criteria c1 0.304 c2 0.218 c3 0.162 c4 0.132 c5 0.117 c6 0.067 criterion c1 has the highest weight coefficient. the difference compared to the least significant criterion (c6) is quite large, which is the result of expert evaluation. criterion c1 has the highest weight coefficient, which presents the expected decision of the expert because it is the criterion directly related to the execution of the task in which the group of construction machines is engaged for the entire time of the task. unlike criterion c1, criteria c2 and c5 are related to the assessment assuming the occurrence of problems in operation and their solution, criteria c3 and c4 are related only to the part of the task, and criterion c6 presents the assessment of possibilities, which does not have to be used during the task. 3.2. selection of the best alternative the sizing of potential alternative solutions, respectively, the groups of construction machines that would be engaged in enabling mobility of the serbian army units to realize the task of repairing and reconstructing the road section, is performed for the needs of selecting the best alternative. the group generally consists of the following types of machines: dozers, loaders, diggers, motor vehicles for the transport of loose material (self-unloaders), road rollers, compressor stations, pavers, transport vehicles (for transport of machines whose technical capabilities do not allow self-propelled movement over longer distances), and others. practical works on the repair and reconstruction of certain road sections have indicated that the dozers and loaders, based on their performance and mode of operation, can be classified into a group of critical machines. in order to understand more fully the possibility of reducing (eliminating) the impact of the critical machine on the success of the assigned task, the formation of alternatives (groups of construction machines) is performed. the groups are composed of variable and permanent composition, in accordance with the construction machines forming part of the serbian army (table 2). 462 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar table 2 overview of alternatives alternative variable composition of group permanent composition of group a1 dozer (imk 14. oktobar tg-170) loader (imk 14. oktobar 160) digger, self-unloader, roller, compressor station, transport vehicles a2 dozer (caterpillar d5k2 xl) loader (caterpillar 966m) a3 dozer (dressta td-15m) loader (caterpillar 966m) a4 dozer (shantui sd 20-5) loader (jcb 436 ht) a5 dozer (imk 14. oktobar tg-170) loader (caterpillar 966m) a6 dozer (imk 14. oktobar tg-170) loader (jcb 436 ht) a7 dozer (caterpillar d5k2 xl) loader (imk 14. oktobar 160) a8 dozer (dressta td-15m) loader (imk 14. oktobar 160) after the alternatives are defined, the conditions for the application of the frafsi method are met. step 1 in the first step, the initial decision-making matrix (x) is defined. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 5 61 2 4 1 2 3 4 5 6 7 8   237, 40, 44 60,70,75 1422, 25, 27 75,80,85 1443, 45, 49 72,75,80 225,30,33 75,78,80 1537, 40, 44 60,70,75 337, 40, 44 60,65,70 22, c c c m c c a vl aa h vh o vs occ v l c a a a a s h o s l aa m a a a h a x a = ( ) ( ) ( ) ( ) 125, 27 75,80,85 143, 45, 49 65,75,80 h o m s occ s                             considering the existence of the qualitative criteria, by applying fuzzy linguistic descriptors (figs. 4 and 5), their quantification is performed by matrix xk. d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 463 ^ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 5 61 2 4 1 2 3 4 5 6 7 8   237, 40, 44 1,1 , 2 60,70,75 6,7,7 6,7,7 1422, 25, 27 5.5,6,6 75,80,85 3.5, 5,6.5 1,1 , 2 43, 45, 49 3.5, 4, 4.5 72,75,80 1 k c c cc c c a a a a x a a a a = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 14.5,3, 4.5 1,1, 2 3.5,5,6.5) 225,30,33 4,5,6 75,78,80 1.5,3, 4.5 1537, 40, 44 1.5, 2, 2.5 60,70,75 6,7,7 3.5,5,6.5 337, 40, 44 1.5, 2, 2.5 60,65,70 6,7,7 6,7,7 122, 25, 27 4,5,6 75,80,85 3.5,5,6.5 3.5,5,6.5 43, 45, 49 2,3, 4 65,75,8 ( 0 1.5,3( ) ( ) 1, 4.5 1.5,3, 4.5                             step 2 in this step we defined ideal set 𝜉𝐼𝑗 and anti-ideal value 𝜉𝑁𝑗 for every criterion cj (j=1,2,...6):     65,6,100,1,7,15 , 15,1,50,7,1,1 .   = = j j i n according to the defined ideal and anti-ideal points, the interval values of all the criteria are defined, including: ▪ for benefit-type criteria:c1[15, 65],c2[1, 6],c3[50, 100],c5[1, 7], c6[1, 15], ▪ for cost-type criteria: c1[1, 7]. step 3 for making standardized matrix, the relation of the ideal and anti-ideal value of 1:6 (n1=1 and nb=6) is accepted. applying expression (13) standardized decision-making matrix (t) is obtained. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 5 61 2 4 1 2 3 4 5 6 7 8 5   3.2,3.5,3.9 1,1,2 2 ,3, 3. 5 5 .17,6,6 5.17 5 ,6,6 1,36 1.7,2,2.2 5.5,6,6 3.5,4,4. 3.08,4.33, c c cc c c a a a a t a a a a = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .58 1,1,1.83 5,64 3.8,4,4.4 3.5,4,4.5 3.2,3.5,4 1.42,2.67,3.92 1,1,1.83 5,64 2,2.5,2.8 4,5,6 3.5,3.8,4 3.08,4.33,5.58 1.42,2.67,3.92 1,36 3.2,3.5,3.9 1.5,2,2.5 2,3,3.5 5.17,6,6 3.08, 4.33,5.58 6,00 3.2,3.5,3.9 1.5,2,2.5 2,2.( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5,3 5.17,6,6 5.17,6,6 1,71 1.7,2,2.2 4,5,6 3.5,4,4.5 3.08,4.33,5.58 3.08,4.33,5.58 1,00 3.8,4,4.4 2,3,4 2.5,3,5.4 1.42,2.67,3.92 1.42,2.67,3.92 1,00                          step 4 applying expressions (17) and (18), the values of geometric and harmonic means (a=3.5 and h=1.71) are obtained, and by using expression (16) the calculation of normalized matrix (n) is done. 464 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar ( ) ( ) ( ) 3 5 61 2 4 1 2 3 4 5 6 7 8 ,   0.46, 0.5 ,0 .5 6 ,0.14,0.14,0.29 0.29 0.43,0.5 0.1 4 c c cc c c a a a a n a a a a = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0.14,0.17 0.74,0.86,0.86 0.19 0.24,0.29,0.31 0.79,0.86,0.86 0.5,0.57,0.64 0.15,0.2,0.28 0.14,0.14,0.26 0.81 0.54,0.57,0.63 0.5,0.57,0.64 0.46,0.5,0.57 0.22,0.32,0.61 0.14,0.14,0.26 0.81 0.29,0.36,0.4 0.57,0.71,0.86 0.( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5,0.54,0.57 0.15,0.2,0.28 0.2,0.38,0.56 0.19 0.46,0.5,0.56 0.21,0.29,0.36 0.29,0.43,0.5 0.14,0.14,0.17 0.44,0.62,0.8 0.86 0.46,0.5,0.56 0.21,0.29,0.36 0.29,0.36,0.43 0.14,0.14,0.17 0.74,0.86,0.86 0.24 0.24,0.29,0.31 0( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )   .57,0.71,0.86 0.5,0.57,0.64 0.15,0.2,0.28 0.44,0.62,0.8 0.14 0.54,0.57,0.63 0.29,0.43,0.57 0.36,0.5,0.57 0.22,0.32,0.61 0.2,0.38,0.56 0.14                          step 5 final calculation of fuzzy criteria functions of alternatives �̃�(ai) is made by applying expression (19). final ranking is done after the defuzzification of fuzzy criteria functions of alternatives, as in table 3. table 3 ranking of alternatives alternative �̃�(ai) q(a) ranking of alternatives a1 (0.335,0.385,0.448) 0.3871 8 a2 (0.418,0.464,0.509) 0.4637 2 a3 (0.449,0.493,0.589) 0.5018 1 a4 (0.35,0.436,0.516) 0.4351 5 a5 (0.361,0.433,0.502) 0.4326 6 a6 (0.354,0.408,0.455) 0.4068 7 a7 (0.361,0.443,0.526) 0.4435 4 a8 (0.347,0.445,0.563) 0.4484 3 using the frafsi method, alternative a3 was ranked first, while alternative a1 was ranked last. such rank of alternatives is expected when considering the data in the initial decision-making matrix (x), respectively, the quantified decision-making matrix (xk). alternative a3, in addition to alternative a8, has the highest value according to the most important criterion (c1). it has significantly high values according to criteria c3, c4 and c6, and slightly lower than the highest one according to criterion c2. alternative a3 is poorly rated, only by criterion c5. on the other hand, alternative a1 , which is the last in the rank, has the values tending to be minimal by all criteria except criterion c5. therefore, the rank of alternative a1 is expected. overall, the final values of decision preferences do not indicate the absolute dominance of the first-ranked alternative, but still are sufficient to consider it the best one. logically, the last step to be made in the model development is a sensitivity analysis. 4. sensitivity analysis decision-making is a complex process in which various mistakes are possible. due to the above, and before adopting the model, a more detailed analysis is necessary to be performed. a sensitivity analysis is usually performed. the sensitivity analysis can be performed by different approaches including: changes in weight coefficients of criteria, change of measurement units in which the values of alternatives are expressed, change of d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 465 scales presenting linguistic criteria, change of type of criteria (cost/benefit), application of dynamic matrices, comparison with other methods, etc. [65]. in most cases, the authors perform a sensitivity analysis based on the changes in weight coefficients of criteria [6676], as is the case in this paper as well. the objective goal of the sensitivity analysis is to evaluate the influence of the most effective influential criterion on the ranking performance of the proposed model [54]. for the sensitivity analysis by the change of weight coefficients, 20 scenarios are developed. the basis for the change in weight coefficients makes the change in the weight coefficient of the best criterion c1. the changes in the weight coefficients of this criterion are made in interval 𝑤𝐶1 [0.003, 0.292], and the values for which the reduction is made are proportionally allocated to the other criteria by applying the proportion 1 1 * *: (1 ) : (1 )n c n cw w w w− = − (21) where 𝑤𝐶1 ∗ represents the corrected value of the weight coefficient of criterion c1, 𝑤𝑛 ∗ the reduced value of the considered criterion, wn the original value of the considered criterion and 𝑤𝐶1 the original value of criterion c1. the proportion set in this way always provides the condition where ∑ 𝑤𝑗 = 16 𝑗=1 . through every correction of criterion c1, the correction respectively, the reduction is done by 5%. the values of the weight coefficients in all scenarios are shown in fig. 6. applying the developed scenarios, changes in the ranks of alternatives are established. the ranking of alternatives by scenarios is shown in table 4. in table 4 are grouped the scenarios according to which the ranking of alternatives is identical. 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 -c1 -c2 -c3 -c4 -c5 -c6 scenarios c ri te ri a w ei g h ts fig. 6 overview of the changes in the weight coefficients of criteria through 20 scenarios 466 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar table 4 ranking of alternatives by different scenarios alternative s1-s3 s4-s7 s8-s11 s12-s13 s14-s19 s20 a1 8 8 8 8 8 8 a2 2 2 1 1 1 1 a3 1 1 2 3 3 4 a4 5 4 4 4 4 3 a5 6 6 6 6 5 5 a6 7 7 7 7 7 7 a7 4 3 3 2 2 2 a8 3 5 5 5 6 6 the analysis of the results obtained by applying different scenarios shows certain changes in the rank of alternatives. this indicates that the presented model is sensitive enough to register changes in the weight coefficients of the criteria. it is clear from table 4 that the rank of the last two alternatives did not change, regardless of the scenario. it is also observed that the first-ranked alternative (a3) retained its position until the eighth scenario, when its place is taken by alternative a2, which is ranked first until the end. in general, changes in the rank of alternatives occur in only five cases: ▪ the rank of alternatives from scenario s1 to scenario s3 (change of the weight coefficient w1 in interval 0.261 ≤ 𝑤1 ≤ 0.292) is identical to the initial rank; ▪ the rank of alternatives from scenario s4 to scenario s7 (change of the weight coefficient w1 in interval 0.201 ≤ 𝑤1 ≤ 0.246) changed in three positions: alternative a4 was ranked fourth while according to the initial rank it was the fifth, alternative a7 was ranked third while according to the initial rank it was the fourth, alternative a8 was ranked fifth while according to the initial rank it was the third; ▪ the rank of alternatives from scenario s8 to scenario s11 (change of the weight coefficient w1 in interval 0.140 ≤ 𝑤1 ≤ 0.185) changes in the position of the firstranked alternative, which is now occupied by alternative a2 and retains that position until the end; ▪ the rank of alternatives for scenarios s12 and s13 (change of weight coefficient w1 in interval 0.109 ≤ 𝑤1 ≤ 0.125) changes through the replacement of the second and the third alternative position, respectively (alternatives a3 and a7); ▪ in scenarios s14 to s19 (change of weight coefficient w1 in interval 0.018 ≤ 𝑤1 ≤ 0.094) changes are observed in the replacement of the place of the fifth-ranked and the sixth-ranked alternative (alternatives a5 and a8); ▪ scenario s20 (change of the weight coefficient where w1=0.003) brings the change at the positions three and four (alternatives a3 and a4); as can be seen from the previous explanation, the changes are gradual and expected because there is a significant change in the weight coefficient of criterion c1. however, it should be noted that the dominance of alternative a3 is not so significant that it retains the first-ranked position in all scenarios. theoretical analysis is confirmed by the statistical correlation of ranks performed using the spearman's correlation coefficient: 2 1 2 6 1 ( 1) n i i d s n n == − −  (22) d numbers – fucom – fuzzy rafsi model for selecting the group of construction machines... 467 where di presents the difference of the rank according to the given scenario and the rank in the corresponding scenario, and n is the number of ranked elements. the spearman's coefficient takes the values from the interval from minus one ("ideal negative correlation") to one ("ideal positive correlation"). in table 5 the values of the spearman's coefficient are provided, comparing the results obtained by applying different scenarios, as well as the initial rank (si). table 5 the values of the spearman’s coefficient scenarios si s1-s3 s4-s7 s8-s11 s12-s13 s14-s19 s20 si 1 1 0.929 0.905 0.833 0.762 0.667 s1-s3 1 0.929 0.905 0.833 0.762 0.667 s4-s7 1 0.976 0.929 0.905 0.833 s8-s11 1 0.976 0.952 0.905 s12-s13 1 0.976 0.952 s14-s19 1 0.976 s20 1 from table 5, it can be noted that the spearman's coefficient of the rank correlation of the considered strategies ranges within the interval s[0.667, 1], presenting a very high correlation degree. general conclusion that can be reached from this analysis is that the developed model registers changes in weight coefficients, through changes in the range of alternatives, as well as that these changes are not significantly large, which is proven by the spearman's coefficient. as the final rank of alternatives the initial rank can be accepted, taking into consideration that the change of the first-ranked alternative occurred when the weight coefficient of criterion c1 decreased from 0.304 to 0.185, which is a significant decrease. 4. conclusion this paper is dedicated to solving the problem of selecting the group of construction machines composition for enabling mobility of the serbian army units based on structural characteristics of construction machines. in order to solve it, a hybrid model based on several methods including: d numbers, the fucom method and fuzzified rafsi method is used. the use of the mentioned methods provided a good treatment of uncertainty following the problem being solved. by applying d numbers, the input parameters for the calculation of the weight coefficients of the criteria were obtained. experts were engaged to define the criteria and their weight coefficients, who were able, due to using d numbers, to present the dilemmas related to the weighting ratios in a way that is closest to their spoken language. in other words, the experts did not have to decide on crisp values when defining the relations of the criteria, but they presented their dilemmas and uncertainty through several different statements. this approach proved to be very applicable in the process of collecting data from experts. the calculation of the weight coefficients of the criteria was performed by the fucom method. eight alternatives were defined for the selection of the best alternative. the defined selection criteria conditioned the use of some of the areas treating uncertainty well when making decisions. in this particular case, triangular fuzzy numbers, respectively, the 468 d. božanić, a. milić, d. tešić, w. salabun, d. pamučar fuzzified rafsi method, were used to present the values of the alternatives by criteria. using the frafsi method, alternative a3 was selected as the best one, which has the dozer dressta td-15m and the loader caterpillar 966m in its variable composition. the stability of the obtained results was tested through a sensitivity analysis. the sensitivity analysis was performed by changing the weight coefficients of the criteria through 20 different scenarios. the results obtained by the sensitivity analysis show that the model reacts to changes in weight coefficients, respectively, that there are changes in the rank of alternatives. these changes are gradual and small. through the analysis of rank correlation, applying the spearman's coefficient, it was determined that almost all the values tended towards ideal rank correlation. in addition to the stability of the results, the sensitivity analysis indicated that any minor errors in defining the weight coefficients of the criteria did not significantly affect the output results. in future research, the presented model and 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technical courier, 67(1), pp. 93-115. 76. tešić, d., božanić, d., 2018, application of the mairca method in the selection of the location for crossing tanks under, tehnika, 68(6), pp. 860-867. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 315 330 doi: 10.22190/fume170508015k © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper the monospiral motorised cable reel in crane applications udc 621.3/.8 vojkan kostić, nebojša mitrović, bojan banković, milutin petronijević faculty of electronic engineering, university of niš, serbia abstract. the main consideration of any reeling system is the effect it has on cable tensions and hence cable life. this paper explains the relationship of reel torque to cable tensions and the reasons why this relationship is so important. such system is characterized by variable parameters, primarily a variable moment of inertia and a variable diameter of the coiled cable. for these reasons, in order to ensure proper dimensioning of the drive, it is necessary to know the motor torques that need to be developed as a function of the coiled cable. the motor should be able to develop the required torques in a very wide speed range. it is shown that for properly sizing the motor it is necessary take into account the dynamics of the cable reel drive. in this paper monospiral motorized cable reel for winding power cable in crane applications with frequency converter fed induction motor is analyzed. also, the equipment selection procedure for the real crane with concrete data is shown. experimental results are recorded during the crane commissioning in real condition. key words: induction motor drive, cable reel, frequency converter, tension control 1. introduction overhead and gantry cranes are typically used for moving containers, loading trucks or material storage. this type of crane usually consists of three separate motions for transporting material. the first motion is the hoist, which raises and lowers the material. the second is a trolley (cross travel), which allows the hoist to be positioned directly above the material for placement. the third is a gantry or bridge motion (long travel), which allows the entire crane to be moved along the working area. very often, in industrial applications additional drives as auxiliary hoist, power cable reel and conveyer belt are needed. therefore, generally, a crane is complex machinery. that is why the cranes are often the subject of received may 08, 2017 / accepted june 22, 2017 corresponding author: vojkan kostić university of niš, faculty of electronic engineering, aleksandra medvedeva 14, niš, serbia e-mail: vojkan.kostic@elfak.ni.ac.rs 316 v. kostić, n. mitrović, b. banković, m. petronijević analysis in terms of performance improvement and drive modernization. electrical technology for the crane control has undergone a significant change during the last few decades. old solutions with ward-leonard drive systems and induction motors with wound rotor are replaced with more contemporary technology that uses frequency converters and standard squirrel-cage induction motors for all types of motion [1, 2, 3]. general classification divides induction motor control schemes into scalar and vector-based methods. opposite to the scalar control, which allows control of only output voltage magnitude and frequency, the vector-based control methods enable control of instantaneous voltage, current and flux vectors. for electric motor drives on cranes, the vector control methods have to be used, rotor field oriented control (rfoc) or direct torque and flux control (dtfc) [4, 5]. energy saving has become an important aspect of the design and operation of new building machinery. currently, energy-saving research has mainly focused on container ports with the focus mostly on only one aspect of electrical or mechanical energy-saving [6]. some experts have only considered electrical energy saving methods. for example, [7, 8] an energy-saving method based on energy recovery by active front end rectifier (afe) at the input of frequency converter is proposed. the afe provides four quadrant operation of drive, which means in motoring and regenerative operation mode. in [9, 10] to achieve energy savings through frequency control and load sharing between the motors in multimotor drives are proposed. the improvement of electrical drives efficiency can be obtained with the use of storage devices. in recent years new electrochemical storage technologies have been developed and, among these, supercapacitors are becoming more and more interesting. their high power density makes them very attractive for application where high powers for short time are required [11]. on the other hand, some experts only consider mechanical energysaving methods. for example, in [12] the use of flywheel in the operation process of the crane is proposed to perform energy recovery. in the papers [13, 14] dynamic behavior of the carrying structure of the portal cranes excited by the crane motion is analyzed, but it does not include the influence of electric motor driven subsystem and control algorithm on crane performance. dynamic behavior of the drive mechanisms that take into account the impact of variable frequency control of the electric motor on executive parts of mechanisms is presented in [15]. one of the most critical issues in overhead cranes is the swing of a suspended load while the crane starts to move and accelerates, changes the movement direction, breaks or stops. in papers [16, 17, 18] continuous anti-swing tracking schemes for crane systems are presented. methods are based on information about the swing angle. sensorless anti-swing control for overhead cranes is described in [19]. this method is based on measuring the voltage and current of trolley drive motor. using the features of rotor field oriented control algorithm, electrical torque, electrical speed, driving force and swing angle are estimated. movable cranes, as container bridges, other outdoor cranes or indoor cranes, realize the current transfer and data transfer by means of flexible energy cables and control cables. maintaining the transmission requires a permanently available system for storing and releasing the cables which moves as synchronously as possible with the movable cranes. the basic patterns of the movement (distance, direction, acceleration, speed, mass) are being defined only by the use of the movable crane. systems which meet such requirements are the motor driven cable reels. for the rail mounted cranes with a long crane path, power supply is usually implemented through the cable reel that is used for winding/unwinding the power cable. based on the the monospiral motorised cable reel in crane applications 317 available literature, the authors of this paper have noted that problems associated with the cable reel are not sufficiently represented, although they are very important for the operation of the crane. there are published papers related to the tension control in manufacturing industries such as rolling mills and cable industry. due to the similar requirements for tension control, variable winding diameter and moment of inertia, individual experiences can be used to control cable reel drives [20]. in this paper the monospiral motorized cable reel for winding power cable in crane applications is considered. the cable reel, from the perspective of electric drives, can be realized in several ways by using: torque motor, wound rotor induction motor, frequency converter fed induction motor. regardless of the way of realization, basic requirement for cable reel drive is maintaining a constant tensile force in the cable, which enables winding/ unwinding cable uniformly. related to this, there are two contradictory demands which are related to tensile force in the cable: large enough value of tensile force sufficient for cable winding as the first and lower value of tensile force in comparison with maximum permitted cable tensile stress as the second. this paper is organized as follows. in the second section theoretical basis for torque, speed and power calculation of monospiral cable reel is exposed. induction motor selection method is described in the next section. in the fourth section selection of induction motor and frequency converter for concrete example (derrick crane installed at the open mine pit in mb kolubara) is illustrated. in the fifth section experimental results of monospiral motorized cable reel for winding power cable with frequency converter fed induction motor in the derrick crane are presented. at the end, some conclusions and recommendations are drawn regarding the monospiral cable reel drive design. 2. theoretical basis induction motor, which drives cable reel, at its shaft must develop torque tm, which should provide cable set-point tensile force, fc,z. therefore, the motor torque must correspond to torque of cable reel, tw, taking into account the losses in the drive and dynamics of the drive. cable reel torque, based on fig. 1, is equal to: 2 , d ft zcw  (1) where d is the outer diameter of the coiled cable. the outer diameter of the coiled cable, based on fig. 1, can be determined as follows: ccore dkdd 2 (2) where dcore is diameter of the core that the cable is wound around, k is number of cable windings on the cable reel, and dc is outer diameter of the cable. number of winding of the cable on cable reel, k, can be determined based on the length of coiled cable, lc, which is based on fig.1: ( ( 1) )c core cl kd k k d   (3) 318 v. kostić, n. mitrović, b. banković, m. petronijević fig. 1 schematic view of the cable reel the number of the cable windings on the cable reel must be a positive real number, and can be determined from the following expression: c c cccoreccore d l ddddd k 2 4)()( 2    (4) taking into account that the length of the coiled cable must be equal to the distance relative to a reference point in which the coiled cable length is zero, the following applies:  dtvlc (5) where v is the line speed of the winding/unwinding cable which is equal to the speed of the crane. if motor speed of cable reel, nm, is measured, cable winding/unwinding line speed, v, can be determined as follows: 260 21 260 2 2 d i n d n d v t mww     (6) where w is rotational speed of cable reel in rad/s, nw is rotational speed of cable reel in rpm, and it is total gear ratio from the motor to the cable reel. the monospiral motorised cable reel in crane applications 319 in order to adapt the speed and torque according to the load it is necessary to incorporate gear units between the motor and the cable reel (e.g., worm gear, spur gear reduction, chain gear). with gear ratio it, the all gears between the motor and the cable reel are taken into account. set-point cable tension force is determined experimentally during commissioning of the drive, based on the desired cable length in the air between the cable reel and the pad. its value may be estimated based on the desired cable length in the air assuming that there are no losses in the cable movement between the cable reel and pads: glmf vclczc ,,,  (7) where mc,l is mass per unit cable length, lc,v is desired cable length in the air, and g=9.81 m/s 2 acceleration of gravity. during winding/unwinding the total mass of cable reel is changed, and thus the total moment of inertia. the total moment of inertia for cable reel, jt, is: cwt jjj  (8) where jw is moment of inertia for cable reel, and jc is moment of inertia of cable on the cable reel. moment of inertia for the cable reel, assuming that it can be regarded as a hollow cylinder, is: 2 21 ( ) 8 w w w wj m d d  (9) where mw is mass of cable reel, dw and dw are outer and inner diameter of the cable reel, respectively. moment of inertia of the cable on the cable reel, assuming that it can be considered as a hollow cylinder, is: 2 21 ( ) 8 c c corej m d d  (10) where mc is the mass of coiled cable. the mass of coiled cable is: clcc lmm , (11) losses for all elements of the drive are usually provided in the form of efficiency. the dynamics of the drive is taken into account by time derivative in the newton equation. moments of inertia of the motor and gears are constant. the moment of inertia of the cable reel is a variable quantity. newton's equations for the drive cable reel, reduced to the motor shaft, which takes into account losses in the drive, and the dynamics of the drive, wherein the losses are "covered" by the motor, can be written as: 1 1 1 1 ( ) w t w t m g w t m w t t t t d dj d i j j j t t dt i dt dt i                (12) where jm is moment of inertia of the motor, and jg is total moment of inertia of all gears, reduced to the motor shaft. 320 v. kostić, n. mitrović, b. banković, m. petronijević because the moment of inertia is a slow rate variable, it is clear: dt dj dt d j t w w t   (13) based on the analysis the conclusion is that the drive with sufficient accuracy can be modeled by: 1 1 1 1 ( ) w w t m g t m w t t t t d d i j j j t t dt i dt i         (14) based on the eq. (14), and taking into account eq. (1), the motor torque is: , 1 1 ( ) 2 w w m t m g t c z t t d d d t i j j j f dt dt i             (15) if the angular speed is expressed through the line speed, and taking into account the time derivative, the expression for the motor torque can be written as follows: ,2 2 2 2 1 1 ( ) 2 m t m g t c z t t dv dd dv dd d t i j j d v j d v f dt dt dt dt id d                      (16) in eq. (16), at a constant tension force of the cable, the time dependent variables are outer diameter of coiled cable, d, and total moment of inertia of cable reel, jt. also, in acceleration and braking of the drive line speed of winding/unwinding cable is time dependent. equations (12) and (14-16) correspond to the case where motor torque, tm, is opposed by cable reel torque tw which happens during the cable winding, fig. 2. it is accepted that, in the motor mode during the cable winding, the motor torque and the motor speed have positive values. also, during the cable winding, the cable reel speed and the cable line speed of winding/unwinding are assumed to be positive. since the direction of the cable force tension does not change, the cable reel torque always has a positive value. equations (12) and (14-16) are valid during rolling up of the cable, in the quasistationary state, at a constant speed of the crane which is equal to the line speed of the winding. in this case, the change in line speed is zero and the change in diameter over time is a positive slow rate variable close to zero. for this reasons it is: 0 2 2 ,2        d f dt dd v dt dv d d j zct (17) also, this relationship holds true at the crane acceleration i.e. drives of the cable reel, when the variation of line speed is positive and the change of diameter is a positive slow rate variable. in these cases, there is the motor mode, the energy flow is from the motor to the cable reel, and the losses in the drive are "covered" by the motor, which is taking account with 1/t in equations. during the crane braking, the speed variation is negative and the change in diameter is a positive and slow rate quantity and it can happen that the drive from the motor mode goes into the regenerative mode and the energy flow from cable reel to the motor. then there is: the monospiral motorised cable reel in crane applications 321 0 2 2 ,2        d f dt dd v dt dv d d j zct (18) fig. 2 schematic view of the cable reel drive winding/unwinding operation in that case, the losses in drive are "covered" by the cable reel, and in eqs. (12) and (14-16) term 1/t should be replaced with t. the case when motor torque tm, and cable reel torque tw, operating in the same direction happen during the cable unwinding is shown in fig. 2. it is accepted that, in the motor mode during the cable unwinding, the motor torque and the motor speed have negative values. also, during the cable unwinding, the cable reel speed and the cable line speed of winding/unwinding are assumed to be negative. during unwinding of the cable, in the quasi-stationary state, at a constant speed of the crane which is equal to the line speed of unwinding the change in line speed is zero and the change in diameter over time is a negative slow rate variable close to zero. for this reason it is: 0 2 2 ,2        d f dt dd v dt dv d d j zct (19) also, this relationship holds true during the crane and cable reel braking, when the line speed changes are positive and the variation of diameter is a negative slow rate variable. in these cases, there is a generator mode, the energy flow from the cable reel to the motor and the losses in the drive are "covered" by the cable reel. in eqs. (12) and (14-16) term 1/t needs to be replaced with term t. during the crane or cable reel acceleration, the line speed variation is negative and the change in diameter is a negative and slow rate quantity and it can happen that the drive 322 v. kostić, n. mitrović, b. banković, m. petronijević from the generator mode goes into the motor mode operation and the energy flow from the motor to the cable reel. in that case it is: 0 2 2 ,2        d f dt dd v dt dv d d j zct (20) in that case, the losses in the drive are "covered" by the motor and eqs. (12) and (1416) are valid. in order to implement eqs. (12) and (14-16) all operation modes of cable winding and unwinding, term 1/t should be replaced with 1/t rw , where rw is operation mode of the cable reel drive, which can be determined as follows:                    v d f dt dd v dt dv d d jsignrw zct 2 2 ,2 (21) in accordance with eq. (21), the operating mode of cable reel drives during winding/unwinding of the cable can take one of the following values:                                                  0 2 2 for,1 0 2 2 for,0 0 2 2 for,1 ,2 ,2 ,2 v d f dt dd v dt dv d d j v d f dt dd v dt dv d d j v d f dt dd v dt dv d d j rw zct zct zct (22) for example, eq. (16) can be written in the following form: ,2 2 2 2 1 1 ( ) 2 m t m g t c z rw t t dv dd dv dd d t i j j d v j d v f dt dt dt dt id d                      (23) induction motor, which drives cable reel, on its shaft must develop speed, nm, which should correspond to cable reel speed nw, i.e. winding/unwinding line speed of the cable that is equal to crane speed v: ttwtwm i d viinn     2 602 2 60 (24) in eq. (24), when the crane is working at constant speed which is equal to the cable line speed of winding/unwinding, the outer diameter of coiled cable, d, is always a time depending quantity. however, during the acceleration and deceleration of the drive, cable winding/unwinding speed, v, is also a time depending quantity. in accordance with the foregoing, it can be concluded that the speed of induction motor varies in all operation modes of the cable reel drive. induction motor for the cable reel drive on its shaft must develop power, pm, which corresponds to motor torque, tm, and motor speed, nm: 60 2  mmm ntp (25) the monospiral motorised cable reel in crane applications 323 by replacing eqs. (23, 24) in eq. (25) for the motor power is obtained: 2 ,2 2 2 2 1 2 ( ) 2 m t m g t c z rw t dv dd dv dd d p i j j d v j d v f v dt dt dt dt dd d                            (26) finally, based on eq. (26), the motor power is equal to: 2 3 4 1 1 ( )m t m g t wrw rw t t dv dd p i j j j d v v p dt dtd                   (27) on the basis of the eq. (27) it can be concluded that motor power, pm, should correspond to power of cable reel, pw, taking into account the losses and dynamics of the drive. in accordance with the foregoing, the conclusion is that the motor power is changed in all modes of cable reel operation during winding/unwinding the cable. also, power can have both positive and negative values i.e. the motor can operate in motoring and in generating mode. 3. induction motor selection the selection of the induction motor, which drives the cable reel, is performed based on the maximum required values of the motor torque and the motor speed. maximum required motor torque, tm,max, is the maximum value of the motor torque, which should provide set-point cable tensile force in all operation modes for winding and unwinding, and taking into account the losses in the drive and drive dynamics. required maximum motor torque can be determined based on eq. (23). it is expected to occur for the crane position near the connection point (cp) i.e. in the middle of the crane runway (fig. 3), during the winding of the cable in the acceleration regime, when rw=1. for the above mentioned case, in order to simplify the motor selection, the following may be adopted: dt dd v dt dv d  (28) also, at the position of the crane near the connection point i.e. in the middle of crane runway, the total moment of inertia has a maximum value, and in eq. (23) the corresponding addend is dominant. having in mind the above, in order to select the motor, the expression for the torque given by eq. (23), can be written as follows: , 2 1 1 2 m t c z rw t t dv d t j f d dt i         (29) maximum motor speed, nm,max, occurs in the extreme left and right position of the crane in relation to the connection point, or at the ends of the crane runway (fig. 3), and at maximum crane speed, vmax, it can be determined using eq. (24). 324 v. kostić, n. mitrović, b. banković, m. petronijević fig. 3 typical crane positions having in mind the required maximum torque and motor speed, minimum motor power, pm,min, can be determined using the following expression: 60 2 max,max,min,   mmm ntp (30) minimum required motor speed, nm,min, is expected in a crane position near the connection point, i.e. in the middle of crane runway, at a minimum speed of the crane, vmin, and can be determined using the eq. (24). minimum required motor speed is the minimum value of the motor speed in the quasi-stationary state in which the motor must develop maximum required torque. the selection of the frequency converter, which fed induction motor, is performed taking into account rated values of selected motor. based on the section 2 analysis, it may be noted that during the operation of the cable reel drive, the motor can operate in a motor mode but also in a regenerative mode. for this reason, it is necessary to select the frequency converter that can operate in four quadrants. also, the possible operation area is in constant torque region (t = const.) and constant power region (p = const.). 4. case study the selection of an induction motor will be illustrated in a concrete example for derrick crane installed at the open mine pit in mb kolubara (ref. num. dk004). the crane is designed for mounting of the mining equipment. in the fig. 4a the crane with indicated drives is shown. in the fig. 4b the cable reel drive components are shown, and in the fig. 4c the terminal box with all elements, for the crane position left in relation to the connection point, are shown. derrick crane has the following drives:  main hoist: load capacity 60 t; lifting height 46 m, maximum lifting speed 6.27 m/min,  auxiliary hoist: load capacity 12.5 t, lifting height 49.2 m, the maximum lifting speed 6.27 m/min,  jib-boom motion: angle of inclination in the range of 82.5 ° ÷ 31.5 °,  travel motion: maximum speed 14.27 m/min,  cable reel: the length of the crane path 500 m, middle connection point. the monospiral motorised cable reel in crane applications 325 fig. 4 derrick crane dk 004 with: a) indicated drives, b) cable reel drive components, c) terminal box of cable reel installed power of the crane is 318 kw. the crane power supply is with 4x95 mm 2 cable cross section. all drives are equipped with appropriate frequency converters. for crane control the programmable logic controller (plc) with adequate performances was used. in the paper [7] all the hoist drives are described in detail. in this paper, the cable reel drive from the point of motor selection is described. the cable reel drive consists of three-phase induction motor (im), worm, spur and chain gear. principal block diagram of the cable reel drive is shown in fig. 5. the induction motor is powered by a frequency converter (fc). taking into account the values in fig. 5, the total gear ratio from the induction motor to the cable reel is: 2667.318 cgsgwgt iiii (31) the total efficiency from the induction motor up to the cable reel is: 0.5841t wg sg cg w      (32) 326 v. kostić, n. mitrović, b. banković, m. petronijević fig. 5 principal block diagram of the cable reel drive on derrick crane dk 004 set-point cable tension force, in accordance with eq. (7), and for desired length of the cable in the air lc,v=5 m is fc,z=299.2050 n < fc,max. in case of derrick crane dk004, maximum required torque occurs at kmax=34 and a=dv/dt=0.25 ms -1 /5 s, and in accordance with eq. (29) is tm,max=4.8015 nm. maximum required motor speed occurs at kmin=3 and vmax=0.25 m/s and in accordance with eq. (24) is nm,max=1238.9 rpm. minimum motor power, according to eq. (30), has a value pm,min=622.9296 w. minimum required motor speed occurs at kmax=34 at vmin=0.06 m/s, and in accordance with eq. (24) is nm,min=77.6203 rpm. having in mind the values required for maximum motor torque, maximum and minimum motor speed and minimum motor power to run the cable reel drive, with a minimum safety factor 2, one can choose a three-phase induction motor with cage rotor 1le1001-0eb4, the manufacturer siemens. the catalog data for this motor are given in table 1. torque-speed dependence tm=f(nm), obtained on the basis of eq. (24) and eq. (29) shown with a possible working area for motor 1le1001-0eb4 and for s1 operation mode (continuous duty cycle), is given in fig. 6. the analysis of fig. 6 can lead us to conclude that minimum motor overload factor, m,min, for s1 operation mode is in the fields of constant torque, and crane position near the connection point, i.e. in the middle of the crane runway where kmax=34, in the winding of the cable during acceleration. the minimum motor overload factor is: m,min=10 nm/4.8015 nm=2.0827. given that applies m,min  2, the desired minimum safety factor is ensured. the monospiral motorised cable reel in crane applications 327 table 1 catalog motor data 1le1001-0eb4, siemens motor: 1le1001-0eb4 (siemens) voltage un 400 v frequency fn 50 hz rated power pm,n 1.5 kw rated speed nm,n 1435 rpm rated torque tm,n 10 nm rated efficiency n 0.828 power factor cosn 0.79 rated current in 3.3 a locked rotor/rated torque tlr/tm,n 2.6 locked rotor/rated current ilr/in 6.4 break down/rated torque tb/tm,n 3.4 moment of inertia jm 0.0036 kgm2 maximal speed nm,max 4200 rpm -2000 -1500 -1000 -500 0 500 1000 1500 2000 -15 -10 -5 0 5 10 15 -t m,n n m,n -n m,n t = const. p = c o n s t. t = const. p = c o n s t. x: 1435 y: 10 windingunwinding x: 1239 y: 2.575 x: 77.62 y: 4.802 t m [ n m ] n m [ rpm ] t m,n a ss d v max v min v min v max a acceleration ss steady state d deceleration d ss a fig. 6 torque-speed characteristics for vmax (black) and vmin (green) with possible working area for motor (red) having in mind the above, it can be concluded that the three-phase induction motor with cage rotor 1le1001-0eb4, manufacturer siemens, can be used to run the cable reel drive, and with a safety factor 2.0827. the next step in the selection of equipment is the selection of the frequency converter for induction motor feeding. a possible area of motor operation supplied from the frequency converter is shown in fig. 6 and it is indicated with a red line. it may be noted that during 328 v. kostić, n. mitrović, b. banković, m. petronijević the operation of cable reel drive at certain section motor operate in a regenerative mode. for this reason, it is necessary to select the converter that can operate in four quadrants. one option is to select the drive which contains the braking chopper with an appropriate resistor. the other option is to select the drive with regenerative capability [7]. having in mind the rated values of the selected induction motor, one can choose single motor module 6sl3120-1te21-8aa4 (vector control), the manufacturer siemens, series sinamics s120. this inverter module uses common dc bus for power supply (regenerative capability). the inverter is in a torque control mode with encoder speed feedback. minimum required motor speed, nm,min=77.6203 rpm, is 5.4091 % of rated motor speed, which provides high control performance. 5. experimental results the experimental results are recorded during the crane commissioning in real conditions using the siemens software starter for sinamics inverter series. in this paper the experimental results of the monospiral motorized cable reel during the winding of the cable in the acceleration regime for crane position near the connection point, i.e. in the middle of the crane runway, are presented (fig. 7). the induction motor, powered by a frequency converter, is in the torque mode, with about 4.8 nm torque reference, in accordance with the previous analysis. after cable tensile, about 2.5 s from beginning, starts cable winding in the acceleration regime (speed increases). fig. 7 experimental results – cable winding, acceleration regime in case of different losses and/or dynamics of the drive in comparison with their adopted values, the cable tensile force will be different from set-point, which essentially does not affect the functionality. the monospiral motorised cable reel in crane applications 329 6. conclusion the cable reel is present in a large number of applications with cranes. its design and dimensioning are not easy and depend on many factors. depending on the application site, the crane speed and thus the speed of the cable winding/unwinding has a very significant impact both on the motor selection and converters as well as the implementation of adequate control algorithm. this paper presents the dynamic equations of the mechanical part for the cable reel drive, which can be applied for all operating modes (acceleration, steady state and braking regime). in these equations variation of the moment of inertia and the outer diameter of the coiled cable are taken into account. also, the expression for motor power which corresponds to the cable reel power and which takes into account losses and dynamics of the drive is obtained. the conclusion is that the motor power is changed in all modes of the cable reel operation during winding/unwinding the cable. also, power can have both positive and negative values i.e. the motor can operate in motoring and in generating mode. based on these equations, with appropriate simplifications, the expression of required motor torque, speed and power are obtained. as proof of the validity of the obtained relations a real case of cable reel drive in derrick crane is considered. based on actual data, mechanical characteristics of the induction motor that respects power supply conditions and the presence of frequency converter are shown. it may be noted that during the operation of cable reel drive at certain section motor operate in a regenerative mode. for this reason, it is necessary to select the converter that can operate in four quadrants. in order to demonstrate the validity of the proposed procedure for the selection of equipment the experimental results are recorded during the crane commissioning in real conditions. in this paper, the control algorithm is not considered. based on the results, a very good agreement with the results of the calculations can be observed. the proposed method for the induction motor selection is applicable in all the cases of monospiral motorized cable reel drives with the total moment of inertia as dominant value. acknowledgements: this paper is supported by project grant iii44006 financed by ministry of education and science, republic of serbia. references 1. chattopadhyay, a.k., 2010, alternating current drives in the steel industry, ieee industrial electronics magazine, 4(4), pp. 30-42. 2. mitrović, n., kostić, v., petronijević, m., jeftenić, b., 2009, multi-motor drives for crane application, advances in electrical and computer engineering, 9(3), pp. 57-62. 3. niu, c.m., zhang, h.w., ouyang, h., 2012, a comprehensive dynamic model of electric overhead cranes and the lifting operations, proc. of the institution of mechanical engineers, part c: journal of mechanical engineering science, 226(6), pp. 1484-1503. 4. kostić, v., mitrović, n., petronijević, m., banković, b., 2011, experimental analysis of direct torque control methods for electric drive application”, proc. forty-sixth international scientific conference on information, communication and energy system and technologies icest 2011, niš, po8.9, 3, pp. 997-1000. 5. yogesh, n.t., mohan, v.a., 2016, direct torque control of induction motor with common-mode voltage elimination, electric power components and systems, 44(20), pp. 2310-2324. 330 v. kostić, n. mitrović, b. banković, m. petronijević 6. haiwei, l., weijian, m., ning, z., yufei, f., 2015, modeling and simulating the operation of the harbor portal crane, journal of coastal research, special issue 73 recent developments of port and ocean engineering, pp. 89-94. 7. mitrović, n., petronijević, m., kostić, v., jeftenić, b., 2012, electrical drives for crane application, mechanical engineering, dr. murat gokcek (ed.), intech, pp. 131-156. 8. pietrosanti, s., harrison, i., luque, a., holderbaum, w., becerra, v.m., 2016, net energy savings in rubber tired gantry cranes equipped with an active front end, ieee 16th international conference on environment and electrical engineering eeeic 2016, florence, pp. 1-5. 9. mitrović, n., kostić, v., petronijević. m., jeftenić b., 2010, practical implementation of load sharing and anti skew controllers for wide span gantry crane drives, strojniški vestnik – journal of mechanical engineering, 56(3), pp. 207-216. 10. beldjajev, v., lehtla, t., mõlder, h., 2010, influence of regenerative braking to power characteristics of a gantry crane, proc. electric power quality and supply reliability conference 2010, kuressaare, pp. 73-78. 11. iannuzzi, d., piegari, l., tricoli, p., 2009, use of supercapacitors for energy saving in overhead travelling crane drives, proc. international conference on clean electrical power iccep 2009, capri, pp. 562-568. 12. hearn, c.s., lewis, m.c., prarap, s.b., hebner, r.e., uriate, f.m., chen, d.m., longoria, r.g., 2013, utilization of optimal control law to size gride-level flywheel energy storage, ieee transactions on sustainable energy, 4(3), pp. 611-618. 13. andziulis, a., eglynas, t., bogdevicius, m., jusis, m., senulis, a., 2016, multibody dynamic simulation and transient analysis of quay crane spreader and lifting mechanism, advances in mechanical engineering, 8(9), pp. 1-11. 14. vasiljević, r., gašić, m., savković, m., 2016, parameters influencing the dynamic behaviour of the carrying structure of a type h portal crane, strojniški vestnik journal of mechanical engineering, 62(10), pp. 591-602. 15. marinković, z., marinković, d., petrović, g., milić, p., 2012, modelling and simulation of dynamic behaviour of electric motor driven mechanisms, tehnicki vjesnik/technical gazette, 19(4), pp. 717-725. 16. precup, r.e., filip, h.i., rădac, m.b., petriu, e.m., preitl, s., dragoş, c.a., 2014, online identification of evolving takagi–sugeno–kang fuzzy models for crane systems, applied soft computing, 24, pp. 1155-1163. 17. huang, j., xie, x., liang, z., 2015, control of bridge cranes with distributed-mass payload dynamics, ieee/asme transactions on mechatronics, 20(1), pp. 481-486. 18. sun n, fang y, chen h.a, 2016, continuous robust antiswing tracking control scheme for underactuated crane systems with experimental verification, asme, journal of dynamic system, measurement and control, 138(4), pp. 041002-1 041002-12. 19. gholabi, a., ebrahimi, m., reza, g., ghayour, y.m., ebrahimi, a., jali, h., 2013, sensorless antiswing control for overhead crane using voltage and current measurements, journal of vibration and control, 21(9), pp. 1745-1756. 20. subari, h., chong, s.h., hee, w.k., chong, w.y., nawawi, m.r., othman, m.n., 2015, investigation of model parameter variation for tension control of a multi motor wire winding system, proc. 10th asian control conference ascc 2015, kota kinabalu, pp. 1-6. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 93 98 https://doi.org/10.22190/fume180105009p © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd short communication solution of adhesive contact problem on the basis of the known solution for non-adhesive one udc 539.6 valentin l. popov berlin university of technology, berlin, germany abstract. the well-known procedure of reducing an adhesive contact problem to the corresponding non-adhesive one is generalized in this short communication to contacts with an arbitrary contact shape and arbitrary material properties (e.g. non homogeneous or gradient media). the only additional assumption is that the sequence of contact configurations in an adhesive contact should be exactly the same as that of contact configurations in a non-adhesive one. this assumption restricts the applicability of the present method. nonetheless, the method can be applied to many classes of contact problems exactly and also be used for approximate analyses. key words: adhesion, normal contact, heterogeneous media, effective surface energy 1. introduction the present short communication is publication of a short memo written on may, 14 th , 2015, and not published at that time as the field of applications of the obtained results seemed to be very narrow. it was communicated privately to colleagues and published for a restricted set of adhesive contact problems in [1]. since then, it has been applied to a variety of problems including axially symmetric contact ones without a compact contact area [2] just as it has been systematically applied to a large variety of contact problems in the recent handbook on contact mechanics [3]. however, the derivation and results are more general than the cases considered in [2] and [3]. they are based solely on existence of some force-indentation and area-indentation dependencies for non-adhesive contacts. in the present paper we provide a general derivation which is even applicable to the situations where the surface energy is a function of the coordinates. received january 05, 2018 / accepted january 31, 2018 corresponding author: valentin l. popov technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: v.popov@tu-berlin.de 94 v. popov 2. axially symmetric contact problem consider indentation of a profile z = f(r) into an elastic medium with plane surface. the medium is assumed to be homogeneous in-plane, but may be heterogeneous in the zdirection (e.g. a layer or a gradient medium and so on). we assume that indentation depth d is much smaller than the characteristic size of heterogeneity, while this may be not the case for contact radius a. we assume that the non-adhesive normal contact problem for this shape and this medium has been solved, so that the dependences of normal force fn and of the contact area on the indentation depth are known. each of these three quantities determines uniquely two others, so that we can consider normal force fn,n.a.(a) and indentation depth dn.a.(a) as known functions of the contact radius, too. we can further define the potential energy of non-adhesive contact, un.a.(a) and the contact stiffness , . . . d ( ( )) ( ) d n n a n a f a d k a d  . (1) which also can be considered as a known function of the contact radius. now let us consider an adhesive contact under assumptions of the jkr-theory (range of interaction of adhesive forces much smaller than any characteristic size of the problem, so to say zero) and characterize adhesion with the work of detachment of surfaces per unit area, . the solution to this problem is given by the following set of equations: . .( ) ( )n a cd d a l a  , (2) , . . . .( ) ( ) ( ) ( )n n n a n a cf a f a k a l a   (3) with lc(a) given by the following equation: . . 4 ( ) d ( ) / d c n a a l a k a a     . (4) eqs. (2) and (3) give in implicit form the dependence of the normal force (in the adhesive contact) on the indentation depth thus solving the adhesion problem. let now prove the eq. (4). if we indent the profile up to contact radius a, then the potential energy in this state will be un.a.(a) and indentation depth dn.a.(a). the force in this moment will be fn,n.a.(a). now let lift the indenter by l without changing the contact area. during this process the stiffness of the contact remains constant and equal to kn.a.(a). therefore, the force will change according to . . . .( ) ( )n n a n af a f k a l   (5) and the potential energy will be equal to 2 . . , . . . .( ) ( ) ( ) ( ) 2 n a n n a n a l u a u a f a l k a      . (6) solution of adhesive contact problem on the basis of the non-adhesive solution 95 the new indentation depth will be . .( )n ad d a l  . (7) let us now solve (7) with respect to l und insert it into eq. (6): 2 . . . . , . . . . . . ( ( ) ) ( ) ( ) ( )( ( ) ) ( ) 2 n a n a n n a n a n a d a d u a u a f a d a d k a      . (8) the total energy (with consideration of the adhesion energy is equal to 2 2. . . . , . . . . . . ( ( ) ) ( ) ( ) ( )( ( ) ) ( ) 2 n a tot n a n n a n a n a d a d u a u a f a d a d k a a         . (9) equilibrium value of a corresponds to the minimum of this energy with respect to a for a constant indentation depth d. to determine the minimum, we let the derivative be zero: 2 , . . 2. . . . . . . . , . . . . , . .. . . . . . . . , . . . . d ( ) d ( )d ( ) d ( ) d ( ) d ( ) ( ) ( ) d d d d 2 d d ( )d ( ) d ( ) d ( ) d ( ) ( ) ( ) d d d d d tot n n an a n a n a n a n n a n a n n an a n a n a n a n n a n a u a a f au a d a k a d al l f a k a l a a a a a a f au a d a d a k a f a l k a l a a a a a                            2 2 2 0 l a      (10) it is easy to see that the terms in brackets are identically zero, thus we get 2 . .d ( ) 2 d 2 n ak a l a a     . (11) solution with respect to l provides eq. (4). eqs. (5) and (7) coincide with eqs. (3) and (2) and provide the solution to the problem. note that this equation is applicable not only to homogeneous media but to all the media for which the dependence of the contact stiffness on radius is known, in particular of coated, multi-layer or gradient media [3]. in the case of a homogeneous medium dkn.a.(a)/da=2e * , where e * is the effective elastic modulus responsible for a normal contact problem [4]. in this case we come to the known rule of heß [5]. 3. general case (not axis-symmetric or non-compact contact area) if the set of contact configurations of an adhesive contact would repeat that of contact configurations of the normal one for the same shape (which, regrettably, will generally not be the case!), then the adhesive contact could be solved in the following way. for simplicity, we consider here homogeneous media. we assume that the normal contact problem was solved so that the dependence of normal force fn,n.a. and contact area a on indentation depth d is known: 96 v. popov , . . , . ( )n n a n n af f d , (12) . . . .( )n a n aa a d . (13) now we define the incremental stiffness , . . . . d d n n a n a f k d  (14) and the formal "effective contact radius" (which in general case has of course nothing to do with any radius, but is just a formally defined quantity): . . * ( ) 2 n ak d a e  . (15) the normal contact now can be described by the method of dimensionality reduction with the equivalent profile z=g(x), where function g(x) is defined according to ( )d g a , (16) (just by solving eq. (15) with respect to d, for details see [5]) . the condition for the equilibrium of an adhesive contact can be obtained from the standard balance of energy at small variation of the "contact radius". we assume that the boundary springs (in the mdr picture) detach when they achieve critical length lc, which is determined by equating the relaxed elastic energy 2. .d 2 2d n a el c k u x l a      (17) to the change of adhesive energy: d d d d d ( ) d d d d d adh a a d a g a u a x x x a d a d a                 . (18) equating (17) and (18) we get 2. .d d 2 2d d n a c k a x l x a a        , (19) hence     . . . . 2 d / d d / d n a c n a a a l k a     . (20) in the case of axis-symmetric profiles with a compact contact area we have of course trivially a = a 2 , da/da = 2a and * c e/al δγπ2δ  , which coincides with the "rule of heß" [5]. solution of adhesive contact problem on the basis of the non-adhesive solution 97 4. generalization for arbitrary media note that in the systems with a complicated “microstructure”, the surface energy also can depend on the size of the contact, as e.g. illustrated for different shapes with internal damages as well as for “brushes” in [7]. in this case eq. (22)is modified to   na 2 ( ) d ( ) / d ( ) d ( ) / d c a a a a l a k a a     , (21) where (a) is the size-dependent effective surface energy which can be determined by means of the concept of the “filling factor” the use of which has been validated in many examples in [7]. together with eqs. (2) and (3) this solves the problem. 5. applicability of the macroscopic approach to contact of rough surfaces recently, ciavarella has come independently to a very similar approach [7]. he has also provided an extensive and instructive discussion of applicability of the macroscopic approach to adhesive contacts of rough surfaces. the main assumption of the approach is that the sequence of contact configurations of an adhesive contact is the same as in the case of a non-adhesive contact. this condition is clearly not fulfilled even in the case of “asperity models” like greenwood and williamson. there are no reasons to assume that in the case of a more general roughness the contact configurations of an adhesive contact will repeat those of a non-adhesive contact, so that in the general case, this assumption is not fulfilled, either. that the present approach cannot be generally applied to contacts of rough surfaces is already clear from the fact that in the present approach there is no “hysteresis of the force of adhesion” (thus, the force of adhesion does not depend on the loading history which is not the case in real rough contacts as discussed in [9, 10]. however, there can be some situations where the above assumption is fulfilled or approximately fulfilled. for example, if the johnson parameter [11] is overcritical then a complete contact can be realized in spite of roughness [12]. furthermore, the concept can be applied to rough surfaces by using the concept of the filling parameter as discussed in [7]. however, this approach uses the notion of a “real contact area” which is a poorly defined quantity (an excellent discussion of this property and the ways of its proper physical definition can be found in [13]. e.g. one of the “regularizing factors” may be the final range of adhesive forces which substantially modifies the contact situation at a small scale [14, 15]. further investigation of this problem, especially using the now available numerical technique of the boundary element method for adhesive contacts [7] is needed. 6. conclusions equations (2), (3) and (22) provide a simple solution for all the problems that the normal contact problem has been solved for – either analytically or numerically. this includes all the contacts with a homogeneous continuum, coated medium, gradient 98 v. popov material, plates, thin layers, membranes or living cells, and so on. the only restriction of the method is that the sequence of the contact configuration is the same as in the nonadhesive problem. this is valid for compact axisymmetric contacts and for some other cases of axisymmetric contacts analyzed in [2]. acknowledgments: the author acknowledges financial support by the deutsche forschungsgemeinschaft (dfg po 810-55-1). the author acknowledges very useful discussions with m. ciavarella. references 1. argatov, i., li, q., pohrt, r., popov, v.l., 2016, johnson–kendall–roberts adhesive contact for a toroidal indenter, proc. r. soc. a 472, 20160218. 2. willert, e., li, q., popov, v.l., 2016, the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape or concave profiles, facta universitatis-series mechanical engineering, 14(3), pp. 281-292. 3. popov, v.l., heß, m., willert, e., 2017, handbuch der kontaktmechanik: exakte lösungen axialsymmetrischer kontaktprobleme, springer, berlin, 341 p. 4. popov, v.l., 2017, contact mechanics and friction. physical principles and applications. springer, berlin. 5. hess, m., 2011, über die exakte abbildung ausgewählter dreidimensionaler kontakte auf systeme mit niedrigerer räumlicher dimension, cuvillier, berlin, germany 6. popov, v.l., pohrt, r., heß, m., general procedure for solution of contact problems under dynamic normal and tangential loading based on the known solution of normal contact problem, the journal of strain analysis for engineering design, 51(4), pp. 247-255. 7. popov, v.l., pohrt, r., li, q., 2017, strength of adhesive contacts: influence of contact geometry and material gradients, friction, 5(3), pp. 308–325. 8. ciavarella, m. , 2017, an approximate jkr solution for a general contact, including rough contacts, arxiv preprint arxiv:1712.05844 9. afferrante, l., ciavarella, m., demelio, g., 2015, adhesive contact of the weierstrass profile, proc. r. soc. a, 471, (2182), 20150248. 10. ciavarella, m., 2017, a note on the possibility of roughness enhancement of adhesion in persson’s theory, international journal of mechanical sciences, 121, pp. 119-122. 11. johnson, k.l., 1995, the adhesion of two elastic bodies with slightly wavy surfaces, int. j. solids structures, 32(3/4), pp. 423-430. 12. ciavarella, m., papangelo, a., 2018, a generalized johnson parameter for pull-off decay in the adhesion of rough surfaces, phys mes., 21(1), pp. 57-75. 13. ciavarella, m., papangelo, a., 2017, discussion of measuring and understanding contact area at the nanoscale: a review (jacobs, tdb, and ashlie martini, a., 2017, asme appl. mech. rev., 69 (6), p. 060802), applied mechanics reviews, 69(6), 065502. doi: 10.1115/1.4038188 14. ciavarella, m., papangelo, a., 2018, a modified form of pastewka–robbins criterion for adhesion, the journal of adhesion, 94(2), pp. 155-165. 15. papangelo, a., ciavarella, m., 2017, a maugis–dugdale cohesive solution for adhesion of a surface with a dimple, journal of the royal society interface, 14(127), 20160996. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 653 664 https://doi.org/10.22190/fume181116015n © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper determination of the rolling resistance coefficient under different traffic conditions zdravko b. nunić1, mesud ajanović1, dario miletić1, ranko lojić2 1university of east sarajevo, faculty of transport and traffic engineering doboj, bosnia and herzegovina 2university of defense, military academy, department of management, bosnia and herzegovina abstract. in this paper, an experimental study of the determination of the rolling resistance coefficient is carried out. the experiment tests a total of six different types of vehicles and calculates the rolling resistance coefficient depending on the condition of the surface and the type of tires. the main aim of the research is to introduce new values of the rolling resistance coefficient and its impact on fuel consumption in real traffic conditions. motor vehicles are subjected to a "free stop" method on a horizontal road. in doing so, the vehicle speed is registered every 10 seconds from an initial speed to stopping. in order to eliminate an error of possible roadway inclination or wind impact, the experiment is repeated five times on the same road section as well as in the opposite direction. the experimental study was carried out during december 2016 and january 2017. three sets of tires were used for each vehicle, the tires with tread depths of 8 mm, 6-7 mm and 4-5 mm, while the type of surface referred to dry and wet conditions of the roadway. both hypotheses have been confirmed using analysis of variances. the results show that the tread depth of tires and the meteorological conditions affect increasing the values of the rolling resistance coefficient. key words: rolling resistance coefficient, vehicle, traffic safety, fuel consumption 1. introduction the most significant aspect of the way in which the road has an impact on traffic accidents lies in the fact that it affects both the driver and the vehicle [1, 2], creates conditions for the influence of other factors, affects the severity of traffic accidents and, at the same time, determines the circumstances of traffic flow. within the scope of transportation, a special place is taken by ecological impact on people and the environment [3] in which transportation is carried out, where, according to trupia [4], one of the received november 16, 2018 / accepted april 10, 2019 corresponding author: zdravko nunić affiliation, university of east sarajevo, faculty of transport and traffic engineering doboj e-mail: zdravkonunic56@gmail.com 654 z. nunić, m. ajanović, d. miletić, r. lojić factors is the rolling resistance coefficient. the improvement of this impact can be achieved by reducing the levels of negative effects of influencing factors in the movement of motor vehicles. amongst many influential factors, the surface and the tire are of significant importance, that is, the interaction of these two factors implies the reduction of rolling resistance rf, i.e. of rolling coefficient f that contributes to the reduction of fuel consumption, as confirmed by srirangam et al. [5] and wei et al. [6]. this is also pointed out by the authors in the paper [7], who emphasize the knowledge of the surface characteristics and the interaction of the vehicle with the surface as a significant starting point for making a choice between the concept of a new vehicle and that of optimum utilization of an existing vehicle. the tires play a great role in the exploitation of vehicles. their impact is most evident in large fleets of vehicles, when they are exposed to various conditions of exploitation and when they are expected to pass a large number of kilometers unobtrusively. therefore, by experimenting with several types of tires, their pressures, and various loads of vehicles, the optimum rolling resistance can be achieved, in accordance with the road conditions as well as the type and purpose of a motor vehicle automatically saving fuel, tires and, what is most important in every business, costs. the force that prevents the movement of a vehicle is called "rolling resistance". the research has shown that one of five full fuel tanks, i.e. 20% of the total amount of fuel consumed, runs on resistance of the tire rolling on the surface. reducing rolling resistance affects an increase in energy efficiency of road traffic, which reduces carbon dioxide emission and helps drivers reduce fuel costs. the subject of the research is to determine the value of the rolling resistance coefficient and its impact on fuel consumption in real driving conditions on a non-deformable surface. such research studies in literature are very rare; hence this study brings new values of the rolling resistance coefficient that can be used in the future research of the given traffic conditions. the tires play an important role in vehicle safety and environment [8] and usually represent a leading parameter in fleet maintaining costs. fluid under pressure is a basic bearing component of the tires and receives up to 95% of the total external load for modern tires while the carcass and protector take over the remaining 5%. inadequate pressure in the tires, higher or lower, makes the tires lose their performance and reliability. this affects the overall performance of vehicles and creates a possibility for traffic accidents. a reduced coefficient of friction between the wheels and the surface of a driveway causes a large number of traffic accidents [9]. the main goal of the research is to determine the influence of the tires/surface interaction on the value of the rolling resistance coefficient of motor vehicles since, according to ejsmont [10], the rolling resistance coefficient is one of the most important parameters that affect the tires/surface interaction. apart from introduction, the paper is structured in four sections. in section 2 the research method is presented. this section describes hypothesis and eqs. for calculating the rolling resistance coefficient. section 3 represents the description and explanation of the experiment performed in this study. section 4 implies results with an example of calculation while the last section represents conclusions with guidelines for future research. 2. research methods the research consists of two parts, namely, the experimental (field research) and the theoretical (processing of results) ones. the experimental part includes the study of the dependence of the speed of movement on the deceleration time, since, according to soliman [11], the resistance coefficient is not constant and depends on the speed of determination of the rolling resistance coefficient under different traffic conditions 655 movement of the vehicle. the theoretical part of the research includes the processing of the data collected and testing of the hypothesis set. experiment modeling and variance analysis are used to obtain relevant hypothesis data. according to the subject and goals of the research, a research program was also developed. the central topic of this paper is the study of the tires/surface interaction and its influence on the value of the rolling resistance coefficient of motor vehicles. the main part of the research program is related to the study of the dependence of the speed of movement on the time of deceleration; on the basis of this, the modeling of the experiment and the analysis of variances need to be carried out. the research program is divided into two parts: the experimental research and the modeling of experiment and analysis of variances. the experimental research program was carried out on reference vehicles in real conditions whereby, during the driving process, the speed of the vehicle was registered for every 10 seconds (which allows a graphic display of the speed curve in the function of the deceleration time) from an initial speed to stopping. the modeling of experiment and the analysis of variances were performed with the limitations of the following factors: factor "a" tire tread depth and factor "b" meteorological conditions. 2.1. hypotheses hypothesis 1: the tread depth of tires and meteorological conditions affect the increase in the value of the rolling resistance coefficient. the tread depth is an extremely important factor for tires, which, as legally defined, must be at least of 1.6 mm in summer and of 4 mm in winter. the tread wear indicator (twi) is an important tool for assessing the residual depth of the tire tread. the winter tires, for example, have an additional wear indicator at a depth of 4 mm because their performance significantly decreases when the tread depth reaches this limit or is below it. worn tires significantly increase the risk of aquaplaning and poor vehicle braking on a wet driveway. if the driving mode is adjusted to winter conditions, it will help preserve safety, although the winter tires provide additional protection against unpleasant surprises in the winter. compared to the summer tires, they provide greater safety, especially when braking, in cold and wet conditions, as well as in snow and ice conditions. hypothesis 2: the value of the rolling resistance coefficient affects fuel consumption up to 25%. the rolling resistance is the resistance that occurs when rolling a tire on a flat surface. the rolling resistance coefficient is a non-dimensional unit obtained when the force of rolling resistance is divided by a vertical tire load point. according to glavaš [12], road traffic is approximately 23% of the world's total energy demand, from which the importance of the hypothesis can be noticed. by increasing the value of the rolling resistance coefficient, the rolling resistance itself increases as well as total resistances that oppose the movement of a vehicle. it is, therefore, necessary to consume more energy to absorb total resistances, resulting in fuel consumption increase. 2.2. the function and energy efficiency of tires the function of tires is a motor vehicle movable support, absorption of vibrations due to unevenness on a driveway, transfer of kinetic engine energy to a driveway and tracking of the movement direction of a vehicle. the criteria that they should fulfill are safety on wet and dry surfaces (at high speeds and braking), comfort in terms of absorption of unevenness and noise reduction, fuel economy and durability. a tire is the only part of a 656 z. nunić, m. ajanović, d. miletić, r. lojić car that is in contact with a driveway. a modern tire has over 200 individual components in its composition. however, most of these components can be classified into three basic groups: rubber (natural and synthetic rubber), fillers (silicon, etc.) and additives (vulcanizing additives, antioxidants, sun protection waxes, etc.). rubber is a mechanically solid and highly elastic material obtained by caoutchouc vulcanization, natural and synthetic. in the process of vulcanization by sulfur, strong chemical cross-linking occurs between polymeric caoutchouc chains with the opening of some double bonds. caoutchouc then takes on elastic characteristics, and stretching strength is up to ten times higher than for unvulcanized caoutchouc. a mixture of natural (14%) and synthetic rubber (27%) make up approximately 40% of the components of a modern tire. the load distribution on the tire profile changes with respect to the speed of movement. by increasing the speed, the trace made by a tire changes its shape from a circle into a rectangle. at moderate speeds, the loss of energy in a form of heat is largely carried out over a tire surface. at higher speeds, the loss of energy is largely performed through inner lining and lateral sides. some countries and regions have introduced one or more programs of energy efficiency improvement but no country or region has a complete program that covers all aspects of energy efficiency of tires [13]. in order to achieve this, the following recommendations are given: ▪ introduction of tire marking and their ranking; the introduction of tire marking and their ranking is an important first step towards improving energy efficiency of tires, as it allows final users to choose, and sellers and manufacturers to provide energy efficient tires. ▪ setting the standards for energy efficiency of tires; as the introduction of tire marking and their ranking can stimulate the manufacture of high-performance tires, the setting of minimum energy efficiency standards for tires may stimulate the manufacture of energy efficient and economically acceptable tires. ▪ introduction of testing and checking of energy efficient tires; additional tire testing and checking would provide data that could improve accuracy and credibility of tire marking and ranking. ▪ monitoring of pressure in tires; proper pressure in tires is a prerequisite for achieving the maximum energy efficiency of tires, as well as a necessary condition for achieving safety in traffic. ▪ introduction of comprehensive regulations; the introduction of comprehensive regulations, i.e. regulations containing several different standards, prevents compromises and unintended consequences that may arise by setting a requirement to meet a single norm. for example, reducing rolling resistance should not reduce safety. to calculate the rolling resistance coefficient, it is necessary first to set the equation of motion: avufoi rrrrfx +++=→= 0 (1) where fo – the tractive force, ra – the acceleration resistance, rf – the rolling resistance, rv – the air resistance, and ru – climb resistance. setting ru = 0 and assuming deceleration a with respect to the positive x-direction (hence, for a > 0 the inertial force acts in the positive x-direction), the equation of motion takes the following form: determination of the rolling resistance coefficient under different traffic conditions 657 2 2 0 va c cosfgfam rrfrrrrf x o vfoddvfo   −−=− −−=−→=+−−   (2) where rd – the deceleration inertial force, m – the mass of the vehicle, δ – the rotational mass coefficient (δ = 1.04 for excluded transmission), cx – the air resistance coefficient, ρ – the air density, a – the front surface, v – the speed of vehicle, g − the vehicle weight, f – the rolling resistance coefficient, α – the ground inclination. setting fo = 0 because the engine is disengaged (vehicle coasting), and α=0, which implies that the coasting occurs on a horizontal ground, it follows: ( )2vakfg g g a +  =  (3) where g – gravity and k – reduced air resistance coefficient (k = cx·ρ/2). by solving the above equations we obtain deceleration time t as: ( ) ( )21 21 1 vv fg ak vv fg ak arctg akfgg g t −   + −     =  (4) where v1 – the speed of vehicles at the beginning of interval tn and v2 – the speed of vehicles at the end of interval tn, where n is the number of intervals (for more explanation see fig. 1 and table 2). eq. (2) is obtained by using differential equation for deceleration motion. from eq. (4), the rolling resistance coefficient can be obtained as follows: ( ) 21 21 vv f w g vv f w g arctg f w tg g f + −   =  (5) ( )          + −   = 2 21 1 21 1 1 1 1 63 63 . vv f w g. vv f w g arctg f w tg g f  (6) ( )          + −   = 2 21 2 21 2 2 2 2 63 63 . vv f w g. vv f w g arctg f w tg g f  (7) where f1 – the rolling resistance coefficient for first interval t1 and f2 the rolling resistance coefficient for second interval t2, w– the air resistance factor (w = k·a), while the coefficient 3.6 in eqs. (6) and (7) implies that speed is given in m/s. 658 z. nunić, m. ajanović, d. miletić, r. lojić 3. experiment the experiment requires determination of the rolling resistance coefficient for certain motor vehicles under real driving conditions by a "free stop" method on a horizontal road. in doing so, the vehicle speed was recorded every 10 seconds (which enables a graphic display of the velocity curve in the function of the deceleration time) from an initial speed to stopping. in order to eliminate an error caused by possible roadway inclination or wind impact, the experiment was repeated five times on the same road section, as well as in the opposite direction, and then, after interpolation, the values were obtained, which are shown in tables. during the study, transmission was separated from drive wheels. experimental study was carried out during december 2016 and january 2017. three sets of tires were used for each vehicle, the tires with tread depths of 8 mm, 6-7 mm and 4-5 mm, while the type of surface referred to dry and wet condition of the roadway. 3.1. models of vehicles on which the experiment was performed in order to perform the practical part of the experiment, there were used vehicles subjected to an authorized technical inspection station, prior to the experiment where it was confirmed that all the vehicles were technically in a good condition. models of motor vehicles used here are: passenger vehicle 1 – audi a4; passenger vehicle 2 – volkswagen golf 4; delivery vehicle 1 – volkswagen caddy; delivery vehicle 2 – volkswagen caddy; light cargo vehicle 1 – iveco 35s13; light cargo vehicle 2 – iveco 50c14. 3.2. an example of determining the rolling resistance coefficient for a light cargo vehicle – iveco 35s13 – the tread depth of 8 mm for the first experimental study on light cargo vehicles, iveco 35s13 was used, with its following characteristics: the year of manufacturing – 2002, the engine volume – 2800 cm3, the engine power – 92 kw, the mass of the empty vehicle – 3500 kg, the total mass of the vehicle – 3590 kg (total mass of the vehicles implies sum of mass of the empty vehicle 3500 kg and mass of driver 90 kg), vehicle length – 5997 mm, width – 1800 mm, height – 2700 mm. the vehicle had tigar tires, with the dimensions of 225/65 r16. for the experiment on a wet roadway, the temperature was 3.5 °c. by the experimental testing, the dependence of the speed of movement on the deceleration time was obtained, which is shown in table 1. table 1 dependence of the speed of movement on the deceleration time of iveco 35s13, on a wet roadway t (s) 0 10 20 30 40 50 v (km/h) 70 53 42 32 23 16 fig. 1 shows the dependence of the speed of movement on the deceleration time from table 1. from fig. 1 it is necessary to identify the speed interval and time. the vehicle slows down from a speed of 70 km/h to a speed of 16 km/h in 50 seconds and from a speed of 53 km/h to 16 km/h in 40 seconds. determination of the rolling resistance coefficient under different traffic conditions 659 fig. 1 dependence of the speed of movement on the deceleration time of iveco 35s13 on a wet roadway table 2 speed interval and time f1 f2 t=50 (s) t=40 (s) v1=70 (km/h) v1=53 (km/h) v2=16 (km/h) v2=16 (km/h) by adding the values from table 2 into eqs. (1-7), the values of the rolling resistance coefficient are obtained. table 3 equation solutions for iveco 35s13, on a wet roadway w/f 0.09 0.16 0.25 0.36 0.49 0.64 0.81 1 2 f1 0.0314 0.0312 0.0311 0.0310 0.0309 0.0308 0.0307 0.0306 0.0298 f2 0.0266 0.0265 0.0264 0.0263 0.0262 0.0260 0.0259 0.0257 0.0249 w/f 3 4 5 6 7 8 9 10 11 f1 0.0293 0.0285 0.0281 0.0273 0.0264 0.0255 0.0247 0.0238 0.0230 f2 0.0243 0.0240 0.0234 0.0230 0.0227 0.0225 0.0223 0.0222 0.0221 w/f 12 13 14 f1 0. 227 0.0218 0.0215 f2 0.0220 0.0218 0.0216 fig. 2 presents the solutions of eqs. (1-7). at the cross-section of the curves, there is a mutual solution which is f=0.0218. fig. 2 graphic display of equation solutions 660 z. nunić, m. ajanović, d. miletić, r. lojić for the experiment on a dry roadway, the temperature was 9°c. by the experimental testing on the roadway, the dependence of the speed of movement on the deceleration time was obtained, which is shown in table 4. table 4 the dependence of the speed of movement on the deceleration time of iveco 35s13, on a dry driveway t (s) 0 10 20 30 40 50 60 70 v (km/h) 70 59 50 42 38 30 25 20 fig. 3 presents the dependence of the speed of movement on the deceleration time from table 4. fig. 3 the dependence of the speed of movement on the deceleration time of iveco 35s13, on a dry driveway from fig.3 it is necessary to read the interval of speed and time. the readings are shown in table 5. table 5 intervals of speed and time f1 f2 t=70 (s) t=60 (s) v1=70 (km/h) v1=59 (km/h) v2=20 (km/h) v2=20 (km/h) by inserting the values from table 5 into eqs. (1-7), the values of the rolling resistance coefficient are obtained. table 6 solution of equations for iveco 35s13, on a dry driveway w/f 0.09 0.16 0.25 0.36 0.49 0.64 0.81 1 2 f1 0.0207 0.0206 0.0205 0.0204 0.0203 0.0202 0.0200 0.0199 0.0195 f2 0.0186 0.0185 0.0184 0.0183 0.0180 0.0178 0.0175 0.0173 0.0170 w/f 3 4 5 6 7 8 9 f1 0.0188 0.0183 0.0176 0.0168 0.0159 0.0153 0.0148 f2 0.0166 0.0163 0.0159 0.0157 0.0156 0.0153 0.0150 determination of the rolling resistance coefficient under different traffic conditions 661 fig. 4 presents the solutions of eqs. (1-7). at the cross-section of the curves, there is a mutual solution which is f=0.0153. fig. 4 graphic display of equations solutions for iveco 35s13, on a dry driveway 4. results and hypothesis testing after applying the same methodology as presented in the previous chapter for all the vehicles involved in the experiment, the values shown in table 7 were obtained. table 7 the values of the rolling resistance coefficient obtained by the experiment tires vehicle type wet roadway dry roadway tread depth of 8 [mm] audi a4 0.0233 0.0097 vw golf 4 0.0111 0.0112 vw caddy 0.0248 0.0154 vw caddy 0.0301 0.0154 iveco 35s13 0.0218 0.0153 iveco 50c14 0.0248 0.0177 tread depth of 6-7 [mm] audi a4 0.0202 0.0106 vw golf 4 0.0198 0.0182 vw caddy 0.0289 0.0159 vw caddy 0.0277 0.0175 iveco 35s13 0.0254 0.0152 iveco 50c14 0.0251 0.0208 tread depth of 4-5 [mm] audi a4 0.0220 0.0155 vw golf 4 0.0207 0.0194 vw caddy 0.0303 0.0155 vw caddy 0.0308 0.0178 iveco 35s13 0.0284 0.0165 iveco 50c14 0.0270 0.0203 table 7 shows the results of the whole experiment, which include the values of the rolling resistance coefficient obtained depending on the type of vehicle, the tread depth of tires and the condition of the roadway. testing of the hypotheses is shown below. 662 z. nunić, m. ajanović, d. miletić, r. lojić h1: the tread depth of tires and meteorological conditions affect the increase in the value of the rolling resistance coefficient. h2: the value of the rolling resistance coefficient affects fuel consumption up to 25%. the results of the experiment, i.e. the values of the rolling resistance coefficient, as well as the sums of results by types and columns, are shown in table 8. table 8 results of the experiment factors wet driveway dry driveway ∑yi tires a 0.0233 0.0111 0.0248 0.0301 0.0218 0.0248 0.0097 0.0112 0.0154 0.0154 0.0153 0.0177 0.2206 tires b 0.0202 0.0198 0.0289 0.0277 0.0254 0.0251 0.0106 0.0182 0.0159 0.0175 0.0152 0.0208 0.2453 tires c 0.0220 0.0207 0.0303 0.0308 0.0284 0.0270 0.0155 0.0194 0.0155 0.0178 0.0165 0.0203 0.2642 ∑yj 0.4422 0.2879 0.7301 the calculation of the sums of squares is made by the calculation of the total sum of squares, the sum of squares of each factor and the sums of squares of factor interaction. ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0,73012 2 2... 0,0233 ... 0,0203 0, 01604 0, 01481 0, 00123 3 2 61 1 1 22 0,73011 ... 1 2 22 0,4422 0,2879 0,01546 0,01481 0,000656 ... 3 6 3 2 61 1 1 a b yn yt ijkl abcnl i j a y ss y a ibn abni ss b an j ss  − = + + − = − =     = = =  = − =  + − = − =     = = = = ( ) ( ) ( ) ( ) 2 2 0,73011 2 2 22 ... 0,2206 0,2453 0,2642 0,0148 0,01481 0,000073 ... 2 6 3 2 2 yb y j abn  − =  + + − = − =      table 9 shows interaction of factors used in this experiment, while table 10 shows analysis of variance of applied factors. table 9 interaction of factors b a dry driveway wet driveway tires a 0.0847 0.1359 tires b 0.0982 0.1471 tires c 0.105 0.1592 determination of the rolling resistance coefficient under different traffic conditions 663 ( ) ( ) 2 1... 2 2 0,0847 ... 0,1592 0, 01481 0, 000656 6 1 1 0, 000073 0, 01554 0, 01481 0, 000656 0, 000073 0, 0000093 1 a b ss ssab ij a b i j y ss y cn abcn  − =  + + − − +    = = = − − − = = − − table 10 analysis of variances source of variations sum of squares degrees of freedom mean of squares fo fo crit a 0.000656 2 0.000328 20 3.35 b 0.000073 1 0.000073 4.451 3.35 ab 0.0000093 2 0.00000465 0.2835 2.73 error 0.00049 30 0.0000164 total 0.00123 35 based on the model results, it is proven that both hypotheses are accurate. the tread depth of tires and meteorological conditions affect increasing the values of the rolling resistance coefficient. by calculating the values of the rolling resistance coefficient based on the velocities obtained by the interpolation of the velocities measured on a wet roadway, it can be concluded that the value of rolling resistance increases in 83% and decreases in 17% of the cases, while on a dry roadway, the value of the rolling resistance coefficient increases in 75% and decreases in 25% of the cases. the value of rolling resistance coefficient influences an increase in fuel consumption by up to 25%, which can be also confirmed by the research performed in [14], where świeczko-żurek et al. conclude that the rolling resistance coefficient can lead to an increase in fuel consumption by up to over 20%. the biggest increase in fuel consumption on a wet driveway was noticed for the vw golf 4 passenger car and was 22%, and on a dry driveway with the same vehicle was 16%. 5. conclusion this paper presents the calculated value of the rolling resistance coefficient based on the research on dependence of the speed of vehicle movement on its deceleration time, and on this basis confirms the assumptions given in the paper. the number of the used factors in such research projects can be more than two which is one of the limitations. by analyzing the data obtained throughout research using the methods implemented in this paper, it has been proved that the tread depth is an extremely significant factor for tires. the tread wear indicator is an important tool for assessing the residual depth of the tire tread. worn tires significantly increase the risk of aquaplaning and weaken vehicle braking on a wet roadway. fuel savings can achieve significant economic and environmental improvements [15], can be determined on the basis of the rolling resistance coefficient and are assigned to one of seven grades at levels a to g, with a referring to the tire with the highest fuel economy rating and g referring to the tire with the lowest fuel economy rating. by increasing the value of the rolling resistance coefficient, the rolling resistance itself increases as well as the total resistances that oppose the movement of the vehicle. it is, therefore, necessary to consume more energy to absorb the total resistance, resulting in fuel consumption increase. for the purpose of safer traffic, it is necessary to include an efficient social mechanism that will comprehensively achieve a desired goal throughout the measures of social intervention. motor vehicle tires, as a system of vital importance, deserve special attention 664 z. nunić, m. ajanović, d. miletić, r. lojić when checking the technical validity of a motor vehicle in order to increase the safety of road traffic and reduce rolling resistance, total resistance and thus reduce fuel consumption. the main contribution of this study implies introducing new values of the rolling resistance coefficient under different traffic conditions. in future research of this subject, experimental research should be carried out through practical testing of vehicles under different conditions and circumstances. but for such research, a large financial support is needed. future research needs to stimulate a better linkage between scientific research and practical application of research in order to ensure the realization of scientific research in real conditions. also, one of directions for future research can be, for example, application of other approaches to the modeling of engine fuel consumption like adaptive neuro-fuzzy inference system (anfis) [16, 17] that has been processed in [18]. references 1. bunevska talevska, j., ristov, m., malenkovska todorova, m., 2019, development of methodology for the selection of the optimal type of pedestrian crossing, decision making: applications in management and engineering, 2(1), pp. 105-114. 2. nenadić, d., 2019, ranking dangerous sections of the road using mcdm model, decision making: applications in management and engineering, 2(1), pp. 115-131. 3. savković, t., miličić, m., pitka, p., milenković, i., koleška, d., 2019, evaluation of the eco-driving training of professional truck drivers, operational research in engineering sciences: theory and applications, 2(1), pp. 15-26. 4. trupia, l., parry, t., neves, l.c., presti, d. l., 2017, rolling resistance contribution to a road pavement life cycle carbon footprint analysis, the international journal of life cycle assessment, 22(6), pp. 972-985. 5. srirangam, s.k., anupam, k., kasbergen, c., scarpas, a., cerezo, v., 2015, study of influence of 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emissions of a diesel engine using diesel fuel and biodiesel blends, applied thermal engineering, 60(1-2), pp. 24-32. 17. stojčić, m., 2018, application of anfis model in road traffic and transportation: a literature review from 1993 to 2018, operational research in engineering sciences: theory and applications, 1(1), pp. 40-61. 18. najafi, g., ghobadian, b., moosavian, a., yusaf, t., mamat, r., kettner, m., azmi, w. h., 2016, svm and anfis for prediction of performance and exhaust emissions of a si engine with gasoline–ethanol blended fuels, applied thermal engineering, 95, pp. 186-203. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 1 8 https://doi.org/10.22190/fume171220005l © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper adhesive force of flat indenters with brush-structure udc 539.3 qiang li 1 , valentin l. popov 1,2,3 1 berlin university of technology, berlin, germany 2 national research tomsk state university, tomsk, russia 3 national research tomsk polytechnic university, tomsk, russia abstract. we have numerically studied adhesive contact between a flat indenter with brush structure and an elastic half space using the boundary element method. various surface structures with different size, number and shape of the “pillars”, as well as their distributions (regular or random) have been investigated. the results validate the theoretical prediction that the adhesive force in contact of an indenter with discontinuous areas is roughly proportional to the square root of the real contact density (“filling factor”). key words: adhesion, brush structure, filling factor, boundary element method 1. introduction textured surfaces are gaining popularity today due to their special physical properties such as nanotextured surface for improvement of water repellent [1] or bactericidal properties of orthopedic implants [2] or tribological properties in hydrodynamic lubrication [3]. some of these structure designs try to mimic the “mystical” property of nature. one interesting example is adhesion of the geckos. it was found that there are 6.5 million setae on the gecko’s feet which could provide a large adhesive force [4]. it was argued that this contact splitting is the one that enhances adhesion [5, 6]. in the present paper we consider adhesion between a rigid flat surface with columns and an elastic half space (see fig.1) under usual assumptions of the jkr-theory (see e.g. [7]). received december 20, 2017 / accepted january 31, 2018 corresponding author: qiang, li affiliation: berlin university of technology, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: qiang.li@tu-berlin.de 2 q. li, v.l. popov a) b) fig. 1 contact configuration: a) sketch of adhesive pull-off experiment with contact between a rigid structure and an elastic half-space; b) an example of a brush-structured surface the classical theory of adhesion between a flat cylindrical punch and an elastic half space was given by kendall in 1971 [8]. over the last few years, several analytical models have been proposed to provide for understanding of adhesive behavior of microcontacts of a cluster structure [9-11]. a detailed review can be found in [12]. three-dimensional numerical methods, in particular the boundary element method have been recently very frequently used to simulate the pull-off process of adhesive contact of different indenters, for example based on the jkr-model (johnson, kendall and roberts) by pohrt and popov (2015) [13], or using the dugdale potential by bazrafshan et al. (2017) [14], and by molibari et al. (2017) [15]. in this paper we will use the method of [13] to simulate adhesive contacts of the cluster system. this method can be applied to various adhesive contact problems, while some of them have been validated by existing theories or analytical solutions: the case of a spherical indenter validated by the classical jkr theory [16], the case of a cylindrical punch by kendall’s theory, tests with toroidal indenter, can be found in [17], and with elliptical one as well as two circular punches in [18]. recently the bem has been extended for contacts of power-law functionally graded materials [19] and has been validated by the theoretical solution and the method of the dimensionality reduction [20]. this effective numerical tool enables us to carry out simulation of various brush structures. 2. analytical estimation a recent study of the influence of contact geometry on the strength of adhesion states that the filling factor (or ratio of real area to the apparent area) plays an important role in the adhesive contact of a flat-ended punch with internal discontinuities [21]. if the real contact area of the punch is in proportion to the whole apparent area a with a filling factor , areal=a, then the total energy is equal to * 2 tot 12 a u e d a     (1) where e * is effective elastic modulus e * =e/(1-2 ) with e the elastic modulus and  the poisson’s ratio, d is the indentation depth, 12 is the work of adhesion per unit area. the first term in eq. (1) indicates the elastic energy stored in the body where (a/) 1/2 represents the “effective radius” of the cross section of the punch, and the second the adhesion energy. the adhesive force and the critical indentation depth can be derived as adhesive force of the flat indenters with brush-structure 3   3/2* a,upper 128f e a   , (2) 12 c,upper * 2 a d e     . (3) the subscript “upper” means the upper bound of the force. similarly, if we use incircle radius a0 of the apparent area instead of effective radius (a/) 1/2 , then the lower bound of the force and indentation depth is estimated as * 3 a,lower 12 08f e a  , (4) 12 0 c,lower * 2 a d e    . (5) both estimations (2) and (4) show that the adhesive force of discontinuous or structured punch is roughly proportional to the square root of the area of the real contact (and thus square root of the filling factor: fa~1/2 . note that we study the indentation depth-controlled pull off experiment, and the adhesive force is defined as the absolute value of the maximal normal force in the pull-off process. 3. numerical simulation now we numerically calculate the adhesive force for different brush structures. three types of cluster will be investigated: (1) regularly distributed pillars (fig. 2a); (2) randomly distributed pillars (fig. 2b); (3) mixed distribution (fig. 2c). in case of (1), the pillars of the same sizes are placed in 4×4 (up to 32×32) lattices. in case of (2), the positions of the pillars are randomly generated in the whole square area and the circles will not overlap with each other. in the case of (3), the cubes and the pillars are mixed (the proportion is randomly given) and put into the 16×16 lattices. their sizes in all these three distributions can also be changed to obtain various densities. a) b) c) fig. 2 examples of three types of brush structures with = 0.29: a) regular distribution in 12×12 lattices; b) random distribution; c) mixed distribution in 16×16 lattices 4 q. li, v.l. popov 3.1. regular distribution of pillars in this case, the cylindrical pillars with radius a * are regularly distributed in a square area with length l and discretization 1024×1024. 40 densities  varying from 10 -3 to 0.75 are realized through the change of radius and number of the pillars. for each  , we simulate the adhesion process with different numbers of pillars, from n=4×4 to 32×32 pillars. fig. 3 shows an example of adhesive contact of brush with n=16×16 pillars, where the indentation and force are normalized by the kendall’s solution for macroscopic incircle of the square area equal to eq. (4) and (5) with =1:   a a 3* 128 2 f f e l   , (6)   * 122 2 / d d l e  . (7) a) b) fig. 3 adhesive contact of a structured indenter: a) force-indentation dependence; b) change of contact area (black color) during the detachment corresponding to the four positions in subplot (a). the gray color indicates the detached elements. the important part of the curve in subplot (a) is enlarged to observe the detachment behavior in the simulation, the “indentation depth” is controlled during the separation and the normal force is calculated at each step. the obtained dependence of pull-off force on the separation distance is shown in fig. 3a, where the subplot is a three-dimensional contact configuration at the final detachment moment (corresponding to point 4 in fig. 3b). the change of contact area during the detachment can be seen in fig.3b. it is found that the contact points at the corners are always firstly broken from the indenter and detachment expands towards the center with an increasing indentation depth. when the contact approaches the incircle of the square area, the surfaces are suddenly separated. this general adhesive force of the flat indenters with brush-structure 5 behavior has been generalized in [21] for indenters with various odd geometries. now we put an emphasis on the maximal value of the pull-off force, i.e. adhesive force fa. for comparison, the estimation of the upper and lower bounds of adhesive force, eqs. (2)-(5) are also added in fig. 3a. numerical results from simulations of a number of surfaces are shown in fig. 4. for a certain area density , adhesive force fa is almost the same for different structures with few (4×4) or many (32×32) pillars (fig. 4a). the adhesive force is roughly proportional to the square root of the contact area density (fig. 4b). a) b) fig. 4 dependence of adhesive force on the square root of the filling factor: a) for different size and number of the pillars; b) in a plot with error bar 3.2. random and mixed distribution a) b) fig. 5 dependence of adhesive force on the square root of the filling factor: a) for randomly distributed pillars; b) for mixed distribution of pillars and squares. the adhesive force is roughly proportional to the square root of the filling factor 6 q. li, v.l. popov we have investigated other structures as shown in fig. 2b and fig. 2c. for the case of randomly distributed pillars, the radius of pillar a * varies from a * =0.003l to a * =0.03l. for a given radius a * and area density , the adhesive force is averaged by 10 surfaces. the results are shown in fig. 5a, where a few examples of structure geometries are presented. the adhesive force is very slightly dependent on the size of the pillars. it is a little bit larger for smaller circles. fig. 5b shows the results of the case for mixed distribution where the pillars and squares appear randomly on the surface. the force-area density dependence in these two cases is the same as in the case of regular distribution of pillars. for comparison, the curve of fig. 4b for the case of regularly distributed pillars is also plotted in figs. 5a and 5b. 4. comparison with circular area in fig. 3b it is seen that the detachment begins at the corner of the square and then extends toward the center. in the final state of absolute instability (complete detachment), the contact shape is approximately the incircle of the square. numerically we simulate the pull-off adhesive contact of the same structure but macroscopically in a circular area with regularly distributed pillars. one example of force-distance relation is shown in fig.6a. we can see that the pull-off force increases linearly with the separation distance, and the whole circular area is found to be suddenly separated without partial detachment. this behavior is the same as in the case of a complete cylindrical punch studied by kendall. the maximal normal force, i.e. adhesive force can be observed to be smaller than the case of square area with the same area density. the results for varied sizes of pillars presented in fig. 6b give a similar dependence of adhesive force on the area density to the case of square area; however, in comparison with the latter, the adhesive force in the case of circular area is a little bit smaller due to a smaller contact area. a) b) fig. 6 adhesive force of the pillar structure in a circular area: a) an example of force-separation dependence; b) dependence of adhesive force on the area density. the adhesive force in the case of circular area is smaller than in the case of square area adhesive force of the flat indenters with brush-structure 7 5. conclusion in this paper we have numerically simulated the pull-off adhesive contact of surfaces with brush-structures. the pillars are regularly or randomly distributed in a macroscopic square area. it is noted that we consider a rigid brush structure contacting with an elastic half space. for the case of flexible pillars, e.g. mushroom shaped microstructures [22], the behavior of detachment could be very different. as found in ref. [21], the discontinuities of the indenter have no essential influence on the development of the detachment process: the detachment starts at the corner and spreads to the center. the adhesive force defined as the maximal pull-off force is found to be roughly proportional to the filling factor that is the ratio of pillar area and the apparent area, which verified the theoretical prediction. therefore, the contact splitting in the case of rigid pillars does not lead to any increase in the adhesive force. acknowledgements: we acknowledge financial support of the deutsche forschungsgemeinschaft (po 810-55-1). references 1. checco, a., rahman, a., black, c.t., 2014, robust superhydrophobicity in large-area nanostructured surfaces defined by block-copolymer self-assembly, adv. mater., 26(6), pp. 886–891. 2. jaggessar, a., shahali, h., mathew, a., yarlagadda, p.k.d.v., 2017, bio-mimicking nano and micro-structured surface fabrication for antibacterial properties in medical implants, j. nanobiotechnology, 15, pp. 1–20. 3. d. gropper, d., wang, l., harvey, t.j., 2016, hydrodynamic lubrication of textured surfaces: a review of modeling techniques and key findings, tribol. int., 94, pp. 509–529. 4. autumn, k., 2002, mechanisms of adhesion in geckos, integr. comp. biol., 42(6), pp. 1081–1090. 5. arzt, e., gorb, s., spolenak, r., 2003, from micro to nano contacts in biological attachment devices, proc. natl. acad. sci., 100(19), pp. 10603–10606. 6. kamperman, m., kroner, e., del campo, a., mcmeeking, r.m., arzt, e., 2010, functional adhesive surfaces with “gecko” effect: the concept of contact splitting, adv. eng. mater., 12, pp. 335–348. 7. popov, v.l., 2017, contact mechanics and friction, 2nd edition, springer verlag. 8. kendall, k., 1971, the adhesion and surface energy of elastic solids, j. phys. d: appl. phys., 4, pp. 1186–1195. 9. guidoni, g.m., schillo, d., hangen, u., castellanos, g., arzt, e., mcmeeking, r.m., bennewitz, r., 2010, discrete contact mechanics of a fibrillar surface with backing layer interactions, j. mech. phys. solids., 58, pp. 1571–1581. 10. argatov, i.i., 2011, electrical contact resistance, thermal contact conductance and elastic incremental stiffness for a cluster of microcontacts: asymptotic modelling, quart. j. mech. appl. math., 64, pp. 1–24. 11. bacca, m., booth, j.a., turner, k.l., mcmeeking, r.m., 2016, load sharing in bioinspired fibrillar adhesives with backing layer interactions and interfacial misalignment, j. mech. phys. solids, 96, pp. 428–444. 12. argatov, i.i., li, q., popov, v.l., 2018, cluster of the kendall-type adhesive microcontacts as a simple model for load sharing in bioinspired fibrillar adhesives, submitted. 13. pohrt, r., popov, v.l., 2015, adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method, facta univ. ser. mech. eng., 13(1), pp. 3–10. 14. bazrafshan, m., de rooij, m.b., valefi, m., schipper, d.j., 2017, numerical method for the adhesive normal contact analysis based on a dugdale approximation, tribol. int., 112, pp. 117–128. 15. rey, v., anciaux, g., molinari, j.f., 2017, normal adhesive contact on rough surfaces: efficient algorithm for fft-based bem resolution, comput. mech., 60, pp. 69–81. 16. johnson, k.l., kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids, proc. r. soc. a, 324, pp. 301–313. 17. argatov, i.i., li, q., pohrt, r., popov, v.l., 2016, johnson-kendall-roberts adhesive contact for a toroidal indenter, proc. r. soc. london a math. phys. eng. sci., 472. 8 q. li, v.l. popov 18. li, q., argatov, i.i., popov, v.l., 2018, onset of detachment in adhesive contact of an elastic half-space and flat-ended punches with noncircular shape: analytic estimations and comparison with numeric analysis, submitted. 19. li, q., popov, v.l., 2017, boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials, comput. mech., doi: doi: 10.1007/s00466-017-1461-9. 20. heß, m., 2016, a simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, int. j. eng. sci., 104, pp. 20–33. 21. popov, v.l., pohrt, r., li, q., 2017, strength of adhesive contacts: influence of contact geometry and material gradients, friction, 5, pp. 308–325. 22. heepe, l., gorb, s.n., 2014, biologically inspired mushroom-shaped adhesive microstructures, annu. rev. mater. res., 44, pp. 173–203. facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 453 472 https://doi.org/10.22190/fume200306032p © 2020 by university of niń, serbia | creative commons license: cc by-nc-nd original scientific paper fuzzy model of the operational potential consumption process of a complex technical system michał pająk university of technology and humanities in radom, faculty of mechanical engineering, poland abstract. during the operation process of a system its technical state is changed. the changes take place because of the wearing factors impact. the impact depends on the flow and intensity of the operation process what is characterized by the time histories of the working parameters. simultaneously, the changes of the technical state are correlated with the changes of the amount of the operational potential included in a system. in order to avoid the inability state occurrence the amount of this potential should be higher than the boundary value. the amount of the operational potential included in a system is determined by the values of the cardinal features of it but in the case of the real technical system the values cannot always be measured. therefore, the amount of the operational potential and the technical state of the system cannot always be determined online. to solve this problem the model of the operational potential consumption process was created and presented in the paper. the model uses artificial intelligence techniques to calculate the change of the operational potential amount by determining the changes of the cardinal features of the system on the basis of the time histories of the working parameters. the verification of the model quality was performed using the pulverized boiler op-650k-040 operating in the power plant. the description of the conducted research and the results of the verification were presented in the end of the paper proving the adequacy of the model implementation in the case of industrial objects. key words: fuzzy model, operational potential consumption, complex technical system, operation, system feature, working parameter received march 06, 2020 / accepted august 30, 2020 corresponding author: michał pająk university of technology and humanities in radom, faculty of mechanical engineering, stasieckiego st. 54, 26-600 radom, poland e-mail: m.pajak@uthrad.pl 454 m. pająk 1. introduction the object under consideration is a crucial technical system of the strategic importance. in the case of such systems the main objective of the operation and maintenance control is to perform the operational tasks and avoiding the failure occurrence [1,2]. this is a very important issue because the consequences of a failure can be very costly or can pose a health risk. in order to avoid failure the operation processes should be stopped and the service activities should be performed. unfortunately, when the operation processes are stopped the technical system user does not obtain any effect of the system operation. thus, the service activities should be started at the proper moment. the main question of the research is how to determine that moment. the failure is defined as a transition of the technical state of the system from the area of the ability states to the area of the inability states [3]. the technical state is determined by the values of the cardinal features of the system [4] and is correlated with the amount of the operational potential included in it. the inability state occurs when the amount of the operational potential included in a system is lower than the amount corresponding to its boundary state [5]. in order to avoid the inability state occurrence it is necessary to know the amount of the operational potential of the system at every moment of the system operation. this amount, similarly to the technical state, is determined by the values of the cardinal features of the system. but not always it is possible to measure the values online. the changes of the values take place during the operation processes because of the forcing factors influence. the forcing factors can be divided into two groups dependent on and independent of the system operation [6]. the first group of the factors acts as a result of the operation process execution while the second characterizes the impact of the environment [7]. thus, the level of the forcing factors influence is determined by the way of the operation process execution and the intensity of the environment impact. if the intensity of the environment impact is assumed as an independent variable then the forcing factors influence is determined only by the operation process flow which is described by the values of its parameters. in fact, the intensity of the operation process is described by the time histories of the working parameters. so, it was stated that in order to avoid the inability state occurrence it is necessary to apply the model of the operational potential consumption process which can calculate the change of the potential as a function of the time histories of the working parameters. therefore, the analysis of the existing models of the operation and maintenance processes was performed [8-11] in order to identify their adequateness in the case of crucial, complex technical systems of the strategic importance. as a result it was stated that in the case of complex technical objects changes of operational potential amount could not be given with proper accuracy thanks to analytical models usage. therefore, in this paper, the system fuzzy model is proposed to model the process under consideration. in chapter 2 the main idea of fuzzy model application is presented. its structure and the way of operation are described in details in chapter 3. subsequently, in chapter 4 the industrial research is presented which was performed in order to identify the model parameters values (described in chapter 5) and to verify prepared model accuracy (described in chapter 6). finally, in chapter 7, some conclusions can be found. the main contribution and the novelty of the proposed solution are the universal creation method of the system model of the operation potential consumption process. the result of the proposed method is the fuzzy model. thanks to the fuzzy sets theory fuzzy model of the operational potential consumption process of a complex technical system 455 implementation the uncertainty of the initial technical state identification and the approximation of the time histories of the operation parameters are taken into consideration. 2. the idea of the fuzzy model of the operational potential consumption process it was decided that the designed model will estimate the levels of the forcing factors influence on the basis of the time histories of the working parameters. subsequently, on the basis of the levels of the forcing factors influence the changes of the cardinal features values will be identified. eventually, on the basis of the changes of the cardinal features values the change of the operational potential amount will be determined eq. (1) puxx t t tff t t tff t t tff t t tff t t tpo t t tpopum s r idid n dd aop    ),...,( ))(,...,)(,)(,...,)(( ))(,...,)((: 1 1 2 1 21 1 2 1 21 1 2 1 2 1 (1) where: mpu the model of the operational potential consumption process, poi i-th parameter of the operation process i = 1,...,aop, aop the working parameters amount, poi(t) the time dependent value of i-th working parameter, 1 2 )( t t tpoi the time history of i-th working parameter in period [t1, t2], j df f j-th forcing factor depended on the system operation j = 1,...,n, k idf f k-th forcing factor independent of the system operation k=1,...,r, xl the change of the value of l-th cardinal feature of the system l = 1,...,s, pu the change of the amount of the operational potential of the system. the application of the constructed model will be used to determine the technical state of the system in t2 moment taking into account the technical state of the system in t1 moment and time histories of the parameters of the operation process in period [t1, t2] eq. (2). )())(,...,)(),(( 2 1 2 1 2 11 ts t t tpo t t tpots pum aop    (2) where: s(t) the technical state of the system for time t. the first input value of the model is the technical state of the system in moment t1. it results from the markov characteristic of the technical state [12]. according to it, the state of the system in moment t2 is unequivocally determined by the state in moment t1 and known input values for t⸦[t1,t2] and is independent of the system state and input values before t1 moment. the operating point of the technical object is expressed by the values of the object features. in the case of complex technical systems operating in the industry the values of the part of the measurable features still can be identified only by destructive testing [13]. therefore, they cannot be identified at given moment in which the operation processes are 456 m. pająk performed. simultaneously, because of immeasurable features presence, it is impossible to accurately determine the technical state of the system. even though, the moment of the object installation in the operation place is treated as initial moment t1, the technical state depends on conditions during the production phase and pre-operation processes execution. in the course of the technical object designing calculations isotropic structure of the material is assumed. in reality this assumption is fulfilled only approximately. for example, as far as metal is concerned, its non-uniform structure causes up to 20% differences in the mechanical characteristics values. additionally, during the assembling the designing calculations requirements can be violated. what is more, because of the lack of the equipment required to perform non-destructive testing, the majority of the producers of the machinery, devices and constructions do not support the full quality tests at the end of the production phase [14,15]. usually, the partial tests are executed (about 5%) and on this basis the decision about the correctness of the whole series of the products is made [16]. as a result, the devices of approximately identified technical state are designated to operation. subsequently, during the pre-operation processes execution the devices are under the influence of forcing factors. this impact depends on the storage conditions and the intensity of the environment influence, which are tested periodically [17]. therefore, it was stated that the technical state of the system at the start moment of the operation process can be identified only approximately. next group of the input data of mpu model are the time histories of the parameters describing performed operation processes. its values, just like those of the system features, can be measurable or immeasurable. and similarly to the system features, the values can be determined only approximately. inaccuracy of the working parameters values and the approximation of the technical condition of the system make it impossible to accurately transform the initial state of the system s(t1) in resultative state s(t2). therefore, the designed mpu model has to take into consideration the approximate character of the input data. this type of inaccuracy consists in the lack of the possibility to categorize the given phenomenon as true or false [18]. to model this type of inaccuracy the fuzzy sets theory is widely used [19,20]. the application of fuzzy logic is especially appropriate in the case when the mathematical model describing the phenomenon does not exist, the existing model is strongly nonlinear, the existing model is irresolvable or the calculation time required to obtain the resolution is too long to fulfill the demands about the model response time [21]. modeling of the operational potential consumption process can be treated as a case when adequate mathematical model does not exist and tries of the model construction face the problems arising from not clear enough process description and approximate character of the input data. in such cases the fuzzy modeling implementation can be found in industrial application [22-25] as well as in theoretical research papers [26-28]. thus, on the basis of the research conducted by the author and the literature analyze it was decided that the mpu model will be constructed by the fuzzy modeling implementation. the formulated model of the operational potential consumption is the model of the multi input single output (miso) type. the calculation process performed by the model consists of three sequential transformations given in eq. (1) where the last one is executed according to eq. (3) [5]: fuzzy model of the operational potential consumption process of a complex technical system 457    n i ii txtxtpu 1 2 21 ))()(()( (3) where: pu(t) the change of the operational potential amount due to the operation process execution in period t = [t1, t2], xi(t1) the value of i-th feature for time t1, xi(t2) the value of i-th feature for time t2. therefore, the task of creation of the fuzzy mpu model was decomposed into creation the ncf fuzzy models performing transformation eq. (4), where ncf is the amount of the cardinal features of the system. lidriddnd aopl x t t tff t t tff t t tff t t tff t t tpo t t tpoxm   ))(,...,)(,)(,...,)(( ))(,...,)((: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 (4) where: mxl the model of the change of l-th cardinal feature value, 1 2 )( t t tpoi the time history of the i-th working parameter in period [t1, t2], j df f j-th forcing factor depended on the system operation j = 1,...,n, k idf f k-th forcing factor independent of the system operation k = 1,...,r,xl the change of the value of l-th cardinal feature of the system l = 1,...,s. the input data of the models are the time histories of the working parameters in period of the operation process execution while the output is the change of the value of the system cardinal feature. it was decided that the considered models will be constructed and verified using the measurement collections recorded during the conducted operational test, subsequently analyzed and transformed to the form of eq. (5) according to the method proposed by the author in the papers [29,30]. 1 2( ) :[ , ,..., , ]j l nipo lmmc x po po po x   (5) where: xl l-th cardinal feature of the system, poi i-th working parameter i = 1,...,nipo(xl), nipo(xl) the number of the important working parameters for feature xl, mmcj j-th major measurement collection, poi the distance between the time history of i-th working parameter and its zero time history (zth the time history of the working parameter for which the difference between the initial and final value of the feature is the minimum one), xl the change of the value of l-th feature of the system. 3. structure and the operation of the model of the operational potential consumption process each of the models of the system feature value change is of miso type describing the unknown in shape solution space. major measurement collections expressed in the form presented above eq. (5) describe the relation between the models output data (change of the feature value) and the models input data (distances between time histories of the 458 m. pająk significant working parameters and their zero time histories). so, the collections are the inputs-output samples of the considered models. the inputs-output models are constructed employing the fuzzy sets theory. they can use inference of mamdani type [31], of takagi-sugeno-kang type [32] or be of relational type [33]. in order to solve the issue under consideration the fuzzy models with use of the mamdani type inference are chosen because thanks to their generalized structure the accurate knowledge about the structure of the modeled object is not required [34]. moreover, the inference rules of the models are not uniformly distributed along the solution space. regions where the solution space changes its shape are modeled by a bigger number of the rules than the regions where the space is of fixed shape. owing to this, in the mamdani models the amount of the inference rules can be decreased. the structure and the general scheme of the worked out models operation are presented in fig. 1. fig. 1 the general scheme of the fuzzy model operation the input data of the models are crisp values of the distance between the working parameters time histories and their zero time histories: ]...[ 21 nipfm popopopo   (6) where: pofm the input data of the fuzzy model, poi the crisp value of i-th input parameter, nip the number of the input parameters of the fuzzy model. the first step of the fuzzy model operation is the fuzzification process. it is transformation of the crisp values to the fuzzy ones. fuzzification is accomplished using knowledge base of the model [35]. the knowledge base consists of groups of the fuzzy sets defined for each input and output parameter of the model. these fuzzy sets cover the domain of the parameter in a way presented in fig. 2. fig. 2 the exemplary group of the fuzzy sets for the input parameter of the fuzzy model fuzzy model of the operational potential consumption process of a complex technical system 459 the fuzzy sets which cover the domain of the parameter are their linguistic values. first of the sets is of l type, eq. (7) while the last one is of  type, eq. (8). the remaining sets are of  type, eq. (9).              rrsx rrsxrrk rrkrrs xrrs rrkx xfsl 0 1 )( (7) where: fsl(x) the membership function of the l type fuzzy set, x the argument of the fuzzy set, rrk the right range (the maximum value) of the kernel of the fuzzy set, rrs the right range (the maximum value) of the support of the fuzzy set.              lrkx lrkxlrs lrslrk lrsx lrsx xfs 1 0 )( (8) where: fs(x) the membership function of the  type fuzzy set, x the argument of the fuzzy set, lrs the left range (the minimum value) of the support of the fuzzy set, lrk the left range (the minimum value) of the kernel of the fuzzy set.                  rrsxlrk lrkrrs xrrs lrkxlrs lrslrk lrsx rrsxlrsx xfs 0 )( (9) where: fs(x) the membership function of the type fuzzy set, x the argument of the fuzzy set, lrk the left range (the minimum value) of the kernel of the fuzzy set, lrs the left range (the minimum value) of the support of the fuzzy set, rrs the right range (the maximum value) of the support of the fuzzy set. during the fuzzification process for each crisp value of the considered input parameter the values of the membership functions of each of its linguistic values are determined: ])(...)()([ 21 ililili popopoz nlv ipoipoipo    (10) where: zi the fuzzy value of i-th input parameter of the model, poi the crisp value of i-th input parameter of the model, )( il poj ipo   the value of the membership function of j-th linguistic value of i-th input parameter for poi value, j poi l j-th linguistic value of i-th input parameter of the model, nlv the amount of the linguistic values defined for the parameters of the model. 460 m. pająk the second step of the models operation is the fuzzy inference. it is performed using the rule bases of the models. the rule base of each model consists of the inference rules described by eq. (11), which should be read according to the notation from eq. (12): n xi m popo k po j poi fmniponip lylpolpolpofmr    ...: 21 21 (11) where: fmri i-th inference rule of the fuzzy model, poi the crisp value of i-th input parameter of the model, j poi l j-th linguistic value of i-th input parameter of the model, yi the fuzzy output value of i-th inference rule, n xfm l n-th linguistic value of the output parameter of the model, xfm the output parameter of the fuzzy model.        ni ni y mzkzjzfmr out out i nipi 0 1 0)(...0)(0)(: 21 (12) where: zi the fuzzy value of i-th input parameter of the model, iout the number of the linguistic value of the output parameter of the model. the fuzzy inference consists in the premises aggregation, implication and resulting sets accumulation [36]. for each of the rules present in the rule base of the model the premises aggregation is performed according to the formula: ))(),...,(),(min()( 21 mzkzjzfmragr nipi  (13) subsequently, the implication is executed according to formula: nvliiyfmragriy outoutiiouti ,...,2,1))(),(min()(  (14) where: fmri i-th inference rule of the fuzzy model, agr(fmri) the value after premises aggregation of i-th inference rule, yi(iout) the value of the membership function of the linguistic value on position iout for fuzzy value of the output parameter of i-th rule, nvl the amount of the linguistic values defined for the parameters of the model. the final step of the fuzzy inference process is the resultant sets accumulation. the step consists in determination of the fuzzy value of the output parameter according to formula: nvliiyiyiyiy outoutnrbroutoutout ,...,2,1))(),...,(),(max()( 21  (15) where: y(iout) the value of the membership function of the linguistic value from position iout for fuzzy value of the output parameter, yi(iout) the value of the membership function of the linguistic value from position iout for fuzzy value of the output parameter of i-th rule, iout the number of the linguistic value of the output parameter of the model, nrbr the amount of the rules in the rule base of the model. thus, the fuzzy value of the output parameter can be expressed in the following form: ]...[ 21 nvl fmxlxlx lll yyyy   (16) where: y the fuzzy value of the output parameter of the model, i lxl y  the maximum value of the membership function of i-th linguistic value of the output parameter, i xfm l i-th linguistic value of the output parameter, nvl the amount of the linguistic values defined for the parameters of the model. fuzzy model of the operational potential consumption process of a complex technical system 461 the expression above should be interpreted as a fuzzy set with support equal to the output parameter domain. the value of the membership function of this set ought to be calculated according to the formula: )),min( ... ),,min( ),,max(min()( 22 11 nvl fmx nvl fmx fmxfmx fmxfmx ll ll llfm y y yx        (17) where: (xfm) the membership function of the output parameter, , xfm the output parameter of the fuzzy model, i fmxl y  the maximum value of the membership function of i-th linguistic value of the output parameter, i fmxl  the value of the membership function of i-th linguistic value of the output parameter, i xfm l i-th linguistic value of the output parameter, nvl the amount of the linguistic values defined for the parameters of the model. the output of the fuzzy model is the crisp value of the output parameter. it is calculated in the defuzzification process. in the process the fuzzy value of the output parameter determined in the inference process is defuzzified by the centre of gravity defuzzification operator application:          max_ min_ max_ min_ )( )( ))(( fm fm fm fm x x fmfmfm x x fmfmfm fmfm xdxx xdxx xcogx   (18) where: xfm the output parameter of the fuzzy model, cog the centre of gravity defuzzification operator, (xfm) the membership function of the output parameter, xfm_min the minimum value of the output parameter domain, xfm_max the maximum value of the output parameter domain. 4. the performed industrial research in the described studies it was decided to construct the considered fuzzy model using learning data registered during the industrial research. there are two main groups of the learning techniques which can be used to solve this issue. first of them is the connection of artificial neural networks and fuzzy systems which is widely applied decision-making and data classification problems [37-39]. the second one is connection of genetic algorithms and fuzzy logic which is implemented in automatic fuzzy model generation and data classification problems solutions [40,41]. the task of creating models as described by eq. (4) was solved using the automatic iterative method of the mamdani fuzzy model generation [42]. the applied method generates the models using the sample data in input/output form. exactly, the sample data were prepared in the form depicted by eq. (5). in order to collect the data the operational tests were carried out on the biggest polish hard coal fired power plant in kozienice. the production system of the plant includes 8 200mw power units, 2 500mw power units 462 m. pająk and 1 1000mw power unit [43]. the tests were limited to 8 200mw production units. main parts of the units were op-650k-040 pulverized coal fired boilers [44] and 13k215 steam turbosets. the performed operational tests lasted fifteen months and consisted in collecting the working parameters of the units. after the preliminary analysis of the data correctness 23 parameters were selected for further considerations (table 1). as a result of the collecting process 5440008 values for each parameter were obtained. the technical state of the op-650k-040 boiler is described by the values of the system features. these features also describe the quality of the whole system operation. therefore, to identify the cardinal features of specified element of the system the tke method was taken into consideration. tke is a common analysis method of an operation quality of power units’ devices. as a result of the completed analysis deviations q3 (the deviation of the heat consumption caused by the temperature of the reheated steam [kj/kwh]), q4 (the deviation of the heat consumption caused by the pressure in the secondary reheater of the reheated steam [kj/kwh]), q5 (the deviation of the heat consumption caused by the water injections to the reheated steam [kj/kwh]) and q8 (the deviation of the heat consumption caused by the reduced efficiency of the boiler [kj/kwh]) were chosen as cardinal features determining the technical state of the boiler [1]. on the basis of the measurements sets collected during the operational tests, the momentary values of the deviations were calculated. applying the polynomial approximation the time functions of each deviation were formulated. table 1 the set of the selected working parameters of the research object (rap rotary air preheater, hp high pressure, mp medium pressure) no value medium side unit symb. 1. temperature main steam left °c t0l 2. temperature main steam right °c t0p 3. pressure main steam both mpa p0 4. pressure outgoing steam of hp turbine both mpa psahp 5. temperature outgoing steam of hp turbine left °c tahpl 6. temperature outgoing steam of hp turbine right °c tahpr 7. temperature reheated steam left °c tssl 8. temperature reheated steam right °c tssr 9. pressure incoming steam of mp turbine both mpa tsbmp 10. temperature feed water both °c tfw 11. temperature flue gasses 1 °c teg1 12. temperature flue gasses 2 °c teg2 13. contents o2 in flue gasses before rap 1 % o2bp1 14. contents o2 in flue gasses before rap 2 % o2bp2 15. contents o2 in flue gasses after rap 1 % o2ap1 16. contents o2 in flue gasses after rap 2 % o2ap2 17. amount injected water left t/h wsil 18. amount injected water right t/h wsir 19. amount burned mazut both t/h ma 20. amount burned coal both t/h ca 21. amount active load both mw p 22. amount main steam left t/h m0l 23. amount main steam left t/h m0r fuzzy model of the operational potential consumption process of a complex technical system 463 using the calculated values of the boiler’s cardinal features as well as recorded measurements sets 874 intervals of equal time length (24 hours) were obtained for each distinguished deviation. the measurement collection was formulated for each interval. the collection consisted of the time histories of 23 working parameters in range from the start to the end time of the interval and the difference between the values of the specified deviation for the start and the end time of interval. among the formulated measurement collections for each deviation one measurement collection was selected. it was the collection corresponding to the smallest change of the deviation value. on this basis, according to the method presented in literature [29] the measurement collections were transformed into the form described by expression eq. (5) but comprising all 23 parameters. subsequently, according to the method proposed by the author [30], the most significant working parameters in term of the operational potential change have been selected. for deviation q3 parameters 3, 12 and 17 have been selected and so have, for deviation q4, parameters 1, 13, 14 and 15; for deviation q5 parameters 8 and 10 are selected and so are, for deviation q8, parameters 17 and 18. the numbers are used according to table 1. thus, as a result of the performed studies, for each of 4 models of the cardinal feature change as a function of the most significant working parameters time histories, 874 major measurement collections in form described by eq. (5) were obtained. 5. the identification of the parameters of the operational potential consumption process model in order to generate the models of the working parameters time histories impact on the system features values, for each deviation the learning and testing data sets were prepared. each of the sets consisted of the major measurement collections which included input and output data of the model. the form of the learning and testing data for each deviation is presented in tables (table 2 table 5). table 2 the structure of the major measurement collections for q3 deviation (names of the fields according to the symbols from table 1) no. field name symbol unit 1. the distance from zth p0 p0db mpa 2. the distance from zth teg2 teg2db °c 3. the distance from zth – wsil wsildb t/h 4. the change of the deviation value object dq3r kj/kwh 5. the change of the deviation value model dq3e kj/kwh table 3 the structure of the major measurement collections for q4 deviation (names of the fields according to the symbols from table 1) no. field name symbol unit 1. the distance from zth t0l t0ldb °c 2. the distance from zth o2bpl o2bp1db % 3. the distance from zth o2bp2 o2bp2db % 4. the distance from zth o2apl o2ap1db % 5. the change of the deviation value object dq4r kj/kwh 6. the change of the deviation value model dq4e kj/kwh 464 m. pająk table 4 the structure of the major measurement collections for q5 deviation (names of the fields according to the symbols from table 1) no. field name symbol unit 1. the distance from zth – tssr tssrdb °c 2. the distance from zth – tfw tfwdb °c 3. the change of the deviation value object dq5r kj/kwh 4. the change of the deviation value model dq5e kj/kwh table 5 the structure of the major measurement collections for q8 deviation (names of the fields according to the symbols from table 1) no. field name symbol unit 1. the distance from zth – wsil wsildb t/h 2. the distance from zth – wsir wsirdb t/h 3. the change of the deviation value object dq8r kj/kwh 4. the change of the deviation value model dq8e kj/kwh table 6 the values of the parameters of the fuzzy model generation process no. parameter value 1. coverage degree of the measurement collection 1.0 2. compatibility limit of measurement collection and inference rule z 0.05 3. incompatibility limit of inference rule and learning data set kicfmr 0.1 4. threshold value of the compatibility degree sfm 0.25 5. t-norm used in generation process min 6. amount of the genetic algorithm generations in one iteration 50 7. amount of useless mutations of evolution strategy until the process stop 25 8. amount of the chromosomes exposed to the evolution strategy 20% 9. the parameter of the value mutation used in evolution strategy 0.9 10. amount of generations of simplification step 500 11. amount of generations of tuning step 1000 12. population size of simplification step 61 13. population size of tuning step 61 14. parameter of non-uniform mutation 5 15. crossover probability of generation process 0.6 16. crossover probability of simplification process 0.6 17. crossover probability of tuning process 0.6 18. mutation probability of generation process 0.004 19. mutation probability of simplification process 0.003 20. mutation probability of tuning process 0.005 21. coefficient of arithmetic min-max crossover operator g 0.35 22. aggregation operator of the fuzzy model min 23. implication operator of the fuzzy model min 24. accumulation operator of the fuzzy model max 25. defuzzification operator of the fuzzy model cog the learning data set comprised 582 measurement collections (first two-thirds of all measurement collections) while the testing data set comprised remaining 292 measurement fuzzy model of the operational potential consumption process of a complex technical system 465 collections. subsequently, on the basis of learning data set the automatic generation of the model was performed. the generation procedure consisted of preliminary generation, simplification and tuning steps. the adopted values of the model generation process can be found in table 6. as a result of the calculations four models of the working parameters time histories influence on the values of the system features were obtained. each of the model in the aggregation process used min operator, in implication process min-max operator, in accumulation process max operator and in defuzzification process cog operator, eq. (18). the rule bases of the obtained models included the groups of fuzzy sets defined for each input and output parameter of the model. below, as an example, the group of fuzzy sets defined for p0db the first input parameter of the model for q3 deviation is presented (fig. 3), eqs. (19-25). the model obtained for q3 deviation consisted of 63 inference rules, the model obtained for q4 deviation consisted of 82 inference rules, the model obtained for q5 deviation consisted of 20 inference rules and the model obtained for q8 deviation consisted of 40 inference rules. the exemplary form of the obtained inference rule is presented below eq. (26). in all expressions describing the rule and knowledge bases of the fuzzy models the symbols were used in accordance with tables (table 2 table 5). the linguistic variables of input and output parameters of the models are marked by subscript meaning the parameter and superscript meaning the number of the linguistic variable. fig. 3 the group of the fuzzy sets of first input parameter p0db of the model for q3 deviation             000146,0p0db 0 000146,0p0db0,000115 000031,0 0000146,0 000115,0p0db 1 1 0 dbp l dbp (19)                000245,0p0db000145,0 0001,0 0000245,0 000145,0p0db000111,0 000034,0 000111,00 000245,0p0db000111,0p0db 0 2 0 dbp dbp l dbp (20) 466 m. pająk                000293,0p0db000258,0 000035,0 0000293,0 000258,0p0db00019,0 000068,0 00019,00 000293,0p0db00019,0p0db 0 3 0 dbp dbp l dbp (21)                000395,0p0db000305,0 00009,0 0000395,0 000305,0p0db000198,0 000107,0 000198,00 000395,0p0db000198,0p0db 0 4 0 dbp dbp l dbp (22)                000463,0p0db000379,0 000084,0 0000463,0 000379,0p0db000285,0 000094,0 000258,00 000463,0p0db000285,0p0db 0 5 0 dbp dbp l dbp (23)                000527,0p0db000454,0 000073,0 0000527,0 000454,0p0db00038,0 000074,0 00038,00 000527,0p0db00038,0p0db 0 6 0 dbp dbp l dbp (24)             0005,0p0db 1 0005,0p0db000489,0 000011,0 000489,00 000489,0p0db 0 7 0 dbp l dbp (25) 1 3 1 1 2 1 0 3 20 edqwsildbdbtegdbp e=l=>dqwsildb=ldb=ltegdb=lp  (26) 6. the analysis of the model accuracy and the results of simulation experiments finally, the verification of the generated fuzzy models was performed. to do it, for each model the following measures of the operation quality were determined: the relative minimum error, eq. (27), the relative maximum error, eq. (28), the relative mean error, eq. (29), the relative mean square error, eq. (30), and the value of the correlation function, eq. (31). the values of the measures were determined taking into consideration the values calculated using the real collected data and the values calculated by the models for testing data set. fuzzy model of the operational potential consumption process of a complex technical system 467 ammci mmcdqmmcdq mmcdqmmcdq dq irir irie ,...,2,1100 ))(min())(max( ))()(min( [%]min     (27) where: dqmin the relative minimum error [%], dqr(mmci) the value of the output parameter of the model for i-th major measurement collection, dqe(mmci) the value of the output parameter of the object for i-th major measurement collection, mmci i-th major measurement collection, ammc the amount of major measurement collections. ammci mmcdqmmcdq mmcdqmmcdq dq irir irie ,...,2,1100 ))(min())(max( ))()(max( [%]max     (28) where: dqmax the relative maximum error [%]. ammci mmcdqmmcdq mmcdqmmcdq ammc dq irir ammc i irie mean ,...,2,1100 ))(min())(max( )()( 1 [%] 1       (29) where: dqmean the relative mean error [%]. mmci mmcdqmmcdq ammcammc rmvdqrmvdq dq irir ammc i irie mean ,...,2,1100 ))(min())(max( )1( ))()(( [%] 1 2 2        (30) where: dq 2 mean the relative mean square error [%]. 100 )()()()( )()( )( 11 2 1               ammc i ieie ammc i irir ammc i irie mmcdqmmcdqmmcdqmmcdq mmcdqmmcdq dqcorr (31) where: corr(y) the value of the correlation function [%]. the values of the operation quality measures of each model are presented in table 7. table 7 the values of the operation quality measures of the models measure of the operation quality model deviation q3 model deviation q4 model deviation q5 model – deviation q8 relative minimum error [%] 0.2498 1.2772 1.2606 1.3084 relative maximum error [%] 155.3265 92.9864 92.0827 84.574 relative mean error [%] 13.8125 5.129 9.4532 11.4643 relative mean square error [%] 1.1368 0.6205 0.8314 0.9089 correlation function value [%] 63.2596 60.7267 60.7023 82.1394 468 m. pająk analyzing the calculated values of the operation quality measures of the models of the working parameters time histories influence on the values of the system features it was decided that the proposed method of the models generation is efficient enough to apply it to the conducted research. it was also stated that the loss of the amount of the operational potential would be calculated as a length of the vector given by eq. (3) and the estimated changes of the deviations values would be the vector components. in order to verify the adequacy of the constructed model of the operational potential consumption process in case of the real complex technical system the simulation experiments were carried out. the experiments covered 14 days period of op-650k-040 steam boiler operation. the calculations were performed for 25 selected initial states spread uniformly in the range of the disposed amount of the operational potential calculated for all initial states. the time histories of the most significant working parameters for the considered 14 days period and the initial values of the deviations determined on the basis of the data recorded during the operational tests were used as input values of the models of the working parameters time histories influence on the values of the system features. as a result of the models operation the estimated values of the deviations for the end of the simulated operation period were obtained. having determined real initial values of the deviations and estimated final values of them the estimated change of the disposed amount of the operational potential was calculated according to expression eq. (3). similarly, using real initial and real final values of the deviations (obtained as a result of the operational tests) the real change of the disposed amount of the operational potential was calculated. in table 8 for each one of the 25 initial states considered during the simulation experiments the initial values of the deviations, the initial value of the disposed amount of the operational potential and real and estimated final values of the disposed amount of the operational potential can be found. additionally, in the table the values of the relative error of the model operation calculated according to formula eq. (32) are presented. %100)(     real realest rel pu pupu pumerr (32) where: errrel(mpu) the relative error of the operation of the mpu model [%], puest the change of the operational potential amount estimated by the model, pureal the real change of the operational potential amount. analyzing results of the model operation in case of 25 selected periods of the op-650k040 steam boiler operation it was stated that the accuracy of the estimation of the operational potential consumption process by the generated model is very high (up to 99%). thus, it was proved that the presented model can be successfully used to estimate the loss of the operational potential in the case of complex technical systems. therefore, it is planned to apply the proposed solution to the control of the complex technical systems operation in order to decrease the loss of not exploited operational potential and avoid the failure of the system. fuzzy model of the operational potential consumption process of a complex technical system 469 table 8 results of the mpu model operation (p disposed amount of the operational potential [kj/kwh]) no. initial values of the deviations [kj/kwh] [q3;q4;q5;q8] initial p final p estimated value final p real value relative error [%] 1. [-19.3074;-5.8577;-8.5227;-405.9765] 638.8881 128.118 120.1735 6.61 2. [-39.0666;4.4073;56.2565;-263.0552] 490.3015 137.7455 138.8941 0.83 3. [-27.5759;8.9035;59.6207;-223.0393] 449.131 103.6373 99.0042 4.68 4. [-36.0886;8.9555;57.9187;-156.9477] 384.6197 111.7565 111.1584 0.54 5. [-17.9859;4.1348;27.6282;-103.3115] 335.065 115.4199 115.7112 0.25 6. [-13.8601;1.3529;42.9267;-90.7771] 319.9943 115.3871 114.5998 0.69 7. [-9.9071;1.5013;44.2801;-79.7246] 308.6944 118.2544 116.392 1.60 8. [-30.7651;11.8714;70.5773;-64.6891] 291.6223 116.3791 114.7306 1.44 9. [-22.9398;-21.7261;36.5607;-18.2813] 255.517 114.4198 113.0098 1.25 10. [-10.7811;-19.3661;36.2806;17.0488] 220.7192 113.5734 115.2506 1.46 11. [-24.7443;-16.0588;62.7121;39.1986] 195.0298 112.5225 112.5837 0.05 12. [-23.6838;-15.0915;63.2691;48.5913] 185.6962 113.0006 112.6729 0.29 13. [-15.2574;25.7487;75.0135;70.5717] 155.21 112.9432 112.3101 0.56 14. [-10.5887;21.7895;42.7999;78.6968] 154.5821 112.3773 111.9687 0.36 15. [-9.8674;24.6442;44.6542;94.1526] 139.3418 99.1767 98.7857 0.40 16. [-9.5275;25.5339;45.6591;101.7391] 131.903 94.7355 95.0247 0.30 17. [-9.2861;25.9666;46.4051;106.8706] 126.8706 87.563 87.0652 0.57 18. [-18.5995;9.2839;46.0432;118.2013] 120.3831 76.1608 75.7769 0.51 19. [-7.7767;26.9908;50.5532;124.9305] 108.5901 76.0057 76.0297 0.03 20. [-28.3647;19.5421;74.9466;130.5547] 101.4431 72.5974 72.6728 0.10 21. [-1.2134;9.3858;70.4222;138.4202 91.0723 70.2585 70.1301 0.18 22. [-15.15119.2709;67.0263;151.5864] 82.7258 56.9301 56.825 0.19 23. [-0.5087;10.4719;74.4923;157.4561] 72.1249 53.7195 53.81 0.17 24. [-0.3174;10.7947;75.6293;162.7304] 66.9231 50.0816 50.0096 0.14 25. [-0.04222;11.2884;77.2998;170.4669] 59.3524 44.0658 43.9252 0.32 7. summary and conclusions in the paper the studies in the field of the complex industrial technical systems operation and maintenance were presented. as a result of the studies the model of the operation process of the power boiler was prepared. below some conclusions and highlights from the research can be found: 1. the prepared model of the operational potential consumption process should be designed to take into consideration inaccuracy of the working parameters values and approximated knowledge of the technical condition of the considered object, 2. the modeling of the operational potential consumption process is the issue when the adequate mathematical model does not exists and attempts of the development the model come across the problems arising from not enough clearly defined description of the process and the approximate character of the input data, 3. the fuzzy modeling was used to develop the system model mpu of changes of the operational potential included in a technical object, 470 m. pająk 4. the task of the fuzzy mpu model creation was decomposed into creation of ncf fuzzy models of the influence of the working parameters time histories on the changes of the values of the system features, where ncf is the amount of the cardinal features of the system, 5. the set of the working parameters can be very large; thus the set of the parameters of the highest influence on the change of the specified feature value should be defined, 6. during the operational tests of the op-650k-040 steam boiler the list of the most significant parameters in term of the operational potential changes was specified, 7. the fuzzy models of the influence of the working parameters time histories on the values of the features of the considered boiler were generated using the inputsoutput samples according to the iterative method of the automatic generation of the mamdani models, 8. the quality of the models operation was verified using the set of the major measurement collections recorded during the conducted operational tests, 9. analyzing the values of the measures of the models operation quality it was decided that the iterative method of the models generation is efficient enough to implement it into the conducted research, 10. considering the results of the simulation experiments performed it should be stressed that thanks to the implementation of the developed mpu model in the case of the selected technical system very high accuracy (mean error about 1%) of the process estimation was obtained. the major benefit that accrues from the described method application to the complex technical system is the development of a very accurate model of the operational potential consumption process (mean efficiency about 99%) according to the universal manner. its novelty consists in the universal system approach which can be used in the case of a wide range of the technical systems. prior to the application of the proposed method the operation parameters time histories should be registered. additionally, all cardinal features should be identified and their values at the beginning and the end of the operation parameters collecting period have to be accessible for measurement. these two elements are the main limitations of the described solution. the research on the problems of complex technical system operation and maintenance control will be continued. the first direction of future studies will cover the verification of the proposed method in the case of different technical systems. it is planned to construct the model of the operational potential consumption process for systems where the number of the cardinal features is significantly higher than four and to check the method usefulness in the case of the systems with a limited number of cardinal features (less than 4). the second issue is the application of the created model to control the complex technical systems operation 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http://citeseerx.ist.psu.edu/ viewdoc/download?doi=10.1.1.113.3817&rep=rep1&type=pdf [last access: 1 november 2017]. 43. polish agency of energy market, 2015, catalogue of power plants and chp plants, warsaw. 44. http://www.rafako.com.pl/products/boilers/668/pulverised-fuel-fired-boilers-with-steam-drum#type _548 [last access: 12 february 2019]. facta universitatis series: mechanical engineering vol. 16, no 3, 2018, pp. 389 403 https://doi.org/10.22190/fume171121020u © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper smart equipment design challenges for real-time feedback support in sport udc 528.835:796.012 anton umek, anton kos faculty of electrical engineering, university of ljubljana, slovenia abstract. smart equipment can support feedback in motor learning process. smart equipment with integrated sensors can be used as a standalone system or complemented with body-attached wearable sensors. our work focuses on real-time biofeedback system design, particularly on the application of a specific sensor selection. the main goal of our research is to prepare the technical conditions to prove efficiency and benefits of the realtime biofeedback when used in selected motion-learning processes. the most used wireless technologies that are used or are expected to be used in real-time biofeedback systems are listed. the tests performed on two prototypes, smart golf club and smart ski, show an appropriate sensor selection and feasibility of implementation of the real-time biofeedback concept in golf and skiing practice. we are confident that the concept can be expanded for use in other sports and rehabilitation. it has been learned that at this time none of the existing wireless technologies can satisfy all possible demands of different real-time biofeedback applications in sport. key words: smart equipment, motor learning, sport, biofeedback systems, sensors, actuators, wireless networks 1. introduction feedback is the most important concept for learning except practice itself [1]. during the practice, the natural (inherent) feedback information is provided internally through human sense organs. augmented feedback is provided by an external source, traditionally by instructors and trainers [2]. modern technical equipment can help both the performer and the instructor. in many sports disciplines video recording is a classical method for providing additional feedback information for post analysis and terminal feedback. modern technical equipment can provide more precise measurements of human kinetics received november 21, 2017 / accepted may 15, 2018 corresponding author: anton kos faculty of electrical engineering, university of ljubljana, tržaška c. 25, 1000 ljubljana, slovenia e-mail: anton.kos@fe.uni-lj.si 390 a. umek, a. kos and kinematics parameters and can considerably improve the quality of feedback information. modern optical motion tracking systems are using passive or active markers and a number of high-speed cameras. an alternative to optical tracking systems are inertial motion unit (imu) based systems, which use several wearable sensors attached to the human body. both types of motion tracking systems are professional and expensive equipment that can be used not only for biomechanics research in sports, rehabilitation and ergonomics but also as an animation tool in movie industry and virtual reality tracking. augmented feedback supported by technical equipment (sensors and actuators) is defined as biofeedback because a human is inside the feedback loop. the general architecture of a biofeedback system is presented in fig. 1. in a biofeedback system, a person has attached sensors that measure body functions and actions. the sensors are connected to a processing device for sensor signal and data analysis. the results are communicated back to the person through one of the human senses. the person attempts to act on the received information to change the body function or action in the desired manner. the term biofeedback was first described in connection with the human physiological processes and shortly afterwards in terms of the physical body activity in sports biomechanics. according to [3], biofeedback can be categorized into two main groups: physiological and biomechanical. in this paper, the word biofeedback concerns body and with body-related activity in the sense of physical movement; it is classified as biomechanical biofeedback in [3]. sensor(s) processing device actuator(s) user fig. 1 general architecture of a biofeedback system to achieve a widespread use of biofeedback applications, important feedback information concerning knowledge of performance should be provided with less complex and cheaper technical equipment. miniature imu sensors (accelerometers and gyroscopes) are integrated in every modern smartphone. consequently, many motion activity applications for smartphones and wearables using only accelerometer data exist. in many sport disciplines various types of equipment are essential or even indispensable for performing the desired task (tennis rackets, baseball bats, golf clubs, alpine skis, etc.). in fact, some of the most relevant human actions are transferred through the equipment. all this equipment can be supported by different types of sensors, not only accelerometers, gyroscopes and magnetometers. for example, strain sensors are the perfect choice to detect and measure force, torque, and bending in different parts of the equipment. an appropriate sensor fusion algorithm can give precise information on performers’ actions and equipment reactions. a smart equipment design challenges for real-time feedback support in sport 391 feedback from the sport equipment could therefore improve the performer’s skills, especially if it is provided in real-time, that is, without a significant delay. smart sport equipment can include any combination of sensor(s), processing device, and actuators(s) as defined in fig. 1. sport equipment manufacturers have already started embedding the digital technology into their products. some examples of smart sport equipment, which are already available on the global market, are: smart shoes, smart tennis racket, smart basketball, smart baseball bat, and smart golf club [4]. in the near future it is possible to predict many improvements in technology that could drive down the prices and encourage a widespread adoption of smart sport equipment. the real-world application of smart sport equipment as a part of biofeedback systems face several constraints that can represent higher or lower obstacles in their acceptance and use. the space constraint defines the possible locations of biofeedback system elements: (a) personal space system, where all system elements are attached to the user, (b) confined space system, where the elements are distributed within a defined and limited space, and (c) open space system, where elements are not restricted in space. the time constraint defines the timing of the feedback given by the biofeedback system, which can work only if the feedback loop is closed. that means that the user receives, understands, and possibly reacts to the feedback information. the feedback information can be given at different times: (a) terminal feedback is given after the activity has been performed; (b) concurrent feedback is given during the activity. feedback loop delay consists of communication delays for the transmission of sensor and feedback signals, processing delay, and user reaction delay. computation constraint is closely related and dependent on the space and time constraints as well as on the properties of sensors and actuators. processing in the biofeedback loop can be done in real time or in post processing. while the post processing mode does not represent a computational problem to the most of the processing devices and communication technologies, real time operation can many times be a difficult problem because the processing device has to finish the processing within the time frame of one sensor sampling period, which can be as low as 1 ms or even less. another important parameter is the communication delay within the biofeedback loop. this parameter is connected to all of the constraints studied above. communication delay is heavily dependent on the communication technology used. for the real-time biofeedback systems the communication delay must be a fraction of the reaction delay. 2. motivation in majority of research work in sport wearable sensors are used for the purpose of monitoring and for the post processing analysis of signals and data. the feedback information is given with delay after the performed activity, what is defined as terminal feedback. the same is true for the majority of sport applications already available on smartphones; post processing with presentation of some vital or important parameters. the concurrent feedback, which is given in real time within the currently performed action, is very useful for motor learning, but is rarely used. the primary aim of our research is the development of technical equipment that would allow implementation of real-time biomechanical biofeedback systems. we are convinced 392 a. umek, a. kos that such systems would allow a leap in research in this field. the important tasks in our research are the selection of appropriate sensors and the assurance of suitable conditions for sensor signal transmission and processing. as it is evident from research papers, imu sensors are the most often used ones in sports [5-12]. one promising direction of research is the use of smartphones in place of one or several elements of the biofeedback loop. smartphones can bring many important advantages, especially in bringing biofeedback systems closer to amateur users, who cannot afford expensive expert systems. smartphone properties and their suitability for biofeedback applications have been studied in [13-17]. the main focus of our current research is motor learning in sport with the help of feedback information provided by sensors integrated into sport equipment [18, 19]. the results of our research can be implemented also in the field of rehabilitation equipment as motor learning in rehabilitation is generally less demanding from that in sport. research efforts in smart sport equipment, many times coupled with sensors attached to the athlete, are present in many sports: from swimming [20], rowing [21, 22], kayak [23], canoe [24], and precision shooting [25], to golf and skiing that are presented and discussed later in this paper. one of the main research motivations of this paper is also the identification and selection of the most appropriate wireless communication technologies for various biofeedback applications that include smart sport equipment. to date, there is no one-fitsall solution to the above challenge. the sources of the data are sensors that are very heterogeneous in many aspects. they produce data rates from of a few bytes per minute for measuring patient’s temperature to a few mbit/s for a high-resolution and high-speed camera used in sport. sensors can be used for measuring physiological processes of a human (heart rate, glucose levels, blood saturation, etc.) to measuring performance of an athlete or sport equipment (high dynamic movement, high frequency vibrations, bending, strain, etc.). the same heterogeneity is expressed in the variety of sensor network technologies; from technologies that cover body area (ban) to technologies that cover metropolitan area (man), from technologies with bitrates of a few kbit/s to technologies of a few hundreds of mbit/s, from frequencies of 400 mhz to frequencies of 60 ghz, etc. it is obvious that the heterogeneity of sensors, networking technologies and application demands will yield the heterogeneity of the most appropriate solutions. we would like to emphasize that this paper represents an interdisciplinary research in the fields of communication technologies, sensors, feedback systems, and sports. during our study and experiments in the abovementioned fields we have come across important obstacles and challenges that have not yet been addressed properly. the main contributions of our paper are: (a) a systematic approach to the design of smart equipment that acts as a component of feedback systems in sport, (b) setting a guideline for the selection of the most appropriate combination of sensors, actuators and wireless technologies for different variants of real-time feedback systems in sport. to the best of our knowledge, there are no research papers that jointly discuss the topics in this paper. smart equipment design challenges for real-time feedback support in sport 393 3. smart sport equipment sensors, actuators and wireless technologies the integration of sensors and actuators into sport equipment enables not only the acquisition of information about motion, static positions, and acting forces, but also the means of giving appropriate feedback information back to the user. using the sensor signal analysis and specific a priori knowledge, the validation of the movement correctness can be achieved and appropriate feedback information can be communicated within the biofeedback loop. this procedure was tested in practical experiments in the field of training in golf and alpine skiing that is presented in section 4. 3.1. sensors and actuators sensors and actuators used in sports are heterogeneous in their properties. they can be grouped based on different criteria, such as measured quantity, bit rate, sampling rate, accuracy, precision, and similar [26-28]. some of the most popular sensors and actuators used in sport are presented in table 1. table 1 sensors and actuators used in sport sensor/actuator bit rate delay temperature < 100 bit/s not critical heart rate < 100 bit/s seconds ecg 20-100 kbit/s < 1 s accelerometer 1-200 kbit/s < 50 ms gyroscope 1-200 kbit/s < 50 ms strain-gauge 1-50 kbit/s < 50 ms tactile actuator < 100 bit/s < 50 ms audio actuator <1 mbit/s < 50 ms video actuator < 10 mbit/s < 50 ms several sensor groups can be distinguished: (a) sensors for low or high dynamic physiological processes, (b) sensors for low dynamic movement activities and (c) sensors for high dynamic movement activities. the main parameters that correspond to abovementioned groups are: sampling frequency (from below 1 hz to a few khz), precision (from 8 to 16 bit), and produced bit rate (from less than 1 bit/s to tens of mbit/s). actuators show less variety in their type and output, but can vary as much as sensors in sampling frequency and bit rate; from a one-bit tactile actuator (buzzer) to a high-definition and high-speed video screen. by combining several sensors and actuators within one (real-time) biofeedback system, the required bit rates and delay constraints can be very high. the selection of the most appropriate wireless technology is of paramount importance for achieving high quality of service of biofeedback system operation. 3.2. wireless technologies like sensors and actuators, wireless technologies are also very heterogeneous in their properties. table 2 lists the most used wireless communication technologies that are in use today for providing the functionalities of ban, pan (personal area network), lan (local area network), and man. only the range and bit rate properties that are the most 394 a. umek, a. kos important for our discussion are listed. more details about the listed technologies can be found in [26]. the selection of the most appropriate wireless communication technology depends on the type and implementation of the biofeedback systems. the heterogeneity of wireless technologies and the variety of biofeedback system versions suggests the use of multi-radio concepts [26-28]. table 2 standardized wireless technologies with potential use in sport technology range bit rate bluetooth 10 100 m 1 3 mbit/s zigbee 10 100 m 20 250 kbit/s ieee 802.11n 70 m 600 mbit/s ieee 802.11ac 35 m 6.93 gbit/s ieee 802.11ah 1 km 40 mbit/s ieee 802.11af >1 km 1.8-26.7 mbit/s lorawan to 100 km 250 5470 bit/s 3.3. selection of technologies for real-time feedback system a large number of factors and constraints have to be considered when selecting the most appropriate elements of feedback systems in sport. this is particularly true for the feedback systems that operate in real time (time constraint) giving concurrent feedback to the user. for help with the selection process, we have composed fig. 2 that illustrates the available bitrates of wireless technologies from table 2 plotted against their ranges, bitrate ranges of various sensors and actuators from table 1, and space constraints of feedback systems defined in the introduction. computational constraint is not addressed in detail here because it can be controlled to a high degree by the system designer while the other two constraints are mostly the given properties of the feedback system. it should be noted that ieee 802.11ah, ieee 802.11af wireless technologies are trying to fill-in the gap for open space systems with kilometre ranges and bitrates in mbit/s, but at the time of writing they were not yet available in the market. a number of important conclusions for the design of real-time feedback system can be drawn from fig. 2. for example: (a) real-time feedback systems based on physiological parameters, such as temperature and heart rate, can be implemented in personal, confined, and open space by using lorawan wireless technology; (b) usage of audio and video actuators in personal space systems is supported by bluetooth, ieee 802.11n, ieee 802.11ac, ieee 802.11ah, and ieee 802.11af wireless technologies; in open space systems these actuators can be used to some extent by implementing ieee 802.11ah, ieee 802.11af wireless technologies; the problem is that the latter two technologies are standardized, but not yet available in the market; (c) inertial sensors (accelerometer and gyroscope) can be used in personal and confined space systems by using all listed wireless technologies, except lorawan, but that is always true only for one such sensor; if there are more inertial sensors in the feedback system, some of the above technologies can prove insufficient; (d) given the zigbee based system in confined space, sensors for low dynamic physiological processes and a limited number of inertial and strain-gauge sensors with low sample rates can be used. many similar useful conclusions can be made based on information included in fig. 2. smart equipment design challenges for real-time feedback support in sport 395 accelerometer / gyroscope strain-gauge temperature heart rate tactile ecg video audio ieee 802.11ac ieee 802.11n ieee 802.11ah lorawan range [m] bitrate [b/s] personal openconfined zigbee bluetooth ieee 802.11af 102 105 106 104 107 103 108 109 102102101 105104103 fig. 2 illustration of available bitrates of wireless technologies from table 2 plotted against their ranges, bitrate ranges of various sensors and actuators from table 1, and space constraints help with the selection of appropriate feedback system elements 4. smart sport equipment prototypes for the validation of the above presented concept, two original smart equipment prototypes have been developed: smart golf club and smart ski. the development of realtime biofeedback system usage concepts and procedures were started based on the first measurement results. they include a systematic approach by defining the useful short practice lessons, adapting the precision of real-time biofeedback system to the capabilities of the user (amateur vs. professional), the choice of correct feedback modality, and the appropriate amount of feedback information adapted to the limited perception capabilities of users during training. 396 a. umek, a. kos 4.1. smart golf club the smart golf club prototype includes: (a) two strain gage sensors, which measure the golf club shaft bend and (b) 3-axis mems accelerometer and 3-axis mems gyroscope, which measure acceleration and angular speed of the golf club. the latter two sensors are a part of the independent shimmer 3 imu equipped with bluetooth communication interface. strain gage sensor signals are acquired by the professional measurement system (national instruments corporation, austin, tx, usa) with ni crio 9063 base (667 mhz dual-core controller with fpga) with bridge amplifier module ni 9237. imu sensor signals are acquired by the labview™ application running on the laptop. shimmer 3 devices can reliably stream sensor data using the bluetooth up to sampling frequencies of 512 hz, which was used in our experiments. the accelerometer's dynamic range is up to ±16 g0 and the gyroscopes dynamic range is up to ±2000 deg/s. the precision of both is 16 bits per sample. in the experiments the shimmer 3 device is fixed to the club's shaft just below the grip as seen in fig. 3. sensor signals are synchronized and processed by the distributed labview™ application running on the laptop and crio platform. after streamed sensor signals are aligned by their impact samples, they are segmented into separate swings, each containing 1500 samples, with impact sample at index 1000. at the sampling frequency of 500 hz the duration of each swing is 3 s. in the graphs presented in this paper only swing signal samples between indexes 250 and 1000 or 1.5 s time frame are plotted. fig. 3 smart club prototype used in the field figure 4 shows the player signatures produced by the two strain gage sensors orthogonally placed on the shaft of the golf club. fig. 4(a) shows trajectories (signatures) of a perfectly performed straight swing of three different players. it can be seen that their smart equipment design challenges for real-time feedback support in sport 397 signatures are distinctively different. figs. 4(b) and 4(c) show the trajectories of different swing types of player 1 and player 3, respectively. it can be seen that the differences in trajectories of the same swing type of different players is greater than the difference in trajectories of the different swing types of the same player. figure 5 shows the sensor signals with marked points in time that correspond to the distinctive phases of the golf swing. sensor signals are acquired by two strain gage sensors (top graph), 3-axis accelerometer (middle graph), and 3-axis gyroscope (bottom graph). trajectories show high consistency of swings, repeatability, and precision of the measuring system. (a) (b) (c) fig. 4 smart golf club prototype includes a 2d bending sensor; its trajectories confirm high consistency of players swings. average trajectories (n=10) show (a) large differences between three different players’ signatures, and much smaller differences between the perfect swing and faulty swings of the same player (b) and (c) 398 a. umek, a. kos fig. 5 sensor signals and phases of the swing: (a) address, (b) takeaway, (c) backswing, (d) top of the backswing, (e) downswing, (f) impact. in the s train gage graph the red curves represent the response of the side-mounted sensor and the blue curves represent the response of the top-mounted sensor. in accelerometer graph red, blue, and green curves represent the acceleration in 1x, 1y, and 1z axis respectively. in the gyroscope graph red, blue, and green curves represent the angular velocity around 1x, 1y, and 1z axis, respectively. smart equipment design challenges for real-time feedback support in sport 399 4.2. smart ski smart ski prototype includes strain gage sensors for measuring the bend of the ski in several sections of the ski, several force sensors for measuring the force that the skier is applying to the ski, 3-axis accelerometer, and 3-axis gyroscope for measuring the motion. bend and force sensors are integrated into the ski, accelerometer and gyroscope are attached to the skier's torso. two additional shimmer 3 devices are attached to the legs of the skier. the prototype is shown in fig. 6; smart skis are shown on the left hand side and the fully equipped skier to the right hand side of the figure. fig. 6 smart ski prototype sensor signals are collected, synchronized and processed by the labview™ application running on the professional measurement system (national instruments corporation, austin, tx, usa) with ni crio 9063 base (667 mhz dual-core controller with fpga) with bridge amplifier module ni 9237 and analogue to digital converter module ni 9205. the sampling frequency of the system is 100 hz. the accelerometer's dynamic range is up to ±16 g0 and the gyroscopes dynamic range is up to ±2000 deg/s. the accelerometer and gyroscope are using the wi-fi connection (ieee 802.11) to stream data to the crio device. shimmer 3 devices are operating in the logging mode because of incompatibility of wireless technologies between the system elements; the laptop used in golf club prototype cannot be 400 a. umek, a. kos used during the skiing action. sensor signals from shimmer 3 device are synchronized with the rest of the sensors in post processing. laboratory tests for equipment operation testing, calibration and validation were followed by several snow tests in different weather and snow conditions and performed with different expert skiers, some ex world cup racers and some from the slovenian alpine demo team. test skiers performed various skiing tasks and techniques according to the predefined schedule. (a) (b) fig. 7 skiing experiments with smart ski prototype (a) and the corresponding signals and calculated plots (b) smart equipment design challenges for real-time feedback support in sport 401 for example, one of the tasks was to perform the carving turns by equally loading both skis. figure 7(a) shows the test skier during testing and fig. 7(b) the corresponding signals acquired during the test. figure 7(b), observing from top to bottom, is showing the following signals and calculated plots: a pair of signals shoving the flexing of the left and the right ski (narrow red and blue lines), total dynamic load applied to both skis (cyan line), a pair of plots showing the load on the left and the right ski edges (thick red and blue lines), relative load balance in the right-left direction (thick black line), and relative load balance in the front-rear direction (thick green line). the final goal of the research is the development of the user application with real-time biofeedback. the feedback can be given through different modalities. an example of a real-time visual feedback, projected onto the skier's goggles, is shown in fig. 8. the exemplary display includes the binary state indicators (carving) and sliders showing skiers current performance (outer/inner). fig. 8 real-time biofeedback application 5. conclusion biofeedback systems are important in motor learning in sports. given the heterogeneity of sensors, actuators, and wireless technologies, countless scenarios of their use in biofeedback systems in sport are possible. prototypes of a smart golf club and smart ski have been designed and the most important results of experimenting with both prototypes have been presented. with the smart golf club prototype it has been shown that the differences in trajectories of the same swing type of different players is greater than the difference in trajectories of the different swing types of the same player. the action of the skier and the reaction of the skis and terrain at the same time can be precisely and timely measured with the smart ski prototype. the acquired information from the integrated sensors is used in the 402 a. umek, a. kos testing of ski performance and for ski technique improvement or learning. the developed application allows the ski expert to analyze the performance of the skier based on several measured and calculated parameters. the application is currently capable of recognizing different phases of carving technique and diagnoses typical errors in regard to the load distribution during the steering phase of the turn. the development of real-time biofeedback system usage concepts and procedures has been started based on the first measurement results. concepts and procedures include a systematic approach by defining the useful short practice lessons, adapting the precision of real-time biofeedback system to the capabilities of the user (amateur vs. professional), the choice of correct feedback modality, and the appropriate amount of feedback information adapted to the limited perception capabilities of users during training. with the development of real-time biofeedback systems we have been challenged by communication technologies limitations and limitations of the of-the-shelf sensor devices. it is important to accept that at this time none of the existing wireless technologies can satisfy all possible demands of different real-time biofeedback application scenarios. the main limitation here is the required real-time operation of the biofeedback system that at the same time requires a high bit rate, low delay, and long range. this can be a problem with highly dynamical human motion tracking and also with sport equipment that is usually used for enhancing or extending human motion, such as baseball bat, golf club, skis, ice hockey stick, and many others. acknowledgements: this work was supported in part by the slovenian research agency within the research program algorithms and optimization methods in 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survey and outlook, ieee communications magazine, 47(12), pp. 84-93. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 189 204 https://doi.org/10.22190/fume200421022p © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a center manifold theory-based approach to the stability analysis of state feedback takagi-sugeno-kang fuzzy control systems radu-emil precup 1 , stefan preitl 1 , emil m. petriu 2 , raul-cristian roman 1 , claudia-adina bojan-dragos 1 , elena-lorena hedrea 1 , alexandra-iulia szedlak-stinean 1 1 politehnica university of timisoara, dept. automation and applied informatics, romania 2 university of ottawa, school of electrical engineering and computer science, canada abstract. the aim of this paper is to propose a stability analysis approach based on the application of the center manifold theory and applied to state feedback takagisugeno-kang fuzzy control systems. the approach is built upon a similar approach developed for mamdani fuzzy controllers. it starts with a linearized mathematical model of the process that is accepted to belong to the family of single input secondorder nonlinear systems which are linear with respect to the control signal. in addition, smooth right-hand terms of the state-space equations that model the processes are assumed. the paper includes the validation of the approach by application to stable state feedback takagi-sugeno-kang fuzzy control system for the position control of an electro-hydraulic servo-system. key words: center manifold theory, electro-hydraulic servo-systems, stability analysis, state feedback takagi-sugeno-kang fuzzy control systems 1. introduction the systematic design and tuning of the fuzzy control systems is supported by analyses that include stability, controllability, observability, sensitivity and robustness. one of the main classical and modern topics in this regard is the stability analysis of the fuzzy control systems, which enable their stable design in the context of the model-based fuzzy control system design. received april 21, 2020 / accepted june 30, 2020 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania e-mail: radu.precup@aut.upt.ro 190 r.-e. precup, s. preitl, e. m. petriu, r.-c. roman, et al. the general approach to dealing with the stability analysis in the model-based fuzzy control, treated in the main woks [1-3], is to make use of takagi-sugeno-kang fuzzy models of the process and express the stability analysis conditions as linear matrix inequalities (lmis) in terms of the parallel distributed compensation (pdc) approach, which states that the dynamics of each local subsystem in the rule consequents of the takagi-sugeno-kang fuzzy models of the process is controlled using the eigenvalue analysis [2, 3]. recent results on lmi-based stability analysis include the relaxation of stability conditions [4-11], the negative absolute eigenvalue approach [12] and the use of lyapunov-krasovskii functionals [13]. the main idea in relation with the pdc-based approach to the stability analysis and stable design of takagi-sugeno-kang fuzzy control systems based on lmis is an extensive use of quadratic lyapunov function candidates. the effect of various parameters of the fuzzy models are considered resulting in non-quadratic lyapunov function-based approaches as, for example, the membership-function-dependent analysis [14,15], non-quadratic stabilization of uncertain systems, exponential stability with guaranteed cost control [16], piecewise continuous and smooth functions, piecewise continuous exact fuzzy models, general polynomial approaches [17, 18], sum-of-squares-based polynomial membership functions [19-22], superstability conditions, integral structure based lyapunov functions [23], the subspace-based improved sector nonlinearity approach [24], the fractional intelligent approach, and interpolation function-based approaches [25]. the presented short literature survey indicates, as shown in [26] and [27], that the classical approach based on pdc to stabilize fuzzy control systems and the use lmis in the stability analysis may introduce computational burden, complexity and coupling of subsystems. therefore, different approaches to lmi-based ones are justified; such also popular approaches include  bilinear matrix inequalities [28, 29],  popov’s hyperstability theory [30-33],  the limit cycle-based approach [34-36],  the circle criterion [37-40];  the harmonic balance method [31], [41-44], and,  the center manifold theory [45]. many of these non-lmi-based approaches work only with mamdani fuzzy controllers and not with takagi-sugeno-kang ones. a part of the authors’ stability analysis approaches is mainly focused on mamdani fuzzy controllers, avoiding lmi formulation and solving; it is built around lasalle’s invariant set principle [46], barbashin-krasovskii theorem [47], the use of quadratic lyapunov function candidates formulated for discrete-time systems [48], and the popular lyapunov’s direct method [49, 50]. some of these approaches also work with takagi-sugeno-kang fuzzy controllers, but many stability analysis approaches are formulated for continuous-time systems and since the real-world implementation of fuzzy controllers is carried out as digital controllers, there is a real need to either develop stability analysis approaches formulated for discrete-time systems or to transfer the continuous-time approaches to discrete-time ones (which is not a simple task). several alternative approaches to fuzzy control have been recently developed. all of them require the systematic design assisted by appropriate stability analysis approaches. a representative alternative approach to the traditional fuzzy systems is represented by type-2 fuzzy systems, with the ability to better capture nonlinear mechanisms in dynamic a center manifold theory-based approach to the stability analysis of state feedback... 191 systems compared to the classical type-1 fuzzy systems. the additional parameters of these nonlinear systems offer increased flexibility (advantageous in case of optimal fuzzy control) but this is paid by their more complicated design and tuning if they are used as fuzzy models and controllers. some of the important applications of type-2 fuzzy systems to fuzzy control with stability analysis and nature-inspired algorithms that ensure their optimal tuning are reported in [14, 16, 33] and [51-54]. a special type of fuzzy systems with nonparametric vectorized antecedents has been proposed by angelov and yager [55, 56], and is referred to as anya. initially supported by the concept of granules, anya is based on data clouds, which are sets of previous data samples close to each other, and the membership degrees are computed using the relative data density of the current data with the existing cloud. the data clouds are used, similar to the classical data clusters, in partitioning the problem space to detect different operational (nonlinear) conditions in the context of systems modeling and identification. but data clouds do not require an explicit definition of the membership functions, and they do not have or require boundaries; therefore, they do not have specific shapes. several applications of anya systems to control and modeling are reported in [57-60]. anya systems can evolve their structure by adding new data clouds. this relates them to evolving fuzzy systems. as shown in [61], the concept of evolving fuzzy systems was coined by p. angelov back in 2001 and further developed in his later works [62-66]. the specific feature of these systems is the computation of the rule bases by a learning process, i.e. conducting continuous online rule base learning, with some recent results given in [67-69]. the stability analysis of systems based on anya and evolving fuzzy controllers is an important subject. as pointed out in [27], another alternative approach is represented by tensor product (tp)-based model transformation, as a numerical method capable of transforming the linear parameter-varying (lpv) dynamic models into parameter-varying weighted combination of parameter independent (constant) system models under the form of linear time-invariant (lti) systems. the tp models are originally polytopic structures, where lti systems are the vertex models of a convex hull of the model; they may be relatively far from any linearized operation points. in the case of tp models an lti system affects the whole operation domain, not just a local area as in case of fuzzy systems, but according to the weighting functions, which actually replace the membership functions of fuzzy systems. representative applications of tp-based model transformation to modeling and control are discussed in [70-76]. in contrast to model-based control, data-driven control or data-based control avoids the system (process) identification by constructing controllers directly from data without identifying a system model. that is the reason why data-driven control is also referred to as model-free control, i.e. model-free in controller tuning. stability analysis is also treated in the date-driven or data-based control, but it is difficult to carry out this analysis if process models are avoided. a useful discussion on model-based versus data-driven control, that inspires future research directions, is presented in [77]. unlike much of the existing work, as, for example [78-80], the fresh paper [81] does not make the a priori assumption of persistency of excitation on the input; instead, it studies necessary and sufficient conditions on the given data under which different analysis and control problems can be solved; thus it reveals situations in which a controller can be tuned from data even though unique system identification is impossible. 192 r.-e. precup, s. preitl, e. m. petriu, r.-c. roman, et al. the model-free tuning of fuzzy controllers is an alternative approach to the modelbased design of these controllers in order to benefit from the advantages of data-driven control and fuzzy control and to mitigate, if possible, their shortcomings. the indirect model-free tuning of fuzzy controllers has initially been proposed and applied in [82-84], and continued in [85-87] by structures that combine data-driven control and fuzzy control in order to incorporate model-free features in novel fuzzy control system structure. thus, steps forward towards direct model-free tuning of fuzzy controllers are currently taken. starting with the center manifold theory approach to the stability analysis of fuzzy control systems with mamdani fuzzy controllers suggested in [45], this paper highlights the center manifold theory approach as a version to stability analysis and next stable design of fuzzy controllers. in this regard the paper is focused on state feedback takagisugeno-kang fuzzy controllers. the presentation is focused on second-order input-affine nonlinear systems and the application is done on a state feedback controller for an electrohydraulic servo system. the paper treats the following topics: the process models are presented in the next section. section 3 describes the center manifold approach to stability analysis of fuzzy control systems. the application of the stability analysis approach to the state feedback position control of an electro-hydraulic servo system is carried out in section 4. the conclusions are pointed out in section 5. 2. process models the dynamics of the process is described by the state-space equation of an input-affine nonlinear dynamical system , )( upp bxfx  (1) where nt pnppp xxx  ] ... [ 21x is the state vector, b is an n1 dimensional column matrix of constant parameters, u is the control signal, nn :f is the process function and t indicates matrix transposition. the only constraint imposed to f is that it must be a smooth function. for the sake of simplicity, as shown in [45], the formulations given as follows will be particularized to second-order (n=2) input-affine nonlinear systems. therefore, eq. (1) is transformed into , ),( , ),( 22122 12111 ubxxfx ubxxfx ppp ppp     (2) where 2 21 :, ff are smooth scalar functions, and  const, 21 bb . a coordinate transformation is next applied, which depends on the values of b1 and b2, and the state-space equations in (2) become ,),( ),,( 2122 2111 uxxgx xxgx     (3) where 2 21 :, gg are smooth scalar functions, and the new (transformed) state variables are x1 and x2. a center manifold theory-based approach to the stability analysis of state feedback... 193 the simplified models given in (2) are justified because many processes can be transformed to such models by model order reduction. in addition, several models as those specific to sliding mode control and model-free control can be expressed so as to depend on the control error and its derivative as state variables and the rest of nonmodeled process dynamics plays the role of a disturbance term. since the free response is analyzed, i.e. the local asymptotical stability of the control system around the origin (0, 0) is analyzed, zero reference input is next considered. the following state feedback control law is applied: ),,(),( 212211212 xxhxkxkxxgu  (4) where  const, 21 kk are the linear state feedback gains and the nonlinear state feedback smooth function 2:h has the features .0)0,0(d ,0)0,0(  hh (5) using (5) in (3), the closed-loop state feedback control system will be characterized by the state equations ),,( ),,( 2122112 2111 xxhxkxkx xxgx     (6) which are next linearized at the origin (0, 0) leading to the linearized state equations . ,)0,0()0,0( 22112 2 2 1 1 1 1 1 xkxkx x x g x x g x          (7) in order to carry out the feedback stabilization, the state feedback fuzzy control is a particular case of the general problem defined in [88], it is required that [45], [89] ,0)0,0()0,0( 2 1 1 1       x g x g (8) in addition, for the sake of simplicity [45] ,12 k (9) and the higher order terms in the taylor series expansion of g1 and h must depend only on x1. 3. stability analysis approach using the assumptions presented in the previous section and expressing the taylor series expansions of g1 and h, the introduction of the coordinate transformation [45] , , 112 * 2 1 * 1 xkxx xx   (10) the center manifold is 194 r.-e. precup, s. preitl, e. m. petriu, r.-c. roman, et al. ).( * 1 * 2 xx  (11) computing a second-order approximation of φ according to [89] in terms of (10), the local asymptotic stability conditions are [45] ,0)0,0()0,0()0,0( ,0)0,0()0,0( 21 2 21 1 2 2 1 1 2 21 1 2 12 1 1 2               xx h xx g x g xx g k x g (12) where the conditions in (12) hold for .0)0,0( 2 1 1 2    x g (13) equations (12) and (13) are next processed resulting in the stability condition [45] .0)0,0()0,0( 21 2 12 1 1 2            xx h k x g (14) the stability analysis approach is formulated accepting nonlinear input-output static map f of the fuzzy controller ),(ifu  (15) where i is the input vector , 2 1              x ti r i i (16) with t  a 23-dimensional constant transformation matrix that is not necessarily set and r  the reference input. the stability analysis approach consists of the following steps also given in [45] but with a different step 2: step 1 the necessary coordinate transformations are done. step 2 input-output static map f of the fuzzy controller is approximated to achieve the control law in eq. (4). two approaches can be used in this regard:  approximation of triangular and trapezoidal membership functions with exponential functions [90] or using approximation approaches transferred from pi and pid-fuzzy controllers,  least-squares fitting by the proper definition of an optimization problem and its solving by classical [91-94] or nature-inspired [95-98] optimization algorithms. step 3 the stability conditions (8), (9), (13) and (14) are checked. their fulfillment guarantees the local asymptotic stability. a center manifold theory-based approach to the stability analysis of state feedback... 195 4. validation on electro-hydraulic servo system position control the validation of the stability analysis approach given in the previous section is done, as in [45], in terms of the state feedback control system structure applied to the position control of an electro-hydraulic servo system (ehs) with the block diagram presented in fig. 1. the blocks and variables in fig. 1 are [99]: nl 1 … nl 5 – technological nonlinearities, ehs – electro-hydraulic converter, svd – the slide-valve distributor, msm – main servo motor, m 1 and m 2 – measuring instrumentation, u – control signal, y – controlled output, x1 and x2 – state variables, x1m and x2m – measured state variables. the values of all parameters of ehs are [99], [100]: ul = 10 v, g0 = 0.0625 mm/v, ε2 = 0.02 mm, ε4 = 0.2 mm, x1l = 21.8 mm, yl = 210 mm, ti1 = 0.001872 s, ti2 = 0.0756 s, km1 = 0.2 v/mm, km2 = 0.032 v/mm. the three steps of the stability analysis approach are applied as follows. fig. 1 block diagram of electro-hydraulic servo system viewed as controlled process [99] in step 1, due to very large parametric insensitivity and large linear domains of nl 1, nl 3 and nl 5, in the conditions of small variations of the variables, omitting the nonlinearities in fig. 1 leads to the simplified state equations of the process [45] , , 12 * 1 mm m xax ubx     (17) with the parameter values a = 14.05 and b * = 26.04. using xp1 = x1m and xp2 = x2m in eq. (17), a simple linear state transformation leads to the state-space equations of the process given in eq. (3), where ,0),( ,),( 2121 * 211  xxgxaxxg (18) with a * = a b * = 365.862. since y = x2m, the fuzzy control system structure is illustrated in fig. 2, where r – reference input, e – control error, i1 and i2 – fuzzy controller inputs also given in (16), and . , 22 11 m m xri xi   (19) 196 r.-e. precup, s. preitl, e. m. petriu, r.-c. roman, et al. fig. 2 state feedback fuzzy control system the block fc in fig. 2 is a takagi-sugeno-kang state feedback fuzzy controller, which is designed starting with a set of linear state feedback controllers to stabilize the simplified ehs model in (17) and next applying the modal equivalence principle [101] to merge the linear state feedback controllers placed in the rule consequents of fc. the disturbance input is not introduced in figs. 1 and 2 because, as specified in the previous sections, stabilization is targeted; reference tracking and disturbance rejection are not objectives of this paper. in step 2, as pointed out in [102], cosine-type membership functions for the linguistic terms of the input and also scheduling variables are considered to be convenient in the analysis of fc. such membership functions are set, and they are shown in fig. 3. fig. 3 points to the parameters of the input membership functions, b1 > 0 and b2 > 0. fig. 3 input membership functions [102] using eq. (19) with r = 0, the state-space equations of ehs as controlled process are . , 1 * 2 * 1 iai ubi     (20) in step 3, the local asymptotic stability is analyzed as follows with i1 and i2 instead of x1 and x2, respectively, on the basis of the results given in sections 2 and 3. the state feedback control law is expressed in terms of eq. (4). use is made of eq. (19), and the modified control law is [45] ),,( 212211 iihikiku  (21) where k1 = 1, and k2 is set to k2 = 1 + a * = 366.862 in order to fulfill the stability condition (8). the stability condition (14) is thus transformed into [45] ).0,0( 21 2 1 ii h k    (22) a center manifold theory-based approach to the stability analysis of state feedback... 197 the stability condition (22) is related to relatively small modifications of the nonlinear part of the control law. that is the reason why function h has to be a smooth one to be produced by the fuzzy controller; therefore, the rule base of fc will be derived as follows in this regard. three linear state feedback controllers are first designed, and they will be next placed as local controllers in the rule consequents of fc. imposing the linear closed-loop control system poles ,)()()*( 2,1 iii jqpp  (23) with 0)( ip and 0)( iq , and the superscript (i) indicates the index of the linear state feedback controller (or the local controller), i = 1…3, expressed as ,2 )( 21 )( 1 )( ikiku iii  (24) the classical pole placement approach leads to local controller gains )( 1 ik and )( 2 ik .3...1 , )()( , 2 ** 2)(2)( )( 2* )( )( 1    i ba qp k b p k ii i i i (25) the first local controller, i.e. i=1 in (24), is that situated at the biggest distance to the zero control signal line 02211  ikiku (26) in the plane <i1, i2>. this local linear state feedback control system is imposed to be the most oscillatory but also fastest one in terms of ,32 ,2 )1()1( pqpp  (27) where p > 0 is a design parameter that affects the dynamics of the fuzzy control system. the second local controller, i.e. i=2 in eq. (24), is that situated at the average distance to the zero control signal line in (26). this local linear state feedback control system is also imposed to be oscillatory but with a smaller overshoot and larger settling time by imposing . , )2()2( pqpp  (28) the third local controller, i.e. i=3 in (24), is that situated exactly on the zero control signal line in eq. (26). this local linear state feedback control system is imposed to be aperiodic with an average settling time compared to the first two local control systems because of imposing .0 ,2 )3()3(  qpp (29) using the information given before, the rule base of the takagi-sugeno-kang state feedback fuzzy controller is obtained by merging the three local linear state feedback controllers in (24) and placing them in the rule consequents. the rule base is presented in table 1. 198 r.-e. precup, s. preitl, e. m. petriu, r.-c. roman, et al. table 1 rule base of takagi-sugeno-kang fuzzy controller i1 n ze p i2 p u(3) u(2) u(1) ze u(2) u(3) u(2) n u(1) u(2) u(3) the fc block in fig. 2 operates on the basis of the sum and prod operators in the inference engine and the weighted average method for defuzzification. this fuzzy controller structure allows for the determination of nonlinear term h(i1, i2) in eq. (21). as done in [45] and [100] for the mamdani fuzzy control system, the following values of takagi-sugeno-kang fuzzy controller parameters ensure the system stability: b1 = 0.5, b2 = 1, p = 3.5. the stability tests of this state feedback takagi-sugeno-kang fuzzy control system were done by digital simulation considering eight different nonzero initial conditions and r = 0. the state trajectories illustrated in fig. 3 give an indication on the system stability. the regulation and tracking performance of the fuzzy control system were not analyzed; however, the optimal tuning can be carried out in this context, with the results that can be quite different for this application and other challenging ones as well [103-108]. fig. 4 state trajectories of state feedback takagi-sugeno-kang fuzzy control system considering r = 0, x1m (v), x2m (v) 5. conclusions starting with the authors’ application of the center manifold theory to mamdani fuzzy control systems, this paper suggests its application to state feedback takagi-sugeno-kang fuzzy control systems as well. a three-step stability analysis is included in the paper as the main result, and an electro-hydraulic servo system application is treated. the paper shows that the stability analysis is not complicated. however, several stability analysis conditions should be fulfilled. a center manifold theory-based approach to the stability analysis of state feedback... 199 the main limitation of the presentation is that it is focused on second-order inputaffine nonlinear systems. the application to higher order systems is a challenge due to the possible difficult computation of the partial derivatives of fuzzy controller input-output static map f. this is one of the directions of future research in a strong relation with inclusion of several advantageous features specific to fuzzy systems [109-115] so as to modify the control system structure. another direction of future research is the stable fuzzy control system design. that requires the derivation of connections between the fuzzy controller parameters and the control system performance indices. nevertheless, the transition of the approach from continuous-time systems to discretetime ones aiming the real-time implementation is also targeted. however, the analysis of systems poles is needed, which is not simple in the context of lpv dynamic system models while avoiding the use of lmis associated to pdc. acknowledgements: this work was supported by grants from the ministry of research and innovation, cncs uefiscdi, project numbers pn-iii-p1-1.1-pd-2016-0331 and pn-iii-p1-1.1pd-2016-0683, within pncdi iii, by the research grant gnac2018 arut, no. 1348/01.02.2019, financed by politehnica university of timisoara, and by the nserc of canada. references 1. tanaka, k., sugeno, m., 1992, stability analysis and design of fuzzy control systems, fuzzy sets and systems, 45(2), pp. 135-156. 2. wang, h.o., tanaka, k., griffin, m.f., 1996, an approach to 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https://orcid.org/0000-0001-7778-6097 dragan marinković https://orcid.org/0000-0002-3583-9434 žarko ćojbašić https://orcid.org/0000-0002-4581-1048 abstract. the topic of piezoelectric active thin-walled structures has attracted a great deal of attention over the previous couple of decades. lightweight structures with piezoelectric material based active elements, sensors and actuators, offer numerous advantages over their passive counterparts. this explains the motivation of authors to dedicate their work to this enticing research field. accurate and reliable numerical tools for modeling and simulation of this type of structures is still a hot topic in the research community. this paper offers an isogeometric finite element formulation for shell type of structures made of composite laminates including piezoelectric layers characterized by the electro-mechanical coupling. the shell kinematics is based on the mindlin-reissner assumptions, thus including the transverse shear effects. a few examples selected from the available literature are considered to demonstrate the applicability of the developed numerical tool and assess its performance. key words: isogeometric analysis, laminated structure, reissner-mindlin kinematics, shell, piezoelectricity, geometrically nonlinear analysis 1. introduction modern structures are made so as to be fit for their purpose. in order to meet this objective, they are designed to be lightweight. modern fiber-reinforced composite materials enable design and manufacturing of lightweight structures, while improving the structural performance in different areas – the structures are characterized by high stiffness-toweight ratio and become less expensive to operate with. additionally, further improvement received: january 05, 2023 / accepted april 02, 2023 corresponding author: predrag milić university of niš, faculty of mechanical engineering in niš, aleksandra medvedeva 14, 18000 niš, serbia, e-mail: predrag.milic@masfak.ni.ac.rs https://orcid.org/0000-0001-7778-6097 https://orcid.org/0000-0002-3583-9434 https://orcid.org/0000-0002-4581-1048 388 p. milić, d. marinković, ž. ćojbašić of structural performance is provided by incorporating active elements made of sophisticated multifunctional materials such as piezoelectric materials [1], shape memory alloys [2], etc. the purpose of adding the active elements is to make the structures adaptive, and such structures are also often popularly referred to as “smart”. in this way, it is aimed at an essential change in their behavior from being only a passive player in the deformational behavior to being an active player that adapts to the current conditions. the structures based on this new paradigm include elements for monitoring structural parameters (sensors), elements that process signals (controllers), and elements that act onto the structure so as to influence their behavior [3]. the application of this type of structures is quite broad, thus covering various fields, including energy harvesting [4], robotics [5], space technologies [6], automotive [7], etc. a design based on thin walls is the most natural way of producing a lightweight structure, particularly if a structure is made of a fiber-reinforced composite. this approach, on the other hand, may increase lateral flexibility, thus significantly affecting deformational behavior of those structures, especially their dynamic behavior being affected [8]. hence, the most common objective of adding the active elements is to positively influence the dynamic behavior of these structures with the aim of diminishing their vibrations and thus improve their safety and robustness, reduce noise emission, improve their life span, etc. the finite element method is currently one of the most advanced methods in the field of structural analysis. in the beginning, the developed elements were aimed at purely mechanical problems. in recent decades, multidisciplinary approaches gained in significance and the developments needed to cover coupled-field problems. the previous two decades have seen numerous developments of robust, accurate and efficient formulations and finite elements for modeling piezoelectric structures characterized by electromechanical coupled field effects [9, 10, 11, 12]. various 2d theories have been developed and applied to modeling and simulation of thinwalled structures [13]. while the classical laminate theory is appealing due to less nodal degrees of freedom required, the higher order of continuity of the shape functions is the major obstacle in its implementation into the fem. despite of that, the issue was addressed in a discretized form by some authors [14]. most of the authors turned their attention to the firstorder shear deformation theory, which is due to both, the better suitability of the theory for laminated structures (prone to the transverse shear effects) and the required c0-continuity of the shape functions [15, 16]. taking an improved accuracy and a more consistent description of the transverse shear effects as the main objectives, some authors based their developments on higher-order shear deformation theories [17, 18]. the numerical effort associated with these theories, which is not far away from the full 3d approach, is the obvious disadvantage. besides these efforts, the need to cover geometrically nonlinear effects was recognized recently and a number of developments were reported in this direction [19, 20]. recent papers were dedicated to the co-rotational fe formulation for both passive and active shell structures [21, 22]. another relatively recent and quite interesting approach is denoted as isogeometric analysis. it is originally proposed by hughes et al. [23] and aims at seamless integration of the cad geometry into the fe models. the most distinctive property of this approach is that the actual, i.e. cad geometry remains unchanged upon the fe discretization. development of isogeometric fe formulations relying on both the classical laminate theory based on the kirchhoff-love kinematics [24, 25], and the first-order shear deformation theory based on the mindlin-reissner kinematics [26], were reported. the same is valid for geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 389 higher-order theories [27]. isogeometric fe formulations for composite laminates were also reported [28, 29]. this paper aims at a geometrically nonlinear isogeometric formulation for shell type of structures made of composite laminates with embedded piezoelectric layers that may act both as actuators and sensors. it represents an extension of the recently reported isogeometric development for the same type of structures, but done for linear analysis [30]. 2. nurbs geometry, basic functions and mesh properties cad description of geometry and, therewith, the isogeometric formulation of finite elements is based on non-uniform rational b-splines – nurbs. nurbs is a special type of b-spline and, hence, its properties partly depend on the properties of b-spline. to determine the basis functions of the nurbs, it is necessary to define the basis functions of the b-spline first. a clear advantage of nurbs compared to a b-spline resides in its ability of accurate representation of complex geometric shapes. the basis functions of b-spline of the order zero are determined as: ( ) 1 ,0 1 0 otherwise i i in +  =       (1) a b-spline of the order p is defined based on the cox-de boor's recursive formula. for this purpose, a knot vector =[0, 1,..., n+p+1] is needed. higher order basic functions are defined as follows: 1 , , 1 1, 1 1 1 ( ) ( ) ( ) i pi i p i p i p i p i i p i n n n           + + − + − + + + + −− = + − − (2) a set of parametric coordinates i given in a non-decreasing order defines the knot vector. the continuity of the basis functions is of great importance and it is defined by the function order, p, and the knot multiplicity, k. a knot multiplicity equals to k implies cp-k continuity. in addition to this, ni,0() is a stepped function giving non-zero values only in a half-open interval [i, i+1). furthermore, ni,p() is given by a linear combination of two functions of degree (p-1). a basic function of the order p gives non-zero values only in a half-open interval [i, i+p+1) and the sum of all basic functions of the order p at any point  is equal to 1. finally, the basis functions are linearly independent and non-negative. a nurbs curve of the order p is defined by using the b-spline basis functions and the control polygon points with the corresponding weight coefficients or with control polygon points and their corresponding nurbs basis functions: , 0 , 0 , 0 ( ) ( ) ( ) ( ) n i p i i n i i p in i i p i i n w p c r p a b n w      = = = = =      (3) here, pi are the control polygon points, wi the corresponding weights, {ni,p()} the porder b-spline basic functions defined on the non-uniform knot vector ={a,...,a, p+1,..., 390 p. milić, d. marinković, ž. ćojbašić m-p-1, b,...,b}, while ri,p represent the basic rational functions of the order p that form nurbs. typically, the knot vector is normalized. this implies that the first index coordinate of the vector is 0 (a = 0), while the last one is 1 (b = 1). the number of repetitions of the elements a and b in the knot vector defines the order of spline. using this parameter, one cab realize the discontinuity at the spline ends. hence, the following equation defines a nurbs surface of the orders p and q in the and -directions, respectively: , , 0 0 0 0 , , , 0 0 ( ) ( ) ( , ) ( , ) 0 , 1 ( ) ( ) n m i p j q ij ij n m i j ij ijn m i j k p l q k l k l n n w p s r p n n w           = = = = = = = =      (4) where rij are the rational b-spline basis functions of nurbs surface, i.e. the nurbs basis functions: , , , , 0 0 ( ) ( ) ( , ) 0 , 1 ( ) ( ) i p j q ij ij n m k p l q kl k l n n w r n n w         = = =    (5) fig. 1 shows an example of a nurbs surface (patch) given by eq. (4). points of the control polygon in the physical space, index vectors in the index space and the basic functions order define the shown nurbs surface. the element boundaries are set in the parameter space. the surface belonging to an element is defined on half-open intervals [i, i+1) and [j, j+1) and they are recognizable after the mapping to the physical space (fig. 1). fig. 1 nurbs surface – mapping between the physical and parametric space the surface of an element is only a part of the whole patch. it is defined by the necessary mesh parameters, which include control polygon points, weight coefficients and the nurbs basis functions belonging to the half-open intervals [i, i+1) and [j, j+1). the nurbs basic functions belonging to the points of the patch control polygon outside the observed half-open interval yield a zero value. consequently, they geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 391 have no influence on this part of the geometry. in order to keep track of this fact, all the control points that define the element geometry and their basic functions are numbered from 1 to nen, with nen=(p+1)(q+1). the initial mesh of finite elements has a total number of finite elements determined by the number of elements in the index vectors and the degrees of the basic functions. it also implies that there is no repetition of elements in the index vectors except at the beginning and the end of the vector: ( )( )en n p m q= − − (6) here, m and n are the number of points of the control polygon along the and  direction, respectively, while p and q are the basic function degrees in the and  directions, respectively. there are several possibilities to change the initial mesh. knots may be inserted or removed from the knot vectors (h-refinement), the degree of basic functions may be changed (p-refinement) and the degree of basic functions may also be changed followed by insertion of new knots into the knot vectors (k-refinement). the degree of continuity of the basic functions across the elements may be affected by the last mentioned technique. 3. isogeometric shell formulation four coordinate systems are used in the formulation of the shell isogeometric finite element. those would be the global cartesian coordinate system (x, y, z), the natural coordinate system (r, s, t) with coordinate values -1<r, s, t<+1, the local cartesian coordinate system (x’, y’, z’), and the curvilinear coordinate system (, , ) (fig. 2). the tangent plane at any point of the surface is given by vectors 1t 'v and 2t' 'v determined as derivatives of the position vector r with respect to  and  ' 1 ' 2 t 1 ' t 1 ( , ) ( , ) en en n k k k n k k k rr v p rr v p         = =  = =    = =     (7) fig. 2 coordinate systems used in shell element formulation here, rk(,) are the basic functions and pk the control polygon points on the element surface. the resulting vectors, 1t 'v and 2t' 'v are not perpendicular to each other. in order to 392 p. milić, d. marinković, ž. ćojbašić generate an orthogonal coordinate system, the vector product of the obtained two vectors is computed to give a vector normal to the tangent plane. ' ' ' 1 2 ' t tz v v v=  (8) the unity vectors of the coordinate system are then easily determined: 2 1t t' ' 'z v v v=  (9) ' ' ' 1 2 '' '' '' ' ' ' 1 2 1 2 3 t t 1 2 3 t t 1 2 3 = , = , = , '' '' '' z'' '' '' x y z '' '' '' z l l l v v v e m e m e m v v v n n n             = = =                  (10) while the definition of the normal to the mid-surface is a trivial task in classical fem, the isogeometric formulation is somewhat challenging with respect to this aspect. the challenge originates from the fact that the control polygon points are not placed on the actual shell surface and they don’t have a unique projection onto the shell surface. the issue has been addressed in the recent work by milic et al. [30], where several methods from the literature are reported and the approach based on the greville points [31] is adopted. the same approach is also used in this work. a point on the structure that is not on the mid-surface is determined by the position vector of its projection onto the midsurface and with the distance in the direction perpendicular to the mid-surface. the thickness of the shell is determined interpolating the known thicknesses hk at the polygon control points by means of their corresponding nurbs functions. the position of any point is given as: 3 3 =1 3 ( , ) 2              = +                    en k k xn k k k k y k k k z x x v h y r y t v z z v   (11) here, xk, yk, zk represent the control point global coordinates, vk3x, vk3y, vk3z are the components of the vector normal to the mid-surface at control point pk. the same shape functions are applied to interpolate the displacement field and define the element geometry. hence: =1 en r k kn r k k k k r k k u u u v r v v w w w            = +                    (12) here, uk, vk and wk are the control point displacements in the x-, yand z directions respectively, while ur k, vr k and wr k denote the relative displacements caused by rotations at the control point k. the latter are to be expressed via the nodal rotations xk, yk and zk, which represent the global rotations. geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 393 the reissner-mindlin kinematics is applied to determine the relative displacement of any point within the shell with respect to its counterpart on the mid-surface, those are defined with respect to the local coordinate system at control point k, as follows: l 2 l 2 l 2 2 2                 = =                      r k xk x'kkr k k k k k yk k k y'kr k k zk u vh h v t -v ,v t -v ,v v w      (13) here x’k and y’k are the nodal rotations with respect to the x'and y'-axis fig. 4. material properties of composite laminates (fig. 3) are directionally dependent. this fact calls for the definition of the strain field with respect to the local coordinate system. this is commonly done so that a distinction is made between the in-plane components (membrane-flexural strains) and the out-of-plane components (transverse shear), as given below:       ' ε ε ' x'x' ' y'y' mf x'y' ' s y'z' x'z u' x' v' y' u' v' y' x' v' w' z' y' u' w' z' x'                               +       = − − = =      − −     − −             +            +     (14) fig. 3 laminated structure the derivatives of the local displacements with respect to the local coordinates are obtained from the derivatives of the global displacements in the global coordinate system by means of the transformation matrix: 394 p. milić, d. marinković, ž. ćojbašić l 2 l 2 3 3 , , , , , , , , , , , k,x ,y ,x ,x ,x ,x , ' , ,y' k ,y ,y ,y k k k,y ,y , , , ,z ,z ,zk,z ,z ,z u v w v u v w u v w v u v w v v v u v wvu v w                =                    (17) where, for instance, , , ,x u = ∂u/∂x and ,xu = ∂u/∂x. the derivatives are transformed from the natural to the global coordinate system using the jacobian inverse matrix:   , , , , , , =1 , , , 3 3 3 2 2 2 en k k k k k k n k k k k k k k t t t k k k k k x k k y k k z n n n x y z x y z n n n j x y z x y z x y z h h h n v n v n v                             = =                  (18)   , , , 1 , , , ,x ,x ,x ,y ,y ,y ,z ,z ,z ,t ,t ,t u v w u v w u v w j u v w u v w u v w       −         =            (19) upon all the mentioned transformations, the strain field given in eq. (16) has the following form:          0 1 mf tm rf t u s ts r s r s r b b | t b u ' = b u = u + b b | b +t b u                        =                                 (20) the element strain–displacement matrix, [bu] consists of the membrane-flexural, [bmf], and transverse shear, [bs], strain–displacement matrices, which further involve translations and rotations related parts (denoted by t and r in the subscript). 4. constitutive relations of piezolayers the form of piezoelectric constitutive equations depends on the choice of independent variables. within the framework of fem, the strain and the electric filed are the usual choice, thus yielding the matrix form of piezoelectric constitutive equations:                t e e = c e e d = e d e            (21) geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 395 here {σ} and {ε} are the mechanical stress and strain in the voigt notation, respectively, [ce ] is the symmetric material constitutive (hook’s) matrix, {d} and {e} are the dielectric displacement and the electric field vector, respectively, [e] is the piezoelectric coupling matrix, and [dε] is the vector of dielectric constants. for quite thin piezopatches, it has been shown that the electric field may be assumed to be constant over the thickness of the piezolayers [32]. hence, for the kth piezolayer: k k k e h  = (22) where k is the difference of electric potentials between the electrodes of the kth piezolayer and hk is the thickness of the piezolayer. the resulting electric field – electric potential matrix has a quite simple, diagonal form: 1 k b h       =        (23) 5. geometrically nonlinear finite element analysis linear structural analysis resides on the assumption that the displacements are relatively small compared to the dimensions of the modeled structure, while the material behavior is also linear. also, the boundary conditions must remain unchanged over the course of deformation. on the other hand, geometrically nonlinear analysis takes into account the change in structural configuration during the deformation. deformation of the structure leads to a change in structural parameters. considering the fact that that the structural deformation progresses continuously, the structural parameters also change are continuously. in each structural configuration, it is necessary to establish the equilibrium between the generalized internal and external forces. the solution of a nonlinear analysis is sought in a step-by-step approach. if the equilibrium of the system at time t is known, a solution at time t+t is sought with a suitably chosen time step t. even in a static analysis, a label t is established as an auxiliary variable that indicates the progress of the load level. a set of linearized fe equations for the piezoelectric continuum are given on the element level as follows:                 t e e e e t t t tt tt euu e u ext int t t t tt tt eu e ext int k u k f f k u k q q      + +     +  = −       +  = −    (24) here, {fexte}, {finte}, {qexte}and {qinte} are the external and internal mechanical and electrical loads, [kuut], [ku] and [k] are the tangential mechanical stiffness, piezoelectric coupling and dielectric stiffness matrices, respectively, while the vectors {ue} and { e } include the mechanical and electrical degrees of freedom of the element, respectively. all the terms are defined for time t, which is denoted by the left superscript, and  is used to denote the 396 p. milić, d. marinković, ž. ćojbašić increment of the corresponding quantity between two configurations. the external excitations are the applied forces/moments and electrical charges. their internal counterparts are obtained by integrating the mechanical stresses and dielectric displacements over the current configuration of the structure, respectively:       t t e t t t int lv f b dv=  (25)     t t e t t t int a q b d da  =   (26) the integration in eq. (25) runs over the whole structure, v, while in eq. (26) only over the surface of the piezopatches, a. updated lagrangian formulation is used in thgis work and, hence, all the vectors and matrices use the last determined configuration as a reference configuration. the tangential stiffness matrix is computed using the following equation:       t t t t t t t t t te uut u uv v k b c b dv g g dv = +             (27) where t[g] is the matrix of partial derivatives and t[] is the cauchy stress tensor:                       t 0 0 0 0 0 0 g g g g x y z g         =  =           (28)               0 0 0 0 0 0 xx xy xz xy yy yz xz yz zz                             =  =                  (29) the piezoelectric stiffness matrix, [ku], and the dielectric stiffness matrix, [k], are computed in the same manner as in the linear analysis, with the only difference that they are integration over the current volume of the structure: t t t t t v k b d b dv          = −       (30)   tt t t t tt u u uv k b e b dv k        = − =        (31) it should be emphasized that the piezo-electrically induced loads are configuration dependent. they are obtained by multiplying the piezoelectric coupling matrix, eq. (31), with the electric potential. the obtained nonlinear problem is resolved by means of the newton–raphson iterative procedure [33]. geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 397 6. numerical examples in this section, a few selected numerical examples of laminar piezo-electric shells under different load sets are considered. bothe the actuator and sensor role of piezoelectric layers are covered. 6.1. nonlinear analysis of a plate the in-plane dimension of considered plate are 4040 mm, while the thickness varies in the considered cases. the structure was discretized using the following finite elements: shell91 (ansys), acshell9 [34] and the developed formulation (nurbs). in its initial geometry, the structure is flat, but becomes curved after the first loading step. first, the results of linear and geometrically nonlinear analysis of the plate made of only one piezoelectric ceramic isotropic layer pzt g1195 (properties in table 1) with different types of finite elements are compared. the structure is exposed to an external force of 24 kn in all considered cases. in the first two cases, the force is distributed over a 40 mm long line (600 kn/m) and in the third case, it is a concentrated force (fig. 5). table 1 material properties [34] material properties pzt g1195 piezolayer t300/976 graphite/epoxy elastic properties y11 (gpa) 63 150 y22 (gpa) 63 9 12 0.3 0.3 g12 (gpa) 24.2 7.1 piezoelectric properties e31 (10-5 c/mm2) 2.286 0 e32 (10-5 c/mm2) 2.286 0 fig. 5 initial flat structures, boundary conditions and loads figure 5 shows the three models of the considered plate. due to the kinematic and dynamic boundary conditions, the maximal displacement in the first case would be larger than in the latter two for the same material properties and the thickness. the first model is a clamped plate with a thickness of 5 mm with a load at the free end. the thickness of the plate is chosen so that an obvious non-linear effect occurs. the results of linear and nonlinear analysis of this model are given in fig. 6. fig. 6a shows the vertical displacement of the free edge mid-point (point a), while fig. 6b gives its displacement in the x-direction. the latter result is equal to zero in the linear analysis. 398 p. milić, d. marinković, ž. ćojbašić fig. 6 clamped structure-displacement of free edge mid-point in: a) vertical direction, and b) in longitudinal (x-)direction the latter two cases deal with the structure simply supported over the opposite two edges and with the structure simply supported over all four edges. the position of the load was chosen adequately so that, in the first case, it is a line distributed force acting over the middle span width and, in the second case, it is a concentrated force acting at the structure mid-point. due to the boundary conditions, the structural behavior is significantly stiffer compared to the first case. in order to have geometrically nonlinear effects, a smaller thickness of 3 mm is chosen. in the second example, point b, i.e. the free edge mid-point is chosen as the point with the largest transverse displacement. the maximal transverse displacement of point b in linear and nonlinear analysis is given in fig. 7a. the linear result for the displacement in xand y-direction of any point of the plate is zero. however, the nonlinear analysis gives a non-zero displacement in the y-direction, as shown in fig. 7b. fig. 7 2-edge-simply supported structure displacement of free edge mid-point in: a) vertical direction and b) in longitudinal (y-)direction in the case of a structure with four simply supported edges, the mid-point of the plate (point c) was chosen as the reference, fig. 5c. due to the susceptibility to shear locking effect of the ansys shell91 element, the initial mesh (4x4) was corrected to 2020 elements to provide accuracy comparable with the acshell9 and nurbs finite elements, which use a 44 mesh. both, linear and nonlinear prediction yield only the transverse deflection of the observed point fig 8. geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 399 fig. 8 4-edges-simply supported structure: structure mid-point vertical displacement in all three considered cases the initial geometry is a plate and the applied loads are such that the linear prediction involves only flexural and shearing strains (no membrane strains). however, with progressing nonlinearity, the membrane strains are also induced as the configuration changes are taken into account. it is obvious and expected that the effect of nonlinearity is the least pronounced in the case of a clamped structure. the model with the nurbs elements in the previously shown examples has a very good agreement with the results obtained with the other considered elements. the first two plate models are further used to illustrate piezoelectric actuation of the structure in the realm of nonlinear deformations. the observed structures are considered to be made of three layers. the outer layers are made of pzt material (tab. 1) and the middle layer is made of graphite-epoxy material with a fiber orientation of 90 with respect to the global x-axis. given that the piezo layer has a relatively small actuation, the aforementioned models should be modified in order to produce a geometrically nonlinear effect. for this reason, a thickness of 0.15 mm is assigned to the piezo layers have and the thickness of 0.2 mm to the composite layer. the piezolayers are oppositely polarized, so that their actuation forms a moment equally distributed along the edges of the plate. a voltage of 300v is supplied to both piezolayers. the fiber orientation is chosen so that the structure has a lower bending stiffness about the y-axis than the bending stiffness about the x-axis. the induced piezo-electric coupling forces are configuration dependent. the new increment of the actuating bending moments is calculated based on the piezo-electric coupling stiffness matrix by performing its integration over the actual structural configuration. the direction and intensity of the induced loads depends on the current configuration (follower forces). the follower nature of the induced forces/moments demands small increments [33] of the electric voltage. an approximate approach is also used by treating the cases as purely mechanical with the loads that are not of the follower type. to achieve this, a full voltage of 300 v is applied to the initial structure and the bending moments are computed. the calculated moment is then applied in increments.the calculations done with the acshell9 and nurbs finite element yield small differences (in the order of 10-1 %). 400 p. milić, d. marinković, ž. ćojbašić fig. 9 initial geometry and boundary conditions the diagrams in fig. 10 are obtained by means of the acshell9 and nurbs diagrams using incremental voltages. the purely mechanical case with he predefined moments is coputed using the shell91 element. those results are also included in the diagrams. in the clamped structure model (fig. 9a), two points, a and b, on the structure are chosen to monitor their motion with respect to the gradually increasing excitation. figs. 10a and 10b depict the transverse deflection and the displacement in the x-direction of the point a. fig. 11 shows equivalent results to those given in fig. 10, but for point b of the structure. fig. 10 clamped piezoelectric plate, point a displacements: a) y-direction; b) x-direction fig. 11 clamped piezoelectric plate, point b displacements: a) y-direction; b) x-direction the previously discussed example reveals a small nonlinear effect on the displacements in the transverse deflection. fig. 12 gives the results for the transverse deflection in the case geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 401 shown in fig. 9b. obviously, the difference between the linear and nonlinear results is significantly larger in this case, which is the consequence of more pronounced nonlinear effect through the induced membrane strains and stresses. fig. 12 2-edges-simply supported piezoelectric structure: displacement of point c in the vertical direction observing the results of the examples shown, a negligible difference can be observed between the application of elements in which the voltage was applied incrementally (acshell9, nurbs) and elements, in which the precomputed moment was applied incrementally (ansys shell91). this should not lead to a general conclusion in the same direction. 6.2. nonlinear analysis of piezo-laminated semicircular arch in the next example, a composite cylinder arc was considered. the structure is formed from three layers. the middle passive layer is made of metal with a thickness of 5.842 mm (y=68.95 gpa, =0.3, =7750kg/m3). the outer layers are piezoelectric layers with a thickness of 0.254 mm (y=63 gpa, =0.3, =7600 kg/m3, e31=e32=16.11 c/m2, d33=1.6510-8 f/m). fig. 13 piezo-laminated semicircular arch 402 p. milić, d. marinković, ž. ćojbašić only one force f, the magnitude of which is 100 n, acts onto the structure in the middle of the tip of the free edge. the response of the structure in the nonlinear static analysis under the action of the force f was observed through the displacement of the tip of the free edge in the radial direction as well as in the circular direction. the acting force also induces electric voltages in the piezoelectric layers. the voltage of the inner piezoelectric layer is observed for the comparison purposes with the results from the literature. actually, tzou and ye [35] proposed the example originally. it was further modified by zhang [36], and marinkovic et al. [22]. zhang discretized the structure with 10 elements and using one element in the width direction. the model by marinkovic at al [22] model uses 160 triangle elements. the nurbs model was formed with quadratic basic functions, with 4 elements in the width direction and 20 elements in the hoop direction. the displacements are shown in fig. 14 and the induced electric potentials of the inner piezolayer are given in fig. 15. fig. 14 displacement of the free arc edge tip in the radial and hoop directions fig. 15 sensor voltage of the inner piezolayer geometrically nonlinear analysis of piezoelectric active laminated shells by means of isogeometric... 403 minor differences between the results for the induced electric potential in the inner piezolayer can be noticed in fig. 15. this can be attributed to different meshes applied and also the fact that zhang [36] used the rigorous geometrically nonlinear formulation, while marinkovic et al. [22] used a co-rotational formulation. 7. conclusions shell type of structures, i.e. thin-walled structures represent a natural way of achieving lightweight design. in addition to this, novel structural materials, such as fiberreinforced laminates with high stiffness-to-weight ratio, are used to produce even lighter structures with an option of tailoring material properties by a careful selection of the composite material composition, number and sequence of layers as well as the fiber orientations in the layers. this has been the state-of-the-art in many high performance structures for a few decades. a further improvement comes in the form of adaptive structures – structures provided with active elements, i.e. sensors and actuators, and which can proactively react to different conditions they are exposed to. in the specific case addressed in this paper, the active elements are in the form of thin piezoelectric layers embedded in the laminate structure. this paper proposed a novel numerical tool for modeling the mentioned type of structures. the finite element method is addressed as the most powerful method in the field of structural analysis. the development is based on a relatively novel isogeometric approach. the main advantage that it offers is a seamless integration of the cad geometry into the fe model, thus keeping the original, correct geometry of the structure. the proposed formulation covers the mechanical field in the whole structure as well as the electrical field and the piezoelectric coupling in the piezoelectric layers. thin-walled structures are prone to deformations characterized by large local rotations whereby the strains remain small. for that reason, the developed tool covers both linear and geometrically nonlinear analysis. for the mechanical field, an equivalent single layer approach based on the first-order shear deformation theory is applied. the considered cases are selected from the available literature in order to have reference solutions. they involve flat and curved structures. but it should be emphasized that although the considered structure is flat in the first set of examples, it is essentially dealt with a shell structure in the geometrically nonlinear analysis as the deformed structure is curved. the computations were done for both purely mechanical and electric excitations. also, the actuator and sensor function of active layers were covered. it was shown that the proposed formulation successfully covers all the mentioned features and represents a good match for the elements based on the classical fe formulation. in the future work, the performance of the developed isogeometric formulation with the mesh refinement should be addressed. also, the formulation should be extended to cover material nonlinearities in the piezoelectric layers that occur at higher voltages. acknowledgement: this research was financially supported by the ministry of science, technological development and innovation of the 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34(1), pp. 110–115. 36. zhang, s., 2014, nonlinear fe simulation and active vibration control of piezoelectric laminated thinwalled smart structures, phd thesis, fakultat fur maschinenwesen der rheinisch-westfalischen technischen hochschule aachen, germay, 210 p. 10856 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume220714047s © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a study of the use of mixed reality for capturing human observation and inferences in production environments milos stojkovic1, rajko turudija1, milan trifunovic1, marko pavlovic2, ivan jovanovic3, nenad uzelac4, vladeta milenkovic5 1university of nis, faculty of mechanical engineering, nis, serbia 2affine technologies, nis, serbia 3kuvag group, usa 4g&w electric, bolingbrook, usа 5netico-group, nis, serbia abstract. augmented and mixed reality is already considered as needful technology of the modern production systems. it is primarily employed to virtualize proper digital content, mainly related to 3d objects, into the human visual field allowing people to visualize and understand complex spatial shapes, their mutual relations, and positioning. yet, the huge potential of the technology is waiting to be revealed in its usage for collecting and recording human observations and inferences about the context of the production environment. its bi-directional interface makes it the most direct and the most efficient knowledge capturing means to date. the paper presents the challenges and benefits that come from the usage of a conceptual interface of an mixed reality application that is designed to collect data, semantics and knowledge about the production context directly from the man-in-process. as a production environment for the development, implementation, and testing of mixed reality applications for this purpose, various processes for the assembly and maintenance of medium-voltage equipment were used. key words: augmented reality, mixed reality, knowledge capturing, industry 4.0, human-to-machine communication received: july 14, 2022 / accepted december 05, 2022 corresponding author: miloš stojković university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, 18000 niš, serbia e-mail: milos.stojkovic@masfak.ni.ac.rs 2 m. stojkovic, r. turudija, m. trifunovic, et al. 1. introduction augmented reality (ar) and mixed reality (mr) earned the reputation of highly beneficial technology for the production systems of the industry 4.0 environment (i4e) standards [1]. the main benefit it provides to the users in the production environment is a fascinating virtualization of digital content into the human visual and haptic field (vhf) (fig. 1). by mixing the virtualized digital content with the real-world objects, a kind of mixed reality is being created, which provides a much more detailed visual, sonic and potentially tactile experience to the operator wearing ar/mr gear. such functionality is ideal for a series of applications in a production environment from visualization of training instructions [2, 3], through visualization of production operating guidelines [4], to effective visualization of measured magnitudes collected from the sensors of equipment and machines [5]. fig. 1 the space of visual and haptic field of an operator – vhf digital content being displayed in vhf can be data, i.e., alphanumeric or graphical content or 3d digital objects (fig. 2). usually, data are presented on virtual panels projected in the operator visual field covering a corresponding part of real space. in most cases these data are related to the context of vhf of a human equipped with ar/mr gear. when a 3d digital object is being displayed in vhf, mostly it is expected from an ar-app to "overlap" the real object with it (fig. 3 and fig. 4). in some specific applications (e.g., assembly), ar-apps can be designed to merely highlight the boundaries of an object to make it easier for a person wearing an ar/mr gear to spot the corresponding detail on a real object, such as highlighting a hard-to-see screw that should be tightened on a real assembly. of course, ar/mr apps can virtualize a 3d digital object with the data that are related to it in vhf (fig. 3). by developing the ar/mr solution for a series of applications like production, assembly, diagnostics, and maintenance of mediumand high-voltage equipment (figs. 2, 3, 4 and 5), it has been determined that ar/mr could be used not only to virtualize digital content into vhf, but also to collect information from the vhf by a human. even though this direction of information flow is not the primary application of ar/mr, it could provide a lot of benefits to the whole i4e production system. a study of the use of mixed reality for capturing human observation and inferences in production ... 3 fig. 2 digital content (alpha-numerical and graphical data and 3d objects) in vhf. (mrapp for assembling the high-voltage equipment. ms hololens was used as mr-app running hardware.) fig. 3 the virtualized high-voltage switchgear with the magnitudes that are being measured by its sensors. (ar-app for diagnosing the high-voltage equipment status in the field. iphone and android phones were used as ar-app running hardware.) machine-to-machine (m2m) communication, which is focal in i4e production systems [6, 7], however, cannot autonomously recognize semantics of the current production context features without big datasets [8, 9]. semantic categorization of complex datasets 4 m. stojkovic, r. turudija, m. trifunovic, et al. gathered from different sources in a production environment as well as the identification of causal-consequential relations between these data and corresponding phenomena and events that might be anticipated by their profound semantic interpretation seems to still be impossible without human intervention. this insufficiency sparked an idea to employ ar/mr technology as a tool for "capturing" the semantics of the process features and context. of course, a human is the one who can and should do the semantic interpretation and categorization of these data and relate them to the production context that is visible in the vhf (through ar/mr gear). this way, it is possible to facilitate and speed up an i4e information system (which is usually expected to be enriched by artificial intelligence algorithms such as machine learning) to learn about the production process. finally, a human's observations and inferences about the current production context that are being fed to an i4e information system through ar/mr gear and an ar/mr app are very useful information for production planners and managers. 1.1 technology challenges the application of ar/mr technology is usually followed by two classes of challenges. the first is related to precise "overlapping" of a digital virtual object with its real counterpart (object) in space, that is, in vhf of an operator wearing ar/mr gear (fig. 4). fig. 4 coinciding the 3d digital model of high-voltage cable termination with its real counterpart. (ar-app for remote assistance in handling the issues with high voltage cable termination. ipad pro™ was used as ar-app running hardware.) the second is related to the methods of capturing and recording human observations, actions, and reactions about the mixed reality context, which exists in an operator’s vh field. a study of the use of mixed reality for capturing human observation and inferences in production ... 5 the precise "overlapping" (of a virtual object with its real counterpart) is a specific challenge inherent to ar/mr primarily due to the following two reasons. firstly, because of the inconsistency of optical aberrations between the visual fields of a human and an ar/mr gear optical system. this complicates the transformation of a virtual object projection in the visual field of an operator significantly and results in inadequate matching of the perspective of an operator and ar/mr gear causing the distortion of virtual objects. secondly, because of impetuous head movements of an operator who wears ar gear, that is, due to fast changing gaze orientation. it is still difficult for an ar/mr app to follow these rapid movements smoothly, which results in the so-called drifting of the projection of a virtual object. sufficiently precise "overlapping" is important especially for the operations where precise marking of a target location is essential for successful or valid performance of the operation. a typical example is guiding the welder to place two parts correctly before welding, such as positioning a stud onto a proper dent at a steel plate. (fig. 5) fig. 5 guiding the welder towards proper dents in the sheet metal plate. (mr-app handwelding of studs onto tank sheet plates. ms hololens was used as mr-app running hardware.) the challenges related to capturing and recording human observations, actions, and reactions about the mixed-reality content could be considered within two aspects as well. 1. the first aspect is related to the technology of noticing and recognizing appropriate human gestures, voice commands and comments. the technology which is already applied in most of current ar/mr wearables has been proven capable of capturing the hand and fingers gestures and recording human voice [10, 11, 12]. 2. the second aspect is about the challenge of embedding the meaning (semantics) of a captured and recorded human observation. the main benefit that is expected from this capability is to facilitate acquisition of so-called tacit or not-formalized knowledge regarding the mixed-reality content and its context. in the case of a production process, tacit knowledge is related to the process or an operation context, which is usually gained through experience. it is to be expected that the formalization of this kind of knowledge or even just assigning (linking or associating) the right 6 m. stojkovic, r. turudija, m. trifunovic, et al. semantic categorization to the mr context and its constituents could improve the performance of ai algorithms engaged within an i4e information system. modeling the semantics of concepts, data, objects, and the whole context is a well-known research challenge that researchers have been trying to solve for a number of years [13, 14]. however, the use of ar/mr gear (i.e., an ar/mr app) seems to be very convenient for assigning semantics to the objects, data, and features of the mr context since human perception can be directly associated with any of the mr context constituents in a way that was not possible before ar/mr. and yet, it appears that this application of ar/mr has not attracted a lot of research interest until now. fig. 6 classification of main challenges for ar/mr application in a production environment 2. related work a comprehensive review about the usage of ar in the manufacturing industry was done by eleonora bottani and giuseppe vignali [15], who analyzed and reviewed the scientific literature spanning from 2006 to 2017. the survey only included studies written in the english language and published in peer-reviewed international journals (174 papers were included in the survey). the review showed that the interest in ar in the manufacturing industry grew over the years and peaked in 2016, since the number of papers was many times higher than in the years prior to 2013. the most popular areas of ar applications were assembly and maintenance, but closely followed by training/learning, with production design, safety, remote assistance, and robotics being less popular areas of research. we should not neglect the possibilities of applying ar for effective visualization and simulation of engineering analyzes of body deformations in real time [16, 17]. francesco de pace et al. [1] also provided an overview of ar application but in the industry domain, in which five major areas of ar application were depicted: human-robot collaboration, maintenance/assembly/repair, training, inspection and building monitoring. even though all these papers recognize the industry benefits a study of the use of mixed reality for capturing human observation and inferences in production ... 7 stemming from implementing ar technology, and even though all roughly agree on the main areas of ar use in the industry domain (maintenance, assembly, training/education, inspection, etc.), the potential for capturing human observations through ar was not mentioned. this is also the case with feng zhou et al. [18] and kangsoo kim et al. [19] review papers on the trends from the international symposium on mixed and augmented reality (ismar) from 1998 to 2007, and 2008 to 2017, respectively. the most obvious benefit of ar use in maintenance and assembly tasks is the replacement of maintenance guides and assembly instructions. these guides/instructions are often in paper form, which makes them quite difficult for the operator to use, as they need to shift their attention from the guide/instruction to the task at hand. therefore, overlaying the needed information at the right time through ar can make the operator’s job easier and faster. this was showcased in the studies by alessandro ceruti et al. [20] and dimitris mourtzis et al. [4]. in alessandro ceruti et al. [20], an ar application was used for aeronautics maintenance tasks. the case study that was carried out showed that just finding a needed maintenance task through the developed ar application was 27% faster than with the paper manual. dimitris mourtzis et al. [4] developed an ar application for visualizing cam instructions for a bending machine, instructions for the machine setup and safety instructions. riccardo masoni et al. [21] presented an example of a solution for remote maintenance where ar can be of great help. their ar tool uses symbols, free-hand sketches, and text to convey a required instruction from the remote expert to the local operator. roberto pierdicca et al. [22] developed and tested an ar application for training-on-the-job purposes for assembly tasks. main drawbacks of this ar application, as well as the majority of ar applications, are hardware insufficient processing power and poor ergonomics. nevertheless, combining ar with practice books that are traditionally used for learning in academic studies can be implemented relatively easy, and can yield great results. saša ćuković et al. [23] showcased this through an ar integrated cad practice book, which helped students with understanding shapes, manufacturing features, engineering drawings, etc., by superimposing virtual objects over specific illustrations in the practice book (every illustration has a square marker associated with some 3d cad model). ivo mal´y et al. [24] used ar technology for human-robot collaboration, and the paper focuses on experiments where the users are moving around а real or virtual object that is being augmented by the application. different visualization techniques were used and tested, as well as two different devices – ar glasses and a hand-held device. ashish doshi et al. [25] showed that not only handheld devices (phones, tablets, etc.) and vizors can be used, but also that projector-based ar systems can be of great help in industrial environments. a 52% reduction of standard deviations was calculated for spot-welding tasks with an ar system, compared to the same tasks done without an ar system. all of these applications take a relatively long time to build, so some research was also focused on improving the efficiency of creating vr applications (that could potentially also work with creating ar applications). flotynski et al. [26] used semantic modeling techniques and developed a new method of building and managing vr training scenarios. gorski [27] suggested using general rules of available knowledge engineering methodologies for building vr applications. going a step further than merely using ar to replace maintenance manuals, send expert instructions, visualize data or “overlap” the virtual onto the real world, ar systems can also be combined with machine sensors, machine learning technology, and other 8 m. stojkovic, r. turudija, m. trifunovic, et al. pillars of i4.0, to get great results. iñigo fernández del amo et al. [28] presented a framework for ar integration in whole maintenance systems, which means not only using ar for operator support, but also for analyzing their performance and recommending improvements. for this, ar applications need to be connected to sensors of the machines that are being maintained, because gathered sensor data can help an operator diagnose problems, but also assess if operators performed the diagnosis correctly. in addition, for performance valuation, gesture tracking was proposed. dimitris mourtzis et al. [29] developed a framework that enhances shop floor maintenance scheduling using ar. the ar application displays the status of a machine (remaining time between failures) gathered by a monitoring tool, and if needed, allows the operator to immediately call a remote expert or schedule the maintenance tasks for later, through the shop floor scheduling tool. fabio bruno et al. [30] proposed the use of ar for detecting and annotating design changes made during work, in addition to formalizing and automatic collection and transferring of data to the designers, so no information is lost. the proposed ar tool can also be used for assembly activities, as it is capable to overlay virtual objects on real ones. gang zhao et al. [31] used deep learning and knowledge modeling in combination with a mobile ar application to enhance the outdoor learning experience. essentially, the camera from a mobile device is used for obtaining a video stream, the category of the learning scene is than recognized, which is crucial for obtaining the virtual learning resource associated with the scene. then textual and audio instructions, pictures and 3d models are superimposed on the scene to give the user a good outdoor learning experience. if looking at a wider scope, in terms of capturing tacit knowledge, several papers come up. teegan l johnson et al. [32] emphasize why capturing tacit knowledge of industry operators is essential. this paper presented case studies in two manufacturing facilities for inspection tasks. the conclusion was that the use of tacit knowledge in some operations could range from 15%, up to 50%, indicating that losing of this knowledge could cause substantial decrease of performance in terms of productivity quality and safety. mikhail fominykh et al. [33] reviewed of ar and vr technologies used in, as they say, capturing “human experience”. however, the research primarily points out that ar and vr can be used for the purpose of recording physical places and human actions, which, later, can be experienced by someone elsewhere replaying the recorded content. adam dudek et al. [34] presented a model of tacit knowledge transformation that can be used in a manufacturing company. the knowledge transfer is being done through automatic speech recognition and natural language processing by transformation of operator’s observations given by the voice into digitally recorded comments. jason hashimoto and hyoung-june park [35] used smart device augmented reality (sdar) for exploring alternative fenestration design. by applying users’ tacit knowledge and intuition, they generated different virtual fenestration designs by adding/deleting openings, positioning them in different locations in a real space and modifying their size, and sdar helped in visualizing the effects of the design on interior day lighting on surfaces (walls, floors, and objects inside the room). all the papers reviewed above suggest that the use of ar/mr technology for capturing human observation has been, to the best of the authors’ knowledge, poorly explored. a study of the use of mixed reality for capturing human observation and inferences in production ... 9 3. the conceptual solutions 3.1 material, methods and means the feasibility of applying ar technology has been explored through several projects (2017-2019) for the client involved in production and maintenance of highand mediumvoltage equipment such as switchgears, reclosers, circuit brakers, cable couplings and terminations. as a result of the research, a few conceptual solutions of ar/mr-apps were developed and explored for various applications and ar/mr gear. specifically, the research was dedicated to the application cases given/represented in table 1. for all applications (table 1), 3d digital objects were virtualized in vhf and overlapped with its real counterpart. also, for all cases, it is possible to cancel the strict overlapping of the digital and real object and place the digital one in the most convenient area of vhf. this option has been proven as very ergonomic to the operator. the testing of the usage of the mr visor showed that permanently active overlapping reduces the visual cognitive performance of operators and makes it very difficult for them to notice all the necessary details on a real object while it is overlapped by its digital counterpart. table 1 ar application cases that were used for the research and experiments conducting application case ar/mr gear 1 diagnosing the equipment status in the field, which included checking the functionality of the high-voltage switchgear, gathering and visualizing the data about measured magnitudes of interest such as voltage, current, power, etc. phones (ios, android) 2 training the maintenance operators to handle equipment and perform maintenance operations (dry run simulations) tablet (ipad pro), vizor (ms hololens) 3 remote surveillance and assistance in equipment handling and performing the maintenance operations in the field phones (ios, android), tablet (ipad pro), vizor (ms hololens) 4 assembling equipment operations in the shopfloor vizor (ms hololens) 5 performing manufacturing (hand-welding) operations in the shopfloor vizor (ms hololens) 3.2 data stream-oriented interface features there are two data/information/knowledge streaming directions the first is from the information system towards the human, and the second one is from the human-in-process towards the information system (fig. 7). in accordance with it, the ar/mr app interface features have been designed in a way to enable data presenting and data collecting. 10 m. stojkovic, r. turudija, m. trifunovic, et al. fig. 7 two principal directions of data (information and knowledge) stream while using ar/mr 3.2.1 system to human data stream (s2h) concerning the digital content that is streamed to the operator from the process via cloud, aside from the virtualization of 3d digital objects in their vhf, the ar/mr app feeds the operator with the data relevant to the current process context and equipment which is in the operator's vhf. the data are gathered from the sensing equipment and processed in the cloud before being forwarded to the operator. this way, the data become meaningful and valuable information for the operator since they are related to the (process and) ongoing operation context. additionally, this data stream feeds the operator with the current operation execution time, comparing it with normed (standard) operating times (unit, preparatory-final, main, and auxiliary time). at the same time, by tracing the execution of operations in real time it is easy for the information system to interpolate operating times and provide the anticipation of the realization of operations to the operators and production management staff. 3.2.2 human to system data stream (h2s) the main contribution stemmed from the research is the original interface features built to allow operators to feed the system with their observations and inferences. in terms of collecting data from the operator about the process, the ar/mr interface was established predominantly through a series of prepared questionnaires to which the operator should provide appropriate responses, as in the case of a quality assurance check list (fig. 8 the image originates from the ar/mr app developed by the authors of the paper for g&w electric for the case of ar/mr-aided assembling the high-voltage cable joint. ms hololens was used as ar/mr-app running hardware.). a study of the use of mixed reality for capturing human observation and inferences in production ... 11 fig. 8 prepared questionnaire for the quality assurance checklist (the sequence where qa requires from an operator to take a photo as an argument that the operation is done correctly) the operator’s responses to the predefined questions enable the production management system to monitor process-related details in real time, immediately identify challenges and failures, and anticipate their consequences in terms of time, quality, productivity, safety and costs. at the same time, passing through predefined questionaries on the status of an operation completion and process interventions, the operator generates the relevant information about the process to other operators nearby. hence, ar/mr gear may be considered as an ideal platform for notification about the ongoing process status, integrating the kanban and andon notification systems in the vhf in an elegant and ergonomic manner. moreover, through the ar/mr app, notifications are selected regarding their relevancy for each specific process participant. besides a series of prepared questionnaires about (the part of) the process context, which are perceptible in the operator's vhf, a specific interface toolset that was built allows the operator to initiate an observation note, which can be corroborated by a video, photo or audio recording. what is even more important, this feature enables the operator to associate his observation to the object in vhf to which this observation refers (fig. 9). thus, in this way, the operator participates in the creation of additional semantic content that is related to the process' context. in terms of a perpetual effort to continuously improve the process (kaizen), self-initiated context-related observations, that the ar/mr app enables, are very valuable since this content is immediately visible and accessible to production managers and can be further analyzed, reviewed, and acted upon to improve the process. 12 m. stojkovic, r. turudija, m. trifunovic, et al. fig. 9 operator’s observation based on captured data from the sensing equipment can be formalized into an expert rule for reaction (mr-app for the case of assembling three phase recloser. ms hololens was used as mr-app running hardware.) 3.3 measuring the benefits and shortcoming as it was mentioned before, one of the greatest imperatives in i4e is to capture large data about an ongoing manufacturing process. the large data set enables an ai toolset, which is expectedly embedded and is running in the cloud computing service, to try to identify and semantically categorize data patterns that can empower the capacity of the information system to anticipate process realization and possible challenges. however, in real industry practice, it is usually not so easy for the ai toolset to learn causalconsequential relations between collected data and the corresponding process context. the possibility to instruct the ai toolset to interpret data patterns properly by an operator (human) in the process can shorten the learning process remarkably. for the simplest data pattern (built by 3 to 4 data sources), the ai toolset needs minimum of 102 occurrences to learn (be trained), i.e., to recognize the “possible” causal-consequential relation by itself. with an operator who uses an ar/mr app to introduce the ai toolset to a new data pattern that is associated to a specific process context, the ai toolset needs just one (first) or a few occurrences. 3.3.1 process improvement indicators the impact of the usage of ar/mr for capturing human observations and inferences about the process is measured by the following indicators: 1. speed of disseminating important information about the process (sod) between the operators and process management staff. this measure indicates how fast the relevant information reaches the process supervisors in-charge as well as the operators who should be informed (e.g., who operate preceding or succeeding sequences). the typical important information about the process that requires to be disseminated as soon as possible is the identified challenge/issue within the operation (regarding assembly, handling the equipment, maintenance, manufacturing). in comparison to the conventional information flow, which usually involves issue reporting (creating a a study of the use of mixed reality for capturing human observation and inferences in production ... 13 digital or paper file) and later reading/analysing these reports, ar/mr-aided reporting through prepared questionnaires makes dissemination immeasurable faster. the information reaches the ones who should be notified immediately instead of at least dozens of minutes or even several hours later, which is common in conventional reporting. one should also emphasize the high relevancy of the information for the designated process participant in the case of ar/mr-aided reporting. in conventional report files, as it was identified through the interviews with operators in the production and assembling processes (12) and management staff (4), usually only 2025% of the content concerns the designated participant, since the report content is aimed at several (4-5) participants in the process. the direct consequence of the improved speed of dissemination of important information about the process is the improved reactivity which is reflected in focusing on overcoming the process issue and finding a solution faster. the reactivity of those who are in-charge to react is incented additionally by scoring the reaction (by the time and effect of a solution that they built). the indirect consequences are improved quality that comes from reducing the repetition of operating mistakes (reduction of rework), improved productivity that comes from the developed/upgraded and formalized solutions and improved safety. 2. speed of learning the process (sol). the learning is related to both human participants (operators and process management staff) and the ai toolset. regarding the ai toolset, this measure indicates how fast the data processing algorithms, which are usually a kind of machine learning algorithms, will be able to correctly recognize and semantically categorize the data patterns from the process. the smaller the number of data-pattern cases needed for learning, the greater the speed of learning. it should also be mentioned that the time and expertise needed for preparing the data pattern for training should be taken into account as well. there are three approaches to training an artificial intelligence system. the first is supervised machine learning which requires at least one knowledge engineer to design predefined training data patters and expected outcomes. the second is to try to make the system learn by itself autonomously, passing through a two-step learning procedure. in the first step, the system should notice and extract the relevant features of real data patterns without the help of a knowledge engineer. then, in the second step, the system is expected to build the algorithm for semantic interpretation and categorization of real data patterns, again autonomously. the third approach suggests employing an ar/mrapp to allow an operator to directly interpret and categorize the semantics of real process data patterns as they appear. the third approach does not need time and effort of a knowledge engineer to prepare training data patterns, but it still requires a kind of self-initiated engagement of an operator to “build the case” by indicating distinctive features of the process context and outcome, which come or can come out of this context. the research involved the first and the third approach because it was assumed that the second approach could not be sufficiently efficient (comparable with the first and the third). additionally, due to safety reasons and inference traceability poorness, the second approach could not be applied, that is, it was not possible to leave the artificial intelligence system to build inferences completely on its own. 14 m. stojkovic, r. turudija, m. trifunovic, et al. 3.3.2 measuring and results testing the use of ar/mr and its effect on improving operational performance was carried out through a series of experiments in several sectors already denoted in table 1. to measure the “speed of disseminating the important information about the process” (sod) objectively, it is important to identify four discrete activities that are involved in the process of dissemination itself (fig. 10). fig. 10 dissemination route of the operator’s observations through the system the first activity is about generating the operator's observation. within this activity the ar/mr app interface plays a crucial role because the efficiency of generating the observation is related to the intuitiveness of the interface. however, the solely intelligent sub-process of this activity is about the categorization regarding the importance of an occurrence that will be noticed, and it has to be done by the operator in the process through the ar/mr app. the second activity is about sorting out the observations, i.e., identifying the target person(s) to whom an operator's observation should be disseminated (the one or more who supervise and coordinate that part of the process). the cloud computing service does this activity in accordance with the predefined observation's tags that are generated by the ar/mr app (and the operator) regarding the process context. the third activity is about presenting the operator’s observation to the supervisors (who are called to react). in the case of experiments, that was done by using a pop up notice through the already existing process supervision application (sending it to the supervisors via computer or phone). the fourth step is about inducing reaction from the “person-incharge”. this step depends on the incentive procedure that is adopted by the company. in the case of experiments, this was the “descending reaction value through time” procedure which induces supervisors to react as soon as they are able in order to earn high values for the reactivity indicator at the end of the control period. in measuring the impact of ar on sod, only the first activity is relevant. the second is done autonomously and instantly by a study of the use of mixed reality for capturing human observation and inferences in production ... 15 the computer, and the third and fourth are the activities that (can) also exist without using ar/mr. so, the time lapses from generating the observation to noticing it by the process supervisor or coordinator was measured and compared (the conventional approach of typing the file versus the approach featured in the ar/mr usage). the measurements are given in table 2. table 2 features and results of the conducted tests # process ar gear test conditions no nt sod1 (collateral, indirect indicators) q1 p1 p2 conv ar con ar/mr con ar/mr con ar/mr 1 diagnosing the equipment status phone lab. env. 4 >30 n/a n/a 1* 0.5 1 0.25 n/a n/a phone real. env. 6 >100 n/a n/a 1 0.5 1 0.25 n/a n/a 2 training handling and maintenance op. vizor lab. env. 6 >50 n/a n/a 1 0.04 1 0.55 n/a n/a tablet real. env. 4 >20 n/a n/a 1 0.05 1 0.7 n/a n/a 3 remote assistance in equip. handling and maintenance in the field vizor lab. env. 3 >50 1 0.2 1 0.4 1 0.75 1 0.67 phone real. env. 4 >100 1 0.1 1 0.5 1 0.67 1 0.8 tablet real. env. 4 >50 1 0.12 1 0.5 1 0.6 1 0.75 4 assembling the recloser vizor lab. env. 4 >50 1 0.12 1 0.4 1 0.5 1 0.64 5 hand-welding studs vizor lab. env. 2 >30 1 0.2 1 0.85 1 0.75 1 0.9 * all values are normalized to avoid publication of real data. for example, if for some activity conventional approach requires 10 repetitions it is normalized as 1, and if the same activity done with ar/mr requires 2 repetitions it is normalized as 0.2. the legend of table 2 is given below: no: number of operators participating in tests. nt: number of conducted tests. sod1: speed of disseminating time to generate and deliver an observation for a test case by the conventional channel (typing/filling the qa file or similar) and the ar app. collateral benefits that demonstrate the effect of applying ar to disseminate observations of the operator in the process can be seen indirectly through the following operation performance indicators. q1: 1st quality performance indicator number of rework operations within test cases that were counted when operators’ observations were recorded and disseminated through the conventional channel (typing/filling the qa file) and through the ar app. p1: 1st productivity performance indicator operation preparatory time within test cases measured when operators’ observations are recorded and disseminated through the conventional channel and through the ar app. p2: 2nd productivity performance indicator – operation performing delay caused by not getting the info about the process disturbance promptly. 16 m. stojkovic, r. turudija, m. trifunovic, et al. measurements presented in the paper were done mainly in a laboratory environment, but some were done in a real ambient. laboratory measurement involves a predefined scenario regarding test operation and disturbance within a controlled ambient (in special workshop test areas). measuring the speed of learning (sol) was done in the laboratory environment only, and for the cases of 1) recloser assembly and 2) remote assistance in equipment handling and maintenance in the field. the test scenario comprehended feeding the ai toolset with five data inputs, three analogue and two digital signals, whose combination values builds a specific process context, i.e., its features that, if they are semantically interpreted correctly, anticipate potential issues regarding quality and productivity that should be expected to appear. as an alternative solution to this learning procedure, a series of micro-services running within the ar app was designed to allow the operator to formalize knowledge about the process gained through the experience into expert rules which can also anticipate the same potential issues. the speed of learning about the process was measured by the effort invested to enable the ai toolset to semantically interpret the process context features correctly. these two possible routes of teaching the information system about the process are shown in fig. 11 – the example is about recloser assembly. it is about recloser assembly process. fig. 11 two possible routes for teaching the system about the process there are three analog inputs: a1: plunger body displacement, a2: handle angular position, a3: locking spring compression, and two digital signals: d1: status of recloser mechanism position (opened/closed), d2: vacuum interrupter contact (on/off). a certain set of values of these inputs indicates possible issues to be expected: a study of the use of mixed reality for capturing human observation and inferences in production ... 17 1. regarding quality – premature failure of vacuum interrupter should be expected due to fatigue since the assembly is overtightened. 2. regarding productivity – prolonged installation and adjusting of the recloser in the field should be expected since the assembly is adjusted improperly. 3. regarding productivity – production delay should be expected due to the necessity to substitute the middle subassemblies since they are arriving in the final assembly operation improperly adjusted earlier in the process. to make the ai toolset capable of anticipating issues based on an input data set, a knowledge engineer needs to feed the ai toolset with numerous samples of the input data set. it becomes even harder for learning if it is necessary to learn the ai toolset to consider what value appears first and what later because the timeline of values appearing can improve the distinction of the process context and anticipation of potential issues that can be caused (for this research such cases were not included). unlike a knowledge engineer, who must first learn the process context and its features by himself, and then prepare a suitable set of samples for teaching the ai toolset, a process operator who uses mr gear performs implicit semantic categorization and interpretation of process context features in real time, creating the appropriate expert rules (potential reaction). table 3 shows the investment in learning of the ai toolset to be able to recognize the context of the process (based on data patterns) that indicates the forthcoming occurrence. the displayed values are obtained through testing in the lab environment. table 3 efficiency of learning ai toolset about the process – comparison between two learning routes: mr aided learning and conventional ml approach. number of data pattern cases (based on 3 analogue and 2 digital data sources) forthcoming occurrence with ar/mr conv. ml approach quality decline or an accident <10 >102 operation productivity decline <10 >102 operation performing delay <10 >102 it is important to also notice that the results showed in the table do not consider time and costs investment for training the ai toolset by engaging a proper knowledge engineer (which, inevitably, steals the time from the operators in the process and makes even more waste). finally, it should not be neglected that engaging a knowledge engineer to teach the system about the process is a one-time job, and the ar/mr-aided learning is a tool for permanent learning. it is questionable whether it is possible to predict all the combinations of data patterns which can appear in real practice at once. one can argue that the main advantage of ar/mr -aided process learning comes from the ability to narrow the context envelope (vhf) and exclude practically impossible datapattern cases in a few steps. additionally, operators with ar/mr gear impose their inference, that is, semantic categorization of the current process context, generating a kind of expert rule in the form of an if-then logical relation between an input data pattern and consequences (fig 9). in that way, the operator formalizes their tacit knowledge regarding the current process context and embeds it into the information system. 18 m. stojkovic, r. turudija, m. trifunovic, et al. from the perspective of human participants, especially process management staff, an ar/mr-aided bottom-up tacit knowledge stream speeds up the learning about the process, also. this data flow helps with getting more profound knowledge about the process features directly from the shopfloor operators, which is usually missing. however, for this research, we did not measure neither compare the learning speed between operators when the ar/mr -app was used and when it was not. an indirect consequence of the ai-toolset (learned) ability to categorize the process context accurately (relevantly) based on data patterns is its functionality to anticipate forthcoming occurrences fully autonomously and to react preventively, at least by sending a warning message to the participants in charge. finally, these reactions of the ai toolset force the participants in the process to focus on finding the corresponding solutions that should prevent a decline in quality, productivity or extending deadlines for the given cases. as the collateral gains derived from data collecting by the ar/mr -app are: 1. reduction of quasi-stochastic disturbances that cause process/operation performing delays. actually, these disturbances are not random, but the information system, operators and supervisors do not know about their origins, so they are classified as quasi-stochastic. the ar/mr -app helps the system to increase the awareness of the origin of these disturbances. for example, the experiments done with assembly tasks showed that the number of quasi-stochastic disruptions that led to a process performing delay was reduced from 22 to 14 over 96 working hours (refer to table 3, the column denoted by productivity indicator p2). 2. eliminating the time needed for tedious reporting and getting more time for performing the primary job. the time saved by using the ar/mr-app instead of conventional reporting is measured between 40 to 70 %, depending on the application case. for example, reporting by typing/filling the qa file in an assembly operation cell usually requires 20 to 45 min for the operator per shift while the same job – generating the qa file within the ar/mr application interface takes 10 to 15 min (additionally it can involve voice comments, and photos are being made and attached faster). in terms of collecting human observations and inferences that have an impact on speeding up the ai toolset learning about the process, the main observed shortcoming was the possibility of wrong indoctrination of the ai toolset. wrong inferences and observations that an operator can add to the system by associating them to the particular process context can mislead the system about the process, result in the wrong process anticipation and consequently in wrong reacting proposals. it is currently possible to overcome this shortcoming by adding (at least one) observation review step, which should be necessary in adding an observation/inference to the system. 4. conclusion and future work through exploring the possibilities of applying the ar/mr app for collecting data and tacit knowledge about the (production) process context from a human-in-process, a few original conceptual interface solutions were developed for several different applications and different hardware platforms (phones, tablet and vizor – ms hololens) and different a study of the use of mixed reality for capturing human observation and inferences in production ... 19 operative systems (android, ios, windows). concerning the human-to-system data flow, which was in the focus of this research, ar/mr applications were equipped with two conceptually different communication forms: prepared questionnaires and self-initiated observations, semantic interpretation and categorization of the process context features. the results of a series of experiments proved that the ar/mr application is a very powerful tool for capturing data, information, and knowledge about the ongoing process context that cannot be built just on data gathered by sensors. the main benefit that comes from using the specific interface features of ar/mr app developed for collecting human observations and semantic interpretations of process context features is fast and profound learning about the process, both by the information system and the human participants in the process. as an indirect benefit of this feature, a production environment becomes significantly reactive (agile) to prevent any decline in terms of operating productivity and quality or any accident in terms of safety. having on mind the results of this research, we can firmly argue that ar/mr technology provides humans with quite a new role in a modern production environment, as some kind of an intelligence agent in the production process. in the near future, we should expect that ar/mr would figure as a key technology in facilitating the human-supervised learning of artificial intelligence systems. when it comes to future research, we would like to focus on using ar/mr for embedding semantics of production process context features. we believe that ar/mr can be used very efficiently for building or upgrading a semantic network by direct linking the digital content virtualized in vhf and 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recognizing topological analogy in semantic network, international journal of artificial intelligence tools, 24(3), pp. 1550006-1 1550006-25. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 219 226 original scientific paper the path towards achieving a lean six sigma company using the example of the shinwon company in serbia udc 658:005.5 srđan mladenović 1 , peđa milosavljević 1 , nevena milojević 2 , dragan pavlović 1 , milena todorović 1 1faculty of mechanical engineering, university of niš, serbia 2shinwon doo niš, sales department, niš, serbia abstract. in the last twenty years, many companies have realized that the demands of the global market, including more demanding and receptive customers, have set new standards for production flexibility. gradual reduction of mass production, characteristic of a large number of companies, has opened space for the introduction of a new system which focuses exclusively on the customer. the customer-oriented system was created with the idea that process optimization would lead to the production of a relatively cheap product, delivered on time and with the best possible quality. such a system is achieved by applying the lean six sigma concept. the aim of this paper is to identify all of the defects that occur as losses and complicate the process of production in order to achieve the lean six sigma level in the shinwon company. the original data from the shinwon company were identified, collected and analyzed, using the methods and tools of the lean six sigma concept (process mapping, 5s audit, pareto diagram, ishikawa diagram, seven basic wastes and spc analysis), in order to present the effectiveness of the quality management system and to evaluate the possibility of its continuous improvement. key words: lean, six sigma, pareto analysis, ishikawa diagram, waste, statistical process control 1. introduction lean and six sigma methodologies were developed each on its own but globalization and competition as well as the constant need of companies for improvement have recently received march 03, 2016 / accepted july 17, 2016 corresponding author: srđan mladenović faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, niš, serbia e-mail: maki@masfak.ni.ac.rs 220 s. mladenović, p. milosavljević, n. milojević, d. pavlović, m. todorović merged these two methodologies into one [1]. there are many companies today that have realized the necessity of applying the lean six sigma methodology and they have profited from it [2]. among them are: general electric, sony, citibank, whirlpool and many others. successful application of the methodology has been given in many papers so far, including not only implementation but also the positive results that the companies have achieved [3]. this combination of the lean practices with the six sigma has gained huge popularity in recent years [4]. the lean six sigma methodology has been used in a wide range of projects involving specific industrial problems [5-7]. its application in sme is given in [8] relating to manufacturing and production companies, as well as in railway sector [9]. the original lean six sigma method originated from automotive industry; therefore, its application and continuous implementation are directly bound to this sector, as can been seen in [10]. the shinwon company has also been caught by this global economic trend. the system of production fostered in this company must be continuously evaluated, improved and perfected. the goal of this paper is to identify all of the defects that occur as losses and complicate the process of production in this company, as well as the final proposal of improvements in order to move the entire organization towards a lean six sigma organization [11]. further in the paper, the data from the shinwon company are identified, collected and analyzed in order to present the effectiveness of the quality management system and to evaluate the possibility of its continuous improvement. once they are collected, the data are analyzed using the methods of the lean six sigma concepts (process mapping, 5s audit, pareto diagram, ishikawa diagram, seven basic wastes [12], and spc analysis with the aim of improving business quality. 2. production process in the shinwon company in this paper we will particularly pay attention to the production process as one of the basic processes of implementation in the company. fig. 1 schematic view of the production process in the shinwon company the path towards achieving a lean six sigma company on the example of the shinwon company in serbia 221 the production process consists of 4 phases (fig. 1):  the process of drawing wire from 8mm to 2mm;  the process of drawing wire from 2mm to smaller diameters;  the process of stranding wires;  the process of insulation. 3. list of 7 basic types of waste based on the defined parameters through which the production process is monitored in the shinwon company, we can see that most of the waste is included in the following categories: breaks, additional time, loss during manufacturing, specialization and training. the only element that would add value to the product would be the category of the performance of production workers in the form of the amount of produced wire based on which the performance of each worker would be monitored, along with the performance of a team on the machine, the team leader and the entire production. other elements comprise the so-called category of eligible or approved costs and that category increases the performance of workers but reduces productivity. the additional time category includes all the additional time that the worker needs in the execution of work; it is not standardized but not due to the worker’s fault. this group of eligible expenses includes the time required for additional cleaning of machines, change in the color of pvc, ink changes, changes in the wire size, etc. breaks include all delays due to malfunctions of machines. training includes the time that is recognized for the worker who has just started to work on a machine. specialization is recognized for the workers who work for the first time on a new type of wire on the machine or change the working position. all of the listed categories of approved costs, i.e. wasted time as well as the part of the working time up to 100% of the achieved norm, are an area where we can identify waste by the lean methodology [13]. table 1 seven basic types of waste in the shinwon company no. type of waste waste in shinwon (if it exists) 1 overproduction does not exist 2 transport does not exist 3 unnecessary movements workers do not use conveyor belts and transfer the product to the desired location by themselves; workers do not demand material from the shift leader and they take it themselves; unnecessary walking out of the time for a break. 4 waiting waiting for the repair of machines; calibration of machines; waiting because of the technological process; waiting for delivery. 5 processing does not exist 6 inventory does not exist 7 defects defects during the production; defects discovered during the product use by customer. 222 s. mladenović, p. milosavljević, n. milojević, d. pavlović, m. todorović 4. 5s audit in the shinwon company the shinwon company has been implementing the 5s method since its establishment, and is still working on improving it. each month, quality control managers check the application of this method and whether some aspects should be improved (fig. 2). fig. 2 results of internal 5s checking for the month of february every question from the checklists was evaluated on a scale from a to e, where:  a – outstanding results,  b – above average results,  c – minimum acceptable level,  d – minimum efforts,  e – without invested effort. 5. pareto analysis the pareto diagram [14] for the process of production of insulated copper wire is made on the basis of waste (list of 7 basic types of waste), which are analyzed in section 3 of this paper. waste is monitored on a monthly basis, and includes the categories of interruptions in the work process that are not workers’ fault as well as defects and waiting because of machine failure. these are the categories which require improvement in order to increase the efficiency and effectiveness of the production process. the pareto diagram visually shows which production process is the most commonly interrupted, or which process is the most critical (fig. 3). the path towards achieving a lean six sigma company on the example of the shinwon company in serbia 223 the interruptions data in operation monitored on a monthly basis are given in minutes for each process for february 2015. the pareto chart shows that the process in which most interruptions occur is that of isolating copper wire. the reason for this lies in the frequent changes in colors and sizes of the produced wire, and this is the process where it is necessary to implement effective solutions. fig. 3 pareto diagram of the process in which the waste occurs in the form of breaks 6. statistical process control the methods used in the shinwon company are the scatter diagram and process capability indices. the results of measurements based on samples are input by the quality control manager into a control table which is then copied to the internal software for assessing the abilities and performance of the process, which further results in scatter diagrams and abilities indices and performance. special emphasis is on the coefficient of potential processes (cp) and process capability index (cpk), which measure how much the production process in the shinwon company is close to the control limits that are set out in the standards and guidelines for sub-processes. the higher the values of these coefficients the more stable and better process we have. table 2 measurement results for the process of the 0.3 wire insulation standard spec analysis evaluation 7/0.26 (0.3sq) lower limit:1.30mm upper limit:1.50mm no of samples: 679 cp=4.85 cpk=4.80 pp=4.57 ppk=4.53 sigma level 14.4 very stable process 224 s. mladenović, p. milosavljević, n. milojević, d. pavlović, m. todorović fig. 4 graph of stability for the process of insulating the 0.3 wire the final process, the process of insulating copper wire is stable, capable and accurate, which can be seen from the scatter diagram (fig. 4) as well as from cp, cpk, pp and ppk coefficients (table 2). 7. ishikawa diagram based on the previously defined waste appearing in the production process in the shinwon company, a cause-effect diagram for the downtime category is created (fig. 5). fig. 5 ishikawa diagram for the category of interruptions in the working process the path towards achieving a lean six sigma company on the example of the shinwon company in serbia 225 the analysis was performed by the 4m method where the following main categories are identified: material, people, methods, machines [15]. through further analysis of these categories and by parsing each of them into increasingly smaller causes, we can conclude that the main problem is the human factor and human errors (inadequate storage of pvc, lack of motivation, different structure of workers, training of workers, poor production planning, etc.). 8. results and discussion based on the given results of the analysis, we can conclude that the greatest opportunity for improving the production process lies in reducing downtime and avoiding defects. improving the process can be achieved by applying the system of incentive payments of salaries based on group and individual performance, adequate and continuous staff training, as well as improving working conditions, because the human factor, or the problem with the employee, arises as the underlying cause problem in production. workers’ skills (poor working experience, training, education) can be improved by a better selection of personnel by the human resource department, as well as better training of the existing employees. the problems of engagement in the workplace and lack of motivation can be eliminated by introducing special incentives for commitment and dedication. the poor interpersonal relationships and non-collegiality can be influenced by the company by organizing joint nonoperating activities, as sports days, excursions, etc., which strengthen the team spirit of employees. based on the ishikawa diagram, it can be suggested that the process improvement can be achieved by improving the work schedules, improving the material control in production and increased supplier control. it is noteworthy that the production process in the shinwon company represents a viable and stable process in spite of the deficiencies found. the company constantly examines, enhances and improves its process, which is very important in today’s market conditions. the focus of the company is the customer, who demands a cheaper product, on time and with better quality, which imposes a constant need to improve the process. 9. conclusion today’s unpredictable and highly competitive business conditions have added significant innovations and imposed a new business philosophy, where the customer is the focus of the business. in these conditions, the lean six sigma philosophy because of its being quick, innovate, cheap and flexible, represents the success of each company [16]. identifying and eliminating unnecessary and wasteful activities in the business process, concentrating solely on the customer and continuously improving the production process can lead to achieving the maximum quality. since turbulent business conditions drastically shorten the life of a product, the lean six sigma methods and techniques allow the company to rapidly improve and adapt its performance in accordance with the existing demands. 226 s. mladenović, p. milosavljević, n. milojević, d. pavlović, m. todorović references 1. george, m., 2002, lean six sigma: combining six sigma quality with lean speed, first edition, mcgrawhill, new york. 2. holweg, m., 2007, the genealogy of lean production, journal of operations management, 25(2), pp. 420-437. 3. bhamu, j., sangwan, s.k., 2014, lean manufacturing: literature review and research issues, international journal of operation & production management, 34(7), pp. 876-940. 4. shah, r., chandrasekaran, a., linderman, k., 2008, in pursuit of implementation patterns: the context of lean and six sigma, international journal of production research, 46(23), pp. 6679-6699. 5. assarlind, m., gremyr, i., backman, k., 2012, multi-faceted views on a lean six sigma application, international journal of quality & reliability management, 29(1), pp. 21-30. 6. gupta, v., acharya, p., patwardhan, m., 2012, monitoring quality goals through lean six-sigma insure competitiveness, international journal of productivity and performance management, 61(2), pp. 194-203. 7. hilton, r.j., sohal, a., 2012, a conceptual model for the successful deployment of lean six sigma, international journal of quality & reliability management, 29(1), pp. 54-70. 8. timans, w., antony, j., ahaus, k., solingen, r., 2012, implementation of lean six sigma in smalland medium-sized manufacturing enterprises in the netherlands, journal of the operational research society, 63, pp. 339–353. 9. zhang, y., gregory m., neely a., 2016, global engineering services: shedding light on network capabilities, journal of operations management, 42-43, pp. 80-94. 10. habidin, n.f., yusof, s. m., 2012, relationship between lean six sigma, environmental management systems, and organizational performance in the malaysian automotive industry, international journal of automotive technology, 13(7), pp. 1119−1125. 11. marksberry, p., badurdeen, f., maginnis, m.a., 2010, an investigation of toyota's social-technical systems in production leveling, journal of manufacturing technology management, 22(5), pp. 604-620. 12. ohno, t., 1988, toyota productivity system, productivity press. 13. womack, j., jones, d., 2003, lean thinking, simon & schuster inc, new york. 14. grosfiled-nir, a., ronen, b. kozlovsky, n., 2007, the pareto managerial principle: when does it apply?, international journal of production research, 45(10), pp. 2317-2325. 15. ishikawa, k., 1982, guide to quality control, asian productivity organization, tokyo. 16. womack, j., jones, d., roos, d., 1990, the machine that changed the world: the story of lean production, rawson and associates, new york. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 123 131 1experimental and numerical determination of the wheel-rail angle of attack udc 629.4 dragan milković 1 , goran simić 1 , jovan tanasković 1 , živana jakovljević 2 , vojkan lučanin 1 1 university of belgrade faculty of mechanical engineering, rail vehicles department 2 university of belgrade faculty of mechanical engineering, department for production engineering, serbia abstract. angle of attack is an important wheel-rail contact parameter. it serves for estimation of the rolling stock curving performance. together with wheel-rail contact forces, angle of attack influences the wear index. this paper presents experimental ontrack measurements of the angle of attack using a specially designed laser device installed on track. experiments are performed on three types of rail vehicles: shunting locomotive series 631-301, motor unit 412-077 and trailing unit 416-077 of electromotor train 412/416. experimental measurements are compared with multibody system (mbs) simulations using specialized computer package vampire pro. we have found good agreement between the results obtained experimentally and by simulations. using these data, we have also performed relative comparison of wear indices of the outer wheels of the leading wheelsets for each of the tested vehicles. key words: angle of attack, measurements, multibody simulations, wear index 1. introduction one of the most important roles of the railway vehicle’s wheelsets is to provide for safe guidance of the vehicle along track. considering the design characteristics of the wheelsets, profiles of the wheel running treads and the rail heads as well as the design limitations of the suspension system link elements, it is almost impossible to achieve perfect radial steering of the wheelsets in sharp curves. the resulting yaw angle of the wheelset relative to the track longitudinal axis is defined as angle of attack (fig. 1). received january 22, 2015 / accepted april 2, 2015 corresponding author: dragan milković affiliation: faculty of mechanical engineering, university of belgrade, kraljice marije 16, belgrade e-mail: dmilkovic@mas.bg.ac.rs original scientific paper 124 d. milković, g. simić, j. tanasković, ž. jakovljević, v. luĉanin fig. 1 angle of attack during curve negotiation criteria for estimation of the railway vehicle dynamics during curve negotiation, including derailment safety, depend on the ratio between lateral y and vertical q forces in the wheel-rail contact (y/q), as well as on the wheel-rail angle of attack  [1, 2]. these parameters together influence wear intensity expressed using wear indices [3–5]. considering the importance of these parameters for experimental research as well as prototype testing of the railway vehicles in this paper we present a developed device for angle of attack (aoa) measurements and experimental results obtained by its use. the developed device is wayside and mobile, comparing to other devices [6, 7] that are installed on the vehicles and it can be used for measurements only on this vehicle. in order to present possible applications of aoa measurements for wear analyses, in this paper we have included some results of contact forces measurements obtained using another developed device presented in [8]. it has turned out that the maximum performances of both devices can be achieved if they are used in parallel, in order to collect as many as possible reliable parameters for each vehicle and to compare these results with numerical simulations. 2. device for aoa measurements the developed device for aoa measurements identifies position of the wheel relative to some surface or axis using a micro-epsilon optoncdt 1700 laser device (fig. 2). fig. 2 laser device positioning relative to the rail experimental measurements and numerical simulations of the wheel-rail angle of attack 125 the angle between the wheel and the defined reference, which represents longitudinal rail axis, defines wheel-rail aoa . considering the laser’s accuracy and the measurement principle, this device should be positioned perpendicularly to the rail longitudinal direction. perpendicularity verification is done by means of the device itself, by recording the laser-to-rail distance while sliding from one guide end to the other. 3. in-situ measurements of the aoa in this paper we present characteristic results recorded for shunting locomotive 621301 and electromotor unit 412/416. immediately before measurements, the laser beam direction was checked relative to the rail (fig. 3). it appeared that deviation of the laser beam from the perpendicular was 12.4 mrad (0.71). according to analysis performed in [9], such a small deviation has negligible influence on the measurement accuracy (less than ± 1%), so it was not necessary to correct the laser position. in order to present the method for data processing, as an example in this paper we present complete results of measurements performed on electromotor train’s trailing unit 416-077. fig. 3 trailing unit emu 416-077– aoa measurement device without protective cover fig. 4 presents laser-obtained recording of passing of the leading and the trailing wheelsets and the position of the outer wheels relative to rail. for obtaining the wheel-rail aoa, the recorded data were transformed from the time domain to the spatial one by using the measured train speed. the speed was measured by using the device itself based on the laser record in time l and the known distance between two adjacent wheelsets (wheelbase) of the tested vehicles passing by the laser. to avoid influence of the wheelsets distance on speed measurement, it is also possible to use a separate device with two switches installed on predefined (known) distance on the track, activated by the wheel passing for speed measurements. we used the device itself considering influence of the wheelset distance with tolerances and the maximum sampling rate of the laser 2500 hz. 126 d. milković, g. simić, j. tanasković, ž. jakovljević, v. luĉanin fig. 4 trailing unit emu 416-077 outer wheels – recorded data in the case of the maximum expected speed of 100 km/h during curve negotiation (for the curve radius of 400 m and the maximum allowable unbalanced acceleration) and the minimum wheelbase distance p = 1.8 m (for the standard freight bogie), relative error will be 0.18%. passing of each of the two outer wheels in spatial domain after transformation are presented in figs. 5 and 6. from these diagrams aoa and the wheel position relative to rail can be derived. aoa  is estimated based on the linear regression slope of the central segment of the recorded line with 100 mm length, which is far enough from rounded zones caused by flange passing by the laser. fig. 5 trailing unit emu 416-077 outer wheels – recorded data measured aoa was  = 13.1 mrad for outer wheel of the leading wheelset and  = 1.0 mrad for the outer wheel of the second wheelset. relative position between the back surface of the outer wheel and the side of the outer rail in the mid position of the wheel, defined as distance between wheel and the rail measured at 10 mm below tor (top of rail), was 28.5 mm for the leading wheelset and 45.0 mm for the second wheelset of the leading bogie. experimental measurements and numerical simulations of the wheel-rail angle of attack 127 fig. 6 trailing unit emu 416-077 – relative position and aoa of the outer wheel of the second wheelset 3.1. wheel-rail wear indices all three vehicles, shunting locomotive, trailing and motor unit of emu, were tested under the similar or almost the same conditions (same location) and all vehicles, while passing over the measurement section, were pushed, hauled, or with turned off traction. in this subsection we present relative comparison of the wear characteristics of the outer wheels of the leading wheelsets for each of the tested vehicles. for comparison, different authors have defined different wear characteristics, which are called wear indices. starting from expressions for wear index given in [3], which includes work of tangential forces in the wheel-rail contact, for comparison of the wear characteristics of the different vehicles with different wheel profiles, with some approximations, we used product of lateral force and aoa according to the following expression: . . . y y v rw i k y k y k y v           (1) where: k – constant, y – measured lateral force, y – creepage in lateral direction, vy – speed component in lateral direction of the wheelset, v – speed of the wheel forward movement,  – aoa. relation between creepage y and aoa  was established based on kinematics and geometric relation shown in fig. 7. approximation and assumption based on which expression (1) was derived were:  creepage represents the ratio between speed component in the creep direction and speed of the wheel forward movement,  during curve negotiation with turned off traction, wear is influenced dominantly by tangential force action in lateral direction. 128 d. milković, g. simić, j. tanasković, ž. jakovljević, v. luĉanin fig. 7 creepage in lateral direction the obtained results of the relative wear index for the three tested vehicles are presented in tab. 1. as a reference vehicle for comparison we have selected shunting locomotive as the vehicle with the least favourable wear performances. table 1 relative wear index r.w.i. of the tested vehicles v (km/h) y (n)    (mrad) y   (nrad) relative wear index r.w.i. shunting locomotive 631-301 13 26900 16.6 446.54 1.00 motor unit 412-077 14 23300 14.7 342.51 0.77 trailing unit 416-077 13 15400 13.1 201.74 0.45 based on the results presented in table 1 it can be seen that the shunting locomotive has the highest wear index. this is an expected result since the shunting locomotive has the largest axle load (18 t), no bogies, and an axle distance of 3 m, comparing to emu 412/416 with 2.6 m axle distance and with rigid axle guidance (radial steering). by comparing results for the trailing and motor unit of the electromotor train, it can be seen that there is a significant difference in calculated wear indices, although both vehicles have the same axle distance and the same bogie distance. the difference comes: 1) from the larger mass of motor unit wheelsets, for having installed axle transmission, 2) due to larger mass of the bogies, caused by mass of traction motors and total larger mass of the motor unit carbody, and 3) due to differences in the installed equipment. radial steering of both vehicles is similar and almost fully rigid, which is in accordance with close measured values of angles of aoa: 13.1 mrad for trailing, and 14.7 mrad for motor unit. relative comparison between three tested vehicles shows that trailing unit 416-077 has the lowest value of the aoa and the best wear performance. experimental measurements and numerical simulations of the wheel-rail angle of attack 129 4. numerical analysis simulation of the railway vehicle dynamics using some of the specialized computer simulation packages may also serve as a good design tool. depending on the scope of an analysis, it can have very detailed vehicle models and, at the same time, it can include some simplifications of the track model, such as considering it either as rigid or as installed on uniform elastic foundation. some research works [10, 11] focus on the track analysis using 2d models, while the excitation influence of the vehicle is approximated using some simplified vehicle models. there are also simulations [12–14] that consider vehicles and tracks as a multibody system (mbs), while considering only vertical vehicle/track interaction. in our research, we have used computer programme vampire pro [15] for a curving analysis during passing through the sharp curve with a 214 m radius, on the track without superelevation, excluding the influence of any other track geometric imperfections. due to the lack of available data on input parameters for all vehicles, we have performed simulation only for electromotor trailing unit for which we had available data on suspension elements stiffness, dumping characteristics, as well as on the axle guiding stiffness. below we present an overview of the performed simulations and obtained results. fig. 8 shows a non-linear model of the emu series 412/416 indicating detail of one bogie with suspension elements. bogie frame and wheelsets are presented transparently in order to provide for a better overview of the suspension system and link elements. fig. 8 model of emu [9] vampire pro allows for linear stiffness and damping terms to be specified between the rail and the sleeper, and sleeper-to-ground as shown in fig. 9. damping terms are not important for a quasi-static curving analysis. in this case they just support convergence of the simulations results. the most significant track stiffness terms used in the simulations are adopted from ref. [15]. they are: the lateral rail to the sleeper stiffness 43 kn/mm, the lateral sleeper to the ground stiffness 37 kn/mm, and the vertical sleeper to the ground stiffness 50 kn/mm. the rails and sleepers are considered to be massless degrees of freedom, which is appropriate for the steady-state curving analysis. 130 d. milković, g. simić, j. tanasković, ž. jakovljević, v. luĉanin fig. 9 track model [15] all the vehicle parameters used in the simulation are selected based either on performed measurements, or from available technical documentation. for the modelling of the wheel-rail contact, non-linear creep law is used. this creep law uses a pre-calculated contact data table that describes the contact data parameters with respect to wheelset lateral shift across the track at rail level. for that purpose real wheel and rail profile data measured using profilometer are used as input for the computation of the actual creep forces. the results of steady-state curving analysis have shown fairly good agreement having in mind strong dependence of the simulation results on various input parameters of the simulation model. although this programme was benchmarked during several internationally recognized benchmarking exercises and the results have shown fairly good agreement [16], the simulation results should be interpreted with caution. from the analyses performed on the trailing unit of the electromotor train 412/416, it can be seen that aoa of the leading wheelset obtained by simulation is 12.0 mrad while the experimental result is 13.1 mrad. aoa of the second wheelset of the trailing unit obtained using simulation is 0.5 mrad, and the corresponding measured value is 1.0 mrad. 5. conclusions the presented results show that a specially designed laser system installed on the track is able to measure accurately the lateral wheel offset, the vehicle speed, and angle of attack . for the selected measurement principle, the speed of the passing vehicle is the most important yet still low influence on the accuracy of  measurement. the high accuracy is possible if the perpendicularity between the laser beam and the rail longitudinal axis is kept within 5, which is relatively easy to achieve. additionally, the obtained results are used for mutual validation of the experimental measurements and the results of the multibody system (mbs) simulations. it has turned out that there exists good agreement between these results, which further encourages experimental research in the railway vehicle dynamics using the presented measurement system. also the comparison of the results shows that, provided the accurate track stiffness parameters are used, vehicle dynamics simulations can predict angle of attack with sufficient accuracy. experimental measurements and numerical simulations of the wheel-rail angle of attack 131 acknowledgements: the authors express their gratitude to the ministry of education, science and technological development, republic of serbia research grant tr 35045 and tr 35006. references 1. en 14363 railway applications testing for the acceptance of running characteristics of railway vehicles testing of running behavior and stationary tests, june, 2005. 2. guan q., zeng j., jin x., 2014, an angle of attack-based derailment criterion for wheel flange climbing, proc imeche part f: j rail and rapid transit, 228, pp. 719-729. 3. kumar s., prasana rao d.l., 1984, wheel-rail contact wear, work, and lateral force for zero angle of attack—a laboratory study, journal of dynamic systems, measurement and control, 106, pp. 319-326. 4. pombo j., ambrosio j., pereira m., lewis r., dwyer-joyce r., ariaudo c., kuka n., 2011, development of a wear prediction tool for steel railway wheels using three alternative wear functions , wear, 271, pp. 238-245. 5. santamaria j., vadillo e.g., oyarzabal o., 2009, wheel-rail wear index prediction considering multiple contact patches, wear, 267, pp. 1100-1104. 6. iijima h., horioka k., kataori a., momosaki s., doi k., 2011, development of continuous measurement equipment for angle of attack and results of measurements, jr east technical review, 19, pp. 46-49. 7. sugiyama h., tanii y., suda y., nishina m., komine h., miyamoto t., doi h., chen h., 2011, wheel/rail contact geometry on tight radius curved track: simulation and experimental validation , multibody syst dyn, 25, pp. 117-130. 8. milković d., simić g., jakovljević ž., tanasković j., luĉanin v., 2013, wayside systems for wheel-rail contact forces measurements, measurement, 46, pp. 3308-3318. 9. milković d., 2012, the influence of wheel-rail contact parameters on railway vehicle dynamic behaviour (in serbian), doctoral dissertation, university of belgrade, faculty of mechanical engineering, belgrade, 172 pp. 10. grassie s.l., 1992, a contribution to dynamic design of railway track, vehicle system dynamics, 20, pp. 195-209. 11. kerr a.d., 2000, on the determination of the rail support modulus k, international journal of solids and structures, 37, pp. 4335-4351. 12. ripke b., knothe k., 1995, simulation of high frequency vehicle–track interactions, vehicle system dynamics, 24, pp. 72-85. 13. sun y. q., dhanasekar m. a., 2002, a dynamic model for the vertical interaction of the rail track and wagon system, international journal of solids and structures, 39, pp. 1337-1359. 14. zhai w., sun x., 1994, a detailed model for investigating vertical interaction between railway vehicle and track, vehicle system dynamics, 23, pp. 603615. 15. vampire® pro version 5.50 programme, deltarail group ltd., derby 2011. 16. iwnicki s. (ed.), 1999, the manchester benchmarks for rail vehicles simulation, supplement to vehicle system dynamics, crc press, p. 206. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240730043g © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper use of five-parameter optimization of attachment lug geometry to improve its fatigue life vanja stefanović gobeljić1, aleksandar grbović2, aleksandar sedmak2, simon sedmak3, gordana djukanovic4, aleksandar bogojević5, ivana vučetić3 1military technical institute, belgrade, serbia 2faculty of mechanical engineering, university of belgrade, serbia 3innovation center of the faculty of mechanical engineering, belgrade, serbia 4faculty of forestry, university of belgrade, serbia 5university of belgrade, institute of physics, belgrade, serbia abstract. pin-loaded attachment lugs are highly loaded and responsible aircraft component prone to fatigue failure. thus, here the geometry of the wing-fuselage attachment lug was optimized using the fatigue life as the criterion. separating morphing and adaptive remeshing technology (smart) was used in the scope of finite element method (fem) to simulate fatigue crack growth by and to obtain fatigue life for different geometry parameters. five-parameter optimization was applied, by varying two radii (rounding radius and radius of a pin hole), “hump” parameter, outer radius and thickness, using five different models. it was shown that fatigue life can be significantly improved with only modest increase of mass. key words: optimization, attachment lug, smart fem, fatigue crack growth 1. introduction the wing spar is connected to the aircraft fuselage by so-called aircraft fittings with attachment lug as the crucial component, fig. 1. needless to say, eventual failure of such a component leads to inevitable aircraft loss, [1]. one should notice that attachment lugs are designed as safe-life components, [2], so experimental verification of their fatigue life is not required by regulations. yet, many recent events indicate unexpected damages in attachment lugs, including fatigue cracks, [3]. as an example, cracks were found in boeing 737ng pickle received: july 30, 2024 / accepted october 02, 2024 corresponding author: simon sedmak innovation center of the faculty of mechanical engineering, belgrade, kraljice marije 16, 11000 belgrade, serbia ssedmak@mas.bg.ac.rs 2 v. stefanović-gobeljić, a. grbović, a. sedmak, s. sedmak. et al. forks after shorter period of time then mandatory inspection period, [3], forcing airlines to make additional monitoring. as a consequence, about 50 jets were grounded worldwide to find a way to prevent fatigue crack initiation and growth, [3-4]. in this way, a new concept was introduced, based on fail-safe design, which includes possibility of crack growth and procedure how to deal with it, contrary to the safe-life design approach, [5-6]. in other words, even though the safe-life design approach tends to be overconservative, it does not guarantee the safety, so here the fail-safe approach will be considered, to accommodate the fact that cracks are often inevitable and controllable. toward this aim one should consider region around attachment lug hole, since this is the location of the highest stresses. fig. 1 light aircraft wing-fuselage attachment (circled) with two pairs of lugs, [6] fatigue behaviour of different types of aircraft attachment lugs and fittings was analysed recently, as described in the following text. failure of two trailing-edge flap (tef) hinge lugs was attributed to large fatigue cracks found around the bearing hole, [7]. cracks were initiated by extensive pitting corrosion, as well as fretting in one case. crack growth rate was about 10 times higher than the manufacturer predicted. another example of corrosion related damage initiating fatigue failure of a wing-fuselage link was analysed in [8]. it was concluded that the failure was the combination of fatigue and corrosion, as well as the consequence of high stresses in the wing lug area. fatigue failure of an aircraft landing gear due to the outer cylinder lug fracture was investigated in [9], indicating both stable and unstable crack growth due to specific microstructure orientation. the most probable cause of this failure was a combination of corrosion cavities and the high landing loads in a critical area. another failure analysis, [10], was focused on the effect of the lug head, introducing model for estimation of fatigue strength of pin-loaded lug with through-the-thickness crack. results were verified by the experimental ones. constant amplitude loading was used in [11] to validate the nasgro 4 fatigue crack growth rate data, and to simulate real aircraft joint behaviour. faster crack growth was found in the lug surface direction contrary to previous results which had indicated faster growth in thickness direction. in [12], straight attachment lug under oblique loading (maximum 45o) use of five-prameter optimization of attachment lug geometry to improve its fatigue life 3 was analysed using ansys, to study effects of crack length, radius ratio, and pin-load angle (β) on the sifs values. two different material combinations for wing-fuselage lug attachment bracket (steel aisi 4340 and aa 2024 t351 in the first case, and titanium alloy ti6al4v and aa 7075 t6 in the second one) were studied in [13] by numerical simulation. the second combination of materials enabled reduction of total weight of the brackets from 36.8 kg to 24.4 kg. extensive study by msc patran/nastran, presented in [14], provided the maximum tensile stress location in a pin-loaded lug. this study proved that the decreasing radius ratio increases stress concentration factor, thus reducing the maximum number of cycles to failure. stress intensity factors for single and elliptical cracks were evaluated by 3d boundary element method for different crack length and radius ratios to estimate fatigue life of attachment lug, [15]. similar approach was used in [16] to prove that numerical simulation of fatigue crack growth through the thickness is in agreement with the experimental results. studies [17-19] verified numerical simulation results by comparison with analytical and/or experimental results for a cracked pin-loaded lug under cyclic loading with either constant amplitude or an overload. the effect of width, hole diameter and thickness on the lug strength was assessed. in particular, the residual life of an attachment lug was estimated in [19], using finite element modelling compared with analytical expressions for stress intensity factors. similar approach was applied in this paper. similar to the previous three studies, analytical models were used for calculation of stress distribution in a lug, [20], and results were verified by fem. computational model was introduced in [21] to analyse the crack growth in attachment lugs with different cracks by using ansys with 4-noded finite element. numerical results were verified by the experimental data, as well as with analytical expressions for fcg, encouraging further efforts in this direction, as shown here. similar problem was tackled in [22] by applying 3d extended version of fem (xfem, as built in abaqus), two different load ratios were used (0.1 and 0.5) indicating significant effect of different loading boundary conditions on the estimated fatigue life. to summarize briefly, numerical and analytical tools are nowadays available as powerful and reliable tools for residual life evaluation of critical components with complex geometry, such as attachment lug. in this paper numerical simulation of fatigue crack growth in a connection lug is used to make five-parameter optimization of an attachment lug geometry. some aspects of this study were already considered [23-26]. solob et al., [23], used both extended finite element method (xfem) and classical fem with separating morphing and adaptive remeshing technology (smart) to calculate sifs for fatigue crack in attachment lug of a light aerobatic aircraft and to estimate resulting remaining fatigue life. in the scope of xfem analysis, growth of a penny-shaped corner crack was compared with the growth of through-the-thickness crack, located at the same position, to calculate number of cycles to complete failure confirming the fact that the actual attachment lugs must be redesigned using a fail-safe approach. afterwards, through-the-thickness crack was analysed also with classical fem using smart, as built in ansys. these two techniques were compared, indicating higher sif values obtained by xfem and thus the shorter fatigue life, [23]. grbovic et al., [24], used two-parameter optimization (thickness, hole radius) to get the optimal geometry of the wing-fuselage attachment by using the fatigue life as the criterion. toward this aim numerical simulation of fatigue crack growth by xfem and smart was used to obtain 4 v. stefanović-gobeljić, a. grbović, a. sedmak, s. sedmak. et al. fatigue life for different geometry parameters. in the meantime, experimental verification of numerical results was carried out on the full-scale wing, [25], indicating good agreement for calculated and measured strains and confirming that loads acting on the wing-fuselage attachment lug were accurately evaluated. finally, in the recently published review paper, [26], some of the results were reconsidered and analyzed in more details. one should notice that contrary to the xfem, which uses the same mesh through the simulation, crack front mesh adapts with every growth step when smart fem is used, [26]. the latter option is applied here, as provided by ansys, using tetrahedral elements. in this paper five-parameter optimization is presented (thickness, rounding radius and radius of a pin hole, “hump” parameter and outer radius) with the goal to prolong fatigue life of lug without excessive increase of mass (fail-safe approach) and to prevent its premature failure in this way. 2. numerical simulation numerical simulation of the fatigue crack growth is commonly based on the use of xfem, [26-28], or, more recently, on the application the separating morphing and adaptive remeshing technology (smart) in the scope of classical fem, [29]. both methods were used recently to solve different problems, such as fatigue crack growth in standard charpy specimen, [30], and remaining life of cracked welded joints, [31]. fem model, including boundary and loading conditions, is shown in fig. 2, indicating fixed points (in pink color) at the bottom surface and loading area in the upper surface (in red color). more details, including finite element mesh effects, are presented in [6]. in respect to the amplitude loading, stress ratio 𝑅=−1 was used as a representative of real wing load. as for the material, paris coefficients 𝑛=2.26 and 𝐶=7.526∙10−11 were adopted from ansys data base, [29], for the high strength steel used in this case, [6]. finite element mesh with an average element mesh size of 1.7 mm is shown in fig. 3. stress intensity factor (sif) values after the first crack growth step are shown in fig. 4, indicating kimax = 1880 mpa√mm, and all the values of ki along the crack front to be in-between 1850 and 1880 mpa√mm, [24]. fig. 2 finite element model of attachment lug, [32] use of five-prameter optimization of attachment lug geometry to improve its fatigue life 5 fig. 3 finite element mesh used in optimization fig. 4 sif values along crack front after 1st step of propagation (original lug) 3. optimization for the optimization purposes it was required to reduce kimax after the first crack growth step, if possible, as explained in [24]. another optimization criterion was to keep the mass of the attachment lug as close as the initial value (87 grams), [6]. the first step in the optimization process was to model increasing thickness. in this way, number of cycles was increased to nmax=1085 (thickness 17 mm), i.e. more than doubled in respect to the original one (nmax=515 for thickness 12 mm). more detailed results are shown in [24, 32]. the next step was two-parameter optimization, with thickness and radius of the surface varied. in the case the thickness 17 mm and rounding radius 19 mm, sif values along the crack front for lug were significantly reduced. anyhow, at the same time the mass was 6 v. stefanović-gobeljić, a. grbović, a. sedmak, s. sedmak. et al. increased for 75.8% (from 87 grams to 153 grams), which was unacceptable, indicating need to introduce additional parameters in the optimization process. thus, three-parameter and later on, five-parameter optimization was performed, using models as described in the following text. three-parameter optimization was done using two models, one of them already described in all necessary details in [32]. 3.1. model for three-parameter lug optimization model for three-parameter optimization had three geometrical input parameters: thickness (fig. 5), rounding radius and the radius of a pin hole (fig. 6). more details can be found in [24, 32]. fig. 5 thickness modeling normal to the marked face [24] a) b) fig. 6 radii of highlighted surfaces were varied: a) rounding radius, b) radius of pin hole two models were made for three-parameter optimization, one of them already described in [32]. in the second model, the initial lug thickness of 12 mm was kept the same, but the rounding radius was increased to 19 mm, whereas the pin hole radius was 7.5 mm. for this model, the mass was 105 grams and sif values after the first crack use of five-prameter optimization of attachment lug geometry to improve its fatigue life 7 propagation step are over 1400 mpa√mm, fig. 7a. the sif values after crack growth of 7.09 mm are shown in fig. 7b. figure 8 shows the number of cycles for three-parameter model presented here and original lug model up to the crack length 7 mm (crack length reached almost 11 mm in the case of three-parameter model). a) b) fig. 7 sif values along crack front after: a) 1st step of growth, b) 15th step of growth fig. 8 crack length vs number of cycles for thee lug models 3.2. models for five-parameter lug optimization two more parameters, shown in fig. 9, were additionally varied to get five-parameter optimization. one of them, so-called pull up, fig. 9a, was used to make a „hump“, while the second one, fig. 9b, was in charge of the outer radius with initial value 1 mm, which was varied up to 3-5 mm. using the ansys workbench® (design exploration module, the total number of 31 design point were generated, three radii, pull up (hump) and thickness, with the initial combination marked in red, fig. 10. besides 5 input parameters, the resulting mass, as the output parameter, is shown in fig. 10. 0 1 2 3 4 5 6 7 8 0 100 200 300 400 500 600 700 c ra ck l en g th ( m m ) number of cycles original lug three-parameter 2 8 v. stefanović-gobeljić, a. grbović, a. sedmak, s. sedmak. et al. a) b) fig. 9 a) hump parameter, b) outer radius fig. 10 generated design points use of five-prameter optimization of attachment lug geometry to improve its fatigue life 9 figure 11 presents all parameters, including the stress intensity factor, as the second output parameter. its values are given separately in the following text. fig. 11 input and output parameters five models were made and used, with different input parameters providing different geometries, as shown in fig. 12, together with the results for the distribution of stress intensity factors. based on values for the stress intensity factors, the number of cycles is calculated for all models and used to construct diagrams crack length vs. number of cycles, as shown in fig. 13. results are also shown in table 1 for the initial value of kimax and total number of cycles. a) model 1 – mass 80.7 g, sifmax 5271.3 mpa√mm 10 v. stefanović-gobeljić, a. grbović, a. sedmak, s. sedmak. et al. b) model 2 – mass 87.6 g, sifmax 3868 mpa√mm c) model 3 – mass 97.4 g, sifmax 2938.8 mpa√mm d) model 4 – mass 100.1 g, sifmax 2831.1 mpa√mm use of five-prameter optimization of attachment lug geometry to improve its fatigue life 11 e) model 5 – mass 106.5 g, sifmax 2593.7 mpa√mm fig. 12 five models for the five-parameter optimization fig. 13 crack length vs. number of cycles for the five-parameter models table 1 results for five-parameter models mass (g) thickness (mm) initial kimax (mpa√mm) total number of cycles original lug 87 12 1880 515 model 1 80.7 (-7.24%) 13 1883 526 (+2.1%) model 2 87.6 (+0.7%) 13 1788 664 (+28.9%) model 3 97.4 (+12%) 13.5 1641 843 (+63.7%) model 4 100.1 (+15.1%) 14 1580 916 (+77.9%) model 5 106.5 (+22.4%) 15 1479 1026 (+99.2%) 0 2 4 6 8 10 12 0 100 200 300 400 500 600 700 800 900 1000 1100 c ra ck l en g th ( m m ) number of cycles 80.7 g 87.6 g 97.4 g 100.1 g 106.5 g 12 v. stefanović-gobeljić, a. grbović, a. sedmak, s. sedmak. et al. 4. discussion table 1 indicates modest increase in mass and significant increase in fatigue life, generally speaking. in the case of model 1, mass is even reduced slightly, but increase in fatigue life is not remarkable. model 2 has almost the same mass with increase of fatigue life close to 30%, while models 3-5 provide fatigue life increase 63.7-99.2% with increase of mass 12-22.4%, making them all candidates for practical use of presented optimization, especially since modern cnc machines can easily produce any of these shapes. comparison with three-parameter optimization is clearly in favour of five-parameter optimization, both regarding the first three-parameter model, described in [28] and the second one, described here, since they provided smaller increases in life (52.6-60%) with larger increase in mass (20.7-24.1%). in other words, with approximate the same increase of mass (22.4% vs. 20.7%), increase of fatigue life is higher (99.2% for model 5) compared to 60% (three-parameter, 2nd model). one can see from fig. 10 that the shape of a ‘hump’ affects the fatigue life, as well as the outer radius. for easier comparison, previous optimization options are summarized in table 2, indicating the longest fatigue life in the case of the two-parameter optimization model (3.75 times), but at the cost of 75.9% of mass increase, [24]. on the other hand, model 5 (table 1) results in a mass increase of only 22.4%, providing almost 2 times longer fatigue life, so it seems to be the optimal option. anyhow, if the risk of failure is high, then the best option is the one with the highest number of cycles. table 2 comparison of properties for five lug models mass (g) initial kimax (mpa√mm) total number of cycles original lug 87 1880.0 515 thickness 17 mm 123 (+41.4%) 1391.2 1085 (+110.7%) two-parameter 153 (+75.9%) 1256.8 1933 (+275.4%) three-parameter model 1 108 (+24.1%) 1267.6 786 (+52.6%) three-parameter model 2 105 (+20.7%) 1498.5 824 (+60.0%) at this point, it is important to clarify and distinguish between two design principles, safe-life vs. fail-safe. in the first case, cracks are not accounted for, since the essence of safe-life principle is to assure full design life of a component without cracks. optimization in that case is meaningless since its aim is to increase number of cycles from the initial to the critical crack length. i.e. to increase life of a cracked component. on the other side, failsafe principle enables safe operation of a cracked component if fatigue crack growth rate is such that the critical crack length is not reached before the next ndt testing. for this approach one needs reliable experimental data for fatigue crack growth rate and realistic numerical simulation of fatigue crack growth, as demonstrated here. finally, one should keep in mind eventual effect of paris coefficients, c and m, and problems with their evaluation. the best option would be to test material in the standard way, [33], under the conditions as close to real ones as possible, but that was not an option here. any other experimental data, like dependence between crack length and number of cycles, could be also use to calibrate c and m, but that was not an option here as well. therefore, we had to rely on database available in literature. use of five-prameter optimization of attachment lug geometry to improve its fatigue life 13 5. conclusions based on the results presented in this paper, one can conclude the following:  the longest fatigue life in the scope of five-parameter optimization is obtained with model 5 (99.2%) with modest increase of the mass (22.4%). more complex shape of model 5 is not a problem, since modern cnc machines can easily produce it.  compared with others, the five-parameter model 5 provides the best option, unless the longer fatigue life is the only criterion.  optimization of attachment lug significantly improves fatigue life and thus, reduces the risk of failure, so it fits well in the 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university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical modelling of the crosswind influence on vehicle aerodynamics in highway traffic conditions daniela alic1, jonas matijošius2, artūras kilikevičius2 1politehnica university timisoara, faculty of engineering hunedoara, romania 2vilnius gediminas technical university, mechanical science institute, lithuania orcid ids: daniela alic https://orcid.org/0009-0007-5773-5058 jonas matijošius https://orcid.org/0000-0001-6006-9470 artūras kilikevičius https://orcid.org/0000-0002-4039-7300 abstract. this study presents an efficient numerical model for evaluating the impact of crosswinds on vehicle aerodynamics using ansys fluid flow. it offers valuable insights into the qualitative behavior of vehicles under typical highway traffic conditions. the analysis explores airflow distribution when a car overtakes a truck at approximately 100 km/h, considering weak, moderate, and strong lateral winds. a detailed examination of the overtaking phases, with vehicles positioned in three distinct configurations, reveals significant aerodynamic interactions. these interactions influence vehicle handling and stability, as reflected in variations in velocity and pressure distributions, increased drag coefficients for the truck under stronger lateral winds, and significant fluctuations in the car’s drag coefficients based on its position relative to the truck. additionally, reynolds numbers indicate a transition to turbulent flow. this paper provides an in-depth analysis and discussion of these findings. key words: aerodynamic analysis, passing maneuvers, crosswind, computational fluid dynamics, drag and lift coefficients, reynolds number 1. introduction in recent years, the study of vehicle aerodynamics has gained increasing attention, particularly due to the growing emphasis on energy efficiency, sustainability, and safety in transportation systems. aerodynamic drag coefficients are a critical factor influencing the energy consumption of vehicles, especially at highway speeds, where air resistance constitutes a significant portion of the overall drag force. a detailed understanding of the factors that contribute to aerodynamic drag, such as vehicle design, speed, environmental and traffic conditions, is essential for optimizing fuel efficiency and reducing emissions received: december 12, 2024 / accepted february 28, 2025 corresponding author: daniela alic politehnica university timisoara, faculty of engineering hunedoara, revolutiei str. 5, hunedoara, romania e-mail: daniela.alic@upt.ro https://orcid.org/0009-0007-5773-5058 https://orcid.org/0000-0001-6006-9470 https://orcid.org/0000-0002-4039-7300 66 d. alic, j. matijošius, a. kilikevičius [1]. among these factors, the impact of crosswinds on aerodynamic performance has emerged as a significant area of interest [2]. crosswinds, acting perpendicularly to the direction of travel, can dramatically alter the aerodynamic forces experienced by vehicles. on highways, where vehicles travel at high speeds and in close proximity to one another, crosswinds not only increase the drag force but also affect vehicle stability and driver control [3]. the interaction between vehicles under crosswind conditions introduces additional complexity, as aerodynamic wake effects and interference between vehicles can amplify drag forces and influence the vehicle’s lateral dynamics. consequently, understanding the effect of crosswind on the vehicle interaction on highways is critical for improving both fuel efficiency and safety. numerical modeling plays a key role in analyzing the influence of crosswinds on vehicle aerodynamics. through extensive computational fluid dynamics (cfd) simulations, researchers can evaluate the complex flow patterns and pressure distributions that arise when crosswinds interact with multiple vehicles [4], as well as the truck-induced wind gusts [5], that may affect the resilience and safety of highway traffic signals structures. numerical models allow for detailed analysis of various parameters, such as vehicle spacing, crosswind speed, and vehicle geometry, without the need for extensive physical testing. these simulations offer valuable insights into the effect of crosswind conditions on the aerodynamic drag, especially in scenarios involving multiple vehicles traveling in close formation, such as convoys, in platoon [6], or during overtaking maneuvers [7]. the motivation for conducting numerical modeling of crosswind effects lies in the potential benefits for both vehicle design and transportation planning. by reducing aerodynamic drag through optimized vehicle shapes and logistic transportation configurations, manufacturers can improve vehicle fuel efficiency, resulting in lower operating costs and reduced greenhouse gas emissions. furthermore, understanding the impact of crosswinds on vehicle dynamics contributes to highway safety by guiding vehicle control strategies and enhancing driver assistance systems, particularly in regions prone to strong crosswinds. this study aims to investigate the effects of crosswinds on the aerodynamic drag generated by vehicle interaction on highways through numerical modeling. by providing a detailed analysis of drag variations under different crosswind conditions, this research seeks to contribute to the development of more energy-efficient and safer vehicle designs, as well as strategies for mitigating the adverse effects of crosswinds on highways traffic. 2. theoretical basis and numerical setup the numerical modeling of crosswind influence on air drag from vehicle interaction on highways relies on a strong theoretical foundation rooted in fluid dynamics, aerodynamics, and computational methods [8]. in the context of computational fluid dynamics (cfd) simulations using ansys software, several fundamental principles and mathematical models are employed to accurately represent the complex flow behaviors encountered in vehicle aerodynamics under crosswind conditions [9]. the core theoretical framework for modeling airflow in cfd, based on the navierstokes equations, describes the motion of fluid substances. these equations are a coupled set of nonlinear partial differential equations for five unknowns: three velocity components and two state variables (pressure, density or energy), which are connected by an equation numerical modelling of the crosswind influence on vehicle aerodynamics in highway ... 67 of state. these account for the conservation of mass (continuity equation), momentum (newton’s second law), and energy (first law of thermodynamics) [10]. our numerical model employs a viscous shear stress transport (sst) k-ω model, a widely used two-equation eddy-viscosity turbulence framework that serves as an approximation for the reynolds-averaged navier–stokes (rans) equations and has proven to provide results that match well with experimental data [11]. in the rans approach, equations are solved for the time-averaged flow characteristics while capturing the scale of turbulent fluctuations. the time average is defined as 𝑓̅ = 𝑙𝑖𝑚 𝑇→∞ 1 𝑇 ∫ 𝑓(𝑥𝑖 , 𝑡) 𝑑𝑡 𝑇 0 (1) and the instantaneous field flow is represented as 𝑢𝑖 = �̅�𝑖 + �́�𝑖, namely the sum of a timemean component and a fluctuating component. averaging the navier–stokes equations yields the rans equations [12], which contain the reynolds stress tensor −𝜌 𝑢𝑖́ 𝑢�́� ̅̅ ̅̅ ̅ as a key term representing the averaged effects of turbulent convection. rans models are classified into two main categories based on how this stress tensor is computed: eddy viscosity models, which assume that stress is proportional to strain (velocity gradients) and incorporate an effective turbulent viscosity, and reynolds stress models, which provide a more detailed representation of the stress tensor without the assumption of proportionality to strain. eddy viscosity, similar in effect to molecular viscosity, facilitates the diffusion of momentum. unlike molecular viscosity, it is not an inherent fluid property but rather a characteristic of turbulent flow. turbulence is characterized by random, instantaneous fluctuations in both space and time. however, statistical averaging of these fluctuations reveals measurable transport mechanisms. the turbulent kinetic energy (k) equation is used to define the turbulence velocity scale: 𝜕(𝜌 𝑘) 𝜕𝑡 + 𝜕(𝜌 𝑢𝑗 𝑘) 𝜕𝑥𝑗 = −𝜌 �́�𝑖�́�𝑗 ̅̅ ̅̅ ̅ 𝜕𝑢𝑗 𝜕𝑥𝑗 − 𝜌𝜀 + 𝜕 𝜕𝑥𝑗 [ 𝜕𝑘 𝜕𝑥𝑗 (𝜇 + 𝜇𝑡 𝜎𝑘 )] (2) where the first term on the right-hand-side represents the rate of production of turbulent kinetic energy, 𝑃𝑘, driven by mean flow gradients, and ε denotes the dissipation rate, describing the conversion of kinetic energy into internal energy due to viscous effects. physically, turbulent kinetic energy is produced due to the mean flow gradients, and is dissipated by viscous effects. imbalance between the production and the dissipation will cause k either to grow or to decay. the final term is a diffusion term, typically modeled using the gradient diffusion hypothesis or reynolds analogy, which introduces the turbulent prandtl number to quantify turbulent diffusion effects. the motivation behind two-equation models originates from the need to obtain the two scales required to compute the eddy-viscosity. based on dimensional arguments, a lengthscale and a time-scale are required for that purpose. the standard k-ω model is a twoequation turbulence model that solves transport equations for k and ω (specific dissipation rate, ε/k). this model excels in simulating wall-bounded boundary layers, free shear flows, and low reynolds number flows. it is particularly effective for complex boundary layer flows under adverse pressure gradients, separation (as in external aerodynamics), and transitional flows. 68 d. alic, j. matijošius, a. kilikevičius the sst k-ω model is an enhanced version of the standard k-ω model, combining the strengths of both the original k-ω approach for near-wall regions and the k-ε model for regions farther from the wall. the corresponding equations are: 𝜕(𝜌 𝑘) 𝜕𝑡 + 𝜕(𝜌 𝑢𝑗 𝑘) 𝜕𝑥𝑗 = 𝑃𝑘 − 𝐶𝜇𝜌𝑘𝜔 + 𝜕 𝜕𝑥𝑗 [(𝜇 + 𝜇𝑡 𝜎𝑘 ) 𝜕𝑘 𝜕𝑥𝑗 ] (3) 𝜕(𝜌 𝜔) 𝜕𝑡 + 𝜕(𝜌 𝑢𝑗 𝜔) 𝜕𝑥𝑗 = 𝐶𝜔1 𝜔 𝑘 𝑃𝑘 − 𝐶𝜔2 𝜌𝜔2 + 𝜌(1 − 𝐹1)𝐶𝐷 + 𝜕 𝜕𝑥𝑗 [(𝜇 + 𝜇𝑡 𝜎𝑘 ) 𝜕𝜔 𝜕𝑥𝑗 ] (4) with the coefficients set in ansys as: 𝜇𝑡 = 𝜌 𝑘 max(𝜔, 𝐹2𝑆 𝑎1 ) , 𝐶𝐷 = 2 𝜎𝜔 1 𝜔 𝜕𝑘 𝜕𝑥𝑗 𝜕𝜔 𝜕𝑥𝑗 𝐶𝜔1 = 5 9 ; 𝐶𝜔2 = 3 40 ; 𝜎𝑘 = 1 0.85⁄ ; 𝜎𝜔 = 2 ; 𝐶𝜇 = 0.09 ; 𝑎1 = 0.31 the standard k-ε model performs well in free-stream fully turbulent regions but struggles with complex flows involving severe pressure gradients, separation, or strong streamline curvature. the sst hybrid model addresses these limitations by employing a blending function 𝐹1 that transitions smoothly between the k-ω model near walls and the k-ε model in high-reynolds-number regions [12]. additionally, it incorporates a modified turbulent viscosity formulation to account for the transport effects of primary turbulent shear stress, enhancing its accuracy in capturing complex flow behaviors. the governing equations are discretized using the finite volume method and least squares cell-based gradient evaluation [13]. as solution methods, a coupled scheme is employed for pressure-velocity coupling, utilizing a rhie-chow momentum-based flux approach [14]. this method provides a robust and efficient single-phase implementation for steady-state flows, offering better performance than segregated solution schemes. to improve the accuracy of the simulation and better capture the turbulent flow over the vehicles, second-order upwind schemes are applied to the turbulent kinetic energy and specific dissipation rate. additionally, the external-aero favorable settings option is used during the hybrid initialization phase to generate a more accurate initial velocity field for external-aero simulations. the volume mesh consists of poly-hexcore elements, with local refinement regions around the vehicles, as part of the standard ansys fluent meshing watertight geometry workflow. convergence verification is conducted by tracking the continuity equation, velocity components, and the k and ω values, ensuring that the solution residues decrease below 10-4 as convergence criterion. 3. numerical analysis 3.1. numerical modeling numerical modeling with specialized computational fluid dynamics (cfd) software, such as ansys fluent, is currently used for a wide range of applications in the automotive sector, due to its advanced physics modeling and industry leading accuracy [15]. fluent is one of the two computational fluid dynamics (cfd) packages included with the ansys computational mechanical software suite. with increased traffic and advancements in technologies aimed at improving road safety, various driving scenarios are being analyzed to assess aerodynamic drag coefficients (which impact fuel consumption), determine optimal vehicle spacing, and ensure safe travel under diverse conditions. numerical modelling of the crosswind influence on vehicle aerodynamics in highway ... 69 as one vehicle overtakes another, the airflow fields around both vehicles interact, generating transient aerodynamic forces that can adversely affect handling and stability. sudden changes in airflow velocity within a vehicle's wake may cause abrupt variations in the aerodynamic forces acting on the overtaking vehicle. these rapid fluctuations can compromise control, making steering more challenging and increasing the risk of accidents. understanding how light vehicles respond to disturbances caused by larger, heavier vehicles remains a complex challenge. this study investigates the impact of these aerodynamic disturbances on a passenger car when passing a large truck. the car geometry is that of a bmw 1 series [16], namely 4.3 m length, 1.8 m width and 1.43 m height. the truck with the streamlined geometry of a european truck [17] has a length of 10.5 m, width of 2.5 m and height of 4 m. the distance between the vehicles is 1.5 m on the z direction (fig. 1). the vehicles have a speed of 28 m/s (approximately 100 km/h), modeled through an airflow entering the front inlet boundary and flowing in the positive x direction (fig. 1). the velocity of the lateral wind, entering the right inlet boundary and flowing in the positive z direction, is varied from 5 m/s, to 10 m/s and 15 m/s corresponding to weak, significant, respectively strong wind. this approach assumes that the relative velocity of the vehicles has a negligible effect on the flow distribution. the fluid is air, with density 1.225 kg/m3 and viscosity 1.7894x10-05 kg/ms. the turbulent intensity, expressed by the percentage ratio of the flow rate variation with respect to the average flow rate, is set to 1%, while the turbulent viscosity ratio is set to 2 at the boundaries. the interaction between the vehicles is modeled in three traffic phases. as the truck runs on the first lane, the car runs on the second lane and their dynamics is studied in the following phases: car behind the truck (fig. 1a), parallel with the truck (fig. 1b) and eventually in front of the truck (fig. 1c). fig. 1 interaction between vehicles in 3 phases: a) car behind truck, b) car parallel with the truck, c) car in front of the truck the numerical simulation domain consists of a box of 50 m length, 40 m width and 24 m height. the boundaries are set up as velocity-inlets: front yz boundary (flow in +x direction) and right yx boundary (flow in +z direction); pressure outlets: back yz boundary and left yx boundary; stationary shear zero wall: top xz boundary; moving no-slip wall: ground xz boundary; stationary no-slip walls: car and truck surfaces. in viscous fluids, the flow velocity at solid surfaces is zero relative to the boundary, which accounts for the noslip condition. this condition is based on the physical observation that at the wall, the adhesive force of attraction between fluid and solid particles is stronger than the cohesive force between the fluid particles. 70 d. alic, j. matijošius, a. kilikevičius 3.2. numerical results, interpretations and discussions our primary goal is to analyze airflow velocity vectors around vehicle bodies in the three cases, with fig. 2 showing their top view in the xz plane for the first traffic phase. a) b) c) d) fig. 2 distribution and magnitude of the induced airflow velocity vectors and the related magnitude of drag and lift coefficients in phase 1, at different lateral wind speeds numerical modelling of the crosswind influence on vehicle aerodynamics in highway ... 71 in the case with no lateral wind, illustrated in fig. 2a, the typical airflow distribution around both vehicles is visible, with only a slight distortion in the x direction due to the interaction of their air currents as they travel in parallel lanes. as lateral wind velocity increases (fig. 2bcd) the main airflow behaves differently, with the truck acting as a lateral shield for the car, creating an area of low airflow velocity around the car. the drag coefficient values confirm this, indicating a reduction in drag on the car in the x direction, making its movement easier despite the increasing lateral wind, as the truck reduces its exposure to the airflow. in the z direction, the drag on the car is minimal and directed towards the truck (showing negative values), suggesting a slight push of the car towards the truck. meanwhile, the airflow velocity over the truck increases with lateral wind, reaching maximum values which exceed 1 (see discussion at the end of the section and reference [18]), not only in the front windshield area but also along the exposed sides and top of the truck body. this results in higher drag forces in both the x and z directions, indicating that the truck is being pushed backward and toward the car. the reynolds number of the induced airflow is plotted in fig. 3, providing a top view of the xz plane. in the case of zero lateral wind (fig. 3a), the plot shows laminar flow over the vehicles (re < 105) and transitional flow in the wake of the truck. with increasing lateral wind, the airflow pattern changes significantly, showing higher reynolds numbers, up to 1.25×10⁶, in transition to turbulent flow, not only in the rear of the truck but also around the car, as the left side of the truck airflow is largely diverted over the car. a) b) c) d) fig. 3 path lines of the induced airflow turbulence and magnitude of the reynolds number in phase 1, at lateral wind of: a) 0 m/s, b) 5 m/s, c) 10 m/s, d) 15 m/s in the second phase of our study, the vehicles travel side by side. a comparison between the zero lateral wind scenarios in phases 1 and 2 reveals a slight increase in the car's drag coefficients along the x direction (figs. 2a and 4a), as some of the airflow separating from the truck's windshield now reaches the car's windshield, raising the pressure. however, as lateral wind increases, the car's drag in the x direction decreases, since the truck disrupts both the frontal and lateral airflows, redirecting them almost tangentially to the car 72 d. alic, j. matijošius, a. kilikevičius (fig. 4bcd). this creates a partial shielding effect, reducing its drag. in the z direction, drag on the car increases, pushing it away from the truck. a) b) c) d) fig. 4 distribution and magnitude of the induced airflow velocity vectors and the related magnitude of drag and lift coefficients in phase 2, at different lateral wind speeds numerical modelling of the crosswind influence on vehicle aerodynamics in highway ... 73 the reynolds numbers (fig. 5) indicate a similar pattern as analyzed for phase 1, with maxima around 106 in the wake of the truck and around its lateral region, where the separated airflow of the truck interacts with the surface of the car. a) b) c) d) fig. 5 path lines of the induced airflow turbulence and magnitude of the reynolds number in phase 2, at lateral wind of: a) 0 m/s, b) 5 m/s, c) 10 m/s, d) 15 m/s in the third and final phase of our study, the car is fully exposed to both the frontal airflow and lateral wind (fig. 6), resulting in a significant increase in car drag in both the x and z directions due to the higher wind velocity. the drag coefficients on the truck have similar values to phase 2, as its dynamics is not strongly influenced by the position of the car. however, the truck's influence on the car is obvious in all phases and still noticeable in phase 3, in the air velocity distribution formed in the wake of the car. the airflow pattern is once again marked by reynolds number peaks of 106 in the separated flow around the truck (fig. 7), indicating a transitional flow that has not yet fully developed into turbulence (re between 5×105 and 3×106). while accurate turbulence modeling is essential for predicting external vehicle flow since higher reynolds numbers suggest more intense turbulence, which can greatly increase drag the results in this paper are produced using accessible computational resources and aim to provide an estimate of crosswind effects on the vehicle aerodynamics. a more detailed study of reynolds numbers would be a valuable tool for optimizing aerodynamics, from evaluating different vehicle shapes to assessing specific design modifications such as spoilers and diffusers [19, 20]. in examining the cases with zero lateral wind velocity, it becomes clear that in all phases, the velocity plots show the airflow coming to a complete stop at the front of the vehicles before accelerating as it moves over the curved surfaces. this creates a highpressure region at the front, which is more pronounced for the truck than for the car, where the air is deflected and flows along the aerodynamic surface. with lateral wind, this highpressure region extends to the side of the truck, which provides a shielding effect for the car in phases 1 and 2. the main stagnation points remain on the front and lateral wind exposed surfaces where the airflow stops, then turns towards the edges and accelerates as it follows the vehicle contours. at the rear, the flow separates, leading to the formation of a recirculation region (wake). due to the sharp change in the truck’s surface shape and the high reynolds number, the flow separates abruptly, resulting in significantly larger separation regions compared to the car. 74 d. alic, j. matijošius, a. kilikevičius a) b) c) d) fig. 6 distribution and magnitude of the induced airflow velocity vectors and the related magnitude of the truck and car drag coefficients in phase 3, at lateral wind of: a) 0 m/s, b) 5 m/s, c) 10 m/s, d) 15 m/s numerical modelling of the crosswind influence on vehicle aerodynamics in highway ... 75 a) b) c) d) fig. 7 path lines of the induced airflow turbulence and magnitude of the reynolds number in phase 3, at lateral wind of: a) 0 m/s, b) 5 m/s, c) 10 m/s, d) 15 m/s as the airflow separates on the front and lateral of the vehicles, it forms pockets of high pressure and low velocity, which contribute to the pressure drag on the vehicles. this results in a significant increase in drag coefficients in the presence of lateral wind. in crosswind conditions, the airflow around both the truck and the car becomes highly unstable and less streamlined. normally, the airflow over the vehicles is relatively smooth, helping to maintain a lower drag coefficient. however, crosswinds introduce additional lateral forces that disrupt this airflow, which becomes highly disturbed and less efficient, increasing the resistance experienced by the vehicles. this increased resistance raises the drag coefficient, in some cases above one [18]. as the flow moves around the vehicles, it accelerates over their curved surfaces, leading to a reduction in pressure. the velocity and pressure distribution on the vehicles, along with the viscous shear, generate forces on the vehicles: the drag force acting in the direction of the airflow and a vertical force acting perpendicular to the flow. the lower pressure on top of the vehicles also creates a lifting effect. when a vehicle encounters a crosswind, the airflow becomes uneven, disrupting the direct front-to-rear flow. this disruption leads to irregular pressure distributions on the vehicle’s surfaces, which can alter the overall lift force. during phase 1, the truck shields the car from the crosswind, significantly reducing the lift on the car and generating downforce (fig. 1bcd). in phase 2, the crosswind initially increases lift by creating a pressure unbalance between the side facing the wind and the protected side (fig. 4b); however, as the angle of the crosswind shifts, the truck shields the car again, reducing lift (fig. 4cd). in phase 3, the car is fully exposed to the crosswinds, experiencing an initial lift increase due to the pressure differential (fig. 6bc), followed by a decrease as the pressure balances on the side of the car exposed to the lateral wind (fig. 6d). while much of the existing research has focused on the direct impact of aerodynamic forces on isolated vehicles [21, 22], the influence of crosswinds and vehicle interactions on highways remains an underexplored yet increasingly important area of study. numerical modeling provides a powerful tool to investigate these complex aerodynamic phenomena, offering insights that are difficult to control and expensive to obtain through experimental methods alone. 76 d. alic, j. matijošius, a. kilikevičius 4. conclusions in automotive engineering, accurately predicting and minimizing aerodynamic drag is essential for improving vehicle performance, fuel efficiency, and overall safety. understanding the effects of crosswinds on vehicle interactions is crucial, as they directly influence both energy consumption and stability. crosswinds can substantially alter aerodynamic forces, particularly in multi-vehicle scenarios such as close-proximity driving or highway overtaking maneuvers. these interactions can lead to rapid and large fluctuations in air drag coefficients, strongly affecting fuel efficiency, handling, and stability, and potentially compromising road safety. furthermore, with the advent of advanced driver-assistance systems (adas) and the future integration of autonomous vehicles, the ability to predict and mitigate adverse aerodynamic effects becomes even more important. vehicles equipped with such technologies often operate in convoys or platoons, where crosswind-induced drag variations can lead to significant energy inefficiencies and control challenges. thus, developing reliable numerical models that can accurately simulate these scenarios is essential for optimizing vehicle design and improving the safety and efficiency of highway transportation. this study provides a comprehensive numerical analysis of crosswind effects on aerodynamic drag coefficients in vehicle interactions on highways. by leveraging advanced computational fluid dynamics techniques, we quantify the drag forces under varying crosswind conditions and vehicle interaction in real-traffic scenarios, where vehicles travel at high speeds and in close proximity. in such conditions, crosswinds can significantly impact vehicle stability and fuel consumption. optimizing vehicle shapes and transportation configurations can help mitigate these aerodynamic challenges, leading to safer and more energy-efficient designs, and improved traffic management under adverse environmental conditions. furthermore, our approach contributes to strategies for minimizing the adverse effects of crosswinds, offering guidance on setting appropriate highway speed limits in moderate to strong wind conditions, ultimately enhancing both safety and efficiency. future work will focus on exploring a broader range of vehicles and configurations, as our current study is limited to a certain type of vehicles, representative for their class, which could provide deeper insights into crosswind effects. additionally, we will investigate the impact of crosswinds on different vehicle 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https://3dmodels.org/3d-models/mercedes-benz-actros-tractor-3-axis-2011/ facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 107 120 https://doi.org/10.22190/fume190318021m © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  the vehicle routing problem with stochastic demands in an urban area – a case study danijel marković, goran petrović, žarko ćojbašić, aleksandar stanković faculty of mechanical engineering, university of niš, serbia abstract. the vehicle routing problem with stochastic demands (vrpsd) is a combinatorial optimization problem. the vrpsd looks for vehicle routes to connect all customers with a depot, so that the total distance is minimized, each customer visited once by one vehicle, every route starts and ends at a depot, and the travelled distance and capacity of each vehicle are less than or equal to the given maximum value. contrary to the classical vrp, in the vrpsd the demand in a node is known only after a vehicle arrives at the very node. this means that the vehicle routes are designed in uncertain conditions. this paper presents a heuristic and meta-heuristic approach for solving the vrpsd and discusses the real problem of municipal waste collection in the city of niš. key words: vehicle routing problem, stochastic demands, municipal waste, heuristic and meta-heuristic 1. introduction waste and mail collection, delivery of newspapers, milk, bread and postal packages, distribution of buses and planes in their networks of lines, all of these represent problems that researchers and traffic experts face daily in practice. inadequate collection and transport, as functions of municipal waste management, result in enormous economic and environmental losses, and they serve as a major incentive to many researchers in their quest for appropriate systemic solutions. the set of transport means supporting the municipal waste collection and transport process most often comprises the majority of the vehicle fleet of public utility companies, which usually amounts to 50–70% of all transport units. this process is also dominant in those business systems that integrate a number of public utilities. in such systems, 15–40% of transport units support the municipal waste collection and transport process. optimizing and applying a combination of heuristic and meta received march 18, 2019 / accepted may 28, 2019 corresponding author: danijel marković faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: danijel.markovic@masfak.ni.ac.rs 108 d. marković, g. petrović, ţ. ćojbašić, a. stanković heuristic methods in only one such system can lead to the reduction in mechanization fuels of 10 ÷ 25% [1]. collecting municipal waste in urban areas can be observed as solving a vehicle routing problem. such a problem is known in the literature as the waste collection vehicle routing problem. the vehicle routing problem (vrp) is one of the most challenging problems of combinatorial optimization. it was first mentioned in 1959 by dantzig and ramser [2]. since then, the vrp has been increasingly applied in the solution of various problems and it bears a great economic importance for the reduction in operating costs of distribution systems [3, 4]. with the aim of satisfying real problems for the vrp solution, several constraints are commonly introduced in the solution of the problem, such as a larger number of depots, different types of vehicles (homogeneous and heterogeneous), different types of customer demands (deterministic and stochastic), infrastructural limitations (one-way streets, prohibited roads), types of service performance (pickup, delivery and mixed), etc. if all these constraints are taken into consideration, the vrp becomes much more complicated to solve and it falls under the np difficult problems category. because of this, different variants of the vrp problem have been introduced in the literature. the basic model of the vehicle routing problem is the capacitated vehicle routing problem – cvrp [5]. in the cvrp customer demands are deterministic, known in advance and cannot be separated. vehicles are identical and they have a common starting point, while the only constraint is the vehicle capacity. the goal function expresses the demand to minimize total costs. various vehicle routing problems are derived from the cvrp such as: fig. 1 basic vehicle routing problems and their relations for the sake of a simplified presentation, fig. 1 uses abbreviations. the detailed explanation of fig. 1 is as follows: distance–constrained capacitated vehicle routing problem – dccvpr [6]; vehicle routing problem with backhauls – vrpb [7]; vehicle routing problem with time window – vrptw [8], vehicle routing problem with pickup and delivery – vrppd [9]; vehicle routing problem with backhauls and time window – vrpbtw [10]; vehicle routing problem with pickup and delivery and time window – vrppdtw [11] in the above vehicle routing problems, the demands of the given transport network are deterministic, i.e. known in advance. however, in certain cases these demands can be random variables, that is, they become a stochastic quantity, which results in the standard the vehicle routing problem with stochastic demands in an urban area – a case study 109 cvrp being expanded into a capacitated vehicle routing problem with stochastic demands – cvrpsd [12]. in the past few years, numerous researchers have applied heuristic and meta-heuristic methods to solve the cvrpsd [13, 14, 15, 16]. examples of stochastic demands can be found in various transport activities. one such example is the municipal waste collection considered in this paper. namely, here we discuss a capacitated vehicle routing problem with stochastic demands – cvrpsd, for municipal waste collection in urban areas. 2. mathematical formulation of the cvrpsd in practice, municipal waste collection in urban areas can be observed as a capacitated vehicle routing problem with stochastic demands – cvrpsd. this means that the amount of waste in nodes for the given transport network is a randomly variable quantity. the amount of waste may vary depending on the season and it can be known only after a vehicle arrives at a certain node to be served. for this reason, it is very hard to design the routes in the classical manner. solving the cvrpsd for municipal waste collection (cvrpsd-mwc) thus encompasses finding their routes for the given transport network with minimal costs while meeting the following conditions:  there is only one depot and each route begins and ends in that depot,  the locations of the depot and the nodes are known,  the amount of waste in each node is a stochastic variable with normal distribution,  the capacity of the waste collection vehicle is known,  the sum of the amounts of municipal waste in a single route must not be greater than the vehicle capacity,  each node must be visited only once. 2.1 the chance-constrained model of the cvrpsd the chance-constrained (ch-c) method is one of the main methods for stochastic optimization under various conditions of uncertainty. the ch-c method allows for the probability of meeting a certain constraint to be above a necessary level. in other words, this method limits the allowable region thus reaching a high level of solution reliability. in recent times, numerous researchers have applied the ch-c method to solving certain variants of the vrp [17, 18, 19, 20]. in this paper the ch-c model for the cvrpsd observes one level of constraint, and that is the amount of waste. namely, it is expected that the amount of waste per node is smaller than the vehicle capacity, with the level of reliability α. using the ch-c method to solve the cvrpsd, the goal function can be defined as follows:      n i n j ijij xcf 0 0 min (1) with the constraints:    n i ij njx 0 ,...,2,1,1 (2) 110 d. marković, g. petrović, ţ. ćojbašić, a. stanković      n i n i jiij njxx 0 0 ,...,1,0,0 (3)      n j n j jj zxx 1 0 1 00 2 (4)      vi vj iji qxqp )( (5) 0 ≤ qij ≤ q, i = 0,1,..., n; j = 0,1,..., n (6) zi{0, 1}, i = 0,1,..., n (7) xij{0, 1}, i= 0,1,..., n; j = 0,1,..., n (8) the minimization function for the cvrpsd is shown in eq. (1), where cij is the transport costs of the vehicle between node i and node j; i, jv it is assumed that cij=dij the constraint given in eq. (2) shows that each node must be visited only once, while the constraint given in eq. (3) presents the continuation of the vehicle flow, i.e. the fact that after serving node j the vehicle must leave that same node. the constraint given in eq. (4) shows that each vehicle must start its route in the depot and end it there. using the ch-c conditions one can assure that the amount of collected waste on the route is smaller than the vehicle capacity with known probability (p) as shown in constraint (5). the capacity constraint (6) shows that the vehicle load never exceeds the vehicle capacity. furthermore, q is the maximum capacity of the vehicle, while qij is the capacity of the vehicle after visiting node i, and before visiting node j. constraints (7) and (8) define the intervals of variables zi and xij. it follows that: 0, vji therwiseo0, j node the visits and inode the after vehicle the if1, xij     mkvi therwiseo inodethevisitsvehicletheif zi     , ,0 ,1 0 constraint (5) can be solved by applying the ch-c conditions. it is assumed that the amount of waste per each node is a random variable with normal distribution, which can be presented as: qi ~ n(μi , σi 2) (9) where μi is the total expected amount of waste for the i-th node, σi 2 is the standard deviation (variance) from the amount of waste for the i-th node. parameters μi and σi 2 can be written using eqs. (10) and (11):      n i iji n j i xqe 0 0 )]([ (10)      n i iji n j i xqvar 0 0 2 )]([ (11) where e(qi xij) is the mathematical expectation of normal distribution, while var (qi xij) is the variance, i.e. the normal distribution scaling parameter. the vehicle routing problem with stochastic demands in an urban area – a case study 111 if the expected customer demand is presented in the following way: qxqexqe n i iji n j iji     0 0 )]([)( (12) and the standard deviation as:      n i iji n j iji xqvarxqvar 0 0 )]([)( (13) using eqs. (12) and (13) one can rework the ch-c condition with constraint (5) into eq. (14) [21].                             n i iji n j n i iji n j xqvar qxqe p 0 0 0 0 )]([ )]([ (14) it is important to emphasize that eq. (14) holds if and only if eq. (15) holds as well: 0 01 0 0 [ ( )] ( ) [ ( )] n n i ij i j n n i ij i j e q x q var q x            (15) eq. (15) can be written as a deterministic equivalent: 1 0 0 0 0 ( ) [ ( )] [ ( )] n n n n i ij i ij i j i j var q x e q x q         (16) where φ is the standard function of normal distribution, while φ-1 is the inverse function of function φ . as the amount of waste is assumed to be a random variable with normal distribution, this means that the waste collection vehicle routes should be designed under the conditions of uncertainty regarding the amount of waste, i.e. the demand value in nodes. the next section of the paper presents the results of the routing optimization for the observed problem. parameter α takes the value of 0.8 for the optimization of the case study routes [22]. 3. defining the model and method for the cvrpsd the cvrpsd model considered in this paper is defined by a transport network that comprises one depot and 29 nodes (fig. 2). the transport network presents “area” 103 according to the division of the territory of the city of niš by the puc “mediana-niš”. the nodes in the transport network present the locations of the containers as defined by the coordinates, i.e. the latitude and longitude (tab. 1). apart from the coordinates, tab. 2 provides the number of containers per each node for the observed transport network. in the transport network, the first and the final node (the depot) is marked with “1”. the other nodes of the transport network are numbered from 2 to 30. 112 d. marković, g. petrović, ţ. ćojbašić, a. stanković fig. 2 the transport network of container nodes for municipal waste collection table 1 coordinates and number of containers for the observed transport network location number latitude longitude number of containers per location depot 43.319256 21.919682 2 43.322794 21.913082 4 3 43.322464 21.914317 4 4 43.324196 21.914412 2 5 43.324696 21.916001 2 6 43.323830 21.916709 3 7 43.323338 21.917632 5 8 43.322615 21.918220 1 9 43.323829 21.921069 2 10 43.322712 21.920759 2 11 43.322109 21.922545 1 12 43.321878 21.921097 1 13 43.321168 21.917535 4 14 43.322144 21.918179 2 15 43.322471 21.915755 3 16 43.319104 21.920206 1 17 43.319854 21.920548 2 18 43.320413 21.921150 2 19 43.320076 21.922198 3 20 43.320751 21.921739 2 21 43.320552 21.923959 2 22 43.322529 21.910364 1 23 43.322181 21.911683 1 24 43.321311 21.913388 1 25 43.321119 21.913767 1 26 43.320082 21.916993 1 27 43.319579 21.917587 1 28 43.319241 21.918091 1 29 43.319097 21.918627 1 30 43.318779 21.919317 1 the vehicle routing problem with stochastic demands in an urban area – a case study 113 to solve the model, waste containers for municipal waste collection, so-called semiunderground containers, with the capacity of 3m3 are installed in the transport network. this model does not consider the optimal locations and the number of containers. the number and location of containers are selected on the basis of the previous positions of waste containers determined by the puc “mediana-niš”. on the locations where there are two or more containers, their positions are defined by a single node, i.e. a single coordinate. the waste collection vehicle departs the depot and it is always assumed empty. waste collection should be performed with only one vehicle. this vehicle possesses a superstructure with a telescopic crane adapted to semi-underground waste containers. it is predicted that the waste collection in “area” 103 takes place three times a week (on mondays, wednesdays and fridays). the matrix of shortest distances is symmetrical, and its elements represent the shortest possible real distance between the node pairs for the transport network. the amount of municipal waste in each node is stochastic, i.e. randomly variable. for the optimization purposes, the municipal waste collection route was monitored at specific time intervals. the monitoring was performed in ten instances for each node, during different seasons. the assessment of how full the containers were was made at each transport network node and this was recorded in a tab. 2. this table was filled on the basis of the routing card. table 2 intervals of monitoring the amount of waste assessment per transport network node node interval of the amount of waste assessment 1 2 3 4 5 6 7 8 9 10 1 12.8 13.6 12.5 13.8 13.0 12.6 14.2 13.9 14.4 13.3 2 13.3 14.4 13.9 12.6 14.2 13.0 13.8 12.5 13.6 12.8 3 6.9 6.2 6.8 6.4 6.6 7.2 7.0 7.1 6.3 6.5 4 6.4 6.8 6.2 6.9 6.5 6.3 7.1 7.0 7.2 6.6 5 10.3 9.4 10.2 9.6 10.0 10.8 10.4 10.7 9.5 9.7 6 16.6 18.0 17.4 15.8 17.8 16.2 17.2 15.6 17.0 16.0 7 3.2 3.4 3.1 3.4 3.2 3.2 3.6 3.5 3.6 3.3 8 6.9 6.2 6.8 6.4 6.6 7.2 7.0 7.1 6.3 6.5 9 6.6 7.2 7.0 6.3 7.1 6.5 6.9 6.2 6.8 6.4 10 3.1 3.4 3.6 3.7 3.1 3.0 3.4 3.4 3.5 3.2 11 3.4 3.1 3.4 3.2 3.3 3.6 3.5 3.6 3.2 3.2 12 13.3 14.4 13.9 12.6 14.2 13.0 13.8 12.5 13.6 12.8 13 6.2 6.7 7.2 7.4 6.2 6.0 6.8 6.9 7.0 6.3 14 9.6 10.2 9.4 10.3 9.7 9.5 10.7 10.4 10.8 10.0 15 3.3 3.6 3.5 3.2 3.6 3.2 3.4 3.1 3.4 3.2 16 6.9 6.2 6.8 6.4 6.6 7.2 7.0 7.1 6.3 6.5 17 6.4 6.8 6.2 6.9 6.5 6.3 7.1 7.0 7.2 6.6 18 10.3 9.4 10.2 9.6 10.0 10.8 10.4 10.7 9.5 9.7 19 6.6 7.2 7.0 6.3 7.1 6.5 6.9 6.2 6.8 6.4 20 6.4 6.8 6.2 6.9 6.5 6.3 7.1 7.0 7.2 6.6 21 3.3 3.6 3.5 3.2 3.6 3.2 3.4 3.1 3.4 3.2 22 3.1 3.4 3.6 3.7 3.1 3.0 3.4 3.4 3.5 3.2 23 3.4 3.4 3.1 3.4 3.2 3.2 3.6 3.5 3.6 3.3 24 3.3 3.6 3.5 3.2 3.6 3.2 3.4 3.1 3.4 3.2 25 3.4 3.1 3.4 3.2 3.3 3.6 3.5 3.6 3.2 3.2 26 3.2 3.4 3.1 3.4 3.2 3.2 3.6 3.5 3.6 3.3 27 3.4 3.1 3.4 3.2 3.3 3.6 3.5 3.6 3.2 3.2 28 3.3 3.6 3.5 3.2 3.6 3.2 3.4 3.1 3.4 3.2 29 3.4 3.1 3.5 3.4 3.2 3.2 3.6 3.5 3.6 3.3 30 12.8 13.6 12.5 13.8 13.0 12.6 14.2 13.9 14.4 13.3 114 d. marković, g. petrović, ţ. ćojbašić, a. stanković the routing card contains the following information: the numerical marker of the container node; the name of the place where the containers are located; the time of arrival, pickup and departure; the number of containers per node and the assessment of how full the container is; the accessibility of the container; the note. 3.1. stochastic optimization problems in stochastic optimization, optimization parameters have a random character described by the methods from the theory of probability and statistics. when one observes the constraint functions whose parameters have a random character, it is not certain that a selection of control variables will ensure that the function is satisfied. therefore, a new optimization task is defined to demand that the probability of a constraint being met should be greater than a predetermined value. if the parameters in constraint functions have a random character (random quantities), then they can be described using, for example, normal distribution, poisson distribution or gamma distribution. 3.2. heuristics and meta-heuristics for the cvrpsd when solving a vehicle routing problem, heuristic methods are used to construct routes, with the construction and improvement of routes being performed iteratively in relation to the goal function. bearing in mind that this paper deals with the determination of the most favorable (optimal) routes of municipal waste collection vehicles, as well as considering all the observed constraints, it can be safely assumed that the sufficiently good solutions were determined by applying the global optimization methods. on the basis of research, the clarke and wright’s savings algorithm is considered in this paper as the representative of the approaches of the constructive heuristic methods [23]. meta-heuristics is conceived as a means of solution of complex optimization problems where other optimization methods cannot provide an efficient and economical solution to the optimization problem at hand. one of the basic representatives of meta-heuristics is the genetic algorithm [24]. today these methods are regarded as belonging to the most practical approaches to solving various complex problems [25], which is particularly related to the solution of numerous real problems that are combinatorial in nature, such as the vehicle routing problem itself. it can generally be said that meta-heuristics is a higher level of heuristics. out of the group of meta-heuristic methods used for the solution of municipal waste collection vehicle routing problems, this paper employs the 2-opt local search [26] and simulated annealing – sa [27]. 3.3. stochastic simulation for computing the expected value and probability check the first step in the optimization of the cvrpsd is to compute the expected value of the amount of municipal waste (μi) and check probability (β). this step is necessary due to the stochastic character of the amount of municipal waste per transport network node. based on the input data on the assessed amount of waste per node, one can compute the expected values of the amount of municipal waste (μi) for each node of the transport network since the distribution is known, i.e. normal distribution. after the expected value of the amount of municipal waste is computed, variance (σi 2) gets computed as well. to compute the mathematical expectancy and variance procedure 1 was used, and its pseudo code is shown in algorithm 1. the vehicle routing problem with stochastic demands in an urban area – a case study 115 algorithm 1: procedure 1 start define the assessed amount of waste per transport network node; define q; for n = 1; n ≤ nuk; n = n + 1; sumn = 0; for i =1; i ≤ 10; i = i + 1; sumn = sumn + qni; pi = qni / sumn: end for end for for n = 1; n ≤ nuk; n = n + 1; compute en; compute σn; compute φ(α); if   n n qe     probability condition = true; else probability condition = false; end if end for end here, sum is the amount of waste per transport network node, qni is the assessed amount of waste in the node. when these two parameters are computed, then probability (β) is checked. algorithm 1 presents the procedures to check the probability. the last step in this procedure is the checking of the ch-c condition, and if this condition is met, the procedure continues (true). in the opposite case the procedure is stopped (false). 3.4. initial solution the next step in the cvrpsd optimization is the formation of the initial solution. bearing in mind that this is a stochastic problem, in line with the previous explanation, the problem is reduced to the solution of the deterministic problem by applying eqs. (15) and (16). the c-w savings algorithm was used to obtain the initial solution. in the application of the c-w savings algorithm parameter qi was substituted with parameter μi. procedure 2 presents the pseudo code for the c-w savings algorithm (algorithm 2). algorithm 2: procedure 2 start define distance matrix; define q; call procedure 1; compute s; sort s' in a non-increasing sequence; form a partial route; expected demand = µi; for all savings from sequence if (probability condition == true) if met operative constraints 116 d. marković, g. petrović, ţ. ćojbašić, a. stanković if expected demand + µi ≤ q expected demand = µi + expected demand form route end if end if end if end for vehicle fullness = expected demand; print routes; print vehicle fullness; end 3.5. 2-opt search and sa meta-heuristic for the cvrpsd the first improvement of the initial cvrpsd solution was performed by applying the 2-opt local search. during the improvement of the initial solution, the number of iterations was varied (1e3 and 1e6). the initial solution was improved by applying the 2opt local search algorithm. the pseudo code of the 2-opt algorithm for the improvement of the initial solution is presented in algorithm 3. algorithm 3: 2-opt algorithm for the improvement of the initial solution start load initial route; u0 = initial route length; for (i = 1; i ≤ n 2; i = i + 1) for (j = i + 2; j ≤ n; j = j + 1) u' = d(i, j) + d(i+1, j+1) d(i, i+1)d(j, j+1); if (u'< u) u' = u; end if end for end for end the next algorithm used to optimize the cvrpsd was the sa algorithm. the solution obtained by applying the sa algorithm largely depends on adjusting the parameters of the algorithm itself. however, this paper does not consider the selection of optimal parameters for the given problem but uses the recommended parameters. the parameters of the sa algorithm used to solve the cvrpsd are [27]: initial temperature t0 = 100, temperature reduction factor α = 0.8. the pseudo code of the sa algorithm for the solution of the cvrpsd is presented by algorithm 4. algorithm 4: sa algorithm for solving cvrpsd start define model; load initial solution u obtained by c-w savings algorithm; define sa algorithm parameters; best solution = u; t = t0; the vehicle routing problem with stochastic demands in an urban area – a case study 117 for it1 = 1; it1<maxit1; it1 = it1 + 1; for it2 = 1; it2<maxit2; it2= it2 + 1; create adjacent node randomly; compute new route length u' ; if (u' ≤ u); u = u'; else δf = u' u;         t f expp ; if p ≥ r, r is a random number from interval [0,1]; u = u'; end if end if end for t = α∙t; end for end here it is the number of iterations, and maxit is the maximum number of iterations, r is the random number from the interval [0,1], and p is the probability of accepting a new solution. 4. computational results the application of the c-w savings algorithm to the solution of the cvrpsd yielded four routes with the total length of 11.34 km, and this solution represented the initial solution. this solution was improved by applying the 2-opt local search algorithm and the sa algorithm. the initial solution was improved by applying the 2-opt local search algorithm. the application of this algorithm yielded three routes with the total length of 9.92 km. fig. 3 shows the appearance of the transport network routes. fig. 3 graphic representation of the model routes obtained by the 2-opt search algorithm 118 d. marković, g. petrović, ţ. ćojbašić, a. stanković the application of the sa algorithm, with the recommended parameters, yielded three routes with the total length of 9.72 km. fig. 4 shows the appearance of the transport network routes. fig. 5 illustrates a comparison of the vehicle loading on the routes obtained by the 2-opt local search algorithm, on the one hand, and the sa algorithm of the solution of the cvrpsd, on the other. the figure shows that the vehicle is not loaded when it leaves the depot and the change in the vehicle loading can be observed along the route. as already emphasized, the maximum capacity of the vehicle is 60 m3. furthermore, fig. 5 shows that the solution obtained by applying the 2-opt local search algorithm results in the vehicle loading in the range of 59.71 m3, 58.76 m3 and 50.34 m3, which represents around 93.8% of the vehicle capacity. the solution obtained by applying the sa algorithm results in the vehicle loading in the range of 59.25 m3, 59.80 m3 and 49.76 m3, which also represents around 93.8% of the vehicle capacity. this shows that the application of both algorithms for the solution of such problems is justified. fig. 4 graphic representation of the model routes obtained by the sa algorithm fig. 5 graphic representation of the expected vehicle loading the vehicle routing problem with stochastic demands in an urban area – a case study 119 5. conclusion the optimization of the cvrpsd by applying the heuristic and meta-heuristic algorithms yielded the routes of the municipal waste collection and transport vehicles for the observed transport network. the improvements obtained by thus designed routes in comparison with the existing routes determined by the puc “mediana-niš” are related to the reduction in the total length of the path and the reduction in the total operating time of the municipal waste collection vehicles. fig. 6 shows the relation between the total length of the routes per day obtained by solving the cvrpsd and the routes of the puc “mediana-niš” vehicles for municipal waste collection. this relation is presented for six intervals, where one interval takes two months into account. fig.6 below shows that the routes obtained in the solution of the cvrpsd model are shorter in five intervals, while the route currently taken by the puc “mediana-niš” vehicles is slightly shorter only in the second interval. the average length of the routes of the puc “mediana-niš” municipal waste collection vehicles in “area” 103 is 65.33 km/day, while the length 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capacitated vehicle routing problem with two-dimensional loading constraints, european journal of operational research, 265(3), pp. 843-859. https://www.inderscienceonline.com/author/bernal%2c+jose https://www.inderscienceonline.com/author/bernal%2c+jose https://www.inderscienceonline.com/author/paz%2c+juan+camilo https://www.inderscienceonline.com/author/paz%2c+juan+camilo https://www.inderscienceonline.com/author/gatica%2c+gustavo https://www.inderscienceonline.com/author/gatica%2c+gustavo https://www.inderscienceonline.com/author/gatica%2c+gustavo facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 79 89 https://doi.org/10.22190/fume190428006a © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  workspace analysis and optimization of the parallel robots based on computer-aided design approach badreddine aboulissane, larbi el bakkali, jalal el bahaoui team modeling and simulation of mechanical systems, faculty of sciences, abdelmalek essaadi university, tetouan, morocco abstract. this paper provides workspace determination and analysis based on the graphical technique of both spatial and planar parallel manipulators. the computation and analysis of workspaces will be carried out using the parameterization and threedimensional representation of the workspace. this technique is implemented in cad (computer aided design) software catia workbenches. in order to determine the workspace of the proposed manipulators, the reachable region by each kinematic chain is created as a volume/area; afterwards, the full reachable workspace is obtained by the application of a boolean intersection function on the previously generated volumes/areas. finally, the relations between the total workspace and the design parameters are simulated, and the product engineering optimizer workbench is used to optimize the design variables in order to obtain a maximized workspace volume. simulated annealing (sa) and conjugate gradient (cg) are considered in this study as optimization tools. key words: cad software, design parameters, optimization, parallel robots, workspace analysis 1. introduction a parallel manipulator is a mechanism in which there are two or more closed kinematic chains attaching the base to the mobile platform. nowadays, most of the manipulators are serial architecture; parallel robots exhibit many advantages, such as high speeds and accelerations, low mobile masses, high stiffness, and great accuracy. the most notable disadvantage of the parallel manipulators is their relatively small workspaces. one can use the workspace volume or surface as an objective function for optimization. in this sense, the researchers focused on workspace determination as a performance index in order to design robots for specific industrial applications. the calculation of the parallel robots’ workspace received april 28, 2019 / accepted january 25, 2020 corresponding author: badreddine aboulissane team modeling and simulation of mechanical systems, faculty of sciences, abdelmalek essaadi university, bp. 2121 m'hannech ii, tetouan, morocco e-mail: b.aboulissane@gmail.com 80 b. aboulissane, l. el bakkali, j. el bahaoui is a complex problem due to the kinematic modeling difficulty. however, the problem of workspace optimization of the parallel manipulators to obtain a prescribed workspace has been investigated in few articles. the concept of the prescribed workspace is a significant issue to optimize and to synthesize a robot. the actuated joint variables, the range of joints motion and the mechanical interferences between the links essentially influence the parallel manipulators’ workspace. in this paper, we focus on some areas of the space that surrounds the manipulator, and limiting its workspace to the prescribed area. several papers studied this problem based on geometrical techniques, and using optimization algorithms to synthesize the design parameters of the parallel manipulators. gallant and boudreau [1] used a genetic algorithm in order to optimize a 3-dof planar parallel manipulator to obtain a workspace as close as possible to a prescribed one. singularities and workspace of planar 3-rpr parallel mechanism for maximal singularityfree workspace by optimizing the geometric parameters are investigated by jiang and gosselin [2, 3] and yang and o’brien [4]. di gregorio and zanforlin [5] studied the workspace of the 3-ruu and the delta robot. they concluded that these robots could have the same closure equations and workspace when some geometric conditions are satisfied. chablat et al. [6, 7] compared 3-dof parallel kinematic machines using two design criteria: regular workspace shape and a kinetostatic performance index that needs to be as homogenous as possible throughout the workspace ; this technique is based on the interval analysis method. in [8] the workspace optimization of translational 3-upu parallel robot is performed using its parameterization by two design variables, which are the prismatic joint stroke and the distance between the base and the mobile platform. zhao, chu, and feng [9] discussed the analogous symmetry properties between the workspace and the mechanism structure. gao, liu, and chen [10] analyzed the relationship between the shapes of the workspaces and the link lengths for 3-dof planar parallel manipulators; his results are useful for the designers to optimize the robots regarding the workspace index. hay and snyman [11, 12] focused on the numerical multi-level optimization for the synthesis of the 3-dof parallel manipulators for a desired workspace. in [13] the workspace of gough-stewart platform was optimized using the genetic algorithm. the idea is to minimize the areas, which do not belong to the intersection between two areas: the workspace of the robot and the prescribed workspace. a genetic algorithm based method is used also in [14] to deal with the optimal dimensional synthesis of the delta robot for a prescribed workspace. the geometrical approaches have been used to represent the workspace of the parallel manipulators, by assad arrouk et al. [15], aboulissane et al. [21], bonevet al. [16], gosselin [17], and merlet [18]. the principle of these methods is to deduce, from the constraints on each limb, a geometrical entity (sub-workspace) which describes all the possible poses of the tool center point that satisfy the leg constraints. then, the robotic manipulator workspace is generated by the intersection of all the subworkspaces. tsirogiannis et al. [22] presented an overall structural design optimization approach for a robot arm link seeking mass reduction and satisfaction of manufacturability with sls am technique. in this paper, a graphical based technique is addressed for workspace’s determination, analysis and optimization of two parallel robots, which are the 3-rpr planar manipulator, and the delta robot. the first section is dedicated to the description of the proposed manipulators. in the next section, we present the steps to determine the workspace of both robots. the last section is about the comparison of the two optimization methods, applied to the workspace of the delta robot. workspace analysis and optimization of the parallel robots based on computer-aided design approach 81 2. kinematic scheme and description of the 3-rpr and the delta robots 2.1. the 3-rpr planar parallel robot fig. 1 shows the kinematic scheme of the 3-rpr planar robot. this mechanism is a parallel robot with closed loop chains. three actuated prismatic joints are linked to passive joints , , and fixed to the base, and , , fixed to the mobile platform. the actuated prismatic joints coordinates are given by the length of the legs, named , , and . the orientation of the mobile platform is given by angle . the components of points and are respectively and . each limb generates an annular region bounded by two concentric circles with radii of and , the centers of the circles are defined by eq. (1), and eq. (2): cos( ) sin( )ci ai bi bix x x y    (1) sin( ) cos( )ci ai bi biy y x y    (2) fig. 1 kinematic diagram of the 3-rpr planar parallel robot the vector describing the 3-rpr parallel manipulator parameters is defined as follows: 1 min max 1 1 2 2 3 3[ ]t c c c c c cx y x y x y   (3) 2.2. the delta parallel robot the robot under study in this section is the delta parallel robot depicted in fig. 2(a); it is composed of a triangular moving platform linked to a triangular fixed base with three closed parallel chains. each one consists of an actuated rotational joint mounted near to the fixed base; the parallelograms and spherical joints transmit the movement of the mobile platform. in this work, all the three arms of the manipulator are identical in terms of geometrical parameters (fig. 2(b)). 82 b. aboulissane, l. el bakkali, j. el bahaoui (a) (b) fig. 2 (a) geometric scheme of the delta robot; (b) the delta robot parameterization the independent design variables for the delta robot are: 2 1 2[ ]tl l r r  (4) 3. workspace determination of the proposed robots first, we need to determine workspace of the proposed robots. we can define this region as the reachable positions and rotations by the end-effector center point, generally located on the platform of the robots. in this work, we used a geometrical technique for the representation of the workspace through the catia software, and there are no limits for all the revolute joints used for each manipulator. 3.1. workspace of the 3-rpr planar robot for this robot, we obtain the workspace by the intersection of three circular areas, which correspond to the areas accessible by the end-effector point center when each leg is taken as a serial manipulator. fig. 3 shows the steps we followed to generate the workspace of the 3rpr manipulator in the catia. (a) (b) (c) fig. 3 steps of the workspace determination of the 3-rpr robot[15] workspace analysis and optimization of the parallel robots based on computer-aided design approach 83 fig. 3(a) depicts the first step; it consists of creating three annular regions in the sketcher workbench; then the pad and pocket commands are applied to the drawing with a finite thickness in the part design workbench. the second step is performed under the part design workbench; it consists of applying the first intersection boolean operation. fig. 3(b) above shows the obtained result. the final step to determine the workspace of the 3-rpr is to apply a second intersection boolean operation on the two remaining regions in the second step. the obtained shape corresponds to the theoretical workspace of the mechanism shown in fig. 3(c). this step is also done in the design part workbench. the area of the 3d model presented in fig. 3(c) is calculated by using a smart area parameter. since the thickness of this 3d model is neglected, the workspace area of the 3rpr robot is obtained, dividing by two, the area previously calculated. the design parameters used to obtain this workspace are tabulated in table 1: table 1 design parameters of the 3-rpr manipulator design parameters limb 1 limb 2 limb 3 xci (mm) -188,632 132,159 56,473 yci (mm) -43,697 -141,512 185,209 min (mm) 120 max (mm) 270  (degrees) 102 area (cm²) 197,02 for each orientation , the workspace of the robot has different shape and area. table 2 illustrates the workspace of the robot for few orientations of the mobile platform. table 2 variation of the workspace with respect to the orientation of the mobile platform a=70,611 cm² a=156,077 cm² a=197,668 cm² a=144,380 cm² a=29,600 cm² a=22,411 cm² 3.2. workspace of the delta robot the workspace of the delta parallel robot is defined as a three dimensional volume in the cartesian space; this volume is reached by a point on the mobile platform. the 84 b. aboulissane, l. el bakkali, j. el bahaoui equations used to determine the workspace of a parallel robot are generally complex to solve by using the traditional approaches. hence, the cad-based approach is used in this work to determine geometrically the workspace of the delta parallel robot. the parallel robot workspace robot can be quickly generated as an area or a volume using the catia, then the complex technique such the numerical method. discretization based techniques produce an approximate form of a low quality workspace. to improve it, it is necessary to use other graphical methods. by implanting the problem of workspace determination in a cad software, these techniques will become more reachable to industry and more precise. as the first step, the proposed method for workspace determination of the delta robot consists in assuming all legs to be independent serial arms having the mobile platform as tool center point. then, the region swept by the tool center point of each arm is determined for a given orientation of this point. in [14] the workspace of the delta robot is presented by following eq. (5): 2 2 2 2 2 2 2 2 2 1 2 1[( ) ] 4 [( ) ]t t t t tx r y z l l l x r z        (5) with r r r   and: cos sin sin cos t i i t i i t x x y y x y z z             (6) (a) (b) (c) (d) fig. 4 generation of the workspace for a delta robot based on the catia v5. (a) the three torus intersect; (b) first intersection boolean operation; (c) second intersection; (d) the workspace of the delta robot (z < 0) as shown in fig. 4, the workspace of the delta robot is based on three tori. the first step consists of drawing a circle with a radius , and another circle with a radius passing by the center of the first circle (fig. 5). each limb of the delta robot generates a torus, fig.4(a) depicts the intersection of those three volumes, and the rest of the figures shows the intersection boolean operations with the final shape of the workspace of the robot presented by fig. 4(d). fig. 5 torus in sketcher workbench workspace analysis and optimization of the parallel robots based on computer-aided design approach 85 3.3. workspace analysis of the delta robot before starting the optimization problem, an analysis between the reachable workspace and the design parameters is required. we presented examples for each variable and , as well as the resulting volume. first we fixed r = 55 mm, r =30 mm, and = 100 mm, then length is varied from 100 mm to 250 mm. table 3 shows the boundaries of the reachable workspace for length of 100 mm, 150 mm, 200 mm, and 250 mm. table 3 workspace shape versus l2 = 100 mm w = 4,765 dm3 l2 = 150 mm w = 4,688 dm3 l2 = 200 mm w = 4,661 dm3 l2 = 250 mm w = 4,647 dm3 it can be seen from table 3 that the workspace volume of the delta robot increases with reducing length of the forearm. now, to demonstrate the effect of length , forearm is fixed at 250 mm, r = 55 mm, r = 30 mm, and is varied from 100 mm to 250 mm. table 4 workspace shape versus l1 = 100 mm w = 4,647 dm3 l1 = 150 mm w = 15,753 dm3 l1 = 200 mm w = 37,637 dm3 l1 = 250 mm w = 75,040 dm3 as shown in table 4, the workspace volume increases considerably by increasing the length . the volume starts to take a cup-shape. 4. optimization problem in robotics, the designer uses numerous indices to evaluate the performance of a manipulator; among these indices, we can mention the workspace that describes the potential robot utilization. in this work, we are using the reachable workspace as a performance index in order to optimize the design parameters of the delta robot. the optimization problem is formulated as follows: 2 2, ,min 2, 2, ,max ( ) i i i maximize w subject to      (7) 86 b. aboulissane, l. el bakkali, j. el bahaoui where w is the workspace volume, and is the vector defined by eq. (4). the main purpose of the maximization of the workspace is to expand the capabilities of the delta robot. the parameters that have an effect on the volume and the shape of the reachable workspace of the manipulator are: and with (j=1,2,3). the parameterization used in the catia software is shown in fig. 6: fig. 6 parameters of the delta robot on the catia for this optimization problem, we used the catia “product engineering optimizer” workbench in which we can use different algorithms such as: conjugate gradient method (cg) a local algorithm and the simulated annealing (sa) a global algorithm. both the methods are employed in our present study. the simulated annealing is listed as the oldest algorithm among the metaheuristics that had an explicit strategy to avoid local minima; it can be applied to the majority of optimization problems. the behavior of this algorithm is strongly dependent on the problem addressed [19]. the other algorithm, which is the conjugate gradient, is a mathematical approach used on both linear and non-linear systems; this approach can be used as an iterative algorithm and a direct method [20]. to realize this optimization, we choose the last five parameters that are shown in fig. 6; we excluded the angles representing the angular offset between different kinematic chains. the optimization parameters are usually provided with an upper and lower limit. the main goal of this optimization is to maximize the objective function represented by the workspace volume of the delta robot, based on the parameters presented in table 5, which can describe eq. (7). the initial parameters correspond to the workspace shown in table 6(a) with a volume w = 4,765 dm3. the first optimization is done using the (sa) algorithm, optimized values are tabulated in table 5 corresponding to the workspace summarized in table 6(b) with a volume w = 215,712 dm3. secondly, we applied the (cg) method. table 6(c) shows the shape of the workspace with a volume w = 110,265 dm3. this workspace is associated with the values of the design parameters presented also in table 5. workspace analysis and optimization of the parallel robots based on computer-aided design approach 87 table 5 optimization results design parameters initial values simulated annealing conjugate gradient r (mm) 30 193,77 54,538 r (mm) 55 344,198 120,308 l1 (mm) 100 350 283,757 l2 (mm) 100 112,255 250 volume (dm3) 4,765 215,712 110,265 table 6 workspace optimization without optimization (a) simulated annealing algorithm (b) conjugate gradient algorithm (c) the number of iterations made to reach the objective is 523 for the (sa) algorithm, and 602 for the (cg) method. the time needed to achieve these two optimizations is about 10 minutes. the simulations were performed on a computer that has the following characteristics: cpu @2.10ghz, 8.0 gb ram. (a) (b) fig. 7 evolution of the design parameters for (a) sa algorithm; (b) cg method 88 b. aboulissane, l. el bakkali, j. el bahaoui fig. 7 presents the evolution of the design variables for both algorithms. from the parametric analysis previously presented in table 4 and the design variables evolution shown in figs. 7 (a) and (b), we can conclude that l1 have a significant impact on the volume of the workspace. on the other hand, the two algorithms applied in this study have set the l1 variable to a maximum value, while other parameters r, r, and l2 are showing a variation in a large range searching for a maximum volume of the robot's workspace. fig. 8 provides a comparison of the convergence rates of the results; it can be seen that the performance of the (sa) is more superior to that of (cg) method due to the good speed of convergence with few generations, also, the optimal value reached by the (sa) algorithm is greater than that obtained by the (gc) algorithm. fig. 8 comparison of convergence cost for sa and gc algorithms 5. conclusion in this paper, we have focused on the cad based technique to determine the workspace of planar and spatial parallel robots. this study is performed on the 3-rpr planar parallel robot and the delta robot. we considered the determination and the characterization of the workspace of the two manipulators. for this purpose, we have applied a geometrical approach that has been implemented in catia workbenches. we put in evidence the effectiveness of this technique for the workspace optimization of the delta robot, taking into consideration the joint limits. two algorithms were applied to maximize the workspace of the delta robot, the simulated annealing and the conjugate gradient algorithms. the best result is related to the (sa) algorithm in terms of convergence speed and the best optimal value of the workspace volume. the manipulators studied herein for the workspace analysis and optimization illustrate the efficiency and the capability of the graphical methodology for the designers, to avoid complex mathematical equations. workspace analysis and optimization of the parallel robots based on computer-aided design approach 89 references 1. gallant, m., boudreau, r., 2002, the synthesis of planar parallel manipulators with prismatic joints for an optimal, singularity‐free workspace, journal of robotic systems, 19(1), pp. 13-24. 2. jiang, q., gosselin, c.m., 2006, the maximal singularity-free workspace of planar 3-rpr parallel mechanisms, international conference on mechatronics and automation, pp. 142-146. 3. jiang, q., gosselin, c.m., 2007, geometric optimization of planar 3-rpr parallel mechanisms, transactions of the canadian society for mechanical engineering, 31(4), pp. 457-468. 4. yang, y., o’brien, j.f., 2007, a case study of planar 3-rpr parallel robot singularity free workspace design, international conference on mechatronics and automation, pp. 1834–1838. 5. di gregorio, r., zanforlin, r., 2003, 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the past few years, a standard tool in contact mechanics for solving axially-symmetric contacts. the numerical implementation of the integral transformations of the mdr can be performed in several different ways. in this study, the focus is on a simple and robust algorithm on the uniform grid using integration by parts, a central difference scheme to obtain the derivatives, and a trapezoidal rule to perform the summation. the results are compared to the analytical solutions for the contact of a cone and the hertzian contact. for the tested examples, the proposed method gives more accurate results with the same number of discretization points than other tested numerical techniques. the implementation method is further tested in a wear simulation of a heterogeneous cylinder composed of rings of different material having the same elastic properties but different wear coefficients. these discontinuous transitions in the material properties are handled well with the proposed method. key words: normal contact, method of dimensionality reduction, stress, wear 1. introduction the method of dimensionality reduction (mdr) is a simple and convenient tool for the calculation of contact forces between elastic and viscoelastic bodies. it is particularly easy to use for the simulation of axially-symmetric contacts. since it was first proposed in 2007 [1] the mdr has been applied to a wide range of problems [2]. the method maps a given three-dimensional contact problem to an equivalent contact problem of a transformed indentation profile with a one-dimensional elastic or viscoelastic foundation of independent elements. from a numerical perspective, the solution of the contact problem in the transformed mdr domain is then trivial due to the decoupled degrees of freedom. a variety received may 26, 2018 / accepted june 30, 2018 corresponding author: justus benad tu berlin, institut für mechanik, fg systemdynamik und reibungsphysik, str. d.17. juni 135, 10623 berlin, de e-mail: mail@jbenad.com 128 j. benad of problems can be solved directly in this domain after the initial transformation to the equivalent problem was performed (see for example [3]). however, there are other problems which require multiple transformations to the mdr domain and back, for example due to a continuously changing indentation profile as it appears in wear simulations, see [4-6]. with such kinds of problems, the main difficulty in achieving an accurate and efficient numerical simulation is the implementation of the mdr transformations. these are given by abel-like integral equations and it is well known that their numerical treatment is challenging [7, 8]. this work is dedicated to providing a simple and fast numerical method for the implementation of the mdr transformations for axially-symmetric contact problems. the transformations have an integrable singularity which is handled well with the proposed method. the parts of this work are organized as follows: in section 2, the mdr transformations are rewritten using integration by parts to a form which is well suited for numerical implementation. in section 3, this numerical implementation technique is explained in detail. section 4 gives some advice on optimizing the implementation for maximum speed. section 5 shows exemplary results of the newly proposed technique and highlights its advantages and weaknesses. in section 6, a small addition to the method is presented to further improve it. in section 7, the accuracy of the introduced numerical method is compared to other known implementation techniques which rely on the original form of the transformations. in section 8, an exemplary wear simulation is conducted with the newly introduced technique and compared to the results of other numerical implementation methods. a conclusion is presented in section 9. this work can be regarded as a small addition to the paper “method of dimensionality reduction in contact mechanics and friction: a users handbook” [9]. in the following, only homogeneous elastic material is considered. however, the mdr is applicable also to gradient media [10] and to viscoelastic media [11], which can be treated in a similar manner as described in the present paper. 2. formulation of the mdr transformations for simple numerical implementation the general mdr procedure is fully described in [9]. three main transformations occur in the method: the transformation of three-dimensional profile f(r) to a one-dimensional profile is 2 2 0 ( ) ( ) x f r g x x dr x r     , (1) the transformation of one-dimensional foundation displacement w1d(x) to three-dimensional normal surface displacement w(r) is 1d 2 2 0 ( )2 ( ) r w x w r dx r x    , (2) and the transformation of one-dimensional force density q(x) to three-dimensional pressure distribution p(r) is fast numerical implementation of the mdr transformations 129 2 2 1 ( ) ( ) r q x p r dx x r       . (3) the singularity arising at x = r in the numerical summation can be avoided when rewriting eqs. (1), (2) and (3) to 2 2 0 ( ) ( ) atan ( ) 2 x r g x x f x x f r dr x r             , (4) 1d 1d 2 2 0 2 ( ) ( ) atan ( ) r x w r w r w x dx r x           , (5) and 2 21 1 ( ) log( ) ( ) log( ) ( ) r p r r q r x r x q x dx         (6) using integration by parts. the following example shall illustrate a possible numerical implementation of the three transformations (4), (5) and (6). 3. exemplary numerical procedure consider a uniform discretization of r∈ [0,l] and x∈ [0,l] with n points each and the same step size 1 l h n   , (7) so that ( 1), ( 1), , {1,2,..., }n kr h n x h k n k n     . (8) the first and second derivatives of a discretized indentation profile fn = f(rn) can be obtained via central differences: 1 1 2 n n n f f f h     , (9) 1 1 2 2n n n n f f f f h      . (10) some care must be taken at the borders. to obtain f1 ′ and f1 ′′ recall that in the present framework of the mdr profile f is axially-symmetric and f(0) = 0. thus, it is 1 0f  , (11) and 2 1 2 2 f f h   . (12) 130 j. benad at the other border the values for fn ′ and fn ′′ can remain undetermined. one-dimensional profile gk can now be obtained with eq. (4). it is , for 2,3,..., 1 2 k k k k kg x f x t k n      , (13) where tk is the result of the integral in (4). using the trapezoidal rule, it is 1 2 2 2 , for 2 4 atan , for 3,4,..., 1 4 k k k n n k n k n f h k t r f h f h k n x r                            (14) again, some care must be taken at the borders. to obtain g1, recall that in the framework of the mdr it is g(0) = 0. thus, it is 1 0g  . (15) at the other border the value for gn can remain undetermined. in a quite similar fashion, normal surface displacement wn can be obtained: the first derivative in eq. (5) can be obtained as in eq. (9) using central differences, and the integral can be calculated as in eq. (14) using the trapezoidal rule. subsequent smoothing of wn with wn := (wn–1 + wn + wn+1) / 3 increases the accuracy of wn. the third transformation to obtain pn is again similar to the first and second transformation. the derivatives in eq. (6) can once more be obtained as in eqs. (9) and (10) using central differences. then it is 1 1 log( ) , for 2,3,..., 2n n n np r q s n n       (16) where sn is the result of the integral in (6). using the trapezoidal rule, it is 2 2 2 1 log( ) log( ) , for 2,3,..., 3 2 log( ) , for 2 2 n n n k n k k k n n n n x q h x r x q h n n s x q h n n                 (17) note that at kinks of q the term qk ′′ h = (qk+1 – 2qk + qk–1) / h converges to finite values for decreasing step-sizes h. note also that the summation in eq. (17) stops at n – 2 because qn ′′ and qn–1 ′′ are undetermined. this is not problematic because in the framework of the mdr it is q ′′ = 0 for sufficiently large x in any way (x > a, where a is the contact radius). once again, some care must be taken at the borders. one way to approximate p1 is via taylor series. a first order expansion yields simply 3 2 1 2 2 p p p p    . (18) fast numerical implementation of the mdr transformations 131 at the other border the values for pn and pn–1 remain undetermined. again, this is not problematic as long as it is ensured that these last points lie outside the contact area. then they can simply be set to 1 0n np p   . (19) 4. performance often, the mdr transformations need to be performed repeatedly. one example is that of wear simulations where the transformations (1) and (3) need to be performed many times after each other for a changing indentation profile. in such cases, consider optimizing the implementation of the mdr transformations for maximum speed. for example, when using the transformation technique presented in the example above, note that the summation in eqs. (13) and (16) can be regarded as a matrix vector product in which the matrix is a kernel which is independent of the indentation profile and can be predefined. this enables full vectorization of the transformations when they are used repeatedly for changing indentation profiles. also consider the possibility of calculating the derivatives in the transformations such as (9) and (10) via matrix vector multiplication using predefined sparse matrices. 5. exemplary results fig. 1 shows the results of the previously described implementation technique for a conic and parabolic indenter at an exemplary indentation depth d. it becomes apparent that already for as few as n = 51 discretization points a fairly good approximation of the analytical solutions can be achieved. the maximum error of gk, wn, and pn with respect to the analytical solutions for g, w, and p decreases when the number of discretization points n is increased as can be seen in fig. 2. for most n, the maximum error of pressure distribution pn (a thin grey oscillating line in fig. 2) is given by the error at the very last discretization point lying within the contact area (highlighted with a star in fig. 1). index n of this particular point shall be denoted with n = s. at all other points a much better accuracy is achieved: if point s would be disregarded in the assessment of the maximum error, the upper limit of the grey oscillating line would move down from the dotted line to the dashed one. in the exemplary case of n = 51 which is displayed in fig. 1 this relatively high error of pn at the point n = s does not immediately become apparent to the viewer due to the large slope of p at the end of the contact area which puts the numerical value close to the analytical curve even if there is a relatively high error. 132 j. benad a) b) fig. 1 results of the mdr transformations carried out with the numerical procedure described above for n = 51 discretization points, exemplary input parameters of l = 1, e * = 1, d = 0.3 and an exemplary conic indenter (left) given with f(r) = r tan(π/8) and an exemplary parabolic indenter (right) given with f(r) = r 2 /2. the pressure which is obtained at last discretization point within the contact area in this example is highlighted with a star a) b) fig. 2 maximum error of gk, wn , and pn for a discretization of n = 51, 52, 53 … 5000, shown for the exemplary inputs l = 1, e * = 1, d = 0.3 and the exemplary conic indenter (left) given with f(r) = r tan(π/8) and the exemplary parabolic indenter (right) given with f(r) = r 2 /2. the oscillating thin grey line shows the maximum error of pn. its upper limit is marked with the dotted black line. neglecting the error of pn at the point n = s in the assessment of the maximum error would cause a much lower upper limit which is marked with the dashed black line. note also that the maximum error of gk for the exemplary conic indenter lies at around 10 -15 and is thus outside the chosen region displayed in the figure fast numerical implementation of the mdr transformations 133 6. adjustments if needed, one possible way of obtaining better results for ps is by inserting one or more discretization points on a finer grid after point s so that it is no longer the last point in the contact area. for example, one can use a technique such as the one illustrated in fig. 3. here, a new discretization point is added right at contact radius a, which is approximated from gk with a simple linear interpolation 1 1 ( )s s s s s s r r a d g r g g        , (20) another discretization point is added in between rs and a at rs + (a – rs) / 2, and at both ends two more points are added, one at rs – (a – rs) / 2 and one at a + (a – rs) / 2. the values for the one-dimensional profile are interpolated linearly from gk to these points. the resulting five equally spaced points are marked with crosses in fig. 3. fig. 3 detailed view of the graph in fig. 1b, here with additional discretization points at the end of the contact area which are marked with black crosses the desired value for the pressure at point s can now be calculated as in eqs. (16) and (17) using a new refined grid. the three inner points are used for the summation while the two additional outer points are only there to obtain the derivatives with the central difference scheme. note that the above method of obtaining a more accurate value for ps does not practically increase the computational time. it is a simple addition of three values, and the three linear interpolations which are needed are also given with small algebraic equations. compared to the time for the main transformation steps, the time for these additional steps is negligible. however, the small correction reduces the maximum error norm (see fig. 4). at the end of this section it shall be noted that higher order methods for the calculation of the derivatives and for the numerical integration do not necessarily lead to more accurate results. it is observed that the use of more neighboring points than in eqs. (9) and (10) for calculating the derivatives tends to decrease the accuracy of the transformations for the contact of the cone and the hertzian contact. it was also observed that using simpson’s rule to perform the summation in eqs. (13) and (16) instead of the trapezoidal rule decreases the accuracy of the transformations. 134 j. benad a) b) fig. 4 maximum error of gk, wn , and pn as displayed in fig. 2 here however ps is corrected with the technique explained above. the old upper limit of the maximum error of pn from fig. 2 is shown with the grey dotted line. the new upper limit of the maximum error of pn after the correction of ps is shown with the black dotted line. in the case of the cone (a) this upper limit falls onto the dashed line marking the maximum error of pn when the point s is disregarded 7. comparison with other numerical techniques recall that at the beginning of the numerical scheme presented above the mdr transformations were rewritten using integration by parts. however, the mdr transformations can also be implemented numerically using their original form of eqs. (1), (2) and (3) without rewriting them to eqs. (4), (5) and (6). here, two such methods which will be called “method i” and “method ii” shall briefly be discussed. their accuracy will be compared to the partial integration methods introduced above, which are referred to as “method iii” and “method iv” in the following. method i – insertion of h at singularity: one technique for the implementation of the transformations using their original form of eqs. (1), (2) and (3) is to insert a single increment h at the singularity where x = r, as in 1 2 2 1 k n k k k n k n f f g x h hx r                 , (21) where the first derivative is computed as in eq. (9) using central differences. this method, however, delivers only very poor results. as can be seen in fig. 5, the technique requires a number of discretization points which is several orders of magnitude higher in order to reach the accuracy which is achieved by the other implementation techniques. this method is not recommended. fast numerical implementation of the mdr transformations 135 method ii – implementation of the kernel with its antiderivative: a far better technique for the implementation of the transformations using their original form is to implement the kernel of the transformation using its antiderivative. for the transformation to gk, this translates to 1 2 2 2 2 1 1 atan atann nk k k k n k n n n r r g x h x r x r f h                (22) and for the transformation to pn, one can use 2 2 2 2 1 1 1 log( ) log( ) n k n k k n k n k k n x r x x r xp h q h                 . (23) the first derivatives can once more be obtained via central differences. as can be seen in fig. 5, the method ii provides a much better accuracy than method i. method iii – partial integration method: this technique was described in great detail in the first sections of this work. here the singularity at x = r is avoided through partial integration of the transformations. this leads to alternative formulations of the transformations in which the second derivative of the three-dimensional indentation profile and the deformed elastic foundation occur. thus, singularities now occur at kinks of these profiles; however, they disappear in the numerical integration, similarly to method ii where small increment h cancels out in eqs. (22) and (23). recall, however, that the singularity which is overcome in method ii occurs in the kernel. method iii, however, overcomes singularities which may occur through the shape of the indentation profile or the deformed one-dimensional foundation. also, the singularity in method ii always influences the transformation values at all discretization points whereas in method iii the singularities through kinks may leave transformation values at some discretization points uninfluenced. in fig. 5 it can be seen that with method iii the number of discretization points can substantially be reduced to achieve the same accuracy as in method ii. however, it stands out that the maximum error in method iii is still fairly close to the maximum error in method ii. this relatively high maximum error of method iii is generally attained at the end of the contact area. method iv – partial integration method with small adjustment: the previously described relatively high maximum error of method iii is reduced in method iv. the simple adjustment through the insertion of an additional discretization point at the end of the contact area is described in section 6 above. in fig. 5 it can be seen that with the method iv the number of discretization points can further be reduced to achieve a certain desired accuracy. 136 j. benad a) b) fig. 5 upper limits of the maximum absolute error of pn (left graph a) and the mean absolute error of pn (right graph b) compared for the different numerical methods: method i – insertion of h at singularity, method ii – implementation of the kernel with its antiderivative, method iii – partial integration method (technique from this paper), method iv – partial integration method with small adjustment (refined technique from this paper). as before, the curves are displayed for the exemplary inputs of l = 1, e * = 1, d = 0.3 and here only for the exemplary parabolic indenter given with f(r) = r 2 /2 8. exemplary wear simulation apart from a high accuracy, method iii and method iv may also show an advantage when they are used multiple times on a changing indentation profile, such as in wear simulations. as an example, consider a heterogeneous cylinder which is pressed onto an elastic half-space with normal force fn and moves tangentially with velocity v0. the cylinder shall be composed of rings of different material having the same elastic properties but different wear coefficients k1 and k2 (see fig. 6a). this setup has recently been studied with the mdr by li et al. [6] using archard’s law [12] n wear 0 ( ) ( ) f s v k r r   (24) to model the change of the indentation profile due to wear. therein, kwear(r) and σ0(r) are wear coefficient, that is, hardness, and with k(r) = kwear(r) / σ0(r) the linear wear is 0( ) ( ) ( )f r k r p r v t   . (25) in the following, the same procedure is adopted. it shows that the numerical method which is used for the mdr transformations has a significant impact on the quality of the simulation results. the limiting profile and pressure reached after a long enough wear process are both displayed in fig. 6b. profile f is normalized with initial indentation depth d0 = fn / (2ae * ) and the pressure distribution is normalized with p0 = fn / (2πa 2 ). as can be seen in fig. 6b, the use of method ii to perform the transformations leads to an oscillating error in the results for both the profile and the pressure (a thin grey jagged line). this error does not occur when method iii or method iv are used (a smooth bold line). for an increasing fast numerical implementation of the mdr transformations 137 number of discretization points or smaller time steps in the wear simulation the oscillating error which occurs with method ii does not vanish although it can be smoothed out in the post-processing. method iii and iv, however, deliver the undistorted results straight away without the requirement for subsequent corrections. note that these raw results of the exemplary simulation obtained using method iii and method iv also reproduce results obtained for validation purposes in [6] with the boundary element method (bem) [13]. a) b) fig. 6 left graph a): a heterogeneous cylinder composed of rings of different material having the same elastic properties but different wear coefficients k1 and k2 is pressed onto an elastic half-space with the normal force fn and moves tangentially with velocity v0. right graph b): simulation results for the limiting profile and pressure after a long enough running-in process as obtained with method ii (a thin grey jagged line), and the techniques from this paper – method iii and method iv (a smooth bold line) with n = 201 discretization points and k2/k1 = 10 9. conclusion a simple implementation technique for the mdr transformations is presented in this work. it relies on integration by parts of the transformations, a central difference scheme to obtain the derivatives, and the trapezoidal rule to perform the summation. it is shown that the results of the method for the contact of a cone and the hertzian contact converge to the corresponding analytical solutions for an increasing number of discretization points. therein, the highest error occurs at the border of the contact area. a small refinement to the numerical method has been presented to reduce this error. the introduced method and its refinement are then compared to other numerical techniques which rely on the original form of the transformations. for the tested examples, the newly introduced method and its refinement deliver more accurate results at the same number of discretization points (see fig. 5). furthermore, it is shown that apart from a higher accuracy when used once, the presented method and its refinement may have another benefit when used multiple times in 138 j. benad wear simulations. in an exemplary simulation, the wear of a heterogeneous cylinder composed of rings of different material having the same elastic properties but different wear coefficients is modeled. these discontinuous transitions in the material properties are handled well by the newly introduced methods, whereas the tested numerical techniques which rely on the original form of the transformations deliver results with a high oscillating error (see fig. 6). acknowledgments: the author would like to thank v. l. popov, m. heß, q. li, e. willert and p. diercks for many valuable discussions on the topic and critical comments. references 1. popov, v.l., psakhie, s., 2007, numerical simulation methods in tribology, tribology international, 40(6), pp. 916-923. 2. popov, v.l., heß, m., 2016, method of dimensionality reduction in contact mechanics and friction, springer, berlin. 3. popov, m., benad, j., popov, v.l., heß, m., 2015, acoustic emission in rolling contacts, method of dimensionality reduction in contact mechanics and friction, springer, berlin, pp. 207-214. 4. dimaki, a., dmitriev, a., chai, y., popov, v.l., 2014, rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction, international journal of solids and structures, 51(25-26), pp. 4215-4220. 5. dimaki, a., dmitriev, a., menga, n., papangelo, a., ciavarella, m., popov, v.l., 2016, fast high-resolution simulation of the gross slip wear of axially symmetric contacts, tribology transactions, 59(1), pp. 189-194. 6. li, q., forsbach, f., schuster, m., pielsticker, d., popov, v.l., 2018, wear analysis of a heterogeneous annular cylinder, lubricants, 6(1), 28. 7. murio, d., hinestroza, d., mejía, c., 1992, new stable numerical inversion of abel's integral equation, computers & mathematics with applications, 23(11), pp. 3-11. 8. hansen, e., law, p., 1985, recursive methods for computing the abel transform and its inverse, journal of the optical society of america a, 2(4), pp. 510-520. 9. popov, v.l., heß, m., 2014, method of dimensionality reduction in contact mechanics and friction: a users handbook. i. axially-symmetric contacts, facta universitatis-series mechanical engineering, 12(1), pp. 1-14. 10. heß, m., 2016, a simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, international journal of engineering science, 104, pp. 20-33. 11. popov, v.l., heß, m., willert, e., 2017, handbuch der kontaktmechanik, springer, berlin. 12. archard, j., hirst, w., 1956, the wear of metals under unlubricated conditions, proc. r. soc. lond. a, 236(1206), pp. 397-410. 13. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17(4), pp. 334-340. 10797 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 677 691 https://doi.org/10.22190/fume220528043m © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper designing an efficient observer for the non-linear lipschitz system to troubleshoot and detect secondary faults considering linearizing the dynamic error hamed mobki1, morteza homayoun sadeghi2, aydin azizi3, mir mohammad eskandari4 1center of condition monitoring at urmia combined cycle power plant, west azerbaijan power generation management company, iran 2department of mechanical engineering, university of tabriz, iran 3school of engineering, computing and mathematics, oxford brookes university, united kingdom 4department of mechanical engineering, university of tehran, iran abstract. the presence of faults in a system leads to a lower value for efficiency, accuracy and speed, and, in some cases, even a complete breakdown. thus, early fault detection is a major factor in efficiency and productivity of the procedure. in recent decades, many research studies have been conducted on troubleshooting and secondary fault detection. the current work presents an efficient and novel observer design capable of stabilizing the residue and dynamic error for the nonlinear lipschitz systems with faults as well as a troubleshooting analysis and determining the formation of secondary faults in defective systems. the observer is designed based on linearizing dynamic error considering uncertainty, disturbance, and defects by employing non-linear gain factors instead of using state transformation. the dynamic error and residue stabilization of a non-linear faulty system have been discussed as well as the likelihood of secondary fault generation. the results indicate that the observer is able to determine fault-emergence, fault-disappearance and secondary fault formation well and quite fast. key words: observer design, troubleshooting, fault detection, secondary fault, dynamic error linearization, lipschitz systems received may 28, 2022 / accepted october 26, 2022 corresponding author: aydin azizi school of engineering, computing and mathematics, oxford brookes university, wheatley campusox33 1hx, oxford, uk e-mail: aydin.azizi@brookes.ac.uk 678 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari 1. introduction in recent decades, many research studies have been drawn to fault-detection and system identification [1-5]. the presence of faults in a system leads to a lower value for efficiency, accuracy and speed, and, in some cases, even a complete breakdown. thus, early fault detection is a major factor in efficiency and productivity of the procedure. the ability to detect faults quickly, without the need for signal excitation, and to detect faults online have made the observer-based fault detection one of the most popular methods for fault detection in non-linear systems; hence, numerous efforts have been made in recent years to further develop the mentioned method. fault detection can be broadly classified into two categories, namely model-based and signal based. in the model-based category, the accurate model of the system presented and any changes in the system parameters or fault occurrence have been studied by means of comparing of the faulty system with the healthy one. in the signal-processing method, fault signature can be extracted from the measured signal of system [6-7]. fault-detection using sliding-mode observer has been presented in refs. [8-9], and utilizing unknown input observer for fault detection can be found in refs. [10-14]. detecting faults using adaptive observer are studied in refs. [15-17]. high gain observers [18-21] have been used frequently in non-linear systems, and a few cases where the kalman filter has been used are presented in refs. [21-24]. although substantial research has been undertaken on the observer-based fault detection, few or no studies have been conducted to investigate secondary fault generation or troubleshoot detection using observer-based approaches. this "lack of interest" on the part of the researchers is simply due to the fact that the observer's stability is not guaranteed while detecting secondary fault generation or the de-faulting method in a defective system, and as such, it should not be seen as insignificant. in other words, no research has been conducted to ascertain the observer's stability for defective systems or to stabilize the dynamic error and residue in nonlinear systems. the authors of this study think that linearizing dynamic error and residue enables both stabilizing the observer of a defective system and tracking the faulty system's behavior. one of the most essential techniques for observer design is dynamic error linearization, which converts the non-linear system to an observable one by state transformation just as it converts the dynamic error to a linear form. krener and isidori [25] and bestel and zeits [26] pioneered dynamic error linearization. since then, as a topic of significant interest to a large number of researchers, multiple strategies for further developing it to various classes have been offered [27-29]. one of the most compelling arguments in favor of linearization is that it ensures the observer's stability during fault detection. the observer will be able to estimate system parameters even in the presence of a fault (assuming the fault does not result in system failure but just modifies operating parameters) and will be able to forecast fault erasure as well as the appearance of secondary faults. however, linearizing dynamic faults solely through state transformations is a lengthy procedure that requires performing complex and numerous arithmetic calculations (such as solving a system of partial differential equations), and even then, finding the appropriate coordinate transformations is not always possible [29, 30]. the approaches available are restricted to a subset of non-linear systems. the requirement for state transition is the primary argument against such approaches. the disturbance and uncertainty are also transformed during state transformation, and the designing an efficient observer for non-linear lipschitz systems to troubleshoot... 679 knowledge regarding the nature of these variables is lost. the range and form of uncertainty as well as the order of disturbance are critical considerations in setting the fault-detection threshold, protecting the system from disturbance, and increasing the system's sensitivity to suspected problems. as a result, linearized dynamic error observers are rarely utilized for defect detection, state estimation, or parameter estimation in industrial systems, and are instead employed to structure and estimate the states of particular non-linear mathematical systems with limited classes. the authors of this article developed a novel method for linearizing dynamic error that eliminates the necessity for state translation and linearizes non-linear components [31]. to linearize the dynamic error, the non-linear term that emerges from the dynamic error equation is expressed as a factor of the dynamic error in the estimated state term and the system output. stabilizing the dynamic error requires stabilizing the matrix of coefficients in the dynamic error differential equation. as a result, the error differential equation is stabilized, and gain factors are shown as functions of the estimated state and system output. indeed, the error differential equation is linearized using nonlinear gain factors. the stability of such an observer is investigated in this study and in the continuation of ref. [31], and the requisite conditions for stabilizing the observer for residue tracking of a broken system to troubleshoot detection as well as secondary fault detection are demonstrated. additionally, the estimation of fault and troubleshoot thresholds is detailed, and mathematical stabilization of the differential equation of dynamic error in faulty mode is demonstrated. the ability of the observer to detect defects in fault-free and defective systems as well as troubleshoot detection of a tunable micro capacitor subjected to nonlinear electrostatic force, are investigated and so is the behavior of residual and observer gains in faulty and fault-free modes. 2. observer design for the nonlinear lipchitz system 2.1. observer construction and dynamic error linearization this section will demonstrate the aforementioned observer's ability to linearize dynamic error and stabilize the residual in the presence of faults. the whole technique for designing an observer is described in ref [31]. to maintain the integrity and completeness of the fault detection process, a brief description of the observer design will be provided. the procedure is based on the transformation of non-linear terms into elements of the dynamic error in the variable that is dependent on the estimated state and output. this approach converts the system of differential equations governing dynamic error to a matrix of coefficients (in terms of the estimated state and output) and then stabilizes it before constructing the observer and determining its gains. to begin, the technique is provided with the assumption that there are no uncertainties in the system. take the following non-linear system as an example: (1) ( , )x ax x u y cx dx x dt = + = = 680 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari in which, x∈𝑅𝑛, u∈𝑅𝑟, and y∈𝑅m represent state, known input and the system output, respectively. in the above equation a∈𝑅𝑛×𝑛 and c∈𝑅𝑚×𝑛 are system matrices of known and constant value. the function (x,u)∈𝑅𝑛 is a continuous, vector function of non-infinity extremes, i.e. it satisfies the lipschitz condition [32]: (2) in which, γ is a positive value denoting the lipschitz coefficient of (𝑥,𝑢). following observer is considered for estimating the state of the system (1): (3) in which, x∈𝑅𝑛 and y∈𝑅m represent the estimated state and the system output, respectively. l(x,y)∈𝑅m×𝑛 is the matrix of the observer gains, and it should be chosen in such a way that the dynamic error remains steady. (for the purpose of simplicity, l(x,y) will be written as l. naturally, in addition to estimating the state, observer (3) can also estimate the system output. the following requirements must be followed in order to calculate the observer's gains [31]: ▪ matrices a and c must be observable, i.e., matrix [aca … can-1]t must be of full rank. ▪ matrix c must be of full rank. ▪ the non-linear term (𝑥,𝑢) must be uniquely determinable using the system output [31]. now, assuming the state error to be e=x-x, the system of differential equations for error is determined as follows: (4) in order to design the observer, l should be determined so that the dynamic errors are asymptotically stable. lemma. if matrix f(y,x,u)∈rn×n can be determined in such a way that satisfies the below equation, then the error vector is asymptotically stable, given the matrix [alc+f(y,x,u)] is negative-definite [31]. (5) stabilizing the matrix [a-lc+f(y,x,u)], the observer's gains are calculable. the total of the elements on the main diagonal (trace of [a-lc+f(y,x,u)]) must be negative in order to stabilize the observer. even and odd submatrices must have positive and negative determinants, respectively. this enables the determination of the observer gains to be made easily. the observer gains are determinable. under such circumstances, the elements of f(y,x,u) are presented as part of the observer gains, and ultimately, the matrix [alc+f(y,x,u)] is presented as a constant matrix independent of u, x, and y and the differential equation of dynamic error is linearized. the dynamic error approaches zero with an exponential rate and the convergence rate is determined as desired. thus, determining f(y,x,u) seems to be the most challenging aspect of the whole procedure since all other tasks in the observer design are relatively straightforward. two explicit methods for determining f(y,x,u) have been presented in ref. [31]. ( ) ( )1 2 1 2, ,x u x u x x  −  − ( ) ( )( ), ,x ax x u l x y y y y cx = + + − = ( ) ( ) ( ), ,e a lc e x u x u = − + − ( )( ) ( ) ( ), , , ,f y x u x x x u x u − = − designing an efficient observer for non-linear lipschitz systems to troubleshoot... 681 2.2. ability of the observer to determine faults and categorizing the faults in the actuator and dynamic system the preceding section discusses the observer design technique for estimating output and system state. the dynamic error can be linearized using the gain factors that depend on the estimated state and output using the procedure described above. on this basis, the present part examines the stability of the error and residue following the appearance of defects in the system and determines the residue range for the defective condition. these are the detection thresholds for faults. the following conditions are required for system troubleshooting and secondary fault detection: ▪ the dynamical system remains stable notwithstanding the emergence of primary and secondary faults. it is self-evident that no observer can detect an unstable system. ▪ system stability should be guaranteed in the presence of uncertainty and disturbance. in other words, the value of the mentioned parameters should be in a range that does not cause any instability in microstructure. ▪ qualitative form of differential equation of dynamic error or governing dynamic equation of dynamical system should be untouched after fault emergence. for example, the size of a and c matrices should not be changed or primary form of (𝑥) should be fixed, but their magnitude can be changed. for example, neither the size of a and c matrices nor their fundamental form should be changed, but their magnitude can be adjusted. for example, matrix a can be changed from [0 1] to [2 3] but it should not be changed to [0 1 0]. or (𝑥,) can be changed from 2sin(x) to 3sin(x) but it should not be changed from 2sin(x) to x2. now, consider the dynamic system: (6) in which, f1∈rp and e1∈rn×p denotes the fault and characteristics matrix, respectively (note that there are no null or zero elements in f1). if observer (3) is used to determine the output and residue of system (6), the system of differential equations for dynamic error may be presented as: (7) in which, k is a negative definite matrix, and is determined by stabilizing [alc+f(y,x,u)]. note that k is not uniquely determined; t1 and e1 represent the time of fault formation and the value of the dynamic error in that time, respectively. considering y=cx and eq. (7), the residue dynamic equation is derived as the following: (8) in which, d=ck and e1c=ce1, while r1 is the residue at the time of fault emergence. the residue response for the above equation is: ( ) 1 1,x ax x u e f y cx = + + = ( ) 1 1 1 1 e ke e f e t e = + = ( ) 1 1 1 1 cr dr e f r t r = + = 682 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari (9) if e1cf1() satisfies f1d≤e1c≤f1u, the dynamic response limits are determined as: (10) now, since k is negatively definite and t1>t, the residue steady response is bound by d-1f1u and –d-1f1d. in fact, these ranges demonstrate the upper and lower limit for detecting the fault f1 after formation. now, suppose that at time t2, fault f2 is formed in the defective system (6). or: (11) in which, f2∈rq and e2∈rn×q represent the secondary fault and the secondary fault matrix (note that there are no zero elements in f2). if observer (3) is to be used in order to estimate the output and residue of system (11), the system of differential equations for dynamic error and residue may be presented as: (12) in which, e2 and r2 represent the value of the dynamic error and residue for t>t2. also, e2c=ce2. for eq. (12), residue response for t>t2 is: (13) if e1cf1()+e2cf2() satisfies f2d≤e1cf1()+e2cf2()≤f2u, the dynamic response limits are determined as: (14) now, since k is negatively definite and t>t2, the residue steady response is bound by – d-1f2u and –d-1f2d. note that e1f1=-e2f2 denotes de-faulting (troubleshoot). as can be observed, linearizing dynamic error and the residue enables the residue to be stabilized in the presence of a defect, which enables secondary fault identification and troubleshooting. this section established the observer's stability in the presence of faults, assuming no uncertainty or turbulence. when there is uncertainty or disturbance in the system, the problem can be handled without sacrificing its general utility; in this situation, the ranges of disturbance and uncertainty should be indicated in eqs. (6) and (7). in other words, when ( ) ( ) ( ) ( ) 1 1 1 1 1 1exp exp t c t r t d t t r e f d t d      = − + − −            ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 exp exp u u d d r t d t t r d f d f r t d t t r d f d f − − − − = − + −   = − + −   ( ) 1 1 2 2,x ax x u e f e f y cx = + + + = ( ) ( ) 1 1 2 2 2 2 1 1 2 2 2 2 c c e ke e f e f e t e r dr e f e f r t r = + + = = + + =   ( ) ( )  2 2 2 1 1 2 2 2( ) exp ( ) ( ) ( ) exp t c ct r t d t t r e f e f d t d    = − + + − −   ( )  ( ) 1 1 2 2 2 2 1 1 2 2 2 2 ( ) exp ( ) ( ) exp ( ) u u d d r t d t t r d f d f r t d t t r d f d f − − − − = − + − = − + − designing an efficient observer for non-linear lipschitz systems to troubleshoot... 683 computing the residue ranges in eqs. (9) and (10), it is necessary that the ranges for disturbance and uncertainty be added to f1d≤e1cf1()+u+n≤f1u and f2d≤e1cf1()+e2cf2()+u+n≤f2u, in which u and n represent uncertainty and disturbance. 3. results and discussion 3.1. model description of the case study (micro tunable capacitor subjected to nonlinear electrostatic force) a parallel plate capacitor is depicted schematically in fig. 1a [33]. it comprises a moveable electrode hung above a stationary conductor plate. between two electrodes, the major gap is g1. electrostatic attraction caused by the applied bias voltage u pushes the moveable electrode towards the fixed plate. the top view of the movable electrode, which is suspended by four supporting beams, is shown in fig. 1b (two at each side). s and h denote the area and thickness of the moveable electrode, respectively. the width, thickness, and length of beams are denoted by the symbols b, h, and l. each beam has an equivalent rigidity of k=12eil-3 where e and i are, respectively, the young's modulus and the cross section moment of inertia. with a density of ρ, the movable electrode is termed isotropic. (a) (b) fig. 1 micro tunable capacitor; a) capacitor from side view, b) capacitor from top view governing dynamic equation of movable electrode of capacitor is as: (15) where m=ρah,c′, k=4k are mass, damping coefficient and resultant spring constant, respectively, t is time and ε0=8.854×10-12 c2n-1m-2 is permittivity of vacuum; z is deflection of movable electrode. physical and geometrical properties of studied micro capacitor are presented in table 1. for convenience, eq. (15) may be rewritten in a non-dimensional form by defining the following parameters [34]: 22 0 2 2 12( ) aud z dz m c z kz dtdt g z  + + = − 684 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari (16) where x, c,  are dimensionless deflection, damping, and time.  is dimensionless parameter of electrostatic force. therefore, eq. (15) can be written as: (17) table 1 physical and geometrical properties of micro capacitor a=400μm×400μm area of movable electrode e=169 gpa young modulus l=100 μm length of each supporting beam b=5 μm width of supporting beam h=1 μm thickness of supporting beam mμ=4 1g primary gap between two electrodes 3=2300 kg/mρ density eq. (17) can be presented is state space form as: (18) considering disturbance, uncertainty and fault vector, eq. (18) is altered as: (19) where x1=x and x2=dx1/dt, δ[0 αn2u2/(1-x1)2]t, δ′[0 sint]t,α(n2-1)u2/(1-x1)2[0 1]t are uncertainty, disturbance and fault vectors, respectively. the simulated fault is abrupt decreasing of the applied voltage which related by n in fault vector. for the fault-free system n=1 and for faulty system 0≤n<1, following observer is considered for output estimation and residual generation of system (19). (20) 3 1 0 , , , , 2 o az c m t x c t g kkt kg t       = = = = = 2 2 2 2(1 ) d x dx u c x dtdt x  + + = −   1 1 2 2 2 2 1 1 2 0 0 1 1 ' (1 ) 1 0 x x u x xc x x y x            = +      − −      −      =     ( )   2 2 1 1 2 2 2 2 2 2 12 2 1 1 1 2 0 0 0 1 0 0 1 1 ' sin 1 (1 ) (1 ) (1 ) 1 0 x x n u u n u x xc t x x x x y x                   −    = + + + +            − − −            − −        =     1 11 2 2 22 2 1 0 0 1 ( ) 1 ' (1 ) x lx y yu x lcx x r y y              = + + −        − −         −   = − designing an efficient observer for non-linear lipschitz systems to troubleshoot... 685 where l1 and l2 are observer gains. with subtracting of eq. (19) from eq. (20), equation of dynamic error is extracted as: (21) attention toy=x1 and using proposed method for construction of f matrix [31] we have: (22) so eq. (21) can be rewritten as: (23) stabilizing the coefficient matrix in eq. (23) is required to determine the observer gains. through the stabilization of the coefficient matrix (for the stabilizing of the mentioned matrix, the even and odd sub matrices of the coefficient matrix must be positive and negative, respectively). observer gains were calculated using this approach and are shown in the following equation: (24) where ε1 and ε2 are positive numbers. by inserting the observer gains, the matrix of alc+f is extracted as follows: (25) ( ) ( ) 1 1 12 2 2 2 22 2 1 1 2 2 2 2 2 12 0 0 1 1 (1 ) (1 ) 0 0 0 1 sin 1 (1 ) (1 ) e e l y yu u e e lc x x n u n u t x y                 = + − −         −− −       − −        −  + + +     −    −  ( ) ( )( ) 12 2 1 0 0 2 0 1 1 x yf u y x      − +=      − −    ( ) 1 1 1 2 1 22 22 1 2 2 2 2 2 2 1 2 ( ) 1 [(1 )(1 )] 0 0 0 1 sin 1 (1 ) (1 ) l e e x y u l ce e y x n u n u t y y     −       = − +    − − −    − −        −  + + +     −    −  1 1 21 2 2 1 2 1 21 2 2 1 2 1 2 ( ) 1 [(1 )(1 )] 2 ( ) 1 ( c) [(1 )(1 )] l c x y l l c u y x x y l c u y x       = − − + = − − + − − − + = − − − + − − 1 2 1 1 ( ) c a lc f c c c    −  − + =   − + − −  686 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari 3.2. results of output estimation and fault detection in this section, the obtained results for fault detection, fault release, and fault detection of faulty micro tunable capacitor are presented. for this simulation u=3v, ε1=2, ε2=1 are intended. in this paper, the thresholds of a healthy system are shown with dot-dot (….). the thresholds of initial and secondary fault detection are line-line (----) and line-dot (-.-.-), respectively. a fault is considered as reduction of applied voltage. the thresholds for determining of the fault size for |δ|<0.1 and |δ′|<0.003 are determined and shown in table 2. table 2 the upper and lower thresholds for different magnitude of n for |δ|<0.1 and |δ′|<0.003 n upper threshold lower threshold n=1 0.006392 0.006392 n=0.9 0.00016220.01211 n=0.8 0.0056730.01722 n=0.7 0.010540.02173 n=0.6 0.014750.02564 n=0.5 0.018320.02895 n=0.4 0.021230.03166 n=0.3 0.02350.03377 n=0.2 0.025120.03527 n=0.1 0.02610.03617 n=0 0.026420.03647 the results of initial fault detection for various amounts of δ′ are shown in figs. 2 to 4. the fault is considered as voltage decreasing where at t=6.2×10-3s magnitude of n=1 abruptly decreased to n=0.7. as can be seen from these figures, the observer is able to determine the size of the fault for different magnitude of δ′. also, it can be seen from this figure that an abrupt change of residual is observed as soon as a fault occurs. and the residual is settled in the bound of relevant threshold. fig. 2 fault detection result for n=0.7 and δ′=0 designing an efficient observer for non-linear lipschitz systems to troubleshoot... 687 fig. 3 fault detection result for n=0.7 and δ′=0.003 fig. 4 fault detection result for n=0.7 and δ′=-0.003 fig. 5 also shows the time depending variation of gain l2. as shown in this figure, as soon as a fault occurs, the coefficient of l2 changes significantly and after a short time, converges to fig. 5 time dependent variation of l2 for n=1 (t<6.2×10-3s), n=0.7(t≥6.2×10-3s) and δ′=-0.003 688 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari near 0.005 . considering figs. 2 to 5, it is obvious that residual and observer gain have similar shapes. they have considerable change after fault occurrence and also converge to specific magnitude. now consider that at t=6.2×10-3s the applied voltage change from 1u(n=1) to 0.7u (n=0.7) and at t=12.4×10-3s the actuator fault is released and the voltage is increased from 0.7u to 1u (troubleshoot). for this situation, after fault is released, the residual must be settled in healthy threshold zone. fig. 6 illustrates this condition for δ′=0.05. as shown in this figure the presented observer has a good ability for detecting the fault occurrence and troubleshooting. fig. 7 shows change of l2 versus time. also, in this case a meaningful similarity can be observed in the behavior of figs. 6 and 7. fig. 6 fault and troubleshoot detection for n=0.7 and δ′=0.05 fig. 7 variation of l2 for simulation of fig. 6 designing an efficient observer for non-linear lipschitz systems to troubleshoot... 689 in the continuation of this section, we intend to examine the observer’s ability for secondary fault detection. it means that initially a fault has occurred in the system, and then another fault occurs in the faulty system. for example, at t=6.2×10-3s the applied voltage decreases from 1u to 0.7u, and then at t=12.4×10-3s the applied voltage decreases from 0.7u to zero. fig. 8 shows the simulation results for such a situation. as it is obvious from this figure, the observer is able to detect a secondary fault in the system at an acceptable speed and can accurately calculate the secondary fault level. as can be seen in fig. 9, after the formation of the primary and secondary faults, l2 changes substantially. fig. 8 the initial (n=0.7) and secondary (n=0) fault defection using proposed observer fig. 9 time dependent variation of l2 for simulation results of fig. 8 4. conclusion troubleshooting and the creation of secondary defects in already problematic systems were studied in this research utilizing observer-based methodologies due to their relative relevance. to accomplish this, an observer capable of linearizing the differential equation 690 h. mobki, m.h. sadeghi, a. azizi, m.m. eskandari of dynamic error and residue was used. it was demonstrated that the linearization approach used ensures the observer's stability in the presence of faults and enables tracking of the residue's behavior even after the onset of faults in order to assess fault disappearance or the generation of secondary faults. the proposed observer design approach does not require state transformation and is based on rewriting the 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journal of control, automation and electrical systems, 31, pp. 625-635. 32. mobki, h., sadeghi, m.h., rezazadeh, g., 2015, application of thau observer for fault detection of micro parallel plate capacitor subjected to nonlinear electrostatic force, international journal of engineering, 28(2), pp. 270-276. 33. mobki, h., sadeghi, m.h., rezazadeh, g., 2015, state estimation of mems capacitor using taylor expansion, international journal of engineering, 28(5), pp. 764-770. 34. mobki, h., majidzadeh sabegh, a., azizi, a., ouakad, h.m., 2020, on the implementation of adaptive sliding mode robust controller in the stabilization of electrically actuated micro-tunable capacitor, microsystem technologies, 26, pp. 3903-3916. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 1, 2015, pp. 1 2 foreword to the thematic issue: tribology in aerospace applications – damping, wear and structural dynamics in aerospace systems valentin l. popov 1 , sergey g. psakhie 2 , alexander g. chernyavsky 3 1 berlin institute of technology, department of system dynamics and the physics of friction, 10623 berlin, germany 2 institute of strength physics and materials science, russian academy of sciences, 634021 tomsk, russia 3 rkk “energia” editorial dynamics and tribology are of high relevance for aerospace structures. as examples may be mentioned: 1) unfolding systems for antenna designs of spaceships: drives, gears, cylindrical and ball bearing, orientation systems; 2) development of systems for solar panels for spacecraft "progress" and modules of the international space station. in these systems, structural dynamics and tribological problems are closely interrelated (fig. 1). for the description and optimization of such systems theoretical and experimental investigations of the structural dynamics in the context of the mechanics of tribological interfaces taking into account the material behavior under high vacuum and partly even in extreme temperatures are required. these problems have been discussed during the german-russian workshop on “tribology in aerospace applications: damping, wear and structural dynamics in aerospace systems”, which was held at the technische universität berlin october 6-8, 2014. in the center of interest of the workshop were issues at the interfaces between structural dynamics, contact mechanics, material science, friction, wear, modeling and simulation. an important issue is the coupling of simulation methods of different scales. the main topics of the workshop included:  materials science aspects of tribology,  discrete element and molecular dynamics,  method of dimensionality reduction,  coupling simulation methods of different scales,  tribology at low temperatures,  polymer materials for friction systems in vacuum technology, and,  system dynamics and tribology: needs of aerospace technologies. after discussion of the presentations and the round table discussion concerning tribology in aerospace applications, the participants of the workshop came to the conclusion that the dynamic modelling of aerospace structures is a topic of high practical editorial 2 v. l. popov, s. g. psakhie, a. g. chernyavsky and scientific interest. of particular interest is structural damping in joints and inside the material. effective simulation methods for frictional interfaces, such as the method of dimensionality reduction [1], should be combined with fast finite element simulation methods [2] of entire structures as well as material analysis via mesoscopic particle methods such as method of movable cellular automata (mca), [3]. other topics of interest are the degradation and internal friction in composite materials under spaceflight conditions, and testing of bio-inspired adhesive and other tribological materials under space conditions. fig. 1 international space station: the space structures contain thousands of tribological interfaces, © rkk energia the present thematic issue contains a collection of papers presented during the workshop with the main emphasis to adhesion, friction, interrelation of friction and vibrations and energy dissipation in systems with tribological contacts. references 1. popov, v.l., hess, m.., 2015, method of dimensionality rediction in contact mechanics and friction, springer, 264 pp. 2. marinkovic, d., zehn, m., marinkovic, z., 2012, finite element formulations for effective computations of geometrically nonlinear deformations, advances in engineering software, 50, pp. 3-11. 3. psakhie, s.g, horie, y., ostermeyer, g.p., korostelev, s.yu. smolin, a.yu., shilko, e.v., dmitriev, a.i., blatnik, s., špegel, m., zavšek s., 2001, movable cellular automata method for simulating materials with mesostructure, theoretical and applied fracture mechanics, 37 (1), pp. 311-334. 12385 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 293 313 https://doi.org/10.22190/fume240122029f © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper multi-objective optimization research on vr task scenario design based on cognitive load qian-wen fu1, qing-hua liu2, tao hu3 1college of design and innovation, tongji university, shanghai, china 2college of computer science and technology, zhejiang university, hangzhou, china 3college of mechanical engineering, guiyang university, guiyang, guizhou, china orcid ids: qian-wen fu https://orcid.org/0009-0002-1988-2536 qing-hua liu https://orcid.org/0000-0003-4644-5518 tao hu https://orcid.org/0009-0006-8966-5384 abstract. in order to improve the efficiency of information acquisition and task selection in virtual reality (vr) systems, enhance the interactive experience, and reduce cognitive load for users, it is crucial to effectively organize and leverage user cognitive psychology and design elements during the vr scene design phase. this paper focuses on analyzing the low cognitive load requirements of users and the need for a satisfactory user perceptual experience based on the cognitive resource theory. we propose a method for optimizing the design of vr system scenario tasks under low cognitive load requirements. by utilizing human-computer hybrid intelligent assistance for predicting user cognitive load and incorporating intelligent optimization genetic algorithms into the optimization of vr system design elements, we aim to minimize cognitive load as the objective function based on the principle of low cognitive load. important knowledge granularity nodes are used as fitness functions in the optimization process of vr system design resource elements. an application study is conducted, combining the multi-channel cognition in a smart city vr system task information interface, to optimize the system resource features. the study validates and compares the solutions obtained through traditional design processes and the solutions optimized by the method proposed in this paper, using virtual reality eye-tracking experiments for the same design task requirements in vr systems. the results demonstrate that users experience lower cognitive load and better task operation experience when interacting with the optimized solutions proposed in this paper. therefore, the optimization method studied in this paper can serve as a reference for the construction of virtual reality systems. key words: human-computer hybrid intelligence, cognitive load, virtual reality, multi-objective optimization, task scenarios received: january 22, 2024 / accepted may 18, 2024 corresponding author: qing-hua liu college of computer science and technology, zhejiang university, hangzhou 310027, china e-mail: yazawaninico@163.com https://orcid.org/0009-0002-1988-2536 https://orcid.org/0000-0003-4644-5518 https://orcid.org/0009-0006-8966-5384 mailto:yazawaninico@163.com 294 q.-w. fu, q.-h. liu, t. hu 1. introduction virtual reality (vr) interfaces, as a novel form of human-computer interaction, possess significant potential advantages due to their natural and intuitive interaction and immersive three-dimensional environments [1-4]. vr systems with natural interaction capabilities can capture users' perceptual and cognitive behavioral data, making them a focal point in the field of natural human-computer interaction [5, 6]. with the rapid development of virtual reality technology, the research on user experience perception and cognitive load in vr human-computer interaction has garnered attention. predicting users' cognitive load and satisfaction levels is essential during the vr task interface design process in order to accurately manage the perceived user experience and construct a virtual reality system with low load and high efficiency. scholars have extensively studied user cognitive behaviors [7] and user satisfaction analysis [8], which have provided valuable research foundations for virtual reality task selection interfaces. with the goal of enhancing user immersion and reducing cognitive load, it is essential to optimize design elements within the visual, auditory, and haptic perception channels. additionally, the impact of multivariate relationships between multi-channel perception and design resource features on cognitive task learning is investigated, aiming to decrease user cognitive load and improve operational efficiency within vr systems. weng et al. [9] proposed a method of remote manipulation using virtual reality and collaboration with robots, which effectively improved skill optimization based on sensory feedback. chakraborti et al. [10] explored intent recognition and task planning paradigms within a mixed reality workspace. hernandez et al. [11] introduced the high-level augmented reality specification (hars), comprising ar interfaces and motion planning, enabling users to specify high-level objectives and map them to spatial regions using motion planning techniques to preview, approve, or modify robot motion. su et al. [12] presented a framework integrating 3d/2d visual and motion mapping in a mixed reality (mr) setting for immersive and intuitive remote manipulation of a complex mobile manipulator. wang et al. [13] investigated the performance of virtual objects in conveying information when augmented with simple graphic annotations, such as arrows and text, revealing a negative correlation between a sense of control and engagement, contributing insights into the factors influencing the visualization of virtual information. paul et al. [14] proposed a novel vr-based framework for creating ar visualizations and demonstrated its application in improving task performance in synchronized, time-driven human-computer collaboration. currently, there is limited research exploring the optimization of virtual reality design elements in vr scenario contexts based on user cognitive load. moreover, mathematical models for optimizing vr scenario design elements are relatively restricted in the existing research. traditional research in the field of vr system resource optimization lacks a predictive feedback mechanism between resource elements and cognitive behavior, and there is a lack of hierarchical and associative analysis of the relationship between design resource elements and user cognitive behavior. in light of this, to balance information capacity and user cognition in human-computer interaction and thereby reduce users' cognitive load, it is essential to consider the feedback information received by users through physical channels from virtual reality software and hardware, which subsequently generates cognitive behavior. based on this foundation, a research method is proposed for vr system scenario elements driven by cognitive load and based on multidimensional associations. multi-objective optimization research on vr task scenario design based on cognitive load 295 this study is grounded in cognitive resource theory, analyzing the demand for low cognitive load and the need for a positive user perceptual experience. a low cognitive loaddriven optimization method is presented for the design of vr system scenario tasks. the method employs human-computer hybrid intelligence to assist in optimizing design elements and predict user cognitive load. intelligent optimization genetic algorithms are introduced to optimize the design features of the vr system, with the objective function being the minimization of user cognitive load. important knowledge granularity nodes are used as fitness functions in the optimization process of vr system design resource elements. by integrating cognitive resource theory and design optimization techniques, this research aims to bridge the gap between resource elements and user cognitive behavior, ultimately facilitating the development of vr systems that provide a seamless and efficient user experience while minimizing cognitive load. 2 related theories 2.1. factors influencing cognitive load in virtual reality task scenarios 2.1.1. mechanisms of cognitive load generation for users of virtual reality operations, the cognitive load is the cognitive load generated while performing virtual reality tasks and while reading information from the system [15, 16]. cognitive load is an important metric for evaluating the usability of virtual reality systems. cognitive load theory contains the accumulation of two major underlying theories, limited resource theory and schema theory [17, 18]. the core element of the limited resource theory is the belief that the user's cognitive resources are limited and that the user's cognitive activities take up cognitive resources. 2.1.2. the influential factors of cognitive load the cognitive load experienced by users in vr task operations is influenced by a combination of user-related factors, including individual characteristics and action features, as well as the arrangement of design elements within the vr system [19]. from the perspective of the cognitive subject, the level of cognitive load is influenced by factors such as users' knowledge cultural background, action habits, cognitive abilities, prior knowledge, and psychological state [20]. additionally, different educational backgrounds and levels of familiarity with interactive vr operations can impact users' comprehension of the target system [21]. users with higher educational qualifications, positive psychological states, and experience with vr operations tend to exhibit deeper understanding of vr tasks, more accurate information retrieval, and smoother task operations, resulting in lower cognitive load [22]. from the perspective of the cognitive object, factors such as the quantity and complexity of information in vr task operations, the hierarchical structure and external presentation of information, and the alignment between vr system information and users' pre-existing knowledge contribute to cognitive load. 2.2. methods for user cognitive load assessment various assessment methods are currently employed internationally for evaluating digital vr interactions. these methods primarily include subjective evaluation, performance296 q.-w. fu, q.-h. liu, t. hu based evaluation, comprehensive theoretical evaluation, mathematical modeling evaluation, and physiological experimental measurement. however, the first four evaluation methods are prone to subjective bias. in contrast, physiological experimental measurement, utilizing eyetracking experiments and electroencephalography (eeg) techniques, provides a more objective evaluation of the quality of task scenarios designed in virtual reality. measurement of cognitive load can be categorized into subjective and physiological measurements. in the realm of subjective evaluation, baceviciute et al. [23] aimed to develop a questionnaire for assessing the user experience (ux) of vr glasses and investigated the relationships between various ux variables. perceiving the quality of ux in relation to vr hardware became a crucial predictive indicator for anticipating interactive operational performance. on the other hand, in physiological experiments, sipatchin et al. [24] introduced the concept of temporal continuity of visual attention in virtual reality (vr) settings. under conditions of freely exploring vr scenes and task-oriented scenarios, this concept predicts users' focus of attention. moreover, a combination of subjective evaluation and physiological experiments was employed by armougum et al. [25], who created a virtual reality simulation model of a train station. through the measurement of electrodermal activity and the nasa task load index (nasa-tlx), encompassing physiological, subjective, and behavioral aspects, they examined users' cognitive load. one notable mathematical model is the fitts' law, evaluated by clark et al. [26] to explore the impact of vr motion on 3d discrete pointing tasks in the oculus rift system. yan et al. [27] investigated the influence of user interface layout on the mental workload of operators, analyzing operators of nuclear power plant emergency operating procedures through a multi-index evaluation approach, combining performance metrics (i.e., completion time and error rate), subjective ratings (i.e., nasa task load index), and physiological measurements (i.e., eye movements). akyeampong et al. [28] employed nasa-tlx to assess the impact of different types of human-machine interface designs on the workload of hydraulic excavator operators, and the experimental results demonstrated that the proposed enhanced interaction was an effective solution for reducing human-machine errors. additionally, emami et al. [29] investigated the influence of visual distractions on users' cognitive load using brain-computer interfaces (bcis), which are systems that translate neural activity into practical outputs. the efficacy of bcis depends not only on the computer itself but also on whether modifying the task interface can ameliorate the effects of these distractors. in this study, a combined research approach employing subjective measurement and physiological experiments was adopted. the two methods are elucidated as follows: 2.2.1. subjective evaluation method nasa-tlx scale in this research, the nasa task load index questionnaire was employed to assess users' cognitive load. the nasa-tlx scale consists of six load factors, as given in table 1. each load factor is rated using a 100 mm visual analog scale, divided into 20 equal segments corresponding to scores ranging from 0 to 100. the two ends of the scale are labeled with the lowest and highest values of the measured factor. the measurement procedure involves users marking their perceived cognitive load on the scale after performing the experimental task. the corresponding score for the load factor is obtained by measuring the distance between the starting point of the scale and the marked point. the overall cognitive workload of the users is determined by summing the weighted scores of multi-objective optimization research on vr task scenario design based on cognitive load 297 the six load factors. the weight assigned to each load factor depends on its contribution to the total cognitive load. table 1 cognitive load metrics projects maxima and minima (0/100) descriptive cognitive demand low/ high how much cognitive and perceptual activity is required for task operations (e.g., decision-making, reflection, searching tasks, judgment, and memory)? is the task operation relatively easy or difficult, simple or complex, and does it demand strict or lenient requirements? physical demand low/ high how much physical effort is required for task operations (e.g., pushing, pulling, twisting, lifting arms)? are the task operations easy or difficult, are the actions slow or swift, and is the overall perception easy or challenging? temporal demand fast/ slow does the time taken to complete the task seem excessively long or rushed? is the completion speed slow or steady? task performance perfect/ failed how successful do you perceive the task completion outcome to be in achieving the set task goals? how satisfied are you with your task completion performance? effort exertion low/ high how much effort do you need to exert in terms of cognitive and physical resources to achieve the aforementioned execution conditions? frustration level low/ high while pursuing task objectives, how much unfamiliarity, pressure, frustration, and annoyance do you experience? 2.2.2. physiological measurement method virtual reality eye tracking technology virtual reality eye tracking technology enables the recording of participants' eye gaze positions and the paths of their visual exploration. this allows for the exploration of users' cognitive intentions and patterns. in this technique, infrared camera devices are utilized to capture real-time eye movement data [30]. prior to recording users' experimental data, preprocessing steps are required for virtual reality eye tracking [31]. eye movement data processing is done as follows. the raw data obtained from the experiments is processed before its specific use, facilitating subsequent data analysis. after preprocessing the raw experimental data, three commonly used eye-tracking data parameters are obtained: userbased fixation points, saccades, and paths. the visualization methods for presenting postexperimental eye-tracking data consist of three components, including the visualization input space, visualization mapping, and visualization output space [32, 33]. the output format of experimental data varies depending on different types of indicators and tasks. examples include scan path method, heat map method, area of interest (aoi) method, and 3d spatial method [34]. in this study, at the experimental validation level, an eye tracker was employed to capture users' implicit cognition. through visualization processing of eye-tracking data, users' underlying cognitive intentions were represented. the cognitive hierarchy was divided into visual stickiness, complexity, interaction efficiency, and cognitive fit. the elemental hierarchy included basic elemental categories, the quantity of elements, task completion time, accuracy of user interaction operations, and users' cognitive resource investment. corresponding eye298 q.-w. fu, q.-h. liu, t. hu tracking metrics consisted of fixation point heat maps, number of fixation points, sequence of fixation points, average fixation duration, and pupil diameter. 2.3. prediction and optimization algorithms in this study, a convolutional neural network (cnn) was employed as the prediction method. cnn is a type of neural network used for analyzing data with network-like structures [35]. it consists of multiple layers of neural networks, each composed of independent neurons. its essential function is to map input data to output data by learning the mapping relationship between a large set of input and output pairs [36]. cnn can automatically extract features at different levels from images and utilize these features for tasks such as classification and recognition. it exhibits high abstraction and generalization abilities, enabling better representation of the underlying nature of the data [37]. genetic algorithms employ genetic operations to generate one or multiple clusters of design entities from the target design objects. the essence of these clusters lies in their tangible characteristics, and design features can be encoded using the principles of genetic algorithms [38]. through the selection process driven by the fitness function, individuals form a new population. the objective of genetic algorithms is to continuously evaluate each individual, ensuring that individuals better adapted to the environment are given a higher chance of reproduction. furthermore, with each iteration and an increased population size, the distance to the optimal solution is minimized. 3. design methodology 3.1. multi-objective optimization process for vr task scenario resources in the system integrating intelligent algorithms to assist designers in making design decisions can significantly enhance work efficiency. this article employs convolutional neural networks for their nonlinear features and genetic algorithms for their multi-objective optimization characteristics, applying them in design decision-making. the predictive model's logical task flow is structured based on the cognitive multi-channel resources and user cognitive load prediction for evaluating interactive selections within vr system cognition. the specific design process is illustrated in fig.1. firstly, vr system configuration requirements are taken as input, based on which interface scenarios and design resource features of multimodal perceptual channels are selected. these include visual channel design feature libraries, auditory channel design feature libraries, and tactile channel design feature libraries. secondly, the priority ranking of design resource features based on the sensibilityrationality analysis of design elements serves as a reference for design schemes. objective data from the vr system's complex network is used for objective analysis of system elements, while simultaneously incorporating user cognitive demand elements through analytic hierarchy process. multi-objective optimization research on vr task scenario design based on cognitive load 299 fig. 1 predictive model task flow next, the vr task scenario system is designed, and the scheme is inputted into a genetic algorithm for multi-objective selection of design elements, providing the optimal solution for designer reference. simultaneously, the scheme is inputted into a cnn neural network to detect whether the constructed design scheme meets the requirements of low cognitive load and design constraints. if the final scheme satisfies the constraints of the design guidelines, it is saved for implementation. 3.2. prediction of user cognitive load in vr systems 3.2.1. data collection for input set of prediction model the data collection for the input set involves the application of design element analysis to deconstruct and reanalyze the design element features of virtual reality interface samples. the design elements include layout of the operational area, visual browsing sequence, primary color scheme, transparency, auditory cues, haptic feedback, and others, while maintaining consistent interface dimensions, font formats, and sizes as shown in table 2. subsequently, the design elements of the virtual reality interface are extracted, and the distribution of each design category is determined based on their corresponding elements. the visual channel resource features are processed using adobe illustrator (ai) techniques, while the auditory channel and tactile resource features are edited using unreal engine 4 (ue4). 300 q.-w. fu, q.-h. liu, t. hu table 2 table of design elements for vr information interface scenario design projects feature map interpretation interface layout the task action bar is arranged vertically on the left side of the interface, with the specific action buttons located on the upper side of the task bar coded as 1; the taskbar is arranged vertically on the left side of the screen, with the specific buttons on the lower side of the taskbar code 2 the taskbar is arranged vertically on the right side of the interface, with the specific action buttons located on the upper side of the taskbar code 3; the taskbar is arranged vertically on the right side of the interface, with the specific action buttons located on the lower side of the taskbar code 4; the task action bar is located horizontally on the interface and the specific action buttons are located on the left side of the task bar code 5; the task action bar is located horizontally on the interface and the specific action buttons are located on the right side of the task bar code 6; the task action bar is located horizontally in the lower part of the interface and the specific action buttons are located on the left side of the task bar code 7; the task action bar is located horizontally in the lower part of the interface and the specific action buttons are located to the right of the task bar code 8; chamfering of graphic areas square chamfer code 1; round chamfer, code 2 main colors light warm code 1; dark warm code 2; light cool code 3; dark cool code 4 task area with overall color contrast same shade different lightness code 1; different shade different lightness code 2; same shade same lightness code 3 visual browsing sequence text length greater than graphic length code 1; text length equal to graphic length code 2; graphic length greater than text length code 3 transparency of the interface yes 1 no 2 prompt tone yes 1 no 2 vibrating touch yes 1 no 2 during the organization of the input set, the design feature module is considered an explicit knowledge feature and falls under the category of feature classification values. hence, the one-hot encoding method is employed to expand the discrete feature values into euclidean space. the digit '0' represents an irrelevant option, while the digit '1' represents a relevant option. for instance, taking experimental sample 1 as an example, its design features are decomposed into the configuration '21112111', and the corresponding one-hot encoded vector is ('01000000', '10', '1000', '100', '01', '010', '10', '10', '10'). all 39 design elements of the vr system information interface are processed as input set information using the one-hot encoding method. 3.2.2. data collection for prediction model output set the output set data consists of cognitive load values and task response durations for vr system task scenarios. to prepare for the experiment, 39 vr task information interface samples are selected and imported into the ue4 system for scenario setup, with the inclusion of varying parameters. a total of 30 participants, aged between 20 and 26 years, are recruited for the cognitive load experiment. among them, there are 19 male and 11 multi-objective optimization research on vr task scenario design based on cognitive load 301 female participants. all participants have normal or corrected-to-normal vision, without any impairments in visual, auditory, or tactile perception. additionally, all participants are righthanded, with 24 of them having prior experience using vr systems, while 6 participants have no prior vr experience. the experiment involves two tasks: 1) reading interface data information, and 2) selecting the "enter system" option within the designated area. the experimental scenario is recorded to capture the time (in seconds) taken by users to select the task buttons, and cognitive load values are measured using the nasa-tlx scale. the cognitive load values and response durations are averaged for the 16 participants, and the results are presented in table 3. table 3 sample code one-hot code cognitive load reaction time 1 0100000010100010001010101010 59.32 1.162 2 0000100001010010001010010101 60.8 1.239 3 0000100010000110010010101001 64.52 1.262 4 0100000001100001010100010101 71.56 1.252 5 1000000010010011001010101010 46.6 1.152 … … … … 39 0010000010000100101001100101 108.6 1.434 3.2.3. construction of the prediction model based on the feature analysis of the input set and output set data, the convolutional neural network (cnn) model for vr system scenario design elements consists of the following layers: convolutional layers for feature scanning and extraction, pooling layers for feature filtering, and dimension reduction layers for data flattening and dimensionality reduction. the neural network for predicting cognitive load of vr system users has an input data shape of 28 rows by 1 column. the model is constructed with six convolutional layers. the first convolutional layer is a one-dimensional convolution with 2048 filters, a kernel size of 7, and an output dimension of (22, 2048). the input for the second convolutional layer is the output of the first layer, with 1024 filters, a kernel size of 5, and an output dimension of (18, 1024). the third convolutional layer has 512 filters, a kernel size of 5, and an output dimension of (14, 512). the fourth convolutional layer has 256 filters, a kernel size of 5, and an output dimension of (10, 256). the fifth convolutional layer has 128 filters, a kernel size of 3, and an output dimension of (8, 128). the sixth convolutional layer has 64 filters, a kernel size of 3, and an output dimension of (6, 64). pooling layers with a stride of 2 are then applied. a 12 matrix is used to average the information along the 1n data, reducing the parameters and improving learning efficiency. the output dimension of the pooling layers is (3, 64). subsequently, an eighth layer with 192 flattened neurons is followed by a ninth layer with 128 fully connected neurons. the tenth layer consists of 20 fully connected neurons, and the eleventh layer is the output layer with a single neuron. 3.3. multi-objective optimization method for virtual reality scene based on genetic algorithm currently, there is a lack of research in the field of virtual reality (vr) scenario element optimization based on user cognitive load. moreover, the mathematical models for optimizing vr scenario design elements are limited. to address this, we first validate the 302 q.-w. fu, q.-h. liu, t. hu reliability and validity of the nasa task load index (tlx) indicators for vr task scenarios. then, integrating the guidelines of the nasa tlx, we propose design principles that influence user cognitive load. subsequently, we construct a mathematical model for virtual reality interaction scenarios. we employ genetic algorithms to analyze the multi-objective optimization problem of vr scenario design elements. this approach aims to enhance user interaction experience and improve the accuracy and efficiency of designers' work. by considering the cognitive load of users, this research provides technical support to achieve better interaction experiences for users and enhance the accuracy and efficiency of designers' work. 3.3.1. multi-objective optimization method using genetic algorithm in traditional vr scene construction, designs and modifications are typically based on the designer's experience, lacking a comprehensive optimization approach throughout the entire process. therefore, the introduction of genetic algorithms into virtual reality scene construction is proposed. by incorporating cognitive load indicators as constraints and establishing corresponding mathematical models, important knowledge nodes are applied as objective functions in the optimization process of virtual reality scene construction. this approach satisfies design constraints and user cognitive requirements, adhering to the principles of human-computer interaction design in virtual reality scenes. 3.3.2. evolutionary mechanism of design elements in the multi-objective optimization of design elements using genetic algorithms in virtual reality, the design variables are transformed into individual encodings, enabling a more comprehensive representation of design elements within the population of virtual reality scenes. in the initialized population, features such as scene tone, interface layout, and sound presence in the virtual reality system can be represented as specific structural data. each encoding unit corresponds to a class of structural feature parameters, and each chromosome encompasses a set of features. by mapping design elements in virtual reality to the search space of genetic algorithm's optimal combinations, the encoded feature parameters are applied to vr scenario design solutions. in the optimization process based on design scheme, we employ the solving method of multi-objective optimization using genetic algorithm. two crucial steps in this process are crossover and mutation, along with determining the evolution mechanism, elaborated as follows: crossover and mutation: crossover is performed between chromosomes under certain constraints. in genetic algorithms, crossover refers to the exchange of partial genes between two parent chromosomes with a certain probability. once a pair of chromosomes is selected, two crossover points are randomly generated, and the range of exchange is determined. offspring chromosomes are obtained by exchanging chromosome segments. mutation occurs after crossover and acts on independent offspring, altering their module instances. determining the evolutionary mechanism: the method for solving virtual reality scene design using genetic algorithms starts with an initial population of design elements randomly generated through computations. the process of selection, crossover, and mutation is iteratively performed. following the principle of survival of the fittest, the population evolves generation by generation towards the established goals. based on the fitness values of individuals, a roulette wheel selection method is employed to select excellent combinations from the current population. sequential crossover is applied to parent individuals, exchanging partial genes, thereby generating new combinations. multi-objective optimization research on vr task scenario design based on cognitive load 303 3.3.3. validity and reliability testing of objective function indicators based on cognitive load metrics calculation process of nasa-tlx scale utilization: firstly, the applicability of the cognitive load metrics of the nasa-tlx scale in virtual reality (vr) scenarios is explored. the nasa-tlx scale is employed for subjective assessment of users' mental workload. this scale is not only highly accepted by users but also exhibits low variability among participants. to investigate the suitability and feasibility of the nasa-tlx scale in assessing users' cognitive load in realistic task scenarios, a virtual reality cognitive experiment is conducted to measure users' cognitive load after completing cognitive tasks. the study hypothesizes that the nasa-tlx scale is suitable for evaluating cognitive load induced by virtual reality task operations, thereby demonstrating good internal consistency reliability and structural validity. experimental research method: the study involved 20 undergraduate and graduate students, aged between 20 and 26. among them, 11 were male and 9 were female, all righthanded and familiar with basic virtual reality operations. the experimental task required participants to engage in cognitive activities within the virtual reality system, such as browsing, selecting, searching, and interactive actions. when participants subjectively felt fatigued, they filled out the nasa-tlx scale to rate the load levels of the six indicators. the cognitive load levels of each item and the overall cognitive load level were calculated using the aforementioned process. validity and reliability testing results of the nasa-tlx scale indicators: the correlation between the nasa-tlx scale indicators and the overall users' cognitive load levels, as well as the correlation among the indicators, was analyzed using spss software, as shown in table 4. the indicators mental demand, physical demand, temporal demand, performance, effort, and frustration are denoted as l1, l2, l3, l4, l5, and l6, respectively. it can be observed from the table that there are significant correlations between the levels of each indicator element and the cognitive load levels. the analysis of the internal consistency reliability (cronbach's coefficient) among the six indicators reveals a value of 0.79, indicating good reliability of the indicators for evaluating the overall cognitive load in virtual reality task scenarios. the structural validity of the nasa-tlx scale was analyzed using principal component analysis to examine the factor loading of each indicator. with an eigenvalue threshold of ≥1 and an orthogonal rotation, two common factors were extracted, accounting for a cumulative contribution rate of 61.43%. the factor analysis results in the table indicate that mental demand, physical demand, temporal demand, performance, effort, and frustration have high factor loadings on the first factor, while performance belongs to the second factor. thus, it can be inferred that in virtual reality task scenarios, mental demand, physical demand, temporal demand, effort, and frustration intuitively reflect users' cognitive load in operating virtual reality scenes, while performance represents cognitive outcomes resulting from higher load levels (see table 5). therefore, through the testing of scale indicators' reliability and validity, it can be concluded that the nasa-tlx scale has good internal consistency and these six indicators can be used as evaluation tools for judging cognitive tasks in virtual reality cognitive scenarios, thus suitable for optimization research on vr scene design elements. 304 q.-w. fu, q.-h. liu, t. hu table 4 correlation analysis between cognitive load indicators and total load l1 l2 l3 l4 l5 l6 l1 0.19 0.47** 0.03 0.68** 0.39** 0.75** l2 0.36** -0.11 0.22* 0.45** 0.49** l3 0.05 0.45** 0.37** 0.75** l4 0.09 -0.13 0.29** l5 0.39** 0.70** l6 0.68** note: * indicates p < 0.05; ** indicates p < 0.01 table 5 correlation analysis between cognitive load indicators and total load factor load common factor variance f1 f2 l1 0.79 0.25 0.73 l2 0.60 -0.43 0.60 l3 0.66 0.08 0.62 l4 -0.02 0.79 0.70 l5 0.78 0.34 0.75 l6 0.73 -0.45 0.67 characteristic root 2.63 1.21 contribution rate (%) 43.09 18.97 cumulative contribution (%) 43.09 64.32 3.3.4. objective function qualification based on the reliability and validity analysis of the nasa-tlx scale indicators, it is determined that these indicators are sensitive to the assessment of user cognitive load in the context of virtual reality task scenarios. they effectively evaluate users' subjective mental workload and demonstrate good reliability and validity. therefore, when considering the design elements of virtual reality interactive scenarios, the cognitive load of users can be assessed from six dimensions: mental demand, physical demand, temporal demand, effort, frustration, and task performance. task performance represents the outcome of the interaction. thus, the assessment and limitation of cognitive load are based on the dimensions of mental demand, physical demand, temporal demand, effort, and frustration. these dimensions are employed in virtual reality human-computer interaction performance research. cognitive requirements: in virtual reality environments, the alignment and composition of design elements based on gestalt psychology's principle of complementarity can reduce users' cognitive load and enhance their understanding of virtual scenes or interfaces [39]. by aligning the elements to the principles of gestalt psychology, the level of alignment and integration among design elements can be optimized, resulting in reduced cognitive demands for users when interacting with virtual reality environments. the formula expression is as follows: 𝐷𝑠,𝑖 = 1 − (𝑛𝑣+𝑛𝑖) 4𝑛 (1) in the context of vr scene design, nv and ni represent the number of design elements aligned vertically and horizontally, respectively. the variable n represents the total number of design elements within the vr scene, in which ds,i represents the alignment degree of design elements. it is important to consider both the quantity and size of components within multi-objective optimization research on vr task scenario design based on cognitive load 305 vr scene design elements. by employing a limited yet precise set of expressive elements, the corresponding information can be effectively conveyed to users. it can be expressed as follows: 𝐷𝑒,𝑐 = 𝑛𝑠𝑖𝑧𝑒 𝑛 (2) where nsize is the number of different component sizes, de,c represents the proportion of the design element size to the number of elements. physical requirements: on the physical demands level, it is essential to distribute design elements with high frequency of use and significant importance within the user's optimal visual range or the most efficient operational area within their hand's reach. this distribution aims to reduce the user's cognitive load and enhance the operational efficiency of human-computer interaction in virtual reality. according to the existing literature, the visual presentation and positioning of design elements yield significant differences in the observed results. the hierarchical level of interface layout can be calculated using eq. (3): 𝑚𝑎𝑥𝐹𝑎 = ∑ ∑ 𝜆𝑘 𝑠𝑖𝑘𝑔𝑖 𝑠 𝑛 𝑖=1 4 𝑘=1 (3) the expression in eq. (3) represents the layout of the primary and secondary elements in a vr scene, where fa denotes the significance of each element within the vr visual space. here, k takes on the values 1, 2, 3, and 4, representing the upper-left, lower-left, upperright, and lower-right regions of the vr visual space, respectively. λk denotes the importance weight of region k. n represents the total sum of design elements. sik denotes the area of element i within region k. s represents the total area of the visually perceptible space. lastly, eq. (4) is defined as the representation of the priority and frequency of operations for element i calculated as the average of its priority mi and frequency fi. 𝑔𝑖 = 𝑚𝑖+𝑓𝑖 2 (4) temporal requirements: temporal requirements in virtual reality tasks are closely associated with the organizational characteristics of design elements during task operations. the visual stimuli of design elements, with varying degrees of aggregation and dispersion within the user's visual field, can be organized to create a coherent visual cognitive effect of the virtual reality scenario. this organization facilitates users in quickly locating target information and completing interactive operations within the virtual reality scene. designers can guide the construction of virtual reality scenes by utilizing the principles of visual organization based on similarity and proximity within the user's visual perception. for instance, when faced with a large number of design elements and information to be expressed, the proximity principle can be applied to arrange related design elements in close proximity or present related visual information using a consistent color scheme. this approach enhances the visual coherence of the system, enabling users to rapidly search for relevant information. therefore, the correlation between virtual reality design elements can reflect the degree of closeness among design elements and the proximity of design elements representing similar functionalities. the functional information correlation matrix of the design elements is expressed the following two equations: 𝑂 = [𝑜𝑖𝑗] = [ 𝑜11 𝑜12 … … 𝑜1𝑚 𝑜21 𝑜22 … … 𝑜2𝑚 … … … … … … 𝑜𝑛1 𝑜𝑛2 … … 𝑜𝑛𝑚 ] (5) 306 q.-w. fu, q.-h. liu, t. hu 𝑂𝑖 = ∑ ( 𝑜𝑖𝑗 𝑑𝑖𝑗 )𝑚 𝑗=𝑖+1 (6) here, m represents the number of design elements, oij denotes the degree of association between design elements i and j. the mathematical model for the correlation between design element i and other design elements is given by eq. (6). effort level: at the effort level, considering the dynamic nature of users' cognitive processes, it is possible to obtain the optimal combination of vr scene design elements based on users' prior knowledge, cognitive abilities, and individual habits. this optimization facilitates dynamic feedback adaptation to users' cognitive processes. by leveraging prior knowledge, design elements that hold higher priority, involve frequent observation, decisionmaking, and interaction, as well as real-world information, can be distributed within the user's comfortable viewing and interaction range. this approach reduces the complexity of user's cognitive resources and the burden on short-term memory, allowing for effective allocation of existing cognitive resources between intuitive reasoning tasks. this allocation ultimately reduces effort levels and cognitive load. furthermore, prioritized and important design elements should be distributed within the user's optimal control area. the importance level of different design elements can be expressed through an importance matrix, eq. (7), which captures their relative significance: 𝐼 = [𝐼𝑖𝑗] = [ 𝐼11 𝐼12 … … 𝐼1𝑚 𝐼21 𝐼22 … … 𝐼2𝑚 … … … … … … 𝐼𝑛1 𝐼𝑛2 … … 𝐼𝑛𝑚 ] (7) the weights are determined based on the expert assessment, and the expert weight matrix is shown in eq. (8), where wz represents the weight value assigned by experts' ratings: 𝑊𝑧 = [𝑊𝑧1 𝑊𝑧2 … 𝑊𝑧𝑟]𝑇 (8) the importance of design element i is determined according to eq. (9), where zi represents the importance of design element i: 𝑍𝑖 = 𝐼𝑖𝑗𝑊𝑧 (9) the frequency of use of different design elements is given by means of a frequency matrix, eq. (10), where ci represents the frequency of use of the design element: 𝐶𝑖 = 𝐹𝑖𝑗𝑊𝐶 (10) thus, the design elements and the importance and frequency of use can then be expressed as eq. (11): 𝑆𝑖 = 𝑢𝑍𝑖 + 𝑣𝐶𝑖 𝑢 + 𝑣 = 1 } (11) where u and v are the weights of the importance and frequency of use of the design elements. degree of frustration: frustration level is closely related to interface density, where density refers to the tightness of design element arrangement. it calculates the difference between the actual density and the optimal density in the scene interface. a density level of 50% is considered reasonable. if the density of design elements is moderate, users will experience less frustration. multi-objective optimization research on vr task scenario design based on cognitive load 307 3.4. comparison of optimization results according to section 3.2.1, the design element dataset is encoded as the input set, and user cognitive load is the output set for performance testing. after normalizing the data in the output layer, the mean squared error (mse) function is utilized for evaluation. the expression of the mse function is as follows: 𝑀𝑆𝐸 = 1 𝑝 ∑ (𝑦𝑘 − 𝑦𝑘 ∗)2𝑝 𝑘=1 (12) where p represents the number of samples, and the mse function quantifies the average squared difference between the predicted values and the true values, thus providing a measure of the optimization results' accuracy. lower mse values indicate better performance and a closer match between the predicted and true values. if the mse value is less than 0.01, it demonstrates the reliability of the cnn model for the vr task selection scenario. by comparing the user cognitive load test data with the output values of the established cnn model and calculating the mean squared error (mse), a result of 0.00424 is obtained. since the mse value is smaller than 0.01, it indicates that the cnn model performs well in terms of testing performance. the fitting results indicating a close match between the predicted cognitive load values and the actual test data. this confirms that the established model is capable of accurately mapping design features related to user cognitive load and multi-channel behavior analysis. the vr system's information interface resource combinations consist of various categories, resulting in a total of 82433222 = 3072 possibilities. analyzing the results of the genetic algorithm, the program shows a flattened trend after 50 iterations, as depicted in fig. 2. the design feature resource encoding for the minimum predicted cognitive load is: 0001000010010010010010101010. in this configuration, the design features mainly include the following aspects: ▪ the task selection area is predominantly located in the lower-right part of the interface for convenient user task selection. ▪ the interface incorporates rounded corners, providing users with a softer sensation. ▪ the overall color scheme adopts cool and light shades, and there is a contrast between the task selection area and the overall color scheme in terms of brightness, facilitating users' recognition of target tasks. ▪ the interface is designed with transparency to allow users to perceive the surrounding environment, enhancing their sense of immersion. ▪ proper visual expression is achieved by appropriately combining graphics and text on the interface, thereby enhancing its visual appeal. ▪ in terms of multi-channel information settings, auditory and visual channels are emphasized, such as incorporating sound prompts, background music, and controller vibrations, enabling natural interactions for users within the vr system. 308 q.-w. fu, q.-h. liu, t. hu fig. 2 number of iterations of the genetic algorithm on the other hand, when the input encoding is: 0000010010001000110100011001, the predicted cognitive load reaches its maximum value of 125.55457. the design features of these two data sets provide reference for designers in terms of their respective proportions. therefore, by combining the predictions from the cnn model and the analysis results from the genetic algorithm, we can proceed with the optimization design and selection of virtual reality scene information interfaces. 3.5. system application optimization 3.5.1. determining design principles and language the vr map system developed in this application explores the visualization of smart city information. the system integrates multimedia, computer graphics, and simulation technologies to construct a simulated digital city in a virtual three-dimensional environment. the system incorporates features such as regional introductions, industrial planning, layered land-use planning, visual representation of urban information, and realistic architectural landscapes. it also emphasizes user interaction in the vr environment, providing users with greater flexibility and enjoyment. 3.5.2. experimental equipment and scenarios the experimental process involves the construction of a vr task information system for a smart city based on previous research. subsequently, for the same task requirements in the vr system, participants are asked to select target areas and navigate virtual scenes while filling out a cognitive load questionnaire. the operation tasks in the target selection area are validated and compared using virtual reality eye-tracking experiments. a comparison is made between the multi-objective optimization research on vr task scenario design based on cognitive load 309 solutions developed through the traditional design process and those optimized using the methods proposed in this study. the experimental equipment and an experimental scenario are shown in fig. 3 and fig. 4, respectively. fig. 3 experimental equipment fig. 4 experimental scenario 3.5.3. optimization results the designed solutions were applied using the methods described in this paper for virtual reality scene information interface optimization and solution selection. solution 1 represents the initial interface without design optimization, while solution 2 represents the optimized solution using the methods proposed in this paper. by comparing annotated trajectory data, it is observed that users following solution 2 had shorter task search paths, faster completion speeds, and experienced less cognitive load. this indicates that solution 2 is better aligned with the users' desire for lower cognitive load. consequently, this method improves the efficiency of designers and enhances the accuracy of design solutions. optimizing the scenario strategy in vr systems should aim to minimize the external irrelevant load indicators and reduce the intrinsic load indicators, thus allocating sufficient cognitive resources to appropriate loads and improving users' perception efficiency and satisfaction. 310 q.-w. fu, q.-h. liu, t. hu 4. guidelines for constructing virtual reality scenes based on cognitive load in order to achieve the optimization goal of vr system scenarios, it is necessary to optimize the user cognitive load indicators reasonably so that sufficient cognitive resources can be allocated to important design elements, thereby improving user interaction perception efficiency and experience satisfaction. therefore, based on the results of cnn prediction, multi-objective optimization, and experimental verification, the following design principles are summarized for reference by designers. table 5 comparison of results projects option 1 option 2 vr scenes design elements interface layout: neat and balanced operating area: bottom right corner of the interface chamfered shape of the graphic area: round primary colour: dark cool tones harmony with the overall environment contrast of the task area with the overall colour palette: no contrast visual navigation sequence: left to right, top to bottom graphical text form: combined interface transparency:yes interface layout: neat and balanced position of the operating area in the interface: lower middle features of the graphic area chamfer: beveled chamfer dominant colour: deep cool tones harmony with the overall environment contrast between the task area and the overall colour palette: brightness contrast visual browsing order: left to right, top to bottom balance of graphic proportions. transparency of the interface: yes watch tracks number of fixations:1229 number of fixations:938 duration 1.5306 1.457 cognitive load brain power requirements 5.267 3.8 physical requirements 6.267 5.067 time requirements 6 3.933 mission performance 5.603z 3.2 level of effort 5.7 3.733 level of frustration 5.4 2.533 overall cognitive load 86.67 54.067 multi-objective optimization research on vr task scenario design based on cognitive load 311 1) principles of reasonable design element layout: when designing the information interface for vr system task scenarios, the layout of the information interface should be considered first, rationalizing the arrangement of the task selection area and the data information retrieval area. 2) principles of design element classification and summarization: for the design of visual browsing sequences, attention should be given to the frequency of text and graphics display. in the context of visualized design in virtual reality systems, the design elements targeted by visualization are diverse. 3) principles of timely information feedback: setting up auditory and haptic reminders can reduce users' cognitive load and improve the accuracy of their operations under audiovisual coherence. 4) principles of target area color and scene tone adaptation: by leveraging the contrast of color intensity, the brightness and saturation of the target task layer can be contrasted with the environment, facilitating visual search and reducing cognitive load. the contrast between the task area and the overall color tone affects the accuracy of users' information reading and task selection. 5) principles of user pleasure orientation: modifying the shape of design elements in system scene design can influence users' perceived experiences. for example, sharp corners can give users a sense of ruggedness, while rounded corners can create a soft and gentle feeling, avoiding excessively sharp corners that may cause discomfort. furthermore, the selection of background music can enhance users' enjoyment. in summary, adhering to these guidelines for vr system design can contribute to improved efficiency in constructing vr systems and enhance user satisfaction. designers can benefit from the intelligent optimization techniques, such as genetic algorithms, to optimize the design elements, thereby reducing users' cognitive load and achieving a more effective humancomputer interaction in the context of virtual reality. 5. conclusions in this study, we combined previous research to construct a task information system for a smart city vr system. we conducted a virtual reality eye-tracking experiment to compare and validate the effectiveness of the traditional design process and the optimized approach proposed in this paper for the same design task in the vr system. by collecting and organizing data from the eye-tracking experiment, we built a cnn system to predict users' cognitive load and satisfaction within the system. this approach effectively organizes design elements within vr scenes, thereby enhancing the efficiency of vr system development for designers. furthermore, we introduced intelligent optimization genetic algorithms into the optimization of vr system design elements, with the objective function based on the principle of minimizing users' cognitive load. by analyzing the results of the intelligent algorithm, we identified the system configuration with the highest and lowest cognitive load values. these findings serve as valuable references for designers. the results of our study demonstrate the effectiveness of the proposed methods in optimizing vr system design, enhancing the overall user experience, and reducing cognitive load. this research contributes to the advancement of human-computer interaction in the field of vr systems, particularly in the context of smart city applications. future studies can further explore and refine the proposed 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of a single third-body particle between two plates is simulated using the boundary element method. the particle is considered as deformable, and the coulomb’s law of friction is assumed at the contact interface. the normal pressure distribution and tangential stress distribution in contact as well as the macroscopic force and force moment are calculated. several movement modes are shown to be possible: rolling, rotation, or sticking during the loading. it is found that, differing from rigid particles, the state of particle may change during the loading. the particle may stick to the plates initially, but rotation may occur when the load becomes larger. examples with the same and different coefficients of friction are presented to show kinematics of particle. the method can be further applied to simulation of multiple third-body particles. key words: third-body, particle, friction, numerical simulation, boundary element method 1. introduction it is the mechanics of interactions between the surfaces of contacting bodies which determines the tribological properties in contact. the processes occurring in the “interface” are multiple and complex: material transfer, wear, friction, particle formation, oxidation and corrosion, heat transfer, fluid flow and many others [1]. the so-called “third-body” immediate vicinity of an interface, including the surface layers of the bodies and the particles in the interface volume plays a significant role in terms of tribological properties. the interface particles and interfacial materials develop during the frictional process and are usually unmeasurable. therefore, a very common way to study third-body is experimentally measuring the coefficient of friction and wear rate for different materials, loading and system parameters and surrounding environment [2, 3]. experimental studies have shown that the wear particle flow is essential for the formation, localization, and reconstitution of load-bearing structures or films as well as the resulting friction and wear rate [4]. there have received october 02, 2020 / accepted november 25, 2020 corresponding author: qiang, li affiliation: berlin university of technology, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: qiang.li@tu-berlin.de 538 q. li been only few analytical and numerical studies of the third body processes due to complicated interactions among the moving surfaces and the intermedia. the molecular dynamic (md) simulation is usually used for simulating interaction of atom layers and particles in a micro scale [5, 6]. recently the monte carlo method has been used to predict the three-body abrasive wear rate [7], cellular automata for evaluation of wear particle flow and load and temperature distribution [8]. due to the complicated third-body processes sometimes only one aspect for obtaining a basic understanding of its role is considered, for example only adhesion or friction without wear and lubrication [9, 10]. in this study, we focus on frictional contact of deformable particles between the rigid plates which is numerically simulated by using the boundary element method (bem). bem is an effective numerical simulation method for solving contact problems, especially assisted by the fast fourier transform [11]. it has been applied to various tribological problems in the last twenty years, including indentation test, partial sliding and adhesive contact, etc. [12-14]. recently it has been further developed to layered system [15], power-law graded materials [16] and arbitrary two-dimensional shapes [17]. however, it has been rarely used for contact of third-body particles. in this work we consider a single soft particle compressed by two parallel plates. under the loading the particle could roll, rotate or keep unmoved (sticking). the normal and tangential stress distribution as well as its moment will be calculated for determination of particle state. 2. methods we consider a single particle trapped between two plates, as shown in fig. 1. the following assumptions are made in the study: (1) deformation of a particle is elastic; (2) particle is massless (meaning that we consider the quasistatic processes). it is further assumed that due to chemical reaction or mechanical reasons it stays initially in a given state as shown in fig.1; then the plates are pressed under the external loading. we assume that the coulomb’s law of friction is valid in contact with coefficients of friction 1 and 2 at the upper and lower interfaces, respectively. a) b) fig. 1 a single particle trapped between two plates. (a) particle is rigid and concentrated normal and tangential forces fn, ft act on contact points a and b; (b) particle is deformable and loads are distributed stresses p and τ on areas simulation of a single third-body particle in frictional contact 539 under the loading, the particle may roll, rotate, or keep its initial state. if the particle is rigid, then the state depends only on the coefficient of friction and the geometry of the particle, not on the loading process. 2.1. rigid particle let us consider a two-dimensional case as shown in fig. 1a, or three-dimensional but the geometry of the particle is symmetric with respect to the xz-plane, so that only the moment around y-axis (perpendicular to xz-plane) has to be considered. the origin of the coordinates is located at the centroid of the particle. the normal and tangential forces acting on the particle at upper contact point a are defined as fn1 and ft1, and forces at lower point b are fn2 and ft2. simply according to the equilibrium condition for the particle, we have n1 n2 t1 t2,f f f f= − = − , (1) and total moment of these forces around the origin should be zero if the particle is unmoved n1 1 n2 2 t1 1 t2 2 0m f x f x f z f z= + + + = , (2) which gives t1 t2 1 2 n1 n2 1 2 f f x x f f z z  − = = = − . (3) following the coulomb’s law of friction, the tangential force at the sliding state is equal to normal force multiplying the coefficient of friction which is the maximal reachable value of tangential force. therefore, by comparing the local coefficient of friction with the ratio in eq. (3), we can predict the state of particle as follows: 1 2 1 2 1 2 1 2 , , sticking and , rolling sticking, sliding and , rolling sticking, sliding , , rotation around origin                        a b b a o . (4) in this paper we define “rotation” as the rotation of the particle around its centroid, and “rolling” as the rotation around the contact point either in the upper or lower interface. if the coefficients of friction are same 1=2, the particle will either rotate or keep unmoved: , sticking , rotation        . (5) from eqs. (3) and (4), it follows that the state of particle under the loading depends only on the coefficient of friction and the geometry and orientation of particle. the similar behavior was discussed about kinematics of particles in [18]. 2.2. soft particle if the body is deformable, the contact will be different. firstly, the loads are not concentrated forces acting on some points, but distributed stresses on some areas. and the moment arms of the stresses are then related to the location of contact area. furthermore, 540 q. li due to the deformation of particle, there could be multi-contact spots in one contact interface during loading. as sketched in fig. 1b, the equilibrium condition for the deformed particle in this case becomes up up low low up up low low up up low low up up low low ( , )d ( , )d 0 ( , )d ( , )d 0 ( , ) d ( , ) d ( , ) d ( , ) d 0 z x f p x y a p x y a f x y a x y a m p x y x a p x y x a x y z a x y z a      = + =  = + =  = + + + =         . (6) the third equation in (6) is the equilibrium condition for the moment of the normal and tangential stresses around the centroid of particle. it is noted that the centroid may change a bit due to the deformation during the loading. the stresses in contact have to be numerically calculated using the bem. for a clear description of numerical simulation, we mention two available basic functions in the bem: (1) normal pressure distribution p can be calculated with given indentation depth d and the geometry of indenter (particle in this study); (2) tangential stress τ can be calculated with the given tangential displacement. the latter one has been already applied to the partial sliding with the coulomb’s law of friction: under the normal load a deformable body is pressed on a rigid plane and then the plane moves a bit in tangential direction; then the tangential displacement of the body at the contact area can be calculated as well as the tangential stress. as shown in fig. 2 with an example of a tilted ellipsoid, there will be a stick region in the middle of contact area and a relative slip region at the boundary of contact (fig. 2b). the criterion can be formulated as follows: when the tangential stress is smaller than normal stress multiplying the coefficient of friction τ<p, the incremental tangential displacement of the surface in contact area ux is equal to that of indenter ux0. these elements are in the stick region. otherwise, the elements are in a state of slip: 0 , in stick region , in slip region    =  = x xu u p . (7) a) b) c) fig. 2 simulation example of partial sliding of a deformable ellipsoid on a rigid plane (a) illustration of contact: the plane has a contact with ellipsoid under the normal load and then moves a little in tangential direction. there is a stick (black) and a slip (gray) region in contact area (b). the tangential stress distribution is shown in (c) simulation of a single third-body particle in frictional contact 541 we consider a soft particle squeezed by two plates. the external normal loading on plates is given and the two parallel plates are fixed in horizontal direction so they can move only in vertical direction. with the existing functions described above we can further realize the following calculations iteratively: (1) the deformation of particle and normal stresses can be obtained with the given normal force; (2) the tangential stress and a stick and slip region can be calculated with the given tangential force under the coulomb’s law of friction (as shown in fig. 2). in more detail, the simulation of particle contact is the following. under known external load fz the normal stresses and contact areas on upper and lower interfaces pup, plow, aup, alow and deformation of particle can be easily calculated, also the moment of normal pressures p up up low low( , ) d ( , ) dm p x y x a p x y x a= +  . (8) if the coefficients of friction in the upper and lower interfaces are the same, μup = μlow = μ, one can simply check whether the particle would rotate. the moment of normal load urges the particle to rotate, but the frictional force in tangential direction will hinder this movement. because the plates are rigid, the moment arms of tangential stress are the same in each contact interface. the state of particle can be determined by comparing the tangential force needed for rotation fʹx which is equal to ' 1 2 , ( ) p x m f z z = − (9) and maximal achievable tangential force μfz (sliding frictional force): particle will rotate, if  x zf f , (10) where z1-z2 is the moment arm of tangential forces. if the condition (10) is not met, the particle will generally keep unmoved but a partial sliding exists. then the tangential stress distribution and a stick and slip region can be calculated with tangential force fʹx in eq. (9). for different coefficients of friction, the criterion is basically same. the tangential force needed for rotation fʹx is still obtained from the results of normal contact in eq. (9); then by comparing it with the value of μfz for the upper and lower interfaces, respectively, the state of particle is determined: up low up low up low up and , rotation and , rolling around upper contact spot and , rolling around lower contact spot a x z x z x z x z x z x z x z f f f f f f f f f f f f f f                   lownd , sticking x zf f  . (11) it is noted that the contact spot is an area in this case, so if the body size is much larger than the size of contact, one can simply let the particle roll around the center of the sticking area. 542 q. li 3. examples of numerical simulation in this section, we illustrate the above method with a few examples. 3.1. cases with the same coefficients of friction firstly, a simple case with the same coefficients of friction is simulated, μup = μlow = μ. the particle has only two states: rotation or keeping unmoved. in the simulation external normal load fn is set to increase linearly. the particle has a “potato” shape and its geometry is symmetric with respect to the shown plane. it is numerically generated by superposition of ellipsoid and sine waves with comparable wavelength and amplitude. in each compression step, the contact area, a stick and slip region and the tangential force are calculated. fig. 3a shows the dependence of the ratio of tangential force and normal force multiplied by coefficient of friction ft/(fn) on the compression distance. the distance of the horizontal coordinate is normalized by initial gap d0. the value of ft/(fn) indicates the state of particle: ft/(fn)<1 (gray region) for a general sticking state (actually in a partial sticking state), and ft/(fn)=1 (red line) for rotation. it is seen that the particle is in a state of stick at the first contact under a very tiny load. if the particle is rigid, it will always stay in this state no matter how large the load is. but in this case of a soft body, we can see that the particle has a few rotations during the compression. observing states a and b marked in the curve (or similar series c-d-e) and corresponding contact areas in fig.3b, the particle has the following behavior: initially it is in a state of sticking, then the tangential force increases during compression and approaching to the value of fn while the sticking region in the contact area shrinks until rotation occurs. this phenomenon repeats several times and finally the particle stays “stable” between the plates while the tangential force decreases (state f). a) b) fig. 3 simulation of a single particle in compression: (a) dependence of ft/(fn) on the compression distance; (b) contact configurations corresponding to the states marked in fig.3a examples with smaller values of coefficient of friction are shown in fig. 4. other parameters remain unchanged. it is observed that the particle rotates at the first contact: after the rotation by an angle 6 in the case of =0.3 and angle 9 in the case of =0.2 it comes into the state of sticking. for =0.2 the sticking state keeps in the whole compression process after the simulation of a single third-body particle in frictional contact 543 initial rotation. it is noted that the tangential force in this case becomes negative at the larger load, which indicates that the tangential force decreases during compression and will act on the surface in the opposite direction. a) b) fig. 4 examples with smaller coefficient of friction: (a) =0.3; (b) =0.2 3.2. cases with different coefficients of friction now we consider a case with different coefficients of friction, up=0.2 and low=0.4. the external normal loading is the following: the normal force keeps constant till the particle reaches a stable state and then increases linearly. from fig. 5a it is seen that under the load the upper interface is in a slip state where the value of ft/(upfn) is one (blue curve with stars) and lower interface in a sticking state where ft/(lowfn) is smaller than one and decreases (red curve with triangles). this rolling continues to a stable state point “b” where the particle sticks in both interfaces (fig. 5b). with an increasing normal loading the particle is further compressed without rotation (state c). a) b) fig. 5 simulation examples with different coefficient of friction in the upper and lower interfaces. (a) state of particle during the loading; (b) three contact configurations corresponding to the states marked in fig. 5(a) 544 q. li 4. conclusion we studied a simple case of third body particle: a single deformable particle is compressed by two parallel plates. the frictional contact of the particle was numerically simulated by the fft-assisted bem. unlike the rigid particle, the soft particle during the compression may change its state, for example from sticking to rolling. the normal and tangential stress distributions, deformation of the particle, and stick and slip regions were calculated. by comparing the tangential force needed for slipping and the maximal achievable tangential force, the state of particle state was determined. applications of this method were shown with a few examples with the same and different coefficients of friction in the upper and lower interfaces. this method can be further developed for the simulation of kinematics of multiple third-body particles 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19, e201900076. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 1, 2015, pp. 33 38 1experimental investigation of the adhesive contact with elastomers: effect of surface roughness udc 539 lars b. voll berlin institute of technology, institute for mechanics, germany abstract. adhesion between an elastomer and a steel indenter was studied experimentally and described with an analytical model. cylindrical indenters having different roughness were brought into contact with an elastomer with various normal forces. after a “holding time”, the indenter was pulled with a constant velocity, which was the same in all experiments. we have studied the regime of relatively small initial normal loadings, large holding times and relatively large pulling velocities, so that the adhesive force did not depend on the holding time but did depend on the initially applied normal force and was approximately proportional to the pulling velocity. under these conditions, we found that the adhesive force is inversely proportional to the roughness and proportional to the normal force. for the theoretical analysis, we used a previously published mdr-based model. key words: adhesion, elastomer, surface roughness, fracture criterion for elastomers 1. introduction in a variety of applications of elastomers adhesive forces play a significant role, which may be desirable or undesirable. the classical adhesion theory was developed by johnson, kendall and roberts for an elastic ball on the rigid flat surface [1]. however, this theory is based on the balance of energy which is released due to propagation of an adhesive crack and the work of adhesion needed to create free surfaces; it is therefore only applicable to purely elastic bodies. johnson compares the jkr theory with experiments made with gelatin balls [2]. however, this comparison is only valid for very small pulling velocities. for finite velocities, the jkr theory is not valid any more. for viscoelastic adhesive contacts, there is a large number of different approaches (see e.g. the review [3]). in the present paper adhesion between an elastomer and cylindrical indenters of steel is studied experimentally to received january 09, 2015 / accepted march 2, 2015 corresponding author: lars b. voll technische universität berlin, str. des 17. juni 135, 10623 berlin, germany e-mail: lars.voll@tu-berlin.de original scientific paper 34 l.b. voll determine the influence of varying surface roughness. the theoretical background for our study is given by the concepts described in [4, 5]. it is clear that for very small roughness the surfaces will effectively be ideally flat. this case was investigated in detail in our previous study [6]. it could be shown experimentally that under the conditions studied, the regularities of adhesion with elastomers are completely different from predictions of the jkr theory. in particular, the influence of the geometry (e.g. the radius in the case of cylindrical indenters) differs qualitatively from the results of the jkr theory. we have shown that, for sufficiently fast detachment, the rheology of the elastomer plays the most important role. at the same time, the adhesion force shows strong dependency on the velocity and temperature. one of the most interesting findings of the previous study was that the adhesion force with a cylinder was approximately proportional to radius a of the cylinder (and not to a 3/2 according to the jkr theory). for multi-contact problems this would mean that the adhesion force should be proportional to the sum of the radii of all contacts, the so-called contact length [7]. otherwise, the contact length is approximately proportional to the normal load, the coefficient of proportionality being dependent on the roughness [7]. in the present study, we try to prove this hypothesis and to see if under the above conditions the adhesion force is proportional to the applied normal force. in studying the force dependence of adhesion, we have to distinguish two cases: the critical load case occurs when the normal force is so large that during the initial indentation the adhesive force achieves a plateau and does not change any more with further increasing normal force. this means that the sum of the diameters of the microcontacts and the macroscopic diameter of the indenter are of the same order of magnitude. in this case, no dependence of the adhesion force on the normal force and on the holding time was found. our main interest was the sub-critical case. the adhesive force in this case depends on the real contact length. to describe the separation of the indenter from the elastomer, different failure criteria should therefore be used. for the description of the experimental study, a deformation criterion was used [5, 8]. in the subcritical case of loading the real contact length between the elastomer and the rough indenter is timedependent [9, 10]. also, elastomeric friction is determined by the viscosity and the adhesion [11]. understanding the adhesion processes is therefore important for understanding elastomer friction. finally, both viscosity and adhesion are strongly temperature-dependent [12]. these dependencies make a complete experimental characterization difficult, and in this paper only the normal force and roughness of the indenter were varied. the temperature, pull-off velocity and holding time were constant. in the following, the experimental set-up and the measurement results will be presented. 2. experimental apparatus for measuring the adhesive force 2.1. set-up roughness ra was measured using a white light interferometer. for the measurement of the adhesive force an apparatus was developed, which enables measurements in a temperature range of approx. -80 °c to 80 °c. fig. 1 shows the main components and their functions. experimental investigation of the adhesive contact with elastomers: effect of surface roughness 35 fig. 1 principle sketch of the apparatus for the measurement of the adhesive force (temperature regulation not shown) the normal force is produced with fixed weights, which are made with an accuracy of 0.1 g. the maximal normal force is limited by the nominal load of the force sensor, at 50 n. the lower limit of 1.5 n is due to the weight of the indenters and the fixture. the indenter is moved with a stepping motor which drives a thread spindle. a special lever system ensures that the indenter and its holder can only be moved vertically. the lever system also makes sure that the only applied force is in the normal direction. the structural implementation of the motor drive takes the resonance frequency of the force sensor into account. the stepping motor can be operated at variable speed and acceleration, and allows controlling the displacement precisely. cooling is provided by gaseous nitrogen flowing through a conduit system around the sample holder. preliminary tests have shown that there is no difference between direct and indirect cooling with nitrogen. for attachment of the material sample a gimbal bearing was designed, which makes sure that the sample is held horizontally. the measurement procedure starts with the insertion and alignment of the material sample and sealing of the measuring apparatus. subsequently the normal force is adjusted by attaching weights. next, the desired temperature is adjusted. when the material specimen has reached the test temperature, the indenter is set down on the elastomer in a controlled manner. after a pre-defined contact time, the indenter is pulled off by means of the lifting unit. the force necessary to detach the indenter from the elastomer is measured. 2.2. experimental results in this section, we present the measurement results. the experiments were carried out with different rough cylindrical indenters. these were pressed into the elastomer with different normal forces. the adhesion force and the displacement of the indenter were measured, as shown in figs. 2 and 3. 36 l.b. voll 4 6 8 10 12 14 16 18 20 22 0 1 2 3 4 5 6 7 8 normal force, n ad h es io n f o rc e, n r a = 0,028 µm r a = 0,260 µm r a = 2,730 µm fig. 2 dependence of the adhesion force on the normal force and surface roughness fig. 2 shows that the adhesion force is inversely proportional to the roughness and proportional to the normal force. the indenters had a diameter of 14 mm. the pull-off velocity, hold time and temperature were kept constant. 0 2 4 6 8 10 12 14 16 18 20 22 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 normal force, n d is p la ce m en t, m m r a = 2,730 µm r a = 0,028 µm r a = 0,260 µm fig. 3 dependence of the displacement on the normal force and surface roughness in fig. 3, the other main result of this investigation shows the displacement of the indenter as a function of the normal force. the displacement is also proportional to the normal force, and increases with increasing roughness, until saturation is reached. experimental investigation of the adhesive contact with elastomers: effect of surface roughness 37 3. theoretical interpretation in [4] and [5] it was shown that adhesion of elastomers is essentially determined by two factors: the rheology of the elastomer and the "failure criterion". in a previous work [6] we have shown that, at least for some elastomers, the so-called deformation criterion [4] is the suitable failure criterion. during a fast pull-off the rheology of the material can be approximated as a newtonian fluid with viscosity . under these conditions, we have derived and verified a simple relation for the adhesion force 04af dv  , (1) where d is the diameter of cylindrical indenter and v0 is the pull-off velocity. the influence of temperature on the adhesive force could be accounted for by the dependence of the viscosity on the temperature given by the arrhenius equation ( ) b tt ae  . (2) where a and b are empirical constants, which are determined by experiments. the equation (1) can also be written in the general form 04a cf l v  , (3) where lc is the “contact length”. it is defined as the sum of the diameters of all microcontacts. in this form, the equation has also been confirmed for fractal surfaces by direct numerical simulations [9]. for rough surfaces, the contact length for small forces is roughly proportional to the normal force [7] n c f l hg  . (4) here, h is the root mean square of the roughness and g is the shear modulus. substituting this into eq. (3) we obtain the adhesion force 04 a n v f f hg   . (5) the adhesive force is, in this case, inversely proportional to the roughness and the shear modulus and proportional to the normal force. the eqs. (3) and (4) are valid in the subcritical case of loading [5]. these dependencies were confirmed by our experimental results and are shown in fig. 2. it would be interesting to examine whether, in our case, the deformation criterion for the fracture process is valid. as follows from the above analysis, as well as [4] and [5], the deformation criterion means that a cylindrical stamp always breaks off at a certain distance from the undisturbed surface, independently of the previous history. this means that if elastomer is loaded with some normal force so that it is indented by some indentation depth and then is pulled away, it has first to move this path back and then additionally the critical displacement before it detaches. this would mean that the vertical displacement will be approximately proportional to the normal force. this was also confirmed experimentally, as shown in fig. 3. the intersection of all lines fn  0 corresponds to the critical displacement. in our case, the critical displacement 38 l.b. voll is approximately equal to 0.006 mm. the offset (fig. 2) of the adhesion force (fn  0) is outside of the experimentally measurable parameter space. small normal forces were not investigated in this study. 4. conclusion and outlook we have investigated the dependence of the adhesion between an elastomer and a steel cylindrical indenter on the surface roughness of the punch. our results are compatible with the "deformation criterion" for the fracture of adhesive contact and with a viscous nature of the adhesive force. further experiments are planned that would detect the transition between subcritical and critical load. in order to describe the influence of roughness more precisely, experiments with a larger number of different rough indenters will be carried out. to understand the offset of the adhesion force, measurements at smaller normal forces will be needed. for a complete characterization of the adhesion processes, investigation of the adhesion in the complete parameter space “normal force – geometry – pull off velocity – temperature” is needed. this extended study is planned for the near future. acknowledgements: the author is grateful to v.l. popov, m. popov and r. heise for discussions, and deutsche forschungsgemeinschaft for financial support. references 1. johnson, k.l, kendall, k., roberts, a. d., 1971, surface energy and the contact of elastic solids, proc. r. soc. lond. a, 324(1558), pp. 301–313. 2. johnson, k.l., 2003, contact mechanics, cambridge university press. 3. maugis, d., barquins, m., 1978, fracture mechanics and the adherence of viscoelastic bodies. journal of physics d: applied physics, 11(14), pp. 1989-2023. 4. popov, v. l., 2012, basic ideas and applications of the method of reduction of dimensionality in contact mechanics, physical mesomechanics, 15(5-6), pp. 254-263. 5. popov, v.l., hess, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer. 6. voll, l.b., popov, v.l., 2014, experimental investigation of the adhesive contact of an elastomer, physical mesomechanics, 17(3), pp. 232-235. 7. popov, v.l., 2010, contact mechanics and friction – physical principles and applications, springer. 8. hess, m., 2012, on the reduction method of dimensionality: the exact mapping of axisymmetric contact problems with and without adhesion, physical mesomechanics, 15, pp. 264-269. 9. kürschner, s., popov, v.l., 2013, penetration of self-affine fractal rough rigid bodies into a model elastomer having a linear viscous rheology, phys. rev. e, 87, 042802. 10. samoilov, v.n., sivebaek, i. m., persson, b.n.j., 2004, the effect of surface roughness on the adhesion of solid surfaces for systems with and without liquid lubricant, j. chem. phys., 121(19), pp. 9639-9647. 11. heise, r., popov ,v.l., 2010, adhesive contribution to the coefficient of friction between rough surfaces, tribol. lett., 39(3),pp 247-250, doi: 10.1007/s11249-010-9617-1. 12. andrade, e.n. da c., 1934, a theory of the viscosity of liquids, part i, london edinb. dub. philos. mag. j. sci., 17(112), pp. 497–511. facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 357 373 https://doi.org/10.22190/fume200305031a © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper development of a new hybrid multi criteria decision-making method for a car selection scenario yousaf ali 1 , bilal mehmood 2 , muhammad huzaifa 2 , umair yasir 2 , amin ullah khan 1 1 department of management sciences, ghulam ishaq khan institute of engineering sciences and technology, pakistan 2 faculty of mechanical engineering, ghulam ishaq khan institute of engineering sciences and technology, pakistan abstract. increasing competition in the automobile industry has led to a vast variety of choices when buying a car thus making car selection a tedious task. the objective of this research is to develop a new hybrid multi-criteria decision-making technique, with accuracy greater than that of the already existing methods, in order to help the people in decision-making while buying a car. hence, considering a broader spectrum, this study aims at easing the process of multi-criteria decision-making problems in different fields. to achieve the objective, seven different alternatives were evaluated with respect to the enlisted evaluation criteria, which were selected after analyzing the secondary data obtained from pak wheels based on style, fuel economy, price, comfort and performance. these criteria were then analyzed using the proposed full consistency fuzzy topsis method. as the name tells, this method is a unique combination of two techniques. the full consistency method is used to calculate the weights of the criteria while the fuzzy topsis approach is applied to rank the alternatives according to their scores in the selected criteria. the outcomes demonstrate an increase in the consistency ratio of the weight coefficients due to which the ranking of the alternatives by the fcf-topsis is more accurate than the topsis and the analytical hierarchy process. the novelty of the method lies in the fact that this combination has not been used for an alternative selection scenario before. in addition to this, it can be used in various industries where a choice between the available alternatives arises based on a set of evaluation criteria. key words: selection problems, car-selection, decision problems, mcdm, fuzzy topsis, fucom received march 05, 2020 / accepted august 15, 2020 corresponding author: yousaf ali department of management sciences, ghulam ishaq khan institute of engineering sciences and technology, pakistan e-mail: yousafkhan@giki.edu.pk 358 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan 1. introduction today cars have become more than just a luxury in the lives of people comparing to the past when they were luxuries that only higher classes could afford while the majority of other people had to rely on conventional transport services. in the 21 st century, however, it can be confidently said that the trend is changing. according to the world bank collection of development indicators, 13 cars are used per 1000 people in pakistan [1]. this shows that now cars, even though still not accessible to everybody, have become a necessity rather than a luxury. this has been acknowledged by the automobile industries which have now extended their focus from just the rich and powerful to the average middle-class person. great is a variety of cars to be chosen from that come in all kinds of ranges for people belonging to different backgrounds. it might seem like an easy task to just select a car considering its cost and the desired room space, but due to advancements in technology and an increase in the number of manufacturers, it has become a tedious task. before someone buys a car, he seeks advice from friends and experts or reexamines the experiences of users of a certain car that can be easily found on credible websites such as pakwheels and olx. mostly, people rank the safety of a car higher while others care about their performance and mileage [2]. with the changing customer demands, car manufacturers have also improved their designs by introducing new technologies such as hybrid and electric cars, i.e. tesla, prius, etc. thus, with a large number of choices, it is relatively harder to select a specific vehicle and that is why it accounts for a complex decision-making situation. owing to the sudden increase in car prices along with the increased devaluation of pakistani currency, the purchasing power of an average person has decreased which means resources are scarce. thus, it is very important to allocate these resources optimally to achieve maximum satisfaction. the advancement in data sciences and computational problem-solving approaches have led to a vast increase in its scope and made its application in this research possible. in this study, a variety of criteria will be considered such as style, comfort, fuel economy, performance, and price which make it a multi-criteria decisionmaking (mcdm) problem. furthermore, these five criteria are then subjected to assess seven alternatives, thus giving rise to a decision-making scenario. such a scenario accounts for proper and effective decision-making tools to make an informed decision. in these situations, multi-criteria decision-making (mcdm) techniques are taken into account, which have proven to be effective in complex decision-making situations. various mcdm techniques have been used previously to solve decision-making problems, the most common being the analytical hierarchy process (ahp) and the fuzzy ahp. there are a few shortcomings of the ahp which have not been previously emphasized such as its high complexity due to numerous pairwise comparisons; moreover, the results are not precise enough. these limitations are also acknowledged by [3]. according to [4], the ahp technique holds inconsistency when positioning an alternative or a factor as a part of the information set. similarly, the rank of conservation can be unrealistic when various variations of the ahp methodology are being utilized. furthermore, using the fuzzy topsis or topsis alone raises issues such as its reliability, meaningful context, and effective results. there are currently many variations of the fuzzy topsis and such variations make it harder for a straightforward selection of a suitable technique [5]. such limitations count for the development of a new hybrid technique that can produce consistent and reliable results. the requirement of the results being error-free is the need of the hour. hybrid multi criteria decision-making method for a car selection scenario 359 to counter these problems this paper introduces a new hybrid method by combining a pairwise comparison technique, the full consistency method (fucom), and a group decision-making method, the fuzzy technique for order of preference by similarity to ideal solution (topsis), in the car selection problem. the fucom is a relatively new technique for determining the weights of criteria which has never been used before in such a problemsolving. in comparison to other mcdm models, it greatly reduces the mathematical complexity in analyzing the data because of the reduced number of pairwise comparisons. also, according to pamucar, it provides a much more precise result than the ahp, which is reflected by a higher degree of consistency in the fucom [3]. similarly, when the criteria weights are calculated using the fuzzy topsis the deviation of the results from the ideal solution is much higher, as shown by this research. moreover, while using the ahp, it is assumed that the criteria are independent. if it is not the case, the ahp will calculate inaccurate weights for correlated criteria as the model would no longer be linear. as far as the fucom is concerned, the equation model formed is, by default, non-linear, so that it can easily calculate accurate weights for the correlated criteria as well [6]. so, the fcf-topsis accuracy increases further when the accurate fucom values are used as an input for the fuzzy topsis. other than the increased accuracy this method also gets rid of the problem of rank reversal, which exists in many other mcdm techniques. in this problem, the relative ranks of the alternatives are reversed when an additional alternative is introduced or an existing one is removed. hence the fcf-topsis can be used to accurately rank any number of alternatives. based on the above discussion, the other previous mcdm techniques hold inaccuracies in their results, either alone or in hybrid combination with other techniques. on the other hand, the fucom technique proved to be much efficient and reliable in results. to avoid such discrepancies, this study introduces a new hybrid technique known as the fucomfuzzy topsis to address the limitations of the previous methodologies. the fcf-topsis is applied to selecting the best possible car as an alternative, based on seven distinct criteria. apart from this, the study also performs a comparative analysis of two different combinations of techniques, i.e. the ahp-fuzzy topsis and the fuzzy topsis with that of the fcf-topsis to depict the consistency of the results, thus forming to be the objective of this research study. the paper proceeds by giving a literature review on the application of multi-criteria decision-making problems in addition to the scope of the fucom and fuzzy topsis technique; this is followed by a detailed description of the data collection methods, the proposed methodology of the fcf-topsis model and the way of analyzing the data. furthermore, the results obtained are discussed and the accuracy of the model is verified. the research concludes by listing the outcomes, limitations of this study, and recommendations for further research. 2. literature review the society is advancing day by day thus turning the previously considered luxury into a necessity. this is, for instance, the case with motor cars. the possession of cars brings great flexibility to personal mobility. one does not have to worry about the strict schedule of the public transports; instead, he can plan his day according to his routes. moreover, public 360 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan transport goes along a fixed route which in the case of rush hours can cause delays whereas those using personal cars can change the routes to avoid traffic jams. furthermore, it is always less comfortable to travel in overcrowded buses/trains; thanks to cars, one does not have to worry about this issue, either [7]. these reasons have influenced people to change their perception and choose cars over buses and other forms of public transport. nowadays, there is cut-throat competition in almost everything. making a decision immediately, i.e. choosing one alternative out of many, is very difficult due to a simple reason that a lot of choices are available in the market with each having its own advantages and disadvantages. while making decisions, people always try to select the most optimal alternative. such a venture for the optimal solution requires decision-making in complex scenarios which is normally not possible when there are multiple criteria and alternatives to be taken into account [8]. this concept can be related to the objective of this study which requires the selection of an alternative based on five distinct criteria. it involves complex decision-making scenario and thus multi-criteria decision-making (mcdm) techniques prove to be a feasible solution to such a problem. multi-criteria decision methods (mcdm) are widely used as help in making such decision-making processes simpler and selecting the best possible option [9]. mcdm is a technique for evaluating different alternatives based on multiple criteria at a time. it helps the administrator to rank the alternatives according to the qualities of each alternative, and select the worthiest one [10]. a simple example may be of the investors who are in two minds about which projects they should invest in. to address this concern, there is a vast amount of literature dedicated to the selection of the best project alternatives in the past years. one such study discusses various qualitative and quantitative methods of project selection in the paper, namely those which are widely used in the industry today [11]. the scope of the mcdm techniques is not only restricted to large scale projects such as the investor's example given above. it may be used for making daily-life decisions, for example, sakthivel and iiangkumaran used the fuzzy analytical hierarchy process (ahp) in the car selection problem [12]. care should be taken while choosing the criteria based on which the comparison is to be made to obtain accurate results. a study concludes that the selection of the best car model is based on the ranking and efficiency level of the different alternatives [13]. the mcdm applications are not limited; their implementation can be found in various studies and areas of research. one such study that forms the baseline for this research is the selection of hybrid electric vehicles in the scenario of a developing country. the study holds the application of a mcdm technique, i.e. fuzzy topsis for decision-making application [14]. similarly, some of latest research studies that incorporate mcdm techniques include: formulation of maintenance framework for the urea industry [15], using the fuzzy lambda-tau (flt) for the computation of ram parameters along with the comparative performance analysis using fuzzy topsis, fuzzy edas and fuzzy vikor [16], to mention just a few. other studies that incorporated mcdm techniques include: implementation of fuzzy fmea and gra approach for risk assessment of thermal power plant [17], risk failure framework via fuzzy fmea and edas [18], hybrid mcdm techniques such as fuzzy fmea and gra for the operational sustainability of the process industry [19], development of meta-model via edas [20], to mention just a few. the study of administrative situations shows that the project selection involves several different criteria; hence it is necessary to evaluate the relative importance of each criterion compared to other ones. this is called assigning a weight to each criterion. the sum of all hybrid multi criteria decision-making method for a car selection scenario 361 the weights is always equal to one [21]. different operational techniques are used for this purpose. mahmoodzadeh and shahrabi have used the ahp to determine the weights of different criteria used to evaluate different projects [22]. similarly, to calculate the criteria weights for imprecise or vague data, fatemeh torf used the fuzzy ahp which evaluates data in the form of linguistic variables [23]. the weights of the decision criteria highly affect the results of the mcdm techniques. thus, the evolution of the available weight determination methods and the formation of more precise methods are very important. theoretically, in multi-criteria evaluations, both subjective and objective weights are used but in practice, only subjective weights are used commonly [24]. hence pamucar developed a new subjective model to determine the weights of criteria which is the full consistency method (fucom). it is based on a pairwise comparison of the criteria and the total number of comparisons is n-1 where n is the number of criteria. pamucar also proves in his paper that the fucom gives more accurate results, in terms of consistency, as compared to other methods (best worst method (bwm) and ahp) with a reduced number of comparisons. moreover, the fucom is much simpler and more flexible when it comes to the application of the technique. some of the latest fucom technique applications can be seen in the form of selecting a brigade command post, in a hybrid combination with a multi-attributive border approximation area comparison (mabac) method [25]. similarly, another study highlights the ranking of dangerous sections of the road via hybrid approach of the fucom with the weighted aggregate sum product assessment method (waspas) [26]. the same combination of methodology has been utilized in another study for the purpose of selecting a forklift for a warehouse [27]. furthermore, evaluation of criteria for a sustainable supplier selection is also one of the extensions of fucom applications [28]. another study depicts the fucom-waspas application for the purpose of the optimization of a wood company’s supply chain management (scm) [29]. lastly, the fucom in combination with another mcdm technique i.e. ahp, possesses application in the ranking of libyan airlines [30]. all these applications show that the fucom technique has the capability of producing reliable results, both individually and in combination with other techniques. some people have used a single technique for the selection of the best possible alternative. gungor and isler adopted the analytical hierarchy process for the selection of the best alternative among multiple different alternatives [31]. dae-ho byun has also used an extension of the ahp to decide on an automobile model [2] whereas mahmoodzadeh and shahrabi showed how different techniques can be combined for one single task. after calculating the weights for each criterion using the analytical hierarchy process, they used the fuzzy topsis to determine the best possible alternative. the fuzzy technique was used to allow the incorporation of qualitative, incomplete, and variable data into the decision-making model. fuzzy techniques are more accurate when real-life problems are to be solved [22]. similarly, tapas and pritha combined ahp and topsis methods to assist in choosing the best commercially available scooter and then compared their results with the multi-attributive border approximation area comparison method (mabac) [32]. a common issue that occurs in various mcdm techniques is a rank reversal problem. according to yong, when a new alternative which was not thought of when the decisionmaking process was initiated, is introduced or an existing one is removed the relative ranks are reversed [33]. senouci and mushtaq illustrated the problem of rank reversal in the topsis method [34]. similarly, ziemba and kong demonstrated how this problem affects the results of the analytical network process in their respective papers [35] [36]. 362 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan the selection of the most appropriate car on the basic five distinct criteria via a novel technique of the fucom-fuzzy topsis proves to be the novelty of this study. the previous literature has no application of such a hybrid combination of mcdm techniques. furthermore, the application of this novel technique is also compared with the existing mcdm techniques; the assumptions based on the effectiveness of this new hybrid technique are drafted. this confirms to be the novelty of this research study. 3. methodology fig. 1 below illustrates the path followed to obtain the results of this research. fig. 1 flow chart of adopted methodology 3.1. data collection this section elaborates on the approach that this paper used to collect the data for the development of the fucom fuzzy topsis (fcf-topsis) technique. initially, the criteria were decided upon, i.e., those according to which the comparison of the alternatives was done. for this very purpose, data from the pak-wheels website was used to get information about the criteria which are considered important by car owners when they provide feedback. the factors that are mainly discussed in the feedback sections show the five most common and relevant criteria as depicted in table 1. the list of criteria may differ when the same study is conducted in some other country. table 1 below lists these criteria. after this, a questionnaire was drafted. it was floated to the car users all over pakistan, through social media and emails, in which they were asked to rate each criterion, on a scale of 1 to 5 with 1 being least important and 5 being most important one, according to their judgment. a total of 70 responses were collected. this data was used to calculate the absolute rating (r) of each criterion by averaging out the responses of each. next, the most commonly used cars in pakistan were selected which are a1 honda civic (1.8 i-vtec), a2 honda city (1.3), a3 honda br-v (i-vtec), a4 toyota corolla gli (1.3 automatic), a5 toyota aqua, a6 daihatsu mira and a7 toyota vitz for hybrid multi criteria decision-making method for a car selection scenario 363 further analysis. keeping the selected criteria in mind, the performance score for each selected alternative was obtained from numerous reviews present on pak wheels. the sample size was different for each alternative. there were 360 reviews for corolla, 248 for city, 213 for civic, 57 for br-v, 34 for vitz, 12 for aqua, and 10 for mira available on the website. table 1 list of criteria for comparison of different models criteria for comparison of different models c1 style c2 fuel economy c3 price c4 comfort c5 performance the gathered scores were converted into linguistic variables to incorporate the fuzziness that existed due to varying opinions of the reviewers. for example, the prices of the different models were determined as shown in table 2. the value for each alternative was then divided by the maximum value. the obtained results, also shown in table 2, were mapped on the graph shown in fig. 2, and the linguistic variables were assigned for the price criterion according to its position on it. for instance, the linguistic value obtained for br-v is 0.871, as shown in table 2 below, which lies just before the midvalue of high and very high region of fig 2. thus, being closer to the high region, it was assigned the linguistic variable “high.” this was done for all the criteria to assign the linguistic variables. as a result, a comparison matrix of the different alternatives in linguistic terms, as shown in table 3, was obtained. this matrix was further used in fuzzy topsis to determine the final ranking of the alternatives. table 2 conversion of price into linguistic terms car price value a1 br-v 3049000 0.8714 a2 – city 2309000 0.6599 a3 corolla 2849000 0.8142 a4 civic 3499000 1.0000 a5 aqua 2600000 0.7431 a6 – vitz 2300000 0.6573 a7 – mira 1690000 0.4830 maximum value 3499000 fig. 3 linguistic variable graph (author’s self-constructed) 364 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan table 4 rating cars in terms of linguistic variables. criteria a1 a2 a3 a4 a5 a6 a7 price (c2) vh h h h h mh h fuel economy (c2) h h h mh vh vh h performance (c3) h h vh h vh h h comfort (c4) h h h h h h vh style (c5) vh h vh h h h vh 3.2. fucom (full consistency method) determination of criteria weights is one of the important tasks in multi-criteria decision-making. the weights of criteria can greatly affect the outcome of the decision model which is the reason why great attention must be paid to their determination. the full consistency method is a relatively new method for calculating the criteria weights which were developed by dragan pamucar, zeljko stevic, and sinisa sremac in 2018 [3]. it is considered better than much other multi-criteria decision-making (mcdm) methods because of its lower mathematical complexity thus reducing the chances of errors. the fucom method is shown below: ai = represents the alternatives (i=1, 2 … b) cj = represents the criteria (j=1, 2…a) a = number of criteria b = number of alternatives step 1: firstly, criteria cj are ranked according to their ratings, determined after evaluating the questionnaire responses. c1>c2>…>cl, where l represents the rank of the criteria (l = 1, 2, … a), and if any two criteria have the same significance, ">" is replaced with "=" sign. step 2: secondly, the ranking criteria are compared using two-way comparisons, and comparative priority (α) is calculated. the comparative priority is the superiority of the criterion of the cl rank compared to the criterion of the cl+1 rank. these comparative priorities are then represented through a vector. 1/ 2, 2 / 3 / 1 ( )l l     (1) to calculate these priorities, initially, significance sl (l=1, 2, 3…) of the highestranked criterion is determined with respect to the rest of the criteria, by using absolute ratings (r) obtained from the questionnaire. 1 l l r s r  (2) for example, if c1 > c2 > c3 so s2=(w1/w2) and s3 =(w1/w3). the value of s1 is equal to 1 because it is the highest-rated criterion. using the significance values, the comparative priorities are calculated using the formula: 1 1 ll l l s s     (3) hybrid multi criteria decision-making method for a car selection scenario 365 next, certain conditions are incorporated to find the weights in step 3. step 3: using this information, two conditions are applied to form a non-linear programming model. the comparative priorities should be equal to the ratio of weights of respective criteria 1 1 l l l l w w     (4) the condition of mathematical transitivity must be met 1 1 2 1 l l l l l l         hence 1 1 2 2 l l l l l l w w w w w w       (5) which leads us to the equation 1 1 2 2 l l l l l l w w         (6) if both the conditions are met, the requirement for maximum consistency is met, i.e., deviation in full consistency (dfc) is χ =0. the whole point of the fucom method is to minimize dfc for better accuracy in results. using eqs. 4 and 6 a non-linear programming model is made for determining the final values of the weight coefficients. min χ s.t. 1 1 l l l l w w       (7) 1 1 2 1 ( l l l l l l w w          (8) 1 1 a l l w   (9) 0lw  (10) by solving this nonlinear model using lingo software, the criteria weights can be calculated and the degree of dfc (χ) is obtained to see whether the results are consistent or not. 3.3. fuzzy topsis method topsis stands for the technique for order of preference by similarity to ideal solution which was introduced by hwang and yoon in 1980 and is currently one of the most commonly used methods for mcdm problems [37]. in time, changes were made to the originally proposed technique. one such alteration was the incorporation of zadeh’s fuzzy set theory to apply the method to a vague data set as well [38]. 366 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan table 5 showing the importance weights and ratings for fuzzy topsis method linguistic variables importance weights importance ratings low (0,0.1,0.3) (0,0,0.25) medium (0.3,0.5,0.7) (0,0.25,0.5) medium high (0.5,0.7,0.9) (0.25,0.5,0.75) high (0.7,0.9,1.0) (0.5,0.75,1.0) very high (0.9,1.0,1.0) (0.75,1.0,1.0) the following is a step by step procedure on how a problem is solved using fuzzy topsis approach. step 1: firstly, decide upon a group of decision-makers and explain to them the criteria based on which the selection of the alternatives is to be done. for our case, the selection of a motor vehicle, the data was taken from the pak wheels website which evaluates the opinion of many end-users and critics based on their reviews. this data was then converted into linguistic variables. step 2: the average weight of every criterion is to be calculated so that a singular value can be assigned to each criterion. for our case, this step was omitted as the weights were calculated using the fucom technique to fulfill the purpose of this paper, design a hybrid mcdm method. step 3: next comes the selection of appropriate linguistic variables so that the importance rating of each alternative with respect to every single criterion can be found. the criteria are rated based on the importance ratings given in table 4 above. 1 2 31 ( ... )k j j j j jx x x x x k      (11) xij k is the rating of the k th decision-maker, against alternative i and criteria j. step 4: succeeding is the transformation of the linguistic terms into fuzzy triangular numbers and a fuzzy decision matrix is constructed. 11 12 1 21 22 2 31 32 3 ... ... ... n n n x x x a x x x x x x            (12) ( ,   , )ij ij ij ijx a b c (13) where m is the number of alternatives and n is the number of criteria and xij represents the triangular fuzzy number step 5: normalization of the fuzzy decision matrix is done. [ ]ij m nr r  (14) * * * ( , , ) ij ij ij ij j j j a b c r c c c  (15) hybrid multi criteria decision-making method for a car selection scenario 367 where, cj * is the maximum or minimum of cij depending on whether the criteria is a benefit or cost criteria. step 6: next the weights found using the fucom technique were multiplied to calculate the weighted normalized matrix. ij ij ju r w  (16) step 7: moving on, the fuzzy positive ideal solution (fpis) and fuzzy negative ideal solution (fnis) represented by (𝐹∗) and (𝐹−), respectively, is found. * * * 1 2* , ,... nf u u u (17) 1 2, ,... nf u u u    (18) where, uj * =(1,1,1,1) and uj =(0,0,0) for benefit criteria and vice versa for cost criteria’s, j=(1,2,…n) step 8: next step is to find the distance from the positive and negative ideal solution using the following formulas. * * 1 ( , ) n ij j j d d u u   (19) 1 ( , ) n ij j j d d u u    (20) where d is representing the distance between two fuzzy numbers and is given by the following formula 2 2 2 1 1 2 2 3 3 1 ( , ) [ ) ( ) ( ) ] 3 ij ijd a b a b a b a b      (21) step 9: lastly, the closeness factor is calculated for every alternative, using the following formula: * d cc d d     (22) step 10: select the alternative having the highest value for the closeness factor. 3.4. fcf-topsis model this model will be a unique combination of fucom and fuzzy topsis techniques. after deciding upon the criteria based on which the alternatives have to be compared, the most important criterion is to be selected. next comes the comparison of every criterion with respect to the most important criterion and significance values are to be assigned. these two things can be done either by asking an expert for help or by floating a questionnaire and hence evaluating the collected responses. as far as this study is concerned, the questionnaire approach was used. after doing so, the fucom technique is 368 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan applied to construct a non-linear programming model that is solved through lingo software to determine the weights of the evaluation criteria. these weights are used to form the weighted normalized rating matrix of the fuzzy topsis technique. next follow steps 3 to 10 given for the fuzzy topsis technique to select one alternative out of many. 4. results and discussion the collected data was analyzed and used in the calculation of the best alternative, by applying the proposed hybrid method. this section illustrates the calculations done to reach optimal alternatives and discusses the final results. based on the responses gathered through the questionnaire, the absolute rating was calculated for each criterion on a scale of 1 to 5 by averaging out the responses of each criterion. the rating in table 5 shows that the price criteria were considered to be the most important whereas people were least concerned about the style of the car. after this, the significance (sl) of the most important criterion with respect to the rest of the criteria was determined. it is shown in table 5 that the higher the value of significance, the lower the importance of that criterion is with respect to the most important one; hence the value for the price is 1 because it is considered to be the most important criterion. this was followed by pairwise comparisons to calculate the comparative priority values (α). these values were further evaluated by applying two different conditions and the following non-linear programming model was constructed. min χ subject to the following equations: 31 2 4 2 3 4 5 1.1325 , 1.069 , 1.103 , 1.078 ww w w w w w w            31 2 3 4 5 1.211 , 1.179 , 1.189 ww w w w w         1 2 3 4 5 1w w w w w     0lw  this model was solved using the lingo software, which calculated the finalized criteria weights, which are also summarized in table 5 below. table 6 finalized criteria weights and the values used for calculating them criteria c1 c2 c3 c4 c5 rating(r) 4.53 4.00 3.74 3.39 3.14 significance (sl) 1.00 1.13 1.21 1.34 1.44 weights (wl) 0.2408 0.2126 0.1989 0.1803 0.1673 hybrid multi criteria decision-making method for a car selection scenario 369 the weights show that the trend identified by the absolute rating is correct. people consider the car to be the most important factor in selecting a car whereas the car's comfort and style are least significant. apart from the weights, lingo also calculated the value of dfc (χ) to be 0.000112. the low value of dfc highlighted the fact that the deviation of calculated weights from optimal value was negligible. then the fuzzy decision matrix was generated by assigning triangular fuzzy ratings to the data shown in table 3 according to their respective linguistic variables. triangular fuzzy numbers are used because the decision-makers find them instinctively easy to use and calculate. also, using triangular fuzzy numbers is has proven to be an effective way to reduce fuzziness from data. it is also proved in the previous referential literature that mostly in practical situations triangular fuzzy numbers are used [39] [40]. these fuzzy ratings were further evaluated to obtain the final results, summarized in table 6 below. table 7 final results obtained alternatives dj* djccj rank a1 3.6284 1.3944 0.2776 6 a2 3.6213 1.4109 0.2804 5 a3 3.5592 1.4647 0.2916 3 a4 3.6744 1.3606 0.2702 7 a5 3.5514 1.4714 0.2929 2 a6 3.5281 1.5022 0.2986 1 a7 3.5624 1.4620 0.2910 4 the results show that a6 mira is considered to be the most optimal solution because it obtained the highest value of closeness coefficient (cc), the reason being that it had scored either “high” or “very high” in all of the benefit criteria (which are intended to be maximized) and it scored the minimum, which is “medium high”, amongst all the alternatives for the cost criteria (which is desired to be minimized). going by the same analogy, all the alternatives have achieved their ranks according to their respective scores in the benefit and cost criteria. furthermore, for comparison, the weights were also calculated using ahp. the results obtained from both the techniques are given in table 7 below. it can be seen that there is not much difference in values. as far as the accuracy of the results is concerned, the lower consistency ratio of fcf-topsis proves that it is more accurate and the results are closer to the optimal values whereas the consistency ratio (cr) of ahp has a value of 0.000648 which is higher than the value of dfc in fcf-topsis. this shows that the results of ahp deviate more from the optimum value because the higher the value of dfc, the higher deviation in the results will be. in this research, the values used in the criteria comparison matrix of ahp are not consistent with the saaty's scale because for a fair comparison with the proposed hybrid method the values had to be similar for both although the questionnaire could have been designed to get the ratings according to the saaty’s scale due to which the results obtained would have shown a further deviated consistency ratio (cr) of ahp. however, our main objective was not to evaluate the data using ahp, but it was just used to prove that fcf370 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan topsis is more accurate which is already been proven by the higher cr of ahp. hence average ratings in decimal numbers were used as inputs to ahp. so, it can be clearly said that the value of ahp's cr was under approximated whereas the calculated dfc (χ) for the proposed hybrid method is overestimated due to the rounding and approximation of values in the manual calculation; this is the reason why dfc is not equal to zero. both of these reasons converge on the fact that the actual accuracy of fcf-topsis, in ideal conditions, would be much greater. another reason for the higher accuracy of fcf-topsis is that it requires a lower number of pairwise comparisons, (n-1) where n is the number of criteria, whereas ahp requires n(n-1)/2 a number of pairwise comparisons [3]. this means that for this case, only 4 comparisons had to be done for fcf-topsis whereas ahp needed 10. due to this the mathematical complexity of fcf-topsis greatly reduces, which in turn decreases the possibility of errors in the result. table 8 comparison of the results of ahp and fucom criteria weights calculated using ahp weights calculated using fcf-topsis price 0.2407 0.2408 fuel economy 0.2123 0.2126 performance 0.1990 0.1989 comfort 0.1805 0.1803 style 0.1671 0.1673 consistency ratio 0.0006 0.0001 also, this problem was solved using the fuzzy topsis and the results were compared with the proposed method. in table 8, it can be clearly seen that when the value of cr (consistency ratio) increases from 0.000112 (fcf-topsis) to 0.000648 (ahp-fuzzy topsis), the closeness coefficient values increase for each respective criterion. observing the closeness coefficients obtained from fuzzy topsis, the values are much larger than those obtained from fcf-topsis (which had the lowest cr). this shows that, although the same ranks have been obtained, the deviance in the values of fuzzy topsis is larger. table 8 comparison of the closeness coefficient obtained using different techniques alternatives fcf-topsis ahpfuzzy topsis fuzzy topsis a1 0.2776 0.2776 0.4706 a2 0.2804 0.2804 0.4800 a3 0.2915 0.2916 0.5069 a4 0.2702 0.2702 0.4511 a5 0.2929 0.2929 0.5125 a6 0.2986 0.2986 0.5297 a7 0.2910 0.2910 0.5045 hence, the results obtained through the fcf-topsis possess higher accuracy than the fuzzy topsis. the reason for this is that the fuzzy topsis technique is applied with the assumption that all the evaluation criteria are independent of each other whereas in practical situations this can never be true. it operates so that the optimal solution is the hybrid multi criteria decision-making method for a car selection scenario 371 one which has the largest distance from the fuzzy negative ideal solution (fnis) and the smallest distance from the fuzzy positive ideal solution (fnis), which increases its efficiency in handling vague data. however, determination of the closeness coefficient by using euclidean distance, calculated from fnis and fpis, neglects the correspondence of the evaluation criteria. hence the challenge to determine criteria weights while keeping the deviation low. for example, in the car selection criteria, if a car's style is designed with an emphasis on its attractiveness rather than aerodynamics, then its fuel economy will be affected to some extent. this relation is neglected while using topsis but the pairwise comparison technique used in this research, fucom, acknowledges it. summing up, the proposed method incorporates the advantage of both the pairwise comparison method and the group decision-making technique. as the value of criteria weights obtained is accurate, with negligible deviation from optimal value, the final alternative rankings calculated using fcf-topsis are much precise and have lower mathematical complexity than other mcdm techniques used in this field. in addition to accuracy, the method is not just limited to solving independent criteria, like ahp and topsis, but it can evaluate correlated criteria with the same ease as well. such results achieve the main objectives of the study, i.e. the selection of the most feasible vehicle in the case of pakistan and the incorporation ofe the most effective mcdm method by verifying the results in comparison with the other mcdm techniques. 5. conclusion ease of access to transportation is the need of the hour for almost every individual. with the advancement of technology and the work environment, people need to commute quickly from one place to another. this need calls for a perfect vehicle that can serve the individual according to his or her requirements. for this purpose, this research paper takes five criteria into account based on which seven alternatives are to be assessed. to do so, the study incorporates a hybrid combination of the fucom-fuzzy topsis. after the assessment, the results concluded the toyota mira to be the best suitable option for the people living in pakistan. it is concluded that people who look forward to buying the car can have the best option in the shape of the toyota mira while keeping the factors of style, fuel economy, price, comfort and performance in mind. furthermore, another objective of this paper was to develop a new multi-criteria decision-making technique to assist the decision-maker in a car selection scenario which comprises various evaluation criteria. the fucom and fuzzy topsis integration is used in this proposed approach. a case scenario in the light of pakistan was analyzed and compared with another technique, i.e. the ahp to validate the results. the results show that the proposed approach is better than the ahp since it reduces the mathematical complexity by reducing the number of pairwise comparisons yet giving better results which are shown by its lower consistency ratio. in the pairwise comparison problems, sometimes the criteria are correlated due to which the ahp gives inaccurate weights as the model no longer remains linear, while the fucom can easily calculate dependent or independent criteria weights as it is by default a nonlinear model. hence, it can be concluded that the fucom can assess results with minimum possible diversion and thus give concluding reliable results as compared to the rest of the mcdm techniques. 372 y. ali, b. mehmood, m. huzaifa, u. yasir, a. u. khan this research study proves to be beneficial in two aspects. firstly, it can provide a reasonable solution to the individuals who are looking to buy a vehicle based on the aforementioned criteria. secondly, the research study has proposed a new hybrid technique with reliable results that can be beneficial for the researchers to implement in future research studies. research limitations of this study are 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oligocrystals provide a unique material for in-depth analysis of deformation mechanisms in metals and alloys. in this paper, the deformation mechanisms in a cu-7.5 wt.% al oligocrystal are investigated experimentally and numerically. an experimental sample containing several coarse columnar grains is produced by wire-feed electronbeam additive manufacturing. based on the experimental data, a crystal plasticity finite element model is built to consider realistic grain shape and orientations. this study involves a comprehensive analysis of experimental data, numerical results and analytical estimates to reveal the peculiarities of slip activation in oligocrystal grains under compression. experimental and numerical findings have shown that along with slip systems with the highest schmid factors, other slip systems are activated in most grains to accommodate their plastic deformation. to elucidate why the slip systems with nonmaximal schmid factors became active, we examined the stress fields within the grains. it was revealed that the stress state at the grain scale deviated from uniaxial. finally, a detailed numerical analysis of strain rate dynamics demonstrated that plastic deformation in particular grains developed in the form of narrow fronts. these fronts periodically generated near the moving punch and subsequently propagated towards the opposite side of the sample. the study has shown that oligocrystal analysis provides valuable insights into the linkage between the intraand intergrain deformation mechanisms with the overall material behaviour. key words: additive manufacturing, directed energy deposition, oligocrystals, crystal plasticity, slip systems, microstructure-based mechanical simulation received: november 28, 2023 / accepted january 11, 2024 corresponding author: varvara romanova institute of strength physics and materials science sb ras, akademicheskii prospect 2/4, 634055 tomsk, russia e-mail: varvara@ispms.tsc.ru 2 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov 1. introduction advanced tribology puts into play a wide range of challenging problems requiring the joint efforts of experts in physics, mechanics, and materials science. among the key issues is the prediction of the deformation and fracture behaviour under an external load applied to the material surface. classical contact mechanics focuses on the macroscopic quantities relating the stress and strain distributions in a subsurface layer with the parameters of an external load applied to the contact surface [1–3]. while valuable, this approach commonly neglects lower-scale deformation mechanisms which are first to occur in a loaded material. meanwhile, understanding the multiscale deformation mechanisms is crucial in view of fracture prediction since progressive damage accumulation at lower scales is a precursor of larger-scale deformation and fracture [4]. oligocrystals (sometimes called olygocrystals) represent a unique material for the detailed study of deformation mechanisms in metals and alloys throughout length scales. oligocrystals contain several millimetre-sized columnar grains with known orientations that span through the sample thickness [5, 6]. due to the limited number of grains in a sample, the complexity of factors influencing deformation mechanisms is significantly reduced. this reduction facilitates the detailed analysis of the microstructural effects on the deformation process and mechanical properties of a material. the investigation of oligocrystals thus offers a more controlled and insightful approach to understanding deformation mechanisms in materials. despite the significant potential of oligocrystals for fundamental research, a limited number of oligocrystal deformation studies are reported in the literature [7]. the authors [7,8] considered oligocrystal research as an excellent opportunity for the development of advanced experimental characterisation techniques. several research groups focused on the experimental grain-scale validation of the implemented crystal plasticity models considering oligocrystals [9–12], which is particularly interesting given the extremes of oligocrystals with the microstructural and macrostructural spatial scales being of the same order of magnitude. playing a key role in tribological applications, deformation-induced surface roughening represents another important research topic. being an intrinsic phenomenon, multiscale surface asperities evolving in plastic deformation are considered as the evidence of deformation mechanisms acting at different spatial scales [13]. in single crystals, alterations in the surface morphology typically appear as multiple slip bands and, at a lower scale, slip steps created by dislocations leaving the bulk [14,15]. in polycrystals, at different spatial scales surface roughening becomes evident as microscale slip steps, grain scale orange peel phenomenon caused by out-of-plane displacement of individual crystals, mesoscale ridging and roping induced by collective deformation of groups of grains, and macroscopic waviness at the specimen length scale [13,16]. few researchers have analysed surface roughening in oligocrystals [9,11,17]. additive manufacturing (am) has opened up groundbreaking opportunities for fabricating oligocrystals [18]. an advantageous 3d printing technology for manufacturing large metal parts is wire-feed electron beam directed energy deposition am (ded-eb type of am technology; this work will use the abbreviation web-am following [19–22]). web-am utilises a power electron beam to melt metal wire feedstock as it is deposited through a feeder track-by-track and layer-by-layer. typically for web-am, coarse grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 3 columnar grains grow throughout multiple layers up to several millimetres in length due to directional solidification. using a combined experimental and computational approach, this paper considers the grain-scale deformation mechanisms in a web-am aluminium bronze oligocrystal in a compression test. special focus is placed on the analysis of microscale stress and strain fields and their interrelation with the activation of slip systems in surface grains and the evolution of surface morphology. the paper is structured as follows. the experimental procedure of sample fabrication, microstructure characterisation and mechanical testing is detailed in section 2. based on the experimental data, a micromechanical crystal plasticity finite element (cpfe) model is constructed in section 3. experimental and computational results are presented and discussed in section 4. the summary of the main findings is given in conclusion. 2. experimental methods 2.1 sample fabrication a build of 90×20×40 mm was fabricated from cu-7.5 wt.% al wire with a diameter of 1 mm using a wire-feed electron beam directed energy deposition machine (ispms sb ras, tomsk, russia). the web-am printing was performed in vacuum with the following parameters: the electron beam accelerating potential – 30 kv; the beam current – 50 ma; the spot size – 4.5 mm; the beam sweep frequency – 1000 hz; and the heat input – 0.22 kj/mm. the aisi 321 steel base plate was mounted on the table moving under the gun in a zigzag fashion at 5 mm/s to print a layer with a thickness of 2 mm. the time delay between printing subsequent layers was 2-3s. a) b) fig. 1 schematics of web-am printing (a) and compression test (b) 4 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov a sample for compression testing was cut from the middle part of the build in the form of a rectangular bar measuring 3.53.5 mm along the x, y and z, respectively. hereinafter, y is parallel and x and z are perpendicular to the build direction (bd), where x is considered as the scanning direction. the web-am printing and compression testing are illustrated schematically in fig. 1a. the sample lateral side perpendicular to the z-axis was subjected to sandpaper grinding (mirka) and diamond suspension polishing. the final stage of surface preparation was ion beam milling at 10 kv for 15 min using the semprep2 device (technoorg linda co. ltd., budapest, hungary). 2.2 microstructure characterisation the grain structure on the lateral side of the web-am printed sample (fig. 1) was analysed by electron backscatter diffraction (ebsd) using a tescan mira 3 lmu scanning electron microscope (tescan orsay holding, brno, czech republic) equipped with an ebsd detector (oxford instruments, high wycombe, uk). the examined area covered the entire lateral surface with 0.1-0.2 mm margins. the ebsd maps of the grain structure are plotted in fig. 2 in inverse pole figure (ipf) colours for the x, y and z directions. in what follows, grains are referred to by their numbers given in fig. 2a. a) b) c) fig. 2 orientation maps for web-am aluminium bronze cu-7.5 wt.% al in ipf colours for the directions ||y (a), ||z(b) and ||x(c) the analysis of grain misorientation angles revealed that 58% grains are characterised by low angle grain boundaries (< 15°), while 42% grains have misorientation angles larger grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 5 than 15°. the high angle boundaries are highlighted in the ebsd maps (fig. 2) to separate 23 individual grains with 16 distinct orientations of crystal lattices. several large columnar grains, about 7 mm in length and 1 mm in width, span the entire height of the sample (grains 4, 10, 14 and 22 in fig. 2). a set of finer grains appearing near the bottom surface might be the evidence of larger grains located below the surface. finer grains observed in between the coarse grain boundaries might be the part of larger grains growing in parallel xy planes, before or after the plane analysed. the coarse columnar grains tend to grow along the heat flow and tilt to the y-axis (y||bd) at an angle controlled by the table velocity with respect to the electron beam (see schematics in fig. 1a). typically for web-am, stepwise grain boundary regions were formed due to electron beam transition from layer to layer. the 2 mm distance between the boundary steps corresponds to the layer thickness. the web-am processing gives rise to specific features of grain formation where a set of fine grains may solidify in narrow regions between the coarse columnar grains. presumably, the red-coloured grains in the left part of the oligocrystal (fig. 2a) solidified conjointly, while grain 4 (green-coloured in fig. 2a) solidified in a standalone manner. this behaviour enabled some additional grains to nucleate and solidify in between these large columnar crystals. the crystal lattices of the three large grains 14, 10 and 22 have close to cube orientations, <100>||y, with low misorientation angles due to a small amount of crystal lattice rotations about the z-axis. grain 4 has a nearly goss orientation with respect to the compression axis, <110>||y. both cube and goss orientations are characterised by a set of equally loaded octahedral slip systems, which is well documented for single crystals [12,13,23,24]. however, the ebsd data show that in the grains of oligocrystal no slip systems are exactly equally loaded. 2.3 compression test and surface examination the schematic of compression testing is shown in fig. 1b. compression was applied along the vertical direction with the long axes of columnar grains tilted at an angle of ≈18 degrees about the y-axis. the compression test was carried out at a constant strain rate 2.4·10-3 s-1 at room temperature according to astm a 370 using an instron elektropuls e100000 universal testing machine (instron, norwood, ma, usa). after 5% compressive strain, the sample was taken from the testing machine and its surface was analysed using an olympus lext ols4100 laser scanning microscope (olympus corporation, tokyo, japan). 3. microstructure-based cpfem model the ebsd map (fig. 2) was graphically processed to obtain a bitmap image subsequently approximated by a 407×880 voxel mesh. reconstructing a full 3d grain model from the experimental data is a challenging task calling for expensive and timeconsuming experimental procedure (e.g., x-ray diffraction tomography [25,26] or serial sectioning techniques [27,28]). in our simulations, we used a simplified 3d model obtained from the 2d experimental image by translation along the z direction. in the fe implementation, each voxel was treated as a hexahedral finite element (c3d8r). the 3d fe model of 3.2×7.0×0.5 mm constructed on a 407×880×64 mesh is shown in fig. 3a. 6 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov the fe model (fig. 3a) was imported into abaqus/explicit to solve the boundaryvalue problem in a dynamic formulation (see [29] for further detail). the finite elements belonging to particular grains were organised into sets and assigned local (crystal) coordinate systems with the axes lying along the [100], [010] and [001] crystal directions. the rotations of crystal frames about the global (sample) coordinate system xyz were given by a set of euler angles provided by ebsd data (fig. 2) to fully replicate the experimental data. the grain behaviour was described in terms of a crystal plasticity model relating the stress and strain tensor components with the dislocation slip in the face-centred cubic (fcc) crystal lattice. let us dwell briefly on some major points of the model formulation, with the details given at length elsewhere [29]. a) b) fig. 3 experimentally-based grain model (a) and schematics of its loading (b) an important moment of cpfem explicit implementation is that constitutive equations are formulated with respect to the local coordinate systems specific for each finite element. in the case at hand, the local frames are associated with the crystal axes which co-rotate with the material under loading. thus, the generalised hooke’s law is written in the same form for all elements irrespective of their orientations in the global coordinate system: ( )p ij ijkl kl klc    . (1) here, σij denote the stress tensor components, εij and p ij are the total strain and plastic strain tensor components and cijkl is the fourth-order tensor of elastic moduli. the upper dot stands for time derivative. three independent elastic modules of a cu-7.5wt.%al alloy are c11=169 gpa, c12=122 gpa, c44=75 gpa [30]. the strain rate tensor components ij are kinematically related to the velocity vector components iu as grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 7 , ,( ) / 2ij i j j iu u   . (2) the plastic strain rates are calculated as the summary slip on active slip systems ( ) ( ) ( ) ( ) ( )1 ( ) 2 p ij i j j is n s n          , (3) where ( ) is  and ( ) in  are the components of the slip direction and slip plane normal vectors on the α slip system. an fcc crystal lattice has 12 potentially active {111}<110> slip systems. the slip rate ( ) relates to the resolved shear stress τ(α) acting on the α-th slip system as   ( ) ( ) ( ) ( ) ( ) 0 0, if , if crss crss crss sign                     . (4) the reference slip rate 0 linearly depends on the equivalent strain rate eq calculated specifically for each finite element with the coefficient of 0.9. thus, the slip rate is naturally higher in the regions of higher strain rate to ensure eliminating the strain rate sensitivity effects [31]. eq. (4) means that a slip system is activated if the resolved shear stress, τ(α)=sj (α)σijnj (α) reaches the critical value ( ) crss  . the latter is the material property depending on the microstructural features and strengthening mechanisms. a common way to determine the critical resolved shear stress is an approximation of the plastic portion of a stress-strain curve. in this work, ( ) crss  was calculated for each finite element as ( ) 0 p crss eqk     (5) where the initial stress value τ0 assigned the same for all slip systems. the second righthand term is a linear strain hardening function where p eq is the equivalent plastic strain accumulated in the finite element. the values of ( ) 0  = 27 mpa and k = 270 mpa were defined to fit the plastic portion of the experimental stress-strain curve, assuming that the critical resolved shear stress is proportionally related to the yield stress through the schmid factor. figure 3b schematically illustrates the load application. the kinematic boundary conditions were put on the specimen top surface to simulate compression along the y-axis, while the bottom surface displacements along y were restricted. the back surface was treated as a symmetry plane with respect to z direction. the other three lateral surfaces were free of external loading. 8 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov 4. results and discussion 4.1 surface pattern and plastic strain distribution figure 4 shows an experimental surface image obtained using confocal microscopy after 5% compression. slip traces are well identified in the surface grains as a series of thin parallel lines inclined to the compression axis. the angle of inclination varies not only from grain to grain but also in various parts of the grains, which indicates the activation of different slip systems there. fig. 4 experimental surface image of web-am cu-7.5wt.%al under 5% compression and analytically predicted slip traces in four selected grains (for the details of the analytical analysis please refer to section 4.2) the experimental analysis reveals highly nonuniform plastic deformation developing in sample parts as early as 5% compression. most regions demonstrate one or two sets of parallel slip lines, with their density varying from region to region. the variation in density is indicative of the amount of accumulated slip – the denser the slip lines, the larger the plastic strain. a prominent example is grain 10 where several distinct regions with different slip line density and tilt angles are observed. the slip lines tend to change their orientations near grain boundary steps, which indicates that different slip systems are activated. the alignment of slip lines between neighbouring grains is so consistent that it might erroneously suggest slip transfer across grain boundaries. however, recent molecular dynamics simulations [32] showed that slip did not propagate from one grain to another, grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 9 but strain gave rise to stress redistribution in the near-boundary region which, in turn, resulted in generating new slip. a) b) fig. 5 equivalent plastic strain field (a) and surface pattern (b) in the oligocrystalline mechanical model under 5% compression the experimentally based conclusion regarding the plastic strain inhomogeneity is supported by numerical results. figure 5a, b represents the equivalent plastic strain field and corresponding surface pattern at 5% compression, as obtained in the cpfem simulation. despite some simplifications in the computational model, it well captures the key features of the deformation behaviour discovered experimentally. notably, plastic strains tend to accumulate in the vicinity of stepwise grain boundaries, near boundaries between coarse and fine grains, and in the regions contacting with virtual punches (fig. 5a). experimentally, these regions demonstrate a series of more pronounced and dense slip lines, often intersected by another set of slip traces inclined at a different angle (cf. figs. 4 and 5a). as plastic deformation develops, the surface of the oligocrystalline model ceases to be flat and the grain boundaries become clearly visible on the surface illuminated by a light source, much like in the experimental image (cf. figs. 4 and 5b). in general, the model successfully reproduces some experimentally observed features of surface morphology. a case in point is grain 10 which in both experimental and numerical samples is divided into two parts along its long axis to form a surface fold. 10 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov 4.2 analysis of active slip systems the experimental data, numerical results and analytical estimates were brought together to reveal the peculiarities of slip activation in surface grains under compression. slip lines, marking where slip planes intersect the surface, provide insight into active slip systems in plastic deformation. in fcc grains, dislocation glide is limited to four {111} planes. therefore, in most cases the active slip plane can be easily determined by the slope of the slip traces once the grain orientation is known. an exceptional case is when equally loaded slip systems may give identical traces of intersection with the surface. unambiguous identification of the slip direction is a more sophisticated task that requires the knowledge of slip traces on two adjacent sides of the sample. however, in many cases the slip direction can be analytically determined by comparing schmid factors in different slip systems calculated from euler angles. the simplest analytical estimates imply that the schmid factor for the α-th slip system is calculated with respect to the compression axis ( ) cos cosyf     , (6) where φ and θ are the angles between the y-axis and the vectors of slip normal and slip direction, respectively. the slip systems with the highest schmid factors with respect to the loading axis are most likely to be activated (table 1). table 1 slip systems with maximal schmid factors regarding to compression axis grain № slip system yf grain № slip system yf 1 (111)[011] 0.4837 13 (111)[011] 0.4937 2 (111)[101] 0.4651 14 (111)[011] 0.4598 3 (111)[110] 0.4161 15 (111)[110] 0.4458 4 (111)[011] 0.4924 16 (111)[110] 0.457 5 (111)[110] 0.4998 17 (111)[110] 0.4552 6 (111)[011] 0.496 18 (111)[101] 0.3148 7 (111)[011] 0.4924 19 (111)[101] 0.4661 8 (111)[011] 0.4971 20 (111)[101] 0.4867 9 (111)[011] 0.4924 21 (111)[101] 0.4902 10 (111)[110] 0.4586 22 (111)[110] 0.4969 11 (111)[011] 0.4971 23 (111)[011] 0.4604 12 (111)[011] 0.498 the schmid factors for 12 {111}<110> slip systems were calculated for all 23 grains from the set of euler angles provided by ebsd. the slip systems with the highest schmid factors are most likely activated under compression (see table 1). indeed, the slip traces which would result from dislocation glide in the analytically predicted slip systems are grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 11 detected in all grains. as an example, in fig. 4 the slip traces analytically predicted and experimentally observed are compared for 4 selected grains. however, additional sets of slip lines, diverging from the analytically anticipated slopes, are also evident in all grains. the experimental findings align closely with the results of numerical simulations where more than one slip system is activated in most oligocrystal grains. let us demonstrate this with the example of grains 4, 10, 14 and 15. figures 6-9 show the amount of slip accumulated in the four grains at 5% compression, with the slip systems having the largest schmid factors marked by red frames. the amount of slip in the α-th slip system was calculated for each finite element approximating the grain as ( ) ( ) p t dt     (7) where t is the process time. in two large grains 10 and 14, which extend from the bottom to the top punch, plastic deformation occurs primarily due to dislocation gliding in two slip systems, including those predicted analytically (figs. 6 and 7). additionally, one or two slip systems are activated in a few local regions near the grain boundaries to accommodate plastic deformation of adjacent grains. noticeably, all the slip traces predicted in the simulations are found experimentally in the corresponding grains, as shown in the snippets in figs. 6-7. unlike other grains, grain 4 has two unconstrained faces so that dislocations have a better ability to glide. in most of this grain, a single slip system is activated, with the exception of a narrow grain boundary region where another slip system becomes active to ensure cooperative plastic deformation with the adjacent grain (fig. 8). a more complicated scenario is realised in grain 15 located between grains 10 and 14 (fig. 9). here, four slip systems activating in different parts of the grain make nearly equal contributions to plastic deformation. notably, the largest contribution is made by a slip system with a non-maximal schmid factor with respect to the y-axis. again, all slip traces predicted numerically are found in grains 4 and 15 experimentally. fig. 6 amount of slip accumulated in different slip systems in grain 10 at 5% compression 12 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov fig. 7 amount of slip accumulated in different slip systems in grain 14 at 5% compression fig. 8 amount of slip accumulated in different slip systems in grain 4 at 5% compression grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 13 fig. 9 amount of slip accumulated in different slip systems in grain 15 at 5% compression 4.3 grain-scale stress analysis to explain why the slip systems with non-maximal schmid factors become active, let us analyse the stress fields developing in the oligocrystal under compression. the distributions of equivalent stress and stress tensor components acting along and across the loading axis are shown in fig. 10 for 5% compression. in consistency with the plastic strain field (fig. 5a), the equivalent (von mises) stress reveals essential inhomogeneity (fig. 10a). the higher stresses tend to concentrate near the grain boundaries with large misorientation angles. a) b) c) d) fig. 10 equivalent (von mises) stress (a) and stress tensor components acting along (b) and across the loading axis (c, d) at 5% compression 14 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov the analysis of the stress tensor components reveals that the stress state in some local regions deviates from uniaxial. not surprisingly, the stress σ22 acting along the y-axis takes on negative values all over the computational model, making the largest contribution to the stress state in most grains (fig. 10b). in some grains, however, the contributions of stresses σ11 and σ33 acting across the compression axis are comparable to that of σ22, as seen in fig. 10c-d. the red and blue regions in fig. 10c exhibit tensile and compressive stresses. the non-zero stresses σ11 and σ33 may result in activating additional slip systems, disregarded in analytical estimates. 4.4 strain rate dynamics numerical simulations provide an exceptional opportunity to trace the development of plastic deformation with high temporal and spatial resolution. in this study, we analysed numerically the strain rate fields evolving during the entire deformation process, which is a challenging task for experimental research. for illustrative purposes, three selected snapshots of the strain rate fields are given in fig. 11, while the full video is available online (see the supplementary file “strainrate.mp4”). the local strain rate values in fig. 11 are normalised to the average strain rate dictated by the velocity of compression. accordingly, the regions painted with colours located in the colour scale above dark blue deform at strain rates above the average level (the regions with colours between the light blue and red in fig. 11). the analysis shows that first plastic strain emerges near the top and bottom corners of the sample, adjacent to the punches (fig. 11a). in the top corner, slip is initiated in two neighbouring grains 4 and 10 separated by a stepwise high-angle grain boundary. as is discussed above, the slip activation in these grains is dictated by their crystallographic orientations relative to the loading axis. grain 10 has nearly a goss orientation with <110>||y, and grain 4 is of a cube orientation with <100>||y. a) b) c) fig. 11 snapshots of equivalent strain rate fields in the grain model at a compressive strain of 0.0004 (a), 0.018 (b) and 0.05 (c) grain-scale deformation mechanisms in an additively manufactured aluminium bronze oligocrystal 15 another decisive factor affecting the scenario of plastic strain evolution in these grains is their position in the sample. an illustrative example is grain 4 where plastic deformation develops in the form of multiple fronts propagating along the grain body. the plastic strain rate in the narrow zone propagating throughout the grain exceeds the average value several times (red regions in fig. 11 b, c) and abruptly decreases behind the front. the fronts of localised plastic deformation are periodically generated near the top corner adjacent to the moving punch and subsequently propagate towards the opposite punch at a velocity 1.5-2 times higher than the velocity of compression. this behaviour agrees with the experimental observations of zuev and khon [33] and numerical simulations for fcc single crystals [15]. 5. conclusion in this study, we analysed the deformation mechanisms developing in a cu-7.5 wt.% al oligocrystal under compression, using a combined experimental and numerical approach. the experimental sample consisting of a few coarse columnar grains was produced by wire-feed electron beam directed energy deposition. based on the experimental data of ebsd analysis, a crystal plasticity model was built to consider a realistic grain structure. deformation features of oligocrystal grains under compression were analysed experimentally, numerically and analytically. the main findings of the study are summarised as follow: (i) activation of slip systems: consistent with theoretical predictions, the slip systems with the highest schmid factors along the compression axis were activated in all grains. however, up to four additional slip systems were activated in most grains to accommodate their strongly inhomogeneous plastic deformation. (ii) complex stress state at grain scale: the study revealed that the stress state at the grain scale considerably deviated from uniaxial. thus, additional slip systems were at work to balance the non-zero stresses acting across the compression axis. (iii) role of grain boundaries in stress concentration: irregularly shaped grain boundaries are sources of high stress concentration, the relaxation of which also requires the activation of additional slip systems. (iv) strain rate dynamics: the numerical analysis of strain rate dynamics with high spatial and temporal resolution showed that plastic deformation in particular grains developed in the form of narrow fronts generated periodically near the moving punch and subsequently propagating towards the opposite side of the sample. this study has demonstrated that oligocrystal analysis provides valuable insights into the linkage between the intraand intergrain deformation mechanisms with the overall material behaviour. acknowledgement: the work was performed according to the government research assignment for ispms sb ras, project fwrw-2021-0002. ebsd data were obtained using an equipment of tomsk regional core shared research facilities center of national research tomsk state university, supported by the ministry of science and higher education of the russian federation through grant no. 075-15-2021-693 (no. 13.rfc.21.0012). 16 v. romanova, d.v. lychagin, m. pisarev, o. zinovieva, r. balokhonov 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diwaniya, iraq 2 university of al muthana, iraq 3 faculty of mechanical engineering, university of niš, serbia abstract. in the oral and maxillofacial surgery, there is a requirement to provide the best possible treatment for the patient with mandibular fractures. this treatment presumes application of reduction and fixation techniques for proper stabilization of the fracture site. the reduction of the bone fragments and their fixation is much better performed when geometry and morphology of the bone and osteofixation elements (e.g. plates) are properly defined. in this paper, a new healthcare procedure, which enables application of personalized plate implants for the fixation of the mandibular fractures, is presented. geometrical models of mandible and plate implants, presented in this research, were created by means of the method of anatomical features (maf), which has been already applied to the creation of accurate geometrical models of various human bones, plates and fixators. by using such geometrically and anatomically accurate models, orthopedic and maxillofacial surgeons can better perform preoperative tasks of simulating and planning the operation, as well as an intraoperative task of implanting the personalized plate into the patient body. key words: cad, orthopedic, mandible, fracture, plate, parametric models, method of anatomical features received july 10, 2017 / accepted may 05, 2018 corresponding author: nikola vitković faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, niš, serbia e-mail: nikola.vitkovic@masfak.ni.ac.rs 370 k. husain, m. rashid, n. vitković, j. mitić, j. milovanović, m. stojković 1. introduction in the maxillofacial surgery for the treatment of mandibular fractures, reduction and fixation techniques are used [1, 2]. the reduction of the bone fragments and their fixation is better performed when geometry and morphology of the bone and osteofixation material (plates, screws, rods, pins, etc.) are properly defined [3]. in order to accomplish this goal, it is of great importance to clearly define geometrical properties and morphometric parameters of the mandible bone and to establish proper correlations between them [4-6]. the mandible (lower jaw) is the largest and the strongest bone in the face and its shape is very complex [7, 8]. mandible fractures are common facial injuries treated by the oral and maxillofacial surgeons as described in [1, 2]. these fractures can be grouped into the broad categories which are defined as unilateral fractures (double or multiple unilateral), bilateral fractures, fractures with contralateral condyle compromise, and bilateral condyle fractures with symphysis/anterior body compromise [9]. reduction and fixation process of mandible fractures should provide biomechanical stability to the assembly of fractured mandible, bone fragments and adequate implants (e.g. mini-plates, screws) as stated in [10]. biomechanical stability is often analyzed by the use of numerical simulations in adequate software packages (abaqus, ansys, etc.) [10]. in order to conduct such analysis valid geometrical models are required. if the geometry and morphology of the models are better defined, then the finite element analysis (fea) will provide more reliable results, and the process of reduction and fixation will be improved. fixation of the assembly is performed by the use of different kind of plates. in general, they can be divided into two general groups: locking plates and non-locking plates [9, 11]. locking plates provide better stability of the assembly and do not require precontouring of the plates. non-locking plates require pre-contouring, and they can interrupt and destroy the periosteum of the bone [9, 12]. in both cases, it is important to properly adjust shape and position of the plate(s) in accordance with mandible geometry. the geometrical models which conform to the anatomy and morphology of the mandible can be created by the use of volumetric imaging methods (e.g. cone beam computerized tomography – cbct, computerized tomography – ct or magnetic resonance imaging mri), 2d methods (x-ray, 2d ultrasound), and predictive methods (based on predictive models). volumetric methods provide 3d models, which can be used for measuring morphometric parameters and initial placement of implants in medical software (e.g. materialize mimics) [8, 13]. these models do not have proper geometrical definitions and correlations between anatomical entities, so they do not have the ability to change and adapt to various requirements. 2d models acquired from 2d scanning methods do not provide enough information about geometrical properties in 3d space. various transformations can be applied [14], but the resulting models still lack anatomical and geometrical definitions, especially in comparison with volumetric methods. predictive methods enable the creation of bone geometrical models by using various types of parametric (statistical) models. these methods can provide valid geometrical models, but they are limited by the input set of the bone samples, type of the applied method, and by the number and type of the parameters involved [15-17]. in this paper, the procedure for the treatment of patient with mandibular fracture(s) is presented. this procedure encapsulates the whole process from scanning the patient to the geometrical models of mandible fracture and plate implant 371 implantation of an adequate plate implant. the essential parts of this procedure are processes in which accurate geometrical models of the mandible, mandible fracture, and plate implants are created. to construct such models the method of anatomical features (maf) is applied. the main goal of this research study is to achieve a complete geometrical definition of the mandible, mandible fractures, and plate implants in order to enable the orthopedic surgeons to adequately prepare and perform orthopedic interventions. 2. anatomy of mandible the lower jaw (mandible) is the biggest and the most massive bone in the face, which is connected to the skull bones through the temporomandibular joint. it represents the biggest odd bone in the face or the viscecranial bone, which participates in construction of the only mobile head joint. it consists of a mandible body and two rami [18, 19]. the mandible body is of complex shape and represents its horizontal part. it consists of two sides (external and internal) and two edges. the first edge is defined as alveolar part of the mandible which corresponds with inferior dental arch (latin: arcus alveolaris) whereas the second (lower) edge is defined as mandible basis (latin: basis mandibulae). ramus is roughly of a rectangle shape, which is located upward and backward in relation to the mandible body. it forms an angle of 90°–140°, most commonly 120°–130°, to the mandible body. ramus has two sides, external and internal. it also has four edges, upper, lower, anterior, and posterior. the upper edge has two processes: coronoid process (latin: processus coronoideus) and condylar process (latin: processus condylaris) [18, 19]. 3. method of anatomical features and its application maf is created by the authors of this research study whose objective is to enable the creation of various types of geometrical models of the human bones and osteofixation material [17]. one of the most important outcomes of the maf application in medical imaging is the creation of a generic parametric model of the specific human bone. in general, parametric model can be defined as a model whose geometry can be changed by the application of different parameters values, while its topology remains the same. in the cases of human bones (in this case mandible), the parametric model is defined as a set of functions, whose arguments are morphometric parameters, whose values can be measured in medical images. morphometric parameters are geometrical dimensions, which are defined individually for each bone in human body [16]. they are used in order to customize the parametric model to the specific patient [15-17]. by the application of measured values, the parametric model transforms into a 3d personalized geometrical model of the specific human bone. “personalized” means that model geometry, shape and anatomy correspond to the patient bone. the main benefit of this model application is in its possibility to create a complete 3d geometric model of the patient bone, even in the cases when input data acquired from medical images are incomplete. the reasons for lack of data can be single, or not enough 2d image(s) (not enough data for 3d reconstruction), inability to perform ct scanning (patient must not be subjected to radiation, medical institution does not have ct device), too much noise in medical image, etc. the personalized model of the human mandible created in this manner can be used for preoperational planning and simulation in 372 k. husain, m. rashid, n. vitković, j. mitić, j. milovanović, m. stojković orthodontics and maxillofacial surgery, the creation of customized plates and other types of implants and fixators, for educational purposes, etc. in this research, the parametric model of the human mandible is used for the creation of a fracture parametric model, and the personalized model of the human mandible with a fracture is used for the creation of a personalized model of the plate implant. 4. process description the main research goal was to create an improved healthcare procedure in maxillofacial surgery and orthopedics, which will enable the surgeons to improve on their performing of surgical interventions. the presented procedure covers the whole process from the diagnostic to the implantation of the plate. it is important to describe the proposed procedure of the implant placement because only in that way can it be understood why it is important to have accurate geometrical models of the mandible fractures and plate implants. treatment of mandible fractures was chosen as an example of the process because these kinds of fractures are very common [1, 2] in today’s clinical practice. the main process is described by using the structured analysis and design technique (sadt) notation. sadt is a methodology that uses diagrams to describe process functionality [20]. the basic elements of sadt are: input elements, resources, control elements, output elements and various types of arrows and connection elements [20]. the process of fixator customization is presented in fig. 1 and in the sadt notation it is defined as a0 process with the following elements:  input elements: volumetric or 2d image of the patient bone, parametric models of the mandible fracture  control elements: anatomical knowledge about human bones, medical image analysis knowledge, anatomical and morphological rules, rules defined in maf  resources: doctor, designer, software packages (medical imaging software, catia)  output elements: geometrical model(s) of the customized plate implant. the context diagram a-0 is broken down at the level a0 into the subprocesses as shown in fig. 2. in this diagram, the whole procedure for the creation of the customized plate is visually presented. the procedure can be divided into the four sub activities described below:  a1 analysis of the medical image – in this the activity analysis of the acquired medical image is performed. doctors (radiologist, orthopaedists, etc.) use created image of the patient bone to determine the type of fracture, its position and orientation, and to decide which plate implant will be used for fixation of the mandible. one important part of this activity is to measure values of morphometric parameters. these values are used to adapt parametric model of the mandible and fracture to the geometry and shape of specific patient to create personalized model of the mandible and fracture (pmomf). values are measured by using technical features of the applied medical software (e.g. vitrea). all of this knowledge represents output from the process and it is defined as “collected knowledge about mandible fracture”. geometrical models of mandible fracture and plate implant 373 fig. 1 creation of the geometrical models of the plate implant and mandibular fracture – context diagram a-0  a2 application of the measured data in cad software – measured values of morphometric parameters are applied to the parametric model of the adequate entity (mandible, fracture), and personalized models are created. this activity consists of two main sub activities and they are: 1. creation of the polygonal model of the mandible with fracture. this model is based on the parametric model of the mandible, which has already been created and described in [17]. the position of the fracture is defined by the measurements conducted in medical image analysis process – it is conditioned by the values of the morphometric parameters. 2. creation of an adequate solid model of the plate implant by following recommendations and procedures defined in literature [21]. it is possible that the resulting models can have some geometry and topological errors, but they can be fixed by using technical features of cad software, e.g. correction and optimization of the model in a sense of number of triangles, orientation of triangles, triangles reduction, filling holes, etc. these correction steps are very important because only valid closed polygonal models can be later converted to the solid models for the purpose of creating assembly. 374 k. husain, m. rashid, n. vitković, j. mitić, j. milovanović, m. stojković fig 2. the detailed structure of the geometrical models creation process– a1 process geometrical models of mandible fracture and plate implant 375  a3 making assembly of fixator and a bone – a polygonal model of mandible with fracture is converted into a solid model by using technical features of the cad software (e.g. catia closed surface technical feature) and an assembly of the mandible with fracture and plate implant (fixator) can be created.  a4 analysis of the created assembly in this activity, anatomical, morphological, and geometrical analysis of the created assembly of the mandible with fracture and plate implant is performed. anatomical analysis presumes that all the anatomical entities important for the proper positioning of the fixator are present. this means that if some crest exits in a real physical model, then the same crest must exist in a virtual model. morphological analysis presumes that shape of the anatomical entities must be preserved. this means that if the crest exists, then the shape of that crest model should be the same as the shape of the real crest. geometrical analysis implies that if the crest model exists and it has the same shape as the real crest, then dimensional deviations between the real crest and its model must be in the minimum range defined by the orthopedic surgeons. if all the conditions are fulfilled then the assembly is ready for other activities (e.g. testing biomechanics, surgery preparation) and the process is finished. it is important to note, that described procedure can also be used in the clinical cases of massive mandible fractures. in such cases, it is possible to add scaffold [22] component to the assembly of bone and plate, in order to enable better tissue growth, and thereafter, faster recovery of the patient. 5. parametric models and their applications the essential elements of the defined procedure are geometrical models of the mandible with fracture and plate implant. in order to present the whole process of their creation, a specific use case is defined and shown in this paper. the use case represents a clinical situation where fracture type b is formed on the patient mandible bone. to provide stability of the mandible bone with such fracture, tension band plate with four holes is chosen [9]. medical data used in this case are acquired from ct scanner (64-slice ct msct, aquilion 64, toshiba, japan) positioned in the clinical centre, niš, serbia. this data was already used for the creation of the parametric model of the human mandible in previous research [17]. in the following text, methods for the creation of parametric models of the mandible body with fracture type b and solid model of the tension band plate implant will be demonstrated. 5.1. parametric and personalized model of the mandible with fracture it is important to distinct the parametric model of the mandible with fracture and pmomf. the parametric model is a virtual mathematical model, while pmomf is a concrete geometrical model of the specific patient bone with fracture (surface or solid). the parametric model of the mandible with fracture was created by the use of points included in the parametric model of the mandible [17]. as stated in [15, 16] the parametric model of the human bone is a geometrical model defined as point cloud. coordinates of points included in the point cloud are defined by parametric functions, as 376 k. husain, m. rashid, n. vitković, j. mitić, j. milovanović, m. stojković described in [16, 17]. parameters are defined for each bone, and for mandible there are ten defined morphometric parameters [17]. to define the geometrical model of the mandible body fracture it is necessary to select proper points in point cloud set. proper points were selected based on standard classification of mandible fractures described in [9]. for the use case defined in this research, the fracture is classified as b fracture type, so adequate points were chosen and presented in fig. 3. pmomf is created by the application of the parameters values measured on ct scan of the specific patient (or any other source of data), in parametric functions. the created point cloud was tessellated and the polygonal model of the mandible with fracture was created. the surface model of the mandible with fracture was created by the use of technical features of the catia software (automatic surface, multisection surface, etc.), and it is presented in fig. 3. in order to create the most precise model possible, some adjustments were performed: cleaning and healing of the acquired point cloud, finer tessellation, and optimization of the surface model (e.g. softening, points adjustments). 5.2. plate implant based on the created pmomf and the defined procedure for the mandible body fracture reduction and fixation described in [21], a solid model of the tension band plate implant was created (standard 4-hole mandible plate 2.0 with centre space plate thickness 2 mm), and presented in figs. 4a and 4b. fig. 3 surface model of the mandible with defined fracture model and parametric points the created solid model of the plate implant can be called personalized because its geometry and shape are adapted to the specific patient. the procedure for the creation of geometry model of the personalized plate implant contains three important steps: geometrical models of mandible fracture and plate implant 377  creation of the tangent (base) plane the tangent plane is defined on the surface model of the mandible with fracture. the position and orientation of this plane is determined by the part of the surface near the fracture and ramus surface, fig. 4a.  construction of the outer contour of the plate the outer contour of the plate model is created in the tangent plane and projected on the surface of mandible model. the thick surface technical feature (thickness 2mm) is used on the projected contour and a solid model of the fixator is created, fig. 4b.  final modifications screw holes are created on the solid model of plate implant by the application of the hole technical feature. position and number of holes are defined in accordance to specification defined in [21]. a) b) fig. 4 construction process of the personalized plate implant: a) construction plane and outer contour of the tension band plate; b) solid model of the tension band plate implant with 4 holes the geometry model of the personalized plate can be used to produce the real implant by the application of additive or conventional manufacturing technologies. in this way orthopedic surgeons can do pre-operative tasks of simulating and planning the operation with geometrically accurate models of mandible with fracture and plate, and intraoperative task of implanting the personalized plate into the patient body. 378 k. husain, m. rashid, n. vitković, j. mitić, j. milovanović, m. stojković 6. conclusion in this paper, an improved healthcare procedure for the implantation of plate implant for the fixation of mandible fractures is presented. the procedure is based on a newly developed method for the creation of personalized geometrical models of the mandible with fracture and plate implant. personalization of the models is achieved by the application of the parametric model of the human mandible. geometry and morphology of the presented models can be customized to the specific patient, by applying the parameters values acquired from the medical images (ct or xray). the geometrical and anatomical precision of the parametric and other geometrical models is already determined and published in previous research studies [16, 17]. by applying the proposed procedure, maxillofacial and orthopedic surgeons can greatly improve pre-operative planning (precise geometrical models can be used), intra-operative procedures (pre-contouring is already performed) and post-operative recovery of the patient (faster and of better quality). acknowledgements: the paper is part of the project iii41017 virtual human osteoarticular system and its application in preclinical and clinical practice, sponsored by the republic of serbia for the period of 2011-2017. references 1. lee, j.h., 2017, treatment of mandibular angle fractures, archives of craniofacial surgery, 18(2), pp.73-75. 2. prasad, v.n., khanal, a., 2016, computed tomography evaluation of maxillofacial injuries, journal of 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g., wang, h.l., sabalys, g., anatomy of mandibular vital structures. part ii: mandibular incisive canal, mental foramen and associated neurovascular bundles in relation with dental implantology, journal of oral & maxillofacial research, 1(1), article e3. 20. marca, d., mcgowan, c., 1987, structured analysis and design technique, mcgraw-hill, inc. new york, ny, usa, p. 392. 21. https://www2.aofoundation.org/, plate types (last access: 08.10.2017) 22. milovanović, j., stojković, m., trajanović, m., 2015, applicability analysis of additive manufacturing processes in fabrication of anatomically shaped lattice scaffold, facta universitatis-series mechanical engineering, 13(3), pp. 295-305. https://www2.aofoundation.org/ 12699 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 529 545 https://doi.org/10.22190/fume240322020w © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper sliding friction in contacts with oneand two-dimensional viscoelastic foundations and viscoelastic half-space toshiki watanabe1, shintaro hatanaka1, ken nakano2, valentin l. popov3 1graduate school of environment and information sciences, yokohama national university, yokohama, japan 2faculty of environment and information sciences, yokohama national university, yokohama, japan 3department of system dynamics and friction physics, technische universität berlin, berlin, germany orcid ids: toshiki watanabe https://orcid.org/0009-0006-4442-3074 shintaro hatanaka https://orcid.org/0009-0006-4989-6065 ken nakano https://orcid.org/0000-0001-7437-7984 valentin l. popov https://orcid.org/0000-0003-0506-3804 abstract. solid viscoelasticity is one of the essential origins of sliding friction, as every solid exhibits energy dissipation due to it during deformation processes. in this paper, we first show theoretical solutions for one-dimensional (1d) problems of viscoelastic friction with a 1d viscoelastic foundation. then, we extend the 1d model to a two-dimensional (2d) model to find theoretical solutions for 2d problems of viscoelastic friction. finally, we apply the method of dimensionality reduction (mdr) to the theoretical solutions for the 1d problems to discuss three-dimensional (3d) problems of viscoelastic friction. key words: sliding friction, viscoelastic foundations, viscoelastic half-space, mdr 1. introduction solid viscoelasticity is one of the essential origins of sliding friction, as every solid exhibits energy dissipation due to it during deformation processes [1,2]. in theoretical investigation, a viscoelastic foundation consisting of independent rheological elements can be considered an asymptotic model of contacts of a rigid probe with a very thin, compressible viscoelastic layer [3-5]. another limiting case is a very thick layer, namely the viscoelastic half-space [6-9]. real systems are in between these two limits. received: april 03, 2024 / accepted may 09, 2024 corresponding author: valentin l. popov technische universität berlin, strasse des 17 juni 135, 10623 berlin, germany e-mail: v.popov@tu-berlin.de https://orcid.org/0009-0006-4442-3074 https://orcid.org/0009-0006-4989-6065 https://orcid.org/0000-0001-7437-7984 https://orcid.org/0000-0003-0506-3804 530 t. watanabe, s. hatanaka, k. nakano, v. l. popov contact mechanics of viscoelastic foundations is simple [10]. by capitalizing on their simplicity, we can obtain a rough but clear picture of the basics of sliding friction due to solid viscoelasticity [11-14]. for example, we need only five system parameters to describe the sliding contacts of a rigid probe with a kelvin-voigt viscoelastic foundation [15]. this simplicity brings master curves for the velocity dependences of several quantities (e.g., the contact width, indentation depth or normal load, friction force, and friction coefficient) described by reasonable dimensionless numbers, showing the effects of the inlet slope and vertical lift on the friction coefficient [16]. this analytical method has been extended to the standard linear solid viscoelasticity, showing how mechanical and material origins for bellshaped velocity-dependent friction compete [17]. contact mechanics of half-spaces is much more complicated compared with elastic foundations due to non-locality of elastic and viscoelastic interactions. however, for axisymmetric normal contacts, there exist a transformation reducing a three-dimensional (3d) contact into a contact with a one-dimensional (1d) viscoelastic foundation. this transformation is given by the so-called method of dimensionality reduction (mdr) [18]. for normal contact, the mdr provides the exact solution of the initial 3d contact problem [18]. there exists no exact procedure for applying mdr to sliding contacts. however, there are reasons to believe that a formal application of mdr to sliding systems gives qualitatively correct results. since the famous work by grosch [19], it is widely accepted that the energy dissipation in rubber friction is mostly due to internal dissipation in the viscoelastic material. based on this idea and foregoing works [20], persson suggested in 1998 a simple idea of estimating energy dissipation during sliding [21]. he used the hertzian solution and assumed that it is realized dynamically with the frequency corresponding to the time of contact of "asperities". this means that he suggested considering the sliding contact as a "repeated normal contact", which surely is not the exact picture of what happens in sliding but a very good qualitative presentation. this idea was used in a later paper by persson and tosatti [22] for treating more complicated topographies. this basic idea by persson means approximate reduction of sliding contact problem to the normal contact problem. however, the normal contact problem is mapped exactly to a problem with a 1d foundation via mdr. this means, the sliding mdr solution will have the same accuracy as the initial concept by persson. this hypothesis was checked by kürschner [23] by comparing the formal sliding mdr-solution with the complete 3d boundary element method (bem) simulations. for contact with a simple fluid, he has shown that the mdr solution and the "numerically exact" solution differ only by a constant factor (depending on the probe shape). kusche [24] could confirm a very good coincidence of mdr-solution with complete 3d bem simulation for a number of more complicated rheology. these results show that the mdr procedure can also be used to sliding contacts with elastomers after correcting them by a constant factor of the order unity. in this paper, we first show theoretical solutions for 1d problems of viscoelastic friction with a 1d viscoelastic foundation (section 2). then, we extend the 1d model to a twodimensional (2d) model to find theoretical solutions for 2d problems of viscoelastic friction (section 3). finally, we apply the mdr to the theoretical solutions for the 1d problems to discuss 3d problems of viscoelastic friction (section 4). sliding friction in contact with viscoelastic foundations and viscoelastic half-space 531 2. sliding contacts of rigid probes with a 1d viscoelastic foundation 2.1. model for 1d problems in section 2, we consider simple 1d problems. fig. 1 shows the analytical models for sliding contact between a rigid probe and a 1d viscoelastic foundation. we consider triangular and parabolic probes: ( ) tanh x x= , (1) 2 ( ) 2 x h x r = , (2) respectively, where θ is the slope angle of the triangular probe, and r is the tip curvature radius of the parabolic probe. the viscoelastic foundation is a 1d array of independent kelvin-voigt elements along the x-axis, characterized by a stiffness (per unit area) k and a damping coefficient (per unit area) c. the indentation depth is δ, and the drive speed of the foundation is v. in this situation, the normal load (per unit length) w' and friction force (per unit length) f' are [16]: d ( ) d d b a h w k h cv x x  −    = − −     , (3) d d ( ) d d d b a h h f k h cv x x x  −    = − − −     , (4) respectively, where a and b represent the positions of the leading and trailing edges of the sliding contact, respectively. the leading edge is determined geometrically, while the trailing edge is determined dynamically (see eq. (6)). the friction coefficient is obtained by μ = f'/w'. fig. 1 analytical models for sliding contact between a rigid probe and a 1d viscoelastic foundation; (a) triangular and (b) parabolic probes. 532 t. watanabe, s. hatanaka, k. nakano, v. l. popov 2.2 theoretical solutions for 1d problems 2.2.1. triangular probe in this section, we show the theoretical solutions for the 1d sliding problem of the triangular probe. first, under constant-δ conditions (where the indentation depth is controlled to be constant), the dimensionless drive speed is [16]: tancv v k     . (5) then, the normalized outlet contact width, dimensionless normal load, dimensionless friction force, and normalized friction coefficient under constant-δ conditions are given as the following explicit functions of the dimensionless drive speed [16]: 1 for 0 1 0 for 1 v vb a v      −    =   , (6) 2 2 1 1 for 0 1 tan 2 1 for 1 2 v v w w k v v         +     =   +   , (7) 2 2 1 2 for 0 1 2 1 for 1 2 v v v f f k v v         −     =   +   , (8) 2 2 4 for 0 1 2 tan 1 for 1 v v v v v          −   + =    . (9) then, under constant-w conditions (where the normal load is controlled to be constant), based on the relationship between δ and w' given by eq. (7), we obtain the following dimensionless drive speed [16]: 1/2 2 1/2 1/2 1 1 for 0 1 2tan 1 for 1 2 w v v v cv k v k w v v v        − −    +          =       +     . (10) from eq. (10), we obtain the dimensionless drive speed for constant-δ conditions as an explicit function of that for constant-w conditions [16]: sliding friction in contact with viscoelastic foundations and viscoelastic half-space 533 2 2 2 2 2 for 0 32 1 1 2 2 for 2 2 3 w w w w w w w v v v v v v v v     − =   + +   . (11) therefore, we obtain the following explicit forms for the other dimensionless quantities under constant-w conditions [16]: 2 2 2 1 for 0 32 2 0 for 3 w w w w v v vb a v   −   −  =     , (12) 2 1/2 2 1 2 1 for 0 2 3 tan 2 2 for 3 w w w w w w v v k w v v v     −      =     + −  , (13) 2 21 2 4 2 for 0 2 3 tan 2 1 for 3 w w w w w w v v v v f f w v   − −      =     , (14) 2 21 2 4 2 for 0 2 3 tan 2 1 for 3 w w w w w w v v v v v     − −     =    . (15) fig. 2 shows the above theoretical solutions for the sliding problem between a triangular probe and a 1d viscoelastic foundation. the black solid line shows the solution (i.e., the dimensionless master curve). the blue and red broken lines show the asymptotic solutions for the lowand high-speed regions, respectively. when the drive speed is high, we see the outlet peeling of the viscoelastic foundation from the probe surface (i.e., decrease in b) in (a) and (b) and the vertical lift of the probe (i.e., decrease in δ) in (b). the friction coefficient linearly increases for low speed and shows constant for high speed in (a) and (b). 2.2.2. parabolic probe in this section, we show the theoretical solutions for the 1d sliding problem of the parabolic probe. first, under constant-δ conditions, the dimensionless drive speed is [16]: 534 t. watanabe, s. hatanaka, k. nakano, v. l. popov fig. 2 theoretical solutions for the sliding problem between a triangular probe and a 1d viscoelastic foundation under (a) constant-δ and (b) constant-w conditions. 2 cv v k r    . (16) then, the other dimensionless quantities under constant-δ conditions are given as the following explicit functions of the dimensionless drive speed [16]: sliding friction in contact with viscoelastic foundations and viscoelastic half-space 535 21 b v v a    = − + + , (17) ( )3 2 3/2 3 2 1 (1 ) 32 w w v v k r       = − + + , (18) ( )2 4 2 3/2 2 2 2 3 2 2 (1 ) 3 f f v v v v v k          = − − + + , (19) 2 4 2 3/2 3 2 3/2 2 3 2 2 (1 )2 1 (1 ) v v v v vr v v            − − + +  = − + + . (20) then, under constant-w conditions, based on the relationship between δ and w' given by equation (18), we obtain the following dimensionless drive speed [16]: ( ) 1/3 1/3 3 2 3/22 1 (1 ) 3 wv v v v   − −  = − + +    . (21) however, it is not easy to obtain the dimensionless drive speed for constant-δ conditions as an explicit function of that for constant-w conditions. in this case, it is reasonable to take the following implicit form [16]: 1/3 1/3 2 w cv k v v w k rw   −   =    . (22) this leads to the following implicit forms for the other dimensionless quantities [16]: 21 b v v a    = − + + , (23) 1/3 2/32 2/3 2 32 4 39 w rk w w   −     =        , (24) 1/3 2 4/3 4 4 w r k f f f w w   −   =    , (25) 1/3 2 4/34 w r k f w w    −   =    . (26) fig. 3 shows the above theoretical solutions for the sliding problem between a parabolic probe and a 1d viscoelastic foundation. when the drive speed is high, we see the outlet peeling of the viscoelastic foundation in (a) and (b) and the vertical lift (i.e., decrease in δ) in (b). the friction coefficient linearly increases for low speed in (a) and (b). for high speed, it shows constant in (a) but decreases (b). the latter property, termed the velocity-weakening friction, is known for the source of stick-slip instabilities in sliding systems [25-32]. 536 t. watanabe, s. hatanaka, k. nakano, v. l. popov fig. 3 theoretical solutions for the sliding problem between a parabolic probe and a 1d viscoelastic foundation under (a) constant-δ and (b) constant-w conditions. sliding friction in contact with viscoelastic foundations and viscoelastic half-space 537 3. sliding contacts of rigid probes with a 2d viscoelastic foundation 3.1. model for 2d problems in section 3, we extend the 1d problems to consider the 2d problems for conical and paraboloidal probes: 2 2( , ) tanh x y x y= + , (27) 2 2 ( , ) 2 x y h x y r + = . (28) the viscoelastic foundation is a 2d array of independent kelvin-voigt elements in the xyplane, characterized by a stiffness (per unit area) k and a damping coefficient (per unit area) c. the normal load w and friction force f are: ( ) (0) ( ) (0) ( ) d d a b y a a y h w k h cv x y x  − −   = − −      , (29) ( ) (0) ( ) (0) ( ) d d a b y a a y h h f k h cv x y x x  − −    = − − −       . (30) the x-positions of leading and trailing edges are dependent on their y-position: a = a(y) and b = b(y), respectively. the former is determined geometrically, while the latter is determined dynamically (see eq. (31)). the friction coefficient is obtained by μ = f/w. 3.2. theoretical solutions for 2d problems 3.2.1. conical probe in this section, we consider the 2d sliding problem of the conical probe. to determine the integral range of eqs. (29) and (30), we must solve the following nonlinear equation: 2 2 2 4 4 3 2 2 2 2 2 4 2 2 1 2 0 y y y y v v v a a a a         − + + − − − + =    , (31) where a = a(0) and β = β(y) = b(y)/a(0). this difficulty comes from the fact that the vertical section of a cone is (not triangular but) hyperbolic, except for the vertical section by the plane that intersects the cone tip. however, considering the limiting cases (i.e., lowand high-speed cases leading to β = 1 and 0, respectively) enables us to obtain asymptotic solutions. the asymptotic solutions for constant-δ conditions are: 2 3 for low speedtan 3 for high speed w w k v         =    , (32) 538 t. watanabe, s. hatanaka, k. nakano, v. l. popov 3 for low speed tan 2 for high speed 4 v f f k v          =    , (33) 3 for low speed 2 tan for high speed 4 v        =    . (34) the asymptotic solutions for constant-w conditions are: 1/3 2 1/3 1/2 1 for low speed 3 tan for high speed 3 w w w w k v      − −      =           , (35) 1/3 9 for low speed 8 tan for high speed 4 w w v f f w         =    , (36) 1/3 9 for low speed 8 tan for high speed 4 w w v           =    . (37) fig. 4 shows the theoretical solutions for the sliding problem between a conical probe and a 2d viscoelastic foundation, showing similar velocity dependences to those for the 1d problem of the triangular probe (see fig. 2). a distinct difference can be seen in the velocity dependence of the indentation depth for high speed (i.e., δ is proportional to v–1 for the 1d problem and v–1/2 for the 2d problem). 3.2.2. paraboloidal probe in this section, we consider the 2d sliding problem of the paraboloidal probe. first, under constant-δ conditions, we can take the same dimensionless drive speed as that for the 1d problem of the parabolic probe (see eq. (16)): 2 cv v k r    . (38) sliding friction in contact with viscoelastic foundations and viscoelastic half-space 539 fig. 4 theoretical solutions for the sliding problem between a conical probe and a 2d viscoelastic foundation under (a) constant-δ and (b) constant-w conditions. this simplicity comes from the fact that the vertical section of a paraboloid is always a parabola with an identical tip curvature radius. then, we obtain the following explicit forms for the other dimensionless quantities under constant-δ conditions: 2 2 2 1 2 1 1 (1 ) (1 ) arctan( ) 4 2 22 w w v v v v kr        − = + − + + , (39) 2 4 2 2 1 5 5 (1 ) arctan( ) 2 32 f f v v v v v v k r          − = − − + + , (40) 2 2 2 1 2 4 2 2 1 3 6 (1 ) 6(1 ) arctan( )2 6 20 12 12 (1 ) arctan( ) v v v vr v v v v v v                 − − + − + +  = − − + + . (41) 540 t. watanabe, s. hatanaka, k. nakano, v. l. popov then, under constant-w conditions, based on the relationship between δ and w given by eq. (39), we obtain the following dimensionless drive speed: 1/4 2 2 2 11 1 (1 ) (1 ) arctan( ) 4 2 2 wv v v v v v      − −  = + − + +    . (42) however, it is not easy to obtain the dimensionless drive speed for constant-δ conditions as an explicit function of that for constant-w conditions, as in the case for the 1d problem of the parabolic probe (see section 2.2.2). in this case, it is reasonable to take the following implicit form: 1/4 1/4 2 w cv k v v w k rw   −   = =     . (43) this leads to the following implicit forms for the other dimensionless quantities: 1/21/2 0 w wrk w w      −     =        , (44) 1/4 3 5/4 5 8 w r k f f f w w   −   =    , (45) 1/4 3 5/48 w r k f w w    −   =    . (46) fig. 5 shows the theoretical solutions for the sliding problem between a paraboloidal probe and a 2d viscoelastic foundation, showing similar velocity dependences to those for the 1d problem of the parabolic probe (see fig. 3). some distinct differences can be seen in the velocity dependences of the indentation depth, friction force, and friction coefficient for high speed (i.e., δ is proportional to v–1 for the 1d problem and v–2/3 for the 2d problem; f and μ are proportional to v–1/2 for the 1d problem and v–1/3 for the 2d problem). 4. sliding contacts of rigid probes with a 3d viscoelastic half-space 4.1. basics of the method of dimensionality reduction (mdr) in section 4, we consider 3d problems between an axisymmetric rigid probe having the shape z = f(r) and a 3d viscoelastic half-space with the plane surface characterized by kelvin-voigt model with shear modulus g and dynamical viscosity . as most elastomers have poisson ratio close to 1/2, we assume that the medium is incompressible. under these assumptions, the application of the mdr consists of the following two steps [18]: sliding friction in contact with viscoelastic foundations and viscoelastic half-space 541 fig. 5 theoretical solutions for the sliding problem between a paraboloidal probe and a 2d viscoelastic foundation under (a) constant-δ and (b) constant-w conditions. step i: the 3d viscoelastic body is replaced by a 1d viscoelastic foundation consisting of an array of non-interacting kelvin-voigt elements which stiffness k and damping coefficient c are defined as follows: 4k g x=  , 4c x=  , (47) where δx is spacing between adjacent elements. step ii: the 3d profile z = f(r) is transformed into a 1d profile g(x) according to: 2 2 0 ( ) ( ) d x f r g x x r x r  = −  . (48) 542 t. watanabe, s. hatanaka, k. nakano, v. l. popov in particular, triangular and parabolic profiles are transformed as follows [18]: 3 3( ) tan ( ) tan 2 d df r r g x x   =  = , (49) 2 2 3 3 ( ) ( ) 2 d d r x f r g x r r =  = . (50) now the transformed shape g(x) is pressed into the viscoelastic foundation by the depth δ and moved tangentially with a given velocity v. to obtain the results which apply to the initial 3d problem, the coefficients of friction obtained by the above mdr procedure should be multiplied with 0.67 for triangular probes and 0.55 for parabolic probes [23]. this means that the solutions obtained in section 2 can be applied – after corresponding coordinate transformation – to contacts with the 3d half-space. 4.2. theoretical solutions for 3d problems 4.2.1. conical probe comparison of shape and material parameter definitions given by eqs. (1) and (3) with those of the mdr, eqs. (49) and (47), shows that the following variable change must be undertaken: 3tan tan 2 d    , 4k g , 4c  , w w  , f f  . (51) for the main governing parameter, the dimensionless speed (5), we obtain: 3tan 2 dv v g      . (52) applying variable transformation (51) to (9) and using the scaling coefficient 0.67 for conical probes [23] yields: 2 2 3 4 for 0 1 21.05 tan 1 for 1 d v v v v v         −   +    . (53) this equation provides the coefficient of friction of a 3d cone in contact with a kelvinvoigt half-space under conditions of fixed indentation depth. the governing parameter for the case of a fixed normal force, eq. (10), becomes 1/2 32 tan d w gv v g w          , (54) and the coefficient of friction (15) becomes: 2 2 3 1 2 4 2 for 0 2 3 1.05 tan 2 1 for 3 w w w w d w v v v v v    − −    =    . (55) sliding friction in contact with viscoelastic foundations and viscoelastic half-space 543 4.2.2. paraboloidal probe comparison of shape and material parameter definitions given by eqs. (2) and (3) with those of the mdr, eqs. (50) and (47), shows that the following variable change must be undertaken: 3 2 dr r  , 4k g , 4c  , nw f , xf f . (56) the dimensionless drive speed (16) becomes: 3d v v g r     . (57) applying transformations (57) and scaling with factor 0.55 for the parabolic profile, we receive: 2 4 2 3/2 3 2 3/2 3 2 3 2 2 (1 ) 0.55 1 (1 )d v v v v v r v v          − − + + = − + + . (58) for the case of fixed force, the governing dimensionless drive speed (22) becomes: 1/3 3 4 w d v g v g r w         . (59) the scaling (26) of the coefficient of friction becomes: 1/3 2 34 0.55 d w r g w          . (60) with definitions (59) and (60), the results reported above in fig. 3 are applicable to the 3d case. 5. conclusion in this paper, we discussed 1d, 2d, and 3d sliding problems of typical rigid probes (triangular and parabolic probes in the 1d and conical and paraboloidal probes in the 2d and 3d) in contact with 1d and 2d viscoelastic foundations and a viscoelastic half-space, respectively. we found that extending the 1d to the 2d models does not change the master curves qualitatively, confirming that the simplest 1d model is a remarkable tool for understanding sliding friction due to solid viscoelasticity. applying the mdr to the 1d model is expected to estimate the sliding friction more quantitatively. acknowledgement: this study was supported by crest (grant number jpmjcr2193) of the japan science and technology agency (jst), kakenhi (grant number 21h01236) of the japan society for the promotion of science (jsps), and by deutsche forschungsgemeinschaft (grant number 423918021). 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2020, dynamic stiction without static friction: the role of friction vector rotation, physical review e, 102(6), 063001. 31. papangelo, a., putignano, c., hoffmann, n., 2020, self-excited vibrations due to viscoelastic interactions, mechanical systems and signal processing, 144, 106894. 32. pal, a., restrepo, v., goswami, d., martinez, r.v., 2021, exploiting mechanical instabilities in soft robotics: control, sensing, and actuation, advanced materials, 33(19), 2006939. facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 513 524 https://doi.org/10.22190/fume171212010p © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a study of the environmental kuznets curve for transport greenhouse gas emissions in the european union nikola petrović 1 , nebojša bojović 2 , marijana petrović 2 , vesna jovanović 1 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of transport and traffic engineering, university of belgrade, serbia abstract. in view of the european union as one of the main polluters in the word and the fact that gdp per capita in the european union is equivalent to the 282 percent of the world`s average, it is interesting to study the relationship between transport ghg emissions and the economic activity within the european union. in the paper, the authors check the environment kuznets curve hypothesis for members of the eu over the period 2000-2014. the analysis results show that an inverse-u relationship exists between transport ghg emissions and gdp per capita. at the same time, the results indicate that the change of economic structure has influenced the transport ghg emissions in the developed countries, that is, in the countries that record a higher level of gdp per capita. key words: transport, ghg emissions, environmental kuznets curve, gdp per capita, european union 1. introduction the development of the world economy as well as the economic development of the transition countries is accomplished in a very turbulent environment [1]. transport represents an important economic activity in the developed countries as well as in the transition ones. the development of an effective and efficient transport system contributes to the products becoming more competitive in the domestic and global market. firstly, the industrial development and, later, the international trade intensification have contributed to the accelerated development of transport. however, this has, in turn, resulted in transport received december 12, 2017 / accepted april 10, 2018 corresponding author: nikola petrović faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: petrovic.nikola@masfak.ni.ac.rs 514 n. petrović, n. bojović, m. petrović, v. jovanović now becoming an economic activity with the fastest growing impact on environmental degradation [2]. on the one hand, transport plays important roles in socio-economic activities and improvement of people’s well-being, while, on the other, transport represents one of the major emitters of hazardous substances that affect the quality of air and the formation of tropospheric ozone. greenhouse gas (ghg) emissions account for 70% of the total ghg emissions from humans with the major emission sources from fossil fuels of energy supply and transport [3]. the rapid development of transport together with the related environmental problems pertaining to air pollution and ghg emissions have been a caution to the world [4]. following the power generation sector, the global transport sector is the second largest sector generating ghg emission causing 23% of total ghg emissions worldwide [5]. “around 75% of the global greenhouse gas emissions is caused by co2” [6]. transport is responsible for nearly one quarter of global energy related carbon dioxide emissions while 75% of these emissions are due to road transport energy use [7]. since 1991, when economists first reported on a systematic relationship between income changes and environmental quality, the relationship known as the environmental kuznets curve (ekc) has become standard fare in technical conversations about environmental policy [8]. the ekc statistical relationship suggests that as development and industrialization progress, environmental quality decreases due to an increasing emission of pollutants. much later, in the post-industrial stage, cleaner technologies and a shift to information and service-based activities combine with a growing ability and willingness to enhance environmental quality [9]. the literature related to the examination of the ekc validity includes various dependent and independent variables. what is observed as dependent variables are indicators of environmental degradation, i.e. the indicators of environmental quality (such as co2, so2, sulfur, arsenic, lead emissions; deforestation; water pollution; and dark matter), while the following indicators are observed as independent variables together with per capita income: income inequality, trade openness, institutional quality, strictness of environmental regulations, and corruption [10]. bearing in mind that the industrial development in its early stages contributed to an increased environmental degradation as well as that the literature has not paid much attention so far to the analysis of the quantification of the industrial development effect, along with the service sector development influence on transport ghg emissions, the paper draws particular attention to the influence of the industrial share in the gross domestic product (gdp) and the effects of the service sector in gdp on ghg emissions. the environment in the transition countries is cleaned up quickly because of rising energy prices and penalizing of energy-intensive activities [11, 12]. therefore, special emphasis will be placed on the interdependence between gdp per capita and transport ghg emissions in the developed and transition countries that are members of the eu. starting from the abovementioned, the purpose of this paper is to quantify the influence of the economic development on transport ghg emissions in the example of certain eu members. the economic growth includes only quantitative changes while the economic development represents a broader category including both quantitative and qualitative changes [13]. the aims of the paper are: 1) analysis of the interdependence between transport ghg emissions and gdp per capita; 2) analysis of the effects of changes in the economic structure, i.e. changes in the service sector participation in gdp ontransport ghg emissions. the following hypotheses will be tested in the paper: a study of the environmental kuznets curve for transport greenhouse gas emissions in the eu 515 h1 – an inverted-u-shaped relationship exists between gdp per capita and transport ghg emissions; and, h2 – changes in the economic structure have a significant influence on transport ghg emissions. 2. review of related literature three main approaches have dominated the literature on the drivers of pollution emissions and other environmental impacts [14, 15] the ipat identity, convergence approaches and the environmental kuznets curve. ipat (abbreviated from impact, population, affluence and technology) expresses the idea that environmental impact results from the following factors: population, affluence and technology. this identity was proposed by ehrlich and holdren as a way of quantifying human impact on environment [16]. the convergence approaches hypothesize that emissions grow more slowly in emissions intensive countries than in less emissions intensive ones [17]. according to kuznets, per capita income increases and so does income inequality at first, but income inequality starts declining as economic growth continues [18]. this interdependence between per capita income and income inequality is known as the kuznets curve. therefore, the question of whether economic growth causes environmental degradation has become a central issue of discussion since publication of the bruntland report in 1987 [19]. in the 1990s, the kuznets curve took on a new form of existence. instead of an inverted-u-shaped relationship between economic growth and economic inequality that represented the original kuznets curve, what was examined was an inverted-u-shaped relationship between economic growth and environmental degradation that is known as the ekc. the first results of the empirical ekc studies appeared independently in the following working papers: an nber working paper as part of a study of the environmental impacts of nafta [20], the world bank’s 1992 world development report [21] and a development discussion paper as part of a study for the international labour organization [22]. grossman and krueger first pointed out an inverted-u-shaped relationship between so2 and smoke [20], on the one hand, and per capita income, on the other, while panayoton first named this relationship as the ekc [22]. the ekc hypothesis is tested empirically in the cases of many countries and regions. however, the results of the various research studies are different and depend on the observed countries, regions and selected environmental quality indicators. numerous studies have examined the relationship between environmental degradation and per capita income. their findings confirmed the presence of an inverse-u-shaped relationship between environmental quality and per capita income [23-31]. those studies have shown that environmental degradation increases at low levels of per capita income while it later decreases at high levels of per capita income. recently, special attention has been paid to the effect of transport on environmental and air quality in the literature. alshehry and belloumi check for the environmental kuznets curve hypothesis for saudi arabia over the period 1971-2011 [19]. they find that the inverse-u relationship does not exist between transport co2 emissions and economic growth in saudi arabia. the paper will attempt to test the ekc in the example of certain eu members, i.e. examine whether an inverse-u relationship between ghg emissions and gdp per capita exists in the eu. 516 n. petrović, n. bojović, m. petrović, v. jovanović 3. research methodology for the purpose of achieving the defined research objective several methods are applied. namely, this research relies on the following methods:  cluster analysis,  correlation analysis, and,  regression analysis. the cluster analysis comprises a range of methods for classifying multivariate data into a number of clusters based on the observed values of several variables for each individual. by organizing multivariate data into such clusters or segments, clustering can help reveal the characteristics of any structure. this analysis has proven useful in a wide range of areas such as psychology, market research and bioinformatics. it was used to examine the heterogeneity of the observed countries in terms of transport ghg emissions and gdp per capita in the period 2000-2014. the correlation analysis is a method of statistical evaluation used to study the strength of a relationship between continuous variables. the paper employs the pearson correlation coefficients which can only be given the values from -1 to +1. the sign shows whether the correlation in question is positive or negative. the absolute value of that coefficient shows the strength of connection. a perfect correlation, which is either -1 or +1, shows that the value of one variable can be determined with certainty if one knows the value of the other variable. on the other hand, the correlation that is equal to zero shows that there is no connection between the observed variables. by means of the correlation analysis the interdependence between gdp per capita and transport ghg emissions is explored. the regression analysis represents a technique that can be used to examine the connection between a continuous dependent variable and many independent variables. it is based on correlation; yet it enables a more sophisticated study of interrelations within a set of variables. it provides an assessment of the model as a whole and the relative contribution of all the variables comprising it. the regression analysis was used to determine the effect that the levels of gdp per capita, the industry share in gdp and that of service in gdp exert on transport ghg emissions. information base of research presents the statistical pocketbook [32], as well as the data of the world bank [33]. 4. research results and discussion the empirical part of the paper is organized into the following segments:  testing heterogeneity of the eu countries in terms of gdp per capita and transport ghg emissions;  analysis of gdp per capita and transport ghg emissions interdependence; and,  analysis of influence of gdp per capita, the share of service as well as that of industry in gdp on transport ghg emissions. a study of the environmental kuznets curve for transport greenhouse gas emissions in the eu 517 4.1. examining the heterogeneity of the eu countries in terms of gdp per capita and transport ghg emissions using the cluster analysis the eu member states were grouped according to gdp per capita and transport ghg emissions during the 21 st century. the results of descriptive statistics for each cluster are given in table 1 (prepared by the authors statistica 8.0). table 1 members of cluster number and distances from respective cluster center cluster 1 cluster 2 cluster 3 members distance members distance members distance cyprus 1902.826 austria 4150.63 bulgaria 4013.021 greece 1511.792 belgium 5588.99 croatia 757.332 italy 6152.557 denmark 3132.07 czech republic 3665.652 malta 4128.699 finland 3777.62 estonia 1847.923 portugal 3094.260 france 7549.23 hungary 912.204 slovenia 2646.965 germany 6615.41 latvia 659.485 spain 2338.866 ireland 2884.98 lithuania 623.774 luxemburg 31304.73 poland 663.010 netherland 2004.20 romania 3083.448 sweden 1769.98 slovak republic 2124.844 united kingdom 6421.15 in the first cluster are the members of the european union which recorded lower average values of gdp per capita and transport ghg emissions compared to the second cluster, namely: cyprus, greece, italy, malta, portugal, slovenia and spain. up to 2008 the observed countries recorded an increase in gdp per capita, i.e. an increase in transport ghg emissions up to 2007, followed by a decrease in both gdp per capita and transport ghg emissions in the ensuing period. it can be concluded that when the observed countries reached the level of gdp per capita (2008) that could contribute to a decrease in transport ghg emissions, they began to record a decrease in gdp per capita due to the financial crisis. at the same time, it can be shown that the observed countries, contrary to the countries that belong to the second and third cluster, experienced a decrease in gdp per capita from 2008 to 2014. that is, the countries of the second and third cluster suffered a decrease in gdp per capita in 2009 and 2010 as a consequence of the financial crisis, but recorded an increase in gdp per capita after that. in the second cluster there are the countries that recorded the highest average values of gdp per capita and the highest average values of transport ghg emissions, which are: austria, belgium, denmark, finland, france, germany, ireland, luxemburg, netherlands, sweden and united kingdom. it can be concluded that this cluster comprises the highly developed eu countries as well as the biggest polluters of air and environment. however, it has to be pointed out that the observed countries have reached the level of gdp per capita where any next increase in gdp per capita would contribute to a decrease in transport ghg emissions. in the third cluster are the countries that recorded the lowest average values of gdp per capita and transport ghg emissions, which are: bulgaria, croatia, czech republic, estonia, hungary, latvia, lithuania, poland, romania and slovak republic. this cluster includes all the transition countries that are members of the eu, except slovenia that belongs to the first cluster. 518 n. petrović, n. bojović, m. petrović, v. jovanović fig. 1 changes in average values of gdp per capita and transport ghg emissions in the 1st, 2nd and 3rd cluster, respectively, in the period from 2000 to 2014 based on the foregoing it can be concluded that the developed countries recorded a high level of gdp per capita and a high level of transport ghg emissions while the transition countries recorded a low level of gdp per capita and a low level of transport a study of the environmental kuznets curve for transport greenhouse gas emissions in the eu 519 ghg emissions. at the same time, it can be shown that any increase in gdp per capita in the transition countries (slovenia excluded) leads to an increase in transport ghg emissions, i.e. the transition countries have not yet reached the level of gdp per capita whose increase would contribute to a decrease in transport ghg emissions (fig. 1). 4.2. analysis of gdp per capita and transport ghg emissions interdependence tables 2, 3 and 4 (prepared by the authors spss statistics 19.0) contain the results of the correlation analyses. for the first cluster, the results indicate that there is a negative correlation between gdp per capita and transport ghg emissions. correlation is not statistically significant because the value of sig. is not less than 0.05. table 2 correlation analysis between gdp per capita and ghg for cluster 1 gdp per capita transport ghg gdp per capita pearson correlation 1 -0.064 sig. (2-tailed) 0.820 n 15 15 transport ghg pearson correlation -0.064 1 sig. (2-tailed) 0.820 n 15 15 the results for the second cluster indicate that there is a negative correlation between gdp per capita and transport ghg emissions, i.e. an increase of gdp per capita is followed by a decrease in transport ghg emissions of the developed eu member states. correlation is statistically significant because the value of sig. is less than 0.05. table 3 correlation analysis between gdp per capita and ghg for cluster 2 gdp per capita transport ghg gdp per capita pearson correlation 1 -0.753** sig. (2-tailed) 0.001 n 15 15 transport ghg pearson correlation -0.753** 1 sig. (2-tailed) 0.001 n 15 15 ** . correlation is significant at the 0.01 level (2-tailed) the results for the third cluster indicate that there is a positive correlation between gdp per capita and transport ghg emissions, i.e. an increase of gdp per capita is followed by an increase in transport ghg emissions of the transition countries. correlation is statistically significant because the value of sig. is less than 0.05. based on the above, it can be concluded that an inverted-u-shaped relationship exists between transport ghg emissions and gdp per capita in the developed and transition countries. however, it is necessary to point to the countries that belong to the first cluster and that still record a decrease in gdp per capita in the longer run after the global financial crisis. it can be concluded that the inverted-u-shaped relationship exists between gdp per capita and transport ghg emissions for the highly developed and transition countries in the european union, i.e. that hypothesis h1 has been confirmed. 520 n. petrović, n. bojović, m. petrović, v. jovanović table 4 correlation analysis between gdp per capita and ghg for cluster 3 gdp per capita transport ghg gdp per capita pearson correlation 1 0.959** sig. (2-tailed) 0.000 n 15 15 transport ghg pearson correlation 0.959** 1 sig. (2-tailed) 0.000 n 15 15 ** correlation is significant at the 0.01 level (2-tailed) the non-standardized coefficients from column b are used to formulate the regression equation while the beta coefficients are employed in the comparison of the contributions of all the dependent variables. one of the aims of this paper is to compare the contributions of dependent variables, i.e. the impact of gdp per capita as well as the shares of industry and of service in gdp on transport emissions; thus special attention is paid to the analysis of the beta coefficients. table 5 the impact of gdp per capita, the shares of industry and of service in transport ghg emissions in cluster 1 model r r square adjusted r square std. error of the estimate 1 0.952a 0.906 0.881 1.22370 2 0.947b 0.897 0.880 1.22801 a. predictors: (constant), service, gdp per capita, industry b. predictors: (constant), service, gdp per capita c. dependent variable: transport ghg emissions based on the results of the regression analysis given in table 5 (prepared by the authors spss statistics 19.0) it can be seen that the coefficient of determination is 0.906 if we consider the impact of gdp per capita as well as the shares of industry and of service in gdp on transport ghg emissions; but if we consider only the impact of gdp per capita and share of service in gdp on transport ghg emission the coefficient of determination is 0.897. when expressed as a percentage, we can conclude that the combined impact of all the independent variables on transport ghg emissions is 90.6% or 89.7% if we consider only the impact of gdp per capita and the share of service in gdp. it is necessary to point out that the value of sig. is less than 0.05 (0.001), and then the contribution observed is statistically significant. to avoid multicollinearity between the independent variables applied within the regression analysis is the method backward. the results of the analysis of the impact of the observed variables for cluster 1 are given in table 6 (prepared by the authors spss statistics 19.0). the results of the regression analysis indicate that gdp per capita and the share of service in gdp have a statistically significant impact on transport ghg emissions. a study of the environmental kuznets curve for transport greenhouse gas emissions in the eu 521 table 6 the value of regression coefficients influence of gdp per capita, the shares of industry and of service in gdp on transport ghg emissions model unstandardized coefficients standardized coefficients sig. b std. error beta 1 (constant) 197.219 64.936 0.011 gdp per capita 0.000 0.000 0.727 0.000 industry -0.716 0.687 -0.755 0.320 service -1.858 0.665 -2.047 0.017 2 (constant) 130.001 7.165 0.000 gdp per capita 0.0004 0.000 0.724 0.000 service -1.176 0.117 -1.296 0.000 based on the results of the regression analysis for cluster 2 given in table 7 (prepared by the authors spss statistics 19.0) it can be seen that the coefficient of determination is 0.915 if we consider the impact of gdp per capita, the shares of industry and of service in gdp on transport ghg emissions. however, if we consider the impact of the shares of industry and of service in gdp on transport ghg emissions then the coefficient of determination is 0.914; but if we consider only the impact of the share of service in gdp on transport ghg emission the coefficient of determination is 0.895. when expressed as a percentage, we can conclude that the combined impact of all the independent variables on transport ghg emissions is 91.5% or 91.4% if we consider the impact of the share of industry and of service in gdp or 89.5% if we consider only the impact of the share of industry. it is necessary to point out that the value of sig. is less than 0.05 (0.001), and then the contribution observed is statistically significant. table 7 the impact of gdp per capita, of the shares of industry and of service on transport ghg emissions in cluster 2 model r r square adjusted r square std. error of the estimate 1 0.956a 0.915 0.892 0.45234 2 0.956b 0.914 0.900 0.43458 3 0.946c 0.895 0.887 0.46237 a. predictors: (constant), service, gdp per capita, industry b. predictors: (constant), service, industry c. predictors: (constant), industry d. dependent variable: transport ghg the results of the analysis of the impact of the observed variables for cluster 2 are given in table 8 (prepared by the authors spss statistics 19.0). the results of regression analysis indicate that the share of industry in gdp has a statistically significant impact on transport ghg emissions. 522 n. petrović, n. bojović, m. petrović, v. jovanović table 8 the value of regression coefficients influence of gdp per capita, the shares of industry and of service in gdp on transport ghg emissions in cluster 2 model unstandardized coefficients standardized coefficients sig. b std. error beta 1 (constant) -87.660 85.547 0.328 gdp per capita -7.645e-6 0.000 -0.063 0.788 industry 1.865 0.878 2.653 0.057 service 1.125 0.881 1.767 0.228 2 (constant) -70.982 58.116 0.245 industry 1.705 0.634 2.426 0.020 service 0.946 0.574 1.487 0.125 3 (constant) 24.748 1.757 0.000 industry 0.665 0.063 0.946 0.000 the coefficient of determination shows how much of the variance of the dependent variable was explained by the model. the coefficient of determination is 0.976 (table 9) (prepared by the authors spss statistics 19.0). when expressed as a percentage, it can be concluded that the joint impact of the gdp per capita and of the shares of industry and of service in gdp on transport ghg emissions is 97.6 % in cluster 3. it is necessary to point out that the value of sig. is less than 0.05 (0.001), and then the contribution observed is statistically significant. table 9 the impact of the gdp per capita, the shares of industry and of service on the transport ghg emissions in cluster 3 model r r square adjusted r square std. error of the estimate 1 0.988a 0.976 0.970 0.24140 a. predictors: (constant), service, industry, gdp per capita b. dependent variable: transport ghg emissions table 10 the value of regression coefficients influence of gdp per capita, the shares of industry and of service in gdp on transport ghg emissions in cluster 3 model unstandardized coefficients standardized coefficients b std. error beta sig. 1 (constant) -92.016 18.680 0.000 gdp per capita 0.0001 0.000 0.377 0.013 industry 1.247 0.229 0.522 0.000 service 0.967 0.196 0.771 0.000 a. dependent variable: transport ghg emissions the results of the analysis of the impact of the observed variables for cluster 3 are given in table 10 (prepared by the authors spss statistics 19.0). the results of the regression analysis indicate that the gdp per capita as well as the shares of industry in gdp and of service in gdp have a statistically significant impact on transport ghg emissions. a study of the environmental kuznets curve for transport greenhouse gas emissions in the eu 523 the results of the regression analysis indicate that changes in the economic structure, i.e. changes in the industrial participation in gdp, as well as changes in the service sector participation in gdp, exert a significant influence on transport ghg emissions. the above research results confirm hypothesis h2. 5. concluding remarks the results of the cluster and correlation analysis have shown that the inverted-ushaped relationship exists between gdp per capita and transport ghg emissions in the highly developed countries as well as transition in the eu. while the highly developed countries record a decrease in transport ghg emissions with an increase in gdp per capita, the transition countries record an increase in transport ghg emissions with an increase in gdp per capita. the results of the regression analysis indicate the following: 1) an increase in the service sector participation in gdp leads to a decrease in transport ghg emissions in the countries belonging to cluster 1; 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it also succeeds in decreasing the height of microprofile irregularities by several times, increasing the surface layer microhardness and creating a substantial extent of compressive residual stresses. a number of authors [1-7] have studied finishing and hardening burnishing without taking into consideration the possibility of forming nanocrystalline structures. to promote burnishing the papers [8-16] bring into the foreground a finishing technique for forming a nanocrystalline structure in the surface layer, which puts to use the method of severe plastic shear deformation (spd), named nanostructuring burnishing (nsb). the nsb technique makes it possible to impart unique performance attributes to the surface layer, received march 30, 2019 / accepted june 02, 2019 corresponding author: viktor p. kuznetsov ural federal university, institute of new materials and technologies, 19 mira street, ekaterinburg 620002, russia e-mail: wpkuzn@mail.ru 162 v.p. kuznetsov, a.s. skorobogatov, v.g. gorgots and it can be implemented in the process of machining on modern lathes and machining centers. however, the influence of the indentor sliding velocity and the loading repetition factor on the evolution of spd as well as the formation of the nanocrystalline structure in the thin surface layer have not been studied in depth. zhao et al. [17] study the formation of a gradient nano/microstructured layer in a sample of pure copper; the layer is produced by severe plastic deformation with roller burnishing (sprb). the experimental results showed that the gradient layer had a thickness over 100 µm. the significantly increased hardness close to the surface resulted predominantly from decreasing the grain sizes. the nano-dimension grains were randomly oriented; most boundaries turned out to be low angle ones. it was in the nanostructure that dislocation activities prevailed, accompanied by rotating grains in the local area. roland et al. [18] describe a research study of the influence that a nanocrystalline surface layer exerts on fatigue properties of a 3161 stainless steel; this layer is obtained by applying mechanical attrition treatment to the surface (smat). qualitative surface layers containing ultra-fine grains or nano-grains can be formed in the process of cryogenic burnishing of an ai 7050-t7451 alloy with a roller tool [19]; it can be achieved by means of applied severe plastic deformation and dynamic recrystallization (drx) inherent in it. the article compares dry and cryogenic burnishing. it shows that refined layers containing nano-grains come into being in the surface that undergoes cryogenic treatment. the average hardness at a depth of 200 µm is higher after cryogenic burnishing than after dry burnishing by 9.5%, 17.5 % and 24.8 % at a rate of 25, 50 and 100 m/min, respectively. a number of authors [20-26] describe the research that has been done into effects of severe plastic deformation parameters on the dispersion of grain structures in structural materials. they show that for the formation of a nanocrystalline structure it is necessary that the shear strain intensity exceeding 2 and the shear strain rate being above 10 2 s -1 should be ensured. the present research aims at establishing a relationship of the indentor sliding velocity and an integral parameter of loading repetition factor and the shear strain intensity and shear strain rate in the surface layer of x20cr13 steel and the dispersion degree of the formed nanocrystalline structure. 2. materials and methods an experiment-calculated procedure is brought forward to determine the plastic shear strain intensity and rate in the surface layer. the procedure is based on measuring the length of the bulge, which is edged by the tool, applying 3d profilometry and the depth, to which shear strain propagates, by examining the deformed surface layer with scanning microscopy (fig.1). material shear strain  can be determined as a tangent of the shear angle or as a ratio of length lr of the edged metal bulge to thickness hs of the sheared layer: r stg l h    (1) the true plastic shear strain is calculated with the following relation: ln(1 ) ln(1 )r sl h      (2) impact of indentor sliding velocity and loading repetition factor on shear strain and structure... 163 fig. 1 location of relative deformation (a) and scanning microscopy of the deformed layer (b) the value of the true shear strain makes it possible to determine a relation of the mean strain rate to indentor sliding velocity vb and loading repetition factor nc relative to the elementary volume in the material. the duration of a single-time action of the indentor on the elementary volume is determined with a relation of contact spot length ac to sliding velocity vb: 1 c bt a v (3) the contact spot length value was determined by measuring width of single path of indenter made with a giving burnishing force as shown in [14]. the resulting spot length value is 370±6 μm. as the tool is fed, generally the indentor acts upon the elementary volume several times. to take the number of these actions into consideration v.p. kuznetsov’s paper [14] introduces an integral parameter of loading repetition factor nc, which is determined as follows: c c bn a f (4) the total deformation time for nc of the indentor’s actions is found as follows: 1 c c t c b a n t t n v   (5) thus, the mean shear strain rate in nanostructuring burnishing of the surface layer can be calculated in the following way: b t c c v t a n     (6) having found the value of accumulated edged material bulge lra and shifted layer thickness hs, it is now possible to evaluate accumulated shear strain intensity . to perform the present experiment-calculated procedure a scheme for burnishing circular paths on a flat sample with a high-precision stopping of the tool (fig. 2, a) is proposed. these kinematics of the tool lead to forming an accumulated bulge of plastically shifted material at the end of the path (fig. 2, b). 164 v.p. kuznetsov, a.s. skorobogatov, v.g. gorgots fig. 2 burnishing a path with a high-precision stopping (a) for forming an accumulated bulge of plastically edged material (b) the experimental study was done by machining x20cr13 steel samples with a 2mmradius aspm-3 polycrystalline diamond (pcd) indentor, the burnishing force being fb = 340 n. the paths were 10mm wide and 70mm long; three different feed rates were applied fb=0.06, 0.04 and 0.02 mm/rev at sliding velocities vb=3, 6, 8, 11, 14, 20 and 26 m/min. at these feed rates the loading repetition factor was, respectively, nc=4.2, 6.3 and 12.6. in this manner, within the framework of a full factorial experiment 21 paths were machined with a unique combination of feed and velocity parameters fb and vb. machining was done on an okuma ma600bii machining center. geometrical parameters of the bulges produced at each path were studied by optical 3d profilometry on a wyco nt1100 profilometer. after it, a cross-section metallographic sample was prepared from the middle of each path; these samples were examined by scanning electron microscopy with a zeiss auriga crossbeam to estimate the thickness of the sheared layer. the structural analysis was done by transmission electron microscopy with a jeol jem 2100. the foils were taken from the surface in the middle of each path. 3. result and discussions on the basis of the results obtained by optical 3d profilometry and scanning electron microscopy (fig. 3) the sought values of the plastically edged metal bulge length and the sheared layer thickness (table 1) were found. the results thus acquired show that as the sliding velocity rises to 11 m/min, an increase of the accumulated bulge length and the sheared layer thickness regardless of the loading repetition factor can be observed. at the same time, a further increase of the velocity leads to a reverse effect. what is more, the maximum thickness of the sheared layer practically does not correlate with the loading repetition factor and remains within the range from 6.29 to 6.63 µm. meanwhile, the length of the accumulated bulge of plastically edged material grows significantly when the loading repetition factor is increased from 272 µm at nc=4.2 to 348 µm at nc=12.5. inserting the results of the measurements in (2), (3) and (6), the relations of the plastic shear strain intensity and the plastic shear strain rate in the surface layer of x20cr13 steel were found (fig. 4). impact of indentor sliding velocity and loading repetition factor on shear strain and structure... 165 fig. 3 scanning electron microscopy of the surface layer in the paths produced at loading repetition factors of 4.2 (a, b); 6.3 (c, d) and 12.5 (e, f). table 1 results of measurements sliding velocity vb, m/min nc=4.2 nc=6.3 nc=12.5 bulge length lr, μm layer thickness hs, μm bulge length lr, μm layer thickness hs, μm bulge length lr, μm layer thickness hs, μm 3 40 5.1 48 3.24 81 3.66 6 78 5.1 42 5.54 59 5.22 8 229 5.75 183 5.58 174 5.65 11 272 6.29 277 6.47 348 6.63 14 204 6.35 202 6.42 227 5.87 20 179 5.99 165 5.76 205 5.7 26 175 6.04 169 6.08 201 6.01 fig. 4 relation of the plastic shear strain intensity (a) and the plastic shear strain rate (b) to the sliding velocity when burnishing at feed rates fb=0.02 mm/rev (nc=12.5), 0.04 mm/rev (nc=6.3) and 0.06 mm/rev (nc=4.2) 166 v.p. kuznetsov, a.s. skorobogatov, v.g. gorgots the maximum shear strain intensity =3.75…4 is obtained when burnishing is done at an indentor sliding velocity of 11 m/min. at lower velocities the shear strain intensity decreases considerably. at the same time, raising the velocity leads to stabilizing the shear strain intensity at 3.4…3.6. these results turn out to agree well with the experimental data acquired in the papers [14] which describe a similar correlation of the sliding velocity to the microhardness in the surface layer. besides, the values of the shear strain intensity are confirmed by mathematic simulation presented in the paper [12]. another conclusion can be drawn that at higher sliding velocities a surface layer shear strain comparable in regard to its intensity may be obtained at substantially higher shear strain rates. fig. 5 transmission electron microscopy of the surface layer in the paths obtained at loading repetition factors of 4.2 (a, b, c); 6.3 (d, e, f) и 12.5 (g, h, i). impact of indentor sliding velocity and loading repetition factor on shear strain and structure... 167 the microstructural analysis of the foils showed that the nanocryslalline structure evolves with various dispersion degrees depending on different combinations of the parameters of the sliding velocity and loading repetition factor that are under study. in addition to it, a relation can be traced between the dispersion of the structure being formed and the shear strain intensity in the material of the surface layer. for instance, a composite ufgand nanocrystalline structure develops at a sliding velocity of 6 m/min and shear strain intensities below 2.8 regardless of loading repetition factor values (fig. 5, a, d, g). in the cases when the velocity is 11 m/min and the shear strain intensity reaches 3.75…4. a homogeneous nanocrystalline structure is observed in the surface layer; it is proved by a circular shape of the interferogrammas (fig. 5, b, e, h). if the velocity is increased to 20 m/min, the dispersion of the structure significantly decreases as a result of the shear strain intensity going down (fig. 5, c, f, i). 4. conclusions notwithstanding considerable differences in the shear strain intensity and the shear strain rate, each combination of the parameters of the loading repetition factor and indentor sliding velocity within the framework of the research meets the conditions necessary for a nanocrystalline structure to form (≥2,  ≥10 2 s -1 ). the research proves that higher values of the shear strain intensity and rate are achieved in nanostructuring burnishing. this appears to be stimulating for obtaining higher dispersion degrees of the grain structure and ensures formation of a homogeneous surface layer. the results thus obtained explain the differences in the microhardness of nanostructured surfaces occurring in the cases when the sliding velocity is altered; these differences were found earlier in the papers [13, 15]. it is worth mentioning that the proposed procedure enables us only to determine certain mean values of the shear strain intensity and the shear strain rate in the edged layer. actually, the shear strain intensity is expected to be higher on the surface and to gradually go down reaching zero on the boundary of the edged layer. from the point of view of controlling nanostructuring the knowledge of how the shear strain intensity and the plastic shear strain rate are distributed across the edged layer is of great scientific value. to determine it experimentally it is necessary 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treatment of cross-bite and deficiency of transversal dimension of the maxilla in patients with and without cleft of palate and lip. for this procedure, generally, different orthodontic appliances and devices generating significant transversal forces are used. the aim of this study is the finite-element analysis of stresses and displacements of the skull without palate cleft and the skull with unilateral and bilateral cleft after activation of the hyrax orthodontic device. two different constructions of the orthodontic device hyrax with different positions of the screw relative palate are considered. in the first case, the screw is in the occlusal horizontal plane, and in the other, the screw is located near the palate. activation of the orthodontic device corresponds to the rotation of the screw on one-quarter turn. it is established that the screw position significantly affects the distributions of stresses in skull and displacements of the cranium without palate cleft and with unilateral or bilateral palate cleft. stresses in the bone structures of the craniums without cleft and with cleft are transferred from the maxilla to the pterygoid plate and pharyngeal tubercle if the screw displaces from the occlusal plane to the palate. depending on the construction of the orthodontic appliance, the maxilla halves in the transversal plane are unfolded or the whole skull is entirely rotated in the sagittal plane. the stresses patterns and displacements of the skull with bilateral palate cleft are almost unchanged after activation of the orthodontic devices with different positions of the screw, only magnitudes of stresses and displacements are changed. the obtained results can be used for design of orthodontic appliances with the hyrax screw, as well as for planning of osteotomies during the surgical assistance of the rapid maxillary expansion. key words: unilateral palate cleft, bilateral palate cleft, hyrax screw, stresses pattern, displacements received december 10, 2016 / accepted may 03, 2017 corresponding author: sergei bosiakov belarusian state university, department of theoretical and applied mechanics, nezavisimosti avenue 4, 220030 minsk, 4, belarus e-mail: bosiakovsm@gmail.com 518 s. bosiakov, a. vinokurova, a. dosta 1. introduction cross bite is the most common transversal anomaly of the dentition interposition requiring a prolonged active treatment. its frequent cause is violation of maxilla growth, reduced chewing function or chewing on one side, as well as congenital cleft palate. among the defects of the craniofacial complex and maxilla, according to the world health organization, the congenital cleft of the lip and the palate are dominant. rapid maxillary expansion (rme) is one of the treatment stages of transversal discrepancy of maxilla. fixed orthodontic devices (appliances), such as devices with the hyrax screw or palatal distractors are recommended for the rme [2, 18]. among various types of orthodontic appliances the most hygienic ones are hyrax devices. moreover, devices of this type are less traumatic and more comfortable for the patients besides having a low incidence of complications after application. at the same time, the design features of the orthodontic appliances for rme, including hyrax devices, significantly affect the intensity and nature of motions of the skull bone structures and teeth [8, 9, 19, 23]. the finite-element (fe) modeling is the main approach to understanding the influence of rme on the loads distribution in the cranium. the fe analysis of the orthodontic loads distributions during rme of the craniofacial complex without palate cleft was performed in [3, 6, 7, 10-13, 17]. an extensive review of the fe calculations of stresses and displacements of the maxillary complex under the action of different types of the orthodontic appliances was carried out in the recent study [13]. it should be noted that the common simplifying assumption adopted in the above-mentioned studies and other similar research projects is a simulation of the orthodontic appliance impact on the skull bone structures and teeth by means of the application of transversal displacements or forces to the anchor teeth. the fe analysis of rme effect on skull with the unilateral cleft palate was carried out in [16]. in this study the distributions of transversal forces in the craniofacial complex with the unilateral palate cleft were evaluated as well as their influence on the displacements of the naso-maxillary bones. to simulate the clinical situation the displacements of 5 mm in transversal plane to the maxilla premolars and first molars were applied. assessment of the rme impact for patients with unilateral cleft palate using the fe method was performed in [4]. this study was a preliminary step in the development of surgical techniques for rme assistance. in accordance with [4], for effective rme of the skull with unilateral cleft the osteotomy of the median palatal suture and of the lateral buttress is usually required for an effective maxillary expansion. the fe biomechanical analysis of the rme effect on the craniofacial complex in patients with unilateral cleft lip and palate was carried out in [21]. it is assumed that the orthodontic device is a rigid body [21]. to simulate the clinical situation the symmetrical displacements of the anchor teeth corresponding to a certain number of the orthodontic devices the screw turns were employed in [21]. in [21], it was noted that the similar simplifying assumptions were assumed in other analogous studies [10, 11, 17]. in [21] the fe evaluations of stresses were carried out without regard to the periodontal ligament, since due to the action of orthodontic appliance the anchor teeth completely overlay the periodontal gap. one of the conclusions of [21] was that the magnitudes of displacements on the normal side of skull are different from those on the skull side with the palate cleft. in accordance with [21] this may be caused by an asymmetrical disposition of the skull bone structures. one of the few fe studies on the comparative analysis of the stress distributions in the skull without cleft, in the skull with unilateral cleft and with craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 519 bilateral cleft is [5]. in [5] action of the orthodontic appliance quad-helix on the maxilla was simulated without periodontal ligament. according to [5], the smallest expansion is observed between the control points for the skull without palate cleft, and the difference between the stress distributions in the skull without cleft and in the skull with cleft (unilateral or bilateral) during rme is quite substantial. the aim of this study is a comparative analysis of the stresses patterns and displacements of bone structures for the skull without palate cleft and for the skulls with unilateral and bilateral palate cleft after activation of the hyrax orthodontic device during rme. the feature of the present study is that the orthodontic device is a deformable rod construction with different dispositions of the screw relative to the palate. therefore, another objective is to evaluate the effect of the screw location on the stresses and displacements distributions in the craniofacial complex with and without palate cleft. fe method was used to reach these goals. 2. fe modeling of maxillary expansion 2.1. solid models of skull, hyrax device and anchor teeth stereolithography (stl) model of the skull is developed using mimics 14.12 (materialise bv, leuven, belgium) on the basis of 210 tomographic images of the dry cadaveric intact skull of an adult with a well-preserved alveolar bone and teeth. models of the first and the second upper premolars (14, 15 and 24, 25 teeth), and the permanent molars (16 and 26 teeth) are also generated based on the tomographic data. model of hyrax orthodontic device is developed via solidworks 2010 (solidworks corporation, usa). four rods of orthodontic device are affixed to the plates with screw and to the crowns; another two rods are affixed to the crowns and are impacted on the second premolars (15 and 25 teeth). simulation of the periodontal ligament is not carried out since the periodontal ligament has almost no effect on the stresses distribution in the craniofacial bones [22]. cranium sutures are not accounted in the fe model. this is because in the adult human skull the sutures are partially or fully ossified [1]. 2.2. boundary conditions. geometrical and material parameters the fe skull model is fixed in the nodes located around of the foramen magnum [10, 13, 17]. displacement of each plate of the orthodontic device is directed only transversely (along x-axis). boundary conditions are indicated in fig. 1. two constructions of the orthodontic device are considered: construction with disposition of rods and screws in a single horizontal (occlusal) plane (model 1); construction with plates and screw shifted closer to the palate relative to the horizontal plane on 8 mm (model 2). the geometrical dimensions of the orthodontic devices are identical, with the exception of lengths of the rods fastening the plates with first premolars and molars. models 1 and 2 are fixed on the anchor teeth of the skull without cleft (swc), on the skull with a unilateral cleft (sulc) and on the skull with bilateral cleft (sblc). elastic properties of materials for orthodontic device, skull bones and teeth are given in table 1. 520 s. bosiakov, a. vinokurova, a. dosta table 1 mechanical properties of the materials material elastic modulus, gpa poisson’s ratio steel (orthodontic device) 200.0 0.3 cortical bone [20] 13.7 0.3 trabecular bone [20] 8.0 0.3 teeth [20] 20.7 0.3 fig. 1 boundary conditions for the fe skull model (a is front view, b is view from below): the boundary conditions, marked by a and c correspond to displacements of orthodontic device plates in transversal direction (along the x-axis); boundary condition, marked by b corresponds to skull fixing in nodes around foramen magnum the lengths of the rods for models 1 and 2 between the device plates and the crowns vary from 8.15 mm to 11.05 mm and 12.20 mm and 16.45 mm, respectively. the length and width of the plates for models 1 and 2 are 10.0 mm and 4.0 mm, respectively; the cross-sectional radius of the rods is equal to 1.0 mm, thickness of crowns is 0.2 mm. 2.3. parameters of fe models the fe models of the skull are developed after processing the stl-model in 3-matic 6.1 mimics. discrete skull model is converted via fe modeler of ansys workbench14 (ansys inc., usa). fe meshing is made in automatic mode (solid72 type elements are used). the number of elements and nodes for the fe models of skulls, anchor teeth and orthodontic devices are given in table 2. table 2 parameters of the fe models model node number element number swc 77036 185302 sulc 24556 85087 sblc 24494 85138 model 1 15918 7798 model 2 16410 8022 craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 521 contacts between the crowns and the teeth, as well as between the maxilla and the teeth are 'bonded' type without sliding and mutual penetration. 3. fe analysis of craniofacial stresses the fe analysis of stresses (von mises) and displacements for swc, sulc and sblc is carried out after activations of models 1 and 2 by means of the transversal displacements of the orthodontic device plates. transversal displacement of each plate of models 1 and 2 is 0.2 mm (corresponding to the activation of the orthodontic device screw on a quarter turn [2, 8, 15]). 3.1. skull without cleft the stress patterns in swc after activations of models 1 and 2 are depicted in figs. 2 and 3, respectively. in figs. 2-7 the magnitudes of stresses are given in mpa. fig. 2 shows that the sufficient stresses in swc after activation of model 1 appear mostly in the maxilla. high stresses occur in the middle and bottom of the nasal cavity and the bottom of the left orbit. the stresses near only one orbit can be explained by the asymmetry of the craniofacial complex and asymmetric disposition of points of rods fixing on the crowns. the higher stresses in the left infraorbital foramen in comparison with the right one (see fig. 2, a) are also obviously caused by this asymmetry. fig. 2 stress patterns in swc after activation of model 1: а is pattern in front of skull; в is pattern in base of skull stress distribution in swc after activation of model 2 significantly changes when compared to model 1. fig. 3 shows that the maxilla is loaded partially and the highest stresses are reduced approximately to 16.10 mpa. the region with nonzero stresses in the anterior part of swc after activation of model 2 displaces from the maxilla to the nasal cavity and zygomatic process. in the median palatine suture region there are almost no stresses; they only appear in the incisive bone (see fig. 3, b). at the same time, the stresses are observed in the occipital bone near the foramen magnum and in the pterygoid plate, but almost no stresses exist in the median palatine suture. 522 s. bosiakov, a. vinokurova, a. dosta fig. 3 stress patterns in swc after activation of model 2: а is pattern in the front of the skull; в is pattern in the base of the skull the highest stresses in swc after activation both of models 1 and 2 occur in the alveolar bone surrounding the anchor teeth (see figs. 2, b and 3, b). nonzero stresses also appear in the zygomatic arches after activation of models 1 or 2. if the screw of the orthodontic device is displaced to the palate, the region with nonzero stresses of the zygomatic arches increases. 3.2. skull with unilateral cleft the palate cleft is complete and passes on the level of second maxilla incisor. figs. 4 and 5 depict the stresses patterns in sulc after activations of models 1 and 2, respectively. fig. 4 stress patterns in sulc after activation of model 1: а is pattern in front of skull; в is pattern in base of skull fig. 4 shows that after activation of model 1 high stresses occur in the maxilla bone, particularly in the zygomatic processes of the maxilla below the infraorbital foramen. craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 523 high stresses appear in the regions of the nasal cavity and orbits, in the bone of frontal process of the maxilla on the cleft side, in the nasal bones as well as in the regions of the fronto-nasal, inter-nasal and inter-nasal-maxillary sutures. the high stresses in the base of the skull are distributed through the lateral and medial pterygoid plate to the pharyngeal tubercle. stresses also occur in the region surrounding the silcus of the auditory tube. this indicates that the rme can have a significant impact on the increase of the nasal cavity dimension and improve nasal breathing [1, 11, 12, 16, 23] just as it leads to changes of the auditory conductivity in patients with cleft palate. the short-term and long-term impacts of maxillary expansion on the auditory conduction are described in [14]. fig. 5 stress patterns in sulc after activation of model 2: а is pattern in front of skull; в is pattern in base of skull figs. 4 and 5 indicate that the stresses patterns in the anterior part of sulc slightly differ after activations of models 1 and 2. however, the maximum stresses in sulc after the activation of model 2 are significantly lower than the maximum stresses in sulc after activation of model 1. at the same time, high stresses occur in the zygomatic process after activation of both models 1 and 2. significant stresses appear at the skull base in the foramen magnum region after activation of model 2 (see fig. 5, b). in the palatal region of the maxilla after activations of models 1 and 2 there are almost no stresses, with the exception of small regions of the alveolar processes. 3.3. skull with bilateral cleft the bilateral palate cleft is complete and passes at the level of the second maxilla incisors on the left and the right sides of the skull. figs. 6 and 7 depict the stresses patterns in sblc after activations of models 1 and 2. it is seen from fig. 6, a and fig. 7, b, the stresses patterns in the anterior part of sblc after activations of models 1 and 2 are almost the same. at the same time, the magnitudes of stresses in sblc vary significantly in dependence on model 1 or model 2. after activation of model 1 the maximum stresses are approximately equal to 24.0 mpa, while after activation of model 2 the maximum stresses are approximately equal to 1.55 mpa. the highest stresses in sblc after activations of models 1 and 2 are observed in the zygomatic 524 s. bosiakov, a. vinokurova, a. dosta and alveolar processes of the maxilla and in the zygomatic bone. also, the high stresses in sblc compared with swc and sulc after activation of model 1 occur in the sphenoid and nasal bones, as well as in the frontal processes of the maxilla. fig. 6 stress patterns in sblc after activation of model 1: а is pattern in front of skull; в is pattern in base of skull fig. 7 stress patterns in sblc after activation of model 2: а is pattern in front of skull; в is pattern in base of skull the stress patterns are almost the same in the base of sblc after activation of models 1 or 2 (see fig. 6, b and fig. 7, b). however, the region with the nonzero stresses of the pharyngeal tubercle is larger after activation of model 2 compared with model 1. the magnitudes of the stresses in sblc after activation of model 2 (maximal stresses approximately are 1.21 mpa, see fig. 7, b) are significantly less than the stresses in sblc after activation of model 1 (maximal stresses approximately equal to 16.0 mpa, see fig. 6, b). craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 525 4. fe analysis of displacements 4.1. intact skull the vector fields of the total displacements and the distributions of displacements along the coordinate axes for swc points after activations of models 1 and 2 are depicted in figs. 8 and 9. displacements along the x-, yand z-axes are the transversal, sagittal and vertical displacements, respectively. the magnitudes of displacements in figs. 8-13 are given in mm. fig. 8 displacements of swc points after activation of model 1: a is vector field of total displacements; b is distribution of transversal displacements; c is distribution of sagittal displacements; d is distribution of vertical displacements it is seen from fig. 8 that the two maxilla halves after activation of model 1 turn relative to the horizontal axis passing approximately through the nasal aperture region and parallel to the y-axis. the highest components of the total displacements are transversal and vertical displacements (see fig. 8, b and d). the vertical displacements of swc 526 s. bosiakov, a. vinokurova, a. dosta points in the region of the anterior incisors and nasal aperture are directed downwards, while the displacements of rest of the skull are directed upwards. the sagittal displacements of swc points after activation of model 1 are the smallest of the three components of total displacements (see fig. 8, c). the upper part of swc and anterior region of the maxilla after the activation of model 1 are moved backwards in the horizontal direction, while the rest of the maxilla and zygomatic arches slightly moves forwards. it is seen from fig. 9 that the total displacement of swc points after activation of model 2 are directed, basically, along z-axis so that the anterior part of swc is moved downwards, while the posterior part of swc is moved upwards, which leads to rotation of swc counterclockwise (relative to the positive direction z-axis). this conclusion is confirmed by the distribution of vertical displacements (see fig. 9, d). fig. 9, a and fig. 9, d indicate that the maximal (in absolute value) vertical displacements of swc points after activation of model 2 are directed downwards in the anterior part. such direction of the vertical displacements corresponds to the distribution of the sagittal displacements (see fig. 9, c). the displacements distributions in fig. 9 show that swc after activation of model 2 is rotated relative a horizontal axis without passing through the cranium itself. axis of swc rotation is located in the region of the foramen magnum and pharyngeal tubercle and parallel to the x-axis. 4.2. skull with unilateral cleft the vector field of the total displacements and distributions of the transversal, sagittal and vertical displacements of sulc points after activation of models 1 and 2 are shown in figs. 10 and 11. fig. 10, a and fig. 10, b show that the transversal displacements are the largest components of sulc total displacements after activation of model 1. the vector field of the total displacements and the transversal displacements distribution are almost symmetrical (see fig. 10, a and fig. 10, b). for instance, the highest magnitudes of oppositely directed transversal displacements are approximately equal to 0.204 mm and 2.0 mm respectively. fig. 10, c and fig. 10, d show that the distributions of the sagittal and vertical displacements of sulc points after activation of model 1 are almost symmetrical as well. the maximum values of these displacements differ slightly. distribution of sagittal displacement shows that the frontal bone deflects backwards, while the maxillary region containing the inter-maxillary suture and the cleft is moved forwards (see fig. 9, c). the vertical displacements of sulc anterior part are directed upwards, while the displacements the points on sulc posterior part are directed downwards (see fig. 10, d). the largest vertical displacements are observed for points of the zygomatic processes and the zygomatic bones. the maximal total displacement of sulc points after activation of model 2 and model 1 (see fig. 10, a and fig. 11, a) are almost the same. however, basically, the total displacements of sulc points after activation of model 2 are directed downwards unlike the directions of the total displacements of sulc points after activation of model 1. craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 527 fig. 9 displacements of sulc points after activation of model 2: a is vector field of total displacements; b is distribution of transversal displacements; c is distribution of sagittal displacements; d is distribution of vertical displacements after activation of model 2 the maximal transversal displacements (in dependence on direction) are equal to 0.0684 mm and 0.0536 mm (see fig. 11, b), which is significantly less than the maximal transversal displacements of sulc points after activation of model 1. distributions of sagittal displacement of sulc points after activation of models 1 (fig. 10, c) and 2 (fig. 11, c) significantly differ from each other. according to fig. 11, c, the sulc anterior part is moved backwards while the bone structures of maxilla and zygomatic bone are moved forwards. the sagittal displacements direction of sulc points indicates its anti-clockwise rotation (from the positive direction of the x-axis). this is confirmed by the distribution of the vertical displacements (see fig. 11, d). according to fig. 11, d, the facial and occipital parts and of sulc are moved downwards and upwards, respectively. considering symmetrical distribution of the sagittal and vertical displacements, it can be concluded that after activation of model 2 sulc is rotated in yzplane relative horizontal axis located approximately above the foramen magnum and parallel to the x-axis. 528 s. bosiakov, a. vinokurova, a. dosta fig. 10 displacements of sulc points after activation of model 1: a is vector field of total displacements; b is distribution of transversal displacements; c is distribution of sagittal displacements; d is distribution of vertical displacements note that the after activations of models 1 and 2 the transversal displacements of sulc points on the side with cleft are larger than the displacements of the normal side. distributions of the transversal displacements of sulc (see fig. 10, b and fig. 11, b) also indicate asymmetry of the cranium displacements. this is consistent with the results of the [16] that the displacements of side of the skull with cleft are larger than those of the normal side. 4.3. skull with bilateral cleft the vector field of the total displacements and distributions of the transversal, sagittal and vertical displacements in sblc after activations models 1 and 2 are shown in figs. 12 and 13. craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 529 fig. 11 displacements of sulc points after activation of model 2: a is vector field of total displacements; b is distribution of transversal displacements; c is distribution of sagittal displacements; d is distribution of vertical displacements it is seen from figs. 12, a and 13, a, that the directions of total displacements of sblc points are almost identical after activations of both models 1 and 2, while the magnitudes of the total displacements for these two cases are significantly different. the maximal component of the total displacement after activations of models 1 and 2 is transversal. the region of sblc with the inter-maxillary suture does not move in the horizontal direction after activation of model 1. the transversal displacements of this region after activation of model 2 are very small (see fig. 13, b). sagittal and vertical displacements of sblc points are significantly less than transversal displacements after activation of models 1 and 2. directions of the sagittal and vertical displacements indicate that there is a slight rotation of sblc in yz-plane clockwise (relative to the positive direction of the x-axis); the angle of sblc rotation is larger after activation of model 1 compared with model 2. the sblc rotation axis after activation of model 2 is located in the foramen magnum region, and it is slightly displaced towards the occipital bone. 530 s. bosiakov, a. vinokurova, a. dosta fig. 12 displacements of sblc points after activation of model 1: a is vector field of total displacements; b is distribution of transversal displacements; c is distribution of sagittal displacements; d is distribution of vertical displacements 5. discussion higher stresses in swc, sulc sblc occur after activation of model 1 compared with model 2. the smallest difference (in 1.86 times) between stresses after activations of models 1 and 2 is observed in sulc. the highest difference (in 15.5 times) between the stresses is in sblc after activations of models 1 and 2. it should be noted that very low stresses appear in sblc after activation of model 2 (maximal magnitude of stresses is equal to 1.55 mpa). the obtained results indicate that the regions of maximal stresses in swc, sulc and sblc, regardless of the orthodontic device model, occur in the alveolar processes of the maxilla and in the zygomatic bone (except swc after activation of model 2). displacement of the orthodontic device screw to the palate leads to redistribution of stresses in swc from the maxilla to the nasal cavity and to the foramen magnum. the most complicated distribution of stresses is observed in sulc after the activation of models 1 and 2. in these craniofacial stress patterns and displacements after activation of hyrax device: finite element modelling 531 cases, sufficiently high stresses propagate to many bone structures. following displacement of the orthodontic device screw to the palate, the stresses in swc and sulc are transferred to the foramen magnum and the pharyngeal tubercle. the orthodontic device construction has almost no effect on the stresses distribution in sblc bone structures. stresses in the sphenoid bone occur in sulc as well as in sblc after activation of model 2. this indicates the expediency of the osteotomy for reducing of resistance of the sphenoid bone (and the sphenoid plate) during rme. fig. 13 displacements of sblc points after activation of model 2: a is vector field of total displacements; b is distribution of transversal displacements; c is distribution of sagittal displacements; d is distribution of vertical displacements displacements of sblc and swc points after activation of model 1 are some of the smallest. the highest total displacements are observed in sblc after activation of model 1. also, the high displacements of swc points are observed after activation of model 2. the maximal displacements of sulc points are approximately the same after activations of both models 1 and 2. at the same time, the transversal displacements of swc points after activation of model 1 can reach more than 91% of its total displacement, and 95% of the total displacements of sulc and the sblc. the highest transversal, sagittal and 532 s. bosiakov, a. vinokurova, a. dosta vertical displacements are 11%, 92% and 97% of the maximal total displacements, respectively. this indicates that swc moves forwards and downwards after the activation of model 2. similar ratios (13%, 74%, and 92%) between the maximal transversal, sagittal, vertical displacements and maximal total displacements of sulc points are observed after activation of model 2. sblc points displace significantly in the transversal direction after activations of models 1 and 2. magnitudes of the transversal displacement of some sblc points coincide with the total displacement of the same points; the highest sagittal and vertical displacements are not more than 5% of the maximum total displacements. acknowledgements: this paper is the result of the implementation of the project: «trans-atlantic micromechanics evolving research: materials containing inhomogeneities of diverse physical properties, shapes and orientations» 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pp. 1831-1838. 23. zimring, j.f., isaacson, r.j., 1965, forces produced by rapid maxillary expansion iii. forces present during retention, angle orthodontist, 35 pp. 178-186. 12156 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 615 630 https://doi.org/10.22190/fume231011044z © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic algorithms in servo system control iuliu alexandru zamfirache1, radu-emil precup1,2, emil m. petriu3 1politehnica university of timisoara, department of automation and applied informatics, timisoara, romania 2romanian academy – timisoara branch, center for fundamental and advanced technical research, timisoara, romania 3university of ottawa, school of electrical engineering and computer science, ottawa, canada abstract. this paper carries out the performance analysis of three control system structures and approaches, which combine reinforcement learning (rl) and metaheuristic algorithms (mas) as representative optimization algorithms. in the first approach, the gravitational search algorithm (gsa) is employed to initialize the parameters (weights and biases) of the neural networks (nns) involved in deep qlearning by replacing the traditional way of initializing the nns based on random generated values. in the second approach, the grey wolf optimizer (gwo) algorithm is employed to train the policy nn in policy iteration rl-based control. in the third approach, the gwo algorithm is employed as a critic in an actor-critic framework, and used to evaluate the performance of the actor nn. the goal of this paper is to analyze all three rl-based control approaches, aiming to determine which one represents the best fit for solving the proposed control optimization problem. the performance analysis is based on non-parametric statistical tests conducted on the data obtained from real-time experimental results specific to nonlinear servo system position control. key words: reinforcement learning, policy iteration, actor-critic, q-learning, gravitational search algorithm, grey wolf optimizer received: october 11, 2023 / accepted november 25, 2023 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania romanian academy – timisoara branch, center for fundamental and advanced technical research, bd. mihai viteazu 24, 300223 timisoara, romania e-mail: radu.precup@aut.upt.ro 616 i. a. zamfirache, r.-e. precup, e. m. petriu 1. introduction the gap between the machine learning (ml) and the automatic control (ac) fields is becoming a highly interesting research subject in recent years. the main reason behind this interest is the lack of automation in identifying the optimal parameters of controllers specific to ac systems. if the parameters of the controllers are not tuned systematically, one of the main shortcomings is their manual tuning, where a lot of experience is needed, but it is difficult to acquire it if no accurate process models are available or few information on the process is available as well. the optimal values of controller parameters are usually found through manual experiments and observations which take time and other resources. this leads to the need to conduct automatically the optimal tuning of controller parameters. the goal is to start the control and, based on the feedback collected from the controlled process, to automatically adjust (namely tune) the controller parameters so that the process achieves some predefined objective assessed in terms of adequately defined performance indices involved in performance specifications. reinforcement learning (rl) has been considered initially to belong to the field of ml, and it uses only information from interaction with the environment where this technique actually operates. as highlighted in [1], this specific feature also makes ml belong to the field of data-driven model-free control as part of ac. the general rl problem is usually formulated using the mathematical framework of markov decision processes (mdps) to solve an optimization problem that specifies the performance specifications, and it makes use of dynamic programming (dp) to solve that optimization problem at hand. rl operates with agents, which carry out actions in the environment; using the received reward signal (measuring the immediate effect of rl agent’s actions), the rl agents adjust their knowledge about themselves and the environment. as shown in [2], by applying this process incrementally, the rl agents will become better in setting actions that maximize or minimize the rewards. the techniques specific to rl represent good candidates to solve optimal reference tracking problems [3] and fill the gap between ml and ac. in this context, the rl agent is included in a control system, i.e. the rl plays the role of a controller, it automatically learns how to change the values of its parameters in controlling a process using the feedback (or reward) received from the controlled process [4]. besides the agent, the environment and the reward signal, the other important elements of an rl problem are the policy function used by the rl agent to generate actions to be executed on the environment, and the value function that measures the long term effect of the actions. based on the policy function, three types of rl agents are generally used leading to three different rl-based control system structures: value-based, policy-based and actorcritic. the value-based agent iteratively learns a value function in terms of the technique called value iteration (vi) in rl, and stores the value of the state vector or the state-action pair. a concrete implementation of the vi approach is the q-learning technique described in this paper where the goal is to iteratively find the q-function, i.e. the value function. the policy-based agent learns a policy function which is incrementally improved and used to generate actions, a technique is known as policy iteration (pi) in rl. the actor-critic agent learns both the value function (called the critic) and the policy function (called the actor). as shown in [2], all types of rl agents contain a policy function to define their behavior. deep reinforcement learning (drl) is obtained as a combination of rl and deep learning (dl) [2]. one version of drl-based control system structure treated in this paper is deep q-learning (dql) obtained by merging the dl and the q-learning algorithm that q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 617 belongs to the value-based structure [5]. the second rl structure treated in this paper is the policy-based structure, namely the pi rl-based control. the third rl structure used for comparison in this paper is the actor-critic framework, which represents a combination of dql and pi. all these rl-based structures make use of neural networks (nns) and their orientation to control leads to nn-based control system structures. metaheuristic algorithms (mas) are optimization algorithms that have been employed recently in order to mitigate the two major drawbacks of the widely used gradient descent (gd) algorithm to train nns, namely slow convergence which often leads to relatively poor control system performance and impossibility to avoid local optima which can lead to unstable and unreliable systems. this is the motivation of the research results presented in this paper. the most popular mas, also called nature-inspired optimization algorithms, involved in ac and rl problems are the gravitational search algorithm (gsa), the particle swarm optimization (pso) and the grey wolf optimizer (gwo) algorithms. gsa, pso and gwo are also viewed as swarm intelligence algorithms. all these algorithms are used in [6] to carry out the optimal tuning of fuzzy controller parameters. an analysis on the usage of mas in the optimal control of industrial applications is conducted in [7]. a brief discussion on the combinations of mas and rl in the close relation to nn training is given as follows. genetic algorithms (gas) are presented in [8] and [9] as an alternative to the gd algorithm-based training of nns, and other gas and other mas are reported in [10]. the hyperparameters involved in rl are optimally tuned in [11] via pso. the pso algorithm is combined with rl in [12] to train the critic in a fault tolerant tracking actorcritic rl-based control system structure, in [13] to design fault tolerant ac of nonlinear processes, to design fuzzy interpretable rl policies in [14], to solve noisy optimization problems in [15], to build a swarm of rl agents which work together for better performance in [16], and to find optimal actions in [17]. a q-learning is applied in [18] in combination with pso to design optimal paths for mobile robots, in [19] to improve the performance of positioning strategies for underwater vehicles, and in [20] to enhance the quality of predictions of ground responses with respect to tunneling. gwo is employed in training feed-forward multi-layered nns in [21]. dql is combined with a gsa and compared to gwo and pso algorithms in carrying out the nn-based optimal reference tracking control of servo systems in [22]. pi rl is combined with a gwo algorithm and compared to a pso algorithm in the same nn-based optimal reference tracking control problem (ortcp) in [23]. a combination of actor-critic rl and gwo, aiming to build a control structure where the progress of the nn-based actor is guided by a gwo-based critic as provided in [24]. the approaches presented in [22-24] are accompanied by comparisons with the classical gd algorithm. building upon the short discussion of the state-of-the-art presented above and on the authors’ results given in [22-24], this paper carries out the performance analysis of three fresh nnand rl-based control system structures and approaches, (i), (ii) and (iii), which combine rl and mas. the approach (i): the ma is employed to initialize the parameters (namely, the weights and the biases) of the nns involved in dql by replacing the traditional way of initializing the nns based on random generated values. the approach (ii): the ma is employed to train the policy nn in pi rl-based control. the approach (iii): the ma is employed to act as a critic in an actor-critic setup, guiding the progress of the nn-based actor. this contribution is advantageous and important in the context of the 618 i. a. zamfirache, r.-e. precup, e. m. petriu literature discussed above as the discussed structures and approaches ensure the relatively simple and transparent mitigation of the gd algorithm ensuring performance enhancement. seven combined algorithms are discussed, namely gsa, gwo and pso in (i) and gwo and pso in (ii) and (iii). these algorithms are also compared to the popular gd algorithm. the comparison is based on the real-time results and data obtained from experiments concerning the position control of a nonlinear servo system, and all these experiments were conducted in the authors’ laboratories. the authors avoided using too many abbreviations, which makes the readability of the paper poor. however, several abbreviations are used in order to keep a reasonable length of the paper. the paper treats the following topics: the rl-based control problems and the control system structures are described in the next section and the combined rl-mas are briefly presented in section 3. the performance comparison expressed in terms of non-parametric statistical tests conducted on experimental results is given in section 4, and the concluding remarks are pointed out in section 5. 2. reinforcement learning-based control problems and control system structures both in rl and in control theory, in order for an agent to learn how to automatically control a system, it needs to come up with a solution to an optimization problem. the three approaches (i), (ii) and (iii) mentioned in the previous section will be briefly treated as follows in sections 2.1, 2.2 and 2.3, respectively. 2.1. optimal reference tracking control problem in the first approach this approach, (i), focused on dql, is expressed as the optimization problem [22] ),(minarg 1 * aa aa j d = (1) where a is the vector (represented as a column matrix) that contains the sequence of actions on all iterations ,)]( ... )( ... )1([ max max tt d tataa =a (2) max...1 ttd = is the discrete time moment, maxt is the maximum number of iterations, )(1 aj is the cost function, a* is the optimal value of a, namely the solution to the optimization problem defined in eq. (1), it consists of the optimal sequence of elements, i.e. actions )(* dta , max...1 ttd = : ,)]( ... )( ... )1([ max max **** tt d tataa =a (3) and ad is the feasible domain of a. the definition of the cost function in eq. (1), which imposes the performance specifications as the aggregate objective of carrying out a tradeoff to small overshoot and small settling time values, is [22] q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 619 |,))(,1())(,(||))(,(|))((1 ddddoddd ttettettetj aaaa −−+= (4) where o is the weight parameter involved in the tradeoff. the dql-based control system structure is illustrated in fig. 1. according to fig. 1, which is actually a state feedback control system structure, the rl agent generates control actions )( dta (they are control signals )()( dd tatu = in the ac formulation) based on the new state (vector) of the controlled process )1( +dts and on the reward )( dtr (it is the controlled output )()( dd trty = in the ac formulation), reflected in the control error )( dte , which depends on dy – the desired output (or the reference input or the set-point) assumed to be constant. the new control signals or actions will be applied to the process until )(1 aj is minimized, i.e. )( dty tracks dy in terms of eqs. (1-4). as shown in fig. 1, the rl agent represents actually the controller in the ac formulation, the environment is the controlled process in the ac formulation, the disturbance input is absent to simplify the problem setting, and the q-function nn with the nonlinear map wq is referred to as the q-function nn wq . fig. 1 informational structure of dql-based cs [22] 2.2 optimal reference tracking control problem in the second approach this approach, (ii), focused on pi rl, is expressed as the optimization problem [23] ),(minarg 1 * θθ θρ j d = (5) where  is the parameter vector used to build the control policy nn, * is the solution to the optimization problem defined in eq. (5), namely the optimal value of , θd is the feasible domain of , and )(1 θj is the cost function |,),1(),(||),(|)(1 θθθθ −−+= ddod tetetej (6) which is similar to that defined in eq. (4), thus justifying the fair comparison of the two approaches implemented using different algorithms. the pi rl-based control system structure is illustrated in fig. 2, which is similar to the structure in fig. 1. the integrator with the additional state variable )( di ts is inserted in order to guarantee zero steady-state control errors in certain regimes. 620 i. a. zamfirache, r.-e. precup, e. m. petriu fig. 2 informational structure of pi rl-based cs (adapted from [23]) the notation θ is used in fig. 2 for the nonlinear map function of the policy nn. the rest of elements in fig. 2 are described in relation with the control system given in fig. 1. 2.3. optimal reference tracking control problem in the third approach this approach, (iii), focused on actor-critic rl, is expressed as the optimization problem [24] ),(minarg 1 * ωω ωω j d = (7) where  represents the configuration, i.e. the parameter vector of the nn-based actor, * is the optimal value of  and the solution to the problem described in eq. (7), d is the feasible domain of , and j1() is the cost function, which is defined similarly to eqs. (46), leading to a fair comparison between this approach and the ones discussed in sections 2.1 and 2.2, as |,)()(||)(|)( 11 ωωωω −−+= ttot eeej (8) where the parameter o is used to control (weight) the overshoot value. fig. 3 presents the actor-critic rl-based control system structure. the integrator is used for the same reason as the on described in fig. 2. the gwo-based critic is constantly monitoring the nn-based actor and only step in if the actor deviates from achieving the predefined control objectives. fig. 3 informational structure of actor-critic rl-based cs (adapted from [24]) the nn-based actor is represented in fig. 3 as  and the gwo-based critic as w. the rest of elements in fig. 3 are described in relation with the control systems given in fig. 1 and fig. 2. q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 621 3. combined reinforcement learning-metaheuristic algorithms using the presentation in [22] focusing on gsa, the steps of the dql-based control approach combined with gsa are presented in fig. 4. the nn architecture used for to implement the nns involved in the algorithm is presented in fig. 5. the dynamic regime considered to solve the optimization problem defined in eq. (1) is set to zero external disturbances and constant reference input. the optimization process will stop when the control error does not decrease and its absolute value is under a threshold value for a certain number of consecutive steps. the nns involved in the rl-based control process are initialized with all the mas considered in [22]. fig. 4 highlights the case where the parameter vector w0 of the qfunction nn qw and the parameter vector 0 of the target nn ρq̂ are initialized with gsa. the dql algorithm is using the experience replay buffer technique to introduce diversity in the training process [22]. using the presentation in [23] focusing on gwo, the steps of the pi rl-based control approach combined with mas are presented in fig. 6 using the description given in [23] from a control perspective. the algorithm presented in fig. 6 iteratively searches for better configurations, i.e. the parameter vector , for the control policy nn . the algorithm stops when the cp’s output continually follows the reference input for a number of  steps. more details about how the algorithm is implemented are provided in [23]. using the presentation in [24] focusing on gwo, the steps of the actor-critic rl-based control approach combined with mas are presented in fig. 7. the gwo-based critic will step in only if the actor nn deviates from following the predefined optimal control objectives. it will reconfigure the actor nn if the control error value will not decrease during the rl-based control process. the algorithm will stop when the final iteration in the rl process is reached. 4. experimental results and their discussion this section presents the experimental results for the gsa-based dql (gsa-dql) algorithm, the gwo-based dql (gwo-dql) algorithm, the pso-based dql (psodql) algorithm based on (i) and suggested in [22], the gd-based dql (gd-dql) algorithm, the gwo-based pi rl (gwo-pi rl) algorithm suggested in [23], the psobased pi rl (pso-pi rl) algorithm suggested in [17] and also applied in [23] and the gdbased pi rl (gd-pi rl) algorithm. this section also describes the results of the gwobased actor-critic (gwo-ac), the pso-based actor-critic (pso-ac) and the gd-based actor-critic (gd-ac) algorithms discussed in [24]. details on the design of the metaheuristic algorithms are given in [22-24]. 622 i. a. zamfirache, r.-e. precup, e. m. petriu fig. 4 flow diagram of the combination of q-learning and gsa q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 623 fig. 5 q-function nn architecture with two hidden layers [22] fig. 6 flow diagram of the combination of policy iteration and gwo 624 i. a. zamfirache, r.-e. precup, e. m. petriu fig. 7 flow diagram of the combination of actor-critic and gwo all these algorithms are built as nn-based state feedback controllers, and the experiments are conducted using the nonlinear servo system laboratory equipment described in [24] and illustrated in fig. 8. the process model is introduced in [25] as ),( ]01[)( ),( / 0 )( /10 10 )( )( ,)( if,1 ,)( if),/())(( ,|)(| if,0 ,)( if),/())(( ,)( if,1 )( 2 1 tty t tk t ttx tx utu utuuuuutu utuu utuuuuutu utu t p b baaba ac cbcbc b x x =       +      − =               −− − −−−+ −− =    (9) q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 625 where the significance of variables and parameters is [25]: t  0 is the continuous time, kp > 0 is the servo system gain, t > 0 is a small time constant, the pulse with modulated control signal u(t) applied to the direct current (dc) motor has a duty cycle 1)(1 − tu , the state vector of the system is ,)]()([)()]()([)( 2121 tt txtxttstst === xs and ua, ub and uc, 0 < ua < ub, 0 < uc < ub are the parameters of the dead zone static nonlinearity variable (t) modeled in the first relation in eq. (9). the parameter values related to eq. (9), obtained by least squares identification, are ua = 0.15, ub = 1, uc = 0.15, kp = 140 and t = 0.92s [25]. fig. 8 photo of nonlinear servo system experimental setup in authors’ laboratories [22] each algorithm is assessed on how well it solves an ortcp – defined in eq. (1) or eq. (5) – in servo system position control using the following performance indices: the minimum value of the cost function j1min, plus the empirical performance indices the settling time ts(s), the 0-to100% rise time t1(s), the peak time tm(s) and the percent overshoot 1(%). the dynamic regime related to the optimization problems defined in eq. (1) and eq. (5) and employed as ortcps in the ac formulation is characterized by: yd = 40 rad step type modification of the set-point, randomly generated initial state variables, zero disturbance input, and time horizon of 30 s. these values are kept permanently during the controller design and tuning in order to guarantee the correct evaluation and comparison of the cost function values. since the mas are essentially characterized by random features, the experiments conducted with the algorithms that include mas were repeated nine times (i.e. episodes) using the same initial conditions and averaging the results. the nine episodes were organized, as proceeded in [22-24], in three stages, and each stage contains three episodes. the experiments involving the two gd-based algorithms were run only once in each of the three stages they do not operate with randomly generated values. in order to ensure a fair comparison of all algorithms, the statistical results are obtained and formulated after processing the results of running 30 realtime experiments with 3000 iterations/experiment for all nn-based controllers. as specified in [22], in the same context of random features, the initialization step in every experiment conducted with the algorithms based on (i) was executed three times and the resulted values of the parameter vectors , w and  were averaged before using them to train the nn-based state feedback controllers. as considered in [22-24], the parameters involved in the optimization problems were set to tmax = 1500 and o = 1.5. the sampling period for control was set to 0.01 s. the parameter vectors of the nn-based state feedback controllers after training contain 17 elements (weights and biases). the augmented state vector contains randomly generated 626 i. a. zamfirache, r.-e. precup, e. m. petriu values in the state space ]30,0[]100,0[]60,0[ =s at each stage, with the first two intervals corresponding to the state variables in eq. (14) and the third interval to the integrator state. the parameter specific to the stochastic gd-based algorithms was set to 01.0= . the number of agents used in the mas was set to 10=n , the initial value of the gravitational constant specific to gsa was set to 9.10 =gravg , and the other parameter settings of mas are described in [6]. the expressions and the obtained values of the parameter vectors of all algorithms are specified in [22-24]. in this context of ensuring full transparency, the data processed in the statistical analysis conducted in order to compare the performance of all algorithms is freely available in the data_fume.m matlab file [26]. the readers are invited to examine several samples of cs responses in [22-24]. the results obtained after conducting the variance (anova) test of the minimum cost function value j1min evaluated after running these ten algorithms are presented in fig. 9. fig. 9 shows that the best performance is achieved by the gsa-dql algorithm, followed by the gso-ac algorithm and the pso-dql algorithm. table 1 summarizes the results of the non-parametric statistical analysis. fig. 9 anova test of minimum cost function value j1 for all algorithms table 2 gives a comparison of the performance indices as far as the cs behavior is concerned for all seven algorithms under discussion. the bold values indicate the best (namely smallest) performance indices among all algorithms. the best algorithm as far as the settling time is concerned is gwo-ac, followed by pso-ac and gwo-pi rl. the best algorithm as far as the 0-to-100% rise time is concerned is pso-pi rl, followed by gd-pi rl and gd-dql. the best algorithm as far as the peak time is concerned is gd-pi rl, followed by pso-pi rl and gwo-pi rl. the best algorithm as far as the overshoot is concerned is gwo-ac, followed by gwo-pi rl and gsa-dql. these conclusions drawn from table 2 indicate that there are certain differences of the empirical performance indices, which make the control system designers q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 627 choose the right algorithm in order to achieve the performance specifications imposed to the control system, and to ensure a convenient tradeoff to them. these conclusions will be different for other dynamic regimes and might also need to be subjected to a tradeoff. table 1 ranks, statistics and related p-values of ten algorithms algorithm friedman test friedman aligned test quade test gsa-dql 2.2667 62.3667 2.1161 gwo-dql 3.2333 73.8333 2.8731 pso-dql 3.1 74.9667 2.8796 gd-dql 5.2667 113.3667 4.7376 gwo-pi rl 4.4333 128.4333 4.7914 pso-pi rl 4.6 133.6667 4.9634 gd-pi rl 5.1 151.8667 5.6387 gwo-ac 1.1333 17.3667 1.1484 pso-ac 2.3333 56.1667 2.3161 gd-ac 2.5333 62.9667 2.5355 statistic 89.97 95.58 13.90 p-value 1.6710−15 1.1110−16 2.0710−20 table 2 comparison of cntrol system performance indices with controllers tuned by ten algorithms algorithm (s) st (s) 1t (s) mt (%) 1 gsa-dql 24.3027 10.89 15.8133 1.1847 gwo-dql 24.7913 10.466 15.6457 1.51 pso-dql 25.456 9.418 14.6713 1.8077 gd-dql 25.7247 8.7743 13.2433 1.8287 gwo-pi rl 23.4215 9.4544 11.9362 0.7658 pso-pi rl 25.259 6.7314 9.3833 4.8424 gd-pi rl 24.0382 7.6027 6.7273 23.2914 gwo-ac 21.7897 12.041 16.2593 0.7051 pso-ac 22.955 13.4543 16.9083 1.7128 gd-ac 24.5883 13.267 16.6147 1.5147 the integrator is introduced to ensure zero steady-state error; experimental data to confirm this conclusion is also included in [22-24]. a sample of control system response is illustrated in fig. 10, which confirms both the zero steady-state control error and the effects of the dead zone static nonlinearity in the servo system model given in eq. (9). the analysis part of the experimental results does not include the training convergence curves, in order to keep a reasonable length of the paper. however, convergence curves can be relatively easily derived using the information extracted from the system responses as those given in fig. 10. nevertheless, the conclusions of this comparison can be different if other nonlinear processes are subjected to the nn-based ortcp approaches and algorithms discussed in this paper as, for instance, various representative applications that include decision-making processes [27], man-computer symbiosis [28], 3d printing objects [29], medical systems [30]–[33], drilling processes [34], fuzzy control [35]–[37], evolving 628 i. a. zamfirache, r.-e. precup, e. m. petriu controllers [38], fuzzy cognitive maps [39], traffic systems [40], quantum computing [41], and telesurgical applications [42]. fig. 10 real-time experimental system responses y and u when using the pso-pi rl algorithm [23] 5. conclusions this paper carried out the performance analysis of three control system structures and approaches, which combine reinforcement learning (rl) and four representative metaheuristic algorithms (mas) resulting in seven rl algorithms that include the classical stochastic gradient descent algorithms. the algorithms, suggested in the recent authors’ papers [22-24], were implemented as neural network (nn)-based state feedback controllers and applied to the position control of a nonlinear servo system. the main benefit of these approaches is the transparency of presentation in an easily understandable way of the nn training and the controller implementation. the disturbance rejection is ensured by the presence of the integrator in the control system structures. the main limitation of the approaches treated in this paper is the relatively large number of parameters of the nn-based state feedback controllers accounting for the application considered in the previous section. nevertheless, the stability analysis is a sensitive subject, in order to use stability conditions as constraints in the optimization problems, but it is discussed in [1] in the context of data-driven control. future research will be focused on applying the algorithms and controllers to other complicated and challenging processes. the simplification of the nn architectures will be tackled as well, aiming the cost-effective training, design, tuning and implementation. q-learning, policy iteration and actor-critic reinforcement learning combined with metaheuristic 629 acknowledgement: this work was supported by a grant of the romanian ministry of education and research, cncs uefiscdi, project number pn-iii-p4-id-pce-2020-0269, within pncdi iii, and by the nserc of canada. the support of dr. raul-cristian roman in conducting the real-time experiments on the servo system equipment is duly acknowledged. references 1. precup, r.-e., roman, r.-c., safaei, a., 2021, data-driven model-free 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studies of freight wagon fals udc 629.4 svetoslav slavchev, kalina georgieva, valeri stoilov, sanel purgić faculty of transport, technical university sofia, bulgaria abstract. a comparative analysis based on the results from strength-deformation analysis of wagon body, series fals, and on the results from the real wagon test was made. calculations were carried out in the department of railway engineering at technical university of sofia and are based on the finite elements method. two computational models of wagon design were developed. one of them consists of shell elements (triangular), and the second one of solid elements (tetrahedral). experimental studies on real wagon were conducted at the national transport research institute. it was found that the results obtained for the stresses are similar, which proves that the models are appropriate and they can help to solve a wide range of issues, for example those related to lightweight design of railway vehicles. key words: railways, freight wagon, strength analysis, fem, tests 1. introduction the paper is a comparative analysis [1, 2] of the results of the static strength obtained theoretically (fem calculation) and the results from the real tests carried out on the supporting steel structure of freight wagon series fals. the goal of this comparison is to determine which of the theoretical models, that emerged in recent years in the calculation of railway structures, describes more accurately the actual test article. the first model was built up of finite elements type “solid” and the second of “shell” finite elements. theoretical static strength analysis was performed by the method of finite elements [3, 4, 5, 6] in the department of railway engineering of technical university of sofia. the tests on the wagon prototype were carried out in bulgarian national research institute of transport (nrit). after the calculations were carried out and the points with the highest stresses were identified, the same points were proposed for the measurements in the test. both theoretical and experimental studies have been done in full compliance with international received june 10, 2015 / accepted july 15, 2015 corresponding author: sanel purgić faculty of transport, technical university sofia, bulgaria e-mail: s_purgic@tu-sofia.bg original scientific paper 92 s. slavchev, k. georgieva, v. stoilov, s. purgić requirements described in the european standard din en 12663[7], technical specifications for interoperability (tsi) rolling stock [8] and code 577 of the international union of railways (uic)[9]. according to these regulations, 13 static load cases and 8 fatigue load cases were carried out on the wagon structure. in order to achieve a more precise comparative analysis, only the so-called "clear load cases" were selected, i.e. those in which the forces are applied only in one direction of the coordinate axes. this allows us to avoid some subjective factors arising from the improper positioning of the force producing elements (hydraulic cylinders) and strain gauges during the tests. the results of only two of the obligatory (clear) load cases are presented in the paper: “compressive force at buffer height 2000 kn” (provisionally marked lc-1) and “tensile force in coupler area 1500 kn” (provisionally marked lc-2) [7, 8, 9, 10]. in the present research, the comparison was done in a limited number of zones. 2. computational models for the purpose of this study, two complicated calculation models for static strength calculations were developed. the first model (fig. 3) was built up from finite elements type 3d solids (fig. 1) and consists of 697 047 nodes and 349 371 elements. the maximum size of the finite elements is 42,6 mm. all theoretically required ratios between the parameters of finite elements that allow modeling of the structure of the body with 3d solids were fulfilled. fig. 1 finite element type 3d solid fig. 2 shell type finite element analysis of the results of theoretical and experimental studies of freight wagon fals 93 in the second model the mesh was generated of finite elements type shell (fig. 2). this mesh has following parameters: maximum size – 31,8mm; minimum size – 10,6mm; rate of increase – 1,5; number of finite elements – 401 874; number of nodes – 801 408. fig. 3 fem model built up of finite elements type 3d solid fig. 4 fem model built up of finite elements type shell fig. 4 shows the finite element mesh of the calculation model built up with elements type shell. the models were optimized by studying the convergence of the solution [11, 12, 13, 14]. the third major component of this research is the real test of the structure [15]. fig. 5 shows the plan and the locations of the strain gauges on the wagon prototype. the horizontal forces from load cases described above were applied by means of hydraulic cylinders acting on the buffer or draw gear. the values of stress measured in experimental test were used in this report for comparison with stress values obtained in fem calculation with both types of finite elements. 94 s. slavchev, k. georgieva, v. stoilov, s. purgić fig. 5 plan of measuring points on wagon prototype 3. analysis of results in the analysis of the two theoretical models the forces and restrictions [16, 17, 18] were not compared in the boundary areas, as the stress values obtained there might be unrealistically high due to modeling. these stress values do not have to be taken into account in the analysis of the results [13]. it is advisable that the comparison of the results from the two computing models should be performed by elements, not by nodes. the elements cover the area where the strain gauges are located and stress value contains the information of the value of its constituent nodes. this in turn leads to a slight problem with the shell model when deploying strain gauges on the thin side of the profile. the analysis shows that it is practically impossible to make a comparison because of the way the shell model was built with shell elements defined by the mid surface of the profile. analysis of other problems related to modeling and fem analysis with shell elements the authors have published in [19]. table 1 represents the comparison of stress values in areas with the highest stresses calculated in fem analysis with both models and stresses obtained in test of the wagon in both horizontal load cases. the analysis of the results obtained shows relatively high correlation of the stress values obtained in the two models. in most cases, the stress values in the shell model are lower than those constructed with 3d solid. analysis of the results of theoretical and experimental studies of freight wagon fals 95 table 1 comparison of stress values from calculations and test lc1 lc2 fem fem test fem fem test no. shell solid shell solid 1 118,6 105,7 91,4 129,8 132,1 105,3 2 86,2 89,0 80,0 98,3 160,8 149,7 4 48,6 148,4 67,9 52,5 12,4 103,5 6 127,1 139,8 283,1 70,5 51,9 67,9 8 52,1 72,6 67,5 101,9 93,8 55,9 9 64,7 53,6 77,8 133 108,7 107,5 11 138,7 249,2 205,2 107,3 126,5 151,3 12 50,2 58,3 55,1 107,1 89,5 107,3 16 32,3 29,8 69,1 8,3 4,7 25,9 19 29,9 16,5 17,1 15,1 13,8 13,5 figs. 6 and 7 show the stress values measured in the zone of gauge no. 2 for the shell and 3d solid models respectively. fig. 8 shows the arrangement of strain gauges on the wagon structure. fig. 6 stress value in the zone of strain gauge no. 2 shell model fig. 7 stress value in the zone of strain gauge no. 2 solids model 96 s. slavchev, k. georgieva, v. stoilov, s. purgić fig. 8 strain gauge no. 2 – placement on wagon prototype during test figs. 6 and 7 show the stress values measured in the zone of gauge no. 2 for the shell and 3d solid models respectively. fig. 8 shows the arrangement of strain gauge no. 2 on the wagon structure. fig. 9 stress value in the zone of strain gauge no. 16 shell model fig. 10 stress value in the zone of strain gauge no. 16 shell model analysis of the results of theoretical and experimental studies of freight wagon fals 97 fig. 11 strain gauge no. 16 – placement on wagon prototype during test all the results (two theoretical for both types of finite elements and physical measurements) show significant similarities. the analysis for all load cases shows a significant difference between the fem values (figs. 9 and 10) and the values measured by strain gauge no. 16 (fig. 11). possible reasons for the differences, in our opinion, are the defects in the welded joint located underneath the strain gauge no. 16, structural changes in the material because of bending and welding during the manufacturing of the wagon and possible inaccurate signal transmission caused by strain gauge installed in the bent part of the steel sheet. however, the results of fem analysis correctly reflect the general nature and distribution of stresses in the wagon structure. 4. conclusions computational models for strength analysis of the structure of a specialized wagon series fals were developed. a comparative analysis of the results obtained by calculations and those obtained in actual tests of the wagon was performed. the stress values were compared for a limited number of zones and a good correlation between the results from the theoretical models and the actual tests were found. in both theoretical models identical distribution stress values in the wagon structures were observed. this allows the developed computational models to find application in the design of new constructions of the wagons of the same series as well as more effectively to optimize the parameters of the supporting construction of this type of wagon. certain shell model problems were identified in the strength analysis due to modeling, correct mesh generation, stress values etc. detailed information about these and other issues can be found in [18]. based on the analysis and conclusions, a simple answer cannot be given as to which model – shell or 3d solid – should be used for fem analysis of the wagon. it is a matter of experience and personal preference which finite elements – shell or 3d solid – should be used to build a calculation model. the authors recommend the use of 3d solid finite elements, because of greater similarity with test data for high stress values. they have a number of advantages: a more precise and easy construction of the geometry, a clear visualization of the elements of the structure, fewer restrictions in modeling etc. 98 s. slavchev, k. georgieva, v. stoilov, s. purgić construction of the hybrid model (if possible) composed of 2d and 3d finite elements for strength-deformation analysis of the wagon structure is appropriate and will reduce the modeling errors. references 1. slavchev s., stoilov v.., 2008, comparative analysis of the results of the static strength calculations and strength tests of a wagon series lagrs. proc. of scientific conference тrans&motauto’08, sozopol, bulgaria. 2. slavchev s., stoilov v.., 2012, comparative analysis of the results of the static strength calculations and strength tests of a wagon series zans. proc. of scientific conference bulтrans-2012, sozopol, bulgaria. 3. stoilov v., mayster a., 2003, strength analysis of the body of wagon series falns, proc. of scientific conference тrans&motauto’03, sofia, bulgaria. 4. stoilov v., mayster a., 2004, strength calculation of the body of freight wagon series falns by finite element method, proc. of scientific conference тrans&motauto’04, plovdiv, bulgaria. 5. slavchev s., stoilov v.., 2008, strength analysis of the body of wagon series lagrs with finite elements method, proc. of scientific conference тrans&motauto’08, sozopol, bulgaria. 6. slavchev s., stoilov v., purgic s., 2012, static strength analysis of the body of a wagon, series zans. proc. of scientific conference bulтrans-2012, sozopol, bulgaria. 7. din en 12663-2, 2010, railway applications structural requirements of railway vehicle bodies part 2: freight wagons, beuth verlag, berlin, germany. 8. technical specifications for interoperability (tsi) rolling stockfreight wagons, brussels, 2007. 9. uic leaflet 577 – wagon stresses. paris, 2005. 10. report erri b12/ rp17 8th edition, brussels, april 1997. 11. zenkiewicz o.c, 1971, the finite element method in engineering science, mcgraw-hill, london. 12. segerlind l., 1976, applied finite element analysis, new york, usa. 13. tenchev r., 1998, operation manual for cosmos/m, 1st edition, tehnika, sofia, bulgaria. 14. stoychev g., 2000, the finite element method the strength and strain analysis, tehnika, sofia, bulgaria. 15. report on contract for testing the freight wagon fals. 2010, nrit, sofia, bulgaria. 16. cosmos/m user’s guide, revision 1.65, 1991, santa monica, california, usa. 17. software cosmosworks – user manual, 2012. 18. kurowski p., 2005, engineering analysis with cosmosworks 2005, schroff development corporation. 19. slavchev s., georgieva k., stoilov v., 2014, issues of wagon modeling with shell elements. proc. of scientific conference railcon’14, niš, serbia, pp. 33-36. 10112 facta universitatis series: mechanical engineering vol. 23, no 1, 2025, pp. 1 15 https://doi.org/10.22190/fume211105006z © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper load path visualization using u* index and principal load path determination in thin-walled structures shengjie zhao, luyue mao, nan wu, sviatoslaw karnaoukh department of mechanical engineering, university of manitoba, winnipeg, canada orcid ids: shengjie zhao https://orcid.org/0000-0002-1908-1133 luyue mao https://orcid.org/0009-0008-7283-4959 nan wu https://orcid.org/0000-0001-8768-1328 sviatoslaw karnaoukh https://orcid.org/0009-0005-0615-8874 abstract. u* index is used to express the load transfer inside a structure from a global perspective. typically, a load path is defined as the ridgeline of the u* contours. however, it is cumbersome to directly locate the load paths by numerical approaches. this paper presents a streamline method with the fourth-order runge-kutta algorithm to visualize the load paths in thin-walled structures. the load paths can be consistently plotted on the surfaces of two-dimensional plates or three-dimensional shells by path projection. a new concept of principal load path is also introduced by evaluating the importance of load paths using statistical means. the principal load path is conceived as the “spine” of the structure that transfers the greatest internal force. a case study of a simplified vehicle body is presented. it is found that the structural stiffness can be greatly improved by reinforcing the set of principal load paths, which gives engineers an important insight into the development of weight-efficient structures. key words: load path, finite element analysis, thin-walled structures, runge-kutta method 1. introduction load path analysis is an emerging technique in the field of structural design. it is essential to understand how the load transfers through a structure to properly arrange the material distribution, particularly with thin-walled structures [1]. several theories and methods for load path analysis have been introduced by researchers and engineers. the concept of load flow proposed by kelly et al. [2-4] is an early attempt to depict the load paths in a structure from the stress results. however, the stress concentration around holes and sharp corners can produce misleading information for load path visualization using this stress-based method [5]. harasaki and arora also proposed a new concept of transferred received: november 05, 2021 / accepted february 15, 2022 corresponding author: nan wu department of mechanical engineering, university of manitoba, winnipeg, mb, canada r3t 2n2 e-mail: nan.wu@umanitoba.ca https://orcid.org/0000-0002-1908-1133 https://orcid.org/0009-0008-7283-4959 https://orcid.org/0000-0001-8768-1328 https://orcid.org/0009-0005-0615-8874 2 s. zhao, l. mao, n. wu, s. karnaoukh force for load path analysis [6]. load paths in a structure were indicated by the directions of transferred force vectors. a topology optimization algorithm was developed based on the magnitude of the transferred force [7,8]. nonetheless, it is found that this method has low calculation efficiency and poor adaptability [9,10] compared to the stress-based methods. the concept of u* index [11] was developed and introduced by a group of japanese researchers to describe the internal load transfer inside a structure. the u* index theory originated from the concept of relative rigidity introduced by takahashi [12]. the relative rigidity between an arbitrary point and the loading point can be expressed by the value of u* index. the main load path in a structure was defined as the route with the highest relative rigidity. the reasonability of u* index theory was experimentally validated by pejhan et al. [13,14]. three design criteria based on u* index theory were also developed [11] and later applied to the applications of the structural design of vehicle components [15,16]. besides, u* index was recorded to be used in fiber trajectory optimization in composite structures [10,17], damage detection in thin-walled structures [18], and combination with other design methods, like bionic design [19]. although the classic u* index theory is based on linear elasticity, it can be extended to the analysis of nonlinear/plastic deformation cases by using the substitution modulus method [20] or altered u* expression [21]. several u* variants (e.g., u*sum [22], u** [23], and u* m [24]) were also introduced over the past decade to further extend the functionality of u* index. yet, the issue of low computation efficiency is also noticed from u* load path analysis due to the iterative finite element (fe) calculations for nodal u* values. to address this issue, sakurai et al. [24] proposed an inspection loading method that decreases the calculation time by over 90% in large-scale models. this method prevents the fe calculations from reconstructing the stiffness matrix by converting multiple boundary conditions to multiple loading conditions. the applications of deep learning algorithms, such as the multilayer perceptron neural network built by wang et. al [25] and the deep residual u-net developed by zhao et al. [26], further shorten the calculation time of u* index. while u* index is promising in applications of structural analysis and design, there is no standardized procedure for visualizing the u* load paths. according to the u* index theory, a load path should follow the ridgeline of u* contours from the loading point [11]. yet it is tough to trace the load path by numerical methods without a specific objective function. three approaches for load path visualization were identified in the literature. vector plot is a straightforward approach to discretely visualize the internal force of a structure. it was used in early studies [2,6] of load path analysis, but it cannot create continuous load paths. the second approach is by using the load function developed by tamijani et al. [27,28]. the load function is a scalar function using generalized beltrami representation that can show the load paths with contour plots without post-processing. however, this function is only applicable to the load path theory introduced by kelly et al. [2-4]. the third alternative for load path visualization is the streamline method, which is extensively used in tracing the fluid flow [29]. kelly et al. implemented this method to visualize the load paths from the field of pointing stress vectors [30-32] with a fourth-order runge-kutta (rk4) algorithm. even though the streamline plot is a more universal approach for vector field visualization, it is not directly feasible for u* index as a scalar parameter. moreover, the classic rk4 algorithm cannot guarantee the continuity of the load paths on curved surfaces, resulting in additional uncertainty in searching and visualizing successive load paths in thin-walled structures. an alternative approach is necessary to confine the visualized load paths on the surfaces of thin-walled structures without exiting the fe solution domain. load path visualization using u* index and principal load path determination in thin-walled... 3 in this work, a methodology for u* load path visualization in thin-walled structures is proposed. the load paths are represented by the streamlines derived from the gradient field of the u* distribution. rk4 algorithm is used to increase the precision in streamline tracing. by using path projection, load paths can be efficiently plotted on the surfaces of two-dimensional (2d) plates and three-dimensional (3d) shells. a new concept of principal load path is also introduced to determine the most important route for global load transfer. finally, a case study of a thin-walled vehicle body is provided to compare the importance of different load paths. two sets of load paths in the vehicle structure are separately stiffened to evaluate their contributions to the structural stiffness. according to finite element analysis (fea) and experimental tests, the reinforcement of the principal load paths is found to be more effective in reducing the total deformation. the method of load path visualization greatly helps the designers understand the internal load transfer inside a structure which is essential for the design of weight-efficient structures. meanwhile, the concept of the principal load path provides the engineers with a simple solution to enhance the structural rigidity without further complicated analysis such as topology optimization. 2. u* index theory u* index was derived from the concept of relative rigidity defined by takahashi [11, 12]. fig.1 (a) presents a linear elastic structure including an applied load at point a, a support at point b, and an arbitrary point c within the structure [11]. the internal stiffness between any two points is represented by the spring connection. the relationship among points a, b, and c considering three translational dofs is expressed as: . a aa ab ac a b ba bb bc b c ca cb cc c p k k k d p k k k d p k k k d             =                  (1) kij (i=a,b,c; j=a,b,c) in eq. (1) denotes the internal stiffness between points indicated by the subscripts. pi and di (i=a,b,c) are the force and displacement vectors. since point b is constrained in the original system, total strain energy up can be expressed as: 1 1    ( ) 2 2 p a a aa a ac c au p d k d k d d=  = +  (2) as shown in fig. 1 (b), point c is also restrained in the modified system. meanwhile, the displacement at point a is forced unchanged. total strain energy u’p stored in the modified system is expressed as: 1 1   ( ) 2 2 p a a aa a au p d k d d =  =  , (3) where p’a denotes the applied load at point a in the modified system. classic u* index is then defined as [11]: * 1 p p u u u = −  . (4) 4 s. zhao, l. mao, n. wu, s. karnaoukh b c a b c a (a) (b) fig. 1 schematic diagram for u* index calculation: (a) original structure; (b) modified structure [11, 23] it can be rewritten by substituting eqs. (2) and (3) into eq. (4) [11]: 1 * 2 1 ( ) a ac c u u d k d −   = −     . (5) the value of u* ranges from 0 at point b to 1 at point a. it is observed from eq. (5) that the u* value is dependent on stiffness matrix kac. therefore, it can be used to describe the connection strength between the arbitrary point and the loading point. for plates and shells that involve the rotational dofs, the u* index is formulated as [23]: * 1 1p m a a a a p m a a a a u u p d m r u u u p d m r +  +  = − = −    +  +  , (6) where up, um are the external works of the original system by applied load pa and bending moment ma, and u’p, u’m are the external works of the modified system by applied load p’a and bending moment m’a. fig. 2 depicts the u* distribution of an example structure. the route that transfers the greatest loading is defined as the main load path. stiffness decay vector β is hereby defined: *grad .u = − (7) the main load path highlighted as the bold line in fig. 2 can be traced along the ridgeline of the u* contours which has the smallest decay rate. however, since the magnitude of β is always zero along the u* contour direction, the direction of the slowest u* decay can be ambiguous. also, in the occasion of more than one ridgeline, stiffness decay vector β may fail to locate the most important load path. therefore, more efficient algorithms are demanded to automatically generate the load paths. load path visualization using u* index and principal load path determination in thin-walled... 5 f main load path fig. 2 distribution of u* values and main load path of a simple structure [11] 3. load path visualization methodology this section introduces a visualization methodology for u* load paths by using the streamline approach. ansys 2020 is used to perform the u* load transfer analysis with a custom apdl script. the u* distribution is then assigned to the fe model as a temperature field. the u* gradient vector v at each node of the fe mesh can be acquired from the thermal gradient by thermal analysis. to successfully visualize the load paths represented by streamlines, instead of searching the slowest u* descent direction from the loading point, the path plotting initiates from the supporting points by gradient ascent to find the local u* maximum. the rk4 algorithm is used for locating the streamline points to reduce errors in path tracing. as illustrated in fig. 3, given an initial point pi on a plane, the new point pi+1 can be determined from: 1 2 3 1 2 1 2 3 1 2 4 1 1 2 3 4 , , , , 1 ( 2 2 ), 6 i i i i n p n p dp n p dp n p dp i i dp v s dp v s dp v s dp v s p p dp dp dp dp + + + + =  =  =  =  = + + + + (8) 6 s. zhao, l. mao, n. wu, s. karnaoukh where ∆s is the step size along the load path, and vn is the normalized u* gradient vector v with a direction of * * *u u u i j k x y z     + +      . vector vn can be specified at any location in the structure by interpolating the vector values at nearby nodes. a complete load path is generated by consecutively connecting the points from p1 to pn, where n is the total number of points. main load path fig. 3 load path creation by runge-kutta sampling vectors in 3d shell structures with curved surfaces, it is necessary to confine the load path within the solution domain [33]. one feasible approach is to project the load path segment onto the plane of its nearest element. fig. 4. depicts the load path projection diagram for the determination of a new point pi+1 from initial point pi in the orange element. p’i+1 is calculated first by the rk4 algorithm. the next step is to find the element that is closest to p’i+1. two vectors, 1w and 2w , are subsequently defined by using any three nodes (n1, n2, and n3) of the shell element in blue color. unit normal vector n of this element can be calculated by the cross product of these two in-plane vectors: 1 2 1 2     .     w w n w w  =  (9) this normal vector intersects the targeted element at node n2. by defining vector q from n2 to p’i+1, distance d from p’i+1 to the element can be calculated by: .d q n=  (10) finally, the coordinate of new point p’i+1 can be obtained by:  q q dn = − , 1 2 .ip n q+  = + (11) a load path visualization example of a shell structure is presented in fig. 5. this structure is constrained at two bottom corners while a horizontal force is applied at the center of the top edge. two load paths are traced from the fixed points to the loading point load path visualization using u* index and principal load path determination in thin-walled... 7 by the proposed methodology. both load paths as displayed in red solid lines in symmetry perfectly lay on the surface of the shell structure. it can be seen that the load transfer conditions in thin-walled structures can be easily learned by the visualized load paths. n1 n4 n2 n3 fig. 4 schematic diagram of vector projection fig. 5 load path visualization in a shell structure 8 s. zhao, l. mao, n. wu, s. karnaoukh 4. determination of principal load path there can be multiple load paths in some load cases, such as the example displayed in fig. 5. it is essential to develop quantitative criteria to assess the importance of each main load path for load transfer. the load path with the greatest load transfer capability is called the principal load path. three statistics measures are used to analyze the u* distribution of the selected load path: * 1 . u s n  = (12) ( ) ( ) * 1 /2 * * 2 /2 1 /2             if is odd .   if is even 2 n n n u n s u u n + +   =  +   (13) * 1 3 ( ) .iu s s n  − = (14) where s1, s2, and s3 represent the mean, median, and standard deviation of the u* distribution along the load path. n is the total number of points on the path, and u*i is the u* value at the i-th point. since the u* value describes the connection degree between an arbitrary point and the loading point, a load path with a great relative stiffness should have large mean and median values. meanwhile, a smaller u* standard deviation indicates a more consistent load transfer along the path with less force dispersion. thereby, the principal load path in a structure can be determined by these three indicators. table 1 statistical measures for the load paths in the plate structure mean, s1 median, s2 standard deviation, s3 load path 1 0.343 0.333 0.221 load path 2 0.454 0.441 0.167 load path 3 0.357 0.405 0.257 table 2 modified plate structures with reinforced load paths load path element thickness (mm) plate model volume (mm3) maximum deformation (mm) deformation reduction (%) original 3 7200 4.062ee-3 n/a load path 1 4 7378 3.739ee-3 7.948 load path 2 4.483 7378 3.407ee-3 16.108 load path 3 4.047 7378 3.848ee-3 5.275 to demonstrate the rationality of the quantitative criteria, a simple plate with a size of 60 mm × 40 mm and a uniform thickness of 3 mm is presented in fig. 6 (a). young’s modulus of 210 gpa and poisson’s ratio of 0.3 are used in fe simulations. the left side load path visualization using u* index and principal load path determination in thin-walled... 9 corners of the plate are constrained and a 100-n force along the x-axis is applied at the lower-right corner. fig. 6 (b) presents the u* distribution in this load case with two load paths in a solid black line and a dashed red line (load paths 1 and 2) automatically generated by the visualization methodology. it should be mentioned that load paths 1 and 2 are two separate load paths traced from the supporting points, even though their lower-right sections are closely located. meanwhile, there are two apparent ridgelines noticed from the u* map. one of them is identical to load path 1, and the other one is labeled load path 1 load path 2 load path 3 (a) (b) fig. 6 load path analysis on a plate structure: (a) dimensions and boundary conditions of the plate; (b) load paths inside the plate traced by streamlines (load paths 1 and 2) and ridgeline (load path 3) as load path 3 in the blue dot line. the u* decaying curves along these three load paths are drawn in fig. 7, where s is the curvilinear coordinate along the load path starting from the loading point, and l is the length of the path. the statistical measures of the u* distributions 10 s. zhao, l. mao, n. wu, s. karnaoukh are listed in table 1. it is observed that load path 2 has the highest mean and median u* value and the lowest standard deviation compared to the other two load paths. meanwhile, load path 2 has the slowest decay rate as shown by the red dashed line in fig. 7, indicating the great load-carrying capability of the load path. given these data, load path 2 can be determined as the principal load path in this sample structure. an interesting finding is that the principal load path does not necessarily agree with u* ridgelines, which are load paths 1 and 3 in this case. fig. 7 u* decay curves of the load paths in the plate to further verify the load transfer efficiency of the principal load path, the plate in fig. 6 is reinforced by respectively adding the same volume of materials (178 mm3) on each load path. the stiffnesses of the enhanced structures are evaluated by their maximum displacements under the same loading condition. the thicknesses of the modified load paths and the deformation results by fea are summarized in table 2. as expected, reinforcing the principal load path (load path 2) offers the greatest deformation reduction rate of 16.11%, while the numbers for load paths 1 and 3 are 7.95% and 5.28%, respectively. it can be concluded that the members that have a great contribution to the structural stiffness can be effectively located by the principal load path. this is a promising feature for structural enhancement without changing the profile of the structure. 5. case study u* index has been widely used in load path analysis on vehicle structures [15,16]. to further investigate the importance of the visualized load paths, a case study of a vehicle body is presented as displayed in fig. 8. this miniature vehicle structure modeled by shell181 elements is supported at the four corners on the back. a horizontal load of 156 n is evenly distributed on the front hoop. the same material properties as those of the previous plate example are used in fe simulations. the shell thickness of the original design is 1.6 mm. after calculating the u* distribution, two sets of load paths are generated by implementing the rk4 algorithm on the u* gradient field as shown in fig. 8 (b). the u* decay curves of both sets of load paths are drawn in fig. 9. with a slower decay rate and load path visualization using u* index and principal load path determination in thin-walled... 11 more desirable statistical properties listed in table 3, load path set 2 can be confidently defined as the principal path set. as discussed in section 4, the principal load path reinforcement would offer a significant enhancement in structural stiffness. followed by the same strategy, two modified designs with the same increased volume of 1169 mm3 (7.5% over the original design) by separately thickening the elements on different sets of load paths are developed. the load path thickness arrangement and model volume of each design are provided in table 4, where org, lp1, and lp2 represent the original design and the modified designs with load path sets 1 and 2 reinforcement, respectively. it is observed that the maximum displacement of the lp2 design is reduced by over 16% compared to that of the original design, while the reduction rate of the lp1 design is only 7.95%. with the same material usage, lp2 with the enhanced principal load paths leads to more significant stiffness improvement. these numerical results reaffirm the importance of the principal load path in structural design. fig. 8 u* load path analysis of the thin-walled vehicle body: (a) u* distribution; (b) two sets of visualized load paths 12 s. zhao, l. mao, n. wu, s. karnaoukh table 3 statistical measures for the load paths in the vehicle body mean, s1 median, s2 standard deviation, s3 load path set 1 0.619 0.830 0.390 load path set 2 0.761 0.903 0.271 table 4 modified designs with reinforced load paths load path element thickness (mm) shell model volume (mm3) maximum deformation (mm) deformation reduction (%) org 1.6 1.556e+04 1.239ee-2 n/a lp1 3.2 1.673e+04 1.227ee-2 7.948 lp2 3.072 1.673e+04 1.118ee-2 16.108 fig. 9 u* decay curves of the load paths in the vehicle body. lp1 and lp2 in this figure respectively denote the load path sets 1 and 2 in the original design last but not least, the reinforced thin-walled structures are experimentally tested with 3d printed parts. the 3d printed specimens shown in fig. 10 (a) are printed by stereolithography (sla) on a formlab form 3b using tough 1500 resin. three specimens are prepared and tested for each design. the stl models for 3d printing are exported from ansys workbench. four small cylinders are added to the supports of the stl models to avoid excessive constraints from the compression plate of the test machine. mts insight electromechanical testing system is used in this test with a 30 kn load cell. as presented in fig. 10 (b), the specimen is subject to unidirectional compression to simulate the load case in the fea. the design performance is evaluated by the specific stiffness in the linear region which is automatically extracted by the mts testing software. the specific stiffness is calculated by the applied force divided by the vertical displacement. the force-displacement curves of each design are plotted in fig. 11 (a) with different colors. the specific stiffness of the design is expressed by the bold solid line that takes the average slope of the regression lines from all three samples. the solid line of lp2 design has obviously the largest slope compared to the other two. finally, the specific stiffness results are normalized by the case of the original design, and their comparison is provided in the bar graph of fig. 11 (b). the enhancement of specific stiffness in lp2 design is 24.41%, while the result of lp1 design load path visualization using u* index and principal load path determination in thin-walled... 13 is 14.61%. these test results again verify the importance gap between the principal load path and the secondary load path regarding the structural stiffness. this case study also showcases the potential of the load path visualization scheme in structural improvement. it offers the designers a clear picture of the load transfer inside a structure, and the principal load path can be directly used in design modification for better structural stiffness. fig. 10 experimental tests: (a) 3d printed specimens; (b) test setup on the mts testing machine 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 org lp1 lp2 normalized specific stiffness (a) (b) fig. 11 results of the compression tests: (a) load-displacement curves, where ss denotes the fitted line with the slope of specific stiffness; (b) normalized specific stiffness of different designs 14 s. zhao, l. mao, n. wu, s. karnaoukh 6. conclusions this paper presents a methodology for u* load path visualization in thin-walled structures using a streamline method with rk4 algorithm. a path projection scheme is introduced to reduce numerical errors and restrain the load paths on curved surfaces. the rationality of the visualization method and the importance difference among different load paths are investigated to define the principle load path. a case study of a simplified vehicle body is also provided to show the potential of the proposed methodology in structural enhancement. the major contributions and findings of this work can be summarized as follows: ▪ the numerical approach including the path projection scheme for u* load path visualization in thin-walled structures is proposed for the first time. ▪ the importance difference among different load paths is numerically and experimentally investigated. whereafter, a new definition of the principal load path is introduced. ▪ the direct reinforcement of the principal load path can significantly enhance the structural rigidity. it gives the designers a straightforward and effective solution in structural design modification and improvement. while only the thin-walled structures are investigated for load path visualization in this study, the proposed methodology is also expected to work for 3d structures modeled with solid elements. it is worth noting that the accuracy of load paths also highly depends on the fe mesh. it can be a limitation as the computation cost of u* distribution for a large-scale model is tremendous. meanwhile, while u* load path can effectively reveal the stiffness distribution of a structure, it overlooks the effect of stress concentration. thereby, stress-based load paths would be recommended when it comes to local detailed design such as fiber path optimization in composites. acknowledgements: this research is supported by the university of manitoba and natural sciences and engineering research council of canada (nserc egp 523932-18 & nserc rgpin 2021-03356). in addition, the authors want to acknowledge that 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(swansea), 28, pp. 196-214. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 209 228 https://doi.org/10.22190/fume191127014j © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial loads and end moments goran janevski1 , predrag kozić1, ratko pavlović1, strain posavljak2 1university of niš, faculty of mechanical engineering, serbia 2university of banja luka, faculty of mechanical engineering, bosnia and herzegovina abstract. in this paper, the lyapunov exponent and moment lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. the method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. the lyapunov exponent and moment lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. as an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments. the validity of the approximate results for moment lyapunov exponents is checked by numerical monte carlo simulation method for this stochastic system. key words: eigenvalues, perturbation, stochastic stability, thin-walled beam, mechanics of solids and structures 1. introduction in recent years there has been considerable interest in the study of the dynamic stability of non-gyroscopic conservative elastic systems whose parameters fluctuate in a stochastic manner. to have a complete picture of the dynamic stability of a dynamic system, it is important to study both the almost-sure and the moment stability and to determine both the maximal lyapunov exponent and the pth moment lyapunov exponent. the maximal lyapunov exponent is defined by 0 1 lim log ( ; )q t t q t→  = q (1) received november 27, 2019 / accepted march 08, 2020 corresponding author: goran janevski university of niš, faculty of mechanical engineering in niš, a. medvedeva 14, 18000 niš, serbia e-mail: gocky.jane@gmail.com 210 g. janevski, p. kozić, r. pavlović, s. posavljak where 0( ; )t qq is the solution process of a linear dynamic system. the almost-sure stability depends upon the sign of the maximal lyapunov exponent which is an exponential growth rate of the solution of the randomly perturbed dynamic system. a negative sign of the maximal lyapunov exponent implies the almost-sure stability whereas a non-negative value indicates instability. the exponential growth rate e [||q(t;q0, 0q || p ] is provided by the moment lyapunov exponent defined as 0 1 ( ) lim log [ ( ; ) ] p q t p e t q t→  = q (2) where e [ ] denotes the expectation. if q(p) < 0 then, by definition e [||q(t;q0, 0q || p ] → 0) as t →  and this is referred to as the pth moment stability. although the moment lyapunov exponents are important in the study of the dynamic stability of the stochastic systems, the actual evaluations of the moment lyapunov exponents are very difficult. arnold et al. [1] constructed an approximation for the moment lyapunov exponents, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. a perturbation approach was used to obtain explicit expressions for these exponents in the presence of small intensity noises. khasminskii and moshchuk [2] obtained an asymptotic expansion of the moment lyapunov exponents of a two-dimensional system under white noise parametric excitation in terms of the small fluctuation parameter , from which the stability index was obtained. sri namachchivaya et al. [3] used a perturbation approach to calculate the asymptotic growth rate of a stochastically coupled two-degrees-of-freedom system. the noise was assumed to be white and of small intensity in order to calculate the explicit asymptotic formulas for the maximum lyapunov exponent. sri namachchivaya and van roessel [4] used a perturbation approach to obtain an approximation for the moment lyapunov exponents of two coupled oscillators with commensurable frequencies driven by small intensity real noise with dissipation. the generator for the eigenvalue problem associated with the moment lyapunov exponents was derived without any restriction on the size of pth moment. kozić et al. [5] investigated the lyapunov exponent and moment lyapunov exponents of a dynamic system that could be described by hill’s equation with frequency and damping coefficient fluctuated by white noise. the procedure employed in khasminskii and moshchuk [2] was applied to obtain an asymptotic expansion of the lyapunov exponent and moment lyapunov exponents of an oscillatory system under two white-noise parametric excitations in terms of the small fluctuation parameter. these results were used to obtain explicit expressions of an asymptotic expansion of the moment and almost sure stability boundaries of the simply supported beam which was subjected to the axial compressions and varying damping which were two random processes. in [6, 7], kozić et al. investigated the lyapunov exponent and moment lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation. in [6], almost-sure and moment stability of the flexural-torsion stability of a thin elastic beam subjected to a stochastically fluctuating follower force were studied. in [7], moment lyapunov exponents and stability boundary of the double-beam system under stochastic compressive axial loading were obtained. in [9], pavlović et al. investigated the dynamic stability of thin-walled beams subjected to combined action of stochastic axial loads and stochastically fluctuating end moments. by using the direct lyapunov method, the authors obtained the almost-sure stochastic boundary and uniform moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 211 stochastic stability boundary as the function of characteristics of stochastic process and geometric and physical parameters. deng et al. [12] investigated the lyapunov exponent and moment lyapunov exponents of flexural-torsional viscoelastic beam, under parametric excitation of white noise. the system of stochastic differential equations of motion is first decoupled by using the method of stochastic averaging for dynamic systems with small damping and weak excitations. the moment and almost-sure stability boundaries and critical excitation are obtained analytically which are confirmed by numerical simulation. also, deng in [13] studied the moment stochastic stability and almost-sure stochastic stability through the moment lyapunov exponents and the largest lyapunov exponent of flexural-torsional viscoelastic beam, under the parametric excitation of a real noise. stochastic stability of a viscoelastic plate in supersonic flow as well typical example of a coupled non-gyroscopic system through lyapunov exponent and moment lyapunov exponents and are investigated by deng et al. [14]. the excitation is modelled as a bounded noise process. by using the method of stochastic averaging, the equations of motion are decoupled into itô differential equations, from which moment lyapunov exponents are readily obtained. the lyapunov exponents are obtained from the relation with moment lyapunov exponents. the aim of this paper is to determine a weak noise expansion for the moment lyapunov exponents of the four-dimensional stochastic system. the noise is assumed to be white noise of such small intensity that an asymptotic growth rate can be obtained. we apply the perturbation theoretical approach given in khasminskii and moshchuk [2] to obtain secondorder weak noise expansions of the moment lyapunov exponents. the lyapunov exponent is then obtained using the relationship between the moment lyapunov exponents and the lyapunov exponent. these results are applied to study the pth moment stability and almostsure stability of a thin-walled beam subjected to stochastic axial loads and stochastically fluctuating end moments. the motion of such an elastic system is governed by the partial differential equations in [9] by pavlović et al. the approximate analytical results of the moment lyapunov exponents are compared with the numerical values obtained by the monte carlo simulation approach for these exponents of a four-dimensional stochastic system. 2. theoretical formulation consider linear oscillatory systems described by equations of motion of the form 2 1 1 1 1 1 11 1 1 12 2 2 2 2 2 2 2 2 21 1 1 22 2 2 2 ( ) ( ) 0, 2 ( ) ( ) 0, q q q k t q k t q q q q k t q k t q + +  −   −   = + +  −   −   = (3) where q1, q2 are generalized coordinates, 1, 2 are natural frequencies and 21, 22 represent small viscous damping coefficients. the stochastic terms 1( )t and 2 ( )t are white-noise processes with small intensity with zero mean and autocorrelation functions 1 1 2 2 2 1 2 1 1 1 2 1 2 1 2 1 2 2 1 2 2 2 2 1 ( , ) [ ( ) ( )] ( ), ( , ) [ ( ) ( )] ( ), r t t e t t t t r t t e t t t t     =   =   − =   =   − (4) 212 g. janevski, p. kozić, r. pavlović, s. posavljak where 1, 2 are the intensity of the random process 1(t) and 2(t), and ( ) is the dirac delta. using the transformation 1 1 1 1 2 2 3 2 2 4, , ,q x q x q x q x= =  = =  (5) and denoting ij ij jp k=  , (i, j=1,2), (6) the above eqs. (3) can be represented in the first-order form by a set of stratonovich differential equations 1 2 ( ) ( )d dt dt dw t dw t= + +  + 0 1 2x a x ax b x b x , (7) where x = (x1 x2 x3 x4)t is the state vector of the system, w1(t) and w2(t) are the standard weiner processes and a0, a, b1 and b2 are constant 44 matrices given by 1 1 1 2 2 2 11 12 1 22 21 0 0 0 0 0 0 0 0 0 0 0 2 0 0 , , 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p p p p         − −     = =         − −                = =             0 2 a a b b , (8) applying the transformation 1 1 2 1 3 2 4 2 2 2 2 2 2 1 2 3 4 1 2 cos cos , cos sin , sin cos , sin sin , ( ) , 0 2 , 0 2 , 0 2, , p p x a x a x a x a p a x x x x p =   = −   =   = −   = = + + +             −   (9) and employing itô’s differential rule, yields the following set of itô equations for the pth power of the norm of the response and phase variables 1 2, ,    : * * * 1 11 1 12 2 * * * 2 21 1 22 2 * * * 1 1 3 31 1 32 2 * * * 2 2 4 41 1 42 2 ( ) ( ), ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ). p d a dt dw t dw t d dt dw t dw t d dt dw t dw t d dt dw t dw t =  +  +   =  +  +   =  +  +  +   =  +  +  +  (10) in the previous transformations, a represents the norm of the response, 1 and 2 are the angles of the first and second oscillators, respectively, and  describes the coupling or exchange of energy between the first and second oscillator moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 213 in the previous equation we have introduced the following markings * 2 2 2 2 1 1 1 2 22 ( sin cos sin sin )pp = −   +   , * 2 2 2 1 1 2 2( sin sin )sin2 =   −   * 3 1 1sin 2 = −  , * 4 2 2sin2 = −  , * 2 2 11 11 1 22 2[ sin 2 cos sin 2 sin ] 2 pp p p = −  +   * 21 11 1 22 2 1 [ sin 2 sin 2 ]sin 2 4 p p =  −   , * 2 31 11 1cosp = −  , * 2 41 22 2cosp = −  , (11) * 12 12 1 2 21 1 2[ sin cos cos sin ]sin 2 2 pp p p = −   +    , * 2 2 22 12 1 2 21 1 2 1 [ sin cos sin cos sin cos ] 2 p p =   −    , * 32 12 1 2cos cosp tg = −    , * 42 21 1 2cos cos cotp = −    . the itô version of eqs.(10) have the following form 1 11 1 12 2 2 21 1 22 2 1 1 3 31 1 32 2 2 2 4 41 1 42 2 ( ) ( ), ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ), p d a dt dw t dw t d dt dw t dw t d dt dw t dw t d dt dw t dw t =  +  +   =  +  +   =  +  +  +   =  +  +  +  , (12) where i are given in appendix 1 and * ij ij =  , (i, j=1, 2, 3, 4). following wedig [11], we perform the linear stochastic transformation 1 1 2 1 2( , , ) , ( , , )s t p p t s−=    =    , (13) introducing the new norm process s by means of the scalar function t(,1,2) which is defined on the stationary phase processes 1, 2 and  1 2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 2 1 0 1 2 00 01 02 11 12 22 11 21 31 41 1 12 22 32 42 2 ( ) ( ) ( ) ( ) ( ) ( ) , ds p t t dt p t m t m t m t m t m t m t m t m t m t dt p t t t t dw t p t t t t dw t                         =  + +   + + + +      + + + + + + +   +   +  +  +  +   +   +  +  +  , (14) where 0 2 11 21 12 22 1 3 11 31 12 32 2 4 11 41 12 42 2 2 00 21 22 01 21 31 22 32 02 21 41 22 42 2 2 2 2 11 31 32 12 31 41 32 42 22 41 42 , , , 1 ( ), m , m , 2 1 1 ( ), m , m ( ) . 2 2 m m m m m =  +   +   =  +   +   =  +   +   =  +  =   +   =   +   =  +  =   +   =  +  (15) 214 g. janevski, p. kozić, r. pavlović, s. posavljak if the transformation function t(,1,2) is bounded and non-singular, both processes p and s possess the same stability behavior. therefore, transformation function t(,1,2) is chosen so that the drift term, of the itô differential eq. (15), does not depend on the phase processes 1, 2 and , so that 1 2 1 2 1 11 21 31 41 1 1 12 22 32 42 2 ( ) ( ) ( ) ( ) ( ) . ds p s dt s t t t t t dw t s t t t t t dw t −    −      = +  +  +  +  +   +  +  +  +  (16) by comparing eqs. (14) and (16), it can be seen that such a transformation function 1 2( , , )t    is given by the following equation 0 1 1 2 1 2[ ] ( , , ) ( ) ( , , ).l l pt t+    =     (17) in (17) l0 and l1 are the following first and second-order differential operators 0 1 2 1 2 2 2 2 2 2 2 1 1 2 3 4 5 62 2 2 1 2 1 21 2 1 2 3 1 2 , , l l a a a a a a b b b c   =  +         = + + + + + + +          + + + +    (18) where 1a , 2a , 3a , 4a , 5a , 6a , 1b , 2b , 3b and c are given in appendix 2. eq. (17) defines an eigenvalue problem for a second-order differential operator of three independent variables, in which (p) is the eigenvalue and t(,1,2) the associated eigenfunction. from eq. (16), the eigenvalue (p) is seen to be the lyapunov exponent of the pth moment of system (7), i. e., (p) = x(t)(p). this approach was first applied by wedig [11] to derive the eigenvalue problem for the moment lyapunov exponent of a two-dimensional linear itô stochastic system. in the following section, the method of regular perturbation is applied to the eigenvalue problem (17) to obtain a weak noise expansion of the moment lyapunov exponent of a four-dimensional stochastic linear system. 3. weak noise expansion of the moment lyapunov exponent applying the method of regular perturbation, both the moment lyapunov exponent (p) and the eigenfunction t(,1,2) are expanded in power series of ε as: 2 0 1 2 2 1 2 0 1 2 1 1 2 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) , ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) . n n n n p p p p p t t t t t  =  +  +   + +   +    =    +     +     + +     + (19) substituting the perturbation series (19) into the eigenvalue problem (17) and equating terms of the equal powers of ε leads to the following equations moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 215 0 0 0 0 0 1 0 1 1 0 0 1 1 0 2 0 2 1 1 0 2 1 1 2 0 3 0 3 1 2 0 3 1 2 2 1 3 0 0 1 1 0 1 1 2 2 1 1 0 ( ) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) ( )n n n n n n n n l t p t l t l t p t p t l t l t p t p t p t l t l t p t p t p t p t l t l t p t p t p t p t p t      − − − − → =  → + =  + → + =  + + → + =  + + +        → + =  + + + +  + , (20) where each function 1 2( , , ) , 0,1,2,i it t i=    = must be positive and periodic in the range 0 2    , 10 2    and 20 2    . 3.1. zeroth order perturbation the zeroth order perturbation equation is 0 0 0 0( )l t p t=  or 0 0 1 2 0 0 1 2 ( ) t t p t    + =    . (21) from the property of the moment lyapunov exponent, it is known that 2 0 1 2(0) (0) (0) (0) (0) 0n n =  + +  + +  = , (22) which results in (0) 0n = for 0, 1, 2, 3,....n = since the eigenvalue problem (21) does not contain p, the eigenvalue 0 ( )p is independent of p. hence, 0 (0) 0 = leads to 0 ( ) 0p = . (23) now, partial differential eqs. (21) have the form 0 0 1 2 1 2 0 t t   + =   . (24) solution of eq.(24) may be taken as 0 1 2 0( , , ) ( )t    =   , (25) where 0 ( )  is an unknown function of  which has yet to be determined. 3.2. first order perturbation the first order perturbation equation is 0 1 1 0 1 0( )l t p t lt=  − . (26) since the homogeneous eq. (24) has a non-trivial solution given by eq. (25), for eq. (26) to have a solution it is required, from the fredholm alternative, that following is satisfied: * * 0 1 0 1 0 1 0 0( , ) ( ( ) , ) 0l t t p t l t t=  − = . (27) in the previous equation, * 0 0 ( )t =  is an unknown solution of the associated adjoint differential equation of (24), and (f,g) denotes the inner product of functions f (,1,2) and g(,1,2) defined by 216 g. janevski, p. kozić, r. pavlović, s. posavljak 2 2 2 1 2 1 2 1 2 0 0 0 ( , ) f( , , )g( , , )d d df g    =            . (28) taking onto account (25), (26) and (28), the expression (27) has the form 2 2 2 1 0 1 0 0 1 2 0 0 0 ( ( ) ) ( ) d d d 0p l      −       =   , (29) and will be satisfied if and only if 2 2 1 0 1 0 1 2 0 0 ( ( ) ) d d 0p l     −    =  . (30) after the integration of the previous expression we have that 2 0 0 0 1 1 1 0 1 02 ( ) ( ) ( ) ( ) ( ) 0 d d l a b c p dd    =  +  +   −  =  , (31) where ( ) ( ) 2 2 2 2 1 1 1 2 1 2 1 1 1 2 1 2 0 0 0 0 ( , , ) d d , b ( , , ) d d ,a a b      =       =         2 2 1 1 2 1 2 0 0 ( ) ( , , ) d d .c c    =       (32) finally, 1a , 1b and 1c are 2 2 2 2 1 11 22 12 21 2 2 2 212 21 1 2 2 2 2 2 11 22 12 21 2 2 2 2 1 11 22 12 21 2 2 2 2 12 21 1 1 ( ) [ 2( )]cos 4 128 ( )cos 2 16 1 [ 6( )] , 128 1 ( ) ( 1)[ 2( )]sin 4 64 1 ( sin cos cot ) 8 1 16 16 32 a p p p p p p p p p p b p p p p p p tg p  = − + − + − − −  − + + + + +  = − − + − + − −  −   + +  −    2 2 2 2 2 11 22 12 21 2 2 2 2 1 11 22 12 21 2 2 2 2 1 2 11 22 12 21 2 1 2 11 [( 2)( ) 2( 1)( )] sin 2 , 1 ( ) ( 2)[ 2( )]cos 4 128 1 16 16 [( 2)( ) 4( )] cos 2 32 1 64 64 [(10 3 )( 128 p p p p p p c p p p p p p p p p p p p p p  − + − + − −   = − + − + − −  −  − + − − − + + −  −  + + 2 2 2 22 12 21) 2(6 )( )] p p p p+ + + + (33) moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 217 since the coefficients (33) of the eq.(31) are periodic functions of , a series expansion of the function  0 () may be taken in the form 0 0 ( ) cos2 n k k k k =   =  . (34) substituting (34) in (31), multiplying the resulting equation by cos 2k (k = 0, 1, 2 ...) and integrating with respect  from 0 to /2 leads to a set of 2n+1 homogenuos linear equations for the unknown coefficients k0, k1, k2... k1 n 0j jjk k)p(ka = = , (35) where ( ) 2 0 cos(2 ) cos(2 )jka l j k d  =    , k=0, 1, 2, 3, ....n. (36) when n tends to infinity, the solution (34) tends to the exact solution. the condition for system homogeneous linear equations (35) to have nontrivial solutions is that the determinant of system homogeneous linear equations (35) is equal to zero. the coefficients ajk to order n=4 are presented in appendix 3. in the case when n=0, we assume a solution (34) in the form 0() = k0. from conditions that a00 = 0, the moment lyapunov exponent in the first perturbation is defined as 2 2 2 2 1 1 2 11 22 12 21 (10 3 ) (6 ) ( ) ( ) ( ) ( ). 2 128 64 p p p p p p p p p p + +  = −  + + + + + (37) in the case when n=1, the solution (34) has the form 0 0 1( ) cos2k k  = +  , then moment lyapunov exponent in the first perturbation is the solution of the equation 2 (1) (1) 1 1 1 0 0d d +  + = where coefficients (1) 0d and (1) 1d are presented in appendix 4. in the case when n=2, the solution (34) has the form 0 0 1 2( ) cos2 cos4k k k  = + +  , the moment lyapunov exponent in the first perturbation is the solution of the equation 3 (2) 2 (2) (2) 1 2 1 1 1 0 0d d d +  +  + = where coefficients (2) 0d , (2) 1d and (2) 2d are presented in appendix 5. however, for n > 2, it is impossible to obtain the explicit expressions of 1 ( )p and the numerical results must be given, for n = 3 and 4. 4. application to a thin-walled beam subjected to axial loads and end moments the purpose of this section is to present the general results of the above sections in the context of real engineering applications and show how these results can be applied to physical problems. to this end, we consider the flexural-torsional vibration stability of a homogeneous, isotropic, thin walled beam with two planes of symmetry. the beam is assumed to be loaded in the plane of greater bending rigidity by two equal couples and stochastic axial loads and stochastically fluctuating end moments (fig. 1). the governing differential equations for the coupled flexural and torsional motion of the beam can be written as given by pavlović et al. in [9] 218 g. janevski, p. kozić, r. pavlović, s. posavljak 2 4 2 2 2 4 2 2 2 2 2 4 2 2 2 4 ( ) ( ) 0, ( ) ( ) 0, u y p p s u u u u a ei m t f t tt z z z i u i gj f t m t ei t at z z z         + + + + =               + − − + + =       (38) where u is the flexural displacement in the x-direction,  is the torsional displacement,  is mass density, a is area of the cross-section of beam, iy, ip, is are axial, polar and sectorial moments of inertia, j is saint–venant torsional constant, e is young modulus of elasticity, g is shear modulus, u,  are viscous damping coefficients, t is time and z is axial coordinate. fig. 1 geometry of a thin-walled beam system using the following transformations ( ) ( ) 2 4 2 2 2 2 2 1 2 2 2 , , , ( ) , ( ) , , , , , 1 1 , , , 2 2 p t cr cr y s cr cr y t y y p u y p y py i u u z zl t k t f t f f t m t m m t a ei aial f m ei gj k e l ei i il l a gjal l s ei i ei iaei  = = = = =    = = = =  =   =  =   (39) where l is the length of the beam, fcr is euler critical force, mcr is critical buckling moment for the simply supported narrow rectangular beam, s is slenderness parameter, 1 and 2 are reduced viscous damping coefficients, we get governing equations as moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 219 ( ) 2 4 2 2 2 2 12 4 2 2 2 2 2 4 2 2 22 2 2 4 2 ( ) ( ) 0, 2 ( ) ( ) 0. u u u u sm t f t tt z z z u s f t sm t e tt z z z       +  + +  +  =             +  −  − +  + =     (40) taking free warping displacement and zero angular displacements into account, boundary conditions for the simply supported beam are ( ) ( ) ( ) ( ) 2 2 2 2 ( ,0) ( ,1) 2 2 2 2 ( ,0) ( ,1) ,0 ,1 0, ,0 ,1 0. t t t t u u u t u t z z t t z z   = = = =        =  = = =   (41) consider the shape function sin(z) which satisfies the boundary conditions for the first mode vibration, the displacement ( . )u t z and twist ( , )t z can be described by 1( , ) ( )sinu t z q t z=  , 2( , ) ( )sint z q t z =  . (42) substituting ( , )u t z and ( , )t z from (42) into the equations of motion (40) and employing galerkin method unknown time functions can be expressed as 2 1 1 1 1 1 11 1 12 2 2 2 2 2 2 2 21 1 22 2 2 ( ) ( ) 0, 2 ( ) ( ) 0. q q q k f t q k m t q q q q k m t q k f t q + +   − − = + +   − − = (43) if we are defined the expressions 2 4 1 =  , 2 4 2 ( )s e =  + , 4 11 22k k= =  , 4 12 21 ,k k s= =  (44) and assume that the compressive stochastic axial force and stochastically fluctuating end moment are white-noise processes (4) with small intensity 1( ) ( )f t t=  , 2( ) ( )m t t=  , (45) then eq. (43) is reduced to eq. (3). using the above result for the moment lyapunov exponent in the first-order perturbation, 2 1( ) ( ) ( )p p o =  +  , (46) with the definition of the moment stability (p) < 0, we determine analytically (the case where n = 0, 1(p) is shown with eq.(37)) the pth moment stability boundary of the oscillatory system as 4 2 2 1 2 1 2 1 10 3 6 64 32 s e p p s s e + + + +   +    +   +   . (47) 220 g. janevski, p. kozić, r. pavlović, s. posavljak it is known that the oscillatory system (40) is asymptotically stable only if the lyapunov exponent 0  . then expression )(o 2 1 += , (48) is employed to determine the almost-sure stability boundary of the oscillatory system in the first-order perturbation       + + ++ + 2 2 2 1 4 21 s 16 3 32 5 es es1 . (49) in [9], pavlović et al. by using the direct lyapunov method, investigated the almost sure asymptotic stability boundary of an oscillatory system as the function of stochastic process, damping coefficient and geometric and physical parameters of the beam. according to the authors, the condition for almost sure stochastic stability may be expressed by the following expression 8 2 2 2 4 2 2 1 2 1 2 1 2 1 2( ) 2 ( )[ ( )] 4 ( ) 0s s s e s e  +  −   +   + + +   +  . (50) for the sake of simplicity in the comparison of results, in the following we assume that two viscous damping coefficients are equal == 21 , (51) for this case, we determine the almost-sure stability boundary as       + + ++  2 2 2 1 4 s 6 5 es es1 32 3 , (52) and the pth moment stability boundary of the oscillatory system in the first-order perturbation as 4 2 2 1 2 1 [(10 3 ) 2(6 ) ] 128 s e p p s s e  + +   +  + +  + . (53) starting from eq. (50), derived by pavlović et al. [9], the almost sure stability boundary can be determined in the form 4 2 2 1 2( ) 2 s     +  . (54) with respect to standard i-section we can approximately take that ratios h / b  2, b / 1  11,  / 1  1.5, where h is depth, b is width,  is thickness of the flanges and 1 is thickness of the rib of i-section. these ratios give us s  0.01928(l/h)2 and e  1.176. for the narrow rectangular cross section, according to assumption /h < 0.1, for thin-walled cross sections s  1.88(l/h)2 and e  0, which is obtained using the approximation 1 + (/h)2  1. moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 221 a) i-section b) narrow rectangular cross section fig. 2. stability regions for almost-sure (a-s) and pth moment stability for 0.1 = almost-sure stability boundary and pth moment stability boundary in the first-order perturbation for i-section are given in fig. 2a, and for narrow rectangular cross section in fig. 2b. it is evident that stability regions in the present study are higher compared to the results obtained by pavlović et al. [9]. also, the moment stability boundaries (53) are more conservative than the almost-sure boundary (52). it is evident that end moment variances are about ten times higher for i-section than for narrow rectangular section, when stochastic axial force vary only a little. 5. numerical determination of the pth moment lyapunov exponent numerical determination of the pth moment lyapunov exponent is important in assessing the validity and the ranges of applicability of the approximate analytical results. in many engineering applications, the amplitudes of noise excitations are not small so that the approximate analytical methods such as the method of perturbation or the method of stochastic averaging cannot be applied. therefore, numerical approaches have to be employed to evaluate the moment lyapunov exponents. the numerical approach is based on expanding the exact solution of the system of itô stochastic differential equations in powers of the time increment h and the small parameter  as proposed in milstein and tret’yakov [8]. the state vector of the system (7) is to be rewritten as a system of itô stochastic differential equations with small noise in the form 1 1 2 2 1 1 1 2 11 1 1 12 3 2 3 2 4 4 2 3 2 4 22 3 1 21 1 2 , [ 2 ] ( ) ( ), , [ 2 ] ( ) ( ). dx x dt dx x x dt p x dw t p x dw t dx x dt dx x x dt p x dw t p x dw t =  = − −  +  + +  =  = − −  +  + +  (55) for the numerical solutions of the stochastic differential equations, the runge-kutta approximation may be applied, with error r = o(h4 + 4h). the interval discretization is [ 0t , t]: { kt : k=0,1,2,3, ....m; 0t < 1t < 2t .........< mt =t} and the time increment is h = tj+1 − tj. 222 g. janevski, p. kozić, r. pavlović, s. posavljak the following runge-kutta method used to obtain the (k+1)th iteration of the state vector x = (x1,x2,x3,x4) 2 2 4 4 3 2 3 2 ( 1) ( )1 1 11 1 1 1 1 1 1 1 2 2 5 2 2 2 2 2 ( )1 11 1 1 1 1 1 1 2 3 2 5 2 ( ) ( )12 1 2 2 12 1 2 2 3 4 ( 2 ) 1 2 24 2 3 1 1 6 6 9 ( 2 ) , 2 6 k k k k k h h p h h x x h p h h h h x p h p h x x +      +    = − + +  +  +            +  − +  +   − +             +     +  +  2 2 2 2 2 2 ( 1) 1 2 2 ( )1 1 1 2 1 11 1 1 1 1 2 2 4 4 3 2 4 4 2 2 ( )1 1 11 1 1 1 1 1 1 2 2 2 2 2 1 2 (1 2 12 2 3 1 1 1 6 3 6 ( 2 ) 1 2 1 2 24 2 36 3 1 6 6 k k k k h h h x h p h h x h h p h h h h x h h p h x +          = −  − +   − +   − +                     −    + − + +  +   − + − +          +   − −    3 2 ) ( )12 2 2 2 4 ( 2 ) , 2 kp h x   −  +  3 2 5 2 ( 1) ( ) ( )21 2 2 2 21 1 2 2 3 1 2 2 2 4 4 3 2 3 2 ( )2 2 22 2 1 1 2 2 3 2 2 5 2 2 2 2 2 ( )2 22 2 1 2 2 2 2 4 ( 2 ) 2 6 ( 2 ) 1 2 24 2 3 1 1 , 6 6 9 k k k k k p h p h x x x h h p h h x h p h h h h x +   +     =  +  +      +    + − + +  +  +            +  − +  +   −           (56) 2 2 2 2 3 2 ( 1) 1 2 ( ) ( )1 2 21 2 2 2 4 21 2 1 2 2 2 2 2 2 2 1 2 2 ( )2 2 2 2 22 1 2 2 3 2 2 4 4 3 2 4 4 2 2 2 22 1 1 1 2 2 ( 2 ) 1 6 6 2 1 1 1 6 3 6 ( 2 ) 1 2 2 24 2 36 k k k k h h p h x p h x x h h h h p h h x h h p h h h h +      −  =   − − +  +             + −  − +   − +   − +                   −   + − + +  +   − + 2 ( )2 41 . 3 kx    −       random variables i and i (i=1,2) are simulated as 1 ( 1) ( 1) 2 i ip p = − =  = = , 1 1 1 212 12 i ip p    −  = =  = =        . (57) moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 223 having obtained l samples of the solutions of the stochastic differential equations (56), the pth moment can be determined as follows 1 1 1 1 ( ) ( ) l pp k j k j e x t x t l + + =   =    , 1 1 1( ) [ ( )] [ ( )]t j k j k j kx t x t x t+ + += . (58) using the monte-carlo technique by xie [10], we numerically calculate the pth moment lyapunov exponent for all values of p of interest as 1 ( ) log ( ) p p e x t t   =   . (59) 6. conclusions in this paper, the moment lyapunov exponents of a thin-walled beam subjected to stochastic axial loads and stochastically fluctuating end moments under both white noises parametric excitations are studied. the method of regular perturbation is applied to obtain a weak noise expansion of the moment lyapunov exponent in terms of the small fluctuation parameter. the weak noise expansion of the lyapunov exponent is also obtained. the slope of the moment lyapunov exponent curve at p = 0 is the lyapunov exponent. when the lyapunov exponent is negative, system (43) is stable with probability 1, otherwise it is unstable. for the purpose of illustration, in the numerical study we considered set system parameters 1 = 2 =  = 1,  = 0.1, l = 4000, h = 0.0005, m = 10000 and x1(0) = x2(0) = x3(0) = 1/2. typical results of the moment lyapunov exponents (p) for system (43) given by eq. (46) in the first perturbation are shown in fig. 3 for i-section and the noise intensity 1 = 0.1 and 2 = 0.15. the accuracy of the approximate analytical results is validated and assessed by comparing them to the numerical results. the monte carlo simulation approach is usually more versatile, especially when the noise excitations cannot be described in such a form that can be treated easily using analytical tools. from the central limit theorem, it is well known that the estimated pth moment lyapunov exponent is a random number, with the mean being the true value of the pth moment lyapunov exponent and standard deviation equal to np / l , where np is the sample standard deviation determined from l samples. it is evident that the analytical result agrees very well with the numerical results, even for n = 0 when the function  0 () does not depend on  and assumes the form 0() = k0. the moment lyapunov exponents (p) in the first perturbation for narrow rectangular cross section and the noise intensity (1 = 0.15 and 2 = 0.01 are shown in fig. 4. unlike the previous example, it is observed that the discrepancies between the approximate analytical and numerical results decrease for larger number n of series (34). further increase of n number of members does not make sense, because the curves merge into one. 224 g. janevski, p. kozić, r. pavlović, s. posavljak fig. 3 moment lyapunov exponent )p( for i-section (1 = 0.1, 2 = 0.15) fig. 4 moment lyapunov exponent )p( for narrow rectangular cross section (1 = 0.15, 2 = 0.01) if we consider the influence of cross-sectional area of stability boundary, generally speaking, the narrow rectangular cross section has smaller stability regions than the isection. as for the influence of intensity of stochastic force, the end moment variances are about ten times higher for i-section than for narrow rectangular section, while the difference in axial force variances is small. acknowledgments: this research was supported by the research grant of the serbian ministry of science and environmental protection under the number oi 174011. moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 225 references 1. arnold, l., doyle, m.n., sri namachchivaya, n., 1997, small noise expansion of moment lyapunov exponents for two-dimensional systems, dynamics and stability of systems, 12(3), pp. 187-211. 2. khasminskii, r., moshchuk, n., 1998, moment lyapunov exponent and stability index for linear conservative system with small random perturbation, siam journal of applied mathematics, 58(1), pp. 245-256. 3. sri namachchivaya, n., van roessel, h.j., talwar, s., 1994, maximal lyapunov exponent and almost-sure stability for coupled two-degree of freedom stochastic systems, asme journal of applied mechanics, 61, pp. 446-452. 4. sri namachchivaya, n., van roessel, h.j., 2004, stochastic stability of coupled oscillators in resonance: a perturbation approach, asme journal of applied mechanics, 71, pp. 759-767. 5. kozić, p., pavlović, r., janevski, g., 2008, moment lyapunov exponents of the stochastic parametrical hill΄s equation, international journal of solids and structures, 45(24), pp. 6056-6066. 6. kozić, p., janevski, g., pavlović, r., 2009, moment lyapunov exponents and stochastic stability for two coupled oscillators, the journal of mechanics of materials and structures, 4(10), pp. 1689-1701. 7. kozić, p., janevski, g., pavlović, r., 2010, moment lyapunov exponents and stochastic stability of a doublebeam system under compressive axial load, international journal of solid and structures, 47(10), pp. 1435-1442. 8. milstein, n.g., tret’yakov, v.m., 1997, numerical methods in the weak sense for stochastic differential equations with small noise, siam journal on numerical analysis, 34(6), pp. 2142-2167. 9. pavlović, r., kozić, p., rajković, p., pavlović i., 2007, dynamic stability of a thin-walled beam subjected to axial loads and end moments, journal of sound and vibration, 301, pp. 690-700. 10. xie, w.c., 2005, monte carlo simulation of moment lyapunov exponents, asme journal of applied mechanics, 72, pp. 269-275. 11. wedig, w., 1988, lyapunov exponent of stochastic systems and related bifurcation problems, in: ariaratnam, t.s., schuëller, g.i., elishakoff, i. (eds.), stochastic structural dynamics–progress in theory and applications, elsevier applied science, pp. 315 – 327. 12. deng, j., xie, w.c., pandey m., 2014, moment lyapunov exponents and stochastic stability of coupled viscoelastic systems driven by white noise, journal of mechanics of materials and structures, 9, pp. 27-50. 13. deng, j., 2018, stochastic stability of coupled viscoelastic systems excited by real noise, mathematical problems in engineering, article id 4725148. 14. deng, j., zhong, z., li, a., 2019, stochastic stability of viscoelastic plates under bounded noise excitation, european journal of mechanics / a solids, 78, article id 103849. appendix 1 2 2 2 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 11 1 12 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 21 1 11 22 12 2 ( sin cos sin sin ) {cos [( 1)cos sin ]sin }( cos cos cos sin ) 2 {cos [( 1)sin cos ]sin }( cos sin cos cos ) 2 ( 2) ( 16 pp pp p p p pp p p p p p p p p p  = −   +   + +  + − +    +   + +  + − +    +   + − + + 2 21 1 2)sin 2 sin 2 sin 2 ,p    2 2 2 1 1 2 2 11 22 12 21 1 2 2 2 2 2 2 2 11 1 1 1 22 2 2 2 2 2 2 2 2 2 2 2 12 2 1 1 21 1 1 ( sin sin )sin 2 ( )sin 2 sin 2 sin 4 16 1 1 cos sin 2 (cos 2 cos 2 sin ) cos sin 2 (cos 2 cos 2 sin ) 4 4 1 1 cos sin (sin sin 2 cos ) cos cos 2 2 p p p p p p p tg p  =   −   − +    − −    −   +    +   + +     −   −  2 2 2 2(sin sin 2 cos ot ),c   −   http://pjm.math.berkeley.edu/jomms/2009/4-10/p03.xhtml http://www.sciencedirect.com/science?_ob=mimg&_imagekey=b6vjs-4ycwnn7-1-5x&_cdi=6102&_user=1793222&_pii=s0020768310000478&_orig=search&_coverdate=05%2f15%2f2010&_sk=999529989&view=c&wchp=dglbvtz-zskwb&md5=990c61b2ee24db395dc537dc4e637cc4&ie=/sdarticle.pdf 226 g. janevski, p. kozić, r. pavlović, s. posavljak 2 2 2 2 211 12 3 1 1 1 2 1sin 2 cos cos sin 2 2 2 p p tg    = −  −  +        2 2 2 2 222 21 4 2 2 2 1 2sin 2 cos cos cot sin 2 2 2 p p   = −  −  +        . appendix 2 2 2 2 2 2 1 11 1 22 2 12 1 2 21 1 2 1 1 a (p sin 2 p sin 2 ) sin 2 (p cos sin cos p sin cos sin ) 32 2 =  −  +   −    , += 2 2 2 1 2 2 12 1 4 2 11 2 tgcoscos 2 p cos 2 p a , += 2 2 2 1 2 2 21 2 4 2 22 3 cotcoscos 2 p cos 2 p a , ( ) 2 2 2 211 12 4 11 1 22 2 1 12 2 1 21 1 2 p p a p sin 2 p sin 2 cos sin 2 (p cos sin 2 sin tg p cos sin 2 sin 2 ) 4 4 = −  −   −    −    , 2 2 2 222 21 5 11 1 22 2 2 12 2 1 21 1 2 p p a (p sin 2 p sin 2 )cos sin 2 (p cos sin 2 sin 2 p cos sin 2 sin cot ) 4 4 = −  −   −   −     , 2 2 1 2 22116 coscosppa = , 2 2 2 1 1 1 2 2 2 2 11 22 12 21 1 2 2 2 2 2 2 2 2 2 2 2 11 1 1 1 22 2 2 2 2 2 2 12 2 p 1 b ( sin sin ) sin sin 2 (p p p p )sin 2 sin 2 sin 4 16 1 1 p cos sin 2 [cos 2(p 1)cos sin ] p cos sin 2 [cos 2(p 1)sin sin ] 4 4 1 p cos sin [(p 2 − =   −  −  + +   − −    + −   +    + −   − −   − 2 2 2 2 2 2 2 1 1 21 1 2 2 1 1)sin sin 2 cos tg ] p cos cos [(p 1)sin sin 2 cos cot ], 2  +   +   −  +   2 3 2 2 2 2 2 1 1 11 1 1 11 22 1 2 2 2 2 2 2 12 1 2 12 21 1 2 p b sin 2 p sin cos [(p 1)cos sin ] p p cos sin 2 sin 2 1 p p sin 2 cos (p 2 pcos 2 )tg p p cos sin 2 sin , 4 2 = −  +   − −  +   + +   − +  +    2 3 2 2 2 2 3 2 2 22 2 2 11 22 1 2 2 2 2 2 2 21 1 2 12 21 1 2 p b sin 2 p sin cos [(p 1)sin cos ] p p sin 2 cos cos 2 1 p p cos sin 2 (p 2 pcos 2 )cot p p sin 2 cos cos , 4 2 = −  +   − −  +   + +   − −  +    2 2 2 2 2 1 1 2 2 11 22 12 21 1 2 2 2 2 2 2 2 2 2 2 2 11 1 12 2 1 1 2 2 2 2 2 2 2 22 2 21 1 2 p(p 2) c 2p( sin cos sin sin ) (p p p p )sin 2 sin 2 sin 2 16 p (p cos cos p cos sin ){cos [(p 1)cos sin ]sin } 2 p (p cos sin p cos cos ){cos 2 − = −   +   + +   + +  +    + − +   + +  +    2 2 2 2[(p 1)sin cos ]sin },+ − +   moment lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial ... 227 appendix 3 2 2 2 2 00 1 1 2 11 22 12 21 2 2 2 2 2 10 1 2 11 22 12 21 2 2 2 2 2 2 20 11 22 12 21 12 21 2 2 30 12 21 (10 3 ) (6 ) ( ) ( ) ( ) ( ), 2 128 64 2 1 1 ( ) ( 2) ( ) ( ), 4 64 4 ( 2)( 4) 17 2( ) ( ), 256 32 3 ( ), 4 p p p p p a p p p p p p a p p p p p p p a p p p p p p a p p + + = − −  + + + + + + = −  − + + − + − + +  = + − + − +   = − 2 2 40 12 21( ),a p p= − + 2 2 2 2 01 1 2 11 22 12 21 2 2 2 2 2 2 11 1 1 2 11 22 12 21 2 2 2 2 2 21 1 2 11 22 12 21 2 2 31 11 ( 2) ( ) ( ) ( ), 4 64 16 1 7 22 8 10 56 ( ) ( ) ( ) ( ), 2 4 512 256 4 6 8 20 ( ) ( ) ( ), 8 128 32 10 24 ( 512 p p p p a p p p p p p p p p a p p p p p p p p p a p p p p p p a p + = −  − + − − − + − + − = −  −  + + + + + + + + + = −  − + − + − + + = + 2 2 2 2 2 2 22 12 21 41 12 21 10 216 ) ( ), ( ), 256 p p p p p a p p + + − + = − − 2 2 2 2 02 11 22 12 21 2 2 2 2 12 1 2 11 22 12 21 2 2 2 2 2 2 22 1 1 2 11 22 12 21 2 32 1 2 ( 2) [( ) 2( )], 256 2 ( 2)( 2) 2 ( ) ( ) ( ), 8 128 16 1 3 10 16 6 80 ( ) ( ) ( ) ( ), 2 4 256 128 6 7 8 12 ( ) ( 8 512 p p a p p p p p p p p a p p p p p p p p p a p p p p p p p p a − = + − + − + − − = −  − + − + − + − + − = −  −  + + + + + + + + = −  − + 2 2 2 2 11 22 12 21 2 2 2 2 2 2 42 11 22 12 21 18 ) ( ) 16 14 48 14 304 ( ) ( ), 512 256 p p p p p p p p p a p p p p + − + − + + + + = + − + 2 2 2 2 2 03 13 11 22 12 21 2 2 2 2 23 1 2 11 22 12 21 2 2 2 2 2 2 33 1 1 2 11 22 12 21 43 6 8 0 , ( ) 2( ) , 512 4 1 3 ( ) ( 2)( ) ( ) , 8 16 4 1 3 10 36 2 12 312 ( ) ( ) ( ) ( ), 2 4 256 256 p p a a p p p p p a p p p p p p p p p p a p p p p p p a − +  = = + − +   −   = −  − − + − + −    + − + − = −  −  + + + + + + = − 2 2 2 2 2 1 2 11 22 12 21 8 10 16 3 56 ( ) ( ) ( ), 8 128 32 p p p p p p p + + +  − + − + − 2 2 2 2 2 04 14 24 11 22 12 21 2 2 2 2 34 1 2 11 22 12 21 2 2 2 2 2 2 44 1 1 2 11 22 12 21 10 24 0 , 0, [( ) 2( )] , 512 6 2 ( ) ( ) ( ) , 8 16 1 3 10 64 2 12 512 ( ) ( ) ( ) ( ), 2 4 256 256 p p a a a p p p p p p a p p p p p p p p p a p p p p p − + = = = + − + − +  = −  − − − + −    + − + − = −  −  + + + + + 228 g. janevski, p. kozić, r. pavlović, s. posavljak appendix 4 2 2 (1) 2 2 2 2 1 2 11 22 12 211 1 21p 13p 7 11p 3p d p( ) (p p ) (p p ), 32 28 256 16 64 128        =  + + − − + + − − +         ( )( ) ( )(1) 2 2 0 1 2 1 2 2 3 4 2 3 4 4 4 4 4 11 22 12 21 2 3 4 2 3 2 2 11 22 1 1 2 2 3 8 4 13 5 5 5 ( ) ( ) 2048 8192 4096 32768 512 2048 512 8192 3 97 29 37 37 1024 4096 2048 16384 256 1024 2 d p p p p p p p p p p p p p p p p p p p p p p p p p = −  + + +   +     + − + + + + + − + + + +             + + + + + − + +     4 2 2 12 21 2 3 4 4 4 12 21 2 3 4 2 2 2 2 11 21 22 12 2 3 2 3 2 2 1 22 2 11 56 4096 23 7 17 5 ( ) 512 2048 2048 8192 15 7 9 5 ( ) 512 2048 2048 8192 3 37 21 5 5 5 ( ) 64 256 512 64 256 p p p p p p p p p p p p p p p p p p p p p p p p p   + +       + − + + + + +       + − + + + + +       + − − −  + + − −     2 2 1 11 2 22 2 3 2 3 2 2 2 2 1 21 2 12 1 12 2 21 ( ) 512 5 7 3 9 15 3 ( ) ( ). 32 128 256 32 128 256 p p p p p p p p p p p p    + +         + − −  + + − −  +           appendix 5 2 2 (2) 2 2 2 2 1 2 11 22 12 212 3p 5 31p 19p 27 17p 5p d ( ) (p p ) (p p ) 2 32 128 256 16 64 128        =  + + − − + + − − +         ( ) 2 2 (2) 2 2 1 1 2 1 2 2 3 4 2 3 4 4 4 2 2 11 22 11 22 2 1 3 9 3 15 1 2 8 16 4 8 3 47 41 61 35 1 55 153 133 83 ( ) 256 2048 32768 8192 32768 128 1024 4096 4096 16384 15 55 39 64 512 204 p p p p d p p p p p p p p p p p p p p     = − −  + − − −   +               + − + + + + + − − + + + +           + − − 3 4 2 3 4 4 4 2 2 12 21 12 21 2 3 4 2 3 4 2 2 2 2 11 12 22 21 13 3 55 231 71 13 3 ( ) 8 2048 8192 32 256 1024 1024 4096 9 165 109 47 17 3 153 61 35 17 ( ) 64 512 2048 2048 8192 32 256 1024 1024 4096 p p p p p p p p p p p p p p p p p p p p p p     + + + + − − + + +              + − − + + + + − − + +     2 2 2 2 11 21 22 12 2 3 2 3 2 2 2 2 1 22 2 11 1 11 2 22 2 3 2 3 2 2 1 21 2 12 1 ( ) 1 43 25 1 3 19 13 ( ) ( ) 8 8 128 256 8 16 128 256 3 45 7 5 3 63 5 5 ( ) ( 8 32 32 128 8 32 16 128 p p p p p p p p p p p p p p p p p p p p p p  + +         + + − −  + + − + − −  + +               + + − −  + + − + − −            2 2 12 2 21).p p+ 9954 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 307 319 https://doi.org/10.22190/fume211110001s © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper the effect of electron-beam treatment on the deformation behavior of the ebam ti-6al-4v under scratching artur r. shugurov, anton y. nikonov, andrey i. dmitriev institute of strength physics and materials science sb ras, tomsk, russia abstract. the effect of the continuous electron beam scanning (cebs) post-treatment on the microstructure, mechanical properties and scratching behavior of the ti-6al-4v alloy samples produced by electron beam additive manufacturing was studied experimentally and by using molecular dynamics simulation. it was found that the cebs post-treatment resulted in the transformation of the microstructure of the samples from the α′-martensite into the α+β structure. the evolution of the sample microstructure was shown to provide improved mechanical characteristics as well as enhanced deformation recovery after scratching. a mechanism was proposed based on the results of molecular dynamics simulation, which attributed the improved recovery of the scratch groves after passing the indenter to reversible β→α→β phase transformations, which occurred in the vanadium alloyed ti crystallites. key words: ti-6al-4v alloy, electron beam additive manufacturing, scratch testing, molecular dynamics, microstructure, phase transformations 1. introduction titanium alloys are among the most extensively used structural materials, especially in aerospace and biomedical applications [1]. this is due to the beneficial mechanical properties of titanium and its alloys, namely its high specific strength and excellent fatigue resistance, as well as biocompatibility and corrosion resistance. dual phase titanium alloys composed of hcp-α phase and bcc-β phase, particularly ti–6al–4v, which alone occupies about half of the global titanium product market, are the most popular titanium alloys. one of the main drawbacks of the titanium alloys is their poor machinability, which results in a large amount of material waste and significantly increases the cost of manufacturing titanium components from the mill products [2]. therefore, in recent years, additive manufacturing (am) also termed 3d printing, which provides for the building of received november 10, 2021 / accepted january 08, 2022 corresponding author: andrey i. dmitriev institute of strength physics and materials science sb ras, 634055, pr. akademicheski 2/4 tomsk, russia e-mail: dmitr@ispms.ru 308 a.r. shugurov, a.y. nikonov, a.i. dmitirev net shape structures of complex geometry by layer-by-layer adding of a material, has been considered the most promising technology for producing titanium components [3-6]. however, high temperature gradients and a rapid solidification of the molten material lead to the formation of metastable phases in the additively manufactured α+β titanium alloys. in particular, the presence of martensitic α′-phase is usually observed in as-built am ti– 6al–4v samples [7-9]. although such microstructure is characterized by high strength (>1000 mpa), it suffers from poor ductility and low toughness [4, 10, 11]. therefore, postmanufacturing heat treatments are generally required to transform the unfavorable α′microstructure into equilibrium α+β structure [12-14]. a large variety of heat-treatment routes have been proposed to improve the microstructure and mechanical properties of am ti-6al-4v [5, 15-18]. however, these processes require additional equipment that inevitably increases the manufacturing costs. in contrast, the electron beam post-treatment, which is widely used for surface finishing, modification and alloying of metals, can be performed directly in an electron beam additive manufacturing (ebam) machine. this technology involves two different approaches: (i) applying of defocused high-current pulsed electron beams (hcpeb) to the large surface area [19-21] and (ii) continuous electron beam scanning (cebs) of the sample surface with a focused electron beam [22-24]. usually both the methods use rather powerful electron beams that results in melting the surface layer of a material followed by its rapid cooling and solidification. the latter, as in the case of the additive manufacturing, favors the formation of metastable microstructures [20, 22]. however, at lower energy inputs the electron beam irradiation can be used for heat treatment of am components after their production. the cebs process appears to be more suitable for this aim because it provides more homogeneous heating of a sample as well as lower heating and cooling rates compared with the hcpeb process. in this study the continuous electron beam scanning was used for post-treatment of asbuilt ebam ti–6al–4v samples in order to investigate its effect on the microstructure and mechanical properties of the dual phase titanium alloy. scratch testing of the as-built and cebs treated samples was performed to study their ploughing and recovery behavior. scratch testing has been shown to be an adequate method for investigating plastic deformation of metals. it was used to study the effect of crystallographic orientations of grains [25, 26], internal interfaces [27, 28], different phases and inclusions [26, 29] on their plastic behavior of materials. this technique also proved its capability to reveal the development of strain-induced phase transformations [30, 31]. the molecular dynamics (md) simulation of the mechanical behavior of materials subjected to scratch testing has been successfully used for investigating defect nucleation as well as development of plastic deformation. defect-free single crystals with fcc [32-34] and bcc [35-37] crystal lattice as well as hcp crystals have been in the focus of the md simulations. therefore, the experimental results obtained in the present work are supported by molecular dynamic simulation of scratching the αand β-ti single crystals, which revealed the mechanisms of the formation of structural defects and phase transformations in the contact zone between the indenter and crystal. 2. experimental details two rectangular ti–6al–4v bars with dimensions of 25 mm × 25 mm × 70 mm (length × width × height) were obtained by wire-feed electron beam additive manufacturing using an ebm machine 6е400. grade 5 titanium wire 1.6 mm in diameter was used as a feed the effect of electron-beam treatment on the deformation behavior of ebam ti-6al-4v under… 309 material. the chemical composition of the wire corresponds to astm b348-13 [38]. an electron gun with a plasma cathode operated at an accelerating voltage of 30 kv was used to melt the wire. the distance between the source of the electron beam and the baseplate with dimensions of 150 mm × 150 mm × 10 mm was 630 mm. the angle between the baseplate and wire feed was 35°. the feed rate was 2 m/min. 22 layers were formed, each 3.2 mm thick. the first three layers were formed at a beam current of 24 ma followed by its decreasing to 21 ma. the 3d-printing strategy of the samples consisted in travelling the baseplate relative to the electron beam along a meander trajectory with mirror fused layers with a speed of 4 mm/s. the distance between the adjacent tracks within the same layer was ~ 3 mm. after welding each layer the baseplate went down by 3 mm. after cooling to room temperature, one of the bars was subjected to electron beam treatment in the ebm machine. the treatment consisted in continuous scanning the bar with an elliptically shaped beam spot (the major axis, which was perpendicular to the scanning direction, was 27 mm long, the minor axis was ~0.5 mm) moving along the bar with a speed of 10 mm/s. the accelerating voltage and beam current were 30 kv and 10 ma, respectively. during the treatment the bar was heated up to a temperature of 1040 °c and held at this temperature for 5 min followed by its cooling in vacuum to room temperature. the bars were separated from the baseplate and cut along the growth direction into 2 mm thick plates using spark cutting. the rectangular samples 10×10 mm in size and 2 mm thick were cut from the plates. the microstructure of the as-built and cebs treated ebam ti–6al–4v samples was investigated using an axiovert 40 mat optical microscope (carl zeiss, göttingen, germany). the samples for the examination were subjected to mechanical grinding and polishing followed by etching with kroll’s reagent. the phase composition of the samples was studied using x-ray diffraction (xrd) with a shimadzu xrd-7000 x-ray diffractometer (shimadzu corporation, kyoto, japan). the xrd-experiments were performed in the bragg-brentano geometry using cukα radiation (λ = 1.5406 å). the measurements of the mechanical properties of the samples as well as their scratch testing were carried out using a nanotest system (micro materials ltd., wrexham, uk). the nanoindentation was performed in a load controlled mode with a berkovich diamond tip at a maximum load of 50 mn. hardness h and young’s modulus e were determined using the oliver-pharr method [39]. the scratch tests were performed using a conical diamond with an apex angle of 120° and a tip curvature radius of 25 µm. the scratching was carried out with a constant velocity of 10 µm/s. 400 μm long scratching tracks were applied to all samples. the scratching process consisted of three steps. initially surface profiles of the tested samples were scanned (step 1) with a load of 0.1 μn (no wear occurs at this load). during scratching (step 2), the surface profile could be sensed and recorded by the depth sensing system. after scratching, the surface profiles of the samples along the scratch lines were scanned again (step 3) to record the deformation recovery. in the second step, the normal load applied to the indenter was linearly ramped between 0 and 200 µm scratching to a maximum load of 200 mn, while between 200 and 400 µm the load was constant at 200 mn. 5 scratches were performed for each sample. the surface topography of the samples in the vicinity of the scratch tracks was scanned using a solver hv atomic force microscope (afm, nt-mdt co., moscow, russia) operating in a contact mode. a series of 10 cross-sectional profiles of the scratch tracks were made and averaged to determine the residual scratch depth for each titanium sample. 310 a.r. shugurov, a.y. nikonov, a.i. dmitirev 3. model description in order to elucidate the experimentally observed phenomena of deformation of titanium samples with different crystallographic structures, two initially defect-free titanium crystallites were considered. crystallite 1 corresponded to the α-phase of pure ti, while crystallite 2 represented the β-phase of ti–6al–4v alloy as shown in fig. 1. each simulated crystallite had a shape of parallelepiped with dimensions of 30.0 × 26.0 × 12.5 nm along the x, y and z directions, respectively. the total number of atoms in both cases exceeded half of one million. taking into account the experimental data, the spatial orientation of the elementary crystal lattice in the laboratory coordinate system for the crystallite 1 was chosen as [1̅65̅0], [52̅3̅7̅], [3̅122̅], along the x, y and z axes, correspondingly. in the case of crystallite 2, the [100], [010] and [001] directions of the bcc crystal lattice were oriented along the x, y and z axes, respectively. crystallite 2 contained 13 at. % of vanadium that corresponded to a content of 4 wt. % in the model sample, which is typical for ti–6al–4v alloy. fig. 1 md model of the scratch test for ti crystallites with α-phase (hcp) and β-phase (bcc) structures scratching of the samples was realized through the movement of an indenter along the x axis at a fixed depth of 3.5 nm with a constant scratching speed of 15 m/s. the indenter has a spherical shape with a radius r of 6.5 nm. thus, atoms, which distance from the center of the indenter 𝑟 was less than equilibrium radius r, were acted upon by a force directed from the force center and equal to 𝐹 = −𝑘(𝑅 − 𝑟)2, (1) where 𝑘 is the tip stiffness coefficient. in our calculations, coefficient k was chosen to be equal to 0.1 ev/å3 similar to the earlier works [25, 28, 40]. a 1.5 nm thick bottom layer (shown grey in fig. 1) simulated a fixed substrate, while the other surfaces of the sample were considered free. a lateral “incursion” of a previously immersed indenter on the crystallite from one of its ends was modeled as denoted in the scheme. the interaction between ti atoms was described by a potential [41] constructed using the embedded atom method. a potential obtained within the frame of the modified embedded atom method was used to describe the interaction between titanium and vanadium atoms [42]. the model sample was considered as an nvt ensemble that maintains the number of atoms n, occupied volume v and the temperature of system t. all md calculations were implemented using the lammps [43]. to analyze the structure of the samples, the dislocation extraction algorithm (dxa) and common neighbor analysis (cna) algorithms implemented in the open visualization tool ovito were used [44]. the effect of electron-beam treatment on the deformation behavior of ebam ti-6al-4v under… 311 4. results and discussion 4.1 experimental investigation of the scratching behavior of the ebam ti–6al–4v samples typical microstructures observed in the ebam ti–6al–4v samples are shown in fig. 2. it is seen that the as-built sample exhibits a fine acicular α′-martensite lath structure within prior large-body primary β-grains (fig. 2a). the thickness of the laths is 2-3 µm. the cebs treatment resulted in increasing the thickness of the laths up to 5-7 µm and their fragmentation (fig. 2b). (a) (b) fig. 2 typical microstructures of (a) the as-built and (b) cebs treated ebam ti-6al-4v samples the xrd patterns of the samples are illustrated in fig. 3. the xrd pattern of the asbuilt sample shows the presence of only α-ti peaks with a strong (100) texture. the cebs treated ebam ti-6al-4v sample also primarily demonstrates peaks of the α-ti phase in the xrd pattern, but (100) preferred orientation became slightly less pronounced. in addition, the (110) peak of the β-ti phase appears in the xrd pattern, which indicates the β-phase retained in the sample after its electron beam irradiation. an analysis of the results revealed that the volume fraction of β-ti phase in the cebs treated sample was 6.7 %. fig. 3 x-ray diffraction patterns of (1) the as-built and (2) cebs treated ti-6al-4v samples 312 a.r. shugurov, a.y. nikonov, a.i. dmitirev the hardness and young’s modulus of the ebam ti–6al–4v samples are listed in table 1. it is seen that both samples are characterized by similar values of hardness and young’s modulus within experimental error. the h/e ratio of the sample subjected to postmanufacturing cebs treatment, which is usually used to rank ductility and toughness of materials, is about 10% higher than that of the as-built sample. table 1. mechanical properties of the ebam ti-6al-4v samples. sample h, gpa e, gpa h/e 1 3.91±0.41 130±4 0.030 2 4.22±0.18 125±6 0.034 (a) (b) fig. 4 longitudinal surface profiles of scratch grooves in (a) the as-built and (b) cebs treated ebam ti–6al–4v samples: 1 – initial surface profile, 2 – residual scratch profile, 3 – scratch profile at the applied load fig. 4 displays longitudinal surface profiles scanned along scratch lines in the ebam ti-6al-4v samples before, during and after scratching. it is seen that the cebs treated sample is characterized by the smaller indenter penetration depth (~600 nm) and residual scratch depth (~250 nm) compared with the as-built sample (~700 and ~400 nm, correspondingly). this well agrees with the results of the afm-investigations presented in fig. 5. the afm-images and cross-sectional surface profiles of the scratch grooves indicate that the scratching of the ti-6al-4v samples resulted in their ductile ploughing, which led to the formation of pile-ups along the groove flanks. plastic ploughing of the material in the as-built sample resulted in the formation of symmetrical pile-ups along both edges of the track, whereas in the cebs treated sample the pile-up was primarily formed only along one scratch flank. this is attributed to different crystallographic orientations of ti crystallites subjected to plastic deformation in these samples with respect to the surface and the scratching direction, which favor activation of different slip systems [25, 40]. it is clearly seen in fig. 5 that the as-built sample demonstrates wider and deeper residual scratch groove than the cebs treated sample. according to the afm results, the average depth and width of the scratch grooves are 40519 nm and 6.00.1 µm, respectively, in the the effect of electron-beam treatment on the deformation behavior of ebam ti-6al-4v under… 313 as-built sample, while they decrease to 24417 nm and 3.10.1 µm, respectively, in the cebs treated sample. fig. 5 afm-images (a, c) and corresponding cross-sectional surface profiles (b, d) of scratch grooves formed at the applied load 200 mn in (a, b) the as-built and (c, d) cebs treated ebam ti–6al–4v samples. 4.2. md simulation of scratching hcp and bcc ti crystallites fig. 6 displays longitudinal profiles of the scratch grooves in the hcp and bcc ti crystallites at different points of loading, when the indenter passed a distance of 10.7 nm (1, shown black), 18.2 nm (2, shown red) and 24.5 nm (3, shown blue). it is seen in fig. 6 that the scratch depths under loading are the same at different time points and correspond to the penetration depth of the indenter into the sample (~3.5 nm). however, after unloading, i.e. moving the indenter away from the considered point, there is a difference in the recovery of the scratch grooves. the residual scratch depth in the bcc ti crystallite (2.3-2.9 nm) is smaller than in the hcp crystallite (2.8-3.3 nm). this is even more clearly evidenced in fig. 7, which exhibits cross-sectional surface profiles of scratch grooves in the ti crystallites cut at x = 10.7 nm before, during and after loading. substantially more pronounced recovery of the scratch groove is observed in the bcc crystallite compared with the hcp one. thus, the results of the md simulations well agree with the experimental findings. (a) (b) (c) (d) 314 a.r. shugurov, a.y. nikonov, a.i. dmitirev (a) (b) fig. 6 longitudinal surface profiles of scratch grooves in (a) hcp and (b) bcc ti crystallites after different scratching distances: 1 – 10.7 nm, 2 – 18.2 nm, 3 – 24.5 nm. arrows indicate the position of the indenter center at the corresponding time points (a) (b) fig. 7 cross-sectional surface profiles of scratch grooves in (a) hcp and (b) bcc ti crystallites cut at x = 10.7 nm at different points of loading: 1 – before loading, 2 – under loading (scratching distance is 10.7 nm), 3 – after unloading (scratching distance is 18.2 nm) in order to gain insight into the physical mechanisms underlying the stronger recovery of the scratch grooves in the bcc titanium, the evolution of local atomic configurations during scratching the ti crystallites was analyzed. the number of atoms belonging to hcp and bcc arrangement was plotted as a function of scratching distance for both ti crystallites in fig. 8. it can be seen that the crystallites demonstrate a decrease in the number of atoms belonging to the main matrix (hcp for crystallite 1 and bcc for crystallite 2) and an increase in the number of atoms with different local ordering (bcc for crystallite 1 and hcp for the effect of electron-beam treatment on the deformation behavior of ebam ti-6al-4v under… 315 crystallite 2) with increasing scratching distance. the first trend is attributed to continuous increasing the relative volume of the crystallites involved in plastic deformation, which results in disordering and fragmentation of their crystal structure. the second trend indicates the deformation-induced phase transformations hcp→bcc in crystallite 1 and bcc→hcp in crystallite 2), which occurred under loading. it should be noted that the number of bcc atoms in crystallite 1 increases until the end of scratching, whereas the number of hcp atoms in the crystallite 2 reaches the maximum value after ~9 nm of scratching, i.e. when the indenter passed approximately a third of the sample length, and nearly twice drops thereafter. the latter indicates that after unloading the hcp atoms in crystallite 2 can rearrange into the bcc structure, i.e. the reversible β↔α phase transformations can occur in β-ti during scratching. figs. 9a-9c demonstrate evolution of the atomic structure of a fragment of the bcc titanium crystallite during scratching. in order to visualize atoms belonging to different types of the crystal lattice a cna analysis was used. the comparison of figs. 9a and 9b indicates that atomic rearrangement from the bcc lattice into the hcp local configuration occurs under compression induced by the indenter action. in addition, the formation of disordered atomic clusters is observed, which atomic configuration cannot be identified using cna. when the indenter passed through the fragment, the majority of the hcp atoms rearranged again into the bcc lattice (fig. 9c). (a) (b) fig. 8 number of atoms belonging to hcp and bcc crystal lattice in (a) hcp and (b) bcc ti crystallites as a function of scratching distance. atoms with non-identified type of ordering are not shown the possibility of the reversible β↔α phase transformations in vanadium-alloyed titanium is confirmed by the analysis of the total energy per atom performed using md calculations for ti crystallites. fig. 10 shows the total energy per atom as a function of atomic volume in hcp and bcc ti crystallites containing 13 at. % of vanadium, which is typical for the β-phase in ti-6al-4v alloy. it is seen that bcc configuration is energetically more favorable in equilibrium state before loading. however, the hcp structure becomes preferable at decreasing the volume per atom, i.e. under compression. therefore, the transformation of β-ti phase characterized by lower packing density into the more closepacked α-ti phase can occur in the zones of compression beneath and ahead of the moving indenter. when the indenter moves away it results in the development of tensile stresses 316 a.r. shugurov, a.y. nikonov, a.i. dmitirev behind it. therefore, according to fig. 10, the bcc arrangement becomes more favorable again that furthers the development of the reverse α→β phase transformation. the reverse phase transformation results in the decrease in the number of the hcp atoms in the vanadium-alloyed ti crystallite that explains its drop in fig. 8b. (a) (b) (c) fig. 9 evolution of the atomic structure of the fragment of bcc ti crystallite during scratching: a – before loading; b – under compression induced by indenter; c – after unloading. bcc atoms are shown blue, hcp atoms are shown red, and atoms belonging to unidentified local configurations are shown grey fig. 10 energy per atom in hcp and bcc ti crystallites with 13 at. % of vanadium as a function of atomic volume evidently, the reversible β→α→β phase transformations can shed light on the origin of the experimentally observed enhanced recovery of the scratch groove in the cebs treated ebam ti–6al–4v sample. since the as-built ti–6al–4v sample did not contain the β phase, the scratch recovery was only happened by means of relaxation of elastic strains. in contrast, the cebs treated ebam ti–6al–4v sample contained the β phase, so that the reversible β→α→β phase transformations could contribute to the densification of the material under loading and its additional recovery after unloading. the effect of electron-beam treatment on the deformation behavior of ebam ti-6al-4v under… 317 4. conclusion the microstructure, mechanical properties and scratching behavior of as-built ebam ti–6al–4v samples and the samples subjected to post-manufacturing continuous electron beam scanning were investigated. the comparative study revealed that the cebs posttreatment resulted in the transformation of the single phase α′-martensitic structure into the dual phase α+β structure. the microstructure evolution was accompanied by increasing the ratio of hardness to young’s modulus, which indicated the improvement of toughness and ductility of the cebs treated ti–6al–4v. scratch testing of the samples revealed significant improvement of deformation recovery of scratch grooves in the cebs treated ebam ti–6al– 4v sample compared with the as-built sample. molecular dynamics simulation of scratching ti crystallites with hcp and bcc structure was performed to gain insight into the physical origins of the enhanced deformation recovery. the simulation showed that reversible β→α→β phase transformations can be one of the mechanisms responsible for the experimentally observed enhanced recovery of the scratch groove in the cebs treated ebam ti–6al–4v samples containing β phase. these phase transformations can occur in the vanadium alloyed bcc ti crystallites in the zones of compression beneath and ahead the moving indenter, because the more closed packed α-ti phase becomes more energetically favorable. thus, the study showed the possibility to use the post-manufacturing cebs process to improve the microstructure and mechanical properties of ebam ti-6al-4v alloy. acknowledgements: the research was performed according to the government research assignment for ispms sb ras, projects fwrw-2021-0006 and fwrw-2021-0010, and was funded by rfbr and tomsk region, project number 18-48-700009. references 1. froes, f.h., 2015, titanium: physical metallurgy, processing and application, asm international: materials park, oh, usa. 2. bermingham, m.j., kent, d., zhan, h., stjohn, d.h., dargusch, m.s. 2015, controlling the microstructure and properties of wire arc additive manufactured ti–6al–4v with trace boron additions, acta materialia, 91, pp. 289-303. 3. dutta, b., froes, f.h., 2015, the additive manufacturing (am) of titanium alloys, in: qian, m., froes, f.h. 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original scientific paper numerical simulation of dynamics of block media by movable lattice and movable automata methods alexander e. filippov1,2, valentin l. popov2 1donetsk institute for physics and engineering, donetsk, ukraine 2technische universität berlin, berlin, germany abstract. two versions of modified burridge-knopoff model including state dependent friction, elastic force and thermal conductivity are derived. the friction model describes a velocity weakening of friction and elasticity between moving blocks and an increase of both static friction and rigidity during stick periods as well their weakening during motion. it provides a simplified but qualitatively correct behavior including the transition from smooth sliding to stick-slip behavior, which is often observed in various tribological and tectonic systems. attractor properties of the model dynamics is studied also. the alternative versions of the model are proposed which apply a simulation of the motion of interacting elastically connected mesh elements and motion of relatively large solid blocks, utilizing technique of the movable cellular automata. first version of the model was already basically studied before. its advanced version here involves all components of the real system: state-depending friction and changeable rigidity, as well as heat production and thermal conductivity. model based on the movable automata also involves the components included into traditional lattice model. it has its own ad-vantages and disadvantages which are also discussed in the paper. key words: burridge-knopoff model, friction, tectonics, voronoi diagram, delaunay triangulation 1. introduction one of the most important applications in the study of the behavioral patterns of mechanical systems of block structures are the geological faults and block environments. of particular interest, of course, is the study of the possible variations of their deformation, potential prediction or even prevention of the strong seismic shocks. the fact that the statistics of earthquake force and their time correlations meet the laws of gutenberg-richter [1,2] and omori [1,3], typical for self-organized critical systems [4,5], received: march 27, 2022 / accepted april 23, 2022 corresponding author: alexander e. filippov donetsk institute for physics and engineering, r. luxemburg str. 72, 83114 donetsk, ukraine e-mail: filippov_ae@yahoo.com 554 a.e. filippov, v.l. popov is often the basis for the conclusion that earthquakes are a process formed on all spatial scales ranging from microscopic to continental plate scale. during last decades [6,10] on the basis of modeling by movable cellular automatons and full-scale experiments it was shown some possibility of initiation of the fault wing displacement and, consequently, the release of a part of the accumulated elastic energy as a result of local effects on the fault area, including vibration load and watering [11-16]. one of the often applied to study the problems under interest is the well-known burridge-knopoff (bk) model [17,18] initially proposed many years ago to investigate statistical properties of earthquakes. numerical studies by carlson et al. [19,20] have demonstrated that bk model can reproduce characteristic empirical features of tectonic processes such as the gutenberg-richter law for the magnitude distribution of earthquakes, or the omori law for statistics of aftershocks [19-22], both properties stemming from the so called “self-organized criticality” of this system. it has been intensively used to simulate different aspects of the problem [21-40] and to discuss general properties of earthquakes statistics as well as predictability of earthquakes. numerical simulations give evidences that the self-organized criticality and the corresponding fractal attractor of the system is closely related to dynamic structures with “traveling waves” [23], their ordering and specific “phase transitions” [24] controlled by a number of parameters (external driving velocity, springs stiffness, number of blocks, their mutual interaction and so on). it is in particular the dependence of the dynamic properties of the bk model on spring stiffness which makes it necessary to introduce the generalization of the friction law proposed in the works [41,17]. the physical reason for the stick-slip instability in the burridge-knopoff model is the assumed decrease of the friction force with the sliding velocity [19,20]. motion of a single block with this friction law is always unstable which does not correspond to properties of real tribological systems. the real law of rock friction is more complicated [29,34]. in this article we base on the realistic friction laws described in the handbooks [34,35]. the main qualitative pictures of a realistic law are: (a) approximately logarithmic increase of the static friction force as a function of contact time – the property, found already by coulomb, and (b) a logarithmic dependence of the sliding friction on the sliding velocity. both properties can be described in the frame of “state dependent” friction laws by introducing additional internal variables describing the state of the contact [17,41-43] as well as shear band propagation effect on the deformation and fracture [44,45], generation and propagation of slow deformation in geomedia [46-48]. modified version of the bk model with a state dependent friction force, reproduces in the simplest way both the velocity weakening friction and the increase of static friction with time when the block is not moving. 2. method let us briefly remind the original bk model. the blocks of mass m are attached to a moving surface by springs with stiffness k1 and are coupled to each other by springs with stiffness k2. the moving surface has velocity v, and the blocks are in contact with a rough substrate. the friction force f between the blocks and the rough surface is assumed to depend only on velocity. the equations of the bk model can be written in the following form: block-media motion lattice and movable automata numerical models 555 ( ) ( ) 2 2 1 1 12 2 ( ) j j j j j j u m k u u u k vt u f v t + −  = + − + − −  , (1) where v=const is the external driving velocity and vj≡∂uj/∂t is an array of individual block velocities (j=1, … n). the sliding friction force is supposed to decrease monotonously from a constant initial value f0. it is further supposed that the static friction f(vj→0) can possess any necessary negative value to prevent back sliding 0 0, 1 ; / 0 1 2 / (1 )( ) ( ,0] / 0 j jj j f f u t vf v u t  = −    + −=   −   =    . (2) the parameter α defines a rate of friction decrease when block starts to slide, and σ is the acceleration of a block at instant when slipping begins [19,20]. the friction equation (2) is an oversimplification of real properties of static and kinetic friction which is now explained in detail in the books [34,35]. dieterich has proposed friction equations with internal variables [29,30], where the friction force is dependent upon an additional variable that describes the state of the contact zone. this variable is in a sense a “melting parameter.” friction force at non-zero velocity drops down from its initial value due to a “shear melting” effect which may have different physical nature [17-20]. when the motion stops, the surfaces start to form new bonds and the static friction increases with time. the dynamics of systems with state dependent friction has been investigated in a number of papers [29-37]. majority of these studies were devoted to the simple oneparticle version of the model, but it was also done for many-body bk model with a state dependent friction [17], where the friction force is defined by the additional kinetic equation: ( )1 0 2 [ ( )]j j j j f v t f f v t  = − +    . (3) with β2<0, at vj>0, and f(vj)=-∞ in opposite case vj≤0.the parameters β1 and β2 have following meaning. when a block starts to slide with vj>0 its friction force monotonously decreases from an initial value f0. general time scale of this process is determined by the first parameter β1 and an effectiveness of the melting is given by a relation between absolute value of negative parameter β2<0 and β1. it should be noted that the kinetic equations for state dependent friction at sliding are not unique. different forms of this equation have been proposed [17,18] leading to qualitatively similar behavior of the dynamic friction at vj>0. the above form is just simplest and even linear one. the equations of motion in eqs. (1-3) are nonlinear wave equations. last few decades different variants of such equations have been widely studied, starting from the very early implementation of the numerical simulations [38]. in particular, it has been shown that n interacting segments of the nonlinear chain form a collective attractor with energy transfer performed by nonlinear excitations [39,40]. depending on the relations between total energy and strength of the interaction between the blocks k2, the chain can form nearly uniform or strongly non-uniform structures and phase patterns. 556 a.e. filippov, v.l. popov analogous behavior should be expected in the modified burridge-knopoff model as well. instantaneously moving blocks must be involved with stationary ones in the overall motion. this causes a “detachment” wave, propagating along the chain. such waves were found and studied in the original bk model [18], and the analogous process exists in the modified one. there are two possible types of traveling waves: (a) after a transition time all the blocks are provoked to move simultaneously, (b) in steady state some of the blocks can be found instantaneously motionless while others move. one would expect the reaction of the chain to depend on the strength of the interaction between the blocks. this question has been studied for the original bk model in [20], and a specific “phase transition” between correlated and uncorrelated behavior has been found. real contact of two surfaces is two-dimensional. let us generalize the mbk to the case. the generalized model is very similar to the one-dimensional model but incorporates a 2d array of blocks connected by elastic springs in both directions. all other components of the mbk remain unchanged. the system of equations of motion takes the form: ( ) ( ) 2 , , 2 1, 1, , 1 , 1 , 1 , ,2 4 ( ); j n j n j n j n j n j n j n j n j n u u m k u u u u u k vt u f v t t + − + −   = + + + − − + − +    (4) ( ), , 1 0 , 2 , , , , ( ( )) , 0, 0 ( ) 0 j n j n j n j n j n j n j n f v t f f v v t f v v     = − +    = −  . (5) where j=1, …, nx and n=1, …, ny. nx and ny are the numbers of elements in the xand ydirections. for general applicability to tribological problems eq.(4) contains also a viscous term η∂uj/∂t. our calculations show that attractor properties are weakly influenced by the viscous term at sufficiently high velocities vj. for generality, we keep small nonzero value η=0.05. other parameters are: β1=1, β2=125, f0=1, k1=1. k2=36, v=0.2. large scale structure of the wave state in the two-dimensional model with number of dimensionless grid cells (nx = 700, ny = 300) shown by the velocities v=v(x,y;t)≡∂u/∂t is reproduced in fig. 1 by the colored surface (matlab color-map ‘jet’). dark blue corresponds to zero. mutual scattering manifests itself in high sharp peaks of the “events” reproduced here by the intensive red of the corresponding densities. instant structure of the wave state and history of the events are combined in fig. 2. the subplots (a) and (c) here represent snapshots of the instantaneous densities of the local displacements u=u(x,y;t) and velocities v=v(x,y;t)≡∂u/∂t, respectively. time-space representation of this process is shown in the subplot (b), and time dependency of total area of events (d). the information about attractor behavior is accumulated in fig. 3 and briefly described in the caption to this figure. block-media motion lattice and movable automata numerical models 557 fig. 1 large scale structure of the wave state in the two-dimensional model shown by the velocities v=v(x,y;t)≡∂u/∂t reproduced by the colored surface (matlab color-map ‘jet’). dark blue corresponds to zero. mutual scattering manifests itself in high sharp peaks of the “events” reproduced here by the intensive red of the corresponding densities. fig. 2 instant structure of the wave state and history of the events in the model. subplots (a) and (c) represent snapshots of the instantaneous densities of the local displacements u=u(x,y;t) and velocities v=v(x,y;t)≡∂u/∂t, respectively. the colormaps are equidistant and normalized to the intervals between maximum (red) and minimum (blue) values for the displacements [-1,1] and velocities [0 1.5]. time-space representation of this process is shown in the (b) and (d) by the spatial-temporal map and time dependency of total area of events. 558 a.e. filippov, v.l. popov fig. 3 the attractor of the variable effective friction in projection to the plane {v, ffriction}, where <ffriction >=<f[vj(t)]+ ηvj(t)> is shown by color-map ‘jet’ in the subplot (a). the attractor behavior of the system causes quasi-periodic behavior of the mean area of the events. correlation function in the subplot (b) here is calculated for the area of events shown in fig. 2a. the same attractor is projected onto the plane {v,u} in subplot (c). to characterize the difference of the correlated and uncorrelated behavior, an order parameter has been introduced in ref. [25]. if we define a value hj in block number j by the condition: 1 u / 0 (theblock is moving); 0 u / 0 j j j t h t    =    = , (6) then the local density of the order parameter h* j can be written as h* j=hj(hj+1+hj-1). this function takes on unit value h* j=hj(hj+1+hj-1)=1 if block j is moving and exactly one of its nearest neighbors is also moving. further, h* j=hj(hj+1+hj-1)=2 when both nearest neighbors of the moving block are in motion. all other cases yield h* j=0. our observations with the modified model show that even for blocks at vanishing inter-block interaction k2→0 (when the motion of neighboring blocks is almost uncorrelated), the fraction of configurations where sets of neighboring blocks are moving is still relatively high. all that to say, the combination h* j=hj(hj+1+hj-1) does not vanish in such an uncorrelated system. it is therefore convenient to construct another simple combination: 1 1 1 all 3 blocks , 1, 1 in contact are moving 0 otherwise j j j j j j j h h h h+ − + − = =   . (7) this makes hj=hjhj+1hj-1 zero in all cases except when both neighboring blocks of a central sliding block are also in motion. this combination can be used as an “order parameter”. to extract integral quantitative information about the steady ordered and disordered states we calculate also the time evolution of the ensemble-averaged order parameter: h(t>≡ <hj(t)>=1/t ∫hj(t)dt. as an illustration the dependence of the integrated order parameter on the stiffness k2, showing a transition from correlated to uncorrelated block motion was calculated for onedimensional system. it is shown in fig.4. two well defined limiting cases can be seen easily: the order parameter tends to a constant non-zero asymptote <hj(t)>→const≠0 at strong interaction k2>>1 and vanishes <hj(t)>→0 below the transition point k2≈0. block-media motion lattice and movable automata numerical models 559 fig. 4 phase transition from correlated to uncorrelated motion of blocks in a onedimensional system. the order parameter tends to a constant asymptote at high mutual interaction k2>>1 and vanishes below transition point k2= k2,critical≈1. two intermediate fluctuation regions a and b correspond to the states with shortrange and long-range correlated nonlinear excitations, respectively. let us return now to general problem. if the interaction is already varied due to the motion with nonzero velocity, it sounds natural to modify the elasticity (potential channel of the interaction) of the system into the state dependent one, which varies at nonzero velocity and gradually returns to the initial one in static limit: ( ) ( ) 2; , 2 2; 1, 2; 1, 2; , 1 2; , 1 2; , 1 20 2 ,4 j nj n j n j n j n j n k j n k k k k k k k k v t + − + −  = + + + − + − +     . (8) the same note can be done about the thermalization of the system by a heating produced by the mechanical work performed by the external force. local power consumed by the system in every node of the lattice can be calculated as a work per unit of time: q(rj,n)=fj,nvj,n. it can be treated as local source of energy acting in every node and temperature expansion inside the system should be described by the thermal conductivity equation: ( ), 1, 1, , 1 , 1 , ,4 ( ) j n j n j n j n j n j n j n t t t t t t q r t + − − +  = + + + − +   . (9) it is possible to repeat all the simulations of the previous section, reproduce all the results presented in figs.1-3 and demonstrate that these properties are common for all the modifications of the model. so, let us concentrate below on some additions coming from generalized version of the model. large scale structure of the velocity fluctuations analogous to that plotted in fig. 1 accompanied by the variations of the elasticity self-consistently calculated according to eq. (8) is shown by the colored surfaces (mainly red picks and blue smooth relief, respectively) in fig. 5. one can see directly that strong fluctuations of the velocity cause deep valleys in the surface of elasticity and in turn “prefer” to appear in the same places again. 560 a.e. filippov, v.l. popov fig. 5 large scale structure of the velocity fluctuations accompanied self-consistently by the variations of the elasticity are shown by the colored surfaces (mainly red picks and blue smooth relief, respectively). fig. 6 the temperature deviations (bright spots in matlab color-map ‘hot’ normalized in the interval [0 2]) caused by the same instant configuration of the system as in fig. 5. green contour lines show median value of the elasticity profile and mark mutual correlation between its deep valleys and hot regions. the same correlation exists between the spatially dependent elasticity and the temperature deviations. in fig. 6 the temperature deviations (bright spots in matlab colormap ‘hot’) caused by the same instant configuration of the system as in fig. 5 are shown. green contour lines show median value of the elasticity profile and mark mutual correlation between its deep valleys and hot regions. to extract physically meaningful quantitative information about the earthquakes in the system different time moments one can calculate a density of power consumed by the block-media motion lattice and movable automata numerical models 561 system at each time moment and overlap it other densities of the system like: distributed rigidity, friction constant and temperature. the local power in the frames of the model can be calculated as a work produced by the forces acting on each lattice node of the system pij=fijvij during a unit of time. calculating spatially distributed power pi=fivi enables one to evaluate the degree of localization of excitations and hence the fraction of the regions in which the fragments of the lattice move practically as a whole. as at a particular site the energy is absorbed and released at different points in time, it is useful to characterize these sites by the absolute value of the power, |pi|=|fivi|. however, if formally accumulate the statistics of such “events” using all the nodes as independent realizations, strongly overestimated value of the small “events” will be obtained. physically only the quakes with a magnitude higher than a critical one would be treated as the observable ones. moreover, if the amplitude of the power exceeds the critical value for a number of mutually connected nodes, all of them will be treated by the potential observers as one common event with an average amplitude in this connected area. so, we organized the procedure in this manner: all the nodes were found with the amplitudes higher than the critical one; defined all the connected domains of them and calculated mean amplitudes for every such a region. to control visually the above procedure, we plotted these domains by the (bright) ‘jet’ colors over the (blue) map of the rigidity for every time moment. typical instant realization of such a picture is presented in fig. 7. the histogram calculated for these events, as it is shown in fig. 8 in double-logarithmic scale. fig. 7 instant connected regions of the “earthquakes” (bright spots) and valleys of the small rigidity (deep blue) plotted together to illustrate their mutual correlations. the artificial colors for the magnitude and rigidity are shifted to red and blue parts of the spectrum to clearly overlap both densities in one picture. as we already noted above the variations of the friction, elasticity and local velocities are mutually correlated. in other words, new events are preferably generated in the same regions (as it is in the reality). as a result, the system partially memorizes such a common relief for extremely long times. one can calculate, for example, time-averaged relief of effective rigidity, and we did this. it is quite expected that the variations are found to be much smaller than local instant variations of the same value, but they exist. 562 a.e. filippov, v.l. popov fig. 8 scaling of the probability to find an earthquake with a particular value of magnitude calculated for the connected event regions. physically it corresponds to a selection of the rare but intensive “events”, which is compatible with the ideology of the empirical gutenberg-richter law. there are also some correlations between such a memorized structure and preferable channels of the wave propagation. again, the correlation is much weaker than it is observed between instant variations of different densities, but it still exists. these results are illustrated in fig. 9, where instant and almost stationary time averaged elastic coefficient integrated over long time periods ((a) and (b) sub-plots, respectively) shown by the ‘jet’ color-map. black contour line depicts mean value of the long-time living rigidity surface. it marks only partial correlations between the averaged and instant rigidities, analogous (only partial) correlation is found between travelling regions of maximal power production (deep red spots in subplot (b)) and time-averaged relief of the rigidity. fig. 9 instant and almost stationary time averaged elastic coefficient integrated over long time periods ((a) and (b) subplots, respectively) shown by the ‘jet’ color-map. contour line marks only partial correlations between the averaged and instant rigidities, analogous (only partial) correlation is found between travelling regions of maximal power production (deep red spots in subplot (b)) and time-averaged relief of the rigidity. block-media motion lattice and movable automata numerical models 563 3. movable automata model generally, the variant of the model which uses movable system is based on a description of a system of interacting n “particles”. in contrast to described above lattice model which treats system as deformable elastic medium with varied friction and rigidity, here each “particle” simulates a complete block (“solid plate”) in some instant position represented radius-vector ri of its center and the mechanical momentum pi. in simplest approximation, the plates interact with a potential u(ri-rj) depending on the distance |ri-rj| between their centers. it formally leads to the following many-body hamiltonian: 2 1 , 1 ( , ) / 2 (| |) / 2 n n i i i i i j i i j h m u = = = + − r p p r r . (10) in a static configuration the plates are supposed to occupy some equilibrium positions defined by a competition of the repulsion and attraction forces acting between them. in the context of this study, it is convenient to represent the interaction potential by a pair of the gauss potentials: 2 2(| |) exp{ [( ) / ] } exp{ [( ) / ] }i j ij i j ij ij i j iju c c d d− = − − − − −r r r r r r , (11) where cij and dij define the magnitude, while cij and dij the radii of attraction and repulsion, respectively. the minimization of the energy corresponding to a desired equilibrium with some characteristic distance between the centers corresponds to the inequalities between the parameters: cij>>dij, cij<dij. initially, correct equilibrium positions of the plates are not known and should be found by a self-assembling transient process. to obtain it in numerical experiments, the “particles” are initially placed at random on a rectangle with the side lengths of [0, lx] and [0, ly]. during the transient process the system finds some frozen structure which is a compromise between the interactions and boundary conditions. depending on the particular study, the boundary conditions can be chosen differently. for example, for the case of shear deformation, it is convenient to employ periodic boundary conditions on the horizontal axis. it allows simulate potentially infinite run for relatively small limited numerical arrays. according to the periodic boundary conditions, a particle leaving the interval [0, lx] is returned to it at the opposite end and the vectors ri-rj connecting particles either located within the interval [0, lx] or the images of “escaped” particles at the opposite side of the system. on the vertical axis such a system can be limited by the horizontal plates at y=0 and y=ly with reflecting boundary conditions: uup=c·exp[(yly)/c] and udown=c·exp[-y/c]. such a combination of the boundary conditions simulates together quasi-infinite motion along horizontal axis and confined motion between two “external” plates in orthogonal direction. the stronger the inequalities c>>cij and c<<cij, the more rigid and sharp are the walls in relation to other forces and lengths relevant to the system. keeping in mind typical “tectonic” geometry with the moving continents, in the present study we are interested in a configuration when along horizontal axis the system is also limited and moreover, is pressed by a movable “rigid wall”. in analogy with standard burridge-knopoff model, the plate is driven by an external “elastic” force: 0( )ext xf k l v t= − , (12) 564 a.e. filippov, v.l. popov in this case it is natural, in contrast to the periodic one, to take the system which is limited along horizontal axis with reflecting boundary conditions: uright=c·exp[(x-lx)/c] and uleft=c·exp[-x/c], but with the movable right boundary lx= lx(t). since the parameters cij, dij, cij and dij form arrays covering all possible combinations, it can be said without loss of generality that the hamiltonian given by eqs. (10)(12) describes systems with any number of components (plates of the different sizes). what is required is just to specify the parameters cjk and cjk. in a real system different values of all the parameters (and what is even more important, the mutual relations between them) regulate different sizes of the physical plates and their mutual rigidity. below, to conserve a generality and get statistically representative results, we use the values cjk, and cjk randomly varied with 10 times difference between maximal and minimal ones around unit cjk, cjk=1, and reset them from one run to another of every numerical experiment performed at other fixed or systematically varied parameters. the simplicity of the equations of motion / ( , ) /i i i i i im t h  = −  =v x p p f , (13) is deceptive. in majority of the cases, simulation using such equations is extremely timeconsuming because it involves a summation at every time step over all possible neighbors, including very distant ones (and even fictitious image particles in the case of periodic boundary conditions). however, in the case of the solid plates contacting by their boundaries the number of the mutual interactions can be essentially reduced. the reduction is based on the following considerations. it is natural for this particular problem to construct voronoi diagram of the set of effective “particles” and associate each cell of this diagram with some of the mutually contacting plates. according to mathematical definition, a voronoi diagram is a partition of a plane into regions close to each of a given set of objects. in the simplest case, these objects are just set of points in the plane (called seeds). for each seed there is a corresponding region, called a voronoi cell, consisting of all points of the plane closer to that seed than to any other. in our case the central “particles” of every “plate” play a role of such mathematical “seeds”. so, the cells of voronoi diagram give practically the same picture, which we have in mind when building our dynamic model. every such a cell is surrounded by the limited (and in fact quite small) number of the other plates (nearest neighbors). mathematically such neighbors are completely defined by the delaunay triangulation, which is dual to the voronoi diagram. according to its definition, delaunay triangulation for a given set p of discrete points in a general position is a triangulation dt(p) such that no point in p is inside the circumcircle of any triangle in dt(p). as result one gets quite realistic representation of the movable plates and short list of the neighbors interacting with every particular plate. interacting plates exchange momentum, and hence a dissipation channel acting to equilibrate the relative velocities of particles (that happen to be within the distance cv close to the equilibrium one) needs to be introduced. this can be done by including an additive velocity-depending force. 2 1 ( )exp [( ) / ] neigboursn i i j i j v j f c =  = − − − v v v r r , (14) acting on the i-th particle, with some dissipation constant η. as a result, the equations of motion assume the following form: block-media motion lattice and movable automata numerical models 565 i i i im t  = −  r vv f f . (15) they can be integrated by using verlet’s method, which conserves the energy of the system at each time step, provided no energy is supplied externally through mechanical work of the movable right boundary which is driven by a balance of the external force fext=k(lx-v0t) and total interaction with the internal plates repulsed by the boundary potential uright=c·exp[(x-lx)/c] 2 2 1 ( ( )) / n x ext right j x j j l m f u x l t x t =  = −  −    . (16) so, the equations (15) and (16) must be solved self-consistently. moreover, counting on the nature of the problem under consideration, this system has to be accompanied by the equations regulating variation of the effective mutual friction between the plates with negative derivative in the same spirit as it was done above for the standard bk model on the elastic lattice: ( )1 0 2( , ) ; j j j j f f f v t v t  = − +    2 ,<0, 0j nv  and ( ) at 0j jf v v= −  . (17) taking into account an experience which obtained in modification lattice bk model it seems natural to add here also variable intensity of the interactions between the plates at nonzero mutual velocities: ( )1 1 10 1 2( , ) ; j j j j c c c v t v t  = − +    2 ,<0, 0j nv  and 1 10( ) at 0j jc v c v=  . (18) at the same time, here is no need to include directly the temperature conductivity equation into the model because it is already based on the brownian dynamic equations for the plate centres by itself, and in fact automatically involves an exchange by kinetic energy (temperature) between them. so, plotting the kinetic energy of the “particles” by the artificial colors one gets a color-map in the representation which is natural for the model based on the movable automata. the results obtained in the frames of this model are summarized in the figs. 10-15 below. typical instant configuration of the plates pressed from one of the boundaries by an external force is shown in fig. 10. let us remind that the rigid wall presses the system from the right side. as a result, it becomes more compressed form this side and areas of the voronoi cells here become smaller than in average inside the system far from this boundary. the mass of every plate conserves in the frame of the model. the only are of it changes. gray-scale map of the subplot (a) is used to show the individual areas of the plates. the compression and motion of the plates causes nonzero mutual velocities of every plate in relation to their proximities. this fact is well seen in the subplot (b) by the matlab ‘jet’ color-map. according to eq.(17) nonzero mutual velocities result in a variation of the effective friction between every plate and its surrounding. this characteristic of the system is depicted in the subplot (c). it is important to note also that the delaunay triangulation is recalculated at every time step. as a result, the links between the plates moving with different velocities can appear and disappear depending on their physical positions at a particular time moment. however, in sufficiently dense 566 a.e. filippov, v.l. popov system (which is used here) such changes of the neighbors happen quite rarely. so, the model seems quite realistic to describe some features of the tectonic motion. fig. 10 typical instant configuration of the plates pressed from one of the boundaries by an external force. gray-scale map of the subplot (a) shows the areas of the plates more compressed from one side (white color) than inside the system (different intensity of gray color). the compression and motion of the plates causes different (mean) mutual velocity of every plate in relation to its proximity which results in the variation of the effective friction between this plate and surrounding ones. both these characteristics are plotted by the ‘jet’ colors. fine structure of the model is presented in fig. 11. movable voronoi diagram in subplot (a) here show the plates in the frame of the model, as above. however, here they are accompanied by the colored seeds of the voronoi cells. the colors show different instant velocities of the plates. dual to the above diagram delaunay triangulation shows connections between the plates and their neighbors counted in the model. according to its mathematical definition it naturally gives a list of the neighbors for every plate. so, one can calculate the interactions between them, solve the dynamic equations, so on. different block-media motion lattice and movable automata numerical models 567 sizes of the nodes here reproduce different (randomly distributed) interaction parameters. besides, it is convenient to control instant number of them for every cell (symmetry of the plate polygon) and visualize it by the colors from the same standard ‘jet’ map. fig. 11 movable voronoi diagram in subplot (a) describes the plates in the frame of the model. the colors of the seeds show different instant velocities of the plates. dual to the above diagram delaunay triangulation shows connections between the plates and their neighbors counted in the model. different sizes of the nodes here reproduce different interaction coefficients and their colors show different instant number of neighbors. two projections of the phase portrait on different planes (subplots (a) and (b)) are presented in the fig. 13 by the colored circles of different sizes. the colors and sizes of the circles have the same meaning as in the fig. 12a. green line with the arrows shows typical individual trajectory, which every point passes through the phase space with the time and reproduces the evolution of the effective friction due to periods of large or small 568 a.e. filippov, v.l. popov local velocities. generally, the subplot (b) here has the same physical meaning as the phase portrait plotted in fig. 3a in the lattice variant of bk model. fig. 12 the same instant configuration as in fig. 11 reproducing mutual relations between the interactions, local rigidities, velocities and friction coefficient directly marked in the plots. the colors and sizes of the circles in the subplot (a) depict velocities and interaction coefficients of the individual plates, respectively. yellow, cyan, magenta and red lines in the subplot (a) display integrated over y-coordinate averaged xdependencies of the velocity, rigidity, effective friction coefficient density and temperature respectively. the colors of the circles in the subplot (b) correspond to different individual power of the events in every plate. to get the local power in the frames of the discrete variant of the model we calculate the work produced by the forces acting on each plate (seed) of the system pi=fivi. calculation of spatially distributed power pi=fivi enables one to evaluate the degree of localization of excitations and hence the fraction of the regions in which the fragments of the lattice move practically as a whole. as at a particular site the energy is absorbed and block-media motion lattice and movable automata numerical models 569 released at different points in time, it is useful to characterize these sites by the absolute value of the power, |pi|=|fivi|. it is shown in subplot (b) of the fig.12. summation over all particles yields the total power vfext(t)=p(t)=∑fivi. it gives an energy which is consumed during every time unit due to motion of rigid right boundary, which is forced by external flow with constant velocity, v=const. however, the boundary here is moving with some varied velocity v(t) which is determined by a balance of the total force, fext(t) and only its time average coincides with <v(t)>=v0 . fig. 13 two projections of the phase portrait on different planes (subplots (a) and (b)). the colors and sizes of the circles depict velocities and interaction coefficients of the individual plates, respectively. green line with the arrows shows typical individual trajectory, which every point passes in phase space with the time. 570 a.e. filippov, v.l. popov fig. 14 instant configuration of the density (surface in subplot (a)) and time-space diagram of the history of events (colormap in subplot (b)). bold black contours mark intensive “earthquake” events near the propagating right boundary. different speed of the front and some “epochs” of intensive events can be seen from different inclination of the front and horizontal lines in the recorded history. in particular, it causes quasi-periodic oscillations of the velocity of boundary which are typical for the “stick-slip” nature of the processes. it is actually the same phenomenon, which takes place in ordinary bk model, shown above in figs. 2-3. in the discrete model one can additionally observe the waves of the velocities, densities and power of the events, which are accompanied by quasi-periodic “epochs” of intensive “earthquakes”. instant configuration of the density and history of the density integrated along y-axis corresponding to the magenta curve in fig. 12a are shown in fig. 14. the instant surface is plotted in subplot (a) and corresponding time-space diagram of the history of events is presented in the subplot (b). bold black contours mark intensive “earthquake” events near the propagating right boundary. different speed of the front and some “epochs” of intensive events can be seen from different inclination of the front and deeply colored almost horizontal lines in the recorded history. exactly the same history of the events (recorded during the same calculation run) as in fig. 14, is depicted by the intensity of power production (shown by the ‘hot’ color map). the epochs of the intensive earthquakes are much stronger pronounced here than in the previous figure. block-media motion lattice and movable automata numerical models 571 fig. 15 exactly the same history of the events, as in fig.14, depicted by the intensity of power production (shown by the ‘hot’ colormap). the epochs of the intensive earthquakes are much stronger pronounced here than in the previous figure. 4. conclusion it is shown that original burridge-knopoff model allows a number of the modifications which can increase its ability to be adapted to description of the different practically important features of the tectonic phenomena. basic modifications of the model were already made previously. they incorporate naturally introduced state dependent friction where the friction force is defined by the additional kinetic equation as well as extension of the model to more realistic two-dimensional case. in the present paper we continued previous modifications and mainly concentrated on the following extensions of the model. first of all, elastic force and thermal conductivity were added into consideration. besides to the velocity weakening of friction between moving blocks, we combined an increase of both static friction and rigidity during stick periods as well their simultaneous weakening during motion. it is expected to give better and qualitatively correct description of plural transitions from smooth sliding to stick-slip 572 a.e. filippov, v.l. popov behavior, which are normally observed in various tribological and tectonic systems. to get easier comparison with the previous versions of the model we initially did this using in standard approach based on 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mesomech., 24, pp. 375–390. facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 91 106 https://doi.org/10.22190/fume191118007a © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  parametric analysis of a grinding process using the rough sets theory subham agarwal 1 , shruti sudhakar dandge 2 , shankar chakraborty 1 1department of production engineering, jadavpur university, kolkata, west bengal, india 2mechanical engineering department, government polytechnic, murtizapur, maharashtra, india abstract. with continuous automation of the manufacturing industries and the development of advanced data acquisition systems, a huge volume of manufacturingrelated data is now available which can be effectively mined to extract valuable knowledge and unfold the hidden patterns. in this paper, a data mining tool, in the form of the rough sets theory, is applied to a grinding process to investigate the effects of its various input parameters on the responses. rotational speed of the grinding wheel, depth of cut and type of the cutting fluid are grinding parameters, and average surface roughness, amplitude of vibration and grinding ratio are the responses. the best parametric settings of the grinding parameters are also derived to control the quality characteristics of the ground components. the developed decision rules are quite easy to understand and can truly predict the response values at varying combinations of the considered grinding parameters. key words: data mining, rough sets theory, grinding, parameter, rule 1. introduction with the rapid advancements of various data analysis tools and network technology, data mining has now become an emerging area in computational intelligence which offers new concepts and methods to analyze voluminous data. availability of a large volume of data in different forms has significantly accelerated the applications of data mining. data mining, also known as „knowledge discovery from databases‟, thus deals with the application of various competent tools and techniques to refine the extracted knowledge from a large database so as to envisage, categorize and characterize the mined data [1, 2]. it can identify interesting patterns in data to aid in valuable decision-making received november 18, 2019 / accepted february 01, 2020 corresponding author: shankar chakraborty department of production engineering, jadavpur university, kolkata, west bengal, india e-mail: s_chakraborty00@yahoo.co.in 92 s. agarwal, s.s. dandge , s. chakraborty where the applications of the popular statistical and predictive models fail. understanding the patterns inherent in the data sometimes becomes important when the data sources are heterogeneous and differently distributed. data mining mainly consists of the applications of various mathematical tools for machine learning, cluster analysis, regression analysis and neural networks. using a predetermined set of features and a training dataset, regression analysis and neural networks develop a single model. on the other hand, a machine learning algorithm develops a number of models in the form of decision rules while providing the interrelationships between various input features and the final decision. cluster analysis can also create the same decision rules when the set of features included in each rule is independent of all other rules. the rules developed by the data mining tools are always expected to be explicit [3, 4]. rough sets theory (rst), developed by pawlak in 1982 [5], falls under the broad category of machine learning. based on extraction of knowledge from the datasets, it can also provide valuable tools for data analysis and generation of independent decision rules for effective data classification. having a strong mathematical foundation, it is well suited to efficiently solve various decision-making problems. although it has some similarities with the fuzzy set theory, today it has evolved out as a separate discipline in data mining. its main advantage is that it does not require any additional information about the dataset to be mined, like the probability theory in statistical approaches, membership functions in the fuzzy set theory, etc. as a non-statistical approach in data analysis, it thus classifies and analyses imprecise, uncertain or incomplete information and knowledge to generate minimal and non-redundant rule sets [6, 7]. in the contemporary automated manufacturing industries, a huge volume of data related to product design, bill of materials, production planning and control, production processes and systems, monitoring and diagnosis, etc. is being regularly captured and stored using various data acquisition tools. valuable information in the form of rules, patterns, clusters, associations and dependencies are always expected to be hidden in the dataset collected from the manufacturing organizations. thus, it becomes the responsibility of the production engineers to augment effective data mining tools to analyze this huge manufacturing-related dataset to identify potential patterns in various input parameters that control a manufacturing process or quality of the output products. it is observed that the rst has already been successfully applied in various domains of engineering and management decision-making, like manufacturing process control [8], quality assurance [9], supplier selection [10, 11], automotive warranty data analysis [12], operations of security forces [13], forecasting [14], etc. in the present day manufacturing industries, grinding has been successfully applied as an efficient material removal process to almost all types of materials providing an extremely high material removal rate (mrr) (more than 2000 mm3/s) and ultra-precision surface finish (up to nanometer level). the precision and surface finish attained through grinding can be up to ten times better than the other machining processes, like turning and milling. due to high hardness of the abrasives used as the cutting medium in grinding, it has now become the first choice for removing materials from the workpieces. grinding process requires least pressure which makes it easy to hold the work material even during automated process using simple devices. it has been found out that application of grinding includes more than a quarter of total machining processes and is still showing an upward trend. thus, in order to study the material removal mechanism in a grinding process, parametric analysis of a grinding process using the rough sets theory 93 while examining the possible interactions between different grinding parameters and responses, it has become critical to provide guidance for further improving the grinding quality and productivity through the identification of appropriate settings of the considered grinding parameters. on the other hand, the rst has several advantages over the other data mining tools, like it does not require any preliminary or additional information about the data to be analyzed, it provides efficient algorithms for searching out the hidden patterns in the data, it is able to find out minimal sets of data for effective pattern generation, it evaluates significance of the data, it generates sets of decision rules from the data automatically, it is easy to comprehend, it is capable to provide straightforward interpretation of the derived rules, it is suited for parallel processing, etc. thus, in this paper, a maiden endeavor is put forward to apply the concepts of the rst to a grinding process so as study the effects of various grinding parameters on different measured responses and predict the optimal settings of those parameters. 2. literature review chadha and lee [15] developed a new optimization tool in the form of a variable length evolution strategy for a cylindrical traverse grinding process. while considering dressing feed, grinding feed, dwell time and cycle time as the grinding parameters, siddiquee et al. [16] presented an approach combining grey relational analysis (gra) and principal component analysis to attain the most preferred values of surface roughness (sr), cylindricity error and diametric tolerance in a centerless cylindrical grinding process. lee et al. [17] applied the taguchi-sliding-based differential evolution algorithm for optimizing wheel speed, workpiece speed, depth of dressing and lead of dressing for a surface grinding process. based on experimental studies on rough-grinding and finish-grinding processes, it was concluded that the proposed approach would provide better solutions as compared to the already adopted methods. asiltürk et al. [18] proposed the application of an adaptive network-based fuzzy inference system for effectively predicting sr and vibration in cylindrical grinding, while taking into account workpiece speed, feed rate and depth of cut as the input parameters. khan et al. [19] presented the application of gra technique for optimizing an in-feed centerless cylindrical grinding process. the considered grinding parameters were dressing feed, grinding feed, dwell time and cycle time, whereas, sr and cylindricity error were the responses. neşeli et al. [20] combined response surface methodology (rsm) and taguchi method to find out the optimal settings of workpiece revolution, feed rate and depth of cut so as to minimize sr and vibration in an external cylindrical grinding process. rudrapati et al. [21] applied the rsm technique to study the relationships between three grinding parameters (in-feed, longitudinal feed and work speed) and sr in a cylindrical grinding process. it was observed that the considered grinding parameters had no significant influence on sr. using the grey-based taguchi methodology, köklü [22] investigated the influences of workpiece speed, depth of cut and number of slots on sr and roundness error in a grinding process. the optimal parametric mix and the most important grinding parameter were also identified. using rsm technique, pai et al. [23] developed regression models correlating three grinding parameters with two responses, i.e. mrr and sr, during grinding of al6061-sic composites. elitist non-dominated sorting genetic algorithm (enhanced nsga-ii) was later employed to determine the optimal grinding 94 s. agarwal, s.s. dandge , s. chakraborty conditions. pawar and rai-kalal [24] adopted nsga-ii technique for determining the optimal operating levels of wheel speed, workpiece speed, depth of dressing and lead of dressing to minimize production cost, production rate and sr in a grinding process. winter et al. [25] applied geometric programming and a weighted max-min model for single as well as multi-objective optimization of an internal cylindrical grinding process. the corresponding pareto optimal solutions were also identified to enhance the eco-efficiency of the considered grinding operation. khare and agarwal [26] developed an analytical model representing the relationship between sr and chip thickness for surface grinding of aisi 4340 steel material. aleksandrova [27] applied generalized utility function for parametric optimization of different dressing parameters in a cylindrical grinding operation. deng et al. [28] applied genetic algorithm (ga) technique to solve a multi-objective optimization model having minimum processing time and optimal carbon efficiency as the two objectives. the optimal parametric combination of different grinding parameters was also identified. rudrapati et al. [29] considered three input parameters, i.e. in-feed, longitudinal feed and workpiece speed, and later applied multi-objective ga to minimize sr and vibration in traverse cut cylindrical grinding operation of stainless steel material. aydaş and el k [30] combined sm and techniques to optimize speed of the workpiece, depth of cut and number of grooves in cylindrical surface grinding operation of aisi 1050 steel material. the effects of those grinding parameters on sr of the machined components were also investigated. chang et al. [31] conducted an orthogonal test to investigate the influences of wheel speed, workpiece speed and grinding depth on surface integrity of a bearing raceway. based on the experimental data, two support vector machine models were proposed in order to significantly reduce the optimization time and derive the global optimal solution. kuo et al. [32] proposed a multi-criteria model to obtain the optimal parametric setting of a grinding process while taking into consideration sr and mrr as the responses. the effects of minimum quantity lubrication on mrr, surface integrity and temperature while grinding ti6al-4v workpiece material were also studied. considering the stochastic nature of a grinding process and based on orthogonal experiment method, ming et al. [33] optimized different input parameters for a five-axis blade grinding setup. liu et al. [34] integrated signal-to-noise analysis with gra technique to optimize different grinding parameters for attaining the most desired values of mrr and grinding efficiency. it can be observed from the exhaustive review of the past research that the investigation of the influences of different grinding parameters, like grinding wheel speed, workpiece rotational speed, depth of cut, cutting speed etc. on various responses, such as mrr, sr, vibration, cylindricity error, grinding efficiency etc., and determination of the optimal combinations of those grinding parameters have been the main topics of interest amongst the researchers. several optimization tools in the form of ga, nsga-ii, gra, utility theory, etc. have been implemented to fulfill the above-cited objectives. the applications of those tools and techniques often lead to sub-optimal or near-optimal solutions, and identify the optimal parametric settings of the considered grinding processes which are sometimes difficult to set and maintain in the existing machining systems. thus, in this paper, the application of a data mining tool, in the form of the rst, is proposed to analyze the experimental dataset of a grinding process and generate the corresponding „if-then‟ decision rules to visualize the effects of different grinding parameters on three important responses and guide the concerned production engineers to identify the most appropriate parametric mix for the said grinding process for attaining the desired quality characteristics parametric analysis of a grinding process using the rough sets theory 95 of the ground components. these decision rules follow a general structure, i.e. if the machining conditions are met then certain response values can be attained or predicted. they are probably the most interpretable prediction models, semantically resembling the natural language and human thinking process. the developed rules aid in solving complex machining problems, providing explanations of how the final decisions have been arrived at and why a particular decision has been made. this rule generation process has been proved to have high speed and scalability. the developed rules for the considered grinding process can also act a repository and executable knowledge base to facilitate the decision-making process in the domain of grinding technology. 3. rough sets theory in the present day automated manufacturing industries, it has now become quite essential to analyze the huge dataset to correctly estimate the real nature of knowledge inherent in it. for this purpose, „if...then‟ rules have emerged as a reliable tool for decision-making while effectively representing information or bits of knowledge. the expression of „if...then‟ rule attains a form, like „if condition then conclusion‟. in order to demonstrate the importance of „if...then‟ rules in data mining, the data presented in table 1 can be cited. in this data matrix, there are three input parameters, each having two different levels and three responses with three varying levels. for example, in the first experimental run, when all the three input parameters are set at „0‟ (minimum) level, all the responses would have „low‟ observations. table 1 illustrative dataset experiment run input parameter response a1 a2 a3 r1 r2 r3 1 0 0 0 low low low 2 0 1 1 low low low 3 0 0 0 medium high high 4 1 1 1 high high high 5 1 0 0 high low low in the initial dataset containing a large number of observations, it has been sometimes noticed that many attributes and responses are duplicative in nature which may be responsible for unwanted bulkiness of the dataset. hence, it has become mandatory to reduce the numbers of attributes and responses in the original dataset to enable extraction of explicit knowledge with framing of simple rules. the rst technique helps in reduction of number of attributes or responses while estimating the dependency between two or more of them. the attributes/responses having higher dependency indexes as compared to the predefined threshold value are removed from the initial dataset. when the dependency index is greater than the threshold value, either the first or second attribute/response of a common pair is excluded from the dataset without losing any valuable information. the dependency index can be calculated using the following equation [8]: 96 s. agarwal, s.s. dandge , s. chakraborty    * )( ),( jal i ji n la aak (1)  lyayla ii  *)( (2) where ai * and aj * are the equivalence classes of attributes ai and aj respectively (the equivalence class is the set of objects having the same value for attributes ai and aj), l is the equivalence class of aj, y is the equivalence class of ai, n is the total number of objects in the dataset, || is the cardinality of a set (number of elements in the set) and ia (l) is the lower approximation of set l over attribute ai. in rst, a dependency index of k(ai,aj) = 0 signifies that the attributes ai and aj are independent to each other, whereas, k(ai,aj) = 100 implies that they are totally dependent to each other. thus, elimination of any one of them would not affect extraction of the knowledge from the dataset. but, the attributes cannot be eliminated only by observing the value of k(ai,aj), the value of k(aj,ai) also needs to be checked. elimination of the attributes can only be possible if min{k(ai,aj), k(aj,ai)} is greater than the predefined threshold value. thus, determination of the corresponding threshold value plays an important role in generation of the decision rules. if the threshold value is estimated to be high, a greater number of incompetent attributes exists in the dataset, making formation of the rules highly complicated. on the other hand, if its value is low, there remains a high chance of many useful attributes getting eliminated with loss of valuable information. hence, it is always recommended to set the threshold value based on the prevailing situation and experts‟ opinions. traditionally, the threshold value is set as 85-90%. based on the dataset of table 1, and using eqs. (1) and (2), the corresponding dependency level matrix for the considered attributes is developed, as shown in table 2. from this table, it can be noticed that k(a2,a3) = k(a3,a2) = 100, and k(r2,r3) = k(r3,r2) = 100, i.e. input parameters a2 and a3, and responses r2 and r3 are dependent on each other (threshold value is taken as 90). thus, either a2 or a3 and r2 or r3 can be eliminated from the initial dataset, while keeping other attributes remain intact as earlier. in this illustrative example, a3 and r3 are eliminated, and a new dataset is formed in table 3. table 2 calculated values of dependency level attribute input parameter response a1 a2 a3 r1 r2 r3 a1 0 0 100 0 0 a2 0 100 20 0 0 a3 0 100 20 0 0 r1 40 0 0 0 0 r2 0 0 0 60 100 r3 0 0 0 60 100 parametric analysis of a grinding process using the rough sets theory 97 table 3 reduced dataset experiment run input parameter response a1 a2 r1 r2 1 0 0 low low 2 0 1 low low 3 0 0 medium high 4 1 1 high high 5 1 0 high low from the reduced dataset, with the help of k-means or any other clustering algorithm, the considered attributes are now discretized with continuous numeric values. it helps the attributes to be well organized themselves in different clusters/groups to provide the rules more proficiency. now, a decision rule generation algorithm is applied to extract „if…then‟ rules from the reduced set of the attributes categorized into appropriate number of clusters. the decision rule generation algorithm is presented as follows [8]: step 1: initialize: a = {a1,a2,…,an}; r = {r1,r2,…,rm} step 2: evaluate jiij rax  for i = 1,2,…,p; j = 1,2,…,q step 3: for each xij ≠ , a rule is assigned as if a1 = v(ai,a1) and...and an = v(ai,an) then r1 = v(rj,r1) and...and rm = v(rj,rm) [p, q, c, qty] [t] where i ij a x p ; j ij r x q ; n x ij c ; ijxqty ; cqpt  in the above algorithm, the “if” statement contains the input or independent parameters, whereas, the “then” statement consists of the dependent parameters or responses. here, t signifies the total weight (relative importance) assigned to a rule for effective decisionmaking. the higher the value of t, the greater the weight of a particular rule is. the maximum value of t identifies a rule to be the optimal one among all the generated rules, and it has the highest chance of occurrence when the whole system is repeated again and again. using the reduced dataset of table 3, two sets a = {a1,a2} and r = {r1,r2} are initially generated, and based on the steps as presented for the rule generation algorithm, the following rules are formulated. a) rules for single response: for response (r1): rule 1: if a1 = 0 then r1 is low. [p = 66.67%, q = 100%, c = 40%, qty = 2][t = 206.67%] rule 2: if a1 = 1 then r1 is high. [p = 100%, q = 100%, c = 40%, qty = 2][t = 240%] rule 3: if a1 = 0 and a2 = 0 then r1 is medium. [p = 50%, q = 100%, c = 20%, qty = 1][t = 170%] from the above-generated rules, it can be concluded that a value of p = 100% in rule 2 indicates that all the elements present in the dataset with condition a1 = 1 satisfy this rule. on the other hand, a value of q = 100% in rules 1, 2 and 3 implies that all the elements present in the dataset, having response r1 as low, high and medium, satisfy rules 1, 2 and 3, 98 s. agarwal, s.s. dandge , s. chakraborty respectively. it can also be observed that for rules 1 and 2, 40% of the observations in the dataset (c = 40%) are covered by these rules. for rule 1, a value of qty = 2 represents the total number of elements (cases) that follow this rule. amongst the three generated rules, rule 2 with the maximum t value of 240% is supposed to be the strongest rule. although rule 3 encompasses both the input parameters, it has poor strength (t = 170%). for response (r2): rule 1: if a2 = 0 then r2 is low. [p = 66.67%, q = 66.67%, c = 40%, qty = 2][t = 173.34%] rule 2: if a1 = 0 and a2 = 1 then r2 is low. [p = 100%, q = 33.33%, c = 20%, qty = 1][t = 153.33%] rule 3: if a1 = 1 and a2 = 1 then r2 is high. [p = 100%, q = 50%, c = 20%, qty = 1][t = 170%] rule 4: if a1 = 0 and a2 = 0 then r2 is high. [p = 50%, q = 50%, c = 20%, qty = 1][t = 120%] for response r2, the value of t is maximum for rule 1, identifying it as the strongest rule. b) rules for two responses: rule 1: if a1 = 0 then r1 is low and r2 is low. [p = 66.67%, q = 100%, c = 40%, qty = 2][t = 206.67%] rule 2: if a1 = 0 and a2 = 0 then r1 is medium and r2 is high. [p = 50%, q = 100%, c = 20%, qty = 1][t = 170%] rule 3: if a1 = 1 and a2 = 1 then r1 is high and r2 is high. [p = 100%, q = 100%, c = 20%, qty = 1][t = 220%] rule 4: if a1 = 1 and a2 = 0 then r1 is high and r2 is low. [p = 100%, q = 100%, c = 20%, qty = 2][t = 220%] these rules as developed taking into consideration two responses simultaneously are supposed to be more reliable and useful as compared to those generated for only one response. for two responses, the value of t for rules 3 and 4 is the maximum which implies that both these rules can collectively extract the optimal information from the considered dataset. 4. rough sets theory in a grinding process grinding is a machining process where a high volume of unwanted material is rapidly removed from the workpiece surface with the help of the abrasive grinding wheel or the grinder used as a cutting tool [35]. it is principally used as a fine finishing process which results in achievement of high surface quality and dimensional accuracy of the machined parts/components. in this process, each grain of abrasive on the grinding wheel removes material from the workpiece in the form of small chips through shear deformation. a grinding setup usually consists of a bed with a fixture to guide and hold the workpiece, and a power-driven grinding wheel with hard abrasives revolving with the required rotational speed. in order to cool the workpiece during the grinding operation, coolants (e.g. water, light duty oil, wax, heavy duty emulsifiable oil etc.) are also applied. the abrasives commonly used in the grinding wheels are aluminum oxide, silicon carbide, ceramics, diamond and cubic boron nitride. on the other hand, the workpiece materials include parametric analysis of a grinding process using the rough sets theory 99 aluminum, brass, plastics, cast iron, mild steel and stainless steel. in manufacturing industries, there are huge applications of grinding operation, e.g. surface finishing, slitting and parting, descaling, deburring, finishing of flat as well as cylindrical surface, and grinding and resharpening of tools and cutters. keeping in mind the large applicability of grinding operations, in this paper, a dataset is chosen where nine experiments have been conducted on low alloy steel workpiece samples (60 × 40 × 8 mm size) using a vitrified bonded alumina grinding wheel. spindle speed (ss) (in rpm), depth of cut (doc) (in mm) and type of the cutting fluid (tcf) have been considered as the input grinding parameters. on the other hand, sr (ra) (in µm), amplitude of vibration (v) (in µm) and grinding ratio (g-ratio) have been treated as the process outputs (responses). spindle speed is the rotational speed of the grinding wheel and depth of cut is the thickness of material being removed during the grinding operation. higher depth of cut provides more mrr while enhancing productivity of a grinding process. during the grinding operation, material removal takes place by abrasion, resulting in generation of substantial amount of heat. to cool the workpiece, the coolant is used to avoid overheating and meet the dimensional tolerances. the ra (arithmetic mean roughness or centre line average roughness) symbolizes surface quality of the machined components. it is one of the most important parameters for measuring sr. if there are large form deviations in the machined surface, the corresponding ra value would be high; otherwise for smooth surface, lower values of ra are obtained. during the grinding operation, the maximum distance to which the grinding wheel goes from its central position is termed as the amplitude of vibration. the g-ratio indicates the efficiency of the grinding operation and can be defined as the ratio of mrr to wheel wear rate. for each of the grinding parameters, three different operating levels have been considered. the detailed experimental plan along with the measured values of the three responses is provided in table 4. in this table, the numbers enclosed inside the parentheses show the respective operating levels of the considered grinding parameters. now, this experimental dataset for the grinding operation is analyzed using the principle of the rst so as to identify those input parameters which are responsible for controlling the output characteristics of the ground parts/components. at first, data preprocessing in the form of attribute reduction and clustering of the considered attributes are performed. table 5 exhibits the dependency indexes as computed for each pair of the attributes and smaller values of those indexes (all the values are less than the threshold limit of 90%) prove the independency of all the attributes as considered in this grinding process. it is worthwhile to mention that in table 5, the values of two dependency indexes r(ss, g-ratio) and r(g-ratio, ss) are obtained as 33.33% and 100%, respectively. but, as the minimum of them, i.e. 33.33% is less than the predetermined threshold value of 90%, both of them can be treated as entirely independent attributes. along with the data reduction, the measured responses are also grouped into appropriate number of clusters using k-means algorithm to convert their continuous values into separate distinguishable ranges. 100 s. agarwal, s.s. dandge , s. chakraborty table 4 experimental dataset for the grinding process exp. no. grinding parameter response ss doc tcf ra v g-ratio 1 2430 (1) 0.02 (1) coolant (1) 0.48 18.22 0.0253 2 2430 (1) 0.03 (2) water (2) 0.56 21.32 0.0262 3 2430 (1) 0.04 (3) coolant+water (3) 0.57 26.23 0.0232 4 2560 (2) 0.02 (1) water (2) 0.61 22.32 0.0356 5 2560 (2) 0.03 (2) coolant+water (3) 0.65 31.22 0.0323 6 2560 (2) 0.04 (3) coolant (1) 0.77 29.57 0.0476 7 2850 (3) 0.02 (1) coolant+water (3) 0.72 26.45 0.0643 8 2850 (3) 0.03 (2) coolant (1) 0.8 31.56 0.0656 9 2850 (3) 0.04 (3) water (2) 0.65 34.78 0.0781 table 5 dependency indexes for various grinding attributes attribute ss doc tcf ra v g-ratio ss 0 0 0 0 33.33 doc 0 0 0 0 0 tcf 0 0 0 0 0 ra 66.67 0 0 33.33 33.33 v 33.33 33.33 33.33 55.56 33.33 g-ratio 100 0 0 44.44 33.33 in fig. 1, the values of all the considered responses for this grinding process are clustered into two separate groups. for ra and amplitude of vibration (both are nonbeneficial characteristics requiring their lower values), the two formed clusters for them are respectively designated as „low‟ and „high‟. here, lower values of a and amplitude of vibration are always preferred. on the other, for g-ratio (being a beneficial characteristic requiring only higher value), the corresponding clusters are also respectively termed as „low‟ and „high‟. but, for -ratio, higher values are always desired. the number of classes in which the responses are to be segregated also plays an important role in subsequent generation of the decision rules. if the number of clusters is high, each generated rule would encompass a small number of elements. on the other hand, when the number of clusters is too small, interpretation of the rules would then become complicated. thus, it is always recommended to choose the number of clusters in such a way so as to make a compromise between simplicity of the rules and the level of knowledge extraction. the details of the cluster analysis results for the three responses of the grinding process are provided in table 6. in this table, the third and fourth columns respectively denote the mean and range values for each of the clusters formed for the considered responses. on the other hand, the column five represents the specific objects (experimental runs) and the last column denotes the total number of objects in each of the formed clusters. parametric analysis of a grinding process using the rough sets theory 101 (a) (b) (c) fig. 1 clustering of the considered responses table 6 details of the formed clusters for the responses response cluster number mean range of each cluster objects in each cluster total number of objects in each cluster ra cluster 1 0.56 0.40-0.60 1,2,3,4 4 cluster 2 0.72 0.60-0.85 5,6,7,8,9 5 amplitude of vibration cluster 1 20.62 17.00-23.00 1,2,4 3 cluster 2 29.97 23.00-35.50 3,5,6,7,8,9 6 g-ratio cluster 1 0.0317 0.02-0.06 1,2,3,4,5,6 6 cluster 2 0.0693 0.06-0.085 7,8,9 3 now, after performing all the required data preprocessing and clustering tasks, the decision rule generation algorithm is adopted to explore valuable information from the experimental dataset in the form of developed rules. these rules simply depict the relationships between various grinding parameters and responses to effectively control the said grinding operation. the first three sets of rules relate one or more grinding parameters to a single response. in contrast, the last set of rules relates multiple grinding parameters to all the three responses. 102 s. agarwal, s.s. dandge , s. chakraborty rules for ra: rule 1: if ss = 2430 then ra is 0.56 [0.40-0.60]. [p = 100%, q = 75%, c = 33.33%, qty = 3] [t = 208.33] rule 2: if ss = 2560 and doc = 0.2 then ra is 0.56 [0.40-0.60]. [p = 100%, q = 25.00%, c = 11.11%, qty = 1] [t = 136.11] rule 3: if ss = 2850 then ra is 0.72 [0.60-0.85]. [p = 100%, q = 60.00%, c = 33.33%, qty = 3] [t = 193.33] rule 4: if ss = 2560 and doc = 0.3 then ra is 0.72 [0.60-0.85]. [p = 100%, q = 20.00%, c = 11.11%, qty = 1] [t = 131.11] rule 5: if ss = 2560 and doc = 0.4 then ra is 0.72 [0.60-0.85]. [p = 100%, q = 20.00%, c = 11.11%, qty = 1] [t = 131.11] rules for amplitude of vibration (v): rule 1: if ss = 2430 and doc = 0.2 then v is 20.62 [17.00-23.00]. [p = 100%, q = 33.33%, c = 11.11%, qty = 1] [t = 144.44] rule 2: if ss = 2430 and tcf = water then v is 20.62 [17.00-23.00]. [p = 100%, q = 33.33%, c = 11.11%, qty = 1] [t = 144.44] rule 3: if ss = 2560 and doc = 0.2 then v is 20.62 [17.00-23.00]. [p = 100%, q = 33.33%, c = 11.11%, qty = 1] [t = 144.44] rule 4: if ss = 2850 then v is 29.97 [23.00-35.50]. [p = 100%, q = 50.00%, c = 33.33%, qty= 3] [t = 183.33] rule 5: if doc = 0.4 then v is 29.97 [23.00-35.50]. [p = 100%, q = 50.00%, c = 33.33%, qty = 3] [t = 183.33] rule 6: if ss = 2560 and doc = 0.3 then v is 29.97 [23.00-35.50]. [p = 100%, q = 16.67%, c = 11.11%, qty = 1] [t = 127.78] rule for g-ratio: rule 1: if ss = 2430 then g-ratio is 0.0317 [0.02-0.06]. [p = 100%, q = 50.00%, c = 33.33%, qty = 3] [t = 183.33] rule 2: if ss = 2560 then g-ratio is 0.0317 [0.02-0.06]. [p = 100%, q = 50.00%, c = 33.33%, qty = 3] [t = 183.33]. rule 3: if ss = 2850 then g-ratio is 0.0691 [0.06-0.085]. [p = 100%, q = 100.00%, c = 33.33%, qty = 3] [t = 233.33] rules for all the three responses: rule 1: if ss = 2850 then ra is 0.72 [0.60-0.85] and v is 29.97 [23.00-35.50] and gratio is 0.0693 [0.06-0.085]. [p = 100.00%, q = 100.00%, c = 33.33%, qty = 3] [t = 233.33] rule 2: if ss = 2430 then ra is 0.56 [0.40-0.60] and v is 20.62 [17.00-23.00] and gratio is 0.0317 [0.02-0.06]. [p = 66.67%, q = 66.67%, c = 22.22%, qty = 2] [t = 155.56] rule 3: if ss = 2560 then ra is 0.72 [0.60-0.85] and v is 29.97 [23.00-35.50] and gratio is 0.0317 [0.02-0.06]. [p = 66.67%, q = 100.00%, c = 22.22%, qty = 2] [t = 188.89] rule 4: if ss = 2430 and doc = 0.04 and tcf = coolant + water then ra is 0.56 [0.40-0.60] and v is 29.97 [23.00-35.50] and g-ratio is 0.0317 [0.02-0.06]. [p = 100.00%, q = 100.00%, c = 11.11%, qty = 1] [t = 211.11] parametric analysis of a grinding process using the rough sets theory 103 rule 5: if ss = 2560 and doc = 0.02 and tcf = water then ra is 0.56 [0.40-0.60] and v is 20.62 [17.00-23.00] and g-ratio is 0.0317 [0.02-0.06]. [p = 100.00%, q = 33.33%, c = 11.11%, qty = 1] [t = 144.44] from the developed rules, it can be propounded that for response ra (a smaller-thebetter type of quality characteristic), rule 1 emerges out as the strongest rule with a t value of 208.33%. based on this rule, it can be concluded that when the spindle speed is 2430 rpm, all the measured a values are expected to be „low‟ ranging between 0.40 µm and 0.60 µm with a rule confidence of p = 100%. similarly, 75% of all the trials (q = 75%) having ra values between 0.40 µm and 0.60 µm have been experimented while setting the corresponding spindle speed at 2430 rpm, and 33.33% of the experimental trials (c = 33.33%) are covered by this rule (i.e. three trials have ra values between 0.40 µm and 0.60 µm). amongst all the nine experimental trials, there are three runs that satisfy this rule (qty = 3). similarly, for rule 3, when the spindle speed is 2850 rpm, the measured ra values are expected to be „high‟ falling within the range of 0.60-0.85 µm. for ra response, all the remaining rules have less strength with not so much importance in controlling this grinding operation. rules 4 and 5 showing the combined influences of two separate grinding parameters on ra appear to be interesting to the production engineers, but they have also low total strength. these rules state that moderate value of spindle speed and moderate/high value of depth of cut lead to higher ra values causing generation of poor surface finish of the machined components. spindle speed appears in all the developed rules signifying its maximum importance in this grinding operation, followed by depth of cut. it is quite interesting to notice that type of the cutting fluid does not appear in any of the generated rules, revealing the fact that it has no role in controlling the surface characteristics of the ground workpiece samples. for amplitude of vibration, six rules are similarly generated. among them, rules 4 and 5 are observed to be the most decisive ones with the total strength of 183.33% each. they signify that when spindle speed is 2850 rpm or depth of cut is 0.04 mm, amplitude of vibration is high, falling within the range of 23.00-35.50 µm. experiment trial number 9 follows both these rules/conditions (i.e. rules 4 and 5), requiring attention of the concerned production engineers. some of the developed rules also exhibit the conjoint influences of two grinding parameters on amplitude of vibration, but they have low strength with smaller t values. among these rules, rule 2 is supposed to be the interesting one, i.e. it reveals that low spindle speed and water as the cutting fluid lead to reduced amplitude of vibration during the grinding operation. similarly, low/moderate spindle speed and low depth of cut also cause reduced vibration. for g-ratio, three decision rules are also formulated. spindle speed only appears in all these rules. it can be thus stated that when the spindle speed is equal to 2560 rpm or less than it, the corresponding values of g-ratio are low, falling between 0.02 and 0.06. in rule 3, having strength of 233.33%, a spindle speed value of 2850 rpm leads to higher g-ratio, in the range of 0.06-0.085. it is also interestingly observed that depth of cut and cutting fluid type do not affect g-ratio. when all the three responses are taken into consideration while formulating the corresponding decision rules, they become more complicated. amongst the five generated rules, rule 1 has the maximum strength of 233.33%, followed by rule 4 (211.11%). it states that when the rotational speed of the grinding wheel (spindle speed) is set at its highest operating level of 3 (i.e. 2850 rpm), higher values for all the considered responses 104 s. agarwal, s.s. dandge , s. chakraborty are simultaneously achieved. higher grinding wheel speed thus leads to poor machined surface with higher ra values, higher amplitudes of vibration and higher g-ratios. but, rule 4 with the second maximum strength is supposed to be the most interesting one for the concerned production engineers, because it encompasses all the grinding parameters and responses. based on this rule, it can be concluded that when the spindle speed is 2430 rpm, depth of cut is 0.04 mm, and a mixture of coolant and water is applied as the cutting fluid, low values of ra and g-ratio along with high value of amplitude of vibration are observed. thus, higher rotational speed of the grinding wheel always leads to higher g-ratio (grinding efficiency) with poor surface qualities of the machined components. similarly, higher depth of cut causes higher vibration during the grinding operation. the application of coolant and water or ordinary water causes enhanced performance of the grinding operation. but, keeping in mind the additional cost of special purpose coolant, it may be advised to apply simple water as the cutting fluid while grinding low alloy steel work materials. spindle speed plays the most significant role in controlling all the quality characteristics of the considered grinding process, followed by depth of cut and type of the cutting fluid. 5. conclusions in this paper, the rst, a machining learning algorithm of data mining, is employed to analyze the experimental data of a grinding process. based on the generated rules, the effects of three grinding parameters, i.e. spindle speed, depth of cut and type of the cutting fluid on three different responses, i.e. average surface roughness value, amplitude of vibration and grinding ratio are studied. using the calculated dependency indexes, the possibility of reduction of the initial experimental dataset is also explored. depending on the type of the responses, they are subsequently grouped into two different clusters, i.e. „low‟ and „high. it is observed from the decision rules developed for average surface roughness that low spindle speed leads to better surface roughness of the ground workpiece samples. on the contrary, higher spindle speed or depth of cut causes increased amplitude of vibration. similarly, higher spindle speed leads to higher grinding ratio (grinding efficiency). the rules formulated while taking into consideration all the three responses demonstrate that at higher rotational speed of the grinding wheel, higher values for all the considered responses are achieved. type of the cutting fluid does not influence attainment of low surface roughness and higher grinding efficiency; it only affects the vibration generated during the grinding operation. these rules developed based on the application of the rst are easy to comprehend and would guide the concerned production engineers in setting the input parameters of a grinding process so as achieve the desired quality characteristics of the ground components. it is observed that the classical rst approach can only process discrete data. however, in real time machining applications, most of the measured data are continuous. hence, for its successful application, there is always an additional task to discretize the continuous response values with the help of a suitable clustering technique. on the other hand, the generalization ability of rough sets needs to be improved and the probability distribution of sample data requires to be further considered for its effective deployment. moreover, in the rst approach, during data pre-processing, attribute reduction may often lead to over-fitting of a problem. parametric analysis of a grinding process using the rough sets theory 105 with automation of manufacturing industries and availability of high speed data acquisition systems, the manufacturing domain is now flooded with huge volumes of experimental data which if mined, can lead to effective and efficient control of different machining processes. the „if-then‟ decision rules generated using the st approach can be applied to any of the conventional and non-conventional machining processes to visualize the influences of their input parameters on the 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evaluation for nanofluid mql in grinding ti-6al-4v based on grey relational analysis, materials and manufacturing processes, 33(9), pp. 950-963. 35. kalpakjian, s., schmid, s., 2014, manufacturing engineering & technology, prentice hall, nj, usa. 10544 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 211 235 https://doi.org/10.22190/fume220221011h © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper the carbon nanotube-embedded boundary layer theory for energy harvesting ji-huan he1,2,3, nader y. abd elazem4 1school of civil engineering, xi’an university of architecture & technology, xi’an, china 2national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china 3school of science, xi'an university of architecture and technology, xi’an, china 4department of basic science, pyramids higherinstitute for engineering and technology, 6thof october city, giza, egypt abstract. single-walled carbon nanotubes (swnts)and multi-walled nanotubes (mwnts) are gaining appeal in mechanical engineering and industrial applications due to their direct influence on enhancing the thermal conductivity of base fluids. with such intriguing properties of carbon nanotubes in mind, our goal in this work is to investigate radiation effects on the flow of carbon nanotube suspended nanofluids in the presence of a magnetic field past a stretched sheet impacted by slip state. cnts flow and heat transmission are frequently modelled in practice using nonlinear differential equation systems. this system has been precisely solved, and an accurate analytical expression for the fluid velocity in terms of an exponential function has been derived, while the temperature distribution is stated in terms of a confluent hypergeometric function. the impact of the radiation parameter, slip parameter, sloid volume fraction, magnetic parameter, eckart and prandtl numbers on the velocity, temperature, and heat transfer rate profiles are demonstrated using a parametric analysis. when compared to the two types of nanoparticles (cooper and silver) in earlier published articles, temperature profiles for single-walled nanotubes (swnts) and multi-walled nanotubes (mwnts) are revealed to be particularly sensitive to radiation, solid volume fraction, and slip parameters. nanomechanical gears, nanosensors, nanocomposite materials, resonators, and thermal materials are only a few of the present problem's technical applications. key words: nanofluids, single-walled nanotubes, multi-walled nanotubes, radiation, slip condition, stretching sheet received: february 20, 2022 / accepted march12, 2022 corresponding author: nader y. abd elazem department of basic science, pyramids higher institute for engineering and technology,6th of october city, giza 12578, egypt e-mail: naderelnafrawy@yahoo.com, nader_47@hotmail.com mailto:naderelnafrawy@yahoo.com mailto:nader_47@hotmail.com 212 j.-h. he, n. y. abd elazem 1. introduction cnts have obtained a lot of interest in the last decades because of their amazing mechanical and electrical capabilities [1-3]. indeed, they are known to have very high thermal conductivity [4-5], therefore liquids containing cnts are predicted to boost heat transfer near the laminar boundary layer. carbon nanotubes are cylinder-shaped nanostructures, which have exceptional properties that make them helpful in mechanical engineering, optics, electronics, nanotechnology, and materials science [6]. zuo and liu [7] offered various unique concepts for graphene/sic composites for the first time, including the graphene fractal and two-scale porosity. theoretical investigations show that graphene concentration and two-scale fractal dimensions have a considerable influence on the mechanical and electrical properties of graphene/sic composites. cnts are either a graphite sheet coiled into a nanoscale tube (single-wall carbon nanotubes (swcnts)) or multiple graphene tubes wrapped around the core of the swcnt (multi-wall carbon nanotubes (mwcnts)). cnts feature unique physical and chemical qualities [8] such as thermal and magnetic performance, as well as unique optical properties. ref. [9] studied the effect of carbon nanotubes on total heat transfer performance in nucleate boiling, which occurs in power plant boilers and evaporators, as well as refrigeration and air-conditioning equipment, and it was discovered that cnts’ infiltration into the thermal boundary layer can form more bubbles at the surface, and this can enhance nucleate boiling heat transfer. carbon nanotubes (cnts) are one of the most promising nanomaterials [10]. single-walled carbon nanotubes (swcnts) and multi-walled carbon nanotubes (mwcnts) are the most often utilized cnts [11]. xue [12] presented a novel model of cnts effective thermal conductivity to reveal theoretically cnts excellent heat transfer performance. these findings are in good accord with the experimental data. carbon nanotubes (cnts) have received a lot of interest in recent years because of their remarkable electrical and mechanical capabilities [7, 13]. in ref. [14], an analysis was presented to study the actual and theoretical properties of cnts thermal conductivity. according to ref. [15], swcnts have a higher thermal conductivity of up to 15% than other nanoparticle-based nanofluids. the authors of [16] explored the influence of swcnts on the presence of saltwater and water with solar radiation energy under varied stream conditions. cnts have roughly six times the heat conductivity of common materials at normal temperature, according to the findings of a study published in [17]. using two different types of carbon nanotubes (swcnts and mwcnts), ref. [18] studied the peristaltic flow of carbon nanotubes in an asymmetric channel with temperature and velocity slip effects. in terms of magnetic field and solid volume percentage, it is discovered that the velocity field for swcnt is bigger than that of mwcnt. the results of the experimental study in ref. [19] showed the relevance of rheological characterization in establishing the presence and structural information of nanofluid forms, which can assist in the understanding of the thermal properties of such materials. analysis was provided in ref. [20] to discuss the peristaltic flow and heat transfer of carbon nanotubes in a diverging tube. it is found that the velocity field for swcnt is greater as compared to the mwcnt and the size and no of trapped bolus decreases for mwcnt when we compared with the swcnt. khan et al. [21] present a complete analysis of the influence of hybrid nano-fluid (ferrous oxide water and carbon nanotubes) cnts-fe3o4, h2o between two parallel plates with a varying magnetic field. it has been noticed that the increase in the volume fraction of the nano the carbon nanotube-embedded boundary layer theory for energy harvesting 213 material significantly fluctuates the velocity profile near the channel wall due to an increase in the fluid density. in addition, single-wall nanotubes have a greater effect on temperature than multi-wall carbon nanotubes. statistical analysis shows that the fraction of the nano-material significantly fluctuates the velocity profile near the channel wall due to an increase in the fluid density. statistical analysis shows that the thermal flow rate of (fe3o4-swcnts-water) and (fe3o4-mwcnts-water) rises from 1.6336 per cent to 6.9519 per cent, and 1.7614 per cent to 7.4413 per cent, respectively when the volume fraction of nanomaterial increases from 0.01 to 0.04. many authors have studied the flow and heat transmission of ordinary nanoparticles and suspended carbon-nanotube nanofluids. for instance, the authors of ref. [22] studied the heat transfer behaviour of cnt nanofluids flowing through a horizontal tube. they discovered that the increase of convective heat transfer is dependent on the reynolds number and solid volume fraction of cnts. he and abd elazem [23] investigated the impact of partial slips and temperature jumps on heat and mass transport in a boundary layer nanofluid flowing over a stretched or shrinking surface. it is found that the slip parameter improves the temperature field and increases the thermal boundary layer thickness, as well as the concentration function’s boundary layer thickness in the case of a stretched sheet. when the slip parameter is raised in the case of the shrinking sheet, the dual solutions for temperature and concentration functions are reduced. thermal conductivity becomes more important at high working temperatures in common applications such as materials engineering and electronic parts. thermal efficiency is described in physics as a material's purpose of transporting heat. this can be seen clearly from fourier's heat transfer theorem. theoretical and practical problems, such as a continuous supply of thermal energy, are growing rapidly. radiation heat transfer research is crucial in high-temperature industrial applications. abd elazem and elgazery [24] examined the impacts of variable viscosity and thermal radiation for a mixed convective heat and mass transfer nanofluid flow along a vertical wavy surface. it is revealed that when the temperature of the radiation increases, the nanoparticle concentration curves near the wavy surface decrease while the far-field increases. further, when the radiation parameter rises, so should the thermal boundary layer. importantly, the variable viscosity parameter boosts the steady-state axial nanofluid velocity profile at the wavy surface boundaries while decreasing it further away. many additional relevant results are also mentioned in the references [25-28]. as a result, the current work intends to explore the influence of the radiation parameter and the boundary condition of partial slip on the flow of carbon nanotubes suspended nanofluids across a stretched sheet in the presence of a magnetic field. the physical model is solved using an appropriate analytical solution for fluid velocity in terms of an exponential function and a confluent hypergeometric function for temperature distribution, which are searched and compared to previously published findings, notably [29]. the effect of the included physical parameters on the flow and heat transfer characteristics of carbon nanotube suspended nanofluids, as well as cu-water and ag-water nanofluids, is depicted graphically. the results of this study indicate that the thickness of the thermal barrier layer of the heat transfer rate rises as the radiation parameter increases, for the stretching sheet, thereby reducing the heat transfer rate. the findings have a wide range of applications, including thermalmechanical effects in high-speed flights, rockets, nuclear reactors and design of a solar energy collector's heat transfer in refs. [23, 30-31]. 214 j.-h. he, n. y. abd elazem 2. mathematical model offlow and heat transfer the pdes that characterize the flow and similarity transformations are described in detail in ref. [29]; they are not included here to avoid duplication. the flow and heat transfer of an incompressible viscous nanofluid containing singleand multi-wall carbon nanotubes across a linearly semi-infinite stretched sheet will be investigated using the radiation term in the article [32] and the slip condition specified in eq. (3) below to modify in [29]. as a result, the final state is governed by the following system of nonlinear differential equations under the influence of the radiation parameter, slip parameter, and a constant magnetic field: ,0)]()()()([)( 2 2 1 =+−−  f c m fffcf (1) .0]))(()()()()(2[pr)()1( 2 4 3 =−−−+  f c ec gfgfc k k gr nf f (2) the slip modifies the boundary conditions in ref. [29] as follows: ,0)(,1)0(,0)(),0()0(,0)0( ===+== ggffldff (3) where the prime implies differentiation with respect to η, f(η) and g(η) are the dimensionless of the stream and temperature functions, respectively, η is the similarity variable, and l is the velocity slip parameter. furthermore, pr, m and ec are the prandtl number, magnetic parameter, and eckert number, as stated in ref. [29]. also, as indicated in ref. [32], r is the radiation parameter, d = 1 signifies a stretched sheet, and )4,...,1(, =ici are defined as ). )( )( 1()1(), )( )( 1( ),1(),1()1( 5.2 43 2 5.2 1 fp cntp fp cntp f cnt f cnt c c c c c c cc             +−−=+−= +−=+−−= (4) the thermophysical characteristics of singleand multi-wall carbon nanotubes are shown in table 1. furthermore, the heat conductivity of singleand multi-wall cnts is discussed in xue's study as follows [12]: . 2 ln21 2 ln21         +         − +−         +         − +− = f fcnt fcnt f f fcnt fcnt cnt f nf k kk kk k k kk kk k k k   (5) it is known that ϕ is the solid volume fraction of the singleand multi-wall cnts, ρf and ρcnt are the densities, (ρcp)f and (ρcp)cnt are the heat capacitances, kf and kcnt are the thermal conductivities, (k)f and (k)cnts denote the basic fluid and solid fractions, and knf is the thermal conductivity, respectively. the carbon nanotube-embedded boundary layer theory for energy harvesting 215 table 1 water's and nanoparticles' thermophysical characteristics [12] physical properties ρ (kg / m³) cp (j / kg k) k (w /m k) fluid phase (water) 997.1 4179 0.613 swcnt 2600 796 6600 mwcnt 1600 235 3000 engineering flow behavior is very significant in materials treatment processes for the local skin friction coefficient, cf, and the nusselt number, nux. here are the formulations for the local skin friction coefficient, cf, and the nusselt number, nux. ).0( re ),0(2re)1( 5.2 g k knu fc nf f x x xf −=         −=− (6) where rex represents the local reynolds number defined in ref. [29]. furthermore, when r=l=0, the system of eqs. (1) to (3) yields to [29]. 2.1. exact solutions the precise solutions of the system of eqs. (1) and (2) with boundary conditions eq. (3) are found in this section. assume the 𝑓-differential eq. (1) has a solution that fulfils the boundary condition eq. (3) in the following form: .)(  −+= menf (7) from eq. (7), we may derive that the boundary requirements for f(η) are provided in eq. (3), and we obtain ,0=+mn (8) and . (1 ) d m l  = − + (9) on inserting eq. (9) into eq. (8), we get . (1 ) d n l  = + (10) therefore, the exact solution off(η) becomes ( ) (1 ). (1 ) d f e l    −= − + (11) this solution meets the boundary conditions in eq. (3) for 0 which are required for calculating flow and skin friction. the thirddegree algebraic equation shown below is generated by inserting eq. (11), into eq. (1). 216 j.-h. he, n. y. abd elazem ( ) .0 2 1 2 2 1 23 =         +−         −+ c c mcd c c mll  (12) there is only one positive root for according to descartes' rule of signs and the fact 0 which is given by , 22 22 6 1 1 3 1 2 3 1 3 1 1 3 1 3 2             ++−= l  (13) where 12 2 1 1 2 3 2 2 2 1 2 1 2 2 2 2 2 1 2 2 2 27 18 (2 9 (2 3 )) 4 1 3 , 1 3 . c d l c m l c c c l c m d c m l c c c m l c  = − + + +   − + − − +      = + (14) when there is no slip, eq. (11) leads to .),1( 1 )( 2 1 1 m c c cef +=−= −     (15) this solution is consistent with [29].the exact solution f(η) in terms of β is then . 22 22 6 122 22 16 )( 2 1 3 1 2 3 1 3 1 1 3 1 3 2 1 3 1 2 3 1 3 1 1 3 1 3 2 22 22 6 1 1 3 1 2 3 1 3 1 1 3 1 3 2             ++−+             ++−               − =             ++−−   l eld f (16) substituting eq. (11) for eq. (2) results in: ,)(pr )()pr2()()pr()( 22 2 11     − −− − =+++ emec gemgmeng (17) where . )1( , )1( 4 3 231 c c kr k c kr k nf f nf f + = + =  (18) the carbon nanotube-embedded boundary layer theory for energy harvesting 217 eq. (17) is difficult to solve using neither the laplace transform nor the power series approach due to the inclusion of e-βη in this equation, as well as the infinite boundary condition eq (3). then, using the following independent variable, we must convert eq. (17) into polynomial type coefficients: . pr pr,pr 2    =−= −e (19) taking . pr ,pr1, 1 2 22 1      ec l d =−= + = (20) eq. (17) becomes 1 1( ) ( ) ( ) 2 ( ) .g g g           + − + = − (21) with the transformed boundary conditions .0)0(,1)pr( ==− gg (22) then the exact solution of the temperature function in terms of η is 2 (1 ) 0 2 2 0 [ 1 , 2 , ( 1) ] ( ) (1 ( pr) ) [ 1 , 2 , 1] ( pr) ( ) , m e g e m e               − − − − − − − − = − − − − − − + − (23) where !)1)....(1( )1)....(1( ],,[ 0 s y sqqq sppp yqpm s   = −++ −++ = is kummer’s function [33], and 0 . 2(1 ))    = − + when l=r=0 and d = 1, the solution provided by eq. (23) simplifies to the one given in ref. [29] (eq. (36)). furthermore, the present exact solutions may be easily checked by immediately substituting them into the original differential equations. 3. results and discussion because of their extraordinary flow stability, carbon nanotubes have lately advanced. for various values of the controlling l, r, m, pr, ϕ,andec parameters, eqs. (1) and (2) applied to the boundary conditions eq. (3) have been solved analytically [29]. tables 2-4 provide the numerical results of the local skin friction -f''(0) and the local nusselt number -g'(0). figs. 1–22 show the behavior of dimensionless similarity function, streamline contours, velocity, sheer stress, temperature, and local nusselt number profiles in the stretched sheet scenario. 218 j.-h. he, n. y. abd elazem 3.1. numerical results for -f''(0) and -g'(0) table 2 compares the proposed study's results for the local nusselt number -g'(0) at various prandtl numbers to those discovered in refs. [29, 34]. the discovery proved that the pr values are completely correlated. it is worth noting that -g'(0)is an increasing function. furthermore, as shown in table 3, the numerical values of the local skin friction -f ''(0) for each of cu-water, ag-water, swcnt, and mwcnt in the absence of slip and radiation parameters at various values of m and ϕ are perfectly in agreement with those reported in references [29, 35]. the numerical findings reveal that in the absence of a magnetic field, at m=0, the skin friction coefficient for cu-water and ag-water nanofluids rises with increasing solid volume fraction [32]. table 4 illustrates the variation in numerical results of -f''(0) and -g'(0) for each of cu-water, ag-water, swcnt, and mwcnt at different values of l, r, m, and ϕ when ec=0.1, and pr=6.7850 in the current study. table 2 comparison of numerical results for -g'(0) at different values of prandtl numbers in the present study with those obtained in refs. [29, 34] pr ref. [29 ] ref. [34] present results 0.72 1.08852 1.0885 1.088524024 1.0 1.33333 1.3333 1.333333333 3.0 2.50973 2.5097 2.509725227 10.0 4.79687 4.7969 4.796873283 100.0 15.71163 15.7120 15.71196715 it is deduced that -f ''(0) is a monotonic function with regard to cu-water and ag-water, but a decreasing function with respect to swcnt and mwcnt by increasing the l, r, and m parameters. -g'(0) on the other hand, is a declining function for all nanofluids. 3.2. flow and heat transfer facts and interpretations figs. 1 and 2 provide plots of variation of dimensionless similarity functionsf(η) andg(η) at the specified data in the current study and earlier published work [29]. it is important to note that in the absence of slip, i.e., l=0,the profile of f(η) as shown in fig. 1 for the mwcnt is greater than for the swcnt, cu-water, and ag-water, whereas the temperature profiles of g(η) in fig. 2 for the swcnt are the highest. furthermore, as the values of the slip parameter increase, the dimensionless similarity functions f(η) for each of mwcnt, swcnt, cu-water, and ag-water decrease, but vice versa for the temperature profiles g(η) as the values of the solid volume fraction increase. figs. 3-5 depict heat transmission inside the boundary layer region using streamline contours of mwcnt, swcnt, cu-water, and ag-water. these graphsare presented for three distinct values of 𝐿: (a) l=0.0, (b) l=0.5, and (c) l=1 with the other parameters set to m=0.01, ϕ=0.25, ec=0.1, and d=1. again, l=0.0 depicts the merely stretching situation, whereas l≠0.0 indicates the moving case. when l is increased from 0 to 1, these motions become more pronounced within the boundary layer area. as a result, with larger values of 𝐿, the streamlines transport more energy during the fluid flow; hence, l plays an important role in the formation of the boundary layer flow area (see figs. 9 and 10). the carbon nanotube-embedded boundary layer theory for energy harvesting 219 table 3 comparison of numerical results for -f''(0) in the absence of slip and radiation parameters at various values of m and ϕ in the present study with those obtained in refs. [29, 35] ref. [29 ] ref. [35] m ϕ cu-water ag-water cu-water ag-water 0 0.05 1.108919904 1.139659703 1.10892 1.13966 0.1 1.174746021 1.225068143 1.17475 1.22507 0.15 1.208862320 1.272152949 1.20886 1.27215 0.2 1.218043809 1.289788016 1.21804 1.28979 2 0.05 1.728872387 1.74874830 1.72887 1.74875 0.1 1.707892022 1.742888091 1.70789 1.74289 0.15 1.671398302 1.717730276 1.67140 1.71773 0.2 1.621264175 1.675834100 1.62126 1.67583 present results m ϕ cu-water ag-water swcnt mwcnt 0 0.05 1.108919904 1.139659703 0.974860316 0.951967598 0.1 1.174746021 1.225068143 0.944438410 0.902716495 0.15 1.208862320 1.272152949 0.909249444 0.852365612 0.2 1.218043809 1.289788016 0.869757340 0.801035688 2 0.05 1.728872387 1.748748300 1.646101156 1.632647753 0.1 1.707892022 1.742888091 1.558470678 1.533546221 0.15 1.671398302 1.717730276 1.469339554 1.434834942 0.2 1.621264175 1.675834099 1.378892540 1.336609508 the findings shown in fig. 6 regarding the influence of the slip parameter l on velocity variation show that f''(η) is a decreasing function in l. furthermore, the mwcntnanofluids have faster velocities than the swcnt-nanofluids, which is consistent with the results of fig. 8 and the other nanoparticles (cu-water and ag-water) as shown in fig. 7. in contrast, as seen in figs. 6 and 7, increasing the slip parameter l causes the shear stress f ''(η) to increase. physically, it can be shown that as l grows, the wall slips velocity drops while the wall drag force increases. for a larger value of l and a short flow passes into the nanofluid, the nanofluid velocity is slowed down off quickly. the flow can penetrate relatively deeply for a modest value of l. the shear stress profiles at the wall are larger for a large value of l, and they grow fast as the flow enters into the nanofluid. it is worth noting that in figs. 6 and 7, the concavity of the velocity f '(η) and the shear stress f''(η) profiles is shifted from above to down. 220 j.-h. he, n. y. abd elazem table 4 variations of numerical results for -f''(0) and -g'(0) at various values of l, r, m, and ϕ when ec=0.1 and pr=6 .7850 in the present study f ''(0) l r m ϕ cu-water ag-water swcnt mwcnt 1 1 0 0.05 0.451038219 0.456562203 0.425025631 0.420238679 0.1 0.462685934 0.471149914 0.418640297 0.409558845 0.15 0.468463835 0.478749835 0.411006550 0.398063307 0.2 0.469990180 0.481520936 0.402103162 0.385687651 3 5 0.1 0.05 0.225492740 0.226771004 0.219455243 0.218342243 0.1 0.227585942 0.229577545 0.217118779 0.214945178 0.15 0.228412363 0.230869212 0.214443019 0.211251204 0.2 0.228488915 0.231085016 0.211398287 0.207222484 5 10 0.5 0.05 0.160958245 0.161226725 0.159772323 0.159568261 0.1 0.160424833 0.160905202 0.158140630 0.157716259 0.15 0.159686464 0.160343847 0.156358478 0.155695813 0.2 0.158745002 0.159557384 0.154403462 0.15348256 g'(0) l r m ϕ cu-water ag-water swcnt mwcnt 1 1 0 0.05 1.783121169 1.755546964 1.380374490 1.428477746 0.1 1.604982420 1.559269414 1.065252751 1.125818696 0.15 1.457837201 1.398823398 0.865564763 0.930729013 0.2 1.333340229 1.264003676 0.724175740 0.791012026 3 5 0.1 0.05 0.704628283 0.690377389 0.530640248 0.553698212 0.1 0.625380551 0.601905211 0.395335561 0.424797834 0.15 0.561570699 0.531598949 0.312361843 0.344182003 0.2 0.509181816 0.474419170 0.255741049 0.288305237 5 10 0.5 0.05 0.294329004 0.288338153 0.199582973 0.209512817 0.1 0.266266609 0.255860920 0.143272409 0.154802619 0.15 0.243312656 0.229537559 0.111623322 0.123323245 0.2 0.224350296 0.207919858 0.091324210 0.102865153 fig. 1 plots of the dimensionless stream functions for mwcnt and swcnt in this study and prior published work [29] at different values of slip parameter when m=r=0, ec= ϕ=0.1, and d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 221 fig. 2 plots of the temperature profiles of mwcnt and swcnt in this study and prior published work [29] at different values of solid volume fraction when m=r=0, ec=0.1, and d=1 fig. 3 streamline contours at l=0, m=0.01, ϕ=0.25, ec=0.1, and d=1 222 j.-h. he, n. y. abd elazem fig. 4 streamline contours at l=0.5, m=0.01, ϕ=0.25, ec=0.1, and d=1 fig. 5 streamline contours at l=1, m=0.01, ϕ=0.25, ec=0.1, and d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 223 fig. 6 impact of the slip parameter on the velocity and the shear stress profiles for mwcnt and swcnt at m= ec= ϕ=0.1, r=5, and d=1 fig. 7 impact of the slip parameter on the velocity and the shear stress profiles for mwcnt and swcnt at m= ec= ϕ=0.1, r=5, and d=1 because of the importance of this discovery for engineers and researchers in practically every aspect of physics and engineering, several real-world examples are offered to support the stated modelling findings. nuclear power plants, gas turbines, and various propulsion technologies for aeroplanes, missiles, satellites, and space vehicles are examples of such technical specialties'. figs. 8-10 show the impacts of the solid volume fraction parameter on the 224 j.-h. he, n. y. abd elazem velocity and temperature profiles of swcnt and mwcnt suspended nanofluids, which are compared to those obtained in [29]. fig. 8 demonstrates that increasing the solid volume fraction of swcnt and mwcnt nanofluids enhances fluid velocity, but increasing the solid volume fraction of cu-water and ag-water reduces fluid velocity [29]. fig. 8 further shows that mwcnt suspension nanofluids have greater velocities than all the other nanofluids, which may be advantageous in both medical and industrial aspects. the increase in temperature with the addition of additional nanoparticles to the base fluids has been a good result in the field of nanofluids. this is due to an increase in the thermophysical properties of the fluid. as shown in figs. 9 and 10, this conclusion is likewise validated for cnts. furthermore, figs. 9 and 10 reveal that swcnt nanofluids have a greater temperature than mwcnt nanofluids [36]. additionally, figs. 9 and 10 show that increasing l (l=0, 1) improves the temperature profiles of all nanofluids. figs. 11 and 12 show temperature profiles for swcnt, mwcnt, cu-water, and ag-water suspended nanofluids at different values of three parameters, pr, m, and l when r=3, ec=ϕ=0.1, and d=1. physically, it is known that temperature profiles decline at large prandtl numbers pr, as seen in figs. 11 and 12 [37]. fig. 8 impact of the solid volume fraction on the velocity profiles at l=1, m=0.5, r=3, ec=0.1, and d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 225 fig. 9 impact of the solid volume fraction on the temperature profiles at l =0, m=0.5, r=3, ec=0.1, and d=1 fig. 10 impact of the solid volume fraction on the temperature profiles at l=1, m=0.5, r=3, ec=0.1, and d=1 this is consistent with the physical reality that the thickness of the thermal boundary layer decreases as pr increases. in other words, increasing the value of pr accelerates the decay of the temperature field away from the heated surface, increasing the rate of heat transfer and a 226 j.-h. he, n. y. abd elazem decrease in the thickness of the thermal boundary layer [38]. however, when the magnetic and slip factors are coupled, the temperature profiles grow as the magnetic and slip values increase. figs. 11 and 12 show temperature profiles for various values of the m and l parameters, as well as the prandtl number, in the presence of swcnt, mwcnt, cu-water, and ag-water nanoparticles. the uniform thermal radiation [32] clearly shows that in the four examples listed above, the heat of water-based swcnt, mwcnt, cu-water, and ag-water accelerates with increasing magnetization, implying that the external electric field tends to heat the fluid and improves thermal performance from the surface. cos of the composite members' electric and thermal conductivity of swcnt-water and mwcnt–water, the dynamism and thermal boundary layer thickness of swcnt–water and mwcnt–water are significantly greater than those of cu-water and ag-water. it shows that reducing the density of the water-based swcnt, mwcnt elevates the fluid temperature difference and proves that applying magnetism to an electrically conducting fluid provides a skid steer strength that induces acceleration in the nanofluid temperature [16]. furthermore, it is discovered that the action of viscous heating causes a rise in temperature, which is more prominent in the presence of a magnetic field. fig. 11 impact of prandtl number and magnetic parameter on the temperature profiles at l=0, r=3, ec=ϕ=0.1, and d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 227 fig. 12 impact of prandtl number and magnetic parameter on the temperature profiles at l=1, r=3, ec=ϕ=0.1, and d=1 figs. 13-15 in this chapter depict the physical effects of several parameters on the solution property. it is worth mentioning that in the presence of the radiation parameter, raising the emission parameter improves the amount of radiative heat transmitted to swcnt-water, mwcnt-water, and thus the fluid temperature substantially when contrasted tocu-water and ag-water. figs. 16 and 17 depict the effect of the m and ec parameters on the heat transport -g'(η) curves of swcnt, mwcnt, cu-water, and ag-water nanofluids. physically, as demonstrated in figs. 11 and 12, raising the m parameter enhances the temperature profiles of all the aforesaid nanofluids while decreasing the heat transfer -g'(η) profiles, as illustrated in figs. 16 and 17 [38]. furthermore, by raising the eckert number, the heat transfer -g'(η) profiles for these nanofluids become rather deep. the heat transfer -g'(η) profiles for cu-water are found to be greater than those for ag-water, mwcnt, and swcnt nanofluids at all m parameter values. figs. 16 and 17 show that as m increases, the heat transmission -g'(η) profiles decrease and the thermal boundary layer thickens. furthermore, for high ec values, the heat transfer -g'(η) profiles of mwcnt and swcnt reduce fast. figs. 18-20 indicate the effectiveness of various parameters, r and pr, on the heat transfer -g'(η) graphs of swcnt, mwcnt, cu-water, and ag-water nanofluids. it was discovered that the heat transfer -g'(η) profiles for cu-water are larger than those of other nanofluids at all r parameter values. figs. 18-20 indicate that when r grows, the heat transmission -g'(η) profiles of cu-water and ag-water drop and the thermal boundary layer deepens in comparison to mwcnt and swcnt nanofluids. additionally, at high pr values, the cuwater and ag-water heat transfer -g'(η) profiles advance quicker than the mwcntand swcnt nanofluids. this is significant in thermal engineering, mechanics, and science, such as thermal processing, airflow, and conditioning systems, as well as renewable energies. 228 j.-h. he, n. y. abd elazem fig. 13 impact of radiation parameter on the temperature profiles at l=0, m=0.5, ec= ϕ=0.1, and d=1 fig. 14 impact of radiation parameter on the temperature profiles at l=10, m=0.5, ec= ϕ=0.1, and d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 229 fig. 15 impact of radiation parameter on the temperature profiles at l=100, m=0.5, ec= ϕ=0.1, and d=1 fig. 16 impact of magnetic parameter on the heat transfer rate profiles at ec=1, l=10, r=3, ϕ=0.1, and d=1 230 j.-h. he, n. y. abd elazem fig. 17 impact of magnetic parameter on the heat transfer rate profiles at ec=10, l=10, r=3, ϕ=0.1, and d=1 fig. 18 impact of radiation parameter on the heat transfer rate profiles at pr=0.73, l=10, ec =1, ϕ=0.1, and d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 231 fig. 19 impact of radiation parameter on the heat transfer rate profiles at pr=1, l=10, ec =1, ϕ=0.1, and d=1 fig. 20 impact of radiation parameter on the heat transfer rate profiles at pr=3, l=10, ec =1, ϕ=0.1, and d=1 232 j.-h. he, n. y. abd elazem finally, figs. 21 and 22 depict the heat transfer -g'(η) profiles of swcnt, mwcnt, cu-water, and ag-water nanofluids at various values of l and ϕ parameters. fig. 21 impact of the slip parameter on the heat transfer rate profiles at ϕ=0.1, m=r=3, and ec= d=1 fig. 22 impact of the slip parameter on the heat transfer rate profiles at ϕ=0.25, m=r=3, and ec= d=1 the carbon nanotube-embedded boundary layer theory for energy harvesting 233 the profiles of heat transfer -g'(η) of swcnt, mwcnt, cu-water, and ag-water nanofluids decrease as l and ϕ parameters rise. it has been shown that all of these nanofluids are faster decrease when the solid volume fraction is greater. the thickness of the thermal boundary layer profiles of the heat transfer -g'(η) of mwcnt and swcnt is lower than that of cu-water and ag-water nanofluids as the thickness of the thermal boundary layer profiles increases. 4. conclusions the effects of radiation on the flow of carbon nanotube suspended nanofluids across a stretched sheet influenced by slip state in the presence of a magnetic field, as well as their applications in medicine and engineering, were investigated. carbon nanotubes may be used for a variety of purposes, including thermal conductivity control, field emission, conductive properties, energy storage, cnt-based molecular electronics, and thermal materials. cos of their large molecular mass, they are nearly impermeable in all known solutions. some of the applications of interest that involve heat transfer via natural convection include the design of chemical processing equipment, the design of boilers, temperature and moisture ranges over agricultural areas, and thermal safety systems. analytical solutions were found for a system of nonlinear ordinary differential equations with specific physical parameter values of l, m, r, ec,ϕ, and pr. our analyses revealed the following significant discoveries: at varying values for the l, r, and m parameters, it was revealed that -f ''(0) is a monotonic function in terms of cu-water and ag-water, but a decreasing function in terms of swcnt and mwcnt, whereas -g'(0) is a decreasing function applicable to all nanofluids. temperature profiles were discovered to be extremely respectful provided to the solid volume fraction, magnetic, slip, and radiation parameters. so, swcnt nanofluids have a greater temperature than mwcnt, ag, and cu nanofluids (see figs. 9 and 10, figs. 11 and 12, and figs. 13 and 14) when these three parameters increase. however, the mwcnt nanofluids had a greater stream function, streamlines, and velocity profiles than the swcnt, cu-water, and ag-water nanofluids (see figs. 1 and 2, figs. 4 and 5, figs. 6 and 7, and fig. 8) in the case of the increase of the slip parameter. the solid volume fraction, magnetic, slip, and radiation characteristics all improve the temperature field and raise the thickness of the thermal function's boundary layer in the case of a stretched sheet, but they reduce the heat transfer rates of all other nanofluids. it is also concluded that when the slip parameter increases, the heat transfer profiles -g'(η) for cu-water grow larger than those for the other nanofluids (see figs. 21 and 22). because of its numerous uses in chemical manufacturing engineering, this partial slip boundary condition has received a lot of attention. the velocity profiles of the swcnt and mwcnt nanofluids increase as the solid volume fraction increases, whereas the reverse is true for the cu-water and ag-water nanofluids (see fig. 8). furthermore, as the ϕ parameter is raised, the mwcnt and swcnt heat transfer rates are deeper than the cu-water and ag-water nanofluids. as the prandtl number increases, the temperature profiles of cu-water, ag-water, mwcnt, and swcnt nanofluids decrease while heat transfer rates increase. the heat transfer profiles -g'(η) for cu-water, ag-water, mwcnt, and swcnt nanofluids gradually deepen as the eckert number increases. 234 j.-h. he, n. y. abd elazem the findings have applications in a variety of fields, including thermo-mechanical processes, biomedical, and the optimal heat 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of suction or injection on flow of nanofluids past a stretching sheet, z. naturforsch. a, 71(6), pp. 511–515. 38. elgazery, n.s., abd elazem, n.y., 2011, effects of viscous dissipation and joule heating for natural convection in a hydromagnetic fluid from heated vertical wavy surface, z. naturforsch. a, 66(6-7), pp. 427 – 440. https://www.sciencedirect.com/science/journal/00179310 https://www.sciencedirect.com/science/journal/00179310 facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 357 364 https://doi.org/10.22190/fume190507041b © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper intelligent machine vision based railway infrastructure inspection and monitoring using uav milan banić 1 , aleksandar miltenović 1 , milan pavlović 2 , ivan ćirić 1 1 faculty of mechanical engineering, university of niš, serbia 2 college of applied technical sciences, niš, serbia abstract. traditionally, railway inspection and monitoring are considered a crucial aspect of the system and are done by human inspectors. rapid progress of the machine vision-based systems enables automated and autonomous rail track detection and railway infrastructure monitoring and inspection with flexibility and ease of use. in recent years, several prototypes of vision based inspection system have been proposed, where most have various vision sensors mounted on locomotives or wagons. this paper explores the usage of the uavs (drones) in railways and computer vision based monitoring of railway infrastructure. employing drones for such monitoring systems enables more robust and reliable visual inspection while providing a cost effective and accurate means for monitoring of the tracks. by means of a camera placed on a drone the images of the rail tracks and the railway infrastructure are taken. on these images, the edge and feature extraction methods are applied to determine the rails. the preliminary obtained results are promising. key words: computer vision, uav, drone imagery, edge detection, railway infrastructure, intelligent systems 1. introduction with the growing industrial development and population, the role of railways in transportation is going to be crucial in upcoming years. one of the main goals of modern railway transport is to increase its quality, as well its effectiveness and capacity while maintaining a very high level of safety. on the other hand, in recent years, with advance of technology solutions in the field of uav (unmanned aerial vehicle), the new possibilities of using uavs for civil purposes are opened. uavs are commonly used in other industries such as oil and received may 07, 2019 / accepted august 15, 2019 corresponding author: ivan ćirić faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: ivan.ciric@masfak.ni.ac.rs 358 m. banić, a. miltenović, m. pavlović, i. ćirić utilities to inspect their structures, such as pylons and oil rigs. the imagery / video collected using an uav is up to date and provides oblique information about the railway infrastructure unlike satellite imagery. sesar european outlook study (2016) [1] defines that the infrastructures such as railway may be monitored and kept secure by using drones. prediction is that railway inspection will be carried out with long range surveying (primarily bvlos). bvlos are fly drones that can be used beyond the visual line of sight and represent future next step for drone industry. europe market potential is that by 2035 up to 400 000 drones will be in use with the majority flying beyond visual line of sight; out of this number around 180 000 could be used for mapping and surveying where one of the key markets is railway. this will have a great impact in a great number of industry sectors and in railway also. drones could benefit the mobility sector through railway inspection in which approximately 200 000 kilometers on a bi-monthly basis could be monitored. kim [2] developed the structure inspection system to detect and calculate cracks in structure using uav and digital image processing technique. he acquired the digital image in order to evaluate uav applicability and performance and field application of the crack detecting program after targeting the bridges. comparing with the measured values, it is verified that the accuracy above a certain level is secured. smith [3] developed a software package capable to process image that can be used to determine the location of the roads and railroads on the image as well as to detect intersections and generate specific trajectories along them. the road detection algorithm presented was proven to have a run speed of 1.25 hz onboard a single board computer, a high classification accuracy of 96.6%, and that it produces very accurate trajectories along roads. mathe [4] focused on lightweight uavs to detect events in railways such as missing indicators or cabling. he developed visual servicing technique and performed a comparison of several object detection approaches. gemert [5] investigates the combination of small uav with automatic object recognition techniques. they record animal conservation in case of animal detection and animal counting. karakose [6] proposed a computer-based visual rail condition monitoring in which camera is placed on top of the train and neighbor rail images are taken. on these images, the edge and feature extraction methods are applied to determine the rails. the results obtained are given at the end of the study. experimental results show that the proposed method gives effective results. flammini [7] studied preliminarily evaluate drone capabilities in a railway monitoring framework including structural faults and security threat detection as well as investigation of the consequences of natural hazards and intentional attacks. he proved that drone-based sensors can be advantageously integrated in existing security surveillance systems by appropriate sensor integration platforms, middleware and frameworks. mittal [8] proved that deep learning models can be used in detect of track defects like sinking, loose ballasts and railway assets like switches and signals. sigha [9] investigated the possibilities of computer vision-based monitoring with uav imagery. the uav camera provides high quality images that contain large information for monitoring and analysis. inspection by drone does not require dedicated track for inspection hence it does not affect smooth running of trains but the aerial images with the camera oriented downward do not give convergence view of track. intelligent machine vision based railway infrastructure inspection and monitoring using uav 359 jianfang [10] proposed multiple hypothesis tracking (mht) algorithm to track an object (target) in a series of image sequences. to obtain target tracking from uav aerial image sequences, three steps must be performed: break each track set into tracklet (track data subset) at a specific time, estimate association cost of each track set and merge trajectory fragments into a longer iteration. increased use of uavs can provide railway engineers with higher levels of quality information to allow them to make the best possible decisions about the future of our structures assets. as well as being cost effective, this innovation can reduce the need for possessions, track access and roped access, reducing safety risk. as well as delivering a more comprehensive 360 degree view of the structures, in daylight and showing all the defects clearly, the uav images can be stitched together with photogrammetry to create high quality 2d elevations, 3d models and also cloud point surveys (avoiding the need for a separate dimensional survey). aerial inspections cannot fully replace an engineer with a hammer and some degree of tactile inspection is still needed. however, the drone imagery powered by intelligent computer vision algorithms can enable detection of the areas of concern and their targeting. having in mind drone imagery, a novel software algorithm used for railway infrastructure inspection and monitoring according to the railway demands and safety procedures needs to be developed. a novel fail safe and reliable system for rail track detection and inspection on railway mainlines as well as detection and monitoring of some specific railway infrastructure elements should be integrated into railway information system. 2. intelligent computer vision methodology an image can be defined by a set of regions that are connected and non-overlapping, so that the pixels in each partitioned region possess an identical set of properties or attributes [1118]. these sets of properties of the image may include gray levels, contrast, spectral values, textural properties, etc. which can be result of variations in reflectance, illumination, color, shade, texture, orientation and depth of scene surfaces. abrupt changes in gray level can be used for partition an image. the principal areas of interest within this category are the detection of lines and edges in an image. typical image processing module consists of four main parts, image acquisition, image segmentation, image understanding and application specific feature extraction (fig. 1). the vision module developed in this paper consists of the same four parts, but each part is specific due to railway application and drone imagery. image segmentation consists of the main image segmentation algorithm, pre-processing and post-processing of segmented image. the goal was to determine rail tracks in an image acquired from a drone. image understanding is done in order to detect irregularities of the rail tracks, but also different railway infrastructure elements detection through classification is to be done in the future work. the core of the image understanding is neural network classifier that detects rail tracks and other railway infrastructure elements. 360 m. banić, a. miltenović, m. pavlović, i. ćirić fig. 1 computer vision block diagram since focus of this preliminary research was on the rail tracks, an edge detection algorithm was logical choice for the basis of the image processing research presented in this paper. edge, in the physical sense, indicates discontinuities in physical, photometrical and geometrical properties of an object. edges are represented in image by changes in the image intensity function. the difference between edges and lines is reflected in that edge essentially defines boundaries between two distinctly different regions, while a line may be inside of the single uniformly homogeneous region. in the process of edge detection, a significant change in pixel intensity is used for identifying and locating sharp discontinuities in an image, and thus detection of edges. the change in intensity level is measured by the gradient of the image so, since an image is a two-dimensional function, its gradient is a vector [17,18]: [ ] [ ⁄ ⁄ ] (1) the magnitude of the gradient may be computed as follows [13]: [ ] ⁄ [ ] | | | | (2) [ ] {| | | |} eqs. (2) represent three possible ways for the computation of the magnitude of the gradient. however, the direction of the gradient is (θ is measured with respect to x-axis): ( ⁄ ) (3) gradient operators calculate the change in gray level intensities as well as the direction of change. this calculation is performed by the difference in values of the neighboring pixels, i.e. the derivate along the x-axis and the derivate along y-axis. the gradients in a two-dimensional image are approximated by [17,18]: (4) intelligent machine vision based railway infrastructure inspection and monitoring using uav 361 for obtaining the x-direction gradient and y-direction gradient, gradient operators require two masks. however, these two gradients are combined in order to obtain a vector quantity whose magnitude represents the strength of the edge gradient at a point in the image and angle represents the gradient angle. the goal of ideal edge detector is to detect an edge point precisely in the sense that a true edge point in an image should not be missed, but a false edge point should not be wrong detected. however, since quality of detection is dependent on lighting conditions, the presence of objects of similar intensities, density of edges in the scene, and noise, different operator-based edge detectors are used for different purposes. fig. 2 block diagram of feature-based classification feature-based classification (fig. 2) is most common for image understanding. from segmented objects we can extract many features, like height, width, eccentricity color, or same shape descriptors that can help us in classification [14, 15]. the contour-based approach is not as popular as the texture-based approach because of the complexity of detecting extended contours. for classification of the segmented contours mean distance k-nearest neighbor (knn) machine learning approach is proposed, where the average distance between the query and the samples is calculated and the class is assigned taking in account this distance [16]. because of the simplicity, effectiveness and implementation of the k-nn based classification, k-nn has been viewed as one of the top 10 algorithms in data mining. the main idea of using mean distance k-nn is to assign the query sample to the class whose mean distance to the query sample is smaller, instead of assigning it to the most represented class. 3. experimental results and discussion the edge detector presented in this paper is based on a single derivative canny edge detector. the canny edge detector is based on operator that uses multi-stage algorithm, which includes: detection of the edge with a low error rate, the edge point detected should be well localized (located edges must be as close as possible to the true edges) and a single edge point response (detector should return only one point for each true edge point). the canny edge detection algorithm consists of the following basic steps [11, 12, 17-18]: 1. smooth the input image with a gaussian filter in order to remove the noise; 2. calculate the gradient magnitude and angle images; 3. apply non-maximum suppression to the gradient magnitude image; 4. use double threshold to determine potential edges; 5. connectivity analysis to finalize detection by suppressing all the other edges that are weak and not connected to strong edges. 362 m. banić, a. miltenović, m. pavlović, i. ćirić fig. 3 original image acquired by drone, adaptive canny edge detection processing results and rail tracks detected the images were extracted from a 19201080 avi file created by camera mounted on a dji phantom iii advanced drone. the video file was created at 24 fps with a video bitrate of 40 mbps. for canny edge detector, all frames were extracted from the video file. the canny edge detection algorithm was applied to acquired images, while simple adaptive parameter tuning previously developed by authors [11, 12] was used. small edges that are not part of the rail track are removed by the simple filter. in order to determine whether the detected edges are rail tracks or not, the edges were classified using k-nn mean euclidean distances. training data set was formed from 500 frames of video and 3671 edges manually labeled edges, where 4 features were extracted (height, width, eccentricity and linear approximation error). intelligent machine vision based railway infrastructure inspection and monitoring using uav 363 rail track image processing results with developed k-nn classifier with k=7 and detected rail tracks are shown in fig. 3 for 5 video frames. developed system was tested for various height and speed of a drone and various light and weather conditions. developed algorithm showed good results in detecting of rail tracks and various railway infrastructure elements (fig. 3). however, despite the good preliminary results, in order to improve robustness and adaptability, tools from artificial intelligence domain can be additionally used in future work for determining of adaptive parameters for object detection, object classification and distance estimation. 4. conclusions competitiveness, efficiency and operational reliability of european railway infrastructure can be achieved through the development of innovative solutions for measuring and monitoring of railway assets based on machine vision technologies. the primary goal of increasing the quality of rail freight as well its effectiveness and capacity, is in line with european transport strategy 2011-2021 (roadmap to a single european transport area towards a competitive and resource efficient transport system). initiatives proposed have a goal to shift 30% of road freight over 300 km to other transport modes such as rail or waterborne transport by 2030, and more than 50% by 2050. in order to achieve such vision, developing of new infrastructure will be necessary, as well as increasing of efficiency throughout the existing infrastructure. for increasing quality and safety of railway transport, many monitoring systems can be used but, because of the infrastructure, their usage is limited. in this paper, an advanced drone imagery system for autonomous inspection and monitoring of rail tracks is presented. the advanced image processing algorithms were suggested, implemented and tested. canny edge detector has shown promising results for rail tracks detection, while some intelligent algorithms have great potential in adaptive adjustment of parameters of conventional image processing algorithms. based on detected rail tracks, some problem or fault can be detected and therefore a quality and safety can be increased. the presented system is one step forward towards total railway automation. shift2rail multi -annual action plan [shift2rail 2015] outlines numerous advantages of railway automation which include the benefits for quality of service (due to better punctuality), increase of capacity (10 – 50%), reduced system costs (20% energy saving), reduction of operation costs (50% reduction of cost for drivers) and overall efficiency increase of 10 %. all of these benefits will result in a system cost reduction in the three-digit million euro range, as well as great customer benefits [shift2rail 2015]. the railway automation is of highest priority for the future of european rail transportation and it is one of innovations which will lead to turnaround necessary for achievement of transport strategy goals. acknowledgements: this research has been done in the framework of horizon 2020 shift2rail project "smart automation of rail transport smart". the authors would like to thank the serbian railway infrastructure for issuing permit and providing operational assistance for testing. 364 m. banić, a. miltenović, m. pavlović, i. ćirić references 1. 2016, sesar european drones outlook study, accessed on 10.10.2018. 2. kim, j.w., kim, s.b., park, j.c., nam, j.w., 2015, development of crack detection system with unmanned aerial vehicles and digital image processing, proc. 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(eds) similarity search and applications. sisap 2013. lecture notes in computer science, vol 8199. springer, berlin, heidelberg 17. acharya, t., ray, a.k., 2005, image processing: principles and applications, john wiley & sons inc., new jersey, usa 18. nadernejad, e., sharifzadeh, s., hassanpour, h., 2008, edge detection techniques: evaluations and comparisons, applied mathematical sciences, 2(31), pp. 1507 – 1520. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 133 141 1rail traffic volume estimation based on world development indicators udc 656.2 luka lazarević, miloš kovačević, zdenka popović faculty of civil engineering, university of belgrade, serbia abstract. european transport policy, defined in the white paper, supports shift from road to rail and waterborne transport. the hypothesis of the paper is that changes in the economic environment influence rail traffic volume. therefore, a model for prediction of rail traffic volume applied in different economic contexts could be a valuable tool for the transport planners. the model was built using common machine learning techniques that learn from the past experience. in the model preparation, world development indicators defined by the world bank were used as input parameters. key words: rail traffic, prediction, machine learning, world bank, development indicators 1. introduction traffic volume prediction is very important task from the planner`s point of view. it can be used for different activities such as transport market survey and analysis, assessment of the transport market demands and level of provided services, development of appropriate plans, measures and strategies, and decision about new investments in infrastructure. there are several different approaches for the traffic prediction, depending on the planner needs. regarding the time interval, it can be performed as a short-term [1-4] or a long-term prediction [5-7]. in addition, the prediction could be performed on the level of state transport network [4-6, 8-10], urban transport network [2, 3, 7, 11, 12], or particular section in the transport network [1]. previous research showed that the application of neural networks for traffic prediction problems provided good results [2, 3, 9, 11, 12]. neural networks (nn) exhibited the flexibility in modeling complex datasets with possible nonlinearities or missing data [13]. further, support vector machines (svm) provided good prediction results in some cases [6, 7]. received january 30, 2015 / accepted april 15, 2015 corresponding author: luka lazarević faculty of civil engineering, university of belgrade, bulevar kralja aleksandra 73, belgrade, serbia e-mail: llazarevic@grf.bg.ac.rs original scientific paper 134 l. lazarević, m. kovaĉević, z. popović qi et al. used neural tree model for prediction of railway passenger traffic. this model provided better results than nn and svm [10]. the similar approach was applied in the research by zhuo et al [9]. this research applied back propagation nn to predict railway passenger volume resulting in quick convergence and high accuracy. both studies implied time-series analysis. li et al. proposed factors of urban rail transit flow, which were used in the prediction of shanghai metro traffic [7]. research by griskeviciene et al. [5] proposed key macroeconomic indicators that are related to railway traffic. the prediction model was developed using previous statistical data, and applied in three prospective scenarios optimistic, basic (realistic) and pessimistic, for prediction of long-term freight traffic on railway corridors ix and i in lithuania. the aim of this paper is creation of a data driven model for prediction of changes in rail traffic volume based on the changes in economic parameters. world development indicators defined by the world bank [14] were used as inputs to the proposed models. two countries were chosen for the creation of the model, serbia and austria, since they have similar area, population and population density [15]. the freight and passenger traffic were separately analysed for both countries. the models were developed using weka 3 software, a collection of machine learning algorithms for data mining tasks, developed at the university of waikato [16, 17]. 2. data representation the first task was to choose the appropriate data representation and prepare a dataset that will be used for building and validation of the proposed prediction models. as it was stated above, world development indicators were chosen as economic parameters to represent the predictors of the traffic for each year in both countries. world bank (wb) defines 1356 indicators divided into ten groups: education (103), environment (136), economic policy and debt (506), financial sector (50), health (122), infrastructure (36), social protection and labor (148), poverty (22), private sector and trade (151), and public sector (82). more details about these indicators can be found on the wb website (http://data.worldbank.org). two of the infrastructure indicators represent data about freight and passenger rail traffic (target values to predict). wb databank provides rail traffic data from 1980 to 2012. therefore this time range was chosen to build the proposed models. since the indicators have different orders of magnitude and measure units, their values were replaced with relative changes (changes in percentage comparing to the previous year). modifications on the initial dataset enabled better analysis of the correlation between economic changes and changes in rail traffic. from the aspect of machine learning, relative changes of world development indicators represent numerical attributes. on the other hand, relative changes of rail traffic were represented as nominal values (classes):  decrease in rail traffic was denoted as n (relative change is negative), and  increase in rail traffic was denoted as p (relative change is positive). introduction of two classes transformed the traffic volume prediction problem into a simple classification task where the change in rail traffic is classified as positive or negative based on the relative changes of world development indicators in the related country. prediction of rail traffic using machine learning classifiers 135 the next step in data preparation was to eliminate the attributes with more than 50% of missing values. the number of attributes (n) was not the same for serbia and austria, since it depends on availability of data. fig. 1 shows the data representation used to build the initial dataset for each country. fig. 1 the proposed data representation a separate set was created for each traffic type from the country's initial dataset by retaining the appropriate class attribute (freight or passenger) resulting in four datasets di, d1 = serbianfreight (sf), d2 = serbianpassenger (sp), d3 = austrianfreight (af) and d4 = austrianpassenger (ap). although these datasets did not contain same number of attributes for both countries, they were used for assessing the performances of prediction models that were initially built. 3. building and validating the models in order to build and validate the model using a machine learning approach, dataset di should be divided into disjoint training and test sets. since di contained small number of examples a valid statistical protocol for model validation demanded the creation of k disjoint train-test splits. the idea of this procedure is to train k different models tested on k different test sets and to average the obtained performances. in this paper five train-test splits were created for each di containing 80% of examples for training and 20% for testing. the next problem to be solved considered the fact that the number of attributes (world development indicators) in each of the four sets was significantly higher than the number of examples, or j >> i according to fig. 1. therefore, the number of attributes was reduced using the correlation feature subset (cfs) filter. this filter selects a subset of attributes that are highly correlated with the class (n or p in this case), while having low inter-correlation. the filter was applied on each of the five train-test splits, for each di. in this study we applied three commonly used machine learning classifiers to build the models with training data for each di: 136 l. lazarević, m. kovaĉević, z. popović  naive bayes (nb)  a probabilistic classifier based on bayes theorem and the assumption of mutual independence of attributes [18],  decision tree (dt)  a classifier which successively tests attribute values in each internal node until it is possible to deduce about the item`s target value (class) [19],  multilayer perceptron (mlp)  feed-forward neural network that maps sets of input data onto a set of appropriate outputs, trained using back-propagation algorithm [20]. after applying classifiers on each train-test split five confusion matrices were obtained. a confusion matrix cn x n (n is the number of classes) is defined with cij representing the number of cases from actual class i that are classified as class j. when generating train-test splits, we applied cross-validation protocol which ensured that the test parts were mutually disjoint. therefore, the final confusion matrix for the model was obtained by simple addition of separate matrices. since the related classification problem assumed the existence of only two classes, the final confusion matrix has the form given with eq. (1): tp fn fp tn       (1) where: tp the number of instances correctly classified as p, tn the number of instances correctly classified as n, fn the number of instances incorrectly classified as n, fp the number of instances incorrectly classified as p. using data from the confusion matrix, three pieces of information related to both classes (p, n) were derived for each model: precision (π), recall (ρ) and f-measure (eq. 2-5). p n tp tn tp fp tn fn      (2) p n tp tn tp fn tn fp      (3) n np p p n p p n n f 2 f 2               (4) p n tp fn tn fp f f f tp tn fn fp tp tn fn fp           (5) class precision measures the percentage of correct decisions among all decisions for the specified class while the class recall measures the percentage of recognized cases from the class. increasing the precision often leads to decreasing in recall. f-measure represents the harmonic mean between the two describing the overall class performance of the model. in order to compare different models with only one measure, a weighted f is defined with eq. 5. the obtained performances of all models are presented in table 1. prediction of rail traffic using machine learning classifiers 137 table 1 results of the prediction models (best cases underlined) classifier model fp fn f naive bayes (nb) sf 0.632 0.462 0.558 sp 0.560 0.718 0.659 af 0.696 0.222 0.563 ap 0.766 0.353 0.624 decision tree (dt) sf 0.529 0.467 0.502 sp 0.500 0.700 0.625 af 0.708 0.125 0.544 ap 0.632 0.462 0.568 multilayer perceptron (mlp) sf 0.579 0.385 0.494 sp 0.583 0.750 0.687 af 0.571 0.182 0.462 ap 0.732 0.522 0.653 sf, sp freight and passenger traffic in serbia af, ap freight and passenger traffic in austria weighted f-measure of the developed models ranged from 0.56 to 0.69, depending on the country and traffic type. the average performance of the models could be explained with the small dataset and the unequal number of examples per class, except in the case of freight traffic in serbia where the class split was almost 50%. according to the values from table 1, nb provided better results for freight traffic and mlp provided better results for passenger traffic. in addition, predicting the passenger traffic for both countries appeared to be the easier task than predicting the freight traffic. predicting decrease in freight traffic in austria was the most difficult task according to the low fn value. 4. influence of the world development indicators as it was mentioned before, number of attributes (world development indicators) in training sets from each di was reduced using the cfs filter. according to the definition of cfs filter, there were actually created subsets of world development indicators that are mostly correlated with the rail traffic changes. after the analysis of selected wd indicators, it was noted that rail traffic mostly depends on indicators belonging to the two groups: environment and economic policy and debt. fig. 2 shows the distribution of selected wd indicators over the four significant groups. for example, from the environment group, parameters related to the co2 emission can be considered as informative attributes (they often repeated in data subsets). in particular, it was determined that the increase of rail traffic in austria was followed by decrease of co2 emission from liquid fuel consumption and co2 intensity (in 70% of examples). although this complies with the expected effects of the shift to rail transport, this type of analysis should also consider many other aspects of economy. 138 l. lazarević, m. kovaĉević, z. popović fig. 2 distribution of selected wd indicators over the four significant groups from the economic policy and debt group, gross national income and gross national expenditure appeared to be informative. in addition, several wd indicators could not be set in the context of the model, although they were selected by the cfs filter. for example, several indicators originated from the health and education groups. however, application of cfs filter showed which groups of indicators are mostly correlated with rail traffic and which indicators can be used for traffic prediction. among the indicators that were selected by the cfs filter, the ones that repeated in all sets (according to fig. 3) were used for preparation of training and test sets according to the methodology described in section 2. table 2 presents indicators that were chosen for the prediction model. fig. 3 selection of the relevant indicators prediction of rail traffic using machine learning classifiers 139 table 2 chosen indicators for the prediction models attribute (indicator) co2 emissions from transport (million metric tons) energy production (kt of oil equivalent) energy use (kg of oil equivalent per capita) adjusted savings: energy depletion (current us$) gdp per capita (current international $) gni per capita (current international $) net income from abroad (current us$) road sector energy consumption (kt of oil equivalent) after the models were recreated using the attributes from table 2 the obtained performances are presented in table 3. table 3 final results of the prediction models (best cases underlined) classifier model fp fn f naive bayes (nb) sf 0.700 0.500 0.613 sp 0.240 0.513 0.411 af 0.784 0.154 0.607 ap 0.667 0.571 0.631 decision tree (dt) sf 0.698 0.381 0.559 sp 0.638 af 0.766 0.353 0.650 ap 0.718 0.560 0.659 multilayer perceptron (mp) sf 0.500 0.500 0.500 sp 0.320 0.564 0.473 af 0.711 0.316 0.600 ap 0.571 0.182 0.425 sf , sp freight and passenger traffic in serbia af , ap freight and passenger traffic in austria nb provided good result in predicting the increase of freight traffic in serbia. although there was almost equal number of examples from two classes, f-measure for prediction of freight traffic decrease was only 0.5. on the other hand, prediction of passenger rail traffic in serbia was more difficult task. dt provided best result for prediction of traffic decrease, but failed to predict the traffic increase (class p). the main reason for this result is unequal class split in dataset for passenger traffic (72.5% of class p and 37.5% of class n) and large number of missing values for the used attributes. dt provided good results for both traffic types in austria. prediction of freight traffic decrease had significantly low performance due the small number of examples for class n in dataset (about 28%). comparing to table 1, better results were obtained for freight traffic in both countries. result for passenger traffic in austria was not improved, while in case of serbia result 140 l. lazarević, m. kovaĉević, z. popović was significantly lower. therefore, attributes presented in table 2 do not reflect the changes in passenger rail traffic in serbia. it is important to mention that the number of missing values ranged from 35-50% of all values depending on the attribute and country. therefore, obtained results were highly influenced by the large number of missing values. 5. discussion and conclusion countries with strong industry and economy have fully organized and functional railway transport. the main reason is high capacity, efficiency and safety of railways. hence, european transport policies are directed towards the shift to rail transport. the shift to rail transport will be followed by certain economic changes. on the other hand, economic changes can influence the changes in rail traffic volume in countries that strongly rely on rail transport. the research presented in this paper was directed towards the development of rail traffic prediction models based on machine learning techniques which utilize world development indicators defined by the world bank. models were developed and evaluated for two countries, serbia and austria, in order to provide sound basis for model comparisons. performances of the prediction models were assessed using f-measure. considering that minimum required f-measure should be 0.75, obtained performances for all models were below this threshold. however, prediction models provided good estimation of changes in rail traffic regardless of the small dataset and large number of missing values. therefore, these models can be used for preliminary estimations for the purposes of transport market survey and analysis, as well as for development of plans, measures and strategies on the level of network or its sections. further research in this field is directed towards determination of the general indicators using several different countries, which would provide larger dataset and thus more advanced prediction model. the main goal would be to create the general model for railway traffic prediction based on world development indicators. acknowledgement: this paper was supported by the ministry of science and technological development of the republic of serbia through the research project no. 36012: „research of technical-technological, staff and 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potential of the railway infrastructure in serbia, railway transport and logistics, 3, pp. 9-22. 16. hall m., frank e., holmes g., pfahringer b., reutemann m., witten i.h., 2009, the weka data mining software: an update, sigkdd explorations, 1(11), pp. 10-18. 17. http://www.cs.waikato.ac.nz/ml/weka/ (accessed on december 26, 2015) 18. mitchel t., 1997, machine learning, mcgraw hill, columbus, ohio, 414 p. 19. quinlan j.r., 1986, induction of decision trees, machine learning, 1(1), pp. 81-106. 20. haykin s., 1998, neural networks: a comprehensive foundation, prentice hall, new jersey, 842 p. http://data.worldbank.org/ http://www.cs.waikato.ac.nz/ml/weka/ plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 495 506 https://doi.org/10.22190/fume171001027d original scientific paper experimental investigation of the convective heat transfer in a spirally coiled corrugated tube with radiant heating udc 662.6 milan đorđević 1 , velimir stefanović 2 , mića vukić 2 , marko mančić 2 1 university of priština, faculty of technical sciences, kosovska mitrovica, serbia 2 university of niš, faculty of mechanical engineering, niš, serbia abstract. the archimedean spiral coil made of a transversely corrugated tube was exposed to radiant heating in order to represent a heat absorber of the parabolic dish solar concentrator. the main advantage of the considered innovative design solution is a coupling effect of the two passive methods for heat transfer enhancement coiling of the flow channel and changes in surface roughness. the curvature ratio of the spiral coil varies from 0.029 to 0.234, while water and a mixture of propylene glycol and water are used as heat transfer fluids. the unique focus of this study is on specific boundary conditions since the heat flux upon the tube external surfaces varies not only in the circumferential direction, but in the axial direction as well. instrumentation of the laboratory model of the heat absorber mounted in the radiation field includes measurement of inlet fluid flow rate, pressure drop, inlet and outlet fluid temperature and 35 type k thermocouples welded to the coil surface. a thermal analysis of the experimentally obtained data implies taking into consideration the externally applied radiation field, convective and radiative heat losses, conduction through the tube wall and convection to the internal fluid. the experimental results have shown significant enhancement of the heat transfer rate compared to spirally coiled smooth tubes, up to 240% in the turbulent flow regime. key words: archimedean spiral coil, corrugated tube, heat transfer received october 01, 2017 / accepted november 20, 2017 corresponding author: milan đorċević faculty of technical sciences, kneza miloša 7, 38220 kosovska mitrovica, serbia e-mail: milan.djordjevic@pr.ac.rs 496 m. đorđević, v. stefanović, m. vukić, m. manĉić 1. introduction the utilization of modern paraboloidal concentrators for conversion of solar radiation into heat energy requires the development and implementation of compact and efficient heat absorbers. that is why this research is directed toward an innovative design solution that involves the development of the heat absorber made of spirally coiled tubes with transverse circular corrugations. the main advantage of the considered design solution is a coupling effect of the two passive methods for heat transfer enhancement coiling of the flow channel and changes in surface roughness. the presence of the superimposed secondary convection in the curved channel flow suppresses axial propagation of the initial turbulent fluctuations so that the curvature stabilizes the flow while the transition from the laminar to the turbulent flow is delayed. on the contrary, the transversal corrugations act as turbulence promoters since the flow turbulence level is increased by a separation and reattachment mechanism. the corrugations act as roughness elements and disturb the existing laminar sublayer. furthermore, morton [1] found that heated pipes also develop vortices resulting from a combination of the radial-directional and the downward motions of the fluid particles induced by the displacement of the boundary layer and developed along the pipe. the substantial features of each of these effects (curvature, corrugated wall and heating) are increased heat and mass transfer coefficients due to the cross-sectional mixing of fluid elements as well as enhanced frictional losses. assessment of the overall impact of the stated effects is impossible either with analytical models or even with numerical ones without the existence of relevant empirical data for their calibration. spiral tubes or spiral coils were introduced in 19 th century and have been widely used in various thermal engineering applications, such as heat exchangers, electronic cooling, chemical reactors, etc. they have better heat transfer performance and compactness in comparison with commonly used straight tube exchangers, which results in their occupying less space. the transport phenomena occurring in spiral tubes are more complicated than those in straight tubes. secondary flows observed in the flow patterns, induced by the centrifugal force, significantly affect the flow field and heat transfer. most of the studies of thermal-hydraulic processes in the coils have been dedicated to helical coils. much less work has been reported on the hydrodynamics of flow and heat transfer in spiral coils. the property of a continuously varying curvature along the length makes spiral coil flows never fully developed unlike helical ones. secondly, the distinction of the flow regime needs specification of two critical reynolds numbers instead of just one [2]. naphon and associates [3, 4] experimentally and numerically studied heat transfer and flow developments in the spirally coiled smooth tubes. they stated that the nusselt number and pressure drop obtained from the spirally coiled tube are for 1.49 and 1.50 times higher than those from the straight tube, respectively, under constant wall temperature boundary condition. their conclusions could not be considered general since the average nusselt number and pressure drop are not appropriate for evaluation of heat transfer and flow characteristics of the tubes with a constantly varying curvature. several numerical studies [57] have been published, which examined the flow and heat transfer phenomena in spiral tubes. these papers investigate the laminar flow of newtonian fluids in coils, while those that investigate turbulent flow conditions are rare. moreover, all take into account two common thermal boundary conditions – constant wall temperature and constant heat flux. even though the interest in spiral coiled systems is on the rise, there are still very few published articles on spiral coil tubes. they are less popular compared to helical ones, experimental investigation on the convective heat transfer in a spirally coiled corrugated tube … 497 which have attracted major attention in the study of coiled tubes for heat transfer. there is very little information about the nusselt number as well as correlations, and in the absence of appropriate correlations, the traditional approach is to use the correlations developed for circular or helical tubes with an average curvature. examining the research studies on thermal-hydraulic processes in straight tubes with high values of relative roughness it can be concluded that the appropriate correlations for determining the coefficients of friction and heat transfer in the laminar flow regime almost do not exist. furthermore, the results of different authors differ significantly in general conclusions. it can be concluded that there is a lack of data for a range of geometrical parameters and the reynolds number for transversely corrugated straight pipes having large relative roughness. moreover, no reference concerning hydrodynamics and heat transfer in spiral tube coils with transverse corrugations was found. another specific aspect of this investigation is that the heat exchanger in the model under consideration is subjected to a radiant asymmetrical heat flux, while most of the existing correlations for convective heat transfer are valid for flow in channels with direct, uniform heating. 2. experiment the objective of this paper was to experimentally study the distribution of the convective heat transfer in a spirally coiled corrugated tube exposed to radiant heating that is characteristic of parabolic dish solar concentrators. investigation of the influence of hydraulic, physical and thermal conditions, as well as the geometry of the spirally coiled corrugated heat absorber, on the local intensity of heat transfer and pressure drop was conducted using modern experimental methods. the archimedean spiral, with a pitch slightly larger than the maximal outside diameter of the corrugated coiling tube, was selected out of different types of spirals in order to achieve the most favorable ratio of active surface area and the total volume of the heat absorber in the parabolic dish receiver. the geometrical characteristics and experimental configurations of the transversely corrugated straight pipe and the transversely corrugated archimedean spiral coil are shown in figs. 1 and 2, respectively, while table 1 shows geometrical parameters of the tested configuration. fig. 1 profile of transverse corrugated fig. 2 geometry and configuration pipe made of stainless steel aisi 304 [8] of transversely corrugated archimedean spiral coil 498 m. đorđević, v. stefanović, m. vukić, m. manĉić in order to simulate the real working conditions of the heat absorber an experimental installation is constructed. the installation consists of a spirally coiled corrugated tube of heat absorber with an accompanying flow loop and a radiant heating system. the schematic diagram of the hydraulic system is shown in fig. 3. the entire experimental apparatus enables variations of the following operating parameters: 1) intensity of the incident radiant heat flux; 2) flow rate and flow direction of the working fluid, and 3) angle of inclination of the spiral heat exchanger in relation to the horizontal plane. table 1 geometrical parameters of the tested configuration transversely corrugated straight tube transversely corrugated archimedean spiral coil d 9.3 mm minimum internal diameter rmin 25 mm minimum radius of the coil d0 11.7 mm maximum internal diameter rmax 202 mm maximum radius of the coil dmax 12.2 mm maximum external diameter n 13 number of coil turns s 0.25 mm wall thickness l 9.324 m length of the coil e 1.2 mm corrugation depth le 0.5 m length of entrance section pc 4.2 mm corrugation pitch ps 13.6 mm spiral coil pitch fig. 3 schematic diagram of the test loop in addition to water, a mixture of propylene glycol and water (90% and 10% by volume, respectively) was also used to expand the experimental range. the volume flow rate is measured with the flask and the ultrasonic flow meter (kamstrup, type 66w02f1318), while the flow is regulated by the ta-stad balancing valve. pressure drops were measured with a hydrostatic pressure gauge (up to a maximum pressure drop of 20 kpa) and a differential pressure gauge (for pressure drop values greater than 20 kpa). measuring the temperature of the working fluid at the inlet and outlet of the coil was performed with two pt100 temperature sensors positioned in the mixing chambers. in order to measure variations of the tube wall temperature along the axis of the tube, as well as its circumferential direction, a total number of 35 type k thermocouples (chromel-alumel, wire diameter ϕ0.22 mm) were located on the outside surface of the experimental investigation on the convective heat transfer in a spirally coiled corrugated tube … 499 tube in a special arrangement so that there was one thermocouple station in the middle of each turn of the spiral coil. at some thermocouple stations, either two or four thermocouples were welded to the tube surface using a capacitor discharge welding machine, which is specially designed and operated for research purposes. peripheral locations at these stations were specified as: location a (θ=0°) corresponds to the outer side of the tube cross section (the furthest from the curvature center), location b (θ=90°) corresponds to the side of the tube cross section which is directly subjected to radiant flux, location c (θ=180°) corresponds to the inner side of the tube cross section (the closest to the curvature center) and location d (θ = 270°) receives heat only by air convection and conduction through the tube wall. since the tube is corrugated, the thermocouples were located both at the basic, minimum, diameter of the tube, as well as on the tops of the corrugations, at the maximum diameter of the tube. the schematic diagrams of thermocouple arrangements around the circumference of the tube cross section are shown in figs. 4 and 5. the additional 9 thermocouples were located in the middle of the remaining turns of the spiral coil on the minimum tube diameter. after the thermocouples were connected, the coil surface was coated with temperature resistant flat black paint "pyromark 2500", whose minimum guaranteed absorptance is 0.9 [9]. fig. 4 circumferential locations of thermocouples at the minimum diameter of the tube the outputs of thermocouples were connected to two input modules ni 9213, with 16 channels both, and one input module ni 9211 with 4 channels, which allow simultaneous recording of signals from 36 thermocouples. the input modules were connected to the computer via a ethernet chassis ni cdaq-9188, while the software package labview 2013 was used for recording temperature values and generation of reports. the incident heat flux upon the tube external surfaces varies both in the circumferential and axial direction. it was obtained by the radiant heating system, which is specifically designed for the purposes of this experimental research. detailed analyses of the radiant heat flux produced by the quartz heating system and radiant absorption characteristics of the corrugated curved tubes could be found in our previous papers [10, 11]. 500 m. đorđević, v. stefanović, m. vukić, m. manĉić fig. 5 circumferential locations of thermocouples at the maximum diameter of the tube kline and mcclintock [12] method was used to estimate uncertainties involved in the calculation of the heat transfer coefficient and nu number. they vary up to 3.72% and 3.85% for calculated values of the heat transfer coefficient and nu number, respectively. 3. thermal analysis the flow in the heated curved tube with corrugations is very complex, characterized by the existence of the secondary flows with recirculation cells in the plane normal to the direction of axial velocity and formation of vortices in the corrugations. the combination of these factors, along with nonuniformity of the boundary conditions, both in the axial direction and around the circumference of the tube, as well as the uncertainty in determining the local fluid conditions, make solving the problem of heat transfer by analytical methods impossible, thus necessitating an experimental procedure. the method of calculation of each of the quantities necessary for estimating the heat transfer coefficient at the inner surface of the tube is presented. the local values of heat transfer coefficient hi and nu number for any given axial (z) and circumferential (θ) location (denoted as │z,θ) at the inner surface of the tube were calculated according to: i , i , i , b | | | | z z z z q h t t      (1) i , i , f | | nu | |     z z z z h d  (2) where ti is the inner surface local temperature of the tube, di is tube internal diameter and λf is the thermal conductivity of the transport fluid. bulk temperature of the fluid at some axial location is defined as: c b c 1 d  a t tu a va (3) where u is the local velocity vector, t is the local fluid temperature and v is the average velocity at the considered cross-section, whose area is ac: experimental investigation on the convective heat transfer in a spirally coiled corrugated tube … 501 c c c 1 d  a v u a a (4) thermophysical properties of water [13, 14] and a mixture of propylene glycol and water (90% and 10% by volume, respectively) [14, 15] are treated as temperature-dependent and were obtained as polynomial functions of temperature. 3.1. calculation of total heat input the net heat flux at the outer surface at any specific axial location along the spiral and at any angular position (qoǀz,θ) can be considered equal to the net heat flux at the corresponding radial location of the inner surface of the tube (qi ǀz,θ). this assumption is justified by a relatively small thickness of the wall, as well as relatively large values of the heat transfer coefficient that characterize all the flow regimes in the considered geometry. heat losses and gains from the outer surface of the coil are due to convection qconv ǀz,θ and radiation qrad ǀz,θ to the outside environment. these heat losses will also be a function of the axial and circumferential position around the tube cross section, as the tube surface temperature is not uniform. the net heat flux at the outer surface is given by: o , abs , conv , rad ,| | | |z z z zq q q q      (5) where qabs ǀz,θ is absorbed radiative heat flux. analytical expressions for the calculation of qconv ǀz,θ and qrad ǀz,θ are: conv , o o , cav| ( | )z zq h t t   (6) 4 4 rad , o , amb| ( | )  z zq t t (7) where ho is the local value heat transfer coefficient at the outer surface of the tube, tcav and tamb represent the temperature of the air in the cavity of the receiver and ambient, respectively, while ζ and ε denote the stefan-boltzmann constant and emissivity, respectively. heat transfer coefficient at the outer surface of the tube wall ho was determined according to the relation given by churchill and chu [16]: 1/ 6 9 /16 8/ 27 2 o ai air e r kv [0.60 0.387·ra / (1 (0.559 / pr ) ) ] d h     (8) where the grashof and rayleigh number are defined as: 3 2 2 ekv air air airgr · /  d g t   (9) airra=gr·pr (10) in previous equations dekv is the equivalent external tube diameter, g is the earth gravity acceleration, δt is the temperature difference between the boundary layer and the bulk fluid, while λair, ρair, βair, μair and prair are thermal conductivity, density, volumetric temperature expansion coefficient, dynamic viscosity and prandtl number of air, respectively. 502 m. đorđević, v. stefanović, m. vukić, m. manĉić assuming stationary heat transfer, an energy balance can provide a relationship between the net value of the heat flux at the outer surface of the tube and the thermal power of the absorber formulated by the parameters of the working fluid: 2 o , o ave ave out in 0 0 | d d ( )       l zq d z v c t t (11) where ρave, cave, tout and tin are average density, average specific heat capacity, outlet and inlet temperature of transport fluids, respectively. based on the previously defined energy balance, the bulk fluid temperature at some axial location at a distance l from inlet can be calculated as: 2 o , o ave ave b in 0 0 | d d ( | ), 0         l z lq d z v c t t l l (12) 3.2. heat conduction in tube wall the inner surface local temperature of tube ti ǀz,θ could be determined based on the corresponding values of the outer surface local temperature of tube to ǀz,θ and the local net heat flux at outer surface qo ǀz,θ according to the expression: o o , i i i , o , w | | | ln | | | 2              z z z z z z d q d d t t  (13) where λw denotes the thermal conductivity of the tube wall. 4. results based on the experimental tests of thermal-hydraulic processes in spirally coiled corrugated tube of the heat absorber exposed to radiant heating, a systematic study of the thermal characteristics of the considered heat exchanger was carried out. the results presented are based on 146 complete series of measurements for different experimental conditions. the determination of the flow stability criteria and of the critical re numbers has already been shown in our previous work [17]. a mixture of propylene glycol and water (90% and 10% by volume, respectively) was used as a working fluid in the reynolds number range re≈56-1734 and prandl number range pr≈36.4-255, while water was used in the ranges re≈1225-16731 и pr≈3.0-7.0. pr number variation of working fluids is in accordance with the range of temperature that characterizes the experimental conditions. the selected re number ranges guarantee a separate existence of all the three flow regimes for all the values of curvature ratio δ (δ=de/2r, while r is the radius of curvature), which varies in the range δ=0.023-0.186 and is determined by the constructive characteristics of the spiral heat exchanger. the local values of nu number were calculated both at the minimum diameter of the tube, as well as on the corrugation crests. in order to determine the values of the peripherally averaged nu number most accurately, the locations of thermocouples at minimum and maximum diameter on the same coil turn were positioned at the axial distance equal to the half of the corrugation pitch. experimental investigation on the convective heat transfer in a spirally coiled corrugated tube … 503 the correlations between the peripherally averaged nu number and the re and pr numbers and the basic geometrical parameter of spiral (δ) were obtained by a multiple nonlinear regression analysis, assuming simple exponential models. proposed correlations for laminar, transitional and turbulent flow regimes, as well as the ranges of their applicability, are represented by eqs. (14-16), respectively: 0.61 0.174 0.164nu 0.556re pr  (14) 100<re<1200; 40<pr<190; 0.023<δ<0.146 0.641 0.3 0.11nu 0.363re pr  (15) 1250<re<3200; 6<pr<90; 0.023<δ<0.146 0.654 0.43 0.07nu 0.289re pr  (16) 3500<re<15000; 4<pr<7; 0.023<δ<0.146 figures 6-8 show the comparisons of experimental data and the best fit lines obtained by least squares analysis of the points predicted by corresponding correlations. fig. 6 graph of the regression model: fig. 7 graph of the regression model: laminar flow trasitional flow fig. 8 graph of the regression model turbulent flow 504 m. đorđević, v. stefanović, m. vukić, m. manĉić the maximum deviations of the predictions are 8% in the laminar, 7% in the transitional and 14% in the turbulent flow regime. after obtaining power-law dependences, the possibility of applying some of the momentum-heat transfer analogies to spirally coiled corrugated tubes was considered. this is a more fundamental approach since analogies are most physically founded and rest on a more sound theoretical ground than widely used power-law formulas. di piazza and ciafolo [18] indicated that the modified reynolds analogy is applicable in smooth curved pipes with constant curvature. it is valid in the average and only approximately is valid locally (analogy is less applicable for a higher curvature). on the other hand, bergles and morton [19] have shown that for the surfaces characterized with large relative roughness no unique relation between friction coefficient and heat transfer could be formulated. an additional problem in attempting to formulate the momentum-heat transfer analogy in this study is the formulation and determination of the local values of the friction coefficient in the spiral because of its dependence on the constantly variable curvature of the flow channel. using the normalized values of the friction coefficient in the petukhov and gnielinski analogies yields values of the peripherally averaged nu number with significantly larger deviations compared to the predictions obtained by power-law models. this leads to the conclusion that the momentum-heat transfer analogies cannot be applied in general in the tubes with a variable radius of curvature that are characterized by high values of relative roughness. 5. conclusions the empirical power-low formulas have been suggested to indicate the dependency of the peripherally averaged nu number on re and pr numbers and on the curvature in the spirally coiled corrugated tubes exposed to radiant heating. a large database reduces the influence of random errors, thus making a statistical approach highly reliable. the obtained correlations for heat transfer are applicable in wide ranges of flow, physical and geometric parameters. the accuracy of the obtained correlations can be compared with that of the correlation in the literature about research studies of similar processes, which justifies the use of the statistical method. the suggested formulas indicate the effects of the considered parameters for the heat transfer rate. a relatively high exponent value of the re number in the laminar flow regime indicates the dominant influence of the secondary flows on heat transfer which significantly increases the heat transfer rate. the reynolds number exponent in the turbulent flow regime is lower than the value of 0.8, otherwise typical of simple geometries. this is consistent with the previous investigation of ciofalo and collins [20] who consider the value of the exponent for complex flow channel geometries where detachment and reattachment of the boundary layer exist. the exponent value of curvature δ decreases with the increase in the re number, which indicates that the secondary flows in the turbulent regime have a relatively small effect on the heat transfer. the experimentally obtained peripherally averaged nu number was compared to the corresponding one in the smooth spiral coil for identical boundary conditions [21]. the nu number in the corrugated spiral coil is not significantly higher in the laminar flow experimental investigation on the convective heat transfer in a spirally coiled corrugated tube … 505 regime and the increase is slightly more than 50% at the end of the laminar flow range (re≈1300). the enhancement of the heat transfer in the corrugated spiral coil is considerably conditioned by the value of the re number, so that the increase in the nu number at the end of the test range in the turbulent flow regime (re≈15000) is about 240%. developing flow along with secondary flows make the development of the peripherally averaged nu number rather oscillatory. it makes difficult to represent the differences in distribution of the peripherally averaged nu number along the spiral axial coordinate for the cases of inflow to outermost or innermost spiral turn. those differences are relatively small and the proposed power-law formulas cannot account for the complexity of the real functional dependences. the proposed formulas 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improve emissions, performance and noise of the automotive diesel engine sanjoy biswas1, 2, achintya mukhopadhyay1 1department of mechanical engineering, jadavpur university, india 2engineering, tata motors ltd, india abstract. the present study aims at investigating effectiveness of the quadruple (earlypilot-main-after [epma]) injection strategy over three different triple [early-main-after (ema), early-pilot-main (epm) and pilot-main-after (pma)] injection scheduling in terms of emissions, performance [brake specific fuel consumption (bsfc), torque, brake thermal efficiency (bte) and fuel economy] and noise. the experimentation was carried out on a heavy-duty bs-iv diesel engine with 45% egr fraction and fixed main injection (crank-angle) scheduling at eight different rpms and three loads of engine (20%, 60% and 100%) using design of experiments(doe). this comprehensive study showed that the quadruple injection strategy provides optimum results in both performance and emissions compared to the promising three triple injection strategy. the quadruple injection strategy exhibits the best bte at all operating conditions and best bsfc at medium to high-speed zone around 0.5–1% inline to reduce combustion noise (cn) level, especially at low speeds and low to medium load of 0.2–2.2 dba. among triple injections, the pma shows the best performance in bsfc, bte, smoke and thc emissions. the epm is the best in the co emissions and torque performance in the low-speed zone. smoke value is marginally higher for the epma at low to medium speed than the pma, although average smoke emissions were the best. taken together, the overall pm emission level was marginally better than triple injections, due to the impact of double pilots in combination with post-injection. in addition, nox emissions were improved (around 3–6%) significantly with quadruple than with triple injections. the epma injection scheduling also showed improvement in constant speed fuel economy and in pass-by-noise at the vehicle. key words: quadruple injections, emission, brake specific fuel consumption, passby-noise, constant speed fuel economy, brake thermal efficiency received march 29, 2021 / accepted june 15, 2021 corresponding author: sanjoy biswas department of mechanical engineering, jadavpur university, kolkata-700032, india e-mail: s.biswas.me@gmail.com 322 s. biswas, a. mukhopadhyay 1. introduction automotive engine out-emissions, noise and fuel economy have a significant impact on environmental pollution. thus, the challenges are multi-fold for an engine developer/engineer (r&d), especially for the diesel engine as he needs to deal with stringent emissions norms in line with mandated fuel economy norms as per applicability [for example, heavy-duty fuel economy (hdfe) or constant speed fuel economy (csfe) or corporate average fuel efficiency/economy (cafe)]. fuel economy norms are targeted to co2 emissions reduction (in gm/km) to control global warming. currently, the diesel engine combustion noise (cn/radiation) has gained significant attention as it is associated with the passengers and pedestrians’ discomfort along with noise pollution. therefore, the fuel injection strategy plays a key role in the simultaneous reduction of emissions and cn without penalizing fuel economy due to the better control of the combustion process. the fuel injection strategy in combination with the common rail direct injection (crdi) technology and heavy or high egr is also promising low-temperature combustion (ltc) methodology [1]. the crdi technology provides flexibility to experiment with various injection strategies based on the parameters such as injection pressure, fuel quantity and injection timing, which are linked to the engine combustion management. furthermore, engine with a turbocharging system enables power augmentation with improved fuel efficiency at particular brake mean effective pressure [2]. a five-dimensional study has been carried out on the modern diesel engine to simultaneously improve the performance (especially fuel economy and cn) and emissions. several studies [3-7] are focused on the effect of pilot injection (single or double/twin) or post-injection on emissions, combustion or cn. some groups are focused on the impact of injection parameters (fuel injection pressure (fip), injection rate, injection timing and fuel quantity) on combustion characteristics and emissions [8-9]. the influence of alternative fuel on performance, combustion and emissions has also been explored [10-11] with variation in injection parameters or injection strategies. on the other hand [12-14], the predictive model was developed based on quasi-dimensional (1d) or phenomenological and multi-dimensional (i.e., cfd) approach and validated for combustion and emissions of di diesel engine with single and multiple injections (double, triple). a recent study [15-18] assessed the impact of multiple injections on the performance, cn, and emission characteristics of the modern diesel engine. the selective outcomes of these studies have been discussed subsequently. mendez et al. [15] examined the impact of multiple (double, triple and quadruple) injections on a low compression-ratio diesel engine with high egr%. the study concluded that the multi-injections strategy is beneficial along with high rate egr (~46%) to accept ignition delay with improvement in the emission level, cn and tradeoff of fuel economy. d’ambrosio et al. [16] concluded that the pilot-pilot-main-after (ppma) strategy could improve in engine out-nox emissions and trade-off among bsfcnox egr curve compared to both the pilot-pilot-main (ppm) and pilot-main-after (pma) strategies at mid-range loads and speeds. in addition, it reduces cn significantly compared to both triple injections. in a previous study [17], the impact of epma injection was assessed with respect to the performance, emissions and cn level of crdi engine with variables, such as variable main injection timing and eight different speeds and loads, for better and smoother torque, bsfc and reduced cn than baseline pma-e. in another study [18], the superiority of epma injection over triple (pma) and double (pm) assessment of quadruple injection strategy over triple injections to improve emissions... 323 injections was assessed with respect to noise, performance and emissions level with fixed main injection time. currently, only limited literature is available on the quadruple injection strategy consisting of a double pilot and one post-injection event combined with high egr on the heavy-duty diesel engine. studies of the influence of multiple injection strategies upon vehicle level fuel economy and noise performance have not been reported yet. considering this research gap, we have focused on the comprehensive assessment of four types of multiple injections, including the newest quadruple injection (fig. 1). in the present study, the potentialities of quadruple injection schedules are evaluated over three triple injections on a classic six-cylinder heavy-duty crdi engine at three different operating loads and eight speeds (low-to-high) using design of experiments (doe) approach. the results of emissions (nitrogen oxides (nox), particulate matter (pm), smoke and total hydrocarbons (thc) levels) and performance (torque, bsfc and bte) and cn (radiated) with fixed egr% and main injection timing were compared at various engine working conditions. the study also elucidated the impact of these injection strategies on vehicle level pass-by-noise (pbn) and csfe. the current literature provides an insight in terms of in-cylinder characteristics. fig. 1 schematic representation of quadruple and three triple injection strategies in fig. 1 the following is represented: epm, ema and pma-triple injections; epmaquadruple injections; tdc-top dead centre; ca-crank angle; ai-main injection advanced w.r.t tdc; aie-early injection advanced w.r.t tdc; aip-pilot injection advanced w.r.t tdc; aia-after injection w.r.t tdc; dta-after/post-injection dwell; ppilot or pilot injection 2; e-pilot injection 1 or early injection; qm-quantity of fuel at main injection; qpf/qaf-quantity of fuel at pilot/early and after/post-injection. 324 s. biswas, a. mukhopadhyay 2. experimental setup and methodology 2.1. experimental setup the experiment was conducted on a heavy-duty bs-iv diesel engine containing a cooled egr and crdi system. the external cooling system (turbocharged intercooler) is attached to the engine. table 1 shows the specification of a typical engine. table 1 specification of experimental engine parameters value engine type bs-iv 6 cylinder inline, turbocharged total displacement volume 5.67 l max power 130 ps @ 2400 rpm max torque 485 nm @ 1500 rpm max speed and min speed (idle) 2750 rpm and 700 rpm compression ratio 17.5:1 injection system crdi, bosch–edc 17 injection pressure 120–170 mpa egr type short route cooled egr no. of holes at injector and tip angle 8 no. and 148° combustion chamber type shallow bowl/semi-quiescent boost pressure @ max power 214 kpa fig. 2 schematic layout of the experimental setup for performance and smoke emissions assessment of quadruple injection strategy over triple injections to improve emissions... 325 fig. 3 schematic layout of nearby noise test at rig in test rig, the engine is propelled by an eddy current dynamometer using a drive shaft. a 6-speed transmission is mounted with the engine. the driveshaft is connected in between the dynamometer and the transmission/gearbox. properly conditioned and metered air and fuel were used during experiments as shown in the setup and data acquisition system (fig. 2) for emissions and performance. inside the testbed, engine-radiated noise was measured according to the schematic in fig. 3, following the is: 10399 guidelines [19]. 2.2. experimental methodology the typical engine has egr for in-cylinder nox reduction and bosch-make crdi system. it also has a definite fuel-mass torque cycle (fmtc) where the broad outline of fuel demand for any particulate torque and speed was mapped. calibration, diagnostics and validation activities were monitored using inca software. the permittable smoke limit for the base engine has been outlined for partial as well as full load application with pma injection schedule. fuel mass (fm) and engine speed are functions of fuel injection pressure (fip). therefore, fip varies within 120–170 mpa based on the fm and engine speed (rpm). in the present study, egr% was the same as the typical production diesel engine (table 2). herein, the key focus was on the comparative study and understanding the effect of four different multiple injection strategies on performance (mainly bsfc) and exhaust emissions trade-off and cn. the base engine also had pma triple injection schedules (ai variable 6 to -3° ca btdc, aip -19.9° ca btdc, dta 1350 ms), which meets bs-iv norms. to adopt all four injection strategies (i.e., epma, pma, ema and epm), delta optimization was carried out at the base level calibration to reduce variable factors in experimentation and complexity. after-treatment arrangement was carried out similarly as in production/base engine during the experiments. in this test bed, the clutch, gearbox, external cooling/intake system and engine mounts adapted from production models were used in a typical engine. the doe method used in this study for a systematic approach of testing was based on the inputs presented in table 2. the experiments were conducted as per the doe matrix shown in table 3 for performance and emissions. the data were captured at a steady-state condition, considering an average of 20 cycles for each dataset. to deal with uncertainty in measurement data, especially for the sensitive fuel economy and emissions, nbl-141 guidelines (issue 2, amendment no. 3, 2000) were followed in the laboratory. the emission tests were based on european stationary cycle (esc) with a european load response (elr) 326 s. biswas, a. mukhopadhyay for smoke trial and european transient cycle (etc) standards to verify the results in reference to regulatory norms. radiated noise/cn of engine was measured based on the is: 10399 [19] standard on noise measurement methodology of a stationary vehicle. during the noise trial, the microphone was placed as shown in fig. 3. test data were acquired only after stabilization of exhaust gas temperature. furthermore, the ambient noise of the test rig was measured before the start of the trials. pbn trial was conducted as per is: 3028 [20] to understand the vehicle level impact. csfc or csfe performance of vehicle was measured based on ais-149 [21] to evaluate the real-time effect. this standard provides the guidelines to calculate the target csfc at 40 kmph and 60 kmph speed based on engine specification, driveline configuration (e.g., 4×2) and vehicle gvw. table 2 doe matrix inputs: factors, level, and value factors level value triple injection1 a epm triple injection 2 b ema triple injection 3 c pma quadruple injection d epma load (%) l1, l2, l3 20, 60, 100 speed (rpm) n1, n2, n3, n4, n5, n6, n7, n8 1100, 1300, 1500, 1700, 1900, 2100, 2300, 2500 fixed factors egr ai aip aie dta 45% 2 °ca btdc 19 °ca btdc 39 °ca btdc 1100 ms table 3 doe matrix for performance and smoke emissions tests trial combination speed (rpm) n1 n2 n3 n4 n5 n6 n7 n8 epm al1 al1n1 al1n2 al1n3 al1s4 al1n5 al1n6 al1n7 al1n8 al2 al2n1 al2n2 al2n3 al2s4 al2n5 al2n6 al2n7 al2n8 al3 al3n1 al3n2 al3n3 al3s4 al3n5 al3n6 al3n7 al3n8 ema bl1 bl1n1 bl1n2 bl1n3 bl1s4 bl1n5 bl1n6 bl1n7 bl1n8 bl2 bl2n1 bl2n2 bl2n3 bl2s4 bl2n5 bl2n6 bl2n7 bl2n8 bl3 bl3n1 bl3n2 bl3n3 bl3s4 bl3n5 bl3n6 bl3n7 bl3n8 pma cl1 cl1n1 cl1n2 cl1n3 cl1s4 cl1n5 cl1n6 cl1n7 cl1n8 cl2 cl2n1 cl2n2 cl2n3 cl2s4 cl2n5 cl2n6 cl2n7 cl2n8 cl3 cl3n1 cl3n2 cl3n3 cl3s4 cl3n5 cl3n6 cl3n7 cl3n8 epma dl1 dl1n1 dl1n2 dl1n3 dl1s4 dl1n5 dl1n6 dl1n7 dl1n8 dl2 dl2n1 dl2n2 dl2n3 dl2s4 dl2n5 dl2n6 dl2n7 dl2n8 dl3 dl3n1 dl3n2 dl3n3 dl3s4 dl3n5 dl3n6 dl3n7 dl3n8 assessment of quadruple injection strategy over triple injections to improve emissions... 327 3. results and discussion the experimental data are presented in tabular and graphical form. in the current study, three major tests were conducted to evaluate performance, emissions and noise sequentially. the data of comparative trials of bsfc and torque among the strategies are shown below. the measured data points have been mentioned on the epma curve or chart for clarity. 3.1. rig level tests 3.1.1. bsfc, torque and bte performance performance test data are plotted in graphical format for each load and speed combination as per the doe strategy (table 3). torque (tr in nm) and fuel flow rate (ffr in kg/h) were directly measured during experimentation. bsfc (g/kwh) value was calculated using equation (1), where n is engine speed in rpm. )( )35.95491000( nt fer bsfc r   = (1) the average bsfc performance was the best for the quadruple injection strategy among the schemes at 100% load, although at few speeds, pma performed better (fig. 4). in addition, the bsfc curve was smoother with the quadruple injection strategy than triple injections, while epm and ema strategies are the bottom two performers in bsfc. the quadruple injection displays the optimum torque performance among the injection strategies (fig. 5). conversely, epm shows the best torque performance in 1200–1800 rpm zone. herein, the torque performance of pma strategy was second optimum, and the ema strategy was the poorest in torque performance at 100% load at almost all speeds. fig. 4 comparative bsfc graphs at 100% load 328 s. biswas, a. mukhopadhyay fig. 5 comparative torque graphs at 100% load at 60% load, epma exhibits an optimum bsfc performance but is the best between 1800 and 2500 rpm compared to the other three multiple injection strategies (fig. 6). the pma injection strategy performed the best <1700 rpm and thus, deemed as the second best. on the other hand, the epm strategy was the poorest in bsfc performance, and the bsfc pattern/curve of epma was smoother than that of other injection schedules. similarly, the quadruple epma injection shows the optimum torque performance among the injection strategies (fig. 7). however, epm shows the best torque performance from 1100 to 1700 rpm. the second optimal torque performance was assessed using the pma injection strategy. the ema is the worst in torque performance at almost all speeds (fig. 7). fig. 6 comparative bsfc graphs at 60% load assessment of quadruple injection strategy over triple injections to improve emissions... 329 fig. 7 comparative torque graphs at 60% load at a partial load of 20%, the bsfc performance among the four multiple injection strategies could not be distinguished, especially between 2200 and 2500 rpm from the line chart (fig. 8). the bsfc of the pma is the best in the zone of 1100–1900 rpm but the bsfc curve is not smooth. on the other hand, the bsfc performance curve was smoother for the epma and marginally better than the pma considering the average bsfc level. the epm and ema injections are the bottom-level performers in bsfc (fig. 8). similarly, the torque performance is also mixed in nature at a partial load of 20% (fig. 9). the quadruple (epma) injection provides the optimum torque performance among the injection strategies (fig. 9), although it is the third-best in the 1100–1400 rpm zone. the epm also shows the best torque performance between 1100 and 1500 rpm. the second optimum torque performance was detected with the pma injection strategy, and the ema showed the most inadequate torque performance from 1100 to 2000 rpm engine speed. fig. 8 comparative bsfc graphs at 20% load 330 s. biswas, a. mukhopadhyay fig. 9 comparative torque graphs at 20% load bte is calculated based on the measured torque (tr), engine speed (n) and ffr. the calorific value (cv) of bs-iv diesel fuel based on the standardization report was 42.8 mj/kg (termed lhv). the fer was fuel flow rate in kg/h. then, brake power (bp) was calculated using the torque and speed data. cvfer bp bte   = 3600 (2) where: 35.9549 nt bp r = (3) fig. 10 average bte at different loads and overall average bte of injection strategies assessment of quadruple injection strategy over triple injections to improve emissions... 331 fig. 10 shows the trends of average bte at different loads and the overall average bte of each injection strategy. the average and overall average bte was the best for epma, while the epm was the worst. the overall average bte of pma was the best among triple injections, although the average bte was the worst at 20% load. indirectly, this study presented the combustion efficiency of each injection strategy with fixed main injection time and egr%. 3.1.2. emission tests nox, thc and co emissions data have been captured following the sampling method of exhaust gases and analyzing using the horriba analyser. the measured smoke data are represented in terms of filter smoke number (fsn), and this parameter was measured using avl smoke meter (fig. 2). in addition, regulatory emission tests were carried out following the esc with elr and etc standards. the measured regulatory test data among the multiple injection strategies were tabulated for comparative analysis of the results. smoke test as per elr: the measured smoke data are shown in the scatter chart format (fig. 11, 12 and 13). the soot concentration (st in mg/m3) could be calculated further using the below imperial equation 4. similarly, several correlations were available to predict the particulate matter (pm) as the summation of soot and hydrocarbons (hcs) with multiplication factors. 405.0 95.4 38.0 fsn t efsn s  = (4) the smoke (fsn) emission is the highest with epm among the multiple injection strategies at 100% load (fig. 11). on the other hand, the epma shows the best performance from medium to high speed, whereas the pma is the best in smoke reduction from 1100 to 1500 rpm. at 60% load, the smoke (fsn) emission pattern is similar to 100% load except for the values (fig. 12). herein, the epma displays the best smoke emission from 1700 to 2500 rpm zone, whereas the pma is the finest in smoke reduction from 1100 to 1500 rpm, and the epm is the worst in smoke performance with a marginal difference. at 20% partial load, the smoke (fsn) emission is the highest with the epm among the multiple injection strategies but has a narrow margin at maximum speeds (fig. 13). on the other hand, the pma is the best except at few random speeds and optimum in smoke emission, wherein the quadruple injection exhibits the second-best smoke performance. 332 s. biswas, a. mukhopadhyay fig. 11 smoke test results at 100% load fig. 12 smoke test results at 60% load fig. 13 smoke test results at 20% load assessment of quadruple injection strategy over triple injections to improve emissions... 333 the average smoke curves (figs. 14 and 15) indicate that the epma injection strategy is optimum and the best in the smoke reduction for a wide speed zone except from 1100 to 1500 rpm. herein, the high fuel-air mixture and diffusive combustion duration play a vital role in smoke or soot formation. post-injection pulse also has a major impact on soot oxidation, which helps in smoke/soot reduction. thus, the epm injection strategy is found the worst, while the pma is the best in average smoke reduction from 1100 to 1500 rpm at low and medium loads. fig. 14 average smoke results with varying speeds fig. 15 average smoke results with varying loads regulatory emission tests: the emission tests have been carried out according to the esc and etc standards for each injection strategy. the measured data are presented in tables 4 and 5, and the units are in g/kwh. table 4 esc test results epm ema pma epma bs-iv limits bs-v limits pm 0.019 0.018 0.015 0.014 0.020 0.020 nox 3.208 3.312 3.278 2.971 3.500 2.000 thc 0.068 0.056 0.043 0.051 0.460 0.460 co 0.053 0.087 0.077 0.061 1.500 1.500 334 s. biswas, a. mukhopadhyay table 5 etc test results epm ema pma epma bs-iv limits bs-v limits pm 0.026 0.025 0.021 0.019 0.030 0.030 nox 3.263 3.501 3.291 3.014 3.500 2.000 thc 0.092 0.084 0.056 0.063 0.550 0.550 co 0.065 0.090 0.087 0.051 4.000 4.000 the epma injection comprising of twin pilots and one post-injection pulse produces optimum emissions compared to the other three triple injection strategies (tables 4 and 5). (1) the epma injection strategy consisted of both advanced and retreaded pilot injection events combined with a high egr rate producing the lowest nox emission due to low burn gas temperature, early termination of the first injection event and controlled heat release rate. the epm shows the second-best performance due to its similar double pilot feature. the ema is the third in nox emissions due to early injection and prolonged duration between early injection and main injection schedule that causes the lower bulk gas temperature inside the combustion chamber compared to the pma. (2) the double pilot-injection strategies gave rise to higher thc emissions due to the rich fuel mixture during the ignition delay period than the single pilot (or early) injection combustion. in addition, the ignition delay has a significant impact on thc that compensates for the overall results. hence, the thc formation of these injection strategies in ascending order is pma<epma<ema<epm. furthermore, the highest ignition delay may cause the second high thc formation by the ema injection strategy. (3) the co emission is the highest from the ema injections due to the prolonged ignition delay with less gas temperature inside the combustion chamber. on the other hand, the epm produces the lowest co emissions due to the shortest ignition delay with high in-cylinder temperature. the epma is the second-best in co emissions, while the co emission of pma is the second-highest among the injection strategy. (4) the epma reduces the pm maximally among the injection strategies, although it produces higher thc than the pma. this phenomenon could be attributed to the reduced soot content due to improved diffusion combustion duration with quadruple injection with twin pilots and a post-injection pulse. 3.1.3 noise tests the cn data are captured for lower three speeds and represented in table 6 and fig. 16, which show that the twin pilot injection had a significant impact on cn reduction, especially at low loads and idle or lower speeds. the average noise data at 1100 rpm are summarized in fig. 16. herein, both the double pilot (epm and epma) injection strategies show improved results in the nearby noise trial. the epma injection strategy was best in the nearby noise test and superior by 2.2 dba compared to the worst type (i.e., pma) at 1100 rpm and 20% load. this might be due to a rate of combustion pressure in a controlled manner owing to twin pilot injections. assessment of quadruple injection strategy over triple injections to improve emissions... 335 table 6 rig level nearby noise test results injection strategy location/ speed load 20% load 60% load 100% 1100 1300 1500 1100 1300 1500 1100 1300 1500 epm 1 86.4 87.6 89.4 89.3 91.0 91.4 92.4 93.2 94.5 2 86.0 87.2 89.1 89.0 90.7 91.1 92.0 92.9 94.2 ema 1 88.2 89.6 91.0 91.4 92.9 93.3 94.6 95 96.1 2 87.6 89.1 90.6 91 92.6 92.9 94.1 94.6 95.8 pma 1 88.4 89.7 91.2 91.7 93.1 93.5 94.8 95.3 96.4 2 87.7 89.3 90.8 91.3 92.8 93.2 94.4 94.9 96.0 epma 1 86.1 87.4 89.3 89.2 90.8 91.2 92.1 92.9 94.3 2 85.8 87 88.8 88.9 90.5 90.9 91.9 92.7 93.9 fig. 16 average cn reduction plot at 1100 rpm 3.2. vehicle level tests 3.2.1. pbn testing vehicle pbn level is captured successively with all the injection strategies on a typical 16t vehicle with six-speed gearbox and 10×20 size based on is: 3028 [20]. test results (table 7) indicate that the injection strategies with double pilots (epm and epma) are better in vehicle level pbn reduction, similar to the nearby noise trial. the table 7 pbn test results injection strategy gear no. engine rpm vehicle speed average noise (dba) at pos aa’ at pos bb’ at pos aa’ at pos bb’ epm 3rd 1870 2800 20 30 80.3 4th 1870 2700 30 38 80.4 5th 1870 2300 40 45 79.7 ema 3rd 1870 2800 20 30 80.8 4th 1870 2700 30 38 80.6 5th 1870 2300 40 45 79.8 pma 3rd 1870 2800 20 30 81 4th 1870 2700 30 38 80.7 5th 1870 2300 40 45 79.9 epma 3rd 1870 2800 20 30 80.2 4th 1870 2700 30 38 80.4 5th 1870 2300 40 45 79.6 336 s. biswas, a. mukhopadhyay quadruple (epma) injection strategy delivered the best pbn results among the four injection strategies compared to the worst type (pma as per table 7) by 0.8 dba (third gear) to 0.3 dba (fifth gear). 3.2.2. constant speed fuel economy (csfe) testing hdfe has been introduced to india as per the directives of the ministry of road transport and highways (morth) in 2018. it provides guidelines (empirical formula) to calculate the csfc for 12–49 ton gvw vehicle at 40 and 60 kmph based on driveline configuration. similar guidelines were followed for doing the trials on a typical 16 ton 4×2 truck of tyre size 10×20 and rear axle ratio 5.28. the epma displayed the best bsfc performance at medium to high speeds and load combinations (figs. 4, 6 and 8) compared to triple injection strategies thus, this quadruple injection exerts a good impact on csfc trials and displays the best performance to meet the csfe norms (table 8). the pma injection strategy was the second best and also met the csfe criteria (table 8). table 8 csfc trial results injection strategy vehicle speed 40 km/h vehicle speed 60 kmph remarks csfc limit (l/100 km) measured value csfc limit (l/100 km) measured value epm 16.19 16.25 21.79 21.80 pma and epma meet csfc target ema 16.21 21.75 pma 15.99 21.71 epma 15.93 21.65 4. conclusion the quadruple (epma) injection strategy has been compared to the best three triple injection strategies (ema, pma and pma) on a typical six-cylinder bs-iv diesel engine featuring 45% egr rate to assess the potential benefits in performance (bsfc, torque and csfc), emissions and noise (cn and pbn) at eight different speeds and three load conditions (20%, 60% and 100%). prior to this experimental investigation, delta optimization was conducted at the base level calibration on the production engine to adopt all four injection strategies in order to maintain the same percentage of egr. all the testing environments were formulated based on the taguchi doe approach and regulatory norms. also, vehicle level trials have been carried out for real-time performance (fuel economy and pbn). the outcomes of this study were as follows: 1. cn is decreased at about 0.2–2.2 dba with the quadruple injection (epma) strategy compared to all three triple injection strategies (ema, pma and pma) at lower rpm and higher to lower loads. the ema is marginally better than the pma injection strategy in cn reduction. this finding indicates that the rate of cylinder pressure rise is well-controlled due to a shorter ignition delay with double pilots (i.e., e and p). at vehicle level pbn evaluation, the epma exhibits an average of 0.3 dba of noise reduction with respect to the pma and ema injection strategies. assessment of quadruple injection strategy over triple injections to improve emissions... 337 2. the epma injection strategy is optimal in average smoke reduction and reduces the smoke at wide-ranging speed except for 1100–1500 rpm. the high fuel-air mixture, diffusive combustion duration and soot oxidation rate post-injection affect the smoke or soot formation. the pma is the second-best in average smoke reduction and the best at low speeds from 1100 to 1500 rpm. on the other hand, the epm injection strategy shows the worst smoke emissions due to the unavailability of any post-injection pulse. a prolonged aie might cause the second-worst smoke performance by the ema. 3. overall pm emission is marginally better for quadruple injection strategy compared to pma and is also better than the other two (i.e., epm and ema) triple injections. both pilot and post-injection have a significant impact on the combustion phases to control the soot/smoke formation and hc. the pm is a mixture of soot and hc. a better smoke performance may cause marginal improvement in pm reduction by the epma injection strategy. the epm and ema strategies are the least effective in pm reduction. 4. a significant reduction was observed in exhaust-out nox emission level with quadruple injection scheme as this reduces the flame or bulk gas temperature inside the cylinder as well as the heat release rate (hrr). the epma strategy is exhibited at 3–6% nox reduction over epm, ema and pma but fails to meet the bs-v emission target for nox. 5. in epm injections, the combination of both advanced and retreaded injection timing at a high egr rate produced the lowest co emissions compared to epma, ema and pma. the epma reported as second best in co emissions. this becomes the favorable condition for reduced co level due to a relatively high temperature inside the cylinder (in-cylinder) with a shorter ignition delay period and enhanced the oxidation rate of co. reportedly, the ema has the highest co emission among the four strategies due to its advanced injection (i.e., 39° ca btdc) schedule in combination with high egr%. 6. the double pilot (epma) injection strategy formed slightly higher emissions of thc level than the single-pilot injection schedule (ema and pma) due to the availability of richer fuel mixture during the ignition delay phase. at the same time, double pilots reduce the ignition delay, which has a significant impact on thc reduction. thus, the final thc formation sequence is found as pma<epma<ema<epm. 7. bsfc performance is optimal and the best with a smoother curve for the quadruple (epma) injection strategy among all the schemes but marginally inferior at medium to low loads and speeds compared to pma. the pma injection schedule provides the second-best results. taken together, the epma exhibits better results (bsfc) at medium to high speeds and loads combination as proved on the basis of the vehicle-level csfc test results. 8. the average bte and overall average bte is the best for the epma injection strategy, whereas the epm was the worst. among the triple injection strategies, the pma shows the best overall average bte and average bte except at 20% load. the literature indicates that in the presence of two pilot injections with a post-injection pulse, the indicated mean effective pressure (imep) value increases and cov-imep value decreases compared to single-pilot injection, which can be indirectly correlated to the results, indicating an improvement in combustion efficiency. 9. furthermore, the torque value, restricted by exhaust out smoke number of engine, was enhanced by pilot injection with advanced injection schedule/timing in addition to a reduction in cn. interestingly, twin pilot (e and p) injections (epma) with advancement in injection timing display a marked improvement of torque at all loads (especially low to medium loads). the inside-cylinder pressure and hrr are high in twin pilot e and p) injections with advancement in injection timing, which underlies the improvement in the 338 s. biswas, a. mukhopadhyay torque of the engine without crossing the smoke number. the epm produce marginally more smoke than other injection schedules due to the absence of post-injection. the present study showed that the quadruple injection scheme has significant potential over the triple injection strategy to trade-off among nox-pm/smoke, cnbsfc, torque-bsfc, bsfc-nox and optimum outcome to correlate the test results with scientific literature in a specific research domain. furthermore, only working with this epma strategy for optimization of injection timing and injection dwell for shaping might meet further emissions with optimum performance. in addition, predictive quasidimensional or phenomenological combustion and emission model formulation correlated and verified the experimental data and the available scientific literature. acknowledgements: we would like to thank both tata motors and jadavpur university for providing the opportunity to work on this interesting topic in the automotive domain and authorising the usage of available resources, such as engine test facilities. references 1. brijesh, p., sreedhara, s., 2013, exhaust emissions and its control methods in compression ignition engines: a review, international journal of automotive technology, 14, pp. 195-206. 2. sinyavski, v., shatrov, m., kremnev, v., pronchenko, g., 2020, forecasting of a boosted locomotive gas diesel engine parameters with oneand two-stage charging systems, reports in mechanical engineering, 1(1), pp. 192-198. 3. zhang, l., 1999, a study of pilot injection in a di diesel engine, diesel engines: combustion and emissions, sp 1484, sae technical paper 1999-01-3493. 4. suh, k.h., 2014, 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measurement, bureau of indian standards. 20. is: 3028, 2007, automotive vehicles – noise emitted by moving vehiclesmethod of measurement, bureau of indian standards 21. ais-149, 2018, conformity of production (cop) procedure for verifying compliance to constant speed fuel consumption norms for diesel vehicles with gvw/gcw exceeding 3.5 tones, automotive industry standards. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 309 320 https://doi.org/10.22190/fume190415036s © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* artificial neural network application for the temporal properties of acoustic perception miloš simonović 1 , marko kovandžić 1 , vlastimir nikolić 1 , mihajlo stojčić 2 , darko knežević 2 1faculty of mechanical engineering, university of niš, niš, serbia 2faculty of mechanical engineering, university of banja luka, banja luka, republika srpska, bosnia and herzegovina abstract. though acoustic perception is well established in literature, it seems to be insufficiently implemented in practice. experimental results are excellent but a lot of issues arise when it comes to the application in real conditions. using artificial neural networks makes acoustic signal processing very comfortable from the mathematical point of view. however, a great job has to be done in order to make it possible. the procedure includes data acquisition, filtering, feature extraction and selection. these techniques require much more resources than mere artificial neural networks and this represents a limiting factor for the implementation. the paper investigates the complete procedure of acoustic perception, in terms of time, in order to identify limitations. key words: perception, temporal properties, localization, filtering, neural networks 1. introduction acoustic perception is a good alternative to the visual perception in engineering applications, with respect to simplicity, reliability and price. there are a lot of techniques of acoustic observation where each of them assumes specific preconditions to be implemented. in accordance with the preconditions the methods provide limited results. it is necessary to combine several methods of acoustic observation in order to overcome these limitations. filtering is, for example, an inevitable method of signal processing in real conditions (presence of disturbances) [1]. the function of filter is to suppress the disturbances while the signal is left unchanged. in such a manner, the filter enhances discriminative capacity (amount of useful information) in the signal. received april 15, 2019 / accepted july 28, 2019 corresponding author: miloš simonović faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: milos.simonovic@masfak.ni.ac.rs 310 m. simonović, m. kovandžić, v. nikolić, m. stojĉić, d. knežević in order to act against acoustic object, it is necessary to identify it. the procedure is called acoustic recognition because it assumes that what is currently being received corresponds in some way to something that has already been processed in the past [2].the problem belongs to the general category of artificial intelligence problems, namely pattern recognition. in the experiment, artificial neural networks were chosen, as a proven pattern recognition tool, for processing acoustic signals [3]. though neural networks can be trained to perform filtering [4], preprocessing, before the signal is presented to a neural network, is inevitable. except filtering, the procedure includes normalization (scaling) [5], feature extraction and feature selection [6]. the last two of them decrease computational complexity by combining several features in a new one, with a higher discriminative capacity. determination of object position, relative to some reference frame, based on acoustic signals is called acoustic localization. the procedure is implemented, in various forms, in science and practice [7-12]. from the mathematical point of view, acoustic localization belongs to two categories. near field acoustic localization is performed if the sound source is in the surroundings of the microphones and far field localization when the source is far away from them. the second is much simpler (computationally) but it gives less information, only the direction of sound source. the first method provides spatial coordinates of the sound source using time difference of arrival (tdoa) between microphones [13] as a reference. analytical solution requires solving system of hyperbolic equations, which is not a trivial problem even for a minimal configuration [14]. in the experiment, the problem is solved by processing tdoas using feed-forward neural network. again, the signals were previously processed using different techniques including obtaining tdoas [15, 16]. 2. acoustic perception the experiment investigates two separate elementary processes of acoustic perception, namely acoustic recognition and acoustic localization, separately. but, in both, artificial neural network is employed as a signal processing tool because of its simplicity, universality and excellent performance. neural networks are built of simple processing elements, called artificial neurons, which are inspired by biological nervous cells. neurons are spatially organized in layers and different layers may perform different transformation on input signals. the neurons in a layer are not interconnected and, depending on the way they are connected between layers, there are several network topologies. neural networks have ability to establish complex nonlinear relationship between input and output variables, by adjusting weights between neurons. the adjustment is done not by an explicit programming but through an adaptive method called learning algorithm, using training data set as a reference. thanks to the massive parallelism in data processing, neural networks have excellent speed of performance, in the phase of exploitation. the use of neural networks spreads in modern engineering permitting us to investigate a wide range of problems [17], but it also demonstrates a superior accuracy with respect to alternative methods as evident in [18]. the most important temporal characteristic of the artificial neural network is temporal resolution. it is a measure of precision with respect to the operating time. temporal resolution is limited by computational power available and number of calculations. the transfer of signals from the layer with i to the layer with j neurons is described by [19] , ( )j ij i j jy w x x f y  (1) artificial neural network application for temporal properties of acoustic perception 311 according to the matrix algebra, the computational complexity of the first equation is o(i×j) and computational complexity of the second is o(j), as element wise function. the total computational complexity of the transition between two layers is o(i×j). both experiments of acoustic perception deal with corrupted signals. in order to cancel the negative effect of disturbances, a digital iir filter of second order was employed [13]. the transfer function of the filter in general is 1 2 0 1 2 1 2 1 2 ( ) 1 b b z b z h z a z a z          (2) computational complexity of the digital filter is equal to its order. on the whole length of the signal it is o(n × r) or simply o(n), because r << n, where n is the signal length and r is filter order. the experiment searches for an efficient procedure of filter design with the goal of improving the accuracy of acoustic perception. duration of the training phase has no influence on the neural network performance, in the phase of exploitation, but it still requires some resources. most of the literature about the neural networks deals with computing problems but when it comes to the practical implementation the crucial phase of training is acquiring training samples. without accurate and properly sampled data, it is not possible to perform a correct training procedure. there is no general rule for data acquisition and feature selection because any implementation requires specific solutions and innovative approach. the experiment searches for efficient methods of acquiring data for the purpose of acoustic perception. the solutions were limited to the regular, simple and cheap equipment. 2.1. acoustic recognition categorization (taxonomy) is essential for acoustic recognition, as for all cognitive processes. it provides generalization rules that are used for making decisions [20]. categories (classes) are groups of objects that have similar characteristics in some frame of reference. a categorization, or division of objects into classes, enables the observer to make predictions of unobserved characteristic of the object based on previous experience. the process, where general rules are derived from specific examples, is called abstraction. taxonomy of the phenomena is never unambiguous. except from the object properties, it depends on the observer properties and his experience. several clues of a sound can be used for acoustic categorization. in the living world, most of the acoustic sensations are attributed by pitch, timbre, loudness and duration. engineering practice suggests different perceptual qualities of sound for different applications. for instance, temporal characteristics, like variance, zero-crossing rate and silence ratio, in combination with spectral properties, like harmonic ratio and sub-band energy, are used for discrimination between speech and music. the most important perceptual property of sound is surely frequency spectrum (envelope) and its derivates (power spectral density, spectral centroid, spectral irregularity, odd and even harmonics). it is perceived as timbre or tone color by living beings. the easiest way to obtain frequency spectrum of the sample, s[k] is to process it using fast fourier transform [21] 1 ( 2 / ) 0 , 0 1 n ikn n k n n s s e k n        (3) 312 m. simonović, m. kovandžić, v. nikolić, m. stojĉić, d. knežević the first half of the fast fourier transform, for the case of real valued signals, is conjugate complex of the second half. that is why one of the halves can be neglected without loss of information. eq. (3) is standard tool for digital signal processing because it provides frequency response with a lower computational complexity, o(nlogn), in comparison with the standard fourier transform, o(n2). at the same time, the procedure provides satisfactory result. the obtained frequency spectrum was presented to a feed-forward neural network for pattern recognition. 2.2. near field acoustic localization people and animals are able to point to the horizontal direction that the sound is coming from using slightly different signals that arrive at each ear [13]. for the vertical direction, spectrum features, produced by a sound reflector (pinna), is used as the auditory cue. artificial devices perform localization based on different acoustic clues. the working principle is strongly dependent on the number of microphones. monoaural (by one microphone) localization, is performed based on energy drop or spectrum deformation, as the sound propagates through the medium. the localization ability of binaural systems can be established by the learning procedure through the repetition of movement. an alternative is employing head related transfer function (hrt), which captures transformations of a sound wave propagating from the source to the microphone. but the most frequent and most valuable clue for acoustic localization is a lag between the signals collected at different spatial positions (tdoa). the basic approach of estimating tdoa is using cross-correlation function [13] 1 0 1 ( ) [ ] [ ] n i i jk r j l s k s k l n     (4) as argument l that maximizes its value within the range of possible lags max max1 ( [ ], 2 2 ij ij s l l t argmax r l l f      (5) where n is the signal length and is the range of expected lags. it has to be chosen in accordance with the measuring range of the experimental setup. since the expression in the bracket has n2 multiplications, n-1 additions and one division, its computational complexity is o(n2). the sum has to be evaluated t+1 times plus searching for the maximum in the range of possible lags. the final estimation of computational complexity of cross-correlation procedure with maximum allowed lag is o(n2×t). a good approximation of crosscorrelation function can be obtained using inverse discrete fourier transformation 2 1 0 1 ( ) ( ) ( ) j fl k k ij ijk r l f r f e k       (6) where rij(f) is cross-power spectral density (xpsd) ( ) ( ) ( )ij i jr f s f s f (7) since frequency spectrum, s(f) is twice shorter than original signal, s[k], and the computational complexity of cross correlation procedure, o(n2×t) eq. (6) can save a lot of computational effort. to be implemented it has to provide acceptable approximation error. artificial neural network application for temporal properties of acoustic perception 313 function ψ(f) is called windowing function; its role is to highlight some of the spectrum features in order to improve its discriminativity. different windowing functions (table 1) are intended for different purposes but the choice among them is still ambiguous. table 1 windowing functions window window function cross correlation 1cc  roth window 1/ ( )roth iis f  phat window 1/ ( )phat ijs f  scot window 1/ ( ) ( )scot ii jjs f s f  3. experimental setup the experiment investigated two separate experiments. in both cases, the acoustic signals were processed using regular processor intel (r) celeron (r) cpu n3350 @ 1.10 ghz. 3.1. acoustic recognition for the training of the feed-forward neural network in acoustic recognition 500 sound samples were recorded and collected from the internet. the samples were processed by a human listener, using specially designed software with the possibilities of playing, visualizing and separating parts of interest from the rest of the content. at the end, each of the samples, 1 s long at the frequency of 44.1 khz, contained only consistent content that can be uniquely labeled with one label. to simulate different levels of abstraction, sound samples were chosen from three categories. the first category was made of 32 recorded sound samples, all produced by the same cricket. this category was most specific among three in the implemented category-abstraction space. the second category was consisted of 261 sound samples produced by different fly individuals that belong to several subgroups and families. this category represented the middle of category-abstraction space. and the last category was made of 392 acoustic samples of different backgrounds, starting from human voices, animals, natural phenomena up to the different engines, vehicles, machines, musical instruments and other technical devices. the samples were grouped in the category, simply named “sound”, which represented the most abstract category in the category-abstraction space. amplitude-frequency spectrum, as a recognition clue, was calculated by eq. (3) and the recognition was performed using a feed-forward neural network with the sigmoid activation function and the back-propagation algorithm with momentum. the number of neurons in the input layer, for the constant sampling frequency, was determined by the signal length. the number of outputs was equal to 3, which is the number of signal categories. the rest of the network configuration and training parameters were obtained as a result of examination. the network was tested by 200 samples, not used in the training procedure. 314 m. simonović, m. kovandžić, v. nikolić, m. stojĉić, d. knežević 3.2. near field acoustic localization the acoustic source was driven along the training path hanged on three strings. the opposite end of each string was wrapped around a step motor driven pulley. three of these pulleys were geometrically placed in vertices of horizontal, approximately 4.35 m edge long equilateral triangle, above the acoustic source, building a simple routing mechanism (fig. 1). a meaningful winding and unwinding of the pulleys were used for achieving any specified location of the sound source within a selected range. the step motors were driven by arduino cnc driver and the driver is governed using pc through usb connection and atmega328p microcontroller. fig. 1 sketch of experimental setup for near field acoustic localization random number generator was employed for selecting 500 spatial locations within a 1.6 m edge long cubic space, below the three pulleys. the locations were intended for training feedforward neural network in near field acoustic localization. before training was performed, the locations were ordered using genetic algorithm with the objective of minimizing training route. the source was stopped, at each of 500 spatial locations and the sound sample was recorded for a period of 3s. for this purpose, the array of 8, low cost, mini spy microphones, was designed and connected to a pc through 8 channel terratec ews88 mt sound card. these microphones were spatially displaced at vertices of 2m edge length cube, symmetrically around the sound source moving zone, with purposely chosen tolerance of +/ 20 mm. parabolic reflector dishes were applied, on each microphone, as an audio signal mechanical amplifier. the first second, of each acoustic sample, was recorded before the sound source has been turned on as a representative of the noise that exists in the room independently of the sound source activity. the rest of each signal was recorded for a period of 2 s, after the emitter has been activated, as a representative of corrupted signal. the samples were artificial neural network application for temporal properties of acoustic perception 315 recorded by cheap equipment in a highly reverberant room full of interfering sound sources (fans and step motors) so they were expected to be corrupted with a high level of noise. to cancel the negative effect of disturbances on tdoa estimation accuracy, the signals were filtered by a suitable second order iir filter. the filter was designed through the iterative steps of evolutionary strategy in order to minimize the mean absolute tdoa estimation error over the recorded collection of samples. the error was calculated as a difference between tdoa estimated using the preprocessed signal and the theoretical tdoa calculated based on the sound speed and geometrical relations between acoustic components. except for filter coefficients, the chance was taken for the rest of preprocessing procedure to be configured. finally, the genotype of the complete preprocessing consisted of 8 variables. the first five of them were digital filter coefficients while one real variable more was employed for determining optimal range of lags to be tested in cross-correlation procedure. two integer variables were used to make choice among windowing functions and nonlinear operators. the algorithm was started with an initial population of 50 individuals each of them represented one preprocessing configuration. the genotype of the initial individuals was chosen randomly within the logical range of values. the termination condition was formulated as a maximum number of successive evolutions with no improvement in the mean absolute tdoa error. the best configuration was employed for preprocessing in the experiment of acoustic localization. tdoas were processed using a feed-forward neural network. the number of neurons in the input layer was determined by the number of employed microphones. in order to achieve the best accuracy, all redundant pairs that correspond to the certain number of microphones were employed as inputs. ( 1) / 2n m m  (8) where m is number of microphones. the number of outputs was always 3 because the spatial position was determined by 3 independent coordinates. the rest of the network configuration (number of hidden layers and artificial neurons in them) and training parameters were obtained as the result of examination. network performance was tested along 126 spatial locations, from the same space, which were not used in the training procedure. 4. experimental results in both experiments, the processing time of neural network and of filtering duration was hard to notice in comparison to the time it takes for the preprocessing techniques. this is in accordance with the theoretical predictions about the computational complexity of these procedures. 4.1. acoustic recognition the best recognition accuracy was achieved using a feed-forward neural network with 50 neurons in a single hidden layer. the network was trained using learning coefficient 0.025, momentum factor 0.999 and 400 training epochs. the overall accuracy was around 92% (fig. 2). the result was estimated as satisfactory since it was achieved in the presence of disturbances. 316 m. simonović, m. kovandžić, v. nikolić, m. stojĉić, d. knežević fig. 2 confusion matrix as result of the acoustic recognition from the point of duration, the most important phase of preprocessing, for the acoustic recognition, was fast fourier transform. fig. 3. represents the duration of the fast fourier transform procedure with respect to the signal length while fig. 4. represents the recognition error with respect to the same parameter. fig. 3 fast fourier transform duration with respect to the signal length fig. 4 mean squared error with respect to the signal length artificial neural network application for temporal properties of acoustic perception 317 4.2. near field acoustic localization geometrical relations between the experimental components and the realized spatial positions of the acoustic source were precisely measured using total station sokkia set630r. the instrument provides a laser measurement of distances with the accuracy of ±3 mm, at the used range of lengths, memorization and automatic data transfer, through the rs-232 port, to pc. after the realized positions were compared to the given coordinates, the resulting mean absolute error, achieved by the routing mechanism over the whole collection of audio samples, was approximately 10 mm. the accuracy of the mechanism was evaluated as satisfactory regarding the near field acoustic localization because the sound source diameter was approximately 40 mm. since the training positions were randomly chosen, the realized positions, precisely determined by total station, were adopted as reference for calculating theoretical (reference) tdoa-s. the path of the acoustic source, between training positions, was optimized using evolutionary algorithm. the procedure reduced the total length of the training path for 45 times. the shorter training path did not only have an influence on a shorter time required for the routing mechanism to complete it but it also affected a better positioning accuracy. the reason for this is that the routing mechanism used in the experiment, governed by the winding strings, made a higher error with a longer movement. all the audio samples collected contained a part recorded before the sound source was activated and the part after it was started. the amplitude frequency spectrums of these are evaluated directly, using fast fourier transform, and averaged over the whole collection of samples. subtracting frequency spectrum of noise from the frequency spectrum of corrupted signal resulted in the frequency spectrum of a clear signal, without noise. the ratio estimated between the dominant frequencies of the clear signal and noise was around 0.1 which is equal to the snr ratio of -20 db, in the logarithmic scale. according to the literature [22], the minimum snr ratio, which provides meaningful tdoa estimation, is in the range between -13 db and -13.5 db. the snr evaluated suggested that the collected audio samples, in the experiment of acoustic localization, contained too much noise to be useful for tdoa estimation. the same conclusion was obtained based on the dependency of the mean absolute tdoa estimation error with respect to the sample length (fig. 5). the diamonds represent the error obtained using raw signals, without any preprocessing, for 8 different lengths of acoustic sample. the tdoas were calculated by cross-correlation in time domain. the approximation line, between them, was obtained using two terms exponential function. the curve shows increasing of the mean absolute tdoa error with the signal length, which is against logical assumption that more data should provide a better result. the influence of preprocessing, on the level of mean absolute tdoa estimating error, is demonstrated by the cyan curve marked with triangles. the approximation line was obtained in the same way as previous. the line constantly decreases with the length of acoustic samples. the results presented, in fig. 5, proved the necessity of preprocessing in the acoustic localization procedure over employed collection of samples. according to the graph of processed signal, in fig. 5, the length of 80ms was adopted as optimal for obtaining tdoas in the experiment of acoustic localization. further increasing of the signal length, despite higher computational complexity, gave no significant improvement of accuracy. the minimal tdoa estimation error achieved in 318 m. simonović, m. kovandžić, v. nikolić, m. stojĉić, d. knežević the experiment was approximately 0.13ms. for the sound velocity of 334.33 m/s, at the temperature of 5°c that ruled during the experiment. the mean absolute error corresponded to the length of 45 mm. this was estimated as satisfactory, regarding the acoustic source diameter too. the average duration of tdoa estimation, which assumed preprocessing (filtering) and calculating of lags between microphones, was just under 0.1s per location. fig. 8 represents tdoa processing time, for 8 microphones, with respect to the signal length while fig. 6 presents tdoa processing time with respect to the number of microphones. both graphs were obtained with the signal length of 0.8s. fig. 6 tdoa processing time with respect to the signal length after testing different configurations, the artificial neural network was adopted with 10 neurons in a single hidden layer. the network was trained using learning coefficient 0.7, momentum factor 0.9 and 4000 training epochs. the final result is presented in fig. fig. 5 average tdoa estimation error with respect to the sample length artificial neural network application for temporal properties of acoustic perception 319 7. average deviation from the actual path was 35.7 mm which is even lower than the mean absolute error of input tdoas. the accuracy was result of redundant microphones. fig. 7 tdoa processing time with respect to the number of microphones fig. 8 actual path and estimated paths of acoustic source 3. conclusion the most demanding procedure of the acoustic recognition was fast fourier transform. on the described processor, it lasted about 500 times shorter than signal itself which leaves the possibility to employ the rest of computational power for improving recognition accuracy. one of the techniques is overlapping audio signals that result in raising temporal resolution. the processing time of near field localization is conditioned with the complexity of cross-correlation (3). on the employed processor, it was at the limit of real time application. 320 m. simonović, m. kovandžić, v. nikolić, m. stojĉić, d. knežević the experiment confirms theoretical assumption that the temporal resolution of acoustic perception, by artificial neural networks, strongly depends on the feature extraction procedure. the paper indicates crucial implementation problems of the acoustic perception, 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khaddour, h., 2011, a comparison of algorithms of sound source localization based on time delay estimation, elektrorevue, 2(1), pp. 31-37. 17. babic, m., calì, m., nazarenko, i. et al., 2019, surface roughness evaluation in hardened materials by pattern recognition using network theory, international journal on interactive design and manufacturing, 13(1), pp. 211-219. 18. fragassa, c., babic, m., bergmann, c., minak, g., 2019, predicting the tensile behaviour of cast alloys by a pattern recognition analysis on experimental data, metals, 9(5), 557. 19. rojas, r., 1996, neural networks, springer. 20. george, i., cousillas, h., richard, j., hausberger, m., 2008, a potential neural substrate for processing functional classes of complex acoustic signals, plos one, 3(5), pp 1-10. 21. smith, w. s., 1997, digital signal processing, california technical publishing. san diego. 22. dhull, s., arya, s., sahu, o.p., 2010, comparison of time-delay estimation techniques in acoustic environment, international journal of computer applications, 8(9), pp 29–31. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 125 131 https://doi.org/10.22190/fume210103018a © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd minireview an analytical approach to the third body modelling in fretting wear contact: a minireview ivan i. argatov1, young suck chai2 1institut für mechanik, technische universität berlin, berlin, germany 2school of mechanical engineering, yeungnam university, gyeongsan, korea abstract. in fretting wear contact, the third body is defined as the wear debris bed between two contacting bodies. the problem of third-body modelling is considered from a point of view of contact mechanics. this paper is restricted to a discussion of recent developments in analytical modelling of fretting wear contact. key words: fretting wear, gross slip regime, partial slip regime, third-body layer, asymptotic modeling, self-similarity 1. introduction fretting wear is a complex mechanical process of surface damage accumulation in loaded contacts of tribosystems, which are subject to oscillatory tangential displacements at low amplitude [1]. fretting fatigue and wear are frequently encountered in critical engineering sectors, such as nuclear [2,3] and aerospace [4]. in contrast to sliding wear, which is realized, e.g., in pin-on-disc experiments, in fretting wear, which takes place, for instance, in ball-on-disc fretting tests, the wear debris experiences difficulties in escaping from the contact interface established between two contacting bodies (usually called first bodies), thereby forming a debris bed or a third-body layer [5]. taking for granted that the third body is formed from first-body particles, its formation, flow, and elimination, in turn, influence the contact conditions between the first bodies as well as the surface degradation process. in recent years, a number of numerical studies have been carried out to simulate the evolution of third body under different testing scenarios [6‒8]. on the other hand, much attention has been also paid to constitutive modelling of third body [9,10]. nevertheless, there is a lack of simple analytical models of contact interaction via a third body layer that can describe at least approximately all main aspects of the fretting wear contact, including received january 03, 2020 / accepted february 08, 2021 corresponding author: young suck chai yeungnam university, school of mechanical engineering, gyeongsan 712-749, south korea e-mail: yschai@yu.ac.kr 126 i.i. argatov, y.s. chai the modification of the contact pressures, the expansion of the contact area, the growth of the third-body layer, and the accumulation of the worn material. in the present minireview, we consider the problem of third-body modelling from a viewpoint of contact mechanics and adopting a continuum mechanics approach. 2. main contact parameters external loading is apparently one of the major factors that govern the process of wear damage. when two bodies are brought into contact under a normal load, p, a contact interface is established, which redistributes the load over a contact region, 𝜔, with some contact pressure density, 𝑝(𝑥). the equation of equilibrium in the normal direction implies that 𝑃 = ∫ 𝑝(𝑥) 𝜔 𝑑𝑥, (1) whereas the distribution of contact pressures 𝑝(𝑥) strongly depends on the contact geometry and the mechanical properties of the contact bodies [11]. in fretting contact, the contact interface is subjected to the additional tangential loading, which often can be regarded as oscillating of a relatively low amplitude, 𝐹𝑇. this means that, strictly speaking, the fretting contact problem cannot be longer considered as monotonicallyevolving, though the oscillatory nature of tangential loading allows to introduce average characteristics for the evolution of tangential stresses per one-cycle loading, such as the dissipated friction energy, δ𝐸d. in the simplest case, when the coefficient of friction, 𝜇, is assumed to be constant, the quantity δ𝐸d will be proportional to 𝜇, which is a consequence of the coulomb law of friction. the dependence of δ𝐸d on the tangential force amplitude 𝐹𝑇 is strongly influenced by the fretting regime [12], which, to simplify the consideration, can be classified into two primary categories: (i) gross-slip and (ii) partial-slip. while under the gross-slip conditions, the linear proportionality between δ𝐸d and 𝐹𝑇 can be adopted without sacrificing accuracy, the case of partial contact, when the contact zone 𝜔 is subdivided into a stick zone and a slip zone, requires a special consideration [13‒15]. in hertzian contact mechanics, the contact configuration is usually described in terms of the gap function, 𝜑(𝑥), which is defined as the variable distance between the contacting surfaces measured along the normal to the joint contact plane in the undeformed configuration. (we note that, specifically, this definition applies to non-conformal contacts [16], when the normalization condition 𝜑(0) = 0 is usually applied.) the effect of wear manifests itself in the variation of the contact geometry [17], which under fretting contact conditions can be conveniently described by the variable gap function 𝜑𝑁(𝑥), where 𝑁 is the number of fretting cycles (for details see, e.g., [18,19]). as such, the change between 𝜑𝑁(𝑥) and 𝜑𝑁+1(𝑥) will be attributed to the effect of wear, and the total wear volume increment in the 𝑁-th cycle can be evaluated as δ𝑉𝑁 = ∫ δ𝜑𝑁(𝑥) 𝜔𝑁 𝑑𝑥, (2) where 𝜔𝑁 is the current contact region. we underline that the quantity defined by eq. (2) accounts for contributions from both first bodies. in sliding wear conditions, it is customary to assume that the worn material escapes from the contact interface and, thus, the wear debris does not interfere with the contact title of the paper (all the main words start with a capital leter, but not connecting words) 127 conditions existing at the contact interface after the moment of wear damage production. generally speaking, the material removed from one contacting body can be added on the other one, and this picture is natural at the microand nanoscales. in what follows, we consider the fretting wear contact from a macroscale point of view, and therefore, the wear process for first bodies will be regarded as an irreversible operation. 3. third body modelling when the wear debris is retained within the contact region, the contact modelling should take into account the fact that the direct contact between the surfaces of the first bodies is lost partially or completely. in principle, the effect of relatively large wear particles trapped between the contacting surfaces can be modelled using a “discrete” approach (see, e.g., [20‒ 22]). the term “third body”, as it is used hereafter, refers to circumstances, where the wear debris forms a third-body layer, which can be looked at from a continuum mechanics point of view. a general method for modelling friction, wear, frictional heat, and mass efflux from a contact interface was developed by zmitrowicz [23] (see also [24]) in the broad framework of the thermomechanical theory. the third-body layer effect was suggested to be modeled as an interfacial layer of wear products, assuming that its thickness is negligibly small, and the deformation of the interfacial layer is described as that of a two-dimensional micropolar material medium. a detailed review of continuum constitutive models of wear debris was given by zmitrowicz [10] with a focus on micropolar thermoelastic layers. however, while the micropolar layer model suits well for modelling thin lubricating films containing wear debris in sliding conditions, there is an experimental evidence [3,25] that the third-body layer thickness varies significantly in fretting wear. the third-body layer can be properly characterized by its variable thickness, ℎ(𝑥, 𝑁), defined inside the contact region 𝜔𝑁 and subject to the initial condition ℎ(𝑥, 0) ≡ 0. thus, assuming that the contact interaction between the two first bodies occurs through the thirdbody layer, we come to the question of modelling the deformation response of third body. as a first approximation, a winkler-type model can be adopted to describe the normal deformation, which (in the asymptotics of a thin elastic layer [26,27]) besides the layer thickness also requires the aggregate elastic modulus of third body, 𝐸𝐴 tb. the latter quantity determines the third-body layer deformation in confined compression, and, in general terms, it can be a function of the in-plane coordinates (see, e.g., [28]) and the number of cycles. an advanced constitutive model for a third-body interface was developed by krejčí and petrov [29] within the mathematical framework of hysteresis operators, which utilizes some material parameters like lamé elastic constants, viscosity coefficient, and yield limit that may change during the process of motion. now, we arrive at the question that drives this review. how should one model the growth of the function ℎ(𝑥, 𝑁) and the expansion of the contact region 𝜔𝑁 that occur due to wear? taking into account that the third-body layer is formed from a part of the cumulative worn material, the latter question implies the formulation of two problems: (i) the mass balance of retained and ejected worn material, (ii) the law of in-plane evolution of the third-body layer. the question of mass balance in the wear debris layer formation was considered in a number of publications (see, e.g., [30]). in particular, a simple model derived by fillot et al. [31], using 3d discrete element simulations, operates with the mean thickness, 𝐻, of the 128 i.i. argatov, y.s. chai third body. by denoting with 𝜌tb the density of the third body, the total mass of the third body particles trapped at the contact interface can be estimated as 𝑀 = 𝜌tb𝐴𝐻, where 𝐴 is the in-plane contact area. then, according to [31], the global progression in time of the third body that accounts for the degradation mass flow, 𝑄d = 𝐶d(𝐻max − 𝐻), and the ejection mass flow, 𝑄e = 𝐶e𝐻, is described by the balance equation d𝑀 d𝑡⁄ = 𝑄d − 𝑄e, from where it follows that the mean three-body thickness 𝐻 exponentially tends to the stable value 𝐻stab = 𝐶d(𝐶d + 𝐶e)−1𝐻max, where 𝐻max is a so-called threshold thickness, which corresponds to the number of third particles that stops the degradation flow. it is to note here that 𝐶d and 𝐶e are empirical constants that drive the degradation and ejection mass flows, respectively. a shortcoming of the lumped parameter model [31], which was further developed in [32], is that it does take into account neither the advancement of the contact area nor the variability of the thickness of the third-body layer. strictly speaking, the density and mechanical properties of the growing third-body layer evolves during the fretting wear process. the displacement accommodation and plastification process of the third body under tangential loading was considered in [33]. a straightforward hypothesis would be that this phenomenon is similar to the shake-down effect observed under repeated loading [34,35]. we note here that an elasto-plastic model for the two-dimensional deformation of the third-body layer in rolling wheel/rail contact was developed in [36]. a review of earlier approaches for modelling solid third bodies in dry sliding contact was given by iordanoff et al. [9] with an emphasis on the debris flow rheology. a simple way to model the evolution of third body was suggested by arnaud and fouvry [37] (see also [38]), who introduced the concept of conversion factor, 𝛾(𝑥, 𝑁), that governs the pointwise increment of the third-body layer thickness as δℎ(𝑥, 𝑁) = 𝛾(𝑥, 𝑁)δ𝑤(𝑥, 𝑁), (3) where δ𝑤(𝑥, 𝑁) is the total linear wear increment. observe that both increments δ𝑤(𝑥, 𝑁) and δℎ(𝑥, 𝑁) depend on the same variables. however, while the dependence on 𝑁 indicates the cycle-to-cycle evolution of the wear characteristics, the dependence on the coordinate 𝑥 needs to be clarified. by writing eq. (3), it is tentatively assumed that the fretting stroke, δ𝑥, is sufficiently small to identify the “point” of wear debris generation, which produces the linear wear increment δ𝑤(𝑥, 𝑁), with the corresponding point of the third-body layer. loosely speaking, the wear debris can be regarded as “frozen” in the first bodies before the detachment moment, and therefore, the abscissa 𝑥 in δ𝑤(𝑥, 𝑁) can be taken as a lagrange coordinate. on the other hand, the dependence of δℎ(𝑥, 𝑁) on the coordinate 𝑥 should reflect the distance to the boundary, which is traveled by the wear debris that escapes from the contact region. we note that eq. (3) generalizes the hypothesis introduced by goryacheva and goryachev [39] that the thickness of the third-body layer is proportional to the depth of the worn layer. to simplify the semi-analytical fretting wear simulations with the effect of wear debris, it was suggested by done et al. [40] to consider that the third-body layer is attached to the surface of one of the contacting bodies (presumably to that which has grater surface curvature). recently, zhang et al. [41] introduced the volume fraction of ejected loose debris, �̅�, as the ratio of the volume increment of the newly generated loose particles to the total volume increment of the newly worn volume δ𝑉𝑁. in such a way, the volume increment of the trapped debris particles compacted on the third-body layer equals δ𝑉tb = (1 − �̅�)δ𝑉𝑁 and the corresponding increment of the third-body layer thickness is calculated as follows: title of the paper (all the main words start with a capital leter, but not connecting words) 129 δℎ(𝑥, 𝑁) = (1 − �̅�)δ𝑤(𝑥, 𝑁). obviously, the latter equation is a form of eq. (3) with a constant conversion factor 𝛾 = 1 − �̅�, which is independent of the spatial coordinates. to account for the evolution of the third-body layer, which is associated with the layer thickness, a threshold-dependence of �̅� on ℎ(𝑥, 𝑁) was also introduced in [41]. fig. 1 plane contact between elastic cylinder and plate with third body following [37], a quadratic law for the conversion factor 𝛾 in the cylinder-on-plate contact configuration (see fig. 1) was adopted by argatov and chai [42], but represented in the self-similar form, that is the factor 𝛾 is assumed to be a function of the dimensionless ratio 𝑥/𝑎(𝑁), where 𝑎(𝑁) is the half-width of the contact region. in the two-dimensional setting, such a quadratic law can be written as 𝛾(𝑥, 𝑁) = 𝛾0 − 𝜅0 𝑥2 𝑎2(𝑁) , (4) where 𝛾0 and 𝜅0 are constants that may depend on 𝑁. in light of (4), it can be argued [42] that after a relatively short initial period, the contact pressure density (with the exception of the boundary-layer solutions) can be described by a self-similar profile 𝑝(𝑥, 𝑁) ≅ 𝑃 2𝑎(𝑁) 𝑓 ( 𝑥 𝑎(𝑁) ), (5) where the function 𝑓(�̅�) of a dimensionless variable �̅� satisfies the normalization condition that follows from eq. (1). formula (5) allows to estimate the continuous variation of the third-body layer profile, thickness, and volume. it is worth emphasizing that eq. (3) tentatively assumes a certain law of wear, which relates the linear wear increment δ𝑤(𝑥, 𝑁) to the contact pressure 𝑝(𝑥, 𝑁). another issue that deserves further attention is related to the simplified model for the third body evolution via the conversion factor, which is described above. apparently, a promising approach to this problem lies in applying the kachanov–rabotnov damage mechanics-based methodology, which was exploited by ghosh et al. [43] in their analysis of fretting wear. 130 i.i. argatov, y.s. chai 4. discussion and conclusion it goes without saying that, while a substantial progress has been made in recent years in analysis of fretting wear (without third body) both in gross-slip and partial-slip regimes as well as in twoand three-dimensional settings, the problem of third body modelling is still under-researched, in particular, in regard to modelling the in time and in-plane evolution of the third-body layer under partial-slip fretting wear conditions. when considering the contact problem with the effect of wear, the issue of contact geometry adaptation [44] should be addressed first. the third body influences the contact pressure distribution not only via the accommodation of the contact displacements, but also via partially filling the variable gap between the surfaces of the first bodies. this means that the variable thickness of the third-body layer is an important characteristic that links the third body evolution to the geometry accommodation aspect. within this perspective, it looks promising to derive a transient analogue of formula (4) for the conversion factor from the previously developed theoretical considerations, which are discussed above. on the other hand, there is still a lack of modelling efforts on representing the results of numerical simulations in an analytical form that is amendable to analysis. to conclude, one can say that the problem of analytical modelling of the third-bodylayer evolution in fretting wear contact is far from been completely understood. acknowledgement: this work was supported by the national research foundation of korea (nrf) grant funded by the korean government (msit) (no. nrf-2017m2b2a9072449). references 1. vingsbo, o., söderberg, s., 1988, on fretting maps. wear, 126(2), pp. 131-147. 2. chai, y.s., lee, c.y., bae, j.w., lee, s.y., hwang, j.k., 2005, finite element analysis of fretting wear problems in consideration of frictional contact, key engineering materials, 297-300, pp. 1406-1411. 3. blau, p j., 2019, a microstructure-based wear model for grid-to-rod fretting of clad nuclear fuel rods, wear, 426, pp. 750-759. 4. farris, t., szolwinski, m., harish, g., 2000, fretting in aerospace structures and materials, in: hoeppner, d., chandrasekaran, v., elliott, c. 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[3] showed that the mdr is also valid for arbitrarily shaped and non-compact contact areas. received october 24, 2016 / accepted november 29, 2016 corresponding author: markus hess institute of mechanics, berlin institute of technology, strasse des 17. juni 135, 10623 berlin, germany e-mail: markus.hess@tu-berlin.de 252 m. hess, v. popov the enormous technological progress in recent times is closely linked to the development of high-performance materials. in order to meet the increased demands, functionally graded materials (fgm) are used, which include the elastically power-law graded materials. they are characterized by a modulus of elasticity which increases perpendicularly to the half-space surface according to the power law: 0 0 ( ) with 0 1 k z e z e k c         . (1) where c0 denotes the characteristic depth in which elastic modulus e0 prevails independently of the exponent of the elastic inhomogeneity (see fig. 1). fig. 1 axisymmetric contact between a rigid indenter and an elastically power-law graded half-space contact mechanics of such materials were mainly developed by booker et al. [4, 5] and giannakopoulos and suresh [6] for normal contacts without adhesion and chen et al. [7] and jin et al. [8] for adhesive contacts. due to the interest in investigating the behavior of elastically inhomogeneous, biological structures as well as the adhesive material behavior in microand nanosystem technology the latter is still a subject of current research. an analytical solution of the tangential contact has not yet been published, but hess [9] presented the solution at a recent workshop. the basic ideas were also mentioned in a further conference paper [10]. the key to the solution of the tangential contact lies once more in the superposition principle of ciavarella [11] and jäger [12]. analogously to contact mechanics of homogeneous materials, all the above-mentioned contact theories for the calculation of contacts between power-law graded materials can again be suitably transformed by means of mdr, so that equivalent one-dimensional models are created which satisfy simple rules. the general foundations for the mapping of contacts between heterogeneous materials were given by popov [13]. the derivation of all mdr-rules for the exact mapping of non-adhesive and adhesive normal contacts between power-law graded materials goes back to hess [14, 15]. in this paper, all the rules are listed and their easy handling for the solution of normal, tangential and adhesive contacts is explained by means of examples. method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 253 2. two introductory steps of the mdr the basic procedure for solving contact problems by mdr is independent of whether we consider homogeneous or inhomogeneous materials. only the rules look a bit different. again, we would like to assume axisymmetric contacts. furthermore, the exponent of elastic inhomogeneity k and characteristic depth c0 of the contacting bodies should be the same. the two solids should thus be able to distinguish themselves only in the poisson's ratios ν1, ν2 and / or in the moduli of elasticity e01, e02 prevailing in the characteristic depth. 2.1. the first step: mapping of material properties the power-law graded properties of the contacting bodies are taken into account within the mdr by linear elastic springs of suitable stiffness. in addition to a normal stiffness, each spring also has an independent tangential stiffness. the spring stiffnesses related to the distance of springs δx are called foundation moduli cn, respectively ct. these have to be chosen as follows: 1 2 2 1 2 1 01 2 02 0 1 1 | | ( ) ( , ) ( , ) k n n n x c x h k e h k e c                 , (2) 1 1 01 2 02 0 1 1 | | ( ) ( , ) ( , ) k t t t x c x h k e h k e c                . (3) coefficients hn and ht in the foundation moduli according to eqs. (2) and (3) are complicated but well-defined functions depending on poisson's ratio ν and exponent k of the elastic inhomogeneity. they are given in the appendix. the decisive factor at the foundation moduli is that they depend on coordinate x (see fig. 2). both stiffnesses increase with the lateral distance from the center point of the contact to exactly the same power law according to which, in the original problem, the elastic modulus increases perpendicularly to the half-space surface. in the special case of homogeneous half-space k  0, the following holds: 2 (0, ) 1 and (0, ) (1 )(2 ) n i t i i i h h      . (4) then coefficients cn and ct are constant and equal to effective elastic moduli e * and g * [2]. fig. 2 series of infinitesimaly adjacent, linear spring elements whose normal and tangential stiffness increase with the lateral distance from the midpoint of contact 254 m. hess, v. popov 2.2. the second step: transformation of profile the second preliminary step involves the transformation of given three-dimensional contact profile f(r) into an equivalent plane profile g(x). the transformation and reverse transformation for the profile functions are [14]: | | 1 1 2 20 2 ( ) ( ) | | d ( ) x k k f r g x x r x r       , (5)   1 2 20 2 2 2cos ( ) ( ) d ( ) r k k k x g x f r x r x       . (6) for better understanding, fig. 3 visualizes the transformation of the profile. it should be noted that the equivalent plane profile is sometimes called equivalent 1d profile since it belongs to the equivalent 1d system. fig. 3 tranformation of the 3d-profile into an equivalent plane profile 2.3. example for the mdr transformation as an example we consider the transformation of a profile whose shape is described by the power function: f(r)  an r n with nℝ⁺, an  const . (7) application of eq. (5) to the profile according to eq. (7) leads to the following equivalent profile: ( ) ( , ) | | ( , ) (| |)n ng x n k a x n k f x   , (8) with 1 2 2 1 11 1 1 1 20 02 1 ( , ) (1 ) d : , 2 2 2 2 (1 ) n k n k n n n n k n k d t t t                       , (9) where b(x,y) is the complete beta function. eq. (8) clearly indicates that the equivalent profile results from a simple, vertical scaling of the original profile. the scaling factor (n,k) is dependent on the exponent of the power function and the exponent of the elastic inhomogeneity. the scaling factor increases with increasing exponent of the power-law profile (see fig. 4). method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 255 fig. 4 dependence of scaling factor  on power-law exponent n for different exponents k of the power-law graded material (adopted from [14]) in the homogeneous case, the known values (1,0)/2 for the conical and (2,0)2 for the parabolic indenter are obtained. the equivalent one-dimensional profiles of the basic contact profiles are listed in table 1. table 1 basic three-dimensional profiles and their equivalent one-dimensional profiles flat-ended parabolic conical power-law …with exponent n ( )f r 0, , r a r a     2 2 r r tanr  n na r ( )g x 0, | | , | | x a x a     2 (1 ) x k r 1 1 1 b , tan 2 2 2 k x       ( , ) | |nnn k a x 3. calculation rules of mdr for solving normal contacts between power-law graded materials without adhesion the mdr procedure for solving contact problems between power-law graded materials is the same as in the classical mdr for homogeneous materials. 256 m. hess, v. popov fig. 5 equivalent 1d contact problem of the 3d contact problem between two power-law graded half-spaces the one-dimensional profile according to eq. (5) is pressed into an elastic foundation of normal modulus given by eq. (2) (see fig. 5). the normal surface displacement at point x within the contact area results from the difference between indentation depth d and profile form g: ,1d ( ) : ( )zu x d g x  . (10) at the edge of non-adhesive contact |x|a the surface displacement must be zero: ,1 ( ) 0 ( )z du a d g a   . (11) this equation determines the relationship between indentation depth d and contact radius a. the sum of all spring forces must correspond to the normal force in equilibrium: ,1 0 ( ) ( ) ( )d 2 ( )[ ( )]d a a n n z d n a f a c x u x x c x d g x x      . (12) eqs. (11) and (12) provide the penetration depth and the normal force as a function of the contact radius. the pressure distribution in the original three-dimensional system can be determined with the help of the one-dimensional displacement using the integral transformation: ,10 1 2 20 2 ( )( ) ( , ) d ( ) z dn k k r u xc c p r a x c x r         . (13) the foundation modulus at position c0 takes into account the elastic parameters of the elastically inhomogeneous materials in contact. from eq. (2) follows: 1 2 2 1 2 0 1 01 2 02 1 1 ( ) ( , ) ( , ) n n n c c h k e h k e           . (14) the normal surface displacement outside of the contact area is given by the transformation:   ,1 1 2 20 2 2 2cos ( ) ( , ) d for ( ) ka z d z k k x u x u r a x r a r x        . (15) method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 257 3.1. examples for normal contacts without adhesion 3.1.1. parabolic contact as the first example we consider a rigid, parabolic indenter, which is pressed into a power-law graded half-space. the shape function of the parabolic contact is defined by: 2 ( ) 2 r f r r  . (16) using the transformation formula (5) yields the shape function of the equivalent onedimensional profile (see also table 1): 2 ( ) ( 1) x g x k r   . (17) thereby the displacement of the winkler foundation is known, so that the indentation depth immediately emerges from eq. (11): 2 ,1 ( ) 0 ( ) ( ) ( 1) z d a u a d a g a k r      . (18) according to eq. (12), the normal force results from the sum of the spring forces, taking into account the increasing foundation modulus from the center of the contact line according to eq. (2): 2 2 3 0 2 2 0 4 ( , ) ( ) ( ) (1 ) (1 ) (1 ) ( 1) ( 3) a k n n n k a e h ka x a f a c x dx k r k r rc k k                 . (19) to calculate the pressure distribution in the contact area, we need the first derivative of the 1d displacement: 2 ,1 ,1 2 ( ) ( ) ( 1) ( 1) z d z d x x u x d u x k r k r        (20) and the adjusted elastic parameter (one body was assumed to be rigid): 0 0 2 ( , ) ( ) 1 n n h k e c c    . (21) from eq. (13) by taking eqs. (20) and (21) into account, the pressure distribution is: 1 21 2 0 2 2 0 2 ( , ) ( , ) 1 (1 ) ( 1) k k n k h k e a r p r a ac k r                  . (22) the application of eq. (15) provides the normal displacement of the surface outside of the contact area: 258 m. hess, v. popov 2 2 2 2 2 2 2 cos 1 1 3 12 ( , ) b , , b , , ( 1) 2 2 2 2 z k a a k k r a k k u r a k r r a r                           , (23) with the incomplete beta function: 1 1 0 b( , , ) : (1 ) d z x yz x y t t t   ∀x,y ℝ⁺. (24) it is easy to verify that from eqs. (18), (19) as well as eqs. (22), (23) in the particular case k  0 the solutions of the hertzian contact exactly follow. 3.1.2. conical contact for the conical contact, the profile functions of the original and the equivalent system are listed in table 1. they are: 1 1 1 ( ) tan ( ) b , | | tan 2 2 2 k f r r g x x           . (25) table 2 summarizes the solutions of the conical contact, which results from the mdr rules eqs. (10)-(13) and (15). again, one body was assumed to be rigid. table 2 solutions to the normal contact between a rigid conical indenter and a powerlaw graded half-space conical contact ( )d a 1 1 1 tan b , 2 2 2 k a        ( )nf a   211 02 2 2 0 ( , ) tan b , (1 ) ( 1)( 2) kk n k h k e a c k k      ( , )p r a  11 2 02 2 2 2 0 ( , ) tan b , 1 1 b , b , , 2 2 2 24 (1 ) k n k k h k e k k r k k r c a                     ( , )zu r a    11 2 2 2 2 2 2 2 tan b , cos 1 1 3 1 b , , b , , 2 2 2 2 2 k k a a k k r a k k ar r                      method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 259 since the beta function in the pressure distribution contains some negative arguments, we extend the definition according to eq. (24), in which we make use of its representation by hypergeometric series: 2 1( , , ) ( ,1 ;1 ; ) xz z x y f x y x z x     (26) where 2 1 0 ( ) ( ) ( ) ( , ; ; ) with ( ) : ( ) ! ( ) n n n n n n a b z x n f a b c z x c n x         . 4. adhesive normal contact between power-law graded materials for the solution of the normal contact with adhesion between two power-law graded materials by means of mdr, there is only a small change to the non-adhesive normal contact: the rule according to eq. (11) for calculating the indentation depth as a function of the contact radius must simply be replaced by rule [14]: ,1 max max 2 ( ) ( ) with ( ) : ( ) z d n a u a a a c a      . (27) clearly, this means that the equilibrium state of the contact with adhesion is found when the elongations of the springs at the edge of contact reach defined value δℓmax(a) (see fig. 6). fig. 6 illustration of the mdr rule for an adhesive contact between power-law graded materials in addition, the mdr solution provides a simple way to calculate the critical contact radii, and thus the (maximum) pull-off force as well as the minimum indentation depth the critical contact radii must fulfill the following condition: 260 m. hess, v. popov max 2 for fixed-load ( ) ( ) 3 ( ) with ( ) : 2 for fixed-grips 1 c c c a a g a k c k c k a a k             . (28) thus, the slope of the equivalent profile at the contact edge is decisive for reaching the critical states. the different definition of coefficient  c k is linked to whether a fixedload or fixed-grips condition is present. with conditions (27), (28) and the general rules of the mdr procedure (10), (12), (13) and (15), every standard contact problem with adhesion can be easily solved. a few examples are presented below. 4.1. examples for adhesive normal contacts 4.1.1. parabolic contact we refrain from re-calculating the equivalent profile for the parabolic contact at this point and adopt the result of eq. (17): 2 2 ( ) ( ) 2 ( 1) r x f r g x r k r     . (29) considering the definition of the displacement of the winkler foundation from eq. (10), the indentation depth as a function of the contact radius for the adhesive contact follows from the separation criterion (27): 12 0 * 2 ( ) ( 1) ( , ) k k n c aa d a k r e h k      , (30) where we abbreviated e * :e0 /(1ν 2 ). according to eq. (12), the normal force must again correspond to the sum of all spring forces. it differs from the normal force in the case of the non-adhesive contact only by a part which results from an additional rigid-body translation δℓmax(a) of all springs: 2 2 max 0 0 * 3 * 3 2 2 0 0 2 ( ) ( )( ) 2 ( ) ( ) (1 ) 4 ( , ) 8 ( , ) ( 1) ( 3) ( 1) a a n n n k k n n k k f a c x a x dx c x a dx k r h k e a h k e a c k k r k c                  . (31) for the calculation of the critical contact radii from condition (28) (limit stability), only the slope of the equivalent profile at the contact edge is required. from eq. (29) it follows ( ) 2 / [(1 ) ]g a a k r   and with it from eq. (28): 1 2 2 3 0 2 * (1 ) 2 ( ) ( , ) k k c n k r c a c k h k e          . (32) method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 261 inserting the critical contact radii according to eq. (32) into eqs. (30) and (31) provides the critical indentation depths and normal forces: 2 2 2 3 0 2 * (1 )1 2 ( ) (1 ) 2 ( ) ( , ) k k c n k r cc k d k r c k h k e           , (33)   2 3 fixed-load 1 ( )(3 ) 2 2 (1 )(5 3 )( ) (3 ) (fixed-grips) 2(3 ) c k r c k k f r k kc k k r k                   . (34) it is needless to say that eqs. (30)-(34) developed by mdr agree exactly with solutions from three-dimensional theory by chen et al. [7]. they drew attention to the fact that, according to eq. (34), the maximum pull-off force is independent of the elastic parameters and independent of the characteristic depth as in the homogeneous case. for the calculation of the pressure distribution according to eq. (13), we need the one-dimensional displacement respectively its derivative, which we specify here again to clarify the treatment of the finite jump at the contact edge (see fig. 6): 2 2 ,1d max( ) ( ) [ ( ) ( )] with ( 1) z a x u x a x a x a k r            xℝ, (35) ,1d 2 2 max 2 ( ) [ ( ) ( )]... ( 1) ... ( ) [ ( ) ( )] ( 1) z x u x x a x a k r a x a x a x a k r                    . (36) in eqs. (35) and (36) h(…) denote the heaviside function and (…) the delta distribution. after insertion of eq. (36) in eq. (13) and taking into account the filter property of the delta distribution, the pressure distribution results in: 1 1 2 2* 1 * 12 2 2 0 0 2 ( , ) 2 ( , ) ( , ) 1 1 ( 1) k k k k n n k k h k e a h k e ar r p r a a ac k r c                                   . (37) 4.1.2. power-law contact profile the equivalent profile of an indenter whose shape is a power function according to eq. (7) has already been calculated in eq. (8): ( ) ( ) ( , ) | |n n n nf r a r g x n k a x   , nℝ⁺ (38) wherein (n,k) has been defined in eq. (9). the solutions of the adhesive contact between a power-law graded half-space and a rigid indenter whose profile is given by eq. (38) using mdr are summarized in table 3. 262 m. hess, v. popov table 3 solutions to the adhesive contact between a rigid indenter whose shape is a power function and a power-law graded half-space adhesive contact of power-law profile ( )d a 1 0 * 2 ( , ) ( , ) k k n n n c a n k a a e h k     ( )nf a * 1 * 3 2 0 0 2 ( , ) ( , ) 8 ( , ) ( 1)( 1) ( 1) n k k n n n k k h k n k na e a h k e a k n k c k c             ca 1 2 1 0 2* 2 2 2 2 ( , ) ( ) ( , ) k n k n n c h k e c k n k n a         ( )cd a 1 1 2 1 0 * 2 1 1 ( ) ( , ) ( ) ( , ) ( ) n k n kk n n c nc k h k e c k n n k a nc k                          ( )cf a 2 1 0 * 1 3 2 1 1 ( )( 1) 2 (2 ) ... ( , )( )( 1)( 1) 1 ... ( ) ( , ) n k n k n k n k n cc k n k h k ec k n k k c k n n k a                                  5. tangential contact between two power-law graded half-spaces we now consider a partial-slip problem between two power-law graded half-spaces with the same exponent k of elastic inhomogeneity, but different elastic parameters e0 and ν. the solids are initially pressed against each other with a normal force fn and subsequently loaded with a tangential force fx in the x-direction (see fig. 7). the axisymmetric gap function is given by f (r). let us assume that normal and tangential contacts are uncoupled, which is strictly permitted only if either [9]:  both materials are equal: ν1  ν2  ν  e01  e02 : e0,  one material is rigid and the other one has a poisson's ratio equal to the holl-ratio [16]: e0i    νj  1/(2+k) with i  j or  both materials have a poisson's ratio which corresponds to the holl-ratio: ν1  ν2  1/(2+k). method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 263 fig. 7 tangential contact between two power-law graded half-spaces it is well-known that the contact region consists of an inner stick and outer slip region. in the stick domain all points undergo the same tangential displacement x. the (undirectional assumed) tangential stresses are determined by coulomb’s law of friction: stick( ) ( ) for ( , )r p r x y a    , (39) stick( ) ( ) for ( , ) \r p r x y a a    . (40) we denote the radius of the stick domain by c. the equivalent model for the tangential contact is shown in fig. 8. as already mentioned, each spring has normal and tangential stiffness according to eqs. (2) and (3). we note once again that these stiffnesses depend on the lateral coordinate according to a power law. fig. 8 equivalent model for the partial-slip problem between two power-law graded half-spaces; each spring has normal and tangential stiffness, which are independent of each other due to the uncoupled normal and tangential contact, we assume the solution of the normal contact problem according to section 3 as already known. the mdr-rules for the solution of the tangential contact require amonton's law for each spring. the tangential line load is thus defined as follows: 264 m. hess, v. popov ,1 ( ) ( ) for | | (stick) ( ) ( ) ( ) for | | (slip) t x x n z d xzq c x x c q x c x u x c x a           . (41) it takes into account both the rigid-body translation of the stick area and the coulomb friction in the sliding area. the calculation of the stick radius is based on the continuity of the tangential line load at the transition between stick and slip domain: ,1 ( ) lim ( ) lim ( ) ( ) ( ) ( ) n x x x x z d x c x c t c c q x q x q c u c c c         . (42) analogously to the normal contact problem, the tangential force results from the sum of the tangential spring forces. if, instead of the spring forces, the line load from eq. (41) is used, the calculation formula is: ,1 0 ( ) ( ) 2 ( ) 2 ( ) ( ) a c a x x t x n z d a c f a q x dx c x dx c x u x dx         . (43) it is also possible to deduce the pressure distribution of the original contact problem from the equivalent model. for this purpose, the knowledge of the tangential line load is sufficient since: 1 2 2 1 2 ( )1 d ( ) : ( ) d d ( ) k x zx r k x q x r r x r r x r            . (44) it should be noted that the tangential line load according to eq. (41) can also be expressed as a difference of the vertical line loads:  ( ) ( , ) h( | | ) ( , ) h( | | )x z zq x q x a a x q x c c x       . (45) where qz (x,a) is the normal line load actually acting in the equivalent model, and qz (x,c) is one that belongs to a smaller contact radius, stick radius c. 5.1. example: parabolic tangential contact between power-law graded half-spaces in the following the same power-law graded materials are assumed which allow uncoupling of normal and tangential contact. the normal contact problem has already been described by means of the mdr in the examples of section 3. the only difference is that one solid is assumed to be rigid. in order to be able to adopt the solution, we only have to adjust the stiffness, which is half as large, since both bodies are elastic. regardless of it, the stiffness is shown here again: 0 2 0 ( , ) | | ( ) 2(1 ) k n n h k e x c x c         , (46) 0 0 1 | | ( ) ( , ) 2 k t t x c x h k e c         . (47) method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 265 the one-dimensional normal displacement of the winkler foundation has already been determined (see eqs. (20), (18)) so that the tangential line load can be specified: 2 2 ( ) for | | (stick) ( ) ( ) for | | (slip) (1 ) t x x n c x x c q x a x c x c x a k r          . (48) from the continuity requirement eq. (42) at the points | x |c of the tangential line load it follows: 2 2 2 2 ( , ) 1 (1 ) ( , )(1 ) n x t h k a c h k k r a             . (49) the maximum displacement before macroscopic sliding (full slip) begins is thus: 2 ,max 2 ( , ) (1 ) ( , )(1 ) n x t h k a h k k r        . (50) integration of the tangential line load (48) over contact length 2a according to eq. (43) yields: 3 3 2 ,max 1 1 1 k k x x n x f c f a                   , (51) where we have taken into account eqs. (49) and (50) on the right. fig. 9 stick radius as a function of tangential force in normalized representation for different exponents of elastic inhomogeneity k fig. 9 shows the dependence of the stick radius on the tangential force for different exponents k. the tangential force as a function of the tangential displacement for the spe266 m. hess, v. popov cial case ν1  ν2  1/(2+k) is depicted in fig. 10. in this case, the prefactor in the maximum displacement deflates and from eq. (50) it follows: 2 ,max 3 2 x k a r     . (52) in fig. 10 the tangential shift was normalized to the maximum tangential displacement in the homogeneous case. as can be seen from eq. (52), the tangential shift increases with increasing k. when the state of full-slip is reached, a 30% larger displacement is obtained for k0.9 in comparison to the homogeneous case. fig. 10 normalized representation of the dependence between tangential force and tangential displacement for different exponents of elastic inhomogeneity and specification of ν1  ν2  1/(2+k) finally, it should be noted that we have assumed the usual approximations which are already contained in the classic solution of cattaneo [17] and mindlin [18] and have been discussed in detail by ciavarella [11]. with reference to the corresponding paper we therefore waive an explicit listing. 6. conclusions this paper presents all the essential rules of the mdr that allow the solution of contact problems between power-law graded materials. we have carefully distinguished between normal, tangential and adhesive contacts and explained the simple application of the rules by means of examples. it does not need to be mentioned that despite its simplicity the mdr reproduces exactly all the results of the complicated three-dimensional theory. we would like to emphasize that the analytical solution of the tangential contact between power-law graded materials is an absolute novelty, since a derivation from the three-dimensional theory was missing so far. it was only in the run-up to the present pub method of dimensionality reduction in contact mechanics and friction: a user's handbook. part ii 267 lication that this gap could be closed [9]. clearly, all mdr rules for solving contact problems between elastically homogenous materials are a special case of the rules presented here. the presented extension of the mdr is of interest to a number of current research areas since functionally-graded materials are gaining in importance. these include tribology, nanotechnology, biostructure mechanics and medicine. completely analogous to the calculation of wear profiles between homogeneous materials [19, 20], the investigation of fretting between elastically inhomogeneous materials should no longer constitute a barrier. the same applies to the extension of the current numerically simulated impact problems between elastically homogeneous spheres [21, 22] onto elastically inhomogeneous ones. based on the mdr-rules presented here, which are comprehensible to everyone, the development of asymptotic solutions [23] for complicated contact configurations between powerlaw graded materials is also likely to be much easier. we would like to point out that the theory presented here is limited to power-law graded materials. the extent to which the theory can be applied to other laws of elastic inhomogeneity remains a challenging, future task. appendix the coefficients contained in the foundation moduli according to eqs. (2), (3) are defined as follows [9, 15]: 2(1 )cos 1 2 2 ( , ) ( , ) 1 ( , ) ( , )sin 2 2 n k k k h k k k c k k                                    , (53) 2 4 ( , )cos 1 2 2 ( , ) ( , ) 1 (1 )(1 ) ( , )sin 2 ( , )(1 ) 1 2 2 2 t k k k h k k k k k c k k                                                  (54) with 1 3 ( , ) 3 ( , ) 2 2 2 ( , ) (2 ) k k k k k c k k                       (55) and ( , ) (1 ) 1 1 k k k           . 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half space with a constant coefficient of friction: numerical analysis based on the method of dimensionality reduction, zamm‐journal of applied mathematics and mechanics, 96(9), pp. 1089–1095, doi: 10.1002/zamm.201400309 23. argatov, i., li, q., pohrt, r., popov, v.l., 2016, johnson–kendall–roberts adhesive contact for a toroidal indenter, proc. r. soc: a, 472(2919): 20160218 12444 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240208025m © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* new approach to experimental determination of rotational pivot behaviour by photogrammetric measurement kristina marković, maja dundović, željko vrcan faculty of engineering, university of rijeka, rijeka, croatia abstract. this paper discusses an innovative approach to experimentally determine the behaviour of rotational pivot behaviour by photogrammetric measurements. the compliant rotational pivot was selected for research as it is a well understood compliant mechanism whose deformations can be calculated analytically or numerically, allowing easy verification of the results. the mechanism was redesigned for monolithic additive manufacturing with the selection of printing directions that have reduced the influence of material anisotropy. rapid developments in image processing and computer vision have resulted in integration of photogrammetry and digital image correlation into a wide range of applications. the primary contribution of this article is in the custom designed experimental pure bending load testing setup with an additional optical displacement measurement system, which was used to study the behaviour of additively manufactured compliant mechanisms. a high quality, consumer-grade camera was used for image capture, while image processing was performed using ready-made and custom developed matlab tools. also, a redesigned compliant mechanism with sufficient precision for applications in low-cost, single-use compliant precise positioning systems was developed, and it was determined whether the selected experimental method is applicable to the research of monolithic compliant rotational joints. the experimental results obtained using this method have been compared to results obtained using finite element analysis and analytical calculations, and it was shown that the results are in good concordance. therefore, it was concluded that photogrammetric analysis aided by feature recognition is applicable to the measurement of parasitic shift of compliant mechanisms. key words: compliant mechanism, parasitic shift, photogrammetry, dic, feature detection received: february 08, 2024 / accepted may 14, 2024 corresponding author: željko vrcan university of rijeka, faculty of engineering, vukovarska 58, hr-51000 rijeka, croatia e-mail: zeljko.vrcan@riteh.uniri.hr 2 k. marković, m. dundović, ž. vrcan 1. introduction the development and availability of innovative technologies present a constant challenge to design engineers to develop new and innovative machine concepts that will keep them competitive and successful in the targeted market. compliant mechanisms are mechanical assemblies that transmit motion, power or force without friction, wear, and backlash with excellent fatigue resistance. these mechanisms achieve motion in whole or in part through elastic deformation of compliant parts and provide an alternative to sliding and/or rolling designs. compliant mechanisms are now widely used in daily life [1,2], especially in mechanical engineering, precision mechanics [3], micro and nano systems technology, metrology [4], seismology, semiconductor manufacturing [4], aerospace engineering, astrophysics, automotive industry, machine tools, precision transfer tools [5], (bio)medical applications, as well as household appliances and medical aids [6,7]. the use of compliant mechanisms offers important advantages, such as high accuracy, precision and motion resolution, and no contact during motion, so there is no wear and friction and therefore the mechanism does not heat up. consequently, no lubrication is required, the mechanism cannot jam, and no maintenance is required, so the mechanism can operate in almost any environment (e.g., radiation exposure, extreme heat or cold, corrosive, dirty or ultra-clean environments). in addition, compliant mechanisms have a reduced part count, which reduces manufacturing and assembly costs [8,9]. in general, their assemblies are simple, compact, and dependable, especially if they consist of a single part when considered monolithic [10]. previously, high manufacturing costs were the main disadvantage of compliant mechanisms, however, the rapid progress of additive manufacturing technologies has enabled the manufacturing of complex functional products, the manufacturing costs of such mechanisms have been reduced and the manufacturing of complex monolithic and thinwalled variants has become much easier [11]. given the obvious advantages and ease of application of compliant mechanisms, the need arose to research the mechanisms and consequently the behaviour of innovative materials, especially those that enable monolithic design and additive manufacturing of compliant mechanisms [12–14]. therefore, it is necessary to research the applicability of new manufacturing technologies for such constructions so that the mechanisms meet the criteria of strength, stiffness, stability, durability, functionality, and ease of manufacturing, leading to a reduction in energy and raw material consumption during manufacturing [15]. additive technologies use materials with an extensive range of mechanical properties and a wide range of applications, covering their application in compliant mechanisms [16]. therefore, an investigation of the properties and behaviour of innovative materials used in the development of compliant mechanism models is appropriate. rapid developments in the area of image processing and computer vision, marked by a combination of advancements in various algorithms, hardware capabilities, and breakthroughs in machine learning and deep learning, etc., have played a significant role in enhancing the accuracy, efficiency, and versatility of photogrammetric methods, resulting in the integration of photogrammetry into a wide range of scientific and industrial applications. this includes medical image analysis [17,18], mapping [19,20], autonomous vehicles [21], indoor and outdoor monitoring [22–24], robotics [25–27], and much more. a fundamental goal of photogrammetry and computer vision is to obtain reliable 3d information from 2d images. multiple computer vision techniques, such as optical flow, new approach to experimental determination of rotational pivot behaviour by photogrammetric... 3 feature tracking and phase correlation, can be employed for displacement measurement [28–30]. digital image correlation, or dic, is one of the most important photogrammetric methods in engineering, materials science, and structural analysis applications. it is a contactless optical measurement technique for the analysis of the deformation and motion of surface objects by comparison of digital images. the basic processing logic of dic may be summarized as follows: acquisition of the image, selection of a feature subset, followed by the correlation of images and analysis of displacement/deformation etc. in broader words, the movement of small subsets or patterns is tracked within a series of images captured at various stages of deformation or motion. the analysis of the displacement and deformation of these subsets, allows dic algorithms to determine the strain and deformation characteristics of the object or surface being researched. the main advantage of dic is that measurements are made without interfering with the movement or deformation and that measurements are made full field, capturing the deformation information across the entire surface of the object. because of this, dic is suitable for both small and large displacements and deformations, making it applicable in the deformation analysis of a wide range of materials and structures. as with all optical methods, photogrammetry requires calibration to achieve accurate results by accounting for camera distortion, lens characteristics, calibration pattern precision, image quality, lighting conditions, etc. various reliable and efficient methods of calibration have been developed for this purpose [31–33]. the contribution of this article is in the custom-designed experimental pure bending load testing and displacement measurement setup. the setup was then used to study the behaviour of additively manufactured compliant mechanisms under such load conditions. the extended contribution was made through the successful application of a high-quality, consumer-grade camera and lens, while the image processing was performed using readymade toolboxes and proprietary code developed for matlab. the authors are not aware of a similar setup or experimental method being discussed in the available literature at the time of writing this article.2. problem statement the compliant rotational joint was selected for optomechanical measurements as it is a well-understood compliant element that can be used as a building block for more complex mechanisms. its deformations can be calculated analytically, allowing for easy verification and cross-checking of experimental and finite element method-based results [34]. furthermore, the mechanism has the advantage that it can be conveniently redesigned for additive manufacturing as the geometry is simple and printer-friendly with no voids or overlaps, allowing selection of printing directions that will reduce the influence of material anisotropy. photogrammetry covers a range of precise and accurate optical methods applicable to spatial measurements that are widely used due to their non-destructive and contactless nature, making them particularly suitable for applications in compliant mechanisms. the design arrangement of a monolithic compliant rotational joint as discussed in this paper, obtained by using spring-strips, as shown in fig. 1, is often referred to as the crossspring pivot. it is characterised by considerable compliance along the ‘in plane’ rotational degree of freedom and high stiffness along the secondary degree of freedom. 4 k. marković, m. dundović, ž. vrcan fig. 1 rotational cross-spring pivot variants: a) classical two-strip design with strips of equal width; b) conventional monolithic design with intersecting strips; c) three-strip design with equal overall stripe width for both directions the design of the mechanism as shown in fig. 1c is the subject of the research performed in this article. this mechanism variant, as shown in fig. 2a, consists of rigid body b connected by spring-strips 1, 2 and 3, crossing at their midpoints, to block a. in the usual configuration, the used spring-strips have the same length l and thickness t. the width of strips 1 and 3 is equal to one-half of the width of strip 2, b1 = b3 = b2 /2. the mechanism, including the spring-strips, was printed monolithically with constant print settings. fig. 2 three-strip rotational pivot: a) geometry; b) parasitic shift; c) displacement of rigid body a of compliant joint in comparison to conventional rigid joint in practical use, the rigid bodies a and b experience rotation about each other due to the deflection of the spring-strips (fig. 2b). in the case of larger rotation angles θ, the ‘geometrical’ centre of the pivot o moves to point o,’ which is defined by the tangents to the fixed ends of the strips on the movable block, resulting in a parasitic shift of amplitude new approach to experimental determination of rotational pivot behaviour by photogrammetric... 5 d and phase φ. the difference between the position of rigid body a (light blue) in the case of a conventional rigid joint and in the case of a compliant joint (dark blue) is shown in fig. 2c. this parasitic motion is unwanted due to its detrimental influence on the precision of the analysed mechanisms [35,36]. the purpose of the research performed in this article is to develop a redesigned additively manufactured compliant element with sufficient precision to be used, for example, in low-cost, single-use compliant precise positioning systems. another objective of the research is to determine whether the selected experimental method may be applied to the measurement of parasitic shifts of compliant rotational joints and then extended to compliant mechanisms in general. this will be significant for the analysis of compliant mechanisms whose deformations cannot be calculated analytically. 3. manufacture of rotational pivot additive manufacturing processes have become more readily available, and the improved manufacturing processes have resulted in the products advancing from simple dimensional prototypes to usable parts. this has also resulted in the process of additive manufacturing becoming applicable in the production of compliant mechanisms. monolithic compliant mechanisms, until now, have mostly been machined from solid blocks of metal by complex and expensive manufacturing processes. additive technologies have made the manufacture of both monolithic and complex geometry-compliant mechanisms readily available. the compliant mechanism which is the subject of the research in this paper, and its accompanying tensile test samples were manufactured by fused deposition modelling (fdm) using a filament-based process. the tensile test samples and the spring stripes were printed with matched filament laying and load direction. the mechanism was printed on a prusa mk3s printer using pla filament. the process parameters are given in table 1. table 1 printing process parameters process parameter value layer height 0,2 mm infill 100% filament diameter 1,75 mm extruder temperature 215 °c bed temperature 60 °c the mechanism is of the same type and geometry as in [34,35], however, modifications had to be made to accommodate for the constraints of the available additive manufacturing process. the mechanism consists of spring-strips, with the angle between the spring-strips being 2α = 90°. the spring-strips are t = 1,5 mm thick, l = 115 mm long, and the respective widths are b1 = b3 = 10 mm and b2 = 20 mm. the complete mechanism is shown in fig. 3., with the additions made to the rigid bodies visible in plain sight. the additions are required to connect the mechanism to the testing rig. the purpose of the additions is to minimize the number of parts required to connect the mechanism to the testing rig, and the additions only increase the rigidity of the setup, therefore improving the accuracy of the test results. 6 k. marković, m. dundović, ž. vrcan fig. 3 printed mechanism 4. experimental setup and procedures two distinct series of experiments were performed, the first being the tensile testing on standard test samples, and the second being the investigation of the compliant mechanism behaviour during rotational deformation with predefined angles of rotation. 4.1 tensile testing of standardized test specimens the tensile tests were performed according to standard iso 527-2, using universal specimens (fig. 4) on a step labs electro-mechanical testing machine for static and dynamic testing capable of exerting a maximal static load of 16 kn and a maximal dynamic load of 24 kn. the samples were printed from the same filament spool as the mechanism specimen, using the same printing parameters (table 1). the test was repeated on five specimens which were loaded at a speed of 1 mm per minute until failure. young’s modulus was determined as the average value of five tensile tests at 110 n/mm2, and this value was later used for purposes of numerical analysis. the tensile test setup is shown in fig. 4. fig. 4 tensile testing setup new approach to experimental determination of rotational pivot behaviour by photogrammetric... 7 4.2 testing of compliant mechanism specimen the experimental measurements were performed on a testing rig set up on an optomechanical table as displayed in fig. 5 to determine the parasitic shifts of the geometrical centre of the pivot in the deformed position. fig. 5 3d model of the mechanism testing rig the experimental setup was modularly assembled with consideration to the expected loads and displacements, based on prior experience and basic numeric simulation. rigid body a of the compliant mechanism is directly bolted onto a motorized rotational stage through a torque sensor, which only allows the rigid body a to rotate around the vertical axis. the motorized rotational stage with an integrated encoder has a resolution of 0,01° and eccentricity of 10 µm or less, while the torque sensor is rated at 0,5 nm and calibrated to an accuracy of 0,2% of rated torque. the levelling of rigid body b is then performed using a height-adjustable precision pillar with vertical movement of 1 mm per revolution. the combination of a radial bearing and translational stage 1 allows two dofs to rigid body b. those two dofs allow for the parasitic shift to exhibit itself. otherwise, a constrained rigid body b would result in the parasitic shift placing the mechanism under undue strain. translational stage 2 is present only to provide proper alignment of rigid bodies a and b. translational stages 1 and 2 are identical with sensitivity under 1 µm, one inch of travel range (25,4 mm) and an angular deviation under 200 µrad. the measurements were conducted by rotating rigid body a up to θ/2 = 15°, resulting in body b being rotated for the same angle due to the spring-stripes (fig. 2b). this gives an overall range of measurement results from θ = 0° to θ = 30°. the measurement was performed in θ/2 = 2,5° increments. after each increment, the motorized stage was stopped, a torque reading was made, and a picture was taken. the canon r5-c camera with rf 100mm f2.8l macro is usm lens was used in the experimental setup. images for photogrammetric displacement measurements were taken in 8k resolution (45 mp, 8192 x 5464). 8 k. marković, m. dundović, ž. vrcan the testing rig and measurement system, shown in fig. 6, were entirely developed inhouse for the experimental measurements. matlab’s computer vision toolbox and image processing toolbox functions were used, more precisely feature detection and extraction, followed by feature matching, and finally calculation of geometrical transformations. camera calibration, as well as communications and control of the hardware, was also performed using matlab. fig. 6 experimental setup camera calibration must be performed before measurements. the camera calibrator application, a part of the matlab computer vision toolbox is used for this purpose. it is considered dependable as it accounts for pattern errors and optical errors. the algorithms used by the application are based on the zhang calibration method [32,33]. a checkerboard pattern of 7x10 squares with a size of 5.08 mm (2/10 inch) was used for the calibration, as shown in fig. 7. the pattern was adjusted in size for the resolution of the high-quality laser printer used to print the pattern, as well as for the size of the area of the measurement. fig. 7 checkerboard pattern used for calibration calibration is performed with the camera held stationary with the calibration pattern being moved by hand. eleven images are then captured to complete the calibration process. new approach to experimental determination of rotational pivot behaviour by photogrammetric... 9 the camera settings (focus, aperture, and exposure) from the calibration procedure are then maintained throughout the measurement process. the mean reprojection error was established to be 0,53 pixels or 7 µm, fig 8. fig. 8 mean reprojection error per image for verification, an additional calibration was performed by placing the measurement scale at six separate locations on the observed surface of the rigid bodies of the compliant mechanism, fig. 9. three measurements were performed at each position, and the results of eighteen measurements were averaged. based on the results, the pixel size for subsequent measurements was established at 16 µm, which is in concurrence with the matlab calibration. fig. 9 feature recognition calibration 10 k. marković, m. dundović, ž. vrcan measurements are performed after calibration, and image processing must be performed to extract results. in this case, the features are glitter particles which were built into the test sample during the printing process. therefore, conventional dic-based sample preparation (spraying) was not required as the mechanism was printed from a darkcoloured material containing glitter particles. this has also improved the precision and repeatability of the tests as the recognition pattern is built into the sample surface. the matlab functions for feature recognition and extraction utilize the speeded-up robust features (surf) algorithm, which analyses the initial image and detects the which are then recognized in every subsequent image. the surf algorithm was chosen because the local features are identified in a way that is invariant to scale, rotation, and illumination changes [37,38]. the displacements were calculated using functions for geometric transformations derived by comparing the locations of matched point pairs from sequential images. the basic principle is shown in fig. 10, where the software has detected a sequence of geometric transformations, the results of which are marked by yellow lines between the starting locations of the features marked with red circles and the end location of the respective features marked with green crosses. fig. 10 measurement procedure the methodology used is shown in fig. 11. the axes of the image are labelled in pixels. the software is first tasked with finding the point in the green square on the right that experiences the least amount of movement. this point has been detected and marked with a single red cross. this point has zero translation, and it is coincident with the centre of the rotation imparted by the motorized stage. then, the green square on the left is searched for the point with the least amount of movement. this point is marked by the blue cross. this new approach to experimental determination of rotational pivot behaviour by photogrammetric... 11 marks the point which is placed exactly above the radial bearing. that point experiences only linear and rotational movement due to the 2 dof constraint provided by the translational stage and the radial bearing. it has been already mentioned that the linear movement of the translational stage due to the rotation of the compliant mechanism may be used as high-precision displacement. fig. 11 feature detection sample image the sequential red crosses in the centre of the image mark the rotational movement of the joint centre (point o on fig. 2b) as in the ideal case of a traditional rigid joint, under the assumption that rotation is imparted at the red cross in the green square on the right, while the blue crosses in the centre mark the rotational movement of the joint centre (point o’ on fig. 2b) as in the ideal case of a traditional rigid joint, under the assumption that rotation and appropriate translation is imparted at the blue cross in the green square on the left. the parasitic shift is then measured as the difference between matched blue and red crosses in the centre of the image. the measurements were performed in six incremental steps (θ/2 = 2,5°, 5°, 7.5°, 10°, 12,5°, 15°). 5. numerical model the experimentally obtained results were used to validate the results of the developed numerical model. numerical analysis was performed using the previously determined value of young’s modulus e = 110 n/mm2, and poisson’s ratio of ν = 0,42. as it is commonly accepted in the analysis of the behaviour of cross-spring pivots for high-precision applications that it must involve the modelling of large (geometrically nonlinear) deflections of the spring-strips while also including the parasitic shifts of their geometrical centre, that principle was also followed in this case [34,35,39]. finite element 12 k. marković, m. dundović, ž. vrcan analysis (fem) was used for numerical modelling of the kinematics of the cross-spring pivot using non-linear large deflection fem analysis, using 3-d, 10-node elements having three degrees of freedom at each node. the constraints of the numerical model were set up to replicate the pure bending case of the experimental setup. the first constraint was placed with the z-axis coincident with the rotational stage rotation axis, allowing only rotation around the z-axis. the second constraint was placed with its z-axis coincident with the axis of rotation of the radial bearing. this constraint allows only rotation around the z-axis, and movement along the longitudinal axis of the mechanism. the rigid body a was rotated by the aforementioned angles of θ/2 in order to introduce the required loads. the rotation imposed on rigid body a was added in ten equal sub-steps. a stable solution was then iteratively pursued for each sub-step until the convergence criteria were satisfied with fifty iterations allowed for each sub-step. an example of fem analysis can be seen in fig. 12, showing the 3d model of the mechanism in the position θ = 30°, overlaid with the wireframe in the unloaded position. the colour scale shows the overall displacement of the points. it was concluded that the overall displacements are in good concordance with the experimental data. fig. 12 results of fea displacement analysis 6. results and discussion the results of the experimental measurements were compared with the results of numerical analysis and analytically calculated results [39] (fig. 13). the experimental measurement results are marked by blue squares, and they have been connected by a thin dashed line to illustrate the trendline of the experimental results. the new approach to experimental determination of rotational pivot behaviour by photogrammetric... 13 fea parasitic shift results obtained for rotation angles θ = 0…30° are well matched with both the experimental measurement results and analytical results obtained in prior work. the differences between the values are within 2-5% for small rotation angles and progressively increase with the rotation angle. the values of the torque acting on the mechanism have been recorded experimentally using the torque sensor as previously mentioned. the experimentally recorded torque had a discrepancy within 8%. this is attributed to unavoidable minor assembly imprecisions and mechanical losses of the radial bearing and translation stage. the linear movement of the translational stage due to the rotation of the 3d-printed compliant mechanism may be used for high-precision displacement. the order of magnitude of the displacement is around 350 µm with θ/2 = 15°. the correlation of precise linear displacements to the rotational angle of the mechanism may be seen in fig. 14. fig. 13 comparison of parasitic shift results fig. 14 correlation of precise displacement and rotational angle 14 k. marković, m. dundović, ž. vrcan the data presented in fig. 14 shows a correlation between the rotational angle and the linear displacement of the translational stage connected to the rigid body b, and that the correlation is linear, with linearity starting to fall off at rotational angles beyond 30°. based on the analysis of the data presented in figs. 13 and 14, it can be summarized that the selected experimental procedure is applicable to the prediction of the displacement of any point of a compliant element and the behaviour of a custom-compliant mechanism design. 7. conclusion the measurement of the unwanted parasitic shift of the monolithic rotational compliant mechanism based on spring-stripes using the photogrammetric measurement method aided by surf feature recognition was covered in this article. to determine the applicability of this method to the measurement of parasitic shifts of compliant mechanisms, the experimental results obtained using this method have been compared to results obtained using finite element analysis. the test samples and the monolithic mechanism were printed with the same process parameters to keep the material properties consistent. a universal testing machine was used to verify the material properties on standardised test samples. an appropriate testing setup was developed to properly constrain the mechanism to the required translational and rotational motions. the mechanism was deformed to the required rotational angles, and the torque values were recorded. images of the deformed mechanism were taken, and photogrammetric analysis was performed to measure the parasitic shift. the numerical analysis was performed using young’s modulus data gathered by tensile testing of the material samples. the comparison of the results has shown that the testing results are in accordance with both the fea and analytically calculated values, with the parasitic shift values within 5%. the discrepancy of the torque values is a bit higher, but still under 8%, and this can be attributed to small imperfections in the testing setup assembly and internal resistances of the radial bearing and translational stage. during the research into the parasitic shift, it was noticed that the linear movement of the translational stage due to the rotation of the 3d printed compliant mechanism may be used for high precision displacement, as data analysis has shown that a correlation exists between the rotational angle and the linear displacement of the translational stage connected to the rigid body b. this correlation was found to be almost linear, with linearity falling off at increased rotational angles. therefore, it was concluded that the results are in accordance and that the 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marković, k., zelenika, s., 2017, optimized cross-spring pivot configurations with minimized parasitic shifts and stiffness variations investigated via nonlinear fea, mechanics based design of structures and machines, 45(3), pp. 380–394. 12515 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 343 383 https://doi.org/10.22190/fume240216037d © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd review article recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive review of mechanisms and failure criteria balaji devarajan1, rajeshkumar lakshminarasimhan2, aravindh murugan1, sanjay m. rangappa3, suchart siengchin3, dragan marinkovic4,5 1department of mechanical engineering, kpr institute of engineering and technology, coimbatore, tamil nadu, india 2au sophisticated testing and instrumentation centre and department of mechanical engineering, alliance school of applied engineering, alliance university, bengaluru, karnataka, india 3natural composites research group lab, department of materials and production engineering, the sirindhorn international thai-german graduate school of engineering (tggs), king mongkut's university of technology north bangkok (kmutnb), thailand 4tu berlin, department of structural analysis, germany 5institute of mechanical science, vilnius gediminas technical university, vilnius, lithuania orcid ids: balaji devarajan https://orcid.org/0000-0002-9121-6215 rajeshkumar lakshminarasimhan https://orcid.org/0000-0003-1917-5460 aravindh murugan https://orcid.org/0000-0002-6536-6228 sanjay m. rangappa https://orcid.org/0000-0001-8745-9532 suchart siengchin https://orcid.org/0000-0002-6635-5686 dragan marinković https://orcid.org/0000-0002-3583-9434 abstract. the use of lightweight natural fiber functional composites in the manufacturing of ballistic protective materials has garnered significant attention in recent years. this is due to their superior mechanical properties, cost-effectiveness, and environmental sustainability. ballistic panels are constructed using multiple layers of diverse composites, which collectively exhibit excellent mechanical properties. these properties enable them to withstand strong impacts enhancing their capability for different applications in defense, military, and aerospace components. the primary focus of this review is to examine the different influential factors that govern the development of novel polymeric materials for current ballistic applications. it also explores various research approaches, such as received: february 16, 2024 / accepted july 25, 2024 corresponding author: sanjay mavinkere rangppa natural composites research group lab, department of materials and production engineering, the sirindhorn international thai-german graduate school of engineering (tggs), king mongkut's university of technology north bangkok (kmutnb), bangkok 10800, thailand. e-mail: mavinkere.r.s@op.kmutnb.ac.th https://orcid.org/0000-0002-9121-6215 https://orcid.org/0000-0003-1917-5460 https://orcid.org/0000-0002-6536-6228 https://orcid.org/0000-0001-8745-9532 https://orcid.org/0000-0002-6635-5686 https://orcid.org/0000-0002-3583-9434 344 b. devarajan, r. lakshminarasimhan, a. murugan, et al. experimental, analytical, numerical modeling, and empirical techniques. the review highlights both internal factors, such as material composition, and external factors, such as projectile parameters (e.g., nose angles, projectile shape, and projectile size). these factors are crucial for optimizing the robust ballistic performance of natural fiber-based polymer composites. in addition, various valuable insights to develop more effective and sustainable ballistic protective materials for applications in bulletproof helmets, defense, aerospace, and military sectors have also been elaborated. consequently, the article presents a comprehensive review of the impact of utilizing various natural fibers as alternative materials to kevlar for armor structures, offering a state-of-the-art perspective and challenges faced in full-scale implementation. key words: ballistic protection, natural fiber composites, multilayer armor, ballistic mechanism, failure criterion 1. introduction ballistic materials are specially designed materials that can withstand the high-velocity impact of projectiles, including bullets, shrapnel, and other ballistic threats [1]. these materials are used extensively in body armor, vehicle armor, and other protective equipment to prevent serious injury or death in potentially dangerous situations [2]. the development of ballistic materials has played a critical and significant role in the advancement of protective technology, particularly in law enforcement, military, and security applications. they offer enhanced protection against various threats, making them crucial for personnel and assets. commercial fibers like kevlar, aramid, uhmwpe, and twaron are commonly used in defense and military armor systems over the past two decades. however, synthetic fiber composites, derived from petroleum-based materials, cause environmental harm, increase manufacturing costs, and require high energy processing, despite their high specific and impact strength [3].the replacement of kevlar fabric with eco-friendly and sustainable lightweight materials to improve kinetic energy absorption and dissipation has been found to improve the ballistic performance of composite panels [4]. in this context, ballistic materials made from natural fibers have garnered significant interest. natural fiber-based ballistic materials, such as flax, hemp, jute, bamboo, kenaf, sugarcane bagasse, and coconut sheath, have emerged as potential alternatives to synthetic kevlar-based multilayer armor systems in various applications. numerous studies on these natural fibers have revealed their exceptional ballistic properties, demonstrating their suitability for replacing synthetic fibers in body armor and protective equipment and its devices [5–7]. in addition, the densities and costs of different natural fibers used in multilayered armor systems can vary, depending on the specific type of fiber and its processing method. however, in general natural fibers generally have lower densities and costs compared to kevlar. for instance, bamboo fibers have a density of 0.55 g/cc, half of kevlar's density. similarly, amazonian guaruman fibers have a density of 0.6 g/cc, and curaua fibers have a density of around 0.5 g/cc [8], while sisal fiber has a density of 1.45 g/cc, serving as a substitute for kevlar in multi-layered armor systems [9]. the application of natural fibers as precursor materials for ballistic applications has several advantages over synthetic materials owing to their being renewable and sustainable, making them a more environmentally friendly and viable option. in this regard, braga et al [10]. experimented with the replacement of kevlar composites using 30 % natural fiber composites in ballistic applications of mass, revealing superior performance and a 275% cost reduction recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 345 compared to kevlar fibers, particularly in intermediate layer efficiency under 7.62 mm impact ammunition. likewise, garcia filho et al. [11] conducted experiments to explore the replacement of kevlar fiber in mas. they found that using piassava/epoxy composites resulted in a much lesser depth of indentation on the clay body at the back compared to the reference armor with kevlar composites. furthermore, it concluded that 70-80% of the cost is lower than dyneema and kevlar. in addition, luz et al. [12] investigated the coir fiber as an intermediate layer to replace the kevlar fabric-based mas. results indicated that an intermediate layer of coir fiber with a 30% volume fraction significantly increased the ballistic performance, which is like that of kevlar fiber. similarly, monteiro et al [13] assessed the ballistic performance of ramie fiber as a potential replacement for kevlar fiber-based mas. the results stated that the usage of ramie fibers decreased the overall production cost by 95% compared with kevlar-based mas. likewise, monteiro et al [14] found that curaua fiber as reinforcement reduced the fabrication cost by around 31.2% by replacing the kevlar fiber in mas. recently, eco-friendly, renewable, biodegradable, low cost and abundant resources were explored as reinforcement material using a 7.62 mm bullet by silva et al [15]. results revealed that 30 wt. % of arapaima fish scale-based mas produced similar dynamic behavior and failure mode as kevlar fiber. the final impact energy was absorbed by the clay witness. de oliveira et al. [16] discovered that curaua fiber offered a comparable failure mode to kevlar fiber using a 7.62mm bullet. therefore, understanding the importance of lightweight natural fiber-based ballistic impact is crucial for a wide range of applications, including the design of protective materials, the development of new weapons and ammunition, and the investigation of various forensic and accident scenarios [17]. thus, most care must be given to material selection to reduce overall manufacturing cost and allow for greater mobility and reduced fatigue for wearers as well as increase the protective strength for the development of antiballistic devices under different conditions. in general, high-performance applications such as aerospace and defense industries, for instance, composite structures are often subjected to extreme loading conditions through foreign objects such as impacts from hailstones, bird strikes, runaway debris, bullets, and explosive fragments [18]. fig. 1 shows ballistic applications such as defense, military, and aerospace. fig. 1 ballistic applications defense, military, and aerospace [19] 346 b. devarajan, r. lakshminarasimhan, a. murugan, et al. these types of impacts can cause varying degrees of damage, ranging from indentation to partial penetration or complete perforation of the composite structure. such damage can compromise the structural integrity of the composite, leading to catastrophic failure and endangering human lives [20]. therefore, it is essential to thoroughly evaluate and understand the loading conditions particularly those that require protection against high-velocity projectiles. these includes evaluating different influential parameters factors like architecture, stacking sequence, hybridization, ply orientation, shear thickening fluid, and projectile parameters like angle of impact, projectile nose, mass, and velocity. these factors significantly influence the impact of damage mechanisms, energy absorption structure, plate thickness, area density, and fracture toughness. [21, 22]. fig. 2 shows the classification of natural fibers. some examples of natural fibers that are commonly used in ballistic-resistant materials include hemp, flax and jute fibers [23]. hemp is a natural fiber that is known for its high strength and durability, and is often used in the production of textiles, paper, and other products. in ballistic-resistant materials, hemp fibers can be used in combination with other materials to create a composite that is both lightweight and highly resistant to impact [24]. flax, a strong and durable natural fiber, is commonly utilized in various industries including clothing, textiles, automotive, and aerospace. in ballistic-resistant materials, flax fibers can be combined with other materials to create a robust and flexible composite. jute, another natural fiber known for its strength, durability, and moisture resistance, finds applications in products such as rope and twine [25]. in ballistic-resistant materials, jute fibers can be used in combination with other materials to create a composite that is both lightweight and highly resistant to impact [26]. fig. 2 classification of natural fibers by proper understanding of these factors, engineers and designers can optimize the design of the robust mas-based composite structure, incorporating protective features such as sacrificial layers, increased thickness, and different stacking sequences, to enhance its ballistic resistance and protect against penetration by external projectiles. this enhances ballistic resistance and protects against external threats [27]. analyzing impact behaviour of influential parameters using theoretical, analytical, and numerical methods helps select the right material. recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 347 therefore, the utilization of different natural fibers as intermediate layers in ballistic materials presents an exciting opportunity to revolutionize the field of protective technology that is not only highly effective but also sustainable and environmentally responsible. the current review focuses on highlighting various natural fiber composites-based mas and understanding the impact by material designation and damage mechanism on ballistic performance. several different influential parameters have been discussed to explore the ballistic characteristics and dynamic characteristics of the composites concerning different projectile parameters, target circumstances, and target material qualities. 2. ballistic material structure and mechanism multilayer armor systems (mas) are advanced protective structures utilized for defense in bullets, explosive arms, and shrapnel, which are highly prone to impact threats at higher velocities [28]. these systems comprise various materials in laminates while each material serves its specific purpose of absorbing and dissipating the energy from incoming high-velocity threats. the ballistic impact and damage mechanism of the structure is governed by three important factors such as projectile parameters such as nose geometry and angle, target circumstances, and quality of material in the targets [29]. the primary layer of a mas is a hard armor plate meant to deform incoming projectiles, while the secondary layer, often made of high-strength synthetic fibers like kevlar, absorbs the remaining kinetic energy. additional layers, like soft padding for reducing blunt force trauma and a final layer to stop fragments, may enhance protection. mas can be customized to offer protection against a wide range of threats, ranging from small arms fire to improvised explosive devices (ieds). the selection of specific materials and layering in a mas is tailored to provide robust protection against various threats. the ballistic impact mechanism and energy-absorbing mechanism are illustrated in figs. 3 and 4, respectively. when yarn is impacted, tension-induced waves travel along its length, while transverse waves affect warp and weft, creating conical projections until yarn fails. fig. 5 illustrates a conical projectile with various nose angles. 3. testing and standards of ballistic materials ballistic materials, including body armor and vehicle armor, are specifically engineered to protect from high-velocity incoming threats such as shrapnel and bullets. to ensure the effectiveness of these materials, they undergo rigorous testing and must meet specific standards. one widely recognized testing standard for ballistic materials is the national institute of justice (nij) standard-0101.06, which sets forth the minimum performance requirements for ballistic-resistant body armor. this standard outlines criteria such as projectile velocity, mass, shape, and number of shots, as well as panel size for testing. it also specifies maximum allowable back face deformation, measuring armor plate deformation and energy transfer to the wearer [30, 31]. furthermore, it serves as a vital guideline to ensure that such armor meets the necessary standards for effective protection. 348 b. devarajan, r. lakshminarasimhan, a. murugan, et al. fig. 3 ballistic impact mechanism and result: (a) multiple layers by cylindrical projectile [30] (b) schematic of impact response of layer of armor fabric in primary yarn [31], (b) impacted kevlar panel [31], and (c) wave propagation and transmission in woven fabrics yarns [31] fig. 4 mechanism of ballistic protection. yarn configuration before impact (i), (ii), (iii), (iv) [32] and (b) mechanism of energy absorption of single layer [32] recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 349 fig. 5 a) conical projectile with various nose angles [33], (b) variation of energy absorption with nose angle variation [33] ballistic materials, including body and vehicle armor, are crucial for shielding against bullets and shrapnel threats, requiring rigorous testing and compliance with standards like the nij standard-0101.06. these standards detail projectile specifications, panel dimensions, and maximum backface deformation, establishing minimum performance requirements for ballistic-resistant body armor [34-36]. the following testing specifications for ballistic materials are provided by the standard. ballistic-resistant body armor undergoes rigorous testing, including backface signature (bfs), ballistic limit (bl), and v50 tests, ensuring its effectiveness against threats. armor is designed to withstand environmental conditions, with maximum allowable bfs set for each level to minimize blunt force trauma. v50 values measure armor effectiveness, while quality control tests ensure consistency and adherence to standards. environmental conditioning before and after testing maintains performance, and visual inspections prevent damage. ballistic limit velocities are measurements used to determine the ability of a material to resist penetration by a projectile. there are different varieties of ballistic limit velocities, including v0, v50, and v100. v0 is the maximum velocity that a projectile can be fired at a material without causing any penetration [37]. it is the highest velocity that a material can withstand without failing. v0 is typically used as a baseline for ballistic testing, and materials that can withstand higher velocities without penetration are considered to have superior ballistic resistance. v50 is the velocity at which there is a 50% probability of complete penetration. in other words, if a material is struck by a projectile traveling at v50, there is a 50% chance that the projectile will penetrate the material completely. v50 is commonly used in ballistic testing to determine the effectiveness of armor and other protective materials [38]. v100 is the velocity at which there is a 100% probability of penetration. this means that if a projectile strikes a material at v100, it will always penetrate the material completely. v100 is often used to test the vulnerability of structures, 350 b. devarajan, r. lakshminarasimhan, a. murugan, et al. such as buildings or vehicles, to high-velocity projectiles. therefore, the higher the ballistic limit velocity of a material, the better its ability to resist penetration by projectiles. however, other factors like the shape, size, and angle of impact also play a significant role in determining penetration. ballistic limit velocities should be used in conjunction with other testing methods to fully evaluate a material's resistance. by adhering to these testing requirements, manufacturers can ensure their ballisticresistant body armor provides the necessary level of protection against high-velocity threats. the standardization agreement (stanag) 4569, a nato standard, sets the minimum level of protection required for military vehicles, including threat types, projectile velocity, mass, armor plate size and orientation, maximum allowable deformation and penetration, and protection requirements [39]. ballistic materials may also be tested using other methods, such as simulating specific threats or testing under extreme conditions. by adhering to established testing protocols and meeting minimum performance requirements, manufacturers can ensure their products are effective and reliable in real-world situations. table 1 and 2 describe the standard armors and testing conditions for ballistics and the nomenclature of ballistic panel arrangement, respectively. fig. 6 illustrates the configuration of test range of a ballistic setup. table 1 standard armors in ballistic testing [40] armor type bullet description bullet mass (g) no. of rounds of test diameter of the bullet – nominal (mm) testing velocity of armor (m/s) maximum back face signature (mm) ii a fmj rn 8.0 1 9.0 373 44 ii 40,s&w fmj 11.7 2 11.7 352 44 iiia fmj rn 08.0 1 08.0 398 44 jsp 10.2 2 10.2 436 44 tmj 08.1 1 08.1 448 44 jhp 15.6 2 15.6 436 44 iii fmj 09.6 1 09.6 847 44 iv fmj 10.8 1 10.8 878 44 table 2 ballisitc panel arrangements nomenclature notations layers material thickness (mm) a front layer ceramics 10 b intermediate layer aramid or jute, coir, bamboo, mallow, guaruman, curaua 10 c backing layer aluminum layer 5 d rear layer clay witness 50 recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 351 fig. 6 ballistic test setup and test range configuration [19] 4. factor influencing ballistic phenomenon 4.1. hybridization hybridization is a technique that involves combining two or more different types of materials to create a composite material with properties that are superior to those of its individual components [41, 42]. when it comes to ballistic resistance, hybridization of natural fibers can provide several benefits, including improved strength, reduced weight, and improved environmental sustainability. there are two types of hybridization: intraply and interply. as a laminate level process, inter-ply hybridization involves using different materials and stacking sequences. in contrast, intra-ply hybridization involves using different fiber bundles in parallel within a single laminate. a few inter-ply studies have demonstrated with the aid of hybridization technique. sinan üstün et al [43] explored the ballistic performance of multilayer composite laminate using 172 g/m2 aramid with twill woven fibers and 282 g/m2 e-glass with plain woven fibers and v50 test method via 1.1 g fragment simulating projectile. results portrayed that the hybridization of e-glass fibers with the aramid fibers improved the energy absorption capability of the hybrid composite panel. vasudevan et al [44] investigated the mechanical and viscoelastic properties of interply kevlar-e-glass/epoxy with two different stacking orientations such as 30k/0g/30k) and 45k/0g/45k. they concluded that stacking orientation of 30k/0g/30k displayed highest tensile, flexural and modulus, while 45k/0g/45k possessed higher elongation owing to its long fiber placed diagonally. the effect hybridization on mechanical and impact properties of basalt/uhmwpe hybrid composites using low velocity impact and high velocity impact with different loading environment was investigated. it was observed that hybridization displayed higher ballistic properties compared to plain composites [45]. similarly, hybridizing kevlar fibers with glass/polyester composite had different effects under different projectile geometry such as blunt-ended, hemispherical and conical and found that damage mechanism and penetration resistance has highly sensitive to projectile shape. it was concluded that hybridization increased the penetration resistance [46]. different stacking sequences of aramid/glass hybrid composite has been analyzed. it was found that aramid fiber had higher impact energy than glass fiber, which is attributed to the fact that fibrillation and pullout play a significant role in the dissipation of impact energy [47]. the effect of carbon to 352 b. devarajan, r. lakshminarasimhan, a. murugan, et al. kevlar ratio on shear and flexural behavior and strain of 3d braided carbon/kevlar hybrid composites was evaluated. in 3-d braided hybrid composites, absorbed energy and flexural strength were found to be decreased with increasing proportion of carbon fiber. furthermore, hybrid composites made from ductile 3d braided kevlar fabric and stiff carbon fabric exhibit comparable flexural strength and impact damage tolerance to all kevlar composites [48]. additionally, there is ongoing research being conducted by many researchers using 3d orthogonal woven composites with hybridization in intra-ply laminates. a drop weight test was performed by alikurusan [49] with different projectile shapes using a 12 mm diameter. results indicated that a hemispherical, ogival, or conical shape indenter exhibited large homogenous deformations, whereas the flat shape indenter possessed brittle damage, like stamping, with less deformation around the contact area. it was also described that the three-dimensional propagation of stress waves resulted in better resistance of stab attacks for the triaxial fabric interlaced at 60°. in addition to stacking aligned layers of fabric over each other, orthotropic woven fabric layers can also be stacked angularly to create soft armor panels for better dissipation energy [50]. the residual velocity of ballistic performance of kevlar/glass hybrid composite was evaluated using finite element analysis (unit cell modelling) and experimental approach. the ballistic penetration was calculated using a userdefined subroutine vuamt and abaqus/explicit, a commercially available fea code. fea results and experimental results were in reasonable agreement and portrayed the better capability of unit-cell model for the prediction of ballistic properties [51]. the energy absorption of aramid/glass (3d) woven hybrid composites was investigated using modified split hopkinson pressure bar. they examined the strain rate sensitivity of energy absorption by obtaining load–displacement curves of the composites in transverse impact. it was observed that energy absorption of composite increased with increases the impact velocity and exhibited same failure mode in warp and weft directions [52]. the transverse impact behavior of aramid/glass was explored using fea (unit cell modelling) and experimental approach (modified split hopkinson pressure bar). to determine the composite's deformation and damage when impacted by a hemisphere-ended steel rod, they developed a unit-cell model based on the composite microstructure [68]. studies were also carried out by introducing nano al2o3 on kevlar/snake grass hybrid composite by different weight fractions. results revealed that the addition of 2 wt. % alumina nanoparticles enhanced the tensile, flexural impact and interlaminar shear strength by 6.42, 53.11, 11.08 and 6.11% respectively, while 8 wt. % nanofiller demonstrated a 68.80% increase in energy absorption [53]. experimental studies were conducted to explore the energy absorption of hybrid and non-hybrid composites of aramid/veneer with same thickness using 9mm luger ammunition as per nij0108.01. results from their study indicated that aramid/densified veneer ration of 2:1 by volume increased the energy absorption by 78.64% (354.70j) than non-hybrid panel [54]. kenaf fibers were used in the development of multilayer kenaf/aramid hybrid composite to analyze the energy absorption and failure behavior using ballistic limit (v50). it was found that incorporation of kenaf fiber resulted in enhanced in thickness and areal density, thereby increasing the energy absorption from 14.46% to 41.30% via different failure modes including fiber shear, fiber fracture and delamination. as a result of this phenomenon, specific energy absorption was lower for hybrid composites [55]. the mechanical and ballistic properties of polyaramid/vinyl ester composite were evaluated using split hopkinson pressure and ballistic limit (v50) with different fiber content (different layers 4,5,6). results indicated that polyaramid hybrid composite consist of 6 layers enhanced recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 353 energy absorption by 35% compared to other configurations. this is because of increased the layer, enhanced the strength and strain rate [56]. the kenaf/aramid polyvinyl butyral laminated hybrid composite was subjected to low-velocity impact response using v50 ballistic limit. results indicated that the same hybrid volume fraction and thickness of woven kenaf with aramid 29 layers alternated with aramid 29 layers had a lower ballistic limit velocity than woven kenaf together with aramid 29 layers separately. it was concluded that proper configuration kenaf fiber reduced fabrication cost and increased the ballistic performance [57]. the ballistic behavior of kevlar/kenaf hybrid composite was investigated using 1.1 g fragment simulating projectile and v50 ballistic limit. results indicated that a kevlar/kenaf ratio of 14:2 displayed better ballistic performance. furthermore, thickness and areal density were increased with increased ballistic properties [58]. in the energy absorption evaluation experiments conducted using glass/kevlar hybrid composites with the aid of two span support ratios in a quasi-static penetration examination, carbon/kevlar/carbon laminate arrangement performed better in terms of perforation and ballistic resistance under both span ratios of 2 and 5 [59]. it can be observed from earlier studies that, to create a hybrid material that is suitable for ballistic-resistant applications, natural fibers can be combined with other materials such as aramid fibers, ceramic materials, or metals. the specific combination of materials will depend on the desired level of protection and the specific requirements of the application. by combining different materials in this way, it is possible to create a composite material that is both strong and lightweight, while also being environmentally sustainable. 4.2. stacking sequence ballistic material design involves using fibers arranged in a particular weave architecture to create structures that resist high-velocity projectile penetration. several studies have been conducted to evaluate the effectiveness of fiber architecture on the failure mechanisms and energy absorption capabilities of the fabrics [60]. stacking sequences of ballistic composite panels are influenced by weave patterns, ply orientations, optimal reinforcement, type of material of constructed laminates influence their structural response and failure mechanisms [61]. recent developments in new materials and innovative weaves have allowed for the creation of textile structural laminates that are lighter in weight but offer similar mechanical strength as traditional laminates [62]. several publications have explored the ballistic impact behaviour of composite laminates reinforced with various fabric architectures [63]. in unidirectional plies all the fibers within a single layer are aligned in the same direction. this allows for maximum strength and stiffness in a particular direction, which can help to stop projectiles from penetrating the material [64]. unidirectional plies are often used in areas where high levels of protection are required, such as in the front panel of a bulletproof vest. in woven plies fibers are woven together in a pattern that distributes load across multiple directions [65]. this can provide greater flexibility and impact resistance, making it ideal for applications such as vehicle armor or blast-resistant structures. fibers are randomly arranged within each layer in randomized plies. this can provide a more isotropic material, meaning it has similar properties in all directions, which can be beneficial for some applications [66]. various experimenters evaluated natural fiber woven fabrics reinforced polymer composites for ballistic applications using numerical, experimental, and analytical techniques. 354 b. devarajan, r. lakshminarasimhan, a. murugan, et al. the energy absorption capability of interplay stacks with carbon, kevlar, and s-glass fibers hybridized with e-glass fibers was determined using low-velocity impact tests. the outer layer kevlar fibers enhanced energy absorption by 8%, while the s-glass and carbon as outer layers decreased it by 14% [67]. the effects of different ply orientation, layering sequence, and ply combination of aramid-based ballistic composite were investigated. results demonstrated that an angled fabric panel consistently absorbed more energy than an aligned panel, and the energy absorption of angled fabric panels was 15%-20% higher based on the number of plies [68]. researchers investigated the ballistic response of uhmwpebased 3-ply panels impacted by projectiles in both aligned and angular orientations and demonstrated that when compared to aligned fabric panels, angled fabric panels consistently absorbed more energy [69]. they found that angled fabric panels absorbed more energy than aligned panels. experimental validation of woven aramid multilayer composites using abaqus or ls-dyna revealed that 3d angle interlocks are more resistant than plain woven fabrics with 2d angle interlocks. warp and weft directional fibers absorb most energy during ballistic impacts in 3d woven hybrid interlocking composites [70, 71]. similarly, a low-velocity impact test was conducted to compare woven s-glass fabric with twill woven carbon fabric by sayer et al [72]. according to them, material bending stiffness significantly influences rear-side impact behaviour. besides, altering the stacking sequence of fiber laminates altered load paths, creating a hybrid material with better stress distribution and structural behaviour. an analytical model was developed considering inplane, isotropic, and elastic properties of multilayer fibrous material against 5.56 mm fsm hollow projectile and ballistic limit [73]. several experimental tests were conducted to observe the effect of different fabric and layer-wise response via projectile geometry, impact velocity and boundary condition [74, 75]. kevlar fibers were arranged using different fabric architecture and their ballistic performance was explored. results indicated that plain weave fabric had higher energy absorption capability than basket and twill woven fabric composites during low-velocity impact tests [76]. the effectiveness of number of laminates of the material and weight of each layer for begetting an enhanced ballistic performance to stop the against 9 mm parabellium projectile with different grades such as 160gsm, 200gsm and 400gsm for 2d kevlar and ballistic gel plain weave composite was investigated. results indicated that 200gsm kevlar grade was increased the effectiveness of ballistic performance among other grades [77]. analytical modeling was conducted on 2d plain weave fabric with multiple layers composite subjected to ballistic impact. the study demonstrated direct compression beneath the projectile, revealing various failure modes including stretching, shear plugging, and tensile failure of primary yarn layers. additionally, secondary yarn layers experienced conical deformation on the back face. friction between the projectile and fabric, along with in-plane yarn friction, contributed to energy absorption during the impact [78]. lowvelocity impact behavior and mechanical behaviour of 2d and 3d kevlar/polypropylene composite were tested. it was discovered that the study found that 3d weaves exhibited superior energy absorption due to their interlocking arrangement, emphasizing the influence of z-direction yarns. various damage mechanisms, including fiber cracking, breakage, shear plugging, and delamination, contributed to this enhanced performance [79, 80]. the ballistic behavior of kevlar/polyester composites with orthogonal woven arrangement was evaluated using finite element analysis at microstructure level. results indicated that tensile and shear failure mode is the main failure mode at the bottom surface, while compression and shear at top surface during the impact caused by conical and cylindrical projectiles [81]. detailed recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 355 investigations of ballistic response of glass/polyester composite with various reinforcement types, ranging from 80m/s to 160m/s. results showed that cross-ply unidirectional and plain weave composites with thicknesses of 3mm and 6mm provided the best ballistic resistance. however, 3mm composites failed due to tension and shear failure, while 6mm composites failed due to severe delamination [82]. fig. 7 presents different weave arrangements used to manufacture composites and subject them to ballistic applications. fig. 7 weave arrangements (a) plain; (b) basket; (c) triaxial and 3d structures (d) braid; (e) orthogonal woven – 3d; (f) triaxial woven 3d [32] different failure modes of spectra polyethylene using both angle ply unidirectional and woven fabric was investigated. they reported that angle ply and woven fabric reinforced possessed subsequent delamination, combined modes of shear and tensile failure of fibers. furthermore, it was noticed that angle unidirectional composite superior ballistic limit [83]. the perforation behavior of specta, kevlar and twaron with four kind of different architecture such as 2d, 3d woven fabric, 3d braided and needle-punched nonwoven fabric subjected impact velocity from 200 to 1100 m/s was investigated. they demonstrated that 3d braided composite exhibited high perforation, and penetration resistance for increasing the structural integrity [84]. fig. 8 shows hybrid fiber and its corresponding energy absorption ability. different failure mechanism of various glass/polyester composite with different fiber arrangements, including woven and stitched arrangements, was investigated and subjected to impact velocity of up to 571 m/s and found that all kind of fabric pattern exhibited similar failure modes [85]. the ballistic behavior of basket 2×2 aramid fabric composite was evaluated by oblique impact. they found that basket weave composites cause less ricochet than plain weave composites [86]. fig. 9 presents energy absorption by 5 layered panels of kevlar 802 f fabrics [87]. therefore, meticulous design of the various stacking sequence including different weave architecture, ply orientation in ballistic material design becomes essential since the structure of the fabric can significantly affect desired level of protection its performance while maintaining other important properties such as flexibility, weight, and durability [88]. 356 b. devarajan, r. lakshminarasimhan, a. murugan, et al. fig. 8 hybrid fiber configurations a) interply b) intraply and corresponding energy absorption ability fig. 9 variation of energy absorption of kevlar 802f fabric panels with various arrangements [87] 4.3. surface treatment natural fibers like flax, hemp, jute, and sisal are being increasingly used in the field of ballistic applications. however, these natural fibers have several limitations such as poor mechanical properties, low resistance to moisture, and poor adhesion with matrix materials. recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 357 to overcome these limitations, various surface treatments can be applied to natural fibers to improve their mechanical properties and adhesion with matrix materials, which are essential for achieving the desired ballistic performance [88]. surface modification is a process that involves changing the surface of a material to improve its characteristics, such as adhesion and biocompatibility. there are various ways to modify the surface of cellulose fibers, including chemical treatment, enzymatic treatment, corona or plasma treatment, and addition of coupling agents. these techniques primarily target the amorphous region of cellulose, which contains hydroxyl groups that make the fiber polar and difficult to bond with the polymer matrix [89]. fig. 10 shows commonly used lightweight natural fiber in ballistic applications. some of the commonly used surface treatments for natural fibers in ballistic applications are alkali treatment, silane treatment, acetylation treatment, plasm treatment, grafting, and gamma irradiation treatment. in the alkali treatment removes impurities and hemicelluloses from fiber surfaces, increasing surface area and energy. silane treatment uses a silane coupling agent to improve interfacial adhesion between fiber and matrix, enhancing composite material mechanical properties [90]. acetylation treatment reduces fiber hydrophilicity, improves moisture resistance, and increases surface energy, enhancing fiber adhesion. this process reduces the hydrophilicity of the fiber and improves its resistance to moisture. it also increases the surface energy of the fiber, which enhances the adhesion between the fiber and the matrix material [91]. fig. 10 commonly used natural fiber in ballistic applications (a) sugar palm fiber, (b) pineapple fiber, (c) hemp fiber, (d) corn fiber, (e) jute fiber, (f) flax fiber, (g) coir fiber, (h) kenaf fiber, (i) empty fruit bunch fiber [91] plasma treatment involves the use of plasma to modify the surface of natural fibers. the plasma treatment can be used to introduce functional groups on the fiber surface, which improves the wettability of the fiber and enhances the adhesion between the fiber and the 358 b. devarajan, r. lakshminarasimhan, a. murugan, et al. matrix material. grafting treatment involves the grafting of monomers onto the fiber surface. the monomers are selected based on their ability to react with the matrix material and improve the adhesion between the fiber and the matrix material [92]. recently, gamma radiation is a method of modifying materials that involves exposing them to high-energy gamma rays, typically emitted by a cobalt-60 source. this process can cause crosslinking, chain scissions, decomposition, and unsaturation within the polymeric chains of the material, which can result in hardening, toughening, and strengthening of the material [93]. gamma-ray irradiation is commonly used in industrial processes to improve the properties of polymeric materials. this can include improving the durability and resistance to wear and tear of fibers, as well as enhancing their overall strength and toughness. gamma-ray irradiation has been widely used in research on lignocellulosic substrates, with a preferred dose range of 1 to 50 kgy. however, it's important to note that irradiation at higher doses can lead to chain scissions and degradation of biopolymers, which can result in a significant loss of the original properties of the natural fibers. therefore, researchers typically try to limit the irradiation dose to not more than 30 kgy when working with lignocellulosic substrates. this ensures that the desired modifications can be achieved without causing excessive material degradation [94]. moreover, careful selection of the appropriate irradiation dose is critical to achieving the desired properties and avoiding any negative effects on the material. the effect of gamma radiation on the trauma penetration depth and ballistic properties of kenaf hybrid/x-ray composite materials has been investigated to improve the interfacial bonding adhesion between the fibers in the laminate [95]. the study found that the configuration sample using the combination of surfacetreated x-ray and naoh solution produced the optimum formulation with hybrid composite properties for bulletproof vest applications. this formulation exhibited significant improvements in tensile, impact, and flexural strength compared to untreated samples. few researchers found that gamma-irradiated 25 kgy uhmwpe composites showed better ballistic resistance compared to 250 kgy composites. the ballistic performance of kevlar fiber-reinforced composites was also enhanced by the addition of sic nanoparticles [96]. the impact characteristics of alumina/uhmwpe composite irradiated with gamma radiations using different doses were investigated. they found that 50 kgy irradiated with 80 wt.% alumina displayed better ballistic resistance compared to other irradiated composites. gamma irradiation enhances the interface properties of ceramic and polymer interfaces [97]. two different chemical pretreatments, such as silane and potassium permanganate-di cumyl peroxide solution on different failure modes of sisal/epoxy composite were investigated. results indicated that better energy transfer was obtained by performing silane treatment [98]. few experimental results pointed out that the energy absorption of bamboo fiber is governed by coarse fiber bundles, fiber pullout, high volume fraction, and crack deflection [99]. different failure modes with different contents (1 wt. % and 3 wt. %) of alkalitreated micro cork particles with e-glass were explored. results demonstrated that 3wt% of micro cork particles had significant enhancement in energy mechanisms such as crack initiation and propagation improved by 130% and 35% respectively [100, 101]. table 3 presents the chemical composition of different natural fibers. recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 359 table 3 chemical composition of natural fibers [102] natural fiber cellulose (%) hemi-cellulose (%) lignin (%) pectin (%) waxes (%) flax 70.5 16.5 2.5 0.9 hemp 81 20 4 0.9 0.8 henequen 60 28 8 0.5 coir 46 0.3 45 4 bamboo 34.5 20.5 26 areca 53.2 35-64.8 13-24.8 abaca 62.5 21 12 0.8 3.0 alfa 45.4 38.5 14.9 2.0 bagasse 37 21 22 10 banana 62.4 12.5 7.5 4 cotton 89 4 0.75 6 0.6 coir 34.5 20.5 26 curaua 73.6 5 7.5 4 jute 67 16 9 0.2 0.4 kenaf 53.5 21 17 2 kapok 13.16 isora 74 23 sisal 60 11.5 8 1.2 pineapple 80.5 17.5 8.3 4 ramie 72 14 0.8 1.95 pissava 28.6 25.8 45 4.5. aerial density aerial density, also known as areal density or mass per unit area, is an important parameter in the design of ballistic fabrics for use in body armor and other impact-resistant applications [103]. in general, increasing the aerial density of a ballistic fabric will improve its ballistic performance. the higher the aerial density, the more surface area in the path of an incoming projectile, providing greater resistance for increasing the ballistic performance. this can be achieved with the aid of increasing the ply thickness or arranging the more no of the ply layer in a striking direction [104]. however, too high thickness and additional layers lead to increasing weight, which can negatively impact the wearer's comfort and mobility for body armor applications. furthermore, it was observed that simply increasing aerial density is not the only factor that influences the ballistic performance of a fabric. the composition and arrangement of fibers or other materials in the fabric also plays a crucial role in the development of high-performance ballistic body armor. for example, a fabric that uses high-strength fibers, such as aramid or uhmwpe, will exhibit superior ballistic behaviour when compared with fabric made from lower-strength materials [105]. furthermore, the specific arrangement of fibers within the fabric can affect how the fabric responds to a ballistic impact. for example, fabrics that use woven or knitted constructions may have different ballistic performance characteristics than those that use non-woven or laminated constructions [106]. aerial density is an important parameter in the design of ballistic fabrics, and increasing aerial density can generally improve ballistic performance [107]. however, the benefits of increasing aerial density must be balanced against the potential drawbacks, such as increased weight and reduced mobility. additionally, other factors, such as the composition and arrangement of fibers, also play important roles in determining the ballistic performance of a fabric [108]. 360 b. devarajan, r. lakshminarasimhan, a. murugan, et al. investigations were made to assess the influence of areal density on the impact energy absorption of aramid fabric composites using gas guns and drop weight tests. the penetration and perforation limits were determined using energy profile diagrams. it was observed that a thin plate absorbed more energy than a thick one beneath the limit of penetration. in contrast, the trend reverses above the perforation limit [109]. the influence of continuous and discontinuous plain weave fabric reinforcement in the composites was evaluated using ballistics tests. it was found that when compared with the randomly oriented fabric composites, the continuous fabric-oriented composites exhibited 9.5% more ballistic resistance and 20% more energy absorption capability [110]. the influence of areal density on the ballistic limit and failure mechanism of uhmwpe was explored and found that the natural fibers enhanced the energy absorption capability [111, 112]. 4.6. matrix and interface system an important function of a matrix in ballistic composites is to distribute stresses between fibers. furthermore, the matrix provides some level of protection against adverse effects of the environment as well as protection against mechanical abrasion of the fiber surface [113]. the main role of matrix in a composite structure is a load-bearing structure. generally, thermoset and thermoplastic are two types of polymers with different properties and characteristics. in the context of ballistic mechanisms, matrices and interface systems can be used to model and optimize the behavior of these materials. matrices can be used to represent the stress and strain relationships in both thermosets and thermoplastic materials. specifically, a 3x3 matrix can be used to represent the state of stress at a given point in the material, while a 6x6 matrix can be used to represent the full stress-strain relationship. nevertheless, there is a significant effect to increase the interfacial adhesion between fiber and matrix that can influence the high performance of polymer-reinforced composite. this can be achieved with the aid of fullerenes which are zero-dimensional, carbon nanotubes which are one dimensional, graphene nanoplatelets (gnps) which are two-dimensional, nano graphite and nanofibers which are three dimensional are often employed to develop lightweight advanced bulletproof materials [114-117]. interface systems can be used to optimize the performance of both thermosets and thermoplastic materials. thermoset and thermoplastic materials can both be used in ballistic applications, although they have different properties and are used for different purposes. thermoset materials are typically used for applications that require high strength, stiffness, and resistance to heat and chemicals [118]. they are also often used for applications where dimensional stability is important. epoxy is a thermoset resin that is widely used in composites for ballistic protection. it is known for its high strength, stiffness, and resistance to impact and penetration. phenolic resins are commonly used in the production of ballistic composites for use in armor and other protective applications which possess high resistance towards chemicals and heat along with elevated strength and stiffness. polyurethane resins are used in the production of ballistic materials such as helmets and body armor. they are known for their high strength, flexibility, and impact resistance. thermoplastic materials, on the other hand, are typically used for applications that require high-impact resistance, flexibility, and ease of processing [119]. polycarbonate is a tough, transparent thermoplastic that is often used in the production of ballistic lenses and visors for helmets and face shields. polyethylene is a lightweight thermoplastic that is often used in the production of body armor and protective shields. it is known for its high recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 361 impact resistance and flexibility. acrylonitrile-butadiene-styrene (abs) is a versatile thermoplastic that is commonly used in the production of protective equipment such as helmets and shields. it is known for its toughness, impact resistance, and ease of processing [120]. in general, thermosets and thermoplastic materials are used in ballistic applications, but they are chosen for different properties and characteristics based on the specific requirements of the application. han et al [121] investigated the effect of matrix on ballistic performance using the quasi-static test, ballistic test and numerical tests. it was observed that surplus matrix (at least 20 percent) increased composite stiffness, inhibiting delamination and bending deformations like thermosetting matrix composites, and reducing energy absorption. furthermore, the main ballistic damage mechanisms of af/pcc changed from compression-shear failure to delamination and tensile failure as the matrix content decreased from 30% to 10%. fig. 11 shows the physical, mechanical, and thermal properties of common polymer matrices. fig. 11 characteristics of various polymers [122] 4.7. target thickness the thickness of a composite also has a significant effect, as thick composites have a better performance against external loads. the thickness of a target plate can significantly influence its ballistic performance, which refers to how well the plate is able to resist and stop incoming projectiles. generally, thicker plates are more effective at stopping projectiles than thinner plates. the thickness of the target plate affects its ability to absorb the kinetic energy of an incoming projectile [123]. as the projectile hits the plate, it transfers its energy to the plate, causing it to deform and potentially break. the thicker the plate, the more 362 b. devarajan, r. lakshminarasimhan, a. murugan, et al. material there is to absorb the energy of the projectile, and the less likely it is to penetrate through the plate. however, there is a limit to how thick a target plate should be. if the plate is too thick, it may become too heavy or difficult to handle, and may not be as effective at stopping certain types of projectiles. there is also a tradeoff between thickness and other factors, such as cost, weight, and mobility. the specific thickness of the target plate needed for effective ballistic performance depends on a variety of factors, including the size and velocity of the incoming projectile, the material and composition of the plate, and the specific application for which the plate is being used. a unique pattern of deformation or damage along the direction of through-the thickness was observed during the impact process which portrays that the thickness of the composite material and the induced stress are directly proportional along the thickness direction. hence, it could be concluded that the thickness of the composite plays a vital role as it has a very high influence over the failure mechanism, energy absorption and total damage area of the composites [124]. few researchers investigated the relationship between thickness and impact behaviour of woven glass fiber composites using a series of impact tests, which subjected 1 to 5 mm thick laminates to impacts from a hemispherical-nosed projectile propelled by a one-stage gas gun [125-128]. the findings indicate that the ballistic limit velocity of laminates with thickness ranging from 1 to 5 mm increases in a linear fashion. specifically, the ballistic limit velocity of the 5 mm thick laminate was discovered to be 2.8 times higher compared to that of the 1 mm thick laminate. furthermore, as the projectiles impact velocity increases, the energy absorption of the laminated plate initially decreases to approximately 10% before stabilizing [129, 130]. ballistic response of kevlar/epoxy plate with thickness ranging from 4.5 mm to 14.5 mm was evaluated using experimental and numerical. results indicated that plate thickness significantly influenced the failure modes and energy absorption capacity [131]. it was stated by some researchers that thicker plates have a greater effect to increase the penetration capacity using low-velocity impact. therefore, thin ballistic composite plates have a higher possibility of failure risk inferred from thicker composite plate [132]. a few experimenters assessed the effect of thickness on impact characteristics of carbon woven epoxy composite using low-velocity impact. researchers found that laminate thickness significantly influenced peak contact force, threshold energy, and load for primary crack initiation [133]. the study conducted by de morais et al [134] assessed the effect of laminate thickness on the low-energy impact characteristics of carbon/glass/aramid hybrid composites. the thickness of the composite laminate was found to be the most important factor for evaluating the ballistic performance of the hybrid composites by the researchers. furthermore, glass fiber composites exhibited maximum ballistic protection due to the broader area coverage provided by the increased thickness of the laminate. additionally, the isotropic properties of the glass fibers enhanced the ballistic resistance when compared with the anisotropic nature of the aramid and carbon fibers. the authors concluded that thicker laminates could withstand higher impact energies before reaching the failure threshold, resulting in primary failure. the effect of laminate thickness to prevent perforation was investigated using a 7.62 mm ammunition standard test and the minimum thickness required was also estimated. it was observed that increasing thickness led to exhibit high energy absorption to stop the 7.62mm bullet [135]. a few researchers examined the uhmwpe composite with different thickness ranging from 13 mm and 20 mm using fragment simulating projectiles. the thickness of the laminate significantly influenced the contact force at its peak, energy threshold, and damage initiation recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 363 load. the panels demonstrated two stages of penetration failure modes with increased thickness: shear plugging during initial penetration, followed by formation of a transition plane and bulging of the rear panel [136]. the effect of velocity of impact, diameter of the projectile, thickness and mass of the target during the contact and ballistic velocity was investigated. it was discovered that the shear plugging is a significant failure mode in energy mechanism [137, 138]. a major factor affecting perforation is laminate thickness, which is directly relationship with laminate density (composite mass per square meter). several researchers have studied the effects of these parameters on both projectile perforation velocity and perforation threshold energy of laminates with different fiber compositions and resin matrices [139, 140]. both linear and non-linear relationships have been observed to attain the minimum perforation velocity or minimum perforation energy and the thickness in composite plates [141, 142]. fig. 12 shows the effect of different projectile geometry on energy absorption. fig. 12 (a) back face damage of (a, e) fragment simulating, (b, f) hemispherical, (c, g) conical, (d, h) flat projectiles for 6.5mm thick panels (c) energy vs thickness variation for 3.2mm thick panels [143] 5. ballistic impact mechanisms and approaches ballistic impact is a highly complex and dynamic phenomenon in which the ballistic projectile interacts with the target. understanding the mechanics of ballistic impact is crucial for a wide range of applications, including the design of protective materials, the development of new weapons and ammunition, and the investigation of various forensic and accident scenarios. the study of ballistic impact mechanisms involves a multidisciplinary approach, encompassing the fields of mechanics, materials science, physics, and engineering. researchers have developed a variety of approaches to investigate the behavior of materials under ballistic impact, ranging from analytical and numerical models to experimental tests and simulations. this field has seen significant progress over the past few decades, with the development of advanced computational tools and testing methods that have enabled researchers to get a deep insight in complex physics and material behavior that occurs during ballistic impact. in this context, this topic remains an active area of research, with ongoing efforts to refine and improve our understanding of the underlying mechanisms and develop new approaches to enhance the effectiveness and safety of various materials and devices. 364 b. devarajan, r. lakshminarasimhan, a. murugan, et al. 5.1. experimental approach energy absorption mechanisms of fabric composites are best explored with the aid of direct experimental methods. research has been conducted on ballistic impact mechanisms in numerous experimental studies [144, 145]. experimental work will examine back-face signatures, residual velocities, damaged targets and caught bullets [34]. optical microscopes, scanning electron microscopes, and high-speed cameras were used for examinations, further analysis, and interpretations [146]. the absorption and transfer of energy beyond the target was evaluated by the experimental approach. this transferred energy absorption of energy depends on various cone angles, types of projectiles [127, 147]. in addition, the energy performance of the target material was influenced by woven arrangement, geometry of the projectile, impact angle and velocity, density and other fiber properties and boundary conditions [148]. generally, the energy dissipation during the ballistic impact would be due to the energy absorbed by the secondary yarns, the tensile failure energy dissipation of the primary yarns and the kinetic energy of the moving projectile cones [149]. meanwhile, there are several other energy-absorbing mechanisms such as shear plugging, interface friction, matrix cracks and delamination [150]. trauma resistance refers to the target's ability to prevent the occurrence of inner wounds. as a result of the depth and diameter of the trauma, it is possible to evaluate the target energy absorption amount and communicated to the target's back, as well as the overall ballistic impact of the armor solution [95]. moreover, in addition to evaluating the performance effectiveness, the depth of the trauma must be less than 44 mm according to the national institute of justice standards. the projectile will cause fatal damage to the wearer's essential organs at higher values of trauma depth [151, 152]. in general, the high-speed camera has been used by researchers to record and assess the backface signature values on typical back clays in different ways to support the packaging at the target’s back. an experimental ballistic testing procedure that could determine a target armor's v50 ballistic limit [153]. the ballistic limit velocity is the minimum velocity of the projectile at which full penetration of the target is possible and the value of maximum velocity at which the penetration is impossible [154]. the residual velocity of the projectile is necessary to evaluate the composite's energy absorption. numerous researchers have assessed the ballistic limit velocity, energy absorption and structural absorption and other parameters such as wave propagation including local and global damage caused by compression and cone formation. these parameters influence the deformation, bowing stretching, projectile-target friction value, delamination and matrix cracks initiation. ballistic performance of aramid woven composites with different thermoplastic matrix was investigated. it was noticed that composites made from polypropylene composite matrix exhibited better ballistic behaviour when compared with unreinforced composites. different energy absorption mechanisms were associated including secondary yarns’ huge deformation, delamination/debonding of matrix and the reinforcement and cracks in the matrix [155]. according to the study made by karahan et al [156], fabric plies orientation has no significant effect on ballistic behavior, but energy absorption, ballistic limit, and thickness are bi-linearly related. experimental tests were performed to assess the influence of composite ply blocking and spacing on uhmwpe based composite ballistic resistance. it was also found that the specific ballistic limits of thin laminates and the multiple thin laminate stacks of laminated panels were higher than a thick laminate panel made of many thin laminates bonded together [157]. fig. 13 shows the effect of impact velocity on energy absorption. recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 365 fig. 13 (a) various projectile shapes (b) variation of energy absorption of the fabrics with different impact velocities [158] 5.2. analytical approach the analytical approach is the study of the motion of projectiles using mathematical and physical principles. it can be leveraged to forecast the trajectory of a projectile, the penetration depth of a projectile into a target, and the energy transfer between the projectile and the target. a method that can describe the physical and mechanical phenomena during an impact [30]. the ballistic impact of the textile materials was assessed under transverse impact loading conditions by developing an analytical model for the stress-strain relationship of the textile yarn-based composites [159]. the conservation of energy during the impact were assessed through energy method by the aid of various analytical models derived out of it. in this method, the imparted kinetic energy is converted into various forms of energy during impact, including kinetic and strain energy in the deformed area of impact, strain energy in the object being hit, and heat generated by friction [160]. thin woven fabricbased composites were analyzed for various energy absorbing and damage mechanisms using some analytical models which took the shear plugging effect into account [161]. by incorporating the kinetic energy balance between the target and the projectile and the elastic wave theory, an analytical model was developed to assess the ballistic impact. some of the outcomes of the ballistic impact on the target include the deformation in secondary yarn, tensile force in the primary yarn, delamination of the composites, friction, matrix cracking, cone formation at the target’s back face and shear plugging [162, 163]. few researchers concluded that composite mass acceleration, tensile strain, matrix cracking and delamination contribute little to energy absorption [164]. likewise, numerous studies have been studied on the various types of energy dissipation mechanisms of the woven uhmwpe-based composites during the ballistic impact phenomenon [165, 166]. some experiments were carried out to determine the ballistic impact behaviour of plainwoven fabrics reinforced polymer composites, which takes the energy dissipation amount during laminate compression and plate bending into account [167]. it is intended to study the ballistic impact process considering various factors, including materials, mechanism and target system, etc., that affected the process of ballistic impact directly. numerous experimental works have typically employed three strategies: geometrical laminate deformation models, momentum equation models and energy conservation law, which have a direct contribution to the ballistic phenomenon to assess the effectiveness of ballistic performance. in recent years, various researchers have made significant contributions to the development 366 b. devarajan, r. lakshminarasimhan, a. murugan, et al. of material-based analytical modeling formulations to predict such behavior and mechanisms across different length scales, ranging from micro to macro levels [168]. additionally, wave propagation (wave theory) and energy balance based analytical simulations can predict the ballistic behaviour of fiber-reinforced composites and textile materials. overall, the development of material-based analytical modeling formulations has provided researchers with a powerful tool for materials behavior prediction and structures under ballistic impact. these models have helped in the development of new materials and structures that can withstand ballistic impact and have contributed to the advancement of various fields such as military, automotive, and aerospace engineering. fig. 14 shows strain gradient of the contact region of a projectile in a single fabric layer primary yarn present in mas systems. fig. 14 schematic of analytical model of primary yarn in single fabric layer [32] 5.3. numerical approach extensive utilization of finite element (fe) and finite difference (fd) methods occurred as a part of numerical modeling techniques. simulation of projectile fabrics and ballistic impact behaviour of the materials were carried out using various software including ansys, abaqus, ls-dyna and dyna-3d. these numerical approaches aim to develop a more robust ballistic impact model for the composites and aid in predicting the target composite material’s performance. at the micro-level, the behavior of materials under ballistic impact is governed by factors such as material microstructure, phase transformations, and deformation mechanisms. researchers have used techniques such as finite element analysis and molecular dynamics simulations to develop material-based models that can predict the response of materials at this level. these models help in understanding the mechanisms that contribute to the deformation and failure of materials at the micro-level. at the meso-level, researchers have considered the behavior of individual layers of composite materials and their interactions under ballistic impact [169]. similarly, considering the behavior of the entire structure under ballistic impact, including the effect of boundary conditions and the interaction between different components of the structure at the macro-level, researchers have focused on the behavior of composite materials under ballistic impact. the development of material-based analytical models at this level has been motivated by the need to predict the response of composite materials used in armor applications. modeling yarn-fabric interactions and projectile-yarn interactions renders the complex interactions at the fundamental architecture of the yarn, but they tend to be recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 367 computationally expensive. because yarn-level interactions are neglected in homogenization methods, they cannot accurately describe individual yarn failures or deformations. to dodge this complexity, the ballistic impact behaviour of fabric-based composites was assessed using multi-scale modeling techniques. numerous researchers have studied and compared the experimental with numerical results to predict the different failure modes of ballistic performance of advanced composite materials [170, 171]. an fea model was introduced to investigate the impacted region using a 3d solid modeling element to predict the energy mechanism and failure mode precisely. on the other hand, the impact zones which are far away from the target were modeled using 3d shell elements for more efficiency [172, 173]. a sub-yarn numerical model was proposed to observe the various deformation and energy dissipation mechanism of kevlar fiber using conventional finite element formulation [174]. an fe model to evaluate the ballistic behaviour of the single kevlar yarn fabric under the action of impact loading in transverse direction was developed by various researchers [175]. to understand the ballistic impact behaviour of the fabrics and the composite laminates, mat162 which is a progressive failure model developed using ls-dyna was utilized [176]. finite element models were developed to predict impact characteristics and failure evolution of 3d braided and warp-knitted composites, 3d orthogonal and angle-interlock subjected to ballistic impact [177, 178]. some experiments were carried out using fabricbased composites with fabrics aligned in various orientations such as unidirectional, plainwoven and multiaxial arrangements to evaluate the blunt trauma resistance by considering both numerical and experimental results. results showed that the multiaxial fibers exhibited better bunt fracture resistance while all the fabric arrangements had shown similar failure modes [179]. hybrid epoxy composites reinforced with carbon and aramid fibers were subjected to flexural property and ballistic impact tests and their behaviour was also analyzed through numerical simulations. results revealed that the addition of carbon fibers along with the aramid fibers enhanced the flexural behaviour, projectile resisting force and shear resistance of the hybrid composite laminates [180]. the models developed at this level consider the behavior of developed material-based models to predict the response of structures under ballistic impact. the different failure criteria such as matrix damage, fiber matrix damage in view of harsh environments such as static and dynamic loading, various temperature, and pressure through fem. in addition, the micromechanical modeling composites numerically can be solved by representative volume element (rve) to deal with microstructural features such as shape, size, and fiber, and matrix interface, distribution of fibers. the response of micromechanical modeling such as elastic constant displacements, strains, and stress will be established at different distant fabric scales via the equilibrium of forces and potential energy method. besides that, the galerkin method, rayleigh-ritz method, and reissner variational method are also used. in addition, multiscale modeling was governed by the rule of mixtures, voigt and reuss approximation and moritanaja method. 5.4. ballistic performance of natural fiber composites experiments pertaining to the indentation resistance of natural fiber composites with fiber reinforcements including bamboo, sisal, jute, curaua, bagasse and amazonian guaruman were considered equivalent to traditionally used kevlar fiber mat owing to their cost effectiveness and were suggested as potential alternatives to the kevlar fibers in ballistic systems. the lower 368 b. devarajan, r. lakshminarasimhan, a. murugan, et al. density of some of the natural fibers used in the multi-layered armor systems, such as bamboo fibers, guaruman, and curaua fibers, is an advantage because it means that the armor can be lighter and more comfortable to wear for the user, without compromising on protection. the lower density also means that the armor can potentially be thinner while still providing the same level of protection as thicker kevlar layers, which can also increase mobility and comfort. therefore, the use of natural fiber-reinforced polymer composites in mas has the potential to offer a cost-effective and lighter-weight alternative to traditional ballistic materials such as kevlar, while providing similar levels of protection and minimizing body trauma to the user. in terms of cost, natural fibers tend to be less expensive than kevlar. for example, the cost of bagasse fiber composite, which is made from sugarcane waste and has been used in multilayered armor systems, can be as low as $1.35/kg [3]. mallow fibers, which have also been used in armor systems, can be as low as $0.28/kg [4]. jute fibers, which have been used in non-woven mats for armor, can be around $1.50/kg [181]. in comparison, the cost of kevlar is around $63.6/kg [1]. thus, natural fibers used in multi-layered armor systems tend to have lower densities and lower costs compared to kevlar, making them potentially more cost-effective and lightweight alternatives. however, the specific properties and costs of natural fibers can vary widely, and careful consideration should be given to the selection and processing of natural fibers for use in armor systems. fig. 15 shows a comparison of cost and density with existing kevlar fiber. tables 4, 5, and 6 consolidate the use of natural fibers in polymer composites for ballistic applications along with their failure modes fig. 15 comparison of cost and density with existing kevlar fiber recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 369 table 4 penetration depth and failure modes of different natural fibers intermediate layer penetration depth (mm) failure modes ref pineapple leaf fiber/epoxy 26 ±2 fiber bridging, delamination, matrix cracking [14] mallow fiber (aligned)/epoxy 22 ±2 fiber pullout, fiber stretching, matrix cracking, fiber bridging [182] sisal/epoxy 12 ±2 fiber pullout, fiber stretching, matrix crack [10] ramie fiber/epoxy 18 ±2 delamination, fiber stretching, matrix crack [13] coir fiber (aligned) epoxy 32 ±2 fiber pullout, stretching, matrix crack [12] jute fabric/polyester 17 ±3 fiber pullout, stretching, matrix crack [183] jute nonwoven mat/polyester 24 ±3 fiber pullout, stretching, matrix crack [184] curaua mat/epoxy 28 ±3 fiber bridging, stretching, matrix crack [14] piassava fiber/epoxy 15.6 ±3 fiber bridging, stretching, matrix crack [185] fique fiber/polyester 16 ±2 fiber pullout, stretching, matrix crack [186] epoxy-15% fique fabric 20 ±1 fiber bridging, stretching, matrix crack [175] table 5 application of natural fiber-reinforced polymer composites in ballistics thermoset natural fiber (intermediate layer) matrix type energy absorption (j) ballistic limit ref non-woven kenaf and kevlar (30, 50, 70%) epoxy 39-148 [187] jute fabric (10, 20, 30 vol. %) polyester 200-260 [183] jute fabric (10, 20, 30 vol%) epoxy [188] coir fiber (10, 20, 30%) epoxy [12] sisal fiber (30%) epoxy 106 [9] fique fiber epoxy [189] curaua fiber (10, 20, 30%) polyester [190] kenaf/kevlar (30/70) epoxy 148 [55] kenaf/kevlar (vf6/vf47) epoxy 175 [55] coconut sheath/kevlar (wt%5/25) epoxy 240 [6] curaua fiber (30vf) epoxy 197 [191] piassava fiber (vf) epoxy 272 ± 19j [185] fique fiber polyester 155 ± 7 [192] fique fabric polyester 112 ± 7 [186] table 6 application of natural fiber-reinforced polymer composites in ballistics thermopropylene natural fiber matrix type energy absorption (j) ballistic limit ref flax, hemp and jute (46%) polypropylene [193] kevalar/baslat polypropylene 112 [79] kenaf polyethylene 774 kenaf fabric/aramid polyvinylbuturyl 477-621 [57] chonta palm wood high density polyethylene [194] 370 b. devarajan, r. lakshminarasimhan, a. murugan, et al. 6. failure criterion the use of natural fiber composites in the intermediate layer of a mas has been studied by many researchers in recent years. the advantages of using natural fiber composites include their low cost, lightweight, and high specific energy absorption capacity. the intermediate composite layer of the mas was fabricated using various natural fibers including guaruman, bagasse, mallow, fique, sisal, coir, ramie, bamboo and jute fibers and fabrics aligned in different orientations [195]. emphasizes the importance of selecting the appropriate natural fibers, concentration, and manufacturing processes to develop an effective and durable intermediate layer for mas. the development of mas using natural fiber composites natural fiber composites in mas has the potential to offer sustainable, costeffective, and eco-friendly ballistic protection solutions. additionally, jute, curaua, guaruman, and bagasse fibers exhibited improved matrix adhesion, resulting in reduced delamination and fiber breakage upon impact [181]. it was also observed from various experimental studies that the intermediate layer of mas made of natural fiber hybrid composites had undergone failure by different modes including failure of matrix and fiber separately and delamination occurring at the interfacial region of matrix and the fiber and so on. these interlaminar failures significantly reduce the strength and stiffness of the natural finer composite material layers [196]. this type of failure can lead to significant material damage, compromising mas effectiveness against ballistic impact. to address this, optimizing the intermediate layer design and natural fiber composite properties is crucial. key considerations include selecting suitable fibers, matrix materials, fiber orientation, and manufacturing processes. furthermore, post-treatments like surface and chemical treatments, along with interphase modification, enhance fibermatrix adhesion, minimizing delamination and fiber breakage during impact. fig. 16 presents the side view of impact regions and their energy absorption mechanisms. fig. 16 2d schematic of a typical woven fabric composite laminate targets (a) armor pierced by apm2 bullets – side view [196] (b) elevation of impact zones and yarns [32], and (c) mechanism of energy absorption in the layered woven fabric composites [32] with natural fiber composites in the intermediate layer of mas offer promising and sustainable ballistic protection, interlaminar failure must be addressed in design and manufacturing. optimizing the intermediate layer's design and material properties can make natural fiber composites a viable solution for ballistic protection. intralaminar failure recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 371 occurs due to various modes like interfacial disbanding and fiber/matrix breaking, while interlaminar failure involves the failure of multiple plies. intralaminar cracking in multidirectional laminates reduces load-bearing capacity and affects material thermoelastic characteristics, with delamination being a common consequence of in-plane matrix failure [197]. accurately predicting and modeling the failure behavior of composite materials under ballistic impact requires addressing the interaction between parameters causing interlaminar and intralaminar failures. however, these failure modes are typically addressed individually and independently, allowing researchers and engineers to focus on developing specific failure criteria for each mechanism [198]. fig. 17 shows different factors involving energy absorption in ballistic body armor. fig. 17 failure criterion for packages based on fabrics 6.1. intralaminar failure mechanisms there are three main intralaminar failure mechanisms considered during composite material analysis which includes failure of the fiber, shear failure at fiber-matrix interface and matrix cracking. failure modes are typically characterized by two in-plane modes, known as modes 1-2, and one out-of-plane shear mode, known as mode 2-3 [199]. these modes are used to describe the behavior of the material under different loading conditions and are crucial for understanding the failure mechanisms that occur during impact events. in most impact investigations, the most typical intra-laminar failure mechanisms include matrix out-of-plane and in-plane shear failure, fiber breakage and transverse tensile failure [200]. these mechanisms are all related to the deformation and damage that occurs within the composite layers during impact loading. figs. 18 and 19 show failure modes of intra-laminar and inter-laminar and its transition. 372 b. devarajan, r. lakshminarasimhan, a. murugan, et al. fig. 18 failure modes (a) intra-laminar failure modes (b) inter-laminar failure modes (c) transition from intra to inter-laminar failure modes [200] fig. 19 ballistic failure mechanism proposed after the impact: (a, b) partial penetration and (c) complete perforation [198] recent developments in natural fiber hybrid composites for ballistic applications: a comprehensive... 373 fiber breaking occurs when the fibers within the composite material fracture and break apart under tensile or compressive loading [19]. transverse tensile failure happens when composite layers separate and delaminate perpendicular to fiber orientation. in-plane shear occurs when matrix material deforms within the composite under loading, while out-of-plane shear happens perpendicular to layer planes. common failure modes in transversely loaded composite structures include transverse tensile failure, out-of-plane and in-plane shear, and fiber breakage in experimental studies [201]. understanding these failure mechanisms is crucial for designing composite materials capable of withstanding impact loading. by characterizing these modes and creating predictive models, researchers and engineers can design lightweight composite structures that offer excellent resistance to impact damage [202, 203]. 6.2. interlaminar failure mechanisms linear elastic fracture mechanics (lefm) is a widely used analysis technique for interlaminar failure or interfacial fracture in composite materials. delamination failures, such as out-of-plane shear mode, in-plane shear mode, and peel mode (specified as mode i, ii, and iii), are commonly observed, each with unique characteristics requiring specific analysis techniques. however, lefm has limitations in analyzing these three common modes of failure [203]. mode i failure, or peel, arises when a tensile load is applied perpendicular to the laminates' plane, causing delamination of the first layer. this process involves crack opening and separation, accurately described by mode -i lefm. mode ii failure, or in-plane shear, occurs when a shearing force is applied parallel to the laminates' plane, leading to sliding between layers. this mode involves crack sliding and shear displacement, also described by mode -ii lefm [204]. mode iii failure or out-of-plane shear occurs when a shearing force is applied perpendicular to the plane of the laminates, causing the middle layer to slide with respect to the upper and lower layers [205]. the fracture process in mode iii is characterized by crack sliding and out-of-plane shear displacement, and lefm is not suitable for analyzing this mode because it assumes that the crack propagates in the plane of the laminates. lefm is a useful tool for analyzing interlaminar failure or interfacial fracture in composites, but it has limitations when it comes to analyzing mode iii failure [206]. 7. conclusions the article demonstrates the potential of renewable resources in pmcs, the effectiveness of plant fibers as replacements for conventional fibers in ballistic protection and armor applications, and the possibility of using them as a suitable substitute for conventional fibers in such applications. natural fiber materials are such kind of sustainable materials in hard body armor that have revolutionized the field of armor systems. owing to their light-weight nature, they can be readily used as an alternative to heavy armor including metal body armor. the conclusions drawn from the above discussions can be summarized as follows: ▪ the impact behaviour of the multilayer armor systems was greatly influenced by the intrinsic parameters of the system including condition of use, target arrangements and its composition and material characteristics of the ballistic materials. ▪ to design and beget an environmentally friendly and sustainable ballistic protection made of natural fiber composites, numerical and experimental methods are to be combined in order to analyze and comprehend the ballistic impact mechanisms. 374 b. devarajan, r. lakshminarasimhan, a. murugan, et al. ▪ it could also be concluded that the intermediate natural fiber hybrid polymer composite layer fails majorly through fiber fracture, fiber pullout, delamination, fiber bridging and matrix cracking failure modes upon impact with a ballistic projectile. these materials possess high energy absorption capacity and allow for a more even load distribution through the varying fiber architecture, ply thickness and aerial density and different influential parameters. ▪ utilization of natural fiber hybrid composites in multilayer armor systems minimizes the cost of raw materials and fabrication without compromising the ballistic performance much. hybridization of the natural fibers with synthetic fibers enhances the ballistic performance of the materials system. ▪ the development of advanced body armor systems has been aided by recent advancements in the material science field. the evolution in the field of defense and aerospace sector led to further exploring these kinds of applications with aid of using additive manufacturing 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1military academy, university of defence in belgrade, belgrade, serbia 2department of public safety, government of brčko district of bosnia and herzegovina, brčko, bosnia and herzegovina 3decision science & operations management area, calcutta business school, diamond harbour road, west bengal, india 4faculty of mechanical engineering, university of niš, niš, serbia 5tu berlin, department of structural analysis, berlin, germany 6faculty of construction management, union nikola tesla university, belgrade, serbia orcid ids: darko božanić https://orcid.org/0000-0002-9657-0889 igor epler https://orcid.org/0000-0002-1616-5139 adis puška https://orcid.org/0000-0003-3274-0188 sanjib biswas https://orcid.org/0000-0002-9243-2403 dragan marinković https://orcid.org/0000-0002-3583-9434 stefan koprivica https://orcid.org/0009-0006-3949-4310 abstract. this paper presents a multi-criteria decision-making model based on the application of two methods, dibr ii and mabac. the dibr ii method was used to define weight coefficients. the mabac method was used to rank alternatives, and it was applied in a rough environment. four experts were engaged in defining the criteria and alternatives as well as in the relation of criteria. the model was applied for ranking the methods and techniques of lean organization systems management in the maintenance of technical systems of special purposes. at the end of the application was conducted a sensitivity analysis which proved the stability of the obtained results. key words: defining interrelationships between ranked criteria ii (dibr ii), multiattributive border approximation area comparison (mabac), rough number, lean concept received: june 14, 2023 / accepted august 11, 2023 corresponding author: darko božanić military academy, university of defence in belgrade, veljka lukica kurjaka 33, 11040, belgrade, serbia e-mail: darko.bozanic@va.mod.gov.rs https://orcid.org/0000-0002-9657-0889 https://orcid.org/0000-0002-1616-5139 https://orcid.org/0000-0003-3274-0188 https://orcid.org/0000-0002-9243-2403 https://orcid.org/0000-0002-3583-9434 https://orcid.org/0009-0006-3949-4310 102 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica 1. introduction by analyzing basic technological process of technical maintenance (tm) of special purpose technical systems (spts) in a real maintenance workshop in the army of serbia, and its documentation which often lacks the data related to the organization of the process itself, time gaps were observed between the planned (normed) times of tm cycle of spts and the realized times of the spts technical maintenance cycle [1]. the periods of time in which spts wait for operation (considering administration and logistics) are generally not recorded in the workshop documentation, which makes it difficult to discuss about with the purpose of their reduction or total elimination. many spts tm processes are not controlled, causing constant losses in the execution of tm on spts. existing technological tm documentation for the most of the spts is “poor” regarding maintenance process management. for that reason, preparing computer data for software information system is very difficult as it relies only on currently available information from the tm technological process. obviously, under such circumstances the management of spts tm processes must be fundamentally changed, primarily in terms of intensifying research and development of practical methods and techniques based on new concepts of maintenance, which are applied and constantly improved by leading companies in the neighborhood and in the world [2]. however, this is easier said than done. there are many difficulties in the management of tm of spts, and the most frequent and influential are as follows: poor quality and reliability of input data primarily contained in technological documentation; complexity of the maintenance system (com) of spts; many years of lack of investment in the management of tm, lack of exchange of work technologies, knowledge and innovations and inadequate training of top management for tm of spts to make them able to recognize and decide what is necessary and important to change in the maintenance management of spts so as to minimize losses. by eliminating negative effects of the mentioned factors, based on lean organization systems management, current state of the work process and organization of tm of spts would probably change qualitatively. qualitative changes would be reflected in [3]: ▪ shortening of spts tm cycle, ▪ optimal use of available capacities (spatial, human, material), ▪ reduction of costs per every maintained spts (increasing efficiency) and ▪ increased effectiveness of the tm process of spts. however, the application of all methods and techniques of lean organization systems management requires significant investments. considering this, a model for evaluation and ranking of the methods and techniques of lean concept was developed. depending on available resources, these methods and techniques (alternatives) would be applied in practice, primarily the first-ranked ones, and when available resources allow, the other alternatives. previous research in the field of organization systems management [4,5] have proven positive effects of applying the lean concept [6,7] in terms of the reduction of resources, both human and material, respectively increasing effectiveness and efficiency. there are indications that the application of the methods and tools of lean organization systems management [8,9] can contribute to the improvement of current situation considering com of spts. the problem of applying lean organization systems management has been discussed in several studies. in the following text, a part of the published research is analyzed. application of the dibr ii – rough mabac decision-making model for ranking... 103 tamanna and rahman [10], based on the data of periods of operations until failure, criticality and consequences of failure of key equipment for maintenance of simga1 shipyard, wanted to analyze and select the optimal ship maintenance strategy with the goal of increasing the reliability and availability of the maintenance equipment and tools, reducing operating and production costs and increasing the shipyard's competitiveness. they observed the opportunity for improvement of shipyard maintenance in the application of lean management tools. by using the ahp method, it was made a selection of the most effective lean tools out of ten available. singha mahapatra and shenoy [11] indicate the constant need of organizations for maintenance and services providing the elimination of activities and processes that do not contribute to the creation of new value for the customer or user, respectively, to the reduction of the price of the product or service. the authors state that the most managers of maintenance systems practically apply the tools of lean management in different ways, because these proved satisfactory improvements in a relatively short period of use. through their study, the authors strive to improve the methodology of selecting and applying lean tools by identifying unique factors consisting the basis for evaluation of the existing rationality of maintenance in any maintenance organization, and which are the basis for measuring improvements after the application of lean tools in a specific organization. bhebhe and zincume [12] analyzing systematically broad literature and applying general scientific method of deduction concluded that correctly dimensioned, constructed, properly exploited transport network with qualified managers was the key of economic power of every state. based on the same data sources, they claimed that the most countries improved their economies by investing in transportation sector by identifying unnecessary costs generators, respectively, the losses within the maintenance function in transport companies. observed cost generators were eliminated or reduced by systematic selection and application of lean management tools that had already given the best results in the similar systems around the world. the paper provides guidelines for the improvement of overall condition by reducing unnecessary costs within transportation systems maintenance department, but also points out the opportunities for adjustment of application methods and its dynamics taking into account the specificities of as many transport companies as possible. korchagin et al. [13] state that nowadays aircraft manufacturers around the world encounter the need and possibility for improvement of position and competitiveness of the aircraft manufacturing industry after the sale of the aircrafts. they quote that for hitting this goal the crucial thing is the improvement of the aircraft maintenance organization within all the phases of its lifespan by using the concepts such as lean and industry 4.0. in this paper was performed the integration of lean management and industry 4.0 approaches, for the purpose of aircrafts maintenance, their modeling and simulation of their joint impact on the improvement of aircraft maintenance in the anilogic simulation system. the results of the simulation showed that the simultaneous use of lean management tools and industry 4.0 from the beginning of the aircraft exploiting life in aviation industry would provide good results which would reflect in the aircraft maintenance efficiency increase, through the identification of bottlenecks in the process and making the right decisions in the direction of constant management of the maintenance process quality. dragone et al. [14] in their paper emphasize that during the lifetime of the residential and commercial buildings are not carried out all the necessary maintenance activities 104 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica often, which can cause serious damage and accidents. they consider that the maintenance of facilities is essential for the effectiveness of their utility value, as well as for the safety of the facilities. traditional concepts of building maintenance management became ineffective over time, and do not provide good results because these are not oriented towards the processes and increasingly complex users’ demands and of the objects of work themselves – buildings. the authors deem that lean maintenance management, with its principles and methods, can provide good solutions, applicable in the practice of facility maintenance. one of the most important parts of any decision-making process is the selection of the methods to be used [15-18]. of course, additional dilemmas in these processes arise when choosing a possible way to modify standard methods, and for the purpose of better treatment of uncertainty in the decision-making process [19-22]. through the analysis of the available studies, and taking into account the nature of the problem, two methods of multi-criteria decision-making were used in this research. the first method is defining interrelationships between ranked criteria ii (dibr ii), which is used for obtaining weight coefficients of criteria. this method is shown through the process of group decisionmaking. the dibr ii method is presented only in the study by [23] and it has not been used so far for solving complex decision-making problems. however, the simplicity that this method provides in communication with experts, as well as the simple mathematical model, recommended this method for solving the mentioned problem. the second method used in ranking of alternatives is mabac, which proved as a very reliable one in up to date conducted researches. considering the values, which were assigned to the alternatives at the beginning of the decision-making process when these were evaluated according to the criteria, the mabac method was applied in a rough environment. rough numbers were combined with decision-making methods in a large number of studies. badi and abdulshahed [24] used rough numbers to modify the ahp method, while in the study by [25] was presented the combination of rough numbers with dematel method. qi et al. [26] combine rough numbers with vikor method, song et al. [27] with topsis method, and arsić et al. [28] with mairca method. rough numbers are commonly used in combination with fuzzy numbers, as in the studies by [29-31]. the mabac method is modified in literature by using rough numbers in multiple studies as [32-36]. through the analysis up so far, two main contributions of this paper stand out. the first one is the application of the dibr method in multi-criteria group decision-making, and for the first time as a part of the process of solving a real problem. the second contribution of the paper is related to the solving of a case study, respectively, the problem of ranking the methods and techniques of lean concept in order to improve the management of the work process and the organization of the tm of spts in the army of serbia. the paper consists of several parts. in the second part the applied methods are described. through the third part, that is, the case study, the definition and calculation of the weight coefficients of the criteria and the ranking of alternative solutions are carried out. the fourth part deals with sensitivity analysis, and at the end of the paper is provided the conclusion of this research. application of the dibr ii – rough mabac decision-making model for ranking... 105 2. description of the applied methods this section describes the methods of dibr ii rough mabac model. the phases and steps of the model are presented in the fig. 1. phase 1: defining the criteria and calculation of weight coefficients of criteria using dibr ii method step 1. definition of criteria step 2. ranking criteria by their importance/weight step 3. defining significance ratio between criteria step 4. calculation of the relation of the most significant criterion with other criteria step 5. calculation of the value of the most significant criterion step 6. calculation of weight coefficients of other criteria step 7. evaluation of quality of defined significance of the adjacent criteria step 8. aggregation of weight coefficients of criteria for each expert phase 2: identification and ranking of alternatives using rough mabac method step 1. forming of initial decision-making matrix step 2. normalization of the elements of the initial decision-making matrix step 3. the calculation of the elements of the weighted matrix step 4. defining border approximation area matrix step 5. calculation of the matrix elements for alternatives distance from the baa step 6. ranking of alternatives step 7. converting a rough number into the crisp value phase 3: sensitivity analysis through the changes of weight coefficients of criteria fig. 1 dibr ii – rough mabac model as it can be observed from the fig. 1, the model has three phases, where the criteria and their weight coefficients are defined first. next, in the second phase of the model, the selection of the best alternative is made and at the end, it is checked the sensitivity of the model. 2.1. defining interrelationships between ranked criteria ii method the dibr ii method is developed by božanić and pamučar [23]. the method was developed for defining weight coefficients of criteria and consists of eight steps presented below. step 1. definition of criteria. as a part of the first step, a set of criteria is defined c={c1, c2,...cn}, based on which the alternative solutions are ranked. step 2. ranking criteria by their importance/weight. all the criteria in the set c are ranked from the most to the least significant. for simple presentation of the method, the set of criteria is defined so that the criterion 1c is the most significant, while the criterion cn is the least significant, respectively, it is defined as follows c1>c2>c3>...>cn. step 3. defining significance ratio between criteria. step 3.1. defining significance ratio between adjacent criteria. for each two adjacent criteria, their significance ratio is defined (ηj,j+1 where , 1 1,2 2,3 3,4 1,{ , , ,..., }j j n n    + − ). thus, for example, for the comparison of the criteria c1 and c2 the significance ratio η1,2 is defined. for this significance ratio it is valid that ηj,j+1≥1. the value ηj,j+1 shows how much more significant the criterion cj is than the criterion cj+1. according to everything stated, the following relations are defined through this step: 106 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica 1 1 2 1,2 1,2 2 : :1 w w w w  =  = (1) 2 2 3 2,3 2,3 3 : :1 w w w w  =  = (2) ... 1 1 1, 1,: :1 n n n n n n n n w w w w  − − − −=  = (3) step 3.2. defining significance relations between the most and the least significant criteria. in this step, the following relation is defined: 1 1 1, 1,: :1n n n n w w w w  =  = (4) this relation has the role of the control factor in evaluating the quality of the defined significance of the adjacent criteria. step 4. calculation of the relation of the most significant criterion with other criteria. based on eqs. (1-3), the value of the second and the other lower in range criteria is presented through the most significant criterion, as follows: ▪ based on eq. (1), the value of the weight coefficient 2w is obtained: 1 2 1,2 w w  = (5) ▪ based on eqs. (2) and (5), the value of the weight coefficient 3w is obtained: 2 1 3 2,3 1,2 2,3 w w w    = =  (6) ▪ based on eq. (3), the value of the weight coefficient nw is obtained: 1 1,2 2,3 1,... n n n w w    − =    (7) step 5. calculation of the value of the most significant criterion. if it is as follows: 1 1 n j j w = = (8) upon applying eqs. (5-8), the result yields: 1 1 1 1 1,2 1,2 2,3 1,2 2,3 1, ... 1 ... n n w w w w       − + + + + =     (9) application of the dibr ii – rough mabac decision-making model for ranking... 107 from eq. (9), the value of the weight coefficient of the most significant criterion is obtained as: 1 1,2 1,2 2,3 1,2 2,3 1, 1 1 1 1 1 ... ... n n w       − = + + + +     (10) step 6. calculation of weight coefficients of other criteria. applying eqs. (5-7), the remaining weight coefficients of criteria, w2, w3,…,wn, are obtained. step 7. evaluation of quality of defined significance of the adjacent criteria. the relations of significance of the adjacent criteria need to be checked, to avoid as much as possible subjectivity of the decision makers. step 7.1. evaluation of quality of defined significance values. the evaluation of quality of defined significance values is made based on the relation of the significance of the most and the least significant criterion (η1,n). the value of the least significant criterion can be obtained from eq. (4): 1 1, k n n w w  = (11) where k nw presents the control weight coefficient of the criterion cn. the values wn and k nw should be approximately equal. if their deviation amounts to 10% approximately, it can be concluded that the relations of significance of the adjacent criteria are defined with quality, and vice versa. checking of the deviation is done by applying the expression: 1 n n k n w d w = − (12) where dn presents the value of the deviation of the weight coefficients of the criterion cn. if the condition 0 ≤ dn ≤ 0.1 is met, then the evaluations of the significance relations of the adjacent criteria are defined with quality, i.e. these meet the requirements. if dn > 0.1, it is necessary to define new relations between the criteria. however, since the research is usually an extensive process engaging significant resources, an additional step can be applied in order to find an error. in such cases, step 7.2 is applied. step 7.2. additional evaluation of quality of the defined significance of the adjacent criteria. before returning to defining relations of significance of the adjacent criteria, it is possible to make additional quality control. for this purpose, the step from step 7.1 is repeated, in which the relations between the criteria cn-1 and cn-2 are defined. for this procedure, it is necessary for the decision-maker to define new relations: 1 1 1 1, 1 1, 1 1 : :1n n n n w w w w  − − − − =  = (13) 1 1 2 1, 2 1, 2 2 : :1n n n n w w w w  − − − − =  = (14) 108 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica then, it is calculated: 1 1 1, 1 k n n w w  − − = (15) 1 2 1, 2 k n n w w  − − = (16) where 1 k nw − and 2 k nw − are the control weight coefficients of the criterion cn-1 respectively, cn-2. finally, the value is obtained: 1 1 1 1 n n k n w d w − − − = − (17) 2 2 2 1 n n k n w d w − − − = − (18) here, dn-1 and dn-2 are the values of deviation of the weight coefficients of the criteria cn-1 and cn-2, respectively. if the conditions dn−1  [0,0.1] and dn−2  [0,0.1] are met, it can be concluded that there was an error in defining the relations of significance between the most and the least significant criterion (η1,n). in that case, the existing results can be accepted or the values can be defined again (η1,n) and quality checked again as well. briefly said, if dn−1  [0,0.1] or dn−2  [0,0.1], the complete procedure of defining the relations of significance and calculation of the weight coefficients must be repeated. step 8. aggregation of weight coefficients of criteria for each expert. the previous seven steps refer to the definition of weight coefficients when decisions are made by a single decision maker, i.e. an expert. in situations where the decision is made by several people, it is necessary to aggregate their opinions. for the purposes of this model, the weigh coefficients of the criteria were aggregated, which were obtained for each expert separately, using the bonferroni aggregator: , 1 1 ( ) ( ) ( 1) l a e p u q i i i e u e u w w w l l =      =   −      (19) where l is the number of experts, a iw are the aggregated values obtained by applying the bonferroni aggregator, p,q ≥ 0 are the stabilization parameters of bonferroni aggregator, e and u are the e-th or u-th expert, where 1 ≤ e, u ≤ l. 2.2. rough mabac method the mabac (multi-attributive border approximation area comparison) method belongs to the group of newer and frequently applied methods. this method was developed by pamučar i ćirović [37]. in this paper, the method is improved by applying application of the dibr ii – rough mabac decision-making model for ranking... 109 rough numbers. in the next part of the paper, the basic assumptions about rough numbers and the presentation of the steps of the mabac method in rough environment are given. in the 1990s, pawlak [38,39], developed rough sets. some twenty years later, inspired by the idea of rough sets, zhai et al. [40], developed rough numbers. rough numbers prove to be very useful in considering uncertainty. a brief description of rough numbers is given below. definition 1 [40]. let’s assume that there is a set f consisting of (k1, k2, k3,…, kt), which represents all the objects in a particular universe u. at the same time, there is y presenting boundary objects of the universe u. if all the elements form part of the sequence k1<k2<k3<…<kt, where it is valid that , , 1qy u k f q t     , then the following parameters can be defined:  ( ) / ( )q qapr k y u f y k=   (20)  ( ) / ( )q qapr k y u f y k=   (21)       ( ) / ( ) / ( ) / ( ) q q q q bnd k y u f y k y u f y k y u f y k =   =     (22) where ( )qapr k is the lower approximation, and ( )qapr k is the upper approximation, while bnd(kq) is boundary interval of the element kq. definition 2 [40]. the element gq can be presented as a rough number (rn(kq)). the rough number rn(kq) is defined by its lower limit ( ( )qlim k ) and the upper limit ( ( )qlim k ), where q 1 ( ) (y) ( )q l lim k f y apr k n =  (23) q 1 ( ) (y) ( )q u lim k f y apr k n =  (24) ( ) ( ), ( )q q qrn k lim k lim k =   (25) here, nl and nu are the number of objects contained in ( )qapr k and ( )qapr k , respectively. upper and lower limits indicate average value of the elements included into the upper and lower approximation, respectively. the difference between them represents the rough boundary interval (rbnd(kq)): q( ) ( ) ( )q qrbnd k lim k lim k= − (26) rough boundary interval represents uncertainty of the element kq, where a higher number indicates higher imprecision, while a lower number indicates better precision, based on what subjective information only now can be marked with a rough number. by using the rough numbers shown, the classic mabac method is modified. the steps of the rough mabac method are given in the following [32-36]. 110 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica step 1. forming of initial decision-making matrix (x). in the first step, the values of m alternatives are being defined by n criteria. 1 2 11 12 11 221 222 31 323 3 1 2 ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n m m m mn c c c rn x rn x rn xa rn xrn x rn xa rn x rn xx a rn x a rn x rn x rn x        =         (27) step 2. normalization of the elements of the initial decision-making matrix. in this step, the elements of the initial decision-making matrix (x) are normalized, in order to obtain the normalized matrix (n), by using the following expressions: , ij j ij j ij ij j j j j x x x x t t x x x x − − + − + − − − = = − − (28) , ij j ij j ij ij j j j j x x x x t t x x x x + − − + − + − − = = − − (29) by using eq. (28) the normalization of the benefit type criteria is done, while using eq. (29) the normalization of the cost type criteria is made, where:  11 max ( ), , min ( ), for cost typecriteriaj ij ij i mi m x x for benefit typecriteria x+    = (30)  1 1 min ( ), , max( ), for cost typecriteriaj ij ij i m i m x x for benefit typecriteria x−     = (31) step 3. the calculation of the elements of the weighted matrix (v) is performed by using the following equation: ( 1), ( 1)a a ij i ij ij i ijv w x t v w x t= + = + (32) step 4. defining border approximation area matrix (baa). the baa matrix (g) has a form n x 1, that is, the baa s formed for each criterion separately, using the following equations: ( ) ( ) 1/ 1/ 1 1 , m m m m j ij j ij i i g v g v = = = =  (33) step 5. calculation of the matrix elements for alternatives distance from the baa (q) ( )[ , ]ij ij m xn q q q= (34) where: application of the dibr ii – rough mabac decision-making model for ranking... 111 ( , ), ( ) ( ) , ( , ), ( ) ( ) e ij j ij j ij e ij j ij j d v g if rn v rn g q d v g if rn v rn g    =   −    for benefit type criteria (35) ( , ), ( ) ( ) , ( , ), ( ) ( ) e ij j ij j ij e ij j ij j d v g if rn v rn g q d v g if rn v rn g −    =      for cost type criteria (36) 2 2 2 2 ( ) ( ) , ( , ) ( ) ( ) , for cost typecriteria ij j ij j e ij j ij j ij j v g v g for benefit typecriteria d v g v g v g  − + −   =    − + −   (37) belonging of the alternative ai to certain approximation area (g, g+ ili g-) is determined as follows: 0 0 0 ij i ij ij g if q a g if q g if q + −      = =       (38) step 6. ranking of alternatives. final values of criteria functions are calculated by summing the elements of the matrix q by rows: 1 1 , , 1,2,..., , 1,2,..., . n n i ij i ij j j s q s g j n i m = = = = = =  (39) step 7. to convert a rough number ( ) ( ), ( )i i irn k lim k lim k =   into the crisp value, the following equation is used [35]:      min ( ) max ( ) min ( )crisp n i i i i i i ii k lim k k x lim k lim k = + −    (40) for the needs of calculation of individual equation elements (40), the following is used [35]:             ( ) ( ), ( ) ( ) min ( ) ( ) max ( ) min ( ) ( ) min ( ) ( ) max ( ) min ( ) i i i i i i i i i ii i i i i i i ii rn k lim k lim k lim k lim k lim k lim k lim k lim k lim k lim k lim k lim k  = =    −  =  −     −  =  −    (41) 112 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica  ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) i i i in i i i lim k x lim k lim k x lim k k lim k lim k − + = − + (42) 3. case study the decision supporting model, as shown above, is applied to the evaluation of the methods and tools of lean concept of management. in the following two sections is given the description of the criteria and the calculation of the weight coefficients of the criteria, followed by a description of the alternatives and their ranking. 3.1. calculation of weight coefficients of criteria in the following, the weight coefficients of the criteria were obtained by applying the dibr ii method, considering that the dibr ii method is used for the first time in solving a real problem through group decision-making, a detailed description of all calculations is given below. step 1. by analyzing available literature and with the help of four experts who deal with maintenance problems, five key criteria were reached according to which it is possible to evaluate the methods and tools of the lean com management. these criteria are given in the following text. reduction of a spts maintenance cycle time (c1), enables overview of basic factors in failures occurrence, distribution and duration of maintenance preparation and execution, and represents a basis for maintenance procedures establishment and analysis of overall efficiency and effectiveness of spts tm. time preview of spts condition consist of their alternating states of operations and failures, which represent time intervals expressed in kilometers, motor hours or hours of spts work. research in spts maintenance area is reduced to the analysis of causes of stated conditions. the analysis of those causes enables determination of basic parameter values, essential for designing the procedures and com of spts as a whole. failure time consists of a number of time segments. the goal of all improvement procedures is to reduce time of waiting for the operation, by which maintenance cycle time is optimized. optimization level of equipment and working positions arrangement (layout) within the maintenance system structure of technical systems for special purposes (c2). proper and purposeful spatial arrangement of equipment and tools is very important for those who want to apply lean management in organization systems, especially in production or spts maintenance. layout helps the management in defining equipment, tools and work positions arrangement, in order to minimize circulation of spts, tools and equipment, at the expense of higher circulation of spare parts and consumables, resulting in greater system operational readiness, due to shorter logistical and administrative waiting times, and lower transportation costs the assumption is that operators working in such a conceived organization system are qualified and trained to quickly and precisely perform all activities of the basic process of the system [8]. spts effectiveness increase percentage, e(t) (c3) can be mathematically defined as probability that maintenance tasks will be performed in a certain period of time, with the application of the dibr ii – rough mabac decision-making model for ranking... 113 prescribed functioning regime and with the implementation of prescribed maintenance activities preventive/corrective maintenance programs, and it is expressed as the product of reliability p(t), functional suitability (suitability for functional use) fp(t) and availability (readiness) for intended use r(t). each of the effectiveness components can take a value in the interval from 0 to 1. therefore, the effectiveness can have the same values. the components of effectiveness and overall effectiveness of spts, should be projected and adopted already in the development phase, but their values should also be monitored and analyzed throughout the lifetime. improvement is achieved if the effectiveness expressed in percentage is increased after the application of the alternatives. system efficiency increase level efik (c4) can be defined as the ratio between the output and input, i.e. as the ratio between the products and resources (benefits and investments, needs and possibilities, planned and realized). it is also a number between 0 and 1. the mathematical interpretation of efficiency, in general, is as follows: ( ) ( ) fik o output e i input = (43) an improvement is achieved if the reached value is lower after the application of the alternatives, compared to the moment before the application of the alternatives, because the quantity being compared is the maintenance cycle time before and after the application of the alternatives. reduction in number of executors (c5) is the goal of optimization of technological process, based on consideration of combining certain work positions, which are similar in terms of technological procedures, additional education and training of executors to perform more related operations by better planning, scheduling spts maintenance operations and eliminating unnecessary idling losses in tm of spts. step 2. after defining the criteria, the experts performed criteria ranking by their significance. the criteria ranking by four experts ee, where e  (1,2,3,4), is shown as follows: 1 1 2 3 4 5 2 1 3 2 4 5 3 1 2 3 4 5 4 1 2 3 5 4 : : : : e c c c c c e c c c c c e c c c c c e c c c c c          =       step 3. table 1 shows the assessment of the significance ratio performed by the experts. table 1 criteria significance ratios defined by experts expert 1 expert 2 expert 3 expert 4 c1:c2=2.0 c1:c3=2.5 c1:c2=1.8 c1:c2=1.7 c2:c3=1.1 c3:c2=1.2 c2:c3=1.0 c2:c3=1.3 c3:c4=1.6 c2:c4=1.6 c3:c4=1.5 c3:c5=1.5 c4:c5=1.3 c4:c5=1.4 c4:c5=1.1 c5:c4=1.1 c1:c5=5.0 c1:c5=7.0 c1:c5=3.0 c1:c4=3.5 114 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica step 4. in this step, by using eqs. (5-7), the relation of the most significant criterion to the other criteria is calculated. if the weight coefficient is marked with e iw , where e are the experts, e  (1,2,3,4), and i is the criterion, i  (1,2,3,4,5), then the following relations are obtained: e1: 1 1 1 2 2 w w = ; 1 1 1 1 1 3 2 1.1 2.2 w w w = =  ; 1 1 1 1 1 4 2 1.1 1.6 3.52 w w w = =   ; 1 1 1 1 1 5 2 1.1 1.6 1.3 4.576 w w w = =    e2: 2 2 2 1 1 2 2.5 1.2 3 w w w = =  ; 2 2 1 3 2.5 w w = ; 2 2 2 1 1 4 2.5 1.2 1.6 4.8 w w w = =   ; 2 2 2 1 1 5 2.5 1.2 1.6 1.4 6,72 w w w = =    e3: 3 3 1 2 1.8 w w = ; 3 3 3 1 1 3 1.8 1 1.8 w w w = =  ; 3 3 3 1 1 4 1.8 1 1.5 2.7 w w w = =   ; 3 3 3 1 1 5 1.8 1 1.5 1.1 2.97 w w w = =    e4: 4 4 1 2 1.7 w w = ; 4 4 4 1 1 3 1.7 1.3 2.21 w w w = =  ; 4 4 4 1 1 4 1.7 1.3 1.5 1.1 3.647 w w w = =    ; 4 4 4 1 1 5 1.7 1.3 1.5 3.315 w w w = =   step 5. by using eq. (10), the weight coefficient of the most significant criterion is calculated. the detailed calculation for each expert is provided as follows: 1 1 1 0.407 1 1 1 1 1 2 2.2 3.52 4.576 w = = + + + + 2 1 1 0.478 1 1 1 1 1 3 2.5 4.8 6.72 w = = + + + + 3 1 1 0.355 1 1 1 1 1 1.8 1.8 2.7 2.97 w = = + + + + 4 1 1 0.382 1 1 1 1 1 1.7 2.21 3.647 3.315 w = = + + + + step 6. based on the ratios defined in the fourth step of this method, the weight coefficients of the other criteria were calculated: e1: 1 2 0.407 0.203 2 w = = ; 1 3 0.407 0.185 2.2 w = = ; 1 4 0.407 0.116 3.52 w = = ; 1 5 0.407 0.089 4.576 w = = application of the dibr ii – rough mabac decision-making model for ranking... 115 e2: 2 2 0.478 0.16 3 w = = ; 2 3 0.478 0.191 2.5 w = = ; 2 4 0.478 0.1 4.8 w = = ; 2 5 0.478 0.071 6,72 w = = e3: 3 2 0.355 0.197 1.8 w = = ; 3 3 0.355 0.197 1.8 w = = ; 3 4 0.355 0.131 2.7 w = = ; 3 5 0.355 0.12 2.97 w = = e4: 4 2 0.382 0.225 1.7 w = = ; 4 3 0.382 0.173 2.21 w = = ; 4 4 0.382 0.105 3.647 w = = ; 4 5 0.382 0.115 3.315 w = = step 7. the weight coefficient of the least significant criterion was recalculated by applying the last (control) relation from table 1 and the deviation from the obtained values was checked. the calculation was made using eqs. (11) and (12). e1:  1 1 1 5 5 5 0.407 0.089 0.081 1 0.099 0,0.1 5 0.081 w d d= =  = − =   ; e2:  2 2 2 5 5 5 0.478 0.071 0.068 1 0.044 0,0.1 7 0.068 w d d= =  = − =   ; e3:  3 3 3 5 5 5 0.355 0.12 0.118 1 0.017 0,0.1 3 0.118 w d d= =  = − =   ; e4:  4 4 4 5 5 5 0.382 0.115 0.109 1 0.055 0,0.1 3.5 0.109 w d d= =  = − =   . considering the fact that the provided deviation is less than 10%, it can be concluded that the experts were consistent in their opinions. step 8. at the very end, the weight coefficients of the criteria were aggregated using standard bonferroni aggregator and eq. (19), with the value of the stabilization parameters p, q=1, which resulted in obtaining the final values of the weight coefficients of the criteria: 1 2 3 4 50.405; 0.196; 0.187; 0.113; 0.099a a a a aw w w w w= = = = = . 3.2. ranking alternative solutions the methods and tools of lean organization systems management, which also include the systems for tm of spts, represent the alternatives or possible solutions by implementation of which the improvement of the existing spts com condition is expected, reflected through the change (decrease or increase) of defined parameters (values) previously presented as criteria in the paper. by the help of experts, five alternative solutions were defined: visual systems vs (a1) present lean method which with the help of visual aids (notice boards, and on warning lights about status of a process, displays, lines on the floors and walls of spare parts and consumables in the working environment) provides all the needed information for employees [5], related to: ▪ basic system processes procedures; ▪ costs; ▪ required or achieved quality of the product/service; ▪ product/service delivery/completion times; 116 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica ▪ current state of the process; ▪ condition of tools; ▪ number of employees; ▪ workplace markings (yellow, green, red lines on plant or storage floors, etc.); ▪ costs, attendance at work, injuries or safety at work and similar. by using this tool of lean organization systems management, such environment is created where everything is precisely defined, from entry to exit, and where visual information is timely and useful [8]. arrangement of workplace and surrounding area 5 s (a2) is a set of rules for organizing a workplace for each employee. the name 5s was given by five english words starting with the letter “s”. the set of 5 s rules (sort, stabilize, sustain, shine, standardize) refers to good maintenance of the plants in the organization system, offices, workplaces, spare parts and consumables warehouses, and it is useful for the analysis of improvements in an organization system [8]. the goal is for every work position inside organizational system to be arranged in such manner so it is maximally efficient, without unnecessary movement, making faster and easier work for each employee, by organizing all tools at their designed place, clearly visible, clean and ready for use in every moment. one of the most important 5 s method rule is that workers themselves take care of their workplace and thus contribute to the overall effectiveness of the process. five s is the most recognizable method of the lean concept [4] of managing organization systems because it is the easiest one to be applied and the results are immediately visible. by implementing the rules of the 5 s method, employees become encouraged in the easiest way to continue with the “lean“ transformation. five s represents a method by which an excessive use of materials, energy, effort, etc. is reduced, while, on the other side, the quality and productivity of the system are raised to the optimal level. computer maintenance management system – cmms (a3) is reflected in the development, implementation and use of a software informational system that minimally does the following functions: management of work orders, maintenance planning, scheduling of maintenance activities, collection of data on maintenance history, management of approved funds in relation to costs, human resources management, management of spare parts and consumables for maintenance purposes, reporting on maintenance and consumption of spare parts and consumables, as well as on the level of their stock [8]. to achieve its purpose, a cmms must be applied with complete and accurate data about spts, spare parts and consumables, plans, norms for maintenance activities and descriptions of maintenance activities, that is, all spts maintenance procedures. work ticket system – kanban (a4) represents a system for making order in executions of process operations, control and management of spare parts stocks. the most common types of kanban are [3]: ▪ pull kanban circulates between spatially organized production structures and define the sequence of process activities between them; ▪ production kanban circulates between workplaces within spatially organized production structures and define the quantities that must be produced in a given process, following the order of operations; ▪ supplier kanban circulates between the producer and the supplier, and the delivery time to the producer, respectively, the consumer (customer), is pre-defined by instructions and procedures in the organization system. for example, commodity is transported once a day to the customer every day at eight o'clock. the driver who application of the dibr ii – rough mabac decision-making model for ranking... 117 hands over the commodity, that is, the realized kanban for a certain delivery, takes over, at the same place, the kanban for the next delivery. the quantity of goods, defined by the next kanban, must be delivered to the customer the next day, at the same time. kanbans can, with certain modifications, be used as warehouse record cards and identification cards in spare parts and consumables storages [3]. continuous improvement – kaizen (a5) is the way towards continuous process improvements in order to eliminate losses. the word kaizen originates from the japanese language and it is composed of two japanese words: kai to separate and zen to fix. the purpose of this lean tool is recognition, separation, analysis and solution of problems, and then implementation and confirmation of that solution in practice. kaizen bases its methods on the teachings of edwards deming and his pdca (plan-do-check-act) quality circle. small but constant improvements of the maintenance process are the goal of kaizen tool implementation. kaizen method involves all levels of employees, who are engaged in improvement process. participants in kaizen events are [8]: workers, heads of departments, mid-level managers, but also top managers. all of them, together in teamwork, should contribute to achieving gradual step by step improvements every day in every place. the kaizen approach encourages small daily continuous improvements and process that never ends, involving everyone’s progression, from workers to managers, using common sense as a basic principle for survival. reasonable judgement and rational decision-making are essential for kaizen approach. balanced workforce engagement nagara system (a6) emphasizes human factor as very important in all organization systems, because there is no process without it, so it is essential to pay attention to effective and efficient use of human labor power. the lean organization systems management resolves the effective and efficient use of human potential by applying the nagara system tool [5], which requires “interdisciplinarity” from each individual (broader view of the work performed in the system, better understanding of work operations for which worker is not directly responsible), respectively, the ability to perform multiple tasks and serve multiple workplaces to a certain extent as needed, which contributes to increasing overall, not just work efficiency. such practice reduces manpower to optimal without endangering working technology, work distribution is easier, and process flow more economical. the management of the organization system should take care of the timely personnel trainings for various related jobs in the organization systems in order to achieve their interdisciplinary skills. after defining the alternatives, the rough mabac method is applied. firstly, it is defined the initial decision-making matrix, as shown in the table 2. table 2 initial decision-making matrix c1 c2 c3 c4 c5 a1 [6.2,6.5] s-m [13,15] h-vh [1,2] a2 [4.5,5] h-eh [10,12] s-m [3,4] a3 [6.8,7.2] es-m [17,19] h-eh [2,3] a4 [7.3,7.5] h-vh [18,22] es-m [1,2] a5 [3,3.7] m-h [12,16] m-h [2,3] a6 [3.2,4] vh-eh [15,20] vh-eh [4,5] 118 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica as a part of the first step, the quantification of qualitative criteria was performed. it was done by using the scale shown in table 3, consisting of seven linguistic descriptors. table 3 linguistic scale for quantifying of qualitative criteria descriptor name abbreviation numeric value extremely small es 1 very small vs 2 small s 3 medium m 4 high h 5 very high vh 6 extremely high eh 7 by using the scale from table 3, the initial decision-making matrix was quantified, as presented in table 4. table 4 quantified initial decision-making matrix c1 c2 c3 c4 c5 a1 [6.2,6.5] [3,4] [13,15] [5,6] [1,2] a2 [4.5,5] [5,7] [10,12] [3,4] [3,4] a3 [6.8,7.2] [1,4] [17,19] [5,7] [2,3] a4 [7.3,7.5] [5,6] [18,22] [1,4] [1,2] a5 [3,3.7] [4,5] [12,16] [4,5] [2,3] a6 [3.2,4] [6,7] [15,20] [6,7] [4,5] by applying steps 2-7, the ranking of the alternatives is obtained. the final ranking of the alternatives is provided in table 5. table 5 rank of alternatives ( )irn k crisp ik rank a1 [0.15,0.478] 0.106 3 a2 [0.028,0.406] 0.000 5 a3 [0.226,0.647] 0.228 2 a4 [0.317,0.705] 0.307 1 a5 [-0.115,0.281] -0.143 6 a6 [0.103,0.522] 0.096 4 from table 4, it is clearly visible that the alternative a4 is the first-rankedthe best, while the alternative a5 is the worst one. application of the dibr ii – rough mabac decision-making model for ranking... 119 4. sensitivity analysis sensitivity analysis is an indispensable part of the entire decision-making process [41,42]. this segment of the decision-making model is mostly implemented through the changes of weight coefficients of criteria and can be found in a large number of recent researches such as [43-45]. the main purpose of sensitivity analysis is to determine how much the most influential criteria affect the final output whether there are any and what the changes in ranking of alternatives are. in this particular case, the most influential is the criterion c1, which is two times more influential than the second-ranked c2. considering significant difference between these two criteria, only changes to the criterion c1 were considered. for this purpose, 20 strategies with different weight coefficients of criteria were developed. the scenarios were created so that the value of the most significant criterion in each scenario was reduced by 5%, and the value that was reduced from the criterion c1 was evenly distributed among the other criteria. the values of the weight coefficients of the criteria according to the scenarios are given in fig. 2. fig. 2 review of criteria weight coefficients changes according to the reviewed weight coefficients, the rough mabac method was applied again and a new ranking was made for each scenario, as in table 5. table 5 ranking alternatives using different scenarios si s1-s5 s6-s9 s10-s11 s12-s16 s17-s20 a1 3 4 4 4 5 6 a2 5 5 5 5 4 4 a3 2 2 3 3 3 3 a4 1 1 1 2 2 2 a5 6 6 6 6 6 5 a6 4 3 2 1 1 1 considering table 5, it is easily noticeable that there are changes in the ranking of alternatives when the weight coefficients change. this clearly indicates that the model is 120 d. božanić, i. epler, a. puška, s. biswas, d. marinković, s. koprivica sensitive to weight coefficients change. what is important is that it should not be overly sensitive, which is analyzed below: ▪ the alternative a1 is ranked from the fourth to sixth place. in relation to the initial ranking, the change already occurs in the first scenario, but it remains until the scenario s11. from the scenarios s12 to s20, this alternative moves to the fifth and sixth place. ▪ the alternatives a2, a3, a4 and a5 as a part of the scenario analysis have only one change in ranking. the alternative a2, from the scenario s12, moves from the fifth-ranked place to the fourth-ranked alternative. the alternative a3 from the scenario s6, moves from the second-ranked place to the third-ranked alternative. the alternative a4 from the scenario s10 moves from the first-ranked position to the second-ranked alternative. the alternative a5 from the scenario s17 moves from the sixth-ranked place to the fifth-ranked alternative. ▪ the alternative a6 has more significant changes. namely, from the fourth-ranked alternative in the initial scenario (si), it gradually takes the first place from scenario s10. ▪ as it can be observed, greater changes occur from the tenth scenario, when the most significant criterion is already reduced by 50%. ▪ the alternative a4, which was ranked first in the initial scenario, and due to major changes in the weight coefficients, retains the first ranking until the scenario s9, while from the scenario s10 until the end, it is the second-ranked. the lowestranked alternative, a6, retains this position in almost all scenarios. the above analysis clearly shows that there are changes in the ranking of alternatives following the changes in the weight coefficients of the criteria, but these changes are gradual. accordingly, it is very clear that the presented model is quite stable. regardless the stated, mathematical proof of the stability of the model was also conducted. by applying spearman's rank correlation coefficient, it was defined whether the changes in the rankings of alternatives can be considered large and unacceptable. the equation for calculating spearman's rank correlation coefficient is as follows: 2 1 2 6 1 ( 1) n i i rccs n n  == − −  (44) here, ξi is the difference of rank according to the given scenario and the rank in the corresponding scenario, and n is the number of ranked elements. the spearman's rank correlation coefficient values are given in table 6. table 6 spearman's coefficient values si s1-s5 s6-s9 s10-s11 s12-s16 s17-s20 si 1 0.943 0.886 0.657 0.543 0.371 s1-s5 1 0.943 0.829 0.771 0.657 s6-s9 1 0.943 0.886 0.771 s10-s11 1 0.943 0.829 s12-s16 1 0.943 s17-s20 1 application of the dibr ii – rough mabac decision-making model for ranking... 121 the spearman's coefficient of correlation of the ranking of considered strategies falls within the range of srcc  [0.371,1]. the most important values of this coefficient are in relation of the initial strategy to other strategies. there it is observable that the spearman coefficient is very high up to the strategy s11. only from the strategy s12 to s16 it is significantly lower (0.513), but still quite high. from the strategy s17 up to s20, this value decreases more significantly, but still shows there is a certain correlation. this is expected since the most influential criterion is reduced to the minimum. both mathematical and theoretical analysis clearly indicate that changes in alternatives rankings by different strategies, due to the changes in weight coefficients of the criteria, are gradual and expected. everything above mentioned confirms the conclusion of model stability. 5. conclusion this paper deals with the issue of ranking methods and techniques of lean concept, with the purpose of improvement of work process and organization management of spts tm in the army of serbia. resulting conclusion suggests that in order to achieve the best results in conditions of limited opportunities for investment in the work process and organization of spts tm, should be used the best-ranked methods or techniques. for ranking alternative solutions, a hybrid model based on two methods, dibr ii and mabac, was applied in a rough environment. research showed that proposed model could successfully evaluate the methods and techniques of the lean concept. considering that the mabac method was applied in a rough environment, the values from reality were very successfully reflected in the model. in this paper, the dibr ii method was applied for the first time in solving a specific case study. expert evaluation of the criteria carried out in this research, showed that the way of comparing the criteria used in this method was very suitable for understanding and use by the experts who had never encountered this issue before. presented model quality was tested in sensitivity analysis. through this process, it was checked the way the changes in the weight coefficients affect the ranking of alternatives. it is clearly observable, through the analysis, that the model is sensitive enough, but not too sensitive. high correlation of alternatives rankings during different changes in weight coefficients of the criteria indicates the stability of the model. basic limitation of the model is the fact that the experts were only able to use crisp values for comparison when defining the ratio of criteria. in some situations, where experts were not certain, there was a clear space for the application of scientific fields that treat uncertainties well, such as fuzzy numbers, rough numbers, gray numbers, etc. the improvement of this dibr ii method should be the subject of future research. references 1. epler, i.j., 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unmanned aircraft for the needs of tactical units of the armed forces, scientific technical review, 71(2), pp. 26-35. 45. radovanović, m., milić, a., petrovski, a., 2021, analysis of accuracy and precision of shooting with home: made automatic rifles using the ahp method, scientific technical review, 71(1), pp. 30-37. https://oresta.org/ 11485 facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 21 29 https://doi.org/10.22190/fume230108006h © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a good initial guess for approximating nonlinear oscillators by the homotopy perturbation method ji-huan he1,2,3, chun-hui he4,5, abdulrahman ali alsolami5 1national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china 2school of mathematics and information science, henan polytechnic university, jiaozuo, china 3school of science, xi’an university of architecture and technology, xi’an, china 4school of mathematics, china university of mining and technology, xuzhou, jiangsu, china 5department of mathematics, faculty of science, king abdulaziz university, jeddah, saudi arabia abstract. a good initial guess and an appropriate homotopy equation are two main factors in applications of the homotopy perturbation method. for a nonlinear oscillator, a cosine function is used in an initial guess. this article recommends a general approach to construction of the initial guess and the homotopy equation. duffing oscillator is adopted as an example to elucidate the effectiveness of the method. key words: homotopy perturbation method, nonlinear oscillator, periodic solution 1. introduction the nonlinear vibration theory has triggered skyrocketing interest in both nonlinear science and engineering, from vibration isolators [1] to nano materials [2] and micro-electromechanical systems [3,4], and the homotopy perturbation method [5] has laid the foundation for fast and accurate insight into the frequency-amplitude relation of a nonlinear oscillator, which occurs anywhere in engineering and science [6]. the traditional perturbation method [6] is widely used for this purpose, however, it is only valid for the received: january 08, 2023 / accepted february 05, 2023 corresponding author: ji-huan he and chun-hui he affiliation, national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, p. r. china e-mails: hejihuan@suda.edu.cn (j.h. he), mathew_he@yahoo.com (c.h. he) mailto:hejihuan@suda.edu.cn mailto:mathew_he@yahoo.com 22 ji.h. he, c.h. he, a.a. alsolami weak nonlinearity. other effective analytical methods include the variational iteration method [7,8,9] and the variational approach [10,11,12]. the homotopy perturbation method [5] is a universal method for nonlinear vibration systems, it has been considered as a strongly promising and unprecedented method for nonlinear problems [13,14], and many reliable modifications were recommended in literature, among which li-he’s modification [15,16] is much attractive. the homotopy perturbation method is to decompose a nonlinear equation to infinite linear equations where laplace transform can be applied, this modification is called as he-laplace method [17]. the parameterized homotopy perturbation method [18], and the couple of the homotopy perturbation method with kashuri fundo transform [19] or the lindstedt-poincare technology [20] has also been caught much attention. the homotopy perturbation method is extremely effective for fractional calculus [21-24], machine learning [25-28] and imaging process [29-33]. though the method is almost matured, there is still much space to further improvement. 2. homotopy perturbation method the homotopy perturbation method [5] is a powerful tool to nonlinear vibration systems. we consider a nonlinear vibration equation in the form ( ) 0u a u + = (1) where a is a nonlinear function of u, and a/u>0. in order to elucidate the solving process of the homotopy perturbation method, we can construct the following universal homotopy equation  2 2( ) ( ) 0 + + − + =u t u p u a u  (2) where ω is the frequency, p is the homotopy parameter. the homotopy perturbation method is to deform eq. (2) gradually from p=0 to p=1. when p tends to zero, eq. (2) results in a linearized oscillator, while when p tends to1, eq. (2) turns to be the original one. the most important two factors of the homotopy perturbation method are: 1) how to choose a good initial guess with possible unknown parameters; and 2) how to establish an appropriate homotopy equation. all iteration methods are sensitive to the initial guess. a good choice of the initial guess leads to a fast convergence, while an inappropriate choice might result in a wrong result [34]. to show the importance of the homotopy equation in the solving process, if we construct the following homotopy equation for a nonlinear oscillator  2 2( ) ( ) 0 − + + =u t u p u a u  (3) we need an infinite iteration to obtain an approximate solution converging extremely slowly to the exact solution. this is because when p=0, eq. (3) has no any property of oscillation, it should be emphasized that the initial guess must have the basic properties of the solution, the classic homotopy perturbation method always begins with 0 ( ) cos=u t a t (4) a good initial guess for approximating nonlinear oscillators by the homotopy perturbation method 23 in this paper, we improve the homotopy equation instead of eq. (2) and choose a better initial guess. instead of eq. (4), this paper uses the following initial guess 0 0 ( ) cos(2 1) = = + n i i u t a i t (5) where ai (i=0~n) are the constants satisfying the following identity 0= = n i i a a (6) the initial guess given in eq. (4), with an unknown frequency, is widely used in the homotopy perturbation method, while eq. (5) contains more unknown constants and provides a more flexible approach to an accurate identification of the frequency. 3. an effective improvement of the homotopy perturbation method in this section, we adopt the well-known duffing oscillator as an example to elucidate the solving process 3 0u u u + + = (7) with initial conditions (0) , (0) 0= =u a u (8) the duffing oscillator is always used as a good paradigm to elucidate the effectiveness and reliability of a method [35-39]. we choose an initial guess in the form 0 ( ) cos cos3= +u t a t b t  (9) eq. (9) is the exact solution of the following linear oscillator with a forcing term 2 2 0 0( ) 8 cos3 0 + + =u t u b t   , (0) =u a and (0) 0 =u (10) where a and b are unknown parameters. according to the initial conditions, the parameters a and b should satisfy the following identity + =a b a (11) accordingly we recommend the following homotopy equation  2 2 2 3 2( ) 8 cos3 (1 ) 8 cos3 0 + + + − + − =u t u b t p u u b t       (12) it is obvious that when p=0, eq. (12) becomes eq. (10), whose solution is eq. (9); when p=1, eq. (9) becomes the original one. for b=0, eq. (12) is the standard homotopy equation. according to the homotopy perturbation method, the solution is expanded as 2 0 1 2= + + +u u pu p u (13) 24 ji.h. he, c.h. he, a.a. alsolami eq. (12) becomes   2 2 2 2 0 1 2 0 1 2 2 2 2 3 2 0 1 2 0 1 2 ( ) 8 cos3 (1 )( ) ( ) 8 cos3 0   + + + + + + + + + − + + + + + + + − = u pu p u u pu p u b t p u pu p u u pu p u b t        (14) proceeding the standard solving process required by the perturbation method [5,6], we can obtain a series of linear differential equations. the first two equations are 2 2 0 0( ) 8 cos3 0 + + =u t u b t   , 0 (0) =u a and 0 (0) 0 =u (15) 2 2 3 2 1 1 0 0( ) (1 ) 8 cos3 0 + + − + − =u t u u u b t     , 1(0) 0=u and 1(0) 0 =u (16) the solution of eq. (15) is eq. (9). using this result, eq. (16) becomes 2 2 1 1 3 2 ( ) (1 )( cos cos3 ) ( cos cos3 ) 8 cos3 0  + + − + + + − = u t u a t b t a t b t b t          (17) simplifying eq. (17) yields the following equation 2 2 3 2 2 1 1 3 2 3 2 2 2 2 3 3 3 3 ( ) (1 ) ( ) cos 4 4 2 1 3 3 ( ) 8 cos3 4 2 4 3 3 3 1 ( )cos5 cos7 cos9 4 4 4 4    + + − + + +      + + + − + +    + + + + u t u a a a b ab t a a b b b b t a b ab t ab t b t              (18) no term of tcosωt should be involved in u1 for a periodic solution, so the coefficient of cosωt should be zero: 2 3 2 23 3 3 (1 ) ( ) 0 4 4 2 − + + + =a a a b ab  (19) or 2 2 2 2 2 23 3 3 3 3 1 ( ) 1 ( 2 ) 1 ( 2 ) 4 4 2 4 4 = + + + = + + = + + −a ab b aa b a b ba    (20) from eq. (7) we have 3 2(0) (0) (0) (1 )u u u a a  = − − = − + (21) while eq. (9) predicts 2(0) ( 9 )u a b  = − + (22) we, therefore, have 2 2( 9 ) (1 )+ = +a b a a  (23) or 2 2 2 (1 ) (1 ) 9 8 + + = = + + a a a a a b a b    (24) a good initial guess for approximating nonlinear oscillators by the homotopy perturbation method 25 for given a and ε, we can obtain the approximate frequency easily by solving eqs. (20) and (24) simultaneously. 4. discussion in case ε<<1 , we can determine approximately the value of b from the following equation by the perturbation method 2 2 23 (1 ) 1 ( 2 ) 4 8 + + + − = + a a a b ba a b   (25) the perturbation solution for b reads 31 32 =b a (26) we, therefore, obtain 2 2 2 2 1 3 1 1 4 1 4 + = = + + a a a     (27) eq. (27) is just same as that solved by the classic homotopy perturbation method, and it is valid for ε<<1. considering our small assumption of the parameter ε, we cannot fail to appreciate its harmony and intoxicating formula valid for all values of ε>0 . considering another case when ε tends to infinity, eq. (25) becomes 3 2 23 ( 2 ) 4 8 + − = + a a b ba a b   (28) solving for b from eq. (28) results in 0.04943=b a (29) as a result, we have 2 2 2 2(1 ) 1 0.7166 8 1.3955 + = = = + a a a a a b     (30) or 1/20.8465= a  (31) while the exact frequency when ε tends to infinity is [40]: 2 2 /2 2 0.5 0 = 0.8472 2 (1 0.5sin ) ex a a t dt      − = = − (32) 26 ji.h. he, c.h. he, a.a. alsolami fig. 1 comparison between the exact solution (continuous line) and the approximate one (discontinuous line) for different cases: (a) a=1 and ε =1; (b) a=1 and ε=100; (c) a=1 and ε=10000. a good initial guess for approximating nonlinear oscillators by the homotopy perturbation method 27 the relative error is 0.08%. fig.1 gives the comparison between the approximate solution and the exact one for different cases, showing an extremely high accuracy of the approximate solution from small to large ones. 5. conclusion we looked into the effect of the initial guess on the solution accuracy. obviously, the obtained solution has a better accuracy than that by the classic homotopy 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excitation characteristics of the centrifugal pump considering the compound whirl effect wenjie zhou1, dongliang yu1, yifan wang1, junlin shi2, bin gan2 1school of energy and power engineering, jiangsu university, zhenjiang, china 2school of mechanical engineering, sichuan university of science & engineering, zigong, china abstract. in order to study the correlation mechanism between the flow characteristics and the fluid-induced force under the compound whirl motion in the centrifugal pump, the rng k-ε model is selected in this paper to simulate a low specific speed centrifugal pump with impeller eccentricity based on the n-s equation. the changes of fluid-induced force with impeller eccentricity and the unsteady flow characteristics of the internal flow field of centrifugal pump under different flow conditions and rotation speeds are investigated, and the relationship between the fluid-induced force of the impeller and the internal flow field characteristics is discussed. the results show that the trend of fluid-induced force and the pressure coefficient is similar. when the rotation speed changes and when the flow is similar, the pressure coefficient under different rotation speeds almost coincides. with the increase of impeller speed and impeller eccentricity, the dynamic and static interferences between the impeller and the volute tongue are more significant, the uneven distribution of the pressure around the impeller makes the internal flow of centrifugal pump more disordered and increases the fluid-induced force near the volute tongue. the research results can provide important reference value for accurately grasping the internal flow excitation principle of the centrifugal pump. key words: centrifugal pump, compound whirl, fluid-induced force, internal flow characteristics 1. introduction centrifugal pumps are widely used in many fields, such as water supply, irrigation and liquid transportation, etc. as one of the investigation focuses, the internal flow phenomena and the characteristics have been given much attention because the flow in the pump is received may 28, 2021 / accepted october 22, 2021 corresponding author: wenjie zhou school of energy and power engineering, jiangsu university, zhenjiang 212013, china e-mail: zhouwenjiezwj@ujs.edu.cn 224 w. zhou, d. yu, y. wang, j. shi, b. gan complicated and unordered [1-6]. there are many factors affecting the flow characteristics of the centrifugal pump, among which, the wake flow of blade and the dynamic and static interference between the impeller and the volute tongue are the two key factors [7, 8]. actually, due to various assembly errors in centrifugal pumps, the geometric center of the impeller does not coincide with the mass center [9]. this phenomenon could generate fluid excitation force and change the dynamic characteristics of the rotor system. therefore, it is important to study the inner flow characteristics and corresponding fluid-induced force of the centrifugal pump considering the eccentric whirl motion of the impeller. the inner flow characteristics are helpful to reveal the essence of fluid-induced vibration of the centrifugal pump. liu et al. [10] analyzed the pressure fluctuations by cfd tool and found that the three-dimensional inlet guide vane can reduce the amplitude of the impeller pressure fluctuation at a small flow rate and a design flow rate. bai et al. [11] proposed a smaller number of diffuser blades, which could reduce the pressure fluctuation in the diffuser. zhang et al. [12] found that changing the shape of the blade trailing edge could greatly improve the flow field at the outlet of the blade and reduce the pressure fluctuation in the centrifugal pump. yang et al. [13] believed that the time series effect had a great influence on the rotor-stator interaction (rsi) and the change of the inducer blade position could reduce the pressure fluctuation. tao et al. [14] pointed out that increasing the flow area of the volute could reduce the pressure fluctuation in the volute and the radial force imposed on the shaft. chalghoum et al. [15-17] studied the influence of static and dynamic interference between the impeller and the volute tongue on internal flow and fluid excitation effect. zou et al. [18] explored the flow field around the impeller and transient fluid excitation effect during the start-up for a large double suction centrifugal pump. tao et al. [19] studied the pressure fluctuation of the impeller by the separated eddy numerical simulation method and pointed out that the predicted value of the pressure fluctuation could not be in good agreement with the experimental value without considering the eccentricity of the impeller. in addition, the influence of clearance on the flow field and fluid excitation effect in pumps is also studied [20-22]. actually, the numerical simulation has become an effective research method for different scale flow problems in engineering [23, 24]. fluid-induced force is another manifestation of fluid excitation effect, which plays a vital impact on the vibration and stability of centrifugal pumps. yao et al. [25] found that the displacement degree of the centrifugal pump shaft increased obviously with the decrease of the flow rate. during the start-up of the centrifugal pump, the direction of radial force acting on the pump shaft is the same as the displacement direction of the pump shaft under the condition of a small flow rate. tan et al. [26] pointed out that the interaction between the rotor and the stator had a great influence on the fluid-induced force, but the rotation speed had a limited influence on it. li et al. [27] changed the load distribution of the blade and found that the front-loaded impeller made the flow inside the pump more uniform, which could reduce the fluid-induced force. barrio et al. [28, 29] obtained the unsteady pressure distribution through the pressure sensor and radial integration of the pressure on the clearance side. the results showed that the numerical results of the fluid-induced force were in good agreement with the experimental results. karaskiewicz et al. [30] examined the correlation between the fluid-induced force and the axial thrust moment of the centrifugal pump. zhou et al. [31] studied the influence of different flow rates, eccentricities and whirl ratios on the flow induced force of the impeller. it is found that the flow induced force is closer to the experimental data when considering the research on the fluid-induced excitation characteristics of centrifugal pump considering... 225 eccentricity of the impeller, and the fitting model of the flow induced force is established. alemi et al. [32] explored the influence of multi-volute structure on hydraulic performance and the impeller fluid-induced force under off-design conditions. jiang et al. [33] found that the fluid-induced force of the impeller would become larger when the diffuser blade approached the volute tongue while the pressure fluctuation intensity and efficiency were reduced. it is suggested that the volute should be placed between the two blades to improve the performance of the centrifugal pump. besides, the effects of the fluid-induced force on the pump rotor system also attracted more and more researchers with the development of different fluid excitation models in recent years [34-39]. even the fluid-induced force of the impeller has drawn a lot of attention; most of the studies only consider the self-rotation and ignore the impeller’s whirl motion. in fact, the real work condition of the impeller includes not only self-rotation but also the orbit motion whirling around the geometric center of the impeller due to eccentricity. the compound motion plays a vital influence on the fluid-induced vibration characteristics of the centrifugal pump. besides, the relationship between the internal flow structures and the macroscopic fluid exciting force still needs further investigation. therefore, a novel model of the centrifugal pump including the compound motion is proposed in order to explore the fluid excitation characteristics in this paper. in order to verify the accuracy of calculation model, the performance curves of numerical simulation and experimental test are compared. the changing trends of the impeller fluid-induced force under different eccentricities, flow rates and rotational speeds are studied respectively. based on the results, the distribution law of the impeller fluid-induced force is researched and the correlation mechanism between the unsteady flow characteristics and the impeller fluid-induced force is also discussed. the results are helpful to enrich the impeller model of the centrifugal pump and provide an important reference for further coupled vibration of the rotor system. 2. 3-d model of centrifugal pump the research object is a low specific speed centrifugal pump with cylindrical blades. the three-dimensional model of this centrifugal pump mainly includes five parts: inlet, impeller, clearance, volute and outlet. the impeller offsets along the outlet direction with different eccentricities for whirl motion. the corresponding design parameters and assembled 3-d calculation model can be seen in table 1 and fig. 1, respectively. hd is the head, qd is the flow rate, z is the blade number, ω is the rotation speed, ns is the specific speed, d1 is the inlet/outlet diameter, d2 the impeller outlet diameter, d3 is the volute inlet diameter. table 1 main design parameters of centrifugal pump parameters hd(m) qd(m 3/h) z ω(r/min) ns d1(mm) d2(mm) d3(mm) value 20 55 6 1450 69 80 260 290 226 w. zhou, d. yu, y. wang, j. shi, b. gan fig. 1 3-d model of centrifugal pump (a) explosion form and (b) assembly form the impeller rotates around its own rotating center at rotation speed ω; meanwhile it rotates around the center of the volute base circle at whirl speed ω in the same circumferential direction. these two different motions construct the actual motion of the centrifugal pump impeller. fig. 2 precession motion of the centrifugal pump impeller the distance between above two centers is e, which is the impeller eccentricity, ε is the eccentricity ratio of e to the initial clearance width d of the impeller. the precession motion of the centrifugal pump impeller in a period is shown in the fig. 2. 3. numerical calculation method based on the n-s equation, the rng k-ε model is selected to flow analysis because it has higher calculation accuracy for a high strain rate flow and complex flow in the centrifugal pump [40]. the impeller and clearance domains are set as rotating domains, the volute, inlet and outlet domains are kept as static reference frame, the interface between static and rotating domains is applied to exchange flow field data. the boundary of inlet and outlet is set as velocity inlet and free outflow, respectively, to describe the actual flow condition. besides, the wall is set as sliding boundary and standard wall function. the time research on the fluid-induced excitation characteristics of centrifugal pump considering... 227 step in unsteady calculation is set to δt = 1.149410-4 s to ensure the coulomb number is less than 1. in addition, the prediction head is taken as the criterion of grid independence to ensure the accuracy of calculated results before the unsteady flow research. the grids in each domain are divided and merged, and the design conditions with different grids are calculated. when the difference between each two predictions is less than 0.5%, the smaller grid is considered as the best grid because it has high accuracy and can save computing resources [41]. 4. verification of the computational model in order to verify the accuracy of the proposed model, the head and efficiency curves under two different conditions of non-eccentricity and 3%-eccentricity are calculated, respectively. the comparison results between the simulation and experiment [42] from 0.8qd to 1.2qd are shown in fig. 3. it is seen that the minimum error between the simulated head and experimental head when  = 0.03 is 0.32% at design flow while the maximum error is 0.75% at 0.8qd. the minimum and maximum errors without considering the whirl motion are 1.10% and 2.29%, respectively. besides, the comparison for pump efficiency presents the similar results. when eccentricity is considered, the minimum and maximum errors are 0.05% and 0.60%, respectively. while they change to 0.59% and 1.75%, respectively, the self-rotation motion is only considered. the verification implies that the flow analysis considering the compound whirl effect of the impeller has a better calculation accuracy compared with the present self-rotation model. this is because the proposed model is closer to the actual motion status of the centrifugal pump impeller and it can better predict the inner flow characteristics. fig. 3 comparison of the experimental and simulated results (a) head and (b) efficiency 228 w. zhou, d. yu, y. wang, j. shi, b. gan 5. computational results and analysis 5.1. effects of , q and ω on fr and cprmse the fluid-induced force of the impeller can be ignored when the circumferential structure of the volute is symmetry. however, the theoretical hypothesis is not valid for the centrifugal pump impeller. there are two following reasons: the first one is that the structure of the volute for a single centrifugal pump is unequal cross-section area, especially the volute tongue structure; the second one is the whirl motion of the impeller, which could induce n uneven flow and generate periodic fluid-induced force. the fluid-induced force can be integrated by the transient pressure on each impeller wall in the unsteady calculation. the influence of eccentricity ratio , flow rate q and rotation speed ω on the fluid-induced force are analyzed. the 20 monitoring points on the outlet edge of the impeller with a certain interval angle 18°, shown in fig. 4, are selected for better presentation and its corresponding static pressure values are calculated. p is the pressure in the pump which is monitored by the monitoring points. the pressure amplitude coefficient cp and corresponding root mean squared error cprmse can be calculated according to equation (1) and equation (2), respectively. cprmse presents the uneven distribution of the impeller outlet pressure, the higher cprmse means the greater pressure fluctuation. fig. 4 monitoring points around the outlet of impeller 2 20.5 p p c u = (1) 2 e 1 1 ( ) 20 n prms pi p i c c c = = − (2) in actual operation, a variety of flow conditions are always needed to meet the industrial requirements. therefore, the transient fluid-induced forces under three flow conditions of 0.8qd, 1.0qd, 1.2qd are calculated, respectively. as shown in figs. 5(a) ~ 5(c), on the whole, there are six peaks and troughs of fluid-induced force fr under the three eccentricities, which is related to the rotor-stator interaction caused by the periodic sweeping of the six blades of impeller. research on the fluid-induced excitation characteristics of centrifugal pump considering... 229 0 60 120 180 240 300 360 -70 -35 0 35 70 105 140 f r (n ) q /degree fr with ε =0 fr with ε =0.03 fr with ε =0.06 cprmse with ε =0 cprmse with ε =0.03 cprmse with ε =0.06 0.02 0.03 0.04 0.05 0.06 0.07 0.08 c p r m s e area(2) area(1) (c) 0 60 120 180 240 300 360 -30 -15 0 15 30 45 60 f r (n ) q /degree fr with ε =0 fr with ε =0.03 fr with ε =0.06 cprmse with ε =0 cprmse with ε =0.03 cprmse with ε =0.06 0.02 0.03 0.04 0.05 0.06 0.07 0.08 c p r m s e area(2) area(1) (b) 0 60 120 180 240 300 360 -50 -25 0 25 50 75 100 f r (n ) q /degree fr with ε =0 fr with ε =0.03 fr with ε =0.06 cprmse with ε =0 cprmse with ε =0.03 cprmse with ε =0.06 0.02 0.03 0.04 0.05 0.06 0.07 0.08 c p r m s e area(2) a re a( 1 ) (a) area(1) different q 0° 90° 180° 270° fig. 5 effects of different q (a) 0.8qd, (b) 1.0qd and (c) 1.2qd on the fr and cprmse with the increase of the flow rate from 0.8qd to 1.2qd, fr presents obvious stratification phenomenon. under the design flow rate, the fluctuation amplitude is minimum, which means that the change of the fluid-induced force is the lowest. at the same time, the average value of fluid-induced force fr-m in periodic motion is calculated, the sequence of fr-m for different flow rates is fr-m(1.2qd)> fr-m(0.8qd)> fr-m(1.0qd) under the same impeller eccentricity. trending curves of fr and cprmse show a positive correlation. the position distribution and interval distribution of the curves are consistent to a certain extent, which implies that the uneven distribution of the circumferential pressure of the impeller is an important source of the fluid-induced force. it can also be seen that when the impeller is not eccentric, the extreme of the fluid-induced force under the three flow rates are stable. in addition, fr(1.0qd) and fr(1.2qd) in the range of 0°~180°, which is far away from the volute tongue, are obviously smaller than those in the range of 180°~360° when eccentricity is considered. however, fr(0.8qd) in fig. 5(a) presents the opposite changing trend. besides, with the increase of eccentricity, the fluctuation amplitude of fr at non-design flow rate increases significantly and the corresponding distribution is more disordered compared with that at design flow rate. the similar conclusion is also suitable for cprmse, the fluctuation amplitude of cprmse at non-design flow rate is obviously larger than that at design flow rate when eccentricity is considered. the above calculated results imply that the non-design flow rate can significantly intensify the circumferential flow of the impeller and the fluid-induced force should not be regarded as a periodic force with same amplitude when the whirl motion effect is considered. 230 w. zhou, d. yu, y. wang, j. shi, b. gan in order to study the influence of rotation speed on the fluid-induced force, the transient fr at three different rotation speeds and different eccentricities is calculated. it can be seen from fig. 6(a) ~ 6(c) that the number of peaks and troughs of fr and cprmse is also 6 due to the same number of blades. under the same eccentricity, fr curve has a similar shape and trend at different rotation speeds. the maximum value and range of fr grows with the increase of the impeller eccentricity, and the average value of fr in one cycle increases with the rise of rotation speed, which is fr-m(1750rpm)> fr-m(1450rpm)> fr-m(1150rpm). fig. 6 effects of different ω (a) 1150rpm, (b) 1450rpm and (c) 1750rpm on the fr and cprmse the above results show that although the internal flow of the centrifugal pump is similar after similarity transformation, the rotor-stator interaction intensity and turbulence intensity are still various at different rotation speeds, and they all increase with the rise of the rotation speed and the impeller eccentricity. the larger fluid-induced force and the oscillation amplitude mean a greater energy loss, which means that the design of the centrifugal pump parameters needs to coordinate various parameters to ensure higher efficiency and a more reasonable flow condition. meanwhile, as fig. 6(a) ~ 6(c) show, the trend of cprmse curve is positively correlated with the fluid-induced force curve, and the size distribution and interval distribution of the two curves under each working condition are very similar to a certain extent. in addition, the range and the average value of cprmse under different eccentricities and rotation speeds are the same as the law of fluid-induced force fr. the average value of cprmse grows with the increase of rotation speed, which is the law of cprmse-m(1750rpm)> cprmse-m(1450rpm)> research on the fluid-induced excitation characteristics of centrifugal pump considering... 231 cprmse-m(1150rpm). it shows that with the increase of the rotation speed, the dynamic and static interference between the impeller and the volute tongue is more intense, and the circumferential pressure distribution at the impeller outlet is more disordered at each moment, which means that the internal flow field distribution of the centrifugal pump is more uneven. the great consistency between the characteristics of cprmse and fluid-induced force fr shows that the dynamic and static interference between the impeller and the volute tongue is one of the important sources of the fluid-induced force. 5.2. effects of , q and ω on pressure fluctuation through the above calculation and analysis, it is known that the impeller eccentricity has an obvious effect on the internal flow and unsteady characteristics of the centrifugal pump. moreover, in order to better investigate the correlation mechanism between the unsteady flow characteristics and the fluid-induced force under the compound whirl motion, the pressure fluctuations considering the compound whirl effect at the monitoring points under different flow conditions and rotation speeds are further studied. the monitoring point 2, which is closest to the volute tongue, and other monitoring points 3, 6, 12 and 17 are selected for the pressure fluctuation study. in the following figures, fr and fbpf mean the rotation frequency and blade passing frequency, respectively. in fig. 7, it is seen that there are a lot of low frequency signals between 0 and fbpf, and the shaft frequency increases significantly with the increase of the impeller eccentricity for 0.8qd. that means the dynamic and static interference between the impeller and the volute tongue is more significant with the increase of the impeller eccentricity, which makes the internal flow of the centrifugal pump more disordered and increases the uneven distribution of the circumferential pressure in the impeller. consequently, the macroscopic manifestation fluid-induced force would increase synchronously, which corresponds to the rule that fr presents fr-m (6%) > fr-m (3%) > fr-m (0%) in fig. 5(a). =0 =0.03 =0.06 fbpf fbpf fbpf fr fig. 7 comparison of pressure spectrum of different monitoring points under 0.8qd as we known, the farther away from the monitoring point of the volute tongue, the smaller impact of the static and dynamic interference on it. in figs. 8 and 9, fbpf at the all monitoring points shows the same change of fp2 > fp3 > fp6> fp12> fp17 in general. therefore, the pressure fluctuation on the side near the volute tongue is larger, and the non-uniformity of the impeller circumferential pressure is more significant, which is also consistent with the calculated results in fig. 5(b) and (c) that fr shows a trend of fr (6%) < fr (3%) < fr (0%) from 0°~180°and fr (6%) > fr (3%) > fr (0%) from 180°~360°. 232 w. zhou, d. yu, y. wang, j. shi, b. gan =0 =0.03 =0.06 fr fbpffbpf fr fbpf fig. 8 comparison of pressure spectrum of different monitoring points under 1.0qd =0 =0.03 =0.06 fr fbpf fbpf fr fbpf fig. 9 comparison of pressure spectrum of different monitoring points under 1.2qd besides, fr, fbpf and 2fbpf at the monitoring point p2 are given more special attention and these data are extracted to research the mechanism, the corresponding amplitude are listed in table 2. on the whole, when the impeller is eccentric, fr increases significantly with the increase of the impeller eccentricity. at the same flow rate, with the increase of the impeller eccentricity, fr enlarges and becomes more significant, fbpf and 2fbpf are reduced with the increase of the impeller eccentricity. in addition, fr shows an inverse proportional relationship with the flow rate when the impeller is not eccentric. on the contrary, fr, fbpf and 2fbpf increase with the flow rate when the compound whirl effect is considered. table 2 amplitude of fr, fbpf and 2fbpf of p2 in different  and q  q/qd fr (pa) fbpf (pa) 2fbpf (pa) 0 0.8 31.777 3971.309 1212.424 0 1.0 22.395 6324.310 1549.156 0 1.2 15.641 8575.916 2036.486 0.03 0.8 759.772 3612.386 1058.963 0.03 1.0 868.504 6024.788 1375.688 0.03 1.2 1223.527 8372.829 1831.697 0.06 0.8 1872.443 3654.332 1043.536 0.06 1.0 1893.877 5955.756 1367.366 0.06 1.2 2780.961 8271.600 1764.368 the effects of the compound whirl motion on the pressure fluctuation of the centrifugal pump at different rotation speeds are also calculated. it can be seen from fig. 10 to fig. 12 that the amplitude of the pressure fluctuation at the blade frequency of monitoring point p2 shows an upward trend with the increase of the impeller eccentricity, which corresponds to the rule that fr presents fr (6%) > fr (3%) > fr (0%) from 180°~360°in fig. 6(a) to fig. (c). furthermore, compared with the calculated data for different rotation speeds, the amplitudes of the blade frequency and its octave frequency at each monitoring point become large with the rotation speed increase. research on the fluid-induced excitation characteristics of centrifugal pump considering... 233 =0 =0.03 =0.06 fbpf fbpf fbpf frfr fig. 10 comparison of pressure spectrum of different monitoring points under 1150rpm =0 =0.03 =0.06 fr fbpffbpf fr fbpf fig. 11 comparison of pressure spectrum of different monitoring points under 1450rpm =0 =0.03 =0.06 fbpf fr fbpf fr fbpf fig. 12 comparison of pressure spectrum of different monitoring points under 1750rpm fr, fbpf and 2fbpf at the monitoring point p2 are also compared and analyzed in table 3. at the same ω, fr increases significantly with the impeller eccentricity, and fbpf and 2fbpf with eccentricity are less than those without eccentricity. in addition, the change of fr without the impeller eccentricity is limited, which means that fr is not sensitive to the rotation speed if the compound whirl effect is not considered. table 3 amplitude of fr, fbpf and 2fbpf of p2 in different  and ω  ω fr (pa) fbpf (pa) 2fbpf (pa) 0 1150 28.2 4037.8 1011.9 0 1450 22.4 6324.3 1549.2 0 1750 25.5 9280.3 2303.7 0.03 1150 603.9 3722.8 853.6 0.03 1450 868.5 6024.8 1375.7 0.03 1750 1373.8 8652.7 1978.2 0.06 1150 1286.7 3730.9 861.6 0.06 1450 1893.9 5955.8 1367.4 0.06 1750 2747.7 8626.7 1970.1 fbpf and 2fbpf enlarge with the rotation speed increase when the impeller is eccentric, but it is almost the same after dimensionless in table 4. due to the similarity of flow, the 234 w. zhou, d. yu, y. wang, j. shi, b. gan different impeller eccentricities and rotation speeds have little influence on the amplitude of fbpf at the monitoring point, which can also explain the trend that the pressure coefficient cprmse in fig. 6 is almost the same for three different eccentricities. the results can help us to understand the correlation mechanism between the unsteady flow characteristics and the fluid-induced force of the centrifugal pump with the compound whirl motion more intuitively. table 4 dimensionless amplitude of fr, fbpf and 2fbpf of p2 in different  and ω  ω fbpf 2fbpf 0 1150 0.0330 0.0083 0 1450 0.0325 0.0080 0 1750 0.0327 0.0081 0.03 1150 0.0304 0.0070 0.03 1450 0.0310 0.0070 0.03 1750 0.0305 0.0070 0.06 1150 0.0304 0.0070 0.06 1450 0.0306 0.0070 0.06 1750 0.0304 0.0069 5.3. effects of , q and ω on pressure distribution finally, the influence of the compound whirl motion for different flow rates and rotation speeds on the pressure distribution are researched. it can be seen from fig. 13 that the turbulence in the centrifugal pump gradually develops fully with the change of the impeller eccentricity and the outlet pressure decreases with the flow rate increase. moreover, the pressure distribution gradient at the volute tongue presents the rule of dp/dl(1.2qd)> dp/dl(0.8qd)> dp/dl(1.0qd) under the same impeller eccentricity, which is similar to the rule of fr-m(1.2qd)> fr-m(0.8qd)> fr-m(1.0qd) in fig. 5(a) ~ 5(c). the calculated results verify that the dynamic and static interference between the impeller and the volute tongue becomes more intense when the eccentricity of the impeller increases, which makes the flow condition of the centrifugal pump more disordered and increases the fluid-induced force near the volute tongue. it also shows that the dynamic and static interference between the impeller and the volute tongue is one of the important sources of the fluid exciting force. fig. 13 comparison of pressure distribution for different impeller eccentricities and flow rates research on the fluid-induced excitation characteristics of centrifugal pump considering... 235 1150rpm 1450rpm 1750rpm 145918 96939 47959 -1020 -50000 [pa] 243469 165102 86735 8367 -70000 341429 238571 135714 32857 -70000  = 0  = 0.03  = 0.06 fig. 14 comparison of pressure distribution for different impeller eccentricities and rotation speeds in fig. 14, under the same impeller eccentricity, the uneven degree of pressure distribution around the impeller increases with the rise of rotation speed. the calculation results show that dp/dl(1750rpm)> dp/dl (1450rpm)> dp/dl (1150rpm), which verifies that the average value of the fluid exciting force in fig.6(a) ~ 6(c) presents the distribution law of fr-m(1750rpm)> fr-m(1450rpm)> fr-m(1150rpm). in addition, at the same rotation speed, with the increase of the impeller eccentricity, the flow near the volute tongue is more turbulent. the flow near the volute tongue shows that dp/dl(ε=0.06) > dp/dl (ε=0.03) > dp/dl (ε=0), which is the same as the rule of fr(ε=0.06)> fr (ε=0.03)> fr (ε=0) in 180°~360°. this shows that the dynamic and static interference between the impeller and the volute tongue intensifies when the impeller eccentricity increases, which makes the uneven pressure distribution around the impeller more significant. 6. conclusion in this paper, a novel model considering the compound whirl motion is proposed to describe the actual motion of impeller and research the internal flow characteristics of centrifugal pump. the effects of eccentricity, flow rate and rotation speed on the fluid-induced force, pressure fluctuation and pressure distribution are mainly calculated. the mechanism correlation between the fluid-induced force and the internal flow characteristics are also analyzed and discussed. the main conclusions are as follows: 1. the proposed model considering the compound whirl effect can better describe the actual motion of the centrifugal pump compared with the present calculation model. the maximum errors of the head and the efficiency calculated by the novel model are 0.75% and 0.6%, respectively, while they are 2.29% and 1.75% calculated by the present model. 2. the fluid-induced force is sensitive to the impeller eccentricity. the fluid-induced force curve would change from smooth to steep when  increase from 0 to 0.06. in addition, the fluid-induced force and the pressure coefficient of the circumferential pressure present a similar changing trend. 236 w. zhou, d. yu, y. wang, j. shi, b. gan 3. the dynamic and static interference between the impeller and the volute tongue becomes more evident with the increase of the impeller eccentricity, which makes the internal flow of the centrifugal pump more disordered and increases the uneven distribution of pressure around the impeller. therefore, fr near the volute tongue is larger than that away from the volute tongue. 4. compared with the amplitudes of fbpf and 2fbpf, the amplitude of fr is more sensitive to the impeller eccentricity than flow rate and rotation speed. this is because the precession motion of the impeller, which has the same frequency with the rotation speed, increases the periodic dynamic and static interference between the impeller and volute tongue. acknowledgements: this work was supported by the national natural science foundation of china (grant no. 51706087), the project funded by china postdoctoral science foundation (grant no. 2018m642177), the sichuan provincial key lab of process equipment and control (grant no. gk201902), the zhejiang postdoctoral preferential foundation (grant no. zj2018009) and the key research and development program of zhenjiang (grant no. gy2018023). references 1. dong, l., shang, h., zhao, y., liu, h., dai, c., wang, y., 2019, study on unstable characteristics of centrifugal pump under different cavitation stages, journal of thermal science, 28(4), pp. 608-620. 2. li, w., zhou, l., shi, w. d., ji, l., yang, y., zhao, x., 2017, piv experiment of the unsteady flow field in mixed-flow pump under part loading condition, experimental thermal and fluid science, 83, pp. 191-199. 3. zhang, j., li, g., mao, j., yuan, s., qu, y., jia, j., 2020, numerical investigation of the effects of splitter blade deflection on the pressure pulsation in a low specific 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(in chinese) facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 17 27 https://doi.org/10.22190/fume190104013p © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  tire wear particle hot spots – review of influencing factors roman pohrt technische universität berlin, germany abstract. automotive tires have played an important role in land-based transportation and will probably continue to do so for many years to come. during their service lifetime, parts of the outer protector layer are worn off and discarded into the environment. a typical passenger car emits about 120 micrograms of rubber per meter but the exact current value depends on a multitude of influencing factors and varies greatly. we review available data on the wear rate (or inverse expected lifetime) of automotive rubber tires and extract qualitative estimations on how the most important parameters alter the deposition rate on a given road section. local hot spots of increased tire wear particle occurrence can be identified from these parameters. it is concluded that generally subjecting tires to milder usage conditions can reduce tire wear by substantial amounts. reducing vehicle speeds is identified as the most effective general measure. key words: tire, wear rate, road particles, non-exhaust emissions, hot-spots 1. introduction transportation on land has relied on the use of rubber tires for many years. being the component of the vehicle that is actually in contact with the ground, the tires support the vehicle weight and transfer loads such as lateral or horizontal accelerations. the number of passenger cars in the world was 1.02 billion in 2016 and is expected to reach 1.3 billion by 2024 [1]. in recent years, the focus of scientific research has shifted away from the emission directly related to the internal combustion. instead, the so called non-exhaust-emissions have gained increasing attention [2-6]. this shift of focus is sensible because with increasing electrification of traffic, these emissions will remain relevant [7]. part of the non-exhaust-emissions is the particles generated from the wear of the tires and the road. because of their comparably large size, most of the particles emitted are not received january 04, 2019 / accepted march 16, 2019 corresponding author: roman pohrt technische universität berlin, sekr. c8-4, straße des 17. juni 135, 10623 berlin, germany e-mail: roman.pohrt@tu-berlin.de 18 r. pohrt airborne. instead, the particles are transported to the side of the road where they enter the soil or they are washed away with rain, either entering surface waters or going to waste water treatment, depending on the sewage water system [8, 9]. thus, one possible approach to reducing the emission of wear particles from tires is to devise local collection systems in the sewage drains using filters that can be cleaned at regular intervals. however, some knowledge about the local emission rate from traffic is required. average emission rates are often obtained when testing the product life span of tires. such test can be found in academic context as well as in consumer magazines. however, most such tests provide an emission rate for a specific vehicle and a specific tire, averaged over a given course. momentary emission rate w (expressed in µg/m) of that tire during the course is likely to vary far more than different tires vary over the same course. we are instead interested in obtaining the deposition rate at a specific road section, averaged over an ensemble of vehicles that have passed this section. the following review paper aims at giving qualitative dependencies that allow comparing the amount of emitted tire wear particles between different road sections. we define deposition rate d in µg/(md) as the mass of tread material that all passing vehicles have lost during one day per meter road length. the following approach will be taken. as a base comparable value, we will compile available data on the dependency of the momentary wear rate of a passenger car from literature. this data can be taken from dedicated test drives, laboratory test benches or will be deduced from rough calculations of the vehicle forces. for vehicles other than passenger cars, it will be assumed that the qualitative dependencies on properties of the road section will remain the same and the wear rate will only differ by a constant factor. this approach deliberately excludes many influencing factors that alter the wear rate of any specific vehicle, some of which are discussed in section 3.5. 2. particles of tire wear tread rubber of tires is emitted in the form of elongated particles. in real life, these particles do not consist of pure tread material but are instead a mixture including wear particles from the road and possibly from other sources. therefore, the particles found on roadsides are usually referred to as tire and road wear particles (trwp) and contain a multitude of materials stemming from other traffic-related or environmental sources [10]. the identification of particles as being trwp is not trivial and still subject of current research [11, 12]. different chemical markers and procedures have been applied [13]. recently a standard was given to measure twrp concentrations based on the pyrolysisgc/ms method [14]. most studies agree that only 0.55…10% [12] of particle mass is below 10 µm. instead the characteristic size is in the order of 65-80 µm [15, 16] and does not get airborne [17, 18]. fig. 1 shows typical particles. see [19] for a recent review on trwp research. because the aim of this study is to give only quantitative relations, we will focus solely on wear rate w defined as the amount of tread material removed from the vehicle per unit road length travelled. the resulting amount of trwp will be higher because of additional material encrusted. tire wear particle hot spots – review of influencing factors 19 fig. 1 rem images of tire and road wear particles, from [20] 3. wear rates of automotive tires the wear rate of an automotive tire is the amount of mass lost (δm) per distance covered (d) and is given in [µg/m]. m w d   . (1) during its service life, a typical modern tire of the passenger car loses around δm=1.4 kg of mass within approximately d=50000 km. the typical overall wear rate of the passenger car (with 4 tires) is, therefore, approximately 112 µg/m. see [6, 9] for an extensive discussion of this average value. however, wear rate w of a tire is a momentary quantity for a specific tire or vehicle in a specific driving situation and varies greatly over time. because no measurement technology is available to capture the wear rate directly on the vehicle during operation, researchers rely on measuring the remaining mass of the tire after a certain operational duration. depending on the influencing factor in question, the following tests can be performed:  vehicle tests use actual cars driving either on public roads or test facilities. the course covered is recorded and used to calculate w based on distance. this type of test is very useful for distinguishing w for different cars, but data for specific driving situations is rarely available because the cars are usually used in a multitude of manoeuvres before the measurement (averaging) takes place. see [2123] for examples of such tests.  bench tests use single tires in a specially designed test machine that emulates driving. the bench test can either be a wheel-in-a-drum [24, 25] or carousel configuration [26]. often times, the tire load and torque as well as the steering angle can be controlled. see [27, 28] for examples of such tests. the advantage of such tests is that very specific situations of tire loading can be investigated.  model tests do not use actual tires but employ the test machine with small rubber samples rolling on a rough counterpart surface. these tests are inexpensive; they allow for parameter studies and are often used to compare chemical compositions 20 r. pohrt of the tread material. different machines have been designed to fit special purposes [29, 30] but the most common test rig was developed by grosch. see [31] for a description of the machine and results. 3.1. traffic profile the intensity and type of traffic that circulates on a given road have a direct influence on d. it is reasonable to assume that individual emissions from all vehicles will sum up to give the total deposition. the simplest approach is to assume a constant emission rate (such as the above-mentioned 112 micrograms/km) multiplied by the number of vehicles during a day. when the vehicles are categorized according to their weight, the deposition rate can be calculated as i i id nw , (2) where ni is the number of vehicles from category i per day and wi is the average emission rate for that category. regrettably, literature on wi for vehicles other than passenger cars is scarce. average values of 107 1500 micrograms/m per lorry alone are listed [32]. this scattering is in part due to a great variety of vehicle types and their configurations. in a rare investigation, gebbe et al. [22] measured the profile depth of tires and calculated the wear rates for a large number of vehicles in berlin. fig. 2 shows the average wear rates of vehicles within different ranges of maximum admissible vehicle weight m. please note that these values should be used carefully because they represent urban traffic. for instance, the wear rate of a city bus can be significantly different from a longdistance coach. if the central value in each category is considered, then the correlation is almost linear. this is consistent with [7, 33, 34]. a rough approximation can be given as 103 67 10 m m w .   . (3) equations (3) and (2) can be used to take into account the influence of traffic intensity and its distribution over different categories of vehicles. 0 50 100 150 200 250 300 350 400 450 up to 1200 1201 1400 1401 1700 1701 2000 2001 3500 3501 7500 7501 12000 12001 18000 commercial vehicles passenger cars maximum admissible weight of vehicle (kg) w ea r ra te w ( m g /k m ) fig. 2 experimentally obtained mean values for wear rates of vehicles as a function of the maximum admissible weight. data taken from gebbe et al. [22] tire wear particle hot spots – review of influencing factors 21 3.2. influence of characteristic vehicle manoeuvres lengthwise load when driving on straight roads, the car tires must overcome the resistive force directed backwards. fresist consists of multiple parts [23] which can be estimated for a given road section based on the likely vehicle operation and as a function of vehicle weight. the inertia force applies to the road sections where vehicles are likely to stop and accelerate (e.g. traffic lights). an average force of 21 4 m sinertiaf . m   (4) is a good estimate for normal operation, where m is the mass of the vehicle. the negative sign represents braking. the slope force applies to inclined roads. angle α is positive for uphill driving and the resulting force simply reads 29 81m s (α)slopef . sin m   (5) the rolling resistance of the passenger car tires increases with speed. for low to medium speeds, it is approximately 1.3% of the vehicles gravitational force, so 20 13m srollf . m  (6) is a good estimate [35]. at very high speeds, this value can be more than twice as high. however, the most relevant force at high speeds is the aerodynamic drag force of the vehicle, which is given by 21 2 drag wf c a v    . (7) here ρ=1.2kg/m 3 is the density of air, cw≈0.4 is the drag coefficient and a is the face area of the vehicle. the total resistive force, the sum of the four components, must be transferred by the vehicle’s driven tires. a tire that is unloaded and rolling freely cannot transfer any forces. only the combination of normal load to the surface and a certain amount of slip enables this. the horizontal slip is defined as difference of the rotational speed of the tire surface and the actual vehicle speed. if an identical unloaded tire b is placed next to the loaded tire a, then slip s can be calculated from the revolutions per minute (rpm) of the tires b a b rpm rpm s rpm   (8) the traction (defined as the horizontal force divided by the load) generated by a tire generally depends on the slip as shown in fig. 3. normal operation of vehicles takes place in the region of s=1…3%, where the relation is linear. the maximum traction is attained for s≈15…20%. 22 r. pohrt fig. 3 dependency of the tire traction on the slip (reproduced from [35]) at increasing vehicle speeds, the traction is decreased for all values of s, as shown with the dotted curves. the same is true for wet surfaces. for almost all road surfaces at low slip rates, the wear rate is proportional to the square of the transmitted force or the slip respectively [35, 36]. for this reason, the total resistive force should be squared to give a qualitative understanding of the local wear rate 2 lengthwi ress istew k f , (9) where klengthwise (unit µg∙m -1 ∙n -2 ) is a constant factor. table 1 lists the characteristic sum of forces squared for different road sections for a passenger car weighing m=1400 kg and having face area a≈2 m 2 . table 1 values of the resistive force squared for selected road scenarios in kn 2 the greatest wear rate is thus to be expected in the short sections where vehicles must come to a stop. considerable values are also to be expected on motorways with very high speeds. the latter case is due to the fact that the drag resistance effectively includes the vehicle speed with power four. please note: you may find a significantly weaker dependency on the rolling speed in some studies [37], but these do not consider the increased drag resistive force. some studies also report a negative correlation between mean trip speed and tire wear rate, meaning that slower driving cars experience more tire wear than faster driving cars [6, 38, 39]. in all cases known to the authors, this is due to the hidden variable of driving in an urban environment (slow but more cornering and acceleration) vs. driving long distance trips. tire wear particle hot spots – review of influencing factors 23 lateral load similar to the lengthwise direction, a rolling tire cannot transfer lateral forces without experiencing a certain amount of slip. this is realized by imposing a certain slip angle defined as the angle between the direction in which the tire is pointing and the direction in which it is actually travelling. the slip angle in normal driving situations is β=-3°…3°. for a turning manoeuvre at speed v and radius or curvature r, centripetal force fcent reads 2 cent v f m r  . (10) in addition to the centripetal force, there is the force related to the cross slope of the road, fcs. let α be the angle of the cross slope (positive for banked turn) 29 81m/s ( )cs csf . sin m.    (11) the total lateral force to be carried by the tires reads lateral cent csf f f  . (12) the dependency between β and flat is nonlinear [35, 40, 41]. determining experimentally the wear rate as a function of β or flateral for a real tire is not trivial because great care must be taken not to overstress it thermally. for instance, lupker et al. [42] had to discard multiple truck tires that were subjected permanently to β=0.2°, because these would wear intensively and in an unrealistic fashion. for the influence on wear rate w, [36] reports a quadratic dependency similar to eq. (9): 2 lateral lateralw k f (13) this is consistent with [40] and with the nonlinear dependency found in [43] against abrasive paper. veith [44] adds that despite of the similar dependency, lateral forces lead to a significantly higher wear rates, translating into klateral ≈ 7 klengthwise. table 2 lists some values for the square of the centripetal force in different road sections. in contrast to table 1, these values are closer to each other, which is not surprising. drivers tend to choose their cornering speeds so that the centripetal acceleration is at an acceptable level for both driver and vehicle. table 2 values of the centripetal force squared for selected road scenarios in kn 2 24 r. pohrt the values obtained for lateral load are all much higher than those for lengthwise load. it can be concluded that most of the tire wear happens during cornering and that cornering section of roads are most likely to accumulate high quantities of tire wear particles. this is in agreement with [36] and with the bench test results of [38] where simulated city driving accounted for 63 % of the tire wear, although it accounted for only 5 % of the distance driven, the rest being a motorway scenario. 3.3. influence of road surface the road surface and in particular its texture influence the wear rate. in an extreme case, [21] reports w to increase by more 10 times on the german nürburgring course compared to concrete/asphalt roads at the same speed. most of european and american road surfaces consist of open asphalt. it has 15-25% hollow space which can retain wear particles [9]. asphalt roads with varying density were tested in [26, 45] and it was found that lower-density asphalt had a higher wear rate. authors assume this was due to the well maintained microstructure of this type. asphalt roads are also found to have higher rolling resistance [23] and higher wear rate than concrete roads [46]. because asphalt is an aggregate of particles with a bitumen binder, a distinction between the macro texture (size, distribution and geometrical configuration of particles) and the micro texture (of the individual particles) can be made. with the wearing away of the bitumen binder and the resulting increase of surface voids, the macro texture tends to increase over time. in contrast, the micro texture tends to diminish due to polishing [36]. in an extensive test using a series of test pavements, lowne [47] showed that the roughness of the micro texture is the main driver to increased wear of a road surface, while the macro texture has minor influence. 3.4. seasonal changes multiple studies investigate the influence of seasonal changes on vehicle wear rates but conclusions vary. during the summer months, roads tend to have smaller micro texture and friction due to polishing compared to winter [36], which is generally associated with increased wear. indeed, le maitre et al. [33] report a 1.6 to 1.7 fold increase in wear rates during the winter season. in contrast, it is found that wear rates increase by 1…4% per °c increase in temperature [37, 48]. this appears to be confirmed by investigations from [21], where an increase of tire lifespan by 31% for passenger cars and 9% for commercial vehicles during the winter season was found. 3.5. other factors because this study focuses on the amount wear deposited at any given road section, averaged over total traffic, influences on wear rate w that are vehicle-specific were not considered in detail. however, they may have great impact. some of the main factors include  exact brand/model: ~4 fold difference in w [28]  tire geometry: wider tires have slightly lower w [33]  tire age: new tires have ~10% higher w [49] tire wear particle hot spots – review of influencing factors 25  balancing of wheels: imbalances cause increase in w [21]  tire pressure: lower pressure increases w [21, 34, 37]  temper of driver: 6.2 fold in w between drivers on the same course [33], or 1.4 fold within the same convoy [21] as a consequence, even identical vehicles with identical tires may have different average wear rates. in the study of gebbe [22], authors identified 4 recurring passenger cars and 3 commercial vehicles with up to 20 units each and compared their wear rate. the ratio between minimum and maximum w was typically 1.3 – 2 with the exception of fiat cinquecento vehicles at 3.7. standard deviations were not given. studded tires are known to have greatly increased wear rates in the tire and the road [15]. they are excluded from this study. because modern cars are equipped with powerassisted steering, it is common for drivers to steer the wheels when the vehicle is stationary, e.g. during inner-city parking manoeuvres. it is expected that these rare operations are particularly harmful to the tire but are excluded from this study. other limitations apply. the combination of different loading and varying surface topography and/or seasonal changes is not strictly linear. for instance, some road surfaces are known to alter the exponent in the force-vs.-wear equation (13) from 2 to 3 or higher values. indeed, different approaches to give an overall severity rating exist [44, 50]. due to the lack of available data for vehicles other than passenger cars, a constant factor can be used to characterize those. even for passenger cars and their typical tires, reliable experimental date for specific loading is sparse. more experimental data on test benches is required that reproduces acceleration and cornering manoeuvres but does so with pausing intervals as not to overstress the tire thermally. 4. discussion the main influencing factors on the estimated amount of tire wear deposition at any given road section have been reviewed. for an average passenger car, the rough dependencies are given in the corresponding sections and can be used in order to compare two otherwise identical road sections qualitatively. it is found that by far the greatest emission is to be expected at road section where changes of direction happen. assuming that a particular term is most prominent in every case where an increased wear is observed, one can identify the most dominating quantities. in curves without cross slope, measure v 4 r -2 (eqs. (10) and (13)) is the most dominating quantity on a one-vehicle-basis. here, wear can be reduced greatly by reducing local vehicle speeds because of the power four in the velocity dependence. for instance, lowering the allowed speed from 50 km/h to 30 km/h in a curve is expected to reduce tire wear emission down to (3/5) 4 =13%! another means is adding a suited cross slope to counteract the centripetal force. at straight road sections, increased wear is only expected at stops or where vehicles travel at very high speeds. in the latter case, the speed again enters the amount of wear in power four (eqs. (7) and (9)); therefore, a motorway speed limit can be an effective measure. we conclude that road sections can differ in their expected deposition rate by orders of magnitude. besides the obvious aspect of low vs. intense traffic, the surface type, the typical vehicle speed and the radius of curvature have a great influence. the relations 26 r. pohrt given in the paper can be used to identify so called “hot spots” of elevated emission exposure for local counter-measures. acknowledgements: the paper is part of the research done within the project “reifenabrieb in der umwelt (rau)”, grant no. 13nkw011a of german ministry of education and research. the author would like to thank dr. rauterberg-wulff for useful discussion and providing refs.[22,35]. references 1. cinaralp, f., 2017, etrma statistics report n.9, technical report, european tyre and rubber industry. 2. kwak, j., kim, h., lee, j., lee, s., 2013, characterization of non-exhaust coarse and fine particles from onroad driving and 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2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization sarfaraz hashemkhani zolfani 1 , morteza yazdani 2 , dragan pamucar 3,4 , pascale zarate 5 1 school of engineering, catholic university of the north, larrondo, coquimbo, chile 2 department of management, universidad loyola andalucia, seville, spain 3 informetrics research group, ton duc thang university, ho chi minh city, vietnam 4 faculty of civil engineering, ton duc thang university, ho chi minh city, vietnam 5 university of toulouse, irit, toulouse, france abstract. decision and policy-makers in multi-criteria decision-making analysis take into account some strategies in order to analyze outcomes and to finally make an effective and more precise decision. among those strategies, the modification of the normalization process in the multiple-criteria decision-making algorithm is still a question due to the confrontation of many normalization tools. normalization is the basic action in defining and solving a madm problem and a madm model. normalization is the first, also necessary, step in solving, i.e. the application of a madm method. it is a fact that the selection of normalization methods has a direct effect on the results. one of the latest normalization methods introduced is the logarithmic normalization (ln) method. this new method has a distinguished advantage, reflecting in that a sum of the normalized values of criteria always equals 1. this normalization method had never been applied in any madm methods before. this research study is focused on the analysis of the classical madm methods based on logarithmic normalization. vikor and topsis, as the two famous madm methods, were selected for this reanalysis research study. two numerical examples were checked in both methods, based on both the classical and the novel ways based on the ln. the results indicate that there are differences between the two approaches. eventually, a sensitivity analysis is also designed to illustrate the reliability of the final results. key words: multiple attributes decision-making, madm, normalization, logarithmic normalization, vikor, topsis received november 29, 2019 / accepted april 12, 2020 corresponding author: dragan pamucar informetrics research group, ton duc thang university, ho chi minh city, vietnam; faculty of civil engineering, ton duc thang university, ho chi minh city, vietnam e-mail: dragan.pamucar@tdtu.edu.vn 342 s. h. zolfani, m. yazdani, d. pamucar, p. zarate 1. introduction in a decision-making problem, there are several elements that influence the preciseness and accuracy of the final solution. in other words, every madm technique has a different functionality, specification and applicability, which undoubtedly affect the decision-making process, the evaluation system rout and the final priority of alternatives. the point distinguishing between those techniques is the composition of the elements, such as decision variables, normalization tools, their attributes and weights and the computation of the final solution. in the same manner, every optimization problem is recognized as having different structures and elements, incommensurable variables, conflicting development objectives and constraints. thus, multi-criteria optimization techniques represent an appropriate tool in the ranking or selection of proper alternatives out of a pool of feasible choices in the presence of multiple, and usually conflicting, criteria. as the most significant elements of a typical madm problem, criteria are naturally stated based on different units of measurement and an optimization direction, such as a meter, a capacity, a litre, a dollar, etc. moreover, the “benefit” or “non-benefit” orientation of criteria depends on their nature. all madm techniques are applied in the normalization process [1]. the normalization process is a process of making comparable scales for criteria values, and different methods utilize different approaches to normalization. multiple attribute decision-making (madm) frameworks vary from the simple approaches based on a small amount of data to the methods based on mathematical simulation and programming techniques, requiring extensive information for each criterion, and on the decision-maker’s preferences as well [2,3,4,5]. a typical decision matrix in a madm problem contains alternatives and criteria. criteria have different scales and optimization objectives. in order to avoid difficulties caused by their different dimensions, such criteria values are transformed (or normalized). when the normalization process is completed, it is possible to evaluate criteria by weighting factors. any madm model may lack in the delivery of the absolute optimum solution, and normalization norms within the solution methods may fail to reveal the actual decision. in addition to this, different normalization techniques may yield different solutions and, therefore, may cause a deviation from originally recommended solutions [6]. all in all, the problem of how these essential elements (for instance, attributes or criteria) act and how any modifications in the structure and the anatomy will generate more efficient and better outcomes is still a question to answer. experts and professionals in this field have made attempts to fill this gap and provide reasonable answers through several academic research studies. in this study, a new normalization tool is applied to the two madm methods called vikor and topsis in order to check the consistency of the results. many theories have been established so as to express transformation through the normalization procedure. vector normalization, linear normalization, non-monotonic normalization, weitendorf’s linear normalization (wln) method, the jüttler-körth normalization (jkn) method and the peldschus non-linear normalization (nln) method [2,7,8,9] are the normalization tools most applied and employed by scholars. vafaei et al [10] applied six different normalizations methods to evaluate topsis in all the possible ways. they didn’t consider ln in the study. zadeh sarraf et al [11] used statistical normalization for evaluating topsis method in a more statistical situation. a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization 343 mcdm methods are categorized based on their performance. the methods based on a reference point or goal, such as the reference point methods as topsis and vikor, the methods based on an initial qualitative assessment as the ahp and the methods based on quantitative measurements using a few attributes to compare alternatives (the comparison preference method) as electre [12] and promethee are the three basic classifications of the madm methods. apparently, with its special characteristics and definitions, every model contains different normalization methods. for the topsis method, lai et al [13] proposed vector normalization, only to be followed by a proposal for employing linear normalization in the same multi-criteria method. it is a fact that normalization procedures may affect the final madm solution. promethee [14,15] and gra normally solve a decision problem by using weitendorf’s linear normalization tool, whereas vikor [16,17] uses linear normalization to obtain the ranking of alternatives. in comparison with the other normalization tools, the logarithmic model may be used in the cases when the values of criteria differ considerably from each other and when the solving of problems segregates normalized values more effectively than other methods. a normalized matrix created by the normalization rule yields more stable results in solving multi-criteria decision problems. furthermore, while in vector normalization (such as topsis & vikor) the ratio of the values remains constant in the interval [0,1], the sum of the normalized values of criteria in the logarithmic model is always equal to 1 [7]. in this paper, a novel normalization tool called logarithmic normalization is applied to vikor and topsis. the paper intends to apply this normalization to both vikor and topsis in order to develop these methods as more powerful methodologies in solving madm applications. given the fact that vikor’s and topsis’ final results for the evaluation of alternatives are based on [0,1], this new normalization allows us to absolutely obtain more precise results in the cases of alternatives being higher than 4, the results of the latter method being very near to those of the two former ones. the authors suggest this new framework and extended vikor and topsis as the extended models and consider them to be applicable in solving multi-criteria decision-making issues. 2. topsis and vikor methodologies as a traditional multi-criteria decision-making method and a madm problem-ranking tool, the topsis methodology was developed with the aim of reaching non-inferior solutions [18,19]. it has satisfactorily been implemented in different application fields. its user-friendly anatomy and easy computational algorithm make a decision problem more reliable, thus leading to optimum solutions. in topsis, the best alternative should have the shortest distance (d*) from the positive ideal solution (v*) (which is made up of the best quantity of each criterion regardless of alternatives) and the largest distance (d ) from the negative ideal solution (v ) (which is made up of the worst quantity of each criterion regardless of alternatives). topsis computations begin with an initial pay-off matrix, including criteria and the alternatives accompanying the weights of each criterion. thereafter, normalized and weighted normalized matrices are detected. the positive ideal solution (pis) and the negative ideal solution (nis) should be obtained. a separation measure from pis and nis is generated, after which relative closeness to both pis and nis is computed. in the final step, 344 s. h. zolfani, m. yazdani, d. pamucar, p. zarate the priority ranking order is derived based on such relative closeness to the ideal solutions. the summary account of the steps of the procedure of the topsis method is given in figure 1. the vikor method was developed in 1998 so as to determine a compromise ranking list of several alternatives with non-commensurable and conflicting criteria and particular weights stability intervals in order to make adjustments to preference stability and the given weights [17,20]. the specific structure of vikor introduces it as an applied tool used when decision experts are not able to explain their preferences in the system designing process. in vikor, each alternative is measured based on an aggregate function, so the compromise ranking of alternatives is implemented by comparing the measure of closeness to the ideal solution [21]. any exclusion or inclusion of an alternative could influence the vikor ranking results. the vikor algorithm prepares the minimum of individual regret and the maximum group utility for the opponent and the majority, respectively [22,23]. in the vikor algorithm, v stands for the weight of the strategy of the majority of attributes and is sometimes a useful means to examine the performance of vikor. one of vikor’s characteristics reflects in the fact that the aggregation function is always closest to the best solutions, whereas in topsis, the cc of materials are not necessarily always very close to the ideal values [24,25]. the interactivity of vikor allows decision-makers to participate in and control the decision-making process (by means of weights). the aggregating (compound) function is used extremely cautiously since it includes a comparison of potentially incomparable quantities (non-commensurable criteria or indicators). normalization in vikor is performed in order to eliminate the units of indicators; so, all indicators are dimensionless [26]. every mcdm method will start from a decision matrix including alternatives and criteria, and the mathematical steps of topsis and vikor are presented in figure 1. topsis has become a key and strategic means and has been launched so as to investigate complex decision-making problems. several applications acquired topsis as the sole method, the integrated approach, whereas in others, the same was amended for the purpose of obtaining excellent outcomes. in the literature, the reformed type of topsis is presented by practitioners. the extended topsis model appeared in a study [19], determining the weights of decision-makers in a group decision-making environment, in which decision information is expressed by an interval value matrix. the traditional topsis model begins with the normalization process and then defines a weighted normalized decision matrix. in fact, the traditional model believes that the influence of weights should occur in this step in a normalized matrix. however, how do weights exert an influence on a decision matrix and where is the catch? on the one hand, the overall performance of alternatives is estimated by using the euclidean distance, whereas on the other, a notion is made that weights should affect this distance, for which reason criteria weights should be incorporated in the distance measurement. the extended vikor model and, actually, a generalization of vikor, are presented by [27,28,29,30]. in order to verify the extended topsis and vikor models, a logarithmic normalization tool is applied as follows: 1 ln( ) ln ij ij n ij i x f x          (1) a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization 345 all in all, the previous studies claimed the extended and modified vikor and topsis models differently from the weighting process, the aggregation process and the normalization tool can improve the quality and accuracy of a decision. so, the application of a new normalization tool to a madm problem is examined in this paper in order to observe the results. 3. case study in order to verify the proposed approach, the two numerical examples based on a real case are illustrated in this section. the proposed model is implemented in the mhv construction company, located in madrid, spain. for approximately 20 years, the company has been participating in the adaptation, designing and construction of buildings, residential apartments and commercial centers projects in madrid and its urban areas. now, due to the new rule of the regional government and the ministry of industry, it is compulsory for small firms that they should assure and guarantee the procurement process. therefore, all further legalization and formal permission of the mhv company is bound by the establishment of the supply evaluation system. the company should address pieces of evidence in order to approve a safe, secure and environmentally friendly project. currently, mhv performs its purchasing operations through the official suppliers who are supposed to meet the main ten requirements, such as the foundation, consulting and geometrical measurement, electricity equipment and raw materials, such as wood, cement, iron, etc. for various projects. among them, a pre-evaluation was carried out, and a decision was made on releasing a list including 4 suppliers (a1 to a4) to be evaluated through a questionnaire. the questionnaire was designed and distributed among the experts anonymously. the supplier evaluation team consisted of the ceo of the company, two architects (with more than 10 years of experience each) and a purchase assistant. the experts were responsible for making a judgment and offering their score rating for each candidate supplier by applying certain criteria. the evaluation criteria were the offered price, the perceived quality, the experience, the delivery conditions, and the environmental certificates. the primal decision-making matrix was developed as shown in table 1 after the data had been collected and after the analysis had been made. table 1 the decision matrix alternatives c1 c2 c3 c4 c5 a1 8 7 7 9 8 a2 7 9 8 7 8 a3 8 8 8 6 9 a4 9 6 7 8 7 table 2 shows the nature and weight coefficients of each criterion and the computation of vikor and the extended model of vikor. based on figure 1, a mathematical computation of the methods was done. according to the results of the vikor and the extended vikor methods, the differences between their respective results are possible to see. the new logarithmic normalization [7] presented the obtained results in a more precise manner in the 346 s. h. zolfani, m. yazdani, d. pamucar, p. zarate circumstance in which the alternatives in the initial decision-making matrix are of approximately the same values, which will help to rank alternatives more precisely. the computation should be performed for topsis. the procedure and the results are illustrated in table 3. unlike the vikor ranking scores, the topsis algorithm output for both the classical and the extended models is the same in this example. the brief algorithm of both methodologies is shown in fig. 1. fig 1 the construction of the normalized decision matrix the decision matrix is the same for both vikor and topsis, as shown in table 1. in this section, both the vikor and the topsis calculations are presented in the two mentioned manners, the first related to the classical way to solve the madm model, and the second being based on logarithmic normalization (ln). all the information about the vikor calculations is given in table 2 and that about the topsis calculations in table 3. as is illustrated in tables 2 and 3, the priority of the alternatives has changed, but no changes are perceived for topsis. in the second example, the different condition of the decision matrix is proposed, implying six criteria and eight alternatives. table 4 shows the decision table for example 2. likewise example 1, the topsis and the vikor methods were adopted as the classical algorithm, including logarithmic normalization as well. the results for each method are presented in tables 5 and 6, respectively. in this example, we may notice that the topsis ranking changed slightly after the new normalization tool had been applied. however, this never happened in the case of vikor, as the same ranking resulted from both the classical and the modified vikor algorithms. 1. construct the weighted normalized matrix 2. define the maximum & the minimum vectors 3. define the utility & regret measures 4. find the vikor ranking index 1. identify the weighted normalized decision matrix 2. determine the positive and the negative ideal solutions 3. calculate the distances from the ideal solutions 4. find the topsis ranking index topsis and vikor brief algorithm the topsis algorithm the vikor algorithm a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization 347 table 2 the final results of vikor crit. c1 c2 c3 c4 c5 c1 c2 c3 c4 c5 max max max max max max max max max max weight 0.197 0.163 0.176 0.197 0.267 0.197 0.163 0.176 0.197 0.267 original vikor method modified vikor alt. normalized matrix normalized matrix a1 0.498 0.461 0.465 0.593 0.498 0.250 0.243 0.242 0.274 0.250 a2 0.435 0.593 0.532 0.461 0.498 0.235 0.274 0.258 0.243 0.250 a3 0.498 0.527 0.532 0.395 0.560 0.250 0.259 0.258 0.224 0.265 a4 0.560 0.395 0.465 0.527 0.435 0.265 0.224 0.242 0.259 0.235 f* 0.560 0.593 0.532 0.593 0.560 0.265 0.274 0.258 0.274 0.265 f0.435 0.395 0.465 0.395 0.435 0.235 0.224 0.242 0.224 0.235 si ri v qi rank si ri v qi rank a1 0.514 0.176 0.5 0.256 2 0.474 0.176 0.5 0.203 3 a2 0.462 0.197 0.5 0.274 3 0.396 0.197 0.5 0.197 2 a3 0.349 0.197 0.5 0.098 1 0.344 0.197 0.5 0.116 1 a4 0.671 0.267 0.5 1 4 0.654 0.267 0.5 1 4 table 3 the topsis different ranking for both the classical and the extended models criteria c1 c2 c3 c4 c5 c1 c2 c3 c4 c5 max max max max max max max max max max weights 0.197 0.163 0.176 0.197 0.267 0.197 0.163 0.176 0.197 0.267 alt. original topsis method modified topsis normalized matrix normalized matrix a1 0.041 0.039 0.04 0.06 0.041 0.25 0.243 0.242 0.274 0.25 a2 0.036 0.050 0.045 0.047 0.041 0.234 0.274 0.258 0.243 0.25 a3 0.041 0.044 0.045 0.04 0.046 0.25 0.259 0.258 0.224 0.265 a4 0.046 0.033 0.04 0.054 0.036 0.265 0.224 0.242 0.259 0.234 weighted normalized matrix weighted normalized matrix a1 0.008 0.006 0.007 0.012 0.011 0.049 0.04 0.043 0.054 0.067 a2 0.007 0.008 0.008 0.009 0.011 0.046 0.045 0.045 0.048 0.067 a3 0.008 0.007 0.008 0.008 0.012 0.049 0.042 0.045 0.044 0.071 a4 0.009 0.005 0.007 0.011 0.01 0.052 0.036 0.043 0.051 0.063 d+ dcc rank d+ dcc rank a1 0.003 0.004 0.623 1 0.008 0.012 0.609 1 a2 0.004 0.003 0.489 2 0.009 0.01 0.527 2 a3 0.004 0.004 0.461 3 0.011 0.011 0.506 3 a4 0.004 0.003 0.443 4 0.012 0.009 0.43 4 348 s. h. zolfani, m. yazdani, d. pamucar, p. zarate table 4 the initial decision matrix of the second example alternatives c1 c2 c3 c4 c5 c6 a1 4 8 8 7 9 8 a2 6 7 7 8 9 6 a3 7 6 5 8 7 4 a4 6 6 4 6 5 4 a5 9 9 4 6 6 7 a6 7 9 8 8 7 8 a7 8 8 9 8 6 9 a8 9 4 7 5 8 6 table 5 the topsis results and ranking for the second example c1 c2 c3 c4 c5 c6 c1 c2 c3 c4 c5 c6 max max min min max max max max min min max max weight 0.12 0.2 0.16 0.32 0.15 0.05 0.12 0.2 0.16 0.32 0.15 0.05 alt. original topsis method modified topsis normalized matrix normalized matrix a1 0.010 0.022 0.027 0.020 0.026 0.027 0.090 0.134 0.142 0.126 0.141 0.142 a2 0.015 0.019 0.023 0.023 0.026 0.020 0.117 0.126 0.133 0.135 0.141 0.122 a3 0.018 0.017 0.017 0.023 0.021 0.013 0.127 0.116 0.110 0.135 0.125 0.095 a4 0.015 0.017 0.013 0.017 0.015 0.013 0.117 0.116 0.095 0.116 0.103 0.095 a5 0.023 0.025 0.013 0.017 0.018 0.023 0.143 0.142 0.095 0.116 0.115 0.133 a6 0.018 0.025 0.027 0.023 0.021 0.027 0.127 0.142 0.142 0.135 0.125 0.142 a7 0.020 0.022 0.030 0.023 0.018 0.030 0.136 0.134 0.150 0.135 0.115 0.150 a8 0.023 0.011 0.023 0.014 0.024 0.020 0.143 0.090 0.133 0.104 0.134 0.122 weighted normalized matrix weighted normalized matrix a1 0.0012 0.0044 0.0043 0.0063 0.0040 0.0013 0.0108 0.0269 0.0227 0.0403 0.0212 0.0071 a2 0.0018 0.0039 0.0037 0.0073 0.0040 0.0010 0.0140 0.0252 0.0213 0.0431 0.0212 0.0061 a3 0.0021 0.0033 0.0027 0.0073 0.0031 0.0007 0.0152 0.0232 0.0176 0.0431 0.0188 0.0047 a4 0.0018 0.0033 0.0021 0.0054 0.0022 0.0007 0.0140 0.0232 0.0152 0.0371 0.0155 0.0047 a5 0.0027 0.0050 0.0021 0.0054 0.0026 0.0012 0.0172 0.0284 0.0152 0.0371 0.0173 0.0066 a6 0.0021 0.0050 0.0043 0.0073 0.0031 0.0013 0.0152 0.0284 0.0227 0.0431 0.0188 0.0071 a7 0.0024 0.0044 0.0048 0.0073 0.0026 0.0015 0.0163 0.0269 0.0240 0.0431 0.0173 0.0075 a8 0.0027 0.0022 0.0037 0.0045 0.0035 0.0010 0.0172 0.0179 0.0213 0.0333 0.0200 0.0061 d+ dcc rank d+ dcc rank a1 0.0033 0.0031 0.488 4 0.0122 0.0113 0.481 5 a2 0.0035 0.0028 0.438 6 0.0125 0.0102 0.451 7 a3 0.0035 0.0028 0.437 7 0.0121 0.01 0.451 6 a4 0.0029 0.0035 0.547 2 0.0096 0.0124 0.563 2 a5 0.0017 0.0046 0.735 1 0.0055 0.0165 0.750 1 a6 0.0037 0.0032 0.465 5 0.0128 0.0122 0.488 4 a7 0.0041 0.0027 0.397 8 0.0139 0.011 0.442 8 a8 0.0033 0.0036 0.523 3 0.0123 0.0129 0.511 3 a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization 349 table 6 the vikor results and ranking for the second example c1 c2 c3 c4 c5 c6 c1 c2 c3 c4 c5 c6 max max min min max max max max min min max max wj 0.12 0.2 0.16 0.32 0.15 0.05 0.12 0.2 0.16 0.32 0.15 0.05 alt. original vikor method modified vikor normalized matrix normalized matrix a1 0.1971 0.3871 0.4193 0.3491 0.4386 0.4205 0.0904 0.1344 0.1421 0.1259 0.1412 0.1419 a2 0.2956 0.3388 0.3669 0.3990 0.4386 0.3154 0.1168 0.1258 0.1330 0.1345 0.1412 0.1222 a3 0.3449 0.2904 0.2621 0.3990 0.3412 0.2102 0.1269 0.1158 0.1100 0.1345 0.1251 0.0946 a4 0.2956 0.2904 0.2097 0.2993 0.2437 0.2102 0.1168 0.1158 0.0948 0.1159 0.1034 0.0946 a5 0.4434 0.4355 0.2097 0.2993 0.2924 0.3679 0.1433 0.1420 0.0948 0.1159 0.1152 0.1328 a6 0.3449 0.4355 0.4193 0.3990 0.3412 0.4205 0.1269 0.1420 0.1421 0.1345 0.1251 0.1419 a7 0.3941 0.3871 0.4717 0.3990 0.2924 0.4730 0.1356 0.1344 0.1502 0.1345 0.1152 0.1499 a8 0.4434 0.1936 0.3669 0.2494 0.3899 0.3154 0.1433 0.0896 0.1330 0.1041 0.1337 0.1222 si ri v qi rank si ri v qi rank a1 0.511 0.213 0.5 0.5691 4 0.522 0.229 0.5 0.6286 4 a2 0.598 0.320 0.5 0.9299 6 0.577 0.320 0.5 0.9321 6 a3 0.645 0.320 0.5 0.9862 7 0.615 0.320 0.5 0.9811 7 a4 0.499 0.150 0.5 0.4013 3 0.484 0.150 0.5 0.3776 3 a5 0.239 0.113 0.5 0.0000 1 0.243 0.124 0.5 0.0000 1 a6 0.581 0.320 0.5 0.9095 5 0.565 0.320 0.5 0.9165 5 a7 0.657 0.320 0.5 1.0000 8 0.630 0.320 0.5 1.0000 8 a8 0.364 0.200 0.5 0.3598 2 0.365 0.200 0.5 0.3518 2 as may be observed in both examples, the top alternative (the 1 st -ranked) remained the same. thereafter, except for the vikor results in case 1, the 2 nd and the 3 rd -ranked alternatives were the same. in the second example, vikor produced a much more stable ranking (the same ranking order). this issue was somehow different, although still acceptable, for topsis, where the first, the second and the third items are similar, whereas the others are replaced in pairs (ranked the 4 th and 5 th , 6 th and 7 th ). in the cases in which alternatives are so close to each other, this normalization type can be considered as effective. in those cases, prioritizing is more complicated due to the fact that there are a larger number of alternatives whose values are approximately the same in the initial matrix. this research study shows how this normalization may improve the decision-making process and enhance the quality of the results obtained both in vikor and topsis by applying logarithmic normalization. the reliability of a decision-making problem can be judged by a number of alternatives, a number of criteria, the type of the criteria (either qualitative or quantitative, deterministic, interval or fuzzy), and the type of the normalization tool used [31,32,33,34]. logarithmic normalization for a decision-making matrix yields more stable results in solving decisionmaking problems [7]. in some cases, solutions are similar; in some other cases, however, very different results are obtained. given the ranking orders of modified vikor and topsis compared with the original methods, the contribution of this study reflects in its providing help to experts in the establishment of different normalization tools for accomplishing optimal solutions and achieving further effective objectives. the normalization rules are a vital part of each madm method [34]; normalizations and their applications are important for the formation of a new anatomy of madm tools. 350 s. h. zolfani, m. yazdani, d. pamucar, p. zarate 4. comprative analysis and discussion the analysis of the stability of the obtained results was carried out in two parts. in the first part, the sensitivity analysis of the ln/original vikor and topsis models was performed by changing the weight coefficients of the criteria. the analysis of the influence of the change in the weight coefficients of the criteria was made through 21 scenarios. in the second part, the analysis of the influence of the dynamic decision-making matrices on the change in the rank of the alternatives was performed. a more detailed overview of the sections of the discussion on the results is presented in the next part of the paper. 4.1. changing the weights of the criteria the first part of the discussion is based on the weights sensitivity analysis according to kahraman [35]. after identifying the most important criterion on the basis of the weights estimated in example 1 and example 2, the weights sensitivity analysis was performed by varying the weight of the most important criterion so as to observe its effect on the ranking performance of the proposed model. according to kahraman [35], the weight coefficient of the elasticity (s) of the most important criterion (for example 1 – c5 and for example 2 – c4) was assumed to be one. for the other criteria, the coefficients of elasticity (c) were estimated and shown in table 7. the elasticity coefficient was used to express the relative compensation of the values of the other weight coefficients in relation to the changes in the weight of the most important criterion. table 7 the coefficient of elasticity for changing weights criteria c (example 1) c (example 2) c1 0.269 0.176 c2 0.222 0.294 c3 0.240 0.235 c4 0.269 1.000 c5 1.000 0.221 c6 0.074 thus, the limit values of c5 and c4 were obtained, being -0.267≤ x ≤0.733 (for example 1) and -0.320≤ x ≤0.680 (for example 2). on the basis of the defined limit values of the change in the weight coefficients of the most important criteria (c5 and c4), the sensitivity analysis scenarios were defined. the intervals -0.267≤ x ≤0.733 and 0.320≤ x ≤0.680 were divided into 21 sequences, based on which a total of 21 scenarios were formed. for every scenario, the new values of the weight coefficients were formed, so 21 new groups of the weight coefficients were obtained, as is shown in table 7. the new values of the weight coefficients were applied to the analyzed models: vikor (original), vikor (ln), topsis (original) and topsis (ln). the influence of the new values of the weight coefficients on the change in the ranks of the alternatives was analyzed by using spearman’s correlation coefficient (fig. 2). a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization 351 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11s12 s13 s14 s15 s16 s17 s18 s19 s20 s2 1 scc (topsis ln) scc (vikor ln) scc (topsis original) scc (vikor original) b) example 2 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11s12 s13 s14 s15 s16 s17 s18 s19 s20 s21 scc (topsis ln) scc (vikor ln) scc (topsis original) scc (vikor original) a) example 1 fig. 2 the correlation of the ranks through the 21 scenarios – a) example 1, and b) example 2 spearman’s correlation coefficient (scc) was used to determine the statistical significance of the difference between the ranks obtained through the scenarios in the examples 1 and 2. by analyzing the obtained correlation values in the example 1 (figure 2a), we notice that there is a significantly better correlation of the ranks in the vikor ln and topsis ln models in comparison with the original vikor and topsis models. these results confirm the fact that the logarithmic model of normalization gives more stable results than vector and linear normalization do [33]. in scenarios s6-s21, the results of all three normalizations are similar, which is expected, as there are no drastic changes in the c5 criterion in these scenarios. similar results were obtained in example 2 (fig. 2b). for s1-s5, changing the weight of the c5 criterion resulted in the different ranks of the intermediate alternatives. in the first five scenarios, the logarithmic model of normalization yielded the stable results, with a high correlation of the ranges. this was confirmed by the average value of the correlation coefficient, which was scc=0.823 for the first five scenarios. unlike the logarithmic model, the vector and linear models showed a very small value of the scc, which was 0.508 in the first five scenarios. in the remaining scenarios, i.e. in the scenarios s6-s21, the results of all three normalizations were similar, i.e. all three normalization models showed a high correlation of the ranks. this was confirmed by the average values of spearman’s correlation coefficient: 1) the logarithmic normalization – scc=0.969; 2) the vector normalization – scc=0.942; 3) the linear normalization – scc=0.958. these results confirm the stability of the results obtained by using the logarithmic normalization model. 4.2. dynamic matrices when solving a problem, researchers encounter a number of internal and/or external factors, which may change the conditions or limitations of the problem by their actions. internal changes in a decision matrix, such as the introduction of a new alternative or the elimination of an existing one from a set of the considered alternatives, may lead to a change 352 s. h. zolfani, m. yazdani, d. pamucar, p. zarate in the final preferences [36]. accordingly, in this paper, the performance analysis of the proposed examples (i.e. examples 1 and 2) was studied in the conditions of a dynamic initial decision matrix. for both examples considered, separate scenarios were formed, and within each scenario, the results of the application of the multi-criteria models were analyzed. for each scenario, a change in the number of the alternatives was made and the ranks obtained were analyzed. the scenarios were formed by removing one inferior (worst) alternative in each scenario from subsequent considerations. within the scenarios, the remaining alternatives were simultaneously ranked according to the newly matched initial decision matrix. the initial solution in example 1 was generated as: (1) original vikor – a3>a1>a2>a4; (1) ln vikor – a3>a2>a1>a4 and original topsis = lntopsis – a1>a2>a3>a4. it is clear that the alternative a4 is the worst option; so, in the first scenario, the alternative a4 was eliminated from the set and a new decision-making matrix was obtained with three alternatives. the new decision matrix was solved again by using mcdm models, and the new rankings of the alternative were obtained (figure 3). the ranking in the first scenario shows that, for ln topsis and conventional topsis, the alternative a1 was still the best alternative, and the alternative a3 was the worst. the further implementation of the described procedure resulted in the ranks of the alternatives as shown in fig. 3. 1 2 3 4 s1 s2 s3 s4 vikor ln a2 a1 a3 a4 1 2 3 4 s1 s2 s3 s4 conventional topsis a1 a2 a3 a4 0 1 2 3 4 s1 s2 s3 conventional vikor a3 a1 a2 a4 1 2 3 4 s1 s2 s3 s4 topsis ln a1 a2 a4 a3 fig. 3 the ranks of the alternatives within the dynamic decision matrices – example 1 based on the obtained results, it is noted that, when the worst alternative is eliminated, there is no change in the best-ranked alternative in the reorganized matrix for ln topsis, ln vikor and conventional topsis. in conventional vikor, there is a rank reversal problem. after having ranked the third, the alternative a2 became the first-ranked alternative after the second scenario. in the third scenario, the two first-ranked alternatives (namely a3 and a2) appeared. for this reason, the fourth scenario was not implemented. also, the alternative a1 changed its rank in the scenario s2. in the second scenario, the alternative a1 became third-ranked. this example shows that logarithmic normalization provides stable solutions and is resistant to the rank reversal problem. a similar analysis was carried out for example 2 (figure 4). the analysis was conducted through a total of eight scenarios. in the first scenario, the alternative a7 was eliminated as a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization 353 the worst. the analysis has shown that the ln vikor and ln topsis models are stabile in a dynamic environment. in the modifications of the initial matrix, which were made through the elimination of the worst-case option, it was noticed that the ln model did not lead to changes in the ranking (a rank reversal) among the alternatives. the alternative a5 remained the best-ranked across all the scenarios, which confirmed the robustness and accuracy of the ranking alternatives in the dynamic environment. however, in both conventional models (topsis and vikor), there is a rank reversal through the scenarios. 1 2 3 4 5 6 7 8 s1 s2 s3 s4 s5 s6 s 7 s 8 topsis ln a5 a4 a8 a6 a1 a3 a2 a7 1 2 3 4 5 6 7 8 s1 s2 s3 s4 s5 s6 s7 s8 vikor ln a5 a8 a4 a1 a6 a2 a3 a7 1 2 3 4 5 6 7 8 s1 s2 s3 s4 s5 s6 s7 s8 conventional topsis a5 a4 a8 a1 a6 a2 a3 a7 1 2 3 4 5 6 7 8 s1 s2 s3 s4 s5 s6 conventionalvikor a5 a4 a8 a1 a6 a7 a3 a2 fig. 4 the ranks of the alternatives within the dynamic decision matrices – example 2 these results show the stability of the logarithmic normalization model and its resistance to the rank reversal problem. on these two examples, the logarithmic model of normalization showed significant stability in comparison with the conventional normalization models (the vector and the linear normalization models). in a large number of tests, the ranks obtained by applying ln were equal to the final values (tables 2, 3, 5 and 6), i.e. there is no rank reversal. this allows us to conclude that logarithmic normalization generates more reliable results than the traditional (vector and linear) normalization models do. also, it is necessary that the fact that this is a new normalization model which has yet to show its advantages through further empirical research in the field of multi-criteria decision-making should be recognized. 5. conclusion the madm methods, such as topsis, vikor and other methods, are still being developed and modified by the addition of other perspectives and approaches. the anatomy of each madm method can be rebuilt by varying the weight replacement process, the aggregating process, and change in the normalization tool. normalization allows for a direct comparison of diverse criteria by eliminating the dimensional units that are different. the accuracy of the normalization process is very significant because it must address diverse criteria and objectives in real madm cases. 354 s. h. zolfani, m. yazdani, d. pamucar, p. zarate as one of the latest normalization methods, the logarithmic normalization method was first presented in 2008. this research study is an attempt to carry out a reanalysis of the two madm methods based on this new normalization. topsis and vikor were selected as the two well-known madm methods for the purpose of achieving this goal, and the numerical examples were presented in order to illustrate the comparative results. the fact that values are exactly equal to 1 is the advantage of this new normalization method that triggered off the conducting of this research study. the authors believe that, by applying ln, results can be different in the cases in which alternatives are close to one another. in the first example, the ranking results obtained by applying the vikor method changed; for the ranking results obtained by applying the topsis method, however, they were still the same, whereas the rank reversal problem occurred in the second example. ln seems to have had a positive effect on both vikor and topsis. technically, this normalization tool can achieve considerable outcomes, comparing to other normalization approaches. using both vikor and topsis on other examples and other cases in future research studies would be highly appreciated. this research study shows that the accuracy of the normalization methods is effective and sensitive to results and methodologies. 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analysis, serbian journal of management, 11(1), pp. 1-27. 35. kahraman, y.r., 2002, robust sensitivity analysis for multi-attribute deterministic hierarchical value models, storming media, ohio. 36. jankovic, a., popovic, m., 2019, methods for assigning weights to decision makers in group ahp decision-making, decision making: applications in management and engineering, 2(1), pp. 147-165. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 307 333 https://doi.org/10.22190/fume201123004g © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a structured framework for reliability and risk evaluation in the milk process industry under fuzzy environment nand gopal, dilbagh panchal department of industrial and production engineering, dr b r ambedkar national institute of technology jalandhar, punjab, india abstract. this paper aims at proposing a novel integrated framework for studying reliability and risk issues of the curd unit in a milk process industry under uncertain environment. the considered plant’s complex series-parallel configuration was presented using the petri net (pn) modeling. the fuzzy lambda-tau (λ-τ) approach was applied to study and analyze the reliability aspects of the considered plant. failure dynamics of the curd unit has been analyzed with respect to increasing/ decreasing trends of the tabulated reliability indices. availability of the considered plant shows a decreasing trend with an increase in spread values. for improving the system’s availability, a risk analysis was done to identify the most critical failure causes. using the traditional fmea approach, the fmea sheet was generated on the basis of expert’s knowledge/experience. the fuzzycomplex proportional assessment (fcopras) approach was applied within fmea approach for identification of critical failure causes associated with different subsystem/components of the considered plant. in order to check the consistency of the ranking results, the fuzzy technique for order of preference by similarity to ideal solution (ftopsis) was applied within the fcopras approach. ranking results are compared for checking consistency and robustness of critical failure causes related decision making which would be useful in designing the finest maintenance schedule for the considered curd unit. overheating/moisture lead to winding failure (mscp5), visible sediment of milk jam in filter (mbfp3), improper quality of oil (h4), blade breakage (ctk4), wearing in gears (pfm11), and cylinder leakage (cfm7) were recognized as the most critical failure causes contributing to system unavailability. the analysis results were supplied to the maintenance manager for framing a suitable time-based maintenance intervals policy for the considered unit. key words: milk process industry, lambdatau, reliability, fmea, fcopras, ftopsis received november 23, 2020 / accepted january 12, 2020 corresponding author: nand gopal department of industrial and production engineering, dr b r ambedkar national institute of technology jalandhar, punjab, india 144011 e-mail: nandgopal504@yahoo.co.in 308 n. gopal, d. panchal 1. introduction dairy milk products are an emerging food industry in the world. at present, the indian government faces an immense pressure to overcome the issues of youth unemployment. milk process industries are of supreme importance for overcoming this issue as these industries heavily depend upon the rural people because of their dependency on agriculture and dairy sectors. dairy based food products such as curd, ghee, ice-cream and cheese, etc. are big contributors to a balance diet and due to this, their consumption has increased to manifold [1]. curd is one of the important dairy products which is a semi-solid product obtained from the pasteurized milk by souring, using bacterial cultures. it is prepared by feeding milk received from the chiller, stored into storage tank at 50c and pumped to balance tank. then, it passes through heat exchanger (pasteurizer) where the temperature of milk is raised to 450c 500c, and then it is subjected to homogenization. homogenization is done at 2000 to 2500 psi to mix all ingredients thoroughly. after that, it again passes through the heating zone of the heat exchanger where the temperature is raised to 900c. then, it is pumped to the holding tank through the booster pump and the milk from the holding tank enters to the cooling zone where heat exchange takes place and milk temperature is reduced to 450c 500c. further, the milk enters different culture tanks where it is cooled to 400c 450c; then culture is added and pouch/cup filling takes place in the specified section. the milk is further held at 43°c in the incubation room where fermentation of curd is done and the curd is cooled at below 5°c in order to stop the fermentation process in the cold store room. the curd processing unit in milk process industries consists of a large number of subsystem/components arranged in series/parallel configuration. any failure in these subsystem/components will lead to a production loss; besides, plant will not be able to meet the customer demand on time. for such type of complex configuration-based units’ failure is an inescapable phenomenon which results in a heavy operational loss. to overcome the issue of operational loss, reliability of subsystem/components should be at top priority. as per the survey in europe for heavy process industries, it is the maintenance cost which contributes nearly 15 percent to the total production cost [2]. minimization of this percentage by designing and implementing an optimum maintenance policy could directly result in reducing the total operational cost, which means bigger profit to the considered industry. for the development of optimal maintenance policy, correctness of analysis result is of supreme importance. crisp set theory based integrated framework has been already developed and applied by many researchers for studying failure behavior of various complex industrial system. these frameworks consider only crisp data obtained from different sources, which has an element of uncertainty in input data results in biased results responsible for poor maintenance schedule. for developing optimum maintenance schedule, it is essential to study and analyze the qualitative and quantitative failure behavior of the complex systems with the consideration of uncertainty in the input data. therefore, to deal with this challenge, performance analysis in terms of reliability and risk analysis under uncertainty is highly important for correct evaluation of failure behavior of the considered complex system for which application of the fuzzy methodology (fm) with in performance evaluation tool is of supreme importance. thus, the current work presents a novel fuzzy methodology (fm) based structured framework utilizing mcdm approaches for analyzing the performance in terms of reliability and risk analysis under uncertainty for correct evaluation of failure behavior of the considered complex system. a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 309 2. literature background reliability and risk analysis are regarded as a noteworthy sustainable prerequisite for any process industry. therefore, all the working equipment must be available for full-time so that maximum productivity can be achieved. for that, it is very essential to maintain the performance of subsystem/components in terms of reliability and risk. milk plant curd unit needs to be evaluated to uphold its high availability. from the available literature, it has been noted that a number of researchers have done work in the direction of studying stochastic failure behavior of real complex industrial systems in terms of reliability and risk parameters. researchers were motivated to develop different mathematical techniques based structured framework and resources to analyze reliability and risk parameters. for instance, aggarwal et al [3] developed a mathematical model based on the chapman-kalmogorav birth-death method for tabulating reliability parameters in order to study performance issues related to skim milk powder system in a dairy plant. gowid et al [4] applied time dependent markov methods based on crisp set theory to obtain the reliability parameters results for production plant of lng. the markovian model so implemented in the above reported work considering crisp data only means it does not consider uncertainty/vagueness in the raw data/information collected from experts. therefore, an element of uncertainty persists with the results obtained from the markovian model. to overcome this drawback fuzzy methodology-based approaches gained strength and were implemented by various researchers in different work. knezevic and odoom [5] developed a fuzzy λ-τ approach to address failure dynamics of the repairable system under uncertainty. gupta et al [6] suggested fourth order runge kutta method for measuring the system's reliability. aksu et al [7] proposed a technique for reliability assessment that complements failure mode effect analysis (fmea), fault tree analysis (fta) and markov analysis and illustrated it with application in the propulsion system of the pod. qiu et al [8] combined probabilistic and non-probabilistic methodology to find the bounds of system’s structural reliability. sharma et al [9] developed structural framework by using the fuzzy methodology (fm) which is valuable for the plant maintenance manager to envisage the behavior of system. sharma et al [10] used the genetic algorithms (ga) based λ-τ method to compute reliability measures of paper mill. zhang et al [11] applied a method of extended the grey relational analysis (gra) to solve multi criteria decision-making (mcdm) problems with triangular fuzzy numbers valued at intervals with an example of hiring a system analysis engineer for software company. durmić et al [12] applied the full consistency method (fucom) and the rough simple additive weighting (saw) methods to obtain criteria weights for sustainable supplier selection. bozanic et al [13] proposed integrated fucom – z-number – multi-attributive border approximation area comparison (mabac) approaches based framework for the selection of command post location. badi and pamucar [14] presented grey theory-based measurement of alternatives and ranking according to compromise solution (marcos) approaches based model for selecting the best supplier in a libyan iron and steel industry. kishore et al [15] proposed analytic hierarchy process (ahp) – saw approaches for the selection of sub-contractors. vesković et al [16] applied fuzzy based pivot pairwise relative criteria importance assessment (piprecia) method for finding individual importance of each criteria in the selection of reach stacker based handling facility. chatterjee and chakraborty [17] developed a meta-model for obtaining technological value of cotton fiber. maity et al [18] implemented grey copras approach for the selection of tool material so as to enhance the machining performance. these mcdm approaches are found to be very 310 n. gopal, d. panchal useful in solving the decision problem of different areas. considering the importance of these approaches, many authors make use of these approaches in studying the reliability and risk analysis of real industrial systems. sharma et al [19] proposed ga based fuzzy λ-τ method to calculate different reliability parameters of the washing system in paper industry. deveci et al [20] applied the fmea approach to evaluate potential errors in the cutting process of electric cable. garg et al [21] expounded an artificial bee colony-based lambdatau (abcblt) hybridized methodology to analyze butter-oil processing plant reliability indices. panchal and kumar [22] carried out reliability analysis of compressor house unit (a subsystem of thermal power plant) using λ-τ approach. fuzzy λ-τ approach and fuzzy fmea approach were also implemented to study and analyze the performance issues of chlorine gas plant in a chemical process industry [23]. in the above reported work, the authors developed reliability and risk parameters based structured framework, which consider matlab toolbox software-based analysis. in the matlab toolbox softwarebased analysis if-then rules are generated and for effectivity of analysis results common if-then rules are required to be eliminated. the elimination of common if-then rules is a difficult task for the analyst and due to this problem biasness in the analysis results still persists. also, consideration of equal weightage to three risk factors under this approach raises serious concern related to accuracy of ranking results. to overcome these limitations, there is a gap for a novel integrated framework for studying the reliability and risk issues under uncertainty in an unbiased manner. to bridge this gap, a novel fm based structured framework utilizing mcdm approaches has been proposed in this work and is presented with its application on the curd unit in the milk process industry located in northern part of india. the flow chart for the proposed structured framework is presented in fig. 1. in the proposed framework, firstly, a reliability analysis was carried out in which failure and repair time data collected from maintenance logbook integrated with maintenance experts was used. pn modeling as per series-parallel arrangement of the considered unit was done. further, crisp raw data of considered unit were converted into a fuzzified form for considering the vagueness of the collected information and reliability indices were tabulated at different spreads (± 15%, ± 25%, ± 60%). output values were converted into crisp form and reliability indices-based failure behavior of the considered unit was studied and analyzed as per increase/decrease trends. in the second phase, for enhancing the system’s availability, a risk analysis was carried out. for that, using the fmea approach risky failure causes investigation was carried out for the listed failure causes under different subsystem/components. risk factors, namely, occurrence (of), severity (s) and detection (od) probability, relationship was obtained from maintenance experts and their weights were calculated by using a fuzzy extent analysis method. mcdm approaches, namely, fcopras and ftopsis were implemented and on the basis of their output scores each listed failure cause was ranked. the ranking results were further compared for an effective and intelligent decision-making related to their criticality which contributes to the system’s unavailability. a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 311 fig. 1 proposed framework flow chart 3. fuzzy concept, reliability and risk modeling based approaches 3.1. notions of the fuzzy set theory this section only deals with those fuzzy concepts which were used in the proposed framework [22]. 3.1.1. crisp and fuzzy set a cisp set is defined as grouping of elements x x that are countable and finite; where each element can either belong to or not belong to the set. in 1965, zadeh introduced the theory of the fuzzy which can be defined as [22]: ]1,0[:)(~ →ux a  (1) where u represents the universe of discourse and ( ) a x is membership function of x in a ~ fuzzy set. 312 n. gopal, d. panchal 3.1.2. membership function (mf) in the literature, different forms of mf such as triangular, trapezoidal, gaussian, piecewise linear and singleton, etc. have been used by different researchers to consider vagueness in the collected information [22, 23]. many researchers were extensively using triangular membership function (tmf) in the past to consider uncertainty/vagueness in the collected data/information for carrying reliability and risk analysis for real industrial systems [21, 24, 25]. in the present work, tmf is used because of its easiness in computation. a tmf is defined mathematically as:           − − =  − − = otherwise txs st xt sx sxr rs rx x a ,0 , ,1 , )(~ (2) 3.1.3. linguistic variables linguistic variable means a variable whose values are words/sentences in a natural/artificial language. it is tough/hard to suggest a justified definition, which states the intricacy of the problems. so, these variables are beneficial for collecting views in many circumstances and could accept words from usual language, which are then wellexpounded by a fuzzy set in the range recommended by variables [26]. 3.2. reliability and mcdm approaches 3.2.1. lambdatau (λ-τ) approach fuzzy λ-τ approach is a powerful tool for evaluating reliability parameters under uncertainty developed by knezevic and odoom in 2001 [5]. since then, this approach has been widely implemented by various researchers for studying the failure behavior of different complex industrial systems based on tabulated reliability parameters at different spreads. since the fuzzy λ-τ approach considers uncertainty in raw data (provided by experts) which is one of the main limitations of the already existed markovian approach, the so-obtained analysis results with the implementation of the fuzzy λ-τ approach are highly effective in terms of accuracy. due to this advantage, the fuzzy λ-τ approach has been applied by many researchers in different process industries like paper mill, thermal power plant, and urea fertilizer industry, etc. [21, 23]. various steps of the fuzzy λ-τ approach are discussed as follows: step 1: use and/or gate develop pn model for representing series-parallel complex arrangement of the considered system. step 2: collect failure and repair time data of different subsystem/components as represented in pn model from various sources, namely, maintenance experts and maintenance logbook, etc. step 3: using tmf as defined by eq. (2), fuzzify the collected crisp failure and repair time data related to each sub-system/component. step 4: using and/or gate transition expressions for series-parallel arrangement as shown in table 1, develop mathematical modeling for the top event of the pn model. a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 313 table 1 basic expression of λ-τ methodology [5] gate λand λor τand τor expressions              =  = n i ji j n j n j j 1 1   = n i i 1     =  = =             n j ji i n i n i i 1 1 1     = = n i i n i ii 1 1   the developed mathematical equations for and/or gate transition from the basic expression (table 1) are represented as: and gate transition expression [5]                               +−−+−−           +−+− =     =  == =  == n j i n ji i iiii n i i n j i n ji i iiii n i i 1 3 1 2332 1 3 1 1 1 1211 1 2 })({}.)({ ,}){(}.){(    (3)                               +− +−−           +−− +− =       =  = = =  = = n j i n ji i ii i n i ii n j i n ji i ii i n i ii 1 1 1 12 3 1 23 1 3 1 23 1 1 12 }){( })({ , })({ }){(      (4) or gate transition [5]       +−−+−=   = = n i n i iiiiii 1 1 323112 })({},){(  (5)                                 +− +−+−− +−− +−+− =     = = = = n j iii n i iiiiii n j iii n i iiiiii 1 112 1 323323 1 323 1 112112 }){( })}.{()({ , })({ })}.{(){(      (6) step 5: using reliability expressions as shown in table 2, tabulate various reliability parameters for the considered mission time (t) at different α cut values varying from 0-1 314 n. gopal, d. panchal table 2 various reliability parameters [5] reliability indices expression mean time to failure c cmttf  1 = mean time to repair c cmttr  1 = mean time between failure ccc mttrmttfmtbf += availability t cc c cc c c cceta )( )(      +− + + + = reliability t c cetr − =)( expected number of failures ]1[ )( )( 2 2 t cc c cc cc cce t enof      +− − + + + = step 6: using center of area (coa) expression as represented in eq. (7) [27], tabulate crisp values for various reliability parameters to study and analyze the failure behavior of the considered system dss dsss ys d c y d c y )( )( ) ~ ( ~ ~ 0   =   (7) 3.2.2. fmea approach fmea is a widely used tool that helps in listing the failure causes, effects, and modes related with different subsystem/components of the complex industrial system [28]. a system or process may have several failure modes/failure causes and failure effects. in that condition, it is necessary to assess each failure cause and prioritize them accordingly. this approach considers the risk priority number (rpn) for prioritizing failure causes; the scores for these causes are computed by multiplying of, s and od. however, the crisp rpn score based ranking results were disparaged by many researchers because of many reasons [29, 30, 31]. significant disparaged results include: ▪ different failure causes may give same rpn score. ▪ equal weightage consideration for three risk factors under traditional fmea approach. ▪ consideration of only crisp values in the form of expert’s feedback means consideration of uncertainty is missing. 3.2.3. fuzzy copras method zavadsas and kaklauskas, first introduced this method in 1996. it is a renowned multi attribute decision-making method for finding the most appropriate alternative among all alternatives. this method is applied vigorously in numerous disciplines of decision making such as in critical infrastructures risk ranking [32], evaluating performance measures [33], hybrid wind farm [34], problems in selecting material [35], renewable a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 315 energy sources [36], selecting maintenance strategy [37], because of its simplicity and consideration of both ideal and ideal-worst solutions [38]. it is based on characteristics of the alternatives where the characteristics are contradictory. although characteristics and expert’s decision may contain uncertainty and imprecise data, the traditional approaches are still inadequate to model complex problems. so, their deftness is enhanced by integrating fuzzy logic into this approach. in comparison to the other mcdm approaches, the copras method deals with the conflicting criteria for solving the decision problem. due to this advantage, in the present work copras approach, has been incorporated within the fmea approach for ranking the listed failure causes under three conflicting risk factors. the various steps involved in the fuzzy copras approach are as follows. step 1: develop the initial decision matrix ) ~ (m for the three risk factors. the developed decision matrix is mathematically represented by eq. (8).               == pqpp q q ij mmm mmm mmm mm ~...~~ ~...~~ ~...~~ ~~ 21 22221 11211  (8) where i = 1, 2….p and j = 1, 2…q; ],,[~ ub ij mb ij lb ijij mmmm = is a tfn and lb, mb, ub are the lower bound, middle bound and upper bound. step 2: using eq. (9), convert a fuzzified values-based matrix into a crisp valuesbased decision matrix. 6 4 ub ij mb ij lb ij ij mmm m ++ = (9) where the developed crisp values-based matrix is represented by eq. (10).               == pqpp q q ij mmm mmm mmm mm ... ... ... 21 22221 11211  (10) where i = 1, 2, ….p and j = 1,2,…q step 3: develop a normalized decision matrix using eq. (11).  = = p i ij ij ij m m m 1 (11) the obtained normalized decision matrix is represented as:               == pqpp q q ij mmm mmm mmm mm ... ... ... 21 22221 11211  (12) where i = 1, 2, ….p and j = 1,2,…q 316 n. gopal, d. panchal step 4: using eq. (13), develop a weighted normalized decision matrix ijjij mwm =ˆ (13) where the weights for the three risk factors are calculated by using the fuzzy extent analysis method [39, 40] and the developed weighted normalized decision matrix is represented by eq. (14).               == pqpp q q ij mmm mmm mmm mm ˆ...ˆˆ ˆ...ˆˆ ˆ...ˆˆ ˆˆ 21 22221 11211  (14) where i = 1, 2, ….p and j = 1,2,…q step 5: calculate the sum for beneficial (bc+i) and non-beneficial (bc-i) criteria values by using eqs. (15) and (16).  = ++ = q j iji mbc 1 ˆ (15)  = −− = q j iji mbc 1 ˆ (16) step 6: using eq. (17), tabulate relative output significance (ri) for listed failure causes.   = − − = − + += p i i i p i i ii bc bc bc bcr 1 1 1 (17) step 7: calculate performance scores (ui) for each failure cause by using eq. (18) and rank the listed failure causes in a descending order. %100 max = r r u i i (18) 3.2.4. fuzzy topsis among many mcdm approaches, the topsis method, established by hwang and yoon in 1981, has become quite widespread owing to its simplicity, completeness and ease in results computation [17]. it is employed to get the ideal result among the similar decisions [41, 42]. this technique permits a compromise amongst several decision considerations where bad effect in any one of the factors can be balanced with the beneficial effect of other factor [43]. since uncertainty occurs almost in all decision data the topsis can be simply extended with a fuzzy set concepts for handling the vagueness/uncertainty of the raw data for high accuracy in decision results [44]. in the past, for checking the consistency of ranking results obtained from many other novel decision-making approaches, the fuzzy topsis approach was integrated and found the a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 317 application of integrated model in different fields [23, 45, 46, 47, 48, 49, 50]. in the present work, the ftopsis approach is integrated with the fcopras approach for evaluating the consistency of ranking results because of its mathematical modeling similarity with many steps of the fcopras approach. the first four steps of ftopsis are similar to the fcopras approach and the remaining different steps are discussed as follows: step 5: calculate ideal best solution (vvj +) and ideal worst solution (vvj -) for the three risk factors. here, in this work, of and s are considered as non-beneficial criteria, and od are beneficial criteria, so (vvj +) is calculated by taking minimum value for non-beneficial criteria and maximum value for beneficial criteria. ideal worst solution (vvj -) is calculated by taking maximum value for non-beneficial criteria and minimum value for beneficial criteria. step 6: tabulate the euclidean distance from ideal best (ssj +) and euclidean distance from ideal worst (ssj -) by using eqs. (19) and (20) as:  = ++ −= n j jiji vvvvss 1 2)( (19)  = −− −= n j jiji vvvvss 1 2)( (20) step 7: using eq. (21), tabulate final performance score (fpsi) for each failure cause and rank the failure causes in a descending order. −+ + = ii i i ssss ss fps (21) 4. case study the proposed framework is presented with its application to the curd unit of a milk production plant located in the northern part of india. the curd unit is one of the most critical units of the considered milk process industry consisting of various subsystem/components, namely, milk storage tank, centrifugal feed pump, milk balance tank pasteurizer and culture tank, etc. which are arranged in series-parallel complex configuration as shown in fig. 2. the main subsystem/components of the considered unit are discussed as follows: (i) milk storage tank: used for storing milk and set in series arrangement. (ii) centrifugal pump: arranged in series configuration and used for pumping the milk from milk storage tank to balance tank. (iii) milk balance tank: it is a standardization process which is done to balance fat and solids-not-fat (snf). it is also arranged in series configuration. (iv) pasteurizer: used to exchange the heat and has three different plates namely, regeneration-1, regeneration-2, regeneration-3 connected in series configuration with the unit. (v) homogenizer: used to reduce the formation of cream by its high pressure 2000 to 2500 psi to mix all ingredients thoroughly and arranged in series configuration with the unit. 318 n. gopal, d. panchal (vi) booster pump: used to pump the milk from pasteurizer to holding tank and culture tank. it is also arranged in series configuration. (vii) holding tank: arranged in series and it is used to hold the heated milk (900c) so that contaminates, and microbes can be killed. (viii) culture tank: used to cool down the milk up to 400c 450c and then culture is added. for pouch filling one unit of culture tank is arranged in series configuration and for cup filling two units of culture tanks are arranged in parallel configuration. (ix) pouch filling machine: used to fill the pouch of curd and two units are arranged in parallel set up. (x) cup filling machine: used to fill the cup of curd and two units are arranged in parallel set up. one unit is operative and other remains in stand-by mode. fig. 2 curd unit schematic diagram 4.1. proposed framework application 4.1.1. reliability analysis-fuzzy λ-τ approach application using and/or gate symbol, develop pn model (fig. 3) representing series-parallel arrangement for the considered curd unit of the milk process industry. a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 319 fig. 3 pn model in fig. 3: cu: curd unit, p: pasteurizer, ct: culture tank, pf: pouch filling, cf: cup filling on the basis of expert opinion and maintenance log book record failure rate (λi) and repair time (τi) data for each subsystem/component was collected as shown in table 3. table 3 λi τi data of considered unit component failure rate (λi) (failures/hr) repair time (τi) (hrs) milk storage tank (n=1) 2.31 x 10-4 4 centrifugal feed pump (n=2) 3.37 x 10-4 5 milk balance tank (n=3) 1.16 x 10-4 2 centrifugal feed pump (n=4) 3.37 x 10-4 5 pasteurizer plant (n=5,7,8,9,10) 2.31 x 10-4 8 homogenizer (n=6) 4.62 x 10-4 1 culture tank (n=11, 12, 13) 2.31 x 10-4 4 pouch filling machine (n=14, 15) 4.62 x 10-4 3 cup filling machine (n=16, 17) 2.31 x 10-4 3 using eq. (2) for tmf, the collected λi and τi data were fuzzified and converted into triangular fuzzy numbers at different spread (± 15 %, ± 25 %, ± 60 %) in order to consider the vagueness/uncertainty in the raw data. using eqs. (3-6), mathematical modeling for the top event as per the developed pn model was generated. fuzzified data were used in the developed mathematical modeling and reliability indices at different 320 n. gopal, d. panchal spread (± 15 %, ± 25 %, ± 60 %) for α -cut values varies between 0-1, were tabulated using reliability expressions (table 2). here, for illustration, reliability indices at left and right spread for 15 % are shown in tables 4 and 5, respectively. table 4 reliability indices left spread values at ± 15 % dom failure rate repair time mtbf reliability availability 1 0.002294 0.006595588 436.00222 0.680230 0.999985 0.9 0.002259 0.006331246 429.54961 0.676304 0.999984 0.8 0.002225 0.006076471 423.28481 0.672400 0.999983 0.7 0.002190 0.005830808 417.19973 0.668519 0.999982 0.6 0.002156 0.005593833 411.28674 0.664660 0.999981 0.5 0.002121 0.005365154 405.53864 0.660823 0.999980 0.4 0.002087 0.005144404 399.94862 0.657008 0.999979 0.3 0.002052 0.004931242 394.51024 0.653214 0.999978 0.2 0.002018 0.004725348 389.21742 0.649443 0.999977 0.1 0.001984 0.004526426 384.06437 0.645693 0.999975 0 0.001949 0.004334196 379.04564 0.641964 0.999974 table 5 reliability indices right spread value at ± 15 % dom failure rate repair time mtbf reliability availability 1 0.002294 0.006595588 436.00222 0.680230 0.999985 0.9 0.002328 0.006869987 442.65123 0.684178 0.999986 0.8 0.002363 0.007154971 449.50573 0.688149 0.999986 0.7 0.002397 0.007451111 456.57541 0.692143 0.999987 0.6 0.002431 0.007759025 463.87059 0.696160 0.999988 0.5 0.002466 0.008079379 471.40221 0.700200 0.999989 0.4 0.002500 0.008412898 479.18199 0.704263 0.999989 0.3 0.002535 0.008760368 487.22239 0.708349 0.999990 0.2 0.002569 0.009122641 495.53674 0.712459 0.999990 0.1 0.002604 0.009500648 504.13929 0.716593 0.999991 0 0.002638 0.0098954 513.04531 0.720750 0.999992 similarly, fuzzified values are tabulated for ± 25 % and ± 60 %. due to space limitations these values are not shown here. here, on the basis of the operational engineer feedback, mission time (t) was considered as 168 hrs for tabulating various reliability indices. using eq. (7), the fuzzified reliability indices values are converted into crisp values as shown in table 6 and are graphically shown in fig. 4(a-g). table 6 trends of reliability indices parameters crisp value spread at ± 15 % spread at ± 25 % spread at ± 60 % trend failure rate 0.002294 0.00229367 0.00229380 0.00229475 increasing trend repair time 0.006595588 0.006942 0.007617 0.017243 mtbf 436.0022241 442.697722 455.394389 599.625582 enof 0.38531913 0.38533034 0.38535010 0.38548316 reliability 0.6802295 0.68098113 0.68231838 0.69230835 availability 0.99998487 0.9999834 0.99998064 0.99994043 decreasing trend unreliability 0.3197704 0.31901886 0.31768161 0.30769164 a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 321 (a) (b) (c) (d) (e) (f) (g) fig. 4(a-g) trends of reliability indices 322 n. gopal, d. panchal 4.1.2. reliability analysis-based failure behavior from table 5, it is noted that the reliability indices, namely, repair time, failure rate, mean time between failure (mtbf), expected number of failure (enof), reliability and unavailability are showing increasing trends with the increase in spread from ± 15 % to ± 25 % and ± 25 % to ± 60 %, respectively. on the other hand, availability shows a decrease in trend as the spread increases from ± 15 % to ± 25 % and ± 25 % to ± 60 %, respectively. since the system availability shows a decreasing trend, in order to improve the availability and maintainability aspects, it is essential to carry a risk analysis of the considered unit as presented in section 4.2. 4.2. risk analysis 4.2.1. fmea application failure mode effect analysis (fmea) sheet entails a failure mode; its effects and causes were prepared with the help of three maintenance experts’ feedback. here, due to space limitation, the fmea detailed sheet with one expert’s ratings feedback is shown in table 10. linguistic scales (tables 7-9) were developed and provided to the experts in order to put their feedback against each listed failure causes under three risk factors: of, s and od. table 7 linguistic variables for of linguistic variables probability of failure tfn very high (vh) 0-2 months (8,9,10) high (h) 2-5 months (6,7,8) medium (m) 5-8 months (4,5,6) fair (f) 8-10 months (3,4,5) low (l) 10-12 months (2,3,4) very low (vl) >1 year (1,2,3) table 8 linguistic variables for s linguistic variables severity effects tfn very serious (vs) the level of severity is very high without warning (8,9,10) very extreme (ve) the level of severity is high with warning (7,8,9) major damage (md) system is unavailable with component damage (6,7,8) moderate (mo) performance of system is affected and maintenance is required (4,6,7) low (l) minor effect on system performance and minor maintenance is required (2,3,4) no effect (n) no effect on system performance (1,2,3) table 9 linguistic variables for od linguistic variables likelihood of detection of failure tfn uncertain (u) detection by opening the sub-system/component & failure not detected visually & required to be replace (7,8,10) very remote (vr) detection by opening the sub-system/component & failure seen visually (6,7,8) remote (r) detection with the help of automatic devices (4,5,6) moderate (mo) detection by display (3,4,5) certain (c) visual detection of operator (1,2,3) a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 323 table 10 fmea detailed sheet with one expert’s ratings feedback sl no components failure mode failure effect failure cause of s od 1 milk storage tank & centrifugal feed pump (mscp) milk valve leakage milk loss mechanical seal failure (mscp1) h l vr cip valve proper closing/opening failure cleaning stops low pressure of air (mscp2) h ve c pump failure of impeller milk flow rate loss impeller blade breakage (mscp3) f ve vr mechanical seal leakage seal wear seals run dry, the friction will cause heat to accumulate (mscp4) h mo c motor electrical failure supply breakdown overheating / moisture lead to winding failure (mscp5) l md u 2 milk balance tank, duplex filter & pasteurizer (mbfp) butterfly valve closing/opening failure supply breakdown over tightened (mbfp1) h l c seal packing failure (mbfp2) vh l c stainless steel filter choke flow rate reduces visible sediment of milk jam in filter (mbfp3) l mo c heat exchanger plates mechanical failure efficiency loss crack in steel plate (mbfp4) vl ve vr corrosion (mbfp5) l ve vr gasket leakage operational loss erosion (mbfp6) vh mo c 3 homogenizer (h) piston mechanical failure supply breakdown insufficient lubrication (h1) vl ve u thermal fatigue (h2) l ve u piston seal oil leakage pressure loss excessive wear (h3) vh mo vr crankcase oil contamination operation loss improper quality of oil (h4) vl md vr ball spring mechanical failure oil flow breakdown spring breakage (h5) f md mo main drive motor mechanical and electrical failure operation loss overheating (h6) l md u winding failure (h7) f md u v-belt belt slippage power transmission interrupted insufficient tension (h8) vl md vr hydraulic pressure setting system leakage drop in pressure air and water contamination (h9) vl mo vr low fluid level (h10) l mo vr 324 n. gopal, d. panchal sl no components failure mode failure effect failure cause of s od 4 culture tank (ctk) tank leakage loss of milk improper insulation (ctk1) vl md c agitator motor mechanical and electrical failure supply breakdown overheating (ctk2) f md r winding failure (ctk3) f md r agitator blade mechanical failure no proper mixing blade breakage (ctk4) vl l c agitator gear box mechanical failure breakdown improper lubrication (ctk5) l ve u wearing in gears (ctk6) vl ve u 5 pouch filling machine (pfm) element electric failure sealing interrupted too high temperature (pfm1) vh md c head gasket leakage operational loss erosion (pfm2) h mo c head spring mechanical failure machine breakdown spring breakage (pfm3) vh md vr injection coil mechanical failure machine breakdown coil breakage (pfm4) l md r injection sheet mechanical failure milk continue to supply i.e. milk loss bearing failure (pfm5) l ve vr rubber failure which is below the sheet (pfm6) l ve vr servo motor electrical and mechanical failure supply to pouch breakdown overheating (pfm7) f ve r too much moisture (pfm8) f ve r grippers mechanical failure fail to hold pouch wear & tear (pfm9) f md c gear box mechanical failure breakdown improper lubrication (pfm10) m md r wearing in gears (pfm11) l md c cam mechanical failure breakdown fatigue & wear due to corrosion (pfm12) l md u belt belt slippage power transmission interrupted insufficient tension (pfm13) m mo u a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 325 sl no components failure mode failure effect failure cause of s od 6 cup filling machine (cfm) proximity sensor diagnostic bit idle cup will not be detected sense gap between sensor face and trip face (cfm1) h mo c vacuum rubber mechanical failure cup will not be pulled out rubber breakage (cfm2) m md vr spring breakage (cfm3) m md c sealing die high temperature sealing interrupted element breakage (cfm4) vl md c injection rod mechanical failure cup filling problem gasket seal breakage (cfm5) vl md vr cylinder air valve excessive heat air control will get interrupted due to heat (cfm6) f mo vr air cylinder mechanical failure sealing die breakdown cylinder leakage (cfm7) l mo vr plc controller input/output module failure machine breakdown voltage supply (cfm8) vl md c 4.2.2. fcopras application fcopras approach was implemented within the traditional fmea approach in order to overcome its limitations as discussed in section 3.2.2. under the fcopras implementation, using fuzzy linguistic rating scales for three risk factors (tables 6-8), the initial fuzzy decision matrix for the set of listed failure causes is generated using eq. (8). here, for illustration, the initial fuzzy decision matrix for the set of failure causes listed under first subsystem/component (mscp) is generated as:                 = )10,8,7(),6,5,4(),10,8,7()8,7,6(),8,7,6(),8,7,6()4,3,2(),3,2,1(),4,3,2( )8,7,6(),3,2,1(),3,2,1()4,3,2(),7,6,4(),7,6,4()10,9,8(),5,4,3(),8,7,6( )8,7,6(),3,2,1(),8,7,6()8,7,6(),8,7,6(),9,8,7()6,5,4(),6,5,4(),5,4,3( )3,2,1(),6,5,4(),3,2,1()9,8,7(),9,8,7(),9,8,7()8,7,6(),10,9,8(),8,7,6( )8,7,6(),6,5,4(),8,7,6()7,6,4(),8,7,6(),4,3,2()8,7,6(),6,5,4(),8,7,6( ~ ijm the initial fuzzy decision matrix for the set of failure causes listed under other subsystem/components was generated in the same way. for considering the effect of all three experts, their average was taken, and a modified initial fuzzy decision matrix for the set of listed failure causes was generated. here, for illustration, the modified initial fuzzy decision matrix for the set of failure causes listed under first subsystem/component (mscp) is represented as:                 = )6667.8,7,6()8,7,6()6667.3,6667.2,6667.1( )6667.4,6667.3,6667.2()6,5,3333.3()6667.7,6667.6,6667.5( )3333.6,3333.5,3333.4()3333.8,3333.7,3333.6()6667.5,6667.4,6667.3( )4,3,2()9,8,7()6667.8,6667.7,6667.6( )3333.7,3333.6,3333.5()3333.6,3333.5,4()3333.7,3333.6,3333.5( ~ ijm 326 n. gopal, d. panchal the modified initial fuzzy decision matrix for other set of failure causes listed under fmea sheet was generated in the same manner. using the modified initial fuzzy decision matrix values in eqs. (9-10), a crisp values-based decision matrix for set of listed failure causes was developed. here, for illustration, the crisp values-based decision matrix for the set of failure causes listed under first subsystem/component (mscp) is represented as:                 = 1111.776667.2 6667.38888.46667.6 3333.53333.76667.4 386667.7 3333.627775.53333.6 ijm the crisp values-based decision matrix for other set of failure causes was also generated in the same way. furthermore, using the crisp values-based decision matrix values in eqs. (11) and (12), a normalized decision matrix for set of listed failure causes was generated. here, for illustration, the normalized decision matrix for the set of failure causes listed under first subsystem/component (mscp) is represented as:                 = 2795.02154.00952.0 1441.01504.02381.0 2096.02256.01667.0 1179.02462.02738.0 2489.01652.02262.0 ijm for other set of failure causes a normalized decision matrix was generated in the same way. using the normalized decision matrix in eqs. (13-14), a weighted normalized decision matrix was generated for each set of listed failure causes under fmea sheet. here, weights for three risk factors are tabulated by implementing the fuzzy extent analysis method [39, 40]. wang’s scale as shown in table 11 was used for collecting feedback from the plant’s maintenance manager in order to develop a comparison matrix for weight calculation. table 11 wang’s scale for comparison matrix uncertain judgment tfn scale approximately equal (0.50, 1, 2) approximately m timesa more important (m 1, m, m + 1) approximately m times less important (1/(m + 1), 1/m, 1/(m 1) between n and o timesb more important (n, (n + o)/2, o) between n and o times less important 1/o, 2/(n + o), 1/n in table 11: am= 2 to 9 and bn, o=1 to 9, n < o. based on the feedback, a comparison matrix was developed; following the mathematical modeling of the fuzzy extent analysis method [39, 40] weights for three risk factors were tabulated as shown in table 12. a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 327 table 12 of, s and od weights of s od weight of (1 1 1) (2 3 4) (1 2 3) 0.5405 s (0.25 0.3333 0.5) (1 1 1) (1 2 3) 0.3074 od (0.3333 0.5 1) (0.3333 1 1) (1 1 1) 0.1520 due to space limitations, weight calculation for the three risk factors is not shown here. using the tabulated weights for three risk factors in eqs. (13) and (14), the weighted normalized matrix for the set of failure causes listed under first subsystem/component (mscp) is represented as an illustration:                 = 0425.00662.00515.0 0219.00462.01287.0 0319.00694.00901.0 0179.00757.01480.0 0378.00499.01223.0 ijm the weighted normalized decision matrix for other set of failure causes was generated in the same way. further, using eqs. (15) and (16), the sum for beneficial (bc+i) and nonbeneficial (bc-i) criteria-based risk factors values were tabulated for each listed failure causes. for illustration, beneficial (bc+i) and non-beneficial (bc-i) criteria-based risk factor values for the set of failure causes listed under first subsystem/component (mscp) are represented in table 13 as: table 13 relative output significance and performance scores based ranking results subsystem’s components bc+i bc-i ri ui rank mscp1 0.0378 0.1722 0.1979 71.5325 3 mscp2 0.0179 0.2237 0.1411 51.0163 5 mscp3 0.0319 0.1594 0.2047 73.9922 2 mscp4 0.0219 0.1749 0.1795 64.8628 4 mscp5 0.0425 0.1177 0.2767 100 1 beneficial (bc+i) and non-beneficial (bc-i) criteria-based risk factor values for other set of failure causes were tabulated in similar manner. using beneficial (bc+i) and nonbeneficial (bc-i) criteria-based risk factors values for each set of listed failure cause in eqs. (17) and (18), relative output significance (ri) and performance scores (ui) were calculated and ranking was done in a descending order on the basis of ui scores. table 12 shows ui based ranking results for the set of listed failure causes for first subsystem/component (mscp) as illustration. similarly, ui score based ranking results were obtained for other set of failure causes listed under fmea sheet and shown in ranking comparison table 14. 328 n. gopal, d. panchal table 14 ranking comparison for failure causes based on fcopras and ftopsis outputs sr.no. failure cause copras output copras rank topsis output topsis rank 1 mscp1 71.5325 3 0.3688 3 2 mscp2 51.0163 5 0 5 3 mscp3 73.9922 2 0.5643 2 4 mscp4 64.8628 4 0.3071 4 5 mscp5 100.0000 1 0.8336 1 6 mbfp1 60.0240 4 0.5184 4 7 mbfp2 49.9655 5 0.2438 5 8 mbfp3 100.0000 1 0.9537 1 9 mbfp4 74.5548 2 0.7941 2 10 mbfp5 74.3876 3 0.7822 3 11 mbfp6 42.0030 6 0.1775 6 12 h1 87.7278 3 0.9064 3 13 h2 82.0554 5 0.8880 4 14 h3 43.3299 10 0.0751 10 15 h4 100.0000 1 0.9820 1 16 h5 51.0587 9 0.4093 9 17 h6 74.4051 6 0.8413 6 18 h7 70.1905 8 0.7952 7 19 h8 73.7672 7 0.7623 8 20 h9 88.3805 2 0.9256 2 21 h10 82.7810 4 0.8747 5 22 ctk1 70.7957 3 0.5910 3 23 ctk2 52.6152 6 0.1381 6 24 ctk3 55.0623 5 0.1862 5 25 ctk4 100.0000 1 0.7179 1 26 ctk5 65.1985 4 0.3792 4 27 ctk6 79.9817 2 0.6534 2 28 pfm1 44.0356 13 0.1098 13 29 pfm2 48.4538 12 0.2198 11 30 pfm3 53.2796 11 0.1689 12 31 pfm4 91.4308 4 0.8810 3 32 pfm5 96.0816 2 0.8900 2 33 pfm6 77.8006 6 0.7362 7 34 pfm7 73.7532 9 0.7241 8 35 pfm8 77.4411 7 0.7382 6 36 pfm9 67.0269 10 0.6561 10 37 pfm10 76.6158 8 0.6998 9 38 pfm11 100.0000 1 0.9094 1 39 pfm12 92.3045 3 0.8770 4 40 pfm13 78.4270 5 0.7396 5 41 cfm1 46.4148 8 0.0250 8 42 cfm2 69.3448 6 0.3808 6 43 cfm3 65.2547 7 0.3577 7 44 cfm4 99.3818 3 0.8158 3 45 cfm5 99.6685 2 0.9187 2 46 cfm6 74.4761 5 0.4957 5 47 cfm7 100.0000 1 0.9207 1 48 cfm8 80.2527 4 0.7602 4 a structured framework for reliability and risk evaluation in milk process industry under fuzzy... 329 4.2.3. ftopsis application for consistency check and effective decision-making of critical failure causes, the well known existing ftopsis approach was applied within the fcopras approach. using the weighted normalized matrix values as tabulated under the fcopras approach, the ideal best solution (vvj +) and ideal worst solution (vvj-) values for the three risk factors are computed. the ideal best solution (vvj +) and ideal worst solution (vvj-) values for the set of failure causes listed under first subsystem/component (mscp) are represented in table 15 as an illustration. table 15 vvj + and vvjfor the three risk factors subsystem’s components of s od mscp1 0.122257 0.049925 0.037848 mscp2 0.147997 0.075676 0.017928 mscp3 0.090085 0.06937 0.031872 mscp4 0.128693 0.046247 0.021912 mscp5 0.051478 0.066217 0.042496 vvj + 0.051478 0.046247 0.042496 vvj0.147997 0.075676 0.017928 similarly, vvj + and vvjfor other set of failure causes were tabulated. further, using vvj + and vvjvalues from each set of listed failure cause under different subsystem/component in eqs. (19) and (20), the euclidean distance from ideal best (ssi +), euclidean distance from ideal worst (ssi -) were tabulated and the final performance score (fpsi) was obtained by using eq. (21). for illustration, the tabulated euclidean distance from ideal best (ssi +), euclidean distance from ideal worst (ssi -) and final performance score (fpsi) with ranking for the set of listed failure causes under first subsystem/component (mscp) are shown in table 16. table 16 ranking results based on final performance score subsystem’s components ssi + ssi fpsi rank mscp1 0.0710 0.0415 0.3688 3 mscp2 0.1039 0.0000 0.0000 5 mscp3 0.0462 0.0599 0.5643 2 mscp4 0.0799 0.0354 0.3071 4 mscp5 0.0200 0.1000 0.8336 1 similarly, for other set of listed failure causes, the euclidean distance from ideal best (ssi +), euclidean distance from ideal worst (ssi -) and final performance scores (fpsi) were tabulated and ranking was done for the set of failure causes listed under fmea sheet as shown in ranking comparison in table 14. 4.2.4. result discussion from table 14, it has been found that with the application of fcopras & ftopsis approaches failure causes mscp5, mbfp3, h4, ctk4, pfm11, and cfm7 associated with different subsystem/components of the considered curd unit with outputs 100,100, 100,100,100 and 100 & 0.8336,0.9537,0.9820, 0.7179, 0.9094 and 0.9207 are prioritized as 330 n. gopal, d. panchal most critical failure causes; responsible for decrease in system’s availability. furthermore, failure causes h2, h7, h8, h10, pfm2, pfm3, pfm4, pfm6, pfm7, pfm8, pfm10 and pfm12 shows little variation in the ranking results with the implementation of both fcopras & ftopsis approaches. from table 13, it is also noted that out of the total 48 listed failure causes, 36 listed failure causes show same ranking results, which indicates towards the consistency and effectivity of the proposed integrated framework. 5. conclusion, limitation and future scope of the work an integrated framework for studying the curd unit performance of a milk process industry has been proposed. tmf was used for the fuzzification of reliability and risk-based input data/information for achieving high accuracy in the performance-based results. availability of the considered unit shows a decreasing trend with an increase in spread. failure causes mscp5, mbfp3, h4, ctk4, pfm11, and cfm7 associated with different subsystem/components of the considered unit are found to be most critical on the basis of their fcopras and ftopsis output scores. fcopras and ftopsis output scores based ranking results were compared for evaluating the consistency or robustness of the proposed integrated framework. the limitations of the traditional fmea approach are covered effectively and an element of uncertainty in the accuracy of decision results was suppressed. 5.1. managerial implications the result outcome of this work was supplied to the maintenance manager of the considered milk process industry and was asked to implement these results for further testing. with the result outcomes the maintenance manager showed his keenness and expressed that once the top management made a decision to implement the results the usefulness of the proposed framework could be evaluated further. from the proposed work, the following managerial implications are derived and communicated to the maintenance manager/reliability engineer of the curd unit of the considered milk process industry; he is able to make decisions related: ▪ to analyze and study the failure dynamics of the subsystem/components because of an increasing/ decreasing trend of reliability parameters. ▪ to consider the uncertainties/ vagueness associated with the raw data by incorporating a fuzzy based approach. ▪ to develop an optimum maintenance schedule for the considered unit in order to improve system’s availability over long duration. 5.2. limitations of the work the analysis results so obtained with the implementation of the proposed framework will depend upon the quality of information/data provided by the maintenance experts. therefore, the obtained results may be subjective in nature. however, by incorporating fuzzy set theory 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by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper microstructure influence on friction behavior of the ti6al4v biomedical alloy at low loads fatima zivic1, slobodan mitrovic1, petar todorovic1, dragan adamovic1, nenad grujovic1, marko spasic2, ivan stojadinovic3 1university of kragujevac, faculty of engineering, kragujevac, serbia 2university of kragujevac, faculty of medical sciences, department of surgery, kragujevac, serbia 3clinic for orthopedics and traumatology, university clinical center, kragujevac, serbia abstract. dynamic friction coefficient (cof) between ti6al4v and al2o3 was analyzed under low loads (100 mn, 250 mn, 500 mn, 750 mn, 1000 mn), sliding speed (4 mm/s, 8 mm/s, 12 mm/s) at dry contact and in the ringer's solution. different ti6al4v microstructures were studied: sample 1 fully lamellar; sample 2 martensitic; sample 3 equiaxed; and sample 4 globular microstructure. the maximum cof values varied as: 0.4 1.23 (sample 1), 0.5 – 2.8 (sample 2), 0.4 – 1.1 (sample 3), and 0.4 – 2.3 (sample 4). lamellar and martensitic microstructures were not beneficial for the tribological response since they exhibited severe wear and very high cof values. the globular ti alloy microstructure showed extremely high cof and wear under dry conditions. in general, water quenching was not a favorable treatment for tribological behavior. the lowest cof values and wear volumes were exhibited in the case of equiaxed microstructure. key words: ti6al4v, friction coefficient, tribochemical wear, reciprocating sliding 1. introduction biomedical applications of ti alloys are broad and extensive due to their high strength, low density, high corrosion resistance, excellent biocompatibility and elastic modulus that can be tailored close to the one of the bone [1, 2]. however, even with extensive research on processing parameters during production and different thermo-mechanical processing [3, 4, 5, 6], ti alloys still exhibit certain drawbacks that prevent their wider application as implant materials. the major problem with these alloys is a low wear resistance [7], received: march 08, 2023 / accepted may 04, 2023 corresponding author: fatima zivic university of kragujevac, faculty of engineering, sestre janjić 6, 34000 kragujevac, serbia e-mail: zivic@kg.ac.rs 2 f. zivic, s. mitrovic, p. todorovic, et al. especially under variable contact conditions including elevated temperatures. high temperatures significantly change the mechanical properties of ti alloys [8]. thermal softening, during the contact, depending on the load and velocity, is proven for ti alloys [9], thus greatly influencing friction and wear response of the material. mechanical properties of ti alloys (tensile behavior and especially the yield strength) are closely related to their microstructure, including multi-scale microstructural properties (like precipitates, regions of different microtextures and their distribution along with dynamic complex changes under external stresses) [10, 11, 12, 13]. changes in the microstructure due to the heat treatments, or some other dynamic process, can result in rather different mechanical behavior even in the case of similar initial microstructures [10, 14, 8, 15]. variation of heat treatment parameters can enable tailoring of different microstructures [5, 4, 16]. dynamic processes at α/β interfaces under deformation have significant influence on the resulting mechanical behavior of ti alloys [17]. tribological contact also provokes different microstructural changes within the surface layers [18, 19, 20, 21]. different contact environments introduce additional influential factors from chemical and tribochemical processes that develop during the tribological contact [22]. ti alloys are highly reactive and prone to different chemical reactions, especially in corrosive and aggressive environments [14, 23]. for example, presence of hydrogen can cause severe embrittlement under deformation [24]. altogether, this makes a very complex and difficult task to predict and design good mechanical response of the ti alloys for all load bearing applications where the tribological contact is present, thus limiting the use of this otherwise excellent biomaterial. standard ti6al4v alloy is widely used in total hip arthroplasty [25, 26], but challenges related to the stress distribution (stress shielding) and poor contact wear resistance, largely focus its application to the stationary femoral stem and acetabular components [27, 28, 29]. during the last decades several astm standards were adopted, pertaining to ti6al4v alloy for medical applications: astm f1472 – 14, astm f1108 – 14, astm f136 – 13, astm f2885-17, astm f2146-13, astm f620-11(2015), astm f3001-14, astm f2193-20. they provide guidelines and recommendations on the optimal processing routes and parameters to obtain the best microstructure, from aspects of its applications. however, friction and wear mechanisms are still under investigations, focusing on the possibilities to surpass the previously listed shortcomings related to the wear resistance. comprehensive review of ti alloy properties is given in [1, 2]. review of different ti6al4v microstructures depending on the processing parameters is presented in [30], along with mechanisms that govern tribological behavior under various dry conditions. they analyzed tribo-oxidation layers and wear mechanisms associated with it. they also reviewed research related to strain rate response, also in conditions of elevated temperature and frictional heating, which leads to significant changes of ti6al4v microstructure in surface layers and further contribute to changes of wear mechanisms. extensive research has been done related to production routes and processing of ti alloys, in order to obtain the most favorable microstructures from aspect of wear resistance [2, 3], and new additive manufacturing technologies have emerged to enable better tailoring of properties [31, 32, 33, 27, 29]. friction causes high flash temperatures in the contact zone [34, 35], especially for the dry sliding, thus decreasing the resistance to plastic shearing in ti alloys; different models have been investigated [36, 37, 38]. surface oxides created during the contact, due to reactions with counterbody material and environment, greatly influences wear and friction microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 3 behavior. in some cases tio2 passive film can be formed on ti6al4v that protects the surface from wear [39]. other ti oxides can be found (ti2o3, tio2) and depending on the counterbody material, very hard and sharp tial intermetallics have also been proven (ti3al, tial, and tial3) that increase wear rate [40, 41]. distribution and size of α and β phases govern the damage mechanisms during the loading while the changes in the microstructure result in different material responses [42, 23]. high flash temperatures that appear within the limited contact zones, due to friction, rapidly decrease ti6al4v yield strength and result in localized material softening [43, 41]. it is further favorable for mechanical alloying of the materials in contact. altogether, it results in complex dynamic changes of ti6al4v microstructures in surface layers during the tribocontacts. both wear and friction can exhibit transition feature during certain contact conditions [39, 40]. research on underlying mechanisms to these complex dynamic processes is still ongoing. this paper presents experimental data and analysis pertaining to the friction and wear mechanisms during reciprocating sliding of ti6al4v with four different microstructures against al2o3, under low loads and different sliding speeds. we varied five values of load, three values of sliding speed and two different contact environments (dry contact and contact in the ringer’s solution), and analyzed friction and wear mechanisms. the efficient friction coefficient (cofeff) that numerically represents the whole cof curve was introduced in the evaluation of friction results. 2. materials and experimental methods 2.1 materials ti6al4v alloy flat samples were cut from commercial ti6al4v alloy bars and subjected to four types of heat treatments, as described in [42, 23]. ti6al4v eli (extra low interstitials) (grade 23) was used as one of the main materials for orthopedic implants, with good fracture toughness due to the lowered oxygen content in comparison to the commercial ti6al4v alloy (grade 5). four types of samples were produced, by annealing them in ar for 1h, at two different temperatures, followed by two different cooling methods and rates (slow cooling in furnace and rapid cooling by water quenching), as follows:  sample 1 annealed at 1000°c; furnace cooled, with fully lamellar microstructure,  sample 2 annealed at 1000°c; water quenched, with martensitic microstructure,  sample 3 annealed at 750°c; furnace cooled, with equiaxed microstructure,  sample 4 annealed at 750°c; water quenched, with globular microstructure. final polishing and cleaning were done in accordance with astm f86, as described in [7]. the final roughness was ra=0.07µm. the material in contact with all ti samples was alumina ball (al2o3), commercially supplied by anton paar company. aluminum oxide is a chemically inert material, with excellent abrasion resistance and significantly harder than ti alloys, which should provide that the material changes during tribocontacts are to be attributed only to the ti alloy. 2.2 tribological tests comprehensive tribological tests were realized, by using linear reciprocating sliding module (tribometer from anton paar, model csm nanotribometer), with details of the 4 f. zivic, s. mitrovic, p. todorovic, et al. device setup and parameters given in [7]. all ti alloys were subjected to 30000 cycles (0.5mm stroke), in contact with alumina ball (1.5 mm diameter) and with variation of the following test parameters: five different loads (100mn, 250mn, 500mn, 750mn, 1000mn), three different maximum sliding speeds (4mm/s; 8mm/s; 12mm/s) and two different contact environments (dry contact and contact in the ringer’s solution), at the room temperature. composition of the ringer solution used is as follows (g per 1 liter of water): nacl 8.6; kcl 0.30; cacl2 0.33; na+ 147.00 mmol; k+ 4.00 mmol; ca+ 2.25 mmol; cl+ 155.60 mmol. in the case of the wet contact, the whole contact zone was fully immersed in fresh ringer's solution, for each of the tests. 30000 cycles accounted for approximately up to 3.6h depending on the nature of the friction coefficient of the investigated sample. during each test, a nanotribometer-recorded dynamic friction coefficient values in real time, with 100hz sampling frequency. the device also recorded values of the penetration depth of alumina ball into flat ti samples, during the contact, with the same sampling frequency. details of the penetration depth parameter changes and wear are given in [7, 42]. images of the worn tracks on ti samples were taken by optical and sem microscopy (meiji mt8530, meiji mc 50t microscopes and jeol jsm-6610lv sem microscope). the dynamic friction coefficient is represented by the continuous curve showing values of friction coefficient at discrete points of time during the test, depending on the adopted sampling frequency. for 100hz sampling frequency, there are numerous recorded values of the discrete friction coefficients for one test (100 values each second during the test), as output results from the nanotribometer. the sine wave shape of the cof curve (fig. 1) enables processing of the raw data signal in several ways aiming to filter the signal for easier interpretations. it is possible to calculate one numerical value, the efficient friction coefficient (cofeff) that will numerically represent the whole cof curve, by applying averaging and root mean square of the cof function. in such way, the complex cof curve is reduced to one number, cofeff that corresponds well with frictional behavior of the contact pair and can rapidly show specific trends with changes of the contact load and speed. detailed discussion of cofeff is given in [44] and it is calculated according to the following equation: n cofcofcofcof cof n eff 22 3 2 2 2 1 ...  (1) where, cofeff is the numerical value (root mean square of the cof function for one test), cofi is the numerical cof value recorded by the device at i-point of time and n is the total number of points at which cof was recorded during one test. 3. results and discussion the dynamic friction coefficient (cof) for 10 cycles (laps), as extracted from one test (sample 1, dry contact, v=4 mm/s; fn=100mn) is shown in fig. 1. it can be seen that, in general, the dynamic friction coefficient curve is of sinusoidal shape. the peak values show a strongly oscillating behavior. all the friction curves at all the tests have similar dynamic nature, with a few peaks at the maximum cof values over time, as shown in fig. 1. microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 5 fig. 1 dynamic friction coefficient for 10 cycles (test duration of 3.93s) (sample 1, dry contact, 4 mm/s, 100mn) the peak values of the friction curve represent the maximum values of the friction coefficient, at the moment when the friction force is of the maximum value. it corresponds closely to the change of sliding direction, considering that this is a linear reciprocating motion of the ball over the flat ti alloy surface (simulation of the point contact, or hertzian contact). the shape of the friction coefficient curve and its peaks reflect the stopping and moving of the ball in opposite direction. accordingly, it exhibits stick-slip effects of the local running-in for a very short period, when the ball changes its direction of sliding. this is a very short time when the system transits from static state to kinetic state. when the ball is stopped for a very short period of time, microscopic junctions that are continuously formed and broken during the dynamic contact between the surface asperities of the contact pair, become stronger and stick-slip occurs, as clearly shown by the peaks in fig. 1. these brief local running-in periods continuously occurred during the steady-state of the dynamic friction coefficient, for all test conditions. general sine wave shape of the dynamic friction coefficient allows different methods of processing of such signal [44]. calculations of average and maximum values, along with some degree of filtering, are common practice with sine waves. on the other hand, dynamic coefficient curves of the whole test can be analyzed from different aspects, such as initial running-in period, steady state, occurrence of abrupt variation of values in steady state, or different periods within steady state (mean and maximum cof values within these periods or increasing/decreasing trends of maximum cof values). each of these indicates some phenomena during the sliding contact, thus altogether resulting in complex interpretation of the whole dynamic friction coefficient behavior, especially at micro and nano scale. selected real time diagrams of the dynamic friction coefficient, with related wear tracks, for each of the four samples, are shown in figs. 2-5. regimes of low load and low sliding speed, in comparison with the high load and high speed, are shown and compared in these images, as representatives of the two opposite contact conditions. also, the comparison of dry contact to the contact in the ringer’s solution, for the same speed and load, is shown in each of the figs. 2-5. the relevant wear mechanisms of these samples 6 f. zivic, s. mitrovic, p. todorovic, et al. and contact conditions are discussed in our previous papers [7, 42]. wear mechanisms and corrosion induced damage behavior during the contact between the same materials as in our study, under higher loads and different contact geometry, were analyzed in these articles [45, 46]. comprehensive review of wear mechanisms related to ti6al4v alloy, under different contact conditions, is given in [30]. frictional behavior is closely related to the wear and damage mechanisms that govern the contact. the cof curves in figs. 2-5 appear as a surface due to a long test period (30000 cycles), but when zoomed to a short time period, the shape of each curve is similar to the one shown in fig. 1. nanotribometer simultaneously provides total time [s], total distance [mm] and number of cycles [lap], as shown in fig.1. especially the total time is a useful parameter because it indicates the friction force level, in general. namely, for 30000 cycles, each test condition had different total times, strongly depending on both the sliding speed and normal load that were applied in the test. in figs. 2-5, horizontal axes show the total time needed to complete the 30000 cycles of the reciprocating linear sliding tests. each cof curve is shown next to the related wear track after the test is finished. each of the figures shows dry contact (figs. 2-5a, b) and contact in the ringer’s solution (figs. 2-5c, d), for one sample. this was a comprehensive experimental study that was realized during one year to complete all tests and these results are representative of the observed trends for the sliding contact between ti6al4v and alumina, under micro-scale loads. the maximum cof values for all samples were significantly larger under low loads and low speeds, then under high loads and high speeds. the cof values significantly decreased for the highest load and speed (1000 mn, 12 mm/s). for all the samples, except in the case of sample 2, wear tracks were significantly larger under low load/speed than under high load and speed. however, regarding wear, uniform trend for all loads cannot be defined, as discussed in [7, 42]. contact conditions with the lowest load (100 mn) and the highest load (1000 mn) exhibited significantly different values for both the friction coefficient and wear volumes, as shown in figs. 2-5. the smallest wear track together with the lowest wear volumes were exhibited for the highest load (1000 mn) and sliding speed (12 mm/s), for all samples. for the sliding in the ringer’s solution, all the cof curves were fully uniform without variation of maximum values, from the start until the end of test, almost without the running-in period, except for a very short initial period (order of a few seconds). maximum cof values (for both dry and wet contacts) varied within the ranges: 0.4 1.23 (for sample 1), 0.5 – 2.8 (for sample 2), 0.4 – 1.1 (for sample 3), and 0.4 – 2.3 (for sample 4). in the case of the sample 1, under the lowest load and speed (100 mn, 4 mm/s), the maximum cof values exhibited a slight increase over time, with periodic small variations within very short periods. wear tracks were of similar sizes, except in the case of dry sliding under 100 mn load and 4 mm/s speed, which is significantly larger (fig 2a), indicating severe abrasive wear, with coarse wear debris particles surrounding the wear track. microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 7 fig. 2 sample 1: dynamic friction coefficient (left) and related wear tracks (right): a) dry, v=4 mm/s; fn=100mn, b) dry, v=12 mm/s; b fn=1000mn, c) ringer, v=4 mm/s; fn=100mn, d) ringer, v=12 mm/s; fn=1000mn for sample 2, low load and speed (100 mn, 4 mm/s) resulted in an extremely high increase of the cof values over longer time, especially in dry conditions (fig. 3a, c). 8 f. zivic, s. mitrovic, p. todorovic, et al. fig. 3 sample 2: dynamic friction coefficient (left) and related wear tracks (right): a) dry, v=4 mm/s; fn=100mn, b) dry, v=12 mm/s; b fn=1000mn, c) ringer, v=4 mm/s; fn=100mn, d) ringer, v=12 mm/s; fn=1000mn increase of load decreased the cof values. same conditions in the case of sample 2 resulted in lower wear rate for wet contacts, especially under higher loads (wear rates were discussed in [7, 42]). this is in accordance with experimental tests in [45] where sample 2 subjected to block-on-disk contact configuration in the ringer’s solution, at high loads and microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 9 speeds (20, 40, 60 n and 0.26, 0.5, 1.0 m/s) resulted in lower friction coefficients (0.40– 0.77). fig. 4 sample 3: dynamic friction coefficient (left) and related wear tracks (right): a) dry, v=4 mm/s; fn=100mn, b) dry, v=12 mm/s; b fn=1000mn, c) ringer, v=4 mm/s; fn=100mn, d) ringer, v=12 mm/s; fn=1000mn our tests also showed a significant decrease of the cof values with the load increase (fig. 7, sample 2). it can be seen from fig. 3 that the smallest wear track was exhibited under 10 f. zivic, s. mitrovic, p. todorovic, et al. the highest load and speed (1000 mn, 12 mm/s) for the wet contact. severe abrasive wear can be noticed in all the wear tracks. fig. 5 sample 4: dynamic friction coefficient (left) and related wear tracks (right): a) dry, v=4 mm/s; fn=100mn, b) dry, v=12 mm/s; b fn=1000mn, c) ringer, v=4 mm/s; fn=100mn, d) ringer, v=12 mm/s; fn=1000mn the lowest cof values, as well as wear volumes, and the most uniform behavior, mainly without abrupt variations were produced for sample 3, what can be seen in fig. 4. however, severe increase of the cof values, almost 2-fold increase, also appeared after microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 11 some time, in the case of the presence of the ringer’s solution in the contact zone, under low load and speed (fig. 4c). it must be noted that the analysis of wear rates presented in [7, 42], showed that with this sample, wet contact produced opposite effect in comparison with the other three types of ti alloy samples: a lower wear rate was exhibited under the highest load and speed. simultaneously exhibited high cof values with low wear rates indicate probable occurrence of the plastic flow within surface layers of the ti alloy during the sliding, with severe plastic deformation of layers. from fig. 4c, it can be seen that the wear track is uniform and small, with deep abrasions only along one inner edge of the track, even though a significant cof increase occurred after approximately 20000 cycles. sample 4 (fig. 5) exhibited rather different cof behavior than the other three samples. in dry conditions, under low load, the cof values were drastically higher during the running-in period, followed by a significant abrupt decrease of the cof values that remained low until the end of the test (fig. 5a). in wet contact, low load produced drastic increase of the cof values over time, without any steady state period, and wear track was the largest one of all the tests. extremely high wear was observed under 100mn and 4 mm/s (fig. 5c), with calculated wear rate of 2.37x10-3 mm3/nm [7, 42] and it was significantly larger for wet contacts for all conditions. all ti samples had wear rates within the range of 10-4 10-5 mm3/nm, for almost all regimes of normal loads and sliding speeds, except in these few cases. in general, higher loads produced lower wear rates. alumina balls also exhibited extensive wear, what is in accordance with research from other authors who studied contacts between alumina and ti alloys at higher loads than in our study [40, 41]. values of the efficient friction coefficient, cofeff, were calculated for each of the realized tests and their changes are given in figs. 6 and 7. variations of the efficient friction coefficient (cofeff) values, with normal load, for each of the sliding speeds, in dry and wet contacts, are shown in fig. 6. changes of the efficient friction coefficient (cofeff) values, with both normal load and sliding speed, for dry sliding and sliding in the ringer’s solution, for all samples, is shown in fig. 7, as 3d plots with mapping of the zones is shown in fig. 8. from fig. 6, it can be seen that extremely high cofeff values were produced for the lowest load (100 mn) and the highest sliding speed (12 mm/s), in the case of sample 2. the largest differences of cofeff for each sample, were obtained for the lowest load (100 mn), where changes of the sliding speed resulted in a wide range of friction coefficient values, especially pronounced in the cases of sample 2 and sample 4. the lowest values were exhibited in the case of sample 3, in general. increase of load resulted in decrease of efficient friction coefficient values for all the samples, and under the highest load (1000 mn), they had very similar values (around 0.3), for all conditions. samples 1 and 2 both exhibited very high cofeff values, especially sample 2, thus indicating that, in general, annealing of ti alloy at 1000 °c is not favorable for tribological performance. also, sample 4 had high cof values, especially for low loads which indicated that quenching in water (rapid cooling) is not favorable from tribological aspects. 12 f. zivic, s. mitrovic, p. todorovic, et al. fig. 6 variation of the efficient friction coefficient (cofeff), with normal load, for each of the sliding speeds for all samples in dry contact and contact in the ringer’s solution from figs. 7 and 8, it can be seen that the sliding speed exhibited complex influence on the cofeff values, for all samples. in the case of sample 1, the sliding speed had negligible influence on the friction coefficient for all test conditions. a very mild transition can be observed for sample 1. for the other samples (samples 2, 3, 4), the sliding speed had pronounced effects under the low loads (100 mn, 250 mn). it can be seen that the change of load resulted in complex responses from aspect of cofeff, similarly to the complex wear behavior as discussed in [7, 42]. the transition nature of the cofeff with the load increase can be seen for all the samples, except in the case of sample 1, and especially pronounced in the case of sample 3, for both dry and wet contacts (figs. 7, 8). in the case of sample 4, the low cofeff values were exhibited for 100 mn, increasing with the load increase up to 500 mn, after which it rapidly decreased with a further load increase to 1000 mn. in the case of sample 3, the highest cofeff values were exhibited for 500 mn. this transition behavior is similar to the results of [40], in which dry contact at higher loads (5 80 n) and speeds (0.0625 – 1 m/s) is studied. they elaborated that the transition characteristic of load and speed dependence of the dynamic friction coefficient is governed by the elevated temperatures in the contact zone (flash temperatures) and the formation of ti intermetallics and oxides due to tribochemical reactions. on one side, the increased temperature provokes increase of cof and wear and on the other side, oxides and accumulation of wear debris within the contact zone result in decrease of the contact surface by forming intermediate layers between the ti alloy and alumina ball. yang et al., investigated synthesis of tial intermetallics on ti6al4v by laser modifications and showed that ti6al4v reacts with al to form major hard phases as ti3al, tial, and tial3, under high temperature [47]. they also showed that wear mechanism associated with tial microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 13 intermetallics is abrasive wear due to their enhanced micro-hardness. instability of the friction coefficient and the formation and influence of the third body presence and surface cracks have been also reported for the contact of tic with alumina ball, where hard debris affected the formation of micro cracks on the surface [48]. fig. 7 variation of the efficient friction coefficient (cofeff), with normal load and sliding speed, for dry sliding (left) and sliding in the ringer’s solution (right), for all the samples 14 f. zivic, s. mitrovic, p. todorovic, et al. fig. 8 mapping of different zones (2d view of the 3d curves shown in fig. 7): variation of the efficient friction coefficient (cofeff), with normal load and sliding speed, for dry sliding (left) and sliding in the ringer’s solution (right), for all samples microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 15 different microstructures of four samples, as shown in fig. 9, clearly indicate that their contact behavior was expected to be quite different [23]. the presence of the ringer's solution additionally changed contact conditions as shown in fig. 10 and table 1. in the case of dry contact, the real time unprocessed cof diagrams (figs. 2-5) have many isolated peaks, or sudden changes of the current friction coefficient, thus indicating abrupt changes of the contact conditions at micro scale, considering the nature of the contact pair (hertzian contact). these sudden changes can result from changes of surface roughness, accumulation (or delamination) of larger quantities of wear debris particles that changes the contact geometry, and appearance of ti intermetallics and oxides in the contact zone or other chemical processes developing within the contact zone that changes the surface layers of materials in contact. it is obvious that presence of the ringer’s solution rather prevented these sudden changes of the friction coefficient, since all the cof diagrams are uniform during the whole test (figs. 2-5). fig. 9 microstructures of four samples: a) sample 1 annealed at 1000°c; furnace cooled, with fully lamellar microstructure, b) sample 2 annealed at 1000°c; water quenched, with martensitic microstructure, c) sample 3 annealed at 750°c; furnace cooled, with equiaxed microstructure, d) sample 4 annealed at 750°c; water quenched, with globular microstructure research showed that lamellar microstructures have better fracture toughness and creep strength, while equiaxed microstructures have better fatigue strength and ductility [12]. alpha phase is harder than beta phase which is more ductile, but more stable at 16 f. zivic, s. mitrovic, p. todorovic, et al. temperatures higher than 882°c degrees [13]. we can elaborate that in the case of the microstructures where beta phase is more present (samples 2, 4), wear behavior of the alloy will be governed by the beta phase wear since it is proven that crack propagation is easier in beta phase [3, 49]. similarly, alpha phase wear will dominate in the case of the microstructures with a higher amount of alpha phase (samples 1, 3). however, contact loading together with high flash temperatures during the contact will influence dynamic redistribution of phases during the contact, whereas initial microstructure highly influences the resulting wear mechanisms as well as its extent. below 882°c degrees, the alpha phase should bear the load as the stronger phase, hence micro-abrasive wear will occur and formation of wear debris. above 882°c degrees, the beta phase is stable and should bear the load [4, 6], meaning that more micro-deformation of surfaces should occur accompanied with the material flow and lower amount of wear debris. fig. 10 sem image of the wear track: a) dry contact, b) contact in the ringers' solution, with positions where eds analysis was done, as shown in table 1. table 1 eds analysis realized for zones shown in fig1, for dry and wet contact: all elements analyzed (in weight %)) dry contact contact in ringers' solution position/ spectru m o al ti v o al cl k ca ti v 1 45.6 8 3.75 48.64 1.93 33.50 1.71 0.54 0.63 0.5 6 61.3 1.75 2 6.94 90.21 2.85 6.60 90.7 2.67 3 24.7 3 4.45 67.56 3.26 30.64 2.92 0.20 0.2 4 64.2 1.79 for the dry contact under low loads and speed (also meaning lower flash temperatures), fully lamellar microstructure (sample 1) will favor micro-delamination of alpha phase when the direction of contact motion is in line with lamellas, beside the high abrasive wear, what can lead to catastrophic total wear (as shown in fig. 2a). high load and speed increase flash temperatures, what can cause instability of the alpha phase when beta phase should begin to bear the contact load, thus lowering the total wear, as shown in fig. 2b. martensitic microstructure (sample 2) exhibits irregular alpha/beta distribution what favors high microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 17 abrasive wear, preventing micro-deformation or plastic flow, resulting in extensive amounts of wear debris (as shown in fig. 3a). equiaxed microstructure (sample 3) exhibits the highest amount of equiaxed large alpha phase grains where direction of the contact sliding has the lowest influence, with governing abrasive wear at the lowest load and speed and plastic flow at highest load and speed (as shown in fig. 4a,b). globular microstructure (sample 4) exhibits smaller globular alpha phase grains than sample 3, thus favoring high abrasive and micro-delamination wear, slightly lower under high loads (as shown in fig. 5a,b). longer contact time contributes to grain refinement and densification within the surface layers, thus favoring larger wear, what can be seen also in the case of sample 3 (as shown in fig. 4a). from the aspect of wear debris accumulation, the presence of the fluid influences its regular elimination from the contact zone, thus preventing its larger accumulation within the contact zone. however, the ringer's solution is beneficial for the development of complex chemical interactions between the two materials in contact. that is not a fluid that would promote lubrication and smoother contact, due to the salt crystals (nacl, cacl2) within its composition that can act like small abrasive particles and third body within a contact zone. it might be elaborated that changes of the cof values during the test under dry conditions is mainly influenced by the wear debris behavior within the contact zone and the chemical changes of the surface layers, such as formation of oxides. formation of oxides during the loaded contact between these ti alloy samples and alumina has been proven by eds analysis, shown in table 1, as well as by other authors [23]. the penetration depth was continuously recorded during the test. changes of this parameter have a good correspondence with the presence (or absence) of the wear debris within the contact zone, as discussed in [7]. in the case of sample 2, dry contact, penetration depth steadily rose during the whole test, thus indicating that sudden changes of the cof values (fig. 3a) could be attributed to chemical changes of the surface layers that promoted strong micro-galling between larger portions of surface asperities of ti alloy and alumina, followed by sudden delamination wear of larger particles (or patches). in the case of sample 3, dry contact (cof diagram in fig. 4a), penetration depth quickly increased up to around 3500 cycles (to the maximum of 23.8 µm), after which it started to decrease slowly down to 15.6 µm at the end of the test. altogether, this indicates accumulation of wear debris followed by its periodic elimination away from the contact zone, what is also confirmed by the wear track surrounded by the coarse wear debris (wear track in fig. 4a). it corresponds to the onset of irregular fluctuations of the cof values after approx. 3500 cycles, as shown in fig. 4a. after approx. 7000 cycles, cof values started to slowly increase until the end of the test. in the case of wet sliding (fig.4b, v=12 mm/s; b fn=1000mn), cof values were uniform during the whole test for sample 3. however, penetration depth in this case showed constant decrease from the beginning (starting value of 0.27 µm), until the end of the test when it reached 1.65 µm. nanotribometer software denoted penetration depth distance by the negative sign when it shows the distance above the flat surface of the sample. accordingly, in this case, alumina ball did not penetrate into the ti sample during the whole test, except very slightly at the very beginning. it might be explained by almost instant formation of the separation layers between the contacting surfaces from the beginning of the test. real time cof diagrams (figs. 2-5) can be analyzed in terms of specific periods during one test, especially if compared to the changes of penetration depth during that same test. in the case of dry sliding of sample 3, three periods can be distinguished: 1) both cof and 18 f. zivic, s. mitrovic, p. todorovic, et al. penetration depth increases (dominant abrasive wear), 2) transition period near the maximum penetration depth values with the onset of cof fluctuations (severe delamination starts to dominate, together with tribochemical wear) and 3) cof slowly increased and penetration depth slowly decreased, indicating a strong combination of abrasive, adhesive and tribochemical wear with the formation of wear debris and tribochemical layers between the contacting surfaces and periodical ejection of wear debris particles away from the contact zone. for the wet sliding, only uniform cof behavior was exhibited, except during the brief running-in period, thus indicating that, in this case, a short period of abrasive wear is followed by the dominant tribochemical wear, due to formation of oxide films on the flat ti sample surface. shape of cof curves corresponds to the appearance of wear tracks as given in figs. 11-14, indicating different wear mechanisms during specific periods/phases during one test, what is in consistence with the results of other authors [39, 40], who have also showed periodic changes in wear mechanisms over time. it can be concluded that, in general, different wear mechanisms appeared during one test, with complex influences of coupled abrasive, adhesive and tribochemical wear mechanisms. levels of wear rates (discussed in [7, 42]), indicated different wear mechanisms in the case of dry and wet sliding. for cof values higher than 1, it is obvious that frictional welding and micro-galling occurred, thus indicating governing adhesive wear. evidence of abrasive wear is clear in figs. 2-5 and figs. 11-14. formation of new layers between the contacting surfaces that decreased the contact area, was clearly indicated by the decrease of the penetration depth in some periods during the test, as recorder by the nanotribometer. a high friction coefficient under a low sliding speed produced severe wear in the case of both dry and wet sliding, except in the case of sample 3 for which the wear rates were rather low, especially for wet contact. tribochemical wear had a significant role under all conditions. wear tracks on sample 1 and sample 3, for dry and wet sliding under the highest load and speed (12 mm/s and 1000 mn) are shown in fig. 11. deep abrasive grooves and large delamination zones are clearly seen. adhesive wear is more pronounced for sample 3 where large delamination zones are visible. in the case of sample 1, abrasive wear with strong influence of third body wear (for dry contact) and tribochemical wear (for wet contact) can be observed. main difference between dry and wet sliding is the appearance of ragged wave-like forms in the central zone of the wear tracks in the case of dry sliding, which does not exist in the case of the wet sliding for both samples. this difference supports the conclusion that wear debris particles were not present in larger quantities within the contact zone during the wet sliding. for dry sliding, the central zone is probably influenced by the strong third-body wear from wear debris particles within the contact zone. serrated appearance of the central zone for dry sliding (fig. 11a, c) is in accordance with periodic abrupt elimination of the wear debris away from the contact zone during a very brief period of time, afterwards replaced with newly formed wear particles. wear track width for sample 1 appears smaller than the track width on sample 3. wear rates were low for these contact conditions, as shown in [42], being significantly lower for sample 3 than for sample 1, thus indicating that plastic flow plays an important part. this is in accordance with influence of high temperature that occurs within the contact zone (flash temperature), especially under high sliding speeds. increase of temperature has significant influence on the ti alloy structure favoring plastic flow instead of abrasion and delamination of surface layers. it also promotes mechanical alloying and the formation of the oxide layers which microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 19 further promote delamination and surface spalling. several authors investigated the influence of flash temperatures and tial intermetallics on tribological behavior during the contact between ti alloys and alumina [40, 50, 43]. fig. 11 wear tracks under 12 mm/s and 1000 mn, on sample 1: a) dry, b) in the ringer's solution and sample 3: c) dry, d) in the ringer's solution for similar contact conditions to our tests, contact temperatures within 200 350 ºc range was proved [41]. high temperature significantly influence ti6al4v alloy by rapidly decreasing yield strength and resulting in material softening within the limited contact zone [43], what is visible in wear tracks in our tests, too. plastic deformation was exhibited during the contact between pure ti and alumina in experimentals performed by [51], who elaborated that the behavior was similar to forming. they also showed that friction coefficient increased over time and associated it with material transfer that occurred during the contact. studies of the tribo-contact between ti alloys and steel showed that surface layers of the ti alloy undergoes significant transformations, with resulting several zones [52]. the authors in this study showed that higher loads resulted in plastic deformation with significant strain levels in localized zones near the ti alloy surface, which influenced severe wear. localized plastic strains induce unstable shear and inhomogeneous plastic deformation, thus rapidly decreasing the alloy resistance to deformation and resulting in severe wear onset. our tests also proved that plastic deformation has an important role in tribological behavior. additionally, material transfer had significant influence on the contact behaviour, thus influencing complex frictional behavior during the test. 20 f. zivic, s. mitrovic, p. todorovic, et al. influence of the sliding speed on tribological behavior was the most prominent under the lowest load (100 mn), where the friction coefficient exhibited widely different values. wear tracks on sample 2 in the ringer’s solution and sample 3 under dry contact for two opposite sliding speeds are shown in fig. 12. friction coefficient of sample 2 in the ringer’s solution was extremely high (around 1.4 in dry sliding and 2.7 in wet sliding) and resulting wear tracks are also large (fig. 12a, b). deep abrasive grooves can be seen, with severe pitting throughout the abrasions. evidence of the plastic flow is visible for wet sliding, while for dry sliding numerous small delamination zones are dominant. fig. 12 wear tracks on sample 2 at 100 mn load, in the ringer's solution (a) 4 mm/s, b) 12 mm/s) and sample 3, dry contact (c) 4 mm/s, d) 12 mm/s) appearance of the pitting, especially pronounced in the case of wet sliding is caused by the tribochemical reactions that together with material changes related to mechanical wear, further promote corrosion within surface layers. both friction coefficients for sample 2, under 4 mm/s, and 12 mm/s, are larger than 1, thus indicating severe galling and frictional welding between the contacting surfaces. higher wear rate would be expected for 12 mm/s (with cof value of 2.7), but it was lower in comparison to the test at 4 mm/s. we can elaborate that such strong adhesive bonding between asperities influenced rearrangement of the surface layers due to plastic flow, rather than its wear, in the case of wet sliding, thus keeping lower wear rates with such high cof values. also, there was probably lack of third-body wear due to the presence of fluid within the contact zone that regularly ejected away wear debris particles. this is also in accordance with existing martensitic microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 21 microstructure of sample 2, having hard surface layers that are prone to delamination as shown in [45]. localized corrosion of ti6al4v during the tribocontacts has been studied because it has significant influence on wear mechanisms [18]. complex corrosion degradation mechanisms were observed on ti6al4v surface, as influenced by the material transfer between contacting surfaces, nano and micro frictional welding of asperities during the contact and chemical reactions of ti alloy with counterbody material, leading to mechanically assisted crevice corrosion, pitting and localized corrosive degradation of surface layers that ultimately can result in fractures. all of these processes are dynamically developing over time during the contact; they are subject of investigations since the mechanisms that influence degradation of hard materials (such as alumina) by softer ti alloy, are not fully understood. some researchers showed that significant redistribution of material occurred in subsurface layers of ti6al4v during the tribocontacts and produced increase in nano-hardness of transformed layers [53, 54]. micro-cracking and localized surface strain hardening appeared as the consequence of adhesion and surface plastic deformation preceding wear debris formation [54]. oxidative processes and chemical reactions between materials in contact additionally influence the wear debris, leading to the formation of fine fully oxidized particles. these microstructural changes within surface layers of ti6alv can make them significantly harder than bulk alloy. sauger et al. [54] showed formation of nano-crystalline structures in surface layers of ti alloy. silva et al. [53] showed formation of martensite-rich thin layers in ti6al4v under the influence of localized high temperatures. licausi et al. [55] studied tribocorrosion mechanisms (at 3n load) and showed that wear is in consistence with the archard law only under initial high contact pressures. for lower contact pressures and presence of the third body they showed that the archard law is not valid for wear, thus indicating transition and changes of wear mechanisms during the contact, due to tribochemical response. experimental tribocorrosion study at 8 – 100 n load range that was focused on the influence of the surface roughness, showed that smooth surfaces exhibited more pronounced surface damage and plastic deformation than the rough surfaces due to corrosion attacks [56]. this research also showed that fretting contact initiated crevice corrosion in all the cases of surface roughness, with transition characteristic of the current density during tribocorrosion tests. transition behavior of wear was also proven for the contact of ti6al4v with steel [52]. for sample 3, influence of the speed is obvious if wear tracks in fig. 12c, d, are compared. low sliding speed resulted in very low friction coefficient (around 0.27) and high sliding speed exhibited friction coefficient of around 0.72. it was similar also for wet sliding. wear track at 4 mm/s was small (fig. 12c) with clear dominant influence of abrasive and tribochemical wear. adhesive wear can be noticed, but delaminated zones appear as roughened fracture surfaces, probably due to numerous detachments of small wear debris particles and influence of chemical processes that developed during the contact. for higher speed (12 mm/s), delamination zones are significantly larger and almost without roughened fracture surfaces within it (as in the previous case), even though they can be noticed. direct consequence of the flash temperature within the limited contact zone, beside extensive plastic deformation, is also frictional welding of asperities in contact along with mechanical alloying and galling, what further promotes plowing and delamination. deep abrasive grooves are broken by delaminated surface layers of larger sizes. influence of load was significant on sample 3, showing transition nature, especially under low sliding speed (fig. 7). wear tracks on sample 3 at the lowest sliding speed 22 f. zivic, s. mitrovic, p. todorovic, et al. (4mm/s), for different loads, under dry and wet sliding, are shown in fig. 13. transition of cof values occurred at 500 mn, as shown in fig. 7. for dry contact, wear track under the lowest load (100 mn) was small, as shown in fig. 4a and fig. 12c, and cof value was low, as shown in figs. 6 and 7 (around 0.27). under 500 mn, adhesive wear started to dominate along with abrasive and tribochemical wear (fig. 13a). increase of load to 1000 mn resulted in lower influence of severe adhesive wear which was mainly exhibited in the central zone of the wear track (fig. 13b). fig. 13 influence of load on wear at 4 mm/s sliding speed. wear tracks on sample 3: dry contact (a) 500 mn, b) 1000 mn), in the ringer's solution (a) 500 mn, b) 1000 mn) for the highest load, abrasive grooves became smooth indicating onset of the plastic flow and smaller amounts of the wear debris in the contact zone. for sample 3, presence of the ringer's solution lowered cof values, in general. however, at 500 mn, cof values were rather high (around 0.6), even though wear rate was low [42]. corresponding wear track (fig. 13c) showed shallow abrasive grooves broken by numerous delamination zones appearing as rough fracture surfaces. the highest load (1000 mn) again resulted in low cof values and very small wear rates. corresponding wear track (fig. 13d) exhibits dominant mild abrasive wear, even though small delaminated zones appeared (mainly in the central zone). in the case of samples 1 and 2, cvijovic-alagic et al. [46] showed that, under high loads (40 n), strain hardening of the surface layers occurred. this is in accordance with visible delamination of the larger patches in the wear tracks, thus indicating that surface layers had different hardness than those beneath, thus making microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 23 them prone to delamination (figs. 11, 12). riaz et al. [57] showed that anisotropic surface texture significantly influences levels of wear and corrosion of biomedical ti6al4v. piao et al. [9] investigated thermal softening effect at different strain rates and showed that ti6al4v alloy exhibit hardening effect with strain rate increase. surface texture has significant influence on the resulting friction coefficient, as studied by several authors [58, 59]. they reported significant friction reduction for textured surfaces, up to 50-85% reduction of friction coefficient for differently textured ti6al4v contact surfaces. in our tests, there was a significant redistribution of surface layers, thus provoking dynamic changes of ti6al4v surface texture within the contact zone which surely had significant influence on friction coefficient, as shown in cof diagrams. alumina ball was selected as a very hard material that is chemically inert, opposed to ti alloy that becomes rapidly softer under elevated temperature. yield strength of ti6al4v promptly decreased with temperature increase, as reported in [43]. however, in our tests, alumina ball also endured severe abrasive wear and pitting, as shown in fig. 14. an exceptionally high friction coefficient was obtained for these contact conditions (sample 2 in the ringer’s solution, 100 mn, 12 mm/s). severe transfer of ti alloy onto the alumina ball developed during this contact. strong adhesive bonds, galling and frictional welding of contact asperities probably developed under these experimental conditions. under the lowest load (100 mn), similar wear behavior was exhibited for all contact conditions of sample 2. central contact zone of the ball (fig. 14a) showed severe spalling of its surface layers and deep abrasive grooves can be seen in the vicinity of the central zone (fig. 14b), also with evidence of mild tribochemical wear. this strongly indicates that very hard tial intermetallic were formed during the sliding (as suggested by different authors), and accumulated within the contact zone causing abrasive effects. contact geometry of the contact pair allows the largest accumulation of the worn particles within the close vicinity of the central contact point. from fig. 14b, it can be seen that severe pitting occurred along the rim of alumina ball, thus indicating that it was the line where the most prominent tribochemical reactions appeared between alumina and ti alloy. this is the thin zone where accumulated wear debris particles were probably arrested during the sliding, even though, at certain moment, they were eliminated from the contact zone, probably when a critical amount had been reached. our results are in accordance with other authors who experimentally showed that during the contact between ti alloy and alumina, very sharp and hard tial intermetallics formed (ti3al), along with ti oxides (tio, ti2o3, tio2) [40, 41]. elevated frictional heat supports development of these complex tribochemical reactions. dominant abrasive behavior was exhibited for the sliding of ti6al4v against alumina ball, with strong presence of adhesive and tribochemical wear [39, 40]. for dry sliding, dong and bell [40] observed delamination of plates on both ti6al4v sample and alumina ball, with appearance of both intergranular and transgranular fracture. they also showed transition nature of load and speed dependence, what is in consistence with our study. several authors elaborated that wear debris, together with ti oxides, can form thin layer between the surfaces in contact and lower the wear rate [39, 40, 41]. contact under the highest load (1000 mn) in our study exhibited the smallest wear tracks, as shown in previous images. 24 f. zivic, s. mitrovic, p. todorovic, et al. fig. 14 wear track on the alumina ball, under 100 mn load and 12 mm/s sliding speed; wet contact with sample 2: a) central contact zone and b) zone in the vicinity of the central contact zone in the case of samples 1 and 2, wear debris around the wear track was coarse, while finer debris particles were generated in the case of sample 4, with very fine wear debris in the case of sample 3, thus indicating that heat treatments of ti alloy samples had very significant effects on tribological behavior. different heat treatments of ti6al4v alloy resulted in completely different microstructure (fig. 9), as discussed in details in [23]. our results showed that lamellar and martensitic microstructures were not favorable, neither for wear or frictional response, under low loads and hertzian contact (ball-on-flat). namely, both sample 1 (lamellar microstructure) and sample 2 (martensitic microstructure) exhibited catastrophic wear and very high friction coefficients in both dry and wet contacts. lamellar and martensitic microstructure both has lower elastic modulus and yield strength than equiaxed microstructure (sample 3). this is obviously reflected on friction and wear capabilities of the alloy. on the other hand, a higher presence of al2o3 and v2o5 oxides in surface passive layers of sample 3 and sample 4 in electrolytic solution is proven [23]. the amount of al2o3 oxides was lower in the case of sample 3 than sample 4, leading to the conclusion that these particles also have influence on the friction and wear (fig. 7), partly due to forming of the passive layer (and increase of corrosion resistance) and in some extent as the third body (and negatively contributes to friction and wear). tribochemical reactions have important role in tribological behavior of ti6al4v – alumina contact and presence of the passive oxide film is of significant influence, as well as the influence of the sharp abrasive tial intermetallics. sample 4 (globular microstructure) exhibited high cof values, especially for low loads which indicated that quenching in water (rapid cooling) is not favorable from tribological aspects. fan et al. [16] studied the influence of different microstructures of ti6al4v on fatigue strength and showed that equiaxed microstructure has the highest fatigue strength, followed by bimodal, lamellar and acicular martensite microstructures, what is in consistence with our results concerning wear. wear is governed by the damage mechanisms and fatigue strength is of the utmost importance for the endurance of surface layers subjected to external loads. poor ductility of martensite and lamellar microstructures supports development of cracks under loading, unlike equiaxed and globular microstructures that are beneficial for fatigue resistance. equiaxed microstructure (as obtained in sample 3) has the highest ductility, thus promoting plastic flow beside crack initiation and propagation. microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 25 series of thermomechanical treatments on ti-6al-4v alloy along with artificial neural network method to correlate microstructure and tensile properties realized by shi et al. [60], showed that microstructural changes of α and β phases had strong influence on resulting mechanical properties, whereas, β phase had a more prominent effect on tensile properties than α phase. strong correlation between ti alloy microstructure and wear resistance, as well as of the energy inputs during the contact, is also showed in experimental study under low load (2 n) [61]. they used first-principles calculations for gibbs free energy, shear and bulk modulus, in order to evaluate brittle or ductile behavior of different oxides that were formed during the tribo-contact (tio, tio2, al2o3 and v2o5 oxides) and showed that these oxides have detrimental influence on the wear resistance of ti6al4v alloy in the case of certain microstructural features. in our tests, ti6al4v alloy annealed in ar at 750 °c and quenched in water, showed the best tribological behavior, with low wear rates, especially in wet contact (of order of 10-7 mm3/nm under high load and sliding speed and with cof values mainly around 0.4). wet contact in our study with the ringer’s solution should represent especially aggressive environment for ti alloys. however, for sample 3, presence of the ringer's solution was beneficial related to friction and wear. wear of 10-4 mm3/nm belongs to a mild to moderate wear and 10-7 mm3/nm wear rate usually produces wear debris with fine particles and can be associated with tribochemical wear. microstructural influence is complemented by the appearance of third body in the contact zone. mechanical alloying and material diffusion between ti alloy and alumina produce particles that can be incorporated within the soft ti surface layers, thus promoting dynamic surface modifications during the contact. licausi et al. [55] showed that wear cannot be modeled by the archard law under low contact pressures and with presence of the third body. dissipative forces, such as friction, depend on the path and are non-conservative forces hence the energy that they take away from the system cannot be restored back. hence, for conditions of the low loads and very small real contact area, properties of the material surface is of the utmost importance, what, on the other hand, serves as the input to the design process of the anatomical bone structures that are used for modeling, additive manufacturing technologies and reverse engineering [62]. one of the first methods to model wear from the energy point of view was the reye’s method, which proposed that the removed volume of material is proportional to the dissipated energy during the contact [63]. carpinteri and pugno [64] proposed wear model based on fractal theory where removed volumes of material has direct dependence on density and hardness of materials in contact. they assumed self-similar distribution of the energy dissipations and developed a model for both brittle and ductile damage mechanisms. their fractal coupled erosion theory and generalization of the reye’s hypothesis (statistical three-dimensional approach) introduced hardness, density and wear erosion coefficients at micro level, as governing factors for the macroscopic volume of removed materials during the contact. they established clear dependence of the wear resistance as the macroscopic property to the microscopic material constants (hardness, density and wear erosion coefficient). this is fully in consistence with our experimental results where microscopic changes of the material structure in surface layers have essential influence on the wear and friction. we can anticipate that, at any time, at least three different materials, at microlevel, are in contact during the sliding (ti6al4v, alumina, ti oxides, tial intermetallics, ti6al4v surface nano-crystals, wear debris, the ringer’s solution) with changes over time, due to the complex tribo-chemical reactions. additionally, ti6al4v alloy has pronounced changes of properties with elevated temperature within microscopic zones. accordingly, 26 f. zivic, s. mitrovic, p. todorovic, et al. material wear would depend on each of these densities, hardness and wear erosion coefficients, what altogether makes a highly complex system from aspects of wear and friction prediction over time. it could be assumed that microscopic changes of hardness within the localized surface zones on ti6al4v, together with its microstructural changes due to the flash temperatures and tribo-chemical processes represent the most influential factors for wear and friction development. further research should study the possibility to apply the fractal coupled erosion theory proposed by 64. carpinteri, and pugno [64], to the multi-particle system contact at micro-level (consisting of ti6al4v, alumina, ti oxides, tial intermetallics, ti6al4v surface nano-crystals, wear debris, the ringer’s solution), considering dynamic changes over time, aiming for predictive wear and friction models and a more detailed understanding of the evolving processes during the contact. artificial intelligence-based models should offer efficient tool for wear and friction prediction over time [65]. 4. conclusions four types of ti6al4v alloy exhibited different friction and wear behavior, with more pronounced difference in the case of the dry sliding. maximum cof values in all tests varied within the range of 0.4 – 2.8, indicating that some contact conditions and microstructures produced severe galling and wear. both friction and wear rate showed transition features, more pronounced for the changes of load than sliding speed, especially in the case of equiaxed microstructure (sample 3). dynamic friction coefficient exhibited almost no initial running-in period for all test conditions. the variation of cof with material condition and testing conditions were significant. the applied load had more effect on the overall results compared to sliding speed. sliding speed had lower influence on cofeff for the highest load, for all samples. sliding in the ringer’s solution, produced fully uniform cof curves, unlike significant fluctuations of maximum cof values in the case of the dry sliding. contact conditions at the lowest load (100 mn) and the highest load (1000 mn) exhibited significantly different values for both the friction coefficient and wear. maximum cof values for all ti alloy samples were significantly larger under low load than at high loads and high speeds. the smallest wear track and the lowest wear volumes were exhibited for the highest load (1000 mn) and sliding speed (12 mm/s), for all samples. alumina ball also endured severe abrasive wear and pitting. the lowest cof values (0.4 – 1.1) were exhibited for the ti6al4v sample with equiaxed microstructure (sample 3), whereas presence of the ringer's solution was beneficial. however, equiaxed microstructure in sample 3 also exhibited full transition characteristic for both load and sliding speed, with rather high cof values for some contact regimes. extremely high increase of cof values over time, appeared in the case of martensitic ti alloy microstructure (sample 2), in dry sliding at low load and speed, with cof values reaching 2.8. in general, water quenching was not a favorable treatment for tribological behavior. lamellar and martensitic microstructures were not beneficial, either for wear or frictional response. wear mechanisms were governed by the abrasion, delamination, plastic deformation and tribo-chemical reactions. microstructure influence on friction behavior of ti6al4v biomedical alloy at low loads 27 acknowledgements: this paper is funded through the eit’s hei initiative smart-2m project, supported by eit rawmaterials, funded by the european union. ti6al4v samples were supplied by the vinca institute of nuclear science, 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udc 534.2+681.5 ulrich gabbert, fabian duvigneau, stefan ringwelski otto-von-guericke-university magdeburg, magdeburg, germany abstract. the paper presents an overall simulation approach to control the noise emission of car engines at a very early stage of the design process where no real prototypes are available. the suggested approach combines different physical models and couples different software tools such as multi-body analysis, fluid dynamics, structural mechanics, magneto-electrodynamics, thermodynamics, acoustics and control as well. the general overall simulation methodology is presented first. then, this methodology is applied to a combustion engine in order to improve its acoustical behavior by passive means, such as changing the stiffness and the use of damping materials to build acoustic and thermal encapsulations. the active control by applying piezoelectric patch actuators at the oil sump as the noisiest part of the engine is discussed as well. the sound emission is evaluated by hearing tests and a mathematical prediction model of the human perception. finally, it is shown that the presented approach can be extended to electric engines, which is demonstrated at a newly developed electric wheel hub motor. key words: acoustics, combustion engines, active noise reduction, sound absorption, engine encapsulations, electric wheel hub motor, finite element method, measurements 1. introduction in recent years, increasing attention has been paid to vibration and acoustic noise control in many industrial applications, especially in automotive industries. the control of noise and vibration is essential since it contributes to comfort, efficiency and safety. there are two different approaches to achieve noise and vibration attenuation. on the one hand, there is a widely used passive approach. passive control techniques mostly reduce vibration and sound emission of the structures by modifying the structural geometry and by applying additional damping materials. these methods are best suited for a higher received june 15, 2017 / accepted july 11, 2017 corresponding author: gabbert, ulrich affiliation: otto-von-guericke-universität magdeburg, universitätsplatz 2, d-39106 magdeburg e-mail: ulrich.gabbert@ovgu.de 184 u. gabbert, f. duvigneau, s. ringwelski frequency range. a drawback of the damping materials is the increase in weight. an overview of the developments in structural design optimization for the passive noise control can be found in marburg [21]. on the other hand, there are active control techniques available, which provide an alternative way to minimize unwanted structural vibrations and noise. active methods are of increasing interest to designers since they are powerful and do not increase structural weight considerably. a first comprehensive overview of the active control of sound is given by nelson [25]. a very beneficial active vibration control technique was first introduced by fuller et al. [7]. in his concept actuators are directly attached to a structure in order to reduce sound radiation by changing its vibration behavior. piezoelectric ceramics are widely used as sensors and actuators because they can easily be mounted onto the vibrating structure [30]. active control techniques are usually employed in a low frequency range. over the past years, several researchers have studied different control strategies for active structural control [9, 10, 13, 18]. li et al. [19] and ringwelski et al. [30-32] applied a velocity feedback control algorithm to a rectangular fluid-loaded plate with piezoelectric layers, which has been proven to be a robust and effective control strategy. ruckman and fuller [33] applied a feedforward control approach to reduce actively the acoustic radiation of a fluid-loaded spherical shell structure. orszulik et al. applied a feedforward/feedback control design for a piezoelectric nanopositioning platform [28]. a linear quadratic optimal control (lq) technique combined with additional dynamics was proposed by nestorović et al. [23] and successfully applied to control the vibration of a car roof [24]. orszulik and gabbert [27] have proposed a data interface for coupling commercial finite element software and control design software. this allows hardware in the loop approaches to design active controlled structures. in comparison to the active approaches the passive damping of structures to reduce the noise radiation by design modifications and the application of additional damping materials is much more pronounced, due to its simpler and cheaper application and a more robust noise reduction effect even in changing environmental conditions. the development and industrial application of noise reduction techniques require efficient and reliable simulation tools to design an acoustically optimized system. virtual models are of particular interest in the design process since they enable the designer to optimize a technical system by a so called virtual twin. the focus of the paper at hand is to improve the acoustic behavior of vehicle engines, where passive methods as well as active ones are taken into account. besides the engine noise there are several other noise sources of vehicles such as the noise excited from the interaction between the tires and the road surfaces (rolling noise) and the excitation of the car body by the air stream around the vehicle. such sources are not discussed in this paper. the paper is organized as follows. at first the overall acoustic simulation approach is briefly presented. then, it is applied to combustion engines, where several simulation results are presented to underline the complexity but also to demonstrate the advantages of the developed simulation approach. it is briefly shown that the active control methods are also advantageous for noise radiation reduction, especially in a lower frequency range. the calculated sound pressure distribution in the air surrounding the engine does not represent the human perception of the sound, which can lead to very different results in the sound design. therefore, an approach is presented, which enables the designer to develop a virtual sound design on simulation results only, without having a real prototype at hand. as an outlook to the ongoing research activities it is shown that acoustic problems noise control of vehicle drive systems 185 are not automatically solved by a change to electro drive systems. measurements at prototypes of a newly developed wheel hub motor have shown a significant high frequency noise radiation, which was the motivation to extend and apply a developed overall simulation approach to electrical machines as well. 2. concept of an overall acoustic simulation approach in today’s design process of engines acoustics is taken into account in a very late development stage, where first acoustics measurements can be performed at the final prototype of an engine. at this development stage the possible changes of the engine design are very limited, and, consequently, it is impossible to receive an optimal acoustic solution. to overcome this problem acoustics should be evaluated very early in the design process, where no prototype exists. but this requires a holistic virtual engineering workflow of the design process, including models to simulate (i) the excitation of the engine, (ii) the vibration of all required engine parts, (iii) the acoustic pressure wave propagation in the surrounding air, and, (iv) the psychoacoustic evaluation of the sound. such an overall simulation approach for a combustion engine is shown in fig. 1. the only input is the gas pressure in the cylinders, which excites the movement of the pistons. by applying an elastic multi-body simulation the vibration excitation of the crankcase can be analyzed. the elastic multi-body simulation (mbs) results in bearing forces as well as in deformations of the cylinder walls. the deformations result from lateral and tilting motions of the piston due to the combustion process respective to the thereby generated gas forces. the resulting superposed load on the cylinder walls is not measureable. consequently, the presented approach provides, on the one hand, an opportunity to substitute the experimental determination of the engine structure excitation and, on the other hand, it enables consideration of a more realistic vibration excitation. based on a complex finite element model (fem) the dynamic behavior of the crankcase can be analyzed, which results finally in the surface velocity, which excites the surrounding air volume. based on the solution of the acoustic wave equation the pressure waves in the surrounding air volume can be calculated. from our experiences, the a-weighted acoustic pressure is not sufficient to evaluate the hearing sensation. consequently, as a final step a model based psychoacoustic assessment is performed. based on the briefly presented methodology the acoustic behavior of an engine can be evaluated and improved in a virtual development step without the existence of a real prototype, which is shown in the next chapters. 186 u. gabbert, f. duvigneau, s. ringwelski (e) (f) (g) subjective o b je ct iv e t n t s (a) (b1) (c1) (b2) (c2) (d) fig. 1 draft of the overall simulation workflow to evaluate the acoustic behavior of a combustion engine: (a) excitation analysis of the engine by the gas forces with the help of an elastic mbs approach including elasto-hydrodynamic interactions [2] (b) vibration analysis with the help of a complex finite element model (b1 – passive methods, b2active methods) (c) analysis of the pressure distribution in the surrounding air with the help of acoustic fem approach (c1 – passive methods, c2active methods) (d) auralization of the acoustic simulation results (e) mathematical analysis of the sounds to calculate psychoacoustic parameters (f) psychoacoustic evaluation of the radiated noise with the help of hearing tests and hearing models (g) generation of a robust regression model to predict the human auditory perception with the help of the best suited calculable parameters noise control of vehicle drive systems 187 3. experimental basis there is a large selection of the devices available in our laboratories to measure vibration and noise of structures. the measurements are mainly applied to evaluation and improvement of developed numerical simulation approaches. the structure vibration is mainly measured contact free with the help of one, two or three laser-scanning vibrometers. in fig. 4 three laser heads are used to measure the three-dimensional vibration of a car body. for the sound pressure measurements the acoustic holography and visualization through the acoustic camera is applied. likewise, several microphones and microphone arrays are available for noise source localization purposes. for the acoustic measurements two anechoic chambers can be used (see fig. 2). the testing of active controlled structures is performed with help of a dspace system. fig. 2 acoustic engine test bench fig. 3 laser vibrometer, © ikam 4. passive concepts for reduction of the vehicle engine noise in the following section the acoustic behavior of combustion engines is analyzed and improved by applying the proposed holistic virtual engineering approach, which is shown in fig. 4. the engine vibration is caused by the gas pressure forces of the combustion process, which excites the piston motion. the gas pressure curve does not have to be obtained by measuring the current engine prototype. it can be taken out from a representative data base of previous engines. alternatively, the gas pressure curve can be taken from the combustion process design because this curve is available very early in the development process and furthermore virtually coincides with the real gas pressure curve of the final design. the piston motion causes the excitation of the cylinder crankcase by the internal cylinder pressure, the forces in the crankshaft main bearings and the piston contact with the cylinder walls. the calculation of the piston lateral motion and the piston tilting requires the consideration of the hydrodynamic fluid film reactions as well as the solid contact of piston and cylinder [2, 17]. the multi-body simulation (mbs) has to be carried out with at least five operating cycles of the engine, from which only the last one is taken into account as input for the subsequent vibration analysis to avoid initial disturbances in the calculated time signals of the excitation loads. the resulting signals are periodical. for this reason, it is possible to extend the time signals by repeating the representative 188 u. gabbert, f. duvigneau, s. ringwelski excitation from one operating cycle into the next. a longer time signal is advantageous for the fast fourier transform (fft), which is necessary for transforming various excitation sources into the frequency domain. both contact parts (piston and cylinder) are modeled as elastic bodies in the mbs. therefore, they are able to capture local deformations, which result from the elasto-hydrodynamic contact [2]. the main bearing forces are received by the solution of the reynold’s differential equation. finally, the gas forces, the contact forces and the main bearing forces are applied as loads of the cylinder crankcase model (see fig. 1a and 1b). the feedback of the crankcase vibrations to the crankshaft and the piston motion can be neglected, as shown in [2]. pcyl(ω) fb(ω) pgas(t) (a) elastic mbs (b) fe vibration analysis (c) fe acoustic analysis p(ω) fb(t) pcyl(t) vs(ω) fig. 4 overall acoustic simulation approach of a combustion engine the finite element method (fem) is used for the subsequent vibration analysis of the engine. it is executed exclusively in the frequency domain in order to reduce the computational costs in comparison to a time domain approach. the bearing reactions are considered as forces and the excitations of the cylinder walls are considered as pressures to facilitate the application of the loads to the fe-mesh. the discretisation of the cylinder walls in the mbs and the fe-analysis are not coincident. the nonlinear elastic mbs of the crank drive requires the minimization of the number of degrees of freedom as far as possible due to computational costs. therefore, in the mbs a rough discretization of the cylinder walls is used. in contrast, in the vibration analysis a detailed model of the whole cylinder crankcase and oil pan is necessary. thus, a much finer discretization is required. consequently, a coincident discretization of the cylinders in the mbs and in the vibration analysis has to be omitted. the resulting forces of the mbs have to be transformed into the nodes of the discretization of the vibration model. to create a static equivalent load by nodal forces on a finer discretization is simple but an energetic equivalent load requires much more effort. in the current investigations the pressure values are used as surface loads to create an equivalent load for the vibration analysis of the engine. in the vibration analysis model this surface load is defined at the midpoints of the finite element surfaces, which form the inner contour of the cylinders. the amplitude of each surface load is obtained by the elastic mbs with the help of the shape functions used for describing the pressure distribution. therefore, the midpoints of the element surfaces of noise control of vehicle drive systems 189 the vibration analysis model are provided for the elastic mbs, where the shape functions of the pressure as primary variables are used to calculate the pressure values at these points. hence, the applied loads of the vibration analysis model match exactly with the calculated loads of the elastic mbs without an additional error by the transformation between different discretizations. the transformation of the excitation of the cylinder walls in the frequency domain is executed after the calculation of the pressure values at the midpoints of the element surfaces of the vibration analysis model. subsequently, the vibration analysis is executed with help of these excitations to get the surface velocities of the whole engine (see fig. 4b). these surface velocities are required as excitation of the surrounding air volume in the following acoustic analysis (see fig. 4c). an uncoupled acoustic analysis is implemented, neglecting the feedback of the air volume to the vibrating structure, which is a common assumption because the engine is made of aluminum and consequently much stiffer than the air. the excitation of the air volume is applied as boundary condition in the calculation of the sound radiation [6]. the degrees of freedom of the structure (displacement) and the fluid (sound pressure) are coupled by special interface elements. these elements are shell type elements without stiffness or mass. they require a coincident discretization of the engine and the air volume at the fluid structure interface. as already mentioned, the previously calculated surface velocities of the engine are applied as boundary conditions at the nodes of the interface elements. the spherical air volume is modeled with a discretization, which becomes coarser to the periphery due to the computational costs (see fig. 1c). generally, tetrahedrons with quadratic shape functions are used to discretize the whole engine and the air volume. the splitting of the large multiphysics system of equations into smaller problems causes a significant reduction in the computational effort. this is due to the fact that the computational effort required increases nonlinearly with the number of degrees of freedom considered. the sommerfeld radiation condition is fulfilled by using absorbing boundary conditions [12], which is a simple and sufficient approach in comparison to the other methods, such as infinite finite elements or the perfectly match layer method. to evaluate this approach a plate like structure was investigated by a comparing the numerical solution with an analytical reference solution based on the rayleigh integral. it was shown that the three different modeling approaches to including the boundary conditions in the far field do not differ significantly. the proposed model has been evaluated experimentally as shown in fig. 2 and a good agreement has been recognized. consequently, it has been used to investigate the influence of different design studies on the engine’s acoustic behavior. as an example fig. 5 compares the influence of the piston contact with the cylinder walls to the noise radiation. but, many other design configurations have been studied as well, such as the gas curves, the shape of the piston, the different axial offsets and several geometrical changes of the crank shaft and the oil sump to figure out their influence on the engine’s acoustic behavior. from fig. 5 it is seen that the oil sump at the bottom of the engine is an important noise radiator due to its large and thin surface. therefore, several investigations to reduce the noise radiation from the oil sump have been performed. in fig. 6 a new functionally integrated oils sump is shown and fig. 7 presents the results of some design configurations to stiffening the bottom of the oil sump by applying an optimized configuration of rips. 190 u. gabbert, f. duvigneau, s. ringwelski 126.0 124.4 122.9 121.3 119.8 118.3 116.8 115.3 113.9 112.4 111.0 109.6 108.2 106.9 105.5 104.2 102.9 101.6 100.3 99.1 97.8 96.6 95.4 94.2 93.0 (b) sound pressure level (spl) in db(a) (a) fig. 5 sound pressure distribution without and with consideration of the cylinder forces: a) only bearing reactions, b) additionally the cylinder excitations are considered but by changing the stiffness and the mass distribution only, the broad band noise level of an engine cannot be reduced properly. therefore, as an alternative approach the development of an optimized engine encapsulation has been tested in order not only to reduce the noise radiation but also to improve the engine’s heat storage capacity. from a tribological point of view the heat storage of the motor oil is of utmost importance since the oil temperature is directly related to the fuel efficiency and it contributes also to pollution reduction as well. the same overall simulation strategy can be employed by adding an additional model of the encapsulation (as seen in fig. 1(c1)). the modeling of the encapsulation materials is not discussed here in detail, see [3]. detailed numerical analyses have been conducted without and with different types of thermo-acoustic engine encapsulations; some results are presented in [3, 4]. the results have also been experimentally evaluated with acoustic measurements in an engine test bench as seen in fig. 2. fig. 6 finite-element-model of the crankcase and the surrounding air-volume (left) with the new developed functionally integrated oil pan (middle, right) noise control of vehicle drive systems 191 fig. 7 surface velocities of the oil sump with different types of rips from studies with different encapsulations we have learned that it is very important to avoid acoustic leakages. also additional seals made of silicone and vibration-decoupling modifications have improved the encapsulation. in fig. 8 (right) the measured sound pressure level shows the performance of the encapsulation. fig. 8 encapsulation (left); sound power level at the thrust side: red: without encapsulation, black: with encapsulation the acoustic effect of the encapsulation (see fig. 8 right) mainly occurs at frequencies higher than 100 hz. additionally, the stored heat improves the cold starting behavior of the engine, meaning that the optimal operating points of the engine at which wear and tear and, therefore, exhaust emissions are reduced as well, are more quickly reached. of course, the engine overheating and overloading of the cooling circuit have to be excluded. 192 u. gabbert, f. duvigneau, s. ringwelski for thermal consideration, an fe-simulation was conducted as well. the finite element mesh was directly taken from the acoustic simulations. only the engine oil inside the oil pan was added. it was also modeled with quadratic tetrahedrons. fig. 9 shows comparison of the acoustic pressure of the engine without (left) and with the developed encapsulation. during the investigations several other parts of a combustion engines have been also investigated and improved by applying the overall virtual engineering approach, e.g. the exhaust systems, the oil sumps, the air charge systems, etc. the investigations presented above are focused on the application of insulation materials like foams, microfibers and mass layers, which are used together as multilayer systems and mounted on the vibrating surfaces in order to create a damping effect that reduces the sound radiation. the influence of the material thickness on the damping behavior was investigated as well as the influence of a thin foil adhered to the surface. we have carried out experiments and simulations which have provided us with generalized guidelines for optimizing damping materials geometry and design according to the specifications of their applications. 116.0 114.1 112.3 110.4 108.5 106.6 104.8 102.9 101.0 99.1 97.3 95.4 93.5 91.6 89.8 87.9 86.0 84.1 82.3 80.4 78.5 76.6 74.8 72.9 71.0 sound pressure level in db(a) fig. 9 sound pressure around the engine: left: without, right: with encapsulation 5. active piezoelectric noise control passive damping methods mostly reduce the vibration and the sound emission of structures by modifying the structural geometry and by applying additional damping materials. these methods are well suited in a higher frequency range (see e.g. fig. 8 right). unfortunately, damping materials are increasing the weight. on the other hand, there are active control techniques available, which provide an alternative way to minimize unwanted structural vibrations and noise [7, 9, 11, 20, 31]. active methods are of increasing interest to designers since they are powerful and do not considerably increase the structural weight. for the development of active systems the overall simulation methodology (see fig. 1) can also http://www.dict.cc/englisch-deutsch/consideration.html noise control of vehicle drive systems 193 be applied. in the following the active noise reduction approach is demonstrated at the oil sump of a combustion engine. as we have seen the oil sump is the main noise radiator of an engine. to actively influence the vibration amplitudes of the structure piezoelectric materials are attached to the structure, and by a properly designed electric excitation the vibration amplitudes are reduced. this results additionally in a reduction of the excited amplitudes of the acoustic waves. as sensors microphones in the surrounding or structural integrated sensors (acceleration sensors or strain gauges) can be used. for modeling the structural behavior the electromechanical coupled field equations have to be taken into account in the finite element model [8, 11, 22]. but for design purposes it is also meaningful to include the control as well in the finite element simulation [30]. but also software in the loop approach, as it is proposed by orszulik and gabbert [27], can be applied. as an advantage of this approach the extensive control strategies can be designed in matlab/simulink and applied during the overall simulation process. also the prior knowledge about the excitation, such as the gas forces, can be used to design an optimal control technique with additional dynamics [23]. in fig. 10 left the finite element model of the crankcase with the oil pan is shown. the pictures show that two collocated actuators and a sensor are applied to the bottom of the oil sump. the placement of the actuators and the sensor are numerically identified with the help of the most dominant mode shapes [31]. in fig 10 (right) a photo of the outer surface of the oil sump is shown, which is used for experimental investigations. at the photo the two applied piezoelectric patch actuators are shown together with extra acceleration sensors for additional test reasons. in fig. 11 the computed and the measured sound pressure distributions of the uncontrolled and the controlled engine are plotted at the most dominating resonance frequency of 975 hz. the measurements have been carried out in a free-field room with the help of a uniformly distributed microphone-array. from fig. 11 it can be seen that the simulation results correlate very well with the experimental ones. in the active damping case a velocity feedback control was used in the simulation as well as in the measurement, where the applied voltage was about 10 v [32]. fig. 10 finite element model of the crankcase with the oil sump with two collocated pairs of actuators and sensors (left); right: photo of the oil sump outer surface equipped with two piezoelectric actuators (a1, a2) and additional acceleration sensors (as) the simulations and the measurements have been performed at a stripped engine. at the most dominating first eigenfrequency a noise reduction of about 16 db and at the second eigenfrequency of about 10 db was achieved in comparison to the uncontrolled case. in order to evaluate the quality of the developed actively controlled oil sump 194 u. gabbert, f. duvigneau, s. ringwelski additional experimental tests were performed on an acoustic engine test bench (fig. 1) under real operating conditions [20]. the behavior of the controlled and the uncontrolled case was measured at engine run-ups from 900 to 3000 rpm. the measurements reveal that due to the controller influence the amplitudes of the sound pressure in the resonance frequency regions of the oil pan are reduced by approximately 4 db, which indicates the noise reduction potential of the designed system. fig. 11 sound pressure distribution around the engine: simulation (left), measurement (right) 6. psychoacoustic evaluation instead of measuring the acoustic behavior of a real prototype [1, 26, 29], the presented overall numerical simulation methodology is used to receive audible sounds of the engine [5]. these audible sounds are used to carry out paired comparison listening tests, which finally lead to an interval scaled ranking of the stimuli. these data from listening tests are used to create a psychoacoustic mathematical model which shows the highest correlation with the subjective evaluation of engine sounds. this psychoacoustic model is a function, which consists of a weighted combination of well-chosen classical psychoacoustic parameters, such as loudness and sharpness. the resulting psychoacoustic model describes and predicts the auditory sensation of the noise quality on the basis of a signal processing of the sound signal only without any further auralization or hearing test. the advantage of such a concept is that the perceived quality of an engine can be optimized before a real prototype is built because only simulation results are needed as input for the psychoacoustic model. for the details of the approach the reader is referred to [14], [15, 5]. the psychoacoustic approach has been developed and applied first to noise control of vehicle drive systems 195 impulsive vehicle sounds such as the door locking and closing noise and the flasher noise [14]. this approach has been recently extended to engine noises [5]. we have found out that for generating a proper psychoacoustic model three parameters are sufficient to generate a robust and quite precise mathematical model by linear regression. in this example these three parameters are (i) maximum gradient of percentile loudness n5, (ii) maximum sharpness smax and (iii) duration of sharpness ts [5]. with such a model different types of combustion engines and different types of engine encapsulation have been evaluated in order to optimize their acoustic behavior. fig. 12 shows the distribution of the sound pressure around a combustion engine enclosed with an encapsulation. the a-weighted sound pressure is compared with the results received by the developed psychoacoustic model for the perceived quality of the engine sounds. it has been carefully proved by additional hearing tests that the developed model is very well suited to describing the human sensation of the sound quality of different engines without further hearing tests. it is very important to consider more complex psychoacoustic models to evaluate the acoustics of engines properly; an evaluation with only single basic parameters is not sufficient enough to describe the sound quality. max min fig. 12 the a-weighted sound pressure level (left) and the perceived sound quality (right) 7. extension to electric drives the presented overall simulation approach can also be extended and successfully applied to electric drives for electro-cars. recently, an electric wheel hub motor for electrically driven cars has been developed [16], which shows an extraordinary power-to-weight-ratio, as it combines two different types of winding to boost torque sharing the same magnetic circuit; one is an air gap winding and the other one is a slot winding, see fig. 13. 196 u. gabbert, f. duvigneau, s. ringwelski fig. 13 design of the electric wheel hub motor [16] in the development process of an electric engine acoustics is usually not in the focus of interest. but it has been proven that the acoustic characteristics of electric engines are a very important topic, which should be taken into account at an early stage of the development process. in contrast to combustion engines in electrical engines the first engine orders are related to much higher frequencies (up to 1250 hz) and the resulting sound is not so noisy. in general it seems that the radiated sound is caused by a few different frequencies only, as the second and the third engine order are the most important engine orders beside the first order, but even their amplitudes are with about 5% and 2% of the first one still comparably small. hence, the emitted sound of an electric engine is more like a single high frequency tone. unfortunately, the human auditory perception is very sensitive with respect to such high frequency sounds. consequently, the noise emission of electric engines is more annoying than the noise emission of combustion engines, even if the amplitudes of the electric sound are lower. for this reason, it is important to consider the acoustic behavior as early as possible in the product development process of an electric engine. with the aid of our overall simulation workflow the acoustics of electric engines can also be optimized before the first prototype is built. the first part of the workflow (see fig. 1) is changed because the excitation is now caused by magnetic forces. therefore, the electromagnetic behavior is modeled first, where it is common to neglect the differences in the direction of the rotation axis to increase efficiency. it is sufficient to use a two dimensional model only (see fig. 14, right upper corner), as the attenuation of the tangential magnetic forces is less than 5% at the boarders. further, the magnetic forces in radial direction are non-linear and cause stability problems if they are linearized. the electromagnetic forces resulting from the first step are used to calculate vibration behavior of the wheel hub motor. this results in the surface velocity, which is used to excite the surrounding air and to calculate the air pressure at any point of the surrounding air volume under free field conditions. noise control of vehicle drive systems 197 numerical vibration analysis calculation of the electromagnetic excitation forcesdesign of the wheel hub motor numerical acoustic analysis fig. 14 holistic simulation workflow for calculating the sound radiation of an electric wheel hub motor based on the electromagnetic excitation forces the vibration and acoustic analyses can be solved in an uncoupled manner, as the feedback of the vibrating air on the much stiffer engine housing can be neglected. for all three solution steps, the electrodynamics, the structural dynamics and the acoustics the finite element method are used. in the vibration and acoustic analyses tetrahedral elements with quadratic shape functions are used due to the complex geometry and the required accuracy which is tested by convergence studies. the acoustic simulations can be done with a much coarser mesh due to the larger wave length in the air. but, at the interface between the structure and the air volume it is appropriate to use a coincident mesh, which is coarsened with increasing distance from the structure to reduce the computational effort of the approach. the developed and numerically tested overall workflow is finally also validated by measurements. the presented overall virtual engineering methodology can be used to optimize the design of the wheel hub motor in further steps in order to fulfill it acoustic requirements. furthermore, it would be a promising future development to extend the overall workflow by a psychoacoustic post-processing as shown in section 6 in order to include the special properties of human hearing sufficiently, instead of determining the classical acoustic parameters such as the sound pressure level. 198 u. gabbert, f. duvigneau, s. ringwelski 8. conclusions in the process of engine development, the reduction of the sound emission is a major objective together with the optimization of both the power and the fuel efficiency. hence, in the paper at hand the aim was to present a recently developed overall virtual prediction methodology that can evaluate the sound quality of engines based on auralized simulation results. this methodology consists of numerical and psychoacoustic analyses. it has been originally developed and tested for combustion engines but it is also capable to be applied to electric engines. the numerical analysis begins with the calculation of the excitation forces. then, the determined excitation forces are used to calculate the vibrations of the engine with the help of the finite element method. subsequently, the resulting acoustic behavior is calculated and rendered audible. at this point, the psychoacoustic part of the analysis begins, where the signal processing of the resulting time signals from the numerical simulations is performed in order to calculate appropriate psychoacoustic parameters. the numerically calculated engine sounds are also used to carry out hearing test with human participants, who evaluate different engine sounds with respect to their perceived sound quality. finally, the results of the signal analysis and the hearing test are compared by regression and correlation analyses. the objective parameters with the best correlation between the results of the signal analysis and the hearing test are used to generate a psychoacoustic prediction model of the perceived sound quality. the generated psychoacoustic prediction model and the numerical results are validated by experimental investigations. the remarkable feature of the presented approach is that the auralized simulation results are used within the hearing tests instead of the sounds of real engines or prototypes. this makes it possible to execute the presented workflow early in the product development process. this means that no hardware or real prototypes are required in order to evaluate the result of engine modifications on the acoustic behavior and its corresponding human perception. hence, the presented concept is suitable for computer based optimization of the engine with respect to the perceived sound quality. to demonstrate the developed workflow the results of the holistic approach applied to combustion engines are presented. it is shown that the application of engine encapsulations can reduce the sound emission of combustion engines significantly. with the help of the developed approach it is possible to make decisions regarding the geometry of the engine structure, a proper damping material for an encapsulation, its layup and thickness, as well as the optimal shape of the encapsulation. besides passive means, which are preferable in the higher frequency range, additionally, active control methods are also briefly discussed. finally, it is shown how the overall simulation approach is extended to electric machines and applied to simulate the acoustic behavior of a recently developed new wheel hub motor. acknowledgements: the presented work is part of the joint project como iii (competence in mobility), which is financially supported by the european union through the european funds for regional development (efre) as well as the german state of saxony-anhalt (zs/2016/04/78118). this support is gratefully acknowledged. furthermore, we would like to thank dr. christian daniel for subfigure 1(a). moreover, we want to acknowledge the cooperation with the chair of technical 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element-boundary element approach, smart materials and structures, 19(10), 105009, 13pages. 31. ringwelski, s., luft, t., gabbert, u., 2011, piezoelectric controlled noise attenuation of engineering systems, journal of theoretical and applied mechanics, 49(3), pp. 859-878. 32. ringwelski, s., zornemann, m., luft, t., gabbert, u., 2012, active control of noise and vibration on a combustion engine, proceedings of the icsv19, vilnius, lithuania, july 08-12, 2012, 8 pages. 33. ruckman, c.e., fuller, c.r., 1994, numerical simulation of active structural-acoustic control for a fluid-loaded spherical shell, journal of the acoustical society of america, 96, pp. 2817–2825. facta universitatis series: mechanical engineering vol. 16, no 3, 2018, pp. 285 296 https://doi.org/10.22190/fume170618018r © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a cad-based conceptual method for skull prosthesis modeling udc 004.922.8:616-089.843 marcelo rudek 1 , yohan b. gumiel 1 , osiris canciglieri jr 1 , naomi asofu 1 , gerson l. bichinho 2 1 production and system engineering graduate program – ppgeps, brazil 2 graduate program in health technology – ppgts, pontifical catholic university of parana – pucpr, brazil abstract. the geometric modeling of a personalized part of the tissue built according to individual morphology is an essential requirement in anatomic prosthesis. a 3d model to fill the missing areas in the skull bone requires a set of information sometimes unavailable. the unknown information can be estimated through a set of rules referenced to a similar yet known set of parameters of the similar ct image. the proposed method is based on the cubic bezier curves descriptors generated by the de casteljou algorithm in order to generate a control polygon. this control polygon can be compared to a similar ct slice in an image database. the level of similarity is evaluated by a meta-heuristic fitness function. the research shows that it is possible to reduce the amount of points in the analysis from the original edge to an equivalent bezier curve defined by a minimum set of descriptors. a study case shows the feasibility of method through the interoperability between the prosthesis descriptors and the cad environment. key words: prosthesis modeling, cad, bezier curves, 3d image 1. introduction the aging of the world population and thereby caused increasing demands for medical services set new challenges in the field of biomedical engineering. this is especially true in the field of image processing, design of personalized prosthesis and automated manufacturing. in the context of machining process, the additive technologies are capable of building complex structures in different materials geometrically compatible with the received june 18, 2017 / accepted november 21, 2017 corresponding author: marcelo rudek pontifical catholic university of parana, pucpr/ production and system engineering graduate program – ppgeps, imaculada conceicao, 1155, 80215-030, curitiba, brazil. e-mail: marcelo.rudek@pucpr.br 286 m. rudek, y. b. gumiel, o. canciglieri jr, n. asofu, g. l. bichinho tissues of human body. the reason why the additive technologies are so much applied in the personalized prosthesis production is the fact that it is possible to build a part of arbitrary complexity once you have its 3d geometric model. the congenital failure or trauma in the skull bone requires surgical procedures for prosthesis implant as functional or esthetical repairing. in this process, a personalized prosthesis built according to individual morphology is an essential requirement. normally in bone repairing, the geometric structure is unrepeatable due to its “free form” [1]. due to the complexity in geometry, free form objects do not have a mathematical expression in a close form to define their structure. however, numerical approximations are a feasible way to the geometric representation. the link between the medical problem and the respective manufactured product (i.e. prosthesis) is the geometric modeling; thus different approaches in the bone modeling have opened new research interests as in [2, 3, 4]. in the prosthesis modeling, we face different levels of information handling from a low level of the pixel analysis in image to the automated production procedure. in general, there are following levels: a “preparation level” (containing ct scanning, segmentation, feature extraction, i.e. entire image processing) and a “geometric modeling level” (containing the polygonal model, curve model, extraction of anatomic features, i.e. entire cad based operations) [2]. the cad systems are important tools in the design of these complex products because they can be used for three-dimensional (3d) modeling of the shapes of bones and respective scaffolds to machining [5]. in our strategy, we need to generate geometric representation of the bone without enough information (e.g. neither mirroring nor symmetry applicable). a common image segmentation procedure is executed as pre-processing at the preparation level. moreover, from the segmented images we define a set of descriptors based on bezier curves [6] in order to describe the skull edge geometry on a ct image. this approach is applied in order to reduce the amount of points capable of representing the skull bone curvature. it was adapted from the method of [4] now using the de casteljau algorithm to define the bezier parameters. the paper explores the accuracy of the prosthesis modeled through the balanced relationship between curve fitness versus number of descriptors. it extends the work of [3] to demonstrate the relationship of descriptors within a cad system in order to build a 3d prosthesis piece. 2. the proposed method 2.1. the conceptual model in our study, the main question is related to the information recovering for automation of the prosthesis modeling process. sometimes it is possible to reconstruct a fragmented image by using information of the same bone structure, e. g. by mirroring, that is, using body symmetry from the same individual. however, in many cases, there is not enough information to be mirrored. a handmade procedure can be performed by a specialized doctor by using a cad system [7, 8, 9]. in order to circumvent mirroring limitations and the user‟s intervention, we are looking for an autonomous process of geometric modeling of skull prosthesis. thus, the basis of our hypothesis is to find compatible information from different healthy individuals from image database. the problem addressed here is the method for finding a compatible intact ct slice to replace the respective defective ct slice. when working with medical images [10], a lot of a cad-based conceptual method for skull prosthesis modelling 287 information is needed to be handled mainly after image segmentation and edge detection, where the total of pixels in edge are still too much information to be processed. our approach is a content-based retrieval procedure contrary to the pixel-by-pixel comparison that is a hard processing task that we need to avoid. in order to optimize the search by similarity, we propose to define shape descriptors by cubic bezier curves. in this way it is possible to reduce the amount of data-to-process to a few parameters. the next important issue is to find the descriptors capable of describing the edge shape as well as possible. thus, we also look for a balance between accuracy and the minimum quantity of information. the next section will explain our approach in curve modeling. 2.2. the curvature representation the curve modelling adopted in this research is based on the de casteljau algorithm applied in calculation of the points of a bézier curve [3, 6]. the de casteljou method [11, 12] operates by the definition of a “control polygon” whose vertices are respective “control points” (or “anchor points”) used to define the shape of the bezier curve. a bezier curve of degree n is built by n+1 control points. the cubic bezier curve has two endpoints (fixed points) and two variable points. they define the shape (flatness) of curve. figure 1 shows an example of a cubic bezier curve where {p0, p1, p2, p3} are the vertices of the control polygon. points {p0, p3} are fixed and they are the beginning and the ending of the curve, respectively; these points belong to the curve. {p1, p2} are variable points occupying any random position in  2 . fig. 1 graphical representation of de casteljau method [11] according to fig. 1, for all points i r i ptp )( , we have a )(0 tpn as a point on the bezier curve. bezier curve b n (t) with degree 'n' is a set of points )(0 tpn , t[0, 1] , i.e. ]}1,0[);({)( 0  ttptb nn . then the polygon formed by n vertices {p0, p1,…, pn} is so called “control polygon” (or bezier polygon) [12]. through the de casteljau algorithm each line segment results in (n1) baselines as 10pp , 21pp , 32pp which are recursively divided to define a new set of control points. by changing 't' value as defined in eq. (2) we obtain the position of the point in the curve:          rni nr tpttpttp r i r i r i ,...,2,1 ,...,2,1 )()()1()( 1 1 1 (1) 288 m. rudek, y. b. gumiel, o. canciglieri jr, n. asofu, g. l. bichinho 02 01 tt tt t    (2) the control points for p[t0 t1](t), are npppp 0 2 0 1 0 0 0 ,...,,, , and the control points for p[t1 t2](t) are 02 2 1 10 ,...,,, n nnn pppp  . in order to avoid misunderstanding in representation, fig. 2 shows the control points, and the recursive subdivision of the de casteljau algorithm labelled as p, q, r and s, where s is the final position of a point in the curve for different values of t. in fig. 2a the value of t = 0.360 and in fig. 2b the value of t = 0.770. a) b) fig. 2 position of the control points and its respective bezier curve adapted from [11]1 as presented in literature, for practical applications, the most common one is to apply the cubic bezier curves (n=3) due to the large possibilities in adapting the shape (flatness) according to our necessities. also in our proposal, the bezier with n=3 is more suitable to fit the skull contour in tomographic cuts. an example of a segmented ct slice is shown in fig. 3. a) b) c) d) fig. 3 a ct slice sample and respective bezier representation: a) a quadratic bezier curve (n=2); b) a quadratic bezier curve on small region; c) a cubic bezier curve (n=3); d) the cubic bezier curve on small region in fig. 3a a quadratic bezier curve (b n (t) with n = 2) adjusted on the skull edge is presented. in this case we have three control points and only two baselines. note that the adjustment in the outer edge seems satisfactory but in the inner edge the result is poor. in 1 under an attribution-noncommercial-sharealike cc license (cc by-nc-sa 3.0) a cad-based conceptual method for skull prosthesis modelling 289 the same way, fig. 3b shows the de casteljau algorithm applied to the smallest segment of the inner edge; in this case, then, the curve representation is improved. in fig. 3c a cubic bezier curve is presented. now more control points exist and the resulting adjustment looks very good for both the outer and the inner edge. also, in fig. 3d the method applied in a small section (the inner edge) is more accurate. the question that we intend to discuss in the next section concerns the similarity measurement. in other words, how good is the quality of a bezier curve that represents a ct skull edge? this is essential for our approach because we need to define the best curve based descriptor. good descriptors will permit us to retrieve compatible ct images to produce the skull prosthesis. 3. application of the method the aim of this research is to define a small set of descriptors to represent the bone curvature. the strategy is to use the cubic bezier curve method calculated through the de casteljou algorithm. in our previous section, we state that the accuracy of curve fitting in our approach by the bézier depends on its degree n value and the length of the edge section. the edge sectioning is defined as follows: 3.1. the sectioning of the edge as presented above in fig. 3, the curve generated on the edge seems to fit better to the smallest length region (i.e., the shape of the curve looks similar to the original edge shape). the first question is about the best number of sections to produce the best-fitted curve. as an example, the edges of a ct image can be sectioned as in fig. 4. a) b) fig. 4 a ct slice sample with: a) k=10 sections; b) k=20 sections figure 4a shows the total of k=10 cuts (with p0 to p10 fixed points) whose section edges lengths are bigger than sections of fig. 4b with k=20 cuts (with p0 to p20 fixed points). for each section, fitness value f is calculated using eq. (3), defined in [3] as: 290 m. rudek, y. b. gumiel, o. canciglieri jr, n. asofu, g. l. bichinho    n i bb iyiyixixkf 1 2 0 2 0 ))()(())()(()( (3) where f(k) is the fitness value for each section k. fitness f calculates the error between the bezier coordinates (xb, yb) and the original edge coordinates (x0, y0) for each pixel „i‟ in the edge. the sectioning procedure and the control points calculation are fully covered in [3]. table 1 shows the average of fitness (error) to respective 5, 10, 15 and 20 sections cuts. table 1 relationship between number of cuts and respective fitness (error) # of section f(5) f(10) f(15) f(20) 1 61.81640 26.03820 8.96430 6.22810 2 87.16330 22.63830 17.94760 9.27160 3 99.14170 21.17040 20.07230 9.34800 4 78.18280 25.94340 16.91680 10.25800 5 68.10270 26.46990 18.00790 10.09840 6 30.86040 17.19840 9.31330 7 28.55470 19.37670 14.06260 8 24.36930 22.17160 9.27060 9 36.52540 17.10820 9.50420 10 48.82190 19.46810 11.26640 11 19.99220 12.80600 12 21.37690 7.48890 13 18.15550 10.65490 14 27.96400 12.36890 15 40.85040 8.37890 16 9.40440 17 9.74180 18 17.47880 19 15.90830 20 25.4706 σ (error) 394.40690 291.39190 305.57090 228.32270 fitness (avg.) 78.88138 29.13919 20.37139333 11.416135 table 1 shows the cumulative error evaluated by eq. (3) and the average of fitness for different values of sectioning. as expected, the error is minimized with larger values of k. the graph in fig. 5 presents the relationship between the number of sections and the calculated error (difference between the original edge and the calculated bézier). as presented in fig. 5, the average of error calculated from the fitness equation goes down as the number of sections is increased. then, in this condition, maybe we could define the k value as the maximum possible, i.e. the length of total of pixels of edge. however, the computational cost of the cubic bézier curve calculation for hundreds of sections also increases. the same proportion of error occurs for all ct slices from different images. from the graph, selecting the value of k=20 is enough to match a relatively good fitness with a small error and give us an adequate balance between precision and computational cost. thus, for k=20 we have in the de casteljau algorithm, 20 “fixed points” and another 20 “variable points”, (i.e. 4 points per section) calculated as in [3]. now, it is possible to represent the total length of each edge (inner and outer) in a a cad-based conceptual method for skull prosthesis modelling 291 ct slice with 80 points descriptors each instead of ≈ 1250 in the original edge (around 15 times information reduced). 5 10 15 20 25 30 35 40 0 20 40 60 80 sections x error number of cut sections f it n e s s v a lu e ( a v e ra g e ) fig. 5 the relationship between the number of cuts in the edge and respective error from the fitted bézier curve in each section 3.2. the curve fitting procedure the curve fitting procedure is applied to each ct slice of defective skull. the same procedure is also applied to each searching image on database. the compatible answer image is retrieval as in the example presented in fig. 6. fig. 6 the defective set of slices and respective compatible ct recovered from medical image database in fig. 6 some samples are shown of the defective ct from the original dataset and respective retrieved ct with compatible descriptors (i. e., minimum error in descriptors). fig. 6 also shows error value (e) for each images pairs. the error is the cumulative difference between the original bone and the calculated bezier curve by applying eq. (3). 4. evaluation of the method a handmade testing failure is built-in in a skull through the fiji software [13]. it is an open source java suitable for a medical image analysis. a set of toolboxes permits us to handle ct slices from the dicom file [10]. the edge from individual slices can be cut in 292 m. rudek, y. b. gumiel, o. canciglieri jr, n. asofu, g. l. bichinho sequence in order to build a failure in a region. thus, after 3d reconstruction, we obtain a synthetically built failure in the skull as in the example in fig. 7a. a) b) fig. 7 testing image: a) a handmade testing failure built on original image; b) region filled with prosthesis modeled fig. 7b shows a failure region filled with compatible ct slices from the medical database. the piece is cut from different slices where bezier descriptors are compatible, i.e. all those cts retrieved with minimum error. the retrieved slices numbers and respective patient are shown in table 2. table 2 retrieved set of ct slices original slice # 280 282 284 286 288 290 292 compatible individual # 7 6 6 6 5 7 6 compatible ct image # 278 285 282 285 284 286 290 fitness value 130.2566 143.4175 128.9885 139.1415 189.7201 226.1416 140.9704 note that the slices retrieved are never from the same patient. the set of retrieved slices (good slices) are recovered from healthy individuals (intact skull) whose descriptors match with the original image in each ct slice (defective slice). from table 2 it is possible to see that many cts are coming from the individual #6. in fact, the individual #6 has similar morphological characteristics with the testing patient of the same gender and of similar age. the filled region is evaluated through the geomagic® software [14]. the differences between the original bone and the prosthesis piece are presented in fig. 8. the software permits overlapping of 3d structures and provides a colored scale to show the spatial difference whose lower error values are represented in green while the higher error tends to red. as shown in fig. 8, the region a008 has an error closest to zero because it is the original skull (skull of patient). the other colored identifications are in the prosthesis area like a001, a003, a005 and a006; they are below 1mm difference and the maximum error is in the region a004 with the value of about 1.7087 mm. a cad-based conceptual method for skull prosthesis modelling 293 fig. 8 3d evaluation of filled region 5. the geometric modeling in the cad system a cad system can be used for modeling of the bone shape as well as for generating prosthesis profiles used in machining preparation. the main objective of the presented approach is the verification of each ct slice by slice in order to create curvature descriptors; then the slices of this new 3d image are the profiles exported to a cad for the creation of the virtual (missing) bone piece. due to the geometrical complexity of the individual human bones, the model cannot be automatically solved in cad systems, but a preliminary step is needed to bring guaranties that the geometry stays totally closed in order to apply cad functions. only after this preliminary step, the superposing of the created surfaces can be transformed into a solid geometry [15, 16, 17]. this model is what a medical team could examine in order to check whether it is suitable to be implanted in the patient and thus be translated into a cam system for manufacturing. assuming that the solid geometry can be generated by the cad system, based on the tomography image, the dimensions of the raw material needed for the cam process (machining) can be determined. a solidworks® [16] example is presented in order to demonstrate the method‟s feasibility. a cut region is handmade and built on a testing skull image. after the method of descriptors is applied, a similar set of cts are retrieved from image database; the cloud of points is imported to solidworks and plotted in 3d as presented in fig. 9a. despite the fact that the cloud of points is enough to set machining procedures, a visualization resource by surfaces can improve a visual evaluation of the piece. then, in the next step, we apply for each 2d plane its respective mesh and surface as presented respectively in the two samples in figs. 9b and 9c. they are the top and bottom limits of the prosthesis surface. also, another viewing possibility is the shape contour as a convex hull in each area generated by spline curves in fig. 9d. 294 m. rudek, y. b. gumiel, o. canciglieri jr, n. asofu, g. l. bichinho a) b) c) d) fig. 9 the imported calculated data to cad system: a) cloud of points; b) mesh for each ct layer; c) surface based on imported data; d) splines applied in edge limit in each ct slice the spline curve around each 2d slice contour is the reference to the “loft” instruction in cad. the loft is a tool to create a 3d solid from cross sections, i.e., in our case it generates the 3d visualization based on the superimposed splines from all ct surfaces. the resulting image of this process is presented in figs. 10a and 10b. the resulting final image of prosthesis piece is presented in figs. 10c and 10d. a) b) c) d) fig. 10 the views of 3d modeled piece after loft in the cad system a cad-based conceptual method for skull prosthesis modelling 295 6. conclusions the paper presents a method for generating skull shape descriptors based on the bezier curves whose parameters are generated by the de casteljou algorithm. edge sectioning in k=20 sections with the same length permits us to define two markers as respective “fixed points” in the bezier curve generator. two more “variable points” calculated by the de casteljau define the total of 4 descriptors for each section. thus, it is possible to reduce all edge size in ct to be represented by a set of 80 descriptors. the descriptors are used to look for compatible ct images whose bone edge shape versus bezier curve calculated by their descriptors have a minimum error. the example shows the result with maximum error in image around 1.7mm. we show it is possible to represent a missing region of the patient‟s skull by a set of similar cts from healthy individuals selected by a reduced descriptors group. all those descriptors can be exported to a cad system to build a prosthesis piece. an example within the cloud of points, mesh and 3d surface is presented to illustrate the integration between data and its respective geometric model. in addition, the example shows that the retrieved slices are from individuals with similar characteristic as to age and gender. in a future work, the database searching engine can group individuals with these characteristics before proceeding to calculating descriptors. acknowledgements: the authors would like to thank the production and system engineering graduate program – ppgeps and the graduate program in health technology – ppgts from pontifical catholic university of parana – pucpr by collaboration in providing all ct image data. also, the 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[2] presented a model of friction in a system of two rigid plates connected by links (springs) and experiencing an external drive. it is shown that macroscopic frictional properties of the system are directly related to the dynamics of rupture and formation of microscopic bonds. different modes of motion are characterized by different rates of rupture and bond formation with respect to the velocity of motion. in particular, it is shown that the stick-slip regime corresponds to a cooperative bond breaking. elmer [3] studied the dynamical behavior caused by dry friction for a spring-block system pulled with constant velocity over a surface. the dynamical consequences of a general type of phenomenological friction law (stick-time-dependent static friction, velocity-dependent kinetic friction) are investigated. three possible types of motion are shown: sticking-sliding, continuous sliding, and oscillations without sticking. wojewoda et al. [4] considered problems with dry friction for various models that did not require determining the beginning of sliding. lyashenko [5] used the dry friction model to study the vibration of a body that has an elastic link with a fixed support, while the body is affected by the friction force of another body by a given law of motion and an externally applied load. popov et al. [6] devoted their studies to the theoretical analysis of sliding friction under the action of oscillations perpendicular to the plane of sliding for a massless system. the influence of the stiffness of tribological contact is analyzed in detail, and the case of large amplitudes of oscillations is considered, at which the contact is lost during a part of the oscillation period, as a result of which the specimen starts to “jump”. it is shown that the macroscopic friction coefficient is a function of only two dimensionless parameters-dimensionless sliding velocity and dimensionless vibration amplitude. kluge et al. [7] considered various models of interaction between the base and the structure and also noted that the friction force is unknown in the absence of sliding. harouz et al. [8] conducted a comparative analysis of the friction and wear rate for three segments nb, nc, and nd with tic addition ratios of 5%, 10%, and 15%, respectively. the comparison was performed concerning a reference sample na, with no tic addition, for dry rotational sliding friction tests against al2o3 ball, at two sliding velocities 0.5 m/s and 0.75 m/s and a fixed temperature of 450℃. brzakovic et al. [9] presented theoretical and experimental analyses of the kinetic friction coefficient of a ball bearing under rotational motion initiated by dynamic impact force. a method and measuring system were developed to determine the value of the kinetic coefficient of friction by measuring the angular acceleration. recently, a trend has been developed towards the widespread use of seismic isolation of buildings and structures using the principle of dry friction between the grillage and the foundation. dynamic problems in the presence of dry friction forces are nonlinear. mirzaev et al. [10-12] obtained a numerical solution to the nonlinear problem with the additional use of a special logical algorithm to determine the direction of the dry friction force between two masses. a logical algorithm determines the beginning of sliding of the upper mass relative to the bottom one, and the beginning of their joint motion. mamatov et al. [13] devoted their studies to the analysis of the current state of residential buildings with sliding supports in the foundation, in bishkek city. based on the results of the research of residential buildings with seismic protection systems were recommended to use energy experimental determination of dynamic coefficient of amonton-coulomb dry friction 505 absorbers. the energy absorbers consist of rubbing surfaces of steel, fluoroplastic and a layer of bulk material placed between surfaces. buckle et al. [14] recommended to use of sliders for seismic isolation of bridges. banović et al. [15] conducted laboratory experiments on the shaking table with the implementation of real seismograms and showed that different earthquakes act differently on a body with a seismic isolating layer. mkrtychev et al. [16,17] presented the results of studies of spatial structures of buildings with seismic isolation under the influence of real earthquakes using the ls-dyna software package. sagdiev et al. [18] defined the main principles of modeling multi-storey buildings with seismic isolation and characteristics of seismic isolation devices. in addition, they presented the results of experimental studies on the laboratory shaking table setup under dynamic impacts arising due to the initial deviation from the equilibrium position. mogilevsky et al. [19], using the method of characteristics, solved unsteady problems for a rod with external dry friction. isakov and kondratenko [20] described the mathematical model of the vertical penetration of a steel pipe in soil by a shock pulse generator. the influence of the external parameters on the attenuation of the shock pulse propagating along the pipe was described, and the generalities of the process were found. today, in practice, there are various methods for determining the dynamic coefficient of friction. bragov et al. [21] and espinosa et al. [22] used the modifications of kolsky’s method for determining the dynamic coefficient of friction. nikitin’s method described by mogilevsky et al. [19] also refer to the measurement of the wave propagation through a section with dry friction, and the dry friction coefficient is determined based on the solution to a one-dimensional problem for a rod with lateral dry friction. miljojković et al. [23] presented the results related to the development of a modular didactic laboratory set for the experimental study of friction and its implementation in engineering education. this article describes the experimental setup and research methodology using the analytical solution of the corresponding problem of free vibrations of the system to determine the value of the dynamic coefficient of dry friction on the steel-on-steel contact with and without lubrication and on the fluoroplastic-on-fluoroplastic contact. 2. material and methods 2.1. the principle of conducting experimental research to determine the value of the dynamic coefficient of dry friction on the steel-on-steel and fluoroplastic-on-fluoroplastic contact surfaces, experimental studies were conducted on the small shaking stand for various frequencies and intensities of external impact. the external impact was generated using a special device of the platform pullback on the shaking stand, changing the length of elastic vertical plates and the amplitude of the pullback (fig. 1a). the shaking table of the setup was fixed on four elastic vertical plates, the length of which could be adjusted vertically using a movable mount, and the required stiffness of elastic plates could be set. a box made of steel sheets with dimensions of 0.45×0.46×0.17 m was rigidly installed on the shaking table. the steel box was filled with sand to a height of 0.1 m. the steel sheet with dimensions of 0.45×0.46×0.005 m was installed above the sand using a hydraulic level. the total mass of the platform is 131.6 kg. the steel plate with dimensions of 0.355×0.363×0.0224 m is located above the steel plate. the mass of the upper steel plate 506 i. mirzaev, kh. sagdiev, a. yuvmitov, m. turdiev, b. egamberdiev is 22.4 kg. in the experimental study, the weight of the upper steel plate was changed by the treatment of the steel-on-steel contact surfaces, and different options with the lubrication of the contact surfaces, another one-with the insertion of a pair of fluoroplastic gaskets 4. the laboratory setup is shown in figs. 1a and 1b. (a) (b) (c) (d) fig. 1 laboratory shaking stand (a) general view of the setup (b) side view of the setup (c) setup scheme with main elements (d) simulation scheme of the setup: 1 – moving part of the shaking stand; 2 – plate made of steel sheet for studying the dynamic coefficient of dry friction; 3 – vertical steel plates with viscoelastic property; 4 – fluoroplastic layer; 5 – displacement strain gauge; 6 – movement clock gauge; 7 – rigid base of the shaking stand; 8 – steel box; 9 – screw for shaking table pullback; 10 – movable mount for changing the working length of the plates when conducting experiments on the laboratory setup, the amplitudes of vibration of the shaking table during pullback reach up to 0.006 m, 0.009 m, and 0.012 m, which is due to the restrictions of using a displacement strain gauge designed to record small amplitudes of vibration (fig. 1b). to record vibrations of the shaking table relative to the stationary support of the shaking stand, the displacement strain gauge with preliminary calibration was installed at the level of the shaking table, and the second strain gauge connected with the steel box was installed to the upper steel plate. the vibrations of the shaking stand were generated by deflecting (pulling back) the movable shaking table from the zero position and instantly releasing it. the setup scheme with main elements and the simulation scheme experimental determination of dynamic coefficient of amonton-coulomb dry friction 507 of the setup are shown in figs. 1c and 1d. the main purpose of the experimental research is to study vibrations of the upper steel plate relative to the bottom steel plate located on the shaking table and to measure the value of displacement of the upper steel plate at its different masses relative to the bottom plate due to dry friction after the oscillation of the system has stopped. 2.2. oscillations of the system with two degrees of freedom during interaction of masses according to the amonton-coulomb’s law of dry friction in this paragraph, we construct a theoretical solution to the corresponding mathematical problem. in the simulation model (fig. 1d), we consider the plates rigid and ignore the rolling friction of the rollers. let us see the case when the mass of the bottom plate m1 (which includes the mass of the shaking table with the box and a half of the unit mass of four vertical viscoelastic plates (fig. 1c)) and the mass of the upper plate m2 moves without sliding. the motion is described by the equation of motion of a system with one degree of freedom, the solution of which is well known at t=t0 u1=u00, u2=u00+u02, v1=v2=v00 (ui, vi are the displacements and velocities, respectively, u00, u02, are the displacements of the bottom plate and upper plate relative to the bottom plate at time t=t0). in this case, the damping of the system oscillations is taken into account by introducing into the equation a term proportional to the oscillation velocity with damping coefficient µ, determined from the graphical representation of the experimental results by calculating the logarithmic damping decrement and measuring the oscillation period. the damping of the system oscillations without sliding occurs due to the loss of energy in four viscoelastic vertical plates and through the points of their connection with the base. the damping of the oscillations of the system occurs additionally at sliding friction due to the dry friction force, included in the equations of motion. at the initial time t0=0, the system deviates from the state of static equilibrium by u1=u2=u00 and the system is released without initial velocity v00=0. after sliding, for each subsequent joint motion of two masses without sliding, the beginning is set at t=t0, and the displacement and velocity of masses are denoted by u00 and v00, respectively. the resulting solution is true until the plates begin to slide relative to each other. if the plates slide at t=t1, determined by the special logical algorithm described below, then for each plate there is its equations of motion: frfucuum =++ 1111   (1) frfum −= 22  (2) 2 1 2( )fr df sign u u m g f= −    (3) where m1, m2, u1, u2 are the masses and displacements of the bottom and upper plates, µ is the damping coefficient, с is the total stiffness of elastic plates of the shaking stand, ffr is the dry friction force, g is the gravitational constant. let us introduce the generally accepted designations for velocities v1= 1u , v2= 2u and accelerations a1= 1u , a2= 2u of plates, respectively. at t=t1, u1=u01, u2=u01+u02, v1=v2=v01, where u01, v01 are the displacement and velocity of the bottom plate at the beginning of sliding at time t=t1. the sign of the friction force is determined by the difference in velocities of the plates, denoted by f=sign(v2 – v1)·fd·g, where fd is the dynamic coefficient of dry friction between the plates. 508 i. mirzaev, kh. sagdiev, a. yuvmitov, m. turdiev, b. egamberdiev the equations of motion of rigid plates in eqs. (1-3) are written in the following form: 121 2 2121 2 m/fmukunu =++  (4) fu −=2  (5) where 1 1 22( ) n m m  = + , 1 2 2m n  = are the damping constants, 21 1 mm c k + = , 1 2 m c k = are the natural circular frequencies of the system without and with slip. then the solutions of the system of eqs. (4) and (5) for the displacements, velocities and accelerations of the two plates (if the sign of the difference of their velocities is retained) are as follows:     2 1( ) 2 1 01 2 1 2 2 01 01 2 2 1 2 ( ) cos ( ) 1 ( ) sin ( ) n t t m f u e u k t t c m f m f v u n k t t k c c − −  = +   − +      + + +    − −     (6)       2 1( ) 2 2 1 01 01 2 2 1 2 2 01 2 2 1 01 2 1 ( ) sin ( ) ( ) sin ( ) cos ( ) n t t n m f v e v u n k t t k c m f u k k t t v k t t c − −    = −  + +    − −       − +    − +   −   (7)   2 1 2 1 2 ( ) 2 2 1 01 01 2 2 2 2 01 2 2 01 2 1 ( ) 2 22 01 2 01 2 2 2 1 ( ) ( ) sin{ ( )} 2 ( ) ( ) cos ( ) n t t n t t n m f a e v u n k c m f n u k k v k t t c m f e v n u n k k t t c − − − −    =  + +  +       +  +  −    − −      −   + +  +   −      (8) 2 2 01 02 01 1 1 1 ( ) ( ) ; 2 u u u v t t f t t= + +  − +   − );tt(fvv 1012 −+= fa =2 (9) these solutions hold until the time when two plates move together without sliding. the time t=t0 of the beginning of motion of plates without sliding is determined when the direction of sliding is changed. the process of nonlinear oscillations is very complex, it is impossible to predict in advance the direction of the dry friction force, as well as to determine the time of the beginning of sliding of the upper plate on the bottom plate. to solve this problem, we will build an algorithm for determining the time points of the beginning and end of the sliding of the upper plate on the bottom plate [10]. to do this, we will perform calculations with a small step (about 0.0001 s) to determine the displacements and velocities using the formulas for the well-known solution in the case of joint motion of masses without sliding and eqs. (6-9). at any given time the system can be in one of the following states: experimental determination of dynamic coefficient of amonton-coulomb dry friction 509 i – for the case of joint motion of the plates without sliding; ii – by formulas given in eqs. (6-9) for f > 0; iii – by formulas given in eqs. (6-9) for f < 0 (the latter means the opposite direction of the dry friction force). if solutions for states ii and iii in relative velocities have different signs, then the relative inertia force of the top plate is less than the limiting value of the dry friction force. it means that the friction force between the plates is less than the limiting value of dry friction force, therefore, the plates at this time move without sliding and the true solution to the problem corresponds to state i. if solutions for states ii and iii in relative velocities have the same signs, then the true solution to the problem at time t=t1 is the solution that corresponds to the minimum absolute value of the velocity difference of the plates. it follows from the fact that the force of dry friction is always directed against the relative motion of the plates and against each plate separately. therefore, having conducted calculations according to the specified formulas and using the described decision-making algorithm at each time step, we obtain the solution to the nonlinear problem in the given time interval. 3. results and discussion we will analyze the effect of the treatment of rubbing plate surfaces and their masses on the oscillatory process, and give a method for determining the dynamic coefficient of dry friction by comparing the analytical solution with the experimental results. the interpretation of the series of experiments shows the following (table 1): abbmi, a=1 is the value of pullback of the shaking table which equals to 0.012 m to create oscillations; a=2 is the value of pullback of the shaking table which equals to 0.009 m to create oscillations; a=3 is the value of pullback of the shaking table which equals to 0.006 m to create oscillations; bb=ff is the friction of fluoroplastic-on-fluoroplastic contact between the steel plates; bb=ss is the friction of the steel-on-steel contact; bb=ssl is the friction of the steel-on-steel contact with lubricant of industrial oil i-20; m0 is the mass of the upper steel plate without additional load; m1 is the mass of the upper steel plate with additional mass of 5 kg; m2 is the mass of the upper steel plate with additional mass of 10 kg; m3 is the mass of the upper steel plate with additional mass of 20 kg; fs is the static coefficient of dry friction; t is the period of oscillation of the system without sliding; us is the relative displacement between the upper and bottom plates after oscillations. the results shown in table 1 are based on experimental studies of the static and dynamic coefficients of dry friction between the upper and bottom plates for different materials under the effect of various additional masses acting on the upper plate. the period of oscillation, the oscillation damping during the period of time the plates are moving without sliding, and the displacement of the upper plate relative to the bottom plate after the oscillation stops were measured from the experimental displacement curves of the system. 510 i. mirzaev, kh. sagdiev, a. yuvmitov, m. turdiev, b. egamberdiev table 1 results of experimental studies № series of experiments m2, [kg] c, [n/m] t, [s] us, [m] n1, [1/s] n2, [1/s] fs fd vmax, [m/s] amax, [m/s2] 1 1ffm0 22.4 6667 0.32 -0.010 0.12 0.17 0.2 0.1 0.22 4.35 2 1ffm1 27.4 6667 0.34 -0.0096 0.07 0.08 0.2 0.1 0.22 4.46 3 1ffm2 32.4 6667 0.36 -0.0092 0.07 0.08 0.2 0.1 0.22 4.52 4 1ffm3 42.4 6667 0.38 -0.0089 0.05 0.07 0.2 0.1 0.23 4.8 5 2ffm0 22.4 10000 0.24 -0.0088 0.16 0.22 0.2 0.12 0.29 9.98 6 2ffm1 27.4 10000 0.26 -0.0083 0.12 0.26 0.2 0.12 0.24 6.6 7 2ffm2 32.4 10000 0.27 -0.0076 0.05 0.07 0.2 0.12 0.3 10.7 8 2ffm3 42.4 10000 0.30 -0.0062 0.16 0.27 0.2 0.12 0.2 4.8 9 3ffm0 22.4 18182 0.20 -0.006 0.23 0.35 0.2 0.12 0.19 6.68 10 3ffm1 27.4 18182 0.21 -0.0058 0.24 0.31 0.2 0.12 0.19 6.84 11 3ffm2 32.4 18182 0.22 -0.0056 0.25 0.32 0.2 0.12 0.19 6.9 12 3ffm3 42.4 18182 0.23 -0.0054 0.26 0.38 0.2 0.12 0.2 7.4 13 1ssm0 22.4 6667 0.30 -0.0078 0.12 0.26 0.4 0.13 0.39 13.5 14 1ssm1 27.4 6667 0.38 -0.0047 0.13 0.16 0.4 0.13 0.18 3.29 15 1ssm2 32.4 6667 0.43 -0.0046 0.1 0.13 0.4 0.13 0.14 2.04 16 1ssm3 42.4 6667 0.47 -0.004 0.13 0.17 0.4 0.13 0.25 6.13 17 2ssm0 22.4 10000 0.25 -0.0054 0.1 0.18 0.4 0.15 0.19 4.37 18 2ssm1 27.4 10000 0.28 -0.0052 0.1 0.13 0.4 0.15 0.19 4.44 19 2ssm2 32.4 10000 0.31 -0.0045 0.13 0.17 0.4 0.15 0.16 3.29 20 2ssm3 42.4 10000 0.33 -0.0042 0.17 0.23 0.4 0.15 0.19 4.7 21 3ssm0 22.4 18182 0.21 -0.0045 0.15 0.23 0.4 0.2 0.19 6.6 22 3ssm1 27.4 18182 0.22 -0.0042 0.24 0.33 0.4 0.2 0.19 6.7 23 3ssm2 32.4 18182 0.23 -0.0045 0.28 0.38 0.4 0.2 0.19 6.9 24 3ssm3 42.4 18182 0.25 -0.0037 0.27 0.37 0.4 0.2 0.19 7.12 25 1sslm0 22.4 6667 0.33 -0.0132 0.11 0.15 0.3 0.07 0.19 3.3 26 1sslm1 27.4 6667 0.35 -0.0105 0.1 0.14 0.3 0.07 0.19 3.4 27 1sslm2 32.4 6667 0.38 -0.0101 0.07 0.09 0.3 0.07 0.19 3.5 28 1sslm3 42.4 6667 0.45 -0.0092 0.08 0.1 0.3 0.07 0.19 2.27 29 2sslm0 22.4 10000 0.25 -0.0095 0.07 0.09 0.3 0.08 0.19 4.46 30 2sslm1 27.4 10000 0.26 -0.0091 0.1 0.13 0.3 0.08 0.19 4.57 31 2sslm2 32.4 10000 0.32 -0.0088 0.05 0.07 0.3 0.08 0.2 4.7 32 2sslm3 42.4 10000 0.34 -0.0075 0.05 0.07 0.3 0.08 0.2 4.9 33 3sslm0 22.4 18182 0.20 -0.0074 0.15 0.18 0.3 0.09 0.19 6.7 34 3sslm1 27.4 18182 0.21 -0.0064 0.09 0.11 0.3 0.09 0.19 6.9 35 3sslm2 32.4 18182 0.21 -0.0074 0.102 0.128 0.3 0.09 0.20 7.12 36 3sslm3 42.4 18182 0.23 -0.0065 0.15 0.18 0.3 0.09 0.20 7.5 then, using known relationships, the frequencies of oscillations and the stiffness of four vertical plates were calculated for different vibration values of the shaking table. next, using a specially developed program based on solutions without/with sliding according to eqs. (6-9), the main indices of the dynamic process were calculated, i.e. dynamic coefficient of dry friction, and absolute maximum values of velocity vmax and acceleration amax of the system. to determine the value of the dynamic coefficient of dry friction, theoretical results were compared with experimental results by gradually decreasing the value of this coefficient until the results coincided with displacements with acceptable accuracy. the mass of the bottom plate with the shaking table when measuring vibrations of the two-mass system was 131.6 kg. experimental determination of dynamic coefficient of amonton-coulomb dry friction 511 from table 1 and fig. 2, it follows that a change in the mass of the upper plate and a change in the oscillation velocity within the considered limits, have a slight effect on the dry friction coefficient, and the properties of the rubbing surfaces and lubrication greatly affect the value of the dry friction coefficient. fig. 2 histogram of values of static and dynamic dry friction coefficients for various materials the comparison of the vibrations of the bottom plate according to the result of experiments conducted for 1ffm0 (table 1) and calculated with the same data by the eq. (6) is shown in fig. 3a. good agreement of these results is due to the fact that the simulation model parameters µ and k1 were determined by measuring the logarithmic decrement of damping and by accurate measuring the period of oscillation of the system without slip. the graph shows that up to time t=2.5 s, the system oscillates more intensively due to slip. in addition, to determine the value of the dynamic coefficient of dry friction, we perform a computational experiment by comparing the theoretical results with the experimental result. the graphs of vibrations of the bottom and upper plates are shown in fig. 3b. it shows how the plates oscillate; sliding starts at the initial time of oscillation and occurs until t=2.5 s with very short periods of movements without sliding. (a) (b) fig. 3 graphs of the displacements depend on time (a) vibrations of the bottom plate (1 – experiment, 2 – analytical calculation) (b) vibrations of the bottom plate u1 and the upper plate u2 (analytical calculation) 512 i. mirzaev, kh. sagdiev, a. yuvmitov, m. turdiev, b. egamberdiev the graph of relative vibrations of the upper plate with account for the sliding on the bottom plate for the same initial data obtained from the experiment for 1ffm0 is shown in fig. 4a. it can be seen that the relative displacement of the upper plate occurs until time t=2.5 s, then the plates oscillate without sliding. the vibration velocities of the bottom (v1) and upper (v2) plates are shown in fig. 4b, with the maximum values of the velocity modulus of the bottom plate vmax=0.22 m/s and the upper plate vmax=0.13 m/s are reached at t=0.1 s. the maximum velocity value of the upper plate is less compared to the upper plate due to their sliding. the period of oscillation of the bottom plate in the case of sliding t=0.4 s is larger compared to the period without sliding between the plates t=0.32 s. (a) (b) fig. 4 graphs of relative vibrations of the upper plate (a) and oscillations of velocity of the bottom v1 and the upper v2 plates (b) the calculated graph of accelerations of the bottom (a1) and the upper (a2) plates are shown in fig. 5a. maximum values of accelerations for the bottom plate amax=4.35 m/s2 are reached at t=0 s, and for the upper plate amax=3.75 m/s2 are reached at t=0.3 s. the acceleration of the upper plate at certain time intervals has a constant value; these time intervals correspond to the time intervals of oscillations of the system with sliding. the relative acceleration of the upper plate, i.e., the difference between the accelerations of the upper and the bottom plates is shown in fig. 5b. at that, its maximum value amax=4.65 m/s2 is reached at t=0.3 s. for time t>2.5 s, the relative acceleration of the upper plate is zero since the plates oscillate without sliding. (a) (b) fig. 5 graphs of accelerations of the bottom a1, upper a2 plates (a) and relative acceleration of the upper plate (b) experimental determination of dynamic coefficient of amonton-coulomb dry friction 513 from eq. (5) and fig. 5a, it follows that the transition from joint motion of masses to their motion by sliding occurs when the absolute value of the acceleration of the system exceeds g·fd. when the velocities are equal (figs. 4b and 5a), a transition from sliding to the joint movement of masses without sliding occurs. 4. conclusion the dynamic coefficient of dry friction between different materials was experimentally determined. the value of the dynamic coefficient of friction was determined by comparing the numerical solution of the problem with experimental data. in the experiment, the mass values varied depending on the surface treatment of the steel-on-steel contact and a separate option with lubrication of the contact surface, and with fluoroplastic 4-on-fluoroplastic 4 gaskets. changing the mass of the upper plate (from 22.4 kg to 42.4 kg) did not affect the value of the dynamic coefficient of dry friction. the maximum velocity of the platform in the experiments varied from 0.14 m/s to 0.39 m/s, which corresponds to a 9-point earthquake on the msk-64 scale; it also slightly affected the value of the dynamic coefficient of dry friction. experimental values of the dynamic coefficient of dry friction on the fluoroplastic-on-fluoroplastic contact were up to two times less compared to the static value, and at the steel-on-steel contact, they were three times less. the use of a contact lubricant for rubbing masses greatly reduced the dynamic coefficient of dry friction. it was determined that under the horizontal movement of two plates, one of which lies on the other, sliding with dry friction occurs when the acceleration begins to exceed the values of the dry friction coefficient multiplied by the acceleration of gravity. references 1. popov, v. l., 2017, contact mechanics and friction, berlin-heidelberg: springer-verlag, 352 p. 2. filippov, a.e., klafter, j., urbakh, m., 2004, friction through dynamical formation and rupture of molecular bonds, physical review letters, 92, 135503. 3. elmer, f.j., 1997, nonlinear dynamics of dry friction, journal of physics a: mathematical and general, 30, pp. 6057-6063. 4. wojewoda, j., stefanski, a., wiercigroch, m., kapitaniak, t., 2008, hysteretic effects of dry friction: modelling and experimental studies, phil trans r soc a, 366, pp 747-765. 5. lyashenko, i.a., 2011, tribological properties of dry, fluid, and boundary friction, technical physics, 56(5), pp. 701–707. 6. popov, m., popov, v.l., popov, n.v., 2017, reduction of friction by normal oscillations. i. influence of contact stiffness, friction, 5(1), pp. 45-55. 7. von kluge, p.n., germaine, d.k., crépin, k.t., 2015, dry friction with various frictions laws: from wave modulated orbit to stick-slip modulated, modern mechanical engineering, 5, pp. 28-40. 8. harouz, r., lakehal, a., khelil, kh., dedry, o., hashemi, n., boudebane, s., 2022, dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds, facta universitatis series mechanical engineering, 20(1), pp. 37-52. 9. brzakovic, l., milovanovic, v., kocovic, v., simunovic, g., vukelic, d., tadic, b., 2022, relation between kinetic friction coefficient and angular acceleration during motion initiated by dynamic impact force, tehnički vjesnik, 29(5), pp. 1622-1628. 10. mirzaev, i., yuvmitov, a.s., turdiev, m.s., shomurodov, j.f., 2021, influence of the vertical earthquake component on the shear vibration of buildings on sliding foundations, e3s web of conferences, 264, 02022. 11. mirzaev, i., turdiev, m.s., 2022, vibrations of buildings with a sliding foundation having lateral yielding contact under real seismic impacts, aip conference proceedings, 2432, 030050. 12. mirzaev, i., turdiev, m.s., 2023, the effect of the size of the horizontal gap between the foundation and the sliding grillage on the oscillation of the building during an earthquake, aip conference proceedings, 2612, 040033. 514 i. mirzaev, kh. sagdiev, a. yuvmitov, m. turdiev, b. egamberdiev 13. mamatov, z., ensebekov, a., taylyakova, z., 2023, the condition of existing residential buildings with a sliding support in the foundation in the city of bishkek, e3s web of conferences, 410, 03027. 14. buckle, i., constantinou, m., dicleli, m., ghasemi, h., 2016, seismic isolation of highway bridges, university at buffalo, the state university of new york, 194 p. 15. banović, i., radnić, j., grgić, n., buzov, a., 2023, performance of geotechnical seismic isolation using stone pebble geogrid layer: experimental investigation, soil dynamics and earthquake engineering, 171, 107941. 16. mkrtychev, o.v., mingazova, s.m., 2020, study of the seismic isolation sliding belt: the case of a monolithic reinforced concrete building, journal of physics conference series, 1425, 012161. 17. mkrtychev, o.v., bunov, a.a., 2014, calculation on displacements features of seismic isolated building, vestnik mgsu, 9(6), pp. 63-70. 18. sagdiev, kh.s., yuvmetov, a.s., egamberdiev, b.o., toshpulatov, s.u., toshmatov, e.s., 2023, simulation of the structure of a multistory building with seismic isolation and the testing technique on a laboratory bench under dynamic ((seismic) impacts), e3s web of conferences, 402, 07024. 19. mogilevsky, r.i., ormonbekov, t.o., nikitin, l.v., 1993, dynamics of rods with interfacial dry friction, journal of the mechanical behavior of materials, 5(1), pp. 85-93. 20. isakov, a.l., kondratenko, a.s., 2021, propagation of shock pulse along a pipe during vertical penetration in soil, journal of mining science, 57(2), pp. 177–183. 21. bragov, а.м., konstantinov, а.yu., lomunov, а.k., 2008, determining dynamic friction using a modified kolsky method, technical physics letters, 34(5), pp. 439-440. 22. espinosa, h.d., patanella, a.j., fischer, m., 2000, dynamic friction measurements at sliding velocities representative of high-speed machining processes, journal of tribology, 122(4), pp. 834-848. 23. miljojković, j., kočović, v., luković, m., živković, a., šimunović, k., 2022, development of a modular didactic laboratory set for the experimental study of friction, tehnički vjesnik, 29(1), pp. 269-277. https://www.scopus.com/authid/detail.uri?authorid=7801655616 12795 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240725044s © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper development and testing of a novel diabetic orthosis for plantar pressure reduction gurpreet singh1, dhruv bose1, arnab chanda1,2 1centre for biomedical engineering, indian institute of technology delhi, new delhi, india 2department of biomedical engineering, all india institute of medical sciences, new delhi, india abstract. the prevalence of diabetes has been increasing worldwide, and this has resulted in a rise in diabetic foot ulcers. several assistive devices, such as custom shoes, unloading orthoses, and insoles, have been developed to address these issues. a modular insole was developed to off-load the peak plantar pressures in diabetic patients. plantar pressure values were measured using an indigenously fabricated pressure mapping insole integrated with high precision force sensitive resistor (fsr) sensors. the performance of the modular insole was quantified quantitatively and qualitatively to assess its effectiveness. for the healthy participants, the results showed that the maximum average zonal pressure (azp) was reduced by 97% from 122.39 ±3.62 kpa to 3.27 ±0.50 kpa during standing and by 98% from 195.71 ±3.19 kpa to 3.57 ±0.31 kpa during walking. similarly, for the diabetic participants, azp values were reduced by 96% from 67.92 ±12.45 kpa to 3.09 ±0.32 kpa for standing and by 97% from 130.96 ±4.27 kpa to 3.67 ±0.54 kpa for walking conditions, respectively. the results confirmed the effectiveness of the modular insole in off-loading the plantar pressure. the development of such devices holds great promise in improving the quality of life of diabetic patients and reducing the risk of complications such as diabetic ulceration. key words: diabetic foot, modular insole, pressure off-loading, intervention 1. introduction diabetes is a chronic illness that affects millions of people worldwide. in fact, there are currently 382 million cases of diabetes globally, and it's estimated that this number will increase to 592 million by 2035 [1–3]. one of the consequences of diabetes is diabetic foot ulceration, which is becoming more common as the population ages and obesity rates received: july 25, 2024 / accepted october 04, 2024 corresponding author: arnab chanda centre for biomedical engineering, indian institute of technology delhi, new delhi 110016, india department of biomedical engineering, all india institute of medical sciences, new delhi 110029, india e-mail: arnab.chanda@cbme.iitd.ac.in 2 g. singh, d. bose, a. chanda continue to rise [4]. diabetic foot ulcers occur in around 5% of all patients with diabetes and can be the precursor to 80% of lower limb amputations due to diabetes [5–7]. these ulcers usually occur in areas of the foot that experience high levels of mechanical stress. if not treated promptly, foot ulcers can become infected and may ultimately require amputation [8]. early diagnosis and timely medication can help prevent and mitigate ulceration in diabetic patients. however, when peripheral neuropathy occurs in the foot, which is a common occurrence in diabetic patients, diagnosing the condition becomes extremely complicated and difficult. when it comes to treating protruding diabetic ulcers, conventional interventions include using pressure off-loading techniques such as crutches, special foot orthotics, and custom pressure off-loading insoles [9–11]. the main objective of medical intervention is to redistribute the pressure exerted on the plantar fascia, which is the underlying cause of ulcers [12,13]. customized foot orthoses are regarded as a better treatment because they provide complete contact with the plantar portion of the foot, unlike prefabricated orthoses [14]. however, effective treatment of diabetic ulcers requires a comprehensive approach, including wound care and glycemic control management, to prevent further complications. therefore, diabetic patients are advised to seek medical attention immediately after they notice any signs of foot ulceration to prevent the condition from worsening [15,16]. previous studies have shown that it is essential to lower plantar pressures in diabetic patients to reduce this risk [17–19]. in a study by caselli et al. [20], they observed that the forefoot and heel areas were the most affected in the diabetic population, with the highest levels of plantar pressure. furthermore, increased pressure was positively correlated with increased severity of neuropathic disease. a force-measuring device was used by cock et al. [21] to determine the center of pressure and plantar pressure of the foot. they found significant variation in the results for different gait phases across all the participants. the recent development of sensor-based insoles has made it possible for patients to collect data outside of clinics, making it easier to monitor their progress and detect any potential issues early on [22,23]. several studies have used shoe-based pressure-measuring devices while walking or running. these devices typically have numerous sensing sites (piezoresistive materials or force-sensitive resistors) and use wireless data transmission methods [24,25]. force-sensitive resistors or piezoresistive materials are often used to provide flexibility to the sensors [26]. after conducting a thorough review of existing literature, it was observed that custom pressure off-loading insoles offer similar assistance to total contact cast (tcc) but with fewer drawbacks. the fabrication of such customized insoles is a laborious and intricate procedure that demands detailed attention to every aspect [27,28]. also, the process of traditional fabrication might be time-consuming, as it may need several days to effectively address the issue of isolating and relieving pressure from the area prone to ulceration [29– 31]. to address these challenges, there is a need to develop a tailored and economical insole that is almost universally applicable and can efficiently alleviate pressure from any specific area on the sole of a diabetic foot. additionally, the insole should possess both comfort and shock absorption characteristics, particularly at the heel, while also providing support to the whole plantar area of the foot. the reported work focuses on developing and validating a customized insole designed to off-load plantar pressure for diabetic patients. this particular insole can effectively intervene for ulceration during post-operative medical care and wound care. development and testing of a novel diabetic orthosis for plantar pressure reduction 3 2. materials and methods the modular diabetic pressure off-loading insole was designed to alleviate pressure on ulcerated areas of the foot. removable supporting segments were made to create a void under the affected area, which helps to distribute weight more evenly and reduce pressure on the ulcer. the fabrication process involves using high-quality materials and advanced manufacturing techniques to ensure that the insole is durable, comfortable, and effective. a pressure measurement device was used to measure the pressure distribution and offloading capabilities of the insole. the testing process involves monitoring the results to verify that the insole is effective in reducing pressure in diabetic patients and corresponding on ulcerated areas (in future studies) and improving the overall comfort and mobility of the individual. 2.1. designing and modeling of modular insole a standard uk-8 insole size was considered for designing and fabricating the modular insole. the outline of the insole’s mid-frame was plotted as per the mondopoint system using solidworks 2021 (dassault systèmes, france). the modular insole comprised two main components: the mid-frame (fig. 1b) and fourteen uniquely molded silicone rubber inserts (fig. 1a). to accommodate the specific needs of each region, molded silicone inserts were designed to be individually removable and re-attachable. this segmentation was based on previous literature studies, which highlighted the most common areas where diabetic patients experience abnormal or peak plantar pressures [15,16]. due to the significant stretching and bending strains experienced when walking, thermoplastic polyurethane (tpu) material with a shore hardness of 95a was chosen for the 3d printing of mid-frame (0.6mm thickness) to withstand these stress conditions [32,33]. the inserts were created with grooves of width and depth of 0.6mm and 3mm, respectively (fig. 1c), to facilitate the proper attachment of the inserts to the matching segments in the mid-frame. these inserts (thickness of 6mm each) were designed to provide maximum comfort and support to the foot, making it easier to walk, run, and stand for prolonged periods of time. the focus of this work was to improve the cushioning properties of insoles; therefore, the cylindrical voids were strategically placed in a homogenous pattern, designed, and incorporated into the insoles (as shown in fig. 1a). the inserts were made using a polymeric solution of shore hardness 10a (liquid silicone rubber), a bio-inert and wear-resistant material that was easy to mold. figure 2 shows the 3d-printed molds for fabricating the modular insole and the fabricated silicone inserts. figure 3 illustrates the 3d-printed mid-frame of the insole and the assembled silicone inserts with the mid-frame. the final modular insole was then packed in the pouch for the participant’s trial. the material used for the pouch was soft and stretchable, which means that the pouch will adjust itself according to the foot's movements, providing a smooth and comfortable experience for the individual. the assembled insole, designed specifically for diabetic patients, is a novel technology that aims to improve the individual's comfort while reducing the risk of ulceration. 4 g. singh, d. bose, a. chanda fig. 1 (a) solidworks design of modular insole, (b) 3d-printed mid-frame for holding the inserts, (c) grooves geometry in inserts fig. 2 (a) 3d-printed mold wall with grooves, (b) mold base with circular voids, (c) assembled mold for fabricating silicone inserts, (d) fabricated silicone inserts development and testing of a novel diabetic orthosis for plantar pressure reduction 5 fig. 3 (a) 3d printed mid-frame, (b) assembled silicone inserts and mid-frame, (c) modular insole in the pouch, (d) mid-foot 1 insert removed from the insole (for representation only) 2.2. pressure measurement of plantar fascia a wearable pressure measurement insole was developed to quantify the pressure values at the foot plantar, with the aim of validating the effectiveness of modular insole in pressure off-loading. the insole was comprised of force sensitive resistor (fsr) sensors manufactured by interlink electronics, inc., california, united states. these fsr sensors (ø12.5 mm) have proven to be highly effective in sensing variations in pressure levels, and they were placed in the centers of eleven different regions of the modular insole. however, three regions (toe 3, heel 2, and heel 3) were not measured due to their small size or because they showed insignificant pressure levels that would not contribute to the occurrence or progression of ulceration [15,16,34]. table 1 shows the location of each sensor with respect to the segment heel as the origin point. the negative coordinates represent the lateral part, and the positive coordinates represent the medial part of the foot (fig. 1b). table 1 sensor location on the in-house fabricated pressure measuring insole sensor location coordinates (in mm) abbreviated word abbreviation x-axis y-axis heel heel 0 0 heel 4 heel medial region 22.5 21 heel 1 heel lateral region -22.5 21 arch arch 14.5 86 mid-foot 2 midfoot region 2 -19 58 mid-foot 1 midfoot region 1 -22 110 fore-foot 3 forefoot region 3 -37 143 fore-foot 2 forefoot region 2 -6 161 fore-foot 1 forefoot region 1 25 158 toe 2 toe region 2 -9 207 toe 1 toe region 1 (hallux) 16.5 203 6 g. singh, d. bose, a. chanda fig. 4 (a) pressure measuring insole with microcontroller board for data recording, (b) eva-pla sandwich with fsr sensors, (c) diabetic volunteer with the modular insole (inside the shoe) figure 4 shows a detailed view of the pressure-measuring insole, designed and fabricated to monitor the plantar pressure values of the foot. the fsr sensors were affixed to one side of a 2mm thick ethylene-vinyl acetate (eva) foam sheet, which was precisely shaped into the dimensions of a uk8-size insole, serving as the base for the fsr sensors. on the opposite side, a 3d-printed pla sheet of 0.6mm thickness was used to cover the sensors, providing fsr sensors with a solid surface for the accurate recording of the data. a voltage divider circuit connected the sensors in the insole to the microcontroller board, which reduced the voltage from the sensors to a level that could be read by the microcontroller [35,36]. a box was 3d printed to fix the microcontroller board, which was development and testing of a novel diabetic orthosis for plantar pressure reduction 7 tied onto the volunteer's leg using a velcro strap during all trials in this study to record the data via a 20-band flat ribbon cable (fig. 4c). by keeping the microcontroller board in a fixed position, the data obtained from the sensors could be accurately recorded and analyzed. 2.3. measurement of the recorded data during the data collection, the pressure-measuring insole was positioned underneath the modular insole inside the shoe. arduino integrated development environment (ide) was used to code pressure measuring insoles with complete scans of 5000 per second to collect sensor data. a complete scan was operationally defined as the time required for the microcontroller to execute one full loop of the whole code and display the processed values of each sensor. the study included the mean value of plantar pressures, calculated at a frequency of 20 milliseconds for each scan, as well as the highest pressure values recorded by each sensor at any point throughout the trial period. the recorded data values were shown on the laptop in the arduino ide program (fig. 4c) and full computed datasets were saved from the microcontroller board to the laptop via a usb connection for each trial. 2.4. performance of modular insole the modular insole was tested quantitatively and qualitatively on a group of fifteen healthy and fifteen diabetic participants (all males) with a uk8 foot size under the approval of the ethical committee of the indian institute of technology delhi (iit-delhi). table 2 listed the details for healthy participants (height, weight, age) and diabetic participants (height, weight, age, blood glucose) of the present study. the modular insole was tested using various performance parameters, including comfort, fit, durability, and pressure offloading, to ensure that it meets the highest safety and efficacy standards for diabetic patients. table 2 details of the healthy and participants with diabetes for the study sr. no. healthy participants participants with diabetes height (in cm) weight (in kg) age (in years) height (in cm) weight (in kg) age (in years) blood glucose (mg/dl) 1 165.1 72 28 170.4 72 62 178.3 2 185.4 66 33 174.2 78 58 141.4 3 180.3 83 31 175.3 81 47 120.5 4 175.4 78 42 170.2 65 51 126.5 5 180.3 86 50 165.4 76 39 130.5 6 177.1 74 52 173.7 77 42 138.2 7 182.9 68 47 165.1 89 60 174.2 8 164.6 91 62 175.5 71 53 230.4 9 171.4 57 30 173.7 87 55 142.4 10 166.3 63 26 170.2 79 49 164.7 11 182.9 61 29 180.3 59 36 136.2 12 185.4 74 35 166.3 64 59 152.1 13 173.7 88 36 177.6 82 58 211.3 14 164.6 76 32 181.1 71 46 132.8 15 175.4 62 35 179.3 83 51 146.5 8 g. singh, d. bose, a. chanda 2.4.1 qualitative tests the study aimed to evaluate the performance of a modular insole using the quebec user evaluation of satisfaction with assistive technology (quest 2.0) questionnaire [37,38]. this questionnaire measures the subjective parameters of an assistive device, such as comfort, ease of use, and effectiveness. the questionnaire has eight items, and each item was scored from 1 to 5, where 5 corresponds to completely satisfied and 1 means completely dissatisfied. the modular insole was tested for healthy participants (hp) and diabetic participants (dp) in two cases, i.e., with all inserts in the midframe in case 1 (q1) and removal of inserts one by one in case 2 (q2). in this way, there were two individual cases for healthy participants (hp-q1 and hp-q2) and diabetic participants (dp-q1 and dp-q2). 2.4.2 quantitative tests the testing setup consists of a pair of shoes with a pressure-measuring insole under the modular insole in one of the shoes by replacing the original insole. the testing trials were carried out under two conditions: standing (stated as case cs) and walking (stated as case cw). initially, each participant trial was conducted with all inserts connected to the modular insole, which is denoted as cs (full) and cw (full). subsequently, nine out of the fourteen inserts (namely, toe 1, fore-foot 1, fore-foot 2, fore-foot 3, mid-foot 1, midfoot 2, heel 1, heel 4, and heel) were individually removed to record the plantar pressure values for each sub-case per participant. toe 2 and arch were not included in the removal experiments due to their significantly low-pressure magnitudes in both cs and cw scenarios. however, pressure for these regions was recorded for the full insole during standing and walking. each participant had a total of 10 trials, consisting of one trial with a complete insole and nine trials with the inserts removed for each instance (cs and cw). the azp values were recorded using the running average algorithm with pressure measuring insole. during the trial in case cs, participants were instructed to maintain an upright posture, as they typically would, for a period of 5 seconds. for walking cases (i.e., cw), the participants were instructed to walk naturally and without any restrictions for a period of 20 seconds. the effectiveness of the modular insole was validated based on its azp values for the off-loaded regions, confirming the efficient pressure off-loading in the desired region. 3. results and discussion 3.1. quest-based qualitative analysis for healthy participants, the quest score was calculated to be 4.63 ±0.08 for the hpq1 case and 4.61 ±0.09 for the hp-q2 case. for diabetic participants, the quest score for case dp-q1 was 4.80 ±0.06, indicating high satisfaction with the insole's comfort and other features. similarly, for case dp-q2, the quest score was 4.81 ±0.17, demonstrating consistent user satisfaction. figure 5 displays the quest scoring for the healthy participants (fig. 5a) and diabetic participants (fig. 5b), providing a detailed view of the individual’s satisfaction levels. development and testing of a novel diabetic orthosis for plantar pressure reduction 9 fig. 5 quest scoring for: (a) healthy participants, (b) diabetic participants according to the quest results, the participants were very satisfied with the performance of the insole, as it scored between 4 and 5 on the scale. during the qualitative trials for healthy (hp-q1, hp-q2) and diabetic participants (dp-q1, dp-q2), certain feedback points regarding the modular diabetic insole were noted. the participants did not notice the truncated regions at fore-foot 1, fore-foot 2, fore-foot 3, heel 1, and heel 4 during q2 (hp-q2 and dp-q2), probably due to support from the surrounding surface of the insole. the empty space left by inserts toe 1 and toe 2 in q2 (hp-q2 and dp-q2) was only felt by participants when they lowered their toes voluntarily. the voids created by removing inserts toe 3, heel 2, and heel 3 were entirely imperceptible due to their minimal 10 g. singh, d. bose, a. chanda areas. the foot area at region toe 3-5 did not add my significant contribution to the plantar pressure values and hence excluded from the design of circular voids for simplicity. the simplified insole design and fabrication made it easy for everyone to use, and participants were comfortable removing and reattaching any insert from the mid-frame. all participants acknowledged that the insole offered a substantial level of comfort to the foot. moreover, the insole can be recommended as a comfort insole that can be used in cases of non-diabetic foot ulceration as well. 3.2. evaluation of pressure off-loading as discussed earlier, a pressure mapping insole was used for the measurement of plantar pressures under eleven segments of the modular insole for healthy and diabetic participants for standing and walking conditions, respectively. these values were then averaged (from the individual values of all participants) to calculate the values of average zonal pressure (azp). the azp values provide a comprehensive understanding of the distribution of pressures across the different regions of the foot. 3.2.1 pressure off-loading for healthy participants table 3 comprehensively summarizes the azp values (mean ±s.d.) for nine cases that were off-loaded during standing conditions for healthy participants. the values for full insole and no insole were also recorded to compare the effectiveness of the off-loading in different cases. similarly, table 4 corresponds to the summary of azp values for plantar pressure off-loading during walking conditions for healthy patients. figure 6 provides detailed information about the pressure values at various regions for healthy participants in standing and walking cases. the pressure values were recorded at eleven different locations (nine off-loaded), and the corresponding azp values (in kpa) were noted for all fifteen participants. these values were then averaged to obtain a representative value (mean ±s.d.). to provide a more comprehensive understanding, figs. 6a and 6c show azp values for healthy participants in standing and walking cases with full modular insole, i.e., without removing any insert from the insole. in contrast, figs. 6b and 6d illustrate the off-loaded azp values obtained by removing one of the inserts and keeping the rest during trials during standing and walking conditions, respectively. for standing condition with the full insole, the maximum value for average zonal pressure was observed at the heel region with a magnitude of 122.39±3.62 kpa (fig. 6a). on the other hand, the minimum magnitude of average zonal pressure was observed at the arch region with a value of 3.86±3.66 kpa. the pressure magnitudes at inserts toe 2 and arch were quite low, accounting for only 1.83% and 0.76% of the total pressure recorded. therefore, the azp values for toe 2 and arch were not considered for the study. in the off-loaded case, as shown in fig. 6b, the plantar pressure values decrease from 97.52% to 99.98% at mid-foot 1 and heel 1, respectively. the maximum azp value was quantified at fore-foot 1 (4.89±1.38 kpa), confirming the off-loading at other regions during standing conditions across all the participants. development and testing of a novel diabetic orthosis for plantar pressure reduction 11 table 3 average zonal pressure values (in kpa) for healthy participants (standing) in different off-loading conditions removed inserts ↓ toe 1 toe 2 forefoot 1 forefoot 2 forefoot 3 midfoot 1 midfoot 2 arch heel 1 heel 4 heel no insole 63.52 7.40 98.34 24.35 78.66 32.27 25.73 3.09 116.64 17.77 133.44 std. dev. 4.40 5.88 4.06 5.27 5.59 6.78 6.50 3.11 3.41 4.98 3.54 full insole 47.52 9.20 73.74 18.55 54.82 22.67 23.55 3.86 105.96 19.31 122.39 std. dev. 3.97 4.89 3.80 3.95 4.78 5.61 5.25 3.66 3.51 6.06 3.62 toe 1 removed 3.52 10.53 74.78 31.87 42.41 21.26 24.45 3.17 95.50 12.08 82.82 std. dev. 0.38 3.51 4.01 3.60 4.76 6.29 5.46 3.14 4.03 4.36 4.25 fore-foot 1 removed 51.86 25.22 4.89 18.73 87.51 50.05 60.60 3.53 98.93 44.23 147.58 std. dev. 4.30 5.45 1.38 4.19 3.98 4.78 4.06 3.47 3.58 4.47 3.65 fore-foot 2 removed 34.71 9.19 53.13 3.31 66.95 22.47 40.72 3.59 115.28 30.70 148.95 std. dev. 3.86 5.48 4.77 0.30 3.79 5.62 4.89 3.39 3.30 6.06 3.56 fore-foot 3 removed 84.69 20.10 87.35 15.28 3.82 15.59 19.22 13.18 86.16 23.30 144.67 std. dev. 3.10 3.95 3.37 3.44 0.34 3.61 3.68 3.73 3.33 3.68 3.29 mid-foot 1 removed 67.52 21.29 88.78 16.40 47.05 3.97 34.81 4.06 109.65 42.74 141.72 std. dev. 3.26 3.56 3.38 3.50 3.94 0.87 5.98 3.41 3.57 4.69 3.47 mid-foot 2 removed 55.69 41.65 34.16 29.71 91.86 60.48 4.41 3.47 117.52 33.77 155.36 std. dev. 4.14 3.84 12.24 4.02 3.87 3.54 1.96 3.37 3.31 4.66 3.35 heel 1 removed 46.71 5.97 111.58 40.03 89.06 29.83 27.35 10.85 3.86 58.52 115.08 std. dev. 3.31 3.29 3.19 3.41 3.52 3.68 4.19 3.48 0.02 3.69 3.38 heel 4 removed 46.71 5.97 111.58 40.03 89.06 29.83 27.35 10.85 58.52 3.09 115.08 std. dev. 3.31 3.29 3.19 3.41 3.52 3.68 4.19 3.48 3.69 0.09 3.38 heel removed 65.62 13.41 106.73 35.25 105.09 43.31 36.13 9.05 52.68 16.63 3.27 std. dev. 4.36 5.12 3.72 4.34 4.01 4.35 3.58 6.01 5.52 4.28 0.50 * bold values represent the removed/off-loaded inserts. figure 6c shows the azp values for healthy participants with the full insole in walking condition. the highest value of average zonal pressure was observed at the heel region with a magnitude of 195.16±3.19 kpa, indicating the maximum pressure applied at that point. conversely, the lowest average zonal pressure value was observed at the arch region with a magnitude of 22.32±6.28 kpa. the areas with high azp values are considered to be at greater risk of skin breakdown and injury, while the areas with low azp values are less susceptible to injury. it should be noted that the azp magnitudes at inserts toe 2 and arch were observed to be very low, contributing only 1.44% and 1.88% of the total recorded pressure. they were not considered for evaluation based on the fact that their impact on the overall pressure was insignificant, and removing these inserts would not have a significant effect on the outcome 12 g. singh, d. bose, a. chanda of the trials. in fig. 6d, the off-loaded cases showed a significant decrease in pressure, ranging from 92.71% to 98.17% at segments heel 4 and heel, respectively. the highest pressure value occurred at segment fore-foot 3, with an azp value of 5.70±1.25 kpa. it was also observed that the fore-foot 1 and fore-foot 2 segments followed the highest pressure value after the fore-foot 3 segment with a magnitude of 4.36 ±0.48 kpa and 4.07±1.36 kpa, respectively. although these values were minor in magnitude, evidence of similar contact during the heel strike was observed at segments heel, heel 4, and heel 1. after considering the observed results and participant feedback, it was concluded that the modular insole provided sufficient pressure off-loading in all nine regions, with a range of 92.71% – 98.17%. table 4 average zonal pressure values (in kpa) for healthy participants (walking) in different off-loading conditions removed inserts ↓ toe 1 toe 2 forefoot 1 forefoot 2 forefoot 3 midfoot 1 midfoot 2 arch heel 1 heel 4 heel no insole 67.16 6.55 114.50 30.53 69.23 22.26 21.64 4.60 116.88 19.58 138.43 std. dev. 5.96 7.35 4.71 7.08 7.41 7.72 9.10 5.96 3.60 6.63 4.31 full insole 128.10 17.06 171.69 93.98 201.21 78.32 80.87 22.32 143.63 48.66 195.16 std. dev. 3.91 5.42 3.42 3.29 3.52 3.44 3.58 6.28 3.23 3.93 3.19 toe 1 removed 3.33 17.79 80.88 36.58 163.14 103.99 106.44 3.19 104.56 71.88 195.55 std. dev. 0.34 7.58 4.10 4.35 3.57 3.33 3.59 3.10 3.61 4.54 3.52 fore-foot 1 removed 77.12 21.64 4.36 52.35 192.36 85.48 102.29 10.03 130.71 71.09 186.05 std. dev. 4.85 5.74 0.48 4.55 4.14 3.42 3.48 11.02 3.75 4.01 3.52 fore-foot 2 removed 130.07 58.66 150.93 4.07 122.55 62.83 87.95 3.26 97.04 50.44 150.24 std. dev. 3.12 3.41 3.34 1.36 3.44 3.33 3.30 3.06 3.15 3.52 3.16 fore-foot 3 removed 132.77 40.54 32.36 57.86 5.70 45.55 63.84 15.02 123.82 55.21 171.36 std. dev. 4.66 3.97 15.62 4.08 1.25 3.62 3.76 14.89 3.75 3.43 3.97 mid-foot 1 removed 129.03 16.72 171.36 88.96 196.58 3.51 67.99 15.07 140.75 47.41 193.46 std. dev. 3.83 6.13 3.51 3.56 3.53 0.23 3.66 5.29 3.35 4.11 3.27 mid-foot 2 removed 92.42 25.12 118.24 27.95 71.85 55.48 3.43 11.89 128.78 59.29 170.78 std. dev. 4.90 3.88 4.39 6.22 5.22 3.52 0.15 13.09 3.45 3.51 3.64 heel 1 removed 93.55 77.55 92.07 58.93 233.12 121.44 115.74 3.27 3.83 79.89 182.08 std. dev. 4.63 3.26 9.22 5.63 3.12 3.41 3.45 3.23 0.45 3.87 3.35 heel 4 removed 75.79 15.84 159.45 85.37 167.68 69.96 73.07 12.21 133.69 3.55 193.68 std. dev. 3.38 4.02 3.13 3.13 3.08 3.32 3.25 3.36 3.13 0.26 3.11 heel removed 68.02 22.30 125.31 50.83 132.90 71.30 44.75 6.58 28.67 11.93 3.57 std. dev. 4.52 7.71 4.15 4.58 4.12 4.22 4.56 7.10 3.50 4.03 0.31 * bold values represent the removed/off-loaded inserts. development and testing of a novel diabetic orthosis for plantar pressure reduction 13 fig. 6 azp values (in kpa, mean ±s.d.) for healthy participants: (a) full insole with all inserts during standing, (b) off-loading of regions by removing their respective insert in standing position, (c) full insole with all inserts during walking, (d) off-loading of regions by removing their respective insert in walking condition 3.2.2 pressure off-loading for diabetic participants the recorded azp values (mean ±s.d.) for the fifteen diabetic participants during standing and walking conditions are tabulated in tables 5 and 6, respectively. additionally, the values were recorded for both the full insole and no insole cases to compare the offloading results for both cases. during the diabetic participants in standing condition with a full insole, the highest azp was found at the heel region, reaching a magnitude of 40.92 ±12.45 kpa (fig. 7a). conversely, the lowest magnitude of azp was recorded at the arch region, with a value of 4.88 ±4.53 kpa. similar to the healthy participants, the pressure magnitudes at the inserts toe 2 and arch were relatively low, accounting for only 2.78% and 2.19%, and excluded from the off-loaded trials. for the off-loading case, the azp values of plantar pressure were decreased, ranging from 87.11% to 97.82% at fore-foot 3 and heel 1, respectively (fig. 7b). the azp values for the rest of the segments were lower than those at fore-foot 3, which confirmed the pressure off-loading across all the other regions. the data presented in fig. 7c shows the azp values for diabetic participants with full insole while walking. the heel region showed the highest azp value of 130.96 ±4.27 kpa, indicating the maximum pressure applied at that specific point. conversely, the lowest average zonal pressure value was observed at the arch region with a magnitude of 12.45±9.51 kpa. for diabetic patients, the plantar region with high azp values is more susceptible to skin breakdown and injury (ultimately the ulceration), while the areas with 14 g. singh, d. bose, a. chanda low azp values are less vulnerable to injury. similar to the previous cases, it should be noted that the azp magnitudes at inserts toe 2 and arch were observed to be very low, contributing only 2.21% and 2.06%. therefore, these were excluded from off-loading cases during the trials. figure 7d illustrates the results of the pressure off-loading trials, which showed a significant decrease in pressure ranging from 81.52% to 97.19% at segments heel 4 and heel, respectively. table 5 average zonal pressure values (in kpa) for participants with diabetes (standing) in different off-loading conditions removed inserts ↓ toe 1 toe 2 forefoot 1 forefoot 2 forefoot 3 midfoot 1 midfoot 2 arch heel 1 heel 4 heel no insole 26.07 3.26 40.00 16.83 41.78 10.40 8.64 3.09 29.26 5.10 57.77 std. dev. 9.64 3.33 16.48 7.74 9.37 6.34 6.50 3.06 6.62 4.21 7.17 full insole 24.40 6.19 33.43 24.34 30.74 13.63 17.89 4.88 47.40 15.23 67.92 std. dev. 9.37 5.24 11.69 6.63 9.37 8.37 8.45 4.53 9.99 5.69 12.45 toe 1 removed 2.42 14.88 90.12 37.40 80.62 24.33 26.39 4.23 118.69 12.53 104.10 std. dev. 2.46 4.18 7.63 5.25 6.03 7.43 8.26 4.63 5.73 5.74 10.12 fore-foot 1 removed 56.85 19.09 3.74 23.63 115.56 45.61 66.93 3.75 84.53 36.64 125.87 std. dev. 3.76 4.88 1.66 3.72 4.11 5.41 4.50 3.55 3.73 4.55 3.45 fore-foot 2 removed 38.24 8.43 14.39 2.47 76.49 16.93 32.95 3.68 102.99 16.25 123.96 std. dev. 3.33 3.21 3.38 0.53 3.28 3.38 3.41 3.19 3.12 3.62 3.28 fore-foot 3 removed 70.68 13.30 57.66 12.93 3.96 14.91 18.94 9.14 90.81 23.65 152.14 std. dev. 3.98 6.03 5.81 4.38 0.38 3.85 3.95 5.88 3.39 3.60 3.28 mid-foot 1 removed 62.18 18.64 84.40 16.59 57.46 1.10 51.84 3.95 118.96 56.87 153.89 std. dev. 3.14 3.22 3.24 3.17 3.30 0.08 3.11 3.17 3.26 3.23 3.14 mid-foot 2 removed 90.81 24.42 116.27 27.32 69.96 53.77 2.26 12.32 126.54 57.33 172.35 std. dev. 4.76 3.85 4.56 6.27 5.39 3.47 0.06 13.99 3.47 3.55 3.70 heel 1 removed 67.83 3.60 122.84 18.34 73.24 16.82 16.53 3.14 1.03 7.72 83.44 std. dev. 4.16 3.57 3.92 4.47 4.30 4.37 4.25 3.11 0.02 4.11 4.27 heel 4 removed 42.63 11.02 81.93 29.09 61.90 42.73 32.58 7.61 68.93 1.58 106.83 std. dev. 3.34 3.26 3.25 3.14 3.21 3.32 4.15 3.31 3.27 0.37 3.37 heel removed 27.76 11.44 54.96 15.17 19.14 11.41 14.02 3.08 60.15 6.55 3.09 std. dev. 3.56 3.39 3.50 3.49 3.60 3.97 3.83 3.08 3.32 3.46 0.32 * bold values represent the removed/off-loaded inserts. development and testing of a novel diabetic orthosis for plantar pressure reduction 15 table 6 average zonal pressure values (in kpa) for participants with diabetes (walking) in different off-loading conditions removed inserts ↓ toe 1 toe 2 forefoot 1 forefoot 2 forefoot 3 midfoot 1 midfoot 2 arch heel 1 heel 4 heel no insole 7.75 5.51 28.68 17.20 14.71 13.45 18.07 3.07 89.63 35.84 105.68 std. dev. 3.50 3.47 4.00 3.63 3.98 3.61 3.64 3.07 3.23 3.31 3.13 full insole 55.76 13.35 66.20 37.13 72.20 49.66 40.25 12.45 98.84 27.07 130.96 std. dev. 6.74 5.26 9.01 6.73 8.18 4.84 6.38 9.51 4.25 6.40 4.27 toe 1 removed 3.44 10.84 62.83 44.17 111.20 31.62 44.12 3.62 93.55 25.18 114.42 std. dev. 0.27 6.27 5.23 4.39 4.29 7.50 5.05 3.44 3.60 4.85 3.42 fore-foot 1 removed 88.05 23.22 4.32 61.19 216.91 87.55 105.50 12.18 138.52 78.71 188.00 std. dev. 4.35 5.88 0.52 3.42 3.29 3.39 3.42 11.06 3.58 3.41 3.55 fore-foot 2 removed 101.82 27.17 75.74 6.86 70.55 39.88 29.44 8.95 137.52 44.20 112.35 std. dev. 4.96 3.61 5.54 1.44 6.87 3.68 5.20 13.51 3.41 3.80 5.60 fore-foot 3 removed 90.17 17.42 70.66 24.43 5.66 23.10 21.88 8.51 111.64 31.09 114.06 std. dev. 4.53 5.67 5.30 4.86 1.11 4.15 3.51 6.38 3.69 4.27 4.37 mid-foot 1 removed 65.98 25.61 32.42 25.58 26.23 4.32 18.54 3.14 128.29 11.99 107.97 std. dev. 3.92 5.60 4.19 8.88 5.14 0.57 3.87 3.15 3.76 5.87 3.64 mid-foot 2 removed 163.14 35.49 176.90 77.94 127.77 76.46 3.52 23.70 139.88 71.48 180.34 std. dev. 3.45 3.46 3.51 3.25 3.57 3.75 0.28 16.23 3.36 3.21 3.84 heel 1 removed 139.47 21.16 156.73 80.90 159.99 72.74 71.58 5.86 3.58 36.24 153.71 std. dev. 3.67 7.41 3.79 3.95 4.37 3.39 3.50 6.09 0.31 4.86 4.00 heel 4 removed 113.38 28.53 122.55 41.19 92.60 43.17 38.77 7.45 97.71 3.19 110.94 std. dev. 3.59 5.24 3.84 4.82 3.49 4.25 3.61 3.78 3.41 0.08 3.32 heel removed 48.68 8.43 131.26 50.21 139.47 54.02 39.03 17.64 32.80 16.05 3.67 std. dev. 3.60 3.39 3.45 3.59 3.35 3.61 3.63 4.05 3.55 3.70 0.54 * bold values represent the removed/off-loaded inserts. after analyzing the results for both healthy and diabetic participants, certain important observations were made. first, diabetic patients have lower azp values with full insole than healthy participants for both standing and walking conditions. for the standing case, the maximum azp values were recorded at the heel region. the calculated value for healthy participants was 122.39 ±3.62 kpa (mean ±s.d.) and 67.92 ±12.45 kpa for the diabetic participants. similar to the standing case, the maximum azp values were recorded at the heel region for the walking. the azp value for healthy participants was found to be 195.16 ±3.19 kpa (mean ±s.d.) and 130.96 ±4.27 kpa for the diabetic participants. the segments of the forefoot region, i.e., fore-foot 1, fore-foot 2, and fore-foot 3, also showed comparable results. this is due to the reason that diabetes affects the stiffness of the plantar fascia of diabetic patients, making it swollen and achy, hindering them from 16 g. singh, d. bose, a. chanda placing the entire foot while standing and walking properly. second, the azp values closely resembled across all the nine off-loading cases for healthy and diabetic participants in both conditions, i.e., standing and walking, confirming the effectiveness of the modular insole. third, based on the quest results, diabetic patients found the insole as a conformable solution for their walking and pressure off-loading. this statement can also be confirmed from the off-loading values during standing and walking in nine different cases for diabetic patients. fourth, the heel region is most prone to ulceration occurrence, and the modular insole can be a preventive measure to off-load the pressure from the region, particularly for diabetic patients. fig. 7 azp values (in kpa, mean ±s.d.) for diabetic participants: (a) full insole with all inserts during standing, (b) off-loading of regions by removing their respective insert in standing position, (c) full insole with all inserts during walking, (d) off-loading of regions by removing their respective insert in walking condition 4. conclusions the novel modular insole has proved to be an effective way of off-loading the pressure from the foot's plantar fascia. it has the potential to improve the quality of an individual’s life suffering from diabetic ulcers by off-loading pressure from the plantar region of the foot, thus reducing pain and improving mobility. the findings of this study would be valuable for researchers and medical practitioners to better understand the progression of ulceration in the diabetic foot. the modular insole can help prevent the progression of ulceration, which can lead to serious complications if left untreated. development and testing of a novel diabetic orthosis for plantar pressure reduction 17 there are some limitations of the modular diabetic insole that should be acknowledged. first, it was observed from the azp values of the heel region at the ball of the foot that the heel was mostly in contact with the shoe during the heel striking of the gait phase. to address the issue, the radius of the silicone insert of the heel region could be decreased and, subsequently, increase the insert areas of heel 1, heel 2, heel 3, and heel 4. second, the insole trials (particularly walking) should be for longer duration to validate all nuances in the performance and comfort of the modular diabetic insole. in conclusion, this study will benefit diabetic patients by providing an analysis of plantar pressure, which can help prevent ulceration or re-occurrence. overall, this study emphasizes the importance of accessible and innovative solutions for managing diabetic foot ulcers. acknowledgement: gurpreet singh is grateful to the ministry of education, government of india, for awarding the prime minister’s research fellowship (ref: iitd/admission/ph.d./pmrf/202021/4062) for pursuing his doctoral research programme at iit-delhi, india. references 1. hingorani, a., lamuraglia, g.m., henke, p., meissner, m.h., loretz, l., zinszer, k.m., driver, v.r., frykberg, r., carman, t.l., marston, w., mills, j.l., murad, m. h., 2016, the management of diabetic foot: a clinical practice guideline by the society for vascular surgery in collaboration with the american podiatric medical association and the society for vascular medicine, journal of 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cross-cultural adaptation and rasch validation of the slovene version of the orthotics and prosthetics users’ survey (opus) client satisfaction with device (csd) in upper-limb prosthesis users, annals of physical and rehabilitation medicine, 62(3), pp. 168–173. 7837 facta universitatis series: mechanical engineering vol. 21, no 2, 2023, pp. 175 190 https://doi.org/10.22190/fume210822069s © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper minimization of deflection error in five axis milling of impeller blades mohsen soori, mohammed asmael department of mechanical engineering, eastern mediterranean university, famagusta, north cyprus, turkey abstract. the 5-axis cnc machine tools are used for manufacturing free form surfaces of sophisticated parts such as turbine blades, airfoils, impellers, and aircraft components. the virtual machining systems can be used in order to analyze and modify the 5-axis cnc machine tools operations. cutting forces and cutting temperatures induce deflection errors in thin-walled structures such as impeller blades through machining operations. thin-walled impeller blades' flexibility can result in machining errors such as overcutting or undercutting. so, decreasing the deflection error during machining operations of impeller blades can achieve the desired accuracy in produced parts. optimized machining parameters can be obtained to minimize the deflection of machined impeller blades. in terms of precision and efficiency enhancement in component production processes, a virtual machining system is developed to predict and minimize deflection errors of 5-axis milling operations of impeller blades. the deflection error in machined impeller blades is calculated by using finite element analysis. the optimization methodology based on the genetic algorithm is applied to minimize the deflection error of impeller blades in machining operations. to validate the integrated virtual machining system in the study, the impeller is milled by using a 5axis cnc machine tool. the cmm machine is used in order to measure and analyze deflection error in the machined impeller blades. as a result, by using the developed virtual machining system in the study, accuracy and efficiency in 5-axis milling operations of impellers can be increased. key words: deflection error, impeller, 5-axis cnc machine tools, virtual machining received august 22, 2021 / accepted november 15, 2021 corresponding author: mohsen soori department of mechanical engineering, eastern mediterranean university, famagusta, north cyprus, via mersin 10, turkey e-mail: mohsen.soori@emu.edu.tr 176 m. soori, m. asmael 1. introduction in order to increase machining performance, virtual machining systems are recently introduced to the process of part manufacturing using cnc machine tools. the systems will reach into the development process of machining operations. virtual machining systems can be built into machining process optimizations to increase efficiency in milling processes. to minimize deflection error in machining operations of thin-walled element, optimization techniques can be implemented to obtain optimized machining parameters. so, virtual machining systems can be applied to develop the part manufacturing by using cnc machine tools in order to increase added value in the processes of part production. the integral impeller and bladed disk are extensively used in various aircraft jet engines. jet engine components are produced with a high degree of precision and reliability by considering the aviation safety issues to present a good performance in working environments. machining operations of sophisticated parts with free form surfaces, such as impeller blades are implemented by using 5-axis cnc machine tools. the high precision required in 5-axis machining operations of complex parts like jet engine impeller blades is filled with complexities and complications. so, to increase accuracy and efficiency in 5-axis milling operations of free form surfaces, the process of part production is investigated by indifferent research works. cutting forces and cutting temperature induce deflection error during machining of thin-walled structures like impeller blades. thin-walled impeller blades' flexibility can result in machining errors such as overcutting or undercutting. as a result, deflection error in machined parts can cause inaccuracy, which should be analyzed and minimized. thus, a virtual machining system which can analyze and minimize deflection error in 5-axis machining operations of thin-walled impeller blades can provide an effective device to increase the accuracy and efficiency in the process of part production. zhao et al. present a method for assessing milling accuracy of thin-walled narrowvane turbine impellers made of nial-based superalloys in order to increase the precision and reliability of machined turbine impellers [1]. huang et al. developed an optimization algorithm for the cutting tool direction in order to reduce friction and deformation in ballend milling of thin-walled impeller blades [2]. chaves-jacob et al. present an efficient strategy for finishing operations in the final phase of the machining program to enhance precision in 5-axis machining operations of impeller blades [3]. huang et al. examine and reduce machining operations errors in 5-axis adaptive flank milling of thin walled impeller blades to integrate the method of machining operations of flexible components [4]. liu et al. analyze an improved technique to predict the deflection error in milling thin-walled impeller blades in order to evaluate and reduce the deflection error [5]. the application of homotopy analysis method is presented by zargar et al. to determine the thermal response of convective-radiative porous fins with temperature-dependent properties [6]. dynamical behavior of the vehicle structure is optimized by zargar et al. to obtain vehicle dynamic characteristics such as natural frequencies and mode-shapes [7]. khandagale et al. evaluate the quantitative methods of the thin-walled workpiece to minimize the deflection error in milling operations of flexible parts due to cutting forces [8]. in order to enhance machined component precision, kang and wang present an effective way of deflection error prediction in peripheral milling of thin-walled workpieces [9]. del sol et al. analyzed milling operations of light alloys with thin walled structures to eliminate deflection errors caused by instability and deformation in machined components [10]. minimization of deflection error in five axis milling of impeller blades 177 bolar and joshi investigate the impact of cutting tool variables such as tool helix angle on the precision and efficiency of machined parts in terms of precision enhancement of machined parts using milling operations [11]. to minimize deflection errors in machined components, de oliveira et al. explored the impact of machining parameters on surface roughness and shape errors in 4-axis milling of thin-walled free-form parts [12]. bolar and joshi present an efficient statistical modeling system to model and analyze cutting tool and workpiece deflection error in machining operations of low rigidity thin walled components [13]. to compensate and reduce deflection error in thin walled component during flank milling operations, li and zhu constructed an advanced cutting tool path optimization approach [14]. to improve the surface precision and reliability of machined impellers using 5-axis cnc machine tools, wang et al. present an integrated machining parameters optimization system [15]. to increase surface properties of machined thin walled parts by reducing workpiece deformation errors, ratchev et al. investigate an integrated machining parameter optimization method [16]. soori et al. expressed virtual machining systems and applications to evaluate and modify machining operations in virtual environments [17-22]. altintas and merdol also introduce a virtual machining method and application in order to obtain excellent milling conditions [23]. an innovative virtual machining system is proposed in the study to predict and minimize deflection errors in 5-axis milling operations of thin-walled impeller blades. by measuring the cutting forces and cutting temperatures at each position of the cutting tool along machining paths, a virtual machining is developed to predict and minimize the deflection error of machined thin-walled impeller blades. the deflection errors in the machined impeller blades are calculated by using finite element analysis (fem). to attain the optimized machining parameters, an optimization technique based on the genetic algorithm is performed in order to minimize the deflection error. to validate the effectiveness of the established virtual machining system in the research work, machining operations of the sample impeller blades are milled by using a 5-axis cnc milling machine tool. the machined impellers are then measured by using cmm machine in order to obtain and analyze the deflection error. thus, by using the developed methodology in the study, accuracy and efficiency in 5-axis machining operations of impellers can be enhanced. sections 2 and 3 discuss the equations for obtaining the cutting forces in milling operations of thin-walled parts, and cutting temperature calculation. section 4 describes the deflection error of thin-wall workpieces during machining operations. in section 5, the developed virtual machining system to predict and minimize the deflection error is discussed. eventually, section 6 is presented to validate the integrated virtual machining system in order to minimize the deflection error of machined impeller blades. 2. cutting force model in milling operations of thin-walled parts the cutting force theory proposed by li et al. is used to determine cutting forces in 5axis machining operations of thin walled components. [24]. a moving feed coordinate will be used to identify the cutting tool position of the (cl) as, 𝑥𝑓 = 𝑦𝑓×𝐴(𝑢) ‖𝑦𝑓×𝐴(𝑢)‖ , 𝑦𝑓 = 𝐴(𝑢)×�́�(𝑢) ‖𝐴(𝑢)×�́�(𝑢)‖ , 𝑧𝑓 = 𝐴(𝑢), (1) 178 m. soori, m. asmael where �́�(𝑢) is the feed vector at the cutting tool tip which is due to guiding curve 𝑃(𝑢). moreover, 𝑥𝑓 is described in the plane defined by the feed direction and axis vectors of cutting tool along machining paths and 𝑦𝑓 is normal to this obtained plane. also, 𝐴(𝑢) presents the unit vector along the cutting tool axis [24], 𝐴(𝑢) = 𝑄(𝑢)−𝑃(𝑢) ‖𝑄(𝑢)−𝑃(𝑢)‖ (2) the cutting entering angle can be presented as [24], 𝜃𝑒𝑛,𝑖𝑗 ’ = 𝜃𝑒𝑥,𝑖𝑗 − acos ( 𝛿𝑡,𝑖𝑗+𝛿𝑤,𝑖𝑗+𝑅𝑖𝑗.𝐶𝑜𝑠(𝜃𝑒𝑥,𝑖𝑗−𝜃𝑒𝑛,𝑖𝑗) 𝑅𝑖𝑗 ) (3) where 𝜃𝑒𝑛,𝑖𝑗 is the entry angle of the cutting tool along machining paths, 𝛿𝑡,𝑖𝑗 and 𝛿𝑤,𝑖𝑗are the deflection errors of the cutting tool and workpiece for the normal direction of the 𝑖𝑡ℎ cutter element surface on the 𝑗𝑡ℎ flute of the cutting tool, respectively. moreover, 𝑅𝑖𝑗 is the rotation radius of the 𝑖𝑡ℎcutter element on the 𝑗𝑡ℎ flute of the cutting tool. in five-axis milling operations of thin-walled components, the cutting tool is within the cutting entry angle for the 𝑖𝑡ℎcutting portion of the cutting tool as shown in fig. 1 [24]. fig. 1 the entry angle for the 𝑖𝑡ℎ cutting element of cutting tool during the five axis milling of thin walled workpiece [24] cutting tool orientation changes constantly during five axis milling process of thinwalled parts due to cutting tool paths on free form surfaces and new angles and locations of flexible parts through cutting forces of machining operations. as a consequence, the cutter runout should be taken into account when calculating cutting forces. [24]. minimization of deflection error in five axis milling of impeller blades 179 as a result, the uncut chip thickness for the 𝑖𝑡ℎcutter element of the 𝑗𝑡ℎflute by considering the 𝑘𝑡ℎ previous flute at angular position 𝜃𝑖𝑗and cutter location u can be obtained as, ℎ𝑖𝑗,𝑘(𝜃𝑖𝑗 , 𝑢) ≈ 𝑘.𝑓𝑡 ‖�́�(𝑢)‖ (�́�(𝑢) + 𝑧𝑖𝑧�́�(𝑢)) 𝑛 + 𝑅𝑖𝑗 − 𝑅𝑖(𝑗−𝐾) (4) where 𝑛 = sin 𝜃𝑖𝑗𝑥𝑓(𝑢) + cos 𝜃𝑖𝑗𝑦𝑓(𝑢) (5) and where 𝑅𝑖𝑗 is the radius of rotation for the 𝑖𝑡ℎcutting tool element on the 𝑗𝑡ℎ flute and 𝑓𝑡 is the feed per tooth for the cutting tool during machining operations. the instant uncut chip thickness is calculated by taking cutter runout into account during five axis milling operations of thin walled components is shown in fig. 2 [24]. fig. 2 the instant uncut chip thickness is calculated by taking cutter runout into account during five axis milling operations of thin-walled components[24] so, the applied cutting forces in the directions of radial, tangential and axial for the 𝑖𝑡ℎ axial disk element of the 𝑗𝑡ℎ flute of cutting tool can be obtained as [24], 𝐹𝑞,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) = 𝑔(𝜃𝑖𝑗)[𝐾𝑞𝑐ℎ𝑖𝑗(𝜃𝑖𝑗 , 𝑢)𝑏𝑖 + 𝐾𝑞𝑒 ∙ 𝑆𝑖], 𝑞 = 𝑟, 𝑡, 𝑎 (6) where the shear and edge force coefficients are presented as 𝐾𝑞𝑐(𝑞 = 𝑟, 𝑡, 𝑎) and 𝐾𝑞𝑒(𝑞 = 𝑟, 𝑡, 𝑎), respectively. the differential chip width and flute length of the 𝑖𝑡ℎcutter element is presented by 𝑏𝑖and 𝑆𝑖, respectively. also, in order to obtain the 𝑖𝑡ℎ cutter element of the 𝑗𝑡ℎ flute which is engaged in the cutting process, the 𝑔(𝜃𝑖𝑗) window function is determined as [24], 𝑔(𝜃𝑖𝑗) = { 1, 𝜃𝑒𝑛,𝑖𝑗 ≤ 𝜃𝑖𝑗 ≤ 𝜃𝑒𝑥,𝑖𝑗 0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (7) 180 m. soori, m. asmael thus, the computed cutting forces for each cutting item in the radial, tangential, and axial directions can be converted into the feed or workpiece coordinate system as [24], 𝐹𝑖𝑗(𝜃𝑖𝑗 , 𝑢) = [ 𝐹𝑥,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) 𝐹𝑦,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) 𝐹𝑧,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) ] = 𝑊𝐻𝑖𝑗 [ 𝐹𝑟,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) 𝐹𝑡,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) 𝐹𝑎,𝑖𝑗(𝜃𝑖𝑗 , 𝑢) ], (8) where 𝐻𝑖𝑗 = [ −sin (𝜃𝑖𝑗) −cos (𝜃𝑖𝑗) 0 −cos (𝜃𝑖𝑗) sin (𝜃𝑖𝑗) 0 0 0 −1 ] (9) and 𝑊 = { [ 1 0 0 0 1 0 0 0 1 ] , 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑒𝑒𝑑 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑓𝑟𝑎𝑚𝑒 [𝑥𝑓 𝑦𝑓 𝑧𝑓], 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑤𝑜𝑟𝑘𝑝𝑖𝑒𝑐𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑓𝑟𝑎𝑚𝑒 } (10) 3. cutting temperature model in machining operations, chip forming produces heat in the cutting region, raising the temperature of the cutting tool and workpiece. shaw [25] presents the cutting temperature equation in terms of the cutting parameters as, 𝜃 = 𝐶𝜃𝑉𝑏1𝑓𝑧 𝑏2𝑎𝑒 𝑏3𝑎𝑝 𝑏4 (11) where θ is the temperature of cutting process (degree celsius), 𝐶𝜃 is temperature coefficient obtained by material of workpiece, machine tool, and cutting tool geometry parameter. the 𝑏1, 𝑏2, 𝑏3, and 𝑏4 are exponents influencing the machining parameters v, 𝑓𝑧, 𝑎𝑒 and 𝑎𝑝, as cutting speed (m/minutes), feed rate (mm/z), radial feed and axial feed (mm), respectively. santos et al. [26] present formulas for cutting tool and workpiece temperatures in al alloy machining operations consisting of multiple nonlinear regression analysis. 𝜃𝑡 = 116.503 𝑉0.211𝑓𝑧 0.181𝑎𝑒 0.0464𝑎𝑝 0.00391 (12) 𝜃𝑤 = 43.319 𝑉0.365𝑓𝑧 0.244𝑎𝑒 0.0423𝑎𝑝 0.019 (13) 4. deflection error analysis of thin-wall workpiece in machining operations the volume of difference between the real machined surface and the nominal surface of the workpiece cad model is called dimensional error of machined surfaces. cutting forces and cutting temperature contribute to create the deflection error during machining operations of thin-walled components. cutting forces and temperatures cause the workpiece to deflect to a new location, resulting in dimensional error and inaccuracy during machining operations. eq. (14) can be used to represent the corresponding surface dimensional error [27], 𝑒𝑝 = 𝛿𝑡,𝑝 + 𝛿𝑓,𝑝 + 𝑅𝑁 − 𝑅𝐴 (14) minimization of deflection error in five axis milling of impeller blades 181 where, 𝛿𝑡,𝑝 and 𝛿𝑓,𝑝 are the normal projections of the cutting force and cutting temperature induced deflection error for point p, respectively. also, 𝑅𝑁 and 𝑅𝐴 are the nominal and specified radial depth of cut, respectively. 5. virtual machining system to predict and minimize the deflection error the visual basic programming language is utilized in the research work in order to develop applications of virtual machining systems to predict and minimize deflection error during milling operations of thin-walled components. the device receives the nominal machining path, the geometry and material parameters of the cutting tool, as well as the cad model of the workpiece. then, the cutting forces for each position of the cutting tool along machining paths regarding the cutting tool information and machining process parameters are calculated by using the developed virtual machining system in the study. fig. 3 shows the dialog box for the cutting force calculator. fig. 3 dialogue box of cutting force calculator furthermore, the cutting temperatures at each position of the cutting tool along machining paths are calculated by the developed software in the study. then, the obtained cutting force and temperature data are transferred to the abaqus r2016x fem analysis program, in order to measure the deflection error of machined impeller blades [28]. to calculate deflection error due to cutting force and temperature during milling operations of impellers, mesh is applied to a cad model of sample impeller blades. then, the calculated cutting forces and cutting temperature at each position of the cutting tool along machining paths are applied to each node of the meshed model of sample impeller in order to determine node displacement. as a consequence, the deflection error due to cutting forces and cutting temperatures at each position of the cutting tool along machining paths of sample impeller can be assessed and displayed. 182 m. soori, m. asmael the genetic technique is used in order to calculate the optimized machining parameters in terms of deflection error minimization. in the optimization process, the natural process of evolution and the collection of chromosomes are integrated. the binary encoding process is used to construct the initial population for the optimal solution. the fitness function is determined as shown in eq. (15) to provide measurement parameters in the optimal solution and to rank the chromosomes in the population [29]. 𝐹(𝑥) = 1 1+𝑓(𝑥) (15) f(x) and f (x) denote the fitness and objective functions, respectively. the algorithms' key operators are regeneration, crossover, and mutation. the operator’s reproduction, crossover, and mutation are added to the original population of the optimization problem in order to create a quicker migration to the optimal machining parameters. fig. 4 displays the flowchart of virtual machining system in order to calculate cutting force and temperature and obtain the deflection error of machined parts. fig. 4 the flowchart of virtual machining system to calculate cutting forces, cutting temperature and deflection error eq. (16) is presented to calculate the surface roughness of machined parts using mathematical theorems [30]. 𝑅𝑎 = 318 𝑓2 4𝑑 (16) minimization of deflection error in five axis milling of impeller blades 183 where f is the feed rate and d is the cutting tool diameter. furthermore, in order to reduce the cost of machined parts, the machining process time should be reduced. eq. (17) shows the mathematical equation for machining time. [30]. 𝑡𝑚 = 𝑘 𝑓 (17) where k is the cutting tool's distance from the operating region, and f is the feed rate. in order to reduce the cost of machining, cutting tool life should also be maximized during the optimization phase. eq. (18) can be used to represent the cutting tool life. [30]. 𝑇𝐿 = ( 60 𝑄 ) [ 𝐶( 𝐺 5 ) 𝑉(𝐴)𝑤] 1 𝑚 (18) where q is the cutting edge's contact ratio with the workpiece per rotation, c is 33.98 for hss tools and 100.05 for carbide tools, g=0.14, v is cutting speed (mm/minutes), w=0.28, m is 0.15 for hss tools and 0.30 for carbide tools, g and a are the slenderness ratio and chip cross-section, which can be seen as eqs. (19) and (20), respectively [30]. 𝐺 = 𝑎 𝑓 (19) 𝐴 = 𝑎 ∙ 𝑓 (20) where 𝑎 is depth of cut and 𝑓 is feed rate in machining operations. the cutting temperature can be decreased by lowering the feed rate and spindle speed, according to eqs. (12) and (13). also, according to eq. (16), by reducing the feed rate, the surface roughness can be decreased. however, according to eq. (17), reducing the feed rate can increase the machining time. furthermore, according to eqs. (8) and (18), increasing the spindle speed can decrease cutting forces as well as cutting tool life during machining operations. it is clear that the mathematical models of cutting forces, cutting temperature, machining time, surface roughness, feed rate, and cutting tool life presented here are interconnected and should be treated as an optimization problem. as a result, to determine the optimized feed rate and spindle speed in terms of deflection error minimization during machining operations, an optimization technique of machining parameters should be implemented. in order to minimize the deflection error, the matlab programming language is used to obtain the optimized machining parameters. thus, optimized feed rate and spindle speed are obtained in milling operations of thin-walled impeller blades in order to minimize deflection errors. the objective function of the optimization process is the minimization of deflection error by obtaining the optimized machining parameters as spindle speed and feed rate along machining paths of impeller blade. so, accuracy as well as efficiency in machining operations of impeller blades can be increased. the method of machining parameter optimization is shown in fig. 5, while the flowchart of the study is shown in fig. 6. 184 m. soori, m. asmael fig. 5 the method of machining parameter optimization fig. 6 the flowchart of the study minimization of deflection error in five axis milling of impeller blades 185 6. validation in order to validate the proposed virtual machining system in the study, the sample impeller is machined by using a 5-axis cnc milling machine tool kondia hm 1060. the material of sample impeller is al 7075. the dimensions of the sample impeller in millimeter unit are shown in the fig. 7. fig. 8 shows the cutting tool path and machining procedures of sample impeller. fig. 7 dimensions of sample impeller in millimeter unit the mastercam software [31] is used to generate cutting tool paths and strategies in machining operations of sample impeller by using a 5-axis cnc milling machine tool. fig. 8 the cutting tool path and machining procedures of sample impeller the 5-axis kondia hm 1060 cnc machine tool is then used in milling operations of test impellers. the cutting tool in the experiment is carbide end mill with 8 mm diameter, 30° helix angle, flute number 4, total length 60 mm, and flute length 35 mm. the feed rate is 200 mm/min and the spindle speed is 200 m/min. fig. 9 shows the 5-axis milling operation of the test impeller. 186 m. soori, m. asmael fig. 9 the 5-axis milling operation of test impeller fig. 10 shows a real al impeller which has been machined. fig. 10 the real machined al impeller the cmm machine is used in order to measure the deflection error in the machined impeller blades. the renishaw rsh 250 probe is used [32] to calculate the deflection error of a machined impeller blades with a 1 𝜇𝑚 repeatability in touching directions. fig. 11 shows the process of measuring the machined impeller by using the cmm machine. fig. 11 the measuring process of the machined impeller blades using the cmm machine minimization of deflection error in five axis milling of impeller blades 187 fig. 12 shows the procedure of impeller blade measuring by using the cmm machine. fig. 12 procedure of impeller blade measuring by using the cmm this study uses the cutting force model of 5-axis cnc machine tools in machining operations of thin-walled parts presented by li et al. [24]in order to calculate cutting forces in virtual environment. the kistler dynamometer type 9129aa is used to determine the coefficients of the basic cutting force for milling operations of thin-walled blades by using 5-axis cnc milling machine tool kondia hm 1060. the feed per tooth and feed rate are 0.5 mm and 100 mm/min, respectively. the spindle rotates at 3000 rpm. as a consequence, the specific cutting force coefficients are obtained as table 1. table 1 the specific cutting force coefficients the specific cutting coefficients krc (n/𝑚𝑚2) kre (n/mm) ktc (n/𝑚𝑚2) kte (n/mm) kac ((n/𝑚𝑚2) kae (n/mm) 406.1 17.68 810.42 16.93 241.28 0.74 the deflection errors of the impeller blade due to cutting forces and temperature are measured by using the abaqus software [28]. as a result, fig. 13 shows the fem-calculated deflection error of a machined thin-walled impeller blade. fig. 13 the fem-calculated deflection error of a machined thin-walled impeller blade 188 m. soori, m. asmael thus, the measured and fem simulated deflection errors in the machined impeller blade are obtained as shown in fig. 14. fig. 14 the measured and predicted deflection errors in the machined impeller blade when the experimental test results are compared to the expected deflection error of the machined thin-walled impeller, a good compatibility is obtained. the developed optimization techniques based on the genetic algorithm in the study are used to measure optimized machining parameters. the population size of 39 is chosen during the optimization process, which is iterated for 306 generations. crossover probability of 0.68 and mutation probability of 0.001 are also chosen. as a result, the feed rate is 170 mm/min, and the spindle speed is 260 m/min. fig. 15 shows the measured and fem simulated deflection errors in the machined impeller blade by using optimized machining parameters. fig. 15 the measured and predicted deflection errors of machined impeller blade by using optimized machining parameters minimization of deflection error in five axis milling of impeller blades 189 7. conclusions application of virtual machining system is developed in the study to predict and minimize the deflection errors in 5-axis milling operations of thin-walled impellers. as a result, by using the cad model of the workpiece, cutting conditions and cutting tool geometries in the developed virtual machining system, cutting forces and temperatures at each location of the cutting tool along the machining path are determined. to calculate the deflection error in a thin-walled workpiece, the finite element method is used. a real impeller is milled by using a 5-axis cnc machine tool in order to validate the integrated virtual machining system in the study. the machined impeller blades are measured by using a cmm machine in order to obtain the deflection error. the obtained results from the virtual machining system and experimental tests are evaluated in order to determine the reliability rate in the established system. in comparison to the experimental and virtual machining system results, there is a 91.6 percent compatibility. optimized machining parameters are obtained by using the genetic algorithm to minimize the deflection error in 5-axis machining operations of impeller blades. the optimized machining parameters resulted in a 24.4 percent decrease in deflection error of the machined impeller blade. as a result, the developed virtual machining system in the study can minimize the deflection error in 5-axis milling operations of thin-walled impeller blades by using the optimized machining parameters. it is also concluded that using optimization techniques in virtual machining systems will improve the accuracy as well as reliability of machined parts. moreover, the developed methodology in the study can be used for increasing the accuracy of 5-axis cnc milling operations of turbine blades by minimizing the cutting tool and workpiece deflection errors. also, deformation 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644-655. 30. tolouei-rad, m., bidhendi, i., 1997, on the optimization of machining parameters for milling operations, international journal of machine tools and manufacture, 37(1), pp. 1-16. 31. about mastercam software. https://www.mastercam.com/. accessed 06 may 2020. 32. about the renishaw rsh 250 probe. https://www.renishaw.com/cmmsupport/knowledgebase/en/rsp2--22120. accessed august 11 2020. 7431 facta universitatis series: mechanical engineering vol. 23, no 1, 2025, pp. 95 108 https://doi.org/10.22190/fume211208025t © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper development of a procedure for increasing the accuracy of the reconstruction and triangulation process of the cranial vault geometry for additive manufacturing paweł turek, grzegorz budzik faculty of mechanical engineering and aeronautics, rzeszów university of technology, poland orcid ids: paweł turek https://orcid.org/0000-0002-5926-4815 grzegorz budzik https://orcid.org/0000-0003-3598-2860 abstract. ct scanners installed in clinics used different slice thicknesses, which usually produce data with an anisotropic structure of voxels. the low visual quality results are due to the discontinuous interpolation between neighboring voxels, resulting in a very “blocky” appearance of the reconstructed surfaces (stair-step artifact). this structure can also directly affect the volume, geometry, and linear accuracy of digital and physical 3d models. the article presents a method that improves the design of cranial vault models for additive manufacturing after the staircase artifact has occurred. the research was performed on 14 different patients (seven males and seven females). changing the slice thickness from 2.4 mm to 4.8 mm generated over 90% errors in reconstructing the cranial vault area in the range of 0.830 mm +/1.364 mm (mean deviation +/expanded uncertainty) for males and 0.780 mm +/1.338 mm for females. to increase the spatial resolution of the digital imaging data, an interpolation process was performed on 2d radiographic images. after using the data interpolation procedure (lanczos filter), deviations were mainly in the range of 0.465 mm +/1.038 mm for males and 0.328 mm +/0.842 mm for females. the last stage of the improved process involved mesh optimization. utilizing laplacian smoothing surface and isotropic polygonal remesh, this procedure decreased global error, especially in regions with high curvatures. over 90% of the analyzed points after using the lanczos filter and optimization mesh procedure are within the range of 0.338 mm +/1.014 mm for males and 0.301 mm +/0.806 mm for females. key words: reverse engineering, triangulation, cranial vault, accuracy, interpolation, additive manufacturing received: december 08, 2021 / accepted may 12, 2022 corresponding author: paweł turek faculty of mechanical engineering and aeronautics, rzeszów university of technology, al. powstańców warszawy 8, 35-959 rzeszów, poland e-mail: pturek@prz.edu.pl https://orcid.org/0000-0002-5926-4815 https://orcid.org/0000-0003-3598-2860 96 p. turek, g. budzik 1. introduction traditional modeling of elements and parts of machines is carried out using computeraided systems, which are presently widely used in designing and manufacturing industrial products [1-3]. the accuracy of the manufactured models is verified using contact [4, 5] and optical measurement systems [6-8]. thanks to the development of coordinate measuring systems [9, 10], data processing systems [11], and modern manufacturing techniques [12, 13], it is possible to obtain the geometry of any object which does not have technical documentation [10, 11, 14], including a geometric model which matches the anatomy of soft or hard tissues [14-16]. the geometry of a piece of tissue or an entire organ is used in tissue engineering [17], surgery planning [18], surgical templates [19], and the manufacturing of implants and prostheses [19-21]. models of anatomical structures are mainly used in orthopedics [22, 23], maxillofacial surgery[18-20], and cranioplasty [24, 25].the leading causes of bone defects in the cranial vault area are swelling of the skull bones, infection of the bone flap, and decompression operations in the case of uncontrolled swelling of the brain substance. to avoid the risk of postoperative complications, it is necessary to accurately design [26-28] and manufacture the implant or bone defect [24, 25]. modern implants for cranioplasty must meet several requirements, including biocompatibility, and sterility. each stage of the reconstruction affects an object's dimensional and geometric accuracy. however, the contribution of each step to the final error remains unclear. the selection of the ct scanner type [14, 29], measurement parameters (e.g., tube potential or current) [29, 30], and image reconstruction parameters (e.g., voxel size, slice thickness, or convolution kernel) [29-32] plays an essential role in the further modeling process based on digital imaging data (dicom). then, a chosen anatomical structure is separated from the dicom data by applying various segmentation methods, based mainly on detecting edges and identifying image areas with some standard features [31, 33]. computer systems can visualize volumetric data in a three-dimensional geometry using, e.g., the marching cube (mc) and delaunay algorithm [14]. final models are mainly manufactured using additive techniques [16, 34, 35]. due to the difficulties resulting from the estimation of errors at the acquisition and reconstruction stages of the geometry of anatomical structures from dicom data, scientists are currently trying to conduct several studies in this area [31-33, 36]. the most common assessment is the deviation in linear dimensions. the process is carried out using a caliper [37], measuring arm [38], or coordinate measuring machine [39] on the natural bone structure provided by medical institutions. then, the realistic bone model is digitalized using the ct systems [37, 39]. based on the obtained results, the change in linear dimension is assessed, which occurred at the acquisition and reconstruction stages of the geometry, using a tomographic diagnostic system to the data obtained on the same research model using a caliper, coordinate measuring machine, or measuring arm. in assessing 3d geometry errors related to the acquisition data, geometry reconstruction, and manufacturing stage, a structural light scanner was also used [40]. a challenging task is to improve the accuracy of digital model geometry at the stage of acquisition and data processing. it is impossible to obtain the highest spatial resolution during ct measurements owing to the need to protect the patient's health. low spatial resolution is related to data characterizing a voxel where the slice thickness is incomparably more significant than the pixel dimensions [29, 30]. during data analysis characterized by irregular voxel dimensions (a partial volume effect artifact is created), there is a limited development of a procedure for increasing the accuracy of the reconstruction and triangulation 97 possibility of successfully segmenting the data. the poor quality of mapping the geometry is due to the lack of continuity of data interpolation, which in turn generates the block structure of the standard tessellation language (stl) model (which produces the so-called stair-step artifact) [14, 32]. this artifact can directly affect the volume, geometry, and linear accuracy of digital and physical 3d models. the anisotropic nature of many datasets can be a severe quality issue for image analysis and visualization techniques. in addition to this, reconstruction algorithms such as an mc or delaunay have their limitations. the generated triangle mesh based on algorithms contains many errors, which are related, among others, to inverted normal vectors, twisted surfaces, or overlapping triangles. these reconstructed geometries require additional mesh editing and optimizing [41-43]. the skull is a significant bone tissue, being a natural shield for the delicate structure of the brain. from an anatomical point of view, there is a discernible difference between the male and female skulls. the forehead is slightly sloping and receding in males, while in females, the forehead is vertical. additionally, the cranial vault is more rounded in males, while the cranial vault is flattened in females [44]. the differences in the anatomical structure of the male and female skulls are most noticed in the sagittal and coronal planes. unfortunately, partial volume effect artifacts in 2d images significantly hinder the correct segmentation of geometry outlines in the presented planes [45]. imaging thick ct layers can produce a stair-step artifact on the 3d model. working on 3d models of male or female skulls containing geometry errors, the process of cad modeling of the skull defect is significantly more difficult. this aspect results from the difficulty of correctly reconstructing the realistic outline of the implant and the problem of fitting it to the entire skull [46]. currently, no methods of improving the accuracy of the reconstruction and triangulation process of cranial vault geometry after stair-step artifact has occurred are available. the article presents dicom data processing and geometry reconstruction methods that may enhance the design of cranial vault models and extend the methodology of analytical accuracy determination at the data processing stage. 2. method the research was performed on 14 different patients. in this group, there were seven males aged from 41 to 72 years old and seven females aged from 34 to 69 years old. dicom data were obtained for all patients on the siemens somatom sensation open 40 scanner installed in the regional clinical hospital no. 1 at the frederic chopin in rzeszow, with a commonly used scanning protocol for the cranial area. patients were scanned twice using different slice thicknesses to carry out the research process (tab. 1) the increase in the layer thickness influenced the creation of partial-volume effect artifacts in the 2d image data. reconstruction geometry was performed in the amira software. in the segmentation process of the cranial vault, a lower threshold of 200 hu was used. the final surface was saved in stl format (fig. 1). 98 p. turek, g. budzik table 1 scanning protocol name of parameters somatom sensation 40 (sequence mode) value of parameters (first measurements) value of parameters (second measurements) kv 120 120 mas 380 380 rotation time 1 second 1 second acquisition 24 × 1.2 mm 24 × 1.2 mm slice collimation 1.2 mm 1.2 mm kernel h31s h31s matrix size 512 × 512 512 × 512 pixel size 0.4 mm × 0.4 mm 0.4 mm × 0.4 mm slice thickness 2.4 mm 4.8 mm to increase the spatial resolution of the dicom data, and to limit the influence of the artifact partial volume effect, an interpolation process was applied to the traverse plane performed for the second measurement. this is a process of creating a new synthetic pixel from adjacent pixels so that it is optically best-matched to the transformed image. the interpolation process is often included to convert data to an isotropic grid. fig. 1 the surface representing cranial models of 14 patients there are many algorithms available that can accomplish the interpolation [47-49], but they differ in their quality and computational effort. however, the current research shows that the lanczos filter is more effective than other methods in minimizing the development of a procedure for increasing the accuracy of the reconstruction and triangulation 99 impact of the stair-step artifact [50, 51]. the algorithm considers neighboring points in 4 x 4, 6 x 6, and 8 x 8 blocks. lanczos convolution kernel l(x) is defined as: sin ( )sin ( / ) , ( ) 0 . c x c x a if a x a l x otherwise −   =   (1) the parameter is a positive integer, usually 2 or 3, that specifies the kernel size. the lanczos kernel has 2 to 1 lobe: a positive one in the center and a 1 alternating positive and negative cams on each side. this filter has excellent characteristics. when correctly applied, it yields a perfect equilibrium between detail preservation (sharpness) and smoothness. this method produces slight overshoot, high edge sharpness, good signal continuity in the smooth region, and decreases artifact partial-volume effect. the interpolation procedure using the lanczos filter did improve the accuracy of reconstruction to reformat a voxel’s structure from 0.4 mm × 0.4 mm × 4.8 mm. to the iso-voxel 0.4 mm × 0.4 mm × 0.4 mm using (fig. 2). then, the dicom data were subjected to data segmentation and geometry reconstruction under the same methods and parameters as the non-processed models. fig. 2 the view on 2d image of the first patient, voxel size: 0.4 mm × 0.4 mm × 4.8 mm (left), and improved using the lanczos filter (right) moreover, the process involved mesh optimization. the mc algorithm generates holes, aliases, saw-toothed paths within the voxel connection, and increased global error, especially in the regions with high curvatures (fig. 3a). using variable-density triangular meshes, the optimization procedure was performed (fig. 3b). this procedure involves two steps: ▪ laplacian smoothing surface by shifting its vertices. each synthetic vertex is shifted to the average position of its neighbors. the laplace function is the sum of the squares of the lengths of the edges having a common node: ( )2 2 1 ( , ) ( ) ( ) k i i i f x y x x y y = = − + − (2) where k is the number of neighbors nodes. node optimization is performed for each mesh node. the new node coordinates (x', y') are calculated from the formulas:  = = k i ix k x 1 1 '  = = k i iy k y 1 1 ' (3) 100 p. turek, g. budzik ▪ remesh the surface by creating small dense triangles in the high curvature region and large sparse triangles in the low curvature region in regard to the coronal and sagittal plane by using an isotropic vertex placement algorithm. a) b) fig. 3 the optimization procedure of stl mesh, a) before, b) after in the process of assessing the accuracy of reconstruction (first step) and triangulation (second step), a model with a voxel structure of 0.4 mm × 0.4 mm × 2.4 mm (first measurements) was selected as the reference model for comparison and the model accuracy assessment (fig. 4). this model was chosen because it obtained a higher spatial resolution than the model with a voxel structure of 0.4 mm × 0.4 mm × 4.8 mm. fig. 4 the comparison process development of a procedure for increasing the accuracy of the reconstruction and triangulation 101 the comparison process was performed using the best-fit method. this algorithm is most appropriate for analyzing deviations in the models with complex shapes. a best-fit alignment is an iterative process that minimizes the square of the distance between the nominal and measured data to converge on a solution. adjustment of point clouds is made using the best-fit to an accuracy of 0.001 mm. 3. results and discussion to examine the accuracy of geometry reconstruction, the models of the 14 cranial vaults with a voxel size of 0.4 mm × 0.4 mm × 4.8 mm were compared and improved using a lanczos filter (first step) and triangulation procedure (second step) with the same cranial vault based on modeling with a voxel size of 0.4 mm × 0.4 mm × 2.4 mm (the gold standard). the statistical parameters and distributions of the reconstructed models from dicom data are presented in figs. 5-7, tab. 2 and 3. in the case of averaged results presented in figs. 5a, 6a, and 7a, the artifact partialvolume effect significantly increases the positive deviations to the gold standard model. the poor quality of geometry reconstruction using the mc algorithm is due to the lack of continuity of data interpolation, which generates the model's block structure (stair-step artifact). for an averaged seven male cranial vaults, the deviations are mainly in the range between 0.357 mm and 0.714 mm, and for seven females, in the range between 0.714 mm and 1.071 mm. the averaged results for 14 patients confirm that more than half of the analyzed deviations range from 0.357 mm to 1.071 mm. there are also places where negative deviations can be observed (for males – 15 % and for females 8 % of analyzed points). over 90 % of the analyzed points are within the range of 0.830 mm +/1.364 mm for males and 0.780 mm +/1.338 mm for females. in the case of skew and kurtosis, all distribution is more peaked than a gaussian distribution (leptokurtic distribution) and characterized by medium asymmetry (value close to 0.5). considering the averaged results obtained after using the lanczos filter, an increase in the accuracy of the cranial vault was observed (fig. 5b, fig. 6b, and fig. 7b). the artifact's partial-volume effect has been minimized (fig. 2). by improving the spatial resolution of the dicom data, an increase in the number of points representing the geometry of the cranial vault was observed. the concentration of deviations has changed. for male and female cranial vaults, the deviations are mainly between 0 mm and 0.357 mm values, respectively 26.80 % and 38.10 %. however, there has been an increase in negative deviations, mainly in the range between -0.357 mm and 0 mm. over 90 % of the analyzed points are within the range of 0.465 mm +/1.038 mm for males and 0.328 mm +/0.842 mm for females. in the case of skew and kurtosis, the parameter values were decreased. all distributions are leptokurtic, characterized by small (for males) and medium (for females) asymmetry. the obtained averaged statistical parameters for 14 patients are consistent with the average results for males and females. after optimizing the triangle mesh, the errors of the structure, which resulted from the use of the mc algorithm, were removed (fig. 3). additionally, the number of points representing the geometry of the cranial vault was reduced, which was doubled due to the lanczos interpolation method (fig. 5c, fig. 6c, and fig. 7c). 102 p. turek, g. budzik fig. 5 histogram representing the average results of the 7th males, a) not improved, b) improved using interpolation, c) improved using the interpolation and triangulation procedure development of a procedure for increasing the accuracy of the reconstruction and triangulation 103 fig. 6 histogram representing the average results of the 7th females, a) not improved, b) improved using the interpolation, c) improved using the interpolation and triangulation procedure 104 p. turek, g. budzik fig. 7 histogram representing the average results of the 14th patients, a) not improved, b) improved using interpolation, c) improved using the interpolation and triangulation procedure development of a procedure for increasing the accuracy of the reconstruction and triangulation 105 table 2 statistical results of not improved models parameters not improved male female average number of points 233335 220757 227046 mean deviation [mm] 0.830 0.780 0.805 standard deviation [mm] 0.682 0.669 0.676 skewness 0.335 0.472 0.404 kurtosis 4.709 4.559 4.634 after using the mesh optimization process, the obtained number of points is similar to the one that defines the model not subjected to any optimization procedure. there was an increase in points within a tolerance of +/0.350 mm. for males, the percentage of points within the tolerance of +/0.350 mm is approximately 56%, and for females, about 62%. compared to the models not subjected to the procedure, the males were around 22% and females around 12 % (fig. 5a, fig. 6a, and fig. 7a). over 90% of the analyzed points after using lanczos filter and optimization mash procedure are within the range of 0.338 mm +/1.014 mm for males and 0.301 mm +/0.806 mm for females. table 3 statistical results of improved models parameters improved using interpolation improved using interpolation and triangulation male female average male female average number of points 380345 406661 393503 207527 295238 251383 mean deviation [mm] 0.465 0.328 0.397 0.338 0.301 0.320 standard deviation [mm] 0.519 0.421 0.470 0.507 0.403 0.455 skewness 0.235 0.407 0.321 0.206 0.210 0.208 kurtosis 3.802 4.201 4.002 4.432 4.060 4.246 4. conclusion the influence of the lanczos filter on the accuracy of reconstruction cranium geometry is very similar for the 14 presented patients (seven males and seven females). when using the lanczos filter, one may observe an improvement in the accuracy of the reconstruction of the cranial vault geometry. also, the time of segmentation was shortened. the presented optimization mesh procedure adopted the resolution of triangles according to the local curvature. large triangles are used to define planar regions in this procedure whereas small triangles are used for curved areas. thanks to this, the process of generating a triangle mesh has been accelerated, without 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1(1), pp. 393-420. 49. lehmann, t.m., gonner, c., spitzer, k., 1999), survey: interpolation methods in medical image processing, ieee transactions on medical imaging, 18(11), pp. 1049-1075. 50. budzik, g., turek, p., 2020, the impact of use different type of image interpolation methods on the accuracy of the reconstruction of skull anatomical model, biomedical engineering: applications, basis and communications, 32(01), 2050008. 51. budzik, g., turek, p., 2018, improved accuracy of mandible geometry reconstruction at the stage of data processing and modeling, australasian physical & engineering sciences in medicine, 41(3), pp. 687-695. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 595 610 https://doi.org/10.22190/fume200416041b © 2020 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper mathematical modeling of the influence parameters during formation and propagation of the lüders bands tin brlić1, stoja rešković1, zoran jurković2, gordan janeš3 1university of zagreb, faculty of metallurgy, croatia 2university of rijeka, faculty of engineering, croatia 3university of rijeka, center for advanced computing and modeling, croatia abstract. in this study, an analysis of the influence parameters measured by the static tensile test, thermography and digital image correlation was performed during formation and propagation of the lüders bands. a new approach to the prediction of stresses, maximum temperature changes and strains during the lüders band formation and propagation is proposed in this paper. application of the obtained mathematical models of influence parameters gives a clear insight into the behavior of niobium microalloyed steel at the beginning of the plastic flow, which can improve product quality and reduce costs during the forming of microalloyed steels with the appearance of the lüders bands. the obtained models of influential parameters during formation and propagation of the lüders bands have been developed by the regression analysis method. the proposed mathematical models showed low deviations of calculated results ranging from 1.34% to 12.37%. the local stress amounts, important in the forming of microalloyed steels since indicating surface roughness and plastic flow possibilities during the lüders band propagation, are obtained by the mathematical model. it was found that stress amounts increase during the lüders band propagation in the area behind the lüders band front. the difference in stress amount between the start of the lüders band propagation and advanced lüders band propagation is 25.53 mpa. key words: lüders bands, regression analysis, microalloyed steel, thermography, digital image correlation received april 16, 2020 / accepted september 02, 2020 corresponding author: tin brlić university of zagreb faculty of metallurgy, aleja narodnih heroja 3, 44 000 sisak, croatia e-mail: tbrlic@simet.unizg.hr 596 t. brlić, s. rešković, z. jurković, g. janeš 1. introduction mathematical modeling [1] and optimization of process parameters [2] are often used for improving the quality of products as well as reducing costs of the production process for various metallic materials. the appearance of the lüders bands presents a problem for the product quality during the forming process [3]. consequently, mathematical modeling was used in order to predict behavior of the lüders bands during cold deformation of metallic materials [4]. it is well known that niobium microalloyed steels show inhomogeneous deformations, i.e. lüders bands, at the start of plastic flow during cold deformation [5], which can be a significant problem during the deformation process of microalloyed steels. it is possible to improve product quality and reduce production process costs by developing mathematical models of influential parameters. given that, it is not necessary to carry out additional experimental tests that require additional costs since the values of significant parameters obtained by the mathematical models can be predicted. the formation and propagation of the lüders bands in metallic materials showed a difference in the values of influential parameters of stress, temperature changes [6] and strains behind the lüders band front [7]. since changes of influence parameters can be more or less pronounced, it is important to analyze them in detail and describe with mathematical models in order to give a clear insight of value changes of the parameters during the formation and propagation of the lüders bands. static tensile test, thermography and digital image correlation are confirmed as reliable methods for determining the stress, temperature changes and strain amounts [8] during the formation and propagation of the lüders bands. it is well known that stress values can be measured from the force-elongation diagram by static tensile test. values of temperature changes and their distribution in lüders bands can be determined with high accuracy using thermography. strain values and their distribution can be determined by digital image correlation during static tensile testing. therefore, it can be concluded that using thermography and digital image correlation can determine temperature changes and strain values which can be compared with the stress values obtained by the static tensile test. previous research has shown that mathematical modeling is often used to describe processes during deformation under various metal forming technologies [9]. bhirud and gawande [10] used the regression analysis method to determine the mathematical model for temperature increase of work piece during final milling. results of the obtained mathematical model are compared with the measured results of temperature increase of the work piece and a confirmation test is conducted for the validation of the predicted results of temperature increase. the regression analysis method proved to be suitable for a better insight into the tensile behavior of aisi 316 stainless steel during which temperature change and strain rate have been associated by corresponding empirical equations [11]. nasri et al. [12] have shown usability of the multiple regression method during sheet metal forming formability for comparison between experimental results and model predictions. jurković et al. [13] conducted the tests of rolling steel sheets where the output parameter, forming force-roll load, are mathematically modeled by the regression analysis of various input parameters such as tensile strength, sheet thickness and sheet width where regression analysis proved to be a reliable and accurate method. the research [14] confirms the regression analysis method as a very good method for obtaining mathematical models in determining the influence of cutting speed, feed rate and depth of cut on cutting tool vibration in different directions on carbon steel. the obtained regression mathematical models for vibration of cutting tool with machining parameters have shown a very good match between the results predicted from the developed models and the experimental ones. other studies [15] mathematical modeling of the influence parameters during formation and propagation of the lüders... 597 confirmed regression analysis as a reliable method for obtaining appropriate mathematical models of selected output parameters. the significance of the process parameter influence of spindle rotation speed, feed speed and rolling-beating mode during cold rolling-beating forming process on astm 1045 steel for obtaining mathematical models of forming forces and forming qualities objectives by regression analysis is determined in [16]. mathematical multiple linear regression analysis of mechanical properties of potential weld joints, such as yield strength, ultimate tensile strength and percentage elongation, on aluminum alloy aa 6061 during manufacturing conditions is conducted in [17]. models have proven to be very accurate in prediction of mechanical properties which can greatly assist engineers in proper planning and simultaneously improve efficiency of aluminum alloy welding operations. the phenomenon of the lüders bands tends to be mathematically described for different steels [18]. studies [4, 19] used developed models by regression analysis for different influencing parameters in low carbon steel with lüders bands. geometrical properties such as length, width and depth of the lüders bands were examined which appeared on the tinplates during the cold stamping process. the predicted model results of influencing parameters showed good agreement with the measured results obtained by experimental tests. the tendency of sadowski et al. [20] to model the phenomenon of lüders bands by regression analysis is manifested where they sought to model different yield plateau gradients for different structural carbon steels. rešković et al. [21] used factorial experiment with repeated measurements to determine influence of different factors at the proportionality limit after which lüders bands appear in niobium microalloyed steel during cold deformation. they concluded that the deformation degree has the greatest influence on the proportionality limit during cold drawing of niobium microalloyed steel tubes. in the local area of maximum temperature changes and maximum strains behind the lüders front detailed influence of individual influencing parameters is not mathematically described. therefore, the aim of this paper is to develop mathematical models for influential parameters δtmax (maximum temperature change), σ (stress) and ɛmax (maximum strain) to predict the behavior of niobium microalloyed steel in the area behind the lüders front which shows the greatest temperature and strain changes during cold deformation. 2. experimental work measured experimental results of influencing parameters during the formation and propagation of the lüders bands were obtained on microalloyed steel with 0.048% nb. the chemical composition of niobium microalloyed steel is shown in table 1. influential parameter of the stress was obtained by static tensile testing at the stretching rates of 5 mm/min, 20 mm/min and 50 mm/min corresponding to strain rates of 0.0018 s-1, 0.007 s-1 and 0.0185 s-1. static tensile tests were carried out on the tensile machine wpm eu 40 mod with a nominal force value 400kn. determination of the local maximum temperature changes and maximum strain values was carried out by infrared camera jenoptik variocam®m82910 and digital camera blackfly s color (flir). values of influential parameters were measured using the aforementioned research methods during formation and propagation of the lüders bands. the parameters of maximum localized temperature change (δtmax), maximum strains (ɛmax) and stress values (σ) were measured using thermography, digital image correlation and the static tensile test. 598 t. brlić, s. rešković, z. jurković, g. janeš in addition to the influential parameters, different strain rates during static tensile testing were also considered. the measurements of parameters δtmax and ɛmax were performed in the area behind the lüders band front during formation and propagation of the lüders bands since the largest changes of the analyzed parameters were measured in this area. other parameters during the experimental tests that may have influenced the accuracy of the mathematical model were kept constant during the static tensile test. the levels of the minimum and maximum values of the input parameters were determined during formation and propagation of the lüders bands when determining σ as output parameter, table 2, and δtmax, ɛmax as output parameters in tables 3 and 4. 2.1 regression analysis method this paper presents modeling of output parameters σ, ɛmax and δtmax using a mathematical statistical method of regression analysis. it is possible to define the relationship between the output effects of process (y) and input variables (xi) by regression analysis. mathematical modeling of influential parameters during the formation and propagation of the lüders bands by the regression analysis is performed by the second order polynomial model using eq. (1): y=b0x0+b1x1+b2x2+b3x3+b11x1 2+b22x2 2+b33x3 2+ b12x1x2+b13x1x3+b23x2x3+b123x1x2x 3 (1) where xi are the variable parameters of process i= 0, 1, 2,..., k and bi are the model coefficients. table 1 chemical composition of niobium microalloyed steel (wt%) element c mn si p s al nb n nb steel 0.12 0.78 0.18 0.011 0.018 0.020 0.048 0.0080 table 2 physical values of input parameters for determining output parameter σ during the formation and propagation of the lüders bands at v = 0.007 s-1 parameters minimum value mean value maximum value formation of the lüders band δtmax (°c) -0.47 1.26 2.05 ɛmax (mm/mm) 0.0072 0.0204 0.0336 propagation of the lüders band δtmax (°c) 1.37 2.64 3.91 ɛmax (mm/mm) 0.0300 0.0396 0.0492 table 3 physical values of input parameters for determining output parameter ɛmax during the formation and propagation of the lüders bands parameters minimum value mean value maximum value formation of the lüders band δtmax (°c) -0.59 0.98 2.55 σ (mpa) 484.70 524.60 564.50 v (s-1) 0.00180 0.01015 0.01850 propagation of the lüders band δtmax (°c) 0.350 2.285 4.220 σ (mpa) 498.30 529.95 561.60 v (s-1) 0.00180 0.01015 0.01850 mathematical modeling of the influence parameters during formation and propagation of the lüders... 599 table 4 physical values of input parameters for determining output parameter δtmax during the lüders bands propagation parameters minimum value mean value maximum value ɛmax (mm/mm) 0.02790 0.03855 0.04920 σ (mpa) 498.30 529.95 561.60 v (s-1) 0.00180 0.01015 0.01850 multiregression coefficient, r, determined by eq. (2) is very important for the quality of the mathematical models obtained by the regression analysis method: r=√1∑ (yj e -yj r ) 2n j=1 ∑ (yj e -y̅j e ) 2 n j=1 (2) where 𝑌𝑗 𝐸 are the experimental values, 𝑌𝑗 𝑅 the calculated values and �̅�𝐸 = ∑ 𝑌𝑗 𝐸𝑁 𝑗=1 𝑁 the arithmetic mean of all experimental results. determination coefficient, r2, is determined by calculated multiregression coefficient r. the general form of a second order polynomial model with interacting factors will take the following forms with respect to the parameters analyzed in this paper. model for output parameter σ: ▪ for the formation and propagation of the lüders band at v = 0.007 s-1: y (σ)=b0+b1∆tmax+b2εmax+b11∆tmax 2 +b22ɛmax 2 +b12∆tmaxεmax (3) the general form of a mathematical model for output parameters δtmax and ɛmax denoted by z: ▪ for the formation and propagation of the lüders band: y(z)=b0+b1z+b2σ+b3v+b11zmax 2 +b22σ2+b33v2+ b12zσ+b13zv+b23σv+b123zσv (4) 3. results and discussion mathematical modeling of influence parameters (ɛmax, δtmax, σ) using regression analysis was performed during the formation and propagation of the lüders bands at the start of the plastic flow in niobium microalloyed steel during cold deformation. the experimental results of the maximum local temperature changes and strains in the area behind the lüders band front during formation and propagation of the lüders bands were performed in the area according to fig. 1. experimental results obtained by using static tensile testing, thermography and digital image correlation are shown in table 5 for output parameter σ, and in tables 6-8 for output parameters (ɛmax, δtmax). 600 t. brlić, s. rešković, z. jurković, g. janeš a) b) fig. 1 the area of measurements of: a) maximum temperature change and b) maximum strain in the area behind the lüders front table 5 experimental results of input parameters (δtmax, ɛmax) and output parameter σ obtained by measured values during formation and propagation of the lüders bands at v = 0.007 s-1 n u m b er o f m ea su re m en t formation of the lüders band propagation of the lüders band input parameters output parameter input parameters output parameter δtmax (°c) ɛmax (mm/mm) σ (mpa) δtmax (°c) ɛmax (mm/mm) σ (mpa) 1 -0.47 0.0076 543.3 1.37 0.0336 498.3 2 -0.35 0.0072 530.6 1.72 0.0374 498.3 3 0.17 0.0137 497.5 1.97 0.0378 500.6 4 0.17 0.0094 535.8 2.01 0.0316 531.1 5 0.42 0.0113 552.0 2.05 0.0300 522.6 6 0.67 0.0211 506.9 2.16 0.0412 512.3 7 1.07 0.0267 499.0 2.19 0.0388 510.0 8 1.23 0.0205 523.2 2.22 0.0398 510.8 9 1.37 0.0336 498.3 2.46 0.0386 540.7 10 1.67 0.0234 536.0 2.71 0.0373 521.3 11 2.01 0.0316 531.1 2.73 0.0416 545.9 12 2.05 0.0300 522.6 2.76 0.0434 549.3 13 2.80 0.0441 554.4 14 3.04 0.0427 528.5 15 3.30 0.0479 558.7 16 3.42 0.0439 524.5 17 3.65 0.0481 526.7 18 3.91 0.0492 521.2 mathematical modeling of the influence parameters during formation and propagation of the lüders... 601 table 6 experimental results of input parameters (δtmax, σ, v) and output parameter ɛmax obtained by measured values during formation of the lüders bands n u m b er o f m ea su re m en t input parameters output parameter n u m b er o f m ea su re m en t input parameters output parameter δtmax (°c) σ (mpa) v (s-1) ɛmax (mm/mm) δtmax (°c) σ (mpa) v (s-1) ɛmax (mm/mm) 1 -0.59 489.4 0.0018 0.0057 19 0.67 506.9 0.0070 0.0211 2 -0.55 484.7 0.0018 0.0073 20 0.74 555.9 0.0185 0.0199 3 -0.47 493.3 0.0018 0.0060 21 0.80 512.6 0.0018 0.0308 4 -0.47 543.3 0.0070 0.0076 22 0.81 539.8 0.0018 0.0350 5 -0.44 508.3 0.0018 0.0069 23 0.99 545.3 0.0185 0.0227 6 -0.35 530.6 0.0070 0.0072 24 1.07 499.0 0.0070 0.0267 7 -0.31 545.6 0.0185 0.0059 25 1.23 523.2 0.0070 0.0205 8 -0.29 515.9 0.0018 0.0127 26 1.37 498.3 0.0070 0.0336 9 -0.28 516.9 0.0185 0.0080 27 1.51 539.5 0.0185 0.0325 10 -0.24 509.4 0.0185 0.0086 28 1.66 549.1 0.0185 0.0324 11 0.02 510.9 0.0018 0.0172 29 1.67 536.0 0.0070 0.0234 12 0.17 497.5 0.0070 0.0137 30 1.69 564.5 0.0185 0.0324 13 0.17 535.8 0.0070 0.0094 31 2.01 531.1 0.0070 0.0316 14 0.34 536.7 0.0018 0.0198 32 2.05 522.6 0.0070 0.0300 15 0.39 509.1 0.0018 0.0279 33 2.06 535.0 0.0185 0.0325 16 0.42 552.0 0.0070 0.0113 34 2.16 536.4 0.0185 0.0380 17 0.5 512.1 0.0018 0.0215 35 2.53 528.1 0.0185 0.0400 18 0.62 536.1 0.0018 0.0280 36 2.55 540.3 0.0185 0.0412 mathematical models for the influence parameters were obtained by using measured experimental values of δtmax, ɛmax and σ during formation and propagation of the lüders bands. the experimental measured values were used to develop the equations of the second order polynomial model. the results of the data processing by regression analysis performed in microsoft® excel are shown in tables 9-12. regression analysis of influence parameters (ɛmax, δtmax, σ) based on measured values during the formation and propagation of the lüders bands revealed high values of multiregression coefficient (r) and determination coefficient (r2) which confirms that the obtained mathematical models can be considered reliable. the results of the multiregression coefficients and determination coefficients obtained by regression analysis of the influencing parameters are shown in table 9. high values of the multiregression coefficients and determination coefficients from the regression analysis were obtained. mathematical models for the influential parameters during formation and propagation of the lüders bands are shown by the following equations: ▪ mathematical model of the influential parameter σ during formation of the lüders bands at v = 0.007 s-1: σ=622.9334+75.945∆tmax-12078.3εmax+26.281∆tmax 2 +281771ɛmax 2 -4063.16∆tmaxεmax (5) ▪ mathematical model of influential parameter σ during propagation of the lüders bands at v = 0.007 s-1: σ=640.894-90.392∆tmax-1049.07εmax-86.8013∆tmax 2 -399310ɛmax 2 +13425.7∆tmaxεmax (6) 602 t. brlić, s. rešković, z. jurković, g. janeš ▪ mathematical model of influential parameter ɛmax during formation of the lüders bands: ɛmax=1.0398+0.178δtmax-0.004σ+4.866v-0.00078δtmax 2 +0.00000401σ2+144.558v2 0.00031δtmaxσ-8.807δtmaxv-0.01577σv+0.0169δtmaxσv (7) table 7 experimental results of input parameters (δtmax, σ, v) and output parameter ɛmax obtained by measured values during propagation of the lüders bands n u m b er o f m ea su re m en t input parameters output parameter n u m b er o f m ea su re m en t input parameters output parameter δtmax (°c) σ (mpa) v (s-1) ɛmax (mm/mm) δtmax(° c) σ (mpa) v (s-1) ɛmax (mm/mm) 1 0.35 508.6 0.0018 0.0309 28 2.27 510.8 0.0018 0.0405 2 0.39 509.1 0.0018 0.0279 29 2.46 540.7 0.0070 0.0386 3 0.61 509.8 0.0018 0.0325 30 2.55 535.6 0.0185 0.0392 4 0.75 511.4 0.0018 0.0331 31 2.71 521.3 0.0070 0.0373 5 0.80 512.6 0.0018 0.0308 32 2.73 545.9 0.0070 0.0416 6 0.81 539.8 0.0018 0.0350 33 2.76 549.3 0.0070 0.0434 7 0.98 543.3 0.0018 0.0379 34 2.80 554.4 0.0070 0.0441 8 1.00 509.7 0.0018 0.0336 35 2.86 529.0 0.0185 0.0378 9 1.11 510.1 0.0018 0.0344 36 3.03 526.7 0.0185 0.0414 10 1.18 549.0 0.0018 0.0432 37 3.04 528.5 0.0070 0.0427 11 1.19 547.3 0.0018 0.0409 38 3.08 539.2 0.0185 0.0408 12 1.37 498.3 0.0070 0.0336 39 3.30 558.7 0.0070 0.0479 13 1.46 545.3 0.0018 0.0448 40 3.35 540.5 0.0185 0.0429 14 1.50 510.6 0.0018 0.0363 41 3.39 533.4 0.0185 0.0424 15 1.55 510.5 0.0018 0.0368 42 3.42 524.5 0.0070 0.0439 16 1.72 498.3 0.0070 0.0374 43 3.60 527.9 0.0185 0.0432 17 1.82 509.7 0.0018 0.0377 44 3.62 545.5 0.0185 0.0449 18 1.91 510.8 0.0018 0.0386 45 3.65 526.7 0.0070 0.0481 19 1.97 500.6 0.0070 0.0378 46 3.74 531.3 0.0185 0.0442 20 2.01 531.1 0.0070 0.0316 47 3.84 535.7 0.0185 0.0429 21 2.05 522.6 0.0070 0.0300 48 3.86 557.2 0.0185 0.0476 22 2.06 534.6 0.0185 0.0402 49 3.91 521.2 0.0070 0.0492 23 2.07 549.0 0.0018 0.0455 50 4.05 546.0 0.0185 0.0463 24 2.16 512.3 0.0070 0.0412 51 4.08 561.6 0.0185 0.0480 25 2.16 528.1 0.0185 0.0324 52 4.12 537.4 0.0185 0.0436 26 2.19 510.0 0.0070 0.0388 53 4.18 530.9 0.0185 0.0473 27 2.22 510.8 0.0070 0.0398 54 4.22 544.7 0.0185 0.0447 ▪ mathematical model of influential parameter ɛmax during propagation of the lüders bands: ɛmax=3.036+0.0732δtmax-0.01168σ-16.120v+0.00107δtmax 2 +0.000011σ2+ 50.721v2-0.00014δtmaxσ+4.2917δtmaxv+0.0287σv-0.00836δtmaxσv (8) ▪ mathematical model of influential parameter δtmax during propagation of the lüders bands: mathematical modeling of the influence parameters during formation and propagation of the lüders... 603 δtmax=-309.167+48.762ɛmax+1.1811σ+2310.75v+1210.47ɛmax 2 -0.00113σ2 10906.6v2-0.03997ɛmaxσ-77240.3ɛmaxv-3.848σv+146.69ɛmaxσv (9) table 8 experimental results of input parameters (ɛmax, σ, v) and output parameter δtmax obtained by measured values during propagation of the lüders bands n u m b er o f m ea su re m en t input parameters output parameter n u m b er o f m ea su re m en t input parameters output parameter ɛmax (mm/mm) σ (mpa) v (s-1) δtmax (°c) ɛmax (mm/mm) σ (mpa) v (s-1) δtmax (°c) 1 0.0309 508.6 0.0018 0.35 28 0.0405 510.8 0.0018 2.27 2 0.0279 509.1 0.0018 0.39 29 0.0386 540.7 0.0070 2.46 3 0.0325 509.8 0.0018 0.61 30 0.0392 535.6 0.0185 2.55 4 0.0331 511.4 0.0018 0.75 31 0.0373 521.3 0.0070 2.71 5 0.0308 512.6 0.0018 0.80 32 0.0416 545.9 0.0070 2.73 6 0.0350 539.8 0.0018 0.81 33 0.0434 549.3 0.0070 2.76 7 0.0379 543.3 0.0018 0.98 34 0.0441 554.4 0.0070 2.80 8 0.0336 509.7 0.0018 1.00 35 0.0378 529.0 0.0185 2.86 9 0.0344 510.1 0.0018 1.11 36 0.0414 526.7 0.0185 3.03 10 0.0432 549.0 0.0018 1.18 37 0.0427 528.5 0.0070 3.04 11 0.0409 547.3 0.0018 1.19 38 0.0408 539.2 0.0185 3.08 12 0.0336 498.3 0.0070 1.37 39 0.0479 558.7 0.0070 3.30 13 0.0448 545.3 0.0018 1.46 40 0.0429 540.5 0.0185 3.35 14 0.0363 510.6 0.0018 1.50 41 0.0424 533.4 0.0185 3.39 15 0.0368 510.5 0.0018 1.55 42 0.0439 524.5 0.0070 3.42 16 0.0374 498.3 0.0070 1.72 43 0.0432 527.9 0.0185 3.60 17 0.0377 509.7 0.0018 1.82 44 0.0449 545.5 0.0185 3.62 18 0.0386 510.8 0.0018 1.91 45 0.0481 526.7 0.0070 3.65 19 0.0378 500.6 0.0070 1.97 46 0.0442 531.3 0.0185 3.74 20 0.0316 531.1 0.0070 2.01 47 0.0429 535.7 0.0185 3.84 21 0.0300 522.6 0.0070 2.05 48 0.0476 557.2 0.0185 3.86 22 0.0402 534.6 0.0185 2.06 49 0.0492 521.2 0.0070 3.91 23 0.0455 549.0 0.0018 2.07 50 0.0463 546.0 0.0185 4.05 24 0.0412 512.3 0.0070 2.16 51 0.0480 561.6 0.0185 4.08 25 0.0324 528.1 0.0185 2.16 52 0.0436 537.4 0.0185 4.12 26 0.0388 510.0 0.0070 2.19 53 0.0473 530.9 0.0185 4.18 27 0.0398 510.8 0.0070 2.22 54 0.0447 544.7 0.0185 4.22 table 9 calculated values of multiregression coefficient (r) and coefficient of determination (r2) of mathematical models of influential parameters during lüders bands formation and propagation influential parameters σ (mpa) ɛmax (mm/mm) δtmax (°c) formation of the lüders band r 0.881 0.977 r2 0.776 0.954 propagation of the lüders band r 0.871 0.948 0.968 r2 0.759 0.899 0.938 604 t. brlić, s. rešković, z. jurković, g. janeš table 10 calculated results of influence parameter σ during formation and propagation of the lüders bands at v = 0.007 s-1 number of measurement formation of the lüders band propagation of the lüders band output parameter output parameter σ (mpa) σ (mpa) 1 532.0 525.6 2 539.7 520.9 3 535.9 533.8 4 530.4 516.6 5 514.6 537.3 6 498.8 525.0 7 496.6 486.1 8 501.5 494.5 9 537.5 515.5 10 541.5 525.6 11 524.5 524.4 12 523.4 514.4 13 529.4 14 532.7 15 537.3 16 540.7 17 542.2 18 553.1 deviation 1.34% 1.65% experimental values of the influence parameters during formation and propagation of the lüders bands are added into the obtained mathematical models. the following calculated values of the influence parameters from the predicted models are shown in tables 10-12. mathematical models of the influence parameters show low deviations of calculated values compared to the experimental results. this confirms the applicability of the obtained mathematical models in monitoring the observed influence parameters during the formation and propagation of the lüders bands during cold deformation in niobium microalloyed steel. the functionality and reliability of mathematical models of influential parameters during formation and propagation of the lüders bands were verified by selecting random values of the influencing parameters in the area of formation and propagation of the lüders bands. mathematical modeling of the influence parameters during formation and propagation of the lüders... 605 table 11 calculated results of influence parameter ɛmax during formation and propagation of the lüders bands formation of the lüders band propagation of the lüders band n u m b er o f m ea su re m en t output parameter n u m b er o f m ea su re m en t output parameter n u m b er o f m ea su re m en t output parameter n u m b er o f m ea su re m en t output parameter ɛmax (mm/mm) ɛmax (mm/mm) ɛmax (mm/mm) ɛmax (mm/mm) 1 0.0060 19 0.0276 1 0.0369 28 0.0373 2 0.0133 20 0.0328 2 0.0390 29 0.0373 3 0.0242 21 0.0037 3 0.0416 30 0.0363 4 0.0266 22 0.0106 4 0.0426 31 0.0345 5 0.0283 23 0.0231 5 0.0408 32 0.0388 6 0.0050 24 0.0292 6 0.0443 33 0.0415 7 0.0096 25 0.0055 7 0.0298 34 0.0430 8 0.0179 26 0.0207 8 0.0297 35 0.0457 9 0.0242 27 0.0334 9 0.0309 36 0.0490 10 0.0079 28 0.0364 10 0.0315 37 0.0357 11 0.0257 29 0.0408 11 0.0333 38 0.0391 12 0.0303 30 0.0103 12 0.0369 39 0.0401 13 0.0053 31 0.0291 13 0.0316 40 0.0438 14 0.0153 32 0.0354 14 0.0339 41 0.0441 15 0.0251 33 0.0080 15 0.0365 42 0.0473 16 0.0324 34 0.0230 16 0.0392 43 0.0374 17 0.0122 35 0.0311 17 0.0396 44 0.0418 18 0.0188 36 0.0403 18 0.0427 45 0.0441 19 0.0340 46 0.0454 20 0.0388 47 0.0453 21 0.0405 48 0.0447 22 0.0441 49 0.0388 23 0.0457 50 0.0410 24 0.0500 51 0.0421 25 0.0341 52 0.0436 26 0.0376 53 0.0471 27 0.0390 54 0.0485 deviation 12.37% deviation 3.37% therefore, a confirmation test was performed to compare randomly selected experimental values of the influencing parameters with the calculated results obtained by predicted mathematical models, tables 13-15. conducted confirmation tests have established high accuracy of the mathematical models in obtaining calculated values of the influence parameters from randomly selected experimental values during formation and propagation of the lüders bands. 606 t. brlić, s. rešković, z. jurković, g. janeš table 12 calculated results of influence parameter δtmax during propagation of the lüders bands propagation of the lüders band n u m b er o f m ea su re m en t output parameter n u m b er o f m ea su re m en t output parameter δtmax (°c) δtmax (°c) 1 0.90 28 2.41 2 1.09 29 2.54 3 1.25 30 2.75 4 1.46 31 1.81 5 1.89 32 2.41 6 1.78 33 2.63 7 0.42 34 2.75 8 0.69 35 2.58 9 0.89 36 2.96 10 0.99 37 2.03 11 1.00 38 2.68 12 1.37 39 3.09 13 0.78 40 3.36 14 1.09 41 3.59 15 1.31 42 4.05 16 1.50 43 3.05 17 1.58 44 3.36 18 1.81 45 3.46 19 1.74 46 3.59 20 2.49 47 3.79 21 3.18 48 4.08 22 3.33 49 2.90 23 3.93 50 3.14 24 4.01 51 3.48 25 1.38 52 3.82 26 1.71 53 4.18 27 1.88 54 4.13 deviation 11.90% based on the previously obtained results and validation of the mathematical models for the influential parameters, it can be concluded that the mathematical models showed high reliability since high coefficients of multiregression and determination coefficients and low deviations of calculated results were obtained. mathematical models are evidence that output parameter stress (σ) can be described by maximum temperature change (δtmax) and maximum strain (ɛmax) considering good agreement between the experimental results and the calculated ones from polynomial models during the formation and propagation of the lüders bands. therefore, maximum temperature change (δtmax) can be considered as a stress change that occurs during the formation and propagation of the lüders bands. mathematical modeling of the influence parameters during formation and propagation of the lüders... 607 table 13 validation of mathematical models by the confirmation test for σ during formation and propagation of the lüders bands at v = 0.007 s-1 n u m b er o f m ea su re m en t formation of the lüders band propagation of the lüders band e x p er im en ta l re su lt s c al cu la te d re su lt s o f th e m o d el e x p er im en ta l re su lt s c al cu la te d re su lt s o f th e m o d el σ (mpa) σ (mpa) σ (mpa) σ (mpa) 1 557 558.7 525.2 523.8 2 533.4 515.5 514.2 522.4 3 534.6 560.4 557.3 547.0 4 535.6 524.5 deviation 2.64% 1.24% table 14 validation of mathematical models by the confirmation test for ɛmax during formation and propagation of the lüders bands n u m b er o f m ea su re m en t formation of the lüders band propagation of the lüders band e x p er im en ta l re su lt s c al cu la te d re su lt s o f th e m o d el e x p er im en ta l re su lt s c al cu la te d re su lt s o f th e m o d el ɛmax mm/mm ɛmax mm/mm ɛmax mm/mm ɛmax mm/mm 1 0.0166 0.0215 0.0455 0.0417 2 0.0199 0.0232 0.0355 0.0341 3 0.0415 0.0441 0.0393 0.0403 4 0.0392 0.0362 0.0491 0.0485 5 0.0404 0.0363 6 0.0457 0.0478 7 0.0467 0.0466 8 0.0450 0.0450 9 0.0480 0.0485 deviation 15.04% 3.59% mathematical models have shown that the values of other influencing output parameters (ɛmax, δtmax) during the formation and propagation of the lüders bands can be described with high accuracy. it means that the behavior of niobium microalloyed steel can be predicted with models of influencing parameters in the region behind the lüders front where maximum local temperature changes, i.e. stress changes, and strain changes are present during the cold deformation. 608 t. brlić, s. rešković, z. jurković, g. janeš table 15 validation of mathematical models by the confirmation test for δtmax during propagation of the lüders bands n u m b er o f m ea su re m en t propagation of the lüders band e x p er im en ta l re su lt s c al cu la te d re su lt s o f th e p o ly n o m ia l m o d el δtmax (°c) δtmax (°c) 1 2.02 2.07 2 1.15 1.22 3 2.07 1.71 4 3.91 4.06 5 2.22 2.72 6 3.07 2.66 7 4.05 3.92 8 4.12 3.83 9 4.01 4.15 deviation 8.82% lüders bands, i.e. surface roughness, can often occur due to the effort of continuously improving of the mechanical properties (increasing yield strength and ductility) of microalloyed steels. therefore, in order to modify and improve the metal forming process, it is important to determine the stress amounts during the lüders band propagation. stress changes at different deformation degrees are visible during static tensile test in the area behind the lüders band front by applying a mathematical model according to eq. (6) during the propagation of the lüders band. from the obtained results of the mathematical model, it can be concluded that the stress amount in the area behind the lüders band front increases with increasing of deformation degree from point 1 to point 2 according to: σpoint 1, lüders band propagation=640.894-90.392∙0.76-1049.07∙0.0234-86.8013∙0.762 399310∙0.02342+13425.7∙0.76∙0.0234=517.63 mpa σpoint 2, lüdersbandpropagation=640.894-90.392∙3.42-1049.07∙0.047-86.8013∙3.422 399310∙0.0472+13425.7∙3.42∙0.047=543.16 mpa comparing the stress amounts at the start of the lüders band propagation (point 1) and during the lüders band propagation (point 2), a higher stress amount of 543.16 mpa was determined at point 2 at a higher deformation degree. a higher stress amount indicates a more pronounced surface roughness in this area during lüders band propagation which is a significant problem during forming of microalloyed steels. the possibility of the stress amounts prediction with the obtained mathematical models is proven. for example, during cold drawing and bending of microalloyed pipes with inhomogeneous deformations, the significance of the surface roughness can be predicted and so can the possibilities of plastic flow of niobium microalloyed steel from obtained mathematical modeling of the influence parameters during formation and propagation of the lüders... 609 stress amounts. predicting of the stress amounts in steels, with the lüders band appearance, can also be significant during sheet forming, especially sheet metal drawing, and deep drawing in automobile production. it has been proved that the obtained mathematical models can improve the product quality and reduce the costs of the forming process of microalloyed steels since the calculation of the values of the influential parameters by using predicted models do not necessarily require additional experimental tests requiring additional costs. 4. conclusion mathematical models of the influence parameters during the formation and propagation of the lüders bands in the area behind the lüders band front in niobium microalloyed steel are developed in this paper. the proposed mathematical models give a new approach to the prediction of stresses, maximum temperature changes and strains during the lüders band formation and propagation in niobium microalloyed steel. high reliability of the obtained mathematical models are determined with high coefficients of multiregression within the range from 0.871 to 0.977, low deviations of calculated results from 1.34% to 12.37% and low deviations of confirmation results from 1.24% to 15.04%. it has been proved that output parameter stress (σ) can be described by maximum temperature change (δtmax) and maximum strain (ɛmax) where maximum temperature change (δtmax) can be considered as a stress change during the formation and propagation of lüders bands. stress amounts at different deformation degrees, in the deformation zone during lüders band propagation, were determined for modification and improvement of steel forming processes. using the obtained mathematical model, it was observed that the stress changes at different deformation degrees during static tensile tests in the area behind the lüders band front. at the start of the lüders band propagation, a lower stress amount of 517.63 mpa, is determined with respect to the advanced lüders band propagation stress amount of 543.16 mpa. a higher stress amount, obtained by the proposed mathematical model, indicates a more pronounced surface roughness in the area of the lüders band propagation. the proposed mathematical models can find their application in the industry for improving the forming process by prediction of the surface roughness and plastic flow of the tested steel using the obtained amounts of influence parameters, especially stress amounts. the significance of the surface roughness as well as the possibilities of plastic flow of niobium microalloyed steel can be predicted by the 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engineering viewpoint. a method based on ct replacement is proposed in order to circumvent the related problems with the missing information to modeling. the method is based on digital image processing and swarm intelligence algorithm. in this approach, a missing region on the defective skull is represented by curvature descriptors. the main function of the descriptors is to simplify the skull’s contour geometry; and they are defined from the cubic bezier curves using a meta-heuristic process for parameter’s estimation. the artificial bee colony (abc) optimization technique is applied in order to evaluate the best solution. the descriptors from a defective ct slice image are the searching parameters in medical image databases, and a similar image, i.e. with similar descriptors, can be retrieval and used to replace the defective slice. thus, a prosthesis piece is automatically modeled with information extracted from distinct skulls with similar anatomical characteristics. key words: prosthesis modeling, computed tomography, cubic bezier curves, content retrieval, abc optimization 1. introduction skull prosthesis is a mechanical piece artificially produced with similar geometric characteristics of the natural bone structure. its production process has been modernized over time due to being supported by digital image processing and cad tools. from the viewpoint of engineering, prosthesis can be considered as a complex product-feature because of its own geometry as well as because of all the processes required for its manufacture. despite specific anatomical characteristics of individuals, it is possible that its production follows standard steps as long as it is characterized by clearly defined received october 15, 2015 / accepted november 20, 2015 corresponding author: marcelo rudek pontifical catholic university of parana, pucpr/ production and system engineering graduate program – ppgeps, imaculada conceicao, 1155, 80215-030, curitiba, brazil. e-mail: marcelo.rudek@pucpr.br original scientific paper 284 m. rudek, y. b. gumiel, o. canciglieri jr methods. the basis of the production process is the geometric modeling in the early stage of design. the construction of a virtual 3d model allows us to evaluate the prosthesis from medical perspective in order to plan a surgical procedure as well as from product development perspective in order to plan its manufacturing process. this process involves different actions starting from the ct image acquisition and respective problem identification (failure in the bone) via the virtual model generation to, finally, the prosthesis production (real model). according to [1] the bones have the similar semantic representation of free-form objects, where the semantic features can contribute in the parameterization of bones‟ models. this approach solves the subjectivity to achieve custom pieces and overtakes the prosthesis modeling limitations as before addressed in [2, 3]. at the level of implementation, the key point is the parameterization of shapes. recent researches have proposed the automation of bone‟s modeling process in some steps, as example, the sub pixel cubes applied in [4], the ellipse adjustment algorithm (eaa) proposed by [5] and its modified approach by superellipses in [6], as well as the anatomic features generation experienced by [7] through a reverse engineering approach. the well known methods like splines [7, 8, 9] and active shape models [10, 11] are frequently used to describe regions, and they can be, for instance, adapted to represent bones. additional difficulties in the bone modeling are present over the open edges problems due to the missing information. in order to overcome limitations while dealing with open contours the new approaches have been proposed such as the ellipse adjustment by [5], which is focused on finding an ellipse with a similar curvature to the bone edge. thus, a circumference fits the bone edge over a specific ct slice in order to represent an incomplete edge. the main limitation is that the bone contour is not always a regular circumference and, for some slices, the adjustment is weak. the recent research about prosthesis modeling is oriented in the same direction with the aim of finding new methods for the generation of the anatomically-adjusted prosthesis models. among these computational processes, it is possible to classify the methods due to strategy in 3d or 2d approaches. from 3d approach, the virtual model is built straight on 3d image, through manual intervention where the prosthesis is drawn on its own 3d surface like in [12, 13, 14] or generated by specialized algorithms that mathematically create a model of implant as in [4]. from 2d image, the virtual model is created from the tomographic cuts. a failure in the skull surface generates an open contour in the slices of respective failure region. in this case, all the open sections on the slices are closed oneby-one through image processing methods in order to complete the open curvature in those bone images, as in [5, 6, 16]. beyond these different approaches in modeling, we can also classify the reconstruction process according to the symmetry criteria. the failure position can be symmetric or non-symmetric. for the symmetric case, the most evident hypothesis is that the healthy side of skull can be mirrored [16] to the appositive side in order to cover the failure region by using the symmetry of body [17]. however, this is only possible when the failure region occurs in the right or left lateral position considering a sagittal cut. in the case of 3d surface reconstruction tools, as in [12], for instance, it is possible to add or remove image fragments from the mirroring region to build the prosthesis piece by a handmade management of the boolean operations; surely the finest adjustment is also required in all the link positions between prosthesis and skull image. the mirroring can be automated as experienced by [16] through a slice by slice procedure in order to avoid manual intervention, but it also addresses problems in the autonomous ct replacement method for the skull prosthesis modelling 285 linkage frontiers between the model and the skull. in the non-symmetric case, the failure sometimes occurs in the frontal region as well while the symmetry-based process cannot be applied because the same proportion of the known image to mirroring does not exist. in this case, the closing of contour can be performed by computational methods as curve adjustment applied over the ct images. for this situation, some different techniques as fourier curves from [18, 19], and active shape models [10, 11], can be used to graphically represent a closed contour region but they have restrictions when it comes to representing open sections because they cannot be adjusted on the missing area. our proposal is to perform an autonomous content retrieval search in a lot of medical image database in order to find healthy bones with shape-similarity to replace missing regions. the method is based on a ct-by-ct approach. the problem concerned with content retrieval is the amount of data to be compared; then our approach is to define a small set of parameters, i.e. descriptors of the curvature shape. we are applying the cubic bezier curves because its principle is the basis for other curve adjustment methods as bsplines for instance [8, 20]. the adjustment is evaluated by abc optimization algorithm [21-25] as a useful tool for evaluating the best of the descriptors parameters. 2. the proposed method the method implies the determination of the control points of a cubic bézier curve fitted on the bone edge curvature [25]. the control points are a set of shape descriptors because they permit us to build a synthetic edge similar to the original contour. the whole method‟s representation is presented in fig. 1. modelling of missing region failure position identification ct slices segmentation parametric modeling of curvature curvature descriptors contour sectioning abc optimization (fitness evaluation) bezier curve adjustment search on image database create the 3d model create the stl file find compative skulls with similar descriptors fig. 1 flow of operation of the proposed method 286 m. rudek, y. b. gumiel, o. canciglieri jr the operation flow in fig. 1 shows the method as having three main layers: image processing, curve descriptors calculation and missing region modeling. this research addresses the curve descriptors calculation. 2.1. curve descriptors calculation the process starts with the image processing. all the ct slices must be extracted from dicom file [26]; this is followed by a threshold process [27]; it is an image processing method to perform the segmentation, i.e. transformation of the original grey level image into an equivalent binary (black & white) image. thus, it enables us to separate the interesting objects of image from its background. within this procedure, we are able to access the bone information only, meaning separated from the remaining tissues. once the bone edge has an outer side and an inner side with their respective shapes we need to apply an edge detector to separate them. after this procedure all the selected ct slices contain binary data whose value „1‟ is a pixel of inner and outer edges and „0‟ is the background. the curve calculation is compounded by two processes, where the first is the failure position identification whose aim is look for the slices with an open contour; and the second is the parametric modeling of curvature whose objective is to determinate its descriptors. 2.1.1. failure position identification once these images are processed, we need to find only the ct slices whose bone edge comprehends the failure region. an automatic procedure is proposed by [5] in order to perform the selection of the set of slices with uncompleted edge. the method operates through the polar coordinates scanning the edge‟s pixels on each ct slice, and it produces a relationship between an angle theta versus the radius (distance from center to pixel) as presented in fig. 2.a. thus, when a pixel exists in that theta direction we get its polar coordinate, and otherwise when no pixel is present (gap position) an infinite radius value is found; then it is possible to mark the initial and final position of failure in each image. following this, from the set of open slices, we proceed to the modeling for each one. 2.1.2. sectioning of the edges the bézier method requires a set of control points to produce an adjusted curve; in this case, some of them are known while the others must be calculated. the known control points p belong to the edge. our method applies the edge sectioning procedure to sub-divide the edge in small sections ck. each section defines two initial parameters for curve generation. fig. 2.b presents the sectioning on edge where each pair [pk pk+1], with k=1,..,n, defines the range of n divisions. the value of n is experimentally determined by gumiel in [25] and in this example, the n value represents 10 cut sections where the sections are c0=[p0 p1], c1=[p1 p2], …, c9=[p9 p10]. the number of descriptors necessary to represent the curvature implies an equivalent number of sections. the difficulty is to determine the balance between n value and the precision required because a higher number of divisions (n) cause a big set of parameters and, consequently, a big amount data for processing. otherwise, small n causes weak adjusted curves due to the lack of information. thus, the main problem is to find equilibrium between the best accuracy of autonomous ct replacement method for the skull prosthesis modelling 287 the results with the minimum quantities of descriptors. this optimization is presented in the following section 2.2.4. fig. 2 a) polar coordinate mapping to identification of the limits in missing region b) sectioning of contour for fixed control points (pk) determination 2.1.3. parametric modeling of curvature the main objective in this phase is to generate a synthetically built curve over original bone edge in order to describe the bone curvature in each ct slice. thus, we get a small set of points to represent the same shape of bone curvature instead those all of edge pixels. the parameters to be calculated by the bézier method are presented in fig. 3. fig. 3 a) a sample of a bézier method showing the curve defined by its respective control points; b) the sample of a curve generate on a piece of bone edge and respective position of control points the aim is to define the “variable control points” that lie between two known fixed points p1 and p2. as presented in fig. 3.a, the fixed control points p1 and p2 define the beginning and the end of the bezier curve in a section, and its shape is given by p3 to p10. by general form we have: 1 2 3 4 5 6 7 8 9{[ ];[ , , , , , , , ]}ck k k k k k k k k k kb p p p p p p p p p p         (1) a) b) a) b) 288 m. rudek, y. b. gumiel, o. canciglieri jr where bck is the respective bezier curve for section ck. ps are known and ps are unknown. the bézier technique needs a “control polygon” in order to generate the curvature whose vertices are the set of control points. as presented in the example in fig. 3.a, the bezier curve is built from four control points so called p1, p2 (as the fixed points lying on edge) and p3, p4 (as variable points) whose coordinates define a “control polygon”. the polygon lines determine a base support line for the variable control points (p5, p6, p7). changing those points, p8, p9 are moved along the support line and define p10. this point is the resulting point and it defines the pixel position to draw a curve. by changing all variable p points it is possible to obtain all curve points. fig. 3.b presents an example of the bezier curve adjusted (blue line) on the bone edge to the outer contour (green line). the quality of adjustment depends on the position of all control points, and the challenge is to calculate the possible coordinates of the “variable points” (red points) once the fixed points are known because they are on the bone edge. in fig. 4.a there is a good bezier adjustment and in fig. 4.b a sample of a wrong adjustment. fig. 4 examples of bezier curves adjusted on edge fig. 4 shows in (a) fixed points p1 and p2 and respective variable points p3 and p4. in these positions, the bezier curve is on the original edge. in fig. 4.b, p1 and p2 are in the same position as (a) but p3 and p4 are not correctly positioned, and then by consequence the bezier curve does not represent the original edge curvature. the process of finding the best position for the variable points for each bck curve is determined by application of the artificial bee colony optimization (abc) as described further on. 2.1.4. the artificial bee colony optimization (abc) the abc optimization algorithm proposed by karaboga [21, 22] or its modified approach by abro [23, 24] is based on concept of collective behavior of insects in nature (bees, for instance), modified to solve optimization problems. it is an algorithm of swarm intelligence (si) where the individuals interact locally among them in order to solve a problem (global objective) [28]. in this case, we are applying the abc algorithm in order to determinate the coordinate of the variable control points (p). karaboga et al [22] have compared the abc performance face the ga (genetic algorithm), de (differential evolution), pso (particle swarm optimization) and demonstrated its applicability. because of the experience acquired in this research, the abc approach is selected as more suitable for our optimization problem. a) b) autonomous ct replacement method for the skull prosthesis modelling 289 the abc algorithm is based on the bees searching for a source of food [21], and in our context, the “source of food” location is a possible solution of the variable control points. in addition, the amount of nectar corresponds to the solution‟s quality, i.e. the quality of the calculated bezier curve. the quality of the calculated curve is evaluated by fitness function, f, as presented in eq. 2. 2 2 1 ( ) ( ( ) ( )) ( ( ) ( )) n b o b o i f k x i x i y i y i      (2) fitness function f(k) is calculated for each section ck=1,2,...,l with l as the total number of sections of the edge, and n as the maximum number of control points for comparison. the value of f defines the difference between calculated values xb and yb (values of coordinates x and y from the bezier curve) and those of all xo and yo (values of original contour pixels). following the abc algorithm from [22], the initial population is a set of the possible solutions randomly generated and in our notation pk and pk+1 are respectively the starting point (first point) and ending point (second point) of the section on the bone edge. then, they form a rectangular polygon that limits a searching area in order to look for the variable points (p3 and p4) as in fig. 5.a. p1 and p2 are known because of lying over the edge at the cut position; the challenge is to find the best p3 and p4 candidate points whose coordinates are more suitable to produce the bezier curve in each section. in figs. 5.b and 5.c, they present a sample of the optimization result applied on contour through ck=10 sections. in total, we have about 20 variable points from 11 fixed points. fig. 5 descriptors calculation a) searching area for section ck; b) fixed and variable control points; c) descriptors to the inner and outer contour the fitness of curvature is evaluated by a function f that measures the difference between the original edge and the generated curve. the best values after the convergence of method are selected as answer points. then, the total of points {pbck} is the curvature descriptors. a) b) c) 290 m. rudek, y. b. gumiel, o. canciglieri jr 3. results and discussion for demonstrating our purpose, a synthetic irregular failure is hand-made built through imagej plugging from [16], [30]. thus, the original cut piece is to be compared with modeled one. from image of fig. 9, a total of 26 ct slices with their respective open contours are extracted as samples in fig. 9.b. now, n=10 sectioning of the edge gives us 20 set of descriptors. the process is repeated for images in database and the parameters are calculated again for each one. the compatible ct slices are extracted from the image database within cts set about 25 patients. the selected are all whose descriptors which match with searching parameters as shown in table i. fig. 6 defective skull and respective failure region as presented in fig. 6, the skull‟s failure region can be decomposed by its respective ct slices. in fig. 6.a an example of a frontal failure is presented. note that in this case the modeling does not apply symmetry by mirroring, like show images i, ii and iii in fig. 6.b. otherwise, only the image iv could be solved by mirroring. then the proposed method attends both symmetric and non-symmetric cases. table i shows the recovered compatible slices for each testing ct image from patient#1. note that the prosthesis range can be replaced by compatible information from 3 distinct patients (patients #05, #06, and #10). these three have similar morphological characteristics in the specific tomographic cut. the best ct is selected within minimum error for each tested slice. due to the fact that the roi position is known on the original image, the same part of ct can be extracted from compatible cts. a sample of some compatible pieces is presented in fig. 7. for example, the first slice number #249 from patient #06 can replace the slice number #249 in patient #01, the second slice number #250 in patient#01 can be replaced by the slice number#245 from patient#05, and so on. all of the recovered slices can be superimposed to form a 3d piece as presented in fig. 8. the new 3d piece fills the failure region as presented in fig. 9. a) b) autonomous ct replacement method for the skull prosthesis modelling 291 table 1 compatible slices whose descriptors match with testing images testing image (ct-patient #1) compatible ct image (cts recovered) fitness (similarity measure) slice # 249 patient no 06, slice no 249 31,0711 slice # 250 patient no 05, slice no 245 63,5262 slice # 251 patient no 05, slice no 247 73,5553 slice # 256 patient no 06, slice no 246 76,5050 slice # 257 patient no 06, slice no 245 81,3127 slice # 258 patient no 05, slice no 247 98,6153 slice # 259 patient no 10, slice no 249 92,8940 slice # 267 patient no 10, slice no 248 166,4766 slice # 268 patient no 06, slice no 244 174,0587 slice # 269 patient no 10, slice no 244 171,3661 slice # 270 patient no 05, slice no 274 81,0014 slice # 271 patient no 06, slice no 271 47,4940 slice # 272 patient no 06, slice no 273 44,2646 slice # 273 patient no 05, slice no 275 46,2553 (249, 6) (247, 5) (246, 6) (249, 10) (248, 10) (244,6) (274,5) (273, 6) fig. 7 the samples of compatibles pieces that match the original missing region, labeled here as („slice#’, „patient#’) fig. 8 three views of modeled prosthesis 292 m. rudek, y. b. gumiel, o. canciglieri jr fig. 9 the piece anatomically adjusted on the roi fig. 10 measurement of error between original skull and the replaced region evaluated through geomagic® system [29] in fig. 9, there is a visual sense of the prosthesis piece covering the missing region on the testing skull image. the linkage between the prosthesis piece and the skull seems as discontinuity because the 3d view tool (from imagej/fiji plugging [30]) caused shadows on the edges of these two objects. the better visualization is given in fig. 10 by presentation of a colored scale for 3d error measured by geomagic® system [29]. the values show the differences among ≈1mm until ≈2mm between the original skull and the replaced region inside the prosthesis range. 4. conclusion and future work the paper addresses the prosthesis modeling problem. it presents a method to fill a defective region on the skull by a ct replacement strategy. the proposed method is compounded by activities involving image processing, optimization and curves adjustment. the aim of proposal is the creation of the contours‟ descriptors based on the cubic bezier autonomous ct replacement method for the skull prosthesis modelling 293 curves. the concept behind the method is to perform a content retrieval from a medical image database in order to look for compatible cts slices from other patients with similar morphological characteristics. the good compatible slices with a closed contour are used to replace the defective slices with an open contour. thus, a piece of image from a good bone edge is superimposed on the position of the missing area of testing image. all the processes for descriptor‟s generation and its respective geometric prosthesis modeling are performed automatically. a testing image is prepared for demonstration. an irregular hole is handmade built on the frontal position of a skull image. in this way, we have enough information to compare the modeled prosthesis with the real bone image. the geomagic software is applied to evaluate the difference in measure. the presented result shows that it is possible to create a respective 3d prosthesis model with minimum error without the user‟s intervention. from the engineering point of view, in terms of geometry, we have demonstrated that it is possible to create anatomical bone models. however, for future work, it is necessary to complete the other phases of process such as printing as well as all medical evaluation. this proposal is an important step to prosthesis modeling in our ongoing study, but in order to comply with further practical application, it is also necessary to enlarge the testing image database to improve the selection of more compatible answers. acknowledgements: the author would like to thanks the graduate program in 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9(7), pp. 676-682. 13629 facta universitatis series: mechanical engineering https://doi.org/10.22190 fume250508031x © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper scenario-level hierarchical energy management control strategy for hybrid electric vehicle feng xiao1, ruiqiang zhang1, chunyang qi1,2 1college of automotive engineering, jilin university, china 2state key laboratory of advanced design and manufacturing technology for vehicle, hunan university, china abstract. to address the challenge that single-scenario solutions often fail to adequately handle the complex and dynamic mixed environments that drivers regularly navigate, this study aims to leverage learning-based algorithms to design energy management control strategies tailored to the unique characteristics of different driving scenarios. the goal is to achieve precise matching and efficient execution of the driving strategies. the main research focus of this study is developing a scene-level hierarchical energy management control strategy (shems) framework for hybrid electric vehicles in mixed driving environments. in the car-following scenario, to address the challenges of reward function design and the impact of environment and driver habits, an adaptive strategy learning strategy with imitation learning is proposed. to overcome issues of suboptimal expert knowledge and the curse of dimensionality, optimization factors are added. for the intersection scenario, aiming at the challenge of reward sparsity caused by the scarcity of traffic signals and safety incentives, an additional reward mechanism is innovatively proposed to enrich the reward function. the experimental results demonstrate that the shems significantly reduces fuel consumption. specifically, the ppo, a2c, a3c, and dqn-based strategies achieved 7.52%, 5.29%, 9.6%, and 5.93% reductions, respectively. taking the dqn algorithm as an example, the emissions of harmful gases co, hc, pm and nox are reduced by 18.3%, 14.23%, 16.94%, and 20.9%, respectively, after layering. key words: hybrid electric vehicle, mixed scenarios, intersection scenarios, reinforcement learning received: may 08, 2025 / accepted september 29, 2025 corresponding author: chunyang qi national key laboratory of automotive chassis integration and bionics, jilin university, changchun 130025, china e-mail: qichunyang@jlu.edu.cn 2 f. xiao, r. zhang, c. qi 1. introduction among various factors influencing vehicle energy consumption, the driving behavior patterns of drivers have been widely recognized to have a significant impact of up to 30 % on the fuel economy of vehicles [1-3]. further in-depth research by the national renewable energy laboratory in the united states has further consolidated this finding, revealing that the implementation of reasonable and efficient driving behavior strategies can achieve a significant effect of around a 20 % reduction in fuel consumption [4, 5].when addressing the energy management control strategy problem in the intersection scenario, yu et al. [6] utilized car-following models and real-time optimal control to optimize the trajectories of connected and automated vehicles (cavs), thereby minimizing the total fuel consumption of each cav during the current control period. the fuel economy of hybrid electric vehicles is closely related to the vehicle's speed distribution, which directly affects the required total energy consumption [7]. the latest research surveys have shown that by combining energy management control techniques and speed planning, energy consumption can be effectively reduced, achieving a 15 % energy-saving effect [8]. it not only helps enhance the market attractiveness of products, but also can bring more substantial economic benefits [9]. hu et al. [10] developed an integrated optimal controller for hybrid electric vehicles equipped with vehicle-infrastructure communication, which achieved up to 16.9% improvement in fuel efficiency on rolling terrain. liu et al. [11] employed a combination weighting method and k-means clustering to develop an ecodriving evaluation model based on multi-source data. in this case, calculating the reference speed curve to avoid idling before the intersection can ensure relatively good energy-saving performance [12]. in recent years, with the rapid development of artificial intelligence technology, many studies have introduced learning-based methods into the energy management control strategy problem, which have also demonstrated the strong adaptability and nonlinear system optimization capabilities of learning-based methods [13-16]. yeom et al. [17] cleverly integrated deep reinforcement learning and model predictive control techniques, significantly improving the energy efficiency and economics of electric vehicles. this innovative hybrid approach, particularly the precise optimization of the speed control strategy for electric vehicles through the model predictive control algorithm, not only brings technological innovation to the current electric vehicle field, but also provides valuable theoretical guidance and practical examples for the future planning of autonomous. addressing the dual objective of reducing fuel consumption during the driving process and simultaneously enhancing driving comfort. he et al. [18] developed a multi-objective intelligent eco-driving strategy for plug-in hybrid electric vehicles based on a multi-head deep q-network deep reinforcement learning approach, which achieves energy consumption optimization while ensuring autonomous driving safety. ozatay et al. [19] proposed a simplified reinforcement learning method, which learns the trends and characteristics of the driver's reference speed online and dynamically adjusts the speed limit to increase the driver's likelihood of following the reference speed. in the frontier field of exploring hierarchical energy management control strategies, wang et al. [20] took an important step forward by carefully conceiving and designing an online control strategy for hybrid electric vehicles. shems aims to comprehensively improve fuel economy, significantly reduce exhaust emissions, and simultaneously ensure a high standard of driving safety. at the low-level execution layer of the strategy, the focus scenario-level hierarchical energy management control strategy for hybrid electric vehicle 3 is on the optimization of power allocation between the internal combustion engine and the battery module, ensuring the rationality and efficiency of energy flow through precise control. at the high-level decision-making layer, the intelligent system is fully responsible for the accurate control and optimization of the vehicle's speed. guo et al. [21] combined energy management control strategies with reinforcement learning algorithms, proposing a hierarchical control framework-based efficient and coordinated optimization strategy for autonomous hybrid power tracked vehicles, which achieved simultaneous optimization of precise path tracking and energy management. aiming at safety and comfort, liu et al. designed an algorithm with a two-layer structure. the upper-level controller uses a grey neural network to predict the future speed trend of the lead vehicle, while the lower-level adopts an adaptive equivalent consumption minimization strategy to implement the hierarchical energy management control strategy [22]. peng et al. [23] proposed a multilane hierarchical optimization algorithm based on a predictive control framework, which plans the optimal speed and lane-changing behavior by considering the vehicle's power demand, driving comfort, and safety at the upper level. at the lower level, dynamic programming is used to design the energy management by tracking the optimal speed. li et al. [24] studied a data-driven optimal energy management control strategy for plug-in hybrid electric vehicles, which has good performance and computational efficiency in speed planning and energy management. this algorithm also has a hierarchical structure and introduces two data-driven algorithms based on high-fidelity neural networks as the evaluation model and system model for speed optimization, reducing the requirement for complex powertrain system models. although there has been extensive research on energy management control and driving style, in the context speed optimization for long-distance urban roads, are relatively limited [25-27]. chen et al. proposed an assistance system that uses a model predictive control design to create a simplified powertrain model-based advisor system, which not only provides speed profiles but also real-time recommendations for advanced driving modes such as cruising and coasting [28]. considering the diversity of driver behavior, ma et al. proposed a novel fuel-efficient driving strategy and investigated its impact on the fuel efficiency of a human-driven vehicle fleet. in mixed traffic, where multiple vehicles share the road, the strategy of the vehicles is realized through vehicle connectivity and longitudinal dynamic control, thereby significantly reducing the vehicle's fuel consumption by avoiding unnecessary braking and acceleration operations [29]. from the current research on energy management control strategies, a significant trend is that a large number of studies have focused on optimization strategies for specific scenarios, such as intersection management and energy management control strategies for highways [30-32]. however, drivers' daily commutes often traverse through complex and ever-changing mixed scenarios, and solutions for a single scenario may prove inadequate in addressing comprehensive road conditions, thus failing to ensure the optimality and adaptability of the strategies. therefore, in the construction of intelligent transportation systems, the development of an algorithm that can flexibly cope with mixed driving scenarios has become increasingly urgent and important. this paper addresses this challenge by delving into the energy management control strategy for hybrid electric vehicles in mixed scenarios, and innovatively proposing shems. the core of this method lies in the customization of energy management control strategies according to the characteristics of different driving scenarios, thereby achieving precise matching and efficient execution of the strategies. 4 f. xiao, r. zhang, c. qi the specific contributions of this paper are as follows: 1. facing the actual situation of hybrid electric vehicles driving in mixed scenarios, shed framework for mixed scenarios has been constructed. 2. regarding the challenge of reward function design, in the car-following scenario, an adaptive policy learning strategy based on imitation learning and a vehicle car-following strategy incorporating driving style are proposed. considering the challenges of nonoptimality of expert knowledge and high-dimensional features, an optimization factor addition strategy is also proposed. 3.to address the sparsity of the reward function in the intersection scenario, a method of adding supplementary rewards to compensate for the sparse reward function and a state prediction calibration method are proposed to solve the problems of sparse reward functions and the uncertainty of the traffic scenario and the learning environment of the intelligent agent. the remaining parts of the paper are as follows: section 2 is the problem description, which includes the modeling of hybrid electric vehicles. section 3 addresses the influence of the traffic environment and driver habits on the driving strategy, and proposes an adaptive policy learning method based on imitation learning and a vehicle car-following strategy incorporating driving style. section 4 discusses the energy management control strategy in the intersection scenario. section 5 verifies the algorithm advantages from both simulation and hardware-in-the-loop experiments. section 6 summarizes the entire paper and looks ahead to the future. 2. problem description 2.1 hev modeling the research focuses on a parallel hev with a p2 configuration, as depicted in fig.1. the p2 hev model offered by the matlab/simulink 2020b community is utilized and modified in the control component to enhance simulation precision and repeatability. table 1 contains detailed information about the model parameters. table 1 other parameters of the vehicle symbol parameter values engine maximum power 92 kw maximum torque 175 nm maximum speed 6500 rpm motor maximum power 30 kw maximum torque 200 nm maximum speed 6000 rpm battery capacity 5.3 ah voltage 266.5 v scenario-level hierarchical energy management control strategy for hybrid electric vehicle 5 gear box torque coupler engine motor controller battery charge differential drive charge fig. 1 vehicle transmission configuration 2.1.1 vehicle dynamics modeling the longitudinal force balance equation of the vehicle is shown in eq. (1). the equation is divided into four parts: rolling resistance ff, aerodynamic drag fw, slope resistance fi, and inertial force fa. { 𝐹 = 𝐹𝑓 + 𝐹𝑤 + 𝐹𝑖 + 𝐹𝑎 𝐹𝑓 = 𝑊𝑓 = 𝐺 ⋅ 𝑓 𝐹𝑤 = 1 2 𝜌 ⋅ 𝐴𝑓 ⋅ 𝐶𝐷 ⋅ 𝑣 2 𝐹𝑖 = 𝐺 ⋅ 𝑖 𝐹𝑎 = 𝛿 ⋅ 𝑚 ⋅ 𝑎 (1) where g represents the vehicle weight, f is the rolling coefficient, ρ indicates the air density, af represents the frontal area of the vehicle, cd donates the drag coefficient, i is the slope angle, δ means the rotational mass conversion factor, and a indicates the longitudinal acceleration of the vehicle. the power of the hybrid electric vehicle is provided by the engine and the electric motor, which can be calculated as: 𝑃dem = (𝑃𝑒𝑛 + 𝑃𝑏𝑎𝑡 ⋅ 𝜂𝑚)𝜂𝑇 (2) where pen represents the engine power, pbat is the battery power, 𝜂𝑚 donates the electric motor efficiency, and 𝜂𝑇 indicates the transmission efficiency. 2.1.2 hev components modeling the engine modeling adopts a static map approach. based on the engine's speed and torque parameters, the corresponding fuel consumption at that speed and torque can be directly obtained from the map chart. in this paper, the transient response can be neglected, and the specific formula is: 6 f. xiao, r. zhang, c. qi 𝑚𝑓𝑢𝑒𝑙 = ∫ �̇�𝑓𝑢𝑒𝑙(𝑇𝑒𝑛𝑔𝑖𝑛𝑒 , 𝜔𝑒𝑛𝑔𝑖𝑛𝑒) 𝑡 0 𝑑𝑡 (3) where mfuel and �̇�𝑓𝑢𝑒𝑙 represent the fuel consumption and the fuel consumption rate, respectively, and 𝜔𝑒𝑛𝑔𝑖𝑛𝑒 is the engine speed. the harmful gas emissions from the engine are modeled with the rsm model. the electric motor has two operating modes: motor drive mode and regenerative braking mode, which functions as a traction motor and a generator, respectively. the power output is represented as: 𝑃𝑚𝑜𝑡𝑜𝑟 = { 𝑇𝑚𝑜𝑡𝑜𝑟𝜔𝑚𝑜𝑡𝑜𝑟 𝜂𝑚𝑜𝑡𝑜𝑟 , 𝑇𝑚𝑜𝑡𝑜𝑟 ≥ 0 𝑇𝑚𝑜𝑡𝑜𝑟𝜔𝑚𝑜𝑡𝑜𝑟𝜂𝑚𝑜𝑡𝑜𝑟 , 𝑇𝑚𝑜𝑡𝑜𝑟 < 0 (4) where pmotor, ωmotor, and ηmotor indicate the electric motor output power, output speed, and efficiency, respectively. the rate of change of the state of charge (soc) is given by: 𝑑𝑆𝑂𝐶 = 𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦(𝑡) 𝑄𝑏𝑎𝑡𝑡𝑒𝑟𝑦 = − 𝐸𝑜𝑐−√𝐸𝑜𝑐 2 −4𝑅𝑏𝑎𝑡𝑡𝑒𝑟𝑦𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦 2𝑅𝑏𝑎𝑡𝑡𝑒𝑟𝑦𝑄𝑏𝑎𝑡𝑡𝑒𝑟𝑦 (5) where pbattery represents the battery charge and qbattery is the total battery capacity. 2.1.3 driver modeling the forward simulation modeling approach is applied, which requires the establishment of a driver model to obtain the acceleration and braking signals. the driver model is constructed based on pid control, by adjusting the pid parameters to calculate the pedal opening based on the target vehicle speed and the actual vehicle speed, as shown in eq. (6): { 𝛷𝑟𝑒𝑞 = 𝐾𝑝 ∙ 𝑒(𝑡) + 𝐾𝑖 ∙ ∫ 𝑒(𝑡)𝑑𝑡 + 𝐾𝑑 𝑑𝑒(𝑡) 𝑑𝑡 𝑒(𝑡) = 𝑣𝑇𝑎𝑟 − 𝑣 (6) where, e represents the deviation between the target vehicle speed and the actual vehicle speed, and kp, ki, and kd mean the parameters of the pid controller, respectively. 2.2 strategy assumptions and scenario classification to make the problem more specific and complete the task better, the following assumptions are made: (1) the connected and automated vehicle (cav) can communicate in real-time with the infrastructure on the roadside during driving, and the vehicle can obtain traffic information, location information, and signal light information at intersections through communication technology. (2) traffic violations by pedestrians and interference from non-motorized vehicles are not considered. (3) the state of the yellow light is not considered. in the smart connected traffic environment, the traffic lights at the intersection have the ability to communicate with the vehicles, and the communication range threshold between the signal light and the vehicle can be represented as rc. when the distance is less scenario-level hierarchical energy management control strategy for hybrid electric vehicle 7 than rc, the signal light needs to provide timing information, and the strategy at the intersection is triggered. when the distance is greater than rc, the strategy at the intersection is not needed, and the vehicle drives under its own car-following behavior. when there is no leading vehicle and no intersection ahead, the vehicle is in a free-driving state. in summary, the scenario classification strategy of this paper is as follows: { 𝐸𝑐𝑜 − 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑦 𝑓𝑜𝑟 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑠𝑐𝑒𝑛𝑒𝑠: 𝑅 < 𝑅𝑐 𝐸𝑐𝑜 − 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑦 𝑓𝑜𝑟 𝑡𝑦𝑝𝑖𝑐𝑎𝑙 𝑠𝑐𝑒𝑛𝑒𝑠: 𝑅 ≥ 𝑅𝑐 𝐸𝑐𝑜 − 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑦 𝑓𝑜𝑟 𝑓𝑟𝑒𝑒 − 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑠𝑐𝑒𝑛𝑒𝑠: 𝑛𝑜 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑜𝑟 𝑛𝑜 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑎ℎ𝑒𝑎𝑑 (7) 2.3 shed modeling under reinforcement learning in the free-driving scenario, the following vehicle's speed should not exceed the maximum speed limit of the lane the leading vehicle is in, nor should it exceed the maximum constraint speed of the vehicle. once the following vehicle accelerates to the maximum permitted speed, it should enter a stable speed control mode and maintain that speed until a different traffic situation is encountered. the speed limit is given by: 𝐹𝐹𝑟𝑒𝑒 = 𝑣𝑙𝑖𝑚(𝑘) − 𝑣(𝑘) (8) 𝑣𝑙𝑖𝑚(𝑘) = 𝑚𝑖𝑛(𝑣𝑙𝑎𝑛𝑒 , 𝑣𝑚𝑎𝑥) (9) the value of vlim(k) is selected as the smaller of the two speeds, which are the lane speed limit vlane and the maximum allowed speed of the following vehicle vmax, over the time step k. the reward function for the strategy in the free-driving scenario is as follows: 𝑅𝑓𝑟𝑒𝑒𝑑𝑜𝑚 = 𝛼1𝐹𝑢𝑒𝑙 + 𝛼2𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦 + 𝛼3𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛 (10) if ffree is negative, the reinforcement learning reward function is directly set to -1 to receive a negative reward. security represents the condition where there is no collision with other traffic environments. the reinforcement learning state vector is: 𝑆𝑡_𝑓𝑟𝑒𝑒𝑑𝑜𝑚 = [𝑉𝑒𝑔𝑜 , 𝐷𝑒𝑔𝑜 , 𝑖, 𝑆𝑂𝐶, 𝐷𝑉2𝑆] 𝑇 (11) where vego indicates the reference vehicle speed, dego represents the distance to the leading vehicle, i means the road slope, soc donates the state of charge of the battery, and dv2s is the distance to the destination. under the typical scenario condition, the car-following model algorithm can integrate environmental characteristics and driver behavior characteristics while ensuring safety. by incorporating energy management and pollutant emissions, the following reward function is obtained: 𝑅𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 = 𝑅𝑓𝑟𝑒𝑒𝑑𝑜𝑚 + 𝐹𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 (12) the reinforcement learning state vector is: 𝑆𝑡_𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 = [𝑉𝑒𝑔𝑜 , 𝐷𝑒𝑔𝑜 , 𝑉𝑝𝑟𝑒 , 𝑎𝑝𝑟𝑒 , 𝐷ℎ , 𝑖, 𝑆𝑂𝐶, 𝐷𝑉2𝑆] 𝑇 (13) 8 f. xiao, r. zhang, c. qi where, vpre represents the speed of the leading vehicle, apre indicates the acceleration of the leading vehicle, dh means the distance to the leading vehicle. in the intersection scenario, the optimization function is composed of objectives for safety, energy management, and harmful substance emission reduction. the reinforcement learning reward function is as follows: 𝑅𝑠𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 = 𝛽1𝐹𝑢𝑒𝑙 + 𝛽2𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦 + 𝛽3𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛+𝛽4𝑇𝑟𝑎𝑓𝑓𝑖𝑐𝑙𝑖𝑔ℎ𝑡 + 𝐹(𝑠𝑡 , 𝑎𝑡) (14) the reinforcement learning state vector is: 𝑆𝑡 = [ 𝑉𝑒𝑔𝑜 , 𝐷𝑒𝑔𝑜 , 𝑉𝑝𝑟𝑒 , 𝑎𝑝𝑟𝑒 , 𝐷ℎ , 𝑖, 𝑆𝑂𝐶, 𝐷𝑉2𝑆, 𝑡𝑟_𝑟𝑒𝑚 , 𝑡𝑔_𝑟𝑒𝑚, 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛, 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 ] 𝑇 (15) that is the hierarchical scenario structure for the shed algorithm. the flow chart of the algorithm is shown in fig. 2. the energy management control strategies can be divided into three typical scenarios: free-driving, car-following, and intersection. in the free-driving scenario, the factors affecting consist of fuel consumption, driving safety, and emissions. the reward function only considers these three factors. for the typical car-following scenario, the strategy is built upon the free-driving case, with additional factors such as the adaptive factor, mutual information, high-dimensional features, and driver's style. the corresponding reinforcement learning state vector is expanded accordingly. under the intersection scenario, the strategy extends the freedriving case by incorporating an additional reward that combines self-supervised internal reward functions, calibration formulas, and traffic signal status. the reinforcement learning state vector is further expanded to capture these intersection-specific elements. 3. adaptive vehicle following strategy based on imitation learning the adaptive strategy learning approach that combines imitation learning, as well as an imitation learning strategy that incorporates the driver's style are proposed. the factors influencing vehicle car-following can be broadly categorized into two aspects: the impact of the environment itself on the driver's behavior and the inherent influence of the individual driver's behavioral habits on the car-following strategy. 3.1 environmental factors influencing driver behavior: adaptive strategy learning for complex and ambiguous environments, imitation learning strategies, which overcome the obstacle of manually specifying an appropriate reward function, have the potential to outperform reinforcement learning strategies [33-35]. in real-world tasks, drivers may exhibit multi-intent behavior when following a lead vehicle, and different drivers have varying driving habits [36,37]. these diverse driving habits make it challenging to select a suitable reward function. the car-following model with adaptive scenario-level hierarchical energy management control strategy for hybrid electric vehicle 9 fig. 2 scene-level hierarchical energy management control strategy factor that addresses the inherent phenomenon of environmental influences on driver behavior is introduced in this section. the main idea is to construct an adaptive gan framework based on the principles of gan, which trains the car-following policy directly without estimating the reward function. as is shown in eq. (16): 10 f. xiao, r. zhang, c. qi 𝑚𝑖𝑛 𝜋   𝑚𝑎𝑥 𝐷∈(0,1)𝑠×𝐴  𝔼𝜋[𝑙𝑜𝑔 𝐷(𝑠, 𝑎)] + 𝔼𝜋𝐸[𝑙𝑜𝑔(1 − 𝐷(𝑠, 𝑎))] − 𝜆𝐻(𝜋) (16) where, 𝜋𝐸 represents the expert policy，𝜋 indicates the policy that needs to be learned, d serves as the discriminator, and 𝐻(𝜋) means the causal entropy, which is shown in eq. (17): 𝐻(𝜋) ≜ 𝐸𝜋[− 𝑙𝑜𝑔 𝜋( 𝛼 ∣ 𝑠 )] (17) despite the adaptive policy is model-free, it still requires interacting with the environment during training. due to the presence of stochastic factors, the resulting trajectories can exhibit significant variations across different individuals. even the same individual may make different decisions when facing the same situation, leading to the generation of multiple distinct policies. in this scenario, a set of expert policies: 𝜋𝐸 = {𝜋𝐸 0, 𝜋𝐸 1 ,… } and the process of generating expert trajectories: 𝑠0 ∼ 𝜌0𝑐 ∼ 𝑝(𝑐), 𝜋 ∼ 𝑝(𝜋 ∣ 𝑐)0𝑎𝑡 ∼ 𝜋(𝑎𝑡 ∣ 𝑠𝑡), 𝑠𝑡+1 ∼ 𝑃(𝑠𝑡+1 ∣ 𝑎𝑡 , 𝑠𝑡) are defined, where the adaptive variable 𝑐 with a prior probability distribution 𝑝(𝑐) are also introduced. the goal of the algorithm is to recover the policy 𝜋(𝑎 ∣ 𝑠, 𝑐), under the adaptive variable 𝑐. to ensure a tighter coupling between the adaptive factor and the policy, and to increase their correlation, we incorporate mutual information into the optimization function. 𝐼(𝑐; 𝜏) = 𝐻(𝑐) − 𝐻( 𝑐 ∣ 𝜏 ) = 𝐻(𝑐) + ∫   𝑐  ∫   𝜏  𝑝(𝑐, 𝜏) 𝑙𝑜𝑔 𝑝(𝑐, 𝜏) 𝑑𝑐𝑑𝜏 = 𝐻(𝑐) + ∫   𝑐  ∫   𝜏  𝑝(𝑐)( 𝜏 ∣ 𝑐 ) 𝑙𝑜𝑔 𝑝( 𝑐 ∣ 𝜏 ) 𝑑𝑐𝑑𝜏 = 𝐻(𝑐) + 𝔼𝑐∼𝑝(𝑐),𝑎∼𝜋( ⋅∣∣𝑠, 𝑐 )[𝑙𝑜𝑔 𝑝(𝑐|𝜏)]] (18) incorporating the adaptive factor into the optimization objective function, the equation can be formulated as: 𝑚𝑖𝑛 𝜃−𝜓  𝑚𝑎𝑥 𝜔  𝔼𝜋𝜃[𝑙𝑜𝑔 𝐷𝜔(𝑠, 𝑎)] + 𝔼𝜋𝐸[𝑙𝑜𝑔(1 − 𝐷𝜔(𝑠, 𝑎))] − 𝜆1𝐼(𝑐; 𝜏) − 𝜆2𝐻(𝜋𝜃) (19) given the difficulty of computing the posterior probability p(c|τ) during the optimization process, the q-value is applied directly as a replacement, as shown in the following equation: 𝐿1(𝜋, 𝑄) ≤ 𝐼(𝑐; 𝜏) (20) 𝑚𝑖𝑛 𝜃−𝜓  𝑚𝑎𝑥 𝜔  𝔼𝜋𝜃[𝑙𝑜𝑔 𝐷𝜔(𝑠, 𝑎)] + 𝔼𝜋𝐸[𝑙𝑜𝑔(1 − 𝐷𝜔(𝑠, 𝑎))] − 𝜆1𝐿1(𝜋, 𝑄) − 𝜆2𝐻(𝜋𝜃) (21) 3.2 surpassing expert-level performance and learning high-dimensional features in the car-following task, the limitations of the expert trajectories input to the network can lead to the situation where the error of the learned policy exceeds that of the expert policy. for the car-following task, it is difficult to define a suitable reward function. from the perspective of the policy, the problem becomes relatively more straightforward. the car-following task model from the policy standpoint is shown in eq. (22): scenario-level hierarchical energy management control strategy for hybrid electric vehicle 11 𝜂(𝜋𝜃) = 𝔼𝑠~𝜀𝐴[𝑟(𝑠)] (22) the objective optimization function can be transformed as follows: 𝑚𝑖𝑛 𝜃−𝜓  𝑚𝑎𝑥 𝜔  𝔼𝜋𝜃[𝑙𝑜𝑔 𝐷𝜔(𝑠, 𝑎)] + 𝔼𝜋𝐸[𝑙𝑜𝑔(1 − 𝐷𝜔(𝑠, 𝑎))] −𝜆𝜃𝜂(𝜋𝜃) − 𝜆1𝐿1(𝜋𝜃 , 𝑄𝜓) − 𝜆2𝐻(𝜋𝜃) (23) where 𝜆 represents a hyper-parameter. the reinforcement function for policy optimization consists of two components: the surrogate reward and the discriminator, which aims to mimic the expert. this approach allows to overcome the limitation of imitation learning. to effectively learn strategies for high-dimensional inputs, the original gail framework has been improved as shown in eq. (24). to address traditional gan networks suffer from issues such as gradient vanishing and mode collapse, the objective function has been further enhanced by incorporating wasserstein gan techniques, as follows: 𝑅𝐹 = &𝑚𝑖𝑛 𝜃−𝜓  𝑚𝑎𝑥 𝜔  𝔼𝜋𝜃[𝐷𝜔(𝑠, 𝑎)] − 𝔼𝜋𝐸[𝐷𝜔(𝑠, 𝑎)] −𝜆𝜃𝜂(𝜋𝜃) − 𝜆1𝐿1(𝜋𝜃 , 𝑄𝜓) − 𝜆2𝐻(𝜋𝜃) (24) where the discriminator network 𝐷𝜔 and the posterior approximation network are treated as separate networks. 3.3 optimizing car-following model by integrating driver's driving style during vehicle operation, it is important to consider not only the impact of the driving environment on the model, but also the need to satisfy the driver's comfort and safety in the car-following task. in this section, the essential characteristics of driving behavior is explored under different driving modes, taking into account the vehicle's own dynamics. the optimization strategy also incorporates the driver's stylistic characteristics in the carfollowing task. the driver is the core of the vehicle-road-traffic system. although the carfollowing behavior describes the relationship between the lead and the following vehicles, its essence is a description of the driver's behavior in a given traffic environment. 3.3.1 driver style clustering analysis to establish a personalized car-following model, the driving data that can reflect the driver's style should be classified based on different driving styles. combining the intervehicle distance and the velocity of the lead vehicle, the time-to-collision inverse (ttci) is defined as follows: 𝑇𝑇𝐶𝐼 = ∆𝑣 𝑑 (25) where, ∆𝑣 represents the difference in velocity between the following vehicle and the lead vehicle, and d donates the distance between the vehicles. in summary, the behavior features consist of the time-to-collision (ttc), time headway, and the absolute values of acceleration and deceleration indirectly reflect the driver's style. the driver's style can be broadly categorized into two main groups: conservative and aggressive. the k-means clustering strategy [38] can be used to cluster the driving data into these two driving style categories. 12 f. xiao, r. zhang, c. qi 3.3.2 integrated car-following strategy considering driver style features given that individuals have different perceptions of comfort, studying the car-following model alone is clearly insufficient. in this section, integrating the driver's style into the carfollowing strategy is introduced, which enables the car-following model to have personalized characteristics, improving the overall car-following task. this strategy can effectively solve the multi-objective optimization problem in the car-following task. drivers with the same style tend to maintain similar time head-ways in typical carfollowing scenarios, but drivers with different styles can have significant differences in time headway. the formula for the inter-vehicle distance can be expressed as: ∆𝑑𝑑𝑒𝑠 = 𝑣𝑓𝑡ℎ + 𝑑0 (26) where vf is the traveling speed of the following vehicle, th represents the progression of time, and d0 means the safe distance when the vehicle is traveling at very low speeds. the goal of vehicle car-following is for the driver to adjust the vehicle's speed to match the speed of the lead vehicle, and maintain the inter-vehicle distance around the desired value. inter-vehicle distance error ∆𝑑𝑒𝑟𝑟(𝑘) is defined as: ∆𝑑𝑒𝑟𝑟(𝑘) = ∆𝑑 − ∆𝑑𝑑𝑒𝑠 (27) where ∆𝑑𝑒𝑟𝑟(𝑘) → 0, ∆𝑣(𝑘) → 0. to provide comfort to passengers during car-following, the absolute values of both acceleration and jerk should also be minimized as much as possible. jk, which represents the rate of change of acceleration is defined as: 𝑗(𝑘) = 𝑎(𝑘)−𝑎(𝑘−1) ∆𝑡 (28) where |𝑎(𝑘)| → 0, |𝑗(𝑘)| → 0. when solving the optimization problem, certain constraints must be satisfied. to avoid colliding with the lead vehicle, the inter-vehicle distance must satisfy the minimum distance constraint. ensuring the following vehicle remains in the car-following scenario, this distance should be less than the maximum car-following distance. the minimum and maximum values of velocity, acceleration, and jerk must also be constrained. model predictive control can handle multi-variable and constrained problems, making it suitable for incorporating the car-following model within an mpc framework. based on the principles of mpc, the cost function can be defined as: 𝐽(𝜉(𝑘), 𝑢(𝑘 − 1), 𝛥𝑢(𝑘)) = &∑ ∥∥𝜂( 𝑘 + 𝑖 ∣ 𝑘 ) − 𝜂ref ( 𝑘 + 𝑖 ∣ 𝑘 )∥∥𝑄 2𝑁𝑃 𝑖=1 +∑ ∥ 𝛥𝑢( 𝑘 + 𝑖 ∣ 𝑘 ) ∥𝑅 2𝑁𝑐−1 𝑖=1 (29) where 𝜼𝐫𝐞𝐟(𝑘 + 𝑖 ∣ 𝑘) represents the reference vector, and q and r are the weighted matrices for the control and target variables, respectively, where r is a one-dimensional vector with a value of 10, and q is a fourth-order square matrix, respectively, which, in the car-following task, can be defined as follows: 𝜂𝑟𝑒𝑓( 𝑘 + 𝑖 ∣ 𝑘 ) = [𝛥𝑑𝑑𝑒𝑠( 𝑘 + 𝑖 ∣ 𝑘 ) 0 0 0]𝑇 𝛥𝑢( 𝑘 + 𝑖 ∣ 𝑘 ) = 𝑗( 𝑘 + 𝑖 ∣ 𝑘 )𝛥𝑡 (30) scenario-level hierarchical energy management control strategy for hybrid electric vehicle 13 the optimization objective and the constraints for the model predictive control are defined as: 𝑠. 𝑡. { 𝑑0 ≤ 𝛥𝑑(𝑘) ≤ 𝑑𝑚𝑎𝑥 , 𝑣𝑚𝑖𝑛 ≤ 𝑣(𝑘) ≤ 𝑣𝑚𝑎𝑥 𝑎𝑚𝑖𝑛 ≤ 𝑎(𝑘) ≤ 𝑎𝑚𝑎𝑥 𝑗𝑚𝑖𝑛 ≤ 𝑗(𝑘) ≤ 𝑗𝑚𝑎𝑥 (31) considering driving styles, the aggressive strategy should have safety and comfort, while the conservative driving strategy should have higher safety and comfort. the final cost function is defined as: 𝐹𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 = 𝑚𝑖𝑛 𝜃−𝜓  𝑚𝑎𝑥 𝜔  𝔼𝜋𝜃[𝑙𝑜𝑔 𝐷𝜔(𝑠, 𝑎)] − 𝔼𝜋𝐸[𝐷𝜔(𝑠, 𝑎)] − 𝜆0𝜂(𝜋𝜃) −𝜆𝐼𝐿𝐼(𝜋𝜃 , 𝑄𝜗) − 𝜆2𝐻(𝜋𝜃) − 𝜆3𝐽(𝜉(𝑘), 𝑢(𝑘 − 1), 𝛥𝑢(𝑘)) (32) 4. reinforcement learning optimization of energy management control strategies in intersection scenarios the factors affecting the driving control strategy at intersections can be divided into two main aspects. the first is the impact of sparse reward functions on the driving control strategy, and the second is the impact of the complexity and uncertainty of intersections on the driving control strategy. to address these issues, two approaches: self-supervised additional reward function and calibration equation for adjusting the training direction are proposed. 4.1 reconstruction of the safety reward function during the actual training process, the safety reward function often suffers from sparsity, which is detrimental to the learning of the intelligent agent. given the above issues, it is crucial to redesign the safety reward function considering the potential impact of the current behavior on future states. in the context of intelligent and connected vehicles, the vehicle itself can receive information about the phase and distance of the next traffic light. drivers generally expect the next traffic light to be green, allowing them to quickly pass through the intersection. based on the remaining time of the red light and the duration of the green light, the speed boundaries at the intersection can be determined as: 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 = 𝑚𝑖𝑛 { 𝑉𝑙𝑖𝑚𝑖𝑡 , 𝐷𝑉2𝑆 ÷ [𝑡𝑟_𝑟𝑒𝑚 + 𝑘(𝑡𝑔_𝑐𝑦𝑐𝑙𝑒 + 𝑡𝑟_𝑐𝑦𝑐𝑙𝑒)] } (33) 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛 = 𝐷𝑉2𝑆 ÷ [𝑡𝑟_𝑟𝑒𝑚 + 𝑡𝑔_𝑐𝑦𝑐𝑙𝑒 + 𝑘(𝑡𝑔_𝑐𝑦𝑐𝑙𝑒 + 𝑡𝑟_𝑐𝑦𝑐𝑙𝑒)] (34) where, vlimit represents the speed limit of the road. dv2s indicates the distance to the stop line at the intersection. tr_cycle shows the remaining red light time. tg_cycle means the duration of the green light. tr_rem is the duration of the red light. 𝑘 = 0,1,2… ,∞ represents the delay period. when the vehicle can pass through the next red-green light during the green light time tw, k = 0. if the lower bound of the green light speed boundary exceeds the road speed limit, it means the vehicle cannot pass through the next red-green light. in this case, we can adjust the delay period 𝑘 to select a suitable green light. 14 f. xiao, r. zhang, c. qi similarly, when the traffic light is green, the speed boundary can be derived as: 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 = { 𝑉𝑙𝑖𝑚𝑖𝑡, 𝑘 = 0 𝑚𝑖𝑛 { 𝑉𝑙𝑖𝑚𝑖𝑡, 𝐷𝑉2𝑆 ÷ [𝑡𝑔_𝑟𝑒𝑚 + (𝑘 − 1)(𝑡𝑔_𝑐𝑦𝑐𝑙𝑒 + 𝑡𝑟_𝑐𝑦𝑐𝑙𝑒) + 𝑡𝑟_𝑐𝑦𝑐𝑙𝑒] } 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 { 𝑘 = 0 𝑘 = 0,1,2… ,∞ (35) 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛 = 𝐷𝑉2𝑆 ÷ [𝑡𝑔_𝑟𝑒𝑚 + 𝑘(𝑡𝑔_𝑐𝑦𝑐𝑙𝑒 + 𝑡𝑟_𝑐𝑦𝑐𝑙𝑒)] where, tg_rem is the remaining green light time. if the ego vehicle's speed exceeds the speed boundaries, there is a risk of running the red light. the traffic light reward function can be represented as a piece-wise function: 𝑟𝑙𝑖𝑔ℎ𝑡 = { 𝛽1 + 𝑙𝑛(𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥) 2 , if 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 < 𝑣(𝑛) and 𝛽1 < (𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥) 2 (𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥) 2 , if 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 < 𝑣(𝑛) and (𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥) 2 ≤ 𝛽1 0, if 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛 ≤ 𝑣(𝑛) ≤ 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑎𝑥 (𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛) 2 , if 𝑣(𝑛) < 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛 and (𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛) 2 ≤ 𝛽2 𝛽2 + 𝑙𝑛(𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛) 2 , if 𝑣(𝑛) < 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛 and 𝛽2 < (𝑣(𝑛) − 𝑉𝑙𝑖𝑔ℎ𝑡_𝑚𝑖𝑛) 2 (36) where, 𝛽1 and 𝛽2 represent the boundary points of the green light speed boundary, respectively. when the vehicle speed exceeds the green light speed boundary, we can calculate a negative reward based on the proposed traffic light reward function, rather than only calculating a sparse reward when the vehicle runs the red light. 4.2 design of additional reward functions the reward function is the guiding direction for the entire task, determining the objectives the agent needs to achieve. the reward function can be simply designed as a combination of fuel consumption, safety, and pollution emissions. additional reward functions to train the neural network are designed innovatively, guiding the vehicle's actions and enabling the hybrid-electric vehicle agent to have the capability of selfexploration, while preventing actions that the agent may regret later. the general approach to improving the reward function is: 𝑟′(𝑠𝑡 , 𝑎𝑡) = 𝑟(𝑠𝑡 , 𝑎𝑡) + 𝐹(𝑠𝑡 , 𝑎𝑡) (37) where, f(st,at) represents additional reward. 𝑟(𝑠𝑡 , 𝑎𝑡) = 𝛽1𝑟𝐹𝑢𝑒𝑙 + 𝛽2𝑟𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦 + 𝛽3𝑟𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛 is basic reward function. the recursive equation for the expected cumulative reward of the revised reward function can be expressed as: 𝑄𝑛𝑒𝑤 𝑧 (𝑠𝑡 , 𝑎𝑡) = 𝑟 ′(𝑠𝑡 , 𝑎𝑡) + 𝛾𝐸𝑟𝑡,𝑠𝑡,1 − 𝐸[𝑄𝑛𝑒𝑤 𝑧 (𝑠𝑡+1, 𝑎𝑡+1)] (38) according to eq. (38), after modifying the reward function, the new q-value of the markov decision process can be derived as: 𝑄𝑛𝑒𝑤 𝜋 (𝑠𝑡 , 𝑎𝑡) = 𝔼𝑟𝑖≥𝑡,𝑠𝑖>𝑡~𝐸 [∑ 𝛾𝑖−𝑡𝑇 𝑖=𝑡 𝑟𝑖 ′ (𝑠𝑖 , 𝑎𝑖)] = 𝔼𝑟𝑖≥𝑡,𝑠𝑖>𝑡~𝐸[∑ 𝛾𝑖−𝑡𝑇 𝑖=𝑡 (𝑟(𝑠𝑡 , 𝑎𝑡) + 𝐹(𝑠𝑡 , 𝑎𝑡))] = 𝑄𝜋(𝑠𝑡 , 𝑎𝑡) + 𝔼𝑟𝑖≥𝑡,𝑠𝑖>𝑡~𝐸[∑ 𝛾𝑖−𝑡𝑇 𝑖=𝑡 𝐹(𝑠𝑡 , 𝑎𝑡)] (39) scenario-level hierarchical energy management control strategy for hybrid electric vehicle 15 as discussed, a self-supervised strategy to compute the additional rewards is designed, allowing the agent to self-explore and complete the ecological tasks at intersections. selfsupervised learning can be viewed as a subset of unsupervised learning, where the agent learns from the inherent relationships in the data without the need for extensive labeled datasets. in the shems, the rewards need to be constructed through the automatic generation of pseudo-labels. specifically, the agent predicts the next state based on the current state and action, and combines the predicted state with the current state to construct the 𝐿𝑜𝑠𝑠 function. the addition of the self-supervised module gives the model a certain level of generalization capability. the self-supervised internal reward functions are shown in eq. (40) and eq. (41), where the model refers to a generic deep learning prediction model: 𝑆𝑡+1 ∗ = 𝑀𝑜𝑑𝑒𝑙(𝑠𝑡 , 𝑎𝑡) (40) 𝐹(𝑠𝑡 , 𝑎𝑡) = 𝑙𝑜𝑠𝑠(𝑆𝑡+1, 𝑆𝑡+1 ∗ ) (41) the essence of reinforcement learning lies in the ability to predict the next action based on the current state. by executing the action for the next time step, the agent obtains the state for the following time step. the uncertainty in the environment will directly impact the agent's choice of actions, and the agent's actions will also directly influence the state vector. the key to maintaining a stable state is the accuracy of the action values. the accuracy of the action values is regarded as a factor in the calibration formula: 𝑟𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑅𝑀𝑆𝐸(𝑎𝑡+1, 𝑎𝑡+1 ∗ ) (42) where 𝑎𝑡+1 ∗ ,which represents the action for the next time step, can be calculated by the formula 𝑎𝑡+1 ∗ = model2(𝑎𝑡 , 𝑠𝑡). a represents the true action values. by integrating the reinforcement learning calibration and the additional rewards generated through selfsupervision, a new additional reward function is updated as eq. (43): 𝐹(𝑠𝑡 , 𝑎𝑡) = 𝑙𝑜𝑠𝑠(𝑆𝑡+1, 𝑆𝑡+1 ∗ ) 𝑅𝑀𝑆𝐸(𝑎𝑡+1, 𝑎𝑡+1 ∗ )⁄ (43) the final reward function for the self-supervised reinforcement learning calibration method is shown in eq. (44): 𝑅𝑠𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 = 𝛽1𝐹𝑢𝑒𝑙 + 𝛽2𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦 + 𝛽3𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛 + 𝛽4𝑇𝑟𝑎𝑓𝑓𝑖𝑐𝑙𝑖𝑔ℎ𝑡 + 𝐹(𝑠𝑡 , 𝑎𝑡) (44) 5. experimental results 5.1 results of the car-following experiment 5.1.1 data process in this study, the car-following model training utilized the high-resolution waymo open dataset [39]. given the measurement errors of the equipment, which may have a significant impact on the analysis of trajectory data, it is necessary to establish an appropriate noise filtering mechanism in the numerical calculation process. we opted to apply the widely16 f. xiao, r. zhang, c. qi used savitzky-golay filter [ 40], which calculates the smoothed data by employing polynomial fitting and interpolation methods to address this issue. as expresses in eq. (45): 𝑔𝑖 = ∑ 𝑐𝑛𝑓𝑖+𝑛 𝑛𝑅 𝑛=−𝑛𝐿 (45) where, fi+n represents the smoothed signal, nl denotes the number of data points to the left of the current data point, nr indicates the number of data points to the right of the current data point, and cn means the weight values. the specific data processing approach involves using a second-order savitzky-golay filter to filter each speed data point. the acceleration and deceleration are calculated based on the filtered speed data. further, an additional smoothing method is employed to eliminate any residual noise. 5.1.2 results of the simulation experiments to verify the accuracy and reasonableness of the car-following results, an analysis of the simulation outcomes is conducted. 80 % of the waymo data is selected as the training set to train the model, and the remaining 20 % of the data is designed for testing. the training parameters are shown in table 2. table 2 car-following model training parameters parameters value learning rate 0.001 discount factor 0.99 training episodes 400 analysis of the time-speed and time-distance curves presented in fig. 3, depicting three distinct operating scenarios, reveals that the proposed algorithm exhibits highly satisfactory car-following behavior. critically, the algorithm consistently maintains the following vehicle within a safe distance envelope relative to the leader. furthermore, it demonstrates a dynamic and responsive capability, as the following vehicle's speed continuously adapts to fluctuations in the leading vehicle's speed. this observed performance effectively confirms the model's operational validity. to quantitatively assess and clearly illustrate the differences in speed tracking accuracy between the following vehicle and the leading vehicle across the evaluated models, the sum of squared errors (sse) was calculated for each dataset. the comparative results highlight significant performance variations: the average sse values for the optimal speed model, the idm (intelligent driver model), the stimulus-response model, and the safe distance model are 24.2, 15.08, 7.52, and 3.1, respectively. crucially, the proposed model achieves a substantially lower sse of 2.89. this demonstrably smaller error value underscores the superior effectiveness of the novel approach in accurately replicating the nuanced car-following strategies characteristic of expert human driving. scenario-level hierarchical energy management control strategy for hybrid electric vehicle 17 (a) (b) (c) fig. 3 time-distance and time-velocity plots for car-following behavior: (a) steady-state condition; (b) acceleration condition; (c) deceleration condition to further evaluate the conformity of the car-following model with the expert algorithm, the time-speed and time-distance curves for the algorithm proposed in this chapter, the expert algorithm, and the leading vehicle are presented in fig. 4. the proposed algorithm is closer to the leading vehicle in certain regions compared to the expert algorithm. in some road segments, the proposed algorithm even surpasses the expert algorithm, and the consistency between the vehicle speed and the leading vehicle is also more closely aligned than the expert algorithm. 18 f. xiao, r. zhang, c. qi (a) (b) (c) fig. 4 comparison of the algorithm in this paper to an expert-level algorithm: (a) steady-state condition; (b) acceleration condition; (c) deceleration condition in order to quantify the results shown in fig. 4, the speed difference between the vehicle model trained with the two different algorithms and the leading vehicle are calculated, as shown in table 3. to more clearly illustrate the performance gap between the proposed algorithm and the expert algorithm, the average values of the speed differences are calculated for each case with 0.38, 0.0026, 0.21. the results indicate that the algorithm developed in this chapter can sometimes outperform the expert algorithm. scenario-level hierarchical energy management control strategy for hybrid electric vehicle 19 table 3 quantitative comparison of speed metrics between the algorithm in this paper and an expert-level steady-state condition (m/s) acceleration condition (m/s) deceleration condition (m/s) time (s) expert paper difference expert paper difference expert paper difference 0 0.099 0.148 0.049 0.000 0.050 0.050 0.752 -0.071 -0.823 0.5 0.049 0.248 0.200 0.573 0.112 -0.461 -0.032 -0.780 -0.748 1 -0.001 0.298 0.299 -0.773 0.120 0.893 -0.832 -1.135 -0.303 1.5 -0.001 0.347 0.348 -0.021 0.110 0.131 -0.355 -0.851 -0.496 2 -0.050 0.396 0.446 0.204 -1.210 -1.414 -0.177 -0.780 -0.603 2.5 0.049 0.446 0.397 -0.001 1.210 1.211 -0.390 -0.851 -0.461 3 0.000 0.544 0.544 -0.240 -1.410 -1.170 -0.774 -0.993 -0.218 3.5 0.100 0.541 0.442 0.891 -1.220 -2.111 -0.832 -1.348 -0.516 4 0.199 0.538 0.339 -0.376 1.630 2.006 -0.142 -0.922 -0.780 4.5 0.150 0.637 0.487 -0.509 1.800 2.309 -0.567 -0.922 -0.355 5 0.200 0.537 0.338 -1.233 -1.500 -0.267 -0.893 -1.206 -0.313 5.5 0.299 0.587 0.288 1.598 0.430 -1.168 -0.761 -0.496 0.265 6 0.300 0.588 0.288 -1.054 -1.100 -0.046 -0.176 -0.355 -0.179 6.5 0.349 0.639 0.290 -0.177 0.200 0.377 -0.113 -0.271 -0.158 7 0.300 0.590 0.290 -0.200 1.200 1.400 -0.170 -0.229 -0.059 7.5 0.398 0.640 0.242 -0.101 0.800 0.901 -0.170 -0.350 -0.180 8 0.399 0.640 0.242 1.679 -1.000 -2.679 -0.228 -0.294 -0.066 8.5 0.399 0.640 0.241 0.075 -0.200 -0.275 -0.026 -0.166 -0.140 9 0.351 0.838 0.487 -0.211 0.120 0.331 -0.108 -0.265 -0.158 9.5 0.450 0.839 0.389 0.514 0.760 0.246 -0.315 -0.315 0.000 10 0.297 0.941 0.644 0.644 1.23 0.586 -0.600 -0.600 0.000 10.5 0.309 0.730 0.421 0.421 -2.1 -2.521 -0.550 -0.500 0.050 11 0.136 0.660 0.524 0.524 0.79 0.266 -0.500 -0.200 0.300 11.5 0.192 0.966 0.774 0.774 -0.82 -1.594 -0.600 -0.300 0.300 12 0.200 0.814 0.614 0.614 -3.06 -3.674 -0.470 -0.300 0.170 5.2 simulation results in a mixed scenario the simulation is conducted on a pc with an i5-12400 cpu and 16 gb of memory, running in matlab and sumo environments. the following vehicle is modeled as a plug-in hybrid electric vehicle (phev) with intelligent connected vehicle (icv) capabilities. the initial conditions for the vehicle are a battery soc of 0.65 and a full fuel tank. the simulation results with the proposed car-following algorithm are presented in fig. 5. the blue solid line represents the speed of the leading vehicle, while the red dashed line depicts the speed of the following vehicle after the optimization of the car-following algorithm. the speed-tracking results demonstrate that the proposed algorithm is able to closely follow the leading vehicle's velocity profile. 20 f. xiao, r. zhang, c. qi fig. 5 speeds of the lead vehicle and the following vehicle after the implementation of the global car-following algorithm, the strategy at intersections is further integrated, resulting in the engine torque distribution under the given driving conditions, as shown in fig. 6. the torque distribution of the engine, generator, and electric motor in the non-hierarchical case are presented in fig. 6(a). in contrast, the torque distribution of the engine, generator mg1, and electric motor mg2 under the hierarchical control strategy is illustrated in fig. 6(b). (a) (b) fig. 6 engine torque, mg1 torque, and mg2 torque: (a) before layering; (b) after layering the soc trajectory under the given driving conditions is illustrated in fig. 7. the overall trend shows a decrease in soc over time. in the non-hierarchical case, the battery scenario-level hierarchical energy management control strategy for hybrid electric vehicle 21 soc exhibits a significant drop around the 1100-second mark. in contrast, the hierarchical control strategy is able to maintain a more stable soc profile throughout the driving cycle. (a) (b) fig. 7 comparison of battery soc before and after hierarchical control: (a) before layering; (b) after layering as shown in table 4, under similar initial and final soc conditions, the fuel consumption per 100 kilometers is reduced across different reinforcement learning algorithms when the hierarchical control strategy is employed, compared to the nonhierarchical approach. the bar chart in fig. 8 provides a clearer visualization of these fuel consumption improvements. the figure shows that the hierarchical control strategy resulted in reductions of 7.52 %, 5.29 %, 9.6 %, and 5.93 % in fuel consumption, respectively, when compared to the non-hierarchical case. as shown in table 5, with dqn algorithm as an example, the strategy with hierarchical control resulted in reductions of 18.3 %, 14.23 %, 16.94 %, and 20.9 % in the emissions of various harmful pollutants such as co, hc, pm, nox, compared to the non-hierarchical approach. fig. 8 fuel consumption comparison for different algorithms 22 f. xiao, r. zhang, c. qi table 4 fuel consumption per 100 km for different algorithms with similar initial and final soc strategy initial/final soc fuel consumption per 100 km (l/00 km) depletion rate ppo shems (offline) 0.65/0.51 5.314 ppo shems (hil) 0.65/0.51 4.917 7.52 % a2c shems (offline) 0.65/0.51 5.212 a2c shems (hil) 0.65/0.502 4.936 5.29 % a3c shems (offline) 0.65/0.514 5.34 a3c shems (hil) 0.65/0.51 4.826 9.6 % dqn shems (offline) 0.65/0.489 5.23 dqn shems (hil) 0.65/0.495 4.92 5.93 % table 5 pollutant emissions with the dqn algorithm pollutant dqn strategy (before layering) dqn strategy (after layering) decrement rate before and after layering dynamic programming co 10.25 8.34 18.3 % 8.24 hc 2.81 2.41 14.23 % 2.40 pm 0.098 0.0814 16.94 % 0.0809 nox 2.68 2.12 20.9 % 2.14 5.3 hardware-in-the-loop experiment 5.3.1 hardware-in-loop experiment to verify the effect of the strategy in real controllers, we have built a hardware-in-theloop (hil) platform. as shown in fig. 9, the experimental system consists of a hybrid power model, a driver operation system, a virtual scenario system, a sensor system, an ni real-time system, and a vehicle control unit. the virtual scenario system provides the driver with a realistic driving environment, making the driving experience more similar to reality. furthermore, it provides traffic environment information, road information, and geographic location information through data interaction. the main function of the vehicle control system is to implement the proposed strategy and output control parameters to the actuators. the driver's operation information is all fed back to the steering system, while the vehicle's speed status information and the status of the electromagnetical system are provided by the real-time simulation system. scenario-level hierarchical energy management control strategy for hybrid electric vehicle 23 fig. 9 hil test bench the integrated system is shown in fig. 10(a). in fig. 10(b), the data acquisition system is combined with the driver operation system, displayed below the driver. based on the existing configuration and technical conditions, can communication technology is applied to achieve data interaction, and real-time data of the steering wheel angle, acceleration, and brake pedal is obtained. these data are input into the vehicle control unit (vcu). (a) (b) fig. 10 integrated system and driver operating system: (a) integrated system; (b) driver operating system 24 f. xiao, r. zhang, c. qi 5.3.2 hardware-in-the-loop experiment data analysis to further validate the hierarchical strategy proposed in this work, hardware-in-theloop (hil) experiments based on the test driving cycle are conducted. the validation process is composed of three key aspects: verifying the vehicle following algorithm, analyzing the shems, and validating the harmful emissions reduction. as shown in fig. 11, the vehicle following performance can be observed. from the figure, it is evident that the vehicle is able to closely follow the lead vehicle without any collision incidents, demonstrating good driving safety. fig. 11 performance of vehicle following in hil the range of soc variations during the simulation and hil tests is shown in fig. 12. the blue line represents the battery soc changes under the offline simulation, while the red line shows the battery soc changes in the hil test. from the figure, it can be seen that both strategies are able to maintain a good range of battery soc under the real-time operation. the battery performance and state remain in a favorable condition, indicating that the battery is functioning normally, which ensures the overall reliability and stability of the strategy. fig. 12 performance of battery soc changes in hil the comparative plots of the engine torque, engine mg1 torque, and electric motor mg2 torque under the test driving cycle are depicted in fig. 13. the inset plot displays the detailed view of the local variations. it can be concluded that there are some differences scenario-level hierarchical energy management control strategy for hybrid electric vehicle 25 between the hil test data and the offline simulation data. however, these differences are within a reasonable range. overall, the variations in the real-time environment between the two sets of data are not significant, and the alignment between them is relatively satisfied. (a) (b) (c) fig. 13 performance in hil: (a) engine torque, (b) mg1 torque (c) mg2 torque as shown in table 6, by comparing the emissions of harmful substances between simulation and hil testing using the dqn algorithm, it can be summarized that the layered dqn algorithm significantly reduced pollutant emissions in the hil experiment, with the emissions approaching the levels achieved under the dynamic programming approach. 26 f. xiao, r. zhang, c. qi furthermore, the results from the hil testing closely matched those from the simulation, with only minor differences. table 6 comparison of emissions of harmful substances between simulation and hil under dqn algorithm harmful gasses dqn (before layering) simulation/hil dqn (after layering) simulation/hil dynamic programming simulation/hil co 10.25/10.02 8.43/8.31 8.24/8.23 hc 2.81/2.95 2.41/2.40 2.40/2.40 pm 0.098/0.094 0.0814/0.0801 0.0809/0.0810 nox 2.68/2.67 2.12/2.13 2.14/2.13 6. conclusion the shems refers to driving techniques and strategies that aim to reduce the environmental impact of vehicles by improving vehicle operation and driving behavior. based on a learning-based approach, a layered ecological driving strategy targeting a hybrid electric vehicle is proposed, with the objectives of driving safety, energy management, and pollutant emission reduction. simulation and hil testing are conducted to evaluate the proposed ecological driving strategy, and the performance is assessed. the main research work of this paper is as follows: the driving scenarios for the shems can be classified into free driving, car-following, and intersection scenarios and the optimization objectives are formulated for each scenario. to account for the impact of traffic conditions and driver behavior on driving strategies, an imitation learning–based approach is introduced as an innovative solution. this strategy constructs an adaptive learning framework, which not only reduces the complexity of reward function design, but also enables the model to exceed expert demonstrations and capture high-dimensional environmental features. the simulation verification shows that the algorithm performs excellently in car-following, vehicle distance maintenance, and speed regulation. the average differences between the expert algorithm and the proposed algorithm are 0.38 m/s, 0.0026 m/s, and 0.21 m/s under steady-state, acceleration, and deceleration conditions, respectively. for the complex intersection scenario, two strategies are proposed creatively, which consists of incorporating an additional reward mechanism and a self-supervised reinforcement learning module to effectively compensate for the sparsity of the ecological driving reward function. to address the challenges posed by the changing intersection environment, a reinforcement learning calibration strategy is introduced to correct the algorithm, significantly improving its efficiency. the reward function not only converges rapidly but also has a denser reward distribution. based on the simulation results, the strategy proposed in this paper has significant advantages in terms of reward value, battery soc, and engine operation. under different algorithms, the fuel consumption is significantly reduced through the layered strategy, with reductions of 7.52 %, 5.29 %, 9.6%, and 5.93 % for the ppo, a2c, a3c, and dqn strategies, respectively. taking the dqn algorithm as an example, the emissions of harmful gases co, hc, pm and nox, are scenario-level hierarchical energy management control strategy for hybrid electric vehicle 27 reduced by 18.3 %, 14.23 %, 16.94 %, and 20.9 %, respectively, after layering. the hil experiment further verifies the safety and effectiveness of the strategy, demonstrating the algorithm's efficiency in reducing harmful gas emissions. acknowledgement: this work was supported by the guangxi science and technology major program (grant no. guike aa23062067), in part by the open fund for state key laboratory of advanced 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engineering, university of niš, serbia 2faculty of technical science, university of novi sad, novi sad, serbia 3“gheorghe asachi” technical university of iaşi, romania abstract. the development of mathematical models by using experimental data is of great importance for modelling and optimization of the laser cutting process. motivated by the lack of research regarding the use of genetic programming (gp) for deriving empirical mathematical models that describe the laser cutting process, the present study discusses the application of gp to the development of a kerf taper angle mathematical model. the aim was to quantify the relationship between three selected input parameters (cutting speed, laser power and assist gas pressure) and kerf taper angle using gp in the co2 laser cutting of aluminium alloy almg3. to obtain the experimental database for the gp model evolution process, a laser cutting experiment was planned as per standard full factorial design where all three selected parameters were varied at three levels. the fit between the experimental and the gp model prediction values of kerf taper angle was found to be appropriate. finally, by using the derived gp mathematical model, the analysis of the effects of input parameters on the change in kerf taper angle values was performed by generating 3d surface plots. key words: kerf taper angle, genetic programming, laser cutting 1. introduction laser cutting technology is one of the leading non-conventional technologies used in modern industry for contour cutting of different materials, with co2 and nd:yag lasers being the most used [1]. numerous benefits and advantages of this technology have led to it becoming an area of great and continuous industrial and scientific research. a number of experimental, theoretical, modelling and optimization studies have been performed aiming at better understanding and optimization of the laser cutting process with respect received august 10, 2021 / accepted january 09, 2022 corresponding author: miloš madić faculty of mechanical engineering, university of niš, aleksandra medvedeva 13, 18000 niš, serbia e-mail: madic@masfak.ni.ac.rs 666 m. madić, m. gostimirović, d. rodić, m.radovanović, m. coteaţă to different criteria such as quality performance, cost, productivity, cutting time, etc. an essential element in these studies is the development of mathematical models for exact quantification of the relationships between the laser cutting parameters (inputs) and performances (outputs). the identification of empirical mathematical models for the approximation of the laser cutting process performance is of great practical importance considering that the complexity of the laser cutting process limits the practical use of analytical models. moreover, the development of analytical models usually requires certain assumptions and generalizations and the modelling process itself is time-expensive and requires considerable expert domain knowledge. on the other hand, for the purpose of empirical mathematical modelling of the laser cutting process only a set of input-output data is needed. input data refer to process variables (factors) such as laser power, cutting speed, assist gas type/pressure, etc., and output data, which can be experimentally measured or calculated, refer to process performances such as material removal rate, surface roughness, kerf width, cost, etc. in order to develop a reliable and accurate empirical model, a number of experimental trials with associate measurements are to be taken, where the application of design of experiments (doe) saves time and resources while ensuring systematic and comprehensive investigation. once the input-output data are available, the aim of empirical mathematical modelling is to develop the functional dependence between dependent variables (performance) and independent variables (factors), which is at the same time the best approximation of the actual laser cutting process. these dependencies can be modelled using polynomials as in the regression analysis (ra) approach, a combination of nonlinear functions and matrix calculus as in the artificial neural network (ann) approach, fuzzy numbers and membership functions as in the fuzzy logic approach, or they can simply be modelled by power, exponential or other more specific mathematical models. in the field of the laser cutting process modelling, ra, ann, fuzzy logic and adaptive neural fuzzy inference system (anfis) models are predominantly used for the development of empirical mathematical models. nassar et al. [2] used an ra model for the prediction of surface roughness in order to optimize laser cutting parameters such as cutting speed, laser power and assist gas pressure in co2 laser cutting of stainless steel 307. the empirical mathematical model was developed based on the 33 full factorial experimental design. singh and gangwar [3] conducted an experimental analysis in co2 laser cutting of aisi 321 stainless steel by using taguchi’s orthogonal array design. in order to develop the mathematical relationship between laser cutting parameters (cutting speed, frequency and assist gas pressure) and surface roughness, an ra approach was adopted. subsequently, a genetic algorithm was used to provide a set of optimum values for input parameters for surface roughness minimization. parametric modelling and optimization of nd:yag laser cutting of aisi 316l stainless steel was conducted by gadallah and abdu [4]. taguchi’s l27 orthogonal array design was adopted as the experimental matrix where laser power, cutting speed, assist gas pressure and frequency were varied at three levels. empirical mathematical models for the prediction of surface roughness, kerf taper and width of the heat affected zone (haz) in terms of considered parameters were developed using response surface methodology (rsm). abhimanyu and satyanarayana [5] conducted an optimization study of cut quality during pulsed co2 laser cutting of mild steel. to this aim they developed two ra models relating surface roughness and hardness with independent parameters such as laser power, cutting speed and material thickness. with the development of ann mathematical modelling of the co2 laser cutting process using genetic programming 667 models, klancnik et al. [6] investigated the effects of laser power, cutting speed and assist gas type on kerf width and surface roughness in co2 laser cutting of the tungsten alloy. among 42 experimental results, 34 data sets were chosen for training the ann model, whilst the remaining results were used as test data. experimental analysis of the effects of laser power, laser repetition rate, and laser scanning speed on surface roughness in laser micro-cutting of polymer plate was conducted by bachy and al-dunainawi [7]. the mathematical model relating surface roughness and independent parameters was developed using ann after performing a number of experimental tests. good agreement was observed between theoretical results and ann predictions. multi-objective optimization of pulsed nd:yag laser cutting of an aluminium alloy through integration of ann models and non-dominated sorting genetic algorithm (nsga-ii) was performed by chaki et al. [8]. a full factorial experimental design was conducted where cutting speed, pulse energy and pulse width were considered as input parameters while kerf width, kerf deviation, surface roughness and material removal rate were considered as process outputs. for the purpose of ann model development, a bayesian regularization algorithm was used. syn et al. [9] developed a fuzzy logic model to predict surface roughness and dross inclusion in co2 laser cutting of incoloy alloy 800. a set of training and testing comprised 125 data. the model was developed in terms of three input parameters, i.e. laser power, assist gas pressure and cutting speed. for the prediction of dross formation in co2 laser oxygen cutting of mild steel, madić et al. [10] proposed a fuzzy logic model. the model was developed based on data from taguchi’s experimental design. the developed fuzzy logic model was based on the use of mamdani-type inference system. with the use of fuzzy and ra models, rajamani and tamilarasan [11] predicted kerf deviation and metal removal rate in nd:yag laser cutting of a titanium super alloy. pulse width, pulse energy, cutting speed and assist gas pressure were considered as independent variables. the experimental trials were performed according to the standard box-behnken experimental design. zhang and lei [12] developed anfis models for the prediction of kerf width and surface roughness in fiber laser cutting of aisi 201 stainless steel. the models were developed in terms of laser power, cutting speed and assist gas pressure. it was noted that anfis models had a superior performance in comparison to the ann model considering error dimensions, training speed and convergence precision. a hybrid approach of ann and fuzzy logic models was applied to develop a fuzzy expert system to predict kerf width and kerf deviation in pulsed nd:yag laser cutting of a titanium alloy [13]. the predicted results were compared with the experimental data and found appropriate. from the above summary of studies it can be observed that empirical mathematical modelling was mainly performed with the application of linear, quazi-linear and nonlinear regression models, while in the case of anns, multi-layer perceptron (mlp) type models were mostly used. the thing which is common for both approaches, as well as for fuzzy logic and anfis models, is that the mathematical model constants (coefficients) are determined based on experimentally collected data where the functional form of the mathematical model (approximation structure) is defined by the decision maker [14]. for example, in the case of ann based mathematical modelling, the number of hidden layers and neurons as well as the selection of transfer functions, which largely determine the (non)linearity of the mathematical model, is determined by the decision maker. apart from the aforesaid approaches, genetic programming (gp) is an empirical mathematical modelling approach intended to identify the model structure as well as constants (model parameters) at the same time. in that sense it represents a more advantageous approach 668 m. madić, m. gostimirović, d. rodić, m.radovanović, m. coteaţă for mathematical modelling without requiring the specification of the mathematical model structure in advance. regression methods and rsm may require the use of statistical tests for the assessment of significant model terms, whereas gp self-prunes the insignificant terms because of the inherent evolutionary traits [15]. in addition, as noted by mitra et al. [16], gp can incorporate very diverse data sets that contain markedly different types of variables and can also handle missing values in the data, where missing data in regression methods and rsm may make impartial estimates of the parameters of interest difficult or impossible. to this aim the focus of the present paper is the application of gp to the development of empirical mathematical models for describing the co2 laser cutting process. to the best of the authors’ knowledge, the gp empirical mathematical modelling approach has not been previously applied to co2 laser cutting process modelling. recent studies promote the use of gp for modelling laser drilling [17], laser microdrilling [18] and laser cladding [19]. in that sense the novelty of the manuscript is reflected in fulfilling the research gap in the co2 laser cutting modelling domain, as well as in the analysis of the obtained results using the developed gp model through consideration of two-factorial interaction effects. in the present study gp was implemented using the data from an experimental investigation of co2 laser cutting of an aluminium alloy. the gp mathematical model for the prediction of kerf taper angle was developed in terms of three laser cutting parameters, namely, laser power, cutting speed and assist gas pressure. 2. experimental details the experimental investigation of the co2 laser cutting process was conducted on an aluminium alloy (almg3) sheet with the thickness of 5 mm. this is magnesium alloyed aluminium (aa5754) with a maximum of 3.0% mg. it is a standard sheet metal alloy with good mechanical properties, good corrosion resistance, excellent weldability and anodizing, and as such has a wide application in construction industry. experimental trials were performed using the cutting head with a focusing lens with a focal length of 5 in (127 mm). the nitrogen gas was used as assist gas and it was passed through a conical shape nozzle with the nozzle diameter of 2 mm. the stand-off distance was set at 0.8 mm. the entire experimental investigation was performed using a prima industry 4 kw co2 laser cutting machine operating in the continuous wave mode. the focus position was always defined to be on the bottom surface of the sheet. based on the literature review, pre-analysis and pilot experimentation, three laser cutting parameters were selected for variation during experimentation, i.e. laser power (p), assist gas pressure (p) and cutting speed (v). for the purpose of experimentation, a full factorial experimental plan (33) l27 oa was adopted where the considered parameters were varied at three levels as given in table 1. this design had 27 rows corresponding to the number of tests (26 degrees of freedom) with 13 columns at three levels [20, 21]. laser power, assist gas pressure and cutting speed were assigned to columns 1, 2, and 5, respectively. it has to be noted that in trial 9 (laser power 3.2 kw, assist gas pressure 15 bar and cutting speed 2 m/min) no throughout cut was achieved, and this may be attributed to the laser power to cutting speed ratio and high reflectivity and thermal conductivity of the workpiece material. therefore, only measurements from 26 trials were considered as the basis for mathematical modelling. kerf width and kerf taper are one of the most important cut quality parameters in the laser cutting process that determine the geometrical accuracy of the finished parts. due to the mathematical modelling of the co2 laser cutting process using genetic programming 669 converging–diverging shape of the laser beam profile kerf taper always exists in laser cutting (fig. 1). table 1 laser cutting parameters and ranges used in the experiment laser cutting parameter unit level 1 2 3 laser power, p kw 3.2 3.6 4 assist gas pressure, p bar 10 12.5 15 cutting speed, v m/min 1.6 1.8 2 fig. 1 laser cut kerf geometry recorded using q-spark image processing software let ku and kb designate the upper and lower kerf widths, respectively. for the workpiece thickness of d the resulting kerf taper angle (kt) in the laser cutting operation can be calculated using the following equation:  180 2 )(        − = d kk k bu t  , (1) the upper and lower kerf widths were measured using the optical coordinate measuring device mitutoyo (type: qsl-200z) with the resolution of the length measuring system of 0.5 µm. the kerf widths were measured at three equally distanced positions along the picture of the kerf, which covers the distance of 3.15 mm (90.35 mm), taken approximately in the middle of the cut. for the purpose of mathematical modelling, the average values of kerf taper angle were considered for each experimental trial. 3. genetic programming (gp) 3.1. gp overview gp is an artificial intelligence (ai) method aimed at intelligent and adaptive evolution of empirical mathematical model structures. each mathematical model corresponds to a given program structure (i.e. individual), which is determined by the unique combination of basic elements-genes. an individual program is a tree-like structure and as such there 670 m. madić, m. gostimirović, d. rodić, m.radovanović, m. coteaţă are two types of genes, primitive functions and terminals [22]. the set of primitive functions consists of arithmetic operators, mathematical functions, boolean logical operators and special functions. the set of terminals, which also constitute the chromosome structure, are usually input parameters (independent variables) and different numerical constants [23]. starting from the random population of programs (chromosomes, models), having various tree-shapes and forms, and implementing the darwinian concept of natural selection through application of selection (reproduction), crossover and mutation operators, the programs evolve until the final solution. during the evolution process each program in the population is monitored via the fitness function, which represents a certain measure of the program appropriateness. actually, the fitness function is one of the key elements which to the great extent control the evolution process, i.e. the improvement of the solution correctly evaluating all the improvements which are being made during the evolution process. the evolution process lasts until the pre-specified number of iterations is achieved, the given evolution process time is expired or an appropriate solution is found. during the evolution process the selection operator controls the selection process of programs which are to be transferred to the next population. koza allowed 10% of the population to reproduce [24]. the essence of this genetic operator is the survival of chosen individuals and the expected consequence is an increase in the mean value of the fitness function of the entire population, i.e. transfer of better genetic material from generation to generation. the crossover operator is intended to combine the genetic material from two, randomly selected, individuals in the randomly chosen point of intersection. as a result of the crossover operation two offsprings are obtained having genetic material from both parents. koza suggested a crossover of 90% of the population since it provides the source of new (and eventually better) individuals [23, 24]. the mutation operator represents a mean for introducing new genetic material into the population. the goal is to introduce certain random changes of individuals so as to maintain diversity in the population, prevent problems of local optimum convergence and/or enable an escape from the local optimum. the mutation is carried out by choosing an individual at random, followed by random selection of the node which is to be mutated, where function is being replaced by function and terminal by terminal. 3.2. gp control parameters an efficient implementation of gp in empirical mathematical modelling depends on a careful selection of genes, i.e. sets of functions and terminals, as well as the specification of main controlling parameter values. the main control parameters are population size, maximum number of generations, probability of crossover and probability of reproduction [22]. other parameters are summarized in [23] and discussed by koza [24]. population size and maximum number of generations depend on the complexity of the problem being solved. generally, a population of 500 or more individuals gives better chances for finding the global optimum. for a small number of independent variables, a starting population of 100 may be sufficient [25]. optimal selection of main gp parameter values is reflected both on the quality of the determined mathematical models as well as the performance of the evolution process. the correct use of gp parameters represents a difficult task, and is primarily affected by the end user knowledge and experience [26]. mathematical modelling of the co2 laser cutting process using genetic programming 671 4. gp mathematical model for the prediction of kerf taper angle implementation of the gp approach in the development of mathematical models implies making a number of modelling decisions related to different parameters including: selection of functional set, definition of fitness function, selection of population size and mechanism for its initialization, definition of terminal set, i.e. specification of independent variables, adjustment of genetic operators, specification of selection mechanism, etc. after comprehensive pilot experimentation with mathematical operators and other gp parameters, the gp parameters, as given in table 2, were used so as to obtain an acceptable error for the mathematical model that is yet quite simple. for the assessment of the gp mathematical model quality, the fitness function in the form of the mean absolute error (mae) was used: 1 ( ) ( ) mae n i e i m i n = − =  , (2) where n is the number of trials, e(i) represents the experimentally obtained value and m(i) is the gp modelled value for the i-th trial. table 2 gp parameters used in mathematical modelling number of data for modelling: 26 fitness function: mae terminal set: laser power (p), assist gas pressure (p), cutting speed (v) probability of crossover: 0.8 functional set: +,-,*,/,power of 2, polynomial terms probability of mutation: 0.1 population size: 500 probability of reproduction: 0.1 initialization: ramped half-and-half algorithm selection: roulette wheel it has to be noted that some solutions with better fitness function values were rejected because of their excessive length. namely, as noted by alvarez et al. [27], when the complexity of the gp mathematical model is increased its ability to generalize can be affected by the risk of over-fitting the data. in many trials during the gp program evolution it turned out that the final solution did not include the cutting speed (v), indicating its possible small influence on the change of kerf taper angle. if one considers the change of this parameter in the covered experimental hyper-space (table 1) this indication might be justified. after reaching 500 generations the following gp mathematical model for the prediction of kerf taper angle emerged: ))29.19)13.3((50/(571 ))21.005.8()49.720/)161((749/())92.3(100( −−− −++−= vp ppppkt (3) the derived gp model showed the mean absolute error of 0.128 over the entire set of training data consisting of 26 pairs of input/output data. this is an indication that the developed gp model has a considerably good prediction accuracy and that it can be used for the analysis of the effects of laser cutting parameters on kerf taper angle. 672 m. madić, m. gostimirović, d. rodić, m.radovanović, m. coteaţă 5. results and discussion the change of kerf taper angle values was analyzed by changing two parameters at a time, while keeping the third parameter constant at the centre level. the change of kerf taper angle values is given as a function of 3 interaction effects as given in fig. 2. from fig. 2 it can be seen that kerf taper angle has no negative values, indicating that the upper kerf is always wider than the lower kerf. as noted by genna et al. [28], since aluminium and its alloys are highly reflective metals, laser cutting of almg3 needs more energy input to initiate the cut, which results in widening the upper kerf. the results of tahir and rahim [29], in the case of co2 laser cutting of ultra high strength steel, also reported this observation. in co2 laser cutting of al6061/sicp/al2o3 composite material with the thickness of 4 mm, a higher degree of melting was found at the top surface of the work material than at the bottom surface [30]. when comparing co2 and fiber laser cutting of aisi 304 stainless steel sheets in the thickness range between 1 and 10 mm, stelzer et al. [31] observed that the kerf width was continuously decreased from top to bottom. however, it was noticed that, in the case of fiber laser cutting, the narrowest kerf width was typically found in the middle of the sheet. in the present experimental investigation the highest ratio of the upper and lower kerf of 2.72 was obtained when using the laser power of 3.2 kw, assist gas pressure of 10 bar and cutting speed of 1.6 m/min. in these conditions the upper and lower kerf widths of 0.734 and 0.27 mm were obtained. the lowest ratio of the upper and lower kerf of 1.72 was obtained under the same conditions except that the laser power of 3.6 kw was used. considering this observation one may argue that laser power predominantly affects kerf taper angle. from figs. 2a and 2b it can be observed that the interactions of laser power and assist gas pressure and laser power and cutting speed are negligible and produce a kerf taper angle of about 2.2º in all parameter combination values. in co2 laser cutting of austenitic stainless steel, ozaki et al. [32] reported that the cross-sectional shapes of kerf were almost the same though the laser power or the cutting speed were varied. slight increases in kerf taper angle with an increase of cutting speed or decrease in laser power may be explained considering that with a decrease in the linear energy density, the angle at which a layer of melt is produced through the thickness of workpiece (inclination angle) is increased. in the present study, in the case of using low laser power level (3.2 kw) there is a sudden rise in kerf taper values. namely, the heat input during the cutting operation is primarily determined with laser power. thus, when the low laser power level was used, there was no sufficient heat to melt the material along the entire workpiece thickness and this resulted in increased kerf taper angle values, i.e. formation of a wide upper kerf and a narrow lower kerf. as noted by tahir and rahim [29], by increasing the laser power the greater material removal rate was obtained which results in reducing the taper formation. the widening of the lower kerf width at high power levels, and a consequent decrease in taper angle, was observed by yilbas et al. [33] in co2 laser cutting of 7050 al alloy reinforced with al2o3 and b4c composites. finally, for the constant laser power of 3.6 kw (figure 2c), an increase in assist gas pressure as well as cutting speed results in a negligible increase of kerf taper angle values. this is in accordance with the results reported by madić et al. [34]. in that research, it has been reported that for certain levels of laser power the effects of assist gas pressure and cutting speed on the change in kerf taper angle may be negligible. mathematical modelling of the co2 laser cutting process using genetic programming 673 a) b) c) fig. 2 change of kerf taper angle with respect to: a) interaction effect of laser power and assist gas pressure, b) interaction effect of laser power and cutting speed, c) interaction effect of cutting speed and assist gas pressure 674 m. madić, m. gostimirović, d. rodić, m.radovanović, m. coteaţă apart from kerf taper angle, another relevant information for industrial practitioners regarding perpendicularity of the cut is angularity or perpendicularity tolerance, which is defined in the iso 9013 (2002) standard [35]. this standard, based on the given workpiece thickness, classifies laser cuts into five classes according to angularity tolerance (u). based on obtained experimental values of upper and lower kerf widths, it was observed that different combinations of laser cutting parameters in the conducted experiment produced angularity tolerance values in the range u=0.145÷0.205mm, which covers classes 2 and 3 according to iso 9013. finally, it is necessary to point out that the perceived observations and obtained results are valid for the covered experimental domain and used cutting conditions. for different ranges of cutting parameters and cutting conditions one may expect different effects of the laser cutting parameters on kerf taper angle, both quantitatively and qualitatively. 6. conclusion this paper reviewed some of the main approaches to mathematical modelling of the laser cutting process with the emphasis on the application of gp. in the present study a gp mathematical model was developed to predict kerf taper angle as a function of laser cutting parameters such as cutting speed, laser power and assist gas pressure in co2 laser cutting of an aluminium alloy using nitrogen as assist gas. from the analysis of the gp model development process and its analysis within the covered experimental hyper-space it was observed that the laser power has the most dominant effect on kerf taper angle and this is particularly pronounced in the case when there was no sufficient heat to melt the material along the entire workpiece thickness. the influences of cutting speed and assist gas pressure on kerf width are much smaller. it was also observed that in all experimental trials kerf profiles exhibited the typical slight v-profile. modelling results using gp indicated that the mathematical model evolution process is highly affected by the selection of main gp parameters, where there is no universal rule for setting these parameters. during the mathematical model development it was observed that there is no repeatability of the final solution even though the same initial conditions and gp parameter settings were used. as an advantage/disadvantage of gp one needs to point out that in some model evolution processes certain independent variables might be excluded from the final model. in conclusion, gp mathematical models proved to be able to adequately represent mathematical relationships between laser cutting parameters and kerf taper angle. increasing the number of training input/output sets, using a bigger initial population and fine tuning of other gp parameters would provide a means for the development of more accurate mathematical models. the development of gp mathematical models can improve the laser cutting process via an appropriate selection of process parameters through optimization. finally, the ability of gp to work with incomplete data obtained from experimental design is particularly beneficial for the laser cutting process modelling considering that some combinations of laser cutting parameters in the experimental matrix might not be able to produce complete cuts. acknowledgement: this research was financially supported by the ministry of 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computations, 6(1), pp. 33-42. 35. en iso 9013:2002(e): thermal cutting – classification of thermal cuts – geometrical product specification and quality tolerances, international organization for standardization, geneva. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 2, 2014, pp. 95 106 on influence of boundary conditions and transverse shear on buckling of thin laminated cylindrical shells under external pressure  udc 539.3 gennadi i. mikhasev, ihnat r. mlechka belarusian state university, belarusian state university, department of bio and nanomechanics, mechanics mathematics faculty minsk, belarus abstract. buckling of a thin cylindrical sandwich shell composed of elastic isotropic layers with different elastic properties under normal external pressure is the subject of this investigation. differential equations based on the assumptions of the generalized kinematic hypothesis for the whole sandwich are used as the governing ones. two variants of the joint support conditions are considered at the shell edges: a) there are the infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges, b) the diaphragms are absent. using the asymptotic approach, the critical pressure and buckling modes are constructed in the form of the superposition of functions corresponding to the main stressstrain state and the edges integrals. as an example, a three-layered cylinder with the magnetorheological elastomer (mre) embedded between elastic layers under different levels of magnetic field is studied. physical properties of the magnetorheological (mr) layer are assumed to be functions of the magnetic field induction. dependencies of the buckling pressure on the variant of boundary conditions and the intensity of applied magnetic field are analyzed. key words: sandwich cylindrical shell, buckling pressure, diaphragms, magnetorheological elastomer 1. introduction thin multi-layered shells are used in many engineering structures, such as airborn/ spaceborne vehicles, underwater objects and cars [1]. application of new materials with different physical properties allows one to design sandwich structures fulfilling up-todate requirements such as good buckling resistance and noiselessness. buckling as well received june 11, 2014 / accepted july 26, 2014  corresponding author: gennadi i. mikhasev belarusian state university, department of bioand nanomechanics, nezavisimosti avenue 4, 220030, minsk, belarus e-mail: mikhasev@bsu.by original scientific paper 96 g.i. mikhasev, i.r. mlechka as vibroprotection of thin–walled structures are of great practical interest for mechanical engineers who develop and model similar structures. it is generally known that the critical value of an external load resulting in buckling of a thin shell depends on a great number of factors such as geometrical characteristics of a structure (thickness, radius, and length), physical properties of a material (young’s and shear module, poisson’s ratio), the boundary conditions, the way of loading (if it is combined) [2, 3]. if a laminated shell is assembled from layers having different properties, this dependence becomes more complex. one of the principal characteristics of the sandwich shell is shear compliance (or the reduced shear rigidity for a whole sandwich). it is influenced by a correlation between thicknesses and mechanical properties of layers composing the sandwich. as shown in [4, 5], taking into account the transverse shears results in an appreciable decrease of both the natural frequencies and the critical axial load of thin laminated shells. a less noticeable effect of transverse shears on the buckling external pressure of a three-layered cylindrical shell has been revealed in paper [6]. in these and other papers, the authors consider as a rule the simplest variant of boundary conditions the joint support conditions when the edges have infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges. within the scope of the accepted kinematic hypothesis for laminated shells [4], this type of boundary conditions does not permit us to include the edge effects generated by shears, but gives us a possibility to find the buckling modes and the critical load in the explicit form. the present paper mainly aims at studying the influence of shears on the critical pressure for a thin sandwich cylinder when its simply supported edges do not have diaphragms. because a similar variant of the boundary conditions does not allow us to find an exact value of the critical pressure, the asymptotic approach is applied to predict the shell buckling. the specific goal defined herein is to analyze the influence of a magnetic field on the buckling pressure of a three-layered cylinder with the magnetorheological elastomer (mre) embedded between the elastic bearing layers. the mres belong to the group of active materials whose physical properties (viscosity, young’s and shear module) can vary when subjected to different magnetic field levels [7-10]. the application of an external magnetic field is expected to permit us not only to suddenly change the rheological properties of the mre-based sandwich and suppress its vibrations [11] but to considerably increase the total stiffness of the structure and prevent its buckling. 2. setting a problem a thin middle length cylindrical sandwich shell consisting of n transversely isotropic elastic layers characterized by length l, thickness hk, young’s modulus ek, and poisson’s ratio k is considered, where k = 1,2,...n. the middle surface of any fixed layer with radius r is taken as the original surface. coordinate system 1, 2, 3 is illustrated in fig. 1, where 1, 2, are axial and circumferential coordinates, respectively. the sandwich shell is assumed to be under normal external pressure qn. two variants of the joint support conditions are considered at shell edges 1 = 0, l: (a) there are infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges, (b) diaphragms are absent at the edges. the problem is to find buckling conservative pressure q * n at different variants of the boundary conditions. buckling of thin laminated cylindrical shells under pressure 97 fig. 1 thin cylindrical sandwich shell and a curvilinear co-ordinate system we accept here the unique kinematic hypotheses of timoshenko for the whole package of the sandwich stated in [4]. from these hypotheses the principal ones relate to the distribution laws of the transverse shear stresses ),()(),()(),,( 21 )( 21 )0( 021 )(  k iki k i zfzfz , (1) and tangential displacements ),()(),(),(),,( 21212121 )(  iii k i zgzuzu (2) across the thickness of the kth layer, where f0(z), fk(z) and g(z) are continuous functions of coordinate 3 = z introduced as follows 0 0 12 2 1 1 ( ) ( )( ), ( ) ( )( )n k k k k f z z z f z z z h h          , 0 0 ( ) ( ) z g z f x dx  , (3) k is the distance between the original surface and the upper bound of the kth layer, h is the total thickness of the sandwich, ui (k) are the tangential displacements of points of the kth layer, i are the angles of rotation of the normal n about the vectors ei (see fig. 1), i = 1, 2; k = 1,2,..., n, and the functions i (0) , i (k) may be found in paper [4]. the remaining two hypotheses concerning normal deflection w and normal stresses are the same as in the classic kirchhoff–love hypothesis. we note that at g  0 hypothesis (2) turns into the kirchhoff–love ones. based on the foregoing hypotheses, the system of five differential equations with respect to w, ui, i have been derived in book [4]. if buckling occurs with formation of large number of hollows although in one direction at the shell surface, these equations may be essentially simplified. introducing functions i appearing in (2) by 1 ,1 ,2 2 ,2 ,1,a a       (4) where a and  are the shear functions defined from equations 2 2 2 1 1 , 2 h h a               , (5) the following compact system 98 g.i. mikhasev, i.r. mlechka 0 1 1 )1(12 2 2 2 0 22 12 2 2 2 3 3                     w t r heh , 0 12 2     w r eh ,            2 1 h w (6) is obtained [4] with respect to the displacement and force functions  and , respectively. in eqs. (6), t2 0 = rqn is the hoop membrane stresses, qn is the external normal pressure. other parameters appearing in (5) and (6) are introduced as follows: 2 3 2 1 1 1 k k k k e h e h       , 1 3 3 2 2 11 1 1 kk k k k k kk k k e h e h                , kk hh  , n nh  , 3 1 2 1 1 12 1 3k k k k c       , 3 1 2 2 13 12 1 3k k k k c c       , 3 2 2 3 1 13 1 4 ( 3 ) 3k k k k k c         , 2 2 1 3 1      , 1 3 2 2 11 1 k k k k k k k k e h e h              , 3 1 12 3 1 k k k k c       , 3 13 1 1 ( )k k k k c       , 2 2 3 0 21 3 0 44 2 3 1 10 1 k k k kk k k k kkk k k k kk q g g                               , 2 44 1 12(1 ) q eh      , 2(1 ) k k k e g    , (7) 0 1 ( ) k k k f z dz       , 1 ( ) ( ) k kn k n k f z f z dz       , 3 21 1 ( ) 12 k k k h g z dz      , 3 2 1 ( ) 12 k k k h zg z dz      . in eqs. (5) (7), the magnitudes e,  are the reduced young’s modules and poisson’s ratio of the sandwich, and parameters , , i characterize the reduced shear stiffness of the whole packet. at  1  0, eqs. (6) degenerate into the well-known equations of the semi-moment theory of isotropic shells (see, for instance in book [3]). in terms of the functions , , the joint support conditions at the shell edges are written as 0,0 1 2     at l,01  , (8) 0 at l,01  (9) for the case when there are the infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges. if these diaphragms are absent, then the boundary conditions for  become more complicated: 2 2 2 2 1 1 0, 1 0 h h                      at 1 0, l  (10) buckling of thin laminated cylindrical shells under pressure 99 2 2 2 2 2 2 2 2 2 2 1 2 1 21 2 1 2 2 0, (1 ) 0                                      at 1 0, l  , (11) while the boundary conditions for force function  are the same as in eqs. (9). we call the boundary-value problems (5), (6), (8), (9) and (5), (6), (9) (11) as the bv problems “a” and “b”, respectively. the problem is to find the minimum absolute value of hoop stress t2 0 for which every from these problems has a nontrivial solution. note that the second eq. (5) has a solution like the edge effect being a rapidly damped function from the shell edges. when considering the bv problem “a”, the shear function  is found from the second eq. (5) and the last boundary condition (8). to estimate the influence of this function on the critical value of t2 0 , it is necessary to consider the complete system of five differential equations with respect to w, ui, i [4]. so, solving the bv problem “a” based on the simplified equations (5), (6), we can assume  = 0. as concerns the bv problem “b”, the boundary conditions (11) show that functions  and  are linked. 3. solution of the bv problem “a” the form of a solution of eqs. (5), (6) depends on the shell sizes. here we consider a middle length sandwich cylinder for which l ~ r . if the shell edges have diaphragms (the bv problem “a”), then the buckling mode is the same as for an isotropic one-layered simply supported shell (without taking into account transverse shears) [3] and may be found in the explicit form [4-6] 0 0 1 2( , ) ( , ) sin( / )sin( / )n l m r       , (12) where n and m are natural numbers, and 0, 0 are constants. substituting (12) into (6) results in the following equation for hoop stresses t2 0 : 0 0 8 4 2 2 2 ( , ) nmt t n m hem      , (13) 4 2 2 2 4 4 4 2 2 1 , 1 nm nm nm nm nm k n h k k r                    , 2 2 2 2nm n m m          , 2 8 3 2 2 , 12(1 ) h l rr        . the minimization of 0 2t over n and m gives the buckling pressure rtqn * 2 *  , where * 0 0 0 * 2 2 2 2 , min ( , ) min (1, ) (1, ) n m m t t n m t m t m   . (14) 4. solution of the bv problem “b” if the simply supported edges do not have diaphragms, eqs. (5), (6) do not admit a solution in the form (12). however, the introduction of additional assumptions for 100 g.i. mikhasev, i.r. mlechka parameters appearing in the governing equations permits us to apply the asymptotic approach. let  be a small parameter. as shown in [12], for a three layered cylinder with the core made of the mre, parameters k /  2 , k /  2 are also small and their orders depend on the correlation between thicknesses of the layers composed a cylindrical sandwich and its radius. in this study, it is assumed that  4242 , kk at 0 , (15) where ,   1. let us introduce dimensionless coordinates x,  and load parameter as follows: 0 6 1 2 2, ,rx r t eh        . (16) functions ,  and  are sought in the form: 1 4 2 1 1( )sin( ), ( )sin( ), ( )sin( )rx x p ehr f x p rs x p              , (17) where p is a wave number, and x(x), f(x), s(x) are unknown functions of x. the substitution of (15) – (17) into eqs. (5), (6) gives the governing equations written in the dimensionless form: 2 4 4 2 2 4 2 (1 ) (1 ) 0 d f x p x dx               , 2 4 2 4 2 (1 ) 0 d f x dx         , (18) 41 2 s s      , (19) where  = d 2 / dx 2   2 p 2 is the differential operator. the boundary conditions (9)-(11) at x = 0, l = l / r are rewritten as follows: 4(1 ) 0,x    2 4 2 (1 ) 0 d x dx     , (20) 2 2 2 2 (1 ) 0 d ds p x p dxdx              , 2 2 2 2 2 0 dx d s p p s dx dx     , (21) 0f  , 2 2 2 2 0 d f p f dx    . (22) as seen, eqs. (18), (19) are singularly perturbed equations. the solution of the boundary value problem (18)-(22) may be presented in the form [3]: )()()()( , edbsedbs fffxxx  , (23) where the superscripts (bs) and (ed) denote functions describing the main stress-strain state and the integrals of the edge effects, respectively. it is well known that under buckling of shells of zero gaussian curvature under external pressure the following asymptotic estimates are valid [3]: )()()()( ~,~ bsbsbsbs fxfxxx  at   0. from eq. buckling of thin laminated cylindrical shells under pressure 101 (19), it may be seen that sxs 2~  at   0. let 1~, )()( bsbs fx . then the asymptotic analysis of eqs. (18), (19) permits us to determine the estimates ( ) ( ) 2 4 ( ) 2 ( ) ( ) 2 ( ), ~ , ~ , ~ , ~ed ed ed ed ed edx f s x x x f x f        at 0  . (24) 4.1. edge effect solutions we assume that (x (ed) , f (ed) ) = 2 4 ˆˆ ˆ( , ),x f s s  , where ˆˆ ˆ, , ~ 1x f s . when taking into account these correlations as well as above estimates for functions with the superscripts (bs), one obtains the following equations describing the edge effects: 2 4 4 8 2 4 ˆ ˆ ˆ 0, d x d x f dx dx     2 2 4 4 2 2 ˆ ˆ ˆ 0 d f d x x dx dx      , (25) 2 4 2 2 2 ˆ 2 ˆ 0 (1 ) d s m s dx             . (26) eqs. (25), (26) have the following solutions 2 2 2 2 3 3( ) ( ) 2 2 1 1 ˆ ˆe e , ( 1) e e i i i ix l x x l x i i i i i i i i x a b f a b                                        , (27) 4 4 2 2 ( ) 1 2 ˆ e e x l x s c c          , 2 2 4 2 (1 ) m       (28) decreasing far from the shell edges. here ai, bi, c1, c2 are complex constants which will be determined from the boundary conditions below, and 1, 2, 3 are complex roots (having positive real parts) of the algebraic equation 6 4 2 1 0      . (29) if ,   0, then the system of equations (25) degenerates into a simple edge effect equation, and eq. (29) has two complex conjugate roots. so, transverse shears may distort the edge effect integrals. 4.2. main stress state solution we will construct functions x (bs) , f (bs) , and load parameter  in the form of series: ( ) 2 ( ) 2 0 2 0 2, ,bs bsx x x f f f       (30) 2 0 2      (31) the substitution of (23), (24), (30), (31) into eqs. (18) and the boundary conditions (20)-(22) results in the consequence of the boundary-value problems. 102 g.i. mikhasev, i.r. mlechka in the first-order approximation, one gets the following boundary-value problem 4 8 60 0 04 ( ) 0 d x p p x dx    , 2 0 2 40 1 dx xd p f  , (32) 0 0 0x f  at 0,x l . (33) it has the simple solution 0 sin( )x a x l (34) if 0(p) = p 2 +  4 / p 6 , where  =  / l. minimizing this function, one obtains the zeroorder approximation of the load parameter 2/18/104/30 00 0 0 3,34)()(min   ppp p . (35) it should be noted that parameter  0 0 does not take into account transverse shears in the shell. eqs. (35) are the same as in the classical shell theory [3]. in the second-order approximation, one has the non-homogeneous differential equation 4 0 8 0 0 6 02 0 2 0 24 [( ) ( ) ] ( , , ) d x p p x az p dx      (36) with the non-homogeneous boundary conditions 2 1 2 2 2 3 2 4(0) , ( ) , (0) , ( )x x l x x l        , (37) where a is an arbitrary constant, the prime denotes differentiation over variable x, and 0 8 0 10 2 0 6 0 2 6 0 2 4 0 6 0 2( ) ( ) 2 ( ) 2( ) ( ) ( )z p p p p p p            , (38) 3 3 2 2 1 2 1 1 ( 1) , ( 1) ,i i i i i i a b           3 3 0 4 2 2 0 4 2 2 3 3 1 1 ( ) (1 ) , ( ) (1 )i i i i i i i i p a p b             . (39) to determine constants ai, bi appeared in eqs. (27), (39), we substitute (23), (27), (34) into the boundary conditions (21) and the second condition (20). collecting coefficients at  2 , one gets the following non-homogeneous system of six algebraic equations 3 3 3 3 2 2 4 4 1 1 1 1 0, 0, 0, 0,i i i i i i i i i i i i a b a b                  3 1 3 1 , i ii i ii abaa (40) with respect to constants ai, bi (i = 1,2,3) it may be seen that ai, bi and i from (39) are directly proportional to a. hence, without loss of generality, one can set a = 1 in the boundary-value problem (36), (37). for the non-homogeneous boundary-value problem (36), (37) to have a solution the following equation buckling of thin laminated cylindrical shells under pressure 103 0 0 2 0 2 0 1 0 2 0 3 0 4 0 0 ( , , ) (0) ( ) (0) ( ) l z p x dx x x l x x l            (41) is required to hold. from this equation, we obtain the correction for the load parameter 1/ 4 1/ 4 4 2 1 2 3 4 2 3/ 4 2 [4 3 3 3 ( )] 2 ( ) 2( ) 3                    , (42) which takes into account the transverse shears (the parameters ,  from eqs. (5), (6)). it should be noted that function ss ˆ4 , with ŝ defined by (28), does not affect parameter 2. to find an appropriate correction for  introduced by function s it is required to consider the next approximation. however, the order of an expected correction equals h / r and are the same as the order of an error of the shell theory applied here. thereby, if the shear parameters satisfy estimates (15), then the influence of shear function  on the buckling pressure is negligibly small. so, in the bv problem “b” as well as in the problem “a”, we can set  = 0. 5. example: buckling of three-layered cylinder with mre core as an example, we have considered a three-layered cylinder with mre core of length l = 1m and radius r = 0,5m. we have performed calculations of q * n vs. induction b of the magnetic field for the case when the external supporting surfaces having thicknesses h1 = h3 = 0,5m are made of the abs-plastic sd-0170 with parameters e1 = e3 = 1510 9 pa, 1 = 3 = 0,4, and the internal layer of the thickness h2 = 8mm is the mre with poisson’s ratio 2 = 0,42 and young’s modules e2 specified by equation [11] kpa040,45230,13)(2 bbe  . (43) this equation are valid at b < 200 mt. it approximates the experiment date obtained for mre by means of the rheometer physica mcr 301 (anton paar) at the frequency of the external impact 10 hz [9]. now we can analyze the influence of the magnetic field induction on the buckling pressure for two variants of simply supported edges. if the edges have the diaphragms, critical pressure q * n will be calculated by eqs. (13), (14). if the diaphragms are absent, then the following asymptotic formula * 6 1 0 2 4 0 2[ ( )]nq ehr o       (44) with the parameters  0 0, 2 defined from (35), (42) will be applied. it should be noted that in our example parameters k, , , ,  appearing in eqs. (13), (29), (42) are functions of the magnetic field induction. table 1 demonstrates the dependence of dimensionless parameters 2, ,  and the critical pressure upon various factors: the variant of boundary conditions, transverse shears and the intensity of a magnetic field. it may be seen that assumption   1 (see eqs. (15)) holds weakly at a very low intensity of a magnetic field (at least for b  20 mt). but at b > 20 mt, the constructed solutions (17), (23), (30), (31) are asymptotically correct. 104 g.i. mikhasev, i.r. mlechka as follows from the asymptotic constructions, wave parameter p 0 and zero approximation  0 0 are not influenced by the transverse shears and the magnetic field induction. for geometrical parameters and materials accepted above, one has p 0 = 1,437,  0 0 = 2,756. the negative values of 2 at 0  b  60 mt point out that accounting transverse shears results in decreasing load parameter  (see eq. (31)). but at a very high intensity of the magnetic field the influence of shears on correction 2 decreases. however, in spite of the complex dependence of the load parameter on a magnetic field, the increase of its intensity results in an increase of both the reduced young’s modules for whole sandwich and the critical pressure. comparing data in the last two columns, one can conclude: the buckling pressure for the cylindrical sandwich shell having diaphragms on both edges is higher than the critical pressure for the shell without diaphragms. however, it is necessary to bear in mind that the data in the last column are obtained as the result of the exact solution of bv problem “a”, whereas the critical pressures for the shell without diaphragms are the asymptotic estimates which are probably slightly understated. table 1 dimensionless parameters 2, ,  and critical pressure q * n vs. magnetic induction b for two variants of the boundary conditions b, mt 2   critical pressure q * n, pa edges without a diaphragm edges with a diaphragm 0 11,598 1,664 4,298 4174 7937 10 7,462 2,054 3,260 5160 8935 20 4,961 2,290 2,628 5762 9697 30 3,286 2,448 2,203 6168 10300 40 2,088 2,560 1,898 6463 10789 50 1,188 2,645 1,668 6687 11195 60 0,487 2,710 1,489 6865 11538 70 0,074 2,763 1,345 7009 11752 80 0,533 2,806 1,227 7129 11906 90 0,915 2,842 1,129 7232 12042 100 1,239 2,872 1,045 7320 12162 6. conclusions governing equations, based on the assumptions of the generalized kinematic hypothesis of timoshenko’s, are used for the prediction of buckling of a thin composite sandwich cylindrical shell under external normal pressure. two variants of the joint support conditions have been taken into consideration. if the shell edges have diaphragms preventing shears in the edge plane, the approximate solutions of the governing equations and the estimate for buckling pressure are found by using the asymptotic method; when the diaphragms are absent, the appropriate solution and the buckling pressure are determined in the explicit form. buckling of thin laminated cylindrical shells under pressure 105 as an example, the three-layered cylinder with mre core has been considered. it is observed that the presence of the diaphragms at the shell edges leads to increasing the critical pressure. other conclusions of this study concern the influence of the applied magnetic field and transverse shears on the shell buckling. when the induction of the magnetic field does not exceed 60 mt, accounting transverse shears results in decreasing dimensionless load parameter . however, for both variants of the boundary conditions, applying the magnetic field with the induction varying in the interval from 0 to 100 mt gives an increase of both the reduced young’s modules and the buckling pressure. references 1. qatu, m.s., sullivan, r.w., wang, w., 2010, recent research advances on dynamic analysis of composite shells: 200-2009, composite structures, 93, pp. 14-31. 2. librescu, l, hause, t., 2000, recent developments in the modeling and behavior of advanced sandwich constructions: a survey, composite structures, 48, pp. 1-17. 3. tovstik, p.e, smirnov, a.l., 2001, asymptotic methods in the buckling theory of elastic shells. singapore: world scientific. 4. grigolyuk, e.i, kulikov, g.m., 1988, multilayer reinforced shells: calculation of pneumatic tires. moscow: mashinostroenie, 288 p. [in russian]. 5. korchevskaya, e., mikhasev, g., marinkovich, d., gabbert, u., 2003, buckling and vibrations of composite laminated cylindrical shells under axial load, in: kasper r., editor. entwicklungsmethoden und entwicklunsprozesse im maschinenbau: 6 magdeburger maschinenbau-tage. berlin: logos-verlag, pp. 183-189. 6. mikhasev, g., seeger, f., gabbert, u., 2001, local buckling of composite laminated cylindrical shells with oblique edges under external pressure: asymptotic and finite element simulation, technische mechanik, 21(1), pp. 1-12. 7. deng, h-x, gong, x-l, 2008, application of magnetorheological elastomer to vibration absorber, communications in nonlinear science and numerical simulation,13, pp.1938–1947. 8. gibson, r.f., 2010, a review of recent research on mechanics of multifunctional composite materials and structures, composite structures, 92, pp. 27932810. 9. korobko, e.v, mikhasev, g.i, novikova, z.a., zurauski, m.a., 2012, on damping vibrations of threelayered beam containing magnetorheological elastomer. journal of intelligent material systems and structures, 23(9), pp. 1019–1023. 10. yeh, j-y., 2013, vibration analysis of sandwich rectangular plates with magnetorheological elastomer damping treatment. smart material structures, 22, 035010. 11. mikhasev, g.i., botogova, m.g., korobko, e.v., 2011, theory of thin adaptive laminated shells based on magnetorheological materials and its application in problems on vibration suppression. in: altenbach h, eremeyev va, editors. shell-like structures. advanced structured materials, vol. 15. berlin: springer, pp. 727–750. 12. mikhasev, g.i., altenbach, h., korchevskaya, e., 2014, on the influence of the magnetic field on the eigenmodes of thin laminated cylindrical shells containing magnetorheological elastomer, composite structures, 113, pp. 186-196. o uticaju graničnih uslova i poprečnog smicanja na izvijanje tankih laminiranih cilindričnih panela pod spoljnim pritiskom predmet ovog istraživanja je izvijanje tankog cilindričnog sendvič panela koji se sastoji od elastičnih izotropskih slojeva sa različitim elastičnim svojstvima pod normalnim spoljnim pritiskom. kao vodeće jednačine koriste se diferencijalne jednačine zasnovane na pretpostavkama o uopštenoj kinematskoj pretpostavki za ceo sendvič. dve varijante zajedničkih uslova potpore se razmatraju na ivicama panela: a) da postoje dijafragme beskonačne krutosti u relativnim smicanjima 106 g.i. mikhasev, i.r. mlechka slojeva duž ivice panela, i b) da su dijafragme odsutne. uz pomoć asimptotskog pristupa, kritični pritisak i modusi izvijanja se konstruišu u obliku superpozicija funkcija koje odgovaraju glavnom stanju naprezanja/deformacije i integrala ivica. daje se primer troslojnog cilindra sa magnetoreološkim elastomerom (mre) umetnutim između elastičnih slojeva pod različitim nivoima magnetnog polja. fizička svojstva magnetorološkog sloja (mr) se uzimaju kao da su funkcije indukcije magnetnog polja. zavisnosti pritiska izvijanja na varijantu graničnih uslova se proučavaju kao i intenzitet primenjenog magnetnog polja. ključne reči: sendvični cilindrični panel, pritisak izvijanja, dijafragme, magnetoreološki elastomer bending vibration and stability of a multiple-nanobeam system influenced by temperature change facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 75 88 original scientific paper bending vibration and stability of a multiple-nanobeam system influenced by temperature change  udc 534.1 danilo karličić 1 , sanja ožvat 2 , milan cajić 3 , predrag kozić 1 , ratko pavlović 1 1 faculty of mechanical engineering, university of niš 2 faculty of technical sciences, university of novi sad 3 mathematical institute of the serbian academy of science and arts abstract. in this study, we analyzed the bending vibration and stability of a multiplenanobeam system (mnbs) coupled in elastic medium and influenced by temperature change and compressive axial load. the mnbs is modeled as the system consisting of a set of m identical and simply supported nanobeams mutually connected by winkler’s type elastic layers. according to the euler bernoulli beam and nonlocal thermo-elasticity theory, the system of m coupled partial differential equations is derived and solved by means of the method of separation of variables as well as the trigonometric one. analytical solutions for natural frequencies and critical buckling loads of elastic mnbs are obtained. the effects of nonlocal parameter, temperature change and the number of nanobeams on the natural frequencies and the buckling loads are investigated through numerical examples. thus, this work can represent a starting point to examine dynamical behavior and design of complex nanobeam structures, nanocomposites and nanodevices under the influence of various physical fields. key words: nonlocal elasticity, vibration, stability, multiple-nanobeam system 1. introduction the nonlocal continuum theory has recently been widely used for the study of the mechanical behavior of nanobeams such as bending, vibration and buckling. such theoretical observations of nanostructures may be important for nanoengineering practice received september 30, 2015 / accepted january 12, 2015 corresponding author: danilo karliĉić faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia, e-mail: danilo.karlicic@masfak.ni.ac.rs mailto:danilo.karlicic@masfak.ni.ac.rs 76 d. karliĉić, s. ožvat, m. cajić, p. kozić, r. pavlović in the development of new nanodevices. other methods of nanostructure examination, such as experimental [1-3] or atomistic simulations [4-7] can be expensive, taking into consideration time or computational resources needed to gather important data about the mechanical behavior of nanostructures. thus, the modified continuum theories are increasingly gaining in importance for the mathematical modeling of dynamical behavior of systems and structures on small-scales [8-10]. in this study, the nanobeam may represent various types of tube-like nanomaterials such as carbon nanotubes (cnt) [11], zinc-oxide nanotubes (zno) [12], boron-nitride (bn) nanotubes [13, 14], etc. according to the experimental analysis and atomistic theories, it has been shown that size-effects play an important role in describing the physical and mechanical properties of structures and systems on the nano-scale level. due to the existence of such effects, the classical continuum theories need to be reformulated to take into account these smallscale effects. aforementioned can be done by introducing the nonlocality in the space domain, modifying the corresponding constitutive equation and introducing the material parameter which takes into account the effects of length scale and influence of interatomic forces. one of the first scale-dependent continuum theories, which takes into account nonlocal effects, is proposed by eringen and co-workers [15-17] also known as nonlocal elasticity theory. according to eringen [18], the nonlocal theory gives great approximation for a large class of problems in nano-systems, where the influence of length-scales is very noticeable. the nonlocal elasticity theory in nanomechanics was first considered in the paper by peddieson et al. [19] on the static behavior study of nanobeams. the authors have investigated the nonlocal euler-bernoulli beam deflection equation for different boundary conditions and applications on the micro actuators. reddy and pang [20] reformulated the equations of motion for euler-bernoulli and timoshenko beam theories, by using the nonlocal theory, and applied them to evaluate static bending, vibrations, and buckling response of the nanobeams with various boundary conditions. recently, nonlocal models for the euler-bernoulli, timoshenko, reddy and levinson beam theories are formulated by reddy [21] based on the hamilton’s principle and nonlocal constitutive relation of eringen. thermal effects studies are important to round up the knowledge of the mechanical behavior of structures, regardless of scale. zhang et al. [22, 23] derived equations of motion and stability equation for double-walled and multi-walled carbon nanotubes, respectively, and solved them by using the separation of variables method. the authors obtained analytical solutions for natural frequencies and critical axial buckling strain, and examined the effects of thermal and other physical and geometrical parameters on the dynamic and stability behavior of the proposed system. in the work by janghorban [24], the static analysis of microbeams was conducted, based on the nonlocal thermal elasticity theory and using the differential quadrature (dq) and harmonic differential quadrature (hdq) methods. the author has presented a variety of numerical results to show the effects of temperature change on bending of microbeams, by choosing different combinations of aspect ratios and boundary conditions. murmu and pradhan [25, 26] have used a nonlocal constitutive relation in modeling of carbon nanotubes embedded in an elastic medium with thermal effects included. they obtained approximated solutions for the natural frequencies and the critical buckling force by using the differential quadrature method for simply supported carbon nanotubes. in addition, they performed numerical experiments for two types of the thermal expansion parameters corresponding to the bending vibration and stability of a multiple-nanobeam system influenced by temperature change 77 lower and higher environmental temperature. it has been concluded that in the case of low or room temperature, the nonlocal critical buckling load and the natural frequencies increase as temperature rises. in the case of higher temperatures, the nonlocal critical buckling load and the natural frequencies decrease for an increase of temperature. in this paper, we used the nonlocal thermo-elastic theory to investigate the free banding vibration and stability behavior of mnbs embedded in winkler’s type of elastic medium. it is assumed that all nanobeams in mnbs are simply supported, and are with the same material and geometric properties. the set of m equations of motion is obtained by using the newton second low and nonlocal thermo-elastic constitutive relation based on the euler-bernoulli beam theory. the closed form solutions for the natural frequencies and the critical buckling loads are derived by using the separation of variables method and the trigonometric method for different number of nanobeams in mnbs. special attention is given to the effects of temperature changes and nonlocal parameter on the vibration and stability behavior of mnbs. by searching the literature, one can find that the dynamic behavior of systems, consisting of an arbitrary number of nanobeams, embedded in a certain type of medium, considering temperature effects, has not been investigated yet. this work can be used as an analytical benchmark study in the future research on vibration and stability behavior of other types of complex multi-nanostructure systems such as coupled photonic crystal nanobeam cavities [32] and nanocomposites [33]. 2. problem formulation 2.1. constitutive relation in this section, we will consider the basic equations of nonlocal elasticity and viscoelasticity in the general and two-dimensional case. eringen [15] derived a constitutive relation for nonlocal stress tensor at a point x in an integral form, based on the assumption that the stress at a point is a function of strains at all other points of an elastic body. the fundamental form of the nonlocal elastic constitutive relation for a three-dimensional linear, homogeneous, isotropic body is expressed as:   ),()(,)( xdvxcxxx klijklij    (1a) ,0)(, xjij (1b)  , 2 1 )( ,, ijjjiij uux  (1c) where cijkl is the elastic modulus tensor for classical isotropic elasticity; ij and ij are stress and strain tensors, respectively, and ui is the displacement vector. with (|x  x'|, ) we denote the nonlocal modulus or attenuation function, which incorporates nonlocal effects into the constitutive equation at a reference point x , produced by the local strain at a source x'. the above absolute value of difference |x  x'| denotes the euclidean metric. parameter  is  = (e0a)/l , where l is the external characteristic length (crack length, wave length), a describes the internal characteristic length (lattice parameter, 78 d. karliĉić, s. ožvat, m. cajić, p. kozić, r. pavlović granular size and distance between c-c bounds) and e0 is the constant appropriate to each material that can be identified from atomistic simulations or by using the dispersive curve of the born-karman model of lattice dynamics. however, some difficulties may arise in searching the analytical solutions for structural mechanics problems when the constitutive relation is in integral form. thus, for the sake of simplicity, eringen [18] proposed the constitutive relation in a differential form, which, for one-dimensional problem, is of the form: , 2 2 xx xx xx e x        (2) where e and g are the elastic and shear modulus of the beam;  = (e0a) 2 is the nonlocal parameter, xx is the normal nonlocal stress and xx = zw / x is the axial strain. in the present work, free vibration and the critical buckling load analysis of mnbs is carried out by assuming e0a to be in the range 0-2 [nm]. the nonlocal thermo-elastic constitutive relation is a combination of nonlocal elasticity and thermo-elasticity theory, as given in the paper [25]. for one-dimensional case, the nonlocal thermo-elastic constitutive relation is of the following form: , 212 2           x xx xx xx e e x (3) where x is the coefficient of thermal expansion in the direction of x axis,  is the change of temperature and  is the poisson’s ratio. for  = 0 there is no influence of temperature and we return to the constitutive relation of nonlocal elasticity. 2.2. equations of motion now we consider the system of m axially loaded nanobeams, coupled through an elastic medium, as shown in fig. 1. further, we assume that mnbs is made of m identical nanobeams of the same length l, elastic modulus e and mass density , uniform crosssection of area a and moment of inertia i . each nanobeam is under the influence of the same compressive axial load f . in addition, for the i-th nanobeam we assume only the transverse deflection denoted as wi(x,t) where mi ...2,1 . further, we limited our analysis only to the case where the nonlocal euler-bernoulli beam theory is used for simply supported boundary conditions. according to the way of coupling of the first and the last nanobeam in mnbs, we can distinguish between two different types of mnbs, “clamped-chain and “free-chain” system [29, 30]. in this work, we used only “clamped-chain” mnbs, in which the first and the last nanobeam are coupled with fixed base through winkler’s type elastic layers of stiffness k0 and km, respectively, as shown in fig. 1b. the coupling conditions are w0 = wm+1 = 0. other nanobeams in mnbs are coupled through elastic layers of stiffness k0 = k1 = ... = ki = ... = km1 = km = k representing the winkler’s type of elastic medium (see fig. 1b). according to the d’ alembert’s principle, from the equilibrium equation for differential element of the i-th nanobeam we can write the following differential equation: ,,...,2,1,)( 2 2 12 2 2 2 mi t w aqq x w fn x m i ii if            (4) bending vibration and stability of a multiple-nanobeam system influenced by temperature change 79 where mf and n are stress resultants defined as: 0 0 , . 1 2 a a x f xx xx ea m z da n da      α θ σ σ ν (5) and the influence of the winkler’s type elastic medium expressed as external load is: 1 1 1 1( ), ( ).i i i i i i i iq k w w q k w w       (6) governing equations of motions can be expressed in terms of displacement wi(x,t) for nonlocal thermo-elastic constitutive relation eq. (3). fig. 1 the multiple-nanobeam system embedded in the winkler’s type of elastic medium: a) the physical model of the system of multiple swcnts embedded in a polymer matrix with atoms of swcnts fixed in such a manner that can be represented by simply supported boundary conditions [31], b) mechanical model of multiplenanobeam system coupled in the “clamped-chain” system introducing eqs. (4-6) into eq. (3) we obtain the following system of equations of motion: 2 2 4 1 1 12 2 4 ( ) ( ) ( )i i i i i i i i i w w w a f n k w w k w w ei t x x                 ρ 2 22 1 1 12 2 2 ( ) ( ) ( ) , 1,2,.. ,i i i i i i i i w w a f n k w w k w w i m x t x                    μ ρ (7) or in the dimensionless form: 80 d. karliĉić, s. ožvat, m. cajić, p. kozić, r. pavlović 2 2 4 1 1 12 2 4 ( ) ( ) ( )i i i i i i i i i w w w f n k w w k w w                θ τ ζ ζ 2 22 2 1 1 12 2 2 ( ) ( ) ( ) , 1,2,.. ,i i i i i i i i w w f n k w w k w w i m                    η ζ τ ζ (8) where the dimensionless parameters are defined as: ,,,, 2 2 2 4 l w w ei l nn lei l kk i iii     ,,, 2 2 ei l ff a ei l t l x    (9) the initial conditions in the general form and the boundary conditions in the dimensionless form for a simply supported i-th nonlocal euler-bernoulli nanobeam are given as: 0 0 ( ,0) ( ,0) ( ), ( ),i i i i w w w v     ζ ζ ζ ζ τ (10a) (0, ) (1, ) 0, (0, ) (1, ) 0, 1,2,..., .i i fi fiw w m m i m    τ τ τ τ (10b) 3. solution of the equation of motion assumed solution of the system of partial differential equations eq. (8) is in the form of trigonometric series, and this solution must satisfy the given boundary conditions eq. (10 b), as: 1 ( , ) sin( ) , 1, 2,... ,nj i in n n w w e i m      τ ζ τ α ζ (11) where 1j , ,...)2,1(  nnn  , win(i = 1,2,...m) and n are the amplitudes and natural frequencies, respectively, in the dimensionless form. we substitute the assumed solution from eq. (11) into the system of equations (8) and take into account the coupling conditions for "clamped-chain" systems in the dimensionless form 0 1( , ) ( , ) 0mw w  ζ τ ζ τ . further, assume that all stuffiness of the elastic layers is equal k0 = k1 = ... = ki = ... = km1 = km = k. finally, we obtain the system of m algebraic equations as: ,1,021  iwvws nnnn (12a) ,1,...3,2,011   miwvwswv nininnnin (12b) ,,01 miwswv mnnnmn   (12c) or in the matrix form as: bending vibration and stability of a multiple-nanobeam system influenced by temperature change 81 , 0 0 0 ... 0 0 0 ... 0 0 0 ... ... 0...000...000 0...000...000 ................................. 000...0...000 000......000 000...0...000 ................................. 000...000... 000...000...0 1 2 1 1 3 2 1                                                                                                                 mn nm nm ni in ni n n n nn nn nn nnn nn nnn nn w w w w w w w w w sv vs sv vsv vs vsv vs (13) where: 2 4 22 ( ) ,n n n n n n ns v f n      θξ α α ξ (14a) ,nn kv  (14b) ,1 22 nn   (14c) analytical solution of the homogeneous system of algebraic equations (13) is available only for the case when mnbs is composed of identical nanobeams and coupling layers. based on the methodology presented in the papers karliĉić et al. [29, 30], solution of the ith algebraic equation can be assumed in the following form: cos( ) sin( ) , 1, 2,... ,in cc ccw n i m i i m    (15) substituting eq. (15) into the i-th algebraic equation of the system (13), we obtain two trigonometric equations, by assuming that constants m and n are not simultaneously equal to zero: { cos[( 1) ] cos( ) cos[( 1) ]} 0, 2,... 1,n cc n cc n ccn v i s i v i i m          (16a) { sin[( 1) ] sin( ) sin[( 1) ]} 0, 2,... 1,n cc n cc n ccm v i s i v i i m          (16b) wherefrom we obtain the following system of equations: ( 2 cos ) cos( ) 0, 2,... 1,n n cc ccs v n i i m     (17a) ( 2 cos ) sin( ) 0, 2,... 1,n n cc ccs v m i i m     (17b) from the above system of equations we can conclude that, n  0, cos(icc)  0 or m  0, sin(icc)  0 when the system has an oscillatory behavior for i = 2,...m1. finally, we can obtain the frequency equation: .cos2 ccnn vs  (18) 82 d. karliĉić, s. ožvat, m. cajić, p. kozić, r. pavlović where cc is unknown parameter determined by replacing the solutions from eq. (15) in the first and the last equation of the system of equations (13). using the methodology presented in the papers proposed by rašković [27], stojanović et al. [28] and karliĉić et al. [29 and 30], we obtain the solutions for unknown parameter cc,s in “clamped-chain” systems as: .,...,3,2,1, 1 , ms m s scc      (19) introducing the expression for cc,s and eqs. (14) into eq. (18), and assuming axial compressive load 0f , we get the natural frequencies in the dimensionless form as: 4 2 , , 2 (1 cos ) , 1,2,.., . n cc s n n n ncc s n v n s m            (20) the analytical form of the critical buckling load under the influence of temperature change can be obtained by introducing the expression for cc,s and eqs. (14) into eq. (18) and assuming that the natural frequencies of system n are equal to zero. then the critical buckling load is of the form: 4 , , 2 2 (1 cos ) , 1,2,.., . n cc s n ncc s n n v f n s m     θ α ξ   (21) finally, in the case when the number of nanobeams in mnbs increases towards infinity i.e. taking m   into eqs. (22) and (23), we obtain asymptotic or critical natural frequencies and critical buckling loads in the following form: , 24 , n nnn m sncc n      (22) , 2 4 ,    nf nn n m sncc   (23) now, we can conclude that the asymptotic natural frequencies and the critical buckling loads represented by expressions (22) and (23), respectively, are dependent only on the system of material parameters and temperature change. 4. numerical results in order to examine the effects of material parameters, such as stiffness coefficient of elastic medium, temperature change and nonlocal parameter on natural frequencies and critical buckling loads of the nonlocal elastic mnbs, numerical examples are presented and discussed here. in addition, the influence of temperature change at low and high temperatures on the vibration and stability response of mnbs is studied. the physical and geometrical parameters used in numerical simulations are adopted from the paper by murmu and pradhan [25, 26]. in the following parametric study, we observe changes of the lowest natural frequency and the critical buckling load of the system for a single vibration mode. bending vibration and stability of a multiple-nanobeam system influenced by temperature change 83 fig. 2 the influence of the nonlocal parameter and stiffness of the winkler elastic medium on a) natural frequencies and b) critical buckling load for different number of nanobeams the first natural frequency and the critical buckling load of mnbs versus the nonlocal parameter and stiffness of the winkler elastic medium, for different numbers of nanobeams, are shown in fig. 2. as expected, an increase in the nonlocal parameter leads to a decrease of the natural frequency and the critical buckling load of the system. from the same figure, one can observe that the stiffness of the winkler’s elastic medium k has a very strong “hardening” effect on the natural frequency and the critical buckling load. further, one can see that an increase of the number of nanobeams (m = 3, 5, 10) in mnbs leads to a decrease of both the natural frequency and the critical buckling load towards certain critical values. this is in line with expressions (22) and (23) and the results obtained in papers by karliĉić et al. [29, 30]. fig. 3 the influence of the nonlocal parameter and temperature change at low or room temperatures on the first a) natural frequency and b) critical buckling load 84 d. karliĉić, s. ožvat, m. cajić, p. kozić, r. pavlović fig. 4 the influence of the stiffness coefficients and temperature change at low or room temperatures on the first a) natural frequency and b) critical buckling load fig. 5 the influence of the nonlocal parameter and temperature change at high temperatures on the first a) natural frequency and b) critical buckling load fig. 6 the influence of the stiffness coefficients and temperature change at high temperatures on the first a) natural frequency and b) critical buckling load figs. 3-6 shows the influence of temperature change  at low temperature (x = 1.610 6 k 1 ) and high temperature (x = 1.110 6 k 1 ) environment (see murmu and pradhan [25, 26]) on the natural frequency and the critical buckling load for different bending vibration and stability of a multiple-nanobeam system influenced by temperature change 85 values of nonlocal parameter and stiffness coefficient of elastic medium. it should be noted that we consider the first natural frequency and the critical buckling load of mnbs composed of ten elastically connected nanobeams (m = 10) coupled in the “clamped chain” system. to illustrate the effect of temperature change at low temperature environment with the natural frequency and the critical buckling load of mnbs, numerical experiments are carried out for different values of nonlocal parameter and stiffness coefficients and three different values of temperature change  = 0.50 and 100k. from fig. 3 it can be observed that an increase of temperature change leads to an increase in both the natural frequency and the critical buckling load. fig. 4 illustrates the influence of a changing stiffness coefficient of elastic medium with the natural frequency and the critical buckling load of mnbs for three different values of temperature change. from this figure, it is also found that the natural frequency and the critical buckling load are nearly linearly dependent on stiffness coefficient k. in addition, it can be observed that an increase of stiffness coefficient increases the natural frequency and the critical buckling load. moreover, the influence of temperature changes is small, but not negligible. the similar behavior can be observed in the previous case presented in fig. 3. finally, one can conclude that the temperature change at low temperature environment causes an increase of the system’s stiffness, which leads to an increase of the natural frequencies and the buckling loads of the system. the obtained results are also in line with the results obtained in papers by murmu and pradhan [25, 26]. the effects of temperature change at higher temperature environment with the natural frequency and the critical buckling load of mnbs embedded in elastic medium where m = 10 and s = 1 are shown in figs. 5 and 6. it can be noticed that temperature change has a different influence on the system’s natural frequency and its critical buckling load than in the case of low temperature environment. here, one can notice that an increase of temperature change leads to a decrease of both the natural frequency and the critical buckling load. from the physical viewpoint, this means that the total stiffness of mnbs decreases for an increase of temperature. regarding the influence of nonlocal parameter and stiffness coefficient of elastic medium on the vibration and stability response of mnbs it is similar as in the case of low temperature environment. molecular dynamic simulation studies analyzing the dynamical and stability behavior of systems with a larger number of coupled nanobeams with thermal effects are not yet published in the literature. however, the authors have found md simulation results for vibration and vibration of a single-walled carbon nanotube presented by ansari et al. [34]. for mnbs it is well known that the lowest natural frequency and the buckling load of such system represent the fundamental frequency and critical buckling load and it is independent of the influence of a number of nanobeams in the system and chain coupling conditions, see karliĉić et al. [33]. the fundamental frequency and the critical buckling load are equivalent to the natural frequency and the buckling load of a single nanobeam. thus, our results for the lowest natural frequency and the critical buckling load of mnbs without thermal change can be used to validate them with the results obtained for free vibration of a single-walled carbon nanotube via molecular dynamics simulation in ansari et al. [34]. this is also in line with expressions for asymptotic natural frequency given in eq. (22). the mechanical properties of single-walled carbon nanotube considered in this comparative study are adopted from ansari et al. [34]. table 1 shows the fundamental natural frequencies obtained by using the molecular dynamic simulation and nonlocal continuum mechanics approach for different values of the aspect ratios l/d. from table 86 d. karliĉić, s. ožvat, m. cajić, p. kozić, r. pavlović 1, it can be noticed that the results obtained by using the trigonometric method are in excellent agreement with the results presented by ansari et al. [34]. table 1 comparison fundamental frequencies (thz) presented analytical solution with results obtained in the ansari et al. [34] timoshenko beam theory ansari et al. [34] presented analysis (trigonometric solution) l/d md simulation classical theory stress gradient theory strain gradient theory strain/ inertia gradient theory nonlocal theory 8.3 0.5299 0.5306 0.5302 0.5302 0.5299 0.5485 10.1 0.3618 0.3606 0.3604 0.3604 0.3603 0.3707 13.7 0.1931 0.1972 0.1971 0.1971 0.1971 0.2016 17.3 0.1103 0.1240 0.1240 0.1240 0.1240 0.1264 20.9 0.0724 0.0851 0.0851 0.0851 0.0851 0.0860 24.5 0.0519 0.0620 0.0620 0.0620 0.0620 0.0630 28.1 0.0425 0.0471 0.0471 0.0471 0.0471 0.0479 31.6 0.0358 0.0373 0.0373 0.0373 0.0373 0.0379 35.3 0.0287 0.0299 0.0299 0.0299 0.0299 0.0303 39.1 0.0259 0.0244 0.0244 0.0244 0.0244 0.0247 5. conclusions in the present paper, the vibration and the stability properties of mnbs embedded in winkler elastic medium for different number of simply supported nanobeams are studied. the system of m partial differential equations is derived using the d’ alembert principle and nonlocal thermo-elastic constitutive relation. the possibility of using nanostructures such as cnts, zno or bn nanotubes as nanoresonators, nanosensors and in other nems devices demands exploration of natural frequencies of single or multiple nanostructure systems. however, external field effects, such as temperature, or the influence of surrounding medium, can significantly shift the values of natural frequencies. thus, such effect needs to be taken into account to obtain, as much as possible, realistic models to simulate the mechanical behavior of nano-scale systems. nonlocal continuum theory, which considers size-effects, is one of the most convenient theories to deal with a large nano-scale system with various external field effects, easily included into a model. the nonlocal eulerbernoulli beam theory takes into account the length-scale i.e. nonlocal parameter, where thermal effects are also introduced through constitutive relation. the bernoulli fourier and trigonometric methods are used to obtain closed form analytical solutions for thermal vibration and stability response of the proposed system. the influence of temperature changes and nonlocal parameter with natural frequencies and critical buckling loads are considered. in addition, the influence of stiffness of the elastic medium k, on both of mechanical responses of mnbs, is investigated through numerical examples. it is found that the natural frequency and the critical buckling load of mnbs are strongly influenced by nonlocal and stiffness coefficients. further, significant influence of the temperature change on the natural frequency and the critical buckling load can also be recognized. from the obtained results, one can conclude that the nonlocal parameter and the temperature change at bending vibration and stability of a multiple-nanobeam system influenced by temperature change 87 high temperature environment are having dampening properties on the natural frequency and the critical buckling load. on the other hand, an increase of stiffness coefficients of the elastic medium and temperature change at low temperature environment leads to an increase of total stiffness of mnbs and consequently to an increase of the natural frequency and the critical buckling load. acknowledgements: this research was supported by the research grant of the serbian ministry of science and environmental protection under the number of on 174001 and on 174011. references 1. bezryadin, a., verschueren, a. r. m., tans, s. j., dekker, c., 1998, multiprobe transport experiments on individual single-wall carbon nanotubes, physical review letters, 80(18), pp.40364039. 2. gogotsi, y., libera, j. a., güvenç-yazicioglu, a., megaridis, c. m., 2001, in situ multiphase fluid experiments in hydrothermal carbon nanotubes, applied physics letters, 79(7), pp. 1021-1023. 3. lu, j. p., 1997, elastic properties of carbon nanotubes and nanoropes, physical review letters, 79 (7), pp. 1297-1306. 4. werder, t., walther, j. h., jaffe, r. l., halicioglu, t., koumoutsakos, p., 2003, on the 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infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 209 222 1 fem based parametric design study of the tire profile using dedicated cad model and translation code udc 519.6+629 nikola korunović, miloš stojković, dragan mišić, miroslav trajanović university of niš, faculty of mechanical engineering in niš abstract. in this paper a finite element method (fem) based parametric design study of the tire profile shape and belt width is presented. one of the main obstacles that similar studies have faced is how to change the finite element mesh after a modification of the tire geometry is performed. in order to overcome this problem, a new approach is proposed. it implies automatic update of the finite elements mesh, which follows the change of geometric design parameters on a dedicated cad model. the mesh update is facilitated by an originally developed mapping and translation code. in this way, the performance of a large number of geometrically different tire design variations may be analyzed in a very short time. although a pilot one, the presented study has also led to the improvement of the existing tire design. key words: tire, parametric design study, steady-state rolling, fem, fea 1. introduction the tire plays an important role in vehicle behavior and traffic safety in general. it is also a very challenging structure regarding its design, due to its complexity, specific materials and adverse conditions of exploitation. the tire design process is still mostly based on tacit knowledge and experience of the designer, so it assumes that the final design contains features of a rather arbitrary nature. constant efforts are taken to improve and shorten this process by application of the latest cad/cam/cae technologies. the tire design process consists of several main phases [1], which may be summarized as: identification of goals/requirements, selection of design features and verification of performance. the first phase implies setting up of performance targets; it is very much determined by tire purpose, customer expectations, manufacturing requirements and governing standards. these requirements also dictate the choice of primary design parameters like received september 23, 2014 / accepted november 8, 2014 corresponding author: nikola korunović faculty of mechanical engineering, department of production engineering, a. medvedeva 14, 18000 niš, serbia e-mail: nikola.korunovic@masfak.ni.ac.rs original scientific paper 210 n. korunović, m. trajanović, m. stojković tire diameter, tire width or section height. the second phase includes the tire profile shape design (mold contour) and its structural components, the design of tread pattern and the choice of tire materials. a large number of secondary design parameters may be varied in this phase, and although their influence on the tire performance is generally known, the sum effect and intensity of mutual parameter changes is hard to predict. the performance of tires is verified through a series of criteria, mainly related to different types of tests, which they undergo in the laboratory or outdoor conditions [1]. if only the physical prototypes were used for performance verification, the tire design process would become lengthy and expensive. for all above reasons, the computer based simulations of tire behavior, mostly based on the finite element method (fem), have become a standard part of the tire design process. verification of the most important aspects of the tire performance may successfully be conducted using a combination of the static finite element analysis (fea) and the steadystate rolling (ssr) fea based on the mixed eularian/lagrangian approach, which, as shown by koishi and kabe [2], represents an optimal approach to simulation of the rolling tire behavior. this statement also relies on the fact that the steady-state rolling represents the primary operational mode of the tire since the transient states in the tire service tend to be of a relatively short duration comparing to the steady-state rolling states [3]. the fem based parametric design studies of the tire design available in the literature [4,5,6] mainly investigate the impact of changes in structural parameters such as tire materials, the number of belts, the density of cords and the like on tire behavior in service. in all those studies, the parameters are chosen in such a way that they may easily be modified by a simple change of certain numerical values within the finite element (fe) model. none of those modifications causes the shape of the finite element mesh to change because the geometric features those parameters relate to are approximated by section or material properties, associated to corresponding finite elements. for example, not every wire in belts or carcass is modeled separately; rather, they are represented by continuous media which has the density that is controlled by that of the cords (number of wires per unit length) and input as a numerical value. the approach presented in this paper offers greater flexibility. it relies on a specially designed cad model, which contains the geometric entities that facilitate the creation of the finite element mesh, and it, therefore, allows for a simple change of the structural parameters that are geometric by nature. the propagation of changes from cad model to fe mesh is done automatically, thanks to the originally developed mapping and translation code. an algorithm has also been defined according to which, using the finite element model, optimization of the tire performance, i.e. parametric studies that investigate the impact of changes in geometry and material parameters on the tire behavior are performed [7]. its use is demonstrated here through a pilot parametric design study of the tire profile shape and belt width, which shows the advantages and benefits of the proposed approach. the dedicated cad model described in this paper can also be used within singleand multi-objective design optimization studies [8, 9]. it this case it would bring even greater benefits as the number of fe model instances that have to be created during such studies typically gets very large. fem based parametric design study of tire profile using dedicated cad model and translation code 211 2. cad based fe tire models to perform the fea of tires, fe models of different complexity are used. the most common ones are fe tire models without detailed tread (with circular channels only) [3-7, 10] and fe models with detailed tread [10-13]. before this study was performed, the response of fe models with and without detailed tread to application of vertical load as well as to braking, accelerating and cornering, had been compared in order to decide which of the two should be used within the study [10]. the comparison showed that although the contact pressure distribution at the tire footprint is locally different, the tire forces and moments obtained using the two models did not differ significantly. one could therefore conclude that it was, in most cases, sufficient to use the fe model without detailed tread to simulate the operational behavior of the tire. thus, such a model was as well chosen for this study. 2.1. cad model the geometry of cad model is constructed in a manner that is very familiar to tire designers, i.e. its construction circles and curves are based on the traditional way the tire profile is designed (fig. 1). such an approach enables them to familiarize with the model and easily make design changes, without thorough knowledge of the cad system itself. when the model is built, one of the main goals is to separate the influence of important tire parameters on tire shape. thus, for example, the width of the crown area may be controlled separately from the belt width. fig. 1 2d cad tire model with some of the construction circles and curves shown cad model is divided into zones that correspond to twelve structural components of the tire, which have associated parameters that may be changed independently of profile shape. apart from basic geometric forms, the cad model contains geometric elements (points and lines/curves) that serve as the basis for the finite element mesh creation and automatically adapt to the changes in the tire geometry [6]. 212 n. korunović, m. trajanović, m. stojković the 2d cad model described here is based on 3d cad model developed earlier, as presented by stojković et al. [14,15], with which it may be combined in cases when the tire tread is going to be modeled in detail. this 3d model has already been used as the base for various cad/cam/cae applications [16,17]. 2.2. translation a methodology for automatic update of the finite element mesh has been designed and implemented, which enables the mesh to adapt to the changes of geometric parameters of the cad model. at first, an axisymmetric fe model is constructed based on the cad model. this is done only once for a certain type of tire, using surfaces and curves exported from the cad model [7]. after the building of the fe model is finished, the mapping is performed between its nodes and the points of the cad model, based on their current coordinates. the mapping, which implies that every point in the cad model has a corresponding node in fe model, is saved and, embedded into the translation code. from that moment on, an easy and fast creation of fe models based on cad models with modified tire parameters is enabled. for all tire types that have similar inner structure, the same fe model may be used. 2.3. fe models fe models used in the parametric study are described in detail in paper by korunović et al [7]. at first, the axisymmetric tire model is built, which may be used for analysis of the tire inflation process. after the initial analyses are conducted, the 3d fe model based on the axisymmetric one is built, which serves as the basis for all the analyses to follow. the algorithm that describes the whole process, including the sequence of analyses to be conducted, the results that may be obtained by those and the actions to take if they are satisfactory or unsatisfactory is also given by korunović et al [7]. the 3d fe model is adjusted in order to simulate the cornering of a tire rolling on the drum and verified through comparison with experimental results obtained on the drum machine [18]. 3. parametric design study of tire profile simultaneous use of the dedicated cad model and the fe tire model will be illustrated here through a pilot parametric design study, performed on a model of existing 165/70 r13 tire. 3.1. chosen parameters and design variations the study examines the impact of changes in sidewall radius and belt width on the tire performance. three different numerical values are in turn assigned to each of the two chosen parameters. sidewall radius, located close to the crown, is adjusted in order to obtain different variations of the tire profile shape (fig. 2). the belt width is changed in increments of 10mm in order to get three variations: a narrow-belted, normal and widebelted tire. the values of the parameters are given in table 1. fem based parametric design study of tire profile using dedicated cad model and translation code 213 a) b) c) fig. 2 three tire instances obtained by changing the upper sidewall radius (r1): a) r1=53.8mm, b) r1=64.4mm and c) r1=75.0mm. the change affects the sharpness of shoulder area and the width of the crown (without affecting the belt width) table 1 values of tire design parameters varied in the study variation 1 variation2 variation3 upper sidewall radius r1 (mm) 53.8 64.4 75.0 belt width (mm) 121.4 131.4 139.4 214 n. korunović, m. trajanović, m. stojković 3.2. types of fe analyses performed within the study each of the tire instances obtained by variation of the chosen parameters has been analyzed according to the proposed algorithm [7]. the results are compared in order to assess the influence of the parameter changes on stresses in the tire structure, contact pressure distribution and forces and moments generated on the footprint during inflation, application of vertical load, acceleration, braking and cornering on dry surface. in this pilot study, the arbitrary friction coefficient of 0.6 between the tire and the surface and tire velocity of 10km/h are chosen so as to minimize the time needed for the analyses to finish. a real-life study requires a more realistic description of friction and a range of velocities to be used, as shown in paper by korunović et al [7]. 3.2. results 3.2.1. inflation fea of tire inflation may be done on an axisymmetric or 3d tire model. the first one represents the logical choice, as fea of the axisymmetric model needs much less time to finish. in fig. 3 carcass stress, obtained as the result of axisymmetric analysis, is compared for three tire variations with different sidewall radii. tire models are inflated to the pressure of 2 bar (0.2 n\mm 2 ). as the value of radius rises, the difference between the lowest and the highest value of carcass stress becomes larger. at point 2 a sudden drop of the stress is present, because the returning portion of carcass takes over a part of carcass load. outwardly returning portion of the carcass is not taken into account. 0 50 100 150 200 0 10 20 30 40 50 60 5432 a x ia l s tr e s s i n c a rc a s s [ m p a ] meridional distance from wheel plane [mm] r 1 =53.8 r 1 =64.4 r 1 =75.0 1 meridional direction 1 2 3 4 5 fig. 3 carcass stress comparison for tire variations that differ by value of r1. the characteristic points on carcass are: 1 – midpoint, 2 – point closest to the end of returning portion, 3 – point on the edge of filler, 4 – point on the upper edge or bead wire and 5 – point on the lower edge or bead wire fem based parametric design study of tire profile using dedicated cad model and translation code 215 difference in the carcass stress between three tire variations that have different belt width is less obvious and thus will not be shown. the stress in the crown area is very similar for all belt widths, while in the bead area it gets slightly higher for tire having wider belts and lower for tire having narrower ones. 3.2.2 static fea of vertically loaded tire load-deflection curve and footprint stress distribution are the most common types of results that tire designer wants to obtain from fea of a vertically loaded tire. the study shows that the sidewall radius variation does not affect the load-deflection curve in a noticeable way, while the widening of the belts makes the tire noticeably stiffer (fig. 4). 0 2 4 6 8 10 12 14 16 18 20 0 500 1000 1500 2000 2500 v e rt ic a l lo a d f z [ n ] deflection [mm] r 1 =53.8 r 1 =64.4 r 1 =75.0 0 5 10 15 20 0 500 1000 1500 2000 2500 v e rt ic a l lo a d f z [ n ] deflection [mm] original wide belts narrow belts fig. 4 comparison of load-deflection curves for all tire variations 216 n. korunović, m. trajanović, m. stojković the most obvious effect of the sidewall radius variation may be seen in fig. 5, which shows its influence on the contact stresses along the footprint at the given vertical load. load intensity is chosen to be equivalent to the load acting on a tire mounted on a small passenger car, with four passengers aboard. the variation that contains the smallest sidewall radius is characterized by a distinct rise of the contact pressure in the area of tire shoulders. in comparison with two other variations, maximum value of contact stress at the footprint is also significantly higher which may lead to uneven and excessive tire wear. a) b) c) fig. 5 contact stress distribution at vertical load of 2750n for a) r1=53.8mm, b) r1=64.4mm and c) r1=75.0mm fem based parametric design study of tire profile using dedicated cad model and translation code 217 3.2.3 braking and acceleration the changes of the chosen parameters do not significantly affect the behavior of the tire rolling on the straight line with action of acceleration or braking moment. the influence of the sidewall radius changes on the braking-to-acceleration curve, obtained by steady-state rolling analysis [7], may be seen in fig. 6. the difference between the curves that correspond to the tire variations with different sidewall radius is very modest, but the braking stiffness of variations with a higher radius tends to be slightly larger. in the case of the belt width changes, the responses of corresponding tire instances are even closer to each other. -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -2000 -1500 -1000 -500 0 500 1000 1500 2000 l o n g it u d in a l fo rc e [ n ] slip ratio =0.6, v=10km/h r1=53.8 r1=64.4 r1=75.0 fig. 6 the effect of sidewall radius variation on full-breaking–to–full-acceleration curve the shape of the footprint and distribution of contact stress in the state of full braking, for different variations of sidewall radius, are shown in fig. 7. as well as in the case when vertical load is applied, the distribution of contact stress is more even for second and third variation. similar results are obtained when accelerating tire is analyzed. 3.2.4 cornering the results of the cornering analysis are also compared for all the proposed tire variations. the cornering curves for slip angle varying from -8 to 8 degrees are shown in fig. 8. the cornering stiffness is higher for models with larger values of sidewall radius, i.e. they are more responsive to steering input. nevertheless, the difference between the responses is very small. in the case of the belt width variation, it may be concluded that the belts widening produces greater cornering stiffness, while at higher slip angles the cornering response seems to be reversed. narrowing of the belts produces the opposite effect. fig. 9 shows the comparison of footprint shape and contact stress distribution for the tire cornering at 2 and 4 degrees, respectively, for different variations of the sidewall radius. stress distribution is more even for two variations with larger radius. the effect of the belt width changes is also noticeable but less significant. 218 n. korunović, m. trajanović, m. stojković a) b) c) fig. 7 tire footprint shapes at full braking for a) r1=53.8mm, b) r1=64.4mm and c) r1=75.0mm fem based parametric design study of tire profile using dedicated cad model and translation code 219 -8 -6 -4 -2 0 2 4 6 8 -2000 -1500 -1000 -500 0 500 1000 1500 2000 v=10km/h, =0.6 r1=53.8 r1=64.4 r1=75.0 s id e f o rc e f y [n ] slip angle  [°] a) -8 -7 -6 -5 -4 -3 -2 -1 0 200 400 600 800 1000 1200 1400 1600 v=10km/h, m=0.6 normal belts wide belts narrow belts s id e f o rc e f y [ n ] slip angle  b) fig. 8 cornering curves for models with three different values of a) sidewall radius and b) belt width 220 n. korunović, m. trajanović, m. stojković fig. 9 tire footprint shapes; first row: r1=53.8 mm, second row: r1=64.4 mm, third row: r1=75.0 mm. left column corresponds to cornering at 2 degrees and right column to cornering at 4 degrees 3.2.5 best design from the results presented above it may be concluded that tire variation having sidewall radius equal to 64.4 mm shows the best overall performance in terms of maneuverability and tire wear. while maintaining the smoothest distribution of contact pressure, it is characterized by improved cornering and braking stiffness. widening of the belts also has a positive effect on the two considered criteria for evaluation of tire performance. 4. conclusion a parametric design study of tire profile shape and belt width is presented in order to illustrate the use of the proposed approach to the fem application in tire design. the earlier reported studies have faced a bottleneck in the preprocessing phase and thus have taken into account only the influence of those design parameters that did not affect the shape of the finite element mesh. the new approach is more flexible as it facilitates the easy and automated changes of the fe mesh through the use of a specially designed cad model and an originally developed mapping and translation code. fem based parametric design study of tire profile using dedicated cad model and translation code 221 the presented study is a pilot one but it clearly demonstrates the advantages of the proposed approach. on one hand, it shows how a large number of design variations, each one geometrically different, could be analyzed in a very short time. on the other, it also leads to the improvement of the existing tire design in terms of footprint stress distribution and maneuverability. in further research, dedicated cad tire model and translation code will be used in the design of experiment phase of the tire design optimization process, to quickly and easily create the required number of fe tire models. references 1. lindenmuth, b., 2005, an overview of tire technology, in gent, a. n. (ed.), the pneumatic tire, national highway traffic safety administration, u.s. department of transportation, washington dc, pp. 1-27. 2. kabe, k., koishi, m., 2000, tire cornering simulation using finite element analysis, journal of applied polymer science, 78, pp. 1566–1572. 3. kelsey, s., branca, t., vossberg, s., 2000, method for designing pneumatic tires for rolling conditions, us patent 6083268 (to bridgestone/firestone inc, usa) 4. olatunbosun, o., bolarinwa, o., 2004, fe simulation of the effect of tire design parameters on lateral forces and moments, tire science and technology tstca, 32, pp. 146-163. 5. ghoreishy, m., 2006, finite element analysis of steady rolling tyre with slip angle: effect of belt angle, plastics rubber and composites, 35, pp. 83-90. 6. ghoreishy, m., malekzadeh, m., rahimi, h., 2007, parametric study on the steady state rolling behaviour of a steel-belted radial tyre, iranian polymer journal, 16, pp. 539-548. 7. korunović, n., trajanović, m., stojković, m., 2008, finite element model for steady-state rolling tire analysis, journal of the serbian society for computational mechanics, 2, pp. 63-79. 8. serafinska, a., kaliske, m., zopf, c., & graf, w., 2013, a multi-objective optimization approach with consideration of fuzzy variables applied to structural tire design, computers & structures, 116, pp. 7-19. 9. yang, x., behroozi, m., olatunbosun, o. a., 2014, a neural network approach to predicting car tyre micro-scale and macro-scale behaviour, journal of intelligent learning systems and applications, 2014. 6, pp. 11-20 10. korunović, n., trajanović, m., stojković, m., vitković, n., trifunović, m., milovanović, j., 2012, detailed vs. simplified tread tire model for steady-state rolling analysis, strojarstvo: časopis za teoriju i praksu u strojarstvu, 54(2), pp 153-160. 11. cho, j., kim, k., yoo, w., hong, s., 2004, mesh generation considering detailed tread blocks for reliable 3d tire analysis, adv eng softw, 35, pp. 105-113. 12. kim, k., 2006, finite element analysis of a steady-state rolling tire taking the effect of tread pattern into account, international journal of automotive technology, 7 pp. 101-107. 13. ghoreishy, m., 2009, finite element modelling of the steady rolling of a radial tyre with detailed tread pattern, iranian polymer journal, 18 p. 641-650. 14. stojković, m., trajanović, m., parametric design of automotive tyre, 2001, in asme greek section, first nat. conf. on recent advances in mech. eng. (patras, greece) 15. stojković, m., manić, m,. trajanović, m., korunović, n., 2003, customized tire design solution based on knowledge embedded template concept, in twenty-second annual meeting and conference of the tire society (akron, ohio, usa) 16. korunović, n., trajanović, m., stojković, m., 2007, fea of tyres subjected to static loading, journal of serbian society for computational mechanics, 1, pp. 87-98. 17. milovanović, j., stojković, m., trajanović, m., 2009, rapid tooling of tyre tread ring mould using direct metal laser sintering, journal of scientific & industrial research, 68 pp. 1038-1042. 18. korunović, n., trajanović, m., stojković, m., mišić, d. , milovanović, j., 2011, finite element analysis of a tire steady rolling on the drum and comparison with experiment , strojniški vestnik journal of mechanical engineering, 57(12), pp. 888-897. http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=137870 http://hrcak.srce.hr/strojarstvo http://hrcak.srce.hr/strojarstvo 222 n. korunović, m. trajanović, m. stojković parametarska studija profila pneumatika bazirana na mke, uz upotrebu namenskog cad modela i prevodioca podataka u radu je prikazana parametarska studija oblika profila i širine pojaseva pneumatika,bazirana na metodu konačnih elemenata (mke). jedna od glavnih prepreka na koju su naišli autori sličnih studija odnosi se na nemogućnost automatske promene mreže konačnih elemenata nakon promene geometrije pneumatika. da bi se ovaj problem prevazišao ovde se predlaže novi pristup. on podrazumeva automatsku promenu mreže konačnih elemenata koja prati promenu konstruktivnih parametara na namenskom cad modelu. ažuriranje mreže moguće je zahvaljujući originalno razvijenom programu za mapiranje i prevođenje podataka. na ovaj način moguće je za veoma kratko vreme analizirati performanse velikog broja geometrijski različitih varijanti konstrukcije pneumatika. iako je u pitanju pilot studija, ona je u isto vreme doprinela i poboljšanju postojeće konstrukcije pneumatika. ključne reči: pneumatik, parametarska studija, stacionarno kotrljanje, mke, analiza primenom mke 12816 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 187 198 https://doi.org/10.22190/fume240603034h © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper periodic solution of a micro-electromechanical system ji-huan he1,2 1school of jia yang, zhejiang shuren university, hangzhou, zhejiang, china 2national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china orcid id: ji-huan he https://orcid.org/0000-0002-1636-0559 abstract. this paper examines the periodic motion of the micro-electro-mechanical system (mems), which is governed by a singularity that makes it challenging to find an exact solution and to understand its dynamical properties. this paper applies the frequency formulation to gain insight into the frequency-amplitude relationship of the system. it is found that when the amplitude reaches a threshold value, the periodic motion becomes pull-in instability. this finding simplifies the warning system for the system’s unsafe operating conditions, and the frequency-amplitude relationship can be used for optimal design of the system with high accuracy and high reliability. key words: mems systems, variational theory, nonlinear oscillator 1. introduction micro-electro-mechanical systems (mems) have been a driving force in the technological revolution. they are employed in a multitude of cutting-edge applications, including fifthgeneration (5g) mobile networks [1], chips [2], ultra-sensitive sensors [3], robots [4], tsunami monitoring [5] and wearable smart fabrics [6,7]. this is due to their extremely simple structure, ultra-finely diminutive size, ultra-light weight, minimal energy consumption, tremendously high reliability, and enormously low cost. the mems system is generally open periodically. however, when the applied voltage exceeds a specified threshold value, the pull-in instability occurs, resulting in system failure. a substantial body of literature exists on the subject of pull-in instability. mikhasev, et al. studied pull-in instability of carbon nanotube nano-tweezers [8], tian and her colleagues [9] suggested a new mathematics concept that can control the pull-in instability, and yang found the pseudo-pull-in stability [10]. received: june 03, 2024 / accepted july 07, 2024 corresponding author: ji-huan he school of jia yang, zhejiang shuren university, hangzhou, zhejiang e-mail: hejihuan@suda.edu.cn https://orcid.org/0000-0002-1636-0559 188 j.-h. he when the system opens near the pull-in voltage, any minor environmental change may result in a significant shift from normal periodic motion to pull-in motion. this property is widely used for ultrasensitive mems-based sensors, which are sensitive to even a small number of nanoparticles in air, or viruses in a room, or even minor environmental changes in, e.g., air velocity, air pressure, poisonous gases, and gravity [3]. it can monitor precisely the propagation of a nano-crack, ground subsidence, the terrestrial plate motion before an earthquake, and other advanced applications, for examples, gas sensors [11], microphones [12], biosensors [13], micro-coils [14], odorant sensors [15], ocean physics [16] and monitoring systems [17]. it is therefore crucial to elucidate its dynamical properties prior to the onset of pull-in motion. this paper will now turn its attention to their periodic motion. a plethora of analytical techniques exists for the analysis of nonlinear oscillators. these include the variational iteration method [18,19], the homotopy perturbation method[20], the hamiltonian-based frequency formulation [21], the energy balance method [22], and others. this paper applies the frequency formulation [23] to investigate the periodic properties of the mems system. 2. mathematical model consider a mems system as illustrated in fig. 1, where the micro/nano beam can be a polyvinylidene fluoride (pvdf) nanofiber [24] or metal wire, it locals at the middle of the two symmetrical current-carrying wires. according to the biot–savart law [25], the micro/nano beam is under a magnetic force 0 1 2 1 1 ( ) 2 i i f h w h w   = + − + (1) where f represents the magnetic force per unit length, while 𝜇0 denotes the magnetic constant. the variables i1 and i2 correspond to the direct currents flowing through the wires, while h denotes the distance between them. fig. 1 mems system with a simple structure periodic solution of a micro-electromechanical system 189 the governing equation can be expressed as 2 0 1 2 2 1 1 ( ) 2 i i ld w m kw h w h wds   + = + − + (2) where m is the mass of the micro/nano beam, k is its elastic coefficient, s is time. eq. (2) can be written in a dimensionless form: 2 2 2 1 1 2 ( ) 1 1 1 d u ku u k u udt u + = − = − + − (3) where u=w/h, t=sω0, ω0=√(k/m), k= 𝜇0 li1 i2/(2πh2k). the initial conditions are (0)u p= , (0)u q = (4) where p and q are constants. the variational formulation for eq. (3) is  2 21 1 ( ) ( ) ln(1 ) ln(1 ) 2 2 du j u u k u u dt dt   = − − − + +     (5) the variational principle for a mems system provides an energy approach to insight into the energy conservation during its operation [26]. it is also a useful mathematical tool for complex systems, such as nano-lubrication [27], solitary waves [28], and singular waves [29]. according to the variational formulation of eq. (5), the following hamilton function can be obtained:  2 21 1 ( ) ln(1 ) ln(1 ) 2 2 du u k u u h dt + + − + + = (6) where h is hamilton constant, which depends on the initial conditions of eq. (4). after identification of h, eq. (6) becomes    2 2 2 21 1 1 1 ( ) ln(1 ) ln(1 ) ln(1 ) ln(1 ) 2 2 2 2 du u k u u q p k p p dt + + − + + = + + − + + (7) taking the derivative of this equation with respect to t, we have 2 2 1 1 0 1 1 du d u du du u k dt dt u u dtdt −  + + + = − +  (8) this equation leads to eq. (3). eq. (7) can be used for analysis of the dynamical properties by the hamiltonian-based frequency formulation [21]. 190 j.-h. he 3. frequency formulation considering the following general nonlinear oscillator ( , , ) 0u h u u u  + = (9) for a periodic solution of eq. (9), it requires [23] (0, , ) 0h u u  = (10) and ( , , ) / 0h u u u u   (11) we assume that approximate solution is given ( ) cos( )u t a t = + (12) where ω is the frequency, a and ϕ can be determined from the initial conditions: 2 2 2 q a p  = + (13) according to the frequency formulation [23], we have 2 2 2 1 ( , , ) u na u a n u na h u u u u    = =− − =−   = (14) where a is the amplitude, 0<n<1, in this paper we set n=√3/2 =0.8660. chun-hui he and chao liu suggested a modification [30] /4 2 0 /4 0 t t hdt udt    =   (15) where ξ is the weighting function. the frequency formulation is the simplest approach to nonlinear vibration systems, it has been successfully applied to various complex oscillators, for examples, fractal vibration systems [31], fractal–fractional oscillators [32], and nonlinear oscillators [33]. for eq. (3), h can be expressed as 2 2 1 ku h u u = − − (16) according to the frequency formulation given in eq. (14), we have 23 2 4 2 2 23 /2 3 3 4 4 1 22 2 (1 ) 1 1 1 1u a a kk k u a a  = − − = − = − = − − − (17) for a periodic motion, ω must be positive. when ω2<0, the pull-in motion occurs. so ω=0 leads to the following critical relationship between the voltage and the amplitude. periodic solution of a micro-electromechanical system 191 23 4 1 * 2 * 0a k− − = (18) where a* and k* are, respectively, the critical amplitude and the critical voltage. 4 8 * * 3 k a − = (19) we can also write eq. (3) in the form 2 3 2 0u u u u u ku − + − − = (20) where h can be expressed as 2 3 2h u u u u ku= − + − − (21) according to the frequency formulation given in eq. (14), we have 2 2 2 2 2 2 3 /2 3 /2 3 3 1 2 1 2 4 4 u a u a u u u k a a k   = =−  = − + − − = + − −  (22) eq. (22) is equivalent to eq. (17). in view of eq. (13), we have the following frequencyamplitude relationship 2 2 23 234 4 2 2 2 1 1 1 1 ( ) k k a q p   = − = − − − + (23) or 2 2 2 2 2 2 2 2 (4 3 3 8 ) (4 3 3 8 ) 12(4 3 ) 2(4 3 ) q p k q p k p q p  + − − + + − − − − = − (24) the relationship between the critical voltage k* and the initial conditions is 2 2 2 2 2(4 3 3 8 *) 12(4 3 ) 0q p k p q+ − − − − = (25) in order to verify the results, we consider two cases when p=a=0.1, q=0 and p=a=0.5, q=0. for the first case, according to eq. (18), the pull-in voltage can be calculated as 23 4 1 * (1 0. 541 ) 2 0. 962k = −  = (26) while the exact value is k*=0.4951 (see fig.2a) with a relative error of 0.23%. for the case of a=0.3, we have 23 4 1 * (1 0. 543 ) 2 662 0.k = −  = (27) while the exact value is k*=0.3751 (see fig.2b) with a relative error of 6.75%. for the case of a=0.5, we have 23 4 1 * (1 0.5 ) 0.40625 2 k = −  = (28) while the exact value is k*=0.5 (see fig. 2c) with a relative error of 8.3%. 192 j.-h. he a) b) c) fig. 2 phase diagram. (a) a=0.1; (b) a=0.3 (c) a=0.5 periodic solution of a micro-electromechanical system 193 to improve the prediction accuracy for k*, we can use eq.(7) to give an exact relationship between the critical voltage (k*) and the critical amplitude (a*):  21 * * ln(1 *) ln(1 *) 2 a k a a h+ − + + = (29) where h is determined by the initial conditions fig.3, fig.4 and fig.5 compare the approximate solutions with the numerical solutions for a=0.1, 0.3 and 0.5, respectively,, and a good agreement is found, showing the method is reliable. it can be found that the errors become larger for a larger k. this is because when k> k*, the pull-in instability occurs, so when k tends to k*, the periodic motion tends gradually to the pull-in instability. a) b) fig. 3 exact solution (continnued line) vs approximate solution (discontinous line) for a=0.1. (a) k=0.4; (b) k=0.49 with k*=0.4951 194 j.-h. he a) b) c) fig. 4 exact solution (continued line) vs approximate solution (discontinous line) for a=0.3. (a) k=0.2; (b) k=0.3; (c) k=0.32 with k*=0.50 periodic solution of a micro-electromechanical system 195 a) b) c) fig. 5 exact solution (continued line) vs approximate solution (discontinuous line) for a=0.5. (a) k=0.2; (b) k=0.3; (c) k=0.32 with k*=0.3751 196 j.-h. he 3. a generalized mems oscillator in fig.1, the wires can be thin plates, in that case, the governing equation can be expressed a s[22] 4 2 2 2 2 4 2 2 2 20 1 1 ( ) 2 2 ( ) ( ) lw w es w w b ei a n dx l xx t x h w h w          + = + + +     − +     (29) where w is the displacement, ei the stiffness, ρ density, a area, n applied force, l length, s cross-section area, b width, ε dielectric constant, and ν the poisson ratio. in eq.(29), the last term is the electrostatic excitation. for the fixed-fixed micro/nano thin plate, the boundary conditions are (0) ( ) 0w w l= = and (0) ( ) 0 dw dw l dx dx = = (30) considering the boundary conditions of eq. (30), we assume that w can be expressed as 2 2 2 1 2( , ) ( ) (1 ) (1 )w t x u t x x c x c x= − + + + (31) where ci (i=1,2,3,…) are constants. submitting eq. (31) into eq. (29) and integrating the resultant equation with x from 0 to l, we finally have 2 8 6 4 2 3 5 7 9 8 6 4 2 0 1 3 5 7 92 ( ) 0 d u a u a u a u a u a a u a u a u a u a u dt + + + + + + + + + = (32) where ai (i=0~9) are constants. we re-write eq. (32) in the form 3 5 7 92 1 3 5 7 9 2 8 6 4 2 8 6 4 2 0 0 a u a u a u a u a ud u dt a u a u a u a u a + + + + + = + + + + (33) the frequency formulation can be modified as /4 3 5 7 9 1 3 5 7 9 2 /4 8 6 4 2 8 6 4 2 0 ( ) cos( ) ( ) cos( ) t t a u a u a u a u a u t dt a u a u a u a u a u t dt          + + + + + + + = + + + + +   (34) submitting eq. (12) into eq. (34), after simple calculation, we have 2 4 6 83 5 3 7 5 3 9 7 5 3 1 3 5 7 92 4 6 4 8 6 4 10 8 6 4 6 6 4 29 7 5 3 7 5 3 5 3 3 8 6 4 2 010 8 6 4 8 6 4 6 4 4 a a a a a a a a a a a a a a a a a a  + + + + = + + + + (35) additionally, we can write eq. (32) in the form  3 5 7 9 8 6 4 2 1 3 5 7 9 8 6 4 2 0 1 ( ) 0u a u a u a u a u a u a u u a u u a u u a u u a     + + + + + − + + + = (36) periodic solution of a micro-electromechanical system 197 by eq. (14) we have 2 2 4 6 8 7 5 3 2 1 3 5 7 9 8 6 4 2 3 /20 3 /2 2 4 6 8 8 6 4 2 23 9 27 81 81 27 9 3 1 3 5 7 9 8 6 4 24 16 64 256 256 64 16 4 0 ( ) ( ) u a u a a a u a u a u a u a u u a u u a u u a uu a a a a a a a a a a a a a a a a a a a    = =−    + + + + − + + + = + + + + − + + + = (37) after simple operation, eq. (37) leads to the following frequency-amplitude relationship 2 4 6 83 9 27 81 1 3 5 7 92 4 16 64 256 8 6 4 281 27 9 3 0 8 6 4 2256 64 16 4 a a a a a a a a a a a a a a a a a a  + + + + = + + + + (38) eq. (35) or eq. (38) unlocks the frequency-amplitude relationship, and in view of eq.(13), we can also reveal the effect of the initial conditions on the frequency property. 4. conclusion this paper studies the frequency property of mems systems, its frequency-amplitude relationship is revealed, and the effect of the initial conditions on the frequency property is also elucidated. when the obtained frequency is zero, we can easily obtain the threshold value for k in eq. 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10.22190/fume170203005a © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper modelling and control of h-shaped racing quadcopter with tilting propellers udc 681.5 ahmed alkamachi 1,2 , ergun ercelebi 2 1 alkawarizmi college of engineering, university of baghdad, iraq 2 department of electric and electronic engineering, gaziantep university, turkey abstract. traditional quadcopter suffers terribly from its underactuation which implies the coupling between the rotational and the translational motion. in this paper, we present a quadcopter with dynamic rotor tilting capability in which the four propellers are allowed to tilt together around their arm axis. the proposed model provides leveled forward/backward horizontal motion and therefore, ensures a correct view of the onboard camera, and increases the vehicle speed by reducing the air drag. the rotor tilt mechanism also provides an instant high speed in the forward or reverse direction and offers a quick and solid air brake to restrain that fast moving speed. the nonlinear dynamical model for the quadcopter under consideration is derived using newtoneuler formalization. a control strategy is then proposed aimed to control the altitude, attitude, and the forward speed of the obtained model. finally, a numerical simulation is used to integrate the system model with the controller and to test the system performance. simulation results are reported to demonstrate the advantages of the proposed novel configuration. key words: genetic algorithm, newton-euler formalization, pid controller, racing quadcopter, tilt rotor 1. introduction in the recent years, unmanned aerial vehicles (uavs) have gained a considerable attention for their applications in scientific, civilian, and military fields. quadcopter uav is defined as a small vehicle with four rotor-propeller sets distributed around its body [1]. it is a highly nonlinear, multi input multi output (mimo), extremely coupled and underactuated system [2]. the quadcopter has become one of the most popular uav designs due to its vertical take-off received february 3, 2017 / accepted april 27, 2017 corresponding author: ahmed alkamachi affiliation: alkawarizmi college of engineering, university of baghdad, katir alnada st., aljadiriyah, baghdad, iraq e-mail: amrk1978@gmail.com 202 a. alkamachi, e. ercelebi and landing (vtol) property, simplicity in mechanical configuration, and ease of maintenance. furthermore, the use of four smaller propellers in the quadcopters reduces the danger posed by the propellers if they touch an external object as compared with one big propeller in helicopter or airplane [3]. the racing quadcopters with h-shape configuration have been gaining an increasing attention due to their use in several critical applications like surveillance, search, rescue, firefighting, and uav-based delivery. classical quadcopters can control their position and orientation by altering the rotors' spinning speeds. for example, if the quadcopter is required to roll, it would make a difference between the right and left propellers' speeds. in a similar manner, the desired pitch angle can be achieved by changing the relative speeds of the front and rear rotors [4]. for the translational motion, the quadcopter needs to roll for lateral motion and to pitch for forward/backward motion. this coupling between the quadcopter states has the undesired influence of changing the onboard surveillance camera viewing axis, so it limits the quadcopter ability to do some vision-based tasks [5]. considerable academic references have been reviewed to understand the current state of the art in the quadcopter design. for the purpose of coping with the underactuation problem, several prospects have been suggested through the reviewed literature spanning different techniques in thrust vectoring concepts, and new mechanical configurations. many gaps have been filled and several new novel designs have been proposed aiming to improve the traditional quadrotor performance. in [6], the authors propose a quadcopter with four tiltable wings distributed under its rotors. the modified quadcopter can change its mode from quadcopter to aircraft and vice versa by tilting its wings all at once. m. k. mohamed et al. propose a novel tri-tilt-rotor uav in which the three rotors can be tilted independently to gain a full authority and control over the vehicle states [7]. following a similar concept, the rotor tilt mechanism has been utilized in several studies to resolve the coupling problem between the quadcopter position and orientation. for instance, m. ryll et al. design and construct a quad tilt rotors mini quadcopter that converts the traditional quadcopter into an overactuated air vehicle [8]. another solution to resolve the fundamental underactuation limitations of the quadrotor is made by adding one degree of freedom to each of the quadcopter rotors [9]. looking for further improvement, fernandes presents a quadcopter with 2–axis tilting mechanism added to two of its rotors [10]. for the purpose of more maneuverability and robustness, the authors in [5, 11, 12] propose a quadcopter with a novel 2–axis tilting mechanism in which the rotors have two degrees of freedom. badr et al. introduce a modified quadcopter that allows each rotor to tilt about the axes perpendicular to its arm [13]. in this paper, an h-shaped racing quadcopter with tiltable rotors is introduced. to the authors’ best knowledge this configuration is novel and its dynamical model has not been obtained yet. the four rotors; which are ordinarily fixed, are allowed to rotate around their arm axes simultaneously. with this tilting capability, the number of control inputs is increased to five (the four rotors spinning speeds plus the tilt angle). the additional control input is used to govern the forward/ backward speed of the proposed model. the main contributions of this work are: (1) to derive a complete dynamical model for the h-shaped racing quadcopter with tilting propellers, (2) to propose and design a tracking control aimed at exploring the rotor tilt advantages, (3) to conceptualize the tilt mechanism and thrust vectoring concept. modelling and control of h-shaped racing quadcopter with tilting propellers 203 the paper is outlined as follows: the mathematical model is derived in section 2. the controller is designed and tuned in section 3. a comprehensive set of numerical simulation tests is then carried out in section 4. finally, the concluding remarks are given in section 5. 2. system modeling the aim of this section is to develop a dynamical model for the h-shaped tilt rotors racing quadcopter so as to define the ratio that relates the quadcopter states with the propellers spinning speed and the rotor tilt angle. at this stage, it is important to state some acceptable hypotheses that are used to simplify the process of obtaining the system dynamics. without these hypotheses, the system mathematical model will be complicated and difficult to obtain [14]. these hypotheses are: 1. the system is assumed to be symmetric and rigid. 2. the quadrotor body frame origin and the centre of gravity (cog) are coinciding. 3. the actuators are assumed to have sufficient fast response, so their dynamics are neglected [15]. 2.1. model configuration the tilt rotor h-shaped racing quadcopter will be shortened as (hcopter) throughout the following sections. the proposed model consists of an h-shaped frame with four rotors distributed at the frame tips as shown in fig. 1a. the rotors are allowed to tilt simultaneously around the arms connecting them to the main frame in the range of –π/2 < α < π/2. fig. 1b shows the rotor tilt angle. a) b) fig. 1 hcopter configuration: a) model cad drawing; b) tilt rotor angle 2.2. coordinate systems first, let f e :{x e , y e , z e } being the earth frame and f b :{x b , y b , z b }being the base frame (see fig. 2) with its center coinciding with the vehicle's cog. it is necessary to define these frames since some quantities should be expressed in f b (for instance the rotors' generated thrusts) while the other should be defined in f e (the gravitational force). a superscript letter "b" is assigned to the variables that belong to f b , while "e" lettered superscript is used to denote the variables resolved in f e . 204 a. alkamachi, e. ercelebi fig. 2 the system model schematic showing the reference frames, euler angles, and axes to go from f e to f b , a rotation matrix should be introduced. the rotation matrix (also called the direction cosine matrix), is a result of multiplying three canonical rotation matrices rx(φ), ry(θ), and rz(ψ) with a specific sequence [16], where φ (roll), θ (pitch), and ψ (yaw) are the nasa based euler angles [17]. it follows that: ( ) ( ) ( )* *b e x y zr r r r              c c c s s c s s s c c c s s s s c s s c s c s c c s c c c                                               (1) where b re is the orientation matrix of the variables in f e with respect to f b . the c and s are the sine and cosine functions, respectively. for the reverse operation (transferring the variables from f b to f e ), an inverse matrix e rb =( b re ) –1 should be used. since the orientation matrix is a result of orthogonal matrices multiplication, then its inverse is just its transpose [16]. 1 ( ) ( ) te b b b e er r r   (2) modelling and control of h-shaped racing quadcopter with tilting propellers 205 2.3. static and dynamic model in this section, all the dominant forces and torques that act on the hcopter body are discovered. 2.3.1. forces rotors' generated forces b ftr: assume that the i th propeller spinning speed is denoted by i. then, according to the blade element theory [18, 19], the generated force in the z–direction is given by k (i) 2 , where k is the rotor thrust coefficient. when the rotor tilts with an angle α, then the generated force can be resolved into its components along the x and z axes. it follows that the i th rotor generated force is: 2 2 ω sin( ) 0 ω cos( ) i b ri i k f k              (3) and the total generated force due to the four spinning propellers is: 4 1 b b tr ri i f f k u    (4) where 0 0 0 0 κ 0 0 0 0 0 0 0 0 0 0 0 0 k k k k k k k k           is the thrust coefficient matrix, and 2 2 2 2 2 2 2 2 1 1 2 2 3 3 4 4[ω * ,  ω * , ω * , ω * ,ω * , ω * , ω * ,  ω * ]tu s c s c s c s c        is the control input vector, sα and cα are the sin(α) and cos(α) function respectively. gravitational force e fg: according to the newton's law of universal gravitation, this force tries to pull down the vehicle toward the earth and it can be expressed in f e by: 0 0e gf mg           (5) where g is the gravitational constant, and m is the vehicle total mass. drag reluctant force b fd: it is a result of the air friction with the quadcopter body in motion and it is directly proportional to the vehicle moving speed. b b d df k p  (6) where kd is the 3×3 aerodynamic coefficient matrix, and [ ,  ,  ]b tp x y z is the hcopter velocity vector which represents the time derivative of quadcopter body position vector p b . 206 a. alkamachi, e. ercelebi total force b ft : the total force acting on the quadcopter body is the vector sum of the above three individual forces. *b b tr e g b b e t df r ff f  (7) 2.3.2. torques rotors' generated torque b mtr: it is a result of the four rotor's generated force around the cog. at this point, it is assumed that the origin of f b and quadcopter cog coincide. the total moment due to the rotors' generated forces is: 4 1 ( )b i ritr b i l fm   (8) where li being a vector directed from the cog to the i th rotor, i.e.: l1=[lx,ly,0] t , l2=[–lx,ly,0] t , l3=[–lx,–ly,0] t , and l4=[lx, –ly,0] t , and lx, ly are shown in fig. 3. fig. 3 hcopter schematic aerodynamic drag torque b mdt : it is the counter rotating torque due to the air drag caused by propeller spinning [20]. according to the blade element theory [18, 19], the i th rotor's drag torque around the z–axis can be expressed as (–1) i b(i) 2 , where b is the rotor drag coefficient and the factor (–1) i is negative for the propellers rotating clockwise (cw) (rotors 1 and 3) and positive for those rotating in a counter clockwise (ccw) direction (rotors 2 and 4). recall that the rotors can be tilted with angle α, then the drag torque of the i th propeller is resolved into x and z components as follows: 2 2 ( 1) ω sin( ) 0 ( 1) ω cos( ) i i i b di i b b m             (9) so the total drag torque for the four propellers is: 4 1 b b dt di i m m   (10) modelling and control of h-shaped racing quadcopter with tilting propellers 207 total torque b mt: expressed in f b , the total torque acting on the hcopter body is the vector sum of the above two individual torques. b b b t tr dtm m m b u   (11) where β 0 0 0 0      y y y y x x x x y y y y b kl b kl b kl b kl kl kl kl kl kl b kl b kl b kl b                  is the moment coefficient matrix, and u is the control input vector as before. 2.3.3. virtual control vector v at this phase of the mathematical modelling and for the purpose of better understanding, it is important to define a virtual control vector v =[v1, v2, v3, v4] t . first virtual control input v1 is in charge of controlling the vehicle altitude since it represents the resultant lifting forces generated by the rotors in the upward positive z direction. 2 2 2 2 1 1 2 3 4(ω ω ω ω )cos( )b trzv f k      (12) second virtual control input v2 is the total torque around x–axis; thus it is responsible for controlling the roll angle. 2 2 2 2 2 2 2 2 2 1 2 3 4 1 2 3 4( ω ω ω ω )sin( ) ω ω ω ω cos(( ) )b tx yv m b kl           (13) third virtual control input v3 represents the total torque around y–axis so it controls the pitch angle of the hcopter. 2 2 2 2 3 1 2 3 4ω ω ω ω cos )( ) (b ty xv m kl       (14) fourth virtual control input v4 is responsible for adjusting the yaw angle since it represents the total torque around z–axis. 2 2 2 2 2 2 2 2 4 1 2 3 4 1 2 3 4ω ω ω ω sin( ) ( ω ω ω ω( ) )cos( )b tz yv m kl b            (15) in addition to the above virtual control signals, rotor tilt angle α is used to control the forward/backward speed of the vehicle. when the rotors tilt, they provide the required horizontal force to move the hcopter along the x–axis direction. equations (11) through (14) can be combined into one matrix equation. it follows that: v u  (16) where γ is the coefficient matrix and is equal to: 0 0 0 0 0 0 0 0 y y y y x x x x y y y y k k k k b kl b kl b kl b kl kl kl kl kl kl b kl b kl b kl b                      208 a. alkamachi, e. ercelebi 2.3.4. model dynamics with a view to obtain the quadcopter dynamic and the equations of motion, we exploit the typical newton-euler formalization. recall that p b represents the hcopter position vector expressed in f b , then the newton-euler equations are: b b tmp f (17) and ( )b b b b tj j m     (18) where jr 3×3 is the moment of inertia tensor, ωb is the body angular velocity vector, and b is the angular acceleration of the quadcopter body expressed in f b . substituting eq. (7) in eq. (17), we can get the linear acceleration vector expressed in f b as: 1 1 1 ( ) 0 b trx b b b b b e g d g e e e d f p k u r f k p r f k v p m m                  (19) angular acceleration b can be obtained by substituting eq. (11) in eq. (18): 2 1 1 3 4 ( ) (( ))b b b b b bo j bu vj v j v j                       (20) where o b represents the vehicle orientation vector expressed in the body coordinate ( [ ,  ,  ]b to    ), therefore ( [ ,  ,  ]b to    ). to this end, we have obtained the hcopter dynamical equations that govern its operation. 3. controller design in this work, the (matlab/simulink) environment is used to integrate and examine the obtained model. it is also used to tune the pid parameters using the genetic algorithm (ga) and to carry out all the subsequent tests. the simulation environment is set up on a personal computer with 2.5 ghz processing speed and 6 gb ram, running on windows 10. the complete control system block diagram for the hcopter is shown in fig. 4. the list of model physical parameters used through all the tests is given in table 1. fig. 4 the proposed controller block diagram modelling and control of h-shaped racing quadcopter with tilting propellers 209 table 1 the proposed hcopter model physical parameters parameters value m 1.2 kg. lx, ly 20 cm k, b 3e-6 n.sec 2 /rad 2 , 1e-7 nm.sec 2 /rad 2 j diag[0.02, 0.02, 0.04] * kg.m 2 kd diag[0.3, 0.3, 0.5] * min, max ** 100, 2000 rad/sec * diag[ ] is a diagonal matrix ** it is the rotor's upper and lower speed limit respectively for efficient surveillance tasks, it is important that the quadcopter has a precise control on its attitude, speed, and altitude [11]. the control problem addressed in this work is an output tracking problem. a five output pid control system is proposed to control the orientation (roll, pitch, and yaw), altitude, and the forward speed of this novel air vehicle. 3.1. forward/backward speed control the additional control input (tilt angle α) is used to control hcopter forward speed which in turns improves the surveillance based tasks. the speed error signal is obtained by subtracting desired x speed ( dx ) from real hcopter speed ( x ). the error signal is then fed to a pid controller that determines the required tilt angle (α) to achieve the desired speed. 3.2. altitude control the altitude error signal is formed by subtracting the measured altitude (z position) from desired elevation zd. it is then applied to the pid controller that adjusts the value of control input v1 to achieve the required altitude. 3.3. orientation control the orientation controller is the core of the control system and it is of critical importance. it consists of three pid controllers to keep the quadcopter attitude to the required roll (φd), pitch (θd) and yaw (ψd) angles by controlling the three virtual control signals v2, v3, and v4, respectively. 3.4 virtual control to rotors' speeds in this block, the determined rotor tilt angle (α) and the virtual control vector (v) are used to determine the proper rotors' spinning speeds with the aid of eq. (16). 3.5 pid parameters tuning using ga genetic algorithm is a search heuristic process that simulates the natural selection procedure [21]. the algorithm is used to tune the five pid parameters by minimizing the following cost function: (ω ω ω           1,2,3,4)| min i max for iobj err    (21) where err is the difference between the desired and the actual output response. 210 a. alkamachi, e. ercelebi the tuning process is subjected to actuator upper and lower constraints as noted in eq. (21). it means that if the selected pid parameters lead to excessive control signal output that cause the actuators to exceed their upper and lower limits, then the cost function value for these selected parameters is assigned to an extremely large weight so that it will be excluded from the next iteration. the genetic algorithm takes the response data from the simulink model to evaluate the cost function and to select the optimal pid parameters. the obtained pid parameters with their associated step response characteristics (settling time ts and percentage peak overshoot mp) are tabulated in table 2. table 2 pid parameters and step response characteristics controlled state (initial final) kp ki kd ts sec. mp % (0 1 .)dz m 22.2 22.4 7.1 1.18 1.5 (0 5deg.)d  17.5 1.04 0.78 0.16 0.3 (0 5deg.)d  17.5 1.04 0.78 0.16 0.3 (0 10deg.)d  1.63 0.04 0.25 0.43 0.4 (0 2 / sec.)d mx  0.99 0.99 0.05 2.11 7.8 4. simulation results in order to check the validity of the proposed model and controller, a simulink model is developed. the purpose of the simulation is to highlight the hcopter capabilities and the dynamic rotor tilting advantages. the simulation process falls into two phases. the first simulation assumes the ideal case and the second simulation assumes the existence of sensor noise, parameters uncertainties, and external disturbances. 4.1. ideal case altitude and attitude tracking: in this test, it is assumed that the hcopter is initially at rest (p e =[0,0,0] t , o b =[0,0,0] t ) and it is then ordered to climb to 1 meter (z = 1m.). the vehicle is then commanded to follow the following desired attitude (φd =5deg. at t=1sec., θd = –5deg. at t=2sec., ψd =10deg. at t=3sec.). the simulation results for the desired step inputs are shown in fig. 5, while the step input characteristics for this test are given in table 2. hcopter speed control: in this part of the simulation, the hcopter is assumed to be hovering at z = 1 meter, and it is required that the body roll, pitch and yaw angles should be kept at zero degree. test 1: the hcopter in this test is examined for its ability to maintain a constant speed while keeping its body leveled φd = θd = 0deg. the speed throttle is applied gradually simulating a ramp input from rest to 10 m/sec(36km/h) in 5 seconds as shown in fig. 6a. the simulation results in figs. 6b and 6c show that the hcopter can track the desired speed efficiently while maintaining zero attitudes which is impossible for conventional quadcopter to do [22]. test 2: the aim of this test is to find the maximum forward moving speed that our proposed hcopter can reach. the tilt angle is increased gradually from 0 deg to its maximum modelling and control of h-shaped racing quadcopter with tilting propellers 211 allowable limit (45 deg.) in 10 sec. it can be seen from fig. 7 that the maximum reachable speed is approximately 39 m/sec (140km/h). a) b) fig. 5 step input test simulation results: a) altitude response; b) attitude response a) b) c) fig. 6 speed control test simulation results: a) desired speed; b) speed error; c) attitude behavior fig. 7 hcopter maximum speed test simulation result 212 a. alkamachi, e. ercelebi air braking: from the previous test, it can be noticed that the proposed quadcopter could reach a very high forward speed in a relatively short period; thus, a solid brake is required to rein the vehicle efficiently at the proper time. fortunately, the rotor tilt mechanism offers an instant braking system that reduces the vehicle speed to zero in a very short period. in this test, the hcopter forward speed is increased to 10 m/s and then suddenly reduced to zero. from fig. 8a, it can be seen that the rotors tilted with a positive angle during the acceleration period and instantaneously flipped to the opposite side to provide the necessary horizontal opposite force to stop the vehicle. the hcopter speed behavior is shown in fig. 8b. a) b) fig. 8 air braking test simulation results: a) tilt angle behavior; b) vehicle speed 4.2. non-ideal case in the non-ideal case, three tests were made to examine the model validity and the controller robustness with the existence of white gaussian sensor noise, external disturbances, and model parameters uncertainty. sensor noise: the hcopter is assumed to be hovering horizontally at z = 1 meter. a white gaussian noise with zero mean (shown in fig. 9a) is applied to the model in the feedback loop to simulate the sensor noise. the simulation results in figs. 9b and 9c reveal the controller ability of suppressing the sensor noise successfully. the maximum error in the altitude is just a few millimeters and the maximum drift in the orientation is bounded by +/– 0.01 degree. external disturbance: the hcopter is assumed to be hovering horizontally at z = 1 meter and it is commanded to travel with 10 m/sec(36km/h) in the forward direction with the existence of opposite light wind with a fixed speed of 1.2 m/sec(4.5km/h) [23]. the simulation result shown in fig. 10 asserts the controller effectiveness in coping with the external disturbances. modelling and control of h-shaped racing quadcopter with tilting propellers 213 a) b) c) fig. 9 noise suppression test simulation results: a) noise signal; b) altitude drift; c) attitude drift fig. 10 hcopter speed error with and without the effect of a light wind disturbance model parameters uncertainties: to check the controller effectiveness in dealing with the model uncertainties, the vehicle is ordered to change its altitude from 1 to 2 meter with and without the existence of a 0.5 kg. payload. the payload is suspended from the cog of the model. a comparison between the results (see fig. 11) for both cases shows that the controller performed well against the external additive weight. fig. 11 uncertainty test simulation results 214 a. alkamachi, e. ercelebi 5. conclusion in this paper, we have addressed the modeling and control of a novel tilt rotor h-shaped racing quadcopter. compared to the traditional quadcopter, the proposed model offers higher moving speed with instant air brake. the new design resolves an important part of the underactuation problem in the traditional quadcopter where the translational and rotational motion cannot be controlled independently. the hcopter, in contrast, allows controlling the rotors thrust direction, therefore granting additional control input results in the uncoupling between the pitch and the forward motion. several simulation tests have been carried out and the results have been demonstrated and discussed to evaluate the proposed controller validation. by obtaining these encouraging results, the next stage is to design, build, and control the hcopter prototype. references 1. salih, a.l., moghavvemi, m., mohamed, h.a.f., gaeid, k.s., 2010, modelling and pid controller design for a quadrotor unmanned air vehicle, ieee int. conf. on automation quality and testing robotics (aqtr), 1, pp. 1–5. 2. mian, a.a., wang, d.b., 2008, dynamic modeling and nonlinear control strategy for an underactuated quad rotor rotorcraft, journal of zhejiang university-science a, 9(4), pp. 539-545. 3. nemati, a., kumar, m., 2014, 2014, modeling and control of a single axis tilting quadcopter, ieee american control conference (acc), pp. 3077-3082. 4. bouabdallah, s., noth, a., siegwart, r., 2004, pid vs lq control techniques applied to an indoor micro quadrotor, ieee/rsj international conference on intelligent robots and systems (iros 2004), 3, pp. 24512456. 5. şenkul, f., altuğ, e., 2013, modeling and control of a novel tilt—roll rotor quadrotor uav, ieee international conference on unmanned aircraft systems (icuas), pp. 1071-1076. 6. çetinsoy, e., dikyar, s., hançer, c., oner, k.t., sirimoglu, e., unel, m., aksit, m.f., 2012, design and construction of a novel quad tilt-wing uav, mechatronics, 22(6), pp. 723-745. 7. mohamed, m.k., lanzon, a., 2012, design and control of novel tri-rotor uav, ieee international conference on 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kabeel, a.e., 2016, a novel modification for a quadrotor design, ieee international conference on unmanned aircraft systems (icuas), pp. 702-710. 14. alkamachi, a., erçelebi, e., 2017, modelling and genetic algorithm based-pid control of h-shaped racing quadcopter, arabian journal for science and engineering, pp. 1-10, doi: 10.1007/s13369-017-2433-2. 15. hua, m. d., hamel, t., morin, p., samson, c., 2013, introduction to feedback control of underactuated vtolvehicles: a review of basic control design ideas and principles, ieee control systems, 33(1), pp. 61-75. 16. gruber, d., 2000, the mathematics of the 3d rotation matrix, xtreme game developers conference. 17. bayrakceken, m.k., yalcin, m.k., arisoy, a., karamancioglu, a., 2011, hil simulation setup for attitude control of a quadrotor, ieee international conference on mechatronics (icm), pp. 354-357. modelling and control of h-shaped racing quadcopter with tilting propellers 215 18. fernando, h.c.t.e., de silva, a.t.a., de zoysa, m.d.c., dilshan, k.a.d.c., munasinghe, s.r., 2013, modelling, simulation and implementation of a quadrotor uav, 8th ieee international conference on industrial and information systems (iciis), pp. 207-212. 19. mahony, r., kumar, v., corke, p., 2012, multirotor aerial vehicles, ieee robotics and automation magazine, 19(3), pp. 20-32. 20. elkholy, h.m., 2014, dynamic modeling and control of a quadrotor using linear and nonlinear approaches, phd dissertation, master thesis, the american university in cairo. 21. mitchell, m., 1996, an introduction to genetic algorithms, mit press. 22. von frankenberg, f., 2016, development of an autonomous omnicopter aerial vehicle, master dissertation, institute of technology/ university of ontario. 23. beck, h., lesueur, j., charland-arcand, g., akhrif, o., gagné, s., gagnon, f., couillard, d., 2016, autonomous takeoff and landing of a quadcopter, ieee international conference on unmanned aircraft systems (icuas), pp. 475-484. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 2, 2014, pp. 157 169 experimental determination of bearing loads in rotating platform drive mechanisms of hydraulic excavators udc 621.22 vesna jovanović, dragoslav janošević, nikola petrović university of niš, faculty of mechanical engineering, serbia abstract. the paper presents a procedure for experimental determination of slewing bearing loads in the rotating platform drive mechanisms of hydraulic excavators with an excavating manipulator. a mathematical model is defined which enables the determination of the vectors of force and moment of bearing loads, on the basis of the measured quantities of the state of an excavator in operation under exploitation conditions. the measured quantities of the state of an excavator relate to the position of the kinematic chain and pressures of the hydrostatic system in the actuator ducts of the excavator drive mechanisms. as an example, the paper provides the research results obtained in the experimental determination of slewing bearing loads in the rotating platform drive mechanism of a hydraulic excavator with the mass of 16t. key words: hydraulic excavators, rotating platform bearing loads 1. introduction hydraulic excavators are mobile machines whose primary function is non-continuous excavation and transport of soil within a variable working range. the structural support of the primary function of an excavator is a kinematic chain, which comprises the following members, independently of the excavator size: support and movement mechanism l1 (fig. 1a), rotating platform l2 and the manipulator with boom l3, stick l4 and tool l5 – most often in the form of a bucket. in the spatial manipulation of an excavator, the rotating platform can rotate in both directions for a desired number of full rotations driven by a drive mechanism which principally comprises: a hydraulic motor 2.1 (fig. 1b), a reducer 2.2 and a slewing bearing 2.3 [1]. the slewing bearing consists of received march 04, 2014 / accepted june 3, 2014 corresponding author: vesna jovanović university of niš, faculty of mechanical engineering, department of transport and logistics, aleksandra medvedeva 14, 18000 niš, serbia e-mail: vesna.nikolic@masfak.ni.ac.rs original scientific paper 158 v. jovanović, d. janošević, n. petrović outer and inner races, out of which one has a toothed edge on the outer or inner side. between the races, in one or multiple rows, there are rolling bodies (balls, rollers) in appropriate grooves [2]. a bolted joint connects the untoothed bearing race to the frame of the rotating platform, while the toothed race is connected to the frame of the support and movement mechanism. the gear on the output shaft of the drive reducer is coupled with the toothed race of the slewing bearing, thus producing the desired rotation of the platform [3]. the manipulator drive mechanism actuators are double acting hydraulic cylinders. former research studies connected to the rotating platform drive of hydraulic excavator are related to the dynamics of hydrostatic part drive [4] and the regulation of platform management rotation system [5] and optimal synthesis of a manipulator of mobile machines[6, 7]. work cycles of hydraulic excavators, regardless of the size of an excavator, characteristically take place in a variable working area by cyclical repetition of the following operations: excavating the material, transporting the material from the plane of excavation to the plane of unloading, unloading the material, and returning to the plane of excavation. the aim of this paper is to determine the cycle operation in which the slewing bearing of an excavator rotating drive is subjected to the highest loading, which can be applicable to the reliable selection of the bearing. 2. mathematical model in this paper, a mathematical model is developed to determine the slewing bearing loads in a platform rotating drive of hydraulic excavators, on the basis of the measured quantities of the state of the kinematic chain and excavator drive mechanisms (table 1) (fig. 1) when the machine operates under real-exploitation conditions. the mathematical model covers the general five-member configuration of an excavator which consists of: support and movement mechanism l1 (fig. 1), rotating platform l2, and the three-member planar manipulator with: boom l3, stick l4 and excavating bucket l5. members of the kinematic chain of an excavator create fifth-class kinematic pairs – rotary joints with a single degree of freedom. joints axes are the axes of relative turning (rotation) of the members which constitute the kinematic pairs of the chain. the support and movement member of the excavator and the support surface create a third-class zero joint with potential movements in the plane of the surface. the center of joint o2 of the support member-rotation member kinematic pair is the point of perpendicular intersection of the vertical axis of the joint through the horizontal plane which contains the centers of rolling elements of the slewing bearing that connects the support and movement member to the rotation member of the chain. centers of the manipulator joints (oi, i = 3,4,5) are points of intersection of the axis of joints through the plane of symmetry of the manipulator chain members. the manipulator chain contained in the model of the excavator is of planar configuration. the axes of joints are parallel, while the centers of joints lie in the same plane – the plane of the manipulator. the intersection of the bucket cutting edge through the plane of the manipulator represents the center of bucket cutting edge ow. experimental determination of bearing loads in rotating platform drive mechanisms of hyd. excavators 159 table 1 measured quantities of the state of the kinematic chain and excavator drive mechanisms measuring spot name of the measured quantity symbol dimension м1 lifting of the support and movement mechanism c1 m м2 platform rotation angle c2 о м3 boom hydraulic cylinder motion c3 m м4 stick hydraulic cylinder motion c4 m м5 bucket hydraulic cylinder motion c5 m м6 pressure in one duct of the hyd. motor for platform rotation drive p21 mpa м7 pressure in other duct of the hyd. motor for platform rotation drive p22 mpa м8 pressure in the boom hydraulic cylinder on the piston side p31 mpa м9 pressure in the boom hydraulic cylinder on the connecting rod side p32 mpa м10 pressure in the stick hydraulic cylinder on the piston side p41 mpa м11 pressure in the stick hydraulic cylinder on the connecting rod side p42 mpa м12 pressure in the bucket hydraulic cylinder on the piston side p51 mpa м13 pressure in the bucket hyd. cylinder on the connecting rod side p52 mpa the assumptions of the mathematical model of the excavator kinematic chain are: the support surface and kinematic chain members are modeled using rigid bodies, the contact between the support and movement member and the excavator support surface is taken as the first joint which has a variable position and form, thus having the form of a translatory-sliding joint along the contact between the support and movement member and the surface, while having the form of rotary joints o11,o12, whose axes represent potential excavator rollover lines, on the edges of the contact, the kinematic chain of the excavator has an open configuration, bearing in mind that even though it has a closed configuration during the digging operation, it is still observed as an open configuration chain, whose final member – the bucket, is subjected to technological digging resistances w, during the manipulation task, the kinematic chain of the excavator is subjected to gravitational, innate and external (technological) forces – digging resistances, the position of the mass center of a hydraulic cylinder is in the middle of the current length of that hydraulic cylinder, the influence of friction resistances is neglected in the kinematic chain and excavator drive mechanism joints. the area of the excavator model is determined by an absolute coordinate system oxyz (fig. 1) with unit vectors k,j,i rrr along the coordinate axes. the excavator support surface lies in the horizontal axis of oxz absolute coordinate system, while vertical axis oy of the same system falls on the axis of the support member-rotating member kinematic pair when the excavator is positioned on the horizontal surface. a member of kinematic chain li, in its local coordinate system oi xi yi zi, with unit vectors iii k,j,i )r)r)r along the coordinate axes, is defined by a set of geometric, kinematic and dynamic parameters [8]: 160 v. jovanović, d. janošević, n. petrović { , , , , }i i i i il e s t m j= i )) ) )rr r (1) where ie )r is the unit vector of joint oi axis which connects member li to the previous member li-1 (fig. 1), is )r is the vector of the position of joint oi+1 center which is used to connect chain member li to next member li+1 (vector si magnitude represents the kinematic length of the member), it )r is the vector of the position of member li mass center, mi is member mass, ij ) is the tensor of the member moment of inertia. fig. 1 mathematical model of an excavator: a) excavator kinematic chain, b) excavator platform rotation drive experimental determination of bearing loads in rotating platform drive mechanisms of hyd. excavators 161 vector quantities marked with a ‘cap’ relate to the local coordinate system, while those without a ‘cap’ relate to the absolute coordinate system. the inner (generalized) coordinates of the mathematical model of the excavator kinematic chain are represented by angles θi of the relative position of member li in relation to previous member li-1 upon rotation around joint oi axis. the lifting angle of movement mechanism θ1 is determined on the basis of the measured relative vertical movement c1 of the support and movement member l1 in relation to the support surface. angles θi (i=3,4,5) of the relative position of manipulator member li in relation to previous member li-1 are determined depending on measured length ci of the hydraulic cylinders of the manipulator boom, stick and bucket drive mechanisms [8]. unit vector ie r of joint oi axis of the excavator kinematic pair in the absolute coordinate system is determined by using the equation: i ioe a e= i )r r (2) unit vector 1e r of the first joint axis is directed along potential longitudinal x-x or transverse z-z (fig. 1) excavator rollover lines. vector ir r of joint oi center of the excavator kinematic pair in the absolute coordinate system is determined by using the equation: 1 1 2,3,4,5 i i jo j j r a s i − = = ∀ =∑ )r r (3) vector wr r of the center of the bucket cutting edge in the absolute coordinate system is determined by using the equation: ∑ = = 5 1i iiow sar )rr (4) vectors tir r of the center of excavator kinematic chain member li mass in the absolute coordinate system are determined by using the equation: iioiti tarr )rrr += (5) where aio is the transfer matrix used to transfer the vector quantities from local coordinate system oi xi yi zi of member li to absolute coordinate system oxyz [8]. kinematic quantities for the center of chain member li mass are: linear vi and angular ωi velocity and linear wi and angular εi acceleration, where the movement of previous member li-1 is taken as transferable, while the movement of observed member li in joint oi is taken as relative. to determine the kinematic quantities of chain member li in relation to the absolute coordinate system, recursive equations are used [8]: iiii e r&rr 1 θ+ω=ω − (6) 1 1(i i i i i i ieε ε θ ω θ− −= + + × r r r )er r&& & (7) 1 1 1 1( ( )) (i i i i i iv v s t tω− − − −= + × − + × )ω r rr rr r r (8) (9) 1 1 1 1 1 1 1 1( ( )) ( ( )) ( ) (i i i i i i i i i i i i iw w s t s t t tε ω ω ε ω− − − − − − − −= + × − + × × − + × + × × r r rr r r r rr r r r )iω rr where i are angular velocities and angular accelerations of member li ,θθ &&& i in joint oi. 162 v. jovanović, d. janošević, n. petrović dynamic quantities of member li are: innate force fi, which is determined by newton’s second law: iii wmf rr −= (10) and the moment of innate forces mi, which is determined by euler’s dynamic equations: ( ) ; ui i i i i i ui io uim j j m aε ω ω= − + × = m ) )r ) ) ) r r) )r r r (11) the total force related to the center of member li mass, also taking into account the influence of gravity, is equal to: jgmff iiui rrr −= (12) bearing in mind the assumption that the vector of digging resistance w acts in the center of the bucket cutting edge, the fictive interruption of the manipulator kinematic chain in two different joints oi and oj (i ≠ j, i,j=3,4,5) can set the equilibrium conditions, for the removed chain parts, by using the equation (fig. 1) [9]: ( ( ))i w i i rie w r r e m 0⋅ × − + ⋅ = r rr r r r (13) ( ( ))j w j j rje w r r e m 0⋅ × − + ⋅ = r rr r r r (14) where rjri are the resulting moments in the center of joints om,m rr i and oj, jiw r,r,r rrr the vectors of the position of bucket cutting edge center ow and joints oi and oj . supposing that, during the digging process, the excavator manipulator is positioned within oxy plane, the use of the previous two equations can determine the quantities of digging resistances wx and wy components according to the equations: ( ) ( ) ( )( ) ( )( ) rzi w j rzj w i x w j w i w i w j m x x m x x w x x y y x x y y − − − = − − − − − (15) ( ) ( ) ( )( ) ( )( ) rzi w j rzj w i y w j w i w i w j m y y m y y w x x y y x x y y − − − = − − − − − (16) where mrzi, mrzj are the rotary moments around the axes of joints oi and oj. the quantities of rotary moments around the axes of joints oi and oj are determined by using the equations: 5 5 ( ( ))rzi i ri ci i uk tk i i uk k i k i m e m m e f r r e m = = = ⋅ = + ⋅ × − + ⋅∑ ∑ r rr r r r rr (17) 5 5 ( ( ))rzj j rj cj j uk tk j j uk k j k j m e m m e f r r e m = = = ⋅ = + ⋅ × − + ⋅∑ ∑ r rr r r r rr (18) where mci, mcj are the moments of drive mechanisms of the excavator manipulator in joints oi and oj. the quantities of the moments of drive mechanisms of the excavator manipulator are determined by using the equations: experimental determination of bearing loads in rotating platform drive mechanisms of hyd. excavators 163 2 2 2 1 1 2 1 2 3 4 ( ) ( ) i 3,4,5, k 1, 1 4 4 i i i ci i ci ci i i cmi d d d m sign k r n p p k k π π η ⎡ ⎤− = ⋅ ⋅ ⋅ − ⋅ ∀ = = = =⎢ ⎥ ⎣ ⎦ 5 − (19) where rci is the transfer function of the drive mechanism of the excavator manipulator [9], di1,di2 are the diameters of the piston and connecting rod of the hydraulic cylinder, nci is the number of hydraulic cylinders in the mechanism, pi1,pi2 are the measured pressures in the hydraulic cylinder on the piston side and on the connecting rod side of drive mechanism, ηcmi is the mechanical degree of the hydraulic cylinder efficiency. the quantity of the lateral component of digging resistance wz is determined by using the equation: 5 5 2 21 22 c2 2 c2 2 2 2 2 2 ( ) 1r ( ( )) 2 c z c uk tk uk k k w d p p w n e f r r e m x η π = = ⋅ −⎡ ⎤= ⋅ ⋅ ⋅ + ⋅ × − + ⋅ ⋅⎢ ⎥ ⎣ ⎦ ∑ ∑ r rr r r (20) where rc2 is the transfer function of the drive mechanism of the excavator platform rotation, dc2 is the specific flow of the hydraulic motor of the platform rotation drive, nc2 – the number of rotating platform drives, p21,p22 are measured pressures in the working ducts of the hydraulic motor, ηc2 is the degree of efficacy of the excavator platform rotation drive. the fictive interruption of the kinematic chain of the excavator in joint о2 of rotating platform l2 and the reduction of all loads, of the removed part, into its centre, yield: the resulting force which subjects the slewing bearing to loading: ∑ = += 5 2 2 i uifwf rrr (21) and the resulting moment which subjects the slewing bearing to loading: 5 5 2 2 2 2 2 (( ) ) (( ) )w w ui ui i i m r r w r r f m = = = − × + − × +∑ ∑ r r rr r r r r (22) where is the vector of the position of the joint centre (slewing bearing) o2r r 2. components of force f2 of joint o2 along the coordinate axes: 222222222222 , , kfafjfafifaf ozoyox )rr))rr))rr) ⋅=⋅=⋅= (23) components of moment m2 of joint o2 along the coordinate axes: 2 2 2 2 2 2 2 2 2 2 2,x o y o z o 2m a m i m a m j m a m k= ⋅ = ⋅ = ⋅ )) ) rr r r) s ) r ) (24) where ao2= a2o t is transfer matrix from the absolute to local coordinate system o2 x2 y2 z2 [9]. components of slewing bearing loads of the excavator rotating platform are axial force f2a, radial force f2r, and moment m2r: ya ff 22 ) = ; f f2 2 0,5 2 2 2( )r x zf= + ) ) 2 2 0,5 2 2 2( )r x zm= +; m m ) ) (25) the size of the bearing is selected on the basis of the determined equivalent spectrum of bearing loads and diagrams of bearing loading capacity, which are provided by the specialized bearing manufacturers. 164 v. jovanović, d. janošević, n. petrović the equivalent spectrum of bearing loads consists of an equivalent force and an equivalent bearing load moment determined by the equations for equivalent force fe and equivalent moment me [10]: 2 2( )e a r sf a f b f f= ⋅ + ⋅ ; 2e s rm f m= ⋅ (26) where a is the factor of the axial force influence, b is the factor of the radial force influence, fs is the factor of the bearing working conditions. values of factors a,b,fs are provided by the bearing manufacturers depending on the type of bearing (single-row, multi-row, ball, roller), type and size of machines and their working conditions. 3. program according to the previously defined mathematical model for determining the slewing bearing loads in a drive mechanism of an excavator rotating platform, it is necessary to measure the quantities of state (table t1) (fig. 1) of the excavator kinematic chain and drive mechanisms in operation under real-exploitation conditions. a program (eaol) is developed for computer processing and analyzing measured quantities. by employing measured quantities ci,pi1,pi2 as input data, the program firstly determines, in the function of the duration of the work cycle, the geometric and kinematic quantities: generalized coordinates θi, coordinates of joint centers and mass centers of chain members, angular velocities i and angular accelerations i , and linear vθ& θ&& i and angular ωi velocity and linear wi and angular εi acceleration for the mass center of the excavator kinematic chain members. quantities of angular velocities and angular accelerations of the excavator kinematic chain members are determined on the basis of the double numerical differentiation by using the equations: ( ) ( ) 2 i t t i t t i t θ θ θ +δ −δ− = δ & (27) ( 2 ) ( ) ( 2 ) 2 2 4 i t t i t i t t i t θ θ θ θ + δ − δ− + = δ && (28) where: θi(t) is the generalized coordinate at moment t of the duration of the digging operation, θi(t+δt), θi(t-δt), θi(t+2δt), θi(t-2δt) are generalized coordinates (angles) in the moment of time which is larger or smaller for one or two intervals of time δt than time t, δt is the interval of time between two subsequent measurements of quantities. the program further determines transfer functions rci and drive moments mci of individual drive mechanisms on the basis of the quantity parameters of actuators di1,di2 and measured pressures pi1,pi2 in their working ducts. finally, after determining innate forces fui and moment mui of chain members, the program determines the vector of digging resistance w and components of force f2 and moment m2 of the bearing load. experimental determination of bearing loads in rotating platform drive mechanisms of hyd. excavators 165 4. example as an example, the determination and analysis of the bearing loads are conducted on the basis of the results of the testing of a continuous tracks excavator bgh 600c, manufactured by imk 14 october – kruševac, with the mass of 16t and the power of 70 kw, equipped with an excavating manipulator with a bucket of 0,6 m3 in capacity. penetrometer measurements show that during the testing the excavation of category iii and iv soil is performed [8]. symbols, names and dimensions of measured quantities are given in table 1, and the measuring chain is shown in fig. 1c. the initial state of the measured quantities is defined on the horizontal contact surface of the support and movement mechanism, with the plane of the manipulator symmetry parallel to the transverse plane of the support and movement mechanism symmetry, with retracted hydraulic cylinder connecting rods and unloaded drive mechanisms. the sampling of measured quantities is conducted in the time interval δt = 0,032 s. during the testing, forty-two full cycles are measured, from the digging operation, through the transfer and unloading of soil, to the operation of returning to the new beginning of excavation, with different manipulation tasks within the entire working range of the excavator. out of the total number of measurements, this paper separates and analyzes the measurements conducted during the digging of a canal, equal in width to the bucket, from the excavator support surface. for the spatial orientation of the measured quantities and obtained results of the analyzed cycle, a path of the center of the bucket cutting edge is given (fig. 1) in оxy and oxz plane of the absolute coordinate system. part of the obtained results is presented in the diagrams of measured (fig. 2) and determined quantities (fig. 3) in the function of the duration of the excavator work cycle. out of the measured quantities the diagrams show changes in: the movement of the support and movement mechanism c1 (fig. 2a), the angle of platform rotation �2 and the motions of hydraulic cylinders c3, c4, c5, and pressures pi1,pi2 (fig. 2b, c) in the ducts of the excavator drive mechanism actuators. diagrams of drive mechanism actuator motion (fig. 2a) show that the selected manipulation testing task is conducted with the simultaneous movement of at least two members of the excavator kinematic chain. depending on the position, the digging process is performed with a separate or simultaneous movement of the manipulator bucket and stick. at the beginning and the end of the digging process the lifting (movement) of the support and movement member occur, while the angle of platform rotation is relatively small. the operation of material transfer from the plane of digging into the plane of unloading is performed by a simultaneous raising of the boom and platform rotation. the operation of material unloading is performed by moving the bucket and stick, and the operation of returning to the plane of digging by moving the boom and stick with platform rotation. during the digging operation, a gradual increase in pressures occurred in the working chambers of the actuators up to their maximal values (fig. 2b, c). due to the discontinuation of the digging process, drastic changes appear in pressures in the drive mechanism actuators. abrupt changes in pressures in the hydraulic motor ducts of the platform rotation drive take place when the platform start and stop moving during the operation of material transfer. 166 v. jovanović, d. janošević, n. petrović fig. 2 measured quantities of the state of the excavator: a) movement of the rotating mechanism, angle of the platform rotation and motions of manipulator hydraulic cylinders, b, c) pressures in the excavator drive mechanism actuators out of the determined quantities, obtained by using the developed program, the paper shows the change in the quantities of platform drive mechanism during the cycle, based on the components of force f2 and moment m2 (fig. 3a, b) of the bearing loads in the static and dynamic mathematical model of the excavator. as the diagrams show, bearing loads determined by using the static and dynamic mathematical model of the excavator experimental determination of bearing loads in rotating platform drive mechanisms of hyd. excavators 167 fig. 3 loading of the platform rotation drive bearing: a) bearing load force components, b) bearing load moment components, c) equivalent bearing load force and moment. vary only slightly during the major part of the digging process, which shows that the dynamic influence due to the movement of the excavator kinematic chain members during the digging process is small since the digging process itself takes place relatively slowly. the dynamic influence on the loading of the bearing occurs at the beginning and 168 v. jovanović, d. janošević, n. petrović the end of the digging process when the lifting (movement) of the support and movement member takes place as well, which also causes the appearance of increased dynamic forces and moments in all members of the excavator kinematic chain. the increased dynamic loadings of the bearing occur at the beginning and the end of the operation of soil transfer and returning to the new plane of digging due to the movement of the excavator platform when the masses of the manipulator kinematic chain members, carried by the excavator rotating platform, accelerate and decelerate. dynamic changes in the loading of the bearing also occur during the unloading operation due to the abrupt change in the dynamic parameters by reducing the mass of soil when the bucket is emptied. during the work cycle of the excavator, the greatest values of the components of force (fig. 3a) and moment (fig. 3b) of the slewing bearing loads, as well as the greatest values of the equivalent force and the equivalent moment of bearing loads, applicable for the selection of the bearing size, occur during the digging operation. 5. conclusion the conducted research, whose part is presented in this paper, represent a contribution to the analysis of defining the character of change in the bearing loads in the rotating platform drive mechanism of hydraulic excavators during the digging process with an excavating manipulator. the analysis shows that the greatest loads, applicable for the adequate bearing selection, according to the criteria of global bearing manufacturers, occur during the digging operation. the importance of knowledge of bearing load vectors is the basis of necessary mechanical, energy and structural simulations and analyses with the aim of optimizing the structure and drive mechanisms of the excavator. the developed software and the set of measured quantities obtained during the conducted testing of the hydraulic excavator can be used not only to define the bearing load vectors but also for other dynamic analyses of the excavator. references 1. janošević, d., mitrev r., andjelković b., petrov p., 2013, quantitative measures for assessment of the hydraulic excavator digging efficiency, journal of zhejiang university-science a (applied physics & engineering), 13(12), pp. 926-942. 2. tomović, r., miltenović, v., 2008, impact of rolling bearing structural parameters on the balanced rigid rotor’s oscillation frequency, facta universitatis series: mechanical engineering 6(1), pp. 5766. 3. jovanović, v., janošević, d., petrović, n., 2013, analysis of axial bearing load of a rotating platform drive in hydraulic excavators, tehnički vjesnik/technical gazette, 2(21), pp. 263-270. 4. jin, k., park, t., lee, h., 2012, control method to suppress the swing vibration of a hybrid excavator using sliding mode approach, journal of mechanical engineering science, 1, pp. 1237-1253. 5. jin, z., 2011, analysis on dynamic characteristics of closed slewing system for large-scale hydraulic excavator, fluid power and mechatronics (fpm), international conference, pp. 663-668. 6. pavlović, j., jovanović, m., milojević, a., 2014, optimal synthesis of a manipulator using two competitive methods, facta universitatis series: mechanical engineering 12(1), pp. 61-72. 7. janošević, d., petrović, n., nikolić, v., 2010, mechanism synthesis of manipulator of mobile machines, machine design, faculty of technical sciences novi sad, pp. 55-58. 8. janošević, d., 1997, optimalna sinteza pogonskih mehanizama hidrauličkih bagera, doktorska disertacija, mašinski fakultet univerziteta u nišu. 9. janošević, d., 2006, projektovanje mobilnih mašina, univerzitet u nišu, mašinski fakultet. 10. slewing bearings, 2007, rothe erde gmbh, dortmund. experimental determination of bearing loads in rotating platform drive mechanisms of hyd. excavators 169 eksperimentalno određivanje opterećenja ležaja pogona obrtne platforme hidrauličkih bagera u radu je dat postupak za eksperimentalno određivanje opterećenja aksijalnog ležaja pogonskog mehanizma obrtne platforme hidrauličkih bagera sa dubinskim manipulatorom. definisan je matematički model koji omogućuje da se, na osnovu merenih veličina stanja bagera pri radu u ekploatacionim uslovima, posredo, odrede vektori sile i momenta opterećenja ležaja. pri čemu se merene veličine stanja bagera odnose na položaj kinematičkog lanca i pritiske hidrostatičkog sistema u vodovima aktuatora pogonskih mehanizama bagera. kao primer, dati su rezultati istraživanja dobijeni pri ersperimentalnom određivanju opterećenja aksijalnog ležaja pogona okretanja obrtne platforme hidrauličkog bagera mase 16000 kg. ključne reči: hidraulički bageri, opterećenja ležaja obrtne platforme experimental determination of bearing loads in rotating platform drive mechanisms of hydraulic excavators( vesna jovanović, dragoslav janošević, nikola petrović 1. introduction facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 253 269 https://doi.org/10.22190/fume210111044b © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd review paper excel vba-based user defined functions for highly precise colebrook’s pipe flow friction approximations: a comparative overview dejan brkić1, zoran stajić2 1it4innovations, vsb – technical university of ostrava, ostrava, czech republic 2research and development centre “irc alfatec”, niš, serbia abstract. this review paper gives excel functions for highly precise colebrook’s pipe flow friction approximations developed by users. all shown codes are implemented as user defined functions – udfs written in visual basic for applications – vba, a common programming language for ms excel spreadsheet solver. accuracy of the friction factor computed using nine to date the most accurate explicit approximations is compared with the sufficiently accurate solution obtained through an iterative scheme which gives satisfying results after sufficient number of iterations. the codes are given for the presented approximations, for the used iterative scheme and for the colebrook equation expressed through the lambert w-function (including its cognate wright ωfunction). the developed code for the principal branch of the lambert w-function has additional and more general application for solving different problems from variety branches of engineering and physics. the approach from this review paper automates computational processes and speeds up manual tasks. key words: hydraulic resistance, colebrook flow friction, lambert w-function, excel macro programming, visual basic for applications (vba), user defined functions (udfs) 1. introduction the colebrook equation from 1939 [1], eq. (1), is an informal standard widely accepted in engineering practice for calculation of turbulent darcy’s fluid flow friction factor. it is an empirical equation based on an experiment with air flow through a set of smooth to fully rough pipes performed by colebrook and white in 1937 [2]. the moody diagram [3] in its turbulent part represents a graphical interpretation of the colebrook equation. received january 11, 2021 / accepted april 08, 2021 corresponding author: dejan brkić it4innovations, vsb – technical university of ostrava, 17. listopadu 2172/15, ostrava, czechia e-mails: dejan.brkic@vsb.cz, dejanbrkic0611@gmail.com 254 d. brkić, z. stajić 1 √f = − 2 · log 10 ( 2.51 re · 1 √f + ε 3.71 ) ≈ − 0.8686 · ln ( 2.51 re · 1 √f + ε 3.71 ) (1) in eq. (1), dimensionless turbulent darcy flow friction factor is given as f, the dimensionless reynolds number as re, the dimensionless relative roughness of inner pipe surface as ε, while briggsian decimal logarithm (to base 10) is given as log10 and napierian natural logarithm (to base e, where e≈2.718) as ln. as shown in eq. (1), the colebrook equation is given in an implicitly entangled logarithmic form which cannot be solved in terms of elementary functions. it can be solved; 1) iteratively (such solution can be treated as accurate after sufficient number of iterations) [4,5], or 2) using one-step approximate formulas specially developed for such purpose (maximal error of such formulas can be estimated in advance). therefore, for solving the colebrook equation, various explicit approximations [6-11] can be used to avoid long computing times caused by iterative schemes during the numerical simulations of pipelines for transport of various fluids [12,13]. also, the colebrook equation can be analytically expressed through the lambert w-function [14-18], but anyway the lambert w-function itself is an implicit function which can only be solved either iteratively or approximately using specially developed one-step formulas [19,20]. fast and accurate execution of codes during calculation of pipe flow friction is essential for calculation of pressure drop and flow rate in oil and gas industry, water distribution, in chemical engineering, etc. to facilitate use of the colebrook equation in spreadsheet solver ms excel [21,22], codes written in visual basic for applications (vba) based on the available highly precise explicit approximations [23-30] are given as user defined functions (udfs) and compared in this review paper. such approach automates computational processes and speeds up manual tasks. 2. visual basic for applications excel user defined functions the codes for solving the colebrook equation used in this review paper are shown through macros for ms excel written in visual basic for applications (vba). in essence, a macro is a term that refers to a set of programming instructions that automates tasks by creating custom calculations that can be used repeatedly throughout workbooks and which can be called by the host application as user defined function (udf). the vba is closely related to visual basic programming language, but on the contrary, vba codes can only run within a host application, and not as a standalone program. the here presented codes are compiled to a proprietary intermediate language that can be executed by ms excel, which is the host application in this case. an udf should be placed in module following the appropriate syntax of the vba programming language as shown in fig. 1. to prepare the udfs for the explicit approximations of the colebrook equation, the following steps in ms excel need to be followed: 1) keyboard shortcut “alt + f11” should be pressed to open the visual basic editor (a screen similar as in fig. 1 should appear), 2) in the visual basic editor, a module for udfs, should be opened using “insert” button from the ribbon, and by choosing “module” from the drop menu, 3) in the opened module, the udf should be written using appropriate syntax of the vba programming language, excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 255 4) using “debug” button from the ribbon, the current project should be compiled by choosing “compile vbaproject” from the drop menu, and 5) finally, the current udf should be saved with extension “xlam”, using “file” button and then “save as” from drop menu (it will be saved by default in: 'c:\users\[user name]\appdata\roaming\microsoft\addins\[name of the document].xlam'). fig. 1 visual basic editor the syntax of any udf for ms excel written in vba programming language has few main parts, such as: ▪ every function starts with “function” and finishes with “end function”, ▪ specific name of the function should be defined (designated by user and avoiding reserved names), ▪ after the designated name of the function, inputs should be specified in parentheses, ▪ data type of inputs and output should be defined using “as” (the data type of other used parameters with “dim” and “as”), ▪ in the body of the function, after the part with calculation but before “end function”, a return value should be assigned to the name of the function. ▪ like any other excel function, an udf can be called from any excel cell (if it is properly loaded). the syntax of the ms excel and of the vba programming language are different. for example, in-built function which returns value for the napierian natural logarithm, in the vba programming language is “log” while in ms excel is “ln” (“ln” is not reserved name in the vba). however, until recently, reusable udfs could be implemented only through scripts written using different syntax than for excel formulas, using vba or using javascript. now, excel users can use a new feature called “lambda” which introduce the ability to create custom functions using excel's formula language [31]. 256 d. brkić, z. stajić 3. solutions to the colebrook equation with their software codes using iterative schemes [4,5,22], the colebrook equation in its native implicit form can be solved with high accuracy after sufficient number of iterations. on the other hand, very accurate explicit approximations of the colebrook equation introduce certain small error which can be predicted in advance [6-11] and which is analyzed in further text. alternatively, the colebrook equation can be transformed analytically in an explicit form through the lambert w-function [14-18]. this approach provides the same accuracy as obtained through an iterative solution, but with a constraint that an overflow error can occur in certain computational approaches for the high values of the argument of the lamber w-function if the calculation is performed as usually in a computer with standard registers [32,33]. the lambert w-function is itself an implicitly given function that needs to be further evaluated iteratively or using specially developed approximate formulas [34] (such solutions of the lambert w-function have wide application in engineering and physics [19]). after thorough examination of the approximations of the colebrook equation from available literature [6-11], nine most accurate explicit approximations [23-30] were selected for analysis and for comparisons performed in this review paper. the examined approximations are ranked in table 1 in terms of 1) accuracy, and 2) time taken for execution: 1) the relative error is calculated as │(f-fi)/fi│·100%, where fi is the friction factor from an iterative scheme, while f is calculated using the observed approximation. 2) approximations require computational resources in terms of the number of floating points for execution and therefore simpler approximations are faster in computer simulations [37-41] (speed of nine selected approximations are evaluated using methodology from [42,43]). computational effort for the mathematical operations was determined by performing 100 million calculations for each mathematical operation using random input each repeated five times, with the average computational time recorded. the results are [44]: addition-23.40sec, subtraction27.50sec, multiplication-36.20sec, division-31.70sec, squared-51.10sec, square root-53.70sec, fractional exponential-77.60sec, napierian natural logarithm-63.00sec, and briggsian decimal logarithm to base 10-78.80sec. accuracy is checked using 2 million quasi-random and as well 90 thousand and 740 uniformly distributed samples, as in [9,23,35,36], which covers the whole domain of the reynolds number, re and of the relative roughness of inner pipe surface, ε, which are commonly used in engineering practice; 2320<re<108 and 0<ε<0.05. explicit approximations of the colebrook equation should be not only accurate but also simple for computation (fig. 2 shows error distribution for the four most accurate approximations). the approximations of the colebrook equation from table 1 are shown in further text, while the related algorithms and codes are given for the four most accurate approximations. the distribution of the maximal relative error is irregular for all available approximations of the colebrook equation. excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 257 fig. 2 distribution of the maximal relative error of the extremely accurate explicit approximations (the error less than 0.01%) 258 d. brkić, z. stajić table 1 results of accuracy and efficiency of the examined approximations with the relative error lower than 0.1%. approximation maximum relative error (%) execution time (sec) ratio of maximum relative error ratio of execution time praks and brkić -sr1 [23] – eq. (4) 0.001204 450.7 1 1.02 serghides [29] – eq. (5) 0.002560 906.4 2.13 2.06 vatankhah [25] – eq. (6) 0.005952 760 4.94 1.73 romeo et al. [28] – eq. (7) 0.007468 817.2 6.20 1.86 buzzelli [27] – eq. (8) 0.019944 667.8 16.56 1.52 praks and brkić -se2 [23] – eq. (9) 0.058517 573.9 48.60 1.30 offor and alabi [26] – eq. (10) 0.062704 477.1 52.08 1.08 shacham [30] – eq. (11) 0.083068 567 68.99 1.29 lamri [24] – eq. (12) 0.097438 440.2 80.93 1 1sr symbolic regression of the wright ω-function, 2se series expansion of the wright ωfunction; ratios are given in respect to the most accurate and fastest approximation. 3.1. iterative solutions iterative schemes are most suitable for the implicitly given types of equations, such as for the colebrook equation [4,5]. the lambert w-function which is used to express the colebrook equation in explicit form, is also implicitly given and therefore is also suitable for evaluation using iterative methods [34]. 3.1.1. simple fixed-point iterative scheme the colebrook equation for flow friction is suitable for calculation in its native form through an iterative method. a chosen simple starting point x0=5 for the fixed-point iterative scheme as from the algorithm in fig. 3 is in the range of applicability of the colebrook equation and can assure fast convergence, while the final solution can be reached after 2 to 11 iterations using xi = −0.8686 · ln ( 2.51·xi−1 re + ε 3.71 ), where x = 1 √f . yes no error yes no simple fixed-point iterative scheme fig. 3 algorithm for simple fixed-point iterative scheme for solving the colebrook equation excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 259 the vba code for solving the colebrook equation through simple fixed-point iterative scheme is given as follows: 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐶𝑂𝐿𝐸𝐵𝑅𝑂𝑂𝐾𝐼𝑇𝐸(𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒, 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒) 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐷𝑖𝑚 𝑥, 𝑥𝑐𝑜𝑛𝑡 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐼𝑓 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 < 2320 𝑂𝑟 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 > 100000000# 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 < 0 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 > 0.05 𝑇ℎ𝑒𝑛 𝐶𝑂𝐿𝐸𝐵𝑅𝑂𝑂𝐾𝐼𝑇𝐸 = 𝐶𝑉𝐸𝑟𝑟(𝑥𝑙𝐸𝑟𝑟𝑉𝑎𝑙𝑢𝑒) 𝐸𝑥𝑖𝑡 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑛𝑑 𝐼𝑓 𝑥𝑐𝑜𝑛𝑡 = 0 𝑥 = 5 𝐷𝑜 𝑈𝑛𝑡𝑖𝑙 𝐴𝑏𝑠(𝑥 − 𝑥𝑐𝑜𝑛𝑡) < 0.000000001 𝑥𝑐𝑜𝑛𝑡 = 𝑥 𝑥 = −2 / 𝐿𝑜𝑔(10) ∗ 𝐿𝑜𝑔(2.51 ∗ 𝑥 / 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 + 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 3.71) 𝐿𝑜𝑜𝑝 𝑥 = 𝑥 ^ − 2 𝐶𝑂𝐿𝐸𝐵𝑅𝑂𝑂𝐾𝐼𝑇𝐸 = 𝑥 𝐸𝑛𝑑 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 3.1.2. lambert w-function to date, the only way to transform analytically the colebrook equation from its native implicit form into an explicit form is through the lambert w-function [18]. a version from [23,36] is given in eq. (2), with the related algorithm in fig. 4 and the code as follows (to use this code, additional udf “lambert” should be defined as explained in further text). 1 √f = z 2.51 · (b+y) z = 2·2.51 ln(10) a = re z · ε 3.71 b = ln(re) − ln(z) x = a + b y = w(ex) − x } (2) yes no error lambert w-based solution of the colebrook equation fig. 4 algorithm for solving the colebrook equation expressed through the lambert w-function 260 d. brkić, z. stajić 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐶𝑂𝐿𝐸𝐵𝑅𝑂𝑂𝐾𝐿𝐴𝑀𝐵𝐸𝑅𝑇(𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒, 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒) 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐷𝑖𝑚 𝑧, 𝐴, 𝐵, 𝑥, 𝑦, 𝑓 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐼𝑓 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 < 2320 𝑂𝑟 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 > 100000000# 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 < 0 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 > 0.05 𝑇ℎ𝑒𝑛 𝐶𝑂𝐿𝐸𝐵𝑅𝑂𝑂𝐾𝐿𝐴𝑀𝐵𝐸𝑅𝑇 = 𝐶𝑉𝐸𝑟𝑟(𝑥𝑙𝐸𝑟𝑟𝑉𝑎𝑙𝑢𝑒) 𝐸𝑥𝑖𝑡 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑛𝑑 𝐼𝑓 𝑧 = 2 ∗ 2.51 / 𝐿𝑜𝑔(10) 𝐴 = 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 ∗ 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / (3.71 ∗ 𝑧) 𝐵 = 𝐿𝑜𝑔(𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆) − 𝐿𝑜𝑔(𝑧) 𝑥 = 𝐴 + 𝐵 𝑦 = 𝐿𝐴𝑀𝐵𝐸𝑅𝑇(𝐸𝑥𝑝(𝑥)) − 𝑥 𝑓 = ((𝑧 / 2.51) ∗ (𝐵 + 𝑦)) ^ − 2 𝐶𝑂𝐿𝐸𝐵𝑅𝑂𝑂𝐾𝐿𝐴𝑀𝐵𝐸𝑅𝑇 = 𝑓 𝐸𝑛𝑑 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 in some of the cases such as in [18,32], the argument of the lambert w-function is fastgrowing and for the values of x>e709.7827, e≈2.718, an overflow error can occur [33] while the colebrook equation expressed in that way cannot be solved always in a computer due to its limited capacity of registers (see fig. 5). however, a version from [23,36] as given in eq. (2) uses the lambert w-function with a shifted argument which allows computation avoiding the explained overflow error. fig. 5 constraints for using the lambert w-function for solving the colebrook equation (based on [18,32]) the halley iterative scheme, eq. (3), is used here for evaluation of the principal branch of the lambert w-function. the lambert w-function function in suitable form is given as li-1, with its first and second derivative given as l’i-1 and l”i-1. this solution is valid for real values for x>-1/e, where e≈2.718. to start the halley iterative scheme for solving the principal branch of the lambert wfunction, a simple and sufficient starting point x0=1 can be chosen, which makes the algorithm from fig. 6 fast for execution. the principal branch of the lambert w-function is used often in engineering and physics [19] meaning that the algorithm from fig. 6 can have much wider application aside for the colebrook equation. excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 261 wi(x) = wi−1(x) − li−1 l′i−1− li−1·l"i−1 2·l′i−1 li−1 = wi−1(x) · e wi−1(x) − x = 0 l′i−1 = ewi−1(x)(wi−1(x) + 1) l"i−1 = e wi−1(x)(wi−1(x) + 2) } (3) yes no error yes no principal branch of the lambert w-function fig. 6 algorithm for the principal branch of the lambert w-function the code for solving the principal branch of the lambert w-function is given as follows: function lambert(x as double) as double ′ computes the principal branch for x > −exp(−1) and for real values only dim wx,wxcont, l, lprim, lsec as double dim iter as integer if x < −exp(−1) then lambert = cverr(xlerrvalue) exit function end if wx = 1 do until abs(wxcont − wx) < 0.000000001 wxcont = wx l = wx ∗ exp(wx) – x lprim = exp(wx) ∗ (wx + 1) lsec = exp(wx) ∗ (wx + 2) wx = wx − l / (lprim − (l ∗ lsec / (2 ∗ lprim))) loop lambert = wx end function 3.2. explicit approximations of the colebrook equation explicit approximations of the colebrook equation which introduce a maximal relative error less than 0.1% are given in table 1. four of them, praks and brkić based on symbolic regression and on the wright ω-function [23], serghides [29], vatankhah [25] and romeo et al. [28], introduce a relative error of less than 0.01% and can be classified as extremely accurate, while those five, buzzelli [27], praks and brkić based on series expansion of the 262 d. brkić, z. stajić wright ω-function [23], offor and alabi [26], shacham [30] and lamri [24], with the maximal relative error between 0.01% and 0.1% can be classified as very accurate approximations. algorithms and vba codes are given here only for extremely accurate approximations while coding of the further approximations is not shown [44]. 3.2.1. praks and brkić approximation based on the wright ω-function and symbolic regression praks and brkić approximation [23], given in eq. (4) with the algorithm in fig. 7, is based on the wright ω-function and on symbolic regression. the wright ω-function is a cognate of the lambert w-function where w(ex)-x=ω(x)-x, which is used to eliminate fastgrowing term ex from calculation. the shown approximation y of ω(x)-x is very accurate within the domain valid for the colebrook equation, i.e., between 7.51<x<619 (symbolic regression is used to approximate ω(x)-x in this domain). 1 √f ≈ 0.8685972 · (b + y) a ≈ re·ε 8.0897 b ≈ ln(re) −0.779626 x ≈ a + b c ≈ ln(x) y ≈ c x−0.5588·c+1.2079 − c } (4) yes no error praks and brkić (symbolic regression) eq. (4) fig. 7 algorithm for the praks-brkić (symbolic regression) approximation excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 263 the vba code based on algorithm from fig. 7 is given as: function praksbrkic(reynolds as double, epsilon as double) as double dim a, b, x, c, y, f as double if reynolds < 2320 or reynolds > 100000000# or epsilon < 0 or epsilon > 0.05 then praksbrkic = cverr(xlerrvalue) exit function end if a = reynolds ∗ epsilon / 8.0897 b = log(reynolds) − 0.779626 x = a + b c = log(x) y = c / (x − 0.5588 ∗ c + 1.2079) − c f = (0.8685972 ∗ (b + y)) ^ − 2 praksbrkic = f end function 3.2.2. serghides approximation the serghides approximation [29] is based on steffensen iterative scheme [4], and the shown version, eq. (5) is improved by genetic algorithms [35,45,46]. it is given in eq. (5), with related algorithm in fig. 8. 1 √f ≈ a − (b−a)2 c−2·b+a a ≈ −0.8686 · ln ( ε 3.71 + 12.585 re ) b ≈ −0.8686 · ln ( ε 3.71 + 2.51·a re ) c ≈ −0.8686 · ln ( ε 3.71 + 2.51·b re )} (5) yes no error serghides eq. (5) fig. 8 algorithm for the serghides approximation 264 d. brkić, z. stajić the vba code based on algorithm from fig. 8 is given as: 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑆𝐸𝑅𝐺𝐻𝐼𝐷𝐸𝑆(𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒, 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒) 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐷𝑖𝑚 𝐴, 𝐵, 𝐶, 𝑓, 𝑙𝑛 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐼𝑓 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 < 2320 𝑂𝑟 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 > 100000000# 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 < 0 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 > 0.05 𝑇ℎ𝑒𝑛 𝑆𝐸𝑅𝐺𝐻𝐼𝐷𝐸𝑆 = 𝐶𝑉𝐸𝑟𝑟(𝑥𝑙𝐸𝑟𝑟𝑉𝑎𝑙𝑢𝑒) 𝐸𝑥𝑖𝑡 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑛𝑑 𝐼𝑓 𝑙𝑛 = 2 / 𝐿𝑜𝑔(10) 𝐴 = −𝑙𝑛 ∗ 𝐿𝑜𝑔(𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 3.71 + 12.585 / 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆) 𝐵 = −𝑙𝑛 ∗ 𝐿𝑜𝑔(𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 3.71 + 2.51 ∗ 𝐴 / 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆) 𝐶 = −𝑙𝑛 ∗ 𝐿𝑜𝑔(𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 3.71 + 2.51 ∗ 𝐵 / 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆) 𝑓 = 𝐴 − (𝐵 − 𝐴) ^ 2 / (𝐶 − 2 ∗ 𝐵 + 𝐴) 𝑓 = 𝑓 ^ − 2 𝑆𝐸𝑅𝐺𝐻𝐼𝐷𝐸𝑆 = 𝑓 𝐸𝑛𝑑 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 3.2.3. vatankhah approximation the vatankhah approximation [25] is given in eq. (6) with the related algorithm in fig. 9 (few versions of this approximation are available in [25], where for one of those approximations, brkić and praks [36] estimate its maximal relative error of no more than 0.0028%). this approximation is related to [47,48]. 1 √f ≈ 0.8686 · ln ( 0.3984·re (0.8686·a) a a+b ) a ≈ 0.12363 · re · ε + ln(0.3984 · re) b ≈ 1 + 1 1+a 0.52·ln(0.8686·a) − a 1+a } (6) yes no error vatankhah eq. (6) fig. 9 algorithm for the vatankhah approximation excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 265 the vba code based on algorithm from fig. 9 is given as: 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑉𝐴𝑇𝐴𝑁𝐾𝐻𝐴𝐻(𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒, 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒) 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐷𝑖𝑚 𝐴, 𝐵, 𝑓, 𝑙𝑛 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐼𝑓 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 < 2320 𝑂𝑟 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 > 100000000# 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 < 0 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 > 0.05 𝑇ℎ𝑒𝑛 𝑉𝐴𝑇𝐴𝑁𝐾𝐻𝐴𝐻 = 𝐶𝑉𝐸𝑟𝑟(𝑥𝑙𝐸𝑟𝑟𝑉𝑎𝑙𝑢𝑒) 𝐸𝑥𝑖𝑡 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑛𝑑 𝐼𝑓 𝑙𝑛 = 2 / 𝐿𝑜𝑔(10) 𝐴 = 0.12363 ∗ 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 ∗ 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 + 𝐿𝑜𝑔(0.3984 ∗ 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆) 𝐵 = ((1 + 𝐴) / (0.52 ∗ 𝐿𝑜𝑔(𝑙𝑛 ∗ 𝐴))) − (𝐴 / (1 + 𝐴)) 𝐵 = 1 + (1 / 𝐵) 𝑓 = 𝑙𝑛 ∗ 𝐿𝑜𝑔(0.3984 ∗ 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 / ((𝑙𝑛 ∗ 𝐴) ^ (𝐴 / (𝐴 + 𝐵)))) 𝑓 = 𝑓 ^ − 2 𝑉𝐴𝑇𝐴𝑁𝐾𝐻𝐴𝐻 = 𝑓 𝐸𝑛𝑑 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 3.2.4. romeo et al. approximation the romeo et al. approximation [28] is given in eq. (7) with the related algorithm in fig. 10. eq. (7) is improved by genetic algorithms [35,45,46]. 1 √f ≈ −0.8686 · ln ( ε 3.7106 − 5·b re ) a ≈ 0.4343 · ln (( ε 7.646 ) 0.9685 + ( 4.9755 206.2795+re ) 0.8759 ) b ≈ 0.4343 · ln ( ε 3.8597 − 4.795·a re ) } (7) yes no error romeo et al. eq. (7) fig. 10 algorithm for the romeo et al. approximation the vba code based on algorithm from fig. 10 is given as: 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑅𝑂𝑀𝐸𝑂(𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒, 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒) 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐷𝑖𝑚 𝐴, 𝐵, 𝑓, 𝑙𝑛 𝐴𝑠 𝐷𝑜𝑢𝑏𝑙𝑒 𝐼𝑓 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 < 2320 𝑂𝑟 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆 > 100000000# 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 < 0 𝑂𝑟 𝐸𝑃𝑆𝐼𝐿𝑂𝑁 > 0.05 𝑇ℎ𝑒𝑛 𝑅𝑂𝑀𝐸𝑂 = 𝐶𝑉𝐸𝑟𝑟(𝑥𝑙𝐸𝑟𝑟𝑉𝑎𝑙𝑢𝑒) 𝐸𝑥𝑖𝑡 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐸𝑛𝑑 𝐼𝑓 𝑙𝑛 = 1 / 𝐿𝑜𝑔(10) 𝐴 = 𝑙𝑛 ∗ 𝐿𝑜𝑔((𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 7.646) ^ 0.9685 + (4.9755 / (206.2975 + 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆)) ^ 0.8759) 𝐵 = 𝑙𝑛 ∗ 𝐿𝑜𝑔((𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 3.8597) − (4.795 ∗ 𝐴 / 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆)) 𝑓 = −2 ∗ 𝑙𝑛 ∗ 𝐿𝑜𝑔((𝐸𝑃𝑆𝐼𝐿𝑂𝑁 / 3.7106) − 5 ∗ 𝐵 / 𝑅𝐸𝑌𝑁𝑂𝐿𝐷𝑆) 𝑓 = 𝑓 ^ − 2 𝑅𝑂𝑀𝐸𝑂 = 𝑓 𝐸𝑛𝑑 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 266 d. brkić, z. stajić 3.2.5. buzzelli approximation the buzzelli approximation [27] is given in eq. (8). 1 √f ≈ a − ( a+0.8686·ln( b re ) 1+ 2.1018 b ) a ≈ 0.7314·ln(re)−1.3163 1.0025+1.2435·√ε b ≈ ε 3.71 · re + 2.51 · a } (8) 3.2.6. praks and brkić approximation based on the wright ω-function and series expansion the praks and brkić approximation [23] are based on the wright ω-function and on its series expansion. it is given in eq. (9). 1 √f ≈ 0.8686 · (b + y) a ≈ re·ε 8.0897 b ≈ ln(re) −0.779626 x ≈ a + b c ≈ ln(x) y ≈ c · ( 1 x−1 + c−2 2·x2 ) − 0.0014} (9) 3.2.7. offor and alabi approximation the offor and alabi approximation [26] is given in eq. (10). 1 √f ≈ −0.8686 · ln ( ε 3.71 − 1.975·a re ) a ≈ ln (( ε 3.93 ) 1.092 + 7.627 re+395.9 ) } (10) 3.2.8. shacham approximation the shacham approximation [30] is given in eq. (11) and is known also as zigrang and sylvester approximation [49]. 1 √f ≈ −08691 · ln ( ε 3.7027 + 5.0605·b re ) a ≈ 0.4343 · ln ( ε 3.7027 + 12.543 re ) b ≈ 0.4343 · ln ( ε 3.7027 + 5.0605·a re ) } (11) 3.2.9. lamri approximation the lamri approximation [24] is given in eq. (12). excel vba-based user defined functions for highly precise colebrook’s pipe flow friction... 267 1 √f ≈ a + 0.8686 · ( 0.8686 b − 1) · ln(b) a ≈ 0.8686 · ln ( re 2.51 ) b ≈ a + re·ε 9.3125 } (12) 4. conclusions nine explicit approximations of the colebrook equation are examined in this review paper. they are divided in two groups: 1) extremely accurate approximations with the relative error of no more than 0.01% and 2) very accurate approximations with the relative error between 0.01% and 0.1%. the most complex approximation is executed using the here presented vbaexcel code only 2.06 times slower compared with the code for the simplest approximation of nine presented in this review paper. therefore, using balance between the smallest relative error and the speed of execution in computers as a criterion for choosing the appropriate approximation for use in large computing simulations, the praks and brkić approximation [23] based on the wright ω-function and on symbolic regression, given in this review paper in eq. (4), is the most suitable and can be recommended for use. almost equally suitable are approximations by serghides [29], vatankhah [25], and romeo et al. [28]. udfs written in the vba, a common programming language for ms excel spreadsheet solver prepared for the presented approximations to the colebrook equation are suitable for use of those engineers who use spreadsheet solvers in their everyday work. also, the shown udf for the principal branch of the lambert w-function [50] can find much wider use in engineering than those for solving of the colebrook equation. acknowledgement: the authors acknowledge a support from the technology agency of the czech republic through the project ceet –“center of energy and environmental technologies” tk03020027 and the ministry of education, science, and technological development of the republic of serbia. references 1. colebrook, c.f., 1939, turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws, journal of the institution of civil engineers, 11(4), pp. 133-156. 2. colebrook, c.f., white, c.m., 1937, experiments with fluid friction in 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colebrook-white equation, journal of hydraulic engineering, 142(2), 04015050. 49. zigrang, d.j., sylvester, n.d., 1982, explicit approximations to the solution of colebrook's friction factor equation, aiche journal, 28(3), pp. 514-515. 50. kesisoglou, i., singh, g., nikolaou, m., 2021, the lambert function should be in the engineering mathematical toolbox, computers & chemical engineering, 148, 107259. elastic damper based on the carbon nanotube bundle facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 1 12 https://doi.org/10.22190/fume200128011r © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper * elastic damper based on the carbon nanotube bundle leysan kh. rysaeva 1 , elena a. korznikova 1,2 , ramil t. murzaev 1 , dina u. abdullina 3 , aleksey a. kudreyko 4 , julia a. baimova 1 , dmitry s. lisovenko 5 , sergey v. dmitriev 1,2 1institute for metals superplasticity problems of ras, russia 2institute for molecule and crystal physics, ufrc, ras, russia 3bashkir state university, russia 4department of medical physics and informatics, bashkir state medical university, russia 5ishlinsky institute for problems in mechanics of the russian academy of sciences, russia abstract. mechanical response of the carbon nanotube bundle to uniaxial and biaxial lateral compression followed by unloading is modeled under plane strain conditions. the chain model with a reduced number of degrees of freedom is employed with high efficiency. during loading, two critical values of strain are detected. firstly, period doubling is observed as a result of the second order phase transition, and at higher compressive strain, the first order phase transition takes place when carbon nanotubes start to collapse. the loading-unloading stress-strain curves exhibit a hysteresis loop and, upon unloading, the structure returns to its initial form with no residual strain. this behavior of the nanotube bundle can be employed for the design of an elastic damper. key words: carbon nanotube bundle, plane strain conditions, lateral compression, elastic damper 1. introduction carbon nanotubes (cnts) attract each other due to relatively weak van der waals interactions and can create molecular crystals or cnt bundles [1-3]. cnt forests can be produced by various techniques [4-7]. they have excellent mechanical properties since individual cnts possess very high tensile strength in the range from 11 to 63 gpa, tensile young's modulus of 1.0 to 1.3 tpa, and high ultimate fracture strain of about 10% [8-11]. moreover, cnts are flexible, lightweight, good heat and electricity conductors, thus being received january 28, 2020 / accepted march 02, 2020 corresponding author: sergey v. dmitriev institute for metals superplasticity problems of ras, khalturin st. 39, ufa 450001; institute for molecule and crystal physics, ufrc, ras, oktyabrya av., 151, ufa 450075, russia e-mail: dmitriev.sergey.v@gmail.com 2 l.kh. rysaeva, e.a. korznikova, r.t. murzaev, d.u. abdullina et al. useful in a number of applications [12-15]. the most important mechanical applications of cnts are high-strength ropes [2,16], fibers [17-21], polymerand metal-matrix composites [22-24], solid lubricants [24,25], etc. it should be pointed out that tension [2,16-21] and compression [26-33] of vertically aligned cnt brushes and forests have been extensively studied. note that cnts grow in vertical direction and compression of vertically aligned cnts means that they are compressed along their axis, while compression of horizontally aligned cnts means that they are compressed normal to their axis. on the other hand, lateral compression of isolated cnts or cnt bundles [34-38] is of interest, but this loading scheme has received much less attention of the researchers. horizontally aligned cnt bundles can be obtained from the vertically aligned cnt arrays by winding, drawing, micromechanical rolling, and shear pressing [39-42]. mechanical properties of cnts, evaluated in the experimental and computational studies, have been outlined in the review [43]. cnt bundles are linear elastic up to hydrostatic pressure of 1.5 gpa; the volume compressibility is 0.024 gpa-1; the hydrostatic deformation of cnt lattice is reversible up to the pressure of 4 gpa [44]. with the help of the x-ray diffraction and raman scattering, it has been found that deformation of cnt bundles under non-hydrostatic pressure is reversible below 5 gpa [45]. computational studies greatly contribute to a better understanding of mechanical properties of cnt bundles. mesoscopic modeling of transformation of a low-density vertically aligned cnt forest into a horizontally aligned cnt forest under pressure has been addressed in [46,47]. a continuum shell model has been used to describe different morphological patterns of cnts subjected to large deformations [48]. applicability of the continuum beam model and nonlocal beam, plate and shell theories to the modeling of the mechanical properties of nanomaterials has been discussed in [49,50]. with increasing cnt diameter, the rigidity of cnt crystal does not decrease [1]. if cnt diameter is above a threshold value, it can be found in two stable configurations, circular and collapsed ones [51-53]. according to the experimental and molecular dynamics investigation, triple-wall cnt bundles under pressure exhibit irreversible transformation in the range of pressure from 60 to 72 gpa, depending on the temperature [34]. mechanical response and failure of cnt bundles can be efficiently studied with the help of the nonlinear coarse-grained potentials developed for cnts [54]. the chain model has been developed in [55] in order to study structure and properties of secondary, van der waals conformations of carbon nanoribbons, such as folds and scrolls [55-59] as well as dynamics of surface ripplocations [60]. the chain model has been adopted to the study of cnt bundles under lateral compression in plane strain conditions in the work [61]. vibration and shock protection applications require development of the materials possessing properties of elasticity and damping. such properties are demonstrated, for example, by rubber or porous metallic rubber [62,63]. such analogs have been proposed for nanomaterials as well. the armchair graphene nanoribbons under tension, at certain value of tensile strain, split into domains with large and small elastic strain [64]. nucleation and motion of domain walls separating such domains produces considerable dissipation of elastic strain energy, and the loading-unloading stress-strain curves demonstrate hysteresis loop [64]. cyclic loading of suspended mos2 sheets during the nanoindentation experiments has also revealed the hysteresis behavior [65]. damping properties of the compressed vertically aligned cnt brushes have been analyzed in [26]. in the present work, damping properties of the laterally compressed cnt bundle are analyzed in frame of the model developed in [61]. elastic damper based on carbon nanotube bundle 3 2. simulation details we consider the nanotube bundle with the zigzag cnts of equal diameter aligned along the z-axis, see fig. 1. indices i=1,...,i and j=1,...,j number the cnts. in fig. 1 the case of i=j=2 is shown, while in this work we take i=10, j=12. atoms within each cnt are numbered by the index n=1,...,n anti-clockwise, starting with the most right atom, as shown by red symbols in fig. 1. in this work, lateral compression of the cnt bundle is considered under plane strain conditions, meaning that each carbon atom stands for a rigid atomic row oriented along the z-axis, which moves as a rigid body on the (x,y) plane. each atom has two degrees of freedom, namely, the components of the displacement vector on the (x,y) plane, thus the dimensionality of the problem is reduced from 3d to 2d. total number of atoms in our model is i×j×n=3600. periodic boundary conditions are imposed in both directions. fig. 1 geometry of the computational cell, which includes i×j cnts (i=10, j=12 in this work and i=j=2 in this figure) numbered by the indices i=1,...,i and j=1,...,j, arranged in cross-section in a triangular lattice. within each cnt atoms are numbered by the index n=1,...,n=30 anti-clockwise, starting from the most right atom. atoms have two degrees of freedom, the components of the displacement vector in the (x,y) plane. atomic chains normal to the (x,y) plane move as rigid bodies. the computational cell size is defined by the distance between centers of neighboring cnts, a, which is the sum of cnt diameter d and distance between cnt walls d. distance between atoms in the cnt wall is a. periodic boundary conditions are imposed the valence bond length in graphene is =1.418 å. the distance between neighboring atomic rows in the zigzag cnt is a=31/2/2=1.228 å. cnt diameter is d=a/sin(/n). the distance between cnt walls is d, then the centers of neighboring cnts are at the distance a=d+d. the computational cell has the form of parallelogram with the sides i×a and j×a. for the case of n=30 considered here, one has d=11.75 å and equilibrium value of d=3.088 å. this choice of cnt diameter guarantees that an isolated cnt has single equilibrium state. if cnt diameter is greater than ~3 nm, then it has another equilibrium state, the collapsed one [51-53], and such cnts may not open after unloading. the elastic damper discussed in this work operates due to cnt collapse during loading and opening during unloading. 4 l.kh. rysaeva, e.a. korznikova, r.t. murzaev, d.u. abdullina et al. the hamiltonian of the chain model employed in this study together with the model parameters can be found in [61]. it includes four terms, h=k+ub+ua+uvdw , (1) where the first term in the right-hand side gives the kinetic energy of the carbon atoms, the second to the fourth terms stand for the energy of valence bonds, the energy of valence angles and the energy of van der waals interactions between cnts, respectively. the model parameters were calculated based on the savin interatomic potential developed for sp2-carbon [66] and successfully used for solving various problems [66-71]. three strain-controlled loading schemes are considered: (i) uniaxial compression along close-packed cnt rows (i.e., along the x-axis with xx<0, yy=0), (ii) uniaxial compression normal to the close-packed cnt rows (i.e., along the y-axis with xx=0, yy<0), (iii) biaxial compression (with xx=yy<0). in all three cases compression up to certain level is followed by unloading to the zero strain. the strain state is characterized by the absolute value of volumetric strain ||=|xx+yy|. perturbation-relaxation molecular dynamics at zero temperature is carried out aiming to obtain the equilibrium structures at different values of strain. the strain is applied by increments ij = -0.0025 during compressive loading and ij = 0.0025 during unloading. compression up to ||=0.3 is analyzed for all three loading schemes, which is sufficient for the purpose of our study. each strain increment is followed by perturbation of atomic positions with small random displacements along xand y-coordinates, uniformly distributed in the range from -10-6 to 10-6 å. then the gradient method is applied to find the equilibrium structure by minimizing potential energy of the system. it is assumed that the energy has reached the minimal value when the absolute value of the maximal force acting on atoms becomes smaller than 10-10 ev/å. 3. simulation results stress components xx and yy as the functions of the absolute value of the volumetric strain, ||=|xx+yy|, are shown in figs. 2 to 4 for the uniaxial compression along the xaxis, along the y-axis, and biaxial compression, respectively. black (blue) lines are used for xx (yy) stress, and solid lines are for loading, while dashed lines for unloading. shear stress is not presented in the plots because it is either zero or, in some regimes, which will be discussed below, it is nonzero but still three orders of magnitude smaller than the normal stress components. compression up to ||=0.3 was simulated for all three loading schemes, but in the plots the most interesting range of strain, ||≤0.25, was shown. in figs. 5 to 7, we present the evolution of the cnt bundle structure for the uniaxial compression along the x-axis, along the y-axis, and biaxial compression, respectively. the top rows stand for compression and the bottom rows for unloading. the absolute values of volumetric strain are given for each panel, and they correspond to the dots indicated in figs. 2 to 4. note that the computational cell has the shape of a parallelogram but here, without loss of information, we present the structures in a rectangular widow using the fact that the cell is under periodic boundary conditions. elastic damper based on carbon nanotube bundle 5 fig. 2 stress components as the functions of the absolute value of volumetric strain, ||=|xx+yy|, for the case of uniaxial compression along the x-axis. blue and black colors are used for xx and yy, respectively. loading and unloading is shown by solid and dashed lines, respectively. dots indicate the values of elastic strain for which the structures are shown in fig. 5 fig. 3 same as in fig. 2, but for uniaxial compression along the y-axis. dots indicate the values of elastic strain for which the structures are shown in fig. 6 6 l.kh. rysaeva, e.a. korznikova, r.t. murzaev, d.u. abdullina et al. fig. 4 same as in fig. 2, but for biaxial compression. dots indicate the values of elastic strain for which the structure is shown in fig. 7 fig. 5 structure of the cnt bundle at different absolute values of volumetric strain, ||=|xx+yy|, for the case of uniaxial compression along the x-axis. the top row is for compression and the bottom row for unloading. the values of strain are given for each panel, and they correspond to the dots indicated in fig. 2 elastic damper based on carbon nanotube bundle 7 fig. 6 same as in fig. 5, but for the case of uniaxial compression along the y-axis. the values of strain are given for each panel, and they correspond to the dots indicated in fig. 3 fig. 7 same as in fig. 5, but for the case of biaxial compression. the values of strain are given for each panel, and they correspond to the dots indicated in fig. 4 4. discussion and conclusions the stress-strain curves, presented in figs. 2 to 4, look similar for all three studied loading schemes, though some differences can be noticed. during compression from the initial state up to about | |=0.07, the components of stress xx and yy are practically equal. this is expected for the case of biaxial loading due to the hexagonal symmetry of the structure, see fig. 7(a), top row. on the other hand, this is quite unusual for uniaxial 8 l.kh. rysaeva, e.a. korznikova, r.t. murzaev, d.u. abdullina et al. loading of an elastic body. the equality xx=yy means that, in this regime of deformation, the poisson’s ratio of the cnt bundle under lateral compression is nearly equal to 1, which is at the border of the thermodynamic stability for the two-dimensional elastic material [72,73]. as it can be seen in the panels (a) of figs. 5 to 7, for | |<0.07 the structure of the cnt bundle is very regular with all cnts having the same slightly deformed cross section shape, and this is true for both loading (top row) and unloading (bottom row). primitive translational cell of the cnt crystals in this regime includes single cnt. when, on loading, the value of strain exceeds about 7% threshold, structural changes start to happen and the equality between components of normal stress is destroyed due to occurrence of the second-order phase transition. for compression along the x-axis, period doubling in both directions results in the primitive translational cell that includes four cnts, see fig. 5(b) and (c), top row. for compression along the y-axis and for biaxial compression, period doubling in one direction takes place and the primitive translational cell includes two cnts, see fig. 6(b) and (c), top row, as well as fig. 7(b) and (c), top row. both these structures have been described in [61]. in this regime all cnts have elliptic, non-collapsed cross section for all three loading schemes. as for the normal stress components in this regime, one has xx<yy for compression along the x-axis (see fig. 2) and the opposite is true for compression along the y-axis and for biaxial compression (see figs. 3 and 4). on loading, appearance of cnts with collapsed cross section is accompanied by a sudden drop of the normal stress components (first order phase transition). this drop takes place at | |=0.13, 0.19, and 0.16 in the case of compression along the x-axis, the y-axis, and biaxial compression, respectively, see figs. 2 to 4. further compression after the drop of stress results in increase of stress and increase of the portion of collapsed cnts. structures with collapsed cnts that appeared on loading can be seen in figs. 5(d) to 7(d), top row. during unloading from | |=0.30, portion of the collapsed cnts gradually decreases but they can be seen in bottom row even at | |=0.125 in fig. 5(c), at | |=0.17 in fig. 6(c), and at | |=0.15 in fig. 7(c), while in the corresponding top panels all cnts have non-collapsed cross section. in the bottom rows of figs. 5(b) to 7(b) all cnts have non-collapsed elliptic cross section but the structures remain irregular, while corresponding top panels show regular structures. during unloading, at about | |=0.06, for all three loading schemes, structure of the cnt bundles becomes regular, so that complete unloading results in complete restoration of the initial structure. structure difference observed in certain strain interval for the same values of strain on loading and unloading is the reason of the hysteresis loops that can be seen in figs. 2 to 4. the area of the loop gives the energy dissipated by the structure during one loadingunloading cycle. this energy is maximal for the case of compression along the y-axis and minimal for loading along the x-axis. we can conclude that the cnt bundle under lateral compression acts as a perfectly elastic damper. the efficiency of the energy absorption is highest when cnt bundle is compressed normal to the close-packed rows and it is intermediate for biaxial compression. practical design of elastic dampers based on cnt bundles requires further research [74]. in particular, the effect of cnt diameter is important to analyze. also in many cases bundles of multi-walled cnts are produced. it is tempting to investigate if they are more efficient absorbers of energy as compared to single-walled cnts. these problems will be the topics of the future studies. study of novel graphene-analogous 2d nanomaterials [7579] is also of high importance for the development of nano-sized dampers. elastic damper based on carbon nanotube bundle 9 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mechanical properties of the 1570c aluminum alloy were studied after equal channel angular pressing (ecap) to the strain of 3 at 325°c and subsequent warm and cold rolling with near 80% reductions at 325°c and 20°c, respectively. even containing a partially recrystallized bimodal structure with a volume fraction of ultrafine grains of 0.3 and their size not exceeding 2 m, the alloy after ecap demonstrated an excellent balance of room temperature static strength parameters (yield strength (ys)  300 mpa, tensile strength (uts)  400 mpa and elongation (el)  26%), and high strain rate superplasticity (with maximum elongation exceeding 2500% at 520°c and a strain rate of 1.4 × 10-2 s1). subsequent warm and cold rolling resulted in an increase in ys to 340 and 430 mpa and uts to 415 and 485 mpa amid el decreased to 24 and 11%, respectively. despite the difference in the deformation structures formed in both rolling states, similar superplastic behavior was observed with maximum elongations of up to 3000% at temperatures of 500-520°c and strain rates of about 10-2 s-1. it was concluded that the initial processing of the alloy to relatively low ecap strains before warm/cold rolling, leading to bimodal structure with a low fraction of ultrafine grains, is sufficient to ensure a favorable combination of both service and technological properties of the sheets obtained. key words: aluminum alloy, equal channel angular pressing, rolling, superplasticity 1. introduction it is known that the balance of technological and service properties of commercial alloys, involving superplastic and static strength characteristics, can be significantly enhanced by processing an ultrafine-grained (ufg) structure (grain size less than 1 µm) received june 01, 2020 / accepted july 16, 2020 corresponding author: elena avtokratova institute for metals superplasticity problems ras, ufa, 450001, russia e-mail: avtokratova@imsp.ru 256 e. avtokratova, o. sitdikov, o. latypova, m. markushev using methods of severe plastic deformation (spd) [1,2]. in pilot productions they are usually implemented by several techniques, such as equal channel angular pressing (ecap), multidirectional isothermal forging, etc (e.g., [1-3]). these techniques can also be used in the ufg sheet manufacture by combining with conventional processes, such as rolling. regarding the superplastic properties, it is known that there are some challenges by employing the number of ufg commercial aluminum sheets in superplastic forming, because of low stability of their microstructure under static and dynamic (deformation) annealing. the strong enhancement of stability of ufg structure is frequently achieved by simultaneous additions of few transition metals, such as mn, cr, zr, sc, etc., forming high densities of nanosized aluminides [4-6]. for instance, unique characteristics of highstrain rate superplasticity were reported for the al-mg-sc(zr) alloys [5,7] with the nearly uniform ufg structures processed by spd with high strains. meanwhile, it was shown [8, 9] that such structures became unstable upon annealing after subsequent cold rolling due to a high driving force for the grain boundary migration in the heavily workhardened structures. in this way, decrease in spd strain, allowing the ufg structure be partially developed by dynamic recrystallization to meet superplasticity in the early stages of subsequent hot deformation, would be a very promising technical approach. at the same time, the effect of markedly lowered straining on high-strain rate superplasticity in the al-mg-sc(zr) alloys, is not so obvious. meanwhile, the spd processed ufg alloys were found to demonstrate not only excellent superplasticity [5, 7], but also a unique balance of strength and ductility at ambient temperature after subsequent warm and cold rolling [10]. therewith, the significance of the spd conditions is ambiguous owing to a number of structural factors that concurrently affect the alloy mechanical behavior. for instance, the strengthening effect due to grain refinement may be less than that from workand/or dispersion hardening due to the rearrangement of the dislocation structures and/or particle coarsening. thus, the aim of the present study was to evaluate the effect of the warm ecap to relatively low strains and subsequent warm and cold rolling (wr and cr), imitating two main routes of ufg sheet processing, on the static strength and superplastic behavior of the al-mg-sc-zr alloy. 2. material and methods the semi-direct cast ingot of the commercial aluminum alloy 1570c (al-5mg0.18mn-0.2sc-0.08zr-0.002be (wt%)) was homogenized at 360°c for 6 hours, and machined into plates of 150×150×30 mm3. ecap was performed by the route bcz (90° rotation around the normal axis to the plate plane between passes) at 325°c to a strain of about 3 using a die with a rectangular cross section and a channel inner angle of 90°. wr and cr were performed along the last pressing direction at 325°c and 20°c with total reductions 85% and 80% (e=1.9 and 1.6, respectively). the room-temperature hardness was measured by the vickers method using the metrotest itb-1-m device. thermal stability of the structure processed was determined by 1-hr annealing in the temperature range of 350-520°c. tensile tests at room and elevated (up to 520°c) temperatures were performed with instron 1185 testing machine using dog-bone shape specimens with a gauge part 3x6x1 mm3 cut along the last pressing/rolling direction. effect of bimodal structure processed by ecap and subsequent rolling on static strength... 257 optical metallography (om), scanning and transmission electron microscopy (sem and tem) were used to analyze the alloy microstructure. om was performed with a nikon l-150 optical microscope on the samples after standard mechanical polishing and etching in a keller's reagent. samples for sem and tem were electropolished at 20 v in a 30% hno3 and 70% ch3oh solution at -28°c using a tenupol-5 unit. the sem, including electron backscatter diffraction (ebsd) analysis, was carried out using a tescan mira 3 lmh field emission electron microscope equipped with the oxford instruments hkl channel-5 system. tem was performed with a microscope jeol2000ex. on ebsd maps, the low(2°≤θ<15°) and high-angle (θ15°) boundaries (labs and habs) were marked with gray and black lines, respectively. the scanning area reached 200×200 m2 at scanning step varied from 0.05 to 0.2 m, depending on the structure analyzed. the structural angular parameters, including the average misorientation angle of the intergranular boundaries θave and the fraction of habs fhabs, were derived from the ebsd data after a standard noise-reduction procedure [11]. (sub)grain boundaries with θ<2° were not taken into account. the sizes of grains and subgrains were measured by the “equivalent diameter” technique (upon conversion of the measurements of the area of crystallites into "equiareal circle diameter"), realized in the ebsd software. more details on methods of analysis are reported elsewhere [7, 9]. 3. results and discussions 3.1. microstructural changes the starting alloy state was characterized by an equiaxed grain structure with an average grain size of about 25 μm (fig. 1a, b) and the predominantly high-angle grain boundary spectrum with fhabs of about 0.9 and θave of about 40o. inside the grains, uniformly-distributed nanosized al3(sc, zr) precipitates with a diameter of about 15 nm and a number density of 104 μm-3 were observed (fig. 1c). ecap led to a partial grain refinement and formation of a heterogeneous bimodal structure with the (ultra)fine grains developed in the vicinity of original boundaries, called as the mantle regions (fig. 2a,d,g,j). the size of new grains was varied from 1 to 2 µm, while their volume fraction was as low as 0.3. consequently, the majority of the material volume was still represented by almost equiaxed fragments of original grains of 15-20 µm in diameter, containing subgrains of about 1 µm in size, bounded by labs. (sub)grain boundary spectrum in this structure was characterized by two peaks in the lowand high-angle diapasons with the fraction of labs of about 0.6 and the θave of only 18o. thus, the structure processed was predominantly (ultra)fine polygonised and the size of (ultra)fine grains developed in the mantle regions was roughly the same as that of subgrains. it can be, therefore imagined that the new grains were formed via an increase in the misorientation of strain-induced labs, which progressively transformed into habs without a notable growth of new grains. such a feature of the structure developed could be conditioned by continuous dynamic recrystallization [12], which can first occur near original grain boundaries during hightemperature ecap of the investigated alloy [9,13]. 258 e. avtokratova, o. sitdikov, o. latypova, m. markushev fig. 1 om (a); sem (b) and tem (c) structure of the homogenized alloy fig. 2 om (a-c), sem-ebsd (d-i) and tem (j-l) structures and corresponding (sub)grain boundary spectrums of in the alloy after ecap (a,d,g,j); ecap and subsequent wr (b,e,h,k); and cr (c,f,i,l): pressing/rolling direction is horizontal effect of bimodal structure processed by ecap and subsequent rolling on static strength... 259 wr subsequent to ecap resulted mainly in pancaking and flattening of coarse remnant original grains in accordance to the macroscopic straining of a billet (fig. 2b). therewith, it might be expected that the alloy structure could exhibit at least more homogeneous distribution of fine grains owing to continuation of recrystallization [14]. however, their fraction in the mantle regions only slightly increased to 0.35-0.40, whereas the shape and sizes of both fine grains and subgrains were scarcely changed (fig. 2b, e, h, k). also the angular parameters of the structure remained close to those after ecap (fig. 2g, h). this may imply that the straining occurred mainly in the mantle regions by grain boundary sliding (gbs), which was attributed to the ultrafine grain size and a relatively high deformation temperature, resulting in dissipation and almost complete relaxation of the deformation energy introduced by rolling [12, 14]. meanwhile, the microstructural development in the coarse grains can be ascribed to the formation of a dynamically equilibrium substructure stabilized by coherent aluminides of sc and zr (fig. 1c) that effectively suppressed dynamic recrystallization [15, 16]. besides, an additional assumption was that ecap realized a simple shear deformation mode, being particularly important for grain refinement at high temperatures, while the pure shear implemented in a conventional process, such as wr, could not complete recrystallization under the considered deformation conditions [17]. cr, in turn, resulted in the formation of a heavily-deformed structure with a lower crystallite size (about 0.3 m) and an increased (up to 0.7) fraction of labs (fig. 2c,f,i,l). tem analysis (fig. 2l) revealed also high-density dislocation structures arranged in cells and deformation bands. note also that the deformation microstructures evolved were apparently non-uniform on the mesoscopic level, as the necklace-like structure processed by ecap (fig. 2a,d) led to more heterogeneous and dense dislocation structures near the mantle regions of fine grains, where the grain boundaries served actively as sources and/or barriers for lattice dislocations. in contrast, the coarse remnant grains containing labs transformed into less developed and more uniform deformation structures, stabilized by the al3(sc,zr) precipitates. 3.2. room-temperature mechanical properties the room-temperature hardness and tensile strength parameters of the alloy before and after ecap and subsequent rolling are represented in table 1. besides, the appropriate data for the alloy with almost fully recrystallized structure with the grain size of about 1 m processed by ecap at 325oc to e=10 and subsequent wr and cr [18], are also shown for comparison. evaluation of the mechanical properties showed that the processed alloy demonstrated a favorable balance of high strength at reasonably high ductility. namely, in comparison to the initial alloy state, hardness (hv), the yield and ultimate tensile strength (ys and uts) were noticeably increased after processing, especially after cr (in about 1.4, 1.8 and 1.4 times, respectively). such an alloy strengthening is quite typical of work-hardening materials [6,19] and mainly caused by grain refinement and development of the dislocation/ (sub)grain structures described above. meanwhile, no alloy ductility decrease due to ecap and further wr was found. to the contrary, the alloy ductility significantly decreased after cr, even although its level, however, still remained quite high exceeding 10%. 260 e. avtokratova, o. sitdikov, o. latypova, m. markushev table 1 room temperature mechanical properties of the alloy condition h v ys, mpa uts, mpa ys/ut s el, % initial 105 240 355 0.67 28 ecap (e=3) 110 295 390 0.76 26 ecap (e=3) + wr 120 340 415 0.81 24 ecap (e=3) + cr 145 430 485 0.89 13 ecap (e=10) [18] 310 390 0.80 31 ecap (e=10) + wr[18] 295 400 0.74 25 ecap (e=10) + cr[18] 520 550 0.94 13 it is well known that the main feature of the work-hardened material is a low elongation to failure (uniform elongation) because of a faster increase in ys than uts, resulted in restricted capacity of further strain-hardening [20]. so even warm straining resulted in the sense increase of the ratio ys/uts from about 0.7 to about 0.8, the main reason of which was the formation of quite equilibrium partially recrystallized structure with ultrafine grains and subgrains. a further decreased gap between ys and uts in the cr processed alloy (to ys/uts0.9) signaled the lower strain hardening capability of the structure and, hence, lower stability of the plastic flow in the material, as its lower uniform elongations, compared to the other conditions investigated. on the other hand, as also mentioned above, the structures, developed in coarse grain interiors under wr and cr, were apparently near homogeneous on the mesoscopic level. in the both cases this homogeneity was attributed to a uniform distribution of the al3(sc,zr) precipitates in the matrix (fig. 1c). the latter are known to effectively interact with the lattice dislocations and homogenize the dislocation slip during straining [15, 16, 21], thereby promoting relatively high uniform elongations. this may increase to some extent the alloy ductility even in the highly deformed condition. table 1 also shows that despite more intense grain refinement with increasing ecap strain to e=10 [18], neither processing the present alloy by ecap nor subsequent wr improved significantly both static strength and ductility at ambient temperature. this suggests that the grain boundary (hall-petch) strengthening [19], which should occur upon grain refinement via the transformation of the subgrain structure into an ultrafine recrystallized one, was almost completely compensated by the simultaneous softening caused by consumption of the dislocation/sub-grain structure. also concurrent alloy ductility increase up to 30% was conditioned by the formation of a more homogeneous fine grain structure after ecap. however, this advantage disappeared after the subsequent wr, when the alloy ductility fell to 25%. in contrast, cr after ecap to e=10 resulted in more significant alloy strengthening than cr after ecap to e=3 with extremely high ys and uts values for non-age hardenable alloys, amid nearly the same total elongation. such a balance resulted from superposition of several factors, including both grain boundaryand dislocation strengthening. namely, ecap provided remarkable grain refinement, while the subsequent cr introduced, in turn, highly deformed substructures with increased dislocation density into the ufg structure. effect of bimodal structure processed by ecap and subsequent rolling on static strength... 261 3.3. annealing behavior it was found that under post-deformation annealing, the structures developed by ecap and subsequent both warm and cold rolling (fig. 2) were replaced with statically recrystallized/recovered ones. regardless the annealing temperature, the structures in all the alloy states remained bimodal and consisted of larger grains, containing substructure, that were surrounded by areas with smaller grains (fig. 3). as seen in fig. 3a, b, d, e, g, h and table 2, under annealing, performed after warm straining, a normal grain growth took place in areas of fine grains, while coarse grains remained fairly stable. according to table 2, the grain size in the mantle regions almost did fig. 3 structures and (sub)grain boundary misorientations developed after 1-hr annealing at 425°c (a,b,c), 475°c (d,e,f,j,k,l) and 520°c (g,h,i) in the alloy after ecap (a,d,g,j), ecap and wr (b,e,h,k), ecap and cr (c,f,i,l). pressing/rolling axis is horizontal 262 e. avtokratova, o. sitdikov, o. latypova, m. markushev table 2 effect of annealing temperature on the grain size in the mantle regions condition as-processed annealing temperature 425°c 450°c 475°c 520°c ecap 1.4 2.0 2.1 2.8 4.2 ecap + wr 1.3 1.6 1.9 2.5 3.5 ecap + cr 2.2 2.4 2.5 3.9 not change during annealing to 425-450°c and increased at higher temperatures (for example, twice after annealing at 475°c c and about threefold at 520°c). however, even after annealing at 520°c, the absolute grain size in the mantle did not exceed 4.5 m. besides, the labs formed after ecap and subsequent wr were preserved in the interiors of coarse grains and were not ubiquitously converted to habs (fig. 3). therewith, angular structural parameters (fhabs and θave) somewhat decreased at high annealing temperatures, probably mostly due to a decrease in the total length of habs upon grain coarsening [12]. it is safe to assume that the microstructural stability observed in both recrystallized and non recrystallized regions is mainly conditioned by the uniform spatial distribution of dispersed al3(sc,zr) phases. the latter, as known, effectively pin grain boundaries, as well as individual dislocations and dislocation structures, thereby prohibiting their rearrangement and, consequently, preventing grain growth and occurrence of static recrystallization [4, 5]. in the cold-rolled alloy, in contrast, structural changes in the mantle regions, where the original fine-grain structure possessed a much higher dislocation density (fig. 2), were caused mainly by static recrystallization followed by normal grain growth (fig. 3c,f,i). as the most possible way, static recrystallization occurred there in a continuous manner due to the presence of a large number of pre-existing (ultra)fine grains produced by ecap, which were then severely strained during rolling. according to [22], some of them containing a lower dislocation density served as potential nuclei for static recrystallization and could further recover and grow during annealing to consume neighboring fine grains with a higher dislocation density. at the same time, more homogeneous and less dislocation-profuse areas containing labs, which were formed in grain cores, still underwent static recovery and were characterized by the development of a polygonized structure stabilized by nanosized precipitates. unlike the mantle regions, only static recovery was operated there as the main restoration mechanism during annealing. thus, the development of the new fine grains around the coarse remnant ones gave back a necklace-like grain structure in the spd processed and annealed material [9]. it is also interesting to note that both the (sub)grain boundary spectrum and the average parameters of the structure that developed during annealing of the alloy subjected to ecap and cr were close to those obtained at the corresponding annealing temperatures after ecap, as well as after wr (fig. 3 and table 2). thus, despite the different operating structural mechanisms, quite similar structures were formed during annealing in the alloy processed to different treatment routes. this may indicate that there would be no significant difference in the microstructures that developed in these alloy states upon its heating and soaking prior to straining at elevated temperatures, involving superplastic forming. effect of bimodal structure processed by ecap and subsequent rolling on static strength... 263 3.4. superplastic properties tensile tests, carried out in a wide temperature range from 350 to 520°c (fig. 4), showed that even with such a low volume fraction of ultrafine grains as 0.3, the alloy after ecap displayed both high strain rateand low temperature superplasticity [2] with strain rate sensitivity coefficients, m > 0.35 (fig. 4 a,b). maximum elongations, up to 2600% and 400%, were observed at a strain rate of 10-2 s-1 at 520°c and 350°c, respectively. after the subsequent wr, the alloy demonstrated even better superplastic behavior with elongations up to 3000% at 500°c and a strain rate of 5.6 × 10-2 s-1 (fig. 4c,d). fig. 4 superplastic characteristics of the alloy at various temperatures and strain rates in the states after ecap (a,b); ecap and wr (c,d); ecap and cr (e,f) after cr the superplastic characteristics were also extremely high: at a strain rate of 10-2 s-1, the elongation reached 3030% at 520°c and 350% at 350°c with a maximum m varying from 0.3 to 0.5 (fig. 4e,f). such unique properties can be discussed as follows. as can be imagined from the previous section, 1-hr heating/soaking of the material at the 264 e. avtokratova, o. sitdikov, o. latypova, m. markushev testing temperatures investigated, led to the development of quite similar bimodal structures with relatively stable sub-grain regions and about 30% of (ultra)fine grains underwent the normal grain growth (fig. 3). despite the above changes, the fine grains in such structures, as well as the subgrains, still remained fairly dispersed just before testing, even at the highest temperature of 520°c, which signaled their high thermal stability. structural analysis in the gage section of samples after maximum elongations (fig. 5) showed in turn that superplastic deformation led to the transformation of the bimodal structures into fully recrystallized ones with an average grain size of about 8.5 µm in the warm deformed states and 10.5 µm after cr. in addition, the evolved grain structures were characterized by similar (sub)grain boundary spectrums and mean values of angular parameters. thus, new grain structures were mainly developed in all alloy states via recrystallization, occurring in the early stages of tensile deformation [23]. the new grains maintained an almost equiaxed shape (fig. 5 a, d, g) even at high elongations (up to 3000%), which explicitly referred to a large contribution of gbs to the total deformation. this conclusion was also supported by the deformation relief formed on the surfaces of the tensile samples, where all the signs of gbs were clearly evident (fig. 5c, f, i). the data fig. 5 structures, (sub)grain boundary misorientations and deformation relief developed upon tension at 1.4 10−2 s−1 to maximum elongations in the specimens after ecap at 520°c (a-c); ecap and wr at 500°c (d-f); ecap and cr at 520°c (g-i): (a,b,d,e,g,h) sem-ebsd; (c,f,i) sem. tensile axis is horizontal effect of bimodal structure processed by ecap and subsequent rolling on static strength... 265 obtained also allow concluding that complex additions of sc and zr to the aluminum alloy were quite effective for stabilizing the grain structure, ensuring extensive operation of gbs following dynamic recrystallization. the strong pinning effect of al3(sc,zr) dispersoids limited the grain growth at elevated temperatures and prevented the degradation of superplastic characteristics. this allowed a sustained behavior of the alloy, contributing to extremely high elongations to failure. 4. conclusions the structure and mechanical properties of the commercial 1570c (al-5mg-0.18mn0.2sc-0.08zr-0.01fe-0.01si, wt.%) alloy were studied upon combination of ecap of a homogenized ingot with the strain of e=3 at 325°c and subsequent warm and cold rolling with the total reduction of 85% (e=1.9) at 325°c and 80% (e=1.6) at 20°c, respectively. the main results can be summarized as follows: 1. under warm ecap conditions, the alloy grain refinement occurred with the formation of a bimodal structure, which was a mantle of new (ultra)fine grains not exceeding 2 m in size with a volume fraction of about 0.3 around fragments of coarse original grains containing subgrains. subsequent wr did not significantly change the type and parameters of as-ecaped structure. during cr after ecap, high-intense dislocation structures developed in regions of fine grains, while more uniform cell-type structures were obtained within the remnant original grains. 2. analysis of the alloy room temperature tensile behavior showed that due to the formation of the above structures, the alloy in all processed states after ecap with e=3 exhibited a favorable balance of moderate-to-high strength and a reasonably high ductility. such a balance did not change much with further grain refinement upon increasing the ecap strain to e = 10 before wr. however, the mechanical properties obtained in the cold rolled alloy subsequent to ecap with e=3 conceded to those after e=10, when much higher static strength was achieved amid roughly the same ductility. 3. one-hour annealing of the alloy in the temperature range up to 520°c after ecap, as further wr, led to minor normal grain growth in the fine-grain mantle regions, along with maintaining a stable subgrain structure in the cores of the remnant original grains. on the contrary, when annealing a cold-rolled alloy, intense and uniform dislocation structures in the mantle and core regions, respectively, were transformed into arrays of fine statically recrystallized grains and statically recovered subgrains. therewith, despite the operation of different structural mechanisms, quite similar structures were formed in all three states of the alloy after annealing. 4. the formation of bimodal/partially recrystallized (ultra)fine-grained structures with a predominant fraction of labs by ecap and subsequent cold/warm rolling was sufficient to achieve high strain rate superplasticity in the 1570c aluminum alloy with elongations exceeding 2500%. therewith, the (ultra)fine-grain structure mainly developed via dynamic recrystallization occurring at the early stages of superplastic flow, and remained stable due to the stabilizing effect of nanoscale al3(sc,zr) dispersoids present in the alloy. 266 e. avtokratova, o. sitdikov, o. latypova, m. markushev acknowledgements: the work was supported by the ministry of science and higher education of russian federation under the state assignment of imsp ras no. аааа-а19-119021390107-8. some graphs and micrographs in figs. 2, 4 and 5 were reproduced from avtokratova, e., latypova, o., sitdikov, o., markushev, m., 2019, superplastic behavior of the al-mg-sc-zr alloy with bimodal structure processed by equal channel angular pressing and subsequent rolling, aip conference proceedings, 2167, 020022. references 1. valiev, r.z, islamgaliev, r.k, alexandrov, i.v., 2000, bulk nanostructured materials from severe plastic deformation, progress in materials science, 45, pp. 103-189. 2. mulyukov, r.r., imayev, r.m., nazarov, a.a., imayev, m.f., imayev, v.m., 2014, superplasticity of ultrafine grained alloys: experiment, theory, technologies, moscow: nauka, 284 p. 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alloys under high pressure torsion, journal of alloys and compounds, 743, pp. 773-779. 22. belyakov, a., sakai, t., miura, h., kaibyshev, r., tsuzaki, k., 2002, continuous recrystallization in austenitic stainless steel after large strain deformation, acta materialia, 50, pp. 1547–1557. 23. yang, x., miura h., sakai t., 2002, continuous dynamic recrystallization in a superplastic 7075 aluminum alloy, materials transactions, 43, pp. 2400-2407. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 141 151 potential strain energy surfaces at the crack tip vicinity  udc 531-3:621 dragan b. jovanović faculty of mechanical engineering, university of niš, serbia abstract results of the potential strain energy surfaces reconstruction in the region of crack as well as the basic procedures of the reconstruction method are presented. relevant limitations and specifications of the method are pointed out while evaluating the best fitting curves and the best fitting three-dimensional surfaces of stress components and potential strain energy. examples of cracks named in the paper as prz-16 and pr-p-22, with exactly the same geometry of crack and plate are analyzed. the same finite element lattice for calculation in both cases is selected. the first example of a plate is loaded with  in y direction on the outer border of the plate, perpendicularly on the plane of crack, while in the example pr-p-22 the plate is loaded with  continually and perpendicularly on the crack contour surface. the reconstruction procedure of the potential strain energy surfaces for two sections of plates y = 0 and z = 0, and the graphical presentation of these surfaces, are carried out. based on the reconstructed potential strain energy surfaces, isoenergy lines in the region in front of the crack tip for both the analyzed examples, are obtained. it is observed that the distribution of stresses as well as that of potential strain energy are very similar in both cases, which indicates considerable influence of the geometric shape of the crack's contour surfaces. for comparison, stress state and potential strain energy state, which are obtained by using relations from literature, are shown. key words: crack, stress components, potential strain energy surfaces reconstruction 1. introduction, framework experimental research which initially led to this idea was realized in 1992, 1995, 1996 and 1997 in the laboratory for experimental mechanics of the university of waterloo in canada, under the supervision of j. t. pindera[7], [8] and [9]. during 1997, 1998, 1999, received november 30, 2013 corresponding author: dragan b. jovanović faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: jdragan@masfak.ni.ac.rs acknowledgements: this work was supported by the ministry of education and science, republic of serbia, through the project on174011, and the faculty of mechanical engineering, university of niš. 142 d. b. jovanović due to the impossibility of conducting conduct an experiment under real physical conditions, i decided to carry out a numerical experiment using the finite elements method and algor software package that was at that time available at the faculty of mechanical engineering in niš. on the basis of the results of numerical experiments providing for an assessment of the stress and strain state in the vicinity of the crack in the plate, the aim was to confirm or refute some of the assumptions that i made during the experimental research in canada. the first results of this research on the basis of numerical experiments have been published in the papers [4], [5], and before that in [3]. during the review of the work [5], which was published in 2002, i learned about the work of parsons, hall, and rosakis [6] from the university of pasadena (california institute of technology), dealing with a three-dimensional stress state and the state of deformation around the crack tip, where the finite element method was applied. this paper did not analyze or reconstruct energy distribution of deformation around the crack tip, but only stress and strain state. the authors used a multigrid method in paper [6], but it was not the same as the method that i used. all of that indicated that there were thoughts that went in a similar direction. at the same time, i learned about the work of g. c. sih and lee, which dealt with a three-dimensional stress state and condition of volumetric energy density (volume energy density) in the vicinity of the crack. 2. reconstruction procedure of the strain energy surface the reconstruction procedure of the potential energy surface had previous steps, the first of which was the modeling of a crack in the plate. that means a choice between different forms of an initial crack (figs. 1, 2), definition of boundary conditions on the crack contour, the use of symmetry that exists in geometry, and load distribution, and the choice of finite elements that will be used in the analysis of the problem. the second step in the work was the calculation of the distribution of stress components in the plate, for the assumed global stress state and the assumed shape of the crack. software packages were used for this. data obtained in this way were used to graphically display the obtained stress distribution (figs. 3, 4, 5, 6, 7, 8) as the basis for determining the value of stress components and component displacements in selected directions and in selected sections. the third step in the work was editing of data in units that were needed for further work. the fourth step was a graphical representation of curves that describe the distribution of stress components x, y, z, xy , yz , zx on selected directions. the fifth step was the calculation of the potential strain energy, where equation (1) was used, and which was built into the program defen. 2 2 2 2 2 21 [( ) 2 ( ) 2(1 )( )] 2 def x y z x y x z y z x y x z y za e                         (1) the sixth step was to edit the data of the potential strain energy and graphical representation of its distribution for selected directions and intersections. the seventh step was the reconstruction of the potential strain energy surface by the process of networking (gridding) of points obtained in the previous procedure. the polynomial regression procedure was also used. the procedure was repeated iteratively to obtain the spatial surface, in which the influence of the selected parameters for networking was eliminated potential strain energy surfaces at the crack tip vicinity 143 or imperceptible. figs. 11 and 12 show a step in the process of reconstruction, where the density of points with predetermined values of the potential strain energy affects the result of the reconstruction. periodic changes in the strain energy, which you can see in these figs., starting at a distance the size of a double thickness of middle plate, or 1.5 thickness of the crack tip, are consequences of the insufficient density of points, which determined the value of the potential deformation energy, interacting with the reconstruction method. consequently, an increase in the number of data points resulted in the ironing of surface in this part. the appearance of the surface waves or curves in the reconstruction as a result of uneven density of data points, was observed previously [7], [8], [9]. the experience from experimental research has been used to successfully overcome and resolve the problem of uneven density of the measurement points.  ~elo prsline ~elo prsline  fig. 1 elliptical crack in the plate, loaded by normal stress  to tension in the direction perpendicular to the crack plane, which is processed in the file pr-z-16 fig. 2 elliptical crack in the plate loaded by normal stress  to pressure in the direction perpendicular to the crack plane, processed in the file pr-p-22 fig. 3 pr16 normal stress distribution x , for a flat cross-section z = 0, detail fig. 4 pr16 normal stress distribution y, for a flat cross-section z = 0, detail. crack front crack front 144 d. b. jovanović fig. 5 pr16 normal stress distribution z, for a flat cross-section z = 0, detail fig. 6 pr16 shear stress distribution xy, for a flat cross-section z = 0, detail fig. 7 pr16 normal stress distribution x , for a flat cross-section y = 0, detail fig. 8 pr16 normal stress distribution y , for a flat cross-section y = 0, detail figs. 9 and 10 show the distribution of the potential strain energy for selected directions z = 0 mm, z = 1 mm of the intersection y = 0 mm. then, data files are merged into a single file with data on potential strain energy of the flat section defined by y = 0 mm. then, i made a series of iterative steps, which reconstructed the surface describing the potential energy distribution in the plane y = 0 mm, as shown in figs. 11 and 12. fig. 9 diagram of the specific strain energy in the direction determined by coordinates z = 0 mm and y = 0 mm fig. 10 diagram of the specific strain energy in the direction determined by coordinates z = 1 mm and y = 0 mm potential strain energy surfaces at the crack tip vicinity 145 fig. 11 diagram of the surface of the specific strain energy of the flat section for coordinate y = 0 mm, in front of the crack's front line (the previous step of reconstruction) fig. 12 diagram of the surface of the specific strain energy of the flat section for coordinate y = 0 mm, in front of the crack's front line (the final form of reconstruction) fig. 13 shows a diagram of the potential strain energy in direction x = 4.5 mm, and plane cross-section y = 0 mm. this diagram shows that minimum of potential strain energy ahead of the crack in the middle of the plate cross-section, and maximums near the outer plate surfaces. this fact suggests that the points on the crack's front line, which are located near the outer plate surface will have crack propagation before the middle of the cross section of the plate. fig. 14 shows the iso energy lines in the cross section at z coordinates. these lines show the constant value of the potential energy of deformation, illustrating the distribution and concentration of potential strain energy, and hence point to the places where crack propagation can be expected. fig. 15 shows a step in the iterative procedure. fig. 13 diagram of the specific strain energy in direction determined by the coordinates x = 4.5 mm and y = 0 mm fig. 14 diagram of the isoenergy lines in front of crack in plane section of plate, coordinate y = 0 mm 146 d. b. jovanović fig. 15 diagram of the specific strain energy for flat section of plate, coordinate z = 0 mm (the previous step reconstruction) fig. 16 diagram of the specific strain energy for flat section of plate, coordinate z = 0 mm (the final step of reconstruction) fig. 16, which represents the distribution of the potential strain energy in the middle plane, shows peak energy a'def = 1451710 3 j/mm 3 in front of the front line of crack, which is, like in the diagram in fig. 12. also noted is the existence of local extreme strain energy at a distance 12 mm from the origin. this local peak energy is a consequence of the change in the law of normal stress, which at that point has a local extreme. at this point the value of the potential strain energy is much smaller than the value at the crack tip. figs. 17 and 18 show the isoenergy lines for middle plane of plates, where you can see the concentration of strain energy ahead of the crack tip and depression of energy field above the crack in the case pr16. fig. 17 pr16 isoenergy lines in the vicinity of the crack tip, plane of section (xy) for z = 0 mm (shown in enlarged detail) fig. 18 pr22 isoenergy lines in the vicinity of the crack tip, plane of section (xy) for z = 0 mm (shown in enlarged detail) potential strain energy surfaces at the crack tip vicinity 147 3. stresses and strain energy in front of the crack tip according to the literature compare the results with graphical display of stress and deformation potential energy obtained from the relation to the refs. [1] and [2]. d. broek on page 95 ref. [1] elementary engineering fracture mechanics gives relations for stresses, in the polar system of coordinates:                    2 3 sin 2 sin1 2 cos 2    r k i x (2)                    2 3 sin 2 sin1 2 cos 2    r k i y (3) 2 3 sin 2 sin 2 cos 2     r k i xy (4) 0z , for the plane stress state and )( yxx   , in the case of plane deformation (5) broek says that in many cases is aki  , regardless of the size effect of the object, where:   tensile stress of the external load, a – greater semi-axis length of ellipse as shown in fig. 19. fig. 19 axially loaded plate with crack length 2a and coordinate system (x, y) and (r, ) (in front of the crack tip from ref. [1] ). firstly, it should be indicated in these solutions that they start from an assumed plane stress or strain in the plate at the crack tip vicinity. all these solutions are based on the assumption that the stress state in front of the crack is critical for crack propagation, not the potential strain energy status or the energy density distribution state in the vicinity of the crack. 2a  2 a  2 a x y x y r  2 a x 2a y 2a xy 2a 148 d. b. jovanović m m m m fig. 20 distribution of normal stress x, in coordinate system (x, y) fig. 21 distribution of normal stress y, in coordinate system (x, y) m m m m fig. 22 distribution of normal stress z, in coordinate system (x, y) fig. 23 distribution of shear stress xy, in coordinate system (x, y) based on the stress values obtained from equations (2), (3), (4) and (5) and by using the expression for the potential strain energy (1) from the literature a diagram of the potential strain energy is obtained, shown in fig. 24. pronounced linearity of this graph does not correspond to the real physical conditions, because the potential strain energy distribution around the crack tip as well as distribution of stresses is non-linear. nonlinearity of stresses has been demonstrated in experimental [7], [8], [9] and analytical studies. stress concentration should no longer be seen as a criterion for determining the crack propagation direction and site, but the potential strain energy concentration. m fig. 24 distribution of potential strain energy adef ahead of the crack tip in coordinate system (x, y). potential strain energy surfaces at the crack tip vicinity 149 we can compare the results of reconstruction with the graphical presentation of potential strain energy obtained from the relations (6), from refs. [1], [2]. gdoutos on page 34 of ref. [2] fracture mechanics an introduction, gives relations for stresses, in the polar system of coordinates of the form of crack propagation type i (opening):                                                     2 3 sin 2 sin1 2 cos 2 ),(    r k r i x                                                     2 3 sin 2 sin1 2 cos 2 ),(    r k r i y 0),(  rz                                                   2 3 cos 2 cos 2 sin 2 ),(    r k r i xy .0),(,0),(   rr yzxz (6) by applying previous relations (6) we get the distribution of potential strain energy ahead of the crack tip shown in fig. 25. (a, b, c). adef adef adef fig. 25 a) surface of potential strain energy obtained by relations (6), (type i) b) the projection of the potential strain energy surface at x-y plane. c) isoenergy lines in front of the crack tip. by superposition of type i and type ii forms of crack propagation, relations are obtained for the combination of the two cases of stress states, from refs. [1], [2]:                                                                                                          2 3 sin 2 cos2 2 sin 22 3 sin 2 sin1 2 cos 2 ),(      r k r k r iii x                                                                                                          2 3 sin 2 cos 2 sin 22 3 sin 2 sin1 2 cos 2 ),(      r k r k r iii y 150 d. b. jovanović 0),(,0),(,)),(),((),(   rrrrr yzxzyxz                                                                                                            2 3 sin 2 sin1 2 cos 2 2 3 cos 2 cos 2 sin 2 ),(      r k r k r ii i xy (7) by applying previous relations (7) we obtain distribution of the potential strain energy ahead of the crack tip, for the combination of type i and type ii form of crack propagation, shown in fig. 26 (a, b, c). adef adef adef fig. 26 a) surface of potential strain energy by relations (7), (type i and type ii) b) the projection of the potential strain energy surface at x-y plane. c) isoenergy lines in front of the crack tip 4. conclusions the results of applying the idea of reconstructing the spatial surface, that describes the potential strain energy distribution in the crack vicinity, are presented. in particular, i have analyzed the examples of cracks identified in the paper as pr-z-16 and pr-p-22. geometry of the cracks and plates are completely the same. i have selected the same finite element mesh for stress calculation in both cases. the only difference between the examples is the way of loading the plate. in example of pr-z-16 plate is loaded in the axial direction of y axis and perpendicular to the direction of the crack plane. while, in the case of pr-p-22 the plate is loaded with a continuous load on the crack surface contour. based on the reconstructed potential strain energy surface i got the isoenergy lines in front of the crack tip, for both analyzed examples. it is observed that the distribution of stresses, as well as that of the potential strain energy are very similar in both cases, indicating the main role of geometric shape of contour surface of the crack. finally, i have showed the comparison of stress and potential strain energy surfaces, which are based on the relations from the literature. potential strain energy surfaces at the crack tip vicinity 151 references 1. broek d., 1982, elementary engineering fracture mechanics, martinus nijhoff publishers, the hague. 2. gdoutos e. e., 1993, fracture mechanics, kluwer academic publishers, dordrecht, 369 p. 3. jovanović d. b., jovanović m., 2000, local stress and strain state in the region of crack for different global stress states in a plate, yusnm, niš 2000, facta universitatis, series mechanical engineering, vol. 1, no. 7, pp 925-934. 4. jovanović d. b., jovanović m., 2001, stress state and strain energy distribution at the vicinity of elliptical crack with compression forces acting on it's contour, facta universitatis, series mechanics, automatic cont. and rob., vol. 3, no.11, pp. 223-230. 5. jovanović d. b., 2002, stress state and deformation (strain) energy distribution ahead crack tip in a plate subjected to tension, facta universitatis., series mechanics, automatic control and robotics, vol. 3, no. 12, pp. 443-455. 6. parsons i. d., hall j. f., rosakis a. j., 1986, a finite element investigation of the elastostatic state near a three dimensional edge crack, sm report 86-29, division of engineering and applied science, california institute of technology, pasadena, california. 7. pindera j. t., josepson j., jovanović d.b., 1996, electronic techniques in isodyne stress analysis, abstract proceedings of the viii international congress on experimental mechanics, nashville, usa. 8. pindera j. t., josepson j., jovanović d.b., 1997, electronic techniques in isodyne stress analysis: part 1. basic relations, experimental mechanics, vol. 37, no. 1, 33-38. 9. pindera j. t., josepson j., jovanović d.b., 1997, electronic techniques in isodyne stress analysis: part 2. illustrating studies and discussion, experimental mechanics, vol. 37, no. 2, 106-110. površine potencijalne energije deformacije u okolini vrha prsline u radu su prikazani rezultati i osnovni elementi postupka rekonstrukcije površi potencijalne energije deformacije u okolini prsline. ukazano je na izvesna ograničenja i specifičnosti metode u procenjivanju krivih i trodimenzionalnih površi koje se najbolje uklapaju sa podacima za komponente napona i potencijalnu energiju deformacije. analiziraju se primeri prsline potpuno istih geometrija prsline i ploče, koji su u radu imenovani kao pr-z-16 i pr-p-22. izabrana je ista mreža konačnih elemenata za proračun u oba slučaja. prvi primer ploče je opterećen naponom σ u y pravcu na spoljašnjoj ivici ploče, upravno na ravan prsline, dok je u primeru pr-p-22 ploča opterećena naponom σ kontinualno i upravno na površ konture prsline. realizovan je postupak rekonstrukcije površi potencijalne energije deformacije za dva preseka ploče y = 0 i z = 0, a urađen je i grafički prikaz ovih površi. na osnovu rekonstruisanih površi potencijalne energije deformacija dobijene su izoenergijske linije u oblasti ispred čela prsline, za oba alalizirana primera. uočava se sličnost rasporeda napona, kao i rasporeda potencijalne energije deformacije za oba slučaja, što ukazuje na značajan uticaj geometrijskog oblika konturne površi prsline. prikazani su za upoređenje stanje napona i stanje potencijalne energije deformacije koji su dobijeni korišćenjem relacija iz literature. ključne reči: prslina, komponentni naponi, rekonstrukcija površi potencijalne energije deformacije 10425 facta universitatis series: mechanical engineering vol. 22, no 1, 2024, pp. 125 138 https://doi.org/10.22190/fume220114015m © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper wear load capacity of crossed helical gears aleksandar miltenović1, milan banić1, jovan tanasković2, jelena stefanović-marinović1, damjan rangelov1, marko perić1 1unversity of niš, faculty of mechanical engineering, serbia 2university of belgrade, faculty of mechanical engineering, serbia orcid ids: aleksandar miltenović https://orcid.org/0000-0002-1453-2548 milan banić https://orcid.org/0000-0001-8684-042x jovan tanasković https://orcid.org/0000-0002-2936-5535 jelena stefanović-marinović https://orcid.org/0000-0001-8823-5362 damjan rangelov https://orcid.org/0000-0003-3071-893x marko perić https://orcid.org/0000-0001-9350-0351 abstract. the paper presents a wear load capacity of crossed helical gears (gear pair consists of a worm and a helical gear). crossed helical gears are similar to worm gear pairs, and it is logical to extend the load capacity calculation approach of worm gears for the case of crossed helical gears. one of the disadvantages of crossed helical gear sets is very high running-in wear. the paper explains the transition between running-in and steady-state wear and proposes a joint wear calculation model. the paper outlines a proposal to extend the calculation method for wear load capacity from a worm gear pair to the case of crossed helical gears (a worm with a helical gear). this calculation method for determining the wear load capacity of crossed helical gears is verified for the cases when the wheel is of bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo. key words: crossed helical gears, wear, bronze, sintered steel, load capacity 1. introduction gears belong to the group of the most common parts used for power transmission in various types of machines [1, 2, 3]. they are exposed to cycles of normal and shear forces that give rise to wear. as one of the most important limiting factors for the lifetime of gears, wear load capacity needs to be given a special attention. gear tooth surface wear is common in the gears together with bending fatigue, pitting, micropitting and scuffing. wear for crossed helical gears is the most common damage since the contact between the gear teeth is of a ‘point type’, which is disadvantageous compared to other gear types that have a so-called ‘line contact’. crossed helical gears are less researched than the other gear types since they have lower load capacity, and, therefore, industry did not show greater interest in investigating them. mostly, industry application is of the crossed helical gears with a smaller size that do not need to have higher load capacity. the application primarily transmits some lower power and motion like in actuators and received: january 14, 2022 / accepted march 25, 2022 corresponding author: aleksandar miltenović university of niš, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: aleksandar.miltenovic@masfak.ni.ac.rs https://orcid.org/0000-0002-1453-2548 https://orcid.org/0000-0001-8684-042x https://orcid.org/0000-0002-2936-5535 https://orcid.org/0000-0001-8823-5362 https://orcid.org/0000-0003-3071-893x https://orcid.org/0000-0001-9350-0351 126 a. miltenović, m. banić, j. tanasković, j. stefanović-marinović auxiliary drives. the advantage of crossed helical gears is that their assembling is much easier and does not affect their operation characteristics compared with worm gears. for the proper operation of worm gear drives and achievement of prescribed values of load capacity is necessary to have precise assembling of the worm wheel. wear for gears is described, in literature, as the process of continuous removal of surface material from tooth flank. in the case of crossed helical gears, removal is intensified initially since running-in wear is in the circumstance of contact in point. running-in is followed by steady-state wear where wear rates are lower since the contact is now in the line that is growing with the progression of wear. vullo [4, 5] gave a review of state of the art for wear generation mechanism and crossed helical gears kinematics. he showed different approaches in calculating gear wear like hertz theory, holm's, and archard's wear calculation method for wear linear progression. shilko [6] used an advanced discrete-element-based mechanical model for modeling wear of ductile materials that considers fracture and surface adhesion. cao [7] proposed fracture-induced adhesive wear criterion. the idea of the wear criterion is to predict the wear process of a mixed lubricated point contact in sliding motion that can be used in the simulation. he compared simulation and experiment measurements that had a reasonable agreement. wagner [8] studied the effect of contact pressure and surface texture on the running-in behavior in the case of carburized steel under boundary lubrication. he demonstrated running-in of the surface within one thousand load cycles "regardless of the pressure or initial composite surface roughness". vdi 2736-3 [9] provide load capacity calculation for crossed helical gear sets with material combination steel and plastics. din 3996 [10] provides the calculation of the load capacity of cylindrical worm gear pairs with rectangular crossing axes. one of the calculations is for wear load capacity. wendt [11] investigated load capacity for worm with helical gear set combination steel/sintered steel. he used sintered steel fe1.5mo0.3c with different material density where he proved that increased material density brings higher wear resistance. in addition, he provides approximate equations for the calculation of safety factors for wear and pitting. boehme [12] presented a calculation method for crossed helical gears with optimized hertzian contact pressures and sliding paths that are designed to increase the load capacity. norgauer [13] presented the calculation and simulation for the design of crossed helical gear sets. simulation is integrated into software schrad2 to calculate contact patterns without and under load. for under load conditions, he determined the stiffness of the gearbox. liang [14] proposed a generation approach of gear pair with small-angle based on the curve contact element to improve load capacity and transmission characteristics of crossed-axis helical gear drive. he introduced spatial curve meshing relationships for contact principle. he also produced a prototype and carried out basic tests that show well contact characteristics. the load capacity of worm gears has been developed over a long period on standards based on industrial experience, and they were published in din 3996. iso tr 14521 standard – calculation of load capacity of worm gears is based on din 3996. wear is one of the typical damages for worm gears, and it usually appears on the tooth flanks of worm wheels [15]. chernets [16] presented the computation method of studying worm gears with archimedean and involute worm. he showed that the correction coefficient of wheel teeth decreases maximal contact pressures and the wear of wheel teeth. wear load capacity of crossed helical gears 127 daubach [17] developed a simulation model for the abrasive wear of worm gears based on the energetic wear equation. this simulation model includes a tooth contact analysis and tribological calculation to determine friction and wear. ivanov [18] analyzed the wear of worm gears in the running-in operation. he showed that wear in the runningin phase in uniform over the gear width influences efficiency, and after the running-in stage is recommendable to replace the lubricant. also, in the running-in phase, variations of load and velocity are undesirable. weisel [19] researched experimentally a worm with a helical gear wheel. he demonstrates that the load capacities of crossed helical gear sets after the running-in phase are like worm gears. miltenovic [20, 21, 22] investigated the wear resistance of worm with helical gear set combination steel/sintered steel where he used fe1.5cr0.2mo with different additional treatments. sintered steel fe1.5cr0.2mo is an interesting material with good wear resistance characteristics, suitable for mass production, while sinter-hardening treatment had the best wear resistance characteristics. krol [23] investigated the parameters of worm gearing with increased wear resistance in the specific application. as a result, he succeeded in influencing the wear rates by changing the basic parameters of the gear set. the novelty of this paper lies in a calculation approach for the wear load capacity of crossed helical gears in applications that are like worm gears. as there is a high geometrical similarity between these two gear pairs, it is logically possible to use the basics of the wear load capacity calculation of worm gears for the case of crossed helical gears. for this approach, verification is performed in an experiment and the calculation validates this method. the investigation is conducted for two materials: bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo with sinter-hardening treatment. bronze cusn12ni2-c-gcb is the standard material for worm wheel, and it covers with necessary data for load capacity calculation according to din 3996. 2. test conditions experiments have been carried on test benches with center-to-center distance of 30 mm. on the input side of the transmission is on-worm-mounted asynchronous motor, and on the output side, on the wheel is connected with a magnetic particle brake. the torque measurement system is mounted on the input and output sides, and measurement was made with a torque gauge bar via a slip ring transmitter. the crossed helical gear set data is given in table 1, and the test bench is presented in fig. 1. the wear resistance of crossed helical gears is investigated for input speeds of 1500, 5000 and 10000 min-1. at the beginning of the test, output torque is 12 nm for speeds of 1500 and 5000 min-1 and 8 nm for 10000 min-1. after 40h, output torque was increased by 4 nm up to the maximal output torque that these crossed helical gears can resist. every test was repeated three times. measurement of wear depth was carried out every 20 h at klingerberg pnc 65 measuring machine. module m of the bronze gears is 1.25 mm while for sintered steel gears is 1.252 mm, which is the consequence of material increase during the sintering process. the test equipment is better explained in [24]. 128 a. miltenović, m. banić, j. tanasković, j. stefanović-marinović fig. 1 test bench table 1 data of the test gear pair parameter data worm wheel center distance 30 mm shape of worm flank zi module 1.25 (1.252) mm transmission ratio 40 number of teeth 1 40 pressure angle 20o profile shift coefficient 0 lead direction left gear width 25 mm 10 mm helix angle at reference circle 82.493o 7.507o reference diameter 9.57 mm 50.43 mm tip diameter 12.09 mm 52.95 mm root diameter 6.54 mm 47.41 mm transverse contact ratio 1.852 material fe1.5cr0.2mo cusn12ni2-c-gcb 16mncr5 input speed 1500, 5000 and 10000 min-1 output torque 12–36 nm for 1500 and 5000 min-1 8–20 nm for 10000 min-1 number of starts 1 per 20h lubrication oil klüber gh6 1500 wear load capacity of crossed helical gears 129 3. calculation method wear describes the loss of material from the surface in the contact. in the case of gears, this leads to a reduction of the tooth thickness. according to din 3996, wear load capacity is determined with safety factor against wear sw calculated from the permissible wear on flank δwlimn and wear depth δwn in (n) normal section (1). lim min 1.1w w n wn ws s =  = (1) wear depth δwn is based on experiments, and it depends on the material and lubricant combination that is used in tests. wear depth directly defines wear resistance, and material and lubricant combinations with lower wear depth have greater wear resistance and higher wear load capacity. wear depth δwn directly depends on wear intensity jw and wear path swm (2). wn w wmj s =  (2) wear intensity jw depends on basic wear intensity j0t for the specific material and factors that bring closer basic wear intensity and specific operation characteristics. according to din 3996 [10], wear intensity jw eq. (3) depends on material lubrication factor wml, which values are given in the appropriate table and the factor for starts wns that is calculated by proper equitation defined in the standard [10]. 0w t ml nsj j w w=   (3) wml values depend on lubrication and material of worm wheel and are limited only to some materials. this means that for material fe1.5cr0.2mo would be impossible to choose the value for this factor. the factor for starts wns considers the influence of repeated starts on wear and can be determined by eq. (4), depending on the number of starts/hour ns. 1 0.015 3ns sw n= +   (4) wendt [11], in his research, proposed eq. (5), where he introduced material factor ytρ that considers material density without using wml and wns. in this case, the introduction of the material factor is logical since he used the same sintered materials for the gear wheel with different material densities (6.9 and 7.2 kg/dm3) in his research. 0w t tj j y =  (5) the authors of this paper choose not to use wns, wml, and ytρ because of the following reasons: ▪ wml – material lubrication factor is practically unusable for these materials, and it has only character of simple magnification that does not affect character of wear progression by crossed helical gear. ▪ wns – factor for number of starts does not have influence in the test case since its value would be 1.00075 and specific test for this factor are not carried out. there is a possibility for this factor to be used in the case of crossed helical gear set in the form of eq. (4) or some other form, but additional tests are necessary. ▪ ytρ – material factor is unsuitable since there are no same materials with different densities. 130 a. miltenović, m. banić, j. tanasković, j. stefanović-marinović according to the tests for crossed helical gears, authors propose that wear intensity jw (eq. (6)) depends on basic wear intensity and crossed helical factor wsr where crossed helical factor wsr include specific character of wear progression for crossed helical gear set. 0w t srj j w=  (6) according to experiments presented in section 4, basic wear intensity j0t has values for bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo that can be calculated with eq. (7). 2 0 1 b t wj b k=  (7) table 2 data of the test gear pair material coefficient b1 b2 fe1.5cr0.2mo sinter-hardening 320.5×10-9 -0.355 cusn12ni2-c-gcb 143×10-8 -0.74 characteristic value for lubrication gap kw is calculated according to din 3996, and it gives connections with load, size, materials, etc. coefficients b1 and b2 for bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo with sinter-hardening treatment are shown in table 2. table 3 coefficients of eq. (8) material coefficient a1 a2 a3 fe1.5cr0.2mo sinter-hardening -0.05359 3.57093x10-4 -2.92964x10-8 cusn12ni2-c-gcb -0.21911 3.42276x10-4 -1.98877x10-8 crossed helical gears factor wsr – eq. (8) includes the wear increase during the running-in period and regular operation of the crossed helical gear set. it depends on lifetime lh and input speed n1 and experimentally determined coefficients a0, a1 and a2 given in table 3 for bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo. 0.02 2 0 1 1 2 1 (log ) 1 ( ) h sr h l w l a a n a n =  +  +  (8) eq. (8) is proposed by the authors because it follows the character of wear progression of crossed helical gear set and it can be used for rotation speed n1 > 1500 min-1 and n1 < 10000 min-1. the crossed helical gears factor can be determined only experimentally. 4. verification of the calculation method due to its geometry, a crossed helical gear has two phases during operation: the running-in phase and the regular operation that comes after the running-in phase. wear load capacity of crossed helical gears 131 fig. 2 shows the comparison of teeth flanks of crossed helical gears made from sintered steel fe1.5cr0.2mo and cusn12ni2-c-gcb for input speed n1 = 5000 min-1 and different output torque from 12 to 32 nm. fe1.5cr0.2mo cusn12ni2-c-gcb t2 = 12 nm t2 = 16 nm t2 = 20 nm t2 = 24 nm t2 = 28 nm t2 = 32 nm fig. 2 comparison of wear at teeth flanks of crossed helical gears for input speed n1 = 5000 min-1 and different output torque every helical gear is measured at reference diameter on klingelberg pnc 65 after 20h long test. еach measurement was performed on three teeth, which are located approx. at 132 a. miltenović, m. banić, j. tanasković, j. stefanović-marinović 120o and the value of wear depth that is used in further research is an average value of wear depth at three teeth. example of measurement protocol is given in fig. 3. fig. 3 measurement protocol for measuring wear depth of helical gear 4.1. wear during the running-in phase the running-in wear occurs immediately at the beginning of the operation due to the high flank pressure at the contact point. the running-in wear is a complex process, and, in some cases, running-in wear rates can be relatively large for such a short operation time. since the contact, in the beginning, is at a point and there is no oil between the flanks, the direct steel-to-steel contact occurs. this can lead to short scuffing or very high abrasive wear due to a direct steel-to-steel contact, which is followed by lower wear rates with the lubrication film. fig. 4 shows wear depth δwn for different input rotation speeds and identical load and operation time. fig. 4 running-in wear for speed n1= 1500, 5000 and 10000 min-1 for output torque 12 nm and operation time 20 h scuffing is a specific damage form and compared with wear it is much more dangerous for the flanks. the scuffing load capacity is not sufficiently explored yet to specify standardized calculation. temporarily occurring scuffing damage to bronze wheels can be "healed" again. this healing is possible only through wear, but it cannot currently be considered in estimating the wear life of the din 3996 standard. scuffing depends wear load capacity of crossed helical gears 133 significantly on the material in contact, and it is much more unstable to predict than wear. this means that running-in wear can encompass short running-in scuffing. rotation speed plays a significant role and has a dual influence on running-in wear. 1. the thickness of the lubrication film depends directly on the speed, and with an increase in the rotation speed, the lubrication film thickness also rises. a higher lubrication film thickness decreases the running-in wear. this means that the running-in wear could be lower at high speeds. 2. a higher speed increases the working temperature, decreasing the viscosity of oil and increasing the running-in wear. this means that the running-in wear could increase more at high speeds. the lowest wear depth is for the speed of n1 = 5000 min-1 in both cases. bronze has a higher wear rate for all speeds, and its running-in wear is higher for n1 = 1500 min-1 than for n1 = 10000 min-1. this process is followed by an increase in the contact line and a decrease in the contact pressure. therefore, running-in wear is the lowest in the case of n1 = 5000 min-1 for both materials. this means that scuffing in the running-in wear phase is the smallest or does not exist at a speed of n1 = 5000 min-1. after the running-in phase comes to the operation with an approximately constant increase in wear depth δwn. the worm acts like a cutter that creates a globoid flank and spreads contact from the contact point to the contact in line that grows as the wear depth grow. fig. 5 compares the calculated (eq. 2) and the measured wear depths during the running-in process for bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo. fig. 5 experimental and calculation results for running-in wear 4.2. experimental results figs. 6 to 11 represent the comparison of calculated and measured values of wear depth δwn for the helical gear made of sintered steel fe1.5cr0.2mo with sinter-hardening treatment and cusn12ni2-c-gcb and for input speeds of n1 = 1500, 5000 and 10000 min-1. the results at the input speed n1 = 5000 min-1 are more stable than with n1 = 1500 and 10000 min-1 and due to the running-in wear phase issue and dual influence of rotation speed are explained in 4.1. 134 a. miltenović, m. banić, j. tanasković, j. stefanović-marinović fig. 6 shows a comparison of calculated and measured wear depth δwn for the helical gear made of cusn12ni2-c-gcb for n1 =1500 min-1. bronze has a highest running-in wear for n1 = 1500 min-1. fig. 6 calculated and measured wear rates δwn for the helical gear made of cusn12ni2-c-gcb for n1 =1500 min−1 fig. 7 shows a comparison of calculated and measured wear rates δwn for the helical gear made of cusn12ni2-c-gcb for n1 =5000 min−1. in this case, the values of wear rate are most minor and repeatability gave the best repetition of results. fig. 7 calculated and measured wear rates δwn for the helical gear made of cusn12ni2-c-gcb for n1 =5000 min−1 wear load capacity of crossed helical gears 135 fig. 8 shows a comparison of calculated and measured wear rates δwn for the helical gear made of cusn12ni2-c-gcb for n1 =10000 min−1. fig. 8 calculated and measured wear rates δwn for the helical gear made of cusn12ni2-c-gcb for n1 = 10000 min−1 fig. 9 shows a comparison of calculated and measured wear rates δwn for the helical gear made of fe1.5cr0.2mo for n1 =1500 min−1. fig. 9 calculated and measured wear rates δwn for the helical gear made of fe1.5cr0.2mo for n1 =1500 min−1 136 a. miltenović, m. banić, j. tanasković, j. stefanović-marinović fig. 10 shows a comparison of calculated and measured wear rates δwn for the helical gear made of fe1.5cr0.2mo for n1 =5000 min−1. fig. 10 calculated and measured wear rates δwn for the helical gear made of fe1.5cr0.2mo for n1 =5000 min−1 fig. 11 shows a comparison of calculated and measured wear rates δwn for the helical gear made of fe1.5cr0.2mo for n1 =10000 min−1. sintered steel has a highest running-in wear for n1 = 10000 min-1. fig. 11 calculated and measured wear rates δwn for the helical gear made of fe1.5cr0.2mo for n1 = 10000 min−1 wear load capacity of crossed helical gears 137 5. conclusions this paper presents the experimental and theoretical results of the worm with the helical gear made of bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo with sinter-hardening treatment. the running-in wear is a complex process that occurs immediately at the beginning of the operation due to the high flank pressure at the contact point. since the contact, in the beginning, is where there is no oil between the flanks, the direct steel-to-steel contact occurs. this can lead to short scuffing or very high abrasive wear due to the direct steelto-steel contact, followed by lower wear rates with the lubrication film. scuffing depends significantly on the materials in contact, and it is much more unstable to predict than wear. this means that running-in wear can encompass short running-in scuffing. rotation speed plays a significant role and has a dual influence on running-in wear. the thickness of the lubrication film depends directly on the speed, and an increase in the rotation speed decreases the running-in wear. on the other hand, higher speeds increase the working temperature, reducing the oil viscosity and increasing the running-in wear. for the running-in phase, the moment of transition from the non-oil film contact to the contact with the oil film is crucial. the lowest wear depth is for speed of n1 = 5000 min-1 for both materials due to the dual influence on the running-in wear. bronze has a higher wear depth for all speeds, and the bronze running-in wear is higher for n1 = 1500 min-1 than for n1 = 10000 min-1. the paper presents a calculation method for the wear load capacity of crossed helical gears in applications that are similar to worm gears. as there is a high geometrical similarity between these two gear pairs, the paper proposes a possible expansion of the wear load capacity calculation of worm gears to the case of crossed helical gears. it gives some form of guidelines for cases of crossed helical gears. for this purpose, the paper introduces coefficient wsr that includes the wear progression during the running-in wear phase and after it; it depends on the material type, lifetime, and input speed. it can be determined only by experiment. for this approach, verification is given in an experiment and the calculation that validates this method. this is done for two materials: bronze cusn12ni2-c-gcb and sintered steel fe1.5cr0.2mo. acknowledgements: this research was financially supported by the ministry of education, science and technological development of the republic of serbia (contract no. 451-03-9/202114/200109 and 451-03-9/2021-14/200105). references 1. xue, q., zhang, x., geng, c., teng, t., 2021, gear shift coordinated control strategy based on motor rotary velocity regulation for a novel hybrid electric vehicle, applied sciences, 11(24), 12118. 2. wang, a., li, y., du, x., zhong, c., 2021, diesel engine gearbox fault diagnosis based on multi-features extracted from vibration signals, ifac-papersonline, 54(10), pp. 33-38. 3. marinković, z., marinković, d., petrović, g., milić, p., 2013, modelling and simulation of dynamic behaviour of electric 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mechanical engineering vol. 23, no 3, 2025, pp. 459 478 https://doi.org/10.22190/fume221126010k © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a modified fmea approach based integrated decision framework for overcoming the problems of sudden failure and accidental hazards in turbine and alternator unit dinesh kumar kushwaha1, dilbagh panchal2, anish sachdeva1 1department of industrial and production engineering, dr b r ambedkar national institute of technology jalandhar, punjab, india 2department of mechanical engineering, national institute of technology kurukshetra, haryana, india orcid ids: dinesh kumar kushwaha https://orcid.org/0000-0002-4263-3012 dilbagh panchal https://orcid.org/0000-0002-9153-7182 anish sachdeva https://orcid.org/0000-0002-7884-9982 abstract. the proposed work presents a novel integrated decision framework, based on intuitionistic fuzzy (if) failure mode & effect analysis (if-fmea), and if-technique for order of preference by similarity to ideal solution (if-topsis) approaches for analysing the failure risk issues of turbine and alternator unit (tau) in a chemical treatment-based sugar process industry. the proposed novel if-fmea approach-based modelling overcomes various demerits of traditional fmea approaches, which are faced during the identification of critical failure causes based on the risk priority number (rpn) outputs. on the basis of detailed qualitative information related to plant operation, fmea sheet was developed and linguistic ratings were collected against three risk factors such as probability of occurrence (o), severity (s), and detection (d). if-hybrid weighted euclidean distance (ifhwed) score has been computed to rank all listed failure causes under three risk factors. the ranking results based on if-fmea approach has been compared with the well existed if-topsis approach for evaluating the accuracy of the proposed modelling results. sensitivity analysis has also been done to check the robustness of the framework. the analysis results were provided to the maintenance executives of the tau unit in order to frame the optimal maintenance plan for overcoming the problems of sudden breakdown. the analysis results are also applicable to tau systems, which are installed in other chemical process industries globally. key words: sugar process industry, if-fmea, if-topsis, failure causes, sudden breakdown, maintenance schedule received: november 26, 2022 / accepted march 07, 2023 corresponding author: dilbagh panchal department of mechanical engineering, national institute of technology kurukshetra, haryana, india 136119 e-mail: panchald@nitkkr.ac.in https://orcid.org/0000-0002-4263-3012 https://orcid.org/0000-0002-9153-7182 https://orcid.org/0000-0002-7884-9982 460 d. k. kushwaha, d. panchal, a. sachdeva 1. introduction tau is one of the crucial functionary units, especially in all chemical process industries like petrochemical, fertilizer, paper and pulp, and chemically treated sugar industry, etc. in order to fulfil the power and heat requirement for various processing stages (heating, atomization, cleaning, sterilization, moisturization, humidification, drying, etc.) failure free operation of tau system is highly important. to ensure the failure free operation of industrial system, like tau, over long duration, development of an optimum maintenance strategy is needed. for the development of an optimum maintenance schedule, recognising most critical failure causes of different subsystem/component are of supreme importance [1,2]. furthermore, due to availability of vague operational data from different sources like mill maintenance log book and maintenance personnel of the considered system, the job of system analyst for identifying riskier failure causes with high accuracy has become highly challenging. in order to resolve this task, the development of a mathematical model, which could cover the uncertainty/vagueness in the operational data of a system, proves to be very useful in the identification of critical failure causes with high accuracy, and thus leads to development of correct maintenance policy. if-modelling based concept proves to be useful in considering the uncertainty/vagueness of the collected data and, hence, results in highly accurate outputs. in chemical process industries, tau system is required to operate 24×7 hours as it is commonly used to drive all rotating equipment such as pumps, compressors, etc., and supply heat to heat exchangers, evaporators and ovens. even a minor failure in the tau system results in an interruption in the continuous functioning of all these components/subsystems, thus causing a sudden failure in the plant. therefore, for ensuring continuous functioning of the considered tau system, each and every associated component/subsystem must function in the most upstage conditions, for which various modes of failure are responsible for sudden failure/breakdown. failures can be caused, namely, by abrasion, corrosion, electrical pitting, fatigue, overheating, inclusion of foreign particles in oil, and chemical attack of reactive agents (electrolytes/ organic acids) must be addressed with high accuracy [3]. around 50% of sudden failures of turbines are due to inclusion of contamination of moisture, dirt, etc., in the lubrication systems [4,5]. in a tau system, improper functioning of a lubrication system often takes place even due to a minor failure of its component/subsystem, which may lead to system’s unavailability or poor operational safety, especially when tau system is used in the chemical process industries. as tau is one of the important functionary unit responsible for continuous and safe operation of different chemical process industries, in considering its importance in the context of accidental safety and loss prevention, it is indispensable to analyse the operational risk issues associated with the considered unit. 2. literature background over the past few years, scholars have presented many works to diagnose failure causes and study riskrelated issues in various process industries and services sectors – geothermal power plant [6], marine industry [7], anaesthesia process [8], health care industry [9], gas processing plant [10], hydraulic turbine generator system [11], cogeneration power plant [12], etc. liu et al. [13] proposed hybrid fuzzy fmea visekriterijumska optimizacija i kompromisno resenje (vikor) – decision making trial and evaluation laboratory (dematel) – analytic hierarchy process (ahp) approaches to perform risk analysis of a turbocharger. panchal and kumar [14] proposed fmea to detect the riskier components a modified fmea approach based integrated decision framework for overcoming the problems… 461 of a power generating unit (pgu) under uncertainty. adar et al. [15] implemented fuzzy fmea for finding the critical failure modes of a water gasification system (wgs). ahn et al. [16] proposed fuzzy-based fmea to present a risk analysis of liquefied hydrogen tankers to encounter the uncertainties involved in crisp based fmea. failure risk analysis of transmission system of an automobile industry was conducted to study the unavailability of considered system [17]. fmea often provides the basis of multi criteria decision making (mcdm) approaches, and is applied in combination with mcdm approaches. also, various limitations associated with the traditional fmea approach are well covered with the incorporation of mcdm approaches into fmea approach. considering this advantage, a new fuzzy fmea-based extended multiple multi-objective optimization by ratio analysis (multimoora) – ahp was introduced by fattahi and khalilzadeh [18], and application of the framework was illustrated for a steel factory. panchal et al. [19] addressed the shortcomings of the if-then rule based fuzzy fmea approach, and proposed an integrated framework with introduction of an improved fmea approach, for identifying the riskier failure causes of a chlorine gas plant. boral et al. [20] proposed application of various fuzzy mcdm, such as interval type-2 fuzzy dematel and modified fuzzy multi-attribute ideal real comparative analysis (fmairca) methods within a fuzzy fmea domain, to encounter the risk relevant to gearbox of a power plant. gopal and panchal [21] proposed a fuzzy fmea approach-based integrated framework for prioritizing the most critical components on the basis of rank of failure causes in a milk process plant. fuzzy based risk analysis approaches have the drawback that it does not take into account the element of hesitation inherent in the qualitative feedback obtained by the experts. intuitionistic fuzzy set (ifs) concept-based techniques overcome this shortcoming of fuzzy theory approach by considering the hesitation element. with a paradigm shift in uncertainty theory concept, researchers incorporated the ifs concept in fmea and mcdm techniques-based risk model to achieve a high degree of correctness of results in decisionmaking approaches. moreover, the application of if based hybrid mcdm techniques has increased encapsulating a wide range of modifications and applications. wang et al. [22] presented a hybrid interval-valued based mcdm approach to perform a risk analysis of hospital service under if environment. baig et al. [23] proposed full consistency method (fucom) and fuzzy quality function deployment (qfd) to identify capabilities, which confirm protection against the vulnerabilities involved in oil supply chain. muhammad et al. [24] proposed grey edas approach to select the best alternatives for treatment of waste in nigeria. efe [25] proposed an integrated framework based on qfd and vikor approach for risk assessment based on fmea of a ship building industry under the intuitionistic fuzzy environment. kushwaha et al. [26] applied ifwa operator based fmea approach identify the riskier components of cutting system by implementing fmea approach under if environment. ilbahar et al. [27] applied fmea-ahp approaches to access the risk involved in the investment of renewable energy utilising if concept. from the above reviewed literature, it is clear that the fuzzy set theory has been applied for analysing the risk involved in various industrial systems. the implemented fuzzy based risk models were insufficient to consider high level of vagueness/ uncertainties involved in the qualitative data/information collected from expert judgement. due to its limitation of not considering the non-determinacy or hesitation, the results are highly uncertain/inaccurate. therefore, to overcome this gap, ifmodelling based novel integrated framework has been proposed in this work for studying and analysing the risk issues of tau in a sugar mill industry. 462 d. k. kushwaha, d. panchal, a. sachdeva 3. newly proposed integrated framework in the first phase, fmea sheet containing various subsystem/components, its potential mode of failures, effect, failure causes, and linguistic variables under o, s, d, has been generated on the basis of feedback obtained from three mill’s personnel within the if-fmea approach. in the second phase, the if-topsis approach has been incorporated into the iffmea approach for tabulating a relative coefficient score based ranking results. sensitivity analysis was done to evaluate the robustness of the proposed integrated framework. the sequential steps of two-phase integrated decision framework are as shown in fig. 1. fig. 1 two-phase integrated decision framework 4. basic ifs notions and risk approaches 4.1. basic ifs notions the various notions, which are used in developing the proposed integrated framework, are discussed as follows [28, 29] a modified fmea approach based integrated decision framework for overcoming the problems… 463 definition 1: an ifs is given by eq. (1) as: (1) where is a membership function and is a non-membership function given by eqs. (2) and (3) and both must lie in the closed interval [0,1]. (2) (3) ifs must satisfy the following condition: (4) definition 2: degree of indeterminacy/ hesitation is defined by eq. (5) (5) definition 3: intuitionistic fuzzy distance (ifd) between two intuitionistic fuzzy number (ifn) is computed by eq. (6). let β1 = (µ1,ν1) and β2 = (µ2,ν2) are two ifns, then ifd between β1 and β2 is given as (6) 4.2. if-fmea approach fmea is one of the most extensively applied preliminary investigation techniques used in industrial systems to detect the riskier components. it has proved to be an effective and efficient measure in tabulating the failure risk associated with different systems like oxygen combustor system [30], water treatment plant [31], water gasification system [15], liquefied natural gas unloading facility [32], power distribution industry [33], milk processing process plant [34], gas refinery [35], electro-chemical industry [36]. the fmea approach so applied by different researchers is very useful for listing the detailed qualitative information related to operation of a system, due to which it has widely gained importance in studying the failure risk, but this approach has several limitations as identified by different researchers [31,21,36]. to overcome the demerits noted by these scholars, a novel if-fmea modelling has been developed for conducting the risk analysis. the novel if-fmea modelling encapsulates membership and non-membership functions together, along with indeterminacy/ hesitation element in expert’s judgement for a high degree of accuracy in diagnosing the critical failure causes of an industrial system. with a trait to address the indeterminacy/hesitation using membership and non-membership function, the proposed if-fmea modelling prove to be highly effective in comparison to other fmea approaches. the steps followed in carrying if-fmea are: step 1: generate fmea sheet for the tau subsystem considering all the failure causes under o, s, and d.  , ( ), ( ) a a a x x x x x =  ( ) a x ( ) a x    : 0,1 , ( ) 0,1 a a s x x x →  → →    : 0,1 , ( ) 0,1 a a s x x x →  → → ( ) ( ) 1 for all a a x x x x +   ( ) 1 ( ) ( ) a a a x x x  = − + 1 2 1 2 1 2 1 2 1 ( , ) , ( ) 2ifdd        = = − + − 464 d. k. kushwaha, d. panchal, a. sachdeva step 2: using tables 1-3 allocate if ratings to all failure causes listed for tau system. table 1 linguistic terms for occurrence (o) linguistic terms failure probability ifns very low (vl) rare failure 0.25,0.70 low (l) failure occurs seldom 0.30,0.60 medium low (ml) few failures 0.40,0.50 medium (m) failure occurs often 0.50,0.50 medium high (mh) almost every time 0.60,0.30 high (h) failure occurs every now and then 0.70,0.20 very high (vh) very frequently 0.75,0.20 table 2 linguistic terms for severity (s) linguistic terms failure probability ifns unaffected (ua) system remains continue in operation without much affecting the production 0.15,0.80 very low (vl) system unavailability with little damage 0.25,.70 low (l) loss of raw material 0.30,0.60 medium (m) system operable with little low quality of finished product 0.40,0.50 medium high (mh) system operable with lower grade of finished product 0.60,0.30 high (h) system has to stop for few hours. 0.70,0.20 very high (vh) system has to stop which leads to much loss of production 0.75,0.20 very very high (vvh) entire system has to halt 0.80,0.1 table 3 linguistic terms for detection (d) linguistic terms failure probability ifns almost uncertain (au) failure is unrecognized 0.10,0.90 uncertain (uc) almost not sure 0.15,0.80 very remote (vr) very inaccessible to find 0.20,0.75 remote(r) inaccessible 0.25,0.70 very low (vl) very low possibility 0.30,0.60 low(l) low possibility 0.40,0.50 moderate (m) medium possibility 0.50,0.50 medium high (mh) medium high possibility 0.60,0.40 high(h) recognized 0.65,0.30 very high (vh) highly recognized 0.70,0.20 certain(c) sure 0.80,0.10 step 3: applying eqs. (7) to (9) aggregate an fmea expert’s feedback associated with the set of listed failure causes of the considered system. (7) ( ) ( ) 1 2 1 1 ( , ...... ) 1 1 1 ,k k o o o o ijo ij ij ijl ij l l o o ij ij k k l ifwa k k          = = = =  =   = − −      a modified fmea approach based integrated decision framework for overcoming the problems… 465 (8) (9) where i=1,…m; j=1,….n; ψk are weights of relative importance of fmea experts (summation of weights should be less than equal to one); ifwa is the intuitionistic fuzzy weighted averaging (ifwa) operator; δijo, δijs, and δijd are the aggregated ifns for failure causes under o, s, and d, respectively; δij o, δij s, δij d are the ifns against linguistic terms according to experts’ opinion for failure causes under three risk factors. step 4: using table 4 aggregate the expert’s opinion for subjective weights for o, s, and d as per eqs. (10) to (12). table 4 linguistic terms for rating the three risk factors linguistic terms failure probability ifns very low (vl) (0.1,0.85) very low (vl) low (l) (0.25,0.7) low (l) moderate (m) (0.5,0.5) moderate (m) high (h) (0.75,0.2) high (h) very high (vh) (0.9,0.05) very high (vh) (10) (11) (12) ( ) ( ) 1 2 1 1 ( , ...... ) 1 1 1 ,k k s s s s ijs ij ij ijl ij l l s s ij ij k k l ifwa k k          = = = =  =   = − −      ( ) ( ) 1 2 1 1 ( , ...... ) 1 1 1 ,k k d d d d ijd ij ij ijl ij l l d d ij ij k k l ifwa k k          = = = =  =   = − −      ( ) ( ) 1 1 2 1 1 ( , ...... ) 1 1 1 ,k k o o o o j j ji j i i o o j j k k i k ifwa k k k kk k      = = = =  =   = − −      ( ) ( ) 2 1 2 1 1 ( , ...... ) 1 1 1 ,k k s s s s j j ji j i i s s j j k k i k ifwa k k k kk k      = = = =  =   = − −      ( ) ( ) 3 1 2 1 1 ( , ...... ) 1 1 1 ,k k d d d d j j ji j i i d d j j k k i k ifwa k k k kk k      = = = =  =   = − −      466 d. k. kushwaha, d. panchal, a. sachdeva j=1,….n; k1, k2, k3 are the aggregated subjective opinions for o, s, d; kj o, kj s, kj d, are the experts opinion in form of ifns against risk factors. step 5: using eqs. (13) and (14) compute the subjective weights values for o, s, d. (13) (14) are the subjective weights (for o, s, d), satisfying the condition . step 6: select the objective weights values for (n=3) for o, s, d from table 5 as defined by xu [37]. table 5 selection criteria for objective weight sr. no (𝑛) no of experts mean (𝜇) standard deviation (𝜎) orness (𝑤) disp (𝑤) objective weight (𝑂�̅�) 1 2 1.5 0.5 0.5 0.6391 (0.5,0.5) 2 3 2 √2 3⁄ 0.5 0.8473 (0.243,0.514,0.243) 3 4 2.5 √5 4⁄ 0.5 1.3122 (0.1550, 0.3450, 0.3450, 0.1550) step 7: develop reference series under o, s, d for minimum value using eq. (15). (15) step 8: using eq. (16) compute ifhwed score values. (16) i=1,….m, where, is the objective weight value derived from table 6; di is the ifhwed score for the failure causes; ϕ is the restriction coefficient. step 9: rank all the failure causes of system against ifhwed score in the order of decreasing magnitude. 1,.... 1 j j jj jj j n n j jj jj j               +  +   = =     +    + =    1j j j  = − − jω j j=1 =1ω n  1 1 2( , ,....., ) ( , ,....., )nr r r r r r r− − − − − − −= = ( ) ( )  ( )  2 1 2 ( ) ( )1 , , (1 ) , m i i j ifd ij jj m j ifd i j jj d ifhwed r r d o d          − − = − =  = = +     −      jo a modified fmea approach based integrated decision framework for overcoming the problems… 467 4.3. iftopsis approach topsis was first proposed by hwang and yoon in 1981 [38]. it is a well-established and extensively applied mcdm approach. the various traits of topsis approaches are (i) very simple in applying under various sets of conflicting criteria, (ii) measures of the relative performance of failure causes with simple mathematical formulations, and (iii) high computational accuracy, which makes this method more and more versatile. fuzzy topsis and if-topsis approaches were applied by many researchers in various fields foundry industry [2], tribological process [39], loader selection [40], sugar mill industry [41], smartphones selection [42], working condition assessment [43], etc. moreover, positive ideal solution (pis), negative ideal solution (nis) and relative coefficient scores are calculated to rank the listed failure causes in topsis approach. over the time, mathematical uncertainties concepts like fuzzy and intuitionistic fuzzy were incorporated in crisp based tosis approach to cover uncertainties/vagueness/hesitations in the expert’s domain of knowledge. in this work, if-topsis approach is incorporated within the iffmea approach due to its ability to provide ranking results based on minimum and maximum reference series values, which proves to be very useful for testing the novel iffmea approach-based results. in if-topsis, the first seven steps to compute the ranking are similar to if-fmea modelling based approach, and hence these steps are not repeated here. the remaining steps involved in if-topsis approach are as follows: step 8: develop minimum and maximum reference series value using eqs. (17) and (18) respectively. (17) (18) step 9: tabulate if-nis and if-pis values using eqs. (19) and (20), respectively. (19) (20) step 10: compute relative coefficient score (χ+) of all the listed failure cause using: (21) step 11: rank all the failure causes against relative coefficient outputs. 1 1 2( , ,....., ) ( , ,....., )nr r r r r r r− − − − − − −= = 1 1 2( , ,....., ) ( , ,....., )nr r r r r r r+ + + + + + += = ( ) ( )  ( )  2 1 2 ( ) ( )1 , , (1 ) , 1,..... m i i j ifd ij jj m j ifd i j jj r ifhwed r r d o d i m          − − − = − =  = =     + −    =   ( ) ( )  ( )  2 1 2 ( ) ( )1 , , (1 ) , 1,.... m i i j ifd ij jj m j ifd i j jj r ifhwed r r d o d i m          + − + = + =  = =     + −    =   i i i r r r  − + + − = + 468 d. k. kushwaha, d. panchal, a. sachdeva 5. industrial case studyapplication of the proposed integrated framework 5.1. if-fmea modelling based results the proposed integrated framework is illustrated with its application to carry risk analysis of a tau in a sugar mill industry, geographically located in the western uttar pradesh, india. tau is the most important unit, which fulfils the electricity demand of mill machinery. the schematic arrangement of the tau is shown in fig. 2. fig. 2 schematic arrangement of tau a preliminary investigation was conducted with three maintenance experts of tau for detailed preparation of fmea sheet. the fmea sheet consists of all the listed components, potential mode of failure, effect of failure, and failure causes. on the basis of the designed linguistic rating scales (table 1-3) for o, s and d, the experts were invited to put the linguistic ratings on the basis of knowledge/experience, which is given in table 6. table 6 fmea sheet sl no components potential mode of failure effect of failure failure cause o s d 1 rotor of turbine increase in wheel chamber pressure vibration in rotor scaling on rotor (rt1) mh vh h blades crack due to prolong operation low power generation scaling (rt2) m h vl corrosion (rt3) l h vl 2 solenoid valve deposition of sediments hamper turbine safety wearing of piston (sv4) m mh m wearing of diaphragm (sv5) ml mh h loose electrical connection (sv6) m h vh a modified fmea approach based integrated decision framework for overcoming the problems… 469 3 thrust and journal bearing rise in temperature tripping of turbine rise in temperature of cooling water (tj7) l h vh rise in lube oil temperature (tj8) m vh vh rise in ambient temperature (tj9) l mh c sintering (tj10) m h r 4 pumps auxiliary oil pump (aop) supply failure stop supply oil to turbine gear electrical fault (ap11) mh h vh emergency oil pump (eop) pressure drops stop back up to aop battery discharge (ap12) m vh vl auxiliary control oil pump (acop) pressure drops poor lubrication unhealthy electrical connections (ap13) m h vh 5 valves emergency stop valve (esv) drop in pressure of lube oil turbine trips stop oil supply (es14) l h m quick control non-return valve (qcnrv) leakage back flow of steam gland scaling (es15) l h l grid valve leakage performance of mill is affected gland leakage (es16) l h l 6 main oil pump (mop) & throttle valve pressure drops low lubrication of turbine parts poor quality of lubricating oil (mp17) m h vl contamination entrapped (mp18) m h r gland leakage (mp19) l h l 7 alternator rotor loud vibrations from bearings hamper the operation of alternator mechanical fault (ot20) m m c poor bearing lubrication (ot21) mh m vh armature winding deterioration of winding low emf generation high temperature (ot22) m vh vh 8 oil tank & valves leakage loss of supply of oil poor welding (ov23) h m c low grade of material (ov24) h h l gland leakage (ov25) m mh m seat rupture (ov26) l h h 9 pump leakage poor cooling of winding mechanical & electrical fault (pp27) l vh vh impeller failure (pp28) m vh vh bearing housing failure (pp29) m h c gasket rupture (pp30) vh vh c overheating of motor (pp31) m h c 470 d. k. kushwaha, d. panchal, a. sachdeva here, the first expert has twenty years of experience, second expert seven, while the third expert has four years of experience in the field of operation and maintenance of the tau unit. the feedback obtained from only the first expert is given in table 6 due to space limitation. taking the weights of three experts as ψ1=0.4, ψ2 =0.3, ψ3=0.25 on the basis of their experience, aggregated ifns values for the set of listed failure causes were tabulated as per eqs. (7) to (9), and the aggregated ifns values for all failure causes of the turbine rotor (rt1) under o, s, d are given in table 7. table 7 aggregated ifns values for set of failure causes listed under rt failure cause δijo δijs δijd μij ϑij μij ϑij μij ϑij rt1 0.4763 0.4457 0.7323 0.1823 0.6938 0.2144 rt2 0.4734 0.4648 0.6983 0.2168 0.2965 0.6162 rt3 0.4399 0.4900 0.6983 0.2168 0.4212 0.5000 using the same equations and experts’ weights, aggregated ifns values for the other set of failure causes listed under different subsystem/component of tau were computed. due to space constraints, these calculations are not shown here. using the linguistic scale (table 4), responses from the three experts were obtained for calculating subjective weights for o, s, and d and are given in table 8. table 8 linguistic ratings, aggregated ifns and subjective weights against o, s, d risk factor occurrence (o) severity (s) detection (d) linguistic ratings vl, l, m vh, h, h vh, h, m subjective weights 0.1538 0.4320 0.4142 using the feedback-based values from table 8 in eqs. (10-12), the aggregated subjective ifn values for o, s, and d were calculated but sample calculation only for o is shown here: using the calculated aggregated subjective ifn values, the values of degree of indeterminacy is calculated using eq. (14). using the above calculated ifns in eq. (13), subjective weights calculation for the risk factors o is given below (other subjective weights are calculated in same manner). all the subjective weights are given in table 8. using table 5, appropriate objective weights were selected according to the three experts as 0.234, 0.514, and 0.234. as per eq. (15), reference series under o, s, and d, for the minimum ( ) ( ) ( )  ( ) ( ) ( ) 0.4 0.3 0.25 0.4 0.3 0.25 1 1 1 0.1 1 0.25 1 0.5 , 0.85 0.7 0.5 {0.2605,0.7080} k = − −  −  −   = 1 0.2605 0.2605 0.0315 0.2605 0.7080 0.2605 0.8143 0.2605 0.0315 0.8143 0.0612 0.2605 0.7080 0.8143 0.1245 0.7713 0.7713 0.0648 0.1538 0.7713 0.1639    +   +  =     +  + +     + +      + +  =  +  a modified fmea approach based integrated decision framework for overcoming the problems… 471 series value, were established for the set of listed failure causes under different subsystem using ifd operator and aggregated ifns values (table 7). a sample calculation for the reference series of turbine rotor, rt1 (first failure cause) under o is computed as: similarly, the reference series under other risk factors and other listed failure causes were calculated, but are not shown here, and the reference series matrix was generated for the set of failure causes of the turbine rotor rt as: reference series matrices for other set of failure causes of tau components are tabulated in the same manner. lastly, ifhwed scores values were calculated for each listed failure causes, using the reference series values and taking the restriction constant value ϕ = 0.6 for subjective weights (table 8), and the objective weights (table 5) in eq. (16). for illustration, sample calculation for tabulating the ifhwed score for rt1 is given as: similarly, ifhwed scores (di) for all other listed failure causes were calculated, and ranking has been done against ifhwed scores, as shown in table 9. 5.2. if topsis based risk results using the aggregated ifns values (as tabulated under if-fmea) for the set of listed failure causes, reference series were established using eqs. (17) and (18) for both the minimum reference series value and the maximum reference series value. the reference series for the rotor turbine rt1 under o is shown below. reference series for minimum value under o reference series for maximum value under o other reference series under s and d were calculated in the same manner. similarly, reference series were established for other listed failure causes, of the considered unit, for both the minimum and maximum values. using the reference values for both the minimum and maximum values in eqs. (19) and (20), for all listed failure causes, if-pis and if-nis values were calculated taking the same value of restriction constant as that of if-fmea. ( )    1 0.4763,0.4775 ( 0.4763 0 ) ( 0.4775 1) 0.7534 2oifdd = − + − = 1 1 2 3 0.7534 1.1412 1.0866 0.7410 1.0900 0.4884 0.6949 1.0900 0.6712 rt rt r rt rt −         = =           1 2 2 2 2 2 2 0.6 0.1538 0.7534 0.4320 1.1412 0.4142 1.0866 (1 0.6) 0.243 0.7534 0.514 1.1412 0.243 1.0866 1.0588 rtd =  +  +  + −  +  +  = )1,0(~ = − r ( )    1 0.4763,0.4775 ( 0.4763 0 ) ( 0.4775 1) 0.7534 2oifdd = − + − = )0,1(~ = + r ( )    1 0.4763,0.4775 ( 0.4763 1) ( 0.4775 0 ) 0.7466 2oifdd = − + − = 472 d. k. kushwaha, d. panchal, a. sachdeva taking the same value of restriction constant as that of if-fmea in eq. (21), the relative coefficient score for all failure causes were calculated and are shown in table 9. table 9 failure causes, ifhwed score, iffmea based rank, if-nis, if-pis, relative coefficient and if-topsis based rank failure causes iifhwed score if-fmea based rank if-nis if-pis relative coefficient if-topsis based rank rt1 1.0588 1 0.2121 0.4736 0.6910 1 rt2 0.8590 3 0.5851 0.7346 0.5390 3 rt3 0.8940 2 0.4568 0.6619 0.5746 2 sv4 0.8953 2 0.3950 0.6324 0.5860 2 sv5 0.9030 1 0.3526 0.6042 0.5991 1 sv6 0.8799 3 0.4227 0.6403 0.5788 3 tj7 1.0811 1 0.1986 0.4662 0.6987 1 tj8 0.9908 2 0.2705 0.5383 0.6480 2 tj9 0.8829 3 0.4611 0.7150 0.5525 3 tj10 0.8228 4 0.6224 0.7648 0.5183 4 ap11 1.0439 1 0.2089 0.4704 0.6893 1 ap12 0.8549 3 0.6158 0.7514 0.5322 3 ap13 0.8799 2 0.4227 0.6403 0.5788 2 es14 0.9794 1 0.3298 0.5895 0.6243 1 es15 0.8529 3 0.4560 0.6769 0.5575 3 es16 0.8574 2 0.4469 0.6686 0.5619 2 mp17 0.8836 1 0.5455 0.7171 0.5520 3 mp18 0.8684 2 0.5043 0.6873 0.5582 2 mp19 0.8611 3 0.4410 0.6676 0.5633 1 ot20 0.8741 3 0.4233 0.6780 0.5632 3 ot21 0.8871 2 0.4254 0.6798 0.5661 2 ot22 1.0128 1 0.2417 0.5076 0.6661 1 ov23 0.8654 4 0.5122 0.7368 0.5401 4 ov24 1.0184 2 0.2339 0.4819 0.6788 1 ov25 0.8953 3 0.3950 0.6324 0.5860 3 ov26 1.0338 1 0.2354 0.5009 0.6736 2 pp27 0.9883 5 0.2881 0.5607 0.6380 5 pp28 1.0661 2 0.2002 0.4617 0.6978 2 pp29 1.0490 4 0.2189 0.4866 0.6831 4 pp30 1.1523 1 0.1193 0.3537 0.7652 1 pp31 1.0568 3 0.2133 0.4804 0.6875 3 5.3. comparative analysis results table 9 shows rt1, sv5, tj7, ap11, es14, mp17, ot22, ov26 and pp30 categorised under turbine rotor, solenoid valve, thrust and journal bearing, auxiliary oil pump, emergency stop valve, main oil pump and throttle valve, alternator rotor, oil tank and valves, and pump with their corresponding ifhwed output scores: 1.0588, 0.9030, 1.0811, 1.0439, 0.9794, 0.8836, 1.0128, 1.0338 and 1.1523 were found to be the most critical failure causes with rank first. when if-topsis approach is implemented within the proposed if-fmea, the failure causes found to be the most critical ones are: rt1, sv5, tj7, ap11, es14, mp19 ot22, ov24 and pp30. they were ranked as first with the magnitude of relative coefficient as: 0.6910, 0.5991, 0.6987, 0.6893, 0.6243, 0.5633, 0.6661 0.6788, 0.7652, respectively. it has been found that four failure causes, mp17 mp19, ov24 and ov26, were ranked differently with the if-topsis approach. a modified fmea approach based integrated decision framework for overcoming the problems… 473 high degree of compatibility among both the approaches has been observed as majority of the failure causes were ranked similar. the similarity in ranking results prove the robustness of the proposed novel if-fmea approach based ranking results. the most critical component as identified with the proposed model will be useful for developing the accurate maintenance policy for the considered unit, and prove to be useful in avoiding the effect of reactive agents during the system operation and, hence, contribute to the system’s operational safety. 5.4. sensitivity analysis-based simulation results if-topsis approach has been chosen to check the stability of the ranking derived against the relative coefficient. as the value of restriction constant, ϕ, has been selected in eqs. (19) and (20) to find the value of if-nis and if-pis, and hence the relative coefficient, so, it becomes necessary to conduct the sensitivity analysis. eleven simulations for each failure cause were run for ϕ =0-1 with an increment of 0.1, to check the variation in the relative coefficient values, and, hence, change in ranking of identified failure causes. when the simulation is run for a set of failure causes of turbine rotor such as rt1, rt2 and rt3, no change in ranking results have been observed with the change in the ϕ values, as shown in table 10. table 10 ranking of failure causes at different values of restriction constant, ϕ  values 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 failure causes ranks rt1 1 1 1 1 1 1 1 1 1 1 1 rt2 3 3 3 3 3 3 3 3 3 3 3 rt3 2 2 2 2 2 2 2 2 2 2 2 sv4 3 3 3 2 2 2 2 2 2 2 2 sv5 1 1 1 1 1 1 1 1 1 1 1 sv6 2 2 2 3 3 3 3 3 3 3 3 tj7 1 1 1 1 1 1 1 1 1 1 1 tj8 2 2 2 2 2 2 2 2 2 2 2 tj9 4 4 4 3 3 3 3 3 3 3 3 tj10 3 3 3 4 4 4 4 4 4 4 4 ap11 1 1 1 1 1 1 1 1 1 1 1 ap12 3 3 3 3 3 3 3 3 3 3 3 ap13 2 2 2 2 2 2 2 2 2 2 2 es14 1 1 1 1 1 1 1 1 1 1 1 es15 3 3 3 3 3 3 3 3 3 3 3 es16 2 2 2 2 2 2 2 2 2 2 2 mp17 2 2 2 3 3 3 3 3 3 3 3 mp18 1 1 1 1 1 2 2 2 2 2 2 mp19 3 3 3 2 2 1 1 1 1 1 1 ot20 3 3 3 3 3 3 3 3 3 3 3 ot21 2 2 2 2 2 2 2 2 2 2 2 ot22 1 1 1 1 1 1 1 1 1 1 1 ov23 4 4 4 4 4 4 4 4 4 4 4 ov24 1 1 1 1 1 1 1 1 2 2 2 ov25 3 3 3 3 3 3 3 3 3 3 3 ov26 2 2 2 2 2 2 2 2 1 1 1 pp27 5 5 5 5 5 5 5 5 5 5 5 pp28 2 2 2 2 2 2 2 2 2 2 2 pp29 4 4 4 4 4 4 4 4 4 4 4 pp30 1 1 1 1 1 1 1 1 1 1 1 pp31 3 3 3 3 3 3 3 3 3 3 3 474 d. k. kushwaha, d. panchal, a. sachdeva the first change in the rank is noted for the set of failure causes under solenoid valve, notated as sv4 and sv6, while the rank of sv5 remains unchanged for the entire simulation. the rank of sv4 changes from 3rd to 2nd when ϕ value changes from 0.2 to 0.3, and the rank of sv6 changes from 2nd to 3rd for the similar change in value of ϕ as that of sv4. also, the change in ranking is observed for tj9 from 4th to 3rd when the value of ϕ is changed from 0.2 to 0.3. the failure cause tj10 change in rank is seen from 3rd to 4th for the same change of ϕ. likewise, for the listed failure causes under other subsystems, namely thrust and journal bearing (tj), auxiliary oil pump (ap), emergency stop valve (es), main oil pump and throttle valve (mp), alternator rotor (ot), oil tank and valves (ov) and pumps (pp), the ranking results were analysed with the change in ϕ values represented graphically in figs. 3-11. it has also been observed that, out of total 31 listed failure causes, only 9 failure causes show a change in ranking, while the remaining ones show similar ranking results with the change in the ϕ value. hence, 29.03% of the failure causes illustrate change in the ranking result for the entire run of simulations, whereas 70.97% show similar rank for failure causes when simulation was run for different values of ϕ. the effect of the restriction constant value on the ranking trend of the listed failure causes under each subsystem are represented graphically in figs. 3-11. fig. 3 sensitivity analysis: rotor fig. 4 sensitivity analysis: solenoid valve fig. 5 sensitivity analysis: thrust and fig. 6 sensitivity analysis: auxiliary oil and journal bearing pump 1 1 1 1 1 11 1 1 1 1 3 3 3 3 3 33 3 3 3 3 2 2 2 2 2 22 2 2 2 2 rt1 rt2 rt3 3 3 3 2 2 22 2 2 2 2 1 1 1 1 1 11 1 1 1 1 2 2 2 3 3 33 3 3 3 3 sv4 sv5 sv6 1 1 1 1 1 11 1 1 1 1 2 2 2 2 2 22 2 2 2 2 4 4 4 3 3 33 3 3 3 3 3 3 3 4 4 44 4 4 4 4 tj7 tj8 tj9 tj10 1 1 1 1 1 11 1 1 1 1 3 3 3 3 3 33 3 3 3 3 2 2 2 2 2 22 2 2 2 2 ap11 ap12 a modified fmea approach based integrated decision framework for overcoming the problems… 475 fig. 7 sensitivity analysis: emergency stop fig. 8 sensitivity analysis: main oil valve pump fig. 9 sensitivity analysis: alternator rotor fig. 10 sensitivity analysis: oil tank and valves fig. 11 sensitivity analysis: pump 5 5 5 5 5 55 5 5 5 5 2 2 2 2 2 22 2 2 2 2 4 4 4 4 4 44 4 4 4 4 1 1 1 1 1111 1 1 1 3 3 3 3 3 33 3 3 3 3 pp27 pp28 pp29 pp30 pp31 476 d. k. kushwaha, d. panchal, a. sachdeva 6. conclusions a novel if-fmea approach based integrated framework was proposed to study and analyse the operational risk associated with tau in sugar mill industry, and, therewith, to minimize or eliminate the problem of plant’s unavailability and accidental hazards. the iffmea approach has been applied to investigate the critical failure causes, and ranking as per the ifhwed score value was done. the proposed model overcomes the limitations of fmea approach in an effective decision making of critical components. the if-topsis approach has also been implemented within the novel if-fmea approach to check the stability of the ranking results on comparative basis. from the comparative analysis, it has also been found that most of the ranking results of failure causes, as calculated, were found the same with the implementation of both the approaches, with exception of failure causes such as mp17, mp19, ov24, ov26. the highest ranked failure causes are often responsible for creating the problem of corrosion inside different parts of the tau system. those are responsible for sudden failure and accidental hazards. hence, utmost remedial actions during maintenance of the unit should be implemented as per the criticality of the listed failure causes, for example, the most critical failure cause under the rotor turbine is scaling of rotor blade and thus, parameters which form scaling should be monitored closely to avoid scaling. moreover, all the parameters which are responsible for the critical failure cause should be kept within permissible limit, and are closely monitored by operators of tau system to avoid failure to enhance the availability of the unit. the sensitivity analysis results also show less change in ranking of failure causes with the change in ϕ values under the if-topsis approach, which indicates robustness of the proposed integrated framework methodology. the accuracy of 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h.d., naithani, a., 2022, significance of topsis approach to madm in computing exponential divergence measures for pythagorean fuzzy sets, decision making: applications in management and engineering, 5(1), pp. 246-263. 43. tutak, m., brodny, j., 2022, evaluating differences in the level of working conditions between the european union member states using topsis method, decision making: applications in management and engineering, 5(2), pp. 1-29. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 445 454 https://doi.org/10.22190/fume190510040s © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper surface morphology and microhardness behavior of 316l in hap-pmedm gurpreet singh 1 , yubraj lamichhane 2 , amandeep singh bhui 2 , sarabjeet singh sidhu 1 , preetkanwal singh bains 1 , prabin mukhiya 2 1mechanical engineering dept., beant college of engineering and technology, india 2mechanical engineering dept., amritsar college of engineering and technology, india abstract. the development of biomaterials for implants nowadays requires materials with superior mechanical and physical properties for enhanced osseointegration and sustained longevity. this research work was conducted to investigate the influence of nano hydroxyapatite (hap) powder mixed electrical discharge machining (pmedm) on surface morphology and microhardness of modified 316l stainless steel surface. the chosen process parameters were discharge current, pulse on/off duration and gap voltage in order to analyze the selected output responses. hap concentration (15 g/l) along with reverse polarity was kept constant for current experimentation. the experimental results testified that surface morphology of pmedm surface was significantly improved along with augmentation of 79% in microhardness (hv) of hap modified surface of medical grade stainless steel. furthermore, xrd and sem characterization confirmed the deposition of calcium, phosphorous and inter-metallic compounds on ha-pmedmed surface. the surface thus produced is expected to facilitate better bone-implant adhesion and bioactivity. key words: pmedm, 316l ss, copper tool, hap powder, microhardness, bioactivity 1. introduction biomaterials are artificial organs or implants preferably from the family of ceramics, polymeric and metallic biomaterials (titanium alloys, stainless steel 316l and cr-co alloys) for the substitution of the damaged human body organ. prominent properties such as cell proliferation, bone-implant adhesion, osseointegration, corrosion and wear behavior in fluidic environment within the individual primarily affect the acceptance or rejection of the selected biomaterials [1, 2]. the demand for orthopedic implants is rising day by day; fascinating the researchers and engineers to improve its surface integrity alongside offering better resistance towards wear and corrosion [3-6]. as a result, implant surface must be modified with received may 10, 2019 / accepted august 08, 2019 corresponding author: gurpreet singh department of mechanical engineering, beant college of engineering & technology, gurdaspur – 143521, punjab, india e-mail: singh.gurpreet191@gmail.com 446 g. singh, y. lamichhane, a.s. bhui, s.s. sidhu, p.s. bains, p. mukhiya bioactive coating, avoiding the release of harmful ions causing corrosion and loosening of joint due to wear [7, 8]. among all the established coating techniques, such as sol-gel, dip coating, ion implantation, pvd, cvd, laser coating, etc., the execution of edm as an alternative to other surface modification methods is still in its beginning era [9]. in this method, bioactive powder is mixed in dielectric medium keeping workpiece as cathode (-) and tool electrode as anode (+) or simply negative polarity [10]. during the process, the produced thermal energy heated up the working area and with the flow of charge from positive to negative, the powder particles strike the surface and solidify in pulse-off duration and thus modify the substrate surface [11]. different experimentations and techniques were explored by researchers in order to modify the surface characteristics of metallic biomaterials. hubler et al. [12] studied wear and corrosion resistance of the 316l ss femoral implants by depositing the ceramics thin films of the transition metals nitrides and ti/n and cr/v layers. it was found that the coatings significantly improved the surface integrity, microhardness and corrosion resistance of the surface. kumar et al. [13] investigated the electrochemical behavior and in-vitro bioactivity of polypyrrole/tio2 ceramic nano-composites on 316l stainless steel. it was found that addition of tio2 exhibits improved corrosion resistance properties of the substrate and offered better biocompatibility. microhardness and in-vitro wear behavior of tio2 treated 316l stainless steel was examined by singh et al. [14] via electro-discharge treatment. they found that material transfer rate of edm is appropriate for the surface modification of bare metal. addition of tio2 exhibits improved microhardness of 233% and protective efficiency of 80% compared to the substrate material. chang et al. [15] inspected microhardness, corrosion resistance and protein adsorption properties of cualo2 deposition on 316l stainless steel. with the use of nano-indentation, it was found that corrosion resistance and microhardness were significantly enhanced with an increment of 46% after the process. optimal ed machining parameters were investigated by bhui et al. [16] for the surface modification of ti6al4v with graphite tool. deposition of bioactive layer and formation of intermetallic compounds were examined using sem and xrd respectively. additionally, apatite growth was observed on machined sample confirming the bioactivity through sem and eds analysis. harun et al. [17] in their study deeply reviewed the application of hydroxyapatite coatings for metallic biomaterials. it was found that hydroxyapatite powder dominantly improves the surface adhesion strength, biocompatibility and corrosion resistance of the metallic biomaterials [18-20]. based on the literature survey and previous studies, it was seen that the use of edm for modifying the biomaterials surface is still in its early stages [21, 22]. as there is no need for vacuum chamber or any special arrangement, the use of ed machining for the surface modification is a better choice when compared to other pricy techniques [23-25]. in the present study, stainless steel 316l; a well-known dominating biomaterial in the field of orthopaedics and joint replacement was investigated for modified surface integrities and microhardness in hydroxyapatite (hap) powder mixed edm with copper tool. 2. experimental work 2.1. materials metallic biomaterial 316l stainless steel was procured as workpiece material for the current experimentation having thermal conductivity 21.5 w/m.k at 500 °c; melting point 1371-1399 °c; and density 7.99 g/cc. tables 1 and 2 show the elemental composition of 316l ss and hydroxyapatite powder, respectively. a pure copper circular electrode (of 10mm surface morphology and microhardness behavior of 316l in hap-pmedm 447 in diameter) was used to machine the workpiece, whereas the hap was in nano-size with true density 3.219 g/cm3 and average particle size of 20-45 nm. table 1 chemical composition of 316l ss element si c mn n p cr ni mo s fe % 0.41 0.01 1.07 0.10 0.02 16.13 10.15 2.05 0.01 balance table 2 composition of hydroxyapatite powder element ca5(oh)(po4)3 al2o3 fe2o3 % >99.5 <0.06 <0.02 2.2. method design of experiments (doe) was generated according to taguchi’s l18 orthogonal array employing mixed level design (2134) with the help of minitab-17. five machining parameters, i.e. dielectric medium (dm), current (ip), pulse-on time (p-on), pulse-off time (poff) and voltage (v), were chosen to vary at three levels (table 3) for the output responses. based on the levels of input parameters and experimental design, each experiment was performed on two different plates and an average of both is plotted for the result analysis. machining time of 30 minutes and reverse polarity were kept constant throughout the experimentation. the following table 4 illustrates the experimental design based on l18 orthogonal array. table 3 input machining parameters input parameters (symbol) units level 1 (low) level 2 (medium) level 3 (high) dielectric medium (dm) hydrocarbon oil hydrocarbon oil + hap current (ip) a 20 24 28 pulse-on time (p-on) µs 60 90 120 pulse-off time (p-off) µs 60 90 120 voltage(v) v 40 60 80 table 4 experimental design according on l18 orthogonal array exp. run dielectric medium (dm) current (ip) pulse-on time (p-on) pulse-off time (p-off) voltage (v) 1 hydrocarbon oil 20 60 60 40 2 hydrocarbon oil 20 90 90 60 3 hydrocarbon oil 20 120 120 80 4 hydrocarbon oil 24 60 60 60 5 hydrocarbon oil 24 90 90 80 6 hydrocarbon oil 24 120 120 40 7 hydrocarbon oil 28 60 90 40 8 hydrocarbon oil 28 90 120 60 9 hydrocarbon oil 28 120 60 80 10 hydrocarbon oil + hap 20 60 120 80 11 hydrocarbon oil + hap 20 90 60 40 12 hydrocarbon oil + hap 20 120 90 60 13 hydrocarbon oil + hap 24 60 90 80 14 hydrocarbon oil + hap 24 90 120 40 15 hydrocarbon oil + hap 24 120 60 60 16 hydrocarbon oil + hap 28 60 120 60 17 hydrocarbon oil + hap 28 90 60 80 18 hydrocarbon oil + hap 28 120 90 40 448 g. singh, y. lamichhane, a.s. bhui, s.s. sidhu, p.s. bains, p. mukhiya 2.3. experimentation out of total 18 experimental runs, nine were performed in pure medium i.e. hydrocarbon oil in znc-edm (oscarmax, s645) dielectric tank itself whereas the following powder mixed trials were conducted in an in-house fabricated dielectric tank of capacity 12 liters. hap was mixed at 15 g/l to the hydrocarbon oil and continuously circulated using a stirrer and pump to avoid the settling down of powder particles as shown in fig. 1. fig. 1 (a) schematic experimental setup and (b) indigenously developed dielectric tank 2.4. investigation of machined samples as it is evident from the previous studies [26-28] that biomaterial surface must be porous and must possess bioactive compounds to portray the bioactivity within the individual, the machined samples were investigated for porous microstructure, powder deposition and formation of new compounds using sem and xrd analysis respectively. furthermore, mitutoyo microhardness tester (fig. 2) with diamond indenter was used to scrutinize the improved hardness of the ed machined specimens. a load of 0.98 n was fig. 2 microhardness testing of ed machined samples surface morphology and microhardness behavior of 316l in hap-pmedm 449 applied for a dwell time of 10 seconds to profile a pyramidal imprint on the specimen. three readings were taken on each machined sample for both the plates and showed in table 5 (rep 1 for mean of plate 1 and rep 2 for mean of plate 2). prior to measurement, the microhardness of substrate was computed at three different points and an average value of 291.80 hv was noted. 3. results and discussion based upon the experimentation performed, the following table 5 demonstrates the output response values and s/n ratios for both the workpiece plates. further, the output responses were statistically analyzed through anova to evaluate the percentage contribution of each input parameter and subsequently their rank. table 5 response table for edmed 316l stainless steel exp. run mrr (mg/min.) s/n ratio (mrr) microhardness (mh) s/n ratio (mh) rep 1 rep 2 rep 1 rep 2 1 2.81 2.53 8.4944 386.5 419.9 52.0881 2 4.65 4.01 12.6585 228.2 342.4 48.5804 3 4.84 6.50 14.7914 319.1 406.3 51.0022 4 10.02 10.63 20.2664 347.8 372.7 51.1165 5 5.21 6.12 14.9799 538.9 369.5 52.6890 6 6.32 5.68 15.5259 515.9 612.2 54.9313 7 11.50 11.61 21.2551 475.2 453.6 53.3308 8 13.16 8.84 20.3220 374.3 404.2 51.7853 9 6.44 8.21 17.1055 459.7 581.8 54.1531 10 6.23 6.42 16.0183 819.4 758.7 57.9228 11 4.15 7.19 14.1225 557.8 665.1 55.6268 12 4.36 4.97 13.3213 859.4 770.9 58.1863 13 10.64 8.98 19.7400 636.2 578.3 55.6377 14 10.86 7.13 18.5156 663.1 567.7 55.7048 15 5.31 4.03 13.1408 904.3 796.9 58.5425 16 17.49 20.52 25.4945 904.6 769.8 58.3720 17 6.98 5.67 15.8812 779.5 823.2 58.0668 18 11.80 12.86 21.7952 958.3 796.9 58.7556 (rep 1 and rep 2: repetitions of experimentations on two separate plates) digital weighing machine (wensar, model: pgb 200) having least count of 0.001g was utilized for measuring the change in workpiece weight after each experimental run for evaluating the material removal rate of 316l ss using the following equation: initial weight -final weight mrr= ×1000mg/min. machiningtime (1) the hardness of biomaterial plays a key role during the cyclic loading on the implanted part particularly in the case of knee and hip joint. for that reason, the signal-to-noise (s/n ratios) analysis for microhardness as well as mrr was calculated according to equation (2) using taguchi’s criteria for larger-is-better for the current experimentation. 450 g. singh, y. lamichhane, a.s. bhui, s.s. sidhu, p.s. bains, p. mukhiya 2 1 1 1 10log r ilb i s n r y                 (2) where r is the repetition of responses and yi the value of response at ith trial. 3.1. analysis of material removal rate evaluation of mrr is a primary output response during the ed machining of the workpiece material. minitab-17 was used to analyze the output values from table 5 for both the workpiece plates in terms of signal-to-noise ratios and percentage contribution of input parameters. fig. 3 and table 6 illustrate the main effects plot for s/n ratios and analysis of variance (anova) for mrr of current experimentation. superior material removal rate (19.01 mg/min.) was witnessed at higher value of peak current (28a) and pulse-on-time (120µs). current depicts the highest percentage contribution of 52.66% followed by pulse-off (14.28%) and pulse-on (8.99%). based on the responses, it is discovered that with an increase in the current intensity, the rate of material removal is sharply augmented and similar results can be observed from the s/n ratios plot. table 6 analysis of variance for s/n ratios of mrr source df seq ss adj ms f-value p-value % contribution dielectric medium (dm) 1 8.862 8.862 1.27 0.292 3.10 ip (a) 2 150.409 75.205 10.81 0.005* 52.66 p-on (µs) 2 25.686 12.843 1.85 0.219 8.99 p-off (µs) 2 40.784 20.392 2.93 0.111 14.28 voltage 2 4.241 2.120 0.30 0.746 1.48 residual error 8 55.673 6.959 19.49 total 17 285.656 100.00 *most significant at 95% confidence level; rank 1: current; rank 2: pulse-off; rank 3: pulse-on fig. 3 s/n ratios plot for material removal rate surface morphology and microhardness behavior of 316l in hap-pmedm 451 3.2. analysis of microhardness analysis of variance was performed to check the dominance of hydroxyapatite powder and other chosen parameters; associated results for microhardness of edmed 316l stainless steel surface are shown in table 7. superior microhardness of 877.60 hv is illustrated at the utmost values of current intensity (28a) and pulse-on-time (120µs) in the presence of hap mixed dielectric (trial 18) with an increment of 79% and 160% comparative to the samples machined in hydrocarbon oil and substrate material, respectively. at a higher value of discharge current and pulse-on, the spark generation between tool and workpiece acts more rapidly permitting the deposition of ha powder mixed in the dielectric medium. the breakdown of electrolyte (hydrocarbon oil) also formed intermetallic compounds reacting with substrate elements and facilitates improved hardness. similar results can be observed from fig. 4 and table 7, dielectric medium (% contribution: 74.06%) portray as the most prominent factor directly influencing the microhardness of ed machined 316l ss surface. table 7 analysis of variance for s/n ratios of microhardness source df seq ss adj ms f-value p-value % contribution dielectric medium (dm) 1 123.447 123.447 59.38 0.000* 74.06 ip (a) 2 10.199 5.100 2.45 0.148 6.12 p-on (µs) 2 14.373 7.186 3.46 0.083 8.63 p-off (µs) 2 0.683 0.341 0.16 0.851 0.41 voltage 2 1.341 0.670 0.32 0.733 0.80 residual error 8 16.633 2.079 9.98 total 17 166.675 100.00 *most significant at 95% confidence level; rank 1: dielectric; rank 2: pulse-on; rank 3: current fig. 4 s/n ratios plot for microhardness 452 g. singh, y. lamichhane, a.s. bhui, s.s. sidhu, p.s. bains, p. mukhiya 3.3. surface evaluation of ha-pmedmed 316l stainless steel the machined surface with maximum value of microhardness (trial 18) was further examined for porous structure and deposition of powder particles through scanning electron microscopy. fig. 6 (b) showed the microstructure of ha powder mixed dielectric depicting porosity in conjunction with surface modification of the substrate surface and deposition of powder particles. apart from this, sample exhibiting maximum hardness (trial 8) in pure dielectric medium was also examined using sem (fig. 6a) and illustrates the cracks, craters on its surface. the modified 316l stainless steel surface in ha powder mixed edm was then analyzed for the changed elemental composition using xrd technique. fig. 5 demonstrating the xrd pattern with the existence of various bioactive (calcium, phosphorus, calcium carbonate) and intermetallic (manganese silicide, chromium carbide, manganese silicide carbide) compounds on the ha-pmedmed surface. as a result, modified surface not only restrict the fluidic reactions but also promotes the bioactivity offering better cell proliferation, biological fixation, etc. [29, 30]. fig. 5 xrd pattern for ha-pmedmed surface (trial 18) fig. 6 sem for maximum microhardness (a) machined in pure dielectric (trial 8); (b) porous surface with white powder layer (trial 18) (b) (a) surface morphology and microhardness 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david liović1, marina franulović1, luka ferlič2, nenad gubeljak2 1university of rijeka, faculty of engineering, croatia 2university of maribor, faculty of mechanical engineering, slovenia orcid ids: david liović https://orcid.org/0000-0002-3608-0052 marina franulović https://orcid.org/0000-0002-6148-9001 luka ferlič https://orcid.org/0000-0002-3985-4711 nenad gubeljak https://orcid.org/0000-0002-3276-8431 abstract. controlling the surface roughness of materials manufactured by laser powder bed fusion (l-pbf) is critical for achieving functional performance of components and improving their mechanical properties. this is important for components whose surfaces cannot be post-treated using subtractive methods. in this study, the surface roughness has been investigated by applying different laser power and scanning speed combinations. furthermore, potential effects of different locations on the build platform have been considered as well. the regression models have been developed using significant predictor variables, with their levels defined using face-centered central composite design. the analysis of variance (anova) procedure has been used to evaluate the statistical significance of factors and model performances for each prediction variable. it has been found that the average surface roughness of l-pbf ti6al4v alloy can be described with high fitting accuracy using laser power and scanning speed as predictor variables. the position of specimens on the build platform showed no statistically significant effect on the average surface roughness. the experimental research and statistical analysis reported in this paper will contribute to a better understanding of how position, laser power, and scanning speed influence the average surface roughness of l-pbf ti6al4v alloy. key words: ti6al4v alloy, l-pbf process parameters, surface roughness, model 1. introduction the possibility of achieving targeted surface roughness of components is of high interest in the scientific community and industry. it is well known that the surface roughness of materials manufactured using laser powder bed fusion (l-pbf) can be influenced by many factors involved. therefore, the surface roughness analysis of l-pbf ti6al4v alloy has already been the subject of many publications [1–9]. however, the models that relate l-pbf process received: july 19, 2023 / accepted september 16, 2023 corresponding author: marina franulović university of rijeka, faculty of engineering, vukovarska 58, rijeka 51000, croatia e-mail: marina.franulovic@riteh.hr https://orcid.org/0000-0002-3608-0052 https://orcid.org/0000-0002-6148-9001 https://orcid.org/0000-0002-3985-4711 https://orcid.org/0000-0002-3276-8431 mailto:marina.franulovic@riteh.hr 64 d. liović, m. franulović, l. ferlič, n. gubeljak parameters and surface roughness are rarely available, especially when titanium and its alloys are considered. surface roughness and product quality are influenced by numerous interacting factors in l-pbf process and of particular significance are: laser power, scanning speed, orientation and position of components, applied heat treatment, as well as granulation, shape and distribution of powder stock [1, 5, 10, 11]. vaglio [2] has found that laser power and scanning speed have a stronger effect on the average surface roughness of the side surfaces, than the hatch distance. the orientation of components on the build platform highly affects surface roughness [8] and residual stresses [12], which directly reflects on the product quality. the staircase effect, which occurs due to layer-wise manufacturing is related to the surface inclination angle, highly affect the surface roughness [13–15]. the granulation, shape and distribution of the used metal powder stock have an impact on the flowability, which affects the l-pbf recoating process [8, 16, 17]. these numerous influencing factors limit the understanding of how physical processes during l-pbf affect the surface roughness. in addition, the surface roughness of topologically simple additively manufactured components can be reduced by subsequent surface treatments. however, additive manufacturing technologies are often used in the production of topologically complex products [18–21]. hence, subtractive post-treatments such as milling, turning and grinding, are not applicable in that case [22]. moreover, the processing of titanium alloys is challenging [23–25]. therefore, it is necessary to gain as much information on the influence of l-pbf process parameters on the surface roughness. to investigate and evaluate the effect of the influential factors on the surface roughness, it is necessary to have reliable and accurate measurements. however, the surface roughness measurements of metallic materials manufactured using l-pbf are challenging. more specifically, contact measurements are often performed using profilometers whose stylus scratches the surface and thus record the surface profile [26]. considering that metals manufactured using lpbf technologies are covered with surface-bonded particles, the movement of the stylus may be hindered due to jamming or undesired separation of loose powder particles [3]. in that case, the stylus is not able to reach the lowest and narrow surface gaps due to tip radius, and thus it is not able to completely record the surface profile, which adversely affects the measurement results. to overcome these restrictions the non-contact measurements are preferable, as they are not sensitive to contact problems. however, they can be influenced by surface reflectivity or by insufficient magnification [22, 26]. despite significant advancements in customization of surface roughness of the l-pbf ti6al4v alloy have been made in recent years, the knowledge in this field remains incomplete. the studies that provide regression models for the laser power – scanning speed effect on the surface roughness and provide information on the influence of position on surface roughness are rarely available. therefore, further research is needed to extend the existing knowledge on the possibility of influencing the surface roughness by changing the position at the build platform, or by applying different laser power and scanning speed combinations. detailed experimental procedures are going to be carried out to investigate the surface roughness of ti6al4v alloy produced at different positions at the built platform by applying different laser powers and scanning speeds. for that purpose, nonlinear response surface models will be developed using the face-centered central composite design. all experimental results and nonlinear regression models will be statistically analyzed and evaluated to obtain valid results and interpretations. surface roughness of ti6al4v alloy produced by laser powder bed fusion 65 2. materials and methods the manufacturing conditions have been selected according to previously reported findings described within this chapter. the specimens have been oriented vertically on the build platform during manufacturing, followed by annealing heat treatment. in this way, the warping caused by residual stresses has been minimized. moreover, the vertical orientation eliminates the staircase effect [27], allowing investigation of the position, laser power, and scanning speed on average surface roughness. the chosen thickness of the powder layer has been set to 25 μm, to minimize the porosity as much as possible and to achieve high dimensional accuracy. the laser power and scanning speed levels have been set in the range of 200–250 w and 1000–1500 mm/s, respectively. surface roughness measurements have been carried out using contact and non-contact methods to compare and evaluate the differences between results obtained using these two methods. in order to contribute to the accuracy of the results and their repeatability, applied design of experiments, utilized powder stock, l-pbf process and its parameters, specimen dimensions and surface roughness measurements are detailed. 2.1. design of experiments and model development face-centered central composite design (fc-ccd) of experiments has been used to define the laser power and scanning speed levels utilized in the l-pbf process. this design has star points at the center of each face of the factorial space [28]. hence, it requires three levels for each predictor variable as shown in table 1. central composite design has proven to be effective for modeling the influence of process parameters of many manufacturing technologies [29–32]. in addition, specimens have been divided into four groups representing their positions on the build platform (pos 1, pos 2, pos 3 and pos 4). this experimental design provides a reasonable balance between the number of experiments required and the information obtained. table 1 parameters and their levels used in design of experiments parameters symbol levels -1 0 1 laser power (w) p 200 225 250 scanning speed (mm/s) v 1000 1250 1500 for each prediction variable, response surface model coefficients have been found using ordinary least square (ols) method. the 95% confidence intervals have been determined for each prediction variable using the following equation [33]: 0 2 1 0 /2, ) 0| (( ) ( )ˆ .ˆ y x df errory x t  −=   0x x x x (1) where �̂�(𝑥0) is the estimated mean response at the grid point, 𝐱𝟎 is a vector containing grid points, x is a model matrix consisting of the levels of the independent variables expanded to model form, �̂�2 is the estimate of the error variance, and tα/2,df(error) is the t-value related to the desired level of confidence and the residuals’ degrees of freedom. since regression models are used for approximations of complex physical phenomena, there is a high possibility that more than one model will fit data with a suitable accuracy [34]. in general, models with higher complexity have better fitting performances. however, 66 d. liović, m. franulović, l. ferlič, n. gubeljak they also have a higher degree of interdependence between model parameters, when compared to more simple models [34]. therefore, it is justified to increase the number of parameters only in the case when the more complex model provides significantly better fit than the simpler model. in that context, two models will be given for each prediction variable, tested using multiple statistical tests, compared, and their fitting performance and complexity will be described in detail. the first model given for each variable has better fitting performance, but it also has a higher complexity. on the other hand, the second model has a lower complexity, which is desirable, but it has a lower fitting performance as well. using this approach, the influence of model parameters will be investigated, and significant model parameters will be identified for each prediction variable. this will result in four regression models (m1 – m4), which will be compared, evaluated, and discussed in detail. it is worth noting that the second proposed models (i.e., models with lower complexity), will be designed by eliminating the non-significant terms. 2.2. test specimens the tensile test specimens have been used for surface roughness measurements. a total of 36 test specimens (fig. 1a) have been used, ensuring 4 repetitions for each utilized l-pbf process parameter combination. specimen groups and their respective positions on the build platform are shown in figure 1b. the dimensions of tensile test specimens have been modified based on rectangular test specimen geometry stated in astm e8 / e8m 16a [35], by maintaining the recommended ratio of gauge length and width of 5:1. therefore, the gauge length and width have been 20 mm and 4 mm respectively, while the length of the reduced parallel section has been 22 mm, as can be seen in fig. 1a. the surface roughness measurements have been performed on the reduced parallel section (22 × 4 mm2) marked with a blue square. fig. 1 a) test specimen dimensions and highlighted region where surface roughness measurements have been performed, b) position, orientation and corresponding ids of test specimens 2.3. laser powder bed fusion and heat treatment the test specimens have been positioned vertically on the build platform, as can be seen in fig. 1b. the coordinates of each specimen placed on the build platform are given in surface roughness of ti6al4v alloy produced by laser powder bed fusion 67 table 2. all reported coordinates represent the position of specimen respect to the origin (mid-point) of the build platform. the specimens have been rotated around z-axis by 45° to ensure that the recoater does not encounter a straight wall while moving across all specimens, as can be seen in fig. 1b. table 2 coordinates of each specimen respect to the origin (mid-point) of the build platform position 1 position 2 a1 b1 c1 d1 e1 f1 g1 h1 i1 a2 b2 c2 d2 e2 f2 g2 h2 i2 x -70 -70 -70 -70 -70 -70 -70 -70 -70 -40 -40 -40 -40 -40 -40 -40 -40 -40 y 88 66 44 22 0 -22 -44 -66 -88 88 66 44 22 0 -22 -44 -66 -88 position 3 position 4 a3 b3 c3 d3 e3 f3 g3 h3 i3 a4 b4 c4 d4 e4 f4 g4 h4 i4 x -10 -10 -10 -10 -10 -10 -10 -10 -10 20 20 20 20 20 20 20 20 20 y 88 66 44 22 0 -22 -44 -66 -88 88 66 44 22 0 -22 -44 -66 -88 in this research the extra low interstitial ti6al4v (eli) grade 23 powder with spherical particles has been used (fig. 2a). to determine the particle size distribution of the powder, 530 measurements have been performed using keyence vhx 7000 microscope. diameters of powder sample particles have been in range from ~4 μm to ~48 μm (fig. 2b) with a median diameter of 27.06 μm. the 10th and 90th percentiles were 12.98 μm and 38.92 μm, respectively. fig. 2 utilized ti6al4v (eli) grade 23 powder: a) powder particles, b) size distribution of powder particles test specimens have been manufactured using concept laser m2 cusing machine equipped with a 400 w single-mode cw ytterbium-doped fibre laser. the l-pbf process has been performed under argon atmosphere to keep oxygen level below 0.2%. the l-pbf process parameters used in this study for manufacturing of test specimens are listed in table 3. each group of specimens is denoted using an id in the form of a capital letter in the range a – i, which represents a specific combination of utilised l-pbf process parameters. laser power and scanning speed levels have been defined using face-centred central composite design to investigate their effect on surface roughness. 68 d. liović, m. franulović, l. ferlič, n. gubeljak table 3 l-pbf process parameters utilized for l-pbf of ti6al4v alloy variation levels laser power (p), w 200 225 250 scanning speed (v), mm/s 1000 1250 1500 1000 1250 1500 1000 1250 1500 energy density (ed), j/mm3 88.9 71.1 59.3 100 80 66.7 111.1 88.9 74.1 linear energy density (el), j/mm 0.2 0.16 0.13 0.23 0.18 0.15 0.25 0.2 0.17 specimen group (id) a b c d e f g h i number of test specimens 4 4 4 4 4 4 4 4 4 constant l-pbf process parameters layer thickness (t) 0.025 mm hatch distance (h) 0.09 mm laser spot diameter (d) 0.1 mm scanning strategy bi-directional, single pass, 90° rotation of scan vector between layers laser power (p) and scanning speed (v) levels have been selected according to previous research findings on appropriate range of linear energy density (el = p/v) that ensures successful l-pbf of ti6al4v alloy [36]. as stated in [36], linear energy densities in 0.1 – 0.2 j/mm range provide the best candidates for successful production of bulk parts. therefore, the laser power and scanning speed levels have been carefully defined to ensure that the center point of the utilized fc-ccd remained within the specified range. the remaining points have been set either inside or close to the recommended interval. these process parameters ensure a stable l-pbf process, thus enabling development of reliable regression models. it is important to note that a wider range of parameters could result in completely different melting modes [36, 37], which would influence surface roughness values differently [9]. in such cases, employing a single regression model may not be appropriate, as different melting modes would have different effects on surface roughness. therefore, the laser power and scanning speed levels have been carefully selected to ensure consistent melting modes for each combination and to enable the manufacturability of bulk parts. the annealing heat treatment under argon atmosphere has been applied with a heating rate of 3.5 °c/min until the temperature reaches 840 °c. the specimens have been held at the temperature of 840 °c for 120 minutes. specimens have been cooled in the furnace under argon atmosphere down to 150 °c with corresponding cooling time from 800°c to 500 °c (δt8/5) equal to 290 min. the annealing heat treatment has been selected, since it is the most preferred heat treatment used for l-pbf ti6al4v alloy due to its effectiveness in residual stress relaxation and ductility increase [38–40]. the heat treatment parameters have been chosen based on the guidelines provided by the manufacturers of the l-pbf machine. 2.4. surface roughness measurements surface roughness generally refers to the deviations in surface height relative to a reference plane [41]. the average surface roughness (ra) measurements have been performed on test specimens using tesa rugosurf 10-g profilometer. prior to surface roughness measurements, all specimens have been cleaned in an ultrasonic bath for 10 minutes in distilled water to remove loose particles from the surface that may adversely affect the measurement reliability. measurements have been repeated 6 times along the gauge length of specimen’s surface (fig. 3a) to make results reliable. the profile filter has surface roughness of ti6al4v alloy produced by laser powder bed fusion 69 been selected according to iso 4288:1996 and iso 3274:1996 standards [42, 43]. therefore, traverse length (lt) has been set to 15 mm, evaluation length (ln) to 12.5 mm, cut-off (λc) and sampling length (lr) to 2.5 mm, and short-wave profile filter (λs) to 8 μm. the area average surface roughness (sa) measurement parameters have been obtained using keyence vhx 7000 microscope. the s-filter value has been set to 25 μm (~3×pixel size) when magnification of 1000× has been used. the l-filter value has been set to 2.5 mm according to the findings stated in [26, 44]. prior to the area average surface roughness (sa) measurements, sensitivity analysis has been performed by scanning the same area using different vertical pitch and magnification levels, as shown in fig. 3b. note that the error bars are omitted in fig. 3b, since the results represent the identical area, but scanned using different magnification and vertical pitch values. fig. 3 a) measurement locations and specimen orientation for sa and ra, b) the influence of the magnification and vertical pitch values on sa in general, higher magnifications generate more robust and reliable results since the quality of the details of the scanned surface is higher. however, in that case the microscope scanning time and data processing are extremely time consuming. therefore, a certain compromise between the appropriate measurement accuracy and time resources should be found. from the results shown in fig. 3b it is evident that the magnifications equal or higher than 500× give similar sa values which is in accordance with findings stated in [22]. the vertical pitch had the smallest influence on the results when a magnification of 1000× has been used. therefore, based on results from the sensitivity analysis, magnification of 1000× and vertical pitch of 1 μm have been selected for further measurements. the area surface roughness measurements, used in response surface analysis, have been performed three times on each specimen’s gauge length, by scanning and evaluating the area of 2 × 2 mm2 at three different locations as shown on fig. 3a. 70 d. liović, m. franulović, l. ferlič, n. gubeljak 3. results and discussion all reported values within this study are expressed in the form of mean value (standard deviation). furthermore, on all response surface figures, models with higher r2 values have been shown to enable a visual assessment of utilized laser power and scanning speed levels on the prediction variables. in that way, a visual assessment for both predictor variables will be enabled. the surface roughness measurements have been performed on one specimen (h2 specimen) before and after ultrasonic cleaning to consider its potential influence on results. before ultrasonic cleaning the average surface roughness has been 6.670 μm (std. 0.406 μm), and after ultrasonic cleaning 6.788 μm (std. 0.389 μm). the ultrasonic cleaning showed no effect on the average surface roughness measurements, given that means and standard deviations have been almost identical. the dependence of ra and sa values on specimen positions has been evaluated using the kruskal-wallis rank sum test as ra and sa were not normally distributed in all specimen groups. the equality of variances between specimen groups has been tested using levene’s test, both on ra and sa data. the variances have been equal, as the calculated p-values for ra and sa data were 0.84 and 0.79 respectively. the calculated p-values using kruskalwallis test for ra and sa values have been 0.80 and 0.90, respectively. it indicates that the different positions do not have statistically significant effect on ra and sa values, which can be seen in fig. 4 as well. this result is in accordance with findings stated in [2], where it was demonstrated that the laser power and scanning speeds are the most relevant factors influencing the side surface roughness. therefore, further regression models for surface roughness can be developed using only p and v as predictor variables. fig. 4 influence of different positions on the variation of: a) ra values, b) sa values by testing models m1 and m2 against each other using anova procedure with respect to their fitting ability (i.e., testing whether the difference between residual sum of squares of each model is statistically significant) it has been found that model m1 does not provide a better fit than model m2, since the calculated p-value using anova procedure was 0.581 which is higher than a level of significance (α = 0.05). therefore, the influence of p and v on average surface roughness (ra) can be described efficiently without term v2. this can be seen in table 4, where p-values have been reported for each parameter used to develop model m1. surface roughness of ti6al4v alloy produced by laser powder bed fusion 71 table 4 parameters of m1 full quadratic model with corresponding p-values m1 source p-value remark intercept < 0.001 significant p < 0.001 significant v < 0.001 significant p2 < 0.001 significant v2 0.58 not significant pv < 0.001 significant both models developed for ra have homoscedastic variance in the error term (p-value > 0.05 in both cases) which has been confirmed using non-constant variance (ncv) score test, as shown in table 5. it indicates that utilized levels of predictors do not affect the variance of model errors, further validating the applicability of the ols regression method. this test also confirms that ra can be efficiently modelled using both models across the whole p and v range specified within design of experiments (doe), since equal model error variance has been present. furthermore, both models have normally distributed studentized residuals as p-values determined using shapiro-wilk (s-w) normality test have been in both cases larger than 0.05, which is a prerequisite for a valid application of the ols regression method. as can be seen in table 5, both models have high r2 and adjusted r2 values. the influence of p and v on ra has been shown on fig. 5a with a superimposed 95% confidence intervals calculated using eq. (1). table 5 regression models for ra and their statistical properties average surface roughness (ra) r2 adj. r2 p-value (ncv) p-value (s-w) m1 2 6 2 a 0.0021 2.721 10 0.7686 0.0563 0.0002 65.154r p v p v pv−= −  − + − + 0.852 0.827 0.417 0.074 m2 2 a 0.0021 0.7686 0.0495 0.002 69.2928r p p v pv= − + − + 0.850 0.831 0.297 0.057 fig. 5 response surfaces describing the influence of p and v on the: a) line average surface roughness (ra) values with superimposed 95% confidence intervals visible as gray surfaces, b) area average surface roughness (sa) values with added experimental data visible as blue dots 72 d. liović, m. franulović, l. ferlič, n. gubeljak testing of the area average surface roughness (sa) models against each other using the anova procedure revealed that model m3 does not have a better fit than model m4 (pvalue = 0.406). therefore, the influence of p and v on sa can be described efficiently without term v2. the p-values for each parameter used to develop model m3 are reported in table 6. however, in this case both models have heteroscedastic variance of error term since p-values are lower than 0.05, as can be seen in table 6. hence, heteroskedasticity-robust procedure (hc3 method) is applied to evaluate model coefficients obtained using ols method. the analysis confirmed the validity of the previously determined model coefficients stated in table 6. consequently, confidence intervals have been omitted from fig. 5b since the assumption of equal variance of residuals has been violated. therefore, on fig. 5b only the scatter plot has been added to the response surface plot to visualize both experimentally measured data and corresponding nonlinear regression model m3. table 6 parameters of m3 full quadratic model with corresponding p-values m3 source p-value remark intercept < 0.001 significant p < 0.001 significant v 0.004 significant p2 0.007 significant v2 0.41 not significant pv < 0.001 significant table 7 regression models for sa and their statistical properties area average surface roughness (sa) r2 adj. r2 p-value (ncv) p-value (s-w) m3 2 6 2 a 0.0014 4.123 10 0.4884 0.0505 0.0002 37.256s p v p v pv−= −  − + − + 0.802 0.769 < 0.001 m4 2 a 0.0014 0.4884 0.0402 0.0002 43.526s p p v pv= − + − + 0.797 0.771 < 0.001 as expected, a strong linear relation between experimentally measured ra values using contact profilometer and sa values measured using microscope is present (r = 0.948). relation between ra and sa values can be visualized by comparing the shapes of the response surfaces, on fig. 5a and b. it is evident that the shape and the curvature of the response surfaces are almost identical, which confirms the relation between those two observed predictions. both ra and sa reach their maximum values for the c group of specimens, i.e., when p = 200 w and v = 1500 mm/s have been used for the l-pbf of ti6al4v alloy. in that case, ra and sa values have been 12.937 μm (std. 1.039 μm) and 11.966 μm (std. 1.821 μm), respectively. this occurrence is attributed to the stacking and incomplete melting of powder particles near the specimen’s surfaces due to the lowest utilized energy input (59.3 j/mm3), which is in accordance with findings stated in [2]. when surface topographies of the c group of specimens (i.e., specimen group with the highest mean ra and sa values) and i group of specimens (i.e., specimen group with the lowest mean ra and sa values) are compared in fig. 6a and b, it is evident that the c group of specimens has higher peaks with a maximum value up to 119.4 μm. surface roughness of ti6al4v alloy produced by laser powder bed fusion 73 fig. 6 a) group of specimens c characteristic by the highest measured ra and sa values, b) group of specimens i characteristic by the lowest measured ra and sa values high values of coefficients of variation (cov), specific to c group of specimens (8.03% for ra and 15.22% for sa), also imply that more incomplete and unequal melting of powder particles is present in contrast to i group of specimens that has lower cov values (3.26% for ra and 4.36% for sa). more specifically, i group has ra and sa values of 7.612 μm (std. 0.248 μm) and 7.195 μm (std. 0.314 μm), respectively. this leads to the conclusion that the surface roughness of specimens in group c is lower and more uniform than surface roughness of specimens in the i group. except for the used l-pbf process parameters, surface roughness of products manufactured using the l-pbf technology predominantly depends on the powder particle size distribution [45], inclination angle with respect to the build platform [5], and heat treatment [11]. thus, in the literature, wide ranges of resulting average surface roughness values of l-pbf ti6al4v specimens have been reported by different groups of authors [1, 9, 46, 47]. when process parameters are considered, mierzejewska et al. [11] found that energy densities in the range from 88 to 113 j/mm3 lead to surface roughness decrease, while energy densities in the range from 44 to 63 j/mm3 causes surface roughness increase. this is valid in the case when only one predictor variable is considered within doe, as demonstrated in [11]. however, similar or identical values of energy densities can be obtained using completely different combinations of p and v, as can be seen in table 3. hence, energy density cannot be isolated and independently used for interpretation of results when two predictor variables (i.e., p and v) are involved [48]. therefore, results have been interpreted by separately observing the effect of p and v on considered prediction variables, as shown in fig. 6a and b. when the influence of scanning speed on the ra and sa values is considered, its effect is more pronounced when the lowest utilized laser power level has been used (i.e., 200 w). more specifically, the increase in laser power levels reduces the effect of the scanning speed on the ra and sa values. it is also evident that the decrease in scanning speed levels reduces laser power’s influence on the ra and sa values. therefore, when low surface roughness of the ti6al4v l-pbf parts is desired, the higher laser power values should be considered. this finding is consistent with the previously published studies, where the decrease in ra and sa with an increase in laser power has been found [2, 49]. elsayed et al. [50] have reported the same effect at substantially lower laser powers ranging from 35 to 50 w, where significant drop of ra with laser power increase has 74 d. liović, m. franulović, l. ferlič, n. gubeljak been found. as reported in [51, 52], increasing the laser power leads to an increase in energy density, which enhances the wettability of the melt pool. this occurrence reduces the differences in surface tensions and subsequently decreases the balling effect [50, 51]. as a result, there is a decrease in the formation of irregular beads or spherical drops, leading to a reduction in average surface roughness. in order to decrease the average surface roughness of the top and side surfaces, the higher laser powers should be used [52, 53]. as stated by mumtaz et al. [52], the higher laser powers promote the flattening of the melt pool surface and enhance its wettability which reduces the chances of the balling effect to occur. 4. conclusions the average surface roughness of the l-pbf ti6al4v alloy has been systematically studied through experimental tests, followed by comprehensive regression analysis and model verification. consequently, nonlinear regression models have been developed to relate laser power and scanning speed with average surface roughness. moreover, the influence of position, laser power, and scanning speed on average surface roughness has been interpreted. the findings of this study are relevant to the l-pbf manufacturing of ti6al4v alloy, specifically when employing laser powers ranging from 200 to 250 w and scanning speeds ranging from 1000 to 1500 mm/s. the influence of laser power and scanning speed on the average surface roughness of the l-pbf ti6al4v alloy can be interpreted and described with high precision using nonlinear regression models that include two predictor variables (i.e., laser power and scanning speed). more specifically, laser power and scanning speed significantly influence the average surface roughness values. by using higher laser powers, it is possible to reduce average surface roughness and decrease the effect of scanning speed. the influence of specimen position on the build platform showed to be non-significant for the average surface roughness. the results gained as part of this systematically conducted experimental research, supported by comprehensive statistical analysis, will contribute to a better understanding of the influence of l-pbf process parameters on the surface roughness of the widely used ti6al4v alloy. acknowledgement: this work has been supported by croatian science foundation under the project number ip-2019-04-3607 and by university of rijeka under project number uniri-tehnic-1834. part of the work presented in this research has been supported by using 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silicate hydroxide combined with amphiphilic molecules bin wang, qiu ying chang, kai gao school of mechanical, electronic and control engineering, beijing jiaotong university, beijing, pr china abstract. this paper reports the synthesis of magnesium silicate hydroxide (msh) nanoparticles and their synergistic tribological properties combined with amphiphilic molecules (ams) as additives in base oil. this combination reduces wear losses substantially due to the formation of a double well-arranged molecular layer or tribofilm on the rubbing surfaces under certain test conditions. from the results of nonequilibrium molecular dynamics (nemd) simulations, lamellate msh nanoparticles provide a medium for the adsorption of ams thus further decreasing the contact of rough peaks. in addition, with the increase of load, a tribofilm containing element mg, si, o forms on the worn surfaces and greatly improves the anti-wear property of base oil. key words: synthetic magnesium silicate hydroxide; amphiphilic molecules; friction and wear; nonequilibrium molecular dynamic simulation 1. introduction to obtain more excellent tribological properties, a variety of approaches including new wear resistant materials [1,2], coating [3], new lubricant additives [4–6], various low-viscosity base oil and others have been explored widely. among these, finding out new additives with outstanding tribological and dispersive properties which can replace or partly replace the traditional lubricant additives such as zinc dithiophosphate (zddp) [7], is always the focus of research. magnesium silicate hydroxide (msh) is a new type of lubricant additive with an ideal chemical formula of mg3si2o5(oh)4. because of the weaker van der waals forces and hydrogen bonds between si-o tetrahedral sheet and mg-o/oh octahedral sheet, it is easy to received january 20, 2019 / accepted march 15, 2019 corresponding author: bin wang affiliation: school of mechanical, electronic and control engineering, beijing jiaotong university e-mail: 15116341@bjtu.edu.cn 66 b. wang, q.y. chang, k. gao decompose and release unsaturated groups such as si–o–si、o–si–o、oh–mg–oh(o)、 oh and o–h–o [8–12]. actually, because of these features, scholars have systematically investigated the anti-wear behavior of serpentine powders whose composition is mainly msh, as additives in oil under different friction conditions and concentrations [13–15]. the results show that a tribofilm mainly consisting of fe, c, o and si elements forms on the rubbing surfaces, which indicates that serpentine particles decomposed under the combination of local high pressure and temperature. however, as a type of natural mineral, serpentine contains a small quantity of aluminum oxide, calcium oxide, iron oxide and other compounds which work against the clarification of its tribological mechanisms. meanwhile, by the method of high-energy mechanical ball-milling, only micro-sized natural serpentine particles can be obtained, which is not suitable for their applications as lubricant additives. inspired by the above research studies, we have synthesized msh nanoparticles and confirmed its outstanding tribological properties in base oil [16–18]. in order to disperse the nanoparticles uniformly and steadily in oil, amphiphilic molecules (ams) like oleic acid or stearic acid are commonly used as a modification agent [19–21]. these molecules do not only adsorb around the nanoparticles and prevent the agglomeration effectively, but also reduce friction and wear by itself [5]. early in 1920s, organic friction modifiers (ofms) based on ams were introduced to increase the energy efficiency of equipment, and are always one of the most important additives in lubricant oil. the main friction-reducing mechanism of ofms is that it can form a vertically oriented, close-packed monolayer on the sliding surfaces [22–24]. to our knowledge, although there have been many literatures on the surface modification effects and the friction properties of ofms, few research studies have been focused on the synergistic tribological behaviors of nanoparticles combined with ams. so as to improve the dispersity of natural serpentine particles in oil, xu et al. [13–15] used oleic acid or other ams as a surface modifier during the process of ball-milling and studied the tribological performance of these surface-coated particles without considering the anti-wear effect of ams. song et al [21] prepared surface-modified znal2o4 nanoparticles by heating and drying a solution containing particles and oleic acid, then also tested their tribological properties by ignoring the influence of oleic acid. in this paper, we synthesized lamellate msh nanoparticles hydrothermally and explored their synergistic tribological behaviors combined with amphiphilic organic molecules while improving its dispersity in polyalphaolefin base oil (pao). at the same time, techniques of nonequilibrium molecular dynamics (nemd) simulations, scanning electron microscopy (sem) and energy dispersive spectroscopy (eds) were used to explain the anti-wear mechanisms. 2. experiment and materials the synthesis of msh nanoparticles was carried out in autoclaves by the method of hydrothermal reaction. nano-sized magnesium oxide (mgo, >99.9wt%, 40nm) and silicon dioxide (sio2, >99.9wt%, 40nm) were used as precursors and their reaction concentrations were 2.7×10 -7 mol/l and 1.8×10 -7 mol/l, respectively. the mixture was hydrothermally treated in a reactor containing naoh aqueous solution at temperature of 200°c, pressure of 1.6mpa and experiment duration of 12h. the resulting powders were washed in distilled water for three times to remove sodium and dried in furnace at 80°c for 10h. table 1 lists synergistic tribological properties of synthetic magnesium silicate hydroxide combined... 67 the x-ray fluorescence spectrometer (xrf) result of the synthetic msh particles. its crystal formula can be expressed as mg2.329si2.202na0.469o5(oh)4. the morphology of the synthetic msh nanoparticles was investigated by sem in fig. 1. table 1 chemical elements of synthetic msh nanoparticles elements content (wt %) magnesium (mg) 24.19 silicate (si) 26.72 sodium (na) 2.23 synthetic msh nanoparticles were ultrasonically dispersed in pao base oil with a viscosity of 73cst at 40℃ and using ams as a dispersant agent (termed as oil+ msh+ ams). the weight percentage of the powder and ams in the oil-powder suspension was 1% and 2%, respectively. to make clear the influence of ams on the tribological performance of additives, pure oil only added 2wt% ams (termed as oil+ ams) also prepared at the same time. fig. 1 sem morphology of synthetic msh nanoparticles. tribological experiments were carried out by a four-ball friction and wear test machine (mrs-10a). gcr15 balls with 62~67 hrc hardness and 0.02m roughness (ra) were used. the radius, poisson's ratio and young's modulus of ball specimens are 6.35mm, 0.3 and 208gpa, respectively. the experimental conditions were: normal loads 200 and 600n (corresponding to maximum pressures of 2.71 and 3.91gpa, respectively), rotational speeds 400rpm (corresponding to speeds of 0.153m/s), duration two hours at room temperature. once the test was finished, the wear scar diameter of ball was obtained by using an optical microscope (accuracy is 0.01mm) and all sets of the experiment were repeated three times. the morphologies and chemical elements of the worn surfaces were characterized by sem and eds. 68 b. wang, q.y. chang, k. gao 3. results and discussion 3.1. lubricant additives from fig. 1, we can see that the synthetic msh nanoparticles are mostly lamellate and have an average lateral dimension of approximately 50 nm10 nm. there are also some msh nanoparticles curled to a certain extent, which is attributed to the hydrothermal conditions. and specific synthesis mechanism of msh can refer to our previous study [17]. 3.2. dispersion property to illustrate the influence of ams on the dispersive property of synthetic msh nanoparticles in oil, sedimental tests lasted 7 days from oil sample preparation were carried out (fig. 2). ams make msh nanoparticles dispersed in oil homogeneously and stably, however, a layer of sediment at the bottom of bottle was formed when only added msh nanoparticles in oil (fig. 2(b)). fig. 2 oil-additive suspensions after 7 days from preparing (a) pure oil and (b) oil only added msh and (c) oil added both msh and ams. in this sedimental test, pure oil only added synthetic msh nanoparticles with a weight percentage of 2% (termed as oil+ msh) prepared at the same time 3.3 friction and wear fig. 3 shows the average wear scar diameters (wsds) and the three-dimensional profiles of samples experimented in three oil samples under different conditions. obviously, the anti-wear property of base oil was significantly improved by the addition ams, and better results were further obtained after suspending synthetic msh nanoparticles in oil under both experimental conditions. with respect to the average wsds, the anti-wear rates of oil+ msh+ ams relative to pure oil were 45.56% and 32.44% under 200n and 600n, respectively. however, it can be seen from fig. 3 (b) that there happened precious little wear when used synthetic msh as additives in oil, meanwhile, the volume wear rate of friction specimens subjected to oil+ msh+ ams decreased by the order of magnitude with respect to that of specimens lubricated by pure base oil or oil+ ams. under the test condition of 200n and 400rpm, the average wsds of 285.8m is close to the hertz contact diameter of 240m calculated by hertz contact radius formula (eq. (1)) which also means little wear happened when lubricated by oil+ msh+ ams. this is because the four-ball friction and wear tests are based on a point contact mode, and the contact pressure is high to synergistic tribological properties of synthetic magnesium silicate hydroxide combined... 69 gigapascals level attributed to the small lubrication area. therefore, the surface deformation cannot be ignored. hertz contact theory is one of the starting points of electrohydrodynamic lubrication (ehl) theory considering the elastic deformation of the surface. in our study, it is likely that the ball material will be removed due to the maximum experimental pressure of 2.91 or 3.71gpa, and the wear scar diameter is greater than that of hertz contact diameter. besides, the d-value between them can be used to evaluate the oil’s anti-wear property, that is, the smaller the difference, the higher the anti-wear. 2 2 1 2 1 2 3 1 2 1 1 3 1 14 e ef a          (1) among them, a is the hertz contact radius; f is the applied load; 1 and 2 are the poisson's ratio of ball specimens; e1 and e2 are the young's modulus of ball specimens; 1 and 2 are radius of ball specimens. although the coefficient of friction (cof) value obtained by the four-ball friction and wear machine cannot be used seriously to explore the friction property of oil, the variation trend of cof with different oil samples under same conditions is of great significance. in this study, all sets of the experiment were repeated three times, and fig. 4a shows the average cof value during friction tests for samples lubricated with different oil samples. under the condition of 200n, if the error is considered, there is no significant difference in cofs of samples lubricated in pure oil, oil+ ams and oil+ msh+ ams. with the load increased to 600n, oil only added ams also had similar stable value of cof than that of oil added synthetic msh particles. however, pure oil has a lower average stable cof value of 0.062. to further analyze the friction property of three different oil samples, figs.4b and c shows the evolution of the cof for one of the three repeated friction tests under experimental conditions of 200n, 400rpm and 600n, 400rpm. both in figs. 4b and c, at the loading stage, the cofs of the samples lubricated in pure oil increased first and then decreased fig. 3 average wear scar diameter values (a) and the three-dimensional profiles (b) of samples tested in pure base oil, oil+ ams and oil+ msh+ ams under different test conditions 70 b. wang, q.y. chang, k. gao which means there happened severe wear and the actual contact area increased to a certain extent. in other words, the higher stable cofs of samples lubricated in oil+ msh+ ams do not mean oil containing msh nanoparticles have worse friction-reduction property than that of pure oil, and it may be attributed to the distinctive frictional performance of tribofilm formed on worn surfaces. fig. 5 shows the sem morphologies of worn surfaces lubricated with base oil and oil+ msh+ ams under different loads. there are a great amount of scratches and furrows on the surfaces for pure base oil, which indicates severe wear occurred on the contact surfaces under all experimental conditions. in contrast, the furrows become shallower and less when adding synthetic msh nanoparticles to oil. the sliding surfaces are extraordinary smooth under the experimental condition of 200n, which is in accordance with the anti-wear results in fig. 3. meanwhile, a dark tribofilm formed on the substrate surface when the experimental load increased to 600n. in fig. 6, the eds analyses of friction surfaces lubricated by oil+ msh+ ams show that the smooth worn surfaces under test condition of 200n mainly consist of fe which agrees with eds result of original substrate (table 2). however, with the increase of experimental load to 600n, a dark tribofilm containing high content of o, mg and si elements formed on the sliding surfaces. mg and si elements come from synthetic msh additives. meanwhile, the chemical compositions (table 2) reveal that the molar rates between mg and si are different from that of synthetic msh nanoparticles, indicating a decomposition of msh occurred under the combination of mechanical and thermal energy during the tribological tests. fig. 4 (a) average coefficient of friction (cof) value, and (b, c) evolutions cof during friction and wear tests for balls lubricated with pure base oil, oil+ ams and oil+ msh+ ams under experimental condition of (b) 200n, 400rpm and (c) 600n, 400rpm. because the load of 600n was imposed stepwise from 200n with a step of 100n every 2.5min, the experimental time in fig. 4c is longer than that in fig. 4b synergistic tribological properties of synthetic magnesium silicate hydroxide combined... 71 fig. 5 sem morphologies of the worn surfaces lubricated with (a) (b) pure oil and (c) (d) oil +msh+ ams under experimental conditions of 200n and 600n (the red frames in this figure are the component analysis areas of eds) fig. 6 eds spectra of the worn surfaces (marked areas in fig. 5) lubricated with oil added both msh and ams under experimental conditions of (a) 200n (b) 600n table 2 chemical compositions of the worn surfaces for different experimental conditions samples content (at %) fe o mg si original substrate 82.3 0.3 smooth area in fig.5c 78.0 0.2 0.6 dark tribofilm in fig. 5d 20.9 45.8 13.4 6.1 72 b. wang, q.y. chang, k. gao 3.3 nonequilibrium molecular dynamic simulations to fully understand the performance of lubricant molecule and additives, nemd simulation including initial configuration composed of hexadecane molecules, stearic acid monolayers, and lamellate iron nanocluster confined between smooth steels was carried out. (100) surface of -iron, stearic acid molecules and iron nanocluster were used as sliding tribo-surfaces, ams, and nanoparticles, respectively. although silicate would be a more accurate representation of an additive in this system, there is no classical md force-field. this size domain was optimized considering the effects of itself and the simulation time. a brief explanation of simulation setup and snapshots after 300ps of sliding are shown in fig. 7. periodic boundary conditions were applied in the x and y directions with size of 31.5331.53å 2 and z dimension varied from 94~135å according to the initial setup of models. both simulations were performed at 353k which is representative of experimental temperature and controlled by a langevin thermostat. the density of lubricant liquid was 0.76g/cm 3 and the surface coverage of stearic acid on all surfaces was 4.5nm -2 . the applying load and shear rate were 3gpa and 10m/s, respectively. all-atom force fields were used in the nemd simulations which enables the structure and large molecular systems to be reliably analyzed. compass force field was applied for hexadecane and ams molecules, and embedded atom model (eam) potential was represented the iron-iron interactions within the slabs. the lennard-jones (lj) potential with cut-off distance of 12.5å was used for van der waals and long-range columbic interactions between the lubricant and the surfaces. the potential parameters can refer to fig. 7 simulation setup and snapshots after 300ps of sliding (a) confined three layers of hexadecane molecules in between two stearic acid monolayers adsorbed on (100) surfaces of -iron (b) confined three layers of hexadecane molecules in between each pair of stearic acid monolayers adsorbed on (100) surfaces of -iron and lamellate iron nanocluster. fe, c, o, and h atoms are presented as orange, cyan (terminal c in yellow), red, and white colors, respectively synergistic tribological properties of synthetic magnesium silicate hydroxide combined... 73 literatures [25, 26]. the simulation procedure can be divided into three main stages: i) optimization of structure, ii) compression in z direction under applying load of 3gpa, and iii) friction and wear in x direction with shear rate of 10m/s. a time relaxation constant of 0.1fs was used and total simulations time was 550ps. under the confinement and sliding motion of the slabs, stearic acid molecules formed a vertically arranged solid-like layer on both surfaces of slab and iron nanocluster which can improve the bearing capacity of base oil to a great extent [22, 27–29]. the thickness of oil added stearic acid and oil added both stearic acid and flake iron cluster reduced from initial 55å and 118å to compressed 37.8å and 82.8å, respectively. lamellate particle in lubricating oil provided a medium to the adsorption of ams; meanwhile, one more pair of vertical arranged layers would be formed between the sliding surfaces as shown in fig.7 b which isolated the friction surfaces completely to achieve “zero wear” (consistent with the three-dimensional profiles of test samples shown in fig. 3b). however, with the increase of applying load or shearing velocity, these vertical arranged layers would be destroyed and lose their bearing capacity [22]. under this situation, the features of nanoparticles and their tribological properties become substantially important. in the nemd simulation, an iron nanocluster was used as the representation of nano-additive but this does not mean any flake nano-material combined with ams has the same anti-wear effect as msh. because of the unique constitution and layered structure, msh nanoparticles are easy to spread and decompose under the condition of certain pressure and temperature meanwhile forming a tribofilm consisting of mg, si, and o on the friction surface instead of acting as abrasive particles. this tribofilm will serve as a secondary anti-wear protection once the lubricant molecules and ams have been crushed. 4. conclusions in conclusion, we have investigated the synergistic tribological properties of synthetic lamellate msh nanoparticles combined with ams in pao base oil. this combination not only makes nanoparticles dispersed in oil homogeneously but also improves the anti-wear property of base oil substantially. under relatively slight conditions, lamellate msh particles provide media for the adsorption of ams to form arranged layers, thus reducing the contact of rough peaks effectively. on the other hand, with the increase of applying load, a tribofilm containing element mg, si and o forms on the sliding surfaces and ensures a secondary anti-wear protection. acknowledgments: the authors would like to thank the national natural science foundation of china (grant no.51075026) and the national defense pre-research foundation of china (grant no.h092013b001). references 1. gershman, i., gershman, e.i., mironov, a.e., fox-rabinovich, g.s., veldhuis, s.c.,2016, application of the self-organization phenomenon in the development of wear resistant materials-a review, entropy, 18(11), 385. 2. lee, k., hsu, j., naugle, d., liang, h., 2016, multi-phase quasicrystalline alloys for superior wear resistance, mater des, 108, pp. 440–7. 3. bhushan, b., gupta, b.k., 1991, handbook of tribology: materials, coatings, and surface treatments, mcgraw-hill, new york, united states. 74 b. wang, q.y. chang, k. gao 4. ali, m.k.a., xianjun, h., 2015, improving the tribological behavior of internal combustion engines via the addition of nanoparticles to engine oils, nanotechnol rev, 4, pp. 347–58. 5. gulzar, m., masjuki, h.h., kalam, m.a., varman, m., zulkifli, n.w.m., mufti, r.a., et al, 2016, tribological performance of nanoparticles as lubricating oil additives, j nanoparticle res, 18, pp. 1–25. 6. dai, w., kheireddin, b., gao, h., liang, h., 2016, roles of nanoparticles in oil lubrication, tribol int, 102, pp. 88–98. 7. rokosz, m.j., chen, a.e., lowe-ma, c.k., kucherov, a.v., benson, d., paputa peck, m.c., et al., 2001, characterization of phosphorus-poisoned automotive exhaust catalysts, appl catal b environ, 33, pp. 205–15. 8. mookherjee, m., stixrude, l., 2009, structure and elasticity of serpentine at high-pressure, earth planet sci lett, 279, pp. 11–9. 9. veblen, d.r., buseck, p.r., 1979, serpentine minerals: intergrowths and new combination structures, science, 206, pp. 1398-400. 10. sclar, c.b., carrison, l.c., thomas, p., et al., 1966, high-pressure reaction and shear strength of serpentinized dunite, science; 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films adsorbed on iron oxide surfaces in squalane, langmuir, 30, pp. 186–95. facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 75 85 https://doi.org/10.22190/fume190115006w © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper energy loss and wear in spherical oblique elastic impacts emanuel willert technische universität berlin, berlin, germany abstract. percussive and erosive wear by repetitive impacting of solid particles damages surfaces even at low impact velocities. as the impact wear is often directly related to the energy loss during the collision and therefore to the coefficients of normal and tangential restitution, in the present study the oblique low-velocity impact of a rigid sphere onto an elastic half-space is analyzed based on the known respective contact-impact solution and with regard to the energy loss during the impact. simple analytic expressions are derived for the total impact wear volume. it is found that the portion of kinetic energy lost in frictional dissipation has a well-located maximum for configurations with weak forward pre-spin. the distribution of frictional dissipation over the contact area has a complex dependence on the impact parameters. for pronounced local slip (e.g. due to a small coefficient of friction) the dissipation accumulated over the collision is localized in the center of impact whereas for dominance of sticking, most energy is lost away from the center. key words: elastic impacts, friction, energy dissipation, wear 1. introduction impact wear, i.e. material degradation due to the repetitive impacting of solid particles onto a surface is a serious source of damage and failure in several technical systems like steam generator tubes [1], mining machinery [2] and others. several studies, starting more than fifty years ago, have been dealing with the erosion of a surface by a stream of solid particles [3-6]. however, due to the complexity of the occurring wear mechanisms and the mathematical difficulty of the contact mechanical description for the impact itself, the problem remains far from being fully understood. received january 15, 2019 / accepted march 07, 2019 corresponding author: emanuel willert technische universität berlin, sekr. c8-4, straße des 17. juni 135, 10623 berlin, germany e-mail: e.willert@tu-berlin.de 76 e. willert the first rigorously analyzed contact-impact problem is the low-velocity normal impact of perfectly elastic, perfectly smooth spheres, solved by hertz [7]. the hertzian normal impact solution has later been generalized to incorporate adhesion [8], plasticity [9], wave propagation [10] and surface roughness [11]. the frictional oblique impact problem of elastic spheres was solved by maw, barber & fawcett (mbf, [12]), based on the contact solution for the frictional tangential contact of elastic spheres under varying oblique loading histories by mindlin and deresiewicz [13]. the outcomes of the mbf theory were demonstrated to be in very good agreement with experimental results if the impact behavior is close to being perfectly elastic [14, 15]. an equivalent but computationally simpler formulation of the mbf solution within the framework of the method of dimensionality reduction [16] was published recently [17]. moreover, there are several rigid-body models of oblique impacts [18, 19]. however, as there are only two regimes of contact in rigid-body dynamics – stick and gross sliding – these models are necessarily wrong in the partial slip regime. a common approach to characterizing the impact wear behavior is to study the loss of kinetic energy during the impact [1]. thereby good agreement has been reported between energy-based models and erosive wear results from the literature [20, 1]. this is in correlation with the classical wear law by archard and hirst [21] for adhesive and khrushchov and babichev [22] for abrasive wear, according to which the wear intensity is proportional to the normal load and the relative tangential velocity of the contacting surfaces. if we additionally apply the amontons-coulomb law of friction, the wear intensity is proportional to the power of frictional energy dissipation, as reported by honda and yamada [23]. the energy-based approach to studying erosive wear was also used by argatov et al. [24], who, nevertheless, only considered sliding contact during the impact and neglected the contribution of spin. thus, in the present study, the loss of kinetic energy due to frictional dissipation will be analyzed for the oblique impact of elastic spheres, based on the mbf model for the contact-impact problem. first, in a general analysis, the rebound velocities and the loss of kinetic energy are calculated in terms of the tangential coefficient of restitution. after that, this restitution coefficient is determined within the framework of the mbf approach. after a visualization of the main results for the total loss of kinetic energy, the distribution of frictional dissipation over the contact area is analyzed. a short discussion closes the manuscript. 2. general analysis let us look into the problem depicted in fig. 1: a rigid sphere of mass m, radius r and moment of inertia j impacts onto an elastic half-space with initial velocities vx0 and vz0 and initial angular velocity ω0. without loss of generality we can assume that vx0 > 0; for brevity let us introduce spin s = rω. the impact shall be short and the macroscopic dynamics shall be determined by point contact forces in normal and tangential direction at the point of first contact k. note that the more general case of colliding elastic spheres in 3d exhibits no additional features if wave propagation can be neglected [25] (i.e. the velocities are small compared to the speed of sound in the elastic bodies) and the surfaces of the contacting energy loss and wear in spherical oblique elastic impacts 77 bodies obey the restrictions of the half-space approximation. moreover, it should be pointed out that in the present work the dissipation due to plastic or viscoelastic deformations, which also can be of utmost importance in erosive wear, are disregarded to simplify the analysis. fig. 1 2d oblique impact of a rigid sphere onto an elastic half-space. schematic representation and notations under the assumptions stated above, the tangential velocity and spin after the collision can be calculated in terms of the coefficient of tangential restitution ex, 0 0 : ,xe e x x v s e v s     (1) and the non-dimensional gyration parameter 2 : j j mr    (2) as follows: 0 0 0 0 0 0 (1 )(1 )( ), (1 )( ). e x x xe x x x s s e s v v v e s v            (3) hence, the loss of kinetic energy during the collision is given by 2 2 0 0( ) (1 ). 2 x x m e v s e     (4) according to the energy-based approach, the total worn volume after one impact will be 0 , e v k     (5) with material hardness σ0 and non-dimensional wear coefficient k. 78 e. willert obviously, all the macroscopic impact characteristics are known if the tangential restitution coefficient can be determined. this is done in the following section. 3. the coefficient of tangential restitution maw, barber & fawcett [12] have shown that, if the friction force is calculated based on the contact solutions by mindlin & deresiewicz, the coefficient of tangential restitution will be a function of only two dimensionless parameters 0 0 0 tan 2 2 : , : , with : and tan : , 2 2 x z v sm m m v               (6) with poisson ratio ν of the elastic half-space and the coefficient of friction μ. m is the ratio of tangential to normal stiffness of the cylindrical flat punch contact, often attributed to mindlin, and α the generalized impact angle accounting for pre-spin of the sphere. to spare the less important parameter, we will assume a homogeneous sphere (i.e. κ = 2/7) and a constant poisson ratio of ν = 1/3. hence, χ = constant = 1.4. fig. 2 coefficient of tangential restitution ex as a function of ψ for the oblique impact of elastic spheres with κ = 2/7 and ν = 1/3, together with the analytic expressions from eqs. (7) and (8); also shown is the relevant term for the loss of kinetic energy the solution for ex = ex(ψ) resulting from mbf theory needs to be calculated numerically and is shown in fig. 2. there are three different regimes: for ψ < 1 the impact starts with a configuration of local stick; for 1 < ψ < 4χ – 1 the impact starts with a phase of gross slip, which ends during the collision; for ψ > 4χ – 1 the contact is fully sliding during the whole impact and the coefficient of tangential restitution is therefore elementarily given by energy loss and wear in spherical oblique elastic impacts 79 fs 4 1, if 4 1.xe         (7) in the other two regimes, for χ = 1.4 the solution can be approximated very well by the analytic expressions 2 4 3 2 0.0193 0.2896 0.1937 if 1, 0.0081 0.115 0.6585 1.6541 1.0152 if 1 4.6. x x e e                     (8) 4. selected results for the lost portion of kinetic energy in many applications the most relevant quantity will not be the absolute value of dissipated energy but rather the portion of initial kinetic energy, which is lost during one impact. normalizing eq. (4) for initial kinetic energy e0 we obtain 2 2 0 0 2 2 20 0 0 0 ( ) (1 ) . 1 x x x z e v s e e v v s           (9) to characterize the mode of the contact point’s tangential motion and thus the effect of pre-spin one can introduce the ratio 0 0 0 : .x x v v s    (10) pre-spin plays an important role for the dissipated energy – which is self-evident for everybody who has seen a tennis ball bouncing off a court with what in sports is called “back spin” (i.e. s0 > 0; as the tangential velocity of the contact point due to the rotation in this case is positive, the notion “forward spin” seems more appropriate) – because it enters both eqs. (4) and (6). ξ = 1 corresponds to no pre-spin at all whereas for ξ = 0 the tangential motion of the contact point results from pure rotation. values ξ > 1 correspond to weak backward spin (contact point k is still moving in positive x-direction) and values ξ < 0 to strong backward spin (the direction of motion of k changes). eq. (9) can be rewritten in the form 2 2 2 20 (1 ) . cot (1 ) 1 x e e e              (11) with all other parameters fixed (especially generalized impact angle α), this expression takes its maximum value at .c  (12) which corresponds to weak forward spin (note, that 0 < κ < 0.5). in fig. 3 the portion of lost kinetic energy is shown for ν = 1/3, κ = 2/7 and ξ = κ as a function of the two remaining parameters, the coefficient of friction and the generalized 80 e. willert impact angle. as one would expect, the portion is increasing if both the impact angle and the coefficient of friction increase. however, for any fixed value of the coefficient of friction the portion of lost energy has a maximum at some certain impact angle and vice versa. fig. 3 contour isoline diagram of the loss of kinetic energy normalized for the initial kinetic energy before the impact as a function of the coefficient of friction and the generalized impact angle for ν = 1/3 and κ = 2/7 at the extremal configuration of tangential motion ξ = κ fig. 4 contour isoline diagram of the loss of kinetic energy normalized for the initial kinetic energy before the impact as a function of the configuration of tangential motion and the generalized impact angle for ν = 1/3, κ = 2/7 and μ = 0.1 energy loss and wear in spherical oblique elastic impacts 81 fig. 4 and 5 visualize the normalized dissipated energy as a function of ξ and the generalized impact angle for ν = 1/3, κ = 2/7 and two values of the friction coefficient. one can clearly see how the higher losses are localized in the regions of weak forward pre-spin, around the critical value given by eq. (12). fig. 5 results as in fig. 4 for μ = 0.5 5. distribution of frictional dissipation over the contact area not only can the total lost energy be of interest, but also its distribution over the contact area. if the energy-based wear law is valid in local form, then the distribution of frictional dissipation will give the distribution of wear and therefore the form of the impact pit after the collision (note again that plastic deformation, which will also result in a residual impact pit, is disregarded here). during the impact the contact area (with radius a) will in general consist of an inner stick area (radius c) surrounded by an annulus of local slip. frictional energy dissipation requires relative motion between the contacting surfaces. hence, energy is dissipated only in the slip area. the area density of power of the frictional dissipation is given by rel( , ) ( , ) ( , ) , ( ) ( ),w r t p r t v r t c t r a t   (13) with the hertzian pressure distribution * 2 22 ( , ) ( ) , ( ). e p r t a t r r a t r    (14) e * = e / (1 – ν 2 ), with young’s modulus e, is the effective young’s modulus of the elastic half-space. the relative velocity between two slipping points on the surfaces of the contacting bodies, vrel, can be calculated from the solution of the impact problem within 82 e. willert the framework of the method of dimensionality reduction (mdr; see [17] for a detailed description of the impact solution algorithm) according to   1d rel rel 2 2 ( , )2 ( , ) d . r c t v x t v r t x r x    (15) the superscript “1d” denotes the respective quantity in the mdr model. when evaluating the abel transform (15) numerically, it is useful to follow an idea by benad [26] and rewrite the transform via integrating by parts,   1d 1d rel rel rel 2 d ( , ) ( , ) arcsin ( ( , ))d , d r c t x v r t v r t v x t x r x          (16) to avoid the singularity in the integrand at x = r. fig. 6 area density of frictional energy dissipation normalized for the average value as a function of the radial coordinate for full-slip collisions fig. 6 shows area density w of the total frictional energy dissipation during the collision normalized for the average value 0 2 max , e w a   (17) with the maximum contact radius during the impact, amax, as a function of the normalized radial coordinate r / amax, if the contact is fully sliding during the whole impact. in these dimensionless variables the distribution of frictional loss of energy is completely independent of the impact parameters (provided the collision indeed takes place in the full-slip regime) and can be approximated well by the expression energy loss and wear in spherical oblique elastic impacts 83 2 fs 0 max max ( ) 2.1208 0.2488 1.9093 . w r r r w a a          (18) fig. 7 area density of frictional energy dissipation normalized for the average value as a function of the radial coordinate for several values of parameter ψ and χ = 1.4. the thin black line denotes the full-slip case fig. 8 area density of frictional energy dissipation normalized for the average value as a function of the radial coordinate for several values of the parameter χ. red lines correspond to ψ = (4χ – 1)/3. black lines correspond to ψ = (4χ – 1)/5 generally, the normalized distribution of frictional distribution will depend on the impact parameters χ and ψ. in fig. 7 the distribution is shown for the case χ = 1.4 (which, as said 84 e. willert before, corresponds to a homogeneous sphere and ν = 1/3) and several values of ψ. for small values of ψ (i.e. dominance of stick) the dissipation is localized away from the impact center whereas for large values (i.e. dominance of slip) the localization tends more to the center. the closer the impact configuration is to the full-slip case, the more the dissipation density is only depending on the parameter ψ / (4χ – 1). this is demonstrated in fig. 8. the red lines corresponding to ψ = (4χ – 1)/3 and different values of χ are very close together. on the other hand, the black lines, denoting configurations with ψ = (4χ – 1)/5 and various values of χ, differ significantly from each other, especially because for χ = 1.2 the respective value of ψ is smaller than one, i.e. the impact does not start with a phase of complete sliding. 6. discussion the wear loss in impact wear has been reported in the literature to be proportional to the amount of energy dissipated during the impact [1]. thus, in the present study the contact-impact solution for the spherical oblique elastic impact has been applied to calculate the amount of kinetic energy lost during one collision to provide an easy-to-evaluate impact wear measure. according to the analysis the main quantities that determine the impact wear volume are the coefficient of friction, the impact angle and the mode of tangential motion of the contact point (due to rotation or translation of the spheres). thereby the portion of kinetic energy, which is lost during the collision due to frictional dissipation, has a welllocalized maximum for weak forward pre-spin. the distribution of frictional dissipation over the contact area shows a complicated dependence on the impact parameters. for pronounced local slip (e.g. due to a small coefficient of friction) the dissipation is localized in the center of impact whereas for dominance of sticking, most energy is lost away from the center. it should be noted that the coefficient of tangential restitution and hence the dissipated energy can be significantly influenced by elastic parameter χ. the case χ = 0.5 is especially interesting because it can be completely energy-conserving (i.e. practically wear-less), provided that ψ < 1. whereas χ = 0.5 is impossible to realize with elastically homogeneous media due to thermodynamic stability restrictions for poisson’s ratio, dissipation-less configurations are within reach for materials with a functional elastic grading [27]. last but definitely not least, plastic deformations during the impact will gain importance for intermediate or high impact velocities. this aspect has been neglected here. although some approaches to tackle the elasto-plastic oblique impact problem have been published in the past years [28, 29], contact mechanically rigorous models for this problem are still scarce and thus require further investigation. references 1. souilliart, t., rigaud, e., le bot, a., phalippou, c., 2017, energy-based wear law for oblique impacts in dry environment, tribology international, 105, pp. 241-249. 2. tarbe, r., kulu, p., 2008, impact wear tester for the study of abrasive erosion and milling processes, 6th international daaam baltic conference industrial engineering, tallin. 3. finnie, i., 1960, erosion of surfaces by solid particles, wear, 3(2), pp. 87-103. 4. neilson, j.h., gilchrist, a., 1968, erosion by a stream of solid particles, wear, 11(2), pp. 111-122. energy loss and wear in spherical oblique elastic impacts 85 5. engel, p.a., 1978, percussive impact wear: a study of repetitively impacting solid components in engineering, tribology international, 11(3), pp. 169-176. 6. talia, m., lankarani, h., talia, j.e., 1999, new experimental technique for the study and analysis of solid particle erosion mechanisms, wear, 225-229 (part 2), pp. 1070-1077. 7. hertz, h., 1882, über die berührung fester elastischer körper, journal für die reine und angewandte mathematik, 92, pp. 156-171. 8. johnson, k.l., pollock, h.m., 1994, the role of adhesion in the impact of elastic spheres, journal of adhesion science and technology, 8(11), pp. 1323-1332. 9. thornton, c., ning, z., 1998, a theoretical model for the stick/bounce behaviour of adhesive, elasticplastic spheres, powder technology, 99(2), pp. 154-162. 10. hunter, s.c., 1957, energy absorbed by elastic waves during impact, journal of the mechanics and physics of solids, 5(3), pp. 162-171. 11. ghanbarzadeh, a., hassanour, a., neville, a., 2018, a numerical model for calculation of the restitution coefficient of elastic-perfectly plastic and adhesive bodies with rough surfaces, powder technology, accepted manuscript, doi: https://doi.org/10.1016/j.powtec.2018.12.079 . 12. maw, n., barber, j.r., fawcett, j.n., 1976, the oblique impact of elastic spheres, wear, 38(1), pp. 101-114. 13. mindlin, r.d., deresiewicz, h., 1953, elastic spheres in contact under varying oblique forces, journal of applied mechanics, 20, pp. 327-344. 14. gorham, d.a., kharaz, a.h., 2000, the measurement of particle rebound characteristics, powder technology, 112(3), pp. 193-202. 15. lyashenko, i.a., willert, e., popov, v.l., 2018, mechanics of collisions of solids: influence of friction and adhesion. i. review of experimental and theoretical works, pnrpu mechanics bulletin, 2, pp. 4461, doi: http://dx.doi.org/10.15593/perm.mech/2018.2.05 . 16. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer-verlag, berlin heidelberg, 265 p. 17. willert, e., popov, v.l., 2016, impact of an elastic sphere with an elastic half space with a constant coefficient of friction: numerical analysis based on the method of dimensionality reduction , zamm zeitschrift für angewandte mathematik und mechanik, 96(9), pp. 1089-1095. 18. doménech-carbó, a., 2013, analysis of oblique rebound using a redefinition of the coefficient of tangential restitution coefficient, mechanics research communications, 54, pp. 35-40. 19. pishkenari, h.n., rad, h.k., shad, h.j., 2017, transformation of sliding motion to rolling during spheres collision, granular matter, 19:70, doi: https://doi.org/10.1007/s10035-017-0755-0 . 20. brach, r.m., 1988, impact dynamics with applications to solid particle erosion, international journal of impact engineering, 7(1), pp. 37-53. 21. archard, j.f., hirst, w., 1956, the wear of metals under unlubricated conditions, proceedings of the royal society of london, series a, 236, pp. 397-410. 22. khrushchov, m.m., babichev, m.a., 1960, investigation of wear of metals, russian academy of sciences, moscow, 252 p. 23. honda, k., yamada, k., 1925, some experiments on the abrasion of metals, journal of the institute of metals, 33(1), pp. 49-68. 24. argatov, i.i., dmitriev, n.n., petrov, y.v., smirnov, v.i., 2009, threshold erosion fracture in the case of oblique incidence, journal of friction and wear, 30(3), pp. 176-181. 25. hunter, s.c., 1957, energy absorbed by elastic waves during impact, journal of the mechanics and physics of solids, 5(3), pp. 162-171. 26. benad, j., 2018, fast numerical implementation of the mdr transformations, facta universitatis, series mechanical engineering, 16(2), pp. 127-138. 27. willert, e., popov, v.l., 2017, the oblique impact of a rigid sphere on a power-law graded elastic halfspace, mechanics of materials, 109, pp. 82-87. 28. ghaednia, h., marghitu, d.b., 2016, permanent deformation during the oblique impact with friction, archive of applied mechanics, 86(1-2), pp. 121-134. 29. wu, y.-c., thornton, c., li, l.-y., 2009, a semi-analytical model for oblique impacts of elastoplastic spheres, proceedings of the royal society of london, series a, 465, pp. 937-960. application of the performance selection index method for solving machining mcdm problems facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 97 106 doi: 10.22190/fume151120001p © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper application of the performance selection index method for solving machining mcdm problems udc 519.8:621.9  dušan petković 1 , miloš madić 1 , miroslav radovanović 1 , valentina gečevska 2 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of mechanical engineering, university in skopje, macedonia abstract. complex nature of machining processes requires the use of different methods and techniques for process optimization. over the past few years a number of different optimization methods have been proposed for solving continuous machining optimization problems. in manufacturing environment, engineers are also facing a number of discrete machining optimization problems. in order to help decision makers in solving this type of optimization problems a number of multi criteria decision making (mcdm) methods have been proposed. this paper introduces the use of an almost unexplored mcdm method, i.e. performance selection index (psi) method for solving machining mcdm problems. the main motivation for using the psi method is that it is not necessary to determine criteria weights as in other mcdm methods. applicability and effectiveness of the psi method have been demonstrated while solving two case studies dealing with machinability of materials and selection of the most suitable cutting fluid for the given machining application. the obtained rankings have good correlation with those derived by the past researchers using other mcdm methods which validate the usefulness of this method for solving machining mcdm problems. key words: multi criteria decision making, performance selection index, machining, optimization, machinability, cutting fluid 1. introduction machining is one of the most important and widely used manufacturing processes. it is a process of material removal using cutting tools and machine tools to accurately obtain the required product dimensions with good surface finish [1]. there are many different machining processes while each of them represents a very complex and highly received november 20, 2015 / accepted may 20, 2016 corresponding author: dušan petković university of niš, faculty of mechanical engineering, а. меdvedeva 14, niš, serbia e-mail: dusan.petkovic@masfak.ni.ac.rs 98 d. petković, m. madić, m. radovanović, v. geĉevska nonlinear multi-input multi-output process. the machining process is performed in the specific tribo-mechanical system consisting of five basic groups of input parameters such as workpiece, machine tool, cutting tool, interface and cutting conditions. in machining processes practically any parameter within these groups can be varied in a wide range. this results in the change of machining performances including cutting forces, surface roughness, temperature in the cutting zone, chip formation, noise, tool wear, etc [2]. appropriate selection of workpiece and tool materials, machine tools, cutting fluids, cutting conditions and sequences is a key factor in achieving minimum cost of production or a maximum production rate, or an optimum combination of both, along with better product quality in machining [1]. numerous manufacturing firms typically select machining parameters i.e. machining conditions based on the machinist’s experience, handbook recommendations or the use of time consuming trial and error approach. however, this type of selection does not guarantee achieving optimum machining performances. the more advanced approach is based on the integration of empirical models based on the regression analysis or artificial neural networks with an optimization method. typical application examples include the application of classical optimization methods and modern meta-heuristic algorithms (such as genetic algorithms, simulated annealing, harmony search, etc.) for determining optimal machining conditions with the ultimate aim to improve surface finish, minimize tool wear, maximize productivity, minimize cost or achieve some combinations of these goals. the application of the taguchi method is also an attractive alternative, particularly in the case of multi-objective optimization problems; however, in this case one needs to integrate it with grey relational analysis, principal component analysis or utility approach. in essence, the application of the taguchi method is particularly popular since it determines robust machining conditions, i.e. machining conditions which are insensitive with respect to all causes of variation (noise factors). noise factors (external conditions, manufacturing imperfections, etc.) are unwanted sources of variation and can be uncontrollable or too expensive to control. the taguchi method application has become very popular because it represents a model-free optimization method that can be readily applied avoiding, in some cases, time consuming empirical model development. integration of empirical models and optimization methods allows for continuous single and multi-criteria optimization of machining processes. in this case, values of independent variables (machining parameters) are defined in continual or integer domain and the goal is to determine the best solution which satisfies all previously set constraints and ranges of independent variables. on the other hand, there are single and multi-criteria optimization problems where there exist a finite number of pre-known solutions (alternatives). for solving these types of optimization problems, a number of methods have been developed and these are in literature known as multi criteria decision making (mcdm) methods. in open literature, solving machining mcdm problems using mcdm methods is mainly related to machinability of materials and selection of cutting fluids [1, 3-7]. past research investigated the applicability of a number of mcdm methods including graph theory and matrix approach (gtma), technique for order preference by similarity to ideal solution (topsis), weighted aggregated sum product assessment (waspas), and preference ranking organization method for enrichment evaluation (promethee). while solving different mcdm problems there is a continuous need for a systematic and simple mathematical method for efficient and effective evaluation of solutions, i.e. alternatives. this paper is focused on the investigation of applicability and suitability of application of the performance selection index method for solving machining mcdm problems 99 the performance selection index (psi) method for solving machining mcdm problems. to the authors’ best knowledge, the psi method has not been previously applied to solving machining mcdm problems. the unique characteristic of this method lies in a fact that is not necessary to determine criteria weights as in other mcdm methods; this was the main motivation to use this mcdm method. two real time case studies were solved using this method and the obtained complete rankings were compared with the previously obtained ones by other mcdm methods. 2. preference selection index (psi) method the psi method was proposed by maniya and bhatt [8] in 2010 for solving material selection mcdm problems. unlike the most mcdm methods, the psi method does not require determination of the relative importance of the criteria, and, therefore, it is not necessary to determine criteria weights. thus, the method is particularly useful in the cases where a conflict in deciding the relative importance among criteria appears [9]. actually, the psi method determines criteria weights only by using information provided in the decision matrix, i.e. it uses an objective approach to determine criteria weights like standard deviation or entropy method. the main application procedure of the psi method for solving mcdm problems includes several steps [8-11]: step 1: determine the objective and identify the relevant criteria for evaluation of the alternatives. step 2: set the initial decision matrix, x: 11 12 1 21 22 2 1 2 . . . . . . [ ] ... ... ... ... . . . n n ij m n m m mn x x x x x x x x x x x                (1) where xij is the assessment value of i-th alternative in respect to j-th criterion, m is the number of alternatives and n is the number of criteria. step 3: determine the normalized decision matrix in which the elements of the matrix are calculated using the following equations:  for maximization (beneficial) criteria: mi x x x ij ij ij ,...,1, max  (2a)  for minimization (non-beneficial) criteria: mi x x x ij ij ij ,...,1, min  (2b) step 4: determine the mean values of normalized performances in relation to each criterion using the following equation:    m i ijx n n 1 1 (3) 100 d. petković, m. madić, m. radovanović, v. geĉevska step 5: determine the values of the variation of preferences in relation to each criterion using the following equation: 2 1 ( ) m j ij i x n    (4) step 6: determine the deviations of the value of the preference in relation to each criterion using the following equation: jj  1 (5) step 7: determine the criteria weights using the following equation:      n j j j jw 1 (6) step 8: determine the preference selection index of alternatives using the following equation: j n j iji wx   1 (7) step 9: based on the preference selection index values of the alternatives, determine the complete ranking of alternatives. the alternative which has the largest preference selection index represents the best ranked alternative. 3. case studies in this paper, an attempt is made to prove and validate the applicability of the psi method by taking into consideration two real time machining mcdm problems, i.e. analysis of materials machinability and selection of cutting fluids for a given machining application. the main objective is to compare the performance of the psi method with the other known mcdm methods for solving machining mcdm problems. 3.1. machinability of materials case study 1 the machinability of a material represents complex technical characteristic, expressed by the measure in which the material can be satisfactorily machined [7]. the machinability aspect is of considerable importance since it is necessary for production engineers to know in advance the machinability of work materials so that machining can be planned in an efficient manner. the study can also be a basis for cutting tool and cutting fluid performance evaluation and machining parameter optimization [1]. enache et al. [7] proposed a new mathematical model for the establishment of the partial and global machinability of materials. the mathematical model is verified experimentally in the case of turning of titanium alloys using different cutting tools of different geometries. turning experiments have been performed without the use of cutting fluid, using cutting speed of 150 m/min, feed rate of 0.15 mm/rev and depth of cut of 0.5 mm. six workpiece-tool material combinations (tial6v4-p20, timo32-p20, application of the performance selection index method for solving machining mcdm problems 101 tial5fe2.5-p20, tial6v4-p20 (tin), tial6v4-k20 and tial6v4-k20*) and three evaluation criteria such as tool wear rate (twr), specific energy consumed (sec) and surface roughness (sr) are considered. experimental results are given in table 1. table 1 decision matrix for case study 1 [7] work-tool combination twr (m/min) sec (j) sr (µm) (1) tial6v4-p20 0.061 219.74 5.8 (2) timo32-p20 0.093 3523.72 6.3 (3) tial5fe2.5-p20 0.064 2693.21 6.8 (4) tial6v4-p20 (tin) 0.028 761.46 5.8 (5) tial6v4-k20 0.034 1593.48 5.8 (6) tial6v4-k20* 0.013 2849.15 6.2 in this study all three criteria are minimization criteria i.e. lower attribute values of alternatives are preferred. in other words, the materials machinability is better if during machining low values of twr. sec and sr occur. enache et al. [7] determined criteria weights as wtwr=0.688, wsec=0.231 and wsr=0.081 and the same were used by rao and gandhi [12], rao [3] and chakraborty and zavadskas [4] while applying graph theory and matrix approach (gtma), topsis and waspas methods, respectively. 3.2. selection of cutting fluids case study 2 during machining lubricants/cutting fluids have different effects including cooling, lubricating, mechanics, diffusing and washing. since cutting fluids serve many useful functions during machining, and, on the other hand, incur one portion of the total manufacturing cost, the appropriate selection of cutting fluids is very important. rao and patel [6] applied promethee method for solving cutting fluid selection problem for a cylindrical grinding operation. four cutting fluids, i.e. alternatives, were evaluated based on eight criteria such as wheel wear (ww), tangential force (tf), grinding temperature (gt), surface roughness (sr), recyclability (r), toxic harm rate (th), environment pollution tendency (ep) and stability (s). alternative attributes with respect to r, th, ep and s were expressed linguistically and subsequently were transformed into real crisp values using fuzzy scales [1]. decision matrix for this case study is given in table 2. among the eight criteria, only r and s are the maximization criteria where higher attribute values of alternatives are preferred. by using the ahp method, rao and patel [6] determined criteria weights as: www=0.3306, wtf=0.0718, wgt=0.1808, wsr=0.0718, wr=0.0459, wth=0.126, wep=0.126 and ws=0.0472. this set of criteria weights was also used by chakraborty and zavadskas [4] whereas the waspas method was applied. table 2 decision matrix for case study 2 [6] cutting fluid ww (mm) tf (n) gt (ºc) sr (µm) r th ep s 1 0.035 34.5 847 1.76 l (0.335) a (0.5) aa (0.59) aa (0.59) 2 0.027 36.8 834 1.68 l (0.335) h (0.665) h (0.665) h (0.665) 3 0.037 38.6 808 2.4 aa (0.59) aa (0.59) ba (0.41) a (0.5) 4 0.028 32.6 821 1.59 a (0.5) aa (0.59) aa (0.59) ba (0.41) l – low; ba – below average; a – average; aa: above average; h – high 102 d. petković, m. madić, m. radovanović, v. geĉevska 4. results and discussion 4.1. application of the psi method the detailed computational procedure of the psi method for solving the afore-stated machining mcdm problems is as follows. since decision matrices for both case studies are already given, the application of the psi method starts with step 3. by using eq. 2 the normalized decision matrix for case study 1 is obtained as shown in table 3. the purpose of normalization is to obtain dimensionless values of alternative attribute values so that all these alternatives can be easily evaluated. table 3 normalized decision matrix for case study 1 0.2131 1.0000 1.0000 0.1398 0.0624 0.9206 0.2031 0.0816 0.8529 0.4643 0.2886 1.0000 0.3824 0.1379 1.0000 1.0000 0.0771 0.9355 using the data from table 3, and by applying eq. 3, mean values of normalized performances in relation to each criterion are obtained as n=[0.4004, 0.2746, 0.9515]. subsequently by applying eq. 4 values of the variation of preferences in relation to each criterion are calculated (table 4). by using eq. 5 deviations of the value of the preference in relation to each criterion is obtained as j=[0.4942, 0.3336, 0.982]. subsequently, by using eq. 6 criteria weights are determined as w=[0.273, 0.1843, 0.5426]. finally, by using eq. 7 preference selection indexes of alternatives are determined upon which complete ranking of alternatives is obtained (table 5). table 4 values of the variation of preferences for case study 1 0.0351 0.5262 0.0024 0.0679 0.0450 0.0010 0.0389 0.0372 0.0097 0.0041 0.0002 0.0024 0.0003 0.0187 0.0024 0.3595 0.0390 0.0003 table 5 performance selection index values of alternatives and complete ranking for case study 1 work-tool combination tial6v4 -p20 timo32 -p20 tial5fe2.5 -p20 tial6v4 -p20 (tin) tial6v4 -k20 tial6v4 -k20* i 0.7851 0.5492 0.5333 0.7226 0.6724 0.7949 rank 2 5 6 3 4 1 from table 5 it is observed that the ranking of machinability of titanium alloy-based work materials determined by the psi method is 2-5-6-3-4-1. thus maximum machinability is obtained with work-tool combination tial6v4-k20*. application of the performance selection index method for solving machining mcdm problems 103 again, the detailed computational procedure of the psi method for selection of cutting fluids is as follows. by using eq. 2 the normalized decision matrix is obtained (table 6). table 6 normalized decision matrix for case study 2 0.7714 0.9449 0.9540 0.9034 0.5678 1.0000 0.6949 0.8872 1.0000 0.8859 0.9688 0.9464 0.5678 0.7519 0.6165 1.0000 0.7297 0.8446 1.0000 0.6625 1.0000 0.8475 1.0000 0.7519 0.9643 1.0000 0.9842 1.0000 0.8475 0.8475 0.6949 0.6165 using the data from table 6, and by applying eq. 3, mean values of normalized performances in relation to each criterion are obtained as n=[0.8664, 0.9188, 0.9767, 0.8781, 0.7458, 0.8617, 0.7516, 0.8139]. subsequently by applying eq. 4 values of the variation of preferences in relation to each criterion are calculated (table 7). table 7 values of the variation of preferences for case study 2 0.0090 0.0007 0.0005 0.0006 0.0317 0.0191 0.0032 0.0054 0.0179 0.0011 0.0001 0.0047 0.0317 0.0121 0.0182 0.0346 0.0187 0.0055 0.0005 0.0465 0.0646 0.0002 0.0617 0.0038 0.0096 0.0066 0.0001 0.0149 0.0103 0.0002 0.0032 0.0390 by using eq. 5 deviations of the value of the preference in relation to each criterion are obtained as j=[0.9449, 0.9861, 0.9988, 0.9333, 0.8617, 0.9684, 0.9136, 0.9172]. subsequently, by using eq. 6 criteria weights are determined as wj=[0.1256, 0.1311, 0.124, 0.1145, 0.1287, 0.1214, 0.1219]. finally, by using eq. 7 preference selection indexes of alternatives are determined upon which complete ranking of alternatives is obtained (table 8). table 8 performance selection index values of alternatives and complete ranking for case study 2 cutting fluid 1 2 3 4 i 0.8457 0.8463 0.8539 0.8725 rank 4 3 2 1 as could be seen from table 8 by applying the psi method, the ranking of cutting fluids is obtained as 4-3-2-1. in other words, cutting fluid 4 is observed to be the most appropriate for this machining application. cutting fluid 3 has the second preference and cutting fluid 1 is the least favored. 4.2. comparison of complete rankings the afore-stated machining mcdm problems were solved by past researchers using different mcdm methods. in this section the comparison of the obtained complete rankings is presented. it has to be noted that the past research while applying different mcdm methods used different set of criteria weights, which significantly affected final rankings. 104 d. petković, m. madić, m. radovanović, v. geĉevska rao and gandhi [12], rao [3] and chakraborty and zavadskas [4] solved case study 1 applying gtma, topsis and waspas methods, respectively. the comparison of the complete rankings obtained by different mcdm methods and the psi method is given in fig. 1. the results show some difference between the psi and other mcdm methods. the reason for this is the fact that the other methods performed the analysis using the criteria weights as wtwr=0.688, wtwr=0.231 and wsr=0.081 while the criteria weights determined by the psi method were wtwr=0.273, wtwr=0.1843 and wsr=0.5426. thus, a drastic change in the priorities among criteria is evident so that one can explain differences in the obtained rankings of alternatives. however, it can be observed that work-tool combination 6 (tial6v4-k20*) received the highest ranking by all methods as well as by the psi; hence, it may be unequivocally regarded as the most appropriate for the given machining application. this mcdm model for the analysis of the machinability of materials shows a high degree of robustness since prioritization of alternatives is not significantly affected by drastic changes in the criteria weights. 1 2 3 4 5 6 1 2 3 4 5 6 r an k work-tool combination topsis gtma waspas psi fig. 1 comparative rankings of different mcdm methods for case study 1 rao and patel [6], rao [1] and chakraborty and zavadskas [4] solved case study 2 while applying promethee, topsis and waspas methods, respectively. the comparison of the complete rankings obtained by different mcdm methods and the psi method is given in fig. 2. 1 2 3 4 5 6 1 2 3 4 r an k cutting fluid topsis waspas promethee psi fig. 2 comparative rankings of different mcdm methods for case study 2 application of the performance selection index method for solving machining mcdm problems 105 from fig. 2 it can be observed that alternatives rankings obtained by the psi method perfectly match with those obtained by the waspas method. on the other hand, promeethe and topsis method obtained the same rankings of alternatives. however, it is evident that cutting fluid 4 has been identified as the best choice for the given machining application by all mcdm methods. also, cutting fluid 1 received the worst ranking by all mcdm methods. 5. conclusion in order to improve total machining performances and get the most from machine tool, production engineers formulate different machining optimization problems. this paper presents the application of a new mcdm method i.e. psi method for solving discrete machining optimization problems. the detail computational procedure of the psi method is demonstrated while solving two case studies dealing with machinability of materials and selection of the most suitable cutting fluid for the given machining application. the conclusions drawn can be summarized by the following points:  the obtained rankings have good correlation with those derived by the past researchers using different mcdm methods which validate the usefulness of the psi method for solving machining mcdm problems. in both case studies, it is observed that the topranked alternatives exactly match with those derived by the past researchers.  the main advantage and benefit of the psi method is that it is not necessary to determine criteria weights as in other mcdm methods. actually, by using only data from decision matrix, the method calculates criteria weights in an objective manner. this unique characteristic has double benefit. first, the cases where there is a conflict in deciding the relative importance among criteria are avoided. second, the decision making process is accelerated.  computational procedure of the psi method, although performed in nine steeps, is relatively simple and can be easily traced by the decision maker. solving different mcdm problems by using the psi method does not requires the use of specialized software packages since the method can be easily implemented in ms excel.  by calculating preference selection index values complete ranking of alternatives is enabled.  evaluation of alternatives is not affected by the introduction of any additional parameters as it happens in case of other mcdm methods, like λ in the waspas method. finally, it should be noted that further investigation will include possibilities for the psi method upgrading and comparisons with other mcdm methods for solving different mcdm problems in manufacturing environment. acknowledgements: the paper is a part of the research done within the project tr35034 supported by ministry of science and technological development of the republic of serbia. 106 d. petković, m. madić, m. radovanović, v. geĉevska references 1. rao, r. v., 2007, decision making in the manufacturing environment: using graph theory and fuzzy multiple attribute decision making methods, springer-verlag london. 2. radovanović m., madić, m., 2010, methodology of neural network based modeling of machining processes, international journal of modern manufacturing technologies, 2(2), pp. 77-82. 3. rao, r.v., 2006, machinability evaluation of work materials using a combined multiple attribute decision making method, international journal of advanced manufacturing technology, 28(3–4), pp. 221–227. 4. chakraborty, s., zavadskas, e.k., 2014, applications of waspas method in manufacturing decision making, informatica, 25(1), pp. 1–20. 5. petković, d., madić, m., radenković, g., 2015, selection of the most suitable non-conventional machining processes for ceramics machining by using mcdms, science of sintering, 47, pp. 229-235. 6. rao, r.v., patel, b.k., 2010, decision making in the manufacturing environment using an improved promethee method, international journal of production research, 48(16), pp. 4665–4682. 7. enache, s., strajescu, e., opran, c., minciu, c., zamfirache, m., 1995, mathematical model for the establishment of the materials machinability, cirp annals, 44, pp. 79–82. 8. maniya, k., bhatt, m.g., 2010, a selection of material using a novel type decision-making method: preference selection index method, materials and design, 31, pp. 1785–1789. 9. singh t., patnaik a, gangil b., chauhan r., 2015, optimization of tribo-performance of brake friction materials: effect of nano filler, wear, 324-325, pp. 10-16. 10. khorshidi r., hassani a., 2013, comparative analysis between topsis and psi methods of materials selection to achieve a desirable combination of strength and workability in al/sic composite, materials and design, 52, pp. 999–1010. 11. vahdani, b., zandieh, m., tavakkoli-moghaddam, r., 2011, two novel fmcdm methods for alternative-fuel buses selection, applied mathematical modelling, 35, pp. 1396–1412. 12. rao, r.v., gandhi, o.p., 2002, digraph and matrix methods for the machinability evaluation of work materials, international journal of machine tools and manufacture, 42(3), pp. 320–330. 10562 facta universitatis series:mechanical engineering vol. 20, no 1, 2022, pp. 177 197 https://doi.org/10.22190/fume220215012d © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper impact of the number of vehicles on traffic safety: multiphase modeling milanko damjanović1, željko stević2, dragan stanimirović3, ilija tanackov4, dragan marinković5 1university of montenegro, faculty of mechanical engineering, podgorica, montenegro 2university of east sarajevo, faculty of transport and traffic engineering, doboj, bosnia and herzegovina 3ministry of transport and communications of republic of srpska, banja luka, bosnia and herzegovina 4university of novi sad, faculty of technical sciences, novi sad, serbia 5faculty of mechanical engineering and traffic systems, tu berlin, germany abstract. traffic safety is one of the key issues nowadays, given the fact that a large number of people lose their lives in traffic accidents every day. there are various influential factors in the occurrence of traffic accidents, the number of vehicles being one of them. this paper assesses the traffic safety in montenegro in the period 1998-2020 by applying the multiphase modeling with a purpose to obtain comparative results which enable implementation of adequate strategies. a total of six scenarios were formed with two inputs and two outputs in a dea (data envelopment analysis) model, with the number of registered vehicles per year being an input in all scenarios. in addition, as inputs, the scenarios included aadt (annual average daily traffic), passengers in road transport, passenger-km by road transport, goods transported by road, tone-km by road, and passengers in local transport. the number of traffic accidents with casualties, the number of traffic accidents with material damage, the number of fatal cases and the number of injured persons, depending on a scenario, were observed as outputs. after the dea model, imf swara (improved fuzzy stepwise weight assessment ratio analysis) was applied to determine the weights of inputs and outputs, while the final state of traffic safety by years was determined using the marcos (measurement of alternatives and ranking according to compromise solution) method. key words: vehicle, traffic safety, imf swara, marcos, traffic accident received february 15, 2022 / accepted march 16, 2022 corresponding author: željko stević university of east sarajevo, faculty of transport and traffic engineering, vojvode mišića 52, 74000 doboj, bosnia and herzegovina email-address zeljko.stevic@sf.ues.rs.ba, zeljkostevic88@yahoo.com 178 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković 1. introduction globalization, as one of important factors nowadays in every field, affects the course of traffic and transport processes as well as the risks of possible occurrence of undesirable situations. certainly, for years, traffic safety has been a burning issue that is a subject of daily professional and scientific analysis. its significance does not need to be emphasized and described too much since every participant in traffic strives to make it as safe as possible. it is necessary to find adequate measures and balance between frequently conflict situations. modern motor vehicles with exceptional performance, excellent mechanical characteristics, equipped in a way to provide greater comfort and safety represent, on the one hand, a significant contribution to this area. however, on the other hand, if it is added a shorter time needed to reach high speeds, which also frequently leads to unsafe overtaking, we have an increased risk of accidents. it reflects the interaction between vehicles and people as important factors that can influence the emergence of risky traffic situations. this paper analyzes traffic safety in montenegro for a period of 23 years through creating multiphase modeling. it involves creating different scenarios with different impact parameters and integrating multiple approaches into a single model. according to podvezko and sivilevičius [1], a system of road transport involves vehicles, roads, traffic participants and freight that are interconnected. transportation infrastructure and logistics are core elements supporting trade facilitation efforts at the local level [2] and, consequently, mobility and an increasing number of vehicles involved in traffic. therefore, this paper analyzes the impact of various factors: the number of registered motor vehicles, aadt, passengers in road transport, passenger-km by road transport, goods transported by road, tonne-km by road, and passengers in local transport. the number of traffic accidents with casualties, the number of traffic accidents with material damage, the number of dead persons and the number of injured persons, depending on a scenario, were observed as outputs. motivation for writing this paper can be explained through the necessity of existing original and quality quantitative model which can be base for bringing adequate strategies which should increase traffic safety field. the goal of creating an integrated model implies the overall quantification of the safety for a specified period and the possibility of identifying a benchmark year according to which further strategies will be created. the main contribution of the study is the developed integrated dea-imf swara-marcos model presented for the first time in literature which can be used for solving various transport, traffic, and others problems. another contribution is the fact that it is through the applied model that the possibility for preventive engineering in traffic safety can be created. the rest of the paper is described through the following five sections. section 2 provides a brief review of the application of different methodologies in the field of traffic safety. section 3 presents materials and methods, giving a detailed overview of all the data used in this paper. additionally, scenarios with a description of influential input-output factors are formed, and the methods that make up the developed integrated model are presented in detail. in section 4, the results obtained for all scenarios are presented and discussed. section 5 is the analysis of the impact of the number of motor vehicles on the occurrence of traffic accidents using regression analysis. the final, section 6, summarizes the conclusions along with guidelines for continuing the research. impact of the number of vehicles on traffic safety: multiphase modeling 179 2. literature review the dea model has often been applied in the field of traffic safety, sometimes individually, and very often in integration with other approaches to ensure the most accurate decision-making. this section of the paper presents the studies that used dea in a whole model or a phase of a model for determining traffic safety. 2.1. the application of dea model in traffic safety mozaffaria et al. [3] applied the dea model to assess traffic safety culture in order to achieve desired safety performance. they observed a three-year period and analyzed a total of 31 provinces in iran. based on the results obtained, they concluded that road safety culture increased on average. the double-frontier dea was applied in [4, 5] to assess the efficiency of iranian safety programs in order to reduce the number of traffic accidents with fatalities. a serial two-stage additive dea model [6] was applied to analyze traffic safety performance in 31 provinces in china. the results showed certain differences between the regions in the level of traffic safety, and suggested certain procedures for adequate traffic safety management. the authors of the paper [7] have evaluated 197 municipalities in terms of traffic safety, applying a dea model, which consists of several phases. in the same paper, different scenarios with two inputs and 14, eight and six outputs, respectively, were modeled. fancello et al. [8] compared ccr and bcc models in order to support traffic management in terms of urban road safety. the goal was to identify the critical roads that have the greatest need for intervention and increase in traffic safety. the fuzzy form of dea method can be successfully applied in the traffic safety field. for example, fuzzy dea has been applied in [9] to evaluating road safety index in iran. 2.2. the application of integrated models in traffic safety infrastructure improvement is one of significant instruments for increasing traffic safety, as stated in the paper [10] in which dea and gis (geographical information system) were combined to assess the risk level of problematic road segments with a length of 100 km. in that way, traffic safety is improved through locating and visualizing problematic points on the observed road segment. in [11], a combination of pca-dea model was used considering undesirable input and output indices. the authors state that the advantage of the applied approach is the benchmarking of the safest roads in order to best allocate budget funds in the field of traffic safety. stanković et al. [12] extended the marcos method with fuzzy numbers to determine traffic safety in bosnia and herzegovina on defined road sections. a total of 38 short sections of 200 m each were evaluated based on six influential factors. the original critic (the criteria importance through intercriteria correlation), fuzzy fucom (full consistency method), dea, and fuzzy marcos model were created in [13] which evaluated nine sections of the road network based on eight criteria divided into four inputs and four outputs. the methodology for assessing road sections is similar to that in this paper because a dea method was applied first, and then the others depending on their purposefulness. the integration of bwm (best worst method) with a dea model being modified to be applicable and adaptable in the field of traffic safety has been applied in [14]. a dea-rs (road safety) model has been defined and verified through a case study in iran. an integrated dea and monte-carlo simulation prioritizing approach is proposed 180 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković in [15] to determine the prioritization of traffic safety management projects. stević et al. [16] have created an original dea-critic-marcos model for evaluating traffic safety on 17 important roads of south africa city. in comparison to other studies, the authors have used dea for calculation criteria weights instead of initial safety performance. 3. materials and methods 3.1. research description and problem setting a research flow diagram is shown in fig. 1. it consists of forming a total of six scenarios where different inputs and outputs are modeled. a multi-phase model of traffic accident analysis, which includes a dea-imf swara-marcos model has been applied. dea ccr model has been selected because of its simplicity and previous exploration in literature. however, the power discrimination of the dea model can be low in many cases. for that reason, we have applied imf swara and marcos model in order to obtain final results with clear differences between the variants. fig. 1 a research flow diagram inputs are results of integration mostly literature review and dialogues with experts. the first scenario s1 includes aadt (annual average daily traffic) [13, 16] and the number of registered vehicles (nrv) [5,17,19] as inputs; and traffic accidents with casualties (tac) and traffic accidents with material damage (tamd) [20]. overall data for the first scenario are presented in table 1. =1 y e s no outputs: s1, s4 and s5: tac, tamd s2, s3 and s6: dp, ip impact of the number of vehicles on traffic safety: multiphase modeling 181 the second scenario s2 implies the number of passengers in road transport as the first input, and the number of registered vehicles (which are one of the inputs in each scenario). in this scenario, the number of dead persons (dp) [21] and the number of injured persons (ip) [14, 20] are outputs. the data for the second scenario are presented in fig. 2. table 1.data for the dea model for the first scenario s1 aadt nrv tac tamd 2009 7388 183,441 1718 8394 2010 7164 187,913 1520 7618 2011 7140 196,419 1451 7068 2012 5593 197,826 1217 6886 2013 4733 203,266 1266 3998 2014 6440 196,059 1334 4197 2015 7471 198,772 1554 3390 2016 7912 209,098 1698 3531 2017 7969 219,378 1831 3847 2018 8953 235,385 1855 4017 2019 4078 249,301 1924 4286 2020 3052 240,611 1490 3102 fig. 2 data for the dea model for the second scenario s2 the third scenario implies only one change compared to the previous, second scenario, and it is reflected through the change of the first input. instead of passengers in road transport, the results with passenger-km by road transport [20, 22] were modeled and the data are presented in fig. 3. 182 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković fig. 3 data for the dea model for the third scenario s3 fig. 4 shows the data for the fourth and fifth scenarios since the only difference is in the first input. transport according to sénquiz-díaz [23] remains a key development factor in any country and has an influence on the transport of goods. in the fourth scenario, the first input is goods transported by road [6] in thousands, while tkm [14] is the first input in the fifth scenario. fig. 4 data for the dea model for the fourth (s4) and fifth scenario (s5) impact of the number of vehicles on traffic safety: multiphase modeling 183 fig. 5 shows the data for the last, sixth scenario (s6) for the dea model in which the inputs are passengers in local road transport and the number of registered vehicles, and the outputs are the number of dead persons and the number of injured persons in traffic accidents. fig. 5 data for the dea model for the sixth scenario after that, the dea model was applied for each scenario separately. in the first scenario, it was observed the period 2009-2020 considering the availability of data for aadt, while the period from 1998 to 2020 was observed in other scenarios. data on traffic accidents sorted by categories are available for the whole period, while data on the number of registered motor vehicles, passengers in road transport, passenger-km, goods transported by road, tonne-kilometers by road and passengers in local transport are available for limited periods, maximum for 15 years. in order to obtain the most relevant analysis for each of these elements, the annual increase or decrease was calculated, and based on that, the average annual trend was calculated. finally, data for the previous historical period that was missing were obtained by applying a linear model based on the calculated annual trend. it is important to note that 2020 is not taken into account in these calculations due to the conditions of covid-19 and the limitations caused by the pandemic. also, the goods transported by road were not taken for 2012 as an input since there was a drastic decline compared to the previous two years. clear advantages of the applied methods are presented in [24] and [25], respectively. 3.2. dea method here, a dea (data envelopment analysis) ccr input model [26-28] is formed for traffic safety evaluation. an input oriented model is: 184 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković 1 1 1 1 max : 0, 1,..., 1 0, 1,..., m input i i input i m m s i ij i ij i i m m s i i output i m i dea w x st w x w y j n w y w i m s − = + = = + + − = + = −  = =  = +     (1) in this model, a decision-making unit (dmu) consists of m inputs for each xij, while s represents outputs for each yij. 3.3. imf swara method the imf swara method is a recently developed method presented byvrtagić et al. [24]. it consists of the following steps [29, 30]. step 1: the criteria were arranged in descending order based on their expected significance. step 2: starting from the previously determined rank, the significance of the criterion (cj) was determined in relation to the previous one (cj−1) according to the scale represented in table 2, and this was repeated for each subsequent criterion. this relation is marked with j . table 2 linguistics and the tfn scale for comparing criteria in the imf swara method linguistic variable abbreviation tfn scale absolutely less significant als (1, 1, 1) dominantly less significant dls (1/2, 2/3, 1) much less significant mls (2/5, 1/2, 2/3) really less significant rls (1/3, 2/5, 1/2) less significant ls (2/7, 1/3, 2/5) moderately less significant mdls (1/4, 2/7, 1/3) weakly less significant wls (2/9, 1/4, 2/7) equally significant es (0, 0, 0) step 3: the fuzzy coefficient was determined j : (1,1,1) 1 1 j j j j   = =   (2) impact of the number of vehicles on traffic safety: multiphase modeling 185 step 4: the calculated weights were determined j : 1 (1,1,1) 1 1 jj j j j   − =  =    (3) step 5: the fuzzy weight coefficients were calculated using the following eq. (4): 1 j j m j j w   = =  (4) where wj represents the fuzzy relative weight of criteria j, and m represents the total number of criteria. 3.4. marcos method the measurement alternatives and ranking according to compromise solution (marcos) method [25] is based on defining the relationship between alternatives and reference values (ideal and anti-ideal alternatives). the marcos method is performed through the following steps [31, 32]. step 1: formation of an initial decision-making matrix. step 2: formation of an extended initial matrix with the ideal (ai) and anti-ideal (aai) solution. 1 2 1 2 11 11 12 2 21 22 2 1 22 21 ... ... ... ... ... ... ... ... ... ... ... n aanaa aa n n m m mn aiai ain c c c xx xaai xa x x a x x x x a x xx ai xx x         =            (5) min maxij ij i i aai x if j b and x if j c=   (6) max minij ij ii ai x if j b and x if j c=   (7) step 3: normalization of extended initial matrix (x). ai ij ij x n if j c x =  (8) 186 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković ij ij ai x n if j b x =  (9) where elements xij and xai represent the elements of matrix x. step 4: determination of weighted matrix [ ]ij m nv v = . ij ij jv n w=  (10) step 5: calculation of the utility degree of alternatives ki. i i aai s k s − = (11) i i ai s k s + = (12) where si (i=1,2,..,m) represents the sum of the elements of weighted matrix v, eq. (13). 1 n i ij i s v = = (13) step 6: determination of the utility function of alternatives f(ki). ( ) ; 1 ( ) 1 ( ) 1 ( ) ( ) i i i i i i i k k f k f k f k f k f k + − + − + − + = − − + + (14) where ( )if k − represents the utility function in relation to the anti-ideal solution, while ( )if k + represents the utility function in relation to the ideal solution. utility functions in relation to the ideal and anti-ideal solution are determined by applying eqs. (15) and (16). ( ) i i i i k f k k k + − + − = + (15) ( ) i i i i k f k k k − + + − = + (16) step 7: ranking the alternatives is based on the final values of utility functions. it is desirable that an alternative has the highest possible value of the utility function. 4. results this section presents in detail the results obtained by applying multiphase modeling of the impact of various factors, primarily motor vehicles on traffic safety in montenegro over a period of 23 years (1998-2020). impact of the number of vehicles on traffic safety: multiphase modeling 187 4.1. results after applying the dea model the results of the dea model for all six scenarios are shown in table 3. table 3 dea model results for all scenarios s1 s2 s3 s4 s5 s6 1998 0.415 0.391 0.719 0.948 0.555 1999 0.577 0.544 0.756 0.921 0.591 2000 0.703 0.664 0.892 1.000 0.712 2001 0.711 0.672 0.976 1.000 0.712 2002 0.777 0.734 0.952 0.987 0.786 2003 0.857 0.811 1.000 1.000 0.840 2004 0.853 0.808 0.977 0.962 0.823 2005 0.787 0.746 0.903 0.876 0.764 2006 0.693 0.657 0.799 0.765 0.678 2007 0.573 0.544 0.676 0.640 0.576 2008 0.663 0.630 0.735 0.688 0.627 2009 0.874 0.677 0.644 0.769 0.712 0.658 2010 0.958 0.818 0.778 0.887 0.815 0.761 2011 1.000 0.885 0.823 0.900 0.850 0.884 2012 1.000 1.000 1.000 1.000 1.000 1.000 2013 1.000 0.996 0.976 0.995 0.992 1.000 2014 1.000 1.000 0.930 0.910 0.939 1.000 2015 1.000 1.000 0.888 0.805 0.900 1.000 2016 1.000 0.941 0.772 0.832 0.852 1.000 2017 0.951 0.886 0.768 0.779 0.775 0.952 2018 1.000 1.000 0.990 0.788 0.786 0.955 2019 0.822 1.000 1.000 0.807 0.802 0.953 2020 1.000 1.000 1.000 1.000 1.000 1.000 the results after applying the dea model show that in the period 1998-2010, 2003 has a satisfactory situation in terms of traffic safety only in scenarios s4 (inputs are nrv and goods) and s5 (inputs are nrv and tkm). in other years, there is no adequate level, which is in a way understandable because over time it is resorted to modern preventive engineering that gives certain results. in the first scenario when nrv and aadt were considered as inputs in the period 2009-2020, a significant number of years (eight out of 12) show a satisfactory level of safety. taking this into account, modeling using other inputs in combination with the number of registered motor vehicles proves to be justified. the two years that can be singled out as a benchmark based on the dea model are 2012 and 2020. however, it is necessary to make their comparative analysis. aadt in 2020 is lower (3052) compared to 2012 (5593), surely due to the limitations caused by the pandemic, so 2012 is certainly better in such conditions. the situation is similar with the scenarios s2, s3 and s5 when it comes to passengers and passenger-km, while the situation in s4 is different since in 2012, compared to all other years, the lowest amount of goods transported by road was recorded, namely only 398 thousand tons. in the sixth scenario, the values of these two observation years are approximate when it comes to the number of local passengers. 188 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković fig. 6 shows a statistical analysis that involves the calculation of the spearman [33] and ws [34] correlation coefficients. fig. 6 scc and ws after applying the dea model fig. 6 shows large deviations in ranks using different scenarios (s2-s6). the second, third and sixth scenarios have the highest correlation, which is very high according to the scc, while it is lower according to the ws coefficient because these are changes in the initial ranks. other correlations are very low. 4.2. results of determining the weights of the criteria using the imf swara method after the application of the dea model, the final results have not been obtained since a large number of dmus, i.e. observation years, have a value of one, depending on a scenario, so it is necessary to apply the mcdm model for their final ranking. this section presents the results of determining the significance of the criteria using the imf swara method (table 4). impact of the number of vehicles on traffic safety: multiphase modeling 189 table 4 calculation process and weights of criteria applying the imf swara method s1 crisp value tac 1.00 1.00 1.00 1.00 1.00 1.00 0.30 0.31 0.32 0.31 aadt 0.22 0.25 0.29 1.22 1.25 1.29 0.78 0.80 0.82 0.23 0.25 0.26 0.25 nrv 0.00 0.00 0.00 1.00 1.00 1.00 0.78 0.80 0.82 0.23 0.25 0.26 0.25 tamd 0.22 0.25 0.29 1.22 1.25 1.29 0.60 0.64 0.67 0.18 0.20 0.21 0.20 3.16 3.24 3.31 s2 crisp value dp 1.00 1.00 1.00 1.00 1.00 1.00 0.32 0.33 0.34 0.33 ip 0.25 0.29 0.33 1.25 1.29 1.33 0.75 0.78 0.80 0.24 0.26 0.27 0.26 nrv 0.22 0.25 0.29 1.22 1.25 1.29 0.58 0.62 0.65 0.19 0.21 0.22 0.21 pass. road 0.00 0.00 0.00 1.00 1.00 1.00 0.58 0.62 0.65 0.19 0.21 0.22 0.21 2.92 3.02 3.11 s3 crisp value dp 1.00 1.00 1.00 1.00 1.00 1.00 0.30 0.31 0.32 0.31 ip 0.29 0.33 0.40 1.29 1.33 1.40 0.71 0.75 0.78 0.21 0.23 0.25 0.23 nrv 0.00 0.00 0.00 1.00 1.00 1.00 0.71 0.75 0.78 0.21 0.23 0.25 0.23 pass. road 0.00 0.00 0.00 1.00 1.00 1.00 0.71 0.75 0.78 0.21 0.23 0.25 0.23 3.14 3.25 3.33 s4 crisp value tac 1.00 1.00 1.00 1.00 1.00 1.00 0.30 0.31 0.32 0.31 nrv 0.22 0.25 0.29 1.22 1.25 1.29 0.78 0.80 0.82 0.23 0.25 0.26 0.25 tamd 0.00 0.00 0.00 1.00 1.00 1.00 0.78 0.80 0.82 0.23 0.25 0.26 0.25 goods 0.22 0.25 0.29 1.22 1.25 1.29 0.61 0.64 0.67 0.18 0.20 0.21 0.20 3.16 3.24 3.31 s5 crisp value tac 1.00 1.00 1.00 1.00 1.00 1.00 0.30 0.31 0.32 0.31 nrv 0.22 0.25 0.29 1.22 1.25 1.29 0.78 0.80 0.82 0.23 0.25 0.26 0.25 tamd 0.00 0.00 0.00 1.00 1.00 1.00 0.78 0.80 0.82 0.23 0.25 0.26 0.25 tkm 0.22 0.25 0.29 1.22 1.25 1.29 0.61 0.64 0.67 0.18 0.20 0.21 0.20 3.16 3.24 3.31 s6 crisp value dp 1.00 1.00 1.00 1.00 1.00 1.00 0.33 0.34 0.35 0.34 ip 0.22 0.25 0.29 1.22 1.25 1.29 0.78 0.80 0.82 0.26 0.27 0.29 0.27 nrv 0.22 0.25 0.29 1.22 1.25 1.29 0.60 0.64 0.67 0.20 0.22 0.23 0.22 pass. local 0.22 0.25 0.29 1.22 1.25 1.29 0.47 0.51 0.55 0.15 0.17 0.19 0.17 2.85 2.95 3.03 from the results obtained using the imf swara method it can be concluded that the most important factors are tac in the first, fourth, and fifth scenarios, while dp is the js jk jq jw js jk jq jw js jk jq jw js jk jq jw js jk jq jw js jk jq jw 190 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković most important criterion in the second, third and sixth scenarios. nrv is the second or third most important criterion in all the scenarios, while the other factors related to local passengers, goods and tkm are less important criteria, but not with significant differences. 4.3. determining the final rank by applying the marcos method the years with a value of 1.000 after the application of the dea model, shown in table 3, represent alternatives in the further implementation of the marcos method. this section of the paper will provide the results for all the scenarios, and a detailed presentation of the calculation only for the first scenario in which the criteria are aadt, the number of registered motor vehicles, the number of traffic accidents with casualties and the number of traffic accidents with material damage. out of a total of 12 observation years, eight have a value of one, which means that in those years, traffic safety in relation to the observed data set is at a relatively satisfactory level. in 2009, 2010, 2017 and 2019, there is a large number of traffic accidents of both classifications, and 2019 stands out in particular, with 1924 traffic accidents with casualties, despite the very low aadt (4078) compared to other years. in the first step of the marcos method, the initial matrix is formed, while in the second step, by applying eqs. (6) and (7), the ideal and anti-ideal solutions are determined and, based on that, an extended initial decision matrix shown in table 5 is formed. table 5 initial extended matrix aadt nrv tac tamd antiideal 3052.000 196059.000 1855.000 7068.000 2011 7140 196,419 1451 7068 2012 5593 197,826 1217 6886 2013 4733 203,266 1266 3998 2014 6440 196,059 1334 4197 2015 7471 198,772 1554 3390 2016 7912 209,098 1698 3531 2018 8953 235,385 1855 4017 2020 3052 240,611 1490 3102 ideal 8953.000 240611.000 1217.000 3102.000 it is important to note that the first and second criteria belong to the benefit criteria where a maximum value is desirable, while the third and fourth criteria belong to those for which the minimum value is desirable. in the third step, the data presented in table 5 are normalized based on eqs. (8) and (9) as follows. 13 1217 0.839 1451 ai ij ij x n if j c n x =   = = 11 7140 0.797 8953 ij ij ai x n if j b n x =   = = other normalized values are obtained in an identical way depending on the orientation of the criteria and are presented in table 6. impact of the number of vehicles on traffic safety: multiphase modeling 191 table 6 normalized matrix aadt nrv tac tamd antiideal 0.341 0.815 0.656 0.439 2011 0.797 0.816 0.839 0.439 2012 0.625 0.822 1.000 0.450 2013 0.529 0.845 0.961 0.776 2014 0.719 0.815 0.912 0.739 2015 0.834 0.826 0.783 0.915 2016 0.884 0.869 0.717 0.879 2018 1.000 0.978 0.656 0.772 2020 0.341 1.000 0.817 1.000 ideal 1.000 1.000 1.000 1.000 in the fourth step of the marcos method, the normalized values are weighted with the weights of the criteria that have been calculated in the previous section of the paper for all the scenarios using the imf swara method. the weight coefficients of the criteria for the first scenario are: w1=w2=0.247; w3=0.309 and w4=0197. the weighted decision matrix is shown in table 7. table 7 weighted decision matrix aadt nrv tac tamd antiideal 0.084 0.201 0.203 0.086 2011 0.197 0.202 0.259 0.086 2012 0.154 0.203 0.309 0.089 2013 0.131 0.209 0.297 0.153 2014 0.178 0.201 0.282 0.146 2015 0.206 0.204 0.242 0.180 2016 0.218 0.215 0.221 0.173 2018 0.247 0.242 0.203 0.152 2020 0.084 0.247 0.252 0.197 ideal 0.247 0.247 0.309 0.197 the rest of the calculation using the marcos method is given below, and the results for the first scenario are presented in table 8. in the fifth step, using eq. (13), the value of saai is calculated as follows: saai = 0.084+0.201+0.203+0.086=0.575 s1 = 0.197+0.202+0.259+0.086=0.744 etc. by applying eq. (11), the following is calculated: 1 0.744 1.295 0.575 i i aai s k k s − −=  = = , i.e. by applying eq. (12): 1 0.744 0.744 1.000 i i ai s k k s + +=  = = 192 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković in the sixth step, the utility function is calculated in relation to the anti-ideal solution by applying eq. (15): 1 8 0.744 ( ) ( ) 0.365 0.744 1.295 i i i i k f k f k k k + − − −+ − =  = = ++ i.e. in relation to the ideal solution by applying eq. (16): 1 8 1.295 ( ) ( ) 0.635 0.744 1.295 i i i i k f k f k k k − + + −+ − =  = = ++ the utility function of all the alternatives is calculated by applying eq. (14): ( )1 0.744 1.295 ( ) 0.615 1 0.635 1 0.3651 ( ) 1 ( ) 11 0.635 0.365( ) ( ) i i i i i i i k k f k f k f k f k f k f k + − + − + − + + =  = = − −− − + ++ + table 8 results for the first scenario after applying the dea-imf swara-marcos model ai si aai 0.575 kiki+ f(k-) f(k+) f(ki) rank 2011 0.744 1.295 0.744 0.365 0.635 0.615 8 2012 0.755 1.314 0.755 0.365 0.635 0.624 7 2013 0.789 1.373 0.789 0.365 0.635 0.652 5 2014 0.806 1.403 0.806 0.365 0.635 0.667 4 2015 0.832 1.449 0.832 0.365 0.635 0.688 2 2016 0.827 1.440 0.827 0.365 0.635 0.684 3 2018 0.843 1.468 0.843 0.365 0.635 0.697 1 2020 0.781 1.358 0.781 0.365 0.635 0.645 6 ai 1.000 based on the results presented in table 8, it can be concluded that the range of differences in final values among alternatives is very small (0.082). it means that, regardless of the fact that there are certain differences, traffic safety according to the first scenario in all years is approximate, i.e. there are very slight nuances. it is confirmed by the fact that there is no alternative that tends to one. the results for the remaining five scenarios are obtained in the same way, as shown in tables 9-13. table 9 results for the second scenario after applying the dea-imf swara-marcos model ai si scenario 2 – s2 aai 0.630 kiki+ f(k-) f(k+) f(ki) rank 2012 0.906 1.438 0.905 0.386 0.614 0.728 1 2014 0.823 1.307 0.822 0.386 0.614 0.661 5 2015 0.854 1.357 0.854 0.386 0.614 0.687 4 2018 0.889 1.412 0.889 0.386 0.614 0.715 3 2019 0.897 1.425 0.897 0.386 0.614 0.721 2 2020 0.801 1.272 0.801 0.386 0.614 0.644 6 ai 1.000 impact of the number of vehicles on traffic safety: multiphase modeling 193 table 10 results for the third scenario after applying the dea-imf swara-marcos model ai si scenario 3 – s3 aai 0.683 kiki+ f(k-) f(k+) f(ki) rank 2012 0.947 1.387 0.946 0.406 0.594 0.741 1 2019 0.908 1.330 0.907 0.406 0.594 0.710 2 2020 0.768 1.126 0.768 0.406 0.594 0.602 3 ai 1.000 table 11 results for the fourth scenario after applying the dea-imf swara-marcos model ai si scenario 4 – s4 aai 0.561 kiki+ f(k-) f(k+) f(ki) rank 2003 0.865 1.540 0.864 0.359 0.641 0.719 1 2012 0.656 1.168 0.656 0.359 0.641 0.546 3 2020 0.827 1.472 0.826 0.359 0.641 0.688 2 ai 1.000 table 12 results for the fifth scenario after applying the dea-imf swara-marcos model ai si scenario 5 – s5 aai 0.548 kiki+ f(k-) f(k+) f(ki) rank 2000 0.808 1.474 0.808 0.354 0.646 0.677 2 2001 0.807 1.471 0.807 0.354 0.646 0.676 3 2003 0.825 1.505 0.825 0.354 0.646 0.691 1 2012 0.654 1.193 0.654 0.354 0.646 0.548 5 2020 0.788 1.437 0.787 0.354 0.646 0.659 4 ai 1.000 table 13 results for the sixth scenario after applying the dea-imf swara-marcos model ai si scenario 6 – s6 aai 0.703 kiki+ f(k-) f(k+) f(ki) rank 2012 0.906 1.289 0.906 0.413 0.587 0.702 1 2013 0.780 1.110 0.780 0.413 0.587 0.605 6 2014 0.813 1.157 0.813 0.413 0.587 0.631 4 2015 0.863 1.228 0.863 0.413 0.587 0.669 3 2016 0.799 1.137 0.799 0.413 0.587 0.620 5 2020 0.894 1.272 0.894 0.413 0.587 0.693 2 ai 1.000 fig. 7 shows a comparative analysis of the ranks through the complete dea-imf swara-marcos model 194 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković fig. 7 comparative analysis of the ranks through all the scenarios in fig. 7, it can be seen that by applying the dea-imf swara-marcos model, the ranks vary in relation to the set criteria, but here, 2012 can also be singled out as a benchmark. 5. impact of the number of registered vehicles on traffic accidents in this section of the paper, a short regression analysis is performed of the impact of the number of registered vehicles as an independent variable on the number of dead persons in traffic accidents (fig. 8), traffic accidents with casualties (fig. 9) and traffic accidents with material damage (fig. 10). fig. 8 the impact of vehicles on the number of dead persons in traffic accidents impact of the number of vehicles on traffic safety: multiphase modeling 195 in fig. 8, it can be seen that 57% of the relation between the number of registered vehicles and the number of dead persons can be described by a linear model. the regression is negative, which means that with the increase in the number of motor vehicles, the number of dead persons in traffic accidents decreases. the reason for this regression can be found in the fact that competent institutions have taken certain measures every year due to preventive engineering when it comes to the number of dead persons in traffic accidents in montenegro. fig. 9 influence of vehicles on the number of traffic accidents with casualties in fig. 9, it can be seen that 25% of the relation between the number of registered vehicles and the number of traffic accidents with casualties can be described by a linear model. the regression is positive, which means that with the increase in the number of motor vehicles, the number of this type of traffic accidents also increases. compared to the previous regression, this shows us that the number of motor vehicles affects the increase in the number of traffic accidents with casualties, but that these are primarily injured persons. fig. 10 influence of vehicles on the number of traffic accidents with material damage fig. 10 shows a negative regression described by a polynomial model. the number of motor vehicles does not affect the increase in traffic accidents with material damage. 196 m. damjanović, ž. stević, d. stanimirović, i. tanackov, d. marinković 6. conclusions in this paper, it was developed an integrated dea – imf swara – marcos model for assessing traffic safety in montenegro over a period of 23 years. different scenarios have been formed using different input and output factors, with the number of vehicles being an irreplaceable input parameter. the contribution of this paper is reflected in the development of an integrated multiphase model for the analysis of traffic safety in montenegro. the first phase involves the application of a dea model for all the scenarios, after which dmus with a value of 1.00 are further implemented in the mcdm model. the imf swafa method was applied to determine the significance of input and output parameters in each scenario, and the final ranking of alternatives was performed using the marcos method. the results obtained show that in terms of traffic safety in montenegro, compared to the end of the previous and the beginning of this century, the situation has improved with certain oscillations over the years. a regression analysis of the impact of the number of motor vehicles on traffic accidents and of its consequences was also performed. with the increase in the number of motor vehicles, the number of traffic accidents with casualties also increases. the benchmark years that should serve as an example of implementing measures and creating a traffic safety strategy are 2012 and 2020, but it should be taken into account that mobility was reduced during the pandemic. therefore, it may be better to take 2012 as a parameter year. limitations of this study can be manifested through the following. some of the uncertainties appear in the year 2020 which has been part of observed years due to covid-19. we have tried to eliminate this uncertainty by creating a linear model in considering data that has been explained in the paper and giving the advantage of the 2012 year in comparison to 2020. future research related to this paper refers to the expansion of influential factors, the application of uncertainty theories in the whole model. additionally, the implementation of adequate preventive engineering measures after this analysis is one of the future steps. references 1. podvezko, v., sivilevičius, h., 2013, the use of ahp and rank correlation methods for determining the significance of the interaction between the elements of a transport system having a strong influence on traffic safety, transport, 28(4), pp. 389-403. 2. sénquiz-díaz, c., 2021, the effect of transport and logistics on trade facilitation and 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© 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a modified frequency formulation for nonlinear mechanical vibrations chun-hui he1,3, ji-huan he2,4,5,6, jinfeng ma7, abdulrahman ali alsolami8, xiao-jun yang3 1 school of civil engineering & transportation, foshan university, foshan, china 2department of mathematical sciences, saveetha school of engineering, simats, chennai, tamil nadu, india 3school of mathematics, china university of mining and technology, xuzhou china 4school of jia yang; zhejiang shuren university, hangzhou, china 5school of information engineering, yango university, fuzhou, china 6school of mathematics and big data, hohhot minzu college, hohhot, china 7college of civil engineering and architecture, jiaxing university, jiaxing, china 8department of mathematics, faculty of science, king abdulaziz university, jeddah, saudi arabia orcid ids: chun-hui he https://orcid.org/0000-0003-0810-5248 ji-huan he https://orcid.org/0000-0002-1636-0559 jinfeng ma https://orcid.org/0009-0000-2348-9142 abdulrahman ali alsolami https://orcid.org/0000-0002-8189-4987 xiao-jun yang https://orcid.org/0000-0003-0009-4599 abstract. the paper elucidates the characteristics of nonlinear oscillators that contain quasi-linear terms arising in mechanical engineering. analytical solutions for this type of nonlinear oscillator are often difficult to obtain; they may involve singular and discontinuous terms, which can make them significantly more difficult to obtain. in this paper, a modified frequency formulation for this type of nonlinear oscillator is presented. while the mathematical proof is not provided, illustrative examples are included to demonstrate the remarkable simplicity and reliability of the approach. this paper challenges the conventional wisdom in analytical methods and offers a promising new direction for further investigation. key words: nonlinear vibration, non-perturbative approach, quasi-linear oscillator, discontinuous oscillator, fractional potential, zigzag oscillation received: may 26, 2025 / accepted july 03, 2025 corresponding author: ji-huan he, jinfeng ma school of jia yang, zhejiang shuren university, hangzhou, zhejiang, china college of civil engineering and architecture, jiaxing university, jiaxing, china email: hejihuan@ygu.edu.cn (j.h. he), majinfenglz@163.com(j. m) https://orcid.org/0000-0003-0810-5248 https://orcid.org/0000-0002-1636-0559 https://orcid.org/0009-0000-2348-9142 https://orcid.org/0000-0002-8189-4987 https://orcid.org/0000-0003-0009-4599 198 c.h. he, j.h. he, j. ma, a.a. alsolam, x.j. yang 1. introduction in the ever-evolving landscape of contemporary science and technology, nonlinear vibration systems [1, 2] have risen to prominence as a foundational building block with far-reaching consequences across a diverse array of disciplines. specifically, they have provided substantial support to the domain of mechanical engineering [3], particularly the field of micro-electromechanical systems (mems) [4]. it is imperative to comprehend the frequency-amplitude relationship to identify novel prospects and propel the advancement of related technologies. for instance, the capacity to discern the slightest tremors of starquakes, a capability of paramount importance in the realm of astronomy [5], exemplifies this phenomenon. these seismic events offer invaluable insights into the internal structure and composition of stars. additionally, high-frequency vibrations have been observed in the coupling of excitons in π-conjugated molecules [6]. the nonlinearity exhibited by oscillators can have various origins. cubic-quintic nonlinearity is a frequently employed phenomenon in the domain of mechanical engineering [7-9]. however, there are also nonlinearities of the form un), with n < 1. for instance, the integer n could be either 1/3 or -1. these nonlinear oscillators consist of both singular oscillators and discontinuous oscillators. the procurement of approximate solutions for these oscillatory phenomena poses a substantial challenge in the realms of mathematics and mechanical engineering. the complexity inherent to this phenomenon stems from the fact that singular oscillators frequently possess unique properties, thereby rendering traditional solution methods ineffective. conversely, discontinuous oscillators pose significant challenges due to their sudden fluctuations in behavior. confronting these challenges necessitates the development of innovative methodologies and a profound comprehension of both mathematical principles and mechanical systems. a paucity of known periodic functions exists for the approximation of nonlinear oscillators. in the domain of nonlinear vibration theory, the cosine function and the sine function are the most frequently employed. while it is possible to theoretically model a zigzag periodic motion through an infinite series of cosine or sine functions, this approach is mathematically intricate. engineers generally prioritize the key factors that influence the period or frequency of these oscillators. however, contemporary analytical methodologies predominantly center on the pursuit of an approximate cosine or sine solution for a complex nonlinear vibration system, with the objective of deriving an expression for the frequency. in order to make a more precise prediction of the impact of parameters on the frequency, it is necessary to undertake a cumbersome derivation process. however, this approach frequently falls short in providing highly accurate results due to the inherent limitations of using cosine and sine functions. alternative methods, such as numerical simulations or advanced experiment techniques [10], have the potential to serve as an auxiliary means of vibration analysis. however, these approaches are costly and cannot provide more accurate predictions or overcome the complexity associated with traditional methods. another periodic function that must be considered is the jacobi elliptic functions [7]. these functions are particularly well-suited for duffing-like oscillators due to their specific mathematical properties, which can accurately capture the behavior of such oscillators. however, the differential operations of these mechanisms are intricate, posing a challenge for engineers to comprehend and utilize. the primary concern of an engineer is not the approximate solution per se; rather, the emphasis is placed on the precision of the frequencyamplitude relationship. this relationship is of paramount importance, as it constitutes the a modified frequency formulation for nonlinear mechanical vibrations 199 foundation for the optimal design of a nonlinear vibration system. a precise understanding of the frequency-amplitude relationship is essential for engineers to fine-tune the parameters of the system, ensuring its optimal performance and reliability. absent a reliable estimate of this relationship, the design process becomes a speculative endeavor, with the potential to yield suboptimal results and inefficiencies. 2. problem formulation in a previous publication, ji-huan he [11] considered the following nonlinear oscillator: 2 1 1 2 1 1 0 n n n n u a u a u + + =  + + = (1) and proposed the following frequency formulation: 2 2 1 2 1 1 3 2 2 n n n n a a a n + = = + +  (2) where a2n+1 (n=0~n) are constants, a is the amplitude. the frequency formulation is effective for nonlinear oscillators with high order nonlinearities, for example, the cubicquintic-septic duffing oscillator [12]: 2 2 1 2 1 2 1 1 ( ) n n n n n a a u+ + = = + (3) this is to choose a location point for each nonlinear term, u2n+1: 1/2 2 1 3 ( ) 2 2 n nu a n + = + (4) this frequency formulation holds immense importance in the realms of both mathematics and nonlinear vibration theory. it offers a swift and effective means of understanding the two main factors of oscillators and their intricate relationship. this knowledge is essential for delving deep into the vibration properties and enabling advanced applications. take, for example, for the most energy harvesting devices or vibration dampers, their low-frequency property implies a large amplitude motion. as an illustration, we now consider the following nonlinear oscillator [13]. 2 2 2(1 ) (1 ) 0au u auu u u + + − − = (5) where α is a constant. it can be re-written as 3 2 2 0u u u auu au u  − + + + = (6) by the frequency formulation of eq. (3), we have 2 2 21 u au auu = − + + + (7) the location point is chosen as 200 c.h. he, j.h. he, j. ma, a.a. alsolam, x.j. yang 3 2 u a= (8) its derivatives can be approximately calculated as 2 2 1 2 u a u a = − =  (9) 2 23 2 u u a = − = −  (10) so we have 2 2 2 2 2 2 2 2 23 1 3 3 1 1 1 4 4 4 4 2 a a a a a a aa= − + + − = − + +    (11) after a simple calculation, we have 3 2 4 1 2 2 1 1 a aa − + = +  (12) which is exactly same as that in ref. [14]. there are an abundant number of publications on the applications of the original frequency-amplitude formulation and its various modifications. he and liu [15] gave a rigorous mathematical perspective for he’s frequency formulation. ismail et al. [16] and hashemi [17] found that this formulation is extremely well-suited for strongly nonlinear oscillators. their studies have demonstrated its effectiveness in analyzing and understanding the complex periodic properties. kawser et al. conducted a comparison between the frequency formulation and numerical simulation for the analysis of the jet engine vibration system. they discovered that the former is not only simple to use but also highly effective [18]. tsaltas [19] concluded that the one-step frequency formulation holds great promise as a new approach for nonlinear oscillators. additionally, other applications of the frequency formulation are equally captivating [20,21]. these applications span different fields and highlight the versatility and importance of the frequency formulation in understanding and solving a wide range of problems related to nonlinear oscillators. although significant achievements have been attained, there is still ample room for further improvement, particularly for nonlinear oscillators with a nonlinear term of the form un where n<1. this type of oscillator encompasses singular oscillators and zigzag oscillators, which are extremely challenging to solve analytically. 3. frequency formulation for quasi-linear oscillator and singular oscillators the aforementioned frequency formulation is applicable to n>1. it is evident that: 2 1 , for 1/ 2nu a n+   . (13) however, this is incompatible with the characteristics of a conservative oscillator, which exhibits vibrations from u=a to u=-a. there exists a large set of nonlinear oscillators a modified frequency formulation for nonlinear mechanical vibrations 201 with an irrational term [22] or a term of the form u1/(2n + 1) [23]. though there were some effective modifications of the frequency formulation [24, 25], here a more effective formulation will be recommended. the following nonlinear oscillator is now considered. 1 0, 2i n n i i i u b u n =  + =  (14) where bi (i=1~n) are constants. for periodic solutions, it requires ( 1) 1 0i n n i i b u − =  (15) the previous approaches to this type of nonlinear oscillators are some famous analytical methods, e.g., the variational iteration method [26] and the homotopy perturbation method [27, 28], all these methods require sophisticated technology with complex calculation, however, in the practical applications, the frequency-amplitude relationship is the most important factor. a simple yet reliable approach is always highly welcome. now, this paper presents the following frequency formulation: ( 1)2 1 6 5 i n n i i i b a n − = = +  (16) it can be also written in the form ( 1)2 1 ( ) i i n n i n i b u − = = (17) where the location point for each nonlinear term is given as ( 1)1/6 ( ) 5 i i n n i u a n − = + (18) the new frequency formulation is extremely simple. a more straightforward approach is always advantageous for engineers. the frequency formulation demonstrates considerable promise for a diverse array of prospective applications. this novel approach has the potential to transform analytical techniques within the domain of vibration theory. the method provides a rapid yet reliable comprehension of intricate vibration systems through a straightforward step that is more efficient than traditional analytical methods, which entail complex and detailed calculations. it can be utilized for the optimization of complex vibration systems without the necessity for onerous computations. the remarkable versatility of this new formulation represents a significant advancement in the field of nonlinear vibration theory. it is adaptable to a variety of situations and requirements, thereby facilitating the realization of novel possibilities and opportunities. as research and development continue, the potential applications of this approach are likely to expand, with the prospect of substantial benefits for the field of nonlinear science. the following examples demonstrate the efficacy and simplicity of this approach. 202 c.h. he, j.h. he, j. ma, a.a. alsolam, x.j. yang 4. examples 4.1. example 1. nonlinear oscillator with fractional potential nonlinear oscillators with fractional potential are also called as quasi-linear oscillators, the nonlinear term is always in the form, u1/(2n+1), where n is a natural number. in this example, we consider the following oscillator. 1/3 0u u + = (19) where β is a positive constant. its variational formulation is 2 4/31 3 ( ) 2 4 j u u u dt   = −      (20) the potential is represented by a fractional exponent, and thus eq. (20) is designated as a nonlinear oscillator with fractional potential. this nonlinear oscillator has been the subject of extensive attention in the open literature, as evidenced by references [29, 30]. the nonlinear nature of the system presents a considerable challenge in attempting to find a general closed-form solution through analytical means. the nonlinear characteristics introduce intricate interactions and behaviors that render the derivation of an approximate solution an exceptionally challenging endeavor. following meticulous and comprehensive calculations, a notable accomplishment is the derivation of the frequency-amplitude relationship for eq. (20), as documented in [29, 30], which is: 1/3 1.0768a−= . (21) this relationship, with a relative error of 0.59%, provides a crucial link between two fundamental properties of the oscillator and serves as a valuable resource for further studies and applications. this provides a basis for analyzing and predicting the behavior of the oscillator in different scenarios, thereby opening up new avenues for research and practical applications in a variety of fields. notwithstanding the significant advancements achieved through meticulous and intricate computations, engineers have been keenly awaiting a straightforward yet efficacious methodology to swiftly and dependably comprehend the vibration characteristics. the time has now come to seize this opportunity. by employing the frequency formulation of equation (17), the desired results can be obtained with immediate effect. 2 2/3 2/3 1 3 6 18 5 16 a a− −= = +    (22) or 1/2 1/3 1.060660 a−=  (23) it is noteworthy that it exhibits the same form as that of eq. (22). given its simplicity, the frequency formulation represents a significant accomplishment in the field. the accuracy of this method is comparable to that attained by the variational iteration method or the homotopy perturbation method. the one-step approach is a highly attractive proposition. the simplicity and efficiency of this approach render it an attractive option, particularly in situations where rapid insights and the frequency-amplitude relationship are required without the intricacy of more elaborate methods. https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-2 a modified frequency formulation for nonlinear mechanical vibrations 203 the exact frequency is [29, 30] 1/2 1/31.070451exact a−=  (24) the relative error is 0.91%, which is deemed to be within an acceptable range for engineering applications. 4.2. example 2. discontinuous oscillator with sign function the study of nonlinear oscillators with discontinuous terms represents a fascinating and rich area of inquiry within the broader fields of physics and mechanical engineering. discontinuous terms may originate from impacts, switches, or sudden alterations in system parameters. the aforementioned discontinuities present considerable difficulties in the analysis and comprehension of the oscillator's behavior. for example, a nonlinear term with the sign function frequently plays a pivotal role in such systems, as it can represent abrupt changes in direction or state. the study of nonlinear oscillators with discontinuous terms represents a vibrant and multifaceted research domain, with implications that extend far beyond the immediate field of inquiry and have the potential to inform a vast array of practical applications. an investigation of the frequency-amplitude relationship may facilitate the discovery of new possibilities for engineering and scientific advancement. researchers are engaged in ongoing efforts to develop new methods for the analysis and control of these oscillators. this includes the creation of novel theoretical models and experimental techniques, with the aim of gaining deeper insights into their behavior and the design of more efficient and reliable systems. in this example, we consider a nonlinear oscillator subjected to a constant force, as illustrated in fig. 1. fig. 1 discontinuous oscillator with discontinuous restoring force the restoring force follows: when u>0, f(u)=1; when u<0, f(u)=−1. the governing equation can be expressed as sgn( ) 0u u + = (25) where β=1/m, m is the mass of the oscillator, and sgn(u) is defined as 0 0 , 0 1, 0 sgn( ) 0, 0 0, 0 1, 0, 0 u u u u u u uu u      = = = =   − −   . (26) 204 c.h. he, j.h. he, j. ma, a.a. alsolam, x.j. yang considerable effort has been dedicated to seeking an accurate analytical approximation for equation (26), as referenced in [31-33]. the primary mathematical tool utilized is the homotopy perturbation method. in the application of this method to nonlinear vibration systems, the construction of the homotopy equation, the selection of the initial guess, and the elimination of secular terms during the lengthy calculation process represent the primary focal points [34]. these aspects are of paramount importance in guaranteeing the efficacy and precision of the method in analyzing and approximating nonlinear vibration phenomena. after a complex calculation, belendez et al. [30] obtained high-accuracy analytical approximations by the homotopy perturbation method, resulting in the following frequencies 1/2 1 1.128379= a− , (27) and 1/2 2 1.107452= a− . (28) the relative errors are 1.59% and 0.29%, respectively. liu obtained the same result of eq. (28) by the modified lindstedt–poincaré method [33]. although the achievement has been attained, engineers are currently seeking a straightforward yet impactful methodology. by employing the frequency formulation, we have promptly derived the following outcome. 2 1 16 6 0 5 5 a a− −= = +    , (29) or 1/2 1/21.0954 a−=  . (30) this result is comparable to those produced by the homotopy perturbation method [31] and the modified lindstedt–poincaré method [33]. the exact frequency is: 1/2 ex 1.1107= a− . (31) the relative error is 1.37%, which is comparable to the results presented in refs. [37, 39], where a length calculation and specialized skills are required. in contrast, the present approach is notable for its simplicity. in nonlinear theoretical analysis, a minimal calculation with one or two lines can have maximal influence in the academic community, as ludwig mies van der rohe (1886–1969) said, "less is more". 4.3. example 3. discontinuous oscillator with absolute value symbols the study of discontinuous oscillators with absolute value symbols represents a fascinating area of inquiry within the field of nonlinear dynamics. the introduction of absolute value symbols can result in the emergence of intricate dynamics, which in turn presents significant analytical challenges. researchers employ a range of analytical techniques to examine these oscillators, undertaking significant efforts in this regard. such techniques may include numerical simulations and analytical approximations. in this example, we consider a discontinuous oscillator with absolute value terms, as illustrated in fig. 2. a modified frequency formulation for nonlinear mechanical vibrations 205 fig. 2 schematic of a discontinuous oscillator with absolute value terms it is assumed that the equivalent elastic coefficient is proportional to the magnitude of displacement, and the oscillator can be expressed as: 0u u u + = . (32) by the frequency formulation, we have: 2 6 6 2 5 7 a a= = +  , (33) or 0.92582 a= , (34) while the exact one is [35]: 0.914681exact a= . (35) the relative error is 1.21%, which is an acceptable level of precision given the simplicity of the method employed. wang and he [36] achieved a notable result through a combination of sophisticated techniques and meticulous calculations, which is 8 0.9214 3 a a= =  . (36) this achievement was of tremendous significance because, prior to this, there was no analytical method that could effectively solve eq. (33). reference [36] made significant advancements in nonlinear science and nonlinear vibration theory. it provides an excellent illustration of the efficacy of meticulous analysis and innovative thinking in addressing complex problems and makes a valuable contribution to the expanding body of knowledge in the relevant discipline. the current approach allows for a straightforward and effective solution, demonstrating the value of simplicity in achieving comparable outcomes. the new frequency formulation provides a novel avenue for future research in nonlinear vibration theory and nonlinear studies. the optimal approach is to employ a method that is both straightforward and efficacious. 4.4. example 4. singular oscillator a singular oscillator represents a distinctive category of dynamical systems that display a range of intriguing and frequently intricate behaviors. these oscillators possess singularities, which give rise to distinctive characteristics that are not observed in traditional oscillators. the study of singular oscillators is of significant importance in a number of fields, including physics, engineering, and mathematics. it facilitates comprehension of complex 206 c.h. he, j.h. he, j. ma, a.a. alsolam, x.j. yang systems and may yield novel insights and applications. researchers are engaged in ongoing investigations into the properties and behaviors of singular oscillators, with the objective of unlocking their potential for innovative technologies and scientific discoveries [37]. the following example considers a singular oscillator: 1 0u u− + = (37) according to the frequency formulation , we have: 2 2 2. 6 1 5 1 5a a− −= = − +    , (38) or 1/2 11.2247 a−=  . (39) the exact one is [37]: exact 1.2533 = a  . (40) the relative error is 2.28 %. the singularity presents a challenge for both analytical and numerical methods. however, the one-step frequency formulation offers a solution that produces optimal results with minimal calculation. 4.5. example 5. nonlinear oscillator with fraction term the study of nonlinear oscillators with fraction terms represents a fascinating and promising area of mathematical inquiry. it presents a challenge to traditional analytical methods, as it may not conform to the standard mathematical models that are typically employed. researchers employ sophisticated techniques to comprehend and anticipate the behavior of these oscillators. consider the following nonlinear oscillator with a fraction term: 2 0 2 1 0 n n n n u u c c u =  + = + . (41) in order to utilize the aforementioned frequency formulation, we assume that a tends to infinity. under this assumption, eq. (41) becomes: 1 2 2 1 0 n n n n u c u − =  + = . (42) its frequency-amplitude relationship is 2 2 2 2 2 1 1 6 3 1 2 5 3 n n n n n n n n b a b a n n − − = = = = − + −   (43) for eq. (42) the frequency formulation should be modified as: 2 2 0 2 1 1 3 3 n n n n n c b a = = − +  . (44) a modified frequency formulation for nonlinear mechanical vibrations 207 in this modification, the case of c0=0 and n=1 can be transformed into the form of eq. (44). so it is suitable for large amplitude oscillation. in order to show its effectiveness, we consider a simple case: 2 0 1 u u u  + = + . (45) by the frequency formulation of eq. (45), the following result is obtained: 2 1 2 1 3 a = +  . (46) the exact period can be calculated as: 0 2 2 1 4 ln(1 ) ln(1 ) a t du a u = + − +  . (47) for the case a tends to infinity, we have: 0 2 2 0 2 2 1 lim 4 lim ln(1 ) ln(1 ) 1 4 2 2 ln( ) ln( ) a a a a t du a u du a a u → → = + − + = = −    (48) so we have: 1 12 lim lim 1.2533 2 exact a a a a t − − → → = = =    , (49) while from eq. (28), we have: 1 2 1 l 4im 1.22 2 3 7 a a a − → = = . (50) the relative error is 2.28%. it should be note that eq. (48) is valid for large amplitude. for the small values of a, eq. (46) can be approximated as: 3 0u u u + − = . (51) by the frequency formulation given in eq. (2), we have: 2 2 23 3 1 1 ( ) 2 4 a a o a= − = − + . (52) considering eqs. (48) and (49), we recommend the following matching one: 2 1 2 3(1 ) 1 3 2(1 ) m m a m m = + − + + −  , (53) 208 c.h. he, j.h. he, j. ma, a.a. alsolam, x.j. yang where m is the weighting factor, m=0 for a<1 and m=1 for a>1, and the value of m can be determined by matching eq. (53) with the exact one when a=1. now this paper challenges the following nonlinear oscillators: 1/3 1/5 3 5 1 2 3 4 5sgn( ) 0u u b u b u b u b u b u + + + + + + = , (54) with arbitrary initial conditions: (0) , (0)u u= =  . (55) and this paper gives the following result without proof: 2 2/3 4/5 1 2 4 1 2 3 4 51 1 3 5 2/3 4/5 1 2 4 1 2 3 4 5 6 6 6 3 3 1 5 5 0 5 2 2 2 4 9 15 6 3 1 1 8 13 5 4 2 b a b u b a b a b a b a b u b a b a b a − − − − − − = + + + + + + + + + + = + + + + +  (56) and 2 2 2 2 a+ =    . (57) 5. discussion and conclusion dozens of research groups around the world are engaged in the development of analytical methods for the study of the frequency-amplitude relationship of complex nonlinear oscillators [38]. the nonlinear vibration theory is cheaper and greener than other artificial methods, including sophisticated experiments, precise measurements, and time-consuming simulations. the nonlinear vibration theory has the potential to enhance the efficiency of energy harvester devices or to facilitate the conservation of mechanical vibrations, thereby reducing the emission of greenhouse gases. furthermore, the frequency formulation can assist in the identification and search for optimal parameters for the devices currently under development. to date, a number of analytical methods have been proposed for this purpose, with the homotopy perturbation method and the variational iteration method representing the most commonly used mathematical tools. the frequency formulation represents a particularly simple method. this paper identifies the most fascinating aspect of the frequency formulation: its ability to link two primary factors that control periodic motion in a complex oscillator. the aforementioned examples illustrate the efficacy of the straightforward yet impactful frequency formulation in the context of scientific and technological advancement. the new frequency formulation presented in this paper represents a continuous process of improvement and expansion of its applications. the discovery constituted a profound breakthrough that astonished the scientific community. while the effectiveness is illustrated by examples, it remains in its nascent stages, and there are numerous challenges and prospects for future research. a rigorous mathematical proof of the frequency formulation is required to confirm its effectiveness. the formulation may encounter counter-examples in practical applications, which could lead to further technological developments. 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1department of civil engineering, university of calabria, via bucci, rende, italy 2university of belgrade, faculty of transport and traffic engineering, belgrade, serbia 3yuan ze university, college of engineering, department of industrial engineering and management, taoyuan city, taiwan 4department of computer science and engineering, college of informatics, korea university, seoul, republic of korea 5institute of mechanical science, vilnius gediminas technical university, lithuania abstract. traffic accidents cause huge financial, human, and emotional losses every year. with increased urbanization and population growth, safety problems are also increasing. this study develops a novel approach by integrating support vector machine (svm) with advanced meta-heuristic algorithms to build a classification model of the number of vehicles involved in a traffic accident and the ranking of factors contributing to the accident, which is essential for improving transportation safety management. for this purpose, nine factors contributing to impacting road safe transport in urban areas of cosenza, southern italy, are assessed and ranked through the application of a stochastic approach. these features were determined using machine learning techniques, including svm, combined with some metaheuristic optimization methods involving the cuckoo search algorithm (csa), multi-verse optimizer (mvo), and whale optimization algorithm (woa). with the svm-woa algorithm offering a great accuracy value, the results reveal that these approaches are successful in detecting elements influencing traffic safety management in road transportation. key words: traffic safety, accident factors, urban areas, machine learning, metaheuristic optimization methods received: november 28, 2024 / accepted april 15, 2025 corresponding author: vladimir simic university of belgrade, faculty of transport and traffic engineering, vojvode stepe, belgrade, serbia e-mail: vsima@sf.bg.ac.rs 2 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. 1. introduction traffic accidents pose a severe challenge globally in terms of great economic losses, human fatalities, injuries, and even serious emotional and psychological effects on people and society. growing urbanization and population seem to increase the need for ensuring better road safety and adopting effective traffic management strategies [1, 2]. basically, improvement in road safety relates to the identification and understanding of various factors leading to traffic accidents. the value of such knowledge is quite substantial for decisions and steps transportation professionals need to take while designing and implementing appropriate measures to minimize the risk of accidents and make roads safer [1, 3]. conventional methods for traffic accident prediction, most of them rooted in statistical models and regression analysis, normally operate under the assumption of linear relationships between contributing factors and accident occurrences. these models assume linear relationships, which often fail to capture the nonlinear dynamics of real-world traffic systems. these limitations have consequently made advanced computational methods, including techniques of artificial intelligence (ai) and machine learning (ml), increasingly embraced by researchers in their struggle to improve predictive modeling capability [4]. the nature of complexity in such approaches means they are very good at unraveling complex patterns and relationships that may exist yet remain obscure in high-dimensional data, an attribute particularly appropriate in the context of traffic accident prediction. among these, support vector machines (svms) have drawn significant attention due to their capability to deal with nonlinear and high-dimensional input effectively. svms are grounded in statistical learning theory. they also perform with a great deal of consistency compared to conventional statistical models that not only stress better accuracy but also generalized capability. other researchers have focused on improving the prediction capability of the svms using metaheuristic optimization algorithms [5]. the metaheuristic algorithms are inspired by natural phenomena such as evolutionary processes and swarm behavior that provide ways of improving the parameters used by the svms with the aim of increasing their predictive accuracy and efficiency. a wide range of metaheuristic methods, each with distinct advantages and constraints, have been effectively used to optimize svm models in several areas, including the crucial space of traffic accident prediction [6-8]. this is notwithstanding the fact that all these highly remarkable achievements have been recorded over the recent years in the area of traffic accident prediction using svms combined with metaheuristic optimization. besides, many challenges persist. first, the computational complexity of developing processes for the svm models makes it difficult to deploy them in real situations and further scalability is hard, especially when united with metaheuristic optimization [5]. second, the volume and quality of the input data, including real-time traffic flow and weather, fundamentally influence the accuracy of prediction models [9]. consequently, this implies that, with increasing urgency for finding a solution to such problems, computation-efficient algorithms that can emerge patterns from big quantities of data in real-time and provide predictions with real timescales are in need [10]. this study addresses known limitations in traffic accident prediction research and offers a multifarious contribution. at first, the research improves the ability to predict by combining svms with a number of metaheuristic optimization techniques. by systematically comparing various algorithms, their pros and cons are made clear, making it easy to pick the most appropriate optimization method for any given traffic situation. enhancing support vector machine performance in forecasting the number of vehicles … 3 the study also looks at urban roads in cosenza, which is in southern italy, in a way that is specific to that area. this shows how local factors can affect predicting accidents. as a result, the most important factors that can cause accidents can be found and ranked, meeting the need for safety solutions that are tailored to the specific situation. lastly, this study fills in a gap in the existing body of work by applying new developments in theoretical modeling to real-world problems that have to do with scalability and high-dimensional computational complexity. this means that there are now even better algorithms that can handle large amounts of traffic data while still being accurate. these algorithms can give policymakers and transportation engineers useful information that they can use to make the roads safer. 2. literature review from traditional statistical methods to state-of-the-art ai systems, this literature review investigates the development of methodologies in traffic accident analysis and prediction, therefore stressing important trends and possible future directions of study. the application of conventional methods, such as traditional statistics and economic methods, has proved helpful in understanding traffic. sipos et al. [11] performed a spatial econometric analysis, especially spatial autocorrelation and geographically weighted regression, to find the geographical trend of traffic accidents. this would show how the spatial factors affect the probability of occurrence of accidents. lord and mannering [12] conducted a thorough study and evaluation of the different statistical methods used to look at crash-frequency data. this showed how important it was to choose the right methods for each situation. vedagiri and killi [13] used surrogate safety measures like time-to-collision and postencroachment time in analyzing uncontrolled intersections in mixed traffic streams. this illustrates how the more general approaches may be used to evaluate traffic safety across particular situations. also, sawalha and sayed [14] looked at it from the temporal point of view. they applied regression analysis on crash data from major arterial roads to develop how safety varies over time and the impacts of actions on that variation in safety changes. traditional statistical methods can be applied under various circumstances, as shown by dell'acqua and russo's [15] work in developing the safety performance function for lowvolume roads by negative binomial regression. all these functions take into account the difficulties arising from fluctuating road and traffic conditions. the introduction of data mining and ml has opened up new possibilities for forecasting and analyzing traffic accidents. li et al. [16] used data mining techniques such as association rule mining and decision trees to determine the causes of one category of pointless catastrophic traffic accidents, thereby demonstrating the usefulness of this type of method in processing crash data. pourroostaei ardakani et al. [17] elaborated further on this by showing the practicality of ml techniques with the use of models like random forest (rf) to carry out predictions on road car accidents. they underlined how these techniques raised forecast accuracy, which would help to produce more accurate and efficient traffic safety rules. wang et al. [18] proposed a more advanced strategy based on multi-view multi-task spatiotemporal networks, incorporating convolutional neural networks and graph convolutional networks, where various kinds of data and various tasks are effectively combined for a global prediction. 4 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. researchers have also concentrated on improving predicting models and handling natural data difficulties in traffic accident research. guido et al. [19] studied feature selection methods including information gain and correlation-based feature selection as well as screening the possible factors regarding safety management, indicating their contribution to the improvement of predictive accuracy. advanced data modeling, such as that using, for example, the sarima model of deretić et al. [20], has made it possible to forecast traffic accidents based on historical patterns. dimitrijevic et al. [21] used the rfbased classifier to include both proactive and reactive accident data in their models as well as improve the accuracy and location-specific forecasts of short-term crash risk prediction. danesh et al. [22] developed data leveling methods including random under-sampling and smote and applied metaheuristic optimization approaches, including the genetic algorithm, to guarantee unbiased model performance, thereby addressing the issue of imbalanced datasets in crash prediction. a growing area of study in recent years is also the combination of traditional statistical methods with ai techniques. the hybrid model proposed by dong et al. [23] combined a statespace framework with support vector regression. this model showcased the potential of merging statistical and ml techniques to enhance the prediction of traffic accidents. the model took into consideration both temporal and spatial dynamics present in the data. mannering and bhat [24] provided a review of the analytic methods in accident research, including conventional and upcoming techniques. they also provided a glimpse of future directions for the same. the practical implementation of ml in real-world applications is reflected in a case study carried out by sufian et al. [25] on traffic accident prediction using heterogeneous urban data, where they employed various ml models and showed their practical implementation. 3. methodology traffic accident prediction and factor analysis involve navigating complex, often nonlinear relationships within high-dimensional datasets. while traditional statistical methods are indeed useful, they capture the underlying complexity rather inadequately. so, in this study, the power of ml is used; i.e. svms that have high recognition for handling nonlinear and high-dimensional data with great efficiency. to enhance the predictive capabilities of svms, we couple them with nature-inspired metaheuristic optimization techniques to tune model parameters for efficient performance optimization. the theoretical underpinning of svm and the adopted metaheuristic algorithms in this paper, namely, the cuckoo search algorithm (csa), multi-verse optimizer (mvo), and whale optimization algorithm (woa), will be included in the following sections. this also encompasses the methodological framework related to data preparation, model construction, and evaluation. the approach, with its integration of svms and metaheuristic optimization, shall provide a robust and accurate means toward the prediction of the number of vehicles involved in a traffic accident and the identification of those significant factors that determine such accidents for the development of appropriate road safety measures. 3.1. support vector machine artificial intelligence (ai) and ml approaches are being used by many academics and industry sectors, which are changing how we approach complex problems and enhancing operational efficiency [26-29]. svm is a successful ml algorithm developed by cortes enhancing support vector machine performance in forecasting the number of vehicles … 5 and vapnik in 1995 [30]. it is a supervised learning algorithm used in regression and classification tasks. svms are linear classifiers that maximize the margin between two classes by creating a classification hyperplane in the center of the maximum margin. the objective is to identify the optimal hyperplane in an n-dimensional space. two labels are assigned for classification: +1 for cases above the hyperplane and -1 for those below. eq. (1) displays a collection of sample sets utilized in the process of learning data for categorization [31]: 𝑆 = {(𝑥𝑖 , 𝑦𝑖)𝑖=1 𝑛 |𝑥𝑖 ∈ 𝑅𝑁 , 𝑦𝑖 ∈ {−1,1}, 𝑖 = 1,2, . . . . . . , 𝑙} (1) here, xi marks the data and yi is the target variable for the i-th sample. determined by applying current support vectors and constraints, the ideal hyperplane is the one with the biggest margin among the produced hyperplanes. eqs. (2) and (3) express constraints [32]: 𝑀𝑖𝑛 1 2 ‖𝑤‖2 (2) 𝑠. 𝑡. 𝑦𝑖(𝑤𝑥𝑖 + 𝑏) ≥ 1 (3) where w denotes the weight vector and b denotes the bias vector [30]. subsequently, after taking into account an error coefficient, the constraints are reformulated and rectified based on eqs. (4) and (5). the purpose of this error coefficient is to enhance the precision of categorization [33]: 𝑀𝑖𝑛 1 2 ‖𝑤‖2 + 𝑐 ∑ 𝜀𝑖 (𝜀𝑖 ≥ 0)𝑛 𝑖=1 (4) 𝑠. 𝑡. { 𝑦𝑖(𝑤𝑥𝑖 + 𝑏) ≥ 1 − 𝜀𝑖 𝑐 ≥ 0 (𝑖 = 1,2,3, . . . . . . , 𝑛) (5) the penalty coefficient is denoted by the symbol c. next, the lagrange approach is employed to address classification difficulties in svm. these problems are formulated as a dual optimization problem, which is based on eq. (6) [34]: { 𝑊(𝑎) = ∑ 𝑎𝑖 − 1 2 𝑛 𝑖=1 ∑ 𝑎𝑖 𝑛 𝑖,𝑗=1 𝑎𝑗𝑦𝑖𝑦𝑗𝐾(𝑥𝑖 , 𝑥𝑗) 𝑠. 𝑡. ∑ 𝑎𝑖𝑦𝑖(0 ≤ 𝑎𝑖 ≤ 𝑐; 𝑖 = 1,2,3, . . . . , 𝑛)𝑛 𝑖=1 (6) the function k is a mathematical function referred to as the kernel function. various types of kernel functions are available including the linear (lin), radial basis function (rbf), and polynomial (pol). parameters gamma (𝛾) and d define the several kinds of kernels for rbf and pol. here, d denotes the degree term of the polynomial (pol) kernel function, while gamma (𝛾) is used for both rbf and polynomial kernels [35]. the main responsibility of the kernel is to convert the input data into the needed structure. the accuracy of categorization could change depending on the knowledge about the application of several kernel functions in pertinent situations. eqs. (7-9) provide respectively the formulas of certain kernel functions, including lin, rbf, and pol [36]: 𝐺(𝑥𝑖 , 𝑥𝑗) = 𝑥𝑖 𝑡𝑥𝑗 (7) 𝐺(𝑥𝑖 , 𝑥𝑗) = 𝑒𝑥𝑝(−𝛾‖𝑥𝑖 − 𝑥𝑗‖ 2 ) (8) 𝐺(𝑥𝑖 , 𝑥𝑗) = (−𝛾𝑥𝑖 𝑡𝑥 + 1)𝑑 (9) 6 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. 3.2. whale optimization algorithm the metaheuristics algorithms are considered very important in the area of engineering, especially because many practical problems require complex system and process optimization for which other methods have proven inefficient [37-40]. the whale optimization algorithm (woa) is a metaheuristic algorithm that uses swarm behavior to solve continuous optimization challenges. it is straightforward to develop and robust, comparable to nature-inspired algorithms [41]. the woa model involves the humpback whale population in a complex search space, with individual whale positions as choice variables and distance between them and food as the objective cost. the movement of an individual whale is determined through three processes: encircling prey, spiral bubble-net feeding technique, and prey search. this method is comparable to nature-inspired algorithms due to its simplicity and robustness [42]. humpback whales surround their prey after sensing where it is. the woa method supposes the target prey, or in close approach to the optimal solution, is the current best candidate solution. while the remaining search agents change their locations to be closer to the ideal search agent, it aims to identify the best search agent. mathematically, this behavior is expressed by the following equations [41]: �⃗⃗� = |𝐶 . 𝑋∗⃗⃗ ⃗⃗ (𝑡) − 𝑋 (𝑡)| (10) 𝑋 (𝑡 + 1) = 𝑋∗⃗⃗ ⃗⃗ (𝑡) − 𝐴 . �⃗⃗� (11) 𝐴 = 2𝑎 . 𝑟 − 𝑎 (12) 𝐶 = 2 ∙ 𝑟 (13) where 𝑋∗⃗⃗ ⃗⃗ denotes the current best position, 𝑋 represents the position of the current whale, t indicates the recent iteration, 𝐴 and 𝐶 are coefficient vectors, 𝑎 represents a linearly decreasing value ranging from 2 to 0 during the iterations, and 𝑟 is a random number uniformly distributed between 0 and 1 [40]. the symbol ‘‘| |’’ denotes the absolute value, and element-wise multiplication is shown by “⋅” [41]. a mathematical equation is used to simulate the spiral movement pattern observed between humpback whales and their prey, resembling the helix-shaped motion of these whales [42]: 𝑋 (𝑡 + 1) = 𝐷′⃗⃗⃗⃗ 𝑒𝑏1. cos(2𝜋𝑙) + 𝑋∗⃗⃗ ⃗⃗ (𝑡) (14) 𝑋 (𝑡 + 1) = { 𝑋∗⃗⃗ ⃗⃗ (𝑡) − 𝐴 . �⃗⃗� 𝑖𝑓 𝑝 < 0.5 𝐷′⃗⃗⃗⃗ 𝑒𝑏1. cos(2𝜋𝑙) + 𝑋∗⃗⃗ ⃗⃗ (𝑡) 𝑖𝑓 𝑝 ≥ 0.5 (15) where 𝐷′⃗⃗⃗⃗ represents the distance of the ith whale to the prey, b denotes a constant defining the spiral shape, l is a random number in [–1, 1], whereas p is a randomly chosen value that falls within the range of -1 to 1 [42]. in the exploration phase of prey search, global optimizers are utilized. if the value of a is greater than 1 or less than -1, the search agent is updated according to a randomly selected search agent, replacing the current best search agent [42]: �⃗⃗� = |𝐶 . 𝑋𝑟𝑎𝑛𝑑 ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ − 𝑋 | (16) 𝑋 (𝑡 + 1) = 𝑋𝑟𝑎𝑛𝑑 ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ − 𝑋 . �⃗⃗� (17) enhancing support vector machine performance in forecasting the number of vehicles … 7 where 𝑋𝑟𝑎𝑛𝑑 ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ is selected arbitrarily from the whales in the current iteration [42]. 3.3. cuckoo search algorithm the cuckoo search algorithm (csa) is a metaheuristic optimization technique inspired by the brood parasitism behavior observed in certain cuckoo species. in nature, some cuckoos lay their eggs in the nests of other host birds, relying on the host to raise their offspring. however, there is a probability that the host bird may detect and discard the foreign eggs [43]. three basic guidelines defined by csa help to replicate cuckoo behavior [44]: 1. each cuckoo lays one egg in a randomly chosen host nest; the number of available nests remains constant. 2. high-quality eggs will be reserved for the next generation. 3. host birds run the chance of seeing and discarding alien eggs. a constant switch parameter pa balances the global search (exploration) and local search (exploitation) that define csa. levy fly [44] is proposed to implement it for the global exploration: 𝑥𝑖 𝑡+1 = 𝑥𝑖 𝑡 + 𝛼 ⊕ levy(𝑠, 𝜆) (18) where α is the scale factor, ⊕ denotes entry-wise multiplications, λ (1 < λ < 3) is the power coefficient, levy (s, λ) is the characteristic scale, which may be computed as 𝑥𝑖 𝑡 and 𝑥𝑖 𝑡+1, respectively. levy(𝑠, 𝜆) ≈ 𝜆γ(𝜆).sin(𝜋𝜆 2⁄ ) 𝜋 1 𝑠1+𝜆, (𝑠 ≫ 𝑠0 ≫ 0) (19) formulated in eq. (19), γ symbolizes the gamma function, s0 is the initial step size of levy flight, and s represents the step size of levy flight [45]: 𝑠 = 𝑈 |𝑉|1 𝜆⁄ (20) u and v denote random values following normal distribution: 𝑈~𝑁(0, 𝜎2), 𝑉~𝑁(0,1) (21) where σ is the variance of the normal distribution followed by u, which is computed as follows [45]: 𝜎2 = [ γ(1+𝜆) 𝜆γ((1+𝜆) 2⁄ ) ∙ sin (𝜋𝜆 2⁄ 2(𝜆−1) 2⁄ ] 1 𝜆⁄ (22) one may arrange the local search for csa as follows [45]: 𝑥𝑖 𝑡+1 = { 𝑥𝑖 𝑡 + 𝛼𝑠 ⊗ 𝐻(𝑝𝑎−∈) ⊕ (𝑥𝑗 𝑡 − 𝑥𝑘 𝑡), 𝑟 > 𝑃𝑎 𝑥𝑖 𝑡 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (23) where 𝐻(𝑝𝑎−∈) is the heaviside function, ∈ is a random integer taken from the uniform distribution, where 𝑥𝑗 𝑡 and 𝑥𝑘 𝑡 are two solutions acquired via random permutation. 8 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. 3.4. multi-verse optimization mvo is a metaheuristic technique exploring search space by simulating wormholes, black holes, and white holes. an inflation rate between exploitation and exploration balances wormhole to black-white hole ratios. in this context, candidate solutions are universes; decision variables are things or components found within those universes. the inflation rate controls the likelihood of finding both white (exploration) and black holes (exploitation); i.e. low rates indicate the best chance of a black hole and high rates indicate the highest possibilities. a roulette wheel selection mechanism is used in mvo to solve the analytical model and exchange the items between the worlds of black and white hole tunnels. the solution space exhibits a random global representation as [46]: 𝑈 = [ 𝑥1 1 𝑥1 2 ⋯ 𝑥1 𝑑 𝑥2 1 𝑥2 2 ⋯ 𝑥2 𝑑 ⋮ ⋮ ⋮ ⋮ 𝑥𝑛 1 𝑥𝑛 2 … 𝑥𝑛 𝑑] (24) in this context, u symbolizes the world, n denotes the frequency of search components, d represents the measurements of control parameters, and 𝑥𝑖 𝑗 represents the jth parameter of the i-th world, which follows the form described below [46]: 𝑥𝑖 𝑗 = { 𝑥𝑘 𝑗 𝑟1 < 𝑁𝐼(𝑈𝑖) 𝑥𝑖 𝑗 𝑟1 ≥ 𝑁𝐼(𝑈𝑖) (25) where ui represents the ith world, ni represents the normalized inflation rate, 𝑟1 represents a random integer between 0 and 1, and 𝑥𝑘 𝑗 represents the jth variable of the kth world, selected using the roulette wheel scenario. the operational mechanism of the transferred items via wormholes is shown as follows [46]: 𝑥𝑖 𝑗 = { { 𝑥𝑗 + 𝑇𝐷𝑅 ∙ [(𝑢𝑏𝑗 −) ∙ 𝑟4 + 𝑙𝑏𝑗] 𝑟3 < 0.5 𝑥𝑗 − 𝑇𝐷𝑅 ∙ [(𝑢𝑏𝑗 −) ∙ 𝑟4 + 𝑙𝑏𝑗] 𝑟3 ≥ 0.5 𝑟2 < 𝑊𝐸𝑃 𝑥𝑖 𝑗 𝑟2 ≥ 𝑊𝐸𝑃 (26) here 𝑥𝑗 is the jth variable of the optimum world; wep (wormhole existence probability) and tdr (traveling distance rate) are coefficients; ubj and lbj indicate the upper and lower limit of jth variable, 𝑥𝑖 𝑗 indicates the jth parameter of the ith world; 𝑟4, 𝑟3, and 𝑟2 are accidental numbers belonging to [0, 1]. wep takes the following form and enhances the exploitation [46]: 𝑥𝑖 𝑗 = 𝑊𝐸𝑃 = 𝑚𝑖𝑛 + 𝑡 × ( 𝑚𝑎𝑥−𝑚𝑖𝑛 𝑇𝑚𝑎𝑥 ) (27) where the variable i represents the current repetition, 𝑇𝑚𝑎𝑥 represents the greatest repetition frequency, and "min" and "max" represent the lowest and maximum values of the controlled variables. the acquisition of tdr is done in the following manner [46]: 𝑇𝐷𝑅 = 1 − 𝑡(1 𝑝⁄ ) 𝑇𝑚𝑎𝑥 (1 𝑝⁄ ) (28) enhancing support vector machine performance in forecasting the number of vehicles … 9 where, p represents the level of precision achieved during the iterations of exploitation. as the value of p increases, the exploitation/local search process becomes faster and more precise. 4. case study the proposed method for analyzing road accidents in italy was tested using a dataset of 564 accident records from the regional road accident center (crisc) of the calabria region, covering the years 2001-2020. crisc acquires on its platform the accidents recorded by istat in the regional territory. istat, the italian national institute of statistics, is the primary source of official statistical data in italy, responsible for gathering and producing information about the italian economy and society. the data is gathered through a monthly survey involving various authorities, such as traffic police, carabinieri, and municipal police, who use the istat ctt/inc model known as "road accidents" to document each accident involving a vehicle on the road network that resulted in injuries [47]. the survey parameters include road accidents, fatalities, and injuries. road accidents involve accidents on a road open to public traffic, resulting in injuries or fatalities to one or more individuals and involving at least one vehicle. fatalities include individuals who perish immediately or within 24 hours following the accident, while injuries are categorized as serious or minor injuries [47]. a comprehensive total of data on accident accidents were meticulously documented and assessed across urban and rural areas of the calabria region, in southern italy (fig. 1). these accidents have been categorized and grouped based on several criteria (table 1). fig. 1 accident sites in the calabria region for the years 2001-2020. (source: crisc regione calabria) 10 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. table 1 accident database fields were analyzed data field type data field description human characteristic driver gender male or female driver age young, middle-age, old vehicle characteristic vehicle type motorcycle, car, bus, truck, other road road type two carriageways, more than two carriageways, one two-way carriageway, one one-way carriageway; road signal absent, horizontal, temporary construction, vertical, vertical and horizontal; road surface dry, wet, icy, snow, covered, slippery; environment date light daylight and nighttime weather conditions clear/sunny, hail, rain, snow, fog, strong wind, other; location urban area intersection, non-intersection, others; accident characteristic number of vehicles number of vehicles involved, one or more than one. 5. modelling process this study used a binary classification method to identify patterns and correlations between different input elements and the number of vehicles involved in accidents. the objective of the binary classification model was to determine whether any correlation between daylight, location, road type, road surface, road signs, weather conditions, types of vehicles, driver age, and driver gender against the number of vehicles as the dependent variable. identifying the control parameters and performance indices are the major steps of algorithms to reach the optimal binary model [48]. in this regard, the current study developed and compared four machine-learning algorithms; i.e. svm, svm-csa, svm-woa, and svm-mvo. the input dataset consisted of nine factors related to the number of vehicles in the urban area of cosenza, southern italy. data were classified by two classes: '1', in the case when at most one car participated in an accident, and '2' in the case with at least two or more involved vehicles. the main class separation criterion considered was based on taking into account the minimum number of vehicles involved in an accident. the best classification models were developed and compared, and the sensitivity analysis was carried out to determine the relevance of an effect independently exerted by each factor. the confusion matrix's accuracy and error in binary classification modeling are practical performance indicators. data normalization is essential for data-driven system modeling methods, as greater scale factors might lead to computational deviations. all data undergoes min-max normalization prior to modeling. enhancing support vector machine performance in forecasting the number of vehicles … 11 5.1. correlation analysis this study highlights that computing and controlling the parametric correlation of independent input datasets are of prime importance in providing correct results and to avoid misinterpretation of the data. pearson's correlation coefficient, sometimes also referred to as the bilateral correlation coefficient or pearson product-moment correlation coefficient, is one of the most useful methods to assess linear connections between variables. eqs. (2932) depict the relationships of pearson's correlation coefficient [49]. 𝜌 = 𝑟 = 𝑆𝑃𝐷𝑥𝑦 √𝑆𝑆𝑋.𝑆𝑆𝑌 (29) 𝑆𝑃𝐷𝑥𝑦 = ∑𝑥𝑦 − (∑𝑥)(∑𝑦) 𝑛 (30) 𝑆𝑆𝑋 = ∑ 𝑥𝑖 2𝑛 𝑖 − (∑𝑥𝑖) 2 𝑛 (31) 𝑆𝑆𝑌 = ∑ 𝑦𝑖 2𝑛 𝑖 − (∑𝑦𝑖) 2 𝑛 (32) here, x and y constitute the independent parameters; ssx and ssy respectively show the standard deviation of x and y. moreover, spdxy expresses the covariance between variables x and y. denoted as ρ(r), pearson's correlation coefficient is a statistical evaluation spanning -1 and +1. the strength of the association between the two independent variables is found by means of these coefficient magnitudes. furthermore, the positive and negative signs of these coefficients point accordingly to the direction of the link, either direct or reverse. while a correlation value close to 0 suggests a poor association between two independent factors, a correlation number close to 1 denotes a significant relationship between them [50]. it should be noted that the python package seaborn [51] was used to obtain the results presented in fig. 2. fig. 2 pearson correlation matrix with circle markers 12 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. fig. 2 shows the pearson correlation coefficient matrix for the dataset, visualized using a grid of colored circles. the strength and nature of the link between variables including daylight (dl), location (l), road type (rt), road surface (rs), road signs (sg), weather conditions (wc), vehicle type (vt), driver age (da), and driver gender (dg) each circle's size and color intensity reflect. red shows positive correlations; blue shows negative correlations; the lack of a circle denotes no noteworthy correlation. we regard as "strong" any ρ whose absolute value exceeds 0.85 for modeling needs. not one of the coefficients in this matrix surpasses this level, though. this graphic depiction highlights important relationships including the modest negative correlation between vehicle type and location (r = -0.485) and the modest positive correlation between vehicle type and daylight hours (r = 0.516), so enabling a quick and easy knowledge of the relationships between variables. without consulting numerical data, the visual signals given by the colored circles help one to recognize patterns and connections in this correlation matrix. the study of the data shown in the table suggests that most correlations are near zero, thereby implying essentially linear connections among them. for example, the rather evident moderate negative correlation between daylight and location (r = -0.311) indicates that the location factor usually tends to drop as daylight rises. correspondingly, the modest positive connection between daylight and driver age (r = 0.349) suggests that older drivers are more likely to be on the road during daylight hours. for initial research, this representation is a useful tool since it helps scientists rapidly identify important relationships and concentrate on areas that might call for more study. though two variables may be related, this does not guarantee one thing causes another. 5.2. svm modelling this work developed a binary classification model in matlab using svm to estimate the involved vehicle count. in this research, the number of accidents was the dependent variable; dl, l, rt, rs, sg, wc, vt, da, and dg were the independent variables. the good performance of an svm model depends on the exact control parameter selection and appropriate architecture of the model. therefore, we aimed to find an optimum svm model for a multi-class classification issue concerning the number of cars involved in accidents by means of an iterative hyperparameter tuning strategy, often known as manual hyperparameter tuning or a guided variation of grid search [52]. in fact, specific relationships for determining control parameters do not exist or are limited, hence; we use the try and error technique to determine them [53,54]. this procedure involved systematic adjustment of important parameters: the kernel function, the box constraint (c), and the kernel scale; i.e. for the rbf kernel. while conventional grid search investigates all hyperparameter combinations within preset ranges, our approach includes flexibility depending on insights obtained from each experimental iteration, therefore matching suggestions in the literature for successful hyperparameter tuning in svm models [55]. model effectiveness was found by performance throughout training, validation, and test sets. the primary goal was to optimize test set accuracy by precisely matching model complexity with generalization capacity, hence lowering the risk of overfitting [56]. the experimental path is summed in table 2, which also offers an understanding of the justification for every hyperparameter change. enhancing support vector machine performance in forecasting the number of vehicles … 13 table 2 empirical analysis of svm hyperparameter tuning no. kernel function box constraint (c) kernel scale (for rbf) train accuracy validati on accuracy test accuracy notes/observations 1 linear 1 (default) 85.73 88.67 80.00 baseline linear kernel; satisfactory but suboptimal performance. 2 rbf 1 (default) auto 89.20 84.67 86.00 transition to rbf kernel; improved test accuracy, yet evidence of mild overfitting (train > validation). 3 rbf 0.1 auto 84.13 80.67 82.00 decreased c to reduce overfitting; successful in reducing overfitting but at the cost of slightly diminished test accuracy. 4 rbf 10 auto 91.07 79.33 88.00 increased c; substantial test accuracy improvement, but overfitting exacerbated. 5 rbf 10 0.1 92.13 74.00 72.00 decreased kernelscale; induced severe overfitting (high train, low validation/test). 6 rbf 10 10 86.00 87.33 84.00 increased kernelscale; successful overfitting mitigation, yet a minor decline in test accuracy. 7 rbf 5 10 85.47 80.67 84.00 further c reduction; overfitting further curtailed, but no test accuracy gain. 8 polynomial 1 n/a 85.47 80.67 76.00 explored polynomial kernel; inferior test accuracy compared to rbf. 9 polynomial 0.1 n/a 90.53 80.67 82.00 decreased c for polynomial kernel; marginal improvement but still lags behind rbf. 10 polynomial 10 n/a 91.60 83.33 78.00 increased c for polynomial kernel; led to overfitting and decreased test accuracy. 11 rbf 7.5 10 84.93 83.33 80.00 further c fine-tuning for rbf; overfitting reduction, but test accuracy slightly lower. 12 rbf 8.75 10 85.47 83.33 74.00 additional c fine-tuning; test accuracy dropped. 13 rbf 10 5 87.60 84.67 84.00 adjusted kernelscale; notable validation accuracy improvement, test accuracy maintained. 14 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. 14 rbf 12 5 86.80 86.00 84.00 slight c increase; minor train/validation improvement, and no change in test. 15 rbf 15 5 87.60 82.00 90.00 further increased c; significant improvement in test accuracy, but a slight decrease in validation. 16 rbf 12 5 86.53 82.00 88.00 decreased c slightly; reduced overfitting, maintained high test accuracy. 17 rbf 13.5 5 86.46 87.33 86.20 further c fine-tuning; improved validation, slight test accuracy decrease. with test accuracy of 86.2% and a solid balance between training and validation accuracy, experiment 17 turned out to be the most fascinating configuration based on the series of tests. this underscores a key characteristic of predictive models: their capacity to generalize appropriately to the test set [57]. furthermore, showing less overfitting than other configurations with the same test accuracy, it suggested a more stable model. analyzing the final model's confusion matrix and receiver operating characteristic (roc) curve would assist us to understand its performance even further as seen in figs. 3 and 4. fig. 3 confusion matrices for the training data (a) and testing data (b) according to fig. 3a, the 17th binary classification model for the train properly identified 165 occurrences of the first class labeled "1," hence implying the participation of only one vehicle in the accident. but it wrongly classified eighteen second-class cases (that is, involving two or more vehicles) as belonging to the first class designated "1". enhancing support vector machine performance in forecasting the number of vehicles … 15 notably, the model was able to classify the train set with a great degree of accuracy, more especially 86%. the binary classification findings for the test data point to 21 instances belonging to the second class with the label "2" and 22 cases as properly classified as belonging to the first class with the label "1" (fig. 3b). under labels "1" and "2," the 17th model, however, incorrectly projected 6 instances from the second class and 1 case from the first class. in test data categorization, the 17th model therefore obtained a reasonable accuracy of 86%. svm is shown in the paper to be a consistent method for system modeling – more notably, for accident prediction. fig. 4 roc curve of best svm model for training dataset (a), and testing dataset (b) the performance of model 17th was evaluated using the roc curve, a probabilitybased curve. the model's auc was higher than other developed models, ranging from 0 to 1. auc values between 0 and 1 indicate the model's performance. auc values less than 0.5 indicate the model's inadequacy, while values above 0.5 indicate the model's effectiveness in training, testing, and total roc curve [58]. this indicates the model's superiority in classification problems. by means of this empirical investigation, we methodically traversed the hyperparameter space employing a directed network search strategy akin to that applied in other studies on svm-based traffic safety assessment models [59]. this resulted in the discovery of an ideal svm configuration balancing generalization with model complexity. based on the chosen configuration, which underlying the rbf kernel, promising performance on the test set indicates its possible efficiency in estimating the number of vehicles involved in crashes depending on supplied criteria. 16 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. 5.3. svm-woa a predictive model was developed in the matlab environment using a mix of woa and svm. the woa algorithm was employed to optimize some parameters of the svm in order to maximize the performance of the svm model. in the modeling process, a random selection is made where 75% of the dataset (375 data points) is designated as the training dataset, while the remaining 25% (125 data points) is allocated as the testing dataset [60]. when using the svm algorithm for modeling, all data with labels "1" and "2" were classified into two classes. defining the control parameters of woa will help one to efficiently establish and maximize the svm model's parameters. the capacity of the model to converge fast and precisely is much influenced by these values. although exact correlations to define these criteria are not known, past studies and professional opinions have helped to create ranges for every one of them. this covers the count of whale populations (between 5 and 60) and the iterations (between 10 and 200). the most appropriate choices then were selected by means of experimentation and improvement [61]. the model also underwent k-fold cross-validation, a technique whereby the data was split into k subgroups. one of the k-1 samples was set aside for validation at each iteration under this strategy; the other k-1 sample was set aside for training. the method was repeated k times, once for training and once for validation using every subset of data. thus, choosing the average result of this k validation helped to define the last estimate. the determination of the k-fold value depends on the volume of data and the opinion of experts rather than on a specific technique. this present work used a k-fold value of 3. furthermore, chosen were three distinct kernel functions: notably rbf, pol, and lin. for svm-woa, a total of 48 models were constructed; 16 models were generated for each kernel function dependent on the number of control parameters. table 3 shows 16 models related to rbf as an alternate category as the most optimal models were linked to the rbf kernel function. table 3 the performance of training and testing models is determined by the different control parameters no. iterations populations training accuracy (%) testing accuracy (%) 1 5 10 86.67 85.60 2 5 50 88.00 84.00 3 5 100 87.73 87.20 4 5 200 87.47 84.80 5 20 10 86.13 83.20 6 20 50 88.00 86.40 7 20 100 86.40 85.60 8 20 200 89.07 84.80 9 40 10 86.67 84.00 10 40 50 86.13 84.20 11 40 100 85.60 84.00 12 40 200 88.53 86.40 13 60 10 86.93 86.20 14 60 50 87.20 84.60 15 60 100 89.33 86.40 16 60 200 89.07 88.00 enhancing support vector machine performance in forecasting the number of vehicles … 17 after building several models and assessing their accuracy, all the models were ranked simply using a basic ranking system [62]. the scores are given based on the accuracy percentage as a performance indicator, in a scale from 1 to 16. here, 16 denotes the highest and 1 represents the lowest performance. these scores will be given according to the performance of each model with respect to other models. after assigning the scores for all performance metrics, the sum of the scores across all indicators gives the total score of each model. then, ranking is done by comparing total scores to determine which model has the highest cumulative score and hence is the best performing. this will ensure that with this scoring and ranking system, multiple dimensions of performance will be considered to properly evaluate all models comprehensively and make a balanced comparison, prioritizing both training and testing accuracy. table 4 lists the ranking results. table 4 ranking of developed models mlpwoa model number iterations populations rating for accuracy of training rating for accuracy of testing total rank 1 5 10 8 12 20 2 5 50 13 8 21 3 5 100 12 15 27 4 5 200 11 11 22 5 20 10 6 7 13 6 20 50 13 14 27 7 20 100 7 12 19 8 20 200 15 11 26 9 40 10 8 8 16 10 40 50 6 9 15 11 40 100 5 8 13 12 40 200 14 14 28 13 60 10 9 13 22 14 60 50 10 10 20 15 60 100 16 14 30 16 60 200 15 16 31 table 4 shows that the best performance of model number 16 has a rank of 31 out of the total 15 training and 16 testing accuracy ratings. therefore, it implies that at rank 31, the model reaches an ideal point for balancing between training and testing accuracy. the total rank of model number 15 comes next with a rank of 30 and performs less compared to model 16. the remaining models have shown various degrees of performances, the least successful mix of training and testing accuracy ratings of models with numbers 5 and 11. with the total ranking of 28, model number 12 has the third highest total ranking, indicating that it has a good balance between its training and testing accuracy. model 16 has the best model in terms of a total rank measure for both training and testing accuracy. based on fig. 5, the 16th binary classification model accurately identified 163 instances of the first class labeled "1" in an accident, indicating the involvement of only one car. however, it incorrectly classified 8 instances of the second class, indicating involvement of two or more vehicles, as "1". the model achieved a high level of accuracy, specifically 18 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. 89.1%, in categorizing the training dataset. the test data showed that 52 cases were correctly identified as "1" and 58 as "2", but the model incorrectly predicted 3 cases and 12 cases from the second class. the 16th model achieved a satisfactory accuracy of 88% in classifying the test data, demonstrating the reliability of the combination of woa and svm for system modeling. fig. 5 confusion matrices for the training data (a) and testing data (b) 5.4. svm-csa a predictive model was developed by combining csa and svm in the matlab environment. in the csa algorithm, the optimum performance of the svm model was achieved after optimizing its parameters. the same datasets used in the svm analysis were used to develop svm-csa modeling, comprising 75% of the dataset containing 375 data as training datasets and 25% as 125 data as a test dataset [59]. all the data were labeled as two classes, which were "1" and "2". the best setting of the svm model had been chosen in the svm analysis section. the determination of control parameters of csa was done, which plays an important role in the convergence of the model. since there is no relation that precisely determines such parameters, the approach to be followed was the same as in the model svm-woa. therefore, the same population number and number of iterations were considered. the best parameters were selected based on a trial-and-error approach. in addition, the three different types of kernel functions including rbf, pol, and lin were used. according to the number of control parameters, a total of 16 models were built and their outcomes are displayed in table 5. following the construction of several models and the evaluation of their accuracy, a straightforward ranking method was employed to rank all the models [62]. the ranking outcomes are presented in table 6. enhancing support vector machine performance in forecasting the number of vehicles … 19 table 5 the performance of training and testing models is determined by the different control parameters no. iterations populations training accuracy (%) testing accuracy (%) 1 5 10 85.07 83.20 2 5 50 87.47 85.60 3 5 100 87.07 86.40 4 5 200 87.73 87.20 5 20 10 85.60 84.80 6 20 50 90.10 88.00 7 20 100 88.80 86.40 8 20 200 88.80 88.80 9 40 10 90.13 87.20 10 40 50 89.60 87.30 11 40 100 87.70 86.40 12 40 200 88.53 85.60 13 60 10 88.27 88.80 14 60 50 88.53 88.00 15 60 100 86.93 84.80 16 60 200 87.07 85.00 table 6 ranking of developed models mlp-woa model number iterations populations rating for accuracy of training rating for accuracy of testing total rank 1 5 10 4 8 12 2 5 50 8 11 19 3 5 100 7 12 19 4 5 200 10 13 23 5 20 10 5 9 14 6 20 50 15 15 30 7 20 100 13 12 25 8 20 200 13 16 29 9 40 10 16 13 29 10 40 50 14 14 28 11 40 100 9 12 21 12 40 200 12 11 23 13 60 10 11 16 27 14 60 50 12 15 27 15 60 100 6 9 15 16 60 200 7 10 17 the ranking of the created models based on their performance in training and testing is presented in table 6. the model with the highest performance is model number 6, which has a total rank of 30. this rank is determined by combining a training accuracy rating of 15 and a testing accuracy rating of 15. the model's exceptional total rank indicates that it has achieved the most optimal balance between training and testing accuracy compared to all the analyzed models. models 8 and 9, which come after model number 6, have a total 20 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. rank of 29, making it the second highest. this model exhibits robust performance in both domains but is slightly lower than model number 6. the remaining models exhibit a variety of performance levels. model number 1 has the lowest total rank of 12, indicating the least effective combination of training and testing accuracy ratings. conversely, model number 10 possesses a total rank of 28, which is the third highest. this suggests that the model achieves a commendable equilibrium between training and testing accuracy. model number 6 is deemed the superior model based on the total rank measure, which considers both training and testing accuracy. the remaining models exhibit different levels of achievement, with certain models demonstrating exceptional training accuracy, others showcasing superior testing accuracy, and some striking a harmonious equilibrium between the two. fig. 6 confusion matrices for the training data (a) and testing data (b) based on fig. 6a, the sixth binary classification model accurately identified 165 instances of the first class labeled "1" (indicating the involvement of only one car in the accident). however, it incorrectly classified 8 instances of the second class (indicating involvement of two or more vehicles) as belonging to the first class labeled "1". it is important to note that the model was able to categorize the training dataset with a high level of accuracy, specifically 90.1%. according to fig. 6b, the binary classification results for the test data indicate that 55 cases belonging to the first class with label "1" and 55 cases belonging to the second class with label "2" were correctly identified. however, the sixth model incorrectly predicted 4 cases from the first class and 11 cases from the second class, with labels "1" and "2" respectively. as a result, the sixth model achieved a satisfactory accuracy of 88% in classifying the test data. the investigation demonstrates that the combination of csa and svm is a dependable methodology for system modeling, specifically in anticipating accident occurrences. enhancing support vector machine performance in forecasting the number of vehicles … 21 5.5. svm-mvo the present work designed a prediction model using the matlab environment, which integrates mvo with svm. parameters of the svm model were tuned using the mvo technique in order to achieve the highest performance. the svm-mvo modeling technique utilizes the same dataset as svm analysis, with 75% of the data allocated for training and the remaining 25% for testing [59]. based on the entire dataset, the two categories were designated as "1" and "2". applying the same approach as the svm-woa model, the control parameters of the mvo were determined through a process of rigorous experimentation. three unique kernel functions were utilized, and a total of sixteen models were produced, depending on the number of control parameters being considered. the results of these models are presented in table 7. table 7 the performance of training and testing models is determined by the different control parameters no. iterations populations training accuracy (%) testing accuracy (%) 1 5 10 85.87 84.00 2 5 50 88.53 86.40 3 5 100 88.80 85.60 4 5 200 86.13 84.80 5 20 10 87.20 84.80 6 20 50 87.73 86.40 7 20 100 88.80 87.20 8 20 200 89.60 87.20 9 40 10 85.87 84.00 10 40 50 88.27 87.20 11 40 100 88.27 86.40 12 40 200 88.00 87.20 13 60 10 86.93 84.00 14 60 50 88.27 87.20 15 60 100 88.80 87.00 16 60 200 89.00 87.20 following the construction of several models and the evaluation of their accuracy, a straightforward ranking method was employed to rank all the models. the ranking outcomes are presented in table 8. it displays the models' relative rankings according to their performance in both training and testing. model 8 attained the highest level of equilibrium between training and test accuracy, resulting in a total rank of 32. model 16, with a rating of 31, demonstrates robust performance in both domains, but slightly below that of model 8. model 8, widely regarded as the leading model, has distinct degrees of achievement. according to fig. 7, which shows the confusion matrix, the 8th binary classification model accurately identified 162 instances of the first class labeled "1" in an accident, with a high level of accuracy of 89.6%. however, it incorrectly classified 6 instances of the second class, indicating the involvement of two or more vehicles, as "1". the model was able to categorize the training dataset with a satisfactory accuracy of 87.2%. the third model achieved a satisfactory accuracy of 87.2% in classifying the test data. 22 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. table 8 ranking of developed models mlp-woa model number iterations populations rating for accuracy of training rating for accuracy of testing total rank 1 5 10 6 11 17 2 5 50 13 14 27 3 5 100 14 13 27 4 5 200 7 12 19 5 20 10 9 12 21 6 20 50 10 14 24 7 20 100 14 16 30 8 20 200 16 16 32 9 40 10 6 11 17 10 40 50 12 16 28 11 40 100 12 14 26 12 40 200 11 16 27 13 60 10 8 11 19 14 60 50 12 16 28 15 60 100 14 15 29 16 60 200 15 16 31 fig. 7 confusion matrices for the training data (a) and testing data (b) 6. result and discussion each year, many people die as a result of road traffic crashes. therefore, knowing the impact of various contributing factors on the number of road accidents and taking the necessary measures to reduce accidents can have a significant impact on increasing the level of road safety. in this research, four ml methods, namely svm, svm-woa, svmcsa, and svm-mvo were employed to conduct the binary classification modeling. after multiple modeling, the best model was selected based on the accuracy of the modeling enhancing support vector machine performance in forecasting the number of vehicles … 23 performance. a comparison was made between the best models based on the accuracy of training and testing, as shown in fig. 8. fig. 8 comparison between the best svm, svm-csa, mlp-woa, and svm-mvo models for training and testing accuracies based on fig. 8, the svm-csa model demonstrated superior performance compared to the other best models in forecasting the number of accidents. it achieved training and testing accuracies of 90.1% and 88%, respectively, which were greater than the accuracies of the other three models. it is important to note that all models demonstrated satisfactory levels of accuracy and robustness. therefore, it can be inferred that they are dependable modeling systems for predicting the occurrence of crashes. these models can serve as valuable tools for analyzing road safety in the field of transportation engineering. the occurrence of road accidents can result in significant economic and human detriment to society. consequently, evaluating the influence of factors that affect the frequency of accidents can offer engineers engaged in road safety management a comprehensive understanding. to evaluate the influence of daylight (dl), location (l), road type (rt), road surface (rs), road signs (sg), weather conditions (wc), vehicle type (vt), driver age (da), and driver gender (dg) on estimating the number of cars involved in accidents, a sensitivity analysis was conducted. this sensitivity analysis utilizes the cosine amplitude approach as described by [63]: 𝑟𝑖𝑗 = ∑ (𝑥𝑖𝑘×𝑦𝑗𝑘)𝑛 𝑘=1 √∑ 𝑥𝑖𝑘 2 ∑ 𝑦𝑖𝑘 2𝑛 𝑘=1 𝑛 𝑘=1 (33) where rij represents the strength of the relationship, n denotes the number of datasets, and xik and yij represent the input parameters and anticipated output, respectively. 24 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. fig. 9 results of sensitivity analysis obtained from the svm-csa model additionally, the individual conditional expectation (ice) plot (fig. 10) was used to achieve a deeper understanding of the relationship between the most influential predictor variable (vehicle type) identified in fig. 9 and the expected result (number of vehicles involved in an accident). ice plots show, for individual instances, how a model prediction would change if the value of one feature varied, while the other features remained constant [64]. fig. 10 individual conditional expectation plot for vehicle type and multi-vehicle accident probability enhancing support vector machine performance in forecasting the number of vehicles … 25 the ice plot visualizes the effect of vehicle type on the probability of multi-vehicle accidents. at an aggregated level, the increase in probability tends to follow the change from "bus/truck/other" via "cars" towards "motorcycles." however, the individual ice lines behave quite differently, showing that this is moderated by other factors that vary across the instances. in the ice plot, the three-colored lines correspond with the three categories of vehicle type: blue for "bus/truck/other", red for "cars", and orange for "motorcycles". also, the partial dependence plot (pdp) (black line) shows the average effect of one feature on model predictions, holding all other features constant. the pdp is an overall picture of the dependence that the model has on a particular feature and is obtained by averaging the lines from ice plots [64, 65]. figs. 9 and 10 demonstrate that the model analysis yields results reflecting the influence of the examined components, hence confirming the dependability of the findings. furthermore, the following remarks can also be made. the vehicle type was found to have the most imperative impact on the number of cars involved in accidents. this points to the indication that some types of vehicles are more prone to accidents and could further enhance the devastating effects of these accidents as well [61]. indeed, further confirmation of this finding has been made through ice plot analysis, which shows that the highest average predicted probability of involvement in multi-vehicle accidents is for motorcycles, followed by cars, and then buses/trucks/other vehicles. the fact that the type of vehicle has a relatively consistent effect on the prediction for motorcycles, given the diversity of instances in this class, would therefore imply inherent characteristics or patterns of traffic that put them at higher risk. this is especially present in multi-vehicle accidents. with the increase in electric motorbikes and scooters, the demand for focused interventional policies, such as customized training programs and improvements in the road structure, also increases [66-68]. therefore, we can conclude that this would form a possible basis on which specific traffic patterns may be examined as a function of the vehicle type. motorcycles, having the highest relative risk, might be expected to have accidents due to their lane changing and higher speeds through congested traffic. such accidents, in particular, could be radically minimized through the introduction of advanced technologies related to adaptive cruise control and warning systems. such an approach would result in fewer road accidents, more even flow conditions of traffic, and reduced costs arising from these road accidents. furthermore, location is also a very important factor; i.e. topography and features of any area or a part of the road can result in more frequent accidents within these areas, requiring further research and focused solutions. bad weather, like rain or fog, also has a great impact on accident rates; that is the meaning of teaching drivers how to behave in traffic management systems capable of adapting to diverse situations [66, 69]. driving factors, such as gender and age, further influence involvement in accidents; this might be due to differences in either driving style or the tendency toward risk exposure. these factors could be addressed through strategic teaching programs and sensitization processes effectively [66]. the nature of the road is another factor, with different roads being risky to different measures. improved road design, traffic control, and signage can help drastically reduce these risks [66]. in general, diurnal daylight conditions, with increased visibility at night, accentuate the occurrence of accidents. therefore, priorities regarding improved street lighting, marks for visibility, and awareness campaigns are needed [70]. additionally, road surface conditions and signage, although having a relatively smaller impact compared to other factors, still play a role in road safety. guaranteeing safety to 26 s. s. haghshenas, m. h. m. seraji, s. s. haghshenas, et al. drivers actually depends mainly on proper maintenance, appropriate signs, and an unobstructed view [71-73]. one of the significant limitations in the proposed approach is that it cannot operate with incomplete data; therefore, full and accurate data availability is crucial for the successful implementation of this approach. moreover, the results and interpretations obtained from this study relate only to the urban road conditions of calabria; thus, applications in other areas or other types of road scenarios are not directly possible. nevertheless, this research proved that classification techniques can be a strong tool in the prediction of the number of vehicles involved in traffic accidents, thus providing important information to transportation engineering. finally, some of the future directions of research could be extension of applicability by testing the proposed approach in a wide range of regions and road conditions to enhance generalizability. future studies may investigate the implications of emerging trends faced by accident prediction models with respect to electric scooters and autonomous vehicles. the interaction between environmental factors, vehicle types, and driver characteristics would provide deeper insights into the causation of an accident. longitudinally, evaluation studies that determine the long-term effectiveness of interventions such as infrastructure improvement, training programs, and publicity should help revise and prioritize road safety measures. 7. conclusions this study applied a binary classification approach to model the relationship between various factors and the number of vehicles involved in road accidents in the urban area of cosenza, southern italy. variables such as dl, l, rt, rs, sg, wc, vt, da, and dg were considered as input factors for the analysis. for this purpose, four machine learning models, namely svm, svm-csa, svm-woa, and svm-mvo, have been developed and compared to choose the best predictive model. the classification process distinguished between single-vehicle and multi-vehicle accidents. model performance was assessed using confusion matrices and roc curves. among the evaluated models, svm-csa achieved the best performance, with a 90.1% increase in training accuracy and a high test accuracy of 88%, outperforming the other techniques. the sensitivity analysis revealed that the factor "vehicle type" had the highest influence, while "road signs" had the least. these findings underline the effectiveness of advanced machine learning techniques in establishing the key 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https://orcid.org/0000-0001-7076-2519 abstract. in multi-criteria decision analysis (mcda), data normalization is essential for ensuring the comparability of heterogeneous and often conflicting evaluation criteria. conventional normalization techniques, although methodologically straightforward, are predominantly tailored for monotonic criteria, rendering them ineffective for nonmonotonic criteria characterized by extrema within the interval rather than at its boundaries. this limitation significantly undermines their applicability in the reidentification of decision models, as they fail to adequately account for the complexity and variability inherent in non-monotonic evaluation approaches. this paper presents a study on the application of stochastic fuzzy normalization (stfn) in combination with popular mcda methods such as vikor, topsis, and mabac in addressing engineering problems. the study evaluates the effectiveness of this approach in re-identifying decision models, emphasizing its capability to manage nonlinearities and nonmonotonic criteria, mitigate rank reversal phenomena, and adapt to dynamic decision-making scenarios. in this work, the fuzzy reference model (frm) is leveraged as a robust simulation framework to evaluate the performance of stfn in re-identifying decision models, enabling comprehensive benchmarking of mcda techniques by providing detailed preference information for each decision option. through a practical case study involving the selection of an optimal energy source for an industrial plant, the study illustrates how fuzzy normalization supports reliable re-identification of decision models. these comparative analyses reveal potential outcomes and highlight notable differences when stfn is applied in conjunction with various mcda methods, demonstrating the value of this approach in decision-making contexts. key words: mcda, re-identification, stfn, topsis, vikor, mabac received: september 16, 2024 / accepted january 23, 2025 corresponding author: wojciech sałabun national institute of telecommunications, ul. szachowa 1, 04-894 warsaw, poland e-mail: w.salabun@il-pib.pl https://orcid.org/0000-0001-5736-4014 https://orcid.org/0000-0001-7076-2519 512 b. kizielewicz, w. sałabun 1. introduction multi-criteria decision analysis (mcda) is a comprehensive decision-support methodology instrumental in situations where multiple, often opposing, criteria must be considered [1]. mcda methods enable systematic and transparent decision-making by integrating a variety of criteria and decision-maker preferences, allowing a more accurate understanding of the compromises between different options. mcda is widely used in many fields, such as management [2], engineering [3], environmental protection [4], urban planning [5], and public health [6]. in management and business, for example, mcda supports companies in evaluating and selecting strategies, investments, and suppliers, considering both financial and non-financial aspects [7,8]. dealing with problems that involve multiple opposing criteria is problematic. therefore, techniques are constantly being developed to facilitate such decisions. one of the critical elements of mcda is data normalization, which makes it possible to compare different criteria by converting them to a standard scale [9]. with normalization, criteria become comparable, greatly facilitating the analysis and selection of optimal solutions. in addition, normalization can immune mcda methods to rank reversal paradox and allow modeling of more favorable subjective preferences of decision makers. normalization, a pivotal process in the re-identification of mcda models, becomes even more significant when expert knowledge is lacking or when decision-making models need updating. in this article, we present a study that not only compares the effectiveness of various normalization methods, including the stochastic fuzzy normalization (stfn) based on triangular fuzzy numbers method [10], in the context of mcda model reidentification, but also provides practical insights into how the stfn method performs in re-identifying models such as visekriterijumska optimizacija i kompromisno resenje (vikor), technique for order preference by similarity to ideal solution (topsis) and multi-attributive border approximation area comparison (mabac). these findings are of direct relevance to researchers and practitioners in the field of mcda. indeed, our study makes an essential contribution to the field of mcda by providing insights into when and what normalization method to use to achieve decision models that closely approximate real-world conditions. the primary novelty of this work lies in its detailed comparative analysis of normalization methods for mcda model re-identification, addressing a gap in the existing literature. specifically, the application of stfn to re-identify models such as vikor, topsis, and mabac distinguishes this study from prior research. the study evaluates the conditions under which stfn excels and demonstrates its potential to mitigate challenges like rank reversal and the absence of expert input. moreover, the sensitivity analyses conducted in this study represent a significant advancement by systematically assessing the robustness of the re-identified models against variations in core values and boundary extrapolation. these analyses provide nuanced insights into the stability and adaptability of stfn-based approaches, offering valuable guidance for applications where decision models must contend with uncertainty and fluctuating data conditions. the practical case of selecting an optimal energy source for an industrial plant further highlights the real-world applicability of these methods. by demonstrating how stfn can effectively balance robustness and adaptability in complex, multidimensional decision-making scenarios, this work bridges the gap between theoretical development and practical implementation. benchmark study of re-identification methods based on stochastic fuzzy normalization... 513 through these contributions, this study advances the state-of-the-art in mcda, providing practical tools and strategies to enhance decision model robustness and adaptability. the results have significant implications for improving the quality and efficiency of decision-making processes across various domains, particularly those characterized by complexity and uncertainty. the rest of the paper is organized as follows. section 2 presents a review of the literature related to the normalization approaches used and their comparison methods. section 3 discusses preliminary assumptions related to fuzzy sets, fuzzy reference models, mcda methods, and the stfn re-identification approach. section 4 conducts a comparative study of re-identification methods. section 5 introduces a practical example related to an engineering problem, specifically the selection of an optimal energy source for an industrial plant, demonstrating the application of stfn-based methods in a real-world context. section 6 presents a discussion of the compared methods for re-identification. finally, section 7 concludes the paper and provides directions for future research. 2. literature review in the mcda literature, many normalization methods are used to compare and evaluate different criteria. normalization is a critical step in the mcda process, as it allows different criterion scales to be converted to a standard scale for direct comparison [9]. the most commonly used normalization methods include: ▪ min-max normalization: converts criteria values to the interval [0,1] based on the minimum and maximum value of a given criterion [11]. ▪ z-score normalization: based on the mean and standard deviation, transforming values in a way that eliminates units of measurement [12]. ▪ vector normalization: transforms values by dividing each value by the norm of the vector, which is particularly useful in cases where the units of measure are different [11]. ▪ sum normalization: involves transforming the values of the criteria by dividing each value by the sum of all the values of the criterion [13]. in the context of mcda, different approaches use different standardization methods to address specific requirements and challenges in the decision-making process. an example of a classic normalization method is the topsis approach [14], which uses vector normalization. another classic method using min-max normalization is the vikor method [15]. however, in addition to these traditional approaches, new decision-making methods also use normalization, among which are combined compromise solution (cocoso) which uses min-max normalization [16], combinative distance-based assessment (codas) which is based on linear normalization [17], multi-attributive border approximation area comparison (mabac) which uses min-max normalization [18], and multi-attributive ideal-real comparative analysis (mairca) also based on min-max normalization [19]. min-max normalization is widespread because of the numerous studies associated with it and its ability to deal with the phenomenon of reverse rankings. therefore, approaches immune to this phenomenon often use similar functions for normalization. an example of such an approach is stable preference ordering towards ideal solution (spotis) [20], which is based on normalization based on boundary values (i.e., min-max). another example is the reference ideal method (rim) [21], which uses a similar mechanism based on cutoff values for criteria for normalization. due to their robustness against the rank 514 b. kizielewicz, w. sałabun reversal paradox, these approaches are essential in decision-making when there is a risk of such a phenomenon. the growing complexity of decision-making problems has significantly contributed to the development of mcda methods, where normalization plays a crucial role in enabling robust decision processes. recent advancements highlight the role of fuzzy-based methods in addressing uncertainty and complexity in data. eti et al. [22] developed an innovative fuzzy decision-making framework to enhance electric vehicle charging infrastructure, showcasing how fuzzy modeling can address nonlinearities in data. similarly, kizielewicz and sałabun [23] introduced the stochastic identification of weights (sitw) method, reidentifying multi-criteria weights to dynamically adjust models to evolving decisionmaking conditions. these developments emphasize the importance of normalization in adapting mcda methods to dynamic environments where data variability and problem multidimensionality are critical. moreover, extensions of fuzzy methods have enabled their integration into intelligent decision support systems. hussain and ullah [24] demonstrated the application of spherical sugeno-weber operators to enhance the practical usability of fuzzy models, while narang et al. [25] proposed a fuzzy extension of the method based on the removal effects of criteria (merec) using parabolic measures, improving precision in criterion classification. these innovations underline the potential of combining traditional mcda approaches with fuzzy frameworks to address complex problems in business and technical contexts. applications of fuzzy methods extend beyond logistics and infrastructure. tešić and marinković [26] employed fermatean fuzzy weight operators to select combat systems based on efficiency, illustrating the potential of these methods in defense-related decision-making. similarly, kannan et al. [27] developed the linear diophantine fuzzy codas method to improve specialist selection processes in logistics, while asif et al. [28] applied hamacher operators to pythagorean fuzzy sets, broadening their applicability to multi-attribute decisionmaking. gazi et al. [29] employed the pentagonal fuzzy decision making trial and evaluation laboratory (dematel) methodology to identify key criteria in empowering women in sports, highlighting the societal benefits of fuzzy mcda approaches. further, kara et al. [30] applied a hybrid merecweighted euclidean distance-based approach (wedba) methodology to evaluate the performance of turkish universities, and kurtay [31] utilized the fuzzy evaluation based on distance from average solution (edas) method to select military vehicles, emphasizing the need for domain-specific adaptations of mcda techniques. mifdal and saracoglu [32] conducted a classification analysis using analytic hierarchy process (ahp) and activity-based costing (abc) to optimize inventory management, while yushuo and ling [33] proposed a prospect theory-based model to evaluate logistics enterprises' safety standardization performance, contributing to operational management. the importance of normalization techniques has also been highlighted in various comparative studies. for instance, [11] explored the impact of normalization techniques on decision rankings, illustrating how different methods can lead to varied outcomes. similarly, [34] examined normalization techniques in mcdm, emphasizing factors like decision-maker preferences and data structures. in [35], researchers identified suitable normalization methods for integration with the preference selection index (psi) method, while [36] proposed an assessment framework enriched with metrics. this framework, employing the simple additive weighting (saw) method, offers a structured approach to evaluating and selecting normalization techniques, providing practical guidance for ensuring reliable decision-making across diverse domains. benchmark study of re-identification methods based on stochastic fuzzy normalization... 515 3. preliminaries 3.1. fuzzy reference model the fuzzy reference model (frm) is a multi-criteria decision maker's preference function that has information about the preferences of each decision option. sałabun proposed this approach to estimate the accuracy of topsis normalization methods [37]. using a fuzzy reference model, multi-criteria decision-making techniques can be compared. in order to create the frm model, the following steps should be followed [37]: ▪ step 1. select the number and monotonicity of criteria, where criteria can have monotonicity such as profit, cost, or be non-monotonic. ▪ step 2. create a membership function for each criterion, where the function values should be in the domain [0, 1]. ▪ step 3. provide an evaluation value for each combination of information grains. this can be a random value but must be consistent with the previous assumptions (criterion type). ▪ step 4. create a rule base for the frm model based on the modus ponens tautology [38]. the fuzzy reference model is visualized in fig. 1. fig. 1 fuzzy reference model 3.2. topsis the technique for order preference by similarity to an ideal solution (topsis) operates within a structured framework centered on reference points and employs a systematic approach to evaluate alternatives [39]. fundamental to this method are two pivotal reference points: the positive ideal solution (pis) and the negative ideal solution (nis). topsis assesses alternatives by measuring their proximity to these reference points. the complete topsis procedure can be represented by fig. 2. fig. 2 topsis procedure 516 b. kizielewicz, w. sałabun 3.3. vikor the viekriterijumsko kompromisno rangiranje (vikor) method was developed to address discrete problems involving conflicting criteria [15]. the core concept of this approach is to identify compromise solutions, rank the decision alternatives, and select the optimal alternative. in the vikor technique, compromise ranking is achieved by assessing the proximity measure relative to the ideal alternative. the steps involved in the vikor method can be represented by fig. 3. fig. 3 vikor procedure 3.4. mabac the multi-attributive border approximation area comparison (mabac) method, introduced by [18], addresses practical multi-criteria decision-making challenges. its notable feature is the ability to manage conflicting criteria and diverse data units through an embedded normalization algorithm. the method's mathematical basis involves assessing alternatives' distances from the boundary approximation area (g). its simplicity contributes to its widespread adoption in decision-making contexts, as evidenced by [40]. moreover, the method offers extensions to handle uncertain data. the mabac procedure in combination with fuzzy normalisation can be represented by fig. 4. fig. 4 mabac procedure 3.5. stochastic fuzzy normalization based on triangular fuzzy numbers this section will introduce a method for re-identification that utilizes fuzzy normalization, termed stochastic fuzzy normalization (stfn) based on triangular fuzzy numbers. initially proposed by kizielewicz and dobryakova in their paper [10], this method employs stfn to re-identify a continuous topsis model. the core concept involves utilizing a stochastic optimization algorithm to search for triangular fuzzy number (tfn). in the context of this study, the stochastic differential evolution (de) method was employed. moreover, this study will delve into a broader spectrum of mcda approaches, including topsis, vikor, and mabac. the entire re-identification process employing the stfn approach can be delineated into the following steps: benchmark study of re-identification methods based on stochastic fuzzy normalization... 517 ▪ step 1. select a dataset. the dataset should contain criteria vectors (c), a criteria weights vector (w), and a ranking vector (r). ▪ step 2. select a stochastic optimization method. in this step, choose a stochastic method for solving the optimization problem and select its parameters. ▪ step 3. model training. training the model is done using the stochastic optimization algorithm and the fitness function, which can be represented fig. 5. fig. 5 stfn fitness function 4. comparative study of re-identification methods this study will compare the effectiveness of multi-criteria model re-identification methods. this area is significant in decision-making in various fields, such as management, engineering, and economics. we will use synthetically generated random data for this purpose, allowing us to control the experiment conditions and obtain reliable results. in order to present the re-identification problem more comprehensively, we will focus on comparing two decision criteria, which often occur in real decision-making scenarios. this way, we can study how different re-identification methods deal with various criteria. the study framework is detailed in fig. 6, which shows the critical steps of the experiment. the first will divide the decision matrix into two parts: a training matrix, used to teach reidentification models, and a test matrix, used to evaluate these models. an essential part of this process will be ensuring that there are enough alternatives in both matrices to ensure that the data is representative. after the split, both matrices will be evaluated using the frm, allowing us to assess the re-identification methods' effectiveness objectively. fig. 6 framework for the comparative study of methods of re-identification of decisionmaking models the scores obtained from the frm will then be converted into rankings: a training ranking and a test ranking. this step will allow us to compare the results of different reidentification methods. this study will focus on three main re-identification methods: 518 b. kizielewicz, w. sałabun stfn-topsis, stfn-vikor and stfn-mabac. each of these methods will be trained using a stochastic evolutionary difference method, and we will select the appropriate parameters based on a literature review and preliminary experiments. for the purposes of this study, the implementation was carried out using the pymcdm library along with the pymcdm-reidentify extension [41, 42]. after training, we will proceed to evaluate the performance of each of these methods by comparing their results with a reference model. in addition to evaluating based on rankings, we will also analyze the frm characteristic objects evaluated by the re-identification models. this analysis will allow us to understand better what decision-making aspects are most relevant to each method and how well they reflect decision-makers preferences. for this reason, metrics such as the weighted spearman rank correlation coefficient and error metrics were utilized. weighted spearman correlation coefficient was primarily employed because the output results were rankings, making it a natural choice for evaluating the consistency of rank order between re-identification methods and the frm. additionally, error metrics such as mean squared error (mse) and mean absolute error (mae) were used to analyze how stfn-based methods, including topsis, vikor, and mabac, evaluate characteristic objects and to quantify the differences between their assessments and the frm. this combination of metrics provides insights into both ranking alignment and numerical discrepancies, ensuring a comprehensive evaluation of the methods. however, our study used a frm, a nonlinear evaluation function for the two criteria under consideration. this model is crucial because it allows us to account for the nonlinear relationships between the alternatives and the decision criteria. the decision plane, as evaluated by the frm, is shown in fig. 7, which allows the complexity of the decision space to be visualized. in order to create the frm model for the two criteria, 14 linguistic values were taken and evenly distributed in the value space. based on these values, 196 characteristic objects were generated, which were used to evaluate the decision matrices. the evaluation process followed the frm methodology, similar to characteristic object method (comet), where each alternative was evaluated concerning the nearest characteristic objects. this evaluation was carried out using the mamdani model, which allows uncertainty and ambiguity to be taken into account in the decision evaluation. fig. 7 decision plane for fuzzy reference model used in the study. after obtaining preferences for the selected decision matrix, training the re-identification methods, i.e., stfn-topsis, stfn-vikor, and stfn-mabac, was started. the stochastic benchmark study of re-identification methods based on stochastic fuzzy normalization... 519 optimization method used to find the boundary values of tfn for normalization was based on the evolutionary difference method. the values of the decision matrix, drawn from the range [0, 1] for both criteria from uniform distribution, were also used as boundary values for tfn. the learning process lasted 1000 epochs, during which, for re-identification by the stfntopsis method, similarity was obtained on the training set with a value of rw equal to 0.93722, for stfn-vikor the value of rw was 0.88446, and for stfn-mabac it reached 0.92212. the decision grids obtained from training the models are shown with the help of fig. 8. it is noticeable that in the case of the stfn-vikor approach, the formation of the function is slightly more distorted than in the case of stfn-topsis and stfn-mabac. this results from a more complex decision function based on two components. in addition, unlike the stfn-topsis and stnf-mabac methods, the stfn-vikor approach uses the v parameter, which also affects the shape of the decision grid. in the case under study, a parameter v of 0.5 was used, and q values were used to carry out the ranking process. the decision lattices obtained using the stfn-topsis and stfn-mabac approaches are closer to the reference model, the frm. fig. 8 obtained decision grids by selected re-identification methods learned on the training matrix the resulting core values for the re-identification approaches for the triangular fuzzy numbers used for normalization are presented using a table 1. this table shows the core values for each re-identification approach for the two decision criteria. it is worth noting that the differences between the obtained core values and the reference core are relatively small. for example, for criterion c1, all re-identification approaches achieved core values close to 0.9, indicating a high degree of similarity to the reference model. however, for criterion c2, some differences in the core values between the approaches can be seen, which may suggest differences in the effectiveness of the reference model mapping in the context of this criterion. table 1 obtained cores for triangular fuzzy numbers intended to normalize criteria in reidentification models approach c1 c2 stfn-topsis 0.903885 0.417369 stfn-vikor 0.903885 0.396207 stfn-mabac 0.903885 0.413124 after obtaining the stfn-topsis, stfn-vikor, and stfn-mabac, re-identification models were tested on a test matrix of 20 alternatives. the first step in this testing was to 520 b. kizielewicz, w. sałabun compare the rankings obtained for the decision test matrix. for this purpose, each alternative was evaluated by all three re-identification approaches, and the evaluations were then ranked according to the selected criteria. for the stfn-topsis and stfn-mabac methods, the higher the evaluation value, the higher the ranking, while stfn-vikor reversed the situation. the resulting rankings are shown in fig. 9. analysis of these rankings showed that the ranking positions for the test set differ slightly between the stfn-topsis and stfn-mabac approaches. there are only two positions of difference for alternative a2 and alternative a10. however, the most significant differences appeared in the comparison between stfn-topsis and stfn-mabac and the stfn-vikor approach. for 11 ranking alternatives, differences were observed between the methods. fig. 9 comparison of the rankings of the test decision matrix by the obtained re-identification models after obtaining the rankings for the test decision matrix, a comparison was made between these rankings and the ranking derived from the frm model. this comparison is shown in fig. 10. the comparative analysis considered two leading indicators: weighted spearman correlation coefficient (rw) and the similarity coefficient of the rankings. a weighted spearman correlation coefficient of 0.91106 was obtained for the stfn-topsis method, indicating a high degree of agreement with the frm model. in addition, the similarity coefficient of the rankings was 0.95431, which further confirms the effectiveness of this method in mapping the preferences of the reference model. for the stfn-vikor method, weighted spearman correlation coefficient was 0.89424, indicating a slightly lower degree of agreement with the frm model than stfn-topsis. however, the similarity coefficient of the rankings reached 0.96000, suggesting high agreement in the order of the alternatives' ratings. the stfn-mabac method obtained a weighted spearman correlation coefficient of 0.90304 and a ranking similarity coefficient of 0.95408. these results indicate good agreement with the frm model and high stability in evaluating the rankings. benchmark study of re-identification methods based on stochastic fuzzy normalization... 521 fig. 10 comparison of rankings derived from redientfication models with a reference ranking (test decision matrix) the analysis of these results allows us to conclude that all three reidentification methods showed high efficiency in reproducing the preferences of the reference model. at the same time, the differences in the results between the methods may suggest their different properties and the degree of adaptation to the specifics of the decision-making problem under study. the main difference is shaping the decision grid by functions, which mcda methods have within themselves. therefore, stfn-vikor has less similarity with the reference ranking than the stfn-topsis and stfn-mabac approaches. this study also compared preferences for characteristic objects. these objects are the main base of the frm model, which plays a reference role. their ratings are crucial to the reidentification process, so we focused on analyzing the preferences of characteristic objects, which play an essential role in our study. however, due to the different rating scales used by the re-identification methods and the consequent differences in the visualization of the decision grid, we decided to normalize the preference values of characteristic objects using a min-max approach. after normalizing the reference preference vectors, we calculated their differences and the preferences of characteristic objects obtained by re-identification methods. the results of this comparison are shown in the table 2. table 2 comparison of normalized preferences of characteristic objects derived from reidentification approaches with the preferences of frm characteristic objects metric stfn-topsis stfn-vikor stfn-mabac mse 0.008166 0.019175 0.007460 mae 0.069528 0.093929 0.069160 r2 0.855731 0.661245 0.868212 analysis of these values allows us to assess the degree of correspondence between the preferences of characteristic objects obtained by the re-identification methods and the reference frm model. higher mse and mae indices and lower values of r2 may suggest more significant discrepancies between the preferences calculated by the re-identification methods and the reference preferences. the stfn-mabac method shows the lowest mse and r2 values and the highest mae value, suggesting that it is least consistent with the reference model. the stfn-topsis method, on the other hand, achieves the highest mse and r2 values and the lowest mae value, suggesting that it is most consistent with 522 b. kizielewicz, w. sałabun the reference model. based on these results, it is possible to direct the selection to the stfn-topsis method as the best fit for the frm model in the analyzed context. fig. 11 presents a comparative analysis of tfn cores obtained from 1,000 optimization runs for the stfn-topsis, stfn-vikor, and stfn-mabac methods. fig. 11 boxplots of tfn core values for c1 (top) and c2 (bottom) across 1,000 runs of stfn-topsis, stfn-vikor, and stfn-mabac, with manhattan distances to the expected solution point derived from the frm (espfrm) the medians of the core values for the criteria (c1 and c2) and their deviations from the expected solution point (esp) derived from the frm model are highlighted. for criterion c1, the median core values were 0.9035 for stfn-topsis, 0.8686 for stfn-vikor, and 0.9057 for stfn-mabac, with manhattan distances from the esp measured at 0.0574, 0.0224, and 0.0596, respectively. criterion c2 showed median core values of 0.4176, 0.3962, and 0.4131 for the same methods, with respective manhattan distances of 0.033, 0.0116, and 0.0285. the results indicate that stfn-vikor demonstrates the closest alignment to the frm model for both criteria, reflecting its excellent consistency in identifying cores closer to the expected reference point. while all approaches produced cores with minor discrepancies, the stfn-topsis method exhibited a noticeable spread of values for the second criterion, with a standard deviation of 0.0021. after a comparative analysis of stfn-based approaches as to the reference model, sensitivity analyses were conducted. fig. 12 illustrates the sensitivity analysis of the stfn benchmark study of re-identification methods based on stochastic fuzzy normalization... 523 topsis, stfn-vikor, and stfn-mabac methods with respect to changes in tfn cores (c1 core and c2 core) across the range [0,1]. fig. 12 comparison of sensitivity analysis for stfn-topsis, stfn-vikor, and stfnmabac: correlation with frm and stfn-obtained rankings across core values (c1 core and c2 core). for c1 core, stfn-topsis demonstrates the highest correlation with the frm model at optimal configurations (max rw = 0.9370) but exhibits greater variability overall, as reflected by its higher standard deviation (sd = 0.1873) and lower mean correlation (mean rw = 0.7250). similarly, stfn-mabac achieves a comparable mean rw = 0.7210, though it shows slightly better robustness with a lower standard deviation (sd = 0.1890). stfnvikor, while maintaining a consistent performance (mean rw = 0.7180, max rw = 0.8840), is marginally less aligned with the frm for c1 core compared to the other methods. for c2 core, stfn-vikor exhibits the highest robustness, with the lowest variability (sd = 0.1719) and a relatively high mean rw = 0.6850. in contrast, stfn-mabac and stfntopsis show greater sensitivity, with stfn-mabac displaying a slightly lower mean correlation (mean rw = 0.6420) and higher variability (sd = 0.2280). notably, stfntopsis achieves the highest peak correlation with the frm model (max rw = 0.9370) but also has significant deviations at non-optimal points, indicating lower robustness overall. the algorithm presented in fig. 13 demonstrates a systematic approach to conducting a sensitivity analysis for the extrapolation of tfn boundaries in stfn-topsis, stfnvikor, and stfn-mabac models, focusing on rank reversal phenomena. 524 b. kizielewicz, w. sałabun fig. 13 python implementation of sensitivity analysis for extrapolated boundaries in stfn-topsis, stfn-vikor, and stfn-mabac models the procedure begins with generating reference samples uniformly distributed within the original tfn boundary interval [0,1] for both criteria (c1 and c2). subsequently, the tfn boundaries are extrapolated beyond their original range using a grid of values from [-0.1, 1.11] for both criteria. however, extrapolated samples falling within the original boundary region (1 ≥ c1 ≥ 0 and 1 ≥ c2 ≥ 0) are excluded, as rank reversal is not observed in this domain. new samples are randomly generated within the extended boundaries, adhering to specific conditions based on the position of c1 and c2 relative to zero. the reference samples are then combined with these additional samples to form augmented datasets, and preference rankings are computed for the reference samples only. weighted spearman rank correlation coefficient (rw) is calculated between rankings derived from the original and extrapolated tfns, with the process repeated 1,000 times for robustness. these experiments are conducted for extended datasets of varying sizes (5, 10, and 25 alternatives), allowing for a comprehensive evaluation of the sensitivity of each method to boundary extrapolation. this sensitivity analysis for the extrapolation of tfn boundaries highlights the robustness of the methods in handling deviations beyond the modeled boundary conditions and provides critical insights into the stability and reliability of rankings under such perturbations. the graphs in fig. 14 illustrate the sensitivity analysis for boundary extrapolation of tfns in stfn-topsis, stfn-vikor, and stfn-mabac models across varying numbers of added alternatives (5, 10, and 25). the axes represent the extrapolated boundaries of criteria c1 and c2, while the color gradient indicates mean weighted spearman rank correlation coefficient (rw) between rankings derived from original and extrapolated tfns. higher mean rw values (lighter regions) signify robustness to boundary changes, while lower mean rw values (darker regions) highlight rank reversal sensitivity. the striped diagonal region corresponds to the excluded domain where extrapolated boundaries overlap the original benchmark study of re-identification methods based on stochastic fuzzy normalization... 525 tfn boundaries, ensuring stability as rank reversal does not occur. stfn-topsis and stfn-mabac demonstrate greater robustness in the extrapolated regions, maintaining lighter color gradients and higher rw values across configurations. in contrast, stfnvikor exhibits lower rw values in the extrapolated regions (closer to darker blue), indicating greater sensitivity to boundary manipulations. these findings emphasize the relative stability of stfn-topsis and stfn-mabac under boundary extrapolation and highlight the increased rank instability of stfn-vikor in such scenarios. fig. 14 sensitivity analysis of stfn-topsis, stfn-vikor, and stfn-mabac methods for extrapolated triangular fuzzy number boundaries the presented graphs in fig. 15 illustrate the standard deviation of weighted spearman rank correlation coefficient (rw) across extrapolated tfn boundaries (c1 and c2) for stfntopsis, stfn-vikor, and stfn-mabac methods under varying numbers of added alternatives (5, 10, and 25). the color gradient represents standard deviation magnitude, where darker regions indicate more excellent stability, and lighter, yellow regions reflect higher variability. stfn-topsis and stfn-mabac demonstrate similar patterns of robust ranking stability with low standard deviation values across most extrapolated regions, showing minimal sensitivity to boundary changes. stfn-vikor, while exhibiting slightly higher variability, maintains consistent sensitivity patterns across the tested scenarios. increasing the number of alternatives from 5 to 25 does not lead to significant changes in standard deviation distributions, suggesting that the methods remain stable regardless of dataset size. these results emphasize the resilience of stfn-topsis and stfn-mabac under boundary extrapolation and highlight that stfn-vikor, while slightly more sensitive, also maintains stable behavior across varying dataset sizes. 526 b. kizielewicz, w. sałabun fig. 15 standard deviation analysis of weighted spearman rank correlation coefficient (rw) for stfn-topsis, stfn-vikor, and stfn-mabac across extrapolated tfn boundaries 5. practical example this section presents a practical application of decision support methodologies to solve a complex engineering problem: reidentifying the model for selecting the optimal energy source for an industrial plant. using synthetic data inspired by studies on energy sources [43,44], the analysis evaluates seven alternative energy options based on six key criteria. these criteria include: c1 (operating cost, $/mwh), c2 (co2 emissions, kg co2/mwh), c3 (installation cost, $/mw), c4 (energy supply stability, %), c5 (environmental impact beyond co2 emissions, measured on a scale of 1 to 10, where 10 indicates minimal impact), and c6 (installation lifespan, years). the decision matrix, presented in table 3, provides a comprehensive comparative framework for these alternatives, enabling a systematic evaluation of their relative performance across economic, environmental, and operational dimensions. table 3 decision matrix for evaluating energy source alternatives for an industrial plant alternative c1 c2 c3 c4 c5 c6 natural gas (a1) 70 400 900000 90 5 25 wind energy (a2) 50 20 1500000 70 8 20 nuclear energy (a3) 100 5 5000000 99 6 60 solar energy (a4) 60 30 1200000 50 9 25 biomass energy (a5) 80 10 1800000 85 6 30 coal (a6) 40 1000 800000 95 3 30 hydroelectric energy (a7) 55 10 3000000 98 10 50 benchmark study of re-identification methods based on stochastic fuzzy normalization... 527 after applying the subjective model to evaluate the energy source alternatives, the ranking obtained was a5 > a2 > a7 > a4 > a6 > a3 > a1. using this ranking, the model was reidentified through the stfn-topsis, stfn-vikor, and stfn-mabac approaches. in this study, equal weights were assigned to all criteria, ensuring that the sum of the weights equaled 1. regarding the nature of the criteria, it would typically be necessary to specify whether each criterion represents a profit or a cost. however, in this case, the use of stfn-based approaches eliminates this requirement. this is because the tfns inherently capture the non-linear trends associated with each criterion, rendering the explicit declaration of profit or cost unnecessary. the reidentification process employed the evolutionary differential algorithm for core identification, as in the previous study. for the tfns used in normalization, the boundary values were set according to the smallest and largest values present in the decision matrix. the identified cores obtained for each method are displayed in table 4. the identified cores reveal notable differences and similarities among the methods. stfn-topsis assigns the highest core values for operating cost (c1 = 100) and co2 emissions (c2 = 1000), emphasizing these criteria more strongly than the other methods. in contrast, stfn-vikor places greater importance on installation cost (c3 = 2780147), while stfn-mabac provides a balanced approach, with the highest emphasis on energy supply stability (c4 = 74.90) and comparable values for environmental impact beyond co2 (c5 = 9.93) and lifespan (c6 = 49.90). despite differences in prioritization, all methods show alignment in valuing sustainability, particularly in terms of minimal environmental impact (c5) and long-lasting installations (c6), as evidenced by their relatively similar cores in these categories. table 4 identified cores for energy source selection using stfn-based methods method c1 c2 c3 c4 c5 c6 stfn-topsis 100 1000 1883156 71.97 9.99 50.18 stfn-vikor 40 48 2780147 62.50 8.93 48.06 stfn-mabac 52 109 2170837 74.90 9.93 49.90 from the models reidentified based on the decision matrix, rankings were obtained and presented in fig. 16. the figure illustrates the weighted spearman rank correlation (rw) between the reference ranking and the rankings produced by the stfn-topsis, stfnvikor, and stfn-mabac methods, along with the positional shifts in alternative rankings. stfn-topsis achieves the highest alignment with the reference ranking (rw = 0.987), indicating strong consistency, while stfn-vikor and stfn-mabac show slightly lower alignment (rw = 0.946), suggesting minor deviations. alternatives a5, a2, and a7 maintain consistent positions across all methods, reflecting stability and agreement with the reference. in contrast, significant positional shifts are observed for a3 (e.g., moving from position 3 in the reference ranking to position 7 in stfn-topsis) and a6 (e.g., shifting from position 6 in the reference to position 3 in stfn-topsis). alternatives a1 and a4 remain stable across all methods. overall, stfn-topsis demonstrates the closest alignment with the reference ranking, with fewer positional deviations, while stfnvikor and stfn-mabac exhibit similar performance but slightly greater variability in rankings. 528 b. kizielewicz, w. sałabun fig. 16 weighted spearman rank correlation and positional changes between the reidentified models and the reference ranking 6. discussion the analysis results highlight the significant role of reidentification using the stochastic fuzzy normalization (stfn) based on triangular fuzzy numbers method in improving decisionmaking processes. stfn application enables precise modeling of complex decision models, mainly when expert knowledge is unavailable or when existing models must be adjusted to changing conditions or decision-maker preferences. reidentification through stfn has proven to be an effective tool in mcda methods such as topsis, vikor, and mabac, enhancing the accuracy of preference representation and the hierarchy of alternatives. the results confirm a high level of alignment between models reidentified with stfn and reference models, making this process an essential element of decision support in dynamic environments. the resilience of stfn-based methods to rank reversal and their ability to maintain ranking stability during boundary extrapolation further underscores their value in dynamic decision-making scenarios. both stfn-topsis and stfn-mabac exhibit consistent performance across various conditions, particularly excelling in scenarios with stable datasets. however, stfn-vikor, while effective, displays higher sensitivity to changes in boundary conditions, suggesting it may be more suited for decision-making problems that require fine-tuned preference modeling rather than broad adaptability. importantly, the flexibility of the stfn reidentification process allows for iterative updates to decision models, enabling quick adaptations to evolving data or decision-maker priorities without requiring a full model redesign. this adaptability, combined with the reidentification process's inherent resilience to rank-reversal phenomena, significantly enhances the stability and reliability of multi-criteria analysis. as a result, stfn-based approaches are particularly appealing in rapidly changing domains like environmental policy, urban planning, or supply chain management, where decision criteria and conditions frequently shift, emphasizing their critical role in enabling robust and adaptable decision-making solutions. the specific case of energy source selection for an industrial plant further exemplifies the efficacy of stfn reidentification. the rankings derived from the decision matrix benchmark study of re-identification methods based on stochastic fuzzy normalization... 529 revealed strong alignment between the reference model and the reidentified models using stfn-topsis, stfn-vikor, and stfn-mabac. stfn-topsis demonstrated the closest correlation with the reference ranking, while stfn-vikor and stfn-mabac maintained slightly lower but comparable alignment. these methods effectively preserved stability in ranking critical alternatives such as a5, a2, and a7, while also capturing meaningful deviations for alternatives like a3 and a6. this case highlights the practical utility of stfn methods in balancing robustness and adaptability, even under conditions of significant variability in criteria or preferences. despite its many advantages, reidentification with stfn faces some challenges. firstly, the effectiveness of this process depends on a clearly defined normalization process within mcda methods. methods that lack an explicit normalization step may require adaptation or additional mechanisms to enable stfn application. secondly, using triangular fuzzy numbers limits the system's ability to model more complex uncertainty structures. expanding reidentification to include other types of fuzzy numbers, such as trapezoidal fuzzy numbers or more advanced membership functions, could significantly broaden its applicability. it is also worth noting that the reidentification process with stfn can be computationally demanding, especially for large datasets. proper parameter selection and accurate interpretation of results are crucial for achieving high effectiveness and precision. despite these challenges, stfn demonstrates exceptional potential in adapting and improving decision models, as evidenced by high correlation coefficients with reference models and the stability of rankings. 7. conclusions this study presents a comprehensive comparison of re-identification methods for mcda models, including stfn-topsis, stfn-vikor, and stfn-mabac, in relation to the frm. the findings confirm that these methods exhibit commendable performance in accurately reflecting the preferences of the frm, with the application of stochastic fuzzy normalization (stfn) based on triangular fuzzy numbers proving highly effective in handling nonlinearities inherent in decision models. stfn-topsis and stfn-mabac demonstrate similar performance, achieving high robustness and stability across various scenarios, particularly in maintaining rank stability during boundary extrapolation. in contrast, stfn-vikor, while effective, shows greater sensitivity to boundary changes and rank reversal, making it less consistent in conditions of significant uncertainty or extrapolated data. sensitivity analyses reveal that all methods maintain stability regardless of dataset size (number of added alternatives: 5, 10, or 25), though rank instability slightly increases near boundary extremes, with stfn-topsis and stfn-mabac showing greater resilience compared to the more variable stfn-vikor. these findings highlight the capability of stfn-based methods to enable robust normalization, mitigate rank reversal phenomena, and ensure reliable decision-making outcomes, particularly in scenarios where expert knowledge is unavailable or dynamic adjustments are required. the practical application of these methods in the selection of optimal energy sources for an industrial plant reinforces their effectiveness in real-world scenarios. the reidentification process highlighted strong alignment between reidentified models and the reference model, particularly with stfn-topsis, which demonstrated the highest correlation and stability. this case study further underscores the utility of stfn-based approaches in balancing robustness 530 b. kizielewicz, w. sałabun and adaptability, even under conditions of significant variability in criteria or preferences, showcasing their value in addressing complex, multidimensional decision problems. in future works, a more extensive comparison of re-identification techniques across diverse decision scenarios could enhance the versatility and efficacy of decision-making models. expanding the repertoire of fuzzy numbers, including moving beyond triangular fuzzy numbers to other types, could better capture nonlinearity and improve accuracy. additionally, adapting the stfn approach to effectively handle uncertain and high-dimensional data is crucial, as realworld decision-making often involves imprecise information and complex datasets. incorporating probabilistic methods, interval-based fuzzy numbers, or dimensionality reduction techniques (e.g., principal component analysis (pca)) could enhance robustness and computational efficiency. advanced machine learning approaches may further refine stfn, enabling it to address complex decision spaces and broaden its applicability to domains like finance and healthcare, ultimately facilitating more informed and adaptive decision-making processes. acknowledgement: the work was supported by the national science centre 2021/41/b/hs4/01296. references 1. cinelli, m., kadziński, m., miebs, g., gonzalez, m., 2022, recommending multiple 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bączkiewicz, a., wątróbski, j., 2022, multi-criteria temporal assessment of afordable and clean energy systems in european countries using the daria-topsis method, procedia computer science, 207, pp. 4442-4453. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 1 30 doi: 10.22190/fume170315001v © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper 1multilayered plate elements with node-dependent kinematics for the analysis of composite and sandwich structures udc 539.3:624.01 stefano valvano, erasmo carrera mul2 group, department of mechanical and aerospace engineering, politecnico di torino, turin, italy abstract. in this paper a new plate finite element (fe) for the analysis of composite and sandwich plates is proposed. by making use of the node-variable plate theory assumptions, the new finite element allows for a simultaneous analysis of different subregions of the problem domain with different kinematics and accuracy, in a global/local sense. in particular higher-order theories with an equivalent-single-layer (esl) approach are simultaneously used with advanced layer-wise (lw) models. as a consequence, the computational costs can be reduced drastically by assuming refined theories only in those zones/nodes of the structural domain where the resulting strain and stress states present a complex distribution. on the contrary, computationally cheaper, low-order kinematic assumptions can be used in the remaining parts of the plate where a localized detailed analysis is not necessary. the primary advantage of the present variable-kinematics element and related global/local approach is that no ad-hoc techniques and mathematical artifices are required to mix the fields coming from two different and kinematically incompatible adjacent elements, because the plate structural theory varies within the finite element itself. in other words, the structural theory of the plate element is a property of the fe node in this present approach, and the continuity between two adjacent elements is ensured by adopting the same kinematics at the interface nodes. according to the unified formulation by carrera, the through-the-thickness unknowns are described by taylor polynomial expansions with esl approach and by legendre polynomials with lw approach. furthermore, the mixed interpolated tensorial components (mitc) method is employed to contrast the shear locking phenomenon. several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present plate element, including comparison with various closed-form and fe solutions from the literature. key words: multilayered plate elements, node-dependent kinematics, equivalent-single-layer, global/local analysis, layer-wise received march 15, 2017 / accepted april 02, 2017 corresponding author: stefano valvano affiliation: department of mechanical and aerospace engineering, politecnico di torino, corso duca degli abruzzi, 24, 10129 torino, italy e-mail: stefano.valvano@polito.it mailto:stefano.valvano@polito.it 2 s. valvano, e. carrera 1. introduction the development of new materials for advanced engineering applications leads to a complex analysis of layered structures in practice. this is mainly due to the complex anisotropy that characterizes this kind of structures and that leads to intricate mechanical phenomena. in some cases, structures may contain regions where three-dimensional (3d) stress fields occur. to accurately capture these localized 3d stress states, solid models or higher-order theories are necessary. the finite element method (fem) has a predominant role among the computational techniques implemented for the analysis of layered structures. the majority of fem theories available in the literature are formulated by axiomatic-type theories. the conventional fem plate model is the classical kirchhoff-love theory, and some examples are given in [1, 2], whose extension to laminates is known as the classical lamination theory (clt) [3]. another classical plate element is based on the first-order shear deformation theory (fsdt), which rely on the works by reissner [4] and mindlin [5]. to overcome the limitations of classical theories, a large variety of plate finite element implementations of higher-order theories (hot) have been proposed in the last years. hotbased c 0 finite elements (c 0 means that the continuity is required only for the unknown variables and not for their derivatives) were discussed by kant et al. [6] and kant and kommineni [7]. many other papers are available in which hots have been implemented for plates, and more details can be found in the books by reddy [8] and palazotto and dennis [9]. the hot type theories presented are esl models; the variables are independent of the number of layers. differently the lw models permit to consider different sets of variables per each layer. finite element implementations of layer-wise (lw) theories in the framework of axiomatic-type theories have been proposed by many authors, among which noor and burton [10], reddy [11], mawenya and davies [12], rammerstorfer et al. [13]. however, the high computational costs represent the drawback of refined plate theories or three-dimensional analyses. in recent years considerable improvements have been obtained towards the implementation of innovative solutions for improving the analysis efficiency for a global/local scenario. in this manner, the limited computational resources can be distributed in an optimal manner to study in detail only those parts of the structure that require an accurate analysis. in general, two main approaches are available to deal with a global/local analysis: refining the mesh or the fe shape functions in correspondence with the critical domain; formulating multi-model methods, in which different subregions of the structure are analyzed with different mathematical models. the coupling of a coarse mesh and a refined one can be addressed as single-theory or single-model methods, and many techniques are present in literature [14, 15, 16]. in general, multi-theory methods can be divided into sequential or multistep methods, and simultaneous methods. in a sequential multi-model, the global region is analyzed with an adequate model with a cheap computational cost to determine the displacement or force boundary conditions for a subsequent analysis at the local level. the local region can be modeled with a more refined theory, or it can be modeled with 3-d finite elements, see [17, 18, 19, 20]. the simultaneous multi-model methods are characterized by the analysis of the entire structural domain, where different subregions are modeled with different mathematical models and/or distinctly different levels of domain discretization, in a unique step. one of the simplest types of simultaneous multi-model methods for composite laminates analysis is the concept of selective ply grouping or sublaminates [21, 22, 23]. in the literature, the local region (i.e., the region multilayered plate elements with node-dependent kinematics for the analysis of composite.. 3 where an accurate stress analysis is desired) is generally modeled by using 3-d finite elements in the domain of a selective ply grouping method. recently, the authors have developed multi-model elements with variable through-the-thickness approximation by using 2-d finite elements for both local and global regions [24, 25, 26]. in this approach, the continuity of the primary variables between local and global regions was straightforwardly satisfied by employing legendre polynomials. another well-known method to couple incompatible kinematics in multi-model methods is the use of lagrange multipliers, which serve as additional equations to enforce compatibility between adjacent subregions. in the three-field formulation by brezzi and marini [27], an additional grid at the interface is introduced. the unknowns are represented independently in each sub-domain and at the interface, where the matching is provided by suitable lagrange multipliers. this method was recently adapted by carrera et al. [28, 29, 30] to couple beam elements of different orders and, thus, to develop variable kinematic beam theories. ben dhia et al. [31, 32, 33, 34] proposed the arlequin method to couple different numerical models by means of a minimization procedure. this method was adopted by hu et al. [35, 36] for the linear and non-linear analysis of sandwich beams modeled via one-dimensional and two-dimensional finite elements, and by biscani et al. [37] for the analysis of beams and by biscani et al. [38] for the analysis of plates. reddy and robbins [39] and reddy [40] presented a multiple-model method on the basis of a variable kinematic theory and on mesh superposition in the sense of fish [41] and fish and markolefas [42]. coupling was obtained by linking the fsdt variables, which are present in all the considered models, without using lagrangian multipliers. the coupling of different kinematics model in the framework of composite beam structure, using the extended variational formulation (xvf), is presented in [43], sinus model and classical kinematics are coupled into non-overlapping domains. in the present work, a new simultaneous multiple-model method for 2d elements with node-dependent kinematics is developed. this node-variable capability enables one to vary the kinematic assumptions within the same finite plate element. the expansion order of the plate element is, in fact, a property of the fe node in the present approach. therefore, between the finite elements, the continuity is ensured by adopting the same expansion order in the nodes at the element interface. this node-dependent finite element has been used by the authors in [44] using classical and hot-type theories; taylor polynomials were used with an esl approach. the novelty of the present work lies in the combination of hot-type and advanced lw theories in the same finite element. in this manner, global/local models can be formulated without using any mathematical artifice. as a consequence, computational costs can be reduced assuming refined models only in those zones with a quasi-three-dimensional stress field, whereas computationally cheap, low-order kinematic assumptions are used in the remaining parts of the plate structure. in this paper, the governing equations of the node-variable kinematic plate element for the linear static analysis of composite structures are derived from the principle of virtual displacement (pvd). subsequently, fem is adopted and the mixed interpolation of tensorial components (mitc) method [45, 46, 47, 48] is used to contrast the shear locking. the developed methodology is, therefore, assessed and used for the analysis of multilayered cantilevered plates with concentrated loads, cross-ply plates with simply-supported edges and subjected to a localized pressure load, and asymmetric laminated sandwich plates simply-supported and subjected to a localized pressure load. the results are compared with various theories and, whenever possible, with exact solutions available from the literature. 4 s. valvano, e. carrera 2. refined and hierarchical theories for plates in this paper, different kinematic assumptions in different subregions of the problem domain are made by a new finite element which allows a simultaneous multi-model analysis, without ad-hoc techniques and mathematical artifices that are usually required to mix the fields coming from two different kinematic models. the present plate structural theory varies within the finite element itself. to highlight the capabilities of the present formulation, a four-node plate element with node-dependent kinematics is shown in fig. 1. the element proposed in this example makes use of a layer-wise theory of the first order at node 1. on the other hand, a second-order refined theory is employed at node 2. at node 3, a third order expansion is adopted. finally, a layer-wise theory of the second order is assumed at node 4. as will be clear further in the paper, the choice of the nodal plate theory is arbitrary and node-variable kinematic plate elements will be used to implement multi-model methods for a global-local analysis. before discussing the present formulation, a brief overview of the refined and advanced plate theories is given below or the sake of completeness. plates are bi-dimensional structures in which one dimension (in general the thickness in the z direction) is negligible with respect to the other two dimensions. the geometry and the reference system that are adopted throughout the present work are shown in fig. 1. fig. 1 example of sandwich structure described by plate element with node-dependent kinematics 2.1. higher-order theories in order to overcome the limitations of classical theories, a large variety of plate higher-order theories (hot) have been proposed in the past and recent literature. as a general guideline, it is clear that the richer the kinematics of the theory, the more accurate the 2d model becomes. hot-type theories can be expressed by making use of taylor-like multilayered plate elements with node-dependent kinematics for the analysis of composite.. 5 expansions of the generalized unknowns along the thickness to formulate equivalent-singlelayer (esl) models. in the case of generic expansions of n terms, hot displacement field can be expressed as in eq. (1). for example, if a parabolic expansion order is taken into account, a graphical representation of a deflection can be depicted as in fig. 2a. moreover, fig. 2b pictorially shows the capabilities of a generic hot model, which can address complex kinematics in the thickness direction. 0 1 0 1 0 1 ( , , ) ( , ) ( , ) ... ( , ) ( , , ) ( , ) ( , ) ... ( , ) ( , , ) ( , ) ( , ) ... ( , ) n n n n n n u x y z u x y z u x y z u x y v x y z v x y z v x y z v x y w x y z w x y z w x y z w x y             (1) w0 x z u0 a) hot-theory with parabolic expansion order n = 2 b) hot-theory with generic expansion order fig. 2 geometrical representation of the higher-order theories for the sake of completeness, the classical models, classical lamination theory (clt) [1, 2, 3] and first-order shear deformation theory (fsdt) [4, 5] kinematics, are particular cases of the full linear expansion, obtained from 1 imposing n = 1. for more details see [49]. therefore, it is well known in literature that linear models are affected by the problem of the poisson locking (pl) phenomenon. the remedy for the poisson locking, apart from using higher-order theories, is to modify the elastic coefficients of the material. the pl phenomenon originates from constitutive laws which state the intrinsic coupling between in and out-of-plane strain components. classical plate theories correct the locking phenomena by imposing that the out-of-plane normal stress is zero. this hypothesis yields reduced material stiffness coefficients which have to be accounted in the hooke’s law. therefore, in literature, the correction of the material coefficients does not have a consistent theoretical proof. this means that the adoption of reduced material coefficients does not necessarily lead to the exact 3d solution, as shown in [50]. for the sake of clarity and simplicity of the present method explanation, the results presented in this work, with the full linear expansion kinematics, are not corrected for the pl phenomena. 2.2. the unified formulation framework according to unified formulation by carrera [49, 51, 52, 53], refined models can be formulated in a straightforward manner by assuming an expansion of each of the primary variables by arbitrary functions in the thickness direction. thus, each variable can be treated independently of the others, according to the required accuracy. this procedure becomes extremely useful when multi-field problems are investigated such as thermoelastic and piezoelectric applications [54, 55, 56, 57]. in a displacement-based formulation, in fact, the three-dimensional displacement field is the combination of through-the-thickness functions weighted by the generalized unknown variables: 6 s. valvano, e. carrera 0 0 1 1 0 0 1 1 0 0 1 1 ( , , ) ( ) ( , ) ( ) ( , ) ... ( ) ( , ) ( , , ) ( ) ( , ) ( ) ( , ) ... ( ) ( , ) ( , , ) ( ) ( , ) ( ) ( , ) ... ( ) ( , ) n n n n n n u x y z f z u x y f z u x y f z u x y v x y z f z v x y f z v x y f z v x y w x y z f z w x y f z w x y f z w x y             (2) similarly, in a compact form one has: ( , , ) ( ) ( , ) 0, 1,...,s sx y z f z x y s n u u (3) where u(x, y, z) is the three-dimensional displacement vector, u(x, y, z) = [u, v, w]; fs are the thickness functions depending only on z; us is the generalized displacement vector of the variables; s is a sum index; and n is the number of terms of the theory expansion. depending on the choice of thickness functions fs, and the number of terms in the plate kinematics n, various theories can be implemented. 2.3. advanced theories the esl models formulated with taylor-like thickness functions, however, may not be sufficiently accurate to describe adequately the multilayered structures in which, due to their intrinsic anisotropy, the first derivative of the displacement variables in the zdirection is discontinuous. nevertheless, it is possible to reproduce the zig-zag effects in the esl models by modifying opportunely fs functions, for example by adding the murakami functions [58, 59]. on the other hand, plate models with layer-wise (lw) capabilities can be implemented by describing the displacement components at the layer level, possibly by using a combination of lagrange and legendre-like polynomial as fs thickness functions [60, 61]. in the case of layer-wise (lw) models, the displacement is defined at k-layer level: ( , , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) , , 2, ..., k k k k k t t b b r r s sx y z f z x y f z x y f z x y f z x y s t b r r n       u u u u u (4) 2 1010 22      rrrbt ppf pp f pp f (5) in which pj = pj (ζk ) is the legendre polynomial of j-order defined in the ζ k -domain: −1 ≤ ζ k ≤ 1; p0 = 1, p1 = ζ k , p2 = (3ζk 2 − 1)/2, p3 = (5 ζk 3 − 3 ζk)/2, p4 = (35 ζk 4 − 30ζk 2 + 3)/8. the top (t) and bottom (b) values of the displacements are used as unknown variables and one can impose the following compatibility conditions: 111   l k b k t n,kuu (6) for example, if a parabolic expansion order is taken into account, a graphical representation of a deflection can be depicted as in fig. 3. multilayered plate elements with node-dependent kinematics for the analysis of composite.. 7 w0 x z u0 fig. 3 geometrical representation of a parabolic layer-wise model deflection on a two layered plate 3. finite element approximation 3.1. constitutive and geometrical relations for plates the definition of the 3d constitutive equations permits to express the stresses by means of the strains. the generalized hooke’s law is considered, by employing a linear constitutive model for infinitesimal deformations. in a composite material, these equations are obtained in material coordinates (1, 2, 3) for each orthotropic layer k and then rotated in the general reference system (x, y, z). therefore, the stress-strain relations after the rotation, calculated for each layer k, are: kkk εcσ  (7) where the stress and strain vectors have six components: [ , , , , , ] [ , , , , , ] xx yy xy xz yz zz xx yy xy xz yz zz               σ ε (8) and c is the material elastic coefficients matrix defined as follows:                      33362313 4445 4555 36662616 23262212 13161211 00 0000 0000 00 00 00 cccc cc cc cccc cccc cccc c (9) for the sake of brevity, the expressions that relate material coefficients cij to young’s moduli e1, e2, e3, shear moduli g12, g13, g23 and poisson ratios 12, 13, 23, 21, 31, 32 are not given here. they can be found in many reference texts, such as [11]. the geometrical relations enable one to express strain vector s in terms of displacement vector u for each layer k: 8 s. valvano, e. carrera k g k udε  (10) where dg is the geometrical vector containing the differential operators defined as follows: t 000 000 000                                        zyx zxy zyx gd (11) 3.2. node-dependant kinematics for plate finite elements by utilizing an fem approximation, the generalized displacements of eq. (3) can be expressed as a linear combination of the shape functions to have: s sj( , ) ( , ) 1, ...,jx y n x y j u u (nodes per element) (12) where usj is the vector of the generalized nodal unknowns, nj can be the usual lagrange shape functions and j denotes a summation on the element nodes. since the principle of virtual displacements in used in this paper to obtain the elemental fe matrices, it is useful to introduce the finite element approximation of the virtual variation of generalized displacement vector u , i( , ) ( , ) 1, ...,ix y n x y j   u u (nodes per element) (13) in eq. (13),  denotes the virtual variation, whereas indexes  and i are used instead of s and j, respectively, for the sake of convenience. in this work, and according to eqs. (3), (12) and (13), thickness functions fs and f , which determine the plate theory order, are independent variables and may change for each node within the plate element. namely, the three-dimensional displacement field and the related virtual variation can be expressed to address fe node-dependent plate kinematics as follows: ( , , ) ( ) ( , ) 0, 1, ..., 1, ..., (nodes per element) ( , , ) ( ) ( , ) 0, 1, ..., 1, ..., (nodes per element) j j s j sj j j j sj x y z f z n x y s n j x y z f z n x y n i         u u u u (14) where subscripts , s, i, and j denote summation. superscripts i and j denote node dependency, such that for example f i is the thickness expanding function and n i is the number of expansion terms at node i, respectively. as example, the displacement field of the node-variable kinematic plate element as discussed in fig. 1 is described in detail hereafter. the global displacement field of the element is approximated as follows:  node 1 plate theory = lw with n 1 = 1 eq. (4)  node 2 plate theory = hot with n 2 = 2 eq. (1)  node 3 plate theory = hot with n 3 = 3 eq. (1)  node 4 plate theory = lw with n 4 = 2 eq. (4) multilayered plate elements with node-dependent kinematics for the analysis of composite.. 9 according to eq. (14), it is easy to verify that the displacements at a generic point belonging to the plate element can be expressed as given in eq. (15). in this equation, only the displacement component along x-axis is given for simplicity reasons: 1 1 2 2 2 3 3 3 3 4 4 4 2 0 1 1 0 1 2 2 2 2 3 0 1 2 3 3 0 1 2 4 1 1 ( , , ) ( , ) ( ) ( , ) 2 2 1 1 3( 1) ( ) ( , ) ( , ) 2 2 2 u x y z u u n x y u zu z u n x y u zu z u z u n x y u u u n x y                                                       (15) it is intended that, due to node-variable expansion theory order, the assembling procedure of each finite element increases in complexity with respect to classical monotheory finite elements. in order to simplify the description of the assembling procedure, the governing equations are developed in form of fundamental nucleus, as described below. 3.3. governing equations and fundamental nucleus the governing equations for the static response analysis of the multi-layer plate structure can be obtained by using the principle of virtual displacements, which states: e a t ldzd    σε (16) where the term on the left-hand side represents the virtual variation of the strain energy;  and a are the integration domains in the plane and the thickness direction, respectively;  and  are the vector of the strain and stress components; and le is the virtual variation of the external loadings. by substituting the constitutive equations for composite elastic materials (eq. (7)), the linear geometrical relations (eq. 10) as well as eq. (14) into eq. (16), the linear algebraic system in the form of governing equations is obtained in the following matrix expression: i s sij ji u:   pku  (17) where k sij and p i are the element stiffness and load fe arrays written in the form of fundamental nuclei. the explicit expressions of the fundamental nuclei for node-dependent variable kinematic plate elements are given in [44]. it must be added that, in this study, an mitc technique is used to overcome the shear locking phenomenon, see [57]. the fundamental nucleus is the basic building block for the construction of the element stiffness matrix. in fact, given these nine components, element stiffness matrices of arbitrary plate models can be obtained in an automatic manner by expanding the fundamental nucleus versus indexes , s, i, and j. in the present fe approach, the node-dependent variable kinematic model, it is clear that both rectangular and square arrays are handled and opportunely assembled for obtaining the final elemental matrices. in the development of esl and lw theories, the fundamental nucleus of the stiffness matrix is evaluated at the layer level and then assembled as shown in fig. 4. this figure, in particular, illustrates the expansion of the fundamental nucleus in the case of a 9-node lagrange finite element with node-by-node variable kinematics, as in this paper. however, for more details about the explicit formulation of the unified formulation fundamental nuclei, interested readers are to refer to the recent book by carrera et al. [49]. 10 s. valvano, e. carrera fig. 4 assembling scheme of a 9-node finite element with node-dependent kinematics. highlights of the influence of the lw contribution of other layers in the fe stiffness 4. numerical results in the numerical section some problems have been considered to assess the capabilities of the proposed node-variable kinematic plate elements and related global/local analysis. these cases of analysis comprise composite laminated and sandwich plate structures with different boundary conditions and loadings. whenever possible, the proposed multi-theory models are compared to single-theory refined elements. the acronyms for the esl models are indicated with the first letter e, the second letter indicates the polynomial kind, t stands for taylor polynomials and l for legendre polynomials. the lw models are indicated with the letters lw. the third letter indicates the number of the theory approximation order. if the analytical navier solution type is employed, subscript a is added. moreover, analytical solution with higher-order single models, and multi-model theories present in literature are given for some cases. for the sake of clarity, present multi-model theories are opportunely described for each numerical case considered. multilayered plate elements with node-dependent kinematics for the analysis of composite.. 11 4.1. eight-layer cantilever plate the first structure case taken into account is a simple example that easily permits to describe, through the results, the main capabilities of the present node-dependent plate element. a cantilever eight-layer plate is analyzed as shown in fig. 5. the structure is loaded at the free end with a concentrated load equal to pz = −0.2 n. the geometrical dimensions are: a = 90 mm, b = 1 mm, h = 10 mm. the mechanical properties of the material labeled with the number 1 are: el = 30 gpa, et = 1 gpa, glt = gtt = 0.5 gpa, lt = tt = 0.25. on the other hand, the mechanical properties of the material labeled with the number 2 are: el = 5 gpa, et = 1 gpa, glt = gtt = 0.5 gpa, lt = tt = 0.25. as it is clear from fig. 5, the material stacking sequence is [1/2/1/2]s. fig. 5 eight-layered plate with concentrated loading. reference system and material lamination scheme first, a convergence study on single-theory plate models was performed. for both lw4 and et4 models, as shown in table 1, a mesh grid of 12×2 elements is enough to ensure convergent results, for transverse mechanical displacement w, in-plane stress xx and transverse normal stress xz. various node-variable kinematic plate models have been used to perform the global/local analysis of the proposed plate structure, and they are depicted in fig. 6. these models are compared in table 2 with lowerto higher-order single-theory models as well as with various solutions from the literature, including an analytical solution based on the 2d elasticity as presented in lekhnitskii [62]. it can be observed for transverse displacements w that mono-theory lw models show a good accuracy solution independently of the polynomial order, differently for singlemodel esl with taylor polynomial yielding good results only with higher-order expansion et3 and et4. moreover multi-theory esl models case a, case b and case c show an intermediate solution accuracy for all the three considered cases without relevant differences. for the multi-theory esl-lw models case d, case e and case f the solution is very accurate, due to the partial lw approximation, and exactly the same solution is obtained for the three considered cases. regarding in-plane stress xx the accuracy solution is not sensitive for all the considered single and multi model theories, except for the case a configuration where the transition elements are acting at the evaluation position. for transverse shear stress xz similar comments respect to the transverse displacement can be drawn. single theory lw models show a good accuracy solution independently of the polynomial order, otherwise higher-order mono-model esl theories with taylor polynomial, et3 and et4, are required to obtain sufficient solution accuracy. nevertheless, accurate solutions in localized regions/points can be obtained by using the multi-theory esl model case b, and with multi-theory esl-lw models case d and case e. 12 s. valvano, e. carrera fig. 6 eight-layered plate. mesh scheme of the adopted multi-theory models with node-dependent kinematics table 1 convergence study of single-theory models of the eight-layer cantilever plate. transverse displacement w = −10 2 ×w(a, b/2, 0), in-plane principal stress xx = 10 3 ×xx (a/2, b/2, +h/2), transverse shear stress xz = −10 2 × xz (a/2, b/2, 0) mesh 2 × 2 4 × 2 6 × 2 8 × 2 10 × 2 12 × 2 lw4 w 3.031 3.032 3.031 3.030 3.030 3.030 xx 651 690 716 725 728 730 xz 2.991 2.797 2.792 2.791 2.790 2.789 et4 w 3.029 3.029 3.029 3.028 3.028 3.028 xx 684 723 729 730 731 731 xz 3.054 2.829 2.820 2.821 2.822 2.822 some results in terms of transverse displacement w and transverse shear stress σxz along the thickness are represented in figs. 7a and 7b, 8a and 8b, respectively. some more comments can be made:  as shown in fig. 7a, the through-the-thickness distribution of transverse displacement w, evaluated at the free tip of the plate, is correctly predicted by a third-order esl model et3. the same accuracy cannot be reached by the proposed esl models with node-variable kinematics. differently, as depicted in fig. 7b, both lw single theory and esl-lw theory accuracy are not sensitive of the chosen model, except for the single linear model lw 1. multilayered plate elements with node-dependent kinematics for the analysis of composite.. 13 table 2 eight-layer cantilever plate. transverse displacement w = −10 2 × w(a, b/2, 0), in-plane normal stress xx = 10 3 × xx (a/2, b/2, +h/2), transverse shear stress xz = −10 2 × xz (a/2, b/2, 0) by various singleand multi-theory models w xx xz dofs reference solutions nguyen and surana [63] 3.031 720 davalos et al. [64] 3.029 700 xiaoshan [65] 3.060 750 vo and thai [66] 3.024 lekhnitskii [62] 730 2.789 present singleand multi-theory models lw4 3.030 730 2.789 12375 lw3 3.030 731 2.788 9375 lw2 3.030 731 2.795 6375 lw1 3.022 731 2.775 3375 et4 3.028 731 2.822 1875 et3 3.027 731 2.822 1500 et2 2.980 731 2.005 1125 et1 2.981 729 2.000 750 case a 3.004 808 2.375 1320 case b 3.010 737 2.781 1365 case c 3.002 731 2.030 1305 case d 3.028 732 2.799 4035 case e 3.028 729 2.799 4425 case f 3.028 731 2.818 3645  figure 8a shows that transverse shear stress xz, evaluated at the mid-span of the plate, is very sensitive to the position of the transition variable-kinematic elements. case b model has the same accuracy as mono-model et3 and et4. on the contrary, the case c configuration has poor accuracy like mono-models et1 and et2. finally, case a model presents an intermediate compromise between the other two multitheory cases. all the esl models are not able to reproduce the accurate behavior of the reference 2d elasticity solution lekhnitskii, presented in [62]. on the contrary, as depicted in fig. 8b, the lw single models are able to reach an accurate solution as the reference solution lekhnitskii, except for the linear model lw1. multi-theory esl-lw (et3-lw2) models have a good approximation of the solution where the verification point is described by lw theories, case d and case e models, therefore case f show the same accuracy solution of model et3. by the evaluation of the various node-variable kinematic models, it is clear that an accurate representation of the stresses in localized zones is possible with dofs reduction if an accurate distribution of the higher-order kinematic capabilities is performed in those localized zones. differently, the displacements values are dependent on the global 14 s. valvano, e. carrera approximation over the whole structure. the dofs reduction can be moderate or stronger, depending on the structure and the load case configuration. a) esl single and multi model with taylor polynomials b) lw single model, and multi-theories with esl model by taylor polynomials combined with lw model by legendre polynomials fig. 7 eight-layer composite plate. transverse displacement w(x, y) = −10 2 ×w(a, b/2). a) esl single and multi model with taylor polynomials b) lw single model, and multi-theories with esl model by taylor polynomials combined with lw model by legendre polynomials fig. 8 eight-layer composite plate. transverse displacement xz(x,y) = −10 2 ×xz (a/2, b/2) multilayered plate elements with node-dependent kinematics for the analysis of composite.. 15 4.2. composite plates simply-supported a simply-supported composite plate is analyzed. the geometrical dimensions are: a = b = 0.1 m, the side-thickness ratio is a/h = 10. a symmetric [0°/90°/0°] stacking sequences is considered. the material employed is orthotropic with the following properties: el=132.5 gpa, et =10.8 gpa, glt = 5.7 gpa, gtt =3.4 gpa, lt = 0.24, tt = 0.49. the plate is simplysupported and a localized uniform transverse pressure, pz = −1 mpa, is applied at top face on a square region of side length equal to a/5 × b/5 and centered at the point (a/2, b/2), see fig. 9. in order to take into account other solutions present in literature [38] a non-uniform mesh grid of 20×20 elements ensures the convergence of the solution, taken from [44], and it permits a fair comparison of the results. the non-uniform adopted mesh and the various node-variable kinematic models, with global/local capabilities used to perform the analysis of the proposed plate structure, are depicted in fig. 9, where the mesh grid of a quarter of the plate is analyzed. due to the symmetry of both the geometry and the load, a quarter of the plate is analyzed and the following symmetry and boundary conditions (simply-supported) are applied: boundary symmetry ( ,0) 0 ( ,0) 0 ( / 2, ) 0 (0, ) 0 (0, ) 0 ( , / 2) 0 s s s s s s u x w x u a y v y w y v x b       (18) the results are given in terms of transverse displacement w and in-plane normal stresses xx, yy evaluated at (a/2, b/2, −h/2), transverse shear stress xz evaluated at (5a/12, b/2, 0), and transverse normal stress zz evaluated at (a/2, b/2, +h/2). for the three-layered plate structure with [0°/90°/0°], mono-theory models are compared with those from the present global/local approach in table 3. the table shows that mono-theory esl models with lower expansion order, et1 and et2, are not able to describe appropriately transverse displacements w and in-plane stresses xx and yy; otherwise lw mono-models represent these variables with a good accuracy solution for every expansion order. to accurately describe shear transverse stresses xz, esl higherorder theories are required, or lw mono-models theories. transverse normal stress zz needs higher-order theories to be well described; both linear esl and lw single-models are not sufficient. table 3 also shows solutions for variable kinematic multi-model theories; the cases taken into account are named from case a to case h, and they are explained in fig. 9. the cases named as case a, case b and case e are equivalent to the models (et1 −et4) a , (et3 − et4) b and (et1 − lw4) e taken from [38] and in which, via the arlequin method and 4-node lagrangian plate elements, a fourth-order plate theory is used in correspondence of the loading; firstand third-order kinematics is used outside the loading zone, respectively. 16 s. valvano, e. carrera fig. 9 non-uniform adopted mesh on quarter of the plate, and graphical representation of the multi-theory models of the cross-ply plate structure multilayered plate elements with node-dependent kinematics for the analysis of composite.. 17 table 3 composite plate with [0°/90°/0°] lamination. transverse displacement w = (−10 5 ) × w(a/2, b/2, −h/2), in-plane normal stresses xx = xx(a/2, b/2, −h/2) and yy = yy (a/2, b/2, −h/2), transverse shear stress xz = (−10)×xz(5a/12, b/2, 0), and transverse normal stress zz = −zz (a/2, b/2, +h/2) by various singleand multi-theory models w xx yy xz zz dofs reference solutions [38] 3d 1.674 11.94 2.019 6.524 lw4a 1.675 11.94 2.020 6.523 39 lw4 1.672 11.83 1.983 6.464 9984 et4a 1.660 11.95 2.005 5.865 15 et4 1.657 11.85 1.985 5.830 3840 (et1 − et4)a 1.609 11.92 1.962 5.848 2448 (et3 − et4)b 1.657 11.84 1.985 5.831 3936 (et1 − lw4)e 1.617 11.91 1.953 6.481 3984 present singleand multi-theory models lw4 1.6745 11.9547 2.0232 6.5557 1.0000 17199 lw3 1.6745 11.9624 2.0302 6.5613 1.0108 13230 lw2 1.6719 11.9141 2.0458 6.3903 1.0731 9261 lw1 1.6369 11.3621 2.1465 6.5881 1.4679 5292 et4 1.6596 11.9556 2.0078 5.8473 0.9905 6615 et3 1.6590 11.9867 2.1164 6.0147 1.2443 5292 et2 1.5625 10.1942 1.7935 3.8521 1.0377 3969 et1 1.4954 10.2867 2.1002 3.7554 1.8261 2646 case a 1.6040 12.0084 1.9821 5.8510 0.9910 5247 case b 1.6596 11.9556 2.0077 5.8473 0.9905 6159 case c 1.5257 11.7328 1.9453 4.9414 0.9938 4167 case d 1.5770 11.8056 1.9510 4.9970 0.9909 4983 case e 1.6103 12.0107 1.9923 6.5254 1.0000 12183 case f 1.6670 11.9699 2.0263 6.5524 1.0108 10494 case g 1.5274 11.7105 1.9474 5.3212 1.0009 8223 case h 1.6613 11.9305 2.0198 6.3616 1.0118 8334 some results in terms of transverse displacement w, and transverse shear stress xz along the thickness are represented in figs. 10a, 10b, 11a and 11b. the following remarks can be made:  transverse displacement w behavior can change sensitively depending on the distribution of the kinematic enrichment within the structure plane. figure 10a shows that case b has the same accuracy as full higher-order et 4 mono-model with a 8% dofs reduction, and an accuracy close to multi-model case h with a 26% dofs reduction. it is noticeable that the choice of the esl or lw model for the loaded zone is not decisive for the correct description of the transverse mechanical displacement, as shown for case c and case g. on the contrary a global more refined approximation get better accuracies, as in the case of the multi-models case a and case e. 18 s. valvano, e. carrera  for the evaluation of transverse shear stress xz, higher-order models are necessary in the regions close to the considered evaluation point. in fig. 10b mono-model lw4 is used as a reference solution. it is evident that esl single-models, for every expansion order, are not able to correctly describe the transverse shear stress. the a) single and multi models b) esl and lw single model, and esl multimodel with taylor polynomials fig. 10 composite plate. transverse displacement w(x, y) = −10 5 × w(a/2, b/2), and transverse shear stress xz (x, y) = −10 ×xz (5a/12, b/2) a) b) fig. 11 composite plate. transverse shear stress xz (x, y) = −10 × xz (5a/12, b/2). multi-theories with esl model by taylor polynomials combined with lw model by legendre polynomials multilayered plate elements with node-dependent kinematics for the analysis of composite.. 19 esl multi-model case a has the same poor accuracy of the theory et4. linear model lw 1 is clearly not sufficient to describe the transverse shear stress, differently from the single value reported in table 3 taken in z = 0. in fig. 11a the multi-model case e and case g, where in the boundary regions a et1 model is used and in the loaded zones a lw4 model is adopted, the accuracy on the transverse shear stress is not completely guaranteed by the lw model. in particular for the case g the evaluation point is close to the transition element while this position is perturbation of the accuracy solution. on the contrary for the case e the evaluation point is not more close to the transition element and the solution accuracy is like the full lw model. finally in fig. 11b the multi-model case f and case h are not suffering any perturbation problem, due to the third-order esl model of the boundary regions. results in terms of in-plane stress xx, transverse shear stress xz and transverse normal stress zz along the in-plane x-axis are represented in figs. 12a, 12b and 13a, respectively. for in-plane stress xx, see fig. 12a, the mono-models lw4 and et4 show the same accuracy solution. multi-models with esl approach with taylor polynomials, case a and case c, produce small oscillations in the transition zone. on the contrary, multi-theories with esl model by taylor polynomials combined with lw model by legendre polynomials, case e and case g, show big fluctuations in the transition elements. moreover, it has to be noticed that if the refined polynomials are limited to the loading zone, case c and case g, the solution accuracy in the loading zone is lower with respect to the reference lw4 solution. for transverse shear stress xz, see fig. 12b, the et4 mono-model has an accuracy close to the mono-model lw4 in the loaded zone; differently the et4 model reaches a maximum value of the shear stress for 9% lower than the reference lw4 solution. for multi-model theories the same comments made for the in-plane stress can be applied for the behavior description of the transverse shear stress. for transverse normal stress zz, see fig. 13a, mono-models lw4 and et4 show the same accuracy solution. for multi-model theories the same comments made for the inplane stress can be applied for the behavior description of the transverse normal stress. it has to be noticed that the oscillations of the transition elements are smaller than those of the in-plane stress and the transverse shear stress. finally, a three-dimensional distribution of transverse shear stress xz is given on a quarter of the plate to underline the global/local capabilities of the present finite element on the whole domain of the analyzed plate structure. the reference single-model solution lw4 is depicted in fig. 14a. for a fair comparison of the results, the extremes of the color bar values of the lw4 model are used to limit the color bar of the other solutions. the single-model et4 is not able to correctly describe the transverse shear stress behavior – it is clear from fig. 14b that the interlaminar continuity of the transverse shear stress is not satisfied. in fig. 15a the multi-model named case e, (et1-lw4) is represented. it is evident that the transverse shear stress is well represented in the lw4 zone only. the multi-model case h, (et3-lw3) is represented in fig. 15b, the small lw3 zone is able to correctly describe the transverse shear stress; on the contrary, the et3 zone has a comparable behavior as the single-model et4. 20 s. valvano, e. carrera a) xx b) xz fig. 12 composite plate. in-plane stress xx (y, z) = xx (b/2, −h/2), and transverse shear stress xz (y, z) = −10 × xz (b/2, 0) along the in-plane direction x, the x-axis is expressed in [mm]. single and multi-theory models a) zz fig. 13 composite plate. transverse normal stress zz (y, z) = −zz (b/2, +h/2) along the inplane direction x, the x-axis is expressed in [mm]. single and multi-theory models multilayered plate elements with node-dependent kinematics for the analysis of composite.. 21 a) lw4 b) et4 fig. 14 composite plate, three-dimensional view of a quarter of the plate. transverse shear stress xz for single models a) case e b) case h fig. 15 composite plate, three-dimensional view of a quarter of the plate. transverse shear stress xz for multi-models 4.3. sandwich rectangular plates simply-supported a simply-supported asymmetrically laminated rectangular sandwich plate is analyzed. the geometrical dimensions are: a = 100 mm, b = 200 mm, the total thickness is h = 12 mm, the top skin thickness is htop = 0.1 mm, the bottom skin is thick hbottom = 0.5 mm, and the core thickness is hcore = 11.4 mm. the two skins have the same material properties: e1 = 70 gpa, e2 = 71 gpa, e3 = 69 gpa, g12 = g13 = g23 = 26 gpa, 12 = 13 = 23 = 0.3, moreover, the metallic foam core has the following material properties: e1 = e2 = 3 mpa, e3 = 2.8 mpa, g12 = g13 = g23 = 1 mpa, 12 = 13 = 23 = 0.25. the plate is simply-supported and localized uniform transverse pressure, pz = −1 mpa, is applied at top face on a square region of side length equal to (a = 5 mm)×(b = 20 mm) and centered at the point (a/2, b/2), see fig. 16. 22 s. valvano, e. carrera fig. 16 reference system of the sandwich plate. three-dimensional deflection representation of a quarter of the plate boundary symmetry ( ,0) 0 ( ,0) 0 ( / 2, ) 0 (0, ) 0 (0, ) 0 ( , / 2) 0 s s s s s s u x w x u a y v y w y v x b       (19) the present singleand multi-model solutions are compared with other solutions present in literature, three-dimensional analytical and three-dimensional fem nastran [67], esl and lw analytical higher-order by the use of fourier series expansions [68], esl and lw fem higher-order [69]. a non-uniform mesh grid of 38×54 elements ensures the convergence of the solution with a lw4 single-model, see fig. 17. for the sake of brevity the study of the convergence is here omitted. the adopted refined mesh is necessary to study the behavior of the mechanical variables along the whole plate domain, and not in one single point. the difficult task is to obtain a good behavior of the mechanical stresses, and in particular of transverse normal stress zz along the in-plane directions avoiding strange oscillations due to the changing of the element size. for the asymmetrically laminated rectangular sandwich plate, mono-theory models are compared with those from the present global/local approach in table 4. esl models are not able to correctly describe all the variables; therefore, lw theories are necessary to match the reference analytical and 3d results. table 4 also shows solutions for variable kinematic multi-model theories. as emerged in the previous numerical sections, the primary variables (displacements) depend on the global domain approximation, in particular transverse displacement w is better described in the case b configuration with a dofs reduction of 34 % respect to the case a multi-model. on the contrary, the postprocessed variables (stresses) are dependent on the local approximation. multilayered plate elements with node-dependent kinematics for the analysis of composite.. 23 fig. 17 non-uniform adopted mesh and graphical representation of the multi-model cases, for a quarter of the sandwich plate 24 s. valvano, e. carrera table 4 asymmetrically laminated rectangular sandwich plate. transverse displacement w, in-plane normal stresses xx and yy, and transverse normal stress zz evaluated at (a/2, b/2) by various singleand multi-theory models z w xx yy zz dofs top skin 3d analytical [67] top -3.78 -624 -241 bottom 580 211 3d nastran [67] top -3.84 -628 -237 bottom 582 102 lwm2 analytical [68] top -3.8243 -619.49 bottom 577.36 lwm2 fem [69] top -3.7628 -595.56 -223.93 bottom 556.00 196.37 top skin lw4 top -3.7774 -622.48 -233.39 -0.9649 327327 bottom 578.60 203.25 -0.8738 lw3 top -3.7723 -618.14 -232.33 -1.0143 251790 bottom 574.87 202.36 -0.8270 lw2 top -3.7552 -601.46 -228.13 -0.9813 176253 bottom 559.72 198.73 -0.8710 lw1 top -3.3896 -562.86 -286.15 -242.69 100716 bottom 530.98 262.78 240.82 et4 top -2.5498 -248.99 -38.930 256.87 125895 bottom 184.89 -1.7709 -275.80 et3 top -0.5995 -121.19 -56.428 -21.706 100716 bottom 59.439 8.9946 -19.349 et2 top -0.0238 -29.573 -28.178 -30.655 75537 bottom -27.989 -27.470 -29.934 et1 top -0.0191 -29.740 -25.448 -25.404 50358 bottom -29.444 -25.211 -25.248 case a top -2.1386 -622.21 -220.95 -0.9649 245619 bottom 567.44 198.35 -0.8738 case b top -2.4177 -609.14 -217.40 -0.9654 161007 bottom 563.79 196.16 -0.8663 some results in terms of transverse displacement w, and transverse normal stress, zz, along the thickness of the sandwich plate are represented in figs. 18a, and 18b. the transverse displacement w behavior can change sensitively depending on the distribution of the kinematic enrichment within the structure plane. fig. 18a shows that esl monomodels can vary sensitively their accuracy depending on the expansion order; differently the lw mono-models have almost the same accuracy independently of the adopted expansion. moreover, for the multi-models, it is noticeable that the choice of the lw higher-order models for the loaded zone is not decisive for the correct description of the transverse mechanical displacement, as shown for case a and case b. on the other hand, for transverse normal stress zz, see fig. 18b, lw higher-order models are able to correctly predict a good behavior along the plate thickness. multi multilayered plate elements with node-dependent kinematics for the analysis of composite.. 25 models theories case a and case b show the same accuracy of the reference solution lw4 in the considered evaluation point. a) w b) zz fig. 18 unsymmetrically laminated rectangular sandwich plate. transverse displacement w(x, y), and transverse normal stress zz (x, y) evaluated at (a/2, b/2) by various singleand multi-theory models results in terms of the three-dimensional representation of in-plane stress xx and its behavior along the in-plane x axis are represented in figs. 19a and 19b, respectively. in fig. 19a it is noticeable that the maximum values of the in-plane stress are located in the loading zone and its surroundings. furthermore, the behavior of in-plane stress xx along the in-plane x axis and evaluated at (y, z) = (b/2, +h/2) is depicted in fig. 19b. monomodels lw4 and et4 and multi-models case a and case b show almost the same accuracy solution. multi-models case a and case b, produce small oscillations in the transition zone. it is noticeable that the oscillations are small. this is due to a finer mesh with respect to the case of the previous numerical section. finally, a three-dimensional distribution on a quarter of the sandwich plate of transverse normal stress zz is given to underline the global/local capabilities of the present finite element on the whole domain of the analyzed sandwich plate structure. referential singlemodel solution lw4 is depicted in fig. 20a. for a fair comparison of the results, the extremities of the color bar values of the lw4 model are used to limit the color bar of the other solutions. the single-model et4 is not able to correctly describe the transverse shear stress behavior, as shown in fig. 20b. 26 s. valvano, e. carrera a) lw4 b) single and multi-models fig. 19 asymmetrically laminated rectangular sandwich plate. in-plane stress xx, three-dimensional view of a quarter of the plate, and in-plane stress along the inplane axis direction x evaluated at (y, z) = (b/2, +h/2), for single and multi-models a) lw4 b) et4 fig. 20 asymmetrically laminated rectangular sandwich plate, three-dimensional view of a quarter of the plate. transverse normal stress zz for single models multilayered plate elements with node-dependent kinematics for the analysis of composite.. 27 multi-model case a and case b are shown in figs. 21a and 21b, respectively. it is evident that the transverse normal stress is well represented in the lw4 zone only, close to the loaded zone. a) case a b) case b fig. 21 asymmetrically laminated rectangular sandwich plate, three-dimensional view of a quarter of the plate. transverse normal stress zz for multi-models 4. conclusions a new simultaneous multi-model approach for a global/local analysis of composite and sandwich plates by means of a two-dimensional finite element with node-dependent kinematics has been introduced. the finite element governing equations are formulated in terms of fundamental nuclei, which are invariants of the theory approximation order. in this manner, the plate theory can vary within the same finite elements with no difficulties. no ad-hoc techniques and mathematical artifices are required to mix the fields coming from two different and kinematically incompatible adjacent zones, because the plate structural theory varies within the finite element itself; therefore, the same kinematics at the interface nodes between kinematically incompatible plate elements is 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(eds.), proceedings of the fifth international conference on sandwich constructions, vol. 1, pp 37–48, emas publishing, zurich. 68. carrera, e, ciuffreda, a, 2005, bending of composites and sandwich plates subjected to localized lateral loadings: a comparison of various theories, composite structures, 68, pp. 185–202. 69. carrera, e., demasi, l., 2003, two benchmarks to assess two-dimensional theories of sandwich, composite plates, aiaa journal, 41(7), pp. 1356–1362. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 295 306 doi: 10.22190/fume170503007d © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper simulation of frictional dissipation under biaxial tangential loading with the method of dimensionality reduction udc 531.4 andrey v. dimaki 1 , roman pohrt 2 , valentin l. popov 2,3,4 1 institute of strength physics and materials science sb ras, tomsk, russia 2 berlin university of technology, germany 3 national research tomsk polytechnic university, russia 4 national research tomsk state university, russia abstract. the paper is concerned with the contact between the elastic bodies subjected to a constant normal load and a varying tangential loading in two directions of the contact plane. for uni-axial in-plane loading, the cattaneo-mindlin superposition principle can be applied even if the normal load is not constant but varies as well. however, this is generally not the case if the contact is periodically loaded in two perpendicular in-plane directions. the applicability of the cattaneo-mindlin superposition principle guarantees the applicability of the method of dimensionality reduction (mdr) which in the case of a uniaxial in-plane loading has the same accuracy as the cattaneo-mindlin theory. in the present paper we investigate whether it is possible to generalize the procedure used in the mdr for bi-axial in-plane loading. by comparison of the mdr-results with a complete three-dimensional numeric solution, we arrive at the conclusion that the exact mapping is not possible. however, the inaccuracy of the mdr solution is on the same order of magnitude as the inaccuracy of the cattaneo-mindlin theory itself. this means that the mdr can be also used as a good approximation for bi-axial in-plane loading. key words: friction, dissipation, tangential contact, biaxial in-plane loading, circular loading, cattaneo, mindlin, mdr 1. introduction friction is a dissipative process transforming mechanical energy into heat and material changes of the contacting partners. the energy dissipation may be connected with material dissipation (wear) [1] or utilized for structural damping [2]. studying both wear and received may 03, 2017 / accepted june 20, 2017 corresponding author: andrey v. dimaki institute of strength physics and materials science sb ras, akademicheskii av. 2/4, 634055 tomsk, russia e-mail: dav@ispms.tsc.ru 296 a.v. dimaki, r. pohrt, v.l. popov damping requires the solution of a tangential contact problem. the simplest case of a tangential loading is an increasing uni-axial tangential loading at a constant normal force. this problem has been solved first by cattaneo [3] and later independently by mindlin [4]. they have shown that a tangential stress distribution can be represented as a superposition of two solutions for the normal contact problem of the same geometry, only multiplied with the coefficient of friction. this reduction to the normal contact problem is exactly the feature which allows the application of the method of dimensionality reduction (mdr) [5], (see also chapter 5 devoted to tangential contact in [6]). however, cattaneo and mindlin have not noticed a small inconsistency in their solution. in their theory, it is assumed that the frictional stresses in the slip domain are all directed in the direction of the applied tangential force. with the exception of the unrealistic case where both the contacting materials have poisson ratio zero, this assumption violates the condition that at every position in the slip domain, the slip is directed in the direction opposing the tangential stresses. the reason for this is the presence of an additional slip motion perpendicular to the direction of the applied force. this was first pointed out by johnson [7] who showed that the maximum inclination of slip angle is on the order of magnitude ν/(4-ν) which is equal to 0.09 for ν=1/3 and 0.14 for ν=1/2. he concluded that the error is not large and that the cattaneo-mindlin solution is a good approximation. later comparisons with numerical solutions have shown that the above mentioned inconsistency may have an important influence on the distribution of wear but has almost no impact on the macroscopic forcedisplacement relations [8]. a detailed analysis can be found also in [9]. in the present paper we consider a more complicated problem of bi-axial oscillating loading (superimposed loading in two in-plane directions). the aim of the paper is twofold: on one hand, we are interested in a better understanding of the energy dissipation in biaxially loaded contacts; on the other hand, we would like to check the applicability of the dimensionality reduction method to this class of problems. at present, there are only a few numerical studies providing the dependencies of dissipated friction energy on the parameters of loading [10]. the applicability of the mdr would provide a simple tool for simulating arbitrary loading histories with applications in the dynamics of structures with frictional contacts. 2. energy dissipation in a single-point contact for circular movement let us start by considering a single isotropic linearly elastic massless element which can deform in normal direction as well as in two tangential directions. we will call this element a “spring”. the spring should have out-of-plane stiffness kz and isotropic inplane stiffness kx=ky. it is first pressed against a rigid half-plane with a normal force fz and then moved in the direction of the x-axis. we will assume that at the immediate contact point between the spring and the substrate, there is friction characterized by a constant coefficient of friction μ. when the free end of the spring is moved horizontally, it first deforms elastically until the in-plane displacement achieves the critical value 0 /z xl f k  . (1) after this, the lower contact point starts sliding and the force remains constant. if the spring is moved on a circle with radius r<l0, then it remains in the stick state at any time. however, if the radius of movement exceeds critical value, r≥l0, the contact simulation of frictional dissipation under biaxial tangential loading... 297 point will slip. in the stationary state, it will move in a circle with a smaller radius rc, while the in-plane displacement of the spring remains constant and equal to l0. the frictional force is assumed to be opposite to the elastic force and at the same time it has to be directed opposite to the velocity vector. therefore, the contact point between the spring and the half-plane will move in the direction of the elastic displacement. on the other hand, this velocity will be directed tangentially to the inner circle with radius rc, which means that the elastic displacement of the spring is directed tangentially to this circle, as shown in fig. 1. the dissipation power is then obviously given by the equation 2 macro macro 0cos 1 ( / )z zw v f v f l r       , (2) where vmacro is the absolute velocity of the spring motion. for one cycle of motion with radius r>l0 the value of the dissipated energy is 2 cycle 02 1 ( / )zw w t r f l r      , (3) where δtcycle is the time needed to perform one cycle of circular motion. if the initial position of the spring does not correspond to the stationary one, it moves on a spiral asymptotically approaching the circle with radius rc as shown in fig. 1b. fig. 1 a) the scheme of a circular motion of a single spring; b) the results of the numerical simulation: the evolution of the trajectory of a single spring during a circular motion 3. energy dissipation in a curved contact for circular movement generally, a non-conforming contact between elastic solids cannot be modeled with a single spring. in the case of uni-axial in-plane loading, the contact problem can be reduced to a contact of a rigid plane profile with a series of independent springs. this method is known as the method of dimensionality reduction [5, 6, 11]. it replaces a contact between two continuum bodies with an ensemble of independent one-spring problems and thus reduces the general contact problem to the above one-spring problem (see fig. 2). 298 a.v. dimaki, r. pohrt, v.l. popov fig. 2 mapping of a three-dimensional contact into one-dimensional one if the mdr-procedure was applicable to the bi-axial in-plane loading, then we could compute the energy dissipation rate just by summing eq. (2) over all effective springs of the mdr-model. let us assume at this point that this is indeed possible and calculate the dissipation in a circularly moving and curved contact. later we will check and discuss the accuracy of this procedure. we consider the movement of a parabolic indenter having the shape z=f(r)=r 2 /(2r0). according to the mdr-rules [5, 6], in the equivalent mdr model it is to be replaced by the plane profile 2 2 2 00 ( )d ( ) x f r r x g x x rx r      . (4) this profile is brought into contact with an elastic foundation consisting of independent springs, each spring having normal stiffness δkz and equal tangential stiffnesses δkx and δky for the displacements along the x -axis and y -axis (not shown in fig. 2) which are defined according to the rules * *, z x yk e x k k g x        , (5) where 2 2 1 2 * 1 2 1 11 e ee     and 1 2 * 1 2 (2 ) (2 )1 4 4g gg     , (6) with e1 and e2 being the young’s moduli, g1 and g2 the shear moduli and ν1 and ν2 the poisson’s ratios of the contacting bodies. further, throughout the paper, we assume that the contacting materials satisfy the condition of “elastic similarity” 1 2 1 2 1 2 1 2 g g      , (7) which guarantees the decoupling of normal and tangential contact problems [12]. if the indentation depth is d, then the vertical displacement of an individual spring at position x is given by ,1 ( ) ( )z du x d g x  (8) and the normal force of a single spring equals to *( ) ( ( )) ( ( ))z zf x k d g x e x d g x       . (9) simulation of frictional dissipation under biaxial tangential loading... 299 the dissipation power in one spring at the position x is given by eq. (2) which we rewrite here as 2 2 * * macro * ( ( )) 1 ( ( )) 1z z x f e d g x w f v e x d g x r k rg                        . (10) let us assume that we have a situation with partial slip. radius c of the stick region is determined by the condition * * 1 ( ) g d g c r e    (11) whence 2 * * 0 1c g d r r e    . (12) the whole dissipation power is thus equal to 2* 2 2 2 2macro 2 2 0 2 ( ) 1 d a c v e a x w a x x r a c           , (13) where 0a r d is the contact radius. evaluation of the integral yields macro 3 ( ) 2 zw v f c   , (14) where 21 2 2 2 1 ( ) (1 ) 1 d 1 c c c              (15) with c c / a . function ( )c is shown in fig. 3. from (14) we see that the energy dissipation power is given by the formally calculated "nominal power" vmacroμfz multiplied with function 3 2 ( )c , which only depends on the reduced radius of the stick area. fig. 3 dependence  c 300 a.v. dimaki, r. pohrt, v.l. popov 4. calculation of stresses in the framework of mdr the above mdr-solution is based on the coulomb criterion for sticking and sliding for the springs of the effective one-dimensional elastic foundation. this mdr model gives the correct solution to the three-dimensional problem only if the conditions for sticking and sliding are fulfilled also for in-plane stresses in relation with normal stresses in the initial (truly three-dimensional) problem. we thus begin our analysis by checking the fulfillment of the sticking conditions and go later to an additional validation by comparison with results of direct 3d simulation given in [10]. according to the mdr rules, the distribution of normal pressure p in the threedimensional problem may be calculated using the following integral transformation [11]: 2 2 ( )1 ( ) dz r q x p r x x r        , (16) where qz(x)=δfz(x)/δx is a linear density of the normal force. a similar transformation is valid for the tangential stress: 2 2 ( )d1 ( ) x x r q x x r x r         , (17) where qx(x)=δfx(x)/δx is a linear density of the tangential reaction force, respectively. the proof for these rules can be found in appendix d of ref. [5]. this proof can be easily generalized to an arbitrary two in-plane dimensions and shows that the transformation (17) can be applied separately to each component of tangential stress, so we can obtain tangential stresses in y-direction similar to eq. (17): 2 2 ( )d1 ( ) y y r q x x r x r         . (18) thus, for calculating the stress component we have to determine first the linear force densities qx(x)=δfx(x)/δx and qy(x)=δfy(x)/δx. let us denote the coordinates of a spring tip as (ux,tip, uy,tip) and the coordinates of the upper point of the spring as (ux, uy). assume that in an iteration step the coordinates of the spring ux and uy, are changed by δux and δuy, so that x x x y y y u u u u u u        . (19) if new coordinates xu and/or yu now lie outside a circle having a central point (ux,tip, uy,tip) and a radius l0(x): 0 ( ) ( ) /z xl x f x k   , (20) then the spring tip will start to slide in the direction of the tangential reaction force (see fig. 4) until it reaches the point , ip ,tip( , )x t yu u : 2 2 , , 0( ) ( ) ( )x x tip y y tipu u u u l x    . (21) simulation of frictional dissipation under biaxial tangential loading... 301 in other words, the new equilibrium point lays on the straight line connecting the points (ux,tip, uy,tip) and ( , )x yu u , at distance l0(x) from ( , )x yu u (see fig. 4). fig. 4 the slip displacement of a single spring in xy plane under lateral motion the components of the tangential reaction force of the spring can be found as follows: , , ( ) ( ) ( ) ( ) x x x x tip y y y y tip f x k u u f x k u u          . (22) we have studied the frictional energy dissipation for the parabolic indenter with the following fictive parameters: r0=1 m, e * =1 gpa, d=0.001 m, ν=0.28, μ=0.3. the indenter was initially moved to the point (ux0, 0) and then subjected to an in-plane harmonic displacement 0 0 ( ) cos( ) ( ) sin( ) x x y y u t u t u t u t      . (23) controlling the tangential reaction forces in ox and oy directions, it is possible to introduce the force-dependent governing parameters, following the paper of ciavarella [10]: / zq f and /m x yr q q , (24) where 2 2max ( ) , max ( ) ,x x y y x yq f t q f t q q q    . (25) note that the value of q, defined in eq. (25), does not correspond to any real tangential force acting on the indenter, but it serves only as a governing parameter in the parametric study of the problem under consideration. with eq. (22) we determine linear force densities qx(x)=δfx(x)/δx and qy(x)=δfy(x)/δx. we then calculate the tangential stress components given by eqs. (17) and (18) and finally the absolute value of the tangential stress: 2 2( ) ( ) ( )mdr x yr r r     (26) 302 a.v. dimaki, r. pohrt, v.l. popov the corresponding dependencies are presented in fig. 5 together with the normal stress distribution multiplied with the coefficient of friction and the formal mindlin solution with the same radius of stick region (dashed lines in fig. 5). one can see that the obtained stress distributions do not exactly fulfill the conditions for stick and slip. in most ranges of radii smaller than the stick radius, the tangential stress is smaller than the normal stress multiplied with the coefficient of friction; there is only a small region inside the stick radius with  too high. thus the stick condition is fulfilled not exactly but in good approximation. however, for radii moderately larger the stick radius, the tangential stress is higher than the normal stress times the coefficient of friction, which means that the sliding condition is not fulfilled. at even larger radii, the condition that in the sliding region the tangential stress must be equal to normal stress times the coefficient of friction is fulfilled with good accuracy. thus, the tangential stress distribution has a qualitatively correct shape but it does not exactly match the stick und slip conditions. the mentioned discrepancy is observed only in a relatively narrow interval of radii. thus, the integral influence of this error may be moderate. this situation can be compared with the solution by cattaneo and mindlin which also has a local error, but the global error in the force-displacement relations is moderate and is generally tolerated. fig. 5 the distributions of normal pressure and the absolute value of tangential stress. p0 is the pressure under the axis of the indenter. the dashed line indicates the formal mindlin solution with the same radius of stick region. rm = 1. a) q/μfz=0.5; b) q/μfz=0.9 in order to estimate the possible global error, let us determine the integral discrepancy between the obtained stress distribution and the cattaneo-mindlin distribution [3] (which fulfils the stick and slip conditions): mdr cm cm 0 0 0 100% 2 ( )rd 2 ( )rd / 2 ( )rd a a a r r r r r r            , (27) where τcm(r) corresponds to the cattaneo-mindlin solution. this discrepancy is shown in fig. 6. the integral difference between tangential stresses, predicted by the theory of cattaneo-mindlin, and the mdr results, is about two percent for low values of q/μfz and rm. this means that the above mdr theory has a good accuracy at least for oscillations with small amplitude comparable to the full slip displacement. simulation of frictional dissipation under biaxial tangential loading... 303 further, let us compare the results of mdr simulation with the full three-dimensional calculations. the tangential stresses are calculated using mdr as described above for the following set of parameters: rm = 1, q = 0.9, which correspond to the same values as used in ref. [10]. comparison of the mdr results with results of full three-dimensional simulations is presented in fig. 7. in fig. 7, on the left hand side, the stress-field simulated by the mdr is presented and so is, on the right hand side, the stress field from the three-dimensional simulation [10]. while both results are in a good qualitative agreement, one can also see some differences. firstly, the stick radius in the mdr results does not decrease after the start of the in-plane rotation, which can be connected to the application of the tangential displacement instead of tangential forces in 3d simulation. secondly, the tangential stresses in the stick area in the mdr solution are higher than those in the full 3d calculation. however, the mentioned discrepancies between the mdr results and full 3d calculations are moderate. we can conclude that the mdr can be also used with “engineering accuracy” for contact problems with bi-axial in-plane loading. fig. 6 the integral difference (27) between tangential stresses, predicted by the cattaneo-mindlin theory, and the mdr results 5. numerical simulation of dissipation under non-circular motion in this paragraph we apply the mdr within its range of accuracy for studying energy dissipation in a contact subject to biaxial tangential loading with different oscillation amplitudes in two perpendicular directions. in order to normalize values of dissipated energy we use the solution of mindlin [3] for friction energy dissipation during one cycle of a uniaxial tangential loading: 2 5 2 3 3 * 2 0 9 2 5 1 1 1 1 10 6m z c z z z r f q q q w w a f f fg                               . (28) 304 a.v. dimaki, r. pohrt, v.l. popov in the performed calculations we have varied the governing parameters (24) in a wide range of values. the results of simulation, accompanied with the corresponding results of the full 3d simulations, given in ref. [10], are shown in fig. 8a. it can be seen that the mdr results are in a good agreement with the results of the full 3d simulations, except for the curve for rm = 1 which also is in a qualitative agreement but shows distinctive quantitative differences. fig.7 the distributions of tangential stresses in the contact area the results of the mdr simulation in the left column, the results of the full 3d simulation, from ref. [10], in the right column: a) after the initial displacement; b) after one revolution; c) after two revolutions; rm=1, q=0.9.the inner circle indicates the stick area simulation of frictional dissipation under biaxial tangential loading... 305 fig. 8 a) the normalized dependencies of the energy, dissipated during one cycle of the circular motion, compared with the data from ref. [10] (indicated by the crosses); b) the normalized dependencies of the energy, dissipated during one cycle of the circular motion, reduced into the universal curve. the normalizing factor w1,c is given by eq. (29) we have found that for various values of rm, the dependencies of w on q/μfz can be reduced to a universal curve (fig. 8b). the results are normalized to the value of the energy w1,c dissipated during one cycle of uniaxial loading with q/μfz=1: 2 1, * 2 0, 3 10m zr q f z c f w w ag      . (29) the universality of the given curve holds for relatively small amplitudes of oscillations. when the amplitude of oscillations becomes comparable with the value of amplitude needed for gross slip transition, the deviations from the universal curve appear (see fig. 8b). we suggest a power-law approximation of the data shown in fig. 8b as follows: 3.5 1,0.45 ( / )c zw w q f  , (30) which fits the results of numerical simulations well for q/μfz < 0.7. note that in fig. 8b the curve for rm = 0 coincides with the results of cattaneo and mindlin. 6. conclusions by analyzing the stick and slip conditions and comparing with three-dimensional calculations we have explored the question whether the mdr is applicable for the simulation of bi-axial in-plane loadings. we have found that the corresponding mapping is not exact (there are local violations of stick and slip conditions) but has an acceptable accuracy comparable with the accuracy of the cattaneo and mindlin solution for tangential contact. comparison with three-dimensional simulations shows a good qualitative agreement but some quantitative deviations. we have found that the dependencies of the dissipated energy on the amplitude of loading, obtained for various values of rm, fit into one universal curve. this curve may be approximated by a power law in the range of small values of q/μfz < 0.7. the obtained results may be helpful for a better understanding of the mechanics of tangential contacts under bi-axial loading. 306 a.v. dimaki, r. pohrt, v.l. popov acknowledgements: a.v. dimaki is thankful for the funding of the fundamental research program of the state academies of sciences for 2013–2020. r. pohrt and v.l. popov thank v. aleshin for helpful discussions. references 1. dimaki, a.v., dmitriev, a.i., chai, y.s., popov, v.l., 2014, rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction, int. j. solids and struct., 51, pp. 4215-4220. 2. ginsberg, j.h., 2001, mechanical and structural vibrations: theory and applications. wiley, 704 p. 3. cattaneo c., 1938, sul contatto di due corpi elastici: distribuzione locale degli sforzi, rendiconti dell'accademia nazionale dei lincei, 27, 342-348, 434-436, 474-478. 4. mindlin, r.d., mason, w.p., osmer, t.f., deresiewicz, h., 1951, effects of an oscillating tangential force on the contact surfaces of elastic spheres, proc. first us national congress of appl. mech., pp. 203–208. 5. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, berlin, heidelberg: springer, 265 p. 6. popov, v.l., 2017, contact mechanics and friction. physical principles and applications, 2nd edition, berlin, heidelberg: springer, 391 p. 7. johnson, k.l., 1955, surface interaction between elastically loaded bodies under tangential forces, proceedings of the royal society, a230, p. 531-548. 8. munisamy, r.l., hills d.a., nowell d., 1994, static axisymmetric hertzian contacts subject to shearing forces, asme journal of applied mechanics, 61, pp. 278-283. 9. gallego, l., nelias, d., and deyber, s., 2010, a fast and efficient contact algorithm for fretting problems applied to fretting modes i, ii and iii, wear, 268.1, pp. 208-222. 10. ciavarella, m., 2013, frictional energy dissipation in hertzian contact under biaxial tangential harmonically varying loads, j. strain analysis, 49(1), pp. 27–32. 11. popov, v.l., hess, m., 2014, method of dimensionality reduction in contact mechanics and friction: a user's handbook. i. axially-symmetric contacts, facta univ. mech. engng. 12, pp. 1-14. 12. johnson, k.l., 1985, contact mechanics, cambridge: cambridge university press, 452 p. models for intralaminar damage and failure of fiber composites a review facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 1 19 review article models for intralaminar damage and failure of fiber composites a review  udc 539.4 klaus rohwer dlr, institute of composite structures and adaptive systems, braunschweig, germany abstract. in order to fully exploit the potential of structures made from fiber composites, designers need to know how damage occurs and develops and under what conditions the structure finally fails. anisotropy and inhomogeneity cause a rather complex process of damage development which may be one reason for an exceptionally large number of existing models. this paper intends to provide an overview over those models and give some hints about current developments. as such it is an updated version of a recent publication [1]. the survey is limited to laminates from unidirectional layers out of straight continuous fiber polymer composites under quasi-static loading. furthermore, focus is laid on intralaminar damage. many failure models smear out the inhomogeneity between fibers and the matrix. simply limiting each stress component separately can lead to surprisingly good results as documented in the first world-wide failure exercise. interpolation criteria consider mutual influence of normal and shear stresses, predominantly through a quadratic failure condition. traditionally one distinguishes between interpolation criteria and physically based ones. as an important physical effect the difference between fiber failure and inter-fiber failure is considered. furthermore, stress invariants are taken as a basis, increased shear strength under compression is accounted for, and characteristic failure modes are captured. fibers and the matrix material are characterized by a large disparity in stiffness and strength. micromechanical models consider this inhomogeneity but suffer from the difficulty to determine relevant material properties. compressive strength in fiber direction has attracted special attention. however, the role of kink band formation, which is observed in the failure process, seems to be not yet fully understood. in summary it must be concluded that despite the tremendous effort which has been put into the model development the damage and failure simulation of fiber composites are not in a fully satisfying state. that is partly due to lack of accurate and reliable test results. key words: fiber composites, unidirectional layers, strength, failure conditions received february 23, 2015 / accepted march 10, 2015 corresponding author: klaus rohwer dlr braunschweig, lilienthalplatz 7, 38108 braunschweig, germany e-mail: klaus.rohwer@dlr.de 2 k. rohwer 1. introduction for an efficient design the structural engineer needs to know as accurately as possible under what conditions the designated material will develop damage and finally fail. only then is it possible to fully exploit the potential of the structure while maintaining the required safety. in any case, some information on damage and failure must be obtained by suitable tests. these tests are usually performed on coupons and aimed at determining the material strength under a specific single state of stress, be it pure tension, compression or shear. the general state of stress in a loaded structure, however, consists of several, if not all, components of the stress tensor. thus, a criterion is needed which maps the actual state of stress to the limited number of test results. this paper reviews different possibilities of formulating criteria and points out development tendencies limited to laminated continuous fiber-reinforced polymer composites. as an updated version of a paper previously published by the author [1] this review was prepared for and presented at the nafems world congress in san diego, ca, in june 2015. it then was adequately formatted and modified for publication in fu mech eng. similar reviews have been performed earlier, for instance by nahas [2] or thom [3]. since then, however, models have been developed further. in parts that is due to the tremendous increase in computational power which allows for more and more complex models. besides, the world-wide failure exercises, wwfe-i [4], -ii [5] and – iii [6], have demonstrated deficiencies in the existing failure theories and therewith fired new developments. the large number of existing theories prohibits recognizing them all; rather only those will be assessed which in the opinion of the author have reached some level of acceptance. furthermore, not every detail of the respective theory can be outlined; only those aspects will be referred to which the author regards important. fig. 1 damage development the scope of this review is focused on intralaminar fracture of laminates made from unidirectional layers which are subjected to quasi-static loading. delaminations, woven fabrics, and effects resulting from fatigue or impact loads are not covered. further, the material behavior after the first appearance of damage is of interest, especially for fiber models for intralaminar damage and failure of fiber composites a review 3 composites. that is because in case of matrix failure the fibers are often able to carry much higher loads. a typical development of cracks and delaminations in a cross-ply laminate under axial tension is depicted in fig. 1. it shows why the crack density is limited, finally forming a characteristic damage state. effects of the progressive failure of fiber composites have been extensively studied by knight [7]. he differentiated between ply discounting approaches and continuum damage mechanics methods. libonati and vergani [8] have recently tested fiber composite behavior before and after failure onset using thermography. they have identified three regions: an initial region without damage, a second region where micro-damages appear which may be initiated by pre-existing defects, and a third region with an extended damage size. considering these results, within this paper the main focus is laid on the second and third region. different failure criteria and damage progression models will be outlined, pros and cons be mentioned, and tendencies in the development will be identified. a vast majority of existing failure criteria is formulated in stresses, and there are good arguments to do so. christensen [9], for instance, mentioned that such a formulation would be more suitable in order to fit with fracture mechanics or dislocation dynamics. in addition he pointed out that viscous material can fail under constant stress, but not under constant strain due to relaxation. a major point of criticism against a stress-based criterion is related to strength measurements. usually strength is obtained as the load carrying capacity at final failure. many tests, however, show a rather nonlinear stress-strain behavior, which is, at least in parts, due to progressive damage. there is a need to clearly define failure of composites. from comparative studies between deterministic and probabilistic analyses of cross-ply laminates under tension sánchez-heres et al. [10] concluded that an increased understanding is required regarding the effects of progressive matrix cracking in order to reach a safer structure. during the design phase ‗quick and dirty‘ methods are needed which are fast, simple to use and lead into the right direction, but do not claim to be highly accurate. among these is the netting theory, where only the fibers are accounted for carrying loads. quite popular is a limitation of strains to a fixed amount. further, there is the 10% rule by hart-smith [11], predicting the strength and stiffness of fiber–polymer composites on the basis of simple ruleof mixtures. though very useful, such methods will not be considered in the following. 2. homogeneous models 2.1. shape of failure envelope of course fiber composites are not homogeneous; however, the overall behavior of the material can be appropriately described by smeared out properties. also, a large number of failure models are based on the assumption of a homogeneous anisotropic material, specifying a failure envelope in stresses or strains. there is a general agreement that the failure envelope should be convex. otherwise, unloading from a certain state of stress may indicate failure. it is under discussion, however, as to whether the failure surface should be open or closed. christensen [9] stated: ―all historical efforts to derive general failure criteria used the condition that the isotropic material would not fail under compressive hydrostatic stress‖, which means that the failure surface is assumed open. in 4 k. rohwer his treaties on failure surfaces for polymeric materials tschoegl [12] pointed at ―the common sense requirement that the surface should be open in the purely compressive octant (because hydrostatic compression at reasonable pressures cannot lead to failure in the ordinary sense)‖. for fiber composites the situation is different. because of the stiff fibers an external hydrostatic load causes matrix stresses which differ considerably from hydrostatic ones. comparing theories and experiments of the wwfe-ii exercise kaddour and hinton [13] mentioned ―the diversity exhibited between the theories as to whether certain failure envelopes are ‗open‘ or ‗closed‘‖. however, this discrepancy should not exist, and christensen [9] has provided reasonable arguments why fiber composites cannot sustain unlimited hydrostatic pressure. 2.2. non-interactive criteria the easiest criterion limits every stress component separately, not accounting for any interaction. fig. 2 illustrates that in this case the failure surface is necessarily closed. fig. 2 maximum stress astonishingly enough, this rather crude approach has been applied quite successfully by zinoviev et al. [14] in wwfe-i. the failure criterion was supplemented by a special model characterizing the progressive damage under transverse tension and in-plane shear of a ud ply within a multidirectional laminate. this model describes the loading as linear elastic — ideal plastic, and the unloading as linear elastic with a smaller module. a comparatively favorable performance was highlighted by hinton et al. [15]. some discrepancies between theoretical predictions and test results zinoviev et al. [16] traced back to the assumption about the fatal impact of ultimate transverse compressive stresses in a single ply on the failure of the whole composite laminate. hart-smith [17–19] applied modified maximum strain as well as maximum stress criteria in the wwfe-i. the modification affects a truncation of the failure envelope in the biaxial tension–compression quadrant. differences between analysis and test results were explained by deficiencies with respect to matrix-dominant failure. the maximum strain criterion in conjunction with plasticity used by bogetti et al. [20, 21] delivered good results in the wwfe-i; the strengthening effect that appears under tri-axial loading or hydrostatic pressure, however, is obviously not well captured as has been admitted by bogetti et al. [22]. furthermore, bogetti‘s theory predicts a completely closed failure envelope even for isotropic materials. models for intralaminar damage and failure of fiber composites a review 5 nahas [2] has referred to further non-interactive theories which to some degree account for the strength of the constituents. in general, however, these theories have not been used very often in practice. it is but the maximum strain model which because of its simplicity is still applied especially in the initial design phase. 2.3. interpolation criteria following yield conditions for isotropic and orthotropic materials, hoffman [23] proposed a quadratic fracture condition accounting for the difference between tensile and compressive strength in fiber and transverse directions. based on the idea that a tensor polynomial can describe the failure surface, tsai and wu [24] came up with a similar approach. these popular failure criteria consider interactions between different components of the stress tensor. a general formulation is given in eq. (1): 1ff iijiij   (1) where fi and fij are strength coefficients. most of their values are easily determined from the measured strengths in fiber and transverse direction. only the interaction terms fij for i  j linked to the product of two normal stress components require difficult tests under biaxial load; and these terms are important since they may indicate implausible strength levels above those in fiber direction as can be seen from fig. 3. the interpolation criteria suffer from a further drawback. distinguishing between fiber breakage, matrix cracks, or interface failure, is not possible by a smooth mathematical function. fig. 3 elliptic failure surface by comparing with test results under plain stress conditions narayanaswami and adelman [25] concluded to rather set the terms f12 to zero. liu and tsai [26] underlined that the failure surface must be closed, and they gave an overview over different possibilities for the interaction terms. further, they have outlined a procedure for determining progressive laminate failure using reduced moduli which in the end leads to last ply failure. deteresa and larsen [27] have proposed relations between the interaction terms and the strengths in fiber and transverse direction which fit to an open failure surface. a test under hydrostatic pressure has shown no damage. there are a number of other interpolation criteria with certain inconveniences or restrictions. the criterion proposed by norris [28] does not explicitly account for differences in tensile and compression strength; on application the user must check the sign of the different stress components and use corresponding strength values. the same 6 k. rohwer holds for the tsai–hill criterion as described by azzi and tsai [29], which differs from the norris criterion only in the interaction between the axial and transverse normal stress. the proposal by yamada and sun [30] is sometimes looked upon as a degeneration of the above mentioned criteria, a view which ignores the intention to determine the final failure of a laminate. further, the shear strength to be used in this criterion must be determined in tests with crossply laminates leading to much higher values than those obtained from a single ply. it is also worth mentioning that yamada and sun stressed the need to account for statistical distributions of the strength values. the criterion by rotem [31, 32] differentiates between failure in the fibers or in the matrix. fiber failure (ff) is modeled by a maximum stress criterion in fiber direction with some modifications accounting for effects of transverse stresses, whereas matrix failure is predicted using a quadratic interaction of axial, transverse, and shear stresses. by means of comparing with test results, kaddour and hinton [13] stated that there are indications ―that the theory does not discriminate adequately between initial and final failure‖. several other interpolation criteria have been mentioned by nahas [2], which to the author‘s knowledge have not reached much public attention. 2.3. physically based criteria distinguishing between interpolation criteria and physically based ones is a bit artificial and a traditional classification. neither are the interpolation criteria free of some physical background nor are the physically based ones free of some simple interpolation aspects. there is rather a gradual transition between both the categories which makes it somewhat arbitrary where to draw the line. in his model development, hashin [33] pointed out that using a formulation quadratic in stresses is based on curve fitting considerations rather than on physical reasoning. he looked at the stress invariants and differentiated between four failure modes: tension or compression in fiber or in transverse direction. for the inter-fiber failure he mentioned the idea to hold the stresses acting at the failure plane responsible. that implies to determine the most probable crack direction which is computationally costly. hence he settled for the quadratic formulation which leads to not fully satisfying solutions. building up on hashin‘s original idea puck [34, 35] formulated a criterion which yielded rather accurate results in the wwfe-i. he strictly distinguished between ff and iff, where the latter comprises matrix cracks and fiber–matrix debonding. puck, too, regarded the stresses in the fracture plain responsible for iff. if the normal stress on the fracture plain is positive (tensile), then all three stress components foster the failure, whereas compressive stress increases the strength by means of internal friction. the different behavior under tension and compression requires additional material parameters which describe the inclination of the fracture master surface at zero normal stress as depicted in fig. 4. recommendations for these inclination parameters are provided by puck et al. [36]. based on puck‘s model the strength degradation of laminates which suffer from an iff within a certain layer was investigated by knops and bögle [37]. also the german engineering guideline [38] regarding the analysis of components from fiber reinforced plastics relies on puck‘s failure criterion. dong et al. [39] complemented puck‘s theory by adding effects of ply thickness and ply angles of neighboring laminae. models for intralaminar damage and failure of fiber composites a review 7 fig. 4 inter-fiber failure after puck [34] the failure mode concept (fmc) as set up by cuntze and freund [40] aimes at capturing the behavior of five different failure modes. based on stress invariants the model provides one failure condition each for two ff modes and three iff modes. corresponding to puck‘s inclination parameters two curve parameters are to be determined by multi-axial tests. possible interactions between failure modes are accounted for by a probabilistically based series spring model approach. the fmc was subsequently improved by cuntze [41, 42]. in connection with the behavior of isolated and embedded laminas special emphasis is put on the difference between the onset of failure and the final failure of composite laminates. furthermore, cuntze [43] carefully examined the tests provided for the wwfe-ii and after certain corrections obtained rather good agreements. at nasa langley research center, dávila et al. [44] have proposed failure criteria for fiber composite laminates under plane stress conditions which were extended to threedimensional stress states by pinho et al. [45] and eventually improved with respect to matrix compression failure by pinho et al. [46]. as with hashin‘s [33] approach the failure model considers four different scenarios: tension and compression in fiber and transverse direction. for compression in fiber direction the effect of fiber undulation is regarded. nali and carrera [47] compared this approach against some interpolation criteria for planestress problems and found good agreement with test results. in a detailed analysis catalanotti et al. [48] described certain pitfalls of existing 3d failure criteria. they pointed to the requirement of using in situ strength properties in order to account for the ply thickness effect. however, by means of micromechanical analysis, herráez et al. [49] found that strength must be independent of ply thickness. the pitfalls could be avoided by an improved criterion for transverse matrix failure. longitudinal tension failure is predicted using a maximum strain criterion, and longitudinal compression failure accounts for fiber kinking. building on this proposal and on the three-dimensional plasticity model for composite laminates developed by vogler et al. [50] camanho et al. [51] formulated new criteria where transverse failure and kinking models are invariantbased. for validation in case of complex three-dimensional stress states computational micromechanics turned out to be a useful tool. 8 k. rohwer 3. damage mechanics approach damage mechanics does not provide conditions at which a certain type of damage occurs; rather, it uses internal variables to describe the progressive loss of rigidity due to damage of material. an example is given in eqs. (2a) to (2d), where d and d denote the damage variables characterizing the behavior after initial damage under transverse tension and in-plane shear, respectively: 11 221211 11 e     (2a) if 22  0 (tension): 22 12 22 11 2 2(1 )e d e       (2b) if 22  0 (compression): 2 111222 22 e     (2c) 12 12 122 (1 )g d     (2d) ladevèze and le dantec [52] have applied damage mechanics to set up a model which describes ply-wise matrix microcracking and fiber/matrix debonding. reaching the maximum mean stress or a maximum of the load-deflection curve specifies the laminate failure. this model was adopted by payan and hochard [53] to study the behavior of ud laminates from carbon fiber-reinforced plastics (cfrp) under shear and transverse tension. they found elastic behavior up to brittle failure in fiber direction, and gradient loss of rigidity due to damage under shear and transverse tension. based on these results they developed a model which covers the damage state by means of two scalar-damage variables describing the loss of rigidity under shear and transverse tension loading, respectively. the model has proven to be valid for a "diffuse damage" phase where micro-cracks occur and it is limited to the first intralaminar macro-crack. hochard et al. [54] have further extended the model to problems with stress concentrations. the approach is based on a fracture characteristic volume which is a cylinder defined at the ply scale where the average stress is calculated and compared to the maximal strength of the material. barbero and de vivo [55] presented a damage mechanics approach where the damage surface has the shape of the tsai–wu [24] criterion. but it goes beyond a failure criterion by "identifying a damage threshold, hardening parameters for the evolution of damage, and the critical values of damage". these parameters are all related to known material properties but not directly measurable (cf. barbero and cosso [56]). van paepegem et al. [57] performed tension tests with [±45]2s laminates and used the results to determined one parameter each for shear modulus degradation and the accumulation of permanent shear strain. the same authors [58] applied these parameters to a mesomechanical model which did not account for time-dependent effects like strain rate or viscoelasticity. nevertheless, they were able to describe the nonlinear behavior up to failure of glass-fiber reinforced composite laminates under various loads rather accurately. time and temperature dependency of fracture strengths both in tension and compression models for intralaminar damage and failure of fiber composites a review 9 were thoroughly studied by miyano et al. [59]. they found out that the strength master curves can be set up successfully by using the reciprocation law between time and temperature. a majority of models for damage progression in laminates are based on the unrealistic assumption that each ply behaves independently of its neighbors. in order to account for the interaction between adjacent layers williams et al. [60] developed a continuum damage approach for sub-laminates. therewith it is not intended to predict details of damage at the ply level, rather to capture the sub-laminate‘s overall response. the idea was further upgraded by forghani et al. [61] considering several aspects specific for damage progression in multidirectional composite laminates and applied to the open hole problems of the wwfe-iii. the open hole tension strength of composite laminates was also studied by ridha et al. [62]. they found a significant interaction between delamination and in-plane damage, so that neglecting delamination would overestimate strength. frizzell et al. [63] developed a numerical method based on continuum damage mechanics that is capable of describing sub-critical damage and catastrophic failure mechanisms in composite laminates. they proposed a ―pseudo-current‖ damage evaluation approach which avoids convergence problems even for complex damage mechanisms. 4. inhomogeneous models 4.1. strength of constituents fibers and the matrix material are characterized by a large disparity in stiffness and strength. though smeared out in the models reviewed above, this fact certainly influences the failure process and thus it is reflected in certain features. in this section, approaches will be discussed which account for the inhomogeneity in one way or the other. to this end strength properties of the constituents are needed. measuring them, however, encounters difficulties. resin strengths are typically measured in appropriate tests with neat material. an overview over models with relevance to resin failure was given by fiedler et al. [64]. these authors have proven that the type of resin failure depends not only on the material itself but also on the state of stress. they found out that ―ductility is a function of the amount of tri-axiality and explains why ductile polymers behave brittle when used as a matrix in fiber reinforced composites‖. such an effect was detected and analyzed already by asp et al. [65]. on the other hand, pae [66] has found that brittle epoxy develops yielding when hydrostatic pressure is superimposed on the loading. because of these intricacies, properties determined from tests with neat resin must be handled with caution when used in a micromechanical failure analysis. shear strength of the fiber-matrix interface can be obtained from fiber pullout or pushout tests. kerans and parthasarathy [67] proposed a procedure for extracting interface parameters from the test data. an analytical model describing the fiber pushout was developed by liang and hutchinson [68]. more involved is the determination of interface strength under transverse loads since secondary transverse stress perpendicular to the primary transverse compression affects the threat of fiber-matrix interface fracture. correa et al. [69] found out that secondary tensile stress increases the risk whereas compression decreases it. 10 k. rohwer measuring fiber tensile strength seems to be a relatively easy task. when performing the tests, however, it becomes apparent that the results depend on the specimen length. the longer the specimen, the lower is the measured tensile strength. even more questionable is the determination of the compressive strength in fiber direction. in a composite the compressed fibers usually do not suffer a material failure but a loss of stability. thus the composite strength limit will depend on the matrix properties. wang et al. [70] have proposed a tensile recoil method to obtain the fiber compressive strength and a microbond fiber pull-out test for the interface shear strength. 4.2. models with some effect of inhomogeneity in this section, approaches will be discussed which to some extent consider inhomogeneity but still show relations to the homogeneous models mentioned above. this evidently holds for the discrete damage mechanics approach as proposed by barbero and cortes [71]. by means of fracture mechanics applied to the inhomogeneous material they determined parameters for stiffness reduction of the homogenized structure. barbero and cosso [56] showed that this approach can be successfully applied to model damage and failure of laminates from cfrp. inhomogeneity plays an important role in tests of inplane shear strength. there is as yet a deep disagreement as how to obtain reliable values. odegard and kumosa [72] have thoroughly investigated the standard iosipescu test with 0° specimens as well as the 10° off-axis test. they found good agreement only if the iosipescu tests are accompanied by fully nonlinear finite element analyses including plasticity and premature cracks, and the 10° off-axis test must be carefully machined to avoid micro-crack at the specimen edges. the growth of cracks in a ud fiber reinforced lamina was modeled by cahill et al. [73]. by means of the extended finite element method (xfem) for heterogeneous orthotropic materials where material interfaces are present as well as a modified maximum hoop stress criterion for determining the direction of the crack propagation at each step they found out that for a material with a large stiffness rate between fiber and transverse direction the crack will propagate along the fiber direction, regardless of the specimen geometry, loading conditions or presence of voids. matrix cracking and fiber–matrix debonding seem to impair each other. by means of shear load nouri et al. [74] generated fiber–matrix debonding and observed its effect on crack density under transverse load. the authors developed a modified transverse cracking toughness model. in order to accomplish the tasks put forward in the wwfe-i, gotsis et al. [75] used the computer code ican by murthy and chamis [76], which determines material properties using micromechanics and accounts for laminate attributes like delamination or free edge effect. in addition to the maximum stress criterion a modified distortion energy failure criterion determines the ply failure. comparison with the test results as provided by gotsis et al. [77] revealed reasonable results in cases of fiber dominated failure, but rather large discrepancies when matrix failure was predominant. analysis methods were further improved to a full hierarchical damage tracking and applied in the wwfe-iii challenge by chamis et al. [78]. therewith constituent properties determined by inverse model application were used for the micromechanical analysis part. models for intralaminar damage and failure of fiber composites a review 11 4.3. tensile strength in fiber direction some effort was put on developing models for the determination of tensile strength in fiber direction from constituent properties. considering the standard composite design with an extension to failure of the matrix much higher than that of the fibers, the composite failure stress can be roughly estimated by the rule of mixture from the failure stress of the fiber and the matrix stress at fiber rupture. however, that does neither account for varying fiber strength along each single fiber nor for strength variation between fibers. a number of hypotheses accounting for these variations have been proposed, e.g. by rosen [79] and zweben [80], but results from their application are not very convincing. more recent developments along this line are the global load sharing scheme by curtin [81], the simultaneous fiber-failure model by koyanagi et al. [82] and statistical models for fiber bundles in brittle-matrix composites by lamon [83]. 4.4. compressive strength in fiber direction models for compressive strength in fiber direction were first set up by studying the buckling of fibers on an elastic support. depending on the fiber volume fraction dow and rosen [84] differentiated between an extension and a shear failure mode. their results, however, proposed too high strength values. xu and reifsnider [85] extended the model by assuming slippage between fibers and the matrix over certain regions and therewith determined a good agreement with test results. following a thorough review of the models developed until then lo and chim [86] proposed to improve the microbuckling concept by considering transverse isotropy of the fibers and the effects of resin young‘s modulus, fiber misalignment, a weak fiber matrix interface as well as voids. they also pointed out that in case the strain to failure of the fibers is reached prior to buckling, then the compressive strength should be determined by the rule of mixture between fibers and the matrix. the effect of fiber misalignment and resultant kinking was studied by budiansky and fleck [87]. their model, however, was not able to predict the width of the kink band and its inclination. micromechanical analyses of the kink band formation after fiber buckling including the effect of fiber misalignment were performed by kyriakides et al. [88] and by jensen and christoffersen [89]. after a thorough derivation of a stress based model for fiber kinking, ataabadi et al. [90] pointed to certain drawbacks of the model. in order to alleviate them they proposed an improvement based on strains and used it to predict the compressive strength depending on the fiber misalignment. on validating this strain based model against test results ataabadi et al. [91] found that for specimens with an off-axis angle greater than 0° this model can predict the compressive strength of ud laminated composites with acceptable accuracy. gutkin et al. [92, 93] distinguished between two different failure mechanisms: shear-driven fiber compressive failure and kinking/splitting. similar to that approach prabhakar and waas [94] studied the interaction of kinking and splitting by means of a 2d finite element model. with a perfect interface the stress–strain curve shows a typical instability behavior with a sharp peak and a snap-back branch afterwards. since local strains then exceeded the strain to failure for polymer matrix material discrete cohesive zone elements were applied at the fiber–matrix interface. it turned out that it is important to know especially the mode-ii cohesive strength of the interface in order to determine the compressive strength and failure mode of ud laminates accurately. the same authors [95] further extended the micromechanical model of failure under compression to multidirectional laminates considering delaminations. that allowed studying the effect of stacking sequence on 12 k. rohwer the compressive strength. mishra and naik [96] used the inverse micromechanical method to calculate fiber properties and applied them to determine the compressive strength for a composite with a different fiber volume fraction. a formulation capable of obtaining the maximum compression stress, and the post-critical performance of the material once fiber buckling has taken place was proposed by martinez and oller [97]. dharan and lin [98] questioned the role of initial fiber waviness and kink band formation on the compressive strength in fiber direction. like lo and chim [86] did earlier, they rather extended the micro-buckling model of dow and rosen [84] by accounting for an interface layer around the fibers, the thickness and shear modulus of which have to be adjusted to test results. zidek and völlmecke [99] used a simple analytical model introduced by wadee et al. [100]. they improved it by accounting for initial fiber misalignment. furthermore this model allows for predicting the kink band inclination angle. obviously, there is not a generally accepted view yet as to whether kink band formation is a failure mode that limits the compressive strength or rather a secondary effect which appears after buckling. 4.5. normal strength in transverse direction tensile and compressive strength in transverse direction was studied by asp et al. [101, 102]. they used a micromechanical approach with a representative volume element, which thereafter became more and more popular. not accounting for fiber–matrix debonding they have found that the fiber modulus has a significant effect on the failure caused by cavitation in the matrix. this brittle failure occurred earlier than yielding. a thin interphase of a rubbery material improves the transverse failure properties. tensile and compressive strength with perfect fiber–matrix adhesion on the one hand and complete debonding on the other hand was compared by carvelli and corigliano [103]. assuming periodicity for rather small fiber volume fraction they determined finite strength under biaxial tension only with debonded interfaces. transverse tensile failure behavior of fiber–epoxy systems was also studied by cid alfaro et al. [104]. they pointed to a strong influence of the relative strength of the fiber–epoxy interface and the matrix. vaughan and mccarthy [105] found out that in case of a strong fiber–matrix interface residual thermal stresses improve the transverse tensile strength. 4.5. shear strength several authors applied micromechanical means for analyzing fiber composite shear strength. king et al. [106] determined the composite transverse shear strength, mainly to predict the effect of fiber surface treatment and sizing on the interfacial bond strength. they found out that the predicted composite shear strength strongly depends upon the type of matrix and the interface strength, and is not significantly dependent on the fiber properties. axial tension tests on [±45°]s laminates are often used to determine the composite shear stress–strain response. comparing the shear behavior of cfrp with epoxy and peek matrix, lafarie-frenot and touchard [107] determined a pronounced plastic deformation but no visible damage in the low loading range. higher load levels led to increased damage in the epoxy matrix and early failure whereas the peek material exhibited even larger plastic deformation in connection with a considerable change of the fiber angle. the detectability of microcracks, however, may have been limited due to the fact that the contrast agent for x-ray inspection was applied to the free edges only. on the models for intralaminar damage and failure of fiber composites a review 13 contrary, by means of tests with dog bone specimens and micromechanical analyses ng et al. [108] found out that it is micro-cracking rather than plasticity, which brings about the observed nonlinear softening. in v-notched rail shear tests on cross-ply laminates reinforced with hs fibers totry et al. [109] did not find any evidence of damage in the mtm57 epoxy resin after a shear deformation of 25%. if the same resin was reinforced with hm fibers, however, intraply damage occurred at γ12=15%. it seems rather unlikely that such large strains can appear without any damage. for laminates out of glass-fiber reinforced epoxy giannadakis and varna [110] determined viscoelasticity and viscoplasticity as the major cause for nonlinearity, whereas the effect of microdamage is very small. until verifying what really happens in the shear tests it seems to be unreasonable to invest further effort into modeling it. 4.6. strength under combined loading micromechanics were also used for strength prediction under combined loading. the influence of interface strength on the composite behavior under out-of-plane shear and transverse tension was studied by canal et al. [111]. they concluded that homogeneous models like those proposed by hashin or puck cannot accurately predict the failure surface. transverse compression and out-of-plane shear was analyzed by totry et al. [112], which led to the finding that interface decohesion must be taken into account for composites in matrix-dominated failure modes. also, for transverse compression and longitudinal shear totry et al. [113] discovered that the interface strength plays an important role for the composite strength. fig. 5 shows some of their results which compare quite well with the tests, especially the strength increase due to internal friction comes out nicely. ha et al. [114] proposed a micromechanics based model which used the maximum stress criterion for ff, a modified von mises yield criterion for matrix failure and a simple quadratic criterion for failure of the fiber–matrix interface. in order to simulate the tasks of the wwfe-ii huang et al. [115] complemented these criteria with a progressive damage model taking care of the nonlinear matrix behavior. a damage factor of 0.4 was assumed for final rupture of the damaged material. huang et al. [116] further adapted the approach to the test results by using a quadratic ff criterion, a fiber kinking model, and a reduction of stress amplification factors for inplane shear terms. melro et al. [117, 118] developed an elasto-plastic damage model suitable for epoxy matrix material which accounts for different behavior under transverse tension, transverse compression, and longitudinal as well as transverse shear. fig. 5 micromechanic strength analysis after troty et al. [113] 14 k. rohwer 5. conclusions and outlook considerable effort has been put into the development of suitable models to reliably predict damage and failure of fiber composites. in spite of the inhomogeneity of the material homogeneous models were first choices for quite some time. they have developed from simple maximum stress or strain criteria via interpolation criteria to physically based ones. on looking at the frequency of publications in this field the development seems to have passed the top. there are quite a number of them available now. what is missing, however, is a reliable statement as to which one should be applied in the respective case at hand. damage mechanics accounts for the residual strength after initial damage. in general that is done by stiffness reduction smearing out local effects and therewith simulating a material nonlinearity of the affected layer. there are indications that interactions between adjacent layers can have a considerable influence on the laminate strength, which also can be accounted for by means of damage mechanics models. closer to the behavior of fiber composites are heterogeneous models. talreja [119] has carefully analyzed ambiguities and uncertainties in classical failure predictions and provided remedies to overcome them, including a comprehensive analysis strategy. chowdhury et al. [120] have compared the reliability of matrix failure prediction between criteria at the lamina level and a micromechanical approach and found out that the latter is more accurate. the greater computational effort required with heterogeneous models is no longer a major handicap thanks to the rapid increase of computational power and storage capacity. it is more the difficulty to determine relevant material properties. that especially holds if the model considers an interface layer between fibers and the matrix. inverse methods cannot be considered as the general solution to that problem since they require the choice of a micromechanical model in the first place. compressive strength in fiber direction has attracted special attention. however, the role of kink band formation, which is observed in the failure process, seems to be not thoroughly understood. all in all, it must be concluded that models for predicting fiber composite damage and failure have not yet reached a fully 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exoskeletons udc 617.58-77 aleksandr poliakov 1 , vladimir pakhaliuk 1 , nikolay lozinskiy 2 , marina kolesova 1 , pavel bugayov 1 , petro shtanko 3 1 sevastopol state university, sevastopol, russian federation 2 st. petersburg state technological institute, st. petersburg, russian federation 3 zaporizhzhya national technical university, zaporizhzhya, ukraine abstract artificial knees play an important role in transfemoral prostheses, lower extremity exoskeletons and walking robots. their designs must provide natural kinematics, high strength and stiffness required in the stance phase of gait. additionally, modern artificial knee is the principal module by means of which the prosthesis control is performed. this paper presents a prototype of an artificial polycentric knee, designed on the basis of the hinge mechanism with cross links. in order to increase strength and stiffness, the elements of the joint have curved supporting surfaces formed in the shape of centroids in relative motion of links of the hinge mechanism. such construction is a mechanical system with redundant links but it allows for providing desirable characteristics of the artificial knee. synthesis of the hinge mechanism is made by a method of systematic study of the parameter space, uniformly distributed in a finite dimensional cube. stiffness of bearing surfaces elements of knee was determined by solving the contact problem with slippage of surfaces relative to each other. key words: transfemoral prosthesis, artificial knee, polycentric hinge mechanism, contact problem with slippage surfaces 1. introduction in the late 19 th century, lesgaft [1] gave an accurate description of joints of living organisms, according to which "... by the analysis of the form, you can define all existing movements in the joint, and vice versa, by movements observed in the living, it is received october 18, 2016 / accepted november 25, 2016 corresponding author: aleksandr poliakov sevastopol state university, universitetskaya str., 33, 299053, sevastopol, russian federation e-mail: a.m.poljakov@sevsu.ru 322 a. poliakov, v. pakhaliuk, n. lozinskiy, m. kolesova, p. bugayov, p. shtanko possible to mathematically determine the form, lying at the base of the movement". from this viewpoint, the knee joint is considered as one of the most complicated and sophisticated human joints. artificial knee joints (ak) play an important role in man-machine systems such as transfemoral prosthesis, exoskeletons of lower limbs, two-legged robots, etc. their structure firstly must provide natural walking of a person or a device similar to a person. however, up to now a reliable ak which contributes to the implementation of biologically natural movement has not been created yet. this is due to many reasons and, in particular, to genetic factors. for example, it was noted in [2], that the natural knee joint (nk) has 16 critical characteristics, which (according to the theory of evolution) were caused due to random and rare genetic errors, called mutations. in this regard, nk represents the so-called irreducible mechanism. elimination or modification therein at least of one of critical characteristics leads to serious problems. therefore, designing of an ak which is fully similar to the nk is only possible in the case of providing therein all critical features, which, however, is virtually impossible due to many objective reasons. nevertheless, a large number of attempts have been made to create ak, which allows to approximately reproduce nk functions. here, first of all, we should note the use a polycentric ak (pak) in the transfemoral prosthesis instead of a single axis ak (sak). it allows us to bring closer the gait of disabled person to the natural one and to provide a necessary clearance between the artificial foot and the ground for a safe walking. but, as is well known, conventional designs of paks and their characteristics differ significantly from nk [3]. the basic kinematic difference consists in the form of centroids in relative motion of femur and tibia. centroids of nk are fully placed inside the joint, while centroids of conventional pak – outside the joint. this suggests that relative movements of femur and tibia in nk and pak are (in most cases significantly) different. more natural movements may be obtained in cases of paks whose structures make use of hinged mechanisms with cross links like cruciate ligaments (cl) in nk [4, 5]. such paks can be considered from different points of view as most promising, primarily because they are able to provide the kinematics of elements limb movement similar to that of biologically natural ones. however, paks with cross links are characterized by the worst stability in stance phase as compared with conventional ones. this disadvantage can be eliminated by means of obligatory use of real-time controlled dampers. it should also be noted that the cross links in paks can only partly be considered as analogs of cl as their deformations during joint work are negligible in comparison with the cl deformations. at the same time, their role in the ak is the same as that of cl in nk. however, this fact indicates that any pak with cross links can be only partly considered as analogous to nk. the main goal of this work is to create a reliable pak, capable of providing kinematics of femur and tibia in relative motion close to the biologically natural. in this sense, such ak can be considered as a biosimilar joint. biosimilar artificial knee for transfemoral prostheses and exoskeletons 323 1. material and methods the optimal synthesis of the polycentric mechanism with cross links was made taking into account the proximity of the movable centroid of ak to the desired centroid. as the desired centroid the surface of condyles medialis femoris has been selected (fig.1a). a) b) fig. 1 conditional centroids in relative movement femur and tibia (a); scheme of the designed mechanism in the initial and final configurations (b) the second synthesis criterion takes into account the proximity of trajectories of basic points, selected arbitrarily on femoral components of ak and nk, respectively. in fig.1b, a kinematic scheme is shown which was used to calculate kinematic parameters satisfying criteria formulated above. mechanism synthesis is made on the basis of the method of systematic probing of the parameter space on uniformly distributed net points in a multidimensional cube [6]. the methodology of synthesis is described in [7]; it is not presented in detail in this paper. as the first approximation, an artificial knee joint was constructed in which the load is transmitted from the femoral element to the tibia element only through links of hinge mechanism. but, as the finite element analysis shows, it is virtually impossible to realize such design, with account of requirements of reliability. this is due to the fact that the increasing sizes of links, in order to ensure their strength and rigidity, leads to the undesirable increasing of sizes of the whole artificial knee. therefore, the contact surfaces transmitting the main part of the load were introduced in the design. designed biosimilar pak using magnetorheological damper with real-time adjustable stiffness is a mechatronic system. the basic principles of its control system as a part of transfemoral prosthesis are presented in [8, 9]. general view of the pak is shown in fig. 2. 324 a. poliakov, v. pakhaliuk, n. lozinskiy, m. kolesova, p. bugayov, p. shtanko fig. 2 general view of biosimilar artificial knee with cross links and bearing surfaces 3. results and discussions it is known that the femur, relative to the tibia, performs a motion in the sagittal plane including an axial rotation, rolling and sliding [10]. in the considered case, one of nk contact surfaces is convex (femoral condyle), while the other is concave (tibial condyle). at the same time, both pak centroids are convex and during joint flexion/extension roll relative to each other without sliding. this is a significant difference of the pak from the nk. the hinged mechanism that underlies the pak provides biosimilar joint kinematics, but its links, when loaded, show significant deformations. in order to reduce cross links deformations, the load in pak, for the most part, is transferred from the femur to the tibia directly via bearing contact surfaces, following the centroids in relative elements movements. this design allows for providing the required joint stiffness without altering its kinematic characteristics. the material of lower bearing surface is titanium alloy with young's modulus e1 =1.5·10 5 mpa and poisson's ratio ν1=0.3; the material of upper bearing surface the titanium alloy coated with hard rubber of thickness δ=0.005 m with e2 =15.0 mpa and ν2=0.48. such choice of materials is to ensure permanent contact in the kinematic pair and to prevent of backlashes in swing phase of limb. on the other hand, the rigidity of bearing surfaces needs to be selected so that it does not impede the flexing/extension of the joint. in order to ensure these conditions, the pak has special a load device, which allows for pre-deforming of the soft surface, by introducing into it a rigid surface by a small amount δ. according to the hertz theory [11], a half of total width of the contact area, approximated as a rectangular, can be determined by the formula: biosimilar artificial knee for transfemoral prostheses and exoskeletons 325 1 2 1 2 1 2 4 ( )p k k r r a r r    , (1) where p is the loading per unit of length of the contact line, 2(1 ) /i i ik e   and ri are the radii of curvature in the contact point of surfaces, i=1,2. for example, for the preliminary load p=7500 n/m, r1=0.07 m, and r2=0.06 m, we will obtain: a=0.0043 m. the pak load device by kinematic approach δ allows us to provide various areas s of the contact surfaces. despite the fact that bearing surfaces of the ak have a complicated shape with variable curvature, in the zone of action of maximum forces in stance phase of invalid gait, they can be approximately regarded as cylinders with constant radii of curvature r1 and r2. in that case the problem is reduced to the contact interaction of elastic and rigid cylinders. it is well known that relative tangential displacement of contacting partners leads to appearance of a slip at the outer border of the contact while the inner part of the contact remains in the stick state. with increasing the macroscopic relative displacement, the sticking area shrinks until the state of complete sliding (gross slip) is achieved. the detailed stress distribution determines the frictional losses and is also one of the factors determining the rolling resistance. to assess mc in the first approximation we assume, that in the process of convergence of the contact surfaces, the distribution of normal pressure in perpendicular direction to the contact area is changed parabolically (fig. 3) [12]: 2 2 0 ( )nq q a x  , (2) where bsa f q n 30 8 3  , fn is the normal force applied to the thigh element of ak (including the force providing preliminary deformation), b is the half length of a rectangular contact area, and s the intensity coefficient of the contact surface (in this case taken s=1). note that the true stress distribution in the hertz-contact is proportional to 2 2a x . however, as is shown in the method of dimensionality reduction (mdr) [13], the tree-dimensional contact problem can be mapped to a one-dimensional one with the parabolic normal stress distribution of the form (2). the following estimation can thus be understood in the sense of the mdr mapping. fig. 3 loading scheme of the ak contact surfaces 326 a. poliakov, v. pakhaliuk, n. lozinskiy, m. kolesova, p. bugayov, p. shtanko assume that the tangential displacements are described by linear relationship: ( )u a x  , (3) where 01 2 2  r rc  and  22 rrc is the dynamic radius of the elastic cylinder [12]. then the specific tangential force is: ( )q u x a     , (4) where λ is the coefficient of tangential stiffness of the elastic cylinder. from the contact geometry follows that λ lies in the range:    020 qa  . (5) if we assume that: nqq   , (6) where µ is the coefficient of dry friction, 0 lim qq   , then the coordinate, which separates regions of clutch and friction, is determined by the following expression [14]: 0q axb    . (7) since xb  –a, then the moment on the hard cylinder caused by the action of tangential forces can be represented as follows: 1 lim 11 2             a x x a c b b dxqdxqbrmm  . (8) after integration of eq. (8), we obtain: 2 3 2 3 1 0 1 2 ( ) (2 3 ) 2 3 c b b bm br a x q a a x x             , and taking into account eq. (6): 2 2 2 2 2 1 0 0 2 2 0 (12 18 5 )1 3 c br a q a q m q          . (9) by substitution the actual loading data, the given characteristics of materials, the value of the external load and the inequality (5) into eq. (9) we can verify that the moment of rolling resistance with slippage of surfaces relative to each other is of the order 10 -6 nm and has virtually no influence on the performance of the ak. another factor that may potentially affect the resistance of relative motion of the ak elements is a hysteresis in the material of the elastic bearing surface. if we consider the deformation of surface f(x) and the rate of its change df(x) / dt = df(x)v / dx, then for the material with a coefficient of hysteresis losses β, the energy loss due to hysteresis can be defined by the expression: biosimilar artificial knee for transfemoral prostheses and exoskeletons 327 ( ) 2 a h n a df x p b q vdx dx      . (10) if we take the parabolic law given in eq. (2), then:            2 2 1 a x xf  . (11) after substituting eq. (11) into eq. (10) and integrating, we obtain:                                       a a a avqab ph           0 4 1 0 4 1 0 4 1 0 4 1 4 2 2 2 2 0 2 . (12) for realistic values of parameters of an ak, the eq. (12) leads to an estimate: vph 6108.2  . (13) thus, it could be argued that for all possible values β and v, the energy lost due to hysteresis is very low. 4. conclusions the designed biosimilar artificial knee joint shows all the desirable features inherent to a natural knee joint. its main advantage is the reproduction of the natural kinematics. at the same time, it has high strength, high stiffness and a compact design. the ak kinematics provides hinged mechanism with cross links like cruciate ligament; the strength and stiffness is provided due to the transfer of the external load through contact surfaces. in the ak design an auxiliary device is used that allows adjusting the pre-load required for deforming one of the surfaces. in this study it is shown that such load has no significant effect on the ak performance but it ensures consistency of the kinematic pair and elimination of possible backlashes. accomplished estimates are made for very approximate models. however, it is expected that a more accurate simulation will lead to similar results. this is supported by preliminary experiments carried out on the ak model which was produced by 3d printing method. acknowledgements: this work has been funded by the ministry of education and science of the russian federation in the framework of the base part of state order in the field of scientific activity with the registration no. 115041610028. 328 a. poliakov, v. pakhaliuk, n. lozinskiy, m. kolesova, p. bugayov, p. shtanko references 1. lesgaft , p.f, dolbnya, i.p., 1887, theory of simple joints, proc. of the st. petersburg biol lab, 2(2), pp. 22-44. 2. burgess, s., 1999, critical characteristics and the irreducible knee joint, journal of creation, 13(2), pp. 112-117. 3. michael, j.w., 1999, modern prosthetic knee mechanisms, clin orthop relat res, 361, pp. 39-47. 4. etoundi, a.c., vaidyanathan, r., burgess, s.c., 2012, a bio-inspired condylar knee joint for leg amputees and for knee implants, wit transactions on ecology and the environment, 160, pp. 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pavlović 1 , ratko pavlović 1 , goran janevski 1 , nikola despenić 1 , vladimir pajković 2 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of mechanical engineering, university of montenegro, podgorica, montenegro abstract. this paper investigates the almost-sure and moment stability of a double nanobeam system under stochastic compressive axial loading. by means of the lyapunov exponent and the moment lyapunov exponent method the stochastic stability of the nano system is analyzed for different system parameters under an axial load modeled as a wideband white noise process. the method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. key words: double nanobeam, lyapunov exponent, moment lyapunov exponent, regular perturbation method, white noise process 1. introduction continuum modeling of nanorods, nanobeams and nanoplates has attracted attention due to its simplicity and computational efficiency. recently, they have been extensively utilized as nanostructure components for nanoelectromechanical and microelectromechanical systems. based on the theory of eringen, nanostructures have been widely studied by many researchers. the bending analysis of embedded nanoplates based on the integral formulation of the eringen’s nonlocal theory is presented in [1]. zhang et al. [2] developed a hencky barnet model for circular and annular single layer graphene sheets and calculated real buckling loads of such sheets with clamped and simply supported restraints. pavlović et al. [3] investigated the dynamic instability of coupled nanobeams. by using the direct lyapunov method, bounds of the almost-sure asymptotic instability of a received april 15, 2019 / accepted december 10, 2019 corresponding author: ivan r. pavlović faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: pivan@masfak.ni.ac.rs 220 i. pavlović, r. pavlović, g. janevski, n. despenić, v. pajković coupled nanobeam system were determined as a function of different system parameters. an accurate and analytical method for investigating the dynamic instability of nanobeams based on the nonlocal continuum mechanics is given in [4]. using the same beam theory, jena and chakraverty [5] investigates free vibration of nanobeams by applying the semi analytical-numerical technique called differential transform method (dtm). the effects of the compressive axial load on the properties of forced transverse vibrations of a double-beam system were investigated by zhang et al. [6]. a systematic study of the moment lyapunov exponents is presented in the works of xie [7, 8]. the method of regular perturbation was applied to obtain weak noise expansions of the moment lyapunov exponent. by using the moment lyapunov exponents, bounds of the almost-sure stability were determined for bounded noise [7] and real noise [8] parametric excitation. kozić et al. [9] investigated the lyapunov exponent and the moment lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation. the results from this study were further used for the study of the almost-sure and moment stability of a double-beam system under stochastic compressive axial loading. similarly, the moment lyapunov exponents of the stochastic parametrical hill’s equation were investigated by the same authors in [10]. pavlović et al. [11] used this method to compare the analytically obtained results of the moment lyapunov exponent for a simple nanobeam numerically determined for the same system. within the concept of the lyapunov exponent, by using the stochastic averaging method, the dynamic stability of a viscoelastic double-beam system under parametric excitations was investigated in [12]. the effect of the van der waals forces on axial buckling of a double-walled carbon nanotube was presented in [13], where an elastic model was used to study infinitesimal buckling of a double-walled carbon nanotube under axial compression. the present paper investigates the almost-sure stability of a double nanobeam. in section 2 the governing differential equations of transverse motion of two elastically connected nanobeams are presented. using the galerkin method the system is discretizated in section 3, where the analyzed system is reduced to two ordinary differential equations representing only the time varying part of the solution. for the obtained discretizated form, the appropriate analytical expression of the moment lyapunov exponent is given in section 4. the lyapunov exponents are then obtained using the relationship between the moment lyapunov exponents and the lyapunov exponent. in section 5 numerical results and stability analysis for different system parameters are presented along with an appropriate conclusion. concluding remarks are given in section 6. 2. analyzed system following the eringen’s nonlocal elasticity theory [14] and the euler–bernoulli beam theory, the partial differential equations of the two elastically connected nanobeams are given by [3] 0)()( 4 1 4 11212 1 2 1 1 12 1 2 11                      x w iewwk x w tf t w c t w al , 0)()( 4 2 4 22122 2 2 2 2 22 2 2 22                      x w iewwk x w tf t w c t w al , (1) dynamic behavior of two elastically connected nanobeams under a white noise process 221 where i and ai are the mass densities and the cross-sectional areas of the beams, k is the stiffness of the elastic medium, wi= wi(x,t) are the transverse displacements, x is the axial coordinate, t is the time, ci are the viscous damping coefficients, eiii are the bending stiffnesses of the beams, and fi(t) are the time-dependent stationary stochastic processes and operator l is .)(1 2 2 2 x ea   l (2) the boundary conditions for the simply supported edges are . x w ,w, x w ,w lx x 0000 0 2 2 2 22 1 2 1             (3) following the work of murmu and adhikari [15], we assume that both nanobeams are identical ρ1a1= ρ2a2= ρa, e1i1= e2i2= ei, c1=c2=c. (4) now, the system given with expression (1) is non-dimensionalized using the following parameters ei a lklxxlwwkt,t tii 2,,  , )2,1(,))(( 20  i a lk k, al f tff, a ck 2 4 i ii 1/2t       , (5) where ε is a small fluctuation parameter. after applying (5) in (1) the following nondimensional form of eq. (1) is obtained 0)())((2 4 1 4 212 1 2 101 2/11 2 1 2                   x w wwk x w tff t w t w l , 0)())((2 4 2 4 122 2 2 202 2/12 2 2 2                   x w wwk x w tff t w t w l , (6) and ,,1 2 2 2 l ea x     l (7) where a and e are the characteristic length scale and the nondimensional constant, respectively. 222 i. pavlović, r. pavlović, g. janevski, n. despenić, v. pajković 3. discretization of the equations of motion now, by using the galerkin method system, (6) is reduced to two ordinary differential equations representing only the time varying part of the solution. the first mode of the transverse motion of the beams can be described by 21sin)()( ,ix,tqtx,w ii   . (8) by substituting (8) in (6), the following discretizated form of equations (6) is obtained 0)()(2 1 2 1 2/1 1 2 211  qtfqqqkqq1  , 0)()(2 2 2 2 2/1 2 2 122  qtfqqqkqq2  , (9) where εβ represents the small viscous damping coefficient, ε 1/2 fi(t) is the white-noise process with small intensity and 22 4 2 1      . (10) 4. lyapunov exponents and stability conditions basic definitions of stochastic stability are the sample or almost-sure stability and the stability in the mean of the p-th order, which is based on the concept of the lyapunov exponent given in arnold et al. [16]. the almost-sure stability is described by the maximal lyapunov exponent defined as );(ln 1 lim 0qtq tt q   , (11) where q(t;q0) is the solution process of a linear dynamic system. it gives the exponential growth rate of the solution. if λq < 0, then, by definition, 0);( 0  p qtq as t, the solution is almost-surely stable, and λq >0 implies the instability of the solution in the almost-sure sense. the exponential growth rate, p qtqe );( 0 , is provided by the moment lyapunov exponent defined as p t q qtqe t qpλ );(ln 1 lim),( 00   (12) where e denotes the expectation. if 0),( 0 qpλq , then by definition,  p qtqe );( 0 as t, and those conditions are referred to as the p-th moment stability. the moment lyapunov exponent provides us with finer stability properties of the random dynamic system. to have a complete picture of the dynamic stability of a stochastic system it is important to study both the sample and moment stability. dynamic behavior of two elastically connected nanobeams under a white noise process 223 following the results from the authors’ previous works [9-11], where the regular perturbation method is performed, the analytical expression for the moment lyapunov exponent is             p pp pλ 24 264 )103( )( , (13) where σ 2 is the intensity of the white noise process. using a property of the moment lyapunov exponent, the lyapunov exponent of the double nanobeam system (3) is            24 2 0 32 5)( p dp pdλ . (14) 5. numerical results this paper investigates the dynamic stability of two elastically connected nanobeams under white noise excitation. according to the lyapunov exponent and the moment lyapunov exponent method, bounds of the almost-sure stability of the observed nanosystem are presented for different system parameters. when the lyapunov exponent is negative, system (1) is almost-surely stable with probability 1. firstly, according to equation (13), numerical results are presented in the plane of the moment lyapunov exponent (p) and the norm degree p. figs. 1 and 2 present the almostsure and p-th moment stability boundaries in the function of the damping coefficient β and parameter ε, respectively. figs. 3 and 4 show the boundaries of the almost-sure stability for (p)=0. firstly, in fig. 3 in the plane of σ 2 and β the bounds of the almost-sure stability are given in the function of the norm degree p. as shown in the previous figures (fig. 1 and fig. 2), this parameter growth leads to the system instability. similarly, in fig. 4 the reduction in stability regions is presented for different values of the nanoscale coefficient. this figure is given in the function of the damping coefficient. as already presented in fig. 1, this parameter growth significantly enlarges stability regions. thus, a negative influence on the system stability produced by the nonlocal parameter growth can be compensated for higher values of the damping coefficient. at the end of this study, numerical results of the bounds of the almost-sure stability are given according to the lyapunov exponent λ presented by (14). the results from fig. 5 are given in the plane of nanoscale coefficient μ and the intensity of white noise process σ 2 , in the function of damping coefficient β. as in fig. 4, where numerical results were obtained according to the moment lyapunov exponent, it can be seen that an increase in the nanoscale coefficient is followed by a remarkable reduction in stability regions especially for smaller values of the damping coefficient. 224 i. pavlović, r. pavlović, g. janevski, n. despenić, v. pajković fig. 1 stability regions in the function of damping coefficient β fig. 2 stability regions in the function of small fluctuation parameter ε dynamic behavior of two elastically connected nanobeams under a white noise process 225 fig. 3 stability regions obtained from the moment lyapunov exponent (14) for (p)=0 in the function of norm degree p fig. 4 stability regions obtained from the moment lyapunov exponent (14) for (p)=0 in the function of damping coefficient β 226 i. pavlović, r. pavlović, g. janevski, n. despenić, v. pajković fig. 5 stability regions obtained from the lyapunov exponent result (14) for λ = 0 in the function of damping coefficient β 5. conclusion by means of the moment lyapunov exponent, the stochastic stability of two elastically connected nanobeams under white noise excitation is studied in this paper. the regular perturbation method is applied to obtain analytical results for the moment lyapunov exponents and lyapunov exponent in terms of small fluctuation parameter ε. moment lyapunov exponents are important characteristic numbers for describing the dynamic stability of a stochastic system. when the p-th moment lyapunov exponent is negative, the p-th moment of the solution of the stochastic system is stable. regions of almost-sure stability are obtained as a function of the moment lyapunov exponent , intensity of white noise process σ 2 , damping coefficient β, norm degree p and nanoscale coefficient μ. as in the authors’ previous studies in this field, it is confirmed that nonlocal effects significantly reduce stability regions. the numerical study shows that the negative influence of the nanoscale coefficient on stability regions can be successfully compensated with a growth in the damping coefficient parameter. finally, with the aim of comparing the results obtained from the moment lyapunov exponent with the results obtained from the lyapunov exponent method, fig. 4 and fig. 5 present the influence of the most important parameters for the system stability, β and μ. acknowledgements: this paper was supported by the ministry of education and science of the republic of serbia, through the project no 174011. dynamic behavior of two elastically connected nanobeams under a white noise process 227 references 1. ansari, r., torabi, j., norouzzadeh, a., 2018, bending analysis of embedded nanoplates based on the integral formulation of eringen's nonlocal theory using the finite element method , physica b: condensed matter, 534, pp. 90-97. 2. zhang, h., wang, c.m., challamel, n., pan, w.h., 2020, calibration of eringen's small length scale coefficient for buckling circular and annular plates via hencky bar-net model, applied mathematical modelling, 78, pp. 399-417. 3. pavlović, i., pavlović, r., janevski, g., 2016, dynamic instability of coupled nanobeam systems. meccanica, 51, pp. 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axial instability of double nanobeam-systems, physics letters a, 375(3), pp. 601–608. 16. arnold, l., doyle, m.m., sri namachchivaya, n., 1997, small noise expansion of moment lyapunov exponents for two-dimensional systems, dynamics and stability of systems, 12(3), pp. 187–211. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 507 516 https://doi.org/10.22190/fume170920028z original scientific paper influence of the changing local climate on wind potentials of mount kopaonik udc 621.7 predrag živković 1 , mladen tomić 2 , dragana dimitrijević 1 , ivana kecman 3 , mirko dobrnjac 2 1 university of niš, faculty of mechanical engineering, niš, serbia 2 university of novi sad, faculty of technical sciences, novi sad, serbia 3 university of banjaluka, faculty of mechanical engineering banjaluka, republic of srpska, bosnia and herzegovina abstract. obtaining all acceptable locations is one of the main tasks for siting wind turbines. the economic factors are usually very limiting. very thorough analyses are needed in order to ensure the project finalization. nevertheless, even after all the steps are made, some problems may occur. one of them is the real status of the winds in the so-called climatology period. this paper focuses on the influences of the changing winds after the preliminary estimations are done. the estimations are obtained using the wasp simulation software. the results are compared in terms of quality and quantity of the wind data and capacity factor. finally, an economic analysis is done. key words: wind power assessment, complex terrain, wasp, climatic changes 1. introduction energy, especially electrical, is of vital importance in the world today. many assessments of fuel resources, mostly fossil, clearly indicate that these resources, especially oil ones, are almost exhausted. the need for energy constantly rises so that the introduction of new resources is inevitable. all these facts point to the necessity of transition to the sustainable development, especially to the usage of renewable energy sources. in that sense, wind energy clearly takes a prominent place considering its large potentials, purity and availability. the present constrains are mostly of financial nature. the most important task is the siting of wind turbines (obtaining the best possible locations for installing the turbines, considering received september 20, 2017 / accepted november 28, 2017 corresponding author: predrag ţivković university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, niš, serbia e-mail: pzivkovic@masfak.ni.ac.rs 508 p. ţivković, m. tomić, d. dimitrijević, i. kecman, m. dobrnjac the possibility for energy production and minimization of losses). for that purpose, the wind atlas method is developed as an easy one for usage in view of the rapid computer development. the position of a wind turbine is in strong correlation with energy production. according to the previous research [1-3], linear models cannot estimate correctly wind energy potentials in the terrain where the ruggedness index (index that represents the terrain slope value) exceeds 0.3. in such a case, using full cfd models, followed by experimental validation is necessary. even if all of the above mentioned is satisfied, there is still a possibility that the wind farm is not going to have the predicted output during the project lifetime, which is 25-30 years, which is about the same as the climatic period, which lasts for 32 years. a number of researchers have estimated serbia’s wind potentials [3-6]. important research was done on atmospheric turbulence [7] and the methods of measuring wind potentials in the complex mountainous terrains of serbia [8]. there are some papers considering the wind influence on other renewable energy sources [9], energy storage [10, 11] and advanced simulation models [12, 13]. 2. mathematical model cfd models are more precise but they need much more computational time. considering the need to obtain the results as soon as possible, the best micro model was extracted from the larger macro model using the fast linear software [14]. then the best wind turbine locations were obtained by using cfd software[15]. 2.1. linear model a linear model is expressed by the following set of equations:  continuity equation ( ) 0i i u x     (1)  logarithmic vertical wind profile            0 * ln z zu u z (2)  weibull distribution equations 1 ( ) exp k k k u u f u a a a                   (3) ( ) exp k u f u a            (4) where xi are the coordinates in index notation, ρ air density, ui are the velocity components, uz is the vertical wind speed component at height z above ground level, u* is the friction velocity, κ is the von kármán constant (κ = 0,4), z the height above ground level, z0 is the surface roughness length, ψ the empirical function, k and a are factors for the weibull influence of the changing local climate on wind potentials of the mountain kopaonik 509 distribution, f is the frequency of the wind speed occurrence and f is the cumulative weibull distribution. representative of the linear software packages is wasp [14]. it calculates the speedup effects of the hills, taking into consideration the effect of energy redistribution in the flow from the component in the flow direction to the vertical component. 2.2. nonlinear model a nonlinear model solves the full set of governing equations of the steady fluid flow by using:  continuity equation: ( ) 0i i u x     (5)  momentum equations: ii j j i eff jj i j x p x u x u xx u u                        1 (6)  turbulence model equations:                              k jk t jj j p x k xx k u (7)          21 cpc kxxx u k j t jj j                          (8) where: j i i j j i tk x u x u x u p                 (9) teff   (10)   /2kct  (11) where p is the body acceleration, pk production of turbulence, υ, υeff and υt are the kinematic, effective and turbulent viscosity, respectively, k is the turbulent kinetic energy, ε turbulence kinetic energy dissipation. the modified set of model coefficients reads: cμ=0,0324, cε1 =1,44, cε2 =1,92, σk=1,0, σε=1,85. the set of the nonlinear partial differential equations is solved by the windsim [15] software package. 510 p. ţivković, m. tomić, d. dimitrijević, i. kecman, m. dobrnjac 3. combined methodology cfd models are more precise but they need much more computational time. considering the need to obtain the results as soon as possible, the best micro model was extracted from the larger macro model. the differences in wind energy estimations while using these different approaches are considerable. many investigations were done on this subject, dealing with different aspects of the software operation. test model of mount seliĉevica [3] was chosen due to its adequate orography as can be seen in fig. 1. it is shown that the wasp predictions are about 30% larger than windsim [15] ones (estimated wind speed is in range 7,75-15,54m/s for wasp, and 4,9612,64m/s for windsim, as shown in fig. 1), due to neglecting of the second-order terms in the momentum equations, eq. (6). fig. 1 mean wind speed fields obtained by simulations in wasp (left) and windsim (right) for obtaining the results the nesting technique is used. it assumes the boundary conditions in the initial simulation to be the terrain on the bottom, geostropic wind on the top and logarithmic wind profiles on the lateral sides of the domain at the macro level. reducing the size of the domain in the following simulations, the lateral boundary conditions are results of the previous simulation, and not the theoretical profile, which significantly increases the quality of the results for the final micro location. simulations were done for the enercon e82 wind turbine. it is very appropriate to use wasp as the initial software at the mezzo level estimations, and windsim for more precise micro level estimations, as the computational time for wasp is about 20 times less than for windsim. in the previous papers [4, 5] results obtained by numerical simulation on over a dozen micro locations are presented. the considered locations mainly cover the mountainous regions of southern and eastern serbia. in this paper, the influence of the changing wind potentials from the main meteorological station kruševac on the potentials of mount kopaonik was investigated. size of the mezzo model was about 5000km 2 . although there is a meteorogical station on mount kopaonik, kruševac station (about 50 km away, fig. 2) was chosen for having more data available for analysis. another influence of the changing local climate on wind potentials of the mountain kopaonik 511 reason was that, as it is known that the harsh climate conditions on kopaonik can cause the cup anemometer to freeze, the quality as well as the quantity of the data in the kruševac station is better. 4. kopaonik wind potentials mount kopaonik (also known as silver mountain) region is situated in the central southern part of serbia. it is the largest mountainous region in serbia, stretching in a northwest-to-southwest direction for about 75km, with the widest part in the mid-section for about 40km. a large part is conserved as the kopaonik national park. it is the largest skicenter in serbia. the highest altitude is panĉić's peak at 2017m asl. there is a large highland area with many peaks, as karaman (1934m), gobelja (1834m), etc. south of panĉić's peak, there are many single rises, namely, ĉardak (1590m), šatorica (1750m), oštro koplje (1789m). with about 200 sunny days annually, kopaonik is also called sunny mountain. its southern, high and open position averts the clouds, keeping the cold air in the surrounding low areas, so that the winter temperatures are not too low, which is very important considering the possibility of wind turbine blade frosting. average yearly temperature is 3,7°c. snow lasts from mid-november to may, about 160 days annually. precipitation levels are over 1000mm annually. the kopaonik ski center has about 62 km of tracks and 25 ropeways with overall capacity of 32000 skiers per hour. chosen wind turbine type is enercon e-82, with unit power of 2mw. considered micro model was chosen by former simulation on the bigger model, from which, using the nesting technique, the named model is obtained. for the turbine siting the method of wake loss minimization and maximal annual energy production was used. also, the recommendations about distance between wind turbines for the siting were as follows: in the wind direction minimally 7d (d – rotor diameter) and in the normal direction 4d. fig. 2 macro model with wind rose (left) and weibull distribution (right) 512 p. ţivković, m. tomić, d. dimitrijević, i. kecman, m. dobrnjac after the simulation is done, using the nesting technique, the micro model is obtained, and the field of annual energy production (aep) is presented in the following figure (range being 4,483-13,087 [gwh]): fig. 3 nesting technique (left) and aep field with 15 turbine wind farm disposition (right) from fig. 2 one can notice that only the best locations are considered though there are possibilities for more locations to choose. table 1 aep data for the 15 turbine kopaonik wind farm parameter total average min max gross aep [gwh] 188,79 12,59 12,26 12,99 net aep [gwh] 188,65 12,58 12,26 12,99 wake loss [%] 0,08 stable wind data (for 15 year period, minimum needed is 13 years) were obtained from the main meteorological station kruševac. as the turbine acceptable wind speed is in the range of 2÷25m/s, only 38,16% of the wind data are in the acceptable range. considering this, as well as the overestimation by the used software, the capacity factor (ratio between possible aep and max aep) is calculated to be cp=0,21. it is considered that the economically acceptable locations are with cp=0,25 or higher. on the basis of the data obtained by simulation, and the known turbine power, capacity factor cp can be calculated for the considered wind farm. the capacity factor is the ratio of the overall power output of a wind farm to its nominal output (if working on the full potential) for a period of time or for the case considered, annually. this can be presented in the form: 318760 38160 ,pt ,aep cp    (12) where pt is installed power of the farm with 15 wind turbines enercon e-82, 0,3816 is the percentage of accepted data (all the others can be considered as calm, i.e. <2m/s) and factor 1,3 is as the wasp software overprediction, as shown earlier [3]. influence of the changing local climate on wind potentials of the mountain kopaonik 513 5. changing local climate after the preliminary analysis was done [6], the wind data was obtained for the period 2014-2016. for this period the wind potentials were estimated for each annual data in order to compare the potentials of each year in the mentioned period. results are presented in the form of the wind roses and weibull distributions for each year, as well as the aep for the same farm, for each year. a) b) c) d) e) f) fig. 4 wind roses (a, b, c) and weibull distributions (d, e, f) for seasons 2014, 2015 and 2016, respectively one can notice that there is a slight, but notable difference in the wind data. the slight decrease of the potentials can be noted as is presented in the fig. 4.d-f. there is also a change of the wind speed distributions, shown on the wind roses, fig. 4.a-c. in fig. 5, aep fields for each year 2014, 2015 and 2016 are presented. one can notice that not only the potentials have slightly decreased, but even the best locations are slightly shifted to the east. as it is impossible to move the turbines after the installation, this shift is not going to be discussed in more detail. 514 p. ţivković, m. tomić, d. dimitrijević, i. kecman, m. dobrnjac a) b) c) fig. 5 aep fields for the year of: a) 2014, b) 2015 and c) 2016 it is already mentioned that capacity factor cp was 0,21 for long-term measured data. in the similar manner, the annual values for cp can be calculated for the considered period 2014-2016. the summary results are presented in table 2. table 2 capacity factor cp change for the period 2014-2016 period long term 2014 2015 2016 cp [-] 0,2056 0,2197 0,1854 0,2041 accepted data [%] 38,16 50,70 44,06 47,44 there are some fluctuations but the decreasing trend is obvious. the differences are mainly in the region of the lower wind speeds, which are increased, but the overall decrease is up to 10%, compared to the long-term data. in order to improve the predictions in the presented research, validation of the data should be done by on-site measurements [9, 10, 11]. as the potentials are near the acceptable margin of potentials (cp~25%), there is a possibility for combining the system with solar or other types of res [12, 13], in order to increase efficiency. some new approaches to the problem of the wind farm production estimations can also be implemented [14]. 6. economic analysis economic analysis is one of the most important parts of every project. renewable energy, including wind energy, is not an exception. having in mind current prices of wind turbines, state of the global and local financial markets, and the fact that the local infrastructure is not very developed, a preliminary financial analysis was done. the initial assumptions are: the farm will operate for 25 years; initial investment is 49,5 million eur; subventions will be 10%; annual discount rate will be 10%; annual inflation will be 7%; increase of the electricity price will be 12% per annum. expected electricity price is 94eur/mw, for the period of the current feed-in tariff. the basic financial indicators, well known from the literature, such as income rate in the first year, a simple and dynamic payback time, a net present value of the investment, an internal rentability rate, a benefit/cost ratio and lifelong cost savings were chosen for analysis. the estimated financial indicators are shown in table 3. influence of the changing local climate on wind potentials of the mountain kopaonik 515 table 3 financial indicators financial indicators 2014-2016. 2014 2015 2016 rate of income (year 01) roi [%] -0,46 0,18 -1,08 -0,36 simple payback time spb [year] 16,75 15,18 18,63 16,49 net present value npv [eur] 53109782 61197811 45235053 54349251 benefit/cost ratio b/c [-] 14,09 14,68 13,44 14,18 lifelong cost savings lcs [eur/year] 5851002 6742045 4983459 5987552 using the above listed financial indicators, it is calculated that the annual income of the kopaonik wind farm could be about 2,71 million eur annually over the chosen period 2014-2016. it shows that the project payback time is over 15 years, which makes this project hardly affordable. one can notice from table 3 that the best production would be for 2014, the lowest for 2015, while slightly increased for 2016. it would be interesting to see its value at the end of the current year (2017). 7. conclusion wind energy is one of the fastest growing renewable energy resources. most of the eu members are using it widely. yet, the available usable locations are not limitless. this gives an opportunity to the less developed countries to use available funds within the scope of 20% of energy in europe to be produced by renewable sources. koponik region is relatively far from the power sources. installing wind turbines could significantly improve the quality of the energy supply in this area, which is very desirable, having in mind the extensive ski-center, the largest in serbia, and one of the largest in europe, with tendency of expanding the capacities. according to the data presented in table 2, it is obvious that the potentials of this site, which are already barely profitable, are actually decreasing. a more thorough financial analysis is needed in order to confirm such findings. such findings are very valuable, as the first wind turbines are currently being installed in the northern part of serbia. nevertheless, the excess energy production compared to the consumption in this area is almost 80% for the summer and about 60% for the winter season [8]. such excess energy can be sent to the distribution network, towards other consumers. as there is no other power source in the 100km radius, any new source can drastically improve the power supply quality, especially having in mind the mentioned ski-center. closeness of the large energy consumer as the kopaonik ski-center can reduce network losses, as the consumers are almost on-site. also, the power supply quality is very important for the people living in the area. acknowledgements: this paper is part of the work within the national program of energy efficiency, project number: tr33036, funded by the government of republic of serbia. i extend my gratitude to my dear co-mentor žarko stevanović and the colleagues from the vinča institute, laboratory for thermal engineering and energy, for the help with the simulations in both wasp and windsim, whose licences they possess. 516 p. ţivković, m. tomić, d. dimitrijević, i. kecman, m. dobrnjac references 1. 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mechanical engineering, p. 351 14. website: www.wasp.dk (last access: 10.nov. 2017). 15. website: https://windsim.com (last access: 10.nov. 2017). plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 209 218 original scientific paper adaptive neuro-fuzzy computing technique for precipitation estimation udc 911.2:556 dalibor petković 1 , milan gocić 2 , shahaboddin shamshirband 3 1university of niš, faculty of mechanical engineering, serbia 2university of niš, faculty of civil engineering and architecture, serbia 3university of malaya, faculty of computer science and information technology, malaysia abstract. the paper investigates the accuracy of an adaptive neuro-fuzzy computing technique in precipitation estimation. the monthly precipitation data from 29 synoptic stations in serbia during 1946-2012 are used as case studies. even though a number of mathematical functions have been proposed for modeling the precipitation estimation, these models still suffer from the disadvantages such as their being very demanding in terms of calculation time. artificial neural network (ann) can be used as an alternative to the analytical approach since it offers advantages such as no required knowledge of internal system parameters, compact solution for multi-variable problems and fast calculation. due to its being a crucial problem, this paper presents a process constructed so as to simulate precipitation with an adaptive neuro-fuzzy inference (anfis) method. anfis is a specific type of the ann family and shows very good learning and prediction capabilities, which makes it an efficient tool for dealing with encountered uncertainties in any system such as precipitation. neural network in anfis adjusts parameters of membership function in the fuzzy logic of the fuzzy inference system (fis). this intelligent algorithm is implemented using matlab/simulink and the performances are investigated. the simulation results presented in this paper show the effectiveness of the developed method. key words: precipitation, adaptive neuro-fuzzy system (anfis), estimation 1. introduction accurate estimation of precipitation is important for hydrologic and ecological modeling [1]. several studies on the estimation of precipitation have been carried out in the last few years. polarimetric rate estimation techniques such as zphi [2] and zzdr received october 16, 2015 / accepted january 31, 2016 corresponding author: dalibor petković faculty of mechanical engineering, aleksandra medvedeva 14, 18000 niš, serbia e-mail: dalibortc@gmail.com 210 d. petković, m. gocic, s. shamshirband [3] can be used for quantitative precipitation estimation. in addition, remotely sensed precipitation from radar and satellites can improve precipitation estimates [4-8]. quantitative precipitation estimation provides information of practical importance which can be applied to different fields such as water resources management, agricultural production, planning biodiversity, and flood protection. the precipitation estimation is one of the key elements in an assessment of the global climate change that represents a nonlinear and discontinuous process [9]. in recent years, various artificial intelligence methods have been developed for the estimation of precipitation [10-13]. artificial neural network (ann) offers a promising alternative for finding precipitation characteristics as well as learning complex nonlinear relationships. for example, zhang et al. [9] applied artificial neural network group techniques for precipitation estimation in which average errors for the overall precipitation event fall below 10%. grimes et al. [14] proposed ann with a principal component analysis to precipitation estimation on a daily timescale. freiwan and cigizoglu [15] developed ann models that were trained with the method of back-propagation algorithm to predict precipitation for the next month. chiang et al. [16] took into account the dynamic ann to construct quantitative precipitation estimation by using a three-dimensional radar data structure. nkuna and odiyo [17] and mwale et al. [18] estimated missing precipitation data by using the ann models. nastos et al. [19] developed the ann models for the prediction of the precipitation intensity for the next four months. in general, all of the cited studies have reported an improvement in performance by using anns. even though a number of models have been proposed for modeling the precipitation, they are still suffering from the disadvantages such as their being very demanding in terms of calculation time. ann can be used as an alternative to the analytical approach since it offers advantages such as no required knowledge of internal system parameters, compact solution for multi-variable problems and fact calculation. in this investigation adaptive neuro-fuzzy inference system (anfis) [20], which is a specific type of the ann family, was used to estimate precipitation. for the presently developed neural network, monthly precipitation data from 29 synoptic stations in serbia during 1946-2012 were used as case studies i.e. as anfis training data. the main purpose of this study is to analyze the performances of anfis for monthly precipitation estimation. anfis shows very good learning and prediction capabilities, which makes it an efficient tool for dealing with encountered uncertainties in any system. researchers used anfis as a hybrid intelligent system that enhances the ability to automatically learn and adapt in various engineering systems [21-26]. so far, there are many studies of the application of anfis for estimation and real-time identification of many different systems [27-35]. 2. study area and collected data the study area was serbia, which is located in the central part of the balkan peninsula. northern serbia is mainly flat, while its central and southern areas consist of hilly and mountainous. the climate of the country is temperate continental, with a gradual transition between the four seasons of the year. adaptive neuro-fuzzy computing technique for precipitation estimation 211 series of monthly precipitation data were chosen from 29 synoptic stations (fig. 1) for the period of 1946–2012. the geographical characteristics, mean and standard deviation of annual precipitation time series of the observed stations are listed in table 1. table 1 geographical description, mean and standard deviation of annual precipitation time series of the synoptic stations used in the study station name longitude (e) latitude (n) elevation (m a.s.l.) mean (mm) standard deviation (mm) 1. banatski karlovac 20°48' 45°03' 89 627.0 139.4 2. becej 20°04' 45°37' 78 579.7 130.8 3. belgrade 20°28' 44°48' 132 692.1 132.5 4. crni vrh 21°58' 44°08' 1027 789.9 154.9 5. cuprija 21°22' 43°56' 123 651.8 124.2 6. dimitrovgrad 22°45' 43°01' 450 631.0 118.0 7. kikinda 20°28' 45°51' 81 549.7 123.8 8. kopaonik 20°48' 43°17' 1711 727.3 217.6 9. kragujevac 20°56' 44°02' 185 628.6 112.2 10. kraljevo 20°42' 43°43' 215 748.1 132.7 11. krusevac 21°21' 43°34' 166 643.6 134.3 12. kursumlija 21°16' 43°08' 383 635.4 131.7 13. leskovac 21°57' 42°59' 230 614.6 109.3 14. loznica 19°14' 44°33' 121 820.0 140.7 15. negotin 22°33' 44°14' 42 637.5 132.7 16. nis 21°54' 43°20' 204 578.8 110.5 17. novi sad 19°51' 45°20' 86 613.8 153.8 18. palic 19°46' 46°06' 102 550.1 117.8 19. pozega 20°02' 43°50' 310 746.7 134.2 20. sjenica 20°01' 43°16' 1038 718.7 130.7 21. sombor 19°05' 45°47' 87 588.2 129.1 22. smederevska palanka 20°57' 44°22' 121 635.7 115.7 23. sremska mitrovica 19°38' 44°58' 82 621.9 122.3 24. valjevo 19°55' 44°17' 176 775.9 134.3 25. veliko gradiste 21°31' 44°45' 80 666.5 140.3 26. vranje 21°55' 42°33' 432 601.5 114.5 27. zajecar 22°17' 43°53' 144 603.2 116.1 28. zlatibor 19°43' 43°44' 1028 954.9 151.9 29. zrenjanin 20°21' 45°24' 80 576.4 121.0 the mean annual precipitation varied between 549.7 mm to 954.9 mm. dry areas with precipitation below 600 mm is in the northeast (becej, kikinda, palic, sombor, zrenjanin). the area along the valley of the south morava to vranje has the precipitation to 650 mm during the year. a larger and more compact area to the west and southwest is the wettest region of serbia. in the mountains, such as mt. zlatibor and mt. kopaonik, precipitation may rise up to 1000 mm per year. the detailed analysis of the considered time series is presented in [36-38]. 212 d. petković, m. gocic, s. shamshirband fig. 1 spatial distribution of the 29 synoptic stations in serbia map data were obtained from the republic hydrometeorological service of serbia (http://www.hidmet.gov.rs/), that carried out technical and quality control of these measurements. there are no missing values in the data set. besides, the homogeneity test and the kendall autocorrelation analysis [39] were applied to the monthly precipitation time series of each station. adaptive neuro-fuzzy computing technique for precipitation estimation 213 3. adaptive neuro-fuzzy application fuzzy inference system (fis) is the main core of anfis. it is based on expertise expressed in terms of ‘if–then’ rules and can thus be employed to predict the behavior of many uncertain systems. fis advantage is that it does not require knowledge of the underlying physical process as a precondition for its application. thus, anfis integrates the fuzzy inference system with a back-propagation learning algorithm of neural network. the basic structure of a fis consists of three conceptual components: a rule base, which contains a selection of fuzzy rules; a database, which defines the membership functions (mfs) used in the fuzzy rules; and a reasoning mechanism, which performs the inference procedure upon the rules and the given facts to derive a reasonable output or conclusion. these intelligent systems combine knowledge, technique and methodologies from various sources. they possess human-like expertise within a specific domain – adapt themselves and learn to do better in changing environments. in anfis, neural networks recognize patterns, and help adaptation to environments. anfis is tuned with a back propagation algorithm based on the collection of input-output data. anfis model will be established in this study to estimate participation in serbia according to the monthly precipitation data during 1946-2012. the anfis networks should determine the optimal participation for a given number of data inputs. fuzzy logic toolbox in matlab was used for the entire process of training and evaluation of fuzzy inference system. fig. 2 shows an anfis structure with one input, month in year, x. fig. 2 anfis structure in this work, the first-order sugeno model with two inputs and fuzzy if-then rules of takagi and sugeno’s type is used. the first layer consists of input variables membership functions (mfs). this layer just supplies the input values to the next layer. the input is month in year. in this study, bell-shaped mfs (2) with maximum equal to 1 and minimum equal to 0 are chosen. the second layer (membership layer) checks for the weights of each mfs. it receives the input values from the 1st layer and acts as mfs to represent the fuzzy sets of the respective input variables. every node in the second layer is non-adaptive and this layer multiplies the incoming signals and sends the product out. each node output represents the firing strength of a rule or weight. the third layer is called the rule layer. each node (each neuron) in this layer performs the pre-condition matching of the fuzzy rules, i.e. they compute the activation level of each rule, the number of layers being equal to the number of fuzzy rules. each node of 214 d. petković, m. gocic, s. shamshirband these layers calculates the weights, which are normalized. the third layer is also nonadaptive and every node calculates the ratio of the rule’s firing strength to the sum of all the rules’ firing strengths. the outputs of this layer are called normalized firing strengths or normalized weights. the fourth layer is called the defuzzification layer and it provides the output values resulting from the inference of rules. every node in the fourth layer is an adaptive node with node function. the fifth layer is called the output layer which sums up all the inputs coming from the fourth layer and transforms the fuzzy classification results into a crisp (binary). the output represents an estimated modulation transfer function of the optical system. the single node in the fifth layer is not adaptive and this node computes the overall output as the summation of all the incoming signals. the hybrid learning algorithms were applied to identify the parameters in the anfis architectures. in the forward pass of the hybrid learning algorithm, functional signals go forward till layer 4 and the consequent parameters are identified by the least squares estimate. in the backward pass, the error rates propagate backwards and the premise parameters are updated by the gradient descent. 4. anfis results at the beginning, the anfis network was trained with measured data by above presented experimental procedure. the training data are presented in fig. 3 for one station (becej). the anfis network determines optimal precipitation based on the measured data. fig. 3 experimental data for anfis training for one station (becej) three bell-shaped membership functions were used to fuzzify the anfis input. after training process the anfis networks were tested. fig. 5 shows the result of the testing of anfis network for three distinct regions in serbia. these regions were experimentally determined in [37, 38]. here, two stations are chosen to represent one region as shown in figs. 4-6. adaptive neuro-fuzzy computing technique for precipitation estimation 215 fig. 4 anfis predicted results of the precipitation in serbia for region 1 fig. 5 anfis predicted results of the precipitation in serbia for region 2 fig. 6 anfis predicted results of the precipitation in serbia for region 3 216 d. petković, m. gocic, s. shamshirband finally, we performed precipitation prediction for serbia based on the regions precipitations as shown in fig. 6. the red solid line represents anfis prediction of the serbia participation while the estimated precipitations at all the stations are also depicted in fig. 7. fig. 7 anfis predicted results of the precipitation in serbia the obtained results show that the maximum precipitation is in june. the decreasing trend of precipitation is detected in the summer, which is in an agreement with the results reported in [40]. 5. conclusion the precipitation was investigated by anfis methodology in serbia using monthly precipitation time series from 29 stations during the period of 1948–2012. the anfis method has been presented and tested for developing an alternative method to estimate precipitation. the main advantage of the soft computing scheme has very good learning and prediction capabilities, which makes it an efficient tool for dealing with encountered uncertainties in any system such as precipitation. since there are three distinct regions in serbia with respect to the precipitation [37, 38], we estimated precipitation for these regions. afterwards the anfis was applied to predict overall estimation according to all the stations in serbia from period 1948–2012. simulations were run in matlab and the results were observed on the corresponding output blocks. the main advantages of the anfis scheme are: it is computationally efficient, well-adaptable with optimization and adaptive techniques. the developed strategy is not only simple but also reliable and may be easy to implement in real time applications using some interfacing cards like the dspace, data acquisition cards, ni cards, etc. for the control of various parameters. this can also be combined with expert systems and rough sets for other applications. anfis can also be used with systems handling more complex parameters. another advantage of anfis is its speed of operation; the tedious task of training membership functions is done in anfis. adaptive neuro-fuzzy computing technique for precipitation estimation 217 references 1. 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paper remote control of the mechatronic redesigned slider-crank mechanism in service udc 621.8 miša tomić, miloš milošević, nevena tomić, nenad d. pavlović, vukašin pavlović faculty of mechanical engineering, university of niš, serbia abstract. slider-crank mechanisms are used in many machines where there is a need to transform rotary motion into translation, and vice versa. implementation of the control into a mechanical assembly of the slider-crank mechanism offers a wide range of applications of such controlled mechanism in mechatronic systems. this paper shows an example of the remote control of the angular velocity of the crank in a mechatronic redesigned slider-crank mechanism in order to achieve the desired motion of the slider. the remote control is achieved over the internet connection and the appropriate software which is executed in the user’s internet browser. the aim of this paper is to present the applied control algorithm as well as to explain advantages of the possibility to remotely run a mechatronic redesigned slider-crank mechanism in service. this is done through an example of using a controlled slider-crank mechanism in a remote laboratory experiment. key words: slider-crank mechanism, remote control, mechatronic system, pid controller, ni labview 1. introduction the main function of any mechanism is the realization of motion. the most common case is that of converting rotary motion of the crank (motor shaft rotation) into a rotary motion (rocker motion) or translational motion of the output link (the slider of the slider-crank mechanism). thanks to the fast development of computers and microprocessors, mechatronics as a discipline which is a synergy of mechanical engineering, electronics, computer science and control, offers great opportunities for the development and improvement of complex technical received may 10, 2017 / accepted july 07, 2017 corresponding author: miša tomić faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: misa.tomic@masfak.ni.ac.rs 258 m. tomić, m. milošević, n. tomić, n.d. pavlović, v. pavlović systems [1]. one of the main tasks of mechatronics is to upgrade existing mechanical systems by implementing additional control functions, particularly in precision engineering, in order to make these systems more functional, with better performance, consuming less power, performing safer and so on. likewise, by implementing the mechatronic approach into the design of mechanisms, it is possible to convert classical mechanical assemblies into mechatronic mechanisms which are more useful and adaptive to different tasks in mechatronic systems. typical examples of using mechatronic mechanisms are the following [2]:  use of mechanisms for transforming rotary motion into translation (for example at slider-crank mechanisms) with appropriate controllable actuators, as a simple constructive solution for the generation of a linear translation of the motion patterns that can be controlled,  using of mechanisms for path generation with the degree of freedom f=1, for the generation of specified path and control of the velocity profile along this path by using appropriate controllable actuator,  using of mechanisms for the path generation with the degree of freedom f=2, for the generation of the specified path by a main drive at constant velocity and one auxiliary drive with controlled velocity or by two controlled actuators,  using of appropriate controllable actuators for driving mechanisms with the non-uniform transmission in order to reduce the required driving torque by controlling the drive,  using of appropriate controllable actuators for driving mechanisms with the non-uniform transmission in order to reduce the non-uniformity of motion by controlling the drive (instead of using the flywheel), and,  using of appropriate controllable actuators for driving mechanisms with the non-uniform transmission in order to minimize the effects of the shaking forces and shaking torques on the frame by controlling the drive (instead of using the counterweights). the effectiveness of use of these options requires, already at the planning stage for their application, an integrated observation of the whole system and the kinematics of the mechanism as well as the operating characteristics of the applied actuator, in particular for driving mechanisms with the non-uniform transmission. in this paper a feasibility study for controlling the motion of a mechatronic redesigned slider-crank mechanism is elaborated. for that case, different approaches can be found in references. by using the particle swarm optimization (pso) algorithm, a novel design method for the self-tuning pid control in a slider–crank mechanism system is presented in [3]. the paper demonstrates, in detail, how to employ the pso so as to search efficiently for the optimal pid controller parameters within a mechanism system. in [4] a supervisory fuzzy neural network (fnn) controller is proposed to control a nonlinear slider-crank mechanism where the control system is composed of a permanent magnet (pm) synchronous servo motor drive coupled with a slider-crank mechanism and a supervisory fnn position controller. in [5] “mechatronic redesign’’ of the slider-crank mechanism is carried out in order to perform a variety of motion patterns. a novel design for quick return mechanisms, where the new mechanism is composed by a generalized oldham coupling and a slider-crank mechanism is proposed in [6]. in [7] the mathematical model of the motor-mechanism coupling system is developed. to formulate the equation of motion, the hamilton's principle and the lagrange multiplier method are applied. an adaptive controller for the motor-mechanism coupling system is obtained by using the stability analysis with the inertia-related lyapunov function. [8] shows a similar approach for the control algorithm. the experiment is carried out in the remote control of the mechatronic redesigned slider-crank mechanism in service 259 virtual environment, where solidworks cad design software is used for modeling the mechanism which is then linked to ni labview graphical programming platform that is used to control the mechanism. in [9] the variable structure control (vsc) and the stabilizer design by using pole placement technique are applied to the tracking control of the flexible slider-crank mechanism under impact. the vsc strategy employed to track the crank angular position and velocity, while the stabilizer design is involved to suppress the flexible vibrations simultaneously. in [10] regulation and vibration control of a flexible slider–crank mechanism is presented. the pda controller composed of the traditional proportion and derivative controllers with the feedback of acceleration of the crank are derived by using the lyapunov’s direct method. suppression of the elastodynamic vibrations of a slider-crank mechanism with a very flexible connecting rod is addressed in [11]. a model for the mechanism is derived using euler-lagrange equations and the assumed modes method. the control action uses two feedback signals: the crank angle and the connecting rod coupler midpoint deflection. two control schemes are proposed for the control of the flexible slider-crank mechanism. one scheme is a simple pd control scheme with feedback linearization. the second scheme is based on the μ-synthesis control technique. in [12] a method of solution rectification by means of transmission angle control which can be used to parameterize a problem to prevent the evaluation of invalid linkages is presented. the solution takes into consideration crank driven and slider driven mechanisms as well as a reversible driver mechanism. dynamic behavior of a slider–crank mechanism associated with a smart flexible connecting rod is investigated in [13]. two control schemes are proposed; the first is based on feedback linearization approach and the second is based on a sliding mode controller. in [14] an optimization method is proposed to alleviate the undesirable effects of joint clearance in order to optimize the mass distribution of the links of a mechanism to reduce or eliminate the impact forces in the clearance joint. for a slider–crank mechanism with a revolute clearance joint between the slider and the connecting rod, an algorithm based on pso is used. in [15] a numerically comparative study on dynamic response of a planar slider–crank mechanism with two clearance joints between considering harmonic drive and link flexibility is conducted. the comparative study of optimization design of the rigid and harmonic drive slider–crank mechanism experiencing wear is also presented. the aim of the research presented in this paper is focused on using the control algorithm for the generation of a linear translation of the motion patterns that can be controlled as well as for explaining the possibility of remote running of the given task. this remote experiment shows an example of the control of the crank angular velocity in a slider-crank mechanism in order to achieve the desired motion pattern of the slider. slider-crank mechanisms are used in many machines where there is a need to transform rotary motion into translation, and vice versa. as with most other mechanisms, for driving slider-crank mechanisms motors with the constant angular velocity are generally used. in that case, the driven motion of a slider-crank mechanism represents the return stroke of the slider for each cycle of rotation of the driving member, which is called a crank. for a centric slider-crank mechanism, the working and the return strokes of the slider have the same duration. if an application requires a mechanism with the slower working stroke (e.g. for the realization of operations of cutting, copying, scanning, deep sheet metal drawing, etc), and at the same time, the rapid return stroke (there are not working operations in the return stroke, it is just necessary to bring back the mechanism to its starting position quickly in order to save the time up to the next working operation), the traditional approach to solving this problem offers a complete redesign of the mechanism structure only. 260 m. tomić, m. milošević, n. tomić, n.d. pavlović, v. pavlović such a requirement, however, could be effectively carried out as well by installing appropriate sensors for determining the position of the crank and appropriate controllable actuators that would be able to change the angular velocity of the crank of a slider-crank mechanism. such a "mechatronic redesign", by using appropriate sensors, control systems and controllable drive actuators, adapts a traditional slider-crank mechanism in order to be able for implementing a variety of different motion-controlled transfer functions without any additional changing of the basic mechanism structure. 2. components of mechatronic redesigned slider-crank mechanism 2.1. mechanical assembly of mechatronic redesigned slider-crank mechanism the mechanical assembly of the mechatronic redesigned slider-crank mechanism is shown in fig. 1. it consists of several parts: crank (1), coupler (2), small wheel (3), guide frame (4), crank carrier (5), pin (6), dc motor with rotary encoder (7) and two plates (8). the two plates are connected thus representing the frame for the mechanism. the front plate has three slots. the two longer and narrower slots are used for screwing the motor, while the third shorter and wider one is used for the motor shaft. the crank carrier is connected to the motor shaft. the guide frame is placed on the front plate. the small wheel is actually a ball bearing, that acts as the slider and it slides along the guide frame with neglected friction. it is connected to the coupler by the pin. the crank is connected on one side with to the crank carrier and on another to the coupler. fig. 1 mechanical assembly of mechatronic redesigned slider-crank mechanism 2.2. electronics of mechatronic redesigned slider-crank mechanism for this experiment the servo dc motor faulhaber 3272g024cr shown in fig. 2a is used for driving the mechanism crank. since this motor is intended to rotate in one direction only, the simple electric driver, whose electric scheme is shown in fig. 2b, is used to drive remote control of the mechatronic redesigned slider-crank mechanism in service 261 the motor. this electric controller consists of one resistor of 3kω, one transistor bdx33c, and one diode. the controller is connected to the motor, and to the multifunction i/o device ni usb 6363 whose fast digital output is used to achieve the pulse width modulation (pwm) for controlling the voltage supply for the motor. fig. 2c shows the ni usb 6363 device. on the back side of the motor shaft, the rotary encoder heds 5540 a12 is mounted. fig. 2d shows the encoder which has 500 pulses per revolution. this encoder is directly connected to the ni usb 6363 device on the corresponding fast digital input ports. there is also a web camera connected to the computer for video live streaming during the execution of the experiment. fig. 2 electronic components of mechatronic redesigned slider-crank mechanism a) dc motor, b) electric driver, c) ni usb 6363 device, d) encoder 3. transfer function of slider-crank mechanism as already mentioned, the drive for the crank is the servo dc motor with the embedded rotary encoder. because the control of the motion of the slider is the aim, it is necessary to know current values of the position or the velocity of the slider of the mechatronic redesigned slider-crank mechanism in service. these values are not easy to measure because of complex movable parts; that is why the encoder on the motor shaft that drives the crank is used for that purpose. the encoder measures the angle of the crank position, and by the transfer function of the (centric) slider-crank mechanism (fig. 3) described by equations below, the position of the slider can be determined: ( ) cos coss a c    , (1) where s represents the position of the slider, φ represents the angular position of the crank, а represents the crank length, c represents the coupler length. since angle γ can be calculated by the equation: a) c) b) d) 262 m. tomić, m. milošević, n. tomić, n.d. pavlović, v. pavlović sin arcsin a c    , (2) the complete transfer function of the centric slider-crank mechanism can be represented as: sin ( ) cos cos(arcsin ) a s a c c     , (3) which is used for the following procedure of controlling the mechatronic redesigned slider-crank mechanism. the transfer function enables calculating the current position of the slider in dependence on the current angular position of the crank measured by the rotary encoder on the motor shaft. positions of the slider in two adjacent moments are used for numerical differentiation with respect to time for estimating the current velocity of the slider. fig. 3 kinematic scheme of centric slider-crank mechanism 4. control concept of mechatronic redesigned slider-crank mechanism as already explained, the position of the slider of the mechatronic redesigned slider-crank mechanism can be determined by using the angular position of the crank measured with the rotary encoder and the transfer function of the centric slider-crank mechanism (3). moreover, it is necessary to define the desired velocity profile of the slider. for this example, it is decided to use desired velocity profile v(t) of the slider shown in fig. 4, because such an example can have the most common use in practice. time t1 represents the time that the slider remains in the initial position, t2 is the time of the slider motion in one direction (the operating motion), time t3 is the time of the slider rests after the operation motion, t4 is the time of the slider motion in the other direction (the return motion). time tu represents the time of acceleration and deceleration of the slider during the transition from a steady state to a motion state, and vice versa, and it depends on the motor power, the mass of the members of the mechanism, friction in joints and the guide frame, etc. https://en.wikipedia.org/wiki/time remote control of the mechatronic redesigned slider-crank mechanism in service 263 fig. 4 desired slider velocity profile the difference between the desired and the estimated slider velocity, obtained by numerical differentiation with respect to time of the transfer function (3), as explained previously, returns the error in the slider velocity which should be corrected by the pid control. the equation of the pid controller is the following: 0 ( ) ( ) ( ) t p i d de u t k e t k e d k dt     (4) where kp is the proportional gain, ki is the integral gain, kd is the derivative gain, e(t) is the error and u(t) is the output from the pid controller. the output from the pid controller is percentage for the width of the pwm signal, i.e. a duty cycle of the pwm signal, and on this way the pid controller adjusts the supplying voltage for the motor. the user can set parameters of the pid controller and exactly on these parameters depends how well the slider will perform the desired motion. fig. 5 represents the block diagram of the control algorithm. fig. 5 block diagram of control algorithm https://en.wikipedia.org/wiki/time 264 m. tomić, m. milošević, n. tomić, n.d. pavlović, v. pavlović 5. mechatronic redesigned slider-crank mechanism in service as remote laboratory experiment it is not always feasible to set up a classroom experiment because of lack of funds. nowadays, the internet opens completely new possibilities by allowing users to perform remote dislocated laboratory experiments in very much the same way, or nearly the same way as operating them on the spot. in this paper the example of a remote laboratory experiment is shown on the remote control of the previously described mechatronic redesigned slider-crank mechanism stationed at the mechatronic laboratory of the faculty of mechanical engineering of university of niš, serbia. fig. 6 shows the starting interface with the parameters and control layout where the user can set the parameters of the slider-crank mechanism, the parameters of motion and rest as well as the gains of the pid controller. in this example, the gains of the pid controller are experimentally obtained. on the same interface calculated and measured positions of the slider can be observed. fig. 6 starting user interface for setting parameters for remote control of mechatronic redesigned slider-crank mechanism it should be noted that, due to the use of relative rotary encoder, the considered mechatronic redesigned slider-crank mechanism should be firstly brought into the inner limit position as a starting position (fig. 7), because the transfer function equations are written as relative to this position, and only then it is possible to start the motion. in the camera and calibration layout with the video live streaming there is a possibility of calibration and adjustment of the mentioned mechanism starting position using buttons rough and fine. pressing and holding of the button rough causes the rotation of the crank into the counterclockwise direction at approximately 60 °/s and pressing and holding of the button fine causes the rotation of the crank into the counterclockwise direction at approximately 1 °/s. with the help of the camera stream it is possible to bring the mechatronic redesigned slider-crank mechanism into the necessary starting position. pressing the done button will save the starting position. remote control of the mechatronic redesigned slider-crank mechanism in service 265 fig. 7 calibration mode with inner limit position of mechatronic redesigned slider-crank mechanism as starting position after setup of all parameters and the calibration of the starting position, it is possible to run the experiment by pressing the operation button, which will put the mechatronic redesigned slider-crank mechanism into the operation mode which is shown in fig. 8. if some additional calibration is needed, then it is possible to go back to the calibration mode by pressing the calibration button (fig. 8). fig. 8 operation mode of mechatronic redesigned slider-crank mechanism 266 m. tomić, m. milošević, n. tomić, n.d. pavlović, v. pavlović for testing of the control algorithm, the next example has been chosen for defining the desired velocity profile of the slider: the time that the slider remains in the initial position is, t1=2 s, the time of the slider motion in one direction (the operating motion) t2=5 s, the time of the slider rests after the operation motion t3=1 s, the time of the slider motion in the other direction (the return motion) t4=3 s and the time of acceleration and deceleration of the slider tu=0.5 s, in accordance with fig. 4. these times are chosen to be slightly longer, so that the user can notice the change in velocity of the crank in order to achieve the desired motion of the slider. the red graph in fig. 9 shows the velocity of the slider calculated from the crank position which is measured by the optical encoder during the experiment, and the blue graph represents velocity reference. it can be noticed from the graph that the velocity of the slider corresponds to the defined velocity pattern. fig. 9 measured velocity of slider obtained during running of experiment this experiment can be started remotely from a computer connected to the internet using an internet browser. all the necessary software for running the experiment remotely is installed on the server computer at the mechatronic laboratory of faculty of mechanical engineering of university of niš, serbia, and there is no need for the remote user to do additional installations or settings. thanks to the web camera it is possible to track the motion of the mechanism. this is a huge advantage of this kind of remote experiments because research studies from other institutions all over the world can use it to test their control algorithms without building the whole experimental setup. interested in testing this remote experiment can contact the corresponding author via e-mail, for detail instructions about how to access to the experiment. 6. conclusion in this paper a mechatronic redesigned approach is presented on a representative example of a centric slider-crank mechanism. it has been chosen because the working and the return stroke of the slider have the same duration, so that if an application requires a mechanism with a slower working stroke, and, at the same time, the rapid return stroke the traditional approach requires the complete redesign of the mechanism structure. using a servo dc motor, encoder, corresponding electronics, acquisition devices and appropriate software, as recognizable components of mechatronic systems, it has been shown that it is enabled to achieve a desired profile of the slider velocity, without changes of the basic structure of the slider-crank remote control of the mechatronic redesigned slider-crank mechanism in service 267 mechanism. for that purpose, a pid controller is used as a control algorithm, where the input into the controller is the error represented as the difference between the desired velocity and the estimated current velocity of the slider, and the output from the controller is a duty cycle of the pwm signal for the motor voltage supply. testing of the proposed approach has confirmed that the measured velocity of the slider obtained during running of the experiment corresponds to the defined velocity of the slider. this mechatronic approach enables an expanded usage of traditional mechanisms in many other applications only by adjusting control parameters, without making mechanical changes of the basic structures of mechanisms. since the mechatronic approach of redesigning mechanical assemblies’ opens possibilities for remote controlling over the internet, the mechatronic redesigned slider-crank mechanism is developed as a remote laboratory experiment stationed at the mechatronic laboratory of faculty of mechanical engineering of university of niš, serbia. for the remote purpose, it is firstly necessary to set the parameters and calibrate the mechanism into the starting position, and then, over live video streaming by a web camera, the mechanism motion can be tracked. the control algorithm, as well as remote access and live video streaming, is developed by ni labview software. moreover, the remote control of the experiment enables the testing of the different control parameters to many research studies by simply visiting the web page of the experiment. acknowledgements: this research has been supported by the tempus project 543667-2013: building network of remote labs for strengthening university-secondary vocational schools collaboration nerela. references 1. kao, c.c., chuang, c.w., fung, r.f., 2006, the self-tuning pid control in a slider–crank mechanism system by applying particle swarm optimization approach, mechatronics, 16, pp. 513–522. 2. braune, r., wyrwa, k., 1998, elektronische kurvenscheiben als antrieb von koppelgetrieben (betriebsverhalten – simulation einsatzoptimierung), vdi berichte, nr. 1423. 3. bishop, r.h., 2002, the mechatronics handbook, crc press llc, the university of texas at austin, austin, texas. 4. lin, f.j., fung, r.f., lin, h.h., hong, c.m., 2001, a supervisory fuzzy neural network controller for slider-crank mechanism, mechatronics, 11, pp. 227-250. 5. nagchaudhuri, a., 2002, mechatronic redesign of slider crank mechanism, proceedings of imece 2002 asme international mechanical engineering congress & exposition, pp. 849-854. 6. hsieh, w.h., tsai, c.h., 2009 a study on a novel quick return mechanism, transactions of the canadian society for mechanical engineering, 33(3), pp. 487-500. 7. lin, f. j., fung, r. f. lin, y. s., 1997, adaptive control of slider-crank mechanism motion: simulation and experiments, international journal of systems science, 28, pp. 1227-1238. 8. tomić, n., milošević, m., tomić, m., pavlović, v., milojević, a., 2015, control of slider-crank mechanism in virtual environment, proceedings of the 3rd international conference mechanical engineering in xxi century, pp. 279-282. 9. fung, r. f., sun, j. h., wu, j. w., 2002, tracking control of the flexible slider-crank mechanism system under impact, journal of sound and vibration, 255, pp. 337-355. 10. fung, r. f., shue, l. c., 2002, regulation of a flexible slider–crank mechanism by lyapunov’s direct method, mechatronics, 12, pp. 503-509. 11. karboub, m. a., 2000, control of the elastodynamic vibrations of a flexible slider-crank mechanism using µ-synthesis, mechatronics, 10, pp. 649-668. 12. wilhelm, r. s., sullivan, t., van de ven, d. t., 2017, solution rectification of slider-crank mechanisms with transmission angle control, mechanism and machine theory, 107, pp. 37-45. 268 m. tomić, m. milošević, n. tomić, n.d. pavlović, v. pavlović 13. akbari, s., fallahi, f., pirbodaghi, t., 2016, dynamic analysis and controller design for a slider–crank mechanism with piezoelectric actuators, journal of computational design and engineering, 3, pp. 312-321. 14. varedi, s.m.., daniali, h.m., dardel, m., fathi, a., 2015, optimal dynamic design of a planar slider-crank mechanism with a joint clearance, mechanism and machine theory, 86, pp. 191-200. 15. li, y., chen, g., sun, d., gao, y., wang, k., 2016, dynamic analysis and optimization design of a planar slider–crank mechanism with flexible components and two clearance joints, mechanism and machine theory, 86, pp. 37-57. 6404 facta universitatis series: mechanical engineering vol. 22, no 1, 2024, pp. 13 24 https://doi.org/10.22190/fume200819022z © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper experimental analysis of the effect of the woven aramid fabric on the strain to failure behavior of plain weaved carbon/aramid hybrid laminates sojan andrews zachariah, satish shenoy, dayananda pai department of aeronautical and automobile engineering, manipal institute of technology, manipal academy of higher education, india orcid ids: sojan andrews zachariah https://orcid.org/0000-0002-1337-3100 satish shenoy https://orcid.org/0000-0003-2374-3854 dayananda pai https://orcid.org/0000-0003-0334-8718 abstract. remarkable advances in the research and development of micro/mini unmanned aerial vehicles (uavs) seek thin, lightweight and strong materials for their structural applications. as these structures involve various loading conditions in both the in-plane and through-the-thickness directions during their life cycle, the assurance of the structural stability in each direction is deemed mandatory. the woven aramid fibers as high strain materials (hsm) are known to improve the through-the-thickness impact strength. however, the addition of the hsm can affect the overall tensile behavior of composite laminates. this study investigates the effect of the woven aramid fiber on the in-plane tensile behavior of carbon/ epoxy laminates. laminates are fabricated using an easy and cost-effective vacuum assisted resin infusion molding (varim) setup. a uniaxial tensile test was conducted to analyze the tensile behavior of carbon/aramid hybrid composites. the effect of adding the woven aramid layer and the carbon fabric sequence on the tensile modulus, strain to failure and modulus of toughness are investigated in this study. the results revealed that the presence of aramid has a positive hybrid effect on the failure strain, exhibiting pseudo-ductile behavior with a compromise in the tensile modulus of the virgin carbon/epoxy laminate. key words: hybrid composites, pseudo – ductility, woven fabric polymer composites, ultimate strain to failure 1. introduction composite laminates are widely employed for various structural applications in the engineering sector. they replace conventional metal structures mainly due to their exceptional strength to weight ratio. laminated composites hold a significant share of their received august 19, 2020 / accepted january 30, 2021 corresponding author: dr. dayananda pai k. professor, department of aeronautical and automobile engineering, manipal institute of technology, manipal academy of higher education, manipal 576104, india e-mail: dayanand.pai@manipal.edu https://orcid.org/0000-0002-1337-3100 https://orcid.org/0000-0003-2374-3854 https://orcid.org/0000-0003-0334-8718 14 s. andrews, d. pai, s. shenoy application in aerospace and automotive sectors since they can serve lightweight components better than any other natural materials. however, the strict safety regulations sometimes make these materials suspicious of their ability in ultimate loading conditions as they have a usual tendency to fail catastrophically. this necessitates the designers to maintain higher values of safety factors in their design. advanced composites are the materials where high strength, high modulus reinforcements are used. they are recognized as a promising lightweight solution [1–3]. currently, advanced composites based on carbon fibers are vastly used in high-end structural applications in aviation, uavs, automotive and energy sectors where the cost of a sudden failure is non-comparable [4]. bidirectional (bd) woven fabrics are considered as two-dimensional reinforcement elements in the composite industry, especially for aerospace structures. they have better impact stability than conventional unidirectional (ud) laminates due to fiber yarns in the warp and weft directions. however, the in-plane mechanical properties like the elastic modulus, strength and toughness modulus of the composite laminates are closely related to the fiber architecture [5]. the connections between the yarns and the yarn crimp lead to a complex structure in the woven fabric, which affects the mechanical properties. nevertheless, the enhanced impact response exhibited by fabric reinforced composites and a decline in the tensile properties happen due to the stress concentrations created by weaving undulations [6]. this trade-off among the through-the-thickness and in-plane properties needs to be thoroughly investigated as a part of ensuring safer structural design. therefore, a comprehensive study of the effects of the construction of fabrics on mechanical properties is deemed vital. carbon woven fabrics are essential materials for aerospace industry. many micro/mini uavs employ carbon fiber-based composites for the fuselage and belly constructions. carbon fiber being more brittle proves to be low in impact strength. hybridization is a practical approach to enhance the impact strength and the in-plane strain and toughness properties. high strain materials (hsm) like glass and aramid fibers are the hybrid choices in imparting better toughness and strain properties [7]. the addition of hsm comprises the high strength and modulus properties of carbon composites [8]. the hybrid approach in laminated composites is mainly divided into two methods. the first method is an inter-ply method in which two types of fibers are stacked layer by layer concerning different number, angle and architecture. the other hybrid approach is an intra-ply method in which a mix of different fibers either in single yarn or one type in the warp and other in the weft are weaved in a single layer [9–11]. erklig and bulut experimentally investigated the tensile and impact behavior of kevlar/glass hybrid laminates. the addition of kevlar from aramid class fibers showed an improvement in tensile strength and impact resistance [12]. thus, aramid can be a potential hsm for hybridization with lsm reinforced laminates. khatri and koczak [13] investigated the tensile responses of e-glass and as4 graphite fibers with a pps (polyphenylene sulfide) matrix. experimental studies showed that the strain values of hybrid composites increased by 2–8%. jesthi et al. [14] studied the effect of carbon/glass with inter-ply sequence and reported that the stacking sequence and the position of hybrid layers have an effect on the flexural and impact properties. saka and harding [15] investigated the tensile behavior of carbonglass/epoxy hybrid composites. sun et al. [16] studied the mechanical properties of carbon/basalt fibers and reported evident influence of stacking sequence and hybrid ratio on in-plane and through – the thickness properties. experimental analysis on the effect of woven aramid fabric on strain to failure behavior... 15 however, there are limited comprehensive studies of the effect of the woven aramid fabric on the in-plane mechanical properties of carbon laminates. it seems that for inter-ply woven hybrid composites, in which hybridization is achieved by layer by layer arrangement of woven fabric, the effect of the woven structure on the trade-off in mechanical properties of low strain materials (lsm) laminates has not been satisfactorily addressed in the literature. furthermore, a great majority of the studies of hybrid composites are focused on glass/carbon fiber hybrid fabric composites, but there are only a few studies on the effects of carbon/aramid hybridization on the mechanical properties. several researchers have investigated the impact performance of unidirectional woven carbon/aramid composites [17]. there is a need for additional data on the tensile performances of the woven composites fabricated in effortless and cost-effective methods for the small-scale applications for both academic and industrial applications. the studies based on the thin woven carbon/aramid hybrid laminates for uav applications are not reported widely. hence in this work, plain woven carbon/aramid hybrid composites have been investigated in order to understand the effect of the woven aramid layers in cfrp on its tensile properties. in this study, the carbon and aramid fibers are reinforced in the epoxy matrix using varim. out of the four types of samples investigated, there is a virgin 3-layer aramid sample as well as carbon samples, respectively. two different hybrid samples are studied with the hsm at the outer sides covering both the faces of the laminates. the effect of the aramid layer, when added to the carbon laminate, on the tensile behavior is investigated using a uniaxial tensile test. photographic images are used to visualize the fracture mechanism in attaining a clear understanding of the effect of the hybrid fiber under tensile load. this study anticipates gaining some useful and novel insights into the carbon/aramid hybrid composites to find broad applications in the micro/mini uavs. 2. experimental procedure 2.1. materials and specimen design the details of the raw material and the fabrication process employed in this study in order to investigate the hybrid effect of carbon/aramid epoxy laminate composite are explained in this section. a low viscosity ct/e 1564 epoxy, which is suitable for resin infusion, is used. the procured resin system is cured using two hardeners ct/ph-3486 and ct/ph-3487. the physical properties of the resin system are shown in table 1. table 1 parameters of the resin system type unit resin (ct/e-1564) hardener-1 (ct/ph-3486) hardener-2 (ct/ph-3487) viscosity at 25°c mpas 700 – 1,100 < 50 < 70 epoxy content g/eq 166 – 185 – – density at 25°c g/cc 1.10 – 1.20 0.94 – 0.95 0.98 – 1.00 flash point °c > 185 >120 >120 the reinforcements used in this study are 12k plain-woven bd carbon fabric manufactured by hyosung corporation and alkex 3000d aramid fabric from hyosung 16 s. andrews, d. pai, s. shenoy corporation supplied by composite tomorrow pvt ltd, india. table 2 shows the physical parameters and the mechanical properties of the carbon and aramid fabrics, respectively. table 2 physical parameters and properties of the fabrics type carbon aramid weave plain plain linear density warp/fill (tex) 12k 3000d ends/picks counts, yarns/10 cm 2.5/2.5 6.7/6.7 areal density, g/m2 400 450 fabric thickness, mm 0.42 0.55 fiber diameter, µm 7 10 fiber young modulus, gpa 230 74 fiber strength, mpa 3450 2923 fiber ultimate elongation, % 1.5 3.6 fiber density, g/cc 1.8 1.44 parameters of stacking sequences for neat and hybrid specimen types investigated in this study are illustrated in table 3. virgin carbon was made of 5 layers, and the virgin aramid layer comprised 3 layers. aramid laminates were named as 3a. in the case of carbon samples, 5c have been denoted for five-layer virgin carbon laminates, respectively. table 3 sequential details of the specimens laminate layers stacking sequence sample type 3 [(0/90)a ]3 3a 5 [(0/90)a /(0/90)c/(0/90)c]s h90 5 [(0/90)a /(0/90)c/(±45)c]s h45 5 [(0/90)c ]5 5c of the two different hybrid samples of 5 layers each was prepared using carbon and aramid fabric. in the hybrid sample, aramid layer was kept at the outside. studies suggest that hsm materials at the outer layers improve the impact performance [18] and also the strain to failure rate [8,19,20]. the symmetric layup sequence has been made for all the samples. out of the two varieties of stacking sequence used in experiment, one was with [(0/90)a /(0/90)c/(±45)c]s with a (±45) at the centre named h45 and the other with all the three layers at [(0/90)a /(0/90)c/(0/90)c]s denoted as h90. experimental analysis on the effect of woven aramid fabric on strain to failure behavior... 17 2.2. specimen manufacturing the laminates were fabricated using varim, as shown in fig. 1. all the specimens were made with a dimension of (300 x 300) mm. the resin – hardener mixing ratio was 100:34, wherein two hardeners were mixed at a ratio of 25.5: 8.5. fig. 1 vacuum-assisted resin infusion moulding (varim) set up the woven fabric reinforcements have been placed on aluminium plate mould. vacuum bagging and required tubing for resin inflow, outflow, and vacuum have been made appropriately. during the pre-filing stage [21,22], vacuum has been maintained for 1hr at -710 mm hg for compacting the dry fabric. during that time, it was confirmed that there is no leak. the filling pressure was maintained at – 650 mm hg. after the infusion of resin into the stacked fabric, the tubes have been sealed for holding the vacuum at -720 mmhg to avoid any bending or expansion of the ply during curing. resin impregnated fabric has been cured in an oven at 80°c for eight hours as recommended in the supplier datasheet. density and void fraction of the laminates were calculated using eqs. (1), (2) and (3) and are tabulated in table 6. experimental density ρexp is calculated using eq. (1) where m and v are mass and volume of the laminate, respectively. theoretical ρth is found out using eq. (2). theoretical density ρth is calculated using eq. (2) with weight fractions wfc, wfa and wm of carbon fiber, aramid fiber and matrix, respectively; their values are as shown in table 4. notations ρfc, ρfa and ρm, are the densities of carbon fiber, aramid fiber and matrix, respectively, as given in the manufacturer datasheet listed in table 5. v m =exp (1) 18 s. andrews, d. pai, s. shenoy m m fa fa fc fc th www   ++ = 1 (2) 100*% exp th th void   − = , (3) table 4 micro mechanical properties of the fabricated laminates sample ρfc (g/cm3) ρfa (g/cm3) ρm (g/cm3) wfc wfa wm 3a 1.44 1.15 0.58 0.42 h90 1.8 1.44 1.15 0.283 0.233 0.484 h45 1.8 1.44 1.15 0.304 0.246 0.449 5c 1.8 1.15 0.53 0.47 notation wc is the weight of the composite laminate. the percentage of void denoted by void % is calculated using eq. (3). the calculated values are shown in table 5. table 5 properties of test specimens sample wc (g) area (cm)2 thickness (cm) volume (cm)3 ρexp (g/cc) ρth (g/cc) void fraction ( vv) void % 3a 276 1054 0.211 222.39 1.241 1.302 0.046 4.68 h90 423 985 0.325 320.32 1.320 1.351 0.022 2.29 h45 390 980 0.299 293.28 1.329 1.369 0.029 2.91 5c 325 840 0.282 236.88 1.372 1.422 0.035 3.53 2.3. tensile test the specimen was cut into test coupons as per astm d3039 -17 using waterjet cutting at stonemax industries, cochin. each test coupon had a dimension of (250 x 25) mm. the final samples were visually inspected to assure appropriate quality. a good integrity of the hybrid laminates was confirmed during the testing procedures, and no phase separation was observed. tensile testing of the hybrid laminates was performed under uniaxial tensile loading and displacement control using a crosshead speed of 2 mm/min on a computer-controlled itw biss 50kn rated universal test machine with wedge-type grips. the results of the minimum 3 specimens are considered to check the repeatability of the results and the averaged values of each sample type are considered for the result analysis to impart clarity of discussion. the data on tensile strength, elastic modulus, strain to failure modulus of toughness and specific strength have been analyzed for conducting a comprehensive investigation on the in-plane mechanical behavior of carbon /aramid hybrid composites. the chord modulus is calculated at a strain range from 0.001 to a value that is 25 % of the total strain as per astm d3039 -17. experimental analysis on the effect of woven aramid fabric on strain to failure behavior... 19 3. results and discussion 3.1. tensile stress-strain result the stress-strain curves of the virgin and hybrid composite as shown in fig.2 exhibited an initial knee after a linear growth in the stress. this is the general bilinear behavior observed in bd fabric laminate caused due to the undulations in the weaving pattern and the pre-failure of 90° fibers [23]. the wavy pattern in weft and wrap gets straightened with the increase in strain [24]. beyond the knee point, the curve shows a slight linear growth until the ultimate strength of the composite. the tensile behaviors of the virgin carbon and aramid laminate are discussed in order to make a clear and comprehensive comparison between the virgin and hybrid samples. the virgin carbon sample 5c exhibited a steep slope in the stress-strain variation when compared to the virgin aramid and hybrid samples. however, the virgin aramid layers took the maximum stress when compared to all the samples. a higher ductility combined with a greater tenacity of the aramid fiber when compared to the carbon one is the reason for a high-stress rate of the aramid laminate. fig. 2 overall comparison of the stress-strain variations the hybrid samples h90 with inter-ply hybrid layers of carbon and aramid fibers a stress-strain behavior intermediate to the virgin samples indicates a positive hybrid effect. the high strain aramid layer helped in the gradual stress drop in h90 sample that exhibited a pseudoductile behavior. the pseudoductile behavior can be attributed as a warning before a catastrophic failure. however, the hybrid sample h45 undergoes a sudden drop in the stress-strain slope immediately after the failure of the carbon layer due to the stress concentrations created by ±45 carbon layer. thus, it is the position; the fiber stacking orientation has a significant influence in imparting a positive hybrid effect. the virgin carbon samples 5c showed a highest elastic modulus among the samples under investigation. the virgin aramid sample produced an elastic modulus almost 36% lower than the virgin carbon laminate. this is an obvious behavior due to the excellent modulus of the carbon fiber. however, aramid samples attained the maximum failure 20 s. andrews, d. pai, s. shenoy stress due to the inherent higher toughness and tenacity when compared to the carbon fiber. the overall strength and modulus data from the experiment are shown in fig.3. sample h90 had a significant positive hybrid effect with the e value almost 20% lower and 22% higher when compared to the virgin carbon and aramid samples, respectively. however, the presence of ±45 ply at the midplane of the h45 sample led to the lower values tensile properties when compared with all the samples under study. thus, it is inferred that the ply orientation also has a significant role in deciding the hybrid effect of hybrid composites. fig. 3 elastic modulus and ultimate strength of the tested samples 3.2. strain to failure and toughness modulus analysis strain to failure is a property that is worth studying in composites since the values of strain at failure have a significant dependency hsm fiber used during hybridization. the enhancement of the strain to failure value contributes to the pseudo ductile behavior. increased strain to failure behavior can also improve the energy absorption capability of a laminate which in fact is the modulus of toughness measured for a material. so, it is worth investigating the ultimate strain and the modulus of toughness together to get an inclusive picture of the effect of hybridization in composite laminates. as shown in fig.4, the hybrid approach increased the failure strain of the samples up to 55% in h45 samples and 75% in h90 samples as compared to virgin carbon samples. this increase is due to the presence of the aramid layer that can be justified with higher values of failure strain and toughness modulus. when the virgin 3a sample is compared to the 3c and 5c samples, the 3a sample showed for about 114% and 170% higher values of strain to failure and toughness modulus, respectively. experimental analysis on the effect of woven aramid fabric on strain to failure behavior... 21 fig. 4 failure strain and toughness modulus of the tested samples the hybrid samples h90 and h45 exhibited enhanced performance in the strain to failure value with an increase of 66% and 47 %, respectively, when compared to the 5c virgin sample. the h90 showed an 83% improvement in the toughness when compared to the virgin carbon sample. however, the h45 sample did not exhibit an appreciable change in comparison with the carbon sample. the early failure of ±45 layer creating a drop in the overall stress value during the tensile loading is the reason for unchanged toughness modulus. 3.3. failure mode analysis a study of the fracture morphology can give an insight into the reasons for the inplane tensile behaviors exhibited by the samples. after the tensile tests, the specimens were photographed, and the failure modes were classified in accordance with the threepart failure mode code as per astm d3039. virgin carbon sample, as shown in fig. 9, underwent a perfect brittle failure with the lateral gage middle (lgm), which is as expected due to the lower elongation to failure of the carbon fiber. tow fracture is the most visible failure observed in the fracture end. as shown in fig. 10, fibrillation of fiber has been observed in the virgin aramid samples, and the fibers at the failure section have undergone a ductile elongation with multimode gage middle (mgm) mode of failure. tow fracture and tow splitting were observed in the virgin aramid sample. delamination was observed more in 3a samples because of two reasons, namely, fiber fibrillation and a lower wettability of fabric to the matrix material. the above-mentioned reasons for the ductile failure of aramid are the factors that bring up pseudo-ductile behavior in laminated polymer composites. 22 s. andrews, d. pai, s. shenoy fig. 9 tensile fracture of virgin fig. 10 tensile fracture of virgin 3-layer 5c carbon laminate 3a aramid laminate the fracture images of the hybrid sample exhibited a real combination of the damage morphologies in virgin carbon and aramid sample. the delamination and fiber breakage are observed in the hybrid samples. the mixed behavior accounts for the property enhancements achieved through proper hybridization. as shown in fig. 11-a, a “v” shaped failure intersection is observed with an angle gage middle mode of failure for the h45. it is mainly due to abrupt failure due to stress concentration of the (±45) carbon layer present in the midplane. the nature of the failure is abrupt, with the aramid fibers failed along with the carbon layer. in the h45 sample, the failure in aramide was tow fracture with delamination and tow splitting was found in the carbon layer. fig. 11 tensile fracture of hybrid samples, (a) h 45 and (b) h90 the hybrid sample with only (0/90) layers of carbon as shown in fig. 11-b had a gradual failure as is visible that the aramid layer remained at one side after the separation of the carbon layers. the failure mode was lgb. this indicates that the sample took the load even after the carbon layers failure whereby resembling a pseudo-ductile tensile failure. a) b) experimental analysis on the effect of woven aramid fabric on strain to failure behavior... 23 4. conclusion in this work, the in-plane tensile behavior of bidirectional woven carbon/aramid epoxy composite laminates manufactured using varim with different carbon fiber layup sequence of 0/90 and ±45 fiber orientations are conducted. the study analyzed the effect of hybrid layers in the overall tensile characteristics of the woven hybrid laminates during uniaxial tensile loading. the hsm like aramid fibers are used in the lsm reinforced laminates to enhance the energy absorption capability, strain to failure and impact properties. however, the effort in improving these properties compromises the tensile modulus and strengths. the sample laminates were made with aramid layers at the outside and carbon as inside layers with an inter-ply hybrid approach, and the following conclusions were drawn. 1. the investigated hybrid samples exhibited a positive hybrid effect by increasing the strain to failure when compared to the virgin carbon composite. 2. the hybrid approach enhanced the failure strain of the samples up to 50% in h45 samples and 66 % in h90 samples as compared to the virgin carbon samples. this increase is due to the presence of aramid an hsm. 3. the toughness modulus of the hybrid samples showed influence on the stacking sequence of the carbon laminate; nevertheless, the presence of the aramid layer. the hybrid sample h90 exhibited a significant improvement in toughness modulus as compared to the virgin carbon laminate. however, the h45 sample with an angle ply at the midplane has not produced any significant improvement from the virgin carbon sample. 4. the hybrid samples showed a compromise in the in-plane mechanical properties, namely, the elastic modulus and strength. there has been a decline of 24% and 40% in the elastic modulus by h90 and h45, respectively. nevertheless, the compromise in the modulus value, h90 maintained the ultimate strength almost the same as the carbon laminate. however, the h45 had a reduction of 31% in the ultimate strength when compared to the virgin carbon sample. 5. varim process enabled the fabrication of the hybrid samples with acceptable void percentage. the pre-compaction, post-compaction, and the monitoring of the vacuum pressure throughout the curing time are a factor that decides upon thickness, density, integrity, and fiber-matrix ratio of the final structure. the vacuum pressure maintained during the curing time was -720 mm hg, and it helped in making the laminate with good strength and uniform thickness. the pressure drop at any stage of the process reduces the inter-laminar strength, increases the overall thickness, produces non-uniform ply thickness at different locations, and adversely affects the stiffness of the composite structure. a trend of pseudo-ductile behavior was observed in the h90 hybrid sample. the current study expects to have provided valuable insights for making thinner and robust 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mech. mater, 62, pp. 14–31. 13071 facta universitatis series: mechanical engineering vol. 23, no 3, 2025, pp. 407 432 https://doi.org/10.22190/fume241001052m © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper intuitionistic fuzzy macont method for logistics 4.0 based circular economy interested regions assessment in the agri-food sector arunodaya raj mishra1, pratibha rani2, dragan pamucar3,4,5, ahmad m. alshamrani6, adel fahad alrasheedi6 1department of mathematics, government college raigaon, madhya pradesh, india 2saveetha school of engineering, saveetha institute of medical and technical sciences (simats), chennai, tamil nadu, india 3department of applied mathematical science, college of science and technology, korea university, sejong, republic of korea 4dogus university, istanbul, türkiye 5department of industrial eng. & management, yuan ze university, taoyuan city, taiwan 6department of statistics and operations research, college of science, king saud university, riyadh, saudi arabia orcid ids: arunodaya raj mishra https://orcid.org/0000-0001-9949-5813 pratibha rani https://orcid.org/0000-0002-9186-4167 dragan pamucar https://orcid.org/0000-0001-8522-1942 ahmad m. alshamrani https://orcid.org/0000-0001-7103-8872 adel fahad alrasheedi https://orcid.org/0000-0003-4492-1082 abstract. this study aims to evaluate and prioritize the key interested regions of circular economy (ce) in terms of implementing the industry 4.0 technologies for the performance of logistics activities in the agri-food sector. for this purpose, we introduce a hybrid ranking framework based on relative closeness coefficient (rcc)-based objective weighting model, the ranking comparison (rancom) subjective weighting procedure and the mixed aggregation by comprehensive normalization technique (macont) with intuitionistic fuzzy information (ifi). in this framework, new if-score function and an improved distance measure are proposed in the context of ifi to evade the limitations of existing ones. a hybrid if-rcc-rancom-macont framework is introduced to prioritize the options over defined criteria. to prove the applicability of introduced approach, it is employed on a case study of circular economy interested regions assessment in the agri-food sector, consisting of five alternatives and nine criteria under the dimensions of sustainability. sensitivity analysis is shown to highlight the impact of used parameters on the final outcomes. at last, a comparison with extant approaches is made to demonstrate the robustness of obtained results. received: october 01, 2024 / accepted december 30, 2024 corresponding author: pratibha rani, dragan pamucar saveetha school of engineering, saveetha institute of medical and technical sciences (simats), chennai602105, tamil nadu, india dept. of applied mathem. science, college of science and technology, korea university, republic of korea e-mail: pratibha138@gmail.com, dpamucar@gmail.com https://orcid.org/0000-0001-9949-5813 https://orcid.org/0000-0002-9186-4167 https://orcid.org/0000-0001-8522-1942 https://orcid.org/0000-0001-7103-8872 https://orcid.org/0000-0003-4492-1082 mailto:pratibha138@gmail.com 408 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi key words: logistics 4.0, circular economy, agri-food sector, intuitionistic fuzzy sets, distance measure, score function, macont, multi-criteria decision-making 1. introduction logistics management is the governance of supply chain management (scm) functions that integrates the activities such as planning, packaging, material handling, warehousing, management of forward and reverse flow of goods, transportation, services and associated information between the point of origin and point of consumption to meet customers’ supplies [1-3]. “forward logistics (fl)” consisting of a procedure of moving goods from the point of production to end customer. as a counter-part of fl, “reverse logistics (rl)” refers to the procedure which reverses common flow of raw materials or finished goods along supply chain (sc) [4-8]. the purpose of efficient logistics management is to ensure customer satisfaction, achieve maximum competitiveness and profitability for the business along with the well-planned network of sc, including the end consumer [9, 10]. both the logistics processes are covered by the term “closed-loop supply chain (clsc)”, where goods lifecycle management is incorporated with its logistics solutions [11, 12]. in the past decade, the need for a transition towards a circular economy (ce) has been studied by several user groups and industries. the ce concept is defined as an economic system that aims to minimize the environmental and material footprint together with industrial transformation at different levels [13]. moving towards a ce system offers a chance to bring business profits including advanced competitiveness, resource security, flexible and diverse business models to enable value formation [14, 15]. in the current age of growing resource scarcity, different researchers, policymakers, and business managers have made empirical assessments on the impact of adopting the ce principle, which does not only cover the reverse flow but also manage the forward flow of the sc involving operations of newer materials [16]. as a revolutionary digitalization, logistics 4.0 refers to the implementation of different digital technologies into logistics operations to optimize and streamline the scm [17-19]. the paradigms of ce and logistics 4.0 offers several opportunities to create more sustainable and effective sc procedure. food insecurity can lead to significant productivity losses, lesser cognitive facility and reduced work performance. although, the agri-food sector cannot allow itself to missed the possibilities initiated by the convergence of logistics 4.0 and the ce. in this work, we consider the potentials of employing industry 4.0 (i4.0) for executing the logistics processes in the ce interest regions under the context of agrifood sector. this selection process may also include several des, whose opinions may differ and ambiguous. considering the hesitation levels, knowledge and expertise of des, fuzzy set theory (fst) and its generalizations are preferred to handle the incomplete and imprecise information of such type of problems [20-23]. as one of the extensions of fst, the notion of intuitionistic fuzzy set (ifs) [24] has received much attention to deal with ambiguous and vague data. in ifss, every object is described by the membership grade (mg) and non-membership grade (ng) with their sum is restricted to one. çelik et al. [25] ensured socio-economic sustainability indicators of medical waste management (mwm). in this regard, they presented a unified approach with the weighting and ranking models and applied for mwm in six hospitals, where the data is given in terms of intuitionistic fuzzy numbers. asif et al. [26] gave various form intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 409 hamacher operators to deal with different decision-making problems. bajaj & kumar [27] studied a new correlation measure on ifss and implemented to the pattern recognition, medical diagnosis and clustering problems. further, an integrated decision support tool has been presented to evaluate the alternatives on ifss. majumder et al. [28] gave an ifmultiple criteria decision-making (mcdm) approach for deriving the significance values of criteria. moreover, they used a unique decision support system to assume a feasibility assessment of solar power plant. wan et al. [29] integrated if-best worst method (bwm) with additive consistency. based on this method, they gave a non-linear programming procedure for solving mcdm problems. hussain and ullah [30] presented a model using sugeno-weber aggregation operators on to solve real-life mcdm problem. şahin et al. [31] used two methods, geographic information measure and ifss-based method to evaluate and rank the locations for solar-wind power plant establishment in netherlands. considering the multiple criteria/factors and decision experts (des), choosing and prioritizing an appropriate ce interested regions in agri-food sector can be defined as a multiple criteria group decision-making (mcgdm) problem [32, 33]. to handle the mcdm problems, wen et al. [34] initiated novel idea of mixed aggregation by comprehensive normalization technique (macont) approach to execute and make three normalized assessment degrees into a single normalized degree to estimate the deviations of original data. the process of classical macont approach includes (i) different normalization techniques for finding the normalized performance values of options with respect to considered evaluation criteria; (ii) aggregating three normalized performance values; (iii) determining the virtual reference option; (iv) using two arithmetic and geometric operators to determine distances between each option and reference option and finding the subordinate comprehensive scores of options; (v) deriving final comprehensive degrees of options and accordingly prioritize the options. truong & li [35] proposed an integrated macont approach based on the dempster-shafer doctrine and e-step procedure for multiple attribute tradeoff assessment in transportation budget sharing. with the integration of different strategies, nguyen et al. [36] developed an improved macont method and compared with existing normalization-based techniques. apart from these studies, several other applications of macont method have been established in the literature [37-39]. further, ranking comparison (rancom) is a pairwise comparison structure using a three-value measure [40]. the rancom is considered for less experienced des, is categorized by strength to conflicts in criterion associations, is intuitive, time-efficient for dealing complex mcgdm problems, and handles with imprecisions in evaluations of des while certifying high repeatability of outcomes. the rancom comprises finding criterion preferences, constructing a matrix of ranking comparison (mac) with pairwise assessments, computing summed criteria weights (scw), and estimating overall weights of different criteria. as per authors’ knowledge, there is no work which introduces if-modified rccrancom-macont method integrating an if-score function-based des’ weighting procedure, the if-modified relative closeness coefficient (rcc)-based objective weighting model, the if-rancom subjective weighting procedure and the macont framework on ifi. the main contributions and novelties of the work are given as ▪ new if-score function is developed to evade the drawbacks of existent if-score functions (xu [41] xu et al. [42], zeng et al. [43], feng et al. [44]). ▪ new if-distance measure is introduced to conquer the advantages of prior developed if-distances (shen et al. [45], tripathi et al. [46], wu et al. [47], ejegwa & agbetayo [48], li et al. [49]). 410 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi ▪ a hybrid ranking approach is proposed by integrating the if-score function-based experts’ weighting model, an objective-subjective criteria weight-determining model and the macont method with ifi ▪ to exemplify the feasibility and efficacy of introduced ranking framework, it is implemented to a case study of prioritizing the ce interested regions areas in agrifood sector. rest part of this paper is organized in the following way. section 2 confers the background of the work. section 3 first develops new if-score function and its effectiveness over the existent if-score functions. further, this section proposes a distance measure for ifss and discusses its properties. comparative study is performed to verify the consistency and rationality of introduced measure. section 4 introduces a hybrid multi-criteria evaluation method, namely ifmodified rcc-rancom-macont. section 5 implements the proposed method for prioritizing ce interested regions in the agri-food sector. moreover, this section conducts sensitivity and comparative investigations to validate the determined outcomes. finally, section 6 confers the conclusions of the paper. 2. new score function and distance measure for ifss this part of the study first presents the concepts of ifss. to show comparison diverse ifns, new if-score function is introduced to reveal the effectiveness over the existing ifscore functions. next, a modified if-distance measure is introduced to describe dissimilarity on ifss and further compared with previously introduced measures. 2.1 preliminaries here, we present the fundamental definitions used in this work. definition 2.1 [24]. let v = {v1, v2, …, vt} be the fixed universe of discourse. in 1986, atanassov [24] presented the mathematical definition of ifs b on v = {v1, v2, …, vt}, given by eq. (1) as ( )( ) , ( ), ( ) , = l b l b l lb v v v v v (1) where μb: v → [0, 1] and vb: v → [0, 1] symbolize mg and ng of a member vl ∈ v to b, respectively, satisfying 0 ≤ μb(vl) + vb(vl) ≤ 1. the hesitancy degree is defined as πb (vl)= 1μb(vl) vb(vl), for each vl ∈ v. the term “(μb(vl), νb(vl))” is termed as intuitionistic fuzzy number (ifn) and indicated in this study as ψ = (μ, ν), where μ, ν ∈ [0, 1] and 0 ≤ μ + v ≤ 1. definition 2.2 [41]. for any two ifns ψ1 = (μ1, ν1) and ψ2 = (μ2, ν2), some basic operations are discussed as a) ( )1 1 1, ,  =c b) ( ) ( )( )1 2 1 2 1 2, ,     =   c) ( ) ( )( )1 2 1 2 1 2, ,     =   d) ( )1 2 1 2 1 2 1 2, ,        = + − e) ( )1 2 1 2 1 2 1 2, ,        = + − f) ( ) ( )( )1 1 11 1 , , 0,       = − −  g) ( ) ( )( )1 1 1,1 1 , 0.     = − −  intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 411 definition 2.3 [41]. for an ifn ψ = (μ, ν), if-score function and if-accuracy function are defined via eq. (2) and eq. (3), respectively. ( ) ( ), = −xs where ( )  1,1 ,  −xs ( ) ,  = +a where ( )  0,1 . a xu et al. [42] gave the normalized version of if-score function and defined as follows: ( ) ( ) 1 1 , 2  = −+xws where ( )  0,1 . xws further, feng et al. [44] presented the generalized version of if-score function and defined as follows: for p > 1, ( ) 1 (1 ) , 2     + − =     p p p fzs where ( )  0,1 . fzs in recent times, zeng et al. [43] developed the logarithmic function-based if-score function and defined as follows: 2log (2 ) ( ) (1 ) , 100 zcs        − − = − − − −  where ( )  0,1 . zcs definition 2.4 [41]. to aggregate the ifns, xu [41] introduced an “if-weighted average (ifwa)” operator and an “if-weighted geometric (ifwg)” operator on ifns ψl = (μl, νl), l = 1, 2, …, t, and given as ( ) ( ) ( )1 2 1 1 , ,..., 1 1 , ,     = =   = − −      l l t t w w t l l l l ifwa (2) ( ) ( ) ( )1 2 1 1 , ,..., ,1 1 ,     = =   = − −      l l t t w w t l l l l ifwg (3) where, the set {w1, w2, …, wt} denotes the weights of ifns ψl = (μl, νl), l = 1, 2, …, t, and wl lies between 0 and 1, and w1+w2+…+wt =1. definition 2.5 [50]. let a, b, h ifss(v). a real-valued function d: ifss(v)×ifss(v) → [0,1] is said to be a distance measure for ifss if it holds the following requirements: (i). 0 ≤ d (a, b) ≤ 1, (ii). d (a, b) if and only if a = b, (iii). d (a, b) = d (b, a), (iv). if ,a b h  then d (a, b) ≤ d (a, h) and d (b, h) ≤ d (a, h). 2.2. new score function for ifn to rank the ifns, xu [41] defined the concept of score and accuracy functions. further, many improved score functions have been introduced in the context of ifns, while some of the well-known if-score functions, presented in definition 2.3 (xu [41], xu et al. [42], 412 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi zeng et al. [43], feng et al. [44]) present unreasonable results during the comparison of different pairs of ifns. to this aim, this paper proposes a new score function for an ifn. for an ifn ψ = (μ, ν), we present a new if-score function, defined as ( ) ( ) ( ) ( ) 11 1 1 1 . 2( 1) 2          − + − = + − + + −  +   s e e e (4) using eq. (4), we discuss the given axioms: property 2.1. the if-score function s (ψ) given in eq. (4), increases monotonically w. r. t. μ and decreases monotonically w. r. t. ν. property 2.2. the developed if-score function fulfils s((0, 1)) = 0 and s((1, 0)) = 1. property 2.3. let ψ1 = (μ1, ν1) and ψ2 = (μ2, ν2) be the ifns. if μ1 > μ2 and ν1 < ν2, then s(ψ1) > s(ψ2). the following example shows the drawbacks of existent if-score functions example 2.1. let us suppose a pair of ifns, given as ψ1 = (0.5, 0.5) and ψ2 = (0.3, 0.3). to compare these ifns, we apply the proposed and existing if-score functions and obtain the results as follows. it is clear that existent if-score functions by xu [41], xu et al. [42] and feng et al. [44] are unable to discriminate these two different ifns as sx(ψ1) = 0 = sx(ψ2), sxw(ψ1) = 0.5 = sxw(ψ2) and sfz(ψ1) = 0.5 = sfz(ψ2) (p = 1), while the proposed score function (4) obtains the result as s(ψ1) = 0.341 and s(ψ2) = 0.318. thus, ψ1> ψ2. example 2.2. consider that ψ1 = (0.7, 0.3) and ψ2 = (0.6, 0.2) are two ifns. the score functions by xu [41], xu et al. [42], zeng et al. [43] and feng et al. [44] are unable to make difference between given ifns as sx(ψ1) = 0.4 = sx(ψ2), sxw(ψ1) = 0.7 = sxw(ψ2), szc(ψ1) = 0.4 = szc(ψ2) szc(ψ1) = 0.4 = szc(ψ2) and sfz(ψ1) = 0.7 = sfz(ψ2) (p > 1), while the proposed if-score function gives a reasonable result, which as s(ψ1) = 0.484 and s(ψ2) = 0.505. thus, ψ1 < ψ2. it demonstrates the usefulness of developed if-score function over several extant functions [41-44]. 2.3. new distance measure for ifss to quantify the dissimilarity between ifss, the concept of distance measure has been proposed by szmidt & kacprzyk [51]. in the past decade, several if-distance measures have been introduced with wider perspectives, while some of them (shen et al. [45], tripathi et al. [46], wu et al. [47], ejegwa & agbetayo [48], li et al. [49]) present counterintuitive results in the measurement of dissimilarity between ifss. for e, g ifss(v), we develop a modified distance measure for describing the dissimilarity between ifss a and b:      (   ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) min ( ), ( ) min ( ), ( ) max ( ), ( ) 1 ( , ) max ( ), ( ) mi7 a l b l a l b l a l b l a l b l b l a l a l b l b l a l a l b l b l a l a l b l b l a l a l b l b l a l v v v v v v v v v v v v v v v v v v v v v v v v d a b v vt                           − + − + − + − + − + − + − + = − +    (     )  (   )     ) 1 . n ( ), ( ) min ( ), ( ) max ( ), ( ) max ( ), ( ) min ( ), ( ) min ( ), ( ) max ( ), ( ) max ( ), ( ) t a l b l b l a li a l b l b l a l a l b l b l a l a l b l b l a l v v v v v v v v v v v v v v v v                 =             −    + − +     − + −    (5) intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 413 theorem 2.1. the function d (a, b), presented by eq. (5), is a valid distance measure for ifss. proof. to prove this theorem, we need to verify the properties (i)-(iv) of definition 2.5. (i). for any two ifss a and b, we know that 0 ≤ μa(vl), μb(vl) ≤ 1, 0 ≤ va(vl), vb(vl) ≤ 1 and 0 ≤ πa(vl), πb(vl) ≤ 1, for each vl ∈ v. it implies that each term of eq. (5) ranges between 0 and 1 and thus, 0 ≤ d (a, b) ≤ 1. (ii). consider that d (a, b) = 0. as each term of eq. (5) lies between 0 and 1, therefore, from our assumption, d (a, b) = 0 is only possible when every term of eq. (5) should be equal to zero. it implies that μa(vl) = μb(vl), va(vl) = vb(vl) and πa(vl) = πb(vl), for each vl ∈ v. hence, a = b. conversely, if a = b, then it is obvious from eq. (5) that d (a, b) = 0. hence, d (a, b) = 0 iff a = b. (iii). it is clear from the property of modulus and ifs, so we have omitted the proof. (iv). for any three ifss a, b and h on a finite universe of discourse v = {v1, v2, …, vt}, if , a b h then 0 ≤ μa(vl) ≤ μb(vl) ≤ μh(vl) ≤1, 0 ≤ vh(vl) ≤ vb(vl) ≤ va(vl) ≤1 and 0 ≤ πh(vl) ≤ πb(vl) ≤ πa(vl) ≤1. it implies that for each vl ∈v, we have ( ) ( ) ( ) ( ) ,   −  −a l b l a l h lv v v v ( ) ( ) ( ) ( ) ,   −  −a l b l a l h lv v v v ( ) ( ) ( ) ( ) ,   −  −a l b l a l h lv v v v     0 min ( ), ( ) min ( ), ( ) min ( ),      a l h l a l b l b lv v v v v   ( ) min ( ), ( ) 1,    a l h l a lv v v      0 min ( ), ( ) min ( ), ( ) min ( ), ( )       a l h l a l b l b l a lv v v v v v  min ( ), ( ) 1,  h l a lv v        0 min ( ), ( ) min ( ), ( ) min ( ), ( ) min ( ), ( ) 1,           a l h l a l b l b l a l h l a lv v v v v v v v        0 max ( ), ( ) max ( ), ( ) max ( ), ( ) max ( ), ( ) 1,           a l h l a l b l b l a l h l a lv v v v v v v v        0 max ( ), ( ) max ( ), ( ) max ( ), ( ) max ( ), ( ) 1,           a l h l a l b l b l a l h l a lv v v v v v v v        0 max ( ), ( ) max ( ), ( ) max ( ), ( ) max ( ), ( ) 1,           a l h l a l b l b l a l h l a lv v v v v v v v also, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,       −  −a l h l h l a l a l b l b l a lv v v v v v v v ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,       −  −a l h l h l a l a l b l b l a lv v v v v v v v ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , .       −  −  a l h l h l a l a l b l b l a l lv v v v v v v v v v hence, it shows that d (a, b) ≤ d (a, h). similarly, we can prove that d (b, h) ≤ d (a, h). example 2.3. let us assume some pairs of ifss, which are case-1: {a= (0.7, 0.3), b = (0, 0)}, case-2: {a = (0.4, 0.6), b = (0, 0)}, case-3: {a = (0.5, 0.45), b = (0.55, 0.4)}, case-4: {a = (0.45, 0.4), b = (0.45, 0.32)}, case-5: {a = (0.3, 0.41), b = (0.5, 0.344)} and case-6: {a = (0.41, 0.2), b = (0.22,0.28)}. to compare each pair of ifss, the proposed and existent if-distance measures are applied to the given six pairs of ifss. table 1 presents the required computational results. from table 1, we can observe the unreasonable results of existent measures, while the proposed if-distance measure successfully obtains the discrimination between all the given pairs of ifss. 414 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi table 1 comparative results of developed and extant if-distance measures (highlighted texts denote counter-intuitive results) measures case-1 {(v1,0.7, 0.3)} {(v1,0,0)} case-2 {(v1,0.4, 0.6)} {(v1,0,0)} case-3 {(v1,0.5, 0.45)} {(v1,0.55, 0.4)} case-4 {(v1,0.45, 0.4)} {(v1,0.45, 0.32)} case-5 {(v1,0.3, 0.41)} {(v1,0.5, 0.344)} case-6 {(v1,0.41, 0.2)} {(v1, 0.22, 0.28)} shen et al. [45] 0.533 -0.033 0.05 0.026 0.152 0.155 shen et al. [45] 0.539 0.51 0.052 0.052 0.159 0.192 tripathi et al. [46] 0.582 0.684 0.001 0.005 0.013 0.013 wu et al. [47] 0.351 0.306 0.05 0.046 0.139 0.139 ejegwa & agbetayo [48] 1.0 1.0 0.001 0.006 0.024 0.021 li et al. [49] 1.0 1.0 0.036 0.081 0.156 0.146 proposed measure 0.9143 0.8857 0.0504 0.0851 0.2051 0.1913 3. an integrated if-modified rcc-rancom-macont method for mcgdm problems in this section, we propose an improved version of macont approach that incorporates the des’ weight-determining procedure, the ifwa operator, and an integrated objective-subjective weighting model with ifi. this approach firstly computes the des’ weights through a collective if-score function and rank reciprocal model and further calculates an aggregated if-decision matrix (a-ifdm). next, the weights of criteria are computed with the combination of objective and subjective weights via if-modified rcc and if-rancom models, respectively. finally, the stepwise procedure of macont approach is presented to solve the mcdm problems with ifi. fig. 1 presents the pictorial representation of the proposed framework. fig. 1 flow diagram of the if-modified rcc-rancom-macont framework intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 415 this method involves subsequent procedural steps: step 1: construction of linguistic decision-matrix (ldm). for an ifi-based mcgdm problem, let f = {f1, f2, ..., fm} be a defined set of options over considered criteria set r = {r1, r2, ..., rn}. to make an optimal decision, a committee of ‘p’ des b = {b1, b2, ..., bp} is invited to provide their opinion regarding the assessment of each option over defined criteria. let x = (xij (k))m×n be an ldm, where xij (k) denotes the linguistic performance value of an option fj over a criterion rj presented by kth de. further, the ldm is converted into if-decision matrix (ifdm) using likert scale. step 2: computing the weights of des. for determining the des’ weights, let us consider a linguistic significance degree of each de based on their skills, expertise and knowledge. let bk = (μk, νk) be an if-significance degree of kth de, then the weight of kth de is estimated as follows, where k = 1, 2, 3, ..., p. step 2.1: using developed if-score function, a normalized assessment rating of kth de is calculated, where k = 1, 2, 3, ..., p. ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 1 1 1 2 .                 − − =  + − + − + + −       + − + − + + −      =  k k k k k k k k p k k k k k k k k k e e k e e a (6) step 2.2: estimate the rank of each de using proposed if-score rating and estimate the rank (rk) of each expert. then, rank-based performance rating of each expert using the rank reciprocal weight of de is estimated through eq. (7) as 1 1 , 1,2,..., . (1/ ) r k k t k k r a k p r = = =  (7) step 2.3: based on the combination of eq. (6) and eq. (7) by means of a parameter ‘α’, we present the following formula to derive the overall weight of kth de, given by eq. (8). ( ) (1 )( ), 1,2,..., ,r k k ka a k p  = + − = (8) where α [0,1] denotes the expert weighting strategy coefficient for assigning significance degree to each de. moreover, λk denotes the weight vector of des’ set b = {b1, b1, …, bp} with λk belongs to [0,1] and λ1 + λ2 +…+ λp = 1. step 3: determine the a-ifdm. in this step, all individual opinions of des are collected through to obtain the combined performance degree of each option over defined criteria. to this aim, we use ifwa (or ifwg) operator to create an a-ifdm ( ) ( , ),ij m n ij ijx x  = = where (1) (2) ( ) (1) (2) ( )( , ) , ,..., or , ,..., .( ) ( ) k k p p ij ij ij ij ij ij ij ij ijx xf x x x xi wa ifw xg  = = (9) step 4: determination of criteria weights by if-modified rcc-rancom model during the procedure of mcdm, weight of criterion is an important aspect for the experts. in the present work, we propose new weighting model with the combination of objective weight using if-modified rcc tool and subjective weight by if-rancom tool 416 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi with ifi. we discuss an integrated procedure based on the combination of if-modified rcc and if-rancom models. case i: objective weighting through the if-modified rcc model. this approach involves subsequent steps: step 4a: determine the positive distance matrix (pdm) ( , )ijd x + on an aggregated ifn ijx and if-positive ideal rating (if-pir) ρ+ through eq. (10). ( )                     1 , min , min , max , max , 7 min , min , max , max , min , min ,                                                           + + + + + + + + + + + + + + + + + + + + − + − + − + − + − + −  = + − + −     + − + −    + −  ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij d x     , max , max ,      + +                          + −    ij ij (10) where ( , )ij ij ijx  = and ( , ) (1,0).     + + + = = compute the negative distance matrix (ndm) ( , )ijd x − on an aggregated ifn ijx and if-negative ideal rating (if-nir) ρvia eq. (11). ( )                     1 , min , min , max , max , 7 min , min , max , max , min , min ,                                                           − − − − − − − − − − − − − − − − − − − − − + − + − + − + − + −  = + − + −     + − + −    + −  ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij ij d x     , max , max ,      − −                          + −    ij ij (11) wherein ( , )ij ij ijx  = and ( , ) (0,1).     − − − = = step 4b: using eq. (10)-eq. (11), we make a relative closeness-decision matrix (rcdm) ψ = (βj)1×n, wherein ( , ) , ( , ) ( , ) ij j ij ij d x d x d x     − − + = + (12) and ( 1 1 ( , ) ( , )ij jd x d x m  − −= 2 3( , ) ( , )j jd x d x − −+ + )... ( , ) ,mjd x −+ ( 1 1 ( , ) ( , )ij jd x d x m  + += 2 3( , ) ( , )j jd x d x + ++ + )... ( , ) ,mjd x ++ j = 1, 2, …, n. intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 417 step 4c: finding the objective weight (wj o) from eq. (13) as 1 ,   = =  jo j n jj w j = 1, 2, …, n. (13) case ii: subjective weight of criteria by if-rancom approach. this approach includes subsequent procedure as step 4d: linguistic values of criteria for expert panels are defined and changed into ifns. we create an aggregated matrix to evaluate the performance of each criterion using eq. (2). step 4e: with the use of proposed if-score function, we find an if-score value of each aggregated ifn for each criterion and form a column matrix ω = (ηj)1×n, j = 1, 2, …, n, where ( ) ( ) ( ) 1 11 1 1 , 2( 1) 2          − + −  = + − + + −  +   j j j j j j j j je e e (14) step 4f: considering the if-score ratings, we estimate the preference order of defined criteria. step 4g: based on the pairwise assessment of criteria, create a ranking comparison matrix (rcm) c = (θlj)n×n, j = 1, 2, …, n, as 1 2 1 11 11 1 2 21 22 2 1 2 ,                =       n n n lj n n n nn r r r r r r where ( ) ( ) ( ) ( ) ( ) ( ) 1, , 0.5, , 0, .      = =   j l lj j l j l if s r s r if s r s r if s r s r (15) here, the comparison result is presented by δij. step 4h: determination of summed criteria weight (scw) of criteria as 1 , = = n j lj t q j = 1, 2, …, n. (16) step 4i: finding the subjective weight of criteria as 1 , = =  js j n jj q w q j = 1, 2, …, n. (17) case iii: calculation of combined weight of criteria by if-modified rcc-rancom model. in this process, we integrate objective and subjective weights obtained through ifmodified rcc and if-rancom models, respectively. thus, the combined weighting formula is defined as (1 ) ,o s j j jw w w = + − j = 1, 2, …, n. (18) wherein γ [0,1] is weighting precision coefficient. 418 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi step 5: normalize an a-ifdm. in this step, the normalized a-ifdm (na-ifdm) u = (ԑij)m×n from if-score value of aifdm ( ) nij mz x  = is calculated by the normalization techniques, given as follows: step 5.1. this process abolishes the dimensions of criteria with linear sum-based procedure. thus, the linear na-ifdm u(1) = (ԑij (1))m×n is created by eq. (19), where ( ) ( ) ( ) 11 1 ( ) ( ) , , 1 ( ) 1 ( ) , , m ij ij j b i ij m ij ij j n i x x r r x x r r  = =    =       (19) where rb and rn symbolize benefit and cost-type of criteria, respectively. step 5.2. this process uses the max operator and finds the normalized extended assessment matrix, i.e., na-ifdm u(2) = (ԑij (2))m×n using eq. (20), where ( ) ( ) ( ) ( )( ) 2 ( ) max ( ) , , 1 ( ) max 1 ( ) , , ij i f ij j b ij ij i f ij j n x x r r x x r r       =   (20) where f denotes the set of alternatives. step 5.3. this process utilizes the max-mix operator and gets the normalized extended assessment matrix, i.e., na-ifdm u(3) = (ԑij (3))m×n via eq. (21), where ( )3 ( ) min ( ) , , max ( ) min ( ) max ( ) ( ) , . max ( ) min ( ) ij i f ij j b i f ij i f ij ij i f ij ij j n i f ij i f ij x x r r x x x x r r x x        −  − =  −   − (21) step 5.4. computation of overall normalized assessment matrix. combining the aforesaid normalized values by means of two parameters α and β, the averaged na-ifdm u = (ԑij)m×n is computed, where ԑij is an if-score value obtained using the following expression: ( ) ( ) ( )1 2 3 (1 ) ,ij ij ij ij     = + + − − (22) where ԑij denotes the averaged na-pf-dm. where α ∈ [0, 1], β ∈ [0, 1] and these two balanced coefficients are estimated by des. in this work, we consider α = β = 1/3. step 6: evaluation of reference/average solution matrix (asm). the reference/average solution matrix is calculated, where the reference value ςj is given by eq. (23) as 1 1 ( ), m j ij i m   = =  j = 1, 2, …, n. (23) intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 419 step 7: calculate the subordinate scores for each alternative. this step computes the subordinate scores (pi (1) and pi (2)) of ith alternative by means of the distance between each option and the reference value, where i = 1, 2, …, m. here, we combine the arithmetic and geometric weighted operators for combining distances on each alternative and asm in relation to all criteria. ( )(1) 1 1 1 ( ) ( ) 1 , ( ) j j n w n j ijt i j ij j n w j ij js p w         = = =  −   = − + −   −       (24) from eq. (24), it is combined best and worst performance of options over defined criteria. here, t (t = 1, 2, . . ., n) represents the part of criteria that holds ԑij < ςj, and s (s = 1, 2, . . ., n) represents the part of criteria that holds ԑij > ςj. in addition, ϑ (0 ≤ ϑ ≤ 1) denotes the preference parameter. if expert considers comprehensive performance of options, then higher rating of ϑ is taken, whereas expert considers individual performance of options, then smaller rating of ϑ is taken. ( ) ( )(2) max ( ) (1 )min ( ) ,i j ij j j ij j jj p w w     = − + − − (25) where i = 1, 2, …, m and j = 1, 2, …, n. ϕ (0 ≤ ϕ ≤ 1) is preference parameter. if the des consider the best performance of options, then higher rating of ϕ is taken, whereas des consider worst performance of options, then smaller rating of ϕ is taken. step 8: computation of final comprehensive score (fcs) for each option. in accordance with subordinate scores (pi (1) and pi (2)) of ith alternative, the fcs of ith alternative is calculated by eq. (26), where i = 1, 2, …, m. (1) (2) (1) 2 (2) 2 1 1 1 . 2 ( ) ( ) i i i m m i ii i p p p p p = =    = +        (26) for the precision and consistency of outcomes, we need to utilize normalization procedure to certify that proportions of subordinate scores pi (1) and pi (2) are identical. since values of pi (1) and pi (2) may be negative, therefore, we implement vector normalization procedure, given in eq. (26). step 9: on the basis of decreasing values of fcs (pi), i = 1, 2, …, m, prioritize the options. higher the fcs (pi) determines the best option. 4. experimental results this section first implements the proposed if-modified rcc-rancom-macont method on a case study of ce interest areas evaluation problem in the agri-food sector. further, sensitivity and comparative discussions are provided to test the validity of obtained results. 420 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi 4.1. case study: assessment of ce interested regions in the agri-food sector here, we present a case study related to ce interested regions assessment in the agrifood sector. by utilizing the proposed framework, we recognize ce interested regions that contribute to the sustainability perspectives of ce scheme in the agri-food sector. for this purpose, some ce areas of interest are taken for evaluation, which are resource efficiency (f1), digital transformation (f2), circular business model (f3), supply chain management (f4), product life cycle management (f5). to collect the data, we have conducted online and offline meetings with the des. next, a committee is formed consisting of four des, who are having expertise in the agricultural sustainability, i4.0, ce and logistics management. based on the discussion with experts’ committee and analysis of the relevant literature, nine criteria are taken to evaluate the ce areas of interest in agri-food sector and specified in table 2. table 2 different types of sustainability criteria (krstic et al. [15], verdouw et al. [19]) for assessing ce interested regions in agri-food sector dimensions criteria description economic implementation costs (r1) it comprises all the costs associated with learning how to use the technology and adapting it for specific purposes, purchasing equipment, software development, training of employees, etc. operational costs (r2) it refers to the costs related to logistics activities including ordering, packing, supply, collection, storage, shipping, etc. material value preservation (r3) it refers to the degree of preservation of the value of materials to minimize its value degradation and maximize its residual value, and consequently increase its utilization, life cycle, and the overall value. social health (r4) it refers the degree of impact of logistics activities on the health of people. safety (r5) it refers to improve safety culture throughout the logistics network. labor market (r6) it considers the positive impact of ce interest area on the labor market. environmental waste reduction (r7) it considers the role of the ce area of interest to waste reduction. emissions reduction (r8) it considers the role of the ce interest area to the reduction of emissions. energy resource preservation (r9) it refers the degree of preservation of non-renewable and renewable energy resources. the implementation of if-modified rcc-rancom-macont is discussed as follows. step 1: to assess the ce interested regions concerning the taken evaluation criteria, we adopt table 3 from tripathi et al. [52], which represents lvs and their corresponding ifns. the des are asked to present the importance rating of alternatives using table 3. next, different opinions of four des are collected regarding the performance of each alternative area and created a ldm with regard to each criterion, and presented in table 5. intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 421 table 3 lvs and corresponding ifns for assessing ce interested regions in agri-food sector lvs ifns extremely high (eh) (0.95, 0.05) very very high (vvh) (0.85, 0.1) very high (vh) (0.8, 0.15) high (h) (0.7, 0.2) moderate high (mh) (0.6, 0.3) medium (m) (0.5, 0.4) moderate low (ml) (0.4, 0.5) low (l) (0.3,0.6) very low (vl) (0.2, 0.7) very very low (vvl) (0.1, 0.8) extremely low (el) (0.05, 0.95) table 4 linguistic decision opinions for assessing ce interested regions in agri-food sector f1 f2 f3 f4 f5 r1 (vvl,ml,l,m) (vl,l,ml,ml) (el,vl,l,l) (m,l,l,vl) (ml,el,l,vl) r2 (l,vl,ml,l) (vl,ml,vl,m) (vl,vl,vl,ml) (vvl,ml,l,ml) (l,ml,vl,vl) r3 (mh,l,ml,h) (ml,ml,h,h) (l,vvh,m,m) (m,h,mh,vvh) (l,vvh,ml,h) r4 (vh,m,h,ml) (mh,h,vh,ml) (m,vh,m,ml) (ml,vvh,m,h) (vh,l,mh,m) r5 (ml,m,h,mh) (h,l,mh,mh) (vl,h,mh,m) (h,vvh,h,m) (m,mh,vh,vh) r6 (m,ml,vvh,h) (h,m,ml,vvh) (ml,ml,h,vh) (mh,m,m,h) (mh,h,mh,m) r7 (m,mh,ml,mh) (mh,mh,h,vh) (ml,m,eh,h) (m,mh,ml,h) (h,m,h,mh) r8 (h,ml,vvh,ml) (vh,m,ml,mh) (l,ml,mh,mh) (m,h,m,vh) (m,h,mh,vh) r9 (m,h,mh,m) (l,mh,mh,m) (h,mh,m,vvh) (vvh,h,m,h) (mh,vh,vh,m) step 2: with the utilization of table 2 and table 3, the linguistic significance ratings of des are assigned as per their skills, expertise and knowledge and given in table 4. from eq. (6)-eq. (8), weight of expert is calculated and mentioned in table 5. table 5 weight of des for assessing ce interested regions in agri-food sector des b1 b2 b3 b4 lvs vh vvh vh eh ak 0.228 0.252 0.228 0.291 ak r 0.138 0.241 0.138 0.483 λk 0.183 0.247 0.183 0.387 step 3: to obtain the group decision opinion regarding the performance of each ce interested regions, we use eq. (9) to create an a-ifdm in table 6. step 4: for determining an objective weight of indicators using if-modified rcc model, the first step is to measure pdm and ndm. based on eq. (10), we find pdm ( , )ijd x + between an aggregated element ijx and if-is ρ+ = (1, 0), given as 0.674, 0.658, 0.341, 0.371, 0.357, 0.331, 0.334, 0.356 and 0.332. next, we calculate the ndm ( , )ijd x − between an aggregated element ijx and if-ais ρ= (0, 1) through eq. (11), and presented as 0.366, 0.386, 0.699, 0.668, 422 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi 0.684, 0.709, 0.704, 0.682 and 0.706. the rc-dm is derived via eq. (12) and given as β1 = 0.352, β2 = 0.37, β3 = 0.672, β4 = 0.643, β5 = 0.657, β6 = 0.682, β7 = 0.678, β8 = 0.657 and β9 = 0.68. from eq. (13), an objective weight of criteria is estimated as wj o = (0.0653, 0.0686, 0.1246, 0.1193, 0.1219, 0.1265, 0.1258, 0.1219, 0.1261). to find subjective weight of each criterion, an if-rancom tool is used. in the following, each expert presents his/her views concerning assessment of each considered criterion and changed into an aggregated ifn using eq. (2). next, the score values of criterion are determined via eq. (14) and accordingly ranked them. table 7 presents an ifscore value and preference of each criterion, respectively. table 6 aggregated ifdm to identify the most potential ce interest areas in the agri-food sector criteria f1 f2 f3 f4 f5 r1 (0.381, 0.517) (0.343, 0.556) (0.235, 0.678) (0.307, 0.591) (0.227, 0.69) r2 (0.297, 0.603) (0.379, 0.519) (0.284, 0.615) (0.335, 0.563) (0.273, 0.626) r3 (0.557, 0.334) (0.596, 0.297) (0.605, 0.306) (0.734, 0.187) (0.665, 0.244) r4 (0.587, 0.321) (0.616, 0.291) (0.572, 0.342) (0.685, 0.226) (0.559, 0.35) r5 (0.568, 0.328) (0.564, 0.33) (0.539, 0.354) (0.692, 0.22) (0.719, 0.213) r6 (0.656, 0.251) (0.705, 0.215) (0.655, 0.265) (0.606, 0.29) (0.594, 0.303) r7 (0.551, 0.347) (0.710, 0.213) (0.722, 0.218) (0.598, 0.297) (0.620, 0.278) r8 (0.59, 0.315) (0.599, 0.311) (0.51, 0.386) (0.691, 0.231) (0.703, 0.219) r9 (0.577, 0.32) (0.517, 0.381) (0.73, 0.192) (0.71, 0.2) (0.676, 0.249) table 7 an aggregated ifn and preferences of criteria for assessing ce interested regions in agri-food sector criteria b1 b2 b3 b4 aggregated ifn crisp value rank r1 vvh m ml l (0.528, 0.378) 0.391 9 r2 vh h m ml (0.6, 0.307) 0.452 8 r3 vh m h m (0.615, 0.294) 0.464 7 r4 mh ml vh vh (0.702, 0.229) 0.535 3 r5 l m h vvh (0.696, 0.222) 0.536 1.5 r6 h mh m vh (0.698, 0.225) 0.536 1.5 r7 l vh ml h (0.64, 0.269) 0.487 6 r8 m h vvh mh (0.676, 0.234) 0.521 5 r9 mh vh m h (0.686, 0.228) 0.529 4 next, an rcm is constructed using eq. (15). in accordance with an rcm, the scws are computed using eq. (16). lastly, we obtain subjective weight of criteria with eq. (17). table 8 presents the required computational steps of the if-rancom model. by utilizing eq. (18), the combined weight of criteria is computed as a sum of γ multiplied by objective weight of and the complement of γ i.e., (1 − γ) multiplied by the subjective weight of criteria. for γ = 0.5 combined weight of criteria is estimated and presented as wj = (0.0388, 0.0528, 0.0932, 0.1399, 0.1597, 0.162, 0.1061, 0.1165, 0.131). intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 423 table 8 results of if-rancom model for assessing ce interested regions in the agrifood sector criteria rcm qj wj s r1 r2 r3 r4 r5 r6 r7 r8 r9 r1 0.5 0 0 0 0 0 0 0 0 0.5 0.0123 r2 1 0.5 0 0 0 0 0 0 0 1.5 0.037 r3 1 1 0.5 0 0 0 0 0 0 2.5 0.0617 r4 1 1 1 0.5 0 0 1 1 1 6.5 0.1605 r5 1 1 1 1 0.5 0.5 1 1 1 8 0.1975 r6 1 1 1 1 0.5 0.5 1 1 1 8 0.1975 r7 1 1 1 0 0 0 0.5 0 0 3.5 0.0864 r8 1 1 1 0 0 0 1 0.5 0 4.5 0.1111 r9 1 1 1 0 0 0 1 1 0.5 5.5 0.1358 next, fig. 2 exhibits the significance values of considered criteria for assessing ce interested regions in the agri-food sector. labor market (r6) (0.162) is the most significant criterion, safety (r5) (0.1597) is the second most significant criterion, health (r4) (0.1399) is third, energy resource preservation (r9) (0.131) is fourth and emissions reduction (r8) (0.1165) is fifth most vital criteria for assessing ce interested regions in the agri-food sector. fig. 2 weights of considered criteria for ce interested regions in agri-food sector step 5: from eq. (19)-eq. (22) and table 6, we determine the first, second, third and averaged normalized a-ifdm, and presented in tables 9 and 10. for averaged na-ifdm, we took α = β = 1/3. step 6: based on table 10 and eq. (23), we calculate reference/average solution matrix f0 = (ςj)1×9 and rating ςj, (j = 1, 2, …, 9) of options over each criterion as follows: ς1 = 0.5131, ς2 = 0.5709, ς3 = 0.4809, ς4 = 0.4691, ς5 = 0.4956, ς6 = 0.511, ς7 = 0.5436, ς8 = 0.5383 and ς9 = 0.5474. 424 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi table 9 first and second normalized a-ifdm for assessing ce interested regions in agrifood sector first normalization second normalization f1 f2 f3 f4 f5 f1 f2 f3 f4 f5 r1 0.154 0.1698 0.2404 0.188 0.2483 0.619 0.684 0.968 0.7558 1.0 r2 0.208 0.1659 0.2159 0.186 0.2239 0.929 0.741 0.964 0.8316 1.0 r3 0.175 0.1881 0.1881 0.238 0.2114 0.733 0.791 0.791 1.0 0.888 r4 0.193 0.2047 0.1867 0.233 0.1827 0.832 0.881 0.803 1.0 0.786 r5 0.183 0.1814 0.1726 0.228 0.2353 0.776 0.771 0.734 0.9695 1.0 r6 0.205 0.2218 0.2018 0.188 0.1835 0.924 1.0 0.91 0.8480 0.827 r7 0.169 0.2245 0.2256 0.187 0.194 0.751 0.995 1.0 0.8268 0.86 r8 0.189 0.1916 0.1623 0.226 0.231 0.82 0.83 0.703 0.9776 1.0 r9 0.177 0.1568 0.2315 0.226 0.2087 0.765 0.677 1.0 0.9754 0.902 table 10 third and combined normalized a-ifdm for assessing ce interested regions in agri-food sector third normalization combined normalization f1 f2 f3 f4 f5 f1 f2 f3 f4 f5 r1 0.0 0.248 0.946 0.475 1.0 0.258 0.367 0.718 0.473 0.749 r2 0.782 0.0 0.894 0.42 1.0 0.64 0.302 0.692 0.479 0.741 r3 0.0 0.215 0.215 1.0 0.582 0.303 0.398 0.398 0.746 0.561 r4 0.214 0.442 0.079 1.0 0.0 0.413 0.509 0.356 0.744 0.323 r5 0.158 0.141 0.0 0.886 1.0 0.372 0.365 0.302 0.694 0.745 r6 0.558 1.0 0.477 0.121 0.0 0.562 0.741 0.529 0.386 0.337 r7 0.0 0.979 1.0 0.305 0.438 0.307 0.733 0.742 0.44 0.497 r8 0.393 0.427 0.0 0.925 1.0 0.467 0.483 0.288 0.709 0.744 r9 0.273 0.0 1.0 0.924 0.695 0.405 0.278 0.744 0.708 0.602 step 7: from eq. (24)-eq. (25) and table 10, we compute two subordinate scores pi (1) and pi (2), i = 1, 2, 3, 4, 5, of each option (for ϑ = 0.5 and ϕ = 0.5) as p1 (1) = 0.1288, p2 (1) = 0.3409, p3 (1) = 0.6275, p4 (1) = 0.5826, p5 (1) = 0.6728, p1 (2) = -0.0084, p2 (2) = 0.001, p3 (2) = -0.0026, p4 (2) = 0.0091 and p5 (2) = 0.0058. step 8: based on eq. (26), the final comprehensive score (fcs) pi, i = 1, 2, 3, 4, 5, of each ce interest area is computed as p1 = -0.2451, p2 = 0.1835, p3 = 0.1814, p4 = 0.5909 and p5 = 0.5026. step 9: rank the alternatives based on obtained fcss p1 = -0.2451, p2 = 0.1835, p3 = 0.1814, p4 = 0.5909 and p5 = 0.5026. thus, “supply chain management (f4) is the most potential ce interested regions contribute to the sustainability development of ce structure in agri-food sector and the prioritization ordering of ce interested regions is f4 f5 f2 f3 f1. 4.2. sensitivity investigation this section analyses the impacts of criteria weighting parameter on the final results. to this aim, we present the following cases case i (objective weight shuffling): considering the if-modified rcc model in place of integrated if-modified rcc-rancom, i.e., we are taking γ = 1.0 in eq. (18) to assess the ce interested regions in the agri-food sector. using an if-modified rcc tool, the fcss intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 425 of ce interest areas are calculated and presented as p1 = -0.3258, p2 = 0.0425, p3 = 0.1808, p4 = 0.5079 and p5 = 0.4703. corresponding to decreasing ratings of fcss, the ranking order of ce interested regions as f4 f5 f2 f3 f1, and the “supply chain management (f4)” is the most potential area among a set of ce regions of interest with respect to nine criteria. case ii (subjective weight shuffling): this case obtains the fcss of ce interested regions in agri-food sector by taking subjective weight instead of combined objectivesubjective weight of indicators, i.e., we are putting γ = 0.0 in eq. (18). using the subjective weighting if-rancom method, the fcss of ce interested regions in agri-food sector are computed and given as p1 = -0.163, p2 = 0.3142, p3 = 0.0898, p4 = 0.6017 and p5 = 0.4847. thus, the prioritization order of considered ce interested regions in the agri-food sector is f4 f5 f2 f3 f1, and the “supply chain management (f4)” is the most suitable choice in relation to given evaluation criteria. case iii (integrated weight shuffling): considering the combined if-modified rccrancom approach, i.e., γ = 0.5 in eq. (18), the fcss of ce interested regions in agrifood sector are calculated as p1 = -0.2451, p2 = 0.1835, p3 = 0.1814, p4 = 0.5909 and p5 = 0.5026. by means of decreasing values of fcss, the prioritization order of ce interested regions in agri-food sector is f4 f5 f2 f3 f1 and hence, “supply chain management (f4)” is the most optimal choice for the given data set. fig. 3 present required outcomes in relation to the weighting factor (γ). fig. 3 variation of fcss of ce interested regions w.r.t. different values of factor (γ) 4.3. comparative study this subsection performs comparative study to analyze rationality of proposed ranking approach in comparison with existing if-mcdm models, given by mardani et al.’s ifcopras [53], komal’s waspas method [54], qin et al.’s if-topsis model [55] and 426 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi tripathi et al.’s if-cocoso [52]. to this aim, we have applied these extant methods on aforesaid case study of sep sites selection problem. 4.3.1. if-copras method using the if-copras [53] on abovementioned case study of ce interested regions assessment, we first compute the maximization and minimization indices, which are denoted as si and ti, respectively. the computed values are s1 = (0.554, 0.347), s2 = (0.586, 0.320), s3 = (0.590, 0.323), s4 = (0.639, 0.268), s5 = (0.616, 0.297), t1 = (0.554, 0.347), t2 = (0.068, 0.911), t3 = (0.090, 0.885), t4 = (0.075, 0.902) and t5 = (0.093, 0.882). with the use of score function (xu et al. [42]), the score values of maximization and minimization indices are computed as sxw (s1) = 0.604, sxw (s2) = 0.633, sxw (s3) = 0.633, sxw (s4) = 0.686, sxw (s1) = 0.659, sxw (t1) = 0.085, sxw (t2) = 0.078, sxw (t3) = 0.103, sxw (t4) = 0.087 and sxw (t5) = 0.106. further, relative degree (rd) of each interested region is obtained as 0.3505, 0.3695, 0.3571, 0.3906 and 0.369, and finally, a ud of alternative is estimated as 89.75%, 94.6%, 91.43%, 100.00% and 94.48%. corresponding to the decreasing ratings of uds, prioritization of ce interested regions in the agri-food sector is 4 2 5 3 1f f f f f and thus, “supply chain management (f4)” is the best region of interest among the set of five ce interested regions in agri-food sector. 4.3.2. if-waspas method by applying if-waspas method [54] on the aforesaid case study, the additive importance (ci (1)) using weighting sum model is computed as c1 (1) = (0.587, 0.313), c2 (1) = (0.614, 0.292), c3 (1) = (0.627, 0.286), c4 (1) = (0.666, 0.242) and c5 (1) = (0.652, 0.262). next, the multiplicative importance (ci (2)) using weighted product model is determined as c1 (2) = (0.585, 0.316), c2 (2) = (0.604, 0.300), c3 (2) = (0.614, 0.296), c4 (2) = (0.66, 0.247) and c5 (2) = (0.645, 0.267). using xu et al.’s score function (xu et al. [42]) the score values of additive and multiplicative importance values are calculated and shown as sxw (c1 (1)) 0.637, sxw (c2 (1)) = 0.661, sxw (c3 (1)) = 0.67, sxw (c4 (1)) = 0.712, sxw (c5 (1)) = 0.695, sxw (c1 (2)) = 0.635, sxw (c2 (2)) = 0.652, sxw (c3 (2)) = 0.659, sxw (c4 (2)) = 0.707 and sxw (c5 (2)) = 0.689. lastly, total significance of each ce interest area is estimated as the arithmetic mean of additive and multiplicative significance values and given as 0.6358, 0.6564, 0.6647, 0.7096 and 0.6919. thus, the ranking order of ce interested regions in agri-food sector is 4 5 3 2 1f f f f f and the “supply chain management (f4)” is the best choice for considered data set. 4.3.3. if-topsis method by applying if-topsis method [55] on abovementioned case study of ce interested regions assessment in agri-food sector, we compute the best and worst values from aifdm and obtains as ϕj + = {(0.227, 0.690), (0.273, 0.626), (0.734, 0.187), (0.685, 0.226), (0.719, 0.213), (0.705, 0.215), (0.722, 0.218), (0.703, 0.219), (0.730, 0.192)} and ϕj + = {(0.381, 0.517), (0.379, 0.519), (0.557, 0.334), (0.559, 0.350), (0.539, 0.354), (0.594, 0.303), (0.551, 0.347), (0.510, 0.386), (0.517, 0.381)}. using the similarity measure, relative closeness rating (rcr) of each ce interested region is estimated as 0.2504, 0.3849, 0.4524, 0.7242 and 0.6278, respectively. hence, the preference order of ce interested intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 427 regions in agri-food sector is 4 5 3 2 1f f f f f and “supply chain management (f4)” is the best choice among a set of five ce interested regions in agri-food sector. fig. 4 presents the acquired ranking results by the proposed and extant if-mcdm methods. as per the obtained results, we found that the preference ranks of ce interest areas by the developed if-modified rcc-rancom-macont approach is slightly different from the mardani et al.’s if-copras [53], komal’s waspas method [54], qin et al.’s if-topsis model [55] and tripathi et al.’s if-cocoso [52], while the most suitable choice “supply chain management (f4)” is the same by all the if-mcdm models. from fig. 5, the spearman's rank correlation coefficient (srcc) is higher than 0.9 for each existing approach with the developed ranking framework. moreover, the ws-coefficient (sałabun and urbaniak [56]) is are higher than 0.85 for each existing approach with the developed ranking framework. the benefit of ws-coefficient describes the similarity degree of preference order of options, which illustrates the consistency of prioritization of ce interested regions in agri-food sector, is high (mishra et al. [57], biswas et al. [58], jafar et al. [59], ullah and shah [60]). accordingly, it can be observed that there is very robust association between prioritization outcomes. also, the generalization and flexibility discussed by the developed if-modified rcc-rancom-macont framework are lacking in the existing methods. fig. 4 ranking orders of ce interested regions in the agri-food sector by different ifmcdm models 428 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi fig. 5 variation of srcc and ws-coefficient of the proposed method with different existing approaches 4.3.4. if-cocoso method this model first computes the additive and multiplicative importance of alternatives, which are similar as if-waspas model [54]. based on additive and multiplicative importance, the relative compromise ratings are determined as t1 (1) = 0.1893, t2 (1) = 0.1955, t3 (1) = 0.1979, t4 (1) = 0.2113, t5 (1) = 0.206, t1 (2) = 0.7773, t2 (2) = 0.8026, t3 (2) = 0.8127, t4 (2) = 0.8676, t5 (2) = 0.8459, t1 (3) = 0.8959, t2 (3) = 0.925, t3 (3) = 0.9367, t4 (3) =1.0 and t5 (3) = 0.975. finally, the overall compromise ratings of five ce interested regions in agri-food sector are estimated and given as 0.5649, 0.5832, 0.5906, 0.6305 and 0.6147, respectively. based on decreasing ratings of overall compromise ratings, thus, the ranking order of ce interested regions in agri-food sector is 4 5 3 2 1f f f f f and “supply chain management (f4)” is the best interested region among a set of five ce interested regions in agri-food sector. next, we present the advantages of developed if-modified rcc-rancom-macont over the existing if-mcdm models as follows: ▪ the developed if-modified rcc-rancom-macont model utilizes new if-score function, which avoids shortcomings of extant if-score functions (xu [41], xu et al. [42], zeng et al. [43], feng et al. [44]). ▪ the proposed method uses new if-distance measure, which overcomes drawbacks of extant if-distance measures (shen et al. [45], tripathi et al. [46], wu et al. [47], ejegwa and agbetayo [48], li et al. [49]). the developed distance measure is implemented to compute distance matrix in criteria weighting procedure and further employed to determine the deviation from the normalized weighted evaluation values of alternatives to the reference point. ▪ the introduced model calculates the des’ weights through new score function and rank reciprocal-based weighting procedure, while qin et al.’s if-topsis [55] and komal’s if-waspas [54] ignores the significance weight of des in the procedure of group decision-making. ▪ the proposed model computes weight of indicators through a collective objectivesubjective weighting procedure combining the if-modified rcc and the if-rancom models for objective and subjective weights of indicators, which does not only estimate weight of indicators with given data but also involve the des’ opinions during the criteria weighting procedure. intuitionistic fuzzy macont method for logistics 4.0 based circular economy... 429 5. conclusion this paper aimed to develop a ranking framework for identifying the ce interested regions which are most affected by implementing the i4.0 technologies during the assessment of logistics process in agri-food sector. to this aim, we have assessed the performance of five ce interest areas including resource efficiency, digital transformation, circular business model, supply chain management, product life cycle management with respect to nine criteria under the three dimensions of sustainability. in this framework, we have derived the weights of involved des using an integrated score function and rank reciprocal-based procedure. in the following, new if-score function has been developed on ifss with its efficiency over existing ones. next, the group decision opinions have been aggregated into a single opinion regarding the performance value of each ce interest area on the basis of given criteria. further, the criteria weights have been computed with integrated if-modified rccrancom approach in which objective weight has been derived from if-modified rcc tool, while the subjective weights have determined via if-rancom procedure. to do so, new if-distance measure has been given for ifss with its efficiency over existing distance measures. finally, a stepwise procedure of if-modified rcc-rancom-macont methodology has been presented to rank the five ce interested regions in agri-food sector, which proves its effectiveness and practicality. the acquired result shows that a region “supply chain management” has highest final comprehensive score among a set of five ce interested regions over economic, social and environmental aspects of sustainability. sensitivity assessment has been conducted to see the effect of criteria weighting coefficient on the final result. lastly, a comparison with extant approaches has been made to test the effectiveness and robustness of proposed ranking framework. this paper has some limitations, which are i) the developed if-modified rccrancom-macont approach does not replicate the way to find the importance rating of des to determine the weight of des, and ii) one of key limitation of proposed method is its computational procedure, which is the outcomes of executing the modified rcc, rancom and macont approaches using the proposed if-distance measure, if-score function to estimate subordinate comprehensive scores of options. further, deriving final comprehensive degrees of options and determine the prioritization the options. this complexity of if-modified rcc-rancom-macont is a potentially protective characteristic for implementing by diverse des. in future studies, the if-modified rccrancom-macont framework can be extended with several mcgdm models such as stochastic identification of weights (sitw), reference ideal method (rim), characteristic object method (comet), stable preference ordering towards ideal solution (spotis), and so on. the developed if-modified rcc-rancom-macont method can be extended with diverse uncertain environments, such as q-rof-rough sets, linear diphantine fuzzy, type-2 fuzzy sets, pythagorean hypersoft sets and different types of linguistic term sets. in future, we can develop new methodologies to solve the interactive ce interested regions assessment in agri-food sector. funding: this research was conducted under a project titled “ongoing research funding program”, funded by king saud university, riyadh, saudi arabia under grant number (orf-2025-533). acknowledgement: the authors extend their appreciation to king saud university for funding this work through ongoing research funding program (orf-2025-533), king saud university, riyadh, saudi arabia. 430 a. r. mishra, p. rani, d. pamucar, a. m. alshamrani, a. f. alrasheedi references 1. nurhasanah, n., machfud, mangunwidjaja, d., romli, m., 2020, a literature review on the design of intelligent supply chain for natural fibre agroindustry, international journal of supply chain management, 9, pp. 182-197. 2. li, l., 2024, mechanism analysis of smart 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for circular economy using uncertain mcdm approach, management science advances 1(1), pp. 31-52. 59. jafar, m. n., muniba, k., saqlain, m., 2023, enhancing diabetes diagnosis through an intuitionistic fuzzy soft matrices-based algorithm, spectrum of engineering and management sciences, 1(1), pp. 73-82. 60. ullah, f., shah, w. s., 2024, optimizing multi-attributive decision-making: a novel approach through matrix theory and fuzzy hypersoft sets, spectrum of engineering and management sciences, 2(1), pp. 100-109. https://doi.org/10.1007/978-3-030-50417-5_47 https://doi.org/10.1007/978-3-030-50417-5_47 facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 473 498 https://doi.org/10.22190/fume210505061o © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper evaluation of logistic flows in green supply chains based on the combined dematel-anp method nikita osintsev1, aleksandr rakhmangulov1, vera baginova2 1nosov magnitogorsk state technical university, magnitogorsk, russia 2russian university of transport, moscow, russia abstract. supply chains and transport corridors have a significant impact on the socioeconomic and environmental situation in the regions where the elements of the logistics infrastructure are located. the achievement of the goals of the concept of sustainable development in these regions is ensured, among other things, as a result of the formation of green supply chain management (gscm), that is, as a result of changes in existing approaches to supply chain management. analysis of the practice of supply chain management showed a wide variety of parameters and indicators of logistics flows used in decision-making at different stages of managing these flows. the authors propose a universal system of the logistic flows parameters and indicators for the gscm, corresponding to the principles of the concept of sustainable development. a methodology for ranking indicators of logistics flows based on a combined dematel-anp method has been developed. the results of a case study on the evaluation of logistics flows for the gscm are presented. the ranks of logistics flow indicators obtained in the study are proposed to be used in gscm to adjust of the logistics flows actual parameters to achieve the goals of the concept of sustainable development. key words: green supply chains, logistic flows, mcdm, dematel, anp, sustainable development, gscm 1. introduction supply chains, transport and logistics systems have become the determining factors in the development of the global economic system. companies around the world are striving to build sustainable supply chains that deliver products to market more efficiently and more environmentally than their competitors. the volume of trade flows in the world economic system since 2000 shows a positive trend [1]. the expansion of the global market for logistics services since 2016 is 3.48% and, according to estimates [2], by 2022 it will amount to about 12.25 trillion us dollars. received may 05, 2021 / accepted august 31, 2021 corresponding author: aleksandr rakhmangulov, nosov magnitogorsk state technical university,lenin street, 38, 455000 magnitogorsk, russia e-mail: ran@magtu.ru 474 n. osintsev, a. rakhmangulov, v. baginova the requirements for reliability, environmental friendliness, and social responsibility of organizations elements of supply chains are constantly growing [3]. in such conditions, the authors were motivated by the need to improve the existing supply chain management (scm) methods [4, 5] for assessing logistics flows to select green technologies and ensure sustainable development of supply chains. sustainable development of supply chains requires the use of methods for making management decisions to change the parameters of logistics flows based on the measurement and evaluation of their indicators. the complexity of green supply chain management (gscm) stems from the lack of research on the relationship between various parameters and indicators of logistics flows [6]. moreover, there is no comprehensive approach to evaluating these parameters and indicators [7] against the background of an increase in the number of criteria and alternatives due to the need to achieve sustainable development goals [8]. multi-criteria decision-making methods (mcdm) are an effective tool for solving these problems. the scientific area for improving mcdm is actively developing currently. a literature review [9] showed an increase in the number of mcdm-related publications over the past ten years. mcdms are actively used to solve various problems in the field of climate change [10], sustainable engineering [9], green logistics [11], gscm [12, 13], reverse logistics [14]. however, the selection of the appropriate mcdm method for a specific situation requires additional research [15]. the most used mcdms are ahp, anp, dematel, topsis, electri, promrethee, and combinations of these methods. the lack of a comprehensive and systematic evaluation of all types of logistics flows, as well as the relationship between indicators and parameters of flows from the point of view of the concept of sustainable development, is the main drawback of most of the existing gscm methods and models [6]. this study aims to develop a methodology for evaluating logistics flows for systemic supply chains management in accordance with the goals of the concept of sustainable development. the main contribution of this study is a new universal system of parameters and indicators of logistics flows in green supply chains. a feature of the proposed system is the ability to comprehensively assess all logistics flows for compliance with aspects of the concept of sustainable development and the quality of green supply chain management. the combined dematel-anp method used in the study considers the relationship between parameters and indicators of logistics flows. the authors hypothesize that this approach contributes to improving the quality of managerial decision-making to adjust the actual parameters of logistics flows in accordance with the goals of the concept of sustainable development. the remainder of this paper is organized as follows. section 2 contains a literature review of research of the gscm based on the evaluation of the logistics flows parameters and indicators using the dematel and anp methods. section 3 presents the original system of parameters and indicators of logistics flows in the gscm, as well as the methodology and algorithm of the hybrid dematel-anp. section 4 contains the results of a case study on the application of the hybrid dematel-anp method to evaluate flows in green supply chains. in the conclusion, the main results of the study are presented, and the directions of its development are discussed. evaluation of logistic flows in green supply chains based on the combined dematel-anp method 475 2. literature review 2.1. evaluation of logistic flows the complexity of decision-making in gscm is associated with insufficient knowledge of the system of indicators and parameters of logistics flows. moreover, there are practically no methods of comprehensive evaluation of green supply chains for compliance with the principles of the concept of sustainable development [7]. the system of logistics flows is an object of management in supply chains and includes material (product) flows, information flows, financial flows, and value (services) flows. [16] currently, a universal system of parameters and indicators of logistics flows is not presented in scientific publications. however, this system is necessary to evaluate the compliance of supply chains with the goals of the concept of sustainable development. different researchers suggest using a variety of sets and systems of parameters and indicators for all or for specific logistics flows. transport mass, transport route and transport time are the main indicators of material flow in the scm according to [17, 18]. additional parameters include the flow map (set of points, path, length), flow speed and time, flow intensity [19]. tyapukhin and coworkers proposed four groups of the logistics flows parameters: quantity, quality, costs, and time [20]. the authors [19] combine the logistics flows parameters into two groups. the first group includes physical parameters that reflect the spatio-temporal properties of logistic flows. the second group unites the statistical parameters that characterize the patterns of change in physical parameters. kozlov proposed to evaluate logistics flows using their average values and the indicator of their disorganization [21]. the vector (direction of movement) and scalar (number of resources) values of logistic flows are studied in [22]. material use indicators and beneficial output indicators are proposed in [23]. these indicators can be used to calculate various material productivity or material intensity indicators. the relationships between the quantitative parameters of flows and stocks in scm are established in [24], and in [25] – the relations between the parameters of various logistics flows. turki and coworkers proposed a discrete flow model for optimizing the closed-loop supply chain based on the criterion of minimum total costs. the model optimizes capacities of manufacturing stock, purchasing warehouse and the vehicle, the value of returned used end-of-life products [26]. several researchers have focused on examining specific logistics flows. for example, material flow theory principles applied to logistics and scm in the context of sustainable development are used in [27, 28]. it is shown that material flow management should be carried out considering the development of a particular country (region). “owner”, “region”, “time”, as well as “flow rate”, “flow chart”, “flow direction”, “flow capacity” are proposed to be used as the main attributes of material flows [28]. “cooperation” indicators for information flow, “costs” for financial flow, and “delivery times” for material flow are proposed to evaluate the performance of the supply chain in [29]. bröcker and coworkers [30] proposed an estimate of trade and transportation flows considering economic growth, globalization, and changing commodity composition of trade flows, along with the evolution of value-to-weight ratios for commodity groups. gerini and sciomachen consider the performance indices of the system as the main indicator of the evaluation of the cargo flow at a warehouse logistic [31]. a performance indicator for material flow effectiveness in production systems is proposed in [32]. martinico-perez and coworkers have proposed two groups of indicators for evaluating 476 n. osintsev, a. rakhmangulov, v. baginova material flow to achieve sustainable development goals. the extensive indicators group includes domestic extraction, direct material input, domestic material consumption, physical trade balance, net addition to stock, and the intensive indicators group consists of resource efficiency or resource intensity, resource productivity, material flow with respect to size of territory [33]. jong and coworkers point to a lack of research on impact of changes in transport costs and times (by mode) on the trade flows and suggest a new model for trade flows in europe that is integrated with a logistics model for transport chain choice through logsum variables [34]. porkar and coworkers propose two groups of indicators for evaluating material flows in the green supply chain with the aim of increasing total profit, depending on the direction of these flows: forward (quality and green design indicators) and backward (green scrap score indicators) [35]. a system of twenty-six universal indicators of logistics flows for evaluating and forming a “resource balance” in green supply chains are proposed in [7]. a complex of thirty-five indicators for evaluating the total costs (financial flows) arising from the formation of innovation flows in the logistics system is proposed in [36]. the “metric” of information flow in logistics is studied in [37]. kolinski and coworkers studied integration of information flow for greening supply chain management. a set of indicators is proposed to evaluate the effectiveness of distribution processes, considering operational and economic aspects, as well as information flow. information flow is evaluated by the following indicators: reliability of information flow, return of delivery rate due to erroneous data, average time for analysis of data on delivery plans [38]. a literature review of research in the field of logistics flows evaluation allows to conclude that at present, a universal system of parameters and indicators of logistics flows is not presented in scientific publications. however, this system is necessary to evaluate the compliance of supply chains with the goals of the concept of sustainable development. 2.2. logistic flows in green supply chains scientific research over the past 15-20 years has increasingly focused on the gscm. this is because, on the one hand, the impact of elements of the supply chains on the environment is objectively increasing, and on the other hand, environmental legislation is being tightened in almost all countries. against this background, there is an increase in the number of publications devoted to the theory and practice of green supply chain management. we have identified six main subject areas in the field of gscm as a result of the analysis of current scientific publications: policy, synthesis, purchasing, manufacturing, green logistics, and reverse logistics (table 1). the authors reached the following conclusions from a review of gscm research: ▪ at present, the conceptual and terminological apparatus of the gscm, the principles of sustainable development and green logistics have been formed. various gscm indicator systems have been developed and used, ▪ the factors of sustainable development of supply chains were identified and systematized. various solutions are proposed for the implementation of green technologies in logistics. the mcdm apparatus is actively used in the gscm, ▪ the complexity of managing green supply chains lies in the insufficiently studied interconnections of indicators and parameters of logistics flows. there is no universally accepted universal system for assessing logistics flows in green supply evaluation of logistic flows in green supply chains based on the combined dematel-anp method 477 chains. assessing the sustainability of supply chains and making decisions on the selection and implementation of green technologies is carried out, as a rule, relating to individual functional elements or areas of the logistics system: purchases, production, warehousing, transport, and marketing. this approach reduces the productivity of green technologies and does not effectively achieve sustainable development goals. table 1 research in gscm field of study characteristic references policy issues of business ethics and corporate social responsibility, environmental audits, as well as solving problems related to environmental protection, compliance with the requirements of legislation and the state in the field of ecology [39-43] synthesis gscm literature reviews, research, and tutorials [16], [44-56] purchasing environmental issues related to supplier-buyer relationships, environmental decisions, certification, and environmental quality standards [13], [57–65] manufacturing problems of design, development, and production of ecological products to reduce harmful emissions and waste [66–70] green logistics environmental issues related to the sustainable transportation, handling and storage of hazardous materials, inventory management, warehousing, the choice of locations for transport and logistics infrastructure, the use of packaging [11], [70–76] reverse logistics problems of separation of reverse flows (material flows from consumers to sources) into recyclable and waste [14], [77–79] 2.3. dematel and anp methods in green supply chain management multi-criteria decision making (mcdm) is widely used today to solve complex multifactorial design and green supply chain management problems. mcdms can be used to quantify trade-offs between economic, social, and environmental goals for sustainable supply chain development [6]. the authors chose a combination of decision making trial and evaluation laboratory method (dematel) [80] and analytic network process (anp) [81, 82] in this study. dematel is used to identify the interdependencies between the criteria under study and to develop a map of the network relationships between the criteria. the anp method [83] is a generalization of the ahp (analytic hierarchy process). anp is used to define dependencies and feedbacks between criteria, structuring these relationships between criteria in the form of a network. the combination of dematel with anp is widely used today to solve various problems. gölcük and baykasoğlu propose to distinguish the following four groups of such combinations: network relationship map of anp; inner dependency in anp; clusterweighted anp and dematel-based anp (danp) [84]. the merit of integrating dematel and anp is the ability to determine the degree of dependence between the dematel criteria and use them to normalize the unweighted supermatrix in anp. 478 n. osintsev, a. rakhmangulov, v. baginova the performed analysis shows that the dematel method in gscm was applied to solve various problems. for example, to evaluate the factors of sustainable development of scm, to select environmental suppliers, to implement green initiatives or best green practices, to choose a development strategy and in other fields (table 2). table 2 research and practice of the dematel and anp methods application in gscm field of study references dematel evaluation of factors influencing the implementation of initiatives in gscm [85] evaluations of factors for choosing environmentally friendly logistics companies [86] evaluation and selecting sustainable suppliers [87–89], [90] identifying critical factors in gscm [91] selecting suppliers with competencies in supply chain carbon management [92] greenfield analysis [93] evaluation of the municipal logistics sustainability within the framework of the concepts of industry 4.0 and logistics 4.0 [94] prioritizing green supply chains implementation within the technologyorganization-environment (toe) approach [95] green corporate social responsibility evaluation [96] evaluation of the gscm practices [97] evaluation of the key success factors (ksfs) for implementing networked scm [98] predicting and measuring the likelihood of success of the gscm implementing [53] dematel-anp green project management in supply chains [99] renewable energy selection [100] assessment of the risk and reliability of the implementation of oil and gas construction projects [101] financial statement supply chain assessment [102] evaluation of the effectiveness of the flight safety management system [103] choosing a company strategy [104] assessing the competitiveness of a green supply chain [99] analysis of consumer demand when choosing a supplier in green supply chains [105] assessment of sustainable development of small and medium-sized enterprises [106] the main identified disadvantages of using the dematel and anp methods in the gscm are: ▪ the generally accepted universal system of parameters and indicators of logistics flows has not been developed, ▪ the methods of complex evaluation of the set of logistics flows indicators are underdeveloped, ▪ the relationship of indicators and parameters of logistics flows from the standpoint of the concept of sustainable development is insufficiently studied, ▪ the evaluation of logistics flows is considered in relation to specific (isolated) elements of the gscm, ▪ the evaluation of all logistics flows in gscm to achieve sustainable development goals is carried out insufficiently comprehensively and systematically. evaluation of logistic flows in green supply chains based on the combined dematel-anp method 479 the use of the combined dematel – anp method in the gscm has the following advantages. first, the method makes it possible to identify relationships both between groups of parameters of logistics flows (see section 3.1) and between indicators of logistics flows within groups of parameters. groups of parameters are more needed to assess the achievement of sustainable development goals and the implementation of a strategy for the development of supply chains, while indicators of logistics flows are used in the operational management of flows in green supply chains. secondly, dematel – anp allows you to assess the relationship between indicators of logistics flows. the variety of properties of logistic flows requires considering the influence on them of many external and internal factors. accounting for this impact is a prerequisite for the effectiveness of the gscm. thirdly, the dematel method allows visualizing causal relationships between parameters and indicators of logistics flows in the form of network maps. the division of indicators into groups of "causes" and "effects" allows you to better understand the structural relationships in the system of indicators of logistics flows and increases the efficiency of management by the decision maker. finally, the use of network maps in the anp method allows one to assess the impact of changes in parameters and indicators of logistics flows on the achievement of gscm goals. 3. methods 3.1. system of parameters and indicators of logistic flows research on improving existing or developing new systems for logistics flows evaluation is motivated by the need to solve the problems considered and eliminate the shortcomings of known evaluation methods. the complexity of gscm lies in the insufficiently studied relationships between indicators and parameters of logistics flows. currently, there is no methodology for a comprehensive assessment of parameters and indicators of logistics flows. for example, making decisions to ensure on-time delivery can lead to an increase in the irregularity of freight traffic, which will negatively affect energy intensity and the volume of greenhouse gas emissions. on the other hand, the desire to increase the discreteness of the flow (decreasing the order size) allows you to achieve a more uniform flow, but also leads to an increase in transport costs. in addition, gscm uses different sets of logistics flow parameters (indicators) and decision-making methods at different levels of management. logistic flows at the operational management level are considered as a collection of flow elements, for example, vehicles, orders, or logistics operations in the service flow. the objects of control of such discrete logistics flows are their individual elements. logistic flows at the strategic management level are considered as continuous, characterized by the average values of their parameters and indicators. finally, the existing scm methodology focuses mainly on economic criteria and does not consider environmental and social aspects. 480 n. osintsev, a. rakhmangulov, v. baginova the authors have developed an original universal system of the logistics flows parameters and indicators [6]. the proposed system is focused on logistics flows evaluation for compliance with the principles and goals of sustainable development. the logistic flow evaluation system (lfes) consists of five groups of parameters (fig. 1): ▪ economic parameters (p1) characterize the efficiency of using all types of resources in the logistics system, as well as the degree of its economic viability, ▪ energy-ecological parameters (p2) characterize the efficiency of energy use during the movement of logistics flows and their impact on the environment, ▪ quality parameters (p3) characterize the safety and timeliness of movement and processing of logistics flows, as well as the quality of their management, ▪ statistical parameters (p4) reflect the patterns of change in the controlled parameters of logistics flows, ▪ controlled (physical) parameters of flows (p5) characterize the intensity of logistics flows and their spatio-temporal changes. goal – sustainability of the logistics flows in green supply chain management economic parameters (p1) energy-ecological parameters (p2) quality parameters (p3) statistical parameters (p4) flow's physical parameters (p5) p ro fi t (i 1 ) o p er at in g e x p en se s (i 2 ) f ix ed i n v es tm en t (i 3 ) t h e en er g y i n te n si ty ( i4 ) g re en h o u se g as e m is si o n s o f c o 2 ( i5 ) s af et y o f ca rg o t ra n sp o rt at io n ( i6 ) t im el in es s o f ca rg o t ra n sp o rt at io n ( i7 ) t h e co ef fi ci en t o f fl o w c o n tr o ll ab il it y ( i8 ) t h e m as s (q u an ti ty ) o f fl o w ( i1 3 ) t h e sp ee d o f fl o w ( i1 4 ) t h e le n g th o f th e ro u te ( i1 5 ) t h e co ef fi ci en t o f fl o w s ir re g u la ri ty ( i9 ) t h e co ef fi ci en t o f co m p le x it y s tr u ct u re o f fl o w ( i1 0 ) t h e co ef fi ci en t o f fl o w s d is cr et en es s (i 1 1 ) t h e co ef fi ci en t o f d if fe re n ti ab il it y o f fl o w (i 1 2 ) fig. 1 logistic flow evaluation system (lfes) lfes is two-tiered. the logistics flow parameters (criteria p1-p5) form the first level of the hierarchy. the parameters are used to assess the compliance of logistics flows with the sustainable development goals. the indicators of logistics flows (sub-criteria i1-i15) form the second level of the hierarchy. they are used to monitor and manage the logistics flows of a specific supply chain. bridging the gap between the actual values of indicators and those required in accordance with the sustainable development goals is carried out through the selection and application of green logistics instruments [107]. a brief description of the indicators (sub criteria) for assessing logistics flows in green supply chains is presented in the table. 3. evaluation of logistic flows in green supply chains based on the combined dematel-anp method 481 table 3 characteristics of logistics flows indicators sub-criteria characteristic profit (i1) difference between total revenue and operating costs operating expenses (i2) the sum of all costs associated with converting investments into profits fixed investment (i3) cash flow for the formation of fixed assets the energy intensity (i4) the amount of energy spent on the movement of the logistics flow greenhouse gas emissions of co2 (i5) the total volume of greenhouse gas emissions from all sources involved in the movement of the logistics flow safety of cargo transportation (i6) comprehensive indicator of the material flow movement without damage, pollution, and loss timeliness of cargo transportation (i7) comprehensive indicator of the material flow movement by the appointed date, regularly, or at the required speed the coefficient of flow controllability (i8) the ratio of the number of information messages on compliance with the indicators of safety and timeliness of transportation to the total number of management decisions the coefficient of flows irregularity (i9) deviation of the logistics flows physical parameters of from their average values the coefficient of complexity structure of flow (i10) the number of streams within the logistic flow the coefficient of flows discreteness (i11) the number of elements of the logistic flow in the stream the coefficient of differentiability of flow (i12) changing the structure of the logistics flow (number of streams) in the process of movement the mass (quantity) of flow (i13) the total number of elements in the logistics flow the speed of flow (i14) the speed of movement of the logistics flow elements the length of the route (i15) distance traveled by a logistic flow element while moving along a route 3.2. decision making trial and evaluation laboratory method (dematel) the dematel method includes five main stages. stage 1. building an initial direct-relation matrix. the initial data for the construction of this matrix are expert evaluating of the strength of the influence of the parameter and indicator (criteria, in general and sub criteria) i on the criterion j, where i, j = 1,2, ... n, n is the number of evaluating criteria. the power of influence is evaluating using a fivelevel scale: 0 – non influence, 1 – low influence, 2 – medium influence, 3 – high influence, 4 – extremely high influence. then the initial average matrix a of size n×n is formed, containing the average estimates of experts eq. (1): 482 n. osintsev, a. rakhmangulov, v. baginova 11 1 1 1 1 j n i ij in n nj nn a a a a a aa a a a        =         (1) where aij is the average expert evaluation of the i-th criterion impact on the j-th criterion, aij = 0, for i = j. stage 2. calculate the normalized initial influence matrix. the initial average matrix a is used to calculate the normalized matrix of direct relations x according to eqs. (2) and (3): x a= (2) 1 1 1 1 1 1 min , max max n n ij ij i n j n j i a a      = =      =         (3) stage 3. compute the total influence matrix eq. (4). 2 1lim( ... ) (1 )k k t x x x x x − → = + + + = − (4) stage 4. computing the levels of influence and effect. the level of influence and effect of each criterion, as well as its importance in comparison with other criteria, are determined depending on the values of the calculated vectors di and rj. the di and rj values show the direct and inverse influence of each criterion on other criteria and allow establishing causal relationships in the system of criteria and sub criteria. the vector di is calculated as the sum of the columns of the common matrix of relations t, and the vector rj – as the sum of the rows of the matrix t. the elements of the matrix t and the vectors di and rj are calculated using the following eqs. (5), (6) and (7): , , 1, 2,...,ij n n t t i j n   = =  (5)   1 1 1 , 1,2,..., n i ij i n j n d t t i n  =    = = =     (6) 1 1 1 , 1, 2,..., n j ij j n i n r t t j n  =     = = =       (7) the value (di + rj), for i = j, shows the importance of the i-th criterion in relation to other criteria, that is, the strength of the relationship. the larger the value (di + rj), the greater the number of relationships of the i-th criterion with other criteria. evaluation of logistic flows in green supply chains based on the combined dematel-anp method 483 the value (di – rj), for i = j, shows the effect of the i-th criterion on other criteria. if the value (di – rj), with i = j is positive, then the i-th criterion can be attributed to the “influence” group. if (di – rj), with i = j is negative, then the i-th criterion is influenced by other criteria and belongs to the “effect” group. stage 5. obtaining the causal diagram, network relation map. a causal diagram shows the structure of the relationship between the studied criteria. the basis of the diagram are points, the ordinate of which is the value (di + rj) in a rectangular coordinate system, and the abscissa is the value (di – rj). each point on the diagram corresponds to the i-th criterion. the points of the diagram are connected by relationships. the chart displays only those relationships between criteria that satisfy the condition eq. (8): , ,ijt i j  (8) where α is the threshold value, which is set by experts or calculated by the eq. (9): 1 1 2 n n ij i j t n  = = =  (9) filtering relationships using a threshold value eliminates insignificant relationships between criteria in the decision-making model. 3.3. analytic network process the anp method includes five main stages. stage 1. formation of the network structure of the criteria system model. criteria are clusters of model elements. the elements of the model are sub criteria. indicators are classified as sub criteria for the investigated lfes model. accordingly, the clusters are parameters in the lfes model. elements and clusters of the network model are connected by relationships. the initial structure of the network model can contain all possible links between elements and clusters, or only pre-selected links in the original criteria model, such as in lfes (fig. 1). moreover, the network structure of the model can only include those relationships that are determined to be significant using various methods, such as dematel (section 3.2., stage 5). stage 2. construction of a pairwise comparison matrices and priority vectors. pairwise comparison of elements is carried out with the involvement of experts and using a nine-point rating scale by analogy with the ahp [83] the priority vector is constructed by normalizing the eigenvectors of the local priorities of the judgment matrix. eigenvectors are calculated as geometric mean of the judgment matrix elements. the priority vector shows the strength of the influence of each element on other elements in the model. stage 3. unweighted supermatrix formation. an unweighted supermatrix is constructed based on the results of the previous stage. the unweighted supermatrix includes the priorities obtained because of various pairwise comparisons. objectives, criteria (sub criteria) and alternatives are placed in the rows and columns of the supermatrix. the order of the elements in the supermatrix is irrelevant. if there are no interconnections between the elements of the criteria system, then zero is set at the intersection of the corresponding row and column of the supermatrix. the relationships are determined according to the criteria system model or using different methods, for example, dematel. the unweighted 484 n. osintsev, a. rakhmangulov, v. baginova supermatrix shows the influence of each criterion on other criteria in the studied model. the general equation for the unweighted supermatrix tc eq. (10):is:                                       = nn c nj c n c in c ij c i c n c j cc nnm n n n iim i i i m nnmn n jjmj j m c ttt ttt ttt c c c x c c c x c c c x c...c x c...c x c...c x t            1 1 1111 2 1 2 1 11 12 11 1 111111 1 (10) where xn is the n-th cluster; сnm is the m-th element in the n-th cluster; tc ij is the vector of the priority of the elements influence which compared in the j-th cluster with the i-th cluster. stage 4. weighted and limit supermatrices formation. the unweighted supermatrix tc is transformed into a weighted tw supermatrix by normalizing the sum of elements in any of its columns to one [83]. this is because the clusters are usually interdependent on the network, and the items in the columns are separated by the number of clusters. then the weighted supermatrix tw is transformed into the limit supermatrix tl by raising it to a large power eq. (11): lim k l w k t t → = (11) where k is an arbitrarily large number. the exponentiation of k is performed until all elements of each row of the supermatrix are identical. the final weights of the criteria and subcriteria of the model under study (lfes) are the result of the calculations presented. 3.4. combined dematel-anp the general scheme of the combined dematel-anp method is shown in fig. 2. evaluation of logistic flows in green supply chains based on the combined dematel-anp method 485 phase 1 define the criteria initial direct relation matrix phase 2 dematel normalize the initial influence matrix network relation map phase 3 anp compute total influence matrix network structure formation construct pairwise comparison matrices limit supermatrix formation unweight and weighted supermatrix formation criteria weight academic experience practical experience strategies, programs, and projects experience literature review fig. 2 combined dematel-anp algorithm evaluation of logistics flows parameters and indicators in gscm is carried out in three stages using the combined dematel-anp (fig. 3). phase 1. the selection and justification of logistics flows parameters and indicators for a specific transport system or supply chain is performed based on a literature review and management practice. this case study uses the logistics flows parameters and indicators system presented in section 3.1. phase 2. relationships between groups of the logistics flows parameters and indicators are analyzed using dematel. a network map of the relationship between parameters and indicators is the result of this analysis. phase 3. the results of phase 2 are used to build the network structure of the logistics flow sustainability model in gscm, that is, to achieve the main goal of the lfes. the weight and rank of each logistic flow parameter and indicator in the gscm is calculated using anp. 486 n. osintsev, a. rakhmangulov, v. baginova justification of the logistics flows parameters and indicators in green supply chains phase 1 justification of parameters and indicators using dematel to analyze relationships between logistics flows parameters phase 2 dematel phase 3 anp using anp to rank logistics flow indicators in green supply chains 2.1 initial direct relation matrix of the logistics flows parameters and indicators 2.2 normalize the initial influence matrix 2.3 compute total influence matrix 2.4 calculation of the strength of the relationship and influence between the logistics flows parameters and indicators 2.5 obtaining the causal diagram academic experience green programs, strategies, and projects best green practices literature review using dematel to analyze relationships between logistics flows indicators 3.1 formation of a logistic flow evaluation system (lfes) 3.2 construct pairwise comparison matrices of the logistics flows parameters and indicators 3.3 supermatrix formation 3.4 limit supermatrix formation 3.5 calculation of the weight of the logistics flows indicators weights of the logistics flows indicators in green supply chains fig. 3 algorithm of the combined dematel-anp method for evaluating the parameters and indicators of logistics flows in green supply chains 4. case study of logistics flow evaluation for green supply chain management the authors used in the case study a system of 5 groups of parameters and 15 indicators of logistics flows, formed in accordance with aspects of the concept of sustainable development (fig. 2). academic experts (5 people, table 4) performed evaluation of the logistics flows parameters and indicators. an example of the results of evaluating the parameters of logistics flows is presented in table. 5. table 4 expert data academic degree number of experts average work experience, years professor, doctor (technical sciences) 2 34 assistant professor, phd (technical sciences) 3 18.5 evaluation of logistic flows in green supply chains based on the combined dematel-anp method 487 table 5 the results of an expert evaluation of the logistics flows parameters parameters p1 p2 p3 p4 p5 experts 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 p ar am et er s p1 0 0 0 0 0 3 3 4 3 3 3 3 4 4 4 1 3 2 2 2 4 3 3 2 3 p2 4 3 1 4 2 0 0 0 0 0 2 2 1 3 0 1 2 0 3 0 2 3 2 2 0 p3 4 4 4 4 3 3 4 2 2 2 0 0 0 0 0 3 3 1 2 2 3 4 3 1 2 p4 1 0 1 2 2 0 0 0 2 1 1 0 0 2 4 0 0 0 0 0 1 0 0 2 1 p5 4 4 3 3 3 3 4 2 3 4 3 4 3 2 1 3 4 1 2 1 0 0 0 0 0 the authors calculated the initial average matrix a, normalized matrix of direct relations x, and total influence matrix t using eqs. (1-4). the results of calculating the matrices a and t for the logistics flows parameters are presented in tables 6 and 7. table 6 initial average matrix a of logistics flows parameters parameters p1 p2 p3 p4 p5 p1 (economic) 0 0.27119 0.30508 0.169492 0.254237 p2 (energy-ecological) 0.237288 0 0.13559 0.101695 0.152542 p3 (quality) 0.322034 0.22034 0 0.186441 0.220339 p4 (statistical) 0.101695 0.05085 0.11864 0 0.067797 p5 (physical parameters) 0.288136 0.27119 0.22034 0.186441 0 table 7 the total influence matrix t of logistic flows parameters parameters p1 p2 p3 p4 p5 p1 (economic) 0.9278 1.0423 1.0291 0.7980 0.9300 p2 (energy-ecological) 0.8261 0.5709 0.6691 0.5427 0.6339 p3 (quality) 1.1314 0.9732 0.7634 0.7832 0.8777 p4 (statistical) 0.4477 0.3698 0.4118 0.2548 0.3460 p5 (physical parameters) 1.1122 1.0097 0.9433 0.7836 0.6978 the authors are computing the levels of influence and effect between the parameters of logistic flows in accordance with eqs. (5-7). the calculation results for d, r, (d + r), and (d – r) are presented in table 8. table 8 results of the dematel calculation for logistic flows parameters parameters d r d + r d – r p1 4.7272 4.4452 9.1724 0.2820 p2 3.2426 3.9660 7.2086 -0.7234 p3 4.5290 3.8167 8.3457 0.7122 p4 1.8302 3.1622 4.9924 -1.3321 p5 4.5466 3.4854 8.0320 1.0612 the authors performed similar calculations for 15 logistics flows indicators. initial average matrix a, the total influence matrix t, and the results of calculating the levels of influence and effect between logistics flows indicators are presented in tables 9-11. 488 n. osintsev, a. rakhmangulov, v. baginova table 9 initial average matrix a of logistics flows indicators indicators i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i1 0 2.2 3.4 i2 3.8 0 2.2 i3 3.6 3 0 i4 0 3.2 i5 1.8 0 i6 0 2 1.4 i7 1.2 0 1.8 i8 2.2 3.2 0 i9 0 1.8 1.6 1.8 i10 2.6 0 2 2.4 i11 3 2.6 0 1.6 i12 2.2 3.2 1.8 0 i13 0 3.2 1 i14 0.8 0 0.4 i15 1 2.4 0 table 10 the total influence matrix t of logistic flows indicators indicators i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i1 1.5606 1.4439 1.6838 i2 2.0437 1.3185 1.7536 i3 2.0742 1.6424 1.5301 i4 1.2857 2.2857 i5 1.2857 1.2857 i6 0.5138 0.9885 0.7220 i7 0.6754 0.6872 0.7375 i8 1.0170 1.4025 0.7312 i9 1.1223 1.2748 1.0137 1.0900 i10 1.6628 1.3682 1.2649 1.3719 i11 1.7144 1.6299 1.0749 1.3228 i12 1.6764 1.7073 1.2837 1.1755 i13 0,1573 0,7735 0,2619 i14 0,1858 0,1558 0,1157 i15 0,2863 0,6335 0,0964 table 11 results of the dematel calculation for logistic flows indicators indicators designation d r d + r d – r profit i1 4.6883 5.6785 10.3668 -0.9902 operating expenses i2 5.1158 4.4047 9.5205 0.7110 fixed investment i3 5.2466 4.9675 10.2141 0.2792 the energy intensity i4 3.5714 2.5714 6.1429 1.0000 greenhouse gas emissions of co2 i5 2.5714 3.5714 6.1429 -1.0000 safety of cargo transportation i6 2.2243 2.2062 4.4304 0.0181 timeliness of cargo transportation i7 2.1001 3.0782 5.1783 -0.9781 the coefficient of flow controllability i8 3.1507 2.1906 5.3413 0.9600 the coefficient of flows irregularity i9 4.5008 6.1759 10.6768 -1.6751 the coefficient of complexity structure of flow i10 5.6678 5.9802 11.6480 -0.3124 the coefficient of flows discreteness i11 5.7420 4.6372 10.3792 1.1048 the coefficient of differentiability of flow i12 5.8429 4.9601 10.8030 0.8828 the mass (quantity) of flow i13 1.1927 0.6293 1.8221 0.5634 the speed of flow i14 0.4573 1.5628 2.0200 -1.1055 the length of the route i15 1.0161 0.4740 1.4901 0.5421 evaluation of logistic flows in green supply chains based on the combined dematel-anp method 489 parameters p1 (economic), p3 (quality), p5 (physical parameters) and indicators i2 (operating expenses), i3 (fixed investment), i4 (the energy intensity), i6 (safety of cargo transportation), i8 (the coefficient of flow controllability), i11 (the coefficient of flows discreteness), i12 (the coefficient of differentiability of flow), i13 (the mass (quantity) of flow), i15 (the length of the route) are assigned to the “influence” group in accordance with the values (di – ri) in table 8 and 11. parameters p2 (energy-ecological), p4 (statistical) and indicators i1 (profit), i5 (greenhouse gas emissions of co2), i7 (timeliness of cargo transportation), i9 (the coefficient of flows irregularity), i10 (the coefficient of complexity structure of flow), i14 (the speed of flow) are assigned to the “effect” group. the authors have developed a network relation map of logistics flows parameters and indicators at the final stage of dematel (fig. 4). the developed structure of relationships between logistics flows parameters and indicators of is used to calculate the weight of each parameter and indicator based on the anp method (section 3.3, stage 3). the authors used super decisions software (http://www.superdecisions.com/) to build the anp model. the unweighted supermatrix, weighted and limited supermatrix (appendix 1-3) are constructed based on pairwise comparison of the model elements on the saaty’s nine-point scale. economic parameters energy-ecological parameters quality parameters statistical parameters flow's physical parameters fig. 4 network relation maps of logistics flows parameters and indicators 490 n. osintsev, a. rakhmangulov, v. baginova the results of ranking the logistics flows indicators in gscm are presented in table 12. table 12 weights and ranks of the logistics flows indicators indicators designation weight rank profit i1 0.16405 1 operating expenses i2 0.01481 14 fixed investment i3 0.12434 2 the energy intensity i4 0.09495 4 greenhouse gas emissions of co2 i5 0.07913 7 safety of cargo transportation i6 0.09098 5 timeliness of cargo transportation i7 0.09698 3 the coefficient of flow controllability i8 0.02335 11 the coefficient of flows irregularity i9 0.06597 8 the coefficient of complexity structure of flow i10 0.02301 12 the coefficient of flows discreteness i11 0.04357 10 the coefficient of differentiability of flow i12 0.01174 15 the mass (quantity) of flow i13 0.06085 9 the speed of flow i14 0.02081 13 the length of the route i15 0.08545 6 the authors have identified interrelationships both between groups of parameters of logistics flows, and between indicators of logistics flows within groups of parameters. the relationship between economic (p1), quality (p3) and physical (p5) groups of parameters is shown in fig. 4. within these groups, profit (i1), the speed of flow (i14) and timeliness of cargo transportation (i7) have the greatest influence. the group of energy-ecological parameters (p2) does not affect the physical parameters and quality parameters. however, a relationship has been established between environmental and economic parameters. decisions to change the values of physical parameters will affect the values of energy and environmental indicators (p2). statistical parameters (p4) group has little effect on the other parameter groups. this group is influenced by physical (p5) and economic parameters (p1). at the same time, the study [108] substantiates the influence of the coefficient of flows irregularity (i9) and the coefficient of complexity structure of flow (i10) on the efficiency of logistic flows management. the predominance of economic and quality indicators is justified by the need to achieve the key goals of scm increasing efficiency and quality. however, study [105] shows the possibility and necessity of improving mcdm by eliminating strong economic criteria at the first stage of ranking. the authors propose to implement this approach relates to assessing the indicators of logistics flows in green supply chains in future studies. the authors recommend using the results of ranking the logistics flows indicators in gscm to select green logistics instruments. a detailed description of these instruments, as well as the methodology for their application, are presented in [107]. the obtained ranks of the logistics flows indicators with green logistics instruments are proposed to be used in the gscm to adjust the actual parameters of the logistics flows to achieve the goals of the sustainable development concept. evaluation of logistic flows in green supply chains based on the combined dematel-anp method 491 5. conclusion a new universal system of the logistics flows parameters and indicators in supply chains has been developed and justified. a feature of the proposed system is its compliance with the principles of the concept of sustainable development and focus on use in green supply chains management. the authors chose a combined dematel-anp method for evaluating and ranking indicators of logistics flows in the gscm. a methodology of applying the combined dematel-anp method for ranking of the logistics flows indicators has been developed. the results of estimating the parameters and indicators of logistics flows using the dematel showed that the parameters p1 (economic), p3 (quality), p5 (physical parameters) and indicators i2 (operating expenses), i3 (fixed investment), i4 (the energy intensity), i6 (safety of cargo transportation), i8 (the coefficient of flow controllability), i11 (the coefficient of flows discreteness), i12 (the coefficient of differentiability of flow), i13 (the mass (quantity) of flow), i15 (the length of the route) are assigned to the “influence” group. parameters p2 (energy-ecological), p4 (statistical) and indicators i1 (profit), i5 (greenhouse gas emissions of co2), i7 (timeliness of cargo transportation), i9 (the coefficient of flows irregularity), i10 (the coefficient of complexity structure of flow), i14 (the speed of flow) are assigned to the “effect” group. ranking of indicators of logistics flows using anp method showed priority i1 (profit) > i3 (fixed investment) > i7 (timeliness of cargo transportation) > i4 (the energy intensity) > i6 (safety of cargo transportation) > i15 (the length of the route) > i5 (greenhouse gas emissions of co2) > i9 (the coefficient of flows irregularity). the least important indicators: i13 (the mass (quantity) of flow) > i11 (the coefficient of flows discreteness) > i8 (the coefficient of flow controllability) > i10 (the coefficient of complexity structure of flow) > i14 (the speed of flow) > i12 (the coefficient of differentiability of flow) > i12 (the coefficient of differentiability of flow). the authors propose to use the obtained results in the gscm to adjust the actual logistics flows parameters in accordance with the goals of the sustainable development concept. the authors intend to improve the proposed approach in two directions. the first direction is based on the combination of mcdm with simulation modeling. the use of a simulation model will allow evaluating the effectiveness of decisions in the gscm, predicting changes in the parameters and indicators of logistics flows, as well as choosing the optimal sequence for implementing green logistics instruments to adjust of the logistics 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the combined dematel-anp method 497 a p p en d ix 2 w ei g h te d s u p er m at ri x t w i1 5 0 0 0 0 0 1 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i1 4 0 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .5 i1 3 0 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .5 i1 2 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 .4 0 3 8 0 .0 9 6 1 0 0 0 0 0 i1 1 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 .4 0 3 8 0 .0 9 6 1 0 0 0 0 0 i1 0 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 .2 8 8 9 0 .1 1 4 9 0 0 .0 9 6 0 0 0 0 i9 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 .5 0 0 0 0 i8 0 0 0 0 .5 0 0 0 0 0 0 0 0 .3 4 3 7 0 .1 5 6 2 0 0 0 0 0 0 0 0 i7 0 0 0 1 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i6 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 i5 0 0 1 .0 0 0 0 0 0 0 0 1 .0 0 0 0 0 0 0 0 0 0 0 i4 0 0 1 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i3 0 0 .5 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i2 0 1 .0 0 0 0 0 0 .3 8 1 2 0 0 .1 1 8 7 0 0 0 0 0 0 0 0 0 0 0 0 i1 0 0 .5 0 0 0 0 0 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 p 5 0 .3 3 3 3 0 .0 7 6 4 0 .1 1 1 5 0 .1 1 8 0 0 .0 2 7 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .1 6 0 6 0 .0 5 4 9 0 .1 1 7 8 p 4 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .2 5 6 2 0 .1 0 4 3 0 .0 7 3 3 0 .0 6 6 0 0 0 0 p 3 0 .3 3 3 3 0 .0 6 7 8 0 .0 7 5 4 0 .0 8 3 7 0 .0 2 4 1 0 .0 8 2 1 0 0 0 0 0 0 .1 7 9 3 0 .1 0 3 4 0 .0 5 0 4 0 0 0 0 0 0 0 p 2 0 .3 3 3 3 0 .3 3 3 3 0 0 0 0 0 0 0 0 .2 5 0 0 0 .0 8 3 3 0 0 0 0 0 0 0 0 0 0 p 1 0 .3 3 3 3 0 .0 4 9 5 0 .0 7 5 0 0 .0 9 5 8 0 .0 2 1 5 0 .0 9 1 4 0 .2 1 1 5 0 .0 3 2 5 0 .0 8 9 1 0 0 0 0 0 0 0 0 0 0 0 0 s d 0 0 .1 4 8 5 0 .2 2 5 0 0 .2 8 7 4 0 .0 6 4 7 0 .2 7 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 s d p 1 p 2 p 3 p 4 p 5 i1 i2 i3 i4 i5 i6 i7 i8 i9 i1 0 i1 1 i1 2 i1 3 i1 4 i1 5 498 n. osintsev, a. rakhmangulov, v. baginova a p p en d ix 3 l im it ed s u p er m at ri x t l i1 5 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i1 4 0 .1 8 2 7 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.0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i9 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i8 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i7 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i6 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i5 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i4 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i3 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i2 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 i1 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 p 5 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 p 4 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 p 3 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 p 2 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 p 1 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 s d 0 .1 8 2 7 0 .1 3 1 7 0 .1 1 0 0 0 .1 3 4 0 0 .0 4 1 8 0 .1 0 9 7 0 .0 4 7 5 0 .0 0 4 2 0 .0 3 6 0 0 .0 2 7 5 0 .0 2 2 9 0 .0 2 6 3 0 .0 2 8 1 0 .0 0 6 7 0 .0 1 9 1 0 .0 0 6 6 0 .0 1 2 6 0 .0 0 3 4 0 .0 1 7 6 0 .0 0 6 0 0 .0 2 4 7 s d p 1 p 2 p 3 p 4 p 5 i1 i2 i3 i4 i5 i6 i7 i8 i9 i1 0 i1 1 i1 2 i1 3 i1 4 i1 5 facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 415 424 https://doi.org/10.22190/fume181219003t © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper redesign and topology optimization of an industrial robot link for additive manufacturing evangelos ch. tsirogiannis, george-christopher vosniakos national technical university of athens, school of mechanical engineering, section of manufacturing technology, greece abstract. design optimization for additive manufacturing is demonstrated by the example of an industrial robot link. the part is first redesigned so that its shape details are compatible with the requirements of the selective laser sintering technique. subsequently, the simp method of topology optimization is utilized on commercially available software in order to obtain the optimum design of the part with restrictions applicable to additive manufacturing, namely member thickness, symmetry and avoidance of cavities and undercuts. mass and strain energy are the design responses. the volume was constrained by a fraction of the initial mass. the desired minimization of maximum strain energy is expressed as an objective function. a 7% reduction in the mass of the part was achieved while its strength and stiffness remained unchanged. the process is supported by topology optimization software but it also involves some trial-and-error depending on the designer’s experience. key words: topology optimization, robot link, lightweight, design for additive manufacturing, additive manufacturing, selective laser sintering 1. introduction in the last few years, topology optimization (to) has emerged as a valuable tool for developing new design proposals within the framework of lightweight engineering, e.g. in the automotive industry [1], in the aircraft industry [2], but also in robotic mechanical structures, e.g. industrial [3], dlr [4] and humanoid robots [5]. a lightweight industrial robot implies safer close collaboration between human and robot in addition to energy efficiency, high positioning accuracy, payload capacity and lower requirements of the pertinent connection structure. in topology optimization problems in real life, each finite element within the design domain is defined as a design variable, allowing a variation in density (homogenization, received december 19, 2018 / accepted february 02, 2019 corresponding author: george-christopher vosniakos national technical university of athens, school of mechanical engineering, section of manufacturing technology, heroon polytechniou 9, 15780 zografou, athens, greece e-mail: vosniak@central.ntua.gr 416 e. tsirogiannis, g.-c. vosniakos simp) [6] or void-solid representation (beso) [7]. additional well-known topology optimization methods are: homogenization [8], ground structure [9], the level-set [10] and the genetic method [11]. in the past, lightweight structures resulting from topology optimization were meant to be produced by material removal or other conventional manufacturing technologies, but, more recently, additive manufacturing (am) methods are focused on [12, 13]. am, a layer-wise material addition process family, may enable complex geometry and material distribution [14] with increases in strength and stiffness, and, at the same time, reduced weight of the part. different reasons concerning fabrication time and cost minimization, including decentralized and on demand manufacturing, modifications and redesigns without penalties, possibility to make any complexity of geometry at no extra cost and time, increased supply chain proficiency as well as reduced environmental footprint are pointed out [14]. certainly, the materials should comply with the design and manufacturing techniques. printing of long fiber composites is not achievable yet but functional parts can be printed directly with metal powders [15]. multiscale structures (foam, 2d / 3d lattice) [16] and multi-material design are also fostered by dfam and can be used in combination with structural optimization [13, 17]. this work advocates the combination of to and design for additive manufacturing (dfam) for developing lightweight industrial robot links. as an example, for a particular loading case, the forearm link of an existing robot is redesigned and topologically optimized for sls. the principal selection criteria of the latter were total cost, accuracy and surface quality in comparison to other am process capable of printing metal [14]. in section 2 the principles of design for am are presented. in section 3 redesign of the particular robot link addressed is outlined to conform to manufacturing by sls. in section 4 to as applied in this case is presented and the pertinent results are discussed. conclusions are summarized in section 5. 2. design for sls: principles a collection of principles to be followed in applying design for sls is presented next within the framework of lightweight engineering. lightweight engineering for am mainly refers to requirements, which are used for thickness distribution. it comprises: (a) design, referring to the creation of optimum geometries and shapes, (b) material, which addresses high stiffness-to-weight ratio and (c) manufacturing, which, in the current study, will take advantage of the sls am technology. 2.1. dfam constraints following specific design rules, robust geometry can be generated for sls [18], an am variant providing high accuracy and surface quality at affordable cost [14] as follows. in sls there is no need for support structures since the powder bed provides for it. as a result, overhangs may be blamed for lower heat conduction but not for the lack of support [12]. a negative or zero inclination angle between a layer and its previous layer denotes the lack of overhangs and minimal distortion of the part. an inclination angle between approximately 45° and 90° leads to larger but usually tolerable geometrical distortion of the part. for angles smaller than approximately 45° not only is geometrical distortion large but surface quality of the overhanging structure is also low, the more so when redesign and topology optimization of an industrial robot link for additive manufacturing 417 inclination approaches 0° [19]. thus, building orientation should be chosen properly, to achieve an inclination angle larger than approximately 45° or even better close to 90° in most layers. if no building orientation gives acceptable results, part design may have to be modified. a similar approach applies to undercuts, as well. moreover, in sls the minimum feature size should be respected. this is related to the minimum section size as constrained by the laser spot diameter. in addition, the bridging distance, i.e. the maximum physical gap that can be tolerated, should also be respected. in both cases suitable modifications in part design may be necessary [12]. the use of curves instead of corners on a layer profile could minimize high acceleration and deceleration stages of metal deposition which cause variations of the deposited material height [12]. note that the design optimization for am is affected by the lack of full specification of mechanical and thermal properties of the powder materials. in addition, anisotropy often results due to preferential crystal growing directions [20, 21]. 2.2. topology optimization for am having achieved a shape compatible with am, topology optimization for am follows. to is usually applied to create lighter and stiffer structures by changing the material and the thickness distribution within the allowable limits dictated by the am process. thickness reduction as well as the creation of so-called ‘multiscale’ structures (e.g. foam, 2d and 3d lattice) results in lower residual stresses and lower distortion during and after the am process. however, constraints and restrictions pertaining to am should be taken into account. taking into account that to is most commonly implemented on a finite element mesh, the main constraint is mesh resolution. a refined mesh implies the emergence of further detail and the improved topology. furthermore, each section of the component should comprise at least 2-3 finite elements to obtain accurate calculation of the displacement, leading to a large number of design variables. alleviating actions could be: hard-kill element elimination, iterative re-meshing and boundary-based to methods [13]. after το, complexity is high and the current methods cannot convert the model into a cad file accurately. hence, the model is usually converted into a stereolithography (.stl) file. restrictions on to apply to the design variables and may refer to (a) ‘frozen’ regions from which material cannot be removed, usually for reasons of interfacing of the part in an assembly, (b) the maximum and minimum member thickness, the former being due to functional reasons and the latter to am process capability, (c) symmetry required for mass balancing purposes, and (d) avoidance of cavities (voids) or undercuts, due to the am capabilities, too. 3. redesign for am 3.1. original design in the present study, the forearm of stäubli rx90bl robot arm is dealt with. it is considered to be a high speed, low payload (4kg), high accuracy and repeatability (50 μm) arm. the link was reverse-designed on catiatm v5 cad system, see fig. 1(a), from in-situ measurements. several points can be noted with respect to the shape of the forearm link. it is monocoque for high stiffness, without holes or gaps to conform to protection class ip65. since rotary joints generate linear accelerations increasing with the distance from the joint axis, a 418 e. tsirogiannis, g.-c. vosniakos tapered cross section or wall thickness is presumed to reduce the associated inertial loads. functional (mating) surfaces with the neighboring elbow and wrist links on either side of the forearm respectively, have high accuracy and surface finish. fig. 1 forearm (from left: isometric, front, left, back, top-bottom views) (a) original, (b) redesigned 3.2. modified design the forearm link (fig. 1(a)) should be redesigned, at least locally, so that it can be produced by sls process, see fig. 1(b). the building direction is selected as parallel to the direction of its largest dimension as demonstrated in fig. 2(a). fig. 2 redesign for sls details (a) inclination angle (b) modification of the upper end based on that, the inclination angle at its upper end is selected as 40° from the horizontal (fig. 2(a)) in order to obtain acceptable overhangs without support. furthermore, intricate features have been replaced by simpler ones. for example, some corners are replaced by particular curves (fig. 2(b)) to minimize overhanging structures. note that in this case overhangs are formed in the interior of the link and not on its external shape. simplified geometry is expected to yield better dimensional accuracy and surface quality in sls production as well as lower stress concentration. redesign and topology optimization of an industrial robot link for additive manufacturing 419 3.3. material selection available alloys used in the sls process include 17-4 and 15-5 stainless steel, maraging steel, cobalt chromium, inconel 625 and 718, aluminium alsi10mg, and titanium ti6al4v [22]. alsi10mg alloy is selected since it is often used for products with thin walls and complex shapes, exhibiting good strength, strength to weight ratio, toughness, dynamic properties and recyclability [21]. mechanical properties of the material are selected equal to those in z (building) direction since they represent the worst case scenario of isotropic material being inferior to the properties along x and y directions. the properties of interest are presented in table 1. safety factor for the alsi10mg alloy on the basis of yield strength is selected at 1.25 [23]. thus, the forearm mass is calculated at 5.717 kg. table 1 properties of alsi10mg alloy [22] property value unit young modulus 62000 μpa yield strength 230 mpa density 2.68 gr/mm3 3.4. analysis the static analysis of the forearm structure was performed in abaqustm, in order to calculate reference stress and strain distribution to which the distribution that is expected to result from to should be compared. three partitions are created by two planes, see fig. 3(a), for generating different meshes as necessary as well as for determining the design area in to. fig. 3 (a) partitions p1-p2 (b) load application points (rp1: wrist and tool center of gravity, rp4: forearm center of gravity, rp2: elbow joint torque center) the loads on the robotic forearm are estimated for a random position of the robot arm movement. for the sake of brevity, the center of gravity between the tool (a 3d printer head in this case), the end effector and the wrist link were calculated and the corresponding reference point rp1, which represents the center of gravity, is defined as shown in fig. 3(b). similarly, 420 e. tsirogiannis, g.-c. vosniakos another two reference points are also defined, i.e. rp4, representing the center of gravity of the forearm, and rp2 located on the bottom flange of the forearm, where the torque of the elbow joint motor is applied, see fig. 3(b). continuum distributing couplings (as defined in abaqustm) were used to connect these points with the pertinent areas of the forearm. the loads applied are shown in table 2. a moment resulting by the motor of the wrist joint is also applied. the magnitude of that moment is 57 nm. accordingly, a boundary condition is applied for 5 dofs (fx, fy, fz, my, mz) on the bottom of the forearm. table 2 loads applied fx (n) fy (n) fz (n) mx (nm) my (nm) mz (nm) rp1 300 250 50 200 175 350 rp4 700 500 100 300 250 1000 rp2 100 abaqustm c3d10 (10-node quadratic tetrahedron) element type was used in meshing. the density of the mesh is specified as 5 by creating seeds along the edges of the model. the free meshing technique is selected as it is the most flexible top-down methodology fully associated with the geometry of the model. during the linear static analysis, it was noticed that the use of connections for pre-process modeling of the loads increased memory allocation on a 8 gb ram, intel i7 2.90 ghz computer, eventually making it impossible to proceed to the solution. hence, as few connections as possible were used. afterwards, the connections were divided into smaller subconnections by selecting the pertinent sub-surfaces for each sub-connection, ultimately decreasing memory allocation to 7973 mb. approximately 30 attempts were needed with a mean duration of 15 minutes each in order to solve the problem in a trial and error fashion. the results obtained concerning von mises stresses and deformations are shown in fig. 4. the results present a maximum stress value of 10.73 mpa and deformations up to 0.09 mm, both being considered low and, thus, acceptable. fig. 4 analysis results before to (a) von mises equivalent stress (b) deformation redesign and topology optimization of an industrial robot link for additive manufacturing 421 4. topology optimization for am simp (solid isotropic material with penalization) to method was used due to its reliability, efficiency and ease of setup, including low storage space and computational load [24]. 4.1. setup initially, the model was partitioned by two planes p1 and p2, see fig. 3(a) between which the design area is defined, whereas the rest had to stay unchangeable. the design variables are the densities of the elements in the design area, which take discrete 0-1 values. two design responses were considered, namely strain energy and volume. the objective function was specified so as to minimize the maximum strain energy calculated for all the elements. the limit applied to the value of volume response is specified as a maximum reduction by 7% from its initial value. consequently, the mass of the forearm’s design section would also be reduced by a maximum of 7%. next, restrictions are taken into consideration. since to in abaqustm does not support am-specific restrictions, the most akin ones are selected. these include: (a) restrictions on cavities and undercuts (b) minimum member thickness at 1 mm, equal to the laser beam diameter [13] (c) planar symmetry between the right and the left side of the part for the forearm to comply with the design rules of robot arms. the global stop condition was obtained by trial and error at 22 design cycles. the general to algorithm applied adjusts density and stiffness while trying to satisfy the objective function and the constraints as outlined in detail in [25]. the same computer and memory allocation as those described in section 3.4 were used in to, too. 4.2. procedure next, the steps leading to the optimization results are explained and shown diagrammatically in fig. 5. in the first phase, attempts were made to achieve a modified forearm design with reduced mass retaining in parallel the strength and stiffness characteristics of the initial design. the objective function to minimize the strain values and the solid volume/mass constraint are defined by the user. thus, after repeated trials, each lasting about 15 min, it was noticed that for mass reduction higher than 10% the optimization problem could not be solved. in the second phase, apart from the mass, strength and stiffness considerations the symmetry and member thickness restrictions, see section 4.1, were introduced with objective function as: "minimize strain energy values". five runs were needed to ascertain that the manufacturing restrictions, in particular the minimum thickness, could not be satisfied with 10% mass reduction, leading to a lower figure of 7%. in the third phase, the target of the objective function was set as to "minimize the maximum strain energy values", the rest of the setting being kept as in the second phase. the resulting robot arm geometry was better compared to the second phase, due to the more homogenous spread of thickness reduction. note that subsequent execution of the optimization for 8% mass reduction was not successful due to the minimum thickness restriction and, as a result, the highest mass reduction was again 7%. in the fourth phase, the restriction on cavities and undercuts was added, see section 4.1. the resulting geometry did not have cavities and undercuts anymore and, as a result, it was better in terms of manufacturability compared to the third phase. again, minimum thickness restriction impaired increase of mass reduction to 8%. 422 e. tsirogiannis, g.-c. vosniakos fig. 5 topology optimization procedure the first four phases amounted to about 50 runs, each lasting about 7 hrs, the optimization maximum number of cycles being defined as 20. the latter, definitely affect quality of optimization results, thus the optimum number of cycles was further explored in the fifth phase. it was observed that the best results were obtained with 22 cycles, with duration of approximately 8 hrs. in the sixth phase, an attempt to reduce the forearm mass by 8% by increasing the maximum number of cycles to 50 was not successful. 4.3. results after progressive removal of the necessary elements to create voids in the internal and the external shape of the forearm, the modified forearm was obtained, see fig. 6. fig. 6 removal of elements: (a) and (b) external shape (c) and (d) internal shape redesign and topology optimization of an industrial robot link for additive manufacturing 423 equivalent von mises stresses and deformations of the forearm after to are shown in fig. 7. the results present a maximum stress value of 10.72 mpa and deformations up to 0.09 mm. based on mass, stress and deformation comparison, the topologically optimized link is better than the original one: its mass is reduced by 7%, it sustains the same mechanical load while respecting the same maximal displacement and maximal von mises stress, which is obvious by comparing fig. 4 and fig. 7. fig. 7 results after to: (a) von mises equivalent stress (b) deformation 5. conclusion and perspectives this study presents an overall structural design optimization approach for a robot arm link seeking mass reduction and satisfaction of manufacturability with sls am technique. mass reduction achieved is deemed moderate at 7% but it is worth noting that the maximum deformation and the maximum von-mises stresses of the customized link have remained the same. at the early design stage, fea and to simulations are very effective as the designer can consider the capabilities of am and obtain the optimal geometry in terms of stress, deformation and weight response. still, to is a trial-and-error process, depending to some extent on designer experience in problem modeling. however, to does restrict extensive trial-and-error, which would otherwise be necessary at a much higher level of part design until the final shape is reached. the use of am and dfam methods is expected to increase substantially in the industrial sector in the years to come. thus, robot industrialists could benefit from short lead times, fast iteration cycles and low costs as well as the to capabilities of am. nevertheless, to is currently a time-intensive task, which can be significantly improved by multiple core processors or special hardware for accelerating computations. as future work, it would be most advantageous to extend to towards rigid body dynamics in order to consider all the possible robot poses and not only one, thus incorporating a multitude of loading cases. in addition, to could be followed by shape optimization in order to obtain smoother geometries. last but not least, further development of the existing το software programs would be useful in order to adapt to am technologies, introducing am specific restrictions. 424 e. tsirogiannis, g.-c. vosniakos references 1. raugei m., morrey d., hutchinson a., winfield p., 2015, a coherent life cycle assessment of a range of lightweighting strategies for compact vehicles, j clean prod, 108, pp. 1168–1176. 2. saleem w., yuqing f., yunqiao w., 2008, application of topology optimization and manufacturing simulations a new trend in design of aircraft components, 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cc by-nc-nd original scientific paper effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine zhibin zhao1, 2, wenjie zhou1, peijun liu2, zhirui liu2 1school of energy and power engineering, jiangsu university, zhenjiang, china 2institute of engineering thermophysics, chinese academy of sciences, beijing, china abstract: sensor aging and sensor failure are the common phenomena due to the high temperature and pressure environment for gas turbines, which can lead to hysteresis of monitoring signals. in this paper, a kind of aero-derivative gas turbine is taken as the research object. the hysteresis effects of single monitoring signal and coupling of multiple monitoring signals on speed control are mainly studied, and the analysis is carried out from the perspective of adjustment time, overshoot, fuel quantity and fuel quantity regulation output. the analysis results show that the pressure signal hysteresis will lead to speed suspension. the speed signal hysteresis will change the speed regulation into a multi-step mode. when the monitoring signal hysteresis is coupled, the effect of pressure signal hysteresis is greater than that of speed signal hysteresis. the results of this paper can provide a reference for the optimal design of speed control of aero-derivative gas turbine. key words: aero-derivative gas turbine, pressure signal hysteresis, speed signal hysteresis, hysteresis coupling, multi-step mode 1. introduction as the core equipment of the power generation system, the aero-derivative gas turbine is an indispensable energy-converting device [1]. the aero-derivative gas turbine is developed based on the mature air engine technology. it retains most of the structure of the air engine and only adds the power turbine at the end of the gas generator. therefore, the aero-derivative gas turbine inherits most of the advantages of the air engine and has the advantages of large power per unit volume, light weight, fast starting, good acceleration and high reliability [2, 3], it has become an important energy conversion equipment and power plant in the industrial field. at present, the aero-derivative gas turbine has been received: december 05, 2022 / accepted may 05, 2023 corresponding author: wenjie zhou1, peijun liu2 1 school of energy and power engineering, jiangsu university, zhenjiang 212013, china 2 institute of engineering thermophysics, chinese academy of sciences, beijing 100190, china e-mail: zhouwenjiezwj@ujs.edu.cn, liupeijun@iet.cn z. zhao, w. zhou, p. liu, z. liu 2 widely used in various fields [4, 5], such as distributed energy, pipeline compression, marine power, mobile power station, power plant and offshore platform. gas turbine control technology is one of the core technologies in gas turbine, and it is also an important guarantee for the economy of gas turbine units. from the perspective of development, gas turbine control system has experienced the stages from the mechanical hydraulic control, analog electronic control to digital electronic control [6]. with the application of low emission combustion control technology, adaptive control technology, remote network control technology, intelligent sensors and actuators in the control system [7-10], the gas turbine control system is gradually developing towards integrated control, network control and active control. pid algorithm is the earliest practical control algorithm, it is still the most commonly used control law in the control system. with the development of control theory, fuzzy control, particle swarm optimization algorithm, genetic algorithm and neural network model combined with pid algorithm are widely used in the field of control engineering to adapt to more complex controlled objects [11-14]. less attention has been paid to the problems that the pid algorithm iteration results are biased due to the hysteresis of the monitoring signal, which causes the control system to send wrong instructions to the controlled object, and then leads to the slow adjustment of the control system and the deterioration of robustness. the hysteresis of monitoring signal mainly comes from two aspects: on the one hand, it is caused by sensor failure, on the other hand, it comes from the control system. gas turbines usually operate under complex conditions, such as high temperature, high pressure, etc. for monitoring and control, the gas turbine is equipped with many sensors. due to long working hours under adverse conditions, the sensor may output abnormal signals, and then the monitoring signal will be delayed [15]. the typical five types of sensor faults are step fault, pulse fault, periodic fault, noise fault and drift fault [16-18]. in addition, the gas turbine control system is usually composed of measuring elements (sensitive elements), electronic controllers and actuators. the problems such as long sampling time, communication failure and long response time of actuators may occur during its work, which will result in delayed monitoring signal of the gas turbine. the research of monitoring signal hysteresis mainly includes the use of fault-tolerant control and the establishment of fault diagnosis system. since fault-tolerant control has the ability to reconstruct monitoring signals and can better maintain the stability of the control system, hadroug [19-24] analyzed and discussed the application of fault-tolerant control. fault tolerant control is mainly divided into two types: data-based fault-tolerant control and model-based fault-tolerant control. in terms of the current application of fault-tolerant control, it is difficult to realize data-based fault-tolerant control in engineering application. however, the existing model-based fault-tolerant control will degrade its fault-tolerant performance when the object does not match the observer model, and could not adapt to the characteristics of gas turbine with variable load. on the other hand, the robustness and reliability of the control system can be improved by establishing a fault diagnosis system [25]. the researches on fault diagnosis were also been paid more attentions, rahmoune [26, 27] established a fault diagnosis system based on neural network and tested some faults. ogaji [28] established a fault diagnosis system to solve the problem of sensor fault and reorganized the fault data into a fault-free form, this reconstructed data could be used to accurately perform sensor calculation. considering that the performance degradation signs of the main components of the gas turbine are different at different operating points, effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 3 mohammadi [29] proposed a fuzzy based fdi system with the help of load parameters as augmented input, which had robust performance against measurement uncertainty. monitoring signal hysteresis is a common problem in the operation of gas turbine. it can reduce the influence of monitoring signal hysteresis on the control system through fault tolerant control and fault diagnosis system. however, the influence way and principle of monitoring signal hysteresis on the control system are fuzzy. in this paper, an aeroderivative gas turbine for power generation is taken as the research object. aiming at the problem of monitoring signal hysteresis, a monitoring signal hysteresis model is established, which is embedded into the digital simulation model to form a digital simulation model with the function of monitoring signal hysteresis. the control performance of this type of aero-derivative gas turbine is optimized. the above research results can provide an important reference for this type of aero-derivative gas turbine to reduce the impact of monitoring signal hysteresis. 2. establishment of digital simulation system using matlab/simulink platform to establish digital simulation model is one of the important means of gas turbine research. the digital simulation model established in this paper is shown in fig 1. the establishment process mainly includes four steps: firstly, based on the mechanism modeling method, the aero-derivative gas turbine model is established; secondly, based on pid algorithm, the fuel quantity algorithm model is established; thirdly, aiming at the problem of monitoring signal hysteresis, a monitoring signal hysteresis model is established; finally, the three models are combined into a digital simulation model with the function of signal hysteresis monitoring. fig. 1 structure diagram of digital simulation model 2.1 the model of the aero-derivative gas turbine the research object of this paper is an aero-derivative gas turbine for power generation. its simplified structure is shown in fig. 2, mainly including compressor, combustion chamber, turbine, power turbine, generator and other components. gas turbine modeling methods are mainly divided into two categories: mechanism modeling and data modeling. z. zhao, w. zhou, p. liu, z. liu 4 in the modeling process of aero-derivative gas turbine, due to the lack of measurement data such as high-temperature component measurement points and air side flow measurement points, the data modeling method is limited. however, the mechanism modeling method mainly depends on the physical mechanism of gas turbine components. therefore, based on the mechanism modeling method, this paper establishes the model of aero-derivative gas turbine. fig. 2 structural diagram of aero-derivative gas turbine 2.2 the model of fuel quantity algorithm fuel quantity is the most important control variable to change the operating state of gas turbine. by changing fuel quantity, the gas turbine can be controlled to complete acceleration or deceleration. the fuel quantity algorithm model mainly includes fuel quantity control algorithm and fuel quantity optimization link, which is used to receive the monitoring signal provided by the gas turbine model, calculate the fuel quantity, and feed back the fuel quantity to the gas turbine model. 2.2.1 fuel quantity control algorithm due to the simple structure, good stability and reliable operation of the pid control algorithm, the pid control algorithm is used to construct the fuel control algorithm. the logic of the fuel algorithm consists of nine regulations, one being the primary one and the other eight being the limiting one, as shown in table 1. it can be divided into speed control, safety limit control and fuel control according to its functions. based on the aforementioned fuel quantity algorithm logic, the fuel quantity pid control logic is designed. the control equation of each pid controller is shown in eq. (1). different from the traditional integral link, the variable integral gain coefficient is used in this paper, and the fuel increment in recent operation cycles is mainly used as the adjustment term to prevent excessive overshoot. 𝑊𝑓𝑥 = 𝐾𝑝𝛥 − ∫ 𝐾𝐼 (𝜏−𝑡)𝜏 0 𝑊𝑓(𝑡 − 1)𝑑𝑡 (𝑡 = 1,2,3, . . . , 𝜏) (1) effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 5 where wfx is the fuel quantity difference, x is the different control variables substituted, kp is the proportional gain coefficient, ki is the integral gain coefficient, ∆ is the deviation between the limit condition value or the target value of the control variable and the current state value, wf is the fuel increment fed back after priority selection and τ is the number of control logic operations that have been run. table 1 composition of fuel quantity algorithm logic number fuel quantity control sub circuit abbreviation type 1 gas generator speed regulation pid_gg main 2 gas generator maximum speed limit regulation pid_ggh limit 3 gas generator acceleration limit adjustment pid_nca limit 4 gas generator outlet temperature limit regulation pid_t5 limit 5 compressor outlet pressure limit regulation pid_p3 limit 6 compressor outlet temperature limit regulation pid_t3 limit 7 maximum deceleration rate limit regulation pid_ncd limit 8 minimum fuel limit regulation pid_wmin limit 9 maximum fuel limit regulation pid_wmax limit 2.2.2 fuel quantity optimization logic since the min-max selector has multiple input items and only one output is executed, which can effectively limit the fuel quantity within the safe range, the fuel quantity optimization logic is designed by using the min-max selector. the designed fuel quantity optimization logic is shown in fig. 3. it contains 9 input items and is selected in three steps, and finally outputs the fuel quantity that meets various restriction conditions. fig. 3 logical structure of fuel quantity optimization the fuel priority selection logic is designed as a three-level priority with the following main functions: a. the first priority uses the minimum selector selection to ensure that the fuel increment obtained does not exceed the safety limit. b. second priority uses maximum selector selection to ensure fuel increment is greater than pid_ wmin to avoid lean oil extinguishing. c. the third priority uses the minimum selector to ensure that the final output fuel increment is less than the pid_ wmax to prevent the gas turbine from rising too fast. z. zhao, w. zhou, p. liu, z. liu 6 2.3 hysteresis model of monitoring signal in order to achieve the effect of monitoring signal transmitting real data periodically, a monitoring signal lag model is established based on the monitoring signal lag principle described in eq. (2.a) and eq. (2.b) establishment of a hysteresis model by adding a zeroorder retention module after the monitoring signal output module of the gas turbine model and providing a rate conversion module to ensure consistent sampling time. the sampling time of the zero-order hold module is hysteresis time, which can be set to 0.1s, 0.2s, etc. as needed. 𝑛𝑇 ≤ 𝑡 ≤ (𝑛 + 1)𝑇 𝑛 = 0,1,2,3 … (2.a) 𝑢(𝑡) = 𝑢(𝑛𝑇) 𝑛 = 0,1,2,3. .. (2.b) where t is the hysteresis time, t is the sampling time, u(t) is the signal value at the current sampling time and u(nt) is the signal value at the last sampling time. 3. experimental scheme and model verification 3.1 experimental scheme the monitoring signal hysteresis mainly comes from two aspects: sensor fault and control system fault. although either cause has little impact on the speed control, the impact after the superposition of the two cannot be ignored. in order to explore the influence of single monitoring signal hysteresis on speed control, the experimental scheme is designed with pressure signal and speed signal as variables. in addition, the experimental scheme is designed by coupling speed signal and pressure signal and combining the analysis results of single signal hysteresis. 3.1.1 experimental scheme of pressure signal the proportional gain coefficient (kp) in the fuel quantity algorithm is selected through the difference table, and the pressure signal is the input of the difference table. as the main influence factor of kp, the pressure signal will lead to deviation in the selection of kp once the hysteresis phenomenon occurs. therefore, it is necessary to study the influence of the pressure signal hysteresis on the speed control. in order to obtain the influence law of pressure signal hysteresis on speed control, the selected pressure signal hysteresis time t_p as shown in table 2, there are 10 experimental schemes. table 2 pressure hysteresis time (t_p) plan 1 2 3 4 5 6 7 8 9 10 t_p(s) 0.1 0.7 1.4 3.5 7 14 21 28 35 50 3.1.2 experimental scheme of speed signal the start-up stage of a gas turbine is usually judged by the gas generator speed. once the speed signal is delayed, it will inevitably affect the judgment of the gas turbine start-up process. therefore, it is necessary to study the effect of speed signal hysteresis on speed effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 7 control. in order to obtain the law of the influence of speed signal hysteresis on speed control, the selected speed hysteresis time t_s shown in table 3, there are seven experimental schemes. table 3 temperature hysteresis time (t_s) plan 1 2 3 4 5 6 7 t_s(s) 0.1 0.2 0.3 0.4 0.5 1 3 3.1.3 experimental scheme of coupling pressure signal and speed signal as the gas turbine usually works in the environment of high temperature and high pressure, it is very easy to have problems such as sensor performance degradation and sensitivity deterioration. therefore, it is necessary to consider the influence of various monitoring signal hysteresis, couple the pressure signal and speed signal, and design the coupling scheme according to the influence law of single signal hysteresis. the designed coupling scheme is shown in table 4, including 9 coupling schemes. table 4 hysteresis time coupling scheme of monitoring signal plan 1 2 3 4 5 6 7 8 9 t_p(s) 0.1 0.1 0.1 14 14 14 35 35 35 t_s(s) 0.1 0.5 1 0.1 0.5 1 0.1 0.5 1 3.2 model validation in order to verify the function of the hysteresis model of the monitoring signal, the hysteresis time of a single monitoring signal is calculated as a variable and compared with the parameters without hysteresis. the comparison results are shown in fig. 4. fig. 4 monitoring signal hysteresis curve: (a) speed hysteresis, (b) pressure hysteresis from fig. 4, it can be seen that the speed hysteresis curve is similar to the normal operation curve, and the hysteresis phenomenon is obvious. the pressure hysteresis will result in a smaller pressure output value, and the difference between the pressure hysteresis curve and the normal operating curve is a necessary phenomenon. therefore, the z. zhao, w. zhou, p. liu, z. liu 8 established monitoring signal hysteresis model has a good ability to reproduce the phenomenon of monitoring signal hysteresis. 4. effect of single monitoring signal hysteresis 4.1 performance index 4.1.1 adjustment time adjustment time is the shortest time that a controlled variable changes from a previous stable state to a new stable state when the control system is affected by external disturbances. in fact, the shortest time required for a control variable to enter a new steadystate value of ± 5% is called the adjustment time of the control system. the adjustment time is calculated as shown in eq. (3): 𝑇𝑆 = 𝑇2 − 𝑇1 (3) where ts is the adjustment time, t2 is the time when the controlled variable reaches a new balance state, and t1 is the time when the controlled variable begins to adjust. 4.1.2 overshoot the ratio of the maximum deviation of the adjustment parameter to the steady state value is called overshoot during the adjustment of the control system, and the maximum deviation refers to the deviation between the maximum value of the adjustment parameter and the steady state value. in this paper, for a clearer comparison and analysis, the maximum deviation of the adjustment parameters is defined as overshoot, and the overshoot is solved by eq. (4): 𝜎 = 𝐺𝐺𝑀𝐴𝑋 − 𝐺𝐺𝑆𝑆𝑉 (4) where σ is overshoot, ggmax is the maximum output value of the adjusted parameter, ggssv is the steady-state value of the adjusted parameter. 4.2 effect of pressure signal hysteresis in order to obtain the changing trend of gas generator speed when the pressure signal is hysteresis, the calculation is carried out according to the 10 pressure signal hysteresis schemes described in table 2, and the calculation results are shown in fig. 5. it can be got from fig. 5 that when the pressure signal hysteresis exists, the gas generator speed mainly presents four phenomena: a. the gas generator speed has obvious overshoot (the target speed is 6800rpm), and the overshoot gradually disappears with the increase of t_p. b. when the t_p exceeds 3.5 seconds, the speed of the gas generator remains at 4600 rpm, which is called the speed suspension phenomenon. c. when the t_p exceeds 35 seconds, the speed decreases. in the actual operation process, this phenomenon will trigger the downtime process. d. when the t_p exceeds 50 seconds, the speed of the gas generator cannot reach the speed demand value. effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 9 fig. 5 contour chart of gas generator speed when pressure signal is hysteresis in order to further determine the influence of pressure signal hysteresis, the influence degree of pressure signal hysteresis on speed control is analyzed by taking performance parameters such as overshoot and adjustment time as evaluation criteria. the adjustment time and the solution of the overshoot follow eq. (3) and eq. (4) and the results are shown in fig. 6. as shown in fig. 6, on the one hand, with the increase of t_p, the speed adjustment time gradually increases, and finally tends to 60 seconds, which fails to reach the target speed of 6800 rpm. on the other hand, with the increase of t_p, the speed overshoot decreases gradually, and eventually tends to -800rpm, which means that the speed fails to reach the expected target. therefore, the pressure signal hysteresis will cause higher adjustment time for gas generator speed, and even make the speed cannot reach the target speed. fig. 6 performance parameters for pressure signal hysteresis: (a) adjustment time, (b) overshoot since the fuel quantity is the main variable to control the speed, and the fuel quantity is selected by the nine types fuel quantity regulation output through the fuel quantity optimization logic, in order to further analyze the reasons for the above phenomena, the selected data are shown in fig. 7 and fig. 8. z. zhao, w. zhou, p. liu, z. liu 10 fig. 7 fuel quantity curve with pressure signal hysteresis: (a) t_p=0.1s, (b) t_p=3.5s, (c) t_p=35s, (d) t_p=50s since the fuel quantity is the main variable to control the speed, and the fuel quantity is selected by the nine types fuel quantity regulation output through the fuel quantity optimization logic, in order to further analyze the reasons for the above phenomena, the selected data are shown in fig. 7 and fig. 8. as shown in fig. 7, fuel quantity and t_p is inversely proportional, and there is a step supply phenomenon. the main manifestation is that when t_p is small, the fuel quantity curve fluctuates sharply. when t_p is large, the fuel quantity curve gradually flattens and presents a step supply phenomenon, which ultimately results in a slow increase in gas turbine speed. as shown in fig. 8, with the increase of t_p, pid_gg and pid_wmax show different degrees of hysteresis phenomenon, and pid_p3 is always larger than pid_gg and pid_wmax, which has no effect on the results of fuel optimization. further analysis of pid_gg and pid_wmax reveals that the main reasons for this phenomenon are: a. the proportional gain factor (kp_gg) of the pid_gg is determined by the outlet pressure of the compressor. the principle is that the pressure at the outlet of the compressor is used as the input of the two-dimensional difference table, kp_gg is selected, and pid_gg is calculated. once the outlet pressure of the compressor is hysteresis, the selected value of kp_gg cannot meet the speed adjustment requirements, and the speed of the gas generator rises slowly. b. compressor outlet pressure is directly involved in the calculation of pid_wmax. effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 11 when the pressure at the outlet of the compressor is lagged, the calculated value of pid_wmax is smaller, which results in a smaller result of fuel quantity optimal selection, insufficient fuel supply, and the gas generator speed is difficult to reach the target value. fig. 8 fuel quantity regulation output curve with pressure signal hysteresis: (a) t_p=0.1s, (b) t_p=3.5s, (c) t_p=35s, (d) t_p=50s 4.3 effect of speed signal hysteresis in order to obtain the changing trend of gas generator speed when the speed signal is hysteresis, the calculation is carried out according to the 7 speed signal hysteresis schemes described in table 3. the calculation results are shown in fig. 9. in order to further determine the influence of speed signal hysteresis, the influence of speed signal hysteresis on speed control is analyzed by taking performance parameters such as overshoot and adjustment time as evaluation criteria. the corresponding adjustment time and overshoot, which are settled according to eq. (3) and eq. (4), are shown in fig. 10. it can be seen from fig. 10 that when the speed signal is hysteresis, the gas generator speed mainly exhibits two phenomena: z. zhao, w. zhou, p. liu, z. liu 12 a. with the increase of t_s, the gas generator speed gradually tends to steep and the peak gradually increases. b. with the increase of t_s, the gas generator speed curve shows a stepped upward trend in varying degrees. when t_s is 3 seconds, the gas generator speed curve cannot reach a steady state, and fluctuates repeatedly in the range of 6000 rpm to 7800 rpm. fig. 9 gas generator speed curve with speed signal hysteresis fig. 10 performance parameters for speed signal hysteresis: (a) adjustment time, (b) overshoot on the one hand, the adjustment time is inversely proportional to t_s, on the other hand, with the increase of t_s, the overshoot increases gradually. when t_s is 3 seconds, the overshoot and adjustment time are difficult to calculate due to the sharp fluctuation of speed, so the adjustment time is the maximum of the expected adjustment time. using the same analysis method as pressure signal hysteresis, the influence mechanism of speed signal hysteresis is analyzed. the selected data are shown in fig. 11 and fig. 12. from fig. 11, it can be seen that with the increase of t_s, the amount of fuel increases effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 13 gradually, the degree of chaos decreases, and the cycle regulation mode tends. from figure 12, with the increase of t_s, the step phenomenon becomes obvious, the step number decreases, and the chaos degree of fuel optimization results decreases. the main reason for the above phenomenon is that when there is a velocity signal hysteresis, the speed adjustment is equivalent to the multiple step adjustment mode. the number of steps is inversely proportional to t_s, the period of steps is positively proportional to t_s. a. when t_s is small, the number of steps is large and the step cycle is short. in order to meet the requirements of speed adjustment, fuel adjustment changes frequently, resulting in sharp fluctuations in fuel quantity. b. with the increase of t_s, the number of steps decreases and the period of steps increases. at this time, the speed adjustment is closer to the target, and the fluctuation of fuel quantity and fuel quantity regulation is reduced. c. when t_s is 3 seconds, the number of steps is the smallest and the step cycle is the longest. as shown in figure 9, during the fourth step cycle, it is always considered that the speed does not reach the target value, the fuel quantity and fuel quantity adjustment continue to increase and exceed the demand value. during the fifth step cycle and sixth step cycle, the rotation speed starts to revert, and the fuel and fuel adjustments continue to decrease and fall below the demand value. rotation speed adjustment shows cyclic regulation, which is manifested by repeated fluctuations of rotation speed in the range of 6000rpm to 7800rpm. fig. 11 fuel quantity curve with speed signal hysteresis: (a) t_s=0.1s, (b) t_s=0.3s, (c) t_s=0.5s, (d) t_s=3s z. zhao, w. zhou, p. liu, z. liu 14 fig. 12 output curve of fuel quantity regulation in case of speed signal hysteresis: (a) t_s=0.1s, (b) t_s=0.3s, (c) t_s=0.5s, (d) t_s=3s 5. influence of hysteresis coupling of monitoring signal in order to obtain the speed control performance of multiple monitoring signals with hysteresis coupling, a coupling scheme is designed according to the influence rule of single monitoring signal hysteresis, which is used to calculate the gas generator speed as shown in table 4. deep analysis is made from the angle of control performance parameters, fuel quantity and fuel quantity adjustment. the calculation results of the gas generator speed are shown in fig. 13. the main variation rules of the speed are as follows: a. when t_p is constant, the gas generator speed curve tends to steep with the increase of t_s, and the more obvious peaks occur when t_p is 0.1 seconds. b. when t_s is constant, the gas generator speed curve tends to be flat with the increase of t_p, and secondary regulation occurs when t_s is 35 seconds. effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 15 fig. 13 monitor the gas generator speed when the signal is hysteresis coupling fig. 14 speed performance when monitoring signal hysteresis coupling: (a) adjustment time, (b) overshoot the adjustment time and overshoot are calculated and analyzed by eq. (3) and eq. (4). the calculation results of adjustment time and overshoot are shown in fig. 14. as shown in fig. 14, on the one hand, the adjusting time is mainly affected by t_p, and the adjusting time is proportional to t_p, the adjusting time decreases slightly when t_s increases. on the other hand, pressure signal hysteresis has a great effect on overshoot, and the overshoot is inversely proportional to t_p, the overshoot increases slightly with t_s increases. z. zhao, w. zhou, p. liu, z. liu 16 fig. 15 fuel quantity curve under hysteresis coupling of monitoring signal: (a) plan 4, (b) plan 5, (c) plan 6 in order to further analyze the mechanism of the effect of the hysteresis coupling of the monitoring signal on the speed control, a discussion is made from the angle of fuel quantity and fuel quantity regulation, which is shown in fig. 15 and fig. 16. as shown in fig. 15, the fuel quantity curves of the three coupling schemes have the same trend with only different oscillations. both t_p and t_s in plan 4 are smaller, resulting in smaller proportional gain factor, more steps and shorter step cycle, and sharp fluctuation of fuel quantity curve. as t_p increases, the fluctuation of the fuel quantity curve decreases. therefore, the influence on the speed adjustment is the overlap of pressure hysteresis and speed hysteresis after the monitoring signal hysteresis coupling. the main reasons for the above changes are as follows: firstly, the coupling of pressure signal hysteresis and speed signal hysteresis will not only affect the selection of proportional gain coefficient and the calculation value of maximum fuel quantity in the output of fuel quantity regulation, but also change the speed regulation into a multi-step regulation mode. secondly, the pressure signal is one of the main parameters in the fuel quantity algorithm. compared with the speed signal, the pressure signal has a more direct influence on the fuel quantity control. thirdly, the speed signal hysteresis only affects the speed adjustment mode, making the speed more tend to the multi-step adjustment mode, and the impact on the fuel quantity control is indirect. effects of monitoring signal hysteresis on speed regulation for the aero-derivative gas turbine 17 fig. 16 output curve of fuel quantity regulation under hysteresis coupling of monitoring signal: (a) plan 4, (b) plan 5, (c) plan 6 6. conclusion this paper takes an aero-derivative gas turbine for power generation as the research object and carries out two aspects of research. on one hand, the influence of pressure signal hysteresis and speed signal hysteresis on speed control is studied. on the other hand, the influence of monitoring signal hysteresis coupling on the speed control is also considered. the main conclusions are as follows: 1．hysteresis of pressure signal will affect the selection of proportional gain coefficient and the calculation of maximum fuel quantity limit value, resulting in deviation of fuel quantity supply, speed suspension of gas turbine and even trip failure. 2．the speed signal hysteresis changes the speed control into a multi-step regulation mode, which results in frequent changes and confusion of fuel supply, serious overshoot of gas generator speed and even speed oscillation. 3．the effects of pressure signal hysteresis and speed signal hysteresis will be z. zhao, w. zhou, p. liu, z. liu 18 combined when the coupling monitoring signal hysteresis is considered, and pressure signal hysteresis is the main influencing factor, resulting in very complex speed adjustment and a longer time for gas generator speed adjustment. acknowledgement: this work was supported by the project funded by china postdoctoral science foundation (grant no. 2018m642177) and the zhejiang postdoctoral preferential foundation (grant no. zj2018009). references 1. wang, g., ge, n., zhong, d., 2020, numerical investigation of the wake vortex-related flow mechanisms in transonic turbines, international journal of aerospace 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s., singh, r., probert, s. d., 2002, multiple-sensor fault-diagnoses for a 2-shaft stationary gasturbine, applied energy, 71(4), pp. 321-339. 29. mohammadi, e., montazeri-gh, m., 2015, a fuzzy-based gas turbine fault detection and identification system for full and part-load performance deterioration, aerospace science and technology, 46, pp. 8293. https://webvpn.ujs.edu.cn/http/77726476706e69737468656265737421e3e40f862f3972587b06c7af9758/nav/mag/info?mags=73cd2a30ff64d4c8f5d0808561a1b0df https://webvpn.ujs.edu.cn/http/77726476706e69737468656265737421e3e40f862f3972587b06c7af9758/nav/mag/info?mags=1b1834d0b59f5aa09a1969290c7186b2 13386 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume250425032m © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case of nuzzle production line in petroleum industries ali mahmoodirad1, dragan pamucar2,3,4, sadegh niroomand5 1department of mathematics, bab. c., islamic azad university, babol, iran 2department of operations research and statistics, faculty of organizational sciences, university of belgrade, belgrade, serbia 3department of industrial engineering & management, yuan ze university, taoyuan city, taiwan 4department of applied mathematical science, college of science and technology, korea university, sejong, republic of korea 5department of industrial engineering, firouzabad higher education center, shiraz university of technology, shiraz, iran abstract. in this study an assembly line configuration is obtained for the nuzzle production line in petroleum industries. this is an important product which is widely used petroleum industries. therefore, applying an optimization scheme for obtaining an optimal configuration is necessary. for this aim, a cost-based mathematical formulation is proposed to obtain the optimal assembly line configuration. in this model, overall station establishment cost, fixed salary, and variable wages are optimized simultaneously. in order to be close to real-world situations, the problem is formulated in a triangular fuzzy environment, where the costand time-based parameters are represented by fuzzy values. the proposed fuzzy formulation is converted to a crisp form using a me measure of fuzzy sets and numbers. then, in order to evaluate the proposed crisp formulation, a case study from the petroleum industries of iran is considered. based on the performed experiments and obtained results, the best configuration of the assembly line is obtained, and a sensitivity analysis is performed as well. key words: assembly line balancing, nuzzle production, petroleum industry, fuzzy sets and numbers, mathematical modeling received: april 25, 2025 / accepted october 07, 2025 corresponding authors: sadegh niroomand, dragan pamucar shiraz university of technology, shiraz, iran; korea university, sejong 30019, republic of korea e-mail: niroomand@sutech.ac.ir, dpamucar@gmail.com mailto:niroomand@sutech.ac.ir 2 a. mahmoodirad, d. pamucar, s. niroomand 1. introduction assembly line is a type of production systems that are widely used in rapid production of the products consisting of several sub-assemblies and raw parts. this type of production system is widely used in the discrete type of production systems in order to make a faster production with more qualitative products [1]. an assembly line consists of several consecutive workstations where in each workstation some operational tasks are to be performed by one or more operators using manual or semi-automated operational equipment. a main characteristic of an assembly line is its cycle time. this is the time that all tasks of each station should be performed as a cycle. for producing a product on an assembly line, when assigning the required operational tasks of the product to the stations of the line, the precedence relationships of the tasks should be respected in the consecutive stations too. here, respecting the limitations of the cycle time and the precedence relationships of the tasks and optimizing an objective function like minimization of the number of stations forms a problem called assembly line balancing problem [2,3,4,5,6,7]. according to boysen et al. [8] the assembly line balancing problem is divided into two types of (1) simple assembly line balancing problem (salbp), and (2) general assembly line balancing problem (galbp). the salbps are divided to some types such as salbp1, salbp-2, salbp-e, and salbp-f, where, the galbps are divided to some types such as mixed model assembly line balancing problem (malbp) and u-shaped assembly line balancing problem (ualbp). the salbp-1 considers a constant cycle time and minimized the number of established stations. the salbp-2 considers a constant number of stations and minimizes the cycle time. the salbp-e simultaneously maximizes the line efficiency and minimizes the cycle time and number of the established stations. the salbp-f generates a feasible assembly line for a combination of a given cycle time and a given number of stations. in the malbp, more than one type of product is assembled on one assembly line, while in the ualbp, the physical configuration of assembly line is ushaped. in recent years, the topic of assembly line balancing has gain many interests from in the areas of industrial engineering and mechanical engineering. adeppa [9] studied the basic models of assembly line balancing problem. sungur and yavuz [10] studied the assembly line balancing problem with hierarchical worker assignment. heydari et al. [11] proposed an entropy-based mathematical formulation for straight assembly line balancing problem. abdullah make et al. [12] performed a review of the two-sided assembly line balancing problems. pereira and álvarez-miranda [13] proposed an exact approach for the robust assembly line balancing problem. a multi-objective assembly line balancing problem with worker's skill and qualification considerations in fuzzy environment was studied by zamzam and elakkad [14]. fathi et al. [15] performed a study based on comparative evaluation of heuristics and computational assessment of objectives for the assembly line balancing problems. abdous et al. [6] introduced an uncertain multi-objective assembly line balancing problem and solved it by a credibility-based fuzzy modeling approach. şahin and tural [16] proposed an effective hybrid fuzzy programming approach for an entropybased multi-objective assembly line balancing problem. álvarez-miranda and pereira [13] provided study and focused on the complexity nature of the assembly line balancing problems. a systematic review of research themes and hot topics in assembly line balancing through the web of science within the years 1990–2017 was performed by didden et al. [17]. liu et al. [18] introduced some classical and hybrid meta-heuristic an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case... 3 algorithms to solve a new cost-oriented assembly line balancing problem. el abidine and koltai [19] introduced a new and effective hybrid goal programming approach for multiobjective straight assembly line balancing problem with stochastic parameters. boysen et al. [1] introduced a new hybrid artificial electric field algorithm for assembly line balancing problem with equipment model selection possibility. fink et al. [20] introduced several hybrid meta-heuristic algorithms for u-shaped assembly line balancing problem with equipment and worker allocations. hou and zhang [21] introduced some new criteria and mathematical formulations for workload smoothing in straight assembly line balancing problem. li et al. [22] introduced a sustainable uncertain integrated supply chain network design and assembly line balancing problem with u-shaped assembly lines and multi-mode demand. sheibani and niroomand [23] developed an optimization model for sustainable multi-product multi-echelon supply chain networks with u-shaped assembly line balancing under uncertainty. for more about this topic, the study of niroomand and vizvari [24] for mathematical formulation, the study of mahmoodirad and niroomand [25] for uncertain modelling, the study of sing et al. [26] for fuzzy modelling, the study of imran et al. [27] for decision-making approaches, and mishra and rani [28] for fuzzy modeling and decision-making can be referred. in this paper a novel study is performed on optimization of the assembly line of the nuzzle for petroleum industries. generally, optimization of industry related problems is very important issue [29,30,31]. for this aim a cost-based mathematical formulation is proposed for such assembly line balancing problem. in this model, overall station establishment cost, fixed salary, and variable wages are optimized simultaneously. in order to be close to real-world situations, the problem is formulated in a triangular fuzzy environment, where the costand time-based parameters are represented by fuzzy values. the proposed fuzzy formulation is converted to a crisp form using a me measure of fuzzy sets and numbers. this is for the first time in the literature that such measure is used in fuzzy assembly line balancing problems. then, in order to evaluate the proposed crisp formulation, a case study from the petroleum industries of iran is considered. based on the performed experiments and obtained results, the best configuration of the assembly line is obtained, and a sensitivity analysis is performed as well. the rest of this paper is organized in some sections. section 2 represents the nuzzle production system and the case study. section 3 presents the fuzzy and crisp mathematical formulations of the nuzzle production system. section 4 includes the computational study and the obtained results. section 5 represents some concluding remarks. 2. problem description nuzzle production assembly line as mentioned earlier, a typical nuzzle production line is to be balanced in this study. the considered nuzzle is made of several parts and needs several assembly operations to be completed. as a case study this nuzzle is to be produced in petroleum industries of iran and it has a wide range of applications in that environment. all information and data of this assembly line is obtained from the petroleum industries of iran. this type of nuzzle requires the raw parts described by table 1. 4 a. mahmoodirad, d. pamucar, s. niroomand table 1 the raw parts required for producing a unit of the nuzzle no. raw part 1 fuel connection 2 mounting flange 3 feed arm 4 shroud 5 deflector valve 6 spring locking ring 7 plastic part on the other hand, some assembly operations (tasks) are required to complete one unit of the nuzzle on an assembly line. these operations and their operating times are described by table 2. furthermore, the precedence relationship graph of these assembly tasks is presented by fig. 1. table 2 the assembly tasks for producing a unit of the nuzzle and their operating times no. assembly task triangular fuzzy operating time (minutes) triangular fuzzy variable cost (wage) ($) 1 fuel connection preparation (4, 5, 6) (0.1, 0.2, 0.3) 2 mounting flange preparation (5, 6, 7) (0.1, 0.2, 0.3) 3 feed arm preparation (3, 4, 6) (0.1, 0.2, 0.4) 4 shroud preparation (3, 4, 5) (0.1, 0.3, 0.4) 5 deflector valve preparation (4, 6, 7) (0.1, 0.2, 0.3) 6 spring locking ring preparation (5, 6, 8) (0.2, 0.3, 0.4) 7 plastic part preparation (3, 4, 5) (0.1, 0.2, 0.3) 8 part 1 to part 2 assembly (5, 7, 8) (0.3, 0.4, 0.5) 9 part 3 to part 2 assembly (9, 10, 11) (0.2, 0.3, 0.4) 10 parts 4 and 5 to part 1 assembly (4, 5, 7) (0.2, 0.3, 0.4) 11 part 6 to part 5 assembly (2, 4, 5) (0.3, 0.4, 0.5) 12 part 7 to part 6 assembly (4, 6, 8) (0.3, 0.4, 0.5) fig. 1 the precedence diagram of the nuzzle an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case... 5 the triangular fuzzy cycle time of (20, 23, 27) minutes is considered for this assembly line where in each station of the line one operator with approximate triangular fuzzy salary of $(1000, 1100, 1300) per month will work. on the other hand, each station can be established by triangular fuzzy average cost of $(10000, 11000, 12000). the managers aim to balance and establish this line by considering establishment cost, operators salaries, and workload smoothness. this is notable to mention that the fuzzy data of this case study are estimated from the historical data of the existing production systems of the considered product. in the next section, a mathematical formulation is proposed for balancing this assembly line according to the preferences of the managers. 3. proposed mathematical formulations in this section, the problem of section 2, first, is formulated as a fuzzy model. then its equivalent crisp formulation is derived. 3.1. fuzzy formulation in order to formulate the assembly line of the nuzzle production line described in section 2, the below assumptions and the notations of table 3 are considered in advance.  the overall cost such as station establishment cost, fixed salary, and variable wage of all stations is to be minimized.  each task is assigned to only one station.  the variable wage of the worker of a station is rated based on the task with highest rate in that station.  all cost and time based parameters are represented by triangular fuzzy values. table 3 the notations used in the formulations of the paper notation nature description i(i) index index used for task (number of tasks) k(k) index index used for station (maximum number of stations) �̃�𝑖 = (𝑡𝑖 (1) , 𝑡𝑖 (2) , 𝑡𝑖 (3) ) parameter triangular fuzzy processing time of task i 𝑐�̃� = (𝑐𝑡(1), 𝑐𝑡(2), 𝑐𝑡(3)) parameter triangular fuzzy cycle time 𝑠𝑡�̃� = (𝑠𝑡𝑏(1), 𝑠𝑡𝑏(2), 𝑠𝑡𝑏(3)) parameter triangular fuzzy establishment cost of each station 𝑠𝑎�̃� = (𝑠𝑎𝑙(1), 𝑠𝑎𝑙(2), 𝑠𝑎𝑙(3)) parameter triangular fuzzy salary paid to the worker of each station �̃�𝑖 = (𝑐𝑖 (1) , 𝑐𝑖 (2) , 𝑐𝑖 (3) ) parameter triangular fuzzy variable processing cost of task i 𝑃𝑅𝑖 parameter predecessor set of task i 𝑋𝑖𝑘 binary variable 1, if task i is assigned to station k 0, otherwise 𝑊𝑘 binary variable 1, if station k is opened 0, otherwise 𝑉𝐶𝑘 variable variable cost (wage) of station 𝑘 6 a. mahmoodirad, d. pamucar, s. niroomand therefore, based on the above-mentioned assumptions and the notations of table 3, the below fuzzy formulation is presented assembly line balancing of the nuzzle production line of section 2.    1 min k k k k of stb sal w vc     (1) subject to       , ,i jrj pr r k ik i x x i k pr (2)    1 1, , k ik k x i k (3)  , ,k ikw x i k (4)    1 , i i ik i t x ct k (5)  , ,i ik kc x vc i k (6)   , 0,1 , ,ik kx w i k (7)  0,kvc k (8) in the above formulation, the objective function of eq. (1) minimizes overall cost of the line including establishment cost of each station, salary paid to the worker of each station, and variable wage paid to the worker of each station. the constraint presented by eq. (2) guarantees that a task can be assigned to a station if its predecessors are assigned to either that station or previous stations. the constraint presented by eq. (3) respects the assumption that a task can be assigned to only one station. the constraint of eq. (4) ensures that a station is established if it contains at least one task. the constraint given by eq. (5) guarantees that the workload of each station cannot exceed the cycle time of the line. the constraint presented by eq. (6) calculates the variable wage rate of each station according to the above-mentioned assumptions. the constraints provided by eqs. (7-8) are the sign constraints of the model. as formulation presented by eqs. (1-8), is a fuzzy formulation, we cannot directly solve it. therefore, first we obtain its crisp form, and then the crisp form is solved by any optimization solver. the equivalent crisp formulation of fuzzy formulation presented by eqs. (1-8) is introduced in the next section. 3.2. equivalent crisp formulation the possibility theory is an effective approach to deal with optimization problems with fuzzy objective function and constraints [32]. in this theory, there are three classical measures of possibility (pos), necessity (nec), and credibility (cr) measures for an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case... 7 converting a fuzzy constraint to its equivalent crisp form. these measures are of pessimistic, optimistic, and average points of view, where decision makers cannot be flexible when using these measures (see [1]). instead, a more flexible measure of this theory called me measure [33] is defined as below, where the weighted average of the possibility and necessity measures of the fuzzy constraint a are considered there.        1me a pos a nec a    (9) in the above formula, the value of γ = 0, results in me{a}=nec{a}, the value of γ = 1 results in me{a}=pos{a}, and γ = 0.5 results in me{a}=cr{a}. the below theorem can clearly explain the use of me measure for a fuzzy constraint with triangular fuzzy parameters. as an advantage, this measure can crisp a fuzzy event from any possibility degree in addition to the necessity, possibility, and credibility measure degrees. theorem 1. for triangular fuzzy variable ξ = (a, b, c) and real number r, the following inequalities are defined for any confidence levels β (from the range of 0 < β ≤ 1) and γ (from the range of 0 < γ ≤ 1) (see [32]).                                                                                                         ,                  1 ,           1 1 ,                   1 ,           1 1 a b r me r b c r b c r me r a b r      (10) therefore, the fuzzy model of eqs. (1-8) is converted to a crisp form using the abovementioned me measure. for this aim, first the below model is obtained. it is notable to mention that objective function given by eq. (1) is considered as a constraint, and the me measure of its constraint form is considered then. minof f (11) subject to    1 k k k k stb sal w vc f     (12)       , ,i jrj pr r k ik i x x i k pr (13)    1 1, , k ik k x i k (14)  , ,k ikw x i k (15) 8 a. mahmoodirad, d. pamucar, s. niroomand    1 , i i ik i t x ct k (16)  , ,i ik kc x vc i k (17)   , 0,1 , ,ik kx w i k (18)  0,kvc k (19) therefore, the below model is obtained by considering the me measure of the fuzzy constraints of formulation presented by eqs. (11-19). minof f (20) subject to    1 k kk k me stb sal w vc f             (21)       , ,i jrj pr r k ik i x x i k pr (22)    1 1, , k ik k x i k (23)  , ,k ikw x i k (24)            1 , i i ik k i me t x ct k (25)      ,0 ,i ik k kme c x vc i k (26)   , 0,1 , ,ik kx w i k (27)  0,kvc k (28) now, applying the relationships given by theorem 1, the below crisp formulation is obtained, which is an equivalent crisp formulation of the fuzzy formulation presented by eqs. (1-8). minof f (29) subject to                             (1) (1) (1) (2) (2) (2) 1 1 , k k k k k k k k stb sal w vc stb sal w vc f        (30) an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case... 9                                  (2) (2) (2) (3) (3) (3) 1 1 1 , 1 1 k k k k k k k k stb sal w vc stb sal w vc f        (31)       , ,i jrj pr r k ik i x x i k pr (32)    1 1, , k ik k x i k (33)  , ,k ikw x i k (34)                             (1) (3) (2) (2) 1 1 0, , i i k k i ik i ik k i i t x ct t x ct k (35)                              (2) (2) (3) (1) 1 1 1 0, , 1 1 i i k k i ik i ik k i i t x ct t x ct k (36)                (1) (2) ,, ,k k i ik i ik k kc x c x vc i k (37)                    (2) (3)1 ,, , 1 1 k k i ik i ik k kc x c x vc i k (38)   , 0,1 , ,ik kx w i k (39)  0,kvc k (40) the crisp formulation presented by eqs. (29-40) is considered as the crisp form of the fuzzy formulation presented by eqs. (1-8). in order to implement this formulation for the case study of section 2, the flowchart of fig. 2 is presented. this notable to mention that, according to the relationship between the confidence levels α, λk, θk, and γ, only one constraint from each pair of the constraints presented by eqs. (30-31), (35-36), and (37-38) is selected in the final model. 4. results and sensitivity analysis as mentioned earlier, the crisp formulation presented by eqs. (29-40) is solved for the nuzzle production line data of section 2 in order to obtain the results and configure the related assembly line. for this aim, the formulation presented by eqs. (29-40) is coded in gams and solved by its cplex solver. all required experiments are run on a pc with core i7-1165g7@ 2.80ghz processor and 16 gb ram. 10 a. mahmoodirad, d. pamucar, s. niroomand fig. 2 the flowchart of implementing the proposed formulations for balancing the nuzzle production line in order to obtain the assembly line configuration, according to the preferences of the managers, the values of α,λk,θk = 0.8 and γ = 0.6 are considered. based on these values, the final crisp formulation consisting of eqs. (29), (31-34), (36), and (38-40) is obtained and solved for the data of section 2 and the results of table 4 are obtained. according to the obtained results, the schematic representation of fig. 3 is considered to show the related assembly line configuration. in table 4, the tasks assigned to each established station is marked. the variable wage of each station is reported. the value of f is the optimal value obtained for objective function (29) and the value of f is obtained by the objective function presented by eq. (1) and the values of the variables obtained in this solution. fig. 3 schematic optimal configuration of the assembly line for the nuzzle production line of section 2 an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case... 11 table 4 the optimal results obtained for the nuzzle production line for α,λk,θk = 0.8 and γ = 0.6 task established stations optimal f equivalent f 1 2 3 4 1 ✔ 70485 (61050, 67155, 73815) 2 ✔ 3 ✔ 4 ✔ 5 ✔ 6 ✔ 7 ✔ 8 ✔ 9 ✔ 10 ✔ 11 ✔ 12 ✔ variable wage 0.30 0.45 0.35 0.45 furthermore, the sensitivity of the crisp formulation of eqs. (29-40) to the confidence level values is studied here. for this aim, the data of section 2 is considered, and the confidence level values are changed (totally 16 experiments are considered). the results for these experiments are obtained by solving the crisp formulation of eqs. (29-40) are represented by table 5. according to the results presented by table 5, the sensitivity of formulation of eqs. (29-40) to the confidence level values are studied. the following remarks can be made for this aim.  in the experiments with the same value of γ, by increasing the value of α, λk, and θk, the value of f is increased. this issue happens for each of the elements of the fuzzy objective function value f separately.  considering the results of table 5, by increasing the value of γ, the value of f is increased. this issue happens for each of the elements of the fuzzy objective function value 𝑓 separately.  the results obtained by the experiments 1-4, is related to the necessity measure of formulation presented by eqs. (29-40).  the results obtained by the experiments 9-12, is related to the credibility measure of formulation presented by eqs. (29-40).  the results obtained by the experiments 21-24, is related to the possibility measure of formulation presented by eqs. (29-40). from the managerial point of view, the proposed problem of this study is developed and solved to help the managers of the industry to design an optimal assembly line for the nuzzle considering the important costs. on the other hand, the fuzziness of the proposed model can help the managers for more robust and near real-world situation decisions. 12 a. mahmoodirad, d. pamucar, s. niroomand table 5 the results obtained for the sensitivity analysis experiment , ,k k   𝛾 optimal f equivalent f 1 0.1 0 64155 (57750, 63525, 69825) 2 0.4 0 68686 (60060, 66066, 72618) 3 0.7 0 73499 (62480, 68728, 75544) 4 1 0 93100 (77000, 84700, 93100) 5 0.1 0.3 43193 (41800, 45980, 50540) 6 0.4 0.3 65739 (58928, 64821, 71250) 7 0.7 0.3 71965 (61914, 68105, 74860) 8 1 0.3 93100 (77000, 84700, 93100) 9 0.1 0.5 42187 (41360, 45496, 50008) 10 0.4 0.5 46807 (43340, 47674, 52402) 11 0.7 0.5 68686 (60060, 66066, 72618) 12 1 0.5 93100 (77000, 84700, 93100) 13 0.1 0.7 41759 (41171, 45288, 49780) 14 0.4 0.7 45019 (42585, 46844, 51490) 15 0.7 0.7 48400 (44000, 48400, 53200) 16 1 0.7 93100 (77000, 84700, 93100) 17 0.1 0.9 41522 (41066, 45173, 49653) 18 0.4 0.9 44551 (42655, 46921, 51574) 19 0.7 0.9 46631 (43266, 47593, 52313) 20 1 0.9 93100 (77000, 84700, 93100) 21 0.1 1 29108 (28820, 31702, 34846) 22 0.4 1 43700 (42020, 46222, 50806) 23 0.7 1 46020 (43010, 47311, 52003) 24 1 1 48400 (44000, 48400, 53200) 5. conclusions in this study a fuzzy mathematical model was introduced and solved for balancing and configuring the nuzzle production line in petroleum industries. this is an important product which is widely used petroleum industries. for this aim, first a cost-based mathematical formulation considering overall station establishment cost, fixed salary, and variable wages was proposed to obtain the optimal assembly line configuration. in order to be close to realworld situations, the problem was formulated in a triangular fuzzy environment with triangular fuzzy costand time-based parameters. the proposed fuzzy formulation was converted to a crisp form using a me measure of fuzzy sets and numbers. then, in order to evaluate the proposed crisp formulation, a case study from the petroleum industries of iran was considered. based on the performed experiments and obtained results, the best configuration of the assembly line was obtained, and a sensitivity analysis was performed as well. the most important limitation of this study was the high uncertainty nature of the problem that applying fuzzy theory and its extensions, a framework was developed for this difficulty. an optimization scheme for the fuzzy cost-based assembly line balancing problem for the case... 13 references 1. boysen, n., schulze, p., scholl, a., 2022, assembly line balancing: what happened in the last fifteen years?, european journal of operational research, 301(3), pp. 797-814. 2. chutima, p., 2023, assembly line balancing with cobots: an extensive review and critiques, international journal of 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geared transmission, contact ratio, dynamic synthesis, gear efficiency 1. introduction today gears are present in all possible fields. their key advantage lies in their providing for a very high working efficiency. additionally, gears can transmit large loads [3, 13]. regardless of their size, gears must be synthesized carefully considering the specific conditions. this paper intends to present the main conditions that must be met for a correct synthesis of the gear [8-11]. the pinnacle of using sprocket mechanisms is to be sought in ancient egypt at least some thousand years before christ. for the first time in human history, transmission wheels "spurred" to irrigate crops were used and so were worm gears for cotton processing [1-2]. in the year of 230 b.c., in the city of alexandria in egypt, multi-lever wheels with gear rack were used. such gears were constructed and used from primeval times, above all for lifting heavy anchors of vessels and for winding catapult arms on the battlefields. they were later introduced in the vehicles with wind and water in order to reduce or enlarge the pump originating from windmills or water (see fig. 1). the antikythera mechanism is the name given to an astronomical calculating device, measuring about 32 by 16 by 10 cm, which was discovered in 1900 in a sunken ship just off the coast of antikythera, an island between crete and the greek mainland. several kinds of evidence point unquestionably to around 80 b.c. as the date of the shipwreck. the device, made of bronze gears fitted in a wooden case, was crushed in the wreck, and parts of the faces received march 02, 2014 / accepted march 31, 2014  corresponding author: florian ion petrescu bucharest polytechnic university, faculty of transport, theory of mechanisms and robots, bucharest, romania e-mail: petrescuflorian@yahoo.com original scientific paper 52 f.i. petrescu, r.v. petrescu were lost, "the rest then being coated with a hard calcareous deposit at the same time as the metal corroded away to a thin core coated with hard metallic salts preserving much of the former shape of the bronze" during almost 2000 years of its underwater existence (fig. 2). fig. 1 transmissions wheeled "spurred" to irrigate crops and worm gears used for cotton processing fig. 2 the "antikythera" mechanism – an astronomical calculating device modern adventure began with the spur gear wheel spurred created by leonardo da vinci, in the 15 th century. he founded the new kinematics and dynamics stating inter alia the principle of superposition of independent movements (fig. 3). fig. 3 the spur gear wheel spurred created by leonardo da vinci, 15 th century high efficiency gears 53 benz had engine with transmissions sprocket gearing and gear chain (patented in 1882, fig. 4), [4-6] but the first gear patent (the drawings of the first gear transmission patented) and gearing wheels with chain were made in 1870 by the british starley & hillman [12]. fig. 4 the benz patent it is in 1912, in cleveland (usa), that the production of industrial specialized wheels and gears (cylindrical, worm, conical, with straight teeth, inclined or curved; see fig. 5) started. fig. 5 in cleveland, the production of industrial specialized wheels started in 1912 54 f.i. petrescu, r.v. petrescu today, the gears are present everywhere in the world of mechanics, such as the car industry, electronics and electro-technique equipments, power industry, etc. (fig. 6). fig. 6 gear today 2. determining gear efficiency as a function of the contact ratio one calculates the efficiency of a geared transmission, having in mind the fact that, at one moment, there are several couples of teeth in contact, and not just one [3]. hence, the initial model incorporates four pairs of teeth in contact (4 couples) concomitantly [7-11]. the first couple of teeth in contact has the contact point i, defined by ray ri1, and pressure angle i1. the forces which act at this point are: motor force fmi, perpendicular to position vector ri1 at i and the force transmitted from wheel 1 to wheel 2 through point i, fti, parallel to the path of action and with the direction from wheel 1 to wheel 2. the transmitted force is practically a projection of the motor force onto the path of action. the defined velocities are similar to the forces (having in mind the original kinematics, or the precise kinematics adopted). the same parameters are defined for the other three points of contact – j, k and l (fig. 7). the quantities of interest are: rb1 – the base radius of drive wheel 1; 1 – the circular velocity of wheel 1; z1 – the number of teeth of wheel 1;  – the pressure angle, so that 1 is the pressure angle for wheel 1 and 0 is the normal pressure angle on the pitch circle. high efficiency gears 55 fig. 7 four pairs of teeth in contact concomitantly as a starting point, we write the following relations between the velocities as [8-11]: 1 1 1 1 1 1 1 1 1 1 1 1 cos cos cos cos cos cos cos cos i mi i i i b j mj j j j b k mk k k k b l ml l l l b v v r r v v r r v v r r v v r r                                                 (1) from eqs. (1), one obtains the equality of the tangential velocities (eq. (2)), and, furthermore, the motor velocities are explicitly obtained (eq. (3)): 1 1i j k l bv v v v r         (2) 1 1 1 1 1 1 1 1; ; ; cos cos cos cos b b b b mi mj mk ml i j k l r r r r v v v v                 (3) the forces transmitted concomitantly at the four points must be equal [8-11]: i j k lf f f f f        (4) the motor forces are given as: 56 f.i. petrescu, r.v. petrescu ; ; ; cos cos cos cos mi mj mk ml i j k l f f f f f f f f            (5) the momentary efficiency can be written in the following form: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 cos cos cos cos 4 1 1 1 1 cos cos cos cos 4 4 i i j j k k l lu i c m mi mi mj mj mk mk ml ml b b b b b i j k l i j k l i f v f v f v f vp p p p f v f v f v f v f r f r f r f r f r tg tg                                                                        2 2 j k ltg tg   (6) eqs. (7) and (8) are auxiliary relations [7-11]: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ; ; ; ( ); 2 2 ( ); 2 2 2 2 ( ); b i b j b k b l b j i b j i b k i b k i b l i k i r tg k j r tg k k r tg k l r tg k j k i r tg tg k j k i r tg tg z z k k k i r tg tg k k k i r tg tg z z k l k i r tg tg k l k i                                                          1 1 1 2 2 3 3b l ir tg tg z z                              (7) 111 2 3; 2 2; 2 z tgtg z tgtg z tgtg ilikij       (8) one keeps eq. (8), with the sign plus (+) for the gearing where drive wheel 1 has external teeth (regardless if it is external or internal gearing), and with the sign (-) for the gearing where drive wheel 1 has internal teeth (the drive wheel has a ring form only for the internal gearing). the relation of momentary efficiency (eq. (6)) uses auxiliary eq. (8) and takes the form of eq. (9). in eq. (9), one starts with eq. (6) where four pairs are in contact concomitantly, but then one generalizes the expression by replacing number 4 (four pairs) by variable e, which represents the whole number of the contact ratio +1, and after restricting the sum expressions, variable e is replaced by contact ratio 12, as well. high efficiency gears 57 the mechanical efficiency offers more advantages than the momentary efficiency, and will be calculated approximately, by replacing pressure angle α1 in eq. (9) with normal pressure angle α0, so that eq. (10) is obtained, where 12 represents the contact ratio of the gearing, and it will be calculated by eq. (11) for the external gearing, and by eq. (12) for the internal gearing: 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 11 2 2 2 2 1 111 4 4 4 2 2 2 4 ( ) ( 2 ) ( 3 ) 4 4 2 4 4 (0 1 2 3 ) 2 (0 1 2 3) 1 4 2 1 ( 1) 2 ( 1) 1 1 i i j k l i i i i i i e e i i i i tg tg tg tg tg tg tg tg z z z tg tg zz tg i tg i e ze z tg                                                                        2 2 1 1 2 11 2 2 1 1 2 11 2 2 1 1 12 12 122 11 44 ( 1) (2 1) ( 1) 6 2 1 2 ( 1)2 ( 1) (2 1) 1 3 1 22 1 ( 1) (2 1) ( 1) 3 i tge e e e e e ze z tg ee e tg zz tg tg zz                                                                            (9) 2 2 0 0 12 12 122 11 1 22 1 ( 1) (2 1) ( 1) 3 m tg tg zz                     (10) 2 2 2 2 1 0 1 2 0 2 1 2 0. . 12 0 sin 4 4 sin 4 4 ( ) sin 2 cos a e z z z z z z                     (11) 2 2 2 2 0 0 0. . 12 0 sin 4 4 sin 4 4 ( ) sin 2 cos e e i i i ea i z z z z z z                     (12) the results of the performed calculations for various gear transmission parameters are summarized in table 1 [8-11]. 58 f.i. petrescu, r.v. petrescu table 1 gear efficiency for different sets of gear transmission parameters the summarized results z1 0 [°] z2 12 ae 12 ae 21 ae 12 ai 12 ai 21 ai 42 20 126 1.79 0.844 0.871 1.92 0.838 0.895 46 19 138 1.87 0.856 0.882 2.00 0.850 0.905 52 18 156 1.96 0.869 0.893 2.09 0.864 0.915 58 17 174 2.06 0.880 0.904 2.20 0.876 0.925 65 16 195 2.17 0.892 0.914 2.32 0.887 0.933 74 15 222 2.30 0.903 0.923 2.46 0.899 0.942 85 14 255 2.44 0.914 0.933 2.62 0.910 0.949 98 13 294 2.62 0.924 0.941 2.81 0.920 0.956 115 12 345 2.82 0.934 0.949 3.02 0.931 0.963 137 11 411 3.06 0.943 0.957 3.28 0.941 0.969 165 10 495 3.35 0.952 0.964 3.59 0.950 0.974 204 9 510 3.68 0.960 0.970 4.02 0.958 0.980 257 8 514 4.09 0.968 0.975 4.57 0.966 0.985 336 7 672 4.66 0.975 0.980 5.21 0.973 0.989 457 6 914 5.42 0.981 0.985 6.06 0.980 0.992 657 5 1314 6.49 0.986 0.989 7.26 0.986 0.994 3. efficiency of helical gears as a function of the contact ratio in praxis, helical gears are used very often. for helical gears, the calculations show a decrease in yield with increasing tooth inclination angle (β). for angles not exceeding 25°, the efficiency of gears is rather good. however, when the inclination angle exceeds 25°, the gears will suffer a significant drop in yield. new calculation relationships can be given in eqs. (13-15): 2 2 1 2 2 2 2 4 2 1 0 0 1 cos 2 ( cos ) cos ( 1) (2 1) 2 cos ( 1) 3 m z z tg tg z                             (13)   2 . . 2 3 1 0 1 2 3 2 0 2 1 2 0 1 [( 2 cos ) ] 4 cos ( cos ) 2 [( 2 cos ) ] 4 cos ( cos ) ( ) a e tg z tg z z tg z z z tg                                    (14)   2 . . 2 3 0 2 3 0 0 1 [( 2 cos ) ] 4 cos ( cos ) 2 [( 2 cos ) ] 4 cos ( cos ) ( ) a i e e i i e i tg z tg z z tg z z z tg                                    (15) 4. validation all the presented relationships have been validated by using the "inventor" software package; a very good compliance is confirmed. several examples have been computed for both external and internal gearing. the corresponding gear pairs have been drawn automatically by the "inventor". figs. 8 and 9 depict the gear pairs together with the main high efficiency gears 59 parameters of the considered gears for external and internal gearing, respectively. for angle 0 ranging between 10° and 20°, the contact line is perfect. for values greater than 25°, the software confirms that such a design is no longer safe, and for 0 taking values lower than 10°, the software has no details necessary for verification because it relies on the experimental values that no longer exist for such small values of angle 0. fig. 8 validated examples of external gearing fig. 9 validated examples of internal gearing 60 f.i. petrescu, r.v. petrescu 5. conclusions the best efficiency is obtained with the internal gearing when drive wheel 1 is a ring. the minimum efficiency will be obtained when drive wheel 1 of the internal gearing has external teeth. for the external gearing, the best efficiency is obtained when the bigger wheel is the drive wheel. with decreasing normal angle 0, the contact ratio increases and efficiency increases as well. efficiency increases too, when the number of teeth of drive wheel 1 (z1) increases. references 1. lei, x. m., ge, y. z., zhang, y. c., liu, p., 2011, design and analysis for high-speed gear coupling, applied mechanics and materials, 86, pp. 658-661. 2. lin c., hou, y., j., zeng, q. l., gong, h., nie. l., qiu, h., 2011, the design and experiment of oval bevel gear, applied mechanics and materials, 86, pp. 297-300. 3. maros, d., 1958, cinematica roţilor dinţate. editura tehnică, bucureşti. 4. matos, l., marinho, b., 2011, a comparison of the delay spread obtained with different power delay profiles de-noising techniques, engevista, 13(2), pp. 129-133. 5. nogueira, o. c., real, m. v., 2011, estudo comparativo de motores diesel maritimos atraves da analise de lubrificantes usados e engehharia de confiabilidade, engevista, 13(3), pp. 244-254. 6. oliveira, m. m. p., lima, f. r., 2003, analogias entre o desenho instrumental e o dedenho computacional, engevista, 5(9). 7. pelecudi, c. h. r.., maros, d., merticaru, v., pandrea, n., simionescu, i., 1985, mecanisme, editura didactică şi pedagogică, bucureşti. 8. petrescu, v., petrescu, i., 2002, randamentul cuplei superioare de la angrenajele cu roţi dinţate cu axe fixe, proceedings of the 7th national symposium prasic, braşov, vol. i, pp. 333-338. 9. petrescu, r., petrescu, f., 2003., the gear synthesis with the best efficiency, proceedings of esfa’03, bucharest, vol. 2, pp. 63-70. 10. petrescu, r. v., petrescu, f. i., popescu, n., 2007, determining gear efficiency, gear solutions, pp. 19-28. 11. petrescu, f.i., 2012, teoria mecanismelor – curs si aplicatii (editia a doua), create space publisher, usa, p. 284. 12. rey g. g., 2013, influencia de la lubricacion en la eficiencia de engranajes de tornillo sinfin , ingineria mecanica, 16(1), pp. 13-21. 13. stoica, i. a., 1977, interferenţa roţilor dinţate, editura dacia, cluj-napoca. zupčanici visoke efikasnosti rad predstavlja originalnu metodu za određivanje efikasnosti zupčastih prenosnika, kao i sila, brzina i snaga u prenosniku. analizira se način na koji određeni parametri utiču na efikasnost prenosnika. takođe, ukratko je predstavljena originalna metoda za određivanje efikanosti zupčastih prenosnika u funkciji stepena sprezanja. pomoću predstavljenih relacija moguće je sprovesti dinamičku sintezu zupčastih prenosnika u cilju postizanja veće efikasnosti mehanizama. ključne reči: zupčanik, zupčasti prenos, stepen sprezanja, dinamička sinteza, efikasnost zupčanika fume 7343 facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 115 124 https://doi.org/10.22190/fume210112027g © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper bubble dynamics-based modeling of the cavitation dynamics in lubricated contacts thomas geike beuth hochschule für technik berlin, berlin, germany abstract. cavitation is a common phenomenon in fluid machinery and lubricated contacts. in lubricated contacts, there is a presumption that the short-term tensile stresses at the onset of bubble formation have an influence on material wear. to investigate the duration and magnitude of tensile stresses in lubricating films using numerical simulation, a suitable simulation model must be developed. the chosen simulation approach with bubble dynamics is based on the coupling of the reynolds equation and rayleigh-plesset equation (introduced about 20 years ago by someya).following the basic approach from the author’s earlier papers on the negative squeeze motion with bubble dynamics for the simulation of mixed lubrication of rough surfaces, the paper at hand shows modifications to the rayleigh-plesset equation that are required to get the time scale for the dynamic processes right. this additional term is called the dilatational viscosity term, and it significantly influences the behavior of the numerical model. key words: cavitation, mixed lubrication, oil stiction, negative squeeze motion, bubble dynamics, negative pressure, wear 1. introduction cavitation is a common phenomenon in fluid machinery and lubricated contacts. in the context of fluid machinery, the focus is on cavitation erosion, in which pressure shocks lead to material failure [1]. in lubricated contacts, on the contrary, there is a presumption that the short-term tensile stresses at the onset of bubble formation have an influence on material wear. to investigate the duration and magnitude of tensile stresses in lubricating films using numerical simulation, a suitable simulation model must be developed. the author proposed such a cavitation model for numerical simulations of mixed lubrication that includes tensile stresses in the lubrication film at the onset of bubble growth [2]. received january 12, 2021 / accepted march 07, 2021 corresponding author: thomas geike beuth hochschule für technik berlin, luxemburger str. 10, 13353 berlin, germany e-mail: thomas.geike@beuth-hochschule.de 116 t. geike generally speaking, simulation models for the cavitation dynamics in lubricated contacts can be roughly clustered into two groups: either without or with bubble dynamics, the first one being the standard case for most fluid film bearing calculations. cavitation is a crucial phenomenon in the calculation of the load carrying capacity of plain bearings. the literature available on this subject is extensive. for most bearing calculations, the tensile stresses present for a short time are not relevant; in this respect, “pressure equals zero” is used for calculations when there is no positive pressure. the presented simulation model belongs to the second group – having the bubble dynamics as one key feature included in the model. the aim here is to calculate, besides macroscopic forces between moving rough surfaces, also the tensile stresses that may affect the wear in mixed lubrication applications. also, for adhesion problems in lubricated contacts (the so-called oil stiction problem), the tensile stresses in the lubricating film are necessarily taken into account when two surfaces separate. the original model of the author yielded qualitatively reasonable results, but three key issues remained [3]. 1. the characteristic time for the decline of tensile stress seems rather short. there is no reliable source of information on the characteristic time scale for the processes studied but results from the oil stiction problem and experiments hint to larger characteristic times than those observed with the original model. this will be discussed in detail in sections 2.2 and 4.3. 2. numerical simulations work well with circular plates, but spherical plates cause numerical stability problems in the original model. 3. experimental evidence with quantitative results for time evolution of pressure on very short time scales are missing for the squeeze motion studied here. the paper at hand deals with the first two open issues from this list. 2. cavitation in lubrication films – some basics 2.1. cavitation means formation and destruction of cavities in lubricated contacts, conditions can occur in which the lubricant becomes temporarily discontinuous. in the lubricating film, a coexistence of liquid and cavities is created and perishes. the phenomenon called cavitation can be divided into gas and vapor cavitation [4,5]. cavitation occurs when the pressure locally falls below a certain limit. gas cavitation can be observed in aerated lubricants at pressures below the ambient pressure. the cavities then contain dissolved gases. vapor cavitation occurs at dynamic load conditions when the pressure falls below the vapor pressure. vapor of the lubricating fluid is then present in the cavities. vapor cavitation is a dynamic process; vaporization does not occur abruptly. therefore, lubricants can also transmit tensile stresses for a short time. 2.2. tensile stresses are observed in negative squeeze motion the classical solution for normal force f and pressure p when squeezing a viscous fluid (dynamic viscosity ) between two parallel circular plates (fig. 1, velocity v, plate radius l, distance h) is bubble dynamics-based modeling of the cavitation dynamics in lubricated contacts 117 𝐹 = 3𝜂𝜋𝑉𝐿4 2ℎ3 (1) and 𝑝 = 𝑝amb − 3𝜂𝑉 ℎ3 (𝐿2 − 𝑟2) (2) with ambient pressure 𝑝amb [6] and radial coordinate r. when the upper plate approaches the base plate, the pressure is always positive (squeeze motion). when the plates are separated (negative squeeze motion), the pressure at the center of the plate falls below the vapor pressure 𝑝v for 𝑉 > ℎ3 3𝜂𝐿2 (𝑝amb − 𝑝v) (3) and cavitation starts. fig. 1 negative squeeze motion for the setup with circular plate (pull-off experiment) time-dependent cavitation in a simple arrangement of parallel plates has been experimentally studied by hays and feiten [7], parkins and may-miller [8], and chen et al. [9]. hays and feiten considered the case of constant velocity, while the other two groups studied the case of periodic motion. the experiments confirm that the lubricating film can transmit a tensile stress. due to the tensile stress, bubble growth occurs and finally the macroscopically visible disruption of continuity. the tensile stress approaches zero within a short time. in the experiments, the cavitation region arises in the center and disappears there. all experiments focus on the characterization of cavitation patterns; in particular, no data are provided that can be used to check numerical investigations sufficiently well. the experiments under practical conditions have shown that cavitation can occur in bearings, e.g. in connecting rod bearings in internal combustion engines, and that this is sometimes associated with considerable damage [10]. 118 t. geike 2.3. tensile stresses are reproduced in simulation models with bubble dynamics an overview of research results on the simulation of cavitation dynamics can be found in the publications [2,3]. there is one common feature of all simulations known to the author: they do not reproduce the fine spatial structures as observed in experiments. as mentioned before, simulation models for the cavitation dynamics in lubricated contacts can be roughly clustered into two groups: either without or with bubble dynamics. approaches without bubble dynamics usually use the reynolds equation complemented with some additional conditions to define the cavitated areas, and do not reproduce tensile stresses in the fluid. the approach with bubble dynamics, on the other hand, reproduces tensile stresses in the fluid film in negative squeeze motion. the building blocks of bubble dynamics-based simulation models are the reynolds equation and rayleigh-plesset equation. the idea of combining the two equations for lubrication problems goes back to someya about 20 years ago [11]. it has been used for journal bearings and squeeze film dampers; it is essentially required for correct numerical calculations of the negative squeeze motion (i.e. the separation of two plates) or the oil stiction problem. 3. modeling the cavitation dynamics in the squeeze motion 3.1. the idea in a nutshell: reynolds equation and bubble growth as said before, a simulation model that reproduces tensile stresses must account for the dynamics of bubble growth and decay. starting from the rayleigh-plesset equation [12,13] for the bubble radius r 𝑅 𝑑2𝑅 𝑑𝑡2 + 3 2 ( 𝑑𝑅 𝑑𝑡 ) 2 = 𝑝v − 𝑝 𝜌liq − 4𝜂liq 𝜌liq𝑅 𝑑𝑅 𝑑𝑡 − 2𝛾 𝜌liq𝑅 (4) and the reynolds equation for compressible fluids 1 𝑟 𝜕 𝜕𝑟 ( 𝑟𝜌ℎ3 𝜂 𝜕𝑝 𝜕𝑟 ) = 12ℎ 𝜕𝜌 𝜕𝑡 + 12𝜌𝑉 (5) partial differential equations can be derived for density  and pressure p in the pull-off experiment. both quantities depend on time t and radial coordinate r. the subscript liq indicates the liquid state.  characterizes the surface tension. this approach for the cavitation problem in lubricating films in the pull-off experiment was first presented in [2]. the work of someya [11] was not known to the author at that time. instead, the approach was inspired by the work of sauer on cavitation in fluid machinery [14]. the spatial discretization of the partial differential equations with the differential quadrature method [15] leads to a system of differential and algebraic equations solved with a solver for differential-algebraic systems of equations (dae). as shown by numerical experiments, a simulation model based on the above equations yields extremely short times for the pressure drop [2] and numerical stability problems for the spherical cap (unpublished research results from the author). it will be shown that adding an additional term (dilatational viscosity term) solves both problems. the stability problem will be discussed in a subsequent publication, dealing with the spherical cap in more detail. the dilatational viscosity term has been used by different authors in the context bubble dynamics-based modeling of the cavitation dynamics in lubricated contacts 119 of cavitation in lubricants, beginning with someya [11]. according to someya, the dilatational viscosity term is related to the marangoni effect. when the bubble expands under negative pressure, the dilatational viscosity increases the resistance against bubble growth and rupture. in the paper by gehannin et al. [16], two terms change compared to the above-mentioned rayleigh-plesset equation. first, an additional term is added with the dilatational viscosity (following someya’s approach), and second, the constant vapor pressure is replaced by a more complex expression. with respect to the dilatational viscosity, gehannin et al. follow the explanations of someya [11] and add the term − 4𝜅 𝜌liq𝑅 2 𝑑𝑅 𝑑𝑡 to the right-side of the rayleigh-plesset equation, with  characterizing the dilatational viscosity term. adding this term to the simulation model has a significant impact on the time scale on which tensile stresses exist during the negative squeeze motion. numerical experiments with a single bubble and the additional term, recently undertaken by the author, give three main insights: (1) the characteristic time scale for the bubble dynamics is about 1000 times larger than in earlier numerical experiments. (2) the inertia term with the second derivative of the bubble radius and the surface tension term can be neglected. (3) when solving the rayleigh-plesset equation for the first derivative of the bubble radius, catastrophic cancellation may occur if not dealt with appropriately. 3.2. continuum with microstructure coupling bubble dynamics with the reynolds equation the rayleigh-plesset equation describes the behavior of a single, spherical bubble in an incompressible fluid of infinite extension. it applies to the growth of the bubble in the first phase, where mechanical effects dominate: inertia, pressure difference, viscosity, and surface tension. modeling cavitation by bubble growth assumes that the bubbles in the lubricating film are far enough apart for the rayleigh-plesset equation to be applicable in good approximation. individual bubbles, characterized by position and radius, are not described. instead, a density  or a vapor fraction  is assigned to a location, more precisely to a radial coordinate r, in the framework of a continuous description method. between density and vapor fraction exists the relation 𝜌 = 𝛼𝜌vap + (1 − 𝛼)𝜌liq (6) where index vap indicates to vapor state. correspondingly, for the time derivative 𝜕𝜌 𝜕𝑡 = 𝜕𝛼 𝜕𝑡 (𝜌vap − 𝜌liq) (7) formally, via 𝑅 = √ 3 4𝜋𝑛0 𝛼 1 − 𝛼 3 (8) each location is assigned a bubble radius, with 𝑛0 characterizing the concentration of bubble nuclei. from the rayleigh-plesset equation follows the time evolution of the bubble radius and hence the time evolution of the density. 120 t. geike 4. negative squeeze motion for the circular plate now the pull-off experiment (negative squeeze motion) shown in fig. 1 will be studied in detail. a circular plate is pulled upward with constant velocity in the simplest case. at the boundary, the pressure is always the ambient pressure. this simple arrangement is studied by boedo and booker [17] using the finite element method and a very simple cavitation model. the arrangement has practical significance for the study of the adhesion behavior of lubricated contacts (oil stiction problem), as it is relevant in the simulation of rapidly opening valves. 4.1. dimensionless model equations: coupling of the rayleigh-plesset equation and reynolds equation the equations constituting the simulation model are made dimensionless as follows. all quantities with a bar on top are corresponding dimensionless quantities. for the lengths, the reference quantity is the plate radius. there is one exception: for the bubble radius is 𝑅 = 𝛬 𝑅 𝐿 (9) although in the practical implementation =1 has always been used so far. for pressure, the ambient pressure is taken as reference value. for the reference time 𝑇ref = 𝜂liq 𝑝amb (10) is set. for the dimensionless bubble radius and the dimensionless concentration of bubble nuclei, we have 𝑅 = √ 3 4𝜋𝑛0   𝛼 1 − 𝛼 3 (11) and 𝑛0  =   𝑛0 𝐿  3 𝛬3 (12) the dimensionless velocity is �̅� = 𝑉 𝑇ref 𝐿 (13) the differential equation for the bubble radius, neglecting the term with the second derivative of the bubble radius and the surface tension term, can be reduced to the form 𝑅′ = − 4 3𝑅 (𝜁1 + 𝜁2 𝑅 ) + √ 16 9𝑅 2 (𝜁1 + 𝜁2 𝑅 ) 2 2 3 𝜁1 + (𝑝 v − 𝑝) (14) eq. (14) is the dimensionless form of eq. (4) considering all mentioned simplifications to the rayleigh-plesset equation. reynolds equation (5) is not repeated here, as its form does not change by making the equation dimensionless. the dimensionless constants are defined as 𝜁1 = 𝜂liq 2 𝛬2 𝑝amb𝐿 2𝜌liq (15) and bubble dynamics-based modeling of the cavitation dynamics in lubricated contacts 121 𝜁2 = 𝜅𝜂liq𝛬 3 𝑝amb𝐿 3𝜌liq (16) for practical implementation, again note that for small bubble radius, the difference is formed from two numbers of nearly equal size. to avoid the catastrophic cancellation effect, the program code needs to be written appropriately. for the time derivative of the vapor volume fraction the relation 𝜕𝛼 𝜕𝑡 = 3𝛼(1 − 𝛼) 1 𝑅 𝑑𝑅 𝑑𝑡 + ℎ2 12𝜂 𝜕𝑝 𝜕𝑟 𝜕𝛼 𝜕𝑟 (17) is used according to eq. (12) from [2]. this gives the change in density over time needed for the reynolds equation. 4.2. dynamics of tensile stresses are plausibly reproduced selected simulation results for the circular plate are discussed below. for the selected dynamic viscosity, 𝑇ref = 0.35 μs. initial distance is ℎ = 10−2at 𝑡 = 0. the lubricant film thickness is thus 1% of the plate radius. the velocity is constant 𝑉 = 10−4. for the start value of vapor fractionα0 two different values, 10-8 and 10-3 are chosen. in diagrams with the spatial coordinate as horizontal axis (i.e. fig. 2 and fig. 4), the curves show different time moments between 0 to 200 as indicated in the diagrams. for 𝑡 = 0 the system starts with the theoretical pressure distribution without cavitation (fig. 2 for α0 = 10−8). the pressure distribution is not given as an initial condition but results from the given α distribution as part of the solution of the dae system. with increasing time, the tensile stress decreases more and more. the massive decrease of the tensile stress happens in a time of the order of 0.1 millisecond. in the context of this work, the characteristic time for pressure drop is defined as the time required for the decay of force to 1% of the initial value. the initial value α0 = 10−3 gives a similar pressure distribution (quite contrary to previously published calculations without dilatational viscosity). fig. 2 pressure distribution for the negative squeeze motion 122 t. geike for 𝑡 = 400, the dimensionless pressure in the center is about -1.4. the dimensionless force drops from -463 at 𝑡 = 0 to -3.7 for 𝑡 = 400 (0.14ms); the drop in force to 1% of the initial value occurs within 0.13ms (fig. 3).thus the characteristic time for pressure drop is here 0.13 ms. fig. 4 shows the corresponding time evolution of vapor fraction and bubble radius as function of radial coordinate and time. vapor ratio, bubble radius and thus density do not change significantly during the phase of rapid decrease of tensile stress. essentially, there are two phases. in the first phase the tensile stress drops rapidly, then in the second phase the density changes appreciably, while the pressure is nearly zero. fig. 3 macroscopic force as function of time for the negative squeeze motion fig. 4 spatial distribution of vapor fraction (left) and bubble radius (right) bubble dynamics-based modeling of the cavitation dynamics in lubricated contacts 123 4.3. additional term in the rayleigh-plesset equation leads to significantly larger characteristic times in the earlier simulations, the pressure drop occurs in a time interval of order 𝑡 ≈ 10−1 or 𝑡 ≈ 0.035  μs [2]. in the new simulations considering dilatational viscosity, a comparable pressure drop occurs about 1000 times slower! the now observed characteristic times of the tensile stress drop in the order of 0.1ms matches the times reported by resch and scheidl for the oil stiction problem [18]. a characteristic time of the order of 0.1 ms is also more plausible when thinking of the experimental observations [7-9]. thus, including dilatational viscosity in the simulation model is key. 5. conclusions an existing model for the numerical simulation of cavitation in mixed lubrication contacts that reproduces tensile stresses on short time scales has been developed further. as assumed earlier adding the dilatational viscosity term to the rayleigh-plesset equation significantly increases the characteristic time of the existence of tensile stresses in the negative squeeze motion. the larger time scale seems more realistic when looking at the related problem of oil stiction and early experimental evidence. furthermore, initial simulations for the spherical cap indicate that the previously stated numerical stability problems can be overcome by the additional term with the dilatational viscosity. further work needs to be done to study the spherical cap in negative squeeze motion in detail, including a comparison of density evolution on larger time scales. from today’s perspective, the two major directions of further research are (1) the simulation of the elasto-hydrodynamic problem of rough surfaces including the cavitation dynamics and (2) studies on the effect of short-time tensile stresses in lubricants on the material wear. for the first topic, the boundary element method (bem) seems to be the right choice for modeling the elastic part. only the surface is discretized in the bem. hence the method allows to model surface roughness with very fine meshes and it is often more efficient for contact problems than the methods requiring the discretization of the entire volume. to make bem simulations with cavitation dynamics efficient, it is also advisable to undertake further studies of the negative squeeze motion regarding the relevance of terms (inertia, viscosity, surface tension). references 1. prandtl, l., 1957, strömungslehre, friedr. vieweg & sohn. 2. geike, t., popov, v.l., 2009, a bubble dynamics based approach to the simulation of cavitation in lubricated contacts, asme j. tribol., 131(1), 011704. 3. geike, t., 2020, review on the bubble dynamics based cavitation dynamics for the negative squeeze motion in lubricated contacts, front. mech. eng., 6, 33. 4. dowson, d., taylor, c.m., 1979, cavitation in bearings, annu. rev. fluid mech., 11, pp. 35-66. 5. szeri, a.z., 2005, fluid film lubrication, cambridge university press. 6. popov, v.l., 2015, kontaktmechanik und reibung, springer vieweg. 7. hays, d.f., feiten, j.b., 1964, cavities between moving parallel plates, in: davies, r. (ed.), cavitation in real liquids, elsevier, new york, pp. 122-127. 124 t. geike 8. parkins, d.w., may-miller, r., 1984, cavitation in an oscillatory oil squeeze film, asme j. tribol., 106, pp. 360-367. 9. chen, x., sun, m., wang, w., sun, d.c., zhang, z., wang, x., 2004, experimental investigation of time dependent cavitation in an oscillatory squeeze film, sci. china ser. g, 47, pp. 107-112. 10. optasanu, v., bonneau, d., 2000, finite element mass-conserving cavitation algorithm in pure squeeze motion validation/application to a connecting-rod small end bearing, asme j. tribol., 122, pp. 162-169. 11. someya, t., 2003, negative pressure in the oil-film of journal bearing, rotrib 03 national tribology conference, university of galati, galati, pp. 215-220. 12. plesset, m.s., prosperetti, a., 1977, bubble dynamics and cavitation, annu. rev. fluid mech., 9, pp. 145-185. 13. feng, z.c., leal, l.g., 1997, nonlinear bubble dynamics, annu. rev. fluid mech., 29, pp. 201-243. 14. sauer, j., 2000, instationär kavitierende strömungen ein neues modell basierend auf front capturing und blasendynamik, phd thesis, universität karlsruhe, karlsruhe, germany. 15. shu, c., 2000, differential quadrature and its applications in engineering, springer, berlin. 16. gehannin, j., arghir, m., bonneau, o., 2009, evaluation of rayleigh-plesset equation based cavitation models for squeeze film dampers, asme j. tribol., 131, 024501. 17. boedo, s., booker, j.f., 1995, cavitation in normal separation of square and circular plates, asme j. tribol., 117, pp. 403-410. 18. resch, m., scheidl, r., 2014,a model for fluid stiction of quickly separating circular plates, proc. imeche c j. mech. eng. sci., 228, pp. 1540–1556. 7444 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 657 670 https://doi.org/10.22190/fume210914002g © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders alexander gritsenko1,2, vladimir shepelev1, semen fedoseev3, tatyana bedych4 1department of automobile transport, south ural state university, chelyabinsk, russia 2department of engineering and technology, south ural state agrarian university, chelyabinsk, russia 3military-air academy named after professor n. e. zhukovsky and y. a. gagarin, chelyabinsk, russia 4m. dulatov kostanay engineering and economic university, kostanay, kazakhstan abstract. in fuel economy, a rising level of interest in heavy duty diesel engines that industry has witnessed over the last few years continues to go up and this is not likely to change. lowering the fuel consumption of all internal combustion engines remains a priority for years to come, driven by economic, legislative, and environmental reasons. according to statistics, the share of operating expenses to ensure transport operations in industrial production is 15-20%, wherein 16-30% of the total volume of transport operations concerns a car, tractor, and trailer. during transport operations, the engine load by the torque, in most cases, does not exceed 40-50%. the paper investigates the increase in fuel efficiency of cars and tractors by disconnecting some of the engine cylinders operated in low-load and idling modes. the research has led to the establishment of the theoretical dependencies between the effective power, engine efficiency, mass of the transported cargo, speed of the car (tractor) and the number of disconnected engine cylinders. results of experiments suggest the interdependencies of the performance parameters of the car (tractor) when disconnecting some of the engine cylinders. it has also been established that the maximum reduction in the hourly fuel consumption occurs in the idling mode while it decreases along with an increase in the load. key words: engine, fuel shutoff, efficiency, environmental friendliness, control received september 14, 2021 / accepted january 08, 2022 corresponding author: vladimir shepelev affiliation, department of automobile transport, south ural state university, 76, lenin prospect, 454080 chelyabinsk, russia e-mail: shepelevvd@susu.ru 658 a. gritsenko, v. shepelev, s. fedoseev, t. bedych 1. introduction the objective of current research on internal combustion engines is to further reduce exhaust emissions while simultaneously reducing fuel consumption [1]. the issues of increasing the fuel efficiency of cars and tractors by disconnecting some of engine cylinders have not been sufficiently studied despite a significant number of publications dealing with this topic [2-9]. patrahaltsev et al. [2] carried out the comparison of opportunities to raise the diesel economy during some testing cycles by disconnecting some cylinders or cycles. it has been established that during reduction of the loading coefficient from 0.6 to 0.36, the rising of economy changes from 3.1 to 7.2%. liu and kuznetsov [3] determined that the valve system in deactivated cylinders has a significant effect on the specific fuel consumption. authors investigated the engine performance in partial load modes at different rotational speeds and torques and with a varying number of deactivated cylinders. the obtained results demonstrate the characteristics of the engine working process under cylinder deactivation and enable a more accurate estimation of the effect of this engine control method. berdnikov et al. [4] proposed a method of forming the order of disconnection of the cylinders of the engine in order to obtain the necessary engine power for an efficient and economical operation of the engine. gosala et al. [5] demonstrated that it is possible to operate a diesel engine at low loads in cylinder deactivation without compromising its transient torque/power capabilities, a key finding in enabling the practical implementation of cylinder deactivation in diesel engines. mo et al. [6] presented results which show that, when the mean effective pressure of the engine is lower than 3.5 bar, cylinder deactivation decreases the brake specific fuel consumption by 0-17% and by 26% at idle if the intake valves and the exhaust valves are kept closed at the same time. however, the engine and the supercharger do not match well after deactivation and the mass of intake air decreased greatly, which also resulted in a large decrease in the nitrogen oxide emissions. thees et al. [7] implemented a new cylinder activation concept (“3/4-cylinder concept”) with the aim of reducing fuel consumption. a fully variable valve train was developed for this engine, which both improves the functionality of the 3/4-cylinder concept and can have a positive influence on exhaust emissions through internal exhaust gas re-circulation. a comparison of this engine concept with its series reference based on measurement data showed a fuel economy advantage of up to 5.2% in the low load field cycles of the dlg powermix. the maximum fuel consumption benefit in the low load engine regime exceeded 15% in some of the operating points. vinodh et al. [8] deals with maintaining the efficiency of the engine at different loading conditions. in this paper a 6-cylinder engine is converted into a 4-cylinder engine. for this, the crankshaft is divided into several segments, so that it will be possible to disengage two of the cylinders from the other four cylinders. during this conversion, servo motor is used to vary the crank angle according to the number of cylinders, which in turn is controlled by ecu (electronic control unit). the engine will be operated at its maximum efficiency. pillai et al. [9] established that the modified baseline model that includes cylinder deactivation maintains comparable emission levels through the optimization of exhaust gas recirculation and variable geometry turbocharger. the results demonstrated reductions in brake specific fuel consumption and higher exhaust gas temperatures for low and part load operating points. disabling fuel injectors and the valve train on half of the engine's cylinders increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders 659 allowed for the implementation of cylinder deactivation. lower engine pumping work and reduced heat transfer to the cylinder walls resulted in reduced fuel consumption. the works of scientists [10-15] deal with complex issues of a more complete additional loading of the cylinders remaining in operation, a decrease in specific and liter fuel consumption, and an increase in fuel efficiency. the studies carried out by the above authors deal with the issues of increasing the fuel efficiency of engines by fuel shutoff in the idling ice cylinders. but the sources do not provide methods for the analytical calculation of the fuel efficiency of cars and tractors during transport operations when some of the engine cylinders are off [16-19]. the strengths and weaknesses of the methods used are summarized in the table 1. table 1 the strengths and weaknesses of the used methods deactivation method strengths weaknesses fuel-off with exhaust gas transfer [20-24] preservation of the thermal conditions of the cylinders high costs to change the vehicle configuration fuel-off without exhaust gas transfer [25-28] no need for reconstruction costs change in the thermal conditions of the deactivated cylinders disconnection of the drive of the gas distribution mechanism [29] no gas exchange losses violation of the thermal conditions of the deactivated cylinders а) valves closed uneven wear of the deactivated cylinders b) valves open increase in the toxicity of exhaust gases after reactivation c) exhaust gas transfer from the working cylinders through the deactivated cylinders problems with the accumulation of lubricating oil in the deactivated cylinders d) circulation of gases in the deactivated cylinders from the outlet to the inlet increased design features deactivation of pistons [23, 24] no mechanical losses increased noise and vibration, disbalancing а) breaking of the rigid connection between the crankshaft and the piston no major structural changes needed complicated maintenance and repair knowing that the prices of hydrocarbon fuels in russia (despite a decrease in the global prices) are constantly growing, and the rated power of tractor engines sustains the upward trend, the topic of the work aimed at increasing the fuel efficiency of cars and tractors is relevant. 660 a. gritsenko, v. shepelev, s. fedoseev, t. bedych 2. theoretical research disconnection of several engine cylinders is advisable when the car (tractor) moves in the transport mode with a small load along a horizontal supporting surface (a road with asphalt concrete or improved surfacing), as well as downhill [30-32]. during the tests, we used a ki-5543 run-in and test stand. the structural diagram of the stand is shown in fig. 1. the loading device of the stand allows us to load the ice with a maximum torque of 400 nm and the range of ice crankshaft speeds of – 1,000÷3,200 rpm. fig. 1 a simplified diagram of the test bench with d-240 (1 – regulator for the ice loading; 2 – balancing mechanism; 3 – torque meter; 4 – weighting device to control fuel consumption; 5 – d-240 engine; 6 – diffuser; 7 – u-shaped manometer; 8 – stand base) we installed two solenoid valves in the fuel system of the d-240 engine to prepare for the tests (fig. 2). two solenoid valves are installed on the fuel system shown in fig. 2. the solenoid valves allow us to deactivate the working cylinders of the ice in real time. when the ice cylinders are off, the incoming fuel is drained back into the fuel tank. the drive of the gas distribution mechanism was deactivated by removing the pushers, after which the inlet and outlet valves of the deactivated cylinders were closed. to perform calculations, we made the following assumptions: we consider the transmission efficiency to be constant at each of the actual gears; a car (tractor) is moving at a constant speed (vm=const), uniformly (j=0), on a horizontal surface (α=0º), without longitudinal vibrations affecting the changes in the tractive effort and torque of the engine, without slipping (δ= 0); we neglect aerodynamic drag force рw because of the low movement speed in the case of a tractor [33-37]. taking into account the accepted assumptions, the engine load factor by the torque (kl) is determined by the expression [16, 18]: 310 ( )e w l trac trac trail trail en t t en m r k g f g f m i m  = =  +    (1) where ме is current engine torque, nm; меn is the engine torque at the nominal power, nm; gtrac is the weight of the car (tractor) with the load (or without load), t; gtrail is the increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders 661 weight of the trailer with load (or without load), t; ftrac, f trail is the coefficient of the rolling resistance of the car (tractor) and trailer; rw is the dynamic radius of the drive wheels of the car (tractor), m; it is the transmission number; ηt is the mechanical transmission efficiency for the selected gear. fig. 2 fuel system of the d-240 engine with solenoid valves to turn off the fuel supply of two ice cylinders the degree of the variation of engine crankshaft speed (kn) depends on actual movement speed (vm) of the car (tractor), the transmission number, the dynamic radius of the drive wheels, and the nominal speed of engine crankshaft (nn): nw tm n n nr iv n n k   == 05.0 (2) where n is current engine crankshaft speed, rpm. when choosing the main estimated indicator characterizing the fuel efficiency level, we took into account that the economic effect when disconnecting some of the engine cylinders takes place only at low loads (while the specific effective fuel consumption has overestimated values, which are not exponential). thus, the hourly fuel consumption is adopted in this work as the main estimated indicator affecting the fuel efficiency of a car (tractor). the function of the dependence of hourly fuel consumption gf on the engine operating mode, which changes when some of its cylinders are off, can be analytically presented as follows:       +   = w h ml en l iu n nf z v p m k h n kg  955 312.0 (3) where hu is the lower calorific value, мj/kg; ηi is the indicated efficiency; pml is the average pressure of mechanical losses, pa kw; vh is the engine volume, l; zw is the number of working cylinders; τ is the engine cycle. an analysis of the above dependencies shows that the car (tractor) engine fuel consumption is affected by the engine operating mode determined by the degree of the variation on the 662 a. gritsenko, v. shepelev, s. fedoseev, t. bedych crankshaft speed, the indicated efficiency, the engine load factor by the torque, as well as the number of working cylinders and the nominal average pressure of mechanical losses. to this end, we should establish regularities of changes in these parameters when performing transport operations, as well as when disconnecting some of the engine cylinders. to solve these problems, we calculated the weighted average values of the engine load factors at n=const for transport transmissions; then, we aligned these values with the operating modes of the transport-tractor unit (ттu): 1) engine idling kl=0; 2) tractor idling: 0>kl>0.14; 3) ттu idling: 0.14≥kl>0.211; 4) ttu operating mode: kl≥ 0.211. when disconnecting the cylinders to maintain the required engine operating mode, the remaining working cylinders should be supplied with a larger amount of fuel at an increased cyclical supply, consequently, the indicated pressure increases in them. when the engine load changes, the fuel combustion quality characterized by the indicated efficiency is adequately described by the equation of the quadratic parabola depending on the indicated pressure: cpbpa iii ++= 2 (4) where а, в, с are the coefficients; рi is the average indicated pressure in the cylinder, мpа. this dependence was obtained in the course of the analysis of the load characteristics of the engine at a constant speed, as well as the thermal calculation of the engine for different load conditions. in our work, we found the following empirical coefficients for the four-cylinder d-240 engine: а = –0.872; в = 0.953; с = 0.256. to determine the change in mechanical losses when the cylinder is off, which are usually characterized by mechanical efficiency, we introduced the coefficient of variation of mechanical losses km of the engine when some of the cylinders are off: iml zpml m n n k _ _ = (5) where nml_zр is the power of mechanical losses when some cylinders are on, kw; nml_i is the power of mechanical losses when all i cylinders are on, kw. fig. 3 shows the dependence of the change in km on the number of working cylinders calculated for a four-cylinder engine when using two ways of disconnecting cylinders. fig. 3 the dependence of the coefficient of the power of mechanical losses variation on the number of working cylinders of a 4-cylinder ice increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders 663 the ratio of the nominal effective engine power when some of the cylinders are off to the nominal effective engine power when all the cylinders are on is denoted as the coefficient of the nominal effective power variation when some of the cylinders are off kn:         −−  == 1 1 max_max_max_ max_ im m im p ie zpe n k i z n n k  (6) having calculated the values of this coefficient for different ways to disconnect the cylinders by the example of a 4-cylinder engine, we obtain dependencies of kn on the number of working cylinders (fig. 4). fig. 4 the dependence of the coefficient of maximum effective power variation on the number of working cylinders of a 4-cylinder ice we can see from the dependency graph shown in fig. 5 that when the fuel and valves are off, the engine will develop a larger power than if only the fuel supply is off. fig. 5 shows the dependence of the change in the hourly fuel consumption calculated for the d-240 engine with a different number of working cylinders on the engine load factor. fig. 5 the estimated dependences of the fuel consumption of the d-240 engine on the load factor at a different number of working cylinders (n=2,200 rpm) 664 a. gritsenko, v. shepelev, s. fedoseev, t. bedych consumption lines for different numbers of working cylinders have intersection points, conventionally called “zero-economy points”. the engine load factor at these points is such that the fuel consumption is identical both when all the cylinders are on and when some of the cylinders are off. an analysis of the presented dependencies shows that when the engine is loaded less than at the zero-economy points, the fuel consumption is higher when all the cylinders are on than when one or several cylinders are off, so it is advisable to disconnect the engine cylinders. in case of a larger load, the fuel consumption is higher when the cylinders are off than when the cylinders are on, so it is not advisable to disconnect the cylinders. the maximum value of decreasing the hourly fuel consumption δgf: zwfiff ggg __ −= (7) where gf_i is the hourly fuel consumption when all the cylinders are on, kg/h; gf_zw is the hourly fuel consumption when some of the cylinders are off, kg/h, which corresponds to the maximum economy, at kl=0 (engine idling mode), while at kl=0.32, δgf=0. with a further increase in the engine load, the fuel consumption further grows to a certain value kl=kn, above which the engine does not work at a given frequency and with a given number of working cylinders. the dependence of the decrease in the hourly fuel consumption δgf calculated for the d-240 diesel engine depending on the engine load factor kl is shown in fig. 5. in fig. 6 the dashed lines show the engine load levels “хх_tp” – when the car (tractor) is moving without a trailer, “хх_ттu” – when the car (tractor) is moving with an empty trailer. in these modes, the expected fuel economy is: when two cylinders are on 7%, when three cylinders are on 4% (for xx_ttu) and when two cylinders are on 12%, when three cylinders are on 5% (for xx_tp). obviously, to ensure the minimum fuel consumption, the engine should operate according to the abcd curve with a cut-off, depending on the engine load and the corresponding number of cylinders. the calculation results are summarized in table 2. fig. 6 the estimated dependences of the decrease in the fuel consumption δgf of the d-240 engine on load factor kl when a different number of cylinders is on (n=2,200 rpm) increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders 665 for different car (tractor) operating modes, we calculated the traction and power indicators and the decrease in the fuel consumption, based on which we selected the optimal number of cylinders to be disconnected. table 2 the results of calculating the parameters of the car (tractor) with disconnecting two engine cylinders, during standing and moving in the transport mode on an unsurfaced road parameter engine idling tractor idling ttu idling ttu operating mode load factor kl 0 0.138 0.211 0.370 cargo mass mcar, kg 0 0 0 4000 traction resistance,r, kn 0 0 0.68 2.28 effective engine power nе, kw 0 7.66 11.52 20.61 decrease in the hourly fuel consumption δgf, kg/h сonsumption δgf, % 1.08 28.67 0.82 17.1 0.62 11.64 0.25 3.88 number of working cylinders, zw 1 2 2 2 thus, we found the desired analytical and graphical dependencies of the fuel consumption on the engine load and the number of working cylinders when the fuel supply and the gdm drive are off, based on which we determined a rational number of cylinders to be disconnected for various operating modes of the car (tractor). 3. experimental research procedure the experimental research program included: 1. preparation of control instruments, recording instruments, and equipment for bench and field tests; 2. development of experimental research methods. the experimental research was carried out in two stages: 1. bench tests of the d-240 diesel engine in laboratory conditions; 2. field tests during the operation of the car (tractor) during three engine operation variants: a) all cylinders are on; b) only the fuel supply is off; c) the fuel supply and the gmd drive (valves in the closed position) are off. during the field tests of the car (tractor), the cylinder disconnection process was controlled through an updated spring-loaded tow bar mounted in the hitch bar of the trailer and equipped with limit switches activated at certain traction resistance values. we used an electronic control signal damper to eliminate false alarms at an oscillation frequency of the traction resistance of over 1 hz. fig. 7 shows a developed block diagram of an automatic engine operation control unit taking into account the load. to reduce the number of experiments while maintaining sufficient accuracy and reliability of the results, we carried out the experimental research ac-cording to the plan based on the theory of experimental design and the experience accumulated during similar works. the basis of the experimental research plan is a three-level box-benkin second-order design. 666 a. gritsenko, v. shepelev, s. fedoseev, t. bedych fig. 7 diagram of an automatic engine operation control unit taking into account the load of the transport unit 4. results of the experimental research when the engine operates according to the load characteristic (fig. 8) at a constant engine crankshaft speed, it is necessary to ensure the specified engine load value for all variants of its operation. fig. 8 load characteristics of the d-240 engine (n=2,200 rpm) the change in the engine performance (cyclic fuel supply, hourly fuel consumption, actual air flow, excess air ratio, coefficient of admission, temperature of exhaust gases, indicated efficiency, specific indicated fuel consumption) is explained by the same causes, as during idling. however, the energy of the used fuel is consumed, in contrast to increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders 667 the changes in the engine mechanical losses (proportional to the crankshaft speed), to overcome the moment of resistance and to realize the effective power. the power of the d-240 engine is limited to 33 kw because at a higher load, the economic effect is not observed when the cylinders are off. when two cylinders of the d-240 engine are off, the power of mechanical losses at the rated frequency decreases by 11%. during idling, when the fuel supply and the gdm drive of the second and third cylinders are off, the hourly consumption rate at the nominal crankshaft speed is reduced by 24.4%. when the d-240 engine is running under load at the nominal speed: а) when only the fuel supply is off, at a load from 0 to 8% of the nominal value, the fuel economy decreases from 8% to 0%; b) when the fuel and the valves of a half of the cylinders are off with an increase in the load from 0 to 39%, the fuel economy decreases from 24.4% to 0%. the value of the maximum engine power when working on two cylinders and at a nominal speed is 37–39%. the results of the design and experimental studies of the d240 engine of the mtz-82 tractor during field tests are compared in fig. 9. fig. 9 the dependencies of the hourly fuel consumption on the operating conditions of the car (tractor) 668 a. gritsenko, v. shepelev, s. fedoseev, t. bedych the analysis of the theoretical and experimental dependences allows us to conclude that the fuel efficiency of ttus can be improved when operating with a low engine torque load characteristic of transport operations. this can be achieved by increasing the load of some engine cylinders while deactivating other cylinders. it is assumed that an increase in the efficiency of fuel combustion in working cylinders and a decrease in mechanical losses of the engine in deactivated cylinders will allow us to reduce the total fuel consumption, which, in turn, will lead to a decrease in specific energy consumption for the implementation of the transport process. this is particularly relevant with regard to the optimization model for: ▪ the relationship between the effective power, the engine economy, the weight of the transported cargo, the speed of the ttus, and the number of deactivated ice cylinders; ▪ the relationship between the energy and fuel-economic performance of the ttus and the engine of the transport tractor in different modes during transport operations. for one of the options of a device for disconnecting some of the cylinders, we calculated the cost of re-equipping the mtz-82 tractor for operation in the economy mode, which comprised 53 thousand rubles. based on the statistical data, we calculated the ttu engine operating time in the modes ensuring fuel efficiency. the payback period of the device for disconnecting some of the engine cylinders is 2.3 years. the annual economic effect of the introduction of the research results and the developed devices is 21 thousand rubles per one mtz-80/82 tractor. 5. conclusions an increase in the fuel efficiency of the tractor and transport unit operating under a low engine load by the torque, which is typical of transport operations, can be achieved by increasing the load of several engine cylinders of a transport tractor at a simultaneous disconnection of other ice cylinders. based on the established regularities of changes in the engine performance: the power of mechanical losses, the effective power, and hourly fuel consumption depending on the number of working cylinders, we developed a mathematical model for the operation of a car (tractor) engine when some of its cylinders are off, which allows us to determine analytically with sufficient reliability the rational number of working engine cylinders and fuel consumption depending on the values of traction effort, traction power, the mass of the transported cargo in various operating modes and road operating conditions of the car (tractor). we established that when performing transport operations of a car (tractor) as part of the mtz-80/82 transport tractor and 2pts-4 trailer, it is expedient to disconnect one or two engine cylinders. in this case, fuel consumption depends on the ice load determined by the road conditions, the transmission number, and the degree of the trailer loading. so, when the car (tractor) moves in the eighth speed position of the gearbox at a speed of 18.5 km/h along an unsurfaced traffic-compacted road, the disconnection of two engine cylinders (by turning off the fuel supply and the gdm drive) allows us to reduce the fuel consumption when the tractor moves without a trailer by 17.1 %; when the car (tractor) moves without cargo by 11.6%; in the “zero-economy point” operating mode it corresponds to the mass of the transported cargo of 2.45 tons. the annual effect of the introduction of the research results and the developed devices expressed in monetary terms is 21 thousand rubles per one mtz-80/82 tractor. increase in the fuel efficiency of a diesel engine by disconnecting some of its cylinders 669 references 1. shatrov, m.g., sinyavski, v.v., dunin, a.y., shishlov, i.g., vakulenko, a.v, 2017, method of conversion of highand middle-speed diesel engines into gas diesel engines, facta universitatis-series mechanical engineering, 15(3), pp. 383-395. 2. patrakhaltsev, n.n., kamyshnikov, r.o., anoshina, t.s., skripnik, d.s., 2014, regulation of the yamz-238 diesel 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parameters with oneand two-stage charging systems, reports in mechanical engineering, 1(1), pp.192-198. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 135 145 original scientific paper software tool for the laser cutting process control – solving real industrial case studies udc 621.9.01+681.5:004.4 miloš madić 1 , marko kovačević 2 , miroslav radovanović 1 , vladislav blagojević 1 1faculty of mechanical engineering, university of niš, serbia 2faculty of electronic engineering, university in niš, serbia abstract. laser cutting is one of the leading non-conventional machining technologies with a wide spectrum of application in modern industry. it order to exploit a number of advantages that this technology offers for contour cutting of materials, it is necessary to carefully select laser cutting conditions for each given workpiece material, thickness and desired cut qualities. in other words, there is a need for process control of laser cutting. after a comprehensive analysis of the main laser cutting parameters and process performance characteristics, the application of the developed software tool “brutomizer” for off-line control of co2 laser cutting process of three different workpiece materials (mild steel, stainless steel and aluminum) is illustrated. advantages and abilities of the developed software tool are also illustrated. key words: co2 laser cutting, process control, software tool, mild steel, stainless steel, aluminum, cut quality 1. introduction the last decade has witnessed an increasing industrial use of contemporary materials with improved properties. consequently, non-conventional machining technologies gain an ever increasing importance since in many concrete applications they represent the only possible, or at least, the only economically viable machining technology. in a large number of non-conventional machining technologies that are widely used in modern industry the laser cutting stands out as a technology that offers a number of advantages such as: high productivity, high cut quality, low waste, ability to cut a wide range of materials, low operational costs when it comes to cutting speed, etc. [1, 2]. received february 3, 2016 / accepted june 28, 2016 corresponding author: madic@masfak.ni.ac.rs faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: madic@masfak.ni.ac.rs 136 m. madić, m. kovaĉević, m. radovanović, v. blagojević however, effective application of the laser cutting technology requires excellent knowledge of the actual process as well as the determination of the most suitable laser cutting conditions for each purpose. in manufacturing companies it is a common practice for machine tool operators and/or process planning engineers to select laser cutting conditions based on their acquired knowledge and previous experience as well as machining handbooks. although this is essentially a subjective approach, and selected laser cutting conditions are not even near optimal ones, in most cases this approach is sufficient [3]. as noted by rao and pawar [4], determination of the most acceptable machining conditions is essential as non-conventional machining technologies incur high initial investment, tooling, operating and maintenance costs. the situation is even more complex when attempting to satisfy multiple objectives (productivity, cut quality characteristics, cost, etc.), since the laser cutting conditions that are suitable for one performance characteristic may deteriorate other performance characteristics [5]. in the situations when there is a need to produce a large number of pieces, cut workpieces made of expensive materials, or to produce finished parts from the limited amount of available workpiece material, a comprehensive analysis and detailed process planning and control of the laser cutting process is necessary and preferred. in that sense, in order to ensure the achievement of the required processing quality, costs and productivity, it is necessary to quantify the relationships between the laser cutting process parameters and the process performance characteristics through mathematical modeling. on the basis of the established relationships it is possible to perform a detailed analysis of the effects of the process parameters on the process performance characteristics, identify near optimal values of the process parameters and control the actual laser cutting process. from relevant literature one can observe that the selection of the laser cutting parameters on the basis of the established relationships between inputs and outputs is performed by formulation of a classical optimization problem with or without some constraints. subsequently, an optimization method/algorithm is applied aiming to find extreme values of the objective function and corresponding laser cutting parameter values [6-9]. in contrast to the afore-mentioned approach this paper discusses the selection of the laser beam cutting parameter values based on the process control approach, i.e. situation when one needs to determine laser beam cutting parameter values so that the objective function has the target value, while, at the same time, other objective functions and possible constraints need to be taken into consideration. there are generally two approaches to the machining process control. one is adaptive control and the other is model-based simulation and control. the principal difference between these two approaches is that the adaptive control is a feedback method, whereas the model-based simulation and control is feed-forward in its characteristics [10]. the model-based simulation and control is basically an off-line strategy performed at the machining process planning stage, and its efficiency depends largely on the availability of accurate and reliable mathematical models [3]. in this paper a comprehensive analysis of the main laser cutting parameters and process performance characteristics shows the complexity of the laser cutting process, thus demonstrating the need for process control. thereafter, the application of the developed software tool for off-line control of co2 laser cutting process is illustrated. three real case studies dealing with co2 laser cutting of stainless steel, mild steel and aluminum alloy are considered. software tool for the laser cutting process control – solving real industrial case studies 137 2. complexity of the laser cutting process laser cutting represents a cutting technology in which material removal is carried out by the focused beam movement across the surface of the material. the laser beam heat melts/vaporizes the workpiece throughout the thickness in a fraction of second, thus creating a cutting front [11]. the molten material is removed using a coaxial jet of a pressurized assist gas. in fig. 1 the schematic of the co2 laser cutting process is given. depending on the nature of the assist gas, it may also help in enhanced material removal through chemical reactions such as oxidation of the material. if high productivity is the main criterion, then oxygen-assisted melt shearing is the chosen cutting mechanism. alternatively, if a high edge quality is more important, then a slower high pressure inert gas melt shearing mechanism is used [2]. in mild steel laser cutting the most frequently used is oxygen as assist gas, whereby due to the exothermic reaction, additional energy is provided and used in the cutting process. in the case of stainless steel, laser cutting is commonly realized using nitrogen as assist gas in order to achieve a cut of high quality. the main reason is that some oxides such as chromium oxide, which is normally created when cutting with oxygen, have a high viscosity and are difficult to be ejected from the cutting zone [1]. aluminum can be cut with both oxygen and nitrogen as cutting gases, but the cutting speed with oxygen is not significantly higher than with nitrogen. this is because solid or viscous aluminum oxide forms a seal on the cut front preventing the oxygen from penetrating into the metal itself [12]. fig. 1 schematic diagram of co2 laser cutting the laser cutting process is difficult to control due to the nature and complexity of the phenomena that occur in the interaction of the laser beam and the workpiece material (absorption of radiation, local heating, melting, evaporation, heat dissipation, removal of melted and vaporized material, etc.). in addition, as majority of the machining processes, the laser cutting process is a complex multi input multi output (mimo) machining process in which a number of controllable and uncontrollable parameters (inputs) have an 138 m. madić, m. kovaĉević, m. radovanović, v. blagojević essential role in the process performance characteristics (outputs). moreover, it is a diffuse process, as there are significant interactions between inputs, and has a stochastic behavior [13]. for successful realization of laser cutting for a given machining application it is necessary to consider a broad set of inputs, i.e. laser cutting parameters that directly determine laser cutting conditions as well as a set of outputs, i.e. laser cutting process performance characteristics (fig. 2). fig. 2 complexity of laser cutting process as mimo machining process as shown in fig. 2, a set of inputs include parameters related to laser cutting machine, workpiece characteristics and laser process parameters. it should be noted that the set of laser process parameters consists of variable process parameters such as laser power, cutting speed and assist gas pressure, and constant process parameters such as focus position, type and purity of the assist gas, nozzle diameter and nozzle-workpiece standoff distance. in a real production environment, laser cutting machine operators usually consider afore-mentioned parameters as means of intermediate intervention for achieving desired effect during the laser cutting process. software tool for the laser cutting process control – solving real industrial case studies 139 as shown in fig. 2 the laser cutting involves a number of process performance characteristics and it is difficult to perceive each of them for a given machining application. moreover, it is important to notice that the process performance characteristics for a given machining application are in most cases interrelated, therefore there is a need to make certain trade-offs. for customers, quality performance characteristics and price are of the prime importance, while process and productivity performance characteristics as well as production costs are in the forefront for manufacturers. ecological performance characteristics are closely related to different criteria defined by international environmental and ecological standards. in addition to energy, the assist gases consumed in the process and waste generated significantly contribute to the total environmental impact of the use stage of a laser cutting machine tool [14]. as clearly indicated in fig. 2, a set of different parameters affect laser cutting process, wherein the character of the influence of certain parameter can be different taking into account interaction with other process parameters. among others, laser power, cutting speed, assist gas pressure, focus position and characteristics of lasers and laser beam significantly affect the performance of the laser cutting process in terms of cut quality, productivity and efficiency for each workpiece material and thickness [1]. the afore-mentioned performance characteristics in laser cutting can be viewed as technological and techno-economic objective functions, which, for a given machining application, are to be met to the greatest extent. 3. developed software tool desktop application “brutomizer” is a software tool developed for mathematical models optimization and off-line model-based simulation and control. the off-line control is based on exhaustive iterative search of input values combinations and corresponding mathematical model outputs. input parameters of developed software tool are the following: a discrete set of possible input values, mathematical model for off-line control and arbitrary number of constraints. both regression and artificial neural network (ann) models are supported. “brutomizer” is able to determine the input values that correspond to the user-defined output with desired accuracy, and at the same time satisfy given constraints. as there can be many viable solutions in the case of off-line control, the developed software tool enables exploitation of solutions hyperspace starting from the smallest values as well as starting from the highest values of input parameters. main benefits of using developed software tool "brutomizer" are the following:  user is able to define limits and search step for each input variable, according to the appropriate machine tool limitations,  by using exhaustive iterative search algorithm, optimality of obtained solutions in given discrete search space is guaranteed,  although brute force algorithm is used, the use of multiple processor cores enables very efficient application of the developed software tool even to very complex problems, and,  intuitive user interface. developed software tool "brutomizer" is publicly available at http://www.virtuode. com/?page=softwaresolutions. a screenshot of the main window is shown in fig. 3. in this window the user can set input parameters for both optimization and off-line control. a screenshot of the window for off-line control is given in fig. 4. http://www.virtuode.com/?page=softwaresolutions http://www.virtuode.com/?page=softwaresolutions 140 m. madić, m. kovaĉević, m. radovanović, v. blagojević fig. 3 main window of the developed software tool “brutomizer” fig. 4 window for off-line control of the developed software tool “brutomizer” software tool for the laser cutting process control – solving real industrial case studies 141 4. case studies the application of the developed software prototype is demonstrated considering experimental data and developed mathematical models from previous investigations [1, 15, 16]. three case studies have been considered, that is co2 laser cutting of stainless steel, mild steel and aluminum. it should be noted that prerequisites for the application of the developed software prototype for off-line control of co2 laser cutting process are: planning and realization of experimental investigation of co2 laser cutting process and development of mathematical relationships between process parameters (inputs) and different performance characteristics (outputs). case studies considered, used experimental plans, and details regarding developed mathematical models for the purpose of control of co2 laser cutting process are given in table 1. table 1 control of co2 laser cutting process: details of case studies workpiece material experimental design mathematical model for control case study 1 stainless steel taguchi l27 ann case study 2 mild steel taguchi l25 regression case study 3 aluminum full factorial design regression in all the considered case studies, search step for each laser cutting parameter is set to 0.01. the search step defines the number of possible values for each parameter inside the parameter limits which are defined in experimental design. 4.1. co2 laser cutting of stainless steel case study 1 consider the following situation in job shop manufacturing using co2 laser cutting technology. on the basis of technical drawing, manufacturing engineers in production planning need to determine laser cutting conditions for a large batch cutting. the main requirement is to obtain surface roughness of the cut edge of ra=1.6 m. it should be noted that an extremely high quality cut surface can be produced at the cost of higher production costs. co2 laser cutting is to be done on workpieces with thickness of 3 mm made of stainless steel by using nitrogen as assist gas. experimental investigation of co2 laser cutting process was performed by using taguchi’s l27 experimental design using different combinations of laser cutting parameters around the laser cutting conditions as recommended by manufacturer or as usually used. in the experiment four laser cutting parameters such as laser power (p), cutting speed v (m/min), assist gas pressure (p) and focus position (f) were varied at three levels in the following ranges: p=1.6-2 kw, v=2-3 m/min, p=0.9-1.2 mpa, and f=-2.5-2 mm [1]. using the obtained experimental data, different ann mathematical models were developed to establish relationships between laser cutting parameters and different performance characteristics. mean absolute percentage error between experimentally measured values of surface roughness and ann model predicted was about 8.5 %. this ann model was then incorporated in the developed software prototype. on the basis of ann mathematical models, which are valid for the covered experimental hyperspace, the software prototype can generate a set of potential solutions which satisfy the aforementioned requirement. 142 m. madić, m. kovaĉević, m. radovanović, v. blagojević based on the given target values for surface roughness and the desired accuracy, δ, (maximum difference between the desired and calculated values of surface roughness) the software prototype can determine appropriate values of the laser cutting parameters. as explained previously, the software prototype can determine these values by two options: (1) starting from the smallest values of inputs i.e. laser cutting parameter values, and (2) starting from the highest values of inputs. which option will be selected in the software prototype depends on the nature of the problem being solved and some other technotechnological aspects. the values of laser cutting parameters that satisfy the condition that surface roughness of the cut edge is ra=1.6 m are given in table 2. table 2 control of co2 laser cutting process of stainless steel option p [kw] v [m/min] p [mpa] f [mm] ra [µm] 1 1.83 2 0.9 –0.5 1.600 2 2 2.83 1.18 –0.5 1.601 desired accuracy, δ=0.001 if, for some reason, the total gas consumption is not so important, the laser cutting process should be realized using laser cutting conditions as determined by option 2, since a higher cutting speed ensures higher productivity. otherwise, the laser cutting condition as determined by option 1 are recommended. 4.2. co2 laser cutting of mild steel case study 2 in the laser cutting process there are often situations when one needs to simultaneously satisfy a number of criteria, i.e. performance characteristics such as surface roughness, kerf taper angle, kerf width, material removal rate (mrr), heat affected zone, cost, dross, etc. for example, when determining laser cutting conditions that will ensure that the desired surface roughness is achieved, one can take other criteria into account, such as kerf width (kw). the application of the developed software tool for control of co2 laser cutting process in these situations is illustrated in the following case study. co2 laser cutting is to be done on workpieces with thickness of 2 mm made of mild steel by using oxygen as assist gas. in an experimental investigation of co2 laser cutting process taguchi’s l25 experimental design was applied, using different combinations of laser cutting parameters around the laser cutting conditions as recommended by manufacturer. in the experiment three laser cutting parameters such as laser power (p), cutting speed v (m/min) and assist gas pressure (p) were varied at five levels in the following ranges: p=0.7-1.5 kw, v=3-7 m/min and p=0.3-0.7 mpa [15]. using the obtained experimental data, two mathematical models for the prediction of ra and kw in the form of regression equations were determined as: 2 2 1.704 1.495 5 1.13 0.72 0.091 0.842 0.136 2.495 ar v p p v p v p p p v p                      (1) 2 2 2 0.306 0.00097 0.576 0.0894 0.011 0.379 0.00286 0.0331 0.01976 wk v p p v p p v p v p                    (2) software tool for the laser cutting process control – solving real industrial case studies 143 mean absolute percentage errors between experimentally measured values of surface roughness and kerf width and model predicted values of surface roughness and kerf width were about 5 and 6 %. let us consider that it is necessary to determine such values of the laser cutting parameters so as to obtain surface roughness of the cut edge of ra=1.6 m and kerf width less that 0.25 mm (kw<0.25 mm). ten laser cutting conditions, i.e. laser cutting parameter value combinations, as determined by options 1 and 2 of the developed software tool are given in table 3. table 3 control of co2 laser cutting process of mild steel option p [kw] v [m/min] p [mpa] ra [µm] kw [mm] 1 0.75 3 3 1.607 0.230 0.75 3 3.05 1.5957 0.232 0.8 3 3.4 1.6086 0.245 0.8 3 3.45 1.5995 0.247 0.8 3 3.5 1.5903 0.250 2 1.15 7 6 1.6032 0.232 1.05 7 5.85 1.5922 0.231 1 7 5.85 1.6036 0.233 0.95 7 5.85 1.6026 0.236 0.9 7 5.9 1.6024 0.241 desired accuracy, δ=0.001 from table 3 it can be observed that for a constant cutting speed there exist a number of different combinations of laser power and assist gas pressure which satisfy given set of conditions. as in the previous case study, one can observe that higher productivity implies higher gas consumption, but also more laser power. 4.3. co2 laser cutting of aluminum case study 3 laser cutting of aluminum and its alloys with co2 lasers is considered difficult because of high reflectivity and thermal conductivity of aluminum alloys. in essence it represents a specific laser cutting process since performance characteristics (outputs) are highly sensitive to the laser cutting parameters and their interactions. moreover, the same performance characteristics can be obtained by a number of different combinations of laser cutting parameter values [16]. the application of the developed software tool for control of co2 laser cutting process in these situations is illustrated in the following case study. co2 laser cutting is to be done on workpieces with thickness of 3 mm made of aluminum alloy by using nitrogen as assist gas. for the purpose of experimental investigation of co2 laser cutting process, a 33 full factorial experimental design using different combinations of laser cutting parameters was applied. in the experiment three laser cutting parameters such as laser power (p), cutting speed v (m/min) and assist gas pressure (p) were varied at three levels in the following ranges: p=3-4 kw, v=3-3.5 m/min and p=0.6-1 mpa [16]. using the obtained experimental data, the mathematical model for the prediction of kerf width (kw) in the form of regression equation was determined as: 144 m. madić, m. kovaĉević, m. radovanović, v. blagojević 2 2 2 0.81791 1.5983 0.32929 1.72637 0.23837 0.00612 0.36 +0.01072 0.00622 0.07778 wk p p v p p v p p p v p v                       (3) mean absolute percentage error between experimentally measured values of kerf width and model predicted values of kerf width was about 5 %. let us consider that it is necessary to determine the values of laser cutting parameters so as to obtain and kerf width less than 0.5 mm (kw<0.5 mm) while maximizing material removal rate (mrr). higher mrr is always preferred, particularly in large batch production. mrr can be calculated as the product of cutting speed, workpiece thickness and kerf width. different cutting conditions, i.e. laser cutting parameter value combinations, as determined by options 1 and 2 of the software tool are given in table 4. table 4 control of co2 laser cutting process of aluminum option p [kw] p [mpa] v [m/min] kw [mm] mrr [mm3/min] 1 3 6 3.1 0.499 4640.7 3 6 3.11 0.499 4655.67 3 6 3.12 0.5 4680 3 6.05 3 0.499 4491 3 6.05 3.01 0.499 4505.97 2 3 6.05 3.09 0.5 4635 3 6.05 3.08 0.5 4620 3 6.05 3.07 0.5 4605 3 6.05 3.06 0.499 4580.82 3 6.05 3.05 0.499 4565.85 desired accuracy, δ=0.001 as can be observed from table 4, among different combinations of laser cutting parameter values that produce almost the same kerf width, the third combination of the laser cutting parameter settings as determined by option 1 stands out and ensures the maximal mrr. 5. conclusion complexity of the laser cutting process imposes the need to control the process in order to fulfill product specifications, improve materials processing and decrease processing costs. in this paper, the application of developed software tool “brutomizer” for offline control of co2 laser cutting process is illustrated. three real industrial case studies were considered showing a number of advantages of using developed software tool for co2 laser cutting process control. among others, the most important are guaranteed optimality of the selected laser cutting conditions, ability to consider machine tool limitations in terms of allowed laser cutting parameter values and ability to obtain a set of solutions. particular advantage of the developed software tool is its ability to be used as a laser cutting technological processor which can be further upgraded by incorporating new mathematical models which describe various co2 laser cutting processes. as the presented approach for selection of process parameters values allows consideration of machine tool limitations and specific constraints, future research will be directed towards software tool for the laser cutting process control – solving real industrial case studies 145 the application of developed software tool for off-line control of other machining processes. also, the application of the developed software tool for solving machining multi-objective optimization problems with a number of constraints of equality and/or inequality type will be investigated. acknowledgements: the paper is a part of the research done within the project tr35034 supported by ministry of science and technological development of the republic of serbia. references 1. madić, m., 2013, mathematical modeling and optimization of laser cutting process using artificial intelligence methods, (in serbian), phd thesis, faculty of mechanical engineering in niš, university of niš. 2. ion, j., 2005, laser processing of engineering materials principles, procedure and industrial application, butterworth-heinemann. 3. madić, m., radovanović, m., blagojević, v., kovaĉević, m., 2014, off-line control of co2 laser cutting process using software prototype, xii international saum conference on systems, automatic control and measurements, niš, serbia, november 12th-14th, pp. 124-127. 4. rao, r.v., pawar, p.j., 2009, modelling and optimization of process parameters of wire electrical discharge machining, proceedings of the institution of mechanical engineers, part b: journal of engineering manufacture, 223(11), pp. 1431-1440. 5. ciurana, j., arias, g., özel, t., 2009, neural network modelling and particle swarm optimization (pso) of process parameters in pulsed laser micromachining of hardened aisi h13 steel, materials and manufacturing processes, 24(3), pp. 358-368. 6. adelmann, b., hellmann, r., 2011, fast laser cutting optimization algorithm, physics procedia, 12(1), pp. 591-598. 7. pandey, a.k., dubey, a.k., 2012, simultaneous optimization of multiple quality characteristics in laser cutting of titanium alloy sheet, optics and laser technology, 44(6), pp. 1858-1865. 8. wahab, h., gröninger, j., 2014, optimization of laser cutting quality with design of experiments, laser technik journal, 11(5), pp. 27-31. 9. gadallah, m.h., abdu, h.m. 2015, modeling and optimization of laser cutting operations, manufacturing review, 2(1), pp.1-20. 10. childs, t., maekawa, k., obikawa, t., yamane, y., 2000, metal machining: theory and applications, butterworth-heinemann. 11. dahotre,n. b., harimkar, s. p., 2008, laser fabrication and machining of materials, springer. 12. https://www.boconline.co.uk/internet.lg.lg.gbr/en/images/laser-cutting410_39413.pdf 13. han, a., gubencu, d., pillon, g., 2005, a generalized structure based on systemic principles of the characteristic variables of material laser processing, optics and laser technology, 37(7), pp. 577-581. 14. duflou, j.r., kellens, k., devoldere, t., deprez, w., dewulf, w., 2010, energy related environmental impact reduction opportunities in machine design: case study of a laser cutting machine, international journal of sustainable manufacturing, 2(1), pp. 80-98. 15. madić, m., radovanović, m., 2013, application of rcga-ann approach for modeling kerf width and surface roughness in co2 laser cutting of mild steel, journal of the brazilian society of mechanical sciences and engineering, 35(2), pp. 103-110. 16. madić, m., radovanović, m., kovaĉević, m., modeling and optimization of kerf width obtained in co2 laser cutting of aluminum alloy using discrete monte carlo method, journal of production engineering, 18(1), pp. 39-42. 11621 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 737 749 https://doi.org/10.22190/fume230826048m © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fractal space based dimensionless analysis of the surface settlement induced by the shield tunneling yuan mei1, xinyu tian1, xuejuan li2, khaled gepreel3 1school of civil engineering, xi'an university of architecture and technology, xi'an, china 2school of science, xi'an university of architecture and technology, xi'an, china 3department of mathematics, college of science, taif university, taif, saudi arabia abstract. the surface settlement during the tunneling process is becoming increasingly difficult to forecast as its surroundings become more and more erratic, and the maximal surface settlement raises risks posed suddenly by various uncertain factors. this paper proposes a novel approach to prediction of the surface settlement and analyzes the stability of tunnel construction. the dimensionless analysis and buckingham’s π-theorem are adopted for this purpose, and some useful dimensionless quantities are found, which can be used to determine the surface settlement’s main properties. in this manner, the paper offers new ways of predicting surface settlement in various cases, and it sheds a new light on the tunnel’s design and safety monitoring. key words: surface settlement, shield tunneling, dimensionless analysis, buckingham’s π-theorem, fractal space 1. introduction due to the rocketing urbanization in china, and generally in the world, the shield tunneling becomes the main tool for the subway construction. however, various accidents were reported, among which the surface settlement is the most serious problem, which might lead to an abrupt disaster of buildings damage and injuries. the shield tunneling induce accidents with fatal outcome to a similar amount as traffic accidents. so far there is neither a universal approach to reliable preventing the disaster, nor an effective method to predict the maximal surface settlement at the initial stage. the surface settlement is an extremely complex process with many influencing factors, making it impossible to estimate and control received: august 26, 2023 / accepted november 30, 2023 corresponding author: xuejuan li school of science, xi'an university of architecture and technology, xi'an, china e-mail: lxj_zk@163.com 738 y. mei, x. tian, x. li, k. gepreel the settlement process. the influencing factors include mainly the geological structure, the shield tunneling machine’s parameters and the tunnel geometrical property. though there are some empirical methods for prediction of surface settlement, the theoretical analysis was rare and rather preliminary [1-4]. a modified empirical peck formula was used to predict the surface settlement in water-rich sandy cobble strata [5]. lu et al. [6] suggested a gaussian function model based on a large number observation data of surface settlements, which can describe the geometrical shape of surface settlement. based on the mair's theory, yang et al. [7] proposed a computational method for long-term settlement of the soil at surface and subsurface, while macklin [8] used the load factor parameter to predict the volume loss. all empirical methods have obvious limitations that they need all tunneling conditions, which are difficult to be obtained. it is extremely difficult to establish a theoretical model, though many scientists have been trying to develop a universal theory [9-11], without a definite success. an alternative approach is provided by versatile numerical methods [12-14], but the unknown boundary conditions and unknown ground properties prevented successful numerical analyses in practical applications. the big data theory and the machine learning became a hot topic because of their versatile applications to majority of complex problems [15-19]. though some success has been achieved in forecasting the surface settlement [20-22], the machine learning method is not a method of choice for the tunneling process, as the missing data makes the real-time prediction impossible. hence, the research roads diverged in the tunneling process, and currently there is no generally accepted approach for prediction of the surface settlement. therefore, this paper sets the objective to propose a new approach – dimensionless analysis. though the method is generally known in engineering [23], it has never been used for this specific purpose. using the dimensionless analysis, estrada-diaz et al. [24] found a very useful mathematical model for predicting the dynamical properties of the electrospinning process, and the complex dynamic spinning process of critical importance for nanotechnology [2527] became suddenly accessible to engineers. he et al. [28] obtained an important bond stress-slip relationship for 3-d printed concretes. kong and he [29] found a totally new friction law for porous surfaces, and coulomb’s empirical law was abandoned completely in the textile engineering [30]. prakash et al. [31] found the main factors affecting the rotorbearing system’s dynamical property. ma [32] elucidated successfully the transient thermal properties. silva et al. [33] established a relationship among linguistic variables. recently, he and liu [34] pointed out a possible link between the fractal dimensions and the exponents involved in the dimensionless analysis, pushing the dimensionless analysis to the research frontiers in various fields, e.g., mathematics, physics, mechanical engineering and material science. in this paper, we apply the dimensionless analysis to the analysis of a seemingly complex problem of the surface settlement during the tunneling process, taking into account the geological structure, the shield tunneling machine’s property and the tunnel’s geometrical property. fractal space based dimensionless analysis of the surface settlement induced by the shield... 739 2. dimensional analysis method engineers have been facing intractable settlement behaviors [35]. wang et al. [36] established a mechanical model and solved it analytically, and in order to obtain the analytical solution, some unnecessary assumptions had to be made, thus rendering the model to be mainly of mathematical interest. pichler, et al. [37] gave a thermo-chemomechanical assessment to improve the tunneling process. tong et al. [38] suggested an improved model for a shallow tunnel in a semi-infinite plane. though the above models can account for out some properties of the settlement and for some special cases, there is no universal one applicable to a general tunneling process. the shield tunneling process is an interdisciplinary subject among civil engineering, mechanical engineering and mechanics. the engineering failure analysis [39] and the typical structural elements of shield tunnels [40] are widely used in the process. in this paper, the dimensionless analysis, which is widely used in mechanical engineering and fluid mechanics, is adopted to analyze the main factors affecting the tunneling-induced settlement. the dimensionless analysis is an approach widely used in science and engineering for complex problems. the most famous achievement is the reynolds number [41] in fluid mechanics, which has been widely used to identify the turbulent flow. other famous dimensionless quantities include nusselt number and prandtl number for the boundary layer theory [42], eckart number for heat dissipation [43], hartmann number for electromagneto-nanofluids [44, 45]. the dimensionless analysis is also widely applied to deal with intractable problems arising in mechanical engineering, e.g., the leakage performance of a seal [46], plastic deformation [47], and non-inertial stopping performance [48]. it is used to study the relation among physical quantities based on their units and dimensions, offering a totally new window for analyzing a complex problem with a simple mathematical treatment. in the engineering problems, it is necessary to keep the same units, and it helps understand physical laws and relationship among various factors [49]. the fundamental dimensions are mass (m), length (l), and time (t). 3. influencing factors analysis for the surface settlement during the tunneling process, we have to consider the geological feature, the shield tunneling machine’s parameters and the tunnel’s geometrical shape. in this section, we will analyze the main influencing factors from the above three aspects. 3.1. geological parameters the surface settlement depends heavily on the soil types. fig. 1 shows some fundamental soils. during the tunneling process, the soil’s dynamical parameters are needed to study the surface’s settlement. in this paper, we choose 11 physical quantities of soil dynamics, e.g., dynamic viscosity, particle diameter, pore water pressure, permeability, density, elastic modulus, effective stress, for the dimensional analysis in tunneling processes. their symbols, units, and dimensions are shown in table 1. 740 y. mei, x. tian, x. li, k. gepreel table 1 physical quantities of soil dynamics in the dimensional analysis physical quantity symbol units dimensions dynamic viscosity η pa.s ml-1t-1 particle diameter d mm l pore water pressure pp mpa ml-1t-2 permeability k m2 l2 face pressure pf mpa ml-1t-2 density ρ kg/m3 ml-3 elastic modulus ee mpa ml-1t-2 stress σ mpa ml-1t-2 the soil volume vg m3 l3 pore volume vp m3 l3 gravity acceleration g m/s2 lt-2 fig. 1 geological structure 3.2. shield tunneling machine parameters various shield modes can be chosen according to the soil’s property. taking the earth pressure balance shield as an example, we can identify legitimately the important parameters of the shield tunneling machine as given in table 2. table 2 physical quantities of the shield tunneling machine in the dimensional analysis physical quantity symbol units dimensions jack thrust f t(9.8kn) mlt-2 penetration rate v mm/min lt-1 grout filling pressure pj mpa ml-1t-2 grouting quantity qj m3/ring l3 soil bin pressure pb mpa ml-1t-2 cutter torque nc kn.m ml2t-2 cutter rotate speed wc rpm t-1 cutter diameter dc m l 3.3. tunnel geometrical parameters generally, the tunnel geometrical parameters are unchanged. in this research, only the cover depth and the tunnel’s radius are considered as influence factors. fig. 2 shows the fractal space based dimensionless analysis of the surface settlement induced by the shield... 741 diagram of the tunnel geometric characteristics and table 3 gives the information of physical quantities for the dimensional analysis. table 3 physical quantities of the tunnel’s geometrical shape in the dimensional analysis physical quantity symbol units dimensions cover depth h m l the tunnel radius r m l fig. 2 the diagram of tunnel geometric characteristics 4. dimensional analysis model in order to investigate the relationship among the variables involved in the soil dynamics (11 physical quantities), the shield tunneling machine (8 physical quantities) and the tunnel’s geometrical shape (2 physical quantities) by buckingham’s π-theorem [49], all parameters in the tables 1, 2 and 3 are chosen as the independent physical quantities. we can assume that max( , , , , , , , , , , , , , , , , , , , , , ) 0j j b c c c p f e g pf v p q p n w d d p k p e v v g h r s =    (1) according to eq. (1) and buckingham’s π-theorem, the calculation formula of the π term is 3 5 6 7 8 3 5 6 7 81 2 4 1 2 4 9 10 11 1 2 max a a a a a b b b b ba a a b b b j j b c c c p f e b b b c c a g p f v p q p n w d d p k p e v v g h r s                          = (2) where π is the dimensionless quantity; a1~a8, b1~b11, c1, c2 and a are the exponents corresponding to all physical parameters. substituting all dimensions into eq. (2), we rewrite eq. (2) in the form 742 y. mei, x. tian, x. li, k. gepreel         1 2 3 4 5 6 7 8 1 3 4 5 62 7 8 9 10 11 1 1 1 2 3 1 2 2 2 1 1 1 1 2 2 1 2 3 1 2 1 2 3 3 2 a a a a a a a a-2 b b b b bb b b b b b c mlt lt ml t l ml t ml t t l ml t l ml t l ml t ml ml t ml t l l lt l l − − − − − − − − − − − − − − − − − − −                                                                            2 0 0 0c a l l m t = (3) according to the principle of dimension homogeneity, we have [l]: 1 2 3 4 5 6 8 1 2 3 4 5 6 7 8 9 10 11 1 23 2 2 3 3 3 0a a a a a a a b b b b b b b b b b b c c a+ − + − + + − + − + − − − − + + + + + + = (4) [m]: 1 3 5 6 1 3 5 6 7 8 0a a a a b b b b b b+ + + + + + + + + = (5) [t]: 1 2 3 5 6 7 1 3 5 7 8 112 2 2 2 2 2 2 2 2 0a a a a a a b b b b b b− − − − − − − − − − − − = (6) for the convenience of analysis, we rewrite eqs. (4-6) as follows 0y =a (7) where y=(a1, a2, a3, a4, a5, a6, a7, a8, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, c1, c2, a)t is the column vector of the exponents and the dimensional matrix a is 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 1 1 1 3 1 2 0 1 1 1 1 2 1 3 1 1 3 3 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 0 0 2 1 2 0 2 2 1 0 1 0 2 0 2 0 2 2 0 0 2 0 0 0 a a a a a a a a b b b b b b b b b b b − − − − − − − −       − − − − − − − − − − − −  1 2 unknowns c c a . we can solve the general form of eq. (7) and obtain a system of the fundamental solutions in the analysis procedure. it is clear that the rank of a is 3. that is, eq. (7) includes 3 independent and 19 dependent variables, which can form 19 linearly independent fundamental solutions. different orders of column vectors in dimensional matrix a might lead to different structures of solution, but the solution space is always the same. generally, we tend to have a specific purpose in dimensional analysis for deriving some useful dimensionless qualities. for example, if we want to emphasize the importance of dynamic viscosity (η), pore water pressure (pp) and permeability (k), these three parameters will be chosen as independent variables, while others are the dependent variables. accordingly, the dimensional matrix is rewritten as a1 1 3 4 7 2 5 6 8 9 10 11 1 2 3 4 5 6 7 1 1 2 1 1 1 3 1 3 3 1 1 1 1 3 1 2 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 2 0 2 0 2 0 2 0 0 2 2 1 2 0 2 2 1 0 0 0 0 b b b b b b b b b b b a a a a a a a − − − − − − − −       − − − − − − − − − − − −  8 1 2 unknowns a c c a . fractal space based dimensionless analysis of the surface settlement induced by the shield... 743 let y1=(b1, b3, b4, b7, b2, b5, b6, b8, b9, b10, b11, a1, a2, a3, a4, a5, a6, a7, a8, c1, c2, a)t, then solve a1y1=0. if a column vector ξj (j=1,2,...,19) is a fundamental solution, then 19 fundamental solutions form a matrix ξ, which can be divided into two parts: a 19-order identity matrix i and a 3×19 affecting factors matrix b, ξ=[b i]t. the matrix b reads: 3 3 3 31 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 0 0 0 2 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 2 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 −    − − − − − − − − − −   − − − − − − − − − − − −  . let ξij (i=1,2, ..., 22) be an element of ξj. a power law behavior between variables yields the following equation ( ) 22 1 ij j i i var   = = (8) where var=(varj)=(f, v, pj, qj, pb, nc, wc, dc, η, d, pp, k, pf, ρ, ee,σ ,vg, vp, g, h, r, smax)t is a column vector including 21 influence factors and the maximum settlement smax. for example, according to the fourth column, the function expression can be written as 2 4 pk p −=   (9) 5. dimensionless quantities and experimental verification the dimensionless quantities are extremely helpful for a deep insight into a complex problem arising mechanical engineering. zhao suggested a dimensionless number for a beam’s dynamic plastic response [50], lesur and longaretti [51] showed that the dimensionless numbers can identify fast and reliably the magnet-rotational instability with ease, while zinola [52] adopted the dimensionless theory to study an extremely complex reaction process. scientists word to decipher how a complex problem is resolved using the dimensionless analysis, without introducing some reliable dimensionless quantities. the success presented in [50,51] is questionable. 5.1. dimensionless quantities now we turn back to our analysis and introduce three important dimensionless numbers, a soil dimensionless number (s), a dimensionless number for the shield tunneling machine (tm), and a geometrical dimensionless number for the tunnel (g). 1) the soil dimensionless number is defined as 1 a b c d d p a b c d d e e f p s e p v d r   + + − = −（ ） (10) 2) the dimensionless number for the shield tunneling machine ( ) ( ) ( )6 6 2 4 6 4 2 3 2 4 4 2 q s t u b c c c s p u t p q s t u u tp j j p n w d tm f q p v − + − + − + + + = (11) 744 y. mei, x. tian, x. li, k. gepreel 3) the geometrical dimensionless number m r g h   =     (12) where a, b, c, d, e, p, q, s, t, u and m are indexes which can be determined experimentally, and they satisfy a+b+c-d>0, 6s-6p+2u-t>0, 6p+4q-2s+3t+2u>0, 4u+t>0; κ is the soil permeability coefficient, and r is the porosity k g  = , p g v r v = (13) the soil dimensionless number is used to determine the type of surface settling pattern as safe or unsafe during the tunneling process. according to the definition given in eq. (10), a higher permeability coefficient (κ) of strata and a larger porosity (r) will lead to a larger value of the soil dimensionless number, implying a larger surface settlement. when the soil dimensionless number reaches a threshold value, an unsafe surface settling process occurs. the indexes involved in eq. (10) might be relative fractal dimensions of the porous soil. for example, the index of the porosity can be calculated by the following he-liu’s fractal dimension formulation [34]: 3e r= (14) here, when the porosity r=1, it becomes an empty volume, and when r=0, the soil can be considered as a continuum. the dimensionless quantity can be used for the drilling process by setting r=0. when the porosity tends to 1, the soil dimensionless number becomes infinitely large, and the surface settlement is totally unsafe. the machine number of eq. (11) shows the effects of the tunneling process on the surface settlement. a larger grouting quantity (qj), or a larger grout filling pressure (pj) leads a smaller value of the machine number (tm), on the other hand, a larger cutter torque (nc) results in a larger value of tm. the geometrical number (g) in eq. (12) shows the importance of the ratio of r/h, when the cover depth (h) tends to infinity, the geometrical number becomes zero, and the surface settlement is almost zero. when the ratio (r/h) increases gradually, the geometrical number increases gradually, too, and an unsafe surface settlement is predicted when the ratio reaches a threshold value. in eq. (12), the index can be calculated by he-liu’s fractal dimension formulation [34] 3(1 )m r= − (15) the geometrical number becomes 3 3r r g h −   =     (16) 5.2. experiment verification to verify our derivation, we consider an extreme case for an elastic beam under uniform load, its maximal displacement can be easily calculated, and it follows [53] max q w e  (17) fractal space based dimensionless analysis of the surface settlement induced by the shield... 745 where wmax is the maximal displacement, q is the uniform load, e is the elasticity modulus. eq. (17) implies the case when a=1, b=c=d=0, and r=0 in eq. (10), that is p e p s e = (18) this implies that the soil dimensionless number given in eq. (10) is reasonable. the geometrical dimensionless number also gives a better insight into the tunnel’s geometrical property. in practical applications, the tunnel is considered to be a shallow tunnel when h<4r. if h>50, the tunnel is considered to be a deep tunnel. the soil’s porosity was not considered. this has changed now as we can use the geometrical dimensionless number to judge accurately if a tunnel is a shallow one or a deep one. as an example, we consider the case when r=0.3 and r=3.0 m. fig. 3 shows the variation law of the dimensionless number g with the ratio (r/h) for different values of the porosity. instead of the value of r/h, the geometrical dimensionless number should be used to assess the tunnel’s geometrical property. fig. 3 variation law of the dimensionless number g when other parameters are set to constants, the machine dimensionless number (tm) is inversely proportional to the penetration rate (v), that is max ' max 1 1 1 1 i tm s s tm v  = + = +    (19) where i=4u+t>0, according to the experimental data, we can obtain the coefficients of eq. (19) by using the least square method. similarly, tm is proportional to the cutter torque (nc), that is max ' max 2 2 2 2 s c tm s s tm n  = + = +    (20) fig. 4 shows the variation law of tm with v. it can be seen that tm decreases with the increase of v. moreover, the variation law of tm with nc is shown in fig. 5. it is clear that 746 y. mei, x. tian, x. li, k. gepreel tm increases with the increasing of nc. according to eq. (11), the machine number shows that a larger penetration rate leads to a smaller value of the machine number, while, on the other hand, a larger cutter torque results in a larger value of tm. in addition, it can be seen from the right parts of figs. 4 and 5 that the dimensionless analysis results are consistent with the experimental data. fig. 4 the variation law of tm (left) and smax (right) with the v fig. 5 the variation law of tm (left) and smax (right) with the nc 5. conclusion we have obtained some dimensionless numbers by dimensional analysis, which enable us to gain insight into the surface settlement with ease for various different cases, and it sheds a bright light on the tunnel’s design and safety monitoring. this paper is expected to trigger much interest in the dimensionless method for various complex problems, and it can be used as a good paradigm for further applications. this paper proposes successfully three dimensionless numbers dealing with an extremely complex problem of the surface settlement, which is crucial for the tunneling process, and the dimensionless numbers are extremely important for engineers dealing with the tunnel design and monitoring the tunneling process. of course, the authors realize fully that, regardless of the rigor of the theoretical analysis, experimental verification is strongly required to determine the fractal space based dimensionless analysis of the surface settlement induced by the shield... 747 indexes involved in the dimensionless numbers and their threshold values for the unsafe surface settlement. the dimensionless analysis provides deep insight for the analysis of the surface settlement induced by the shield tunneling. the given dimensionless numbers give reliable criteria for civil and mechanical engineers of tunneling machines as well. acknowledgement: the authors would like to acknowledge the deanship of scientific research, taif university, for funding this work. references 1. chen, r.p., zhang, p., 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https://doi.org/10.22190/fume210505053b © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the turkish airline industry mahmut bakır1, şahap akan2, emircan özdemir3 1samsun university, samsun, turkey 2anadolu university, eskişehir, turkey 3eskişehir technical university, eskişehir, turkey abstract. aircraft selection is an important issue in achieving long-term goals in the airline industry. for this issue in which multiple conflicting criteria are involved, the extant literature points to the use of multi-criteria decision-making (mcdm) methods. in this respect, this study aims to propose a systematic and comprehensive framework with a focus on the regional aircraft selection perspective. to achieve this, an integrated fuzzy pivot pairwise relative criteria importance assessment (fpiprecia) and fuzzy measurement alternatives and ranking according to the compromise solution (f-marcos) approach was employed. in this study, in which six regional aircraft alternatives were evaluated according to 14 criteria, data were collected from five decision experts. as a result, it was found that the most pivotal criterion is c33 (operational cost), and the least important criterion is c12 (nox). in addition, crj1000 was identified as the most promising regional aircraft alternative. the results of the application were further validated by applying a three-stage sensitivity analysis. the proposed structure is anticipated to assist airline managers in aircraft selection decisions under uncertainty by offering a robust and systematic tool. key words: fuzzy sets theory, piprecia, marcos, regional aircraft selection, passenger perceptions 1. introduction the airline industry has been growing steadily since deregulation, and this trend is expected to continue in the future [1]. accordingly, the number of passengers, which was 4.5 billion in 2019, is projected to rise to 8 billion by 2039 [2]. similarly, icao [3] predicts that the number of passengers may reach 10 billion by 2040. this development received may 05, 2020 / accepted july 10, 2021 corresponding author: mahmut bakır samsun university, school of civil aviation, ballıca campus 55420, samsun, turkey e-mail: mahmut.bakir@samsun.edu.tr 424 m. bakir, ş. akan, e. özdemi̇r in the airline industry covers a wide range of potential customers. passengers living in remote areas with relatively low populations are also one of these potential customer groups. it should be ensured that these passengers are transported to the hub airports or to various destinations via the hub airports. moreover, airline service should be provided from these regions to destinations with high passenger potential through point-to-point flights. at this point, regional airlines come to the fore, and it is regarded that regional aviation is a suitable business model for providing airline services to the aforementioned target group [4]. regional airlines must make some decisions to match the increasing passenger demand with airline capacity in current market conditions. one of these decisions is to select the most suitable aircraft types in line with their strategies [5]. however, it is challenging for airline managers to make such complex decisions. this difficulty stems from the fact that choosing the appropriate aircraft type often depends on taking into account a large number of conflicting criteria that are difficult to assess simultaneously [5, 6]. there are many factors that need to be considered when selecting regional aircraft, both from airline and passenger perspectives. in terms of airline management, factors such as suitable payment terms, inclusion in the fleet at the right time, and the low operational cost come to the fore. from the passenger perspective, factors such as the appropriate schedule, flight comfort, and service quality are important. therefore, not only operational factors should be taken into consideration in regional aircraft selection, but these factors should also be evaluated together with passenger expectations. in addition, the environmental impact of the aircraft to be selected and their compliance with governments’ policies need to be considered [7]. in this context, airlines should consider all factors and make the optimum choice according to their business model, customer profile, and strategies [8]. however, aircraft selection relies on different criteria, while some of them can be expressed numerically and others can be described qualitatively [8]. therefore, it is crucial to employ an appropriate methodological approach to the problem of aircraft selection. the extant literature suggests the use of mcdm methods in aircraft selection [5, 9]. mcdm methods are tools that allow decision-makers (dms) to make appropriate choices in complex decision problems where there are several conflicting criteria [10, 11]. on the other hand, it should be noted that human judgments are often uncertain and ambiguous [9]. for this reason, mcdm methods are often adapted to fuzzy sets theory to deal with subjective and qualitative judgments under uncertainty [12]. given that aircraft selection is an mcdm problem involving a large number of subjective criteria, this present study aims at proposing a systematic and comprehensive framework for regional aircraft selection. another specific purpose of this study is to examine the applicability of the proposed framework in the context of the turkish airline industry. to this end, an integrated f-piprecia and f-marcos approach was employed. there are a few motivations behind this study. firstly, given the increasing passenger demand [2] and the anticipated need for regional aircraft in the next 15 years [13], it is clear that the regional airline business model will show great development. therefore, the problem of aircraft selection, which is strategically important for regional airlines, is worth investigating. however, there is a paucity of studies in the literature specifically examining regional aircraft selection [7, 8]. secondly, previous research addressing the aircraft selection problem has mainly focused on technical, operational [5, 8], or environmental [7] criteria. in addition to these criteria, this study also considers the passenger perspective criteria. thirdly, the airline industry in turkey has grown dramatically in the last two decades [14]. in this regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 425 regard, it is anticipated that this study will provide valuable insights to airline managers regarding the development of regional transportation in turkey. this study provides a few contributions to the existing literature: a) a systematic and comprehensive framework is proposed for regional aircraft selection, b) this is the first study to investigate regional aircraft selection in turkey, and, c) the robustness of the application is achieved through a rigorous three-stage sensitivity analysis. the remainder of the paper is structured as follows. section 2 provides further information about the extant literature on regional airlines and aircraft selection criteria. then, section 3 details the step-by-step procedures of the proposed methodology. section 4 begins by explaining the proposed framework and then presents an empirical study with sensitivity analysis. finally, section 5 provides a summary of the study’s results, implications, limitations, and directions for future research. 2. literature review 2.1. regional aviation activities regional airlines are of great importance to communities in rural and remote areas. airlines operating in this business model contribute to economic and social development by connecting low population settlements to air transportation hubs. they represent a critical lifeline for stakeholders in these areas [15]. this business model of critical importance in the airline industry has undergone some changes over time. as a result of these changes, different definitions have been made for this airline business model. in the previous definitions, regional airlines were defined as airlines with lower seat capacity (up to 90), but their seat capacity has increased even more over time [4]. especially, after aircraft manufacturers such as embraer started to produce regional jet aircraft with a capacity of 120 seats, the definition of the regional airline has expanded [16]. in this study, regional airlines are considered as short-haul scheduled carriers with seat capacity below 120. nowadays, regional airlines operate in different ways of business strategy. the first of these is that the regional airlines operate under code share agreements with one or more major carriers [17]. the second group is that they operate independently with their own flight code [18]. regional airlines have a characteristic that feeds hubs in both business strategies. they operate mostly in countries with high gdp. it is predicted that regional aviation will also become more widespread in developing countries in the forthcoming years. embraer [13] stated that there would be a need for 10,550 regional jets worldwide between 2019-2038, and the demand for regional jets in certain parts of the world would increase. therefore, one of the important research areas on regional airlines has been identified as the problem of aircraft selection. it is anticipated that such studies will provide valuable contributions to the stakeholders of the airline industry. 2.2. aircraft selection problem airline managers have to make many decisions in order to fulfill a number of timeconstrained processes due to the nature of their work. these cover a wide range from simple procedural decisions to large investment decisions. one of the risky investment decisions is aircraft selection, which can affect the sustainability and competitive ability of airlines [19]. aircraft selection problem, which has a quite high investment amount, is 426 m. bakir, ş. akan, e. özdemi̇r a process that is affected by many different factors such as business strategies, passenger needs, government policies [7]. therefore, optimum decisions should be made for the airline by considering all factors. at this point, a number of tools that enable dms to make effective and efficient decisions are highly beneficial. due to the multiple criteria nature of the aircraft selection problem, mcdm techniques come to the fore in this regard [19]. mcdm techniques have been frequently used in aircraft selection problems in the aviation industry. in the past studies, many selection problems on fighter jet [20, 21], aerobatics aircraft [22], training aircraft [23, 24], and commercial aircraft [7, 8, 25, 26] have been handled with these techniques. in addition, cargo aircraft [27], long-haul aircraft [28], medium-haul aircraft [9, 25, 29], and short-haul aircraft [26] selection studies have been conducted by detailing the aircraft selection problem. methodologically, prior research employed numerous techniques, including ahp and topsis in interval type-2 fuzzy sets [9], fuzzy ahp [7, 8, 26], fuzzy topsis [23, 30], fuzzy ahp and topsis [24], ahp and even swaps method (esm) [5], swara and smaa-2 method [22], an integrated ahp, copras, and moora methodology [25], fuzzy ahp and fuzzy grey relational analysis (gia) [27], linear physical programming [19], ahp [20], naiade (novel approach to imprecise assessment and decision environments) [31], fucom and aras [21], fuzzy ahp and fuzzy anp [29], electre, saw, and topsis [28], anp [32], and fuzzy reference ideal method (frim) [33]. in table 1, the aircraft selection literature is depicted by detailing research purposes and criteria sets. the problem of aircraft selection in regional airlines has found a limited area in the literature. ahmed et al. [7] addressed an aircraft selection problem of regional airlines in canada. it has addressed the aircraft selection problem of regional airlines in canada by focusing on the sustainability issue. bruno et al. [26] used a hybrid approach based on ahp and fuzzy set theory in order to evaluate jet engine regional aircraft alternatives. dožić and kalić [5] conducted a study limited to six criteria using ahp and esm methods. in a similar study, dožić et al. [8] mainly focused on technical and economic factors among aircraft alternatives by using fuzzy ahp. gomes et al. [31] used the naiade method for regional aircraft selection problem using aircraft alternatives with up to 26 seats. when the studies conducted are evaluated in general, the methods used, criteria, and aircraft alternatives differ from each other. in terms of methodology, fuzzy ahp was predominantly used in the studies. however, as the number of criteria increases in the fuzzy ahp, the number of binary comparisons increases, so consistency problem may occur [34]. therefore, we believe that there is a need for methods that will provide more consistent and robust results in regional aircraft selection problems. in terms of the criteria, it can be stated that different approaches were adopted in previous studies. accordingly, some studies mainly focused on economical and technical criteria, while some studies focused on today’s important issues such as sustainability. hence, putting forward a comprehensive framework that includes many criteria may be an important contribution to the existing literature. in terms of aircraft alternatives used and their seat capacities, quite different aircraft engine types (e.g., turbojet and turboprop engines) and various seat capacities were evaluated in the past. therefore, it is necessary to consider quite wide and different aircraft options with low and high seat capacity, considering different needs. consequently, this study, which focuses on regional aircraft selection, taking into account the limitations mentioned above and focuses on the perspective of the turkish airline industry. in doing so, using a comprehensive set of criteria shown in regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 427 fig. 2, an integrated f-piprecia and f-marcos approach has been adopted that can yield powerful results methodologically. table 1 aircraft selection criteria in the literature author(s) o b je ct iv e r an g e s c s p ee d m t o w m c o c a p a s l c iq s p a b a c o 2 n o x n o is e bruno et al. [26] r kiracı & akan [9] c wang & chang [23] t dožić et al. [8] r sánchez-lozano et al. [24] t dožić & kalić [5] r durmaz & gencer [22] a kiracı & bakır [25] c akyurt & kabadayı [27] ca ahmed et al. [7] r ilgın [19] c ali et al.[20] f gomes et al. [31] r van hoan & ha [21] f ozdemir & basligil [29] c sun et al. [28] c yeh & chang [30] c ozdemir et al.[32] c sánchez-lozano & rodríguez [33] t our proposed framework r notes r: regional, c: commercial, t: training, ca: cargo, f: fighter, sc: seat capacity, mtow: maximum take-off weight, mc: maintenance cost, oc: operational cost, ap: aircraft price, asl: aircraft service life, ciq: cabin interior quality, sp: safety perception, aba: aircraft body appearance, co2: carbon dioxide emission, nox: nitrogen oxide emission. 3. methodology 3.1. preliminaries of fuzzy sets many real-world examples include ambiguity and vagueness that cannot be articulated precisely. zadeh [35] proposed the fuzzy sets theory to express and model such uncertainties more partially. within the scope of fuzzy sets, the membership function ( ) : [0,1] f x r → of a triangular fuzzy number (tfn) f̃ is found as follows [36]. 428 m. bakir, ş. akan, e. özdemi̇r 0, , ( ) , 0, f x l x l l x m m l x u x m x u u m x u   −     − =  −    −    (1) from eq. (1), l and u represent the lower and upper bounds of fuzzy number f̃, and m shows the most promising value of f̃. if we consider that f̃=(l1, m1, u1) and k̃=(l2, m2, u2) are two tfns, the basic operations for the tfns are given below [37, 38]: 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )f k l m u l m u l l m m u u = + = + + + (2) 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )f k l m u l m u l l m m u u =  =    (3) 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )f k l m u l m u l u m m u l− = − = − − − (4) 1 1 1 1 1 1 2 2 2 2 2 2 ( , , ) , , ( , , ) l m u l m uf l m u u m lk   = =     (5) 1 1 1 1 1 1 1 1 1 1 1 1 ( , , ) , ,f l m u u m l − −   = =     (6) let us assume again that f̃=(l1, m1, u1) is a tfn. the defuzzified (crisp) value of f̃ can be obtained as follows [39]: 4 6 crisp l m u df + + = (7) 3.2. f-piprecia method the piprecia method is derived from the swara method to measure the cognitive attitudes of dms [40]. the piprecia approach is beneficial for group decisions involving a large number of criteria or experts, as it decreases the cognitive burden of dms by reducing pairwise comparisons. the piprecia method has been extended to fuzzy numbers by stević et al. [11] to handle ambiguous judgments more effectively. the f-piprecia method has existed in many successful applications in the literature. stević et al. [11] introduced the f-piprecia method for the barcode technology implementation case. furthermore, this approach has been successfully applied in many areas, including selecting reach stacker [41], road transportation risk analysis [42], green supplier selection [43], determining the safety level in railway crossings [44], and the highperformance computing problem [45]. the f-piprecia method needs the following steps for calculation [11]: step 1 identify the dms and define the criteria to be used. dms rank criteria from first to last, regardless of the significance of the criteria. regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 429 step 2 starting with the second criterion, dms evaluate the criteria in order to obtain the relative importance of the criteria. 1 1 1 1 1 1 j j r j j j j j if c c s if c c if c c − − −    = = =   (8) where r js denotes the evaluation of the criteria by decision-maker r. to construct a matrix js , the geometric mean of matrix r js has to be calculated. dms use the linguistic scales in table 2 and table 3 when evaluating the criteria compared to the previous criteria [11]. table 2 the scale 1-2 for the evaluation of the criteria linguistic terms l m u dfv almost equally important 1.000 1.000 1.050 1.008 weakly more important 1.100 1.150 1.200 1.150 moderately more important 1.200 1.300 1.350 1.292 more important scale 1-2 1.300 1.450 1.500 1.433 strongly more important 1.400 1.600 1.650 1.575 very strongly more important 1.500 1.750 1.800 1.717 absolutely more important 1.600 1.900 1.950 1.858 table 3 the scale 0-1 for the evaluation of the criteria l m u dfv linguistic terms 0.667 1.000 1.000 0.944 weakly less important 0.500 0.667 1.000 0.694 moderately less important 0.400 0.500 0.667 0.511 less important scale 0-1 0.333 0.400 0.500 0.406 really less important 0.286 0.333 0.400 0.337 strongly less important 0.250 0.286 0.333 0.288 very strongly less important 0.222 0.250 0.286 0.251 absolutely less important if the marked criterion is more important than the previous one, the evaluations are completed by following table 2. conversely, if the marked criterion is less important than the previous one, the linguistic expressions in table 3 are adopted. the tables also include defuzzified values (dfv) of each comparison, making it easier for dms to evaluate [11]. step 3 calculate coefficient jk . 1 1 2 1 j j if j k s if j = = =  −  (9) step 4 determine fuzzy weight jq . 1 1 1 1 jj j if j qq if j k −  = =  =    (10) 430 m. bakir, ş. akan, e. özdemi̇r step 5 obtain the relative weight of criteria jw . 1 j j n j j q w q = =  (11) the f-piprecia approach dictates the use of the inverse methodology after assigning coefficients jw . step 6 to provide inverse pairwise comparisons, the evaluation is performed starting with a penultimate criterion in this round. 1 1 1 1 1 1 j j r j j j j j if c c s if c c if c c + + +     = = =   (12) where r js  represents the evaluation of criteria made by a decision-maker r. it is again necessary to aggregate the inverse comparison values by means of the geometric mean. step 7 determine coefficient jk  . 1 2 j j if j n k s if j n  = = =  −  (13) step 8 determine fuzzy weight jq  . 1 1 j j j if j n q q if j n k +  = =  =    (14) step 9 obtain the relative weight of criteria jw  . 1 j j n j j q w q =   =  (15) step 10 fuzzy values jw and jw  should be defuzzified in order to achieve the final weights of criteria jw  . 1 ( ) 2 j j jw w w = + (16) step 11 using spearman’s rank correlation coefficient (src), the consistency of the weights is confirmed in the final step. regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 431 3.3. f-marcos method the marcos method by stević et al. [46] is a utility function-based mcdm method. the basic computational principle of the technique relies on the ideal (id) and anti-ideal (ai) solutions. accordingly, it takes into account the relationships between ideal and anti-ideal points and alternatives [37, 47]. the marcos method, like the topsis method, defines the alternative that is nearest to the ideal point and farthest from the anti-ideal point as the best-ranked solution [48, 49]. stanković et al. [37] introduced the f-marcos methodology to provide more precise results in an uncertain environment. this extended version both overcomes uncertainty and gives more stable results in larger decision-making matrices [37]. the f-marcos approach was first used to analyze road traffic risk in the literature [37]. following this, it has been successfully applied in many studies such as determining the level of safety on the roads [50], selecting sustainable suppliers [48], evaluating the competitiveness of spa centers [47], and evaluating the safety levels of railway crossings [44]. the f-marcos method includes the following steps [47, 48]: step 1 construct the initial fuzzy decision matrix. as with other mcdm techniques, a decision matrix is constructed in which the m alternatives and n criteria are present. table 4 can be used for dms to evaluate alternatives [47, 48]. table 4 the linguistic scale for rating of the alternatives linguistic terms tfns very poor (0, 0, 1) poor (0, 1, 3) medium poor (1, 3, 5) medium (3, 5, 7) medium good (5, 7, 9) good (7, 9, 10) very good (9, 10, 10) step 2 extent the initial fuzzy decision matrix. the initial decision matrix is extended by adding fuzzy ideal ã(id) and fuzzy anti-ideal ã(ai) solutions. while ã(id) indicates the most desirable alternative, ã(ai) indicates the most undesirable one. considering the type of criteria, solutions ã(id) and ã(ai) are found as follows. ( ) mini îja ai x= if j b and maxi ijx if j c (17) ( ) maxi îja id x= if j b and min i ijx if j c (18) where b represents the maximization type criteria and c represents the minimization type criteria. step 3 formulate the normalized fuzzy decision matrix. the elements of the decision matrix including solutions ã(id) and ã(ai) are standardized. ( , , ) , , l l l l m u id id id ij ij ij ij u m l ij ij ij x x x n n n n x x x   = =       if j c (19) 432 m. bakir, ş. akan, e. özdemi̇r ( , , ) , , l m u ij ij ijl m u ij ij ij ij u u u id id id x x x n n n n x x x   = =       if j b (20) where elements , ,l m u ij ij ijx x x come from step 1, and ,l u id idx x are retrieved from ã(id). step 4 construct the weighted fuzzy decision matrix. normalized values are multiplied by the weight coefficients, thus obtaining matrix ṽ. ( , , )l l m m u u ij ij ij ij j ij j ij jv n w n w n w n w=  =    (21) step 5 obtain fuzzy summation matrix (s̃i). the row elements of the weighted fuzzy decision matrix are summed. 1 n i ij i s v = = (22) step 6 calculate the utility degrees of the alternatives. matrices ik− and ik+ are constructed according to the total values of ideal and anti-ideal solutions. , , l m u i i i i i u m l ai ai aiai s s s s k s s ss −   = =     (23) , , l m u i i i i i u m l id id idid s s s s k s s ss +   = =     (24) step 7 create fuzzy matrix t̃i. ( , ) ( , , )l m u l l m m u u i i i i i i i i i i i i it t t t t k k k k k k k k− + − + − + − += = + =  = + + + (25) then a new fuzzy number (d̃) is derived from the elements of matrix t̃i. ( , , ) maxl m u i ijd d d d t= = (26) then, fuzzy number d̃ is defuzzified using eq. (7), yielding dfcrisp. step 8 identify utility functions of alternatives. using number dfcrisp, solutions ( )if k+ and ( )if k− are created from the results of step 6. ( ) , , l m u i i i i i crisp crisp crisp crisp k k k k f k df df df df − − − − +   = =       (27) ( ) , , l m u i i i i i crisp crisp crisp crisp k k k k f k df df df df + + + + −   = =       (28) at this step, a defuzzification procedure must be applied for each of ik− , ik+ , ( )if k− and ( )if k+ . step 9 create final utility function f(ki). regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 433 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) i i i i i i i k k f k f k f k f k f k + − + − + − + = − − + + (29) it should be noted that the most optimal alternative is the one with the highest utility function f(ki). 4. empirical study for the turkish airline industry after the liberalization of the aviation market in 2003 in turkey, the number of major components in the aviation system such as airlines and airports has increased rapidly [51]. by increasing passenger accessibility and lowering ticket prices, the number of passengers on domestic flights has increased significantly [52]. the main distinguishing feature of regional airlines compared to other airline business models is that they connect small communities to different destinations with short and medium-distance domestic flights [53]. today, domestic passenger demand in turkey still maintains its increasing trend. therefore, future fleet planning studies of regional airlines are important. the aircraft selection problem, which is included in the fleet planning activities of regional airlines, is a strategic decision that is extremely effective in the success of airlines. in the study, the regional aircraft selection problem for the turkish airline industry has been addressed using the f-piprecia and f-marcos methods. 4.1. framework for the regional aircraft selection the study is divided into four phases, as shown in fig. 1. the first of these is the preparatory stage. the second stage is the f-piprecia application, where regional aircraft selection criteria weights are calculated. the third stage is the f-marcos application, where alternative regional aircraft types are ranked. in the fourth and last stage, a three-stage rigorous sensitivity analysis of the empirical study is given. in the first step of phase i, the research problem was determined as selecting aircraft type for the regional airlines in turkey. instead of specifying an airline company for aircraft selection, the regional airline business model is tackled within the scope of the turkish airline industry characteristics. the solution to be produced in this context is intended to be suitable for all regional airline activities in turkey. considering the aircraft selection literature and regional aviation characteristics in turkey, four main criteria that are effective in regional aircraft selection have been determined: technical performance, economical performance, passenger perception, and environmental impact. technical performance, economical performance, and environmental impact criteria have been addressed in many previous aircraft selection studies [7, 26, 31, 54]. to the best of the authors’ knowledge, no previous study has discussed aircraft selection problem by addressing the passenger perception criterion. on the other hand, it is well-established that the aircraft body appearance [55], cabin interior design [56], and safety-related factors [31, 57, 58], which are directly related to the aircraft type, have an effect on the perception of the passengers. for this reason, passenger perception is an important criterion to be taken into consideration in the selection of aircraft type of regional airlines. 434 m. bakir, ş. akan, e. özdemi̇r creating the set of suitable alternatives forming team of decision-makers (dms) training dms about method implementation evaluating of criteria and alternatives by dms formulating criteria and sub-criteria vectors aggregating of dms evaluation using geometric mean. calculating final criteria weights comparing results with other fuzzy methods running the f-marcos algorithm ranking of alternatives phase i: preparatory stage running the f-piprecia algorithm phase ii: f-piprecia application testing robustness with simulated weights creating the fuzzy initial matrix checking the rank reversal effect phase iv: sensitivity analysis defining of research problem and goals identifying a set of criteria for selection of regional aircraft literature review experts opinion finalizing criteria and sub-criteria phase iii: f-marcos application fig. 1 the phases of the research methodology fig. 2 presents the proposed framework based on the aircraft selection literature and expert opinions. all criteria are taken into account as beneficiary types, and considered with the maximization function in this research. in order to demonstrate the practical applicability of the proposed framework, the airport network in turkey has been analyzed, and the constraints of the regional aviation system have been determined. in this way, determining feasible aircraft alternatives for the constraints is aimed. the present status of the airports in turkey is given in fig. 3. regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 435 technical performance (c4) economical performance (c3) passenger perception (c2) environmental impact (c1) mtow (41) speed (c42) seat capacity (c43) range (c44) aircraft service life (c31) aircraft price (c32) operational cost (c33) maintenance cost (c34) aircraft body appearance (c21) safety perception (c22) cabin interior quality (c23) noise (c11) nox (c12) co2 (c13) selection of regional aircraft fig. 2 the proposed framework for regional aircraft selection fig. 3 present status of airports in turkey there are 56 airports in turkey by the year of 2021. 40 of them are international airports, and the remaining 16 are domestic airports. the distance between the two farthest airports is approximately 1650 kilometers, based on the current state of the airport network. in light of current conditions, aircraft alternatives in the regional aircraft selection problem have been determined to have a minimum range of 1650 kilometers. in addition, taking into account expert opinions based on regional passenger traffic in turkey, a maximum seat capacity limit of 120 is established for aircraft alternatives. all of the short-haul narrow-body regional aircraft alternatives are compatible for operating at all airports in turkey in terms of maximum take-off weight (mtow), maximum landing weight (mlw), and take-off run distance features. in consequence, there are no constraints for operational features such as mtow, mlw, and take-off run distance. for building aircraft alternatives set, several aircraft types used for regional aviation purposes in different geographies around the world have been identified. as a result of the examination, bombardier crj1000, atr 72-600, de havilland canada q400 nextgen, embraer erj-190, sukhoi ssj100, comac arj21-700 aircrafts are included in the alternative 436 m. bakir, ş. akan, e. özdemi̇r set. operational characteristics of alternatives have been acquired from the manufacturers’ documentations. economical performance cost data is mainly taken from the airline data project [59]. emission values of aircraft alternatives vary depending on the engine type and model used. in this context, emission values are taken from icao aircraft engine emissions databank [60], considering the most widely used engine model for each alternative aircraft. manufacturer documents and various aviation platforms have been consulted for the missing data in the relevant references. in order to evaluate six aircraft alternatives in line with four main criteria and 14 subcriteria, consisting of five different experts, who are well-informed on airline fleet planning, aircraft selection, and airline management, a dm team was formed. dm team was informed about the criteria, alternatives, and application steps before the evaluation process. the following information was presented to each dm during the evaluation of alternatives: manufacturer company/country, maiden flight date, range (km), cruise speed (km/h), mtow (kg), maintenance cost ($ per block hour), operational cost ($ per block hour), aircraft price ($), aircraft service life, co2 emission (kg per landing/take-off cycle), nox emission (kg per landing/take-off cycle), noise pollution (db). in addition, fuselage and cabin photographs of aircraft types were presented so that dms could evaluate aircraft alternatives from the passenger perspective in line with the passenger perception criteria. the aircraft selection application for the turkish regional aviation case made based on the comparison data obtained from the evaluations of dms is presented in the following sections, which correspond to phase ii, phase iii, and phase iv. 4.2. f-piprecia results in this section, using the responses collected from five dms, the weights of the criteria and sub-criteria are presented. because both the f-piprecia and f-marcos methods are well-established in the literature, we omitted comprehensive calculation procedures from this paper. the dms determined the weights of the criteria and sub-criteria using the linguistic scales in tables 2 and 3. accordingly, in the first step of ordinary f-piprecia, each dm compared the main criteria using eq. (8). then, as shown in table 4, an aggregated matrix js was created by using the geometric mean. using eq. (9), js values were subtracted from the number 2, thus jk matrix was obtained. in the next step, based on the elements of matrix jk , matrix jq was created with the help of eq. (10). criteria weights were obtained by using eq. (11) as the last step of the ordinary f-piprecia method. in addition, the defuzzified values (dfv) of the criteria weights were given, as eq. (7) dictates to facilitate the calculation. the fact that the f-piprecia method produces the final weights of the criteria depends on employing the inverse f-piprecia methodology applied in eqs. (12)-(15). in contrast to ordinary f-piprecia, this approach can be accomplished by following the previously mentioned calculation procedures while taking into account the penultimate criterion. following eqs. (12)-(15), inverse criteria weights were calculated using eq. (15) in the last step of the inverse f-piprecia method. these procedures were applied first for the main criteria, and then for the sub-criteria. to use this paper sparingly, this section clarifies only the calculation procedures for the main criteria. table 5 gives the main criterion weights and the dfv values derived from them. regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 437 table 5 weights of the main criteria (c1-c4) through f-piprecia piprecia js jk jq jw dfv rank c1 * (1.000, 1.000, 1.000) (1.000, 1.000, 1.000) (0.131, 0.153, 0.197) 0.156 4 c2 (1.173, 1.255, 1.305) (0.695, 0.745, 0.827) (1.209, 1.342, 1.439) (0.158, 0.205, 0.284) 0.226 3 c3 (1.297, 1.444, 1.494) (0.506, 0.556, 0.703) (1.719, 2.412, 2.845) (0.224, 0.368, 0.562) 0.376 1 c4 (0.488, 0.662, 0.803) (1.197, 1.338, 1.512) (1.137, 1.803, 2.376) (0.148, 0.275, 0.469) 0.286 2 sum (5.065, 6.557, 7.660) piprecia-i js  jk  jq  jw  dfv rank c1 (0.488, 0.662, 0.803) (1.197, 1.338, 1.512) (0.484, 0.641, 0.927) (0.112, 0.172, 0.277) 0.18 4 c2 (0.447, 0.580, 0.833) (1.167, 1.420, 1.553) (0.731, 0.858, 1.110) (0.169, 0.231, 0.331) 0.237 3 c3 (1.119, 1.179, 1.229) (0.771, 0.821, 0.881) (1.135, 1.217, 1.296) (0.262, 0.328, 0.387) 0.327 1 c4 * (1.000, 1.000, 1.000) (1.000, 1.000, 1.000) (0.231, 0.269, 0.298) 0.268 2 sum (3.350, 3.716, 4.334) in order to determine the final weights of the criteria, the criteria weights arising from ordinary f-piprecia and inverse f-piprecia applications should be averaged. in this case, the main criterion weights (c1-c4) assigned by dms were found as follows: 1cw = 0.168, 2cw = 0.224, 3cw = 0.351, 4cw = 0.277. the first-order global weights were also calculated in order to transfer the sub-criteria weights to the f-marcos application. table 6 presents the global criteria weights by multiplying the weight of each sub-criterion by the weight of the corresponding main criterion. table 6 final criteria weights derived from the f-piprecia method criteria criteria weight sub-criteria local weight global weight rank c1 0.168 c11 0.346 0.058 12 c12 0.274 0.046 14 c13 0.386 0.065 11 c2 0.224 c21 0.322 0.072 7 c22 0.380 0.085 4 c23 0.303 0.068 9 c3 0.351 c31 0.224 0.079 5 c32 0.216 0.076 6 c33 0.302 0.106 1 c34 0.268 0.094 2 c4 0.277 c41 0.250 0.069 8 c42 0.202 0.056 13 c43 0.323 0.089 3 c44 0.244 0.067 10 as shown in table 6, c33 (operational cost) is the most pivotal criterion in terms of its impact on regional aircraft selection, followed by c34 (maintenance cost) and c43 (seat capacity). c12 (nox), on the other hand, is the least important criterion. finally, we examined the main criteria and sub-criteria rankings obtained from fpiprecia and inverse f-piprecia application with src by using eq. (30). the src 438 m. bakir, ş. akan, e. özdemi̇r coefficient is a non-parametric correlation test used when at least one of the variables is not normally distributed or there are no tied ranks [61, 62]. 2 2 6 1 ( 1) i s d r n n = − −  (30) where di is the difference between in the paired ranks and n is the number of cases. since the src coefficient did not fall below 0.80, the rankings are largely consistent. 4.3. f-marcos results following the determination of the criteria weights, this section employs the fmarcos approach for selecting regional aircraft alternatives. dms first evaluated the alternatives by adopting the linguistic scale in described table 4. table 7 shows the fuzzy aggregated decision matrix formed as a result of the judgments of five dms. in the aggregation process, the arithmetic average was used due to the nature of the judgments. table 7 fuzzy aggregated decision matrix c11 c12 … c43 c44 crj1000 (5.40, 7.20, 8.40) (4.20, 6.20, 7.80) … (6.20, 7.80, 9.00) (6.60, 8.40, 9.40) atr 72-600 (5.80, 7.80, 9.20) (5.80, 7.80, 9.40) … (1.60, 3.40, 5.40) (1.60, 3.40, 5.40) q400 nextgen (4.20, 6.20, 7.80) (5.40, 7.40, 8.80) … (3.80, 5.80, 7.60) (4.20, 6.20, 8.20) erj-190 (3.80, 5.80, 7.40) (1.40, 2.60, 4.40) … (7.80, 9.20, 9.80) (8.20, 9.40, 9.80) ssj100 (5.80, 7.60, 8.80) (5.00, 7.00, 8.80) … (5.40, 7.40, 9.00) (6.60, 8.40, 9.40) arj21-700 (5.80, 7.80, 9.20) (5.40, 7.40, 9.00) … (2.60, 4.60, 6.60) (4.60, 6.60, 8.40) then, using eqs. (17)-(18), the fuzzy aggregated evaluation matrix was extended to include fuzzy ideal ã(id) and fuzzy anti-ideal ã(ai) solutions. in the study, since all criteria are used as beneficial criteria due to the structure of the linguistic scale, ã(id) represents the most desirable and ã(ai) represents the most undesirable values. moreover, the extended fuzzy initial decision matrix was normalized by applying eqs. (19)-(20) (table 8). table 8 fuzzy normalized decision matrix c11 c12 … c43 c44 ai (0.41, 0.63, 0.80) (0.15, 0.28, 0.47) … (0.16, 0.35, 0.55) (0.16, 0.35, 0.55) crj1000 (0.59, 0.78, 0.91) (0.45, 0.66, 0.83) … (0.63, 0.80, 0.92) (0.67, 0.86, 0.96) atr 72-600 (0.63, 0.85, 1.00) (0.62, 0.83, 1.00) … (0.16, 0.35, 0.55) (0.16, 0.35, 0.55) q400 nextgen (0.46, 0.67, 0.85) (0.57, 0.79, 0.94) … (0.39, 0.59, 0.78) (0.43, 0.63, 0.84) erj-190 (0.41, 0.63, 0.80) (0.15, 0.28, 0.47) … (0.80, 0.94, 1.00) (0.84, 0.96, 1.00) ssj100 (0.63, 0.83, 0.96) (0.53, 0.74, 0.94) … (0.55, 0.76, 0.92) (0.67, 0.86, 0.96) arj21-700 (0.63, 0.85, 1.00) (0.57, 0.79, 0.96) … (0.27, 0.47, 0.67) (0.47, 0.67, 0.86) id (0.63, 0.85, 1.00) (0.62, 0.83, 1.00) … (0.80, 0.94, 1.00) (0.84, 0.96, 1.00) following that, the criteria weights obtained from the f-piprecia application were included in the calculation. using eq. (21), the fuzzy normalized decision matrix elements were multiplied by their respective criterion weights. the fuzzy weighted normalized decision matrix created is shown in table 9. regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 439 table 9 fuzzy weighted normalized decision matrix c11 c12 … c43 c44 ai (0.02, 0.04, 0.05) (0.01, 0.01, 0.02) … (0.01, 0.03, 0.05) (0.01, 0.02, 0.04) crj1000 (0.03, 0.05, 0.05) (0.02, 0.03, 0.04) … (0.06, 0.07, 0.08) (0.05, 0.06, 0.06) atr 72-600 (0.04, 0.05, 0.06) (0.03, 0.04, 0.05) … (0.01, 0.03, 0.05) (0.01, 0.02, 0.04) q400 nextgen (0.03, 0.04, 0.05) (0.03, 0.04, 0.04) … (0.03, 0.05, 0.07) (0.03, 0.04, 0.06) erj-190 (0.02, 0.04, 0.05) (0.01, 0.01, 0.02) … (0.07, 0.08, 0.09) (0.06, 0.06, 0.07) ssj100 (0.04, 0.05, 0.06) (0.02, 0.03, 0.04) … (0.05, 0.07, 0.08) (0.05, 0.06, 0.06) arj21-700 (0.04, 0.05, 0.06) (0.03, 0.04, 0.04) … (0.02, 0.04, 0.06) (0.03, 0.05, 0.06) id (0.04, 0.05, 0.06) (0.03, 0.04, 0.05) … (0.07, 0.08, 0.09) (0.06, 0.06, 0.07) following the weighting procedure, fuzzy summation matrix (s̃i) was created by applying eq. (22). one of the unique elements of the f-marcos method is the utility degree. therefore, using eqs. (23)-(24), ik− and ik+ matrices, which are utility degrees according to ideal and anti-ideal solutions, were constructed. then, a fuzzy matrix t̃i is created by summing the utility degrees with the help of eq. (25). moreover, an fmarcos-specific fuzzy number (d̃) is derived from this matrix using eq. (26). however, it should be noted that this new and all fuzzy values obtained after this step will be defuzzified by using eq. (7). based on defuzzified value dfcrisp, ideal ( )if k+ and anti-ideal ( )if k− solutions were obtained using eqs. (27)-(28) in the next step. as mentioned earlier, ik− , ik+ , ( )if k− and ( )if k+ values suggested by the f-marcos method were defuzzified by applying eq. (7) to create the f(ki) vector. the finalized f-marcos results and observed ranking are shown in table 10. table 10 defuzzified values and results of f-marcos application ( )if k− ( )if k+ ik− ik+ ( )if k− ( )if k+ ( )if k rank crj1000 (0.20, 0.29, 0.42) (0.32, 0.64, 1.35) 2.05 0.86 0.30 0.70 0.77 1 atr 72-600 (0.15, 0.24, 0.37) (0.24, 0.52, 1.19) 1.71 0.71 0.24 0.59 0.50 6 q400 nextgen (0.15, 0.24, 0.38) (0.24, 0.52, 1.20) 1.71 0.71 0.24 0.59 0.51 5 erj-190 (0.19, 0.27, 0.39) (0.31, 0.60, 1.25) 1.92 0.81 0.28 0.66 0.67 4 ssj100 (0.19, 0.28, 0.42) (0.31, 0.62, 1.33) 2.00 0.84 0.29 0.68 0.72 2 arj21-700 (0.18, 0.27, 0.41) (0.30, 0.60, 1.32) 1.96 0.82 0.28 0.67 0.69 3 as shown in table 10, the best-ranked alternative was found as crj1000. that is, crj1000 is the most promising aircraft type for regional flights in the turkish airline industry. in contrast, it is observed that the worst-performing alternative was identified atr 72-600, with the lowest utility function. 4.4. sensitivity analysis the credibility and robustness of the results depend on how resistant they are to changes in input parameters. therefore, this section is devoted to validating the results employing a threefold rigorous sensitivity analysis. in the first part, we tested the effect of changing criteria weights on the results using simulated weights. using eq. (31), we generated 25 scenarios based on the most critical criterion c33 [50, 63–65]. 440 m. bakir, ş. akan, e. özdemi̇r (1 ) (1 ) n n n w w w w = − −    (31) in that formula, wn denotes the scenario-adjusted weights of the criteria, and wn reflects the reduced values of the most important criterion (c33). w denotes the original weight assigned to the marked criteria, while wn is the original weight of the most important criterion (c33) [66]. at this point, the rate of reduction in wn was determined as 4%. the weight scenarios generated are shown in fig. 4. furthermore, scenario-based application rankings are portrayed in fig. 5. in scenario-based rankings, the lowest level of src was measured as 0.829, thus achieving a high level of consistency. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 s1 s3 s5 s7 s9 s11 s13 s15 s17 s19 s21 s23 s25 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 fig. 4 weights of criteria through 25 scenarios fig. 5 alternative rankings generated through 25 scenarios regional aircraft selection with fuzzy piprecia and fuzzy marcos: a case study of the... 441 in the second part, f-marcos findings were compared to those obtained using seven well-established fuzzy-based mcdm techniques [67]. fig. 6 illustrates the effects of the comparative analysis through a correlation map. as a result, the lowest src between fmarcos and the outcomes of other techniques is 0.943, indicating that the rankings are almost identical and the implementation is satisfactorily credible. fig. 6 comparative results of different fuzzy-based mcdm methods in the last part, any rank reversal risk was checked considering the matrix size [68]. in doing so, the application is continued until the last alternative remains by eliminating the worst listed criteria in each scenario. we tested the rank reversal effect in five scenarios. as shown in fig. 7, there is no change in the rankings, and the results gave quite stable results. in summary, the results of the three-stage rigorous sensitivity analysis established the credibility and robustness of the application. fig. 7 scenario results on rank reversal test 442 m. bakir, ş. akan, e. özdemi̇r 5. conclusion fleet planning, one of the most important strategic decisions made by airlines, has a significant impact on airline productivity and efficiency [69]. selecting the most suitable aircraft type for the airline’s business model and business strategies is an important step in the fleet planning process [70]. regional airlines, on the other hand, differ from other airline business models in terms of flight operation characteristics. some of these are features such as having cross flights, low passenger volume, feeding certain hubs, code-sharing with major airlines [71]. the unique characteristics of regional airlines require different approaches to the aircraft selection problem from other business models. although the criteria sets of many aircraft selection problems are similar, criteria weights change according to the airline business model. focusing on the turkish airline industry, this study revealed that the most important criterion in aircraft selection is operational cost (c33). the second and third most important criteria are maintenance cost (c34) and seat capacity (c43), respectively. among the main criteria, the most important regional aircraft selection criterion was determined as economic performance (c3). the results corroborate previous research in which economical efficiency was a significant factor in aircraft selection [9, 26, 31]. on the other hand, technical performance (c4) ranked second. passenger perception (c2) and environmental impact (c1) came third and last, respectively. in addition, it was found out that the importance of the passenger perception criterion, which is evaluated for the first time within the scope of the aircraft selection problem, is very close to the technical performance. specifically, according to global criterion weights, safety perception (c22) was found as the fourth most important criterion. in this regard, the study contributes to the literature by proposing a framework regarding regional aircraft selection and evaluating passenger perceptions in this framework. in terms of managerial contributions, the study offers valuable insights for regional airline managers in the turkish airline industry, which continues to grow rapidly. the implementation of regional aircraft selection, which is considered within the current state of the country’s aviation, reveals the criteria that the industry should pay attention to. it is also seen that the crj1000 is the most preferable aircraft type among alternatives. lastly, the approach employed yields consistent results within the case study. despite its novelty, this study has some limitations. there are many constraints and criteria that affect the aircraft selection problem, which is very complex. in this study, the regional aviation problem has been discussed 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truitt, l.j., haynes, r., 1994, evaluating service quality and productivity in the regional airline industry, transportation journal, 33, pp. 21–32. http://web.mit.edu/airlinedata/www/default.html http://web.mit.edu/airlinedata/www/default.html 11408 facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 137 150 https://doi.org/10.22190/fume221215005h © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fractal dimensions of a porous concrete and its effect on the concrete’s strength chun-hui he1, chao liu1,2 1school of civil engineering, xi'an university of architecture and technology, xi’an, china 2school of science, xi'an university of architecture and technology, xi’an, china abstract. all mechanical properties of a porous medium depend upon its fractal dimensions, however, how to measure the fractal dimensions is still an open issue. this paper adopts the two-scale fractal theory to calculate fast and effectively the fractal dimensions of a porous concrete. of the concrete's properties that have been fascinating engineers and scientists, by far the most perplexing is the effects of its porosity and pore size on concrete's strength. though there were many ad hoc empirical formulae for predicting the strength, much deviation arose for practical applications. here a dimensionless model and the fractal theory are adopted to insight theoretically into the effects, and for the first time ever, some physically relative and mathematically reliable formulations are proposed. additionally nano/micro particles’ size and distribution can also be used for theoretical prediction of the concrete’s strength, it shows that the boundary-induced force occurs when the particles tend to micro/nanoscales. the present theory sheds new light on the optimal design of various functional concretes. key words: two-scale fractal, geometric potential, dimensionless analysis, hall-petch effect, porosity 1. introduction a porous medium always behaves extremely attractively compared to its continuum partner, the latter is focus of the continuum mechanics, which has matured into a fully-fledged theory, and has laid the foundation for the mechanical engineering, however, there is no universal theory for porous problems. xue and liu [1] found that a porous medium with a hierarchical structure has an excellent heat insulation. xo, et al. [2] revealed the mechanism of heat prevention for cocoon-like hierarchy. xue, et al. [3] further elucidated cocoon's biomechanism using the fractal theory. hierarchical porous materials are now received: december 15, 2022 / accepted february 01, 2023 corresponding author: chao liu school of civil engineering, xi'an university of architecture and technology, xi’an, china e-mail: chaoliu@xauat.edu.cn 138 c.h. he, c. liu widely used for high-rate electrochemical capacitive energy storage [4], supercapacitors [5] and energy harvesting [6,7]. nano/micro scale porous membranes have extremely high permeability and extremely small pressure drop [8-11], the diffusion process in a porous medium (e.g. water) has attracted much attention in the academic community, because the seemingly stochastic diffusion process is actually deterministic in a fractal space, making the impossible possible [12,13,14]. the vibrating process in air can also be considered in a fractal space, and a new discipline, the fractal vibration theory, has been skyrocketing [15,16,17]. the fractal theory and the artificial intelligent have been successfully applied to investigate the hardness properties of tool steel alloys [18,19,20]. some phenomena arising in porous media cannot be explained by continuum mechanics, where the smooth space is the footstone. this paper focuses on the most commonly used porous material on the earth, that is the concrete [21,22,23], using the fractal theory [24]. 2. fractal dimensions all mechanical properties of a continuum medium are relative to its dimensions, for example, its volume scales with the cube of the measured size 3v r (1) where v is the volume, r is the measured size. similarly for a porous medium, its volume can be written as v r (2) where  is the fractal dimensions. in a fractal space with a fractal dimensionality α, the volume is a measurement of the measured size. the relation of the fractal dimensionality α and the volume can be expressed in eq. (2). when α=3, it becomes a continuum, and when α=0, it is an empty pore. zuo and liu elucidated that the mechanical and electrical properties of a composite depend upon the fractal dimensions [25], mandelbrot, et al. revealed that fracture property of metals can be effectively explained by the fractal dimensions [26]. babič, et al. elucidated that the fractal dimensions are relative to the material’s surface characteristics and mechanical property [18,19,20]. however how to calculate the value of the fractal dimensions is a difficult problem, and mathematicians and engineers will be captivated by an effective and reliable measurement. there are many methods to calculate fractal dimensions, among which the hausdorff dimensions are the most used one, its definition is [24, 26] ln ln n r  = (3) where n is new measured units when we measure the fractal pattern using a reduced 1/r scale. taking the cantor set as an example, when we use a reduced 1/3 scale, we find two new units, n=2, so α=ln2/ln3. but for a porous medium, we might know only its porosity, then how to calculate its fractal dimensions? we consider a sierpiński-like porous area as shown in fig. 1. when the porosity is zero, the area is two dimensional; while when the porosity equals to one, the area fractal dimensions of a porous concrete and its effect on the concrete's strength 139 is zero dimensional, so it is obvious that 0<α<2. the sierpinski carpet is a pure mathematical concept, the first cascade is similar to fig. 1, however, each unit can continue iteration to form a hausdorff dimensions of α=ln8/ln3 when c=l/3. feng, et al. [27] suggested the following formulation to calculate its fractal dimensions 2 2ln( ) ln l c l  − = (4) where l2 and c2 are the areas of the measured unit and the porosity, respectively. fig. 1 a sierpiński-like porous area according to the definition of eq. (4), we have 2 2 0 0 ln( ) lim lim 2 lnc c l c l  → → − = = (5) and 2 2ln( ) lim lim lnc l c l l c l  → → − = → − (6) though eq. (5) meets the continuum assumption for a continuous medium, eq. (6) is not physically inconsistent. kong [28] suggested the following modified one 2 2ln( / 1) ln( / ) l c l c  − = (7) it is interesting to note that most natural materials have fractal dimensions closed to 1.618, the golden mean [29,30]. the fractal dimensions are also the key factor affecting a porous concrete’s properties, rieu and sposito suggested the following formulation [31]: pore3min max 1 ( ) r r   − = − (8) 140 c.h. he, c. liu where φ is the porosity, rmin and rmax are the largest and smallest pore radiuses, respectively, αpore is the fractal dimensions of the pore space. yu [32] pointed out that eq. (8) is physically inconsistent, because for a continuum medium, we have αpore =0 and φ=0; on the other hand, for the full porosity, αpore =3 and φ=1. for the both cases, eq. (8) gives wrong results. yu suggested that following one [32]: 3min max ( ) pore r c r   − = (9) for a porous concrete, c=1, and eq. (9) becomes 3min max ( ) pore r r   − = (10) eq. (10) is physically consistent and mathematically reliable. but in practical applications, we have difficulty in determining rmin and rmax, and its fractal dimensions can not be calculated through the porosity. in this paper we adopt the two-scale fractal dimensions [33-35], which uses two scales, l and c, to measure the area for fig. 1. when we use the scale of l, any pores with sizes less than l are ignored, so the area is l2 with a two-dimensional property; when we measure it using a scale of c, its area becomes l2-c2. according to the definition of the two-scale fractal dimensions [33-35], we have 2 2 2 2 l l c = − (11) or 2 2 2 2( )l c l  − = (12) for a porous concrete, when using a large scale, we can obtain its volume, v, with a three-dimensional property; while if we use a micro scale, the porous structure can be found, and the two-scale fractal dimensions for the concrete can be calculated through the following relationship [33]: pore 3 v v v − = (13) where vpore is the total volume for pores. the porosity can be expressed as porev v  = (14) so the two-scale fractal dimensions of the porous concrete can be calculated as 3(1 ) = − (15) fractal dimensions of a porous concrete and its effect on the concrete's strength 141 the fractal dimensions for the porous space in the concrete can be expressed as 3 3pore  = − = (16) the two-scale theory has become an effective tool to various discontinuous problems [36-40]. 3. concrete’s strength vs. fractal dimensions concrete is a porous material, and the porosity and the pore size significantly affect the concrete's properties, especially its strength. there are many empirical formulae to express the relationship between the strength and its pore structure. the most famous one is [41, 42, 43] 0 (1 )mf f = − (17) where f is the concrete's strength, f0 is the strength of its continuum partner with zero porosity, m is an empiric constant, φ is the porosity. eq. (17) reflects only the effect of porosity, and the pore size is not considered. kumar and bhattacharjee suggested the following one [43] 0 1/2 (1 ) f f k r − = (18) where k is a constant, r is the pores' average radius. in eq. (17), the physical understanding of the parameter m lacks, and there is no practical criterion for choosing its value. though eq. (18) considers the effects of porosity and pores' size, the parameter k also lacks its physical meaning. the main problem of eq. (18) is that the parameter k is dimension-related and physically irrelative. we re-write eq. (18) in the form 1/2 0 (1 )f k f r − = (19) when the porosity φ tends to zero, we have r=0, the concrete becomes a continuum medium and eq. (19) implies that f/f0 becomes infinitely large instead of f/f0=1, so this is physically irrelative. furthermore the left side of eq. (19) is dimensionless, so the dimension of k has to be m1/2. the value of k is different if r uses different dimensions, e.g., nanometer or micrometer. in order to resolve this apparent contradiction, the dimensionless analysis [44,45] can be powerfully applied, which is the central dogma of complex problems. using the dimensionless analysis, estrada-diaz, et al. [44] found a useful mathematical formulation for electrospinning, he, et al. [45] established a bond stress-slip model for 3-d printed concretes, and kong [28] found a totally new friction law for porous fabrics. according to the dimensionless analysis, eq. (15) can be modified as 0 0 (1 ) ( )a b dimensionless m rf k f r = − (20) 142 c.h. he, c. liu where kdimensionless is a material constant, a and b are constants to be determined later, rm is the average radius, r0 is the porosity size of a reference pore, it can be the minimal/maximal porosity size. eq. (20) is a dimensionless formula, so it is physically relative and mathematically reliable. in order to determine the value of a, we write down the concrete's strength in the form f a= (21) where  is the stress, a is the contacted section area. considering the porosity, the whole volume can be calculated as 0 (1 )v v = − (22) where v0 is the sample's total volume. the section area scales approximately with 2/3a v (23) so we have 2/3 2/3 2/3 0 (1 )dimensionlessa kv k v = = − (24) according to the above relationships, we obtain 2/3 2/3 2/3 0 0(1 ) (1 )dimensionless dimensionlessf k v k f  = − = − (25) where f0 is the strength with zero porosity. if we have the section's area porosity, p, the actual area can be written as 0= (1 )a a p− (26) where a0 is the section area when p=0. eq. (21) becomes 0 (1 )f a p= − (27) according to eq. (23), the area porosity and the volume porosity have the following approximate relationship 2/3(1 ) (1 )p −  − (28) after a simple calculation, we have 2/3 0 0(1 ) (1 )dimensionless dimensionlessf k a p k f = − = − (29) which is the same as eq. (25). according to the above analysis, the value of m in eq. (17) and a in eq. (20) should be approximately 2/3. eq. (17) should be corrected as 2/3 0 (1 )dimensionlessf k f = − (30) fractal dimensions of a porous concrete and its effect on the concrete's strength 143 in ref. [42], m=8.15, this large deviation is due to the ignorance of the effect of pores' size. when the pores' size tends to micro/nano scales, the size effect [46] becomes enormous. now we correct eq. (20) as 2/3 0 0 (1 ) ( )b dimensionless m rf k f r = − (31) to understand the parameter b, we explain the size effect [46] through the geometric potential theory [47]. when the pores' size tends to nano/micro scales, high surface energy (geometric potential) [47] can be produced. the geometric potential theory assumes that any surface produces a boundary-induced force, and it can explain many complex phenomena, for examples, fangzhu's absorption of water molecules from the air [48,49], the nanofiber's wetting [47], and the cell orientation [50]. in order to use the geometric potential theory [47], we modify eq. (31) in the form 2/3 0 0 (1 ) ( )b particle m rf k f r = − (32) where kparticle is a geometrical parameter, rm is the average radius of the particles in the concrete, r0 is the reference size. the geometric potential of particles can produce a surface force [47]: m d f dr  = − (33) where π is the geometric potential produced by the particles. generally, it can be expressed as [47] 1 ( )mr    (34) where β is the geometrical parameter. for a sphere like the sun, β=1, which leads to newton's gravity. the concrete's strength due to nano/micro particles can be expressed as ( ) particle b m k f r  (35) where b=β+1. generally b=1/2 for the qualitative analysis as that in hall-petch effect [51]. 2/3 1/20 0 (1 ) ( )particle m rf k f r = − (36) generally we have rm scales with rm, so eq. (36) can be expressed as model i: 2/3 1/20 0 (1 ) ( )dimensionless m rf k f r = − (37) 144 c.h. he, c. liu using the experimental data given in ref. [43], kdimensionless and r0 in eq. (37) can be identified, and finally for the studied concrete of ref. [43], we have 2/3 1/2 0 1 52. ( 3. 9 1 ) ( ) 64 m f f r = − (38) fig. 2 shows the comparison between the theoretical prediction of eq. (38) and experimental data given in ref. [43], and a relative agreement is seen. the deviation arises in various factors, the main factor is the pore size distribution because, in our theory analysis, only average pore size is considered. fig. 2 comparison between the theoretical prediction of eq. (38) and experimental data given in ref. [43] as discussed above, b=1/2 is only used for the qualitative analysis. to understand the parameter b, the fractal theory has to be adopted. as the concrete's strength is reflected by the contacted area, in a fractal space, the area and the volume have the following scaling relationship: ( 1)/a v  − (39) when α=3, we have the well-known 2/3 scaling law: 2/3a v (40) for a 4-dimensional space, we have 3/4a v (41) this 3/4 scaling law plays an important role in life science [52]. in a fractal space, eq. (39) holds exactly, and eq. (23) is approximate one, so eq. (37) can be further improved as model ii: fractal dimensions of a porous concrete and its effect on the concrete's strength 145 ( 1)/ 1/20 0 (1 ) ( ) m rf k f r     −= − (42) where kα is a geometric parameter, α=3(1-φ). using the experimental data given in ref. [43] to determine kα and r0 in eq. (42), we have ( 1)/ 1/2 0 19.47 44.0(1 ) ( ) m f f r   −= − (43) fig. 3 shows the comparison between the theoretical prediction of eq. (43) and experimental data given in ref. [43]. fig. 3 comparison between the theoretical prediction of eq. (43) and experimental data given in ref. [43]. we write the concrete's strength in the form 2/3 0 0 1 1 ( ) (1 ) ( )c d c c d particle particle m m f k v k f v r r  −= = − (44) where kparticle is a geometrical constant, c and d are constants. according to the dimensionless analysis, the following equation should be satisfied: 2 3( ) 0 3 c d− − = (45) 146 c.h. he, c. liu so we have model iii: 2/3 (3 2) 0 0 1 (1 ) ( ) , 2 / 3c c c particle m f k v c f r − −= −  (46) and 2/3 (3 2) 0 0 1 (1 ) ( ) , 2 / 3c c c pore m f k v c f r − −= −  (47) eq. (46) or eq. (47) reveals that the concrete's strength depends also upon its volume. using the experimental data given in ref. [43], we have approximately the following formulation: 40.798 0 0.391 136.2(1 ) ( ) m f f r = − (48) fig. 4 shows the comparison between the theoretical prediction of eq. (48) and experimental data given in ref. [43]. fig. 4 comparison between the theoretical prediction of eq. (47) and experimental data given in ref. [43]. fractal dimensions of a porous concrete and its effect on the concrete's strength 147 table 1 comparison of our two models with experimental data [43] porosity (%) average pore radius (nm) strength (mpa) k-b model[43] (mpa) model i (mpa) model ii (mpa) model iii (mpa) 12.96 34.3 18.3 30.3 30.4 30.4 30.1 11.93 38.7 28.4 28.9 28.9 28.8 29.0 10.87 58.7 26.8 23.7 23.6 23.6 24.8 11.1 41.3 22.7 28.2 28.1 28.1 28.4 13.53 42.3 21.5 27.1 27.3 27.3 27.6 12.75 26.6 27.5 34.5 34.6 34.6 33.4 10.8 39.3 29.7 29.0 28.9 28.8 29.1 10.83 52.9 26.8 25.0 24.9 24.8 25.9 11.8 45.8 30.3 26.6 26.6 26.5 27.1 11.22 31.2 35.3 32.4 32.3 32.3 31.7 11.5 30.4 40.3 32.7 32.7 32.6 32.0 9.26 28.1 43.2 34.9 34.6 34.4 33.7 10.38 41.9 38.7 28.2 28.1 28.0 28.5 16.55 34.2 28.3 29.1 29.6 29.8 29.1 9.5 23 42.5 38.5 38.1 38.0 36.4 9.63 30.3 39.3 33.5 33.2 33.1 32.6 33.7 146.9 14.2 11.2 12.3 13.1 13.6 33.14 126.7 16.4 12.1 13.3 14.1 14.5 11.22 41.6 15.5 28.1 28.0 27.9 28.3 12.04 35.4 24 30.2 30.2 30.1 30.0 11.39 71.3 23.2 21.4 21.4 21.3 22.9 12.23 31.2 14.9 32.1 32.1 32.1 31.5 15.37 49.6 13.6 24.5 24.8 24.9 25.4 12.01 30.5 23.7 32.5 32.5 32.5 31.8 10.38 47.5 25.7 26.5 26.4 26.3 27.1 10.4 68.3 23.9 22.1 22.0 21.9 23.5 11.3 43 30.7 27.6 27.5 27.5 27.9 13.55 45 33.8 26.3 26.4 26.5 26.9 11.85 29.3 37.7 33.2 33.2 33.2 32.4 9.9 36.9 35.4 30.3 30.0 29.9 30.1 9.92 43.6 28.8 27.8 27.6 27.5 28.1 13.31 36.9 24.2 29.1 29.3 29.3 29.1 9.28 35 36.2 31.3 31.0 30.9 30.9 9.54 35.9 36.3 30.8 30.5 30.4 30.5 33.6 122.1 17.7 12.3 13.5 14.3 14.7 31.7 109.7 19.6 13.3 14.5 15.3 15.7 fig. 4 and table 1 show the deviation of eq. (48) becomes much less than those of eq. (38) and eq. (43), showing the reliability of our theoretical model given in eq. (47). 4. discussion and conclusions if data for the porosity size is segmented, the following one can be considered: 2/3 0 0 1 (1 ) ( ) i n i r i i im r f k f a r  = = −  , 1 1 n i i a = = (49) 148 c.h. he, c. liu where rim is the i-th segment’s average radius, ri0 is the i-th segment’s reference radius, which can be the segment’s largest radius, ai is the weighting factor. if the pores are distributed continuously, then we have max min 2/3 0 0 1 ( ) (1 ) n r nr r f kf dr r  + = −  (50) this paper suggests some conformable formulations for estimating the strength of a porous concrete, making it applicable to various cases, and shedding new light on the optimal design of the porous concrete with a given strength. in our theory, particles’ size distribution might be useful for practical applications. though our mathematical dimensionless model is mathematically correct and physically relevant, experimental verification is very much needed in future, and the fractal-fractional 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walled frc structures – a numerical study  udc 539.4 tobias rademacher, manfred zehn berlin institute of technology (tu berlin), department of structural and computational mechanics, germany abstract. this paper presents a novel method for detecting locations of damages in thin walled structural components made of fiber reinforced composites (frc). therefore, the change of harmonic distortion, which is found by current research to be very sensitive to delamination, under resonant excitation will be derived from femsimulation. based on the linear modal description of the undamaged structure and the damage-induced nonlinearities represented by a nonlinear measure, two spatial damage indexes have been formulated. the main advantage of this novel approach is that the information about the defect is represented mainly by changes in the modal harmonic distortion (mhd), which just needs to be measured in one (or few) structural points. the spatial resolution is given by the pairwise coupling of the mhd with the corresponding mode shapes. key words: structural health monitoring, composites, damage location, nonlinear acoustics, harmonic distortion, nonlinear vibration 1. introduction due to their high stiffness and strength combined with design versatility, frcs have massively gained in importance in recent years. however, as a consequence of their heterogeneous structure, the thin-walled structural components made of frc are particularly susceptible to damages in the laminate, which are in most cases characterized by separation of the laminate layers (delamination). these local delaminations can already arise in the manufacturing process. furthermore, they can be a consequence of a low and moderate energy input, such as bird strikes on aircraft or carelessness in the maintenance. received february 1, 2016 / accepted march 16, 2016 corresponding author: tobias rademacher tu berlin, fachgebiet für strukturmechanik und –berechnung, str. des 17. juni 135, 10623 berlin, germany e-mail: tobias.kaempf@tu-berlin.de 22 t. rademacher, m. zehn the particularly critical aspect of this type of damage is the fact that it is visually difficult to detect, spreads as a result of operating loads, and can give rise to total failure. therefore, for safety-related applications, it is of crucial importance to be able to detect, locate and assess the defects and damages caused by imperfections in production and/or that appears in operation. the ultrasound and x-ray based inspections are conservative and highly accurate methods. however, they are also quite demanding in terms of the equipment, time and accessibility of the examined structural components. therefore, especially for larger structures, the methods that are relatively easy to implement are needed in order to enable a coarse localization of potential damages. in this context, a great number of methods for structural health monitoring (shm) have been published in the past decades (e.g. reviews in [1-3]). many of the developed methods are based on a damage-related alteration in the dynamic structural behavior [4, 5]. these methods have also been applied to the structures made of fiber-reinforced composites [6] and extensively studied [7, 8]. 1.1. linear methods the idea that a local damage affects the global vibration response has led to a series of modal-based approaches to structural health monitoring [9]. most of these “classical” methods are based on the fact that the damage locally reduces the stiffness, thus affecting the natural frequencies [10] and mode shapes, including derived quantities (e.g. modal strain) [11]. a simple method of damage identification has been proposed by montalvão, ribeiro and duarte-silva [12]. they have extended the principle of damage indicators based on eigenmodes in this case defined as the mode-shape-curvatures by weighting the eigenmodes with a relative change in the modal damping, ηr, induced by the damage. they invented the damping-damage-indicator (dadii) which can be obtained by superposition of strain based modal shape functions weighted by δηr. montalvão, ribeiro and duarte-silva could prove that it is possible to locate a defect in the composite panel made of carbon fibre prepreg by using the damping-damage-indicator. however, experience has shown that especially the modal damping coefficients in frc structures strongly depend on temperature. additionally, the suitability of modal damping as a damage indicator is limited by the diversity of physical causes of damping and the interaction between the structure and the environment, which influences the damping in a way that is hard to forecast and/or control [13]. 1.2. nonlinear methods methods of nonlinear analysis of the dynamic system response open up many possibilities for structural health monitoring. however, they are often strongly related to a specific problem [14]. the analysis of higher harmonic components of the vibration response in the frequency domain provides a relatively practical approach to damage detection and identification, which is sometimes superior compared to the classical linear methods. considering a damaged beam, prime and shevitz [15] demonstrated that the higher harmonic components in the spectrum and the associated mode shapes are much more sensitive to damages than the modal parameters. modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 23 considering thin-walled structural components made of frc, delamination is the most frequent type of damage. therefore, in the context of structural health monitoring, the research is focused on its impact on the dynamic behavior [16]. experimental investigations carried out on a delaminated beam show that the alternating opening and closing of a delamination gap (delamination breathing) particularly changes the vibrational response compared to the undamaged case [17-19]. the beating normal contact between the gap's opposite surfaces and the resulting direction-dependent stiffness leads to a non-linear structural behavior as a consequence of an abrupt state change. hence, when such a system is exposed to a harmonic excitation, the response spectrum is characterized by the occurrence of higher harmonics, obtained as integer multiples of the excitation frequencies [20, 21]. these nonlinearities can be quantified, for instance, in an integral form by means of the clapping factor [22]. 2. the idea of nonlinear damage index the new method presented in this paper aims at the coarse localization of damages within thin-walled structural frc components made of fiber-reinforced composites in order to get an overview for more detailed investigations. for this purpose, usually weak non-linear effects in the vibration response, which are the consequence of the damage, should be quantified by means of a suitable quantity λr. in the present work this will be a value derived from harmonic distortion. the basic idea is that the structure is exposed to a resonant vibration excitation. it is also necessary that, despite of the damage, the original modal representation can still sufficiently describe the overall dynamic behavior of the structure, which implies that the damage induced changes in eigenmodes and eigenfrequencies are still negligible. the validity of this assumption has been confirmed in the literature (for example in [18, 23, 24]) in the case of slighter damages of the laminates. hence, a damage-induced nonlinear measure λr is assigned to the r th mode (see fig. 1). fig. 1 coupling of nonlinear-measure, modal strain and damage using a suitable combination of linear modal description of the original structure and nonlinear measure λr associated with damage-induced changes, a spatial damage index, nonlinear damage index nldii, could be formulated in a similar way as was done by montalvão, ribeiro and duarte silva [12]: . max : ~ here where, ~ ~ : :,:, ,, , 1 , 1 , iii iiiii irn r ir n r irr inldi              (1) 24 t. rademacher, m. zehn fig. 2 flow chart of the suggested damage localization approach the necessary steps of the proposed method are depicted in fig. 2. a prerequisite for successful damage localization is a valid linear fe model of the structure not including damages. this enables the determination of optimal measurement and excitation points for the experimental determination of λr. on the other hand, each eigenmode or a derived quantity, e.g., the curvature-dependent shape function used here as defined in eq. (1), can be represented with a high spatial resolution. nonlinear measures λr associated to the modes, are obtained by means of comparative measurements carried out on the original and potentially damaged structure. change δλr, is supposed to serve not only as a damage indicator, but also for the localization by means of proposed damage index nldi. the formulation given in eq. (1) takes advantage of an increased distortion when the referred mode shape is triggering the local damage induced nonlinear mechanism. likewise, the linearity of each mode can be used to define a reciprocal nonlinear damage index: . ~ ~1 : 1 1 , 1 ,                      n r ir n r ir r irnldi  (2) 3. numerical proof of the concept this section describes how the nonlinear response is obtained for test purposes by a numerical simulation of an experimental set up. all calculations stated below have been done using matlab. modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 25 3.1. finite element model of a thin fiber reinforced composite plate for the numerical proof of the concept a rectangular glass fiber-reinforced polymer plate (gfrp) is modeled using the finite element method. for the sake of convenience it is assumed that the balanced and symmetric cross ply (0°/90°) laminate is used. thus, by applying the classical laminate theory (see [25]) the membrane and bending problem is decoupled. because of being easily stimulated and measured in the experiment, only bending modes will be considered here. using the first order shear deformation theory the displacement field can be written similarly to the reissner-mindlin plate model as: .]),(),(),([ ][),,( t xy t t yxwyxzyxz w y w z x w zwvuzyx              u (3) by applying derivation to the displacements the strain field becomes: t yxxyt xyyyxx yxyx                   ][κ (4) and ,][ t xy t yzxz y w x w              γ (5) where the vectors for curvatures κ, resulting from bending moments m, and transverse shear strain γ, due to shear forces fs, can be written separately. the equation of motion can be derived by using hamilton’s principle:   2 1 1 2( ) ( ) 0 ( ) ( ) 0 . 2 1 t t ' i i i i i t t δ t p , p π p dt δwdt with δp t δp t i       (6) the generalized coordinates identified as independent “displacements” are: .][][ 321 yxwppp p (7) including the assumptions above, kinetic energy t follows to: 3 2 2 2 21 1 1 ( ) 2 2 2 12 x y v a a t t ρu dv tρw da ρ da      (8) applying the classical laminate theory (see [25]) and calculating the shear correction factors [26], strain energy π can be integrated over the thickness: 1 1 ( ) d ( ) d 2 2 clt tt t t s v a π v a    m κ f γ κ dκ γ sγ (9) 26 t. rademacher, m. zehn where with the above given assumptions the laminate bending and shear stiffness matrices have the form: . 0 0 0 0 and . 0 0 11 11 here 22 11 66 22 1211                         s s s s dsym d dd sd (10) virtual work δ’w of non-conservative and external loads acting on plates surface a and its edges s becomes: .ddd' savw sav   pfpppr  (11) for discretization of plate surface a a planar four-node (ne=4) quadrilateral element with bilinear shape functions ni is used. thus, element solution u e of the displacement model for the plate is expressed as: . w 00 00 00 11                        ne i yi xi i i i ine i ii ee n n n  qnnqu  (12) with substituting the displacement fields in eq. (6) with these interpolation functions for one element domain the dynamical equilibrium reads: ,0ddd ~ ee            tδaa e t t e int e ext e s t sb a t b e a 2 1 ee qffqbsbbdbqcqnmn t  (13) where dissipative loads are assumed to be proportional to the nodes velocity. because times t1 and t2 and virtual node displacements δq e can be chosen arbitrarily this finally yields to: ,dd ~ t e int e ext e s t sb a b eee a aa ee ffqbsbbdbqcqnmn t    (14) ( ) ,e e e e e e e e e b s ext int    m q c q k k q f f (15) where consistent element mass matrix m e follows from numerical integration of the shape functions with the diagonal kronecker product .]1[: ~ 2 12 12 12 1 44 ttt  im  (16) the bending component of the element stiffness matrix is set up by using straindisplacement matrix bb according to modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 27 . 0 00 00 11                                          ne i yi xi i ii i i ne i ibi e b e w y n x n y n x n  qbqbκ (17) in order to avoid shear locking and hourglass modes (which might cause instabilities in time integration), the shear component of the element stiffness matrix is calculated by using the assumed natural strain approach (see [27]). therefore, the out-of-plane shear strains at edge centers are firstly calculated. subsequently, these values are used to interpolate shear strain field γ: .e e sγ b q (18) due to the integral form of eq. (6) the dynamical equilibrium for the whole plate domain follows with eq. (15): ext ne e e int e ext ne e ee b eeee fkqqcqmffqkqcqm     11 (19) which can be integrated in time for a given external load vector fext and the initial conditions for node displacements q and velocities q (see section 3.3). in the present work an unbounded plate with the characteristics as shown in table 1 was discretized using a regular quadrilateral mesh with 12x24 or 14x28 elements. both configurations show a maximum error for the desired eigenfrequencies of 1% with respect to abaqus model (24x48 quadratic elements type s8r). in order to suppress rigid body motion and to achieve reasonable damping for the elastic modes a viscoelastic foundation is set up. so the frequencies of significantly damped rigid body modes are smaller than 1% of the first elastic mode and the modal damping of the elastic modes become approximately 0.1%, which is in good agreement with experimental studies. table 1 characteristics of the glass fiber-reinforced polymer plate length a width b thickness t density ρ d11 d22 d12 d66 s11= s22 0.24 m 0.48 m 3.5 mm 2100 kg/m3 71.5 nm 98.6 nm 8.6 nm 13.3 nm 99.7 mnm-1 3.2. modeling the time dependent behavior of delamination despite a great research effort, it is still not very well understood how the physical mechanisms appearing in delaminated structures affect the nonlinearities in dynamic response. for the wavelength range investigated in this paper, the phenomena on macroscopic scale [28], namely the clapping mechanism (see [29, 20]) and bilinear stiffness due to opening and closing the delamination [30], will affect the vibrational response mostly. modeling the former would require a very fine spatial and time resolution of the delaminated area to represent the governing contact mechanisms. taking 28 t. rademacher, m. zehn into account that for the method proposed in this paper many time integration steps (for every mode) over a quite long period of time (essential to obtain a sufficient frequency resolution) need to be calculated, this phenomenon is therefore not feasible here. whereas the state-dependent impedance mismatch, caused by a slight local stiffness reduction in the case of an open-state delamination, can be easily modeled with a lumped parameter model [31, 32]. in this paper the local curvature at the damage position is assumed to determine whether the delamination is open or closed and, accordingly, the local stiffness is reduced or not. therefore, one characteristic value  , representing an overall curvature based on the local values given in eq. (17), has to be defined. therefore, the major curvature calculated by solving the eigenvalue problem: 0det !           yyxy xyxx (20) 2 2 / / 1 1 ( ) ( ) . 2 2 i ii i ii xx yy xx yy xy                  (21) the large magnitude is supposed to govern the delamination state (see fig. 3). hence, characteristic curvature value  is defined by: . openondelaminati0 closedondelaminati0 :          iii (22) fig. 3 major curvatures all element stiffness matrices k e* corresponding to delaminated area are assumed to undergo a curvature-dependent degradation like: 2 1* 2 0 for 0 (1 ( )) where ( ) ˆ 1 else. e r r r e                          k k (23) where reasonable values for maximum degradation r  range between 0 (no reduction) and 1 (no remaining element stiffness). as shown in fig. 4 the course of degradation is a gaussian smoothed step function where the smoothness is controlled by value ̂ . this modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 29 smoothed shape is chosen instead of a sharp heavy side function for two reasons: on the one hand, delamination with its natively non-regular topology will not act upon a strict all-or-nothing law. on the other hand, the numerical time integration becomes less critical for discretization errors near the reversal point. this can finally improve the (physical) stability of the integration process. fig. 4 curvature dependent stiffness reduction r (left) and its time response 3.3. time integration to achieve the slightly nonlinear vibrational response of the damaged plate, the discretized equation of motion given in eq. (15) has to be integrated in time until a steady state (or limit cycle) is reached. to avoid numerical overhead caused by transient oscillation with stimulated eigenfrequency fr the linear steady state solution is set up as the initial condition. to obtain a steady solution of the distorted response 32 periods of main frequency (tpre=32/fr) are processed until the time span (tana=32/fr) used for spectral analysis starts. to ensure a sufficient resolution of the degradation process one period is sampled with 128 time steps. hence the sampling data for each mode r follows as: 1 1 1 1 (32 32) . 128 total pre ana s r r δt and t t t f f f       (24) to provide comparability between the modal triggered nonlinearities the excitation level has to be equalized for all the modes with respect to the related mode shape derivate. here the curvature is assumed to be the governing characteristic related to generation of nonlinearity. hence, each modal excitation force fext,r has been tuned in such a manner that for every mode the same rms-value of the characteristic curvature is achieved: ..ˆwhere)2cos(ˆ 2 ,,, rconsttf rrextrrextrext   fff (25) the time integration is carried out by using the implicit hht-α method (see [33]). in this generalization of the newmark-β method, the equations of motion are modified, using a parameter α, which causes a numerical lag in internal and external forces. this method is chosen for two reasons. on the one hand, using the correct parameters it becomes at least second-order accurate and unconditionally stable. on the other hand, in this way it tends to dampen-out the response at high frequencies which is desirable for physical stability. 30 t. rademacher, m. zehn 3.4. extracting the nonlinear measure from time series as described above the nonlinear measure is extracted from a spectrum of a steady state response in one or only a few testing points. therefore, the last 32 periods of the corresponding velocity signals are analyzed using the fast fourier transform. in order to minimize the amplitude error the flat-top window hft116d as suggested in [34] is used to process linear power spectra lps. fig. 5 typical whole time response of nodes velocity (left) including a detailed view. 64 periods of third mode (330.3 hz) are shown, where the last 32 are used to calculate the linear power spectrum (right). the higher harmonics are marked as shown in fig. 5 the peaks of higher harmonics (fhn=nfr with n = 2, 3, …) of (linear) frequency fr can be clearly determined. starting with the 7 th harmonic a significant alternating behavior – low amplitudes for odd and high for even orders – can be seen. this is the reversal to the pattern which can be observed analyzing a sdof-system having a bilinear stiffness (see [28]) and might be caused by the more complex modeling of damaged changed stiffness. in the present work nonlinear measure λr , which characterizes the nonlinearity, is estimated by the sum of the first four peaks of higher harmonics: 5 2 1 1 ( )[ ] and m i i r i r r r n i λ : lps w nf λ λ m     (26) if more than one measurement point is used, then the overall nonlinear measure is simply averaged mode wise. of course, a more complex and maybe more significant estimation rule could be deduced, e.g., discrimination between even and odd higher harmonics. however, as stated below, even this simple way leads to very promising results. modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 31 4. results in the present work the suggested approach for damage localization has been tested by a numerical simulation of experiments using the assumptions made in section 3. the first 42 elastic modes of the undamaged structure, ranging between 135.6 hz and 6272 hz, are stimulated and analyzed as described above. 4.1. damage index for rectangular frc-plate with a single damaged area a typical distribution of the plain (eq. (1)) and the reciprocal (eq. (2)) nonlinear damage indexes is displayed in fig. 6. it is obvious that both of them show a symmetric pattern, which is a consequence of the fact that underlying shape functions (see eq. (1)) are symmetric as well. however, despite this, both the indexes can sufficiently localize the defective area with the use of only one measurement point, where nonlinear measure λr is determined. this result varies only slightly when this measurement point is changed or if more points are taken into account. this is a major advantage of the proposed method: only a few measurement points have to be equipped with sensors while the structure is under test. while using the plain formulation of the damage index the shape functions seem to be “showing through” and some local maxima become apparent, but the reciprocal form gives a very clear result. this difference can be observed for most of the studied damage positions. but in some cases rnldi tends to highlight whole lines or columns of elements (compare to fig. 9). this might be caused by the strong repetitive characteristics of the mostly purely sine-like eigenforms of the plate under investigation. fig. 6 linear nldi (left) and reciprocal rnldi (right) nonlinear damage index. excitation at (0, 0) (green) and the nonlinear measures are taken at (0.1 m, 0.07 m) (blue). only one element (0.05 m, 0.12 m) is damaged (red) 32 t. rademacher, m. zehn 4.2. influence of position and characteristics of the damaged area it can be seen in fig. 7 that the value of modal nonlinear measure λr scales very well with the squared maximum degradation. in case of heavy stiffness reductions (25% and higher), significantly distorted modes tend to slightly lower values for scaled λr. however, the influence of the maximum degradation to relative pattern of λr is negligible. hence, it does not affect the validity of the damage index. but it has to be noted that the observed quadratic dependency of λr with respect to degradation might cause resolution problems in experimental studies. fig. 7 scaled nonlinear measures 2r/λr  for different values of maximum degradation r  . excitation at (0, 0). nonlinear measures are taken at (0.02 m, 0.04 m). only one element is damaged to ensure that a local damage can be found for any location, a huge amount of complete simulation cycles have been carried out. for every single element of the lower left quadrant of the plate (x in [0, 0.5a], y in [0, 0.5b]) damage is applied and the complete time integration procedure is performed. subsequently, the spatial damage indices nldi and rnldi have been calculated. in order to decide whether the damage could be uniquely identified, the damage index ratio is computed between the real damage position and the maximum index value of the plate excluding the immediate surroundings of the damage. the result can be seen in fig. 8. it can be observed that both the plain and the reciprocal nonlinear damage indexes succeed in the identical (excluding effects of symmetry) location of the damaged area in most cases. the distribution of values and, therefore, the few damage positions in which no identical localization was possible seems to be randomly distributed. however, using both proposed damage indices should ensure a coarse localization of damage for the engineering praxis as a preliminary check for more elaborate local investigations. modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 33 fig. 8 uniqueness of the nonlinear damage index and its reciprocal representation for the lower (left) quadrant. damage positions were the damage index could not uniquely identified (values<1) are marked with a cross. excitation at (0, 0). nonlinear measures are taken at (0.02 m, 0.04 m) an increase of the spatial spread of the delamination is implemented by involving more damaged elements. it is found that with an increased damaged area the values of the modal nonlinear measure rise as well although the significant differences between them which can be observed for small damages (see fig. 7) seem to vanish. hence, as shown in fig. 9, the significance of the damage index is reduced. while for one damaged element fig. 9 reciprocal damage index for different spatial extends of damage. damaged elements (degradation r = 12.5%) are highlighted red. excitation at (0, 0) (green). nonlinear measures λr are taken at (0.08 m, 0.08 m) (blue) 34 t. rademacher, m. zehn rnldi, regarding effects of symmetry, matches the damaged position, the affected area in the case of nine damaged elements is overestimated. however, the largest value is still located in the damaged area. in case of 13 damaged elements the supposed method fails completely – which can be slightly improved by using more points to estimate the nonlinear measure. this behavior is inherent for this approach. the larger the damaged area becomes, the more mode shapes are triggering the nonlinearities. so the weights for the superposed shape functions get equalized. 4.3. damage index for a single damaged asymmetric frc-structure finally, the proposed method is tested for asymmetric structures, which might be more relevant for real world applications. even symmetric structures can be easily misbalanced by a slight change in the boundary conditions or by attaching (known) masses while the device is under test. a simple asymmetric geometry, based on the rectangular plate as described above, can be seen in fig. 10. it appears that both the indices mark the defective element with the highest value. while the linear damage index shows some areas having falsely high values in the lower part, the reciprocal nonlinear damage index fits perfectly. once more it has to be emphasized that the nonlinear measure is the only value related to the damaged structure and this is achieved with only one (!) measurement point. fig. 10 linear and reciprocal damage index for asymmetric flat geometry. damaged elements (degradation r  =12.5%) are highlighted red. excitation at (0, 0) (green). nonlinear measure λ is taken at (0.19 m, 0.3 m) (blue) modal triggerd nonlinearities for damage localization in thin walled frc structures – a numerical study 35 5. conclusions and future work the results presented here are showing that combining the linear modal description of a structure and slight nonlinearities triggered by resonant excitation is a suitable approach for damage localization. by simulating an experimental set up it is found that the two suggested formulations of a spatial nonlinear damage index are able to reliably identify a moderately large defective area. especially to be emphasized is the fact that the nonlinear measure, which is the only value associated with the damage, can be determined at any structural point. in the present paper the nonlinear damage indexes were processed in a very straightforward way – all first 42 bending modes were taken in account and the nonlinear measure is estimated as a sum of the first four higher harmonic amplitudes. in order to improve the validity of the damage localization more complex definitions, e.g., an appropriate selection of modes and/or discrimination 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https://doi.org/10.22190/fume180227021c © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper determination of the wall variables within the zonal model of radiation inside a pulverized coal-fired furnace udc 536.3, 621.6 nenad crnomarković, srđan belošević, stevan nemoda, ivan tomanović, aleksandar milićević university of belgrade, vinča institute of nuclear sciences, serbia abstract. determination of the wall variables (wall emissivities, wall temperatures, and heat fluxes) when the zonal model of radiation is used in numerical simulations of processes inside a pulverized coal-fired furnaces is described. two methods for determination of the wall variables, i.e., a repeated run of numerical simulation (rrns) and a temporary correction of the total exchange areas (tctea) are compared. investigation was carried out for three values of the flame total extinction coefficient and four values of the initial wall emissivities. differences of the wall variables were determined using the arithmetic means (ams) of the relative differences. the ams of the relative differences of the wall variables increased with an increase in the flame total extinction coefficient and changed a little with an increase in the initial values of the wall emissivities. for the selected furnace, the smallest differences of the wall variables were obtained for kt=0.3 m-1 and w,in=0.7. although both methods can be used for determination of the wall variables, the rrns method was recommended because the manipulation with files was easier for it. key words: zonal model, pulverized coal, boiler furnace, numerical simulation, wall variables received february 27, 2018 / accepted june 09, 2018 corresponding author: nenad crnomarković university of belgrade, vinča institute of nuclear sciences, mike petrovića alasa 12-14, serbia e-mail: ncrni@vin.bg.ac.rs 220 n. crnomarković, s. belošević, s. nemoda, i. tomanović, a. milićević 1. introduction pulverized-coal fired furnaces of the utility boilers are basically rectangular-shaped constructions, inside which complex processes of reactive turbulent two-phase flows with radiative heat exchange occur. values of the thermo-fluid variables, such as velocity components, temperature, component concentrations, and others, in every point of furnaces are determined by numerical simulations. except for thermo-fluid variables, the numerical simulations should reveal values of the wall variables, such as wall temperatures, wall emissivities, and heat fluxes. as the wall temperatures and wall emissivities are found on the basis of the heat fluxes, the wall variables determination is clearly a task of the radiation model. the objective of this investigation is determination of the wall variables when the zonal model is used as a radiation model in numerical simulations. in numerical investigation of processes inside pulverized coal-fired furnaces, several radiation models are used: discrete ordinate model [1], discrete transfer model [2], spherical harmonics model [3], six-flux model [4], monte carlo [5], and zonal model [6]. each of them except for the zonal model (and monte carlo, which is based on the same concept as the zonal one) easily takes into account changes of all wall variables during the calculation procedure of the numerical simulation. the zonal radiation model is characterized by a high level of accuracy, independent of radiative properties and other conditions of the radiative heat exchange calculation. the model is based on division of furnace space into volume zones and furnace walls into surface zones [7]. all zones are considered isothermal. radiative heat exchange is found in interaction of every zone with all zones. the total extinction coefficient and scattering albedo are assigned to all volume zones to determine the direct exchange areas (deas) for every pair of zones. for the gray furnace walls, net radiative heat exchange is found using the total exchange areas (teas). the calculation procedure for the teas requires the values of the wall emissivities for all surface zones. one set of the teas is used in the case of gray medium. to take into account the non-gray behavior of the gas phase, the concept of the weighted sum of gray gases model, by which a real gas is replaced by a mixture of gray gases [7], is used. parameters of that model, temperature dependent weighting coefficient and constant absorption coefficient of each gray gas, are used to calculate the directed flux areas, which are then used to find net radiative exchange of zones. in the classical application of the zonal model, radiative properties of the medium (total extinction coefficient and scattering albedo) and boundary walls (wall emissivity) are not changed during the calculation procedure. in that or similar way, the zonal method was used for numerical investigations of industrial furnaces [8-10], to find incident radiative fluxes on the freeboard walls of a bubbling fluidized bed [11], and for numerical investigations of a pulverized coal-fired furnace [6, 12]. nonclassical applications of the zonal model were described for nonhomogeneous radiatively participating medium [13-16]. these methods were developed for the black-walled systems [13, 14, 16], or for nonscattering medium [15], while the walls of pulverized coal-fired furnaces, inside which is a scattering medium, are gray. papers which describe determination of wall variables by numerical simulations when radiative heat exchange is solved by the zonal model are very rare. crnomarkovic et al. [17] described the method of the zonal model application, here called the repeated run of numerical simulation (rrns). the new method, here called the temporary correction of teas (tctea), is described in this paper. values of the wall variables obtained by rrns and tctea methods for various flame radiative properties and initial wall emissivities are determination of the wall variables within the zonal model of radiation inside a pulverized coal-fired furnace 221 determined and compared. the investigation is expected to provide the recommendation for one of the methods. the investigation was carried out for the pulverized coal-fired furnace of the 210 mw monoblock thermal unit, located in obrenovac, serbia. the furnace was fired by kolubara lignite. geometry of the furnace and coal properties were described [18]. in the following text, the mathematical model, the results of the investigation and conclusions are described. 2. mathematical model of the process and methods of the zonal model application mathematical model of the process inside the furnace describes a two-phase reacting flow with radiative heat exchange. model was described in detail in 6, 18. here, only the main characteristics are described. the gas phase is described by the time averaged differential equations of conservation of the momentum, enthalpy, the concentrations of the gas-phase component, particle concentrations, turbulent kinetic energy, and the rate of turbulent kinetic energy dissipation, in eulerian reference frame. the general form of the gas-phase equation is the following: ,pdiv( ) div( grad ) s s       u (1) where  is gas-phase density (kg m -3 ), u is gas-phase velocity (m s -1 ),  is the variable of the gas phase,  is the transport coefficient for variable  (kg m -1 s -1 ), s is the source term, and s,p is the source term due to the presence of particles. the set of equations was closed by an appropriate turbulence model 18, which connects the turbulent kinetic energy and rate of its dissipation. the differential equation for the pressure field is obtained from the combination of the continuity and momentum equations, by the simple algorithm. the flame temperatures are solved from the enthalpy equation for the condition of the thermal equilibrium between the gas and dispersed phases 6: f f f f ,r ,cdiv( ) div( grad( )) gradh h hp s sc t c t    u u (2) where cf is the specific heat capacity of flame (j kg -1 k -1 ), tf is flame temperature (k), h is enthalpy (j kg -1 ), p is pressure (n m -2 ), sh,r is the source term due to radiation (w m -3 ), and sh,c is the source term due to combustion (w m -3 ). the dispersed phase is described by the differential equations of motion and change of mass and energy in the lagrangian reference frame. motion of the particles is tracked along the trajectories with constant flow of particles. the particle velocity vector is the sum of the convective and the diffusion components. heterogeneous reactions of the coal combustion are modeled in the kinetic-diffusion regime, as described in 19. radiative heat exchange is solved by the zonal model of radiation. the heat flux of a surface zone is determined from the following relation: 4 4 4 f , f , w, w, 1 1 w, , 1, , m n m i m n i n i i i m n i i g s t s s t a t q i n a             (3) 222 n. crnomarković, s. belošević, s. nemoda, i. tomanović, a. milićević where tw is wall temperature (k), m ig s is volume-surface tea (m 2 ), n is s is surfacesurface tea (m 2 ), m is the total number of the volume zones, n is the total number of the surface zones,  is stefan-boltzmann constant (w k -4 m -2 ), w is wall emissivity (-), and a is the surface area (m 2 ). the sum of the first two terms in the numerator of formula (3) is the heat transfer rate of the gained energy whereas the third term is the heat transfer rate of the energy loss due to emission of radiation. the wall emissivity is a function of wall temperature: w w w( )t  . the furnace wall is a composite wall, consisting of a metal wall layer and an ash deposit layer. one boundary surface of the furnace wall, i.e. the metal wall boundary surface, is in contact with the flow of steam-water mixture and the other boundary surface, i.e. boundary surface of the ash deposit layer, is in contact with the flame. the wall temperature and the wall emissivity are the properties of the ash deposit layer boundary surface in contact with the flame. the ash deposit layer is treated as a gray emitter of radiation [20], although there is evidence that does not support such an assumption [21]. the thickness of the metal wall is 4.0 mm whereas the thickness of the ash deposit layer is chosen according to the objectives of the investigations. the temperature of metal wall boundary surface tm,sw (k) which is in contact with the steam-water mixture is determined on the basis of the convective heat transfer: w, m,sw sw iq t t h   (4) where h = 14.0 kw m -2 k -1 is the convection transfer coefficient 22 and tsw = 615.0 k is the temperature of the steam-water mixture. the temperature of metal wall boundary surface tm,a (k) which is in contact with the ash deposit layer is determined on the basis of the one-dimensional heat conduction: m m,a m,sw w, m i l t t q k   (5) where lm is metal wall thickness (m) and km is the thermal conductivity of metal wall (w m -1 k 1 ). the thermal conductivity of the metal wall depends on temperature m m m( )k k t , where mt (k) is the arithmetic mean (am) of metal wall boundary surface temperatures, m m,sw m,a( ) / 2t t t  . wall temperatures tw (k) are determined by formula (5) replacing tm,sw, mt , lm, and km, by tm,a, at , la, and ka, respectively. here, at is the am of boundary surface temperatures (k), la is thickness (m), and ka is effective thermal conductivity (w m -1 k -1 ), all of the ash deposit layer. dependences of ka and w on temperature are in the polynomial forms: 7 a a 0 1000 j j j t k a          (6a) 7 w w 0 1000 j j j t b          (6b) determination of the wall variables within the zonal model of radiation inside a pulverized coal-fired furnace 223 using the diagrams given in 20. coefficients aj and bj can be found in 23. dependence of the metal wall thermal conductivity on temperature is that of carbon steel 24, 25. thermophysical properties of the gas phase are determined from the equation of the state, tabulated values and empirical relations. the set of equations is solved by the finite difference method. discretization and linearization of the equations are achieved by the method of the control volumes and hybrid difference scheme. stability of the iterative procedure is provided by the under-relaxation method 26. fig. 1 flow charts of the methods: a) rrns, b) tctea the flow charts of the rrns and tctea methods are shown in fig. 1a,b. the rrns method starts with the determination of the teas using the initial wall emissivities and flame radiative properties. during the calculation procedure, wall temperatures (tm,sw, tm,a, and tw) are determined on the basis of the heat fluxes and thermophysical properties of the wall. when the numerical simulation is completed, new values of the wall emissivities are determined in accordance with the wall temperatures, formula (6b). the teas are determined again and the numerical simulation is repeated with all variables (except wall emissivities) starting from their initial values, which are the same as in the previous numerical simulation. the tctea method is similar to the rrns method. the main difference is in that the next run of the numerical simulation starts with the values of all variables obtained by the previous numerical simulation. 224 n. crnomarković, s. belošević, s. nemoda, i. tomanović, a. milićević the tctea method actually behaves as one numerical simulation. for the rrns method, every new numerical simulation is independent of the previous one and contains the impact of the surface zone emissivities on the results. the necessary condition for applying the methods is the convergence, which must be shown using the variable values obtained by the numerical simulation. 2. results and discussion radiative heat exchange was solved on the coarse numerical grid composed of the cubic volume zones of the edge dimension of 1.0 m. the surface zones are squares of the same edge dimension. the furnace was divided into 7956 volume zones and 2712 surface zones. flow field was solved on the fine numerical grid which was obtained by dividing every volume zone into 64 control volumes. the fine numerical grid contained 620 136 control volumes. agreement with experimental data and the grid independence study were already shown 6. deas of the close zones were determined using correlations given in 10. teas were calculated by the method of original emitters of radiation 7. improvements of the values of the teas were accomplished using the generalized lawson’s smoothing method 14. one numerical simulation consisted of 4000 iterations. the investigation was carried out for three values of the total extinction coefficient: (1) kt = 0.3 m -1 , (2) kt = 1.0 m -1 , and (3) kt = 2.0 m -1 , and for four values of the initial wall emissivities: 0.60, 0.70, 0.80, and 0.90. the scattering albedo was 0.5 for every value of kt. such values of the total extinction coefficient were selected because the previous investigation 6 conducted for the same furnace showed that the heat transfer rates of the absorbed radiation (with constant wall emissivities and temperatures) and heat fluxes were maximal and almost constant for the values of kt in the interval 0.2-2.0 m -1 and scattering albedo not bigger than 0.5. the thickness of the ash deposit layer of 0.6 mm was uniform for all walls. the wall variables were determined for surface zones. the analysis of the convergence of the methods included flame temperatures tf. the convergence of the methods was shown through the relative differences expressed by formula (7): 3 ,cn 3 100% nr         (7) where nr designates the number of the numerical simulation run,  designates the variable, and  designates the am of the variable: 1 n i i n     (8) in formula (8), n is the total number of surface zones that represent a solid wall (for w, tw, and qw) or total number of control volumes (for tf). for the determination of the convergence, the numerical simulations were run eight times for every set of conditions. the similar values of the variables were obtained for both methods. the results obtained for kt = 1.0 m -1 are shown in table 1. for nr  3, the changes of the ams become very small and although the oscillations appear, both methods provide convergent results. the determination of the wall variables within the zonal model of radiation inside a pulverized coal-fired furnace 225 results obtained for the fourth run (nr = 4) of the numerical simulation were used for determination of the wall variable differences. table 1 relative differences ,cn (%) obtained for kt = 1.0 m -1 , w,in = 0.80 nr w (-) tw (k) qw (kw m-2) tf (k) rrns tctea rrns tctea rrns tctea rrns tctea 1 0.26 0.26 0.34 0.25 1.80 1.54 0.21 0.35 2 0.0 0.0 0.003 0.002 0.009 0.22 0.003 0.22 4 0.0 0.0 0.002 0.009 0.12 0.44 0.003 0.003 5 0.0 0.0 0.004 0.001 0.23 0.007 0.001 0.10 6 0.0 0.0 0.0009 0.004 0.001 0.14 0.002 0.009 7 0.0 0.0 0.001 0.0002 0.004 0.0004 0.002 0.003 8 0.0 0.0 0.0009 0.009 0.002 0.36 0.004 0.14 in the following analyses, the changes and differences of the wall variables are determined through the ams of the relative differences of the wall variables and the relative differences of the ams of the heat fluxes, wq . as the surface zones are squares of the same edge dimension, the ams of the heat fluxes represent the heat transfer rates through the furnace walls. the changes of the wall variable values with the change of the flame total extinction coefficient were found using the relative differences, determined by comparison of the wall variables with the values obtained for total extinction coefficient kt = 0.3 m -1 : t t , 0.30, , , 0.30, 100% k i i k i i         (9) the ams of the relative differences of the wall variables and relative differences of the ams of the heat fluxes are given in table 2. table 2 ams of relative differences t,k (%) of the wall variables and the relative differences of the ams of the heat fluxes wq (%) kt (m -1) , rrns , tctea w tw qw wq qw a w tw qw wq qw a 1.0 1.192 2.667 12.13 0.84 1.993 1.295 2.851 12.74 1.01 1.731 2.0 1.777 3.828 17.04 0.55 1.425 1.991 4.183 18.36 0.50 0.328 a – by formula (9), without the absolute value sign the results show that the ams of the relative differences of the wall variables increase with the increase in the total extinction coefficient of the flame. the values are similar for both methods. the biggest values of the ams of the relative differences were obtained for heat fluxes. on the other hand, the relative differences of the ams of the heat fluxes are much smaller. there are two reasons for such a difference. the first reason is in the use of an absolute-value sign which prevents the canceling out of positive and negative terms. 226 n. crnomarković, s. belošević, s. nemoda, i. tomanović, a. milićević the ams of the relative differences of heat fluxes determined by formula (9) without the absolute value sign are shown in table 2. they are much smaller than the ams determined with the absolute-value sign. the second reason is in distribution of the relative differences along the furnace walls. the values of the heat fluxes determined by the tctea method for kt = 0.3 m -1 and relative differences for kt = 1.0 m -1 and kt = 2.0 m -1 are shown in fig. 2a-c. it is evident that for some surface zones, the relative differences of the heat fluxes are big whereas the values of the heat fluxes are relatively small. although such zones affect the relative differences of the ams of the heat fluxes very little, they considerably influence the ams of the relative differences of the heat fluxes. the initial value of the surface zone emissivities must be determined in advance, as explained beforehand. the changes of wall variables with the change of the initial wall emissivity were found from the relative differences determined by comparison with the values obtained for the initial wall emissivity of 0.60: w,in w,in , 0.60, , 0.60, 100% i i i i          (10) the ams of relative difference w,in are presented in table 3 and show that the influence of the initial wall emissivity on the wall variables and their distribution along the furnace walls is very small. the wall variables are almost not affected by the initial wall emissivity if it is in the interval 0.60-0.90. table 3 ams of relative differences w,in (%), kt = 0.3 m -1 w,in (-) , rrns , tctea w tw qw w tw qw 0.70 0.050 0.101 0.526 0.060 0.115 0.500 0.80 0.042 0.084 0.434 0.108 0.201 0.909 0.90 0.067 0.133 0.689 0.094 0.194 0.897 the difference of the wall variable values caused by the selection of the method were found for three values of the total extinction coefficient: kt = 0.3, 1.0, and 2.0 m -1 , and four initial wall emissivities: 0.6, 0.7, 0.8, and 0.9 (only for kt = 0.3 m -1 ). the relative differences of the variables were determined by formula (11): rrns, tctea, ,m, tctea, 100% i i i i         (11) determination of the wall variables within the zonal model of radiation inside a pulverized coal-fired furnace 227 fig. 2 heat fluxes and relative differences, tctea method: (a) the heat fluxes for kt =0.3 m 1 , (b) relative differences qw,kt for kt = 1.0 m 1 , (c) relative differences w t,q k for kt = 2.0 m 1 228 n. crnomarković, s. belošević, s. nemoda, i. tomanović, a. milićević the ams of the relative differences are shown in table 4. the ams increase with an increase in the total extinction coefficient, and so the smallest ams of the relative differences were obtained for kt = 0.3 m -1 . the ams of the relative differences of the wall variables for the selected initial wall emissivities are presented in the right half of table 4 (for w,in = 0.80, the results are presented in the first column of the left half of table 4). the ams of the relative differences are almost constant for w,in from 0.6 to 0.80, and the smallest values are obtained for w,in = 0.70. it is in the agreement with the previous result that the selection of the initial wall emissivities has small influence on the difference of the wall variables. the biggest ams of the relative differences are obtained for the heat fluxes, for the same reason as previously. and again, the relative differences of the ams of the heat fluxes are much smaller than the ams of their relative differences. table 4 ams of relative differences ,m (%) of the wall variables and the relative differences of the ams of heat fluxes wq (%)  kt (m -1) w,in (-) 0.3 1.0 2.0 0.60 0.70 0.90 w 0.332 0.544 0.732 0.331 0.329 0.365 tw 0.837 1.321 1.765 0.828 0.815 0.901 qw 3.963 6.480 8.684 3.900 3.802 4.294 wq 0.110 0.061 0.148 0.191 0.020 0.261 the wall variable values determined by the rrns method and for kt = 0.3 m -1 are shown in fig. 3a-c. it is shown that the biggest wall temperatures and heat fluxes are obtained for the surface zones with the smallest emissivities. to further investigate their relation, the ams of the wall emissivity, wall temperature, and heat fluxes were determined by the rrns method for three thicknesses of the ash deposit layer. the results are shown in table 5. it is clear that the increase of the ash deposit layer thickness reduces the heat exchange between the flame and furnace walls not only because it increases the wall temperatures but also because it reduces the wall emissivities. on the other hand, the fly-ash particles are the main contributor to the flame radiative properties. without them, the radiative heat exchange inside the furnace would not be so intensive 27, 28. table 5 ams of the wall emissivities, wall temperatures, and heat fluxes, kt = 0.3 m -1 la (mm) w (-) wt (k) wq (kw m-2) 0.3 0.812 772.85 81.595 0.6 0.777 870.20 74.379 1.0 0.733 960.04 65.748 determination of the wall variables within the zonal model of radiation inside a pulverized coal-fired furnace 229 fig. 3 wall variables determined by the rrns method, kt = 0.3 m -1 : (a) heat fluxes, (b) wall temperatures, (c) wall emissivities 230 n. crnomarković, s. belošević, s. nemoda, i. tomanović, a. milićević this investigation shows that any of the developed models, rrns and tctea, can be used for determination of the wall variables. the methods are simple and can be easily applied. the rrns method is recommended for the application in numerical simulations, because the manipulation with the files is easier for that method. the main drawback of the methods is the repetition of the numerical simulations, which is a consequence of the ash emissivity dependence on temperature. in the case of constant wall (that is ash layer) emissivity, the single run of numerical simulation would be enough to determine wall temperatures and heat fluxes. the objective of the further investigation is the formation of the new method by which the wall variables could be obtained from the single run of the numerical simulation. the main contribution of such investigation would be to improve the application of the zonal model in numerical simulations. 4. conclusions two methods for determination of the wall variables: rrns and tctea, for use in numerical simulations of processes inside pulverized coal-fired furnaces are compared. the radiative heat exchange is calculated using the zonal model of radiation. the changes and differences of the wall variables are determined for various conditions of the radiative heat exchange. the following conclusions are attained.  for both methods, the ams of the relative differences of the wall variables increase with an increase in the total extinction coefficient of the flame. the biggest increases of the ams of the relative differences are obtained for the heat fluxes, whereas the relative differences of the ams of the heat fluxes are much smaller. that is a consequence of the method of calculation and distribution of the relative differences along the furnace walls.  the initial values of the wall emissivities influence the ams of the relative differences of the wall variables very little, for the initial wall emissivities in the interval 0.60-0.90.  the differences of the wall variables caused by the selection of the method increase with an increase in the flame radiative properties and only slightly depend on the wall emissivities. for the selected furnace, the smallest differences of the wall variables are obtained for kt = 0.3 m -1 and w,in = 0.70.  as the difference of the wall variables obtained by both methods is small, the rrns method is 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connecticut. 26. sijercic, m., 1998, mathematical modeling of complex turbulent transport processes, yugoslav society of thermal engineers and vinca institute of nuclear sciences, belgrade, (in serbian). 27. brkic, lj., zivanovic, t., tucakovic, d., 2002, thermal calculation of steam boilers, faculty of mechanical engineering, belgrade, (in serbian). 28. blokh, a.g., 1988, heat transfer in steam boiler furnaces, hemisphere, new york. 12923 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240930019w © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper making more informed decisions in multi-criteria problems under uncertain conditions of criteria weights: drone selection study case jakub więckowski1, wojciech sałabun1,2 1national institute of telecommunications, warsaw, poland 2research team on intelligent decision support systems, department of artificial intelligence and applied mathematics, faculty of computer science and information technology, west pomeranian university of technology, szczecin, poland abstract. the growing demand for robust decision support systems (dss) highlights the need for methods that enhance decision-making under uncertainty, particularly when criteria weights in multi-criteria evaluations are unknown. this study presents a novel approach for addressing multi-criteria decision-making (mcdm) problems where precise criteria weights are uncertain, focusing on drone selection for challenging, remote locations. using the multiattributive ideal-real comparative analysis (mairca) method, the research explores a range of criteria weight scenarios with various distributions, enabling a more comprehensive evaluation. a key innovation is the integration of fuzzy ranking techniques to aggregate results from multiple assessments, enhancing the robustness of decision outcomes. this approach overcomes the limitations of traditional ranking aggregation methods, providing a more reliable understanding of the stability and reliability of recommendations. by offering a more adaptive and uncertainty-resilient framework, this study advances multi-criteria decision analysis and improves reliability of recommendations provided by decision support systems in problems in which criteria weights remain unknown. furthermore, it equips decisionmakers with deeper insights into the reliability of recommendations, empowering more confident decision-making in complex, uncertain environments. key words: drone selection, uncertain weights, decision support, fuzzy ranking, decision-making received: september 30, 2024 / accepted may 03, 2025 corresponding author: wojciech sałabun national institute of telecommunications, ul. szachowa 1, 04-894, warsaw, poland e-mail: w.salabun@il-pib.pl 2 j. więckowski, w. sałabun 1. introduction in the complex landscape of decision-making today, operational research is fundamental for effective management [1]. given the multitude of factors that must be considered in intricate decision problems, decision support systems (dss) provide essential tools to aid decision-makers [2,3]. multi-criteria decision analysis (mcda) methods are commonly employed to navigate the complexities of real-world scenarios in a structured manner [4]. these methods enable the construction of decision models tailored to specific criteria, including their significance adjusted to the given requirements [5]. the effectiveness of mcda methods has been repeatedly demonstrated in practical applications, including business analytics [6], sustainable development [7,8], and engineering problems [9,10]. despite the widespread use and ongoing development, mcda methods present notable challenges for decision-makers [11]. a primary issue is selecting an appropriate evaluation method, as different methods may yield varying results for the same decision problem, as stated in the literature [12,13]. another significant challenge is determining the relevance of criteria [14]. criteria weights are crucial input data for many mcda methods and have a visible influence on the final results [15]. various approaches were developed to simplify identifying criteria relevance [16,17], but their reliability can be inconsistent. objective weighting methods, which rely solely on data from the decision matrix, may lead to misleading results [18]. conversely, subjective weighting methods can suffer from biases or inaccuracies in expert judgments, potentially skewing obtained outcomes [19]. a major difficulty arises when the relevance of criteria is unknown, reflecting scenarios where decision-makers lack information about the multi-criteria problem or cannot precisely define criteria importance but still seek to understand potential outcomes. new approaches are needed to provide a robust and reliable analysis of decision problems despite incomplete input data. the problem of unknown criteria weights is common in real-world scenarios, especially in fields related to engineering problems [20,21]. decision-makers and stakeholders often lack a full understanding of the complex nature of decision problems, which limits their ability to make informed choices [22]. this uncertainty is further compounded when there is hesitation about the relevance of different criteria, complicating the decision-making process [23,24]. as a result, there is a clear need for comprehensive approaches that explore the decision problem space under various criteria weight vectors. by analyzing the criteria weight space in depth, the robustness of recommendations can be improved, and decisionmakers' awareness of possible outcomes can be enhanced. furthermore, this approach generates multiple individual results based on different weight distributions, which must be aggregated into a final ranking reliably. however, current methods for aggregating results obtained from multiple assessments scenarios are limited [25]. while they produce a single consensus ranking [26], they fail to provide a detailed overview of how individual rankings change, leaving decision-makers without a full understanding of the implications of each scenario [27]. this raises the question of whether a more effective method of presenting results from diverse evaluation scenarios could better support informed decision-making. thus, to enhance the comprehensiveness of supporting decisions in complex problems, it is important to provide robust frameworks for evaluating decision variants. exploring decision problems with criteria weight vectors with different distributions of values could making more informed decisions in multi-criteria problems under uncertain conditions 3 lead to a more nuanced understanding of the outcomes, especially in problems where uncertain conditions of criteria weights occur. existing approaches for aggregating results from multiple evaluation scenarios often fall short in presenting insights from individual recommendations, highlighting the need for more effective result representation techniques. for this purpose, this paper is directed toward evaluating multi-criteria decision problems under unknown criteria weight values. the proposed approach extensively explores the space of criteria weight scenarios and uses the multiattributive ideal-real comparative analysis (mairca) method to evaluate considered decision variants [28]. the problem of selecting drone for accessing remote and challenging locations is used to verify the effectiveness of the presented evaluation procedure. moreover, this study applies a novel fuzzy ranking concept to examine the robustness of evaluation outcomes across assessment scenarios. the comparative analysis with selected weighting methods and compromise solution techniques are presented to examine the stability of the results. the proposed work aims to support decision-makers and enhance the understanding of the reliability of recommendations, leading to equipping decision-makers with deeper insights into complex decision problems under uncertain conditions of criteria weights. the main contributions of the study are:  providing a comprehensive framework for assessing multi-criteria decision-making problems under uncertain conditions of criteria weights;  demonstrating the effectiveness of the proposed approach in a practical problem of drone selection for accessing remote and challenging locations;  conducting a comparative analysis of selected weighting methods and compromise solution techniques to examine the stability and reliability of the results. the rest of the paper is organized as follows. section 2 shows the preliminaries of the methods used in the study with the procedure used for criteria weights generation, the main assumptions of the mairca method, and the description of a fuzzy ranking approach. section 3 presents a case study with the evaluation of drones for accessing remote and challenges locations under uncertain conditions of criteria weights. section 4 presents the results obtained, while section 5 comparatively analyzes the outcomes from selected methods. finally, section 6 draws conclusions from the research and indicates future development directions. 2. preliminaries 2.1 criteria weights generation procedure exploring the decision problem space by including all possible vectors of criteria weights in the evaluation process allows for a comprehensive assessment of each criterion's potential relevance within the considered problem. given the requirement that criteria weights must sum to 1, each of the generated weight vector must satisfy this condition. the criteria weights generation procedure that could be used to determine the weights vectors that represent different importance of criteria in the problem is presented in fig. 1. 4 j. więckowski, w. sałabun fig. 1 visualization of the procedure used for criteria weights generation the provided procedure can be adjusted to set the precision of criteria weight vector generation. the parameter s controls the step between weight values and can influence both the number of generated vectors and the variation of values distribution between consecutive vectors. this parameter directly affects the accuracy of the results: smaller values produce a larger number of scenarios to evaluate, improving the precision of the analysis. however, this also requires more computational resources, increasing the time complexity. 2.2 the mairca method the multiattributive ideal-real comparative analysis method was developed to promote alternatives with the smallest combined gap, which are treated as the closest to the ideal [29]. it proves to be highly stable, even when criteria are adjusted [30]. the method's effectiveness relies on measuring the gap between ideal and actual weights. the mairca method can be applied through a five-step process, illustrated in fig. 2. fig. 2 visualization of subsequent steps of the mairca method making more informed decisions in multi-criteria problems under uncertain conditions 5 2.3 fuzzy ranking approach in contrast to traditional methods of aggregating mcda results from multiple evaluation scenarios, the fuzzy ranking approach is based on calculating how often each alternative appears in specific ranking positions and then determines the membership degrees, which reflect the robustness of the results obtained [31]. the fuzzy ranking produces a twodimensional matrix that represents the membership degrees of alternatives, giving decision-makers a detailed view of the uncertainty and stability of the ranking positions. by addressing the shortcomings of conventional aggregation methods, this approach offers a more comprehensive and informed perspective in multi-criteria decision analysis. the procedure used for calculating the fuzzy ranking is presented in fig. 3. fig. 3 flowchart presenting steps for calculating the fuzzy ranking 3. study case this study presents a methodology aimed at enhancing decision-making in multi-criteria problems where criteria weights are uncertain. the proposed approach utilizes a systematic procedure to generate a range of criteria weight scenarios that thoroughly explore the decision space. the level of precision in this exploration can be adjusted by modifying the step parameter, allowing for a balance between detailed analysis with finer weight increments and improved computational efficiency with larger steps that produce fewer weight combinations. these weight scenarios are then applied within the mairca method to assess the alternatives across different weight distributions. while various mcda methods are available, the mairca method is chosen due to its proven performance across multiple applications and problem domains [32,33]. its robustness has been verified in numerous studies, establishing solid foundations for developing reliable decision models within the mcda field [34,35]. the resulting rankings are analyzed using a fuzzy ranking approach, which offers a more comprehensive and nuanced presentation of the findings, helping decision-makers make better-informed decisions. the proposed work addresses the growing need for effective solutions in accessing remote and challenging locations, a significant concern in today's landscape. in problems connected to planning and management, particularly site accessibility, drones offer a 6 j. więckowski, w. sałabun transformative approach to reach areas that are otherwise difficult or unsafe to access, enhancing operational efficiency and safety. this research provides insights into optimizing drone selection, which is essential for improving logistical operations and expanding capabilities in various industrial and engineering applications. the criteria considered in the drone selection problem are presented in table 1, with name of the criterion, its label, unit and type. table 1 criteria set considered in the problem of drone selection evaluation name label unit type price c1 eur cost maximum flight distance c2 km profit maximum flight time c3 min profit battery power c4 mah profit megapixels of camera c5 mp profit maximum iso c6 iso profit weight c7 grams cost width c8 mm cost based on the identified criteria, the specifications of six selected drones were gathered and stored in the decision matrix, presented in table 2. table 2 decision matrix showing specification of selected drones used for the multi-criteria evaluation alternative c1 c2 c3 c4 c5 c6 c7 c8 a1 5477.17 22.5 32 3350 48 6400 898 440 a2 3084.30 7.0 30 5870 20 12800 1390 300 a3 4728.48 30.0 46 5000 20 6400 899 283 a4 3791.06 7.0 30 5870 20 6400 1380 225 a5 5415.41 32.0 45 5000 48 25600 920 347 a6 3656.86 32.0 45 5000 20 6400 915 347 using the defined input data, the evaluation of drones was conducted through multicriteria decision analysis. traditional assessments within the mcda field assume that criteria weights are accurately defined and known, which could simplify the evaluation process as it could not reflect the actual importance of criteria. moreover, significant challenges arise when the importance of criteria weights is uncertain. standard approaches in the literature often involve objective weighting methods that derive criteria relevance from the decision matrix data. a major limitation of this approach is that it produces a single vector of criteria weights, which may not reflect the true importance of the criteria and covers only one of multiple possible criteria weight vectors. to address this challenge, a more effective solution could be to analyze multiple scenarios to evaluate the robustness of the results under varying conditions. it also enables assessing a given decision problem, even with the criteria relevance remaining completely unknown. to bridge this gap, the proposed research extensively explores the decision problem space by generating combinations of criteria weights with a defined resolution, covering the whole decision problem space. it allows for examining the single decision problem with multiple weighting scenarios which translates into more robust outcomes, making more informed decisions in multi-criteria problems under uncertain conditions 7 especially in problems with unknown criteria weight values. for this analysis, the weight generation step was set to 0.05 to ensure high precision in exploring the decision space. the generated weighting scenarios were then applied within the mairca method to assess the alternatives. the resulting multiple rankings from different scenarios were used to calculate a fuzzy ranking matrix, providing a detailed analysis of the robustness of the ranking order under performed evaluations. 4. results following the procedure outlined in fig. 1, combinations of criteria weight vectors were generated for 8 criteria with a step size of 0.05. a sample of these resulting weight vectors is shown in table 3. the procedure produced a total of 50,388 scenarios to cover all possible combinations for the 8 criteria vector. it can be seen that the number of obtained scenarios is substantial, as the procedure generates criteria weights for the examined decision space with high resolution. with bigger values of steps, the number of scenarios could be reduced. nevertheless, the proposed extensive approach allows for a thorough evaluation of the decision problem space under various criteria weight distributions, providing in-depth insights into the outcomes across different conditions. table 3 sample of criteria weights vectors generated with the presented procedure using step of 0.05 scenario c1 c2 c3 c4 c5 c6 c7 c8 s1 0.05 0.50 0.05 0.05 0.05 0.05 0.05 0.20 s2 0.10 0.45 0.05 0.05 0.05 0.05 0.05 0.20 s3 0.06 0.40 0.05 0.05 0.05 0.05 0.05 0.20 s4 0.08 0.35 0.05 0.05 0.05 0.05 0.05 0.20 s5 0.10 0.30 0.05 0.05 0.05 0.05 0.05 0.20 ... ... ... ... ... ... ... ... ... s50388 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.65 the evaluation process utilized the generated criteria weight scenarios to assess the decision matrix shown in table 2 using the mairca method. each weight vector from the possible combinations was applied individually, and the resulting rankings were gathered for further analysis. in total, 50,388 rankings were generated from this evaluation process. these rankings were used to calculate a fuzzy ranking, which was achieved by counting the occurrence of each alternative at specific ranking positions and normalizing these values to determine membership degrees. these degrees indicate the robustness of the results. fig. 4 illustrates the fuzzy ranking matrix derived from the calculations, represented as a heatmap. this matrix displays the membership degree of each alternative for each ranking position, ranging from 0 (indicating no membership degree for that position) to 1 (indicating strong confidence in that position). 8 j. więckowski, w. sałabun fig. 4 heatmap representing results of the calculated fuzzy ranking based on 50,388 evaluation scenarios using step of 0.05 from the obtained results, it could be seen that the alternative a5 was the most robust selection under the examined space of decision problem with a given step of 0.05. this decision variant visibly outperforms other drones, obtaining a membership degree value of 1.00, while the second-best drone (a6) for the 1st position reached the value of 0.11. for the second position in the ranking, it could be seen that two alternatives reached high membership degree values. alternative a6 obtained a value of 1.00, while alternative a3 reached the certainty of 0.70 regarding the recommendation to be placed in the 2nd position. similar cases can be seen for the rest of the positions in the ranking where small differences between subsequent positions and classified alternatives are notices regarding the obtained membership degree values. the smallest difference between the alternatives can be observed for the 4th position, where a1 reached 1.00 membership degree and a2 obtained a value of 0.90, also showing high certainty of being placed in this position. 5. comparative analysis the obtained results were analyzed with a critical evaluation of the proposed methodology's efficiency and effectiveness. to validate the robustness of the results, further experiments compared the outcomes generated using different step sizes in the criteria weight scenarios. specifically, scenarios with step sizes of 0.10 and 0.04 were contrasted with the primary step size of 0.05. this comparison assesses how variations in step size affect the number of generated scenarios and, consequently, the precision of the decision space exploration. by analyzing these different resolutions, the study aimed to making more informed decisions in multi-criteria problems under uncertain conditions 9 determine the impact of step size adjustments on the overall accuracy and computational efficiency of the evaluation process. additionally, the comparative analysis involves benchmarking the proposed approach against selected weighting techniques and compromise solution methods. this comparison includes evaluating traditional weighting methods and various aggregation techniques to determine how well the proposed methodology performs in aggregating and interpreting results. the effectiveness of the fuzzy ranking approach is tested against these conventional methods to highlight its advantages and limitations. by integrating results from different weighting and aggregation techniques, the study examined how the proposed approach holds up in practice in multiple scenarios evaluation, providing a comprehensive understanding of its effectiveness and reliability in multi-criteria decision-making scenarios. 5.1 impact of the step size in weights generation procedure the substantial difference in the number of generated vectors with varying step sizes highlights the trade-off between precision and computational efficiency. with a step size of 0.10, only 36 vectors are generated, providing a simplified but computationally manageable overview of the decision space. using a step of 0.04 to generate the weighting vectors provided 346,104 scenarios, offering finer resolution and more detailed exploration of the criteria weighting space, albeit at the cost of slightly increased computational complexity and time required. however, with today's computational capabilities, this is not a demanding task for personal use machines, as long as the results are not to be obtained in real time. on the other hand, striving for greater accuracy using step 0.02 would produce 85,900,584 scenarios which could be challenging to process. while the finer step size improves the precision of the analysis and potentially yields more robust results, the higher computational burden may limit its practical applicability in large-scale problems. fig. 5 heatmaps of fuzzy rankings obtained based on generated weighting scenarios using steps of 0.10 and 0.04 despite the significant difference between the number of weighting scenarios generated with the examined steps of 0.10 (36) and 0.04 (346,104), in fig. 5, it can be seen, that the 10 j. więckowski, w. sałabun obtained fuzzy rankings do not differ substantially. the most visible difference can be observed for the 3rd position and the case of alternative a6, where it obtained the membership degree values of 0.25 and 0.78 for steps of 0.10 and 0.04, respectively. for the more general evaluation using step 0.10, the same alternatives reach the highest membership degree values as in the case of assessment with step 0.04. the alterations can be observed for the 4th and 6th positions, where for the step of 0.10 the alternatives a2 and a4 obtained the highest membership degree and were ranked 4th and 6h, respectively. for the lower step of 0.04, the alternative a1 was recommended as the most robust choice for 4th and 6th place, showing that with evaluated weighting scenarios, it can be stated with the same certainty that it could be ranked in those positions. 5.2 comparison of weighting techniques to thoroughly evaluate the proposed approach, the study compared it against several established weighting methods, including angle [36], entropy [37], gini [38], method based on the removal effects of criteria (merec) [39], and integrated determination of objective criteria weight (idocriw) [40]. these methods were used to determine the criteria weights and assess their impact on decision outcomes. for calculating consensus rankings, the study employed borda count [41], dominance directed graph (ddg) [42] and rank position [43] methods. the calculations were performed with the pymcdm tool [44] and pysensmcda library [45]. by comparing these traditional consensus techniques with the fuzzy ranking results, the study aimed to analyze how well each method aligns with the proposed approach and its ability to handle uncertain criteria weights. this comparison helps to provide insights into the robustness and reliability of the proposed methodology relative to established practices. table 4 rankings calculated based on using selected objective weighting methods for the problem of drones evaluation alternative angle entropy gini merec idocriw fuzzy ranking a1 4 3 4 4 2 5 a2 5 5 5 5 5 4 a3 3 4 3 3 4 3 a4 6 6 6 6 6 6 a5 1 1 1 1 1 1 a6 2 2 2 2 3 2 table 4 presents the rankings of considered drones evaluated using five objective weighting methods. the obtained rankings revealed general consistency across the methods but also highlighted several differences. alternative a5 was consistently ranked highest across all methods, indicating strong performance regardless of the weighting approach. conversely, alternative a4 was ranked lowest in all cases, reflecting its relatively weak performance. alternatives a2 and a1 showed variability, with a2 placed highest in all methods except idocriw, where it was ranked second, while a1’s positions ranged from second to fourth. alternatives a3 and a6 varied regarding assigned positions, with a3 performing slightly better in the angle, entropy, and idocriw methods compared to a6, which has a more uniform placement across examined methods. the results obtained making more informed decisions in multi-criteria problems under uncertain conditions 11 from the proposed approach, which involved generating a comprehensive set of weighting scenarios to explore the entire decision space, aligned closely with those derived from traditional objective weighting methods. however, the proposed approach provided deeper insights by offering membership degree values that reflect the stability and reliability of the recommendations. the fuzzy ranking, calculated with a step size of 0.05, revealed that multiple alternatives were likely to be ranked in the 2nd, 3rd, 4th, 5th, and 6th positions, highlighting the potential variability in rankings. in contrast, the crisp rankings from the selected objective weighting techniques offered a more simplified view, lacking the nuanced information about ranking stability and certainty that the fuzzy ranking approach provides. table 5 shows the consensus rankings for the considered problem using three different compromise solution methods: borda count, dominance directed graph, and rank position. the compromise rankings were calculated based on the individual rankings obtained from the evaluation using generated weighting scenarios from the proposed procedure and the application of the mairca method. table 5 consensus rankings calculated with selected compromise solution methods alternative borda ddg rank position fuzzy ranking a1 5 5 5 5 a2 4 4 4 4 a3 3 3 3 3 a4 6 6 6 6 a5 1 1 1 1 a6 2 2 2 2 across the examined methods, alternative a5 was ranked highest, reflecting its superior performance relative to the others. conversely, alternative a4 was consistently ranked the lowest. it is worth noticing that all alternatives were ranked in the same positions despite applying different compromise solution methods. this consistency underscored the similar manner of performance of those techniques. nevertheless, compared to the results obtained from the fuzzy ranking approach, it also highlighted that these methods provide a simplified view compared to the detailed membership degree values, which can be particularly useful when multiple scenarios of assessments are considered. 6. conclusion making reliable recommendations in decision-making problems characterized by uncertainty and unknown criteria weights poses a significant challenge. the evaluation approach presented in this study, which involves an extensive exploration of the decision problem space with varying criteria weights, offers a robust and comprehensive solution. by utilizing a defined step size for generating criteria weights, the presented approach ensures that recommendations account for diverse input conditions regarding the importance of criteria. the integration of a fuzzy ranking concept provides decisionmakers with a nuanced perspective, enhancing their ability to make well-informed decisions. the results from the drone selection problem demonstrate that this approach is 12 j. więckowski, w. sałabun effective for dealing with unknown criteria weights and offers a more detailed analysis compared to traditional weighting techniques and compromise solution methods. however, the study is not without limitations. one significant limitation is the computational complexity associated with generating and analyzing a large number of criteria weight scenarios, which can be resource-intensive and time-consuming. additionally, the effectiveness of the proposed method may vary with different types of decision problems and criteria. future research should focus on optimizing the procedure to enhance its efficiency for scenarios involving a large number of criteria. additionally, exploring the application of various mcda methods could reveal whether different techniques yield significantly different results from 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support tool based on fuzzy multimoora, journal of project management, 9(3), pp. 197-212. 43. altuntas, s., dereli, t., yilmaz, m. k., 2015, evaluation of excavator technologies: application of data fusion based multimoora methods, journal of civil engineering and management, 21(8), pp. 977-997. 44. kizielewicz, b., shekhovtsov, a., sałabun, w., 2023, pymcdm—the universal library for solving multicriteria decision-making problems, softwarex, 22, 101368. 45. paradowski, b., więckowski, j., sałabun, w., 2024, pysensmcda: a novel tool for sensitivity analysis in multi-criteria problems, softwarex, 27, 101746. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 169 180 https://doi.org/10.22190/fume190312023b © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* on the problem of strain localization and fracture site prediction in materials with irregular geometry of interfaces ruslan balokhonov, varvara romanova institute of strength physics and materials science, sb ras, tomsk, russia abstract. the interfacial mechanisms of the stress-strain localization in non-homogeneous media are investigated, using a steel substrate iron boride coating composition subjected to tension as an example. a dynamic boundary-value problem in a plane-strain formulation is solved numerically by the finite-difference method. the curvilinear substrate-coating interface geometry is assigned explicitly in calculations and is in agreement with experiment. constitutive relations accounting for an elastic-plastic response of the isotropically-hardened substrate and for a brittle fracture of the coating are employed. three stages of the plastic strain localization in the steel substrate are found to occur due to the irregular interface geometry. distributions of the stress concentration regions in the coating are shown to be different at different stages. the stress concentration in the coating is demonstrated to increase nonlinearly during the third stage. the location of fracture is found to depend on the strength of the coating. key words: mesomechanics, interfaces, plasticity, fracture, coated materials 1. introduction stress-strain localization phenomena have been much studied both experimentally and theoretically (see, for instance, [1-3]) and may be due to different physical processes operating at different scale levels. at the microlevel, dislocations, slip bands, dislocation cells, fragmented structures, etc., are formed in single crystals and in grains of polycrystals. it is ab initio geometry associated with the lattice discreteness which is responsible for the stress-strain localization in this case. the macroscopic stress-strain localization may be due to the geometry of mechanically contacting elements or specimens even at the elastic stage of loading [4], for instance, due to the received march 12, 2019 / accepted june 24, 2019 corresponding author: ruslan balokhonov affiliation: institute of strength physics and materials science, sb ras, 634055 tomsk, russia e-mail: rusy@ispms.tsc.ru mailto:rusy@ispms.tsc.ru 170 r. balokhonov, v. romanova shape of indenters [5]. a prominent example of plastic strain localization is a familiar necking phenomenon. in simulating this deformation instability observed in homogeneous specimens under uniaxial loading, the form of the plastic potential [6] generally plays a decisive role. changes in the yield surface under plastic deformation result from competitive strain hardening and softening processes and can be described with the use of a phenomenological approach or physically-based dislocation theories. for example, in [7], strain localization occurs where the strain hardening coefficient reaches its critical value, and in [8] the softening rate is a decisive factor for the onset of localization. another famous example of the macroscopic strain localization is the propagation of luders bands. a physical substantiation of the process is found to be of microscopic origin and is attributed to the dynamic strain aging effects. a gradual involvement of local regions of the material in plastic deformation is associated with the mechanism for dislocation locking by interstitial atom atmospheres. the dislocations pinned in this way require extra energy to tear away and continue in motion. a macroscopic collective effect of the dislocation behavior gives rise to formation and slow motion of a localized plastic deformation front, resulting in a yield drop and a yield plateau in the macroscopic stress-strain curve. a large number of experimental (see, e.g., [9-12]) and theoretical studies [12-16] have dealt with the luders band and portevin–le chatelier effects. the mesoscopic stress-strain localization is associated with different interfaces between microstructural components: interfaces between a matrix and reinforcing particles in composite materials, different phases of alloys, a coating or a hardened surface layers and base material, grain or pore boundaries, etc. the curvilinear interface is a major factor responsible for the occurrence of rotational deformation modes and local geometrical stress concentrations. a number of papers was devoted to analytical modeling of, e.g., interface curvature effects [17]. an abundance of investigations deals with numerical simulations where an explicit account is taken of the material microstructure (see, e.g., [18-21]). different constitutive models have been developed to describe the mechanical response of individual microstructural components. the goal of these and related investigations is to study and gain insight into the mechanisms for and the special features of the stress-strain localization in the vicinity of interfaces and to examine their effects on the macroscopic mechanical properties of materials. in this work, we investigate a special feature of the mesoscopic stress-strain localization associated with a self-consistent evolution of stress-strain patterns in a nonhomogeneous material. particular emphasis is placed on the fact that the geometrical stress concentration regions appearing in the vicinity of curvilinear interfaces and initially randomly distributed over the bulk of the material tend to interact with each other. due to an ongoing competition of strain hardening and stress relaxation during plastic deformation, a system of stress concentration regions evolves under the action of external forces, approaching equilibrium. the evolution gives rise to stress redistribution. new regions favorable for a rapid stress growth are formed, and there are other regions where the previously formed stress concentrations are suppressed. this is not believed to be a stochastic or spontaneous process; rather, it is predetermined from the outset and governed by the characteristics of the non-homogeneous medium. three controlling factors of critical importance are involved here: (1) the difference in the mechanical properties of contacting materials (reinforcing particles, matrices, coatings, substrates, interlayers, polycrystalline grains, etc.), (2) the interfacial curvature, and (3) the parameters of external loading. the challenges of mathematical modeling and numerical simulations are to on the problem of strain localization and fracture site prediction in materials with irregular... 171 reveal and investigate the individual effect of each of the factors on the location of maximum stress-strain concentration regions in which fracture occurs at a certain instant of loading time. the aim of the present paper is to investigate the mesoscopic stress-strain localization phenomenon, using uniaxial loading of a coated material as an example. the major factor responsible for the evolution of a deformable system such as this is a curvilinear coatingsubstrate interface. special attention is given to the influence of the plastic strain localization in the substrate on the location and evolution of maximum stress concentrations in the elastic-brittle coating. 2. statement of the problem let us consider the microstructure of a steel specimen subjected to surface hardening by a diffusion borating technique. this technology enables high-strength coatings with needleshaped high-curvature profiles to be produced (fig. 1a). the technique is used for repairing and strengthening the surface of machine parts and structural components. the coated steels take on increased surface hardness and high resistance to impact loads, friction and abrasive wear. in our earlier works [15, 22], we have examined microstructures of this type in some detail. studies were made on different aspects of deformation and fracture of coated materials, including luders band propagation, coating thickness and strain rate effects. fig. 1 experimental [24] (a) and model microstructures (b) and a calculated microscopic stress-strain curve (c) for a specimen with a curvilinear coating-substrate interface in this contribution, particular emphasis is placed on an analysis of the evolution of stress concentration in the vicinity of the interface. a total system of equations for simulating the deformation of a coated material includes mass and momentum conservation laws, strain relations and constitutive equations describing the material response [15, 23]. in the case at hand, the use is made of models for elastic-plastic behavior of the steel substrate and for a brittle fracture of the iron-boride coating. a dynamic boundary-value problem in a plane strain formulation is solved numerically by the finite difference method [22, 23, 25]. the boundary conditions for the leftb1 and right-hand surfaces b3 of the computational domain simulate uniaxial tension of the coated material along the х direction, while those 172 r. balokhonov, v. romanova for the top b2 and bottom surfaces b4 correspond to the conditions for free surface and symmetry, respectively, (fig. 1b). the steel substrate exhibits an elastic-plastic behavior. the plastic flow rule ij p ij s  is associated with a yield surface: )( р eqeq  , (1) where ζeq is the equivalent stress and ε p eq is the cumulative equivalent plastic strain, ε p ij is the plastic strain tensor components and λ is a scalar parameter equal to zero in the elastic region. the following function satisfying the experimental data is used to describe the isotropic hardening of mild steel: 0( ) ( )exp( / )p p p eq s s eq r          , (2) where ζs and ζ0 are the ultimate strength and the initial yield point, respectively, and ε p r is a characteristic value of the equivalent plastic strain. cracking of the coating was examined, using a maximum distortion energy criterion. in [15, 22, 23] the foregoing modified fracture criterion employed in combination with an explicit account of the material microstructure is shown to provide an adequate direction of crack propagation in brittle compounds. according to the criterion, fracture occurs where the equivalent stress reaches limiting values cten or ccom depending on the type of the stressed state (tension or compression) found in a given local region:       0for,c 0for,c kkcom kkten eq (3) here cten and ccom are the strength of iron boride under tension and compression. the fracture criterion given by eq. (3) accounts for the following factors. a local coating region subjected to tension (εkk>0) fails where the local equivalent stress reaches a value cten. it is assumed that for the failed coating regions both the deviator sij=0 and the pressure p=-kεkk=0. for compressive regions (εkk<0), the limiting fracture surface in stress space is restricted by ccom. in this case, the failed coating material offers no resistance to shear alone (sij=0). the mechanical properties of steel substrate and iron boride coating are listed in table 1. note that k and µ denote the respective bulk and shear moduli. table 1 the mechanical properties of the steel and iron-boride [24, 26] k, gpa , gpa s, mpa 0, mpa p r cten, gpa ccom, gpa substrate 133 80 395 174 0,093 – – coating 200 140 – – – 1.1 4 on the problem of strain localization and fracture site prediction in materials with irregular... 173 3. computational results figure 1c depicts a homogenized stress-strain curve for a coated material subjected to tension. the stress is calculated as an equivalent stress averaged over the computational domain:    n,1k k n,1k kk eq ss , where n is the number of computational cells and s k is the k-th cell area. strain ε corresponds to the elongation of the computational domain along the x-axis: ε =(l-l0)/l0, where l0 and l are the initial and the current specimen lengths. there are two main deformation stages in the stress-strain curve: e and p denote elasticity and plasticity, respectively. at the elastic stage, both the steel substrate and the boride coating are strained elastically. because of the difference in the elastic moduli between the coating and the substrate, stress and elastic strain distributions exhibit a nonuniform pattern. local regions of stress concentration occur in the coating material near the coating-substrate interface (fig. 2, regions 1−9). the stress value in these regions is dictated by the local interfacial geometry. at the plastic stage, the coating still exhibits an elastic behavior, whereas the substrate deforms plastically. fig. 2 equivalent stress concentrations in the vicinity of the coating-substrate interface at the elastic deformation stage: the total strain of the coated material ε is 0.03 % (see fig. 1c) let us examine the evolution of the most powerful stress concentrations (peaks 1–5 in fig. 2) formed in the elastic coating. figs. 3 and 4 show the initial evolution period including stage e and substages p1 and p2. the plots in fig. 3a show the manner in which the equivalent stresses at the peaks ζeq i vary with the specimen elongation. by and large, the growth rate is different for different peaks (1−5), but the stresses are seen to exhibit a nearly linear growth. it should be pointed out that both in fig. 3a and in the stress-strain curve (fig. 1c) the deformation stages and substages are not clearly visible. however, they are quite discernible in the case where the relative stress level is examined (fig. 3b) 174 r. balokhonov, v. romanova i eq i eq i eqi eq    , where   5 1i i eq i eq 5 1 is the magnitude of the equivalent stress averaged over the 5 peaks. as seen from fig. 3b, δζeq is nearly constant during elastic deformation of the coated steel. this means that the stress values at peaks 1–5 increase at the same rate, and the stress patterns remain qualitatively unchanged (compare figs. 2 and 4a). at stage e, there is an average stress level in regions 1−3, whereas the stress values ζ 4 eq and ζ 5 eq are by 5 % higher and by 5% lower than 〈ζ i eq〉, respectively (see view a in figs. 2 and 3b). three stress peaks are formed in region 4 of the highest stress concentration. a maximum equivalent stress is found to be in local region 4.1 (fig. 2). the elastic stage continues until the total strain amounts to 0.06 % (figs. 3b and 1c). fig. 3 evolution of maximum equivalent stresses (see fig. 2) in regions 1−5 (a) and their deviations from the average value 〈ζ i eq〉 (b) during elastic stage 1 and plastic substages p1 and p2 of the coated material deformation on further loading, the steel substrate deforms plastically, with substages p1 and p2 being well-pronounced in strain ranges of 0.06–0.12 and 0.12–0.24 %, respectively (fig. 3b). at substage p1, the steepest rise in the maximum stress is observed in region 4 (figs. 3b and 4). remarkably, out of 3 peaks located in this region the equivalent stress increases where its magnitude is at a minimum at the elastic stage (fig. 2, arrows 4). to the contrary, in regions 4.1 and 4.2, the local stress rate slows down (fig. 4). the same conclusion suggests itself for region 5, where the stress-strain localization is suppressed (figs. 3 and 4). in regions 1−3, the stress growth rate is still close to the average level 〈ζ i eq〉, slowing down late at substage p1. at substage p2, the qualitative evolution pattern is, on the whole, retained: stresses δζ 2 eq and δζ 3 eq continue to decrease at a higher rate, whereas δζ 4 eq and δζ 5 eq continue to increase and decrease, respectively, but the rate of change is lower (fig. 3b) than at substage p1. the only exception is that a characteristic feature inherent to this deformation stage is a change in the on the problem of strain localization and fracture site prediction in materials with irregular... 175 slope of the curve for stress peak 1. this means that the stress-strain localization in region 1 is enhanced (fig. 3b, boxes). fig. 4 evolution of the equivalent stress pattern at substages p1 and p2 of the coated material deformation. the total strain  = 0.046 (a), 0.06 (b), 0.07 (c), 0.11 (d), and 0.2% (e) (see figs. 1c and 3). video data file “fig4.avi” is available online the simulation results were analyzed to show that the substages of the stress peak evolution in the boride coating were associated with a specific character of plastic strain localization in the steel substrate (figs. 5 and 6). at substage p1, the plastic strains nucleating near the interface cover the substrate material in a step-by-step manner. first, plastic shear strains arise in the steel material at the roots of boride teeth (fig. 5a) and propagate deep into the material, filling up the space between the teeth (fig. 5b−d), with the major part of the substrate material being in the elastic state. then localized shear bands start to form in the bulk (fig. 5e−f). the bands originate near the interface asperities (mainly at the tooth humps) and are localized in conjugate directions at an angle of ≈45 degrees to the tensile direction, causing an abrupt change in the slope of the macroscopic stress-strain curve (fig. 1c). substage p1 is over when most of the substrate material transforms into a plastic state, with the band system being formed completely. at substage p2 (fig. 5f, fig. 6a), the band distribution changes but only slightly, while the plastic strain localization in the bands is enhanced to form a clearly visible shear band system. this ordering of the shear bands is due to the competing stress relaxation and strain hardening processes in local regions of the substrate material. 176 r. balokhonov, v. romanova to summarize the foregoing simulation results for the initial evolution period, the stress growth rate in local near-interface regions of the elastic iron boride coating is constant at stage e, linearly increases in region 4 and slows down in region 5 at substages p1 and p2 due to plastic strain nucleation and shear band formation in the steel substrate. fig. 5 equivalent plastic strain patterns for total strain  = 0.06 (a), 0.07(b), 0.08 (c), 0.09(d), 0.1, (e) and 0.11 % (f); substage p1 quite a different evolution pattern is seen at substage p3 characterized by nonlinear stress-strain localization in the vicinity of the coating-substrate interface. much as at substages p1 and p2, this is due to the special features of the plastic strain localization in the substrate material. analysis of the obtained results shows that the conjugate shear bands intersecting at an angle of 45 degree are smeared at substage p3. the plastic strain localization mechanism in the steel substrate changes: due to a decrease in the local strain hardening coefficient in near-interface regions, the localization in the shear bands (fig. 6a) gives way to the localization along the coating-substrate interface alone (fig. 6b). as a consequence, the evolution of the stress concentration acquires a reverse pattern. a fast nonlinear rise of the stress peak is observed in region 5 (figs. 7 and 8), i.e., in the region where the stress-strain localization was hitherto suppressed (figs. 3 and 4). at the same time, the seemingly most powerful stress peak in region 4 that dominated at previous stages is seen to weaken rapidly. at a certain instant of loading time (see inverted triangles in fig. 8b), the stress growth rate for ζ 4 eq becomes the lowest among the ζ i eq rates, being inferior in magnitude even to the rates of initially less-developed ζ 2 eq and ζ 3 eq. equivalent stress ζ 1 eq increases steadily at an average rate of 〈ζ i eq〉 and will exceed ζ 4 eq on further loading when the curves 1 and 4 in fig. 8b intersect. on the problem of strain localization and fracture site prediction in materials with irregular... 177 fig. 6 equivalent plastic strains for the total strain  = 0.24 (a) and 2.41% (b); substage p3 fig. 7 evolution of the equivalent stress pattern at substage p3 of the coated material deformation. the total strain  = 0.24 (a), 0.51(b), 0.81(c), 1.71(d) and 2.61% (e). unlike the quasi-linear stress growth in regions 1−4, the dynamics of the stress-strain localization in region 5 is nonlinear (fig. 8a) with the highest local stress growth rate (fig. 8b). it is in this region where the local interfacial geometry is seen to be the most favorable for the plastic strain localization along two neighboring boride tooth sides. in other words, these sides are oriented in the directions lying most closely to those of maximum tangential stresses. this fact enhances the straining in region 5 and causes the most rapid increase in the stress concentration in the coating as compared to regions 1–4. 178 r. balokhonov, v. romanova fig. 8 evolution of maximum equivalent stresses (see figs. 2 and 8) in regions 1−5 (a) and their deviations from the average value <σi eq> (b) at substage p3 of the coated material deformation fig. 9 equivalent stress patterns showing the fracture localization for varying coating strength of 1.1 (a) and 21gpa (b) thus the maximum stress concentration developing in the coating along the coatingsubstrate interface is found at different points depending on the stage of the coated material deformation: maximum equivalent stresses are observed in regions 4.1, 4 and 5 at stage e, substages p1-p2 and substage p3, respectively. this means that the fracture site can vary with the deformation stage. in other words, for a given microstructural geometry and mechanical properties of the constituent materials, the coating strength determines the fracture location. the numerical simulation results illustrating this conclusion are presented in fig. 9. two calculations for varying coating strength were performed to show the difference in the fracture patterns. a crack in the coating originates in a near-interface region of a maximum stress concentration and propagates perpendicular to the tensile direction towards the free surface that corresponds to the experiment (fig. 1a). for a low-strength coating, cracking occurs in region 4 (fig. 9a) at substage p2 for the total strain  = 0.2 % (figs. 3 and 4), whereas a high-strength coating fails (fig. 9b) late at substage p3 for  = 2.2 % under the maximum local stress conditions in region 5 (figs. 7 and 8). on the problem of strain localization and fracture site prediction in materials with irregular... 179 4. conclusion a mesomechanical analysis of the stress-strain localization and fracture in iron boride coating – steel substrate composition under uniaxial tension has been performed. a dynamic boundary-value problem was solved numerically by the finite-difference method. the elastic-brittle and elastic-plastic properties were assigned for the coating and substrate materials, respectively. a curvilinear coating-substrate interface corresponded to the configuration found experimentally and was accounted for explicitly in calculations. due to the difference in the elastic moduli between steel and boride, local regions of the stress-strain localization were shown to arise along the interface even at the elastic stage of the coated material deformation. the stress concentration in the coating regions depends on the local interfacial geometry. redistribution of the stress concentration peaks discussed in this paper is not found to have occurred at the elastic stage. three stages of plastic deformation in the steel substrate have been revealed. the first stage corresponds to the nucleation of plastic strains in local near-interface regions and their propagation deep into the steel material to involve the internal regions in plastic flow. the second stage represents formation of well-defined shear bands oriented at an angle of 45 degrees to the loading direction and plastic strain localization in the bands. at the third stage, the shear bands are smeared, and the plastic flow is localized along the curvilinear interface alone. the stage-by-stage change in the mechanisms for the plastic strain localization in the steel substrate is responsible for the stress concentration redistribution in the near-interface coating regions, the location of maximum equivalent stress is found to vary with the deformation stages. consequently, the fracture localization depending on the critical equivalent stress was found to be affected by the coating strength. notably, the stress-strain localization at the third stage was found to develop in a nonlinear manner and was seen to occur where it was suppressed at previous stages. the evolution of the stress-strain localization suggests the general conclusions that are not limited to the particular case of the coated material considered in this work. to prove or reject this assumption, further investigations are warranted into the effects of the mechanical properties of constituents, interfacial geometry and loading conditions on a special character of the nonlinear stress-strain localization in composite materials. acknowledgements: this work is supported by the fundamental research program of the state academies of sciences for 2013-2020, line iii.23. references 1. vinogradov, a., estrin, y., 2018, analytical and numerical approaches to modelling severe plastic deformation, progress in materials science, 95, pp.172–242. 2. antolovich, s.d., armstrong, r.w., 2014, plastic strain localization in metals: origins and consequences, progress in materials science, 59, pp.1–160. 3. lunt, d., xu, x., busolo, t., quinta da fonseca, j., preuss, m., 2018, quantification of strain localisation in a bimodal two-phase titanium alloy, scripta materialia, 145, pp. 45–49. 4. popov, v.l., 2010, contact mechanics and friction: physical principles and foundations, springer, berlin 5. li, q., popov, v.l., 2016, indentation of flat-ended and tapered indenters with polygonal crosssections, facta universitatis-series mechanical engineering, 14(3), pp. 241-249. 180 r. balokhonov, v. romanova 6. becker, r., needleman, a., 1986, effect of yield surface curvature on necking and failure in porous plastic solids, j. appl. mech., 53, pp. 491–499. 7. dequiedt, j.l., 2013, localization in heterogeneous materials: a variational approach and its application to polycrystalline solids, international journal of plasticity, 48, pp. 92–110. 8. xue, l., 2010, localization conditions and diffused necking for damage plastic solids, engineering fracture mechanics, 77, pp. 1275–1297. 9. beukel, a.v.d., kocks, u.f., 1982, the strain dependence of static and dynamic strain-aging, acta metall. mater, 30, pp. 1027-1034. 10. neuhäuser, h., hampel, a., 1993, observation of luders bands in single crystals, scripta metallurgica et materialia, 29, pp. 1151-7. 11. casarotto, l., tutsch, r., ritter, r., weidenmüller, j., ziegenbein, a., klose, f., neuhäuser, h., 2003, propagation of deformation bands investigated by laser scanning extensometry, computational materials science, 26, pp. 210-218. 12. anjabin, n., karimi taheri, a., kim, h.s., 2013, simulation and experimental analyses of dynamic strain aging of a supersaturated age hardenable aluminum alloy, materials science & engineering a, 585, pp. 165–173. 13. mccormick, p.g., ling, c.p., 1995, numerical modeling of the portevin-le chatelier effect, acta metall. mater, 43, pp. 1969-1977. 14. hähner, p., rizzi, e., 2003, on the kinematics of portevin-le chatelier bands: theoretical and numerical modeling, acta materialia, 51, pp. 3385-3397. 15. balokhonov, r.r., romanova, v.a., martynov, s.a., schwab, e.a., 2013, simulation of deformation and fracture of coated material with account for propagation of a lüders-chernov band in the steel substrate, physical mesomechanics, 16(2), pp. 133-140. 16. balokhonov, r.r., romanova, v.a., schmauder, s., makarov, p.v., 2003, simulation of meso–macro dynamic behavior using steel as an example, computational materials science 28, pp. 505–511. 17. li, y., ortiz, ch., boyce, m.c., 2013, a generalized mechanical model for suture interfaces of arbitrary geometry, journal of the mechanics and physics of solids, 61, pp. 1144–1167. 18. panin, v.e., (eds.), 1998, physical mesomechanics of heterogeneous media and computer-aided design of materials, cambridge international science publishing, cambridge. 19. needleman, a., 2000, computational mechanics at the mesoscale, acta materialia, 48, pp. 105-124. 20. schmauder, s., schäfer, i., 2016, multiscale materials modeling: approaches to full multiscaling, walter de gruyter gmbh & co kg, berlin. 21. balokhonov, r.r., romanova, v.a., panin, a.v., kazachenok, m.s., martynov, s.a., 2018, strain localization in titanium with a modified surface layer, physical mesomechanics, 21(1), pp. 32-42. 22. balokhonov, r., zinoviev, a., romanova, a., zinovieva, o., 2016, the computational micromechanics of materials with porous ceramic coatings, meccanica 51(2), pp. 415-428. 23. balokhonov, r.r., romanova, v.a., schmauder, s., schwab, e.a., 2012, mesoscale analysis of deformation and fracture in coated materials, computational materials science 64, pp. 306–311. 24. koval, a.v., panin, s.v., 2000, mesoscale deformation and cracking of surface-hardened low carbon steel, theoretical and applied fracture mechanics 34, pp. 117-121. 25. richtmyer, r.d., morton, k.w., 1967, difference methods for initial-value problems, interscience publishers, john wiley & sons, new york – london – sydney. 26. grigorieva, i.s., meilihova, e.z., (eds.), 1991, physical values, reference book, energoatomizdat, moscow. http://www.begellhouse.com/authors/44b9b20811938366.html http://www.sciencedirect.com/science?_ob=articleurl&_udi=b6tw8-3ydg01n-7&_user=10&_coverdate=01%2f01%2f2000&_alid=794541343&_rdoc=9&_fmt=high&_orig=search&_cdi=5556&_sort=d&_docanchor=&view=c&_ct=10&_acct=c000050221&_version=1&_urlversion=0&_userid=10&md5=91e8121782a519eaaa223c342ccff127 http://www.begellhouse.com/authors/543eb2bd131552e8.html plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 1, 2015, pp. 39 46 1analysis of impact on composite structures with the method of dimensionality reduction udc 539.3 valentin l. popov berlin university of technology, germany national research tomsk state university, russia abstract. in the present paper, we discuss the impact of rigid profiles on continua with non-local criteria for plastic yield. for the important case of media whose hardness is inversely proportional to the indentation radius, we suggest a rigorous treatment based on the method of dimensionality reduction (mdr) and study the example of indentation by a conical profile. key words: contact mechanics, impact, plasticity, non-local constitutive relations, roughness, reduction of dimensionality 1. introduction constitutive relations for materials are mostly formulated in terms of stresses and deformations. correspondingly, the critical behavior of materials is generally described by parameters having the dimension of stress such as yield stress, fracture stress or hardness. further, the constitutive relations are in most cases assumed to be local relations. however, it is well known that the processes of plastic deformation, damage and fracture, independently of the detailed mechanism, are always non-local processes. for dislocation plasticity, this immediately follows from the mechanism of deformation by formation of shear zones [1]. each elementary event of plastic deformation is a nonlocal process [2, 3]. the same is valid for fracture processes: both the initial concept of griffith [4] and its later microscopic [5] and macroscopic [6] generalizations are intrinsically non-local. in the past, there was much effort to address the non-locality in the frame of gradient generalizations of both theory of elasticity and plasticity [7, 8, 9, 10]. practitioners often take the non-locality into account by introducing critical stresses received january 02, 2015 / accepted february 1, 2015 corresponding author: valentin popov berlin university of technology, 10623 berlin, germany; national research tomsk state university, 634050 tomsk, russia e-mail: v.popov@tu-berlin.de original scientific paper 40 v.l. popov depending on the size of the system or the size (or depth) of indentation. thus, in [12], it is shown that the strength of micro contacts of au-au and au-pt is proportional to the contact radius for the contact radiuses between 10 and 100 nm, meaning that the “fracture stress” is inversely proportional to the contact radius. while for metals this size dependence is observed only at a sufficiently small scale, for composites it can be valid already at the macroscopic scale. the same is valid for plastic yielding: the nonlocal nature of plastic deformation leads to the size dependence of the “yield stress”. in [11], it is shown that the indentation hardness of polydimethylsiloxan (pdms) is inversely proportional to the indentation depth over three decimal orders of magnitude. similar dependencies are observed by nanoindentation of au [13]. such dependence of the characteristic parameters on the size of the system shows of course an inconsistency in the theory and should be replaced by an appropriate non-local formulation. in the following we confine ourselves to the processes of plastic deformation and damage prior to the complete fracture of a structure. the whole spectrum of non-local yield criteria and thus the dependence of the hardness on the size of an indenter can be very roughly divided into three classes: (a) constant hardness σc, (b) hardness, which is inversely proportional to indentation radius a : σc = qc / a, (c) hardness, which is inversely proportional to indentation area: σc = fc / a2. in the cases (b) and (c), the hardness is in reality not a proper material parameter. it is rather quantity qc having the dimension of surface energy density in the case (b) and quantity fc having the dimension of the force in the cases (c), which now characterize unambiguously the plastic properties. from the point of view of mechanisms of plasticity, the cases (a), (b) and (c) correspond to the situations where the plastic flow is governed by: (a) volume processes, the characteristic critical quantity having the dimension of energy per volume or stress, (b) surface processes, the characteristic critical property having the dimension of energy per area, (c) line processes, the characteristic critical property having the dimension of energy per unit length. all known criteria for plastic yielding either coincide with one of these classes or can be considered as their combinations. in the present paper, we concentrate our attention only on the “intermediate” class of constitutive relations (b) and describe how the impact on materials with such non-local plastic criteria can be described using the method of dimensionality reduction (mdr) [14, 15, 16, 17]. 2. method of dimensionality reduction at the first glance, the non-local plasticity criterion (b) seems to be more complicated than the local criterion (a). however, the non-locality in three-dimensional systems can sometimes lead to a much simpler theoretical description than in the case of local relations. in a series of publications by popov, hess and co-authors (see e.g. [20, 14, 15, 16]), it is shown that the mechanical properties which in a three-dimensional system are analysis of impact on composite structures with the method of dimensionality reduction 41 proportional to the contact diameter, can be easily mapped onto a contact with a onedimensional elastic foundation. for example, this is the case for the contact stiffness of an arbitrary contact with an elastic half-space. the contact stiffness is proportional to the diameter of the contact and thus can be described naturally by a one-dimensional model. the same property is present in contact conductivity (both electrical and thermal) which thus can be described in the framework of mdr [21]. in the publications [14] and [22], it is shown that the mapping of the contact properties from 3d to 1d is exact for arbitrary bodies of revolution provided some rules are considered for recalculation of the material properties and profiles of the contacting bodies. in the case of plastic deformation with the indentation hardness inversely proportional to the contact radius (and thus the indentation force proportional to the radius), we again have a property, which is directly proportional to the contact radius. it is therefore logically to assume that the indentation with such a yield criterion will be correctly described within a one-dimensional model. in the following, we shortly recapitulate the basics of the method of dimensionality reduction and then formulate its extension to the description of plasticity. if applied to a contact of a body with an elastic or viscoelastic half-space, the mdr consists of two main independent steps: i. first, a viscoelastic half-space is replaced by a one-dimensional series of parallel springs with stiffness zkδ and dash pots with damping constant δγ (fig. 1): , * zk e xδ = δ 4 xγ ηδ = δ , (1) where is the effective elastic modulus *e 2 2 1 * 1 2 1 11 v e ee 2v− − = + , (2) e1 and e2 are the young’s moduli of contacting bodies, ν1 and v2, their poisson-ratios, and η the dynamic viscosity of the medium. fig. 1 equivalent one-dimensional visco-elastic foundation ii. secondly, the form of the indenter must be recalculated according to the following rule of hess: if a body of revolution is described by equation z = z(r), then the equivalent one-dimensional profile is defined as 42 v.l. popov 1 2 2 0 ( )( ) d x d z rz x x r x r ′ = − ∫ . (3) it is proven in [14] and [22] that the contact of the modified 1d profile will provide exactly the same relations between the normal force, the indentation depth, and the contact radius as in the initial full three-dimensional problem. if the three-dimensional profile is described by a power-function with an arbitrary positive power n , (4) ( ) n nz r c r= the equivalent one-dimensional profile is a power-function with the same power, but a modified coefficient: 1 1 2 2 0 ( ) d x n n d n n nrz x c x r c x x r − = − ∫ %= nc , (5) where n nc κ=% (6) and 2 1 2 2 ( ) 2 ( ) n n n nγπ κ = γ + . (7) ( )nγ is the gamma-function . (8) 1 0 ( ) dn tn t e ∞ − −γ = ∫ t in particular, for a cone (n = 1) we get κ1 = π /2 and for a parabolic profile (n = 2) κ2 = 2. this last case is known as the rule of popov [14] (fig. 2). r fn fn r 1 d a a b fig. 2 (a) a three-dimensional contact and (b) its one-dimensional representation in the mdr the solution of the one-dimensional problem provides not only correct relations between the global properties (total force, indentation depth and contact radius), but allows to determine the exact three-dimensional distributions of stress. in the elastic foundation, normal forces f (x) of separate springs are immediately determined for any contact configuration. we can define linear force density q(x) by dividing these forces with spacing xδ : analysis of impact on composite structures with the method of dimensionality reduction 43 ( )( ) f xq x x = δ . (9) in [14], it is shown, that normal stress σzz(r) in the contact area can be obtained from linear force density q(x) by the following integral transformation: 2 2 1 ( ) ( ) dzz r q xr x r σ π ∞ ′ = − ∫ x . (10) 3. plasticity criterion in the method of dimensionality reduction as stated above, basically all properties which in three dimensions are proportional to the contact radius, can be easily mapped to an appropriate one-dimensional system. for plasticity, this is the case if the indentation force is proportional to the indentation radius: 2 n c cf a qπσ π= = a . (11) it is easily seen that we reproduce this behavior by introduction of the following criterion for plastic yielding of the one-dimensional model described in the previous section: 2c cf q xπ = δ . (12) note that while the three-dimensional criterion is a non-local one, the corresponding criterion in the equivalent one-dimensional model occurs to be local. further numerical investigation of the model could lead to another coefficient of proportionality in eq. (12), but they cannot change the functional form of this relation. with criterion (12), the complete problem of an indentation in a “visco-elastic, nonlocally plastic” medium is reduced completely to a contact problem with a one dimensional elastoplastic foundation with independent elements. 4. impact of conical profiles on the material with a non-local yield criterion as an illustration, let us consider an impact of a conical profile of the form tanz rθ= ⋅ (13) with an elastoplastic medium with the non-local yield criterion (11). according to the rule of hess, the equivalent one-dimensional profile is given by tan 2 z π θ x= ⋅ . (14) for indentation depth d, the displacement of a spring having coordinate x will be tan 2zu d xπ θ= − ⋅ . (15) 44 v.l. popov the contact radius is obtained from condition zu ( a ) d= , hence, 2 tan da π θ = . (16) the spring forces which are still in the elastic state are equal to *( ) tan , for 2 2 cf x e x d x f q xπ θ⎛ ⎞= δ − ⋅ < δ⎜ ⎟ ⎝ ⎠ π (17) after achieving the critical value, the spring force remains constant and equal to ( ) 2 cf x qπ x= δ . (18) up to the indentation depth *2 c c qd e π = , (19) there will be no plastic yielding of any spring. thus, there exists a critical indentation force fc for starting the plastic yielding: 2 * * * 0 0 12 ( )d 2 ( ( / 2)tan )d tan2 c ca a c c z c qf e u x x e d x x e π π θ θ = = − ⋅ =∫ ∫ . (20) after exceeding the critical force, the inner part of the contact will be in the state of plastic yield. radius c of the plastically deformed area is given by the condition f (c) = π qcδx / 2, hence * 1 2 tan cqc d eθ π ⎛ ⎞= −⎜ ⎟ ⎝ ⎠ , * tan cqa c e θ − = (21) the normal force is given by * 22 ( )d ( tan a c n z c c c c c c q )f e u x x cq f cq f d dπ π θ = + = + = +∫ − . (22) after achieving the critical state, the normal force increases linearly with the indentation depth. we do not consider at this point the complete dynamic problem of an impact, which generally can also depend on the structure stiffness. we limit ourselves to the case of a very rapid impact, when the indentation depth is much larger than the critical one. in this case we can write fn ≈ 2qc d /tanθ. the work of plastic deformation up to the maximum indentation will be max 2 max 0 2 d( ) tan tan d c cq qw d d d θ θ ≈ =∫ . (23) analysis of impact on composite structures with the method of dimensionality reduction 45 the area of contact will be equal to 2 2 max max max2 4 tan a a dπ π θ = = . thus, the damaged area will be proportional to the impact energy: max 4 tanc a w q ≈ π θ . (24) this differs from the result for the media with the local plastic criterion, where the impact energy is proportional to the expelled volume [18, 19]. 5. conclusion in the present paper, we considered an indentation of a rigid profile into an elastoplastic medium with a non-local yield criterion. we considered only the case where the non-locality leads to the inverse proportionality of the indentation hardness to the indentation radius. for such media, we have suggested a generalization of the method of dimensionality reduction and analyzed the case of indentation of a rigid conical indenter. as the method of dimensionality reduction is also valid for tangential contacts [23], the proposed method can be easily generalized for the case of impacts with a tangential component of impact velocity. acknowledgements: many valuable discussions with s. psakhie, m. ciavarela, s. dubinsky and a. ushakov are gratefully acknowledged. this work is supported in part by tomsk state university academic d.i. mendeleev fund program. references 1. popov l. e., kobytev v. s., kovalevskaya t. a., 1982, concepts of strengthening and dynamic recovery in the theory of plastic deformation, russian physics journal, 25, pp. 520-541. 2. popov l. e., slobodskoi m. i., kolupaeva s. n., 2005, simulation of single slip in fcc metals, russian physics journal, 49, pp. 62-73. 3. l. e. popov, kolupaeva s.n., vihor n.a., puspesheva s.i., 2000, dislocation dynamics of elementary crystallographic shear, computational materials science, 19(1), pp. 267-274. 4. griffith a.a., 1921, the phenomena of rupture and flow in solids, philos. trans. r. soc. lond. ser. a, 221, pp.163–198. 5. prandtl l., 1933, ein gedankenmodell für den zerreißvorgang spröder körper, zamm, j. appl. math. mech., 13, pp. 129–133. 6. rice j.r., 1968, a path independent integral and the approximate analysis of strain concentration by notches and cracks, journal of applied mechanics, 35, pp. 379–386. 7. aifantis e.c., 1999, strain gradient interpretation of size effects, int. j. fracture, 95, pp. 299–314. 8. nix w.d., gao h., 1998, indentation size effects in crystalline materials: a law for strain gradient plasticity, journal of the mechanics and physics of solids, 46(3), pp. 411-425. 9. gao h., huang y., nix w.d., hutchinson j.w., 1999, mechanism-based strain gradient plasticity—i. theory, journal of the mechanics and physics of solids, 47(6), pp.239-1263. 10. popov v.l., 1992, gauge theory of plastically incompressible medium without dissipation. 1. dispersion relations and propagation of perturbations without dissipation, international journal of engineering science, 30, pp. 329-334. 11. wrucke a.j., han c.-s., majumdar p., 2013, indentation size effects of multiple orders of magnitude in polydimethylsiloxane, j. appl. polym. sci., 128, pp. 258–264. 46 v.l. popov 12. budakian r., putterman s.j., 2002, time scales for cold welding and the origins of stick-slip friction, phys. rev. b, 65, pp. 235429. 13. corcoran s.g., colton j., 1997, anomalous plastic deformation at surfaces: nanoindentation of gold single crystals, phys. rev. b, 55 pp. r16057. 14. hess m., 2012, on the reduction method of dimensionality: the exact mapping of axisymmetric contact problems with and without adhesion, phys. mesomech., 15(5-6), pp. 264-269. 15. popov v.l., hess m., 2013, methode der dimensionsreduktion in kontaktmechanik und reibung – eine berechnungsmethode im mikround makrobereich, springer-verlag. 16. popov v.l, 2013, method of reduction of dimensionality in contact and friction mechanics: a linkage between micro and macro scales, friction, 1(1), pp. 41-62. 17. popov v.l., hess m., 2014, method of dimensionality reduction in contact mechanics and friction, springer. 18. popov v.l., 2010, contact mechanics and friction. physical principles and applications, springerverlag, 362 pp. 19. kleis i., kulu p., 2008, solid particle erosion, occurence, prediction and control, springer, 206 pp. 20. popov v.l., psakhie s.g., 2007, numerical simulation methods in tribology, tribology international, 40, pp. 916–923. 21. hess m. and popov v.l., 2013, wärmeleitung und wärmeerzeugung, in: popov, v.l. and hess, m.. methode der dimensionsreduktion, springer, pp. 115-131. 22. hess m. and popov v.l., 2013, exakte lösungen in drei dimensionen für den normalkontakt rotationssymmetrischer körper, in: popov v.l. and hess m., methode der dimensionsreduktion, springer, pp. 227-239. 23. hess m. and popov v.l., 2013, exakte lösungen in drei dimensionen für den tangentialkontakt rotationssymmetrischer körper, in: popov v.l. and hess m. (eds.), methode der dimensionsreduktion, springer, pp. 227-239. analysis of impact on composite structures with the method of dimensionality reduction 1. introduction 2. method of dimensionality reduction 3. plasticity criterion in the method of dimensionality reduction 4. impact of conical profiles on the material with a non-local yield criterion 5. conclusion references facta universitatis, series: mechanical engineering open journal systems journal help user username password remember me notifications view subscribe language select language cрпски english journal content search search scope all authors title abstract index terms full text browse by issue by author by title other journals information for readers for authors for librarians current issue home about login search current archives indexing downloads ethics author's rights & permissions home > vol. 23, no 4, 2025 facta universitatis, series: mechanical engineering facta universitatis, series: mechanical engineering (fu 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optimization and performance analysis of bimorph magneto-electro-elastic energy harvesters meisam moory shirbani, sayed ehsan alavi pdf 667-686 machine learning of structure – property relationships: an application to heat generation during plastic deformation filip nikolić, marko čanađija pdf 687-707 vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak foundation nikola nešić, milan cajić, danilo karličić, mihailo lazarević, sondipon adhikari pdf 709-733 homogenised properties of lattice metal composite cell stefano valvano pdf 735-756 prediction of contact distribution on rough surfaces using deep learning algorithms ilkwang jang, yong hoon jang pdf 757-785 a fast insight into high-accuracy nonlinear frequency estimation of stringer-stiffened shells chun-hui he, mostafa mohammadian pdf 787-806 innovative design strategy for an internal combustion engine with improved output characteristics jovan dorić, nebojša nikolić, stjepan galamboš, dalibor feher, boban nikolić pdf 807-825 porosity effects on nonlinear static performances of functionally graded shells considering thickness stretching hanen mallek, hana mellouli, lotfi ben said, mondher wali, fakhreddine dammak, muapper alhadri pdf 827-860 mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals minjia chen, xingqi luo, wanbo chen, jianjun feng, xiaohang wang pdf 861-880 a rolling bearing fault diagnosis method based on extreme learning machine optimized by improved whale optimization algorithm xin li, shiliang guo, dejie sun, lijun cao, cong li, shuyao tian, peng liu, yadong qi pdf 881-907 effect of electric pulsing on the structure, texture and hardness of cryorolled fine-grain copper mikhail markushev, irshat valeev, aygul valeeva, rafis ilyasov, elena avtokratova, stanislav krymskiy, oleg sitdikov pdf 909-920 cu-cu mechanical bonding for 3d integration of the next generation electronic chips: interfacial mechanisms, surface engineering, and emerging low-temperature strategies sooyong choi, chih-ming chen, byungil hwang pdf 921-943 reliability prediction and process parameter optimization of welded joints: artificial neural network and fuzzy logic jaganathan gokulachandran, mohanavelu thenarasu, bhadrinath pothkanoori, madhavarao seshadri narassima, erfan babaee tirkolaee pdf 945-969 issn: 0354-2025 (print) issn: 2335-0164 (online) cobiss.sr-id 98732551 zdb-id: 2766459-4     facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 499 514 https://doi.org/10.22190/fume201224039a © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper measuring efficiency change in time applying malmquist productivity index: a case of distribution centres in serbia milan andrejić, milorad kilibarda, vukašin pajić university of belgrade, faculty of transport and traffic engineering, serbia abstract. in the last decade, more and more attention has been paid to the efficiency of logistics systems not only in the literature but also in practice. the reason is the huge savings that can be achieved. in a very dynamic market with environmental changes distribution centers have to realize their activities and processes in an efficient way. distribution centers connect producers with other participants in the supply chain, including end-users. the main objective of this paper is to develop a dea model for measuring distribution centers’ efficiency change in time. the paper investigates the impact of input and output variables selection on the resulting efficiency in the context of measuring the change in efficiency over time. the selection of variables on the one hand is a basic step in applying the dea method. on the other hand, the number of basic and derived indicators that are monitored in real systems is increasing, while the percentage of those used in the decision-making process is decreasing (less than 20%). the developed model was tested on the example of a retail chain operating in serbia. the main factors changing the efficiency have been identified, as well as the corresponding corrective actions. for measuring efficiency change in time malmquist productivity index is used. the developed approach could help managers in the decision-making process and also represents a good basis for further research. key words: distribution center, efficiency, logistics performance, data envelopment analysis, malmquist productivity index 1. introduction survival in the logistics market has become increasingly challenging in recent years. competition is becoming fiercer, service users are becoming more demanding, social responsibility is increasing. in such circumstances, more and more companies recognize the efficiency of operations as a key factor of success and a prerequisite for business received december 24, 2020 / accepted march 30, 2021 corresponding author: milan andrejić university of belgrade, faculty of transport and traffic engineering m.andrejic@sf.bg.ac.rs 500 m. andrejić, m. kilibarda, v. pajić improvement. distribution centers (dcs) of trading companies and dcs, in general, represent complex logistics systems with a very important place and role in the supply chains [1-3]. they connect producers with other participants in the chain including endusers. in that manner logistics performances are very important [4-6]. due to a complex structure, estimating their efficiency is a very complicated process. „single ratio“ indicators have been used for a long time to estimate the efficiency of dcs. recently, an increasing number of authors have been advocating the use of approaches such as the data envelopment analysis (dea) method [7, 8]. the dea method is used for estimating the efficiency of homogeneous decision-making units (dmus). starting with the initial papers [9, 10] and making the foundations of the dea method, as well as the introduction of the dmu concept, an expansion of papers in this field, can be observed. the universality of applicability and quality of obtained results have influenced the usage of this method in various profit and non-profit organizations [11, 12]. the dea method is widely used in logistics. for estimation of third-party logistics providers’ (3pl) efficiency both from the provider’s perspective [7] and from a user’s perspective [13]. zhou et al. [14] used the dea method to define benchmark values of performances for 3pl providers in china. they also discuss the change of efficiency in time as well as the mutual influence of certain factors on performances. dea method is applied for estimating the efficiency of 3pl providers with an emphasis on warehouse operations [15]. they compare the results of two dea models with and without weight restrictions. certain papers analyze the efficiency of reverse logistics channels including solid waste [16] and also container terminals [17]. dea is used for estimating container port efficiency [18], as well as dcs efficiency, as a part of complex supply chains [19]. they also analyze efficiency change in time. de koster and balk [20] used the dea method for benchmarking and monitoring international warehouse operator’s performances. a model with multiple inputs and outputs to evaluate the efficiency of warehouse systems is proposed by hackman et al. [21]. they also confirm conclusions concerning the relation between warehouse size, level of technology and efficiency. cook et al. [22] applied the dea method for estimating efficiency in supply chains. the dea method is often combined with other methods. combining dea and analytic hierarchy process (ahp) method it is possible to evaluate the warehouse provider from the aspect of qualitative and quantitative criteria [23]. park and lee [24] used the dea method to assess the efficiency of large logistics providers in korea. a combination of dea and ahp can be used for different problems in logistics [25, 26]. the stochastic frontier analysis (sfa) is also combined with the dea method [27]. pca-dea model is used for estimating dc efficiency [8, 28]. momeni et al. [29] used a fuzzy network slacks-based dea model for evaluating the performance of supply chains with reverse logistics. mihajlović et al. [30] used ahp and a weighted aggregated sum-product assessment (waspas) for the logistics distribution fruit center location selection in the southern and eastern serbia region. pamučar and božanić [31] used the neutrosophic mabac model to locate multimodal terminals. malmquist productivity index is used for technical efficiency analysis of container terminals in india [32]. lei et al. [33] investigated the impact of logistics technology progress on employment structure based on the dea-malmquist method. mavi and mavi [34] applied the malmquist method for the analysis of the energy and environmental efficiency. shahverdi and ebrahimnejad [35] used dea and malmquist productivity indices in order to measure group performance in two periods. measuring efficiency change in time applying malmquist productivty index: a case of distribution centres... 501 based on the previously described and extensive review of the literature, it can be concluded that most papers focus on specific examples of efficiency measurements, but not examining the impact of variable selection on the resulting efficiencies as well as their change over time. to the best of our knowledge, there is a lack of papers in the literature concerning the field of logistics which analyze the impact of input and output variables selection on the resulting efficiency. in this paper, the impact of input and output variables selection on the applicability of the model and efficiency change in time is analyzed. the main objective of this paper is to develop a model which would provide the efficiency change evaluation of dcs that represent the distribution network of one trading company in serbia. the paper describes the impact of input/output variables selection on the resulting efficiency as one of the most important steps in the process of applying the dea method. this paper also analysis efficiency changes in time as a result of a dynamic environment. developed models are tested on real data and the model which successfully describes the dc's operations is selected. the main contribution and novelty of the developed approach are reflected in the identification of main (elimination of insufficiently authoritative) indicators, development of a model for measuring changes in efficiency over time, identification of factors influencing changes in efficiency, and defining corrective actions from the manager’s perspective. based on the literature research, there are no papers that integrate all the mentioned aspects into a unique methodological procedure applicable in real logistics systems. the model's concept could provide easier decision-making of the company management on corrective actions that would improve dc's operations. the paper consists of seven sections. after the introduction, the second section describes the dea method. the third section describes the distribution center’s efficiency as well as the developed methodology. developed models for estimating dcs efficiency are described in the fourth section. after that, the orientation of the model is analyzed. corrective actions of developed models are described in the sixth section. malmquist productivity index was used for analyzing efficiency and productivity change of distribution centers. at the end of the paper, the concluding remarks and directions of future research are described. 2. dea method dea is a non-parametric linear programming technique that enables the comparison of efficiencies of different dmus, based on multiple inputs and outputs. the efficiency is relative and relates to the set of units within the analysis. charnes et al. [10] proposed a non-parametric approach for efficiency estimation, where they reduce multiple inputs to a single virtual input and multiple outputs reduced to a single virtual output using weighting coefficients. in the set of homogeneous units, the dea finds the most efficient dmus and according to them, it defines the efficiency of other units. this method is also used for obtaining information about corrective actions of inefficient dmus. obtained efficiencies are relative since they relate only to a set of observed dmus and they cannot be considered as absolute. the dea method was chosen primarily because of the large number of advantages, as well as the specificity of the problem that was solved in this paper. this non-parametric approach provides, among other things, the possibility of an objective assessment of efficiency over time. the approach completely excludes the subjectivity of experts. also, the dea approach allows quick and easy integration of multiple outputs and inputs into a 502 m. andrejić, m. kilibarda, v. pajić single measure of efficiency. an additional advantage of this approach is reflected in the relatively simple application that would allow wider application in practice and help improve logistics systems. the basic ccr [10] model presents the basis of all present models. in the original form, this model presents the problem of fractional programming. according to the appropriate transformations, the model is reduced to the linear programming problem. in order to estimate dmu efficiency, it is necessary to have data of consumed input and realized output variables. in the process of dea method application, the ccr model is preferable as the initial model. as in linear programming problems, the ccr model also has two formulations: primal and dual. a dual formulation of the ccr model was used in this paper [10]. the mentioned formulation is well known, and it is not necessary to describe it in more detail. 3. methodology for measuring dc efficiency there are numerous problems with measuring the efficiency of dc and logistics systems in general. one of the basic ones is their complexity. for successful evaluation of dc efficiency, it is first of all necessary to define activities that are realized within it, and then to quantify them [36]. according to the process approach, proper definition of all activities and processes in the logistics system enables managers to create a clear image of the system operating and to identify any possible failures and defects. according to aminoff et al. [37] the main activities in dc, among others, are: receiving, shipping, control, packing, storage, order picking, order processing, etc. to assess the efficiency of each of them, it is necessary to define certain inputs/outputs that best characterize them. as mentioned earlier according to the process approach dc efficiency depends on the subsystems, process and activity efficiencies, and therefore, it is even more difficult to assess the overall efficiency. dc is characterized by a number of different input/output variables. in this paper, the efficiency of a dc of a trading company is observed, with a special emphasis on the warehouse subsystem. in observed dcs, as well as in most real systems, performances are evaluated by „single ratio“ indicators such as: turnover per employee, turnover per pallet place, warehouse utilization, etc. mentioned variables are not good indicators of dc efficiency since they do not provide enough information about their operating style. dea method provides the possibility of integrating a large number of different indicators into a unified measure of efficiency [12]. the development of appropriate models for estimating dc efficiency is an iterative process. defining an acceptable model requires fewer or more iterations. for each iteration, it is necessary to analyze the obtained results. the methodology of model development and its application for measuring dc efficiency change in time is given below: ▪ step 1 – defining potential input and output variables; ▪ step 2 – a selection of input and output variables and model defining: model 1, model 2 and model 3; ▪ step 3 – model testing; ▪ step 4 – model selection; ▪ step 5 – testing model orientation. the process of model testing and result analysis was made on the example of one trading company with dcs located in different parts of serbia. there are some recommendations in the literature for dmu selection and the relation of the number of dmu and the number of input measuring efficiency change in time applying malmquist productivty index: a case of distribution centres... 503 and output variables. some authors recommend that the minimum number of dmus is at least twice the total number of inputs and outputs in the proposed dea model [12, 14, 38]. for these reasons, in this paper, smaller models are developed. for estimating the efficiency of seven dmus (in this case dcs) models with two input variables and one output variable ("2+1"), and models with one input and two output variables ("1+2") were developed. observed dcs are in larger cities where there are competitive companies and customers with different demands and characteristics. 4. selection of input and output variables and model development for the successful application of the dea method, one of the key steps is the selection of input and output indicators. the choice of indicators itself greatly affects the resulting efficiencies and discriminatory power of the model. the results of the model are efficiency scores of observed dcs. according to these values, it is possible to define corrective actions for input and output variables in order to improve the efficiency of every dc. on one hand, it is necessary to determine a set of values that in the best way describe the system operating and provide obtaining real operating indicators of dcs efficiency. on the other hand, the objective is to select variables that are appropriate for applying corrective actions and which can be changed in real conditions [8]. in this paper, the initial list of input and output variables was reduced after consultations with managers in dcs, quantitative analyses and preliminary results obtained by applying potential models. during the preliminary analysis, all those variables that did not provide new information and represented duplication of indicators were eliminated. in this way, the wellknown problem of excessive indicators that are monitored in dc, but are not used in the decision-making process, has been overcome. based on the research conducted in this paper, it was found that over 80% of the indicators monitored are not used in the decision-making process. also, preliminary tests have shown that one part of the indicators has no effect on the discriminatory power of the model. these indicators were also excluded from further consideration. the selected input and output variables are shown in table 1. these variables are used as the basis for creating different dea models for estimating dc efficiency. three models were tested in this paper: model 1, model 2 and model 3. three suggested models have the same mathematical formulations but they represent different combinations of input and output variables. all models are input-oriented. model 1 is based on indicators that are most commonly used in the literature. input variables in the model are warehouse floor space and number of employees, and the output variable is the warehouse utilization. models similar to this one in literature are applied to estimate the efficiency of banks, libraries, etc. [39]. a similar model is used for estimating the efficiency of 3pl providers [15]. in model 1, warehouse floor space is taken for the first input variable. a number of employees represent the total number of employees in dc where the largest number of employees is engaged in the warehouse and on receiving, shipping, order picking procedures, etc. the output variable is the warehouse utilization which is obtained as the ratio of the number of the occupied pallet places and a total number of pallet places. this variable is expressed in percentage (%). 504 m. andrejić, m. kilibarda, v. pajić table 1 summary of input and output variables dmu warehouse floor space (m2) employees (no.) utilization (%) forklifts (no.) turnover (106 mon. unit) pallet places (no.) retail stores (no.) realized deliveries (no.) dmu 1 14856 107 81.33 24 483.13 6775 1285 11232 dmu 2 750 14 100.00 2 52.42 548 386 4458 dmu 3 8147 114 98.24 24 522.90 4486 934 11834 dmu 4 10609 82 100.00 28 333.72 6286 876 9491 dmu 5 4272 64 100.00 13 146.11 3234 688 6198 dmu 6 6993 68 91.68 15 216.61 5241 733 4982 dmu 7 5708 32 78.92 9 89.70 4824 551 5705 the increasingly intensive application of financial indicators led to model 2 [23, 40]. the input variables are warehouse floor space and the number of forklifts, and the output variable is the turnover of dc. the number of forklifts represents one of the equipment indicators that are used for the realization of basic logistics activities. it is possible to take other equipment indicators instead of this variable: energy consumed, number of working hours, etc. the output variable is the turnover. turnover is the most frequently used variable not only in logistics but in all other areas. this variable is expressed in the monetary units (m.u). the idea of developing model 3 is to define a model that best describes all aspects of dc functioning. the main idea was to select variables that describe the operation of dcs of trading companies in a good way from a large number of variables. three typical variables are: a number of pallet places, the number of retail stores that dc supplies, as well as the number of successfully realized deliveries. unlike the variables in previous models (warehouse floor space, number of employees, number of forklifts, turnover) that are strategic, in model 3 the operational variables are included. such variables are more appropriate for measuring the efficiency and implementation of appropriate corrective actions in dc. a number of retail stores represent some kind of gravity area. this variable is determined by the way dc operates, by the position and competitors in the region. all retail stores are similar in size. the number of realized deliveries is the total number of successfully realized customer's demands. the number of pallet places provides more information on the facility capacity than warehouse floor space since the height of the facility is taken into account. in literature, some authors put an emphasis on the lack of warehouse floor space as a space indicator [15]. testing and selection of models for further analysis of changes in efficiency over time are described in detail in chapter 5. 5. model selection and orientation testing all previously described models were tested on a real example. a detailed analysis of the results was done in accordance with the real situation in the company. the results of all three models are shown in table 2. those dmus that have a value of 1 in table 2 can be considered completely efficient. measuring efficiency change in time applying malmquist productivty index: a case of distribution centres... 505 table 2 dcs efficiencies according to different models dmu model 1 model 2 model 3 model 4 dmu 1 0.1064 0.7680 0.6899 0.6899 dmu 2 1.0000 1.0000 1.0000 0.9115 dmu 3 0.1206 0.9183 1.0000 1.0000 dmu 4 0.1707 0.4547 0.8551 0.8551 dmu 5 0.2188 0.4893 0.7129 0.7110 dmu 6 0.1888 0.5509 0.5364 0.5364 dmu 7 0.3453 0.3802 0.8172 0.8172 the universality of model 1 is reflected in the fact that it uses the variables that are most common in the literature. however, the results of this model do not correspond to the real state of observed dcs. for example, dc with the most modern equipment – dmu 3 according to this model, has an efficiency of only 12%. the first reason is the larger number of employees in this than in other dcs. the other reason is the large throughput which this dc realizes. throughput is an important element of efficiency that is not taken into consideration by this model. the main disadvantage of applying this model in practice is the implementation of corrective actions. the change of surface and number of employees are more strategic than operational decisions that are difficult to implement in the case of complex logistics systems such as dc. in order to overcome the problem, model 2 gives greater focus to financial indicators. the results of this model are more appropriate for the real state of observed dcs. there are different opinions in the literature on the application of financial indicators. there are authors who advocate the use of these variables as the key elements of efficiency. the use of financial indicators is often overemphasized in logistics systems, especially in those companies whose main activity is not the provision of logistics services (as is the case in the considered trading company). on the other side, there are authors who propose the use of non-financial indicators that better describe the state of the system. due to the company’s core activity (trade company), financial variables can present suitable input and output variables in the developed models. in order to include more authoritative indicators that better describe the functioning of logistics systems, model 3 was developed. this model estimates the efficiency according to variables that describe dc operating in a good way (the number of pallet places, number of retail stores and number of realized deliveries). comparing to the first two models, the efficiencies scores of model 3 are the most appropriate for the real state of dc. dmu 3 and dmu 2 are the most efficient and the least efficient is dmu 6. model orientation is a very important step in the efficiency measurement process. in dea terminology, there are input and output-oriented models. input orientation involves minimizing input variables with the same or greater outputs, and output orientation maximizing output with the same or fewer inputs. model orientation does not change efficiency value but only the way of achieving those values. the application of input and output models depends on the type of system, specific conditions and management decisions about the variables that are appropriate for corrective actions. based on the resulting efficiencies it is possible to define appropriate corrective actions that reduce input values and increase output values. output-oriented model 4 was developed on the basis of the input-oriented model 3. this model features one input and two output variables. the input variable is the number of retail stores that dc supplies and the output variable is the number of realized deliveries 506 m. andrejić, m. kilibarda, v. pajić and turnover. model 4 results, regardless of the change of orientation and number of input and output variables, correspond to model 3 results and the real state of dc (table 2). tables 3 and 4 show values of virtual inputs and outputs for model 3 and model 4 i.e., target values of input and output variables that enhance the efficiency of observed dcs. in the dea approach, the target values for each observed variable represent the values that the observed dmu must achieve in order to improve its efficiency. they are the result of the model and show the extent to which a particular dmu must implement corrective actions. in that sense, these values show to what extent it is necessary to reduce the input variables, i.e., to what extent it is necessary to increase the output variables. table 3 target values – model 3 dmu pallet places retail stores realized deliveries target value % target value % target value % dmu 1 4257.80 37.15 886.49 31.01 11232 0 dmu 2 548.00 0.00 386.00 0.00 4458 0 dmu 3 4486.00 0.00 934.00 0.00 11834 0 dmu 4 3597.82 42.76 749.08 14.49 9491 0 dmu 5 2305.60 28.71 490.49 28.71 6198 0 dmu 6 1888.56 63.97 393.21 46.36 4982 0 dmu 7 2162.64 55.17 450.27 18.28 5705 0 table 4 target values – model 4 dmu pallet places realized deliveries turnover target value % target value % target value % dmu 1 1285 0 16281.25 44.95 719.40 48.91 dmu 2 386 0 4890.71 9.71 216.10 312.24 dmu 3 934 0 11834.00 0.00 522.90 0.00 dmu 4 876 0 11099.13 16.94 490.42 46.96 dmu 5 688 0 8717.12 40.64 385.17 163.63 dmu 6 733 0 9287.28 86.42 410.37 89.45 dmu 7 551 0 6981.30 22.37 308.47 243.91 by applying model 3 and model 4 slack values of input and output variables are obtained and they show the possibility of their change with the aim of improving dc efficiency (table 5). table 6 shows reference sets of inefficient dmus in both models. by analyzing the results, it is possible to define appropriate corrective actions for every dc which will improve their efficiency. table 5 slack values dmu dmu 1 dmu 2 dmu 3 dmu 4 dmu 5 dmu 6 dmu 7 model 3 416.10 0.00 0.00 1777.41 0.00 922.88 1779.46 model 4 19.09 158.59 0.00 100.16 179.68 6.58 198.71 measuring efficiency change in time applying malmquist productivty index: a case of distribution centres... 507 table 6 reference sets dmu 1 dmu 2 dmu 3 dmu 4 dmu 5 dmu 6 dmu 7 model 3 dmu 2 / 1.0000 / / 0.0385 / / dmu 3 0.9491 / 1.0000 0.8020 0.5093 0.4210 0.4821 model 4 dmu 3 1.3758 0.4133 1.0000 0.9379 0.7366 0.7848 0.5899 6. corrective actions and managerial implications one of the main advantages of applying the dea method is information on the necessary corrective actions of inefficient units. as mentioned in previous chapters, this is one of the reasons for choosing this method. model 3, as input-oriented, strives to minimize the number of pallet places and the number of retail stores that dc supplies with the same or more successfully realized deliveries. in general, dcs can perform some of three corrective actions: reducing the number of pallet places, reducing the number of retail stores that dc supplies, or increasing the number of realized deliveries. based on the results of model 3, inefficient dmus in order to improve their business must implement corrective actions such as reducing the number of pallet places and the number of retail stores. according to this model, increasing the number of realized deliveries does not present the necessary corrective action for achieving efficiency. the number of pallet places presents a type of resource that dc uses in order to realize the delivery. common to all inefficient dmus is the fact that they can realize the same number of deliveries with a smaller number of pallet places. according to the discussion with managers of observed dcs, it was concluded that the implementation of these corrective actions was justified. from a mathematical perspective (table 3), it can be concluded that inefficient dmus can increase their efficiency by reducing the number of retail stores. this means that there are efficient dcs that realize a larger number of deliveries while supplying a small number of retail stores. this information may be useful to managers of inefficient dcs. however, the implementation of these corrective actions, in this case, is not justified. regardless of the fact that this corrective action will not be performed, managers of inefficient dcs know very well that their customers have a relatively small number of delivery requests. it is necessary that the management of the company analyzes in detail the reasons for such a situation. some of the potential reasons may be the structure of customers, the existence and functioning of competition, etc. the results in table 3 unequivocally show that only 2 dcs have an efficiency value of 1 (dmu 2 and dmu 3). this further means that they form an envelope. dmu 2 and dmu 3 present dcs with the best combination of input and output values. common to all inefficient dcs is the need for reducing the number of pallet places with the aim of improving efficiency. the results analysis indicates that dmu 3 is an example of good practice for all inefficient dmus. this result can be fully explained by the real situation in this system. dmu 3 is a modern distribution center that was planned for this purpose, unlike dmu 1, which was converted from a production plant to a warehouse facility. it is equipped with modern technology that enables a better flow of information and goods. model 4, as an output-oriented model, strives to reach the maximum number of realized deliveries and turnover with the same or smaller number of retail stores. in general, one can expect the following corrective actions: reducing the number of retail stores that dc supplies, increasing the number of realized deliveries, or increasing the turnover. 508 m. andrejić, m. kilibarda, v. pajić by applying model 4, and according to target values (table 4), it can be concluded that inefficient dcs can become efficient by increasing the number of realized deliveries and turnover which was confirmed by dc's managers. according to this model, it is not necessary to perform corrective actions which relate to reducing the number of retail stores which is consistent with the result analysis of model 3. the results of measuring efficiency using model 4 indicate that only one unit of dmu 1 is effective. the efficiency scores of other dmus are relatively similar to the scores from model 3, except that the efficiency of dmu 2 is smaller and it is 91.15% (table 2). the fact is that the reference set is made only of dmu 3 and that all other dmus have deficits in realized turnover (table 5). in this paper, it was found that the results of model 3 and model 4 are fundamentally different in the way corrective actions are taken to improve efficiency. namely, dmu 4 can improve operating in two ways. in the first case, dmu 4 can improve operating by reducing the number of pallet places, without changing the number of realized deliveries, while in the second case dmu 4 can improve operating by increasing the number of realized deliveries and turnover, without reducing the number of retail stores. it is possible to define the efficiency of other inefficient dcs in the same way. orientation in these models has not influenced the dc efficiency, but it greatly defines the corrective actions. in the case of output-orientated models, better results are obtained by including two outputs and one input, while for input-oriented models it is better to use two inputs and one output. in this sense, model 3 is developed for input-oriented models while model 4 is developed for output-oriented models. 7. efficiency analysis over time a very important aspect is the analysis of the change in efficiency over time under the influence of various internal and external factors. an extremely dynamic environment affects changes in dc efficiency. this paper analyzes the changing efficiency trend over time as well as changes in multifactor productivity with the aim of determining the factors that influence the change of dc efficiency. analyses were done for the period of 12 months of the observed year, applying the malmquist index (mi). 7.1. trend analysis previously obtained model testing results were performed in the relevant month, december. in the long run, measuring efficiency is a much more complex problem. the change of efficiency of seven dcs in the observed year according to model 3 is shown in table 7. according to table 7 there are three groups of dcs. the first group consists of efficient dcs with stable performances – dmu 2 and dmu 3, which were efficient in the observed period. it should be mentioned that dmu 3 had certain decreases in efficiency, but these decreases were not below 95%, and they were the consequence of certain technological and organizational changes as well as of redistribution of tasks between dmu 1 and dmu 3. the second group consists of dcs whose efficiency varies significantly in time – dmu 6 and dmu 1. large efficiency increases of dmu 6 are achieved in july, summertime when the turnover of most dcs increases. during the holiday season, increased sales are recorded and that period, together with the period of new years’ holidays (december) presents a „peak“ in the trade. the efficiency of dmu 1 in the first half of the year increases measuring efficiency change in time applying malmquist productivty index: a case of distribution centres... 509 while in the second half it decreases. this phenomenon has a real explanation since in the second half of the year part of the tasks is taken by dmu 3, and the efficiency of dmu 1 decreases. at the very end is the third group consisting of inefficient dcs with relatively stable performances: dmu 4, dmu 5, dmu 7. these dcs can to some extent provide useful information on stability and resistance to various factors that affect their functioning. table 7 trends of dc efficiency change jan. feb. mar. apr. may june july aug. sep. oct. nov. dec. dmu 1 0.28 0.76 0.8 0.78 0.8 0.8 0.76 0.73 0.78 0.71 0.63 0.69 dmu 2 1 1 1 1 1 1 1 1 1 1 1 1 dmu 3 0.43 1 1 1 1 1 1 0.98 0.97 1 1 1 dmu 4 0.27 0.67 0.6 0.62 0.68 0.75 0.8 0.84 0.75 0.69 0.7 0.86 dmu 5 0.39 0.72 0.68 0.68 0.62 0.65 0.66 0.68 0.62 0.57 0.61 0.71 dmu 6 0.24 0.46 0.46 0.47 0.49 0.56 0.78 0.74 0.57 0.54 0.49 0.54 dmu 7 0.17 0.67 0.69 0.73 0.72 0.78 0.83 0.8 0.81 0.76 0.71 0.82 7.2. analysis by malmquist productivity index in order to obtain a complete and true picture of the change in efficiency in the observed time period, the mi index was applied. mi index was presented by the authors [41, 42]. later mi has been used frequently [35, 43, 44]. productivity can be presented as the ratio between input and output, while the change of productivity presents the change of that ratio in time. over time, system productivity may change due to a frontier shift which appears as the consequence either of technological progress which has happened or due to the change of relative efficiency of dc. a detailed procedure for calculating the main components is shown below. to calculate mi, it is necessary to determine four indexes of relative distance, i.e., to solve four linear programming problems. mi of the company’s productivity change is calculated as: 𝑀𝐼 = √ 𝐷0 𝑡(𝐵𝑡+1) 𝐷0 𝑡(𝐵𝑡) 𝐷0 𝑡+1(𝐵𝑡+1) 𝐷0 𝑡+1(𝐵𝑡) (1) where d0 t(bt) is the efficiency of point bt in the moment t, d0 t+1 is the efficiency of point bt in the moment t+1, d0 t(bt+1) is the efficiency of point bt+1 in the moment t, and d0 t+1(bt+1) marks the efficiency of point bt+1 in the moment t+1. the previous formulation can be decomposed into an index of relative technical efficiency change (tec) and index of frontier shift (fs) in the following way: 𝑀𝐼 = 𝑇𝐸𝐶 ∗ 𝐹𝑆 = 𝐷0 𝑡+1(𝐵𝑡+1) 𝐷0 𝑡(𝐵𝑡) √ 𝐷0 𝑡(𝐵𝑡+1)𝐷0 𝑡(𝐵𝑡) 𝐷0 𝑡+1(𝐵𝑡+1)𝐷0 𝑡+1(𝐵𝑡) (2) the efficiency of seven dcs is estimated in this paper over the period of 12 months. for every period three values were calculated: mi, tec and fs. obtained results are shown in table 8. the values were calculated based on the eqs. (1) and (2). the period of change (t; t+1) is one month as indicated in the second column of table 8. 510 m. andrejić, m. kilibarda, v. pajić table 8 indicators of efficiency change in time period dc dmu 1 dmu 2 dmu 3 dmu 4 dmu 5 dmu 6 dmu 7 jan/feb 0.955 0.872 0.984 0.969 0.909 1.057 1.018 feb/mar 1.180 1.083 1.060 1.008 1.055 1.126 1.154 mar/apr 0.947 1.000 0.982 0.997 0.967 0.990 1.026 apr/may 1.023 0.979 0.999 1.094 0.920 1.051 0.978 malmquist may/jun 0.946 0.985 0.975 1.039 0.981 1.075 1.025 index jun/jul 0.902 0.936 0.977 1.015 0.975 1.330 1.016 (mi) jul/avg 0.928 1.066 0.956 1.017 0.984 0.902 0.930 aug/sep 1.112 1.020 1.027 0.925 0.945 0.799 1.049 sep/oct 0.948 1.000 1.016 0.958 0.957 0.974 0.969 oct/nov 0.949 1.000 1.000 1.010 1.040 0.955 0.967 nov/dec 1.057 1.041 0.992 1.183 1.148 1.062 1.115 jan/feb 2.757 1.000 2.312 2.483 1.854 1.905 3.855 feb/mar 1.050 1.000 1.000 0.897 0.937 1.001 1.027 mar/apr 0.979 1.000 0.998 1.030 0.999 1.023 1.060 apr/may 1.026 1.000 1.002 1.097 0.923 1.054 0.981 techical may/jun 1.001 1.000 1.000 1.100 1.038 1.138 1.085 efficiency jun/jul 0.945 1.000 1.000 1.064 1.022 1.395 1.065 change jul/aug 0.968 1.000 0.977 1.060 1.026 0.941 0.970 (tec) aug/sep 1.069 1.000 0.992 0.889 0.909 0.769 1.009 sep/oct 0.899 1.000 1.033 0.918 0.916 0.949 0.939 oct/nov 0.900 1.000 1.000 1.020 1.082 0.912 0.934 nov/dec 1.087 1.000 1.000 1.217 1.166 1.093 1.147 jan/feb 0.346 0.872 0.426 0.390 0.490 0.555 0.264 feb/mar 1.124 1.083 1.060 1.124 1.126 1.124 1.124 mar/apr 0.968 1.000 0.984 0.968 0.968 0.968 0.968 apr/may 0.997 0.979 0.998 0.997 0.997 0.997 0.997 frontier may/jun 0.945 0.985 0.975 0.945 0.945 0.945 0.945 shift jun/jul 0.945 0.936 0.977 0.945 0.945 0.945 0.945 (fs) jul/aug 0.959 1.066 0.979 0.959 0.959 0.959 0.959 aug/sep 1.040 1.020 1.036 1.040 1.040 1.040 1.040 sep/oct 1.055 1.000 0.984 1.044 1.045 1.027 1.032 oct/nov 1.054 1.000 1.000 0.990 0.961 1.047 1.035 nov/dec 0.972 1.041 0.992 0.972 0.985 0.972 0.972 values of the frontier shift are relatively stable and up to july they have a value less than 1, and after that period values become higher than 1 which presents technological progress. the technological progress of the observed sample is necessary according to market conditions. it is likely that this trend will continue in the future. variations in these values are directly caused by competition and lower prices of services and products, and thus lower profitability. the established technological progress corresponds to the results obtained in research on examples of other industries such as [45, 46]. dmu 2 and dmu 3 do not change significantly since the ratio of efficiency in certain time intervals is 1. the efficiency of other dcs changes over time to some extent. the multifactor productivity index is shown in the first part of table 8. mi values higher than 1 indicate a positive change and values lower than 1 indicate a negative change of multifactor productivity. dmu 3 records a positive change of multifactor productivity measuring efficiency change in time applying malmquist productivty index: a case of distribution centres... 511 which largely corresponds to the real situation. mi values for dmu 1 vary significantly which corresponds to the real situation. this phenomenon is a direct consequence of operation changes. dmu 3 takes over a large number of retail stores of dmu 1. dmu 1 has specialized for specific groups of retail stores and products. common for most of the dcs is the positive change of mi in a period of „peak“: november-december. one of the shortcomings of this analysis is a relatively short period of observation. despite the large number of time sections (in this case 12 months), the total period of observation is only one year. during this period some significant and radical technological developments cannot be expected. however, some changes in the way of operating of some dcs quickly reflect on the positive change of mi. for example, dmu 3 records positive changes of mi in the long period of six months, as a direct result of the introduction of better and more convenient warehouse management system (wms). these and similar changes are prerequisite for successful operation of dc. the limitation of this research is reflected in the relatively small number of dcs considered. more authoritative conclusions could be obtained if the developed model were applied to other companies and in other market conditions (other countries). this could fully explain the impact of markets, environments and organizational changes on the efficiency of logistics systems. a special aspect of future research is sustainability efficiency [47]. 7. conclusion one of the most important factors for market survival is measuring, monitoring and improving efficiency change in time. the main objective of this paper is the development of a model which measures dc efficiency in an appropriate way and defines appropriate corrective actions that can be applied in real logistics systems. models are tested on real data of one company in serbia. by analysing the results, the models which did not adequately describe the operation of dcs were rejected. after testing, model 3 was further analysed and varied with the aim of improvement. model orientation was studied and in the case of output orientation. a more detailed analysis of input and output variables shows that variables which are mostly used in dea method application are not suitable for analysing dc efficiency. the first problem that has been successfully solved in this paper relates to the choice of variables to be used in the model. the problem of a large number of indicators in logistics systems that are monitored but not used in the distribution process has been successfully overcome. it was found that less than 20% of the indicators monitored are used in the further decision-making process. preliminary testing also eliminated those variables that do not affect the discriminatory power of the model. in this paper, among the great number of potential input and output variables which in the best way describe dc operation, the following were chosen: number of pallet places, number of retail stores and number of realized deliveries. the selection of input and output variables greatly affects efficiency scores and corrective actions. malmquist productivity index was used for measuring efficiency change over time. the results show efficiency changes in a relatively short period (12 months). in the process of model development, the assumption from the literature was confirmed. smaller dcs are more efficient than larger dcs [21]. in almost all models presented in the paper, the smallest dc in the sample, dmu 2, had an efficiency of over 90%, which confirms the assumption. the frontier shift was found to be stable until july when values greater than 1 were observed 512 m. andrejić, m. kilibarda, v. pajić and technological progress was recorded. mi values show that some decision units have positive and some negative changes in multifactor productivity. a positive change in the efficiency of dmu 3, which fully corresponds to the technological changes (introduction of more advanced wms) and redistribution of activities with other dcs. a positive change in mi during the peak period (november-december) was found. this is a direct consequence of better resource utilization and higher turnover. in literature, there is a lack of case studies, i.e., model testing in the concrete dc examples. this fact indicates the insufficient amount of researches in this area. this paper shows how a theoretical model can be applied in practice. the developed methodology represents support in the decision-making process. models presented in this paper, with minor adjustments, can be used for measuring and improving the efficiency of providers, warehouses, suppliers, etc. presented models are a good basis for the development of future models with a larger number of input and output variables. in future 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12(4), 1367. 46. wang, c.n., tibo, h., nguyen, h.a., 2020, malmquist productivity analysis of top global automobile manufacturers, mathematics, 8(4), 580. 47. klumpp, m., 2017, do forwarders improve sustainability efficiency? evidence from a european dea malmquist index calculation, sustainability, 9(5), 842. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 147 158 original scientific paper cad/cam design and the genetic optimization of feeders for sand casting process udc 621.7 nedeljko dučić 1 , žarko ćojbašić 2 , radomir radiša 3 , radomir slavković 1 , ivan milićević 1 1 faculty of technical sciences, ĉaĉak, university of kragujevac 2 faculty of mechanical engineering, university of niš 3 research and development institute lola l.t.d. abstract. the paper proposes methodology of feeder design and optimization for sand casting process. casting part is a part of excavator buckets, i.e. holder of the cutting tooth. process of design and optimization is based on the application of the rules, which are the result of many years of work researchers in the field of metal casting. computer aided design (cad) is used as a methodology in the design of feeders. genetic algorithm (ga) as an artificial intelligence technique is used in the optimization process of the feeder geometry. computer aided manufacturing (cam) is used as methodology that involves numerical simulation of the casting process. numerical simulation is used to verify the validity of the optimized geometry of the feeding system. key words: sand casting, feeder system, optimization, genetic algorithm, numerical simulation 1. introduction cooling of casting part causes shrinkage of material, therefore, the basic function of feeders is to compensate for the deficiency of metal during casting part solidification. unlike filling as a relatively short process, feeding is a necessary, time-consuming and slow one. casting part cooling involves three phases of volume contraction (shrinkage) – liquid shrinkage, solidification shrinkage and solid shrinkage [1]. volume contraction is manifested through some adverse phenomena, i.e. internal shrinkage voids, surface deformation or dishing, and received august 15, 2015 / accepted january 28, 2016 corresponding author: nedeljko duĉić faculty of technical sciences ĉaĉak, university of kragujevac, svetog save 65, 32000 ĉaĉak e-mail: nedeljko.ducic@ftn.kg.ac.rs 148 n. duĉić, ţ. ćojbašić, r. radiša, r. slavković, i. milićević surface puncture. the elimination of the above side effects is obtained by designing feeders which are removed after cooling the casting part. optimal design of feeders, in addition to providing higher quality products, also provides consumption savings of molten metal in the casting process. the total volume of feeder should be minimized to improve the casting yield and productivity [2]. jacob et al. (2004) presented the integration of genetic algorithm (ga) and cad in the design and optimization of feeders in casting process. the synergy of cad and genetic algorithms generates a class of optimal feeders that also satisfy the standard criteria set up in literature [3]. tavakoli and davami (2007) focus on the design of the feeding system in the sand casting process using topology optimization (solid isotropic material with penalisation method) [4]. tavakoli and davami (2009) combine finite-difference analysis of the solidification process with evolutionary topology optimization to systematically improve the feeding system design [5]. kotas et al. (2010) summarize the findings of multi-objective optimization of a gravity sand-cast steel part for which an increase of the casting yield via feeder optimization is considered. this is accomplished by coupling a casting simulation software package with an optimization modulus [6]. sutaria et al. (2011) present a new approach to evaluating and optimizing casting feeding system design using feed-paths. the feed-paths are computed by vector element method (vem) [7]. choudhari et al. (2013) describe the parameters to be considered while designing a feeder of an optimum size to get higher casting yield. theoretical design model has been analyzed in ansys 12.0 simulation software to ensure that shrinkage porosity is completely eliminated from casting [8]. campbell (1991, 2004 & 2011) propose seven conditions for feeding casting part: 1. do not feed (unless necessary); 2. the heat-transfer requirement; 3. mass-transfer (volume) requirement; 4. the junction requirement; 5. feed path requirement; 6. pressure gradient requirement and 7. pressure requirement [9]. those conditions were the starting points for the design and optimization of feeders in the research studies. this paper presents a methodology of designing and optimizing the feeders for casting a cutting tooth holder. based on the rules given in [9], using the computer aided design/ computer aided manufacturing systems and genetic algorithm (as a technique of the artificial intelligence), the optimal feeding system for casting part is obtained. the paper includes three crucial rules proposed by campbell: (2) the heat-transfer requirement, (3) mass-transfer (volume) and (5) feed path requirement. those rules are included in the optimization process through the initial setup of the feeding system, definition of fitness function and definition constraints. numerical simulation confirm the validity of the implemented methodology for feeder design and optimization. 2. feeder system design casting part is a part of excavator buckets, i.e. a cutting tooth holder (its cad model is shown in fig. 1). production technology of the cutting tooth holder is sand casting. molds for single use are made of a mixture of sand, binders and additives. a quartz sand (sio2) is usually used in making molds. the core is more exposed to the molten metal and high temperatures. therefore, the core is made of sand with rough grains. cad/cam design and the genetic optimization of feeders for sand casting process 149 fig. 1 cad models: cutting tooth, cutting tooth holder and an entire excavator bucket fig. 2 presents the casting part (a cutting tooth holder) and gating system with its marked elements. fig. 2 gating system for casting a cutting tooth holder the first step in the feeding system design is to define the type of feeder, the number of feeders and their position in the system of casting. the selected type of feeders is a cylindrical open feeder because of the largest modulus (the cylinder has the smallest surface area per volume). place of feeders is determined so as to provide "directional solidification", thus achieving the solidification of thinner to thicker cross sections of the casting part and finally in the feeders. the number of feeders depends on the feeding distance. there has been much experimental effort to determine feeding distances. the early work by pellini and his co-workers (summarized by beeley (1972)) at the us naval materials laboratory is a classic investigation that has influenced the thinking on the concept of feeding distance ever since. the number of feeders and feeding distance are defined on the basis of the rules: “feed path requirement feeding distance”. therefore: 150 n. duĉić, ţ. ćojbašić, r. radiša, r. slavković, i. milićević 4.5dl t (1) where: ld – feeding distance, t – thickness of the casting. approximate dimensions of casting part are 295x320x40 mm, so the recommended feeding distance ld=180 mm. given the dimensions of the base of casting part (295x320 mm), it is necessary to use two feeders positioned to meet the requirement of distance feeding ld. namely, the proposed solution comprises two cylindrical symmetric feeders intended to compensate for volume shrinkage of metal (fig. 3). fig. 3 the proposed gating system with feeders the next challenge in the design of the feeding system is the geometry feeders‟ optimization. 3. ga optimization of feeder optimization of feeders is based on the use of genetic algorithm (ga) and rules: (2) the heat-transfer requirement and (3) mass-transfer (volume) requirement. genetic algorithms (gas) are a family of adaptive search algorithms described and analyzed in references [10-14]. gas derive their name from the fact that they are loosely based on models of genetic change in a population of individuals. these models consist of three basic elements: (1) a darwinian notion of „fitness‟, which governs the extent to which an individual can influence future generations; (2) a „mating operator‟, which produces offspring for the next generation; and (3) „genetic operators‟, which determine the genetic makeup of offspring from the genetic material of the parents [13]. the main objective of the optimization is to find the value of diameter (d) and height (h) of the feeder, with minimizing volume of the feeder and respecting constraint specified on the rules. cad/cam design and the genetic optimization of feeders for sand casting process 151 fig. 4 algorithm of optimization and design of feeders 3.1. fitness function defining the fitness function is one of the major steps of optimization via ga. creating the fitness function is based on the rule: the heat-transfer requirement. the heat-transfer requirement for successful feeding can be stated as follows: the freezing time of the feeder must be at least as long as the freezing time of the casting [9]. casting solidification time can be predicted using chvorinov‟s rule shown by the equation: 2 s v t c a        (2) where st is the total solidification time of the part or feeder, c is a mold constant, v is the volume of metal, and a is the total surface area of the part or feeder. as the feeder serves to compensate for the volume shrinkage in the casting part, it needs to be the final link of the solidification chain, i.e. it should be the last to solidify in directional solidification process. the longest solidification time for a given volume is the one where the shape of 152 n. duĉić, ţ. ćojbašić, r. radiša, r. slavković, i. milićević the part has the minimum surface area. the cylinder has the smallest surface area per volume and can easily be made, which was crucial when designing the feeder. feeder modulus is taken to be a minimum of 20 % higher than the modulus of casting part, i.e. modulus of part who feeds: _1.2feeder casting partm m  (3) the modulus for a cylindrical feeder is given by the following equation: 2 4 feeder dh m d h   (4) based on the values of volume and surface of the casting part, obtained by analyzing cad model (vc=3661113.8 mm 3 and ac=253716.66 mm 2 ), the casting part modulus is: mcasting_part = 14.43. as the feeders are symmetrical and designed that each of them should feed half of the casting part, relation (5) gives the final expression of fitness function that minimizes: _ 1.2 8.66 0 2 2 4 casting part feeder m dh f m d h       (5) 3.2. defining constraints constraint is based on the rule mass-transfer (volume) requirement. the fulfillment of the conditions given in the rule “the heat-transfer requirement”, does not guarantee the fulfillment of the conditions given in the rule “mass-transfer (volume) requirement”. although the feeder may have been provided of such a size that it theoretically would contain liquid until after the casting is solid, in fact it may still be too small to deliver the volume of feed liquid that the casting demands. thus it will be prematurely sucked dry, and the resulting shrinkage porosity will extend into the casting. this is because they are themselves freezing at the same time as the casting, depleting the liquid reserves of the reservoir. effectively, the feeder has to feed both itself and the casting. we can allow for this in the following way. if we denote efficiency ε of the feeder as the ratio (volume of available feed metal) / (volume of feeder, vf) then the volume of feed metal is, of course   vf. if the solidification shrinkage of the liquid is α, during freezing, then the feed demand from both the feeder and casting together is α(vf,vc), and hence [15]: ( )f f cv v v   (6) solving for the feeder volume yields: 2 0.4 0.2 2 4 c c f c v v d h v v          (7) where: α – solidification shrinkage of the liquid during freezing ( 4  , for steels the fcc structure applies above 0.1% carbon where the melt solidifies to austenite), 14%  – for a normal cylindrical feeder, vc/2 – because one feeder feeds half of the casting part. the relation (7) is a constraint in the optimization process of geometry feeders. cad/cam design and the genetic optimization of feeders for sand casting process 153 3.3. optimization results search domains of size values that are being optimized are given in table 1. the upper and lower limits are set in accordance with the proposed geometry of the casting part and mold. feeder height is limited by the mold height. table 1 optimal values search domain d (mm) h (mm) lower bound 10 10 upper bound 150 80 the population range is 100 individuals, with 100 generations at maximum. stochastic universal sampling algorithm is used in the selection. the population is seen as mapped on the roulette-wheel, larger parts of the wheel belonging to strings with lower fitness (since it is a minimization problem). n pointers are evenly placed on the roulette, n being the number of individuals in a population. a population is generated by rotating the roulette once. uniform crossover is used as the operator of the crossing. the coefficient of the „crossover fraction‟ is 0.8. crossover fraction defines a portion of the new population derived from a crossing (non-elite individuals), its value being between 0 and 1. no more than two elite individuals are to be transferred to the next generation. the operation of ga is terminated as early as the 92 generation due to the violation of specified limits. the obtained optimized values are as follows: d=105 mm and h=80 mm. fig. 5 shows the position of the proposed feeders. fig. 5 the proposed gating system with feeders and sand mold 4. numerical simulation in addition to practical knowledge applied in foundries, simulation programs are used to better understand the casting process and manage it more easily. the primary objective of simulation is to confirm the validity of the methodology applied in the optimization and design of the feeding system, and reduce the cost and ensure reliable and quality production 154 n. duĉić, ţ. ćojbašić, r. radiša, r. slavković, i. milićević accordingly. the entire casting process, from filling the mold cavity with liquid metal and all the way to solidification and feeding is available as a good representation of the physical model. computer „cold casting‟ allows improving the process step by step [16]. geometry casting is substantially facilitated, i.e. it is easy to perform changes in the geometry of the casting part, the position of feeder and core, etc. as well as in the parameters of the casting process (temperature of the casting part, filling speed, etc.). modern software package magmasoft 5 (version 5.2) is used to simulate the casting process [17]. model of casting part, as well as models of the optimized gating system and feeders‟ geometry, are used in the stl graphical format. in our paper, the alloy which is being the subject of the casting simulation is not available in the software database, therefore it is required that the data be loaded into the database. a material with chemical composition and properties similar to the one to be cast are chosen from the database. the chosen material has its name changed and loaded into the user‟s part of the database whereupon it is possible to load a new composition of chemical elements and features for the material to be cast. the properties of the new material remain in the database. chemical composition of an alloy affects its castability and behavior during the casting process, as well as the properties of the final casting part. the paper describes simulation of casting steel of chemical composition 0.5 % c, 1.4 % mn, 0.035 % p, 0.035 % s, 0.4 % si, 0.1 % v. compared to standard content of chemical elements in steel (designation gs30mn5), contents of carbon, manganese and vanadium are changed. initial temperatures of steel, sand mold and coldbox are 1600 °c, 40 °c and 20°c respectively. basic limiting conditions at casting are as follows: liquidus temperature 1498°c; solidus temperature of 1406 °c; criterion temperature 1: 1415.2 °c. at a temperature named "criterion temperature 1" feeding of the cast part is calculated as the default value; criterion temperature 2: 1500 °c. criterion temperature 2 is calculated temperature based on the cooling rate and gradient of change in temperature over time; latent heat 254 kj/kg; feeding effectivity 30 %. feeding effectivity criterion shows the volume of material to which macroscopic feeding is possible; surface tension coefficient 1.496 n/m. subsequent to defining groups and types of materials in the casting system, it is necessary to determine heat transfer coefficients – htc for pairs of materials in contact. in sand casting the mold and the core surfaces are coated so as to affect thermal behavior of the mold and core. fig. 6 heat transfer coefficient in low alloy (la)-steel-coat pair cad/cam design and the genetic optimization of feeders for sand casting process 155 the database provides data on htc coefficients for different material pairs. fig. 6 shows the htc for low-alloy steel and the mold coated with a layer of a particular thickness. mold cavity and core are coated with a layer of 0.1 mm thick. heat transfer coefficients – htc applied in the simulation are used from the database software (in sand-core we use constant htc=800 w/m 2 k, in feeder-sand and gate-sand we use the temperature-dependent steel-sand coefficient). each htc can be modified according to the specific conditions in the drive, which includes the additional thermal conductivity. properly defined geometry network is in the basis of any simulation and is essential for the accuracy of the simulation results. the fineness and density of the network enables more accurate results but the simulation requires longer time. the total number of final volumes is one million, whereby the number of volume elements pertaining to metal 157611, without critical elements present. to bring the simulation into correlation with the physical process in reality requires that at least 3 elements of network volume are provided between two adjacent walls. in our study, the number of volumes between two adjacent walls on the axes x/y/z is 154/130/49, respectively. to define solidification, i.e. cooling of the casting part, parameters and limiting conditions are required. primarily, time or temperature at which the solidification simulation ends need to be specified first. in our study, the final temperature in the simulation is 200 °c. the feedmod criterion (fig. 7) is specifically designed for sand casting. the objective is to change the value of „feedmod‟ to allow directional solidification to feeders. the highest values should be in the feeder; they gradually become lower in the casting part. the values of the thermal module confirm the validity of the proposed optimized feeders. fig. 7 feedmod criterion liquidus to solidus criterion shows the time needed for a material to pass from liquid to solid state. it helps us to discover areas in the casting part that solidify later, particularly the volumes which solidified after the feeding time (isolated volume). fig. 8 confirms the correctness of the designed gating system and feeders by this criterion, given that the solidification period is the longest in the feeding area. hot spot result shows the volumes of metals that solidify last. it helps us to identify the areas containing residual 156 n. duĉić, ţ. ćojbašić, r. radiša, r. slavković, i. milićević liquid metal. it also enables us to foresee possible macroscopic feeding of the volume from the feeder. based on fig. 9, there is a volume that solidifies in the feeder, which is a prerequisite for a regular cast. fig. 8 time elapsed from liquidus to solidus temperature fig. 9 hot spot criterion total porosity result shows cumulative porosity and microporosity in a single result, i.e. maximum values of porosity and micro-porosity. fig. 10 shows the x-ray total porosity in the casting part and that through the mid-section of the feeder. total porosity is expressed through percentage (%). the x-ray view of the casting part renders conclusion that alloy casting technology provides required quality. cad/cam design and the genetic optimization of feeders for sand casting process 157 fig. 10 porosity criterion 5. conclusion the aim of this study is to present a methodology for optimizing the geometry of the feeding system for sand casting process based on the application of genetic algorithm. the result of the optimization are diameter and height of feeders (d=105 mm and h=80 mm), obtained as function of minimizing volume of feeder. based on the obtained results of the optimization, the entire feeding system is designed by means of cad software. performed numerical simulations confirm the correctness of the designed optimized feeding system in view of all the criteria. the proposed methodology of design and optimization provides a complete filling of the mold cavity, directional solidification and eliminate porosity. acknowledgements: the paper is a part of the research done within the project tr35037 and tr35015. references 1. asm handbook committee, 2002, asm metals handbook, asm international, 1246 p. 2. ravi, b., srinivasan, m., 1996, casting solidification analysis by modulus vector method, int. j. cast. met. res., 9(1), pp. 1–7. 3. jacob, e., sasikumar, r., praveen, b., gopalakrishna, v., 2004, intelligent design of feeders for castings by augmenting cad with genetic algorithms, journal of intelligent manufacturing, 15(3), pp. 299–305. 4. tavakoli, r., davami, p., 2008, optimal riser design in sand casting process by topology optimization with simp method i: poisson approximation of nonlinear heat transfer equation, structural and multidisciplinary optimization, 36(2), pp. 193-202. 5. tavakoli, r., davami, p., 2009, optimal riser design in sand casting process with evolutionary topology optimization, structural and multidisciplinary optimization, 38(2), pp. 205-214. 158 n. duĉić, ţ. ćojbašić, r. radiša, r. slavković, i. milićević 6. kotas, p., tutum, c. c., hattel, j. h., snajdrova, o., thorborg, j., 2010, a casting yield optimization case study: forging ram, international journal of metalcasting, 4(4), pp. 61-76. 7. sutaria, m., joshi, d., jagdishwar, m., ravi, b., 2011, automatic optimization of casting feeders using feedpaths generated by vem, proceedings of the asme 2011 international mechanical engineering congress & exposition, 2011, denver, colorado, usa. 8. choudhari, c. m., dalal, n. s., ghude, a. p., sankhe, p. p., dhotre a.s., 2013, design and analyisis of riser for sand casting, international journal of students research in technology & management, 1(2), pp. 176-191. 9. campbell, j., 2011, complete casting handbook, 1st edition, elsevier, 661 p. 10. holland, j. h., 1992, adaptation in natural and artificial systems, mi: university of michigan press, ann arbor, 82 p. 11. de jong, k., 1980, adaptive system design: a genetic approach, ieee transactions on systems, man, and cybernetics, 10(9), pp. 556-574. 12. de jong, k., 1985, genetic algorithms: a 10 year perspective, proceedings of the first international conference on genetic algorithms and their applications, pittsburgh, pa: lawrence erlbaum. 13. de jong, k., 1988, learning with genetic algorithms: an overview, machine learning, 3(2-3), pp. 121-138. 14. goldberg, e. d., 2006, genetic algorithms, pearson education, 62 p. 15. kotas, p., 2011, integrated modeling of process, structures and performance in cast parts, ph.d. thesis, technical university of denmark department of mechanical engineering, denmark, 50 p. 16. fang, y., hu, j., zhou, j., yu, y., 2014, numerical simulation of filling and solidification in exhaust manifold investment casting, international journal of metalcasting, 8(4), pp. 39-45. 17. jeong, s., jin, c., k., seo, h., y., kim, j., d., kang, c., g., 2015, mold structure design and casting simulation of the high-pressure die casting for aluminum automotive clutch housing manufacturing, the international journal of advanced manufacturing technology, doi 10.1007/s00170-015-7566-4. facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 361 380 https://doi.org/10.22190/fume210214031p © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making: application in logistics dragan pamučar1, mališa žižović2, sanjib biswas3, darko božanić1 1university of defense in belgrade, military academy, belgrade, serbia 2faculty of technical sciences in čačak, university of kragujevac, serbia 3decision sciences and operations management area, calcutta business school, india abstract. logistics management has been playing a significant role in ensuring competitive growth of industries and nations. this study proposes a new multi-criteria decision-making (mcdm) framework for evaluating operational efficiency of logistics service provider (lsp). we present a case study of comparative analysis of six leading lsps in india using our proposed framework. we consider three operational metrics such as annual overhead expense (oe), annual fuel consumption (fc) and cost of delay (cod, two qualitative indicators such as innovativeness (in) which basically indicates process innovation and average customer rating (cr)and one outcome variable such as turnover (to) as the criteria for comparative analysis. the result shows that the final ranking is a combined effect of all criteria. however, it is evident that in largely influences the ranking. we carry out a comparative analysis of the results obtained from our proposed method with that derived by using existing established frameworks. we find that our method provides consistent results; it is more stable and does not suffer from rank reversal problem. key words: logarithm methodology of additive weights (lmaw), bonferroni aggregator, operational performance, logistics service providers, rank reversal, sensitivity analysis 1. introduction logistics management (lm) encompasses an uninterrupted flow of materials, services, and information related to the movement through seamless integration of all stages of the supply chain connecting the points of source and use [1]. the broader spectrum of lm includes various activities like material handling and storing, inventory received february 14, 2021 / accepted march 18, 2021 corresponding author: dragan pamučar university of defence in belgrade, military academy, pavla jurišića šturma 33, belgrade, serbia e-mail: dpamucar@gmail.com 362 d. pamučar, m. žićović, s. biswas, d. božanić optimization and management, network planning, transportation arrangement, order processing, distribution planning, channel management, and management of returns [2]. in this era of globalization, lm bridges the interrelated and interdependent supply chains of different partnering organizations and industries spreading over a wide geographical region. lm enables the industries to consolidate their resources for optimization of cost, generate supply chain surplus and offer utmost service quality to the customers [3]. a country’s competitive growth especially for the developing nations like india is significantly contributed by lm activities. according to a recent market research [4], organizations across the globe are increasingly focusing on creating a global production base which largely depends on effective lm. india as a fastest growing economy in the south-east asia with surpassing demographic dividend and tremendous market size and variety, is significantly positioned as a potential driver of global operations in the coming decades. an effective lm planning and execution can bolster the ambitious initiatives like “make-in-india” led by the government of india (goi). a very recent report [5] has estimated a cagr of 10.5% from 2019 to 2025 for the logistics sector in india which shall draw a notable foreign direct investment (fdi) and cash inflow to the country. hence, it is quite imperative to mention that logistics is under the spotlight from industrial and country’s growth perspective and as a result, a lot of research works are being conducted by the practitioners and scholars on lm. in this context, logistics service providers (lsp) play a crucial role. in this era of extreme competitions, the organizations are putting more emphasis on strengthening their core competencies for improving performance, reducing operational costs, and capital investments, optimally utilizing resources, and, finally, providing better quality products and services to the customers, thereby increasing return on investment for the shareholders [6-7]. hence, the importance of lsps has been increased in the last two decades. most of the organizations outsource their lm activities to the lsps. however, as lsps have become strategic partners to the firms, selection of an appropriate vendor is of paramount importance to the supply chain managers. selection of a lsp is a complex task that depends on multiple aspects (both subjective and objective) which quite often are conflicting in nature [8]. there have been a sizeable number of research contributions towards developing a measurement framework for assessing lm performance of the service providers. some of the parameters that are mentioned in extant literature include order fulfillment, on time delivery, faster response, reduction in lead time, improved service quality for customer delight, flexibility and adaptability, convenience, sharing of information, seamless coordination and cooperation, optimization of operational cost, innovativeness, adoption of new technologies, reputation building, and the ability to withstand uncertainties [9-19]. it is evident from the discussions and observations on the past work that the comparative performance assessment of the lsps is a mcdm issue. for solving real-life complex problems, the decision-makers (dm) are confronted with the requirement of consistent decision-making through rational evaluation of the possible alternatives subject to the influence of conflicting criteria [20]. mcdm frameworks enable the dms to evaluate available possibilities under the effect of different criteria in a structured and cost effective way with reasonable precision and accuracy to arrive at an acceptable solution [21-22]. as a result, mcdm techniques are frequently used by the researchers and dms for solving variety of complex problems, for example, related to facility location selection [23], supply chain performance [24-26], investment decision-making a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 363 [27]. over the years researchers have developed various mcdm methods which are dissimilar in nature. the features that differentiate various mcdm methods are formulation of decision matrix, choice of normalization, functionality and applications, type of information (subjective and objective) and computational algorithms. as a result, the selection of an appropriate mcdm technique for solving a given problem is essential to find out optimum solution [28]. the literature is rife with a significant number of valuable contributions by several researchers pertaining to the mcdm domain. the evolution of the stated field has been supported by several algorithms. some of the popularly used mcdm frameworks are simple additive weighting (saw) [29], elimination et choice translating reality (electre) [30], analytical hierarchy process (ahp) [31], multicriteria optimization and compromise solution (serbian: više kriterijumska optimizacija i kompromisno rešenje (vikor)) [32-33], technique for order preference by similarity to ideal solution (topsis) [34], preference ranking organization method for enrichment evaluation (promethee) [35-36], multi-attribute utility function based mcdm [37], complex proportional assessment (copras) [38], analytic network process (anp) [39], multi-objective optimization by ratio analysis (moora) [40], and its subsequent extension (with full multiplicative form) known as multimoora [41], additive ratio assessment (aras) [42], step‐wise weight assessment ratio analysis (swara) [43], multi-objective optimization on the basis of simple ratio analysis (moosra) [44], weighted aggregated sum product assessment (waspas) [45], kemeny median indicator ranks accordance (kemira) [46], multi-attributive border approximation area comparison (mabac) [47], evaluation based on distance from average solution (edas) [48], combinative distance-based assessment (codas) [49], pivot pairwise relative criteria importance assessment (piprecia) [50], full consistency method (fucom) [51], combined compromise solution (cocoso) [52], level based weight assessment (lbwa) [53], measurement of alternatives and ranking according to compromise solution (marcos) [54], and ranking of alternatives through functional mapping of criterion sub-intervals into a single interval (rafsi) [55]. in this paper, we introduce a new mcdm algorithm such as lmaw. the lmaw method presents a new multi-criteria decision-making framework that has a methodology for determining the weight coefficients of the criteria. the lmaw method showed greater stability compared to the topsis method, which is based on similar principles, respectively, the definition of the distance of alternatives in relation to reference points. compared to the topsis method, the lmaw method showed robustness of results when changing the number of alternatives in the initial decision-making matrix. the topsis model showed that eliminating the worst alternatives from the decision-making matrix led to the change in the existing rank, respectively, to the occurrence of the rank reversal problem. on the other hand, the lmaw method did not cause rank reversal problems. thus, the lmaw method showed significant stability and reliability of results in a dynamic environment. it is also important to note that in numerous simulations the lmaw method showed stability when processing larger data sets. this was confirmed also by the case study discussed in this paper. in addition to the above mentioned, the following advantages of the lmaw method can be highlighted: (1) mathematical framework of the method remains the same regardless of the number of alternatives and criteria; (2) a possibility of application in the case studies considering a number of alternatives and criteria; (3) a clearly defined range of alternatives expressed in numerical values, which makes it easier to understand the results; and 364 d. pamučar, m. žićović, s. biswas, d. božanić (4) the presented methodology allows the evaluation of alternatives expressed by either qualitative or quantitative types of criteria. the rest of the paper is structured as follows. in section 2, we summarize some of the related work in the field of performance evaluation of lsps. in section 3, we elucidate the new methodology and define the computational steps. section 4 presents the case study of comparative evaluation of logistics service providers in the indian context wherein we apply the new methodology. section 5 exhibits the analysis and findings related to validation and sensitivity analysis of the proposed model. finally, section 6 concludes the paper while highlighting some of the implications of this research and future scope. 2. literature review we notice that several mcdm techniques are applied for comparative performance analysis of the lsps in umpteen occasions. for instance, in [56] anp was applied for selection of lsp from growth perspective for a medium-scale fmcg organization. a combination of anp and topsis was considered in the work of [57]. some researchers (for example, [11]) have considered qualitative information and applied delphi method in conjunction with anp. optimization is also given due consideration by the contributors. as example, data envelopment analysis (dea) was used in the work of [58] while in [59], a combination of ahp and goal programming (gp) was applied. bajec and tuljaksuban [19] used a combination of ahp and dea to solve lsp selection problem. however, andrejić [60] mentioned the difficulty of precise assessment of logistics performance due to the presence of many conflicting aspects. it is evident from the literature that researchers put due diligence to the issue of impreciseness. we find that a good number of works have been carried out in uncertain environment. in this regard, we observe three strands of literature: the first one applied fuzzy concepts; the second one worked with rough numbers and the final one used grey theory based models. apart from these, some contributions included a combined approach also. the study of [61] used an integrated fuzzy ahp and integer gp while the authors [62] relied on a combined fuzzy ahp-topsis framework. on a different note, we observe that in [63] logic and rule based reasoning, and compromise solution based algorithms were used for the comparative analysis. in this category, liu and wang [64] put forth an integrated delphi, inference system and linear assignment based framework for solving the lsp selection problem. causal mcdm techniques like interpretive structural modeling (ism) have also been used to delve into the interrelationship among the criteria along with the outranking algorithm like fuzzy topsis for the selection of suitable third party lsp (3pl) for the return channel for a battery manufacturer [65]. akman and baynal [66] conducted the research on selection of 3pl for a tire manufacturing unit using fuzzy ahptopsis model. for selecting a reverse logistics partner, prakash and barua [67] took help of fuzzy ahp and vikor while in a recent work, li et al. [17] introduced the concept of the prospect theory and applied fuzzy topsis. in the work of [16], we observe that an expert decision-making framework has been used wherein the authors used fuzzy swara and copras approach. the combination of fuzzy ahp-topsis is seen as a popular framework [18]. however, some contributors (e.g., [68]) have also considered the degree of indeterminacy and carried out a more granular analysis using hesitant and intuitionist fuzzy sets. for enhancing clarity and preciseness in analysis, the concept of rough numbers has also been used significantly. for instance, sremac et al. a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 365 [15] used rough swara–waspas model while pamucar et al. [69] applied interval rough number based best worst method (bwm)-waspas-mabac framework for ranking of 3pls. nevertheless, in some cases fuzziness cannot be determined realistically (e.g., opinion based analysis when varying levels of measurement and considerable amount of information is not available explicitly or information loss is present) [70]. under those circumstances, the grey theory [71-72] has been considered by many scholars while applying mcdm models. for instance, in [14] a grey forecasting based analysis was carried out. mercangoz et al. [73] devised a grey based copras scheme for evaluating competitiveness of lm performance of european union (eu) member states. 3. new mcdm framework: logarithm methodology of additive weights (lmaw) in the following section, the new logarithm methodology of additive weights (lmaw) is presented as implemented through six steps: step 1: forming initial decision-making matrix (x). in the first step, it is performed the evaluation of m alternatives a = {a1,a2,...,am} compared to n criteria c = {c1,c2,...,cn}. the weight coefficients of criteria wj (j = 1,2,...,n) are defined also meeting the condition where 1 1 n j j w = = . it is assumed that the evaluation of the alternatives is performed by k experts e = {e1,e2,...,ek} based on a predefined linguistic scale. then, for every expert what is obtained is matrix [ ]e e ij m nx  = (1  e  k), where e ij presents the value from the defined linguistic scale. applying bonferroni aggregator through the expression (1), aggregated initial decision-making matrix x = [ij]mn is obtained: 1 ( ) ( ) 1 1 1 ( ) ( ) ( 1) p q k k x p y q ij ij ij x y y x k k    + = =     =  −      (1) where ij presents the averaged values obtained by applying bonferroni aggregator (1); p,q  0 present stabilization parameters of the bonferroni aggregator, while e presents the e-th expert 1  e  k. step 2: standardization of the initial decision-making matrix elements. standardized matrix 11[ ]m n  = is obtained by applying the expression (2). is benefit, is cost. ij j ij j j ij ij j ij j ij if c if c          + + −  + =  =  + =  (2) where max( )j ij i  + = , min( )j ij i  − = , while ij presents the standardized values of the initial decision-making matrix. 366 d. pamučar, m. žićović, s. biswas, d. božanić step 3: determining weight coefficients of the criteria. the experts from the group e = {e1,e2,...,ek} prioritize criteria c = {c1,c2,...,cn} based on the value from the predefined linguistic scale. prioritizing is performed by adding a higher value from the linguistic scale to the criterion with higher significance, while adding a lower value from the linguistic scale to the criterion with lower significance. in this way what is obtained is priority vector 1 2( , ,.., )e e e e c c cnp   = , where e cn presents the value from the linguistic scale assigned by expert e (1  e  k) to criterion tc (1  t  n). step 3.1: defining absolute anti-ideal point ( aip ). absolute anti-ideal point is defined in relation to the minimum values from the priority vector and should be lower than the smallest value from the priority vector. we can define  aip value as  aip =  e min / s, where min 1 2min{ , ,..., }e e e e c c cn   = , and s is a number greater than the base of logarithm (a). if we take ln as a logarithmic function, then s = 3. step 3.2: applying the expression (3), the relation is determined between the elements of the priority vector and absolute anti-ideal point ( aip ). e e cn cn aip    = (3) thus we obtain relation vector 1 2( , ,.., )e e e e c c cnr   = , where e cn presents the value from the relation vector which is obtained by applying the expression (3), while re presents the relation vector of expert e (1  e  k). step 3.3: determining the vector of weight coefficients wj = (w1, w2,...,wn)t. applying the expression (4), the values of the weight coefficients of the criteria are obtained for expert e (1  e  k): log ( ) , 1 log ( ) e e a cn j e a w a b  =  (4) where e cn presents the elements of relation vector r, while 1 n b e cn j b  = = . such obtained values of the weight coefficients meet the condition where 1 1 n e jj w = = . applying bonferroni aggregator as in the expression (5), we obtain the aggregated vector of weight coefficients wj = (w1, w2,...,wn)t. 1 ( ) ( ) 1 1 1 ( ) ( ) ( 1) p q k k x p y q j j j x y y x w w w k k + = =     =  −      (5) where p,q  0 present stabilization parameters of bonferroni aggregator, while e jw presents the weight coefficients obtained based on the evaluations of the e-th expert 1  e  k. step 4: calculation of weighted matrix (n). the elements of weighted matrix [ ]ij m nn  = are obtained by applying the expression (6): a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 367 2 (2 ) j j j w ij ij w w ij ij     = − + (6) where 1 ln( ) ln ij ij m ij i    = =        (7) while ij presents the elements of standardized matrix 11[ ]m n  = , while wj presents the weight coefficients of the criteria. step 5: calculation of the final index for ranking alternatives (qi). the rank of alternatives is defined based on value qi. the preferable alternative is with as high as possible value of qi. 1 n i ij j q  = = (8) where ij presents the elements of weighted matrix [ ]ij m nn  = . 4. comparison of performance of selected logistics service providers in india 4.1. the case study in our case study we consider six large scale multimodal integrated supply chain and logistics service providers in india providing the services like ftl (full truck load), ltl (less than truckload), phh (project & heavy haul), and rail (for different organizations), people transport, cfs (container freight stations), and warehousing. all these lsps are having all india presence. many of them operate worldwide including neighboring countries. these service providers are significantly old. for confidentiality of information, their names are not disclosed in this paper. let us code the names of these lsps as a1, a2, … a6. our objective is to carry out a comparative analysis of their performances using both objective operational metrics and subjective factors. the following table (see table 1) lists the criteria considered for the comparative analysis. table 1 criteria for evaluation of alternatives criteria code uom effect direction turnover (to) c1 rs. cr. (+) innovativeness (in) c2 scale value (+) annual overhead expenses (oe) c3 rs. cr. (-) annual fuel consumption (fc) c4 1000 lit (-) cost of delay (cod) c5 rs./hr. (-) average customer rating (cr) c6 scale value (+) 368 d. pamučar, m. žićović, s. biswas, d. božanić here, we consider six criteria. the first criterion (to) signifies business growth on the basis of revenue generated by providing services to the customers. in other words, it is a proxy measure of customer satisfaction. the growth prospect is not a single day affair. the firm needs to stay agile, flexible, adaptable to changes, and responsive. most importantly, organizations need to anticipate the changing scenario and customer requirements and be capable to promise service. therefore, organizations need to be innovative in terms of meeting the changing requirements as well as staying cost effective for providing services at an affordable price. hence, the second criterion (in) is of notable importance to the lsps. next, we consider criteria related to operational cost (c3 and c4). on time delivery and speed of operation are mandate for the success for the lsps. therefore, we include the fifth criterion (cod). finally, perception of performance among the customers plays a significant role in retaining existing and/or attracting new business opportunities. hence, customer rating (cr) is an important aspect that we, with due consideration, include in our analysis (c6). as evident, criteria c1, c3, c4 and c5 represent quantitative criteria, while criteria c2 and c6 belong to the group of qualitative criteria. in order to describe the quantitative group of criteria (c1, c3, c4 and c5) we have used the real indicators collected during the research, while the qualitative group of criteria (c2 and c6) is presented on the basis of expert preferences. a seven-point scale was used to present expert preferences: 1 absolutely low (al), 2 very low (vl), 3 low (l), 4 medium (m), 5 medium high (mh), 6 high (h) and 7 very high (vh). 4.2. results the evaluation of alternatives was performed by applying new logarithm methodology of additive weights (lmaw) which was implemented through six steps presented in the next section. step 1: the evaluation of alternatives was performed in relation to the six criteria presented in table 2. since criteria c2 and c6 present qualitative criteria, four experts evaluated the alternatives in relation to criteria c2 and c6. research-based unique values are defined for quantitative criteria. applying bonferroni aggregator from the expression (1), the values of the qualitative criteria are aggregated; thus we obtain the initial decision matrix: table 2 decision matrix   1 2 3 4 5 6 1 647.34 6.24 49.87 19.46 212.58 6.75 2 115.64 3.24 16.26 9.69 207.59 3.00 3 373.61 5.00 26.43 12.00 184.62 3.74 4 37.63 2.48 2.85 9.35 142.50 3.24 5 858.01 4.74 62. 6 c c c c c c a a a a a x a = 8 5 45.96 267.95 4.00 222.92 3.00 19.24 21.46 221.38 3.4 9 max max min min min max                    a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 369 the values at the position a6-c6 are obtained by averaging expert preferences 1 66 3, = 2 66 4, = 3 66 4 = and 4 66 3 = . applying bonferroni aggregator, as in the expression (1), we obtain the averaged value:   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 66 1 1 1 1 1 1 1 1 1 1 ,   1 3 4 3 4 3 3 4 3 4 4 4 31 3,4,4,3 3.49 4(4 1) ... 4 4 4 3 3 3 3 4 3 4 p q  + =    +  +  +  +  +  = = =    − + +  +  +  +  +    the remaining values of the qualitative criteria in matrix x are obtained in a similar way. step 2: applying the expression (2), we perform the standardization of the elements of initial decision matrix x ; hence we obtain the standardized matrix, table 3: table 3 standardized decision matrix   1 2 3 4 5 6 1 1.75 2.00 1.06 1.48 1.67 2.00 2 1.13 1.52 1.18 1.96 1.69 1.44 3 1.44 1.80 1.11 1.78 1.77 1.55 4 1.04 1.40 2.00 2.00 2.00 1.48 5 2.00 1.76 1.05 1.20 1.53 1.59 6 1.26 1.48 1.15 1.44 1.64 1 c c c c c c a a x a a a a = . 52 max max min min min max                    the values at the positions a1-c1 are obtained by applying the expression (2) as follows: 11 11 647.34 1.75 858.01 858.01 j j     + + + + = = = where the values are 1 1 647.34, 115.64, 373.61, 858.01 1 max( ) max 37.63, 858.0 , 222.92 i i  +   = = =    the remaining elements of the standardized matrix are obtained in a similar way. step 3: in the following section are calculated the values of the weight coefficients of the criteria. four experts prioritized the criteria based on the following scale: 1 absolutely low (al), 1.5 very low (vl), 2 low (l), 2.5 medium (m), 3 equal (e), 3.5 medium high (mh), 4 high (h), 4.5 very high (vh) and 5 absolutely high (ah). considering that the evaluation is performed by four experts, four priority vectors are defined: 370 d. pamučar, m. žićović, s. biswas, d. božanić 1 2 3 4 ( ), ( ), ( , , , , , ), ( , , , , , ). , , , , , , , , , , p p p h l l vl h l ml vl mh ah eh vl m e e ah e p a l a h h vl l l m l h h = = = = step 3.1: absolute anti-ideal point aip is arbitrary defined as value 0.5aip = . step 3.2: based on the data from the expert priority vectors and aip = 0.5, by applying the expression (3), the relation is determined between the elements of the priority vector and absolute anti-ideal point (aip). in the following section the relations are presented between the elements of the priority vector and the aip: 1 2 3 4 8, 4, 5, 3, 7, 10 9, 3, 5, 2, 6, 9 8, 4, 4, 3, 6, 10 8, 3, 4, 2, ( ), ( ), ( . 7, 8 ), ( ) r r r r = = = = the elements of vector 1r are obtained by applying the expression (3) as follows: 1 1 4 8 0.5 c = = , 1 2 2 4 0.5 c = = , 1 3 2.5 5 0.5 c = = , 1 4 1.5 3 0.5 c = = , 1 5 3.5 7 0.5 c = = and 1 6 5 10 0.5 c = = . the elements of remaining vectors r2, r3 and r4 are obtained in a similar way. step 3.3: applying the expression (4), the values of the weight coefficients of the criteria by experts are obtained: 1 2 3 4 ( ), ( ), ( ), ( ). 0.200, 0.133, 0.154, 0.105, 0.187, 0.221 0.229, 0.115, 0.168, 0.072, 0.187, 0.229 0.207, 0.138, 0.138, 0.109, 0.178, 0.229 0.224, 0.118, 0.149, 0.075, 0.21, 0.224 j j j j w w w w = = = = the elements of vector 1 jw of the first expert are obtained by applying the expression (4) as follows: 1 1 ln(8) 0.200 ln(33600) w = = , 1 2 ln(4) 0.133 ln(33600) w = = , 1 3 ln(5) 0.154 ln(33600) w = = , 1 4 ln(3) 0.105 ln(33600) w = = , 1 5 ln(7) 0.187 ln(33600) w = = , 1 6 ln(10) 0.221 ln(33600) w = = . where 1 8 4 5 3 7 10 33600b =      = . a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 371 the obtained values of the weight coefficients meet the condition where 6 1 1 1jj w = = . the elements of remaining vectors w2 j , w3 j and w4 j are obtained in a similar way. applying the expression (5), we obtain the aggregated vector of the weight coefficients 0.215, 0.126, 0.152, 0.09, 0.19, 0.22( )6 t jw = . the value of weight coefficient 1 0.215w = is obtained by averaging values we j (1  e  4) for every expert, respectively, by averaging values w1 j = 0.200, w2 j = 0.229, w3 j = 0.207 and w4 j = 0.224. applying the expression (5), we obtain the averaged value: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ,   1 1 {0.200,0.229,0.207,0.224} 0.200 0.229 0.200 0.207 0.200 0.224 ...1 0.215 4(4 1) 0.207 0.224 0.224 0.200 0.224 0.2229 0.224 0.207 p qw + == =    +  +  + + = =    −  +  +  +    the remaining values of the weight coefficients vectors are obtained in a similar way. step 4: applying the expression (6), the elements of weighted matrix (n) are calculated, table 4: table 4 weighted matrix   1 2 3 4 5 6 1 0.81 0.87 0.72 0.88 0.77 0.78 2 0.65 0.84 0.80 0.91 0.77 0.71 3 0.76 0.86 0.77 0.90 0.78 0.73 4 0.55 0.82 0.93 0.91 0.80 0.72 5 0.83 0.86 0.71 0.85 0.75 0. 74 6 0.71 0.83 0.7 9 0.88 0.77 = c c c c c c a a a a a x a 0.72                    the values at the positions a1-c1 are obtained by applying the expression (6) as follows: 1 1 1 0.215 11 11 0.215 0.215 11 11 2 2 0.28 0.81 (2 0.28) 0.28(2 ) w w w      = = = − +− + where value 11 presents additive normalized weight of the elements of the normalized decision-making matrix at the positions a1-c1, while w1 presents the weight coefficient of criterion c1. additive normalized weight of elements a1-c1 is calculated as follows: 11 11 1 1 ln( ) ln(1.75) 0.28 ln(7.52) ln( ) m i i    = = = =  where 1 1 1.75 1.13 1.44 1.04 2.00 1.26 7.52 m i i  =     ==  . the remaining weighted decisionmaking matrices are obtained in a similar way. 372 d. pamučar, m. žićović, s. biswas, d. božanić step 5: applying the expression (8), the final indices of alternatives are calculated based on which is performed the ranking of alternatives: 1 4.840 2 4.681 3 4.799   4 4.733 5 4.736 6 4.704 iq a a a a a a           =          since it is preferable for the alternative to have as high as possible value of iq , we can define the rank: a1>a3>a5>a4>a6>a2. 5. validation and discussion of the results 5.1. comparison of the results with other multi-criteria techniques in the following section, the comparison of the results of the lmaw method with other traditional multi-criteria techniques is presented. the comparison is made with the topsis [34], vikor (multi-criteria compromise ranking) [32-33], rafsi [55], copras (complex proportional assessment) [74], and mabac [47] multi-criteria models. all multi-criteria techniques are applied to the same initial data from the initial decision-making matrix and with the same values of the criteria weights. numerous studies showed that the application of different models for data normalization could influence the change of the ranking results [75-79]; thus in this analysis are selected the multi-criteria methods, which apply different ways of data normalization. the results of the application of the mentioned methods are presented in fig. 1. a1 a2 a3 a4 a5 a6 1 2 3 4 5 6 alternatives r a n k copras topsis lmaw mabac vikor rafsi fig. 1 comparison of the lmaw method with other multi-criteria methods a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 373 the vikor, topsis and rafsi methods confirmed the ranks of the lmaw method, with a high correlation, as the spearman’s coefficient (scc) for all three methods amounted to 0.943. in the mabac method there was a slightly lower correlation, compared to the vikor, topsis and rafsi methods, in which the scc = 0.886. the lowest correlation of results appeared in the copras method where the scc = 0.714. however, all models confirmed the rank of the first-ranked alternative a1, and the last two ranked alternatives {a2, a6}. for the remaining three alternatives, a3, a4 and a5, different ranks were proposed, with the greatest similarity in the ranks of the mabac, vikor, rafsi and lmaw methods. the largest deviations in the ranges of alternatives a3, a4 and a5 occurred in the copras and topsis methods. such a result was the consequence of the application of different data normalization methods, respectively, vector normalization (topsis) and additive normalization (copras). in order to confirm this fact, an experiment was performed in which the same way of data normalization as in the lmaw model was applied in both copras and topsis models. at the same time, the rest of the algorithm of the copras and the topsis model remained unchanged. after changing the way of normalization, identical ranks were obtained in all models. based on the presented results, we can conclude and confirm robustness of the lmaw model as well as that the lmaw model provided credible and reliable results. 5.2. rank reversal problem robust multi-criteria models provide stable solutions in the conditions of changing the number of alternatives, respectively, by introducing new alternatives to the set or by eliminating bad alternatives from the set. in such conditions, the model is not expected to show logical contradictions that may appear in the form of unwanted changes in the ranks of alternatives. if such anomalies occur, then reasonable fear can be expressed indicating a problem with the mathematical apparatus of the applied method. rank reversal problem (rrp) is one of the most significant problems in multi-criteria decision-making that can lead to illogical and controversial decisions [80]. significant attention has been paid to the research of the rrp in the literature [55, 75-76]. therefore, the resistance of the lmaw model to the rrp is analyzed in the following section. the experiment was conducted through five scenarios. in every scenario, one of the worst alternatives from the set of considered alternatives was eliminated and the influence of the change in the number of alternatives on the change of ranks and criteria functions of the alternatives was analyzed. the ranks of the alternatives are presented through five scenarios in table 5. table 5 ranking of alternatives by scenarios lmaw model alt. s0 s1 s2 s3 s4 s5 a1 1 1 1 1 1 1 a3 2 2 2 2 2 a5 3 3 3 3 a4 4 4 4 a6 5 5 a2 6 374 d. pamučar, m. žićović, s. biswas, d. božanić it can be clearly noted from table 5 that the lmaw model provides valid results in a dynamic environment. at the same time, the mabac, vikor, rafsi, copras and topsis models were applied in the same experiment. the results showed that mabac, vikor, rafsi and copras models provided stable results, while the rrp appeared in the topsis method. the results of the topsis method application are shown in table 6. the topsis, vikor and copras models were used under the same conditions. all three models showed stability and resistance to rank changes. however, in all four models, the values of the criteria functions changed through the scenarios. accordingly, it can be concluded that for other values in the initial decision-making matrix, in the topsis, vikor and copras models, changes in ranks can be expected, which is analyzed in the second experiment presented in the next section of the paper. based on the presented analysis, it can be summarized that there is a rank reversal problem in the topsis model, which can lead to the appearance of illogical results in the conditions of variable input parameters in the initial decision-making matrix. at the same time, it can be concluded that the mabac, vikor, rafsi, copras and lmaw models show resistance to the rank reversal problem in the presented experiment. from this analysis it can be concluded that the lmaw model contributes to a realistic and stable assessment of alternatives in solving real world problems. table 6 ranks of alternatives by scenarios topsis model alt. s0 s1 s2 s3 s4 s5 a1 1 1 1 1 1 1 a5 2 2 3 3 a3 3 3 2 2 2 a4 4 4 4 a2 5 5 a6 6 5.3. influence of changing parameters p and q on ranking results mathematical formulation of the bonferroni function clearly indicates that the change in the values of parameters p and q affects the change in the aggregated values [81], and thus the change in the final values of the indices of alternatives of the lmaw model. therefore, in order to validate the results, in the following section is analyzed the impact of changes in parameters p, q on the ranking results. the analysis of the change in the value of parameters p and q was performed through a total of 300 scenarios during which the change of parameters p and q in the interval was simulated. the limit for variation of the values of parameters p and q were the values of p = 300 and q = 3000. based on a large number of simulations of the values of parameters p and q, it was noticed that for the values of parameters over 300 there were no significant changes in the ranks of alternatives. the results of the influence of parameters p and q on the ranking results are shown in fig. 2. as the values of parameters p and q increase, the bonferroni function becomes more complex since several relations between the criteria are considered at the same time. the decision makers choose the values of these two parameters according to their preferences. when making decisions in real conditions and in real time, it is recommended for the value of both parameters to be p = q = 1. this simplifies the decision-making process and a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 375 at the same time allows the consideration of internal relations between attributes. fig. 2 shows that when parameters p and q have different values, the score function changes, but these changes do not cause any changes in the ranks of the alternatives. this confirmed that there was sufficient mutual advantage between the alternatives just as it confirmed the initial ranking. 0 50 100 150 200 250 300 4.55 4.6 4.65 4.7 4.75 4.8 4.85 scenarios: 1 ≤ p,q ≤ 300 s c o re f u n c ti o n o f l m a w m o d e l a1 a2 a3 a4 a5 a6 fig. 2 influence of parameters p and q on the ranking results 5.4. influence of changing criteria weights on the ranking results the next section presents the analysis of the influence of the change of the most significant criterion (c6) on the ranking results. the change of the weight coefficient of criterion c6 through 50 scenarios was simulated. the scenarios were made based on the proportion: * * 6 6: (1 ) : (1 )n nw w w w− = − (1) where w* 6 presents corrected value of the weight coefficient of criterion c6, w* n presents reduced value of the considered criterion, wn presents original value of the considered criterion and w6 presents original value of criterion c6. in the first scenario, the value of criterion c6 is reduced by 1%, while the values of the remaining criteria were proportionally corrected applying the shown proportion. in every subsequent scenario, the value of criterion c6 was corrected by 2%, while correcting, at the same time, the value of the remaining criteria. thus, 50 new vectors of weight coefficients were obtained, as in fig. 3. once the new vectors of the weight coefficients of the criteria (fig. 3) were formed, the values of the indices of the alternatives of the lmaw model were obtained, as in fig. 4. it can be observed from fig. 4 that the change in the value of criterion c6 affects the change in the index value of the lmaw model alternatives. in the scenarios s1-s40, the initial rank of alternatives a1>a3>a5>a4>a6>a2 was retained. in the scenarios s40376 d. pamučar, m. žićović, s. biswas, d. božanić s50, there was a change in the ranks of the first two-ranked alternatives, a1 and a3, respectively, the rank a3>a1>a5>a4>a6>a2 was obtained. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 criteria weights s c e n a ri o s c1 c2 c3 c4 c5 c6 fig. 3 weight coefficients of the criteria through 50 scenarios 0 5 10 15 20 25 30 35 40 45 50 4.65 4.7 4.75 4.8 4.85 4.9 scenarios s c o re f u n c ti o n o f l m a w m o d e l a1 a2 a3 a4 a5 a6 fig. 4 influence of the change of criterion c6 to the change of the indices of the alternatives of the lmaw model by the above-presented analysis it is shown that the changes in the values of the weight coefficients significantly affected the change in the value of the index of alternatives of the lmaw model, which further confirmed the sensitivity of the lmaw model. based on the presented analysis it can also be concluded that the initial rank of the alternatives is confirmed and that alternatives {a1, a3} are indicated as good solutions, with the confirmed advantage of alternative a1 over alternative a3. a new logarithm methodology of additive weights (lmaw) for multi-criteria decision-making... 377 6. conclusion in this paper, we present a new additive mcdm approach using logarithms and the bonferroni function. we apply the proposed methodology for solving a real-life problem such as a comparative performance analysis of lsps in indian context. we observe that our method performs well as compared with the widely popular mcdm framework such as topsis. our method provides a more stable result and may be applied for solving complex real-life issues which involve a considerable number of conflicting criteria. however, this work has some limitations which may be treated as the scopes for future work. for example, we have only considered the operational metrics related to turnover and cost. in a typical complex scenario, one may include the criteria like order fulfillment, disruption risk loss, human resource productivity, market innovation, r&d expense, etc. further, we have considered only six alternatives. one may check the robustness of this method considering a large set of alternatives and criteria. further, the other functions like einstein aggregation, heronian mean function may be used to check the results. lmaw is proposed in this paper only. therefore, one may be curious to develop some extended models in uncertain domain using fuzzy and rough sets. nevertheless, we believe that these future scopes do not undermine the usefulness our proposed method. this easy-to-use methodology can be 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beijing, p.r. china 4department of mechanical engineering, shiraz university, shiraz, iran 5department of mechanical engineering, college of engineering in wadi addwasir, prince sattam bin abdulaziz university, ksa 6department of mechanical engineering, faculty of engineering, university of khartoum, sudan 7lehrstuhl für thermodynamik, technische universität münchen, garching, germany abstract. nano-encapsulated phase change suspension is a novel type of functional fluid in which the nanoparticles undergo phase change that contribute to heat transfer. thus, the working fluid carries heat not only by sensible heat but also in the form of latent heat stored in the particles. the natural convection and heat transfer of nanoencapsulated phase change materials (nepcms) suspensions within a boundary layer along a heated flat surface are theoretically investigated in this work. the nanoparticles are core-shell structured with the core fabricated from pcms covered by a solid shell. a similarity solution approach along with the finite element method is employed to address the phenomena. the outcomes indicate that a decisive factor in boosting the heat transfer is the temperature at which nepcm particles undergo the phase transition. the heat transfer parameter can be enhanced by about 25% by just adding 5% of nepcm particles, compared to the case with no nepcm particles. key words: nano-encapsulated phase change materials; phase change materials; boundary layer heat transfer enhancement; similarity solution. received: june 03, 2022 / accepted july 30, 2022 corresponding author: dognsheng wen technical university of munich, institute of thermodynamics, boltzmannstr. 15, 85747 garching, germany e-mail: d.wen@tum.de 520 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen 1. introduction heat transfer is one of the important major demands of recent technological advancement in electronic packaging, aerospace and avionics systems, power engineering, and highpower x-ray devices. these devices produce a tremendous amount of waste heat in confined space, requiring efficient cooling systems to extract the produced heat, and transfer it to the surrounding. the most challenging aspect of thermal control of surfaces with high heatflux-density is heat removal from the surface. moreover, many of high-power devices are sensitive not only to the high temperatures but also to the temperature gradients across the surface of the device. the temperature gradient within a device would produce mechanical stress that may damage the sensitive structures of the device. therefore, new working fluids or new techniques capable of enhancing heat transfer or inducing a uniform temperature distribution are of high interest. in this regard, phase change materials, nanofluids, and nepcms are introduced as promising passive ways to facilitate heat transfer and improve thermal management. a phase change material (pcm) benefits from a large latent heat thermal storage at a fusion temperature. this remarkable quality has drawn researchers' attention towards potential applications of pcms in heat storage and heat sink technologies. construction [1, 2], homes and offices air conditioning systems [3], space heating/cooling, and waste heat recovery in different industries are just a few areas that using pcm can exceptionally be beneficial. about two decades ago, nanofluid was introduced as a stable suspension of nanoparticles into a base fluid to enhance heat transfer [4]. although nanoparticles can notably boost the heat transfer of a base fluid, they also have the drawback of reducing the working fluid's heat capacity, which is not good for many applications [5]. when the heat capacity of fluid is low, the temperature gradient along the cooling fluid increases. the nano-encapsulated pcm (nepcm) are nanoparticles comprising a core and a shell. a phase change material such as paraffin, n-tetradecane, or octadecane is used in manufacturing the core, and the shell is usually made of a polymer [6, 7]. a stable suspension of nepcm nanofluid can effectively remove a large amount of heat from a hot surface using the latent heat of nepcm particles. the other important advantage of nepcms is the temperature control of a coolant at the fusion temperature. indeed, nepcms tend to remain at their fusion temperature and go through a phase change. due to important applications of pcms and recent advancement in the improvement of these materials using nanotechnology, sarkar et al. [8] and sidik et al. [9] explored the thermal advantages of using nanofluids and phase change materials for energy storage applications. sidik et al. [9] addressed the key factors that affect the enhancement of heat transfer of pcms, including the shape and size of nepcms, and shell fraction of nanoparticles. sarkar et al. [8] concluded employing encapsulated pcm in building components for cold storage applications could improve the energy saving. shah [10] reviewed the nanotechnology aspect of pcms by focusing on the synthesis technique and materials/morphology of pcms and their influence on thermal-conductivity enhancement. the boundary layer theory for heat and fluid flow is an important and hot topic in applied mathematics and engineering applications. as a matter of fact, the formation of a boundary layer over a solid surface is a common phenomenon in a viscous flow. so, it has been the subject of various researches for decades. considering heat transfer within the boundary layer of nanofluids, manjunatha et al. [11], sharma and mishra [12], roy et al. [13], munir et al. [14], and bilal et al. [15] addressed the boundary layer flow and heat convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 521 transfer flat surfaces. yasin et al. [16] studied the stagnation point flow and boundary layer heat transfer over a stretching sheet. rashed et al. [17] utilized a two-phase model to investigated the boundary layer heat transfer of nanofluids over a plate in motion. madhu et al. [18] investigated the non-newtonian effects of nanofluids over a stretching sheet. they noticed that the local nusselt number varies inversely with the power-law index of the non-newtonian fluid. reddy et al. [19] studied the boundary layer heat transfer of nanofluids over an inclined vertical surface. they reported that the nanoscale effects of brownian motion and thermophoresis tend to increase the plate's boundary layer thickness. reddy and chamkha [20] analyzed the natural convection boundary layer heat transfer of nanofluids over a cone in the presence of chemical reactions. the outcomes reveal that the presence of a chemical reaction decreases the amount of the temperature gradient of nanofluid at the surface. however, the boundary layer heat transfer of a suspension of nepcms has not been explored yet. as mentioned, the suspension of a nepcms shows promising properties as a heat transfer working fluid. in this regard, several studies addressed the effect of nano/microencapsulation of pcms in micro channel heat transfer applications. seyf et al. [21] numerically studied the flow field and thermal performance of a slurry of water-nepcm over an isothermal unconfined square-cylinder in a channel. the core of the nepcm was produced from n-octadecane, and the average size of capsules was 100 nm. in another research, seyf et al. [22] investigated the thermal responses of octadecane nepcm suspended in polyalphaolefin (pao) in a microtube heat sink. the results reveal that using nepcms considerably enhances the heat transfer rate, while it tremendously increases the pressure drop. they also reported that when the massconcentration of the nanoparticles increases, the heat transfers more evenly, and it accelerates temperature uniformity. in another numerical investigation conducted by rehman et al. [23] the thermal enhancement and hydrodynamics characteristics of a confined slot jet impingement for a mixture of nepcm as a coolant were examined. the water-base coolant was composed 100 nm n-octadecane nepcm particles. they found that dispersing nepcm particles in the base fluid resulted in notable enhancement of the thermal performance. however, the presence of nepcms also increased the viscosity of the slurry, consequently increasing the pressure drop. in a recent study, edalatifar et al. [24] employed an artificial intelligence approach to estimate the free convection heat transfer of nepcms in an enclosure. ho et al. [25] conducted an experiment to inspect the impact of using encapsulated pcms on the heat transfer enhancement of micro channels. their report shows up to 52% improvement in heat transfer in some cases. however, in some cases, the possibility of a reduction in heat transfer due to utilizing encapsulated particles is detected. wang et al. [26] analyzed the thermal performance of encapsulated pcm particles flowing through microchannels. they suggested that using 2% of encapsulated nanoparticles can boost the rate of heat transfer up to 1.36 times higher than that of distilled water when the particles underwent the phase transition. ho et al. [27, 28] synthesized samples of nanofluids (water-al2o3 nanofluids) and samples of a slurry of encapsulated pcms. they examined the thermal performance of the samples in a tube [27] and a mini channel heat sink [28]. the results reveal that the performance of each type of nanofluid or slurry depends on variables such as the flow rate, heat transfer, and the downstream or upstream position. although there are noteworthy studies addressing the flow and thermal behavior of nepcms in channels, there is no study so far, which explores the motion and thermal behavior of nepcms in boundary layer flows. the main goal of the present investigation is to theoretically study the free convection boundary layer flow and heat transfer of nepcms over a hot flat plate for the first time. 522 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen 2. mathematical model consider a stabilized dilute mixture of nepmc particles suspended in a pure liquid, which flows over a flat plate forming a boundary layer. these particles are capsules of pcms comprising a shell layer and a pcm core. the plate is placed in a quiescent suspension of a base fluid and nepcms. the plate is at the constant temperature of tw, and the surrounding fluid is at the cold temperature of t∞. fig.1 illustrates the details of the coordinate system and the physical model. the gravity acts in a downward direction. the thermal buoyancy force gives rise to natural convection, which flows across the plate in the upward direction. the transient temperature of nepcm particles (tf) is lower than the temperature of the plate, even though it remains above t∞ the cold ambient temperature. the nepcms move with the natural convection flow of the base fluid. when nepcm particles reach hot regions (t>tf), a part of the thermal energy is absorbed as the latent heat through the phase transition of the nanoparticles from solid to liquid. accordingly, when they reach cold places (t<tf), they cool down and undergo solidification by releasing their energy. due to the significance of heat of transformation, a considerable advance in the convection can be expected by using a suspension of nepcm nanoparticles. however, the presence of nepcm particles changes other thermophysical properties of the suspension, including the density, the thermal conductivity, the dynamic viscosity, and the thermal volume expansion of the suspension. consequently, these thermophysical properties can alter the convection of the suspension. thus, modeling and systematic inspection of convective heat transfer of nepcm suspensions are essential to evaluate the thermal behavior of nepcms. fig. 1 schematic view of the physical model and the coordinate system when it comes to modeling the boundary layer and heat transfer of nepcms, considering some presumptions and simplifications is inevitable. in the present study, the deviation of the wall's temperature from ambient is reckoned limited. hence, aside from density which is modeled using the boussinesq model and prompts the buoyancy forces, the temperature does not affect the other thermophysical properties. the temperature of the nanoparticles and the fluid around them is the same due to very tiny size of nanoparticle. the volume fraction of convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 523 nepcm particles is considerably small (ϕ), and hence, the suspension of nepcms nanoparticles is a dilute suspension, obeying the newtonian fluid model. finally, the particles are considered uniformly dispersed within the base fluid, and the suspension remains homogeneous and stable. by employing the buoyancy effects, the governing equations for the nepcm's suspension within a base fluid can be introduced as [23, 29]: 0, u v x y   + =   (1)   2 2 b b b b b2 2 u u p u u u v g t t x y x x y               + = − + + + −               (2a) 2 2 b b b 2 2 v v p v v u v x y y x y            + = − + +               (2b) ( ) ( ) 2 2 b 2 2b b t t t t cp u cp v k x y x y          + = +              (3) where u and v are the velocity components in the x and y directions, respectively. t represents the temperature and p the pressure. in the aforementioned equations, ρ and μ are presenting the density and the dynamic viscosity, respectively. the specific heat capacity (cp), and the thermal conductivity (k) are also thermal characteristics of the suspension. the gravitational constant is denoted by g, and the coefficient of thermal volume expansion is represented by β. the subscript of b indicates the bulk properties attributed to a suspension of nepcm fluid. by employing the usual boundary layer approximations, the control equations (1)-(3) are simplified as: 0, u v x y    + =     (4)   2 b b b b b2 u u p u u v g t t x y x y             + = − + + −          (5a) 0 p y  =  (5b) ( ) ( ) 2 b 2b b t t t cp u cp v k x y y        + =         (6) following the representation of the physical model, the appropriate boundary conditions are introduced as: w0 : 0, y u v t t= = = = (7a) : , 0, .y u v t t→ → → (7b) 524 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen 3. bulk properties of the suspension the thermophysical properties in the governing equations of eqs. (4-6) with the subscript of b represent the suspension (a mixture of nepcms and the pure liquid) thermophysical properties. thus, some thermophysical models are required to evaluate the thermophysical features of the mixture. following [30], it is possible to obtain the density of nepcms' mixture and the base fluid using: ( )b f p1   = − + (8) here, ρ indicates the density, and ϕ represents the nanoparticles concentration. the subscripts of f and p represent the pure liquid and the nanoparticles, respectively. the nepcm nanoparticles are consist of a shell and a core pcm [30, 31]. therefore, the density of nepcm composed of a shell and a core pcm can be evaluated as [30, 32]: ( ) c s p s c 1       + = + (9) where c and s as subscripts indicate the core and the shell, respectively, and ι signifies the weight ratio of the core-shell. in eq. (9), the density of the core pcm is considered as the mean of its solid and liquid densities. the subscript p also denotes the nanoparticles. as discussed by khanafer and vafai [5], the assumption of thermal equilibrium between the particles and the pure liquid makes it possible to drive the effective heat capacity of both a suspension of nanoparticles and the pure liquid from the energy equations as follow: ( ) f f p p b b 1 cp cp cp     − + = (10) however, many of the previous researchers have preferred to use a more simplified relation as [30, 32]: ( )b m f m p1cp c cp c cp= − + (11) where cpp is the specific heat of encapsulated particles. cm represents the mass fraction of nepcm particles, which is related to the particles concentration by ϕ = cm×ρb/ρp [32]. here, we use the definition of eq. (8) for the evaluation of the bulk specific heat capacity of the suspension. when there is no phase transition, cpp is calculated using the following relation [32]: ( ) ( ) c,l s c s p s c p cp cp cp       + = + (12) where the core pcm's specific heat capacity (cpc,l) can be taken through the mean of its solid and liquid specific heat capacities. providing that the phase transition is taken into account in evaluating the heat transfer and the latent heat of the confined particle is at hand, it is possible to employ an approximate profile with an integral value equal to the latent heat of phase change. for example, a rectangular, a triangle, or a sinusoidal profile is feasible for modeling the temperature dependent specific heat of encapsulated particles [22, 30, 33, 34]: convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 525 sf c c,l mr h cp cp t = + (13a) 1 , , sin 2 sf c c l c l mr mr h t t cp cp cp t t       − = + −          (13b) ( ) , , 1 2 2 fs c l c c l mrmr h cp cp cp t t tt   = + − −    (13c) where tmr is the phase-change temperature range (tmr = t2 t1). indeed, the phase change takes place in the range of t1 to t2 instead of the exact fusion temperature of tf. taking the fusion temperature as the middle of the transition temperature range, t1 and t2 can be introduced as t1 = tf tmr/2 and t2 = tf + tmr/2. in the case of temperatures out of the melting range, the magnitude of the specific heat of the nanoparticle is matching cpc, and regarding temperatures within the range, the specific heat of the particle is estimated to solve the aforementioned equation. here, following the literature studies [22, 30, 33, 34], it is presumed that the specific heat of the molten and the solid pcm is similar. as stated by alisetti and roy [34], the divergence between applying different profiles for evaluation of the specific heat of pcm in a circular duct is under 4%. thus, in the present study, the simple rectangle profile of eq. (11a) is employed to be a representative of the latent heat of nepcm as a function of temperature. as a result, by considering the phase change latent heat capacity and the sensible heat capacity, the total heat capacity of the suspension of nepcm capsule can be evaluated as: ( ) ( ) c s c s p s c p cp cp cp       + = + (14a) where 1 sf c c,l 1 2 mr 2 0 0 t t h cp cp t t t t t t     = +      (14b) the coefficient of the thermal volume expansion, βp, is evaluated through the upcoming equation, which was discussed by khanafer and vafai [5] for a base fluid containing nanoparticles: ( )b f p1   = − + (15) now, closure models for the estimation of the dynamic viscosity and the thermal conductivity of the suspension are required. many of the previous studies have evaluated the thermal conductivity of micro/nano-epcm as two parts of static and dynamic thermal conductivity. in most of the literature-works, the mixture's static thermal conductivity was estimated by using the maxwell model [5, 35], and then the dynamic thermal conductivity of the mixture was modeled using the fluid shear rates [22, 29, 30, 36]. however, buongiorno [37], using a scale analysis approach, demonstrated that the shear rate effects are not of importance in the nanoscales. hence, it can be concluded that the thermal conductivity 526 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen relations for the microencapsulated pcms are not applicable to the case of nano-encapsulated suspensions. in most of the literature studies, the dynamic viscosity of the suspension of micro-encapsulated pcms (mepcms) was approximated using the vand [38] model as [22, 30]: ( ) 2.5 2b f 1 a     − = − − (16) in eq. (16), the parameter of a depends on the size of particles and can be evaluated using a curve-fitting on the experimental data. for example, the value of a has been reported as a=1.16 for 0.13mm diameter glass spheres [38], a=3.4 (10–30 μm diameter mepcm particles) [39], a=3.7 (6.3μm average diameter) [40], and a=4.45 (about 10 μm diameter mepcm particles) [41]. it should be noted that the model of vand [35] has been introduced for the mepcm particles and not for nano-epcm particles. regarding nanoparticles, buongiorno [42] and venerus et al. [43] addressed the effects that the presence of nanoparticles can lay upon the dynamic viscosity and the thermal conductivity of dilute suspensions of nanofluids in two comprehensive benchmark studies. more than thirty authors around the world have studied the aforementioned properties of nanofluids in different experimental setups and through various types of measurement devices. the outcomes demonstrate the possibility of expressing the dynamic viscosity and the thermal conductivity of fluids containing nanometer-sized particles by linear relations. following this outcome, ghalambaz et al. [44] utilized the upcoming linear equations to study the heat and mass transfer of nanofluids:  b f 1 k nc k    = +    (17a)  1b f nv      = +      (17b) here nc is the thermal-conductivity number, and nv indicates the dynamic-viscosity number. the magnitudes of nc and nv depend on several parameters, including the shape of the particles, the nanoparticles material, the pure liquid, and the operating temperature [44]. it is obvious that the higher the value of nc or nv, the greater the amplification of the thermal conductivity or the dynamic viscosity when the flow contains nanoparticles. nc and nv are attainable by applying linear curve fitting over gathered experimental data for the suspensions thermophysical properties. a list of values of nc and nv is evaluated and reported in [44]. it is noteworthy to mention that these relations are valid for a diluted suspension of nanoparticles, and more advance relations or actual experimental data shall be used for high volume fractions of nanoparticles. here, eqs. (17a) and (17b) are adopted to determine the thermal conductivity of the suspension and the dynamic viscosity within the flow field. 4. similarity solution for the purpose of transforming the equations governing the boundary layer flow field along with the attributed boundary conditions into a selection of ordinary differential equations, some similarity variables are required to be proposed. first, in order to satisfy convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 527 the continuity equation, the stream function is introduced into it as u=∂ψ/∂y and v=∂ψ/∂x. the pressure between equations of (2a) and (2b), is excluded between momentum equations in x-direction and y-direction by taking the cross. for obtaining a similarity solution through which the equations can be rearranged as a scaled expression, the similarity variables are proposed as: 1 4η ra x y x = (18a) (18b) where η is the similarity variable, s and θ are the scaled flow and temperature. here, the local rayleigh number (rax) is introduced as: ( ) 3 f f f f ra x g t t x    − = (19) by positioning the similarity variables of eq. (18) and using the thermal conductivity and dynamic viscosity relations of eq. (17), the partial differential equations of eqs. (5) and (6) along with the boundary conditions of eq. (7) are explained into a series of nonlinear ordinary differential equations as: ( ) ( ) bf f b b f 1 2 3 1 4pr s s s s nv s           − = + + (20) ( ) ( ) ( ) b f 3 1 0 4 c nc s c       + + = (21) where pr=μf /(ρf αf). the density ratio of ρb/ρf can be simply evaluated using eq. (7) as (1-ϕ)+ϕρp/ρf. the thermal expansion ratio of βb/βf can be rewritten as ϕβp/βf +(1-ϕ) by using eq. (15). the other important thermophysical property would be the heat capacity ratio (ρcp)b/(ρcp)f which is evaluated using eqs. (8), (10) and (14) as follow: ( ) ( ) ( )b f 1 1 cp f cp ste       = − + + (22) where θf represents the phase-change temperature, δ is the parameter of fusion range, which should be a very small number. when δ approaches zero, the phase change temperature approaches the scaled phase change temperature. λ is the scaled heat capacity introduced as the proportion of the sensible heat capacity of nepcm nanoparticles to the sensible thermal capacity of the base fluid, where ste is the stefan number. these scaled parameters and numbers are introduced as: ( ) f f w t t t t    − = − ( ) mr w t t t   = − , ( ) ( ) ( ) c,l s c s s cf cp cp cp        + = + , ( ) ( )( )w s cf sf c s cp t t ste h      − + = (23) 1 4 , w x t t s t t ra      − = = − 528 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen and f f 2 othe 0r 2 wi1 sef if f        = −   + =    (24) by substituting the final scaled form of the thermo-physical ratio parameters, the governing equations of the flow and heat transfer are rewritten as: ( ) ( ) ( ) ( ) p p f f 1 1 2 3 1 1 4pr s s s s nv s                  − + − = + + − +        (25) ( ) ( ) 3 1 1 1 0 4 nc f s ste           + + − + + =    (26) in the above equations, it should be noticed that f is not a constant, and it a function of scaled temperature as defined by eq. (24). the transformed form of the boundary conditions of eq. (7) are: ( ) ( ) ( )0: 0 0, 0 0, 0 1,at s s = = = = (27a) ( ) ( ): 0, 0.as s   → → → (27b) the important characteristics of heat transfer of the nepcms suspension are the heat transfer at the surface. the balance in the heat transfer at the surface of the plate is given using the convective heat transfer coefficient (h) as: ( )b w t k h t t y   − = −  (28) the local nusselt number is achieved through using the above equation and implementing the similarity variables brought in eq. (18), and equals to: ( ) 1 b 4 f 0 rx x k nu a k  = − (29) where, using eq. (17a), the local nusselt number can be rewritten as: ( ) ( ) 1 4nu ra 1 0x x nc  − = − + (30) it is worth noticing that the term nuxrax -1/4 is known as the reduced nusselt number (nur). 5. numerical method and code validation eqs. (25) and (26) are a set coupled, high order, and non-linear ordinary differential equations. it is very hard to obtain an analytical solution for non-linear high-order differential equations. hence, a numerical approach is required to integrate the equations. here, bvp4c finite element code with automatic grid adaptation and error control is employed for the purpose of finding a solution for the governing equations along with the boundary conditions. in order to implement the ode equations in bvp4c code, they need to be written in the form of a set of first-order coupled equations as follow: convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 529 ( ) ( ) ( ) ( ) ( ) ( ) p p2 2 1 3 4 f f 1 2 2 3 3 1 5 4 5 5 1 1 2 3 1 4 pr , , , 1 1 1 3 , 4 1 y y y y y y y y y nv f y y ste y y y nc                   − + − − − +         = = = +   − + +    = = − + (31) f 5 f1 otherwise 0 2 2 f if y f       = −   + =    (32) accordingly, the corresponding boundary conditions using the new notation are: ( ) ( ) ( )1 2 4η 0: 0 0, 0 0, 0 1,y y y= = = = (33a) ( ) ( )2 4η : η 0, η 0.y y→ → → (33b) two of the boundary conditions are approaching infinity. therefore, the governing equations have to be solved in a large domain to reach an asymptotic solution. in the present study, the value of η∞ is increased starting from η∞=10 to higher values, and the reduced nusselt number is monitored to ensure that the asymptotic solution is reached and it is independent of the selected value of η∞. the absolute error was set as 10-8, and the absolute relative error was set as 10-6. more details about the utilized code can be found in [45]. the reliability of the present code is evaluated by comparing the obtained results with the literature. in fig. 2 the temperature profiles brought by bejan [46] are compared with those reported in the present research. fig. 2 a comparison between the current data and those of bejan [46] 530 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen table 1 shows a comparison between the outcomes of the present study and the results that can be found in the literature for the reduced nusselt number (nur) when ϕ=0. table 1 comparison nur calculated in the present investigation and the results available in literature when ϕ=0 pr present study bejan [46] 0.72 0.3874 0.387 1 0.4010 0.401 2 0.4260 0.426 10 0.4650 0.465 100 0.4900 0.490 6. results and discussion here, for estimating the range of the scaled parameters, n-octadecane [47] for the core pcm, poly (methyl methacrylate) pmma for the shell [48], and water for the base fluid [47] are adopted. the core-shell weight proportion is selected as ι=0.7. in this case, λ and ste (stefan number) can be approximated as 0.4 and 0.313, respectively. the volume fraction of the nepcm nanoparticles within the water is adopted to be fixed as ϕ=5%. the fusion temperature is picked in the range of 0<θf<1 where the scaled ambient temperature and the scaled hot wall temperature equal to zero and unity, respectively. to illustrate the results, the scaled parameters are adopted in the following range: 0<nc<6 and 0<nv<6 for a mixture where nanoparticles are distributed throughout a base fluid. since the pcm core is less dense than water [49, 50], the density ratio is implemented as ρp/ρf=0.9, and the buoyancy ratio of ϕ×βp/βf is assumed negligible because of the inconsiderable volume fraction of nanoparticles (0.05) and very low thermal expansion of solid particles compared to a liquid. the prandtl number (pr) is fixed 6.2, which is about the prandtl number of water. for a fixed value of pr=6.2, the asymptotic value of η∞ is about η∞=20. as a summary, the preset values of the scaled parameters would be assigned as pr=6.2, θf=0.2, ι=0.7, λ=0.4, ste = 0.313, nc=nv=3.0, (ρp/ρf)=0.9, and (ϕ×βp/βf)=0. in this study, the results are presented as the preliminary values of the scaled parameters, and in other circumstances, the value of the parameter will be stated. for the first part of the numerical calculations, it is examined how the average nusselt number may vary by the changes in the value of δ. the delta parameter (δ) is the scaled fusion range, and it should be a small parameter. the calculations for various values of δ were carried out. the outcomes indicate that the results are independent of this parameter for δ<0.025. it is important to keep in mind that very small values of δ lead to a very low fusion temperature range. since the latent heat of phase change varies according to a temperature within the fusion process, the quality of the utilized grid is of utmost importance for capturing the phase change within acceptable accuracy. the utilized bvp4c code in the present study enjoys the advantage of automatic error control and automatic grid adaptation. although the automatic error control and automatic grid adaptation can significantly reduce the computational cost of the calculations to deal with a small value of δ, using a small value of δ results in a dramatical increase of computational costs. hence, for the sake a balance between the accuracy and computational cost, the appropriate value of δ=0.025 is adopted for the calculations. convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 531 since all of the presented results are derived from scaled governing equations, the scaled suffix is left out for brevity. figures 3 and 4 depict the effect of the fusion temperature (θf) on the velocity and temperature profiles. the computational domain of solution is about η∞=20; however, here, the results are plotted in a smaller domain for the sake of graphical clarity. as mentioned, the velocity profiles equal to zero at the surface and tend to zero at the infinity, which satisfies the hydrodynamic boundary conditions. there is a maximum velocity peak next to the heated surface. any rise in the phasechange temperature elevates the maximum value of the maximum-velocity peak. it also shifts the maximum-velocity peak to the right. attention to the temperature profiles of fig. 4 illustrates that when the fusion temperature rises, the temperature gradient dwindles. moreover, in the case of θf=0.1, a sharp change in the path of the temperature profile can be observed at the location of about η=2.4. the similar trends of the path changes but smoother can be seen for other fusion temperature profiles. the surge of the fusion temperature shifts the mentioned path changes towards the surface. indeed, the location of this path change is the place that the change phase of the nepcm particles occurs. in the case of θf=0.1, the path change occurs at η=2.4 corresponding to the temperature of θ=0.1. indeed, when the fusion temperature elevates, the magnitude of the temperature profile extends. consequently, the increase in temperature corresponds to the increase of the buoyancy forces. hence, as it was observed in fig. 3, the elevation of the fusion temperature promotes the fluid velocities and the maximum-velocity peaks. the reason behind the shift of the maximum-velocity peaks toward the surface (by the rise in the fusion temperature) is that the velocity profiles and the buoyancy forces tend to follow the trend of the change path in the temperature profiles due to the growth of the transient temperature. as mentioned, a surge of the fusion temperature shifts the temperature profiles toward the hot plate. fig. 3 the velocity profiles for various values of the scaled fusion temperature (θf) when nc=nv=3.0 532 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen fig. 4 the temperature profiles for various values of the scaled fusion temperature (θf) when nc=nv=3.0 figures 5 and 6 show the effect of fusion temperature (θf) and the thermal conductivity number (nc) and the dynamic viscosity number (nv) on the magnitude of the temperature gradient and the reduced nusselt number at the wall surface. for each set of nc and nv numbers, fig. 5 shows that there is a maximum-peak for the temperature gradient about θf=0.2. starting from a low value of θf ~ 0.025, the temperature gradient and nusselt number raise by the increase of θf until about θf=0.2, which corresponds to a peak for the maximum temperature gradient. after that, whenever the fusion temperature (θf) ascends, the nusselt number and the temperature gradient follow a descending trend. a smooth local-minimum can be seen about θf = 0.975. in fact, the growth of the fusion temperature physically means that the phase transition at the core of nanoparticles will occur in a temperature closer to the wall temperature. when the fusion temperature and the wall temperature are close to each other, the nepcm particle undergoes phase change just next to the wall, and hence, it would keep the temperature of the liquid at a high level next to the wall. however, it should be heeded that the phase transition occurs, a substantial amount of energy would be absorbed because of the latent heat of the phase change, which results in the increase of the heat transfer rate. consequently, when the phase changes next to the wall, the wall temperature gradient decreases, which eventually drops the magnitude of heat transfer enhancement. although this transition next to the wall declines the wall temperature gradient, still the other thermophysical properties, including the enhancement of conduction, can result in an overall improvement of the heat transfer in the presence of nepcm particles. hence, as seen in fig. 6, generally, the heat transfer rate is higher within the nepcm suspension than the pure fluid. the nusselt number of a pure base fluid with no nepcm particle is depicted as a gray narrow dashed line in fig. 6. convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 533 fig. 5 the surface temperature gradient vs the scaled fusion temperature (θf) and for different nc and nv fig. 6 the reduced nusselt vs the scaled fusion temperature (θf) and for different nc and nv at very low fusion temperatures, the phase of nepcm particles changes at the boundary layer's edge, which is quite far from the hot surface. a small increase in the fusion temperature shifts the phase transition of the nanoparticles inside the boundary layer, which leads to an increase in the heat capacity of the cold fluid at a cool temperature. as it was observed in fig. 4, as an example for θf=0.1, the phase transition occurs at η=2.4. it was also observed that any rise in the fusion temperature (θf) would shift the phase change of nepcm particles towards the wall. the phase change tends to uniform the temperature profiles about the fusion temperature. so, when it occurs with a sufficient distance from the 534 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen wall, indeed it decreases the actual the thermal boundary layer's thickness, and the hot wall sees a uniform cold region in nearby. this is where the higher fusion temperature improves the reduced nusselt number. attention to the case of nc=nv=0 in figs. 5 and 6 show that the temperature gradient and nusselt number are higher than the pure fluid (the gray line) in most cases. the case of nc=nv=0 indeed neglects the major effect of the increment of thermal conductivity and dynamic viscosity of the nepcm suspension because of existing particles. in this case, only the thermal capacity ratio, the density ratio, and the buoyancy ratio parameters remain effective. in these figures, it is obvious that the heat transfer of nepcm suspension is higher than the pure fluid for the case of θf<0.8. however, for the fusion temperature higher than 0.8, the heat transfer would be less than that of the pure fluid since the phase transition affects the surface temperature gradient. the change of nc and nv only shifts the observed trend of behavior for temperature gradient and nusselt number. the growth of the thermal conductivity (nc) number declines the magnitude of the temperature gradient but boosts the nusselt number. indeed, when nc increases, the thermal conductivity of the suspension surges in the presence of nepcm particles. the increase of conduction tends to uniform the temperatures next to the wall, which causes a drop of the temperature gradient. however, the heat transfer can be calculated as the thermal conductivity multiple by the temperature gradient. hence, although the magnitude of the temperature gradient of the surface drops, a better thermal conductivity would boost the overall heat transfer rate. following any rise in the dynamic viscosity (nv) number, the suspension's viscosity in the presence of the particles grows, and the rate of heat the flow accordingly plummets. the increase of viscosity would diminish the fluid velocity, and as a consequence, the overall rate of heat transfer. based on the results of fig. 6, improvement in the heat transfer is expected in most cases of regular suspensions of nepcm particles. fig. 7 depicts the reduced nusselt number as a function of stefan number. as seen, the increase of stefan number reduces the thermal performance of nepcms. it should be pointed out that the phase transition's latent heat is reversely related to stefan number. this means that a large value of stefan number represents a very low latent heat of phase change, and in contrast, a small value of stefan number indicates a large amount of latent heat of phase change. fig. 7 reveals that nepcm particles with low stefan number are very effective on the increase of heat transfer. it also confirms that not only the stefan number of nepcm particles but also the fusion temperature of nepcm particles are important. at fusion temperatures close to the hot wall temperature, the presence of nepcm particles only shows a minor enhancement in heat transfer rate regardless of the stefan number. table 2 lists a brief summary of the impact of the scaled parameters on the amount of the temperature and the velocity gradients at the wall. adopting the first row of the table as a reference and comparing the results of the other rows with this row, table 2 shows that the magnitude of the wall temperature gradient (-θ'(0)) is a decreasing function of λ, θf, and ste. the effect of the variation of the density ratio (ρp/ρf) is negligible. the surface velocity gradient is a degreasing function of λ and (ρp/ρf) and an increasing function of θf and ste. convective flow and heat transfer of nano-encapsulated phase change material (nepcm)... 535 fig. 7 the reduced nusselt number vs stefan number for various scaled fusion temperatures (θf) table 2 the magnitude of temperature gradient (-θ'(0)) and the velocity gradient (s''(0)) for various value of scaled parameters when ϕ=0.05, (ϕ×βp/βf)~0, and nv=nc=3.0 ste λ θf (ρp/ρf) -θ'(0) s''(0) 0.3 0.4 0.2 0.9 0.4469 0.8823 0.3 0.4 0.2 0.8 0.4469 0.8825 0.3 0.3 0.2 0.9 0.4463 0.8830 0.3 0.4 0.3 0.9 0.4456 0.8931 0.3 0.4 0.1 0.9 0.4464 0.8696 0.2 0.4 0.2 0.9 0.4606 0.8640 7. conclusions the flow and heat transfer behavior of nepcm particles dispersed in a pure fluid along a heated vertical plate was studied. the boundary layer governing equations were introduced and rearranged as a group of non-linear ordinary differential equations and solved numerically. the influence of some important scaled parameters, including prandtl number (pr), stefan number (ste), dynamic viscosity number (nv), thermal conductivity number (nc), and fusion temperature (θf), as well as density ratio of the nepcm (ρp/ρf), were investigated, as summarized below: ▪ generally, the presence of nepcm nanoparticles enhances the natural convection. ▪ the decrease in the stefan number enhances the heat transfer rate. ▪ at the optimum scaled fusion temperature, θf ~0.2, the heat transfer rate reaches its maximum value. at θf = 0.2, the reduced nusselt number is increased from nur=0.445 for the case of the base fluid to 0.56 at 5% vol. of nepcm particles, showing 25% enhancement in heat transfer. 536 m. ghalambaz, h. jin, a. bagheri sarvestsani, o. younis, d. wen ▪ as the fusion temperature elevates, the velocities of the nepcm within the boundary layer rises as well, which, however, does not always lead to the enhancement of heat transfer. for instance, the velocity corresponding to the case of θf=0.2 is lower than that of θf=0.5, but corresponding to a large heat transfer enhancement. this indicates the significance of how the latent heat of nepcm particles can affect the heat transfer behavior of suspension. it shall be noted that one assumption made in this work is that the suspension of nepcm is diluted (i.e., ϕ<5%). hence, the interactions among the nepcm particles and the non-newtonian effects were neglected. it was also assumed that nepcm particles remain homogeneous in the fluid. however, in practice, some slip mechanisms such as the brownian motion and thermophoresis effects could change the local 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engineering, national academy of science, ukraine 4 national research tomsk state university, russia 5 national research tomsk polytechnic university, russia abstract. recent progress in space research and in particular appearance of complex movable constructions with a number of components exposed to the extreme conditions of open space causes a strong demand for development of new tribological and adhesion systems which are able to resist such conditions. in the last few years, many engineering solutions in the field of tribology and adhesion have been found based on “biomimetics approach” that is searching for ideas originally created by living nature and optimized during billions of years of natural selection. surprisingly some of the living creatures are found to be optimized even for survival for a long time in the conditions of open space. such ability is very promising from the point of view of development of new adhesives for future space applications. in this paper we discuss what we can learn in this context from the so-called “water bears” (tardigrades) in a combination with some other features, already adopted to reversible technical adhesives from other animals, such as insects and gecko lizards. key words: adhesion, tribology, space, bio-inspired systems, gecko, tardigrade 1. introduction a continuous demand for the development of new tribological and adhesion systems which are able to resist extreme conditions of the open space exists practically from the very beginning of astronautics. in the last few years, this demand was additionally motivated by a long-time and extensional exploitation with the international space station (iss) which contains a large number of mechanical members and complex movable components with received april 8, 2015 / accepted july 15, 2015  corresponding author: valentin l. popov berlin institute of technology, strasse des 17. juni 135, 10623 berlin, germany e-mail: v.popov@tu-berlin.de original scientific paper 242 filippov a.e., gorb s.n., popov v.l. active surfaces exposed directly to the extreme conditions of open space. here we will concentrate only on a single one yet interesting question of adhesion in spatial conditions based on ideas from biological systems (biomimetics). in the biomimetic approach new systems are designed using ideas taken from living nature. such strategy can be successful because the living prototypes have been optimized by natural selection during billions of years. surprisingly, some of the living creatures are found to be optimized to survive even for a long time under conditions of open space, being exposed to the radiation, wide temperature variations, vacuum, etc. such unexpected ability is promising from the point of view of space applications. in this paper we will mainly limit ourselves by one famous example of such “extremals”, combining this information with that obtained from adhesive systems of other animals, such as insects, spiders and lizards. it is so-called “water bear” belonging to the tartigrade, a group closely related to other arthropods. 2. practical lessons from lizards and tardigrades 2.1 lizards the idea to use adhesive attachment systems for the space applications is associated mainly with the micro-gravitation on the board of manned spacecraft on the orbit which allows applying a much weaker attaching system than is necessary to hold body weight of the astronaut in strong gravitation near the earth surface. another field of possible applications of artificial adhesives is that of small robots inspired by the biological prototypes. the latter could be employed in microand nanosatellites which represent a very rapidly growing segment of the satellite launch industry. for example, development activity in the 1–50 kg mass range of the space apparatus has been significantly exceeding that in the 50–100 kg range. in the 1–50 kg range alone, there were fewer than 15 satellites launched annually in 2000 to 2005, 34 in 2006, then fewer than 30 launches annually during 2007 to 2011. the number of launches increased to 34 in 2012, and 92 in 2013 [1]. for maintenance of functioning or repairing of autonomous devices, small robots with adhesive surfaces and manipulators could be employed. one of important ideas here is to mimic the adhesive pads of the animal feet to allow small robots to climb up the wall of a spacecraft when maintaining and repairing it. such repair robots could extend the lives of expensive spacecraft and perhaps minimize risky spacewalks for astronauts. the nature of bio-inspired adhesive devices was studied extensively in the last decade on the example of lizards, first of all geckos. it was established [2, 3] that the function of gecko’s sticky feet is based on the van der waals forces. these forces are only effective on very small scales, when atoms are in close contact. however, absolute majority of apparently smooth surfaces, where the climbing robots have to work, are actually quite rough on the micro and nanoscales [4]. the gecko’s tiny foot hairs on earth conditions fill those gaps thus maximizing the contact area between foot and wall, and make the van der waals force effective. gecko foot uses a "dry adhesive" technique and does not rely on sticky glues, but on the bunches of extremely tiny hairs on their feet with ends just 100 to 200 nanometers. the ventral side of gecko toes bears so-called lamellae with arrays of 3–5 µm thick setae, which are further subdivided at their tips into 100–1000 single nanofibers ending with flattened tips (spatula) of both width and length of about 200 nm [2, 3] what can we learn from “water bears” for adhesion systems in space applications? 243 and thickness about 15 nm [4, 5]. such subdivision of large adhesive contact into many single separate contacts leads to the enhancement of adhesive force of this fibrillar system due to the variety of reasons [6, 23]. this effect is also enhanced by the specific spatulalike shape of singe contacts [7]. all the above properties of gecko inspired manufacturing of adhesives with similar microscopic structure and are potentially useful to let robots climb on rigid surfaces in space. just as in the case of the real gecko foot, the directional asymmetry of the synthetic adhesive allows the sticking power to be turned on and off using a shear force. because the adhesive relies on van der waals forces to adhere to surfaces, it is potentially insensitive to temperature, pressure, and radiation. an important aspect of practical usability of the gecko type structures in space is the change of properties of the filament material with time and temperature: generally, it can gradually degrade, loose elasticity and become too stiff being cooled too much, or in the conditions of vacuum and strong radiation. it is therefore interesting to look for other biological examples to find possible solutions. in this relation, it is very interesting to look both to the adhesive properties and sustainability to severe conditions of a small animal called tardigrade, which provides an example of an unexpectedly good adaptation even to extreme conditions of open space [8, 9], or at least, recovery of their materials at good conditions. 2.2 tardigrades tardigrades, water bears, are microscopic water-dwelling, segmented microscopical animals, with eight legs [10-14]. the name “water bear” is given because they visually resemble a bear in microscope. let us briefly describe the main features of their body (shown schematically in error! reference source not found.). tardigrades have barrelshaped bodies with four pairs of stubby, poorly articulated legs. most of them are from 0.3 to 0.5 mm in length, although the largest species may reach 1.2 mm. the body consists of a head, three body segments with a pair of legs each, and a caudal segment with a fourth pair of legs. the legs are without joints while the feet have four to eight claws each. the cuticle contains chitin and protein. a) b) fig. 1 two sketches of different projections of a tardigrade, presenting the main features of its body: head, three body segments having pair of the legs each, and a caudal segment with an additional pair of legs 244 filippov a.e., gorb s.n., popov v.l. water bears have somewhere over 1000 cells at average length about 500 micrometers and can be used as a model organism to teach a wide range of principles in life science. scientific name tardigrada means a "slow stepper". since their discovery in 1778, over 1150 tardigrade species have been identified. they live practically everywhere on our planet. in a specific inactive state (so-called cryptobiosis) tardigrades are capable of surviving 20 h at -273 c° and 20 months at -200 c°. they can survive at +150 c°, 6000 atmospheres of pressure, in a pure vacuum, excessive concentration of many different suffocating gases, under x-ray and ultraviolet radiation, and can be reanimated after 150 years [10]. cryptobiosis is an ametabolic state of life entered by an organism (tardigrade in this case) in response to desiccation, freezing, and oxygen deficiency and any other unfavorable environmental conditions. in this state, all metabolic processes stop, preventing reproduction, development, and repair. probably they can neither adhere in this state, however, their organism in a cryptobiotic state can essentially live indefinitely until environmental conditions return to being hospitable. in this case it returns to its previous metabolic state of life. under dry and cold conditions the tardigrades form a so-called tun. briefly, the main forms in which tardigrades can exist are as follows: (a) active form in which it can eat, grow, move and reproduce itself; (b) in oxygen deficit it makes osmoregulation causing “swelling” and “turgidity”; (c) in cold, extreme salinity, desiccating it turns to shriveled dry tun. these forms are schematically plotted in fig.2. a) b) c) fig. 2 main forms of the tardigrade: (a) active, (b) in oxygen deficit, (c) in cold and extreme salinity the tun forms as the animal retracts its legs and head and curls into a ball, which minimizes the surface area. they can dry almost completely turning practically to a powder of their ingredients without water. while in the cryptobiotic state, the tardigrade's metabolism reduces to less than 0.01% of the normal state, and its water content can drop to 1% of normal. when rehydrated, tardigrades return to their active state in a few minutes to a few hours. when the animal undergoes freezing, it can be revived. it is important to note for a generality and in a context of the possible applications, that in principle many biological and artificial materials can also "revive". some of the tardigrades have similar shape of the adhesive hair tips as mentioned above and already were artificially made due to the inspiration from insects and gecko hairs. for example, the batillipes tardigrade has a leg with six finger-like setae, each equipped with a tiny adhesive disc, schematically illustrated in fig.3. what can we learn from “water bears” for adhesion systems in space applications? 245 fig. 3 schematic view of adhesion toe of batilipes tardigrade. one of the legs with few adhesive toes is shown in the subplot this animal is much smaller than insects, spiders, and geckos and has therefore a much more simple construction of the adhesive system. it contains only few elements: a stiff hollow cylindrical shaft and sticky ending at the edge of the toe disc which has the form of a leaf with a central ripple. the 'leaf' is very elastic and thin and conceptually it resembles terminal spatula of other adhesive systems of insects, spiders, and geckos [6] independently evolved due to the natural selection. the tardigrades are small and one cannot exclude that tartigrades produce adhesive fluid for the purpose of adhesion, as the insects do. so, for their attachment system survival and further recreation is important. it is commonly acknowledged that survival of tartigrades is supported by the release or synthesis of cryoprotectants. these agents may change the tissue freezing temperature, slowing the process and allowing an orderly transition into cryobiosis, and they may suppress the nucleation of ice crystals (which normally destroy cell boundaries), resulting in an intermediate form that is favorable for subsequent revival with thawing. one of the important candidates to this role is trehalose [15]. trehalose is a natural alpha-linked disaccharide formed by a bond between two glucose units. molecular formula of it is c12h22o11. its structure is schematically shown in fig. 4. trehalose can be synthesized by bacteria, fungi, plants, and invertebrate animals. this substance provides plants and animals with the ability to withstand prolonged periods of desiccation. it has high water retention capabilities. the sugar is thought to form a gel phase as cells dehydrate, which prevents disruption of internal cell organelles, by effectively splinting them in position. rehydration then allows normal cellular activity to be resumed without the major, lethal damage that would normally follow a dehydration/rehydration cycle. the previously described presence of soft hydrated adhesive hair tips, containing high proportions of resilin, a rubber-like protein, in the setal tips of beetles [16, 17], allows us to assume that similar kind of material might be present in fig. 4 trehalose is a natural alpha-linked disaccharide formed by a bond between two glucose units 246 filippov a.e., gorb s.n., popov v.l. adhesive pads of tardigrades. it is also well known that resilin can be completely dehydrated and that, after rehydration, it can recover its mechanical properties [18]. also tanned arthropod cuticle, which is definitely present in adhesive structures of tardigrades, is much softer in hydrated condition, which explains a stronger adhesive and frictional performance of spiders at certain relative humidity [19]. also keratin, which is the main component of gecko setae, is softer at higher humidity, which is a part of explanation of stronger adhesive forces of gecko setae at higher humidity [20]. even if we do not exactly know the materials composition of tardigrade adhesive pads, it is plausible to assume that after complete rehydration at space conditions, this biological material or material composition will be able to recover its adhesive properties after rehydration, due to the material softening after water absorption. however, since cuticle of tardigrades does not contain porous channels [21] in contrast to the majority of arthropods, one can also assume that its dehydration rate will be extremely small. tardigrades are the first multicellular animal which survived exposure to the lethal environs of outer space [13, 14]. in 2007 european researchers launched an experiment exposed cryptobiotic tardigrades directly to solar radiation, heat and the vacuum of space on the european space agency’s biopan 6/foton-m3 mission. it was done with the tuns of tardigrades orbited 260 kilometers above the earth. a container with tardigrade tuns inside was opened and they were exposed to the sun radiation. when the tuns were returned to earth and rehydrated, the animals moved, ate, grew, shed and reproduced. they survived. in the summer of 2011 in project biokis colonies of tardigrades were exposed to different levels of ionizing radiation. some limited damage was studied later to learn more about the ways the cells react to radiation and, perhaps, how tardigrade cells keep off their damage. sustaining intense radiation suggests an especially effective dna repair system in an active organism. effective osmoregulation in extreme salinity implies a vigorous metabolism – osmoregulation in the face of high environmental salinity is energetically extremely expensive as metabolic transactions go, requiring the pumping of ions against steep osmotic and ionic gradients. thus, we see in tardigrades two opposing responses to environmental demands [8]. 3. conclusion some perspectives of using biologically motivated adhesive systems in open space are discussed. the main requirement to such systems is that they should not apply liquid adhesives, but only use dry van der waals forces. one of the key aspects of the gecko inspired artificial polymer structures made of polyurethane, polyvinylsiloxane, and polystyrene [22] is that they can gradually degrade, loose elasticity and become too stiff in the conditions of vacuum, or due to strong temperature variations and radiation. in searching the proper material for adhesives for space applications, we came to tartigrades. they provide an astonishing example of adaptation to extreme conditions of open space [13, 14]. using mechanical and chemical mechanisms, it can transform itself into a cryptobiotic state conserving many properties which are necessary for revival at any moment, when appropriate conditions will be restored. learning from some of its capabilities will be certainly useful to generate new biologically motivated artificial adhesive films and robotic systems. what can we learn from “water bears” for adhesion systems in space applications? 247 acknowledgements: this work was supported in part by the ministry of education of the russian federation, the german academic exchange service and the tomsk state university academic d.i. mendeleev fund program. references 1. annual market assessment series. 2014, nano/microsatellite market assessment. atlanta, georgia: sei. january 2014. p. 18. retrieved 18. february 2014. 2. hiller, u., 1968, untersuchungen zum feinbau und zur funktion der haftborsten von reptilien. z. morphol. tiere, 62, pp. 307-362. 3. autumn, k., liang, y.a., hsieh, s.t., zesch, w., chan, w.p., kenny, t.w., fearing, r., full, r.j., 2000, adhesive force of a single gecko foot-hair, nature 405, pp. 681-684. 4. persson, b.n.j., gorb, s.n., 2003, the effect of surface roughness on the adhesion of elastic plates with application to biological systems, j. chem. phys., 119, pp. 11437-11444. 5. huber, g., gorb, s.n., spolenak, r., arzt, e., 2005, resolving the nanoscale adhesion of individual gecko spatulae by atomic force microscopy, biol. lett., 1, pp. 2–4. 6. varenberg, m., pugno, n.m., gorb, s.n. 2010, spatulate structures in biological fibrillar adhesion. soft matter, 6, pp. 3269–3272. 7. filippov, a.e., popov, v.l., gorb, s.n., 2011, shear induced adhesion: contact mechanics of biological spatula-like attachment devices. j. theor. biol., 276, pp. 126–131. 8. guidetti, r., rizzo, a.m., altiero, t., rebecchi, l., 2012, what can we learn from the toughest animals of the earth? water bears (tardigrades) as multicellular model organisms in order to perform scientific preparations for lunar exploration, planetary and space science, 74(1), pp.97-102. 9. bertolani, r., rebecchi, l., joensson, k.i., borsari, s., guidetti, r., 2001, tardigrades as a model for experiences of animal survival in the space, mssu: microgravity and space station utilization, 2, pp. 211-212. 10. william r.m., 1997, tardigrades: bears of the moss, kansas sch. naturalist, 43 (3). 11. thorp, j.h., rogers, d.ch., 2014, freshwater invertebrates: ecology and general biology. elsevier, i(iii). 12. kinchin, i.m., 1994, the biology of tardigrades, portland press, london, p. 186. 13. zernkevich. l.a., 1969, life of animals, (in russian), 3rd edition, prosveŝenie, p. 637. 14. mcinnes, s.j., norman, d.b., 1996, tardigrade biology, zoological journal of the linnean society, 1-2, pp. 1-243. 15. hengherr, s., heyer, a.g., köhler, h.r., schill, r.o., 2008, trehalose and anhydrobiosis in tardigrades-evidence for divergence in responses to dehydration. febs journal, 275 (2), pp. 281-8. 16. peisker, h., michels, j. and gorb, s.n., 2013, evidence for a material gradient in the adhesive tarsal setae of the ladybird beetle coccinella septempunctata. nature communications, 4(1661) (doi: 10.1038/ncomms2576) 17. gorb, s.n. and filippov, a.e., 2014, fibrillar adhesion with no clusterisation: functional significance of material gradient along adhesive setae of insects. beilstein journal of nanotechnology, 5, pp. 837–845. 18. andersen, s. o., weis-fogh, t., 1964, resilin. a rubberlike protein in arthropod cuticle, advances in insect physiology, 2, pp. 1–65. 19. wolff, j.o., gorb, s.n., 2012, the influence of humidity on the attachment ability of the spider philodromus dispar (araneae, philodromidae). proceedings of the royal society of london b, 279(1726), pp. 139-143. 20. puthoff, j.b., prowse, m.s.,wilkinson, m., autumn, k., 2010, changes in materials properties explain the effects of humidity on gecko adhesion. journal of experimental biology, 213, pp. 3699–3704. 21. baccetti, b., rosati, f., 1971, electron microscopy on tardigrades. iii. the integument. journal of ultrastructure research, 34(3–4), pp. 214–243. 22. heepe, l., gorb, s.n., 2014, biologically inspired mushroom-shaped adhesive microstructures. annual review of materials research, 44, pp. 173-203. 23. filippov, a.e., popov, v.l., 2006, to optimal elasticity of adhesives mimicking gecko foot-hairs, phys. lett. a,358, pp.309-312. 13259 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 807 825 https://doi.org/10.22190/fume241218013d © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper innovative design strategy for an internal combustion engine with improved output characteristics jovan dorić1, nebojša nikolić1, stjepan galamboš1, dalibor feher1, boban nikolić2 1university of novi sad, faculty of technical sciences, serbia 2university of niš, faculty of mechanical engineering, serbia orcid ids: jovan dorić https://orcid.org/0000-0002-4532-4880 nebojša nikolić https://orcid.org/0000-0003-4657-3640 stjepan galamboš https://orcid.org/0000-0003-3491-2860 dalibor feher https://orcid.org/0000-0001-7548-9697 boban nikolić https://orcid.org/0000-0002-9694-6719 abstract. the paper describes the concept of a new internal combustion engine. presented ic engine concept possesses a variable displacement, variable compression ratio, combustion at constant volume, improved intake and exhaust processes, as well as more complete expansion of the working fluid. a comparison of some performance of this new concept with conventional ic engine and electric motors is provided. the results indicate that the performance of ic engine can still be improved. the achieved efficiency of the engine is over 40%. the ricardo wave program was used for simulating the characteristics of the engine. cad model of concept engine was made with catia v5. key words: ic engine, concept, efficiency, variable displacement, constant power 1. introduction when in 1877 otto, in collaboration with gottlieb daimler, designed and patented the four-stroke internal combustion engine on german soil, he probably had no idea what role his invention would play in the future development of technology and society. compared to the steam engines and their performance, especially when it came to efficiency which was around 2%, otto's engine appeared quite superior to all technical solutions of that time. today, almost 150 years later, it can be said that it is difficult to imagine the perception of the modern world without the use of such engines. the problems brought about by these engines are more than obvious and have become very relevant in recent years. the issue of received: december 18, 2024 / accepted march 25, 2025 corresponding author: nebojša nikolić university of novi sad, faculty of technical sciences, serbia, novi sad, trg dositeja obradovića 6 e-mail: nebnik@uns.ac.rs 808 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić improving the efficiency of ic engines is highly relevant today, both in theoretical research and in practical and experimental work. there's a noticeable increase in the use of sophisticated software packages for analyzing engine processes and predicting their performance, such as one-dimensional gas dynamics programs like ricardo/wave, avl boost, and gt power. alongside the development of these programs, their verification through experiments has also progressed. many authors described the application of wave in the early stages of engine design. meanwhile, some authors presented the calculation and optimization of variable intake valve opening using 1d-cfd (one dimensional-computational fluid dynamics) code. as for simulating the efficiency, performance, and emissions of the engine-vehicle system, it was presented in [1]. apart from basic modules for detailed engine process analysis, some authors have gone further to simulate gtdi engine with the intention of maximizing the engines efficiency and minimizing the emission [2]. comprehensive simulations and accurate predictions intended for highly stressed engines have also been carried out in references [3]. as different software solutions have evolved, the idea of integrating two or more software packages for more comprehensive and precise analysis of complex phenomena in the intake and exhaust systems of ic engines has emerged. such integration of numerical codes has enabled the connection of 1d software with 3d analyses of individual engine model segments, allowing for detailed monitoring of fluid flow effects in specific engine components. the results of these simulations aim to optimize the observed components better. simulations of air path systems and intake manifold problems are presented in references [4, 5]. simulations of flow through the intake manifold of a v6 engine were described in [6] coupled with 3d cfd vectis code. there are several papers where 1d code for simulating engine model processes was linked with promo 3d cfd code, and intake manifolds, coolant flow, and exhaust manifolds were subjected to simulation to observe the advantages between one-dimensional and three-dimensional process observations. simulation of a turbocharged engine also was carried out in many publications. from the aforementioned research and research that can be found in worldwide literature, it is evident that all significant aspects of the classic engine concept have been simulated, with many publications successfully verifying these results through experiments. however, all research conducted so far has focused on engine operation based on a predetermined kinematics i.e. in cfd code programs a characteristic law of volume change or piston motion was not provided, only the geometry of elements defining the change in engine displacement of the conventional piston mechanism was defined. this practically means that, with knowledge of piston stroke length, piston diameter, crankshaft radius, and crank mechanism deaxiality if it exists, the program deals with volume change at any given moment. from a theoretical perspective of studying thermodynamic cycles in engines, numerous publications critically review volume changes in conventional piston motion. primarily, research in finite time thermodynamics repeatedly demonstrate heat addition at constant volume as the most efficient method of heat addition [7, 8], such heat addition has been specifically studied in the otto cycle [9]. then, variable compression ratio as a modernization of piston mechanism kinematics is well known and explained from various aspects [10, 11]. similarly, the advantages and efficiency of variable displacement engines are detailed in [12]. not to be forgotten are the significant drawbacks of working fluid discharge resulting from conventional piston motion, inevitably leading to premature expansion process interruptions [13, 14]. likewise, the classic piston mechanism kinematics leads to increased pumping work and efficiency degradation [15]. innovative design strategy for an internal combustion engine with improved output characteristics 809 from all this, the authors believe that there is a lack of analyses in the field of ic engine efficiency that would describe the output parameters of engine operation to some extent when using a modified piston motion law. all previous research has been extensively conducted, with expected results proven on test benches and documented in the mentioned publications. therefore, in addition to their experimental research, the authors rely on certain principles verified by numerous eminent researchers in the analyzed field. however, the emphasis in this study will also be on simulating engine operation in the case of unconventional piston motion laws. in this paper the integration of modified piston motion laws was performed by inserting the laws into the cfd code of the 1d ricardo/wave program. the modified piston motion laws aim to achieve variable engine displacement and variable compression ratio, as well as the peculiarities of piston rest in tdc (top dead center) and bdc (bottom dead center). 2. new concept of ic engine the basic components of the new ic (internal combustion) engine concept with certain similarities and differences to today's engines will be presented. it should be noted that the illustrations presented are merely a basic description of the appearance and elements that are there to essentially describe the mechanism, rather than provide insight into the complete technical documentation of the engine. fig. 1 basic components of a new concept ic engine 810 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić in fig. 1, the basic components of the engine assembly are presented. some elements can be observed that are present in modern automotive internal combustion engines, as well as elements that appear for the first time as part of the ic engine mechanism. the following elements are shown in the indicated positions: 1-torsional block, 2-engine head, 3-torsional piston, 4-intake manifold, 5-exhaust manifold, 6-camshaft, 7-valve, 8-valve spring, 9-motion generator housing, 10-flywheel, 11-14-non-circular gears, 15-electric motor for changing engine displacement, 16-electric motor for changing compression ratio, 17-crankcase. to provide a clearer and more comprehensive understanding of the concept and the formation of the toroidal combustion chamber, fig. 2 is presented below. in addition to the mentioned elements in the previous figure, the following are shown: 7-camshaft of exhaust valves, 8-exhaust valve, 18-camshaft of intake valves, 19-intake valve, 20-engine head cover. it is clearly observed how combustion chamber is formed, limited by elements 1 (torsional block), 2 (engine head), 3 (piston), 8 (intake valves i.e., intake valve heads) and 19 (exhaust valves i.e., exhaust valve heads). the analogy of piston movement through the cylinder in a standard engine is here similarly executed with a toroidal piston and toroidal engine block through which the piston moves. several factors determine the size of the toroidal section, the first being that the ratios of piston stroke and diameter remain within the limits used today in conventional engines. another important aspect is undoubtedly the ability to perform too large stroke of the piston through the toroidal section. it is observed from fig. 2 that in the described case, a toroidal section of about 100 degrees is chosen. it could be said that this size also represents almost maximum values, because by increasing the size of this section, it would be very difficult to construct an intake manifold. fig. 2 engine cross-section (formation of the toroidal combustion chamber) since it is a variable displacement and variable compression ratio engine, the piston will move through the torus with different angular displacements. this practically means that at low active volumes, the piston will oscillate, i.e., angularly move in the upper parts of the toroidal section, while for example, at very high volumes it will move through almost innovative design strategy for an internal combustion engine with improved output characteristics 811 the entire toroidal volume. of course, if it is about regimes that require large displacement and small compression ratios the piston will not reach the top of the torus because that part of the space will be reserved for the compression volume. today there are engines that change displacement not by changing the piston stroke, but by deactivating cylinders. in such engines, regardless of deactivation, the piston continues to move through the cylinder, only fuel combustion does not occur in it. therefore, even though the cylinder is not active, it still incurs certain mechanical losses. when changing the engine volume due to changes in piston stroke, there will not be unnecessarily much greater piston movement as is the case with cylinder deactivation, thus reducing losses will be influenced. it is known that there are many different concepts of internal combustion engines, and for many there is a challenge in designing the combustion chamber sealing due to the piston different shape. in this concept, sealing is not an issue because the piston rings are cylindrical in shape. the torus itself represents a kind of cylinder, i.e., the cylinder would be a special case of the torus where the diameter of the tube tends to infinity. in figs. 1-3, an approximate appearance of an unconventional engine with variable piston motion is shown. all basic elements and mechanisms of the engine can be observed on the same images. the engine flywheel serves as an energy accumulator, as in a classic engine, only it is connected to the piston mechanism through the motion generator with two pairs of non-circular gears. in the shown case, such a transmission ratio is implemented that the flywheel rotates three times about its axis during one working cycle. in contrast to a classic engine where the flywheel rotates as many times during the working cycle as the crankshaft, i.e., twice in four-stroke engines and once in two-stroke engines. the basic setup of the system for the distribution of working fluid, exhaust system, and intake system can be observed on the same image. the most demanding mechanism highlighted is the motion generator with non-circular gears, all other engine elements are more or less similar to the elements of today's engines. the described concept is not presented to provide a final image of the engine assembly, but only to outline a rough picture of the new mechanism and the possibilities of realizing such an engine. the presented mechanism can be realized in several ways, for example, by adding another section of pistons, or by a different execution of the complex mechanism with non-circular gears in order to better withstand high torques. the phase of the two pairs of non-circular gears is crucial for achieving the required angular oscillation, with an increase in the phase shift the amplitude of oscillation increases but not the frequency. to increase the frequency of oscillation, it is necessary to increase the angular velocity of the non-circular gears or the flywheel. for easier understanding of the way the phase shift is formed, two cross-sections of the engine are presented in figs. 4 and 5. the used sections are outlined in fig. 3, where certain elements are removed for better clarity. although of a complex profile, it is observed that the engagement of the presented pairs of gears always occurs at points lying on the line of the shortest distance from the axes of rotation of the non-circular gears. it has been observed that two adjacent, identical, non-circular gears overlap within the mechanism. this design feature enables the piston to stop at both the bottom dead center (bdc) and top dead center (tdc) positions. as a result, the combustion process occurs with a smaller change in volume, closely approximating an isochoric heat addition. furthermore, this configuration allows for the later opening of the exhaust valve, which facilitates a more complete expansion of the working fluid. 812 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić fig. 3 approximate appearance of an unconvetional ic engine with marked cross sections (electric motors 15 and 16 shown in fig. 1 are removed here) electric motors 15 and 16 (fig. 1) are connected through gears and levers and can control two important parameters: the compression ratio and the volume. electric motor 15 is responsible for changing the engine displacement (by adjusting the oscillation angle of the piston), while electric motor 16 is responsible for changing the compression ratio. fig. 4 cross-section of the first pair of non-circular gears (cross-section 1 from fig. 3) innovative design strategy for an internal combustion engine with improved output characteristics 813 fig. 5 cross-section of the second pair of non-circular gears (cross-section 2 form fig. 3.) 3. simulation model in the following section the most important input data for the engine model will be presented. first and foremost, basic data given in table 1 should be mentioned. it is observed that the data are very similar to those of a standard engine. when it comes to the valve mechanism, it is almost exactly the same as in a conventional engine. although this involves a toroidal-shaped piston, the piston stroke actually represents the mean diameter of the toroid, and it is therefore expressed in millimeters, even though it would be more accurate to express this quantity in terms of an angle. also some values of valve lift are diffirent in order to optimize the fluid flow. table 1 main engine data engine type otto four stroke number of cylinders 2 number of valves per cylinder 4 piston bore 120 [mm] piston stroke 30-177 [mm] intake valve diameter 44 [mm] exhaust valve dimaeter 40 [mm] valve lift 15 [mm] exhaust valve open duration (evdur) 235 [deg] iintake valve open duration (ivdur) 230 [deg] exhaust valve maximum open point (evmp) 253.3-245 [deg] inteake valve maximum open point (ivmp) 479.3-471 [deg] compression ratio 8-16 814 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić it is also noted that the valve lift lengths defined by the previous table are very similar to the values of conventional engines. this is a result of the alignment of the kinematics of the valve mechanism and the piston mechanism. in addition to the classic parameters, it is observed that the compression ratio in this model is represented within certain limits. since this is a variable compression ratio engine, this value must also be defined within the limits where it changes during the simulation process. the engine displacement, as well as the compression ratio, represents one of the variable parameters, and therefore the stroke is also defined within specific limits. the values of evmp and ivmp actually represent the angle when the valve is fully open. this concept allows for the operation of the engine under variable conditions of displacement and compression ratio, so the analysis of such an engine will result in much more complex outcomes compared to a conventional engine. for this research the ricardo/wave software package was used for making engine model (fig. 6). unlike conventional engine modeling where piston motion is defined using classical kinematic parameters, in this study, piston motion or changes in working fluid volume were defined using the profile editor. results from kinematic analysis were individually inserted into the editor, ensuring accurate distribution of piston positions relative to the crankshaft position. although the presented concept does not feature a conventional crankshaft, changes in displacement had to be defined relative to a reference angle. therefore, the traditional crankshaft rotation angle was replaced with the term equivalent angle for easier interpretation of results. so in this study a 720-degree motion profile was used (fig. 7), even though the flywheel actually rotates three times, not twice. fig. 6 engine model innovative design strategy for an internal combustion engine with improved output characteristics 815 to generate diagrams in the wave post processor, the input piston motion law was defined with a step for every degree of crankshaft rotation, requiring 720 points for one complete cycle. this mechanism allows for an infinite number of values of angular displacements of the toroidal piston. one such diagram of possible angular displacements is shown in fig. 7. fig. 7 complex angular motion of piston (18 cases were examined) the use of one-dimensional cfd simulation for engines is now one of the most significant tools during the early stages of engine development. engine design through simulation drastically reduces the time that would otherwise be required for extensive experiments and prototypes as such simulations can predict a large number of experiments. one of the main advantages of this approach to the development of internal combustion engines is the relatively high correlation between simulation results and experiments, which, with proper definition and shaping of all engine components, can achieve accuracy of approximately +/5%. the main problem to address in this case is the synchronization of the valve mechanism and the piston mechanism, defining the ignition timing angle, and limiting engine operation due to the risk of knocking, given that this is a virtual engine and there are no predefined values for parameters that have a significant impact on engine performance. in this case, we are dealing with an unconventional piston motion law, so it is crucial to determine the ignition timing angle, which will differ from standard values. the reason for this is the piston position near tdc, which allows possibility of avoiding larger negative work (ignition before tdc). in this concept, the piston is held around tdc for a longer period, which allows the spark to occur later (closer to tdc). this way, negative work is reduced by decreasing pressure buildup before tdc. the parameter evmp holds the same significance as ivmp, but in this case, the simulation process is related to the exhaust valves. although the timing of the intake valves is always marked as having a very sensitive impact on process efficiency due to cylinder charging, it is observed that the timing of the exhaust valve openings also significantly affects efficiency. one of the main reasons for this efficiency trend is that the valves are designed to open only when the piston reaches bdc. unlike in conventional designs where valves open before the piston reaches bdc, such valve movement is not desirable in the described concept. this approach allows the engine model to achieve greater efficiency by enabling more complete expansion of the working fluid. 816 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić besides the selected engine parameter values, it is necessary to define the wall temperatures that enclose the combustion chamber as initial quantities. the defined values are taken from the ricardo/wave recommendations. these values are presented in table 2. table 2 combustion chamber temperatures wall/border temperature [k] piston head 520 cylinder head 500 cylinder wall intake valve exhaust valve 430 450 520 when all simulation parameters are defined, the modeling of the engine model segments can begin. generally speaking, the basic engine model is relatively simple; it consists of the intake system, the fuel-air mixture formation system, the engine itself, the exhaust system, and the working environment. the working environment does not necessarily have to be the same for the intake and exhaust sections, but standard values are usually adopted, which is also the case in this model. the intake system is designed to meet the conditions of minimal losses and to avoid undesirable air flow oscillations that reduce the cylinder filling coefficient. in this analysis, the efficiency coefficient of the coupled gear pair is assumed to range from 0.98 to 0.99, depending on the type of coupling. specifically, higher values are chosen for internal couplings, as recommended, while external couplings and couplings of non-circular gears are slightly less favorable, with a coefficient of 0.98 adopted for these connections. using these input values, only a rough estimation of the mechanical losses in the motion generator can be determined. since the ricardo/wave software includes an option for defining these losses through multiple parameters, it was necessary to align the losses in the new concept with the classical setup. it can be stated that the losses associated with the working fluid distribution mechanism remain unchanged in this new concept. likewise, the losses in the crankshaft-to-engine block connections can be considered unchanged. for further analysis, a modified chenflynn model was used. this correlation includes several terms, among which one accounts for engine equipment losses, another is a function of cylinder pressure, a third term is linearly dependent on the mean piston speed (calculating hydrodynamic losses), and the fourth term considers aerodynamic losses through a quadratic dependence. each cylinder contributes to the overall engine losses, which are based on the actual pressure in the cylinder, known to vary from one cylinder to another. by adjusting input parameters, friction losses in the piston assembly can be reduced, while mechanical losses in the piston motion generator may increase. this approach provides an approximate estimation of total mechanical losses. although not ideal, it represents the only viable method for defining mechanical losses in unconventional designs within the ricardo/wave software. 4. results and discussion when it comes to the characteristics of full-load performance for various stroke ratios, the curves obtained can be seen in fig. 8. it is observed that maximum efficiency is achieved at small stroke values and high engine speeds. conversely, for large strokes, innovative design strategy for an internal combustion engine with improved output characteristics 817 maximum efficiency is achieved at lower engine speeds. this is primarily due to cylinder charging, excessively large piston strokes cannot achieve effective cylinder charging at higher engine speeds, whereas smaller strokes have the opposite effect. figure 8 highlights the curve of maximum efficiency as well as the curve for the typical efficiency of conventional internal combustion engines. in general, it can be concluded that increasing the compression ratio leads to a rise in the maximum cylinder pressure, which is expected. it should be noted that these results applied to cases where the ignition timing and corresponding compression ratio were aligned, and the results clearly indicated that the maximum pressure approached tdc with increasing compression ratio. in fig. 8, areas of constant efficiency can be identified. it is observed that high efficiency is achieved across all engine speeds but for different stroke to bore ratios or stroke/diameter (s/d). this practically means that the engine can operate very efficiently under full load if the necessary s/d ratios are met. similarly, it is unrealistic to expect high engine speeds with a large piston stroke. by changing the s/d ratio, the benefits of variable displacement can be achieved with this ic engine [16]. fig. 8 engine efficiency in relation to s/d and rpm in comparison to standard si engine (dotted line) it should not be forgotten that at very high engine speeds in a conventional engine concept, there is a significant increase in the average piston speed, which negatively impacts mechanical losses. in the case of variable piston stroke, at very small strokes and high engine speeds, the average piston speed does not increase significantly compared to regimes with large strokes and low speeds. this is one reason why the impact of engine speed on mechanical losses is expected to be smaller for engines with variable piston stroke (vpm), as evidenced by the results. analyzing the obtained results leads to the idea of developing a concept that allows for the realization of maximum power with high efficiency across all operating regimes. this means that the engine can deliver high power at low speeds when using large displacements and high power at high speeds with small displacements. as expected, the engine exhibits exceptionally high power at low speeds (80 kw at slightly over 2000 rpm). of course, excessively high power at low angular speed of the drive wheels is impractical because it leads to large torque, resulting in significant force on the drive wheels. such high forces can cause wheel slip, making it inefficient to realize excessively high power at low engine speeds. finally, the characteristics of full818 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić load power for different s/d ratios can be presented in fig. 9. it has been observed that a relatively high level of power is reached quickly. following this initial surge, the power curve increases at a slower rate. this gradual rise is primarily due to the possibility of achieving a slightly higher compression ratio under certain operating regimes. fig. 9 engine power in relation to s/d and rpm it is well-known what the torque curve of a conventional internal combustion engine looks like. when considering a design that has the property of varying the stroke-to-bore ratio (s/d), it is concluded that such an engine possesses a multitude of torque curves corresponding to each specific s/d ratio. this can be illustrated by the results shown in fig. 10. due to this change in torque, it is possible to realize a curve known as the ideal hyperbola of torque. it should be noted that this curve is achieved without compromising power at specific operating regimes, particularly at high engine speeds. the curve of maximum torque over a broad range of engine speeds is a logical outcome of the operational cycle in a variable piston motion engine. from previous analyses it is evident that at high stroke-to-bore ratios, high charging coefficients are achieved at low engine speeds. this positively affects torque and allows for relatively high maximum power to be maintained even at very high engine speeds. as engine speeds increase, the degree of cylinder charging deteriorates with such large strokes, leading to a reduction in torque. therefore, at medium engine speeds, conditions are created to reduce piston stroke in favor of increased charging efficiency, which helps maintain constant power. if engine speeds were to increase further, the newly established regime would no longer meet the criteria for effective cylinder charging, necessitating a reduction in piston stroke again. in this manner, innovative design strategy for an internal combustion engine with improved output characteristics 819 a constant power curve is achieved, or, if considered in terms of torque, this behavior of engine displacement as a function of engine speed will produce an ideal torque hyperbola. fig. 10 engine torque in relation to s/d and rpm an ideal torque hyperbola is not characteristic of conventional internal combustion engines but can be realized in some diesel engines within narrow ranges of engine speeds. however, having a torque hyperbola in a narrow operational range is positive but not sufficient for significant advantages. the concept described here actually offers the possibility of eliminating the need for a traditional gearbox. it is not enough to simply have a torque hyperbola, a sufficient broad range of engine speeds where power remains constant is also necessary. the described concept is capable of meeting both these necessary and sufficient conditions for operating a vehicle without the need to change the current transmission gear. now we can compare the performance of this concept with an electric motor which is used in electric vehicle. first, the performance of electric motor efficiency is presented in fig. 11. to obtain such diagrams, it is necessary to first measure the observed parameters across the entire operating range of the engine, at a sufficient number of different engine speeds and load conditions [17]. in this case, since it is a virtual engine model, the data was obtained by analyzing a multitude of load conditions, which consisted of many operating points. here, the load refers to the corresponding stroke to bore ratio, or the current engine displacement. as described, changes in displacement can be used to form a power curve that remains constant over a large portion of the engine's operating speeds. naturally, for well-known reasons, power cannot remain constant throughout the entire range, as at very low speeds, the engine would generate excessively high torque, which would not be practical for transmission to the road surface. nonetheless, this engine can generate constant power over a wide range of speeds. finally, we can see the efficiency map similar to those of electric vehicles (fig. 12). 820 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić fig. 11 electric motor efficiency map fig. 12 efficiency map of unconvetional ic engine it is interesting to compare this unconventional engine with a conventional engine (fig. 13). innovative design strategy for an internal combustion engine with improved output characteristics 821 fig. 13 comparison of power and torque between the conventional and unconventional ic engine the overall impact of the described piston mechanism kinematics on engine processes is comprehensively presented in this study. for clarity, table 3 provides a comparison of the characteristics of these two concepts. table 3 comparison of conventional and unconventional engine conventional ic engine unconventional ic engine fixed compression ratio varibale compression ratio fixed displacement variable displacement combustion during volume changes constant volume combustion exhaust valve open before btc more complete expansion (exhaust valves open almost at bdc) ignition before tdc (negative work) ignition near tdc gearbox needed (non-constant power engine) gearbox not needed (constant power engine) table 4 outlines the essential differences and similarities between the basic systems and elements in these two different engine design approaches. it is evident that the proposed concept closely resembles a conventional engine. the primary distinction lies in the motion generator, which consists of non-circular gears that can partially replace the conventional transmission system. additionally, it can be observed that the cylinders in this concept exhibit a different design due to their angular motion. table 3 provides a comprehensive overview of the key characteristics of the newly proposed concept. the engine is designed with a variable compression ratio, allowing for operation on multiple fuel types [18, 19] or with different biodiesel fuel [20] ensuring optimal efficiency across various operating conditions. furthermore, it features an enhanced heat supply mechanism and an improved working fluid distribution system. these advancements collectively contribute to an increase in the thermal efficiency of the ic engine. last but not least, the engine features a variable displacement, which can be utilized to achieve constant power over a wider range of engine speed. 822 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić table 4 comparison of basic parts of conventional and unconventional engine conventional ic engine unconventional ic engine innovative design strategy for an internal combustion engine with improved output characteristics 823 exhaust gas emissions will not be presented in this paper, but they were also analyzed as part of the study on the impact of piston motion, variable displacement, variable compression ratio and heat addition under small volume changes during combustion. almost constant volume during combustion and variable piston motion contributes to a overall reduction in co and hc emissions. with slower piston motion around tdc the engine can achieve better performance during cyclic variations, which are quite common in conventional internal combustion engines [21]. slower piston movement near tdc allows more time for better mixture formation during injection, which benefits the combustion process and engine emissions [22]. 5. conclusion in this paper the analysis of an ic engine with modified piston motion compared to standard piston mechanisms is presented. differences primarily lie in the implementation of combustion at almost constant volume, variable displacement and variable compression ratio. the study simulated output parameters and engine characteristics across all operating regimes, with some results presented in this article due to the extensive nature of the research. an ic engine represents a heat engine, and its characteristics depend on how the engine interacts with the working fluid inside it. it is well known theoretically that the most efficient cycle for describing the operation of an internal combustion engine is the otto cycle, provided unconventional cycles such as the miller and atkinson are disregarded, which involve different or longer expansion strokes. increasing compression ratio enhances the efficiency of the otto cycle, reducing the need for the atkinson cycle since higher compression ratios increase the expansion ratio of the working fluid, thereby reducing pressure at the end of the expansion stroke. however, achieving extremely high compression ratios with current combustion technology and fuel chemistry is impractical. hence, the ideal otto cycle remains the most economical for a constant compression ratio. therefore, in this paper research focus was on realizing thermodynamic cycles in real engines that approach the ideal otto cycle model by altering the working fluid. the otto cycle becomes particularly interesting with increasing compression ratio, aligning with the mechanism described in this concept. leonardo da vinci, one of the greatest renaissance artists, had ideas far ahead of his time, and many of his inventions were never constructed. his versatility meant that few of his ideas were realized, as he often moved from one idea to another. the brilliant ideas of this polymath mostly remained sketches. in the document "codex madrid," leonardo first presented sketches of non-circular gears. the idea created by leonardo da vinci half a millennium ago and the thermodynamic cycle described by nikolaus otto are creatively linked and presented in this work. changing the gear ratio between two gears has enabled achieving heat input at constant volume to the working fluid in the engine cylinder, previously unattainable, thereby increasing the efficiency of converting chemical energy into mechanical work, which was the goal of this research. this study aimed to bring real cycles closer to ideal ones. simulations and predictions unequivocally demonstrate that efficiency can be significantly increased under certain conditions. additionally, this research has enhanced the overall effectiveness of the engine, thereby improving a broader motor-vehicle system. the described research represents a modest scientific contribution to analyzing potential directions for the development of 824 j. dorić, n. nikolić, s. galamboš, d. feher, b. nikolić internal combustion engines characterized by increased efficiency and environmental friendliness. these two characteristics are now the primary goals of all human systems as we search for new solutions in hopes of a better future. acknowledgement: this research has been supported by the ministry of science, technological development and innovation (contract no. 451-03-136/2025-03/200156) and the faculty of technical sciences, university of novi sad through project “scientific and artistic research work of researchers in teaching and associate positions at the faculty of technical sciences, university of novi sad” (no. 01-50/295). references 1. mccollum, l.d., thornton, j.m., teaylor, d.j., 2004, application of wave 1-d engine models with vehicle simulation tools to investigate efficiency, performance, and emission impacts of advanced engine operation, presented at the ricardo software 9 th annual international users conference, southfield, michigan, usa, march 12, 2004. 2. lethwala, y., sharma, n., jain, r., 2019, engine performance simulation of ricardo wave for gtdi optimization, international journal of engineering and advanced technology, 8(5), pp. 1444-1448. 3. farrugia, m., 2004, fsae: engine simulation with wave, oakland university formula sae team. 4. derrico, g., onorati, a., ferrari, g., 1999, an integrated 1d-2d fluid dynamic model for the simulation of wave action in i.c. engine manifolds, ice99 – international conference on internal combustion engine, capri-naples, italy. 5. stockar, s., 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845-849. 22. biswas, s., mukhopadhyay, a., 2022, assessment of the quadruple injection strategy over triple injections to improve emissions, performance and noise of the automotive diesel engine, facta universitatis-series mechanical engineering, 20(2), pp. 321-339. 12109 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume231112007s © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper investigations on the influence of laser surface treatment on hardness and wear behaviour of al-mmc prepared through powder metallurgy route arvind sankhla1, kaushik patel1, mayur makhesana1, anand patel1, kapil gupta2 1mechanical engineering, institute of technology, nirma university, ahmedabad, india 2mechanical and industrial engineering technology, university of johannesburg, johannesburg, south africa abstract. aluminium metal matrix composites (al-mmcs) are extensively used in various industrial sectors, including aerospace, automotive, construction, and electronics, owing to higher hardness, low density, higher fatigue and specific strength. powder metallurgy is an effective method for manufacturing composite materials. compared to pure metals and alloys, the mechanical characteristics of the sicreinforced mmcs are enhanced. the al-mmc surface can undergo metallurgical changes due to the laser treatment, which can also strengthen the binding between the matrix material and the reinforcement particles. therefore, the current work investigates the impact of sic particle addition and laser surface treatment on the hardness and wear characteristics of aluminium metal matrix composite (al-mmc). the al-mmc is initially fabricated using a powder metallurgy process, and then the mmc is treated with a laser. compared to the untreated mmc, the laser surface treatment increased the hardness by almost 12%. additionally, the addition of sic content by 10%, 15%, 20%, and 25% in laser-treated al-mmc resulted in increased hardness by 12%, 14%, 15%, and 16%, respectively, compared to untreated al-mmc. furthermore, the wear resistance improved as the reinforcement particles increased. the laser-treated samples exhibited lower wear than untreated ones due to the formation of a new layer on the treated surface, preventing the release of sic particles. the surface treatment of mmc through the laser is a novel approach to fabricating wear-resistant al-mmcs. key words: al-mmc, laser treatment, powder metallurgy, hardness, wear received: november 12, 2023 / accepted february 26, 2024 corresponding author: mayur makhesana mechanical engineering department, institute of technology, nirma university, ahmedabad, india e-mail: mayur.makhesana@nirmauni.ac.in 2 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta 1. introduction laser (light amplification by stimulated emission of radiation) is helpful in all manufacturing sectors like machining, forming, and fabrication [1-3]. composites are specialized materials to fulfil many high-end engineering application requirements. the aluminium-based provide an alternative to iron and copper-based composites due to its improved properties, such as high specific strength, light weight, low density, enhanced tribological properties, etc. [4]. the development of composites can also be complimented by a laser. it is also possible to use laser in many ways to fabricate metal matrix composites (mmc) through an additive manufacturing approach or surface treatment. moreover, surface alloying can also be done. laser sintering is one of the most popular techniques for creating materials like mmc [5]. laser sintering is a name given to one of the rapid prototyping techniques to fabricate mmc and the application of laser for processing mmcs in different ways, such as surface treatment, fabricating porous mmcs by melting layers of powder on one another, or surface alloying. it provides much versatility when working with complicated shapes. dealing with the components of metal matrix composites with higher melting temperatures and harder materials requires laser sintering as well [6]. surfaces of aluminium metal matrix composites (al-mmc) can be treated with the laser by varying laser power and scanning speed. kumar and kruth [7] presented the study of various rapid prototyping processes for composite materials. according to their work, apart from fusion deposition modelling (fdm) for polymers and polymer-based materials, the fdm approach can be extended to metal-based composite, i.e., mmcs, selective laser sintering/melting (sls/slm), laser-engineered net shaping (lens), and ultrasonic consolidation, which are new emerging rapid prototyping techniques [8]. for example, cu-fe-based mmc can be fabricated using laser-based technology, where the purpose of copper is not to enhance the properties of iron. instead, copper can help effectively consolidate the iron powder without causing any hindrance [9]. laser-based methods can also process various reinforcements in the polymer matrix. hence, various types of composite material can be fabricated using laser, e.g., al-filled polyamide powders, which are developed to reduce agglomeration effectively. such powder can be further processed by laser just like in fdm techniques, or this powder can be used in other processes like powder metallurgy or hot isostatic pressing (hip). gill and hon [10] utilized a laser in the form of sls to fabricate sic-based polyamide composites. they fabricated the composite using 50w of laser power. they experimentally found that the power of the laser beam and scan speed are the most critical parameters that can be optimized for developing target mmc with the required properties. the laser-based methods are also used to fabricate tungsten carbide and cobalt-based mmc with the addition of copper, as surface tension in the melt is affected by the presence of copper [11]. in past studies, various composite materials were processed by laser technology, and it can be seen that laser can process a wide variety of materials to fabricate mmc, hybrid mmc and ceramic-based composites [12-14]. the laser energy is also used to control chemical reactions. since the laser melts the material, this phenomenon is known as a laserinduced chemical reaction. the laser beam energy is used to overcome the activation energy needed to form compounds and to trigger the production of sufficient thermal energy to propagate chemical reactions. according to kruth et al. [15], the in-situ formation of mmc involves both types of phenomena. mmc is better than those mmc in which reinforcement material is added conventionally [16], such as stir casting [17]. copper investigations on the influence of laser surface treatment on hardness and wear behaviour... 3 based mmc uses the activation energy-based phenomenon, and tic-al2o3-based mmc uses the second phenomenon, i.e., the release of thermal energy to form compound-based mmc from tio2, al, and c. gaard et al. [18] fabricated tic-based mmc in the matrix of invar 36 through the direct metal laser sintering (dmls) approach. furthermore, liquid-phased sintering was the prime mechanism for forming a tic-based dendritic structure along with a thin shell-like structure. thermal cracking was also observed when the proportion of tic was increased by 20% in the matrix. however, thermal cracking was not observed for the lower proportion of tic, which signifies that laser can affect the bond formation to a maximum extent by thermally affecting hard compounds like tic. the formation of spherical particles confirms the melting and quick recasting of constituents in the compound form. they investigated the wetting properties of the liquid phase of constituents, which are responsible for forming a dense layer in the structure, affecting the strength of the mmc. nofar et al. [19] reported increased wear resistance of al-alti3-based metal matrix composites when processed by hot pressed technique. mishra et al. [20] reported that due to dmls, the bonding of sic particles in the matrix of ti6al4v improved, and tial was formed due to exothermic chemical reactions. they reported that the hardness of mmc improves due to laser treatment, and the laser power was the critical factor for hardness. this resulted in the increased wear resistance of mmc much higher than that of pure ti6al4v alloy. sateesh et al. [21] fabricated coated sic-based inconel mmc through direct metal laser sintering (dmls). they applied a 240w laser to fabricate the mmc at four different scan speeds. they found that the density of mmc increased due to lower scan speed. at lower scan speed, more laser interaction with matrix and reinforcement material occurs. hence, due to favourable metallurgical changes, density improves. they reported that the hardness of laser-processed mmc increased by 13 to 33 %, as due to laser processing, the dislocation density increased, and mmc became more resistant to plastic deformation. they also proposed that the difference between the cte of inconel-625 and sic is also one of the reasons for improving the density and hardness of laser-processed mmc. muvvala et al. [22] conducted an experimental investigation on tic inconel 718 mmc processed through laser and reported that increasing the exposure time of tic to high temperature can result in the decomposition of tic, which further results in a change in the solidification behaviour of mmc and tic particles can form dendritic structure. chang and gu [23] worked on carbon nanotubes (cnt) reinforced ti–mmc through dmls. they reported that micropores were generated at the interfacial regions due to the distribution characteristics of cnts and the balling effect, which affected the density of the resultant mmc. by increasing the laser power, such issues could be resolved, agglomerations could be converted into homo-disperse, and the wear rate of mmc could be reduced significantly. promakhov [24] found that there is a scope for self-propagating high-temperature synthesis during inconel 625–tib2 mmc fabrication by laser application. due to this, the wettability of the matrix material and reinforcement material improved. as a result, the bonding and distribution of reinforcement particles were improved to a large extent. the microhardness of inconel 625–tib2 mmc was 1.5 times more than that of the parent material, and tensile strength was improved by 15% compared to pure inconel. venkatesh et al. [25] synthesized al-mmc using stir casting by reinforcing 0.5 wt%sic, 1.0 wt%sic, 1.5 wt%sic and 2.0 wt%sic nanoparticles. it was found that the maximum hardness of 431 mpa and ultimate tensile strength of 163 bhn resulted with 4 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta 1.5 wt% sic reinforced composite. however, 2 wt% sic reduced the mechanical properties due to brittle agglomeration. similar findings were reported, concluding that adding 2 wt% sic in the aluminium develops the secondary phases due to chemical reactions with neighbouring particles. which further reduced the strength of al-mmc [26]. adding uniform sic nanoparticles to aluminium matrix dispersion improves the bonding strength between the nearby particles, prevents dislocations, and improves mechanical strength. adding more than 2 wt% sic results in slip planes and initiates plastic deformation compared to 1.2 wt% sic [27]. chong et al. [28] found that surface treatment through laser on mo-wc-based mmc showed excellent wear resistance, especially crack-free s laser sintered mmcs were excellent against abrasive wear compared to base metal or conventional alloy. hussain et al. [29] investigated the effect of laser power on the density of tin-ss316 mmc, and they found that with the increase of laser power, density improves for all the compositions of mmc, i.e., tin was used as reinforcement by 10%,15%, and 20%. the intense energy of the laser caused massive consolidation. therefore, density improved and porosity was reduced. hence, the laser treatment can also be considered as an alternative secondary process to reduce the porosity when mmcs are fabricated through the powder metallurgy route [30]. it is understood from the available literature that the fused deposition of powder is one of the techniques by which metal matrix composites can be fabricated. in many cases, the laser can be utilized to fuse the powder on a layer basis to fabricate metal matrix composites. many works are reported with laser processing to fabricate metal matrix composites (mmcs). however, limited research attempts have been reported towards analyzing laser-treated mmcs fabricated through the powder metallurgy route. considering these facts, the current work attempts to conduct an experimental investigation on aluminium metal matrix composite (al-mmc) fabricated through powder metallurgy. the objective of the work is to investigate the impact of sic particle addition and laser surface treatment on the hardness and wear characteristics of aluminium metal matrix composite (al-mmc). the work also aims to investigate the effect of sic particle addition and laser treatment on the coefficient of friction compared to untreated al-mmc. a laser is used to treat the al-mmc surface. it was discovered that applying a laser to mmc's surface increased its hardness compared to untreated mmc, and the wear resistance is improved with the increase in the percentage of reinforcing particles. therefore, with this approach, al-mmcs with target properties can be fabricated. hence, the surface treatment of mmc through the laser is a novel approach to fabricating wear-resistant al-mmcs. further, the enhanced hardness after laser treatment can reduce the requirement of hard reinforcement particles, which is also a significant factor in the optimum usage of reinforcement material. 2. materials and methods the preparation of the metal matrix composite material included using pure aluminium and silicon carbide (sic) powders. the aluminium powder was obtained from central drug house (p) ltd., new delhi, india. the purity of aluminium powder was 99%, and the average size of aluminium and sic particles was 50-75 microns. the al-mmc samples were fabricated through the powder metallurgy route. the methodology followed for conducting the experiments and analysis is shown in fig. 1. investigations on the influence of laser surface treatment on hardness and wear behaviour... 5 fig. 1 procedure followed in the present research work 6 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta in powder metallurgy, a 40 kn utm machine was employed. green compacts were fabricated by compacting powders at a pressure of 150 mpa. after carefully moving to a muffle furnace, the green compacts were sintered for an hour at 595°c. a muffle furnace operating in an inert argon gas environment was used to complete the sintering process. the inert atmosphere of argon gas helps prevent oxide formations of aluminium and achieve adequate bonding of particles. following the sintering process, the specimens underwent water cooling and surface cleaning. hardness and wear testing was carried out before and after laser treatment. the hardness of al-mmc specimens was measured on rockwell b scale at 100 kgf of major load. before the hardness measurement, al-mmc's surface was cleaned using finegrit emery paper. around 5 to 8 measurements for hardness were carried out at each face. the laser treatment of specimens was carried out using a pulse laser with a maximum peak power of 1500w, a pulse width of 0.05-50 ms, and max pulse energy of 15 j. the modulation frequency of the laser machine was 0~5000 hz. the samples were subjected to laser treatment for 27 seconds at 400w of power, the distance between the laser head and specimen surface was 5 mm, the tool path was a concentric circle with a spacing of 0.2 mm, and the scanning speed was 300 mm/s (fig. 2). subsequently, the hardness of the laser-treated samples was measured again. pin-on-disk wear tests were performed on mmc samples before and after laser processing to determine the effect of laser surface treatment. fig. 2 laser processing of prepared samples fig. 3 shows the al-mmc samples before and after the laser treatment. a depression on the surface is created after applying the laser, which shows that the laser affects the surface of al-mmc in real time. the intense energy of the laser causes the aluminum matrix's melting since it has a low melting point compared to sic. due to the quick melting and recasting of the upper layer, porosity at the surface level is eliminated. hence a shallow depression is seen on the surface of al-mmc samples. the depression level at the surface depends on the laser's power. all the al-mmc samples are laser-treated with different laser powers and scanning speeds, and al-mmc's hardness and wear behavior are evaluated. investigations on the influence of laser surface treatment on hardness and wear behaviour... 7 fig. 3 al-mmc samples (a) before (b) after the laser treatment pin-on disc apparatus was used to investigate the wear behaviour of the al-mmc. the samples were machined to the diameter of 10-12 mm to mount them in the sample holder of the pin-on-disc apparatus, as shown in fig. 4. the wear tests were conducted on a ‘ducom’ make computerized pin-on-disk apparatus. the apparatus can apply a maximum 200 n load on specimen during testing and run the disc from 20 rpm to 2000 rpm. a hardened steel disc is mounted for wear testing. the steel disc is 165 mm in diameter and 8 mm thick and hardened to 62 hrc. for wear testing, a pin of 3 mm to 12 mm in diameter is mounted on the specimen holder. an initial mass of samples was noted, and wear tests were performed. the mass of samples is recorded by utilizing a precision digital weighing machine. this procedure is followed for both types of al-mmcs, i.e., laser-treated and untreated ones. fig. 4 wear testing of al-mmc on pin-on-disc setup 8 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta 3. results and discussion laser treatment of al-mmc for the different compositions, i.e., 10%, 15%, 18%, 20%, and 25% of sic reinforcement, was carried out, and the surface hardness was measured. a surface micrograph is obtained through an image analyzer, wear testing is done, and results are presented and discussed below. re-melting and recasting of matrix material, i.e., aluminum, was observed, and thus bonding of matrix material with sic particles was further improved. 3.1 change in hardness of al-mmc through laser treatment the hardness of al-mmc was measured before and after the laser treatment. with the increase in the proportion of sic in the aluminuim matrix, as the hard sic particles take the significant load, bonding is improved due to the fusion of the aluminium matrix under the action of the laser beam. the sic particles are held more rigidly. hence, the hardness of mmc increases after the laser treatment [31]. the intense energy of the laser can remelt the soft matrix easily. simultaneously, because of the high temperature, there is an improvement in the bonding between the hard sic particles and the soft matrix, which is aluminium, as the matrix material melts and is recast. the temperature of the laser beam impacts the surface energy of sic particles, which lowers the total surface tension at the edges of sic particles. this improves the affinity of sic particles with aluminum and results in better bonding. as a result, the hardness of al-mmc improves after laser treatment. fig. 5 illustrates a micrograph of al-mmc's surface following laser treatment, whereas fig. 6 displays an sem image of the same material. in many regions, as presented in fig. 5, small nodular type or tiny spherical-shaped particles can be seen. these are the nodules resulting from the melting and recasting of aluminium, a matrix material with a low melting point compared to sic. since sic particles have sharp edges, i.e., angular grains like structure, bonding between matrix and reinforcement further increases during melting and recasting of matrix material, which completely engulfs the sic particles. an interlocking action is also possible to improve the bonding of particles. hence, the hardness of al-mmc can be further improved by laser treatment. due to the intense action of the laser beam, dislocation generations happen due to a thermal mismatch between the matrix material and sic particles [32]. moreover, the quick recasting of matrix material while engulfing the sic particles cause a reduction in the average particle size resulting in a smaller grain size. the smaller grain size can offer resistance to dislocation mobility. hence, hardness increases [33]. strong interfacial bonds are on the top layer of al-mmc after laser treatment. therefore, load transfer from soft matrix to hard reinforcement particle improves, causing hardness enhancement. the improvement in bond in the matrix material is partially improved due to super-solidus liquid phase sintering. it promotes the in-situ transformation of reinforcement particles into the matrix, which has a comparatively lower melting point than reinforcement material [34]. investigations on the influence of laser surface treatment on hardness and wear behaviour... 9 fig. 5 traces of recast nodules of matrix material fig. 6 sem image of laser-treated al-mmc, showing fine spheroids the action of the laser is very intense and develops much thermal energy due to the concentration at a minimal area. fig. 7 shows the intense effect of the laser. even after removing the top layer from the laser-treated al-mmc sample, traces of the laser path were visible at the macro level, i.e., through the naked eye, the laser traces can be seen very clearly. this serves as evidence that a laser can impact the properties of al-mmc through its thermal energy. it can be viewed that the surface treatment of al-mmc can be considered secondary sintering, by which properties at the surface level can be altered easily, and it is also possible that more than one reinforcement can be introduced in the matrix material through the surface alloying approach. by doing this, hybrid mmcs can be produced. 10 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta fig. 7 traces of laser tracks, showing the effect of the laser beam (a) on the surface (b) laser tracks at 3 mm depth an improvement in hardness by more than 12% is seen through laser treatment. the hardness of al-mmc increases with the increase of reinforcement, which follows the rule of mixture. however, the hardness of al-mmc further improves with laser treatment, and this enhancement is more pronounced when al-mmc contains a higher amount of reinforcement particles. the results show that hardness increases by 12% for the 10% sic content, while the hardness increases by an average of 15% for the 20% sic and 16% for the 25% sic content in al-mmc. the comparison of hardness achieved is graphically shown in fig. 8. fig. 8 comparisons of the hardness of laser treated and untreated mmc a investigations on the influence of laser surface treatment on hardness and wear behaviour... 11 with a higher amount of reinforcement, saturation in the increase in hardness can be observed due to limitations of the sintering of mmc through conventional furnaces. this issue can also be addressed through the use of laser. kundu et al. [35] also concluded through their experimental work that the hardness of mmc does increase with the increase in laser power apart from the reinforcement. when the scanning speed is maintained while the laser power increases, a portion of the molten pool fills the microscopic spaces, resulting in more significant densification. hardness increases as a result of stronger bonds created by more fusion. the laser scan speed may also affect al-mmc's hardness. with faster scan speeds, thermal action lasts a shorter time. therefore, the impact of thermal energy on the modification of surface properties may be another area of research [36]. the energy absorption of the laser beam will be influenced by optical interaction or the laser beam scattering on the al-mmc surface. this will determine the temperature that is produced in the surface area that is being laser-treated. according to gosh et al. [37], higher scan speeds may cause some mmc to remain untreated. therefore, using laser treatment, optimization techniques can be used to optimize the scan speed for the specified range of hardness in the al-mmc. fig. 9 micrographs of the laser-treated surface of al-mmc, showing the distinct alternate region 12 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta the micrographs of al-mmc surfaces treated with laser are shown in fig. 9. this was done using an image analyzer. additional findings from the laser-treated materials show evidence of thermal action due to the laser beam's high energy. furthermore, the fact that the aluminium matrix of al-mmc has melted throughout is confirmed by nodular formations on its top layer. the micrograph, which demonstrates that the laser does affect the surface topography, demonstrates that melting and recasting are the causes of the surface appearance and the creation of sphere formations. after mmc has been lasertreated, the high heat action can also reduce the surface tension at the aluminum/silicon interface, enhancing bonding. 3.2 change in wear resistance of al-mmc through laser treatment fig. 10 shows the specimen before and after the wear tests. it is to be noted from the appearance of the face of the al-mmc sample that even after too much rubbing, hard particles are retained and offer resistance to wear. the steel grey colour appearance can be seen. as discussed in the previous section, the retainment of sic particles and lower mass loss signifies that bonding between the matrix material and reinforcement particles increases due to laser treatment. fig. 10 al-mmc sample before, and after wear testing, the dark color shows the rubbing of sic particles still embedded in the matrix. the results of wear testing are shown in fig. 11 for all four compositions of al-mmc. a load of 40 n is applied during testing. the mass of each sample was measured before and after testing, and it was done for both treated lasers and untreated samples. it was found that laser-treated samples had better resistance to wear than untreated samples, i.e., there was less mass loss with laser-treated al-mmc samples. at higher sliding distances, the wear loss is less, i.e., laser-treated al-mmc shows higher wear resistance. the new layer formed on the top surface of mmc due to laser treatment does not allow sic particles to get released. instead, they are held firmly, with a greater exposure. as a result, the wear resistance increases [38]. in the initial region of wear testing, only the sharp edges of hard sic particles are chopped off. after that, no sign of loss of sic particles was observed. hence, increased wear resistance is mainly due to the thermal action of the laser, which has improved the bonding of al-matrix with sic particles and, at the same time, prevented the detachment of the sic particles. investigations on the influence of laser surface treatment on hardness and wear behaviour... 13 fig. 11 wear of laser-treated and untreated al-mmc with (a) 10% sic (b) 15% sic (c) 20% sic (d) 25% sic at 40 n load as discussed in the previous section, al-mmc that has been laser-treated has a harder surface than untreated al-mmc. this harder surface also contributes significantly to increased wear resistance. forming a stable tribo-layer also helps reduce mass loss during the dry sliding wear test. the dark appearance of the al-mmc once formed does not change during the entire wear test, which is evidence of a stable tribo-layer. the sem images of al-mmc after the wear test have been obtained, and one of the images is shown in fig. 12 (a). it can be observed that sic particles are found firmly held in the matrix and projected on the surface. this confirms that the thermal action of laser bonding has been improved. it may be concluded that along with improved bonding, the hardness of almmc also improves as firmly bonded sic particles offer resistance to deformation under loading, which indicates that hardness has increased. the sub-surfaces of laser-treated al-mmc were examined for microstructural changes. it can be seen from fig. 12 (b) that, after laser treatment, a small grain structure is formed. it shows that the energy liberated from the laser beam could settle the large grain into 14 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta smaller grains due to melting and recasting. except for a few sites, micro-porosity also diminishes significantly, indicating an improvement in bond strength [39]. fig. 12 (c) shows an sem image of overlapping layers seen with the presence of sharp-edged sic particles. these overlapping layers in the microstructure can be referred to as clad layers. the clad layers are not showing any significant separation. instead, they are attached well to each other, which is another sign of confirmation that bonding is improved after the laser treatment and reinforcement particles, i.e., sic, are held and exposed, causing increased exposure to wear resistance. it may be noted that an increased number of projected particles can affect the coefficient of friction also. hence such al-mmc can be considered a replacement for conventional brake lining materials. fig. 12 sem images of (a) al-mmc after wear test with sic particles in the matrix (b) microstructure of subsurface of laser-treated al-mmc (c) microstructure of laser-treated al-mmc with clad layers a similar effect was reported by lorusso et al. [40] when they investigated the tribological behaviour of alsi10mg-tib2 mmc. the mmcs were fabricated through direct metal laser sintering (dmls), and their performance was compared with cast mmcs to see the effect of laser processing. the microstructure of laser-processed al-mmc showed a very fine grain structure against the dendritic structure of cast al-mmc. the total dissolution of reinforcement particles makes the al-mmc strong and harder than the cast al-mmc. the same was observed through the measurement of hardness, i.e., the investigations on the influence of laser surface treatment on hardness and wear behaviour... 15 hardness of laser sintering-based al-mmc was superior to that of cast mmc. in the light of improved hardness, laser or dmls-based al-mmc was proved to be a better wearresistant material, and the micrograph of cast mmc and laser-processed al-mmc revealed that except for plough marks, no other wear tracks were observed. also, there was no evidence of subsurface cracking during the wear test. hence it can be concluded that the laser can help produce wear-resistant al-mmcs, and the results are in line with the work of other researchers. however, there is a limit to which al-mmc can be wear-resistant, and the applied load and pressure intensity can decide this limit. if severe plastic deformation starts under the applied load, then al-mmc can wear out. increased size of wear debris due to predominant sub-surface cracking can cause higher wear rates. hence if laser treatment prevents this, then even at higher loads, al-mmc can be transformed into more wear-resistant mmcs.this approach makes it possible to fabricate a wear-resistant material, even with less reinforcement (for target application). also, surface alloying can make the al-mmc, a hybrid mmc, with enhanced properties compared to conventional materials. the coefficient of friction has also been evaluated using a computerized pin-on-disc setup, and the results are shown in fig. 13. the coefficient of friction increases with sic content in al-mmc, while it is even higher in the case of laser-treated al-mmc for the same composition. due to laser treatment, rearrangement of sic particles happens, and more particles are exposed. as a result, the coefficient of friction increases with the laser treatment of al-mmc, and the high content of sic also contributes to the same. the increased number of exposed sic particles does not lower the asperity of the junction, and no traces of any oxide layers were found. hence, the coefficient of friction increases. hard sic particles do not form a passive layer compared to carbon nanotubes (cnt), where cnts form a carbonaceous layer, reducing the friction coefficient. gupta et al. [41] also reported similar results of increased friction coefficients of the laser-processed al-mmcs. in their study, the coefficient of friction is obtained in the range of 0.193 to 0. 339. the laser power of 400w was found to be sufficient to cause engulfing of b4c and cbn particles in the matrix, and these particles got exposed very well during the wear test. hence, the same hypothesis mentioned above, i.e., when more particles are exposed and firmly held in the matrix, the coefficient of friction is increased [42]. kumar et al. [43] utilized mos2 particles as reinforcement in al-4% mg and observed increased density and microhardness. the wear test revealed a reduction in wear loss by 16.03% for the composite with 4% mg and 6% mos2. the study found that using the powder metallurgy approach to create al-mg-mos2 composites can increase wear resistance. in the case of powder metallurgy, micro-porosity is an inherent issue. in the case of metal matrix composite, it can be a severe issue when significant grain angle boundaries are formed, and poor matric-reinforcement interfacial interaction happens during sintering. this issue can be addressed very well on the surface by applying high-density energy provided by laser. 16 a. sankhla, k. patel, m. makhesana, a. patel, k. gupta fig. 13 variation of coefficient of friction for laser-treated and untreated al-mmc 4. conclusion this work presents the application of laser to alter the mechanical properties of the almmc at the surface level. based on the findings, the following important conclusions are summarized: 1. in laser-treated al-mmc, an average of 12% improvement in the hardness is achieved at the surface level. laser-treated al-mmc could become more wear-resistant due to the improved bonding, i.e., during the dry sliding wear test, laser-treated al-mmc is subject to less wear loss. hence it is possible to develop target service life-based almmc for wear applications. 2. the hardness comparison showed that the hardness of prepared samples increased by 12% for the 10% sic content, while the hardness increased by an average of 15% for the 20% sic and 16% for the 25% sic content in al-mmc. 3. the sem analysis revealed that sic particles are firmly held in the matrix after a dry sliding wear test. hence, along with improved bonding, the hardness of al-mmc also improves as firmly bonded sic particles offer resistance to deformation under loading, which indicates that hardness is increased. 4. the laser-treated al-mmc resulted in a higher coefficient of friction due to laser treatment, rearrangement of sic particles happens, and more particles are exposed. as a result, the coefficient of friction increases with the laser treatment of al-mmc, and the high content of sic also contributes to the same. in future work, the effect of laser treatment on the surface quality of the workpiece can be evaluated. furthermore, the effect of the different laser power and scanning speeds of the laser on the mechanical properties of al-mmcs can be explored. besides, the wear behaviour of laser-treated al-mmcs under different lubricating environments can be analyzed for their practical use in industries. investigations on the influence of laser surface treatment on hardness and wear behaviour... 17 acknowledgement: authors would like to acknowledge the resources and facilities provided by nirma university to conduct the research. references 1. madić, m., gostimirović, m., rodić, d., radovanović, m., coteaţă, m., 2022, mathematical 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triangle shaped auxetic skin graft simulants gurpreet singh1, vivek gupta1, arnab chanda1,2 1centre for biomedical engineering, indian institute of technology (iit) delhi, india 2department of biomedical engineering, all india institute of medical sciences (aiims), delhi, india orcid ids: gurpreet singh https://orcid.org/0000-0001-5910-3213 vivek gupta https://orcid.org/0000-0001-9276-3854 arnab chanda https://orcid.org/0000-0002-6266-078x abstract. the expansion of the skin grafts plays a key role in treating severe burn injuries. split-thickness skin grafting, which is a well-known technique for stretching donor skin samples beyond its capacity, typically produces expansions which are insufficient to cover large burn areas. in this work, the expansion potential of skin grafts with novel rotating triangle (rt) shaped auxetic incision patterns were investigated extensively. a skin simulant was employed and a range of rt configurations, with internal angles varying from 0° to 135°, were tested through the development of skin graft simulants. mechanical testing and digital image correlation (dic) were used to characterize the poisson’s effect, meshing ratios, and induced stresses of the skin graft simulants, up to 50% strains. the 0° model produced the highest negative poisson’s effect and areal expansions. as the internal angle of the auxetic was increased, expansions were observed to decrease significantly. beyond 60°, positive poisson’s effect and contractions occurred with an increasing trend, and its peak at 105°. at 15° and 120°, the induced strains were observed to be significant, posing risks of skin rupture. overall, the expansions were observed to be higher at lower strains. such experimental findings on expansion potentials and estimations of mechanical properties with auxetic skin grafts simulants have not been reported to date, and would be indispensable for further research in skin graft expansion and severe burn injury treatment. key words: skin graft, mechanical properties, simulant, expansion, dic, burns received: february 26, 2022 / accepted september 14, 2022 corresponding author: arnab chanda centre for biomedical engineering, indian institute of technology (iit), delhi, india department of biomedical engineering, all india institute of medical sciences (aiims), delhi, india e-mail: arnab.chanda@cbme.iitd.ac.in https://orcid.org/0000-0001-5910-3213 https://orcid.org/0000-0001-9276-3854 https://orcid.org/0000-0002-6266-078x 80 g. singh, v. gupta, a. chanda 1. introduction the skin, which is the first line of body’s defense, commonly experience burn injuries, caused due to heat, friction, radiation, and hazardous chemical exposure. over 180,000 people die every year globally from burn injuries, majorly from low and middle-income countries [1]. severe burns, if left untreated, may cause hydration, bleeding, and infection. split thickness skin grafting is a common surgical procedure for treatment of such difficultto-heal burn injuries. in split thickness skin grating, a small portion of healthy skin is excised from a donor site and projected with a parallel incision pattern using a skin graft mesher [2, 3]. the meshed skin graft is then stretched beyond its capacity over the large burn region and transplanted to cover the damaged skin. the most common skin grafting modality is of the split-thickness type, in which the entire epidermis and some part of dermis are utilized [1, 4]. previous studies on split-thickness grafts (sgsts) have indicated significant disparity between their claimed meshing ratio and the real meshing ratio [5-8]. real meshing ratios as low as 1:1 was reported, which is insufficient for covering severe burn injury sites [1, 2]. therefore, innovation to produce higher meshing ratios from skin grafts is warranted. auxetic structures, commonly known as metamaterials, exhibit negative poisson’s ratio, and are known to offer high expansion potentials. rotating triangle shaped auxetic structures, in particular, have demonstrated expansion potentials much higher than that offered by traditional skin grafts. grima et al. [9] modeled rotating triangle structures with varying angles and reported a maximum negative poisson’s ratio of -0.90. dudek et al. [10] studied the dynamics of the rotating triangle based auxetic patterns and studied the rotating units for understanding the mechanical characteristics through frictional flexible hinges. they observed convergence in incremental poisson's ratio beyond -1. in addition to the auxetic structure itself, material deformation and shear were found to affect the poisson’s ratio in polymers. to date, there has been no research on the application of such high expansion potential auxetic patterns in skin grafting. in this work, a range of novel rotating triangle based auxetic variants were tested experimentally on skin simulants in order to understand their expansion potential in skin grafts. mechanical testing and digital image correlation (dic) were employed to characterize the biomechanical properties and expansions. the poisson’s ratios, meshing ratios, and induced stresses, were estimated for the skin graft simulants under tensile loading. the findings from this study are anticipated to pave the way for further research in skin graft expansion, for severe burn injury treatment. section 2 will discuss the modeling, fabrication, and testing of skin graft simulants. section 3 will present the results and analysis, followed by the conclusions in section 4. 2. materials and methods 2.1. geometrical modeling of auxetic patterns skin is a multi-layered tissue consisting of the epidermis (top layer), dermis (intermediate layer), and hypodermis (bottom layer or subcutaneous). the epidermis layer mainly consists of the stratum corneum and is the first line of defense from the outer environment, such as uv radiation, infection, etc. the dermis layer is made of different collagens and elastins with different orientations and provides mechanical strength to the skin [11, 12].the hypodermis mechanical characterization of rotating triangle shaped auxetic skin graft simulants 81 layer, i.e., the bottom layer, has heterogeneous material, including the subcutaneous fat covering the muscles [13-15].in this study, the combined mechanical effect of skin’s epidermis and dermis were considered to focus on evaluation of the expansion properties of split-thickness type auxetic skin grafts. solidworks 2020 (dassault systèmes, france) was used to design the unit cell of the rotating triangle (rt) shaped auxetic structures and the corresponding skin graft models (fig. 1). fig. 1 unit cell and rt skin graft models with internal angle ranging from 0° to 135° 82 g. singh, v. gupta, a. chanda the dimensions of all the skin graft models were fixed as 100 mm × 75 mm, with a thickness of 2.5 mm. this specific size was chosen to implement symmetric repeating patterns within the skin graft models. the rt angle was varied from 0° to 135° to investigate the changing mechanical properties, effective poisson’s ratio, and induced stresses within the skin graft models. the first pattern of the rotating triangle, rt-0° represented an internal angle of 0°. for the other rt patterns, this angle was varied with an increment of 15º, from 0º to 135º, generating ten skin graft patterns. all these patterns were symmetrically positioned and allowed to run through the edges of the skin model. a unit cell spacing of 1mm for the slits and gaps was considered based on literature studies [1, 16, 17]. 2.2. development of skin graft simulants the geometrical auxetic models were used to develop negative graft molds with a constant dimension of 100 mm × 75 mm × 2.5 mm (fig. 2). a voxelab aquila 3d printer (flashforge technology ltd, jinhua, zhejiang, china) was used to 3d print the stl models of the auxetic graft molds with pla (polylactic acid) material. these 3d printed molds were employed further to cast the skin graft simulants. fig. 2 3d printed skin grafts molds multiple skin simulant compositions were developed using silicone, based on previous literature works [1, 16, 18, 19] to fabricate the skin graft simulants. the skin simulant compositions were tested under uniaxial tensile loads at a displacement rate of 0.4 mm/s and compared iteratively with the mechanical properties of the cadaveric human skin [20-24] for determining the most suitable composition for mimicking skin. a two-part silicone material with a shore hardness of 5a was mixed in a 1:1 weight ratio with another two-part silicone material with a shore hardness of 30a. the compositions were thoroughly mixed to obtain shore hardness of 15±2a [11], poured into 3d printed molds (fig. 2), and left to cure for 8 hours. the hardness of the fabricated skin graft simulants was measured using shore durometer. the developed skin graft simulants are shown in fig. 3. mechanical characterization of rotating triangle shaped auxetic skin graft simulants 83 fig. 3 skin grafts simulants with varying rt angles 2.3. mechanical testing and digital image correlation the auxetic skin graft simulants were tested at a constant displacement rate of 0.4 mm/s, up to 50% strain. the force-displacement response of the skin graft simulants was measured using a universal testing machine (utm) (finetechno engineering pvt ltd, kolkata, india).the skin graft simulant was clamped on the utm with one end fixed and as the other end was pulled (see fig. 4a). the engineering stress-strain plots were generated from the force-displacement data. on the surface of skin graft simulants, spray paint was used to create dense speckle patterns (fig. 4b) for the digital image correlation (dic) analysis. the dic is a technique to estimate the strain cartographies of a sample during the mechanical testing. in literature, the dic was used by the researchers to evaluate the strain cartographies of peripherical tissues such as skin [25]. however, to date, strain cartographies have not been estimated for any auxetic structures or grafts, to the best of our knowledge. in this work, the tensile tests were recorded using a high-quality camera (20.1 megapixel sony dsc-w830, tokyo, japan) mounted on a tripod stand (fig. 4c). the distance between a test specimen and the camera was considered on the basis that the camera can record the video of full elongation test (i.e., from start to end). the recorded videos were run through an open-source media player (videolan client, france) to get 15-20 images for each tensile test. the images were post-processed to calculate the strain fields on the skin graft simulants using vic-2d software (correlated solutions inc., sc, usa). based on the displacement of spray particles, the lagrangian strain tensor was used for the strain estimation over different deformation phases during tensile testing. 84 g. singh, v. gupta, a. chanda a) b) c) fig. 4 a) mechanical testing setup for skin graft simulant on the utm, b) densely spray paint speckles, and c) dic setup mechanical characterization of rotating triangle shaped auxetic skin graft simulants 85 3. results and discussion 3.1. uniaxial mechanical properties of skin graft simulants across all the skin graft simulants, the stress values were within the range of 0.5 to 19 kpa at 50% strain. the stress versus strain for the skin graft simulants, i.e., rt-0° to rt135°, is presented in fig. 5. the maximum stresses were generated in the rt-75° model, and minimum stresses were observed for the rt-0° model. rt-75° and rt-90° exhibited similar stress-strain values after 20% strain and followed the same trend up to 50% strain. similarly, rt-135°, rt-105°, rt-120°, and rt-60° also showed similar stress-strain behavior. among all the rt skin graft simulants, rt-0° was determined to be a desirable auxetic structure for skin grafting subjected to the low-stress values. rt-75° and rt-90° skin graft simulants appeared to be the least desirable models. fig. 5 stress-strain plot of skin graft simulant models fig. 6 poisson’s ratio of rt skin graft simulants at 50% strain 86 g. singh, v. gupta, a. chanda the effective poisson’s ratio was computed in the range of -0.9 to 0.26 for the skin graft simulants with varying angles at 50% strain (fig. 6). rt-0° skin graft simulant generated the maximum negative poisson's ratio, while rt-105° showed the minimum poisson’s ratio. other skin graft simulants exhibited effective poisson’s ratios in between rt-0° and rt-105°. at 50% strain, rt-0°, rt-15°, rt-30° and rt-45° skin graft simulants showed auxetic behavior, and rt-60°, rt-75°, rt-90°, rt-105°, rt-120°, and rt-135° showed non-auxetic behavior. rt-15° and rt-30° skin graft simulants exhibited similar effective poisson’s ratios of -0.46 and -0.42, respectively. it should be mentioned that due to changes in the internal structure of auxetic patterns, the effective poisson's ratio of skin graft simulants varied with strain [26-28]. for low strain values (rt-0° to rt-45°), the skin graft simulants exhibited auxetic behavior in line with previous findings [29, 30]. 3.2. strain distribution in auxetic skin graft models the skin graft simulants were subjected to stretching up to 50%, and the induced strains were measured using digital image correlation (dic). at the loading rate of 0.4 mm/s, 50% strain was attained in 125 sec in all models. significant strain build-up was observed near the utm clamps across all the skin graft simulants, indicating the possibility of skin rupture (figs. 7 to 9). fig. 7 strain distribution observed through dic in skin graft simulants (rt-0° to 30°) mechanical characterization of rotating triangle shaped auxetic skin graft simulants 87 fig. 8 strain distribution observed through dic in skin graft simulants (rt-45° to 75°) fig. 9 strain distribution observed through dic in skin graft simulants (rt-90° to 135°) 88 g. singh, v. gupta, a. chanda in rt-0° model, the maximum localized strain was very low (i.e., 0.166). a uniform strain distribution was observed across most of the unit cells, except in the upper middle. while rt shaped unit cells were observed closer to the fixed end with uniform stresses all around, triangular unit cells were more prominent at the displacement end, with nonuniform stresses at the edges. the most distorted triangles, transitioning to hexagons, were in the upper middle with wide variation in strains across their edges. in the rt-15° model, several regions exhibited high strain build-ups, including the right edge, top and bottom. the highest strain value recoded was 1.5. all the stretched pattern unit cells were observed to be triangular, with minimal strain build up at the edges. while the high strains at the top and bottom sides could be attributed to the utm clamping, the strain build up at the right side may be due to the lateral asymmetry and excess skin material presence. in rt-30° model, the maximum strain was almost half (i.e., 0.88) of in the case of rt-15° model. the strain distributions were similar, with all stretched unit cells in shape of hexagon in the case of rt-30° model. rt-45° showed strain values (i.e., 0.735) lower than rt-30°. the strain distributions were very uniform, unlike in the cases of rt-15° and rt-30°, with fully developed hexagons of nearly similar dimensions. slight curling at the bottom clamp, and lateral expansion was observed, indicating significant auxetic behavior. in rt-60°, rt-75°, and rt-90° models, fully developed hexagonal unit cells were observed, and the strain distributions were uniform across them. the maximum strains were in the range of 0.2720.465, much lower than that of rt models with angles of 0° to 45°. the stretched rt-105° model also had fully developed hexagonal unit cells, but a nonuniform strain distribution. the upper half of the model showed higher strain localization compared to the lower half, with a maximum value of 1.41. rt-120° and rt-135° models exhibited similar unit cell shape with uniform strain distributions, with maximum values of 1.15 and 0.89 respectively. rt-135° exhibited similar maximum strain value as that of rt-30°. due to lower void area in rt-30°, better auxetic properties and expansions are anticipated in rt30°. overall, rt-15° model exhibited high possibility of skin rupture at the upper clamp site, followed by rt-105°, and rt-120°. rt-30°, rt-45°, and rt-135° models showed moderate skin rupture possibility. rt-60°, rt-75°, and rt-90° exhibited low possibility of skin rupture, followed by rt-0° which was the least strained model. 3.3. expansion potential of skin graft simulants the area estimations of all the skin graft simulants at 0% and 50% strain are shown in table 1. a constant grid size was used to trace the outer boundaries of the skin graft simulants. the initial area of all the skin graft simulants was 7500 mm2. the final area of the skin graft simulants varied from 7651 to 13200 mm2. the maximum area was covered by rt-0° (13200 mm2) model, and rt-135° (7651 mm2) showed the minimum covered area. the final area of the other skin graft simulants was: rt-15° (11400 mm2), rt-30° (10950 mm2), rt-45° (10350 mm2), rt-60° (9700 mm2), rt-75° (8841 mm2), rt-90° (7914 mm2), rt-105° (7806 mm2), rt-120° (7778 mm2), respectively. the final areas of the skin graft simulants were further used to calculate the meshing ratios and quantify expansion potential. mechanical characterization of rotating triangle shaped auxetic skin graft simulants 89 table 1 initial and final area of the skin graft simulants at 50% stretch (units of xand y-axis in mm) skin graft simulant initial area final covered area rt-0° rt-15° rt-30° 90 g. singh, v. gupta, a. chanda rt-45° rt-60° rt-75° mechanical characterization of rotating triangle shaped auxetic skin graft simulants 91 rt-90° rt-105° rt-120° 92 g. singh, v. gupta, a. chanda rt-135° fig. 10 presents the meshing ratios of all the graft simulants with varying rt angles, estimated at 50% strain. the meshing ratio was calculated as the ratio of the final covered area (after stretching) and the initial area (table 1). in decreasing order, the estimated mr values were 1.76, 1.52, 1.46, 1.38, 1.29, 1.17, 1.05, 1.03, 1.02, and 1.01, for the rt-0°, rt-15°, rt-30°, rt-45°, rt-60°, rt-75°, rt-90°, rt-120°, rt-135°, and rt-105° respectively. the mrs of the rt skin graft simulants were found to be higher than the meshing ratios of traditional skin grafts. from a clinical perspective, a skin graft with a high meshing ratio exhibits potential to cover large burn areas with minimum amount of available donor skin. fig. 10 uniaxial meshing ratios for rt skin graft simulants at 50% strain mechanical characterization of rotating triangle shaped auxetic skin graft simulants 93 4. conclusions in this work, a novel experimental framework was developed to study the expansion potential of skin grafts with rotating triangle shaped auxetic incisions. a biofidelic skin simulant material was employed to fabricate auxetic skin graft simulants with a range of rotating triangle variants, using additive manufacturing and biomimetics. the stress-strain behavior, induced strain response, and expansions of the skin graft simulants were characterized using tensile testing and dic analysis. with low induced stresses and high expansions, the 0° model was found to be the most suitable auxetic graft structure compared to other models. these findings are comparable with previously reported observations in rotating triangle based auxetic polymers [31-33]. the 105° model exhibited minimal expansion with high induced stresses, making it the least ideal auxetic for skin graft expansions. it was observed that only four out of ten skin graft simulants, namely the 0°, 15°, 30°, and 45° models showed auxetic and elastic expansion. two additional simulants (i.e., 60° and 75° models) exhibited expansions beyond their size purely due to elasticity. such findings are novel and have not been reported so far on skin graft expansions. a few limitations of this work should be acknowledged. an isotropic biofidelic skin simulant was employed instead of actual skin to the study the effect of rotating triangle shaped auxetic incision patterns on the skin graft expansions. to maintain uniformity in the analyses, the skin graft simulant area and thickness were fixed. in reality, human skin exhibits anisotropic behavior and skin grafts vary widely in shapes and thicknesses. in future studies, cadaveric skin with varying shapes and thicknesses will be projected with the auxetic patterns through innovative meshing techniques for more accurate analyses. in conclusion, the rotating triangle shaped auxetic incision patterns exhibited the potential for expansions higher than in traditional skin grafts. the degree of expansion was found to be tuneable without causing skin rupture through modification of the auxetic’s internal angle. with further clinical trials and validations, skin grafting using the tested auxetic incisions may allow the covering of large burn areas using a small amount of donor skin. acknowledgements: gurpreet singh is grateful to the ministry of education, government of india, for awarding the prime minister’s research fellowship (ref: iitd/admission/ph.d./pmrf/202021/4062) for pursuing his doctoral research program at iit-delhi, india. references 1. gupta, v., chanda, a., 2021, expansion potential of skin grafts with novel i-shaped auxetic incisions, biomedical physics &engineering express, 8(1), 015016. 2. henderson, j., arya, r., gillespie, p., 2012, skin graft meshing, over-meshing and cross-meshing, international journal of surgery, 10(9), pp. 547–550. 3. swaim, s.f., 1990, skin grafts, veterinary clinics of north america: small animal practice, 20(1), pp. 147–175. 4. kanapathy, m., hachach-haram, n., bystrzonowski, n., 2016, epidermal grafting versus split-thickness skin grafting for wound healing (epigraaft): study protocol for a randomized controlled trial, trials, 17, 245. 5. asuku, m., yu, t.c., yan, q., boing, e., hahn, h., 2001, split-thickness skin graft donor-site morbidity: a systematic literature review, burns, 47(7), pp. 1525–1546. 6. serebrakian, a.t., pickrell, b.b., varon, d.e., mohamandi, 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line contact of finite-length cylinders facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 63 71 doi: 10.22190/fume170222003l © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper normal line contact of finite-length cylinders udc 539.3 qiang li, valentin l. popov department of system dynamics and the physics of friction, berlin institute of technology, berlin, germany abstract. in this paper, the normal contact problem between an elastic half-space and a cylindrical body with the axis parallel to the surface of the half-space is solved numerically by using the boundary element method (bem). the numerical solution is approximated with an analytical equation motivated by an existing asymptotic solution of the corresponding problem. the resulting empirical equation is validated by an extensive parameter study. based on this solution, we calculate the equivalent mdr-profile, which reproduces the solution exactly in the framework of the method of dimensionality reduction (mdr). this mdr-profile contains in a condensed and easy-to-use form all the necessary information about the found solution and can be exploited for the solution of other related problems (as contact with viscoelastic bodies, tangential contact problem, and adhesive contact problem.) the analytical approximation reproduces numerical results with high precision provided the ratio of length and radius of the cylinder are larger than 5. for thin disks (small length-to-radius ratio), the results are not exact but acceptable for engineering applications. key words: line contact, boundary element method, finite-length cylinder, contact stiffness, method of dimensionality reduction 1. introduction the contact problem of cylinders with parallel axes or of a flat elastic body with a “lying” cylinder is very common in practical engineering applications, in particular in mechanical elements such as roller bearings, gears and cams [1, 2]. in contrast to the hertz-like contacts of bodies with curvature in two directions which in engineering mechanics are called “point contacts”, the contact of two parallel cylinders is denoted as a “line contact”. being an immediate two-dimensional analog of the hertz contact, the line contact between cylinders with parallel axes is one of basic problems in contact mechanics. received february 22, 2017 / accepted march 15, 2017 corresponding author: valentin l. popov department of system dynamics and the physics of friction, berlin institute of technology, str. des 17 juni 135, 10623 berlin, germany e-mail: v.popov@tu-berlin.de 64 q. li, v.l. popov even if the effective dimensionality of a line contact is lower than that of the point contact (2d vs. 3d), the line contact is in some sense more complicated than its 3d analog since in the line contact the “indentation depth” cannot be defined unambiguously. this is related to the logarithmic divergence of the 2d fundamental solution in infinity. in other words, the properties of a 2d contact (line contact) are not “local” but depend on the macroscopic shape of the body. this dependence, however, is relatively weak (logarithmic). therefore, there exist a large number of approximated solutions [3-5], which have been applied in many further studies, for example in elastic hydrodynamic lubrication [6, 7]. the main structure of all approximate solutions is relatively simple: it reproduces the logarithmic divergence and cuts it up at some characteristic distance, at which the contact ceases to be a line-contact. in the case of a true 2d contact, this is the size of the system (e.g. the radius of contacting cylinders). on the other hand, for any finite contact, e.g. that of a lying cylinder of finite length, the indentation depth can be determined unambiguously. the macroscopic length which in this case limits the “two-dimensionality” of the line contact is the length of the cylinder. one can anticipate that the indentation depth at a given force will be a weak logarithmic function of the cylinder length. finding the exact form of this function is the main goal of this paper. as the solution of the underlying two-dimensional contact problem gives the basis for our consideration, we first provide a brief overview of earlier works on this topic. the contact of cylinders with parallel axes was early studied by prescott [8] and thomas and hoersch [9], and later also investigated by many researchers, for example lundberg et al. [10]. a good review of analytical solution for this contact problem can be found in norden’s report [11], where the deviation of relationship between the normal load and the indentation depth is given in detail; this relationship is still widely used today. the same analysis is also presented in puttock and thwaite’s report [12]. in these solutions, the existing result of pressure distribution for elliptical contact is used for cylindrical contact with parallel axes by assuming one axis of ellipse is infinitely large, and then the contact area is considered as a finite rectangle whose length is much larger than its width. for further reference, here we reproduce their solution of a cylinder with radius r and length l pressed into elastic half space under normal load f [12]: the half width of contact rectangle b is equal to: * 4rf b e l  (1) and indentation depth d is: * 3 2 * * 2 4 1 ln 1 ln f e l f l d le rf le b                   (2) where e* is effective elastic modulus and equal to: 2 2 1 2 * 1 2 1 1 1 e e e      . (3) with e1 and e2 are elastic module of contacting bodies, ν1 and ν2 are poisson’s ratio. later johnson and other authors gave other forms of force-displacement e.g.: * * 4 ln f e lr d const le f          (4) where const=1 in [4], 0.72 and 0.572 in others [13]. normal line contact of finite-length cylinders 65 correspondingly, the experiment investigation consisting of compression of parallel cylinders or indenting a cylinder into an elastic body is carried out in [11, 14]. empirical equations of load-displacement are provided to verify theoretical solutions. in thwaite’s compression test [15] of a contact between a cylinder and a half-space, the comparison of slopes (or contact stiffness) shows that the solution (2) is the closest to the experimental results. in kunz’s experimental investigation [14], it is found that the approach of parallel cylinders is proportional to the loading force: f=ce*ld, where c is constant c=0.175. in the last few decades, some numerical studies of the finite length line contact have been carried out to investigate the effect of contact edge and bone shape of the contact area and on stress distribution [16, 17]. however, a more precise solution for the line contact is rarely provided. recently an analytical solution for contact problem of toroidal indenter is given by argatov et al. [18] and is validated numerically by the boundary element method. in this paper we propose an analytical approximation of contact stiffness of a rigid cylinder and an elastic half space based on the results of [18]. while we overtake the general form of solution, we let some parameters free and determine them finally by numerical simulation of indentation test using the boundary element method. 2. effective mdr-profile for a finite length lying cylinder 2.1. analytical approximation based on an asymptotic solution the main physical result of this paper will be the dependence of normal force on the indentation depth for a contact between an elastic half-space and a “lying” cylinder (with the axis parallel to the surface of the half-space.) however, we will “pack” this dependence in the terms of the method of dimensionality reduction (mdr) [19] where the whole information about the system is compressed into effective plane profile g(x). the advantage of this presentation is that g(x) can be used not only to easy reconstruct the dependency of the normal force on the indentation depth but also to solve a variety of other related problems such as tangential contact problem with friction in the interface, adhesive contact problem, and contact of visco-elastic bodies. in this sense, g(x) is, so to say, the “visiting card” of the profile in question which allows multi-purpose use. note that the possibility of mapping three-dimensional contacts onto contacts with elastic foundation is well known for axis-symmetric indenters with compact contact area [19]. less known is that the same concept can also be used to arbitrary other profiles, as e.g. of a torus (not compact contact area) or a rough surface. the corresponding proof as well as examples of mdr-profiles for a number of non-axisymmetric contacts can be found in [18] and [20]. profile g(x) determines straightforwardly force-indentation dependence f(d). thus the information content of both dependencies is equal: one can either determine f(d) from g(x) or g(x) from f(d). even if the information content of both the functions is the same, it is more convenient to have g(x) as it allows to solve much more various problems than f(d). in [20] the explicit procedure of “extracting” profile g(x) from dependency f(d) is described. the procedure is very trivial: from known dependency f(d) one first determines differential contact stiffness kn(d)=df(d)/dd and then determines the dependence of d as function of variable x=kn/(2e*). dependency d(x) is exactly searched-for function g(x). in the present paper, this procedure is carried out numerically: first, dependency f(d) is determined by direct simulation using the boundary element method. subsequently, the 66 q. li, v.l. popov described procedure is applied to extract g(x). finally, numerically found profile g(x) is approximated analytically. for the analytical approximation we use the form of g(x) found in [18] for indentation of a torus. in [18], it is derived by an asymptotic analysis and verified through bem simulations. the 1d profile for toroidal indenter is given by: 2 2 2 ( ) 1 exp 8ln 2 4 r r r g x x x                    . (5) here, r is the distance from the center of the torus tube to the center of the torus,  is the radius of the torus tube. fig. 1 a rigid cylinder lying on an elastic half space the contact of the torus is very similar to that of a “lying” cylinder: both contacts are basically line ones and thus two-dimensional contacts whose logarithmic divergence is cut at some distance. for the torus, the role of the cut-off length plays the radius of the torus, while in the case of the lying cylinder this is the length of the cylinder. equation (5) found for the torus thus can be used for lying cylinder with radius r and length l (fig. 1) just by replacing rl and r. the differences in two configurations are taken into account by introducing two coefficients c1 and c2 which in the case of the torus are equal to 1, but in the case a lying cylinder are allowed to take some other values: 2 2 2 1 2 1( ) 1 exp 8ln 2 4 l l l g x c c c r x x                   . (6) in a more compact form eq. (6) can be rewritten as: 2 ( ) 1 exp l l l g x r x x                  (7) where we have introduced two other fitting coefficients  and  (instead of c1 and c2). introducing dimensionless parameters: x x l  and 2 ( ) ( ) g x r g x l  , (8) normal line contact of finite-length cylinders 67 eq. (7) can be written in the dimensionless form: ( ) 1 expg x x x                  . (9) 2.2. numerical simulation of the indentation test two unknown coefficients  and  in eq. (7) will be determined by numerical simulation of indentation test using the boundary element method which was developed by pohrt, li and popov for various 3d contact problems including the partial sliding contact [21] and adhesive contact [22, 23]. in the simulation, the whole simulation area was divided into 512512 rectangular elements. the rigid cylinder was modeled as parabolic indenter f(y)=y2/(2r), where y is in-plane coordinate perpendicular to the axis of the cylinder. the cylinder was indented in an elastic half space with controlled indentation depth in 100 steps from zero (first contact) to 0.15r. the pressure distribution as well as the normal load and normal contact stiffness were calculated in each step of indentation. one example of contact configuration and pressure distribution is shown in fig. 2. the concentration of pressure at the contact edge can be clearly observed in fig. 2b. (a) (b) fig. 2 an example of numerical simulation for l/r=5 and d=0.1r: (a) contact state (b) pressure distribution 3. results and discussion we have performed indentation simulations for cylinders with 29 increasing values of l/r ranging from l/r =0.1 to 20 (10 linearly increasing l/r from 0.1 to 1, and 19 from 2 to 20). resulting mdr-profiles g(x) are shown with crosses in fig. 3a. in fig. 3b, all curves are plotted in dimensionless form in coordinates (8). it is seen that all crosses for different l/r collapse to a single curve thus confirming the basic structure of the solution: 2 ( ) l x g x r l        . (10) 68 q. li, v.l. popov fig. 3b provides the numerically determined form of function (.). fig. 3 1d profile of lying cylinder calculated by bem simulation (cross) and fitting with eq. (7) in the following, we provide analytical approximation for this function on the basis of eq (9). the values of coefficients  and  are calculated by the method of least squares. the agreement of fitting (black solid lines) with numerical simulation can be seen in fig. 3. values of  and  are presented in fig. 4, where we can see that both factors are almost constant for large ratios of l/r but change significantly for small ratios. we thus discuss the cases of “long cylinders” and “short cylinders” separately. fig. 4 values of coefficients of α (a) and β (b) for different ratios of l/r obtained by fitting of eq. (7) with numerical results (a) large ratio l/r (“long cylinder”) from fig. 4, one can see that  and  for larger l/r are almost constant: =/2 and =0.4537. thus, for large ratios in the range of about l/r 5 we can give the following approximation: 2 ( ) 0.4573 1 exp , for 5 2 2 l l l g x l r r x x                 , (11) a) b) a) b) normal line contact of finite-length cylinders 69 or in the dimensionless form: ( ) 0.4573 1 exp , for 5 2 2 g x l r x x                 . (12) numerical results and analytical approximations (11) and (12) in this range of l/r are shown in fig. 5. fig. 5 1d profile of cylinder for larger ratios of l/r: (a) for different l/r (b) in dimensionless form (b) small ratio of l/r (“short cylinder”) in practical applications, many line-contact machine elements are thin plates meaning a small value of l/r, as e.g. cams. the results of numerical simulation and fitting for l/r =0.1~4 are clearly shown in fig. 6. coefficients  and  in this range are different (fig. 4); however, from the subplot of fig. 6b we can find that the fittings agree with the numerical results also well, except for very small l/r (=0.1 or 0.2) (subplot in fig. 6b), so we list their values in tab. 1 for the further studies. fig. 6 1d profile of cylinder for small ratios of l/r: (a) for different l/r (b) in dimensionless form a) b) a) b) 70 q. li, v.l. popov table 1 values of coefficients α and β for small l/r l/r α β l/r α β l/r α β 0.1 3.8695 17.1183 0.6 2.0528 1.9139 2 1.7086 0.7936 0.2 2.8664 6.6132 0.7 1.9928 1.6847 3 1.6518 0.6495 0.3 2.4734 3.9560 0.8 1.9400 1.4963 4 1.6231 0.5799 0.4 2.2714 2.8770 0.9 1.8996 1.3589 5 1.6048 0.5376 0.5 2.1455 2.2975 1 1.8668 1.2517 6 1.5930 0.5091 finally, let us compare our results with those following from eq. (2). differentiation of the force with respect to the indentation depth provides the normal contact stiffness: * * 3 lnn rf k le e l    . (13) together with relation (2) we can obtain 1d profile g(x), which has also a dimensionless form similar to eq. (12). this dimensionless form is shown with a dashed line in fig. 7. one can see that it differs substantially from the “numerically exact” result found in the present paper and plotted in fig. 7 with a bold line. fig. 7 comparison of 1d profile obtained from existing solution and in this paper 4. conclusion we have numerically simulated indentation of the finite-length cylinder lying on an elastic half space. the ratio of the cylinder’s length and radius was varied in a wide range from a thin disk (ratio 0.1) to a long pole (ratio 20). based on the results of numerical simulation, the equivalent mdr-profiles containing the whole information about the contact problem was “extracted” and subsequently approximated analytically using an equation inspired by an asymptotic solution of the contact problem. for large length-to-radius ratios (larger than about 5), the fitting coefficients are almost constant and a general form with high accuracy is provided in a closed analytical form. for small length-to-radius ratios, analytical solution is provided which contains two constants provided in the form of a table. comparison of the present solution with the already available analytical approximation shows that the present solution is much more precise. normal line contact of finite-length cylinders 71 references 1. harris, t.a., 2001, rolling bearing analysis, john wiley and sons, new york. 2. norton, r., 2009, cam design and manufacturing handbook, industrial press. 3. popov, v.l., 2010, contact mechanics and friction: physical principles and applications, springer, berlin. 4. jonson, k.l., 1985, contact mechanics, cambridge university press, cambridge. 5. landau, l.d., lifshitz, e.m., 1970, theory of elasticity, course of theoretical physics. 6. venner, c.h., lubrecht, a.a., 1994, transient analysis of surface features in an ehl line contact in the case of sliding, journal of tribology, 116(2), pp. 186–193. 7. hamrock, b.j., schmid, s.r., jacobson, b.o., 2004, fundamentals of fluid film lubrication, marcel dekker, new york. 8. prescott, j., 1946, applied elasticity, dover publications. 9. thomas, h.r., hoersch, v.a., 1930, stresses due to the pressure of one elastic solid upon another: a report of an investigation conducted by the engineering experiment station, university of illinois in cooperation with the utilities research commission. 10. lundberg, g., 1939, elastische berührung zweier halbräume, forschung auf dem gebiet des ingenieurwesens a, 10(5), pp. 201–211. 11. norden, b.n., 1973, on the compression of a cylinder in contact with a plane surface, national bureau of standards, washington d.c. 12. puttock, m.j., thwaite, e.g., 1969, elastic compression of spheres and cylinders at point and line contact, national standards laboratory. 13. nakhatakyan, f.g., 2011, precise solution of hertz contact problem for circular cylinders with parallel axes, russian engineering research, 31(3), pp. 193–196. 14. kunz, j., de maria, e., 2002, die abplattung im kontaktproblem paralleler zylinder, forschung im ingenieurwesen, 67(4), pp. 146–156. 15. thwaite, e.g., 1969, a precise measurement of the compression of a cylinder in contact with a flat surface, journal of physics e: scientific instruments, 2(1), pp. 79–82. 16. glovnea, m., diaconescu, e., 2004, new investigations of finite length line contact, in asme proceedings, special symposia on contact mechanics, pp. 147–152. 17. najjari, m., guilbault, r., 2014, modeling the edge contact effect of finite contact lines on subsurface stresses, tribology international, 77, pp. 78–85. 18. argatov, i., heß, m., pohrt, r., popov, v.l., 2016, the extension of the method of dimensionality reduction to non-compact and non-axisymmetric contacts, zamm journal of applied mathematics and mechanics, 96(10), pp. 1144–1155. 19. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer, berlin. 20. popov, v.l., pohrt, r., heß, m., 2016, general procedure for solution of contact problems under dynamic normal and tangential loading based on the known solution of normal contact problem, the journal of strain analysis for engineering design, 51(4), pp. 247–255. 21. pohrt, r. li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17(4), pp. 334–340. 22. pohrt, r., popov, v.l., 2015, adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method, facta universitatis, series: mechanical engineering, 13(1), pp. 3–10. 23. li, q., popov, v.l., 2016, boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials, arxiv:1612.08395. 12513 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240213030k © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper picture-hesitant fuzzy-based dual-normalized multiple aggregations for material selection in a lift-based converter daekook kang1, krishnan suvitha2, michael sandra2, dragan pamucar3,4,5, vladimir simic6,7, samayan narayanamoorthy2 1department of industrial and management engineering, institute of digital anti-aging health care, inje university, gyeongsangnam-do, republic of korea 2department of mathematics, bharathiar university, coimbatore, india 3department of operations research and statistics, faculty of organizational sciences, university of belgrade, belgrade, serbia 4department of industrial engineering & management, yuan ze university, taoyuan city 320315, taiwan 5department of mechanics and mathematics, western caspian university, baku, azerbaijan 6faculty of transport and traffic engineering, university of belgrade, serbia 7department of computer science and engineering, college of informatics, korea university, seoul, republic of korea abstract. this paper addresses the challenge of material selection for lift-based wave energy converters (wecs) by presenting a novel methodology that integrates fuzzy information representation and multi-criteria decision-making (mcdm) techniques. the aim is to provide decision-makers with a robust framework for optimizing material choices despite uncertainties and limited field data. the methodology combines a material selection approach tailored for lift-based wecs with a new fuzzy information representation technique, probabilistic picture hesitant fuzzy sets (p-phfss). the results underscore the potential of these methods for reducing the risk of machine failures in wec technology. the proposed method contributes to the development of wec technology by providing a systematic and flexible framework for decision-makers to optimize material choices and improve the reliability and performance of wec systems. key words: material life cycle, wave energy converter, decision making, fuzzy logic, probabilistic picture hesitant fuzzy set, preference selection index received: february 13, 2024 / accepted may 18, 2024 corresponding author: samayan narayanamoorthy department of mathematics, bharathiar university, coimbatore, india e-mail: snmphd@buc.edu.in mailto:snmphd@buc.edu.in 2 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy 1. introduction wave energy has seen commercialization in the past decade, driven by government sup port, private investment, and the offshore energy sector’s efforts to combat climate change. wave energy, combined with wind and wave power, can help carbonize coastal energy systems and achieve zero waste energy, and large-scale wecs are expected soon [1]. lift-based wecs show high potential for competitiveness in the offshore sector due to their immersed mode capability and passive stalling in storm conditions stress and potential mechanical failure [2]. numerous efforts have been made to develop methods for quantifying and preventing wear damage to offshore infrastructure. offshore constructions, like wave energy converters, are susceptible to random loading due to the fluctuation of ocean waves. the wecs can be complex due to the unique structural hydrodynamic interactions of each wave energy converter. accurate fatigue damage models can help create a durable and cost-effective structural wec, reducing its leveled energy cost. analytical models for assessing and quantifying fatigue damage in wec structures must be developed during the design stage [3]. the development of hydrodynamic models to predict electrical performance has formed a significant part of the research on lift-based wecs. common risk reduction measures for wave energy converters include digital twins, adaptive control systems, and suitable materials. the development of solutions that minimize the risk of failure is crucial for enhancing the technical readiness level (trl) of lift-based wecs [4]. this study focuses on lift-based wecs. this form of wec generates lift forces using rotating hydrofoils. liftbased wecs can operate submerged and use pitch control to reduce hydrofoil load fluctuations, but loading variations cannot be avoided due to the unpredictable nature of different waves within sea states [5]. the structure of hydrofoils allows for the estimation of fatigue injury to lift-based wecs using a probabilistic approach, rather than a deterministic approach, under specific sea conditions [6]. multi-criteria decision-making (mcdm) is used to make complex decisions based on multiple factors. the objective is to identify the optimal sites for wecs based on their performance across various parameters [7]. numerous studies have demonstrated the effectiveness of mcdm methods in identifying suitable locations for renewable energy sources like wind and solar electricity [8]. the study on wave energy is currently lacking in depth. the lift-based wec material selection process comprises five stages: criteria selection, data normalization, criteria weights, alternative assessment, and outcomes validation. an mcdm, which assesses alternatives based on expert opinions and specific criteria, has gained popularity as a viable study area [9]. the uncertainty and ambiguity in the decision-making process have made it challenging for decision-makers to evaluate and choose the best option in mcdm challenges [10]. interest in fuzzy sets (fss) [11] and their classical expansions, including interval-valued fss (ivfss) [12], intuitionistic fss (ifss) [13], interval-valued ifss [14], hesitant fuzzy sets (hfs) [15] and picture fss (pfss) [16] has been significant in research and development. the concept of hesitant fuzzy sets (hfss), which enable the degree of an element’s membership to be represented by a collection of alternative values [17]. ali et al., [18] introduced the picture’s hesitant fuzzy element (phfss), an extension of hfss consisting of positive membership degree, neutral membership degree, and negative membership degree functions. we introduce probabilistic picture hesitant fuzzy sets (pfile:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark0 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark12 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark13 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark14 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark15 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark16 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark18 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark21 picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 3 phfss) to improve their effectiveness by considering the likelihood of each member in the phfe. p-phfss are highly effective as they can represent membership degrees, their probabilities, and their positive, neutral, and negative probabilities [19]. the study explores the effectiveness and efficiency of the performance selection index (psi) method in resolving machining mcdm issues [20]. this study uses p-phfss and fuzzy dnma techniques to develop material selection characteristics for lift-based wecs. material fitness-for-purpose methodologies are crucial in preventing the failure of innovative technologies and thereby reducing investment loss. the dnma is a utility theory-based method that employs two distinct normalization procedures and three aggregation strategies [21]. the dnma method has been compared to other utility-theory-based mcdm techniques for the first time in terms of its benefits. this study’s implications extend beyond lift-based wecs to other submerged marine energy-collecting methods operating near the ocean’s free surface. the study introduces pphfss, a tool that simplifies complex assessment information representation in decisionmaking situations, including operations, score function, accuracy function, and comparison approach. the framework introduces a distinctive material selection method in lift-based wecs, employing fuzzy double normalized-based multiple aggregations (dnma) as an mcdm. an emphasizes material fitness-for-purpose to avoid technological failure and financial losses. it applies to numerous stages of the product cycle, including lift-based wave energy converters. this study has implications for additional submerged marine energy-collecting devices that operate near the ocean’s surface. the sensitivity analysis is used to evaluate the accuracy of the scenario assessment, considering uncertainties or changes in values. the remainder of this study is organized as follows: the literature review is discussed in section 2. the basic concepts are presented in section 3. section 4 provides the proposed methodology and an evaluation framework. a case study is applied to the method for material selection for lift-based wecs in section 5. further, the results and discussion of the case study are in section 6. finally, a summary of the findings, limitations, and future work is in section 7. 2. literature review the material selection architecture for lift-based wecs is based on seven criteria: structure dependability, hydrodynamic efficacy, offshore maintenance, corrosion resistance, production cost, eco-friendliness, and shelf life. the lift-based wec consortium’s three-year horizon 2020 research project, which concluded in 2023, identified critical criteria that significantly increased the trl of lift-based wecs [22]. the device’s rotor is produced by the hydrofoils and center shaft, while the stator serves as the stationary component of the wec. the lift wec project’s final idea involves a floating prototype, which reduces slamming and high loads through submergence and passive stalls, potentially saving installation and maintenance costs [23]. an mcdm is a crucial operational research method in engineering fields, where problem solutions are determined by multiple independent and often conflicting criteria. although mcdm techniques are aimed at managing quantitative values, their integration with fuzzy set theory can provide answers in situations where managing qualitative values is difficult. this paper presents an mcdm strategy for a new device’s material selection file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark23 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark24 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark28 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark30 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark34 4 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy challenge, assuming a team of specialists conducts the evaluation. based on the primary material selection criteria and lift-based wec development stage evaluate using p-phfs and the fuzzy dnma method. the lift-based wec idea is explained, with hydrofoils as the demonstration case for the framework’s structural component [24]. mzili et al., [25] developed a unique decision-support technology that combines hesitant linguistic term collection, the dnma technique, and a basic decision-making process. the applicability of this method has been demonstrated in estimating geographic sites for retail shopping malls using fuzzy linguistic information. farooq [26] discussed processing and connections of probabilistic hesitation fuzzy elements (phfe) information, proposing a method for ranking phfes based on score and deviation values, with the comparison technique reflecting absolute priority. tian et al., utilized the tomada de decisao interativa multicriterio (todim) approach to model decision-making perceptions in prospect theory, incorporating probabilistic hesitant fuzzy information [27]. song and chen have developed a new madm technique using distance and the complex proportional assessment (copras) method for a phf environment [28]. the preference selection index (psi) method is simpler to comprehend than other mcdm approaches due to its statistical calculation of total preference values without relative weight. the psi technique is beneficial for assessing optimal alternatives in cases of controversy regarding criteria relevance and requires fewer numerical calculations [29] and [30]. the study presents an mcdm material selection for lift-based wecs, demonstrating the device’s hydrofoils to fill a research gap. select the best composition for the optimum performance of the metallic alloy [31]. selected the rank of flexible manufacturing system (fms) flexibility [32]. determined the optimum phase combination of biodegradable composites [33]. the method based on the removal effects of criteria (merec) integrated psi-mcdm approach was used to select india’s green renewable energy source [34]. we introduced p-phfss, a unique method for accurately and effectively describing complex assessment information for decision-makers [35] and [36]. the objective of this work is to use the fuzzy psi-dnma technique to develop an ordered material selection method for lift-based wecs. the fuzzy psi-dnma approach was utilized to rank each criterion and alternate material for hydrofoils, addressing uncertainty. 3. preliminaries 3.1. picture hesitant fuzzy set let x be a fixed set, a picture hesitant fuzzy set (phfs) m on x is defined by  , ( ), ( ), ( ) | m x x x x x x   (1) where, ϕ(x) = {σ | σ ϵ ϕ(x)} , χ(x) = {ς | ς ϵ χ(x)}, and ψ(x) = {τ | τ ϵ ψ (x)} are three sets of possible values in [0,1], indicating the possibility of positive (pmd), neutral (nemd), and negative (nmd) membership degrees. the degrees mentioned above satisfy the requirement of 0 ≤ σ+ + ς+ + τ+ ≤ 1. as a convenience, m = {x, ϕ(x), χ(x), ψ(x)} phfe, denoted by m = (ϕ, χ, ψ). file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark35 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark36 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark40 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark46 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark50 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark61 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark62 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark63 file:///c:/users/karthik/onedrive/desktop/downloads/facta.doc%23_bookmark67 picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 5 3.2. probabilistic picture hesitant fuzzy set let x be a fixed set, a probabilistic picture hesitant fuzzy set (p-phfs) m on x is defined by  , ( ) | ( ), ( ) | ( ), ( ) | ( ) | m x x s x x t x x u x x x   (2) where, ϕ(x)|s(x), χ(x)|t(x) and ψ(x)|u(x) contains a number elements, ϕ(x), χ(x), and ψ(x) denote the possibility (pmd), (nemd), and (nmd) of x ϵ x to the set m, respectively. here s(x), t(x), and u(x) are related with probabilistic information. furthermore, ϕ(x)|s(x), χ(x)|t(x), and ψ(x)|u(x) satisfy the requirement of 0 ≤ σ+ + ς+ + τ+ ≤ 1, and si, tj, uk ϵ [0,1]. # # # 1 1 1 1, 1, 1        i j k i j k s t u    where, si ϵ s(x), tj ϵ t(x), and uk ϵ u(x). the notation of #ϕ, #χ, and #ψ indicate the total number of elements in, ϕ(x)|s(x), χ(x)|t(x), and ψ(x) | u(x) respectively. for convenience, m = {x, ϕ(x)|s(x), χ(x)|t(x), ψ(x)|u(x)} a p-phfe, denoted by m = {ϕ|s, χ|t, ψ|u}. 3.3. basic operations let , m1 = (ϕ1|s1, χ1|t1, ψ1|u1) and m2 = (ϕ2 |s2, χ2 |t2, ψ2|u2) be three p-phfes, ϑ > 0, then m c is the complement of m, and p-phfe operations can be defined as: , , ( | , | , | ) ({ | },{ | },{ | })     cm s t u u t s                 1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 2 2 2 , ,1 2 1 2 1 2 1 2 1 2 , , ({ | },{ | },{ | }         m m s s t t u u                          1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 2 2 2 , ,1 2 1 2 1 2 1 2 1 2 1 2 , , ({ | },{ | },{ | }m m s s t t u u                                       , , ({1 (1 ) | },{ | },{ | })      m s t u               , , ({ | },{1 (1 ) | },{1 (1 ) | })        m s t u               3.4. score and accuracy function let be a p-phfe, then the score s(m) and accuracy functions h(m) are defined by # # # 1 1 1 1 1 1 1 ( ) ( ) ( ) # # # ( ) 2               i i i i i i i i i s t u s m          (3) # # # 1 1 1 1 1 1 1 ( ) ( ) ( ) # # # ( ) 2               i i i i i i i i i s t u h m          (4) 6 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy let m1 = (ϕ1|s1, χ1|t1, ψ1|u1) and m2 = (ϕ2|s2, χ2|t2, ψ2|u2) be two p-phfes, then, if s(m1) > s(m2) then (m1) > (m2), if s(m1) < s(m2) then (m1) < (m2), and if s(m1) = s(m2) then (m1) = (m2). if h(m1) > h(m2) then (m1) > (m2), if h(m1) < h(m2) then (m1) < (m2), if h(m1) = h(m2) then (m1) = (m2). 4. proposed methodology an integrated psi-dnma approach is utilized in the material selection procedure illustrated in fig. 1 for parametric optimization problems. fig. 1 method of solving an optimization problem picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 7 4.1. preference selection index method the psi approach is utilized to address fuzzy decision-making issues in material selection, particularly in addressing debates about the relevance of attribute. the psi approach to change weights, considering the contrast strength of each criterion and the contradictory nature of assessment criteria. step 1: create the initial decision matrix 11 1 1                n ij m n m mn (5) step 2: the normalized decision matrix is determined by calculating its elements using the following equations. max min for beneficial criteria for cost criteria         ij ij ij ij ij (6) step 3: calculate the mean values of the normalized performances in response to each 1 1 n ij in     (7) step 4: calculate the variation of preferences in response to each criterion. 2 1 ( )     m j ij i (8) step 5: calculate the deviations of the response preference for each criterion. 1 j j (9) step 6: compute the criteria weights. 1   j j n j j  (10) 4.2. double normalized based multiple aggregation method liao and wu introduced the dnma approach to the literature, which is an innovative method for ranking alternatives. two normalized approaches and three distinct joining 8 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy functions (full compensation, incomplete compensation, and incomplete compensatory) to implement a compensation system. step 1: the decision matrix's normalization.     1 1 max max , min min ,     ij jn ij ij j ij j ii t t t (11)     2 2 2 1 1 ij jn ij m ij j i t t      (12) where dij = d(ħij, ħj), ħj represents the goal value for the criteria cj. max beneficialcriteria min non-beneficialcriteria     ij i j ij i h the normalized decision matrices are represented as ħ1n and ħ2n, respectively. 1 1 1 2 2 1 max max      n n n ij ij ij i n n n ij ij ij i step 2: compute of criteria weights. the fuzzy psi technique calculates criteria weights based on the decision matrix's information, using an objective method for attributes like entropy. the basic conceptual approach of the psi method for solving mcdm problems is described in subsection 4.1. step 3: determine the subordinate utility values and rank. complete compensatory model (ccm)   1 1 1  n n i j ij j p  (13) un-compensatory model (ucm)    1 2 1 max 1    n i j ij j p  (14) incomplete compensatory model (icm)    1 3  jn i ij j p  (15) step 4: the normalized subordinate utility values picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 9        2 1   q in q m q i i p p p (16) where q=1,2,3 and i=1,2,...,m. the subordinate rank of each alternative is determined by the descending order of ℧ 1(pi) and ℧ 3(pi), and the ascending order of ℧ 2(pi). step 5: rank the alternatives. assigning weights to the (ccm), (ucm), and (icm)         2 2 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 ( ) 1 ( ) / max ( ) (1 ) ( ) ( ) / max ( ) (1 ) ( ) 1 ( ) / max ( ) (1 )                                     n n i i i i n n i s i i i i n n i i i i m p p p m p r p p p m m p p p m             (17) where ξ ϵ [0, 1] indicates the relative significance of the subordinate values and is weighted by ω1, ω2, ω3, satisfying ωi ϵ [0, 1] . the rankings of all alternatives can be determined by utilizing the descending order of rs(pi) for i=1,2,…,m. 5. case study this case study uses a lift-based wec with a fiberglass wave floating structure, consisting of feathers and an essential shaft as the rotor and a stator as the stationary component. structural reliability refers to an asset’s ability to function as planned for a specific period under specific technical and atmospheric conditions. evaluating these criteria can be challenging due to their unclear and quantitative nature. the fuzzy psidnma approach is used to evaluate various criteria, establishing a ranking level based on factors. fig. 2 illustrates the phase of the hydrofoil concept for demonstration purposes. five materials, including aluminum alloys, offshore steel, high-strength steel, cfrc, and gfrp, have been assessed for use in wave energy converters based on seven criteria, including structure dependability, hydrodynamic efficacy, offshore maintenance, corrosion resistance, production cost, eco-friendliness, and self-life. aluminum alloys (m1) aluminum alloys are widely used in modern industrial and consumer products due to their lightweight, corrosion-resistant, and versatile properties, including aircraft, automobiles, building materials, electrical conductors, and packaging materials aluminum silicon alloys are fluid, durable, and castable, suitable for various applications like engine blocks. aluminum is commonly alloyed with copper, zinc, zirconium, lithium, silicon, magnesium, nickel and lead in small proportions. high-strength offshore steel (m2) high-strength offshore steel is specially designed and manufactured for use in offshore constructions including oil rigs, platforms and other marine applications. these steels have superior mechanical qualities, including better 10 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy tensile strength, yield strength, and toughness than ordinary structural steels. the steel industry has developed advanced, ultra-high-strength steels in the last two decades, an offshoregrade structural steel used in the offshore sector. offshore steel (m3) offshore steel is a type of steel material that is developed and manufactured, particularly for use in offshore constructions including oil and gas platforms, drilling rigs, and other maritime facilities. these constructions are subjected to extreme circumstances such as severe weather, corrosive seawater, and the requirement to bear large loads and dynamics. these criteria guarantee that the steel satisfies strict specifications for mechanical qualities, chemical composition, and other variables. carbon fiber reinforced composite (cfrc) (m4) the cfrc is a form of composite material in which carbon fibers are inserted in a matrix material, usually a polymer resin. the combination of carbon fibers and matrix results in a composite material with improved mechanical characteristics, such as high strength-to-weight ratio, stiffness, and durability. carbon steel, a low-cost material used in refineries and petrochemical facilities for pipelines and valves, is primarily composed of carbon and iron compounds, but its poor corrosion resistance makes it less suitable for offshore conditions. glass fiber reinforced plastic (gfrp) (m5) the gfrp, also known as fiberglass reinforced plastic (frp), is a composite material consisting of glass fibers embedded in a polymer matrix. this composite material combines the high strength and stiffness of glass fibers with the plasticity and mold ability of a plastic matrix, usually a thermosetting resin such as polyester or epoxy. gfrp is a fiber-reinforced plastic, similar to graphitereinforced plastic, made from plastic reinforced with thin glass fibers, which can be chopped strand mats (csm) or woven cloth. 5.1. determining the criteria weights the decision matrix (h) is being initiated. the alternative (m) score, as per table 1, corresponds to the attributes (n) of the relevant p-phf element . step 1: experts can express their material selection aspirations using p-phf numbers, which are summarized in the p-phf decision matrix using eq. (2), as shown in table 2. step 2: eq. (6) is used to obtain the normalized result matrix shown in table 3. step 3: the mean values of normalized performances for each criterion are computed using data from table 3 and eq. (7). ℑ = {0.7482, 0.7274, 0.7678, 0.7240, 0.7596, 0.7940, 0.7914} step 4: eq. (8) is used to calculate the results of the variance of preferences according to each criterion value shown in table 4. step 5: the preference value is obtained by dividing the deviation of each criterion value using eq. (9). ℧ j = {0.8633, 0.8714, 0.8729, 0.8719, 0.8713, 0.8664, 0.8637}. step 6: the criteria weights are determined using eq. 10. wj = {0.1420, 0.1432, 0.1435, 0.1433, 0.1436, 0.1424, 0.1419} picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 11 fig. 2 hierarchical organization of the material selection of lift base wec table 1 the p-phf number-based linguistic conversion scale linguistic variable p-phf number very low (0.1, 0.2, 0.4) low (0.3, 0.5, 0.7) medium low (0.4, 0.3, 0.6) equal (0.5, 0.5, 0.5) high (0.8, 0.5, 0.3) absolutely high (1, 0.4, 0.6) table 2 p-phf decision matrix c1 c2 c3 c4 ({0.13 | 0.15, 0.18 | 0.25, 0.21 | 0.6} ({0.25 | 0.12, 0.17 | 0.32, 0.14 | 0.56} ({0.32 | 0.44, 0.1 | 0.33, 0.3 | 0.23} ({0.21 | 0.34, 0.03 | 0.43, 0.05 | 0.23} m1 {0.16 | 0.28, 0.25 | 0.32, 0.04 | 0.4} {0.05 | 0.53, 0.14 | 0.27, 0.23 | 0.2}) {0.1 | 0.46, 0.05 | 0.42, 0.03 | 0.12} {0.46 | 0.61, 0.2 | 0.39}) {0.13 | 0.47, 0.23 | 0.3, 0.08 | 0.23} {0.09 | 0.13, 0.05 | 0.4, 0.09 | 0.47}) {0.1 | 0.15, 0.13 | 0.25, 0.23 | 0.6} {0.2 | 0.32, 0.11 | 0.42, 0.43 | 0.26}) ({0.56 | 0.3, 0.27 | 0.42, 0.53 | 0.28} ({0.08 | 0.25, 0.41 | 0.32, 0.23 | 0.43} ({0.15 | 0.27, 0.32 | 0.55, 0.16 | 0.18} ({0.55 | 0.19, 0.05 | 0.81} m2 {0.23 | 0.2, 0.2 | 0.52, 0.18 | 0.28} {0.14 | 0.38, 0.02 | 0.42, 0.03 | 0.2}) {0.31 | 0.23, 0.16 | 0.77} {0.21 | 0.16, 0.17 | 0.24, 0.25 | 0.6}) {0.43 | 0.36, 0.44 | 0.53, 0.35 | 0.11} {0.02|0.25, 0.09 | 0.75}) {0.2 | 0.16, 0.14 | 0.84} {0.17 | 0.18, 0.11 | 0.4, 0.16 | 0.42}) ({0.28 | 0.15, 0.12 | 0.35, 0.06 | 0.5} ({0.41 | 0.44, 0.22 | 0.56} ({0.06 | 0.68, 0.12 | 0.12, 0.11 | 0.2} ({0.15 | 0.38, 0.22 | 0.36, 0.18 | 0.26} m3 {0.31 | 0.12, 0.17 | 0.32, 0.23 | 0.56} {0.05 | 0.02, 0.03 | 0.78, 0.01 | 0.2}) {0.3 | 0.22, 0.11 | 0.33, 0.15 | 0.45} {0.08 | 0.23, 0.17 | 0.43, 0.16 | 0.34}) {0.27 | 0.56, 0.6 | 0.34, 0.02 | 0.1} {0.11|0.5, 0.03|0.23, 0.07|0.27}) {0.04 | 0.46, 0.23 | 0.54} {0.1|0.4, 0.13|0.26, 0.17 | 0.34}) ({0.13 | 0.9, 0.05 | 0.1} ({0.32 | 0.26, 0.21 | 0.42, 0.11 | 0.32} ({0.31 | 0.59, 0.2 | 0.11, 0.1 | 0.3} ({0.14 | 0.66, 0.1 | 0.14, 0.21 | 0.2} m4 {0.34 | 0.32, 0.24 | 0.42, 0.22 | 0.26} {0.53 | 0.13, 0.46 | 0.43, 0.36 | 0.44}) {0.08 | 0.45, 0.2 | 0.55} {0.32 | 0.74, 0.12 | 0.12, 0.22 | 0.14}) {0.22 | 0.11, 0.11 | 0.33, 0.3 | 0.56} {0.01 | 0.36, 0.27 | 0.26, 0.38 | 0.38}) {0.32 | 0.31, 0.22|0.43, 0.19|0.26} {0.42 | 0.22, 0.38 | 0.78}) ({0.18 | 0.2, 0.21 | 0.31, 0.07 | 0.49} ({0.14 | 0.54, 0.21 | 0.1, 0.03 | 0.36} ({0.11 | 0.2, 0.27 | 0.55, 0.23 | 0.25} ({0.29 | 0.42, 0.16 | 0.32, 0.02 | 0.26} m5 {0.03 | 0.4, 0.1 | 0.12, 0.15 | 0.48} {0.32 | 0.54, 0.28 | 0.22, 0.29 | 0.24}) {0.31 | 0.83, 0.27 | 0.12, 0.29 | 0.05} {0.26 | 0.64, 0.19 | 0.24, 0.18 | 0.12}) {0.18 | 0.27, 0.17 | 0.3, 0.13 | 0.43} {0.32 | 0.75, 0.3 | 0.15, 0.31 | 0.1}) {0.14 | 0.63, 0.13 | 0.37} {0.07 | 0.36, 0.46 | 0.22, 0.26 | 0.42}) 12 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy c5 c6 c7 ({0.43 | 0.43, 0.23 | 0.23, 0.13 | 0.34} ({0.32 | 0.26, 0.15 | 0.43, 0.11 | 0.31} ({0.21 | 0.24, 0.14 | 0.11, 0.02 | 0.65} {0.03 | 0.24, 0.13 | 0.53, 0.11 | 0.23} {0.02 | 0.65, 0.12 | 0.35} {0.5 | 0.14, 0.3 | 0.86} {0.21 | 0.19, 0.17 | 0.36, 0.18 | 0.45}) {0.3 | 0.37, 0.29 | 0.22, 0.28 | 0.41}) {0.02 | 0.25, 0.01 | 0.12, 0.2 | 0.63}) ({0.16 | 0.25, 0.12 | 0.45, 0.17 | 0.3} ({0.19 | 0.28, 0.14 | 0.32, 0.16 | 0.4} ({0.04 | 0.32, 0.3 | 0.68} {0.04 | 0.43, 0.23 | 0.26, 0.2 | 0.31} {0.09 | 0.57, 0.3 | 0.33, 0.15 | 0.1} {0.3 | 0.44, 0.28 | 0.13, 0.29 | 0.43} {0.09 | 0.67, 0.3 | 0.12, 0.2 | 0.21}) {0.16 | 0.37, 0.11 | 0.63}) {0.23 | 0.11, 0.34 | 0.42, 0.2 | 0.47}) ({0.32 | 0.66, 0.36 | 0.21, 0.15 | 0.13} ({0.14 | 0.78, 0.11 | 0.22} ({0.26 | 0.42, 0.29 | 0.11, 0.17 | 0.47} {0.19 | 0.45, 0.28 | 0.12, 0.16 | 0.43} {0.3 | 0.33, 0.21 | 0.4, 0.1 | 0.27} {0.2 | 0.55, 0.31 | 0.21, 0.27 | 0.24} {0.02 | 0.27, 0.05 | 0.32, 0.34 | 0.41}) {0.4 | 0.65, 0.37 | 0.22, 0.29 | 0.13}) {0.19 | 0.27, 0.14 | 0.73}) ({0.24 | 0.54, 0.13 | 0.22, 0.02 | 0.24} ({0.15 | 0.37, 0.07 | 0.16, 0.01 | 0.47} ({0.12 | 0.33, 0.03 | 0.11, 0.04 | 0.56} {0.08 | 0.37, 0.34 | 0.63} {0.3 | 0.28, 0.4 | 0.28, 0.29 | 0.44} {0.07 | 0.55, 0.09 | 0.16, 0.1 | 0.29} {0.31 | 0.27, 0.03 | 0.73}) {0.01 | 0.48, 0.37 | 0.36, 0.38 | 0.16}) {0.34 | 0.44, 0.12 | 0.21, 0.35 | 0.35}) ({0.14 | 0.17, 0.24 | 0.83} ({0.2 | 0.55, 0.19 | 0.3, 0.43 | 0.15} ({0.12 | 0.62, 0.15 | 0.38} {0.22 | 0.37, 0.16 | 0.1, 0.11 | 0.53} {0.02 | 0.27, 0.26 | 0.73} {0.16 | 0.11, 0.07 | 0.89} {0.04 | 0.16, 0.09 | 0.32, 0.18 | 0.52}) {0.3 | 0.51, 0.29 | 0.49}) {0.02 | 0.22, 0.3 | 0.28, 0.29 | 0.5}) table 3 normalized p-phf score decision matrix c1 c2 c3 c4 c5 c6 c7 m1 0.9296 0.8135 1 0.8637 0.8800 1 0.8631 m2 1 0.8707 0.8972 0.8143 0.9202 0.9979 0.9740 m3 0.9068 1 0.8738 0.8183 0.8922 0.9292 0.9672 m4 0.7878 0.8617 0.9231 1 1 0.9072 0.9445 m5 0.8651 0.8184 0.9126 0.8480 0.8652 0.9295 1 table 4 preference variation values c1 c2 c3 c4 c5 c6 c7 m1 0.0329 0.0074 0.0539 0.0195 0.0145 0.0424 0.0051 m2 0.0633 0.0205 0.0167 0.0081 0.0257 0.0415 0.0333 m3 0.0251 0.0743 0.0112 0.0088 0.0175 0.0183 0.0309 m4 0.0015 0.0180 0.0241 0.0761 0.0577 0.0128 0.0234 m5 0.0136 0.0082 0.0209 0.0153 0.0111 0.0183 0.0434 5.2. determining priorities of alternatives step 1: table 5 displays the results of linear normalization using eq. (11). table 6 displays the results of vector normalization using eq. (12). step 2: the fuzzy psi method is used for determining criteria weights. step 3: table 7 presents the values obtained from eqs. (13), (14), and (15) for computing the utility functions ccm, ucm, and icm. step 4: the normalized subordinate utility values are calculated using eq. (16), as illustrated in table 8. step 5: the performance values of the alternative are calculated using eq. (17). experts provide ξ = 0.5, ω1 = 0.5, ω2 = 0.2, and ω3 = 0.3 as appropriate values for considering. picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 13 table 9 displays the estimated performance values and ranking of the alternatives. table 5 linear normalized p-phf decision matrix c1 c2 c3 c4 c5 c6 c7 m1 0.6684 0 1 0.3079 0.1252 1 0 m2 1 0.3068 0.1859 0 0.4434 0.9779 0.8102 m3 0.5607 1 0 0.0260 0.2247 0.2387 0.7610 m4 0 0.2584 0.3909 1 1 0 0.5947 m5 0.3642 0.0261 0.3077 0.2137 0 0.2408 1 table 6 vector normalized p-phf decision matrix c1 c2 c3 c4 c5 c6 c7 m1 0.9954 0.9873 1 0.9916 0.9924 1 0.9915 m2 1 0.9913 0.9934 0.9878 0.9952 0.9998 0.9984 m3 0.9940 1 0.9919 0.9881 0.9933 0.9956 0.9980 m4 0.9860 0.9907 0.9951 1 1 0.9942 0.9966 m5 0.9912 0.9877 0.9944 0.9904 0.9913 0.9956 1 table 7 values of ccm, ucm, and icm ccm ucm icm m1 0.4429 0.4473 0.9592 m2 0.5306 0.4476 0.9666 m3 0.4008 0.4481 0.9617 m4 0.4645 0.4448 0.9633 m5 0.3066 0.4482 0.9519 table 8 normalized subordinate utility values and rank order alternatives descending ascending descending ℧1 n(p ) rank order ℧2 n(p ) rank order ℧3n(p ) rank order m1 0.4549 3 0.4473 4 0.4465 3 m2 0.5449 1 0.4476 3 0.4500 1 m3 0.4116 4 0.4481 2 0.4477 4 m4 0.4770 2 0.4447 5 0.4484 2 m5 0.3148 5 0.4482 1 0.4432 5 table 9 performance values alternatives ranking values ranking order m1 0.4384 3 m2 0.6352 1 m3 0.3675 4 m4 0.5472 2 m5 0.2293 5 14 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy 6. results and discussion this section analyzes the outcomes of solving the fuzzy mcdm problem for material selection in a lift-based wec. the materials used include aluminum alloy, high-strength offshore steel, offshore steel, cfrc, and gfrp. the materials are evaluated for their structural dependability, hydrodynamic efficacy, offshore maintenance, corrosion resistance, production cost, eco-friendliness, and shelf life, as previously discussed in subsection 5.1. subsection 5.2 solves the p-phf-dnma algorithm by reversing the actual ratings of total cost and environmental effect. the results indicate that high-strength offshore steel is a suitable material for hydroelectric power in lift-based wecs and serves as a viable alternative. the conceptual design phase utilizes weighting parameters to select materials with robust dependability structure and high hydrodynamic efficiency. marine steel is a popular choice in offshore oil and gas because of its high exhaustion and maximum power. the material is also resistant to erosion and corrosion, ensuring minimal power generation loss. the study suggests that novel approaches must meet specific criteria such as stability and hydrodynamic efficacy to be considered suitable for their intended purpose and operation. the cfrc is criticized for its high cost and significant environmental impact, which outweigh its structural dependability benefits. the aluminum alloy has emerged as a significant competitor to offshore steel options. the findings align with numerous marine renewable experiments, typically conducted in-house at universities or local research centers, involving small-scale prototype manufacturing. the wec project, based on lifts, requires more thorough scrutiny of decision-making weighing elements as it progresses. as the number of trls increases, it is possible to decrease danger by implementing design changes and protective mechanisms. structural dependability is not equally weighted, but uncertainties in ratings are expected to decrease with more studies and larger-scale devices, focusing on manufacturing costs and environmental impact. the study reveals that the material options for lift-based wecs vary depending on the design stage, as illustrated in fig. 3. composites are the most cost-effective, maintenanceintensive, and environmentally friendly materials for mass manufacturing lift-based wecs due to their higher environmental impact. major wave energy businesses like ocean energy use offshore steel for devices, with weighting variables varying between commercialization and conceptual stages. the future material options for wec’s hydrofoil may include cfrc and gfrp, evaluated for cost-effectiveness, structural reliability, and environmental competitiveness. the reduction in uncertainty in fatigue and ultimate strength properties and the assumption that cfrc and gfrp are more eco-friendly than offshore steel and aluminum options are achieved. material selection is consistent with advances in offshore renewable structures such as wind turbines that demand improved structural reliability and affordability. further research is needed in the comprehensive design stage, considering manufacturing costs, hydrodynamic efficiency, fatigue stress, and carbon footprint assessment, reflecting current developments in offshore renewable frameworks. picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 15 fig. 3 the material life cycle process for lift-based wec 6.1. comparative analysis of different ranking techniques the correlation coefficient between ranking scores can be utilized for pragmatic confirmation and agreement. this proposed method can be considered a global approach for comparing decision outcomes in real-world applications. a comparison study is done using mcdm procedures based on several ranking algorithms to determine the ranking's stability. in many complicated decision-making situations, the robustness and reliability of alternative ranking scores are assessed by comparing the results of one model to those of other accessible and recognized approaches. the psi-dnma-based model’s reliability was evaluated by a comparison of various commonly used methods, including topsis [37], vikor [38], waspas [39], copras [40], cradis [41], and edas [42]. the chosen methods were deemed advantageous due to their numerous advantages, broad applicability, and ability to swiftly identify options in a multi-criteria selection scenario. the ranking results are given in table 10 and fig. 4. the method used slightly influenced the rankings of other quarters. the correlation between findings obtained through different methods is determined using ws coefficient and weighted spearman’s coefficient (rw). figs. 5 and 6 display a comparison of ranks using spearman order correlation. table 10 the comparison results are presented in a ranked order topsis vikor waspas copras cradis edas m1 0.4224 0.5770 0.4629 0.9816 0.5596 0.3959 m2 0.5088 0.0000 0.4734 1.0000 1.0000 0.7085 m3 0.4863 0.5721 0.4711 0.9882 0.5490 0.5053 m4 0.4599 1.0000 0.4325 0.9945 0.5697 0.5367 m5 0.3364 0.5785 0.4594 0.9639 0.3822 0.1144 16 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy fig. 4 comparison of various f-mcdm methods fig. 5 correlation between the ranking values of ws fig. 6 correlation between the ranking values of rw picture-hesitant fuzzy-based dual-normalised multiple aggregations for material selection... 17 6.2. sensitivity analysis sensitivity analysis is a crucial decision-making technique that compares a model’s results to other accessible and well-structured methods to evaluate the resilience and reliability of alternative ranking scores. sensitivity analysis investigates the impact of the δ parameter on the ranking of alternatives in the proposed psi-dnma technique. table 11 provides a concise summary of the results. the alternative’s ranking is determined to be m5 < m3 <m1 < m4 < m2 based on the results. we conclude that when δ changes from 0.1 to 0.9 using proposed method, m2 is the best option. fig. 7 depicts the graphical representation of the δ variation possibilities. the ranking of the alternatives remained the same when the relative importance of the parameter was changed. this effectively explains why the proposed method of ranking is unaffected by parameters. this demonstrates the dependability and durability of the suggested framework. table 11 the psi-dnma method’s findings rank in the order of parameters δ = 0.1 δ = 0.2 δ = 0.3 δ = 0.4 δ = 0.5 δ = 0.6 δ = 0.7 δ = 0.8 δ = 0.9 m1 0.3094 0.3835 0.4028 0.4211 0.4384 0.4550 0.4709 0.4863 0.5011 m2 0.6352 0.6604 0.6515 0.6432 0.6352 0.6277 0.6204 0.6135 0.6068 m3 0.2382 0.2864 0.3154 0.3423 0.3676 0.3749 0.4141 0.4357 0.4563 m4 0.5073 0.5650 0.5570 0.5514 0.5472 0.5404 0.5418 0.5402 0.5390 m5 0.0411 0.0998 0.1484 0.1910 0.2294 0.2482 0.2972 0.3279 0.3568 fig. 7 influence of δ parameter onto the solution 18 d. kang, k. suvitha, m. sandra, d. pamucar, v. simic, s. narayanamoorthy 7. conclusion this study introduces a novel hybrid mcdm technique for solving practical decision making issues, with primary contributions listed below. we introduced a novel method, p phfss, to enhance the comprehension of complex assessment data by decision-makers. p phfss are more effective in managing decision-makers fuzzy judgments due to their ability to express positive, neutral, and negative membership degrees and provide accompanying probabilistic information. optimal material selection based on our approach can result in cost savings with more economical materials without compromising performance. this innovative combination enhances the ability to handle uncertainty and ambiguity in decision-making processes, providing a more robust and flexible framework for material selection in engineering applications. the framework introduces a distinctive material selection method in lift-based wecs, employing fuzzy dnma as an mcdm. expert opinion was used to identify and assess key hydrofoil characteristics, a unique and original activity for this type of device. the team of experts uses fuzzy logic selecting and inability setting to assess essential hydrofoil material selection criteria, allowing for flexible and rigorous evaluation of lift-based wecs. the framework emphasizes material fitness-for-purpose to prevent technological failure and financial losses. this applies to various stages of the product cycle, including lift-based wave energy converters. the study suggests that further submerged marine energycollecting devices operating near the ocean’s surface may be beneficial. the paper presents a probabilistic picture hesitant fuzzy set-psi-dnma strategy for mcdm problem implementation, considering uncertain, indeterminate, and inconsistent information. the proposed strategy integrates the psi method for objective weight and incorporates fundamental operations and the score accuracy function with p-phfs for flexibility during the energy process. the psi-dnma model is utilized to evaluate the material selection for lift-based wecs, focusing on sustainability in p-phfs. we discuss factors such as structure dependability, hydrodynamic efficiency, offshore maintenance, corrosion resistance, eco-friendliness, self-life, and production cost when choosing materials for lift-based wec. future research will explore the dependability of structures 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chen, x., 2023, edas method for multiple attribute group decision making with probabilistic dual hesitant fuzzy information and its application to suppliers selection, technological and economic development of economy, 29(2), pp. 326-352. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 277 288 1 design and thermal performance of the solar biomass hybrid dryer for cashew drying udc 662.6 dhanushkodi saravanan 1 , vincent h. wilson 2 , sudhakar kumarasamy 3 1 prist university, thanjavur, india 2 toc h institute of science and technology, arakkunnam, kerala, india 3 energy centre, national institute of technology, bhopal, m.p, india abstract. drying of cashew nut to remove testa is one of the most energy-intensive processes of cashew nut process industry. for this reason a hybrid dryer consisting of a solar flat plate collector, a biomass heater and a drying chamber is designed and fabricated. 40 kg of cashew nut with initial moisture of 9 % is used in the experiment. the performance test of the dryer is carried out in two modes of operation: hybridforced convection and hybrid-natural convection. drying time and drying efficiency during these two modes of operation are estimated and compared with the sun drying. the system is capable of attaining drying temperature between 50º and 70ºc. in the hybrid forced drying, the required moisture content of 3% is achieved within 7 hours and the average system efficiency is estimated as 5.08%. in the hybrid natural drying, the required moisture content is obtained in 9 hours and the average system efficiency is 3.17%. the fuel consumption during the drying process is 0.5 kg/hr and 0.75 kg/hr for forced mode and natural mode, respectively. the drying process in the hybrid forced mode of operation is twice faster than the sun drying. the dryer can be operated in any climatic conditions: as a solar dryer on normal sunny days, as a biomass dryer at night time and as a hybrid dryer on cloudy days. based on the experimental study, it is concluded that the developed hybrid dryer is suitable for small scale cashew nut farmers in rural areas of developing countries. key words: biomass heater, collector efficiency, hybrid dryer, cashew nut, drying rate 1. introduction within the whole of the world cultivation of cashew nut plants, india occupies the top position with about 8.93lakhs ha while its total annual production is estimated as 6.95lakhs m.t [1]. in addition, india has an enormous experience in processing the plants as indicated by its high export of cashew kernels in comparison with other countries. received october 19, 2014 / accepted november 21, 2014 corresponding author: dhanushkodi saravanan prist university, thanjavur 613403, tamilnadu, india e-mail: dhanushkodi.vs@gmail.com original scientific paper 278 d. saravanan, v. h. wilson, s. kumarasamy besides, it imported raw nuts (about 7.52 lakh mt) during 2009-10 to meet out the requirements of its cashew nut processing industries [2]. the main objective of the cashew processing is to remove a valuable cashew kernel from the shell with as little damage as possible. whole kernels command a higher price than the broken pieces. pale, ivory colored (or) white kernels are preferable. therefore, the processor must finely tune the process in order to achieve the best quality kernels. the cashew nut processing consists of the following five steps: 1. removal of the outer shell called a shelling process, 2. removal of the testa (the thin skin covering the kernel) called a peeling process, 3. grading into different sizes and color in accordance with different grade standards, 4. humidifying or drying to attain a final moisture content of 3-5 percent, and, 5. depending upon the quantity, finally pack into air tight bags or cans. of the above five processes, peeling is a very important and difficult one. it needs temperature around 65°c 70°c with time duration of 5-6 hours. the moisture content of the raw kernel is reduced from 9% to 3% to prevent natural decay [3]. majority of the processing industries carry out this process by using steam drying and electrical drying. the total energy consumption for processing (cashew nut drying, cashew nut steaming and drying of cashew kernels altogether) of 1000 kg of raw cashew nuts is around 5000-6000 mj [4]. in the sun drying, the kernels are uniformly spread out on the floor and the process is heavily reliant on a constant supply of sunshine. although the sun drying does not pose any risk of scorching the kernels, it may be prolonged in the case of bad weather, which can lead to an extended drying process time. other drying methods represent very expensive processes. therefore, the cashew nut farmers are searching for alternative methods for drying cashew kernel. 2. referential literature review mohamad hanif et al [5] have used a dish type solar dryer for drying grapes. debbarma et al [6] have designed and tested a low-cost solar bamboo dryer for drying chillies at manit, bhopal. amer et al [7] have designed and evaluated the performance of a new hybrid solar dryer for banana drying. hossain et al [8] have developed a prototype hybrid solar dryer for tomato drying. chandra kumar and bhagoria [9] have carried out performance evaluation of the mixed mode solar dryer with forced convection. andrew et al [10] have designed and developed an indirect solar dryer with a biomass backup heater for drying pepper berries. azimi et al [11] have carried out an experimental study on eggplant drying by an indirect solar dryer and open sun drying. a mixed mode dryer with a natural convection mode and a biomass backup heater is designed by e tarigan et al [12]. many of them use solar energy in the natural and the forced convection mode (or) a solar biomass heater in the hybrid mode for drying agricultural products with the temperature ranging from 45°c-65°c. this brief review of referential literature clearly indicates that very little information is available about a small scale hybrid dryer for drying cash crops like cashew. cashew drying is a highly energy intensive and expensive process. in order to reduce the energy cost associated with cashew drying, alternate renewable energy based sources of drying need to be explored. hence, the present investigation is carried out with the following objectives in mind: design and thermal performance of the solar biomass hybrid dryer for cashew drying 279 1. to fill the void left by the researchers in this area, 2. to design and develop a solar biomass hybrid dryer for drying cashew, and, 3. to ensure continuous day and night operation of the hybrid dryer. 3. experiments the dryer consists of a biomass heater, a solar air heating collector, a centrifugal blower and a drying chamber with chimney. the schematic view and a photograph of the experimental setup are shown in figs. 1 and 2, respectively. fig. 1 solar biomass hybrid dryer fig. 2 perspective view of the hybrid solar dryer 3.1. solar collector the solar air heating collector system consists of an absorber, a double glass cover, a back plate and insulation. the solar air collector has dimension of (2m×1.1m×0.2m). the whole system is enclosed in a rectangular box made of galvanized iron sheet of 0.99 mm thick. the absorber is made of 2mm thick aluminum plate coated with black paint to absorb incident solar radiation. two toughened glass plates are fixed on the top of the collector at a distance of 0.04m above the absorber plate in order to reduce heat losses from the top side of the collector. during the operation the air flows through the space between the absorber plate and the back plate. the spaces inside the collector are baffled to change the direction of the airflow while two inlet connections are provided at the front end of the collector. one is connected to the blower for running the system in the forced convection mode and another one is opened to the atmosphere for running the system in a free convection mode. 3.2. biomass heater the biomass heater consists of four main parts: inner and outer shells, a cross pipe, a chimney and openings. the base of the biomass heater acts as a combustion chamber. 280 d. saravanan, v. h. wilson, s. kumarasamy the cylindrical ms cross pipe is located in the middle of the inner shell to divert the flame towards the periphery of the inner cylindrical shell. glass wool insulation of 0.08 m thickness with aluminum cladding is provided on the outer shell of the biomass heater. 3.3. drying unit the drying unit consists of four main parts: a base frame, a drying chamber, drying trays and a loading door. this chamber is made up of a mild steel frame and covered with galvanized iron sheet of 1mm in thickness. the drying chamber consists of ten perforated aluminum trays which are arranged from the bottom to the top of the drying chamber. they are evenly placed at a distance of 0.015m. a door is provided with locking arrangement for loading and unloading of the product 3.4. blower the blower is attached to the solar air heating collector to induce and control the airflow rate inside the collector. the blower is connected to the solar air collector and the biomass heater by a pipe line. the maximum operating speed of the blower is 2800rpm. 3.5. chimney a chimney with varying cross-sections (bottom 0.16 m, top 0.2 m, height 0.6 m) is provided at the top of the drying chamber to remove moist air. a sliding door is provided at the top of the chimney to control the exhaust air flow. the design parameters of the hybrid solar dryer are given in table 1. table 1 design parameters of the hybrid solar dryer component specifications solar collector type area glass cover number of glazing absorber plate tilt angle insulation flat plate 2.2m² 4mm 2 aluminum sheet, 2 mm thick 15º glass wool drying chamber size/no of trays tray area chimney tray thickness 0.64×0.6×0.73 m / 10 0.54×0.51 m bottom 0.16 × 0.16 ,top 0.2×0.2 & height 0.6 m 0.003 m blower capacity, speed & voltage 0.37 kw,0-2800 rpm & 440 v(a.c) biomass heater inner shell diameter outer shell diameter height shell thickness 0.34 m 0.42 m 0.94 m 0.003 drying capacity 40kg design and thermal performance of the solar biomass hybrid dryer for cashew drying 281 3.6. operating system two different modes of operation are carried out in order to study the thermal performance and drying characteristics of the system. the drying system is operated in the hybrid forced mode by running the blower at 1400 rpm. an optimum air flow rate of 0.0402 kg/s is maintained continuously during the trail. the air is uniformly distributed to the solar collector and the biomass heater. additional heat input is provided by burning 0.5 kg of fuel wood in the biomass heater. a hybrid forced mode operation. the experiment is repeated in a hybrid natural mode without the use of blower. sun drying operation is also carried out simultaneously in order to get the drying time. 3.7. drying product boiled cashew nut shell weighing 80 kg is procured from a local farmer in cuddalore district, tamilnadu, india. a hand operated shell cutter is used to remove shell. the cashew nut kernel obtained after cutting operation and has a brown skin called testa. to remove the testa from the kernel, a drying operation is performed. the uniformly controlled heating of 60-70°c for a period of 6-8 hours is required to remove the moisture from 9% to 3%. 3.8. experimental procedure the given experiments are conducted in the forced convection solar biomass hybrid dryer under no load condition. at each hour of drying the temperature is measured of the drying chamber at bottom, middle and top trays by using a thermocouple inserted inside the drying chamber. ambient temperatures, collector outlet temperature, drying chamber outlet temperature are all measured by means of rtds. other results observed are solar intensity by solar power meter, relative humidity in ambience and drying chamber by thermo hygrometer. the initial and final weights of the product are measured by using digital weight balance. the air flow rate is calculated by using a hotwire anemometer connected between the blower and collector and the gasifier inlet; energy consumption of the blower is also calculated by energy meter. the experiment is repeated in different solar days by varying the speed of the blower in the range of 100 rpm with minimum speed of 300rpm and maximum speed of 2500 rpm in a total number of 23 days. three optimum flow rates are identified (0.0289 kg/s, 0.035 kg/s, 0.042 kg/s) as suitable for this product. the three optimum flow rates are used to dry 40kg of cashew per batch and the results are notified. similar experiments are repeated in a natural convection solar dryer and a biomass dryer with a forced convection mode and compared with the open sun drying. the comparison is done with the open sun drying on same day and at same time (from 8.00 am to 5.00pm). 3.9. instrumentation solar radiation is measured by means of a solar power meter. three calibrated thermocouples with ± 0.5°c accuracy are fixed at top, bottom and middle of the solar drying chamber to measure the drying air temperature. energy consumption of the blower is measured with an energy meter having ±0.5 accuracy. air velocity at the collector inlet as well as the biomass heater are measured by means of a hot wire anemometer (accuracy of ±0.01m/s). the relative humidity of the ambient air and the drying chamber are measured by using a thermo hygrometer. the cashew nut kernel mass and the biomass fuel are measured with an electronic balance of accuracy 0.01g. the list of instrument used in the experiment is shown in table 2 for ready reference. 282 d. saravanan, v. h. wilson, s. kumarasamy table 2 instrumentations used in the experiment s.no parameter instruments accuracy 1 temperature thermocouple and rtds 0.05°c 2 mass electronic balance 0.01g 3 solar radiation solar power meter ± 1w/m² 4 air velocity hot wire anemometer ±2.5% 5 power consumption of blower energy meter ±0.1kwh 6 relative humidity thermo hygrometer ±2.5% 3.10. efficiency calculation the performance of the system and of the drying characteristics is calculated using the following expression. the moisture content (mc) is expressed as a percentage of moisture present in the product. the instantaneous moisture content at any given time on wet basis and dry basis is calculated using the following expression. ( ) 100 i d c i m m m wet basis m    (1) ( ) 100 i d c d m m m dry basis m    (2) where mi is the initial mass of the sample in kg md is the final mass of the sample in kg drying rate (rd) is formed by a decrease of the water concentration during the time interval between two subsequent measurements divided by this time interval: i d d m m r t   (3) where ‘t’ is the time of drying in sec. collector efficiency (ηc) is defined as the ratio of useful heat gain (qu) over any time period to the incident solar radiation over the same period, with i denoting the solar intensity in w/m 2 and a the collector area in m 2 : . u c q i a   (4) ( ) . p o i c mc t t i a    (5) where ‘m’ is the mass flow rate of air in kg/sec, cp is the specific heat of air in kj/kg k, to is the collector outlet temperature in o c and ti is the collector inlet temperature in o c. dryer efficiency (d) of a system is defined as the ratio of energy used to evaporate the moisture from the product to the energy supplied to the dryer. in the case of forced convection dryer the energy consumption of blower is taken into account. the efficiency of forced convection and natural convection solar biomass hybrid dryer is calculated by the following expression: design and thermal performance of the solar biomass hybrid dryer for cashew drying 283 100 w fg d f v m h iat e m c      (6) 100 w fg d f v m h iat m c     (7) where ‘mw’ is the mass of moisture evaporated at a time in kg, hfg is the latent heat of vaporization of water for the drying chamber in kj/kg, e is the energy consumption of blower in kwh, mf is the mass of fuel used in kg/hr, cv is the calorific value of wood chips in kj/kg. the effectiveness factor can be defined as the ratio of the drying rate in the indirect solar dryer to that in the open sun drying: dryingsuninratedrying emodhybridinratedrying factoresseffectiven  (8) 4. results and discussion 4.1. analysis of temperature profile inside the dryer the results obtained by running the dryer from 8am to 5pm in hybrid natural convection mode are shown in fig. 3. the ambient air temperature at the dryer inlet varies from 28 to 32ºc and solar intensity varies from 600w/m² to 880w/m² on the test day. fig. 3 temperature profile inside the dryer in hybrid natural mode the collector outlet temperature varies from 50 to 80°c and drying chamber temperature varies from 50 to 65°c. average temperature gain in the collector is around 41ºc in the hybrid natural mode. during this period the bottom, middle and top tray temperatures are almost the same. the results obtained for running dryer from 8am to 5pm in the hybrid forced convection mode are shown in fig. 4. the ambient air temperature at the dryer inlet varies from 28 to 33ºc and solar intensity varies from 600w/m² to 960w/m². an optimum air flow rate of 0.042 kg s -1 is maintained during the forced mode operation. the collector outlet temperature varies between 60 and 90°c. the drying chamber temperature remains between 55-70°c 284 d. saravanan, v. h. wilson, s. kumarasamy which is ideally suitable for drying of cashew. average temperature gain in the collector is around 45ºc in the hybrid forced mode, respectively. fig. 4 temperature profile inside the dryer in hybrid forced mode maximum average temperature of 62.5 c is measured at the top tray and temperatures of 63.7°c and 62.5ºc are observed in the middle and bottom tray, respectively (table 3). the temperature recorded inside the drying chamber in the hybrid forced mode is moderately higher than that in the hybrid natural mode. a slight decreasing temperature profile is observed from the bottom to the top tray of the dryer. there are no significant variations in different tray temperature. this ensures uniformity in drying to maintain the commercial value of the product. uniform temperature inside the drying chamber is also essential for avoidance of searing and under-drying. table 3 collector and drier efficiency at different drying modes (%) drying mode average collector outlet temperature drying chamber bottom drying chamber middle drying chamber top hybrid natural mode 71.4 57.5 56.4 55.1 hybrid forced mode 75.6 65 63.7 62.5 4.2. moisture loss and drying rate the initial moisture content of the cashew nut shell is 9.29%.the desired final moisture content is in the range of 3.5 to 4.6 %. fig. 5 shows the variation of drying rate in the sun drying, the hybrid natural mode and the hybrid forced convection mode with time. the average drying rate by the hybrid forced and the hybrid natural drying are 0.0012 and 0.00088 kg/hr, respectively. variation of moisture content with drying time is shown in fig.6. final moisture content of 3.5 % is obtained within 7 hours of drying in the forced convection mode, whereas it takes 9 hours of drying in the natural convection mode. however, it takes more than 14 hrs in the open sun drying. the sun drying performance is purely influenced by the climatologically conditions like ambient temperature, relative humidity and solar intensity. the ambient temperature is fluctuating and not sufficient design and thermal performance of the solar biomass hybrid dryer for cashew drying 285 enough to dry the cashew within permissible time. this prolonged drying time leads to poor product quality which is not acceptable for cash crops like cashew. in all the three cases, the drying rate decreases with the decrease in moisture content. fig. 5 variation of drying rate with time fig. 6 variation of moisture content with drying time 4.3. thermal efficiency of the solar collector and dryer the efficiency of the solar collector depends on the inlet air flow rate and outlet temperature of collector and solar intensity. fig. 7 shows the variation of collector efficiency with time in the hybrid natural and the forced mode. maximum efficiency of 40 55% is obtained in the hybrid natural mode. collector efficiency ranging from 58% and 90% is obtained in the hybrid forced mode. fig. 7 variation of collector efficiency with time 286 d. saravanan, v. h. wilson, s. kumarasamy the solar air heating collector efficiency follows a similar pattern of the solar intensity. overall drying efficiency has been evaluated for the system based on the energy used to evaporate the moisture from the product to the total input energy (solar intensity + biomass fuel) supplied to the drier. the latent heat of vaporization is calculated by means of the average drying chamber temperature from standard steam tables. fuel wood of 0.5 kg/hr is burned during the natural mode operation .the hourly efficiency of the dryer run by in natural convection mode varies from 2% to 4%. fuel wood of 0.75 kg/hr is burned during the forced mode operation. the hourly efficiency of the dryer in forced convection mode varies from 3 to 8 % as shown in fig. 8. also the average dryer efficiency and collector efficiency in both modes of operation is depicted in table 4. fig. 8 variation of dryer efficiency with time table 4 collector and drier efficiency at different drying modes (%) drying mode collector efficiency (%) overall dryer system efficiency (%) hybrid natural mode 46.6 3.17 hybrid forced mode 75.64 5.08 4.4. effectiveness of the drying the variation of drying time versus effectiveness factor is shown in fig. 9. it is observed that the effectiveness factor is always higher than one except during the last phase of the drying time. a high effectiveness factor of 13.25 indicates the usefulness of the hybrid dryer as compared to the sun drying. fig. 9 variation of effectiveness factor with time design and thermal performance of the solar biomass hybrid dryer for cashew drying 287 4.5. quality evaluation and energy conservation benefits the dried cashew nut kernels from both the test conditions are analyzed for their quality. the kernels are graded manually by using a hand/ sieve. the grading is carried out in accordance with the export criteria set by the indian government. kernel dried in both the mode is having size conforming to w 240 (between 485-530 kernels per kg) superior quality grade. there is no significant scorched and splits kernel among the dried samples in the hybrid forced and the natural mode. the conventional drying of cashew nut using steam drying to reduce the moisture content from 10 % to below 3.5% is one of the energy intensive operations in cashew nut processing industry. the hybrid solar biomass drying technology reduces the drying time and cost of energy associated with the conventional drying operation. the system could be one of the most viable options for the cashew nut farmers. 5. conclusions the solar biomass hybrid dryer has been fabricated for the purpose of drying 40kg of cashew nut per batch. the average collector efficiency of the system in the hybrid forced mode is 75.6%. temperature between 5575ºc and 75ºc can be obtained depending on the weather conditions and fuel used. this is a practical technology which can be used for dying of cashew as well as of other agricultural products. this system could reduce drying time by half when compared to the open sun drying and it produces a high quality cashew nut (w240). improvements in the performances of dryer could be achieved through further modification which include (1) providing the parabolic reflector on both sides of the collector, (2) increasing the absorptivity of the absorber plate by replacing copper plate with aluminum one, (4) increasing air flow rates, and (5) providing pvt operated electrical heating coil. it can be concluded that the developed dryer is more suitable for cashew nut farmers in rural areas of developing countries. references 1. senthil a., mahesh, m.p., 2013, analysis of cashew nut production in india, asia pacific journal of marketing and management review 2(3), pp. 106-110. 2. babaganna g.,silas k., ahmed m., 2012, solar dryer an effective tool for agricultural products preservation, journal of applied technology in environmental sanitation 2(1), pp.31-38. 3. atul m., sudhir j., powar, a.g., 2010, energy option for small scale cashew nut processing in india, energy research journal 1(1), pp. 47-50. 4. atul mohod, sudhir jain , ashok powar, 2011, quantification of energy consumption for cashew nut (anacardium occidentale l.) processing operations, international journal of sustainable energy 30, s11-s23. 5. muhammad, h., muhammad, r., muhammad, a., 2012, drying of grapes using dish type solar air heater, journal of agricultural research 50(3), pp.423-431 6. debbarma m., rawat p., sudhakar k., 2013, thermal performance of low cost solar bamboo dryer, international journal of chem tech research. 5, pp. 1041-1045. 7. amer b.m.a., hossain m.a., gottschalk, k., 2010, design and performance evaluation of a new hybrid solar dryer for banana, energy conversion and management 51, pp. 813-820 8. hossain m.a., amer b.m.a., gottschalk k., 2008, hybrid solar dryer for quality dried tomato, drying technology 26, pp. 1591-1601 9. pardhi b.p., bhagoria, j.l., 2013, development and performance evaluation of mixed mode solar dryer with forced convection, international journal of energy and environmental engineering, 4(23) 288 d. saravanan, v. h. wilson, s. kumarasamy 10. andrew ragai henry rigit, abdul qauoom jakhrani, shakeel ahmed kamboh, patrick low tiong kie, 2013, development of an indirect solar dryer with biomass backup heater for drying pepper berries, world applied sciences journal 22 (9), pp. 1241-1251. 11. azimi a., tavakoli t., beheshti h.k., rahimi a., 2012, experimental study on eggplant drying by an indirect solar dryer and open sun drying, iranica journal of energy and environment 3(4), pp. 347-353. 12. tarigan, e., tekasakul, p., 2005, a mixed-mode natural convection solar dryer with biomass burner and heat storage back-up heater, proceedings of the australia and new zealand solar energy society annual conference, pp. 1–9. dizajn i termička performansa hibridne solarne sušilice sa biomasom za sušenje indijskog oraha sušenje indijskog oraha radi uklanjanja ljuske jeste jedan od energijski najzahtevnijih procesa u agro-industriji. iz tog razloga je projektovana i proizvedena hibridna sušilica koja se sastoji od solarnog kolektora sa ravnom pločom, grejača sa biomasom i komorom za sušenje. u eksperimentu se koristi 40kg indijskog oraha sa početnom vlažnošću od 9%. testiranje performanse sušilice se obavilo u dva radna režima: sa hibridno-indukovanom cirkulacijom i sa hibridno-prirodnom cirkulacijom. procenjeni su vreme sušenja i efikasnost sušenja tokom ova dva radna režima i upoređeni sa sušenjem na suncu. sistem je sposoban da postigne temperaturu sušenja između 50 º i 70 º. kod hibridno-indukovanog sušenja, traženi sadržaj vlažnosti od 3% postiže se za 7 sati a prosečna efikasnost sistema je procenjena na 5,08%. kod hibridno-prirodnog sušenja, traženi sadržaj vlažnosti postiže se za 9 sati a prosečna efikasnost sistema je procenjena na 3.17%. potrošnja goriva tokom procesa sušenja je 0,5 kg na sat i 0,75 kg na sat za indukovani i za prirodni režim, respektivno. proces sušenja kod hibridno-indukovanog režima rada je dva puta brži nego kod sušenja na suncu. sušilica se može koristiti pod bilo kojim klimatskim uslovima: kao solarna u normalne sunčane dane, kao sušilica sa biomasom noću i kao hibridna sušilica kada je oblačno. na osnovu ovog eksperimentalnog istraživanja, zaključuje se da je razvijena hibridna sušilica pogodna za male odgajivače indijskog oraha u seoskim oblastima zemalja u razvoju. ključne reči: solarna sušilica sa biomasom, efikasnost kolektora, hibridna sušilica, indijski orah, stopa sušenja 12839 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 227 250 https://doi.org/10.22190/fume240804047g © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a monitoring method of milling chatter based on optimized hybrid neural network with attention mechanism haining gao1, hongdan shen1, caixu yue2, rongyi li2, steven y. liang3, yinlin wang1, wenfu liu1, yong yang1 1huanghuai university, zhumadian, china 2harbin university of science and technology, harbin, china 3georgia institute of technology, atlanta, usa orcid ids: haining gao https://orcid.org/0000-0002-6809-2090 hongdan shen https://orcid.org/0009-0009-7143-1733 caixu yue https://orcid.org/0000-0003-4659-6236 rongyi li https://orcid.org/0009-0004-1113-9777 steven y. liang https://orcid.org/0000-0003-2852-0959 yinlin wang https://orcid.org/0009-0001-8163-1933 wenfu liu https://orcid.org/0000-0003-1371-3619 yong yang https://orcid.org/0000-0002-2248-5323 abstract. machining chatter is a self-excited vibration between the cutting tool and the workpiece, which can reduce surface quality and tool life, and even endanger the safety of operators in severe cases. considering that milling chatter has multi-scale features and the debugging of neural network hyperparameters heavily relies on experience, a milling chatter monitoring method based on an optimized hybrid neural network with an attention mechanism (missa-mscnn-bilstm-atm) is proposed. firstly, the harmonic of the spindle rotation frequency is filtered out using the spindle rotation frequency removal technique (sft). then, an improved sparrow search algorithm (missa) is proposed based on multiple strategies including improved circle chaotic mapping, golden sine strategy, and enhanced lévy flight. subsequently, missa is utilized to optimize the hyperparameters of the milling chatter classification hybrid neural network model, combining multi-scale convolutional neural networks (mscnn), bidirectional long short-term memory (bilstm), and attention mechanism (atm). in numerical simulations with cec2005 complex functions, missa demonstrates better optimization accuracy, stability, and shorter computation time compared to other intelligent algorithms. compared with other milling chatter classification models, the proposed method exhibits significant improvements in accuracy and stability. key words: chatter monitoring, attention mechanism, hybrid neural network, optimization algorithm received: august 04, 2024 / accepted november 10, 2024 corresponding author: haining gao college of energy engineering, huanghuai university, no. 76 kaiyuan avenue, yicheng district, zhumadian, china, 463000 e-mail: 20191908@huanghuai.edu.cn https://orcid.org/0000-0002-6809-2090 https://orcid.org/0009-0009-7143-1733 https://orcid.org/0000-0003-4659-6236 https://orcid.org/0009-0004-1113-9777 https://orcid.org/0000-0003-2852-0959 https://orcid.org/0009-0001-8163-1933 https://orcid.org/0000-0003-1371-3619 https://orcid.org/0000-0002-2248-5323 228 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang 1. introduction aerospace components, such as turbine blades, are characterized by thin-walled structures, complex geometries, and are predominantly manufactured from difficult-to-machine materials such as titanium alloys. chatter frequently occurs during the milling process, resulting in deteriorated surface quality and accelerated tool wear. recent research has demonstrated that chatter monitoring is crucial for ensuring both the integrity of the cutting system and workpiece quality in aerospace manufacturing processes. chatter monitoring consists of four primary components: signal acquisition, signal processing, feature extraction, and pattern recognition. various signal types employed for chatter monitoring include vibration signals [1-3], cutting force signals [4-7], current signals [8-10], sound signals [11-12], and multi-sensor information [13-15]. among these, acceleration signals are particularly advantageous due to their non-intrusive nature, high reliability, and sufficient response bandwidth [4]. however, experimental data acquisition through sensors is characterized by limited processing conditions, high costs, and significant time requirements. to address these limitations, scholars have generated simulated cutting force signals covering a wide range of machining parameters based on dynamic milling process models [16], or used generative adversarial networks to generate rich training data [17]. current multi-sensor fusion approaches for chatter detection lacks comprehensive theoretical frameworks for sensor selection, sampling strategies, and fusion mechanisms. consequently, these methods predominantly rely on empirical parameter settings and case-specific validation, creating significant challenges in establishing unified evaluation criteria and generalized solutions. sensor data acquisition in machining processes encompasses multiple signal components, including spindle rotation frequency with its harmonics, chatter frequencies, local outliers, and measurement noise. in chatter monitoring systems, the presence of local outliers and noise can result in both false alarms and missed detections [18]. while stable components, particularly the spindle rotation frequency and its associated harmonics, do not contribute to chatter phenomena, their removal can significantly enhance chatter monitoring sensitivity [19]. to address these issues, scholars have conducted extensive research. albertelli et al. [18] used order tracking and synchronous averaging methods to eliminate noise and cutting periodicity effects. yan et al. [19] removed periodic components related to spindle rotation using a designed time-frequency filter. wan et al. [20] used a matrix notch filter to filter out the harmonics of the spindle rotation frequency and color noise components. matthew et al. [21] used an adaptive threshold method to reduce the impact of noise, rotation frequency, and harmonics. subsequently, a wavelet based adaptive filter was constructed to reduce the impact of noise [4]. kuo et al. [17] employed fractional-order convolutional kernels to eliminate local outliers in training data. li et al. [22] added an artificial offset after filtering out stable frequency components, avoiding zero division operations. wan et al. [23] used their proposed variable forgetting factor recursive least squares algorithm to filter out chatterindependent components composed of environmental noise and periodic components. feature extraction methodologies in machining processes can be categorized into three primary domains: time-domain, frequency-domain, and time-frequency domain approaches. time-domain feature extraction offers computational efficiency and straightforward implementation. these primarily comprise dimensionless features, including statistical measures such as mean [10,19], standard deviation [10,19], root mean square [4,5,19,24], shape factor [19,23,25], and kurtosis [4,19,23,25]. in the frequency domain, features are derived through discrete fourier transform analysis and non-parametric power spectral a monitoring method of milling chatter based on optimized hybrid... 229 density estimation. key metrics in this domain include spectral mean [19,25] and spectral centroid frequency [19,25]. time-frequency domain methods have gained significant prominence due to their capability to simultaneously capture temporal and spectral characteristics. these methods encompass short-time fourier transform [1,10,21], continuous wavelet transform [1,26], wavelet packet transform [9,14,27,28], variational mode decomposition (vmd) [29,30,31], and multi-synchronous compression transform [4,19,31]. jauhari et al. [30] introduced bayesian optimization to the vmd parameter selection process, achieving automatic optimization of decomposition parameters. in addition to the above features, some special features have also been applied to chatter monitoring and achieved good results. thaler et al. [24] found that deterministic features extracted from recurrence plots can effectively monitor chatter. whether the recurrence plot can accurately reflect the dynamic characteristics of the cutting system depends on the setting of its hyperparameters. to address this issue, chen et al. [32] proposed an adaptive particle swarm optimization algorithm to calculate the hyperparameters and automatically obtain the recurrence plot. threshold methods or intelligent classification algorithms are often used to identify chatter. many chatter indicators have been proposed to detect chatter, such as entropy [14, 23,30], energy ratio [19], and statistical indicators [18,28,29,33]. however, most existing indicators are directly or indirectly related to machining parameters, leading to changes in the indicators with changes in machining parameters and making early chatter detection difficult. albertelli et al. [18] developed an automated chatter threshold determination method based on statistical process control theory. lu et al. [29] selected the correlation coefficient between chattersensitive intrinsic mode functions as a chatter indicator. this indicator is not easily affected by machining parameters. zhao et al. [33] accumulated the information content in the maximum information entropy features calculated from the sequence to obtain a chatter indicator. the threshold based on two risk levels overcomes the limitations of a single threshold and improves the reliability of monitoring. yang et al. [28] established a fractionalorder energy entropy probability model for chatter detection, overcoming the difficulty of manually adjusting thresholds. the most widely used intelligent classification algorithms are support vector machines [15,23,25,29], logistic regression [34], and random forest [19,14,34]. wan et al. [25] improved the support vector machine using the adaptive boosting algorithm, which improved the accuracy of chatter monitoring. deep learning has demonstrated powerful feature learning and classification capabilities, attracting widespread attention and gradually being applied to the field of chatter detection. sener et al. [26] proposed a chatter detection method based on a deep convolutional neural network. this method introduced cutting parameters such as cutting depth and spindle speed into the dcnn model, effectively improving the test accuracy. transfer learning approaches [34,35,36] have been successfully applied to chatter detection to leverage knowledge from source domains to target domains. kounta et al. [35] proposed a transfer learning framework based on pre-trained vgg16 and resnet50 models, which achieves high-precision detection of chatter in industrial machining processes through the extraction and classification of fft image features. the sft is employed to filter out the spindle rotation frequency and its multiples. then, by combining time-frequency domain methods, time-domain features, frequency-domain features, and time-frequency domain features related to the machining state are obtained. subsequently, the missa algorithm is derived by optimizing the sparrow search algorithm through a combination of strategies including improved circle chaos mapping, golden sine strategy, and enhanced levy flight. a mscnn module integrated with spatial attention 230 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang mechanism is constructed to adaptively extract critical spatial feature information at different scales. next, bilstm is utilized to handle long-term dependencies in sequences and capture temporal feature information. then, by integrating the multi-head attention mechanism (msa), the contribution degree of extracted features is effectively grasped, and the features are emphasized and strengthened, leading to the mscnn-bilstm-atm chatter identification model. finally, the missa algorithm is used to optimize the hyperparameters in the model, thereby improving the accuracy of chatter recognition. 2. signal preprocessing and feature extraction 2.1. signal preprocessing the periodic components (spindle rotation frequency and its multiples) generated by spindle rotation and intermittent contact between the tool and workpiece are the main components of the machine tool processing acquisition signals. however, when chatter occurs in the system, non-periodic components caused by regenerative effects will be generated, and these components will gradually approach the natural frequency of the machine tool. but in the early stages of chatter, the frequency amplitude is very small and often submerged in the rotation frequency, making it impossible to detect it from the signal in time. for example, in the initial stage of chatter, the energy proportion of periodic components at low frequencies still accounts for more than 80% [24], greatly increasing the difficulty of monitoring the initial stage of chatter. therefore, in order to achieve highprecision and efficient identification in the initial stage of chatter, it is particularly important to remove the spindle rotation frequency and its multiples. compared to traditional notch filters that only remove a single frequency, the proposed sft can eliminate multiple related frequencies, effectively highlighting the initial chatter frequency submerged in the frequency spectrum. firstly, the spindle rotation frequency and its multiples that need to be removed are identified. secondly, a fast fourier transform (fft) is applied to the collected signal, and the sections of the spectrum corresponding to the spindle rotation frequency and its multiples are removed. finally, an inverse fast fourier transform (ifft) is conducted on the defrequenced spectrum. the specific algorithm steps are outlined below. step 1: perform fft on time series {x{k}, k=1,2,…,m} to obtain spectrum x(i) as shown in eq. (1). the length of the spectrum is m=length(x) 1 ( ) ( ) 1,2, , m ki m k x i x k w k m = = = (1) where, j is an imaginary unit and 2 expki mw j m   = −    . step 2: set the spindle rotation frequency fz that needs to be removed. its doubling frequency is cfz (c=1,2,…,n). the calculation of spindle rotation frequency is shown in eq. (2). 60z sf n= (2) where, ns is the spindle speed. step 3: considering the fluctuation of fft transformation frequency, introduce c1 floating constant with a value of 0.5. set the spectral amplitude of the interval corresponding a monitoring method of milling chatter based on optimized hybrid... 231 to frequency (fz-c1,fz+c1) in the spectrum to zero, and the specific calculation is shown in eq. (3). 1 1 1 1 0 ( ) ( ) ( 1) z z z z cf c i cf c x i x i c f c i cf c −   + =  − +   − (3) step 4: if i a is greater than c∙fz+c1 and less than n/2, make c=c+1 and repeat steps 2 to step 4. otherwise, proceed to the next step. step 5: because the spectrum is symmetric about the center, let x(m+1-i)=x(i), i=1,2,…,m/2, to obtain the complete spectrum after removing the frequency. step 6: perform ifft on the spectrum to obtain the time-series after frequency reduction, and calculate as shown in eq. (4). 1 ( ) ( ) m ki m i x k x i w − = = (4) 2.2. feature extraction comprehensively and accurately obtaining the state characteristics during the machining process is the prerequisite to precise identification of milling chatter. currently, time-domain methods, frequency-domain methods, and time-frequency domain methods are commonly used to obtain state characteristics during the machining process. the vibration signals, cutting force signals, and torque signals of the milling process have nonlinear and non-stationary characteristics [13]. these signals contain a large amount of state information from the machining process and are a comprehensive reflection of various excitation signals. since time-domain and frequency-domain features are only calculated from one domain of the data, important features with high resolution are discarded, while time-frequency domain methods such as wavelet analysis can solve the above problems. in this paper, time-domain features, frequency-domain features, and time-frequency domain features are selected as state features, as shown in table 1. 3. missa the traditional sparrow search algorithm (ssa) has the problem of being easily trapped in local optima. this article optimizes the sparrow search algorithm by improving the circle chaotic mapping, golden sine strategy, and enhanced lévy flight strategy to improve the optimization accuracy. 3.1. improving the circle chaotic mapping the ssa population is initialized using the improved circle chaotic mapping to ensure a uniform distribution, as follows: 1 0.7 mod(3.85 0.4 sin(3.85 ),1) 3.85 n n nx x x  + =  + −    (5) where, xn+1 is the value of introducing singer chaotic mapping, n is the dimension of the solution. 232 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang table 1 time frequency domain characteristics category feature expression feature expression time domain peak to peak value (ptp) 1 max mini ix x x= − root mean square (rms) 2 2 1 1 n i i x x n = =  absolute mean (abs) 3 1 1 n i i x x n = =  standard deviation (std) ( ) 2 4 3 1 1 n i i x x x n = = − kurtosis (kur) 4 3 5 1 4 1 n i i x x x n x=  −  =      skewness (ske) 3 3 6 1 4 1 n i i x x x n x=  −  =      peak factor (paf) 7 2 max ix x x = pulse factor (puf) 8 3 max ix x x = waveform factor (waf) 2 9 3 x x x = clearance factor (cf) 10 2 2 max( )ix x x = frequency domain mean of frequency spectrum (mfa) 11 1 1 ( ) n i i x p f n = =  gravity frequency of frequency spectrum (gffs) 1 12 1 ( ( )) ( ) n i i i n i i f p f x p f = =  =   mean square frequency of frequency spectrum (msfs) 1 13 1 ( ( )) ( ) n i i i i n i i f f p f x p f = =   =   root mean square frequency of frequency spectrum (rmsfs) 14 13x x= frequency spectrum variance (fsv) 2 15 1 1 (( ( )) ( ) n n i i i i i x f fc p f p f = = = −  ） time frequency wavelet energy entropy 2 16 ( , ) 1 n p j i i x x = = 3.2. golden sine strategy add the golden sine strategy when updating the discoverer's position, and the update formula is shown in eq. (6). , 1 2 1 1 2 , 21 , , 2 sin( ) sin( ) , , t t t i j best i jt i j t i j x r r r x x x x r st x x q l r st +   +    −   =  +   (6) a monitoring method of milling chatter based on optimized hybrid... 233 with 1 2 (1 (( 5 1) / 2)) 2 (( 5 1) / 2) 2 x x      = − + − −   = − + −  (7) where, t is the current iteration number, r1[0,2], r2[0,], respectively determine the distance and direction of individual position iteration movement, x1 and x2 are the gold coefficients. 3.3. enhanced lévy flight strategy the lévy flight step size factor is usually a fixed value. when its value is set to a larger number, although the global search ability is enhanced, a high-precision solution cannot be obtained. when its value is set to a smaller number, more iterations are required to find the theoretical value of the algorithm, resulting in reduced efficiency. to address this issue, a lévy flight mechanism with enhanced search is proposed as follows. the calculation of the dynamic step size factor is given below: max max ( ) sinh(1 ) t t t r t t  = −  (8) where, t is the current iteration number, tmax is the maximum iteration number, r is the adjustment parameter. by plotting the function curve of eq. (8), we can observe that the range of α(t) is [0, 0.261]. multiple experimental results have shown that when r=3.86, the variation condition of α(t) at [0,1] can be met, and the enhanced levi flight strategy can find the optimal value. although the new positions generated through lévy flight search can escape from local optima, it cannot be guaranteed that the updated positions are evenly distributed around the optimal position. therefore, a gaussian random distribution function is added as a constraint factor in the lévy mechanism to make the particle population more evenly distributed in the exploration space. the definition of the constraint factor is shown in the following equation: (0, )n = (9) where, δ is a scale parameter with a value of t/tmax. the final enhanced levi's flight expression is shown below: 1 ( )levy t        =   (10) with 1 1 ( ) 2 (1 ) sin( ) 2 1 ( ) 2 2         −    +   =   +       (11) where, β is 1.5, μ and v follow a normal distribution. the position update of the joiner is shown in eq. (12). , 1 2 , 1 1 , exp( ), 2 , t worst i j t i j t t t p i j p x x n q i x i x x x a l levy otherwise + + + +  −   =   + −    (12) 234 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang 4. chatter monitoring model 4.1. mscnn cnn is a feedforward neural network inspired by biology that is mainly composed of convolutional layers, pooling layers, and fully connected layers. the convolutional layer mainly extracts implicit feature information from the data through convolutional operations and then uses activation functions to introduce nonlinear factors into the features, enhancing the expressive ability of the features. the operation process of the convolutional layer is shown in eq. (13). , , 1 ,1 1 ( ) i j i j l ml l l i jj i x x w b+ = = =  +  (13) where, xl+1 i,j is the feature of layer l+1 after convolution calculation, w l i,j is a convolutional kernel, x l i,j is the j-th feature value of the i-th feature map, b is the bias term, and l is the size of the convolutional kernel. the time-domain vibration and cutting force signals in cutting machining often show the characteristics of multiple time scales. when the cutting chatter occurs, its characteristics usually also show the nature of multi-scale. to effectively extract the milling chatter component and enhance the recognition model effect, this paper designs a model based on the inception module and optimizes its model structure. the feature extraction ability under different scales is enhanced, as shown in fig. 1. fig. 1 mscnn structure a monitoring method of milling chatter based on optimized hybrid... 235 the size of the convolution kernel in each layer of the multi-scale convolutional neural network is fixed, and the number of channels is a parameter to be optimized. a channel attention mechanism is added to each convolutional neural network module to generate weights for each channel and continuously optimize and update the network parameters during the training process. the multi-scale convolutional neural network can continuously optimize the processing status data to obtain the importance of different features, strengthen key features based on their importance, suppress interference information, and improve the diagnostic efficiency and effectiveness of the model. 4.2. bilstm bilstm fully considers the relationship between the past and future of time series data and can effectively extract the temporal characteristics of signals for time series data with high correlation and strong periodic changes. the data enters the bilstm through the input layer, and a value is obtained through forward calculation of lstm, while another value is also obtained through backward calculation of lstm. therefore, the value in the hidden layer is determined by these two values. the specific calculation process is shown in eq. (14). 1 1 ( ) ( ) ( ) t x t t h n t x t t h n t y t y t y h f w x h w b h f w x h w b y f w h w h b − −  =  +  +  =  +  +  =  +  + (14) where, f is the activation function, w is the weight and bias term, th is the output of the forward layer, th is the output of the backward layer, and yt is the final output. 4.3. attention mechanism 4.3.1. channel attention mechanism the channel attention mechanism mainly consists of squeeze and excitation operations, as shown in fig. 2. firstly, the feature map undergoes the squeeze operation, compressing the two-dimensional feature map into a real number through global pooling to obtain global features. then, the excitation operation is performed on the global features, and a weight value is generated by passing the obtained global features through two fully connected layers and a sigmoid function. finally, the weight value is multiplied by the original feature map to obtain the final feature. the calculation process is as follows. 1 1 1 ( ) ( , ) h w c sq c c i j z f i j h w   = = = =   (15) 2 re 1( , ) ( ( ))ex sigmoid lus f z w w w z = =  (16) ( , )c scale c c c cx f u s s u= =  (17) 236 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang fig. 2 channel attention mechanism 4.3.2. multihead self-attention (msa) the multi-head attention mechanism is composed of multiple self-attention structures and is used to simultaneously process the same feature information. its output is the concatenation of multiple self-attention results. the formula is as follows. 1( , , ) ( , , ) o hmultihead q k v concat head head w=  (18) ( , , )q k v i i i ihead attention qw kw vw= (19) where, w q i , w k i and w v i are the weights of the mapping matrix, and wo is the output weight matrix. 4.4. missa-mssnn-bilstm-msa chatter monitoring model the structure of the mscnn-bilstm-atm chatter monitoring model constructed in this paper is shown in fig. 3. there are many hyperparameters in the mscnn-bilstm-atm model, such as learning rate, regularization parameter, number of neural network layers, number of convolutional layers, and batch size, which will affect the training effect of the model. therefore, the missa algorithm is introduced to optimize the hyperparameters in the model so as to improve its accuracy. the accuracy of the validation set samples is used as the fitness function maxf for the missa algorithm to optimize the hyperparameters of the mscnn-bilstm-atm model. 1 j min max min max 2 2 min 2 max min max ( ) 100 max numf [numf ,numf ] [ , ] [ , ] [ , ] n i vaild i j j j y y f n lr lr lr l r l r l r layers layers layers =  −   =           (20) a monitoring method of milling chatter based on optimized hybrid... 237 fig. 3 mscnn-bilstmatm structure 5. milling experiment 5.1. experimental setup the vdl-1000e three-axis cnc machine tool produced by dalian machine tool is used. the tool is a flat-bottomed milling cutter with four teeth and a diameter of 10mm. the workpiece material is tc4 titanium alloy, and the workpiece size is 200×200×5mm. a pcb accelerometer with a sensitivity of 10.42mv/g is used, and the donghua dh5922 acquisition system is used to collect the acceleration signals during the machining process. the sampling frequency is 5000hz. the milling mode is down milling and dry cutting. the experimental site is shown in fig. 4. the simulation experiment uses the windows10 (64bit) operating system, with hardware platform parameters of intel(r) core(tm) i9-12900k cpu, nvidia geforce rtx3080, 3.2 ghz clock speed, and 32 gb memory. the computing environment is matlab r2023b. 238 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang (a) experimental site (b) vibration data acquisition fig. 4 experimental setup 5.2. experimental parameter settings the modal test was conducted using a force hammer with a sensitivity of 3.41 pc/n to obtain the natural frequency of the titanium alloy thin-walled component, which was 864 hz. the cutting parameters and machining conditions selected based on the stability lobe diagram drawn in reference [13] are shown in table 2. the radial cutting depth is 0.5 mm, and the feed rate is 0.1 mm/tooth. table 2 machining state with different conditions cutting parameters machining state cutting parameters machining state no. rotating speed (r/min) axial depth (mm) no. rotating speed (r/min) axial depth (mm) 1 750 6.8 stable 13 1050 7.2 slight chatter 2 750 7.2 14 1050 7.5 3 900 5.9 15 1200 5.1 4 900 6.3 16 1200 5.4 5 1050 6.0 17 750 8.2 severe chatter 6 1050 6.4 18 750 8.5 7 1200 4.3 19 900 7.6 8 1200 4.7 20 900 7.8 9 750 7.9 slight chatter 21 1050 8.0 10 750 8.0 22 1050 8.2 11 900 7.0 23 1200 5.8 12 900 7.1 24 1200 6.5 a monitoring method of milling chatter based on optimized hybrid... 239 (a) stable cutting original signal (b) slight chatter original signal (c) severe chatter original signal (d) stable cutting filtered signals (e) slight chatter filtered signals (f) severe chatter filtered signals (g) spectrum of stable cutting (h) spectrum of slight chatter (i) spectrum of severe chatter (j) spectrum of stable filter signal (k) spectrum of slight chatter filter signal (l) spectrum of severe chatter filter signal fig. 5 comparison of data preprocessing results 240 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang 6. result analysis 6.1. data preprocessing results the sft technique is adopted to filter the time-frequency signals obtained from milling, removing the signal components related to the rotational speed and obtaining other disturbance components. the filtering results of the acceleration signals in the x direction under the conditions of n3, n12, and n20 are shown in fig. 5. fig. 5(a), (b) and (c) display the original vibration signals. fig. 5(d), (e) and (f) display the filtered vibration signals. fig. 5(g), (h) and (i) display the fft spectra of the original vibration signals. fig. 5(j), (k) and (l) display the fft spectra of the filtered vibration signals. as can be seen by comparing fig. 5(a), (b), and (c), the machining vibration gradually increases with unstable machining conditions. as shown in fig. 5(d), (e) and (f), the amplitude of the filtered time-domain signals decreases to varying degrees. fig. 5(g), (h), and (i) indicate that the spectra of the original stable cutting signals and slight chatter are mainly concentrated around the spindle rotational frequency and its multiples. the maximum spectral value of severe chatter is concentrated near the natural frequency of the cutting system. as shown in fig. 5(j), (k), and (l), the sft technique can effectively eliminate periodic components. after removing the periodic components, the chatter frequency becomes more prominent. the harr wavelet is selected as the basic wavelet function for wavelet packet decomposition. the original signal is decomposed into 8 frequency bands through 3-level wavelet packet decomposition. the wavelet packet decomposition results of the acceleration signal in the x direction under the n12 condition before and after signal processing are shown in fig. 6. as can be seen from fig. 6, the wavelet packet coefficients of each frequency band are inconsistent. the energy of the original signal is mainly concentrated in the frequency range of (625, 937.5], accounting for 44.94%. the energy of the original signal after removing the main frequency and its multiples is still concentrated in the frequency range of (625, 937.5], accounting for 62.20%. removing the main frequency and its multiples from the signal enhances the significance of chatter characteristics. the time-frequency domain analysis is conducted on the vibration signals with the main frequency and its multiples removed to obtain the time-frequency domain characteristics under different working conditions (see table 2 in detail). the box plot of some timefrequency domain characteristics is shown in fig. 7. it can be seen from fig. 7 that the five indicators of std, rms, abs, ske, mfa, ske and kur basically follow the same pattern, with their mean values gradually increasing as chatter occurs. the data for stable cutting and slight chatter are relatively more concentrated, while the data for severe chatter are more scattered. the two indicators of fsv and gffs follow basically the same pattern, with their mean values gradually decreasing as chatter occurs. the data for slight chatter are relatively more scattered. the values of waf, ptp and paf under slight chatter and severe chatter conditions are basically similar, and are all larger than those under stable milling conditions. therefore, based on the extracted signal characteristics and distribution patterns, it is theoretically possible to identify these three milling conditions through machine learning. a monitoring method of milling chatter based on optimized hybrid... 241 (a) original signal wavelet packet coefficients b) filtered wavelet packet coefficients (c) proportion of original signal energy (d) proportion of filtered signal energy fig. 6 comparison of results before and after signal processing 242 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang fig. 7 distribution pattern of time-frequency domain characteristics 6.2. performance analysis of missa to validate the optimization performance of the missa algorithm, simulations were conducted on benchmark functions in comparison with sparrow search algorithm (ssa) [37], improved grey wolf optimizer (igwo) [38], improved sparrow search algorithm (issa5) [39], aquila optimizer (avoa) [40], and snake optimization algorithm (so) [41]. this study selected f1, f2, f4, f8, f17, f18, and f22 from the cec2005 test suite as the benchmark functions. the population size was set to 30, and the maximum number of iterations was 500. the dimensions of f1, f2, f4, and f8 were 30, while the dimensions of f17 and f18 were 2, and the dimension of f22 was 4. each experiment was independently conducted 30 times, and three indicators, namely the optimal value (opv), mean, and standard deviation (std), were used to evaluate the solution accuracy and stability of the 0 5 10 15 20 25 (a) std slight chatterstable 0.8 1.0 1.2 1.4 1.6 1.8 2.0 (b) waf severe chatter 20 30 40 50 60 70 80 90 100 (c) ptp 3 4 5 6 7 8 (d) paf 2 4 6 8 10 12 14 16 (e) abs 0 5 10 15 20 25 (f) rms 4 6 8 10 12 14 (g) puf -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 (h) ske 150000 200000 250000 300000 350000 (i)fsv 0.00 0.01 0.02 0.03 0.04 0.05 (j) mfa 0 2 4 6 8 10 12 14 16 (k) kur 360 380 400 420 440 460 480 500 520 (l)gffs a monitoring method of milling chatter based on optimized hybrid... 243 proposed algorithm. the comparison results are presented in table 3. the computational time of each algorithm is shown in fig. 8. table 3 comparison of algorithm accuracy and stability function index ssa issa5 igwo avoa so missa f1 opv 1.34e-153 1.52e-182 5.61e-63 0 1.72e-195 0 mean 6.46e-84 2.106e-180 3.80e-61 0 2.34e-192 0 std 1.94e-83 0 5.98e-61 0 0 0 rank 4 3 5 1 2 1 f4 opv 3.5e-80 6.32e-78 5.39e-13 0 3.31e-87 6.75e-148 mean 2.27e-41 3.52e-76 1.512e-11 3.59e-298 2.94e-85 3.74e-130 std 6.73e-41 3.52e-76 2.19e-11 0 3.80e-85 1.12e-129 rank 5 4 6 1 3 2 f8 opv -1063.29 -938.03 -980.2 -1063.29 -1062.52 -1063.36 mean -1063.29 -870.55 -831.51 -1063.29 -1034.05 -1063.30 std 6.46e-4 38.32 80.83 0 34.60 2.278e-2 rank 3 6 5 2 4 1 f17 opv 397.89e-3 397.89e-3 397.89e-3 397.89e-3 397.89e-3 397.89e-3 mean 397.89e-3 397.89e-3 397.89e-3 397.89e-3 397.89e-3 397.89e-3 std 0 0 0 0 0 0 rank 1 1 1 1 1 1 f18 opv 3 3 3 3 3 3 mean 3 3 3 3 3 3 std 0 0 0 9.53e-8 0 0 rank 1 1 1 1 1 1 f22 opv -10.40 -10.40 -10.40 -10.40 -10.40 -10.40 mean -10.40293 -10.40294 -10.40294 -10.40294 -10.40133 -10.40294 std 4.34e-5 0 0 0 2.62e-3 0 rank 2 1 1 1 3 1 according to the simulation experimental results in table 3, the missa algorithm demonstrates the best performance in terms of optimization stability and solution accuracy on the benchmark test functions f1, f4, f8, f17, f18, and f22. on the f1 test function, the missa algorithm has the same optimization capability as the avoa algorithm, but its average computational time is only 40.41% of the avoa algorithm (as shown in fig. 8). on the f17 test function, all six algorithms exhibit the same optimization performance. the average computation time of the missa algorithm is only 25.03%, 78.16%, 5.71%, 38.15%, and 98.17% of the ssa, issa5, igwo, avoa, and so algorithms, respectively. on the f18 test function, all six algorithms also exhibit the same optimization performance. the average computation time of the missa algorithm is only 26.83%, 76.84%, 5.32%, 35.01%, and 90.68% of the ssa, issa5, igwo, avoa, and so algorithms, respectively. on the f22 benchmark test function, the missa algorithm has the same optimization capability as the issa5, igwo, and avoa algorithms. the average computation time of the missa algorithm is only 59.47%, 8.52%, and 53.18% of the issa5, igwo, and avoa algorithms, respectively. on the f4 test function, although the missa algorithm ranks second, its optimization capability is not significantly different from the avoa algorithm. the average computation time of the missa algorithm is only 50.98% of that of the avoa algorithm. based on the comprehensive comparison of the experimental 244 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang results, it can be concluded that the missa algorithm has higher convergence accuracy, better stability, and shorter computation time, which significantly outperforms the other five algorithms. fig. 8 computational time of different algorithms 6.3. detection of milling chatter with a population size of 30 and a population iteration count of 15, the missa algorithm was used to solve eq. (20). the optimal learning rate for the mscnn network was found to be 0.043949. the numbers of channels for the convolutional kernels were determined to be 16, 18, 16, 32, and 32, respectively. the regularization parameter was set to 6.7562×10-4, and the number of hidden layer neurons in the bilstm was set to 31. all data were divided into a training set and a test set, with proportions of 80% and 20%, respectively. the recognition results for the training set and test set under the three states are shown in fig. 9. according to fig. 9(a) and (b), the recognition accuracy of the test set is 99.5413%, with only one slightly chatter data being recognized as stable cutting. from fig. 9(c) and (d), the recognition accuracy of the test set is 96.4286%, with one slightly chatter being recognized as stable cutting and one severe chatter being recognized as slightly chatter. a monitoring method of milling chatter based on optimized hybrid... 245 (a) training data (b) the confusion matrix of the training set (c)test data (d) the confusion matrix of the test set fig. 9 identification results of different processing states the t-sne algorithm was used to visualize the features extracted from different network layers of the missa-mscnn-bilstm-atm model, and the results are shown in fig. 10. fig. 10(a) demonstrates the t-sne visualization effect of the test set data samples at the input layer of the model. it can be seen that various data samples are mixed with each other, resulting in poor clustering effects. fig. 10(b) presents the visualization results after feature extraction by the mscnn layer. although the data is basically separated, there are still many data points mixed together, indicating poor convergence. fig. 10(c) shows the visualization results after feature extraction by the bilstm layer. it can be observed that data samples of the same class have basically converged together, but there are still a few data points mixed in. fig. 10(d) is the visualization result after feature extraction by the atten layer. it is evident that all data samples have been completely classified into three categories, corresponding to the three processing states. therefore, the missa-mscnnbilstm-atm model can effectively classify milling states. 246 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang (a) input layer (b) mscnn layer (c) bilstm layer (d) msa layer fig. 10 visualization results of classification model to validate the reliability and superiority of the model, comparisons were made with dcnn [26], adaboost svm [25], msst-rf [19], and ilr-dnn [7]. accuracy, precision, recall, f1 score, and specificity were used as performance indicators to evaluate the state classification model. to avoid errors, the average results of ten repeated experiments were taken as the final evaluation results. the comparison results for the training set are shown in fig. 11, and the comparison results for the test set are shown in fig. 12. according to fig. 11, the proposed method achieved an average accuracy of 99.48%, which was a 2.1% increase compared to dcnn, a 1.57% increase compared to adaboost svm, a 2.62% increase compared to msst-rf, and a 1.05% increase compared to ilrdnn. additionally, the average precision of the proposed method was 99.59%, which represented a 1.83% improvement over dcnn, a 1.24% improvement over adaboost svm, a 2.43% improvement over msst-rf, and a 1.2% improvement over ilr-dnn. furthermore, the average recall rate of the proposed method was 99.43%, an increase of 2.3% compared to dcnn, 1.69% compared to adaboost svm, 2.73% compared to msstrf, and 0.9% compared to ilr-dnn. the average f1 score of the proposed method was 99.51%, showing a 2.11% improvement over dcnn, a 1.53% improvement over adaboost svm, a 2.6% improvement over msst-rf, and a 1.08% improvement over ilr-dnn. finally, the average specificity of the proposed method was 99.69%, representing a 1.11% increase compared to dcnn, a 0.86% increase compared to adaboost svm, a 1.36% increase compared to msst-rf, and a 0.44% increase compared to ilr-dnn. a monitoring method of milling chatter based on optimized hybrid... 247 fig. 11 comparison of training set results fig. 12 comparison of test set results according to fig. 12, the proposed method achieved an average accuracy of 97.38%, representing a significant improvement of 6.47% compared to dcnn, 6.29% compared to 248 h. gao, h. shen, c. yue, r. li, s. y. liang, y. wang, w. liu, y. yang adaboost svm, 8.49% compared to msst-rf, and 4.85% compared to ilr-dnn. additionally, the average precision of the proposed method was 97.76%, an increase of 7.35% over dcnn, 4.21% over adaboost svm, 10.26% over msst-rf, and 7.28% over ilr-dnn. furthermore, the average recall rate of the proposed method was 97.13%, a notable enhancement of 5.94% compared to dcnn, 9.87% compared to adaboost svm, 4.82% compared to msst-rf, and 2.26% compared to ilr-dnn. the average f1 score of the proposed method was 97.4%, showing a substantial improvement of 6.96% over dcnn, 9.93% over adaboost svm, 9.44% over msst-rf, and 5.73% over ilr-dnn. finally, the average specificity of the proposed method was 98.58%, representing a notable enhancement of 2.81% compared to dcnn, 3.58% compared to adaboost svm, 3.12% compared to msst-rf, and 1.61% compared to ilr-dnn. based on the comprehensive analysis of both the training and test sets, the method proposed in this paper achieved the best performance in terms of machining state classification. 7. conclusions this research first obtained the sensitive features of machining status based on sft technology and time-frequency domain analysis methods. then, an missa algorithm integrating improved circle chaos mapping, golden sine strategy, and enhanced lévy flight strategy was proposed. subsequently, an mscnn-bilstm-atm machining status monitoring model was constructed. finally, the missa algorithm was used to optimize the hyperparameters in the monitoring model. the test and experimental results show that the proposed methods have achieved high recognition accuracy. the specific conclusions are as follows. using sft technology to remove the spindle rotation frequency and its multiples from the vibration signal can effectively improve the significance of chatter features. in the numerical experiments of cec2005 complex functions, compared with other intelligent algorithms, the missa algorithm exhibited the best optimization stability and solution accuracy. the simulation running time was the lowest, only 5.32% of other intelligent algorithms. accuracy, precision, recall, f1 score, and specificity were used as performance indicators to evaluate the status classification model. the proposed chatter monitoring model achieved an average accuracy of 97.38%, an average precision of 97.76%, an average recall of 97.13%, an average f1 score of 97.14%, and an average specificity of 98.58% on the test set. compared with dcnn, adaboost svm, msst-rf, and ilr-dnn, the proposed method improved the average accuracy by up to 8.49%, average precision by up to 10.26%, average recall by up to 9.87%, average f1 score by up to 9.93%, and average specificity by up to 3.58%. further validation of the proposed methodology across diverse machining conditions is necessary. in the current investigation, only vibration signals were employed for analysis, and the development of a multi-sensor feature fusion framework could potentially enhance detection precision. the conventional time-frequency feature extraction approach necessitates manual preprocessing procedures, which become labor-intensive when handling large-scale datasets. a systematic feature importance evaluation framework can be established for automated selection of discriminative diagnostic features. a monitoring method of milling chatter based on optimized hybrid... 249 acknowledgement: this research was funded by the henan province young backbone teachers support program in higher 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crusher mounting facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 73 83 doi: 10.22190/fume161013006m © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper a theoretical-experimental approach for elasto-damping parameters estimation of cone inertial crusher mounting udc 622.7 rosen mitrev 1 , simeon savov 2 1 technical university, mechanical engineering faculty, sofia, bulgaria 2 mining and geology university, sofia, bulgaria abstract. the present paper deals with estimation of the elasto-damping parameters of a cone inertial crusher mounting. the numerical values of these parameters are crucial for accurate reproduction of the machine vibrational behavior and dynamical model adequacy. due to the significant difficulties arising during the purely theoretical determination of the stiffness and damping parameters of the rubber vibroisolators it is well-suited to use a theoretical-experimental approach. the developed approach is based on the theoretical determination of the mounting stiffness parameters as a function of two experimentally measured natural frequencies of the mechanical system. the crusher is represented as a six degrees of freedom system with two planes of symmetry. by using the system characteristic polynomial, the theoretical derivation of mathematical relationships for the mechanical system natural frequencies as a function of stiffness, inertial and geometrical parameters is performed. a good agreement is shown when comparing the experimental and the theoretical results for the system kinematical characteristics. key words: cone inertial crusher, modal analysis, parameters estimation 1. introduction in recent years, in some cases of their application, the cone inertial crushers have been established as the only option for an effective and low energy milling of mineral raw materials [1]. the design feature that makes them particularly preferred is the absence of a rigid kinematical connection between the driving system and the internal crushing cone. that is why this type of crushers is protected against damage in the case of an unbreakable object’s entering their crushing chamber. received october 13, 2016 / accepted december 26, 2016 corresponding author: rosen mitrev technical university, mechanical engineering faculty, kliment ohridski 8 blvd., sofia, bulgaria e-mail: rosenm@tu-sofia.bg 74 r. mitrev, s. savov fig. 1 shows the kinematical scheme of a cone inertial crusher. an electric motor (pos. 7), v-belt transmission (pos. 6), elastic rubber coupling (pos. 5), universal joint (pos. 4) and adjustable eccentric weight (pos. 3) realize the driving of the internal crushing cone (pos. 2). the outer crushing cone is rigidly mounted in the machine housing (pos. 1) placed on four rubber vibroisolators (pos. 8), rigidly connected to the ground. in the machine non-operating condition, the vertical axes of the outer and the internal cones and the eccentric weight axis of rotation are aligned, while in the operating mode this is not true due to the internal cone precession. 1 2 3 7 8 4 5 6 9 10 fig.1 kinematical scheme of the cone inertial crusher generated by the eccentric weight, the centrifugal force causes the internal cone to travel along the inner surface of the outer cone. the inertial force of the internal cone in addition to the inertial force of the eccentric weight generates a crushing force and as a result the two cones are pressed toward each other and the mineral material (pos. 10) is fragmented. the change of the rotational speed of the adjustable eccentric weight causes change of the crushing force and the crusher operational characteristics. when an unbreakable object enters the crushing chamber, the internal cone stops its movement, but the eccentric weight continues to rotate and thus the machine is protected against damage. a general trend in the theoretical investigation of the cone inertial crushers is the study of their motions and kinematical characteristics through dynamical models that allow the mathematical description of the machine behavior in 3d space [2]. a crucial step during the dynamical models generation is the estimation of the numerical values of the elastodamping, geometrical, force and inertial parameters that provide the best match between the model and the real world machine behavior. obviously, to achieve maximum accuracy, numerical values of the parameters should be set appropriately. there is a variety of methods for identification of the mechanical dynamical systems and estimation of their parameters values [3, 4, 5], a significant part of which are based on the study of the system properties in the time or frequency domain. while the geometrical and inertial parameters of the machine are determined relatively easily by use of cad models, direct measurements or calculations, the estimation of the mounting elasto-damping parameters is accompanied by significant difficulties. the results, received by use of theoretical relationships or computer simulation are unreliable, a theoretic-experimental approach for elasto-damping parameters estimation of cone inertial.... 75 especially when the machine mounting is realized by rubber vibroisolators. typically, the characteristics of the rubber vibroisolators are in general: nonlinear [6], changing over time due to ageing [7], dependant on the temperature [8] and vibration frequency [9] or are described by complex models depending on a set of experimentally determined constants [10]. in the engineering practice, the problem is overcome by the use of appropriately organized experimental research for elasto-damping parameters estimation. one commonly used method [11] is the experimental measurement of the most important characteristics of any mechanical oscillating system the values of the natural frequencies and the subsequent estimation of the rigid body properties and stiffness parameters by the use of theoretical relationships derived from the conventional modal analysis of the system. the main goal of the present paper is to develop and validate a theoretical-experimental approach for estimation of elasto-damping parameters of the cone inertial crusher mounting. for this purpose, we should obtain theoretical relationships that relate stiffness, geometrical and inertial parameters of the system with natural frequencies that can be easily determined experimentally. the present paper is structured as follows: section 1 comprises analyzed papers concerning some problems of estimation of the mounting system parameters and the goal of the paper is defined; section 2 presents a developed dynamical model of the crusher which takes into account the geometrical and inertial symmetry of the system; section 3 is devoted to the theoretical investigation of the oscillating system and development of mathematical relationships for natural frequencies of the system. experimental setup, based on industrial crusher of type kid-300 and experimental measurements are presented in section 4. section 5 presents the results of the developed approach together with validation. section 6 comprises the conclusion. 2. dynamical modeling for the purposes of the present study, the cone inertial crusher is represented in 3d space as a rigid body with 6 dof (pos. 1), mounted on four fixed to the ground similar rubber vibroisolators (pos. 2), see fig. 2 а) and b)). the crusher has two vertical planes of geometrical and inertial symmetry and the vibroisolators are mounted symmetrically according to these planes – fig. 2 b). every vibroisolator is represented by a set of three linear elasto-damping elements whose directions are aligned with the global coordinate system axes. to the ground is connected a global immovable cartesian coordinate system оxyz and to the rigid body gravity center is fixed a cartesian coordinate system cxyz whose axes are principal axes of inertia of the body. in the static equilibrium, position origins o and c of the coordinate systems are coincident and the vector of generalized coordinates measured according to the equilibrium position is: [ ]tx y z   q (1) where by x, y and z are denoted the linear translations of body mass center c, and by ψ, θ and φ are denoted body rotations according to the corresponding axes, see fig. 2. 76 r. mitrev, s. savov x ψ 1 y θ φ z lz cz bz cy by cx bx o,c 2 cy by cx bx l xy o,c φ x y l xy l xy l xy а) b) fig. 2 dynamical model of the crusher: а) 3d view, b) top view undamped free vibrations of the system are represented by the following linear system of homogeneous ordinary differential equations of second order: 0 mq cq (2) where m is the mass matrix of the system, c is the stiffness matrix, and by [ ]tx y z   q is denoted the vector of generalized accelerations. kinetic t and potential p energies of the system are positive definite quadratic forms of the correspondingly generalized velocities [ ]tx y z   q and generalized coordinates – eq. (1): 1 2 t  t q mq (3) 1 2 p  t q cq (4) under assumption of small motions, the elements of mass matrix m and stiffness matrix c are computed as: 2 ,i j i j t m q q     (5) 2 ,i j i j p c q q     (6) performing the differentiations in eqs. (5) and (6) and taking into account the geometrical and mass symmetry of the system, for the mass and stiffness matrix we obtain [2]: ( , , , , , )xy xy zdiag m m m j j jm (7) a theoretic-experimental approach for elasto-damping parameters estimation of cone inertial.... 77 2 2 2 2 2 4 0 0 0 4 0 0 4 0 4 0 0 0 0 4 0 0 0 0 4 0 2 4 0 0 4 0 0 0 2 4 0 0 0 0 0 0 4 xy xy z xy xy z z xy z xy z z xy xy z xy z z xy xy xy c c l c c l c c l l c l c c l l c l c l c                      c (8) in eqs. (7) and (8) are used the following notations: m – mass of the crusher housing (pos.1, fig.1) together with the contained movable elements (pos. 2, 3, 4 in fig. 1); jx=jy=jxy and jzthe mass moments of inertia of the crusher according to coordinate system cxyz axes; cx=cy=cxy and cz stiffness coefficients of vibroisolators in x, y and z directions; lz – distance in z direction between the gravity center and the point of connection of the vibroisolator to the ground; lxy distance in x and y directions between the gravity center and the point of connection of the vibroisolator to the ground. 3. determination of the system natural frequencies and mode shapes matrices m and c can be exploited to determine dynamical matrix h of the system and corresponding characteristic polynomial f(λ) [12]:  -1 h m c (9) ( ) det( )f   h i (10) where i is the identity matrix. after performing mathematical operations in eqs. (9) and (10) for the characteristic polynomial we obtain: 2 2 2 2 2 ( 2 ) ... ( ) ( 4 )( 4 ) ... 4 ( ( ) 2 ) xy xy z z z xy xy xy xy z xy z m j l c f m c j l c c j ml l c                   (11) roots λi of characteristic polynomial f(λ) are the eigenvalues of matrix h and are equal to squared natural frequencies ωi of the mechanical system under consideration: 2 2 1 2 1 2 1 2( ) ( )x y xy z k k mj j          (12) 2 3 3 4 ( )z zc m    (13) 2 2 1 2 4 5 4 5( ) ( ) xy z k k mj j           (14) 2 2 6 6 4 ( ) xy xy z l c j    (15) 78 r. mitrev, s. savov where: 2 2 1 (2 ( ) )z xy xy z xy zk j c j ml ml c   , 2 2 2 2 2 2 (2 ( ) ) ... ... 8 xy xy z xy z z xy xy z xy c j ml ml c k j ml c c j          , and 1 x , 2 y , 3 z , 4  , 5  , 6  are the natural frequencies associated to the corresponding coordinates. as can be seen from eqs. (12) and (14), due to the symmetry of the system there are two repeated roots and that is why the system has only four unique natural frequencies. eigenvectors ui corresponding to natural frequencies ωi are determined from the following equation: ( ) 0i i h i u (16) modal matrix ф of the system is composed of the eigenvectors 1 2 3 4 5 6[ ]φ u u u u u u (17) and for the real world machine parameters take the form: 0.93 0 0 0 0.03 0 0 0.93 0 0.03 0 0 0 0 1 0 0 0 0 0.38 0 1 0 0 0.38 0 0 0 1 0 0 0 0 0 0 1                       φ (18) the review of the modal matrix shows that the translation along z axis and rotation around z axis are uncoupled while the translations along x and y coordinates and rotations around y and x axes are coupled. providing experimentally measured values for two unique natural frequencies and knowing the inertial and geometrical parameters of system, eqs. (12-15) can be used for determination of unknown stiffness parameters cxy and cz. a total of six combinations in pairs are available for four unique natural frequencies, namely combinations 1 3 1( , )x z  , 1 4 2( , )x   , 1 6 3( , )x   , 3 4 4( , )z   , 3 6 5( , )z   , 4 6 6( , )   . after the determination of the values of natural frequencies of a particular combination, using eqs. (12-15) the corresponding system of equations can be solved analytically or numerically for unknown stiffness parameters. least computational complexity has combination 3 6 5( , )z   for which from eqs. (13) and (15) we obtain: 2 3( ) 4 z z m c   (19) 2 6 2 ( ) 4 z z xy xy j c l   (20) due to the large weight of the machine and its design features, a particular problem in using this combination are the difficulties during excitation of oscillations along z-axis and around z-axis. the easiest way to excite oscillations is along axis x (or y) and the a theoretic-experimental approach for elasto-damping parameters estimation of cone inertial.... 79 coupled oscillations around y (or x) axis and thus values of the natural frequencies for second combination 1 4 2( , )x   can be determined. these values along with the eqs. (12) and (14) can be used for calculation of cxy and cz. the system has two solutions and due to the vibroisolator design, one must choose the solution for which is fulfilled cz>cxy: 1 2 24 z n n c l m   (21) 1 2 24 xy n n c l m   (22) where 2 1 4 1( )x xyn m j   , 2 2 2 2 2 2 2 2 4 1 4 1( (( ) ( ) ) ) 4 ( ) ( )x x xy xy zn m j j m l       . 4. experimental setup and measurements for determination of natural frequencies 1 x and 4  an experimental study is conducted. the used experimental setup (fig. 3) is based on the industrially used cone inertial crusher of type kid-300. the numerical values of the experimental setup parameters are shown in table 1. table 1 numerical values of the experimental setup parameters parameter value and dimension mass of the crusher vibrating parts m=838.6 kg inertial moments jxy=61.44 kg.m 2 , jz=51.52 kg.m distances lxy=0.486 m, lz=0.3 m, l=0.212 m mass of the eccentric weight mv=43.1 kg radius of rotation of the eccentric weight e=0.03 m angular velocity of the eccentric weight ω=113 rad/s, 157 rad/s x ψ y θ φ z l o a b sax say sby a) b) fig. 3 scheme of the experimental setup and accelerometers layout 80 r. mitrev, s. savov in fig. 3 a) and b) the mounting positions of the used piezoelectric accelerometers are shown. at point a, situated on the housing and on local x axis, is mounted one two-axial accelerometer whose axes are directed along x and y axes of the local coordinate system – fig. 3 b). the second accelerometer is uniaxial and is mounted also on the housing in point b situated above point a. its axis is directed along local y axis. accelerometermeasured linear accelerations sax, say and sby are used for calculation of the generalized accelerations along coordinates x and ψ under the small motion assumption: axx s (23) by ays s l    (24) the vibration signal captured for 4 seconds which occurred as a result of an artificial impulse excitation along x axis is shown in fig. 4 a). in the same fig. the signal after the high frequency filtering is shown. the value of natural frequency 1 x is determined as shown in fig. 4 b) fft spectral analysis of the vibration signal. similarly, two other signals are captured and according to eq. (24) the angular acceleration is computed and natural frequency 4  is determined. the received values for the two frequencies are 1 17.71 rad/sx  and 4 64.10 rad/s  . additionally, the filtered acceleration signal, shown in fig. 4 a) is used to determine vibroisolator damping parameter bx. exponentially decaying consecutive amplitudes of the acceleration suggest predominantly viscous damping in the system and asymptotically stable motion [13]. in fig. 5 the filtered acceleration signal and decaying exponential curve, which is a regression line for amplitudes peak points, are shown. the exponential curve is described by eq. (25), where the constants determined by the least-squares method are a0=1.209 m/s 2 and n=-0.804. 0 nt ex a e (25) 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x a c c e le ra ti o n , m /s 2 time, s experiment filtered experiment 0 1 2 3 4 5 6 7 8 9 10 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 frequency, hz a m p li tu d e a) b) fig. 4 measured vibration signal a) and spectral analysis b) a theoretic-experimental approach for elasto-damping parameters estimation of cone inertial.... 81 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x", m/s2 time, s filtered experiment peak points exponential curve fig. 5 filtered acceleration signal and exponential curve if along x axis the system is considered as that of a single degree of freedom, then the total for all four vibroisolators damping coefficient is determined as: 2xyb mn  (26) and its value is 1348.8 ns/mxyb  and correspondingly, the damping coefficient for one vibroisolator is bxy=337.2 ns/m. similarly, the measured value for damping coefficient in z direction is bz=1039 ns/m. 5. numerical example and discussion to illustrate and validate the developed approach, the stiffness parameters of the experimental setup mounting are determined. if the already experimentally determined values of natural frequencies 1 17.71 rad/sx  and 4 64.10 rad/s  are substituted in eqs. (21) and (22) then the values of the stiffness coefficients are computed as 472.3 kn/mzc  and 74.4 kn/m xy c  for a single vibroisolator. according to eqs. (13) and (15) the stiffness values are used for computation of the two other unique natural frequencies. the computed values are 3 47.46 rad/sz  and 6 36.93 rad/s  which are very close to the experimentally measured values 46.60 rad/s and 36.91 rad/s correspondingly. to validate the dynamical model and check the reliability of the estimated values of the stiffness and damping parameters additional experimental measurements are carried out. the system excitation is performed by the generated due rotation of the eccentric weight force – pos. 3, fig. 1. in this case, the eccentric weight rotation causes a harmonic excitation and the resultant motion of the system is described by the following system of equations:   mq bq cq f (27) where b is the damping matrix, received in a formal way by replacing symbol “c” in c by symbol “b”, and f is a vector of generalized forces: 82 r. mitrev, s. savov [ 0 0 0 0]t x yf ff (28) in eq. (28) fx and fy denote the harmonic excitation forces generated by the eccentric weight rotation: 2 sin( )x vf m e t  (29) 2 cos( )y vf m e t  (30) where mv is the eccentric weight mass, e is eccentricity of the weight, ω is the eccentric weight angular velocity. in fig. 5 the captured experimental vibration signal is shown as well as the numerical solution for acceleration of system of eqs. (27) in which the determined values of the stiffness and damping coefficients are used. 0.40 0.44 0.48 0.52 0.56 0.60 -60 -40 -20 0 20 40 60 a cc el er at io n x a x is , m /s 2 time, s experiment theory a) 0.20 0.24 0.28 0.32 0.36 0.40 -20 -10 0 10 20 30 a cc el er at io n x a x is , m /s 2 time, s experiment theory b) fig. 6 graphs of numerical and experimental vibration signal: a) ω=157 rad/s, b) ω=113 rad/s a theoretic-experimental approach for elasto-damping parameters estimation of cone inertial.... 83 the experiments are performed for two typical values of the angular velocities in the crusher operating range ω=157 rad/s and ω=113 rad/s. as can be seen, the experimental and numerical results show good agreement both in amplitude and in frequency and this is confirmation of the determined values correctness and adequacy of the dynamical model. 6. conclusion the significant difficulties arising during the purely theoretical estimation of the elasto-damping parameters of the cone inertial crusher mounting can be successfully overcome by using the developed theoretical-experimental approach. the single performed experiment allows by capturing vibration signals of the suitable mounted one two-axial and one uniaxial accelerometers to determine two unique natural frequencies of the crusher. by using eqs. (12-15), eq. (21) and eq. (22) along with measured natural frequencies, the stiffness parameters and two other unique frequencies can be calculated. because of good agreement has been shown when comparing the experimental and theoretical results it can be concluded that the developed approach is suitable for estimation of the elasto-damping parameters of a cone inertial crusher mounting. the approach could be easily applied to other mechanical structures with two planes of symmetry according to the vertical planes. references 1. blazy p., zarogatsky l., jdid e., 1994, hamdadou m., vibroinertial comminution — principles and performance, international journal of mineral processing, 41(1-2), pp.33-51. 2. savov s., 2014, research of mechanical and technological parameters of cone inertial crushers type kid, ph.d thesis, mining and geology university “st. ivan rilski”, sofia, 221 p. 3. savov s., 2016, experimental determination of natural frequencies of cone inertial crusher (kid-300), bulgarian journal for engineering design, issue 30, october 2016, pp.5-10. 4. ljung l., 1999, system identification: theory for the user, englewood cliffs, nj: prentice-hall, 672 p. 5. grigorov b., mitrev r., 2016, dynamic behavior of a hydraulic crane operating a freely suspended payload, j zhejiang univ-sci a, in press, doi: 10.1631/jzus.a1600292 6. bruns j.-u., lindner m., popp, k., 2003, identification of the nonlinear restoring force characteristic of a rubber mounting, proc. appl. math. mech., 2, pp. 270–271. doi:10.1002/pamm.200310120 7. dashevskij m., motorin v., mironov e., samoilenko t., 2003, engineering design of rubber pads ageing properties: theory and experiment, constitutive models for rubber iii, proceedings of the third european conference on constitutive models for rubber, 15-17 september 2003, london, uk. 8. tachibana, e., li, k., 1996, temperature dependence of high damping rubber in base-isolated structures, proceedings of the 11th world conference on earthquake engineering, acapulco, mexico. paper no.492. oxford: pergamon. 9. koblar, d., boltezar, m., 2013, evaluation of the frequency-dependent young's modulus and damping factor of rubber from experiment and their implementation in a finite-element analysis, experimental techniques, published online 11 november 2013, doi: 10.1111/ext.12066 10. feng, w., hallquist, j., 2012, on mooney-rivlin constants for elastomers, 12th international ls-dyna users conference 2012, pp.1-10. 11. malekjafarian a., ashory r.m., khatibi n.m., latibari m.s., 2011, a new method for estimation of rigid body properties from output-only modal data, international operational modal analysis conference, iomac’11. 12. strommen e., 2014, structural dynamics, springer series in solid and structural mechanics, vol. 2, 510 p., doi: 10.1007/978-3-319-01802-7 13. meirovitch l., 2001, fundamentals of vibrations, mcgraw–hill higher education, new york, 806 p. http://www.sciencedirect.com/science/article/pii/0301751694900043 http://www.sciencedirect.com/science/article/pii/0301751694900043 http://www.sciencedirect.com/science/article/pii/0301751694900043 http://www.sciencedirect.com/science/article/pii/0301751694900043 http://www.zju.edu.cn/jzus/iparticle.php?doi=10.1631/jzus.a1600292 http://www.zju.edu.cn/jzus/iparticle.php?doi=10.1631/jzus.a1600292 http://www.springer.com/series/10616 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 133 139 reverse engineering of the human fibula by the anatomical features method  udc 658.5+611 milica tufegdžić 1 , miroslav trajanović 2 , nikola vitković 2 , stojanka arsić 3 1 mechanical engineering and electrotechnical school, kruševac, serbia 2 faculty of mechanical engineering, university of niš, serbia 3 faculty of medicine, university of niš, serbia abstract. this paper describes reverse engineering (re) of the human fibula, on the right male bone, by using the method of anatomical features (maf) with the aim to obtain a 3d surface model. the first step in the process of reverse engineering is ct scanning and digitalization of data. ct data are obtained with the toshiba msct scanner aquillion 64 and saved in the dicom format. they are subjected to further processing and imported in the computer aided design (cad) program as a stl file. the process continues in the cad program with identification and determination of the referential geometrical entities (rges) which are crucial for re process. the rges are the basis for defining the axis and planes of intersection. the intersecting polygonal model of the human fibula, namely upper and lower extremities and the body with these planes, results in a set of curves used for determining points on the given planes. through these points the splines are pulled, and with loft function surface models of extremities and the body of fibula is built. joining and merging of these models leads to 3d shape model of fibula. the model accuracy is confirmed by conducting distance and deviation analysis. the model is suitable for rapid prototyping, reconstruction of the missing parts of fibula, orthopedic training and simulation. key words: reverse engineering, geometrical model human fibula, 3d surface model  received november 29, 2013 corresponding author: miroslav trajanović faculty of mechanical engineering, aleksandra medvedeva 14, 18000 niš, serbia e-mail: miroslav.trajanovic@masfak.ni.ac.rs acknowledgements: this paper is part of project iii41017 virtual human osteoarticular system and its application in preclinical and clinical practice, funded by the ministry of education, science and technological development of republic of serbia, for the period of 2011-2014. 134 m. tufegdžić, m. trajanović, n. vitković, s. arsić 1. introduction production of geometrical models of the existing objects is necessary in many industrial areas. unlike the traditional forward engineering (also called "direct engineering"), in which products are manufactured on the basis of the previously prepared technical documentation, re can provide a computer-based reproduction of an object or a product without design [1, 2]. the application of re in the field of medicine and dentistry is resulting in biomedical objects or implants with adequate properties for the biomedical needs. the examples are: different types of implants (personalized, dental, artificial hip joints), external orthopedic prostheses, bony tissue scaffolds, [3]. another field of application of re in medicine includes visualization, diagnostic (diagnosis), surgery planning, surgical templates, production of the artificial organs, training and teaching [4, 5]. the objective of this paper is to present a re process of the human fibula, based on the anatomical features method (maf), described in [5]. the activities involved in our modelling approach are: 1) ct scanning, 2) polygonal model building and healing, 3) determination of rges, 4) creation of a 3d surface model of fibula extremities and body, 5) 3d surface model assembling, and, 6) verification of obtained model. these activities are presented at fig. 1. this process is the result of the improvement of the earlier process presented in [6]. fig. 1 activities in re of the human fibula reverse engineering of the human fibula by using method of anatomical features 135 2. material and methods 2.1. ct scanning re process starts with the acquisition of three dimensional shape data of the human body structures. the common systems used in medical imaging to obtain anatomical information are: x-ray, computer tomography (ct), magnetic resonance imaging (mri), ultrasound system (us), mammography, radiography (plain x-ray), laser digitizer and digital fluoroscopy, as discussed elsewhere [4, 7, 8, 9, 10, 11, 12]. in this case we present re of a 61-year-old healthy male fibula. input data are a series of traverse ct images, obtained on the toshiba msct scanner aquillion 64 (120kv, 150 mas), with the following parameters: thickness 1 mm, in-plane resolution 0.781  0.781 (pixel size), acquisition matrix 512  512; field of view (fov) 400  400 mm. all ct data are saved in the dicom format. 2.2. polygonal model building and healing the initial 3d fibula point cloud is established via masks creating, region growing, calculation, and remeshing of 3d objects [13]. this model is imported in re software in the form of stl (stereolitography) format and subjected to specific operations for eliminating the model errors (isolated triangles, nonmanifold vertices or edges, etc.). after removing all unnecessary entities, polygonal model was created. operations such as healing, optimization and mesh smoothing are conducted with the aim of improving the polygonal model. this model is suitable for determination of rges in the cad program. 2.3. determination of rges 3d shape model of the human fibula is created by using rges described in [14, 15] as the basis for definition of the bone 3d geometry (curves, polygons, surfaces). rges represent the geometrical entities (points, lines, axes and planes) created in accordance to the anatomical landmarks, [5]. fibula is a paired long bone and, and like all long bones, has two extremities (upper and lower) and the body. it is placed on the lateral side of the leg. its upper extremity is articulated with the tibia, and its lower extremity with malleolar surface on the lateral side of the talus. for the purpose of our research, at the fibula we have defined the following rges described in [6]: 1) at the upper extremity (lat. epiphysis proximalis s. extremitas proximalis):  acf (lat.apex capitis fibulae) – apex of the fibular head,  facf (lat.facies articularis capitis fibulae) – articular surface for the articulation with the lateral condyle of tibia; 2) at the lower extremity (lat.epiphysis distalis s. extremitas distalis):  faml (lat.facies articularis malleoli lateralis) – articular surface for the articulation with the talar lateral malleolar surface;  aml (lat.apex malleoli lateralis 1 ) – top of the lateral malleolus, as the most distal point on the lateral malleolus. 1 it is not official term from terminologia anatomica. 136 m. tufegdžić, m. trajanović, n. vitković, s. arsić on the basis of the previously defined rges we have constructed:  the mechanical axis of the fibula which connects the centers on the articular surfaces of the upper (facf) and the lower extremity (faml) of the fibula, and,  a-p (anterior-posterior) plane defined by mechanical axis and acf. 2.4. creating 3d surface model of the human fibula the polygonal model of upper extremity of the fibula is intersected with the planes obtained by rotation of the plane passing through the mechanical axis of the bone and acf. the mechanical axis is taken for the axis of rotation, while the rotation angles are variable. in these intersections a set of curves is obtained and points on them are defined. through these points we have constructed the splines. using the loft function we have got a surface model of the upper extremity, as shown at fig. 2. the same methodology is used for generating a model of lower extremity of the fibula, but the polygonal model of the lower extremity is intersected with the plane which passes through mechanical axis and aml. for the rotation axis, the axis that connects acf and aml is taken, as the most distant points at the fibula. the lower extremity model is presented at fig. 3. fig. 2 surface model of the upper extremity fig. 3 surface model of the lower extremity (with splines) (with splines) for the body of fibula we have used 30 cross-sections perpendicular to the mechanical axis. we have obtained 30 curves of intersections and defined points on them. these points are used for constructing the splines. by loft function we have obtained surface model of the fibular body, presented at fig. 4. fig. 4 surface model of the fibular body reverse engineering of the human fibula by using method of anatomical features 137 3. results and discussion all surface models are joined and merged (merging distance of 0.001mm) and 3d surface model of the right fibula is obtained. this model is shown at fig. 5. fig. 5 3d surface model of the fibula we have created a polygonal model at 3d surface model with the aim of analyzing the differences between the re model and the initial stl polygonal model. the full distance analysis is computed in 3d, with discretization of 50, based on the chosen color range, and shown in selected color range at fig. 6. the obtained statistical results indicate that the most of the area is in the range of 0  0.083 mm, and we consider this result as very good. fig. 6 distance analysis between final fig. 7 deviation analysis between and initial polygonal model initial and final point of cloud 138 m. tufegdžić, m. trajanović, n. vitković, s. arsić for the purpose of deviation analysis we have exported our final surface model as a stl file. this model is suitable for rapid prototyping. the mean deviation is 0.00628 mm, standard deviation is 0.0654 mm. the range of mean deviation values is from 0.0432 mm to 0.0399 mm, while the range of maximum deviation is from 0.601 mm till 0.575 mm, which is another proof that our surface model is satisfactory. these results are presented at fig. 7. 4. conclusion judging by the results of verification which is conducted through distance and deviation analysis, we can conclude that the presented approach provides a 3d surface model of the human fibula with high accuracy and precision. as expected, the maf has proved to be suitable for reverse engineering of the human fibula. the resulting model is convenient for building of the solid model as well as for rapid prototyping of the bone. rges which are defined on the human fibula are important for development of the predictive parametric model, which represents the next step in our further research. references 1. l. m. galantucci, g. percoco, g. angelelli, c. lopez, f. introna,c. liuzzi and a. de donno, reverse engineering techniques applied to a human skull, for cad 3d reconstruction and physical replication by rapid prototyping, journal of medical engineering & technology, vol. 30, no. 2, march/april 2006, pp 102–111 2. l.c. hieu, j.v. sloten, l.t. hung, l. khanh, s.soe, n. zlatov, l.t.phuoc and p.d. trung, medical reverse engineering applications and methods, 2nd international conference on innovations, recent trends and challenges in mechatronics, mechanical engineering and new high-tech products development, mecahitech„10, bucharest, 23-24 september 2010, proceedings, pp 232-246 3. sh choi and hh cheung (2011). digital fabrication of multi-material objects for biomedical applications, biomedical engineering, trends in materials science, mr anthony laskovski (ed.), isbn: 978-953-307-513-6, intech, available from: http://www.intechopen.com/books/biomedical-engineeringtrends-in-materialsscience/digital-fabrication-of-multi-material-objects-for-biomedical-applications 4. pero raos, antun stoić and mirjana lucić, rapid prototyping and rapid machining of medical implants, 4th daaam international conference on advanced technologies for developing countries september 2124, 2005 slavonski brod, croatia 5. vidosav majstorovic, miroslav trajanovic, nikola vitkovic, milos stojkovic, reverse engineering of human bones by using method of anatomical features, cirp annals manufacturing technology 62 (2013) pp 167–170 6. trajanović, m., tufegdžić, m., arsić, s., veselinović, m., vitković, n., reverse engineering of the human fibula, 11th international scientific conference mma 2012 advanced production technologies, novi sad, 2012, pp 527-530 7. b. starly, z. fang, w. sun, a. shokoufandeh and w. regli, three-dimensional reconstruction for medical-cad modeling, computer-aided design & applications, vol. 2, nos. 1-4, 2005, pp 431-438 8. yumi iwashita, ryo kurazume, kahori nakamura, toshiyuki okada, yoshinobu sato, nobuhiko sugano, tsuyoshi koyama and tsutomu hasegawa, patient-specific femoral shape estimation using a parametric model and two 2d fluoroscopic images, accv'07 workshop on multi-dimensional and multi-view image processing, tokyo, nov., 2007, pp 59-65 9. yeon s lee, jong k seon, vladimir i shin, gyu-ha kim, and moongu jeon, anatomical evaluation of ct-mri combined femoral model, biomedical engineering online 2008, 7:6 doi:10.1186/1475-925x7-6 reverse engineering of the human fibula by using method of anatomical features 139 10. g. anastasi, g. cutroneo, d. bruschetta, f. trimarchi, g. ielitro, scammaroto, a. duca, p. bramanti, a. favaloro, g. vaccarino, and d. milardi, three-dimensional volume rendering of the ankle based on magnetic resonance images enables the generation of images comparable to real anatomy , j anat. 2009 november; 215(5): 592–599, epub 2009 aug 12. 11. p kalral, p beylot, p gingins, n magnenat-thalmann, p volino, p hoffmeyer, j fase, and f terrier, topological modeling of human anatomy using medical data, proc. computer animation '95, april 95, geneva, pp.172-180 12. paulo j. s. gonçalves and pedro m . b . torres, registration of bone ultrasound images to ct based 3d bone models, technology and medical science, crc press 2011, pp 245-250 13. sheng zhang, kairui zhang, yimin wang, wei feng, bowei wang, and bin yu, “using threedimensional computational modeling to compare the geometrical fitness of two kinds of proximal femoral intramedullary nail for chinese femur,” the scientific world journal, vol. 2013, article id 978485, 6 pages, 2013. doi:10.1155/2013/978485 14. stojkovic m, milovanovic j, vitkovic n, trajanovic m, arsic s, mitkovic m, (2012) analysis of femoral trochanters morphology based on geometrical model. journal of scientific and industrial research 71(3), pp 210–216 15. vitković, n., milovanović, j., korunović, n., trajanović, m., stojković, m., mišić, d., arsić, s., software system for creation of human femur customized polygonal models, computer science and information systems, vol. 10, no. 3, 1473-1497. (2013) reverzni inženjering ljudske fibule metodom anatomskih karakterstika u radu je prikazan reverzni inženjering (ri) ljudske fibule, na primeru desne muške fibule, pomoću metode anatomskih karakteristika u cilju dobijanja 3d površinskog modela. skeniranje kompjuterskom tomografijom i digitalizacija podataka predstavljaju prvi korak u procesu ri. za dobijanje podataka kompjuterskom tomografijom korišćen je toshiba msct scanner aquillion 64, a podaci su sačuvani u dicom formatu. podaci su obrađeni i uveženi u cad program u obliku stl datoteke. proces se nastavlja identifikacijom i određivanjem referentnih geometrijskih entiteta (rge) u cad programu. rge predstavljaju osnovu za definisanje osa i ravni preseka. presecanje gornjeg i donjeg okrajka, kao i tela poligonalnog modela fibule ovim ravnima ima za rezultat skupove krivih, na kojima su definisane tačke. kroz ove tačke provučene su prostorne krive, te su pomoću loft funkcije dobijeni površinski modeli okrajaka i tela fibule. spajanjem i stapanjem ovih modela dobijen je 3d površinski model fibule. tačnost modela je potvrđena analizama rastojanja i devijacija. model je podesan za brzu izradu prototipova, kreiranje delova fibula koji nedostaju, obuku i simulaciju u ortopediji. ključne reči: reverzni inženjering, rge, fibula, 3d površinski model plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. i i editorial foreword surely each one of you has encountered a situation which made you wonder about the importance of automatic control systems. probably the easiest way to answer this question is to imagine briefly the absence of automatic control. it does not take much to realize that, without all the modern devices and systems for whose operation automatic control is essential (the majority of electronic devices, heating or cooling appliances, means of transport, etc.), our everyday life would be reminiscent of ancient times. and, as a matter of fact, the roots of automatic control systems may even be traced back to ancient times. but it was with the industrial development that its importance actually came to the fore. particularly over the past couple of decades the field reached such a level of maturity that the development of intelligent systems, smart structures and similar systems that were perceived as futuristic not a long time ago, is now a reality. the scientific conference systems, automatic control and measurements (saum) gathers researchers in this attractive field of work. it also recognizes the multidisciplinary character of the field and, hence, works from the fields of modeling, measurements, system identification, robotics, management and industrial processes, etc., are also included in the conference scope. the backbone of this issue of facta universitatis series mechanical engineering are the selected papers presented at the saum 2016 conference hosted by the faculty of electronic engineering, university of niš, and organized in collaboration with the faculty of mechanical engineering, university of niš. the papers have been properly modified and extended to meet the journal standards. further articles originating from regular submissions are included in the issue so that hopefully a rather interesting, challenging and inspiring set of articles is offered to a vast audience. vlastimir nikolić guest editor dragan marinković editor-in-chief facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 639 651 https://doi.org/10.22190/fume200703037c © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper influence of a bird model shape on the bird impact parameters janusz cwiklak military university of aviation in deblin, institute of navigation, poland abstract. one of the factors which significantly exert a negative influence on flight safety is a collision of an aircraft with birds. various parts of an aircraft are subjected to damage. within the conducted analyses, the impact loaded object was a helicopter windshield. apart from the mandatory physical tests, there are various numerical methods for bird strike modeling. among them, in this paper, the smooth particle hydrodynamics (sph) is being used and developed for bird modeling. investigations exploit various geometric figures in order to model the bird shape. few authors present research findings which employ an approximate shape of certain bird species. for comparison three bird models were elaborated upon, one in the shape of a cylinder with hemispherical ends (homogeneous model) and two others as multi-material models, one in the shape of a simplified white stork and the other one close to the real-life white stork. multi-material bird models had various parameters. it must be noted that the maximum value of the resultant windshield displacement varies for different bird models. the bird model close to the real-life white stork caused the smallest deflection, while the bird model in the shape of a simplified white stork and the homogeneous bird model led to the biggest damage, respectively. it is important to add that the models are of the same mass, impact velocity and a different size. this has an impact on the kinetic energy distribution during the collision process, which results in different windshield bending values. key words: bird strikes, bird model shape, numerical simulation, sph 1. introduction flight safety is a vital issue for air transport. one of the factors which significantly exert a negative influence on flight safety is a possibility of an aircraft collision with birds and other objects [1, 2]. various parts of an aircraft are subjected to damage [2]. it appears that damage to the windshield is extremely dangerous. the consequence of penetrating the canopy can cause a serious injury to the pilot, disabling him to continue piloting an aircraft. it is important to underline the fact that apart from the mandatory physical tests, in order to received july 03, 2020 / accepted october 12, 2020 corresponding author: janusz cwiklak military university of aviation, institute of navigation, dywizjonu street 303/35, 08-530 deblin, poland e-mail: j.cwiklak@law.mil.pl 640 j. cwiklak meet certification requirements, there are various numerical methods for modeling bird strikes, for instance, the lagrangian approach, the arbitrary lagrangian eulerian (ale) approach and the smooth particle hydrodynamics (sph) method [3]. each has different advantages under certain circumstances [3, 4]. among them the sph is being developed for bird modeling [5, 6, 7]. the sph technique was exploited in computational fem codes in order to avoid limitations connected with severe mesh distortions during solving problems of large deformations [3, 4, 8]. the fundamental difference between the classic langrangian approach and the sph one is lack of any mesh. the particles themselves create an object and the equations are solved for them. for the accuracy of the sph technique, it is important for the particle distribution to be as regular as possible. besides, too large differences in distances among them should be avoided [3]. the sph method is used for simulating fluid flows. since the bird body contains approximately 90% water and is treated as a soft body, the method was chosen for the examinations presented in this paper. in order to perform the modeling of bird strikes, it is necessary to select a proper bird model. investigations exploit various geometric figures in order to model the bird shape, the most common one being a cylinder, an ellipsoid and a cylinder with hemispherical ends [3, 7, 8]. few authors present research findings, which employ an approximate shape of certain bird species on the basis of biometric data [5, 9, 10]. mccallum and constantinou compared a sph multi-material model of canadian goose weighing 3.6 kg to a hemispherical-ended cylinder [10]. they stated that a target may become pre-stressed from the initial impact of the head and the neck, prior to the impact of the torso. thus the shape of a bird may have an important consequence for damage initiation and failure of the target. the other numerical simulations regarding the above mentioned issue were conducted by nizmpatman [9]. three different multi-material bird models were investigated during the bird impact against a rigid target. the first one was hemispherical-ended cylinder built with two discrete materials of different densities that were randomly distributed throughout the bird torso. the second was of the same shape, but it included, among other features, a torso made of three elements material consisting of homogenous mixture of water and air as well as high density lumps to represent the main bone structure and low density lumps to represent soft tissue and lungs. the third model was the most realistic one. the model consisted of head, neck, torso, bones, lungs and wings. the mass of all bird models equaled 4 kg. to assure comparability to wilbeck`s experimental results the initial velocity was set to 150 m/s. the conclusion withdrawn from this work is that the realistic multi-material bird model provides the most detailed description of the impact load process and gives much more precise information on the contribution of each part of a bird to the impact load spectrum. within the conducted analyses, the impact loaded object was a helicopter windshield. the author selected the windshield of agusta a-109, manufactured by agusta westland concern [12]. the main criterion adopted for its selection was the availability of the geometric model cad, downloaded from grabcad free cross-platform source [13] as well as the accessibility of the strength parameters of the material, with which the windshield was produced. the legal regulations concerning the strength requirements for the helicopter glazing include the specification cs 29.631. according to these specifications, the windshield in the category of a heavy helicopter should withstand a bird strike whose mass equals 1 kg at the velocity of the rotorcraft equal to vne (‘never-exceed velocity’), and an altitude up to 2438 m [14]. an example of a comparison of experimental and numerical investigations regarding windshield (polycarbonate plate of 8 mm in thickness) was presented in studies [21]. a influence of a bird model shape on the bird impact parameters 641 body of dead chicken weighing 622 g was used as an impactor against a target. the initial velocity equaled 128 m/s. the contact force and the displacement`s values were measured and compared to numerical simulations. a strong deflection of the polycarbonate plate and an elastic spring were observed in both. it means that the polycarbonate plate is a very flexible material. a shockwave in polycarbonate plate is generated at the impact and can be observed on the frame of the windshield. generally, the results obtained from the experimental and numerical simulations were similar. the differences could result from using a homogenous bird model. therefore, a multi-material bird model should be investigated in numerical analyses of a bird impact. there are no requirements regarding the categories of light helicopters and airplanes. taking into account the bird strikes, in which the bird mass exceeded 1 kg, the windshield was damaged and the bird remains were in the cockpit [15], the author decided to examine similar cases in a numerical environment. the existing research concerning bird models of a complex shape mostly analyzed events, in which the bird mass was from 0.3 kg to 3.6 kg [5, 8, 10]. the collisions with a specific aircraft part did not undergo testing. the analyses mainly focused on the impact of the model with flat steel or an aluminum plate. therefore, it seems reasonable to conduct a numerical investigation of a selected aircraft element, e.g. a helicopter windshield. 2. methods 2.1. parameters of numerical analyses in order to conduct an analysis of a bird strike into an aircraft windshield, the author used the ls_dyna software package [11]. a quick explicit formulation has been chosen for the investigation [8, 11]. similar to the most of studied papers regarding bird strikes, the windshield was not loaded with initial air pressure and flight speed of bird was not included in the calculations. the velocity of the object impacting the windshield was determined on the basis of the helicopter cruise speed, which was equal to 285 km/h (79.167 m/s). this is the speed of the impacting bird model into a fixed windshield. the analysis time was taken to be from 10 to 20 ms depending from a kind of bird model. other parameters are listed in table 1. table 1 initial simulation parameters initial simulation parameters description element type of bird models sph elements element type of the windshield belytschko-tsay shell elements transformed to 8-node solid elements contact type automatic nodes to surface hourglass control flanagan-belytschko viscous form (ihq=2) coefficient (qm=0.14) bulk viscosity control quadratic viscosity coefficient (q1=2.0) linear viscosity coefficient (q2=0.25) time step 6e-6 s initial velocity 79.167 m/s analysis time 10-20 ms 642 j. cwiklak for all bird models the initial moment of the analysis is illustrated in fig. 1. the boundary particles of the bird models were positioned at an equal distance from the theoretical piercing point on the windshield. fig. 1 position of bird models at the initial moment of analysis for all bird models in order to immobilize the windshield, which corresponds to its fixing to the helicopter frame, all the nodes on the windshield edge were grouped in a set of nodes, and all the translational degrees of freedom in x, y, z directions were constrained. also the rotational degrees towards all the three axes of the global coordinate system were constrained. contact automatic nodes to surface were exploited. the function of the master segment was taken by the windshield whereas the slave segment was a bird model. the analyses took into consideration friction coefficient between the contact objects, which equaled 0.1. in order to avoid instability time step 6e-6 s was applied, which was obtained by the equation 1, based on courant-friedrichs-lewy (cfl) condition for sph method: 0.1 h t u   (1) where, 0.1 is constant factor, h particle spacing in sph, and u is the maximum velocity in the computation. 2.2. windshield model the analyses were conducted using the windshield model, built of 8-node solid elements. for this purpose, a generator of solid components in the pre-processor ls-prepost was used. it created a mesh of solid elements by adding thickness to the existing shell elements. thus, the number of elements was not changed (9339), yet the number of nodes doubled. in the properties, the default type of elements – elform = 1 was declared. table 2 material data used in windshield model [16] data of windshield material density young’s modulus poisson’s ratio yield stress failure strain tangent modulus hardening parameter [kg/m3] [pa] [–] [pa] [–] [pa] [–] 1.19 3.13·109 0.426 6.8·107 0.067 0.00 0.5 influence of a bird model shape on the bird impact parameters 643 the windshield of the helicopter agusta a-109 is one-layer made from acrylic glass [12]. this material is available under different trade names, depending on the manufacturer, e.g.: plexiglas™, perspex™. its main ingredient is polymethyl methacrylate – pmma. the material parameters of the windshield, which are given in study [16], are presented in table 2. an isotropic plastic material model, with the reinforcement modulus equal to 0, was exploited. the elements of the windshield model in which the failure strain for eroding elements exceeded 0.067, were removed from the model during the simulation. 2.3. numerical bird models for the comparison three bird models were elaborated upon presented in fig. 2; one is a homogenic model in the shape of a cylinder with hemispherical ends (a) and two others as multi-material models, one in the shape of a simplified white stork (b) and the other one close to the real-life white stork (c). multi-material bird models had various parameters. the dimensions of the bird models were set on the basis of an assumed bird mass (3.6 kg) and density of the material. for a cylinder with hemispherical ends, the author assumed the ratio of the model length and the diameter equal to 2:1. (d = 142.52 mm and l = 285.04 mm) whereas the number of the sph particles equaled 28784 [17]. for the stork model of a simplified shape, the mass equals 3.6 kg, where 70% of the mass is the bird torso. the densities of the material for the head, neck and torso were assumed on the basis of paper [10], in which the authors considered a goose, disregarding, however, the bird’s beak. the densities of these components should not deviate from each other too much. the densities equaled: 900 kg/m3 – head, 1.5 kg/m3 – neck, 1.15 kg/m3 – torso; moreover, the density of the wings was determined by the total mass of the bird model and was finally equal to 590 kg/m3. when generating the bird model, the following simplifications were assumed: ▪ the bird’s beak, legs, feet and the tail were disregarded; ▪ the shape of the torso and the head were assumed to be cylindrical; ▪ the shape of the neck was assumed to be cylindrical; and, ▪ the shape of the wings was assumed to be rectangular with hemispherical ends. taking into consideration the above data and the used simplifications, it was possible to obtain a bird model, whose weight of 3.604 kg was composed of 29972 sph particles. it was 455.51 mm long. its wing span measured 757.58 mm. when developing the shape of particular parts of the simplified stork model, the work [10] was taken into consideration. next, the author produced a stork model whose parameters were similar to the shape of a natural bird. thus, a shell stork model, which served as a basis for the sph model, was used. its parameters were as follows: ▪ mass = 3.6 kg; ▪ 68 % bird’s torso, legs and feet; ▪ 22 % wings; and, ▪ 10 % neck, head and beak. the developed model consists of 37638 sph elements. the bird’s length, from beak to tail, equals 991.83 mm and the wing span is 1534.15 mm. the biometric parameters correspond to an average stork size [18]. in general, the dimensions of a natural stork (length and wingspan) were twice bigger than the model of a simplified shape. while 644 j. cwiklak generating the sph particles, attempts were made to make similar distances between them in each of the three models. the distances equaled on average 6 mm. the bird model had an initial velocity (79.16 m/s). the vectors of velocity were applied to all particles of the model (sph), grouping the above-mentioned components in the so-called sets. the bird model used a “zero” material model. taking into account previous author's work [17] and results of the other researchers [3, 5, 9, 20] regarding exploiting of equations of state (eos) and material porosity, the author chose the grüneisen's equation of state. the equation defines the pressure in the shock-compressed material, as [11]: 20 2 0 02 2 3 1 2 3 2 1 1 2 2 ( ) , 1 ( 1) 1 ( 1) a p c a e s s s                   + − −        = + +     − − − −   + +     (2) whereas for the expanded material, as: 2 0 0( ) ,p c a e   = + + (3) (2) where: c is bulk speed of sound, 0 is grüneisen gamma, s1, s2 s3 are linear, quadratic cubic, coefficients, a is first order volume correction to 0,  is volume parameter, expressed as  = (/0) – 1,  is actual density, 0 is initial density, e is internal energy per unit of mass. the material data of bird models are listed in table 3. table 3 material and eos data used in bird models [16] material parameters of bird models density cut-off pressure viscosity coefficient relative volume for erosion in tension relative volume for erosion compression [kg/m3] [pa] [pa·s] [-] [-] "density varies for different bird model parts as given in section 2.3 –106 0.001 1.1 0.8 grüneisen's eos parameters bulk speed of sound linear coefficient quadratic coefficient cubic coefficient grüneisen's gamma [m/s] [–] [–] [–] [–] 1.438 1.92 0 0 0.1 influence of a bird model shape on the bird impact parameters 645 fig. 2 bird models used in numerical analyses (a) cylinder-shaped model with hemispherical ends, (b) bird model with a simplified shape, (c) real-life white stork bird model fig. 3 distribution of hugoniot and steady-flow stagnation pressure for various bird models (b) (a) (c) 646 j. cwiklak fig. 4 distribution of hugoniot and steady-flow stagnation pressure from wilbeck experimental test (left side) [19] and for a bird model in the shape of hemisphericalended cylinder (right side) [20] in order to perform a preliminary validation of the developed models, numerical analyses were carried out by making assumptions from wilbeck’s and koh’s works [19, 20]. the obtained courses of pressure are presented in fig. 3. the analysis of the curves (fig. 3) shows that their shape is similar to those obtained in the above-mentioned investigation (fig. 4). it is possible to distinguish hugoniot and steady-flow stagnation pressure. it can be seen that the maximum value of pressure (80 mpa) for koh’s hemispherical ended cylinder model is similar to that obtained in author’s analyses. moreover, it can be noted that the bird model with a simplified shape (b) and the real-life white stork bird model (c) reach the pressure peak of 53 mpa and 32 mpa, respectively. it means that the values of a peak of pressure obtained for the author’s multi-material bird models are closer to wilbeck’s experimental investigation (24 mpa). however, taking into consideration wilbeck’s experimental investigation [20], the hugoniot pressure is about three times smaller than in numerical simulations. wilbeck explains that this situation can be caused by unsuitable pressure transducers that might have been unable to capture the hugoniot pressure in a very short time. moreover, as hedayati states in [5], an additional reason for that can be difference between the initial contact area of the various shapes of bird models. 3. results as a result of the research, various simulations using the ls-dyna software package were conducted, including kinetic energy, impact velocity, displacement, impact forces, pressure and other. fig. 5 depicts a comparison of the windshield deformation resulting from impacts with the analyzed bird models. it can be noted that the character and the grade of the windshield damage depend on the bird model. it can be observed that the maximum value of the windshield displacement varies for different bird models (fig. 6). the bird model (c) caused the smallest deflection (31 mm), while the bird model (b) (34 mm) and the bird model (a) are the biggest ones (35 mm), respectively. it is important to add that the models have the same mass, impact velocity, but different sizes. in particular, the length ranges from 0.28 m (homogenic bird model), 0.45 m (simplified stork), to 0.99 m (real-life stork model). influence of a bird model shape on the bird impact parameters 647 fig. 5 windshield deformations caused by a bird impact, left column – bird model, cylindershaped with spherical endings, middle column – bird model with a simplified shape, right column – real-life white stork bird model fig. 6 resultant windshield displacement depending on a bird model fig. 7 pressure distribution depending on the bird model the courses of pressure for individual models differ among one another (fig. 7). for model (a), the pressure builds up until piercing the glass. in multi-material models (b) and (c), 648 j. cwiklak there are several leaps of pressure values, depending on the impact of individual parts of a bird. in model (b) – the first one corresponds to the impact of the head or rather the neck of an extremely high density, and the second one is related to the impact of the torso. in model (c), there are three characteristic peaks: the first peak occurs after an impact of the beak, the next with the pressure drops as a result of the head impact of lesser density, and finally the pressure grows again after an impact of a more muscular neck, and next of the body and wings. the pressure starts to rapidly fall at a time of glass penetration in each case. generally, the courses differ from one another because of the length of particular models, which is directly related to the time of the impact. as shown in fig. 5, the applied models have a different nature and the course of the windshield deformation. the length factor seems to have a considerable impact on the kinetic energy distribution during the collision process (fig. 8), which resulted in different windshield bending values. fig. 8 kinetic energy distribution depending on bird model table 4 shows maximum velocities, during which there is no windshield penetration, depending on the used model. it is interesting to note that these velocities are higher for multi-material models, especially for the model whose shape is similar to a real stork. in the event of a collision with a cylinder-shaped model with hemispherical ends, this velocity is the lowest and equals 195 km/h. on the other hand, an application of the model in the shape of a simplified stork increases this velocity by 235 km/h, i.e. 20%, and accordingly 266 km/h for the bird model whose shape is close to a real stork. the velocity is higher than for the previous models by 36% and 13%, respectively. table 4 permissible velocities of impact of a dummy bird, for which there is no penetration bird models velocity (km/h) homogeneous model (cylinder with hemispherical ends) 195 multi-material model (simplified white stork) 235 multi-material model (real-life white stork) 266 influence of a bird model shape on the bird impact parameters 649 fig. 9 depicts deflection of the glass depending on the velocity of the bird model whose shape is similar to a white stork. the analyses were carried out at velocities ranging from 74 to 79 m/s. the curve shapes are similar to the time 15 ms – the curves overlap whereas after that time there is a change in the deflection course of the glass, depending on the velocity. fig. 9 displacement of the windshield depending on the bird model velocity it is clear to observe a drop in deflection, which proves that the model is bounced from the windshield. at a velocity of 76, there is windshield penetration, which is expressed by maintaining deflection. however, at a velocity of 78 there is a further increase in the deflection, which means that the process of failure (penetration) of the glass is becoming deeper. the process of the windshield fracture can be well analyzed on the basis of the record of stresses. both by analyzing the map of stresses (fig. 10) and the curves (fig. 11), it is possible to identify the moment of breaking the glass. as seen in fig. 10, the stress values grow for each of the models to the maximum value. this is due to the windshield elasticity property. fig. 10 contours of effective stress depending on time of bird impact 650 j. cwiklak fig. 11 effective stress depending on bird models then, the penetration occurs and the value of stresses rapidly decreases. obviously, there is a difference in the process of the bird strike due to the shape of the models, particularly their length. 4. conclusion the paper presents the results of numerical analyses of three dummy birds, which differ in shape, during the impact process with the helicopter windshield. taking into account an analysis of the available subject literature as well as previous research, conducted by the author, the sph technique was selected for bird modeling. also grüneisen's equation of state was applied for this purpose. the author determined an influence of the bird shape and its dimensions on such analysis parameters as pressure, kinetic energy, windshield deflection, velocity at which the glass was penetrated. by comparing the results of the analyses of the above-mentioned parameters, it must be stated that length of the bird model has large influence on values of particular parameters, which directly affects the time of the bird strike process. in addition, the shape of a bird exerts an impact on the distribution of pressure. generally, the maximum pressures for all models are similar. however, their distribution is different at the time of an impacting process. it is important that the shape of the model affects the velocity of breaking the glass. using a more complex shape in comparison with a basic solid, e.g. a cylinder, leads to an increase in the maximum velocity of penetration from 20 to 36% depending on the model. taking into consideration above-mentioned conclusions and results obtained by other investigators, especially presented in [5, 8, 10, 21] using multi-material bird models with a realistic bird shape in numerical bird strike analyses of large birds provides more details about impact and gives much more precise information on the contribution of each part of the bird to the bird strike process. moreover, it is the only applicable method, since there are no experimental methods for investigating bird strike with a realistic bird model. therefore, a multi-material bird model with a realistic shape seems to be the most representative to analyze the bird strike process of large birds. influence of a bird model shape on the bird impact parameters 651 acknowledgements: the research was financed by the statutory research funds of the military university of aviation. references 1. vogt, r., adamski, m., głebocki, r., 2015, integrated navigation – flight control system of guided projectiles and bombs, journal of theoretical and applied mechanics, 53(1), pp.119-123. 2. dennis, l., lyle, d., 2009, bird strike damage & windshield bird strike final report, report european aviation safety agency, easa.2008.c49. 3. heimbs, s., 2011, computational methods for bird strike simulations: a review, computers and structures, 89, pp. 2093–2112. 4. lavoie, m.a., gakwaya, a., ensan, m.n., zimcik, d.g., 2007, review of existing numerical methods and validation procedure available for bird strike modelling, icces, 2(4), pp. 111-118. 5. hedayati, r., ziaei-rad, s., 2012, a new bird model and effect of bird geometry and orientation on birdtarget impact analysis using sph method, international journal of crashworthiness, 17(4), pp. 445-459. 6. grimaldi, a., sollo, m., guida, m., marulo f., 2013, parametric study of a sph high velocity impact analysis: a bird strike windshield application, composite structures, 96, pp. 616-630. 7. grimaldi, a., 2011, sph high velocity impact analysis a bird strike windshield application, phd thesis, department of aerospace engineering, university of naples federico ii, italy, 25 p. 8. nizampatnam, ls., 2007, models and methods for bird strike load predictions, phd thesis, wichita state university, usa, p. 54, 114. 9. hedayati, r., ziaei-rad, s., eyvazian, a., hamouda, a., 2014, bird strike analysis on a typical helicopter windshield with different lay-ups, journal of mechanical science and technology, 28(4), pp. 1381-1392. 10. mccallum, s.c., constantinou, c., 2005, the influence of bird-shape in bird-strike analysis, 5th european ls-dyna users conference, birmingham, united kingdom, pp. 2c-77. 11. livermore software technology corporation, 2007, ls-dyna keyword user’s manual, usa. 12. hetman, k., agusta westland a.109, available from: https://www.polot.net/en/pzl_swidnik_agusta_ a_109_2005r_/konstrukcja  [last access 10 july 2020). 13. ferreira, m., augusta a-109, available from:  https://grabcad.com/library/augusta-a-109-1 [last access: 15 july 2020]. 14. european aviation safety agency, 2007, certification specifications for large rotorcraft, cs-29, amdt 1, easa, 1-d-3 p. 15. air accidents investigation branch, 2012, serious incident n109tk, aaib bulletin, 3/2012, 12 p. 16. wang, f.s., yue, z.f., yan, w.z., 2011, factors study influencing on numerical simulation of aircraft windshield against bird strike, shock and vibration, 18, pp. 407-424. 17. cwiklak, j., 2020, numerical simulations of bird strikes with the use of various equations of state, journal of konbin, 50(3), pp. 333-345. 18. burnie, d., hoare, b., 2009, bird: the definitive visual guide, dk adult, first edition, london, england, u.k., 159 p. 19. wilbeck, j.s., 1978, impact behavior of low strength projectiles, report no. afml-tr-77-34, air force materials lab., air force wright aeronautical lab’s, wright-patterson air force base, 73 p. 20. chuan, k., c., 2006, finite element analysis of bird strikes on composite and glass panels, phd thesis, department of mechanical engineering, national university of singapore, 24 p. 21. plassarda, f., hereil, p., pierric, j., mespoulet, j., 2015, experimental and numerical study of a bird strike against a windshield, european physical journal web of conferences, 94, 01051. https://www.polot.net/en/pzl_swidnik_agusta_a_109_2005r_/konstrukcja https://www.polot.net/en/pzl_swidnik_agusta_a_109_2005r_/konstrukcja https://grabcad.com/library/augusta-a-109-1 https://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=utf8&field-author=david+burnie&text=david+burnie&sort=relevancerank&search-alias=books https://www.amazon.com/s/ref=dp_byline_sr_book_3?ie=utf8&field-author=ben+hoare&text=ben+hoare&sort=relevancerank&search-alias=books https://en.wikipedia.org/wiki/london plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. i iii © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd editorial foreword to the special issue: interplay of mechanics, tribology and materials science valentin l. popov1, khayrulla khudoynazarov2 1department of system dynamics and friction physics, technische universität berlin, germany 2institute of theoretical and applied mechanics, samarakand state university, uzbekistan this thematic issue contains selected papers related to presentations at the international workshop “interplay of mechanics, tribology and materials science”, samarkand state university, september 4-16, 2023, organized by the samarkand state university and technische universität berlin. 80 participants of the workshop represented germany, uzbekistan, russia, china, usa, and slovenia. the workshop was devoted to interdisciplinary research of mechanical and biological systems considering the interrelation and interplay of their mechanical, tribological and material aspects. in the last years, tribology has expanded to qualitatively new fields of applications, which are at the forefront of global development trends of technology and society, in particular microand nanotechnology as well as biology and medicine. the main challenge of tribology remains its interdisciplinary character which was in focus of the workshop. the goal of the workshop was to review the recently established concepts, tools and research activities and to outline the most important tasks for the future. the selected papers from the workshop cover a broad range of applications of contact mechanics in engineering, biology and medicine. they are devoted to the following topics: ▪ influence of tangential sliding on the contact area of a macroscopic adhesive contact (j. wilhayn, i.a. lyashenko, q. li, v.l. popov). based on numerical simulations with the boundary element method and experiments, the authors argue that the much-debated question of whether shear leads to an increase or decrease of contact area of an adhesive contact and contact strength, cannot be answered unambiguously as the answer depends on preloading history. ▪ in silico analysis of an articular cartilage regenerative rehabilitation under conditions of mesenchymal stem cells implantation and their mechanical stimulation (a. poliakov, v. pakhaliuk). the results of experimental studies and a number of successful clinical practices indicate that the treatment of joint diseases caused by damage to the articular cartilage is possible within the framework of a new medical direction regenerative rehabilitation, which synergistically combines the methods of regenerative and rehabilitation medicine. regenerative rehabilitation of articular cartilage defects involves the use of cellular technologies, the ii v.l. popov, kh. khudoynazarov effectiveness of which is enhanced by mechanical stimulation of chondrogenic cells, which accelerates their proliferation, differentiation, and formation of an extracellular matrix. ▪ triboinformatics: machine learning methods for frictional instabilities (m. nosonovsky, a.s. aglikov). the study of friction is traditionally a data-driven area with many experimental data and phenomenological models governing structure-property relationships. triboinformatics is a new area combining tribology with machine learning (ml) and artificial intelligence (ai) methods, which can help to establish correlations in data on friction and wear. ▪ modeling osteocyte under shock-wave therapeutic loading (a. smolin, g. eremina). dental implants have substantial significance in modern dentistry, but their osseointegration stage remains the most crucial and time-consuming step for compensating for a lost tooth. one promising approach to improve osseointegration rates is the use of extracorporeal shock wave therapy, which has shown efficacy in treating fractures, bone defects, and bone tissue regeneration in surgical and arthroplasty procedures. ▪ self-consistency conditions in static three-body elastic tangential contact (e. willert). the problem of "third body" is one of the central contemporary challenges of tribology 1. in the paper by willert, an elementary process of the third body dynamics is considered: the contact problem for an elastic third-body particle between two elastic half-spaces. the normal and tangential contact problems are analyzed analytically considering partial slip in the contacts and the influence of third-body weight. self-consistency conditions between global equilibrium and the contact solution are formulated to give criteria, under which circumstances static slip and stationary sliding are possible states for the third-body particle. ▪ selection of optimal technological parameters for forming nominally flat surfaces with lubricating microcavities (v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin). this article demonstrates a multi-step process for forming a nominally flat surface with circumferential lubricating microcavities on a tribological assembly of an х38crsi steel shaft. the process includes finish turning, preliminary strengthening burnishing, deformation profiling of microcavities by a honing stone and smoothing of microprotrusions. ▪ longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure (kh. khudoynazarov). three-layer structural elements are widely used in aviation and shipbuilding, construction of buildings and structures, the space industry and other industries. the paper considers a cylindrical three-layer structure of arbitrary thickness made of viscoelastic material. equations have been developed that enable to determine the stress-strain state at any point of an arbitrary section of a circular cylindrical three-layer viscoelastic structure based on the results of solving longitudinal-radial vibrations. ▪ contact interaction of an underground pipeline with soil under dynamic impacts (k. sultanov). underground pipelines serve as essential engineering life support lines for the population and industry, making them crucial for the economy of any country. hence, ensuring the strength and durability of underground pipelines is a critical and relevant problem. the paper shows that the longitudinal stress waves and their amplitudes in underground pipelines strongly depend on the law describing the process of (frictional) interaction of the underground pipeline with soil. ▪ experimental determination of dynamic coefficient of amonton-coulomb dry friction (i. mirzaev, kh. sagdiev, a. yuvmitov, m. turdiev, b. egamberdiev). the purpose of the study was to experimentally determine the dynamic coefficient of dry friction between different materials under vibrations due to the initial deflection. the dry friction between the grillage and the foundation is used for the seismic isolation of buildings and structures. a measuring foreword to the special issue iii complex with corresponding strain-gauge measuring channels was prepared on the laboratory one-component shaking stand to record the relative and absolute displacements of two-mass systems. ▪ interlayer effect of deformation and fracture of dendritic structure formed during wire-feed electron-beam additive manufacturing of al-si alloy (r. balokhonov, v. utyaganova, d. gatiyatullina, a. zemlianov, v. romanova). interfaces and surfaces play an important role in tribology, mechanics and materials science, causing plastic strain localization and stress concentration of different spatial scales. the interfacial inhomogeneity is highly pronounced in 3d printed materials due to thermo-cycling and layer-by-layer building. in this paper, the inlayer and interlayer structure of a eutectic alsi alloy fabricated by wire-feed electron-beam additive manufacturing is investigated by optical and electron microscopy. model structures inheriting the experimental morphology are created, and their deformation and fracture are simulated. ▪ sliding friction in contacts with oneand two-dimensional viscoelastic foundations and viscoelastic half-space (t. watanabe, s. hatanaka, k. nakano, v.l. popov). solid viscoelasticity is one of the essential origins of sliding friction, as every solid exhibits energy dissipation due to it during deformation processes. in this paper, first theoretical solutions for one-dimensional (1d) problems of viscoelastic friction with a 1d viscoelastic foundation have been found. then, the 1d model was extended to a two-dimensional (2d) model to find theoretical solutions for 2d problems of viscoelastic friction. finally, the method of dimensionality reduction (mdr) [2] has been applied to the theoretical solutions for the 1d problems to discuss three-dimensional (3d) problems of viscoelastic friction. references 1. popov, v.l., 2018, is tribology approaching its golden age? grand challenges in engineering education and tribological research. front. mech. eng. 4:16. 2. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer, 253 p. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 313 320 doi: 10.22190/fume1603313l original scientific paper tangential displacement influence on the critical normal force of adhesive contact breakage in biological systems udc (539.612) iakov a. lyashenko 1,2 department of system dynamics and the physics of friction, 1 berlin university of technology, berlin, germany 2 sumy state university, sumy, ukraine abstract. the dependencies of the critical components of normal and tangential forces corresponding to the contact breakage between a parabolic indenter and an elastic half-space have been determined taking into account adhesive interaction. in order to describe the adhesive contact, the method of dimensionality reduction (mdr) and the modified rule of heß taking into account tangential displacements have been used. the influence of the surface energy depending on the indenter separation angle has been studied. key words: adhesion, friction, tribology, shear force, numerical simulation, method of dimensionality reduction 1. introduction adhesive forces play an important role in the processes taking place in biological systems. one bright example of how animals use adhesion is a gecko motion along an inclined surface [1]. during such motion, a firm adhesive contact is established between the gecko’s feet and the surface allowing him to move easily both on vertical and horizontal surfaces (on the "ceiling"). this is possible because the gecko’s feet surface consists of a huge amount of fibers, each of them having very good adhesive properties. the fibers’ elasticity together with their endings’ good adhesion to surfaces allows the gecko to easily attach to the majority of natural surfaces. based on the gecko’s feet structure as a natural prototype, a scientific group around s. gorb [2] has created an artificial material with a similar structure, which "sticks" to almost any surfaces. received october 19, 2016 / accepted november 28, 2016 corresponding author: iakov a. lyashenko sumy state university, department of modeling of complex systems, rimskii-korsakova 2, 40007 sumy, ukraine e-mail: nabla04@ukr.net 314 i. lyashenko it is necessary to emphasize that the adhesive contact can be destroyed not only by an increase in the normal force value, but also by applying a tangential loading as, for example, shown in [3-8]. during the gecko’s motion in a horizontal plane (on the "ceiling"), the normal component of the force plays a decisive role. in the case of motion in a vertical plane the normal force required for the foot to detach is created by its muscular power. however, there is also a tangential force caused by gravity. while moving along an inclined plane with different slopes, the contribution of tangential and normal components to the contact breakage will vary. nevertheless, the gecko’s foot detachment always takes place at a particular ratio between these two force components, which is a function of the tangential (or normal) loading [9]. for example, an increasing tangential force will reduce the normal force required to break the contact. therefore, the aim of the present work is to determine the critical force components that correspond to the adhesive contact breakage taking into account a relationship between the surface energy and the separation angle that was previously determined in [3]. the present research is a continuation of the work [9], in which the surface energy was supposed to be independent of the direction of motion. it is necessary to point out that we consider a case when there is an equivalent contribution of the tangential and normal loading to the adhesive bonds’ breakage. this approach can be applied not only to the gecko’s motion description, but also to the adhesion between atomically flat surfaces or long polymer molecules changing their orientation at a tangential displacement. while describing such processes, it is also important to understand what will happen after the adhesive bonds have been broken. for example, in the case of a pure tangential motion, the bonds will be restored after their breakage. however, in the case of a gecko moving along a surface, such behavior is not observed (after separation, the contact is fully recreated but in a different place). that is why we consider the situation when the adhesive bonds are not restored after their destruction. in addition, we limit ourselves to the description of only the contact breakage phase, because we are interested in the dependencies of the critical normal and tangential force components corresponding to the complete contact breakage [9]. our investigation will be carried out within the framework of the well-known method of dimensionality reduction (mdr) [10], which allows us to reproduce the classical results of the theory by johnson, kendall and roberts [11] (jkr) for the adhesive normal contact using the rule of hess [10]. the present work consists of two parts. in section 2 we briefly describe the adhesive contact modeling procedure within the mdr. section 3 will show the investigation of the tangential loading influence and new results are given. section 4 will be finally dedicated to conclusions. 2. mdr for the adhesive normal contact in order to describe the contact of axially symmetric bodies within the framework of mdr, the following steps should be performed [12]: at first, the initially threedimensional profile z = f(r) is replaced by a one-dimensional function g(x) according to the abel transform: 2 2 0 ( ) ( ) d x f r g x x r x r     . (1) influence of tangential displacement on critical normal force of adhesive contact breakage... 315 in this paper we restrict ourselves to the parabolic profiles in the form f(r) = r 2 / (2r). in this case, eq. (1) gives the equivalent one-dimensional profile: 2 ( ) x g x r  . (2) as a second step, it is necessary to replace the elastic half-space by a one-dimensional elastic foundation of independent linear springs with normal and tangential stiffness: * zk e x  , * xk g x  , (3) where sampling step x is the distance between two springs, and effective elastic moduli e * and g * are determined by the equations: * 2 2 11 e g e     , * 4 2 g g   , (4) with shear modulus g and poisson number ν, leading to the so-called mindlin ratio: * * 2 2 2 g e      . (5) later we will use a criterion of detachment given by eq. (14), which contains equivalent contributions to the adhesive force from both the normal and tangential displacements of the indenter. we should note that eq. (14) is valid only for ν = 0. in this case, the stress concentration factors for the modes ii and iii are equal along the entire boundary line [9, 13]. on the other hand, for a no-slip contact between a rigid indenter and an elastic half space, the condition of elastic similarity (which necessarily has to be fulfilled to ensure the exact correctness of the mdr) corresponds to ν = 0.5. however, accounting for elastic dissimilarity will severely complicate the calculations and the error made by not accounting for it was proven to be small [14]. therefore, in further calculations we have chosen ν = 0 in order to provide the correctness of the detachment criterion in eq. (14). the mindlin ratio in this case will be equal to one. if transformed profile g(x) is pressed into the elastic foundation with penetration depth d, the displacement of an individual spring inside the elastic contact will be determined by the following expression: 2 ( ) ( )z x u x d g x d r     . (6) the adhesive contact size (i.e. its radius a) can be easily found using the rule of heß, which gives the tension level for the boundary springs in contact l = –uz(a), where the value of l is determined by the following equation [12]: * 2 a l e      . (7) 316 i. lyashenko by combining expressions (6) and (7), we get the equation: 2 * 2a a d r e      . (8) as a result, total normal force value fz can be calculated as the sum of forces of all individually stretched and compressed springs [10]: 2 * 3 * 3 * 0 4 ( ) ( )d 2 d 8 3 a a z z a x e a f a u x x e d x a e r r                 . (9) let us now consider a more general case when the indenter also moves in the tangential direction with displacement (0) xu . for convenience, we represent our results in terms of the following dimensionless parameters: 0 a a a  , 0 z z f f f  , 0 d d d  , (0) (0) 0 x x u u d  , 0 z z u u d  , (10) where f0, a0 and d0 are the critical values of the normal force, the contact radius and the absolute value of the indentation depth at the moment of the parabolic indenter’s detachment from the elastic half-space under “fixed load” conditions [15]: 0 3 2 f r   , 1/3 2 0 * 9 8 r a e          , 1/3 2 2 0 *2 3 64 r d e          . (11) in terms of these dimensionless parameters eqs. (8) and (9) take the form: 2 1/23 4d a a  , (12) 3 3/22f a a  , (13) which, of course, just reproduce the jkr solution [11]. 3. influence of tangential displacement let us consider a situation of an actually non-zero tangential displacement (0) xu . in this case, the energy released during the detachment of the two outermost springs will be equal to * 2 * (0)2( )z xe u a x g u x   . by equalizing it with the adhesive work 2πaδxδγ, we get the equilibrium condition in the form [9]: * 2 * (0)2( ) 2z xe u a g u a    . (14) in our previous work, we supposed that adhesive work δγ is independent of the tangential loading [9]. however, some investigations have shown that such dependence may occur [3, 4, 16]. for example, in [3] in order to take into account the influence of the tangential displacement, the surface energy dependence has been proposed in the form: influence of tangential displacement on critical normal force of adhesive contact breakage... 317 * (0)* 2 1 0 * * 1 tan (1 ) tan ( ) x z g ue g e u a                     , (15) with dimensionless parameter λ, introduced in order to determine how the surface energy depends on the motion direction. this equation has been obtained in the work [3] for the crack opening regime; that is why it is valid only for negative normal forces values zf (or in our notations for negative displacement values ( )zu a ). the situation of a surface energy independent of the motion direction is determined by λ = 1. in order to find out how λ influences the contact breakage process, we should rather use energy γ0 (eq. (15)) in eq. (14) instead of standard constant δγ. using dimensionless parameters, the corresponding equilibrium condition takes the form: * (0)* * * 22 (0) 2 1 * * * * ( ) 16 1 tan (1 ) tan ( ) x z x z g ue e e u a u a g g g e u a                   . (16) let us perform numerical simulations of the adhesive contact using this detachment condition at different values for λ. in the case under consideration, function ( )zu a is determined by expression (16). using eq. (6), we can find the relationship between indentation depth d and contact radius a in the form: 2 ( )z a d u r a  . (17) the normal and tangential forces are functions of contact radius a: 3 *2 3 z a f e ad r        , (18) * (0)2x xf g a u  . (19) in terms of the dimensionless parameters in eq. (10), eqs. (17)-(19) can be written in the following form: 23 ( )zd a u a  , (20) 2( ) 2 z a f d a  , (21) * * (0) 2 x x e u g f a  . (22) these equations define the dependencies of the normal force on the indentation depth taking into account the tangential displacement. it is necessary to point out that substitution of the indentation depth (eq. (20)) into the equation for the normal force (eq. (21)) at zero tangential displacement (0) 0xu  leads to the classical solution (eq. (13)) for the normal contact. 318 i. lyashenko let us consider “fixed-grips” and “fixed-loads” loading conditions. in the case of “fixed-grips” conditions, a very stiff external system controls the macroscopic indenter displacement. physically it means that during the system’s motion toward the equilibrium state, the displacement value is kept constant. the “fixed load” conditions can be implemented physically with the help of a very soft spring. as a result, the force value is fixed during the relaxation process. we note that the mdr-relations described before have been successfully applied to the simulation of the adhesion influence at particles elastic collisions under "fixed-grips" loading conditions [17]. in the case of “fixed-grips”, the loss of the contact stability is defined by expression d ( ) / d 0d a a  , which using eq. (20) can be written in the explicit form: d 6 0 d zu a a   . (23) by solving the system of eqs. (16) and (23), we have obtained the dependence of critical radius ,c fga on tangential displacement (0) xu . the substitution of the result into eqs. (20)-(22) allows us to obtain the desired relationship between the normal force (the adhesion force) and applied tangential force  z xf f . fig. 1a shows these dependencies built for different values of λ. note that at zero tangential force 0xf  (or zero displacement (0) 0xu  ) under “fixed-grips” conditions, the critical normal force is (0) 5 / 9zf   for all of the curves [9]. under the “fixed-load” conditions, the instability occurs when the negative normal force reaches its maximum value [10]. consequently, the instability condition is written in the form d / d 0zf a  , resulting in the equation: d 6 0 d z zu u a a a    . (24) we have performed the above described numerical analysis. but instead of eq. (23) we have used expression (24). fig. 1b gives the numerical calculations results. note that all the curves in fig. 1 are shown in the range of negative forces 0zf  , for which eq. (15) holds true. in fig. 1b the critical value of the normal force without the tangential displacement is (0) 1zf   . 0 1.5 3 4.5 -0.4 -0.2 fz ~ f x ~                            0 1.5 3 4.5 -1 -0.8 -0.6 -0.4 -0.2 fz ~ fx ~                            a b a) b) fig. 1 normalized dependencies of critical normal force zf on tangential force xf for e * =g * at different values of λ: (a) “fixed-grips” loading conditions in both directions; (b) “fixed-load” conditions in the vertical direction and “fixed-grips” conditions in the tangential direction influence of tangential displacement on critical normal force of adhesive contact breakage... 319 in accordance with the results shown in fig. 1 in both the considered cases the indenter can detach from the half space at zero normal force at the expense of tangential force xf . these critical tangential forces versus parameter λ at zero normal forces 0zf  are shown in fig. 2 for “fixed-grips” conditions (solid curve) and “fixed-load” conditions (dashed curve). both the curves correspond to the results shown in fig. 1. it can be seen from fig. 2 that these two dependencies ( )xf  have the similar form, but under the “fixed-grips” conditions, the critical value of tangential force required for detaching is smaller than for situations with “fixed load” conditions. 0 0.2 0.4 0.6 0.8 1 2 3 4 f x ~  fz = 0 ~ fixed load fixed grips fig. 2 normalized dependencies of critical tangential force xf on λ at zero value of normal force for the parameters of fig. 1a (solid curve) and fig. 1b (dashed curve). 4. conclusions the adhesive contact between an axially-symmetric indenter and an elastic half-space has been investigated under superimposed normal and tangential loading taking into account the dependence of the surface energy on the separation direction. it has been shown that the presence of the tangential displacement leads to a decrease in the critical value of the normal force that corresponds to detachment of the indenter from the surface. different combinations of “fixed-load” and “fixed-grips” loading conditions have been studied in the normal and tangential direction. for each case, the universal curves have been built in terms of corresponding dimensionless parameters. the investigation results can be applied to the description and simulation of adhesive processes taking place at the adhesive interaction of a gecko’s foot with a surface along which it moves. acknowledgements: the presented work has been performed during scientific visit of the author to institute of mechanics (berlin university of technology). the author is grateful to prof. v. l. popov for the invitation, for the financial support of the work, for the formulation of the problem and his helpful advices during the investigation process. this work was partially supported by ministry of education and science of ukraine under the project no. 0116u006818 “thermodynamic theory of the phase transitions between structural states of the boundary lubricant with spatial inhomogeneity”. 320 i. lyashenko references 1. gao, h., wang, x., yao, h., gorb, s., arzt, e., 2005, mechanics of hierarchical adhesion structures of geckos, mechanics of materials, 37(2-3), pp. 275-285. 2. carbone, g., pierro, e., gorb, s. n., 2011, origin of the superior adhesive performance of mushroomshaped microstructured surfaces, soft matter, 7(12), pp. 5545-5552. 3. hutchinson, j. w., suo, z., 1991, mixed mode cracking in layered materials, advances in applied mechanics, 29, pp. 63-191. 4. waters, j. f., guduru, p. r., 2010, mode-mixity-dependent adhesive contact of a sphere on a plane surface, proc. royal soc. a, 466(2117), pp. 1303-1325. 5. waters, j. f., guduru, p. r., 2011, a mechanism for enhanced static sliding resistance owing to surface waviness, proc. royal soc. a, 467(2132), pp. 2209-2223. 6. waters, j. f., kalow, j., gao, h., guduru, p. r., 2012, axisymmetric adhesive contact under equibiaxial stretching, the journal of adhesion, 88(2), pp. 134-144. 7. waters, j. f., gao, h. j., guduru, p. r., 2011, on adhesion enhancement due to concave surface geometries, the journal of adhesion, 87(3), pp. 194-213. 8. popov, v. l., dimaki, a. v., 2016, friction in an adhesive tangential contact in the coulomb-dugdale approximation, the journal of adhesion, doi: 10.1080/00218464.2016.1214912 9. popov, v. l., lyashenko, i. a., filippov, a. e., 2016, influence of tangential displacement on the force of adhesion between a parabolic profile and plane surface, arxiv:1611.00570 [cond-mat.soft], 12 pp. 10. popov, v. l, heß, m., 2015, method of dimensionality reduction in contact mechanics and friction , springer, 265 p. 11. johnson, k. l., kendall, k., roberts, a. d., 1971, surface energy and the contact of elastic solids, proc. royal soc. lond. a, mathematical and physical sciences, 324(1558), pp. 301-313. 12. popov, v. l., 2013, method of reduction of dimensionality in contact and friction mechanics: a linkage between micro and macro scales, friction, 1(1), pp. 41-62. 13. johnson, k. l., 1997, adhesion and friction between a smooth elastic spherical asperity and a plane surface, proc. r. soc. lond. a, 453, pp. 163-179. 14. borodich, f.m., galanov, b.a., prostov, y.i., suarez-alvarez, m.m., 2012, influence of complete sticking on the indentation of a rigid cone into an elastic half space in the presence of molecular adhesion. journal of applied mathematics and mechanics, 76, pp. 590-596. 15. popov, v. l., 2010, contact mechanics and friction. physical principles and applications, springer, 361p. 16. kim, k. -s., mcmeeking, r. m., johnson, k. l., 1998, adhesion, slip, cohesive zones and energy fluxes for elastic spheres in contact, journal of the mechanics and physics of solids, 46(2), pp. 243-266. 17. lyashenko, i. a., willert, e., popov, v. l., 2016, adhesive impact of an elastic sphere with an elastic half space: numerical analysis based on the method of dimensionality reduction , mechanics of materials, 92, pp. 155-163. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 233 247 https://doi.org/10.22190/fume180117024g © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta winglet-type vortex generators udc 536.2 seyed alireza ghazanfari, mazlan abdul wahid high-speed reacting flow laboratory, faculty of mechanical engineering, universiti teknologi malaysia, johor, malaysia abstract. heat transfer rate, pressure loss and efficiency are considered as the most important parameters in designing compact heat exchangers. despite different types of heat exchangers, fin-and-tube compact heat exchangers are still common device in different industries due to the diversity of usage and the low space installation need. the efficiency of the compact heat exchanger can be increased by introducing the fins and increasing the heat transfer rate between the surface and the surroundings. numerous modifications can be applied to the fin surface to increase heat transfer. delta-winglet vortex generators (vgs) are known to enhance the heat transfer between the energy carrying fluid and the heat transfer surfaces in plate-fin-and-tube banks, but they have drawbacks as well. they increase the pressure loss and this should be considered. in this paper, the thermal efficiency of compact heat exchanger with vgs is investigated in different variations. the angle of attack, the length and horizontal and vertical position of winglet are the main parameters to consider. numerical analyses are carried out to examine finned tube heat exchanger with winglets at the fin surface in a relatively low reynolds number flow for the inline tube arrangements. the results showed that the length of the winglet significantly affects the improvement of heat transfer performance of the fin-and-tube compact heat exchangers with a moderate pressure loss penalty. in addition, the results show that the optimization cannot be performed for one criterion only. more parameters should be considered at the same time to run the process properly and improve the heat exchanger efficiency. key words: compact heat exchangers, parametric study, vortex generators, heat transfer enhancement. received january 17, 2018 / accepted may 30, 2018 corresponding author: seyed alireza ghazanfari high-speed reacting flow laboratory, faculty of mechanical engineering, universiti teknologi malaysia, 81310 utm skudai, johor, malaysia e-mail: alireza.ghazanfari@gmail.com 234 s.a.ghazanfari, m.a.wahid 1. introduction nowadays, tube bank fin heat exchangers are commonly exploited in different industrial tools like cooling systems of locomotive engine and chillers. one of the main drawbacks of such heat exchangers is related to their high energy consumption. thus, in order to mitigate the consumption of energy, there have been efforts on improving the heat transfer in the fin sides. one of the recently known ways of achieving this purpose is making modification on the geometry of the fin surface. based on the previous works on approaches to geometry modification, the most outstanding modifications include the wavy fin [1], the slit fin [2], the louvered fin [3], the interrupted annular groove fin [4], the fin with winglet-type vortex generators (vgs) [5], and some combination enhanced fins [6, 7]. one of the well-established approaches for improving the heat transfer in fin side is creating a secondary flow with vgs. it functions by both making interruption in development of thermal boundary layer and creating longitudinal vortices, which improves the moment and mass transfer of fluid between the area close to the wall and the region remote from the wall. fiebig [8] and jacobi and shah [9] conducted a review study on exploitation of vgs in compact heat exchangers. their studies have shown that plain winglet vgs consisting of delta-winglet [10–15] and rectangular-winglet [16] are considered as the most prevalent applications. based on the result from the comparative study done by tiggelbeck et al. [17], delta winglets and the rectangular winglets are considered as the highest and second highest performance respectively. in a similar way, he et al. [18] made a comparison on the performance of delta-winglet pairs consisting of two layout styles (continuous and discontinuous) with that of the traditional large winglet. based on the results, discontinuous oriented winglets are found to be the best heat transfer improvement mode. in a similar study done by torri et al. [19], delta winglet-type vgs are applied, which has led to eliminating areas with poor heat transfer around wake of the tube. in the other study related to the application of vgs in heat transfer, lin et al. [20, 21] analyzed flow features of heat exchangers which are fitted with wave-type, annular and inclined formed vgs. in another research by leu et al. [22] thermal-hydraulic features of inlined and staggered plate fin-tube heat exchangers were studied through investigation of block styled vgs fitted behind the tubes, which indicated optimization of the vg's span angle and the vg's transverse place. dupont et al. [23] did an empirical study on flow characteristics by exploiting embossedtype vgs which were cyclically arranged. they demonstrated that the application of such forms of vgs could be the striking point in the area of heat transfer enhancement. ye et al. [24] compared the performance of curved trapezoidal winglet vgs with that of conventional plain vgs consisting of rectangular-winglet, trapezoidal-winglet and deltawinglet forms. according to the results, the delta winglet mode is observed as the best form under laminar and transitional flow condition. on the other hand, under the turbulent condition, curved trapezoidal winglet has shown the best performance. it should be noted that in spite of its superiority, it is not commonly used in tube bank fin heat exchangers existing on the market. based on the current research stream on the heat transfer enhancement of the compact heat exchangers, vgs are basically applied to create vortices. besides, in some studies, they have been exploited to put the flow in the right direction. therefore, due to the existence of wake heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta... 235 area behind tubes, vgs can be taken granted with two different functions including generating vortex and directing flows. however, identifying the optimal design should be an important issue requiring sober consideration. for instance, plane vgs are found to be fitted on the fine surface, while other block styled vgs do not fit in this format. therefore, this study has adopted a delta-winglet vg which is fitted on the fin surface as demonstrated in fig. 1. by applying the proposed vg pattern, this study tries to direct fluid flow to the tube wake areas, which is likely to improve the weak heat transfer on the fin surfaces that are in touch with the wake area. yet, it is expected that the produced vortices can improve the transfer in a big area of the fin surface. to achieve the main objective of this study, a numerical method is conducted to examine how delta winglet vortex generators function. in the next section, first, the physical model and numerical formulation are introduced, and then the results from analyses of fluid flow features are explained. 2. numerical setup 2.1. geometry the solution domain describes the approximate location where the solution is performed. the shape of the domain can be rectangular. most of the previous literatures show that rectangular domain is the best for this case [25]. the main parameters of the ftche are specified in table 1 and shown in fig. 1. fig. 1 solution domain (a) top view, (b) side view 236 s.a.ghazanfari, m.a.wahid table 1 detailed geometry parameters of based ftche with delta winglet parameter symbol (unit) value transverse tube spacing pt (mm) 25.4 longitudinal tube spacing pl (mm) 25.4 fin pitch fp (mm) 3.0 fin thickness δf (mm) 0.2 fin length fl (mm) 101.6 fin width fw (mm) 25.4 tube position from the inlet xl (mm) 12.7 number of tubes n 4 angle of attack θ(degree) 30 length of the vg l (mm) 6 horizontal position of vg xv (mm) 5.275 vertical position of vg yv (mm) 5.275 2.2. boundary conditions the complete details of boundary conditions were simulated in this study to investigate the thermal and fluid dynamic characteristics that are used in ansys fluent are described as follows:  inflow: velocity inlet;  outflow: outflow;  side wall: wall;  top and bottom fins: symmetry;  tube walls: no-slip walls  fluid domain: fluid 2.3. governing equations the general form of the continuity and naiver-stokes equations with reynolds averaging [26] are used along with the k-ε model equation as explained below. based on the study conducted by ferrouillat et al. [27], k-ε is well capable of predicting the flow behaviour. ji i t i t i j i j j k f uu u vk k k u v v d t x x x x x x                                  (1) 2 1 1 2 2 ji i t i t i j i j j f uu u vk k k u f c v f c v e t x k x x x k x k x                                     (2) 2 t k v f c   (3) where ui is the mean velocity vector of the flow, vt is turbulent kinematic viscosity, fµ is damping function, cµ is model constant, k is turbulent kinetic energy and ε is energy dissipation rate. in a comparison to the standard closure models, the low-re k-ε equations contain damping functions fµ, f1 and f2, destruction terms d and e, and molecular heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta... 237 diffusion terms. also, the values of model constants, cµ, cε1 and cε2 are to be specified by the user and are different for different models. the model constants for low-re k-ε model are: cµ = 0.09, cε1 =1.50, cε2 =1.90, σk = 1.40 and σε = 1.40. in addition to that of the terms d = 0 and e = 0. the damping functions considered in this model are 2 2 (3/4 1 2 ) 2 2 5 1 1 14 200 1 1 1 0.3 3.1 6.5 k t u t k t ry f exp exp r f ry f exp                                                        (4) where k y y u v  and 2 t k r v  [28]. for more details on low-re turbulence models, model constants and notations readers are advised to refer ansys tm fluent manual [29]. 2.4. mesh generation the next step is the mesh generation in our domain and around the tubes. as can be seen in fig. 2, the mesh is also divided into three parts to reach the maximum accuracy and minimum computational time. the mesh is 3d and except the edge of the winglet where tetrahedral elements are used, all other parts use a structured mesh. also for reaching the desirable accuracy, the result convergence has been checked by using different numbers of elements: 382351, 507256, 725665 and 983683, for a case with an angle of attack b = 30° at reynolds number 400 (chosen arbitrarily). table 2 shows the numerical results and the average deviation for three different meshes for re=400. as expected, this difference (error) decreases as the mesh becomes finer. however, it was found that the error between the results achieved with the mesh with 725665 and 983683 elements was less than 0.1% regarding the friction factor and less than 2% regarding the colburn factor. based on the validation and in order to keep a balanced compromise between computational time and solution accuracy, the mesh with 725665 elements was selected. table 2 summary of the grid independence number of elements friction factor, f different percentage for f mesh 1 382351 0.07447 mesh 2 507256 0.07462 0.21% mesh 3 725665 0.07483 0.29% mesh 4 983683 0.07482 -0.02% 238 s.a.ghazanfari, m.a.wahid fig. 2 mesh generation on the computational model of che with delta winglet vg 2.5. simulating steady-state tests steady-state simulations were performed using fluent software with parameters set as follows. the first order implicit solver with the steady option (k and ε, parameter turbulent kinetic energy=1 m 2 /s 3 , turbulent dissipation rate=1 m 2 /s 3 ) was selected together with the standard wall function, while other options remained as by default, with energy function. coupled algorithm based on [5, 30, 31] was used to calculate pressure-velocity coupling, pressure discretization and the momentum discretization were the first order upwind discrete mode. the force and momentum data are recorded in every step. due to the small fin pitch and low fluid velocity, the incompressible flows in the airside passages turn out as laminar streams [32]. 2.6. validation validation of the numerical study in this research has been done by comparison of the cfd results with other researcher’s experiments. for this phase, the experiment carried by wang et al. [33] was selected. figure ‎3 illustrates the comparison between the fig. 3 comparison of experimental results presented by wang [33] and present work heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta... 239 experimental and numerical results for the colburn factor on the air side. as can be seen, there is a good agreement between the numerical results and experimental data, which reveals the consistency of numerical simulation introduced in this study. the highest difference between the numerical results obtained by the current model and the experimental data for the colburn j factor were about 12%. 3. results and discussion the parametric study was performed in order to investigate the most important design variable in the heat exchanger performance and choose the most appropriate optimization algorithm. the design variables are the angle of attack (θ), length of the vgs (l), horizontal position of the vgs based on the tube center (xv) and vertical position of the vgs based on the tube center (yv). to investigate the degree of importance of each variable before the design optimization is carried out, the effect of each design variable on the pressure drop, nusselt number and overall heat transfer performance was examined by varying only one variable among the baseline parameters. for example, for examining the effect of angle of attack on the heat transfer and flow characteristics of the fin-and-tube compact heat exchangers (ftche) with delta winglets, the angle of attack varies from 10° to 60° and the other parameters remain the same as listed in table 1. 3.1. effect of angle of attack in the present section, a comparative study of the effects of the angle of attack of vortex generator on the performance of compact heat exchangers is evaluated as the parametric design input variable for the optimization algorithm. the angle varies from 10° to 60° and the other parameters are kept the same as in table 1. the graph of nusselt number, friction factor, and overall performance of ches are conducted by numerical method. figure 4 shows the average nusselt number and the friction factor with air frontal velocity for various angles of attack. it can be seen that nusselt number increases with the increase in angle of attack up to 30° and then starts to decrease, while the friction factor increases in all angles. figure 4 shows that the maximum values of convective heat transfer rate occur in the case of angle equal to 30°. the results show the minimum values of the nusselt number created for the angle of 60°. the results have shown that the average nusselt number decreases significantly with the increase of angle. figure 4 shows the effects of the angle of attack on friction factor. the f factor increases with the increase in the angle of attack. the results displayed the minimum values founded for the angle of 0°. it is clearly shown that larger values of angles of attack may lead to the higher friction factor, and smaller angle may lead to smaller flow resistance. 240 s.a.ghazanfari, m.a.wahid fig. 4 effect of angle of attack of vgs on the friction factor and average nusselt number fig. 5 effect of angle of attack of vgs on the on overall performance criteria figure 5 shows the effect of angle of attack of vgs on the overall performance criteria (jf factor). it can be seen that the jf factor decreases with increase in angle of attack. 3.2. effect of the length of the vortex generator the parametric study is performed in order to investigate the importance of wing length of vortex generator as a design variable in the heat exchanger performance when the other parameters are kept the same. the effect of the wing length on the nusselt number and friction factor is shown in fig. 6. the results for various vg length are plotted as a function heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta... 241 of the nusselt number and friction factor. the nusselt number significantly decreases with the increase of the length, whereas the friction factor decreases with the increase of length. the highest values of friction factor occur at the length of 7mm and this due to a blockage that these vgs make in front of the flow. it’s totally logical that most resistance is on the biggest block. the highest nusselt number is at the length of 5mm and this paradox between the friction factor and nusselt number, again shows the significance of optimizing for the vgs. (a) (b) fig. 6 influence of winglet length: (a) on friction factor, (b) on nu number figure 7 shows the variations of the overall performances for different vgs length. the jf factor decreases with an increase in vgs length. it is noted obviously that the augment on pressure loss is smaller than the improvement in heat transfer, which leads to the positive effects of making the vgs smaller. 242 s.a.ghazanfari, m.a.wahid fig. 7 effect of vg length in on overall performance criteria 3.3. effect of the horizontal position of the vortex generator examining the effect of the horizontal position of the vgs based on distance from the centre of the tubes (fig. 8), it is seen that the distance varies from 4.55 mm to 6.4 mm, and the other parameters are kept the same for the designs of compact heat exchangers. figure 9 shows the effect of horizontal position on both the nusselt number and friction factor. it can be seen that the nusselt number decreases when the vgs move away from the tubes, but the rate of decreasing become less at 5.825mm. also, a downward trend in friction factor is achieved at all positions. the only exception is at 5.285mm in which the friction has raised, but then again, it started to decrease for the rest of the positions. according to fig. 9(a), the maximum values of friction factor are achieved when the vgs positioned at 4.725mm from the centre of the tubes. moreover, fig. 9(b) indicated that the maximum nusselt number is achieved for the nearest position of the vgs. then it started to decrease very sharply to 5.825mm and after this point, the variation becomes insignificant. fig. 8 definition of horizontal and vertical position of vgs heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta... 243 (a) (b) fig. 9 effect of the horizontal position of the winglet on (a) friction, (b) nusselt number figure 10 shows the maximum values of j_factor for the different horizontal position from the centre of the tubes. the overall performance of ftche with delta winglet significantly decreased by moving longitudinally from the centre of the tubes fig. 10 variation of values of overall performance based on horizontal position according to fig. 10, at the range of the various horizontal position of the present study, the maximum values of jf factor are achieved for the nearest vgs. for the far 244 s.a.ghazanfari, m.a.wahid positions, the j_factor values are almost same and without considerable changes. this because of the boundary layer and whenever the vgs are in the boundary layer, they disturb this boundary layer more so they have more influence on the total performance of ftche. 3.4. effect of the vertical position of the vortex generator examining the effect of the vertical position of the vgs based on distance from the centre of the tubes (fig. 8), it is seen that the distance varies from 4.55 mm to 6.4 mm, and the other parameters are kept the same for the designs of compact heat exchangers based on table 1. figure 11 shows the effect of vertical position on both the nusselt number and friction factor. it can be seen that the nusselt number decreases when the vgs move away from the tubes, but the rate of decreasing become less at 6.1mm. but there is an upward trend in the friction factor at all positions. according to fig. 11(a), the maximum values of friction factor are achieved when the vgs positioned at the farthest possible position, i.e. 6.375mm from the centre of the tubes. (a) (b) fig. 11 effect of the vertical position of the winglet on (a) friction, (b) nusselt number heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with delta... 245 moreover, fig. 11(b) indicates that the maximum nusselt number is achieved for the nearest position of the vgs. then it starts to decrease very sharply to 6.1mm and after this point, the variation becomes insignificant. figure 12 shows the maximum values of j_factor for the different vertical position from the centre of the tubes. the overall performance of ftche with delta winglet significantly decreased by moving longitudinally from the centre of the tubes. fig. 12 variation of values of overall performance based on the vertical position according to fig. 12, at the range of the various horizontal position of the present study, the maximum values of jf factor are achieved for the nearest vgs. for the far positions, the j_factor values are almost same and without considerable changes. this because of the boundary layer and whenever the vgs are in the boundary layer, they disturb this boundary layer more so they have more influence on the total performance of ftche. in addition, according to fig. 11(b) and fig. 12, the results show that at the same position, the nusselt number and j_f factor have the similar trend, which indicates that the overall performance of the system is dominated by convective heat transfer. 4. conclusions this study presents a numerical investigation of the effect of different parameters on the thermohydraulic performance of compact heat exchangers with vortex generators. the parameters investigated in this study are the angle of attack, length of the winglet, horizontal and vertical placement of the winglet. the main outcomes of this study are as follows:  this study shows the importance of the vgs and how they affect the heat transfer rate and pressure drop.  the heat transfer rate of the compact heat exchangers improves in lower angle of attack as well as lower length. but it decreases by putting the vgs far from the tubes.  the pressure drop has more consistent behavior and shows in all cases a stable trend. increasing the angle of attack and length of the vgs will result in more pressure drop. in vertical positions far from the tubs, also pressure drop increases. and the horizontal position has the least effect on the pressure drop despite a slight improvement in case of a bigger gap. 246 s.a.ghazanfari, m.a.wahid  in all cases, the overall performance of the compact heat exchangers based on j_factor decreases with the increase of the aforementioned parameters.  this study shows that, for improving the thermohydraulic efficiency of the compact heat exchangers with vortex generators, many parameters must be considered simultaneously and it is necessary to use a multi-objective optimizer to reach the optimum configuration. acknowledgement: the authors gratefully acknowledge the support by the faculty of mechanical 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delta-wing vortex generators on a flat plate: vortex interactions with the boundary layer, experimental thermal and fluid science, 14(3), pp. 231–242. 33. wang, c.-c., chi, k.-y., 2000, heat transfer and friction characteristics of plain fin-and-tube heat exchangers, part i: new experimental data, international journal of heat and mass transfer, 43(15), pp. 2681–2691. 11587 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 709 733 https://doi.org/10.22190/fume230419022n © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak foundation nikola nešić1, milan cajić2, danilo karličić2, mihailo lazarević3, sondipon adhikari4 1faculty of technical sciences, university of priština, kosovska mitrovica, serbia 2mathematical institute of the serbian academy of sciences and arts, belgrade, serbia 3faculty of mechanical engineering, university of belgrade, serbia 4james watt school of engineering, university of glasgow, united kingdom orcid ids: nikola nešić https://orcid.org/0000-0001-6237-4735 milan cajić https://orcid.org/0000-0001-5513-0417 danilo karličić https://orcid.org/0000-0002-7547-9293 mihailo lazarević https://orcid.org/0000-0002-3326-6636 sondipon adhikari https://orcid.org/0000-0003-4181-3457 abstract. this study investigates the stability of periodic solutions of a nonlinear nonlocal strain gradient functionally graded euler–bernoulli beam model resting on a visco-pasternak foundation and subjected to external harmonic excitation. the nonlinearity of the beam arises from the von kármán strain-displacement relation. nonlocal stress gradient theory combined with the strain gradient theory is used to describe the stress-strain relation. variations of material properties across the thickness direction are defined by the power-law model. the governing differential equation of motion is derived by using hamilton's principle and discretized by the galerkin approximation. the methodology for obtaining the steady-state amplitude-frequency responses via the incremental harmonic balance method and continuation technique is presented. the obtained periodic solutions are verified against the numerical integration method and stability analysis is performed by utilizing the floquet theory. key words: nonlocal strain gradient theory, functionally graded beams, pasternak layer, duffing oscillator, incremental harmonic balance method, floquet multipliers received: april 19, 2023 / accepted july 29, 2023 corresponding author: nikola nešić faculty of technical sciences, university of priština, 38220 kosovska mitrovica, kneza miloša 7, serbia e-mail: nikola.nesic@pr.ac.rs 710 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari 1. introduction traditional composites are homogeneous mixtures of two or more materials, where compromise is made between the desirable properties of the component materials. on the contrary, a functionally graded material (fgm) is a two-component composite characterized by a compositional gradient from one component to the other and because significant proportions of fgm contain pure forms of each component, the need for compromise is eliminated. in such a case, the properties of both components can be fully utilized [1, 2]. although a significant amount of work is done in the field of fg structures, there is still a lot of space and need for the investigation of mems/nems systems composed of fgms. nanobeams and nanoplates are used in different mems/nems devices [3, 4], such as microactuators [5], microswitches [6], micro sensors [7, 8], nanoscale resonators [9], energy harvesting nanodevices [10], etc. besides that, carbon nanotubes are used as an addition in some mixtures to increase the dynamic stability of the material [11]. the dynamics of such systems can be studied through experiments [12, 13], molecular dynamics simulations [14], and continuum mechanics [15, 16, 17]. the most vastly used continuum theories for studying nanostructures are: nonlocal elasticity theory [17, 18], strain gradient theory [13, 19], modified couple stress theory (or modified strain gradient theory) [20], the surface elasticity theory [21], and the nonlocal strain gradient theory (nlsgt) [22], which includes both nonlocal and length scale effects into consideration. an overview of available theories and methodologies for the analysis of nano-isotropic, nano-functionally graded, and cnt reinforced nanocomposite structures is given by garg et al. [23]. nonlinearity can be introduced in beam vibration problems through geometric imperfections in beam material [24] and foundation [25]. geometric material nonlinearity is usually introduced with von kármán strain-displacement relation [24, 26, 27]. simsek [26] examined the nonlinear vibration behavior of an nlsgt euler-bernoulli fg beam with von kármán’s geometric nonlinearity. liu et al. [24] studied the nonlinear vibration of fg sandwich nlsgt beams in the presence of initial geometric imperfection induced by the von kármán theory and a cosine function. they solved a nonlinear differential equation and obtained nonlinear frequency relation by using he’s variational principle. a nonlocal theory proposed by eringen can be utilized in differential and integral forms. the integral form is more general and accurate but complex to apply when solving advanced problems. the differential form is straighter forward in the application. for most cases of boundary conditions, it is scientifically proven that the differential form is equal to the integral form. however, cantilever beams give contradictory results when employing integral and differential forms. barreta and marotti de sciarra [28] suggested a methodology to bridge two forms by replacing the integral form with differential with additional constitutive boundary conditions which is applied to several simple problems [29, 30, 31]. still, the equation of motion, for a problem similar in complexity to ours, is exclusively derived in differential form as can be found in recently published papers [24, 26, 32, 33, 34]. therefore, we will utilize differential approach, but limit our research on pinned-pinned, clampedpinned and clamped-clamped boundary conditions, since clamped-free boundary conditions produce contradictory results. in recent times, stress-driven, strain-driven, and foundation-driven vibrations have become in research focus. penna et al. [35] investigated nonlinear free vibrations of geometrically imperfect fg nano-beams based on stress-driven nonlocal elasticity. penna and feo [36] studied nonlinear free vibrations of functionally graded porous bernoulli– vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 711 euler nano-beams resting on a winkler elastic foundation through a stress-driven nonlocal elasticity model. vaccaro et al. [37] studied the size-dependent behavior of nonlocal elastic beams by adopting the stress-driven elasticity theory where the kinematics of beams is modeled by the reddy variational third-order beam theory. ansari et al. [38] used straindriven nonlocal formulations of eringen’s theory in both differential and integral forms for a numerical study of free vibration behavior of piezoelectric bernoulli–euler nanoscale beams considering flexoelectric and nonlocal effects. comparative studies between stressdriven and strain-driven nonlocal elasticity theory have been made by several authors [39, 40, 41]. elasto-statical displacement-driven nonlocal problems are studied by vaccaro [42], patnaik [43] and pinnola [44]. moreover, pinnola et al. [44] have studied the elastostatic problems of classical euler-bernoulli beam on a reaction-driven nonlocal foundation. they proposed an integral elasticity mixture considering a convex combination of winkler (local) and wieghardt (nonlocal) laws and applied the proposed methodology to find analytical solutions to simple elasto-static problems in structural mechanics. the dynamic behaviour of nlsgt structures composed of fgm was studied by many authors. the most common approach in their analysis includes applying different perturbation methods. for example, gao et al. [45] studied the nonlinear free vibration of fg circular nanotubes using nlsgt and two steps perturbation method. by using the same methodology, they also studied the vibration of nano arches [46], and fg nanobeams with several different functionally graded distributions [47]. el-borgi et al. [15] investigated the free and forced vibration response of a simply supported fg beam resting on the nonlinear elastic foundation using the perturbation multiple scales method to obtain the amplitude-frequency curves of the system. wang and shen [48] studied the lateral nonlinear vibration of an axially moving simply supported viscoelastic nlsgt beam. for obtaining the steady-state amplitude-frequency responses in the subharmonic parametric resonance regime a direct multiple scales method is used. jafarsadeghi-pournaki et al. [49] investigated the heat-induced nonlinear vibration of fg capacitive nanobeam based on nlsgt. they used averaging perturbation method to obtain the governing equations and study the steady-state responses. li et al. [50] studied analytically the longitudinal vibration of nlsgt rods and discovered that the nlsgt rod model shows a stiffness-softening effect when the nonlocal parameter is larger than the length scale parameter and a stiffness-hardening effect in the opposite case. however, the main disadvantage of perturbation methods is that they can only treat systems with small nonlinearity. nanostructures modeled by nonlocal stress and strain gradient theory are also studied by other methods, for example, the differential quadrature method. li [51] used the generalized differential quadrature method to investigate the vibration of axially fg beams based on nlsgt and euler-bernoulli beam theory. however, in the recent time, the ihb method becomes popular in the investigation of nonlinear dynamical systems with harmonically varied excitation. in an interesting work of bhattiprolu et al. [52], they found periodic solutions of a nonlinear euler-bernoulli beam resting on viscoelastic uniand bilateral foundations by using the incremental harmonic balance method (ihbm). however, they investigated vibration of a linear beam on a nonlinear foundation subjected to multiple concentrated forces and only pinned-pinned boundary conditions. our considered beam is nonlinear with three different boundary conditions. the advantage of ihb over other methods is in its accuracy, the possibility to treat small and huge nonlinearities, and relatively simply implementation. perturbation methods can solve differential equations with small nonlinear terms and rather simple problems. therefore, ihb is used in our work to find periodic solutions when strong nonlinearity is present in the system. 712 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari stability of fg nanobeams has been investigated in the literature by using several different criteria and methods. wang and shen [48] employed the routh–hurwitz criterion to determine the stability of the (non-) zero equilibrium solution of nonlinear laterally vibrating axially moving simply supported viscoelastic nlsgt beam. jafarsadeghipournaki et al. [49] used a shooting technique in conjunction with the floquet theory for capturing the periodic motions and examining their stability of heat-induced nonlinear vibration of fg nlsgt capacitive nanobeam. jalaei et al. [53] investigated the dynamic stability of a temperature-dependent timoshenko fg nanobeam under the axial excitation load and magnetic field in a thermal environment by using navier's and bolotin's method. in this paper, we used floquet theory to determine the stability of our model because it is suitable to be implemented coupled with the ihb method. nešić et al. [27] studied the vibration of a nonlocal nonlinear fg beam on the fractional visco-pasternak foundation based on the amplitude-frequency response analysis and utilizing the ihb, perturbation multiple scales, and newmark methods to find and confirm the periodic solution. in the current paper, the stability of periodic solutions is investigated for the amplitude-frequency responses of a nonlinear nonlocal strain gradient functionally graded euler–bernoulli beam model resting on a visco-pasternak foundation and subjected to external harmonic excitation. therefore, this study can be considered as a special case and natural extension of the work done by nešić et al. in [27] since a special case of the visco-pasternak foundation model without fractional derivatives is adopted and stability of periodic solutions analysis is performed. nonlinear von kármán strain-displacement relation is used together with the nonlocal stress and strain gradient theories. variations of material properties across the thickness direction are defined by the power-law model. the governing differential equation of motion is derived by using hamilton's principle and then discretized by the galerkin approximation. the incremental harmonic balance method and continuation technique are employed to obtain the amplitude-frequency responses and some periodic solutions are verified against the numerical integration runge-kutta method. for each periodic solution, a stability check is performed by utilizing floquet theory. the novelty in presented paper includes introduction of the model of nonlocal nonlinear fg beam vibrating on the visco-pasternak foundation. nonlinearity is introduced via von karman strain-displacement relation and nonlocality through strain gradient and stress gradient constitutive relations. the proposed visco-pasternak foundation is defined with four parameters enabling modeling of vast range of practical foundations. originality is obtaining periodic solutions for such a system by using ihb method and continuation technique and analyzing amplitude-frequency response curves for different parameters. additional novelty is utilizing floquet theory to determine stability of periodic solutions for different excitation frequencies of such vibrating system and usage of basins of attraction to determine stability of the solution based on initial conditions. technical interest in considering such structural schemes includes the development of mems/nems devices composed of small beams that can vibrate on a broad type of elastic supports. in the practical implementation, specific end supports might be preferable. therefore, three different boundary conditions are considered to demonstrate their influence on vibration behavior. vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 713 2. governing equation of fg beam on a visco-pasternak foundation let us consider a mechanical model of a beam as given in fig. 1. relations for young's modulus and density of fg beam, as well as relations for the nonlocal strain gradient theory can be found in the appendix of the paper [27]. fig. 1 model of the simply-supported nonlinear nonlocal strain gradient fg beam resting on a visco-pasternak foundation and excited by traverse load the displacement field of the euler-bernoulli beam is given as: 𝑢𝑥(𝑥, 𝑧, 𝑡) = 𝑢(𝑥, 𝑡) − 𝑧 𝜕𝑤 𝜕𝑥 , 𝑢𝑦(𝑥, 𝑧, 𝑡) = 0, 𝑢𝑧(𝑥, 𝑧, 𝑡) = 𝑤(𝑥, 𝑡), (1) where ux, uy, and uz denote the displacements along with the length, width, and thickness directions, described by x, y and z coordinates, respectively. functions u and w are the axial and transverse displacements of the physical middle surface, respectively. the only nonzero strain component of the euler-bernoulli beam is described in terms of displacements with von kármán's nonlinearity as 𝜀𝑥𝑥 = 𝜕𝑢 𝜕𝑥 + 1 2 ( 𝜕𝑤 𝜕𝑥 ) 2 − 𝑧 𝜕2𝑤 𝜕𝑥2 . (2) following stress resultants are considered in further study: 𝑁𝑥𝑥 = ∫ 𝑡𝑥𝑥𝑑𝐴 𝐴 , 𝑁𝑥𝑥 (0) = ∫ 𝜎𝑥𝑥𝑑𝐴, 𝐴 𝑁𝑥𝑥 (1) = ∫ 𝜎𝑥𝑥 (1) 𝑑𝐴, 𝐴 𝑀 = ∫ 𝑧𝑡𝑥𝑥𝑑𝐴 𝐴 , 𝑀(0) = ∫ 𝑧𝜎𝑥𝑥𝑑𝐴 𝐴 , 𝑀(1) = ∫ 𝑧𝜎𝑥𝑥 (1) 𝑑𝐴. 𝐴 (3) where nxx, n(0) xx and n(1) xx are axial forces due to total stress txx, nonlocal stress σxx and high-order nonlocal stress σ(1) xx, respectively. all stresses are in axial direction. m, m(0) and m(1) are bending moments resulting from the following stresses in axial direction: total, nonlocal and high-order nonlocal, respectively. additionally, the extensional axx and the bending coefficient dxx are defined in the following form: {𝐴𝑥𝑥 , 𝐷𝑥𝑥} = 𝑏 ∫ {1, 𝑧2} 𝐻/2−𝑐 −𝐻/2−𝑐 𝐸(𝑧)𝑑𝑧, (4) where h represents the beam height and c is the distance between physical and geometrical middle surfaces of the beam. note that for homogeneous beam axx = ea and dxx = ei, with 714 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari e denoting young’s modulus, a cross sectional area and i second moment of area. reaction force of the visco-pasternak foundation acting on the beam is defined as [27]: 𝐹𝑚 = (𝑘𝑤 + 𝐾𝑤 𝜕 𝜕𝑡 ) 𝑤 − (𝑘𝑔 + 𝐾𝑔 𝜕 𝜕𝑡 ) 𝜕2𝑤 𝜕𝑥2 , (5) with the elastic pasternak parameters kw, kg and viscose constants kw, kg shown in fig. 1. note that the time derivative in eq. (5) is in the paper [27] replaced with a more general fractional derivative, being a real number, with values between 0 and 1 that have a physical meaning. foundations described with fractional derivatives as in [27] are more general and therefore can describe a broader range of real value applications. however, a disadvantage of the fractional derivative model is that a stability analysis of fractional differential equations cannot be done with straightforward methods presented in this paper. after following of the procedure presented in [27], we obtain expressions for variations of strain, kinetic and external energy. after applying hamilton's principle as given in [27], two equations of motion are obtained. we assume fast dynamics, which means that acceleration in an axial direction is negligible. finally, this leads us to the following equation of motion of a nonlinear fg beam resting on the visco-pasternak foundation: (1 − 𝜇2 𝜕2 𝜕𝑥2) [− 𝐴𝑥𝑥 2𝐿 ∫ ( 𝜕𝑤 𝜕𝑥 ) 2 𝑑𝑥 𝜕2𝑤 𝜕𝑥2 𝐿 0 − 𝑚2 𝜕4𝑤 𝜕𝑥2𝜕𝑡2 + 𝜕2𝑤 𝜕𝑡2 + 𝑏𝑘𝑤𝑤 + 𝑏𝐾𝑤 𝜕𝑤 𝜕𝑡 −𝑏𝑘𝑔 𝜕2𝑤 𝜕𝑥2 − 𝑏𝐾𝑔 𝜕3𝑤 𝜕𝑡𝜕𝑥2 + 𝑄 𝑐𝑜𝑠 𝛺1 𝑡] + 𝐷𝑥𝑥 (1 − 𝑙2 𝜕2 𝜕𝑥2) 𝜕4𝑤 𝜕𝑥4 = 0, (6) where b is the beam width, μ and l are small parameters in nonlocal and strain gradient theory, m2 is the second moment of inertia, q and ω1 are the external force magnitude and frequency, respectively. eq. (6) is non-dimensional with the following relations: 𝑋 = 𝑥 𝐿 , 𝑊 = 𝑤 𝑘𝑥 , 𝛴 = 𝑙 𝐿 , 𝜆 = 𝜇 𝐿 , 𝜁 = 𝑏 𝐿 , 𝜏 = 𝑡 𝑘𝑥 𝐿2 √ 𝐴𝑥𝑥 𝑚0 , 𝛺 = 𝛺1 𝐿2 𝑘𝑥 √ 𝑚0 𝐴𝑥𝑥 , 𝐹 = 𝑄𝐿4 𝐴𝑥𝑥𝑘𝑥 3 , 𝑦 = 𝑚2 𝑚0𝐿2 , 𝑘𝑥 = √ 𝐷𝑥𝑥 𝐴𝑥𝑥 , 𝑘1 = 𝜁𝑘𝑤𝐿5 𝐴𝑥𝑥𝑘𝑥 2 , 𝐾1 = 𝜁𝐾𝑤𝐿3 𝑘𝑥√𝐴𝑥𝑥𝑚0 , 𝑘2 = 𝜁𝑘𝑔𝐿3 𝐴𝑥𝑥𝑘𝑥 2 , 𝐾2 = 𝜁𝐾𝑔𝐿 𝑘𝑥√𝐴𝑥𝑥𝑚0 . (7) note that radius of gyration kx, appearing in eq. (7), for the homogenous beam is kx = (ix /a)1/2. non-dimensional form of equation of motion eq. (6), obtained by substituting eq. (7) in eq. (6) is (1 − 𝜆2 𝜕2 𝜕𝑋 2) [− 1 2 ∫ ( 𝜕𝑊 𝜕𝑋 ) 2 𝑑𝑋 𝜕2𝑊 𝜕𝑋 2 1 0 − 𝑦 𝜕4𝑊 𝜕𝑋 2 𝜕𝜏2 + 𝜕2𝑊 𝜕𝜏2 + 𝑘1𝑊 + 𝐾1 𝜕𝑊 𝜕𝜏 −𝑘2 𝜕2𝑊 𝜕𝑋 2 − 𝐾2 𝜕3𝑋 𝜕𝜏𝜕𝑋 2 + 𝐹 𝑐𝑜𝑠 𝛺 𝜏] + (1 − 𝛴2 𝜕2 𝜕𝑋 2) 𝜕4𝑊 𝜕𝑋 4 = 0. (8) the solution of eq. (8) could be assumed as a sum of products of amplitude and time functions for each mode. nayfeh and lacarbonara [54] have shown in their study that in certain cases one-mode galerkin approximation fails to predict the dynamic behavior of hinged-hinged beams, especially when quadratic type nonlinearity is involved and even modes are observed in certain subharmonic or superharmonic resonance conditions. since we have only a cubic nonlinear term, we will use only a single mode discretization which vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 715 is the most commonly used method in the investigation of nonlinear structural vibration problems [15, 22, 26, 55, 56]. the solution of eq. (8) is assumed as 𝑊(𝑋, 𝜏) = 𝜑𝑛(𝑋)𝑞(𝜏), (9) where φn is the amplitude function, q is the time function and n = 1, 2, ... is the mode number. amplitude function φn, also known as eigenfunction, can be defined as the form of trigonometric functions and should satisfy the boundary conditions. in the present paper, three different boundary conditions are taken into account: pinned-pinned (p-p), clampedpinned (c-p) and clamped-clamped (c-c), and the amplitude function for each case is given in eq. (10), respectively [34], as 𝜑𝑛(𝑋) = | | 𝑠𝑖𝑛 𝛾𝑛 𝑋, 𝛾𝑛 = 𝑛𝜋, (𝑠𝑖𝑛 𝛾𝑛 𝑋 − 𝑠𝑖𝑛ℎ 𝛾𝑛 𝑋) − 𝑠𝑖𝑛 𝛾𝑛+𝑠𝑖𝑛ℎ 𝛾𝑛 𝑐𝑜𝑠 𝛾𝑛+𝑐𝑜𝑠ℎ 𝛾𝑛 (𝑐𝑜𝑠 𝛾𝑛 𝑋 − 𝑐𝑜𝑠ℎ 𝛾𝑛 𝑋), 𝑡𝑎𝑛 𝛾𝑛 = 𝑡𝑎𝑛ℎ 𝛾𝑛 , (𝑠𝑖𝑛 𝛾𝑛 𝑋 − 𝑠𝑖𝑛ℎ 𝛾𝑛 𝑋) − 𝑠𝑖𝑛 𝛾𝑛−𝑠𝑖𝑛ℎ 𝛾𝑛 𝑐𝑜𝑠 𝛾𝑛−𝑐𝑜𝑠ℎ 𝛾𝑛 (𝑐𝑜𝑠 𝛾𝑛 𝑋 − 𝑐𝑜𝑠ℎ 𝛾𝑛 𝑋), 𝑐𝑜𝑠 𝛾𝑛 𝑐𝑜𝑠ℎ 𝛾𝑛 = −1. (10) coefficients s0 s5 are calculated as {𝑠0, 𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5} = ∫ {𝜑, 𝜑2, 𝜑″𝜑, 𝜑𝐼𝑉𝜑, 𝜑𝑉𝐼𝜑, 𝜑′2}𝑑𝑋. 1 0 (11) we replace eq. (9) into eq. (8), and use eq. (11) to obtain nonlinear differential equation as �̈� + 𝛾�̇� + 𝜔0 2𝑞 + 𝜃𝑞3 = 𝑓 𝑐𝑜𝑠 𝛺 𝜏, (12) with the following coefficients 𝛾 = 𝐾1𝑠1−𝐾2𝑠2−𝜆2𝐾1𝑠2+𝜆2𝐾2𝑠3 𝑠1−𝑦𝑠2−𝜆2𝑠2+𝑦𝜆2𝑠3 , 𝜔0 2 = 𝑘1𝑠1−𝑘2𝑠2−𝜆2𝑘1𝑠2+𝜆2𝑘2𝑠3+𝑠3−𝛴2𝑠4 𝑠1−𝑦𝑠2−𝜆2𝑠2+𝑦𝜆2𝑠3 , 𝜃 = − 1 2 𝑠5𝑠2+ 1 2 𝑠5𝑠3𝜆2 𝑠1−𝑦𝑠2−𝜆2𝑠2+𝑦𝜆2𝑠3 , 𝑓 = −𝑠0𝐹 𝑠1−𝑦𝑠2−𝜆2𝑠2+𝑦𝜆2𝑠3 . (13) 3. nonlinear periodic response for obtaining the amplitude-frequency response of a nonlinear problem described by eq. (12), the ihb method and continuation technique will be used. ihb is a vastly used procedure for solving the nonlinear differential equations of duffing and mathieu-duffing type [27, 57]. the continuation technique serves as a supporting methodology for obtaining the periodic solutions in points close to resonant states. 3.1. the incremental harmonic balance method to apply the ihb method, we introduce a new time scale τ ̅ = ωτ into eq. (12) to obtain the system of nonlinear ordinary differential equations in the following form 𝛺2 𝑑2𝑞 𝑑𝜏 2 + 𝛾𝛺 𝑑𝑞 𝑑𝜏 + 𝜔0 2𝑞 + 𝜃𝑞3 = 𝑓 𝑐𝑜𝑠 𝜏. (14) 716 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari for arbitrarily chosen initial values of q0 and ω0 for the steady-state modal amplitude, a neighboring state of motion is incrementally changed to the current state and it can be expressed in the following form 𝑞 = 𝑞0 + 𝛥𝑞, 𝛺 = 𝛺0 + 𝛥𝛺. (15) substituting eq. (15) into eq. (14) and neglecting higher-order terms, we obtain a linearized incremental relation given as 𝛺0 2𝛥𝑞″ + 𝛾𝛺0𝛥𝑞′ + 𝜔0 2𝛥𝑞 + 3𝜃𝑞0 2𝛥𝑞 = 𝑟 − 2𝛺0𝑞0 ″ 𝛥𝛺 + 𝑓 𝑐𝑜𝑠 𝜏, (16) where r is the residual term given as 𝑟 = −(𝛺0 2𝑞0 ″ + 𝛾𝛺0𝑞0 ′ + 𝜔0 2𝑞0 + 𝜃𝑞0 3). (17) to obtain the periodic solutions of the differential equation, q0 and δq are expanded into finite fourier series of n terms as 𝑞0 = 𝑎0 + ∑ [𝑎𝑛 𝑐𝑜𝑠 𝑛 𝜏 + 𝑏𝑛 𝑠𝑖𝑛 𝑛 𝜏]𝑁 𝑛=1 = 𝑪𝑨𝟎, (18) 𝛥𝑞 = 𝛥𝑎0 + ∑ [𝛥𝑎𝑛 𝑐𝑜𝑠 𝑛 𝜏 + 𝛥𝑏𝑛 𝑠𝑖𝑛 𝑛 𝜏]𝑁 𝑛=1 = 𝑪𝜟𝑨, (19) where the vector c, and vectors a0 and δa with unknown constants are given as: 𝑪 = [1 𝑐𝑜𝑠 𝜏 𝑐𝑜𝑠 2 𝜏 … 𝑐𝑜𝑠 𝑁 𝜏 𝑠𝑖𝑛 𝜏 𝑠𝑖𝑛 2 𝜏 … 𝑠𝑖𝑛 𝑁 𝜏], (20) 𝑨𝟎 = [𝑎0 𝑎1 𝑎2 … 𝑎𝑁 𝑏1 𝑏2 … 𝑏𝑁]𝑇 , (21) 𝜟𝑨 = [𝛥𝑎0 𝛥𝑎1 𝛥𝑎2 … 𝛥𝑎𝑁 𝛥𝑏1 𝛥𝑏2 … 𝛥𝑏𝑁]𝑇 . (22) substituting eqs.(18)-(22) into eq. (16) and applying the galerkin procedure leads to the system of linearized algebraic equations in terms of δa given as 𝑲𝜟𝑨 + 𝑽𝛥𝛺 = 𝑹, (23) where, elements of the jacobi matrix k of size [2n+1, 2n+1], the corrective vector r, and vector v, both of size [2n+1, 1], are defined as: 𝑲 = 1 2𝜋 ∫ [𝛺0 2𝑪𝑇𝑪″ + 𝛾𝛺0𝑪𝑇𝑪′ + 𝜔0 2𝑪𝑇𝑪 + 3𝜃𝑞0 2𝑪𝑇𝑪] 2𝜋 0 𝑑𝜏, (24) 𝑹 = − 1 2𝜋 ∫ [(𝛺0 2𝑪𝑇𝑪″ + 𝛾𝛺0𝑪𝑇𝑪′ + 𝜔0 2𝑪𝑇𝑪 + 𝜃𝑞0 2𝑪𝑇𝑪)𝑨𝟎 + 𝑪𝑇𝑓 𝑐𝑜𝑠 𝜏] 2𝜋 0 𝑑𝜏, (25) 𝑽 = 1 2𝜋 ∫ [2𝛺𝑪𝑇𝑪″] 2𝜋 0 𝑑𝜏𝑨𝟎, (26) with ( )' denoting the derivative with respect to non-dimensional time τ̅. if we want to get the solution at a given single frequency, we should set δω to zero in eq. (23). otherwise, we solve eq. (23) for both δa and δω, by inserting δω in the first entry of the vector δa and transforming the system of equations. we initialize the solution process by entering guessed values of a, and calculating δa by using the eq. (23). the solution δa is then added to the current estimated value of a to determine the new vector a, i.e, 𝑨𝒌+𝟏 = 𝑨𝒌 + 𝜟𝑨. (27) we repeat this process until the value of the residuum norm |r| is within the preset tolerance (in our case less than 10-10). vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 717 3.1.1. the continuation method the main benefit of using the continuation method is the possibility to compute periodic solutions at each point. the requirement for starting the continuation process is the determination of the periodic solutions in two successive points by using the ihb method. it is usual that these initial points are taken far from the resonant state since response amplitudes at both points have similar values, which are also small. subsequently, the predictor-corrector method can be used to make point-to-point computations and obtain corresponding branches of the amplitude-frequency responses. we introduce new vectors x = [a, ω]t and δx = [δa, δω]t, a function of vector g(x) = xtx, and an arc-length parameter η to follow direction of the path. an augmented equation is given as 𝑔(𝑿) − 𝜂 = 0. (28) when two previous points xk-1 and xk-2 of the response curve are known, the slope for computation of the next point can be determined as 𝑿′ = 𝑿𝒌−𝟏−𝑿𝒌−𝟐 ‖𝑿𝒌−𝟏−𝑿𝒌−𝟐‖ . (29) the first prediction of the next point is obtained by 𝑿𝒖 = 𝑿𝒌−𝟏 + 𝛥𝜂𝑿′. (30) after extending eq. (23) with eq. (28) we have [ 𝑲 𝑽 𝜕𝑔 𝜕𝑨 𝜕𝑔 𝜕𝛺 ] [ 𝜟𝑨 𝛥𝛺 ] = [ 𝑹 𝛥𝜂 − 𝑔 ]. (31) more information about the continuation method one can find in [52, 58, 59]. 4. stability analysis when the periodic solution is obtained in the form of fourier series as in eq. (18) for a chosen value of ω, we can investigate the local stability of such periodic solution by using the floquet theory [60, 61]. based on this theory, we applied the methodology given by hsu [62] for the approximation of the transition matrix during one period. let us consider that the solution of eq. (14) is given as 𝑞(𝜏) = 𝑞0(𝜏) + 𝛥𝑞 (32) where δq denotes a small perturbation in the proximity of previously determined periodic solution q0. by substituting eq. (32) into eq. (14), and after linearization, one can obtain the linear differential equation with time-periodic coefficients given as: 𝛺2𝛥𝑞″ + 𝛾𝛺𝛥𝑞′ + 𝜔0 2𝛥𝑞 + 3𝜃𝑞0 2𝛥𝑞 = 0, (33) to determine the stability of the obtained periodic solutions using the floquet theory, the transformation of eq. (33) into the state-space form should be performed as 𝑑𝒀 𝑑𝜏 = 𝑷(𝜏)𝒀 (34) where y (τ) = [δq, δq']t and p (τ) denotes the periodic matrix with the period t = 2π. 718 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari the stability criteria based on the floquet theory is used for the determination of the local stability of periodic solutions and it is related to the determination of the floquet multipliers as hsu gave in [62]. the floquet multipliers are represented by the eigenvalues of the monodromy matrix. by solving the corresponding eigenvalue problem, for the case when all floquet multipliers are located inside the unit circle centered at the origin of the complex plane, the determined periodic solutions are stable or asymptotically stable. otherwise, when the values of floquet multipliers are outside the unit circle, the periodic solutions are unstable [60]. depending on where the floquet multipliers or a pair of complex conjugate multipliers cross the complex plane unit circle, different bifurcation points can be detected. introduction of the hsu procedures for the numerical approximation of the transition matrix during one period leads to the numerical determination of floquet multipliers as a solution of the corresponding eigenvalue problem as shen et al. stated in [60]. it is assumed that the period t = 2π of the periodic solution q0 (τ) is divided into nk sub-intervals, where the k-th interval is equal to δk = τk τk-1 for τk = kt / n. furthermore, p (τ) is the continuous periodic matrix with respect to τ and period t, wherein the k-th interval it can be replaced by the constant matrix provided in the case when nk is chosen to be sufficiently large: 𝑷𝒌 = 1 𝛥𝑘 ∫ 𝑷(𝜏)𝑑𝜏. 𝜏𝑘 𝜏𝑘−1 (35) the monodromy (transition) matrix can be written in the following form 𝑀= ∏ 𝑒𝑷𝒊𝛥𝑖 𝑁𝑘 𝑖=1 ≈ ∏ (𝑰 + ∑ (𝑷𝒊𝛥𝑖)𝑗 𝑗! 𝑁𝑗 𝑗=1 ) 𝑁𝑘 𝑖=1 , (36) where nj denotes the number of terms in taylor series, for the approximation of matrix exponent of the constant matrix pk. from the monodromy matrix m, given in eq. (36), one can obtain floquet multipliers as its eigenvalues σ from relation: 𝑑𝑒𝑡(𝑴 − 𝜎𝑰) = 𝟎. (37) for eq. (14), the periodic matrix p (τ) can be obtained in the following form 𝑷(𝜏) = [ 0 1 − 𝜔0 2+3𝜃𝑞0 2 𝛺2 − 𝛾 𝛺 ]. (38) in this study, for approximation of the monodromy matrix the values nk = 5000 and nj = 5 are adopted as the optimal values compromising computational cost and precision. larger values for nk and nj wouldn’t offer some benefit in determination whether point is stable or unstable in this particular numerical example, but would increase computation time significantly. 5. numerical results here, the stability of the obtained periodic solutions of the forced duffing differential equation eq. (12) is checked by using the methodology given in the previous section. the ihb and continuation method are numerically applied to trace the periodic solutions branches of a nonlinear model of a nonlocal strain-gradient beam resting on a visco-pasternak foundation with direct transverse harmonic excitation. moreover, the floquet theory is applied to check the stability of periodic solutions for given amplitude-frequency curves. in the first part of the numerical study, we validate our model by comparing the beam's natural frequencies in two vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 719 simplified cases with the results available in the literature (tables 1 and 2). then, the validity of the results from the ihb method is examined, which is then followed by the parametric and stability study in the frequency domain. the parameters used in simulations are adopted from the paper [33] and extended with the parameters for the pasternak layer, fg material, and excitation force as given in table 3, unless other specified. the number of adopted harmonics in the fourier series is n = 6. the amplitudes corresponding to particular fourier coefficients (eq. (21)) and harmonics (eq. (20)) are computed as 𝐴0 = 𝑎0, 𝐴𝑖 = √𝑎𝑖 2 + 𝑏𝑖 2, (𝑖 = 1,2, . . . , 𝑁). (39) table 1 the first five non-dimensional fundamental natural frequencies of a local eulerbernoulli beam resting on winkler-pasternak foundation for the simply-supported boundary conditions (k1 = 25, k2 = 25) present ref.[63] ref.[64] ref.[65] ref.[66] 19.2133 19.2133 19.2133 19.21 19.2178 50.7002 50.7002 50.7002 50.7 50.7804 100.6767 100.677 100.6767 100.7 170.0281 170.028 170.0281 170.1 258.9868 258.987 258.9868 259.1 table 2 comparison of non-dimensional fundamental natural frequencies of simplysupported nonlocal euler-bernoulli beam with different nonlocal parameters μ (l = 10, h = 1, ρ = 1, e = 30∙106, ν = 0.3) μ present ref.[17] ref.[67] ref.[68] 0 9.8293 9.8696 9.8298 9.8696 1 9.3774 9.4159 8.3814 9.4159 2 8.9826 9.0195 8.9892 9.0195 3 8.6338 8.6693 8.6424 8.6693 4 8.3228 8.3569 8.3329 8.3569 table 3 parameter values of the presented mechanical model parameter symbol value unit young's modulus at top et 390 gpa young's modulus at bottom eb 210 gpa density at top ρt 3960 kg/m3 density at bottom ρb 7800 kg/m3 power-law index k 1 height of the beam h 100 nm width of the beam b 1 μm length of the beam l 10 μm nonlocal parameter μ 10 nm length scale parameter l 100 nm winkler coeff. of viscoelastic layer kw 1e-8 m-1 winkler coeff. of viscoelastic layer kw 1e-8 nsα/m3 pasternak coeff. of viscoelastic layer kg 1e-8 m pasternak coeff. of viscoelastic layer kg 1e-8 nsα/m amplitude of excitation force q 0.002 n 720 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari 5.1. verification table 1 compares the first five non-dimensional fundamental natural frequencies of a local euler-bernoulli beam resting on winkler-pasternak foundation for the simply supported boundary conditions with foundation parameters k1 = 25, k2 = 25, taking the values obtained in [63, 64, 65, 66], where fine agreement is achieved. besides that, we made a comparison of non-dimensional fundamental natural frequencies of simply supported nonlocal euler-bernoulli beam with different values of nonlocal stress-gradient parameter μ with data available in the literature [17, 67, 68]. these results are also in good agreement (table 2). to demonstrate the reliability and accuracy of the presented methodology for the determination of the amplitude-frequency responses and corresponding periodic solutions, the obtained results from the ihb are verified with the direct numerical integration by using the runge-kutta method. choosing the desired ihb periodic solutions at the corresponding excitation frequency (magenta and cyan stars) from fig. 2 (a, b) and extracting the initial conditions for the use in the runge-kutta method, two solutions are compared in fig. 3 where excellent matching of the results from two different methods ihb and runge-kutta can be observed for both excitation frequency examples. periodic solutions in fig. 3 are depicted in the phase plane where the velocity is given on the ordinate axis, while the displacement is given on the abscissa. 5.2. numerical results of stability analysis in this section, we investigate how excitation force, parameters of the functionally graded material, and foundation are affecting the frequency response curves in terms of the first and third amplitudes. for pinned-pinned boundary conditions, fig. 2 (a, b) shows the frequency response curves for the amplitudes a1 and a3, respectively, which are then given on the ordinate axis while the excitation frequency ω is on the abscissa. here, one can observe that the maximum amplitude experiences a rise at two resonance frequencies. two such regions can be noticed at frequencies around ω = 4 and ω = 16-27, depending on q. in the case of the first dominant amplitude a1, this change is weak in the first resonance region while it is very high for the second region, where the system experiences the hardening stiffness nonlinearity. here, one can observe jump-down and jump-up behavior as well as coexisting periodic solutions, which is the characteristic of this nonlinear phenomenon [69]. on the other side, despite the much lower values than a1, the amplitudes a3 show an increase in the first resonance region that is more pronounced compared to those occurring within the second region but displaying only the linear-like behavior. however, in the second resonance region, a strong nonlinear hardening stiffness behavior occurs even for very low values of the amplitude. an increase of the excitation force amplitude q increases the amplitude values and resonant picks. moreover, a forward frequency sweeping due to an increase of the excitation frequency ω, starting from small values, shows changes of response amplitudes up to the point when the periodic response loses its stability. this critical point known as fold point is given as upper brown dot in fig. 2. it can be observed that further forward frequency sweeping produces another branch of stable periodic solutions, which is also known as the lower branch of periodic solutions. vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 721 it can be noticed in fig. 2 that the upper branch of the amplitude grows until the critical moment (response peak) which represents the maximal value of the response amplitude. fig. 2 the amplitude-frequency response curves of the nlsgt beam on visco-pasternak foundation for pinned-pinned (a,b), clamped-pinned (c,d) and clamped-clamped (e,f) boundary conditions. amplitudes a1 and a3 are plotted for different values of external load parameter q. the unstable periodic solutions are represented by the tick solid lines. 722 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari at this peak point, periodic solutions lose their stability leading to the jump-down phenomena of the response amplitude i.e. a sudden jump of the response amplitude to some lower values. further forward frequency sweeping produces stable periodic solutions. on the contrary, the backward frequency sweeping process starts from values higher than the resonant frequency. by moving backward, response amplitude increases slowly until the critical point (another fold point) where the periodic solution loses its stability and the jump-up phenomena occur. if we further decrease the excitation frequency another stable (upper) branch of periodic solutions occurs. the branches of unstable periodic solutions displayed as thick solid lines can be detected in between the two different fold points of the same curve by applying the path following method in combination with the floquet theory. similar observations can be made for the fig. 2 (c, f), for clamped-pinned and clamped-clamped boundary conditions. fig. 3 periodic response obtained by ihb and rk method for two points fig. 4 shows the amplitude-frequency responses for four different values of the functional graded material parameter k. for pinned-pinned boundary conditions, the first (fig. 4 (a)) and the third (fig. 4 (b)) amplitude are traced, displaying both stable and unstable branches of the response. the linear growth of response amplitude a1 (or a3) appears for an increase of the excitation frequency from some small values until the fold point when the periodic response loses its stability. same as in the previous case, further forward frequency sweeping produces a lower branch of stable periodic solutions. therefore, the upper branch of the amplitude grows until the critical moment (response peak) which represents the maximal value of the response amplitude. at this peak point, periodic solutions lose their stability leading to the jump-down phenomena of the response amplitude. again, the backward frequency sweeping process starting from the values higher than the resonant frequency displays a slow increase until critical point where the periodic solution loses its stability and the jump-up phenomenon occurs. a further decrease of the excitation frequency gives the stable (upper) branch of the periodic responses. the branches of unstable periodic solutions can be detected between two instability points (lower and upper brown points in fig. 4) and they are represented by the thicker solid lines. in these unstable points, floquet multipliers exit the unit circle of the complex plane as displayed on the fig. 7. similar observations can be made for clampedpinned and clamped-clamped boundary conditions (fig. 4 (c, f)). vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 723 fig. 4 the amplitude-frequency response curves of the nonlinear nlsgt fg beam on viscopasternak foundation for pinned-pinned (a,b), clamped-pinned (c,d) and clampedclamped (e,f) boundary conditions. amplitudes a1 and a3 are plotted for different values of parameter k. the unstable periodic solutions are represented by the tick solid lines. 724 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari fig. 5 the amplitude-frequency response curves of the nonlinear nlsgt fg beam on viscopasternak foundation for pinned-pinned (a,b), clamped-pinned (c,d) and clampedclamped (e,f) boundary conditions. amplitudes a1 and a3 are plotted for different values of parameter kg. the unstable periodic solutions are represented by the tick solid lines. vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 725 further, fig. 5 shows amplitude-frequency response curves for changes of the parameter kg of the visco-pasternak foundation. one can observe that the change of this parameter yields similar shapes of amplitude-frequency response curves exhibiting the nonlinear hardening stiffness behavior. moreover, an increase of parameter kg, related to viscous properties of the foundation, results in a decrease of pick amplitude values. consequently, this yields shortened unstable branches (thick bold lines) of the periodic solutions located in between the corresponding fold points (fig. 5). this behavior can be observed for both amplitudes a1 and a3. for clamped-pinned and clamped-clamped boundary conditions, an increase of parameter kg can even stabilize the system by making an unstable region (bold line) disappear. a more detailed parametric study of amplitude-frequency responses of the similar system with nonlocal strain gradient functionally graded beam on the fractional visco-pasternak foundation can be found in the paper by nešić et al. [27]. to study the influence of the type of boundary conditions on the amplitude-frequency response curve three different cases (pinned-pinned, clamped-pinned, and clampedclamped boundary conditions) are plotted in fig. 6. it can be observed that reducing degrees of freedom in supports decrease maximal amplitude and shifts resonant frequency to the right, which is expected behavior. fig. 6 different boundary conditions in all previous examples, the unstable periodic solutions are detected by the calculation of floquet multipliers, where at least one of them crosses the unit circle in the complex plane in the +1 direction. fig. 7 displays a unit circle in the complex plane and floquet multipliers for the case of parameters adopted from table 3. it can be observed that floquet multiplies are crossing the unit circle in the complex plane in the +1 direction for the unstable periodic solutions located between the two different fold points of the amplitudefrequency response curve. it is well known that different bifurcation points can be detected depending on where the floquet multipliers are crossing the unit circle. therefore, when floquet multipliers are crossing the unit circle in the +1 direction, one can expect that a saddle-node bifurcation point occurs in that case. we also observe that number of floquet multipliers crossing the unit circle is the highest in the case of the simple supports and that the ones that are crossing the unit circle are the most distant to it compared to other 726 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari boundary conditions. on the other hand, in the case of clamped-clamped supports, the number of floquet multipliers crossing the unit circle and their distance to it is minimal. consequently to this behavior, clamped-clamped boundary conditions contribute to the most and pinned-pinned boundary conditions to the least stable system. therefore the influence of the boundary conditions on stability cannot be neglected. fig. 7 floquet multipliers crossing unit circle (vertical axis is imaginary, horizontal real): a) simply-supported, b) clamped-pinned and c) clamped-clamped boundary conditions fig. 8 shows a pair of floquet multipliers for points with different excitation frequency ω. we can observe in fig. 8 (a, b) that between approximately ω = 14.5 and ω = 23 at least one of two floquet multipliers has modulus greater than one, meaning that this region is unstable. the stability border is reached when floquet multipliers modulus (fmm) is equal to one. the degree of instability is defined by the distance of a point in the instability region from the unit circle (fig. 7) which is equivalent to the distance of slowly growing curve (with small slope) on fig. 8. for example, for the case of simply-supported beam and ω = 16 instability is higher than in the case ω = 22 (fig. 8 (a, b)). we can also observe from fig. 8 that with an increase of the system's stiffness by reducing degrees of freedom in boundary conditions, the main instability region is decreased and shifted to the right towards higher excitation frequencies. the stability of the solution for higher excitation frequencies can be achieved by introducing the pinned-pinned boundary conditions. on the contrary, stability on lower excitation frequencies can be reached with clamped-clamped (a) (b) (c) vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 727 boundary conditions. an unstable region with clamped-free boundary conditions is between unstable regions for another two pairs of boundary conditions. this result can be utilized for the practical realization of such a system. depending on the excitation frequency and operation range of the system, boundary conditions can be selected and physically established so that unstable solutions for amplitudes are avoided, which can prolong the duration of the system. fig. 8 floquet multipliers modulus (fmm) of the first (fmm1) and second (fmm2) floquet multiplier versus excitation frequency ω moreover, in the amplitude-frequency responses, one can notice that the coexisting periodic solutions exist for certain frequency ranges. knowing that the amplitudes of these periodic solutions are dependent on initial conditions and starting positions in excitation frequency sweeping, there are two sets of initial conditions for which the response amplitude will converge towards the lower or upper branch of the periodic solutions forming the basin of attraction as given in fig. 9 and fig. 10. therefore, the system will possess two attractors for the given excitation frequency and system parameters. these two attractors are highlighted in two different colors for two sets of initial conditions. for the presented nonlinear system, the basin of attraction is given in fig. 9 and fig. 10 and for the excitation frequency ω = 20 and ω = 25 and coexisting periodic solutions from the 728 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari amplitude-frequency response curve given in fig. 2 and varying initial conditions. one can observe that the basin of attraction is displaying two colors with two coexisting periodic solutions in fig. 9 and fig. 10. here, the case with the initial condition highlighted in red color converges to the upper stable branch while the case with the initial conditions highlighted in blue converges to the lower stable branch. since excitation frequency is close to the bifurcation point, the solution converges to both upper and lower stable branches of periodic solutions. in the special case of simple-supported beam, the set of initial conditions that converges to the upper stable branch of periodic solutions is dominating over the set of initial conditions that converges to the lower branch of solutions. fig. 9 basin of attraction for ω = 20 fig. 10 basin of attraction for ω = 25 for clamped-clamped boundary conditions to summarize, in this chapter, after a short verification of eigenfrequencies for simplified model with the results available in the literature, stability analysis is performed. for different values of excitation frequency acting in transversal direction, the first (a1) and the third (a3) harmonic of the amplitudes for displacement of the beam in transversal direction are obtained and given in the figs. 2, 4, 5. these figures, for given frequency, give us information about the number of solutions (1, 2 or 3) for each of the amplitudes vibration and stability of a nonlinear nonlocal strain-gradient fg beam on a visco-pasternak... 729 (a1 and a3) and whether the solution is stable or not. all this is to predict the vibration behavior of the beam at certain frequency and set of the parameters. selected solution for amplitude is stable if a pair of floquet multipliers is in the unit circle of the complex plane as presented on the fig. 7. floquet theory is basically a tool for proving stability. in figs 7-8 are graphically given values of floquet multipliers for different values of excitation frequency from the range ω = 0 – 30, enabling reader to determine how far from stability border are solutions at particular excitation frequency. finally, for specific frequency, which is in the range of multiple amplitude solutions, basins of attractions are given in figs. 9-10, describing whether amplitude will be greater or smaller, depending on initial conditions. 6. conclusions in this work, a stability analysis of a nonlinear nonlocal strain-gradient beam resting on the visco-pasternak foundation is performed. the governing equation of the system is derived by using hamilton's principle and then discretized by using the galerkin approximation method. the periodic solutions of the strongly nonlinear differential equation are then sought by using the incremental harmonic balance method in combination with the continuation technique. the results obtained by the proposed approach are verified with the direct numerical integration technique (runge-kutta method) and good agreement is confirmed. a parametric study is performed through the amplitude-frequency responses and the stability of each periodic solution is examined by using the floquet stability theory and hsu procedures. the main contributions of this work can be summarized as follows: ▪ the parametric study is performed and stability of obtained periodic solutions of a nonlocal strain-gradient functionally graded beam on the visco-pasternak foundation is investigated. it is demonstrated that the hardening stiffness nonlinearity becomes more apparent for an increase of the excitation amplitude and for lower values of the parameter of functionally graded material, which consequently leads to an increase of the unstable branches of periodic solutions. opposite to this, an increase of the parameter related to viscous properties of the foundation yields lower picks in the amplitude-frequency response and smaller unstable branches, therefore increasing the stability of the system. ▪ investigation of the floquet multipliers and their behavior in the complex plane revealed the nature of bifurcation points where unstable solution appears. moreover, the exiting of floquet multipliers out of the unit circle in the +1 direction demonstrated the existence of saddle-node bifurcation points. ▪ modulus of floquet multipliers is compared for different values of excitation frequencies and shown that in an unstable region, instability degree is higher for lower excitation frequencies. ▪ to obtain stable solutions on higher excitation frequencies, pinned-pinned boundary conditions should be applied. on the contrary, stability on lower excitation frequencies can be reached with clamped-clamped boundary conditions. an unstable region with clamped-free boundary conditions is between unstable regions for another two pairs of boundary conditions. ▪ the occurrence of coexisting periodic solutions and hardening stiffness effect in the strongly nonlinear functionally graded beam system is confirmed as well as their dependence on the initial conditions and starting positions of the excitation frequency sweeping. therefore, the basin of attraction is obtained for such a system to examine which of the stable periodic solutions will prevail for different sets of initial conditions. 730 n. nešić, m. cajić, d. karličić, m. lazarević, s. adhikari the paper illustrates how the incremental harmonic balance method in combination with the continuation technique and floquet theory can be efficiently utilized to study the stability of strongly nonlinear structural vibration systems with different sizes, boundary conditions, and foundation effects included. the future perspectives of the proposed research and the developed model is application of the model and methodology presented in this paper in the development of mems devices, since the presented model describes oscillating nonlocal beam, and nonlocal theory is used in small scale systems. additionally, when neglecting two nonlocal parameters, presented model of beam oscillating on viscopasternak foundation with four adjustable parameters can be used to describe wide range of foundations used in civil and mechanical engineering and oscillatory behavior of the nonlinear beam resting on it. acknowledgement: authors n. nešić, d. karličić and m. lazarević acknowledge the support by the serbian ministry of education, science, and technological development through the faculty of technical sciences, university of priština in kosovska mitrovica, the mathematical institute of the serbian academy of sciences and art and faculty of mechanical engineering, univerisity of belgrade (grant no. 451-03-68/2022-14/200105). authors m. cajić and s. adhikari acknowledge funding from european union’s horizon 2020 research and innovation program under the marie skłodowska-curie grant agreement no. 896942 (metasink). references 1. ghayesh, m.h., farajpour, a., 2019, a review on the 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jump-up and jump-down frequencies of the duffing oscillator, journal of sound and vibration, 318(4-5), pp. 1250-1261. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 229 239 evidence of the semi-solid formation in the medical grade ti6al4v alloy using induction heating  udc 620 carlos roberto fernandes, beatriz luci fernandes pontifical catholic university of paraná, department of health technology, curitiba, brazil abstract. one alternative for processing cost reduction with simultaneous improvement of the mechanical properties of the medical ti6al4v alloy is to get its semi-solid feedstock with a non-dendritic microstructure for further processing. the purpose of the present work is to evaluate the possibility of obtaining a semi-solid ti6al4v alloy by heating it up from the room temperature to the range temperature between the lines solidus and liquidus, using induction heating. the ti6al4v billets underwent heat treatment and quenching for semi-solid formation using a designed device and specific time pulsed profile. the billet temperature reached 1630 oc, and after the cooling rate of 54 oc/s, some samples formed a globular phase characteristic of the semi-solid alloy. this study shows that it is possible to get a semi-solid microstructure of this alloy starting from its solid state. key words: ti6al4v eli, semi-solid state, induction heating, semi-solid ti alloy 1. introduction titanium alloys are widely used in various fields of engineering and medicine due to their high tensile strength, biocompatibility and corrosion resistance [1]. the medical ti6al4v alloy is the biomaterial of choice for many orthopedic applications as implants and prostheses, manufactured by conventional metallurgical processes and posterior heat treatments for stress relief [2]. hot forging, for instance, is required for the ti6al4v alloy that involves heating of the die to usually about 200 °c, and of the workpiece between 900 °c and 1000 °c to avoid residual stress and defects in the product. moreover, the processing must be fast to prevent cooling of the workpiece with a consequently increasing forging force and risk of damaging of the die [3]. because of the high production costs of the ti6al4v alloy and the difficulties to produce a material with received october 30, 2015 / accepted november 26, 2015  corresponding author: beatriz luci fernandes pontifical catholic university of paraná, department of health technology, r. imac. conceição, 1155, curitiba, brazil e-mail: beatriz.fernandes@pucpr.br original scientific paper 230 fernandes c.r., fernandes b.l. predetermined service properties [4], new processing methods have been suggested and studied. the conventional ti6al4v has the young modulus between 105 – 120 gpa, elongation about 8% and hardness of about 399 hv [5]. the microstructure, and consequently, the mechanical properties of ti alloys are very sensitive to the thermomechanical processing parameters. at room temperature the ti6al4v is a two-phase (α + β) alloy, and although the body-centered cubic (bcc) β phase is desirable to aid deformation of the alloy (young modulus about 73 gp, elongation between 45% and 95%, and 336 hv[5]) the β grains are not abundant (< 10%). therefore, the plastic strain is limited by the slip systems of the hexagonal close-packed (hcp) α phase and the mechanical properties of the alloy reflect the α phase properties. at high temperatures, however, especially above 900 °c, the amount of β phase increases abruptly [6]. the high temperature phase distribution of the ti6al4v alloy is, therefore, interesting to improve its formability. one alternative for processing cost reduction with simultaneous improvement of the mechanical properties of metal alloys is developed by flemings and co-workers at the mit in the 1970s called semi-solid metal forming (ssf). the process was originally developed using lightweight aluminum alloys for further industrial applications and, since then, thixoforming and rheocasting became an alternative to the traditional casting. the main purpose of the process is to get a semi-solid alloy feedstock with non-dendritic microstructure [7]. induction melting process has been widely used as a method for melting high-purity metals. the heating occurs via electromagnetic field that induces an eddy current resulting in the joule heating of the metal. the electromagnetic field is imposed by a coil [8]. until now, the induction heating has been suggested for semi-solid preparation starting from the melted metal and only for aluminum alloys. however, for application of this technology in ti alloys, the melting temperature of about 1670 o c [9] is a limiting factor. the purpose of the present work is to evaluate the possibility of obtaining a semi-solid ti6al4v alloy by heating it up from the room temperature to the range of temperature between the lines solidus and liquidus using induction heating. 2. materials and method to evaluate the possibility of obtaining a semi-solid titanium alloy, the alloy with the following mass composition was used: 89.58% ti; 6.14% al; 4.00% v; 0.16% fe; 0.11% o; 0.003% c; 0.005% n; 0.002% h. the raw material was provided in an extruded bar with 9.53 mm in diameter. six billets were cut from the center of the bar, each one of 40 mm in length, using a metallographic precision cutter. two billets were used for the measurement calibration and the electromagnetic induction equipment and discarded, another two were used as control specimens and for evaluation of the microstructure of the alloy without treatment, and the remaining two underwent a semi-solid treatment. to apply the electromagnetic induction to the billets a specific device was designed and built that allows the correct positioning of the billets within the induction coil without touching them. the device consists of an epoxy-painted stand made of carbon steel and the rubber feet that permit height and level adjustment to align the billet with the induction coil. since, at high temperatures, titanium shows a close affinity to oxygen, the device contains a chamber for an internal atmosphere control with an injection of argon or some other inert gas. the chamber is constructed of 5 mm thick plates of an epoxy resin and fiberglass evidence of semi-solid formation in medical grade ti6al4v alloy using induction heating 231 composite, non-metallic materials, to avoid the magnetic field to influence them. to allow for billet visualization, the upper part of the chamber has a 6 mm thick tempered glass window fixed and sealed with high temperature silicone. at the bottom of the chamber there is a trapdoor to permit the billet to fall quickly into a reservoir containing cold water as soon as the desirable surface temperature is reached. this procedure is necessary in order to freeze the metastable microstructure. the billets are positioned in the chamber as shown in fig. 1 in order to obtain semi-solid microstructure. the reservoir beneath the chamber is loaded with 40 l of cool water (5  1 °c) measured with an alcohol thermometer column. for the billet temperature measurement a laser pistol raytek  model raynger 3i plus is used, with 1 o c accuracy and temperature range from 700 to 3000 o c calibrated with a platinum thermocouple and a temperature controller minipa  model mt-510, suitable for direct reading of a specimen. the chamber is closed and argon is allowed to flow through the tube connecting the chamber to the gas cylinder. the programmable medium frequency induction heating jmmf  , 5 kw maximum power, is turned on and the billet temperature is monitored uninterruptedly with the laser pistol positioned vertically at 300 mm far from the upper surface of the billet and set to the center of it. the sequence of the heating treatment for the samples from billet 1 and billet 2 are presented in table 1. fig. 1 chamber sealed with silicone: (a) trapdoor; (b) argon ingate; (c) billet and (d) coil the billet temperature rises rapidly to 1630 o c which is maintained for 3 s and the trapdoor is opened releasing the billet into the cold water reservoir and freezing the microstructure. the two billets prepared in this way together with the two control billets undergo the metallographic routine to allow the microstructure evaluation in the optical microscope: cutting with a minitom precision cutter struers  with diamond disc; embedding in the cold resin to avoid microstructural changes; grinding sequentially with paper grit 220, 400, 600 and 1200; polishing with soft cloth impregnated with alumina particles of 0.3 µm in diameter; and etching with the keller's reagent. the microstructural evaluation is performed in an optical microscope (om) olimpus  model cx31rbsfa. 232 fernandes c.r., fernandes b.l. table 1 heating treatment sequence of the samples from billet 1 and billet 2 machine power (%) pulse (on/off) time (s) initial temperature ( o c) final temperature ( o c) 75.0 continue 90 32 1300 50.0 intermittent 1s/1s 30 1300 1630 50.0 intermittent 1s/2s 6 1630 1630 3. results the parts cut transversally and longitudinally to the billet axis and embedded in the resin of the first billet are shown in fig. 2. the om images from the ti alloy as supplied can be seen in fig. 3. fig. 2 (a) parts for om evaluation showing the faces analyzed nominated as a1, b1 and c1 for the billet 1 and a2, b2 and c2 for the billet 2; (b) left: sample longitudinal to the billet axis; right: sample transversal to the billet axis. it is possible to see the beginning of the melting process revealing, therefore, that the temperature inside the part achieves the melting point of the ti alloy (1670 o c) while the temperature of the billet surface is 1630 o c, measured by the laser pistol. fig. 3 om images from the billets as supplied. (a) image took from field b (figure 2a) close to the center of the sample,100x and (b) 400x. a) b) a) b) evidence of semi-solid formation in medical grade ti6al4v alloy using induction heating 233 one can see that the microstructure is relatively homogeneous in 400x magnification. fig. 4 presents the om images from the marked fields. from the microstructure shown in fig. 4 it is possible to suggest that with the heat treatment employed, a different microstructure from the as supplied one is formed, indicating that freezing of the metastable microstructure has been performed. the cooling rate is, proximately, 54 o c/s. the darker area inside the globular formations is characteristic for the  phase of the ti6al4v alloy [10]. fig. 5 shows the microstructure of the first billet, cut longitudinal to its axis. fig. 4 (a) samples b1 and c1 from the billet 1 showing the fields analyzed, 400x; (b) field 1, detail of globular formations; (c) field 2, detail of globular formations; (d) field 3 and (e) field 4 a) b) c) d) e) 234 fernandes c.r., fernandes b.l. fig. 5 (a) sample a1 from billet 1 showing the fields analyzed; (b) field 5, detail of globular formations, 400x; (c) field 6, detail of globular formations, 400x; (d) and (e) field 7 and 8, respectively with no apparent globular formation, 400x fig. 6 presents the om images from the marked fields for the second billet. it shows that, in these fields of samples b2 and c2, the globularization of the alloy fails. in fig. 6e it is possible to notice a fully lamellar coarsed microstructure, one of the three microstructures found in the annealed ti6al4v alloy (fully lamellar, fully equiaxed, and bimodal) [11]. probably this field has reached a temperature above 995 o c (β transus) but below the semi-solid range. therefore, an extensive grain growth occurs and large grains are formed which have transformed into lamellar α + β upon cooling [12]. a) b) c) d) e) evidence of semi-solid formation in medical grade ti6al4v alloy using induction heating 235 fig. 6 (a) samples b2 and c2 from the billet 2 showing the fields analyzed: on the left the area taken from the center of the billet and on the right took from its bottom; (b) field 1, apparently with no globular formations, 400x; (c) and (d) field 2 and 3, respectively, with one globular formation with a phase inside, 400x; (e) field 4, detail of grain boundaries, 400x a) b) c) d) e) 236 fernandes c.r., fernandes b.l. fig. 7 displays the om images from the marked fields for the second billet. fig. 7 (a) sample a2 from billet 2 showing the fields analyzed: on the left side the center of the billet and the right side its bottom; (b), (c) and (d) field 5, 6 and 7, respectively, showing globular formations, some in a visible coalescence phase (arrows), 400x; (e) field 8 showing one globular formation with a phase inside, 400x 4. discussion the semi-solid metal is obtained, usually, by heat treatment of the alloy in the muffle furnace with or without stirring or by induction heating. among these processes, induction heating has the advantages of a short period of warm up, versatility in the workpiece sizes and energy efficiency. some alloys such as ti6al4v, however, form a a) b) c) d) e) evidence of semi-solid formation in medical grade ti6al4v alloy using induction heating 237 dense and protective layer of oxides at the semi-solid temperature range requiring atmosphere free of oxygen during the process. in the present work a device is designed and set up to evaluate the possibility of obtaining a semi-solid ti6al4v alloy within a chamber with argon rich atmosphere. during the process the chamber has shown efficiency in protecting the billet against excessive oxidation. the equipment used in this work is a medium frequency induction heater that has the characteristic of heating the workpiece positioned in the center of the coil, from its surface to the center. fig. 5a displays cavities in the center of the billet that seem to have been formed by core melting. therefore, although the temperature read at the top of the billet is 1630 o c, its center may have reached at least 1660 o c [9] the melting point of the alloy. although the intermittent pulses 1s on/2s off during 6 s shown in table 1 guarantee that the temperature at the surface of the billet remains in the range of semi-solid formation it does not happen in its center. this occurrence can be explained by the argon flow at room temperature constantly cooling the surface where the temperature is taken. the homogenization of the temperature in the center could be achieved by maintaining the intermittent pulses for a period longer than 6 s and at lower power also allowing homogenization of the microstructure and maintaining the argon atmosphere with no flow. sample a1, taken from the first billet, form some globular phase characteristic of the semi-solid alloy better seen in figs. 4 (b and c). comparing the microstructure of fig. 3 to fig. 4 to fig. 7, it is possible to suppose that the heat treatment provides different microstructures that are not stable as those of the supplied material, suggesting that the freezing of the high temperature microstructure, the metastable energy state, is achieved by the cooling rate of 54 o c/s. at the room temperature the low concentration β phase is present in the ti6al4v alloy in the lamellar formation of about 200 µm in thickness alternated with the lower concentration α phase. at temperatures above 900 °c the amount of the β phase increases [6, 11]. thus, in the range of temperature for semi-solid, it is expected that spheroid microstructures should appear 13 and almost all α phase has been transformed into β phase. although in a low amount, it is possible to see the β phase globular formation in figs. 4-7. the lowest quantity of the β phase (darker) compared to the α phase (lighter) in the figs. 4e and 5 (d and e) can be attributed to an insufficient residence time at the temperatures above 900 °c. some characteristic semi-solid formation, the globular β phase (darker), can be seen in figs. 4 (b and c), 5 (b and c) and 7 (b, c and d). in the center of sample a2, field 7, shown in fig. 7d, it is possible to notice the β globular phase in formation. as discussed earlier, the central region does not receive the same thermal treatment as the surface regarding heating as well as the cooling process when the billet is dropped in cold water. also in sample a2, fields 5, 6 and 7, fig. 7 (b, c and d) a significant amount of the β phase is observed. the longer the permanence in the semi-solid range of temperature the greater the globular formation [6]. fields 5 and 6 are located near to the sample surface, fig. 7 (b and c) and the field 7, fig. 7d, in the center of the sample but near to the bottom of the billet where the heat dissipation is faster than in the center of the billet, fig. 7e, justifying the coalescenting of the β phase. besides, the coalescent globules in fig. 7 (b, c and d) are an important indicator that the residence time in the semi-solid temperature range is sufficient for the initial formation of a semi-solid alloy, because during the semi-solid formation small globules tend to join each other to form larger ones [14]. the microstructural differences between the samples from billet 1 and billet 2 can only be justified by further tests under more controlled conditions than those used in this 238 fernandes c.r., fernandes b.l. work. the main difficulty of reaching a semi-solid microstructure of the ti6al4v alloy is its narrow temperature range between the solidus and the liquidus lines that is the range where the semi-solid globular structure appears [15]. however, from the results of the present work, despite the narrow range of semi-solid temperature of the ti6al4v alloy, it seems possible to get it in a semi-solid state. 5. conclusion although heat treatments of the ti6al4v alloy have been extensively investigated and some studies have been performed on its semi-solid formation from the melting to room temperature, until now, no investigation has been done regarding the semi-solid structure achievement by heating the alloy from the room temperature to the semi-solid range one. this work has explored this possibility by using heating induction and the equipment that requires low investment and is applicable to many alloys. the authors have realized that induction heating of the cylindrical workpieces creates a heating profile more complex than the plane ones; hence it will be taken into account for further works. despite the limitations of this study, it is new in the metallurgical field and there are no approaches to support our findings. therefore, based on the results, the next steps would be to build a better controlled environment in order to get stronger evidence for the semi-solid acquisition. acknowledgements: the authors would like to thank jamo equipamentos ltda, vick comércio de plásticos e metais ltda and grade institute of basic science (igcb). references 1. sampaio, m., buciumeanu, m., henriques, b., silva, f.s., souza, j.c.m., gomes, j.r., 2016, tribocorrosion behavior of veneering biomedical peek to ti6al4v structures, journal of the mechanical behavior of biomedical materials, 54, pp. 123-130. 2. leyens, c., peters, m., 2003, titanium and titanium alloys. fundamentals and applications,willey-vch verlag gmbh & co. kgaa, weinheim, germany, 513 p. 3. bridges, p.j., magnus b., 2001, manufacture of titanium alloy components for aerospace and military applications, proc. rto avt specialists on cost effective application of titanium alloys in military platforms, loen, norway. 4. gronostajski, z., bandola, p., skubiszewski, t., 2010, argon-shielded hot pressing of titianium alloy (ti6al4v) powders, acta of bioengineering and biomechanics, 12(1), pp. 41-46. 5. vrancken, b., thijs, l., kruth, j-p., van humbeeck, j., 2014, microstructure and mechanical properties of a novel b titanium metallic composite by selective laser melting, acta materialia, 68, pp. 150–158. 6. yang, l-q., yang y-q., 2014, deformed microsctructure and textule of ti6al4v alloy, transactions of nonferrous metals society of china, 24, pp. 3103-3110. 7. zhang, l., li, w., yao, j.p., 2013, microstructures and thermal stability of the semi-solid 2014 aluminun alloy prepared using the pulsed magnetic field process: effects of technological parameters, journal of alloys and compounds, 554, pp. 156-161. 8. jang, b.y., kim j.s., ahn, y.s., 2011, induction melting process using segmented graphite crucible for silicon melting, solar energy materials & solar cells, 95, pp. 101–106. 9. li, j.j.z., johnson, w.l., rhim, w-k., 2006, thermal expansion of liquid ti-6al-4v measured by electrostatic levitation, applied phisics letters, 89(111913), pp. 1-2. 10. zherebtsov, s., murzinova, m., salishchev, g., semiatin, s.l., 2011, spheroidization of lamellar microstructure in ti-6al-4v alloy during warm deformation and annealing, acta materialia, 59, pp. 4138-4150. evidence of semi-solid formation in medical grade ti6al4v alloy using induction heating 239 11. salem, a.a., shaffer, j.b., satko, d.p., semiatin, s.l., kalidindi, s.r., 2014, workflow for integrating mesoscale heterogeneities in materials structure with process simulation of titanium alloys, intregrating materials and manufacturing innovation, 3(24), pp. 1-22. 12. vrancken, b., thijs, l., kruth, j-p., humbeeck, j.v., 2012, heat treatment of ti6al4v produced by selective laser melting: microstructure and mechanical properties, journal of alloys and compounds, 541, pp. 177-185. 13. chen, q., chen, g., han, l., hu, n., han, f., zhao, z., xia, x., wan, y., 2016, microstructure evolution of sicp/zm6 (mgendezn) magnesium matrix composite in the semi-solid state, journal of alloys and compounds, 656, pp. 67-76. 14. xing, b., hao, y., li, y-d., ma, y., chen, t-j., 2013, microstructure control of az31 alloy by self-inoculation method for semisolid rheocasting, transactions of nonferrous metals society of china, 23, pp. 567−575. 15. kliauga, a.m., ferrante, m., 2002, the effect of sn additions on the semi-solid microstructure of an al-7si0.3mg alloy, materials science and engineering, 337(1-2), pp. 67-72. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 41 50 https://doi.org/10.22190/fume171229010d © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper simulation of fracture using a mesh-dependent fracture criterion in the discrete element method udc 539.4, 519.6 andrey dimaki 1 , evgeny shilko 1 , sergey psakhie 1 , valentin popov 2 1 institute of strength physics and materials science sb ras, tomsk, russia 2 berlin university of technology, berlin, germany abstract. recently, pohrt and popov have shown that for simulation of adhesive contacts a mesh dependent detachment criterion must be used to obtain the mesh-independent macroscopic behavior of the system. the same principle should be also applicable for the simulation of fracture processes in any method using finite discretization. in particular, in the discrete element methods (dem) the detachment criterion of particles should depend on the particle size. in the present paper, we analyze how the mesh dependent detachment criterion has to be introduced to guarantee the macroscopic invariance of mechanical behavior of a material. we find that it is possible to formulate the criterion which describes fracture both in tensile and shear experiments correctly. key words: fracture, mesh-dependence, discrete element method, particle size 1. introduction since the work of hertz [1] it is well-known that stress distribution in a contact area is not uniform. stress distribution in the contact between an elastic half-space and sharpedged rigid or elastic counter-bodies of a different shape (e.g. a cylinder, frustum etc.) tends to infinity in a vicinity of the contact area boundary [2, 4-6]. in reality, stresses at the contact area boundary never achieve infinite values due to surface roughness and/or plastic deformation [3]. nevertheless, these stresses are several times higher than in the center of the contact area. the presence of high stress concentrations near contact patch boundaries, notches etc. leads to nucleation of cracks on different scales [7, 8]. in order to describe conditions of fracture in the regions with heterogeneous stress/strain fields, non-local fracture criteria received december 29, 2017 / accepted february 05, 2018 corresponding author: andrey v. dimaki institute of strength physics and materials science sb ras, 634055, tomsk, russia dav@ispms.tsc.ru 42 a. dimaki, e. shilko, s. psakhie, v. popov have been developed. perhaps one of the most popular nonlocal fracture criteria is novozhilov fracture criterion [9] which reads 1 1 0 1 ( ) d y cx dx d    , (1) where σc is the strength of a homogeneous material under uniaxial tension without stress concentrators, σy(x) is the maximum normal stress, x axis corresponds to a “most probable” direction of crack propagation, d1 is a material constant having a dimensionality of length. similar forms of nonlocal criterion (1) were used in [10] for composites and in [11] for notched samples. another way of taking into account stress concentrators is to include both absolute value of stress (either maximal or average value) and its gradient in a crack processing zone: 0( , / )e cf l l   , (2) where l0 is a characteristic size of material structure elements, le is a size of stress concentration zone [12]. in the framework of this approach the size of stress concentration zone depends on a local stress gradient: le = le(grad(σ)). when developing a numerical model of the contact, or, more generally, of a sample with stress concentrators, it is necessary to adequately describe crack nucleation conditions since they significantly depend on stress distribution approximation accuracy that is determined, in turn, by mesh size. correspondingly, variation of the mesh size may have influence on the values of mechanical characteristics of the simulated samples, e.g. their compressive/tensile strength, etc. the traditional way of describing heterogeneous stress/strain fields, based on mesh refinement, is not applicable to problems where “quasi-infinite” local values of stresses can occur. these problems include friction and wear, contact problems with punches having sharp edges, etc. in this case, in order to adequately describe fracture in numerical models, nonlocal fracture criteria of types (1) and (2) are widely used. examples of their applications in the finite-element method (fem) can be found in papers [13, 14] and in many others. recently, popov and pohrt have suggested a mesh-dependent detachment criterion in the boundary element method (bem) for a normal adhesive contact problem of linearly elastic and power-law graded elastic materials [15-17]. the main idea of the suggested approach is a modification of local ultimate stress value instead of stress field correction. the given criterion is local and it allows adequate descriptions of a normal contact between an elastic half-space and a rigid indenter in the presence of adhesive forces. despite a wide application of non-local and mesh size dependent fracture criteria in continuum-based approaches like fem, the size-dependent fracture criteria in dem still remain much less studied [18]. in the paper we study the relation between tensile and shear contact strength and a discrete element size. hereinafter the term “contact strength” means a maximal value of a force divided to the contact square that is required to detach the contacting bodies. based on the obtained results, we suggest a mesh-dependent local fracture criterion which allows us to overcome the mentioned problem, namely to avoid dependence of the contact strength on element size. simulation of fracture using a mesh-dependent fracture criterion in a discrete element method 43 2. formulation of the problem let us consider a contact between an elastic half-space and an elastic punch, which are initially bonded (see fig. 1). we study dependence of the contact tensile strength on element size d. a displacement with constant vertical velocity is applied to the upper layer of the punch while the lower layer of the half-space is fixed in vertical direction. horizontal motions of elements of the top boundary of the punch and the bottom boundary of the halfspace are allowed. the materials of the punch and the half-space are considered linearly elastic. we used the movable cellular automaton method (mca) [19-21] which is a dem family representative with multi-particle distinct element formulation of the equations of elasticity and motion. fig. 1 schematic of the simulated sample; element size d=0.001 m the elastic modulus of the half-space was ehalf-space = 200 gpa, the elastic modulus of the punch epunch was varied in a certain range in order to perform a parametric study of the problem. the values of the poisson’s ratios were νpunch = νhalf-space = 0.3. the value of the ultimate stress in von mises fracture criterion was ymises = 200 mpa that leads to relatively small values of strains in the punch and the substrate at the moment of fracture. the strain rate was of the order of  ≈ 0.3 sec -1 that provides a quasi-static loading regime. we used a finite-size punch and a large counter-body which imitates a half-space. evidently, the boundary conditions produce a contribution to the values of sample strength obtained during simulations. in order to diminish influence of the boundary conditions we studied the convergence of the strength value when thickness and height of a punch and a counter-body simulating a half-space increase. the values of thickness h and width w of the counter-body greater than five widths of the punch 2a0 guarantee convergence of the value of the tensile strength and, consequently, the absence of the boundaries’ influence. punch height h>3a0 guarantees tensile strength being independent of h. the following values of the sizes of the punch and the “half-space” were used in the calculations described below: a0=0.01 m, h=0.03 m, h=0.125 m, w=0.1 m. the closed package of 44 a. dimaki, e. shilko, s. psakhie, v. popov discrete elements was used. it should be noted that the use of a close-packed ensemble of circular discrete elements leads to the appearance of an "artificial roughness" on the boundaries of the sample (see fig. 1). 2.1. description of the discrete element model in the framework of mca a solid body is considered as an ensemble of interacting particles (elements) of finite size. contacts of interacting pairs of elements are considered to be initially bonded that simulates a consolidated solid, while the contacts between crack faces are considered as unbonded. switching between bonded and unbounded states occurs when a given fracture or linkage criterion is satisfied. interaction between the contacting elements (either bonded or unbonded) is realized through their common plane faces. the evolution of an ensemble of discrete elements is defined by a numerical solution of the system of newton-euler equations of motion: 2 2 1 1 ( ) i i n ni i i i ij ij j n i i ij j d r dv m m f f dtdt d mj dt               , (3) where ir , iv and i are the radius-vector, velocity vector and angular velocity of discrete element i, mi is the element mass, ji is the moment of inertia of an equivalent disc or sphere, n ijf and ijf  are the forces of central (normal) and tangential interaction between element i and its neighbor j, and ij ij ij ijm q n f   is the torque. equation (3) describes translational and rotational motion of the discrete elements having a finite size. both the constituents ( n ijf and ijf  ) of the interaction force between discrete elements i and j in eq. (3) include potential ( np ijf and p ijf  ) and dissipative/viscous ( nv ijf and v ijf  ) contributions [19]: n np nv ij ij ij p v ij ij ij f f f f f f         . (4) as follows from eq. (4), the value of torque ijm includes both potential and dissipative constituents. the major features of the mca method are the approximation of a homogeneous stress-strain distribution in a simply deformable element and the postulated form of the relation for the reaction force of a discrete element in response to the action of its neighboring element. in the simply deformable element approximation the state of a discrete element is determined by average stress tensor i  and average strain tensor i  [21] calculated based on the specific values of normal and tangential forces in interacting pairs of discrete elements. tensors i  and i  are assumed to be related by an assigned constitutive law. this law determines a relationship between force and spatial interaction parameters for a pair of elements. the mca method postulates a structural form of relations for the central and tangential interaction forces of discrete elements. we have used a generalized hooke’s law simulation of fracture using a mesh-dependent fracture criterion in a discrete element method 45 for isotropic materials as constitutive relation ( )i i    for the elastic response of a simply deformable discrete element. the developed approach allows using multi-parametric failure criteria (von mises, mohr–coulomb, drucker–prager and others) for simulation of bond breaking between the pairs of interacting discrete elements. in this paper we used the von mises fracture criterion in the following form: 2 2 2 21 ( ) ( ) ( ) 6 2 xx yy yy zz zz xx xy misesy           , (5) written for a pair of interacting discrete elements. 3. results of simulation there is a well-known analytic solution for the normal pressure distribution under an elastic flat punch contacting with an elastic half-space [3, 6]. this solution suggests the presence of peaks of normal pressure near the boundary of the contact area. height and, more generally, shape of these peaks depend on curvature of the corners of the punch, the relation of elastic moduli of the punch and the half-space, the roughness and the coefficient of friction between the contacting bodies, etc. in any case, these peaks are much higher than the pressure on the axis of the punch. this fact leads to a curious result at the dem simulation of detachment of a punch from a half-space. namely, tensile strength σt of a contact between a punch and half-space becomes lower with decrease of the diameter of a discrete element in accordance with the power law: 0 0 s t t d d          , (6) where σt0 is a parameter having the dimensionality of stress, d is a discrete element size, d0 is a normalizing constant and s is the exponent. the dependencies of tensile strength on element size for different relations of elastic moduli of the punch and the half-space e = epunch / ehalf-space are shown in fig. 2. evidently, the upper limit of the element size is a punch size and the lower limit tends to zero. as one can see from fig. 2, there is no convergence (at least, asymptotic) of the values of tensile strength on the lower limit of element size. this means the formulation of the numerical model is physically inadequate and must be improved in a way guarantying convergence of the results of simulation with decreasing of an element size. it is necessary to note the fact that the mentioned effect of element size dependence of tensile contact strength holds for different types of fracture criteria (von mises, drucker-prager, detachment at a given value of normal tensile stress etc). also this effect exists when a quadratic package of elements is used. 46 a. dimaki, e. shilko, s. psakhie, v. popov fig. 2 dependencies of the contact tensile strength on the element size for the size independent local detachment criterion in order to study the influence of the direction of loading, we carried out a series of calculations in which shear loading with constant horizontal velocity was applied to the top layer of the punch, while it was fixed in vertical direction. it was found that shear strength of the contact between the punch and the half-space depends on the element size in the same manner as the tensile strength discussed above (see fig. 3). shear strength of the contact obeys to the power law 0 0 q shear d d          , (7) where τ0 has the dimension of stress, q is the exponent determining the slope of the logarithmical dependence of shear strength on the element size. based on the analysis of the dependencies shown in figs. 2 and 3 we obtained the following estimate of the exponents from eqs. (6) and (7) for e = 1: 0.4 0.01s q   . (8) fig. 3 shear strength of a contact versus element size simulation of fracture using a mesh-dependent fracture criterion in a discrete element method 47 the estimates of s and q for different values of e are summarized in table 1. it is evident that the highest discrepancy between the values of exponents s and q takes place for e = 10 (the stiffest punch); in other cases the difference between them does not exceed few percent. this demonstrates universality of the obtained dependencies of strength on element diameter and their applicability to construction of a mesh-dependent fracture criterion. table 1 values of s and q for different values of e = epunch / ehalf-space e tension shear s q 0.2 0.3 0.31 1 0.4 0.41 10 0.44 0.47 4. discussion the obtained element size dependence originates from the non-uniform stress distribution in the contact area with high values of stresses near boundaries. the peaks of stresses take place for all components of stress tensor and for equivalent (von mises) stress which determines crack formation due to the fact that the von mises fracture criterion was used in the performed calculations. at that, the smaller an element size the higher the peaks and the earlier fracture occurrence (see fig. 4). based on the hypothesis that the power-law dependence of contact strength on element size has nearly the same value of the exponent as the contact stress distribution, we have carried out the following test. in the paper [6] an approximation for normal stress distribution has been proposed: 2 2( ) ( , ) /( )p x fm a a x    , (9) where f – total force, applied on punch, m(λ, a) – a non-dimensional weight function, x – spatial coordinate along the contact patch measured from the punch center, λ – the exponent (see also, [23]). there is an analytic solution for λ, obtained by rao [3] for a punch with an arbitrary angle at the corner θ:   2tan(1 ) (1 )sin 2 sin 2(1 ) 1 cos 2(1 ) (1 ) (1 cos 2 )e                 . (10) for a cylindrical punch which is rectangular in 2d cross-section θ = π / 2 and eq. (10) simplifies to 2tan(1 ) sin(1 ) 1 cos(1 ) 2(1 ) 0e            . (11) it is interesting that eqs. (10) and (11) do not contain poisson’s ratios of a punch and a half-plane that means, in particular, that a value of λ is the same for plane stress and plane strain conditions [3]. 48 a. dimaki, e. shilko, s. psakhie, v. popov fig. 4 spatial distributions of the von mises stress on the surface of the half-space for different values of element size having applied eq. (9) for approximation of the von mises stress distribution in the contact area, we obtained the estimation λ ≈ 0.39 (see fig. 5). this value agrees well with the values of exponents s and q, which enter eqs. (6) and (7) correspondingly. the given fact demonstrates that the character of the dependence of contact strength on element size is determined by stress distribution in a contact area. the analytic estimate of parameter λ, calculated by means of solution of eq. (11), is λ ≈ 0.226 for e = 1. the discrepancy between the given analytic estimate and the value of λ, obtained on the basis of approximation of numerical simulation results (see. fig. 2), has the following reason: we applied eq. (9) for approximation of the von mises stress distribution in the contact area although the given equation was initially obtained for description of normal stress distribution, taking no account of squeezing and bending of a material in the contact area and surroundings. fig. 5 normalized dependence of the von mises stress under the punch and its analytic approximation epunch = ehalf-space. simulation of fracture using a mesh-dependent fracture criterion in a discrete element method 49 based on the results described above, we suggest an equation for the value of ultimate stress in the von mises failure criterion: ( ) 0 0 ( ) e mises d y d y d         , (12) where λ(e) is a function of the relation e = epunch / ehalf-space, the value of y0 depends on material strength, ratio e, contact size, etc. obtaining closed-form equations for λ(e) and y0 requires additional research. application of eq. (12) in fracture criterion (5) allows obtaining almost the same values of tensile strength of samples with different element size (see figs. 6a and 6b). one can see that the loading diagrams of samples under tension are almost identical (fig. 6b). a distinction between them is conditioned by a small difference of stiffness of samples with different element size. the deviation of tensile stress from its sample mean value does not exceed two percent (fig. 6b). it is a well-known fact that macroscopic plasticity of brittle solids is often a consequence of microscopic failures nucleation, motion and healing [22]. this is one of the facts underlining similarity and interconnectedness of plasticity and fracture phenomena. from this point of view it is obvious to make an assumption about an existence of mesh-dependent criterion of plasticity. the latter is the topic for a future work. fig. 6 a) the diagrams of tensile loading for samples with different element size; b) the deviation of tensile strength from its mean value vs. element size epunch = ehalf-space 5. conclusions we have carried out the numerical simulation of a contact between an elastic punch and a half-space within the discrete element method. we have shown that the use of local detachment criterion without accounting for a discrete element size leads to a power-law dependence of contact strength on element size. the value of exponent in this dependence is approximately equal to 0.4±0.01 both for tensile loading and for shearing. based on the obtained results we have proposed a mesh-dependent fracture criterion which explicitly includes the discrete element size. the suggested fracture criterion provides almost size-independent values of tensile and shear strength for a punch detachment from a half-space. although the obtained fracture criterion is not universal, it demonstrates a 50 a. dimaki, e. shilko, s. psakhie, v. popov perspective of development of discrete-element models for brittle and elastic-plastic materials with stress concentrators. acknowledgements: this work was supported in parts by the fundamental research program of the state academies of science for 2013-2020 (russia) and by the deutscher akademischer austauschdienst (daad). references 1. hertz, h., 1881, über die berührung fester elastischer körper, journal fuer die reine und angewandte mathematik, 92, pp. 156-171. 2. johnson, k.l., 1987, contact mechanics, cambridge university press, 452 p. 3. shtaerman, i.ya., the contact problem of the theory of elasticity, moscow-leningrad, 271 p. 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international journal for numerical methods in engineering, 106(8), pp. 623-643. 21. shilko, e.v., psakhie, s.g., schmauder, s., popov, v.l., astafurov, s.v., smolin, \a.yu., 2015, overcoming the limitations of distinct element method for multiscale modelling of materials with multimodal internal structure, computational materials science, 102, pp. 267-285. 22. paterson, m.s., wong, t.-f., 2005, experimental rock deformation – the brittle field, springer-verlag, berlin heidelberg, germany, 348 p. 23. popov, v.l., heß, m., willert, e., 2018, handbuch der kontaktmechanik: exakte lösungen axialsymmetrischer kontaktprobleme, springer vieweg, 339 p. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 103 112 https://doi.org/10.22190/fume190326017o © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper high-frequency vibrations in the contact of brake systems johannes otto, georg-peter ostermeyer technische universität braunschweig, braunschweig, germany abstract. the processes and interactions that occur due to friction in the brake are still not fully understood today. in particular, the processes in the boundary layer have been shown to be responsible for a variation in the coefficient of friction and the associated wear. dynamic contact structures in the boundary layer are made responsible for this behaviour. vibration analyses on brake systems usually concentrate on operating vibrations analyses of the brake system components. in order to gain an understanding of the cause of such phenomena and oscillations, it is necessary to understand the mechanism of origin in the contact area. therefore, highly specialized tribotesters have been developed at the institute for dynamics and vibration to investigate the dynamic processes through experiments and simulative investigations. it can be shown that ultrasonic frequencies are generated in the friction boundary layer. these ultrasonic frequencies could not only be found in pin-on-disc testers, but also in complete vehicle brake systems. it was possible to identify that the vibration signatures between 20 and 80 khz depend on a whole series of different influencing variables and have no dependence on the testing machine. in connection with the friction theories, it is an open question whether these oscillations can be made responsible as a kind of trigger pulse for the squealing of 1 to 20 khz. in addition, it is a problem that the parking sensors installed in the vehicle work on an ultrasonic basis in the same frequency range and can therefore lead to failure due to these frequencies. key words: high frequency, vibrations, ultrasonic, boundary layer, noise 1. introduction the phenomena of friction are still not fully understood and difficult to predict. the friction itself is an essential component between two bodies when they are in contact and moved against each other. as a rule, the dissipation of energy from the system is undesirable as it reduces the efficiency of the system. nevertheless, there are systems, such as the received march 26, 2019 / accepted may 22, 2019 corresponding author: johannes otto tu braunschweig, inst. f. dynamik und schwingungen, schleinitzstraße 20, 38106 braunschweig, germany e-mail: j.otto@tu-bs.de 104 j. otto, g.-p. ostermeyer friction brake, in which high friction is desired. the general goals are that the friction brake can transmit maximum friction power, has low wear rate and high resistance to noise vibrations and harshness (nvh). a large number of research projects have shown that the objectives mentioned above are significantly influenced by the friction boundary layer in the case of high-load contacts, see for example [1]. the friction boundary layer is often referred to as the third body and describes the contact area between the brake pad and brake disc. however, the friction boundary layer is dependent on a number of different parameters, which makes the predictability of friction in contact particularly difficult. in order to make a scientific contribution to this complex topic, this work focuses on the topic of frictioninduced high-frequency oscillations within the boundary layer between brake pad and brake disc. fundamental investigations will be carried out in order to gain a systematic understanding of the high-frequency friction-induced vibrations within the contact. this paper deals with the high-frequency oscillations in the range of 20 to 80 khz. 2. the process in the boundary layer the tribological properties are determined by the processes and interactions that occur between the brake pad and the brake disc. within the boundary layer, highly complex dynamic processes occur which are significantly influenced by hard, load carrying contact patches [2, 3]. the development process of a patch begins with a wear-resistant particle that comes into frictional contact with the brake disc as a result of the progressive wear of the surrounding material. these abrasive particles can often be identified by edx analysis as steel fibers [4], but also as quartz particles [5]. at this point between the particle and the disc, the movement of the wear material is prevented and the wear material accumulates. as a result, the hard particles begin to carry an increased proportion of the frictional and normal force and thus cause a strong and local increase in temperature. the combination of high local temperatures and normal pressure leads to sintering processes within the boundary layer. as a result, contact patches are formed as shown in fig. 1. fig. 1 patch structure on the brake pad surface [6] high-frequency vibrations in the contact of brake systems 105 in the further friction process, the high mechanical and the thermal stresses lead to the formation of cracks within the contact plateaus and to destruction. simulation investigations by ostermeyer using multi-body systems indicate the excitation of high-frequency oscillations within the boundary layer resulting from self-organization of the patches [7]. in addition, it can be seen in [8] that wave propagation can occur in friction contact and excite frequencies in the ultrasonic range. the question now arises whether the postulated high-frequency oscillations can be recorded by metrological investigations. therefore, this paper presents an experimental study of friction-induced high-frequency oscillations within the boundary layer. in the past, a number of authors investigated the influence of external ultrasound on the friction contact [9, 10, 11]. it was found that the excitation amplitude of the high-frequency oscillations applied from the outside have a significant influence on the coefficient of friction. 3. measuring device and investigation of the dynamics of the test unit the test device mainly used for the following investigations is a scaled automated universal tribotester (aut), fig. 2. the aut is a high-precision and fully automated pindisc tribometer. the brake disc is set in rotation by an electric motor and speed characteristics, as they occur in conventional motor vehicles, can be realized by an intelligent motor control. the other friction partner is the brake pad. a brake pad sample of 20 x 10 mm² is sawn from the brake pad and glued into a sample holder, see the brake pad sample in fig. 3. in [12] the comparability between scaled pin-on-disc testers and dynamometer test benches was investigated. it was found that the dynamic variation of the coefficient of friction is the same and that only the absolute magnitude of the coefficient of friction differs slightly. by mounting the specimen holder on the load unit, forces or pressures can be realized as they occur in any vehicle. for further information on the test equipment, see the publication [13]. fig. 2 automated universal tribotester 106 j. otto, g.-p. ostermeyer two different measuring instruments are available at the ids to measure the highfrequency vibrations between the brake pad and the brake disc. the first is the laser doppler vibrometer with the decoder which has a sampling rate of up to 2.5 mhz and the second is an ultrasonic microphone with a sampling rate of up to 0.5 mhz. in order to measure the vibrations synchronously during the brake application, a fully automated measurement chain was realized, see paper [14]. the dynamic behaviour of the experimental system was investigated using complex eigenvalue analysis [14]. it was found that the first eigenfrequencies of the system components are below 1 khz and that in the frequency range to be investigated a large number of eigenfrequencies are close to each other. 4. proof of high-frequency vibrations in friction contact 4.1. comparison between laser vibrometer and ultrasonic microphone the resulting high-frequency vibrations in friction contact between the brake pad sample and the brake disc at the aut are measured by using the laser doppler vibrometer and the ultrasonic microphone. the laser is focused alternately on the tangential and radial friction surfaces of the friction sample. to ensure optimum signal quality, reflective foils are bonded to the friction pad surfaces. the ultrasonic microphone is adjusted at an angle of 45° between tangential and radial coordinates. the measurement setup is shown in fig. 3. the brake applications are performed under constant boundary conditions, which means constant speed, constant pressure, constant starting temperature. before the measurement, a complete bedding procedure is carried out with the aim of ensuring a full-surface contact and stationary behaviour of the friction boundary layer. fig. 3 measurement setup [14] the measured frequency spectra can be seen in fig. 4. the left side shows the frequency spectrum from the laser vibrometer in the tangential friction direction. on the ordinate the amplitude of the velocity is shown in m/s and on the abscissa the frequency between 10 and 200 khz. looking at the spectrum, peaks can be identified at 16 khz, 19 khz, high-frequency vibrations in the contact of brake systems 107 37 khz and 56 khz. a comparison with the frequency spectrum recorded simultaneously by the ultrasonic microphone shows these same frequencies. in addition, a frequency at 24 khz can also be detected in the microphone's spectrum. this frequency can be determined in the radial spatial direction of the friction sample. in conclusion, it can be seen that highfrequency oscillations between brake pad and brake disc can be detected by two different measuring principles. in addition, it can be determined that all frequencies measured in the individual spatial directions of the laser can also be detected by the 45° mounting of the ultrasonic microphone. fig. 4 measurement results of the laser vibrometer and the ultrasonic microphone [14] 4.2. investigation of the influence of the friction direction and the sample holder in this section the influence of the specimen geometry and the specimen holder itself on the high-frequency vibrations shall be examined. for this purpose, two brake pad samples with dimensions 20 x 10 mm² and 10 x 20 mm² were produced. in addition, a special specimen holder was developed and manufactured which allows the brake pad specimen to be glued in place rotated by 90°, see illustration at top right. the tests are carried out under constant boundary conditions. before the actual measurement, a bedding procedure is carried out to achieve a full-surface contact and a constant boundary layer. the laser vibrometer is focused on the surface of both samples in tangential friction direction, see figure 5 in the top for clarification. 108 j. otto, g.-p. ostermeyer fig. 5 investigation of the same brake pad sample in two different friction directions, initial state left and turned sample right if the experiments are now carried out, the frequency spectrum for the initial state results is shown in the figure on the left side. in particular, the high peak at approximately 39 khz can be identified. on the right side of the figure, the frequency spectrum of the rotated sample can be seen. a significant increase in amplitude at 40 khz can also be seen here. in addition, an amplitude increase in the range of 74 khz can be observed for both frequency spectra. thus, this investigation indicates that the specimen holder and the specimen geometry have no influence on the induced vibrations. furthermore, it can be determined that a different friction material shows a different frequency spectrum under the same conditions: compare fig. 5 and fig. 4. 4.3. investigation of the influence of the sample length the longer the period of time that a brake pad is installed in a vehicle, the lower is the brake pad height. therefore, possible influence of the brake pad length (pad height) or the sample length on the high-frequency vibrations is to be investigated here. two characteristic tests are carried out for this purpose. the first test is carried out with the initial brake pad sample. this has a length of 15.2 mm. the measuring direction of the laser is in the tangential friction direction. the resulting frequency spectrum under constant boundary conditions is shown in fig. 6 on the left. significant frequencies at 14.7 khz, 19.3 khz and 45 khz can be identified. this sample is then reduced to 11 mm using the milling manufacturing process and afterwards a running-in procedure is performed. this ensures a constant initial condition. the test is repeated. it can be seen that the frequency at 44 khz is independent of the sample length. the frequency at 14.7 khz in the left output state increases to 15.8 khz. this can be justified by the fact that the eigenvalue analysis and operating vibration shape analyses of the brake pad sample identified a natural frequency in the tangential direction of the brake pad sample in the order of 15 khz [14]. in summary, it can be stated at this point that the high-frequency oscillations are independent of the brake pad sample length. high-frequency vibrations in the contact of brake systems 109 fig 6 investigation of the influence of the sample length, initial state left and worn state right 4.4. investigation of the same friction pairing at different test devices at this point, the extent to which the test equipment has an influence on the highfrequency oscillations shall be investigated. the same friction pairing (disc and brake pad) are investigated on two different pin-disc tribometers. in the first step, the measurements are carried out at the aut, where the previous investigations were also carried out. in the second step, the brake disc and the brake pad are mounted on the vvt (variable velocity tribotester) and the measurements are carried out there. the vvt and the aut do not differ in their test principle, but rather in their design parameters, such as stiffness, damping, inertia, etc. the investigations are carried out at a constant sliding speed and six different normal forces. each normal force consists out of 5 measurements, each with a friction time of 5 seconds. the laser vibrometer measures the vibrations occurring on the brake pad surface in a tangential spatial direction. figure 7 shows the waterfall diagram for the different normal forces at the aut on the left side and on the right side for the vvt. a high correspondence between the two frequency spectra can be seen. the frequency at about 30 khz increases to about 35 khz as the normal force increases. the amplitudes in the frequency range between 50 and 70 khz is combined with increasing normal force to a frequency at 60 khz. this investigation thus shows that the ultrasonic oscillations in friction contact are independent of the test equipment and are much more influenced by the magnitude of the normal force. 4.5. investigation at a complete brake system studies also show that the same high-frequency oscillations can be found in complete brake systems (dynamometer test bench). the vibrations were recorded from the brake pad surface by means of laser vibrometers and significant frequencies in the range from 20 to 80 khz could also be identified here [13]. 110 j. otto, g.-p. ostermeyer fig. 7 comparison between two different testbenches [14] 5. passive vibration reduction in the previous sections it was shown that high-frequency vibrations occur between the brake pad and the brake disc. various factors influencing the high-frequency vibrations could be identified. in order to gain a better understanding of the system, this section will attempt to reduce the vibrations. in the literature there many different passive methods to design systems with low vibration, for example vibration damping, vibration isolation and vibration tuned mass damper are mentioned. with the first two methods mentioned, however, the problem is that they must be used directly at the source of origin and thus in the friction boundary layer itself. therefore, this paper is focused on the use of a tuned mass damper for the high-frequency oscillation. the tuned mass damper, a so-called "sacrificial mass" is placed on the vibrating system, which should be designed in such a way that the initial system performs as small movements as possible and the majority of the energy introduced is "transferred" into the sacrificial mass. for clarification, it can be imagined that the absorber principle is based on the properties of a specially designed dual mass oscillator. the natural frequency of the tuned mass damper ωt should be designed in such a way that it corresponds to the natural frequency of the system mass. however, tuned mass dampers have the disadvantage that they only work in a certain frequency band and that new resonances are created below and above the original resonance. in order to make optimum use of the tuned mass damper, it is placed as close as possible to the friction contact. in order to be independent of the friction pairing to be investigated, a new sample holder is developed. bending beams of different lengths can be attached to the upper side of the new sample holder (see fig. 8). the bending beams should function as absorbers themselves. the frequency of the tuned mass damper can be varied by different lengths of the bending beams. as a basic rule, the longer the bending beam, the lower the natural frequency. the natural frequency of the bending beams was calculated in advance using the finite element method. high-frequency vibrations in the contact of brake systems 111 fig. 8 sample holder with tuned mass damper for the subsequent tests, a friction pairing is selected which has a characteristic frequency in the range of 45 khz. for the experiments, the laser doppler vibrometer is focused on the sample surface in the tangential friction direction and is braked with constant normal force and speed. on the left side of fig. 9, the recorded frequency spectrum of the initial state (without tuned mass damper) can be seen. then the tuned mass damper is mounted on the sample holder and the braking is repeated under the same boundary conditions. a comparison of the two frequency spectra shows that the significant amplitude increase at 45 khz is no longer visible in the right image. however, new frequencies have formed just below and above. this behavior suggests that the tuned mass damper is working as expected and thus that a reduction of the high-frequency oscillations that occur in the friction boundary layer is possible. further analyses will be carried out in the near future. fig. 9 comparison between initial state and tuned mass damper 6. conclusion the underlying physics of friction has not yet been fully understood. the large dependency on a large number of different parameters makes predictability difficult. therefore, this work deals with the friction-induced oscillations that occur in the boundary layer between brake disc and brake pad in order to contribute to the scientific understanding of friction. 112 j. otto, g.-p. ostermeyer it is possible to identify characteristic high-frequency oscillations in each brake pad disc combination, which are in the range between 20 and 100 khz. the following points could be summarized:  the high-frequency oscillations can be recorded with two different measuring principles.  the high-frequency oscillations are independent of the test setup and test equipment.  the high-frequency oscillations are dependent of the friction pairing, normal force, coefficient of friction, speed, boundary layer [14].  the high-frequency vibrations can be recorded on the complete brake & on the pindisc tribometer.  the high-frequency vibrations can be reduced with a tuned mass damper. this allows new studies to be carried out to investigate the friction behavior during a reduction in vibrations. in summary, high-frequency oscillations are excited in the boundary layer. the extent to which these are relevant for the nvh phenomena described and what the decisive excitation mechanism is will be discussed in further investigations. references 1. ostermeyer, g., wilkening, l., 2013, experimental investigations of the topography dynamics in brake pads, sae int. j. passeng. cars mech. syst., 6(3), pp. 1398-1407. 2. eriksson, m., jacobson. s., 2000, tribological surfaces of organic brake pads, tribology international, 33(12), pp. 817-827. 3. ostermeyer, g.-p., 2009, on tangential friction induced vibrations in brake systems, sae int. j. passeng. cars-mech. – mech. syst., 1(1), pp. 1251-1257. 4. eriksson, m., lord, j., jacobson, s., 2001, wear and contact conditions of brake pads: dynamical in situ studies of pad on glass, wear, 249(3-4), pp. 272-278. 5. österle, w., gripentrog, m., gross, t., urban, i., 2001, chemical and microstructural changes induced by friction and wear of brakes, wear, 251(1-12), pp. 1469-1476. 6. ostermeyer, g.-p., 2001, friction and wear of brake systems, forschung im ingenieurwesen, 66, pp. 267272. 7. ostermeyer, g.-p., 2010, dynamic friction laws and their impact on friction induced vibrations, sae int. j. passeng. cars-mech., pp. 1-15, doi:10.4271/2010-01-1717 8. recke, b., ostermeyer, g., 2018, boundary layer dynamics and sound generation, sae technical paper 2018-01-1900, pp. 1-5, doi:10.4271/2018-01-1900. 9. pohlman, r., lehfeldt, e., 1966, influence of ultrasonic vibration on metallic friction, ultrasonics, 4, pp. 178-185. 10. littmann, w., storck, h., wallaschek, j., 2001, sliding friction in the presence of ultrasonic oscillations: superposition of longitudinal oscillations, archive of applied mechanics, 71(8), pp. 549-554. 11. popov, v., starcevic, j., filippov, a., 2009, influence of ultrasonic in-plane oscillations on static and sliding friction and intrinsic length scale of dry friction processes, trib. lett., 39(1), pp. 25-30. 12. perzborn, n., agudelo, c., ostermeyer, g., 2015, on similarities and differences of measurements on inertia dynamometer and scale testing tribometer for friction coefficient evaluation, sae international journal of materials and manufacturing, 8(1), pp. 104-117. 13. ostermeyer, g.-p., schramm, t., raczek, s., bubser, f., perzborn, n., 2015, the automated universal tribotester, presented at eurobrake 2015, pp. 1-10. 14. otto, j., sandgaard, m., ostermeyer, g.-p., 2019, high-frequency vibrations in the friction boundary layer of brake systems, eurobrake 2019, may 2019, dresden, germany, pp. 1-14. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 217 242 https://doi.org/10.22190/fume190227027f © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* methods of the pattern formation in numerical modeling of biological problems alexander e. filippov 1,2 , stanislav n. gorb 1 1 functional morphology and biomechanics, zoological institute, kiel university, germany 2 donetsk institute for physics and engineering, national academy of sciences of ukraine abstract. evolution of different systems can be described in terms of their relaxation to the minimums of some effective potential relief. this observation leads us to face us with a question how to generate corresponding potential patterns which describe adequately various physical and biological systems. in this review, we present a number of different ways of generating such potentials demanded by the problems of different kinds. for example, we reproduce such a generation in the framework of a simple theory of phase transitions, automatic blocking of the growing phase nucleation and universal large scale structure. being frozen at late stages of their evolution they form majority of meta-stable structures which we observe in real world. counting on above-mentioned universality of naturally-generated fractal structures and their further utilization in numerical simulations of biological problems, we reproduce also formal algorithms of generation of such structures based on random deposition technique and fourier-transform approaches. key words: pattern formation, phase transitions, large river effect, nucleation, biological applications, frozen kinetics 1. introduction a large problem of experimental biology is that we are dealing with systems which are very fragile and sensitive and behave statistically, which means that often no kind of experiment is possible, and if so, only at extremely high costs and usually without proper control samples/experiments. this is one of the most important fundamental differences between experimental biology and other experimental disciplines dealing with nonbiological matter. again, the most helpful approach in this context would be modeling, which can then be used as an experimental platform “in silico” in order to predict received february 27, 2019 / accepted may 18, 2019 corresponding author: alexander e. filippov donetsk institute for physics and engineering, national academy of sciences of ukraine, donetsk, ukraine e-mail: filippov_ae@yahoo.com 218 a.e. filippov, s.n. gorb outcomes in real biological experiments. in some cases, a „virtual‟ experiment may even be the only viable possibility. these considerations lead to the insight that in many instances experimental biology can profit from new methods made possible by novel modeling techniques that take advantage of the recent developments in modeling and visualization. in many concrete examples we will need numerical generation of the effective potential. if effective potential is known and dynamic equations are properly written, the system is attracted to the correct configurations by itself. the potential can appear in real space, or in some imaginary space of parameters. in some sense, it does not matter. however, it can matter and it should matter from either physical or biological points of view for particular problems, where the potential has some specific meaning. in this review, we will present some practically useful and technically convenient examples of effective potentials (or patterns, if we are talking more definitely about their formation in 2d or 3d spaces). from the very beginning, one can divide them into two large classes. 2. simple theory of phase transitions and pattern formation one kind the potentials are abstract numerically generated formations which are simple enough and convenient for the theoretical numerical simulations. the other ones are provoked by the studies of the specific, experimentally observed systems. in this latter case, the degree of simplicity of the potential or the procedure of its generation is not so important because in such a case we are more interested in the potential (surface, pattern, density distribution, etc.) by itself at the final stage of the study, or maybe even as a goal of the particular research. for example, if we would like to know how this particular picture could appear self-consistently. keeping this in mind, we will continuously jump between these two opposite limits. sometimes, using very abstract models to illustrate general ideas, or simply to generate the equations, reproducing what we have observed, even without complete motivation. but, sometimes, we will devote almost the whole study to an extraction of the equations or procedures from particular nature of the problem under consideration. the same note can be done about dimensionality of the surface (potential, line, etc.) generated by a procedure. in some cases it is enough to create 1d line with specific properties. it is important to note that even for such “trivial” case our motivation can be different. for example, it can be made just for simplicity of further application of the generated object. however, in many cases, it can be even more correct than to generate and apply 2d pattern and use it. it can be proven that for many problems of contact mechanics, real 3d problem can be exactly transformed to the 1d model! it is so-called method of dimensionality reduction (mdr) invented and actively developed last during decades by valentin popov and coworkers [1]. the cases of the mdr application we will mention in this paper and specially discuss them in corresponding sections. it should be additionally mentioned that the absolute majority of patterns, we have used in studies of biological problems under consideration, were generated by simulations of physical or chemical kinetic processes. below, we basically report on the methods of physical kinetics. for example, the ideas from fluctuation theory, phase separation, and phase transitions can be applied for pattern generation. of course, it is practically impossible methods of the pattern formation in numerical modeling of biological problems 219 to reproduce all the mathematical foundations of this branch of science, which are published in hundreds of the handbooks starting from the 19 th century. thus, in many cases, we will simply suppose that it is either known or can be proven. independently of the accuracy and correctness of the terminology, one can start from a very formal and methodically transparent approach. let us suppose that there is fluctuating density ρ(x) randomly distributed originally at t=0 in one-dimensional space {x}. random initial distribution means that fluctuations δρ(x,t) on average are equal to zero <δρ(x,t)>=0 and originally independent at every point and time of the system <δρ(x,t)δρ(x´,t´)>=dδ(x-x´)ρ(tt´). it is so-called δ-correlated noise. in the majority of real systems the densities of fluctuations interact. these interactions can cause mutual attraction and enforcement of the fluctuations leading to the nucleation of the regions with nonzero density ρ(x) and their expansion in space. one of the most accepted theories of this process is landau theory [2] of phase transitions (which can be treated as a combination of main interactions within the system in the framework of the simplified “mean field” theory). main idea of landau theory is that energy of the system can be expressed as a function of the so-called “order parameter” density. in particular, it can be a functional u(ρ(x)) of spatially distributed real density ρ(x). the simplest variant of the theory corresponds to the case, when local form u(ρ(x))→u(ρ) of the energy (depending on uniform variable ρ) has two minimums corresponding to “ordered” ρ=ρ0≠0 and “disordered” ρ=0 states. depending of the relationship between the energy minimums u(ρ0) and u(ρ=0) one or another state is energetically preferable. for example, if u(ρ0)<u(ρ=0) and the system originally was in another state ρ(t=0)=0, it can be transferred from it to the energetically preferable state with ρ=ρ0≠0. typical form of local energy u(ρ) corresponding to this quantitative discussion is shown in fig.1 in fact, this form can be correctly derived from a microscopic description of the system, but this is outside of our scope here. let us just suppose that we have correct function of the energy and can represent this in as simple form as possible. starting from a beginning of the 20 th century, when physicists were building modern quantum mechanics they told: “if you do not know what to do with a function, expand it into to the series”. we will follow this wise rule and suppose that local energy, depending on the density, can be approximately written in the following form: 2 3 4( ) ( )u b g         (1) symbol o(ρ) means that one part of the energy, which is relatively small in comparison to others, can be neglected. at proper choice of parameters τ, b and g in expansion u(ρ)=τρ 2+bρ 3+gρ 4+o(ρ), the function gets a qualitative form with two different minimums corresponding to the above discussion. 220 a.e. filippov, s.n. gorb fig. 1 typical form of local energy u(ρ) corresponding to the quantitative discussion in the text thus, one can expect that if we start from initially disordered fluctuating field ρ(x) and write proper kinetic equation we would get a process of its ordering ρ(x)→ρ0. unfortunately, it only sounds easily to do this, when function ρ(x) is not uniform. generally, densities ρ(x) in different points are different and interact one with another. thus, the total energy has some nonlocal addition due to the interaction: 2 1 1 1 2 2 21 [ ( )] ( ) ( ) ( )nonlocalu x dx dx x v x x x    (2) as already-mentioned above in the introduction, such nonlocal forms in the numerical modeling because it leads to the time consuming calculations. especially, it is the case, if v(|x1-x2|) represents a long range interaction. however, next simplification can be done, if the interaction is short-ranged enough. we will not present complete proof of this step and only ascertain that it is possible to expand energy unonlocal[ρ(x)] over gradients ∇ ρ=∂ρ/∂x of density ρ(x). in a standard case, it can be transformed into the following form: 2 2[ ( )] [ ( ) ( ( )) ...]nonlocalu x dx a x c x      (3) the first term here joins analogous local term τρ 2 in the expansion eq. (1) and simply renormalizes it (τ+a)→τ. gradient term c(∇ ρ) 2 is qualitatively more important. it conserves information about the interaction between the densities in different points of the space. normally it tends to make the densities in different points close one to another and total density distribution ρ(x) smoother. it is the main reason why originally uncorrelated fluctuations of density <δρ(x,t)δρ(x´,t´)>=dδ(xx´)ρ(tt´) tend to the ordered state with <δρ(x,t)>=0 and <ρ(x,t→∞)>=ρ0=const. in this place, the reader can ask us: why do we need so complex description, if after all everything tends to the simple constant ρ0=const? the main problem here lies in the words “after all”. we are not interested in trivial final of the kinetic process, but in its intermediate state. absolute majority of real surfaces and substances are “frozen” in an intermediate states of the kinetic process and namely at these stages they produce methods of the pattern formation in numerical modeling of biological problems 221 practically important patterns with non-uniform ρ(x) with nonzero gradients ∇ρ. moreover, in many cases (for more complex systems than we study right now) even final stage of the kinetic process are domains and other structures with non-uniform distributions ρ(x). below we will study some of them, but now let us return to the simplest (!) case. combining eqs. (1) and (3) one can write energy functional in the following form [3]: 2 2 3 4 2 [ ( )] [ ( ( )) ( ) ( ) ( ) ...] 2 2 3 4 [ ( ( )) ( ( ))] 2 c b g x dx x x x x c dx x f x                      (4) however, if it is still not enough to write down complete equation for the fluctuating density [4]: ( , ) / [ ( , )] / ( , )x t t x t x t        (5) if ρ(x,t) really fluctuates, it is not only random value <δρ(x,t)δρ(x´,t´)>=dδ(x-x´)ρ(tt´) with intensity of d at the beginning of the process, but it continues to fluctuate further. it means that eq. (5) should include also a source of random fluctuations δ(x,t). mathematically, it has the same properties as the initial condition for the fluctuations of density: ( , ) 0x t  , ( , ) ( ', ') ( ') ( ')x t x t d x x t t       . (6) this random source stays in the right hand side of the equation of motion ( , ) / [ ( , )] / ( , ) ( , )x t t x t x t x t          (7) and cause generation of new fluctuations during the whole process of the density nucleation. eq. (7) is already equation of motion for this simplest (!) model. taking into account explicit form eq. (4) of the functional φ[ρ(x)] and performing the variation δφ[ρ(x,t)]/δρ(x,t) one can write the form of the kinetic equation that is more convenient for the solution: 2 3 4( , ) / [ ( , ) ( , ) ( , ) ( , ) ] ( , )x t t c x t x t b x t g x t x t               (8) where δρ(x) is laplasian operator which in 1d case is equal simply to second derivative δρ(x)=∂ 2 ρ(x)/∂x 2 . starting from the random distribution, the density evolves with time according to the eq. (8). the barrier in the local part of energy f(ρ(x)) does not allow majority of the density fluctuations to grow immediately to the minimum corresponding to the nonzero equilibrium density ρ0=const. the only relatively large fluctuations can pass the barrier and grow. they grow in both senses: their amplitude at maximum tends to ρ0 and they expand in space [5, 6]. the same logics can be easily expanded into 2d space where corresponding pattern of the density will appear. typical intermediate configuration with different nuclei of nonzero density is shown in fig.2. if there is physical reason to freeze this configuration, we would get almost static representation of the non-uniform pattern of density. the simplest possible reason for this may be very small damping constant  which controls 222 a.e. filippov, s.n. gorb rate of the kinetic process in eq. (8). if the characteristic time 1/γ of evolution of the distribution ρ(x,t) to the equilibrium ρ0 is extremely long in comparison with other times of a problem, one can ignore the evolution of the density at all and treat its intermediate distribution as practically static. it has some ideological analogy with the galaxies developing for billions of years. it is much longer than our life and majority of the processes on our planet. so, the galactic patterns can be treated as practically static ones. fig. 2 typical intermediate pattern appearing as a result of solution of eq. (8) however, in many cases the kinetic process may be really stopped by some natural reasons at an intermediate stage. normally it happens when the system is actually open to the external influences or consists of a number of interacting subsystems. in the first case, external forces can cause change of the coefficients of energy expansion eq. (4) already in the course of the density evolution. as result, the system starts its development to some equilibrium at the given form of f(ρ) and finishes in absolutely another equilibrium corresponding to modified functional f(ρ). in the second case, alternating interactions between the subsystems can lead to the growth of different densities in different domains of the space. when such growing domains meet each other, they mutually block further expansion and build a static (or maybe very slowly shifting) domain wall between them. such blocking leads to the fixation of the domain structure and on large scale forms a pattern including many domains of different sizes. the simplest way of obtaining such domain structure is to use expansion f(ρ) with even terms only: 2 2 4 6 2 [ ( )] [ ( ( )) ( ) ( ) ( ) ...] 2 2 4 6 [ ( ( )) ( ( ))] 2 c g u x dx x x x x c dx x f x                      (9) corresponding equation of motion with the functional eq. (9) leads to the typical domain structure shown in fig. 2. methods of the pattern formation in numerical modeling of biological problems 223 fig. 3 formation of the domain structure in the kinetics. two different kinds of domains are shown by red and blue colors, respectively 3. automatic blocking of the nucleation and freezing of the process in both previous cases, telling about almost frozen kinetics, we have supposed that it this process is simply much slower than other ones in a particular model. however, it was shown more than 20 years ago that since the formation of new phase nuclei includes processes that prevent their appearance and growth in other regions in space, it should result in autostabilization of an intermediate mixed state. one can call it “automatic blocking of the nucleation” [7]. normally it is caused by the nonlocal (long-range) interactions which are caused in an ordered process by itself. from the introduction, we remember that numerical modeling does not like nonlocal interactions because of an extreme time consumption required for these calculations. however, it is possible to show that the effect of blocking long-range interaction can be approximately included by some local additions to the above models. various mechanisms for the formation of an effective long-range interaction in such systems can be analyzed. here we will mention only a couple of them. normally, formation of the substance (and the surface, which is needed here) includes the reaction of a crystal lattice (striction) to a change in magnitude of order parameter ρ(x,t) during a phase transition. one can show that in a relatively simple case of an isotropic medium and quadratic striction, the local energy functional of the free energy is modified in the following manner [3]: 2 2 2 2 2 1 1 , [ ( )] [ ( ( )) ( ( )) 2 1 ( ( ) ( ) )] 2 3 u ii ii k k ll l k c x dx x f x k g x u u u u                 (10) if the lattice vibrations manage to follow the variations of ρ(x,t), we can utilize condition δφ[ρ,u]/δuik=0 to eliminate variables uik and get, after some standard mathematical transformations, an effective functional solely in terms of field ρ(x,t): 2 2 2[ ( )] [ ( ( )) ( ( )) ( ) ' ( ') ] 2 2 c x dx c x f x x dx x v            (11) here ( ( ))f x is the renormalized local form of f(ρ(x)) with the same structure as the original function f(p) (we shall henceforth omit the tilde), and constant κ is defined by 224 a.e. filippov, s.n. gorb expression 2 1 1( / 2 )[( / 2 2 /3) ( / 2 2 /3) ]q v k k p      . constant p is determined by the external pressure or other constraints which prevent free expansion of the crystal (twins, defects, etc.) and, in turn, fix the sign of κ. when p>μ, κ>0; otherwise κ<0. nonlocal construction ∫dx[ρ(x) 2∫dx´[ρ(x´) 2 ] in functional φ[ρ(x)] generates a term with a long-range effect in the equation of motion for the field variable 2( , ) / [ / ( , ) ' ( ', ) ] ( , ) 2 x t t c f x t dx x t x t v                   (12) whose presence significantly accelerates or slows down (or even totally stops) the ordering process, depending on both the magnitude and sign of κ. typical picture of blocked almost static domains is shown in fig. 4. fig. 4 formation of the domain structure in the kinetics at the presence of self-blocking. two different kinds of domains are shown by red and blue colors, respectively. it is important to note that the last stage here is the practically final one and does not develop further with time. let us discuss on one more example of an interaction, which leads to a similar model. the local variation of the order parameter is accompanied by the evolution or absorption of heat (depending on whether the transition is to the lowor high-temperature phase). this results from heat conduction in heating (cooling) of the surrounding regions of space, which, of course, slows down the transition process in all cases. this mechanism seems to be universal, and its effectiveness is determined only by the relationship between the rates of the nucleation and heat-conduction processes. the local heating (cooling) of a system in a region, where a nucleus appears, can be taken into account by assuming that quantity τ in expressions (9) is a function of position and time. the kinetic equation for the order parameter should be supplemented by an equation which describes the evolution of τ=τ(x,t). the latter equation should be a heat-conduction equation with the heat removal and with a source β[ρ], whose intensity is proportional to the rate of change of the free energy, i.e. β[ρ]~∂ρ/∂t·δφ/δρ. as a result, we have the following system of the connected equations [7]: ( , ) / [ ( , )] / ( , ) ( , )x t t x t x t x t          ; (13) ( , ) / ( , ) ( , ) / [ ( , )] / ( , ) ( , )x t t x t x t t x t x t x t                (14) methods of the pattern formation in numerical modeling of biological problems 225 before starting discussion of the results of the numerical experiments, we show that with some roughening of the model, the mechanism under consideration can be described in terms of a single field variable ρ(x,t), which evolves in accordance with an equation similar to eq. (12). the physical arguments, which lead to a functional like eq. (11) in this case, too, are fairly simple. each growing domain of the new phase creates a non uniform temperature field τ(x,t) around itself. owing to the heat conduction, the temperature at other points in space deviates from trial temperature altering the conditions for the growth of other domains at those points. this signifies the appearance of an effective long-range field accompanying the nucleation process in the system. relating the variation of the temperature field to the order-parameter field ρ(x,t), we obtain an energy functional φ[ρ] like that in eq. (11). in fact, when the fluctuations of τ(x,t) are "turned on" in a system with a temperature equal to the heat-bath temperature τ0, after a unit of time the mean value of τ(x,t) deviates from τ0 by 1 2 0 0 1 1 /dx t v v            (15) the mutual influence of the domains of the new phase becomes significant, when they become so large (and this is seen from the results of numerical experiments) that the energy of the domain boundaries between the ordered and unordered phases can be neglected. as result, we arrive at a functional like eq. (1), which was obtained to describe striction effects in the kinetics of a first-order phase transition. in some cases, this makes it possible, in principle, to disregard the specific mechanism for realizing the long-range effect accompanying the first-order phase transition and to formally analyze models with nonlocalities of the general form. 4. large scale structure of the fluctuating field. universality and scaling trial structure of the energy functional has direct relation to the microscopic interactions in the system. in many cases, it can be even analytically derived from the microscopic theory. it means that the coefficients of the expansions used in the kinetic equations above have well defined specific values and in turn should completely determine density distributions ρ(x,t) and as result the structure of the contact surface. however, it is well known that absolute majority of real surfaces have practically universal (scaling) structure with the power low distribution of the relief. it means that if there is no special reason to produce another structure, it should appear due to universal kinetic process which makes difference in the initial energy functional negligible. as we saw in the previous section, ordering transition is anticipated by nonlinear excitations, which can be interpreted as nucleation centers. the kinetics of the first-order phase transition in different physical systems has been the subject of intensive studies. as a rule, the ordering of a metastable disordered phase is due to the fluctuation production and finally to the growth of the nucleus of the stable phase. in a first-order phase transition, there is a change in some order parameter between these two phases, which lowers the free energy as the new phase forms. 226 a.e. filippov, s.n. gorb the corresponding local energy density f(ρ) must have a metastable minimum and be energetically favorable. however, the free energy is transformed due to the fluctuations. this change is especially essential in the critical region at a second-order transition. it is well known from the theory of critical phenomena that the fluctuations manifest themselves by renormalization of critical exponents. but, the renormalization group (rg) method allows one not only to perform purely numerical calculations of critical exponents, but also to predict some qualitatively new effects which could not be obtained within conventional approaches, e.g., within the landau approximation, applied above. among them, there are qualitative effects, such as the fluctuation-induced first-order phase transition. this effect takes place in some anisotropic systems, where the renormalized free energy f(ρ) undergoes transformations, which are typical for the first-order phase transition. the same kind of the nucleation centers can be found in a fluctuation-induced first order transition. however, one can expect that even in the situation, when the fluctuations are not strong enough to change the transition order, they manifest themselves somehow. the mean field model is very convenient for analytical study, but it reduces the fluctuation interaction, and it is impossible to control a correction which should be done to the free energy with account of the neglected fluctuations [8]. a more correct approach was developed almost 50 years ago. it states that the fluctuations renormalize energy functional φ[ρ(x,t)] according to so-called renormalization group (rg) equation ˆ/ l r    . here we do not need the complex exact form of the rg equation and use its simplified approximate version: 2( )ˆ [( 2) ( )] ' 2 ( ) ( ) ( ') ( ) ( ') d d r r r d r d r r d r r r r r r                                   (16) where d is the space dimensionality. the first term in eq. (16) corresponds to a simple scale transformation of the density distribution ρ(x,t) and the second term appears due to the integration over internal fluctuations inside small regions after scale transformations. according to the general rg hypothesis in critical point of the phase transition (exactly, where the ordering of ρ(x,t) takes place) the functional φ→φ * tends to the fixed point * *ˆ/ 0l r     . let us study now the time-space evolution of fluctuating density ρ(x,t) in this state. we plan to show that ρ(x,t) produces a well-pronounced large-scale structure in spite of its scale invariance on average. as always kinetic equation can be written in the form: ∂ρ/∂t=−γδφ*[ρ]/δρ+δ. at every time moment, average probability w=<w[ρ]> to find density distribution ρ(x,t) is determined by functional w[ρ]=exp(−φ[ρ]). this probability develops with time according to the equation: [ ] / /d w w t d r t          . (17) by combining eq. (16) with condition *ˆ 0r  and ∂ρ/∂t=−γδφ*[ρ]/δρ+δ, one can obtain that in the critical point time the evolution of probability w=<w[ρ]> is reduced to the simple scale transformation: * / [ ] [( 2) ] 2 d rw t w d r d r              (18) methods of the pattern formation in numerical modeling of biological problems 227 physically it means the following: at the point of the transition energy, the functional becomes the universal one φ→φ * . the kinetic equation based on this functional produces with time new realizations of ρ(r,t) with the same statistical properties, but larger scales. in the limit t→∞, the structure becomes scale invariant. this structure is generally similar to that found in the intermediate stage of the nucleation process at firstorder phase transitions, but it never finishes in static ordering. to calculate practically some particular realization of the density in the critical point, one can solve numerically equation ∂ρ/∂t=−γδφ*[ρ]/δρ+δ with the energy density found from the rg equation in its local approximation: 2[ ( )] [ ( ) ( )] 2 c x dx f      , (19) where ( )f  is normalized to the critical temperature * 2( ) ( 0)f f f       solution of the local version of rg equation *ˆ 0r  : 22 2 2ˆ 0 2 d f f f rf df                   (20) despite of its visual simplicity, eq. (20) is nonlinear and cannot be solved analytically. but, in contrast to functional equation *ˆ 0r  it is already an “ordinary” differential equation and it is easy to find physical branch f * of its solution *ˆ 0rf  numerically with a very high accuracy. discrete data array f * (ρk) has big enough number n≫1 of points k=1,2…n, where each value of {ρk} defines unique value of {f * k}. with a good accuracy, it can be used in the kinetic equation: */ [ / ]t c f            (21) instead of the analytic formulae for the energy, otherwise normally used before. repeating the simulation with random initial conditions and with the random timedependent noise <δ(r,t)>=0, <δ(r,t)δ(r´,t´)>=dζδ(r-r´)δ(t-t´) after long-time runs, we can get unlimited number of the realizations ρ(r,t). direct observations of the simulation results show that every instant density distribution ρ(r,t) contains many nuclei of different size. then, the wider particular maximums of density are formed, the longer time they survive in the general landscape. it means that at large scales the total process of transformation becomes slower and slower. besides, stronger correlation between the densities in different spatial positions appears. in an extremely long run limit (when t→∞), density distribution ρ(r,t) tends to the expected scale invariant structure. one can calculate correlation function g(rr´) =< ρ(r,t)ρ(r´,t´)> at the fixed time moment t and find this scaling correlation function. it means that g(rr´)=<ρ(r,t)ρ(r´,t´)~1/|rr´| β depends as a power function on distance |rr´| between the points. this quite general result favors to the common observation that in many cases, irrespective of the specific features of the system under investigation, the evolution of the surface layer proceeds in a fairly universal manner. 228 a.e. filippov, s.n. gorb 5. chemical appearance of fractal surfaces one more example of automatic generation of fractal surfaces comes from the case, where in the immediate vicinity of the smooth flat interface of the two media in contact, a dense layer of one or more reaction products emerges, as a result of a chemical reaction. it was observed that in the process of growing this layer becomes more and more porous and rough. gradually, an essentially inhomogeneous but, as a rule, scale invariant structure is formed, and the laws governing the growth of this structure are characterized by fractal dimension and growth exponent [9]. in addition to arousing purely scientific interest, the study of corrosion-front growth attracts attention because of its importance from the standpoint of practice, since in some applications the problem is closely linked to that of raising the efficiency of electric batteries. for instance, when a lithium anode is placed into an electrolyte containing socl2 as an additive, due to the exceptionally high reactivity of lithium, a porous twocomponent layer of licl and so2 is formed at the surface of the anode. the presence of such a layer leads to what is known as a lag effect, when the element is stored for a long time. micrographs of the surface layer show that the layer can be considered as a combination of a relatively dense initial layer with a subsequent transition to a fractal structure with an ever increasing porosity. the highly universal properties manifested by different systems suggest that one can use universal growth models based on a combination of the ideas of continuum field theory and kinetic equations with a random source. being fairly common in the theory of phase separation and fluctuation phenomena in phase transitions, the kinetic equations with a source of noise should be used cautiously in describing front growth, the reason being that, in contrast to phase transitions, where generation of the order parameter occurs in the bulk of the system, a random source cannot be considered additive. the generation of a finite density of components forming the front occurs only in the immediate vicinity of the already existing boundary. this means that in the case at hand the corresponding source in the equation must be multiplicative i.e., at least contain the density as a factor. however, in recent publications, devoted to theoretical studies of phase diagrams and transitions in systems with multiplicative noise, it was noted that the presence of such strong noise can have a dramatic effect on the ordered structure and on the phase diagram, and may lead to the emergence of new nontrivial phases. in our case, this means that the model equation should be written in such a way so as to exclude additional difficulties associated with this noise. from an experimental standpoint, the study of fractal corrosion structures is convenient since the corrosion front is observed directly in micrographs and the corresponding two dimensional distributions of density can be studied explicitly. at the same time, the processes involved are very complex, and notwithstanding the continuing efforts, the theoretical models still remain extremely simple, although they presuppose a numerical analysis of the kinetic equations. usually, only the density of a single distributed quantity that is considered the most important in each specific case is involved. this is generally not the case in physicochemical processes, since usually two or more components participate in the reactions. no matter how subtle the description of a system by the single-component approach is, it is sufficient to replace the study of the system by an analysis of purely theoretical models. given contemporary computer modeling methods of the pattern formation in numerical modeling of biological problems 229 techniques, any attempt to reduce the problem to a single equation is more a tribute to the analytic tradition than a real necessity. here we will demonstrate the feasibility of moving in this direction by the example of a two-component model formulated for the description of growth and corrosion of a broad class of porous surface layers initiated by chemical reactions. below we examine the simplest two-component case, assuming, as an example, that we are dealing with chemical reactions that proceed in a system with a contaminated lithium anode. the complete picture of the reactions in such a system is fairly complicated and can be expressed as follows: li→li + +e , 4li + +4e +2socl2→4licl+so2+s. actually, we are interested only in the formation of a front consisting of lithium chloride licl contaminated by the reaction product so2 that concentrates near the surface. bearing all this in mind, we can interpret the bare equation 9 as an initial equation for the evolution of the density of licl which we denote by ρ1(r,t). the corresponding coefficients and the source of noise will be labeled by the index “1”. by ρ2(r,t) we denote the density of so2. we model the local repulsion of the reaction of the products licl and so2 by a fixed-sign additional term in the effective energy of the system, v12(ρ1,ρ2), which in the lowest order that can be written v12(ρ1,ρ2)=bρ 2 1ρ 2 2/2. equation of motion for the first density becomes ∂ρ/∂t=−γ[c1δρ1+ρ1(1−ρ1)][c1+δ1(r,t)]bρ1ρ 2 2. this equation must be augmented with an equation describing the evolution of the second component, ρ2. the second component ρ2, just like the first one, is generated as a result of the same reactions near the free (not contaminated by so2) licl surface. this means that for ρ2, we must use the same generating term ρ1(1−ρ1) as for ρ1: 2 1 2 2 1 1 2 2 2 1 2/ [ (1 )][ ( , )] ( ).t c c r t b f                  here we have allowed for the fact that although both densities, ρ1 and ρ2, emerge as a result of the same reaction, the rate of formation of the dense components in r ρ1 and ρ2 may differ, so that generally c1≠c2. obviously, the terms linear in ρ2 cannot ensure that the increase in ρ2 is stopped and is stabilized ρ2→1 in the static limit. we must also bear in mind that far from the front, there is no spontaneous generation of ρ2, and hence the effective energy v2(ρ 2 2)=bρ 2 2(1−ρ 2 2), whose variation yields the combination proportional to the function f(ρ2)=ρ2(0.5−ρ2) (1−ρ2). it contains a barrier that separates the two similar minima at ρ2=0 and ρ2=1. it can be shown that in the continuum approximation, with only the lowest harmonics in the energy (and hence laplasian terms δρ1,2 written in the equations), one has to add generation terms with the step-function cut-off factor | '| ( ' ( ') ) r r dr r a         . it turns on the generation, when the density in some neighborhood |r-r´|<σ exceeds critical threshold a. in other words, θ→1 when ' ' ( ') r r dr r a      and θ→0 in the opposite limit. finally kinetic equations take the form: 2 1 1 1 1 1 1 1 1 2/ [ (1 )][ ( , )]t c c r t b               (22) 2 1 2 2 1 1 2 2 2 1 2 2 2/ [ (1 )][ ( , )] (0.5 )(1 )t c c r t b                     . in purely local approximation, the system eq. (22) can be reduced to the form of ordinary differential equations: 230 a.e. filippov, s.n. gorb 2 1 1 1 1 1 1 2/ (1 )[ ( , )]t c r t b           (23) 2 1 1 1 2 2 2 1 2 2 2/ (1 )[ ( , )] (0.5 )(1 )t c r t b                 . here the scale of time is normalized to γ=1. the global structure of the phase portrait is depicted in fig. 3 which presents a physically interesting realization of the portrait. the local system of equations eq. (23) actually describes the evolution of the densities at each point in space without allowing for interaction between different points. in this approximation, the interaction is taken into account only via the initial conditions. specifically, as the front arrives at a point in space, both densities ρ1 and ρ2 begin to be generated at that point, so that the physical scenario in the phase portrait in fig. 5 corresponds to the trajectories that emerge in the neighborhood of the trivial point ρ1=ρ2=0. fig. 5 phase portrait of local version of the kinetic equations describing growth of porous surface. fixed points corresponding to possible combinations of local densities are marked by red color. two well pronounced “large rivers” leading to the static final configurations are shown by the blue lines. cloud of small gray points represents instant realizations of the densities numerically found for the exact nonlocal equations. the separatrix connecting this point and the saddle point divides plane {ρ1,ρ2} into attracting basins for the two stable directly seen fixed points. studying the behavior of the trajectories that start near the separatrix, we can predict several results of numerical modeling of the complete equations and, in the final analysis, the properties of real systems. in particular, we can easily predict the role of the source of noise. if the noise is strong, the phase trajectories can pass both above and below the separatrix, irrespective of the scenario according to which the front arrives at the given point. within a certain time interval after the arrival of the front, both densities ρ1,2 increase essentially simultaneously and very fast, to which the first maximum in the evolution rate methods of the pattern formation in numerical modeling of biological problems 231 2 2 2 1 2( ) [( / ) ( / ) ]w t t t       (24) corresponds, as depicted in fig. 4. near the saddle point, there can be no further increase in ρ1,2. the rate w(t) rapidly decreases. at the same time, there is phase separation in the system, with one of the densities, ρ1 or ρ2, expelled from the given region in space. this is followed by a sharp increase in w(t) accompanied by a rapid buildup of the remaining component, a process that is slowed down near one of the stable fixed states. evolution rate w(t) as a function of time is shown in fig. 6. fig. 6 evolution rate w(t) as a function of time. good correlation with the phase portrait (shown in the insert) is clearly seen. the particular trajectory for which w(t) is calculated is marked by red color. low evolution rate corresponds to the vicinities of the fixed points and “large river” leading to final static configuration. discrete set of black points represent the mean rate numerically found for the exact nonlocal equations. the characteristic double-humped curves representing the evolution rate are indeed observed, when the complete system of equations is solved numerically. in accordance with the physics of the problem, the initial condition is selected in the form of a narrow strip of density ρ1(r,t), near one of the boundaries of the two-dimensional system. here the process of generation and separation of the densities ρ1,2(r,t) is accompanied by the formation of characteristic dendritic spatial distributions of both densities. the two densities are generated simultaneously in the vicinity of the front. however, in the absence of noise, the initial density distribution leads to the formation behind the front of a completely filled region. numerical solution shows that further with time the front becomes discontinuous, and the expanding ordered region is transformed into a fractal. in fig. 5, we use a copper scale to show a characteristic fragment of the system with a distribution of the total density that emerges at the intermediate stage in the transition from the homogeneous growth to the fractal growth. contaminated regions and 232 a.e. filippov, s.n. gorb regions of active growth are clearly visible. these are characterized by intermediate values of the densities which correspond to sections in fig. 5 with intermediate shades of copper color. fig. 7 two intermediate stages of the porous surface layer formation (from subplot (a) to (b)). with the time, such system can develop very complex structure with not simply fractal surface, but also with lacunas of a negative curvature, which effect in a strong increase of adhesion of the soft tissue contacting such a surface. in comparison to the study of growth processes with models with only one fluctuating variable, the novel property of the system lies in the possibility of the emergence of voids behind the front, i.e. regions filled with neither of the two components. when there is only one field, i.e. ρ1(r,t) (such lacunae must be thought of as being filled with the other, „„contaminating‟‟ component of density, ρ2(r,t), which in this case is not explicitly present in the equations. the model contains additional information about the second field ρ1(r,t), which makes it possible to distinguish between the regions occupied by ρ2(r,t) and the voids. the mechanism of void formation is clearly seen in fig. 7a, which for a small fragment of the front depicts a typical growth sequence. three characteristic moments in time are singled out: the emergence of a dense initial layer, the emergence of the first dendritic protuberances, and the collapse of the first internal pores in a structure with a total density ρ(r,t)=ρ1(r,t)+ρ2(r,t). well-formed voids are clearly visible in fig. 7b. void formation is closely related to the ability of the „„contaminant‟‟ ρ2(r,t) to block the active sections of the front ρ1(r,t) and, at least in principle, to terminate the growth process. as the pores collapse, the front usually continues to move in both directions. naturally, the outer boundary ρ1(r,t) is almost insensitive to the presence of a pore blocked somewhere inside the system and continues its forward motion. the inner boundary ρ1(r,t) surrounding a pore is qualitatively similar to the outer boundary and can move „„back,‟‟ up to the point, where it is completely blocked by sections with ρ2(r,t). the scenario of the evolution of the system turns out to be exceptionally multifaceted and methods of the pattern formation in numerical modeling of biological problems 233 under a slight variation of the coefficients of the model makes possible a reproduction of very realistic configurations of the densities ρ1(r,t) and ρ2(r,t). 6. formal generation of fractal surfaces it was an example of creation of the fractal surface by growth. now, let us turn to more formal methods of numerical generation of surfaces. depending on a particular problem, we use at least two ways to generate rough surface that is close to the fractal one. one is to apply fourier expansion with the harmonics restricted between maximal and minimal wavelengths. depending on the number of the waves and both limits, the surface will be more or less close to the real fractal one and include or not irregularities with small sizes [10]. another way is to use random deposition of the gaussians. depending on their number, height dispersion, minimal and maximal widths, the surface will be closer to fractal or not. many of biological surfaces have a structure made as a pattern of close objects with some specific scale. they can be more conveniently simulated by a set of deposed objects. that is why, for our studies we usually choose the second method, which easily provides such possibility. in many cases, such surface appears as a result of random deposition of some particles, balls, block of some other objects. we definitely admit that the formal model completely ignores the fine structure of adhesion-like weakening at motion, fracture and restoration of the bonds, so on. often just only two basic mechanisms are involved into our minimalistic models: attraction to the surface and energy dissipation. the first one actually restricts the scales, which we include to the surface by the irregularities close to the “size of contact points” (radius of attraction in the model). in particular, this limitation corresponds to a situation with adhesive terminal elements (spatulae) of attachment organs of different animals, which were studied elsewhere [11]. the spatulae practically ignore irregularities much smaller or larger than their size. as result, such surfaces can be treated as almost flat. that is why we normally use here the surfaces with the irregularities comparable to the spatula. such deposition with fixed interval of scales can be done either naturally or artificially. we will not concentrate on any specific case and discuss only a couple of more or less abstract examples. despite of formal mathematical generality such approach helps in developing a simple, but realistic numerical model that might be useful, for example, in the study of animal spatula interaction with various substrate profiles. formal models can well explain results of experimental studies, and also predict adhesion of animals to the real surface profiles depending on the dimension and stiffness of the spatula. previously, we studied such biological systems using numerically generated surfaces and compared the results obtained with the real 3d surface profiles obtained experimentally by depositing of spheres. in the purely numerical approach, different roughness of the surfaces can be achieved by depositing of spheres with defined radius on originally flat surface. to simulate this process numerically, we organized our procedure as follows. at every given radius r , we have taken an array of equal spheres, or circles in twodimensional (d=1+1) case and placed them successively into the randomly chosen positions xn, where n=1,2,…,n. the number of the spheres n=||l/r|| in the array was taken to be an integer value corresponding to the total number of the circles of radius r 234 a.e. filippov, s.n. gorb which are necessary to cover the system of the length l uniformly. each sphere was added virtually to the corresponding segment of the surface: 2 2( ; ) ( )n n ny x x r x x   . (25) due to the random deposition, the segments can fall either on the top of already existing coverage or onto the empty space. as a result of such a deposition, the total surface gradually accumulates all the segments 1 ( ) ( ; ) n n n n y x y x x   , which are generally speaking placed one on top of another one. this procedure can be easily generalized for d=2+1 surfaces. typical realizations of such surfaces for different radii r are presented in the fig. 8. fig. 8 consequent stages of the formation of rough surface by the random deposition of the hard balls with fixed radius on the originally flat surface in many cases specific form of the deposed particles is not very important (or maybe not important at all). in such a case, one can treat random deposition procedure as an absolutely abstract way of constructing desirable surface. one of the simplest ways to do this in this case is to model rough surface z(x,y) by a random deposition of the gaussians: 2 2 2( , ) ( , ,{ , }) exp[ (( ) ( ) ) / ]n n n n n n n n n z x y g x y x y a x x y y w       . (26) the advantage of this approach is its mathematical transparency. one can regulate everything by simple manipulation with varied positions, amplitudes and widths of the gaussian functions. the only limitation is generally that the characteristic scale of the artificial surface irregularities should be adjusted to be smaller or comparable with the scale of the elements interacting with the surface (spatulae of animals, for example). in particular, the typical distance between the hills and valleys of the randomly accumulated surface z(x,y)=gn can be regulated by the number of gaussians. an additional convenience of the method is in the fact that one can even not control the amplitude of the asperities during accumulation of the sum gn. the amplitude of roughness after accumulation is finally regulated by the normalization z(x,y) → a(z(x,y)−min(z))/(max(z)−min(z)), where desirable amplitude a can be chosen from the limit of the flat surface a=0 to the values comparable with the characteristic lengths of the system under consideration. different variants of the rough surface generated by the random deposition of gaussians are shown in fig. 9. methods of the pattern formation in numerical modeling of biological problems 235 fig. 9 different variants of the rough surface generated by the random deposition of gaussians. the number of gaussians used to generate the surface grows from left to right. one of the widely known reasons to generate numerical surface or potential u is its application to tribology. let us mention, in this context, a commonly used prandtltomlinson model that has proven to be successful in describing shear response in surface force apparatus configuration. this model describes the lateral motion of a driven plate and has the following form: 2 2/ / ( ) / ( ) 0x t x t u x x k x vt          (27) here a driven plate of mass m and the center-of-mass coordinate x is pulled by a spring of a spring constant k. in standard tomlinson equation that is normally applied to study atomically flat surfaces, u(x) is the effective periodic potential u(x)=u0cos(2x/b) experienced by the plate due to the presence of an embedded system. in some sense, such simple potential is also particular realization of the surface pattern appearing on microscopic level, where its creation is regulated by the process of crystallization, which normally leads to an ideally periodic crystal lattice. below for general theoretical study, we use dimensionless units normalized to characteristic microscopic scales and energies of the particular system under consideration. parameter , as usually, is responsible for the dissipation. the spring is connected to a stage which moves with a velocity v. tomlinson equation was well supported microscopically, but usually people do not extend its application even to the mesoscopic scales. however, as we saw in many examples of the formation of many surfaces (may be better to say all the surfaces which we meet in normal life), the surface is dictated by frozen kinetics, which leads to the relief, which is quite far from an ideal periodic structure. as a result, one partially knows how to relate the parameters that appear in eq. (27) to the microscopic characteristics of the system, but can not correlate them with the macroscopic friction. this problem is not solely limited to the friction studies, but appears also when dealing with adhesion. in the previous papers, we have seen how important it is for the study of attachment devices of animals, which have to adapt to real surfaces also on intermediate mesoscopic scales, but not only at the microscopic one, where pattern either can be approximately treated as a flat or even simply periodic [12]. so, what becomes a major problem here is that the mesoscopic structure of frictional surfaces is of a fractal character and thereby cannot be characterized by a certain wave vector (or even few ones), like it is normally used in applications of the tomlinson model. on the other hand, it could be inconvenient or simply senseless to reproduce each time one of the physical or chemical processes described here. it is important to know, 236 a.e. filippov, s.n. gorb how such surfaces appear, or maybe it becomes important, if the studied surface really has specific substructure with the periodicity important for the modeling. however, if we can forget about underlining process and accept a naive hypothesis that the majority of the surfaces are simply fractal, it is natural to extend the model into a directly generated pattern according to very formal mathematical definition of the fractal. let us consider a fractal potential of the form: 2 1 0( ) ( )cos( ) q q u x u dqc q qx   , ( )c q q  (28) here q1 and q2 are characteristic cut off wave vectors and δ(x) is the random phase that we assume δ-correlated <δ(q)δ(q‟)>=2πδ(q-q‟). for the majority of physically interesting systems, index β is close to β≈0.9. below we will keep this value for definiteness. for further study, it is convenient to go over to a discrete representation of the integral in eq. (2) ∫dqc(q)→σ with a discrete step between the wave vectors δq determined by the smallest vector q1 corresponding to an inverse maximal length lmax of the system, which equals normally to its size lmax=l . total number ntot of the terms in the sum is given by ntot=q2/q1≡q2/δq. the discrete approach is natural for our further numerical studies. it allows us to adjust the potential to different scales by a number of the modes included into summation nmodes<ntot. the tomlinson model based on analytical definition of the potential is much simpler for computer implementation. the force is calculated using analytical formula at each time-step and procedure can be formally extended to infinite time-space runs. the modified fractal model operates with the data array ufractal(x)=u{xj}, where j=1,2,…nmodes. it means that its numerical generation has to be extended to an infinite run too. for this sake, instead of eg. (2), we use the following differential definition of the force caused by the fractal potential: 0( ) / ( )sin( );fractal j j j j u x x u x q c q q x      . it allows us to extend ufractal(x) infinitely each time, when x-coordinate runs out of the array bonds. for the numerical procedure, it means that the modified tomlinson equation 2 2/ / ( ) / ( ) 0fractalx t x t u x x k x vt          (29) is actually extended by an additional differential equation defining ufractal(x)/∂x, which is solved in parallel. to study the scale dependence of friction force, one can generate a set of fractal potentials for different number of modes included (nmodes<ntot). this number defines a cutoff wave vector qcutoff=δqnmodes and the corresponding cutoff wave length λcutoff=2π/qcutoff of the potential. the potentials with different nmodes can be treated as to the same potential "measured" with different spatial resolution. all space scales larger then the cutoff wave length are included in the potential ufractal(x) and should be treated explicitly in the frame of the dynamic model like eq. (29). analogous procedure based on fourier transformation can be extended to generate realistic 3d surface covering 2d plane. typical fractal surface generated by the formal application of fourier transformation shown in fig. 10 by the colormap. the red color corresponds to the higher vertical coordinate. the second subplot in the same figure methods of the pattern formation in numerical modeling of biological problems 237 presents 2d cross-section of the same surface along one of the horizontal lines. in some numerical experiments each such a crossection can be used as a particular realization to accumulate statistics. fig. 10 fractal 3d surface generated by the formal application of fourier transformation shown by the colormap. the red color corresponds to the higher vertical coordinate. the lower subplot presents 2d cross-section of the same surface along one horizontal line (the central line in this particular case). both representations of the surface can be used in numerical experiments depending on the particular problem. the generation of the structured surfaces is not limited only by the friction or adhesion problems. in principle, surface modification and functionalization through nanostructures is a well known way of producing desirable properties of various materials. however, it is not an innovation since it was employed by nature a long time before. 7. combination of discrete and continuous techniques the main idea of the simplification using a combination of two techniques is to substitute an original continuous problem by its combination with some preliminary applied discrete approach (which is used to get more or less final or, at least, almost stationary) configuration and further continuous density calculation. combination of the discrete and continuous approaches was applied to many different chemical and physical systems (including such underlying processes in biological systems) [13]. for example, it can be applied to study of superconducting or magnetic ordering in quasi 2-dimensional systems, or to study growth of surface structures, and so on. in all such cases, the real process involves an interaction of many spatially distributed densities and its direct simulation can cause extremely time consuming calculations. a famous example of such an approach is the study of topological phase transition in quasi 2-dimensional superconducting systems, mentioned above, where extremely complex 238 a.e. filippov, s.n. gorb evolution of the superconducting vortices was substituted to some extend by a motion of “charged particles” [14]. this approach was found to be extremely fruitful and was even awarded by the nobel prize in 2016. one important example is the surface nanostructure in the snakeskin that is supposed to reduce both abrasion and friction, as it has been previously shown for similar artificial nanostructures [15]. on the ventral surface of snakes, this nanostructure is typically constituted by arrays of the nano-dimples. one of the most widely spread structural arrangement type in biological systems is the hexagonal arrangement, which provides the highest packaging of the structure and the highest structure density. visually arrays of nano-dimples remind such structure. fig. 11 scanning electron microscopy image of the tail ventral scale in the snake morelia viridis. the black shadowed gray circle marks a typical hexagonal arrangement of dimples, whereas both white and black circles mark fiveand sevenfold symmetrical arrangement of dimples, respectively. however, it is well known that there is no ideal arrangement in biological systems. previously, we performed the analysis of the arrangement imperfections (disorder) using correlation analysis in a way similar to the previously used for other systems appearing as frozen kinetics [6]. correlation analysis is expected to quantitatively reveal the difference in hexagonal nanostructure arrangements, and in the snakeskin. we will also suggest here a simple mathematical model that helped to explain the difference in these correlation functions and obtain some hints related to the mechanisms of formation of nanostructural patterns in biological systems additional to the previously discussed ones. in this model, the structure with the structure arrangement similar to that found in snakeskin appears due to the "freezing" of an overdamped relaxation. it is caused by a weak repulsion between the structures from an initially random distribution. further, analogous methods of the pattern formation in numerical modeling of biological problems 239 correlation analysis is applied to this model and its‟ results are compared with the results of the correlation analysis of the real nanostructure arrangement in the snakeskin. power spectrum and distribution of distances between the nearest neighbors can be calculated for the snakeskin in the same manner as for any crystal microscopic structure. it was done and found that, despite of the structure in the snakeskin looks like hexagonally ordered: the power spectrum here does not demonstrate pronounced peaks, even in the first ring. it is because of the absence of highly-oriented domains with a prominent arrangement. at the same time, there is characteristic distance between the nearest neighbors, around 0.4 µm. because the first ring in the correlation function, matching the maximum probability of the distances, is well pronounced. it can be assumed that isotropy of the correlation function, which does not show specific directions in dimple arrangement in the snakeskin, is important for e.g. maintaining isotropic tribological properties. the presence of a preferential direction implies a stress enhancement in some direction, and, therefore, material failure facilitation, as well as stronger abrasion. standard approach to weakly disordered hexagonal systems normally supposes application of the directly prescribed hexagonal arrangement, which is slightly perturbed from the originally perfect positions by a random noise of the displacements. in this approach, the particular amplitude of gaussian random shifts might be appropriately adjusted to fit numerically found distribution of the distances close to the experimentally observed in real snakeskin. despite this, the arrangement of the snake nano-dimples has obviously different symmetry of its power spectrum compared to the simulated one. this result means that the nano-dimple spatial distribution can not be simulated by such naive slightlyrandomized hexagonal configuration. thus, real arrangement appears to be a result of some self-organization process caused by an interaction between nanostructures. in spirit of the approach presented here, one can perform generate, as an initial condition, an array of the randomly distributed dimples and apply short range repulsion between them with some potential. system of the dimples evolves according to the standard equation of motion for their centers: / /i ij j r t u r      . but, here we are actually interested not in an exact pattern, but rather in the shape of the distribution of distances and in the symmetry of the correlation function. they can be easily calculated at every single step of the simulation. the total number of dimples in numerical array is completely predefined by the dimple number in a given observation area of the real system. as result, in the course of kinetic process, the mean distance between the dimples remains fixed by the restricted size of the observation area and the only parameters that change during the relaxation are the symmetry and shape of the distribution of distances. from the mathematical point of view, it means that the only one fitting parameter remains in the simulation. it is the width of the final distribution of distances. relaxation process runs up to the moment, when its width coincides with the experimental one. it automatically defines a time moment, when the simulation has to be terminated. this stop immediately freezes the final configuration that is statistically close to the one observed in the real system. other well-known example of biologicalу produced structured surfaces are superhydrophobic surfaces that gain this functional feature due to nano-structures of the surface, often due to the surface coatings by crystals or particles. we will talk later about biological inspirations for these kinds of technical solutions. now let us mention that in material 240 a.e. filippov, s.n. gorb science, surface coatings with regularly arranged globular particles have been produced by so-called colloidal lithography, a technique utilizing the self-assembly of nano-particles on the substrate surface. however, the range of producible patterns and properties of these coatings are still quite limited, as well as their durability. therefore, the study of the close problems by numerical modeling seems to be promising for generating new solutions. a very interesting and non-trivial problem, in particular, was raised by the study of water repellence and the ultrastructure of covering granules. here we will be just interested in complex patterns, which can self-organize from frozen kinetics on the spherical surfaces of colloidal particles. direct observation shows that fine structure of distribution of asperities on the surface of spheres is species-specific and probably depends on the interactions responsible for the appearance of these structures. in order to gain a better understanding of the process of self-assembly and selfarrangement of nano-particles on spherical microstructures, we here apply a theoretical approach. in the spirit of previously described kinetic approach, we introduce a numerical model that allows us to test the effect of different interaction properties between the particles on the morphology of the eventual structure. our results show good mutual correspondence with numerically found structures. as in the previous example (snakeskin), the model of the systems under consideration is organized as a combination of both discrete and continuous approaches. from a general (physical, chemical and biological) point of view, one can expect that the observed structures appear in some kinetic process during which continuously an initially more or less uniformly distributed substance redistributes and solidifies in a 3-dimensional space. as always, our goal is to minimize calculation time, from one side, and to get transparent understanding of the process in the frame of relatively simple model. in the particular case of the structuralized colloidal particles, the problem is complicated by the specific topology of the surface, on which the process runs, because the redistribution of the densities and their further solidification takes place directly on spheres [16]. here, one has to deal with a kind of phase transition (or phase separation) inside a spherical layer. as far as we know, such a problem with geometrically frustrated assemblies was not widely studied in conventional physics and chemistry. however, in biological systems, such geometry is widespread, and therefore must be properly studied. topological complexity of the problem forces us to as simple as possible reduction of the numerical model, namely substitute an original continuous problem by a combination of discrete model to find almost stationary configuration and further continues density calculation. let us mention that such an approach is also applicable for the tamme‟s problem, where the study of kinetic ordering on a sphere can be substituted by a finding of equilibrium ordering of “n equal charges” on the sphere. in all these cases, it is important that, in the used approach, one can finalize a discrete distribution of the maximums or minimums of the densities, and only after getting the distributions to “dress” them by the continuous field with an expected density distribution. however, it is important to note that in its classical mathematical form tammes problem is treated as an optimization problem which can be formulated as follows: given a natural number n, place n points on the surface of a sphere in a way that maximizes the shortest distance between any two points. this question normally dictates maximal simplification of the problem (like limiting a number of points by special cases, which allow analytical solution, or limiting of their interaction by the simple coulomb one). our goal here is almost an opposite one. we study which particular structures can appear at different interactions and how they reproduce the observed ones. these methods of the pattern formation in numerical modeling of biological problems 241 structures are certainly not optimal, and in many cases, the distribution of final densities, being generated by frozen kinetics, is even extremely inhomogeneous. it rather reminds study of structural phase transitions between different phases in solid state physics depending on the relations between the interactions. however, it differs from the standard physical studies due to new atypical confinement of the geometrically frustrated assemblies of particles on the spherical surface. according to these ideas, we organize the procedure as follows. at the first stage, we randomly distribute some number of particles (further nucleation centers for the densities) on the surface of the sphere. at this stage, the most important parameters are the relationships between the sphere with radius rs, number of particles n, and the characteristic distance of repulsion interactionr0 between them. more specifically, the model and the numerical procedure was described earlier in one of the previous papers [16]. here we mention only that total interaction can be chosen alternatively as just a pure short range repulsion: 1(| |)j ku r r , or as a combination of both repulsion and attraction 2 (| |)j ku r r terms. the second variant leads to a typical effective potential with minimum 1 2(| |) (| |) (| |)interaction j k j k j ku r r u r r u r r     , which determines an equilibrium distance between the particles reachable in long time asymptotics. the mostly nontrivial aspect of the situation here is that the particles have to move confined to the sphere. from the mathematical point of view, it means that we apply a strong potential, (| |)sphere j su r r which attracts them to the surface of given radius rs. if this potential is strong enough, despite of (relatively weak) repulsion usphere≫u1 between the particles, they practically can not leave the spherical surface. the equation of motion for the “particles” takes the form: /jr t   [ | | | |]/interaction j k sphere j k k k u r r u r r r      . during the routine, the particles dynamically rearrange with a rate, which gradually goes down with time. it is one of the particular implementation of the so-called “a large river effect”. a particular realization of this procedure is shown in fig. 12. as usual, one can control this slow-down calculating mean velocity of the particles and stop the procedure, when velocity becomes negligible. certainly, as we already discussed before, the particles can never reach real “ground state”. especially, it is correct for the spherical surface. but, at long time run, they get some stationary distribution which looks quite realistically. it is what we expect for the “large river” (or frozen kinetics) in a real system. the final stage of the simulation is to “dress” the spherical surface ( )j r and nucleation centers (particles) by the continuous density distributions. fig. 12 combination of the discrete and continual approaches in a single model. at the first stage of the calculation, a discrete array of the interacting points, randomly 242 a.e. filippov, s.n. gorb placed on the sphere (a), dynamically orders itself into a frozen, almost static configuration (b), which is “dressed” by the continuous density (c) the total density around the nucleation centers should be accumulated by a summation of all the particles ( ) ( ) ( )total j s j r r r    . it can be proven that each numerically found pattern is determined by a relation between the area of the sphere and the size of the particles on one side (or density of them on the surface), and the interaction between the particles on the other side. according to general discussion, we can not directly visualize obtained continuous density distributions as a projection on the 2d surface of the plots. but, in standard manner, we can plot the surfaces corresponding to any constant density and compare them with the data of real structures. acknowledgments: this work was supported by the georg forster research award (alexander von humboldt foundation, germany) ukr 1118826 to a.e.f. renate schilling is greatly acknowledged for the linguistic corrections of the manuscript. references 1. popov, v.l., heß m., 2015, method of dimensionality reduction in 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j. phys. c: solid state physics, 6, pp. 1181–1203. 15. kovalev, a, filippov, a.e., gorb, s.n., 2016, correlation analysis of symmetry breaking in the surface nanostructure ordering: case study of the ventral scale of the snake morelia viridis, appl. phys. a, 122(253), pp. 3-6. 16. filippov, a.e, wolff, j.o., seiter, m., gorb, s.n., 2017, numerical simulation of colloidal self-assembly of super-hydrophobic arachnid cerotegument structures, j. theor. biol., 430, pp. 1-8. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 301 313 https://doi.org/10.22190/fume200302027k © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper assessment of velocity accuracy of aircraft in the dynamic tests using gnss sensors kamil krasuski 1 , adam ciećko 2 , grzegorz grunwald 2 , damian wierzbicki 3 1military university of aviation, institute of navigation, dęblin, poland 2university of warmia and mazury in olsztyn, faculty of geoengineering, institute of geodesy and civil engineering, olsztyn, poland 3military university of technology, faculty of civil engineering and geodesy, institute of geospatial engineering and geodesy warsaw, poland abstract. the paper presents a new model for determining the accurate and reliable flight speed of an aircraft based on navigation data from the three independent global navigation satellite system (gnss) receivers. the gnss devices were mounted onboard of a cessna 172 aircraft during a training flight in south-eastern poland. the speed parameter was determined as the resultant value based on individual components from 3 independent solutions of the motion model. in addition, the standard deviation of the determined flight speed values for the cessna 172 aircraft was determined in the paper. the resultant on-ground and flight speed of the cessna 172 aircraft ranged from 0.23 m/s to 74.81 m/s, while the standard deviation of the determined speed values varied from 0.01 m/s to 1.07 m/s. in addition, the accuracy of research method equals to -0.46 m/s to +0.61 m/s, in respect to the rtk-otf solution. the rms parameter as an accuracy term amounts to 0.07 m/s for the presented research method. key words: aircraft, velocity, gnss receiver, flight test, accuracy 1. introduction the basic parameters of flight dynamics include navigation data, among which the most important are coordinates, altitude, flight speed and orientation angles. the use of gnss receivers in aviation allows for the determination of the above-mentioned navigation parameters of the aircraft [1, 2, 3]. the aircraft position established on the received march 02, 2020 / accepted june 19, 2020 corresponding author: kamil krasuski military university of aviation, 08-521 dęblin, dywizjonu 303 nr 35 street, poland e-mail: k.krasuski@law.mil.pl 302 k. krasuski, a. ciećko, g. grunwald, d. wierzbicki basis of the gnss technique is determined in accordance with icao recommendations using blh(b-latitude, l-longitude, h-ellipsoidal height) ellipsoid coordinates [4]. in turn, the flight speed can be estimated based on the difference of: xyz geocentric coordinates or enu (e-easting, n-northing, u-up) local coordinates whereas the hpr (h-heading, p-pitch, r-roll) orientation angles are typically specified in the enu local coordinates [5]. the determination of aircraft navigation parameters is very important for maintaining and ensuring the continuity of flight mechanics [6, 7, 8]. therefore, the realtime monitoring of the aircraft flight navigation parameters is of key importance in terms of the safety of flight operations. the motivation of this work is to determine the flight speed of an aircraft using a gnss sensor. the undertaken scientific problem has already been presented in many research papers. cannon et al. [9] presented a mathematical model of numerical simulation for determining the flight speed of an aircraft based on navigation data from two gnss receivers. szarmes et al. [10] presented a very interesting solution in which the doppler effect based on gnss data was used to determine the flight speed of an aircraft. krasuski [11] described an extended solution using the doppler effect. namely, the gps code measurements and the doppler measurement at l1 frequency were used to determine the flight speed of the aircraft. ćwiklak et al. [12] presented a mathematical model for determining the flight speed of an aircraft based on data from one on-board gps receiver. in this case, the flight speed of the aircraft was determined in the enu local coordinates. however, kozuba and krasuski [13] proposed the solution of the aircraft flight speed model for the on-board glonass receiver in the xyz geocentric coordinates. a very interesting solution for determining speed was published by he [14], who determined the aircraft flight speed by using two on-board gps/glonass receivers. salazar [15] presented two models for determining the flight speed of an aircraft with the use of gps sensors the kennedy model and the eva model. the speed results obtained from both models are convergent for a single on-board gps receiver. van graas and soloviev [16] presented a mathematical model for estimating the flight speed of an aircraft employing gps autonomous code positioning of the spp (single point positioning) mode and code differential dgps (differential global positioning system) mode. various researchers presented interesting research results regarding the determination of flight speed based on a multi-sensor solution. wang et al. [17] and wu et al. [18] showed the results of aircraft flight speed tests based on a solution from a gps sensor and an ins sensor, while the flight speed readings of the aircraft from the gps sensor and pitot tubes were published and compared by the foster et al. [19]. as part of the presented work, a new solution for determining the flight speed of an aircraft was presented based on readings from 3 gnss receivers installed on board the plane. the resultant aircraft flight speed value is determined on the basis of three independent readings. the research experiment used real navigation data from on-board gnss receivers mounted on-board cessna 172 aircraft. the proposed solution is innovative in the aspect of improving the navigation indications of aircraft flight mechanics. presented research considerations were carried out on a large sample of navigation data acquired from three on-board gnss receivers. assessment of velocity accuracy of aircraft in the dynamic tests using gnss sensors 303 2. materials and methods in the study real gnss data collected from on-board receivers, located in the cockpit of cessna 172 aircraft, were used. the flight took part in south-eastern poland near dęblin airport (epde). there were 3 gnss receivers of different brands and configurations on-board the cessna 172 aircraft, as follows [20]:  thales mobile mapper pro receiver using gps l1 code positioning,  thales mobile mapper pro receiver using gps l1 code augmented with egnos,  topcon hiper pro a dual-system gnss receiver using gps/glonass code observations. all 3 receivers recorded navigation data with an interval of 1 second. typical accuracy of position determination for examined gnss solutions was in the range of 1 to 5 m. all gnss receivers were placed in the cockpit of a cessna aircraft, very close to each other, as shown in fig. 1. fig. 1 the on-board gnss receivers in cessna 172 aircraft the location of gnss sensors in the cockpit allowed the determination of basic flight navigation parameters, including position, time and speed. it should be noted that gnss receivers did not have any direct impact on the work of other flight instruments. nor did they disturb the pilot in any way. in flight mechanics, the reliable and accurate recording of flight parameters is crucial. equipping the aircraft with 3 gnss sensors enables the verification of aircraft flight parameters in real time. the effective verification can also be made in post-processing mode. for every second of the flight, the position of the aircraft is determined by each gnss sensor. the three-dimensional position can be given in the form of xyz geocentric coordinates or blh ellipsoidal coordinates. on the basis of collected coordinates of the flight position, the components of the flight speed of the aircraft in the xyz geocentric or enu topocentric coordinates are determined. in the first stage of research, individual components of the aircraft flight speed are determined. in the presented work, the components of the cessna 172 flight speed were calculated based on the xyz geocentric coordinates for all 3 gnss receivers independently, as given below: 304 k. krasuski, a. ciećko, g. grunwald, d. wierzbicki { { (1) { where ( ) are flight speed components along the xyz axes based on readings from the thales mobile mapper pro receiver (gps l1 solution), ( )are flight speed components along the xyz axes based on readings from the thales mobile mapper pro receiver (gps l1 + egnos solution),( )are flight speed components along the xyz axes based on readings from the topcon hiper pro receiver (gps/glonass solution), is an observation interval, , is a coordinate increment along the x axis based on readings from the thales mobile mapper pro receiver (gps l1 solution) for the time interval, is a coordinate increment along the x axis based on readings from the thales mobile mapper pro receiver (gps l1 + egnos solution) for the time interval, is a coordinate increment along the x axis based on readings from the topcon hiper pro receiver (gps/glonass solution) for the time interval, is a coordinate increment along the y axis based on readings from the thales mobile mapper pro receiver (gps l1 solution) for the time interval, is a coordinate increment along the y axis based on readings from the thales mobile mapper pro receiver (gps l1 + egnos solution) for the time interval, is a coordinate increment along the y axis based on readings from the topcon hiper pro receiver (gps/glonass solution)for the time interval, is a coordinate increment along the z axis based on readings from the thales mobile mapper pro receiver (gps l1 solution) for the time interval, is a coordinate increment along the z axis based on readings from the thales mobile mapper pro receiver (gps l1 + egnos solution) for the time interval, is a coordinate increment along the y axis based on readings from the topcon hiper pro receiver (gps/glonass solution)for the time interval, stands for the current epoch of observation and is the previous epoch of observation . based on eq. (1), the resultant flight velocity of the cessna 172 aircraft was calculated based on the xyz coordinates for all 3 gnss receivers independently as presented below: { √( ) ( ) ( ) { √( ) ( ) ( ) (2) { √( ) ( ) ( ) assessment of velocity accuracy of aircraft in the dynamic tests using gnss sensors 305 where is the resultant flight speed based on readings from the thales mobile mapper pro receiver (gps l1 solution), is the resultant flight speed based on readings from the thales mobile mapper pro receiver (gps l1 + egnos solution) and stands for the resultant flight speed based on readings from the topcon hiper pro receiver (gps/glonass solution). in the final step, velocity components of eq. (1) and the resultant values for each gnss receiver, eq. (2), allow for the cessna 172 plane velocity determination based on the entire gnss sensor array, as follows: (3) where is the total resulting airspeed of the cessna 172 and . the parameter means the resultant speed of vessel movement in flight mechanics using gnss sensors. the standard deviation for parameter is also defined according to: √ [ ] (4) where is a standard deviation of the total resultant flight speed of the cessna 172 and is a correction, difference between parameters and , , , according to: [ ]. 3. results the results of the tests are presented in section 3. first, the flight velocity components for xyz axes were determined for 3 gnss sensors independently, according to eq. (1). table 1 presents the results of individual speed components along the xyz axes based on 3 solutions: gps l1, gps l1+egnos and gps/glonass. the results show that the minimum flight speed along the x axis based on data from 3 gnss receivers ranges from 48.79 m/s to -48.72 m/s. whereas, the maximum flight speed along the x axis based on data from 3 gnss receivers is from +56.44 m/s to +56.84 m/s. the minimum flight speed along the y axis based on data from 3 gnss receivers stretches from -61.83 m/s to -61.44 m/s. while, the maximum flight speed along the y axis is between +61.25 m/s and +61.34 m/s. the minimum flight speed along the z axis based on data from 3 gnss receivers is from -49.03 m/s to -48.91 m/s. whereas, the maximum flight speed along z-axis is from +41.11 m/s to +41.54 m/s. 306 k. krasuski, a. ciećko, g. grunwald, d. wierzbicki table 1 results of aircraft velocity along xyz axes for each gnss receiver gnss receiver minimum range of velocity component along x axis [m/s] maximum range of velocity component along x axis [m/s] minimum range of velocity component along y axis [m/s] maximum range of velocity component along y axis [m/s] minimum range of velocity component along z axis [m/s] maximum range of velocity component along z axis [m/s] thales mobile mapper pro (gps l1 solution) -48.72 +56.44 -61.44 +61.25 -49.03 +41.11 thales mobile mapper pro (gps l1 + egnos solution) -48.73 +56.48 -61.45 +61.26 -49.01 +41.14 topcon hiper pro (gps/glonass solution) -48.79 +56.84 -61.83 +61.34 -48.91 +41.54 table 2 displays the resultant flight velocity of the cessna 172 aircraft based on 3 solutions: gps l1, gps l1+egnos and gps/glonass, according to eq. (2). the results show that the minimum resultant flight speed based on the data from 3 gnss receivers ranges from +0.11 m/s to +0.42 m/s. while, the maximum resultant flight speed based on data from 3 gnss receivers stretches from +74.50 m/s and +75.56 m/s. table 2 results of total aircraft velocity for each gnss receiver gnss receiver minimum range of velocity [m/s] maximum range of velocity [m/s] thales mobile mapper pro (gps l1 solution) +0.12 +74.51 thales mobile mapper pro (gps l1 + egnos solution) +0.11 +74.50 topcon hiper pro (gps/glonass solution) +0.42 +75.56 fig. 2 shows the relevant results of the test, i.e. the total resultant velocity for the frame of all gnss sensors installed in the cessna 172, according to eq. (3). based on the obtained test results, the total flight speed of the frame of 3 gnss sensors placed on board the cessna 172 aircraft is between +0.23 m/s and +74.81 m/s. the average flight speed is +48.53 m/s. it can be observed that for the first 200 measurement epochs, the flight speed was less than 20 m/s. starting from the epoch 250, the flight speed increased to over 40 m/s. the maximum speed can be observed in about the 750 epoch and from the 2400 epoch, the flight speed starts to drop down to around 10 m/s. fig. 3 shows the results of the total resultant velocity for a frame of all gnss sensors installed in a cessna 172, as a function of the distance travelled by the aircraft. it can be seen that after passing a point of 3 kilometers, the flight speed increases to over 40 m/s. at a distance of around 40 km, the speed rises to a maximum value of about 75 m/s. up to 130 km of the route, the speed of flight is over 40 m/s, then it starts to fall systematically during approach and landing at the airport. assessment of velocity accuracy of aircraft in the dynamic tests using gnss sensors 307 fig. 2 the total velocity of gnss sensor array installed in cessna 172 aircraft as a function of time fig. 3 the total velocity of gnss receivers array in cessna 172 aircraft as a function of distance 308 k. krasuski, a. ciećko, g. grunwald, d. wierzbicki fig. 4 displays the results of the total resultant velocity for a frame of all gnss sensors installed in the cessna 172, as a function of aircraft flight altitude. it is worth noting that from an altitude of about 200 m, the flight speed increases to over 40 m/s. the highest flight speed values are visible, with a maximum flight altitude of 600-700 m. at altitudes from 200 m to 500 m, the flight speed ranges from 40 m/s to 60 m/s. fig. 4 the total velocity of gnss receivers array in cessna 172 aircraft as a function of flight altitude table 3 results of parameter parameter minimum value [m/s] maximum value [m/s] mean value [m/s] median value [m/s] parameter -0.53 +0.61 +0.01 +0.01 parameter -0.46 +0.62 +0.01 +0.01 parameter -1.24 +0.88 -0.01 -0.01 table 3 shows correction values as the difference between the total speed and resultant speeds for 3 different receivers: , , (see eq. (4)). from the results obtained it can be concluded that the dispersion of the parameter results ranges from -1.24 m/s to +0.88 m/s. it should be noted that the nature of the parameter resembles a white noise model whose mean values are close to 0. in the example under consideration, the mean value of the parameter is ±0.01 m/s. assessment of velocity accuracy of aircraft in the dynamic tests using gnss sensors 309 fig. 5 presents the standard deviation results for the total resultant velocity for the frame of all gnss sensors installed in the cessna 172, according to eq. (4). the value of the standard deviation for all measurement epochs ranges from 0.01 m/s to 1.07 m/s. in addition, the mean and median of the parameter equals 0.08 m/s and 0.06 m/s respectively. in about 74% cases the parameter is less than 0.1 m/sec whereas in over 94% cases the parameter is less than 0.2 m/s. fig. 5 the standard deviation of total velocity of gnss receivers array in cessna 172 aircraft 4. discussion as part of the discussion, the results obtained from the proposed research method were verified. for this purpose, the results of parameter were compared with the flight reference speed determined by the rtk-otf differential technique. the reference flight speed was determined on the basis of precise gps phase observations using the rtk-otf differential technique [21]. first, the reference position of the aircraft from the rtk-otf solution was determined, and then the reference speed of the cessna 172 flight, according to the formula: { √( ) ( ) ( ) (5) where is the reference value of v of aircraft based on rtk-otf solution while ( ) are the flight speed components along the xyz axes based on rtkotf solution. reference value speed was from 0.23 m/s to 74.81 m/s for the entire measuring cycle. comparison of parameter results and enables the determination of speed errors and, additionally, determines the accuracy of the presented test method. resultant speed errors are defined as follows: 310 k. krasuski, a. ciećko, g. grunwald, d. wierzbicki (6) where is the velocity error. fig. 6 shows the results of speed errors as a function of observation time. speed error values are between -0.46 m/s and +0.61 m/s. the average value of speed error is 0.01 m/s; therefore, the nature of the changes in the parameter resembles white noise. it is worth noting that over 91% of the obtained parameter values are within ± 0.1 m/s. fig. 6 the values of velocity error for presented research method a statistical measure of accuracy in the form of an rms mean square error [22] was also determined for the parameter, as follows: √ [ ] (7) where is accuracy and is a number of measurement epochs. the rms error value for the analyzed aviation test is 0.07 m/s. therefore, it can be concluded that, for the presented research method, high accuracy was achieved. as a part of the discussion, the resultant aircraft flight speed was also determined based on the readings from 3 gnss receivers as a weighted average mathematical model as below: (8) where stands for a speed weight using the gps solution, is a speed weight using the egnos solution and is a speed weight using the gps/glonass solution. the implemented calculation scheme assumes that the speed weights are respectively: assessment of velocity accuracy of aircraft in the dynamic tests using gnss sensors 311 { { (9) { where is a number of gps satellites, is a number of gps+egnos satellites and stands for a number of gps/glonass satellites. therefore, it can be concluded that weighting according to eq. (8) takes place as a function of the number of tracked gnss satellites, which were used in the calculation process of determining the aircraft xyz coordinates. table 4 shows the results of determining the resultant aircraft flight speed based on the mathematical model (3) and (8). it can be stated that the results of the flight speed on the basis of both test methods are close to within ±0.03 m/s to ±0.08 m/s. therefore, the resultant speed performance based on eq. (8) including the weighing process is similar to the results of the speed calculated as the arithmetic average for 3 gnss receivers. table 4 comparison results of the total velocity of aircraft based on eqs. (3) and (8) total velocity of aircraft minimum range of velocity [m/s] maximum range of velocity [m/s] velocity model based on eq. (3) +0.23 +74.81 velocity model based on eq. (8) +0.20 +74.73 in the last stage of the discussion, the accuracy of the research method was determined from eq. (8) based on eqs. (6) and (7). fig. 7 shows aircraft flight speed errors calculated as the difference between the weighted average speed and parameter. speed error values fig. 7 the values of velocity error as a difference between weighted average of velocity and the rtk-otf solution 312 k. krasuski, a. ciećko, g. grunwald, d. wierzbicki take results from -0.38 m/s to +0.67 m/s. it is worth noting that over 97% of the obtained parameter results are within ± 0.1m/s. in addition, the rms error is less than 0.05 m/s. therefore, the accuracy of this method is relatively high. 5. conclusions the paper presents the results of the flight speed test of the cessna 172 aircraft during a training flight at dęblin airport (epde) in south-eastern poland. until now, in the investigated research paper, the readings from a single gnss receiver or another measuring sensor have been used to determine the aircraft speed. in the analyzed example, the authors of the paper decided to use real data from 3 gnss receivers placed in the cockpit to determine the flight speed of the cessna 172 aircraft. in the mathematical model of speed, individual components were determined and finally the resultant value was calculated for the whole measurement frame of 3 gnss sensors. the presented research method has its advantages because it is based on a multi-receiver gnss solution, making the result independent of on-board avionics and most importantly it gives pilots additional information and navigation data concerning aircraft flight mechanics in real time. therefore, the presented research method can be applicable in the area of flight technology of manned and unmanned aircraft. the obtained research results show that the flight speed of the cessna 172 aircraft for the entire gnss sensor frame was from +0.23 m/s to +74.81 m/s. in the work the flight speed parameter was determined as a function of time, distance travelled by the plane and flight altitude. in turn, the standard deviation parameter was also determined for the resultant flight speed value. the standard deviation value for all measurement epochs ranges from 0.01 m/s to 1.07 m/s. the paper analyzes the accuracy of the research method presented. the aircraft flight speed errors were determined as between -0.46 m/s and +0.61 m/s. in addition, an rms error was determined, whose value is 0.07 m/s. the article also presents a model for determining the weighted average flight speed parameter. the accuracy of determining the speed from the weighted average model in relation to the rtk-otf solution is from 0.38 m/s to +0.67 m/s, while the rms error is less than 0.05 m/s. in future, the authors plan to develop their scientific research on the use of gnss sensors to determine the flight speed of aircraft. it should be noted that the authors intend to use other methods or systems to determine the flight speed of aircraft. it is planned to use the doppler effect and use the ins system to determine the flight speed of the aircraft. the 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january 2010 07 january 2010 orlando, florida, usa. 20. krasuski, k., 2019, the research of accuracy of aircraft positioning using sppcode method, phd thesis, warsaw university of technology, poland, 106 p. 21. ćwiklak, j., kozuba, j., krasuski, k., jafernik, h., 2018, the assessment of aircraft positioning accuracy using gps data in rtk-otf technique, proc.18th international multidisciplinary scientific geoconference sgem 2018, 02-08 july, bulgaria. 22. przestrzelski, p., bakuła, m., galas, r., 2017, the integrated use of gps/glonass observations in network code differential positioning, gps solutions, 21(2), pp. 627–638. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 193 202 isogeometric structural analysis based on nurbs shape functions  udc 515.3 predrag milić 1 , dragan marinković 2 1 university of niš, faculty of mechanical engineering, niš, serbia 2 berlin institute of technology, department of structural analysis, berlin, germany abstract finite element method (fem) is a tool that is mostly used for the structural analysis. the method is based upon approximation of the actual geometry and displacement field of the structure over the finite element domain. in practice, lagrange polynomials are mostly used as shape functions (approximate functions). they provide only c0-continuity at the element boundaries. this paper elucidates a possible approach based on the so-called isogeometric analysis to improve this property. the isogeometric fem analysis applies the same functions to describe the cad model, the element geometry and the element displacements. the paper represents developed algorithm and the obtained results for a relatively simple test case. key words: isogeometric structural analysis, nurbs, fem 1. introduction a reliable design of sophisticated mechanical systems requires numerous computational analyses and simulations. hence, the engineering design cannot be considered separately from the analysis. they are strongly related because the process of design optimization is evaluated through the analyses and, vice versa, the analyses drive the design process by revealing design weakness. in the 'classical' finite element method (fem), the geometric model suitable for structural analysis is usually not the very same cad geometry model. most often a simplified geometry is used for generating the mesh of finite elements. furthermore, the resulting domain of fem models is only an approximation of the already simplified cad geometry. these approximations give rise to larger or smaller errors which produce several consequences. for example, the stability of thin-walled structures is very sensitive to geometric imperfections and,  received december 5, 2013 corresponding author: predrag milić faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: pmilic@masfak.ni.ac.rs acknowledgements. this work is supported by the project tr35049 financed by the ministry of education and science of republic of serbia. 194 p. milić, d. marinković therefore, without the exact geometry, an adaptively formed fe mesh may yield results of insufficient accuracy. the high accuracy of the geometry description is also important in design optimization. furthermore, the quality of the fem geometry descriptions has a significant impact in contact problems. development of the classical fem is mainly based on the application of isoparametric finite elements, which implies mapping of finite elements from the real geometry into the so-called master element, i.e. from the global coordinate system into the natural coordinate one. these types of elements use the same shape function, typically lagrange polynomials, to approximate both the displacement field and the geometry. the quality of geometry description directly depends on the degree of the selected polynomial and element size, i.e. the quality of the finite element mesh. one of the main disadvantages of this approach is that the lagrange polynomials as shape functions provide only c 0 continuity at the element boundaries, while many elements require at least c 1 continuity, such as beams, plates or shells. in relation to the aforementioned characteristics of the classical fem, a significant step forward would be a fem formulation that allows for geometry description done in exactly the same manner as in the cad software packages. such a fe model would ideally represent the analyzed geometry, regardless of the mesh quality. this fem concept is known as isogeometric analysis [1, 2, 5]. as basis for setting the cad model, different technologies of computational geometry can be used. currently in the engineering design, nurbs (non-uniform rational bspline) is mostly used. the main advantage of nurbs is a convenience for representing free forms, accurate representation of conical sections (circles, cylinders, spheres, ellipsoids), etc. the continuity of p-order nurbs is c p+1 . as opposed to b-spline and nurbs surfaces, a tspline represents a surface based on the points of a control polygon and not on the control mesh [3, 4]. t-spline is non-uniform b-spline surface with t-junctions (for more information see [3]). t-splines are also used to merge multiple patches into one tight part. immersed boundary methods, which do not require body-fitted meshes, may also be applied within the isogeometric approach. this allows a structural analysis of rather complex engineering parts [6]. 2. nurbs geometry definition of complex geometric shapes is usually performed by using splines. splines application in geometric modeling is related to the work of french engineers pierre bezier and paul de faget in the automotive industry in the sixties (fig. 1). fig. 1 example of drawing a spline with lattice isogeometric structural analysis based on nurbs shape functions 195 as the basic type of spline for geometric modeling, b-splines and splines that use the same basis as b-splines are applied. this primarily refers to nurbs and t-splines, the latter representing a generalization of nurbs and also a part of the so-called pb spline (point based spline). a p-order nurbs curve can be represented as a rational function using the bernstein basic functions by means of the following expression [1, 2, 8]: bua wun pwun uc n i ipi n i iipi      0 , 0 , )( )( )( (1) where pi are the control points that form a control polygon, wi are the weights, and {ni,p(u)} are the b-spline basic functions of order p defined on non-uniform knot vector u: },...,,,...,,,...,{ 11 bbuuaa pmp u (2) usually the knot vector is normalized, i.e. a = 0 and b = 1. repetition of elements a and b in the knot vector depends on the order of spline and it is p+1. in this way, the discontinuity is realized at the ends of the spline. rearranging eq. (1), the nurbs curve can be represented by the following expression, where ri,p are the basic rational functions of the p-order nurbs:      n i ipii n i n j jpj ipi purp wun wun uc 0 , 0 0 , , )( )( )( )( (3) surfaces and solids can be defined in a similar way. a nurbs surface of order p in the u-direction and order q in the v-direction can be expressed as: 1,0 )()( )()( ),( 0 0 ,,, 0 0 ,,        vu wvnun pwvnun vus n k m l lkqlpk n i m j ijijqjpi . (4) by substituting the basic rational function: 1,0 )()( )()( ),( 0 0 ,, ,,     vu wvnun wvnun vur n i m j ijqjpi ijqjpi ij , (5) the equation for the surface takes the form:     n i m j ijij pvurvus 0 0 ),(),( . (6) the b-spline basic functions are used to determine the nurbs basic functions and they are determined by means of the cox-de boor's recursive formula and using the basic func196 p. milić, d. marinković tions of lower order. for any given knot span, at least p+1 b-splines and nurbs basic functions are not equal to zero. also, the basic function has values different from zero at p+1 knot spans. in fig. 2, a nurbs surface is shown in the so-called physical space, index space and parameter space together with the corresponding basic functions of b-spline. the area defined by index coordinates ξ and η is known as a patch. a less complex geometry can be described by a single patch [7], while more complex geometric forms need to use more patches. repeating the elements knot vector creates an open knot vector and, hence, in modeling complex geometries, the c 0 -continuity is obtained on the element borders because only the continuity of the polygon control point displacements is realized. as the nurbs basis functions are constructed from the b-spline basis functions, the derivatives of rational functions will clearly depend on the derivatives of their non-rational counterparts as well [1]. in literature, there are several algorithms for computing derivatives of the b-spline basic functions. fig. 2 nurbs surface, index space, and b-spline basic function 3. basics оf isogeometric analysis the isogeometric analysis is based on the fact that the same shape functions are used to describe the cad geometry and the displacement field of the element. it implies that there is a set of basic functions, r, which perform geometric mapping from index space ξ to physical space x, i.e. there is parameter ξ such that mapping x: ̂ is realized as follows:                                 e a e a e ae a a z y x r z y x en 1 )(ξ ξ ξ ξ ξx (7) where xa e , ya e , za e are the components of the a th element of the vector that defines element geometry ωe (the set of control points). in a similar way, other connections between the parameter and the physical space can be formed, such as connections between control point displacements di and the displacements within the element, i.e. rh  ˆ:)(û : isogeometric structural analysis based on nurbs shape functions 197 i n i i h np r d)()(û 1    ξξ (8) for a planar problem, the previous equation takes the form:                    enn i ee ii enen enen r vrvr urur v u 1 )()( 11 11 ... ... rddu (9) where u and v are the components of the element displacement obtained by means of basic functions ri and the corresponding displacements at control points di, and d (e) are the displacements of all the element control points. assuming small displacements (linear approach), strains ε are further given as:               xy y x ε , where x v y u y v x u xyyx             ,, (10) respectively, using the previously presented concept: )( 1 )( 1 e n i e ii n i i i i i i i i i enen v x r u y r v y r u x r bddbε                                (11) where b is the strain-displacement matrix. the element stiffness matrix is computed by integration over the element volume using the strain-displacement matrix, b, and the hooke's matrix of the material, d. element stiffness matrix kij that relates the i-th and j-th point of the control polygon can be expressed as:        1 1 1 1 )()( ||),(),( ddte j t i e ij jdbbk (12) where t is the thickness of the structure. computation of the integral is performed by numerical integration, typically by means of the gaussian quadratures [9]. in order to compute the system stiffness matrix, partial derivatives of the shape functions are needed so that the strain-displacement matrix and the jacobi matrix can be determined for an adequate set of the gaussian integration points. the number of integration points depends on the order of basic functions. for each element, there are at least p+1 main functions different from zero in one direction. this means that, for a solid element with the second-order nurbs basic functions, displacements in the element are determined by means of 27 main functions. this number of basic functions requires significant computational time. as the influence of a basic function spreads not only over the domain of a single element, but over p+1 elements, this means that any effect in one element is reflected in the p neighbouring elements. the consequence of this property is that the system stiffness matrix typically has a wider matrix band compared to the stiffness matrix in the classical finite element method. 198 p. milić, d. marinković 4. software solutions a new software solution has been developed in order to implement the isogeometric fem formulation. there are basically two approaches to structural analysis by means of the isogeometric fem. the first approach implies that the entire structure is described by means of one patch and it comes down to computing the stiffness matrix of a single patch. the second approach means solving a complex structure. in this case, once the stiffness matrices of individual patches are determined, the stiffness matrix of the whole system is assembled. the input data for the developed software are: geometry data (degree of interpolation functions for each direction, knot vectors for all three directions, mesh of control points with coordinates and weights), material properties, loads and constraints. fig. 3 the algorithm of the isogeometric analysis using nurbs shape functions for the example discussed in this paper, the assumption of linearly elastic, homogeneous, isotropic material has been adopted. loads can be defined as concentrated forces, surface loads (pressure or force components on the surface) and volume loads. the software implementation of loads depends on the type of load. the basic steps and information flow in the algorithm are shown in fig. 3. the resulting system of equations yields the displacements of the polygon control points. the computation of element displacements and the corresponding element stress-strain states requires the values of basic functions and their derivatives at certain points of the element. the stresses are determined based on the known strains and the hooke's matrix. isogeometric structural analysis based on nurbs shape functions 199 5. numerical example the application of the developed fem formulation and algorithm will be demonstrated on a relatively simple example of a linear static analysis. the results have been compared with the solution obtained using commercial fe software package ansys. fig. 4a depicts the quarter model of a plate with a hole that has been used as a test example. the constraints in the model are set so as to represent the symmetry of the modeled structure. the structure has been modeled as a solid and the depicted load is defined as a surface load of intensity 10 kn/cm 2 . the cad model of the above described problem (polygon of control points and elements) is given in fig. 4b as a nurbs patch. fig. 4 plate with a hole – a) physical model, b) nurbs geometry the exact analytical solution of the infinite plate with a circular hole under constant in-plane tension exists [10] and it can be represented in polar coordinates (r, θ) as: )2sin(321 2 ),( )2cos(31 2 1 2 ),( )2cos(341 2 1 2 ),( 4 4 2 2 4 4 2 2 4 4 2 2 2 2                                                        r r r r r r r r r r r r r r r r r x r xx xx rr (13) where   rrr ,, are the stress components in the polar coordinate system for corresponding coordinates r and θ. the maximum value of normal stress in the direction of tension computed with previous equations yields 30 kn/cm 2 , i.e. the stress concentration factor is k = 3. for a similar case, but with finite plate dimensions, there is no exact analytical solution, but a higher stress concentration factor is expected. as a reference model for evaluating the accuracy of the isogeometric solution, a rather fine fem model created in ansys software package has been used (fig 5.). 200 p. milić, d. marinković fig. 5 fem model created in ansys with 44235 quadratic elements (plate183) the solution convergence has been investigated for this problem by means of isogeometric analysis based on two different geometry definitions. the geometric models and, consequently, the fem models, differ in the degree of nurbs basic functions and definitions of knot vectors. the initial model has been prepared manually, while further model refinements have been done by using originally developed codes for knot insertion and degree elevation. the first algorithm increases the number of entries in the knot vector and, hence, the number of finite elements. the second algorithm is used to obtain the nurbs basic functions of higher order. the results of analyses are shown in table 1 and fig. 6. table 1 the dependence of results on the interpolation function degree and mesh size case number of elements order number of basis functions translation in x direction [m] 1 2 2 24 -2.54158e-07 2 8 48 -2.69193e-07 3 32 120 -2.74696e-07 4 2 3 48 -2.70553e-07 5 8 80 -2.74405e-07 6 32 168 -2.75427e-07 fig. 6 the result convergence with quadratic and cubic nurbs shape functions isogeometric structural analysis based on nurbs shape functions 201 although the number of elements in the isogeometric model was significantly smaller (128 elements) compared to 44235 elements of the fem model in ansys, the obtained results for displacements (fig. 7) and stresses (fig. 8) differ negligibly. this demonstrates the capability of the nurbs shape functions to describe the stress field very accurately even with a small number of elements. the stress concentration factor computed by means of the reference model is k=3.57 and, as anticipated, it is larger than the value for the theoretical case with infinite dimensions. the classical fem model (the ansys model) needed a high number of elements to recover very accurately the high stress gradients in the vicinity of the hole edge. fig. 7 the displacement field obtained by a) isogeometric structural analysis, b) commercial software package ansys (with 267032 dof) fig. 8 the normal stress component xx (n/cm 2 ) obtained by a) isogeometric structural analysis, b) commercial software package ansys 6. conclusions the isogeometric fem analysis is a relatively new direction of fem development in the field of structural analysis and a rapid pace of its development is expected in the years to come. the world's leading vendors of cad software have already included nurbs as the main tool to define and display more complex geometries. the main problem resides in a seamless integration of cad geometries and fem models for structural analysis. the 202 p. milić, d. marinković isogeometric analysis is a significant step forward in this direction. one of the major disadvantages of nurbs is their inability for local mesh refinement, but this can be overcome by applying a special type of nurbs denoted as t-spline. this will be the focus of the authors' further work. it is expected that the field of structural dynamics will particularly benefit from the advantages offered by the isogeometric fem analysis. references 1. t.j.r. hughes, j.a. cottrell, y. bazilevs, isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement, computer methods in applied mechanics and engineering, volume 194, pp. 4135–4195, 2005. 2. j.a. cottrell, t.j.r. hughes, a. reali, studies of refinement and continuity in isogeometric structural analysis, computer methods in applied mechanics and engineering, volume 196, issues 41–44, pp. 4160–4183, 2007. 3. t. w. sederberg, j. zheng, a. bakenov, a. nasri, t-splines and t-nurccs, acm transactions on graphics (tog), proceedings of acm siggraph 2003 tog homepage, volume 22 issue 3, july 2003, pp. 477-484 4. p. wanga, j. xua, j. denga, f. chena, adaptive isogeometric analysis using rational pht-splines, computer-aided design, 43, pp. 1438-1448, 2011. 5. r. sevilla, s. fernández-méndez,a. huerta, 3d nurbs-enhanced finite element method (nefem), international journal for numerical methods in engineering, volume 88, pp. 103–125, 2011. 6. d. schillinger, l. dedè, m. scott, j. evans, m. borden, e. rank, t. hughes, an isogeometric designthrough-analysis methodology based on adaptive hierarchical refinement of nurbs, immersed boundary methods, and t-spline cad surfaces, computer methods in applied mechanics and engineering, volumes 249–252, pp. 116–150, 2012. 7. e. cohen, t. martin, r.m. kirby, t. lyche, r. f. riesenfeld, analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis, computer methods in applied mechanics and engineering, volumes 199, pp. 334-356, 2010. 8. l. piegl, w. tiller, the nurbs book (monographs in visual communication), springer, 1996, isbn: 3540615458 9. t.j.r. hughes, a. reali, g. sangalli, efficient quadrature for nurbs-based isogeometric analysis, computer methods in applied mechanics and engineering, volume 199, pp. 301–313, 2010. 10. w.pilkey, d. pilkey, peterson's stress concentration factors, third edition wiley, 2008, isbn-10: 0470048247 izogeometrijska strukturna analiza zasnovana na nurbs funkcijama oblika metoda konačnih elemenata je najčešće korišćen alat u strukturnoj analizi. metoda se zasniva na aproksimaciji stvarne geometrije i polja pomeranja strukture u domenu konačnih elemenata. u praksi se kao funkcije oblika (aproksimacione funkcije) najčešće koriste lagrange-ovi polinomi. oni omogućuju samo c0 kontinuitet na granici elementa. u ovom radu će biti predstavljeno rešenje za poboljšanje kontinualnosti na granici elementa zasnovano na tzv. izogeometrijskoj analizi. izogeometrijski pristup u okviru metode konačnih elemenata koristi iste funkcije oblika za definisanje cad modela, geometrije konačnog elementa i za interpolacione funkcije pomeranja u polju konačnog elementa. rad predstavlja razvijeni algoritam i dobijene rezultate ze relativno jednostavan test primer. ključne reči: izogeometrijska analiza, nurbs, fem facta universitatis series: mechanical engineering vol. 16, no 3, 2018, pp. 297 305 https://doi.org/10.22190/fume170620029s © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper 3d digitization of featureless dental models using close range photogrammetry aided by noise based patterns udc 771:778 ţeljko santoši 1 , igor budak 1 , mario šokac 1 , tatjana puškar 2 , đorđe vukelić 1 , branka trifković 3 1 university of novi sad, faculty of technical sciences, serbia 2 university of novi sad, faculty of medicine, serbia 3 university of belgrade, school of dental medicine, serbia abstract. development and improvement of 3d digitizing systems provide for the ability to digitize a growing number of materials and geometrical forms of greater complexity. this paper presents the application of 3d digitizing system using close range photogrammetry on the upper jaw cast in plaster in order to obtain its 3d model. because of the low visual characteristics of gypsum, such as color and texture, many questions arise about the possibility of applying this particular method to this type of physical models. in order to overcome bad visual properties of gypsum, this paper analyzes the possibility of the photogrammetry method application supported by the projected light texture which is based on patterns in the form of noise-obtained mathematically modeled functions. in order to determine the selected image for light texture which gives the better results, an experiment was designed and carried out. only two images were tested. one image is selected based on previous research and the other one was generated by the matlab function for uniformly distributed random numbers. for validation and a comparative analysis of the results, an object of 3d digitization was generated with and without projected light texture. cad inspection was applied for the analysis of the obtained 3d digitizing results. 3d model obtained by approved professional optical 3d scanner as a reference was used. the results in this paper confirm better accuracy of 3d models obtained with the use of light textures, but this approach requires additional hardware and setup adjustment for images acquisition. key words: 3d digitization, close range photogrammetry, noise-based patterns, cad inspection received june 20, 2017 / accepted july 11, 2018 corresponding author: igor budak university of novi sad, faculty of technical sciences, trg dositeja obradovića 6, 21000 novi sad, serbia e-mail: budaki@uns.ac.rs 298 ţ. santoši, i. budak, m. šokac, t. puškar, đ. vukelić, b. trifković 1. introduction 3d technologies, new approaches and applications of innovative methods in the context of reverse engineering (re) and 3d digitizing, lead away mechanical engineering from the conventional product design and toward its increasing search for another role and importance in other fields such as biomedical engineering, protection of cultural heritage, animation, criminal investigation, forensic science, etc. [1-6]. the dentists need the necessary technical resources and advancements that will facilitate their work with patients, and, therefore, provide their patients with a better treatment and healthcare service [7]. a high level of cooperation is especially required in the fields of dental prosthetics for dentures as well as in oral surgery for bone grafts [8-10]. the main task of all dental restorations is to meet both functional and aesthetic requirements. a conventional approach to making dental restoration usually requires a couple of visits to the dentist that can be stressful for the patient, and the dental restorations are produced with less precision when compared to the modern cad/cam manufacturing. application of re in dentistry greatly facilitates the modeling and designing process of dental restorations, as well as procedure planning and their successful implementation. in the field of re several 3d digitizing methods are developed and/or adapted for acquisition of virtual 3d models in dental prosthetics, such as optical 3d digitizing methods based on structured light, laser triangulation or photogrammetric stereovision. in addition to the transmission methods (computed tomography ct) that are suitable for obtaining a complete 3d model of the human anatomy, optical methods are also developed which, because of their simplicity and accessibility, provide good accuracy during 3d scanning of dental impressions. on the basis of optical 3d digitizing methods, different types of extraoral and intraoral digitizing scanners [10-12] are developed. this paper shows application of the advanced mechanical engineering techniques, from the re and cai (computer-aided inspection) fields, to dentistry for the purposes of performing a sophisticated 3d geometrical analysis. its aim is to test two different images used for projection of light texture and their influence on the 3d digitization results. section 2 describes materials and methods used in this paper, where subsection 2.1 briefly describes the working principle of close range photogrammetry while subsection 2.2 shows the used images for projection of light texture. in section 3 experimental workflow and used equipment are described. results and discussion are shown in the section 4 and so are the conclusions in section 5. 2. materials and methods 2.1. close range photogrammetry close range photogrammetry has found its place in the group of non-contact optical 3d digitizing methods [4]. it is based on the structure from motion (sfm) approach [13, 14], that enables reconstruction of physical 3d models on the basis of 2d digital photos that are taken arbitrarily, so that specific points on the objects surface are visible on at least two common photos. this method is based on the principles of multi-view stereovision (fig 1.), where the position of object’s characteristic points in the threedimensional space and the positions of photos are determined simultaneously, with or without previous calibration (auto calibration) [4]. 3d digitization of featureless dental models using close range photogrammetry aided by noise... 299 fig. 1 multi-view stereovision approach, 3d net configuration of rays [4] the main precondition for the application of this method, in addition to the previously mentioned multi-view stereovision approach, is that the surface of the object has a unique and pronounced visual texture. the object’s photos are so captured that the object is covered from all sides. since all photogrammetric measurements are dimensionless, it is necessary to use reference markers in order to properly scale generated 3d models. regarding the errors, they can occur during the scaling of the 3d model. also, errors can occur due to excessive distortions caused by imperfection of optical acquisition system, to an insufficient number of characteristic points, as well as an insufficient or excessive distance between two common images that form a stereopair. the consequence of these errors is incorrect calculation of the characteristic points' positions in the three-dimensional space. with an increasing number of the characteristic points on the object's surface, the captured images carry a lot more information which greatly reduces errors related to calculation of reconstructed photos as well as points to the positions in the three-dimensional space. also, with a great number of detected points it can easily overcome the difference in the baselines distances between images, and in this way, minimize the error. this method provides flexible and fast image acquisition which enables easy coverage of the whole object. 2.2. noise based patterns in order to perform 3d digitization of the object that has no characteristic visual texture, the techniques for improvement of visual characteristics of digitized surface must be applied. in practice, two techniques are applied in order to enhance the visual characteristics of the digitized objects surface. the first is based on physical application of paint or powder on the object, while the second includes projection of light textures [15, 16]. physical application of paint or powder on the objects surface impairs the geometric accuracy to a small degree due to the fact that a thin layer of the filler material is applied to the object. still, it is cheaper and it does not require the use of any additional hardware. on the other hand, for the projection of light texture, additional hardware in the form of a light projector is needed, but there is no influence on accuracy [4]. the type of light texture that is projected onto a 3d object depends on how well digitization will improve. noise-based patterns are visual interpretations of mathematical models or functions, 300 ţ. santoši, i. budak, m. šokac, t. puškar, đ. vukelić, b. trifković where each calculated value represents the intensity of a given pixel on the image, which is arranged in a m  n array. this paper will discuss the application of two different patterns, which are:  uniform random pattern [17], and,  wavelet pattern [18]. the uniform random pattern is selected because of its greatest histogram uniformity and random distribution of pixels, while the wavelet pattern has given the best results in recent research [15]. fig. 2 shows the images of the used patterns and their histograms, on which intensity distribution can be seen. in table 1 calculated statistics such as standard deviation, entropy, pixels minimum and maximum intensity values, pixel median, mode, range and number of colors are given. fig. 2 images of the a) used patterns: random pattern (left), wavelet pattern (right); b) their respective histograms table 1 pattern images' statistical parameters pattern standard deviation entropy min value max value median mode range number of colors random 73.9890 7.9623 0 255 127 / 255 255 wavelet 44.8413 7.4508 3 245 126 81 242 238 in the recent research koutsoudis et al. [15] conducted an analysis of the group of patterns based on mathematical functions and found that the wavelet pattern gives the best results in the case of a case of 3d digitization of a cultural heritage object. a) b) 3d digitization of featureless dental models using close range photogrammetry aided by noise... 301 3. 3d digitization of featureless dental models using the close range photogrammetry approach as an object for 3d digitizing a working model of the upper jaw cast in plaster is chosen with its dimension 908030mm. based on visual inspection it is assumed that the visual characteristics of this object are not suitable for applying photogrammetry based on the sfm approach because of its simple visual texture. hence, it is considered as appropriate for testing of the generated patterns. the experiment was designed by using a video projector that projects a light texture pattern, generated on the basis of mathematical functions, onto the object. the whole setup is adopted according to the available equipment, but also due to the shape and size of the object. images are captured and used to reconstruct and generate 3d models. on the basis of both projected light texture patterns an independent 3d model is created. cad inspection is applied in order to determine which 3d model is better in terms of its geometrical accuracy. the reference 3d model of the plaster jaw was earlier obtained using an optical system for 3d scanning atos ii triple scan from gom with scanning resolution from 0.02 to 0.79 mm. this type of system is considered amongst the most used optical measuring devices for 3d scanning, and it is based on a combination of structured light and stereovision. the workflow of the experiment is illustrated in fig. 3. wavelet pattern random pattern without pattern 3d models generation nominal 3d model acquisition cad inspection fig. 3 experimental workflow 302 ţ. santoši, i. budak, m. šokac, t. puškar, đ. vukelić, b. trifković the photo acquisition for the realization of this experiment is performed by means of a canon 1200d dslr camera equipped with 18-55 mm lens, set at the maximum value of 55 mm. maximum focal length value provides a better field of view of object to camera distance ratio than minimum focal length. photos are captured within distance of 50cm (± 5cm), the aperture size is set to f18, and those settings give approximately 5 cm depth of field. the video projector is set up at a distance of 75cm and at an angle of 50° approximately, compared to the plane of the table. in this setting the gsd (ground sample distance) is between 0.5 and 0.6 mm at a resolution of 1024x768 pixels of the video projector, while the gsd of the captured photos ranges between 0.06 and 0.07 mm. experimental setup is shown on fig. 4a. a) b) fig. 4 a) the experimental setup; b) 3d digitization object because of the specific shape of the plaster jaw (fig. 4b) it is not possible to obtain a complete 3d model from a single set of photographs. therefore, four groups with 8 photographs are made. during the acquisition of each group of 8 photos, projector and 3d digitization object are not moved. during that time the dslr camera acquires stereopair from left and right sides of the projector on two levels. from each camera's position, three photos are captured: one with a random pattern, one with a wavelet pattern and the third one without projected light textured pattern. once the first three groups are captured, each with 8 photos, the object is rotated by 90° in relation to the projector; thus the capturing process continues using the same procedure. alignment of the corresponding groups is carried out using 12 equally distanced reference markers printed on an a4 paper sheet, arranged in a circle with diameter of 160 mm, and placed at the center where the object will be digitized. based on the known distance between the markers, the actual size of the 3d model is determined. after the acquisition of photos, they are processed using agisoft photoscan software based on sfm approach, and all three digitized 3d models of the object are obtained. 3d digitization of featureless dental models using close range photogrammetry aided by noise... 303 4. results and discussion using the cad inspection it is possible to make a comparative analysis between the nominal geometry, which in this case is a 3d model obtained by the atos ii triple scan 3d scanner, and 3d models obtained by using photogrammetry. cad inspection is carried out using gom inspect software and results are shown in fig. 5. once the models are aligned using the best fit method for alignment, the limit is set to search and display tolerances within ±1mm. fig. 5 result of cad inspection: without pattern (upper left), random pattern (upper right) and wavelet pattern (down) the advantage of the cad inspection, in relation to other inspection methods, is in its visual interpretation and easiness of perception of critical areas. based on fig. 5, and the results shown in table 2, it can be seen that there is a very small difference in the results achieved using random and wavelet light projected patterns, while the results achieved without the use of any pattern showed the least accurate results. the advantage of the random and wavelet patterns is in unique textures which allow the photo editing software, which is based on the sfm approach, to easily detect characteristic points on the surface of a 3d object. between the 3d model obtained using the textured light patterns there are small variations in accuracy, while compared to the 3d model generated without a textured light pattern, increasing accuracy is noticed. the reason for the slight variations in the 3d models obtained using the projected pattern can be seen from their histogram in fig. 1, and the statistics in table 1. a high quality pattern must have high randomness and disorder. due to a smaller depth of the projector field 304 ţ. santoši, i. budak, m. šokac, t. puškar, đ. vukelić, b. trifković with respect to the object size, when projecting the image, a slight blur occurs in the pattern that increases gsd of the projector. in the case of the pattern with a large number of intensities, as a result a reduced number of detected points is achieved, even though the subject is within the cameras' depth of field. table 2 results of cad inspection 3d model prealignment best fit standard deviation [mm] max distance [mm] min distance [mm] mean distance [mm] distance standard deviation [mm] without pattern +0.096 +4.211 -9.875 -0.049 +0.184 random pattern +0.081 +4.521 -6.201 -0.036 +0.115 wavelet pattern +0.074 +4.454 -3.897 -0.038 +0.109 5. conclusions methodology based on the textured light pattern projection upon the object with reduced visual characteristics proved to be successful. based on the assumptions and results related to the application of close range photogrammetry to the physical model of the plaster cast used in dental prosthetics, several conclusions come up:  gypsum as the material does not belong to the group of preferred materials for 3d scanning using the close range photogrammetry based on sfm due to its simple and monotonous visual texture,  due to their visual characteristics, objects made of plaster are necessary to undergo additional treatments in order to obtain a unique texture that is the key factor for obtaining high-quality 3d models using the close range photogrammetry,  3d digitizing of plaster models with the close range photogrammetry assisted with the textured light patterns, yields in better results compared to 3d digitizing without the textured light pattern. further research will focus on developing a new pattern designs for application in the close range photogrammetry to the materials with different surfaces and visual qualities. acknowledgements: this paper presents the results achieved within the project no. 114-451-2723 / 2016-03 funded by the province secretariat for higher education and scientific research, and within the project tr 35020, funded by the ministry of education, science and technological development of the republic of serbia. references 1. stojkovic, m., veselinovic, m., vitkovic, n., marinkovic, d., trajanovic, m., arsic, s., mitkovic, m., 2018, reverse modeling of human long bones using t-splines case of tibia, tehnicki vjesnik, 25(6), pp. 1753-1760. 2. sansoni, g., trebeschi, m., docchio, f., 2009, state-of-the-art and applications of 3d imaging sensors in industry, cultural heritage, medicine, and criminal investigation, sensors, 9(1), pp. 568-601. 3. apuzzo, n., consulting, h., 2006, overview of 3d surface digitization technologies in europe, in: corner b.d., li p., tocheri m. 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mechanical engineering https://doi.org/10.22190/fume240828014s © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd review paper sustainable composite materials for electric vehicle applications: a comprehensive review manoj kumar singh1, sathish kumar palaniappan1, indran suyambulingam2, sanjay mavinkere rangappa1, suchart siengchin1, dragan marinkovic3,4 1natural composites research group lab, department of materials and production engineering, the sirindhorn international thai-german graduate school of engineering, king mongkut's university of technology north bangkok, bangkok, thailand 2sophisticated testing and instrumentation centre (stic), department of mechanical engineering, alliance school of applied engineering, alliance university, bengaluru, karnataka, india 3institute of mechanical science, vilnius gediminas technical university, vilnius, lithuania 4tu berlin, department of structural analysis, berlin, germany abstract. the shift towards electric vehicles (evs) plays an essential role in achieving sustainable and eco-friendly transportation. this comprehensive review investigates the incorporation of sustainable composite materials within the realm of electric vehicle applications. examining a spectrum from biocomposites to recycled polymers, the study thoroughly evaluates thermal and environmental characteristics, alongside a life cycle assessment, providing valuable insights into the potential of these materials to elevate the overall sustainability of evs. furthermore, recent steps in manufacturing technologies are highlighted, encompassing the integration of renewable energy sources, implementation of automation, and incorporation of machine learning and artificial intelligence in sustainable composites manufacturing. the application landscape of evs is thoroughly explored, taking into account various influencing factors. in response to the escalating demand for evs, a detailed market analysis and trend discussion are included to facilitate a comprehensive understanding. addressing the multifaceted nature of the review, challenges associated with sustainable composites, innovative solutions, and future prospects are discussed, collectively contributing to the ongoing pursuit of greener and more efficient electric mobility solutions. this review serves as a valuable resource for researchers, policymakers, and industry professionals invested in advancing sustainable practices within the electric vehicle domain. received: august 28, 2024 / accepted march 31, 2025 corresponding author: sanjay mavinkere rangappa natural composites research group lab, department of materials and production engineering, the sirindhorn international thai-german graduate school of engineering (tggs), king mongkut's university of technology north bangkok (kmutnb), bangkok 10800, thailand e-mail: mavinkere.r.s@op.kmutnb.ac.th 2 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. key words: sustainability, advanced manufacturing, environmental impact, life cycle assessment, market trends, challenges 1. introduction a worldwide movement towards sustainability and the reduction of carbon emissions has propelled the electric vehicle (ev) market to the vanguard of the automotive industry's transformation. the increasing popularity of electric vehicles represents a departure from conventional internal combustion engines towards greener options [1]. nevertheless, there will be difficulties throughout this change. producing environmentally friendly and lightweight parts that are up to the required standard in terms of performance and safety is an area where the electric vehicle industry really shines. in the electric vehicle industry, sustainable production is important for two reasons [2]. the first is a moral obligation to protect the environment. an important contributor to the total amount of carbon emissions from vehicles is the carbon footprint of their production, which includes the procurement of materials and the manufacturing processes [3]. the second consideration is the element of performance. one of the most important metrics for electric vehicles' competitiveness market and customer acceptability is the efficiency and range of their batteries, both of which are improved by using lightweight materials [4]. one promising approach to these problems is the use of sustainable composites. these materials, which are usually composed of a matrix (such as resin) and fibers (such as carbon or glass), provide the optimal strength-to-weight ratio for electric vehicles. proper selection of constituents may demand application of multi-criteria decision making methods [5]. when these composites are made using methods that reduce their environmental effect or are sourced from renewable resources, their sustainability is improved. composites offer great adaptability, performance, and positive impact on the environment. high stiffness and strength for structural components and heat resistance for battery enclosures are two examples of the tailored qualities that composites may offer to various ev components [6,7]. sustainable composites, produced from renewable sources and recycling, contribute to lowering the overall carbon emissions associated with ev manufacturing. by utilizing agro, forest and oil waste, the automotive industry can reduce its dependence on virgin materials. this resource-efficient approach helps conserve natural resources, contributing to a more circular and sustainable manufacturing model. sustainable composites support the principles of a circular economy by repurposing waste materials into valuable resources [8]. this approach minimizes the linear "takemake-dispose" model, promoting the longevity and reusability of materials. ongoing research and development in sustainable composites aims to enhance their mechanical, thermal, and flammability properties. a crucial aspect of sustainable materials is their life cycle impact (lca). lca evaluates the environmental footprint of these composites, considering factors from raw material extraction to end-of-life disposal [9–11]. sustainable composites showcase a reduced ecological burden compared to traditional materials. as these materials evolve, they are expected to match or exceed the performance of traditional counterparts, ensuring the feasibility of their adoption in critical ev components. in addition, as environmental regulations become more stringent, the use of sustainable materials aligns with compliance requirements. embracing eco-friendly composites sustainable composite materials for electric vehicle applications: a comprehensive review 3 positions automotive manufacturers favorably in the evolving regulatory landscape. consumers increasingly prioritize environmentally conscious products. ev manufacturers incorporating sustainable materials in their vehicles can appeal to a growing market segment that values green and ethical choices. the adoption of sustainable composites demonstrates the industry's commitment to innovation and leadership. companies at the forefront of integrating these materials position themselves as pioneers in sustainable manufacturing practices. while initial costs for sustainable composites may vary, long-term benefits include reduced environmental impact and potential cost savings as manufacturing processes mature and economies of scale are achieved. the automotive industry's increasing recognition of the importance of sustainability fosters a climate of collaboration. stakeholders across the supply chain collaborate to accelerate the adoption of sustainable composites, contributing to a greener automotive landscape. thus, sustainable composite materials present a compelling solution for achieving the environmental goals of the ev industry. their eco-friendly attributes, coupled with ongoing research and collaborative efforts, position these materials as the key contributors to a sustainable and resilient future for electric vehicles. extensive research is being conducted to explore the potential of different biofiber composites made from various fibers, polymer matrices, and formulations. the demand for biofiber and biodegradable polymer composites in the automotive industry is increasing, driven by their higher specific strength compared to glass fibers and lower energy requirements for production. these composites offer advantages such as higher impact energy absorption, ease of work, and lower health risks [12]. different manufacturing techniques are being employed for biofiber composites, including compression molding, robotics-assisted molding, resin transfer molding, and vacuum-assisted resin transfer molding. these techniques enable the production of biofiber composites with varying fiber loadings and properties. leading manufacturers are actively designing and deploying biofiber composite components in the automotive sector, reflecting the global trend towards sustainable materials. this includes the use of biofiber composites in interior panels and exterior parts of automobiles. the ongoing research priorities and global sustainability initiatives are shaping the future trends in biofiber composites, with a focus on enhancing their performance, expanding their applications, and promoting their commercialization and adoption. electric vehicles face various challenges in terms of heavy curb weight and fire threats, and the need for sustainable materials with enhanced flame retardancy for lightweight ev components. the use of high-performance sustainable polymer composites can address these challenges by reducing weight, increasing travel ranges, minimizing crash impacts, lowering braking distances, and reducing tire wear and tear [13]. sustainable composites, such as recycled carbon fibers and advanced biocarbon, offer advantages over traditional plastics in terms of environmental impact and circular economy principles. the transition to evs requires addressing challenges such as limited range, slow charging, battery damage, and end-of-life issues. materials research plays a significant role in improving the sustainability of battery vehicles, including mitigating thermal runaway through improved thermal stability and sustainable flame retardants. the integration of renewably resourced, recycled, and waste recovered materials in automotive components enables the circular economy and reduces environmental impact [14,15]. the inclusion of sustainable flame retardants in ev components can improve thermal management and delay thermal runaway, enhancing safety [16,17]. composites, such as owens corning's advantex glass, offer significant reductions in fluoride, particulate, and 4 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. nox emissions compared to traditional e-glass [18]. the properties of composites enable manufacturers to incorporate sustainable attributes into projects, increasing energy efficiency, durability, and the use of materials with a lower environmental impact. fiberreinforced polymeric composites (frpcs) are widely used in various industries but are flammable, posing a safety risk to humans and property. flame-retardant treatments are necessary to enhance the fire resistance of frpcs. a high-quality shield consisting of a strong framework (fibers) and dense fillings (chars) can effectively block heat, oxygen, and degraded volatiles, providing flame protection. the integration of fibers and chars results in a shield with a certain strength. carbon fibers decorated by polyelectrolyte complexes can be used in epoxy resin composites to enhance fire safety. biobased layer-by-layer coated ramie fabric-reinforced unsaturated polyester resin composites exhibit improved properties and fire safety. the orientation of natural fiber composites can affect their fire reaction properties, and the kinetics and thermodynamics of thermal decomposition can be determined for polymers containing reactive flame retardants [19]. various flammability test techniques for composites, such as limiting oxygen index (loi), ul 94, nbs smoke chamber, osu test, flame spread tests, cone calorimeter test (cct), single item burning (sbi) test, room coroner test, and room fire test can be used to measure the flammability [19]. fire modeling, such as fire dynamics simulator (fds) developed by nist and firefoam developed by fm global tools, can be used to assess the fire safety of materials in different realistic fire situations [19]. 2. manufacturing processes for sustainable composites in evs different production methods and technologies are being used nowadays to make sustainable composites for automobile manufacturing [20]. various manufacturing processes are discussed in this review, from more conventional ones such as resin transfer molding (rtm) to more advanced techniques such as 3d printing using sustainable materials [21]. environmental effect, efficiency, and viability as an ev production technique are the criteria used to evaluate each approach [22,23]. this review consists of various significant factors which need to be considered by the ev industry. it will provide a comprehensive overview of the status of sustainable composites manufacturing to researchers, policymakers, and experts in the ev sector. this review has the potential to help in speeding the adoption of greener practices in the automobile sector by showcasing the most promising sustainable manufacturing technology and processes. table 1 presents sustainable materials and their application in the automotive industry [12]. table 1 current application of biocomposites in the automotive industry [12]; pp: polypropylene, pur: polyurethane, pla: polylactic acid, pbs: polybutylene succinate, pet: polyethylene terephthalate, sorona® ep: poly trimethylene terephthalate company model biofillers matrix applications audi a2, a3, a4, a6, a8, roadster, coupe, q7 wood fiber, flax, sisal pp, epoxy, pur seat backs, side and back door panels, boot lining, hat rack, spare tire lining bmw 3, 5 and 7 series kenaf, flax, hemp, wood fiber pp door trim panels, headliner panel, boot lining, seat backs, noise sustainable composite materials for electric vehicle applications: a comprehensive review 5 insulation panels, dashboard citroen c5 wood fiber, flax epoxy interior door paneling daimler chrysler sebring flax, sisal, abaca pp door panels, seat cushion, head restraint, door cladding, seatback linings, and floor panels fiat punto, brava, marea, alfa romeo 146, 156 flax, sisal, hemp, cotton, coconut fiber pp seat bottoms, back cushions and head restraints ford motors ford flex, ford focus bev, freestar wood fiber, wheat straw, coconut coir, soy, rice straw pp, pur interior storage bins, load floor, foam seating, headrests, headliner general motors cadillac deville, chevrolet impala, gmc envoy, trail blazer, terrian, opel vectra wood, kenaf, flax, cotton pp, polyester seatbacks, trim, rear shelf, cargo floor, door panels, package trays, acoustic insulator, ceiling liner honda pilot wood fiber n/a floor area parts land rover 2000, others kenaf pp insulation, rear storage shelf/panel, door panels, seat backs lotus eco elise 5 hemp polyester body panels, spoiler, seats, interior carpets mazda hydrogen re hybrid corn pla console, seat fabric mercedes benz a-class, c-class, e-class, m-class, r-class s–class abaca/banana, hemp, flax, sisal, jute pur, pp, epoxy door panels, seat cushion, head restraint, underbody panels, seatbacks, spare tire cover, engine and transmission cover mitsubishi concept car bamboo pbs interior components peugeot 406 hemp, flax pp, pur seat backs, parcel shelf renault clio, twingo jute, coir pp, pur rear parcel shelf saab 9s flax pp spare wheel, door panels toyota lexus ct 200h, prius, raum kenaf, bamboo, corn, starch pet, sorona® ep, pp, pla luggage-compartment, speakers, floor mats, instrument-panel, airconditioning vent, spare tire cover, shelves volkswagen golf, passat, bora, touareg flax pp door panel, seat back, boot lid finish panel, boot liner volvo c70, v70 flax polyester seat padding, natural foams 6 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. 2.1 sustainable composites manufacturing techniques for ev application 2.1.1 resin transfer molding (rtm) to produce high-performance composite parts, one option is the low-pressure, closed molding method known as resin transfer molding (rtm). to impregnate the fiber reinforcement, resin is injected into a closed mold. the method's capacity to manufacture intricate forms with superior surface treatments, qualities crucial to the aerodynamics and aesthetics of evs, has propelled it to the forefront of the ev industry. to meet the varied demands of the automobile industry, rtm has progressed with innovations such as light rtm and high-pressure rtm (hp-rtm). for example, hp-faster rtm's cycle times make it ideal for mass production. rtm is a greener alternative to open mold methods. during the process less resin overflows, which means less waste and decreased emission of volatile organic compounds (vocs). this procedure is even more eco-friendly because it can employ natural fibers and resins that come from plants. structural components such as battery enclosures and chassis elements are the most common uses in electric vehicles [24,25]. composite materials offer these components a significant advantage due to their excellent strength-to-weight ratio. the usage of rtm in the manufacturing of battery housings for a prominent ev producer is one prime example. to produce a one-piece enclosure that is much lighter than conventional metal enclosures, rtm processes can be used. both the vehicle’s range and efficiency can be enhanced by reducing the overall weight. the battery container is made safer by incorporating fire-retardant polymers and strengthened fibers throughout the rtm process. in addition, the production method can be in line with the company's sustainability goals due to the reduced waste and lower voc emissions. 2.1.2 vacuum assisted resin transfer molding (vartm) the vacuum-assisted resin transfer molding (vartm) technique is an expansion of the rtm process. the new and enhanced vacuum technology and more efficient resin systems have made vartm an ideal choice for complicated and large-scale applications. while vartm and rtm are comparable, the former permits lower injection pressures, which is useful when working with complicated or big molds. when compared to highpressure options, this method is better for the environment because it uses less energy and produces fewer emissions. the vartm process is more environmentally friendly since it is controlled, which reduces waste and improves the resin-to-fiber ratio [26]. vartm is perfect for making lightweight electric vehicle chassis components because of its capacity to construct big, complete structures. these parts help reduce the vehicle's overall weight, which in turn increases the effectiveness of the battery and the distance it can travel. to optimize electric vehicle performance and battery efficiency, a lightweight chassis is essential. lightweight chassis components have been made possible with the use of vartm. an integral part of electric vehicles' dynamic performance is their high strengthto-weight ratio, which is made possible by allowing the fabrication of a sophisticated and integrated structure [27]. the vehicle's overall weight can be reduced using vartm, which leads to better battery efficiency and an increased driving range. by reducing energy consumption and material waste, the approach can lower the environmental footprint compared to standard manufacturing processes [28,29]. sustainable composite materials for electric vehicle applications: a comprehensive review 7 2.1.3 3d printing with sustainable materials a composite part for an electric vehicle can be mass-produced with the help of 3d printing, also known as additive manufacturing. small-batch production is ideal for this technology because of its waste-minimizing capabilities in creating complex geometries. new developments in 3d printing technology have allowed for the incorporation of continuous fibers into thermoplastics, increasing their strength and stiffness. in keeping with its environmental objectives, the electric vehicle industry uses sustainable materials such as natural fiber reinforcements, bio-based polymers, and recycled plastics [30]. the additive aspect of 3d printing helps to decrease waste and maximize resource utilization, which in turn reduces the carbon footprint of component manufacturing. without the need for costly mold changes, manufacturers can customize components for individual applications. prototyping and creating new electric vehicle models rely heavily on this adaptability, which speeds up the innovation process. in addition to drastically cutting down on waste, 3d printing's layer-by-layer construction method guarantees that material is only utilized where it is truly needed [31]. combining 3d printing with sustainable materials allows for rapid prototyping and customization, making it perfect for designing new ev components. by utilizing bio-based composites and recycled plastics, the ev industry can prove to be environmentally friendly. 3d printing not only reduces the time to design and test vehicles, but it also demonstrates how the automotive industry might reduce its negative effects on the environment. 3d printing with recycled materials demonstrates a circular economy strategy, which supports environmental responsibility in the automotive industry [32]. 2.1.4 autoclave and out-of-autoclave techniques autoclave and out-of-autoclave (ooa) are processes used to cure composite materials under controlled temperature and pressure. autoclaves are essentially pressure vessels that provide a high-quality finish and high fiber volume contents, ideal for structural components of evs. however, they are energy-intensive and can be costly. ooa techniques, on the other hand, have been developed to reduce energy consumption and process costs, making them increasingly popular for larger components such as body panels. ooa processes are gaining importance in the ev manufacturing sector due to their lower energy requirements and reduced environmental impact. they also offer greater flexibility in terms of the size and shape of components that can be produced, compared to autoclaves. both autoclave and ooa techniques are crucial in the manufacturing of highperformance composite parts for evs. 2.2 technologies enhancing sustainable manufacturing in ev production electric vehicle manufacturers are putting more effort into environmentally friendly production methods by using a range of cutting-edge technologies. minimizing environmental effects, optimizing resource utilization, and promoting energy efficiency are the goals of these technologies. to improve sustainable manufacturing in the production of electric vehicles, some of the important technologies recently being used are listed below. the integration of these technologies into ev production is a giant leap forward in the direction of greener production methods. in addition to helping the environment, they improve efficiency, cut costs, and encourage new ideas in the automotive business. 8 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. 2.2.1 renewable energy integration solar and wind power are becoming more attractive in powering manufacturing plants. this change is in line with international initiatives to fight climate change and lessen the environmental impact of electric vehicle production. sustainable production also includes using renewable energy sources during production. to lessen the environmental impact of their factories, more and more ev manufacturers are turning to renewable energy sources including solar, wind, and hydropower. the overall decrease of greenhouse gas emissions linked to ev production is considerably facilitated by this strategy. sustainable manufacturing in the electric vehicle business is being advanced by integrating ai and automation, developing innovative material technologies, and using renewable energy. in addition to helping achieve the larger environmental objectives of decreasing emissions and increasing the use of renewable resources, these technologies improve the sustainability and efficiency of production operations. 2.2.2 robotics and automation automated systems and robotics in production lines increase efficiency and precision. by making better use of resources (both material and energy), they lessen waste as well. better automobiles with less impact on the environment are a result of these technological advancements. robotic arms, automated cutting systems, and quality control based on sensors are all part of the automation process in composite production. by improving repeatability and decreasing room for human mistakes, these technologies help make production more efficient. to keep up with the ever-increasing demand for evs while minimizing environmental impact, automation is also crucial. 2.2.3 machine learning (ml) and artificial intelligence (ai) ml and ai are utilized for industrial process optimization, quality control, and predictive maintenance. improved overall sustainability is a result of their contributions to less downtime, higher product quality, and less resource consumption. to improve environmentally friendly production methods, automation and ai are becoming more important. ai algorithms are used in ev composite manufacturing to optimize material utilization, forecast how composite materials will behave under various conditions, and improve the accuracy and efficiency of production lines. optimization of rtm and vartm settings is possible with the help of ai-driven systems that sift through numbers of data. these systems provide constant quality while reducing energy consumption and material waste by forecasting the best resin flow pathways, curing durations, and pressure conditions. 2.2.4 eco-friendly or sustainable components electric vehicle manufacturing is increasingly using eco-friendly, recyclable, and lightweight components. vehicles made with these materials are more efficient and have less effect on the environment [8]. to achieve sustainability in electric vehicle production, it is crucial to create novel composite materials. a lot of people are working on producing materials that are stronger, lighter, and made from renewable or recycled materials. some of the examples of these advanced composites are natural fiber composites and carbon fiber reinforced plastics (cfrp). the overall weight of electric vehicles can be reduced with the sustainable composite materials for electric vehicle applications: a comprehensive review 9 help of cfrps because of their low weight and strong strength and stiffness [33]. moreover, natural fiber composites are making a significant impact since they are renewable and have a smaller carbon footprint. modern advancements in the field have led to bio-based resins and the use of nanomaterials, such as carbon nanotubes and graphene, which improve the mechanical characteristics of composites. to achieve the electric vehicle industry's performance targets while also protecting the environment, these innovations are essential. 2.3 examining the use of sustainable materials in composite manufacturing the electric vehicle industry sees biocomposites as a type of material that combines renewable or biological fibers with a polymer matrix as a viable, eco-friendly substitute for conventional composites. a rising awareness of the need to lessen the negative effects of cars on the environment, both during their usage and their production and disposal, is propelling the incorporation of biocomposites into electric vehicles. a growing number of evs are being made using biocomposites, a sustainable material that combines natural fibers with a polymer matrix [34]. the reduced environmental effect, increased strength, and reduced weight make these biocomposites highly desirable. some biocomposites that are being considered for use in electric vehicles are natural fiber-reinforced composites, wood polymer composites, cellulose/lignin-based composites, bio-resin composites, and recycled composites. biocomposites are made by combining natural fibers with a polymer matrix. the lightweight, strong, and environmentally friendly nature of these biocomposites makes them highly desirable [35]. wood plastic composites are manufactured by combining wood fibers or wood grain with polymers. in interior applications such as trim elements and paneling, they are excellent because they offer a good balance between the mechanical qualities and the weight of the material [36]. these composite materials are both durable and lightweight, and they are derived from cellulose fibers. it is possible to utilize them in a variety of interior components, and in some cases, they can even be used in structural applications. components made of composites can benefit from the utilization of lignin, which is a byproduct of the paper industry. these composites are not harmful to the environment and can be developed for the use in a variety of applications within electric vehicles [31]. these are composites in which the matrix is manufactured from bio-based resins, which are derived from plants or other renewable resources, rather than polymers that are derived from petroleum. they can be used in biocomposites that are completely sustainable when coupled with natural fibers [37,38]. materials that are both renewable and biodegradable are used for manufacturing evs. in the case of non-structural components, for instance, a green composite can be created by combining natural fibers with a matrix that is based on starch [39]. the production of evs is becoming increasingly environmentally friendly, and recycled composites play a crucial part in this. the utilization of recycled composites contributes to the reduction of waste, the reduction of the carbon footprint, and the conservation of resources as the automobile industry moves toward more environmentally friendly methods [40]. recycled composites are gaining significance in a variety of industries, including the building industry, the aerospace industry, and the automotive industry, because of the environmental benefits they offer and the growing demand for materials that are sustainable. in most cases, these composites are made up of a polymer matrix that is 10 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. reinforced with recycled fibers or fillers. recycled cfrp is becoming increasingly popular in the electric vehicle production industry due to the excellent strength-to-weight ratio it possesses. carbon fibers that have been recovered can be recycled and employed in the production of new composite materials [41,42]. gfrp, which is comparable to carbon fiber reinforced plastic, is capable of being recycled and utilized in a variety of automotive applications. glass fibers that have been recycled retain a significant amount of strength and are typically more affordable than carbon fibers [43]. when compared to thermosets, thermoplastic composites are more easily recyclable than thermoset composites. as a result of their ability to melt and remold, they are useful for a variety of uses in electric vehicles [44]. to reduce weight without sacrificing strength, recycled composites are utilized in the production of non-structural body panels as well as some structural components. it is possible to construct components of the car, including dashboards, door panels, and seat frames, out of recycled composites, which contributes to the overall sustainability of the vehicle [45]. the use of recycled composites for protective underbody pieces and battery casings helps to reduce the overall weight of the vehicle, which in turn increases its range and efficiency. certain manufacturers are investigating the possibility of incorporating recycled composites into the construction of wheels and suspension components to lessen their overall weight and improve their overall performance. one of the most important strategies for lowering the environmental effect of the automotive industry is the utilization of recycled composites in the production of evs [29,46,47]. it is anticipated that these materials will become more widespread as technology continues to evolve, which will contribute to the development of automobiles that are more sustainable and environmentally friendly [48-50]. fig. 1 illustrates the evolving advanced manufacturing strategy in lightweighting, with a specific focus on ev technology [47]. fig. 1 illustration of advanced lightweight composites manufacturing strategy for ev components [47] sustainable composite materials for electric vehicle applications: a comprehensive review 11 3. importance of weight reduction in evs and the role of lightweight composites weight reduction in evs is an essential factor that has a direct impact on the efficiency, performance, and overall sustainability of these vehicles. it is becoming increasingly important that lightweight composites play a part in attaining this weight reduction. this is because these materials give the same level of strength and durability as traditional materials, but at a weight that is far lower. one of the most significant advantages of lowering the weight of an electric vehicle is the increased driving range it provides [51,52]. considering that bigger vehicles take more energy to move, it follows that a heavier electric vehicle will have its battery depleted more quickly than a lighter one. by decreasing the weight of the vehicle, energy efficiency can be enhanced, which in turn leads to an increase in the number of miles driven on a single charge [53,54]. decreased weight results in improved acceleration and handling characteristics for the vehicle. this is of utmost significance for evs, as the instant torque that is supplied by electric motors can be utilized more effectively with a chassis that is lighter, which ultimately results in superior vehicle dynamics. the operation of a lighter vehicle takes less energy, which can lead to the development of battery designs that are both more compact and more efficient. in addition to lowering the price of the battery, this also lessens the negative impact on the environment that is caused by the manufacturing and disposal of electrical batteries. while it is essential to accomplish weight reduction, it is of utmost importance to preserve structural integrity and safety. composites that are lightweight offer good strength-to-weight ratios, which ensures that the safety of the vehicle is not compromised in the search of reduced weight [55–57]. the components of lighter vehicles, such as brakes and tires, are subjected to less stress, which results in less wear and tear and lower maintenance expenses over the course of the vehicle's lifetime [58,59]. the global lightweight materials market was valued at usd 113.8 billion in 2016 and was expected to grow at a cagr of 8.9% from 2017 to 2024. automotive manufacturers are switching to lightweight materials to reduce vehicle weight and fuel emissions, driving market growth. north america is a key market due to the presence of major auto manufacturers and renewable energy equipment manufacturers. the market is segmented by product (aluminum, high strength steel, titanium, magnesium, polymers, composites, others), application (automotive, aviation, energy, other), and region (north america, europe, asia pacific, latin america, middle east, africa) [60]. fig. 2 shows the market size with respect to the production of lightweight materials in north america from 2013 to 2024 [61]. fig. 2 market of lightweight materials in north america from 2013 to 2024 [61] 12 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. 3.1 role of lightweight composites in evs the development and production of electric cars are both significantly aided by the utilization of lightweight composites evs. the performance, efficiency, and general sustainability of electric vehicles can all be significantly improved with the help of these materials [62,63]. many different aspects can be used to understand the role that lightweight composites play in electric vehicles. composite materials that are lightweight and used in electric vehicles often consist of high-strength fibers such as carbon or glass that are placed in a polymer matrix. strength, rigidity, and lightness are all characteristics that these materials possess in remarkable proportions [64,65]. there are several components of electric vehicles that make use of composites. these components include body panels, chassis components, and even structural frames in some cases. by utilizing them, the overall weight of the vehicle is greatly reduced, and this is accomplished without sacrificing either its longevity or its performance [66,67]. lightweight composites are also utilized in the construction of protective structures and housings for batteries. they offer the essential protection for the battery pack while making a negligible contribution to the overall weight of the vehicle [24,25]. processes such as rtm, compression molding, and automated layup procedures are being refined for mass production as the fabrication of composite materials continues to undergo technological advancements. composite components are becoming increasingly practical for broad usage in electric vehicles because of this innovation in manufacturing, which helps to reduce the cost of composite components [61,68]. the thermal properties of composites can be tailored to be advantageous, which is vital for evs because effective thermal management of electric motors and batteries is essential for achieving maximum possible performance [61,69]. moreover, the aesthetic and design of evs affect the overall economy of the vehicle, while the versatility of composites in terms of molding and shaping makes it possible to create unique vehicle designs that are not only visually beautiful but also aerodynamically efficient [70]. 4. applications in electric vehicles and influencing factors in evs, sustainable composite materials made of natural fibers incorporated in a polymer matrix have multiple possible applications. their meteoric rise to the top of the automotive industry is largely attributable to their low weight, great mechanical features, and acceptance to the environment. electric vehicles often use sustainable composites for interior components such as door panels, dashboards, and seat frames [71,72]. because of their reasonable mechanical properties, light weight, and natural look, biocomposites are perfect for various applications. bumpers, body panels, and other exterior components made of biocomposites are still the subject of active research. problems with durability, longevity, and weather resistance are being addressed by new materials, hybrid composites as well as nanocomposites [73]. for non-essential construction components, biocomposites offer a strong and lightweight alternative [74,75]. their application in structural sections is still in its early stages of development, but the present focus is on improving their loadbearing capacity and impact resistance. by decreasing their total weight, biocomposites aid electric vehicles in achieving higher energy efficiency and longer driving ranges. the acoustic and thermal insulation properties of natural fibers are unparalleled. biocomposites provide a sustainable solution for the electric vehicle sector that is in line with global efforts sustainable composite materials for electric vehicle applications: a comprehensive review 13 to reduce environmental impacts throughout the vehicle's lifecycle. biocomposites have great potential to revolutionize environmentally friendly transportation, even though there are still certain challenges to be solved [76,77]. as a result of continuous research and development and industry collaboration, sustainable composites may one day completely transform the manufacturing process for electric vehicles. the initial generations of commercial evs focused mostly on modifying existing vehicle chassis with battery systems. recent evs include evolving structural designs that are specifically engineered to operate only on battery power from their inception. this design flexibility has led to new potential for polymer composites, including glass-reinforced polyurethanes as reduced weight substitutes for metal components. ev designs employ predominantly linear components in their structure, making them well-suited for the utilization of polymer composites. the incorporation of novel technology and content continuously enhances the total curb weight of motor vehicles. thermoplastic materials provide a favorable combination of robust structural integrity and reduced density. the detailed discussion of ev part applications and influencing factors will be presented in the upcoming sections. 4.1 battery cell separators the enhancement of lithium-ion battery performance is a matter of utmost importance, and there is currently a renewed focus on the development of superior membranes to achieve this goal. the lithium-ion battery separator (libs) is a permeable membrane inserted between a battery anode and cathode. the primary purpose of this ev component is to maintain a sufficient distance between the two electrodes to prevent any electrical short circuits, while simultaneously facilitating the movement of ions for the purpose of both charging and discharging. it is the most crucial electric car component in evaluating battery stability as well as safety. uhmw-pe finds its best grade suitable for libs in the ev industry. 4.2 high voltage battery high voltage traction batteries are the essential source of power for electric propulsion in the automobile sector. they satisfy numerous criteria to offer dependable lifetime operation, including high voltage/current protection and electrical insulation, thermal propagation protection and flame retardancy, thermal shock and crash resistance. in the present day, batteries in evs can enhance overall durability, lightweighting, connectivity, safety and range. htn and pa, polybutylene terephthalate (pbt) and polyphenylene sulfide (pps) serve as effective polymers for evs due to their exceptional electrical insulation, together with superior flame retardancy and thermal shock resistance [78]. various thermoplastic materials are additionally utilized for weight reduction and crash resistance of the battery unit [79]. continuous fiber reinforced thermoplastics, when combined with long fiber reinforced thermoplastics (lfrt), offer superior toughness, stiffness, and strength while achieving maximum weight loss [68]. battery components utilize materials that function as electrical insulators or facilitate heat transmission. hybrid designs integrate materials with distinct qualities into a single component, resulting in reduced weight and cost constraints. polymeric materials provide the necessary characteristics to meet essential criteria for dependable applications, such as flame retardance and thermal expansion, maintain their shape, exhibit high rigidity even with thin walls, withstand thermal shock and hydrolysis resistance. 14 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. due to the growing interest in hybrid cars, the incorporation of a battery management system (bms) has emerged as a vital component [80]. the objectives of bms are to optimize the duration of each discharge cycle and maximize the total number of usable battery life cycles. the management of automotive batteries is challenging due to the necessity of real-time operation under rapidly fluctuating charge-discharge circumstances, such as during vehicle acceleration and break, in a hostile and unregulated environment. furthermore, it is imperative that it establishes a connection with other systems integrated within the vehicle, including the engine management, temperature controls, communications, and safety systems. bms consists of three primary components: the sensor system, the control unit, and the switching unit. bms actively maintains uniform voltage or state of charge across all the battery cells to optimize capacity and prevent localized under-charging or over-charging. this is achieved through a process called balancing. ev battery unit designs have modules that store energy and have their own cooling systems. these modules serve separate purposes, including structural and electrical operations, and are mounted into the vehicle body without prioritizing structural strength. this strategy would effectively tackle issues related to cell cooling, system assembly, chemical bonding performance, safety, and sustainability. a novel prototype of an all-inone battery module utilizes tab cooling, structural thermoplastic cooling channels, and an electrical connecting plate to accomplish this. this provides advantages in terms of increased range, integration of components, and enhanced thermal management. 4.3 battery coolant hoses and quick connectors a variety of components are specifically engineered to enhance the longevity and safety of evs and other automobiles by facilitating efficient management of automotive fluids. polymeric materials exhibit enduring chemical resistance and efficient fluid control, even when exposed to water, glycol, auto gearbox fluids and dielectric fluids. both internal and exterior ev battery systems need a variety of coolant hoses with different levels of flexibility, ranging from flexible to semi-stiff and stiff. these hoses must be able to function efficiently in under-hood settings with lower temperatures (30-90⁰c) and reduced pressure [78]. thermoplastic vulcanizate (tpv) enables the production of coolant system hoses and tubes for electric vehicles that offer enhanced durability, reduced weight, and cheaper prices compared to systems made from ethylene propylene diene monomer (epdm) rubber, polyamide (pa) materials, and polyphenylene sulfide (pps) alternatives [81]. in addition, the utilization of tpv provides distinct sustainability benefits due to its recyclability during the manufacturing process and at the conclusion of its lifespan. longchain polyamides have exceptional resistance to hydrolysis at high temperatures. if employed with thermoplastic olefins such as tpv epdm/pp, they provide significant flexibility for both monoand multi-layer tubing, serving as substitutes for metal pipes and epdm rubber cooling lines. ethylene acrylic elastomers (aem) are a proven elastomer solution for high-temperature and flexible oil cooler hoses and seals and are compatible with numerous types of oils, including new ev fluids. fluid management lines can be complemented with quick connections that are constructed using strong materials such as glass fiber reinforced nylon (pa66 gf), high-temperature nylon (htn), glass fiber reinforced polyphthalamides (ppa gf), or glass fiber reinforced long chain polyamides (lcpa gf). these connectors provide a fully integrated polymer solution. sustainable composite materials for electric vehicle applications: a comprehensive review 15 4.4 power electronics inverters convert dc, generated by a lithium-ion battery, into ac to power electric traction motors. high-voltage converters reduce the voltage of the battery to supply power to various systems in the automobile, including entertainment and safety systems. they also convert dc from a high level (e.g. hv lithium-ion battery) to a lower level (e.g. 12v battery). these processes cause energy losses which are transformed into heat. plastic insulating components used in power electronics must be able to endure elevated temperatures while maintaining their insulation qualities over the whole temperature range. pps and pbt have outstanding electric insulating properties, including dielectric strength, surface and volume resistivity at elevated temperatures. the present process of inversion and conversion generates electromagnetic fields that have the potential to damage delicate onboard devices. current inverter and converter housings are manufactured of aluminum to hide these electro-magnetic forces. diverse plastic and plastic/metal composite options are available for electric cars, such as lfrt steel fiber polymers or metal coated polymers. these options meet emi standards and provide significant reductions in weight. 4.5 hv connectors automotive electric engines work on high electric currents and voltages up to 800v with estimates for even greater voltages in future generations [78]. connectors utilized in high voltage electrical circuits must include reliable insulation to prevent short circuits and dangerous fires. pa, pps, and pbt possess exceptional insulating characteristics that fulfil the demands of hv connections, such as elevated cti, superior dielectric strength, flame resistance, remarkable rigidity, durability, and color persistence. special grades pa with low acidic content, which avoids corrosion of the soft sealings, is used in connection with high voltage connectors. 4.6 traction e-motor to enhance the comfort and efficiency of existing automobiles, future mobility models need a higher quantity of electric motors. materials utilized in under-the-hood applications must possess the ability to endure demanding circumstances without compromising safety, quality, or service life. high-performance polymers offer enhanced efficiency, particularly in compact motors, enabling manufacturers to decrease both cost and weight. pps, pa, and liquid crystal polymer (lcp) grades are components of an extensive range of polymers designed for use in extremely elevated temperature and chemical environments [78]. 4.7 thermal management the thermal management system of electric cars is crucial, since it directly impacts the performance and durability of key components, including battery cells, power electronics, and traction motors. the fundamental components of thermal management systems in electric cars closely resemble those found in combustion engines. hybrid cars incorporate a fusion of both. the need for connections with high-performance engineered materials such as pps, htn, and pa is driven by the need to withstand cooling agents, meet extended lifespan requirements, and achieve high accuracy in coolant pumps, impellers, and thermostat housings [78]. tpv is very compatible with tubes and hoses utilized in the thermal management systems of contemporary evs. 16 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. 4.8 noise, vibration and hardness (nvh) evs offer a smoother and more comfortable journey experience. the issue is in the absence of engine noise, which often serves to conceal the various squeaks, rattles, and buzzes. consequently, the formerly serene ride becomes increasingly boisterous and disruptive. efficiently addressing nvh (noise, vibration, and harshness) can effectively mitigate this problem, hence enhancing the overall experience for both the driver and passengers. many sophisticated materials, such as zytel nvh gen 2, which have been demonstrated to effectively diminish noise and vibration in applications such as engine mounts, offer superior dampening properties. 5. environmental impact and life cycle assessment nowadays automobile industries are focusing on zero-emission transport techniques to combat climate change, which can be achieved using evs. evs are considered due to reduced greenhouse gas emission which is directly correlated to the environmental impact. by using evs powered by renewables, the emission of co2 can be reduced by 80 % as compared to fossil-generated electricity [82]. a traditional diesel/petrol-powered vehicle emits 233.5 % more co2 when compared to renewables-powered evs [82]. the electricity mix significantly influences ev emissions, making them carbon-neutral in jurisdictions such as costa rica with entirely renewable electricity generation. despite lower global warming potential (gwp), evs exhibit higher human toxicity potential (htp) and particulate matter formation potential (pmfp) during battery manufacturing and disposal [83]. factors such as increased vehicle weight, high torque, and accelerated tire wear contribute to environmental concerns, including particulate pollution and ocean plastic waste. additionally, the disposal of li-ion batteries poses environmental and safety risks. the life cycle impact of evs is also influenced by climatic conditions, energy consumption variations due to heating or cooling requirements, assumed lifespan, and the local electricity mix. therefore, understanding the environmental impacts of evs requires considering a combination of factors such as jurisdiction-specific conditions, electricity sources, charging times, and lightweight strategies. table 2 presents the environmental impact of evs in certain jurisdictions. table 2 environmental impact potential of electric vehicles in substantial jurisdictions [15]; gwp: global warming potential; ap: acidification potential; htp: human toxicity potential (non-carcinogenic); pmfp: particulate matter formation potential; *kg pm2.5 eq./km jurisdiction engine power vehicle weight, kg battery weight, kg gwp, kg co2 eq./ km ap, kg so2 eq htp, kg 1,4-dcb eq./ km pmfp, kg pm10 eq./ km germany 100 kw 1615.0 – 0.140 – 0.738 0.0004* czech rep./ poland 2.1 kwh 1180.22 262.0 0.214 0.001 0.306 0.0004 canada – – 312.0 0.160 0.001 0.260 – canada (petrol) – – 0.254 – 0.045 – czech rep./ poland 1400 cc, petrol 1200.0 – 0.284 0.001 0.085 0.0003 sustainable composite materials for electric vehicle applications: a comprehensive review 17 environmental advantages of eco-friendly composites for evs can be easily accessed using life cycle analysis. china, as the world's largest carbon dioxide emitter, has been dealing with energy and environmental problems and has implemented policies and financial incentives to promote the development of evs [84]. researchers have conducted a comparative life cycle assessment of various fueled vehicles, including internal combustion engine vehicles, hybrid electric vehicles, and electric vehicles, to evaluate their environmental impact [85]. seven different environmental impact categories were considered, including abiotic depletion, acidification, eutrophication, global warming, human toxicity, ozone layer depletion, and terrestrial ecotoxicity. electric and plug-in hybrid electric vehicles show higher human toxicity, terrestrial ecotoxicity, and acidification values due to manufacturing and maintenance phases. hydrogen vehicles are found to be the most environmentally benign option due to their high energy density and low fuel consumption during operation [85]. the eco-indicator 99 impact assessment method is used, which includes human health and ecosystem quality as important indicators for decision-making regarding clean transportation vehicles [85]. another work dealt with the development and utilization of new plant-fiber composite materials and microcellular foam molding processes for lightweight, low-carbon, and sustainable composites for automobiles. life cycle assessment was conducted to investigate the resource and environmental impacts of plant-fiber composite material automotive components and microcellular foam molding processes. electricity consumption was the largest influencing factor causing environmental issues throughout the life cycle, accounting for more than 42% of indicators such as ozone depletion, fossil resource consumption, and carbon dioxide emissions [86]. a typical life cycle of a vehicle technology can be divided into two primary stages: the fuel cycle and the vehicle cycle. the fuel cycle involves processes from feedstock production to the utilization of fuel in the vehicle [87]. fig. 3 illustrates the life cycle analysis along with the environmental impact assessment of fuel and vehicles [13]. fig. 4 shows an overview of various impacts of internal combustion engine vehicles (icevs), hybrid electric vehicles (hevs), and battery electric vehicles (bevs) during lca using the electricity mix of 2015 [88]. the analysis was done considering global warming, ionizing radiation, human carcinogenic toxicity, human non-carcinogenic toxicity, land use, and fossil resource scarcity. li-ion batteries are one of the major concerns nowadays in the field of evs. the most influential parameter when it comes to the environment is the use of evs followed by production and transport. fig. 3 life cycle environmental impact assessment of fuel and vehicles 18 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. fig. 4 impact of (a) global warming, (b) ionizing radiation, (c) human carcinogenic toxicity, (d) human non-carcinogenic toxicity, (e) land use, (f) fossil resource scarcity during lca (electricity mix of 2015) for icevs, hev, and bev [88] sustainable composite materials for electric vehicle applications: a comprehensive review 19 6. market trends and economic viability the electric vehicle market is a highly dynamic sector within the dominion of clean energy. in 2021, the sales of evs experienced a remarkable doubling from the previous year, reaching a new record of 6.6 million units. to put this into perspective, back in 2012, the global sales of electric cars amounted to a mere 120,000. fast forward to 2021, and we now witness over 120,000 electric cars sold each week. this surge in popularity translated into nearly 10% of all global car sales being electric in 2021, a staggering fourfold increase from the market share observed in 2019. consequently, the number of electric cars traveling the world's roads has soared to approximately 16.5 million, a threefold increase compared to the 2018 figure. the momentum in global electric car sales has not slowed down in 2022, as the first quarter saw a robust performance with 2 million units sold, marking an impressive 75% increase from the same period in 2021 [89]. several key factors have contributed to the success of evs. foremost among them is the consistent support from policies that encourage their adoption. in 2021, public spending on subsidies and incentives for evs nearly doubled, reaching close to usd 30 billion. many countries have also committed to phasing out internal combustion engines or have set ambitious targets for vehicle electrification in the coming years. moreover, numerous car manufacturers have gone beyond policy requirements and are actively electrifying their fleets. furthermore, the availability of electric vehicle models has significantly expanded, with five times more models on the market in 2021 compared to 2015, enhancing their appeal to consumers. currently, there are around 450 electric vehicle models available [89,90]. china stands out as a major driving force behind the surge in ev sales in 2021, accounting for half of the global growth. china's 2021 sales alone (3.3 million) exceeded the total global sales in 2020. china included new energy vehicles in its national "863" plan in 2001 and launched the "major science and technology project of evs" to kickstart ev research and development [91]. the chinese government has been promoting the use of evs since the 12th five-year plan (2010-2015) to make travel cleaner. however, the economic crisis has affected people's recognition and acceptance of evs, hindering their early market diffusion. europe also witnessed strong growth, with sales increasing by 65% to reach 2.3 million units, following a similar boom in 2020. even in the united states, where sales had experienced a decline for two consecutive years, there was an uptick, reaching 630,000 units. the trends observed in the first quarter of 2022 closely mirrored these patterns, with china leading global growth with more than a doubling of sales compared to the same period in 2021, a 60% increase in the united states, and a 25% increase in europe. despite a challenging year for the car market in 2020, the sales of electric vehicles, including battery electric vehicles (bevs) and plug-in hybrid electric vehicles (phevs), nearly doubled in 2021, reaching a total of 6.6 million units (fig. 5). this substantial growth raised the overall number of electric cars in circulation to over 16.5 million. similar to previous years, bevs continued to dominate the increase, representing approximately 70% of the total. on the flip side, electric vehicle sales are lagging in other emerging and developing economies, primarily due to the limited availability of affordable models for mass-market consumers. in countries such as brazil, india, and indonesia, electric cars constitute less than 0.5% of total car sales. nevertheless, there are promising signs of growth in several regions, including india, where ev sales doubled in 2021. with the right investments and 20 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. supportive policies in place, these regions could potentially experience accelerated ev market uptake by 2030 [89]. fig. 5 growth in evs sale and model availability from 2016-2021. bev: battery electric vehicles; phev: plug-in hybrid electric vehicles [89] the share of global ev battery production capacity reached 57 gwh in 2021. assessing risks in high-capacity battery supply chains, as mandated by an executive order in early 2021, is highly required. the potential of vehicle-to-grid technologies, which allow evs to contribute to grid stability and balancing needs, should be particularly exploited in countries falling under the case of increasing ev load with high variable renewables generation. it emphasizes the need for the development and deployment of smart chargers to enable evs to participate in balancing the system and contribute to grid stability [89]. the automotive sector is facing challenges related to energy consumption and climate change, and reducing vehicle weight is crucial in addressing these challenges. a 10% decrease in curb weight can result in a 6-8% reduction in energy consumption. composite materials with a high strengthto-weight ratio are considered a viable option for lightweight component design and manufacturing in the automobile sector. the use of composite materials in electric vehicles can reduce weight, improve aerodynamic properties, and decrease fuel consumption, leading to a reduction in harmful emissions and particulate matter [92]. the market trends and demand for sustainable composites in the electric vehicle sector are on the rise. the automotive industry is actively seeking ways to reduce the weight of vehicles in order to minimize energy consumption [92]. composite materials, particularly fiber-reinforced polymers, have gained attention due to their high performance and lighter weight [93]. the use of composite materials in electric vehicles not only reduces weight but also improves aerodynamic properties, leading to decreased fuel consumption and emissions [94]. the demand for electric vehicles is increasing globally, driven by concerns about the environment and the depletion of fossil fuel reserves [95,96]. as a result, there is a growing need for sustainable materials in the electric vehicle market. the transition from internal sustainable composite materials for electric vehicle applications: a comprehensive review 21 combustion engines to electric vehicles is accelerating, creating a demand for efficient and powerful electric motors. overall, the market for sustainable composites in the electric vehicle sector is expanding due to the push for cleaner and more sustainable transportation options. 7. challenges and innovations exploration of lightweight materials, such as carbon fiber, sustainable and advanced composites are required to improve energy efficiency and extend driving range. challenges in ev production include thermal events associated with batteries and the increased overall weight, which can affect various performance factors of evs. strategies for addressing these challenges include securing the lithium supply chain, enhancing battery manufacturing hubs, creating new charging infrastructures, and establishing policy mandates towards zero emission transportation. the utilization of high-performance lightweight and sustainable composites with superior flame retardancy can help address the challenges of weight and thermal events in evs. lightweighting ev components can increase travel ranges, minimize crash impacts, lower braking distances, and reduce tire wear and tear [97,98]. despite all these possible advancements, the cost of batteries remains a significant factor in the overall price of evs, posing a challenge to widespread adoption. ongoing research aims to enhance energy density, decrease charging time, and increase the lifespan of batteries, contributing to improved overall ev performance. balancing the power capability and energy density in evs batteries is equally significant. this has practical implications for ev manufacturers, as they need to optimize battery performance while considering factors such as cost and safety. battery shortages, rising raw material costs, high demand from affluent buyers, and insufficient charging infrastructure contribute to the high cost of electric cars. for instance, the average price exceeds $66,000, making evs a luxury item and largely inaccessible to the public in the us. without price regulation or additional subsidies, widespread adoption remains a challenge [99]. the us government’s ambitious goal of having 50% electric cars by 2030 requires a substantial infrastructure investment. furthermore, the expansion of the ev market necessitates the development of robust charging infrastructure and supportive government regulations [100]. in the us, the plan to spend $5 billion on 500,000 public charging stations falls short of the estimated 1.2 million needed, with an associated cost of $35 billion. upgrading the power grid to support a rapid ev shift is another costly endeavor, impacting consumers through potential tax increases. fig. 6 shows the geographical distribution of mineral resources related to battery manufacturing and its associated companies [101]. computer simulation can play a crucial role in accelerating systematic battery design, which has practical implications for battery researchers and engineers. by using atomic-scale simulations and machine learning, battery materials can be selected and system designs can be optimized, reducing design cycles and improving battery performance [100]. measuring and modeling complex material properties accurately in battery cells is useful for practical implications in battery design and testing. accurate measurement and modeling of material properties are crucial for understanding battery failure mechanisms and improving battery safety. the potential cost reduction in battery technology is also possible by using high energy density materials and replacing expensive materials such as cobalt. 22 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. fig. 6 geographical distribution of mineral resources related to battery manufacturing and its associated companies. (a) countries with mineral resources, (b) countries with mineral resources and manufacturing companies, and (c) manufacturing countries [101] the development of the charging infrastructure is another crucial part to address range anxiety and facilitate long-distance travel for ev users. faster and more widespread charging infrastructure is required, including ultra-fast chargers, to reduce charging times and increase the appeal of evs [15]. the driving range of evs on a single charge remains a concern for potential buyers, especially in regions with limited charging infrastructure. deploying smart energy management systems to optimize charging schedules, incorporate renewable energy sources, and minimize overall energy consumption is highly recommended. establishing standardized technologies and charging protocols is essential to ensure interoperability and compatibility among various ev models and charging stations. expanding the range of available ev models on the market is important to address diverse consumer preferences and needs. standardized technologies and charging protocols are needed to ensure interoperability and compatibility across different ev models and charging stations. there is also a need to convince consumers so that they can shift from traditional internal combustion engine vehicles to evs. the main concern of the consumers is related to charging accessibility, range, and initial costs. to make evs more successful, consistent establishment, supportive policies and regulations including incentives, subsidies, and emission standards are required [102]. also, increased awareness and understanding are also needed among consumers about the benefits, capabilities, and the evolving nature of the ev technology. lastly, the environmental implications of battery disposal and the establishment of sustainable recycling practices for ev batteries constitute another crucial element of sustainable development that requires attention. fig. 7 illustrates the model of an ev ecosystem, considering all the factors required for its advancements [103]. sustainable composite materials for electric vehicle applications: a comprehensive review 23 fig. 7 model of the ev ecosystem and advancements in the ev ecosystem along with its elements 8. conclusions and future scope manufacturing evs with sustainable composites is an intriguing new frontier that comes with its own set of problems. one reason composites are finding their way into electric vehicles is the demand for lighter materials, which can improve performance while cutting down on fuel usage. sustainability, performance, and cost-effectiveness are all complicated issues that must be balanced. several technical challenges remain in the way of sustainable composite manufacturing, notwithstanding considerable progress. some of these obstacles include the fact that rtm and vartm are not very scalable when it comes to mass production, maintaining quality across huge batches of products, and incorporating new materials into current production frameworks. one big obstacle is the economics of evs. there is typically a significant up-front expense associated with sustainable materials and innovative manufacturing technology when contrasted with more conventional options. this may discourage broad use, especially in areas with less economic flexibility or among smaller producers. another factor is the difficulty with availability and consistent quality due to supply chain complications. this is especially true for novel and inventive materials. it can be difficult for manufacturers and suppliers to work together efficiently due to the absence of industry-wide standards for sustainable materials and production techniques. sustainability in the production of evs is an exciting and ever-changing topic. the production process is becoming more and more focused on sustainability as the demand for electric vehicles increases. we may anticipate that research into novel composite materials will persist, with an eye toward improving performance while decreasing the 24 m. k. singh, s. k. palaniappan, i. suyambulingam , et al. ecological footprint. some examples of this include investigating novel material formulations, improving bio-based composites, and incorporating nanomaterials. new developments in manufacturing technology should solve the problems of scalability and cost-effectiveness that exist now. more sustainable and widely available composite components may soon be on the horizon, thanks to the developments in 3d printing, automated assembly, and ai-driven process optimization. moving towards a model based on the circular economy will be a major trend in sustainable manufacturing going forward. this entails making items in a way that makes them easier to disassemble, recycle, or repurpose when their lifetime comes to a close. making electric vehicles more recyclable and less harmful to the environment requires research into the development of composite materials and components. moreover, thanks to major advances in material science and, particularly, in modeling tools [104], the adoption of active/smart structures, involving multifunctional materials such as piezoelectric materials, magnetostrictive materials and shape memory alloys, is expected to grow within the automotive industry, thus enhancing the safety, ergonomics, efficiency, and robustness of vehicle structures. with the global trend towards more stringent environmental laws, the electric vehicle sector may witness a surge in efforts led by legislation that encourage sustainable behaviors. incentives for sustainable material use, rules requiring manufacturers to cut emissions, and funding for sustainable technology research and development are all examples of what could fall under this category. composites will have an even smaller environmental impact in the future as more and more factories switch to renewable energy. in line with the environmentally conscious character of evs, solar, wind, and geothermal energy have the potential to become the go-to power sources for factories. there are possibilities and threats at this crossroads in the sustainable composite production for the electric vehicle industry. even though there are a lot of financial and technological hurdles that need to be overcome, there are some exciting developments coming up that might completely transform the sector. the future of electric vehicle production is looking brighter in terms of sustainability, thanks to the developments in materials science, production technology, energy consumption, the adoption of circular economy ideas, and enabling legislative frameworks. a more environmentally friendly and sustainable automotive future is within reach, thanks to this changing landscape and the thrilling potential it offers to industry stakeholders. acknowledgement: this research budget was allocated by the national science, research and innovation fund (nsrf) (fundamental fund 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impact of the oil tank filling level on the aircraft separator tomasz szwarc1,2, włodzimierz wróblewski1, tomasz borzęcki2 1silesian university of technology, poland 2avio polskasp. z o.o., poland orcid ids: tomasz szwarc https://orcid.org/0000-0002-0391-3353 włodzimierz wróblewski https://orcid.org/0000-0002-3617-3515 tomasz borzęcki https://orcid.org/0000-0002-4226-7921 abstract. this paper presents a cfd analysis of an air-oil separator of an aircraft gas turbine engine with a focus on the impact of the oil tank filling level on the separator performance for a selected point of the flying mission. the separator efficiency and the oil quality affect the efficiency of the oil system. new design criteria and standards require a better understanding of the phenomena occurring in the separator. to optimize its structure, the flow of the air-oil mixture must be modeled in the design process. although many papers are addressing the issue of gas-liquid separation, very little knowledge is available on the flow ratio typical of aircraft turbine engines. the separation phenomena were investigated using the volume-of-fluid method. transient calculations were performed at a selected mission point of the separator and compared with experimental data. a mesh independence study using a structural mesh is included to understand the mesh impact on the analysis results. the current analysis results will support further studies focusing on an optimization analysis where a proper mesh, an adequate turbulence model and an appropriate oil level have to be selected. key words: multiphase flow, air-oil separator, cyclone, volume of fluid method 1. introduction secondary air and lubrication systems have a decisive influence on the characteristics and operational capabilities of aircraft engines. the characteristics of these two systems define the mixture volume ratio, which depends on the flight mission conditions. during the engine operation, the oil tank level changes due to the gulping effect, engine oil hiding and attitude. these phenomena affect separation efficiency and change the flow field formation. the separator performance has a decisive impact on the oil quality, directly received: december 14, 2021 / accepted march 27, 2022 corresponding author: tomasz szwarc silesian university of technology, ul.grażyńskiego 141, bielsko biała 43-300 e-mail: tomasz.szwarc@avioaero.it https://orcid.org/0000-0002-0391-3353 https://orcid.org/0000-0002-3617-3515 https://orcid.org/0000-0002-4226-7921 18 t. szwarc, w. wróblewski, t. borzęcki affecting the oil system elements and the engine thermal capability. the oil flow loss can also cause overheating due to the reduced heat transfer in the oil cooler, caused by low thermal capacity of the air. other issues are related to the oil tank. due to foam formation, the oil level can be indicated wrongly. increased foaming can lead to high internal pressures causing seepage from connections or even resulting in the tank rupture in the worst case. the most affected component of the lube system is an oil pump unit. a 1-2% air volume fraction in the oil system can affect the functioning of the gear pump at a high altitude and high-power operation. vapor locked in the lube element area can cause cavitation while the creation of air-oil-metal interfaces inside the engine can be the starting point for corrosion. the influence on volumetric efficiency is still part of contemporary research [1]. oil aeration problems were studied in the past and are investigated nowadays [2]. the oil aeration problem can be resolved in several different ways. a patent solution was proposed in [3] by adding an oil injection passage downstream of the pump to avoid recirculation of air bubbles in the gear meshing area. another way is to decrease the scavenge pump discharge pressure by decreasing inlet losses or by using an oil type with suitable anti-foaming additives [2]. it prevents the formation of fine bubbles, which are the hardest to separate. other mechanical routines of removing the gas entry can be obtained by designing a scavenge line with a velocity below 5 ft/s. the last approach is to increase radial acceleration and centrifugal force in the cyclone separator. it is a lube system component installed inside the engine oil tank. the major advantage of this component is its reliable structure. the deaerator must be designed to keep a good balance between the increase in the oil mixture velocity and the pressure drop. overcoming pressure losses can cause issues during the engine cold start or can affect the design of the scavenge pump. in the cyclone separator, the air-oil mixture flows through a tangential inlet slot. as a result, a swirl is generated causing the air to separate due to the centrifugal force. regarding the separation phenomena on the wall, the air bubbles leave the oil particles and form foam. due to the changes in the oil density and viscosity, small bubbles need more time to reach the free surface. experimental data show that the bubbles do not rise so rapidly in the oil containing appropriate additives. continuing the flow, the oil moves towards the wall and downwards while the air flows to the centre line and exits through the vortex finder. depending on operating conditions, some oil droplets flow with the air and move up towards the air core. this phenomenon is called the liquid carry-over. at the bottom exit, air bubbles can remain in the oil and exit the tank. this phenomenon is called the gas carry-under [4]. compared to typical industrial cyclone separators, those used in an aircraft need to meet more requirements. changing operating conditions during flight missions involve changes in many parameters like the air-oil ratio, the oil tank filling level, the tank pressure, and the aircraft position (attitude). a highly swirling flow inside the separator is frequent and unsteady. the unstable flow field determines the separation performance. another problem of the aircraft separator is its small volume, which makes the device very sensitive to variations in the flow parameters, such as the inlet flow pattern, or tangential and axial velocities. this can degrade the separation performance, which can cause operational problems [5]. considering the existing available publications on aircraft cyclone separators, it can be stated that not many of them focus on performance [6]. the works presenting the state of the art in industrial cyclone separators [7-9] focus on a closed conical shape (stairmand, stern or lapple), which is often impossible to implement due to the limited design space. moreover, the mathematical models of cyclone separators based on such research are limited to the single-phase flow or the dispersed-oil phase. experimental and numerical study of the impact of the oil tank filling level on the aircraft... 19 the studies on the fluid dynamic phenomena related to the air-oil flow in a gas turbine system recognize that the gas-liquid cylindrical cyclones (glcc) used in the oil and gas industry behave in a similar way [10, 11]. through systematic experiments and research performed in the past [12-17], detailed mechanistic models have been developed to describe important separation phenomena. following the above, an approach intended for the air-oil separator used in the aircraft oil system is under preparation. based on experimental data, a theoretical study and a mechanistic model were developed to predict the operational envelope for the liquid carry-over and bubble trajectories [5, 14] for the glcc. as indicated by the literature results, the turbulent swirling flow in cyclone separators is anisotropic [16]. therefore, an anisotropic turbulence model should be used to model such a flow accurately. in other studies [4,16] different turbulence models were used to investigate the sensitivity of the flow field predictions. using a reynolds stress model, which is an anisotropic turbulence model, the simulations presented in [7-9] showed only a slight improvement over the k-ε model prediction of the flow field in the glcc. erdal and shirazi [16] compared experimental results with a 3d cfd simulation. in the simulation, k-ε and rsm turbulence models were used. the analysis shows a general trend of experimental data but both models fail to predict certain parameters. the k-ε model tends to predict a higher rotational flow, whereas the rsm model predicts a different wavelength of the vortex and local axial and tangential velocities. the most often used numerical modeling method aiming to describe the two-phase flow in separators is presented in [18]. the application of an appropriate numerical model is conditioned by the existence of a specific two-phase flow pattern, which should be identified or assumed based on experimental knowledge and maps of the flow pattern. the volume of fluid (vof) is a free-surface modeling technique, which makes it possible to track the air-oil interface [19]. the interface between the phases is tracked using the continuity equation for the volume fraction. the volume fraction equation is not solved for the primary phase. as the volume of a phase cannot be occupied by other phases, the balance calculations are carried out for a given phase considering the volume fractions. a single momentum equation is solved throughout the domain, and the resulting velocity field is shared among the phases. the momentum equation is dependent on the volume fractions of all phases through the mixture density and mixture viscosity. the initial numerical analyses of the oil separator investigated are presented in [20]. the calculations were performed using the vof model and rng k-ε turbulence model. these analyses were conducted for other definitions and discretization of the calculation domain. a tetragonal numeric grid was applied. the results of the calculations were compared with the experimental data for the adopted boundary conditions; it was found out that the chosen numeric models predict oil quality and performance correctly but do not have a complex domain enabling stabilization of the oil level. this paper presents an analysis based on a separator used in the aircraft gas turbine engine. the objective of this study is to investigate tank oil level impact on separator performance. the computed separator parameters – oil quality and efficiency were compared with available experimental data to validate the model. additionally, the impact of the oil level in the tank on the flow field below the separator inlet was investigated. the observed flow field phenomenon was compared to other separator types available in the industry. 20 t. szwarc, w. wróblewski, t. borzęcki 2. models and methods 2.1. mathematical model turbulence was simulated using the rng k-ε model. the model is derived from the instantaneous navier-stokes’s equations using a mathematical technique known as the “renormalization group” (rng) method. it is based on the standard k-ε model but includes refinement. an additional term in the ε equation improves the accuracy for rapidly strained flows. due to that, the effect of swirl on the turbulence is included, enhancing the accuracy for swirling flows. the rng theory provides an analytical formula for the turbulent prandtl numbers, while the standard k-ε model uses user-specified, constant values [21]. the equations for turbulence kinetic energy k and the dissipation rate of turbulent kinetic energy ε are solved: 𝜕 𝜕𝑡 (𝜌𝑘) + 𝜕 𝜕𝑥𝑖 (𝜌𝑘𝑈𝑖) = 𝜕 𝜕𝑥𝑗 (𝛼𝑘𝜇𝑒𝑓𝑓 𝜕𝑘 𝜕𝑥𝑗 ) + 𝐺𝑘 − 𝜌 (1) 𝜕 𝜕𝑡 (𝜌𝜀) + 𝜕 𝜕𝑥𝑖 (𝜌𝜀𝑈𝑖) = 𝜕 𝜕𝑥𝑗 (𝛼𝜀𝜇𝑒𝑓𝑓 𝜕𝜀 𝜕𝑥𝑗 ) + 𝐶1𝜀 𝜀 𝑘 𝐺𝑘 − 𝐶2𝜀𝜌 𝜀2 𝑘 − 𝑅𝜀 (2) in the above equations, αk and αε are the inversed effective prandtl numbers for k and ε, respectively. c2ε and c2ε are constant values of 1.42 and 1.68, respectively. the scale elimination procedure in the rng theory results in a differential equation for turbulent viscosity. in the high range of the reynolds number, effective viscosity µeff is expressed as: 𝜇𝑒𝑓𝑓 = 𝜌𝐶𝜇 𝑘2 𝜀 (3) with cµ=0.0845 derived using the rng theory. it is interesting to note that this value of cµ is very close to the empirically determined value of 0.09 used in the standard k-ε model. the two-phase flow was modeled using the volume-of-fluid method [19]. the model type was chosen due to the possibility of creating a free surface between air and oil. this is an important feature considering the oil reservoir included in the simulation. the free surface will make it possible to determine the oil level in the tank and its influence on the separator operation. the continuity equation is solved: 𝜕 𝜕𝑡 (𝜌) + 𝛻 ⋅ (𝜌�⃗�) = 0 (4) where ρ is the mixture density calculated from: 𝜌 = 𝛼𝑎𝜌𝑎 + (1 − 𝛼𝑎)𝜌𝑜 (5) where ρa, ρo are densities of air and oil, respectively, and αa is the volume fraction of air which is, in this case, the second phase. 𝜕 𝜕𝑡 (𝛼𝑎𝜌𝑎) + 𝛻 ∙ (𝛼𝑎𝜌𝑎�⃗�𝑎) = �̇�𝑜𝑎 − �̇�𝑎𝑜 (6) where �̇�𝑜𝑎, �̇�𝑎𝑜 are mass transfer from oil to air and air to oil, respectively. the volume fraction equation is not solved for the oil phase (primary), and the oil-phase volume fraction is computed based on the following constraint: 𝛼𝑎+𝛼𝑜 = 1 (7) experimental and numerical study of the impact of the oil tank filling level on the aircraft... 21 for this case, the volume fraction equation is solved through an implicit formula. a single momentum equation is solved throughout the domain, and the resulting velocity field is shared among the phases. the momentum equation, shown below, is dependent on the volume fractions of all phases through quantities ρ and µ. 𝜕 𝜕𝑡 (𝜌�⃗�) + 𝛻 ∙ (𝜌�⃗��⃗�) = −𝛻𝑝 + 𝛻 ∙ [𝜇𝑒𝑓𝑓(𝛻�⃗� + 𝛻�⃗�𝑇)] + 𝜌�⃗� + �⃗� (8) 2.2. numerical model the geometry of the calculation domain was created based on the geometry of an existing test bench, where the separator was installed in a cylindrical tank shown in fig. 1.the calculation domain was simplified to improve the mesh quality and to decrease the number of elements. rig connections like bolts and fittings were removed or simplified, and the fillet radius of the edge of the components was eliminated. the temperature of the mixture is constant; there is no heat exchange with the surroundings. considering together the calculations performed with structure maps and the close distance from the scavenge pump, it was possible to assume a uniform phase distribution. at the inlet, the mass flow rates of both air and oil were applied as boundary conditions. these values were set on the rig. the boundary condition at the separator outlet is defined by the scavenge pump characteristic. it is also essential that the outlet boundary conditions are determined correctly. the selected conditions must allow the occurrence of phenomena such as the liquid carry-over and the gas carry-under, where the outflow of air and oil can occur with different ratios. the problem was resolved by defining static pressures at the outlet boundaries since these values were monitored. the oil properties in the simulation were selected for a specific temperature. the test bench parametrized dimensions are listed in table 1. fig. 1 separator with tested oil tank 22 t. szwarc, w. wróblewski, t. borzęcki table 1 cyclone dimensions in reference to inlet tube height (a) cyclone height (h) vortex finder height (h) cyclone diameter (d) vortex finder diameter (d) inlet tube height (a) inlet tube width (b) tank height (w) 3.2 1.4 3.6 1.3 1 1.6 12 the numerical schemes for the current analysis were selected based on the analysis performed in [20, 22], but this analysis was conducted using a transient approach. a coupled scheme was selected as the algorithms for the pressure-velocity coupling. a “noslip” wall boundary condition is assumed. the time step used in the simulations was related to the global courant number. at the beginning of the simulation when the tank was being filled with oil, it was important to keep the courant number low at ~8, which resulted in the time step of 10e-5 s. after a certain number of iterations when the tank level has been stabilized, the simulation was continued with an increased courant number, which resulted in the time step of 10e-3 s. 2.3. numerical grid study a grid independence test was carried out to discover the optimum grid size for the present study. the mesh cross-section of the numerical model and the locations of crosssections of the velocity profiles are shown in fig 2. fig. 2 mesh at the cross-sections of the separator (center and bottom exit) the separator geometry was discretized using the ansys icem 19.2 package and the present analysis was performed using a hexa-type mesh. the mesh pattern was prepared as vertical and coincident with the flow direction. the grid independence study considered four meshes with 238k, 613k, 1113k and 1714k elements. the mesh density was increased uniformly in the whole domain of the separator. the quality of all 3d computational meshes was above 0.3. the value of y-plus for all the meshes was about 30. four parameters were considered for the mesh study: the separator pressure drop, the oil quality, experimental and numerical study of the impact of the oil tank filling level on the aircraft... 23 efficiency and the oil tank filling level. in addition, the axial velocity and the circumferential velocity for cross-section a were compared. the 238k elements mesh showed an about 49% lower oil tank filling level compared to the finest mesh. the mesh with 613k elements showed a 10% difference in pressure drop compared to 1113k elements and higher. it was found that there was no significant change in the separator efficiency and beyond the grid size of 613k, the oil quality was not changed for more than 1%. fig. 3 mesh impact on the analyzed parameters (reference value for 1714k mesh) fig. 4 velocity profiles: a) tangential velocity over separator inlet velocity (cross-section a); b) axial velocity over separator inlet velocity (cross-section a) mesh with 1113k elements showed comparable values of all four parameters compared to mesh 1714k (fig. 3). the velocity results showed that the difference between the 1113k grid and the finest grid for peak velocities near the wall was below 0.3% (fig. 4a-b). 24 t. szwarc, w. wróblewski, t. borzęcki however, a very fine mesh requires significantly larger computational resources and time while there is no experimental data that can be used for validation. considering the difference in parameters, the mesh with 613k elements was selected for the numerical study. since the main objective of this study is to investigate the effects of the gas-liquid free interface on the flow field and performance, the selected mesh guarantees acceptable accuracy of the results in a reasonable computing time. 2.4. test rig arrangement the test bench created for the evaluation of the static deaerator efficiency is shown in fig. 6. the oil used in the test complied with the mil-l-23699 specification. the air-oil ratio can be set as engine conditions [6]. the oil temperature is limited to the engine normal operating conditions (no overheating). altitude conditions cannot be simulated. the separator is located in a dummy tank with a system measuring the oil level inside the tank. the tested object is in its original dimensions with the possibility of being reused in the aircraft engine. the oil flow is managed by the rig gear pump (2) with its rotational speed adjustable by a frequency inverter. airflow is provided by an external system of the test facility. the air mass flow rate and the oil volumetric flow rate are measured. fig. 5 diagram of the test bench in the experiment, the volumetric air/oil ratio at the inlet was equal to 0.4. the oil-air mixture is generated by the mixing tee (7). separated oil returns to the tank (1), which is several times bigger than the master tank (9) to ensure degassing of the oil delivered for separation. the oil level in the reservoir (11) and the time of the level rising are monitored. due to that, the oil loss flow rate can be established. the oil tank filling level (otfl) is defined by: experimental and numerical study of the impact of the oil tank filling level on the aircraft... 25 𝑂𝑇𝐹𝐿 = 𝑊𝑜𝑖𝑙 𝑊𝑡𝑜𝑡𝑎𝑙 (8) where woil is the mass of oil in the calculation domain and wtotal is the total mass possible accumulated in the domain. the performance of the separation process, which is the object of this calculation, is described by two coefficients (cf. eq. (10-11)). oil separation efficiency ηs is estimated by eq. (10) determined by oil trapping in the oil collection bottle (�̇�𝑜,𝑣𝑒𝑛𝑡). 𝜂𝒔 = �̇�𝑜,𝑖𝑛𝑙𝑒𝑡−�̇�𝑜,𝑣𝑒𝑛𝑡 �̇�𝑜,𝑖𝑛𝑙𝑒𝑡 (10) the oil quality coefficient (oq) is defined as the ratio of the oil volumetric flow rate at the separator inlet to the sum of the volumetric flow rate of air at the bottom outlet and the volumetric flow rate of oil at the separator inlet. 𝑂𝑄 = ( �̇�𝑜 �̇�𝑎+�̇�𝑜 ) 𝑜𝑖𝑙 𝑜𝑢𝑡𝑙𝑒𝑡 (11) the test was performed for the selected engine operating conditions. first, the flow of oil was initiated to obtain the right temperature. then, the oil level in the tank was determined. the next step involved providing air to the t-pipe. the tank was refilled with to maintain constant level of the separated oil. 3. results and discussion 3.1. determination of the separator performance during the experimental tests, both the quality of oil and the separator performance were determined depending on the oil level in the tank. while testing, it turned out that a low level of oil (below 50%) caused uncovering the tank oil outlet and drawing in large volumes of air, which led to a significant decrease of the oil quality. fig. 6 separator performance: a) oil quality results vs. tank filling level; b) efficiency vs. tank filling level 26 t. szwarc, w. wróblewski, t. borzęcki due to the above, the analysis was performed for the oil level range between 50% and 100%. in fig.6a, it was observed that the greater the tank filling level, the higher the oil quality. the experimental data showed 10% decrease in the oil quality at the mid-point of the studied range. according to the numeric model calculations, the decrease was ~8% for the entire range. the difference between the experimental results and the computed values was only 1%, in the range with higher filling levels. for the lower tank filling level this difference was greater and was about -5% compared to the model values. the results obtained comply with the model presented in [5], because the greater filling level increases the retention time of oil particles in the tank. during normal operation, air bubbles have more time to separate in the bottom part of the tank. oil sloshing in the tank also affects the oil quality. at a lower oil level, this phenomenon negatively affects the air separation in the tank because the stream of oil which leaves the separator splashes on the bottom, which may cause undesired repeated aerating. the subsequent experiment results showed decreased separator performance as the oil level in the tank grew. the separator performance drop reached ~0.57% in the experiment and was higher than the performance drop determined by the numerical model (~0.24%). at the lower oil level, the values obtained were similar to the model assumptions; however, above the level of 0.9 the deviation from the model value was about ~0.4%. there was also growing dispersion of points when the filling level was high. the numerical model provides similar performance values within the filling range of 0.7-0.8 (fig. 6b). 3.2. characteristic of free surface and oil fraction the performed simulations for different filling levels revealed that a free surface and a layered structure of oil fractions were formed in the tank. it was observed that the lowest oil surface was localized in its axis and the highest was by the tank walls – it was caused by the liquid rotary motion (fig. 7). the author’s observations are consistent with the separator tests performed in the petrochemical industry (fig. 7). the performance drop as a consequence of the increased volume of oil flowing out through the vortex fig. 7 oil volume fraction iso surface experimental and numerical study of the impact of the oil tank filling level on the aircraft... 27 finder (eq. 10) was caused by the approach of the free surface to the vortex finder and the decreased height of the inner vortex. fig. 8 oil volume fraction contours for different oil tank filling levels further analyses focused on the time-average oil fraction contours for 3 different tank filling levels. the oil film pattern inside the cyclone had an asymmetric profile. the oil phase concentrated in the same regions and layers of oil fractions were created in the same way. it was shown that the high oil level otfl 0.97 led to a greater number of oil fractions (0.2-0.6) in the internal parts of the separator (fig. 8a). the increased amount of oil fraction in the separator inner part decreased the performance because part of the oil would be drawn through the outlet. oil flow through the separator vent was non-stationary. for otfl 0.71, there was oil fractions outflow (fig. 8b), but it was visible for otfl 0.59 (fig. 8c). the oil which flows back to the separator disturbed the formation of the oil film on the separator walls only for otfl 0.97 (fig. 8a), whereas for other levels the oil fraction contours were similar and did not show distortions (fig. 8b-c). further observations of the oil fraction contours revealed three concentration sites – the top lefthand corner and the top and bottom right-hand corner. in the top left-hand corners, the concentration of oil results from its accumulation due to tangential velocity and gas flow over the inlet, and it causes oil to rise. this phenomenon is not desired due to mixture recirculation. another concentration site is in the top right-hand corners where the oil fraction layer is the thickest, but air separation is low. in all cases, the highest oil concentration was also observed in the bottom right-hand corner. it results from the separation of oil due to the rotary motion and formation of the spiral flow down the outlet (fig. 8a-c). 3.3. velocity fields and profiles in the next stage, the observation was focused on the tangential and axial velocity field. it was shown that the flow inside the separator was highly dependent on the tangential inlet geometry and the highest tangential velocity was recorded close to the inlet region. it was confirmed in the tests performed by erdal et al. [4]. on the left side of the separator, in turn, the velocity field was uniform at its whole height (fig. 9a-c). 28 t. szwarc, w. wróblewski, t. borzęcki fig. 9 tangential velocity contours for different oil tank filling levels the tangential velocity values in a and b sections were then analyzed. the velocity profile was divided into 4 ranges. the first was close to the wall where the velocity was almost zero due to the wall skin friction. the second was where the forced vortex occurs, in which the velocity grew significantly and reached the maximum. in the third range, the tangential velocity decreased linearly to the value of about 0.5. the fourth range was close to the vortex finder and the velocity dropped to 0. fig. 10 tangential velocity at sections a and b the highest velocity was recorded in the inlet region in the a section, and it decreased near the separator vortex finder (b section). velocity on the left side of the separator section was similar in all cases. the oil level in the tank had no significant impact on the velocity in the second range. the difference in velocity in the last two ranges (-0.8÷0.8 r/r) was noticeable but its value was insignificant (about ~0.25). it should be noted that the tangential velocity profile experimental and numerical study of the impact of the oil tank filling level on the aircraft... 29 presented in fig. 10a-b is nearly symmetrical about the center of the separator is the same as presented in [4]. fig. 11 axial velocity contours for different oil tank filling levels another investigated parameter was the axial speed. differences between the contours, connected with the unstable flow within the separator were observed. it was noted that the highest velocity was where the oil phase concentrated on the wall in the bottom right-hand corners. the gas flow velocity in the core was three times lower than by the walls. there was also an asymmetrical flow in the vortex finder, caused by the non-axial location of the top outlet against the bottom outlet (fig. 11). fig. 12 axial velocity for sections a and b 30 t. szwarc, w. wróblewski, t. borzęcki next, axial velocity profiles for sections a and b were evaluated. the velocity of the mixture by the inner walls was turned downward towards the separator outlet, and as it deviated from the wall towards the separator axis, it turned upwards to the vent line. the values calculated indicated a strong flow along the cyclone walls. it is of key significance for the separation process because this mechanism plays a dominant role in removing the separated oil-air mixture. within the second range, the flow turned into the opposite direction and, as a result, there was z place where the axial velocity was 0 – the so-called locus zero. in the studied geometry the locus zero is found at values r/r of ~-0.8 and ~0.7 (fig. 12a-b). 4. conclusion the publication presents a cfd analysis of the air-oil separator used in a turbine aircraft engine. the simulation was performed for one set of boundary conditions. the vof model and the rng k-ε turbulence model used allow simulating the complex interfacial surface shape in the separator. it refers to both the free surface of the oil in the tank and the surface of the oil structures in the separators. the proposed hexagonal grid was effective in modeling the two-phase flows in the oil separator. the oil quality and the separator performance were calculated numerically for different filling levels and compared with the test results to validate the calculation model. the results obtained show that in the bottom filling range the performance complies with the measurements and there is a difference in the oil quality values, whereas in the top filling range it is the opposite – the oil quality values are similar while separator performance values are different. based on the results obtained, it was concluded that the increase in the oil level above 0.9 may decrease the separator performance by about 0.5% but it helps obtain the oil quality above 95%. in the case presented herein, the optimal filling level which would allow maintaining properly high oil quality and performance is 70-90%. the tangential and axial velocity profiles were examined for the simulated boundary conditions. the tests did not show that the filling level affects the profiles of the average velocity components. due to the lack of experimental data, the profiles obtained were compared to the ones described in glcc literature. the behavior of the velocity profile was consistent with the test results for the separators used in the petrochemical industry. for future tests, it would be important to verify the oil quality experimental results while adopting a different measuring approach. as the next step, a check with other engine conditions is planned. references 1. ippoliti, l., hendrick, p., 2013, influence of the supply circuit on oil pump performance in an aircraft engine lubrication system, proceedings of the asme turbo expo 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analysis of a cylindrical cyclone separator used in aircraft turbine engine, technical science, 23(2), pp. 131-142. 21. yakhot, v., orszag, s.a., 1986, renormalization group analysis of turbulence. i. basic theory, journal of scientific computing, 1(1), pp. 3-51. 22. wang, l.z., gao, x., feng, j.m., peng, x.y., 2015, research on the two-phase flow and separation mechanism in the oil-gas cyclone separator, iop conference series: materials science and engineering, 90, 012075. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 269 280 https://doi.org/10.22190/fume190919003v © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper computed torque control for a spatial disorientation trainer jelena vidaković 1 , vladimir kvrgić 2 , mihailo lazarević 3 , pavle stepanić 1 1lola institute belgrade, serbia 2institute mihajlo pupin belgrade, serbia 3faculty of mechanical engineering, university of belgrade, serbia abstract. a development of a robot control system is a highly complex task due to nonlinear dynamic coupling between the robot links. advanced robot control strategies often entail difficulties in implementation, and prospective benefits of their application need to be analyzed using simulation techniques. computed torque control (ctc) is a feedforward control method used for tracking of robot’s time-varying trajectories in the presence of varying loads. for the implementation of ctc, the inverse dynamics model of the robot manipulator has to be developed. in this paper, the addition of ctc compensator to the feedback controller is considered for a spatial disorientation trainer (sdt). this pilot training system is modeled as a 4dof robot manipulator with revolute joints. for the designed mechanical structure, chosen actuators and considered motion of the sdt, ctc-based control system performance is compared with the traditional speed pi controller using the realistic simulation model. the simulation results, which showed significant improvement in the trajectory tracking for the designed sdt, can be used for the control system design purpose as well as within mechanical design verification. key words: computed torque control, robot, motion control, spatial disorientation 1. introduction the challenge of robot control stems from nonlinear and time-variable coupling effects in the dynamic model. different advanced control strategies based on adaptive control [1], intelligent control [2], soft computing schemes [3, 4], optimization techniques received september 19, 2019 / accepted december 18, 2019 corresponding author: jelena vidaković lola institute, kneza višeslava 70a, belgrade, serbia e-mail: jelena.vidakovic@li.rs 270 j. vidaković, v. kvrgić, m. lazarević, p. stepanić [5], etc. have been used to overcome nonlinearities and uncertainties in robot dynamics. to select a proper control method, different factors have to be taken into account. application for which the robot is designed defines motion (range of velocities, accelerations) and performance requirements. characteristics of the mechanical design, applied actuators and implementation requirements [6] have a great practical value for making a choice of the potential control strategy. robot modeling and control methods can be applied to various multibody systems that are not necessarily flexible in their application. herein, a control strategy for the spatial disorientation trainer (sdt), fig. 1, a flight simulation training device designed to train pilots to avoid and cope with in-flight illusions, is considered. the sdt is modeled as a 4dof robot manipulator with revolute joints [7]. modern combat aircraft are capable of unconventional flight with unusual orientations. spatial disorientation (sd) is one of the major threats to the pilots of combat aircraft [8-10]. according to the most widely used definition, sd refers to: “a failure to sense correctly the position, motion or attitude of the aircraft or of him/her within the fixed coordinate system provided by the surface of the earth and the gravitational vertical” [11]. training within sd simulators of different levels of complexity is considered the most effective countermeasure to spatial disorientation [10]. the sdt considered herein is a robot manipulator specifically designed to examine the pilot's ability to recognize unusual flight orientations, to train the pilot to adapt to them and to persuade the pilot to believe in the aircraft instruments for orientation, and not into his senses [7]. the simplest approach in robot control design is to adopt an lti-model of the process and to consider variable nonlinear robot dynamics as a disturbance. the traditional control method is pid control. this approach can be justified for highly geared manipulators, as the influence of a nonlinear variable dynamics decreases significantly with high gear ratio [12], and also for stiff manipulators realizing slow trajectories, as the stiffer mechanical design enables the adoption of the larger controller gains. in the case of the sdt, direct drive motors are used for three axes, but the device typically does not achieve high values of speed [7]. within a choice of a control strategy for the sdt, the influence of nonlinear coupling effects in the dynamic model on the tracking capability of the considered controllers has to be investigated. the feedforward computed torque control (ctc) method [13] implies the cancelation of nonlinear coupled terms in a robot dynamic model. the use of feedforward control is considered as a solution capable of suppressing the speed error in cases with known disturbances [14]. however, besides the complexity of dynamic modeling for multiple dofs robots [15], the ctc method suffers from drawbacks related to 1) errors due to structured and unstructured uncertainties in a dynamic model; 2) possible difficulties in implementation. the purpose of this study is to compare, using appropriate simulation techniques, the performances of the traditional pi controller and the controller with ctc compensation added to feedback for the case of the sdt. motivation is not only to determine a proper control strategy but also to verify the mechanical structure design of the device. computed torque control for a spatial disorientation trainer 271 2. trajectory planner development in the previous work [7], kinematic and dynamic models of the sdt are derived and implemented into the trajectory planner. for the sdt, the discrete control system is developed with a trajectory planner that calculates joint trajectories in the offline regime. at the path update rate defined by interpolation period δt, reference joint trajectories calculated in trajectory planner are sent to motor controllers. fig. 1 sdt with 4 dof [7, 16] joint trajectories qk, k, k, k=1, 2,..4, fig. 2, that are used as reference values in this study are obtained by the trajectory planner presented in [7]. joint accelerations of the trajectories previously studied in [7], obtained after applying limitations of angular accelerations according to maximum torques that chosen actuators can achieve [7] are used herein, and numerical integration is performed to obtain reference speeds and positions of joints. 272 j. vidaković, v. kvrgić, m. lazarević, p. stepanić fig. 2 reference trajectories of the sdt joints 3. control system design for the sdt in this section, the design of the pi speed controller and the ctc-based controller are presented. the model of the motor’s mechanical subsystem is based on inertia reflected on the rotor’s shaft (effective inertia), obtained from the inverse dynamic (id) model of the sdt. the load torque calculation from the id model is presented. 3.1. model of the motor’s mechanical subsystem from the equation of motion of rigid body rotation about an axis, the nonlinear timevariant model of the motor’s mechanical subsystem in robot’s joint k can be given in the form: eff m m l .k k k ki q    (1) where qmk= qmk (t) is the angular position of the rotor; ieffk= ieffk(q) is effective inertia (resulting from the coupling of motor with inertial load, as seen from the side of the rotor shaft) which is a function of the instantaneous manipulator configuration q=q(t)=(q1(t), q2(t),.., qn(t)); n is the number of degrees of freedom; τmk= τmk(t) is the driving torque generated by the motor; τlk=τlk(t) is the load torque, t is time. when the motor in joint k is coupled with the inertial load using gear train with gear ratio rk, the relation with the angular position of joint k is qmk= rk qk. in eq. (1), the bounded nonlinear friction terms computed torque control for a spatial disorientation trainer 273 are neglected and treated as disturbances [17]. the deterministic part of load torque τlk, τldk, and effective inertia ieffk are calculated from the id model. herein, the robot id model is given in the form of a set of n coupled nonlinear differential equations: 1 1 1 ( ) ( ) ( ) n n n kj j kji j i k k j j i d q h q q g .       q q q (2) each equation (for every joint k) in the presented set of n differential equations contains the torque or force terms classified into four groups: 1) inertialdkk(q) k, 2) reaction terms generated by accelerations of other jointsdkj(q) j, j≠k, 3) reaction-velocity generated (centrifugal and coriolis) terms-hkji(q) j i, 4) force or torque generated at the joint by gravity in the current manipulator configuration-gk(q). in eq. (2), τk is the actuating torque for joint k. the id model of the sdt obtained by the recursive newton–euler method was presented in [7]. the effective inertia for the motor’s mechanical subsystem in joint k, eq. (1) can be calculated for every interpolation period in the form of eq. (3), [13]: 2 eff m( ) ( ( ) / ).k k kk ki i d r q q (3) where imk is the inertia of motor and gearbox, and dkk(q) is calculated from the id model, eq. (2). in this study, the model parameters for motors’ mechanical subsystems are chosen based on motors selected in [7]. it should be noted that the algorithm that calculates the achievable joint trajectories based on maximum torques that motors can achieve presented in [7] is implemented into the trajectory planner. 3.2. decoupling of robot dynamics and its implementation in the simulation models with the decoupling of a robot dynamics, a single joint control that takes into account a dynamic model through the deterministic part of the motor load torque τlk, eq. (1), denoted herein as τldk, can be considered. herein, the method for decoupling of robot dynamics given in [13] is extended to include the case when the motion of other links (actuated by their motors) alleviates the load of the motor in the observed joint. within this method, load torque τldk is calculated for every interpolation period from the id model, eq. (2), rewritten in the following form: coupled coupled 1, 1 1 ( ) , ( ) ( ) ( ). kk k k k n n n k kj j kji j i k j j k j i d q d q h q q g              q q q q (4) torque τcoupledk =τcoupledk (t) in eq. (4) consists of load torque τldk= τldk (t), and torque τalleviatek = τalleviatek (t) produced by the motion of other links actuated by other motors, which contributes to the motion of link k (it acts in the direction of desired angular acceleration k) and reduces the driving torque that the motor in joint k has to generate to 274 j. vidaković, v. kvrgić, m. lazarević, p. stepanić achieve the desired link motion. τldk and τalleviatek are calculated for every interpolation period from the id model, eq. 4, as given in algorithm 1: algorithm 1: if sign ( k)=sign(τcoupledk), then τldk=τcoupledk / rk, τalleviatek=0; else if abs(dkk(q)q k)> abs(τcoupledk), then τldk=0, τalleviatek= -τcoupledk / rk; else τldk= -τk / rk, τalleviatek= (τk -τcoupledk ) / rk; in algorithm 1, abs stands for absolute value. the application of the proposed algorithm in the simulation models used in this study is given below in fig. 3. fig. 3 decoupling of the inverse dynamic model the dynamic saturation is implemented in the following way: if sign ( k(t))>0 then the upper motor saturation limit is increased by value of τalleviatek(t), and if sign ( k(t))<0 then τalleviatek(t) is added to the lower motor saturation limit (τalleviatek(t) and k are always the same in sign). in this way, the motor saturation in the simulation model does not influence τalleviatek, nor does it with the real device. 3.3. design of the pi speed controller for simulation models pi controller is the most commonly used control algorithm in the process control industry [18]. herein, pi controller is selected to obtain a second-order closed-loop system with a characteristic polynomial in a form s2+2ζkωnks+ωnk 2, (ζk is damping factor), for comparison of the natural frequency of closed-loop system ωnk with lowest natural frequency ωr of the mechanical structure. the characteristic polynomial of the closed-loop system is equal to the denominator den(s) of the closed-loop transfer function and for the pi speed controller, it is given in eq. (5): 2 ps eff is effden(s) k / k / ,k k k ks s i i   (5) where kpsk and kisk are proportional and integral gains, respectively. for joint k, k=1, 2..4, the load torque due to the motion of the chain of other interconnected links, τldk, is treated as a disturbance. as said before, ieffk in eq. (5) depends on robot configuration. herein, tuning is performed for the lti-model with the highest load [19], i.e. for the maximum value of effective inertia. given that the structural flexibilities of the system are not modeled, special attention must be paid not to excite structural resonances. a rule of thumb is that the maximal natural frequency of closed-loop system ωnk is at least two times smaller than the lowest natural frequency of mechanical structure ωr (ωnkmax=0.5ωr) [20]. however, if we consider a request that the motion of the robot link is never underdamped computed torque control for a spatial disorientation trainer 275 [12], the value of damping factor ζk=1 for the maximum value of effective inertia ieffkmax, eq. (6), achieves the fastest response without oscillations for all values of ieffk [6].  ps eff isk / 2 ( )kk k k ki  q (6) considering that the sdt is not flexible in application, ieffkmax can be determined beforehand, in the offline regime. here, ieffkmax, for which it applies ζk=1, is obtained from id model simulation using eq. (3), for the motion given in fig. 2. for realistic simulation purposes, to take into account possibilities for a higher value of ieffkmax for different sdt trajectories, choice of pi speed controller gains takes into account the lowest structural natural frequency, with integral gain kisk chosen to be kisk = 0.25ωr 2 ieffkmax. if a basic structure of the pi speed controller is used, overshoots are present for the values ζ>1 [14]. in fig.4 step response for the sdt first axis’ closed-loop system with the lti model process in which ieff1= ieff1max, obtained using the basic structure of pi speed controller with kisk set as 0.25 ωr 2 ieff1max and for the damping factor ζ1=1, is given in red line. the rise time is 0.022, the settling time is 0.16, and the overshoot is 13.53%. in an attempt to achieve smaller overshoot, (in many practical servo control applications, the overshoot for the speed step response is usually limited below 10% of the step level [21]), a different structure of pi controller is used here, fig. 5. proportional gain relocated in the feedback path avoids the overshoot for the values of ζ≥1 due to the closed-loop zero removal, while at the same time keeps the denominator unchanged [14]. the response is now slower and, to obtain a faster response, values of damping factor ζk are chosen to be slightly lesser than 1 for ieffkmax. in fig. 4, step response for the sdt first axis’ closed-loop system, with the applied pi speed controller structure shown in fig. 5 [14] is given in blue line. for the lti model process with ieff1max, with kisk set as 0.25 ωr 2 ieff1max, and for damping factor ζ1=0.95, the rise time is 0.094, the settling time is 0.159, and the overshoot is zero. this type of pi speed controller is adopted for all four axes of the sdt in simulation models. fig. 4 step responses for the sdt’s first axis with pi speed controllers 276 j. vidaković, v. kvrgić, m. lazarević, p. stepanić fig. 5 the pi speed controller with proportional gain in the feedback path [14], mrk is the reference speed for joint k it should be noted that in simulation models that use only pi speed control, load torque τlk=τldk and torque contribution of other links’ motion τalleviatek, eq. (4) and algorithm 1, are simulated, and the dynamic saturation is included as presented in fig. 3. 3.4. feedforward computed torque method in the single joint computed torque control method, the load torque due to the motion of the chain of robot’s interconnected links, τldk= τldk(q), calculated from the id model for every interpolation period, eq. (4) and algorithm 1, is canceled with a feedforward signal. the feedback controller is added to improve the reference-tracking capability (to suppress the errors in dynamic modeling, as well as the effects of stochastic disturbances). for achieving of realistic comparison in a simulation model, to account for modeling errors and stochastic disturbances, the load torque is simulated as τlk= τldk(q)(1+ asinωdkt) where a and ωdk are the amplitude and frequency of the simulated disturbances [6], fig. 6. τldk and τalleviatek, eq. (4) and algorithm 1, are simulated, and the dynamic saturation is included as presented in fig. 3. fig. 6 simulation model for single joint ctc method computed torque control for a spatial disorientation trainer 277 4. simulation results in this section, simulation results for the two single-joint control methods presented in section 3 are given. the performance of the pi speed controller is compared to the same feedback controller with added ctc compensation. from catia software, the lowest structural natural frequency of the sdt is obtained to be ωr =10.5028 hz. for axis 1, an ac motor is chosen with a maximum torque of 203.2 nm and a gearbox ratio 67.2, moment inertia of the motor is im1=1291.10-4 kgm2 [7, 22]. axes 2, 3 and 4 are actuated by torque motors. the motor for axis 2 achieves a maximum torque of 3950 nm and has a moment of inertia im2=173.10-2 kgm2. the motors for axes 3 and 4 achieve maximum torques of 2150 nm and have moments of inertia im3,4=53.1.10-2 kgm2 [7, 23]. the simulated gondola payload is 180 kg [7]. simulink models are designed for all 4 axes of the sdt for processes with maximum loads (maximum effective inertias). the reference speeds are simulated as a series of discrete values obtained from the trajectory planner, fig. 2. in the models with pi speed feedback only, fig. 5, load torque τldk and torque contribution of other links’ motion τalleviatek, eq. (4) and algorithm 1, fig. 3, are simulated for every interpolation period δt=5 ms. in models with pi speed feedback plus ctc compensators, load torque τldk is compensated in every interpolation period, while load torque τlk is simulated as τlk= τldk(q)(1+ asinωdkt), a is chosen to be 0.05 (meaning that the load torque estimation error is about 5 %); τldk and τalleviatek are calculated from eq. (4) and algorithm 1, fig. 6. dynamic saturation presented in section 3.2 is applied at the outputs of controllers for all simulation models. process and controller parameters for pi speed control are given in table 1. variation of the effective inertia in percent is given, and the variation of damping factor ζk for the motion given in fig 2. is presented for the minimum, the median and the maximum value of effective inertia. it should be noted that the oscillation frequency of the closed-loop system time response is smaller than the natural frequency of the closed-loop system 2 o n 1k k k    . for ζk=0.95 for ieffkmax, the oscillation frequency of the closed-loop system time response with the adopted gains is ωok=0.32ωnk=0.16ωr. table 1 process and controller parameters, variation of effective inertia and variation of damping factor ζk for the motion given in fig. 2 joint ieffkmax [kgm2] gain variation of eff. inertia [%] ζk kpsk kisk ieffkmax ieffkmed ieffkmin 1 4.26 267.1 4.64.103 39.95 0.95 1.06 1.23 2 98.01 6. 34.103 1.07.105 31.82 0.98 1.07 1.18 3 796.3 5.2.104 8.67.105 7.29 0.99 1.009 1.03 4 250.28 1.57.104 2.73.105 25.4 0.95 1.017 1.099 in fig. 7, trajectory tracking for axes k=1, 2.. 4 with two considered types of controllers are presented. the reference value (a series of discrete values obtained from the trajectory planner, fig. 2) is given in blue color, the outputs obtained by the pi speed controller are given in red, while the outputs obtained by the pi speed controller with added ctc are given in green color. the errors ek= k rk, k=1, 2.. 4 in obtained speeds are given in fig. 8. 278 j. vidaković, v. kvrgić, m. lazarević, p. stepanić fig. 7 the obtained trajectory tracking results: reference value is in the blue line, tracking obtained by the pi speed controller is in the red line, tracking obtained by ctc compensation added to the pi speed controller is in the green line as can be seen, there is an improvement in trajectory tracking when ctc compensation is added to pi speed feedback for the desired sdt motion. the constant reference speed is simulated in segments for the joints 2 and 4 (denoted in fig. 8), and it can be seen that the small steady-state error is present with the pi speed control as a result of varying disturbance and limited controller gains. the ctc addition achieves zero error in steadystate. the short-lasting high values of error in pi speed control for joint 2 at certain time instants are caused by sudden and large changes in load torque τld2 in those instants. the gains for pi speed controllers are selected for the maximum values of effective inertia ieffk, and control system performance is expected to deteriorate to a certain extent for smaller values of ieffk. to examine the performance decay, the simulation models with the same pi controllers (designed for ieffkmax) and processes with the minimum value of ieffk are developed. the performance deterioration is not significant except for joint 2. for example, the maximum error for joint 2 with the pi speed controller for the process with ieff2max (circled in fig 8.) is 1.08 rad/s, and for the process with ieff2min this error is 1.204 rad/s. when ctc is added, the difference in these errors (for ieff2max and ieff2min) is insignificant. simulation results showed that for the designed sdt the influence of the nonlinear dynamic model on the control system performance is not negligible for relatively small values of joint speeds. with the reduction of inertia/mass of the mechanical structure, the significance of improvements in trajectory tracking achieved by the addition of ctc compensation may be higher (due to gains limitation of general pid controller to avoid unwanted resonant effects). considering that weight reduction is performed to reduce the computed torque control for a spatial disorientation trainer 279 motors’ size (power) (to achieve the design and usage cost reduction), this possibility should be tested by control system performance simulation for the chosen smaller motors and selected pid controller structure with the adopted tuning method. the benefits of the achieved improvements should be weighted with the complexity of practical implementation. 5. conclusion in this paper, the application of the computed torque method for the motion control of the spatial disorientation trainer is investigated using realistic simulation. the sdt device is modeled as a 4dof robot manipulator with revolute joints. models for the motors’ mechanical subsystems used in simulation examples account for robot dynamic model through inertia reflected on the rotor shaft. gains of the applied pi speed controllers are limited taking into account the lowest natural frequency of the mechanical structure obtained from cae software. the dynamic saturation based on the maximum torques for the selected actuators is applied at the outputs of controllers. within ctc compensation, the reasonable error in load torque calculation from the dynamic model is assumed. the addition of ctc compensator to the pi speed feedback controller achieved considerable improvement in trajectory tracking in simulation example. the simulation results are significant regarding the choice of the control method for the sdt, and also in reference to the design of the mechanical structure of the manipulator and the appropriate choice of motors. fig. 8 errors in speed: obtained by the pi speed controller in the red line, obtained by ctc compensation added to the pi speed controller in the green line 280 j. vidaković, v. kvrgić, m. lazarević, p. stepanić acknowledgements: this research has been supported by research grants of the serbian ministry of education, science and technological development under contract numbers 451-03-68/202014/200066, 451-03-68/2020-14/200105 and 451-03-68/2020-14/200034. references 1. slotine, j-je., weiping, l., 1988, adaptive manipulator control: a case study, ieee transactions on automatic control, 33(11), 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tuning, (2nd ed.), isa, research triangle park nc, 52 p. 20. paul, r.p., 1981, robot manipulators: mathematics, programming, and control: the computer control of robot manipulators, mit press, cambridge ma, 200 p. 21. dhaouadi, r., kubo, k., tobise, m., 1993, two-degree-of-freedom robust speed controller for highperformance rolling mill drives, ieee transactions on industry applications, 29(5), pp. 919-926. 22. siemens configuration manual, (pft6), edition 12, 2004, 6sn1197-0ad12-0bp0. 23. siemens configuration manual, 2009, 05/2009, 6sn1197-0ae00-0bp3. 12767 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 601 614 https://doi.org/10.22190/fume240601038c © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper mechanically pressed polymer-matrix composites with 3d structured filler networks for electromagnetic interference shielding application chunghyun choi1, nadeem qaiser2, byungil hwang3 1department of intelligent semiconductor engineering, chung-ang university, south korea 2material science and engineering, physical science and engineering division, king abdullah university of science and technology (kaust), saudi arabia 3school of integrative engineering, chung-ang university, south korea orcid ids: chunghyun choi https://orcid.org/0009-0001-7634-9933 nadeem qaiser https://orcid.org/0000-0001-7417-9857 byungil hwang https://orcid.org/0000-0001-9270-9014 abstract. the increasing demand for miniaturization, high-power density, and highfrequency electronic devices highlights the importance of polymer composites with high electromagnetic interference (emi) shielding. these composites are crucial to maintaining devices, reducing communication errors, and protecting human health. in this study, we developed a mechanically pressed composite of polystyrene, mxene, and boron nitride nanosheets (bnnss) with a 3d structured filler network through electrostatic interactions and the hot-pressing technique. constructing a 3d filler network within the composite led to a significant emi shielding effect, particularly in the low-frequency range. furthermore, it was observed that the bnnss-coated sample contributed to superior emi shielding effectiveness compared to the non-coated sample. this suggests that bnnss improve the emi shielding effectiveness by providing additional interfaces within the composite and help prevent the degradation of mxene. we expect our study to provide valuable insights into the development of 3d structured filler networks within composites while contributing to improvements in thermal conductivity and emi shielding performance. key words: polymer-matrix composites, 3d structured hybrid fillers network, electromagnetic interference shielding, electrical insulation, 2d materials, mxene, boron nitride nanosheets. received: june 01, 2024 / accepted july 27, 2024 corresponding author: byungil hwang school of integrative engineering, chung-ang university, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr https://orcid.org/0009-0001-7634-9933 https://orcid.org/0000-0001-7417-9857 https://orcid.org/0000-0001-9270-9014 602 c. choi, n. qaiser, b. hwang 1. introduction owing to the development of 5g wireless communication, coupled with high-frequency and high-power density devices, advanced electronic devices induce electromagnetic waves, which can adversely impact human health and the environment [1-6]. furthermore, due to the interference of the generated electromagnetic waves, electromagnetic interference (emi) influences the stability and accuracy of devices. this poses a significant issue, especially in areas where devices are densely packed or in high-precision equipment and sensitive electronic systems. for example, malfunctions due to emi in medical equipment, aircraft, and military systems can lead to severe consequences [7-10]. polymers are widely used as matrices to fabricate functional composites due to their advantages such as their lightweight nature, low cost, and processability. however, pure polymers have electrically insulating properties. theoretically, to prepare composites with outstanding emi shielding effectiveness (emi se), high electrically conductive materials are required. therefore, fillers are introduced into polymer matrices to attain high electrical conductivity. selecting an appropriate filler is a crucial factor in optimizing the emi se of composites. in this study, we used two types of 2d materials as fillers: ti3c2tx mxene and boron nitride nanosheets (bnnss). mxene, an emerging 2d material for emi shielding, is the general name of transition metal carbides/carbonitrides and has the general formula of mn+1xntx, where m is an early transition metal (e.g., ti, zr, v, nb, ta, and mo) and x is c or n. tx represents functional groups such as -o, -f, or -oh [11]. in the mxene family, the most studied ti3c2tx has a high specific surface area, high electrical conductivity, and abundant functional groups [12-15]. these groups make mxene polar, making it hydrophilic and allowing it to disperse in water without a dispersant, thus making it easy to react with other materials. despite these excellent properties of mxene, it is known to oxidize easily, which is a significant obstacle for practical use. this deteriorates mxene’s performance over time, dramatically reducing the emi shielding effect. bnnss, another type of 2d materials, have gained prominence as thermally conductive fillers due to their high thermal conductivity, electrically insulation properties, high thermal stability, and chemical resistance [16, 17]. notably, bnnss possess the quality of being oxidation-resistant and inert to most chemical reactions, which can delay the oxidation of mxene [18, 19]. therefore, we expected the synergistic effects between mxene and bnnss to enhance emi se and oxidation stability. to realize durable and outstanding emi se, we proposed polystyrene (ps)@mxene@bnnss microspheres and then fabricated the composite with a 3d structured filler network using the hot-pressing method. to prepare ps@mxene@bnnss microspheres like the core-double-shell structure, we introduced electrostatic force by fabricating positivecharged ps, negative-charged mxene via various functional groups, and positive-charged bnnss by surface modification. after constructing the 3d filler networks through hot pressing, the composite with the 3d continuous networks exhibited excellent emi se. 2. materials and methods 2.1. synthesis of polystyrene (ps) ps was synthesized through the dispersion polymerization reaction of styrene monomer (duksan). dispersion polymerization is a type of precipitation polymerization where the mechanically pressed polymer-matrix composites with 3d structured filler networks... 603 monomer and initiator are entirely dispersed [20, 21]. however, as polymerization progresses and the molecular weight of the polymer exceeds a certain level, it does not disperse and precipitate. first, the styrene monomer was reacted with activated alumina (daejung chemical) for activation. to prepare ps with a positive surface charge, 4 ml of purified styrene monomer, 215.9 mg of polyvinyl pyrrolidone (daejung), 18.9 mg of 2,2-azobisisobutyronitrile (99%, daejung), and 0.1 ml of [2-(methacryloyloxy)ethyl]trimethylammonium chloride solution (dmc, 75 wt.%, sigma-aldrich) were dispersed in a three-necked flask containing 40 ml of ethanol and 4 ml of deionized (di) water. dmc provided a positive functional group on the ps surface, making the surface charge of the entire molecule positive. this solution was reacted at 70 degrees for 30 min while purging with nitrogen and then stirred at 500 rpm for 18 h while continuously supplying nitrogen. the resulting solution was washed thrice with ethanol using a centrifuge. finally, it was dried under vacuum at 50 degrees for 24 h to obtain ps with a positive charge. 2.2. synthesis of ti3c2tx mxene nanosheets ti3c2tx mxene can be obtained by selectively etching the al layers of titanium aluminum carbide (ti3alc2 max, >90%, <40 µm particle size, sigma-aldrich). for etching, acidic solutions of fluorides, such as hydrofluoric acid [22], ammonium hydrogen bifluoride [23], or a solution mixed with hydrochloric acid (hcl) and lithium fluoride (lif) [24], are used. among these, the hcl and lif method provides a high yield, is safe, and can be synthesized quickly and easily. 4.8 g of lif (fujifilm) and 60 ml of 9 m hcl were added into a teflon container and stirred until lif was completely dissolved. the temperature of the solution was raised to 35 degrees, and 3 g of ti3alc2 was slowly added. then, the mixture was reacted at 350 rpm for 24 h. the reacted solution was repeatedly centrifuged at 8000 rpm for 10 min using di water until the ph was about 6. when the ph of the supernatant reached about 6, unreacted substances and impurities were removed, and single-layer or few-layer ti3c2tx mxene was obtained by centrifuging at 1000 g for 20 min. the supernatant that did not settle was filtered using a vacuum-assisted filtration device with cellulose ester membrane filter (mce, pore size: 20 µm, hyundai micro) and dried at room temperature under vacuum for 36 h. the dried mxene was mixed with di water to attain a concentration of 1 mg/ml. the mixed solution was purged with ar gas (99.99%of purity) for over 30 min. subsequently, while continuously supplying ar, it was put into an ultra-sonicator filled with ice and ultrasonically treated. consequently, the al layers were removed, and each layer of the layered structure of mxene was exfoliated. mxene nanosheets with a negative surface charge were thus obtained. 2.3. synthesis of boron nitride nanosheets (bnnss) 1.5 g of bulk hexagonal-boron nitride (h-bn, 98%, sigma-aldrich) was put into a solution mixed with 150 ml of di water and 150 ml of isopropanol (ipa, 99.8%, samchun pure chemicals) and ultrasonically treated for 12 h. h-bn exists in a hexagonal columnar layered structure. after ultrasonication, each layer was exfoliated to obtain thin layers of bnnss. after ultrasonic treatment, it was centrifuged at 2000 rpm for 20 min, and the supernatant that did not settle was vacuum-assisted filtered using an mce filter to obtain sufficiently exfoliated bnnss. the filtered bnnss were dried at a temperature of 100 degrees for 10 h. the surface of the obtained bnnss had a -oh functional group, which was formed when each layer was separated, thus lending it a negative surface charge. to 604 c. choi, n. qaiser, b. hwang change this to a positive charge, it was reacted with (3-aminopropyl)triethoxysilane (aptes, 99 %, sigma-aldrich). aptes reacted with the -oh on the surface of bnnss, and the -nh2 functional group at the end changed the surface charge to positive [25]. 0.015 g of aptes was put into 25 ml of ethanol and reacted at 60 degrees for 30 min. then, 0.5 g of bnnss was added and stirred at 80 degrees for 6 h. after the reaction, it was washed three times with ethanol to remove impurities. finally, it was dried at 100 degrees for 12 h to obtain bnnss-aptes with a positive surface charge. 2.4. fabrication of polystyrene@mxene@boron nitride composite ps has a positive charge and mxene has a negative charge; therefore, when the two materials react, they attract each other due to electrostatic forces. moreover, ps has a spherical 3d structure and mxene has a planar 2d structure, so mxene can stick to ps like a shell, resulting in a core-shell structure of ps@mxene. if a sufficient amount of mxene reacts with ps, mxene covers the surface of ps, giving it a negative surface charge. when these ps@mxene particles with a negative charge react with bnnss-aptes with a positive charge, the same electrostatic forces work to obtain ps@mxene@bnnss microspheres with a double-core-shell structure. we prepared a solution of ps dispersed in di water at 3 mg/ml and a solution of bnnss-aptes dispersed in di water at 5 mg/ml. the previously prepared mxene solution was slowly added to the ps solution at 33.3 ml with varying amounts (3, 5, 7, 10, 13, 15 wt%) and then stirred at 500 rpm for 2 h. at this stage, ps@mxene microspheres were formed, and various mxene concentrations were tested to determine the sufficient amount that could react with bnnss-aptes. subsequently, an amount of bnnss-aptes solution, equal to the amount of the mxene solution, was slowly added and reacted at 500 rpm for 2 h. after the reaction, the particles and di water were separated using vacuum filtration with an mce filter. when dried at 50 degrees for 12 h, ps@mxene@bnnss microspheres were obtained. a composite with a 3d structure of filler network was obtained by adding 0.1 g of ps@mxene@bnnss microspheres into a mold for hot pressing and pressing at a temperature of 130 degrees with a pressure of 10 mpa for 30 min. 2.5. characterization the zeta potential was measured with a particle size and zeta potential analyzer (delsa nano c particle analyzer, beckman coulter) to determine the change in charge on the material surface. x-ray photoelectron spectroscopy (xps, k-alpha+, thermofisher scientific) was used to confirm whether each material was synthesized, and surface modification was performed correctly. a particle size analyzer (psa; s3500, microtrac) was used to confirm whether the polymerization of ps occurred uniformly. field-emission scanning electron microscopy (fe-sem; sigma 300, carl zeiss) was used to analyze the surface morphologies of all the materials. additionally, x-ray diffraction (xrd; new d8advance, bruker-axs) was utilized to analyze the mxene and confirm whether the al layer in the ti3alc2 max phase was completely etched. fourier-transform infrared spectrometer (ft-ir; nicolet 6700, thermo scientific) was used to determine whether aptes worked adequately on the surface of bnnss, and the emi se was measured using a vector network analyzer (vna; 8720d, hewlett-packard, palo alto, usa). mechanically pressed polymer-matrix composites with 3d structured filler networks... 605 3. results and discussion fig. 1 illustrates the fabrication process of the ps@mxene@bnnss composite. dispersion polymerization was used to synthesize ps, incorporating an initiator, monomer, and the positive agent dmc to create ps with a positive surface charge. mxene initially exists in the max phase, where mx layers alternate with a layers. by etching the al layers with a lif/hcl solution, layered mxene was obtained. delaminating these layers produced mxene flakes. bn typically exists in a bulk layered structure, but bnnss was produced by exfoliating bn with di water and ipa. these bnnss contain -oh groups, which reacted with aptes to yield silanized bnnss-aptes. using self-assembly driven by surface charges, the materials were sequentially assembled as mxene and bnnss-aptes onto the ps microspheres, forming ps@mxene@bnnss microspheres. finally, hot pressing these microspheres resulted in the ps@mxene@bnnss composite. fig. 1 schematic illustration of the fabrication of ps@mxene@bnnss 606 c. choi, n. qaiser, b. hwang during synthesizing ps, dmc was added to change the ps surface so that it could be positively charged. the functional group -n(ch3)3 + in dmc imparted a positive surface charge, which was confirmed through xps analysis and zeta potential measurements. fig. 2(a) compares the zeta potential values of the normal ps without dmc and the charged ps with dmc. the zeta potential represents the potential at the slipping plane of a particle [26]. when charged particles are dispersed in a solution, counterions in the solution adsorb onto the particle to neutralize the particle’s charge, forming an electrical double layer. this double layer comprises a stern layer and a diffuse charge layer. the slipping plane is defined as the outer boundary where the particles move, and the potential at this plane is known as the zeta potential, indicating the electrical properties of the interface. normal ps showed a negative zeta potential (-22.94 mv), but with the addition of dmc, the charged ps exhibited a positive zeta potential (26.5 mv), confirming the effective functionalization of ps with dmc. fig. 2(b) displays the xps results for the n 1s region of the charged ps. the peak intensity of around 403 ev corresponds to the -n(ch3)3 + functional group, which is the terminal part of dmc, confirming the presence of dmc on the ps surface. psa and fe-sem analyses were performed to verify the uniform particle size of the synthesized ps via dispersion polymerization. fig. 2(c) shows the particle size distribution of the charged ps measured by psa, indicating that most particles were around 0.7 µm in size, with a d50 value of 0.741 µm. fig. 2(d) provides an fe-sem image, revealing uniformly sized spherical ps particles with smooth surfaces. fig. 2 characterization of the positively charged polystyrene particles: (a) zeta potential measurements, (b) xps analysis, (c) particle size distribution profile, and (d) fesem image mechanically pressed polymer-matrix composites with 3d structured filler networks... 607 when the max phase transformed into the mxene phase, the existing al layers were removed, and various functional groups formed, which altered the surface characteristics. these functional groups, such as -oh and -f, carried negative charges, resulting in the etched mxene having a negatively charged surface. fig. 3(a) shows the zeta potential of mxene, measured at -52.26 mv, indicating a negative surface charge. additionally, the zeta potential provides insights into dispersibility, suggesting that mxene has good dispersion stability in water. fig. 3(b) and 3(c) demonstrate the successful etching from the max phase to the mxene phase. figure 3(b) presents the max and mxene xps analysis. the disappearance of the al peak after transformation to mxene confirms that al was entirely etched away by the acidic solution. the structural change from max to mxene is evident in the xrd patterns (figure 3(c)). as the transformation progressed, the al layers were removed by lif/hcl, causing the characteristic peaks at (101), (103), (104), and (105) to disappear [27]. the broadening of the (002) peak was also observed, which can be attributed to the increased interlayer distance after al removal. the etched mxene exhibited an accordion-like layered structure, which was delaminated through simple sonication. this delaminated mxene was very thin and flexible, as shown in figure 3(d). fig. 3 characterization of ti3c2tx mxene and ti3alc2 max: (a) zeta potential measurements, (b) xps analysis, (c) xrd analysis, and (d) fe-sem image the layered structure of bn transitioned to bnnss as the interlayer bonds broke during sonication in a mixture of ipa and di water. during this process, -oh functional groups formed on the surface of bnnss, imparting a negative charge. to facilitate interaction with ps@mxene, which also had a negative charge, it was necessary to change the surface 608 c. choi, n. qaiser, b. hwang charge to positive. this was achieved by reacting the -oh groups with aptes. figure 4(a) shows the zeta potential measurements of bnnss and bnnss-aptes. bnnss had a value of -36.7 mv, which changed to +37.1 mv after reacting with aptes, indicating that the nh2 + groups of aptes effectively modified the surface of bnnss to a positive charge. figures 4(b) and 4(c) present the xps and ft-ir results of bnnss and bnnss-aptes, respectively. since bnnss consist of boron and nitrogen, no si peak was observed. the xps results show that si, present in aptes, was detectable in bnnss-aptes. additionally, the ft-ir results reveal a -ch2-related peak, which was absent in bnnss but appeared in bnnss-aptes, further confirming the presence of aptes on the bnnss surface [28, 29]. this demonstrates that aptes effectively reacted with the surface of bnnss, converting the surface charge from negative to positive. the fe-sem image (fig. 4(d)) also shows the well-exfoliated appearance of bnnss-aptes, confirming successful exfoliation. fig. 4 characterization of boron nitride nanosheets (bnnss) and bnnss-aptes: (a) zeta potential measurements, (b) xps analysis, (c) ft-ir analysis, and (d) fe-sem image in this study, the surface charge of materials was adjusted through surface modification for use in the self-assembly method. the adjusted surface charge was confirmed by zeta potential measurements, as shown in fig. 5(a). the zeta potential of ps was measured at +26.5 mv, while mxene exhibited a negative charge of -52.26 mv. bnnss-aptes had a positive charge of +37.1 mv. these surface charges facilitated the subsequent reactions. initially, ps and mxene were reacted together, with sufficient mxene added to ensure a mechanically pressed polymer-matrix composites with 3d structured filler networks... 609 significant negative charge for further reaction with bnnss-aptes. the zeta potential of ps@mxene, depending on the mxene concentration, is depicted in fig. 5(b). the zeta potential gradually decreased with an increase in the mxene content, indicating the effect of mxene on the particle surface. the zeta potential saturated around 10 wt% mxene. ps@mxene containing 10 wt% mxene was then used for further modification. the morphology of ps@mxene with 10 wt% mxene is shown in fig. 5(c). the image reveals that the smooth surface of ps was uniformly covered by thin mxene flakes, resulting in a negative surface charge. after reacting with bnnss-aptes, the morphology is shown in fig. 5(d). bnnss-aptes was added at five times the amount of mxene to ensure complete coverage of ps@mxene. the final microspheres, ps@mxene@bnnss-aptes, show bnnss-aptes uniformly covering the negative-charged ps@mxene particles. these results demonstrate that surface charge modification enabled straightforward coating processes using surface charge-driven reactions. this approach offers a simple and effective method for coating materials through self-assembly. fig. 5 (a) zeta potential of the charged ps, ti3c2tx mxene, and bnnss-aptes; (b) zeta potential of ps@mxene as contents of mxene; (c) fe-sem image of ps@mxene; and (d) fe-sem image of ps@mxene@bnnss-aptes the initial electromagnetic wave undergoes intensity modulation through various mechanisms when it encounters the surface and interior of a shield. this phenomenon arises due to the impedance and refractive index difference between the air and the shield material. as these electromagnetic waves pass through the shield, they are attenuated, leading to a decrease in intensity. the impedance difference on the shield’s front and back 610 c. choi, n. qaiser, b. hwang surfaces induces the reflection of electromagnetic waves. the degree to which the shield blocks and reduces the initial electromagnetic wave is defined as emi se (set), which is expressed in logarithmic scale units and represents the reduction in intensity of the transmitted electromagnetic wave compared to the initial wave. it is defined as follows: 𝑆𝐸𝑇 (𝑑𝐵) = 10 𝑙𝑜𝑔 𝑃𝑇 𝑃0 = 20 𝑙𝑜𝑔 𝐸𝑇 𝐸0 (1) where p0 and e0 are the initial power density and electric field strength, respectively, and pt and et represent those of the transmitted electromagnetic waves. according to the schelkunoff formula[30], the emi se occurs through three mechanisms: absorption loss (sea), reflection loss (ser), and multiple reflection loss (sem) [31]. 𝑆𝐸𝑇 = 𝑆𝐸𝐴 + 𝑆𝐸𝑅 + 𝑆𝐸𝑀 (2) as the initial electromagnetic wave passes through the shield, it is absorbed according to the attenuation constant α. the strength of the initial electromagnetic wave decreases exponentially with the shield thickness d, according to α, and is represented through the following formula: 𝐸 = 𝐸0𝑒−𝛼𝑑 (3) the attenuation constant α is defined as follows: 𝛼 = 𝜔√ 𝜇 2 √1 + ( 𝜎 𝜔 )2 − 1 (4) where ω is the angular frequency, μ is the magnetic permeability, ε is the dielectric permittivity, and σ is the electrical conductivity. for effective absorption loss of electromagnetic waves, the shield must possess a high electrical conductivity, large dielectric permittivity, or high magnetic permeability. most of the absorbed electromagnetic waves are converted into thermal energy. the absorption loss sea is defined by the skin depth δ and the shield thickness d as follows [32]: 𝑆𝐸𝐴(𝑑𝐵) = 20 log 𝑒𝛼𝑑 = 20 ( 𝑑 𝛿 ) log10 𝑒 (5) skin depth δ is the distance through which the wave decreases to e-1 of the original incident wave intensity. reflection loss (ser) occurs at the shield’s surface due to the impedance or refractive index difference between the air and the shield material. this loss can be expressed by the fresnel equation, which involves electric conductivity σ, frequency of the electromagnetic wave f, and magnetic permeability μ. 𝑆𝐸𝑅 (𝑑𝐵) = 20 log (𝜂+𝜂0)2 4𝜂𝜂0 = 39.5 + 10 log 𝜎 2𝜋𝑓𝜇 (6) where η and η0 represent the impedances of the shield and air, respectively. reflection loss is influenced by the impedance difference, electrical conductivity of the shield, magnetic permeability, and the frequency of the incident electromagnetic waves. multiple reflection loss is significantly affected by the shield thickness. the electromagnetic wave reflected from the shield’s back surface may encounter the front surface again, causing further reflections. the electromagnetic wave intensity decreases mechanically pressed polymer-matrix composites with 3d structured filler networks... 611 progressively as these reflections continue between the front and back surfaces. multiple reflection loss can be expressed as follows: 𝑆𝐸𝑀 = 20 log10(1 − ℯ−2𝛼𝑑) = 20 log10(1 − ℯ − 2𝑑 𝛿 ) (7) multiple reflection loss becomes negligible if the total emi se exceeds 15 db or if the shield thickness is similar to or greater than the skin depth. the total emi se can be calculated from the scattering parameters obtained from two ports using a vna. s11 represents the ratio of the voltage input to port 1 and the voltage output from port 1, and s21 represents the ratio of the voltage input to port 1 and the voltage output from port 2. similarly, s22 and s12 represent the reflection and transmission parameters for port 2, respectively. for a uniform shield, s11 = s22 and s21 = s12. sea and ser can be defined in terms of these scattering parameters as follows [33]: 𝑇 = |𝑆21|2 (8) 𝑅 = |𝑆11|2 (9) 1 = 𝐴 + 𝑅 + 𝑇 (10) 𝑆𝐸𝐴 = −10 log10 𝑇 1−𝑅 (11) 𝑆𝐸𝑅 = −10 log10(1 − 𝑅) (12) here, a, r, and t represent the absorption, reflection, and transmission power coefficients, respectively. the total emi se measured by vna for ps@mxene and ps@mxene@bnnss is shown in fig. 6. the composites were fabricated through hot pressing, and the emi se of these composites was compared based on the coating of bnnss-aptes. it was observed that in all the frequency ranges, the presence of bnnss significantly enhanced the performance. fig. 6 emi se of ps@mxene and ps@mxene@bnnss 2 4 6 8 10 12 0 5 10 15 20 25 30 e m i s h ie ld in g e ff e c ti v e n e s s ( d b ) frequency (ghz) ps@mxene ps@mxene@bnnss 612 c. choi, n. qaiser, b. hwang this improvement can be attributed to the bnnss layer encapsulating the oxidizing mxene, thereby preventing its oxidation. additionally, the repeated interfaces of various materials increased the emi se due to the impedance difference between each material. as seen in the initial part of the graph, the composites exhibited relatively high emi se. considering that a shielding effectiveness of 20 db or higher corresponds to 99% blockage of electromagnetic waves, the composites developed in this study demonstrated commendable performance, particularly in the low-frequency range. furthermore, the thicknesses of the fabricated composites were relatively thin, at just 0.8 mm. since the thickness of the composite greatly influences emi se, the performance observed in this study can be considered excellent. 4. conclusions in this study, we fabricated a multifunctional composite with electromagnetic wave shielding capabilities by controlling the surface charge of ps, mxene, and bnnss. it is crucial to control the surface charge of each material precisely to achieve an optimal composite. since mxene and bnnss need to bind to the surface of spherical ps particles, it is important to form a nanosheet of the 2d material. for ps, the surface was initially negatively charged after dispersion polymerization, but it became positively charged with the addition of dmc. this allowed us to create ps@mxene microspheres by reacting the positively charged ps with negatively charged mxene. similarly, we modified bnns (with a negative surface charge) with aptes to impart a positive charge, enabling the fabrication of ps@mxene@bnnss microspheres through a self-assembly reaction. these microspheres were then hot-pressed to form the ps@mxene@bnnss composite. the ps@mxene@bnnss composite exhibited excellent electromagnetic wave shielding efficiency in the low-frequency range. furthermore, it consistently demonstrated higher shielding efficiency than the ps@mxene composite, indicating the oxidation prevention effect of bnns. given that bnnss have excellent thermal conductivity and completely encapsulate ps, the resulting ps@mxene@bnnss composite is also expected to possess high thermal conductivity. therefore, the ps@mxene@bnnss composite developed in this study shows significant potential for use in electronic devices requiring electromagnetic wave shielding. it is particularly suitable for applications where insulation, low-frequency electromagnetic wave generation, and thin material thickness are essential. additionally, the method proposed in this study offers a solution for both thermal management and electromagnetic wave shielding, providing a practical approach for manufacturing a dual conduction 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metallic bicrystals has been carried out to investigate the behavior of the symmetrical tilt grain boundaries under shear loading. σ5 and σ9 grain boundaries in ni and α-fe were analyzed. it is found that behavior of the defect depends not only on the structure of boundaries but also on the type of crystal lattice. in particular it is shown that under external stress the grain boundary (gb) behaves differently in the bcc and fcc metal. a comparison of the values of displacement of various types of gb due to their migration caused by shear deformation is carried out. the results can help us to understand the features of the plastic deformation development in nanoscale polycrystals under shear loading. key words: molecular dynamics, symmetrical tilt grain boundary, shear loading, grain boundary migration, non-equilibrium structure 1. introduction getting new knowledge about mechanisms of the plastic deformation development is one of the key challenges of modern materials science. this task has been the subject of intensive experimental and theoretical works. the results they achieve show that for polycrystalline materials the features of plastic deformation are determined not only by the loading conditions, but mainly by the characteristic dimensions of the grain structure. thus, for metals with a relatively large grain size, the main mechanisms for the plastic deformation development are the motion of dislocations, which leads to the appearance of deformation localization bands. when decreasing the grain size and thus increasing the received may 12, 2017 / accepted june 20, 2017 corresponding author: andrey i. dmitriev institute of strength physics and materials science sb ras, 634055, pr. akademicheski 2/4, tomsk, russia. e-mail: dmitr@ispms.ru 286 a.i. dmitirev, a.yu. nikonov total proportion of the grain boundaries, the grain boundary sliding becomes the dominant deformation mechanism [1–4]. as a result, the materials consisting of very small crystal grains are known to show some particular behavior. a typical example is the superplastic deformation of metals under tensile stress, which elongates up to several hundred percent without fracture [5]. that high deformation of bulk materials is considered to be controlled by the slip of grains at the boundary in the rearrangement of the grain configuration. however, this mechanism seems to be much more complicated; new data are required with respect to atomic transfer both inside and outside the grains. it is shown in a number of studies, that it is possible to form interfacial non-equilibrium structures by shear deformation that exists only during the stage of the active loading [6, 7]. at the same time, due to a great variety and transience of the processes as well as the complexity of the experimental observation of deformation mechanisms realized near the interfaces, the methods of computer simulation can be considered as a useful tool [8–11]. in particular, molecular dynamics (md) simulation appears to be a good method to follow the atomic ensemble evolution under such a loading. the advantage of this method is in the possibility to investigate elementary atomic mechanisms realized near the interfaces and then to use the results for modeling at higher scales as well as for interpreting the experimental data. earlier [10] it has been shown by means of md simulation that the external shear deformation leads to restructuring the crystal lattice near the grain boundaries. such a restructuration, in its turn, may result in a preferential growth of only one grain on account of the other neighboring ones. the results in the cited paper are obtained on a sample of the copper containing symmetrical tilt grain boundary σ5 (210)[001]. later, similar studies have been done on copper bicrystals containing tilt grain boundaries different from σ5 (210)[001] ones [10]. studies have shown that the behavior of grain boundaries such as, for example σ9 (1 2 2)[011], under shear deformation may significantly differ from the behavior of the σ5 grain boundary. thus, if the ycoordinate of σ5 gb due to shear deformation changes in the direction perpendicular to applied loading then σ9 gb does not migrate in the bulk of the material and shear deformation provides the formation of structural changes along the plane of the defect. the aim of this study is to check the generality of the observed grain boundary sliding mechanism by modeling the behavior of these defects under shear loading conditions for nickel and α-iron bicrystal samples. 2. numerical model investigations are carried out within the framework of the conventional molecular dynamics method using the large-scale atomic/molecular massively parallel simulator (lammps) software [13, 14]. the visualization of md simulations data and structure analysis is carried out using the open visualization tool ovito [15]. the behavior of bicrystals with the initial face-centered-cubic (fcc) and body-centered-cubic (bcc) structures is investigated by the examples of ni and α-fe crystallites. within the numerical simulation of grain boundary behavior, there is a problem of generating the initial structure of the defect. the peculiarity of the gbs in the general form is their non-periodicity. since only a fragment of the sample is modeled within the framework of molecular dynamics method, which is later repeated due to the use of periodic boundary conditions, it is necessary to consider a the features of σ5 and σ9 grain boundaries migration in bcc and fcc metals under shear loading... 287 sufficiently extended fragment to generate gb of a general form. therefore, the special type grain boundaries that have a periodic structure are most often used with the md method. in both considered cases, the modeled specimens are in the form of rectangular fragments and contain a planar structural defect such as a high-angle tilt grain boundary oriented as shown in fig. 1. tilt grain boundaries σ5 (210)[001] and σ9 (1 2 2)[011] are obtained by means of the coincidence site lattice (csl) principle. the algorithm to design the initial structure is described in details in [16] and it includes the following steps. firstly, the grain orientation in space is that external axes x, y and z correspond to the crystallographic directions determined by the type of defect. then, the second grain is generated by mirroring the first one in plane xoz, which then becomes the plane of the defect. the last step involves the relaxation of the resulting sample. the modeled structure is located between the two loaded layers and subjected to a sliding loading with constant velocities (v) +20 and -20 m/s applied to the upper and lower layers, respectively, as shown schematically in fig. 1. thus, the resulting shear loading rate is 40 m/s. the sample with an initial temperature of 200k is simulated. a certain temperature is achieved by using the velocity rescaling method during the md simulation process from the energy balance described by eq. (1): 2 1 3 2 2 n i i b i m v k t n  (1) where n is the atom number, kb is the boltzmann constant, mi and vi are the i th atom mass and velocity, respectively. fig. 1 schematic representation of the simulated sample a periodic boundary condition is prescribed in the xand z-directions of the specimen to reduce the simulation scale effect. atomic interaction is described in the framework of the embedded atom method [17]. the total number of atoms is about of 100000. the modeled sample is considered as an nve ensemble maintaining the number of particles n, and occupying volume v and the energy of system e. the equations of motion are integrated on the basis of the velocity-verlet algorithm with a time step δt of 1×10 -15 s. 288 a.i. dmitirev, a.yu. nikonov 3. shear loading of ni bicrystal initially, the modeling of grain boundaries behavior in a nickel sample under conditions of shear deformation is carried out. the calculated values of the specific energy of the considered plane defects are as follows: eς5=2,290 j/m 2 , eς9=1,309 j/m 2 . firstly, we investigate the behavior of nickel sample containing the grain boundary σ5. since the atomic lattice of ni is fcc, i.e. similar to that of cu, we expect a similarity in a response of ni crystallite with identical gb. the projection of atoms on the plane xoy of the fragment with initial structure containing a planar defect is shown in fig. 2a. black color indicates the atoms lying in the plane of the defect, dark gray color denotes atoms arranged initially in a plane perpendicular to that of the grain boundary and used for the deformation distribution visualization along the specimen. the results of modeling the behavior of grain boundaries σ5 in nickel sample show that under conditions of shear deformation the relative sliding of interacting grains is accompanied by displacement of boundary position in the direction perpendicular to the applied loading. the resulting structure of the modeled specimen after 200000 steps of sliding is shown in fig. 2b. the magnitude of this displacement is by a factor of 1.5 greater than the relative slip between the grains very much alike it was observed previously for the same gb in cu bicrystal [10, 14]. it should also be noted that the configuration of the atomic lattice near the plane of the defect retains a regular arrangement of atoms throughout the process of moving the grain boundary so that no non-equilibrium structures are formed. (a) (b) fig. 2 the projection of atomic structure near the grain boundary σ5 on the plane xoy at different moments of time a) t = 0ps; b) t = 200ps the next stage of our research is to model the behavior of ni sample containing a grain boundary σ9. the resulting equilibrium atomic configuration near the plane of the defect is shown in fig. 3a. a peculiarity of the σ9 grain boundary structure is its asymmetric character with respect to the defect plane [18]. according to the results of simulation, the shear loading has never been able to move such a boundary in the direction perpendicular to the defect plane. under shear loading, only a relative slippage occurs between the grains along the x axis which is accompanied by curving the defect plane. the latter is confirmed by the the features of σ5 and σ9 grain boundaries migration in bcc and fcc metals under shear loading... 289 resulting distribution of marked atoms initially arranged perpendicular to the gb plane. this is well seen in fig. 3b, where the structure of the central part of the sample is shown at time t = 100ps. (a) (b) fig. 3 the projection of atomic structure near the grain boundary σ9 on the plane xoy at different moments of time a) t = 0ps; b) t = 100ps to analyze the features of the structural location of each atom during shear loading simulation the common neighbor analysis (cna) is used [19]. according to cna, a single hexagonal-close-packed (hcp) coordinated-layer means a coherent twin boundary while two adjacent hcp-coordinated-layers indicate an intrinsic stacking fault. fig. 4a shows the projection of the central fragment of the structure after 100000 integration steps. in fig. 4 bright gray dots mark atoms belonging to the defect-free fcc lattice. atoms defined as those of local hcp structure are marked by large dark gray circles. black dots (a) (b) fig. 4 structure (projection on xoy plane) near the gb σ9 at different moments of time (a) t=100ps and (b) t=160ps. atoms are colored according to cna value 290 a.i. dmitirev, a.yu. nikonov denote atoms whose local structural allocation differs from fcc, bcc and hcp. such atoms are located near the grain boundaries and other structural defects and can be used to identify the position of the border between two grains. further loading of bicrystal with gb σ9 leads to the formation of separate segments crystallographically misoriented with respect to each other as those seen in fig. 4b. again, no non-equilibrium structure formation is observed. comparing the behavior of copper and nickel bicrystals containing σ5 and σ9 symmetrical tilt grain boundaries, we observe a good qualitative agreement between the shear deformation simulation results as was expected due to identical atomic lattice of both metals. therefore, one can expect a similar behavior for other metals with face-centered cubic lattice under shear loading. 4. the behavior of σ5 and σ9 gbs in iron sample to verify the generality of the revealed regularities of the behavior of grain boundaries on other types of atomic lattices, shear deformation of metal with bcc structure was carried out. specimens of α-iron with symmetrical tilt grain boundaries σ5 (120)[001] and σ9 (114)[110] were simulated. the calculated values of the specific energy of the considered plane defects were as follows: eς5=1,122 j/m 2 , eς9=1,309 j/m 2 , respectively. firstly, we investigated the behavior of α-iron sample containing the grain boundary σ5. the algorithm similar to that described above was used to generate the initial structure. firstly, the grain orientation in space was specified in such a way that external axes x, y and z corresponded to the crystallographic orientations determined by the type of the defect. then the second grain was generated by mirroring the first one in plane xoz, which then became the plane of the defect. the next step included finding the minimum of potential energy by relative displacement of grains along the plane of the defect. shift vector for grain boundary σ5 was t = (1,28; 0; 0). the last step contained the relaxation of the resulting structure. the projection on xoy plane of atomic configuration after relaxation of α-iron specimen containing the grain boundary σ5 (120)[001] is shown in fig. 5. the loading as well as boundary conditions were identical to those used for shear deformation of nickel bicrystal. the results show that the response of the simulated crystallite containing the grain boundary σ5 differs from those previously obtained on ni and cu bicrystals. according to the simulation data, the response of the specimen to external shear loading can be divided into three stages as follows. initially, the atomic lattice is deformed elastically outside of the plane of the defect (stage i). this is confirmed by a spatial distribution of atoms marked by dark gray color shown in fig. 6a. next stage includes the formation of crystal structure that is different from the initial one and located near to the original position of gb. in accordance with the simulation results, the thickness of this layer in our example reaches of 4-5 inter-planar distances. in fig. 6b the border of this layer is highlighted by dashed lines. due to action of periodic boundary conditions along x and z axes this structure can be associated with a thin in-plane defect. the features of σ5 and σ9 grain boundaries migration in bcc and fcc metals under shear loading... 291 fig. 5 the initial structure (projection on xy plane) of α-iron sample with gb σ5 (a) (b) fig. 6 the fragment of the structure (projection on the plane xoy) of atomic configuration near the grain boundary σ5 at different moments of time (a) after 40 ps and (b) after 60 ps of shear loading for α-fe bicrystal the observed crystal structure is not stable because its further deformation leads to restructuring. this is accompanied by a mismatch of the ordered structure near the grain boundary, which is analogous to that observed in a copper sample with a grain boundary σ5 at high sample temperatures or with an imperfect initial boundary structure [10, 14]. after that the grain boundary starts moving in the direction perpendicular to the applied external loading (stage iii). analysis of the structure at different moments of time shows that despite the disordered structure near the defect, the boundary moves uniformly with constant relative velocity, which is 1,2 times higher than the rate of external loading. the mentioned above stages are clearly seen on the time dependence of the grain boundary position along y direction shown in fig. 7a. the stage ii was associated with a sharp change in the gb position, which occurred due to restructuring of the non-stable layer that took place between 45 ps and 70 ps of the loading time. the founded lifetime of the stage ii (about 25 ps) is too short to be identified experimentally but we expect that it should be sensitive to the loading conditions. 292 a.i. dmitirev, a.yu. nikonov the resulting configuration of the structure of the sample that contains grain boundaries after 200000 integration steps is shown in fig. 7b. the spatial distribution of marked atoms confirms monotonically migration of the grain boundary at the stage iii. the disordered structure near the gb is also well seen. (a) (b) fig. 7 a) time dependence of boundary displacement s on the initial position; b) the structure (projection of atoms on xoy plane) of α-fe specimen with the grain boundary σ5 after 200 ps of shear loading. the arrow shows the direction of gb migration during shear deformation at stage iii the next stage of our research was to model the behavior of σ9 grain boundary under shear deformation. to generate this type of symmetrical tilt gb, the vector of relative shift was as follows t = (10,17576; 0,5325; 1,04988). fig. 8a shows the resulting structure of the bicrystal containing such a gb after relaxation of the sample. the simulation results show that the behavior of such a defect in α-iron under shear deformation is similar (a) (b) fig. 8 a) the initial structure (projection of atoms on xoy plane) α-fe specimen with gb σ9; b) time dependence of boundary displacement s on the initial position the features of σ5 and σ9 grain boundaries migration in bcc and fcc metals under shear loading... 293 to that of grain boundary σ5 (210)[001] in fcc metals. due to the shear loading a synchronous restructuring of atomic lattice near the plane of the defect occurred that, in its turn, led to the grain boundary motion in the direction perpendicular to the applied loading. at that the structure of the defect is still regular. the simulation results show that under loading the σ9 grain boundary begins to move along y axis with a practically constant velocity (fig. 8b) which is defined by the shear loading rate. thus, the estimations give us about 70 m/s for the grain boundary migration velocity while the relative sliding velocity of two loaded layers is 40 m/s. 5. conclusions using the md simulations the features of the grain boundary sliding mechanism have been studied on the scale of individual atoms by the example of large angle symmetrical tilt grain boundaries σ5 and σ9 in bicrystals of ni and α-fe. it is shown that for a certain orientation of the defect a grain boundary sliding under shear loading may be accompanied by rearrangement of the atomic configuration in the plane of the defect, which leads to an efficient movement of the position of gb in the direction perpendicular to applied loading. at that, the boundary displacement velocity can be several times higher than the shear loading rate. it is found that the dynamic properties of the boundary migration depend both on the features of gb structure and the type of crystal lattice. in particular, the "mobility" of grain boundary σ5 under shear deformation in fcc lattice disappears in the crystallite with bcc structure, and vice versa – "unmovable" grain boundary σ9 in fcc bicrystal begins to move (shift) in the direction perpendicular to the applied loading with changing the type of atomic lattice. note that in this paper we study only an elementary grain boundary slip mechanism that is realized in pure bicrystals with periodic boundary conditions on the stage of active deformation. the 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methods for modeling metallic systems, mrs bulletin, 21(2) pp. 24-28. 18. van swygenhoven, h., farkas, d., caro, a., 2000, grain-boundary structures in polycrystalline metals at the nanoscale, physical review b, 62(2), pp. 831-838. 19. honeycutt, j. and andersen h., 1987, molecular dynamics study of melting and freezing of small lennardjones clusters. journal of physical chemistry, 91, pp. 4950–4963. 10128 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 617 632 https://doi.org/10.22190/fume211016070k © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper geometric analysis by constructal design of stiffened steel plates under bending with transverse i-shaped or t-shaped stiffeners derick m. p. kucharski1, vinícius t. pinto1, luiz a. o. rocha1, elizaldo d. dos santos1, cristiano fragassa2, liércio a. isoldi1 1school of engineering, federal university of rio grande furg, rio grande, brazil 2department of industrial engineering, university of bologna, bologna, italy abstract. several stiffened plates arrangements subjected to bending were configured applying the constructal design method (cdm) and solved by finite element method (fem), aiming through the exhaustive search (es) technique analyze the influence of transverse i-shaped or t-shaped stiffeners in mechanical behavior. considering a nonstiffened plate as reference and maintaining the total steel volume constant, a portion of the reference plate was deducted from its thickness, and transformed into stiffeners through the  volume fraction parameter, which represents the ratio between the steel volume of the stiffeners and the steel volume of the reference plate. assuming  = 0.3, 25 plates with just i-shaped stiffeners in longitudinal and transverse directions and 25 plates with i-shaped stiffeners in longitudinal direction and t-shaped stiffeners in transverse direction were proposed. the results showed that the plates with transverse t-shaped stiffeners are more effective, reducing the maximum von mises stress and maximum deflection, respectively, in up to more than 60% and 50% when compared with the plates with just i-shaped stiffeners. key words: stiffened plates, computational modeling, constructal design, t-shaped stiffeners, i-shaped stiffeners 1. introduction due to the increasingly common engineering trend of building lightweight structures, thin-walled structures have attracted a great deal of attention over the last few decades. numerous researchers have dedicated their work to provide highly efficient numerical tools to model and simulate the elastic behavior of thin-walled structures [1-4], deal with received october 16, 2021 / accepted november 30, 2021 corresponding author: cristiano fragassa department of industrial engineering, university of bologna, viale risorgimento 2, 40136 bologna, italy e-mail: cristiano.fragassa@unibo.it 618 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. their design [5, 6], and buckling issues [7, 8], extend existing methods to isogeometric formulations [9, 10], etc. plates are flat, two-dimensional structural components, characterized by their thickness being much smaller than the other two dimensions. one of the most desirable plate features is the great load support resistance with relative low weight, finding application in several engineering areas, such as bridges, aircraft fuselages and ship hulls [11, 12]. due to the typical high plate slenderness, it is a usual practice to insert reinforcements (stiffeners) in order to increase the flexural rigidity of thin plates. these stiffeners can assume different cross section shapes, being normally welded in the longitudinal and/or transverse directions of the plate [13]. regarding the structural analysis of non-stiffened thin plates, stresses and displacements can be obtained in a relatively simple way through analytical solutions, applying navier or lévy solutions, for instance as well as the reisner's theory for non-stiffened thick plates [12]. however, when it comes to stiffened plates, analytical solutions are complex and often have limitations related to the boundary conditions and loading, leading to inaccurate results. hence, numerical solutions became an effective option to solve problems involving stiffened plates. over the past years, several studies on this topic have been published. mukhopadhyay and satsangi [14] applied the finite element method (fem) in order to analyze stiffened plates under bending, by means a numerical model employing isoparametric elements, allowing insertion of the transverse shear deformations and curvature limits. guo et al. [15] focused on the analyses of plates and bridge decks composed by beam-slab system under transverse loading through a semi-discrete finite element formulation, such that the plate-stiffener interaction was considered in terms of bending deflections. using the galerkin method based on first-order shear deformable theory (fsdt). peng et al. [16] studied the mechanical behavior of stiffened plates subjected to bending, through a meshless approach that allows flexibility in the placement of the stiffeners and expressing the stiffener displacement field as a function of the average surface of the plate. taking advantage of the current high-performance computers, commercial software based on numerical methods, mainly the fem, have been used both in industry and in scientific research. in this sense, a parametric analysis of stiffened plates subjected to bending was carried out by singh et al. [17] using the ansys (fem commercial software package) considering different loads and boundary conditions. khosravi et al. [18] analyzed the mechanical behavior of composite steel plate shear walls by means of abaqus (fem commercial software package), also, souza et al. [19] showed a topology optimization of stiffened plates using a modified adaptive morphogenesis algorithm (mama), solving different plate arrangements by means of ansys. when it comes to engineering, high structural performance is always a target. therefore, regarding steel plates, the constructal design method (cdm) associated with computational modeling by fem, have reached interesting results, as in: isoldi et al. [20, 21], helbig et al. [22-24], da silva et al. [25], and da silveira et al. [26] who investigated the buckling effect on perforated plates; lima et al. [27, 28] who analyzed the buckling of stiffened plates; and cunha et al. [29, 30], de queiroz et al. [31], troina et al. [32], nogueira et al. [33], and pinto et al. [34, 35] who studied the bending of plates with i-shaped stiffeners. given the above, this work focused on the geometric evaluation and optimization of stiffened plates subjected to bending concerning the influence of the stiffeners cross section geometry, with i-shaped stiffeners in the longitudinal direction and i-shaped or t-shaped stiffeners in the transverse direction. to do so, several stiffened plates arrangements were geometric analysis by constructal design of stiffened steel plates under bending with transverse... 619 defined using cdm, solved by fem, and compared among each other through the exhaustive search (es) technique, allowing identifying the transverse stiffener geometry influence on the maximum von mises stress and maximum deflection, as well as to indicate the optimized configuration that minimizes these performance indicators. 2. methodology 2.1. computational modeling in this study, all computational models were solved by fem using the ansys software, considering a linear-elastic material behavior and a geometric linear structural analysis. the discretization of the computational domains was performed applying the two-dimensional finite element shell281 in its quadrilateral version. this finite element is suitable for modeling thin and moderately thick plates, has 8 nodes with 6 degrees of freedom per node: 3 translations and 3 rotations related to x, y and z axes; being based on the first-order sheardeformation theory (or mindlin-reissner shell theory) [36]. 2.1.1. computational model verification the computational model verification was carried out based on the stiffened plate presented at fig. 1, which was previously solved by troina et al. [37] using a regular mesh composed of 3,928 three-dimensional finite element solid95 in its hexahedral version by ansys. fig. 1 stiffened plate used in the numerical model verification (unit: mm) the simply supported stiffened steel plate of fig. 1 has elastic modulus of 206.8427 gpa and poisson's ratio of 0.3, being subjected to a uniform normal loading of 68.95 kpa. the case was solved using 15,200 shell281 finite elements in its quadrilateral version, defined from the convergence test presented in fig. 2. figure 2 also shows the result obtained by troina et al. [37] for the central deflection of the stiffened plate. 620 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. fig. 2 mesh convergence test and computational model verification from fig. 2, it is perceptible that the difference concerning the central deflection uz is about 1%, comparing the present study using the 2-d finite element shell281 and the result presented by troina et al. [37] with the 3-d finite element solid95, verifying the computational model. 2.1.2. computational model validation the experimental case adopted to validate the computational modeling was presented by carrijo et al. [38], as shown in fig. 3. the acrylic stiffened plate was subjected to a uniform normal load of 0.96 kpa, having an elastic modulus of 2.5 gpa and a poisson's ratio of 0.36. with respect to the boundary conditions, just its four corners were considered simply supported. in the present study the case of fig. 3 was numerically reproduced using the shell281 quadrilateral finite element with a mesh of 2,436 computational cells, established according to the mesh convergence test presented in fig. 4, which also shows the result experimentally obtained by carrijo et al. [38]. by analyzing the fig. 4, it is possible to notice that the numerical result for the central deflection of the plate uz = 6.505 mm, represents a difference of 4.58% when compared with the value of uz = 6.220 mm obtained in the experimental study, validating the computational model. geometric analysis by constructal design of stiffened steel plates under bending with transverse... 621 fig. 3 stiffened plate used in the numerical model validation (unit: mm) fig. 4 mesh convergence test and computational model validation 2.2. constructal design method (cdm) the constructal theory brings the comprehension that the design of all flow systems (such as the atmospheric electrical discharges, deltas of rivers, roots and branches of trees, veins and arteries of the human body, social dynamics, engineering and technology and so on) are generated by a physical phenomenon named as constructal law [39]. the constructal law states: "for a finite-size flow system to persist in time (to live), its configuration must evolve in such a way that provides greater and greater access to the 622 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. currents that flow through it" [40]. according to rocha et al. [41], the constructal law is about the evolution direction in time, i.e., the design phenomenon is a dynamic and continuous process. the constructal law is applied in engineering flow systems by means the constructal design method (cdm). the cdm is a geometric evaluation method and its apllication demands the definition of constraints (global or local), degrees of freedom (free to vary, respecting the constraints) and performance indicators that can be maximized or minimized (which represents a view of ease of flow access). however, if the goal is to define the optimized geometric configuration, the cdm must be employed in association with an optimization method (such as exhaustive search or heuristic methods). in this case, the cdm is responsible for creating the search space (containing the possible geometric configurations) while the optimization method is in charge of identify the system geometry that leads to the superior performance [26, 41, 42]. in this work, the cdm application started from a non-stiffened steel plate taken as reference, with length a = 2000 mm, width b = 1000 mm and thickness t = 20 mm. thereafter, maintaining constant the total material volume and the dimensions a and b, a portion of material fully deducted from the thickness of the reference plate was transformed into stiffeners through a parameter named volume fraction  (ratio between the steel volume of stiffeners and the steel volume of the reference plate). according to troina et al. [32], the value of  adopted in this study was 0.3, which means that 30% of the non-stiffened plate volume (reference plate) was converted into stiffeners. due to this, the thickness of the stiffened plate became tp = 14 mm. a total of 50 stiffened plates' arrangements were analyzed: 25 with only i-shaped stiffeners in both directions and 25 with i-shaped stiffeners in longitudinal direction and t-shaped stiffeners in transverse direction, with the main purpose to analyze the stiffeners influence over the maximum out-of-plane displacement (deflection) and the maximum von mises stress. it is important to highlight that the choice to study the transverse stiffeners cross section shape influence is based on nogueira et al. [33], where it was inferred that the transverse stiffeners are more effective for the reduction of the maximum and central deflections of stiffened plates. the mathematical definition of the volume fraction 𝜙 for plates with just i-shaped in both directions and having i-shaped stiffeners in longitudinal direction and t-shaped in transverse direction are, respectively, given by: ( ) ( ) sx s s sy sx s s ss r n ah t n b n t h tv v abt  + − = = (1) ( ) ( ) + + − = = sx s s sy w w f f ins r n ah t n bh t ba t vv v abt  (2) where vs is the material volume of the stiffeners and vr is the material volume of the reference plate. the height and thickness of the i-shaped stiffeners are hs (variable) and ts (kept constant and equal to 8 mm). concerning the t-shaped stiffeners, tw and tf represent the thickness of the web and flange, respectively, both considered with 8 mm. the height of the web is named hw (variable), the length of the flange is af (considered as half of the web height, i.e., af = 0.5hw), and the stiffeners intersection volume is vin. moreover, the number of stiffeners in x (longitudinal) and y (transverse) directions are, respectively, nsx and nsy, being ranged from 2 to 6. geometric analysis by constructal design of stiffened steel plates under bending with transverse... 623 in addition, to identify the stiffened plate arrangements, the following nomenclature was adopted: pi(nsx,nsy) to the i-shaped stiffened plates and pt(nsx,nsy) to the t-shaped and i-shaped stiffened plates. all of these aforementioned parameters can be seen at figs. 5 and 6; while fig. 7(a) shows the search space defined by the cdm application and fig. 7(b) presents a step by step of the methodology adopted in this work. fig. 5 geometric parameters of stiffened plate pi(3,3) fig. 6 geometric parameters of stiffened plate pt(2,3) 624 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. fig. 7 (a) search space defined by cdm; (b) methodology of the study it is important to mention that all geometric configurations of the search space (see fig. 7(a)) were numerically simulated and the results compared to each other, thus performing an optimization procedure by means of an exhaustive search (es) technique. 3. results and discussion as in de queiroz et al. [31] and troina et al. [32], all stiffened plates analyzed in this study (see fig. 7(a)) were considered with boundary conditions of simply supported edges and subjected to a uniform normal loading of 10 kpa. regarding the material, it was adopted the a-36 steel with poisson's ratio 0.30,young’s modulus 200 gpa, and yield strength of 250 mpa. in order to define the suitable size of finite elements used to discretize all computational models, a mesh convergence test was performed considering the more complex geometry pt(6,6) among all proposed stiffened plates, concerning the maximum von mises stress. according troina et al. [32], several meshes were considered taking as reference the plate width b. the result of the mesh convergence test can be observed in fig. 8(a) and the successive mesh refinements are indicated in fig. 8(b). geometric analysis by constructal design of stiffened steel plates under bending with transverse... 625 fig. 8 mesh convergence test: (a) maximum von mises stress variation; (b) mesh refinement criteria observing the results of fig. 8(a), it can be noted that from the mesh number 11, there was a stabilization trend for the stress value. therefore, the refinement b/220 with a finite element size of 4.55 mm was adopted to discretize all computational domains investigated in this study. the results of the stiffened plates, for maximum von mises stress and maximum deflection are presented in figs. 9 and 10, respectively. the bar graph format was chosen to allow a direct observation between variations of stresses and deflections occurred in the plates with only i-shaped stiffeners and with tand i-shaped stiffeners. a first finding emerging from figs. 9 and 10 is that the conversion of a portion of material from the non-stiffened reference plate, keeping the total material volume constant, according to the main cdm premise, causes an increase in the plate stiffness, once all values of maximum deflection are lower than the value reached by the reference plate of uz = 0.697 mm. when it comes to the maximum von mises stress, most of the stiffened plates with just i-shaped stiffeners presented values greater than the value of 13.313 mpa obtained by the non-stiffened reference plate. meantime, concerning the plates with transverse tshaped and longitudinal i-shaped stiffeners, in all cases, the maximum von mises stress value is lower than 13.313 mpa of the reference plate. it is well known that in structural engineering focused on design of flat components, whether in metallic (present study) or reinforced concrete (e.g., slabs) structures, deflection and stress are important parameters. the deflection is a serviceability parameter which not necessarily compromise the safety, as long as, the values presented are in accordance with design standards. on the other hand, values of stress are directly related with safety, i.e., if exceeded the maximum limits the structure is on the verge of collapse [43]. therefore, in this study it was chosen as main parameter the maximum von mises stress following by the maximum deflection, i.e., the criterion that qualify the better stiffened plate arrangement in each group of plates (see fig.7(a)) is the maximum von mises stress. 626 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. fig. 9 results of maximum von mises stress: (a) group a; (b) group b; (c) group c; (d) group d; (e) group e that said, figs. 9(a) and 10(a) (group a) indicate that the plate with better mechanical behavior was pt(2,2), having a maximum von mises stress of 3.581 mpa, which represents a minimization of 48.2% in comparison with the value of 6.920 mpa of the plate pi(2,2). concerning the maximum deflection, the plate pt(2,2) presents 0.023 mm, representing a slight reduction of 8.7% if compared with the plate pi(2,2) that achieved uz = 0.025 mm. in turn, with respect to the group b (see figs. 9(b) and 10(b)) the plate pt(3,2) showed superior performance among all arrangements. comparing the von mises stress value of 4.825 mpa obtained by the plate pt(3,2) with the value of 10.957 mpa found for the plate pi(3,2), it is reached a reduction of 55.96%. still on group b, regarding the maximum deflection the plate pt(3,2) achieved a minimization of 29.4% in comparison with the plate pi(3,2), being the maximum deflection values of 0.026 mm and 0.037 mm, respectively. geometric analysis by constructal design of stiffened steel plates under bending with transverse... 627 for the group c, as showed at figs. 9(c) and 10(c), one can note that the plates containing tand i-shaped stiffeners tend to present a better performance than the plates with only ishaped stiffeners for both performance parameters. the best performance was obtained by the plate pt(4,2), reducing the maximum von mises stress (of 5.976 mpa) and maximum deflection (of 0.035 mm) in 59.3% and 41.2%, respectively, in comparison with the plate pi(4,2) that showed von mises stress equal to 14.678 mpa and maximum deflection of 0.059 mm. fig. 10 results of maximum deflection: (a) group a; (b) group b; (c) group c; (d) group d; (e) group e in sequence, analyzing the results of group d in figs. 9(d) and 10(d), the plate pt(5,2) achieved the best mechanical behavior with a von mises stress of 7.716 mpa and a maximum deflection of 0.045 mm. if pt(5,2) is compared with pi(5,2), that reached 628 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. maximum values of 20.488 mpa and 0.087 mm, reductions of 62.3% for maximum von mises stress and 48.6% for maximum deflection were observed. lastly, when it comes to the group e (see figs. 9(e) and 10(e)), the superior performance was presented by the plate pt(6,2), with maximum von mises stress of 9.231 mpa and maximum deflection of 0.058 mm; meaning a decrease in comparison with the plate pi(6,2) of 63.0% for the von mises stress (of 25.0 mpa) and 52.9% for maximum deflection (of 0.123 mm). among all analyzed groups, the group e achieved the highest percentage reductions in the performance indicators. for this reason, aiming to illustrate these mechanical behaviors, figs. 11 and 12 depicted, respectively, the stress and deflection distributions for the stiffened plates pt(6,2) and pi(6,2). from fig. 11, one can note that the maximum von mises stress of 25.0 mpa occurs in the transverse i-shaped stiffeners region farthest from the plate (fig. 11(b)). in addition, keeping constant the total amount of material and just changing the geometric configuration of the cross section of the transverse stiffeners from i-shaped to t-shaped, the flow of stress is better distributed and a significantly lower von mises stress magnitude is reached (fig. 11(a)). likewise, in fig. 12, it is possible to observe the influence of the geometric configuration in the out-of-plane displacements distributions provided by the cdm application, allowing reducing the maximum deflection by more than 50% when using the t-shaped instead of the i-shaped in transverse stiffeners. fig. 11 distribution of the von mises stresses: (a) pt(6,2); (b) pi(6,2) geometric analysis by constructal design of stiffened steel plates under bending with transverse... 629 fig. 12 distribution of the deflections: (a) pt(6,2); (b) pi(6,2) it is important to emphasize that the superior performance of the plates with transverse tshaped stiffeners when compared with the plates with transverse i-shaped stiffeners (see figs. 9 and 10) can be explained through the influence of the moment of inertia of the stiffeners cross section. the moment of inertia of the t-shaped stiffeners is greater than the moment of inertia of the i-shaped stiffeners, considering the same amount of material according to cdm. to exemplify, the table 1 shows the moments of inertia per stiffener for the aforementioned plates that presents best performance in each group (see figs. 7(a), 9, and 10). table 1 transverse stiffener’s moment of inertia for the plates with the better performance plates moment of inertia (106 mm4) i-shaped stiffener t-shaped stiffener (2,2) 10.585 13.809 (3,2) 4.474 6.582 (4,2) 2.293 3.650 (5,2) 1.328 2.237 (6,2) 0.837 1.473 it is worth to mention that the comparison presented in table 1 serves only to illustrate the difference of the moment of inertia of the stiffeners cross section in a simple way, justifying its direct influence in the overall behavior of the stiffened plates. 4. conclusion several rectangular plates arrangements with i-shaped stiffeners in longitudinal direction and with i-shaped or t-shaped stiffeners in transverse direction were evaluated regarding to the maximum von mises stress and the maximum deflection applying cdm, fem and es technique. 630 d.m.p. kucharski, v.t. pinto, l.a.o. rocha, e.d. dos santos, c. fragassa, et al. initially it was inferred that converting a material portion of a non-stiffened steel plate (deducted entirely of its thickness) into stiffeners, while maintaining the total material volume, is effective to significantly reduce the maximum deflection of the plate. however, concerning the maximum von mises stress, the plates with transverse i-shaped stiffeners can reach values higher than the one reached by the reference plate; while all the plates having transverse t-shaped stiffeners achieved inferior magnitudes for the maximum von mises stress if compared with reference plate. it was also concluded that for the vast majority of stiffened plates analyzed, those with transverse t-shaped stiffeners presented better performance than the plates with only ishaped stiffeners, being able to minimize up to more than 60% the maximum von mises stress and up to more than 50% the maximum deflection, such as plates pt(6,2) and pi(6,2). these aspects show the relevance of geometric evaluation in structural engineering, since the amount of material of the plates was kept constant and the mechanical behavior was considerably improved only due to the change in the geometric configuration of the stiffeners. therefore, based on the positive results achieved here, as well as in the previous studies, it would be interesting to apply the presented approach in future works in order to investigate other values of volume fraction , different aspect ratios a/b for the plate, and another shapes of stiffeners. one may expect that a similar trend of improvement would be achieved. acknowledgement: this work was performed with the support of the: coordenação de aperfeiçoamento de pessoal de nível superior – brazil (capes) – finance code 001, fundação de amparo à pesquisa do estado do rio grande 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(eds.), sustainable energy technologies, crc press. 43. salmon, c.g., johnson, j.r., malhas, f.a., 2009, steel structures: design and behavior, pearson. facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 681 704 https://doi.org/10.22190/fume210331040n © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper investigation of the glued insulated rail joints applied to cwr tracks attila németh, szabolcs fischer department of transport infrastructure and water resources engineering, faculty of architecture, civil engineering and transport sciences, széchenyi istván university, hungary abstract. this article summarizes the research results related to our own conducted extensive laboratory tests of polymer composite and steel fishplated glued insulated rail joints (girjs), namely axial tensile tests as well as vertical static and dynamic tests. the investigation dealt with the examination of girjs assembled with steel and special glass-fiber reinforced plastic (polymer composite) fishplates, both of them for cwr railway tracks (i.e. so-called gapless tracks or, in other words, railway tracks with continuously welded rails). the exact rail joint types were mth-p and mth-ap, consistently. the mth p types have been commonly applied for many years in the cwr tracks in europe, mainly in hungary. the mth-ap rail joints consist of fishplates that are produced by the apatech factory (russia). they are made of a fiberglassamplified polymer composite material at high pressure and controlled temperature. this solution can eliminate electrical fishplate lock and early fatigue failures just as it can ensure adequate electrical insulation. the advantage of such rail joints can be that they are probably able to ensure the substitution of the glued insulated rail joints with relatively expensive steel fishplates currently applied by railway companies, e.g. hungarian state railways (máv). the aim of the mentioned research summarized in this paper is to formulate recommendations on technical applicability and on the technological instructions that are useful in everyday railway operation practice on the basis of the measurements and tests carried out on rail joints in laboratory. key words: glued insulated rail joint, fishplate, polymer composite, steel, laboratory test received march 31, 2021 / accepted may 12, 2021 corresponding author: szabolcs fischer department of transport infrastructure and water resources engineering, faculty of architecture, civil engineering and transport sciences, széchenyi istván university, h-9026 győr, egyetem tér 1., hungary e-mail: fischersz@sze.hu 682 a. németh, s. fischer 1. introduction in the 21st century the railway transportation will be an increasingly important task related to the developed world and society. it started in the 19th and the 20th century with enormous railway construction procedures all over the world. in the past, the main tasks that had to be solved were a high traction force of the locomotives and other hauling vehicles, transition from the steam to diesel and electric hauling as well as a higher and higher speed demand of the vehicles, etc. nowadays the environmental friendly technologies (e.g. lower pollution, lower energy demand, green and greener energy for operation, etc.) and the economic solutions (cost efficient with the lowest life cycle costs, even higher lifetime, sustainability, etc.) came to occupy the foreground [1]. in relation to the above-mentioned technologies and technical solutions, we can mention some special ones such as application of noise and vibration damping layers and elements [2–6]; ensuring lower stress in the railway permanent ways’ that result and ensure lower elastic and plastic deformation in the track as well as in the layer structure [7–8]; techniques which provide special work processes, e.g. tamping of wide sleepers or side tamping that does not disturb the dense, compacted sub-ballast layer [9–10]; new and newer technologies ensuring decreased dynamic effects that can result in subsidiary failures (maybe cracking, fracturing, breaking) in the tracks’ elements [11]; minimizing the wear of track elements (mainly the rails’) [12] and lowering the ballast breakage [13– 16]. the determination of the geometrical deterioration process is also quite important [17–18] for the ability to predict maintenance work. other “trends” can be found regarding the development of railway infrastructure: the improvement of railway network in the aspect of interoperability across the state borders [19–20], accurate surveying of railway tracks, switches and crossings [21–22] as well as checking and measurement of visibility triangles at grade crossings [23]. in the case of dealing with railway infrastructure, the railway traffic, its safety can be also discussed. there are improved, special fuzzy-logic methods and algorithms for safety evaluation [24]. although it is not strictly related to railway engineering (transport infrastructure part), it is worth mentioning the risk management on railway projects [25], where risk is to be taken into consideration in planning, construction and maintenance phases: pareto analysis can be applied. of course, the above mentioned aspects are not the only ones that can be taken into consideration in relation to the developed railway transportation and railway infrastructure, mainly in the 21st century. it can be assumed that more and more brand new technology and technical improvements will be announced and introduced into this area of transportation because of the demand for the ‘best’ or high-tech solutions in the future. this paper deals with a possible new direction of the railway development. in chapter 2 the authors introduce the basis, the base problems related to the topic of their research. they detail why it is important to consider the application of modified, modern structural elements (e.g. polymer composite fishplates), regarding aspects of cwr tracks (tracks with continuously welded rails) as well as signaling and processing (safety) system related to safe railway traffic. investigation of glued insulated rail joints applied to cwr tracks 683 2. technical background, literature review based on railway maintenance experiences of hungarian state railways (máv) as well as raaberbahn, győr-sopron-ebenfurth railway (gysev), glued insulated rail joints prefabricated with steel fishplates need a lot of maintenance source due to rail deformations. the other main problem is a false (railway control) signal due to rail end failures (e.g. creasing, end post cracking, etc.). all this leads to the railway capacity restriction. in hungarian practice, three rail systems (i.e. rail profiles) were considered for new constructions and rehabilitations as well as for renewal, in the application of polymer composite fishplated glued insulated rail joints: máv 48, 54e1 (i.e. uic 54) and 60e1 (i.e. uic 60) rail profiles. the main criterion for the choice of rail quality and rail profile should be the railway traffic (axle load, speed, etc.). it means the rails should fulfill requirements as bended-sheared structures with elastic bedded sleepers or elastic bedded rc slabs, etc. besides, their wear resistance has to be high enough. the opportunities to connect (join) rails are the following: application of fishplated rail joints, rail welding, and rail expansion devices [26–27]. basically, the rail joints with fishplates can be set up as (i) traditional fishplated rail joints without any insulation and adhesive material (they are applied in normal ‘gapped’ railway permanent ways mainly without electric hauling and without special electric railway signaling and processing (safety) systems; (ii) traditional fishplated rail joints with electric insulation (insulated rail joints with fishplates) but without adhesive material (gluing) in normal ‘gapped’ railway permanent ways as well as in cwr tracks mainly with electric hauling and with special electric railway signaling and processing (safety) systems; (iii) glued insulated rail joints with electric insulation and adhesive material mainly in cwr tracks (fig. 1). (it has to be mentioned that application of glued insulated rail joints is also allowed in ‘gapped’ tracks, but it is not a common solution due to its high costs.) fig. 1 glued insulated rail joint assembled by polymer composite fishplates from other point of view, the fishplated rail joints can be supported or suspended joints [26–27]. the supported rail joints fit only to wooden/timber and synthetic sleepers, but the suspended rail joints can be applied with sleepers produced of any material (i.e. wooden/timber, concrete, reinforced concrete, pre-stressed reinforced concrete, steel, synthetic). 684 a. németh, s. fischer the cross sectional geometry of the fishplates can be seen in fig. 2. the cross sections of fishplates for glued insulated rail joints are similar to the common flat fishplates, but the main difference between them is that they fill out the space until the rail web while the outer sides of the fishplates are flat, smooth. this geometry provides that they have a higher moment of inertia than the common flat fishplates. the lengths of fishplates have two common values: the 4-hole types are 650 mm long and they are applied to the supported rail joints; the 6-hole types are 900 long and they fit to the suspended rail joints. fig. 2 common cross sections of fishplates (left: common flat fishplate, middle: angled fishplate, right: bone shape fishplate) (on the basis [27]) insulated rail joints are special types of fishplated rail joints where the rail ends are insulated from each other. in this way a metallic connection cannot arise either at the rail ends or via fishplates [27]. the role of the rail connections (rail joints) is to ensure the continuity of the rails without vertical and horizontal ‘steps’ as well as without horizontal angle. the glued insulated rail joint solution with polymer composite fishplates can eliminate failure caused by early fatigue just as it can ensure electrical conduction insulation between the rails to be connected. one of the advantages of such rail joints is the ‘exchange’ possibility of the glued insulated rail joints (with common steel fishplates) currently applied by máv. it is important to note that with the precise construction process (e.g. size of the rail end gap) in the context of the rail joint’s assembly, the design of the bedding layer with sufficient compactness (dense) greatly influences the change of the condition of the glued insulated rail joints. these rail connections are the weak points of the railway track because their fishplates can compensate only for 60% of the moment of inertia of the original rail profile. the wheel, while passing the gap between the rail ends, hits the forthcoming rail end, which is disadvantageous for the whole railway superand substructure as well as for the railway vehicle. dynamic effects are much higher in the case of vertical and/or horizontal steps (and angles) than in the case of rail joints without geometrical failures [28]. it has to be mentioned that lower bending stiffness of fishplates causes a higher rail deflection, or even a permanent deformation of rail ends. such rail joints are characterized by the impact angle between both the connected rails, which, as a consequence, causes impact forces. glued insulated rail joints can be applied in suspended and supported joints depending on their type in the case of value of sleeper space and wheel/axle load on given railway tracks. high tensile strength bolts have to be applied to press fishplates and rail(s) together. in this way, a significant high friction force can be achieved between the fishplates and rails (rail webs). it causes that the high axial forces in the rails cannot open the rail joints (i.e. there investigation of glued insulated rail joints applied to cwr tracks 685 is not any relative movement/displacement between the fishplates and rails; or, in other words, the gap between the rail ends does not open compared to the original value). plastic profile lining (plate) is incorporated between rail ends to be able to achieve adequate electric insulation in this location, too [26–27]. glued insulated rail joints can be produced and assembled in factory as prefabricated ‘elements’ with given rail length values as well as on site, where they are assembled from collected parts [27]. it can be stated that the quality of glued insulated rail joints that produced in factory circumstances is better than the pieces made on site. as mentioned earlier, there are several relevant problems with glued insulated rail joints [26–27]: (i) railway track geometry faults (results of normal geometrical deterioration of railway tracks); (ii) mechanical/structural failures (due to rolling contact fatigue or fracture of one/some of the elements, i.e. rail, fishplate, bolt; etc.); (iii) failures based on electric insulation problems (i.e. trouble in signaling and processing (safety) systems) due to seeable or non-seeable reasons; (iv) other failures generated due to the (i)…(iii) original problems (water pockets below the support sleepers, etc.). the main reason for application and preferring of glued insulated rail joints with glass-fiber reinforced fishplates is related to the above mentioned (iii) problems. however, the mechanical/structural failures are also important but they are not the general ones. this solution avoids the failures because of the main electric insulation troubles; its basement is the fact that glass-fiber reinforced plastic (polymer composite) material is an electric insulator. in the international literature there are many examples dealing with the topic of insulated and glued insulated rail joints’ failures. in the following paragraphs the authors summarize the main consequences from their literature review, regarding and focusing on the relevant aspects. it can be noted that sandwich [29] and composite, laminated structures and products [30] are applied in a very wide range of industry. in the case of rail joints, the damage is due to a relatively low moment of inertia of the fishplate pair (compared to the whole rail profile) [31] as well as the evolved high stresses that can result in plastic deformation and lipping in the rail steel [32–35]. the lipping phenomenon is a typical failure mode (plastic deformation) at the rail ends because of the ‘closing’ of the rail heads and passing wheels. the other significant damages, namely, failure modes that are prone to happen, are summarized in the following list [36]: battering, shelling and metal flow of rail end; endpost battering, delamination; cracked joint bar (fishplate); railhead spalling; chipping at rail end and gage corner; railhead crushing; opening (pull apart) of insulated rail joint; dipping of insulated rail joint. the maintenance cost of rail joints is relatively high; it is critical from the aspect that avoiding of glued insulated rail joints can be solved in a very complicated manner in cwr tracks. it would need difficult solutions to be able to operate modern railway signaling and processing system [31]. the modified rail head geometry (in longitudinal direction, i.e. rounded/offset rail end) is able to decrease the arising stresses and to increase rail joint lifetime [37–38]. the evolution of lipping phenomenon can be hindered by application of higher rail steel qualities and/or rails with heat-treated heads [32–33]. the choosing of adequate adhesive materials influences the deterioration rate [32, 35] while the rail end post with appropriate material and thickness is a critical point [32]. next to them, the significance of the gluing pattern and the applied adhesive quantity is relatively high [37]. 686 a. németh, s. fischer 3. applied methods, main characteristics of rail joints and carried out laboratory tests 3.1. applied methods there is currently no valid standard that contains requirements related to glued insulated rail joints assembled with glass-fiber reinforced fishplates. the specifications and parameters of the european pre-standard (wg18/dg11) [39] were adopted and applied for the authors’ laboratory tests. this pre-standard has parameters and testing methodologies only for glued insulated rail joints with steel fishplates. however, in international literature there are more connected and relevant standards and specifications, like as 1085.12:2002 standard [40] and austrian railways’ technical specification [41]. yet, these could not be used in the authors’ research due to the main differences between polymer composite and steel materials. steel fishplated structures have significant lower deformations than others assembled with plastic ones because e.g. the high difference in young-moduli as well as bending stiffness values. mth-ap type fishplates can be used in cwr tracks as joint bars in suspended rail joints only in the case of temporary purposes. it means that if a permanent steel fishplated glued insulated rail joint is damaged and it has to be replaced/exchanged, glass-fiber fishplates are able to be installed without adhesive but they can be kept at that place for 6 months at most while the highest allowed speed is 120 km/h [42]. after 6 months, they must be replaced with new, steel fishplated girjs. one of the purposes of the research is to investigate in the period between 2015 and 2020 as well as some years earlier (i.e. between 2010 and 2012) (detailed in [38]) whether the mth-ap fishplates were able to be utilized in cwr tracks or not. based on the previously mentioned specifications, the authors planned and conducted more tests: (i) in laboratory; (ii) on site (i.e. in real tracks); (iii) they made fe models and have run 2d simulations. the laboratory test parameters were defined in accordance with the following standards and specifications: (i) axial pulling tests considering δtrail=90 °c (δtrail is the change in rail temperature in [°c]) according to d.12/h. hungarian prescription [43]; (ii) axial pulling tests with the consideration of δtrail=50 °c according to wg18/dg11 pre-standard [39]; (iii) static and dynamic bending tests according to wg18/dg11 [39]. it has to be noted that axial pulling tests were only conducted for the higher δtrail value while the static and dynamic bending tests were performed for more cases than the european pre-standard required. for the laboratory tests the rail joints were assembled with the application of two types of adhesives (notation ‘a’ or ‘b’ refer to the type of adhesive). the main characteristics of the tested specimens are detailed in chapter 3.2.1. even in the axial pulling tests as well as in the static and dynamic bending tests the authors have applied data analysis and process with the help of ms excel software. the main evaluations are based on engineering and scientific aspects and methods. the authors have tried to formulate the results and the main conclusions from the point of view of their being able to be used in real applications, under real circumstances. up to now there is no published, useful, extensive literature which can be adequate for the glued insulated rail joints with special polymer composite fishplates. investigation of glued insulated rail joints applied to cwr tracks 687 3.2. main characteristics of the rail joints and the conducted laboratory tests 3.2.1. main characteristics of the rail joints in this chapter the main properties of the elements of the tested rail joints are specified. it means that the authors give only the relevant ones, not all the available data. rails: for each of the assembled specimens (for axial pulling tests as well as static and dynamic bending tests) ‘old’ and used rail pieces were applied. it means that they had a relatively low traffic loading in the past; in this way they did not have a significant wear in any direction. for the tested rail joints máv 48, 54e1 and 60e1 (in the following: rail profiles 48, 54 and 60 abbreviations are also applied, respectively) rail profiles were utilized. because all the profiles are standard ones, the geometrical data can be found in standards (e.g. msz en 13674-1 standard [44]) and data sheets (e.g. d.54 specification of máv [45]) or in technical books [36]. fishplates: steel fishplates were mth-p types, the glass-fiber reinforced (polymer composite) fishplates were mth-ap types (produced by the apatech factory in russia). they all are 900 m long and 6-hole fishplates with the hole geometry related to hungarian standard [42–43, 45]. the main parameters of the fishplates can be found in data sheets of the producers (they cannot be reached on the internet; the details can be found in [38]). fishplate bolts were m27 8.8 (in the case of 60 rail profile) and m24 8.8 (in the case of 54 and 48 rail profiles). the main characteristics of the applied adhesive materials are summarized in table 1. it should be mentioned that the adhesive type ‘b’ is a common adhesive material at the austrian state railways (öbb) for glued insulated rail joint with steel fishplates. hence the adhesive type ‘a’ ensured significant high shear strength capabilities during the authors’ previously carried out laboratory tests [46]. table 1 main characteristics of the applied adhesive materials adhesive type text details installation temperature [°c] open time [min] bonding/drying time [min] ‘a’ two-component polyurethane sealant and adhesive. thixotropic. binding mechanism: poly-addition. gluing for up to 5.0 mm thickness layers. +15…+35 10 60 ‘b’ two-component methyl methacrylate based adhesive. gluing for up to 5.0 mm thickness layers. +15…+30 13 720 688 a. németh, s. fischer 3.2.2. main characteristics of the axial pulling tests glued insulated rail joints have to bear very high axial forces in cwr tracks due to change of temperature of the rails compared to the so-called neutral rail temperature. in this case the german method [27] is applied for construction of cwr track (i.e. long rails mainly with rail welding connecting them). neutral rail temperature is ‘set’ during the last rail welding, and it means the rail temperature during this procedure. the neutral temperature range (the allowed interval) is almost between +15 °c and +28 °c. there are three neutral temperature values related to a railway track section that is cwr track: (i) neutral temperature of left rail; (ii) neutral temperature of right rail; (iii) neutral temperature of the track that is calculated as the average value of the previous two values. as above mentioned, these values are only related to a tracks section, not the whole railway line. these values are changing due to the expansion and the contract(ion) of the rails and the lateral movements of the track in the ballast bed, in summer and in winter, respectively. the difference between the real rail temperature values and the neutral rail temperature values is the basis of the calculation of the axial force (as well as axial stress) in rails due to temperature change. it is only one part of the reasons for occurrence of this kind of inner force (stress) in the rails, but it is one of the most relevant parts as well. in the case of the hungarian climate condition, the highest considered rail temperature is +60 °c, the lowest is –30 °c. it means that in the summer a relatively high axial compression force (as well as stress) can be formed and so can, in the wintertime, a relatively high tensile force (as well as stress). eqs. (1) and (2) give the formulas for calculation of the above mentioned force and stress value. railrailsteelsteeltempaxial δtaef =., (1) railsteelsteeltempaxial δte = ., (2) where faxial,temp. is the axial inner force in rail (profile) due to change in rail temperature [n], steel linear thermal expansion coefficient of the rail steel (1.15×10–5 1/°c), esteel is the youngmodulus of rail steel (2.1×1011 n/m2), arail is the cross sectional area of rail profile [m2], and axial,temp. is the normal stress in the rail (profile) due to the change in rail temperature [n/m2]. the wg18/dg11 pre-standard [39] has a requirement related to the considered value of trail=50 °c, as well as a safety factor (s=1.5) that has to apply both equations (eqs. (1) and (2)) as a factor. it means that altogether 75 °c (i.e. 50×1.5) change in rail temperature has to be taken into consideration during the calculations. the authors decided that the inner force and stress values are also determined for 90 °c (the difference between +60 °c and – 30 °c; according to d.12/h [43]) change in rail temperature. table 2 contains all the important parameters and their calculated values as well. investigation of glued insulated rail joints applied to cwr tracks 689 table 2 ultimate axial tensile forces, gap opening at ultimate axial tensile forces, as well as gap opening resistance values of specimens, during axial pulling tests [39] rail profile arail [m 2] faxial,temp. × s [mn] (trail=50 °c; s=1.5) faxial,temp. [mn] (trail=90 °c) máv 48 0.006292 1.13964 1.36757 54e1 (uic54) 0.006977 1.26371 1.51645 60e1 (uic60) 0.007670 1.38923 1.66707 in fig. 4 the principle arrangement of the axial pulling test is represented. the structure that was applied for the axial pulling tests is very strange equipment because the authors had to solve the main problem of the test procedure: the application of high axial forces. in this case tensile force is loaded onto the rail joint and this force has to be symmetrical just as it should act in the mass point of the rail joint. because the application of tensile force with this high value is significant and complicated, the authors decided that compression force(s) will be transformed to tensile force(s). four steel cantilevers were needed for it, namely, those that are fixed to the rail joints’ end with high strength fishplate bolts (m27, 8.8). these cantilevers have additional cross-horizontal screws (8 pieces: 4 pieces on the top (fig. 3) and 4 pieces on the bottom side of the structure). these bolts bear a relatively high bending moment due to the eccentric loading by hydraulic jacks/pistons. during the tests, the loading force and the gap opening are measured and recorded. fig. 3 test set-up of axial pulling test the main parameters of the test and the test set-up are summarized in table 3. it has to be noted that the glued insulated rail joints with only glass-fiber reinforced fishplates were tested, but there were also insulated rail joints assembled by glass-fiber reinforced fishplates (i.e. without adhesive). during the axial pulling tests all of the applied parameters and characteristics were in accordance with the pre-standard wg18/dg11: the environmental parameters (air temperature, humidity, etc.) as well as the other relevant parameters i.e. speed of loading, the value of maximum loading force, etc. 690 a. németh, s. fischer table 3 applied parameters during axial pulling tests parameters value length of specimens ~2500 mm fishplate bolts m24 8.8 (in the case of 48 and 54 rail profiles) and m27 8.8 (in the case of 60 rail profile) quality bolts fishplate types mth-ap type fishplates gap rail end post (4.0 mm) ± 0 mm (this is considered as ‘0’ state) adhesive types type ‘a’ and type ‘b’ force effort 2×200 tons hydrafore hydraulic piston (type hydrafore yg 200100; work range: 0-700 bar) measurement of force with the help/base of hydraulic pressure (700 bar means ~4000 kn, the system was 500 bar, i.e. 500 bar means ~2857.143 kn) measurement of displacement with the help of an inductive transmitter (lvdt) on the rail head (type hbm w50tk; measurement range: 50 mm; output signal: 80 mv/v; precision: 0.4%; shift of ’0’ point: 0.05%) maximum allowed load of the set-up 220.0 tons (~2200 kn) because of the load bearing capacity of the steel cantilevers fix onto the rail joints by bolts) 3.2.3. main characteristics of the static and dynamic bending tests the parameters for the vertical static and dynamic 3-point bending tests are summarized in the following paragraphs. in the first round, laboratory tests were carried out with the use of three rail systems (máv 48: in this case 48.5 kg/m; 54e1 and 60e1); altogether 9 pieces of specimens were assembled for these tests. in the reality conducted tests were as below according to the pre-standard wg18/dg11 on every specimen in this order (from point a) to d)). it has to be mentioned that the deflection values were measured and registered at four points (two in the center of the rail joint to avoid and/or minimize errors of measurements due to twisting, and the other two at the ends of the specimens to be able to consider the uplifting effect). all of tested rail joints (9 pieces) with three rail systems were made by two types of adhesives (6 pieces with adhesive type ‘a’, 3 pieces with adhesive type ‘b’). the specimens were glued insulated rail joints with polymer composite fishplates. a) static bending tests without break with the consideration of three different support bay length values, before fatigue (‘bf’): 1490 mm, 1200 mm and 1000 mm; b) fatigue test (i.e. dynamic bending test) with a load cycle of 3.5 million, the considered-applied bay length value was 1200 mm; c) static bending test without break with the previously mentioned three different support bay length values, after fatigue (‘af’); d) static bending test until break (collapse) using of support bay length value of 1490 mm. investigation of glued insulated rail joints applied to cwr tracks 691 in the second round there were more tasks related to additional 3 pieces of specimens (one specimen from each rail profile) which were glued insulated rail joints with polymer composite fishplates: the static bending tests were carried out without break considering 13 different support bay length values between 900 and 1490 mm with 50 mm steps, before fatigue (‘bf’). the loadings consist of two phases: in the first phase, a preload is applied to achieve the desired position of the tested rail joint. in the second phase, the loading was made up to the calculated maximum force (holding it at this value for 2 minutes) and then decreased the force to the unloaded (‘0’) position (fig. 4); fatigue test with a load cycle 3.5×106 in more steps (support bay length value was 1200 mm) for all the three specimens; after every 0.5×106 cycles, a static bending test was performed until 3.5 million cycles (i.e. between 0.5 and 3.5 million cycles) without break considering 13 different support bay length values (900...1490 mm), (‘af’); static bending tests after 3.5 million cycles in the case of support bay length values 600...850 mm considering symmetrical support arrangement as well as in the case of support bay length values 600…950 mm considering asymmetrical support arrangement. fig. 4 set-up of test arrangement of static bending test, supplemented by the position of the applied lvdts (from 1 to 7) in the third round in the case of glued insulated rail joints with steel fishplates static bending tests were performed after 3.1 million cycles, then fatigue test with 0.4 million cycles and additional static bending tests after fatigue of 3.5 million cycles. related to steel fishplated glued insulated rail joints symmetrical and asymmetrical support arrangements were taken into consideration regarding static bending tests. in the case of second and third rounds the vertical displacement (deflection) values were measured and recorded on the rail head at 7 points. when changing the support bay length values (both symmetrical and asymmetrical support arrangements), the lvdts (linear variable differential transformers) are placed in the calculated new position in each case. the deflection values were measured and recorded with a precision of 0.001 mm and an accuracy of 0.4%. in the bending tests (both the static and dynamic ones) there were 3 specimens with polymer composite fishplates and 3 specimens with steel fishplates. in the 3-3 specimens there were 1-1 specimen assembled with the earlier mentioned three rail profiles. 692 a. németh, s. fischer the staff of máv-thermit ltd. prepared all the glued insulated rail joint specimens in order to conduct laboratory bending tests. each of the rails was cut to a length of nearly 85 cm, i.e. the total length of the rail joint was 1700 mm). in order to make them able for comparison, in the second round the specimens were tested without application of adhesive materials. these measurements were carried out only related to three different support bay length values (1000, 1200 and 1490 mm). all of the bending tests were performed considering test parameters in accordance with the standard wg18/dg11 [39]. the requirements for frequency (f) of the dynamic bending test is 3…10 hz; the considered fvert.,min.=5 kn, fvert.,max. can be calculated regarding the methodology shown in the following paragraphs, 3×106 loading cycles, the temperature of the structural elements during the tests must not exceed 50 °c. using eqs. (3) and (4) the considered maximum vertical loading force can be calculated. max0.125 ( )r c steel railm q e i w=       (3) l m f r max.vert = 4., (4) where mr is the considered bending moment according to pre-standard wg18/dg11 [nm]. c is the safety and correction factor (1.5 for suspended joints), q is the static wheel load (×125 kn), irail is the inertia moment of the rail cross-section with respect to the horizontal axis [m4], wmax is the maximum deflection (vertical displacement) of the glued insulated rail joint (×1.5 mm), fvert.,max. is the maximum vertical loading force [kn] and l is the support bay length value [m]. table 4 contains the effective maximum loading force values that are taken into consideration in the calculations. they differ from the values calculated by using eqs. (1) and (2) because parameter mr was determined in a modified method. table 4 shows the mr values recommended by pre-standard wg18/dg11, but the authors changed them a little bit, using the zimmermann-eisenmann calculation method [26] considering lm71 type vehicle load 4×250 kn with a vehicle axle space of 1.60 m [47]. during the calculation the authors considered the following parameters: ▪ static axle load, 2×q=250 kn; ▪ distance of axles (axle space); x=1.60 m; ▪ speed; v=160 km/h (in the case of 54 and 60 rail profiles); ▪ speed; v=100 km/h (in the case of 48 rail profile); ▪ probability is 99.7%; t=3 (student-distribution); ▪ track condition; =0.15 (average); ▪ effective length of half sleeper; ℓ=950 mm; ▪ effective with of half sleeper; b=300 mm; ▪ considered sleeper space; k=1200 mm (a non-supported sleeper is taken into consideration, k=2×600 mm=1200 mm); ▪ bedding modulus; c=0.10 n/mm3; ▪ dynamic amplificator factor; daf=1.714 (in the case of 54 and 60 rail profiles); ▪ dynamic amplificator factor; daf=1.285 (in the case of 48 rail profiles); ▪ characteristic length; lch=1014.656 mm (in the case of 60 rail profile); ▪ characteristic length; lch=949.834 mm (in the case of 54 rail profile); and, ▪ characteristic length; lch=882.346 mm (in the case of 48 rail profile). investigation of glued insulated rail joints applied to cwr tracks 693 the other details of the calculation are not published; the result of the modified bending moment values (mr) to be considered are 42.63; 40.85 and 34.71 knm, related to 60; 54 and 48 rail profiles, respectively. based on the considered modified bending moment values, calculated fvert.,max. values are determined; they can be seen below (they are only detailed related to l=1.2 m; the other calculation can be performed using eq. (4)): 142.00; 136.20 and 115.70 kn, related to 60; 54 and 48 rail profiles, respectively. as an example a bending moment graph is given in fig. 5 based on the calculated data. -20 -10 0 10 20 30 40 -5.00 -3.00 -1.00 1.00 3.00 5.00 7.00 9.00 b en d in g m o m en t m r [k n m ] distance (position) x [m] fig. 5 bending moment vs. longitudinal position diagram (rail profile: máv 48; calculation method: zimmermann-eisenmann; lm71 vehicle load; maximum positive bending moment mr=+34.71 knm at x=0.00 m and x=4.80 m positions) the lvdts during static bending tests were hbm w50tk types (measurement range: 50 mm; output signal: 80 mv/v; precision: 0.4%; shift of ’0’ point: 0.05%) (see fig. 4). during the static and dynamic bending tests all of the applied parameters and characteristics were in accordance with the pre-standard wg18/dg11 [39]: the environmental parameters (air temperature, humidity, etc.) as well as the other relevant parameters, i.e. speed of loading, etc. 4. results 4.1. results of the axial pulling tests in the case of the axial pulling tests the gap opening parameter was the most important to be analyzed. in figs. 6 and 7 the variation of axial pulling (tensile) force as a function of gap opening values is represented related to all the tested rail joints. in the legends of figs. 6 and 7, the numbers 48, 54 and 60 mean the rail system (rail profile) of the rail joints, letter ‘a’ and ‘b’ are related to the type of the applied adhesive material (type ‘a’ and ‘b’, respectively), ‘wg’ means ‘without gluing’. 694 a. németh, s. fischer 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0.0 2.5 5.0 7.5 a x ia l te n si le f o rc e [k n ] gap opening [mm] 48_a11 48_a12 54_a11 54_a12 60_a11 60_a12 fig. 6 axial tensile force vs. gap opening diagram 1 0 200 400 600 800 1000 1200 1400 1600 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 a x ia l te n si le f o rc e [k n ] gap opening [mm] 48_b11 54_b11 60_b11 60_wg fig. 7 axial tensile force vs. gap opening diagram 2 the authors examined axial tensile force values at 0.1 mm gap opening (initial state). the results showed that the bolts and the adhesive material worked adequate. on the basis of the measurements the following statements can be formulated. in the case of specimens assembled with 48 rail profiles the adhesive type ‘a’ ensured approximately 1.8 to 2.7 times higher initial axial tensile force values than the specimens produced with adhesive type ‘b’ (124 and 185 kn, as well as 68 kn, respectively) at 0.1 mm gap opening, the gap opening values were approx. 5 mm at the ultimate axial tensile forces. regarding specimens with 54 rail profiles and adhesive type ‘a’ provided approx. 1.7 to 2.1 times higher axial tensile forces than ones with adhesive type ‘b’ at 0.1 mm gap opening value, the final gap opening was 20% higher in the case of specimens with adhesive type ‘b’ than with type ‘a’. in the case of specimens assembled with 60 rail profiles the measured axial tensile forces at 0.1 mm gap opening were nearly the same, but compared to the ultimate tensile investigation of glued insulated rail joints applied to cwr tracks 695 force values they were only their 0.037-0.047%. it means that the rail joints opened at relatively low loading states and the 4.00 mm gap evolved at the beginning of the tests. the maximum gap opening value of specimens made with adhesive type ‘a’ was approx. 3.4 mm. in the case of specimens ‘wg’ the axial initial gap opening (0.1 mm) developed at even 17 kn, the ultimate gap opening value was approx. 16 mm. regarding specimen 60_a11 (it was the first measured specimen) the break did not occur even for the third trying; in the case of the first and the second trying the m27 8.8 bolts (that fixed the loading cantilevers to the rail joint specimen) were sheared, the specimen was able to bear the required axial tensile force. the gap opening resistance (unit kn/mm) was calculated related to all of the specimens. this parameter characterizes the stiffness of the glued insulated rail joint against gap opening at a given tensile force. the higher the gap opening resistance, the lower the evolved gap at the same tensile force is, i.e. at a given force interval. this parameter has to be determined for the constant tangent section of the loading diagrams. table 4 shows not only the gap opening resistance values representing the tested specimens, but the ultimate axial tensile forces as well as the gap opening values at these ultimate loading force values. table 4 ultimate axial tensile forces, gap opening at ultimate axial tensile forces, as well as gap opening resistance values of specimens, during axial pulling tests designation of specimen ultimate axial tensile force [kn] gap opening at ultimate axial tensile force [kn] calculated gap opening resistance [kn/mm] 48_a11 1317.30 5.050 460.64 48_b11 1346.00 4.650 344.72 48_a12 1547.20 4.606 499.85 54_a11 1547.20 3.603 582.03 54_b11 1433.90 4.653 369.13 54_a12 1605.30 3.725 535.57 60_a11 1767.80 3.378 556.60 60_b11 1420.90 6.069 383.96 60_a12 1645.10 3.406 545.62 60_wg 1332.20 16.106 146.96 based on the obtained results (see figs. 6 and 7 as well as table 4) it can be concluded that the specimens had practically very low (or almost no) resistance in the initial intervals, although better for specimens made using adhesive type ‘a’). this statement is true not only for glued specimens, but the non-glued (‘wg’) one. it means that approx. only the 0.04% of the ultimate axial tensile forces can be obtained at 0.1 mm gap opening. it has to be noted that the higher the gap opening, the higher the risk for failure of glued insulated rail joints is; it can happen earlier during railway operation. the question can be formulated why adhesive material in rail joints is applied if these joints do not have better behavior against gap opening at the initial stage. (it has to be noted that rail joints with adhesive material will have significantly better behavior during the lifetime under a repeated loading.) the axial tensile stresses were then determined and plotted into fig. 8 the data (stress values) are related to the center line of the rail joint (taking into account the crosssectional area of the fishplate pair). 696 a. németh, s. fischer of course, the ‘style’ and the ‘appearance’ of the axial tensile force and axial tensile stress diagrams are almost the same, but the different cross-sectional area of the fishplates (i.e. related to rail profiles 48, 54 and 60) influences the peak values as well as the tangent of the curves. these stress values are important; they can be compared to the limit values related to the base material. it has to be noted that higher normal stress values evolved in the fishplates due to the reduced cross-sectional area considering the bolt holes. these stress values 51.6%, 42.3% and 36.2% higher than in fig. 8 (related to 48, 54 and 60 rail profiles, respectively). (the effective cross sectional areas are 6416 and 4232 mm2; 7346 and 5162 mm2; as well as 8212 and 6028 mm2; related to the nonweakened and the weakened cross-sections and 48, 54 and 60 rail profiles, respectively). 0 20 40 60 80 100 120 140 160 180 200 220 240 260 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 t en si le s tr es s [m p a] gap opening [mm] 48_a11 48_a12 54_a11 54_a12 60_a11 60_a12 fig. 8 axial tensile stress vs. gap opening diagram 4.2. results of the static and dynamic bending tests in this paragraph only the relevant results are introduced. because of many measurements related to many aspects and many specimens, the authors collected some of the main results which represent the behavior of the investigated glued insulated rail joints. in every figure in this paragraph the abbreviation ‘bf’ means ‘before fatigue’, ‘af’ means ‘after fatigue’. where there is not any supplementary information next to ‘af’ abbreviation, it is related to 3.5 million loading cycle. there are a lot of new introduced calculated parameters which are adequate for evaluation of glued insulated rail joints. in every case they are explained and they are combined and represented by unique diagrams, graphs, or maybe tables. the authors examined and calculated the deterioration rate of the specimens considering the carried out laboratory dynamic bending tests with the fatigue interval up to 3.5 million cycles, supplemented by static bending tests before and after the fatigue steps. these evaluations are based on the assessment of the variation of vertical deflection values of the specimens as a function of fatigue (loading) cycles. the values are able to be compared to the initial values considering each specimen, this initial value (‘0’ state) is the measured state and bending behavior of the specimens (during static bending tests). the calculations were carried out related to bending tests with support bay length values of 1000, 1200 and 1490 mm, and the examined three rail profiles (48, 54 and 60) were taken into account. investigation of glued insulated rail joints applied to cwr tracks 697 linear regression lines were plotted into all of the vertical load vs. deflection functions (diagrams), i.e. linear regression functions were calculated to relevant intervals. the main parameter of these linear regression functions is the tangent (slope) in kn/mm unit. during the calculations the averages of the vertical deflection values were considered related to the maximum vertical loading force at the middle-bay position, regarding each rail profile and each fatigue step. the determined so called ‘stiffness of rail joint’ parameter (its unit is kn/mm/m) considers the support bay length values. they were compared to the initial measurements that are related to the ‘before fatigue’ states. in table 5 the measured peak (ultimate) vertical force values, the peak (ultimate) bending moment are collected. the safety factor is 1.5 in the calculations. table 5 ultimate vertical force, ultimate bending moment as well as the calculated bending moment values considering safety factor, during the static bending tests until break (the support bay length value is 1490 mm) designation of specimen ultimate (peak) vertical force [kn] ultimate (peak) bending moment [knm] calculated bending moment considering safety factor [knm] 48_a1 196.32 73.13 52.065 48_b1 230.28 85.78 52.065 48_a2 244.68 91.14 52.065 54_a1 294.60 109.74 61.275 54_b1 240.24 89.49 61.275 54_a2 306.00 113.98 61.275 60_a1 308.52 114.92 63.945 60_b1 334.32 124.53 63.945 60_a2 430.08 160.20 63.945 in fig. 9 the variation of stiffness of rail joint is represented. fig. 10 gives examples for variation of the stiffness of the rail joint parameter as a function of support bay length between 900 and 1490 mm. 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 0 0.5 1 1.5 2 2.5 3 3.5 s ti ff n es s o f ra il j o in t [k n /m m /m ] fatigue cycle [million] 48_a3_900mm 48_a3_1490mm 54_a3_900mm 54_a3_1490mm 60_a3_900mm 60_a3_1490mm fig. 9 stiffness of rail joint vs. fatigue cycle diagrams; 48, 54, 60 rail profiles and 900 and 1490 mm support bay length values 698 a. németh, s. fischer 0 20 40 60 80 100 120 140 160 180 200 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 s ti ff n es s o f ra il j o in t [k n /m m /m ] support bay length [mm] 60_wg2 60_a3_bf 60_a3_af 60_steel_af (3.1 million) 60_steel_af (3.5 million) 60_a12 60_wg1 60_a2_bf 60_a2_af fig. 10 stiffness of rail joint vs. support bay length; 60 rail profile fig. 11 shows a newly introduced parameter, the so-called ‘stiffness ratio’. this parameter characterizes the rail joints compared to the initial state (i.e. ‘bf’) of the examined glued insulated rail joints; it means the stiffness ratio is 1.0 at 0.0 million fatigue cycle, related to all the investigated rail joints with each rail profile (48, 54 and 60). the values that are represented in fig. 11 are average values of the stiffness of rail joints; the considered values in the calculation of average are related to 900 and 1490 mm support bay length values. there are two relevant phases in each graph. the first part is between 0.0 and 0.5 million fatigue cycles, the second part is related to between 0.5 and 3.5 million fatigue cycles. in this second phase linear regression functions were determined which have relatively high r2 coefficient (>.94). table 6 represents the ‘deterioration speed’ parameter related to specimens with 48 rail profile, its unit is kn/mm/m/fatigue cycle. it is calculated on the basis of ‘stiffness of rail joint’ (unit: kn/mm/m). in the ‘deterioration speed’ the number of fatigue cycles is also considered. actually it is the tangent of the calculated line of the ‘stiffness of rail joint’ between 0.0 and 3.5 million points (it is not regression line). y = -0.0491x + 0.9499 r² = 0.9844 y = -0.0419x + 0.8419 r² = 0.9476 y = -0.0292x + 0.9742 r² = 0.9851 0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 s ti ff n es s ra ti o [ -] fatigue cycle [million] 48_a3 54_a3 60_a3 fig. 11 stiffness ratio diagrams in the case of 48, 54, 60 rail profiles calculated from the average of stiffness of rail joints (900 and 1490 mm) investigation of glued insulated rail joints applied to cwr tracks 699 table 6 deterioration speed values of 48 rail profile specimens designation of specimens deterioration speed [kn/mm/m/fatigue cycle] 48_a1_1000 mm –0.070 48_b1_1000 mm –0.032 48_a2_1000 mm –0.137 48_a1_1200 mm –0.055 48_b1_1200 mm –0.041 48_a2_1200 mm –0.143 48_a1_1490 mm –0.055 48_b1_1490 mm –0.023 fig. 12 contains a graph related to (vertical) deformation line of glued insulated rail joint (with 48 rail profile) measured by 7 pieces of lvdts fixed on the rail head during static bending tests, in the case the support bay length value is 900 mm. the plotted graphs given in fig. 12 refer to the vertical deflection values related to the considered maximum vertical loading force (calculated from the zimmermann-eisenmann method see chapter 3.2.3 and [26]) of course, the measurements are available for each support bay length value and all of the considered and tested three rail profiles. in fig. 13 another special, new parameter is shown. it is the so-called ‘deflection parameter’ and it is determined in mm2 unit. this parameter is calculated as the integral (area) below the deflection line in the whole length of the rail joints. there are two phases in each graph. the first part is between 0.0 and 0.5 million fatigue cycles, the second part is related to between 0.5 and 3.5 million fatigue cycles. in this second phase linear regression functions were determined which have relatively high r2 coefficient (r2>0.80). -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 -850 -680 -510 -340 -170 0 170 340 510 680 850 v er ti ca l d is p la ce m en t [ m m ] measuring points 48_wg2_900 mm 48_a3_bf_900 mm 48_a3_af_900 mm fig. 12 measured deformation line of insulated rail joints assembled by polymer composite fishplates with and without gluing in the case of 900 mm support bay length, as well as 48 rail profile 700 a. németh, s. fischer y = 122.21x + 1752.2 r² = 0.9488 y = 201.04x + 3923.5 r² = 0.8924 y = 188.01x + 2066.4 r² = 0.7206 y = 311.34x + 4424.2 r² = 0.8148 y = 74.904x + 1371.3 r² = 0.9297 y = 83.99x + 3085.5 r² = 0.9423 0 1000 2000 3000 4000 5000 6000 d ef le ct io n p ar am et er [ m m ²] fatigue cycle [million] 48_a3_900 mm 48_a3_1490 mm 54_a3_900 mm 54_a3_1490 mm 60_a3_900 mm 60_a3_1490 mm fig. 13 deflection parameters vs. fatigue cycle 5. conclusions the authors formulated theses that can be read in the following paragraphs based on the results in chapter 4. in the first part they phrased the statements related to static and dynamic bending tests while in the second part there are statements related to axial pulling tests. the authors proved that there are two, well-separated sections, phases in the stiffness ratio vs. fatigue cycle functions related to the tested, investigated glued insulated rail joints assembled by glass-fiber reinforced fishplates. there is a border line between the two phases at the 0.5 million fatigue cycle value. in the second phase the deterioration (variation) can be characterized by linear regression functions that have significant r2 coefficients. the authors introduced parameters connected to the static and dynamic bending tests of glued insulated rail joints in laboratory. these are the ‘stiffness of rail joint’ [kn/mm/m], the ‘deterioration speed’ [kn/mm/m/fatigue cycle], that can help with the evaluation and comparison of the tested glued insulated rail joints. the authors stated the variation of stiffness of rail joints related to the examined glued insulated rail joints made by glassfiber reinforced and steel fishplates as a function of fatigue cycle. in the case of polymer composite fishplated glued insulated rail joints these variation speed (the tangent of stiffness of rail joint graphs as a function of fatigue cycle) is 1.17%, 2.16% and 8.54% higher compared to the tested steel fishplated glued insulated rail joints related to the 60, 54 and 48 rail profiles, respectively. the types of fishplates are mth-ap and mth-p (polymer composite and steel, respectively). the integral function of the deflection graphs (i.e. the area ‘below’ the deflection graphs) was determined as a function of support bay lengths; they are adequate to assess the whole deformation line of the holding device (i.e. the glued insulated rail joint) in the vertical plane. the variation of these integral functions are proved that it can be characterized by quadratic polynomial regression functions in any cases (r2>0.97). based on performed calculations the authors stated that the previously investigation of glued insulated rail joints applied to cwr tracks 701 mentioned integral value is 1.27%, 1.91% and 1.61% higher in the case of glued insulated rail joints with glass-fiber reinforced fishplates than ones with steel fishplates (related to 60, 54 and 48 rail profiles, respectively). the last statements mean (in the case the vertical deflection values at middle-bay are known) that the deformation lines of glued insulated rail joints with polymer composite fishplates are close to a symmetric triangle shape while the steel fishplated ones’ deformation lines are close to a beam without joint (i.e. they have quadratic parabola shapes). based on the axial pulling tests it is proved that adhesive type ‘a’ ensured an average 10% higher ‘gap opening resistance’ than adhesive type ‘b’ in the case of glued insulated rail joints assembled by glass-fiber reinforced fishplates (the effective values are 10.96%, 9.74% and 9.00% related to 60, 54 and 48 rail profiles, respectively). the adhesive type ‘a’ can be applied more universal in the case of the tested rail joints; in the case of using of adhesive type ‘b’ the axial pulling graphs spectacularly differ just as they guaranteed higher deviation related to the peak axial tensile stresses (and forces). the authors would like to summarize the experiences not only of their laboratory tests, but tests on site, too, because this paper is considered as a concluding article. based on the conducted laboratory [46, 48] and tests on site [49] some relevant statements can be drawn which are able to be useful for the application of mth-ap fishplated in glued insulated rail joints: ▪ the behavior and service life of the polymer composite fishplated glued insulated rail joints depend significantly on the accuracy of the preparation of the rail joint. when mounting, no gap is allowed between the end of the rail and the end of the rail end post on either side. ▪ those adhesive types that can only be used in thick layers are not suitable for gluing the rail end post to the end of the rail because the thick layer is able to pug due to traffic (passing wheels) after a relatively short time. ▪ high attention has to be given to the surface and surface design of polymer composite fishplates because the adhesive material may not be able to adhere sufficiently to lacquered and/or surface-coated fishplates. ▪ the fishplates have to fit exactly within specified tolerances to the rail web. ▪ the required torque for fishplate bolts is min. 850 nm. ▪ the applicable support set-up of track considering glued insulated rail joint is suspended joint. ▪ min. 3-3 pieces of supporting sleepers next to the glued insulated rail joints have to be tamped carefully, unsupported sleepers are not allowed. ▪ the appearance of ‘whitening’ visible to the eyes in the lower part of the joint (i.e. fishplates), even in the middle, makes it likely that the fishplate crack will develop nearby. the crack in the fishplate is not followed by a sudden fracture (break), it increases slowly with the repetition of the load. the cracked fishplate must be replaced. ▪ in the case of glued insulated rail joints with mth-ap fishplates, the allowed static axle load is 225 kn and the allowed speed is up to 160 km/h (for 54 and 60 rail profile), as well as 100 km/h (for 48 rail profile). ▪ the rail temperatures during installation should be taken into account in accordance with máv regulation d.12/h [43]. ▪ the relaxation of cwr tracks has to be performed according to máv regulation d.12/h [43]. 702 a. németh, s. fischer ▪ production of glued insulated rail joints assembled by mth-ap fishplates is preferred in factory, not on-site. ▪ the fishplate bolts should be min. 10.9 strength quality. ▪ the connecting cut (trim) of the rails should be perpendicular to the longitudinal axis of the rail. the inclined cut is not preferred. ▪ the application of heat-treated or special rail steel categories is preferred, but not prescribed. ▪ in the case of structural ballast gluing is planned for the glued insulated rail joint, it is necessary to consult with the owners of the patent law. 6. summary and further research possibilities this article summarizes the research carried out in the topic of glued insulated rail joints assembled by glass-fiber reinforced (polymer composite) fishplates. the authors introduced their own performed laboratory tests, as well as the relevant results and the connecting evaluations and recommendations for application. the results were formulated as theses, which were drawn as scientific statements. at the end of the article, the authors drew relevant statements, main aspects and recommendations related to application of mth-ap (produced by russian company, apatech) based on the conducted laboratory tests as well as on-site tests and their results. it has to be mentioned that since 2016 there are four testing (experimental) sites in the international main railway line no. 1 in hungary (kelenföld-hegyeshalom state border). these glued insulated rail joints work well, no special problems there were related to them [38, 49]. the authors formulate some further, research possibilities in this topic: ▪ application of computer tomography related to the laboratory tests (i.e. axial pulling tests, as well as static and dynamic bending tests) to be able to assess the changing of fishplates’ inside material structure regarding fatigue cycles. ▪ there are possibilities for evaluating the deformation characteristics and their changing of fishplates. these technologies are the measurements by gom aramis, gom tritop [50], as examples. ▪ special 2d/3d fe modeling of fishplates as well as the whole rail joint. ▪ continuation of diagnostics and the data processing of field tests (i.e. measurements made by railway track geometry recording cars). ▪ taking into consideration of rail and wheel wear due to girjs. ▪ paying respect to application of fuzzy and fuzzy-random logics or neural networks during assessment of field tests. acknowledgements: the authors would like to thank for the support of máv (hungarian state railways), máv-thermit ltd., as well as universitas-győr nonprofit ltd. the authors have much to be thankful 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fiber-glass reinforced synthetic fishplates using in continuously welded tracks, pollack periodica, 13(2), pp. 77-86. 49. németh, a., fischer, s., 2019, field tests of glued insulated rail joints with usage of special plastic and steel fishplates, nauka ta progres transportu, 80(2), pp. 60-76. 50. németh, a., fekete, i., szalai, s., fischer, s., 2019, supplementary laboratory investigations of modern plastic-polymer fishplates for rail joints, nauka ta progres transportu, 84(6), pp. 86-102. https://pdfs.semanticscholar.org/7c15/18810a7a6f19100d9c90a3aa7f0ffb4bd4ef.pdf https://pdfs.semanticscholar.org/7c15/18810a7a6f19100d9c90a3aa7f0ffb4bd4ef.pdf https://www.saiglobal.com/pdftemp/previews/osh/as/as1000/1000/108512.pdf facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 805 816 https://doi.org/10.22190/fume210415057w © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fft-based implementation of the mdr transformations for homogeneous and power-law graded materials emanuel willert technische universität berlin, institute of mechanics, germany abstract. it is shown how the abel transform solution to the general axisymmetric normal contact problem for homogeneous and power-law graded elastic materials, which is paramount for the solution of different classes of tribological problems with the help of the method of dimensionality reduction (mdr), can be written in terms of explicit convolutions. these can be very efficiently evaluated with the 1d fast fourier transform (fft), which reduces the numerical complexity of the transformations from the order of n2 for the standard matrix-vector-multiplication (mvm) algorithm to the order of n. convergence and performance of the proposed method are studied in detail. as an illustrative example a fretting wear simulation based on the new implementation is shown, the results of which are compared to the standard mvm implementation. key words: method of dimensionality reduction, fast fourier transform, boussinesq problem in contact mechanics, power-law graded materials 1. introduction the boussinesq problem, i.e., the frictionless normal contact problem of elastic bodies, is one of the basic fundaments of contact mechanics. in the case of axisymmetric, convex contacting bodies within the half-space approximation, several qualitatively different classes of contact problems can be reduced to the frictionless, non-adhesive normal contact [1], e.g. the jkr-adhesive normal contact [2] the tangential contact in cattaneo-mindlin approximation ([3, 4]) or – based on the correspondence principle between boundary value problems of linear elasticity and linear viscoelasticity – the normal contact of viscoelastic bodies ([5]). the solution to the general axisymmetric boussinesq problem for homogeneous materials was first discovered by schubert [6]. a very clear and convenient interpretation of the solution is given by the method of dimensionality reduction (mdr, [7]); the solution received april 15, 2021 / accepted august 02, 2021 corresponding author: willert emanuel technische universität berlin, institut für mechanik, sekr. c8-4, straße des 17. juni 135, 10623 berlin, germany e-mail: e.willert@tu-berlin.de 806 e. willert consists of a series of abel-like integral transformations, which will be given in detail below and whose numerical implementation in terms of matrix-vector-multiplications (mvm) is straight-forward and usually does not pose severe problems [8]. however, the computational complexity of the mvm is of the order of n2, where n denotes the number of elements/nodes used for the numerical transformations. if the spatial resolution needs to be very high, or the transformations need to be executed very often, e.g., for the simulation of fretting wear [9] or other highly transient phenomena, this complexity might still lead to undesirably high computational costs. on the other hand, as will be shown below, the transformations can be written in terms of simple convolutions, whose numerical implementation in one dimension can be accelerated with the (discrete) fast fourier transform (fft). the 1d-fft reduces the numerical complexity to the order of n (omitting logarithmic factors). therefore, in the present manuscript it will be shown, how the general solution to the axisymmetric boussinesq problem (in its mdr interpretation) can be obtained extremely fast based on the 1d-fft. for homogeneous materials this is done in section 2. nevertheless, the mdr has also been extended to explicitly incorporate contacts of functionally graded materials (fgm). within an fgm, the mechanical properties vary continuously over the volume, the material is inhomogeneous. examples are hardened surfaces as well as diverse biological and biotechnological systems such as bones, joints or their artificial variants, adhesive devices (e.g., on the feet of geckos) and cell membranes [10]. one of the most important classes of fgm are materials in which the elastic modulus e varies with the depth z. over time, different concrete functions e = e(z) have been considered (see e.g. the treatise by selvadurai [11]); the only case that allows a largely analytical treatment seems to be the power law e ~ zk. the solution to the axisymmetric boussinesq problem for power-law graded materials stems from giannakopoulos & suresh [12] and jin et al. [13]. in the latter work as well as by chen et al. [14] also the respective jkr-adhesive problem was solved. the implementation of these solutions into the framework of the mdr has been demonstrated by heß [15]. moreover, tangential contact problems of power-law graded materials can, within the cattaneo-mindlin approximation, be analysed with the mdr [16]. as the correspondence principle applies to viscoelastic graded materials, if the spatial and temporal variations of the moduli are separable [17], also contacts of viscoelastic power-law graded materials can be studied within the framework of mdr [18]. this is particularly interesting for biological graded materials, since these usually have viscoor poroelastic properties [10]. the axisymmetric normal contact solution for power-law graded materials is most conveniently formulated in terms of generalized abel transforms – the homogeneous case is, of course, always included as a limiting case, if the exponent k of the power-law equals zero – which can also be written as explicit convolutions and thus very efficiently evaluated by the 1d-fft, as will be shown in section 3. the application of this method for the simulation of fretting wear can be very fruitful in several applications, as it has been demonstrated that graded materials can offer a solution to the wear-fatigue dilemma in fretting [19]. finally, section 4 is devoted to discussions and conclusive remarks. fft-based implementation of the mdr transformations for homogeneous and power-law graded materials807 2. transformations for homogeneous materials 2.1. profile transformation consider a rotationally symmetric rigid indenter with the convex profile f(r), that is pressed into an homogeneous, isotropic, linearly elastic half-space with the effective young’s modulus e* = e/(1–υ2), with young’s modulus e and the poisson ratio υ (the determination of f and e* for the contact of two elastic bodies is elementary and can be found in any textbook on contact mechanics [1]). then, the contact solution is most conveniently formulated in terms of the equivalent profile [1] 2 2 0 d ( ) ( ) , ( ) : d . d x g rf r g x g x r x x r = = −  (1) for example, the function g will give the relations between the macroscopic contact quantities – contact radius a, indentation depth δ and normal force fn [1] * * 0 ( ), 2 2 ( )d . a ng a f e a e g x x= = −   (2) to write the transform (1) as an explicit convolution, following bracewell [20] we introduce the substitutions 2 2 ˆ ˆ: , : , ( ) ( ), ( ) ( ).x r f r f g x g= = = =    (3) note that the function g will be given by the scaled derivative ˆ ˆd d d ˆ( ) 2 ( ). d d d g g g x g x = = =      (4) after substitution and integration by parts one obtains 0 0 ˆ1 ( ) ˆˆ ( ) d ( ) d , 2 f g f = = − −              (5) where the prime denotes the first derivative with respect to the given argument. the last integral shall be evaluated numerically on equidistant one-dimensional arrays, 2 : ( 1) , : ( 1) , : , , 1: , 1 i j l i h j h h i j n n = − = − = = −   (6) where n is the number of elements and l is the length of the simulation area for the spatial coordinates x and r. note that the discretization is equidistant in the squares of the spatial coordinates. applying the trapezoidal rule, it is (by convention the sum over an empty set shall be zero instead of ill-defined) 1 3 1 2 1 ˆ ˆˆ 1 . 2 i i j j g h f i f i j − =    = − + −     (7) 808 e. willert to demonstrate the performance advantage of the 1d-fft, the sum in the last equation has been evaluated both with mvm and as an fft-accelerated convolution. the calculation time for the pure transformation step (without build-up of a priori known matrices or vectors) as a function of the number of array elements n is shown in fig. 1 in doublelogarithmic scale. while the absolute value for the calculation time is not too relevant, as it depends on the computational setup, it can be easily seen, that the numerical complexity of the mvm is of the order of n2, while the fft complexity only increases with n. the scatter in the calculation time for the fft is probably due to performance variance of the fft depending on n. note that the numerical error mainly originates from the discretization and the trapezoidal rule. therefore, when comparing the mvm and fft solutions with analytical solutions, both variants exhibit the same convergence behaviour. fig. 1 comparison of the calculation time for the transformation step (7) with matrixvector-multiplication (mvm) and as a fft-accelerated convolution, depending on the number of array elements used for the transformations. 2.2. transformation of local displacements several classes of tribological problems can already be solved with the mdr only based on the one transformation given in the previous subsection, as the relations between the macroscopic contact quantities (contact radii, forces, and macroscopic displacements) are the same in the mdr model as in the axisymmetric original system [7]. this, e.g., already allows the extensive study of different classes of dynamic contact problems, like impacts, with the mdr [21]. however, local quantities, i.e., local displacements and contact stresses, do not exist explicitly within the mdr model but must obtained from the respective mdr quantities by yet different integral transforms. the local displacements (both normal and tangential) in the axisymmetric original system, u(r), and its mdr image, u(x), are connected via the transform [1] 2 2 0 2 ( ) ( ) d . r u x u r x r x = −  (8) fft-based implementation of the mdr transformations for homogeneous and power-law graded materials809 note that formally this is also the inverse transform to eq. (1), to obtain the axisymmetric profile f from the equivalent profile g. introducing ( ) : ( ) (0),u x u x u= − (9) and the substitutions ˆˆ( ) ( ), ( ) ( ),u u r u u x= =  (10) one obtains 0 ˆ2 ( ) ˆ( ) (0) d . 2 u u u= + −          (11) now, there are (among others) two ways to integrate this expression by parts, 0 2 ˆˆ( ) (0) ( )arccos d ,u u u   = +              (12) or, 0 0 ˆ2 ( ) d ( ) ˆ( ) (0) lim ( ) d , ( ) : . dx u x u u u x→     = + − + − =                          (13) obviously, the second variant has several disadvantages: the occurring limit value must exist and β(0) should be finite for the trapezoidal rule to be applicable without problems (that is e.g. not the case if u(x) is parabolic); however, the remaining integral in eq. (13) is a convolution which can be evaluated once again very efficiently with the 1d-fft. on the other hand, if the given restrictions are not met, a matrix-vector-multiplication based on the less restrictive eq. (12) will be preferable. 2.3. transformation between line loads and stresses the line loads in the mdr model, q(x), and the contact stresses in the axisymmetric original, σ(r), are connected via the abel inverse transform [1] 2 2 1 ( ) ( ) d . a r q x r x x r  = − −   (14) note that, once again, this transform is valid for both the normal and the tangential stresses or line loads. introducing the usual substitutions, ˆ ˆ( ) ( ), ( ) ( ),r q x q= =    (15) integrating by parts, 2 2 2 2ˆ1 ( ) 2 ˆ ˆ ˆ( ) d ( ) ( ) d , a a q a q a q    = − = − − + −  −                    (16) and discretizing based on eq. (6) and the trapezoidal rule, one obtains 810 e. willert 2 1 3 1 2 ˆ ˆ( ) , 1: 1, ( 1) , 1 ˆ ˆ . 2 j j j i i j j h q t j a h t h j q q i j − = +  = − − + = − = −      = − + −              (17) the remaining sum will again be evaluated with the 1d-fft. in fig. 2 the results of a convergence study for that algorithm with an analytical solution, 2 2 24 ˆ ˆ( ) , ( ) (2 ) 3 q a a= = − − +       (18) are given. the mean relative error (the star denotes the analytical solution) *1 * 1 ˆ ˆ1 : ˆ1 j j j j      − = −  = −  (19) is shown as a function of the number of elements used for the transformation in doublelogarithmic scale. all derivatives were determined by second-order finite differences. the convergence is obviously fast and stable. fig. 2 convergence study for the mean relative error depending on the number of array elements for the fft-based implementation of the discrete transform (17). 3. transformations for power-law graded materials in this section the convolution formulation of the relevant transformations is shown for power-law graded materials. naturally, all previous results for homogeneous media are recovered, if the exponent k of the power-law of the elastic grading equals zero. fft-based implementation of the mdr transformations for homogeneous and power-law graded materials811 3.1. profile transformation for inhomogeneous materials with an elastic grading in the form of a power-law the profile transformation between the axisymmetric profile f and the profile g reads [1] 1 2 2 1 2 2 1 0 0 ( )d d ( ) , ( ) : ( ) ( ) d . 1 d( ) x xk k k k f r r x g g x x g x f r x r r k xx r − − + −  = = = − +−   (20) introducing the same substitutions as in eq. (3) we obtain 1 0 ˆˆ ( ) ( ) ( ) d ,kg f += −       (21) and 1 ˆ2 d ˆ( ) ( ). 1 d kg g x g k −= = +    (22) with the same discretization as in eq. (6) and the trapezoidal rule we finally arrive at 1 3 1 1 1 2 1 ˆ ˆˆ ( 1) ( ) , 2 i k k k i j j g h f i f i j − + + + =    = − + −     (23) the sum contribution of which is a discrete convolution, which once again can be efficiently evaluated with the 1d-fft. 3.2. transformation of local displacements the transformation rule for the local displacements in the mdr model and the original axisymmetric system in the case of power-law graded materials is given by [1] 2 2 1 0 2 ( ) ( ) cos d . 2 ( ) r k k k x u x u r x r x +   =     −    (24) note that, once again, this formally is also the inverse transform of eq. (20) to obtain the axisymmetric profile f from the profile g. with the same procedure as in subsection 2.2 we obtain two different possible formulations after integrating by parts. the generalization of eq. (12) is given by ( ) 0 1 2 1 2ˆ ˆˆ( ) (0) ( ) cos ; d , 2 1 1 1 3 ( ; ) : , ; ; , 1 2 2 2 k k u u u u h k k k k h y k y f y k +    = + −        + + +  =   +             (25) with the hypergeometric function 2f1. the alternative, more restrictive formulation, which on the other hand includes an explicit convolution, i.e., the generalization of eq. (13), reads 812 e. willert 1 1 10 0 1 2 ( ) ˆ( ) (0) cos lim ( ) ( ) d , 2 ˆd ( ) ( ) : . d k k kx k k u x u u x u − − −→ −     = + − + −            =                    (26) the discretization procedure for eq. (26) works completely analogous to the previously showed one and shall therefore be omitted here for brevity. in fig. 3 the results of a convergence study for that algorithm with an analytical solution, 4 4(1 )(3 ) ( ) , ( ) 8 k k g x x f r r + + = = (27) are shown. the exponent of the power-law was chosen to be k = -0.5 (note that the value of k has practically no influence on the convergence behaviour). the mean relative error (defined as before) is shown as a function of the number of elements used for the transformation in double-logarithmic scale. all derivatives were determined by second-order finite differences. the convergence is stable but slower than the one for the pressure transform in fig. 2, the gradient (in double-logarithmic terms) of the curve in fig. 3 is close to minus one. fig. 3 convergence study for the mean relative error depending on the number of array elements for the fft-based implementation of the transform (26) of the local displacements for power-law graded materials. 3.3. transformation to obtain stresses for power-law graded materials it is unfortunately not that easy to directly transform between the line loads in the mdr model and the contact stresses in the axisymmetric system. however, in the elastic case, transformations exist between the local displacements in the mdr system and the contact stresses. for example, the pressure distribution is [1] 2 2 1 ( )d ( ) , ( ) a n k r c g x x p r x r −  = −  (28) fft-based implementation of the mdr transformations for homogeneous and power-law graded materials813 with a lengthy constant cn, which can be found in the respective literature [1] and which reduces to e* in the homogeneous limit, and a similar transform exists for the determination of the frictional shear stresses in the elastic tangential contact [1]. after completely the same procedure as in subsection 2.3, the discretized pressure distribution, i.e., the generalization of eqs. (17), is given by 1 1 2 1 3 1 1 1 2 ˆ ˆ( ) , 1: 1, ( 1) , (1 ) 1 ˆ ˆ( ) ( ) . 2 k kn j j k k k j i i j c p j h g t j a h k t h j g g i j + + − + + + = +  = − − = − = −   +    = − + −             (29) 4. exemplary fretting wear simulation let us demonstrate the application of the proposed method to a simple fretting wear simulation. fretting is a phenomenon that occurs in dry, partial slip, oscillating contacts; it causes both wear and fatigue and, despite its high localization, is highly relevant for the lifetime of various tribological systems in technology and biology [19]. consider the following contact situation: an initially parabolic indenter with the radius of curvature r is pressed into a half-space (which of the bodies is elastically deformable, does not actually matter) by a constant indentation depth d. moreover, the contact experiences small tangential oscillations with a displacement amplitude ua. these oscillations shall be too small to cause gross sliding of the contacting bodies; however, due to the partial slip in the contact there will be wear, which can be modelled by the archard law; as the wear will happen on a much larger timescale than the period of one fretting oscillation, we can treat one fretting cycle as the time step for the wear simulation. hence, the local change of profile during one fretting cycle is given by [9] wear 0 ( ; ) ( ; ) ( ; ), k f r t p r t u r t =   (30) with the wear coefficient kwear, the hardness of the worn body σ0 and the slip displacement distribution δu, which can be calculated analytically, as is shown in another manuscript [22] by the author. therefore, the wear simulation only requires the two-step transformation f → g – see eq. (1) – and g → p – see eq. (28) with k = 0 – in every fretting cycle. in fig. 4 the results for the time-evolution of the worn profile for different numbers of normalized fretting cycles are shown in normalized variables. the characteristic number of fretting cycles is given by [9] 2 0 0 0 wear 0 , a a d n k f u   = (31) with the initial normal force f0. the number of elements was chosen to be n = 2001. the results on the left have been obtained with the standard matrix-vector multiplication procedure; the results on the right were determined with the fft-based implementation proposed in the present manuscript. there are no notable differences between both variants. however, the fft-based implementation operates faster by a factor of five. 814 e. willert moreover, this speed advantage of the new implementation obviously increases, if more elements are used for the transformation to obtain a finer resolution. (a) (b) fig. 4 time-evolution of the worn profile f, normalized for the fixed indentation depth d, as a function of the radial coordinate r, normalized for the initial contact radius a0, for tangential fretting wear of an initially parabolic indenter and different numbers of normalized cycles, n/n0. the dashed black line denotes the limiting no-wear profile [23]. (a) results for the semi-analytic method with matrix-vector-multiplication (b) results for the semi-analytic method accelerated by the fft 5. conclusions it has been shown, how the abel-like integral transform solution to the general axisymmetric boussinesq problem can be written in terms of real convolutions to enable the evaluation of the integral transforms based on the 1d-fft. both elastically homogeneous materials and those with an elastic grading in the form of a power-law were considered. while the convolution formulation works perfectly for the profile transformation and the determination of the contact stress distribution, it is significantly more restrictive than the standard formulation in case of the determination of the local displacement distribution. moreover, it was demonstrated that the fft-based solution operates faster than the standard formulation, if the number of array elements exceeds 103 (obviously, the advantage of the fft will grow, if the number of array elements further increases). while modern computers are, of course, able to quickly perform the transformations in the traditional form, there might be applications for which the reduction in calculation time and memory space (for a very fine discretization) can come in handy, if the contact simulation is part of a more global (hybrid) numerical routine, e.g., for the coupled analysis of wear and fatigue in fretting. as the main numerical issues in the transformation result from the finite difference derivatives and the trapezoidal rule, the fft-based solution exhibits practically the same convergence and stability as the standard procedure. the fft-formulation can be used for highly transient contact simulations; in this case, one, however, must keep in mind that the one-dimensional arrays used for the fft are equidistant in the squares of the spatial coordinates; this also leads to a finer resolution at the edge of contact, which might be beneficial in several circumstances. fft-based implementation of the mdr transformations for homogeneous and power-law graded materials815 there exists a wide literature on the numerical implementation of the abel transform and its inversion (see the review paper by hickstein et al. [24] and the references therein), mainly in the framework of image processing. however, the algorithms used in that context are optimized to handle very noisy data and therefore prioritize stability over efficiency. this is usually not necessary in the framework of axisymmetric contact mechanics. acknowledgement: this work has been supported by the german research foundation under the project number po 810/66-1. moreover, the author is very grateful to mr. justus benad for valuable discussions on the topic. references 1. popov, v.l., heß, m., willert, e., 2019, handbook of contact mechanics: exact solutions of axisymmetric contact problems, springer-verlag, berlin heidelberg, 347 p. 2. barquins, m., maugis, d., 1982, adhesive contact of axisymmetric punches on an elastic half-space: the modified hertz-huber’s stress tensor for contacting spheres, journal de mécanique théorique et appliquée, 1(2), pp. 331-357. 3. jäger, j., 1998, a new principle in contact mechanics, journal of tribology, 120(4), pp. 677-684. 4. ciavarella, m., 1998, tangential loading of general three-dimensional contacts, journal of applied mechanics, 65(4), pp. 998-1003. 5. lee, e.h., radok, j.r.m., 1960, the contact problem for viscoelastic bodies, journal of applied mechanics, 27(3), pp. 438-444. 6. schubert, g., 1942, zur frage der druckverteilung unter elastisch gelagerten tragwerken, ingenieur-archiv, 13(3), pp. 132-147. 7. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springerverlag, berlin heidelberg, 265 p. 8. benad, j., 2018, fast numerical implementation of the mdr transformations, facta universitatis-series mechanical engineering, 16(2), pp. 127-138. 9. dimaki, a.v., dmitriev, a.i., chai, y.s., popov, v.l., 2014, rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction, international journal of solids and structures, 51(25-26), pp. 4215-4220. 10. liu, z., meyers, m.a., zhang, z., ritchie, r.o., 2017, functional gradients and heterogeneities in biological materials: design principles, functions, and bioinspired applications, progress in materials science, 88, pp. 467-498. 11. selvadurai, a.p., 2007, the analytical method in geomechanics, applied mechanics reviews, 60, 3, pp. 87-106. 12. giannakopoulos, a.e., suresh, s., 1997, indentation of solids with gradients in elastic properties: part ii. axisymmetric indentors, international journal of solids and structures, 34(19), pp. 2393-2428. 13. jin, f., guo, x., zhang, w., 2013, a unified treatment of axisymmetric adhesive contact on a power-law graded elastic half-space, journal of applied mechanics, 80(6), 061024. 14. chen, s., yan, c., zhang, p., gao, h., 2009, mechanics of adhesive contact on a power-law graded elastic halfspace, journal of the mechanics and physics of solids, 57(9), pp. 1437-1448. 15. heß, m., 2016, a simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, international journal of engineering science, 104, pp. 20-33. 16. heß, m., 2016, normal, tangential and adhesive contacts between power-law graded materials, presentation at the workshop on tribology and contact mechanics in biological and medical applications, technische universität berlin, november 14-17, berlin. 17. paulino, g.h., jin, z.h., 2001, correspondence principle in viscoelastic functionally graded materials, journal of applied mechanics, 68(1), pp. 129-132. 18. willert, e., 2020, ratio of loss and storage moduli determines restitution coefficient in low-velocity viscoelastic impacts, frontiers in mechanical engineering, 6, 3. 19. willert, e., dmitriev, a.i., psakhie, s.g., popov, v.l., 2019, effect of elastic grading on fretting wear, scientific reports, 9, 7791. 20. bracewell, r.n., 2000, the fourier transform and its applications, 3rd edition, new york: mcgraw-hill book company, 636 p. 816 e. willert 21. willert, e., 2020, stoßprobleme in physik, technik und medizin: grundlagen und anwendungen, springer vieweg, berlin, 241 p. 22. willert, e., 2021, a simple semi-analytic contact mechanical model for tangential and torsional fretting wear of axisymmetric contacts, symmetry, submitted manuscript. 23. popov, v.l., 2014, analytic solution for the limiting shape of profiles due to fretting wear, scientific reports, 4, 3749. 24. hickstein, d.d., gibson, s.t., yurchak, r., das, d.d., ryazanov, m., 2019, a direct comparison of high-speed methods for the numerical abel transform, review of scientific instruments, 90, 065115. 11924 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 275 292 https://doi.org/10.22190/fume230820049i © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper carbon nanotube under pulsed pressure marat a. ilgamov1,2,3, aigul a. aitbaeva3, igor s. pavlov4,5, sergey v. dmitriev6,7 1blagonravov institute of mechanical engineering, russian academy of sciences, moscow, russia 2institute of mechanics and engineering, ksc, russian academy of sciences, kazan, russia 3institute of mechanics, ufrc, russian academy of sciences, ufa, russia 4institute of applied physics, russian academy of sciences, nizhny novgorod, russia 5lobachevsky state university of nizhny novgorod, nizhny novgorod, russia 6institute of molecule and crystal physics, ufrc, russian academy of sciences, ufa, russia 7department of equipment and technologies of welding and control, ufa state petroleum technological university, ufa, russia orcid ids: marat a. ilgamov https://orcid.org/0000-0002-8766-8964 aigul a. aitbaeva https://orcid.org/0000-0002-4735-1671 igor s. pavlov https://orcid.org/0000-0002-2666-4943 sergey v. dmitriev https://orcid.org/0000-0002-6744-4445 abstract. the radial dynamics of a single-walled zigzag carbon nanotube under pulsed pressure is studied. uniform external pressure is applied instantly, then remains constant for a certain time, and then is instantly released. this loading scheme allows one to consider a carbon nanotube under plane strain conditions and replace it with a circular ring formed by one zigzag row of carbon atoms. the bending deformation of the ring in its plane is described by an equation based on the kirchhoffs hypothesis. an effective parameter is used, including the bending stiffness of the ring and areal density. the regimes of oscillatory motion and exponential growth of radial displacements are investigated depending on the magnitude and duration of the applied pressure. alternatively, the ring is analyzed in terms of a molecular dynamics model with a reduced number of degrees of freedom, taking into account the plane strain conditions. with the help of molecular dynamics, the limits of the thin shell theory are established. for (n, 0) cnts with n > 15, in the regime of small-amplitude vibrations, the discrepancy between the continuum model and molecular dynamics calculations does not exceed 5% and decreases with increasing n. key words: carbon nanotube, molecular dynamics, shell theory, pulsed pressure, natural frequencies received: august 20, 2023 / accepted december 05, 2023 corresponding author: sergey v. dmitriev institute of molecule and crystal physics, ufrc ras, oktyabrya ave. 71, 450054 ufa, russia e-mail: dmitriev.sergey.v@gmail.com https://orcid.org/0000-0002-8766-8964 https://orcid.org/0000-0002-4735-1671 https://orcid.org/0000-0002-2666-4943 https://orcid.org/0000-0002-6744-4445 276 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev 1. introduction free and forced vibrations of carbon nanotubes (cnts) have been analyzed in a number of studies [1-13] because they affect the operation of nano-devices such as sensors, actuators, etc. the physical and mechanical characteristics of cnts, including their behavior under static and dynamic loading, are often investigated using the methods of molecular dynamics and continuum mechanics [1-5]. in the latter case, the effective dimensions of cnts, effective elastic and mass parameters are introduced. in this work, the molecular dynamics and the theory of thin shells with a parameter determined from the comparison of natural frequencies within the framework of these models are used. the effect of a uniform dynamical pressure on a single-walled cnt is considered. the effect of pressures on cnts is discussed in reviews [6, 7]. there are experimental studies of the dynamic, structural and electrical properties of single-walled and multi-walled cnts, depending on the uniform static and dynamic pressure [8-13]. these works also consider theoretical modeling within the framework of molecular dynamics and continuum mechanics. the deformation of isolated cnts under static pressure is analyzed, which is a necessary step in studying the complex behavior of a cnt bundle under dynamic pressure. it has been shown [10, 11] that with an increase in static pressure up to p1≈3d/r, the cnt cross-section acquires the shape of an ellipse (here d is the cnt wall bending stiffness and r is the cnt radius). in the elliptic state, the radial stiffness of the cnt decreases by two orders of magnitude [11]. further increasing the pressure up to p2≈p1[1-ln(f2/f1)] results in a cross-sectional shape with two opposite points of zero curvature. with a further increase in pressure up to p3≈p1[1-ln(f3/f1)], the deformation of the cross-section increases and the electrical properties of the cnt change (transition to semiconductor state takes place). here f1, f2 and f3 are the cross-sectional areas corresponding to the pressures p1, p2 and p3. for a single-walled armchair cnt (10, 10), p1=1.55, p2=1.75 and p3=2.2 gpa [10]. the static stability and deformations of thin elastic circular rings under nonuniform pressures p=p0(1+qcos) were studied and it was found that the ring deforms only in a doubly symmetric fashion, while static buckling into an asymmetric pattern was not observed [14]. vibrational properties of long hollow cylindrical shells interacting with the external medium were analyzed in the works [15, 16]. the structural transformations in cnt bundles during laser compression and the resulting oscillations have been considered [8, 9, 12]. in these and other works [17, 18], raman spectroscopy is widely used in experimental studies. the behavior of cnts was studied experimentally at different distances between them in the bundle (from 1.5 to 1.7 nm). in [13], the change in the properties of a single multilayer cnt under shock compression through the surrounding elastic material was studied. the radial breathing mode frequency of various zigzag and armchair cnts with radii between 3.5 and 8.1 å have been determined using the first-principles calculations [19]. dispersion relations for long-wavelength phonons in graphite and achiral single-walled cnts have been analyzed using a continuum model [20]. analysis of vibration behavior of carbon nanomaterials is important for design of nano-mass, nano-force and strain sensors [21-29] as well as microelectromechanical systems for various purposes [7, 30]. dynamic loading of carbon nanomaterials is considered in solving important problems related to the damping of shocks and vibrations as well as projectile protection. molecular dynamics simulations and experiments were conducted to study energy dissipation in cnt carbon nanotube under pulsed pressure 277 films under ballistic impact [31, 32, 33]. energy dissipation by collapsing cnts in the bundle under shock loading was analyzed in the work [34]. in this work, we analyze the response of single-walled cnts to a uniform dynamic pressure within the framework of the theory of thin shells and molecular dynamics. 2. problem statement it is assumed that a time-varying external uniform excess pressure p(t) is applied at time t=0 to a single-walled zigzag cnt, which at t<0 is under the same external and internal pressure p0 (for example, atmospheric pressure). for a uniform pressure, zigzag cnt can be considered as a ring formed by single zigzag chain of carbon atoms (see fig. 1). as the length of the valence bond, we take l=1.273å which is slightly less than the experimental value in graphene, 1.4 å, due to the effect of curvature of the cnt wall, especially for cnts with a small radius. in the ring, the distance between the carbon atoms, projected onto the xy plane, is equal to a=lcos30º=1.102 å. fig. 1 (a) the structure of a zigzag cnt wall and (b) representation of the zigzag cnt by the chain model. the valence bond length is l=1.273å, the distance a=lcos30º=1.102 å, cnt radius is r. cross section of the zigzag cnt (n,0) is presented by n=2n carbon atoms numbered with the index i=1,…,n; the case of n=30 is shown in (b). a ring of width b=3l/2=1.91 å is considered with periodic boundary conditions along the z axis. the area per one c atom s=2.105 å2 is shown by dotted lines (equilateral triangle). a continuum mechanics model considers a ring formed by two cross sections at a distance b=3l/2=1.91 å symmetrically relative to a zigzag row of carbon atoms. the radius r is determined from the equality 2 sina r n  = , (1) where n is the number of atoms forming the ring. note that n=2n for the zigzag cnt (n,0) the area per atom is equal to s=2.105 å2. the mass of a carbon atom is m=1.99×10-26 kg or m=12 a.m.u. here we use angstroms, picoseconds, and electronvolt as units of measure for distance, time, and energy, respectively. in these units the carbon atom mass is m=1.244×10-3 ev·ps2/å2. the effective area density is 278 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev 2 2 4 ev ps 5.908 10 å m h s − = =  , (2) where  and h are the effective bulk density and ring thickness. 2. elastic ring model before pressure p(t) is applied, a perfectly circular ring performs small-amplitude free oscillations in its plane. these free vibrations will be represented as a sum of a few first harmonics. it is assumed that the pressure begins to act at the time of the greatest deviation for each harmonic. then the initial conditions with respect to the deflection function w(,t), where  is the central angle, are taken in the form 0 2 0 0 2 2 ( ,0) cos( ), ( ,0) 0, ( 1) , 0, m m m m w w m w w w m = − =  =   = = −     (3) where m is the harmonic number, w2 0 is the initial amplitude of the harmonic m=2. the overdot denotes the derivative with respect to time t. the uniform distribution of the amplitudes wm 0 corresponds to =0 and for >0 the amplitude decreases with the harmonic number. the linear equation of motion of the ring in its plane with respect to the deflection function w(,t) has the form [14]: 6 4 2 4 2 4 2 6 4 2 4 2 2 2 3 * 2 2* 3 2 2 2 3 0, 3 , , , . 12(1 ) w w w w w r w w p d d eh p d hp r          + + +  + + − =               = =  = =  −  (4) here, d is the effective bending stiffness, e and  are the effective modulus of elasticity and poisson’s ratio, respectively, p* 2 is the critical pressure at which the loss of stability of the circular shape of the ring with respect to the lowest harmonic m=2 occurs (it is indicated as p1 above [10]). since the width of the ring b enters the values of masses, stiffness and pressure, it is canceled in eq. (4). eq. (4) does not take into account the influence of the surrounding gas environment, which is characterized by the parameter f r/(h), where f is the density of the surrounding medium,  is a complex function of the input quantities [14, 15]. the real and imaginary parts of  are responsible for the added mass of the medium and the radiation of energy from the tube. except in special cases not covered here,  does not increase the value of this parameter. therefore, the influence of the external environment on the motion of the tube can be judged by the dimensionless parameter f r/(h). if its value is much less than one, then the influence of the environment is small. taking into account the above value of h and the equality 2r≈an, this condition can be written as f r~10-6 kg/m2 or f n=4×104 kg/m3. for air at atmospheric pressure f =1.2 kg/m3, and this condition gives r~103 nm or n=3×104. we will consider the dynamics of a ring formed from atoms of carbon nanotube under pulsed pressure 279 order 10 to 100. therefore, we will not take into account the influence of the external environment. with a slow increase in pressure on the tube, the considered influence of the medium on the tube is absent. 3. molecular dynamics model cnt under plane strain conditions can be simulated with the use of the chain model proposed in [35, 36] and adapted for simulation of cnts in [37, 38]. the chain model considers one zigzag row of carbon atoms moving in the xy-plane as shown in fig. 1(b). the atoms are numbered with the index i=1,…,n; their positions are defined by the radius-vectors ri. the hamiltonian of the chain is 2 1 | | 2 n i b a i m h r u u = = + + , (5) where the first term gives the kinetic energy of the chain and the terms ub and ua describe the potential energy of valence bonds and valence angles, respectively. the energy of valence bonds is  = + −−  = n i iib arru 1 2 1 )|(| 2 , (6) where  =25.279 ev/å2 is the valence bond stiffness. the energy of valence angles is  = += n i iau 1 ]cos1[ , |||| ),( cos 11 11 iiii iiii i rrrr rrrr −− −− = +− +− , (7) where  =0.00166 ev is the valence angle stiffness. external forces applied to each atom were added to the equations of motion following from the hamiltonian eq. (5) to simulate a uniform conservative external pressure p(t). the chain model was successfully used for simulation of carbon nanostructures, in particular, structure and thermomechanical properties of cnt bundles were analyzed in the works [37, 40-45], folded and scrolled packings of carbon nanoribbons in [46], crumpled graphene under compression in [47]. chain model can also be applied to the analysis of other two-dimensional nanomaterials [48, 49]. for n carbon atoms of the cnt cross section numbered by the index i (see fig. 1(b)) the radial displacements can be presented as the sum of n/2+1 fourier harmonics 2 1 0 2 1 1 1 2 2 2 cos( ) cos sin n i n m m m mi mi w f f i f g n n n n n − =        = +  + +           , (8) where the coefficients of the expansion are calculated as follows 280 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev 1 0 1 0 2 cos , 0,..., 2, 2 sin , 1,..., 2 1. n m i i n m i i mi f w m n n mi g w m n n − = − =   = =      = = −      (9) the following initial radial displacements are applied 4 0 2 2 cos , 0. i m m i mi w w n w =   =     =  (10) it can be seen that initially only three first bending harmonics shown in fig. 2 are excited (the harmonics m=0 and m=1 are tensile and translational, respectively). we take equal amplitudes for the three harmonics, wm 0=10-3r, m=2,3,4, which is an analog of eq. (3) for =0. fig. 2 the three first radial vibrational modes of the cnt cross section table 1 cnt radius r (in å) and numerical values of  (in å2/ps) for various n and m=2,3 and 4. n r  (m=2)  (m=3)  (m=4) 18 3.179 54.75 52.84 52.71 30 5.278 56.71 56.03 54.78 42 7.382 57.55 57.09 56.45 54 9.487 57.84 57.55 57.18 66 11.59 57.98 57.78 57.49 78 13.70 58.06 57.92 57.74 90 15.80 58.12 58.01 57.86 102 17.91 58.15 58.07 57.95 114 20.02 58.18 58.11 58.02 126 22.12 58.19 58.14 58.07 138 24.23 58.21 58.17 58.11 150 26.33 58.22 58.19 58.14 carbon nanotube under pulsed pressure 281 the amplitudes of n/2+1 fourier harmonics are 0 0 2 2 2 2 | |, , 1,..., 2 1, | | . m m m n n w f w f g m n w f = = + = − = , (11) numerical examples will be given for the cnt with the number of atoms n=200. for such a large value of n we take =58.2 å2/ps, see table 1. from eq. (1) one finds the cnt radius r=an/(2)=35.08 å and from eqs. (2) and (4) d=2.001 ev. the equations of motion of the atoms representing the cnt cross section are integrated numerically by the symplectic stormer method of the sixth order of accuracy [39]. time evolution of the fourier harmonic amplitudes, wm(t), is analyzed. 4. natural vibration frequencies of cnt the theory of thin elastic shells gives the following expression for the natural circular frequencies of bending vibrations of a cylindrical shell [14] 2 2 2 ( 1) 1 m m m r m −   = + , (12) where m≥2 is the harmonic number and r is the shell radius. the numerical values of the parameter  included in eqs. (4) and (12) were determined in [40, 41] by comparing the frequencies m by eq. (12) and by the chain model; they are given in tab. 1. the analytical approximation (n,m) given in [41] gives a satisfactory result only for the mode m=2. here, this approximation is somewhat improved and has the form 5 2( , ) ( )n m a sm q n− = − − , (13) where a=58.2 å2/ps, s=7.2×103 å2/ps, q=8.4×103 å2/ps. for n>150 and m≤4 the value =58.2 å2/ps remains constant. in fig. 3, the dependencies (n,m) according to eq. (13) are given for m=2,3 and 4 together with the numerical data (scattered) obtained with the help of the chain model. the value of  increases with an increase in the number of atoms n and decreases with an increase in the harmonic number m. the ratio of the wavelength around the circumference of the ring 2r/m ≈ an/m to the thickness h is an/(mh). taking into account the value of a=1.102 å and the value of h=3.3 å, which is the interlayer distance in graphite, we obtain this ratio equal to n/(3m). equation of motion eq. (4), based on the kirchhoff hypotheses [14], is valid for n/(3m)>10. it was obtained without taking into account the inertia of rotation of the ring element and transverse shear, which affect the solution for n/(3m)<10. in this case, it is necessary to use the timoshenko model, which takes these factors into account [50, 51]. on the other hand, at n/(3m)<10, there are less than 15 atoms per half-wavelength, and the inaccuracy of the equations of continuum mechanics becomes noticeable. taking into account the inertia of rotation and transverse shear leads to a decrease in the natural vibration 282 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev frequencies in comparison with the result based on kirchhoff hypotheses. this trend is seen in fig. 3. fig. 3 dependence of the parameter  entering eq. (13) on the number of atoms n forming a single-layer ring and the harmonic number m of radial vibrations. the dots show the numerical values from [18]. 5. long-term pressure response at t=0 the external pressure p=const is instantly applied to the cnt and acts for a long time. by a long duration of a pressure pulse of constant intensity, we mean a time interval t<, an order of magnitude or more exceeding the oscillation period 2/m. 5.1. analytical treatment looking for a solution to equation eq. (4) in the form 2 ( )cos( )m m w w t m = =  , (14) from eq. (4) and eq. (14) we obtain 02 =+ mmm ww , (15) were 2 2 2 3 1 1 m m m    =  −  −  , (16) here, m is the natural frequency of the m-th harmonic at zero pressure defined by eq. (12), m is the frequency under the action of excess pressure p. the critical pressure value for the cnt with n=200, r=35.08 å, and d=2.001 ev, with respect to the harmonic m=2, in accordance with eq. (4), is carbon nanotube under pulsed pressure 283 3 * 4 2 3 3 ev 1.391 10 22.28mpa å d p r −= =  = . (17) from the condition m =0 one finds from eq. (16) the critical values of pressure for other harmonics, 2 * * 1 , 3 m m m p p − = (18) and hence, for the considered cnt, * * 4 3 2 8 3.709 10 59.41mpa 3 p p −= =  = (19) * * 4 4 25 6.955 10 111.4mpap p −= =  = (20) next, we analyze the cnt dynamics after an instantaneously applied constant external pressure p. in the case of an abrupt increase in pressure at time t=0 from 0 up to a constant value p, the solution to eq. (15) has the form ( ) m mt t m m mw t a e b e  − = + . (21) satisfying the initial conditions eq. (3), we obtain for the m-th harmonic 2 0 2 cos( ), 3 1,( ) cosh( ), 3 1. mm m m t mw t w t m     − =     − (22) thus, depending on the ratio of the effective pressure to the critical pressure  and the number of the harmonic m, oscillations or/and an exponential growth of the initial deflections can be excited after the load is applied. for a relatively small value of p<pm *=p2 *(m2-1)/3, the first, time periodic solution in eq. (22) is realized for the harmonic m. for example, for harmonic m=3 this condition is p<(8/3)p2 * and for m=4 this condition is p<5p2 *. in the case of a large value of the effective pressure, for one or several lowest harmonics the second solution in eq. (22) takes place. the harmonic number mr separating these lowest modes of motion is determined from the condition m=0 and is equal to 1 3rm  = +    , (23) where ⌊·⌋ denotes the round down operation. solution eqs. (14) and (22), can be written as 0 2 1 ( 1) cosh( ) ( 1) cos( ) cos( ) r r m m m m m m w m t m t m w  − − = = +   = −  + −        . (24) it can be seen from eq. (24) that the first harmonics from the 2nd to mr show an exponential increase in amplitude with time, and the larger m, the faster the growth. harmonics above mr oscillate with frequency m. the fastest growth of the amplitude occurs for the harmonic with the number m=ml, determined from the condition dm /dm=0. in the simplest case of the independence of the 284 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev parameter  from n (n>78) and the same initial deviation amplitudes (=0) this condition gives 2 21 2 12 2 l r r rm m m m    − + +    , (25) where ⌈·⌉ denotes the round up operation. the ml value is somewhat less than the mr value. therefore, the fastest increase in the amplitude is observed for the harmonic at the border of exponential growth and oscillatory motion. what has been said applies to the case of a uniform distribution of the initial deviation over harmonics (=0). at >0, the picture changes. at the beginning of the process, the harmonic with number m=2 prevails due to the nature of the initial conditions eq. (3) and the presence of the factor (m-1)- in solution eq. (24). with increasing time, the second term in eq. (24), 2-cosh(3t) becomes larger than the first, cosh(2t), and so on. thus, during dynamic deformation of a cnt with a decreasing distribution of small initial deflection over harmonics, a rearrangement in m occurs over time. at large time, the harmonic ml, which is determined approximately by eq. (25), becomes dominant. in the presence of limiters of movement along the radius, for example, contacting media, the harmonic ml may not become dominant. nonlinearities can also lead to this limitation. 5.2. chain model results let us first check the accuracy of eq. (16) in predicting the main frequency of cnt vibrations at an external pressure p<p2 *, i.e. for <1. molecular dynamics modeling is carried out as follows. the initial condition eq. (10) is applied to the cnt at pressure p, and the time evolution of the harmonic amplitudes m=2,3 and 4 is calculated from eq. (11). in fig. 4(a) time evolution of the harmonic amplitudes w2, w3 and w4 is presented for cnt with n=200 at pressure p=1.14×10-4 ev/å3 = 18.2 mpa. in view of eq. (17), one finds the analytical value of =p/p2 *=0.816. from the numerically obtained function w2(t) one finds the oscillation frequency of the second harmonic as 2=54.6 ghz. from eq. (12) it follows that the natural vibration frequency of the second harmonic is 2=126.9 ghz. from eq. (16), for m=2 one has =1 – (2/2)2=0.815, which is in a very good agreement with the analytical estimation. similar numerical results were obtained for different pressure values, they are presented in fig. 4(b) by black squares. the red line shows the analytical dependence 2/2=(1-)1/2, which follows from eq. (16) for m=2. it can be seen that the theory predicts very well the dependence of the fundamental oscillation frequency on the applied external pressure in the studied case of <1 amplitudes of all harmonics are periodic functions of time. next, we analyze the case >1, in particular, consider =2,3 and 7. for =2, p>p2 * and p<pm * for m>2. from eqs. (23) and (25) for =2 we find mr=⌊2.6⌋ and ml=⌈2.0⌉. rounded values are mr=3 and ml=2. according to the above theory, in eq. (24) the first sum is taken from 2 to 3, and the second from 4. the harmonic m=2 grows exponentially with time and is the fastest growing harmonic. the remaining harmonics are periodic in time. in fig. 5(a), the time dependence of the harmonic amplitudes is presented. in line with the theoretical prediction, w2 grows exponentially, while w3 and w4 fluctuate over time. carbon nanotube under pulsed pressure 285 fig. 4 (a) amplitudes of fourier harmonics as the functions of time found for cnt with n=200 under external pressure p=18.2 mpa (=p/p2 *=0.816). (b) frequency of the harmonic m=2 normalized to the natural vibration frequency at p=0 as the function of the applied pressure normalized to the critical value p2 *. red line shows the analytical solution eq. (16) for the elastic ring and black squares the numerical results for the cnt. for =3, p>p3 * and p<pm * for m>3. from eqs. (23) and (25) for =3 we find mr=⌊3.4⌋ and ml=⌈2.4⌉. rounded values are mr=3 and ml=3. as can be seen from fig. 5(b), w2 and w3 grow exponentially, while w4 performs oscillations. as predicted by the theory, the fastest growth is demonstrated by w3. for =7, p>p4 * and p<pm * for m>4. from eqs. (23) and (25) for =7 we find mr=⌊4.7⌋ and ml=⌈3.4⌉. rounded values are mr=4 and ml=4. in eq. (24), the first sum is taken from 2 to 4, and the second from 5. the harmonics m=2,3 and 4 grow exponentially, with the fastest growing harmonic m=4. the numerical results are presented in fig. 5 (c) to confirm the theoretical predictions. starting from m=5, oscillations occur with increasing frequency and decreasing amplitude around the motion determined by the harmonics m=2,3 and 4. 286 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev fig. 5 (a) amplitudes of fourier harmonics as the functions of time found for cnt with n=200 under external pressure p corresponding to (a) =2, (b) =3, and (c) =7. 6. reaction to a pressure pulse below we analyze the dynamics of cnts due to a relatively short rectangular pressure pulse. 6.1. analytical treatment in the case of the pressure pulse shown in fig. 6, one can use solution eq. (24) within 0≤t≤, where  is the pulse duration. for t > , when p=0, =0, n=n , the solution to eq. (15) has the form ( ) cos( ) sin( )m m m m mw t a t b t=  +  . (26) the integration constants are determined from the conditions carbon nanotube under pulsed pressure 287 2 0 2 2 0 2 cos( ), 3 1,( ) cosh( ), 3 1, sin( ), 3 1,( ) sinh( ), 3 1. mm m m m mm m m m mw t w m mw t w m      − =      − −     − =       − (27) the first line in wm(t) and ( )mw t corresponds to the case when the effective pressure p<p2 *(m2-1)/3, see fig. 6(a), and the second line is valid in the opposite case, see fig. 6(b). fig. 6 a rectangular pressure pulse with a height (a) less than the critical pressure p2 *, (b) greater than the critical pressure. the pulse duration is . the dynamics of the ring after the termination of the action of the pressure pulse (t>) is determined by the expressions     −+− −−− = ,0)],(sin[)sinh()](cos[)cosh( ,0)],(sin[)sin()](cos[)cos()( 2 2 0 mmmmmm mmmmmm m m tit tt w tw (28) where i is the imaginary unit and we have introduced the notation 2 2 3 1 1 m m   = − − (29) in accordance with eq. (16), so that m 2=m 2m 2. for =0 in both solutions of eq. (28), the motion is a continuation of the initial motion defined by eq. (3), w=w0mcos(mt)cos(m), since there is no pressure drop yet. in the case p=0 (=0, m=±1) from the first solution we obtain the same function, and the second solution eq. (15) has no meaning. in the general case, the motion in the interval 0<t< is determined by eq. (24), and for t> by eq. (28). thus, during the pressure action (0<t<), when 3p<p2 *(m2-1), oscillations occur with frequencies m and amplitudes wm 0. after the cessation of the pressure (t>), oscillations for each harmonic occur with a higher frequency m than m. in the case 3p>p2 *(m2-1), within the pressure pulse (0<t<), an exponential growth of the harmonic amplitudes occurs. at t>, oscillations occur with the same frequencies m, but with increased amplitudes. this solution is valid within the formulation of the problem in the linear approximation. 288 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev example. let us consider in more detail the dynamics of the harmonic m=2 in the cnt with the number of atoms n=200. according to eq. (12), 2=0.127 rad/ps. for p=18.2 mpa (=p/p2 *=0.816, 2=0.429) one has from eq. (16) 2=0.054 rad/ps. since p<p2 *, the first line of eq. (28) applies,     2 0 2 cos(0.054 ), 0 , cos(0.054 )cos 0.127( ) 0.429sin(0.054 )sin 0.127( ) , ( ). t tw t t tw    =   −  +  −    (30) in this case, the oscillation amplitude after the termination of the pressure does not depend on its duration . now we consider the case of >1. let us take =1.1, then 2=(p/p2 *-1)1/2=0.316, 2=22=0.040 rad/ps. for t< the solution is given by eq. (24) and for t> the solution is given by the second line of eq. (28):     2 0 2 cosh(0.040 ), 0 , cosh(0.040 )cos 0.127( ) 0.316sinh(0.040 )sin 0.127( ) , ( ). t tw t t tw    =   −  −  −    (31) it is clear that for p>p2 * there is a strong dependence of the amplitude of oscillations after the termination of pressure pulse on its duration . 6.2. chain model results the time dependence of the harmonic amplitudes w2, w3 and w4, are shown for the cnt with n=200 for the pulsed pressure in fig. 7. in fig. 7(a) p=18.2 mpa (=0.816) and pulse duration is equal to =100 ps. this result should be compared with the one shown in fig. 4(a), where the same pressure is constantly applied. under the action of a pressure pulse, the cnt wall performs periodic oscillations, and the fundamental bending harmonic m=2 has a frequency 2=0.054 rad/ps. after the external pressure is removed, the amplitude of oscillations of the m=2 harmonic decreases somewhat, and the frequency increases in accordance with the theoretical prediction, see the first line of eq. (28). removing the pressure has little effect on the dynamics of higher harmonics m=3 and 4. in panels (b) and (c) of fig. 7, the external pressure is greater than p2 *, namely, it corresponds to =2 and 3, respectively. the duration of the pressure pulse is =25 and 20 ps, respectively. these graphs should be compared with fig. 5(a) and (b) respectively, where the same pressure is constantly applied. it can be seen from fig. 7 that after the removal of external pressure, the exponential growth of w2 in (b) and w2 and w3 in (c) turns into a periodic motion with an amplitude reached by the time . this means that the oscillation amplitude of these harmonics strongly depends on , as it was shown theoretically, see the second line of eq. (28). it can be concluded that the theory of thin shells describes very well the dynamics of cnts under pulsed pressure. this conclusion is supported by comparison of the critical pressure values calculated using molecular dynamics and the shell theory, see fig. 4(b), and also by the dynamics of various harmonics under constant and pulsed pressure, see fig. 5 and fig. 7. carbon nanotube under pulsed pressure 289 fig. 7 (a) amplitudes of fourier harmonics as the functions of time found for cnt with n=200 under external pulsed pressure (a) p=18.2 mpa (=0.816), (b) p=36.4 mpa (=2.0), and (c) p=54.6 mpa (=3.0). pressure drops to zero at time  shown by the vertical dashed line: in (a) =100 ps, (b) =25 ps, and (c) =20 ps. 5. conclusions at present, the static deformation of the cnt cross section under the action of external pressure has been studied and a strong dependence on the cnt diameter has been shown. the spectral properties of the nanotubes were also analyzed depending on the external pressure. to the best of our knowledge, the dynamic behavior of cnts under the action of a pressure pulse of various durations and intensities has been less studied. in this work, to analyze the dynamics of cnts under plane strain conditions, we applied a continuum approach based on the theory of thin cylindrical shells based on the kirchhoff hypotheses. in addition, we used the molecular dynamics method based on the chain model, which describes the deformation of the cnt cross section under a uniform external pressure. by comparing the eigenfrequencies of radial vibrations of a circular ring and cnt wall consisting of a different number of atoms n , the parameter  is determined, which contains the bending stiffness and density of the ring, see table 1. 290 m. a. ilgamov, a. a. aitbaeva, i. s. pavlov, s. v. dmitriev the introduction of the parameter  into the analysis opens up the possibility of a wide application of the models and relations of the well-developed theory of elastic thin shells. in this case, it is not necessary to involve such parameters as effective density, elastic modulus, poisson’s ratio, wall thickness, the determination of which is a difficult task. determining the parameter  from such an integral characteristic as natural frequencies of cnt vibrations seems to be more justified than from the effective parameters, which are given in the literature with a wide spread. for example, the effective thickness of the cnt wall is given in the literature in the range 0.7-3.4 å. from the analysis of the values of the parameter  for cnts of different diameters, determined by the chirality index n (or number of carbon atoms in a cnt cross section n=2n) given in table. 1, one can conclude that for the zigzag cnts (n,0) with n>15, in the regime of small amplitude vibrations, the discrepancy between the continuum model and molecular dynamics calculations does not exceed 5% and decreases with increasing n. as a result of the theoretical analysis carried out within the framework of the linear theory of thin shells, the dynamics of the deflection function of the cnt wall is described under the instantaneous application of a time constant or pulsed external pressure. theoretical results are compared with molecular dynamics calculations, where the expansion of the deflection function in a fourier series was analyzed. in particular, it is shown that the dynamics of bending harmonics with numbers m≥2 changes qualitatively when the external pressure exceeds the critical values pm * determined from eq. (18). at pressure in the range p* m < p < p* m+1, the amplitude of harmonics with numbers less than m+1 grows exponentially, while harmonics with higher numbers exhibit oscillatory motion. interestingly, the fastest increase in the amplitude is observed for the harmonic at the border of exponential growth and oscillatory motion. under the action of a rectangular pressure pulse of short duration (compared to the oscillation period of the fundamental bending harmonic), the dynamics of different modes depends both on the value of the applied pressure p and on the pulse duration . for oscillating harmonics, the amplitude of oscillations after the termination of the pressure pulse weakly depends on , and for harmonics whose amplitude grows exponentially with time, the amplitude of oscillations at t >  increases rapidly with increasing . the results presented in our work deepen our understanding of the dynamics of cnts under the action of pulsed pressure. in the future, similar studies 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239–255. 54. pavlov, i.s., dmitriev, s.v., vasiliev, a.a., muravieva, a.v., 2022, models and auxetic characteristics of a simple cubic lattice of spherical particles, continuum mechanics and thermodynamics, 34(6), pp. 1669–1685. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 1, 2014, pp. 61 72 optimal synthesis of the manipulator using two competitive methods  udc (519.863, 621.86) jovan pavlović, miomir jovanović, andrija milojević university of niš, faculty of mechanical engineering, serbia abstract. this paper defines a program realization for finding the optimal geometry of a planar z-mechanism. the paper shows the mathematical procedure for defining the objective function, the constraint function and the search fields which are used for solving the optimization problem. starting from the solutions in practice, a numerical example is given to determine an optimum design with four optimization parameters. all the optimization parameters are geometric on the mechanism for defining the bucket position. the problem is solved by two different numerical methods the method of formal search of hyperspace (passive scanning method) and the approximate method of sequential quadratic programming sqp (applying the function fminmax of the matlab optimization toolbox). the verification is performed with animation using software for geometric modeling. the results are graphically illustrated. key words: mining machine, wheel loader z – mechanism, passive scanning, optimum design of mechanism, numerical realization, sqp 1. introduction for working with bulk material the wheel loaders have a mechanism that enables the position adjustment an inclination angle of the bucket. at lifting, the bucket should keep an upward position without changing the inclination. this can be achieved by adjusting the geometry of the parts of the bucket movement mechanism. one of the mechanisms providing for it is known as the z mechanism (fig. 1). the task of finding a solution for the special mechanism providing for a special request is known as the optimal synthesis according to the theory of mechanisms. solving this problem is a common example of research and development. the similarity of different plane mechanisms of the machines offers the possibility of using the same optimization procedure. nowadays there is a variety of mathematical methods received november 14, 2013 / accepted february 11, 2014  corresponding author: miomir jovanović university of niš, faculty of mechanical engineering, department of transport and logistics, niš, serbia e-mail: miomir@masfak.ni.ac.rs original scientific paper 62 j. pavlović, m. jovanović, a. milojević in use, from the classical one, based on the authorized procedures of differential programming, gradient methods, the method of polyhedra (nelder-mead methods) [1], approximate methods of programming and sequential quadratic programming [2], up to the modern methods based on the techniques of genetic algorithm [3], aca algorithm (ant colony algorithm) [4] and simulation [5]. this paper first presents the method of formal searching for optimal solutions based on the computer changes of the geometric parameters of the mechanism, followed by the method of sequential quadratic programming. changing the parameters in the method of formal searching is performed incrementally by taking discrete values of independent parameters in an increasing order (scanning technique) [6], [7].for the evaluation of the effectiveness of the passive formal search method, the method of sequential quadratic programming (sqp) is applied in this paper. the complexity of the general approach to the problem can be seen in the work by yang yu [8] where the optimization is performed with twenty-two parameters. besides geometric optimization parameters, the synthesis can also introduce the constraints of internal mechanism forces. such an approach can be found in the work of gao xiuhua [9] (derived using the adams software package), and rongyi zhang [10], using the matlab programming environment with the use of a combination of three optimization algorithms. because of the complexity of the general approach, this paper deals with the mechanism synthesis based on a small number of optimization parameters. 2. geometric model of the z-mechanism a planar mechanism of a manipulator loader consists of an ar m (1) with a bucket (4) and a lifting hydraulic actuator (5), fig. 1. the arm (1) possesses a mechanism for defining the bucket position (mdbp) that does loading, transport position holding and emptying of the bucket. this mechanism consists of a tilting hydraulic actuator (6), a two-arm lever (2) and a coupling rod (3) (fig. 1). solving the optimization problem involves finding a geometry of mechanism for defining the bucket position to ensure minimal change in the tilt of the bucket when the arm of the loader is lifting from the minimum to the maximum slope angle β, min {fc}. from there, it is possible to search for optimization parameters in the mdbp mechanism geometry. the mechanism for the lifting arm is designed from the given operating conditions of the bucket position and is not subject to optimization. the optimization parameters can therefore be the position of the bearing of the hydraulic actuator (6), the geometry of the two-arm lever (2) and the length of the coupling rod (3). in this study, four parameters are chosen to optimize the mdbp mechanism related to the position of the bearing of the hydraulic actuator (6) and linkage o23 on the lever (2). fig. 1 geometric model of the z – mechanism for defining the bucket position (mdbp) optimal synthesis of manuplator using two competitive methods 63 for an efficient solution of multidimensional problems with constraints, numerical methods with passive searching can be used [6]. these methods use a fixed interval of indetermination (h), where the field of change of independent variables 1 2 3 4{ , , , }z z z z z is divided by the selected number of points equal to the number of sub-areas. these points serve as the basis for the construction of n dimensional space. the objective function ( )fc z is calculated at each node of the grid thus formed.with the analysis of all solutions the one that is the most advantageous is chosen. the advantage of the passive search method is in the reliability of searching. a disadvantage of multidimensional problems of passive searching lies in the number of calculation procedures. therefore, this paper employs the procedure of choice (constriction) of the field of search based on the testing of the objective function. the testing determines the solution with the good initial values of ( )fc z based on a large interval of indetermination. after identifying the "good areas" of solution, the subject of solution is the grid with a fine interval of indetermination. such a model with a reduced number of optimization parameters (z1, z2, z3, z4) and numerical implementation can give sub-optimal solutions thanks to the simplicity of the algorithm search. the synthesis procedure for an mdbp mechanism is illustrated in fig. 1. 3. mathematical formulation of the problem the formation of a mathematical model for optimization according to fig. 1 includes defining a planar geometric model of the manipulator. the manipulator is a planar (2d) mechanism with an arm for carrying the bucket and a mechanism for defining the bucket position. a mechanism for defining the bucket position is labeled with darker shades in fig. 1. the whole mathematical model uses the global cartesian x-y coordinate system and local coordinate systems of the parts of the mechanism shown in fig. 2. fig. 2 shows four parts of the mechanism. the planar form of geometry of these two coupled mechanisms with the selected marked positions of parts is given in fig. 3. fig. 2 parts of mdbp: a) the arm of the manipulator, b) the two-arm lever, c) the coupling rod, d) the bucket 64 j. pavlović, m. jovanović, a. milojević let us define the geometry of the rigid mechanical system in fig. 1:the coordinates of linkage o11 are given values x11, y11 of some type of a loader. the coordinates of linkage o12 at the top of the arm are computed as: 12 11 2 3cos cos(90 )x x a a       (1) 12 11 2 3sin cosy y a a     (2) the coordinates of bearings of the two-arm lever, the position of linkage o22 are computed as: 22 11 1 cosx x a    (3) 22 11 1 siny y a    (4) the coordinates of linkage o21 are determined by an intersection of the two circles with the center in linkage o6 whose radius is c6 – the length of the hydraulic actuator (6) and the circle with the center in linkage o22 whose radius is equal to the length of the arm l1 of the lever (2). we consider that the length of the hydraulic actuator (6) is constant. the coordinates of linkage o21 are obtained by solving the equation: 2 2 2 21 22 21 22 1 2 2 2 21 6 21 6 6 ( ) ( ) ( ) ( ) x x y y l x x y y c         (5) fig. 3 geometric model of the manipulator of the loader the distance between the centers of the circles is computed as: 2 2 6 22 6 22( ) ( )d x x y y    (6) optimal synthesis of manuplator using two competitive methods 65 value f and the coordinates of linkage o21 are computed as: 1 6 1 6 1 6 1 6 1 ( ) ( ) ( ) ( ) 4 f d l c d l c d l c l d c            (7) 2 2 22 6 6 22 1 6 22 6 21(1,2) 2 2 ( ) ( ) 2 2 2 x x x x l c y y x f d d          (8) 2 2 22 6 6 22 1 6 22 6 21(1,2) 2 2 ( ) ( ) 2 2 2 y y y y l c x x y f d d         (9) let us determine the coordinates of linkage o23. angle η1 between direction r and d of the two-arm lever is computed as: 3 1 2 1 l arctg l l    (10) variable angle η (between the arm and the two-arm lever), if is x22  x21 > 0 is then computed as: 21 22 22 21 y y arctg x x     (11) if x22  x21 = 0, angle η = 90°, and if x22  x21<0, angle η is computed as: 21 22 21 22 180 y y arctg x x      (12) length d of the arm of the two-arm lever (fig. 2b) is computed as: 2 2 2 1 3( )d l l l   (13) coordinates of linkage o22, are computed as: 23 22 1 cos( )x x d     (14) 23 22 1 sin( )y y d     (15) the introduced optimization parameters z1, z2, z3, z4 correspond to the required geometry of the z mechanism by the following relations: 1 6z x (16) 2 6z y (17) 3 2 1z l l  (18) 4 3z l (19) 66 j. pavlović, m. jovanović, a. milojević the coordinates of linkage o4 (x4, y4) are determined using the intersection of the circle with the center of linkage o23 whose1. the solution of coordinates x4, y4 is obtained from the equation: radius is the length of coupling rod c1 and the circle with the center of linkage o12 whose radius is length b: 2 2 2 4 23 4 23 1 2 2 2 4 12 4 12 1 ( ) ( ) ( ) ( ) x x y y c x x y y b         (20) the distance between the centers of the circles and value f1 are computed as: 2 2 1 12 23 23 12( ) ( )d x x y y    (21) 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) 4 f d c b d c b d c b c d b            (22) coordinates of linkage o4 : 2 2 12 23 12 23 1 1 23 12 4 1(1,2) 2 2 1 1 ( ) ( ) 2 2 2 x x x x c b y y x f d d          (23) 1 2 2 23 12 12 23 1 1 23 12 4 1(1,2) 2 2 1 ( ) ( ) 2 2 2 y y y y c b x x y f d d         (24) bucket tilt angle for the current value of arm slope β is computed as: 4 12 4 12 ( ) y y arctg x x      (25) the back angle of bucket γ is determined by the position of the angle of bucket α and constant α4 (fig. 2d): 4 (26) the objective function is the minimum change in the slope of the bucket while the arm of the loader is lifting: 1 2 3 4 max min( , , , )fc z z z z    (27) the constraint functions are determined by inequalities: 1 1 5 3 2 1 6 3 3 2 7 4 4 2 8 4 0 0 0 0 0 0 0 0 g z a g z e g z b g z f g z c g z u g z d g z h                         (28) where: a, b, c, d, e, f, u, h – are the constants of the search space. optimal synthesis of manuplator using two competitive methods 67 4. algorithm and program fig. 4 presents the algorithm of global numerical procedures developed for the solution of the optimal mechanism synthesis. the algorithm shows that the scanning method is realized by an increment change of the optimization parameters in a permissible fourdimensional hyperspace. the permissible hyperspace is empirically determined. the initial setting of geometry of both mechanisms is provided through the files dat1-dat6. the solutions of objective function fc for discrete values of independent parameters z1 ÷ z4 are entered in the file dat7. by comparing these solutions, the solution that is characterized by the minimum value of objective function fc is chosen. then the obtained solution is checked from the point of numerical accuracy, satisfying the given nominal characteristics of the reach and lift height. after that, the continuity of coupling of the parts of mechanism for defining the bucket position is also checked. the following numerical example illustrates the application of this mathematical model, the efficiency of the algorithm and the practical realization of optimal synthesis. fig. 4 the algorithm of the program for optimal numerical synthesis of mdbp 68 j. pavlović, m. jovanović, a. milojević 5. numerical example the aim of this study is to search for the design of the mechanism for defining the bucket position which minimizes the angle of the bucket tilt during the bucket lifting operation. the initial solution is based on freely assumed geometry which is in this case taken from photographs and sketches [11] (komatsu wa320). four parameters are selected to optimize the synthesis: x, y coordinates of bearings of the hydraulic mechanism for defining the bucket position and x, y local coordinates of linkage o23 of the two-arm lever. the procedure for the algorithm program flow is fast and numerically stable. the stability is based on the simplicity of analytical modeling geometry. angle β is the slope angle of the arm in relation to the horizontals and ranging, in this case, from -24 o to 62 o . parameters z1, z2 represent the search field of bearings position o6, where the geometry is in the following ranges: in the x-axis direction z1min=0.94 m; z1max=1.14 m (δz1=0,2 m) (a=0.94 m; b=1.14 m) and in the y-axis direction z2min=1.645 m; z2max=1.845 m (δz2=0.2 m) (c=1.645 m; d=1.845 m). the allowable field defined by constants a, b, c, d is chosen on the basis of the available space on the arm. by choosing m = 50 points of change of one independent parameter of optimization, the interval of indetermination is: 1 1,14 0,94 2 2 0,00816 [m] 1 50 1 b a h m        (29) parameters z3, z4 represent the search field of the position of linkage o23 in the local coordinate system of the two-arm lever, the size of change is in the area: z3min=1.4 m; z3max=1.5 m (δz3=0.1 m), (e=1.4 m; f=1.5 m) in the x-axis direction, and z4min=0.175 m; z4max=0.275 m (δz4=0.1 m), (u=0.175 m; h=0.275 m) in the y-axis direction. the field is empirically chosen. initial zeros are: z1=1.04 m, z2=1.745 m, z3=1.45 m, z4=0.225 m. these areas of solutions of the optimization task determined the allowable space of search-scanning. the dividing of the permissible field of parameters z3 and z4 by the n = 50 points is selected, as well as the dividing of the permissible field of parameters z1 and z2 by m = 50 points. in this way a number of combinations of k=m 2 •n 2 =6250000 are obtained. from the file dat-8 we can read the optimal parameters of the mechanism for defining the bucket position. based on these parameters, it is possible to draw curves of objective function fc, depending on the angle of the slope of the arm β for the optimal and initial solution (fig. 5). starting from the fixed size of the loader mechanism given in table 1, the mdbp optimal solutions are obtained, given in table 2. the curves of the objective function of the required mdbp are plotted for the optimal solution. table 1 initial data of model a1 (m) a2(m) a3 (m) l1 (m) l2 (m) l3 (m) a1 (m) b1 (m) 1.800 2.550 0.865 0.720 1.450 0.225 0.765 0.370 table 2 values of the obtained optimal solution optimal parameters objective function z1 (m) z2(m) z3(m) z4(m) fc (˚) initial zeros 1.04 1.745 1.45 0.225 7.364 optimal values 0.973 1.678 1.400 0.175 1.707 optimal synthesis of manuplator using two competitive methods 69 fig. 5 curve of initial fc0 and curve of optimal solution fc1 the verification is performed by simulation of the working mode (meshing) of the mechanism, in the solidworks software, by using the option blocks which is suitable for the simulation of planar mechanisms. this achieved the verification of the behavior of the mechanism in successive positions, fig. 6. it provides a functional control study of the mechanism and eliminates errors. fig. 6 simulation of the obtained optimal solution 6. minimization using sqp method in order to assess the benefits from the chosen method of minimizing the objective function, the method of the sequential quadratic programming (sqp) is used. minimization is performed by programming the objective function and constraint function in matlab by using the program function (command) fminimax as part of the matlab optimization toolbox. the general quadratic programming problem is defined as: 70 j. pavlović, m. jovanović, a. milojević 1 min ( ) 2 t tf z c z z h z   (30) where: bza  , z  0 (31) are the constraints in the form of inequality, c – the n-dimensional vector that describes the coefficients of linear terms of the objective function, z – the n-dimensional vector of unknown values, h – the (n×n) symmetric matrix that describes the coefficients of square terms of the objective function, a – the (m × n) matrix of constraints, and, b – the n-dimensional vector of constraints [12]. 5.1. function minimax function minimax (fminimax) is part of the optimization toolbox in matlab, which finds the minimum of problems determined by the function: min max ( )i z i f z such that is ( ) 0 ( ) 0 c z ceq z a z b aeq z beq lb z ub             (32) where the variables on the right side are the constraints: c(z), ceq(z) – vectors of constraints in the form of inequality and equality, a – matrix and b – vector are, respectively, the coefficients of constraints in the form of linear inequalities, aeq – matrix and beq vector are, respectively, the coefficients of linear constraints in the form of equity, lb – vector or matrix of lower bounds of constraints, ub – vector or matrix of upper bounds of constraints. in the model that is described by equations 1-28, we use only constraints lb, ub – the vectors of lower and upper bounds of the search field of the optimization parameters solution (a, b, c, d, e, f, u, h), whose values are listed in the previous numerical example [13]. based on the function fminimax, where it is necessary to define the objective function and the constraints, the desired optimal parameters are obtained: z = fminimax(fun,z0,a,b,aeq,beq,lb,ub) (33) where: z – required parameters, fun – objective function, z0 – initial parameters a, b, aeq, beq, lb, ub – constraints. these values are defined above in equations 1÷28, since in this case constraints a, b, aeq and beq are not used. they are written as an "empty" matrix. the inputs of constraints are in the form of the lower bounds lb and the upper bounds ub. in this case, the line of the code of function fminimax looks like: parameters=fminimax('objectfun',parametri0,[],[],[],[],lb,ub) (34) optimal synthesis of manuplator using two competitive methods 71 sqp has performed the verification of the previous procedure and the optimization achieved by using the passive formal search. the results obtained by the sqp method are shown in the following diagram, fig. 7, and a table of optimal solutions. table 3 values of obtained optimal solutions by using the function fminimax optimal parameters objective function z1 (m) z2(m) z3(m) z4(m) fc (˚) initial zeros 1.04 1.745 1.45 0.225 7.364 optimal solution 1.0176 1.7188 1.4095 0.2002 0.3518 fig. 7 curve of initial fc0 and optimal fc1solutions, obtained by using the minimax function 7. conclusion 1. the mathematical model of optimization operates with four independent parameters out of the potential eight. even four parameters of optimization give a solution of objective function (change of the tilt is less than two degrees) good enough for practical performance. by increasing the number of independent parameters to possible eight, the quality of the optimal solution can be increased, with a complex mathematical procedure. 2. it is recommended to use an optimal model of the mdbp mechanism for factory design because it is an important design requirement. 3. improving the model can be done by introducing the constraint function of internal forces in the mdbp which would require the extension of this task in the domain of static analysis. 4. the sqp procedure has given a slightly more accurate solution which is a result of a freely adjustable step. the formal procedure limits the accuracy (quality of solutions) by selecting the interval of indetermination. the quality of a solution can be improved by increasing the number of search combinations. 5. the computing time of implementation in each case is less than one second. 72 j. pavlović, m. jovanović, a. milojević acknowledgements: the paper is a part of the research performed within the project tr 35049, faculty of mechanical engineering. the paper is also a part of the doctoral studies of the first author. the authors would like to thank the ministry of education and science of the republic of serbia. references 1. radić, m., marinković, z., jovanović, m., 2002, dynamics and optimization of cranes, university of niš, faculty of mechanical engineering, scientific monograph, niš (in serbian) 2. jain, a., issac, k.k., 2003, optimal design of front end loader attachment for tractors, proc. eleventh national conference on machines and mechanisms nacomm 2003, indian institute of technology delhi, new delhi. 3. i̇pek, l., 2006, optimization of backhoe-loader mechanisms, master thesis, middle east technical university, turkey, 71 pp. 4. zhang, r., qi, j., cai, y., 2007, optimum design of the working mechanism of loader with ant colony algorithm (aca), proc. fourth international conference on mechatronics and automation icma 2007, harbin, china, pp 3066 – 3070. 5. cao, x., cleghorn, w., 2011, parametric optimization of eight-bar mechanism of a wheel loader based on simulation, information technology journal 10(9), pp. 1801-1808. 6. papazov, p. s., 1975, optimal design of electromagnetic systems, tehnika, sofia. 7. vujčić, v., ašić, m., miličić, n., 1980, matematičko programiranje, matematical institute, university of belgrade. 8. yu, y., lianguan, s., mujun, l., 2010, optimum design of working device of wheel loader, proc. first international conference mechanic automation and control engineering mace 2010, wuhan, china. 9. xiuhua, g., yunchao, w., erzhong, a., jiawei, h., 2007, optimization of the working device of loader based on adams, computer simulation journal, 1(1). 10. zhang, r., qi, j., cai, y., 2008, intelligent optimum design of the working mechanism of loader, proc. seventh world congress on intelligent control and automation wcica 2008, chongqing, china. 11. komatsu wa320, http://www.komatsu.com/ce/products/pdfs/wa320-5.pdf, (accessed january 25, 2014). 12. arora, s. j., 2007, optimization of structural and mechanical systems, university of iowa, iowa, 595 pp. 13. venkataraman, p., 2009, applied optimization with matlab programming, wiley, 544 pp. optimalna sinteza manipulatora koristeći dve uporedne metode u radu je data programska realizacija za traženje optimalne geometrije ravanskog zmehanizma. rad pokazuje matematičku proceduru definisanja funkcije cilja, funkcija ograničenja, oblasti pretraživanja kojima se rešava zadatak optimizacije. polazeći od rešenja u praksi, dat je numerički primer određivanja optimalnog dizajna sa četiri parametra optimizacije. svi parametri optimizacije su geometrijski na mehanizmu za određivanje nagiba kašike. zadatak je rešavan dvema različitim, numerički metodama metodom formalnog pretraživanja hiperprostora (metoda pasivnog skeniranja) i aproksimativnom metodom kvadratnog sekvencijalnog programiranja – sqp (primenom fminmax funkcije iz matlabovog optimizacionog toolbox-a). verifikacija rešenja je vršena animacijom u programu za geometrijsko modeliranje. rezultati su grafički ilustrovani. ključne reči: pasivno traženje, z mehanizam, utovarivač, rudarska mašina, optimizacija mehanizma, numeričko rešavanje, sqp file:///h:/doktorske%20studije/i%20godina/50668823/intelligent-optimum-design-of-the-working-mechanism-of-loader http://www.komatsu.com/ce/products/pdfs/wa320-5.pdf 7300 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 727 735 https://doi.org/10.22190/fume210101019c © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper on the afferrante-carbone theory of ultratough tape peeling michele ciavarella1, robert m. mcmeeking2,3,4,5, gabriele cricrì6 1politecnico di bari. dmmm department, bari, italy 2materials department, university of california, usa 3department of mechanical engineering, university of california, usa 4school of engineering, university of aberdeen, king’s college, scotland, uk 5inm – leibniz institute for new materials, germany 6department of industrial engineering (dii), università di napoli federico ii, napoli, italy abstract. in a simple and interesting theory of ultratough peeling of an elastic tape from a viscoelastic substrate, afferrante and carbone find that there are conditions for which the load for steady state peeling could be arbitrarily large in steady state peeling, at low angles of peeling what they call "ultratough" peeling (afferrante, l., carbone, g., 2016, the ultratough peeling of elastic tapes from viscoelastic substrates, journal of the mechanics and physics of solids, 96, pp.223-234). surprisingly, this seems to lead to toughness enhancement higher than the limit value observed in a very large crack in an infinite viscoelastic body, possibly even considering a limit on the stress transmitted. the afferrante-carbone theory seems to be a quite approximate, qualitative theory and many aspects and features of this "ultratough" peeling (e.g. conformity with the rivlin result at low peel angles) are obtained also through other mechanisms (begley, m.r., collino, r.r., israelachvili, j.n., mcmeeking, r.m., 2013, peeling of a tape with large deformations and frictional sliding, journal of the mechanics and physics of solids, 61(5), pp. 1265-1279) although not at “critical velocities”. experimental and/or numerical verification would be most useful. key words: soft materials, peeling, adhesion, viscoelastic materials 1. introduction adhesion of two objects can arise due to chemical bonding as by a glue, or by physical interaction such as by van der waals or electrostatic attraction [1]. various tests have been devised to probe the strength and toughness of adhesive bonds, including the peeling of a tape from a substrate or from another tapelike feature [2, 3]. our schematic received january 01, 2021 / accepted february 12, 2021 corresponding author: michele ciavarella dimeg politecnico di bari, v.le japigia 182, politecnico di bari, 70125 bari, italy e-mail: mciava@poliba.it 728 m. ciavarella, r.m. mcmeeking, g. cricrì of the former version of the test is illustrated in fig. 1. rivlin [2] provided a formula for the relationship between the effective adhesion energy, γ, and the force, p, required to peel an inextensible tape from an elastic substrate. the energy of adhesion as the work per unit area of bond requires to progressively separate the objects from each other. kendall [4, 5] then augmented the formula by accounting for the effect of the strain energy of an elastic tape. the importance of adhesion, and its science and technology, can be confirmed by perusal of the references provided by afferrante and carbone [6], and by the reviews by creton and ciccotti [7] and long et al. [8]. fig. 1 geometry of the peeling of an elastic tape from a viscoelastic substrate for the ac theory viscoelastic effects are known to occur in the peeling of polymer and organic materials, e.g. kendall [2, 3] presents such data and attributes the effect to viscoelastic behavior in the separation process. such viscoelastic behavior is addressed in an extensive literature (e.g. [9–12]), suggesting steady state critical crack growth during peeling occurs with an enhanced apparent work of adhesion according to the gentschultz law 0 0 1 n     = +      (1) where γ is the apparent or effective work of adhesion, γ0 is its reference or datum value at a very slow rate of propagation, ν is the rate of peeling, and 1 0 ( )n tka −= in which k and n (0 < n < 1) are (or are supposed to be) materials constants, and n t a is the wlf factor for unifying results from various temperatures (williams, landel and ferry [13]). on the afferrante-carbone theory of ultratough tape peeling 729 gent and schultz [10] describe the reference value, γ0, as the thermodynamic work of fracture, and attribute the enhancement in eq. (2) to viscoelastic dissipation. a model incorporating such a point of view was developed by de gennes [14], described as the “viscoelastic trumpet,” with different mechanisms of deformation and adhesive rupture dominating at different scales at different rates of crack or peel front propagation. however, de gennes [14] finds that the resulting highest effective work of adhesion for very rapid crack or peel front propagation is finite instead of unlimited as in eq. (1), and is given by 0 0 e e   =  (2) where e∞ is the instantaneous modulus and e0 is the relaxed, or equilibrium modulus. note that the ratio in eq. (2) can span 3 or 4 orders of magnitude [12]. the velocity dependence of eqs. (1) and (2) was, at first, difficult to rationalize as cracks and peel fronts in linear viscoelastic materials admit an inverse square root stress singularity, implying that the high strain rate modulus should always dominate at the crack and peel front tip, indicating a lack of velocity dependence [15]. however, introduction of a barenblatt or maugis-dugdale cohesive zone at the crack tip enabled the identification of velocity dependence for crack propagation [16-20], as then the effective modulus at the crack tip depends on the propagation rate. the resulting model predicts the relationship in eq. (2) for mode i crack propagation (i.e. tension), where, in this case, γ∞ is defined to be 2 0/k e , where k∞ is the mode i stress intensity factor at high rates of crack propagation. similarly, 2 0 0 0/k e = . one can expect an interplay among cohesive zone behavior, peel rate, viscoelastic dissipation and specimen sizescale. such an effect was explored in finite element modeling by rahulkumar et al. [21] for t-peeling of a viscoelastic polymer strip. they found that the apparent adhesion energy at low and high rates of peeling effectively obeys eq. (2), but with an intermediate propagation rate where viscoelastic dissipation additionally enhances the apparent adhesion energy. as a result, the relationship between the apparent adhesion energy and the peel rate is non-monotonic, in contrast to eq. (1). afferrante and carbone [6] (henceforth ac) address a similar issue but for an elastic tape being peeled from a viscoelastic substrate as in fig. 1. the substrate is considered to be semi-infinite, the tape thickness is l and the peel angle is θ. to model the transfer of loads to the substrate, ac assumed that they could be treated as uniform tractions applied over a width l at the peel front, i.e. a traction normal to the substrate surface plus a shear traction. the substrate surface is traction free elsewhere, allowing use of the viscoelastic correspondence principle for the substrate. the ac model therefore has features that make it similar to a cohesive zone model with uniform cohesive tractions, and has a size scale, the elastic tape thickness, that can introduce an interplay between the peel rate and viscoelastic dissipation in the substrate. in the next section we identify some aspects of the peeling behavior in the ac model. 2. afferrante-carbone model for an elastic tape of modulus e and a viscoelastic substrate with a single relaxation time τ0, (e.g. the standard model of linear viscoelasticity), the ac equation (14) [6] provides 730 m. ciavarella, r.m. mcmeeking, g. cricrì ( ) 2 0 0 0 1 1 cos 2 e p p f el e l el el            = − + −            (3) where p is the load per unit out-of-plane width of the tape and fν is a function of exponential integrals of its argument ντ0/l, see the ac equation (15) [6]. note that eq. (3) is written as an expression for the intrinsic adhesion energy, γ0. the function fν is zero for both very slow peeling rate, 0 → , and at extreme rates of peeling,  → . otherwise, fν is positive, i.e. consistent with viscoelastic dissipation from which it arises, and has a maximum at an intermediate rate of peeling that depends weakly on e∞/e0 if e∞/e0>15. at very low peel rates, and very high peel rates, with fν 0→ , the result in eq. (3) is simply kendall’s [4], or rivlin’s [2] if the strain energy term containing p2 is neglected, as it can be for high peel angles. this behavior is expected as the system behaves purely elastically at very low and very high rates of peeling with negligible viscoelastic dissipation. as a result, the formula in eq. (3) at very slow and very fast rates of peeling shows that in these cases the intrinsic adhesion energy is the only contributor to the work of adhesion. however, the intrinsic adhesion energy may be rate dependent, as in the case where a soft, lightly cross-linked polymer is being used as an adhesive and its crazing and fibrillation can lead to viscoelastic behavior within the adhesive. the result in eq. (3) is a quadratic equation for p/el with discriminant ( ) 2 0 0 0 4 1 1 cos 2 e f el e l         = − + −      (4) and, therefore, real solutions for p/el exist only for 0  . the solutions are 0 0 1 cos 2 1 p eel f e l    −   =   −     (5) and we accept only positive real solutions. we deduce that the apparent adhesion energy, γ, the sum of the thermodynamic adhesion energy and the contribution of viscoelastic dissipation, is given by ( ) 2 2 0 0 0 1 1 cos 2 p p e p f el el el el e l el            = + − = +              (6) note that the first result on the right-hand side of eq. (6) is the normalized energy release rate as given by kendall [4]. we observe, therefore, that, according to the ac model, the kendall [4] formula for the energy release rate also provides the effective adhesion energy, as expressed by the first equality of eq. (6). from eq. (6) we obtain the ratio 2 0 0 0 0 1 el e p f e l el      = +         (7) where p is obtained from eq. (5) when there are real, positive solutions for it. on the afferrante-carbone theory of ultratough tape peeling 731 for illustration, we consider the case where e/e0=1.5 and e∞/e0=10, γ0/el=10-4 as also used by ac. in fig. 2a, we present the result for γ/γ0 from eq. (7) where the lower positive real result for p/el from eq. (5) is utilized. it can be seen that the result for ϑ=π/64=2.8° can have an enhancement of the apparent adhesion energy that can be very large. although not shown in fig. 2a, the result for γ/γ0-1 for ϑ=π/64=2.8° can be as high as ≈ 140. there are no real solutions for this angle in the range where the results in black are not visible in the plot. this outcome is much larger than e∞/e0=10 as predicted by de gennes [14] for the ratio of the apparent adhesion energy for very fast peeling compared to very slow peeling. the enhancement of the apparent adhesion energy is thus nonmonotonic and has a maximum at peeling rates for which ντ0/l ≈ 1. a) b) fig. 2 enhancement (γ/γ0-1) for the ac peeling theory shown as a function of the peeling velocity : (a) lower load solution (b) higher load solution, both for e/e0=1.5, γ0/el=10-4, e∞/e0=10. peel angles are θ = π/64, π/8, π/4, π/2 (respectively black, blue, red and green). note also that for peel angles equal to ϑ = π/8 = 22.5° and above the enhancement of the apparent adhesion energy is negligible. we note that we find negligible enhancement of the apparent adhesion energy for peel angles greater than approximately 8° for the 732 m. ciavarella, r.m. mcmeeking, g. cricrì parameters used to plot fig. 2a. as noted above, the effective adhesion energy is then essentially equal to the intrinsic adhesion energy, γ0, and is so over the whole range of peel rates. as also noted above, γ0 itself may be rate dependent, leading to a higher peel force at fast rates of peeling than at low rates. however, it is then not obvious that eq. (2) will predict the ratio of effective adhesion energies at fast peel rates versus slow rates, and the behavior in terms of effective adhesion energy may well be given by eq. (1). if the intrinsic adhesion energy lacks rate dependence, then the peel force for peel angles equal to or greater than 22.5° will be insensitive to the rate of peeling, in disagreement with eqs. (1) and (2). for example, if van der waals forces are in control of adhesion, according to the ac model, the peel force can be expected to be insensitive to the rate of peeling for peel angles greater than or equal to 22.5°. as noted by ac, it is possible that the higher magnitude, 2nd solution of eq. (5) is relevant during peeling and that the system can be forced into this regime. to explore this possibility, in fig. 2b we use the higher magnitude, 2nd solution from eq. (5) to evaluate and plot the result from eq. (7) for the same parameters used for fig. 2a. we plot only some of the lower values we obtain as many of the values are extremely high for the parameters utilized. given that, for peel angles greater than or equal to 22.5° the 2nd, higher magnitude solution gives an apparent adhesion energy that is at least 3 orders of magnitude greater than the lower magnitude solution, we conclude that it is unlikely that, for the parameters used in fig. 2, that the 2nd, higher magnitude solution is relevant to peeling. there are no real solutions in fig. 2b for the ϑ = π/64 case. we also find that the high apparent adhesion energy regime of peeling in the ac model depends on the ratio e/e0. as noted above, when e/e0 = 1.5, such high apparent adhesion energy behavior occurs for peel angles below 8°. for e/e0 > 1.5 this range of ultratough adhesion becomes larger. 3. discussion compared to other widely used models, like the gent-schultz law (1), and the ‘viscoelastic trumpet’ developed by de gennes, the ac point of view appears to be, like them, an approximate solution. indeed, the first two give results that are essentially qualitative, and whose validity depends on experimental databases; this point is clearly stated, e.g., by saulnier et al. [22] in the case of the viscoelastic trumpet. on the other hand, the ac model is developed following the mathematical consequences of an energy balance imposed through a well-defined, though inexact, physical model. notwithstanding the great difference between the respective frameworks, the ac results appear to be qualitatively similar to the outcomes of saulnier et al. [22]. in fact, in [22] the energy balance is based on a singular stress field directly derived from the correspondence principle, where such a singular stress is completely absent in ac. more specifically, both models predict that the viscous contribution to the detachment energy rate, γ(ν), is zero for very low or very high peel rates, whereas it can be very large at an intermediate rate. at this intermediate rate the ac results seem to be in contrast with the experimental outcomes from peel experiments in geometries similar to fig. 1 with viscoelastic components. it seems quite likely that the ac model is oversimplified as the interaction between the tape and the substrate is modeled as a uniform traction within a constant interaction length l, whatever the traction angle ϑ, the tape and substrate stiffness and the tape on the afferrante-carbone theory of ultratough tape peeling 733 thickness. to explore this point, let us consider two extreme conditions ϑ = π/2 and ϑ = 2 (see fig. 3). in the first case, the ac model neglects the tape flexural stiffness (although if the tape is very thin its flexural stiffness will be negligible so this is not always a problem), and in the second one, there is an issue about the interaction zone length: the cohesive zone length could be much smaller than l and it might be larger than l. also, if it is friction that determines the interaction length frictional slip zone might be larger than l and it might be smaller than l depending on the friction shear stress. if it is sticking friction, the interaction zone will be very small. on the other hand, eq. (7) shows clearly that the interaction length l is a very important parameter in the evaluation of the viscous contribution to the apparent adhesion energy. for these reasons, one should consider the ac predictions to be only approximate and qualitative, especially for low peel angles. indeed, it seems very unlikely that ‘ultratough peeling’, even if it exists, can be captured with such simplistic assumptions as are utilized by ac. fig. 3 interface interaction for the ac model for ϑ = π/2 and ϑ = 0 (above) with a more realistic one shown below even if the ac result is a straightforward consequence of the energy balance (and a poorly designed model), the interpretation of eq. (7) as a ‘toughness’ may be misleading, notwithstanding its coherence with many widely accepted formulations. this implies that, in some conditions, the power absorbed by the viscous substrate in steady state peeling is unlimited, which is physically an unlikely result. instead of identifying toughness, as is normally what is done using kendall [5], in the ac case one can speak unambiguously of peel load and, ultimately, of ‘ultra-high-strength’ peeling. while ac provides a result that we find to be counter-intuitive, unfortunately we cannot yet present a better model. thus, we can neither prove our claims nor disprove ac. there is a substantial difference between the ac and other models cited. this situation may partially be attributed to the fact that the volume in which viscous energy is dissipated in 734 m. ciavarella, r.m. mcmeeking, g. cricrì the ac model is infinite. a more detailed model, consistent with physical behavior of the interface interaction between the tape and the substrate would help deeper investigation. 4. conclusions we have discussed the fact that the "ultratough peeling" theory of afferrante and carbone [6] leads to two possible equilibrium solutions for the load at a given peeling speed. the low load solution in some regimes produces very high toughness enhancement, possibly already much larger even than that expected in a crack in an infinite system. more problematic appears to be the high load solution which, even considering a limit stress carried by tape and substrate, will not avoid some solutions to have unbounded toughness enhancement. in addition, viscoelastic effects seem to induce an on-off mechanism, where for large angles of peeling it predicts solutions extremely close to the purely elastic kendall case, while below a threshold angle, the toughening abruptly rises to very high levels, and the "ultratough" peeling regime appears. it seems very likely that this strange result is due to approximations present within the model, even if it is quite sophisticated from a mathematical point of view. features of this "ultratough" peeling (e.g. conformity with the rivlin result at low peel angles) are obtained also through other mechanisms [23] although not at “critical velocities”. references 1. israelachvili, j.n., 2011, intermolecular and surface forces, 3rd edition, academic press, 704 p. 2. rivlin, r.s., 1944, the effective work of adhesion, paint technology, 9, pp. 215-216. 3. spies, g.j., 1953, the peeling test on redux-bonded joints: a theoretical analysis of the test devised by aero research limited, aircraft engineering and aerospace technology, 25(3), pp. 64-70. 4. kendall, k., 1971, the adhesion and surface energy of elastic solids, journal of physics d: applied physics, 4, pp. 1186-1195. 5. kendall, k., 1975, thin-film peeling – the elastic term, journal of physics d: applied physics, 8, pp. 1449-1452. 6. afferrante, l., carbone, g., 2016, the ultratough peeling of elastic tapes from viscoelastic substrates, journal of the mechanics and physics of solids, 96, pp.223-234. 7. creton, c., ciccotti, m., 2016, fracture and adhesion of soft materials: a review, reports on progress in physics, 79(4), 046601. 8. long, r., hui, c.y., gong, j.p., bouchbinder, e., 2020, the fracture of highly deformable soft materials: a tale of two length scales, annual review of condensed matter physics, 12, doi: 10.1146/annurevconmatphys-042020-023937. 9. gent, a.n., petrich, r.p., 1969, adhesion of viscoelastic materials to rigid substrates, proceedings of the royal society of london. a. mathematical and physical sciences, 310(1502), pp. 443-448. 10. gent, a.n., schultz, j., 1972, effect of wetting liquids on the strength of adhesion, journal of adhesion, 3(4), pp. 281-294. 11. andrews, e.h., kinloch, a.j., 1974, mechanics of elastomeric adhesion, journal of polymer science: polymer symposia, 46(1), pp. 1-14. 12. barquins, m., maugis, d., 1981, tackiness of elastomers, journal of adhesion, 13(1), pp. 53-65. 13. williams, m.l., landel, r.f., ferry, j.d., 1955, the temperature dependence of relaxation mechanisms on amorphous polymers and other glass-forming liquids, journal of the american chemical society, 77, pp. 3701-3707. 14. de gennes, p.g., 1996, soft adhesives. langmuir, 12(19), pp. 4497-4500. 15. graham, g.a.c., 1969, two extending crack problems in linear viscoelasticity theory, quarterly of applied mathematics, 27, pp. 497-507. 16. knauss, w.g., 1974, on the steady propagation of a crack in a viscoelastic sheet, in: h.h. kausch et al. (eds.), deformation and fracture of high polymers, plenum press, pp. 501-541. on the afferrante-carbone theory of ultratough tape peeling 735 17. schapery, r.a., 1975, the theory of crack initiation and growth in viscoelastic media i. theoretical development, international journal of fracture, 11, pp. 141-159. 18. schapery, r.a., 1975, the theory of crack initiation and growth in viscoelastic media ii. approximate method of analysis, international journal of fracture, 11, pp. 369-388. 19. schapery, r.a., 1975, the theory of crack initiation and growth in viscoelastic media iii. analysis of continuous growth, international journal of fracture, 11, pp.549-562. 20. rice, j.r., 1978, the mechanics of quasi-static crack growth, brown university report coo-3084-63, conf-780608-3. 21. rahulkumar, p., jagota, a., bennison, s.j., saigal, s., 2000, cohesive element modeling of viscoelastic fracture: application to peel testing of polymers, international journal of solids and structures, 37, pp. 1873-1897. 22. saulnier, f., ondarçuhu, t., aradian, a., raphaël, e., 2004, adhesion between a viscoelastic material and a solid surface, macromolecules, 37(3), pp. 1067-1075. 23. begley, m.r., collino, r.r., israelachvili, j.n., mcmeeking, r.m., 2013, peeling of a tape with large deformations and frictional sliding, journal of the mechanics and physics of solids, 61(5), pp. 1265-1279. 7414 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 363 380 https://doi.org/10.22190/fume220131021l © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper multi-scale numerical approach to the polymer filling process in the weld line region xuejuan li1, dan wang1, tareq saeed2 1school of science, xi'an university of architecture and technology, xi’an, china 2nonlinear analysis and applied mathematics (naam) research group, department of mathematics, faculty of science, king abdulaziz university, jeddah, saudi arabia abstract. in this paper, a multi-scale coupling mathematical model is suggested for simulating the polymer filling process in the weld line region on a micro scale. the model considers two aspects: one is the coupling model based on stresses in the whole cavity region; the other is the multi-scale coupling model of continuum mechanics (cm) and the molecular dynamics (md) in a weldline region. a weak variational formulation is constructed for the finite element method (fem), which is coupled with the verlet algorithm based on the domain decomposition technique. meanwhile, an overlap region is designed so that the fem and the md simulations are consistent with each other. the molecular backbone orientation of the whole cavity is illustrated and the position of the weld line is determined by the characteristics of the molecular backbone orientation. finally, the properties of the polymer chain in the weld line region are studied conformationally and dynamically. the conformational changes and movement process elucidate that the polymer chains undertake stretching, entangling and orientating. moreover, the effect of the number of chains and melt temperature on the spatial properties of chain conformation are investigated. key words: multi-scale method, weld line, vms-fem, md, domain decomposition 1. introduction the injection molding is a common polymer processing technology. however, the weld line is a common macro defect in polymer processing [1], and it is very difficult to avoid the occurrence of weld line when the products have an insert or the multi-gate technology is used. from the formation of the weld line, the macromolecular chain parallel to each other and the melt properties are different when two or more melt fronts meet. these may cause loose microstructure and stress concentration in the fusion zone, and further affect the received: january 31, 2022 / accepted april 24, 2022 corresponding author: xuejuan li school of science, xi'an university of architecture and technology, xi’an, 710055, china e-mail: lxj_zk@163.com 364 x. li, d. wang, t. saeed products’ mechanical, optical and thermal properties [2]. in order to control this defect and optimize the actual processing, the formation mechanism in processing should be revealed from the molecular orientation, molecular stretching, molecular chain entanglement and so on. therefore, it is great significance to study the weldline’s mechanical properties based on the characteristics of microstructure to improve the quality of the products. for this purpose, much literature was appeared using experiments alone [3-11], for examples, minh et al. studied the weld line strength by using various venting systems in injection molding process [3]. li, et al. [4] and wang, et al. [5] proved that the rapid heat cycle molding (rhcm) was an effective way to improvement of the mechanical properties of weld lines. mosey, et al. [6] and hashimoto, et al. [7] researched the weldline reduction of injection based on different processing parameters. baradi, et al. [8] and liao, et al. [9] found that the experimental x-ray tomography provides us with an effective tool to ensuring the mechanical and geometrical characters of the weldlines and predict the location of weld lines via molecular orientation distribution. oh, et al. [10] and kalus, et al. [11] showed that the digital image is another useful technique to find weldline defects and can illustrate strain profiles in the weld line region. all these researches are extremely useful to show detailed microstructural characterization of the weld line region experimentally, however only experimental methods are not enough to have a deep physical insight into the polymer filling process in a weld line region. although much experimental achievement had been obtained by various advanced experiments, there are still many problems to be solved because of the limitation of experimental conditions, and many hidden mechanisms cannot be revealed by experiment alone, for examples, the chains’ entangling, orientating and stretching. recent decades had seen the fast progress in computer simulation methods, and now the numerical simulation has become a powerful method to improve the quality and processing of polymer products [12-21]. wang et al. [12] adopted fem and the arbitrary lagrangian-eulerian (ale) method to track the free surface of a polymer melt flow in the filling processing. nguyen, et al. [13] and yang, et al. [14] used numerical approaches to study the fiber orientation of the filling processing, and the finite volume method (fvm) was adopted to deal with the polymer melt process. li, et al. [15] and cao, et al. [16] used fem/fvm and fem to solve the flow-induced stress, respectively. deng, et al. [17] pointed out that the lattice boltzmann method (lbm) can predict the injection molding process accurately. farahani, et al. [18] applied the smoothed particle hydrodynamics (sph) technology to study numerically the hybrid process of sheet metal forming and injection molding. zhang, et al. [19] adopted the boundary element method (bem) to complete the steady-state cooling simulation. pashmforoush used finite element analysis to study carbon fibers/carbon nanotubes reinforced polymer composite [20]. moreover, li et al. [21] used a weak variational formulation based multi-scale finite element method (vms-fem) to predict successfully the polymer melt filling process. these numerical methods have their respective advantages in the application in the injection molding, and can make up for the defects of the experiment. above all, we know that there are many research findings for mechanical properties of a weldline region based on some experimental methods and various numerical methods in the injection molding. however, there is few studies on the numerical simulation of weld line region based on multi-scale modeling and calculation. the numerical simulation using multiple scales can reveal some hidden mechanisms which cannot be found using a single multi-scale numerical approach to the polymer filling process in the weld line region 365 macro scale, for example, a chain’s stretching cannot be simulated by any a numerical method, and a molecule’s scale must be used, which makes the numerical simulation impossible. in our previous work [22], the double-equation extended pom-pom (dxpp) stress equation was adopted to describe the molecular orientation in the flow of polymer melt. it is known that some researches based on the molecule’s scale are widely used in various areas [23-27]. cocker et al. [24] tracked a single molecule’s trajectory on a molecule’s scale by femtosecond orbital imaging. zhang, et al. [25] reported the isotropical expansion ability of macromolecular ferritin crystals with integrated hydrogel polymers. moreover, the molecular dynamics (md) simulation can reveal microstructure of the materials [28-30]. for example, valiullin et al. [28] considered a transient sorption in a mesoporous material by the molecular dynamical technology. zhao et al. applied the all atom molecular dynamics to simulate the hiv-1 capsid structure [29]. zepeda-ruiz, et al. studied metal plasticity by the fully dynamic atomistic simulation, and gave the limits conditions of dislocation-mediated plasticity [30]. however, it is not suitable for the problem of long calculation time due to huge computational cost. based on the above analysis, the numerical simulation of mechanical properties of weld line region based on multi-scale modeling and calculation will be studied in this paper. meanwhile, the coupling model of the dxpp stress equation and governing equations of melt flow based on the stress is used to describe the molecular orientation in the whole mold cavity and help to distinguish the weld line region according to the characterization of molecules’ orientation. moreover, the polymer filling process is a two-phase flow problem have discontinuity of unknowns on the interfaces generated by the different material properties of the fluids. multi-scale method with a pressure-enriched finite element shape function enables to simulate the different behavior of two fluids. so, the vms-fem is used to solve this coupling model. the md method is adopted in the weld line region to obtain numerically the real molecules’ information. moreover, the coupling molding of continuum mechanics (cm) and md is established based on the domain decomposition technology. 2. multi-scale model for polymer filling process for the multi-scale model, the multiple scale concept means two aspects: one is the coupling model based on stresses in the whole cavity region excluding weld line, the other is the multi-scale coupling model of continuum mechanics and the molecular dynamics in the weld line region. details are as below in this section. 2.1. the coupling model of the cm and dxpp model based on stresses the polymer melt filling process follows laws in the continuum mechanics, i.e., the mass conservation law and the moment conservation law [22].the moment equation and the mass equation are t t ( ) 1 + ( ) ( ( )) re 1 =( ) ((1 ) ( ) ) re u uu u u u u τ t p h         −   +   +  −  + + (1) 366 x. li, d. wang, t. saeed 0u = (2) where u, p, τ, re, ρ, η, β and φ are, respectively, speed, pressure, and stress, reynolds number, the density, viscosity, viscosity ratio and level set function. hɛ(φ) [31] is given as 0, 1 ( ) 1 sin ( ) , 2 1, h              −     = + +        (3) the average material parameters in gas-melt mixed zone can be written as ( ) ( ) ( ) ( ) ( ) ( ) g m g g m g h h               = + − = + − (4) the subscripts represent gas (g) and melt (m), respectively. in order to obtain the molecular information in the whole filling cavity, we use the dxpp stress model to describe the molecular orientation (s) and stretch (ʌ). in our previous research [22], we had simulated the dxpp model based on a benchmark problem. the dxpp equations read ( )4 4 2 we + ( ) 2[ : ] 1 1 3 (1 3 tr ) 0 3 t t        −  −   +    −  +  + − −  − =    s u s s u u s d s s λ s s λ s s s i λ (5) ( 1)we( + ) we ( : ) ( 1)re t  −  = − −  λλ u λ λ d s λ (6) where we is the weissenberg number; i denotes the unit tensor; d is the deformation rate tensor; r is the relaxation time ratio; α is a material parameter, defining the amount of anisotropy; ν= 2/q, where q is the amount of arms at the end of a backbone. eqs. (5) and (6) are the molecular orientation and stretch equation, respectively. the stresses tensor can be written as [22] 21 (3 ) we − = −τ λ s i (7) so, eqs. (1), (2), (5)-(7) constitute the multi-scale coupled model based on stresses tensor. our model can not only reveal the macro physics quantities, e.g., the velocity and the pressure, but also illustrate the micro molecular orientation and stretch. 2.2. the coupling model of the cm model and the md model the couple of cm and md simulations is constructed for numerical study of the microstructural characterization of the weld line region in the polymer filling process, the latter is to solve atoms’ motion [32] multi-scale numerical approach to the polymer filling process in the weld line region 367 2 2 i i i d r m f dt = 1,2, ,i n= (8) 1 2( , , , )i nf e r r r= − (9) where mi, ri, fi and e are an atom’s mass, displacement, force and the total potential energy, respectively. e consists of following components [32] nb bond angle torsione e e e e= + + + (10) 2 12 12 6 6 nb 2 bond eq bonds 2 angle eq angles 31 2 torsion [ / 4 ( / / )] ( ) ( ) [1 cos( )] [1 cos(2 )] [1 cos(3 )] 2 2 2 i j ij ij ij ij ij ij ij i j r ii i i i i i e q q e r d r d r f e k r r e k vv v e          = + −   = −   = −   = + + − + +      (11) where the subscripts i, j represent the atoms i and j , the subscript “eq” is the equilibrium point; qi represents an atom’s fixed charge, and rij is the distance between two atoms; the parameters dij and ɛij are, respectively, the zero-energy distance and the minimal interaction energy; r and kr are, respectively, its length and stiffness; θ and kθ represent, respectively, the bond angel and the angle stiffness; vn i ( n=1, 2, 3 ) are fitting parameters, and φi the improper dihedral angle. for the domain decomposition technology, the whole calculation region is decomposed into 2 domains, one for the cm simulation, and the other for the md simulation in fig. 1. from this figure we can know that the shading domain represented by the skew lines is the domain of cm simulation region and the domain represented by the dots is the domain of md. c→p is the left boundary of md region, while p→c is the right boundary of cm region. the domain between of these two boundaries is the overlap region, and we assume that in this overlap region the description of two scales is consistent with the description of md. c p→ c p fig. 1 the domain decomposition between the cm and md simulations 368 x. li, d. wang, t. saeed the continuum assumption requires [33] 1 ( ) ( )i j ij t t n =v u (12) and ( )1 ( ) j i ij d t t n dt = u x (13) where nj is the number of molecules. in order to satisfy this non-holonomic constraint, we modify eq. (8) as follows 1 ( )1 jn i j i i ij d t m n m dt= = − + f u x f (14) and its discrete scheme is 1 1 1 1 1 ( ) ( ) j jn n i i i i j md i ij md j t t t m n m t n= =   = − − − +        f f x x u (15) where δtmd is the time step. 3. numerical method for multi-scale model 3.1. vms fem in the asgs stabilized formulation for a general numerical method, the discontinuities of interfaces cannot be captured by standard interpolation functions. moreover, the convective term of the momentum and the constitutive equations always lead to instabilities. fortunately, a pressure-enriched finite element shape function enables to simulate the discontinuities of interfaces. the variational multi-scale method can stabilize the numerical approximation of the convective term. therefore, a vms-fem method is used in this paper. the detailed derivation process is as follows. we introduce u=[u, p, s, ʌ] and write eqs. (1), (2), (5) and (6) in the form ( ) ( , , ; ) 0t u u+ =u s λ (16) where 0 ( ) : we we t t u t t           =            u s λ multi-scale numerical approach to the polymer filling process in the weld line region 369 and 4 4 2 ( 1) ( )2 ˆ ( ( )( 1) 1) re re ˆ ˆˆ ˆ ˆ ˆ( , , ; ) : we( ( ) ) 2we[ : ] 1 1 3 (1 3 tr( )) 3 ˆwe ( : ) ( 1) t h h p u re            −    − − +   −   +           =  −  −   +   −   +  + − −  −       −  − −  λ u u d τ u u s u s s u u s d s s λ s s λ s s s i λ λ d s u λ λ . the system given in eq. (16) is strongly nonlinear and it is difficult to establish a variational principle. in this paper, a weak variational theory is considered for eq. (16), that is to find u=[u, p, s, ʌ]∈x for all v=[ν, q , b, χ ]∈x. the weak variational formulation of the problem can be written as ( ( ), ) ( , , ; , ) 0t u b u v+  =v u s (17) for all v∈x, where: 4 4 2 2 ( )2ˆ ˆˆ ˆ( , , ; , ) ( ( )( 1) 1) , ( , ) , re re ˆ ˆ ˆ( , ) ( , )+2we[ ]( , ) we( ( ) , ) 1 1 1 3 (1 3 tr( )) , , 3 ˆ+(we ( : ) t h b u v h v v v p v q b b b b               = − +  +  +         −   +   +  −  −   −   +  + − −  −      −  − u s d u u τ u d : s s u s s u u s λ s s λ s s s i λ λ λ d s u λ ( 1) ( 1), )re − −λ λ we decompose u=uh+u’, where uh and u’ are, respectively, numerically solved by fem method and the sub-grid scale method [34]. 1 2 3 4 ( ( ), ) ( , , ; , ) ( ; , ) ( , ) ( ; , )+ ( ; , ) 0 t h h h h h h h h h h h h h h h h h h h u v b u v s u v s u v s u v s u u v +  + + + = u s u u (18) 1 1 , ( )2 ˆ( ; , ) [ ( ( )( 1) 1) ], re re ( )2 ˆ ( ( )( 1) 1) (1 ) re re h h h h h h h h k h h v h h h k h s u u v p h p h v h v q               =  − − +  −  +  − − +  − −  +  u u d τ u d 2 2( , ) [ ],h h h h k k s u v p u v=   3 3 2 4 4 2 4 4 ˆ ˆ ˆwe( ( ) ) 2we[ : ] ( ; , ) , [3 (1 3 tr( )) (1 ) 3 ] ˆ ˆ ˆwe( + + ( ) ) 2we[ : ] [3 (1 3 tr( )) t h h h h h h h h h h h h k h h h h h h h h t h h h h h h h h h h h h h h h h s u u v p b b b s b s s s s         − −   −  −   + =   +  + − −  − −        + +  + − −   u s s u u s d s s λ λ s s λ s s s i u u u d λ λ λ (1 ) 3 ]h k b − − i 370 x. li, d. wang, t. saeed ( 1) 4 4 ( 1) ˆ( ; , ) [we ( : ) ( 1)], ˆwe ( : ) ( 1) h h h h h h h h h h h k h h h h h h k s u u v p re re       − − = −  − − −  − −  λ λ λ d s u λ λ d s u 1 1 1 22 21 hu c c hh   −   = +    , 2 1 2 1 1 h c   = , 1 3 3 4 22 2 h h uc c u h       −    = + +        , 1 5 4 2 c   −   =     . where αi, i=1, 2, 3, 4 are adjusting parameters, and the algorithmic parameters cj, j=1, 2, 3, 4 must be adjusted in the stabilized formulation. the parameters h1 and h2 are the characteristic element lengths, the latter is in the streamline direction. the finite element simulation based on the weak variational formulation given in eq. (17) can reveal the macro properties of the polymer filling process. in the simulation process, we adopt the crank-nicolson-based split (cnbs) algorithm to study numerically the problem, see ref. [15] for detailed description. 3.2. verlet algorithm eq. (8) is discretized as [35]: 2 ( ) ( ) ( ) ( ) 2 i i i i t t t t t t t  + = + + a r r v (19) ( ) ( 2) ( ) 2 i i i t t t t t  +  = + a v v (20) ( ) ( ) ( 2) 2 i i i t t t t t t t  +  +  = +  + a v v (21) where vi and ai are, respectively, the speed and acceleration of the particle i. 1 n 32 1 n 32 1 n 32 1 n 32 1 n 32 1 n 32 1 n 32 1 n 32 1 n 32 fig. 2 the sketch of periodic boundary conditions for the md simulation, the finite and infinite systems are extremely not similar. however, the simulated system is usually much smaller than the real physical system. generally, the simulation takes place in a cube cell with periodic boundary to eliminate the multi-scale numerical approach to the polymer filling process in the weld line region 371 interfacial effects. a periodic system is illustrated in fig. 2, and the simulation chart is given in fig. 3. fig. 3 the flow chart of the coupled numerical method 4. numerical results 4.1. determination of weld lines for a mold filling process with multiple gates or inserts, the weld lines will be introduced where two or more streams of melts meet with a certain angle to each other in injection molding. in this paper, a 101 cavity is used and the two injection gates are symmetrically located on the left and right sides. we use the improved level set method [21] to capture the front interface of polymer melt. fig. 4 shows the positions of the double gates, the flow direction and the front interface. moreover, it can be seen that the position of the interface fusion region is about x=5. fig. 4 the positions of the double gates and front interface at different time 372 x. li, d. wang, t. saeed for the determination of weld lines, we can obtain the molecular backbone orientation by the dxpp model. moreover, the molecular backbone orientation of the weld region is vertical orientation to the flow direction. therefore, we can determinate the position of weld line from the characteristics of the molecular backbone orientation. the ellipse method is used to find the backbone orientation [22]. the molecular backbone orientation in the inlet (a), the boundary (b) and the interface fusion region (c) is shown in the fig. 5. it can be seen clearly that the molecular backbone orientation in different regions is significantly different. moreover, there are clearly horizontal orientation and vertical orientation near x=5. generally, these results indicate that molecular backbones’ direction follows the streamline in the center part, whereas molecular backbones’ direction is almost perpendicular to the streamline at the weldline. so, we can determinate the position of weld line from the characteristics of the molecular backbone orientation. fig. 5 the molecular backbone orientation of different regions the weld lines positions are obtained based on the molecular backbone orientation and the level set method [36], respectively. they are shown in fig. 6 and the weld line positions (blue lines in fig. 6) are consistent. therefore, the molecular backbone orientation method is effective for capturing the weld line in the filling process. (a) (b) fig. 6 the position of the weld lines. (a) the molecular backbone orientation;(b) level set method 4.2. the results analysis in weld region in this sub-section, all variables are the dimensionless forms. the initial chain spatial configurations are zigzag configurations and the number of chains in a cube cell has three cases: one chain, two chains and four chains, as shown in fig. 7. we assume all chains to have the same length including 8 carbon atoms. multi-scale numerical approach to the polymer filling process in the weld line region 373 (a) (b) (c) fig. 7 the initial spatial configurations. (a) one chain; (b) two chains; (c) four chains (a) (b) (c) (d) (e) (f) fig. 8 the spatial configurations of one chain at different times. (a) t=50; (b) t=550; (c) t=1050; (d) t=1550; (e) t=2050; (f) t=2500 the properties of the polymer chain in the weld line region are studied for conformational purposes. the conformational changes and movement process of one chain, two chains and four chains are shown in fig. 8, fig. 9 and fig. 10, respectively, which elucidate that the polymer chain undertakes stretching, orientating, and entangling. moreover, we can see clearly that the entangling polymer chains will separate again, as shown in fig. 10. 374 x. li, d. wang, t. saeed (a) (b) (c) (d) (e) (f) fig. 9 the spatial configurations of two chains at different times. (a) t=50; (b) t=550; (c) t=1050; (d) t=1550;(e) t=2050; (f) t=2500 to study the molecular chain, the spatial properties of chain conformation are usually investigated by<r2> <s2> , and gxy, defined by 2 2 1 2 1 2 3 1 | | 1 ( )( ) n n xy ix x iy y i r r r s g g g g r r r r n =   = −   = + +   = − −   (22) where, r _ is the center of mass; g1, g2, g3 are the three eigenvalues of gxy; the ratios of eigenvalues g21=g2/g1 and g31=g3/g1 are important factors, when they are different from unity, the distribution is non-spherical. fig. 11 shows that the effect of the number of chains and the melt temperature on the bond length (l). it can be seen clearly that a larger number of molecular chains sees a longer bond length, and a higher melt temperature results in a longer bond length as well. the effects of the number of chains and the melt temperature on <r2> and <s2>are shown as in fig. 12 and fig. 13, respectively. we can see clearly that they decrease with the increasing of the number of molecular chains and no significant changes about the melt temperature is observed. multi-scale numerical approach to the polymer filling process in the weld line region 375 (a) (b) (c) (d) (e) (f) fig. 10 the spatial configurations of four chains at different times. (a) t=50; (b) t=550; (c) t=1050; (d) t=1550;(e) t=2050;(f) t=2500 (a) (b) fig. 11 influences of the chains’ number (a) and the melt temperature (b) on the bond length 376 x. li, d. wang, t. saeed (a) (b) fig. 12 influences of the chains’ number (a) and the melt temperature (b) on <r2> (a) (b) fig. 13 influences of the chains’ number (a) and the melt temperature (b) on<s2> (a) (b) fig. 14 influences of the chains’ number (a) and the melt temperature (b) on g21 multi-scale numerical approach to the polymer filling process in the weld line region 377 (a) (b) fig. 15 influences of the chains’ number (a) and the melt temperature (b) on g31 for the ratios of eigenvalues, fig. 14 and fig. 15 show that the values of g21 and g31are far less than 1, which indicates the mass distribution of polymer chain is non-spherical. moreover, the ratios decrease with the increasing of the number of molecular chains while no obvious changes with the melt temperature’s change. 5. discussion and conclusion a multi-scale coupling model is suggested for simulating the weld line region on a micro scale of the polymer filling process and the other part in a macro scale. the macro scale simulation follows the laws in continuum mechanics, while the micro scale simulation uses the molecular dynamics to reveal the physical understanding of the chains’ motion including stretching and entangling. recently the two-scale concept was further developed into a new mathematical branch called the two-scale fractal calculus [37-39], which studies a same problem, e.g., the polymer filling process, using two differential scales, one is macro scale for continuum mechanics and the other is generally the micro scale of the molecule’s size. seeing with a single scale is always unbelieving [40]. now the two-scale fractal theory and two-scale thermodynamics have been widely used to study various discontinuous problems [41-45]. li, et al. established successfully a fractal two-phase flow model for the polymer filling process [46]. zhou, et al. studied the motion of a drop of ren ink in a saline water [47], though the motion can be easily simulated by the molecular dynamics, the two-scale fractal calculus provides us with an effective tool to the analysis of red ink’s motion. the macro scale reveals the main flow property of the studied problem – the red ink will move along the streamlines of the moving fluid, which can be predicted by the laws in continuum mechanics. however, the continuum assumption in fluid mechanics cannot reveal the diffusion of the red ink in water, so another smaller scale of the molecule’s size is needed, and the fluid has to be considered as a porous medium, where conversation laws can be established in a fractal space. in this paper we adopt a weak variational principle, there might not be a genuine variational principle for navier-stokes equations, however, we can establish an approximate variational principle for the polymer filling process by the semi-inverse method [48-51], we will discuss it 378 x. li, d. wang, t. saeed in a forthcoming paper. other numerical methods, for examples, the variational iteration method (vim) [52] and the reducing rank method [53], can also be used for the numerical simulation. in this paper, we propose two coupling models: the coupling model of the cm and dxpp model and the coupling model of the cm and md. moreover, vms-fem and verlet algorithm are used on macro-scale and micro-scale based on the domain decomposition technique, respectively. the molecular backbone orientation can be obtained in the whole filling process by the dxpp model. therefore, we can determinate the position of the weld line by the molecular backbone orientation’s characteristics. finally, the properties of the polymer chain in the weld line region are studied conformationally and dynamically. the conformational changes and movement process illustrate that the polymer chain undertakes stretching, orientating, and entangling. moreover, 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numerical methods for heat and fluid flow, 30(11), pp. 4797-4810. 53. he, j.h., el–dib, y.o., 2021, the reducing rank method to solve third-order duffing equation with the homotopy perturbation, numerical methods for partial differential equations, 37(2), pp. 1800-1808. 13732 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 649 665 https://doi.org/10.22190/fume250528023h © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper variational approach to micro-electro-mechanical systems ji-huan he1,2,3,4, jinfeng ma5, abdulrahman ali alsolami6, chun-hui he7,8 1school of information engineering, yango university, fuzhou, china 2department of mathematical sciences, saveetha school of engineering, simats, chennai, tamil nadu, india 3school of jia yang; zhejiang shuren university, hangzhou, china 4school of mathematics and big data, hohhot minzu college, hohhot, china 5college of civil engineering and architecture, jiaxing university, jiaxing, china 6department of mathematics, faculty of science, king abdulaziz university, saudi arabia 7school of civil engineering & transportation, foshan university, foshan, china 8school of mathematics, china university of mining and technology, xuzhou china orcid ids: ji-huan he https://orcid.org/0000-0002-1636-0559 jinfeng ma https://orcid.org/0009-0000-2348-9142 abdulrahman ali alsolami https://orcid.org/0000-0002-8189-4987 chun-hui he https://orcid.org/0000-0003-0810-5248 abstract. this paper presents a novel approach for formulating a variational principle tailored to microelectromechanical systems (mems) through the utilization of the semiinverse method. the resulting variational principle is that of least action, which is of considerable significance. the newly presented variational principle has the potential to be applied in a number of advantageous ways. one of the primary applications of this approach is the determination of the pull-in voltage. the application of this principle allows for a more accurate and efficient determination of the pull-in voltage. this is of paramount importance for the optimal functioning and optimization of mems devices. the enhanced accuracy in determining the pull-in voltage enables more precise design and greater reliability of memsbased systems. additionally, the enhanced computational efficiency allows for the saving of valuable time and resources during the design process. furthermore, the paper addresses the topic of fractal mems and puts forth a novel approach to fractional differentiation based on two-scale fractal differentiation, which is anticipated to facilitate the discovery of new insights and optimization strategies for mems devices. key words: semi-inverse method, pull-in instability, fractal mems, two-scale fractal derivative based fractional derivative, wearable textiles received: may 28, 2025 / accepted july 08, 2025 corresponding author: ji-huan he, jinfeng ma, chun-hui he school of jia yang, zhejiang shuren university, hangzhou, china college of civil engineering and architecture, jiaxing university, jiaxing, china school of civil engineering & transportation, foshan university, foshan, china email: hejihuan@ygu.edu.cn (j.h. he), majinfenglz@163.com (j. m), mathew_he@yahoo.com (c.h. he) 650 j.h. he, j. ma, a.a. alsolam, c.h. he 1. introduction micro-electro-mechanical systems (mems) have been instrumental in the modern technological landscape, primarily due to their ability to facilitate the development of miniature integrated devices that effectively combine mechanical and electrical functionalities [1]. their applications are found in critical fields, such as wearable textiles [2]. mems sensors have been developed for the monitoring of physiological parameters, including heart rate and body temperature, with exceptional precision. flexible actuators have been incorporated into adaptive clothing, thereby empowering it to respond dynamically to environmental stimuli. in the domain of biomedical engineering, mems devices play a pivotal role in facilitating targeted drug delivery and real-time diagnostics. these devices leverage their microscale size to interact seamlessly with biological systems, a property that has been demonstrated to support inner ear theragnostics [4]. furthermore, mems oscillators function as essential components in communication systems, providing stable timing signals that are crucial for 6g and subsequent generations [5]. roy et al. [6] developed a generative ai-assisted piezo-mems ultrasound device tailored for plant dehydration monitoring. this device merges piezo-mems technology with generative ai to enable efficient tracking of plant water status. liang and lin [7] introduced a mems electrothermal actuator (eta) featuring highly linear actuation and stable characteristics. the actuator under consideration consists of a bilayer cantilever array of au and si₃n₄, and it is compatible with standard cmos manufacturing processes. the device's surface roughness engineering prevents stiction, exhibits exceptional linearity between displacement and squared driving voltage, achieves an ultra-low power consumption of 7.5 mw, and maintains stable performance over 100 cycles. potential applications include optical modulation and medical imaging, among others. these diverse applications necessitate accurate models to predict dynamic behaviors, particularly pull-in instability – a phenomenon where electrostatic forces surpass mechanical restoring forces, leading to device failure and compromising operational reliability [8]. the variational principle has long served as a foundational tenet within the realm of theoretical physics, offering a unifying framework for the analysis of system dynamics through the minimization of energy. from hamilton's principle, which governs classical mechanics, to its extensions in field theory and quantum mechanics, variational methods integrate conservation laws and dynamic constraints into a single formulation [9]. in the field of mems research, early applications of variational principles centered on linear oscillations and elementary geometries [10], drawing on techniques such as the variational iteration method [11] and finite element analysis [12]. however, these approaches encounter challenges due to nonlinear electrostatic-mechanical coupling, microscale effects (e.g., surface tension), and complex fractal geometries. these limitations hinder the accuracy of predictions of critical parameters like pull-in voltage [13, 14]. concurrently, niu has substantiated the robust minimum condition of a variational formulation applicable to mems systems [15]. against this backdrop, the need for a tailored variational principle for mems becomes evident [16]. traditional models fail to capture singularities in displacement fields and overlook fractal-induced dynamics, leading to design inefficiencies. this paper addresses these gaps by developing a novel variational formulation via the semi-inverse method, with applications to pull-in instability analysis and fractal mems modeling. variational approach to micro-electro-mechanical systems 651 2. mems oscillator following the notable developments of the duffing oscillator and van der pol oscillator, the mems oscillator is identified as the subsequent most crucial component in the domain of nanotechnology. a mems oscillator represents a revolutionary timing device that has gained significant prominence within the field of electronics. it employs microelectromechanical systems technology to facilitate precise and reliable oscillatory behavior. the micromachining technology that emerged in the late 1980s enables the fabrication of sensors and actuators with a micron-scale resolution. such micro transducers can be integrated with signal conditioning and processing circuitry to form microelectromechanical-systems, which are capable of performing real-time distributed control. the system operates periodically in general, but the pseudo-pull-in stability is a subject of research study [17, 18]. the dimensionless mems oscillator can be written as [17]: 0 1 m w w w  + − = − (1) where w is the dimensionless displacement, 0<w<1, and m is voltage-related parameter. this equation arises from the dynamic balance between mechanical restoring forces and electrostatic forces in a parallel-plate mems structure. when m is small, eq. (1) has periodic solution, however, when m is larger than a threshold value, m*, the pull-in motion occurs, and m* is often called as the pull-in voltage. this equation possesses distinctive characteristics, namely: 1) zero initial conditions, i.e., w(0)=0 and w’(0)=0; and 2) the singularity at w=1. nevertheless, it can be effectively solved by the homotopy perturbation method [19, 20, 21], the variational iteration method [22, 23], and he-liu’s modified frequency formulation [24], albeit with the aid of certain special techniques. the variational principle in mems systems constitutes a fundamental concept that facilitates comprehension and analysis of the behavior of these complex systems (see ref. [10]). to explain eq. (1) in a mechanical view, we consider the motion for a particle with unit mass in a potential field v: 21 ( ) ln(1 ) 2 v w w m w= + − (2) the variational principle of mems oscillator given in eq. (1) can be written in the form 2 2 21 1 1 ( ) ( ) ln(1 ) 2 2 2 j w w v w dt w w m w dt      = − = − − −          (3) according to the variational formulation given in eq. (3), the following identity 2 21 1 ln(1 ) 2 2 w w m w h + + − = (4) where h is the hamilton invariant, it can be calculated by the zero initial conditions, and it results in h=0, so we have 2 21 1 ln(1 ) 0 2 2 w w m w + + − = (5) 652 j.h. he, j. ma, a.a. alsolam, c.h. he we introduce a new variable, k, defined as: 21 2 k w= . (6) the new variable is the kinetic energy, from eq. (5), we have: 21 ln(1 ) 2 k w m w= − − − . (7) for small displacement, eq. (7) can be approximately expressed as 2 2 2 21 1 1 1 ( ) 2 2 2 2 k w m w w w mw mw= − − − − = − + + , (8) or 22 4 8 (1 ) 2(1 ) m m k m w m + − − = − . (9) eq. (9) implies that the displacement (w) is a function of m and k. 3. criterion for pull-in instability pull-in instability in mems refers to a phenomenon where an electrostatically actuated mems device collapses or snaps to a new stable state. this occurs due to the interaction between electrostatic forces and mechanical restoring forces. as the applied voltage increases, the electrostatic force overcomes the mechanical restoring force, leading to instability. it can cause failure in mems devices and is an important consideration in their design. understanding and predicting pull-in instability is crucial for ensuring the reliable operation of mems. the criterion for pull-in instability in mems is a key aspect in understanding and designing these devices. as mentioned earlier, the criterion for pull-in instability is that when the kinetic energy and the accelerated speed become zero simultaneously [16]. this means that at this point, the device is in a state of unstable equilibrium and is likely to collapse or snap to a new stable state. this criterion provides a quantitative measure for engineers to determine when pull-in instability may occur. for example, in the design of a mems actuator, engineers can use this criterion to calculate the maximum allowable voltage that can be applied before pull-in instability occurs. by understanding this criterion, engineers can design mems devices that are more stable and reliable, and avoid potential failures due to pull-in instability. the criterion for pull-in instability is that when the kinetic energy and the accelerated speed become zero simultaneously. according to eq. (1) and eq. (7), we have: 0 1 m w w − = − (10) variational approach to micro-electro-mechanical systems 653 21 ln(1 ) 0 2 w m w− − − = (11) solving eqs. (10) and (11) simultaneously, we find the threshold values of the displacement and the voltage, * 0.7153 * 0.2036 w m =  = (12) that implies that in the event that the value of w is greater than w* or the value of m is greater than m*, the phenomenon of pull-in instability takes place, w* is called as the pullin displacement, and m* is the pull-in voltage. this criterion for pull-in instability is highly accurate and extremely reliable. the accurate determination of the pull-in voltage provides a valuable benchmark for researchers and engineers working in the field of electrostatically actuated mems/nems. it allows for a more precise understanding and prediction of the occurrence of pull-in instability, enabling better design and optimization of these micro and nano devices. with such a highly accurate criterion, it becomes possible to make more informed decisions regarding the design parameters and operating conditions to avoid or manage pull-in instability effectively. 4. semi-inverse method and a new variational formulation for mems oscillator the semi-inverse method [25] is a powerful approach for establishing a variational principle. it offers a systematic way to solve complex problems in various fields, including mechanics and engineering. though the semi-inverse method, many new variational formulations were appeared in literature for incompressible fluids [26], lubrication problems [27], fractal solitary waves [28] and the simplified navier-stokes equations [29]. we re-write eq. (1) in the following form: 2 0w w w w w m + − − − = (13) the semi-inverse method commences with the formulation of an informed hypothesis regarding the form of the variational formulation. based on physical intuition or prior knowledge, a trial functional is selected that contains some unknown parameters. subsequently, the variational problem is formulated by applying specific constraints and principles. the fundamental concept is to partially invert the problem by selecting a trial functional that satisfies some of the essential properties of the solution. subsequently, the functional is varied with respect to the unknown parameters, thereby obtaining the euler-lagrange equations. the semi-inverse method is advantageous in that it reduces the complexity of the problem by narrowing down the search space. it offers a methodology for the acquisition of approximate solutions when the identification of exact solutions is challenging. moreover, it is capable of handling nonlinear problems and can be extended to multidimensional and time-dependent systems. 654 j.h. he, j. ma, a.a. alsolam, c.h. he in accordance with the semi-inverse method, the following assumption is made: 2 2 2 31 1 1 1 ( , ) 2 2 4 3 j w k w w w w w mw f dt    = − + + + +     (14) where f is unknown yet. equation (14) represents a trial variational principle in our study. in this context, the independent variables play a crucial role in defining the problem. the kinetic energy (k) is considered an independent variable as it reflects the main dynamical property of the mems oscillator. when k = 0, the pull-in motion occurs. additionally, the displacement (w) is another independent variable. when w=1, the pull-in instability occurs. thus, k and w are the main parameters for studying the pull-in property. the variational principle allows us to understand the significance of these variables and how they interact. a comprehensive understanding of these independent variables is essential for a detailed analysis of the variational principle and its applications in our study. the euler-lagrange equation with respect to w is: 2 21 1 ( ) 0 2 4 f w w w w w w m w     − − + + + + + = , (15) or 2 21 0 2 f w w w w w w m w     − − + + + + + = , (16) where δf/δw is the fractional derivative [30], defined as: 2 2 ( ) ( ) f f d f d f w w dt w wdt      = − +     . (17) in view of eqs. (13) and (6), we have: 2 2 21 1 2 2 f w w w w w w m w k w      = + − − − − = − = − . (18) from eq. (18), f can be identified as: 1f kw f= − + , (19) where f1 is an unknown function of k. so the trial-functional can be updated as: 2 2 2 3 1 1 1 1 1 ( , ) 2 2 4 3 j w k w w w w w mw kw f dt    = − + + + − +     (20) the euler-lagrange equation with respect to k is: 1 0 f w k   − + = . (21) that is: variational approach to micro-electro-mechanical systems 655 2 1 2 4 8 (1 ) 2(1 ) m m k mf w k m   + − − = = − . (22) now f1 can be identified as: 3/2 2 1 2 2 1 4 8 (1 ) 2(1 ) 24(1 ) mk f m k m m m  = − − − − − . (23) finally, the following variational formulation is obtained: 2 2 2 3 3/2 2 2 1 1 1 1 ( , ) 2 2 4 3 2 1 4 8 (1 ) 2(1 ) 24(1 ) j w k w w w w w mw kw dt mk m k m dt m m    = − + + + −       + − − −  − −    (24) the new obtained variational formulation given in eq. (24) gives deeper insight into the physical property of the mems oscillator in an energy sense, but it is valid for the small displacement. when w is not small enough, the derivation of the variational principle is much more complex, and we obtain the following variational formulation 2 2 2 3 3 3 2 2 3 1 1 1 1 ( , ) 2 2 4 3 1 ln(1 ) ln(1 ) 6 1 1 ln(1 ) ( ) 2 2 j w k w w w w w mw dt w mw w mw m w dt a k w m w b k w dt    = − + + +      − + − − − −        + + + − + −          (25) where a and b are nonzero constants. now we use the established variational formulation to investigate the pull-in voltage. before pull-in inability, the kinetic energy and accelerated velocity tend to zero, under this extreme case, we have: 2 31 1 ( ) 2 3 j w w w mw dt   = − + +     . (26) the euler-lagrange equation becomes: 2 0w w m− + + = (27) solving m from eq. (27), and submitting the result into eq. (7), we have: 2 21 ( ) ln(1 ) 0 2 w w w w+ − − = (28) this equation can be solved by newton iteration method, here introduces an ancient chinese algorithm [31, 32]. we assume two trial solutions w1=0.7 and w2=0.75, eq. (28) 656 j.h. he, j. ma, a.a. alsolam, c.h. he produces the residuals r1= -0.007834 and r2=0.06249. the ancient chinese algorithm results in the following approximate solution: 1 2 2 1 1 2 0.71 r w r w w r r − = = − . (29) we choose w3=0.71, and the residual is r3=-0.001737, which leads to: 1 3 3 1 1 3 0.7128 r w r w w r r − = = − . (30) the pull-in voltage can be calculated as: 2* 0.2045m w w= − = . (31) the precise value is 0.203632188, as demonstrated in fig. 1, which illustrates the phase-space dynamics of the mems oscillator by plotting dimensionless velocity against displacement for varying voltage-related parameter. fig. 1 phase trajectories for m=0.05, 0.1, 0.15, and 0.203632189 sub-critical voltage behavior manifests when m < m*. for values of m ranging from 0.05 to 0.15, the trajectories exhibit closed, elliptical loops, thereby directly corresponding to periodic solutions where electrostatic and mechanical forces maintain stable equilibrium. these loops remain constrained within the range of values 0 < w < w*, thereby confirming that kinetic energy and acceleration never simultaneously reach zero—a finding that is consistent with the established pull-in criterion. the critical voltage behavior emerges when the parameter m is equal to m*. the innermost trajectory, collapsing to a point near w = w*, visualizes the threshold at which the system teeters on instability. the loop contracts towards a singularity, thereby satisfying the pull-in condition: when m=m*, the kinetic energy and acceleration simultaneously approach zero. this phase transition serves to substantiate the variational principle's forecast of a welldefined boundary between stable oscillation and pull-in motion. variational approach to micro-electro-mechanical systems 657 equation (31) demonstrates a notable level of precision. it is noteworthy that the ancient chinese algorithm attains higher accuracy with a single iteration, underscoring its property of rapid convergence to a substantial level of precision. in addition, the variational principle facilitates precise voltage-controlled manipulation, thereby providing a robust framework for highly accurate voltage regulation. the synergy between the specified pull-in voltage value, the efficacy of the algorithm, and the precision of the variational principle underscores their collective potential for facilitating precise and regulated operations in the domains of mems design and optimization. fig. 2 demonstrates that when m < m* or w < w*, the system exhibits periodic solutions. however, as the voltage parameter m increases, the displacement, w, approaches its threshold value, w*. fig. 2 periodic solution for m<m* once w exceeds w*, pull-in motion occurs, as shown in fig. 3. in contrast to the periodic behavior observed in fig. 2, where the dimensionless displacement oscillates within the bounded range, 0<w <w*, under sub-critical voltages (m<m*), the dynamics depicted in fig. 3 reveal a striking shift when m exceeds the critical threshold, m*. here, w abandons all oscillatory characteristics and instead increases monotonically, advancing relentlessly toward the singularity at w=1. this unidirectional trajectory signifies an irreversible pullin motion, a catastrophic transition driven by the breakdown of the delicate balance between electrostatic and mechanical forces that governs stable operation. in the sub-critical regime (fig. 2), mechanical restoring forces—rooted in the elastic properties of the mems structure—successfully counteract the electrostatic attraction between electrodes, limiting w to periodic oscillations. however, as m surpasses m*, the electrostatic force undergoes a rapid amplification. this force exceeds the mechanical restoring capacity of the system, which weakens as the gap between electrodes narrows, as a result, the electrodes make contact, rendering the device inoperable. 658 j.h. he, j. ma, a.a. alsolam, c.h. he fig. 3 pull-in motion for m>m* notably, fig. 3 illustrates that the steepness of the pull-in curve intensifies with increasing m beyond m*. this phenomenon arises because a higher m corresponds to a stronger initial electrostatic force, which accelerates the narrowing of the electrode gap. as the gap shrinks, the electrostatic force increases further, creating a positive feedback loop that steepens the trajectory. this behavior aligns with the predictions of the variational principle, which emphasizes that beyond m*, the system’s energy balance (governed by eq. (5) is violated, with the potential energy dominated by electrostatic contributions that drive the irreversible collapse. this monotonic, gap-closing motion captured in fig. 3 thus serves as a critical experimental validation of the theoretical pull-in criterion: the simultaneous decay of kinetic energy and acceleration at w = w* marks the point of no return, beyond which the system is pulled inexorably toward the singularity—a hallmark of pull-in instability in electrostatically actuated mems devices. 5. two-scale fractal derive based fractional derivative and its application to mems system the core variational principle, eq. (24), is derived from the classical mems oscillator model, which presupposes idealized, smooth geometries. for fractional mems featuring fractal structures (e.g., rough surfaces or porous microstructures), the electrostaticmechanical coupling is inherently modulated by multi-scale geometric irregularities. to address this disparity, the variational functional ought to be extended to incorporate fractal dimensions, thereby explicitly establishing a connection between the semi-inverse method and fractal dynamics. variational approach to micro-electro-mechanical systems 659 the two-scale fractal, as a geometric concept, is tailored to describe porous media or unsmooth boundaries characterized by two distinct scales [33]. within the domain of fractal mems systems, the two-scale fractal concept finds several pivotal applications: 1) modeling complex geometries: fractal mems devices often have complex geometries that cannot be accurately described by traditional geometric models. the twoscale fractal geometry provides a more realistic representation of these structures. for example, the surface of a fractal mems device may have irregularities and self similar patterns at different scales, which can be modeled using two-scale fractals. this allows for a better understanding of how the geometry affects the device's physical properties such as capacitance, stiffness, and thermal conductivity. 2) understanding physical processes: the two-scale fractal derivative helps in understanding the physical processes that occur within fractal mems devices. it can be used to analyze how mechanical and electrical properties change with respect to the fractal geometry. for instance, in the study of pull in instability in mems, the two-scale fractal derivative can provide a more detailed analysis of how the instability is affected by the fractal structure of the device. this can lead to better design and optimization of mems devices to avoid or control the pull in instability. 3) novel device design: the concept of two-scale fractal can inspire novel designs for mems devices. by incorporating two-scale fractal geometries, it may be possible to achieve enhanced performance in terms of sensitivity, selectivity, or power consumption. for example, a fractal mems sensor with a two-scale fractal structure may have a higher sensitivity to certain physical quantities compared to a traditional mems sensor. this can open up new avenues for the development of advanced mems devices for various applications such as sensing, actuation, and signal processing. the study of mems oscillators often requires a comprehensive understanding of their behavior under different geometric and physical conditions. in this regard, the generalization of the governing equations to account for more complex geometries, such as the two-scale fractal, is of great importance. this section focuses on detailing how the eq. (1) for the mems oscillator can be extended to incorporate the two-scale fractal concept, providing a more accurate and detailed description of the oscillator's behavior. fractal mems systems [34, 35] hold the potential to be a truly revolutionary advancement in the extensive and ever-evolving field of microelectromechanical systems. these remarkable systems draw their inspiration from the captivating concepts such as the gecko effect [36], with its astonishing ability to allow geckos to adhere to various surfaces, and the lotus effect [37], which endows lotus leaves with their remarkable self-cleaning properties. the fractal surface at the nano scale is truly a wonder, as it possesses an extremely high surface energy, known as geometric potential [38, 39], which is widely used to design nano scale materials, demonstrating a creative and forward-thinking approach that could potentially reshape the landscape. tian and her colleagues, in their remarkable research efforts as documented in ref. [40], put forward the highly innovative concept of fractal mems systems. this concept is aimed at overcoming the pull-in instability that often plagues such systems. by introducing this concept, they strive to make the systems significantly more reliable. the governing equation becomes: 2 2 0 1 d d d w m w wdt + − = − . (32) 660 j.h. he, j. ma, a.a. alsolam, c.h. he in this equation, d represents the two-fractal dimensions, which are essential in defining the fractal nature of the geometry. the two-scale fractal derivative is given as follows: 0 0 0 0 ( , ) ( , ) ( ) (1 ) lim ( ) d d dt t t w x t w x tw t d t t t− → − =  +  − . (33) this definition captures the essence of the fractal derivative by considering the limit of the difference quotient as the time interval approaches zero, weighted by the gamma function of the fractal dimension. the two-scale fractal derivative has found extensive application in modeling complex problems. as evidenced by numerous studies [41, 42], it has demonstrated its remarkable effectiveness in this regard. this derivative possesses the following properties: 1 0( ) (1 )( ) d d d w w t d t tt −    +   (34) 2 2 ( ) d d d d d d w w t t t    =    (35) 2 2 , 0 , 1 , 2 d d w d w w d tt w d t   =    = =    =  (36) the generalized equation has several important implications for the study of mems oscillators. firstly, it provides a more accurate model for predicting the behavior of mems oscillators with fractal geometries. this is particularly useful in applications where the fractal nature of the mems structure plays a significant role. for example, in certain sensor and actuator designs, the fractal geometry can enhance the sensitivity and performance of the device. the generalized equation allows us to analyze how the fractal structure affects the oscillator's response to different inputs, such as voltage changes or external perturbations. secondly, the generalized equation allows for a deeper understanding of the physical processes involved in the operation of mems oscillators. by considering the fractal geometry, we can better analyze the effects of various factors on the oscillator's performance. for instance, the surface roughness of the mems device, which is often related to its fractal structure, can influence the dissipation of energy and the damping characteristics of the oscillator. additionally, the distribution of material properties within the fractal structure can affect the mechanical and electrical properties of the oscillator, such as its stiffness and capacitance. the generalized equation enables us to study these effects in more detail, providing insights into how to optimize the design and performance of mems oscillators. in conclusion, the generalization of eq. (1) for the mems oscillator to the two-scale fractal case is a significant step forward in understanding the behavior of mems oscillators in the context of fractal geometry. this generalization not only allows for a more accurate description of the oscillator's behavior but also provides insights into the physical processes variational approach to micro-electro-mechanical systems 661 involved. future research can further explore the applications and implications of this generalized equation for the development of more advanced mems devices. by continuing to study the behavior of mems oscillators under different geometric and physical conditions, we can improve the design and performance of these devices, leading to more efficient and reliable mems based systems. the fractional mems is indeed another highly captivating hot topic in the field of technology. in this domain, the fractional derivative, as demonstrated in references [43], plays an absolutely crucial role. fractional mems systems truly represent a remarkable technological innovation. these systems are an enthralling combination of advanced engineering and profound mathematical concepts. through the incorporation of fractional calculus principles, they manage to achieve not only enhanced performance but also expanded functionality. the utilization of fractional derivatives enables a more accurate description of the intricate behaviors and dynamic processes exhibited by these miniature mechanical and electrical systems. for instance, in the area of stochastic processes [44] and adaptive chaos control [45], the fractional derivative has shown great potential in providing a deeper understanding and better control of these complex phenomena. as research in this field continues to progress, fractional mems systems are expected to bring about even more significant technological breakthroughs and open up new avenues for applications in various industries. other definitions of the fractional derivatives can be modified in a similar way, for example, the caputo-fabrizio derivative and its generalization [46] can be updated as: 0 1 exp( ) 1 1 d t d w w ds t t        = − − −   (37) 0 1 ( , ) 1 ( , ) d t d w k s w ds k tt t        = −   (38) we give a generalized definition of the two-scale fractal derivative based fractional derivative in the forms: 0 ( ) ( , ) ( , ) d t d w w s d t s ds t s        =    (39) 0 ( , ) ( , ) ( ) d t d w d t s w s ds t t        =    (40) where  is the normalization function and  is the kern function. by choosing suitably the normalization function and the kern function, the above definitions can be converted to a known fractional derivative, e.g., abc fractional derivative [47, 48]. for example. if we choose: ( ) ( , ) 1 ( ) d t      = − , (41) ( ) ( ) ( , ) ( ) ( ) 1 t t t s e t s t       = −  −  , (42) 662 j.h. he, j. ma, a.a. alsolam, c.h. he where ξ(ω) is the normalization function, satisfying ξ(0)= ξ(1)=1, eω(t) is the ml function, we can convert eq. (39) and eq. (40) to a modified abc fractional derivative [47, 48]. here, we introduce the two-scale fractal derivative based fractional derivative. in this context, within the known fractional derivative, the traditional derivatives are replaced by the two-scale fractal derivative. for example, the caputo fractional derivative and riemann-liouville fractional derivative are updated respectively in the following manner: 1 0 1 ( ) ( ) ( ) d t d d w w s t s ds dt s     − −  = −  −   , (43) 1 0 1 ( ) ( ) ( ) d t d d w t s w s ds dt t     − −  = −  −   . (44) when d=1, eq. (43) and eq. (44) turn back to caputo fractional derivative and riemannliouville fractional derivative, respectively. 6. conclusions in conclusion, this study has yielded several significant new findings, with particular emphasis on demonstrating the practical utility of the proposed variational principle through a specific mems system. firstly, the application of the semi-inverse method to establish a new variational principle for mems systems has proven to be a groundbreaking advancement. it surpasses certain limitations of traditional variational approaches by offering a more precise portrayal of the physical behavior of mems devices, facilitating enhanced prediction of their performance, and presenting improved computational efficiency, which is vital for design and optimization procedures demanding multiple simulations. this underscores the semiinverse method as a potent tool for handling the intricate nature of mems systems. to illustrate concretely the applicability of this new variational principle, we focused on a specific case: mems pressure sensors integrated into wearable textiles [49]. these sensors are critical for real-time physiological monitoring (e.g., heart rate, respiratory rate, and body temperature) due to their miniaturization, flexibility, and low power consumption. however, their reliability is heavily constrained by pull-in instability, which can lead to sudden performance degradation or failure under dynamic physiological conditions. the proposed variational principle directly addresses this challenge. for the mems pressure sensor, we applied the new formulation to model the electrostatic-mechanical coupling behavior, where the sensor’s diaphragm displacement, w, and voltage-related parameter, m, are the core variables. using the derived criterion for pull-in instability (simultaneous zero kinetic energy and accelerated speed), we computed the pull-in voltage with higher precision compared to traditional methods. specifically, the variational principle enabled us to incorporate microscale effects (e.g., surface tension and fractal surface roughness) that are critical for wearable applications but often overlooked in conventional models. furthermore, the two-scale fractal derivative-based fractional modeling, integrated with the variational principle, enhanced the accuracy of predicting the sensor’s dynamic response under varying fractal surface geometries, which is common in wearable textiles https://zh.wikipedia.org/wiki/%ce%9e https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 https://zh.wikipedia.org/wiki/%ce%9e https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 https://zh.wikipedia.org/wiki/%ce%9e https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 variational approach to micro-electro-mechanical systems 663 due to fabric texture. this integration allowed us to quantify how surface roughness (characterized by fractal dimension d) influences pull-in behavior. secondly, this specific demonstration validates the potential of the new variational principle for future research. it serves as a foundation for further refinement, potentially leading to more inventive and dependable mems designs, particularly for wearable technologies where robustness and miniaturization are paramount. it also opens avenues for deeper investigations into the relationship between variational principles and microscale electromechanical performance. finally, the application of mems in wearable textiles is reaffirmed as highly promising. the specific mems pressure sensor case demonstrates that our variational approach not only improves design efficiency, but also enhances the reliability of wearable systems. in summary, the proposed variational principle, validated through the mems pressure sensor in wearable textiles, advances both theoretical modeling and practical design of mems, paving the way for more robust and versatile microscale devices in emerging applications. specifically, the modeling framework and analytical approach presented herein can be further extended to the study of mems graphene resonators [50], where precise characterization of dynamic behaviors and instability thresholds is equally critical for optimizing performance in high-frequency communication and sensing scenarios. this cross-application potential underscores the generalizability of our variational principle in addressing the core challenges of diverse mems configurations. acknowledgement: the work is supported by guangdong basic and applied basic research foundation (no.2022a1515111135 & no.2025a1515010407). references 1. tang, h. n., wang, y. t., ni, x. q., watanabe, k., taniguchi, t., jarillo-herrero, p., fan, s. h., mazur, e., yacoby, a., cao, y., 2024, on-chip multi-degree-of-freedom control of two-dimensional materials, nature, 632(8027), pp. 1038-1044. 2. 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vietnam abstract. piezo-actuated stick-slip micro-drives (passmds) are often used in microrobotic applications due to their advantages of a straightforward design and good operational characteristics. in this work, influences of fabrication tolerances on operational characteristics of passmds are theoretically investigated. a dynamic model describing the whole macroscopic movement of the driver’s runner and the actuators, and the microscopic behavior of the frictional contacts using the method of dimensionality reduction, was used. three essential parameters of the drives including the angle between working surfaces of the runner, the alignment angle between two actuator blocks on each side of the runner, and the stiffness of an individual actuator, whose values are impacted by the tolerance of the fabrication and assembly process, were considered as the input of the investigation. by performing hybrid-dynamic simulations, influences of these parameters on the drives' operational characteristics, including the repeatability of the step length, the critical amplitude phenomenon, and the maximal-achievable driving frequency/velocity, were evaluated. simulation results show that these parameters significantly influence on the characteristics of the drives. the contribution of this work is so important that several important phenomena of passmds, which are experimentally detected, are physically explained for the first time. the results of this work could help designers to optimize for better generations of passmds. key words: microrobotics, stick-slip micro-drivers, piezo actuators, contact mechanics, dynamic simulation, friction received march 11, 2021 / accepted april 12, 2021 corresponding author: xuan-ha nguyen department of machine design and robotics, school of mechanical engineering, hanoi university of science and technology, hanoi 112452, vietnam e-mail: ha.nguyenxuan@hust.edu.vn mailto:ha.nguyenxuan@hust.edu.vn 110 h. x. nguyen, a. h. nguyen 1. introduction for many techniques of the micro-system technology and the microrobotics, a need for small-scale ultra-high-precision positioning and manipulation devices is extremely necessary [1]. these devices are required to generate motions with a traveling range of several centimeters and a very high resolution of sub micrometer or several ten nanometers. additionally, the devices must operate in special working environments, which are usually small-space, clean and high-rate of vacuum. therefore, the miniaturization and the dustfree capability are also considered. piezo-actuated stick-slip micro-drives (passmds) are a very good candidate to solve the requirements mentioned above. passmds have a very simple design consisting of a mobile part (the runner) guided and actuated by the piezo actuators (the actuator) through frictional guiding contacts attached to the piezo elements [2]. motions of the runner are induced by the deformation of the piezo elements usually supplied by sawtooth-like driving voltages. these motions consist of a sequence of stick and slip phases. during the stick phase, the piezo elements are slowly elongated, which makes the runner to move slowly along with the actuator due to the static friction in the contacts between the runner’s surfaces and the guides of the actuator. this phase is followed by the slip phase when the piezo elements are rapidly contracted. the runner cannot follow this movement and thus slides relatively with respect to the actuators. subsequently, after a small backward motion (back-step), a step of the runner is achieved. although the deformation of the piezoelectric element is small, this principle of motion allows for movements of the runner with a theoretically unlimited range by performing many cycles of the stick and slip phase (stepping mode). also, motions with a very high resolution are obtained by a slight deformation of the piezo elements within the stick phase (scanning mode). there have been many prototypes as well as commercial products of passmds released [3]. also, many theoretical investigations and optimizations for drives have been published [4, 5]. important research and development topics are briefly summarized as follows: ▪ design and miniaturization: the monolithic concept for modular drives to reduce efforts of assembling micro-components [6], exploitation of the flexure-based mechanism for adjusting the preload and thus the friction with the aim of increasing step length and load capacity [7], ▪ waveform and driving electronics optimization: new driving waveforms for maximal velocity [8-10], optimized driving electronics for reducing the slew-rate and thus increasing the step length and velocity [11], ▪ physical modeling and simulation: modeling the friction in the guiding contacts using coulomb model [12, 13], single state friction model [14-16], tangential contact mechanics with the method of dimensionality reduction for hybrid dynamic simulations [3, 10, 17-20]. among these topics, the physical modeling is the most important because it can theoretically explain important characteristics of the stick-slip drives experimentally detected such as: i) the so-called 0-amplitude [10, 19, 21] /critical amplitude [20] which relates directly to the presliding displacement of guiding contacts and the dynamic behavior of the piezo actuators. the state-of-the-art explanation for this phenomenon can be found in [20]; ii) the micro-vibration and the maximal achievable driving frequency [10, 20]; iii) the repeatability – an evaluation criterion for the changing of the step length after each investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 111 step which is believed to relate to the tolerance of the fabrication and assembly of microcomponents of the drives. this phenomenon has not been physically analyzed yet. the fabrication tolerance is determined by the micro-machining of micro-parts as well as their assembly processes typically performed in manual ways. this leads to the fact that the tolerance of manufactured drives is very fluctuated, influencing significantly on the operational characteristics of the drives. to our knowledge, there has not been any investigation considering the influences of fabrication tolerances on operational characteristics of passmds. in this work, the influences of fabrication tolerances on the above-mentioned operational characteristics of the passmds are theoretically investigated. based on previously published works [10, 18-20], an experimental platform having a linear stick-slip micro-drive, in which a runner is guided and driven by six piezo-actuators, was illustrated. a hybrid-dynamic model [10, 18-20], which allows us to simulate the overall macroscopic movements of the runner and the microscopic behavior of guiding contacts, was used. the angle between the working surfaces of the runner, the alignment angle between two actuator blocks, and the stiffness and damping ratio of an individual actuator, which are impacted by the fabrication and assembly tolerances, were considered. influences of these parameters on operational characteristics of the drive, including the fluctuation of the step length, the critical amplitude and the maximal achievable driving frequency are analyzed and presented. 2. experimental implementation 2.1. experimental platform the experimental platform used in this work is the result of the work of edeler [16] and the adaption of our previously published works [3, 10]. some measurement results of these works are shown in this paper for the first time. for the convenience of readers, a brief description of the system is presented. the investigated drive is a linear micro-positioning device. the device consists of a runner which is guided and driven by six piezoelectric micro-actuators, as shown in fig. 1. each actuator is constructed by a ruby hemisphere glued on segments of a structured piezo element. these actuators are arranged on both sides of the runner, four actuators on the left and two actuators on the right (see figs. 1b, c). in each side of the runner, the actuators are aligned to each other with an angle of 90°. with this arrangement, the runner can perform horizontally translational forward and backward movements. the actuator holder on the right is fixed to the housing, while the left one is movable and guided by a very low-friction linear ball bearing. preloads applied to the drive are generated and regulated by a springsystem acting on the left actuator holder. the value of preloads is measured by a force sensor (m17 by ati industrial automation, pinnacle park apex, nc, usa). the position of the runner is measured via a high-resolution laser interferometer (sp-120 by sios meßtechnik gmbh, ilmenau, germany). in the runner's movement direction, a miniature load cell (m31 by sensing and control automation and control solutions honeywell, golden valley, mn, usa) is set up in front of the runner to measure forces generated by the drive. when the runner touches the load cell, the force is measured and recorded. the piezo elements are supplied by saw-tooth like driving voltages whose amplitude, slew-rate and frequency can be varied via driving electronics. when six synchronous sawtooth like driving voltages are applied to the piezo elements, the six hemispheres also performed synchronous saw-tooth like displacements, leading to stick-slip motions for the 112 h. x. nguyen, a. h. nguyen runner. the displacement amplitude of the contact center of each ruby hemispheres/actuator is denoted as the actuation amplitude. this amplitude depends on the radius of the hemispheres, the dimension and the structure of the piezo, as well as the amplitude of driving voltages. the radius of hemispheres, the driving frequency, the amplitude of driving voltages and the preload are varied to characterize the drive’s behavior. in this work, the radii of the ruby hemispheres of 0.25𝑚𝑚, 0.5𝑚𝑚 and 1𝑚𝑚 are considered. fig. 1 setup of the experimental platform: (a) photo of the test stand; (b) photo of the piezo actuators; (c) cad-based schematic front-view [16,10] 2.2. fabrication and assembly tolerance the investigated drive was fabricated and assembled manually, which could make the tolerance of the fabrication and the assembly of micro-parts vary in an uncontrollable manner. the structure of the piezo elements and the assembly of the actuators are schematically shown in fig. 2. the piezo elements are fabricated out of a piezoceramic bulk (pic151) via the laser-based micro-machining. there are two types of the structured piezo. the so-called “a-type” has two segment zones where two ruby hemispheres are glued on. the “b-type” has only one segment zone with one glued hemisphere (see also fig. 1b). each block of the piezo element and the ruby hemispheres are then glued to the actuator holders. for each actuator holder, the two blocks must align together with an angle of 90°. to our practical experiences, the fabrication and the manual assembly of such micro-parts are complicated and thus, the tolerance can vary in a large range. this makes the operational characteristics of the drive case-by-case very different. error resources determining the tolerance of the investigated drive are caused by both the fabrication and the assembly as follows: ▪ for the fabrication: the laser-based micro-machining method cannot ensure the depth of the groove of the piezo elements equal. this makes the dynamic behavior (stiffness and damping coefficients) of an individual actuator different from others. also, the discrepancy of the displacement amplitude of the actuators appears. the fabrication of the runner also has variation. investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 113 ▪ for the assembly: i) in fact, the ruby hemispheres are manually attached to the piezo elements using a glued layer under the support of a light microscope. the thickness and the configuration of the glue layer can randomly vary and cannot be well controlled. the configuration and the thickness of the glue layer between the ruby hemisphere and the piezo element, and their relative position, are proved to strongly impact the stiffness and the displacement amplitude of the actuators [21]. therefore, discrepancies in the stiffness and the displacement amplitude among actuators are visible; ii) the geometrical errors caused by the assembly of each block of actuators and their attachment to the actuator holder, can change the contact angle and contact point between the hemisphere of the actuator and the runner’s surfaces (see fig. 2b). this leads to discrepancies in the displacement amplitude of the actuators as well as the normal force pressed on each contact. in this work, influences of the discrepancy of stiffness/damping, the displacement amplitude among actuators, and the normal force of each contact on operational characteristics of the investigated drive are modelled and simulated. fig. 2 cad-based schematic view of actuator design: (a) the structured piezo; (b) the assembly of the actuators [3] 2.3. measurement results the experimental results shown in fig. 3 are presented in this paper for the first time. the graphs present measurements of the step length of the runner at different contact positions between the runner’s surfaces and the actuators. the level of the preload is also varied. for each level of the preload, measurements of ten measured positions are randomly chosen to be drawn. fig. 3a presents data with the case of the radius of the ruby hemispheres 𝑅 = 1𝑚𝑚 and fig. 3b for 𝑅 = 0.25𝑚𝑚. in fig. 3a, with the case 𝑅 = 1𝑚𝑚 and the level of the preload 𝐹𝑝 = 0.2𝑁, the discrepancy between the maximal step length and the minimal one is 22𝑛𝑚 while the average of the step length is 149𝑛𝑚. the maximal standard deviation is 8.6%. for the higher preload 𝐹𝑝 = 1𝑁 the discrepancy of the step length increases with the standard deviation of 13.4%. thus, for this case the repeatability is reduced. in fig. 3b, for the case 𝑅 = 0.25𝑚𝑚 and 𝐹𝑝 = 0.5𝑁, the repeatability of the step length is very low. the measured step lengths have a large discrepancy with a maximum of 35𝑛𝑚 in an average of 47.2𝑛𝑚. the maximal standard deviation is 41.3%. with 𝐹𝑝 = 0.2𝑁, the repeatability is better. 114 h. x. nguyen, a. h. nguyen fig. 3 the repeatability of the step length: (a) 𝑅 = 1 𝑚𝑚; (b) 𝑅 = 0.25 𝑚𝑚 in fig. 4, the dependency of the critical amplitude on the preload for different ruby hemispheres radii is presented. we consider the error bar in the graphs. it is visible that the error bar becomes larger when the level of the preload increases. also, if the radius of the ruby hemispheres is reduced, the error bar increases slightly. experimental evidence shows that with the case 𝑅 = 0.25𝑚𝑚 and the preload higher than 0.5𝑁 the repeatability of the measurement is very low even the runner cannot perform a reliable movement. a maximal discrepancy between measured values of the critical amplitude is 8.2% at a preload of 0.5𝑁 and 𝑅 = 0.25𝑚𝑚, and 6.8% at a preload of 1.2𝑁 and 𝑅 = 1𝑚𝑚. fig. 4 the repeatability of critical amplitude depending on the preload for different ruby hemispheres radii fig. 5 shows the measurement of the runner’s velocity which depends on the driving frequency for the case the radius of the ruby hemispheres 𝑅 = 0.5𝑚𝑚 and a preload 𝐹𝑁 = 0.4𝑁. at each represented point, an error bar is shown which presents the repeatability of ten measurements. the driving frequency is varied in a range from 4.72 to 34.66𝑘𝐻𝑧. it is clearly seen that the runner’s velocity increases with rising driving frequencies. when the driving frequency reaches to a value of 26𝑘𝐻𝑧, the velocity of the runner is maximal at a value of 1.76 𝑚𝑚/𝑠 with a standard deviation of 7.4%. if the driving frequency continues increasing, the velocity decreases gradually. this can be explained that, with driving frequencies higher than a maximal-achievable value, after each slip phase of the runner, (a) (b) investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 115 the micro-vibration cannot be damped before the runner starts the next stick-phase. as a result, chaos appears which leads to reductions in the runner’s velocity. it is also seen that the higher the driving frequency is, the larger the error bar is. the repeatability of the runner’s velocity reduces with the rising driving frequency. fig. 5 the repeatability of runner’s velocity depending on the driving frequency with a radius of the ruby hemispheres 𝑅 = 0.5𝑚𝑚 and a preload 𝐹𝑝 = 0.4𝑁 we come to the following comments: ▪ due to the fabrication and assembly tolerance of the actuators and the runner, discrepancies among actuators appear, including the stiffness/damping coefficients, the actuation amplitude, as well as the normal force of the guiding contacts. during the operation, the dynamic behavior of an individual actuator and thus the contact force are changed which influences the repeatability of the overall movement of the runner; ▪ the impact of fabrication and assembly tolerance on the repeatability becomes larger when the size of the system is decreased, and the preload and the driving frequency rise. 3. modeling and simulation 3.1. modeling the concept of the hybrid dynamic simulation for passmds was first proposed in [18] and further validated in [3, 10, 19, 22]. the basic idea of the concept is the combination of the macroscopic movement of the runner and the actuators with the microscopic behavior of the guiding contacts using the so-called method of dimensionality reduction [17]. in these works, the authors modeled the actuators as ideally rigid. the dynamic behavior of the actuators was not considered. recently, in [20], the authors extended the model so that the actuators are considered as mass-spring-damper systems. however, all these works assume that the fabrication and assembly tolerance do not influence the characteristics of the drive. therefore, discrepancies in the actuators' dynamic behavior and actuation amplitude and the contact forces were not modeled. in this work, we continue extending the model proposed in [20] so that the discrepancies caused by the fabrication and assembly tolerance are considered. the extended macroscopic model to describe the investigated drive's overall movement is presented in fig. 6 where a free-body diagram is drawn. the runner has a mass of 𝑚𝑟 116 h. x. nguyen, a. h. nguyen whose movements in the horizontal direction are described by the coordinate 𝑥𝑟 under the acting of the normal and friction forces of the guiding contacts. in fig. 6a, three massspring-damper systems of actuators are drawn where each system is depicted for two actuators (two at the top and four at the button). unlike previously published work [20], in this work, due to the fabrication and assembly tolerance, dynamic parameters of the actuators as well as the contacting forces are not identical. therefore, each actuator is modeled by a mass 𝑚𝑎𝑖, a stiffness 𝑘𝑝𝑖 and a damping 𝑐𝑝𝑖. displacements of the actuators are described by 𝑥𝑎𝑖 . the deformation of the piezo elements is 𝑥𝑝𝑖 has a form of saw-tooth like as shown in fig. 6c. for each contact, the normal and friction force are presented by 𝐹𝑁𝑖, and 𝐹𝑓𝑖 (𝑖 = 1 ÷ 6), respectively. the stiffness of the load cell is presented by 𝑘. fig. 6 macroscopic modeling for the investigated drive macroscopic movements of the runner and the actuators are formulated via the newtonian mechanics, in which eq. 1 for the runner and eq. 2 for the actuators. it is noticed that the influence of the gravity force is neglected [10]. the friction forces 𝐹𝑓𝑖 in eqs. (1) and (2) are modelled by a microscopic model using mdr [17] which depends directly on the contact mechanics and the normal forces. since the fabrication and assembly tolerance is considered, the modeling of them is additionally be implemented. 6 1 r r r fi i m x kx f = = − + (1) ( ) ( 1 6)ai ai pi pi ai pi ai fim x k x x c x f i= − − − =  (2) variations of the actuation amplitude of the contact points and the normal force applied on each contact, caused by the fabrication and assembly tolerance, are modeled as in fig. 7. investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 117 • firstly, the fabrication and assembly error of the actuator blocks are modeled in figs. 7a and 7c. fig. 7a depicts the ideal case when the actuator blocks are aligned together with an angle 90° (the continuous, black line) and the manufactured case. the fabrication and assembly tolerance are considered (the dashed, red line) where the alignment angle is not 90° but rather 𝛿. thus, the contact point between the ruby hemisphere and the runner’s surface is shifted from the top of the ruby hemisphere 𝐶𝑖 to 𝐶𝑖 ′ determined by 𝜃𝑖. as shown in fig. 7c, in the direction of the movement 𝑥, when one ruby hemisphere is rotated with a displacement 𝜑𝑖 around 𝛼𝑖, the translational displacement of 𝐶𝑖 and 𝐶𝑖 ′ are 𝑥𝑎𝑖 and 𝑥𝑎𝑖 ′ respectively, whose relation is followed by the eqs. (3) and (4). thus, the actuation amplitude of the actual contact point 𝐴𝑖 ′ can be smaller than the amplitude of the ideal case 𝐴𝑖 determined by the variation of 𝛿 around 90°. in this case, the normal forces applied to the contacts remain unchanged. ' ' ' ' cos cos ai i ai ai i i i i ai i i x r x x a a x r =  =  = =     (3) 45 2 i = −    (4) ▪ secondly, the fabrication tolerance of the runner is considered. in fact, the runner’s surfaces are not perfectly flat and geometrically accurate. as shown in fig. 7b, the angle between the two working surfaces of the runner is modeled by 𝛾. this angle is changed around 90°. for this case, the actual contact point is not 𝐶𝑖 but rather 𝐶𝑖 ′ (the dashed, red line for the real situation). similarly, the actuation amplitude of the contact point is changed and determined by eqs. (3) and (5). consequently, compared to the ideal case with 𝛾 = 90°, the actual normal forces applied in each fig. 7 modeling of the influence of the fabrication and assembly tolerance on actuation amplitudes and normal forces 118 h. x. nguyen, a. h. nguyen contact 𝐹𝑁𝑖(𝑖 = 1 ÷ 6) are formulated by eq. (6), where 𝐹𝑁𝑖 𝑧 (𝑖 = 1 ÷ 6) are the projection of 𝐹𝑁𝑖 on the 𝑧 direction; 𝐹𝑝 presents for the preload (see fig. 7d). 45 2 i = −    (5) 1 2 3 4 5 6 sin 2 z ni ni z z z z z z n n n n n n p f f f f f f f f f  =        + = + + + =  (6) the friction forces 𝐹𝑓𝑖 (𝑖 = 1 ÷ 6) in eqs. (1) and (2) can be modeled and calculated via the mdr [17]. since this model was already described in detail in previously published works [3, 10, 18-20, 22]. thus, we only briefly present the theory in this paper. fig. 8b shows an equivalent one-dimensional contact model from the original three-dimensional model (fig. 8a), where the elastic space is replaced by an elastic foundation with a set of springs. these springs have normal stiffness ∆𝑘𝑧 and tangential stiffness ∆𝑘𝑥, which are chosen according to the following rules: * ; *z xk e x k g x =   =  , (7) where ∆𝑥 is the distance between the spring, and 𝐸∗ and 𝐺∗ are respectively the equivalent elastic and shear modulus. the radius of the one-dimensional profile 𝑅1 must be: 1 2 r r = . (8) using this simplified model, the dynamically changing contact area is estimated and thus, the friction force is computed. at each instant of time 𝑡 of a dynamic simulation, the changing stick and slip area of the contact is determined by the state of the springs in the contact and the friction force is calculated as follows: , 1 ( ) ( ), 1 6 n fi x j j f t f t i = = =  , (9) where 𝑓𝑥,𝑗(𝑡) is the tangential force of the springs in the contact, whose value depends on the state of each spring (stick or slip) related to the normal force 𝐹𝑁𝑖; 𝑁 is the number of springs in contact 𝑖. (a) (b) fig. 8 modeling of the micro-contact using mdr: a) three-dimensional contact model; b) one-dimensional contact model via mdr [17, 18] investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 119 3.2. hybrid dynamic simulation simulations were performed using a hybrid dynamic algorithm (see fig. 5 of [18]), in which a macroscopic model is combined with a microscopic model. thus, in each time step, the runner's overall macroscopic movement is computed and coupled with the microscopic dynamic behavior of the guiding contacts using routines of matlab® version 2015a. basic parameters of the drive used for numerical simulations are collected from [20] and listed in table 1. the dynamic parameters of the actuators, which are depended on the radius of the ruby hemispheres, are referred from [14] and shown in table 2. additionally, the preload range and the actuation amplitude of the guiding contacts were set according to the experimental implementation in [10]. it is noticed that, for the case that the radius of the ruby hemispheres is of 0.25𝑚𝑚, the maximal level of the preload is only 0.5𝑁. simulations with higher levels of the preload would have no meaning since the experimental implementation proves that when the preload level is higher than 0.5𝑁 the drive does not work reliably anymore. table 3 shows variations of simulation parameters caused by the fabrication and assembly tolerance. in the standard case, on one side of the runner, the angle between two actuator blocks 𝛿 and the angle between two working surfaces of the runner 𝛾 are equal to 90° (see figs. 6b and 7a, b). if the fabrication and assembly tolerance are considered, these parameters are varied according to the column “varied value” of table 3. because the fabrication and assembly tolerance are random quantities, values of 𝛿 and 𝛾 can be smaller or larger than 90°. also, the stiffness and the damping ratio of the actuators are varied to reduce to amounts of 10% and 20%, describing for the fabrication tolerance of the piezo and the inequality in the thickness of the glue layer. based on these varied parameters, we perform dynamic simulations and compare results with that of the standard case. consequently, discrepancies in percentage between the standard case and the varied cases are calculated and presented. simulation cases are as follows: ▪ case 1: the angle between two actuator blocks on one side of the runner (𝛿) is varied. the change is assigned to either type-a side or type-b side; ▪ case 2: the angle between two working surfaces on one side of the runner (𝛾) is varied and the change can happen either on type-a side or type-b side; ▪ case 3: the stiffness and damping ratio of actuators in one group (either the typea group with four actuators or the type-b group with two actuators) are varied to reduce. other parameters remain unchanged. table 1 data used for numerical simulations [10, 20] parameters value young's modulus ruby 𝐸1 (𝐺𝑃𝑎) 370 young's modulus steel 𝐸2 (𝐺𝑃𝑎) 210 poisson's ratio 𝜈1, 𝜈2 0.33 friction coefficient µ 0.3 mass of the runner 𝑚𝑟 (𝑔) 3 stiffness of the load cell 𝑘 (𝑁/𝑚) 2.5 × 104 radius of the ruby hemispheres 𝑅 (𝑚𝑚) 0.25, 0.5, 1 120 h. x. nguyen, a. h. nguyen table 2 standard parameters dependent on the radius of hemispheres used in simulations [20] parameters 𝑅 = 0.25 𝑚𝑚 𝑅 = 0.5𝑚𝑚 𝑅 = 1𝑚𝑚 stiffness of actuators 𝑘𝑝 (𝑁/𝑚) 4 × 106 6 × 106 6.5 × 106 damping ratio 𝑐𝑝 (𝑘𝑔/𝑠) 10 14 16.3 mass of actuator 𝑚𝑎 (𝑔) 1.1 × 10−5 1.6 × 10−5 2.2 × 10−5 preload range 𝐹𝑝 (𝑁) 0.1 ÷ 0.5 0.1 ÷ 0.8 0.1 ÷ 1.5 actuation amplitude 𝐴 (𝑛𝑚) 96 160 200 table 3 parameter variation depending on the fabrication and assembly tolerance parameter position standard value varied value angle between two actuator blocks on one side of the runner (𝛿) type-a side type-b side 𝛿 = 90° 𝛿 = 80°/100°, 70°/110° angle between two working surfaces on one side of the runner (𝛾) type-a side type-b side 𝛾 = 90° 𝛾 = 85°, 95° stiffness and damping of the actuators (𝑘𝑝 , 𝑐𝑝) actuator 1, 2 actuator 3, 4, 5, 6 from that of table 2 𝑘𝑝, 𝑐𝑝 reduced 10% 𝑘𝑝, 𝑐𝑝 reduced 20% 4. results and discussion in this section, the operational characteristics of the drive including the repeatability of the step length, the critical/0-amplitude phenomenon, and the maximal achievable driving frequency/velocity are analyzed. based on the simulation results, discrepancies are shown and discussed. also, some analytical formulations are presented. 4.1. the repeatability of the step length discrepancies of the step length depending on the preload when comparing the simulation results of the standard case with the varied cases (changing of 𝛿 and 𝛾) are shown in fig. 9., where fig. 9a is for the case radius of the ruby hemispheres 𝑅 = 1𝑚𝑚 and fig. 9b for 𝑅 = 0.25𝑚𝑚. it is seen in fig. 9a that if 𝛾 or 𝛿 is varied, the discrepancy is increased with the rising level of the preload. when the preload is small, the discrepancy is very low and can be neglected. however, when the preload becomes higher, for example, 1.5𝑁, the maximum discrepancy reaches 7.5%. it is also clear that the influence of 𝛾 is larger than that of 𝛿 because the changing of 𝛾 leads to a change in both the actuation amplitude and the normal force, while the changing of 𝛿 changes only the actuation amplitude. the influence of 𝛿 is quite small and thus can be neglected. discrepancies of the step length depending on the preload when comparing the simulation results of the standard case with the case of reduced actuators stiffness and damping ratio are shown in fig. 10 where fig. 10a is for the case radius of the ruby hemispheres 𝑅 = 1𝑚𝑚 and fig. 10b for 𝑅 = 0.25𝑚𝑚. a similar trend as in fig. 9 is obtained. however, the discrepancy is smaller. comparing the simulation results from figs. 9a and 10a with the experimental data in fig. 3a, it is inferred that the simulation can cover the overall trend of the experiment where the discrepancy slightly increases with rising preloads. however, the level of changing in investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 121 simulations is still smaller than that of the experiment. a similar trend for the case 𝑅 = 0.25𝑚𝑚 is observed. fig. 9 discrepancies of the step length depending on the preload when comparing the simulations with the standard case and the varied cases (changing of 𝛾 and 𝛿): a) 𝑅 = 1𝑚𝑚; b) 𝑅 = 0.25𝑚𝑚 fig. 10 discrepancies of the step length depending on the preload when comparing the simulations with the standard case and the case with reduced actuators stiffness and damping ratio: a) 𝑅 = 1𝑚𝑚; b) 𝑅 = 0.25𝑚𝑚 it can be concluded that when the size of the system is reduced, the repeatability of the step length is decreased and the influence of the fabrication and assembly tolerance becomes more apparent with the increasing discrepancy. the fabrication tolerance of the runner has the strongest influence on the step length because it changes both the actuation amplitude and the normal contact forces. (a) (b) (a) (b) 122 h. x. nguyen, a. h. nguyen 4.2. the critical amplitude the so-called “zero/critical-amplitude” phenomenon of passmds was experimentally detected and modeled in previously published works [10, 14, 16, 18, 20]. if the actuation amplitude of the piezo elements is smaller than a critical value, no measurable step of the runner is achieved. the runner is only oscillated around its initial position without any generation of a net step. this phenomenon relates directly to the maximal tangential deformation of the guiding contact between the runner and the actuators and the tangential dynamic behavior of the piezo actuators. the state-of-the-art explanation for this phenomenon is presented in [20]. in this work, the influences of the fabrication and assembly tolerances of the system on this characteristic are analyzed. based on the variation of the parameters in table 3, simulations of the dependency of the critical amplitude on the preload were performed. in fig. 11a, discrepancies between simulation results of the standard case and the varied case (𝛾 = 85°) of the critical amplitude depended on the preload are shown. two cases of the radius of the ruby hemispheres are considered. it is seen that, for both cases of the radius, the discrepancy varies from 2 to 7% for the overall preload range. this discrepancy qualitatively matched with the experimental data where the maximal values vary from 6.8 to 8.2% (see fig. 4 and the corresponding text). simulations for other varied cases (changes of 𝛿, stiffness and damping ratio) show that the discrepancy is smaller and has a similar trend. for the sake of brevity, the results are not shown here. the obtained results can be explained based on the causation of the critical amplitude. if 𝛾 is changed, the normal contact force and the displacement amplitude of the contact point are changed, leading to a change in critical amplitude. for the case of changing of 𝛿, only the displacement amplitude of the contact point is changed and thus, the changing of the critical amplitude is smaller. similarly, for the case of reducing stiffness and damping ratio, a reduction of 10% or 20% does not much affect the stiffness and damping ratio because their values are still high. therefore, less changing of the critical amplitude is achieved. we come to the conclusions that for actuators with high stiffness, the influence of the fabrication tolerance on the critical amplitude is very small and can be neglected. the influence of 𝛿 is also small since it changes only the displacement amplitude of the contact point in a small amount proportional to the ratio 𝐴/𝐴′. in [20], analytical formulations (eqs. (10) – (14) of [20]) for explaining the critical amplitude were established, which are an extension of eq. (3.26) of [22] when considering the dynamic behavior of the actuators. the formulations fit excellently with experimental data. however, the formulations are established in an assumption that the behavior of an individual actuator in each actuator group is identical. no fabrication and assembly tolerance are considered. thus, in this paper, we continue extending the formulations in such a way that influences of the tolerance are considered (see eqs. (6) and (10) – (13)). in this extension, the critical amplitude is a function of the maximal tangential displacement of the guiding contacts 𝑢𝑥,𝑚𝑎𝑥,𝑗, the tangential deformation of the piezo 𝑢𝑥,𝑎,𝑗 under consideration of the interaction between the two actuator groups, where 𝑗 = 1 for actuators of the type-a side and 𝑗 = 2 for actuators of the type-b side. the influence of the tolerance of 𝛾 and 𝛿 on the changing of the displacement of the contacts is represented by factor 𝐴/𝐴′ in eq. (12). also, the change in normal forces is determined by eq. (6). the reduction of the stiffness is modeled in eq. (11). fig. 11b shows the comparison of dynamic simulations of the critical amplitude depending on the preload and the analytical investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 123 approximation (13) for both cases of the radius of the ruby hemispheres. it is seen that the analytical approximations fit excellently with the dynamic simulations. fig. 11 dependency of the critical amplitude on the preload: a) discrepancies between simulation results of the standard case and the varied case (𝛾 = 85°); b) comparison of dynamic simulation and analytical formation determined from eqs. (6) and (10)–(13). ( ) 2 31 3 , * 1 3 3 4 * 1, 2 34 nj x max,j fe u j g r   = =     (10) , , ( 1, 2) nj r a x a j pj r f m m u j k m + =   = (11) , , ,max, , ,' ' ( 1, 2)x max j x j x a j a u u u j a = + = (12) ,max,1 ,max,2 , , ,max,1 ,max,2 2 ' ' ' ' ' x x x max e x x u u u u u = + (13) 4.3. the maximal achievable driving frequency/velocity one important characteristic of the drive is the maximal-achievable velocity of the runner, depending on the driving frequency. in fact, after each slip phase, there appear micro-vibrations of the overall system. if the micro-vibration cannot be damped before a new cycle of the stick and slip phase is started, it will induce chaos to the system. this prevents us from controlling the drive at high driving frequencies, influencing directly to the velocity of the runner. thus, we can only drive the system at a maximal-achievable driving frequency. above this frequency level, the velocity of the runner is reduced. the micro-vibration depends on the mechanics of the guiding contacts, the preload, and the dynamics of the piezo actuators, which is influenced by the fabrication and assembly tolerance. fig. 12 shows a comparison of the maximal-achievable velocity of the runner (b) (a) 124 h. x. nguyen, a. h. nguyen depending on the preload between the standard case and the varied cases. for each level of the preload, the value of the maximalachievable driving frequency is shown. it is seen in fig. 12a that, for all cases, if the preload rises, the maximal-achievable driving frequencies and thus the maximal-achievable velocity of the runner is decreased. for the case with 𝛿 = 80° or 100° only the displacement amplitude of the guiding contacts is slightly changed and thus, it will not influence significantly on the result when comparing with the standard case. the two plots of these cases are mostly identical. for the case 𝛾 = 85°, the displacement amplitude of the guiding contacts is reduced, and the normal contact forces are increased. compared to the standard case, this case leads to reductions of the maximal-achievable velocity of the runner with a maximal discrepancy of 6%. when both stiffness and damping ratio are reduced to 20%, the velocity is slightly increased. this can be explained that the reduction of the damping ratio leads to the increasing step length. this increasing level is larger than the decreasing step length caused by the reducing stiffness. consequently, the final step length is increased, leading to an increment of velocity for the runner. the maximal discrepancy for this case is 4%. simulation results for the case with 𝑅 = 0.25𝑚𝑚 are shown in fig. 12b. a similar trend as in fig. 12a is detected. the maximal discrepancy is 5%. however, the maximal fig. 12 comparison of the maximal-achievable velocity of the runner depending on the preload between the standard case and the varied cases: a) 𝑅 = 1𝑚𝑚; b) 𝑅 = 0.25𝑚𝑚; c) 𝑅 = 0.5𝑚𝑚 (b) (a) (c) investigation of influences of fabrication tolerances on operational characteristics of piezo-actuated... 125 achievable driving frequency and thus the velocity of the runner is reduced significantly. fig. 12c shows simulation results for the case 𝑅 = 0.5𝑚𝑚. a similar trend is also obtained. at the preload 𝐹𝑁 = 0.4𝑁, the maximal-achievable velocity of the runner reaches 1.62𝑚𝑚/𝑠 at a maximal-controllable driving frequency of 31𝑘𝐻𝑧. when considering the influence of the fabrication and assembly tolerance, the maximal discrepancy is 6.4%. compared to the experimental data shown in fig.5, where the maximal velocity is 1.76𝑚𝑚/𝑠 at a driving frequency of 26𝑘𝐻𝑧 and the maximal discrepancy is of 7.4%, it is observed that the simulation fits very well with the experiment. it can be concluded that the maximal-achievable driving frequency/velocity depends strongly on the size of the system. when the size of the system decreases, the maximalachieve driving frequency/velocity as well as the repeatability reduce. the influence of the fabrication and assembly tolerance becomes larger. 5. conclusion and outlook in this work, influences of the fabrication and assembly tolerance on operational characteristics of passmds have been investigated for the first time. three possible cases which are caused the fabrication and assembly tolerance are analyzed and chosen to investigate. based on a passmd and a model of the previously published work, an extended model has been introduced. the model allows considering parameters representing influences of the fabrication and assembly tolerance. under these influences, three important operational characteristics of the drive, including the repeatability of the step length, the critical amplitude and the maximal-achievable driving frequency/velocity have been evaluated via the more general hybrid-dynamic simulations. it is concluded that the repeatability of the step length is clearly influenced by the fabrication and assembly tolerance. the discrepancy among step lengths is gradually increased if the preload rises and the size of the system is reduced. for all possible cases caused by the tolerance investigated in this paper, the influence of the fabrication tolerance of the runner represented by the angle 𝛾 is largest since it modifies not only the displacement amplitude of the contact point but also the normal contact forces. the influence caused by the assembly tolerance of the actuator block described by 𝛿 is less of impact. the changes in the stiffness and damping ratio of the actuator influence all the characteristics moderately. if the stiffness of the actuators is low, the influence becomes apparent. when the size of the system reduces, the influence of the fabrication and assembly tolerance on the driver’s characteristics become larger. the results of this work would be very helpful for the design and fabrication process of passmds. understanding causations resulting from the changes of operational characteristics and their influencing levels allows to optimize for 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mechanical engineering vol. 19, no 4, 2021, pp. 751 765 https://doi.org/10.22190/fume201029023f © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper calibration of material models for the human cervical spine ligament behaviour using a genetic algorithm marina franulović1, kristina marković1, ana trajkovski2 1faculty of engineering, university of rijeka, rijeka, croatia 2faculty of mechanical engineering, university of ljubljana, ljubljana, slovenia abstract. research of biomaterials in loading conditions has become a significant field in the material science nowadays. in order to provide better understanding of the loading effects on material structures, complex material models are usually chosen, depending on their applicability to the material under consideration. in order to provide as accurate as possible the material behavior modeling of the human cervical spine ligaments, the procedure for calibration of two material models has been evaluated. the calibration of material models was based on the genetic algorithm procedure in order to make possible optimization of material parameters identification for the chosen models. the influence of genetic algorithm operators upon the results in evaluated procedure has been tested and discussed here and the simulated behavior of the material has been compared to the experimentally recorded stress stretch relationship of the material under consideration. since various influential factors contribute to the genetic algorithm performance in calibration of complex material models and identification of material parameters, additional possible improvements have been suggested for further research. key words: material model, behavior modeling, biomaterial, genetic algorithm, inverse analysis 1. introduction research and modeling of the biomaterials behavior have both secured their place in a broader field of material science research. in order to accurately model the behavior of any such material, an adequate constitutive material model must be implemented, regardless of whether using the known one or the one adapted for the investigated material or, in some cases, a developed new material model. the common feature in the biomaterials behavior modeling lies in the calibration of an adequate material model received october 29, 2020 / accepted february 08, 2021 corresponding author: kristina marković faculty of engineering, university of rijeka, vukovarska 58, 51000 rijeka, croatia e-mail: kritina@riteh.hr 752 m. franulović, k. marković, a. trajkovski through its validation. the validation process here is based on the material behavior simulation and its comparison to the material behavior acquired from the experimental procedures where test samples are loaded, usually with constant strain rate, while stresses are recorded until the sample separation in two parts. the data sets recorded through these experiments serve as a basis for acquiring the knowledge of the tissue properties. the basic material models used for modeling of soft tissues were the ones originally developed for rubber-like materials displaying hyperelastic behavior, such as mooney’s model from 1940 [1] and improved version of the same model (mooney-rivlin’s material model) from 1948 [2]. with the development of materials research as well as the other fields where similar models are usually used, the number of available models has grown [3–6]. some of the hyperelastic material models are relatively simple, while most of them consist of complex expressions and, consequently, a large number of unknowns, which are material parameters. the identification of these models’ parameters requires the use of a numerical procedure which is expected to result in an accurate set of parameters while capturing the nonlinearity of the material behavior during loading process. one of the methods that is widely used relies on the principle of evolutionary algorithms, such as the genetic algorithm, whose usefulness in multidimensional search problems has been confirmed [7] as well as in the cases where the objective function is “discontinuous, nondifferentiable, stochastic, or highly nonlinear” [8]. while the beginning of genetic algorithms applications in search of variables domain dates back to 1975 [9] (changed edition 1992) and in some form to an older date, a wider use in the field of optimization was seen in the 1990’s, with the development of computer technology and software. some of the early works in 1900’s were general parameter optimization[10] and application to computation [11], dynamic parameter encoding [12], physical motion analysis [13], identification of cloth animation models [14] and overview of possible uses in optimization and statistics [15], identifying parameters for dynamic simulation [16]. more recent use includes topology optimization [17], solving ivbv problems [18] and optimization of perforated elastic plate properties [19]. in addition to the genetic algorithm, many other evolutionary methods have found their applications in medicine and healthcare. this has provided quicker and easier diagnostics and improved the healthcare systems [20]. examples for the use of the genetic algorithms in tissue mechanics and/or parameter determination include: breast tissues [21, 22], anterior cruciate ligament [23], mass-spring models for soft tissues [24], qlv theory [7, 23], ventricular myocardium [25], liver [26], porcine coronary media and adventitia [27], sheep tendons [28] and human placenta [29]. in this paper the influence of the genetic algorithm operators on the results inassesed procedure have been tested and discussed. also, the simulated behavior of material has been compared to the experimentally recorded stress stretch relationship of the material under consideration, which is the material behavior modeling of the human cervical spine ligaments. 2. mechanical principles in order to model and simulate the behavior of soft tissues, a material model suitable for modeling its hyperelastic behaviour must be implemented. experimental observations of biological material, in presented case, the ligaments of the human cervical spine were conducted. it was found that the maximum load capacity of the calibration of material models for human cervical spine ligament behaviour using a genetic... 753 material is in the direction of their fibers [30]. ligaments are composed of collagen fibers organized uniformly in one direction and integrated together by intercellular material. consequently, resulting mechanical behavior of ligaments depends on the behavior of individual ligaments' constituents as well as on their correlation and relative position. mathematical modeling of biological materials, especially with the presumption of modeling the said ligament material, is based upon the knowledge of the basic mechanical properties of materials. it is also necessary to know the type of interdependence of the stresses and strains. these characteristics can be determined by the experimental procedures that include subjecting samples to tensile loading until their fracture. cca. eighty experiments were performed and fig. 1 shows the load – elongation curve of the anterior longitudinal ligament (sampled at the cervical spine, level c4–c5, from the human donor, age 73) which was loaded in tension until fracture. fig. 1 experimentally determined load – elongation curve of the anterior longitudinal ligament governing equations for calculating stress in hyperelastic material models are based on the strain energy density function, which contains the relationship between stress and strain through the materials life in loading conditions. strain energy density function can be defined as a linking function between the strain energy density and deformation gradient. as stated in [2] and [31], it is possible to express strain energy density function using three strain invariants (i1, i2 and i3). it should be noted that not all invariants are used in every material model. for example, the simplest version of the neo-hookean material model uses only the first invariant and the mooney-rivlin material model uses only the second invariant [32], while the ogden model uses stretches instead of invariants [33]. general expression of a strain energy density function for an isotropic hyperelastic material model is [34, 35]: 1 2 3 1 2 3( , , ) ( , , )w w i i i w   = = (1) where w represents strain energy density function, i1, i2 and i3 represent invariants of the cauchy-green deformation tensors and λ1, λ2 and λ3 represent principal stretches. invariants are related to the principal stretches via the following expressions: 754 m. franulović, k. marković, a. trajkovski 2 2 2 1 1 2 3 2 2 2 2 2 2 2 1 2 2 3 3 1 2 2 2 3 1 2 3 i i i             = + + = + + = (2) the relationship between stress and stretch is derived from the aforementioned strain density function. for this particular case, a complex material model was implemented in the material behavior analysis. the complexity here refers to more than one expression for a relationship between stress and stretch, based on a current value of stretch. in the presented analysis, only the first two parts of the curve (toe region and linear region) were modeled. two complex material models were implemented, the first one (3) from [36] and the second one (4) which slightly differs. the stress-strain relationships in each of them are as follows: 4 ( 1) 3 5 6 [ 0 1 1]cc e c c         −= −      + (3) 𝜎 = { 0 𝐶3[𝑒 𝐶4(𝜆−1) − 1] 𝐶5𝜆 + 𝐶6 𝜆 < 1 1 < 𝜆 < 𝜆∗ 𝜆 ≥ 𝜆∗ 4 1)2 1 3 2 1 ( 2 5 6 1 2 1 1 2 0 [ ] 1 cc c e c c c               −       = − + −         − + +             (4) 𝜎 = { 0 𝐶1 (𝜆 2 − 1 𝜆 )+𝐶3[𝑒 𝐶4(𝜆−1) − 1] 𝐶1 (𝜆 2 − 1 𝜆 ) + 𝐶5𝜆 + 𝐶6 𝜆 < 1 1 < 𝜆 < 𝜆∗ 𝜆 ≥ 𝜆∗ values c1 to c6 represent unknown variables material parameters which have to be identified by the numerical procedures in order to calibrate material model, σ is stress value, λ represents the current stretch value and λ* represents the stretch value on the transition point between toe and linear regions (marked as a in fig. 1). calibration of material models for human cervical spine ligament behaviour using a genetic... 755 3. parameter determination the material model calibration consists of determining the material parameters values, which in the cases of the models consisting of a larger number of parameters means determination of the parameter sets as ensemble. the number of parameters in the parameter set also correlates to the number of the phenomena under consideration that occur within the material structure and, therefore, higher complexity in models often offers requirements fulfillment. as the number and meaning of material parameters differ from one model to another, a short overview is given in this section. from eqs. (3) and (4), it can be seen that the number of material parameters differs depending on the used material model. parameters c1-c5 represent material parameters, while value c6 actually is not a common material parameter but a value that can be expressed in correlation to the other parameters. the expression for value c6 in the case of material model from (3) is the following [36]: ( )4 * 1 6 3 51 c c c e c   −  = − −   (5) on the other hand, when model (4) is used, the expression is: ( )4 * 12 6 3 51 c c c e c    −  = − −   (6) there is also another value that will be treated as a parameter in the genetic algorithm procedure, but which is not a material parameter in a more narrow sense. that value is λ*, which, as aforementioned, represents stretch value in transition point between toe region and linear region, as can be seen in fig. 1. 4. inverse problem inverse problem can be described as an assumption of an unknown parameter set and a reconstruction of the object’s mechanical properties using the known data [13]. in it, the response is observed and the system parameters are calibrated to reach that response [18]. for the identification of the values of the parameters, inverse analysis was used. inverse analysis itself consists of three parts: system characterization, forward modeling and inverse or backward modeling. the first part consists of the definition of model parameters and was done in the previous section for both of the material models. forward modeling deals with foreseeing the system behavior in correlation with the test results and is based on the mechanical principles of the tested material. the third part is founded upon the test results, which are used to calibrate the material model parameters as accurately as possible. 5. material parameters in order to carry out the “inverse modeling” part of the inverse analysis, an appropriate objective function should be set. the basis for the definition of the objective function is the following expression [37]: ( ; )ia=   (7) 756 m. franulović, k. marković, a. trajkovski where  is the mapping function that connects experimental and theoretical stressstretch relationship through the loading process, in the presented case for the first two characteristic regions of the materials life, with the material parameters, where ai represents the material parameters set. parameters set ai for models (3) and (4), respectively, is: ai = [c3,c4,c5,*] (8) ai = [c1,c3,c4,c5,*] (9) 6. objective function the choice of objective function in inverse modeling influences both the accuracy of the results and the velocity for acquiring them. since evolutionary method, genetic algorithm is chosen to combine with the inverse modeling in this particular problem, even simple objective functions are expected to be efficient and to reach global minima in any given material behavior. two different objective functions, which represent fitness functions in the genetic algorithm procedure, were tested here. the first one is a simple least squares method [13]: 2 1 [ ( ; )] n j j i j f a = = −    (10) where tilde (···) refers to the experimental data and n is number of recorded stress – stretch points in data set. the second function is an upgraded least squares method with additional influence of weighting factor, and is based on the one proposed in [37, 38]: 2 1 ( ; )n j j i j j a f =  − =           (11) 7. operators in genetic algorithm in order to obtain the optimal parameter sets for previously described material models, a numerical procedure should be applied. in this case, an evolutionary genetic algorithm is chosen because of its advantageous characteristics, such as applicability in nonlinear systems with a large number of unknowns, robustness considering the choice of objective function, ability to take into account a large number of data, ability to spread the possible results domain in order to avoid local minima and fast convergence to the results. genetic algorithm can be defined as a heuristic search method with the main purpose of optimization of the described system [39], based on the principles of natural evolution. thus the genetic operators in numerical procedure are inspired by the natural evolution principles, such as selection, reproduction, mutation and crossover [11, 19, 23, 39]. together with the definition of its main properties, such as population initialization, definition of results domains, convergence rule and generations progress rule it is a powerful tool for the calibration of material models of nonlinear systems, which are based on experimental data gained from the materials response to the probing signal. calibration of material models for human cervical spine ligament behaviour using a genetic... 757 search space ν and a function g [39] are defined on domain r: :g r (12) the problem is described by the following expression [21, 39]: i iarg min or arg maxa g a g      = = (13) in this particular case, the goal is set in acquiring the minimal value of objective functions given in eqs. (10) and (11). search space (or population domain) as well as the definition of genetic operators is also determined in an endeavour to build an appropriate numerical model for the given problem. 7.1. population initialization the first step in the definition of genetic algorithm should be the creation of the initial population. it is suggested that one or more individuals should be selected heuristically, instead of doing it all stochastically [23]. the domains for each of the parameters defined through the mechanical principles are chosen as follows: 1 3 4 50;100 ; 0;2 ; 0;8 ; 0;100 ; 2;3c c c c      (14) it should be noted that these are the initial domains for the initial population. the values of the final solution can be outside of these domains because of the influence of the genetic operators. another variable is the size of the population (n). larger population will have more solutions included, but would slow down the process of computing, so the user should find the optimal value. madjidi et al. used both n=100 and n=200 individuals, with both populations producing good results [23]. in [7], only the population of n=100 was used. martínez-martínez et al. used populations of n=50, n=100, n=150 and n=200 [26]. 7.2. ranking, selection and elitism in order to proceed with the selection process, individuals are ranked based on a fitness value. those with the better rank should have better chances of further selection. the process of selection in genetic algorithm can be done by various schemes [40]. ktournament selection is here seen as an extension of the “normal” tournament selection from two to k contestants (in other words, “normal” tournament selection can be seen as 2-tournament selection) [41]. this selection randomly picks k number of individuals, all of whom are chosen with the same probability, and the best one is selected for the mating pool (the next generation). the tournaments last until there are as many individuals in the next generation as there were in the previous one. for the presented case of genetic algorithm in material parameter identification problem, k value of four (4) was chosen, based on the previous research in similar problems [37]. although not a genetic operator in a classical meaning of the word, there is also concept of elitism (also called a population overlap). the concept secures that the best individuals survive into the next generation. there is one variable, elite population ratio that is set by the user [39]. the number of individuals that will be replaced is: 758 m. franulović, k. marković, a. trajkovski rn n e n= −  (15) where nr is the number of individuals to be replaced, n is the total number of individuals (population size) and e is the elite population ratio. 7.3. crossover there are also various crossover mechanisms. during the implementation of genetic algorithm for the given problem, intermediate crossover has been used because it is characterized by the children created with the influence of the parents’ weighted average. the application of this kind of crossover is widely used in such cases, but with various crossover ratios. louchet et al. suggested a crossover ratio of 0.3, which is a very low value considering other sources [14]. in this case, several values were tested, but all of them closer to the value of 0.8, as suggested in [10] and [24] or 0.9, suggested in [19]. only martínez-martínez et al. tested many values (from 0 to 1, with steps of 0.1 and finally set on 0.9), with the best one depending on the each case observed and varying greatly [26]. the new generation’s values are calculated here as follows [37]: 1 2 10.9 _ _ (1.1 _ 0.9 _ )child parent k ratio k parent k parent k=  +   − (16) where child represents the new generation values, parent_k1 and parent_k2 are the first and second parent’s values. ratio_k is crossover ratio. one of the possible scenarios is that both parents appear as equal, considering they are randomly selected from the same pool. since this scenario reduces the variety of genetic material, mutation can be additionally implemented on these individuals and then combined through the crossover procedure. in this particular case, the mutation ratio is set at 25% (0.25). mutation procedure here is the following, based on [37], but adapted for this particular case while the domains are given in expression (14): 1 1 1 _ _ _ _ _ 0.25 _ 0; ( _ ) parent k parent k ratio m change m ratio m domain change m random length parent k = +  =  = (17) where ratio_m is mutation ratio and change_m is the size of change. 7.4. mutation the last genetic operator to be defined is a mutation. its importance is significant in the cases where the convergence to unwanted local minima is probable [7]. the most influential variable in the mutation process is mutation ratio. madjidi et al. suggested a mutation ratio of 0.01 [23], louchet et al. suggested 0.2 [14] and wright suggested between 0.005 and 0.3 [10], depending on the algorithm. bianchi and solenthaler also suggested adaptive mutation, but with a different mutation ratio function [24]. in order to achieve faster convergence to global minima, the adaptive mutation is applied to the given problem [37]. it means that the mutation possibility becomes lower with each new generation: calibration of material models for human cervical spine ligament behaviour using a genetic... 759 _ _ _ _ _ _ 1 _ _ 0; ( _ ) change m parent m ratio m change m current generation ratio m last generation change m random length parent m = +  = − = (18) where parent_m is the parent whose values would be mutated while current_generation and last_generation are the ordinal number of current generation and the last generation (or the number of total generations). 8. stopping criteria and adaptation of genetic algorithm as the computing environment for the development of genetic algorithm procedure for the material parameters identification is matlab (version r2017b), default stopping criteria has been used here, including number of generations, time limit, fitness limit, stall generations and stall time limit. in practice, most of the times, the stopping criteria is the number of generations, as the genetic algorithm is mostly effective and able to achieve relatively good results in an early stage of reproduction process. slightly less often, fitness limit was reached just before the last generations, which results in the best solution. sometimes, though, the run was terminated by “stall generations” criteria, i.e. the results would not converge and the final result is mostly not good in that case, which can be overcome by the fine tuning of genetic operators and genetic algorithm properties described above for the particular case. some procedures developed in the published researches use only number of iterations (generations) and threshold value [21, 38]. there are more complex stopping criteria developed, such as those described in [42] or [43], but those were not implemented in this case and present a possible subject for further research. some of the previously mentioned elements of genetic algorithm have variables (crossover ratio, elite population ratio, population size, domains of the parameters and number of generations) that can be set by the user based on previous experiences. during the adaptation of this genetic algorithm, various combinations of those variables were used and fine tuning of their values was deployed in order to acquire an optimal procedure which will result in convergence to the accurate outcomes. as can be expected, some of the developed genetic operators and genetic algorithm properties influence the results more than the others. the only really significant improvement here was accomplished with the increase of the initial population size and even then only to some upper threshold. increase above that threshold did not increase the quality of the final results by much. elite ratio also showed slight influence in the way that ratios of 0.1 and 0.15 mostly showed better results than the ratio of 0.05. representative results of fitness function values are given in tables 1 and 2, where table 1 represents the increase in population size and table 2 the influence of the elite ratio. other operators and variables showed even less influence than the elite ratio and are not shown in tables. however, higher values of crossover ratios (0.7-0.9) showed more consistent results than the lower ones. number of generations did not have big influence, as the problems solved here are relatively simple in terms of optimization. 760 m. franulović, k. marković, a. trajkovski table 1 comparison of the results obtained using different population sizes for material model (3) and objective function (10) population 100 300 500 1000 1500 2500 results (fitness values) 1355.88 1516.84 1405.21 1366.44 1358.36 1319.39 1583.09 1611.93 1430.92 1353.01 1319.80 1323.84 2612.97 1368.23 1322.01 1326.68 1346.62 1329.84 2886.96 1349.44 1328.99 1333.65 1353.89 1318.32 1452.07 1531.86 1422.72 1358.97 1348.87 1334.68 2875.70 1443.55 1467.38 1337.50 1443.37 1319.53 3324.66 1404.70 1341.01 1341.34 1325.27 1323.69 1623.73 2595.61 1324.94 1764.23 1462.29 1322.86 2324.78 1333.25 1322.74 1384.54 1350.09 1320.29 1400.56 1383.87 1350.86 1349.65 1318.26 1319.08 average 2144.04 1553.93 1371.68 1391.60 1362.68 1323.15 median 1974.26 1424.13 1345.94 1351.33 1349.48 1321.58 the best result 1355.88 1333.25 1322.01 1326.68 1318.26 1318.32 the worst result 3324.66 2595.61 1467.38 1764.23 1462.29 1334.68 table 2 comparison of the results obtained using different population sizes for material model (4) and objective function (11) population 500 1000 elite ratio 0.05 0.1 0.15 0.05 0.1 0.15 results (fitness values) 1385.29 1326.55 1405.21 1346.61 1374.62 1366.44 1696.33 1352.92 1430.92 1319.72 1352.06 1353.01 1440.30 1403.69 1322.01 1730.98 1322.93 1326.68 1397.48 1997.37 1328.99 1336.99 1331.02 1333.65 1357.30 1368.02 1422.72 1335.89 1318.18 1358.97 1651.64 1360.86 1467.38 1370.84 1335.30 1337.50 1327.00 1333.33 1341.01 1382.76 1336.31 1341.37 1359.75 1541.09 1324.94 1352.01 1344.73 1764.23 1675.67 1324.30 1322.74 1364.64 1388.52 1384.54 1630.75 1894.56 1350.86 1523.31 1349.95 1349.65 average 1492.15 1490.27 1371.68 1406.38 1345.36 1391.60 median 1418.89 1364.44 1345.94 1358.33 1340.52 1351.33 the best result 1327.00 1324.30 1322.01 1319.72 1318.18 1326.68 the worst result 1696.33 1997.37 1467.38 1730.98 1388.52 1764.23 population 1500 2500 elite ratio 0.05 0.1 0.15 0.05 0.1 0.15 results (fitness values) 1327.36 1339.31 1358.36 1358.25 1332.85 1319.39 1320.30 1320.11 1319.80 1318.42 1323.44 1323.84 1399.55 1381.25 1346.62 1331.24 1319.46 1329.84 1320.11 1328.91 1353.89 1346.60 1325.42 1318.32 1475.95 1326.52 1348.87 1320.28 1355.59 1334.68 1321.64 1322.58 1443.37 1328.33 1366.07 1319.53 1424.25 1325.45 1325.27 1320.50 1321.98 1323.69 1366.10 1345.80 1462.29 1333.57 1333.51 1322.86 1370.43 1411.75 1350.09 1358.28 1321.64 1320.29 1330.50 1327.87 1318.26 1362.46 1319.14 1319.08 average 1365.62 1342.96 1362.68 1337.79 1331.91 1323.15 median 1348.30 1328.39 1349.48 1332.41 1324.43 1321.58 the best result 1320.11 1320.11 1318.26 1318.42 1319.14 1318.32 the worst result 1475.95 1411.75 1462.29 1362.46 1366.07 1334.68 calibration of material models for human cervical spine ligament behaviour using a genetic... 761 9. discussion and parameters identification after the set-up of the numerical procedure is done, and the adaptation of genetic algorithm finished, the comparison of results for the two previously described material models and two given objective functions follows. it should be noted that it is impossible to numerically compare the results obtained using different objective functions and graphical comparison is used instead. two material models for the same objective function are easily compared using both numerical value and graphical representation. in order to provide comparison of the chosen material models and selected objective functions, to validate the model for the observed material behavior and thus provide the meaningful discussion on the matter, the experimental data set, as mentioned in chapter 2, has been used. as stated before, for this calibration, only „toe“ and „linear“ regions were taken into account (stretch values from 1 to 4 as shown in fig. 1). the first comparison is the one using the objective function from expression (10). graphical representation is shown in fig. 2a. it can be seen that both material models offer a good representation of experimental values for low stretch values and higher stretch values. however, it is also visible that the material model (3) significantly outperforms material model (4) for stretch values between cca 2 and 3. the difference in the two material models in this section is also clear when observing the numerical value of fitness. the best fitness value obtained using the material model (3) is 1318.18, while it was impossible to achieve the value better than 3900.21 with the one from (4). in other words, the first material model has almost three times better fitness value than the second, as the smaller value is better. a very similar conclusion can be made by observing the graphical comparison (fig. 2b) between the two models using the objective function (11). in this case, the difference is even more pronounced than in the previous one, as the material model (4) shows a large deviation from the experimental results even for the low stretch values. this time the ratio in fitness values is even for four times in favor of the material model (3). the best value achieved using it was 15.4658 and the best value achieved using the other model was 60.2334. fig. 2(a) stress-stretch chart for objective function (10); (b) and objective function (11) 762 m. franulović, k. marković, a. trajkovski finally, the comparison between the two objective functions can be made. as stated before, it can be done only by visually comparing the results obtained using both objective functions (fig. 2a and 2b). it was already stated that the results for objective function (11) were less accurate than those for the function (10). when comparing the two functions for material model (3), which was proved to be better for both functions, it can be seen that the results are pretty close to the experimental values for low and mid-range stretches. however, function (10) significantly outmatches the other one for higher values of stretches. the main reason for this is the fact that the function (11) divides the difference of squares with the experimental value, thus favoring the lower stress (or stretch) values. the best results achieved for parameters are given in the table 3. table 3 genetic algorithm and material model parameters for the best case genetic algorithm data material parameters population 1000 c3 1.1592 elite ratio 0.1 c4 1.6962 crossover ratio (χ) 0.8 c5 46.5241 material model (3) c6 -107.034 objective function (10) λ* 2.8647 fitness value 1318.18 10. further research and improvements while the genetic algorithm was proven to be an effective tool for optimizing identification of material parameters in highly nonlinear systems, there are various improvements suggested by other authors that could be a subject for further research and implementation and which can additionally improve process of material model calibration. the follow-up in further research could be according to harb et al. who suggested the use of the combination of a genetic algorithm and exact analytical optimization in order to make it shorter and more effective [38]. in [24], it is predicted that further evaluation of adaptive mutation is a key to faster convergence. some works suggested other methods could be used instead of genetic algorithms. han et al. mentioned simulated annealing as a possible alternative [22]. in [26], three methods were compared: an iterative local optimization, scatter search and genetic algorithm. genetic algorithm showed the best result for 50% of cases, scatter search for 30% and iterative local optimization for 20%. in [13], simulated annealing and genetic algorithm were utilized and the genetic algorithm was proven to be more versatile for that use. the marquardtlevenberg nonlinear least squares method was used by pandit et al. and the results they have obtained show that there is no significant difference between that method and the genetic algorithm [27]. in this paper, adaptive mutation rate was used. however, certain works, [8] and [23], also suggest the use of adaptive crossover rate. according to eiben et al, there are some cases in which multi-parent recombination should be considered, although they admitted it was a subject for further investigation [44]. almost 20 years later, akbari and ziarati proposed mleo (multilevel evolutionary optimization algorithm) that features recombination on individual and group level and, although it showed positive results, they also admit future research is needed [45]. also, calibration of material models for human cervical spine ligament behaviour using a genetic... 763 there are many different selection schemes mentioned in [40]. it is reasonable to consider that some of them might outperform current 4-tournament selection. 11. conclusion evolutionary procedure, in this research genetic algorithm, has been used for material parameter identification in modeling material behavior of soft tissues. in order to provide inverse modeling and thus apply presented method for the material models calibration and consequently validation of the chosen parameters set, detailed theoretical basis has been given in the paper. according to the numerous research studies worldwide, which follow a global trend in similar problems, the application of genetic algorithms or similar evolutionary procedures is proven to be generally accepted because of their advantageous properties in acquiring fast and reliable solutions in a very short time. therefore, it has been applied to calibrate two well-known non-linear material models used for modeling of soft tissues behavior. since genetic algorithm efficacy depends on its operators and also on its properties, their detailed representation has been given here in order to describe the fine-tuning for the given problem and to make possible its repeatability for possible additional material models or for the simulation of differently structured soft tissues behavior. for both material models tested, the developed algorithm produced good results with a relatively small calculation time while running on an average one-year-old personal computer. additionally, while comparing the results obtained using both material models, they showed that the application of the proposed method can improve the decision-making in the process of selection of the material model which should be applied in material behavior modeling. as far as the algorithm itself is concerned, the genetic operators and properties were developed and initially tested. the biggest influence of them on the quality of the results has been shown to be a population size, the property which has been chosen for the detailed testing and fine-tuning of algorithm efficacy. the analysis showed that only the increase up to some population size (around 1000) showed significant influence, which turned out as a threshold for the quality increase. although other genetic operators and properties have been taken into account here, their detailed influence has not been investigated, but rather built based on the previous experience. their influence on the results are also expected, but not in the same intensity as the influence on the population size shown here. some of the possible further improvements were also presented, some of them based on the detailed analyses of the influence of the remaining genetic operators and properties, but all of them dealing with the improvement on the genetic algorithm. they take into account operators (use of adaptive crossover, multi-parent recombination and use of selection other than the k-tournament selection). all considered, most of the improvements suggested in the literature would be further researched and, if at all possible, tested and implemented in future genetic algorithms. also, more complicated material models would be tested once the effectiveness of the algorithm has been proven. acknowledgements: this work has been supported by the croatian science foundation under the project number ip-2019-04-3607 and by the university of rijeka under projects number uniritehnic-18-34 and uniri-pr-tehnic-19-21. 764 m. franulović, k. marković, a. trajkovski references 1. mooney, m., 1940, a theory of large elastic deformation, journal of applied physics, 11(9), pp. 582-592. 2. rivlin, r.s., 1948, large elastic deformations of isotropic materials. iv. further developments of the general theory, philosophical 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eiben, a.e., raué, p.e., ruttkay, z., 1994, genetic algorithms with multi-parent recombination, proc. parallel problem solving from nature, ppsn 1994, jerusalem, pp. 78-87. 45. akbari, r., ziarati, k., 2011, a multilevel evolutionary algorithm for optimizing numerical functions, international journal of industrial engineering computations, 2(2), pp. 419-430. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 1, 2014, pp. 1 14 method of dimensionality reduction in contact mechanics and friction: a users handbook. i. axially-symmetric contacts 1 udc (539.3) valentin l. popov, markus hess technical university berlin abstract. the method of dimensionality reduction (mdr) is a method of calculation and simulation of contacts of elastic and viscoelastic bodies. it consists essentially of two simple steps: (a) substitution of the three-dimensional continuum by a uniquely defined one-dimensional linearly elastic or viscoelastic foundation (winkler foundation) and (b) transformation of the three-dimensional profile of the contacting bodies by means of the mdr-transformation. as soon as these two steps are completed, the contact problem can be considered to be solved. for axial symmetric contacts, only a small calculation by hand is required which does not exceed elementary calculus and will not be a barrier for any practically-oriented engineer. alternatively, the mdr can be implemented numerically, which is almost trivial due to the independence of the foundation elements. in spite of their simplicity, all the results are exact. the present paper is a short practical guide to the mdr. key words: normal contact, tangential contact, adhesion, friction, partial slip, stress 1. introduction in the recently published book [1], the so-called method of dimensionality reduction (mdr) is described for the first time in detail. mdr can be traced back to the solution of the normal contact problem by galin (russian academy of sciences) in the 1940s [2]. his results were later published by sneddon and, in this way, made public to the western world [3]. the method of dimensionality reduction takes these results and puts them into such a form that even a layman in the field of contact mechanics can use them for a multitude of contact mechanical problems. in doing this, it merges the ideas and results from cattaneo [4], mindlin [5], jaeger [6], and ciavarella [7] about a close relationship between normal and tangential contacts, the solutions of galanov and borodich [8, 9, 10] received february 20, 2014 / accepted march 18, 2014 corresponding author: valentin popov tu berlin, department of system dynamics and physics of friction, berlin, germany e-mail: v.popov@tu-berlin.de original scientific paper 2 v. popov, m. hess for adhesive contacts of axially-symmetric profiles of power functions (later found independently by yao and gao [11]), as well as the theory of lee and radok about the relationship between elastic and viscoelastic contacts [12, 13]. the book [1] contains all of the necessary evidence and many examples of how to apply the mdr. however, it has proven to be too comprehensive for practical users. there is a need for the fundamental ideas and "recipes" of the mdr to be presented in a concise way without extensive reasoning or proof, a sort of "user's handbook." this work is dedicated to exactly such a practical instruction for the method of dimensionality reduction. 2. two introductory steps of the mdr we consider a contact between two elastic bodies with moduli of elasticity of e1 and e2, poison's numbers of 1 and 2, and shear moduli of g1 and g2, accordingly. in this work, we restrict ourselves to the axially-symmetric profiles, which is not necessarily required. a generalization to profiles that are not axially-symmetric is possible, but is not considered in this work. we denote the difference between the profiles of bodies as z = f(r). in the framework of the mdr, two independent steps are conducted: the first step: first, the three-dimensional elastic (or viscoelastic) bodies are replaced by a one-dimensional linearly elastic foundation. this is considered to be a linear array of elements having independent degrees of freedom and a sufficiently small separation distance x, fig. 1. a) b) c) fig. 1 one-dimensional foundation of different materials: elastic foundation (a), purely viscous foundation (b), and viscoelastic foundation (c) with an example rheology according to kelvin-voigt in the simplest case of the elastic contact, the foundation consists of linearly elastic spring elements that have normal stiffness kz and tangential stiffness kx (fig. 1a): 2 2 * 1 2 * 1 2 1 11 withzk e x e ee        , (1) * 1 2 * 1 2 2 21 with 4 4 xk g x g gg        . (2) starred values e * and g * denote the effective elastic moduli. incompressible linearly viscous materials are presented by a linear damping element with damping coefficient  (fig. 1b), which is dependent on the viscosity  of the viscoelastic partner according to: method of dimensionality reduction in contact mechanics and friction: a users handbook 3 4 x    . (3) arbitrary combinations of these two base elements are also possible in order to satisfy the most complicated elastomers – fig. 1c, for example, shows a viscoelastic foundation built out of elements of in parallel connected springs and dampers (kelvin-voigt model). in this paper, we will restrict ourselves to the case of "elastically similar" materials: 2 2 1 1 2121 gg    , (4) which guarantees the independence of the normal and tangential contact problems [14]. this condition is always met in important cases of contacts between the bodies with the same elastic properties or those between a rigid body and an elastomer. the second step: in the second step, three-dimensional profile z = f(r) (fig. 2, left) is transformed into a one-dimensional profile (fig. 2, right) according to:     x r rx rf xxg 0 22 d )( )( . (5) the reverse transformation is:    r x xr xg rf 0 22 d )(2 )( . (6) fig. 2 the three-dimensional profile is transformed into a one-dimensional profile using the mdr for a less trivial example, we consider the contact of a parabolic profile with a worn tip (fig. 3): 2 2 0 for 0 ( ) for 2 r b f r r b b r a r          . (7) the mdr transformed profile according to eq. (5) is given by: 2 2 0 for 0 ( ) for x b g x x x b b x a r           . (8) 4 v. popov, m. hess fig. 3 parabolic indenter with a "worn" tip: original (solid line) and equivalent (dashed line) profiles for comparison examples for the mdr transformation by inserting profiles into eq. (5) that correspond to a cylinder, paraboloid, cone, or an arbitrary power function z ~ r n , we obtain the mdr transformed one-dimensional profiles which are summarized in table 1, where:                 2 1 2 2 2 n n n kn and    0 1 d:)( tetn tn is the gamma function. table 1 the three-dimensional profile is transformed into a one-dimensional profile using the mdr 3. calculation step of the mdr using the example of a normal contact without adhesion the one-dimensional profile according to eq. (5) is now pressed into an elastic foundation corresponding to eq. (1) with normal force fn (see fig. 4). the normal surface displacement at point x within the contact area results from the difference between indentation depth d and profile form g: method of dimensionality reduction in contact mechanics and friction: a users handbook 5 fig. 4 equivalent model for a hertzian contact )()( xgdxuz  . (9) at the edge of non-adhesive contact x =  a, the surface displacement must be zero: )(:0)( agdauz  . (10) this equation determines the relationship between the indentation depth and contact radius a. it should be noted that this relationship is independent of the rheology of the medium. the force of a spring at the point x is proportional to the displacement at this point: xxuexukxf zzzz  )()()( * , (11) and the sum of all spring forces must correspond to the normal force in equilibrium. in the limiting case of very small spring separation distances x  dx, the summation becomes the integral: * * 0 : ( )d 2 ( ( ))d a a n z a f e u x x e d g x x      . (12) equation (12) provides the normal force in dependence on the contact radius and on the indentation depth, if eq. (10) is taken into account. we now define linear force density qz(x): )( )( :)( * xue x xf xq z z z     . (13) the stress distribution in the original three-dimensional system can be determined with the help of the one-dimensional distributed load using the integral transformation [15]: 2 2 1 ( ) ( ) d r q x p r x x r       . (14) the normal surface displacement (both inside and outside of the contact area) is given by the transformation: 2 2 0 ( )2 ( ) d r z z u x u r x r x    . (15) 6 v. popov, m. hess we point out once more that one-dimensional values contain the independent variable x, while three-dimensional values contain radial variable r; function uz(x) inside the integrand of eq. (15) means the surface displacement of the linearly elastic foundation. we will carry out the last two transformations using the example of a cone. the threedimensional profile in this case is f(r) = rtan; the mdr transformed profile is ( ) tan 2 g x x   . the displacement within the contact area is ( ) tan 2 zu x d x    . the linear force density is *( ) tan 2 q x e d x          and its derivative is *( ) tan 2 q x e     (for positive x). insertion into eq. (14) and eq. (15) results in: 2* * 2 2 d ( ) tan tan ln 1 2 2 a r e x e a a p r r rx r                   , (16) 2 2 2 0 2 ( / 2) tan 2 ( ) d arcsin 1 a z d x d a r r u r x r a ar x                               . (17) equation (17) gives the normal surface displacement outside of the contact area. a similar calculation for a parabolic profile f(r) = r 2 /(2r) initially provides uz(x) = d-x 2 /r and, after insertion into eq. (15), results in: 2 2 ( ) 2 arcsin 1z d r a r u r a r a                              . (18) for the contact of a flattened paraboloid, eq. (7), we obtain the contact radius by using eq. (10): 22)(:0)( ba r a agdau z  . (19) and the normal force using eq.(12): * * * 2 2 2 2 2 2 0 0 2 2 2 d d (2 ) 3 a a n e e f e d x x x b x a b a b r r         . (20) these results correspond of course with those of eijke [18] from the three-dimensional theory. examples for normal contacts insertion of the mdr transformed profiles into eqs. (10) and (12) and an elementary integration provides the results for the "classical profiles" of a cylinder [17], sphere [16], and cone [3] as well as the general power profile [2], which are summarized in table 2. the order of the rows corresponds to the order of the calculation steps. method of dimensionality reduction in contact mechanics and friction: a users handbook 7 table 2 solutions to the normal contact problem for simple profiles 4. adhesive normal contact the mdr rule for the mapping of adhesive contacts will be formulated in the following. as with the non-adhesive contact, the mdr transformed one-dimensional profile is brought into contact with the linearly elastic foundation defined in section 2. now, it will be assumed that all of the springs in the contact adhere to the indenter, then the contact radius remains the same after a successive decrease in the normal force. from the edge of the contact towards the middle, however, more and more springs will be loaded in tension. as soon as the change in length of the outer springs reaches the maximum allowable value: max * 2 ( ) ( ) : a l a l a e        , (21) there will be a state of indifference between adhesion and separation (fig. 5). here,  is the separation energy of the contacting bodies per unit area. this state corresponds exactly to the equilibrium state of the three-dimensional adhesive contact [15]. in contrast to the algorithm for non-adhesive contact, equation (10) must only be replaced by: )()(:)()( maxmax alagdalauz  (22) in order to calculate the indentation depth. the normal force is given as before by eq. (12):               aaa n alaxxgaagexxgadexxgdef 0 max * 0 * 0 * )(d)()(2d)(2d))((2 (23) 8 v. popov, m. hess fig. 5 illustration for the mdr rule for an adhesive contact or         a n alaxxgxef 0 max * )(d)(2 . (24) if the force is controlled during the separation, then critical value ac of the contact radius at the moment of the loss of stability is determined by condition dfn /da = 0: * d ( ) 9 d 2 g a a ae    . (25) inserting the critical radius obtained from this equation into eq. (23) results in the maximum negative force. we will call its magnitude the adhesion force fa: * max 0 2 ( ) ( )d a a c c c f e a l a xg x x          . (26) the simplest example is calculating the adhesion force between a cylinder with the radius a and an elastic half-space. in this case, the integral in eq. (26) is equal to zero and the adhesion force is only given by the first term: * 3 * max2 ( ) 8af e a l a a e     , which corresponds to the result of kendall [20]. calculations for other profiles are just as simple and are summarized in table 3. in order to stress the generality and simplicity of the calculation method, we will now conduct the calculation for a parabolic profile with a worn tip (fig. 6). if we take the equivalent profile from equation (8) into account, the resulting indentation depth is: 2 2 max * 2 ( ) : ( ) ( ) a a d a g a l a a b r e        , (27) fig. 6 qualitative presentation of the adhesive contact of a parabolic profile with a flattened tip method of dimensionality reduction in contact mechanics and friction: a users handbook 9 and from eq. (12), the normal force is: * * 2 2 2 2 3 *2 ( ) ( )d (2 ) 8 3 a n z a e f a e u x x a b a b a e r          . (28) this equation corresponds to the results from the three-dimensional theory [22]. examples for adhesive normal contacts inserting the mdr transformed profiles into eqs. (25) and (26) and conducting an elementary integration provides the results summarized in table 3 for the "classical profiles" of the cylinder [20], sphere [19], and cone [21], as well as the general power function profile [10, 11]. the order of the rows corresponds to the order of the calculation steps. table 3 solutions for adhesive contacts for simple profiles 5. tangential contact we now consider an axially-symmetric indenter that is initially pressed into an elastic half-space with normal force fn and subsequently loaded with tangential force fx in the xdirection (fig. 7). we assume that coulomb's law of friction is valid in the simplest fig. 7 qualitative presentation of the tangential contact 10 v. popov, m. hess form in the contact: as long as tangential stress  is smaller than the coefficient of friction  times normal stress p, the surface is in a state of sticking. after slip sets in, the tangential stress remains constant and equal to p: ( ) ( )r p r  for stick, (29) ( ) ( )r p r  for slip. (30) it is known that for a small tangential force at the edge of the contact, a ring-shaped slip domain develops, which expands inwards for increasing force until the complete slip is exhibited. we denote the inner radius of the slip domain (or the radius of the stick domain) with c. the mdr is applied to the tangential contact as follows. the modified profile g(x) is pressed into the linearly elastic foundation with force fn and then tangentially displaced by ux (0) . the linearly elastic foundation is denoted by the stiffnesses according to (1) and (2). every spring sticks to the indenter and is displaced along with it as long as the tangential force fx = kx ux (0) is smaller than fz. after the adhesion force is reached, the spring begins to slip and the force remains constant and equal to fz. this rule can also be incrementally formulated so that it can be applied for arbitrary loading histories: for a small displacement of indenter of ux (0) , we obtain: (0)( ) , if ( ) ( ) ( ) , in a stateof slip x x x x z z x x u x u k u x f f x u x k          . (31) the sign of the last equation is dependent on the direction of motion of the indenter. by following the incremental changes in the position of the indenter, we can explicitly determine the displacement of all the springs in the contact area; with this, all tangential forces: )()( * xuxgxukf xxxx  . (32) and the linear force density (distributed load): )()( * xug x f xq x x x     (33) are also known. the distribution of tangential stress (r) as well as displacements ux(r) in the original three-dimensional contact are determined as follows [1]:    r x x x xr xu ru 0 22 d )(2 )( . (34) * 2 2 2 2 ( ) ( )1 ( ) d dx x r r q x u xg r x x x r x r               . (35) if the indenter is displaced in one direction from the equilibrium position, then radius c of the stick domain is determined from the condition that tangential force kxux (0) is equal to  times normal force kzuz(c) (fig. 8): method of dimensionality reduction in contact mechanics and friction: a users handbook 11 * (0) *( ( ))xg u e d g c  . (36) the tangential displacement is equal to: (0) * * , for ( ) ( ( )), for x x u x c u x e d g x c x a g              , (37) the distributed load is: * (0) * , for ( ) ( ( )), for xg u x c q x e d g x c x a        , (38) and the resulting tangential force is 2 : * 0 2 ( )d 2 ( ( )) ( ( ))d . a a x x c f q x x e c d g c d g x x             (39) the normal force is still given by equation (12) and ratio fx /(fn) is given by: 0 ( )d ( ) ( )d a x c a n xg x x f f ag a g x x       . (40) the relative displacement ux = ux (0) – ux(x) of the surfaces in contact is obtained by subtracting ux (0) from eq. (37): fig. 8 equivalent model for the classical tangential contact according to cattaneo and mindlin   * * 0, for ( ) ( ) , for x x c u e g c g x c x a g              . (41) 2 we stress once more that all macroscopic values obtained using the procedure described above correspond exactly to the three-dimensional solutions of cattaneo [4], mindlin [5], jaeger [6] and ciavarella [7]. 12 v. popov, m. hess the relative displacement in the original three-dimensional system is calculated using eq. (34) as: * * 2 2 0 2 ( ) ( ) ( ) d , for r x e g c g x u r x c r a g r x             . (42) for example, for a conical, we obtain: 2* * ( ) tan 1 arcsin , 2 x e r c u r c for c r a c rg                                . (43) examples for tangential contacts inserting the mdr transformed profiles into eqs. (36) and (40), results in the relationships between radius c of the stick domain, ratio fx /(fn), and tangential displacement ux (0) . the results for the "classical profiles" of the sphere [4, 5], cone [23], as well as the power function profile are summarized in table 4. the order of the rows corresponds to the order of the calculation steps. table 4 solutions for the tangential contacts of simple profiles 6. conclusions in the present paper, we have limited ourselves to the essential rules and procedures of the method of dimensionality reduction. evidence for the statements herein can be found in the works [1] and [15]. the possibilities of the mdr are much more expansive as presented in this composition. further successful applications have been found for the rolling contact [24, 25], contacts with elastomers [26], contacts of rough surfaces [27, 28], elastomer friction [29], thermal effects in contacts [1], acoustic emission in rough contacts [31], and wear [32]. interested readers are referred to the cited literature as well as the book [1]. in many cases, the mdr allows an analytical solution to the problem, as shown in this work. method of dimensionality reduction in contact mechanics and friction: a users handbook 13 however, it can also be easily implemented numerically and used for the investigation of systems with complex dynamic loadings [33]. let us stress that the presented form of the mdr is only applicable to contacts with homogeneous elastic or viscoelastic half-spaces and it does not take into account size effects [34]. however, extensions to contacts with final bodies or heterogeneous media are also possible [35]. the application of the mdr to rough contacts requires a separate paper. references 1. popov, v.l., hess, m., 2013, methode der dimensionsreduktion in kontaktmechanik und reibung, berlin heidelberg: springer-verlag 2. galin, l.a., 1961, contact problems in the theory of elasticity, north carolina state college, usa 3. sneddon, i.n., 1965, the relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile, int. j. eng. sci., 3(1), pp. 47-57. 4. cattaneo, c., 1938, sul contatto di due corpi elastici: distribuzione locale degli sforzi, rendiconti dell'accademia nazionale dei lincei, 27, pp. 342-348, 434-436, 474-478. 5. mindlin, r.d., , 1949, compliance of elastic bodies in contact, journal of applied mechanics., 16, pp. 259–268. 6. jaeger, j., 1995, axi-symmetric bodies of equal material in contact under torsion or shift, archive of applied mechanics, 65, pp. 478-487. 7. ciavarella m., 1998, tangential loading of general three-dimensional contacts. journal of applied mechanics, 65, pp. 998-1003. 8. galanov, b.a., 1993, development of analytical and numerical methods for study of models of materials, report for the project 7.06.00/001-92, 7.06.00/015-92. kiev, institute for problems in materials science, (in ukrainian) 9. borodich, f.m., galanov, b.a., 2004, molecular adhesive contact for indenters of nonideal shapes, in: ictam04, abstracts book and cd-rom proceedings, warsaw, ippt pan 10. borodich, f.m., 2008, hertz type contact problems for power-law shaped bodies, in: galin, l.a., gladwell, g.m.l. (eds.) "contact problems. the legacy of l.a. galin", springer, 2008, pp. 261-292. 11. yao, h., gao, h., 2006, optimal shapes for adhesive binding between two elastic bodies, journal of colloid and interface science, 298(2), pp. 564-572. 12. lee, e.h., 1955, stress analysis in viscoelastic bodies, quart. appl. math, 13, pp. 183-190. 13. radok, j.r.m., 1957, viscoelastic stress analysis, quart. appl. math, 15, pp. 198-202. 14. johnson, k.l., 1987, contact mechanics, cambridge university press 15. hess, m., 2011, über die exakte abbildung ausgewählter dreidimensionaler kontakte auf systeme mit niedrigerer räumlicher dimension, cuvillier, berlin, germany 16. hertz, h., 1882, über die berührung fester elastischer körper, journal für die reine und angewandte mathematik, 92, pp. 156-171. 17. boussinesq, v.j., 1885, application des potentiels a l'etude de l'equilibre et du mouvement des solids elastiques, gautier-villar, paris, france 18. ejike, u.b.c.o., 1981, the stress on an elastic half-space due to sectionally smooth-ended punch, journal of elasticity, 11(4), pp. 395-402. 19. johnson, k.l., kendall, k., roberts a.d., 1971, surface energy and the contact of elastic solids. proceedings of the royal society of london, series a, mathematical and physical sciences, 324(1558), pp. 301-313. 20. kendall, k., the adhesion and surface energy of elastic solids, journal of physics d: applied physics. 1971, 4, pp. 1186-1195. 21. maugis, d., barquins, m., 1981, adhesive contact of a conical punch on an elastic half-space, le journal de physique lettres, 42(5), pp. 95-97. 22. maugis, d., barquins, m., 1983, adhesive contact of sectionally smooth-ended punches on elastic halfspaces: theory and experiment, journal of physics d: applied physics, 16, pp. 1843-1874. 23. truman, c.e., sackfield, a., hills, d.a., 1995, contact mechanics of wedge and cone indenters, int. j. mech. sci., 37, pp. 261-275. 14 v. popov, m. hess 24. wetter, r., 2012, shakedown and induced microslip of an oscillating frictional contact , physical mesomechanics, 2012, 15(5-6), pp. 293-299. 25. wetter, r., popov, v.l., 2014, shakedown limits for an oscillating, elastic rolling contact with coulomb friction. international journal of solids and structures, 2014, 51(5), pp. 930-935. 26. kürschner, s., popov, v.l., 2013, penetration of self-affine fractal rough rigid bodies into a model elastomer having a linear viscous rheology, phys. rev. e, 87, 042802. 27. pohrt, r., popov, v.l., filippov, a.e., 2012, normal contact stiffness of elastic solids with fractal rough surfaces for oneand three-dimensional systems, phys. rev. e, 86, 026710. 28. pohrt, r., popov, v.l., 2013, contact stiffness of randomly rough surfaces. sci. rep. 3, 3293. 29. li, q., popov, m., dimaki, a., filippov, a.e., kürschner, s., popov, v.l., 2013, friction between a viscoelastic body and a rigid surface with random self-affine roughness, physical review letters, 111, 034301. 30. popov, v.l., voll, l., li,q., chai, y.s., popov, m., 2014, generalized law of friction between elastomers and differently shaped rough bodies, sci. rep. 2014, 4, 3750. 31. popov, m., 2012, contact force resulting from rolling on self-affine fractal rough surface, physical mesomechanics, 15(5-6), pp. 342-344. 32. popov, v.l., 2014, analytic solution for the limiting shape of profiles due to fretting wear, sci. rep., 4, 3749. 33. teidelt, e., willert, e., filippov, a.e., popov v.l., 2012, modeling of the dynamic contact in stick-slip micro-drives using the method of reduction of dimensionality, physical mesomechanics, 15(5-6), pp. 287-292. 34. argatov, i., 2010, frictionless and adhesive nanoindentation: asymptotic modeling of size effects, mechanics of materials, 42 (8), pp. 807–815. 35. popov, v.l., 2014, method of dimensionality reduction in contact mechanics and tribology. heterogeneous media, physical mesomechanics, 17(1), pp. 50-57. metoda dimenzionalne redukcije u kontaktnoj mehanici i tribologiji: uputstvo za korisnike i. aksijalno-simetrični kontakti metoda dimenzionalne redukcije (mdr) predstavlja metodu za proračun i simulaciju kontakta elastičnih i viskoelastičnih tela. suštinski se sastoji od dva jednostavna koraka: (a) zamene trodimenzionalnog kontinuuma jedinstveno definisanom jednodimenzionalnom linearno elastičnom ili viskoelastičom podlogom (winklerova podloga) i (b) transformacije trodimenzionalnog profila kontaktnih tela pomoću mdr-transformacije. nakon izvršenja ova dva koraka, kontaktni problem se može smatrati rešenim. za aksijalno-simetrične kontakte samo mali deo proračuna prevazilazi granice elementarnog računa i taj deo neće predstavljati prepreku za praktično orijentisanog inženjera. kao alternativno rešenje, mdr se može primeniti i numerički, što je gotovo trivijalno zbog nezavisnosti elemenata podloge. i pored jednostavnosti, svi rezultati su egzaktni. rad predstavlja kratak praktični vodič kroz mdr. ključne reči: normalni kontakt, tangencijalni kontakt, adhezija, trenje, delimično klizanje, napon 10469 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 457 478 https://doi.org/10.22190/fume210207007j © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous nanobeams under electro-mechanical loads piotr jankowski faculty of mechanical engineering, bialystok university of technology, poland abstract. the present investigation examines the range of effect of nonlocal parameters on dynamic behavior of a smart beam-like nanostructure modeled as sandwich functionally graded porous nanobeam with piezoelectric layers. therefore, the study is concentrated on determining length of the structure for which nonlocal effects are observed for vibration of nanobeam under in-plane electro-mechanical forces. the nanobeam-based nems device model is obtained based on assumptions of the nonlocal strain gradient theory in conjunction with reddy higher-order shear deformation theory. the investigation present differences in obtained results for nanostructure’s free vibration based on classical and nonlocal assumptions. to study range of application of nonlocal parameters for different length of simply supported nanobeam, defined eigenvalue problem is solved in view of variation of length to thickness ratio, distribution of material properties, as well as electro-mechanical loads. what is more, the study attempts to determine and calibrate values of size-dependent coefficients based on expected natural frequencies, material properties, and applied loads. the results are completed with extensive discussion on the dependence of nonlocal parameters on nanobeam’s dynamic response, thus may be an important step forward to extend understanding of ultra-small structure’s behavior. key words: nanobeam, fgm, porosity, piezoelectric effect, free vibration, detected nonlocal parameters 1. introduction nanoelectromechanical systems (nems) are modern devices that employ ultra-small structures in form of beams, plates, and shells which are powered by electrical circuits [1]. due to extremely small size of the structure, these devices experience unique properties and characteristics. this special kind of structure found applications in various received: february 07, 2022 / accepted february 26, 2022 corresponding author: piotr jankowski faculty of mechanical engineering, bialystok university of technology, wiejska 45c, 15-351 bialystok, poland e-mail: p.jankowski@doktoranci.pb.edu.pl 458 p. jankowski areas, such as logic switches, sensors, actuators, and energy harvesting [2-4]. ultra-small devices operation may be based, inter alia, on electromagnetic [5], piezoelectric [6], and thermo-electric [7] phenomena. what is more, operations principle for diverse smallscaled devices is based on their buckling and vibration characteristics [8]. in view of ultra-small dimensions of the structure, the size-dependent characteristic is fundamental in modelling of mechanical behavior of nanoand micro-sized elements [9]. there are diverse approaches to study small-scaled structures, including molecular dynamics [10] as well as continuum-based nonlocal theories. the latter characterizes lower computational effort for very complex ultra-small systems, and therefore gained much interest [11]. it should be noted, due to size-dependent phenomena, the classical continuum-based theories are insufficient to accurately describe mechanical behavior of ultra-small structures. among advanced continuum-based nonlocal theories, that capture size-dependent effects in microand nanostructures, one can distinguish the couple stress theory [12-14], the modified couple stress theory [15], mindlin’s strain gradient theory [16,17], the modified strain gradient theory [18], gurtin-murdoch surface elasticity [19], the nonlocal elasticity [20-24], the stress-driven nonlocal integral elasticity [25] and a higher-order nonlocal elasticity and strain gradient theory [26]. among these, the straingradient-based ones are established on assumption that ultra-small structures’ mechanical behavior depends on classical strains and higher-order gradient of strains. on the other hand, the stress-based theories indicate that the key role in size-dependent characteristics has classical stresses and higher-order gradients of stresses two entirely different responses, such as stiffness softening and hardening effects, has been seen in nanostructures mechanical behavior [27]. therefore, it has become an important issue to combine both phenomena in one refined theory [26,28]. diverse methods and approaches were used in modelling and analyzing nanostructures [29-41], nevertheless in the current literature survey is focused on dynamics of beam-like nanostructures. wang and varadan [42] presented effect of eringen’s nonlocal parameter on frequency of both single-walled and double-walled carbon nanotubes based on eulerbernoulli beam theory. reddy [43] compared results obtained for diverse beam models for bending, buckling and free vibration of simply supported nanobeams considering nonlocal elasticity of eringen. wang et al. [44] studied influence of small-scale parameter from eringen’s nonlocal theory and diverse length to diameter ratio on vibrations of singlewalled nanotubes modelled by timoshenko beam theory with diverse boundary conditions (bcs). aydogdu [45] used eringen’s nonlocal elasticity to study axial vibration of nanorods with clamped-clamped and clamped-free bcs based on euler-bernoulli model assumptions. roque et al. [46] employed rbf meshless numerical formulation to study bending, buckling, and free vibration of timoshenko nanobeams using nonlocal elasticity theory. li et al. [47] conducted a study on transverse vibrations of simply supported euler-bernoulli nanobeam under axial mechanical load. thai [48], and then thai and vo [49] presented a closed-form solution for deflection, buckling, and vibration of eringen’s nanobeam under transverse loads modelled by diverse refined beam theories. ke et al. [50] investigated piezoelectric effect on nonlinear vibration of timoshenko nanobeam with various bcs using the differential quadrature method (dqm). eltaher et al. [51] used eulerbernoulli beam assumptions and a finite element formulation to study eigenfrequencies of functionally graded (fg) nanobeam based on eringen’s nonlocal theory. berrabah et al. [52] analytically solved bending, buckling and vibration problems of eringen’s nanobeam modelled via various theories. şimşek [53] compared nonlinear to linear detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 459 frequencies obtained for euler-bernoulli nanobeam with various boundary conditions based on a nonlocal elasticity. rahmani and pedram [54] presented analytical solution for natural frequencies of functionally graded nanobeam according to timoshenko beam and eringen’s nonlocal theories. nejad and hadi [55] analyzed impact of bi-directional functionally graded material properties on eringen’s nanobeam vibrational response using the generalized differential quadrature method (gdqm). beni [56] discussed impact of piezoelectric phenomenon on nonlinear vibration and bending characteristics of eulerbernoulli nanobeam using the couple stress theory. arefi and zenkour [57] discussed effect of electric field on vibration and bending of nanobeam with piezomagnetic layers resting on winkler-pasternak foundation based on timoshenko beam model assumptions and eringen’s nonlocal elasticity. lu et al. [58] used the nonlocal strain gradient theory to perform analysis of nanobeams free vibration based on the sinusoidal shear deformation theory. liu et al. [59] examined vibration response of visco-elastic functionally graded porous timoshenko nanobeam with magneto-electro-elastic coupling effect for different boundary conditions. thai et al. [60] analytically solved deflection and vibration problems of eringen’s nanobeams subjected to uniformly distributed load. karami and janghorban [61] proposed novel higher-order shear deformation beam theory and studied natural vibration of functionally graded nanobeam using nonlocal strain gradient theory. liu et al. [62] utilized galerkin approach to study nonlinear vibration of functionally graded sandwich nanobeam with diverse boundary conditions based on the nonlocal strain gradient theory and euler-bernoulli model. jankowski et al. [63,64] presented analytical and numerical approaches to study vibrational behavior of fg and sandwich fg with piezoelectric layers nanobeams based on reddy third-order shear deformation theory and the nonlocal strain gradient theory. nasr et al. [65] studied effect twoparameter foundation and thermoelastic phenomenon on deflection and vibration nanobeam based on euler-bernoulli beam assumptions and eringen’s nonlocal elasticity. 1.1. novelty of the present study based on the best author’s knowledge and presented literature survey there is a lack of investigation focused on the range of nonlocal parameters interaction on dynamic response of functionally graded porous beam-like nanostructures, with simply supported edges, under in-plane electromechanical forces. achieved nano-actuator model based on the nonlocal strain gradient theory in conjunction with the reddy third-order shear deformation theory enables a wide spectrum to analyze nanostructures dynamic response in view of nonlocal parameters as well as geometrical and material properties. the study ensures different point of view to the analysis of vibrational response of nanobeams. in contrast to previous papers, the present investigation is focused on an influence range of nonlocal parameters on dynamic behavior of nanostructure in view of material and geometrical properties together with an applied electromechanical load. the investigation considers length of the nanostructure at which the nonlocal phenomena have crucial influence on its vibrational response. in accordance with this objective, the study displays differences in results obtained based on classical and nonlocal assumptions. what is more, the present study together with solution methodology provides a possibility to estimate values of small-scale parameters depending on expected fundamental frequencies, assumed, or predicted material properties along with mechanical and electrical forces acting in nems devices. the presented approach could be an important tool in better 460 p. jankowski understanding nems structures and consequently, may avoid some of cost-consuming and complex material characterization processes, especially for nems devices in engineering applications. 2. higher-order nonlocal elasticity and strain gradient theory a higher-order nonlocal elasticity and strain gradient theory called as the nonlocal strain gradient theory [26] is a novel hybrid refined theory that enables consider both stiffnesses softening and hardening effects in the nanostructure. thus, such an approach allows taking advantage of eringen’s nonlocal theory and mindlin’s strain gradient theory from a single point of view [66]. according to assumptions of the nonlocal strain gradient theory, the total stress tensor is presented in the form: (1)  = − (1) where  refers to classical stresses (work conjugate of classical strains) and (1) to higher-order stresses (work conjugate of gradients of strains). used vector differential operator ∇ = ei∂/∂xi is coupled with unit vector ei and direction xi of a nonlocal effect (gradient) in nanostructure. the classical and higher-order stress tensors are derived as: ) :( v cx x =  0 0 'α ,', e 'dva (2a) (1) 2 ) :( v x cx =  1 1', 'dv'α ,e a (2b) where α0 and α1 represent principial and additional attenuation kernel functions to describe nonlocality effects in terms of the euclidean distance between the point x and neighboring points x′ in the domain v. additionally, ℓ stands for the length scale parameter form the strain gradient theory, as well as e0 and e1 are nonlocal constants together with a represent internal characteristic length. then, ε′ is the cartesian components of the strain tensor at point x′, and c is the elastic modulus tensor of classical elasticity. a general form of constitutive equations in differential form may be obtained assuming that nonlocal attenuation kernel functions satisfy conditions proposed by eringen [24] and applying linear differentiational operators on both sides of eq. (1). the differential operator is in the form: 2 21 ( ) = − ii e a for i = 0,1l (3) the generalized nonlocal constitutive relations based on the higher-order nonlocal elasticity and strain gradient theory take the form: 2 2 2 2 2 2 2 2 2 21 ( 1 ( 1 ( 1[ ) ][ ) ] [ ) ] : [ ) ]( :c c  −  − − −= −  1 0 1 0e a e a e a e a (4) where ∇2 = ∂2/∂x2 is one dimensional differential operator. assuming e0 = e1 = e the higher-order constitutive relations may be easily reduced to the lower-order nonlocal strain gradient model: 2 2 2 2[ ) ]( :1 [ ]1 c = − − ea (5) detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 461 what is more, based on the present model, eringen’s nonlocal elasticity may be obtained assuming ℓ = 0: 2 2[ ) ]1 ( :c  =− ea (6) from the other point of view, form of mindlin’s strain gradient theory may be achieved by taking ea = 0: 2 2[ :1 ]c −= (7) finally, classical constitutive relations may be obtained while both nonlocal coefficients are assumed to be zero. 3. material properties and piezoelectric effect 3.1. three-layered fgm nanobeam in the study we consider composite nanobeam-like actuator model that consists of porous functionally graded nanobeam with thickness h and piezoelectric layers of thicknesses hp. the total thickness of the nanostructure is assumed h = h + hp, length l and unit width b. between fgm core and piezoelectric face-sheets it is presumed ideal mechanical contact. to imitate environment impact in the nems device, the nanobeam is subjected to an electrical field with external voltage ϕ0 and axial compressive/tensile mechanical in-plane forces ˆ xxn . the cross-section and coordinate system of sandwich nanobeam are presented in fig. 1. fig. 1 three-layered porous fgm nanobeam as a nanoactuator model 3.2. functionally graded core elastic parameters of the functionally graded porous nanobeam core are given by: ( )= c e z, xx (8a) ( , ) 2(1 )  = + e z c xz (8b) 462 p. jankowski e(z,ϒ) stands for young’s modulus of nanobeam core and ϒ(z,ϑ) is porosity distribution function. young’s modulus, as well as an fg porous structure mass density ρ(z,ϑ), vary in the thickness direction according to the power-law distribution. in the study, poisson’s ratio υ is assumed to be constant since its low volatility has negligible influence on mechanical behavior of the structure. properties of functionally graded core are symmetric with respect to the midplane, and presented as: ( ) [ ][1 ( , )] = −(n) c m me z, (e e )v (z,g)+e z (9a) ( ) [ ][1 ( , )]     = −(n) c m mz, ( )v (z,g)+ z (9b) components of the functionally graded material generally consist of ceramic (ec, ρc) and metallic (em, ρm) parts but can be replaced by other materials/nanomaterials with wellknown experimental values of elastic/physical parameters. properties of functionally graded core are assumed to be symmetric with respect to the midplane. the volume fraction function v(n) controls volatility of material distribution through the nanobeam core thickness, and is described as follows: ,0 2     =  −       g (1) h z + h2v for z h 2 (10a) 0, , 2 g (2) h z h2v for z h 2   −  =    −    (10b) where g is the power-law index. in the investigation, we consider three diverse symmetric to the midplane porosity distributions: ( , ) cos     =     πz type 1 : z h (11a) 1 cos ,0 2 2 ( , ) 1 cos 0, , 2 2        +  −         =     −       2z h π for z h type 2 : z 2z h π for z h (11a) 1 cos ,0 2 2 ( , ) 1 cos 0, , 2 2        +  −         =     −       z h π for z h type 3 : z z h π for z h (11a) detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 463 where ϑ refers to averaged volume of voids in the structure called porosity coefficient. distributions as well as volume of voids, together with variation of volume fraction function are shown in fig. 2. a) b) fig. 2 material variation in functionally graded nanobeam core: a) porosity distribution and accumulation; b) volume fraction function 3.3. piezoelectric relations the studied nanobeam is subjected to an electric field. electric potential  is presented as approximated combination of half-cosine and linear variation [67] satisfying maxwell relations: 0( ) cos ( ) p p p p πz 2z x,z,t x,t h h     = −  +      (12) where zp stands for variable that is measured from the geometrical center of the piezoelectric layers, z1 = z-h/2-hp/2 for top face-sheet and z2 = z+h/2+hp/2 for bottom facesheet. additionally, φ(x,t) refers to variation of electric potential in longitudinal direction. electric field components are achieved based on electric potential function: cos p x p πz e x h x    = − =       (13a) 0 2 sin p z p p p πzπ e z h h h    = − = − −      (13b) 4. displacement field and constitutive equations displacement vector components are described in the framework of reddy third-order shear deformation theory [68]: 464 p. jankowski ( ) ( ) ( ) ( ) ( )3 0 x 0 x 1 x w x,t u x,z,t = u x,t + z x,t c z x,t + x          (14a) ( ) ( )z 0u x,z,t = w x,t (14b) where u0 and w0 are axial and transverse displacements of material point on the midplane in an undeformed configuration, φx is rotation of a point on the centroidal axis x of the beam, and c1 = 4/(3h2) for clarity of formulation. the strain-displacement relations are presented as linear and infinitesimal green-lagrange strain tensor:   2 (0) (1) (3) 2 , , , ,        =              0 x x 0 xx xx xx 1 u w ε ε ε -c + x x x x (15a)  (0) (2), ,        =        + +    0 0 xz xz x 2 x w w -c x x (15b) where c2 = 3c1. thus, the strains take the form: (0) (1) (3)= + + 3 xx xx xx xxε ε zε z ε (16a) (0) (2) = + 2 xz xz xz2ε z (16b) based on a higher-order nonlocal elasticity and strain gradient theory, the constitutive relations are: 2 2 2(1 ) 1( )c xx xx xxc ε− = −b (17a) 2 2 2(1 ) 1( )c xz xz xz2c ε− = −b (17b) 2 2 2( )( )(1 ) 1p xx xx xx x zd ε e e = − −b (18a) 2 2 2 )(1 ( )) 1 (2p xz xz xz z xd ε e e− = −b (18b) 2 2 2 )(1 ( )) 1 (2p x z xz x xd e ε e− =  +−b (18c) 2 2 2( )( )(1 ) 1p z x xx z zd e ε e =  +− −b (18d) where ℬ = (ea)2. σc ij, σp ij and dp i are nanobeam core stress tensor, piezoelectric facesheets stress tensor and electric displacement component. additionally, dij are elastic constants for piezoelectric layers, ei are piezoelectric permittivity constants and ∈i are dielectric permittivity constants. detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 465 5. equations of motion and closed-form solution procedure equations of motion are obtained based on modified hamilton’s variational principle including virtual strain energy u, virtual kinetic energy k, virtual electric field e, and virtual work done by external mechanical and electrical forces v in the form ( )    t 0 u k e + v dt = 0 (19) a detailed explanation on the derivation of the equations of motion is presented in previous paper [64]. the equations of motion of fgm sandwich nanobeam are presented by displacements and obtained based on the nonlocal strain gradient theory as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 0 1 3 0 02 3 5 1 3 0 0 05 2 2 2 4 12 2 2 4 4 4 3 14 4 3          + − + +             −      =− + −       0 x x 0 0 xx xx xx x xx x x 0 xx xx x 0 0 u w u a a a + x c b a xx x x x x w a a + b i u u x x x c ib (20a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 1 1 1 4 2 0 2 1 3 3 2 4 4 3 3 6 0 1 3 3 2 2 2 2 2 1 12 2          −       +  − + −                    +     −   + − − +           +    0 0 0 0 xz x xz x xz x xx 0 x x x 0 xx xx xx xx x 0 xx z x x w w w u a + a + a + a x x x x u w a a a a + x x x x x w a + b x c c c c c c cb b x xx ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 3 0 2 42 3 3 3 2 2 2 2 2 2 22 2 2 4 4 4 4 4 1 1 14 4 4 4 4 2 1 5 1 3 2 4 4 5 6 2               + − +                        − + − − +          − −       −      z x 0 x 0 x 0 xz xz xz 0 0 x x x 0 xx xx xx xx xx xx x c c c c c c b x x w w w a + a + a + x x x x x u u w a a a a a + x x x x x x ac ( ) ( ) ( ) ( ) 4 3 3 3 3 1 24 3 3 3 3 2 2 1 2 1 2 2 3 2 3 2 1 1 5 0 1 3 5 2 4 6 2 24 63 2 3 2 2              +               − + − −               +             + − −      x 0 z x x x z 0 0 x x x 0 0x x w + b b b b x x x x x x w w i i + i +c + c c = c c c i x x x w w i + i x x x x b      (20b) 466 p. jankowski ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 3 0 2 4 3 2 1 2 23 3 3 2 2 2 1 2 2 2 4 4 6 0 2 2 4 3 3 0 1 23 3 2 5 32 1 12 2 5 2               + +         + − +                   − −           +   − 0 x 0 x 0 x xx xz xz xz 0 x x 0 xx xx z z 0 x x xx xz u w w a a + a + a + x x xx x x c w w a a + b b x x x x x x u a c c +c cb a c x c c x ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 23 3 3 5 5 4 2 2 2 1 13 5 4 4 4 4 4 2 2 2 1 1 14 4 4 4 2 4 4 4 6 4 6 0 4 6 3 0 4 6 2           − +                +                 + − −                = + −    + 0 x 0 xz x 0 x x 0 xz xx xx z z x x 0 x w w + a + x x x x w w a + a a + b x x x x x x b b i w i i + c x xx c c c c c c x c 2 2 2 2 2 2 3 3 2 1 12 2 3 3 4 4 0 4 4 4 6 4 4 ˆ ˆ ˆ ˆ xx xx        + +           − + − +           + +       0 0 0 0 xx 0 0 0 w w x x w w w i i i + x x x x x n w w n n c c n x x b (20c) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 1 0 3 2 2 2 2 0 12 2 4 4 0 3 4 1 3 32 2 2 3 3 3 3 1 23 43             + +        +   − +          − −               − − +            − −           0 x x 0 x 0 x x z x x 0 0 x z x x x 0 x 0 z x x z x u w w b b b + b + x x x xx x w u b + b b x x x x x w w b + b + b x x x x c c c c c c c ( ) 3 4 4 3 2 2 4 2 4 0   −      =        + x 0 z z x w + x x x x c c (20d) where aij (m), bj (m), ci, ciϕ and ii stand for resultant stiffnesses, piezoelectric coefficients, dielectric coefficients and mass moments of inertias. the terms are also presented in [64]. for simply supported boundary conditions navier solution method is used: ( ) ( ) ( ) ( ) 1 cos cos sin sin          =          =            =   n n n n iω t n iω t n ni 0 0 xx n 0 ω t n iω t n 0 x e x e n l u x x e u w w e (21) detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 467 where 0u ,  x , 0w ,  are maximum values of displacements and electric potential. natural frequency of the nth mode is expressed as ωn. system of governing equations are expressed in a vector form:      0 − =   0 0x t n u wk ω m (22) [k] and [m] are symmetric stiffness and inertia matrices, respectively. components of these are presented in [64]. presented solution procedure and numerical results based on it were comprehensively verified with results from the literature. verification study is also presented in [64]. 6. parametric study the study ensures a novel perspective of analyzing small-scaled structures. the numerical results show the range of influences of nonlocal parameters on a dynamical response of complex nanobeam. to this aim, variation of length of the nanobeam with different nonlocal parameters values is considered to observe when size-dependent effects disappear for its vibration under in-plane electro-mechanical forces. the interaction range parameter is defined as differences in obtained solutions for classical (local) l nω , and nonlocal nl nω theory of elasticity as: ( )% 100% − =  l nl n n l n ω ω ω (23) the top lines are used to show dimensionless frequencies in form: 2 2 010=    i i n n c 2 i ω ω l for i = l,nl e i (24) where ωi n is dimensional frequency. the material and geometrical parameters of the studied sandwich nanobeam are assumed as follows hp = 1.5 nm, h = 7 nm, ρ = 5550 kg/m3, ρc = 3100 kg/m3, ρm = 2700 kg/m3, dxx = 226 gpa, dxz = 44.2 gpa, ec = 380 gpa, em = 70 gpa, υ = 0.3, ex = -2.2 c/m2, ez = 5.8 c/m2 , ∈x = 5.64∙10-9 c/vm, ∈z = 6.35∙10-9 c/vm. not mentioned here coefficients are assumed to be variable and explained in individual studies. what is more, the present section shows selected results for predicting nonlocal coefficients depending on expected free vibration frequencies, material gradation, distributions and volume of voids together with defined in-plane mechanical and electrical forces. 6.1. nonlocal interaction range parameter figure 3 displays influence of diverse length to thickness ratio on the nonlocal interaction range for fundamental frequency of the studied nanobeam with homogeneous (g = 0) core. 468 p. jankowski a) b) c) fig. 3 influence of length to thickness ratio on the nonlocal interaction range parameter for nanobeam fundamental frequencies: a) ψ = 2; b) ℬ = 0; c) pseudorandom ψ ratio detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 469 nonlocal to length scale coefficients ratio is introduced as ψ = ℬ/ℓ and assumed for this study ψ = 2 nm in fig. 3 a). additionally, for fig. 3b, it is assumed constant nonlocal coefficient of eringen’s elasticity as ℬ = 0 (pure mindlin’s strain gradient), and pseudorandom ψ ratio for fig. 3c. the pseudorandom distribution was chosen to clearly present influence of nonlocal coefficients. diverse ratios of nonlocal eringen’s to length scale coefficients are required in modelling nanostructures because values of nonlocal parameters vary due to initial stress, mode shape, rotary inertia, boundary conditions as well as material properties [69-72]. based on fig. 3, it may be concluded that the more lengthened the nanostructure is (higher length to thickness ratio), the lower impact of size-dependent parameters on dynamics of nanobeam. the scale coefficients refer to nanostructure and consequently has nanoscale dimensions. therefore, large dimensions of the structure cause the effect of nanoscale to disappear. in the study, only length is used as a variable since nonlocal constitutive equations consider gradients of strains and stresses in the x-direction. for that reason, using thickness as a variable does not change mechanical response of the structure. on the other hand, increasing values of nano-scale coefficients induced higher gradients of stresses and strains (higher-order derivatives in equations of motion), and consequently, higher differences in results obtained by classical and nonlocal approaches. in addition, the study agrees with previous studies [58] that stiffness hardening and softening phenomena caused by strain and stress gradients disappear while ℓ2 = ℬ because nonlocal parameters reduced each other out, consequently referring to classical elasticity. moreover, greater differences are observed in the case of the strain gradient approach because only stiffness hardening effect is considered. additionally, fig. 3c clearly shows that appropriate detection of nonlocal parameters has a crucial role in predicting dynamic response of nano-scaled structures. the effect of diverse axial in-plane forces on the nonlocal interaction range parameter for eigenfrequencies of homogeneous nanobeam is presented in fig. 4. in the study, we assumed ψ = 1.2 nm and length of the nanostructure l = 100 nm. the study considers impact of both, mechanical forces and electrical loads induced by an external voltage. it should be noticed, applying compressive load (in both cases refer to positive values) relates to higher differences in classicaland nonlocal-based results. this is because additional compression generates strains and stresses in the structure. the nanostructure under compression experiences small shortening and bent. thus, nonlocal parameters causing further gradients of stresses and strains result in greater differences between obtained results. the higher load is applied, the higher difference is achieved. it may be concluded that small-scale parameters have the most important role when the nanostructure is subjected to loads near their critical values. on the other hand, negative values of applied loads decrease the results differences because the tension causes opposite situation and results in a small lengthening of the structure. continuing study on diverse loads acting on sandwich nanobeam, the nonlocal interaction range parameter for natural frequency of nanobeam subjected to complex loadings is examined considering diverse length to thickness ratios. in the study, it is also assumed homogeneous core. figure 5 indicates that despite simultaneous acting tension and compression, the applied loads lead to an increase of differences between results from local and nonlocal approaches. it is also observed, despite applying loads the nonlocal interaction range decrease with increasing length to thickness ratio. it clearly presents, increasing dimensions of the nanostructure decreases applications’ range of size-dependent coefficients by decreasing a long-range interaction. consequently, it may be an important issue to 470 p. jankowski appropriately detect the nonlocal parameters in terms of dimensions of the studied nanostructure. a) b) fig. 4 influence of external axial in-plane forces on the nonlocal interaction range parameter: a) mechanical forces; b) electrical forces fig. 5 influence of complex loadings, length to thickness ratio and small-scale parameters on the nonlocal interaction range parameter detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 471 the effect of varying mechanical properties of functionally graded core in conjunction with nonlocal parameters and diverse length to thickness ratios on the nonlocal interaction range parameters is examined in fig. 6. in this study, we assumed constant compressive force acting on the nanostructure as ˆ xxn = 6 n. according to previously discussed cases, size-dependent coefficients change influence on the nonlocal interaction range parameter depending on l/h ratios. nevertheless, opposite to previous results, higher differences are observed for higher length to thickness ratio when the core structure is functionally graded. what is more, increasing the power-law index increases value of the nonlocal interaction range parameter. varying material properties influence resultant stiffness and mass moments of inertias, consequently coupled with nonlocal parameters and higher-order displacements derivatives have a greater impact on a dynamic behavior of nanostructures. moreover, applied compressive mechanical forces are approaching the critical value for fgm structure with l/h = 15. it confirms the previously stated suggestion that small-scale parameters are significant when the structure is under load that is near to its critical value. fig. 6 influence of length to thickness ratio, material gradation and small-scale parameters on nonlocal interaction range parameter fig. 7 influence of length to thickness ratio, volume of voids and small-scale parameters on nonlocal interaction range parameter 472 p. jankowski figure 7 displays impact of volume of voids (type 1 porosity distribution), size-dependent parameters, and aspect ratio on the nonlocal interaction range parameter. in the study it is assumed that homogeneous nanobeam is subjected to electric field induced by external voltage ϕ0 = 0.95 v. similar to the previously discussed situation, increasing volume of voids decreases nanostructure’s stiffness and mass density, so it affects resultant stiffness coefficients and mass moments of inertia. therefore, applied nonlocal parameters and higherorder derivatives of displacements have a greater impact on free vibration response of nanobeam with lower stiffness and density. this investigation ensures, that regardless of mechanical or electrical forces are applied, loads tending to critical values have a higher impact on the nonlocal interaction range parameter because of more significant strains or stresses gradients. the influence of diverse porosity distribution together with the porosity coefficient as well as length to thickness ratio is presented in fig. 8. in the present study, homogeneous sandwich nanobeam is subjected to compressive mechanical forces ˆ xxn = 15 n, and nonlocal parameters are assumed ℬ = 1 nm2 and ℓ = 4 nm. diverse porosity types are characterized by different voids accumulation through the thickness of nanobeam core. therefore, they have a diverse impact on resultant stiffness and density coefficients, and consequently, on the nonlocal interaction range parameter. increasing volume of voids has a significant impact on the nonlocal interaction range parameter for higher values of length to thickness ratios. weakness of the structure caused by porosity becomes important for structures with a high aspect ratio when the applied force is near the critical value. therefore, regardless of whether the nanostructure stiffness is influenced by material gradation or porosity, the used size-dependent coefficients have a bigger impact on non-homogeneous structures. thus, proper detection of nonlocal parameters is a significant issue, especially for heterogeneous structures. fig. 8 influence of diverse porosity distributions, volume of voids and length to thickness ratio on the nonlocal interaction range parameter 6.2. detection of nonlocal parameters in the current section, the detected values on nonlocal parameters are presented with superscripted *. first of all, figure 9 presents predicted both eringen’s nonlocal coefficient and the length scale parameter for homogeneous and functionally graded nanobeam. the detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 473 figure shows the dependence of small-scale coefficients on assumed nanostructure’s eigenfrequencies. it may be observed that increasing frequencies relate to decreasing eringen’s nonlocal constant. it agrees with the theory assumptions because the parameter is involved in stiffness softening phenomenon. so, the lower predicted frequencies are, the higher nonlocal coefficient is needed, because assuming constant other parameters, the structure must be softer. on the other hand, increasing length scale parameter causing stiffness hardening phenomenon, affects on increasing size-dependent parameter of eringen. in contrast, increasing fundamental frequency values causes increasing in predicted length scale coefficient. higher frequencies are related to stiffer construction (assuming constant mass) therefore higher length scale coefficient is required. likewise, increasing eringen’s nonlocal parameter increases stiffness softening effect, so length scale parameter values are greater. a) b) fig. 9 detection of nonlocal parameters for expected fundamental frequencies: a) eringen’s nonlocal coefficient for homogeneous nanobeam core; b) length scale parameter for fgm (g = 2) core 474 p. jankowski next, relationship between nonlocal parameters and electro-mechanical forces is presented in figure 10. in the study, we assumed ψ = 4/3 nm and expected fundamental frequency as ω1 = 21.4 ghz for homogeneous nanobeam. the structure is complexly loaded, so there are initial gradients of stress and strains. increasing values of mechanical forces together with positive values of external voltage results in compression of the nanostructure. on the other hand, negative values of applied voltage refer to tension of the structure. compression/tension of the structure results in its stiffness decrement/increment. increasing values of nonlocal parameters involves a higher stiffness hardening effect due to assumed ratio of nonlocal to length scale coefficients. therefore, increasing applied forces, with constant predicted nanobeam frequency, requires stiffness enhancement effect from nonlocal coefficients. being knowledgeable about environment acting in nems device, for example, applied loads and operation’s frequency, this solution gives a possibility to estimate/detect nanostructure nonlocal coefficients. fig. 10 detection of nonlocal parameters for expected applied loads figure 11 represents a nonlocal parameters detection in view of porosity distributions and volume of voids. it is assumed homogeneous nanobeam core, nonlocal coefficients ratio as ψ = 1.5 nm, and fundamental frequency ω1 = 24.65 ghz. increasing volume of voids decreases nanostructure stiffness and influences nanobeams mass. nonetheless, diverse distribution affects stiffness and mass in diverse ways. type 1 and 2 are characterized with non-porous bottom and lower surfaces, while type 3 porosity distribution at these points is described with a maximum volume of porosity. therefore, diverse effect on stiffness to mass ratios obtained from diverse porosity distribution is connected with diverse nano-scale effects from nonlocal coefficients. detection of nonlocal calibration parameters and range interaction for dynamics of fgm porous… 475 fig. 11 detection of nonlocal parameters for different porosity distributions and volume of voids 7. concluding remarks the current study gives a different perspective to higher-order nonlocal strain gradient theory. the approach enables an analysis of differences in obtained results from classical and nonlocal theories for dynamic response of sandwich piezoelectric nanobeam under complex loads. the model represents a three-layered fgm porous beam-based nanoactuator subjected to external mechanical and electrical loads. functionally graded material properties along with diverse porosity accumulation types are obtained based on the power-law distribution. the contribution of an electric field is represented by a combination of halfcosine and linear variation of the electric potential. utilized equations of motion obtained based on hamilton’s principle and the nonlocal strain gradient theory, include eringen’s nonlocal coefficient and the length scale parameter to investigate both size-dependent phenomena from a single point of view. the displacement field is obtained based on reddy third-order shear deformation theory that does not require shear correction factor, then it may be used for a wide range of structures. the results and discussion show effect of nonlocal parameters in conjunction with diverse length to thickness ratio, external loads, and material properties on a free vibration response of intelligent nanobeam. defined the nonlocal interaction range parameter clearly demonstrates that smallscale coefficients are dependent on geometrical properties of the structure, its material properties, and applied loads. therefore, employing nonlocal theories is necessary for modelling of nanostructure-based ultra-small devices. what is more, the presented multiparameter solution gives a possibility to estimate/detect nonlocal parameters for nanobeams based on assumed/predicted natural frequencies, material properties, and applied loads. therefore, the conducted investigation, due to a widening understanding of nonlocal effects, may be an important tool in modelling, optimization, and control of nems devices. 476 p. jankowski acknowledgement: the paper has been conducted within wz/wm-iim/3/2020 project and was financed by the funds of the ministry of science and higher education, poland. references 1. rieth, m., schommers, w. 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https://doi.org/10.22190/fume210310035p © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper relation of kinematics and contact forces in three-body systems with a limited number of particles kristin m. de payrebrune institute for computational physics in engineering, tu kaiserslautern, germany abstract. in many tribological systems, an intermediate layer of a limited number of abrasive particles exist. thereby, the resulting wear and friction phenomena are desirable in many manufacturing processes, such as lapping or polishing, whereas in machine elements, they are unwanted due to reducing lifetime or performance. for a better understanding of the contact phenomena and the interaction of tribological systems with an intermediate layer of a limited number of particles, fundamental investigations are carried out on a tribometer test rig. for this purpose, two test scenarios are investigated, a) the kinematics and contact forces of single geometrically defined particles such as dodecahedron, icosahedron and hexahedron, and b) the contact forces and surface roughness of a layer of silicon carbide particles of different sizes. the measured ratio of tangential to normal force can be used as an indicator of the dominating kinematics of the particles and of the generated surface roughness, respectively. the higher the force ratio, the higher the tendency to slide for a given particle type and paring of particle and counter body. for one geometrically defined particle the short-time fourier transform additionally helps to distinguish the state of motion since the excited frequencies during rolling are reduced. for a layer of silicon carbide particles, the velocity and particle size have the strongest influence on the overall motion and the surface roughness produced. larger particles tend to slide and create more scratches, while smaller particles tend to roll and create indentations in the counter body. furthermore, for the same particle size, an increase in velocity causes a transition from sliding to rolling, resulting in an increased surface roughness. key words: three-body contact, particle kinematics, contact force analysis, tangential to normal force ratio, surface roughness received march 10, 2021 / accepted april 12, 2021 corresponding author: kristin m. de payrebrune tu kaiserslautern, erwin-schrödinger-str. 56, 67663 kaiserslautern e-mail: kristin.payrebrune@mv.uni-kl.de 96 k. m. de payrebrune 1. introduction many tribological applications with single or a limited number of particles exist that significantly affect the system. often these penetrating particles are undesired in technical applications and machine elements, as they reduce the service life or impair the performance. however, in other cases, abrasive particles are used for processing, such as for lapping and polishing, to achieve particularly high surface finishes. in order to obtain the desired surface quality and roughness values, the choice of the appropriate parameters (contact pressure, rotational speed) and their interactions with the particles, as well as the contact phenomena are decisive. even though the applications of tribological systems with an intermediate layer of particles are extensive, the area of research often focuses on the dependence of coefficient of friction, wear behavior, or material removal rate on process parameters such as applied force, particle size and sliding speed of the counter body [1-3]. specific investigations on the surface roughness and interaction between particles and counter body of hemanth et al. showed an increased roughness at the exit of particles compared to the entry into the contact [4]. additionally ahn and park measured an increased roughness with larger particles, which they attributed to the reduced number of particles in contact [5], whereas zhang et al. additionally stated that higher wear rate and worn surface features appeared a lower applied loads [6]. especially for lapping processes detailed analysis of the lapping slurry on the cutting depth are carried out by belkhir et al. and wang et al. who stated an increased material removal rate for suspensions that corrode the workpiece surface [7, 8]. additionally cozza et al. were able to relate measured coefficients of friction to the kinematics of abrasive particles from different materials. in general a low concentration of abrasive slurry results in a high coefficient of friction and a grooving motion, whereas a high concentration of abrasive slurry let reduce the coefficient of friction and rolling becomes dominant [9]. the adaptation of the particle motion by special lapping plates was also used in the lapping process of guo et al. to achieve different surfaces roughness [10]. beside analyses of the overall material removal rate, detailed investigations on single particles help to better understand the local penetration mechanism. indentation and scratch tests of single particles are therefore standard procedures in order to correlate the particle kinematics with the material failure and the removed material. thereby, two important findings were established. firstly, the kinematics of the particles can be directly assigned to the generated surface structures (scratches, indentations), secondly the damage behavior of the material has a significant influence on the final surface [3, 1113]. belkhir et al. and buijs and korpel-van houten have observed scratches and cracks starting from the plastic zone below the surface when doing tests on glass samples [7, 14, 15]. they estimated the occurring stresses according to hertz contact law, so that a prediction of the surface condition was possible. in particular, heisel and avroutine developed a model to estimate the kinematics of a particle based on the geometry and contact forces [12]. vangla et al. introduced two shape parameters that allowed them to characterize granular material and to relate the shape of particles to their tendency to roll or slide [16]. with a more recent molecular dynamics model, shi et al. investigated the surface indentation of an ellipsoidal particle and observed a transition from rolling to sliding when increasing the normal load [17]. further studies on elastic particles between two plates with a boundary element model by li revealed local stick and slip regions and relation of kinematics and contact forces in three-body systems with a limited number of particles 97 a subsequent transition from rolling to gliding [18]. in previous works of the author, bilz and de payrebrune found comparable dependencies on the motion of a hard, cubical particle. for a constant coefficient of friction, the cube-shaped particle can roll and slide depending on its orientation (contact with edge or corner of a hexahedron), while an increased velocity leads to a detachment of the upper plate and thus to a strongly changed force ratio [19, 20]. generally, the investigations of single particle contacts have been carried out on defined grits with the aim to relate the particle geometry to the produced surface failure or kinematics. in contrast, lapping investigations focused primarily on the achievable material removal rate and surface quality for different particle properties, lapping slurries and materials of the samples. the results are well established, however, in many tribological systems the geometry of penetrating particles is not defined and measurements of impact parameters are only possible indirect, such as analyzing the overall process forces or the surface roughness after operation. in our case, we aim to generate a specific microstructure through a customized lapping process by controlling the kinematics of the particles through appropriate process parameters and by using no lapping fluid. generally, this requires precise knowledge of the interplay between particle kinematics, set parameters and particle properties (shape, size, quantity), which we investigate on the basis of indirectly measurable quantities during the process, such as the forces and the resulting force ratio. therefore, two fundamental analyses are conducted. firstly, forces are measured of tribological systems with single particles of different shapes and are linked to the observed particle kinematics. secondly, this knowledge is used to correlate measured forces of experiments with one layer of particles to their overall kinematics and generated surface. in the following, the setups and test procedures for individual particles and a layer of particles are explained. based on these experiments, observations on the behavior of the particles are discussed and an additional relation between surface roughness and particle kinematics introduced, before the main findings are concluded. 2. experimental investigation the kinematics of individual particles or a layer of abrasive particles directly influences the material removal and the occurring process forces. in order to study these relations, tribometer tests have been carried out. 2.1. general setup two experimental scenarios were developed to investigate local and global interrelations, namely: i. measurement of tangential forces and motion of individual, geometrically defined particles for preset normal forces on different surfaces, ii. measurement of tangential forces of an abrasive layer of silicon carbide particles for preset normal forces and rotational velocities. subsequently, the achieved surfaces of the samples are tactilely measured and the roughness values correlated with the particle kinematics and force ratio. both test scenarios are carried out on a tribometer test rig with three different base plate speeds of 4 rpm, 12 rpm and 22 rpm (75 mm/s, 220 mm/s and 400 mm/s). via a cantilever 98 k. m. de payrebrune arm mounted on one side, a counter plate is pressed onto the particles with a defined normal force. in order to study the kinematics of individual particles, a metal plate is first attached to a 3-axis force dynamometer (type 9119aa1, force range 4 kn, kister) and then to the cantilever arm. for the tests with a layer of abrasive particles a sample holder is mounted on the force sensor to fix steel samples of 30305 mm. the normal force is set by weights which are attached to the cantilever arm, cf. fig. 1. in this work, the influence of a liquid on the tribological behavior is explicitly not studied. fig. 1 tribometer setup for experimental investigations of individual particles and a layer of silicon carbide particles. photo of the test rig in a), scheme of the setup for one particle in b) and for a layer of particles in c) 2.2. setup for single particles the experiments with single acrylonitrile butadiene styrene particles (plastic dice made of abs) are performed on dodecahedrons, icosahedrons and hexahedrons with an average size of 20 mm. for both, base plate and counter plate, a smooth steel surface and a sand paper surface with grit size k150 are used, so that four surface combinations are tested. the normal force has been set to 50 n throughout all experiments and the motion behavior is recorded with a high-speed camera (chronos 1.4, kron technologies inc.) at 4484 fps and the forces are stored accordingly at a sampling rate of 4484 hz. in addition, the path of the particles is tracked with tracing paper for selected experiments. fig. 2a shows exemplary a force profile of the normal and tangential directions, which also reveals the different phases of the experiment. the jump of the normal force at 5 s results when the cantilever arm is set down on the base plate. at about 7 s the motor is switched on and starts rotating with a delay of 3 s. between the 10th and 13th second the actual experiment takes place, which ends with switching off the engine (at 13 s) and lifting the cantilever arm at 15 s. depending on the particle motion (rolling or sliding) the duration of the measurement varies. however, during the actual experiment, the particle moves once from its initial left position by angle  to the right end of the plate, cf. fig. 2b. the measured tangential force components fx and fy represent clearly the angular influence during the actual experiment (10th to 13th second). first, the particle moves in negative y-direction, which results in a negative force component fy. during the curse of relation of kinematics and contact forces in three-body systems with a limited number of particles 99 motion, the value becomes positive when the particle starts moving in positive y-direction after intersecting the y-axis. the force component fx remains positive during the entire test, but varies as a result of the angle between x-axis and tangential force ft. the applied normal force should actually remain constant, but due to tolerances in the bearing of the cantilever arm a small tilting occurs, which leads to a variance of 1.5 n in fz. nevertheless, the force ratio of tangential and normal force 2 2 x yt n z f ff = f f + (1) is determined and evaluated for various experiments. fig. 2 representative measurement of process forces with an individual particle in a), scheme of the particle's path in relation to the used coordinate system in b), and shapes of the used particles in c) 2.3 setup for a layer of particles the experiments with one particle layer are conducted with particles of silicon carbide of size f12 (~1765 µm), f30 (~625 µm) and f60 (~260 µm), cf. fig. 3b. the base plate is made of steel, which is renewed after each series of nine tests for a specific particle size. the counter plate is a steel sample of 30305 mm, which is renewed after each experiment. the applied normal forces are set to 40 n, 60 n and 80 n and the duration of experiments are adjusted to the rotational speed of the base plate such that the sliding distance remains the same. during experiments, the forces are recorded at a sampling rate of 1000 hz in all three directions. in preparation for each experiment, 50 g of particles are uniformly distributed on the base plate in a ring. during experiments, the clamped steel sample attached to the cantilever arm is first placed on the particle layer and base plate. after the motor has been switched on and started, the test is carried out until a specific sliding distance is reached. in order to detect drifts of the forces, the sample is lifted again at the end of the test, similarly to the test procedure of individual particles. fig. 3a exemplary shows the force profiles of the normal and tangential directions, and the different phases of the experiment. because the sample is placed on the point of intersection with the y-axis (cf. fig. 2b), the force component fy is almost zero. nevertheless, the force ratio is calculated from both tangential force components by eq. (1) and the force ratio is evaluated in relation to the test settings and the generated surface of the sample. for this purpose, 100 k. m. de payrebrune following the experiments, the surfaces of the samples are measured tactilely on a nanoscan (hommel-etamic). fig. 3 representative measurement of process forces for one player of particles in a), and images of investigated silicon carbide particles of sizes f12, f30 and f60 in b) 3. experimental results and discussion the aim of these experimental investigations is to correlate the particle kinematics and the generated sample surfaces with process parameters and measured forces in order to better understand tribological systems with a limited number of particles. to this end, the behavior of the individual particles is investigated first. the results obtained are then used to analyze the behavior of a layer of particles in more detail. 3.1. results for single particles the experiments with single particles are performed on dodecahedron, icosahedron and hexahedron, whose shape changes from spherical to more and more angular, and for which their motion and related forces are recorded. besides analyzing the single force components, the ratio of tangential and normal force ft/fn and the frequency response are evaluated and are set in relation to the motion and the tracked path of the particles by tracing paper1. figs. 4 to 6 show results of an icosahedron, a dodecahedron and a hexahedron moving on sand paper with grit size k150. considering the corresponding high-speed videos, the measured forces can be clearly related to the individual motion of the particles. generally, as soon as the particle starts rolling, the tangential forces drop significantly, whereas during sliding, large amplitudes and vibrations occur. depending on whether the particle rolls over a corner or over an edge, the force fluctuation varies in strength, compare [19]. according to high-speed videos of the dodecahedron (cf. fig. 4), the particle rolls over an edge at motion {1, 4, 7, 10, 13, 15}, which can also be obtained from the lower decrease in the force ratio ft/fn, or a smaller increase in fx, respectively. the 1only for the special measurements for tracking the particle trace was the test rig modified so that the cantilever arm points in the x-direction instead of the y-direction (cf. fig. 2a). this modification has the effect of increasing the normal force when the particle moves in the positive x-direction since the lever arm between the pivot point of the cantilever and the point of contact is reduced. all measurements for the analysis of the force ratio are carried out with the original setup relation of kinematics and contact forces in three-body systems with a limited number of particles 101 force component fy varies similar to fx, except that the sign depends on whether the particle rolls in positive or negative y-direction, cf. figs. 4a and c. fig. 4 force measurements with displayed components fx, fy and fz in a), calculated force ratio ft/fn in b), and tracked path with tracing paper in c) for experiments with an individual dodecahedron and 4 rpm. short-time fourier transform of the force components are shown in d)-f). numbers indicate the beginning of rolling. fig. 5 force measurements with displayed components fx, fy and fz in a), calculated force ratio ft/fn in b), and tracked path with tracing paper in c) for experiments with an individual icosahedron and 4 rpm. short-time fourier transform of the force components are shown in d)-f). numbers indicate the beginning of rolling. for the icosahedron, similar force progressions have been observed with regard to rolling over an edge or a corner, cf. fig. 5. however, when the counter plate has contact with a flat surface of the icosahedron, a short sliding motion occurs before the next rotation over a corner or edge starts, which results in normal force fluctuations. compared to the dodecahedron, the amplitude increase is stronger for the icosahedron, which indicates a higher resistance to rolling. 102 k. m. de payrebrune fig. 6 force measurements with displayed components fx, fy and fz in a), calculated force ratio ft/fn in b) and zoomed forces during sliding with the corresponding motion in c) of experiments with an individual hexahedron and 4 rpm. short-time fourier transform of the force components are shown in d)-f), and and tracked path with tracing paper in g). numbers indicate the beginning of rolling. in comparison to the icosahedron and dodecahedron, the hexahedron is very angular and has a very high resistance to rolling, which results in large vibrations of the measured forces in fig. 6. by help of the high-speed videos, the local oscillating forces can be related to a repeatedly tilting of the hexahedron (peak at 2 in fig. 6c), but instead of proceeding rolling, the hexahedron flips back on its flat face (drop at 3). generally, as soon as the tangential force is large enough to start rolling, the tangential forces decreases significantly and the force ratio becomes ft/fn |min = 0.04, regardless of the shape of the particle. hence, both values, the tangential force component fx and the force ratio ft/fn are valid indicators to distinguish different motions of the individual particles. beside the force analyses, the frequency response of each force component is additionally evaluated. therefore, a short-time fourier transform (stft) is performed of 500 measured values each of every 0.55 s, which results in a frequency resolution of 8.9 hz. the stft of the three particle shapes are displayed as color maps in figs. 4d-f to 6d-f. as soon as the particle starts rolling the stfts of fx show a significant frequency decrease at low values (0-10 hz), accompanied by decreased amplitudes of fy when the particle rolls over an edge and radial motion occurs. furthermore, at higher frequencies between 20 hz and 80 hz, the stft of fz shows a strong excitation during sliding and a drastic decrease during rolling (especially visible for the hexahedron cf. fig. 6). consequently, the stft of the measured forces also helps to distinguish the different states of motion of the individual particles. in order to transfer these findings to a particle layer in which only statements about the overall behavior of all particles can be made, the observed kinematics of the individual particles are correlated to averaged values of the force ratio ft/fn, as displayed in fig. 7a. in here, all experiments are grouped in red where either the rotating plate relation of kinematics and contact forces in three-body systems with a limited number of particles 103 and/or the counter plate is a smooth steel surfaces and the corresponding high-speed videos show a pure sliding movement of the particle. the force ratio of ft/fn  0.2 is almost constant with low variances, compare also with fig. 7b. in contrast, when both plates are covered with sand paper, the average force ratio depends on the rolling ability of the particle, as the results in fig. 7a show in blue. the more frequently a particle rolls and the less the contact duration of upper plate with a flat surface of the particle is, that results in increased force amplitudes of fn, the lower the force ratio becomes. comparing the force ratios for overall rolling of a dodecahedron and an icosahedron, this is particularly evident when comparing the results in fig. 4, fig. 5. when the particles roll to one of their flat sides a short sliding follows. as a result, the force ratio increases to values in the same range as those of the sliding hexahedron (cf. fig. 7c) and the normal forces show increased amplitudes. related to the number of surfaces of the particle geometry, the different contact situations (contact with flat surface, edge or corner) changes more rapidly for the icosahedron with 20 surfaces than for the dodecahedron with 12 surfaces. in addition, due to its more spherical shape compared to the icosahedron, the dodecahedron can more easily roll, which reduces the time intervals with strong force fluctuations. consequently, the influence of the intermediate sliding dominates for the icosahedron and leads to a higher mean force ratio than for the dodecahedron. due to the angular shape of the hexahedron, it slides over long periods of time, which results in a much higher overall force ratio on sand paper with ft/fn = 0.52 than for the other particle geometries. however, when determining the force ratios of pure rolling and pure sliding, the force ratios show a clear distinction between both motions, as indicated in fig. 7a in green. consequently, the force ratio for rolling is not influenced by sliding periods, which results in a lower value compared to the other geometries and the force ratio for sliding with ft/fn = 0.58 exceeds slightly the mean force ratio. so by knowing the lower and upper boundary for rolling and sliding, the value of the force ratio indicates the ratio of both states of motion. fig. 7 force ratio ft/fn of dodecahedrons, icosahedrons and hexahedrons. mean force ratio on different surfaces with shown median value, first quartile, third quartile and whiskers based on the interquartile range in a). force ratio as a function of time during sliding on steel in b) and force ratio during rolling, or sliding and rolling in the case of a hexahedron, respectively on sand paper in c). 104 k. m. de payrebrune table 1 average values and standard deviation  of force components with an applied normal force of 50 n and rotational speed of 4 rpm, and tracked path according to the geometry of the individual particle when both plates are covered with sand paper geometry ft fn ft/fn trace dodecahedron 11.27 n 50.15 n 0.220.12 icosahedron 14.169 n 49.20 n 0.280.08 hexahedron 23.93 n 48.97 n 0.520.4 table 1 lists the average values of the force components the ratio of ft/fn and its standard deviation when both plates are covered with sand paper of grit size k150. generally, we can state that the smaller the average force ratio is, the more frequent the particle rolls, and the results for individual, geometrically defined particles can be summarized as follows: a) the tangential force needs to overcome the resistance against rolling. if the tangential force is too small, the particles slide on smooth surfaces and chatter on rough surfaces. b) the force progression and the short-time fourier transform can be used to distinguish between rolling and sliding of the particle. when the particle rolls, the force ratio significantly drops and the amplitudes in the frequency domain decreases. c) also the mean value of the force ratio can be used to distinguish between different states of motion. when knowing the force ratios of pure rolling and pure sliding, the force ratio indicates the ratio between both states of motion. 3.2. results for a layer of particles the analysis of one layer of silicon carbide particles is analogous to the analysis of individual particles. during the experiments, the normal and tangential force components are recorded, however, the mean force ratio is calculated from the filtered values, which is done by using a first order butterworth filter with a cut-off frequency of 100 hz. in addition, roughness parameters obtained from tactile surface measurements of the steel samples are analyzed. fig. 8 shows the average force ratio ft/fn in terms of all three analyzed parameters: particle size (f12, f30, f60), rotational speed of base plate (4 rpm, 12 rpm, 22 rpm), and applied normal force (f = 40 n, f = 60 n, f = 80 n). here, no dependence on the normal force is noticeable. however, the force ratio depends and partly on the speed of the base plate and strongly on the particle size, whereby the absolute values of the force ratio become smaller with decreasing size. the strongest influence on the rotational speed show particles of size f30 with a force ratio that decreases from ft/fn=0.27 to ft/fn = 0.20, while the other particle sizes lead to more constant values. according to the results of individual particles a high force ratio indicates sliding and chattering. applied to the measurements of one particle layer, this means that the particles of size f12 slide more frequently than the smaller particles. images of the relation of kinematics and contact forces in three-body systems with a limited number of particles 105 sample surfaces confirm this assumption. fig. 9 shows the generated surfaces and tactically measured surface profiles for a normal force of fz = 40 n and various particle sizes and speeds. in accordance to the large force ratio of particles f12, all samples show numerous scratches over the entire surface due to a dominant sliding motion. fig. 8 mean values of force ratio ft/fn in a) and corresponding standard deviation in b) of measurements with one layer of particles and for different parameters. fig. 9 black and white surface images of steel samples generated with different silicon carbide particles, rotational speeds and with an applied normal force fz = 40 n. the arrow indicates the direction of particle motion with the upper side pointing outwards in a). tactile measured surface profiles with standard deviation of profiles at the entry, center and exit at each sample in b). transferring the findings of individual particles to the measured force ratio for a particle layer with size f30, the tendency to roll should increase with rotational speed of the base plate. the corresponding surface images confirm this, as large scratches are only found at low speeds (cf. fig 9a) and with higher velocity the scratches shorten. hence, these particles perform a transition from sliding to primarily rolling as a function of the 106 k. m. de payrebrune rotational speed. in contrast, of the smallest particles of size f60, the mean force ratio and standard deviation, as well as the surface images indicate a predominant rolling motion. accordingly, the surface structure produced by a layer of silicon carbide particles correlates with the state of motion predicted by the force ratio and can also be distinguished in the roughness measures. the surface profiles are measured orthogonal to the direction of particle motion at the entrance, center and exit of particles at each sample (cf. fig. 9a), with the standard deviation of the three profiles shown as a gray ribbon in fig. 9b. the corresponding arithmetical mean deviation ra and the core roughness rk are additionally calculated and averaged over six standard measuring lengths of 12.5 mm at each position, which are placed within the surface profile. as evident from fig. 10, the arithmetic mean deviation ra and the core roughness rk increase with velocity and normal force. in conjunction with the measured reduction of the force ratio at higher speeds for particle size f30, it becomes apparent that higher roughness values are related to an increased rolling motion. even if the experiments with individual particles did not result in any material removal, statements about the contact conditions can be made on the basis of the recorded traces. especially evident at the trace of a hexahedron cf. fig. 6g, sliding causes a wide and brighter trace than rolling, for which dark concentrated dots can be seen when a corner or edge is in contact. these higher contact stresses during rolling results in deeper indents, as already observed from [14, 15]. transferred to the particle layer, for a fixed particle size deeper indentations and a higher roughness result during rolling than sliding, as visible for f30. furthermore, the indents become deeper with particle size and normal force, and the roughness increases, as previously observed [5]. fig. 10 arithmetical mean deviation ra in a) and core roughness rk in b) of steel sample surfaces generated by tribometer tests with a layer of silicon carbide particles as a function of particle size, base plate speed, and applied normal force. shown are the median values with first quartile and third quartile as error bars calculated from surface profiles at all locations on one sample. according to the good correlation of observations on individual particles and one layer of particles, the following statements on the motion behavior can be made: a) the average force ratio ft/fn and its standard deviation  are valid parameters to distinguish different motion tendencies of the particle layer. thereby, the results of a particle layer matches the observations of individual particles. relation of kinematics and contact forces in three-body systems with a limited number of particles 107 b) in general, the states of motion depend less on the applied forces than on the speed and particle size. c) there is a size dependence with regard to the states of motion and the influence of process parameters. for the parameters investigated, the largest particles (f12) show predominantly sliding, the smallest particles (f60) predominantly rolling, and the medium-sized particles (f30) a transition from sliding to rolling. d) according to the state of motion, rolling results in higher localized stresses and consequently in deep indents and a high roughness, whereas sliding results in scratches with a lower roughness. additionally the roughness values increases with normal force and particle size. 4. conclusion the movement of individual, non-abrasive particles and a layer of abrasive silicon carbide particles was investigated by tribometer tests. the tests on single particles were performed with three geometric shapes (dodecahedron, icosahedron, hexahedron) on smooth steel surfaces and on rough sand paper with grit size k150. during experiments with one layer of particles, the particle size, the speed of the base plate and the normal force have been varied. the analyses show that the average force ratio ft/fn of the tangential and normal force components and their standard deviation  are valid indicators to distinguish different states of motion for both test cases. for the same surface pairing and particle shape, a reduction in the force ratio means an increase in rolling, which is associated with a reduction of the standard deviation due to a more uniform motion. if the force ratios for pure rolling and sliding are known, the average force ratio can be used to determine the proportion of sliding and rolling. in both test cases, the tendencies of the force ratio and the standard deviation are similar, so that observations about the state of motion of individual particles can be transferred to systems with an intermediate layer of particles. based on the recorded trace of individual particles, contact with a corner or edge results in darker imprints than when the flat side of the particle is in contact during sliding. consequently, rolling results in deeper indentations, which leads to a higher roughness. additionally, the roughness increases with a larger particle size and higher applied loads. however, due to the complex relationships between particle size, motion type and roughness value, reference measurements must be carried out for each particle type and material paring. in general, the correlation found between the state of motion and the mean force ratio, its standard deviation, and roughness measures can be used as an indicator of motion, and to change the system behavior in a desired way. both findings of the force analysis and the roughness values additionally help to characterize tribological systems and increase the understanding about the complex phenomena in the contact. acknowledgement: the work is funded by the deutsche forschungsgemeinschaft (dfg, german research foundation) – project-id 172116086 – sfb 926. 108 k. m. de payrebrune references 1. stachowiak, g.b., stachowiak, g.w., 2001, the effects of particle characteristics on three-body abrasive wear, wear, 249(3-4), pp. 201-207. 2. agbaraji, c., raman, s., 2009, basic observations in the flat lapping of aluminum and 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simulation of a single third-body particle in frictional contact, facta universitatis-series mechanical engineering, 18(4), pp. 537-544. 19. bilz, r., de payrebrune, k.m., 2019, analytical investigation of the motion of lapping particles. pamm, 19(1), e201900076. 20. bilz, r., de payrebrune, k.m., 2021, investigation of the influence of velocity in a tribological three-body system containing a single layer of rolling hard particles from a mechanical point of view, tribology international, 159, 106948. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 2, 2014, pp. 107 121 an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions  udc 624.01+626.5 gil rama hochtief infrastructure gmbh offshore, hamburg, deutschland abstract. heavy lift jack-up vessels (hljv) are used for the installation of components of large offshore wind farms. a systematic fe-analysis is presented for the hljv thor (owned by hochtief infrastructure gmbh) under extreme weather conditions. a parametric finite element (fe) model and analysis are developed by using ansys®1apdl2 programming environment. the analysis contains static and dynamic nonlinear fe-calculations, which are carried out according to the relevant standards (iso 19905) for in-place analyses of jack-up vessels. besides strategies of model abstraction, a guide for the determination of the relevant loads is given. in order to calculate the dynamic loads, single degree of freedom (sdof) analogy and dynamic nonlinear fe-calculations are used. as a result of detailed determination of dynamic loads and consideration of soil properties by spring elements, the used capacities are able to be reduced by 28 %. this provides for significant improvement of the environmental restrictions of the hljv thor for the considered load scenario. key words: heavy lift jack-up vessels, site-specific assessment, drag/inertia parameter method, thor, offshore industry 1. introduction the ambitious goal of the federal government of germany of having 25 gw wind energy output installed by 2030 (see [1]) could not be achieved without special heavy tools and machinery like hljvs. besides crane capacity of up to 1500 tons, hljvs are characterized by their jacking system that allows them to elevate their hull above the water surface (see fig. 1). in this way the hydrodynamic loads are decreased and all loads are transferred into the ground. as a result, the environmental restrictions for this kind of vessels can be increased in comparison with conventional floating installation vessels. received may 28, 2014 / accepted july 10, 2014  corresponding author: gil rama hochtief solutions ag, civil engineering marine and offshore, fuhlsbüttler straße 399, 22309 hamburg, germany e-mail: rama_gil@gmx.de 1 ansys® is a registered trademark of ansys inc. 2 ansys parametric design language professional paper 108 g.rama environmental restrictions for operations are described by maximum wave height and wind velocity, which are given in the technical specification of a hljv (for thor see [7]). in case these restrictions are exceeded, the vessel has to change into survival mode. in this mode, the crane operations are stopped and it is moved into rest position. if necessary, fig. 1 hochtief hljv thor the safety distance between the hull and the water surface (air gap) is increased. like in the operating mode, the restrictions are defined for the survival one. if these are exceeded, the vessel has to find shelter in a port or it has to be evacuated. the survival restrictions depend on the ultimate limit state (uls) of the current system configuration. the determination of uls is a challenge due to the complex load situation (see [10]). the loads acting on jacked hljvs can be classified into three main categories: 1. deadweight 2. wind and ocean loads 3. inertia loads the determination of these loads is influenced by the configuration of the vessel, its deck load and the operating location. assumptions have to be made (see [9], [11]) for their calculation and application. it is necessary to create a fe-model, which represents the stiffness and dynamic behavior of the vessel. the stiffness of the leg structure can be assumed as soft, compared to the hull structure. this leads to relatively large deformations and hence the geometric nonlinear effects have to be taken into account (see [14]). the soil properties are always site-specific and present only approximations without detailed soil surveys. this uncertainty has to be considered in the analysis model by choosing appropriate boundary conditions (see [12]) or using a conservative approach like a pinned support. under survival conditions the in-place analysis (ipa) is assessed for the extreme storm event. if the critical load headings are unknown, different loads distributed around the circumference directions have to be taken into account. this includes investigating a large number of load cases by using the lrfd-method (load and resistance factor design). an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 109 2. methodology a systematic fe-analysis is presented for the leg structure of the hljv thor under survival conditions. in order to carry out an analysis according to the relevant standards, the deterministic two-stage approach is used. in the first stage, an inertia load set is determined, which is calculated either with a single degree of freedom analogy in combination with the total base shear or with more detailed methods like a random wave time domain dynamic analysis. in the case of applying a detailed method, a simplified fe-model is sufficient to calculate the inertia load set. afterwards the maximums of the environmental loads (wave, current and wind actions) are determined. in the second stage these loads are combined with the inertia load set to find out the response with the detailed structural fe-model including geometrical nonlinear effects in a static analysis. different load directions are considered, where the combination of actions is applied in phase. 3. fe-models of hljv thor two different fe-models are required for the deterministic two-stage approach: 1. detailed fe-model of the thor (static analysis) 2. simplified fe-model of the thor (dynamic analysis) the detailed fe-model (see [1]) mainly consists of shell and beam elements. the decks, longitudinal girders, longitudinal bulkheads, transverse frames, bulkheads, tank walls and the outer skin of the vessel are modeled using shell elements. beam elements are used to model the stringers, deck supports and leg structure. the leg cross sections are modeled as areas and then implemented as cross-sections in ansys. the hull-leg-connection is realized by an analogous model consisting of link and spring elements (see fig. ). the simplified fe-model (see fig. ) consists of beam and pipe elements and is used for the dynamic analysis in irregular sea states. the hull-leg-connection is realized by an analogous model consisting of spring elements. fig. 2 detailed fe-model of hljv thor 110 g.rama fig. 3 simplified fe-model of hljv thor fig. 4 hull-leg connection (detailed fe-model) soil properties are considered through nonlinear spring elements. the nonlinear loaddisplacement relationship for vertical, horizontal forces and overturning moment is described by a hyperbolic curve, which is defined according to [14]. if soil parameters are unknown, the interaction between soil and structure should be assumed as a pinned support. the generation of the fe-model is parameterized by using the ansys-apdl programming environment so that all the necessary configurations including the choice of boundary conditions can be created. 4. eigenvalue analysis in order to characterize the basic dynamic system behavior an eigenvalue analysis is performed. the resulting natural frequencies and mode shapes, which are functions of the structural properties and boundary conditions, allow an evaluation of the system regarding stiffness and mass distribution. furthermore, these values are used to adjust the properties an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 111 of the simplified model. to evaluate the influence of the soil in the eigenvalue analysis, the resulting natural periods for the following boundary conditions are determined: table 1 natural periods of investigated variants no. variants normalized natural periods [s] 1. 2. 3. 1 pinned support 1.00 0.97 0.71 2 spring support 0.67 0.66 0.48 3 spring support (pre-stressed) 0.72 0.71 0.50 the natural periods for the listed variants in tab. 1 are normalized by dividing the values by the first natural period of the first variant (pinned support).additionally, the natural periods are used for the determination of excitation periods within the structural dynamic analysis. the first two mode shapes of jacked hljvs are usually the displacements of the hull in longitudinal and the transverse direction (see fig. 5). fig. 5 first and second mode shape of jacked hljv thor the maximum difference between the variants 1 and 2 of the first natural period is 33 % (see tab. 1), which shows a large influence of the kind of support. the influence of pre-stress has a maximum value of 7 % for the considered load scenario (see tab. 2). 5. loads and their application all relevant loads are calculated in the developed apdl macros. in the following, the used load assumptions and corresponding macro descriptions are given. 5.1 deadweight the deadweight of the hljv can be roughly divided into two groups, permanent and variable masses. the permanent masses are taken into account by means of the modeled structure elements and an assigned density. the variable masses such as cargo, ballast and equipment are taken into account by a single point mass connected to the hull structure. its location and mass are calculated by using the center of gravity principle (eqs. (1) and (2)): 112 g.rama , ,.i trg trg i actual actual i trg actual x m x m x m m   , (1) corr trg actualm m m  , (2) where xactual, xtrg and mactual, mtrg are the actual and target coordinates and masses. 5.2 ocean loads the legs of hljv thor are idealized as slender cylinders. the hydrodynamic loads on submerged line elements are calculated by using the morison equation for moving bodies (according to [5]): 0.5 | |w rel w a rel w d rel rel f a a c a a c d v v l      , (3) where ca is the added mass coefficient, cd is the drag coefficient, w is the water density, d is the cylinder diameter, a is the cylinder cross section area, and vrel and arel are relative velocities and accelerations between the structure and the water particles. two different wave theories are used. for regular waves stokes 5th order wave theory is considered and in the case of random waves the linear wave theory with an empirical modification around the free surface in order to account for free surface effects (wheeler stretching see [15]) is used. the equations of motion for a submerged structure can then be expressed as: ( ) 0.5 | ( ) | ( )a w w a w dm m u cu ku av c av c d v u v u          , (4) with a w a pm c a a , (5) where m and ma are the mass and the added mass matrix of the entire structure. matrix c represents the damping and k the stiffness matrix and ap particle acceleration. the buoyancy force and the hydrostatic pressure on each submerged element are calculated as follows (see [5]): 2 0 0; ( ) 4 ab w i i c df g p g z z p l        , (6) 0 0( ) a i ip g z z p    , (7) where pi and pi a are the inner and outer tube pressure, z0 is the z-coordinate of the water surface, ḡ is the acceleration vector and cb is the buoyancy coefficient. ocean loads include the effects of waves, current, drag, and buoyancy. they are taken into account by the definition of an ocean environment in ansys. in the static analysis of the deterministic two stage approach, the sea state during a storm event is represented by regular waves defined by wave height hmax and wave period tp. for each wave angle of attack, the wave travelling through the structure is simulated with 72 timesteps/period. for every considered time-step, the acting wave loads and the total base shear (tbs) are determined by static calculation and subsequently saved in an array. in the random wave time domain dynamic analysis the wave elevation is modeled as linear random superposition of the regular wave. the sea state is defined by increased significant wave height hs (hspr is the significant wave height) and peak period tp using an ../../local%20settings/appdata/local%20settings/program%20files/ansys%20inc/v150/commonfiles/help/en-us/help/ans_thry/thy_anproc2.html#thyeq1systemsnov2001 an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 113 appropriate spectrum (see section 6). the considered significant wave height is calculated according to [4] and the peak period is calculated according to [3] (see eqs. 8, 9 and 10): max /1.86sprh h , (8) [1 0.5 exp( / 25)]s sprh d h    , (9) 3.6p sprt h , (10) where d is the water depth. the created sea states are to be checked within the limits of the theoretical targets for the mean, standard deviation, skewness and kurtosis according to [4]. if all criteria are fulfilled, the current seed number is saved for further use. this allows reproducing the statistically representative sea state at any time. 5.3. wind loads wind forces and pressures on members above the sea surface are considered as steady loads. the wind force component acting normal to the member axis or surface is calculated with the following equation: 2 0.5wi i wi air ref h s wif pa v c c a  , (11) where ch and cs are the height and shape coefficients, vref is the reference wind velocity, awi is the windage area of a structural component and ρair is the air density. the wind loads are automatically calculated by an apdl macro. the macro determines the resulting wind loads for any configuration of the parameterized fe-model for each considered load direction θ. height coefficient ch is calculated according to [8] with an idealized profile model representing the variation of mean wind speed as a function of the height above the still water level. the windage areas are determined with a surface model developed especially for the calculation of wind forces (see fig. 6.). the created windage areas are divided into stripes (see fig. 7, left). the wind loads on each stripe are calculated and combined to resulting force fw(). fig. 6 surface model for the calculation of windage areas 114 g.rama fig. 7 windage area for wind heading of 45° (left); height coefficient profile (right) 5.4. inertia loads for the determination of the inertia loads two methods were implemented: 1. single degree of freedom (sdof) analogy in combination with the amplitude of tbs qa (see eq. 12) 2. random wave time domain dynamic analysis (detailed method) the sdof analogy is permissible for factored (see eq. 12) values of 1 > fac > 0.5. natural period 0 9 exicitation period n fac w t ω t .    , (12) for excitation periods near the resonance range or for critical load cases with small values (0.5-0.7), it is necessary to use a more detailed method like the random wave time domain dynamic analysis. 5.4.1. inertia load set based on a single degree of freedom analogy for a single degree of freedom system the dynamic amplification factor (daf) is: 2 2 2 1 (1 ) (2 ) daf      , (13) for excitation periods in the resonance range a maximum daf (dafdnv see eq. 15) based on a parametric study, described in [8], is used. 0.650.75st   , (14) 1 2 dnv st daf   , (15) inertia load fi (θ) is determined as given in eq. 16 for all considered load directions: max min( ) ( ) ( ) 2 a tbs tbs q      , (16) , ( ) ( 1) ( )i sdof af daf q     , (17) as recommended in [4], this load is applied at the cog of the hull structure. an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 115 5.4.2 inertia load set based on random wave time domain dynamic analysis the dynamic nonlinear response of the structure is not a gaussian process. for this reason the prediction of the most probable maximum extreme (mpme) value of the response during an extreme storm event needs specific probabilistic models. in order to estimate these values, the drag inertia parameter method is used. in the drag inertia parameter method the maximum value of the dynamic response (mpmd) is expressed as a quasi-static part, an inertia part and an appropriate correlation factor (see eq. 18). 2 2 2 2d i s r i smpm mpm mpm mpm mpm     , (18) the correlation factor r is defined as: 2 2 2 2 rd ri rs r ri rs          , (19) where σrs, σri and σrd are the standard deviations of the static, inertia and dynamic response. the mpme value of the dynamic response (mpmerd) is calculated with the mean value of the dynamic response (μrd) and the maximum value of dynamic response: ,rd rd dmpme mpm  , (20) the most probable maximum extreme value of the static response is determined (see eq. 21) using probability factor crs, standard deviation σrs, and the mean value of static response μrs. rsrsrsrs cmpme  , (21) probability factor crs is calculated according to [4] with the:  standard deviation of the static response with a totally drag dominated morison force [σrs(cm=0)]  standard deviation of the static response with a totally inertia dominated morison force [σrs(cd=0)] 2 22 )]0(7.3)0(8[ )]0(7.3[)]0(8[    drsmrs drsmrs rs cc cc c , (22) the inertia mpmeri is then estimated by the difference between the calculated mpme values for tbs and the overturning moment (otm). rsrdri mpmempmempme  , (23) the inertia load set consists of: tbsii mpmef , , (24) otmii mpmem , , (25) inertia force fi is applied at a location in the horizontal cog of the hljv but at an elevation zi to fulfill the resulting overturning moment value. because large deformations are taken into account, the resulting overturning moment is greater than the calculated one (see eq. 116 g.rama 25) due to the displacement of the cog. by using zi elevation the overturning moment will be overestimated. for the dynamic fe-calculations, the simplified fe-model of hljv thor (see section 2) is used. according to [2] a maximum critical damping of 7 % is adopted. hydrodynamic damping (2% 3%) is considered by the application of the morison equation in combination with taking large deformation into account. the structural and soil damping is included by a proportional damping (rayleigh damping). for this purpose, the coefficients α (mass damping) and β (stiffness damping) are calculated assuming a constant critical damping for the frequency range between the first and second natural frequency. 6. apdl routine the above explained approach for the determination of all relevant loads and the creation of both fe-models were implemented in the developed apdl routine. in addition, all listed assessment checks according to their relevant standards were implemented: 1. structural assessment check according to norsok [6] 2. overturning stability check according to [2] 3. holding capacities check [2] fig 8 (left) shows a flowchart of the created routine. the input, as well as the output is summarized as a text file in ascii format (see fig. 8, right). the input parameters can be set in any conventional editor. fig. 8 apdl routine (left); output text file (some parts) analysis 1 (right) an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 117 application example the following table shows the site specific conditions for the investigated case. table 2 considered load scenario – site specific parameters description value displacement 13500 t cog3 35.00 m / 00.00 m / 14.50 m water depth 41.00 m leg penetration 3.00 m air gap 12.50 m initial spring stiffnesses4 kh = 1340 mn; kv = 1580 mn; krot = 19200 mnm the investigation consists of two analyses: 1. detailed: dynamic fe-calculations are carried out to determine the inertia load set. the structure is fixed at the bottom using springs with nonlinear stiffness curves, which describe the soil mechanical properties. 2. conservative: the sdof method is used to calculate the inertia load set, and the structure is fixed at the bottom with a pinned support. all partial safety factors are set according to [2]. in both analyses 3 load directions are considered, 0°, 45° and 90°. ocean loads a maximum surface current speed of 1.1 m/s and a constant current profile are adopted. the considered maximum wave height hmax is equal to 11.5 m. as a conservative approach, the nearest possible wave period to the structure resonance is considered. for the current configuration the wave excitation period will always be greater than the natural period of the structure. as a result, minimum possible wave period tmin based on steepness criteria gives the most critical period regarding dynamic amplification. the limiting steepness value s of 1/7 (see eq. 26), in combination with maximum considered wave height hmax and gravitational acceleration g, allows to calculate wave period tmin (eq. 27). max max 2 min 2 1/ 7 h s gt   , (26) max min max 2 7.18 h t s g s      , (27) the following figure is an illustration of the resulting ocean loads for  equal to 0°. 3 basis ship coordinate system 4 rounded hochtief project values 118 g.rama fig. 9 tbs (left) und fwave() (right); t = 4.29 s and  = 0° the maximum and minimum tbs values and the resulting qa-values are listed in tab 3. table 3 tbs values for all considered load directions direction [°] tbsmin [mn] tbsmax [mn] qa[mn] 0 0.99 2.79 1.89 45 0.38 1.92 1.15 90 2.06 3.85 2.96 wind loads a maximum reference wind velocity of 36 m/s at 10 m above the water surface is adopted. the resulting wind loads and areas are listed in the following table. table 4 calculated wind loads direction [°] fw [mn] aw [m²] 0 1.41 1369 45 2.51 2460 90 2.10 2043 inertia loads for the time domain random wave analysis a random sea is simulated over 1 hour for environmental headings at 45-degree intervals from 0 to 90 degrees. the sea state (see fig. 10, left) is defined by the increased significant wave height hs equal to 6.75 m and a peak period tp of 8.98 s using the jonswap spectrum. the size of the used time step is t = tn/20. the dynamic (blue) and static (purple) response signal of tbs (right) is illustrated in the following figure (for heading 0°). an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 119 fig. 10 statistically representative sea state (3600 s), dynamic and static tbs response signal of the random wave analysis the resulting inertia loads calculated by the time domain random wave analysis (analysis 1) and sdof-method (analysis 2) are listed in the following table. table 5 inertia load set for analyses 1 and 2 analysis dir. [°] fi [mn] method [] boundary conditions [] 1 0.0 3.98 di 3 45.0 1.05 di 3 90.0 1.46 di 3 2 0.0 5.21 sdof 1 45.0 3.18 sdof 1 90.0 8.15 sdof 1 assessment checks results the calculated utilizations are presented in the following table. table 6 assessment checks two stage analysis hmax = 11.5 m ana. no. [] leg pen. [m] air gap [m] hmax [m] t [s] leg strength check[] holding capacity [] overturning stability [] 1 3 12 11.5 7.18 0.52 0.84 2.71 2 3 12 11.5 7.18 1.12 1.17 1.35 differences of utilizations 53.6 28.2 50.2 the calculated used capacity of hljv thor in analysis 1 is less than 1.0. thus the assessment checks satisfy the followed standards requirements (see [2] and [6]). the second analysis only fulfills the requirement of safety against overturning. both analyses are [2] compliant but are of different computational effort. the first analysis consumes a multiple of computational time compared to the second one. the second analysis using simplified approaches can be carried out within hours, the first analysis within days. 120 g.rama 7. conclusion site specific ipas are needed for hljvs as each location offers new conditions. the created apdl routine allows carrying out of these assessments in a fast and consistent way. the different options, regarding boundary conditions and the determination of the inertia load set enable the user to choose the level of detail. ipas of non-critical load cases can be performed resource-efficient with the sdof method. it is easy to implement but it represents only a rough approximation and it does not necessarily lead to conservative results. for critical load cases the implemented drag inertia parameter method offers the possibility to investigate the dynamic behavior according to its irregular nature, in irregular sea state with and achieving a higher accuracy. by taking soil conditions into account (spring support) and performing a time domain random wave analysis (drag inertia parameter method) the used capacities are reduced by 28 % compared to simplified analysis with sdof-method and pinned support. the calculated reserves can be used to increase the environmental restrictions and thus to achieve a higher working capacity of hljv thor. in order to advance the workability of hljv thor parameter studies based on the created script can be performed to estimate the possible range of conditions. the presented assumptions and implemented strategies of load calculations are in accordance with the relevant standards and generally valid. references 1. bundesamt für seeschiffahrt und hydrographie, 2014, bundesfachplan offshore für die deutsche ausschließliche wirtschaftszone der ostsee 2013 und umweltbericht, bsh (7602). 2. iso international standard, 2011, petroleum and natural gas industry – site-specific assessment of mobile offshore units – part1: jack-ups, 19950. 3. det norske veritas, 2010, recommended practice dnv-rp-c205 environmental conditions and, rpc205. 4. sname, 2008, guidelines for site specific assessment of mobile jack-up units, technical & research bulletin 5-5a. 5. ansys, 2013, ansys 15.0 release documentation, 15. 6. norsok standard norway, 2013, n-004 design of steel structures, n-004. 7. hochtief solutions ag, 2014, operating manual – sps n575-rep-41-001-07. 8. det norske veritas, 2012, recommended practice dnv-rp-c104 self-elevating units, rp-c104. 9. sing-kwan lee, deguang yan, baili zhang, chang-wei kang, 2009, jack-up leg hydrodynamic load prediction a comparative study of industry practice with cfd and model test results, proceedings of the nineteenth international offshore and polar engineering conference (isope), osaka, japan. 10. f. van den abeele and j. vande voorde, 2011, coupled eulerian lagrangian approach to model offshore platform movements in strong tidal flows, proceedings of the asme 2011 30th international conference on ocean, offshore and arctic engineering, rotterdam, netherlands. 11. mike j r hoyle; john j stiff; ruper j hunt, 2011, jack-up site specific assessment – the voyage to an, proceedings of the asme 2011 30th international conference on ocean, offshore and arctic engineering,rotterdam, netherlands. 12. houlsby, g.t. & cassidy, m. j., 2002, a plasticity model for the behavior of footing on sand under combined loading, geotechnique, 52(2), pp. 117-129. 13. det norske veritas, 1992, classification notes no. 30.4. 14. yan jenny lu, youl-nan chen, pao-lin tan, yong bai, 2002, prediction of most probable extreme values for jack-up dynamic analysis, marine structures 15-34 elsevier science ltd. 15. j. d. wheeler, 1970, method of calculating forces produced by irregular waves, journal of petroleum technology, 22, pp. 359-367. an automatized in-place analysis of a heavy lift jack-up vessel under survival conditions 121 automatizovana analiza na lokaciji brodske dizalice za teške terete u uslovima opstanka brodovi-dizalice za prevoz teških tereta (hljv) koriste se za instaliranje komponenti velikih offshore farmi vetrova. u radu prikazujemo sistematsku fe analizu za hljv thor (vlasništvo hochtief infrastructure gmbh) pod ekstremnim vremenskim uslovima. model i analiza parametričnih konačnih elemeneta su razvijeni korišćenjem programskog okruženja ansys®-apdl. analiza sadrži statičke i dinamičke nelinearne fe proračune izvršene prema značajnim standardima (iso 19905) za lokacijske analize brodova-dizalica. pored strategije apstrakcije modela, date su i smernice za određenje značajnih tereta. za izračunavanje dinamičkih opterećenja, korišćena je analogija sa jednim stepenom slobode (sdof) kao i dinamički nelinearni fe proračuni. kao rezultat detaljnog određenja dinamičkih opterećenja i razmatranja karakteristika tla opružnim elementima, korišćeni kapaciteti su se mogli smanjiti za 28%. time smo obezbedili značajno poboljšanje sredinskih ograničenja za hljv thor za razmatrani scenario opterećenja. ključne reči: brodovi-dizalice za prevoz teških tereta, procene za specifičnu lokaciju, metoda parametra vuče/inercije, thor, offshore industrija facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 657 680 https://doi.org/10.22190/fume201009045s © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fourth-order strain gradient bar-substrate model with nonlocal and surface effects for the analysis of nanowires embedded in substrate media worathep sae-long1, suchart limkatanyu2, piti sukontasukkul3, nattapong damrongwiriyanupap1, jaroon rungamornrat4, woraphot prachasaree2 1civil engineering program, school of engineering, university of phayao, thailand 2dept. of civil eng., faculty of engineering, prince of songkla university, thailand 3construction and building materials research center, department of civil engineering, king mongkut’s university of technology north bangkok, thailand 4applied mechanics and structures research unit, department of civil engineering, faculty of engineering, chulalongkorn university, bangkok, thailand abstract. this paper presents a new analytical bar-substrate model for the analysis of an isotropic and homogeneous nanowire embedded in an elastic substrate. a fourth-order nonlocal strain gradient model based on a thermodynamic approach is employed to represent the small-scale effect (nonlocal effect) while the gurtin-murdoch continuum model based on the surface elastic theory is used to account for the size-dependent effect (surface energy effect). the proposed model is derived from the virtual displacement principle, leading to the governing differential equations and its associated natural boundary conditions. the analytical solutions of the sixth-order governing differential equation for the nanowire-substrate element are provided, and were employed in numerical simulations. two numerical simulations are used to demonstrate the performance and to investigate the characteristics of the fourth-order nonlocal strain gradient model on nanowire responses, when compared to the classical model and the second-order nonlocal strain gradient model. the first simulation investigates the influences of nonlocal and surface effects on the responses of a nanowire embedded in an elastic substrate, while the second simulation study assessed the sensitivity of system stiffness on parameters in the nanowire-substrate model. key words: nanobeam-substrate model, displacement-based formulation, nonlocal thermodynamic approach, fourth-order nonlocal strain gradient model, nonlocal strain gradient received october 09, 2020 / accepted december 07, 2020 corresponding author: suchart limkatanyu department of civil engineering, faculty of engineering, prince of songkla university, songkhla, 90112, thailand e-mail: suchart.l@psu.ac.th 658 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. 1. introduction with the rapid development of technology and science, nanotechnology plays a greater role in several applications and has led to a huge effort to study nano-sized structures in the engineering fields. however, the characteristics of these nanoscale structures differ from the large-scale structures, and size or scale affects those behaviors [1-5]. furthermore, the development of numerical models for replacing the results from the experiments, which are expensive and require specific equipment for testing or observation, is a challenging problem to the structural analysis community, motivating also this current study. the available numerical models to characterize the mechanical responses of nanostructures in the structural engineering field fall into two categories, namely, (i) atomistic models; and (ii) continuummechanics models. the atomistic models are established at molecular or atomic level based on several techniques, such as molecular dynamics simulations [6-7], density functional theory [8-9], etc. these models can provide comprehensive details and are beneficial in the analysis of small and simple nanostructures [10]. however, the atomistic simulations are very cumbersome and have high computational costs when applied to complex or large nanostructure systems. for large-scale nanostructures, the continuum-mechanics models are an attractive alternative to simulate the mechanical characteristics of the nanostructures, with good applicability and simplicity. although the classical continuum models do not originally incorporate size-scale effect, this model type can be applied with non-classical continuum theories, and have been named “nonlocal” continuum models [11-12]. according to the nonlocal continuum models, several non-classical continuum theories are employed to account for the size-scale or the so-called “nonlocal” effect, which introduces additional material length-scale parameters in nonlocal elasticity theory [13-14], couple stress theory [15], modified couple stress theory [16-17], strain gradient theory [18-19], or in modified strain gradient [20-21]. in the last twenty years, the nonlocal elasticity theory of eringen [13-14] was the most popular for addressing sizescale effect, and was widely applied in several studies of nanostructures [22-24]. however, there are paradoxical results (the vanishing of nonlocal effect) in the modeled responses of nanostructures, such as for a cantilever beam subjected to an end load [2526]. romano et al. [27] reported that this paradox is caused by the incompatibility of equilibrium equations and constitutive boundary terms. to avoid the ill-posedness of eringen nonlocal elasticity theory, peddieson et al. [25] suggested the nonlocal strain gradient theory as expanded from the general integral constitutive equation of eringen nonlocal elasticity theory [14]. when only the first two terms were considered, the general form of the stress-strain relation based on the nonlocal strain gradient theory was similar to the form of the simplified strain-gradient elasticity model as proposed altan and aifantis [28]. this led to the so-called “second-order” nonlocal strain gradient model [29-30]. the performance of this nonlocal constitutive model has been investigated by several researchers [23, 29, 31-37]. for example, zaera et al. [31] studied the inconsistency of the strain gradient model with the boundary terms. khodabakhshi and reddy [23] presented the governing equations and boundary terms of the strain gradient beam model based on von karman nonlinearity in the euler-bernoulli beam theory, timoshenko beam theory, and thirdorder reddy beam theory. kong et al. [32] used the strain gradient theory to investigate static and dynamic characteristics of euler-bernoulli beams. barretta and marotti de sciarra [33] proposed a strain gradient constitutive model based on a consistent thermodynamic approach for the analysis of carbon nanotubes (cnts). this concept was later applied to fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 659 establish the modified constitutive laws for euler-bernoulli nanobeams [34-35], timoshenko nanobeams [36], and a nanowire [37]. narendar and gopalakrishnan [29] studied an ultrasonic wave dispersion characteristics of a nanorod based on nonlocal strain gradient model. these studies have confirmed the superiority of the strain gradient model. in this work, the nonlocal strain gradient model based on the thermodynamic framework is chosen to improve the proposed model and to eliminate the paradoxical results. the size-dependent effect or the so-called “surface energy” is caused by altered energy of the atoms at a free surface. this effect often plays a greater role in the mechanical responses of small-scale structures, when the surface-to-volume ratio is large, as found in several publications [38-41]. to incorporate the surface energy within the continuum model, gurtin and murdoch [42-43] proposed a surface elasticity theory and established a surface continuum model for isotropic materials based on zero thickness of the surface layer. to confront the surface effect, a lot of literature [24, 44-48] has applied the gurtin and murdoch surface model to nanostructures. for example, liu and rajapakse [44] studied the static and dynamic characteristics of elastic nanobeams under different load conditions. fu et al. [45] later extended the work of liu and rajapakse [44] to cover the nonlinear analysis of both static and dynamic nanobeam problems. limkatanyu et al. [24] investigated the flexural responses of nanobeams based on the eringen nonlocal elasticity theory, while fu et al. [46] studied these responses based on the strain gradient theory. furthermore, lim and he [47] examined the buckling behavior of a micro/nanofilm in relation to the nonlinear buckling characteristics of cylindrical nanoshells based on the classical shell theory proposed by sahmani et al. [48]. in this current work, the gurtin-murdoch model [42-43] is applied to represent the surface surrounding effect within a uniaxial member in the nanoscale. among the several continuum models for the analysis of a nanowire embedded in a substrate, many approaches were used to establish the nanobar-substrate models. for example, limkatanyu et al. [49-50] proposed nonlocal bar-substrate elements based on the nonlocal elasticity theory of eringen [13-14] and the surface continuum model of gurtin and murdoch [42-43] for both displacement-based formulation [49] and forcebased formulation [50]. however, these models are ill-posed for the nanowire system under uniform load when the substrate modulus reaches zero [37]. later, sae-long et al. [37] extended the strain gradient bar model based on the thermodynamic approach of barretta and marotti de sciarra [33] to the bar-substrate model. this model can overcome the ill-posedness in a series of eringen nonlocal models, but this model is categorized into the second-order nonlocal strain gradient model when only the first two terms of the nonlocal constitutive relation are considered. in light of the need to upgrade the bar model for nano-scale problems, the main objective of this work was to develop the bar-substrate model with inclusion of higherorder terms in the nonlocal strain gradient model [25, 29] and the surface energy effect. the first three terms of the nonlocal strain gradient constitutive relation are considered, making this a “fourth-order” nonlocal strain gradient model [29-30]. this kind of model has been confirmed the superiority of performance to represent the complicated characteristics [51]. thus, this work takes an interest in the influence of adding the third term in the nonlocal strain gradient model on the responses of the nanowire-substrate system, when compared to the second-order nonlocal strain gradient nanowire-substrate model [37]. the so-called “exact” displacement shape functions of the proposed sixthorder governing differential equilibrium equation given in this work are adapted from the 660 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. analytical solutions of the beam model on kerr-type foundation, proposed by morfidis [52] and avramidis and morfidis [53]. to the best knowledge of the authors, this is the first time that the fourth-order nonlocal strain gradient model is applied in a displacementbased bar element to study the characteristics of a nanowire embedded in an elastic substrate. the contents of this paper are organized as follows. sections 2 and 3 provide the brief discussions of the nonlocal strain gradient model and the surface elastic theory that are employed to represent nonlocal and surface energy effects in this study, respectively. then, the model formulation is presented in section 4. it includes the governing differential equations such as the compatibility equations, constitutive relations, and equilibrium equation. based on the virtual displacement principle, the governing differential equilibrium equation, its associated natural boundary conditions, and analytical solutions of the proposed model are also presented in this section. next, section 5 shows two numerical simulations of the characteristics of a nanowire embedded in a substrate. the first simulation is used to investigate the higher-order term in the fourth-order strain gradient model, with nonlocal and surface energy effects on the mechanical responses of the nanowire-substrate system, while the second simulation examines the sensitivity of system stiffness on nanowire and substrate properties. finally, the last section gives the conclusions from this study. all symbolic calculations in this work were performed with the computer software mathematica [54]. 2. nonlocal strain gradient model the nonlocal strain gradient model [25, 29] was originally derived from the nonlocal elasticity theory [14], which assumed the stress at a reference point x is a function of the strain field at every point in the body. the original constitutive relation of the nonlocal elasticity theory was represented by the weighted averages of the contribution of the strain tensor at all points in the body to the stress tensor at the given point. this leads to the mathematical problems to get the solution for nanostructure continuum models. to overcome this limitation, eringen [14] presented the kernel function in the form of a modified bessel function for applying in the differential constitutive equation. however, the integral constitutive relation as suggested by eringen [14] can be converted exactly into a corresponding differential form for some kernels [25, 29]. later, peddieson et al. [25] presented an alternative way to avoid the mathematical problem in the nonlocal elasticity theory by using the nonlocal strain gradient model. the constitutive relation of the nonlocal strain gradient model comes from the integral constitutive relation of the nonlocal elasticity theory [14] when the material-length scale parameter is very less than 1 (e0a <<1). thus, the expansion of the elastic energy w based on the following assumption for a uniaxial stress problem is obtained by only considering the first three terms as: 2 1 2 local term 2 2 4 2 0 1 0 2 nonlocal terms 1 [ ( ), ( ), ( )] ( ) 2 1 1 ( ) ( ( )) ( ) ( ( )) 2 2 xx xx xx xx xx w x x x e x e e a x e e a x = + +       (1) fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 661 where exx is the young modulus; e0a is the nonlocal-scale parameter; εxx(x) is the axial strain; ε1(x) and ε2(x) are the axial strain gradients corresponding to the second and the third terms in the nonlocal strain gradient model, respectively. it needs to be emphasized that the first term on the right-hand side of eq. (1) represents the local (conventional) strain energy (as in large-scale structural analysis), while the second and third terms represent the nonlocal strain energy (small-size-scale effect) associated with the second and the third terms of the nonlocal strain gradient model, respectively. the elastic energy time rate of the energy functional w of eq. (1) is: 1 2 1 2 0 1 1 2 2 1 2 [ ( ), ( ), ( )]xx xx xx xx w w w w x x x    = + + = + +                   (2) where a dot on top of a symbol (·) denotes the time derivative; σ0 is the local (conventional) normal stress; σ1 and σ2 are higher-order normal stresses associated with the second and the third terms of strain gradient, respectively. these normal stresses can be defined as: 0 xx xx xx w e  = =     (3) 2 1 0 1 1 ( ) xx w e a e  = =     (4) 4 2 0 2 2 ( ) xx w e a e  = =     (5) it is clear from eqs. (3) to (5) that the local normal stress σ0 is the conjugate-work pair of the axial strain εxx, as shown in eq. (3), while the higher-order normal stresses σ1 and σ2 are the conjugate-work pairs of the axial strain gradients ε1 and ε2 as demonstrated in eqs. (4) and (5), respectively. in order to fulfill the thermodynamics, the elastic energy time rate of the energy functional w of eq. (2) must be satisfied as following condition [33]: xx xx l a l a da dx wda dx      −            (6) where σxx is the nonlocal normal stress. by substituting eqs. (3) to (5) into eq. (6), the relation is: 0 1 1 2 2 l a local term second-order strain gradient term fourth-order strain gradient term σ da dx 0xx xx xx l a l a l a da dx da dx da dx         − − − =                              (7) 662 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. to simplify the variational form in eq. (7) for analysis of a uniaxial problem, it can apply terms in the axial force nxx(x): 0 1 1 2 2 local term second-order strain gradient term fourth-order strain gradient term ( ) ( ) ( ) ( ) 0xx xx xx l l l l n x dx n x dx n x dx n x dx− − − =       (8) where 0 0( ) a n x da=  are the local axial forces; 1 1( ) a n x da=  and 2 2( ) a n x da=  are the higher-order axial forces associated with the second and the third terms of the nonlocal strain gradient model within the thermodynamic approach, respectively. 3. surface elasticity theory the size-dependent effect or the so-called “surface energy” effect stems from different energy density at the surfaces of the bulk material, becoming significant when the surfaceto-volume ratio is large. this is found in various experiments and in atomistic simulations of micro/nano-scale problems [38, 41]. to account for the effect of surfaces and interfaces, gurtin and murdoch [42-43] introduced a non-classical continuum model of surface elasticity under the hypothesis of a zero-thickness surface layer and perfect bonding between a solid core and an outer surface shell, as shown in fig 1. for the analysis of the uniaxial responses in this study, the relation between the surface stress τxx and surface strain εsur (constitutive relation of surface elasticity) along the longitudinal direction are given by gurtin and murdoch [42-43] as: 0xx sur sure− =   (9) where esur and τ0 are, respectively, the surface elastic modulus and residual surface stress when the values of both of these are obtained from experiments or from atomistic simulations [38]. fig. 1 nanowire embedded in substrate with a circular or a rectangular wire cross-section [37] bar bulk bar bulk bar bulk x rectangular sectioncircular section outer surface shell outer surface shell nanowire z z z y y d b h fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 663 4. formulation 4.1. compatibility equations based on the small strain assumption, the sectional axial strain of the nanowire εxx(x), and the sectional axial strain gradients ε1(x) and ε2(x) are related to the axial displacement of the nanowire ux(x) through the compatibility conditions: ( ) ( ) x xx u x x x  =   (10) 2 1 2 ( ) ( ) ( ) xx xx u x x x x   = =     (11) 2 3 2 2 3 ( ) ( ) ( ) xx xx u x x x x   = =     (12) under the hypothesis of perfect bonding between the bulk of nanowire and the surface surrounding outer surface shell [42-43], the surface displacement usur(x) is perfectly compatible with the axial displacement of the nanowire (usur(x) = ux(x)). therefore, the compatibility relation of the surface surroundings involving surface strain εsur(x), surface displacement usur(x), and axial displacement of the nanowire ux(x) is as follows: ( ) ( ) ( ) sur x sur u x u x x x x   = =    (13) like the assumption regarding surface surroundings, the interaction between nanowire and substrate is assumed to be perfect based on the kinematic assumption of the winkler foundation model [55]. thus, the compatibility relation between the substrate deformation δsub(x) and the nanowire axial displacement ux(x) is: ( ) ( )sub xx u x = (14) 4.2. differential equation and its boundary conditions: the principle of virtual displacement in the displacement-based formulation, the principle of virtual displacement is employed to establish the governing differential equilibrium equation and its boundary conditions for the nanowire element embedded in the substrate, as shown in fig. 2. the features, such as the nanowire-substrate interaction, the nonlocality, and the surface energy effect, are incorporated to the model in this procedure. 664 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. fig. 2 ideal nanobar element embedded in substrate a general form of the total virtual work expression is: int ext 0w w w  = + = (15) where δw is the total virtual work of the bar-substrate element system; δwint is the internal virtual work; and δwext is the external virtual work. the internal virtual work δwint and the external virtual work δwext of eq. (15) can be expressed as: int 0 nanowire term surface energy term substrate medium term ( ) ( ) ( ( ) ) ( ) ( ) ( ) xx xx xx sur l a l sub sub l w x da x dx x d x dx d x x dx      = + −         +              (16) ext end boundary terms external load term ( ) ( ) t x x l w p x u x dx= − − u p   (17) where dsub(x) is the substrate interactive force and the conjugate-work pair of the substrate deformation δsub(x); px(x) is the axial distributed load; u = {u1 u2 u3 u4 u5 u6}t is the displacement vector that contains the end displacements of the nanowire element; and p = {p1 p2 p3 p4 p5 p6}t is the force vector that contains the end forces of the nanowire element. considering the variational form of eqs. (7) and (8), the internal virtual works in eq. (16) becomes: local term second-order strain gradient term forth-order strain gradient term int 0 1 1 2 2 nanowire terms ( ) ( ) ( ) ( ) ( ) ( ) ( ) xx l l l sur sur w n x x dx n x x dx n x x dx n x = + + +        surface energy term substrate medium term ( ) ( ) ( )sub sub l l x dx d x x dx+    (18) where 0( ) ( ( ) )sur xxn x x d  = −    represents the equivalent axial force from the surface stresses. by substituting the compatibility equations of eqs. (10) to (14) into the internal fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 665 virtual work δwint of eq. (18) and then combining with the external virtual work δwext of eq. (17), the total virtual work of nanowire-substrate element system of eq. (15) is: 2 3 0 1 22 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 x x x l l l x sur sub x x x l l l t u x u x u x w n x dx n x dx n x dx x x x u x n x dx d x u x dx p x u x dx x    = + +     + + −  − =       u p         (19) to move the all differential operators of the axial displacement ux(x) into the sectional force resultants, integration by parts is applied to eq. (19): 3 2 02 1 3 2 2 2 1 02 0 2 2 1 22 0 0 ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) 0 sur x sub x l x l x sur x x l x l x x x x t n x n xn x n x w u x d x p x dx x xx x n x n x u x n x n x xx u x u xn x n x n x x x x = = = = = =     = − + − − + −         + − + +        + − + +       − =  u p       (20) following the cartesian sign convention, eq. (20) can be expressed as: 3 2 02 1 3 2 2 2 1 1 0 12 0 2 2 1 2 3 2 0 3 0 2 2 4 ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) sur x sub x l sur x x x n x n xn x n x w u x d x p x dx x xx x n x n x u n x n x p xx n x u n x p u n x p x n u = = =     = − + − − + −          − − + + +           − − + + − +         +       1 0 42 2 5 1 5 6 2 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) 0 sur x l x l x l t x n x n x n x p xx n x u n x p u n x p x = = =    − + + −           + − + − + −        − =u p    (21) considering the arbitrary variable of δux(x) in eq. (21), the governing differential equilibrium equation for the proposed nanowire element embedded in elastic substrate is: 3 2 02 1 3 2 ( ) ( )( ) ( ) ( ) ( ) 0sur sub x n x n xn x n x d x p x x xx x    − + − − + − =    (22) 666 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. from eq. (21), the boundary force conditions at the ends of an element are also obtained by considering arbitrary δu: 2 2 1 1 02 0 2 2 1 3 2 0 0 2 2 1 4 02 2 5 1 6 2 ( ) ( ) ( ) ( ) ; ( ) ( ) ; ( ) ; ( ) ( ) ( ) ( ) ; ( ) ( ) ; ( ) sur x x x sur x l x l x l n x n x p n x n x xx n x p n x p n x x n x n x p n x n x xx n x p n x p n x x = = = = = =    = − − + +       = − − + = −        = − + +       = − + =     (23) 4.3. force-deformation relations the force-deformation relations of the axial resultants for nanowire are established by inserting the compatibility equations of eqs. (10) to (12) into the relations in eqs. (3) to (5) and considering stress in term of axial force, resulting in the following relations: 0 ( ) ( ) x xx u x n x e a x  =  (24) 2 2 1 0 2 ( ) ( ) ( ) x xx u x n x e a e a x  =  (25) 3 4 2 0 3 ( ) ( ) ( ) x xx u x n x e a e a x  =  (26) where a a da=  represents the cross-section area. the force-deformation relation of the axial force induced by the surface stress can be found by substituting the compatibility condition eq. (13) into eq. (9), and considering this in terms of the axial resultant. thus, the force-deformation relation of surface surroundings is ( ) ( ) x sur sur u x n x e x  =   (27) where d   =  is the section perimeter of nanowire. similarly, the relation between substrate interactive force dsub(x) and the substrate deformation δsub(x) can be expressed by enforcing the compatibility relation of eq. (14) as: ( ) ( ) ( )sub s sub s xd x k x k u x=  = (28) where ks represents the elastic modulus of the substrate medium. the compatibility relations, equilibrium equation, and force-deformation relations of the proposed nanowire element fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 667 embedded in elastic substrate media as demonstrated above, can be conveniently summarized for the whole model formulation in a tonti’s diagram [56], as shown fig. 3. fig. 3 tonti’s diagram for the fourth-order nonlocal strain gradient model of nanowire embedded in elastic substrate 4.4. “exact” displacement shape functions according to the force-deformation relations in eqs. (24) to (28), the governing differential equilibrium equation in eq. (22) can be rewritten in terms of the axial displacement ux(x) as: 6 4 2 4 2 0 06 4 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x x x xx xx xx eff s x x u x u x u x e a e a e a e a e a k u x p x x x x    − + − + =    (29) where (exxa)eff = exxa + esurγ represents the effective nanowire stiffness accounting for the contribution of the solid core and the outer surface. the effective nanowire stiffnesses (exxa)eff for circular and rectangular sections are given by juntarasaid et al. [57] as: 2( ) for circular section 4 ( ) 2 ( ) for rectangular section xx eff xx sur xx eff xx sur e a e d e d e a e bh e b h = + = + +   (30) prescribed end displacements ( )xp x compatibility ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 1 0 3 2 0 sur sub x n x n x n x x x x n x d x p x x    − + −     − + − =  equilibrium ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 2 3 2 2 3 ; ; x xx x xx xx x sur x sur sub x u x x u x x x x x x x u x x x x u x u x x x u x x x    = = =      = =     = =  =         material constitutive ( ) ( ) ( ) ( ) ( ) 1 2, , , xx sur sub x x x x x     ( )xu x natural bcs prescribed end forces essential bcs ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0 2 1 0 1 4 2 0 2 xx xx sur sur sur xx xx sub s sub n x e a x n x e x n x e a e a x n x e a e a x d x k x = =  = = =      ( ) ( ) ( ) ( ) ( ) 0 1 2, , ,sur sub n x n x n x n x d x 668 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. where d is the diameter of circular nanowire cross-section; and b and h are, respectively, the width and height of a rectangular section. it needs to be pointed out that the sixth-order governing differential equilibrium equation in eq. (29) represents the state equilibrium for the fourth-order strain gradient model of the nanowire embedded in elastic substrate, with inclusion of the surface energy effect. when the third term of the nonlocal strain gradient model in eq. (22) vanishes (n2(x) = 0), the governing differential equilibrium equation in eq. (29) will be reduced to a second-order nonlocal strain gradient model proposed by sea-long et al. [37]. furthermore, the governing differential equilibrium equation in eq. (29) can be transformed into the local bar-elastic medium system when the nonlocal-scale parameter and the surface energy effect are ignored (e0a = esur = 0). the general form of differential equation in eq. (29) is found in several engineering applications. for example, in a beam on elastic threeparameter foundation as reported by morfidis [52] and by avramidis and morfidis [53] etc. the natural boundary conditions (end boundary force terms) in terms of axial displacement ux(x) can be expressed by the force-deformation relations in eqs. (24) to (27) in eq. (23), giving 5 3 4 2 1 0 05 3 local termfourth-order strain gradient term second-order strain gradient term 0 4 4 2 0 4 four ( ) ( ) ( ) ( ) ( ) ( ) ; ( ) ( ) x x x xx xx xx eff x x xx u x u x u x p e a e a e a e a e a xx x u x p e a e a x =       = − − +        = − −  2 3 2 4 0 3 02 3 th-order strain gradient term second-order strain gradient term fourth-order strain gradient term0 0 4 4 0 ( ) ( ) ( ) ; ( ) ; ( ) x x xx xx x x xx u x u x e a e a p e a e a x x p e a e a = =          + = −             = 5 3 2 05 3 local termfourth-order strain gradient term second-order strain gradient term 4 4 5 0 4 fourth-order strain g ( ) ( ) ( ) ( ) ( ) ; ( ) ( ) x x x xx xx eff x l x xx u x u x u x e a e a e a xx x u x p e a e a x =       − +        = −  2 3 2 4 0 6 02 3 radient term second-order strain gradient term fourth-order strain gradient term ( ) ( ) ( ) ; ( )x x xx xx x l x l u x u x e a e a p e a e a x x = =          + =             (31) it can be observed from the boundary conditions of eq. (31) that the end boundary forces associated to the local system, p1 and p4, are not resisted only by the local axial force but also by the higher-order axial forces from the gradient terms. furthermore, the statically equivalent axial force due to the higher-order strain gradient terms n1(0), n2(0), n1(l), and n2(l) must satisfy kinematics of the nanowire element at the end boundaries, thus resulting zeroes for – (∂n2(0) / ∂x) + n1(0), n2(0), – (∂n2(l) / ∂x) + n1(l), and n2(l). in the current work, the analytical solutions of the sixth-order governing differential equilibrium equation in eq. (29) are adapted and applied from the general solutions of the beam problem on the kerr-type foundation, proposed by morfidis [52] and avramidis and morfidis [53]. this leads to the so-called “exact” displacement shape functions: 1 1 2 2 3 3 4 4 5 5 6 6( ) ( ) ( ) ( ) ( ) ( ) ( )x c cu x x x x x x c xc c c= + + + + +      (32) fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 669 where ψ1(x) to ψ6(x) are the functions associated with solutions of the sixth-order differential equation; and c1 to c6 are the generalized coordinates corresponding to the geometric boundary conditions. more details of the exact displacement shape functions are given in appendix a. 5. numerical simulations in this section, two numerical simulations are demonstrated and assessed for the mechanical characteristics of nanowire embedded in an elastic substrate. the first simulation investigates the mechanical responses of that nanowire to illustrate the influences of higher-order terms in the strain gradient, nonlocal, and surface energy effects on the nanowire responses. the second simulation examines the influences of system parameters on the effective young’s modulus of the nanowire system, and the element stiffness at the end of the nanowire model. 5.1. simulation i: nonlocal strain gradient models vs local model in the analysis of a nanowire embedded in an elastic substrate a free-free silver nanowire embedded in the elastic substrate under the tensile force pend of 100,000 nn at the right-hand side of the member end, as shown in fig. 4, is studied in this simulation. fig. 4 simulation i: system with a free-free nanowire embedded in an elastic substrate for response analysis the material and geometric properties of the nanowire come from the publication of juntarasaid et al. [57], with 76 gpa bulk modulus of the silver nanowire exx; and 50 nm diameter d. the length of the nanowire l is assumed to be 1,000 nm. the nonlocal-scale parameter e0a is set at 200 nm. this value has been found in several studies on characteristics of nanostructures [37, 49-50]. the surface energy choice follows the publication of he and lilley [58] with elastic surface modulus esur = 1.22 nn/nm, while the residual surface stress 1,000l nm= elastic substrate medium elastic substrate medium 100,000endp nn= nanowire (silver): nonlocal-scale parameter: surface layer: substrate modulus: 76xxe gpa= 0 200e a nm= 01.22 / ; 0 /sure nn nm nn nm= = nanowire section 214.92 /sk nn nm= 50d nm= bar bulk 670 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. is assumed zero (τ0 = 0 nn/nm). these values correspond to the crystallographic direction [001] as introduced by shenoy [59]. the characteristics of a polymer as the surrounding medium are represented by the elastic substrate modulus ks = 14.92 nn/nm2, given by liew et al. [60]. in order to assess how nonlocal-scale effect, surface energy, and the higher-order terms of the strain gradient affect the nanowire’s responses, three analytical models of such responses are compared. these are the proposed model; the second-order nonlocal strain gradient nanowire-substrate model of sae-long et al. [37]; and the local bar model embedded in elastic substrate. it needs to be noted that the so-called “local” model does not consider the size-scale effect. furthermore, each analytical model gives two systems for the discussion of surface energy effect, namely the system with surface effect and the system without surface effect. thus, there are six cases to analyze by simulations. fig. 5 superimposes the axial displacement along a nanowire’s length obtained from the six analytical cases. this diagram indicates that the influence of the nonlocal effect on nanowire displacement is more pronounced than the surface energy effect, especially at the end of the nanowire under tensile load. the nonlocality makes the system distinctly stiffer, while the additional term in the fourth-order nonlocal strain gradient makes the element slightly stiffer, as clearly observed when compared with the local model or with the second-order strain gradient bar model. the surface energy effect slightly affected system stiffness by stiffening it when nonlocality is included. furthermore, it can be observed that the substrate modulus is in effect nullified when nonlocality is ignored. this matches prior observations in several bar-substrate models [37, 49-50]. similar to the observations in fig. 5, the distribution of axial strain in fig. 6 and the distribution of axial force in fig. 7 are affected by the nonlocality, the higher-order terms in strain gradient, and the surface energy effects. it can see in fig. 6 that the axial strain of the bar-substrate system distinctly differs between the local model and a set of nonlocal strain gradient models, especially at the end of the nanowire under tensile load. the axial strain from the nonlocal strain gradient models tends to spread smoothly from the end load, while the axial strain associated with the local model increases abruptly. the maximum axial strains differ between the local model and the proposed model by approximately about 3.10 times, while the maxima between the second-order model and the proposed fourth-order model differ approximately about 3.28 %. the axial force distribution in fig. 7 clearly shows that the additional terms in higher-order nonlocal strain gradient models influence the axial-force action. there is a change in the axial force in compression (negative value) for both the second-order strain gradient model and the proposed fourth-order strain gradient. these results are associated with the higher-order governing differential equation of each model, seen in eq. (29), and correspond to the static indeterminacy of the nanowire-substrate system. fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 671 fig. 5 nanowire axial displacement versus distance along the nanowire fig. 6 nanowire axial strain versus distance along the nanowire -10 0 10 20 30 40 50 60 70 80 0 200 400 600 800 1000 n a n o w ir e a x ia l d is p la ce m en t (n m ) distance along the nanowire (nm) proposed bar model with surface effect proposed bar model without surface effect second order strain gradient nonlocal bar model with surface effect second order strain gradient nonlocal bar model without surface effect local bar model with surface effect local bar model without surface effect -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 200 400 600 800 1000 n a n o w ir e a x ia l st ra in distance along the nanowire (nm) proposed bar model with surface effect proposed bar model without surface effect second order strain gradient nonlocal bar model with surface effect second order strain gradient nonlocal bar model without surface effect local bar model with surface effect local bar model without surface effect 672 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. fig. 7 nanowire axial force versus distance along the nanowire fig. 8 illustrates the distribution of substrate interaction force along the nanowire for the six cases. the shapes in this diagram are similar to the axial displacement in fig. 5. this result does not surprise because the substrate interactive force is directly obtained from the axial displacement based on the hypothesis of the winkler-foundation model [55]. therefore, the additional higher-order terms in the fourth-order nonlocal strain gradient model, nonlocal, and surface energy effects on this response are similar to the axial displacement in fig. 5. fig. 8 substrate interaction force versus distance along the nanowire -20000 0 20000 40000 60000 80000 100000 120000 0 200 400 600 800 1000 n a n o w ir e a x ia l fo rc e ( n n ) distance along the nanowire (nm) proposed bar model with surface effect proposed bar model without surface effect second order strain gradient nonlocal bar model with surface effect second order strain gradient nonlocal bar model without surface effect local bar model with surface effect local bar model without surface effect -200 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 s u b st r a te i n te r a c ti v e f o rc e ( n n /n m ) distance along the nanowire (nm) proposed bar model with surface effect proposed bar model without surface effect second order strain gradient nonlocal bar model with surface effect second order strain gradient nonlocal bar model without surface effect local bar model with surface effect local bar model without surface effect fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 673 5.2. simulation ii: parametric study this simulation study investigated the sensitivity of the effective young’s modulus on nanowire and substrate parameters of the fourth-order and second-order nonlocal strain gradient models for the problem of a nanowire embedded in the substrate medium. the nanowire-substrate system under the tensile force pend of 10 nn at the free-end right-hand side member, as shown in fig. 9, is employed to study herein. fig. 9 simulation ii: system with a free-free nanowire embedded in an elastic substrate for parametric study the material properties of nanowire and its surface energy come from the publication of assadi et al. [61], with 70 gpa bulk modulus of the anodic alumina nanowire exx; and 5.1882 nn/nm elastic surface modulus esur. the length of the nanowire l is fixed with 1,000 nm while the residual surface stress τ0 is assumed zero. to investigate the sensitivity of the effective young’s modulus, the variation of the nanowire and substrate parameters include the nonlocal-scale parameter e0a with 0 – 200 nm, the nanowire diameter d with 5 – 30 nm, and the substrate modulus ks with 10 – 100 nn/nm2. the method to examine the influence of the size effects on the effective young’s modulus eeff was adapted from he and lilley [58] and jiang and yan [62], and has been widely used [37, 49]. in the first step, the end displacement uend including the nonlocal and surface effects was determined from the proposed model and the second-order nonlocal strain gradient model of sae-long et al. [37], as shown on the left-hand side of eq. (33). then, the effective young’s modulus eeff is solved from eq. (33). coth s end eff end s eff k p l e a u k e a         = (33) 1,000l nm= elastic substrate medium elastic substrate medium 10endp nn= d bar bulk nanowire section nanowire (anodic alumina): nonlocal-scale parameter: surface layer: substrate modulus: nanowire diameter: 70xxe gpa= 0 0, 50, 100, 150, 200e a nm= 05.1882 / ; 0 /sure nn nm nn nm= = 210, 25, 50, 100 /sk nn nm= 5, 10, 15, 20, 25, 30d nm= 674 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. it needs to be pointed out that the right-hand side of eq. (33) is the end displacement based on the classical bar model on elastic foundation [63] (without nonlocal and surface effects). therefore, the meaning of the effective young’s modulus eeff in eq. (33) is the value of young’s modulus that makes the displacement in the local nanowire-substrate system equal to the displacement in the nonlocal nanowire-substrate system with the surface effect. from the parametric study, it is clear that in all the cases shown in fig. 10 the effective young’s modulus eeff increased with the nonlocal-scale parameter e0a and decreased with the nanowire diameter d, especially with larger magnitudes of the substrate modulus ks. the nanowire-substrate interaction with nonlocal and surface energy effects plays an important role in scale effects on the effective young’s modulus eeff. furthermore, it can be observed that the variation of the effective young’s modulus eeff due to the nanowire diameter d is independent on the substrate modulus ks, when the nonlocal effect is not included, as shown in eq. (29). finally, the added higher-order term (fourth-order nonlocal strain gradient term) of the proposed model makes the effective young’s modulus eeff increase when compared to the second-order nonlocal strain gradient model [37]. in order to investigate the effects of the added higher-order term in the nonlocal strain gradient model on the end stiffness of the nanowire-substrate system, this study defined the end stiffness as: andend end str c end end p p k k u u = = (34) where kstr and kc are the end stiffnesses corresponding to the analytical axial displacement uend as obtained from the nonlocal strain gradient models and the local model, respectively. this approach modifies the method to determine the contact stiffness from jiang and yan [62], khajeansari et al. [64], and limkatanyu et al. [49]. fig. 11 presents the normalized end stiffness kstr / kc with varied nanowire diameter d, nonlocal-scale parameter e0a, and substrate modulus ks. based on these results, the influences of small-scale, size-dependent, and nanowire-substrate interaction effects on stiffness matches the observations in the parametric study of fig. 10. the gradient terms are nullified when nonlocality is not included (e0a = 0). furthermore, the influence of the higher-order term in the fourth-order nonlocal strain gradient model is more pronounced than that of the highest term in the second-order nonlocal strain gradient model, especially when the diameter of the nanowire is small and the nonlocal-scale parameter and the substrate modulus are comparatively large. this observation is clear when compared to the end boundary forces of each model, seen in eq. (31). fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 675 fig. 10 sensitivity of the effective young’s modulus to nanowire diameter, nonlocal scale parameter, and substrate modulus obtained from the proposed model and the second-order nonlocal strain gradient model 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 210 /sk nn nm= e (g p a ) ef f b proposed model 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 second-order strain gradient model 210 /sk nn nm= e (g p a ) ef f b 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 proposed model e (g p a ) ef f b 225 /sk nn nm= 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 second-order strain gradient model 225 /sk nn nm= e (g p a ) ef f b 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 250 /sk nn nm= proposed model e (g p a ) ef f b 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 second-order strain gradient model 250 /sk nn nm= e (g p a ) ef f b 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 2100 /sk nn nm= proposed model e (g p a ) ef f b 0 700 1400 2100 2800 3500 4200 3500-4200 2800-3500 2100-2800 1400-2100 700-1400 0-700 second-order strain gradient model 2100 /sk nn nm= e (g p a ) ef f b 676 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. fig. 11 variation of normalized end stiffness with diameter of nanowire for various nonlocal scale parameters and various substrate moduli 6. conclusions this paper presented a new bar-substrate model considering nonlocal and surface energy effects in the analysis of a nanowire embedded in an elastic substrate. the features of the proposed bar model include nanostructure-substrate interaction, size-scale (nonlocal) effect and size-dependent (surface) effect. the nonlocal and surface energy effects are, respectively, represented through a fourth-order nonlocal strain gradient model and the gurtin-murdoch continuum model, while the interaction between bar and substrate is modeled based on the winkler foundation model. the proposed model is in a displacementbased formulation, so the governing differential equilibrium equation and its boundary conditions were determined from the principle of virtual displacement. analytical solutions were directly obtained by solving the sixth-order governing differential equation, and were employed to provide analytical responses of the nanowire system in this study. the performance of the proposed model in characterizing the nanowire-substrate system was assessed from two numerical simulations. proposed fourth-order nonlocal strain gradient model second-order nonlocal strain gradient model e a = 50 nm e a = 0 nm e a = 100 nm e a = 200 nm e a = 150 nm 0 0 0 0 0 0 1 2 3 4 0 5 10 15 20 25 30 210 /sk nn nm= d (nm) k / k st r c 0 1 2 3 4 5 6 0 5 10 15 20 25 30 225 /sk nn nm= d (nm) k / k st r c 0 1 2 3 4 5 6 7 0 5 10 15 20 25 30 250 /sk nn nm= d (nm) k / k st r c 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 2100 /sk nn nm= d (nm) k / k st r c fourth-order strain gradient bar-substrate model with nonlocal and surface effects... 677 the first simulation demonstrates that nonlocality, added higher-order terms in the strain gradient theory, and surface energy effects stiffen the nanowire-substrate system. this matches the second parametric simulation study. furthermore, both simulations indicate that the influence of nonlocality on system responses is stronger than of the surface energy effect, and the substrate modulus is nullified when nonlocality is ignored. finally, the influence of the added term in the fourth-order nonlocal strain gradient model increased with the nonlocal-scale parameter e0a and with substrate modulus ks, but decreased with nanowire diameter d. the system stiffness was larger than with the second-order nonlocal strain gradient model. the next step in developing the proposed nanowire-substrate model is to consider the effects of geometric nonlinearity (geometric stiffness matrix) in the element. moreover, this study will be extended to buckling analysis of a nanostructure embedded in an elastic substrate. finally, the authors hope that the proposed nanowire-substrate model will be useful in the field of nanotechnology. acknowledgement: this study was supported by trf senior research scholar under grant rta 6280012. any opinions expressed in this paper are those of the authors and do not reflect the views of the sponsoring agencies. special thanks go to the copy-editing service of research and development office, and assoc. prof. dr. seppo karrila and assoc. prof. dr. pattamad panedpojaman who dedicated his time to provide valuable comments, and gratefully acknowledge the support and assistance received. appendix a the homogeneous differential equation of eq. (29) can be rearranged into the general form: 6 4 2 1 2 36 4 2 ( ) ( ) ( ) + ( ) 0x x x x u x u x u x u x x x x    + + =       (a1) where the parameters ϑ1, ϑ2, and ϑ3 are: 1 2 32 4 4 0 0 0 ( )1 ; ; and ( ) ( ) ( ) xx eff s xx xx e a k e a e a e a e a e a = − = = −   (a2) for convenience, the parameters in the general solution are defined by: 32 3 21 1 2 31 2 (2 / 27) ( / 3)( / 3) ; ; and 3 2 − +− + = =  = +         (a3) there are many solutions of the homogeneous differential equation of eq. (a1) that depend on sign of the quantity δ = α3 + β2. all the possible solution cases are described by morfidis [52] and avramidis and morfidis [53]. they suggest that the most likely solution case i has the quantity δ larger than zero (δ = α3 + β2 > 0), while in case ii δ is less than zero (δ = α3 + β2 < 0). thus, these cases are assessed in this study. however, there are other types of solutions when δ is equal to zero (δ = α3 + β2 = 0). this case is unlikely and almost impossible. therefore, this study neglected this case. the general form of the homogeneous solution to eq. (29) is 1 1 2 2 3 3 4 4 5 5 6 6( ) ( ) ( ) ( ) ( ) ( ) ( )x c cu x x x x x x c xc c c= + + + + +      (a4) 678 w. sae-long, s. limkatanyu, p. sukontasukkul, et al. for the solution case i (δ = α3 + β2 > 0), the analytical solution can be expressed as: 1 1 1 2 3 4 5 6 ; ; cos[ ]; ( ) sin[ ]; cos[ ]; sin[ ] ( ) ( ) ( ) ( ) ( ) r x r x rx rx rx rx e e e qx x ex qx e q x x x xx e qx − − − = = = = = =       (a5) where ( ) ( ) ( ) 13 3 1 3 3 3 3 1 2 2 2 2 ; ; ; 2 2 3 2 / 3 2 2 ; 3 m n r m m n m r q m n         + + + − = =          = − +  + − −  − − +  + − −  + − +  − − −    = − = (a6) in case ii (δ = α3 + β2 < 0), the analytical solution can be written as: 1 2 1 2 1 2 3 4 5 6( ) ; ( ) ; ( ) ; ( ) ; ( ) ; ( ) r x r x r x r xrx rxx e x e x e x e x e x e − −−= = = = = =      (a7) where 1 1 1 1 31 2 2 2 cos ; 2 cos ; 3 3 3 3 4 2 cos ; cos / 3 3 r r r            − +    = − − = − −        +   = − − = − −      (a8) 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pp. 115-128. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 565 577 doi: 10.22190/fume191014012c © 2020 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper dynamical contact parameter identification of spindle-holder-tool assemblies using soft computing techniques đorđe čiča1, milan zeljković2, saša tešić1 1university of banja luka, faculty of mechanical engineering 2university of novi sad, faculty of technical sciences abstract. in industry, the capability to predict the tool point frequency response function (frf) is an essential matter in order to ensure the stability of cutting processes. fast and accurate identification of contact parameters in spindle-holder-tool assemblies is very important issue in machining dynamics analysis. this work is an attempt to illustrate the utility of soft computing techniques in identification and prediction contact parameters of spindle-holder-tool assemblies. in this paper, three soft computing techniques, namely, genetic algorithm (ga), simulated annealing (sa), and particle swarm optimization (pso) were used for identification of contact dynamics in spindle-holder-tool assemblies. in order to verify the proposed identification approaches, numerical and experimental analysis of the spindle-holder-tool assembly was carried out and the results are presented. finally, a model based on the adaptive neural fuzzy inference system (anfis) was used to predict the dynamical contact parameters at the holder-tool interface of a spindle-holder-tool assembly. accuracy and performance of the anfis model has been found to be satisfactory while validated with experimental results. key words: contact dynamics, parameter identification, soft computing 1. introduction regenerative chatter is a well-known and undesirable machining phenomenon that could result in cutting process instability, poor surface quality, excessive tool wear and reduced material removal rate. many research efforts have made significant contributions to modeling of chatter mechanism [1-3]. their work includes analytical or numerical time-domain techniques for generating stability lobe diagrams as the main tool to identify the stable zone in the machining process. regardless of the approach used, knowledge of received october 14, 2019 / accepted january 28, 2020 corresponding author: đorđe čiča university of banja luka, faculty of mechanical engineering, stepe stepanovića 71, 78 000 banja luka, bih e-mail: djordje.cica@mf.unibl.org 566 đ. čiča, m. zeljković, s. tešić the machine tool dynamics, specifically the tool point frequency response function (frf) is an essential requirement for generation of stability lobe diagrams. the tool point frf is traditionally obtained using experimental modal analysis by simple impact testing. however, due to a large number of holder and tool combinations, these tests are expensive and time consuming. recently, in order to minimize experimentation, researchers have attempted to obtain the tool point frf semi-analytically. schmitz et al. [4, 5] propose a method for predicting the tool point frf using the receptance coupling theory of structural dynamics. park et al. [6] included the rotational degree-of-freedom at the tool holder-tool joint, which was indirectly identified using translational responses measured from a set of short and long blank tools. kivanc and budak [7] modeled complex end-mill geometry in finite elements and practical equations were developed to predict the static and dynamic properties of the tools. erturk et al. [8] proposed a reliable analytical model for predicting the tool point frf by using the receptance coupling and structural modification methods where all components of the spindle-holder-tool assembly were modeled with the timoshenko beam theory. kiran et al. [9] applied inverse receptance coupling substructure analysis approach in order to compensate mass and damping for accelerometer-based impact testing. ji et al. [10] proposed a new receptance coupling substructure analysis methodology to predict the tool tip dynamics solving the accuracy problem in the estimation of the rotational/moment receptance. postel et al. [11] presented a new approach for prediction of tooltip frfs under operational conditions for arbitrary tool-holder combinations. qi et al. [12] presented tool point frequency response prediction based on timoshenko beam model using receptance theory. successful application of these models strongly depends on the accurate identification of dynamical contact parameters at the spindle-holder and holder-tool interfaces. therefore, accurate and fast dynamical contact parameter identification in spindle-holder-tool assemblies has become an important issue for obtaining the accurate tool point frf. most researchers [4, 5, 13] used the nonlinear least square error minimization for identifying the contact parameters in spindle-holder-tool assemblies. schmitz et al. [14] present a finite element modeling approach to determine the stiffness and damping behavior between the tool and the holder in thermal shrink fit connections. ahmadi and ahmadian [15] considered the holder-tool interface as a distributed elastic layer between the holder-spindle and the tool shank part. namazi et al. [16] presents modeling and identification of holder-spindle interface stiffness using translational and rotational springs which were uniformly distributed at the contact zone. ozsahin et al. [17] proposed identification approach where elastic receptance coupling equations previously used for coupling the spindle-holdertool assemblies are rearranged to give the complex stiffness matrix at the holder-tool and spindle-holder interfaces in a closed-form manner. gou et al. [18] developed a static model of bt40 spindle-holder system and presented an identification method based on analysis of the rigid body deformation and elastic modulus of the virtual material layer. gao et al. [19] proposed an analytical method combining classic elasticity theory with the fractal theory in order to estimate the contact stiffness of spindle-holder joint. effects of cutting force on contact stiffness at the spindle-holder interface were also investigated in this study. zhao et al. [20] present a macro-micro scale hybrid model to obtain the contact stiffness at the spindle-holder interface at high speeds. the taper contact surface of spindle-holder joint is assumed flat in macro-scale and the finite element method is used to obtain the pressure distribution at different speeds, while in micro-scale, the topography of contact surfaces is fractal featured and determined by fractal parameters. dynamical contact parameter identification of spindle-holder-tool assemblies... 567 liao et al. [21] developed identification method based on fractal topography theory to determine the contact properties at the holder-tool interface in the shrink-fit connection. some researchers use artificial intelligence methods to identify the contact parameters of spindle-holder-tool assemblies. movahhedy and gerami [22] present two joint models with linear and rotational springs to model the holder-tool connection. the model is then applied to real machine tool cases and an optimization method based on genetic algorithm is used to identify joint parameters at the tool-holder interface. wang et al. [23] proposed an identification method to recognize the connection parameters at the holder-tool interface by using receptance coupling substructure analysis and particle swarm optimization. ganguly and schmitz [24] also implemented a particle swarm optimization technique to automate the identification of the euler-bernoulli beam parameters for each mode. liu et al. [25] introduced an optimization technique based on particle swarm optimization algorithm to obtain the high contact stiffness of btf40 spindle-holder joint at the high speed. having an accurate prediction of contact parameters in spindle-holder-tool assemblies is very important for obtaining correct tool point frf. this research aims to illustrate the utility of soft computing techniques in identification and prediction contact parameters in spindle-holder-tool assemblies. due to high complexity of this optimization problem, three non-traditional algorithms, the genetic algorithm (ga), simulated annealing algorithm (sa), and the particle swarm optimization (pso) have been employed to resolve this problem. firstly, in order to verify the proposed identification approaches, numerical analysis of the spindle-holder-tool assembly has been performed and the results were presented. then, ga identification approach was verified experimentally and obtained results were used to train the anfis model for prediction of translational and rotational stiffness at the holder-tool interface of assembly. 2. mathematical model of spindle-holder-tool assembly it is generally accepted that an analysis of complex dynamical systems can be simplified by disassembling a system into a set of interconnected subsystems. in this sense, the problem referring to dynamic properties of the spindle-holder-tool assembly can be so simplified that instead of viewing it as single, the specified system is regarded as the one composed of three components, namely: a spindle (s), holder (h), and tool (t), as shown in fig. 1. after obtaining the end-point receptance matrices of spindle (s), holder (h) and tool (t) these components should be elastically assembled through the complex stiffness matrices at the spindle-holder (shk) and holder-tool (htk) interfaces of assembly. in this paper, the analytical model proposed by erturk et al. [8] was applied where part of the holder inside the spindle is considered rigidly joined to the spindle and the part of the tool inside the holder is considered rigidly joined to the holder. the following equation represents the elastic coupling of subsystem spindle and subsystem holder frfs to obtain the end point frf of the spindle-holder subassembly at the holder tip: 1 1( )ii ii ic cc cc sh ci − −= −  + + sh h h h s k h (1) where hii, hic, hci, hcc and scc are submatrices of receptance matrices of the holder and spindle, respectively, that include the point and transfer receptance functions of the segment end point (i) and (c). 568 đ. čiča, m. zeljković, s. tešić complex stiffness matrix of the spindle-holder interface has the following form: 0 0 sh t sh t sh sh r sh r k i c k i c   +    =   +    k (2) where: shkt is the translational stiffness, shct is the translational damping, shkr is the rotational stiffness and shcr is the rotational damping at the spindle-holder interface, ω is the excitation frequency and i is the unit imaginary number. fig. 1 elastic coupling of the spindle and holder (left) and elastic coupling of the spindleholder subassembly and tool (right) the receptance matrix of the tool can be coupled with the spindle-holder subassembly (fig. 1) and the resulting matrix of the assembly spindle-holder-tool can be obtained in a very similar manner as follows: 1 1( )ii ii ic cc cc ht ci − −= −  + + sht t t τ sh k t (3) where tii, tic, tci, tcc and shcc are submatrices of receptance matrices of the tool and spindle-holder subassembly, respectively, that include the point and transfer receptance functions of the segment end point (i) and (c). complex stiffness matrix of the holder-tool interface is defined as: 0 0 ht t ht t ht ht r ht r k i c k i c   +    =   +    k (4) here, htkt is the translational stiffness, htct is the translational damping, htkr is the rotational stiffness and htcr is rotational damping at the holder-tool interface. a very important issue in the receptance coupling theory of structural dynamics is proper modeling of dynamical contact parameters (stiffness and damping) at spindleholder and holder-tool interfaces which have a significant impact on the dominant elastic modes of the tool point frf. however, these parameters are very difficult to model adequately and accurately. in the end point frf of the spindle-holder subassembly at the holder tip (eq. 1), shkt, shct, shkr and shcr are all unknowns to be recognized. these parameters can be represented as depicted in eq. 5. dynamical contact parameter identification of spindle-holder-tool assemblies... 569  , , ,sh sh t sh t sh r sh rk c k c=α (5) when ω = ωj, frf of the spindle-holder subassembly at the holder tip is shii(ωj), simplified as shii. meanwhile the measured frf is ( )ii jsh , simplified as iish . if ωj = ω1, ω2, ... , ωm, there are m frequency measurement points, and an error vector she can be constructed by eq. 6. iish ii= −e sh sh (6) in eq. 6 she is the nonlinear function of vector shα to be identified. similarly, in the end point frf of the spindle-holder-tool assembly at the tool tip (eq. 3), htkt, htct, htkr and htcr are all unknown to be recognized and these parameters can be represented as  , , ,ht ht t ht t ht r ht rk c k c=α (7) error vector hte is the nonlinear function of vector htα to be identified: iiht ii= −e sht sht (8) where shtii is frf of the spindle-holder-tool subassembly at tool tip and iisht is the measured frf. 3. formulation of optimization problem the general approach to the problem of parameters identification is to minimize the differences between the results obtained by measuring and by the mathematical model. an appropriate method of identification depends on the assumptions and knowledge about the errors of measurement. classical optimization algorithms are not suitable for solving nonlinear and large scale optimization problems. hence, novel optimization algorithms are emerging to solve these problems. recently a lot of metaheuristic algorithms have been proposed to find the best global solution and to overcome the restrictions and difficulties of classical optimization algorithms. the classical nonlinear constrained optimization problem can be written as minimize f(x) subject to hj(x) = 0 (j = 1, 2, ..., m), gk(x) ≤ 0 (k = 1, 2, ..., n), xi lb ≤ xi ≤ xi ub (i = 1, 2, ..., p). in general, objective function f(x) as well as equality constraint function hj(x) and inequality constraint function gk(x) are nonlinear implicit functions with respect to the design variables. xi lb and xi ub denote the lower boundary condition vectors and upper boundary condition vectors, respectively. based on the presented mathematical model of the spindle-holder-tool assemblies, for accurate prediction of the frf it is necessary to know the exact stiffness matrix at the spindle-holder, and holder-tool interface. the matrix elements are stiffness and damping between these subsystems, and as the specified values cannot be experimentally measured, they need to be determined in other way. therefore, it is necessary to explore the possibilities of applying different methods for identification of the contact parameters in spindle-holder-tool assemblies. the proposed identification procedure of contact parameters in spindle-holder-tool assemblies using soft optimization techniques is presented in fig. 2. 570 đ. čiča, m. zeljković, s. tešić the identification procedure estimates the vector of unknown parameters, so that the deviation between the mathematical model and the real system responses to the same input is minimized. in this paper, considerations are limited to identification of dynamical contact parameters only at the holder-tool interface. however, the same procedure can also be applied for identification of other parameters, such as contact parameters at the spindle-holder interface as well as the dynamical parameters of bearings. in the direct tool point frf (eq. 3), htkt, htct, htkr and htcr are all unknowns to be recognized (eq. 4). these parameters can be represented as depicted in eq. 7. in engineering application of optimization algorithms an important step is to choose the form of the objective function. in this study, the objective function was constructed as euclidian norm of vector differences between the experimental and analytical responses. since both response values are complex, they are divided into real and imaginary parts. fig. 2 general identification procedure of dynamical contact parameter in spindle-holdertool assemblies using soft optimization techniques the presented dynamical contact parameter identification of spindle-holder-tool assemblies can be solved by means of some optimization methods, such as newton and quasi-newton methods, conjugate gradient methods, the simplex method, etc. however, these methods are ill-conditioned, very sensitive to noise, and numerically complicated, resulting in the problem of convergence to a local minimum instead of global one. another drawback is that the optimum identified parameter values strongly depend on the initial guess of the parameter. as an alternative to these methods, non-traditional optimization techniques based on soft computing have been employed to overcome these disadvantages. this paper considers three soft computing techniques, namely, ga, sa and pso, for dynamical contact parameter identification in spindle-holder-tool assemblies. dynamical contact parameter identification of spindle-holder-tool assemblies... 571 4. numerical case study to create conditions that will lead to a successful experiment, it is desirable to analyze the possibility of identifying unknown parameters of the spindle-holder-tool assembly using optimization techniques based on soft computing prior to experimental obtaining tool point frf. therefore, in this section, a numerical case study for identification of the contact parameters in spindle-holder-tool assemblies is presented. in order to be able to use eq. 3 to calculate frf at the tool tip of a spindle-holder-tool assembly, the receptance matrices for each of the components of the spindle-holder-tool assembly are required. in this section, translational and rotational dynamic responses for each of the components of the spindle-holder-tool assembly are obtained through the fem analysis. the spindle-holder-tool assembly of the case study (fig. 3) is modeled using a reliable finite element software ansys. each of the subsystems is composed of several sections with different diameter and lengths which are modeled as multi-segment beams. the dimensions of the subsystems, i.e. spindle, holder and tool, bearings properties, spindle‐holder interface dynamics, and holder‐tool interface dynamics are given by cica [26]. beam element beam188 which is based on the timoshenko beam theory is used for modeling the spindle, holder and tool, since this element supports the effects of rotary inertia and shear deformation. dynamics of bearings and spindle‐holder and holder‐tool interfaces, were modeled using combination element combin14, which is a springdamper element. material properties of assembly are taken as follows: young’s modulus e = 210 gpa, mass density ρ = 7800 kg/m3, poisson’s ratio ν = 0.3. the fem analysis of the spindle-holder-tool assembly is performed in ansys and the tool point frf for the frequency range of 0‐3000 hz with a frequency increment of 1 hz is obtained. in this numerical case study, unknown contact parameters at the holder-tool interface were identified using ga, sa and pso optimization techniques. optimization routines proceeds by considering a ranges of 106 to 109 n/m, 105 to 107 nm/rad, 1 to 300 ns/m and 1 to 300 nms/rad for translational stiffness, rotational stiffness, translational damping, and rotational damping at the holder-tool interface, respectively. the fitness function of optimization algorithms was simply the minimization of the error vector given by eq. 8. it is noted that the error could be either positive or negative. therefore, absolute value of the error is used as a fitness function. fig. 3 the spindle-holder-tool assembly for numerical simulation for each optimization technique a parametric study is carried out so that the value of one parameter is varied at a time, while other parameters have fixed values. in the ga 572 đ. čiča, m. zeljković, s. tešić optimization process, the commonly used ga operation parameters were adopted. the optimal values of evolutionary parameters were 2430 and 420 for number of generations and population size, respectively. the genetic operations reproduction, crossover and mutation were also used. probability of reproduction, crossover and mutation were 0.1, 0.65 and 0.015, respectively. in the pso optimization process, the commonly used pso operation parameters were adopted, namely the population size, number of generations, cognitive acceleration, and social acceleration. the optimal values of number of generations, population size, cognitive acceleration, and social acceleration were 280, 120, 0.5 and 1.35, respectively. the sa optimization process takes place with the values of reannealing interval, and initial temperature. the optimal values of reannealing interval, and initial temperature were 50, and 300, respectively. the number of iterations required to find an optimal solution for ga, pso and sa were 51, 276 and 1020, respectively. the identified contact parameters at the holder-tool interface in the numerical case study using ga, pso and sa based optimization techniques are shown in table 1. table 1 identified dynamical contact parameters at the holder-tool interface in the numerical case study parameter exact value ga identified value rel. error [%] sa identified value rel. error [%] pso identified value rel. error [%] htkt [nm] 2.1·107 2.0999663·107 0 1.8999804·107 9.5 2.0936593·107 0.3 htct [ns/m] 15 16.6 10.7 15.4 2.7 13.6 9.33 htkr [nm/rad] 1.4·106 1.381904·106 1.3 1.540449·106 10 7.02672·106 49.8 htcr [nms/rad] 3 1.2 306.7 89 2866 9 200 as shown in table 1, the accuracy of the identified parameters is more than satisfactory. somewhat larger errors are encountered in the identification of the rotational stiffness and rotational damping, but those parameters have no significant impact in the synthesis of dynamic subsystems. the most dominant factor in the synthesis of dynamic subsystems is translational stiffness (htkt), and this value was most accurately identified. in order to show the accuracy of the ga, sa and pso identification methods, the spindle-holder subassembly receptance matrix is coupled with tool frfs through the forward coupling equation (eq. 3). in coupling of the spindle-holder and tool subsystems, the constant values identified from all three methods (table 1) were used. fig. 4 shows the comparison of frf at the tool tip of a spindle-holder-tool assembly with the identified and real values of contact parameters at the spindle-holder interface. it can be seen from these figure that the ga and pso based identification methods presented in this study give excellent results, while the sa based method yielded somewhat different results. so far, the implementation of the soft computing identification approach proposed in this paper is demonstrated with an example for extracting the holder-tool interface parameters for a typical spindle-holder-tool assembly. it is observed that accuracy of obtained results allows prediction of the response of the spindle-holder-tool assembly with very high accuracy. an experimental application of the identification approaches is given in the following section. dynamical contact parameter identification of spindle-holder-tool assemblies... 573 fig. 4 comparison between numerically and analytically obtained frf with identified contact parameters at the holder-tool interface 5. experimental details in this section, an experimental case study for the ga based identification approach is provided. ga was selected because this method provides the best results in terms of identification of translational stiffness (htkt), which is the most dominant factor in the synthesis of dynamic subsystems. experiments were performed with iso 40 type holder, in which different combination of tool diameters (d = 9-30 mm) and different tool overhang lengths (l = 16-83 mm) were inserted, and assembled to the spindle. the spindle-holdertool assembly shown in fig. 5 was suspended to obtain free‐free end conditions for performing an impact tests. fig. 5 schematic layout of experimental setup for dynamical contact parameter identification of spindle-holder-tool assembly in order to verify the presented identification models, frf of the spindle-holder-tool assembly was measured or obtained with fe analysis for each of these substructures. due to the relative complex geometry of the spindle, and since it does not change over time, it is preferred to obtain the frf of this substructure by modal analysis. on the other hand, the cutting tool usually has a much simpler geometry and frf of this substructure may be 574 đ. čiča, m. zeljković, s. tešić obtained numerically e.g. from an fe analysis. first, frf of the spindle-holder subassembly (without the tool part outside the holder) is obtained by performing an impact test. next, the tool is inserted into the holder and the tool point frf of the assembly is obtained also by performing impact tests. in order to obtain angular displacement or moment-related frfs of spindle-holder subassembly, the method proposed by park et al. [6] was used. finally, the receptance matrices of the tool component in free-free boundary conditions were separately computed using fem modeling. in this way, 178 measurements were made with different combinations of spindle-holder-tool assembly. according to the presented mathematical model of the spindle-holder-tool assembly, accurate knowledge of complex stiffness of holder-tool interface dynamics is necessary for accurate prediction of the dynamic response. the ga optimization routine proceeds by considering a range of 105 to 109 for stiffness and 1-300 for damping coefficients of the holder-tool interface as the feasible range. the obtained experimental results were used to train the anfis model for prediction of contact parameters of spindle-holder-tool assembly for different cases. in the present work, the input variables for anfis were tool diameters and overhang length of the tools, while the output variables were identified data relating to the translational and rotational stiffness at the holder-tool interface. a 148 sets of data were selected from the total of 178 sets obtained using ga identification method for the purpose of training in anfis, while the other 30 sets were then used for testing after the training was completed to verify the accuracy of the predicted values of contact parameters. several anfis models were developed and tested based on the same training data in order to achieve the maximum prediction accuracy. the models were developed using different shapes of input membership functions (mfs) type which were triangular, trapezoidal, gaussian, and bell shapes, with a different number of the mfs. the constant and linear output mfs type were employed to produce the stiffness values, while a hybrid of the least-squares method and the back propagation gradient descent method was used to emulate a given training data set. in order to estimate the prediction capability of the developed anfis models, normalized root mean square error, absolute fraction of variance, mean absolute percent error and maximum mean absolute percentage error were used. in the current work, the best anfis model was obtained with gaussian curve built-in membership functions for each input and a linear output function. the number of fuzzy rules is related to the number of fuzzy sets for each input variable. because the inputs, tool diameters and overhang length of the tools, were classified into 3, and 7 fuzzy sets, respectively, the total number of fuzzy rules formed will be 21. first-order sugeno fuzzy inference system is used in this work with the hybrid learning rules used in the training fig. 6 shows the comparison between the experimental and the anfis model results for translational stiffness for test data set. normalized root mean square error, absolute fraction of variance, mean absolute percent error, and maximum absolute percent error for developed model were 0.09427, 0.99541, 4.5%, and 11.8%, respectively. referring to fig. 7 indicates the comparison in prediction of rotational stiffness obtained using the anfis model. normalized root mean square error was 0.08993, absolute fraction of variance was 0.99591, mean absolute percent error was 0.7%, and maximum absolute percent error was 3.2%. hence, it is obvious that there is good agreement between predicted and experimental values of translational and rotational stiffness. fig. 8 demonstrates a linear regression between the predicted values and corresponding target values. dynamical contact parameter identification of spindle-holder-tool assemblies... 575 fig. 6 the experimental and predicted values of translational stiffness at the holder-tool interface in the test data set fig. 7 the experimental and predicted values of rotational stiffness at the holder-tool interface in the test data set fig. 8 regression plot of actual and predicted translational (a) and rotational stiffness (b) at the holder-tool interface in the test data set 576 đ. čiča, m. zeljković, s. tešić 6. conclusions soft computing techniques are among the fast-growing and promising research topics that have drawn a great deal of attention from researchers in recent decades. these techniques have excellent learning and generalization abilities on handling ill-defined and complex problems. various soft computing techniques have been successfully applied to a wide range of applications, and in this paper three techniques, namely, genetic algorithm, simulated annealing, and particle swarm optimization, were used for identification of contact dynamics of spindle-holder-tool assemblies. numerical and experimental studies were presented to confirm the effectiveness of proposed methodology. first, a numerical case study for identification of the contact parameters in spindle-holder-tool assemblies (with a focus on the holder-tool interface) was presented. it is observed that the ga and pso based techniques give exceptional results, while the sa based technique yielded somewhat different results. furthermore, analysis of the identification results shows that the most dominant factor in the subsystems synthesis of the spindle-holder-tool assembly is translational stiffness, and this value was most accurately identified. slightly larger deviations were observed at identification of rotational parameters, but these parameters do not have significant impact on the synthesis of dynamic subsystems. the identification approach based on ga was also experimentally verified in terms of identification contact parameters at the holder-tool interface of a free-free spindle-holder-tool assembly. finally, the anfis model was used to predict identified dynamical contact parameters at the holder-tool interface. the results revealed that the developed anfis model can very accurately predict the dynamical contact parameters in spindle-holdertool assemblies. references 1. minis, i., yanushewsky, t., tembo, r., hocken, r., 1990, analysis of linear and nonlinear chatter in milling, cirp 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mills for form error and stability analysis, international journal of machine tools and manufacture, 44(11), pp. 1151-1161. 8. erturk, a., ozguven, h.n., budak, e., 2006, analytical modeling of spindle-tool dynamics on machine tools using timoshenko beam model and receptance coupling for the prediction of tool point frf, international journal of machine tools and manufacture, 46(15), pp. 1901-1912. 9. kiran, k., satyanarayana, h., schmitz, t., 2017, compensation of frequency response function measurements by inverse rcsa, international journal of machine tools and manufacture, 121, pp. 96-100. 10. ji, y., bi, q., zhang, s., wang, y., 2018, a new receptance coupling substructure analysis methodology to predict tool tip dynamics, international journal of machine tools and manufacture, 126, pp. 18-26. 11. postel, m., özsahin, o., altintas, y., 2018, high speed tooltip frf predictions of arbitrary tool-holder combinations based on operational spindle identification, international journal 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16. namazi, m., altintas, y., abe, t., rajapakse, n., 2007, modeling and identification of tool holder-spindle interface dynamics, international journal of machine tools and manufacture, 47(9), pp. 1333-1341. 17. ozsahin, o., erturk, a., ozguven, h.n., budak, e., 2009, a closed-form approach for identification of dynamical contact parameters in spindle-holder-tool assemblies, international journal of machine tools and manufacture, 49(1), pp. 25-35. 18. guo, h., zhang, j., feng p., wu, z., yu, d., 2015, a virtual material based static modeling and parameter identification method for a bt40 spindle-holder taper joint, the international journal of advanced manufacturing technology, 81(1-4), pp. 307-314. 19. gao, x., wang, m., zhang, y., zan, t., 2016, a modeling approach for contact stiffness of spindle-tool holder based on fractal theory, proceedings of the institution of mechanical engineers, part b: journal of engineering manufacture, 230(10), pp. 1942-1951. 20. zhao, y., xu, j., cai, l., shi, w., liu, z., cheng, q., 2016, contact characteristic analysis of spindle-tool holder joint at high speeds based on the fractal model, proceedings of the institution of mechanical engineers, part e: journal of process mechanical engineering, 231(5), pp. 1025-1036. 21. liao, j., zhang, j., feng p., yu, d., wu, z., 2017, identification of contact stiffness of shrink-fit tool-holder joint based on fractal theory, the international journal of advanced manufacturing technology, 90(5-8), 2173-2184 22. movahhedy, m.r., gerami, j.m., 2006, prediction of spindle dynamics in milling by sub-structure coupling, international journal of machine tools and manufacture, 46(3-4), pp. 243-251. 23. wang, e., wu, b., hu, y., yang, s., cheng, y., 2013, dynamic parameter identification of tool-spindle interface based on rcsa and particle swarm optimization, shock and vibration, 20, pp. 69-78. 24. ganguly, v., schmitz, t.l., 2013, spindle dynamics identification using particle swarm optimization, journal of manufacturing processes, 15(4), pp. 444-451. 25. liu, z., xu, j., zhao, y., cheng, q., cai, l., 2017, stiffness optimization for high-speed double-locking tool holder-spindle joint using fractal theory, proceedings of the institution of mechanical engineers, part e: journal of process mechanical engineering, 232(4), pp. 418-426. 26. cica, dj., 2010, modelling of dynamical behaviour spindle-holder-tool assembly, ph.d. thesis, university of banja luka, banja luka (in serbian) 8045 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 199 209 https://doi.org/10.22190/fume210722066t © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper application possibilities of the s960 steel in underwater welded structures jacek tomków, michał landowski, grzegorz rogalski faculty of mechanical engineering and ship technology, gdańsk university of technology, poland abstract. in this paper, the application possibilities of the ultra-high strength (uhss) domex 960 steel in the underwater welded structures are analyzed. in the research, the investigated material has been tested in bead-on-plate wet welding conditions with the usage of different heat input values, namely 0.63 kj/mm, 0.72 kj/mm and 0.93 kj/mm. specimens were performed by the manual metal arc (mma) welding method with the usage of rutile covered electrodes. firstly, the nondestructive visual testing (vt) was carried out. in the next step, the metallographic macroand microscopic tests were performed. finally, the hardness of the weld metal and heat-affected zone (haz) was measured by the vickers hv10 method. the performed experiments allow the statement that the domex 960 steel could be welded in a water environment. it also showed that increasing heat input leads to decreasing the hardness in haz by 30 hv10. it may result in decreasing the susceptibility to cold cracking during buttand filet welding in the water environment. key words: underwater welding, ultra-high strength steel, microstructure, macroscopic testing, hardness measurements, cold cracking 1. introduction underwater welding is a special process, which requires special equipment or highqualified welders. underwater processes could be classified in three groups. dry welding allows us to isolate the welding area and welder from the surrounding environment by locating them in a special chamber [1]. the biggest limitation of this method is a high cost of the chamber. following this, it is used only for building underwater structures, e.g., pipelines. another underwater welding method is local cavity welding [2]. it uses a small chamber, which isolates the welding arc and molten metal from the water environment. received july 22, 2021 / accepted october 23, 2021 corresponding author: jacek tomków faculty of mechanical engineering and ship technology, gdańsk university of technology, gabriela narutowicza 11/12 street, 80-233 gdańsk, poland e-mail: jacek.tomkow@pg.edu.pl 200 j. tomków, m. landowski, g. rogalski however, the welding equipment and welder diver are located directly in the water [3,4]. while using this method, the welding area is covered by the chamber. following this, there are problems with controlling the process, which is the biggest disadvantage of local cavity welding. the third and most used method is wet welding [5]. in wet welding, the welder diver and the whole welding area are in direct contact with water. it is mostly carried out by manual metal arc (mma) welding [6,7] and flux cored arc welding (fcaw) [8,9]. wet welding is the most often used underwater welding method, due to cheaper and smaller equipment. water generates some problems during welding. firstly, it creates water bubbles near the welding arc, which leads to limited visibility [10]. this makes it difficult for welders to control the process [11]. another problem is instability of the welding arc, which decreases the quality of the performed joints [12,13]. furthermore, water provides slag inclusions in the weld metal, which decrease the mechanical properties of welded structures [14]. the next problem generated by water is high diffusible hydrogen content in the deposited metal, which is one of the factors responsible for the cracking of steel-welded joints [15]. fydrych and łabanowski [16] showed that a water environment generates at least two times higher amounts of hydrogen than during welding in air. klett et al. [17] performed investigations, which showed that no significant differences in the diffusible hydrogen content could be found between the samples of varying geometry. this amount depends on the used materials. one of the underwater welding problems is the increased cooling rate. a high cooling rate generates brittle microstructures in heat-affected zone (haz) [18]. mentioned factors are responsible for high susceptibility to cold cracking of steel during welding in a water environment [19]. cracking phenomenon leads to decreasing the weldability in wet welding conditions, which is the reason of the poor quality of welded structures [20]. offshore steel structures in marine environment face a high degree of damage due to dynamic environmental and operational factors such as corrosion, fatigue cracking, or cold cracking [16,18,21,22]. many repairs have to be performed in the water environment [23]. underwater repair processes have been described for mild steels and high-strength lowalloy steels [7,11,20]. however, for offshore structures, ultra-high strength steels (uhss) are used increasingly each year [24]. their behavior during underwater welding has not been investigated earlier. the literature analysis showed that there are some problems with welding the s960 uhss in the air, which is popular as a material for offshore structures. welded joint parameters and microstructural characterization strongly depend on the parameters of a welding process [25,26]. improper parameters lead to angular distortion of the joint [27], form brittle structures in the haz [28] and form welding imperfections [29,30]. mician et al. [31] focused on the influence of the cooling rate on the mechanical properties of the joints. they showed that a lower cooling rate leads to decreasing the yield point and tensile strength of s960 steel joints. similar results were observed by szymczak et al. [32]. it was stated that micro-jet cooling, which increases the cooling rate, leads to improve the mechanical properties of welded joints. it suggests that this material may be welded in a water environment, where the cooling rate is very high. however, the s960 steel is characterized by high susceptibility to hydrogen-assisted cracking [33]. schaupp et al. [34] demonstrated that cracks occur in the coarse-grained haz. this area of cracking is typical for underwater processes [16,18]. furthermore, schaupp et al. [35] in their next paper showed that s960 ugss is characterized by a high risk for hydrogen-assisted cracking both in the weld metal and in the haz. these investigations suggest that the s960 application possibilities of the s960 steel in underwater welded structures 201 steel may be characterized by very high susceptibility to cold cracking in underwater conditions, due to a high amount of water during joining in this environment. however, the behavior of uhss during wet welding has not been investigated yet. the aim of the presented research was to assess the possibility of welding the s960 steel in wet welding conditions. moreover, the influence of heat input on the microstructure and hardness in different areas of underwater welded structures were investigated. 2. materials and methods 2.1. materials as a material for the experiment, s960 thermomechanically rolled steel plates with dimensions of 150×100×6 mm were chosen. there is a lack of filler material for welding in water for the materials with grade s960. however, the common e42 2 1ni rr 51 covered electrodes (4.0 mm diameter) for underwater welding were selected. the chemical compositions of the used materials are presented in table 1, and their mechanical properties are shown in table 2. presented values are in accordance with the manufacturer data. table 1 chemical composition of used materials, wt. % material c si mn cr s p ni al carbon equivalent ceiiw s960 steel 0.18 0.50 2.1 17.57 0.010 0.020 0.018 0.50 e42 2 1ni rr 51 electrode 0.05 0.45 0.5 19.00 0.025 0.025 0.3 table 2 mechanical properties of used materials material re [mpa] rm [mpa] a5 [%] s960 steel min. 960 960-1250 min. 8 e42 2 1ni rr 51 electrode deposit min. 540 min. 10 2.2. methods the welding process was performed manually using the mma process. three specimens were bead on plate welded in tap water (20 °c) at 0.25 m depth in the flat position (pa). each specimen was prepared by laying one bead in the middle of the plate. for welding, the negative (dc-) polarity was used following the filler material manufacturer requirements. welding parameters such as welding current (i) and arc voltage (u) were selected in the range required by the covered electrode manufacturer. the welding speed (vsp) was chosen to obtain different heat input (ql) values to show the influence of heat input on the properties of wet welded specimens. welding parameters are presented in table 3. 202 j. tomków, m. landowski, g. rogalski table 3 welding parameters specimen i [a] u [v] vsp [mm/s] ql [kj/mm] 1 196 26.8 8.34 0.63 2 220 28.8 8.80 0.72 3 232 30.5 7.54 0.93 after welding, each specimen was subjected to nondestructive and destructive testing. firstly, the visual test (vt) following the en iso 17637:2017 standard was performed. this test started 48 h after welding, which is necessary to avoid cold cracks after vt. secondly, the specimens were cut, ground, polished, and etched (nital 4 %) for destructive testing. the metallographic macroand microscopic tests were performed in accordance with en iso 17639:2013 standard. macrographs were taken using a canon eos 1200d camera. for microscopic investigations, the olympus bx51 light microscope was used. in the last step, the vickers hv10 hardness measurements were performed. hardness was measured in the measurement line following fig 1. this line was located in the axis of the weld bead. fig. 1 schematic view of hardness measurement points distribution 3. results and discussion 3.1. visual testing the main aim of vt was choosing areas without imperfections, from which the samples were cut for further investigations. results of vt are shown in fig. 2. during welding, some problems typical for wet welding were observed. during specimen 1 performing, there was a problem with burning the welding arc. however, after burning, the arc was stable, and no further problems occurred. the geometry of the performed bead was satisfactory (fig. 2a). sample for further investigations was cut in the middle of the bead. there was no problem with initiating the process in specimen 2. however, the instability of the welding arc was observed near the beginning of the bead (green arrow in fig. 2b). moreover, the shape defects were detected in the middle of the sample (red arrow in fig. 2b). the sample for further investigations was cut from an area without imperfections. no problems were observed during the specimen 3 preparation. the welding arc was stable. no serious imperfections were detected during vt (fig. 2c). moreover, the welds showed differences due to different heat input. the narrow weld morphology can be observed for specimen welded with the lowest heat input (fig. 2a). application possibilities of the s960 steel in underwater welded structures 203 with the increase of heat input, the weld size continues to become wider. similar results were observed by wang et al. [36] and tomków et al. [37]. fig. 2 results of vt; a) specimen 1, b) specimen 2 – results of instability of welding arc (green arrow), shape defects (red arrow), c) specimen 3 3.2. macroscopic testing the exemplary results of the macroscopic test are presented in fig. 3. performed investigations showed a significant influence of welding heat input on the size of haz. increasing heat input leads to increasing the area of haz. similar results were observed for underwater wet welding of s700mc steel [37]. moreover, in specimen 3 (fig. 3c) the coarse-grained haz is the widest, which may result in microcracks typical for welding in the water environment [11,20]. besides assessing the haz size, macroscopic tests were carried out to detect welding imperfections. in specimen 1 two cracks were observed (fig. 3a). these cracks run through the haz perpendicularly to the fusion line. it suggests that the heat input (0.63 kj/mm) is too low for the s960 uhss welding. macroscopic test of specimen 2 confirmed results of vt. the shape defect of the performed bead was observed (fig. 3b). however, no other imperfections were detected. in specimen 3, no macrocracks were observed (fig. 3c), but an undercut was detected. this imperfection is typical for a high value of heat input, which was demonstrated by tomków et al. [37,38]. the macroscopic tests showed differences in weld geometries in different specimens. it can be seen that the weld width, weld penetration, and weld reinforcement approximately increase with increasing the heat input. 204 j. tomków, m. landowski, g. rogalski fig. 3 exemplary macrographs; a) specimen 1 – cracks in haz, b) specimen 2, c) specimen 3 undercut 3.3. microscopic testing the exemplary photographs of the investigated microstructures are shown in fig. 4. during metallographic microscopic tests, the weld metal areas and haz of each specimen were observed. moreover, the base metal (bm) was investigated. bm is characterized by finegrained bainitic-martensitic microstructure. the photos of the microstructure in the weld metal showed a typical dendritic structure (fig. 4a, 4c and fig. 4e). dendrites are arranged with columns. they rise to the axis of the performed bead. the microstructure of the weld metal consists of a mixture of bainitic and martensitic phases. with increasing the heat input, the content of martensite slightly increased. similar results were observed earlier by schaupp et al. [34]. more differences were observed in haz for different specimens. specimen 1 welded with the lowest value of the heat input presented in the haz mostly coarse-grained martensitic microstructure near the fusion line, with a small amount of bainite (fig. 4b). with increasing the heat input value, the content of bainite also increased (fig. 4d and 4f). the width of haz increased with increasing heat input. it confirms the results of macroscopic observations. in the area of haz located near the bm, the partially tempered region with a bainitic-ferritic structure may be observed (fig. 4d). kurc-lisiecka and lisiecki [39] observed the same effect during laser welding of domex 960 steel. however, the width of haz in their specimens was much smaller than during underwater welding. microscopic observations also showed the influence of heat input on the number of cracks in the haz. as well as in the macroscopic tests, the biggest number of cracks was found in specimen 1, welded with the lowest heat input. except the two long cracks detected in macroscopic investigations (fig. 3a), the haz consists of many short cracks parallel to the fusion line (fig. 4b). the location of the presented cracks is typical for underwater welding [11]. wang et al. [40] showed that low heat input values of highstrength steel in underwater conditions lead to an increased number of cracks. it was stated that the investigated materials should be welded with a higher heat input, which allows decreasing of the susceptibility to cracking. a lower number of cracks were found application possibilities of the s960 steel in underwater welded structures 205 during specimen 3 observations. in this specimen, the cracks were located in the coarsegrained region of haz and run perpendicularly to the fusion line (fig. 4f). unexpectedly, the lowest number of cracks was found in specimen 2. no cracks were observed in coarse-grained haz. however, a long crack (800 μm) was found in the region of haz, located near the bm (fig. 4d). fig. 4 exemplary micrographs; a) specimen 1 – weld metal, b) specimen 1 haz, c) specimen 2 – weld metal, d) specimen 2 – haz, e) specimen 3 – weld metal, f) specimen 4 haz 206 j. tomków, m. landowski, g. rogalski 3.4. hardness measurements results of vickers hv10 hardness measurements are presented in fig. 5. before the measurements following scheme (fig. 1), the domex 960 uhss hardness was measured. it was in the range 330-350 hv10. in the next step, the hardness in different regions was measured in each specimen. no significant influence of the heat input value on the weld metal hardness was observed. all measurements were in the range of 262-283 hv10. the hardness in haz was higher than the weld metal hardness for all cases, which is typical for welding steels characterized by high-strength [11,37,40]. performed measurements showed a strong relationship between heat input value and haz hardness. it was observed that increasing heat input leads to decreasing the haz hardness. the highest values were found in specimen 1, which was welded with the lowest heat input (0.63 kj/mm). values significantly exceeded 400 hv10 (the highest 448 hv10), and decreased with the distance from the fusion line, which was observed for each specimen. in other specimens, each measurement was lower than 400 hv10. the lower values were observed in specimen 3 welded with 0.93 kj/mm. however, microscopic tests showed that this specimen was characterized by the cracks presence in the haz. it suggests that heat input 0.93 kj/mm may be too high during welding and can lead to failure of underwater wet welded structures. the lowest hardness in specimen welded with the highest heat input confirmed the results of microscopic tests. the haz of specimen s3 consists of a higher content of bainitic microstructure, which is characterized by lower hardness than martensite [37]. fig. 5 results of vickers hv10 measurements, n1 – 0.63 kj/mm, n2 – 0.72 kj/mm, n3 – 0. 93 kj/mm 5. conclusions in the paper, the application possibility of the domex 960 steel in underwater welded structures was assessed. the assessment was prepared by testing bead on plate welded structures performed with different heat input values. investigations showed a significant influence of welding parameters on the properties and microstructure of wet welded structures made by uhss. it was showed that the domex 960 steel may be considered for underwater welding. however, the heat input values must be carefully controlled. the main conclusions resulting from the experiments are: application possibilities of the s960 steel in underwater welded structures 207 1. the investigated domex 960 ultra-high strength steel may be welded in underwater conditions by covered electrodes. however, there is a range of heat input, which leads to formation cracks in the haz. the lowest number of cracks was found during welding with 0.72 kj/mm heat input. 2. performed measurements showed a strong relationship between heat input value and haz hardness. it was observed that increasing the heat input allows decreasing the hardness in the haz by 60-70 hv10. the lowest hardness was found in the specimen welded with 0.93 kj/mm. however, this value leads to forming cracks 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xingtong zhu https://orcid.org/0000-0002-5450-0356 minchuan huang https://orcid.org/0000-0003-3218-0643 ke chen https://orcid.org/0000-0002-9341-9526 abstract. according to the generalized jaccard coefficient and false degree, an improved approach is proposed by incorporating dempster-shafer proofs for determining the level of confidence in the evidence. it also determines the weight of proof in terms of trust and falsity. then, the base probability of the original evidence is weighted and averaged, followed by the adoption of the combined dempster's compositional rule. it is evident that the above combination can be applied in condition monitoring of bearings up to rupture. firstly, the supporting vibration signal is decomposed by applying the empirical mode decomposition, empirical wavelet transformation and variational mode decomposition approaches. all the vectors of the fault characteristic are extracted by combining the sample entropy. then, the fault probability is obtained by performing preliminary diagnosis using the relevance vector machine, where the obtained preliminary diagnostic result is considered as the primary probability of the dempstershafer evidence theory. finally, it is revealed that an accurate diagnosis could be achieved by performing fusion using the enhanced evidence combination method. specifically, the accuracies of the initial condition monitoring based on the emd, ewt and vmd sample entropies and rvm were found to be 97.5%, 98.75% and 95%, respectively. the closeness and high values of these accuracies show that the selected methods are valid. the obtained condition monitoring results show that the relevance vector machine combined with the dempster-shafer evidence could enhance the efficiency. this theory has the least error and better reliability in supporting failure diagnosis. key words: evidence combination, generalized jaccard coefficient, falsity, bearing condition monitoring, dempster-shafer (d-s) received: april 25, 2023 / accepted august 11, 2023 corresponding author: minchuan huang school of computer science, guangdong university of petrochemical technology, no. 139, guandu 2nd road, maoming city, guangdong province: 525000, china e-mail: h0912259520@163.com https://orcid.org/0000-0002-5450-0356 https://orcid.org/0000-0003-3218-0643 https://orcid.org/0000-0002-9341-9526 148 x. zhu, m. huang, k. chen 1. introduction a bearing is an important part of rotational components of a machine. the failure of the bearing affects the performance of the machine, leading to power loss, more wear and tear of the machine components, and even to break down of the machine. since every bearing usually remains around 30% defective, the vibration signals generated from bearings should be monitored continuously. timely identification and maintenance of bearing defects can ensure the safety of machine components and production personnel. the research on the diagnosis of bearing defects has practical importance to consider the general operations of rotating machine components and safety of operators. the dempster-shafer (d-s) evidence theory [1-3] uses both upper and lower limit probabilities in solving multivalued mapping problems, thus leading to be an inaccurate reasoning theory. such study is initiated from the dempster-shafer evidence theory [1,2]. it is supposed to be an effective method in handling uncertain information, and widely used in information merging, decision-making and many other areas [4-7]. the performance can be enhanced by combining evidences using an average rule. average the base probability distribution value and use the dempster combination rules for the n-1 combination [8]. the credibility of evidences is first obtained by measuring the closeness error between them through the jousselme distance function as per the murphy's method [9]. then, the weightage of evidence is determined by combining the average credibility as per the dempster's combination rule [9]. in this article, a combined method is presented for proving conflict by using the generalized jaccard coefficient [10-12]. the objective of the present work is to replace the jousselme distance with the generalized jaccard coefficient [11]. both the generalized jaccard coefficient and jousselme distance are measures of the distance between two sets of evidence in the dempster-shafer theory. however, the generalized jaccard coefficient is considered superior to the jousselme distance in several ways: (a) handles the empty set: the jousselme distance cannot handle the empty set, which means it is not well-defined when one of the sets is empty. on the other hand, the generalized jaccard coefficient can handle the empty set and provide a meaningful result in such cases. (b) reflects uncertainty: the jousselme distance does not reflect the uncertainty associated with evidence. in contrast, the generalized jaccard coefficient can reflect the degree of uncertainty in the evidence, which is important in applications where the evidence may be ambiguous. (c) considers negative evidence: the jousselme distance does not consider negative evidence, which may be useful in some applications. the generalized jaccard coefficient can handle both positive and negative evidence, allowing for a more comprehensive analysis of the evidence. (d) consistent with probability theory: the generalized jaccard coefficient is consistent with probability theory, which means it satisfies the axioms of probability. this property is important for applications where a probability-like interpretation of evidence is desired. for these reasons, the generalized jaccard coefficient is considered superior to the jousselme distance in several ways, making it a more versatile and reliable measure of distance between sets of evidence in the dempster-shafer theory. song et al. [13] proposed a method of combining evidences based on their levels of confidence with falsity. an accelerated single-value thresholding (asvt) method was considered in [14] for estimating missing values in large scale problems, which was more efficient than a singlevalue thresholding (svt) method. a three-step process was proposed in [15] by combining condition monitoring of roller bearing using enhanced dempster-shafer evidence theory 149 the weighted conflicting evidence with the credibility and uncertainty data of evidence considered according to the hellinger distance along with the belief entropy. an extended version of the evidence theory was proposed in [16], with the application to monitor the position of gas pressure in a gas pressure regulation station, for dynamic fusion of time domain information by combining the time decay factor with the relative conflict factor. an innovative evidential correlation coefficient (ecc) was proposed in [17] for decision making in measuring the conflict distance of a belief function by satisfying various criteria, such as symmetry, non-negativity, insensitivity, boundedness, and extreme consistencies. a combined method of conflict proving, called as the automatic k-means encoder (ae-k means), was proposed in [18], where a composite credibility defined based on an automatic encoder and similarity of evidence integrates both direct and indirect data about the evidence for efficiently obtaining detail data by reducing the involvement of other factors. the d-s evidence theory is used in condition monitoring as an uncertainty algorithm because of its certain benefits. the back propagation (bp) neural network, widely used as the primary method to enhance the vector machine and radial basis function, was used in [19] for preliminary diagnosis, where three preliminary diagnosis results were combined using an improved ds evidence theory. a new weighted evidence fusion method was proposed in [20,21] for monitoring bearing conditions. in [22], features from vibration signals were extracted by using a multi-dimensional feature extraction method that combines entropy feature, holder coefficient feature and improved generalized box dimension feature; followed by the calculation of the basic belief distribution using the gray correlation algorithm; and finally patterns of defects were recognized by merging the basic belief distribution using the yager method. in [23], an enhanced fusion method of the dempster-shafer theory of proof was presented for adjusting sensor weights, as per the diagnostic accuracy of different sensor data, by using the kappa coefficient as the closeness error between evidences. the reader is also referred to a recent review by hakim et al. [24]. it is observed that there is no universally recognized method for combining evidences. hence, researchers focus on improving such methods based on the d-s evidence theory. since the weight distribution of the existing evidence is not reasonable enough, more importance should be given to effective methods for combining conflicting evidences. in this article, a method of combining evidences is proposed based on the generalized jaccard coefficient and false degree, and its effectiveness is demonstrated by diagnosing failure of bearings. the aim of the study is to propose an improved approach for condition monitoring of bearings up to rupture using the dempster-shafer evidence theory and to demonstrate its effectiveness in achieving accurate diagnosis. this paper is written to present a novel approach for condition monitoring of bearings that incorporates the dempster-shafer evidence theory and to provide evidence of its efficiency in supporting failure diagnosis. the difference of the present submitted paper from the previous works is the incorporation of the dempster-shafer evidence theory for determining the level of confidence in the evidence and weighting the base probability of the original evidence, followed by the adoption of the combined dempster's compositional rule. this approach is shown to enhance the efficiency, have the least error, and better reliability in supporting failure diagnosis. 150 x. zhu, m. huang, k. chen 2. improving dempster-shafer (d-s) evidence theory the bayesian method, generalized by using the d-s evidence theory, is primarily a mathematical method to cope up with uncertainty by using the bayesian conditional probability theory. the method needs to know the likelihood probability of a problem in advance. in this case, the d-s evidence theory is more adoptable in handling uncertain data, which can express the uncertainty of a problem without knowing its likelihood probability a priori. in the d-s identification framework, a set of incompatible proposals provide many possible answers, among which one might be the right answer. the concept of "basic probability distribution (bpa)" refers to a subset of the recognition framework and the degree of confidence for each proposal. if the event criteria are met, then the value of the impossible event of the corresponding base probability distribution becomes 0. m(ϕ)=0 indicates 0 value of the probability distribution for the impossible event. it is described in the confidence function as the total probability corresponding to each subassembly in 𝐴. the likelihood function, which represents the uncertainty measure, sums up all the subassemblies bpa that intersect a and the expression of non-repudiation confidence in a. in the trust space [bel(a), pl(a)], bel(a) represents the lower limit function, and pl(a) represents the upper limit function. ∑ 𝑚(𝐴) = 1𝐴⊆θ (1) bel: 2θ → [0,1] (2) bel(𝐴) = ∑ 𝑚(𝐵) (∀𝐴 ⊂ 𝑃)𝐵⊂𝐴 (3) pl(𝐴) = 1 − bel(�̅�) = ∑ 𝑚(𝐵) − ∑ 𝑚(𝐵) = ∑ 𝑚(𝐵)𝐵∩𝐴≠𝜙𝐵⊂�̅�𝐵⊂𝑃 (4) 2.1. dempster composition rules for ∀a ⊆ θ , the rule given by the dempster combination of two different mass functions 𝑚1 and 𝑚2 is defined on θ: k as the conflict coefficient. if there are 𝑛 finite functions for masses m1, m2,…mn defined on θ, then their dempster combination rules can be expressed as follows: 𝑚1⨁𝑚2(𝐴) = 1 𝐾 ∑ 𝑚1𝐵∩𝐶=𝐴 (𝐵) ∙ 𝑚2(𝐶) (5) (𝑚1⨁𝑚2 ⊕∙∙∙ ⨁𝑚𝑛)(𝐴) = 1 𝐾 ∑ 𝑚1(𝐴1) ∙ 𝑚2(𝐴2) ∙∙∙ 𝑚𝑛(𝐴𝑛)𝐴1∩𝐴2∩∙∙∙∩𝐴𝑛=𝐴 (6) 𝐾 = ∑ 𝑚1(𝐴1) ∙ 𝑚2(𝐴2)𝐴1∩∙∙∙∩𝐴2≠𝜙 ∙∙∙ 𝑚𝑛(𝐴𝑛) = 1 − ∑ 𝑚1(𝐴1) ∙ 𝑚2(𝐴2)𝐴1∩∙∙∙∩𝐴2=𝜙 ∙∙∙ 𝑚𝑛(𝐴𝑛) (7) 2.2. evidence combining method based on generalized jaccard factor and false degree the generalized jaccard coefficient, also referred to as the tanimoto coefficient, is a method of measuring similarity, e.g., the similarities of evidences. in this paper, a method is proposed for combining evidences based on the generalized jaccard coefficient and false degree. the important points of this method are described below. ▪ input the base probabilities of 𝑛 evidences. ▪ calculate the generalized jaccard coefficient of proof using the following formula: condition monitoring of roller bearing using enhanced dempster-shafer evidence theory 151 sim(𝑚1, 𝑚2) = ∑ 𝑚1 𝑛 𝑖=1 (𝐴𝑖)∙𝑚2(𝐴𝑖) ∑ 𝑚1(𝐴𝑖)2𝑛 𝑖=1 +∑ 𝑚2(𝐴𝑖)2−∑ 𝑚1(𝐴1)∙𝑚2(𝐴𝑖)𝑛 𝑖=1 𝑛 𝑖=1 (8) ▪ calculate the trust between evidences as follows: crd𝑖 = sup (𝑚𝑖) ∑ sup (𝑚𝑖)𝑛 𝑖=1 ; where, sup(𝑚𝑖) = ∑ sim(𝑚𝑖, 𝑚𝑗)𝑛 𝑗=1,𝑗≠𝑖 (9) ▪ calculate the falsehood between evidences using the falsification formula of evidence. the false degree of evidence is a measure of evidence conflict as proposed by schubert [25]. it is determined according to the conflict coefficient of evidence as expressed by eq. (10), where k0 is the global conflict system and ki is the conflict coefficient between the remaining evidences except mi. the false degree vector of evidence has n elements, m1,m2,m3,…,mn . 𝐹(𝑚𝑖) = 𝐾0−𝐾𝑖 1−𝐾𝑖 ; where, 𝐹 = [𝐹(𝑚1), 𝐹(𝑚2),∙∙∙ 𝐹(𝑚𝑛)]𝑇 (10) ▪ the degree of trust and falsehood of evidence are obtained by using the following weight coefficient: 𝑤𝑖 = crd𝑖 + 1 − 𝐹(𝑚𝑖) (11) ▪ the weight coefficient is normalized as follows: 𝑤𝑖̅̅ ̅ = 𝑤𝑖 ∑ 𝑤𝑖 𝑛 𝑖=1 (12) ▪ after weighting the base probabilities of all the evidences using the normalized weight coefficient, calculate the average value as follows: mae = ∑ (crd𝑖 × 𝑚𝑖) 𝑛 𝑖=1 (13) ▪ combine weighted average bpa for (𝑛-1) times using the dempster synthesis rule. 3. numerical example in this paper, we take an example from the efficient combination approach of conflicting evidences for generating data, and set the identification framework as θ={f1,f2,f3}. in this case, five evidences are involved in the probability distribution, which are given below. e1: 𝑚1(𝑓1) = 0.50, 𝑚1(𝑓2) = 0.2, 𝑚1(𝑓3) = 0.30 e2: 𝑚2(𝑓1) = 0.00, 𝑚2(𝑓2) = 0.9, 𝑚2(𝑓3) = 0.10 e3: 𝑚3(𝑓1) = 0.55, 𝑚3(𝑓2) = 0.1, 𝑚3(𝑓3) = 0.35 e4: 𝑚4(𝑓1) = 0.55, 𝑚4(𝑓2) = 0.1, 𝑚4(𝑓3) = 0.35 e5: 𝑚5(𝑓1) = 0.55, 𝑚5(𝑓2) = 0.1, 𝑚5(𝑓3) = 0.35 table 1 shows the comparison of the methods used to synthesize evidences. 152 x. zhu, m. huang, k. chen table 1 comparison of methods used in evidence synthesis. item 𝑚(𝑓1) 𝑚(𝑓2) 𝑚(𝑓3) bisht and kumar [2] 0.0000 0.1228 0.8772 deng [10] 0.7958 0.0932 0.1110 song [13] 0.9636 0.0109 0.0255 present methodology 0.9703 0.0048 0.0249 4. bearing condition monitoring the proposed d-s evidence theory method for condition monitoring of bearings includes two main steps: construction of evidence body and combining evidences. 4.1. condition monitoring model based on enhanced d-s evidence theory data of the vibration signals of rolling contact bearings is used for monitoring bearing conditions. direct extraction of effective information affects the accuracy of condition monitoring. in order to avoid this problem, inaccurate signals are filtered out by using a single-signal decomposition method. in this paper, the empirical mode decomposition (emd), empirical wavelet transformation (ewt) and variable mode decomposition (vmd) approaches are used for this purpose. the characteristic information of rolling contact bearings is extracted from their signals generated due to vibration by combining the sample entropies using the relevance vector machine (rvm). this information is used for enhancing the preliminary diagnosis by improving the evidence theory. figure 1 depicts the structure of the diagnosis model. specifically, calculations of sample entropy are performed for each of the three methods, followed by mrvm fault diagnosis. thereafter, fusion diagnosis was carried out on the basis of the improved d-s evidence theory. finally, the final diagnosis results are extracted based on the decision rules. fig. 1 methodology for condition monitoring of bearings by using enhanced d-s evidence theory collect the vibration signal of rolling bearing mrvm fault diagnosis emd calculate sample entropy fusion diagnosis based on improved d-s evidence theory output final diagnosis results according to decision rules calculate sample entropy calculate sample entropy mrvm fault diagnosis mrvm fault diagnosis ewt vmd condition monitoring of roller bearing using enhanced dempster-shafer evidence theory 153 4.2. diagnosis of condition monitoring mode the condition monitoring mode is diagnosed in the following steps: ▪ collection of vibration signal of rolling contact bearing. the vibration signals of the inner race, outer race and rolling element in both fault and normal states are collected by an acceleration sensor. ▪ feature extraction. the wavelet packet, empirical mode decomposition, and vmd approaches are used for calculating the sample entropy of signal components by suppressing their vibration signals. ▪ probability is determined by performing preliminary diagnosis using the correlation vector machine. ▪ the input of evidence fusion is obtained in the form of fault probability by performing preliminary diagnosis using rvm, and then the evidence combination method based on similarity and falsehood is used to fuse the evidence bodies, so as to obtain the fused probability and finally to complete the condition monitoring. 5. experiment and result analysis the data used in this work was derived from the bearing data center of case western reserve university, united states [26]. the schematic of the test bench used for collecting experimental data is shown in figure 2, which is mainly composed of a 1.5kw motor, a decoder and a power tester. the drive-end of the motor is equipped with a bearing (model: skf6205), and the fan-end is also equipped with another bearing (model: skf6203). the vibration signal of the rolling contact bearing is measured using an acceleration sensor, which is installed at the drive-end of the motor. the measurement includes the vibration signals of the inner race, outer race and rolling element in both fault and normal states. in this experiment, 2048 sample data points were collected, including 100 samples of vibration signals for each bearing state. fig. 2 schematic of the experimental bearing test rig. 5.1. collection of vibration data of rolling contact bearing the bearing data used in this paper is the vibration data acquired by the acceleration sensor positioned at the drive-end of the motor. data is collected in different states, including the normal state (nc), irf, orf and rbf. the experimental data reveals that the combination of rvm and d-s evidence theory could more precisely complete the condition monitoring of the rolling contact bearing. figure 3 shows the amplitude of the acceleration signals. tenque sensor/decode power test meter 1.5kw[2-hp] electric motorelectionic conroter bearing test rig used for the experimentdata 154 x. zhu, m. huang, k. chen fig. 3 experimental data of amplitude of vibration of (a) irf (b) orf and (c) rbf. (d) vibration signal under normal condition 5.2. feature extraction after breaking down the vibration signal by the emd, ewt and vmd approaches, the sample entropy of the signal component is computed. then, three characteristic subspaces are constructed as the input of the correlation vector. figures 4 to 6 show the waveforms of the component signals of the faulty inner race decomposed by the emd, ewt and vmd methods. the sample entropies corresponding to the component signals obtained by the emd, ewt and vmd decomposition methods, as shown in figures 4-6, are [0.5728, 0.5814, 0.3126, 0.1925, 0.0633], [0.4603, 0.5997, 0.3063, 0.2776, 0.4518] and [0.5902, 0.5938, 0.5663, 0.1034, 0.4921], respectively. condition monitoring of roller bearing using enhanced dempster-shafer evidence theory 155 fig. 4 transformed waveforms of the original signals obtained by emd method fig. 5 transformed waveforms of the original signals obtained by ewt method 156 x. zhu, m. huang, k. chen fig. 6 transformed waveforms of the original signals obtained by vmd method 5.3. preliminary diagnosis results by vector machine for the purpose of diagnosis, the output probability is obtained by using the relevance vector machine. tables 2 to 4 present the preliminary condition monitoring results for few test samples obtained by the rvm method. table 2 results of emd sample entropy and rvm condition monitoring test sample no rolling element failure probability failure probability of inner ring failure probability of outer ring normal state probability test label actual label 1 0.9981 0.0000 0.0013 0.0006 1 1 3 0.3573 0.0412 0.0002 0.6012 4 1 4 0.9906 0.0000 0.0062 0.0032 1 1 14 0.9785 0.0000 0.0027 0.0189 1 1 17 0.9582 0.0000 0.0000 0.0418 1 1 19 0.9830 0.0000 0.0168 0.0001 1 1 43 0.5485 0.0000 0.4351 0.0164 1 3 56 0.0408 0.0001 0.9480 0.0112 3 3 it can be seen that the rolling element failure probability in tables 2 and 3 appear to be approximately inverse to the failure probabilities of the outer and inner rings, respectively, while no correlation was observed in table 4. these preliminary results are useful in elucidating the observations in the next sub-section and table 5. condition monitoring of roller bearing using enhanced dempster-shafer evidence theory 157 table 3 results of ewt sample entropy and rvm condition monitoring test sample no rolling element failure probability failure probability of inner ring failure probability of outer ring normal state probability test label actual label 1 0.9160 0.0834 0.0006 0.0000 1 1 3 0.7632 0.2368 0.0000 0.0000 1 1 4 0.4653 0.5315 0.0032 0.0000 2 1 14 0.8586 0.1176 0.0000 0.0238 1 1 17 0.9920 0.0079 0.0000 0.0001 1 1 19 0.9745 0.0255 0.0000 0.0001 1 1 43 0.0000 0.0012 0.9983 0.0006 3 3 56 0.0002 0.0036 0.9962 0.0000 3 3 table 4 results of vmd sample entropy and rvm condition monitoring test sample no rolling element failure probability failure probability of inner ring failure probability of outer ring normal state probability test label actual label 1 0.9996 0.0003 0.0001 0.0000 1 1 3 0.9982 0.0000 0.0018 0.0000 1 1 4 0.9998 0.0000 0.0002 0.0000 1 1 14 0.1517 0.0001 0.8482 0.0000 3 1 17 0.4605 0.5260 0.0135 0.0000 2 1 19 0.0551 0.5337 0.4112 0.0000 2 1 43 0.0018 0.1450 0.8531 0.0000 3 3 56 0.4336 0.1355 0.4309 0.0000 1 3 5.4. final diagnosis results by improved evidence combination method for fusion preliminary diagnosis is performed by rvm, where the failure probability is obtained as the evidence. the final diagnosis is performed by the method based on the combination of the generalized jaccard coefficient and false degree, where each evidence body is fused to obtain the probability to be used in condition monitoring. table 5 shows the evidence of this combination method, and the results of the combined evidences are shown in tables 2-4. table 5 condition monitoring results obtained by the proposed fusion method test sample no rolling element failure probability failure probability of inner ring failure probability of outer ring normal state probability test label actual label 1 1.0000 0.0000 0.0000 0.0000 1 1 3 0.9954 0.0018 0.0000 0.0028 1 1 4 0.9982 0.0018 0.0000 0.0000 1 1 14 0.9993 0.0002 0.0005 0.0000 1 1 17 0.9979 0.0021 0.0000 0.0000 1 1 19 0.9999 0.0000 0.0000 0.0000 1 1 43 0.0025 0.0003 0.9972 0.0000 3 3 56 0.0018 0.0000 0.9982 0.0000 3 3 158 x. zhu, m. huang, k. chen a total of 80 test samples were considered in this experiment, 20 samples in each state. the accuracies of the initial condition monitoring based on the emd, ewt and vmd sample entropies and rvm were found to be 97.5%, 98.75% and 95%, respectively. finally, the enhanced d-s evidence combination procedure was employed for fusion to complete the diagnosis. 6. conclusion in this paper, taking rolling contact bearing as the object, bearing fault features are studied by different methods, and then the relevant vector machine is used to determine the probability. finally, the types of faults are identified by using an improved method based on the d-s evidence theory for fusion, which does not figure out the fundamental probability distribution. the probability obtained by rvm is applied as the bpa value, and then the enhanced evidence combination method is applied to diagnose the bearing fault. compared with a single method, the diagnosis of the proposed method was found more accurate and reliable. in conclusion, the proposed approach that incorporates dempster-shafer proofs for determining the level of confidence in evidence and determining the weight of proof in terms of trust and falsity has shown promising results in condition monitoring of bearings up to rupture. the combination of the relevance vector machine and the enhanced evidence combination method based on the generalized jaccard coefficient and false degree has enhanced the efficiency and achieved accurate diagnosis with the least error and better reliability in supporting failure diagnosis. the experiment's results on 80 test samples with different states have shown high accuracies of the initial condition monitoring based on the emd, ewt, and vmd sample entropies and rvm. finally, the enhanced d-s evidence combination procedure was employed for fusion to complete the diagnosis. overall, this approach provides a new perspective for condition monitoring that can improve the diagnosis and reliability of the system. acknowledgement: this work was supported by the guangdong basic and applied basic research foundation (grant no. 2018a030307038), projects of phds’ start-up research of gdupt (grant no. bs-xj2022000501), national natural science foundation of china (grant 62073090), and the key research platform and project of universities in guangdong province (2020zdzx3038). we would like to thank the guangdong university of petrochemical technology. project number: 2019rc076 (702-519186, 702-71013303119, 702-72100003102, 702-72200010122, 702-72200010358). references 1. yang, b., gan, d., tang, y., lei, y., 2020, 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physical science and engineering division, king abdullah university of science and technology (kaust), saudi arabia abstract. with the technological development of wearable devices, there are increasing demands for stretchable conductor that have stable electro-mechanical performance. in this study, a stretchable pdms composite electrodes using ternary systems of fillers consisting of poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (pedot:pss) / carbon nanotube (cnt) / silver nanowire (agnw) is explored in a perspective of electromechanical response. pdms matrix is mixed with binary fillers of cnt and pedot:pss, which is followed by agnw peeling-off process. the pdms composite is mechanically reliable especially under tensile deformation, which showed a high rupture strain of ~102% and tensile strength of ~2.7 mpa. in addition, the pdms composites shows the stable electro-mechanical response, where high electrical conductivity is sustained even under stretchable conditions, showing an electrical resistance value of ~11.7 ω/cm under 40% of strain. as a demonstration, a supercapacitor using the pdms composites is demonstrated that shows reliable electrochemical performance. key words: electro-mechanical response, composite, tensile deformation, strength, conductivity 1. introduction stretchable electronics have attracted much attention for futuristic devices such as wearable health care sensors [1-3], energy harvesting or storage systems [4-7], and data processing devices [8, 9]. a key component of developing a stretchable electronic device is a stretchable conductor that is mechanically reliable under repeated tensile or compressive deformation [10]. there are two main trends to fabricate stretchable conductors: 1) utilizing all stretchable materials [11, 12] and 2) bridging rigid components received: december 05, 2022 / accepted: january 15, 2023 corresponding author: byungil hwang school of integrative engineering, chung-ang university, 84 heukseok-ro, dongjak-gu, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr 52 h. kim, n. qaiser, b. hwang with stretchable interconnects [13-16]. the thin metal film electrodes patterned to have serpentine structure is the representative example for the latter case of the stretchable interconnects [17]. the serpentine-structured metal electrodes interconnect the si-based electronic components having excellent device performances while releasing the imposed mechanical strain by accommodating their geometry under deformation, thereby realizing stretchable and ultra-high-performance electronic devices [18]. despite the advantages of the method utilizing patterned metal interconnects, the use of such methods for electronic devices is limited due to the complex fabrication process using a multistep lithography process [19]. on the other hand, the fabrication of composite films by mixing functional fillers with the stretchable matrix is the most widely used strategy to fabricate the stretchable conductor in the former case [12, 20-23]. the composite films with different properties can be fabricated by introducing different types of functional fillers, which allowed the fabrication of various electronic devices [20, 24, 25]. for example, lee et al. fabricated a stretchable conductor by integrating silver nanowires (agnws) and graphene with polydimethylsiloxand (pdms), which showed a rupture strain of 100% [26]. in the work by shi et al, graphene hybridized cnt film with pdms matrix for wearable sensors that resists under cyclic mechanical strain was developed [27]. gallium-based liquid metal alloy (egain) is used as a conductive filler as well. gallium and indium are mixed in a ratio of 3:1 desirably and directly mixed with liquid metal-filled magnetorheological elastomer (lmmre), showing high electrical conductivity and mechanical deformability [28]. however, more research related to electro-mechanical response is needed for use in stretchable electronic systems for wearable devices. in our previous study, the hybrid system of poly(3,4-ethylenedioxythiophene):poly (styrenesulfonate) (pedot:pss), carbon nanotubes (cnts), and agnws was proposed as the supercapacitor materials, which acted as pseudocapacitor, electrical double layer (edl), and current collector materials, respectively [29]. in this system, the advantages of a pseudocapacitor with high energy density and edl with high power density were combined, thereby exhibiting excellent supercapacitor performances [30]. in addition, the high electrical conductivity of the agnw network-based current collector optimized the supercapacitor performances [31]. however, the layers of pedot:pss, cnts, and agnws were deposited on silk fiber as a form of the thin film; thus, the use of the supercapacitor for stretchable devices was limited. in this study, we demonstrated mechanically stable conductive pdms composites using the hybrid system of pedot:pss, cnts, and agnws as functional fillers. as commented above, the hybrid system took benefits of each materials, where pseudocapacitor and edl characteristics were come from pedot:pss and cnt, respectively. a simple blending and peeling-off method were used to fabricate the composites where molten pdms was mixed with pedot:pss and cnts and then was coated with agnws. to improve the conductivity of pedot:pss, ethylene glycol (eg) was added to pedot:pss solution as a doping agent. the water-soluble pedot:pss was able to be mixed with the hydrophobic pdms by reducing the hydrophobicity of pdms aided by the surfactant, triton x-100. tensile tests were performed on the pdms composites with different filler contents, enhancing the composition of the matrix and filler materials. the mechanical failure mechanism of the pdms composites with the varying filler composites was also investigated by microstructural analysis using scanning electron microscopy (sem). the electrochemical performance of the optimized pdms composites with the hybrid fillers of electro-mechanical response of stretchable pdms composites with a hybrid filler system 53 pedot:pss, cnts, and agnws was characterized and confirmed the potential of the demonstrated composites for a stretchable supercapacitor. 2. materials and methods 2.1. fabrication of cnt/pdms/pedot:pss composites cnt (jeno tube 6a, jeio) was first dispersed in a deionized (di) water with the surfactant, sodium dodecylbenzene sulfonate (50.0%, sdbs, samcheon chemical co. ltd, korea). then, a tip sonicator was used to disperse cnt more uniformly at 60% amplitude of 500 w and 20 khz for 10 min with a 3 mm diameter tip. pedot:pss, eg (sigma aldrich), and triton x-100 (daejung, korea) were blended together using a magnetic bar at 450 rpm for 1 h. after that, the mixed pedot:pss, eg, and triton x-100 added to the dispersed cnt was evaporated on a hot plate at 150 c for 20 min. the solvent-removed mixture of functional fillers (cnt/pedot:pss), and pdms (sylgard-184, dow corporation, michigan, usa) was blended in a planetary vacuum mixer (arv-310, thinky corporation, tokyo, japan) at 2000 rpm under 20 kpa for 5 min with a curing agent. agnw solution (flexiowire 2020, sg flexio co. ltd, korea) was coated on a polytetrafluoroethylene (ptfe) casting molds (8.0 (w) × 48.0 (l) × 0.3 mm (t) and 40.0 (w) × 30.0 (l) × 0.3 mm (t)) at 100 c for 1 h in an oven. the previously mixed cnt/pedot:pss/pdms composite blend was then poured on the surface of agnw-coated ptfe mold. prior to the curing process, the cnt/pedot:pss/pdms composite blend was degassed in a vacuum at room temperature under a 15 kpa for 2 h and later cured in a vacuum oven at 100 c for 6 h. the agnw-coated cnt/pedot:pss/pdms composite was fabricated via the peeling-off method from the ptfe mold. 2.2. characterization field-emission scanning electron microscopy (fe-sem; sigma, carl zeiss ag, oberkochen, germany) was utilized to analyze the surface and cross-section morphologies of the fabricated composite. a rheometer (compac-100ii, sun scientific co. ltd., tokyo, japan) was used for tensile testing under a maximum load cell of 100 n with a table speed of 120 mm/min. a high-resolution drop shape analyzer (dsa 100, krüss gmbh) was utilized to evaluate the contact angle of the fabricated sample to analyze its wettability. a battery cycler system (wbcs3000l, wonatech, seoul, korea) was used to measure the electrochemical property of the composite by cyclic voltammetry (cv). 3. results and discussion fig. 1 illustrates the fabrication process of the agnw-coated cnt/pedot:pss/pdms composites. first, cnts were dispersed in di water by using a tip sonicator (fig. 1a). the cnts having hydrophobic surface tend to be agglomerated in hydrophilic di water. therefore, an anionic surfactant, sodium dodecyl benzene sulfonate (sdbs) was added, where amphiphilic sdbs enhanced the dispersion of cnt in di water (inset of fig. 1a) [32]. afterward, the remaining solvent was removed through the drying process. next, pedot:pss solution was mixed with triton x-100 and eg through stirring (fig. 1a). the 54 h. kim, n. qaiser, b. hwang weight fraction of the pedot:pss solution was varied from 0 to 20 wt%, in order to examine the effect of the concentration of pedot:pss on the stretchability of the composites. 15 wt% of pedot:pss solution was used for all the composites of this study because more than 15 wt% of pedot:pss dramatically degraded the stretchability (fig. 2). the high content of pedot:pss caused the mechanical rupture at the much lower strain than those with low content of pedot:pss as shown in fig. 2. the early rupture resulted in the low strength value. eg was added to increase the size of the conductive grain, cause agglomeration, and increase the carrier mobility, enhancing the electrical conductivity of the pedot:pss [11]. moreover, it disconnects the insulating pss part from pedot:pss which results in a further increase of the conductivity [33]. triton x100 serves as a surfactant, which allows for hydrophilic pedot:pss to be mixed with hydrophobic pdms [34]. the mixture of pedot:pss/eg/triton x-100 and the treated cnts were then mixed with the pdms and curing agent by using a planetary vacuum mixer (fig. 1b). agnws were coated on the fabricated polymer blend film via a peeling-off method to further decrease the resistance, leading the film to have higher electrical conductivity. here, agnw solution was first poured on the ptfe casting mold followed by drying in an oven (fig. 1c). lastly, the mixture of cnt/pedot:pss/pdms was poured on the agnw-coated ptfe mold. agnw was embedded to the molten mixture of cnt/pedot:pss/pdms, causing stronger adhesion between the two layers. in order to eliminate the air bubbles in the composites, degassing process was performed, which was then cured in a vacuum oven (fig. 1d). by peeling off the cured composite from the ptfe mold, the stretchable agnw/cnt/pedot:pss/pdms composite electrode was fabricated (fig. 1e). fig. 1 schematic illustration of the fabrication process of the agnw coated cnt/pedot: pss/pdms composite. electro-mechanical response of stretchable pdms composites with a hybrid filler system 55 fig. 2 stress-strain curves of pdms composites with the different pedot:pss contents. fig. 3 shows the cross-sectional sem images of the composites with different fillers. pure pdms (fig. 3a) and pedot:pss/pdms (fig. 3b) showed smooth surface indicating that the pedot:pss was homogeneously dispersed in pdms without severe agglomeration. in the cnt/pedot:pss/pdms composites (fig. 3c), the cnts were distributed through the bulk part of the composites and no severe agglomerated cnts were observed. this confirmed that our dispersion method for cnts and pdms was effective to fabricate the uniform cnt/pedot:pss/pdms composites. in fig. 3d, agnws were embedded on the surface of cnt/pedot:pss/pdms composites. during the curing step, agnws on the ptfe mold were partially immersed in the mixture of cnt/pedot:pss/pdms, which was then bound to the pdms matrix forming the partially embedded agnw networks. fig. 3 cross-sectional sem images of (a) pure pdms, (b) pedot:pss/pdms composite, (c) cnt/pedot:pss/pdms composite (the arrow indicates the presence of cnt), and (d) agnw-coated cnt/pedot:pss/pdms composite (dash line indicates the boundary between agnw and the cnt/pedot:pss/pdms mixture). fig. 4a shows the stress-strain curves of the various composites mixed with a different mixture of fillers, which were pure pdms, pdms/pedot:pss, cnt/pdms/pedot:pss, and agnw-coated cnt/pdms/pedot:pss, respectively. the failure stress and strain decreased as the fillers were added. pure pdms exhibited a strain and stress of ~235% and ~3.7 mpa, respectively, which were decreased to ~165% and ~2.8 mpa, respectively, with the addition of 15 wt% of pedot:pss. the cnt/pdms/pedot:pss showed a further 56 h. kim, n. qaiser, b. hwang decrease in strain and stress values showing ~102% and ~2.7 mpa, respectively. the coating of agnw on the surface of pdms/pedot:pss/cnt composite showed similar stress and strain values compared to those of the pdms/pedot:pss/cnt composites. the addition of the brittle pedot:pss resulted in the decreased volume fraction of the stretchable pdms in the composites. thus, the pedot:pss/pdms composites showed lower stretchability than those of pure pdms. the addition of cnts further decreased the volume fraction of pdms, which resulted in the poorer stretchability of cnt/pedot:pss/pdms composites than those of pedot:pss/pdms composites. in addition, the applied stress can be localized at the interface between cnts and pdms, which further degrades the stretchability. however, the partial embedding of agnws only on the surface of the composites showed no significant change in the volume fraction of pdms in the composites, as well as the limited regional stress concentration, resulted from the agnws, thereby showing similar stretchability to those of cnt/pedot:pss/pdms composites. according to the rule of mixture [35], 𝐸𝑐 = 𝑓𝑚𝐸𝑚 + 𝑓𝑓𝐸𝑓 (1) where ec, em, ef, fm, and ff are the modulus of composite, matrix, filler, and the fraction of matrix and filler, respectively. the elastic modulus of the pdms is expected to increase as the hard fillers such as pedot:pss or cnt are added. fig. 4b shows the change in the elastic modulus of pdms composites depending on the added fillers. the elastic modulus, e, was calculated by using a general equation, 𝐸 = 𝜎 𝜀 , (2) where 𝜎 ans 𝜀 are stress and strain values extracted from the stress-strain curves in fig. 4a. pure pdms showed a modulus value of 0.9 mpa ± 0.10, which was in the range of data sheet reported in the previously reported study [36]. with the addition of pedot:pss, the modulus increased to 1.5 mpa ± 0.11. because the elastic modulus of pedot:pss was reported as ~2.0 gpa [37], the increased modulus values of the composites were confirmed to be due to the addition of pedot:pss. the modulus of the composites was further increased by the addition of cnts, which showed a 2.0 mpa ± 0.12 [38]. the elastic modulus of cnts is reported to be ~900 gpa [39]; thus, even the small addition of cnts (~2 wt%) resulted in a dramatic increase in the modulus of the composites. the partial embedding of agnws only on the surface of the composites showed no significant change in the modulus values; thus, it showed a similar value of 2.0 mpa ± 0.11 to those of cnt/pedot:pss/pdms composites. because of the increased elastic modulus of the composites, the stress values at the same level of a strain of the composites was much higher than those of the pure pdms. meanwhile, the rupture stress of the composites was smaller than those of the pure pdms since the failure of the composites occurred at the much lower strain than those of pure pdms. however, the rupture strain of the agnwcoated cnt/pedot:pss/pdms composites still showed a sufficient value of ~102% for wearable devices, where ~70% strain is considered to be the maximum strain values exerted by human motion [40]. electro-mechanical response of stretchable pdms composites with a hybrid filler system 57 fig. 4 (a) stress-strain curves and (b) young’s modulus of pdms composites using different types of functional fillers. fig. 5a shows the electro-mechanical response of the agnw-coated cnt/pedot: pss/pdms composites as a function of imposed strain. the resistance value of cnt/pedot: pss/pdms composites was too high to be used for electrodes of the stretchable supercapacitor, which showed ~1.5 x 106 ω/cm. the resistance of the composites decreased to ~11.7 ω/cm by coating the agnws. therefore, the coating of agnws on the composite was necessary to fabricate the stretchable composite electrode using cnt/pedot:pss/pdms composites. the agnw-coated cnt/pedot:pss/pdms composite was highly stretchable under tensile strain. there was no obvious change in the resistance up to the strain of 40%. the resistance started to increase from the strain value over 40%, and from the 50% of strain, a rapid increase in the resistance was observed. considering that the conductivity of the composites was largely contributed by the agnws, the increase in the resistance over 50% of strain was attributed to the loss of the percolation of the agnw networks by the stretching. fig. 5b and 5c show the top-view sem images of agnw-coated cnt/pedot: pss/pdms composites under 40% and 60% of tensile strains, respectively. there was no trace of cracks or failure of the composites under 40% of strain as shown in fig. 5b. at the 60% strain, however, the composites clearly showed failures on the surface (fig. 5c). therefore, the agnw networks on the surface of the composite have been disconnected which resulted in the rapid increase in resistance under the severe tensile strain of over 50%. the electrochemical performance of agnw-coated cnt/pedot:pss/pdms composites was characterized by cv using coin cell configuration with 1 m of na2so4 aqueous electrolyte (fig. 6). the cv curve was displayed at a scanning rate of 10 mv/s, 50 mv/s, 100 mv/s, and 200 mv/s under an operation potential window between 0 to 0.8 v. the cv curve was devoid of any redox peaks within a potential range and retained sharp rectangular shapes, demonstrating the typical behavior of an electrical double-layer capacitor. in addition, on increasing scan rate, the shape of the potential window remained similar, and the integrated area of the cv curve increased with the increasing scan rate. this indicates that the supercapacitor has faster charge/discharge capability, verifying the high-rate performance of the supercapacitor. the specific capacitance values were measured to be 248.59 mf/g, 81.77 mf/g, 53.26 mf/g, and 32.55 mf/g at different scan rates of 10 mv/s, 50 mv/s, 100 mv/s, and 200 mv/s, respectively. as the scan rate increases, the specific capacitance value decreased, because of the slower ion diffusion [41]. the agnw-coated surface was hydrophilic, while the 58 h. kim, n. qaiser, b. hwang cnt/pedot:pss/pdms composite part was more hydrophobic than agnw coated surface, due to pdms (figs. 6b-e). therefore, the aqueous electrolyte might not sufficiently permeate through the electrode, causing slower ion diffusion. fig. 5 (a) normalized resistance change of the agnw-coated cnt/pedot:pss/pdms composites as a function of tensile strain values. top-view sem images of the agnw-coated cnt/pedot:pss/pdms composites under the tensile strain of (b) 40% and (c) 60%. fig. 6 (a) cyclic voltammetry (cv) curves for the agnw coated cnt/pedot:pss/pdms composite at different scan rates. (b-e) optical images of water drop shape on (b) pure pdms, (c) pedot:pss/pdms, (d) cnt/pedot:pss/pdms, and (e) agnw coated cnt/pedot:pss/pdms. electro-mechanical response of stretchable pdms composites with a hybrid filler system 59 4. conclusion in this study, the electro-mechanical response of pdms composites with hybrid filler were explored. the cnt/pedot:pss/pdms composite was fabricated via a simple blending method and agnw was successfully coated on the surface of the composite using the peelingoff method. the fabricated composite not only showed high mechanical reliability under tensile deformation but also maintained electro-mechanical stability, showing a gradual increase in resistance under high elongation values. moreover, the conductive functional fillers acted effectively as synergetic materials for supercapacitors, demonstrating better electrochemical properties than those using single fillers. the developed agnw coated cnt/pedot: pss/pdms composites with high 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1938-1955. 41. li, y., zhao, x., xu, q., zhang, q., chen, d., 2011, facile preparation and enhanced capacitance of the polyaniline/sodium alginate nanofiber network for supercapacitors, langmuir, 27(10), pp. 6458-6463. implementation of the lean-kaizen approach facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 145 161 doi: 10.22190/fume161228007k © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper implementation of the lean-kaizen approach in fastener industries using the data envelopment analysis udc 519.85:621.7 sunil kumar 1 , ashwani kumar dhingra 1 , bhim singh 2 1 mechanical engineering department, university institute of engineering & technology, maharshi dayanand university, india 2 mechanical engineering department, sharda university, india abstract. this research paper is an attempt to improve the quality system of ten small scale fastener manufacturing industries through the implementation of the lean-kaizen approach using the data envelopment analysis (dea) charnes cooper & rhodes (ccr) model with constant returns to scale (crs). output maximization is taken as the objective function to identify the percentage scope of improvements. the data is collected by paying personal visits to the selected industries for three inputs (manpower, maintenance, and training of employees) and two outputs (quality, on-time delivery) of their quality system. the dea ccr model is applied to identify efficiency scores of the quality system by taking the most efficient industry as a benchmark for the rest of the organizations. the lean-kaizen approach is applied to identify waste / non-value added activities in outputs of the selected industries. four kaizen events are proposed to eliminate waste / non-value added activities in their quality system. the data collected after the kaizen events are further analyzed by the dea ccr model. the improvements in efficiency scores of the selected industries are presented as findings in this research paper. two fastener industries became 100% efficient while the rest of the organizations reported 8% to 49% improvements in their efficiency scores of the quality system. the conclusions are made as the lean-kaizen using dea is found to be an effective approach to improve the quality system of fastener industries. this study will be beneficial for researchers, practitioners and academicians for tackling the inefficiencies in the organization. key words: lean-kaizen, quality management system, brainstorming, data envelopment analysis received december 28, 2016 / accepted february 19, 2017 corresponding author: sunil kumar mechanical engineering department, university institute of engineering & technology, maharshi dayanand university, india e-mail: sunil.panchal2007@gmail.com 146 s. kumar, a. k. dhingra, b. singh 1. introduction quality is understood as a measure of excellence or a synonym of zero defects, zero deficiencies or absence of variations in the product by many industries. in order to achieve the desired product quality, the quality system performance is continuously monitored and evaluated for the sake of constant improvements of customer satisfaction, morale and reliability. the adoption of the lean-kaizen approach improves the organization output by solving problems through identifying and implementing small improvements in process, product, and system [1-2]. so, the lean-kaizen approach is required to be implemented in order to produce quality products by eliminating waste (muda) in the entire system of the organization [3]. the lean-kaizen technique, as a novel one, is composed of two basic words i.e. lean and kaizen which implies continuous elimination of waste through small-small improvements [4]. it is adopted for waste identification and elimination; it helps industry to be lean [5-6]. it is a systematic way that focuses on continuous improvement of the process, productivity, and quality of the product by suggesting effective and efficient kaizen events [7]. leanness can also be defined in terms of efficiency and effectiveness of the manufacturing system [8]. many methods such as analytical hierarchy process (ahp), data envelopment analysis (dea) and fuzzy-technique for order preference by similarity to ideal solution (topsis) are available for measuring performance of the organization. the multiple databases such as a number of employees, maintenance, training of employees, quality, on-time delivery and many other variables of industry make it complex to measure the quality system efficiency. many analytical tools are available to calculate the efficiency score of the quality system, but dea is one of the simplest and efficient tools which resolve this complexity more easily and effectively than other alternative methods as it does not perform pair wise comparison (ahp) nor does it require expert system for evaluation (fuzzy-topsis). some other advantages of dea as analysis tool [9] are discussed as follows:  multiple inputs and outputs (controllable and non-controllable) variables can be easily analyzed to obtain technical efficiency (te).  each decision-making unit (dmu) is compared with other dmus that provide te with best-performed dmus set as the benchmark/peer for each inefficient unit.  no prior weight of inputs and outputs is required.  both strategies such as input minimization and output maximization can be achieved. because of these advantages, dea has been applied to rank many organizations in order to achieve improvements through benchmarking process. in this research paper, the dea ccr model with crs is applied to assess the efficiency of quality system of ten small scale industries manufacturing fasteners (producing the same sort of products) by taking three inputs (manpower (no‟s), maintenance (%), training of employees (hours)) and two outputs (quality (%), on-time delivery (%)) of the quality system. the data is collected by paying a personal visit to the selected industries. the objective function as output maximization is performed in order to identify the possible percentage of improvements in the quality system outputs. the lean-kaizen concept is applied in order to identify waste; the kaizen events are proposed as a solution to eliminate waste in the quality system. after the kaizen events, the data is collected and further analyzed by the implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 147 dea ccr model and finally, the improvement in efficiency score of the selected industries is recorded. two industries scored 100% efficiency while the rest of them reported 8% to 49% improvements in their efficiency scores of the quality system. 2. literature review 2.1 kaizen and lean manufacturing concept continuous improvement or kaizen implies those small radical changes or the result of innovative ideas which take place over time without investing huge capital. in 1981, kaizen is introduced and implemented by masaaki imai in japanese companies to sustain continuous improvement in process, product, and system by focusing on the elimination of waste, defects, variations and deficiencies by the active participation of workforce. it is comprised of two words, i.e. „kai‟ means „change for‟ and „zen‟ means „betterment‟, simply known as „continuous improvement‟ or „change for the betterment‟ [10-11]. kaizen is carried out by trained and skillful employees in order to achieve potential improvements in the quality performance of the organization. shah and ward [12] discussed the objectives of a kaizen in the workshop which make people's jobs easier by taking innovative actions to improve the industry performance. suarez-barraza et al. [13] proposed personal kaizen for individuals to attain improvement in their quality of life. imai [14] demonstrated kaizen‟s umbrella concept to identify the process for reducing waste [15-16]. taiichi ohno [17] developed toyota production system (tps) and introduced „lean‟ as a process focused on identification and elimination of non-value added activities [1820] or waste (muda) in all systems and processes of the organization. the concept got popularized by the famous book “the machine that changed the world” [21]. kaizen is building a block of lean thinking [22-25]. several researchers [26-29] examined the critical success factors for implementation of lean manufacturing and kaizen within smes. moeuf et al. [30] examined strengths and weaknesses of smes regarding the implementation of lean manufacturing and concluded that the absence of functional organization, deficiency of formal procedures and lack of methodology are major difficulties experienced by smes during lean implementation. eaidgah et al. [31] presented a visual framework based on a lean approach to measure performance of management and continuous improvement systems in the manufacturing industry. 2.2 lean-kaizen with dea lean-kaizen as a simple improvement technique provides a better scope of improvements which helps to tackle all types of inefficiencies in all types of organizations. it also provides a better understanding for the organizations to take part in achieving their goals and to attain continuous improvements in quality of products [32-33]. the dea model also helps managers to tune quality system variables in such a way that the entire system will become efficient and effective. mishra and patel [34] used the dea model in supply chain management to improve the level of customer services and attain continuous improvement in process control. kuah et al. [35] applied dea as a benchmarking tool to measure and evaluate inefficient areas in quality systems for improvements. xie et al. [36] applied dea to measure environmental management efficiency of manufacturing sector. dabestani et al. [37] used dea to rank service quality dimensions using importance148 s. kumar, a. k. dhingra, b. singh performance analysis and to compare the outcomes for the customer groups. lau [38] pointed out that dea is the most powerful tool to access relative efficiency just as it is more sensitive for consideration of input and output variables. the study concluded that dea does not provide any guideline to choose these variables for efficiency analysis and hence researchers are free to choose their own inputs and outputs variables. jafarpour et al. [39] employed dea for evaluating performance of 30 esfahan‟s steel industries based on the suggestion system and concluded that any organization can improve its performance through raising awareness of managers, achieving solutions by suggestion system, through benchmarking, promoting motivation and improving bonus system. azadeh et al. [40] proposed an integrated approach to simulation and taguchi method with the dea model to select an efficient supplier in a closed loop supply chain in which inputs and outputs are selected so effectively as to minimize cost level and maximize a number of highquality products. warning [41] introduced dea model as a tool to support employee selection by the human resource department and used the weights assigned by the managers for individual applicants to calculate efficiency scores. bian et al. [42] proposed centralized dea and evaluated efficiencies of the parallel systems by shared inputs and outputs. wu et al. [43] proposed a two-stage dea model to evaluate sustainable manufacturing performance by using recycled and re-used waste of chinese iron and steel industry. emrouznejad et al. [44] proposed a multiplicative dea model for ranking forecasting techniques which help forecasters to make a decision for choosing the best forecasting methods in order to minimize waste. amirteimoori et al. [45] described variable reduction in dea. in addition, dea does not demonstrate price information in calculating efficiency score for each dmu; thus it can be appropriately useful for nonprofit organizations where price information is not available. the dea model has also been successfully applied to evaluate, measure and compare efficiencies in respective fields such as supply chain management [46], power plants [47], vendor selection [48-50], quality circle [51], transportation [52], education system [53], hospitals [54], etc. after studying referential literature, it is concluded that the application of various lean tools and its associated benefits in industries are well documented in the literature, but very few studies witness the application of lean-kaizen concept using dea. 3. research methodology the present case study is carried out in ten small scale fastener industries situated in the non-capital region that produces homogenous products for the general market. the dea ccr model with crs is selected in order to calculate efficiency score of all selected fasteners industries. the multiple data was collected from the selected industries in which various processes like forging, rolling, heat treatment, plating, final inspection, packing, and dispatch are carefully observed for improvements. the first personal visit is taken for collecting data pertaining to the quality system from all selected dmus. selection of inputs and outputs for the dea model is done on the basis of the mutual coefficient of correlation. then a dea model is constructed and efficiency score of the quality system is calculated for all the selected industries by taking the most efficient dmu as a benchmark for the rest of the organizations. the second personal visit is paid to the benchmark industry. all the factors which make the industry efficient and that can be taken as kaizen events for the inefficient organizations have been noted. in the third implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 149 personal visit, the lean-kaizen concept is applied to all inefficient industries in order to eliminate waste and maximize output. in the last and final visit, all the data is recorded for the result analysis and conclusions. 3.1. data envelopment analysis (dea) charnes et al. [55] developed a mathematical programming technique to evaluate the relative efficiency of organizational units known as decision-making units and identify efficient frontier by evaluating input and output set of objects [56-57]. selected fasteners industries are considered as dmus. n is the number of dmus under analysis and k is the dmu being assessed from the set of dmuk: where k = 1,..., n dmus. in this study, m = number of inputs produce, s = number of outputs, xik = observed input i at dmuk: i = 1,…,m, yrk = observed output r at dmuk: r = 1,…,s, ur = weight of output r for all, r = 1,…,s, vi= weight of input i for all, i = 1,…,m. efficiency e of dmuk is measured as: (p1) 1 1 s r rkr k m i iki u y e v x      (1) to maximize the above fractional programming problem (fpp), the mathematical formulation is: (p2) 1 1 s r rkr k m i iki u y maxe v x      (2) subjected to: 1 1 0 1; 1,...., s r rkr k m i iki u y e k n v x         (3) constraints are: srmiuv ri ,..,1;,....,1;0,  (4) the objective function (p2) is non-linear and fractional by nature and difficult to solve for te, but it can be shortened by converting it into linear programming problems (lpp) through normalization of the denominator. the lpp formula is possible by minimizing the weighted sum of inputs, setting the weighted sum of outputs equal to unity. this dea model refers as a ccr output maximization model with crs. crs assumes that returns are constants in the case of the ccr model which simply means that if input increases, the output will increase in the same proportion. (p3) 1 s r rkr maxz u y   (5) subjected to 1 1 m i iki v x   (6) constraints 1 1 0; 1,.., s m r rk i ikr i u y v x k n       (7) , ; 1,...., ; 1,..,i rv u i m r s    (8) 150 s. kumar, a. k. dhingra, b. singh where e is an arbitrary small positive number that ensures the positive values of input and output weights. the dual model of the objective function (p3) is derived by assigning dual variables to each constraint. (p4) 0 1 1 [ ] m s i ii k minw s s       (9) subjected to 01 n ik k ik ik x w x s     (10) 1 n rk k rk ik y y s     (11) ,0,,  rik ssy (12) where  rik ssw ,,,0  are dual variables. dual variable k limits efficiency of each dmu to not greater than one. the positive value of k in the dual identifies the benchmark or a peer group for inefficient dmu. this dea ccr model assumes to be an input-oriented crs model in which the input level is minimized as much as possible by keeping the current output level. the efficient target can be obtained as follows: misxx iikik ,..,1,'   (13) srsyy irkik ,..,1,'   (14) from eq. (9-14), it is clear that dmuk is efficient if and only if ф = 1, and 0  ii ss for all i and k and inefficient if and only if ф = 0, and 0,0   ii ss for all i and k where an asterisk denotes a solution value in an optimal solution. 3.2. input and output variables selection out of many variables of the quality system, a consensus decision is taken on selecting five most critical variables such as manpower, maintenance, training of employees, quality and on-time delivery that prominently affect the current quality system of selected industries as inputs and outputs for the dea ccr model by conducting brainstorming session [58] with managers, supervisors and skilled manpower of all the selected dmus. the description and unit of selected inputs and outputs are given in table 1. 3.3. data collection and data analysis for selected inputs and outputs, the data is recorded from different concern departments of the all selected dmus over a period of six months (table 2). the analysis of collected data is done by calculating correlation coefficients for all inputs and outputs as shown in table 3. input x3 is excluded due to a high value of correlation coefficient (0.55) against output y2 [59-62]. the finally selected inputs and outputs for the dea model under the study are shown in fig. 1. implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 151 table 1 input and output description and units 3.4. model selection the dea ccr model with crs is proposed to evaluate the quality performance of the selected dmus. the inputs (x1 & x2) and outputs (y1 & y2) are used to compute the efficiency score of the selected dmus, set the benchmark or peer groups for inefficient dmus, and measure the percentage of potential improvements by using output oriented optimization mode (maximized output). the model facilitates the lean-kaizen concept implementation for the selected dmus to be efficient. table 2 value of inputs and outputs dmu names (1) inputs outputs x1 (2) x2 (3) x3 (4) y1 (5) y2 (6) dmu-1 8 4.32 103 55.22 98 dmu-2 11 16.87 92 66.89 85 dmu-3 25 5.59 72 71.03 81 dmu-4 10 18.27 111 79.04 74 dmu-5 27 2.67 92 73.08 73 dmu-6 27 18.98 56 93.22 69 dmu-7 5 1.09 98 95.11 96 dmu-8 27 22.98 81 65.35 75 dmu-9 9 2.98 89 72.98 89 dmu-10 15 11.69 70 71.56 66 maximum 27 22.98 111 95.11 98 minimum 5 1.09 56 55.22 66 average 16.4 10.54 78.5 74.35 80.6 variable type name of variables / notation unit description input manpower (x1) numbers total number of employees in quality department. the manpower data was collected from human resource management (hrm) department of all selected dmus. maintenance (x2) percentage machine maintenance time/ planned production time *100 the machine maintenance data was obtained from production and maintenance departments. training to employees (x3) hours total monthly training imparted in hours/numbers of employees *100 training data was obtained from hrm departments of all selected dmus. output quality (y1) percentage number of quality product in a lot/ lot size*100 the data was collected from final inspection departments of selected dmus. on-time delivery (y2) percentage number of on-time deliveries/ total number of deliveries made*100 the data was obtained from marketing departments of all selected dmus. 152 s. kumar, a. k. dhingra, b. singh table 3 correlation coefficients of selected inputs and outputs inputs/ outputs y1 y2 x1 0.011031087 -0.667480965 x2 -0.00814726 -0.572696089 x3 -0.243145823 0.55394473 fig. 1 selected inputs and outputs for dea model 4. identification of percentage improvement in quality system the collected data shown in table 2 is analyzed by the dea ccr model with crs for objective function as output maximization in order to identify the percentage of improvement scope in the existing quality system of all selected dmus. the efficiency scores of the selected dmus are calculated by maximizing the output variables for the same set of input variables. after the dea analysis, table 4 shows that dmu-7 obtained 100% efficiency score and is the benchmark for rest of all selected dmus. the actual value, target value and potential improvements of chosen inputs/outputs of the selected dmus are clearly recorded in table 5 for maximizing outputs strategy. a brief summary of recorded observations are as the following:  dmu-7 scored 100% efficiency hence improvement recommendation is zero.  dmu-1 and dmu-9 recorded recommended improvement in output (y2) by 56.73% and 94.16%, respectively, which are easy to achieve. similarly, dmu-2, dmu-3, dmu-4, dmu-5, dmu-6, dmu-8 & dmu-10 recorded recommended improvement in output more than 100% which required to pay more attention to achieve these improvements. table 4 dea efficiency scores for all selected dmus dmu name dea efficiency score (%) rank peer/ benchmark dmus total peers dmu-7 100 1 dmu-7 1 dmu-1 63.80 2 dmu-7 1 dmu-9 51.50 3 dmu-7 1 dmu-4 41.60 4 dmu-7 1 dmu-2 40.20 5 dmu-7 1 dmu-5 31.40 6 dmu-7 1 dmu-10 25.10 7 dmu-7 1 dmu-6 18.20 8 dmu-7 1 dmu-3 16.90 9 dmu-7 1 dmu-8 14.50 10 dmu-7 1 implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 153 these opportunities of improvements identified in the selected dmus are taken into consideration for applying the lean-kaizen concept and an attempt is made to make them 100% efficient. table 5 recommendation for improvement in selected dmus dmu input/ output variables objective function = maximize output actual value target value potential improvement percentage dmu-7 x1 5 5 0.00% x2 1.09 1.09 0.00% y1 95.11 95.11 0.00% y2 96 96 0.00% dmu-1 x1 8 8 0.00% x2 4.32 1.78 -59.83% y1 55.22 152.18 175.58% y2 98 153.6 56.73% dmu-9 x1 9 9 0.00% x2 2.98 1.96 -34.16% y1 72.98 171.2 134.58% y2 89 172.8 94.16% dmu-4 x1 10 10 0.00% x2 18.27 2.18 -88.07% y1 79.04 190.22 140.66% y2 74 192 159.46% dmu-2 x1 11 11 0.00% x2 16.87 2.40 -85.79% y1 66.89 209.24 212.82% y2 85 211.2 148.47% dmu-5 x1 27 12.25 -54.64% x2 2.67 2.67 0.00% y1 73.08 513.59 602.78% y2 73 518.4 610.14% dmu-10 x1 15 15 0.00% x2 11.69 3.27 -72.03% y1 71.56 266.25 272.07% y2 66 268.74 307.18% dmu-6 x1 27 27 0.00% x2 18.98 5.89 -68.99% y1 93.22 513.59 450.95% y2 69 518.4 651.30% dmu-3 x1 25 25 0.00% x2 5.59 5.45 2.50% y1 71.03 474.75 568.37% y2 81 479.19 491.59% dmu-8 x1 27 27 0.00% x2 22.98 5.89 -74.39% y1 65.35 513.59 685.91% y2 75 518.4 591.20% 154 s. kumar, a. k. dhingra, b. singh 5. lean-kaizen implementation through the observations recorded in table 5, dmu-1, dmu-2, dmu-3, dmu-4, dmu5, dmu-6, dmu-8, dmu-9, and dmu-10 are identified for deficiency in outputs (y1 & y2) which need to be improved. the benchmark dmu-7 has been visited in search of improvement factors that make the industry 100% efficient. then inefficient dmus are further visited in order to collect data so that the required changes/ improvements in the quality system can be achieved. the „5-why‟ analysis is applied to identify the root cause of deficient output in selected departments of inefficient dmus (table 6). 5.1. kaizen events for output maximization after identifying the root cause of the deficient outputs, four kaizen events are proposed as a solution for output maximization. table 6 root cause analysis for deficient outputs 5-why analysis deficiency in y2 in production planning and control department deficiency in y2 in production planning and control department deficiency in y1 in packing department deficiency in y1 in quality department why 1 improper material storage delay in material movement variable packed quality of finished products complexity in identification of material grade of similar products why 2 conventional drum used for material storage long path for material movement non-equal quantity was packed due to manual weighing identification tags were dissolved in the material during material movement why 3 nonstandardized material storage process followed by selected dmus gangway available for material transfer within departments conventional method was adopted due to heavy quantity to be packed for customer paper material used for identification token why 4 no work instruction was given to trace material for movement in-house work area distribution of selected dmus on-time delivery to customer pressurize manpower for quick packing no alternate provision was found for change of identification token why 5 sharp corners led to delay in receiving material for processing no alternative was presented for the shortest path. lack of technology for packing equal quantity automatically identification tag problem implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 155 two kaizen events were performed by brainstorming technique [63] in production planning and control (ppc) department for maximizing y2 and two kaizen events were conducted by using poka-yoke method [64] in packing and quality departments for maximizing y1 of the selected dmus. kaizen event 1 & 2: maximizing y2 in ppc department of dmu-1 and dmu-9 through brainstorming technique. the brainstorming technique was performed in which all the participants (managers, supervisors, and workers) of dmu-1 and dmu-9 provided ideas of improvements that enhance on-time delivery of products. out of these ideas, rectangular trays with traceable numbers and broken walls for the shortest route were selected and recommended for implementation in respective dmus. implementation of kaizen event 1: provided rectangular trays to improve y2. the activity of material movement was critically analyzed within departments of dmu-1 and dmu-9; it is observed that ppc is using conventional drums for moving material from one department to another due to the lack of management awareness, communication and operator‟s negligence. the stored quantity and traceability of material were major issues while transferring material by ppc supervisors. the problem was fixed by using new rectangular trays of identification number instead of using broken drums to move material within departments (table 7). implementation of kaizen event 2: removing walls for optimized material movement for improving y2. the shop floor of selected organization was critically analyzed in identifying the shortest route for material transportation. the problem was fixed by eliminating the walls across departments in order to eliminate waste such as unnecessary transportation of material and improved on-time delivery within departments. the shortest path was selected by removing the wall of forging department. the broken walls were repaired and made available for the gangway (table 7). kaizen event 3 & 4: maximizing y1 in dmu-2, dmu-3, dmu-4, dmu-5, dmu-6, dmu-8 and dmu-10 through poka-yoke. poka-yoke is the process which activates inventions that can perceive an abnormal situation before it occurs in the process and if it occurs, the system will be stopped. in order to improve on-time delivery in the present case, the manual weighing process is critically observed and an auto counter and feeder machine is recommended for product packing (kaizen events 3). a program of equal quality packets is planned, prepared and installed within the machine in order to pack the same set of packets. in the case of noticing unequal quality of packing in the packet, the machine will automatically stop for manual inputs. in addition, engraved plastic tokens of different colors and names for various grades are recommended to avoid mixing of materials within departments as caused by the workers (kaizen events 4). the colors of plastic tokens are selected as green for „ok‟, yellow for „non-conformities‟ and red for „scrap‟ material for movement within departments. these various grades engraved in golden color on plastic tokens are suggested for the sake of their great visibility by workers. the material will be immediately stopped for sorting if it carries a wrong plastic token (table 7). implementation of kaizen events 3: using auto counter and feeder machine in packing department to improve y1. the personal visit to packing department was made for critical analysis of the manual weighing process. it was found that there is a lack of technology in packing department for packing of an equal quantity of finished products to the customer. the problem was fixed by adding packing program to the auto counter and feeder machine. in order to improve on-time delivery of product, five successful run trails of a 156 s. kumar, a. k. dhingra, b. singh packing program for a set of 10 packets of 100 products (each) were performed. the new process of packing products was included in standardized work instruction and recommended to be followed by the packing department (table 7). table 7 implementation of recommended kaizen events in the selected dmus kaizen event / variables selected dmus/ dept. current situation of selected dmus (before kaizen) improved situation of selected dmus (after kaizen) 1. / y2 1, 9/ production planning and control non-standardized material stored in the drums rectangular trays replaced the drums for material movement 2. / y2 1, 9/ production planning and control long route for material movement. one new door opened from the main gangway side. 3. / y1 2, 3, 4, 5, 6, 8, 10/ packing the conventional weighing balance was used to weigh the packed components. auto counter & feeder installed to pack exact quantity of material during final packing. 4. / y1 2, 3, 4, 5, 6, 8, 10/ quality complexity in the identification of grades of the materials due to a variety of similar products on the same line. this complexity created confusion for packing and inspection personals and led to customer dissatisfaction. engraved plastic tokens (different colors/ name) for identification eliminated the confusion of similar products of different grade. implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 157 implementation of kaizen events 4: providing plastic tokens to improve y1 in the quality department. the mixing of different grade material caused a considerable delay in moving material within the departments. the problem was fixed by implementing colored plastic tokens engraved grades for material movement. the quality supervisors were trained to issue new plastic tokens of the same material grade after each first approval of the products. the new process of issuing colored plastic tokens was added in standardized work instruction and recommended to be followed by the quality department (table 7). all the kaizen events were accomplished within a period of two months in all the selected dmus. table 7 shows the implementation of the recommended kaizen events in all inefficient dmus. 6. results analysis from the data collected and analyzed after each kaizen event, in dmu-1 and dmu-9, the on-time delivery (y2) of products within departments was improved by 33%-35% (with fine material traceability at all levels) which increased the working hours of these selected organizations. in addition, the automatic packing of exact quantity and correct material sent to customer improved quality (y1) by 12% in dmu-1 and dmu-9. after successfully implementing the selected kaizen events, the input and output data of all the selected dmus were recorded. table 8 dea efficiency score after lean-kaizen implementation dmu x1 x2 y1 y2 dea efficiency score improvement percentage rank dmu-7 8 4.32 96 97 100% 0 % 1 dmu-1 5 1.09 98 100 100% 36.20% 2 dmu-9 9 2.98 97.12 100 100% 48.50% 3 dmu-5 11 16.87 89 73 95.30% 63.90% 4 dmu-2 10 18.27 67.72 85 46.50% 6.30% 5 dmu-4 15 11.69 81 74 41.80% 0.20% 6 dmu-3 27 2.67 90.08 82 38.90% 22.00% 7 dmu-10 27 18.98 75 68 26.80% 1.70% 8 dmu-6 25 5.59 97.12 69 21.00% 2.80% 9 dmu-8 27 22.98 78.11 76 15.90% 1.40% 10 after analysis of recorded data and further application of the dea ccr model, the results are presented in table 8 which clearly indicates that dmu-7 is still 100% efficient and is also appeared as a benchmark for rest of dmus. dmu-1 and dmu-9 have become 100% efficient and gained 36.20% and 48.50% improvements, respectively. 158 s. kumar, a. k. dhingra, b. singh from further analysis of collected data, the reasons to become 100%efficient of dmu1 and dmu-9 are found as management awareness and willingness of employees to follow the standardized work instructions which were prepared after the application of kaizen events. additionally, the reasons to gain 8% to 49% improvements in the efficiency score of in dmu-2, dmu-3, dmu-4, dmu-5, dmu-6, dmu-8 and dmu-10 are observed as a lack of management support and worker‟s negligence in order to follow new standardized work instructions while performing forging as well as plating processes. fig. 2 shows efficiency scores of quality system in all selected dmus before and after kaizen events implementation. 7. conclusions in this research paper, the lean-kaizen concept using the dea ccr model is applied in order to improve the quality system of ten small scale fastener industries. four kaizen events have been proposed and implemented to eliminate waste in outputs (quality, ontime delivery) that consequently improved the efficiency score of quality system of the selected dmus. the integration of lean-kaizen concept with the dea ccr model is found to be an effective tool in identifying the potential improvements in inefficient firms which can be learned and achieved from benchmark/peer industries (most efficient). the efforts have been made to minimize or eliminate errors arising out of it. fig. 2 efficiency score of all dmus before and after kaizen events this study can be further explored to optimize the results by collecting data after the kaizen events implementation in all the selected industries. a further comparison can be made by the dea ccr model in order to find new benchmarks/ peers so that new improvements can be achieved by applying strategies of outputs maximization. the main drawback of this approach is that it does not clarify the number of steps that needs to be taken to obtain an optimized result. the identification and implementation of improvement opportunities are dependent on individual skill and experience. this framework can also be implemented to measure the performance of production, maintenance, and quality of the industries producing different products and ready to adopt the lean-kaizen concept across all levels of the organization. the study concluded that the lean-kaizen using dea is an efficient technique for improvement in the quality system of the organizations. implementation of lean-kaizen approach in fasteners industries using data envelopment analysis 159 references 1. panwar, avinash, jain, rakesh, and rathore, a.p.s., 2015, lean implementation in indian process industries – some empirical evidence, journal of manufacturing technology management, 26(1), pp. 131–160. 2. singh, b., garg, s. k., and sharma, s. k., 2010, scope for lean 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license: cc by-nc-nd original scientific paper a novel integrated mcdm-swot-tows model for the strategic decision analysis in transportation company irena đalić1, željko stević1, jovo ateljević2, zenonas turskis3, edmundas kazimieras zavadskas3, abbas mardani4 1university of east sarajevo, faculty of transport and traffic engineering, doboj, bosnia and herzegovina 2university of banja luka, faculty of economics, banja luka, bosnia and herzegovina 3vilnius gediminas technical university, institute of sustainable construction, faculty of civil engineering, vilnius, lithuania 4university of south florida, tampa, muma college of business, tampa, usa abstract. in this paper, based on the strengths, weaknesses, opportunities, and threats (swot) analysis, a matrix of threats, opportunities, weaknesses and strengths (tows) was formed. it represents possible business strategies of the transport company. to choose the right plan, a model based on the integration of fuzzy pivot pairwise relative criteria importance assessment (fuzzy piprecia), full consistency method (fucom) and measurement alternatives and ranking according to compromise solution (marcos) methods, has been formed. a case study was conducted in the transport company from bosnia and herzegovina which provides services on the domestic and the european union market for 20 years and belongs to a group of small and medium enterprises (smes). the swot analysis in this transport company was the basis for forming the tows matrix, which represents a set of possible business strategies. these strategies are the basis for developing five basic alternatives. the transport company should choose the best one of them for future business. the research focuses on forming a model for choosing the best strategy by which the transport company seeks to improve its business. decision-making (dm) is not a straightforward sequence of operations, so the harmonization of methods as well as the verification of their results, are essential in the research. this model is applicable in smes that make these and similar decisions. using this model, companies can adjust their business policies to the results of the model and achieve better business results. this research is the first that allows the use of such a model in making strategic decisions. key words: mcdm, fuzzy piprecia, fucom, swot, marcos, transport received november 25, 2020 / accepted february 05, 2021 corresponding author: irena đalić university of east sarajevo, faculty of transport and traffic engineering, vojvode mišića 52, 74000 doboj, e-mail: i.naric@yahoo.com 402 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani 1. introduction increasing speed and reliability of the freight transport is considered a major transport policy objective in most countries 1, so the company management involves planning, organizing, leading and controlling whereby the management, together with other resources, effectively and efficiently accomplishes the goals of the company. if a company wants to operate successfully, it must define the right goals. to assess a newly developed plan, enterprises need well-backgrounded problem-solving models 2. to achieve its mission the company should provide correlation of internal and external factors. the management finds a way of doing business by getting to know strengths and weaknesses of the company in order to take advantage of opportunities and to master the dangers that come from the environment. the company management must be changing in line with the changing environment in which it operates. chances often come up from the background. the company needs to be able to recognize and exploit them in time. the management should identify, predict and determine the size and strength of chances. the strategy most often refers to us as the way of determining the direction of growth and development of the company in the domestic and international markets. according to 3, the strategy implies the realization of primary long-term goals and tasks of the company as well as acceptance of the direction of activities and distribution of the means necessary to achieve those goals. meanwhile, 4 the defined strategy relates to a critical set of management decision tools, encompassing objectives, policies, and ways of achieving them in economic activity. in the narrower sense, the plan is a business decision that defines basic ways of achieving goals. to achieve its goals in the best possible way, the company needs to determine its strategy. it means that the management must make a decision about what and how many resources need to be engaged in the business process. the strategy shows a more rational way of directing limited resources to achieve a specific objective. a strategy is a tool by which the company defines and realizes its goals. it expresses the way in which business responds to the environment over time. based on the plan, the management defines the relationship between the company and its environment as well as the structure of business competence that will help it to meet the challenges. strategic management is a separate management process and represents the process of directing the activities of the company. the essential factors of the company’s business success are identified based on predicting the chances, dangers, strengths and weaknesses of the company. strategic management includes the links between leadership, entrepreneurship and management, company-wide knowledge, relationship with the environment, expertise in finance, manufacturing, marketing, as well as the experience of people and their work. the management will be able to use the research results and make a decision concerning the best strategy to be applied for future business. after the introduction, the second part of the article presents a literature review. the third part of the article presents the formed decision models and the investigation outcome. to achieve these goals, the authors used three fuzzy mcdm methods, namely marcos, fucom, and piprecia. based on this model, it will be possible to make a decision about the choice of the best strategy. the fourth part of this article presents a case study and a swot analysis of the transport company. in this way, the research describes the current situation in the transport company. this part of the article shows the problem-solving process by using the developed model. sensitiveness of the outcomes follows in the next part. the last part a novel integrated mcdm-swot-tows model for the strategic decision analysis... 403 of the work presents the conclusions of the research with the directions of perspective exploration. 2. literature review the modern business needs reliable solutions, both locally and internationally. business leaders should choose excellent and realistic development strategies instead of looking for ways to resolve conflicts and problems 5. the swot analysis can be a useful tool for analyzing the business. a large number of researchers use this method for analyzing the current situation in the company. the swot analysis helps us make strategic decisions 6. this method helps to solve problems in different fields of business. novikov 7 used swot to analyze high-tech strategic development in manufacturing companies. rauch et al. 8 used this approach to analyze the state of forest fuel supply chains in southeast europe and to find a rational development strategy. živković et al. 9 applied the approach to investigate strategic dm in the technical faculty. bohari et al. 10 and kolbina11 performed a swot analysis in the food industry. swot analysis is applicable in the energy sector 12, 13. this method of analysis helped scholars in many more areas of research 14-19. kuo et al. 20, comino and ferretti 21 and hatefi22 used the swot analysis for strategic planning. valverde et al. 23, yan et al. 24 and jasiulewicz-kaczmarek25 performed swot analysis as a method of helping management with decision-making. a widespread use of combinations of the swot analysis with other methods proves the relevance of model development in this paper. abdel-basset et al. 26 formed a model for strategic planning and dm by combining ahp and swot methods. ruzgys et al. 27 presented an integrated mcdm model to select materials using swot, swara, and todim methods. edrogan et al. 28 solved the problem of construction management. turskis and juodagalvienė 29 integrated ten different mcdm methods to one model and solved the problem of shape assessment. korableva and kalimullina30 created bsc-swot matrix and applied it to the optimization of the organization. and there are many more authors who use using combinations of swot and other methods in their research 31-35. recently, scientific articles using the fucom method have been published frequently. pamučar et al. 36 suggest that managers use a simple algorithm and the fucom method to prioritize criteria, as well as evaluate occurrences against decision-makers (dmr) priorities. the fucom method, in combination with other methods, gives excellent results when forming dm models 37. the authors use the fucom method to evaluate criteria in different studies 38-42. when researchers evaluate particular alternatives, they use fucom method in their research 43-47. in this paper, the marcos method helps to determine the utility functions of alternatives and to make a compromise ranking concerning ideal and anti-ideal solutions. stević et al. 48 recently developed the marcos method. several studies 49-51 used it to solve problems. 404 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani 3. methods the article presents the methodology through five phases. fig. 1 shows them. fig. 1 the decision-making methodology the first phase will help dmrs to collect data from a particular transport company. it describes the current situation in the transport company and determines its internal strengths and weaknesses as well as external opportunities and threats. the data is the basis for the swot analysis. after this in the second phase, the fuzzy piprecia method ranked elements of the swot. the piprecia method allows dmrs to evaluate the importance of criteria without sorting 52. dmrs groups solve most multi-criteria decision-making (mcdm) problems. therefore, the swara method forces dmrs to solve this problem systematically 53, 54, taking into account an integrated opinion of dmrs about criteria significance ranks and ensures full consistency of final assessment of criteria weights. if the number of dmrs participating in the fuzzy piprecia increases, the model works worse. stević et al. 55 developed the fuzzy piprecia method. this method consists of eleven steps. the third phase, the cross-swot analysis helps to form the tows matrix and to define strategies. this phase defines the criteria to evaluate the strategies. a novel integrated mcdm-swot-tows model for the strategic decision analysis... 405 the fucom method helps to rank the criteria in descending order of importance and finally to assess their significance in the fourth phase. the basis of this method is to compare the relative importance of criteria in pairs, from the most important to the least influential, and to present the results, given that the dmrs are sure that the rankings of the requirements given in the previous step are well defined and do not need to be changed 36. fucom requires dmrs to determine the effect of criterion i on criterion j. the fucom method has some benefits, such as relatively few pairwise comparisons (number of criteria minus one) compared to the ahp method. in real life, the fucom method is the same as the swara method. it is the same as the swara method. the fucom method presents a small modification of the swara algorithm. dm in the fucom method directly presents the ratios of pair of criteria importance level instead of describing a distance of significance in a pair of criteria, which later add to 1. it means that in the fucom method one step from the swara method is omitted. the final step of the fucom method (determining values of criteria weights of the criteria) includes the subjective effect of dmrs preferences. however, unlike other criteria weighting methods, the fucom determines criteria weights with smaller deviations (puška et al. 2019). the fucom method consists of three steps. the marcos approach helps to evaluate and evaluate strategies in the fifth stage. the primary basis of marcos is to establish the relationship between alternative and reference values (perfect and anti-ideal alternatives). this approach consists of seven steps 48. based on specific relationships, it determines the utility function's values of options and reaches a compromise on ideal and anti-ideal solutions. utilities show preferences of available options and the position of each choice concerning the perfect and anti-ideal solutions. the best alternative is the one closest to the ideal but furthest from the anti-ideal starting point. finally, based on this assessment and ranking, the choice chosen first is the best strategy. therefore, dmrs have to select and implement it. 3.1. fuzzy pivot pairwise relative criteria importance assessment fuzzy piprecia method step 1 formation of the required benchmarking set of criteria and formation of a team of decision-makers. step 2 in order to determine the relative importance of criteria, each decision-maker individually evaluates the criteria by starting from the second criterion 1 1 1 1 1 1 j j r j j j j j if c c s if c c if c c − − −    = = =   , (1) where sr j denotes the evaluation of the criteria by a decision-maker r. decision-makers evaluate criteria by applying linguistic scales. step 3 determining the coefficient 1 1 2 1j j if j k s if j = = =  −  (2) 406 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani step 4 determining the fuzzy weight 1 1 1 1 jj j if j qq if j k − = =  =     (3) step 5 determining the relative weight of the criterion 1 j j n j j q w q = =  (4) step 6 evaluation of the applying scale defined above, but this time starting from a penultimate criterion. 1 1 1 1 ' 1 1 j j r j j j j j if c c s if c c if c c + + +    = = =   (5) sr j' denotes the evaluation of the criteria by a decision-maker r. step 7 determining the coefficient 1 ' 2 'j j if j n k s if j n = = =  −  , (6) n denotes a total number of criteria. step 8 determining the fuzzy weight 1 1 '' ' jj j if j n qq if j n k + = =  =     (7) step 9 determining the relative weight of the criterion 1 ' ' ' j j n j j q w q = =  (8) step 10 in order to determine the final weights of the criteria, it is first necessary to perform defuzzification wj and w'j 1 '' ( ') 2 j j jw w w= + (9) step 11 checking the results obtained by applying spearman and pearson correlation coefficients. a novel integrated mcdm-swot-tows model for the strategic decision analysis... 407 3.2. full consistency method (fucom) step 1 in the first step, the criteria from the predefined set of evaluation criteria c=(c1,c2,...,cn). the ranking is performed according to the significance of criteria cj(1)>cj(2)>...>cj(k) step 2 in the second step, a comparison of the ranked criteria is carried out and comparative priority (φk/(k+1), k=1,2,...,n), where k represents the rank of the criteria) of the evaluation criteria is determined  = (1/2, 2/3,..., k/(k+1)) (10) step 3 in the third step, the final values of the weight coefficients of evaluation criteria (w1,w2,...wn)t are calculated. 3.3. measurement alternatives and ranking according to compromise solution (marcos) method step 1 formation of an initial decision-making matrix. step 2 formation of an extended initial matrix by defining ideal (ai) and anti-ideal (aai) solution. 1 2 1 2 11 11 12 2 21 22 2 1 22 21 ... ... ... ... ... ... ... ... ... ... ... n aanaa aa n n m m mn aiai ain c c c xx xaai xa x x a x x x x a x xx ai xx x         =            (11) step 3 normalization of extended initial matrix x. ai ij ij x n if j c x =  (12) ij ij ai x n if j b x =  (13) where elements xij and xai represent the elements of matrix x. step 4 determination of the weighted matrix by equation ij ij jv n w=  (14) step 5 calculation of the utility degree of alternatives ki. i i aai s k s − = (15) i i ai s k s + = (16) 408 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani where is (i=1,2,..,m) represents the sum of the elements of weighted matrix v. step 6 determination of the utility function of alternatives f(ki). the utility function of alternatives is defined by equation ( ) ( ) ( ) ( ) ( ) ; 1 1 1 i i i i i i i k k f k f k f k f k f k + − + − + − + = − − + + (17) utility functions in relation to the ideal and anti-ideal solution are determined by applying equations: ( ) i i i i k f k k k + − + − = + (18) ( ) i i i i k f k k k − + + − = + (19) step 7 ranking the alternatives. 4. case study the swot analysis is one of the simplest but most effective ways to determine the situation in a company. fig.2 shows thus identified the internal and external factors, which affect the success of the transport company's business. internal factors are those that the management and the company’s employees may change or affect in whole or in part. and yet the management and employees cannot replace external factors. however, they can significantly improve the company's business success by systematically evaluating external factors and selecting the right business plan on time. fig. 2 swot analysis a novel integrated mcdm-swot-tows model for the strategic decision analysis... 409 fig. 3 defines the factors that represent internal and external factors, i.e. strengths and weaknesses in the transport company as well as opportunities and threats from the environment. the upper left corner of the figure lists the advantages of the transport company. the transport company has a modern fleet of a large number of trucks and therefore, it can meet all customer requirements. the management of the transport company is aware that the employees are the ones who do most of the work, so they have provided rewards to everyone who achieves excellent results in performing their activities. twenty years of a successful business are the result of professionalism and business organization. years of successful business and responsibility have created a recognizable brand of transport services. costs have been reduced to a satisfactory level, although the managers argue that costs that incurred daily are a significant weakness of the transport company. using all these strengths, the transport company strives to minimize threats from the environment. the lower right corner of the figure shows the risks. closing other transport companies on the market, unloyal competition, fluctuation of labor are threats on which the transport company can respond with its experience, professionalism and organization. growth of levies, unexpected problems from the ground and the eu restrictions are threats to which the transport company can react by associating with other transport companies and taking advantages of the association. the lower-left corner of the figure shows the opportunities. using all these opportunities, the transport company strives to minimize the weaknesses of the company. the upper right corner of the picture shows all fallings. dmrs hired independent external experts to obtain objective results. the dmrs act in the fuzzy and dynamically changing environment 56. analysis of swot of transport company helps dmrs assume that organizations achieve maximum strategic success by effectively leveraging and strengthening their strengths and opportunities in a dynamically changing business environment. the use of beneficial and appropriate management tools helps to reduce a company’s shortcomings and threats to the vulnerability of its business. analysis of the impact of internal and external factors on each strategy is also critical. after the swot analysis, the fuzzy piprecia method was used. the approach evaluated and ranked criteria. the weight and rank of each measure were obtained 57. the authors did a complete fuzzy piprecia calculation. for calculation purposes, the factors of swot analysis are marked as criteria: c1 – strengths, c2 – weaknesses, c3 – opportunities and c4 – threats (table 1). table 1 results of fuzzy piprecia criteria weight of criteria rank c1 – strengths 0.337 1 c2 – weaknesses 0.274 2 c3 – opportunities 0.188 4 c4 – threats 0.231 3 strengths and weaknesses are on the first and second according to the results of fuzzy piprecia method with importance values of 0.337 and 0.274, respectively. opportunities and threats are on the fourth and third place with a value of 0.188 and 0.231, respectively. this table helps to conclude that strengths and weaknesses are more critical for the transport company as internal factors with an influence on its business than external factors, i.e. opportunities and threats. dmrs, in each of these groups of elements, each 410 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani element of the swot matrix evaluated and ranked separately. therefore, the total number of listed items into the swot matrix is 23. the first and the second-ranked element is modern trucks and the ability to respond to all requests and brand recognition as factors with the most considerable influence on the business of the transport company. these elements are in the group of strengths. the worst-ranked feature from this group is cost optimization (14th place). the best-ranked component from the group of weaknesses is disloyalty of employees (3rd place). the worst-ranked element from this group is workers' failures (18th place). from the group of opportunities, the best-ranked item is the business expansion, and it takes seventh place. the worst-ranked element from this group is the eu funds, and it takes 22nd. the best-ranked component from the group of threats is the fluctuation of labor, and it takes the sixth place in the total rank. the worst-ranked element from this group is unexpected problems from the ground, which also receives the worst 23rd position in the full status of features. spearman's coefficient 58 helps to determine the correlation between these ranks. the calculated value of it is 1.00. the result shows that these ranks completely correlate. pearson's coefficient 55 helps to determine the correlation between the weights of the criteria. the calculated value of it is 0.985. the tows matrix formed after the ranking of the criteria represents the business strategies of the transport company. table 2 shows the strategies (tows matrix) created by the cross-swot analysis. table 2 tows matrix strategies strategy so strategy wo 1. expanding business based on years of experience and brand. 2. applying for european funds based on responsibility, organization and professionalism. 3. association with other transport companies using business on the territory of the eu. 1. cost rationalization through eco-trainings. 2. increasing loyalty of employees by creating a driver evaluation and reward model. 3. increasing the productivity of disponents by hiring one administrative worker. strategy st strategy wt 1. fight against unfair competition using advantage of modernization and quality. 2. reducing fluctuation of workers using advantage of motivation. 3. reducing levies using the strengths and benefits of association. 1. easier problems solving on the ground by improving communication between workers and management. 2. faster problem solving by reducing closeness and intimacy between owner and worker. 3. increasing the volume of domestic transport using the benefits of infrastructure growth and development. table 2 shows twelve formed strategies. the transport company can offer its services to new customers and gain their trust based on the years of experience and brand. in this way, the transport company can expand its business. the transport company has been functioning well for an extended period, and its main characteristics are responsibility, organization and professionalism. based on the features that embellish a business, provides a good chance of receiving support from some european business funds. the transport company operates in the territory of the eu, where it has its offices. based on this distribution of business, the company can join its forces with other transport companies in a novel integrated mcdm-swot-tows model for the strategic decision analysis... 411 its field of business and in that way, it can use all benefits of the association. eco training enables reducing costs in the transport company by as much as 15%. therefore, the transport company needs such training. if every worker, that is, his work and effort were valued, there would be an increase in loyalty of employees. it would avoid the possibility of putting both good and bad workers in the same basket. for the disponent not to waste time and effort on administrative tasks, it is necessary to employ one administrative worker in the transport company. this move would increase the productivity of the disponents, in this case, who would only do their job. the transport company has a modern vehicle fleet and performs all transport services with high quality, so it should use these advantages in the fight against unfair competition. the transport company motivates employees with a variety of rewards that should reduce the fluctuation of employees. recently it becomes more pronounced. companies from all areas of the economy are struggling with this. companies individually do not have any particular strength to fight the levies that are accumulating more and more every day. still, transport companies together have much more opportunities to act to reduce various taxes in many fields. in this transport company, there are specific problems of delaying information from employees to the management. untimely informing the administration by the employees about the new issues in the field delays the solution of the same creates problems in transport and generates additional costs. if employees understand the importance of the speed of transmission of certain information, it would be more comfortable and faster to solve problems in the field. besides, another way to solve problems faster and easier is that the transport company owners should be less biased and attached to workers because this creates the impossibility of objective reasoning. the last few years are witnesses of the growth and development of road infrastructure in the domestic field. it is beneficial to expand and grow the business. the previously defined strategies are the basis to assess the general strategy of the transport company’s development: 1. expanding business based on the years of experience and brand, 2. applying for european funds, 3. cost rationalization, 4. driver evaluation and rewards program, 5. increasing the volume of domestic transport using the benefits of infrastructure growth and development. the following set of criteria was the basis to evaluate the strategies: c1 the time of strategy realization, c2 the possibility of strategy realization, c3 investment costs for strategy implementation, c4 the necessary resources for realization, c5 the potential benefits of the strategy, and, c6 influence on the economic system. the linguistic scale is the basis to evaluate all criteria. all criteria are equally present from the aspect of criterion orientation. the first, third and fourth criteria need to be minimal (desirable minimum values), while the others maximal (preferable maximal values). based on the fucom method, the criteria rank according to their importance, i.e. according to the strength of the impact on the evaluation of general strategies. first step: c3>c2>c1>c5>c4>c6 412 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani second step: c3 c2 c1 c5 c4 c6 1 1.3 1.5 1.9 2 2.2 the determined preferences of criteria are the basis to calculate relative seniority of criteria (eq. (10)): 𝜑𝑐3/𝑐2 =1.3/1=1.3, 𝜑𝑐2/𝑐1 =1.5/1.3=1.15, 𝜑𝑐1/𝑐5 =1.9/1.5=1.27, 𝜑𝑐5/𝑐4 =2/1.9=1.05, 𝜑𝑐4/𝑐6 =2.2 /2=1.1. third step: a) 𝑤3 𝑤2 =1.3, 𝑤2 𝑤1 =1.15, 𝑤1 𝑤5 =1.27, 𝑤5 𝑤4 =1.05, 𝑤4 𝑤6 =1.1 b) 𝑤3 𝑤1 =1.3×1.15=1.495, 𝑤2 𝑤5 =1.15×1.27=1.461, 𝑤1 𝑤4 =1.27×1.05=1.334, 𝑤5 𝑤6 =1.05×1.1=1.155 the following equation defines the final model to determine the weight coefficients: 3 52 1 4 2 1 5 4 6 3 52 1 1 5 4 6 6 1 min 1.30 , 1.15 , 1.27 = , 1.05 = , 1.10 = , . . 1.50 , 1.46 , 1.33 , 1.16 1, 0,j j j w ww w w w w w w w w ww w s t w w w w w w j           =  − = − = − − −    − = − = − = − =    =     the solution of this model provides decision-makers with the final weights: (0.255, 0.196, 0.170, 0,134, 0.128, 0,116) and deviation from complete consistency χ=0.000. the tags given at the beginning of table 3 shows the values of the criteria. table 3 criteria priorities criteria c1 c2 c3 c4 c5 c6 j 0.170 0.196 0.255 0.128 0.134 0.116 a novel integrated mcdm-swot-tows model for the strategic decision analysis... 413 fig. 3 weights of criteria for evaluating the general strategies obtained by the fucom method table 3 shows the results of the fucom method. for the sake of transparency, the results are also shown in fig. 3. fig. 3 shows that the criterion c3 investment costs for strategy implementation is the most significant criterion; that is, this criterion has the most considerable influence on the evaluation of general strategies. further, the figure shows that the measure is c2 the possibility of strategy realization is the following by importance, and then c1 the time of strategy realization. the following is c5 the potential benefits of the strategy, and then c4 the necessary resources for realization and in the last place is the criterion c6 impact on the economic system. alternatives, i.e. general strategies, are ranked by importance using the marcos method. the main strategies are designated as alternatives. a1 expanding business based on the years of experience and brand, a2 applying for european funds, a3 cost rationalization, a4 driver evaluation and rewards program, a5 increasing the volume of domestic transport using the benefits of infrastructure growth and development. steps 1 and 2: in these steps, the initial extended matrix is formed (table 4), and the perfect and anti-ideal solutions are determined (eq. (11)). 414 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani table 4 initial extended matrix criteria c1 c2 c3 c4 c5 c6 anti-ideal 5.000 5.000 5.000 4.000 5.000 3.000 a1 5.000 7.000 5.000 4.000 9.000 9.000 a2 3.000 7.000 1.000 2.000 5.000 3.000 a3 5.000 5.000 3.000 3,000 7.000 5.000 a4 1.000 9.000 2.000 1.000 7.000 4.000 a5 3.000 5.000 2.000 3.000 5.000 7.000 ideal 1.000 9.000 1.000 1.000 9.000 9.000 step 3: dmrs normalized the cost criterion values using eqs. (12) and (13), from step 3, for example: 𝑛𝑖𝑗 = 𝑥𝑎𝑖 𝑥𝑖𝑗 𝑖𝑓 𝑗 ∈ 𝐶 ⇒ 𝑛14 = 1.000 4.000 = 0.250 the following equation helps to obtain benefit criteria: 𝑛𝑖𝑗= 𝑥𝑖𝑗 𝑥𝑎𝑖 𝑖𝑓 𝑗 ∈ 𝐵 ⇒ 𝑛12 = 7.000 9.000 = 0.778. table 5 shows the complete normalized matrix. table 5 normalized matrix criteria c1 c2 c3 c4 c5 c6 anti-ideal 0.200 0.556 0.200 0.250 0.556 0.333 a1 0.200 0.778 0.200 0.250 1.000 1.000 a2 0.333 0.778 1.000 0.500 0.556 0.333 a3 0.200 0.556 0.333 0.333 0.778 0.556 a4 1.000 1.000 0.500 1.000 0.778 0.444 a5 0.333 0.556 0.500 0.333 0.556 0.778 ideal 1.000 1.000 1.000 1.000 1.000 1.000 step 4: this step extends the normalized matrix by multiplying all the values of the standardized form by the importance of the criteria (eq. 14). table 6 shows the normalized and weighted matrix. table 6 the normalized and weighted matrix criteria c11 c12 c13 c14 c15 c16 anti-ideal 0.034 0.109 0.051 0.032 0.075 0.039 a1 0.034 0.153 0.051 0.032 0.134 0.116 a2 0.057 0.153 0.255 0.064 0.075 0.039 a3 0.034 0.109 0.085 0.043 0.105 0.064 a4 0.170 0.196 0.128 0.128 0.105 0.052 a5 0.057 0.109 0.128 0.043 0.075 0.090 ideal 0.170 0.196 0.255 0.128 0.134 0.116 a novel integrated mcdm-swot-tows model for the strategic decision analysis... 415 the marcos method, applying equations from steps 5 and 6, provides the results in table 7. step 5: the process to obtain the results is as follows: all values (in rows) for alternatives are summed in the following eqs. (15) and (16), from step 5 as follows for saai : 𝑆𝐴𝐴𝐼 = 0.034+0.109+0.051+0.032+0.075+0.039=0.339 the values for the remaining alternatives dmrs similarly calculated. dmrs, using the following equation, calculated the degrees of benefits concerning the ideal solution are. example: 𝐾1 − = 0.520 0.339 = 1.532 while applying the following equation, dmrs calculated degrees of benefits concerning the perfect solution, e.g.: 𝐾1 + = 0.520 1.000 = 0.520 step 6: the utility function in terms of the anti-ideal solution dmrs calculated by applying the following eq.18: 𝑓(𝐾1 −) = 𝐾1 + 𝐾1 + + 𝐾1 − = 0.520 0.520 + 1.532 = 0.253 while the utility function in terms of the ideal solution dmrs determined by applying eq. (19): 𝑓(𝐾1 +) = 𝐾1 − 𝐾1 + + 𝐾1 − = 1.532 0.520 + 1.532 = 0.747 finally, dmrs calculated the utility function of alternative a1 by applying eq. (17): 𝑓(𝐾1) = 𝐾1 + + 𝐾1 − 1 + 1−𝑓(𝐾1 +) 𝑓(𝐾1 +) + 1−𝑓(𝐾1 −) 𝑓(𝐾1 −) = 0.520 + 1.532 1 + 1−0.747 0.747 + 1−0.253 0.253 = 2.052 1 + 0.339 + 2.953 = 2.052 4.292 = 0.479 step 7: table 7 shows the results of the marcos method. table 7 ranks of alternatives ai si aai 0.339 kiki+ f(k-) f(k+) f(ki) rank a1 0.520 1 0.520 0.253 0.747 0.479 3 a2 0.642 1.532 0.642 0.253 0.747 0.591 2 a3 0.440 1.891 0.440 0.253 0.747 0.405 5 a4 0.778 1.296 0.778 0.253 0.747 0.716 1 a5 0.501 2.292 0.501 0.253 0.747 0.461 4 ai 1.000 416 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani table 7 presents alternatives ranked using all seven steps of the marcos method. dmrs determined perfect and anti-ideal solutions, that is, values of 1.000 and 0.339, respectively. the best is the alternative whose value of the utility function is closest to the ideal solution, and it ranks as the first alternative. in this research, it is the alternative a4 driver evaluation and reward program, whose value of the utility function is 0.716. this alternative stands out as the best because the implementation of this strategy is possible; it does not involve the engagement of additional resources, nor it requires much time to realize, as can be seen from the evaluation of the criteria. the worst-ranked alternative is the one whose value of the utility function is closest to the value of the anti-ideal solution, and here it is alternative a3 cost rationalization, with the value of the utility function as 0.405. 5. sensitivity analysis to verify the obtained results, we compared the results obtained by the marcos method with the results of other mcdm methods. therefore, this part of the paper presents a sensitivity analysis of the results obtained by the marcos method. sensitivity analysis compared the effects of ranking obtained by the new marcos method and four other methods: saw 59, aras 60, waspas 61 and mabac 62. table 8 and fig. 4 show the results of the analysis. table 8 sensitivity analysis of results obtained by the marcos method fig. 4 validation of results through the application of other methods marcos saw aras waspas mabac 0.479 3 0.520 3 0.472 4 0.461 4 -0.070 4 0.591 2 0.642 2 0.634 2 0.615 2 0.106 2 0.405 5 0.440 5 0.413 5 0.421 5 -0.142 5 0.716 1 0.778 1 0.780 1 0.758 1 0.354 1 0.461 4 0.501 4 0.486 3 0.492 3 -0.022 3 a novel integrated mcdm-swot-tows model for the strategic decision analysis... 417 table 8 and fig. 4 show that there are no significant changes in the position of the strategies. only the first and fifth strategies change places in some approaches by occupying the third or fourth positions. one of the ways of checking the validity of the solution obtained by the dm model is to create a dynamic matrix and investigate the results of the application of the model under the newly formed conditions. if the answers reveal some logical contradictions related to the undesirable changes in the ranks of the alternatives, this may indicate problems with the mathematical apparatus of the method used. checking the sensitivity of this model to the rank reversal problem is a logical step to validate the model results. to this end, dmrs experimented with assessing the resistance of the model to the rank change problem. dmrs developed three experimental scenarios that simulated changes in problem matrix elements. dmrs changed the number of alternatives for each situation, removing the worst case from further considerations. after defining a new set of choices, dmrs evaluated the remaining options under the newly formed conditions using the proposed model. in the first scenario, the dmr removed the worst third strategy (a3) from further consideration. after receiving the new assessment, they adopted a new set of four alternatives to using the model. fig. 5 shows this. the new decision confirms that the fourth strategy is still the best alternative and the fifth strategy is the worst. furthermore, if the worst-case the fifth strategy is not included in the model, the alternatives rank in the same way. in the third scenario, only two strategies need assessment. based on this confirmation, dmrs concluded that the values of the strategies did not change and the results are relevant. fig. 5 results of the validity of the model concerning dynamic changes in the initial matrix 418 i. đalić, ž. stević, j. ateljević, z. turskis, e.k. zavadskas, a. mardani in the next validation phase, the dmrs analyzed the impact of the change in the most critical criterion (c3) on the rating. the following equation helped to form ten scenarios: (1 ) (1 ) n n n w w w w   = − − here, wnβ represent corrected criteria values c1, c2, c4, c5 and c6, wnα represent the reduced values of criterion c3, wβ is the original value of the considered criterion and wn is the initial value of criterion c3. in the first scenario, the dmrs reduced the value of criterion c3 by 5%, while the values of the other criteria adjusted proportionally using the above equation. in each of these following scenarios, the value of criterion c3 is 10% lower, and the remaining characteristics are adjusted to meet the condition that sum of wj equal to one. fig. 6 shows the results of the model derived from the newly constructed ten criteria weights vectors. fig. 6 results of validity concerning changes in the significance of criteria values fig. 6 helps to conclude that the change in the significance of the criteria values does not play a significant role and that the model is not overly sensitive to the importance of the characteristics. the only difference that emerges is the rotation of the first and second strategy, starting with the sixth to the tenth scenario. the reason is that in the mentioned scenarios, there is a drastic decrease in the values of the most critical third criterion, while the importance of all other measures is increasing. a novel integrated mcdm-swot-tows model for the strategic decision analysis... 419 6. conclusion the authors present the research conducted in a transport company that operates in bosnia and herzegovina and the eu. the decision-makers performed a swot analysis to determine the current situation in the transport company. based on that, the strengths and weaknesses of the transport company were determined as well as the opportunities and threats in the environment of the transport company. a tows matrix was also formed based on the cross swot matrix. in this way, the business strategies of the transport company are determined, among which the management should choose the best one. managers can decide about the plan based on the results of this research. during the study, the authors developed a decision model. this model involves a combination of the fucom, fuzzy piprecia and marcos methods. the authors obtained the results using this model. the results show that the best strategy that the transport company can choose at this moment is a4 driver valuation and reward program, whose value of the utility function equals to 0.716. this strategy does not involve the engagement of additional resources; neither does it require much time to implement. the worst-ranked plan is the a3 cost rationalization, whose value of the utility function equals to 0.405. according to these results, the management should establish a program to evaluate and reward drivers and to provide both rationalization of costs and reduction of emissions in the operation of drivers. this developed model for dm is applicable in small and medium enterprises. following this research, the question that remains for future researchers is: how much cost reduction would be if to implement this strategy? another issue is the productivity of drivers; that is, how much would this increase their productivity? thus, further research could include the behavior and performance of drivers after the evaluation and reward of established programs. of course, future research may also focus on new growth 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stojanović1, jelena janevski1, jovan škundrić2 1university of niš, faculty of mechanical engineering, serbia 2university of banja luka, faculty of mechanical engineering, bosnia and herzegovina abstract. modern society and economic development are completely dependent on various forms of energy while the ever-increasing demand for energy, in combination with significant environmental topics, has resulted in state-of-the-art ideas and solutions for fulfilling these often-contradictory goals, i.e. increasing efficiency or environmental protection and economic goal. the efficiency of the existing operating units for electricity production based on the usage of low-quality coal does not go hand in hand with the requirements of this new concept. one of the most efficient ways to reduce specific energy consumption is using combined heat and power plants. in comparison to classical, separate heat and power plants, the advantage for chp plants comes from their high efficiency. the result of higher efficiency is lower primary energy consumption and lower environmental pollution due to low values of co2 emissions. several revitalization configurations can be applied in order to fit the existing thermal power plants into combined cycles. the idea is to install, at the existing location, one gas turbine to increase the overall efficiency. this paper analyzes the potential of a combined gas-steam facility in the situation where the gas facility is used for heating feed water, which enters the heat recovery steam generator. a comparison of energy efficiency for various operating regimes, with and without heat production, is performed for this option. key words: repowering power plants, steam power plant, gas turbine, energy analysis, efficiency received february 18, 2020 / accepted may 13, 2020 corresponding author: dejan mitrović university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, 18000 niš, serbia e-mail: dejan.mitrovic@masfak.ni.ac.rs mailto:dejan.mitrovic@masfak.ni.ac.rs 54 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić 1. introduction with the increasing global population, and a rise in the living standard, energy consumption rises as well, exhausting energy resources and threatening the environment. a rapid increase in energy consumption could be expected in currently undeveloped economies, due to the industrialization or re-industrialization. the previous statement is particularly true considering the demand for electricity, since 1/3 of energy resources are being utilized (used, consumed) for electricity production [1]. the limits of energy reserves, the necessity of their further usage and the constraints of the atmosphere potential, have led to shifts in energy policies in numerous developed countries with regard to sustainable development. the strategic goal of these revised energy policies is to achieve maximum energy efficiency by using advanced and state-of-the art energy technologies, which provide maximum efficiency and minimum losses during energy conversion. maximum environmental protection during energy production is achieved through minimization of pollutants acting on the atmosphere, soil and water, while preserving flora and fauna. since classically constructed steam blocks have reached their peak considering further improvements and efficiency increase, while gas blocks operate with high specific heat consumption, special attention is given to the construction of combined steam-gas power plants. repowering the old, existing steam power plants is an appropriate method for reducing thermal losses and increasing their overall efficiency. repowering decreases the operation costs of energy production and minimizes environmental pollutions. in addition, repowering enables an increase in the capacity of power plants to a significant extent. therefore, electricity production companies are giving more attention to this issue [1–3]. the reduction in environmental effects and investment expenses is the most important advantage of repowering a steam power plant [4]. repowering a steam power plant may increase the net output power by 200% and may improve the efficiency of the power plant by 30% [5]. methods of repowering steam power plants can be categorized as: full repowering or partial repowering. the most used methods of partial repowering are the feed water heating method, hot wind box method, and supplemental boiler method [6]. in the feed water heating method, the gas turbine exhaust gas is used to heat the current boiler feed water. power plants suitable for the application of this method are relatively new and modern. the method is particularly suitable for large power plants [7]. techno-economical characteristics (simplicity and flexibility of feasible designs, low investment cost and lower specific cost of generated electricity) in comparison to other repowering methods are the main advantages of feed water heating in comparison to other methods of partial repowering [8]. the efficiency of the power plant increases with this type of partial repowering [9, 10]. the importance of repowering power plants is well documented in the literature. in [2], general techniques for the conversion of steam cycles to combined cycles by full recovery are given with a few practical cases described. the technically evaluated integration of gas turbines into steam cycles by one of the partial repowering methods is shown and compared with the combined cycle in [11]. the authors found that the partial repowering methods could improve the quality of power production and reduce emissions in steam power plants. bracco and siri used a mathematical model to optimize a single pressure heat recovery boiler for a combined cycle using the first and second laws of thermodynamics [12]. there are many published papers in which the combined cycles are techno-economically analyzed or optimized. some of these studies examine the effect of each of the main components of the cycle on the economic and technical characteristics energy analysis of repowering steam power plants by feed water heating 55 of the system. bassily studied the influence of parameters on the quality of power generation in a three-pressure combined plant based on the cycle efficiency [13]. the presented papers on repowering are almost always focused on the effects of different methods of full and partial repowering on target functions. feed water is mainly heated in the power plant boiler, but the system feed water is preheated indirectly (via heat exchanger) before entering the boiler. in repowering by feed water heating, part of the heat needed to preheat the feed water is added in heat exchangers; this heat is released by gas turbine(s) hot exhaust gases. this paper analyzes the possibility of repowering a power plant by heating system feed water in a heat recovery steam generator, with an option to keep the steam boiler as part of the power plant. repowering the power plant produces heating energy in addition to electricity production. the developed mathematical model allows for varying heat load on the plant, depending on the actual consumer heat demand. the impact of selected parameters of the chp plant on energy efficiency is analyzed as well. 2. repowering options of steam power plants in the republic of serbia, electricity is dominantly (to a larger extent compared to other sources) produced in steam power plants, most of which have already surpassed their initial operation lifetime. experiences from other countries considering the repowering of steam power plants show that there is an acceptable solution for improvements in power plants and the whole electricity system, in the sense of increasing capacity and overall efficiency, reducing fuel consumption, investment (capital), maintenance and operation costs and reducing environmental impact (footprint). 2.1. options for facility improvements options for modernizing power plants, which include gas turbines, with the largest possibilities are: ▪ using the existing location, ▪ adding a gas turbine with the existing steam boiler modification, ▪ heating feed water with gas turbine exhaust gases, and, ▪ replacing the steam boiler with a gas turbine and heat recovery steam generator. 2.1.1. modernizing by using the existing location this modernizing process encompasses the disassembly of the existing facility on the location and the construction of a new combined gas-steam facility (new parts are a gas turbine, heat recovery steam generator and steam turbo-facility). in this option, the existing cooling system, connections to the grid, buildings, reservoirs and other objects could be kept. often called the “brown field” investment, this option's main advantage is in applying new, state-of-the-art equipment for the combined cycle, without any technological compromises during the assembly and connection to the existing equipment. compared to the “green field” investment, this option has the following advantages: there are no costs for land acquisition and administrative and legal barriers that can occur, the time saved in administrative procedures and socio-economic analysis is significant, and the technical 56 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić performances of the new power plant are practically the same as in the case of repowering the existing power plant. 2.1.2. modernizing by installing gas turbine(s) in this type of modernizing, exhaust gases of one or several gas turbines are connected to the fresh air supply of the existing steam boiler, operating on the same primary fuel. this option can increase the nominal capacity by 10-25% and the efficiency by 10-20% while improving the operation on partial load and reducing nox emissions. this option has good results if applied to larger and relatively newer blocks based on liquid or gas fuel. capacity and efficiency increases are the main advantage, and since the gas turbine has a small share in total power and thus has a small impact on the overall efficiency, larger power plants are good candidates for applying this option. 2.1.3. preheating feed water with gas turbine exhaust gases in this option, the exhaust gases from the gas turbine are used for preheating feed water entering the steam boiler. the steam, usually taken from the turbine for this purpose, can produce additional work and give more power, if the steam turbine and generator have the capacity for this additional power. another solution is to inject this steam in the gas turbine and increase its capacity. the existing regenerative heaters can be used when the gas turbine is off. this configuration of a combined facility should be taken into consideration when the goal is to obtain additional capacity for electricity production. the existing steam turbine is then used for base load while the gas turbine operates in order to get additional power and improve efficiency during maximum peak loads. 2.1.4. modernizing by replacing the steam boiler with a gas block and heat recovery steam generator the most used option for modernizing the existing power plants is to disassemble the existing steam boiler and to replace it with a gas turbine and heat recovery steam generator while keeping the existing steam turbine. this approach leads to the capacity increase of up to 150-200% compared to the classical steam block. heat consumption is reduced by 30-40% and nox emissions are reduced as well. due to a significant capacity increase, this approach is usually applied for older units, with the capacities of up to 250mw. the biggest concern is to optimize the cycle of the existing steam turbine with a new equipment of the combined cycle. 2.2. selected model of the power plant facility the schematic flow diagram of a combined cycle power plant is given in fig. 1. the scheme is of flexible configuration since it allows, depending on the component selection and operation parameters, the analysis of: ▪ operation of the condensing type steam block, ▪ operation of the steam block for combined heat and power production, ▪ operation of the combined gas-steam plant only for the production of power, and, ▪ operation of the combined gas-steam plant for the combined heat and power production. energy analysis of repowering steam power plants by feed water heating 57 the reason for selecting this model is the flexibility for plant configuration. depending on the heat demand, the condensing type steam block operation analysis (power production only) or the operation of the combined heat and power production are enabled. during the combined heat and power operation, the heat output represents an input for the numerical simulation of plant operation. in the case of the gas turbine operation, the plant becomes a combined gas-steam plant with possibilities to analyze both only power production and combined heat and power production. there is also a possibility to adjust the scheme to represent the option of replacing the existing regenerative heaters (steam boiler remains) with a gas turbine, and to represent the option of replacing the existing regenerative heaters and steam boiler with a gas turbine. fig. 1 schematic flow diagram of a combined cycle power plant 3. steam power plant cycle description turbines include a high-pressure turbine (hpt), a medium-pressure turbine (mpt) and two low-pressure turbines (lpt). the steam turbine set is equipped with seven extraction ports. preheating of the condensed steam is done in the low-pressure preheaters of the plant which include four low-pressure heaters (lph4, lph5, lph6, lph7). heating up the boiler (sb) feed water to the final stage at the input of the boiler is done by three high-pressure regenerative heaters (hph1, hph2, hph3). these heaters are supplied with steam from the extraction ports (ep1, ep2, ...). part of the steam from the high-pressure 58 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić turbine is extracted for the high-pressure regenerative heater (hph3), while the other regenerative heaters and deaerator are supplied with steam extracted from the medium and low-pressure turbines. the deaerator (da) removes the non-condensable gases from the steam cycle. used steam from the low-pressure turbine is completely condensed in the surface condenser (cn). for satisfying the heat demand, two heaters are defined (dhh1 and dhh2, for heat output between 0 mw and 170 mw) with controlled steam extraction from the medium-pressure turbine (ep1 and ep2). the plant uses natural gas as a primary fuel with the lower heating value of this fuel mostly above 50020 kj/kg [14]. the gas turbine facility is of classical configuration with a compressor, burning chamber, gas turbine and heat recovery steam generator. the central place in the heat-flow scheme of the combined gas-steam facility belongs to the heat recovery steam generator, which is used for preheating feed water or for the production of superheated steam. additional combustion in the heat recovery steam generator is provided as well. technical specifications are presented in table 1. table 1 total properties of the existing steam power plant nominal electrical power of the facility (mw) 121 nominal heat power of the facility (mw) 170 boiler steam production (kg/s) 121 steam pressure in point 1 (mpa) 12.1 steam temperature in point 1 (ºc) 555 interheated steam pressure (mpa) 2.6 interheated steam temperature (ºc) 363 low/high pressure pre-heater number 4/3 heater number 2 condensing pressure (mpa) 0.0065 natural gas lower heating value (kj/kg) 50020 extraction port pressure ep7 (mpa) 2.6 extraction port pressure ep6 (mpa) 1.84 extraction port pressure ep5 (mpa) 1.02 extraction port pressure ep4 (mpa) 0.42 extraction port pressure ep3 (mpa) 0.23 extraction port pressure ep2 (mpa) 0.069 extraction port pressure ep1 (mpa) 0.0067 4. thermodynamic analysis modeling the repowered cycle the mathematical model of the analyzed plant is a group of nonlinear algebraic equations. for solving these equations, a sequential simulation approach of the facility is adopted in this paper. custom software is developed for solving the mathematical model and performing the simulation of combined heat and power production. selecting the operation parameters allows switching the heat-flow scheme to various options (section 2.2) and simulating nominal and other plant operating regimes. plant operation simulation enables one to check the performances of the analyzed option and to perform diagnosis for increasing energy efficiency. plant simulation is merely the first step toward its optimization. the algorithm that the software follows is shown in fig. 2. energy analysis of repowering steam power plants by feed water heating 59 the model is defined as the network of inter-connected modules: turbines, heat exchangers, pumps, compressor, combustion chamber, gas turbine and heat recovery steam generator. the model has a modular structure, so that it quickly adopts various operating regimes. the mathematical model on the proposed flow sheeting problem formulation is created as a type of steady state simulation. numerical integration was done using the microsoft excel programming platform and the visual basic programming language. the thermodynamic properties of water and steam can be calculated by a group of functions using the equations from the iapws-if97 formulae for industrial applications [15]. the independent variables of the simulation model are defined as mass flow rate, pressure and specific enthalpy. by combining the iapws-if97 formulae with an iterative procedure, the dependent variables of the model can be calculated as a function of pressure and specific enthalpy. thus, using the three variables, the properties of each flow for the different regions can be determined. fig. 2 methodological framework the repowered cycle is separately modeled for each section. the governing equations of each section are as follows. 60 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić 4.1. steam turbine (st) the steam turbines effective efficiencies are evaluated using the spencer, cotton and cannon model [16]. the basic efficiency of the turbine, which is a function of the load, is corrected by factors, considering volume flow, pressure ratio, initial pressure and temperature: basic efficiency  correction factors = effective efficiency the most detailed approach is given by spencer, cotton and cannon [16]. the turbine internal efficiency for any operating regime is calculated by correction factors: * , 0 1 (1 ) n st new i j j= = +   , (1) where: * 0i the “zero” or weighted internal efficiency without a correction for the influential parameters deviation, j the efficiency deviation for the impact of the j-th influential parameter, n – the number of correction coefficients. different correction factors are evaluated in each turbine section, i.e. from one extraction point to the next extraction point. thus, a different effective efficiency is calculated for every section. exhaust losses, mechanical losses and generator losses are considered as well. when the extractions of the steam turbines are closed, the flow rate and state of steam will change at each section. therefore, the steam turbines will no longer work at the designed condition. there are three steam turbines with different operating pressures, and each turbine should be separately analyzed. the following equation is called the stodela equation and it relates to mass flow rates, temperatures and pressures for the first and the new cycles [17-19]: 2 , ,, , 2 , , , , 1 1 in new out newnew out new in new out first in first in firstfirst out first p pp tm p t pm p     −      =   −      , (2) the net steam turbine power is given as below: , ,( )st steam steam in steam outw m h h   = − , (3) where: steamm  mass flow rate of steam (kg/s), h – specific enthalpy (kj/kg). 4.2. heat exchangers the heat exchangers are modeled using the ttd and tdca parameters and pressure losses provided by the manufacturers. ttd is the difference between the temperature of the outlet cold flow and the saturated steam temperature of the inlet hot stream. tdca is the temperature difference of the outlet hot flow, which is subcooled, and the inlet cold flow [20, 21]. energy analysis of repowering steam power plants by feed water heating 61 4.3. pump the pumps (p) are simulated with regard to their efficiency as a function of the mass flow, evaluated from the experimental data. the condensate pump is neglected in the simulator, because its consumption is very small. the condenser, the boiler and the reheater are not simulated, only matter and energy balances are used to evaluate the properties of the unknown streams. in the case of the boiler and reheater, energy efficiency is evaluated from the real plant data. 4.4. compressor (ac) the outlet temperature is obtained assuming an adiabatic process [22]. 1 , , 1 1 air air k k p ac out in ise ac r t t  −   − = +       , (4) , , , out ac p ac in ac p r p = pressure ratio, (5) where air air p air v c k c = for air, pc specific heat capacity at constant pressure (kj/kgk) and vc heat capacity at constant volume (kj/kgk). the required power for the compressor is given by: , , , , (p air out ac in ac ac air ise ac c t t w m     − =       , (6) 4.5. combustion chamber (cc) for this component the following balance equation applies: mass and energy balance equations are written as bellow: ,air f cc gm m m    + = . (7) , , , ,, ,air b g c air f cc g cp air b cc p g cm c t m lhv m c t    + = , (8) in the chamber there is a pressure loss, which is given as an input database parameter, based on which the leaving pressure can be calculated as: , ,(1 )g c cc air bp p= − . (9) where cc pressure loss (%). the boundary condition is given as the exhaust gas temperature leaving the chamber. based on this temperature, the composition and thermophysical properties of exhaust gases are determined. 62 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić 4.6. gas turbine for this component, the following balance equation applies [23]: the temperature of steam leaving the turbine: 1 , , , ,1 (1 ) g g k k iout gt in gt ise gt p gtt t r − −   = − −     . (10) g g p g v c k c = for exhaust gases (11) . , , in gt p gt out gt p r p = pressure ratio, (12) where cc pressure loss (%) the net gas turbine power is related to the turbine and compressor power as below: ,net gt gt acw w w    = − , (13) 4.7. heat recovery steam generator (hrsg) this section is based on the heat exchange between two fluids: the water coming from the condenser and the exhaust gases of the gas turbine(s) combined with the gases produced in the duct burner. the thermodynamic properties at different sections of a double pressure reheating hrsg are obtained by: / , / , / , , , 1( ) ( )(1 ) g water steam gout water steam in water steam p in g out gm h h m c t t e   − = − − . (14) where e1 is the percentage of heat loss in each section, which is taken as 5%. this equation can yield the heat exchanged in every component as well as temperatures in characteristic points of hrsg [24]. 4.8. duct burner (db) applying the energy balance equation to calculate the mass flow rate of the fuel added in the duct burner: , , , , ,, ,d g d g e g f db g ep g d db p g em c t m lhv m c t    + = . (15) the mass balance equation is written as bellow , , ,g d f db g em m m    + = . (16) 4.9. exhaust gases for the mixture of ideal gases (g flue gases can be treated as such), the specific heat capacity was calculated by the rosario-messina method [25]: energy analysis of repowering steam power plants by feed water heating 63   5 * 0 ( ) (ln ) /i p i i c t a t j molk = = , (17) where ia are the coefficients from [25] and * 0 t t t = . (18) 4.10. thermal efficiency the thermal efficiency of the combined cycle (ccpp) is given by [26]: , , , net ccpp dh th ccpp f ccpp w q q     + = , (19) where the net power of the combined cycle equals the sum of the net power of the gas cycle and the net power of the steam cycle: , , ,net ccpp net gt net stw w w    = + . (20) the total fuel consumption equals the sum of the fuel consumption in the gas cycle, the fuel consumption in the steam cycle and, if necessary, the fuel consumption in the heat recovery steam generator: f ,ccpp f ,cc f ,db f ,sbq q q q     = + + . (20) 5. results and discussion the developed software provided numerical simulations of the combined gas-steam cycle for different configurations. in this paper, the simulation of the combined facility is shown in fig. 1. the presented scheme is one of the potential repowering methods for the existing thermal power plants, with the gas-block replacing the regenerative feed water and condensing steam heaters, while keeping the steam boiler in operation. in both cases the plant operates without steam interheating. the steam part of the combined cycle represents the scheme applied in the chp plant novi sad, which uses natural gas as a primary fuel and has been in operation for more than 25 years. the model inputs are given in table 1, representing the design conditions for the chp plant novi sad. based on these parameters and the described mathematical model of the facility, a numerical simulation of the intermittent operation was performed. the simulation was performed by considering the parameters from table 1 and varying some parameters for the intermittent operation as: ▪ steam production ranges from 60 to 110% compared to the nominal value (i.e. 90130 kg/s), ▪ the plant operates in the condensing regime (qdh=0 mw) or in the regime with a controlled extraction of steam (qdh=170 mw), ▪ the compression ratio of the compressor ranges from 10 to 30, and, ▪ the temperature of exhaust gases entering the gas turbine ranges from 1200 to 1600k. 64 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić 5.1. impact of the compression ratio and the entering temperature in the gas turbine on the energy efficiency of the plant figs. 3-11 represent the impact of the compression ratio and the temperature of exhaust gases entering the gas turbine on the thermal efficiency of the plant for the configuration with the steam boiler. the dependency is drawn for three different cases of heat output and maximum, average and minimum steam production in the boiler, and for four different temperatures of exhaust gases entering the gas turbine. fig. 3 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 130 kg/s steam production and 170 mw heat output fig. 4 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 130 kg/s steam production and 102.4 mw heat output energy analysis of repowering steam power plants by feed water heating 65 fig. 5 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 130 kg/s steam production and no heat output fig. 6 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 114 kg/s steam production and 170 mw heat output 66 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić fig. 7 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 114 kg/s steam production and 102.4 mw heat output fig. 8 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 114 kg/s steam production and no heat output energy analysis of repowering steam power plants by feed water heating 67 fig. 9 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 90 kg/s steam production and 170 mw heat output fig. 10 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 90 kg/s steam production and 102.4 mw heat output 68 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić fig. 11 thermal efficiency as a function of the compression ratio (r) and the temperature of exhaust gases entering the gas turbine for 90 kg/s steam production and no heat output in general, for the same compression ratio, thermal efficiency increases with the increasing temperature of exhaust gases entering the gas turbine (figs. 3-11). for a constant temperature of exhaust gases entering the gas turbine, thermal efficiency increases to the maximum value with an increase in the compression ratio, and then decreases. thermal efficiency also depends on the operating regime of the facilities. with a constant heat output it decreases if the steam production increases, and vice versa, for a constant steam production it decreases if the heat output decreases. depending on the heat output, thermal efficiency varies from 35% (condensing operation) to 81% (heat production operation). compared to the results from different authors [27] (fig. 12), the same trend of thermal efficiency occurs. 5.2. influence of the temperature of exhaust gases entering the gas turbine on fuel consumption fuel consumption dependency on the temperature of exhaust gases entering the gas turbine, for the combined gas-steam cycle with the steam boiler, is shown in fig. 13. fuel consumption is given for the average steam production and nominal heat output, but for different compression ratios. fig. 12 chp thermal efficiency as a function of the compression ratio (r) [27] energy analysis of repowering steam power plants by feed water heating 69 fuel consumption slightly increases with an increase in temperature, due to the fact that the total consumption matches the consumption of the gas and steam part of the plant. fig. 13 fuel consumption as a function of the temperature of exhaust gases entering the gas turbine, for steam production of 114 kg/s and heat output of 170 mw 5.3. influence of the temperature of exhaust gases entering the gas turbine and steam production on the electrical power of the plant the influence of the temperature of exhaust gases entering the gas turbine and steam production on the electrical power of the plant, for the nominal heat output and compression ratio, is given in fig. 14. electric power increases with an increase in temperature and steam production. fig. 14 electric power as a function of the temperature of exhaust gases entering the gas turbine, for the nominal heat output 170mw and compression ratio of rpac=30 70 d. mitrović, m. ignjatović, b. stojanović, j. janevski, j. škundrić 5.4. influence of the compression ratio on electrical power fig. 15 show the dependency of the electrical power output on different compression ratios, for a constant steam production and temperature of exhaust gases entering the gas turbine. the same figure represents the change of thermal efficiency for the same conditions. the electrical power output rises with the lowering heat output, since less steam is extracted for satisfying the heat output. at the same time, thermal efficiency of the plant decreases with the lower heat output. the electrical power output increases with an increase in the compression ratio. fig. 15 electrical power output and thermal efficiency as a function of the compression ratio for steam production of 114 kg/s and entering temperature of 1600 k 6. conclusion repowering is an effective method for improving the efficiency and increasing the productive lifespan of an old steam power plant. using the energy analysis method, the repowering of a power plant was investigated in this paper. the partial repowering option with feed water heating in the heat recovery steam generator was analyzed. the repowered power plant was analyzed for several operating regimes. energy efficiency of the repowered cycle was selected as the target function. the best repowering mode at which the maximum energy efficiency was achieved is for steam production of 114 kg/s and nominal heat output of 170 mw. in these conditions, energy efficiency reaches the value of 71%, for the compression ratio of 30. for the same conditions, fuel consumption slightly increases with an increase in the temperature of exhaust gases entering the gas turbine, and ranges from 9.69 to 9.75 kg/s. the electrical power output increases with a temperature increase and with stream production increase, for the nominal heat output. for parameters which lead to maximum energy efficiency, electrical power rises from 166 mw (temperature of 1300k) to 178 mw (temperature of 1600k). electrical power increases with the heat output decrease (less steam is extracted from turbines), but energy energy analysis of repowering steam power plants by feed water heating 71 efficiency decreases. for the compression ratio of 30, the electrical power output varies from 177 mw to 199 mw, with the heat output reduced from the nominal (170 mw) to zero. energy efficiency varies in the range of 0.38 (slightly higher compared to the steam power plant) to 0.72. these values show that the repowering option with the combined gas-steam cycle is very attractive regardless of the type of operation, i.e. whether only power or both heat and 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https://doi.org/10.22190/fume190415018n © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper mechanical structure design to avoid friction-induced instabilities: in-plane anisotropy and in-plane asymmetry ken nakano 1 , naohiro kado 1 , chiharu tadokoro 2 , takuo nagamine 2 1 yokohama national university, yokohama, japan 2 saitama university, saitama, japan abstract. the stability of a two-degree-of-freedom (2dof) sliding system with the velocity-weakening friction was examined by the eigenvalue analysis, where the inplane anisotropy and the in-plane asymmetry were considered. the obtained eigenvalues were organized by using the minimum modal damping ratio as the stability maps. selecting a stable point in the stability map corresponds automatically to embedding the yaw-angle-misalignment (yam) method in the mechanical structure design to avoid the instability. if we accept the mechanical structure design of sliding systems with the in-plane anisotropy and the in-plane asymmetry, we can find new stable conditions spread widely in the two-dimensional space, which are invisible from the conventional point of view. key words: friction, vibration, instability, stabilization, yam method 1. introduction friction-induced instabilities are important problems to be solved since they result in vibration and noise of mechanical products (e.g., brakes, transmissions, wipers, belts, and tyres for automobiles). one of the most famous mechanisms of friction-induced instabilities is the velocity-weakening friction. it has been known that mechanical systems tend to become unstable when the frictional force decreases with decrease in the velocity [1] (e.g., the frictional force of lubricated sliding systems operating under mixed lubrication conditions). received april 15, 2019 / accepted june 19, 2019 corresponding author: ken nakano yokohama national university, yokohama, japan e-mail: nakano@ynu.ac.jp 114 k. nakano, n. kado, c. tadokoro, t. nagamine when engineers have to solve the instability caused by the velocity-weakening friction, they always have two types of options. one is “improving frictional properties” and the other is “improving mechanical structures”. usually, they tend to choose the former since the former probably seems to be the lower-cost solution than the latter. however, in reality, a great deal of effort is required to find proper materials that show proper frictional properties. if they had some guidelines for mechanical structure design to avoid the instability in advance, the situations of the engineers would be improved. recently, as a promising method to stabilize mechanical systems suffering from the velocity-weakening friction, the “yaw-angle-misalignment (yam) method” has been proposed by the authors. the fundamental theory was first shown with an extended onedegree-of-freedom (1dof) sliding system in the context of friction measurements [2]. in the fundamental theory, by observing the two velocities of the sliding surfaces in the “top view”, their angular misalignment about the “yaw axis” was considered. after confirming the validity of the fundamental theory experimentally [2] and numerically [3], based on the structure of disc brakes, it was applied to a pad-on-disc-type sliding system numerically [4] and experimentally [5]. besides, as another application format for rotational machines, the stabilizing effect of parallel misalignment in circular contacts was also shown [6]. from the viewpoint of guidelines for mechanical structure design, to examine the stabilizing effect in more practical sliding systems, the above fundamental theory was extended to a two-degree-of-freedom (2dof) sliding system [7]. first, the stabilizing effect of the yam method in the 2dof sliding system was confirmed by numerical simulations. then, the stability limit was examined by the eigenvalue analysis in a dimensionless form, which clarified the importance of the two quantities: the anisotropy of the in-plane stiffness (termed simply as the “in-plane anisotropy”) and the asymmetry of the in-plane structure (termed simply as the “in-plane asymmetry”). however, as a price of using the dimensionless form, the final format did not become user friendly as a guideline for mechanical structure design. in light of the above situation, in this study, the results of the eigenvalue analysis for the 2dof sliding system are organized in a user-friendly format by focusing on the two most important quantities (i.e., the stiffness quantifying the “in-plane anisotropy” and the misalignment angle quantifying the “in-plane asymmetry”) to embed the yam method in mechanical structure design. by introducing the minimum modal damping ratio, the degree of stability is quantified and shown as the stability maps, which would be helpful for selecting the two quantities in the mechanical structure design of sliding systems. 2. theory 2.1. model fig. 1 shows the analytical model, which is the 2dof sliding system in the top view. a “ball” with mass m is in contact with a “plate” parallel to the xy plane at constant normal load fz. the ball is supported elastically in the xy plane by two springs with no damping. the stiffnesses in the x and y directions are kx and ky, respectively. when kx ≠ ky, they represent the “in-plane anisotropy” of the sliding system. the plate is driven at constant velocity va parallel to the xy plane. misalignment angle of va from the x axis is mechanical structure design to avoid friction-induced instabilities 115 φ. when φ ≠ 0, it represents the “in-plane asymmetry” of the sliding system. note that according to the above definition, the x and y axes are the principal axes of stiffness. fig. 1 analytical model: 2dof sliding system considering with “in-plane anisotropy” (kx ≠ ky) and “in-plane asymmetry” (φ ≠ 0) 2.2. velocities let x and y be the position of the ball (or the elongations of the two springs) as a function of the time t, i.e., x = x(t) and y = y(t). relative velocity vab of the plate to the ball is given by ab a b v v v , (1) where vb is the instantaneous velocity of the ball. the magnitudes of va and vb are given by a a constantv  v , (2) 2 2 b bv x y  v , (3) where ( • ) is the derivative with respect to t. from the velocity triangle made by va, vb, and vab, we obtain 2 2 ab ab a a( cos ) ( sin )v v x yv    v , (4) a ab sin sin yv v     , (5) a ab cos cos xv v     , (6) where θ is the direction of vab, which is defined as the angle from the x-axis. note that as shown in fig. 1, θ is not necessarily equal to φ. 116 k. nakano, n. kado, c. tadokoro, t. nagamine 2.3. equation of motion let f = f(vab) and f = f(vab) be the frictional force vector and its magnitude, respectively. focusing on the ball, we obtain the following equation of motion:  mx kx f , (7) where x y        x , (8) 0 0 m m        m , (9) 0 0 x y k k        k , (10) ab ab ( )cos ( )sin f v f v          f . (11) note that the direction of f is θ since it corresponds to the direction of vab. in steady sliding (i.e., vb = 0), the ball remains at rest at the equilibrium position x = xeq: a eq a ( ) cos ( )sin x y f v k k v f                x . (12) this is because vab = va when vb = 0, which leads to vab = va and θ = φ. 2.4 linearization let x be the displacement disturbance from xeq, that is: eq x x x . (13) substituting eq. (13) into eq. (7) and linearizing it around x 0 and x 0 under the assumption that ax v , we obtain the following linearized equation:   mx cx kx 0 , (14) where xx xy yx yy c c c c        c . (15) the elements of c are given by 2 2 1 2cos sinxxc d d  , (16) mechanical structure design to avoid friction-induced instabilities 117 1 2( )sin cosxy yxc c d d     , (17) 2 2 1 2sin cosyyc d d  , (18) where 1 a( )d f v , (19) a 2 a ( )f v d v  . (20) note that (' ) in eq. (19) is the derivative with respect to vab. therefore, d1 means the slope of the frictional force against the relative velocity at vab = va, which acts as a damper for the sliding system, although there are no dampers in the sliding system. 2.5 eigenvalue equation using state vector x defined as        x x x , (21) eq. (14) is described as x ax , (22) where, by using zero matrix o and identity matrix i, a is given by 1 1          o i a m k m c . (23) hence, by solving the following eigenvalue equation: det( ) 0s a i (24) for s: s j   , (25) the complex eigenvalues are obtained, where j is the imaginary unit. if the real part σ of every eigenvalue s is negative, the equilibrium point is stable, which leads to steady sliding. otherwise, the equilibrium point is unstable, which leads to friction-induced vibrations in the sliding system. 3. method the eigenvalue equation (eq. (24)) was solved numerically under the assumption that kab ab( ) ( ) zf v v f , (26) where μk = μk(vab) is the kinetic friction coefficient: 118 k. nakano, n. kado, c. tadokoro, t. nagamine k k k0 k a b f b a( ) ( )exp v v v              , (27) where μk0 and μk∞ are the dimensionless constants, and vf is the velocity constant. in this study, the values of the three constants were μk0 = 2, μk∞ = 1, and vf = 10 –2 m/s. note that since μk0 > μk∞, the magnitude of frictional force f(vab) shows the negative dependence on relative velocity vab, which means the velocity-weakening friction. 4. results and discussion 4.1 typical results fig. 2 shows the real part σ (upper) and imaginary part ω (lower) of every eigenvalue as functions of drive velocity va. the left column (a) is for the “isotropic” stiffness (kx = ky) and the “symmetric” structure (φ = 0). the middle column (b) is for the “anisotropic” stiffness (kx ≠ ky) and the “symmetric” structure (φ = 0). the right column (c) is for the “anisotropic” stiffness (kx ≠ ky) and the “asymmetric” structure (φ ≠ 0). fig. 2 real part σ (upper) and imaginary part ω (lower) of complex eigenvalue for m = 1 kg, kx = 10 6 n/m, μk0 = 2, μk∞ = 1, vf = 10 –2 m/s, and fz = 10 1 n; (a) isotropic and symmetric (ky = 10 6 n/m and φ = 0°); (b) anisotropic and symmetric (ky = 10 7 n/m and φ = 0°); (c) anisotropic and asymmetric (ky = 10 7 n/m and φ = 45°) four eigenvalues were obtained from the eigenvalue equation since the analyzed model was 2dof. some of them were conjugate: for example, the red curves of (a), the red curves of (b), and the blue curves of (c). besides, for higher va (e.g., va > 10 –2 m/s), the blue curves of (a), the blue curves of (b), and red curves of (c) were also conjugate. focusing on the upper graph of (a), the red curve locates above the zero line for any va, which means that due to the velocity-weakening friction, the sliding system is always unstable. as shown in the upper graph of (b), this situation does not change even if the mechanical structure design to avoid friction-induced instabilities 119 sliding system is anisotropic. however, as shown in the upper graph of (c), if the sliding system is not only anisotropic but also asymmetric, the blue curve changes from positive to negative with increase in va. considering that the sliding system cannot be asymmetric when it is isotropic, we can say that the combination of the in-plane anisotropy (kx ≠ ky) and the in-plane asymmetry (φ ≠ 0) is necessary to stabilize the sliding system. on the other hand, focusing on the lower graphs of (a) to (c), every curve is vertically symmetric about ω = 0. note that the non-zero broken lines in the lower graphs show the natural frequencies defined as n x x k m   , (28) n y y k m   . (29) fig. 3 modal frequency ratio ω * (upper) and modal damping ratio ζ * (lower) for m = 1 kg, kx = 10 6 n/m, μk0 = 2, μk∞ = 1, vf = 10 –2 m/s, and fz = 10 1 n; (a) isotropic and symmetric (ky = 10 6 n/m and φ = 0°); (b) anisotropic and symmetric (ky = 10 7 n/m and φ = 0°); (c) anisotropic and asymmetric (ky = 10 7 n/m and φ = 45°) fig. 3 shows the same complex eigenvalues as fig. 2, which are shown by using the modal frequency ratio ω * and the modal damping ratio ζ * defined as * n i i x     , (30) * 2 2 i i i i        , (31) 120 k. nakano, n. kado, c. tadokoro, t. nagamine respectively. note that by using ω * and ζ * , four eigenvalues are found to be reduced to two values (i.e., red and blue). for the sliding system to be stable, every ζ * needs to be positive, where ζ * takes a value in the range of –1 ≤ ζ * ≤ 1. fig. 4 shows the effect of the misalignment angle φ on ω * and ζ * . note that the ω * and ζ * -values at φ = 45° in fig. 4 correspond to those values at va = 10 –2 m/s in fig. 3(c). when φ < φcr1 ~ 27°, the sliding system is unstable due to the negative ζ * denoted by the red curve. on the other hand, when φ > φcr2 ~ 60°, the sliding system is also unstable due to the negative ζ * denoted by the blue curve. however, only when φcr1 < φ < φcr2, the both ζ * -values are positive, which means the sliding system is stable. as stated above, the combination of the in-plane anisotropy (kx ≠ ky) and the in-plane asymmetry (φ ≠ 0) is necessary to stabilizes the sliding system. however, it is not sufficient. fig. 4 effect of in-plane angular misalignment φ on modal frequency ratio ω * (upper) and modal damping ratio ζ * (lower) for m = 1 kg, kx = 10 6 n/m, ky = 10 7 n/m, μk0 = 2, μk∞ = 1, vf = 10 –2 m/s, fz = 10 1 n, and va = 10 –2 m/s note that the strongest asymmetry does not necessarily mean φ = 45°. as shown in the lower graph of fig. 4, the red curve crosses with the blue curve at φ = φopt ~ 36°. by considering that the stability of the sliding system is governed by the minimum modal damping ratio: * * min min( )i  . (32) φ = φopt is believed to be a favorable choice for the mechanical structure design since ζ * min takes the maximum at φ = φopt. in light of the above discussion, in the next section, the results of the eigenvalue analysis are shown by using ζ * min as the stability maps. mechanical structure design to avoid friction-induced instabilities 121 4.2 stability maps fig. 5 shows the stability maps in the ky-va plane. the value of ζ * min from –1 to 1 is shown by the color from red (unstable) to blue (stable), respectively. the horizontal broken line in each graph shows the “isotropic” condition (i.e., kx = ky). the left and right graphs are for the “symmetric” conditions (i.e., φ = 0° and 90°, respectively), while the middle graph is for the “asymmetric” condition (i.e., φ = 45°). fig. 5 stability maps in ky-va plane with minimum modal damping ratio ζ * min; m = 1 kg, kx = 10 6 n/m, μk0 = 2, μk∞ = 1, vf = 10 –2 m/s, and fz = 10 1 n; red: unstable (ζ * min < 0) and blue: stable (ζ * min > 0) the blue color (ζ * min > 0: stable) can be seen only in the middle graph, which locates above (ky > kx) and below (ky < kx) the horizontal broken line. this means that if the sliding system is asymmetric, not only increasing the stiffness but also decreasing the stiffness is effective for stabilization. however, comparing the upper blue portion with the lower blue portion, the stability limit of the former is wider than the latter. the stability limit for the upper blue portion seems to have a slope of –1 for lower va, while the stability limit for the lower blue portion seems to be determined by va = vf = 10 –2 m/s. this indicates that increasing the higher stiffness (ky > kx) is better than decreasing the lower stiffness (ky < kx) although both these changes strengthen the in-plane anisotropy. note that the red color (ζ * min < 0: unstable) locating for lower va is caused by the provided frictional property (see eq. (27)). the velocity-weakening friction (μk0 > μk∞) makes the damping coefficient d1 negative (see eq. (19)), the value of which is decreased by decreasing va, which enhances the instability for lower va. the other damping coefficient d2 is the essence of the yam method, which is always positive (see eq. (20)). when the sliding system is symmetric (φ = 0), the damping term of d2 dissapears because sin 2 φ = 0 (see eq. (16)). however, when the sliding system is asymmetric (φ ≠ 0), the positive d2 becomes the competitor of the negative d1 (see eq. (16)). this is the fundamental mechanism of the yam method to stabilize the sliding system, which is strongly related to the in-plane rotation of the frictional force vector [8, 9]. fig. 6 shows the stability maps in the ky-φ plane, which consists of nine graphs. the bottom, middle, and top rows are for the normal load fz = 10 –1 , 10 1 , and 10 3 n, respectively. 122 k. nakano, n. kado, c. tadokoro, t. nagamine the left, middle, and right columns are for the drive velocity va = 10 –4 , 10 –2 , and 10 0 m/s, respectively. the ordinate of each graph is stiffness ky representing the in-plane anisotropy of the sliding system, while the abscissa of each graph is misalignment angle φ representing the in-plane asymmetry of the sliding system. the value of ζ * min is shown by the color in the same manner as the previous figure. fig. 6 stability maps in ky-φ plane with minimum modal damping ratio ζ * min; m = 1 kg, kx = 10 6 n/m, μk0 = 2, μk∞ = 1, and vf = 10 –2 m/s; red: unstable (ζ * min < 0) and blue: stable (ζ * min > 0) only the red color (ζ * min < 0: unstable) can be seen in the top-left, top-middle, and middle-left graphs, which means that any combination of ky and φ within the graph cannot stabilize the sliding system. however, in the rest six graphs, the blue color (ζ * min > 0: stable) can be seen. especially, in every graph located on the diagonal from the bottom mechanical structure design to avoid friction-induced instabilities 123 left to the top-right, two deep blue portions surrounded by dense contour lines can be seen above and below the horizontal broken lines. by considering the positions of the deep blue portions, for example, a combination “ky ~ 10 7 n/m and φ ~ 40°” or another combination “ky ~ 10 4 n/m and φ ~ 50°” seems to be favorable for stabilizing the sliding system. note that the sliding system is unstable on the horizontal line (kx = ky: isotropic) and the two vertical lines (φ = 0° and φ = 90°: symmetric), although the bottom-right graph seems to be filled with the blue color. for example, the “h-shaped” red area observed in the bottom-left graph is the area spread from the three unstable lines. friction-induced instabilities have been often discussed with the conventional 1dof sliding model [10]. it is a picture when we see the 2dof sliding system of φ = 0 in the “front view” (i.e., in the y direction). in other words, it is implicitly assumed that the sliding model is isotropic and symmetric. however, based on the stability maps proposed in this study, we realize that the conventional 1dof sliding model just have provided the discussion on the horizontal broken line of the left graph in fig. 5 or the discussion on the vertical line φ = 90° of the graphs in fig. 6. now we can discuss the instabilities caused by the velocity-weakening friction based on the stability maps. if we accept the mechanical structure design of sliding systems with the in-plane anisotropy (kx ≠ ky) and the in-plane asymmetry (φ ≠ 0), we can find new stable conditions spread widely in the two-dimensional space, which are invisible from the conventional point of view. 5. conclusion the stability of the 2dof sliding system (fig. 1) was examined by the eigenvalue analysis, with considering the in-plane anisotropy (kx ≠ ky) and the in-plane asymmetry (φ ≠ 0). the results were organized by using the minimum modal damping ratio ζ * min (eq. (32)) as the stability maps (figs. 5 and 6). especially for the stability maps in the ky-φ plane (fig. 6), selecting a combination of the two quantities (ky and φ) from the deeper blue portions corresponds automatically to embedding the yam method in the mechanical structure design to avoid instabilities caused by the velocity-weakening friction. again, when engineers have to solve friction-induced instability problems, we must not forget that they always have two types of options. as a guideline for “improving mechanical structures”, the authors believe that the stability maps considering the in-plane anisotropy and the in-plane asymmetry can be powerful tools, which would at least reduce the effort for “improving frictional properties” involving the trial-and-error processes. if we accept the mechanical structure design of sliding systems with the in-plane anisotropy and the in-plane asymmetry, we can find new stable conditions spread widely in the twodimensional space, which are 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engineering vol. 16, no 3, 2018, pp. 337 345 https://doi.org/10.22190/fume180824033b © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper experimental investigation of optimal ed machining parameters for ti-6al-4v biomaterial  udc 621.9.011:669.295 amandeep singh bhui, gurpreet singh, sarabjeet singh sidhu, preetkanwal singh bains department of mechanical engineering, beant college of engineering & technology, punjab, india abstract. the present study investigates optimal parameters for machining of ti-6al4v using edm with graphite electrode. herein, another technique of modifying surface properties and enhancing machining rate using electrical discharge machining (edm) was developed. in the present study, design of experiment (d.o.e) was developed using the taguchi’s orthogonal array to examine the effect of the input machining factors on the machining characteristics, and to forecast the optimized edm parameters in terms of peak current, pulse-on time, pulse-off time and applied gap voltage. each experiment was performed to obtain a hole of 1mm depth on the workpiece. from the results, it is found that the discharge current has significant influence on material removal rate (mrr) and surface roughness (sr) followed by other selected parameters, i.e. pulse-on time, pulse-off time. the mrr augmented steeply with the current and was recorded as maximum at 4 amps. in-vitro bioactivity test was conducted in the simulated body fluid to examine bioactivity confirming a significant apatite growth on the surface treated with ed sparks. the surface and chemical alteration were analyzed by using scanning electron microscopy (sem) and x-ray diffraction (xrd) along with the identification of the substantially enhanced morphology for clinical success. key words: electrical discharge machining, material removal rate, surface roughness, bioactivity 1. introduction one of the most investigated modern machining processes in the recent decades is electric discharge machining (edm). the conversion of electric into heat energy takes place in the gap between the electrode and the workpiece [1]. the edm is widely used in received august 24, 2018 / accepted november 01, 2018 corresponding author: gurpreet singh department of mechanical engineering, beant college of engineering & technology, gurdaspur – 143521, punjab, india e-mail: singh.gurpreet191@gmail.com 338 a.s. bhui, g. singh, s.s. sidhu, p.s. bains die making, aerospace and medical industries. during the spark generation, material erosion from both the electrodes takes place and at the end of machine cycle, constituents from both the dielectric and the tool electrodes get deposited on the material surface [2]. the edm is found to be a potential candidate for producing surfaces with better surface topology and precision, compared to other machining methods [3]. titanium alloys have wide applications in aerospace, medical domain (dental and orthopedics) subject to their competitive mechanical, anti-corrosive, and biocompatible properties. ti-6al-4v (α + β) phase alloy has been the most desirable metallic biomaterial for medical implants [4, 5]. however, the presence of amino acids and proteins in the body fluids accelerates the corrosion process by releasing metallic ions which causes poor osseointegration and leads to cytotoxicity resulting in allergic reactions and, ultimately, to failure of the implant [6]. although the formation of metallic ions is harmful, the use of ti-6al-4v alloy would continue due to its excellent properties for use in the human body. for this reason, improvement in surface properties of the alloy is required for prolonged metalbone-tissue adhesion. the experiments were conducted on ti grade-5 alloy by verma et al. [7] using copper electrode in the die sinking edm for optimum input parameters manifested to attain maximum mrr and minimum surface roughness. micro-cracks and craters were observed under the sem on the edmed surface. kumar et al. [8] put forward the investigation of mrr and surface finish of ti-6al-4v eli with the edm using current, pulse-on and voltage as input parameters. the current was concluded as the most significant factor for both mrr and surface roughness. optimum parameters were 18 ampere, 40 volts and 100 µs pulse-on that yielded maximum mrr and minimum sr. peng et al. [9] investigated that ed machining of titanium alloy performed for a short duration resulted in the formation of nano-structured compounds on the surface while a bioactive nano-porous tio2 layer was also observed at the same time. lee et al. [10] studied the edmed layer of ti-6al-4v alloy by varying pulse durations from 10 µs to 60 µs. the new surface thus produced had micro surface roughness and a nano-porous tio2 layer. increased adhesion, proliferation and differentiation of mg63 cells were observed. it was concluded by prakash et al. [11] that edm holds the potential for surface improvement and surface modification of ti alloy in terms of forming oxides and carbides on the surface that inhibit bio-compatibility, improve surface hardness, corrosion resistance and the formation of a nano-porous layer. ayesta et al. [12] conducted a study project on edm applications, effects of input factors on output responses by machining thin slots using electrical discharge machining. they concluded that peak current and pulse-on time were the dominant factors affecting machining time and the tool wear rate and machining time. ristic et al. [13] presented their work on the selection of material for implants using „expert system‟. in their study, they demonstrated a novel technique based on multiple criteria decision-making using fuzzy logic for appropriate selection of biomaterial. similarly, while optimizing the metal removal rate and sr of en31 tool steel in edm machining by das et al. [14], it was found that the discharge current was the most significant parameter for variations in the metal erosion rate. with increase in the value of discharge current and pulse-on-time, the metal erosion rate also increased during the machining process. mahajan et al. [15] reviewed the feasibility of various metallic biomaterials, i.e. titanium alloy, stainless steel, cr-co alloys and their surface modification techniques for enhanced functionality. torres et al. [16] conducted their experiments using copper electrode for better surface properties of machining and concluded that mrr was directly proportional to experimental investigation of optimal ed machining parameters for ti-6al-4v biomaterial 339 the current, i.e. the former enhanced with increase in discharge current due to rise in temperature of the workpiece. furthermore, with increase in pulse time, the electrode wear decreased drastically, whereas surface roughness was greater at higher pulse-on time and higher current intensity. ti and its alloys especially ti-6al-4v alloy gained more attention in recent scenario, due to their superior biocompatibility and excellent mechanical properties, especially low young‟s modulus, high fatigue performance, high corrosion resistance and low density as compared to other metallic biomaterials [17]. this study is dedicated to interpreting the influence of edm process parameters on the machining rate and surface characterization of medical grade titanium alloy. furthermore, the in-vitro test was conducted on the machined province for examining biological behavior of the modified surfaces. 2. material and method 2.1. materials used titanium grade 5 (ti-6al-4v) procured from baoji fuyuan tong industry and trade co. ltd., china, was employed as the workpiece in the form of a rectangular block (size: 80 mm x 80 mm x 5 mm) for conducting experimentation. an electrolytic graphite electrode with diameter of 900μm was preferred as a tool due to its better conductivity for the machining purpose. the chemical composition of ti-6al-4v used for the experimentation is shown in table 1. table 1 chemical composition of ti-6al-4v element fe c o n h al v ti % 0.09 0.03 0.03 0.003 0.001 6.1 4.2 balance 2.2. method the design of experiment was generated using the taguchi‟s l9 (4x3) array by minitab 17 software. table 2 demonstrates the input machining factors, i.e. current, pulse-on time, pulse-off time, gap voltage, with their respective levels selected for the experimentation. the output parameters were analyzed with the condition „larger-is-better‟ for material removal rate and surface finish. the units and levels of experimental factors, i.e. discharge current, pulse-on time, pulse-off time and voltage are shown in table 2. table 2 factors and their levels input factors level 1 level 2 level 3 units current (i) 1 2 4 a pulse-on time (p-on) 30 45 60 µs pulse-off time (p-off) 60 90 120 µs voltage (v) 30 40 50 v 2.3. experimental procedure the experiments were performed on the die sinker type edm machine (make: oscarmax, taiwan, fig. 1) in negative polarity conditions (i.e., the work piece (-) and tool 340 a.s. bhui, g. singh, s.s. sidhu, p.s. bains electrode (+)). during machining, the tool feed was set downward to the workpiece with the servo mechanism used by the edm. magnetic fixtures were used to hold the workpiece and the electrode during the process. for each trial, machining was carried out up to a depth of 1 mm on the workpiece surface and time for each experiment was noted down. surface roughness was measured using the mitutoyo sj-400 surface roughness tester in terms of arithmetic average of absolute value ra (µm). each sample was measured diametrically from three locations on the machined surface and was averaged for further analysis. after the machining, surface roughness was checked using the mitutoyo surface roughness tester. in this experimentation, discharge current (i), pulse-on time (p-on), pulse-off time (poff) and voltage (v) were selected as input factors being the most common and influencing parameters. the response parameters selected were material removal rate (mg/min) and surface roughness (ra) in μm. fig. 1 pictorial view of experimental set-up further, a bone-like apatite formation ability test was conducted to analyze bioactivity of the edm-treated ti-6al-4v alloy. disc type samples of 10 mm × 5 mm were cut from edm-treated ti-6al-4v alloy specimen using wire-edm. then, the samples were immersed in 50 ml ringer‟s solution in a hermetic beaker and kept under thermostatic conditions in a water bath at 37 ± 1°c for 7 days, as per the procedure adopted elsewhere [18]. 3. results and discussion mrr and surface roughness were calculated against the combination of selected input machining parameters. digital weighing machine (citizen, cy220), with reading up to three decimal places, was used to measure weight before and after machining for each sample; surface roughness was measured with the mitutoyo surface roughness tester. the experimental design matrix and responses are shown below in table 3. the responses tabulated in table 3 are represented according to the signal to noise ratio (s/n) experimental investigation of optimal ed machining parameters for ti-6al-4v biomaterial 341 methodology a ratio of strength of signal to the magnitude of error. s/n ratio depends on the type of responses measured such as a “higher-is-better” type given by equation below:                         r 1i 2 ihb y 1 r 1 log10 n s (1) where r is the number of repetitions of responses; yi value of response at i th trial. table 3 experimental design and responses exp. no. current (a) p-on (µs) p-off (µs) voltage (v) responses mrr (mg/min.) sr (µm) r1 r2 s/n ratio (db) r1 r2 s/n ratio (db) 1 1 30 60 30 2.00 2.00 6.0206 0.073 0.080 -22.35 2 1 45 90 40 1.73 1.77 4.8591 0.106 0.050 -23.88 3 1 60 120 50 2.07 1.94 6.0286 0.103 0.103 -19.74 4 2 30 90 50 4.94 5.72 14.4647 0.176 0.210 -14.39 5 2 45 120 30 6.73 7.93 17.2147 0.593 0.499 -5.352 6 2 60 60 40 8.70 9.31 19.0747 0.116 0.086 -20.20 7 4 30 120 40 6.00 6.00 15.5630 0.560 0.426 -6.385 8 4 45 60 50 25.61 23.59 27.7967 0.123 0.120 -18.31 9 4 60 90 30 26.88 24.89 28.2417 1.140 1.156 1.198 r: repetitions 3.1. analysis of variance of s/n ratio for mrr, and sr the responses were analyzed using statistical analysis of variance (anova) of the s/n ratio using the taguchi‟s methodology presented in table 4. for anova the insignificant process parameters are pooled and the most significant process parameters are identified using p-value. in table 4 the current is the most significant factor affecting mrr (% contribution: 77.39) and sr (% contribution: 46.17). the material removal capacity rises steeply with rise in the current and maximum at 4 a. thus, the precise productivity of the selected biomaterials can be enhanced at a high current value (i.e. 4 a) which simultaneously provides the desired surface morphology for requisite bioactivity. the optimum values identified for the desired output are selected from the main effect plot as shown in fig. 2. furthermore, one set of experiment is conducted on these optimum value results in mrr as 25.64 mg/min and sr measured as 16.13 µm. table 4 analysis of variance for s/n ratio of responses factors dof sum of squares variance p-value % contribution mrr sr mrr sr mrr sr mrr sr current 2 507.94 277.4 253.97 138.7 0.048* 0.028* 77.39 46.17 pulse-on 2 55.81 # 27.91 # 0.315 # 4.83 # pulse-off 2 33.72 171.3 16.86 85.66 0.432 0.044* 1.29 27.99 voltage 2 # 127.2 # 63.22 # 0.058 # 20.44 error 25.67 7.87 12.83 3.93 total 623.15 583.5 # pooled factor; * significant factor at 95% 342 a.s. bhui, g. singh, s.s. sidhu, p.s. bains fig. 2 main effects plot for s/n ratio (a) material removal rate (b) surface roughness the edmed surface obtained at optimal values of process parameters was subjected to surface analysis in terms of surface topography, chemical composition to examine the machined surface and in-vitro bioactivity analysis. the morphology of edmed sample was investigated using the sem (jsm-6610 lv joel, japan). the sem (fig. 3) reveals the formation of craters (pores) and material deposition. further, it is affirmed that the formation of porous structure represents favorable surface conditions for the cell growth. fig. 3 sem of edmed surface representing the porous structure at i=4a; p-off= 120µs; v=30v it was depicted in the previous study that the edm spark energy altered the surface morphology and chemical composition thus supporting cell viability. the x-ray diffractometer (xrd) pattern of the edmed surface is shown in fig. 4. the xrd analysis results confirm the formation and deposition of various inter-metallic compounds and carbides as well as silicides on the edmed surface. the formation of silicides and carbide demonstrated high corrosion resistance and contribute to enhanced bioactivity [19]. (a) (b) experimental investigation of optimal ed machining parameters for ti-6al-4v biomaterial 343 fig. 4 xrd spectra of edmed surface showing formation of new compounds 3.2. in-vitro bioactivity analysis the capacity of apatite formation in simulated body solution (clean ringer‟s solution with ph value 7.2) has been widely used to assess bioactivity of biomaterials. the bioactivity of the edmed surface of ti-6al-4v alloy was evaluated by the apatite growth level. the immersion of specimen in the ringer‟s solution in particular environmental conditions allows deposition of calcium phosphates on the surface. fig. 5 shows the growth of apatite layer on the edmtreated substrate in a hermetic beaker and kept under thermostatic conditions in a water bath at 37 ± 1°c for 7 days. the sem micrograph showed the formation of hydroxyapatite (hap: a constituent of natural bone) content on the edm-treated surface, which indicates the apatiteinducing ability, as can be seen in fig. 5. the associated eds spectrum confirms the presence of ca and o elements on the edm-treated ti-6al-4v alloy surface, which conferred the formation of apatite layer, as can be seen in fig. 6. fig. 5 sem showing formation of apatite growth 344 a.s. bhui, g. singh, s.s. sidhu, p.s. bains fig. 6 eds spectrum showing elements on edm-treated ti-6al-4v alloy 4. conclusions the surface of the ti-6al-4v alloy specimen was processed by the edm at negative polarity to explore machining performance and biological responses. the following observations are made:  the material erosion rate is directly proportional to the current, whereas bioactive morphology is obtained at optimal combination of the process parameters such as i=4a; p-off= 120µs; v=30v.  the material removal capacity increased significantly with rise in the current and was witnessed as maximum at 4 a. the same output parameter, on the contrary, declined with decrease in pulse-off duration.  the edmed samples at the optimal spark energy (i.e. i=4a; p-off= 120µs; v=30v) resulted in the development of pores and evolution of carbides and silicides on the machined substrate.  peak current (i), pulse-off duration and voltage have been the significant deciding factors while analyzing the sr as response.  the apatite-inducing capability of the edmed surface has been enhanced in the machined surface, thus bestowing excellent bioactivity. references 1. bains, p.s., sidhu, s.s., payal, h.s., 2016, study of magnetic field assisted ed machining of metal matrix composites, materials and manufacturing processes, 31(14), pp.1889-1894. 2. mahajan, a., sidhu, s.s., 2018, enhancing biocompatibility of co-cr alloy implants via electrical discharge process, materials technology, 33(8), pp. 531-534. 3. bains, p.s., sidhu, s.s., payal, h.s., kaur, s., 2018, magnetic field influence on surface modifications in powder mixed edm, silicon, doi:10.1007/s12633-018-9907-z. 4. singh, p., pungotra, h., kalsi, n.s., 2017, on the characteristic of 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department of transport infrastructure and water resources engineering, hungary 2budapest transport privately held corporation (bkv), hungary 3széchenyi istván university, department of vehicle manufacturing and technology, hungary abstract. the current study aims to define a welding technology that allows for the repair of normal and heat-treated rails in paved tracks with partial dismantling and partial preheating in such a way that the resulting layer is not susceptible to cracking and can be done with minimal dismantling, even during night shifts or while traffic is present. high-elongation austenitic consumables (capable of approximately 2-3 times greater elongation than rails; rails have approximately 8-14% elongation) were used and tested for this purpose. welds and rail tread defects were repaired on hungary's busiest tram line (12-14 million gross tons per year), which also has a high axle load compared to other european tram lines. the repairs were performed on various rail grades using different layer numbers, and the experimental consumables were compared to conventional hardfacing welding methods. welds were continuously monitored after welding, and surface hardness measurements were taken. a hardening function and an applicable technology were defined based on the results. the function compares the through-rolled axle tons to the expected hardness values over the first 6 million tons. partial preheating and partial track disassembly can be used to weld a layer of hardness equal to the hardness of the rails and wheels. key words: rail, build-up welding, austenitic, hardfacing, hardness, running surface, paved track, tramway received: august 28, 2023 / accepted october 03, 2023 corresponding author: szabolcs fischer department of transport infrastructure and water resources engineering, faculty of architecture, civil engineering and transport sciences, széchenyi istván university, h-9026 győr, egyetem tér 1., hungary e-mail: fischersz@sze.hu 2 a. brautigam, s. szalai, s. fischer 1. introduction long-distance train travel became a popular mode of land transportation in the twentieth and twenty-first centuries [1-3]. it is convenient, safe [4], quick, and usually on time; however, this depends on the country and train operator. long-distance train travel is less competitive than air travel beyond 500-1000 km in terms of journey time [5]. travel costs, particularly ticket and season ticket rates, put it at a competitive disadvantage compared to low-cost airlines. railway development in the eighteenth and nineteenth centuries, on the other hand, enabled great cities to build railway and passenger terminals in the inner core, even in old city centers. airports are frequently located outside of cities. it could be anywhere between 10 and 50 kilometers long. tourists and tourism suffer because these long distances necessitate more public or private transportation. risk assessments are required for potential projects. given the impact of risk events on infrastructure project costs, timeliness, and quality, investment in risk management is required to avoid or mitigate negative consequences [6]. rising energy and fuel prices (diesel, kerosene, etc.) will pose significant challenges to countries and public and private (large) public transportation businesses in 2021-2023. in fact, there are additional reasons for this. among the causes and explanations are the global political crisis, the covid-19 epidemic, and the cyclical stock market boom and crash. because no single factor can be overlooked, the most likely answer is a combination of the above. transportation, particularly rail transit and railway lines [7,8], is critical to the national economy. it is worth noting that electric road vehicles have grown in popularity in recent years, primarily for private transportation [9,10] (another "fancy" direction is the group of autonomous vehicles [11]). electric buses can provide urban-suburban bus services in the short term in the case of public transportation [12]. (when considering electric vehicles, battery analysis is also an important research area [13,14].) benip (built environment information platform) is a remarkable initiative in infrastructure planning, development, and research [15]. benip depicts the theoretical and practical aspects of the interconnected and chained world of engineering in the built environment, ranging from the smaller entities of architecture (buildings) to the civil engineering domain (complex structures) to the vast transportation systems. the current paper is about the use of austenitic filler metals in paved railway tracks to repair running surface defects. the authors then compiled relevant findings from the international literature on this specific research field. consumables and base metals with austenitic microstructures are widely used in a variety of industries, from mining to automotive and defense, due to advantageous properties such as high elongation (> 35%), high resistance to cracking compared to rails, and surface hardening ability. austenitic plates have long been known in the mining industry for their advantageous properties and are preferred for their hardening properties, as they can harden from an initial hardness of 200 hv (hv unit means vickers hardness that equals 199 brinell hardness [hb]) up to 580 hv (513 hb) [16]. the military industry also uses consumables for armor plates (miilux protection 500 material). gürol's research [17] focuses on fillet welding methods on high-strength miilux protection 500 steel, used as an armor material in the defense industry. elements mapping methodologies, microstructural analysis, and microhardness testing were used to describe the welded investigation of the application of austenitic filler metals in paved tracks for the repair of the… 3 constructions. the study discovered that using ferritic and austenitic filler metals with gmaw (gas metal-arc welding) at low heat input successfully coupled the weld metal and haz (heat-affected zone) [17]. several foreign studies have already looked into the hardness values of rails and rail welds, as well as the interactions between wheels and rails. molyneux-berry et al. [18] discuss the impact of wheel/rail contact conditions on the microstructure and hardness of railway wheels. the observed hardness and microstructure variations, as well as the stress history at the wheel/rail contact (derived from dynamic simulations), are discussed. the wheel/rail interactions generate stresses that exceed the material yield stress between 1 and 7 mm depth, resulting in work hardening without a macroscopic change in microstructure. the underlying hardness trend is linked to wheel manufacturing procedures as well as microstructural variations like proeutectoid ferrite content and pearlite lamellae spacing. liu et al. [19] provide another example of wheel-rail interaction. an experiment on the wear and damage behaviors of machined wheel/rail materials under dry rolling-sliding conditions is carried out using twin-disc wear testing equipment. rail discs have slightly higher surface hardness after machining than wheel discs, but turned wheel discs have significantly higher surface roughness and plastic deformation layer. wheel discs wear out significantly faster than rail discs. the surface damage morphology of wheel discs differs significantly from that of rail discs, with adhesive wear, spalling, peeling, and fatigue cracks dominating. wheel discs wear out faster than rail discs, but rail material is more wear-resistant. the surface hardness, surface roughness, and plastic deformation layer of wheel/rail discs all have an impact on wear behavior after machining, with friction pairings with the appropriate original surface hardness and roughness providing the most minor wear loss. when the tested #w2 specimens (samples) make rolling-sliding contact with the rail discs, abrasion resistance improves overall. bozkurt [20] investigated the tribological characteristics of the r260 rail quality for the rail head, web, and foot. he discovered that the third material (water, humidity, lubricant, debris, or a combination of these elements) between them has a significant influence on the wheel-rail contact, resulting in a rail coefficient of friction between 0.35 and 0.39. it was also demonstrated that the hardness values decrease continuously as a function of distance from the rail head to the rail foot, with an average hardness of 353 hb on the rail head, 316 hb on the rail web, and 278 hb on the rail foot after the tests. apart from the pearlitic microstructure typical of r260 rails, microstructural investigations revealed that the rail foot contains the most ferrite. protsenko et al. [21] proposed determining the optimum rail grinding parameters after investigating the tribological properties. rail grinding is a preventative maintenance procedure that removes problematic material layers from the rail surface while preserving the size, shape, and surface quality. this work aims to determine the effect of grinding on the rail surface's tribological properties and establish the optimal grinding process parameters to ensure the best rail surface wear resistance. the best process parameters for increasing rail surface wear resistance are 0.007 mm depth of cut, 30 m/s grinding wheel speed, and 20 °c rail temperature (it is best to treat rails during the warm season). additional tribological research revealed that sliding wear of rail and wheel steels is simulated using a pin-on-disc test model system that simulates contact wear between the wheel flange and the rail gauge corner. normal load variation was used in both dry and 4 a. brautigam, s. szalai, s. fischer lubricated trials. the researchers discovered that while pair hardness had no effect on the mass loss of the tribosystem studied under dry conditions, mass loss was significantly lower in lubricated testing than in other tests. this emphasizes the technological significance of implementing wheel flange lubrication, improving rail lubrication operations, and revising wheel standards and/or specifications [22]. cracks appeared in the heat-treated rails during pre-bending before installation during the construction of the zagreb tramway line, which were later tested [23]. despite significant variations in the rail cross-section, the hardness values were higher than the minimum values required (on average, 24% higher). it was also discovered that the heattreated rails' hardness or tensile strength values did not cause the defects. poznyakov et al. [24] investigated crack initiation prevention. the study focuses on the properties of rail steel welded joints in electric arc welding, specifically the effect of the thermo-deformational cycle of welding (tdcw) on structural changes, strength, and ductility qualities of haz metal in rail steel welded joints containing 0.72% carbon. this hardened metal is strong but ductile, making it challenging to prevent cold cracks from forming in joints without preheating to 250 °c. konstantinova et al. [25] investigated the possibility of hardening the side surface of the rail head with plasma surface quenching to avoid frequent rail replacement due to wear during heavy traffic. the surface 2.5-3 mm thick layer was successfully hardened to 450830 hv (419-600 hb) in the experiments, but martensite was already present in the microstructure. the results are expected to increase in-service life by 2-3 times. woodhead [26] concentrated on determining how different material properties affect rolling contact fatigue (rcf) and rail steel wear. hardness is a good predictor of tensile strength, and steel samples from the head and foot of rails have significantly different yield strengths (24%). the examinations showed that the ratio of the product of young's modulus squared and percentage elongation to hardness cubed (e2×pe/h3) had a much better correlation (r2=0.98) to wear data than just hardness (r2=0.89), where e is the young's modulus [pa], pe is percentage elongation [%], and h is the hardness [hb]. the above study confirms that there has been a lot of research on rail hardnesses and their interactions with railway wheels, but no studies have shown the application of austenitic filler materials on the rails' running surfaces. with ever-increasing urban traffic, track maintenance is becoming increasingly difficult. developing technology for repairing running surface defects has become necessary for paved tracks, which can be accomplished through partial preheating and partial track dismantling (removing the embedding pouring to the bottom of the rail head). the complete dismantling, welding, and "re-pouring" of the embedding pouring in a paved track, as well as the embedding pouring's binding time, can take anywhere from 24 hours to a whole weekend. in contrast to the requirements outlined above, welding can be performed even during nighttime shutdowns (and, in some cases, during traffic) because the short period of partial dismantling does not cause structural and geometric stability issues in the track. as rail quality improves, the carbon content of the base material rises (and vice versa), necessitating the highest possible preheating temperature (350-400 °c) to avoid rapid cooling and cracking. given that the embedding pourings in paved tracks already melt at 100 120 °c, a consumable that does not require such a high preheating temperature and does not cause cracking in the weld and base material was required. traditional hardfacing materials used for rail build-up welding do not meet these investigation of the application of austenitic filler metals in paved tracks for the repair of the… 5 requirements but are met by austenitic 18-8-6 materials (18% cr, 8% ni, 6% mn, or higher alloying content). the authors set out to thoroughly investigate the technology described above in a real railway track environment with traffic loads. for this purpose, repair welds were prepared on a tramway track in budapest (hungary), where the annual traffic load on the line is relatively high. the welds were carried out in accordance with the most stringent technological standards. welding repairs for the austenitic build-up and control hardfacing were also carried out. approximately 5.5 million gross tons were rolled through the sections during the five-month test period. hardness measurements were taken 85-120 cm from the welds' center axis at a resolution of 5 cm. the degradation-change functions were computed as a function of the axle tons. the findings may be helpful in railway track maintenance, where a new type of austenitic build-up repair welding may be considered an alternative to traditional repair technologies. the paper is structured in the following way. section 2 discusses the problem statement and additional literature review that are closely or tangentially related to this topic, as well as their relevant findings and results for the current article. section 3 summarizes the materials and methods used in the study. sections 4 and 5 deal with the findings and discussion, respectively. section 6 discusses the main conclusions drawn from the experiments and investigation. 2. problem statement and further literature review the authors have chosen to supplement the brief literature review presented in section 1 with a more detailed analysis in this section to obtain a more complete picture of the relevant research results in the research area. austenitic filler materials have been used for decades in three areas of the railway industry, one of which is the repair of hadfield steels by build-up welding (see table 1 and fig. 1). the second major group is the welding of r220 grooved rails in the factory and on the track to reduce side wear and noise emission [27]. the welding factory mills the most worn parts of the rail head (the area bounded by the running surface and the running edge, the area where the wheel contacts the rail head, and the area where the check rail contacts the wheel sidewall) to prepare the nest into which the austenitic filler material is welded (see table 2 and fig. 2). this layer not only helps to reduce noise, but it is also more resistant to abrasion wear than the base rail grade when hardened by vehicle traffic. figure 3 depicts the layer applied at the rail in the track. furthermore, these filling materials in group 3 act as a buffer layer beneath the hardfacing layer. build-up weldings are used to prepare them for railway crossings and other rail defects. table 1 chemical composition of the austenitic manganese steel electrode chemical composition [%] c si mn p s cr ni v nb min. 0.60 0.50 11.10 max. 1.20 1.10 14.90 0.05 0.01 0.10 0.10 0.10 0.10 6 a. brautigam, s. szalai, s. fischer fig. 1 repair of a turnout frog from hadfield steel. a – defect, b – preparation, c – welded structure table 2 chemical composition of austenitic solid wire chemical composition [%] c si mn cr ni 0.08 0.9 7.0 19.2 9.0 fig. 2 build-up welding on grooved rails to reduce wear the european standards for vignol [28] and grooved [29] rails specify a minimum hardness value for the rail head to indicate steel grade (for example, r260 has a minimum hardness of 260 hb) but allow for a range within the rail head hardness to be met before installation. fig. 3 austenitic layer in the rail surface investigation of the application of austenitic filler metals in paved tracks for the repair of the… 7 table 3 steel grades and their surface hardness values according to en 13674 (vignol rails) [28] steel grade the hardness of the rail running surface center line [hb] r200 200 to 240 r220 220 to 260 r260 260 to 300 r260 mn 260 to 300 r320 cr 320 to 360 r350 ht 350 to 390 r350 lht 350 to 390 r370 crht 370 to 410 r400 ht 400 to 440 table 4 steel grades and their surface hardness values according to en 14811 (grooved rails) [29] steel grade the hardness of the rail running surface center line [hb] r200 200 to 240 r220 g1 220 to 260 r260 260 to 300 r290 ght 290 to 330 r340 ght 340 to 390 international articles closely or tangentially related to this topic, as well as their relevant findings and results for the current article, are detailed in the following paragraphs. kou [30] thoroughly reviewed the literature on detecting and evaluating rail surface defects. surface rail defects can cause wheel wear, increased derailment risk, and safety concerns. early detection is critical for railway safety and service life extension. several approaches to defect inspection have recently been investigated, including ultrasonic and acceleration detection, computer vision, and deep learning. machine learning visual inspection techniques are commonly used to detect railway surface problems. using multiple methodologies to detect discrete defects improves detection. for difficult tasks, more research is needed to detect rail surface crackings with an accuracy of 75% to 85%. long-term rail surface deterioration research on the swedish iron-ore line was conducted, considering maintenance interventions [31]. the fr8rail running gear was 8 a. brautigam, s. szalai, s. fischer found to be superior overall, with cross-bracing and a softer primary suspension being especially noticeable for curve radii greater than 600 m. the possibility of extending the grinding interval and reducing track damage is highlighted. another study investigates using consumable guide enclosed arc welding (cgeaw) technology to arc weld rail profiles [32]. cgeaw uses a specially designed electrode and self-shielded flux-cored wire, making it portable, easy to manufacture, and capable of high filler material deposition rates. the required gap between welded rails is 12-16 mm, which can be used to calculate process efficiency. the research [33] looks into the structural phase transition of rail steel in compression, examining its structure and inadequate substructure to 50% uniaxial compression. cold hardening is discovered to be a multi-stage process, with pearlite grain fragmentation increasing with deformation. carbon atoms are transported out of the cementite crystal lattice and form tertiary cementite particles with sizes ranging from 2 to 4 nm. due to dislocation delay by cementite particles, a non-uniform dislocation substructure emerges during steel deformation. in 2019, the authors of [34] investigated the genesis and progression of contact-fatigue cracks in rail wear. after designing and manufacturing an experimental setup, a two-hour fault detection test program was carried out using fluorescent magnetic particle inspection. contact-fatigue cracks form in the rail head after approximately 200 thousand load cycles, consistent with other studies showing similar damage after 190-290 thousand cycles. the primary cause of rail head contact-fatigue faults is insufficient rail steel contact-fatigue strength. the research [35] also investigates the mechanical properties and fatigue characteristics of two types of wheels and rails, emphasizing how they affect wear and contact fatigue. the ks60-rail and uic60-rail had higher yield strength and hardness with lower elongation than the rsw-wheel and r7-wheel. the crack toughness of the ks60-rail was 73 mpa m–2, 1.8 times greater than the uic60-rail's 41 mpa m–2 crack toughness. the uic60-rail and r7-wheel outperformed the ks60-rail and rsw-wheel regarding wear resistance. the study [36] investigates the dynamics of welded rail gap and rail base hardness using a semi-infinite rod and elastic-based damping. it finds both static and dynamic solutions, indicating that the dynamic factor does not exceed 1.5 during rail breakdowns. on an elastic basis, the paper also generates equations for the thermal deformation of a jointless track with a stiffness-varying insert. although a rigid insert of up to 20% of the rail's length has no effect on the track's stress-strain state (sss), it may increase clearance for unintended rail breaks, reducing traffic safety. according to the study, a 10-12 cm gap is possible at random kink rail. gertsyk and volgina's [37] article investigates the causes and types of rail track degradation, as well as rail manufacturing processes such as heat treatment, end treatment, surface hardening, and anti-flake treatment. several procedures, including volumetric quenching with oil cooling and subsequent tempering, thermal rail hardening with tp. heating (high-frequency current), complicated deoxidizers, high carbon content steels, and continuously cast billets have been proposed to improve rail track endurance. these techniques are intended to reduce flock sensitivity while increasing rail track hardness. fischer et al. [38] investigated the heat-affected zone of various thermit rail weldings. they considered the rail grades r260 and r400ht. their first sample was r260-r400ht with a normal non-heat-treated welding portion for r260 rails, and their second sample investigation of the application of austenitic filler metals in paved tracks for the repair of the… 9 was r260-r400ht with a heat-treated welding portion for r350ht rails. the critical technological directions were strictly adhered to. based on the results of the laboratory hardness, macro-, and microstructural tests, sample #2 performed better in running the hardness profiles. it indicates that if temporary intermediate hardness rails cannot be incorporated, welding in the two-step rail grade leap with the welding section corresponding to the intermediate rail is preferable. to summarize, more laboratory testing will be needed in the future, where it would be worthwhile to investigate several cases between r200 and r400ht (considering special, for example, bainitic rails, etc.), not only for vignole rails but also for grooved rails and block rails. jovanovc et al. [39] investigated the use of an excavator on railway tracks. they examined this mechanical structure using materials science, metallurgy, and mechanical engineering, which frequently overlap in this paper. steels, which are iron alloys containing carbon and other elements, are indispensable in these disciplines due to their versatility and wide range of properties. in materials science, they are studied microstructural to discover the relationships between their composition, processing methods, and properties. metallurgy is the refinement of the extraction and alloying processes to produce steels with specific properties suitable for various applications. mechanical engineers frequently use these well-characterized steels in their designs and projects where strength, durability, and cost-effectiveness are critical factors. welding is a common technique for joining steel parts in these fields. welding is a material-coalescing process required for assembling and repairing many steel structures and components, making it an essential aspect of these fields of study and practice. tica et al. [40] used the sophisticated design software dvobrz to discuss reversible planetary gearsets controlled by two brakes for internal combustion railway vehicle transmission applications. this software can be used to design welding machine components and/or welding processes. a carefully planned design process must be fulfilled before determining the required material (steel and metal) qualities. they are critical in the adequate and appropriate mechanical engineering preparatory process. considering a gearset's wear and resistance to wear, as well as the associated design steps, necessitates similar engineering thinking as analyzing the relationship between a railway wheel and a rail. it can only be considered tangentially for this article up to the point where the base steel material is considered. welding is very rarely used to repair gearset components. however, it is possible to apply a wear-resistant coating to extremely resistant to wear components and elements, and it is also possible that they and their microstructural changes associated with hardness increase may occur during use. monek and fischer [41] investigated the use of digital twins in smart manufacturing. this paper presented a digital twin model of a research and demonstration platform in a des (discrete event simulation) environment that improved the connection between the physical and digital layers through data acquired by a smart workpiece with an embedded imu (inertial measurement unit) sensor. without user intervention, the system may detect anomalies such as a jammed workpiece or a reduced conveyor speed. the authors created an easy-to-use framework with a user interface for visualization, process monitoring, and troubleshooting. the goal is to create a modular framework that can be used by both small and medium-sized businesses and large corporations. merging smart product and plc data with machine learning support enables high-precision workpiece state identification and motion prediction, with 98.7% accuracy on training data and 98.45% on test datasets. the extended digital twin framework for typical des simulations can improve synchronization 10 a. brautigam, s. szalai, s. fischer between the digital twin and the physical system, detect anomalies, and reduce input data collection time. the importance of the cited article stems from the fact that the repair welding presented in this article can be automated and created in a computer environment, serving as its digital twin. however, because the use of imu in welding can be problematic (high temperatures, vulnerable or fragile parts, placement to avoid getting in the way of any technological process, etc.), further development of the solution is definitely warranted. ovchinnikov et al. [42] investigated the possibility of extending rail service life in case of a rail head defect. they discussed how rails are subjected to wear, corrosion, and contact fatigue, all of which cause flaws and damage. the size, location, and direction of flaws determine the residual life of rails. this article uses the finite element method to estimate the maximum permitted crack-size values and investigates crack plane orientation. it is discovered that the stress intensity factor is location-dependent, allowing safe operation on low-activity railroads. the fem simulation model considers internal strains caused by different crack sizes and orientations. pultznerová et al. [43] investigated the effect of railway superstructure conditions on traffic noise on the regional line. in their research, they only looked at slovak railway lines. there is now an effort to upgrade not only major railway lines but also minor ones. this is also true for the poltár-uteká line segment, where the sleepers are in poor technical condition and will be replaced with a y-type sleeper soon. the new construction's impact on the sleeper's noise output should also be assessed. it can be stated that the running surface of the rails, rather than the condition of the sleepers, has the most significant impact on noise emission. taking these factors into account may necessitate additional research. the impact of traction electricity (te) consumption factors on driving speeds and energy consumption in rail transportation is investigated by kampczyk et al. [44]. it employs expert techniques and a novel te consumption calculation to analyze the movement resistances of electric locomotives during braking and starting. the findings show a significant relationship with the speeds of existing catenary-supplied electric vehicles. this innovative technique advances engineering and technical disciplines, particularly civil engineering, surveying/geodesy, and transportation. better quality rail repair weldings, of course, improve the geometric condition of the railway track and slow geometric degradation at these critical locations. the task of infrastructure managers is to ensure that the infrastructure they manage and operate is in the best possible condition at the lowest possible cost. the lower the additional traction energy, the better the condition of the railway track. of course, other influencing factors and aspects should not be overlooked [45]. 3. materials and methods the chosen site is not only the busiest tramway line in hungary (12-14 mgt/year, where mgt stands for million gross tons) but also one of the busiest in europe. the site has r290 ght rail quality installed; the welds and their surroundings bulged at several points on the bridge due to years and traffic loads, which were repaired by build-up welding on the running surface. prior rail grinding was also performed on the entire section in question, resulting in a running surface hardness of 330-350 hb. the parameters of the straightness measuring beam are as follows: investigation of the application of austenitic filler metals in paved tracks for the repair of the… 11  type: secnet dual;  measurement base length: 1 m;  simultaneous measurement of the running surface and the running edge;  measuring range: –2.5…+1.5 mm;  accuracy: 0.05 mm in the measuring range. the welds and their surroundings were measured (fig. 4) and evaluated (fig. 5) using a rail straightness measuring device, and the required weld lengths, typically 0.8-1.2 m, were determined. figure 6 depicts one such failure location. the embedding pouring was removed in the first step of the technology over the length to be welded + 0.1 m in both directions, on both sides of the rail, deep down to the bottom of the rail head (due to grinding and heat input, it is recommended to increase the removal length compared to the welding length fig. 7, part a). part 'b' depicts the surface prepared for welding; the preparation is done with the running surface cut out at 45°. part 'c' depicts the condition following partial preheating (120 °c). part 'd' depicts the welded section with the weld layers applied. the following was the average number of layers deposited on the deposited length: 1-2 layers deposited at the weld's two edges, 2-3 layers deposited in and around the weld, and one layer removed on average during the finishing process (grinding). table 5 shows the chemical composition of the filler metal used. figure 8 depicts the finished welding after grinding. fig. 4 rail straight measurement device fig. 5 rail straight measurement protocol on the running surface 12 a. brautigam, s. szalai, s. fischer fig. 6 wear on the running surface at a welding in paved track table 5 chemical composition of austenitic steel electrode for repairing the running surface chemical composition of weld metal [%] c si mn cr mo ni nb fe 0.1 0.5 6.0 19.0 0.2 8.5 < 0.1 64.5 fig. 7 technology steps. a – removed embedding pouring next to the rail head, b – preparing the surface, c – preheating, d – welding investigation of the application of austenitic filler metals in paved tracks for the repair of the… 13 fig. 8 the completed welding after grinding visual inspection, geometric measurements, and surface hardness testing were used to monitor the sites after welding continuously. visual inspection was performed to ensure that the weld had a solid, homogeneous structure, that no grinding defects had occurred during the finishing process, and that no cracks, material defects, or slag inclusions were visible. for hardness measurement, a sauter hmo dynamic portable hardness tester (fig. 9) was used, the main technical parameters of which are listed below [46]:  complies with relevant din, en, and astm standards, can be subject to accredited calibration;  results can be displayed as hrc, hrb, hv, hb, hs and tensile strength values;  accurate measurement results: ±6 hld;  hardness scales: hl, hrb, hrc, hv, hb, hs;  tensile strength u.t.s.: σb range from 375 to 2639 (steel only),  accuracy: ±1% at 800 hl, ±6 hld;  impact test: d (standard);  optional sensors: dc, g;  max. hardness value: 976 hv;  min. specimen weight: 2-5 kg lifted, 0.05-2 kg attached;  min. specimen thickness when attached: 8 mm;  min. specimen bark thickness: 0.8 mm;  dimensions: 135 mm × 83 mm × 24 mm. 14 a. brautigam, s. szalai, s. fischer fig. 9 hardness measurement device the hardness tester reads hardness values (hld) from the recoil rate using a leeb scale, which can then be converted to the desired hardness value, e.g., hb, hv, and so on, and it can be used in most measuring positions (directions: ↓↙←↖↑). the following observations were made following the hardness measurement tests: the pavement layers were crack-free, hardened quickly under traffic loads, and hardened more than the hardness of the connecting rail and wheels (320-350 hb) (fig. 10). transverse hardness measurements revealed that hardening was higher in areas more exposed to wheels and that transverse values were more dispersed than measurements made on rail base materials. the measurements also revealed that the number of layers influences the hardening degree. hardness values were 15-20% lower in sections where 2-3 layers were welded (weld and its immediate surroundings) than in sections where 1-2 layers were welded (sections connected to the rail). according to the measurements, the welded section reached 90-95% of the final hardness values in the first months. this equates to approximately 2-3 million throughrolled axle tons in this case. the technology and filler material were also tested to repair previously repaired cracks during the experimental welds (fig. 11). if the crack is in the rail head and does not spread to the rail web, it is possible to repair it. in this case, the authors discovered that it is suitable for crack repair at previously repaired locations (even for cracks that begin at these locations). figure 11 depicts one such example. investigation of the application of austenitic filler metals in paved tracks for the repair of the… 15 fig. 10 hardness values on the running surface at location #8 (top – rail, down – welding) fig. 11 welding of a rail crack (left – crack, middle – preparing, right – completed welding) the results were evaluated after the welds were completed and monitored; the main benefits and drawbacks are shown in table 6. 100 150 200 250 300 350 400 450 500 550 600 07.2022 08.2022 10.2022 12.2022 01.2023 03.2023 h a rd n es s [h b ] date of measurement [month.year] min average max 100 150 200 250 300 350 400 450 07.2022 08.2022 10.2022 12.2022 01.2023 03.2023 h a rd n es s [h b ] date of measurement [month.year] min average max 16 a. brautigam, s. szalai, s. fischer table 6 advantages and disadvantages of the results advantages disadvantages can be carried out during a night shift or even during traffic more expensive consumables than hardfacing electrodes (but significantly cheaper overall) lower demolition and restoration costs excessive hardening (uneven wear, impact on the vehicle wheel) less disruption to traffic than in the case of complete demolition limited ndt inspection (magnetic principle instruments) not susceptible to cracking more significant deviation in hardness values in cross-sectional direction than hardfacing technology reduced preheating time and costs cannot be cut by flame the first three advantages demonstrate that the chosen method is unquestionably advantageous from a railway operator and economic standpoint. however, because the hardness values were significantly higher than those of the rails and vehicle wheels, additional research was conducted to determine the number of layers that would result in a hardness close to that of the rails and vehicle wheels while remaining technically and economically feasible. according to the findings, hardening occurs according to a power or logarithmic function. the hardness values for single-layer welds are clearly higher due to the carbon added from the base material, whereas the hardness values for two-layer welds are lower. as a result, the goal was to create a technology that met the following criteria: 1. technology that can be used with partial preheating (~80-120 °c); 2. without or with minimum demolition of the paved track; 3. use of crack insensitive filler (table 5); 4. hardness is still equal to or greater than rails and welds. following that, a testing site was chosen that met the following criteria: sufficient rail wear (on the running surface) to determine different layer numbers and excavation depths and a site with paved track to test an even more limited preheating technology. in this case, only the deepest section (about 70% of the total length) was preheated to allow the less worn section to heat up due to the heat input during welding. furthermore, a section adjacent to this site was chosen where a reference section was repaired using conventional hardfacing technology in addition to stripping. other important considerations included the sections' high traffic volume and the fact that the sites had previously been repaired using hardfacing technology, which is no longer feasible, and track replacement is not feasible. this will demonstrate that the chosen solution can extend the track's technical lifetime. the location was once again on the same line as the previous measurements (fig. 12). the austenitic drift technology received a 1.2 m long section, while the hard-applied technology received a 0.8 m long section. figure 13 illustrates the limited preheating in this section. investigation of the application of austenitic filler metals in paved tracks for the repair of the… 17 fig. 12 the selected location for austenitic build-up welding fig. 13 partial preheating on the selected location because of the weld defects, the authors were able to inspect 1-6 weld layers together on site. the welds were completed on march 14, 2023, with the following technological parameters:  base material: 2.2 according to iso tr 15608 [47];  steel grade: r260 ght according to en 14811 [29]; 18 a. brautigam, s. szalai, s. fischer  welding process: 111 – manual metal arc welding according to iso 4063 [48];  welding position: pa according to iso 6947:2019 [49];  build-up welding;  filler metal: e fe 10 according to en 14700 [50] or 1.4370 material no.;  electric current type: dc (+) / ac;  diameter: 6 mm, length 450 mm;  amperage: 220-250 a;  preheating: 120 °c;  examination: vt (visual testing), geometry measurement. 4. results the hardness values of the completed welds were measured every 5 cm on the running surface. to obtain more accurate results, several measurements were taken weekly at first; then, when the hardening rate began to slow, the measurements were changed to weekly and bi-weekly, and when the hardness values became relatively stagnant, a control measurement was taken after six weeks. this happened again after more than two months. the authors intend to examine this section further in the future. in addition to hardness measurements, the following tests were performed: visual inspection of the weld, straightness measurement with a steel ruler, hardness measurement on the tread transversely every 0.05 m, and a penetration test if necessary. it should be noted that, despite the weld being hardened during forming, there was no change in the initial geometry during hardening. table 7 shows the measurement days and the corresponding through-rolled axle load data in tonnes. table 7 measurement dates and axle loads in tons date [day.month.year] through-rolled axle tons [tons] 14.03.2023 9,450 16.03.2023 70,560 20.03.2023 196,740 24.03.2023 349,020 31.03.2023 589,680 06.04.2023 807,300 21.04.2023 1,311,660 05.05.2023 1,792,980 19.05.2023 2,274,300 29.06.2023 3,679,920 25.08.2023 5,643,270 investigation of the application of austenitic filler metals in paved tracks for the repair of the… 19 based on fig. 2, the average hardness values of the austenitic and control hardfacing welds are shown in figs. 14 and 15, respectively. a consumable with the chemical composition shown in table 8 was used for the control hardfacing, and the preheating temperature was set to 350 °c. the base materials in both cases were r260 ght grooved rails. fig. 14 austenitic build-up welding hardness (average values) fig. 15 hardfacing build-up welding hardness (average values) table 8 hardfacing stick electrode chemical composition chemical composition in % of weld metal c si mn cr fe 0.15 1.10 1.20 0.80 balanced 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 0 10 20 30 40 50 60 70 80 90 100 110 120 h a rd n es s [h b ] length of the welding seam [cm] 14.03.2023 16.03.2023 20.03.2023 24.03.2023 31.03.2023 06.04.2023 21.04.2023 05.05.2023 19.05.2023 29.06.2023 25.08.2023 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 h a rd n es s [h b ] length of the welding seam [cm] 14.03.2023 16.03.2023 20.03.2023 24.03.2023 31.03.2023 06.04.2023 21.04.2023 05.05.2023 19.05.2023 29.06.2023 25.08.2023 20 a. brautigam, s. szalai, s. fischer 5. discussion the findings clearly show that, in the case of a hardfacing technology, deformationinduced hardening in the railway track is negligible, regardless of the number of layers used. hardening of about 15-18% (45 hb) was observed in the first month under an axle load of 600,000-800,000 tonnes, which remained essentially unchanged thereafter, and under further loads significant for the period under study the hardness of the applied layer hardly increased at all, with a maximum increase of 7-10 hb (1-2% increase). previous operator experience confirms that after 20-30 years in service, r260 grade rails harden to this value. the authors obtain much more interesting results for austenitic build-up welding. before delving into the data, it is worth noting that while there were no outliers between the average and maximum values for conventional hard surfacing, the variance was more remarkable for the austenitic layer, with maximum values on the running surface being 48% (15-20 hb) higher than the average values. the running surface here showed a greater divergence between the parts of the running surface that were constantly in contact with the vehicle wheel and the surfaces that were not in contact with the wheel, whereas this phenomenon was scarce on the hardfacing layers. the diagram also clearly shows that high hardness values could be measured on the two edges of the seam as a result of the carbon picked up from the base material and faster cooling, particularly on the left side of the seam, where the depth of wear and preparing was less than on the right side. hardness values of 520-550 hb are not uncommon in the former case, while hardness values in the latter case increased above 400 hb. the section between 30-95 cm roughly corresponded to the previously expected and predicted 340-380 hb values. this range is very close to the hardened normal r260 rail quality and the hardness of the heat-treated rails, and it is also very close to the hardness of the vehicle wheels, resulting in optimal wheel-rail interaction. it is also worth noting that the number of layers used in this width range varied between 2 and 6, but the hardness values did not decrease significantly as the number of layers increased. as a result, with two or more layers of welds, the number of welds has no significant effect on hardening, and there are no significant differences in hardness values. this condition develops after approximately 3.5 million tonnes of axle load. the graph also shows that the initial period (the first 0.5 million tonnes) shows a more significant hardening, followed by a steadily slowing rate. the stirring of the carbon and the rapid cooling caused by partial preheating cause critical hardening of the joint sections, which may remain more prone to cracking in the case of 1-layer welds. laboratory tests to determine the amount of martensite formed could be part of future research. the deposited layer in the multi-layer sections remained austenitic (non-magnetizable). table 9 summarizes the approximate hardening curves determined for each measurement cross-section as a function of traffic load. investigation of the application of austenitic filler metals in paved tracks for the repair of the… 21 table 9 regression functions for the hardness on each cross-section by austenitic welding (he is the estimated brinell hardness on the running surface [hb], t is the through-rolled axle tons [tons]). nr. of the cross-section equation of the regression function r2 #1 𝐻𝑒 = 77.577 × 𝑡0.1289 0.904 #2 𝐻𝑒 = 88.392 × 𝑡0.1122 0.941 #3 𝐻𝑒 = 103.14 × 𝑡0.1035 0.952 #4 𝐻𝑒 = 105.42 × 𝑡0.0967 0.981 #5 𝐻𝑒 = 121.26 × 𝑡0.0789 0.943 #6 𝐻𝑒 = 170.31 × 𝑡0.0530 0.930 #7 𝐻𝑒 = 131.26 × 𝑡0.0716 0.959 #8 𝐻𝑒 = 139.79 × 𝑡0.0667 0.958 #9 𝐻𝑒 = 156.11 × 𝑡0.0570 0.925 #10 𝐻𝑒 = 149.66 × 𝑡0.0588 0.971 #11 𝐻𝑒 = 171.58 × 𝑡0.0488 0.963 #12 𝐻𝑒 = 158.91 × 𝑡0.0550 0.945 #13 𝐻𝑒 = 150.58 × 𝑡0.0571 0.981 #14 𝐻𝑒 = 147.52 × 𝑡0.0575 0.960 #15 𝐻𝑒 = 167.66 × 𝑡0.0483 0.854 #16 𝐻𝑒 = 128.14 × 𝑡0.0671 0.931 #17 𝐻𝑒 = 107.88 × 𝑡0.0804 0.945 #18 𝐻𝑒 = 129.14 × 𝑡0.0666 0.891 #19 𝐻𝑒 = 153.14 × 𝑡0.0560 0.982 #20 𝐻𝑒 = 139.48 × 𝑡0.0638 0.955 #21 𝐻𝑒 = 140.50 × 𝑡0.0636 0.970 #22 𝐻𝑒 = 118.76 × 𝑡0.0870 0.957 #23 𝐻𝑒 = 123.56 × 𝑡0.0779 0.969 #24 𝐻𝑒 = 150.62 × 𝑡0.0679 0.937 #25 𝐻𝑒 = 147.15 × 𝑡0.0715 0.942 the measured hardness functions for all 25 cross-sections are presented in fig. 16, as are the calculated average function and the corresponding power regression function (see table 9). fig. 16 measured hardness values of all cross-sections (from #1 to #25) on the running surface in the case of austenitic welding, as well as the calculated average function and the corresponding power regression function he = 131.43 t0.0729 r² = 0.9712 200 250 300 350 400 450 500 550 600 0.0e+00 1.0e+06 2.0e+06 3.0e+06 4.0e+06 5.0e+06 6.0e+06 h a rd n es s [h b ] through-rolled axle tons [tons] #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24 #25 average regression function 22 a. brautigam, s. szalai, s. fischer best-fit equations were also examined for 3.5 and 5.5 million tons. the power function, rather than the logarithmic approach, provided the best correlation in both cases. the power functions had an average r2=0.951 after 3.5 million tons of axle load, while the logarithmic equations had r2=0.944, a ratio that became even closer after more than 5.5 million tons of loading: power r2=0.946, and logarithmic function r2=0.944. the authors intend to pursue their research in the long term, and the power function may no longer be the best-fitting function in the future. the following formula (eq. (1)) gives the best approximation for multi-layer (at least 2-3 layers) welding up to nearly 6 million tons of load when the hardness values of each cross-section are considered as a function of the axial load. 𝐻𝑒 = 150 × 𝑡0.0575 (1) it should be noted that the equation for one-layer welding is not shown because this solution should be avoided. it should also be mentioned that the equation in fig. 16 is related to the average of all cross-sections, which includes everything from single-layer welds to multi-layer welds. because the remaining seam layers #2...#4 are technically and economically the best solution, only the seams in this range are considered in the equation, which is most typical for seams #9...#17, and a conclusion is drawn from the equations with the best r2 values (see eq. (1)). 6. conclusions based on the findings, it is possible to conclude that a crack-insensitive welding technology can be used for the permanent repair of most rail qualities. based on the above results, a minimum of 2-3 layers (remaining after grinding) can be determined, for which additional parameters can be defined. the preheating temperature has already been mentioned; the grinding angle should be 45° to the rail's running surface (where the machined part meets the intact material's running surface). grinding can be done with a tool, such as a milling (grinding) machine. the ideal preparation depth is 5-10 mm, and it must be uniform across the entire length to be welded. a more significant flaw can be used, but preparation along the entire length is difficult and potentially time-consuming. when welding paved tracks, the embedding pouring must be removed from both sides of the rail head, at least up to the bottom, over a length of –10 to +10 cm. the electrode diameters used are 4, 5, and 6 mm, with a seam height ranging from 2-3 mm to 4-6 mm. due to slower welding, smaller electrode diameters allow more heat to remain in the system, preventing rapid cooling. it should be noted that the first layer deposited is typically slightly thinner than the thicknesses described above. two rows of welds should always be welded at 45° joints to avoid cracking at the edges of the welds. to avoid end craters, weld the trapezoidal mold's edges according to the numbered orange seams shown in fig. 17 (seams 1 to 4, including double seams 1 and 3), and a 1 to 1 seam is sufficient in the longitudinal direction. after welding the prepared part's edges, the rows of red seams can be laid down in layers (5 to n) using one-way welding (for heat conduction) to avoid slag trapping. investigation of the application of austenitic filler metals in paved tracks for the repair of the… 23 fig. 17 welding seams on the running surface finally, the results obtained can provide numerous benefits to 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international organization for standardization, 2019, iso 6947:2019, welding and allied processes. welding positions, 28 p. 50. european committee for standardization, 2014, en 14700:2014, welding and allied processes, 14 p. facta universitatis series: mechanical engineering vol. 16, no 3, 2018, pp. 381 388 https://doi.org/10.22190/fume170623030a © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* device for in vitro wear analysis of biomaterials in the hinged prosthesis configuration udc 620:617.3 josé n. athayde 1 , beatriz l. fernandes 1 , carlos j. de m. siqueira 2 , percy nohama 1 , carlos r. fernandes 1 1 pontifical catholic university of paraná, health technology, curitiba, brazil 2 federal university of paraná, department of mechanical engineering, curitiba, brazil abstract. we have developed a device that, coupled to the tribometer, allows movement simulation of the hinged type knee prosthesis. two tests were performed using the samples designed and one test without the device using a pin-on-flat configuration. for the first and the third tests, the metallic samples were used as machined while for the second one they were electrolytically polished. the test parameters were running length of 0.663 rad and compression load of 22.35 n. the hertzian contact stress of 15.93 mpa obtained between the samples designed is close to that for the real prosthesis. the measured volumetric wear revealed the influence of roughness of the counterpart surface on the wear behavior. the device has allowed its coupling to the tribometer without any interference on its functioning thus making a contribution to the scientific investigations related to wear behavior of a couple of different biomaterials. key words: wear testing, sliding wear, three-body abrasion, joint prosthesis, hinged knee prosthesis 1. introduction a simply hinged prosthesis is one of the configurations used in the knee prosthesis although it cannot provide complex movements of the knee joint. total knee arthroplasty (tka) is a surgical procedure involving the insertion of an artificial joint that replaces the natural one degenerated by osteoarthritis [1], trauma or tumor. received june 23, 2017 / accepted july 11, 2018 corresponding author: beatriz luci fernandes graduate program on health technology , pontifical catholic university of paraná – pucpr, rua imaculada conceição, 1155, 80215-901 curitiba pr, brazil e-mail: beatriz.fernandes@pucpr.br 382 j.n. athayde, b.l. fernandes, c.j. de m. siqueira, p. nohama, c.r. fernandes the total knee prostheses are classified depending on the extension of the tibia and femur resection. the conventional prosthesis is chosen when the resection is limited to the borders of the joint while the unconventional one is chosen when a part of the tibia and the femur has to be removed [2, 3]. a variety of studies reports on the tribological behavior of a couple of biomaterials used in the conventional knee prosthesis. however, very few of them are found to be focused on unconventional knee prostheses (unckp). despite the diversity of models of the last ones and although their use is commonly intended for oncological patients, the new diagnostic techniques and treatments have been raising life expectations of those individuals [4]. therefore, the development of a better design for unconventional prosthesis and its tribological characterization is worth discussion. like conventional prostheses, the unconventional ones are subjected to septic loosening due to infection and aseptic loosening due to excessive wear of the biomaterials in the sliding contact. the aseptic loosening is the main problem that leads to the prosthesis revision, affecting patients from 10 to 29 years after the replacement surgery [2, 5, 6]. in this case, the wear debris generated and released from the biomaterials in the joint prosthesis accumulates in the adjacent tissue, and depending on the size and quantity, it can destroy it [7, 8]. the amount and morphology of the released debris are related mainly to the biomaterials properties and congruency between the surfaces in contact that leads to punctual load conditions [9]. therefore, the contact pressure between the prosthesis parts directly influences wear of the biomaterials and must be evaluated in order to assist to the proper design of the implants [10]. the tests performed in the tribometer can provide initial data about the tribological behavior of the biomaterials couple involved in prosthesis production; however, without the simulation of the real loads, the range of movements and superficial contact area it is not possible to make conclusions about their behavior in vivo. although there are gait simulators for validation of materials and designs of knee prosthesis, the tests are expensive and time-consuming being used mainly on the final products for regulatory reasons. based on the context exposed herein, in the present work we have designed, produced and evaluated a device that, linked to a linear tribometer, can perform a wear test in biomaterial couple simulating the loads and the boundary conditions that occur in vivo in an unconventional total knee prosthesis hinged type. 2. materials and methods the device is designed to fit a linear tribometer csm ® that allows a maximum load of 46 n, the linear sliding amplitude of 20 mm and a maximum speed of 31 cm/s. apart from the reciprocal linear motion module, none of the original mechanisms or electronic sensors of the tribometer was modified or taken away to adapt the device. therefore, the device uses the structure, the rotation mechanism, the sensors system for load and speed and the headstock adjustment of the tribometer. the controlled parameters are speed, frequency, and rotation. the design of the samples manufactured for the tests in the present study is based on the most frequent design of the total unconventional knee prosthesis available on the market. it has one degree of freedom allowing only flexion and extension. since the knee prosthesis device for in vitro wear analysis of biomaterials in the hinged prosthesis configuration 383 causes an oscillatory movement, in order to reproduce it using the tribometer, it was necessary to convert the tribometer rotational mechanism, exposed when the reciprocal linear motion module is detached, into oscillatory one. this was done through a tilting coupling assembled perpendicularly to the tribometer eccentric shaft, which was so conceived as to support a small stainless steel vat where the samples could be tested immersed in lubricant fluid. the tribometer rotational movement was converted into tilting one with a maximum angular amplitude of ± 9,49 o (right and left), limited by the tribometer design. the mechanical support of 12.0 mm 2 section bar was built with aisi 304 stainless steel that suites the requirements of corrosion and mechanical resistances for this application. seeing that the unckp functions as a hinge [11], the samples for the wear test using the device and the tribometer are designed to simulate the hertzian contact that happens in the unckp in vivo. a couple of samples manufactured in the usual biomaterials for a joint prosthesis, ti6al4v alloy and ultra high molecular weight polyethylene (uhmwpe), have the dimensions presented in fig. 1, fixed through the designed parts. in this way, the contact between the surfaces of the titanium alloy and the polyethylene samples is made by a curvature radius of 3.0 mm, corresponding to twice the maximum angular amplitude of ± 9,49 o or 18,98 o . this value is the total course conducted by a contact point between the two samples during the test. fig. 1 samples designed for use in the device: (a) metallic sample in ti6al4v alloy with dimensions in mm and (b) polymeric sample in uhmwpe with dimensions in mm to perform the device assessment, a course of 0.663 rad, resulting in a distance of 1.98 mm, is considered. the configuration of the tribometer csm ® with the adapted device is obtained through the software tribox ® inserting the parameters: load, cycle time corresponding to ½ of the amplitude of the eccentric bearing, properties of material samples, and environment temperature and humidity. the free software hertzwintm with the parameters presented in table 1 is used to simulate the hertzian contact stress that occurs in the unckp in order to compare to the proposed test configuration. the contact between the materials is considered cylindricalcylindrical for unckp (hinge type) [12, 13] and cylindrical-flat for the designed sample couple shown in fig. 2. a compression load of 3 kn is considered for the unckp simulation supposing a subject with 70 kg weight multiplied by a factor of 4.2 [14] while the load of 25.35 n is applied to the designed samples in the simulation, which is limited by the tribometer configuration. 384 j.n. athayde, b.l. fernandes, c.j. de m. siqueira, p. nohama, c.r. fernandes fig. 2 (a) the device designed alone; (b) the device adapted to the tribometer; and (c) detail of the designed samples (a uhmwpe and b ti6al4v) and the stainless steel vat the results from the software hertzwintm show 15.93 mpa of the hertzian contact stress between the samples designed against 15.14 mpa for the unckp simulation, suggesting that the samples designed and the test load applied can simulate the stress in vivo. with these results, the tests with the device were performed. a b c device for in vitro wear analysis of biomaterials in the hinged prosthesis configuration 385 table 1 parameters used to simulate the hertzian contact stress using the software hertzwintm, for the couple uhmwpe-ti6al4v applied in real unckp and in samples designed to use on the tribometer [12, 15, 16] parameters uhmwpe ti6al4v unckp samples designed unckp samples designed young modulus (gpa) 1.2 1.2 110 110 poisson coefficient 0.4 0.4 0.34 0.34 contact geometry cylindrical flat cylindrical cylindrical contact radius (mm) 6.8 ∞ 6.5 3.0 contact length (mm) 40 15 40 15 load (n) 3000 25.35 3000 25.35 the ti6al4v alloy and uhmwpe samples are machined, cleaned and sterilized with ethylene oxide (eto) before the testing procedure. with the purpose of the device assessing, three tests are performed according to the parameters presented in table 2: two tests with the samples designed and one without the device using a pin (15 mm length x 6 mm diameter) and a flat with the same dimensions of the previous tests. table 2 parameters used in tests to evaluate the device parameters test 1 test 2 test 3* contact surfaces cylindrical ti6al4v/ flat uhmwpe cylindrical ti6al4v/ flat uhmwpe ti6al4v pin/ flat uhmwpe constant load 25.35 n 25.35 n 25.35 n number of cycles 500.000 500.000 500.000 frequency 2.0 hz 2.0 hz 2.0 hz lubrication no lubricant no lubricant no lubricant temperature 21.5 ±1.5oc 22.0 ±1.5oc 21.5 ±1.5oc relative humidity 60.0 ± 10% 60.0 ± 10% 60.0 ± 10% the third test is intended to evaluate the importance of the designed samples over the tribometer performance and the influence of the hertzian contact stress over the wear. for the first test, a metallic sample is used as machined without any subsequent finishing as well as the pin for the third test. on the contrary, the metallic sample for the second test receives electrolytic polishing before the test. for test 3, to match the sliding distance between the designed samples and the common ones, the tribometer amplitude is set to ½. the hertzian contact stress in each test from table 2 is determined by hertzwintm using spherical (3,0 mm contact radius) – flat (infinite contact radius): 15.93 mpa for test 1, 15.93 mpa for test 2, and 102.93 mpa for test 3. due to a low wear rate of the uhmwpe, the gravimetric wear determination is strongly affected by intrinsic uncertainty of the measurement [17]. the alternative is to determine the area of the grooves formed in the uhmwpe sample during the test due to adhesive wear mechanisms [17, 18] using a contact stylus profilometer veeco dektak150. the volumetric wear is determined through the area integration (sum of the area of several rectangles that fill the area formed by the curve). 386 j.n. athayde, b.l. fernandes, c.j. de m. siqueira, p. nohama, c.r. fernandes 3. results one example of the 2d profile provided by the contact stylus profilometer veeco dektak-150 is presented in fig. 3. the volumetric wear with the standard deviations for the areas used to calculate them is presented in table 3. it was considered that the student’s t-test distribution with n-1 degree of freedom for the samples and four measurements were made for each sample. fig. 3 one of the uhmwpe groove profiles provided by the profilometer veeco dektak-150 table 3 volumetric wear for each test and the standard deviation of the mean area value test volumetric wear (mm3) area standard deviation (mm2) 1 2.93 x 10-2  9.04 x 10-4 2 3.47 x 10-2  5.06 x 10-4 3 8.41 x 10-2  2.70 x 10-4 the surface roughness of the designed samples for the test 1 is 0.628 µm for ti6al4v alloy (as machined) and 0.265 µm for uhmwpe. for the test 2, roughness is 0.277 µm for the ti6al4v sample (electrolytic polishing) and 0.285 µm for the uhmwpe sample. finally, for the test 3, roughness is 0.628 µm for the ti6al4v alloy pin (as machined) and 0.299 µm for the flat uhmwpe. 4. discussion as one can see from table 3, the wear for the second test (3.47 x 10 -2 mm 3 ) is higher than for the first one (2.93 x 10 -2 mm 3 ). in the test configuration, the only difference is the surface roughness of the metallic samples. the reason for this is that the difference device for in vitro wear analysis of biomaterials in the hinged prosthesis configuration 387 between the roughness of the uhmwpe samples is negligible because the three samples are used as machined (no surface treatment). this behavior is consistent considering that the uhmwpe wear is extremely sensitive to the roughness of the counterpart surface [18] that should be as smooth as possible [19]. the uhmwpe part is abraded when sliding against a harder surface releasing debris, and both surfaces may be abraded by harder third bodies [17, 20]. however, in nolubrication conditions with little hard third bodies as in the tests performed herein, part of the uhmwpe is transferred to the ti6al4v counter face forming a lubricating film [18] suggesting that the differences between the initial surface roughness may not be as significant in the wear analysis as in the test conditions. the tribometer did not suffer any interference in functioning when coupled to the developed device. until now, there is no report about devices that simulate unconventional knee prosthesis. therefore, the device presented in this work brings a significant contribution to the scientific research related not only to this kind of prosthesis but also to the tribological behavior of counter faces of different biomaterials. the device also allows performing tests with samples immersed in lubricant, including bovine serum, which allows comparison with the results obtained from wear tests of the conventional knee prosthesis performed on gait simulators. the angle of 37.96 o that is equivalent to the total sliding trajectory of the sample (back and forth) is the maximum value allowed by the eccentric bearing of the tribometer. although the flexion angle in in vivo knee prosthesis is higher [21], it does not affect the wear performance since, for the hinged prosthesis, the congruence between the surfaces is the crucial parameter. the sliding distance can be compensated by increasing the number of cycles. the higher volumetric wear from the conventional pin-on-flat test (test 3) confirms the influence of the geometry of samples in wear behavior. it is clear that the smaller contact area and consequently a higher hertzian stress is responsible for the poorer performance in wear resistance compared to the designed samples. the higher stress provokes a higher deformation of the uhmwpe part resulting in a higher wear rate, initially by adhesion wear and, after the adhered debris is released, by abrasive wear. both wear mechanisms are commonly identified in the total knee prosthesis [22]. through the hertzwintm free software, the hertzian contact stress value of 15.93 mpa obtained between the designed samples is very close to the value of 15.14 mpa obtained in the simulation of real unconventional knee prosthesis. this result demonstrates that the sample designs are adequate to simulate the contact that occurs in the unconventional knee prosthesis in vivo. 5. conclusion the conceived device allows its adaptation to a conventional linear tribometer without any interference in its functioning. this characteristic proves the functionality of the device in assessing wear resistance between the couples of biomaterials intended to be applied to sliding movements. the samples geometries provide the movement simulation of the unconventional total knee prosthesis, respecting its degree of freedom and hertzian stress as well as allowing realistic performance evaluation of the biomaterials couple before expensive tests in gait simulators. 388 j.n. athayde, b.l. fernandes, c.j. de m. siqueira, p. nohama, c.r. fernandes a profilometer is convenient to obtain volumetric wear, especially when the debris volume or its isolation does not reach an accurate mass measurement. references 1. dieterich, j.d., fields, a.c., moucha, c.s., 2014, short term outcomes of revision total knee arthroplasty, journal of arthroplasty, 29(11), pp. 2163-2166. 2. robertsson, o., lewold, s., knutson, k., lidgren, l., 2000, the swedish knee arthroplasty project, acta orthopaedia scandinavica, 71(1), pp. 7–18. 3. tsal, l.y., garcia, r.j., petrilli, a.s., korukian, m., viola, d.c.m., petrilli. m. de t., godoy, f.a.c., 2007, protocolo fisioterapêutico em pacientes submetidos à endoprótese não convencional de joelho por osteossarcoma: estudo prospectivo, revista brasileira de ortopedia, 42(3), pp. 64-70. 4. mantyh, p., 2013, bone cancer pain: causes, consequences, and therapeutic opportunities, pain, 154, pp. s54–s62. 5. atwood, s.a., currier, j.h., mayor, m.b., collier, j.p., van citters, d.w., kennedy, f.e., 2008, clinical wear measurement on low contact stress rotating platform knee bearings, journal of arthroplasty, 23(3), pp. 431-440. 6. bertin, k.c., 2007, tibial component fixation in total knee arthroplasty, journal of arthroplasty, 22(5), pp. 670-678. 7. abu-amer, y., darwech, i., clohisy, j.c., 2007, aseptic loosening of total joint replacements: mechanisms underlying osteolysis and potential therapies, arthritis research & therapy, 9(1), pp. 1-7. 8. bahraminasab, m., sahari, b.b., edwards, k.l., farahmand, f., arumugam, m., 2013, aseptic loosening of femoral components – materials engineering and design considerations, materials and design, 44, pp. 155-163. 9. au, a.g., raso, j., liggins, a.b., amirfazli, a., 2007, contribution of loading conditions and material properties to stress shielding near the tibial component of total knee replacements, journal of biomechanics, 40, pp. 1410-1416. 10. o’brien, s.t., luo, y., brandt. j-m., 2015, in-vitro and in-silico investigations on the influence of contact pressure on cross-linked polyethylene wear in total knee replacements, wear, 332-333, pp. 687–693. 11. grimer, r.j., carter, s.r., tillman, r.m., sneath, r.s., walker, p.s., unwin, p.s., shewell, p.c., 1999, endoprosthetic replacement of the proximal tibia, journal of bone & joint surgery, 81-b(3), pp. 488-494. 12. manning, d.w., chiang, p.p., freiberg, a.a., 2005, hinge implants revision total knee arthroplasty, 1st ed. new york, springer, pp. 219-222. 13. fabroni, r.h., steverlynck, a.m., aguilera, a.l., zeballos, j.i., 1994, endoprótesis no convencionales fabroni no cementadas, revista de la asociación argentina de ortopedia y traumatología, 59(1), pp. 66-73. 14. wehner, t., claes, l., simon, u., 2009, internal loads in the human tibia during gait, clinical biomechanics, 24, pp. 299-302. 15. tardy, n., chambat, p., murphy, c.g., fayard, j.m., 2014, bilateral custom-fit total knee arthroplasty in a patient with poliomyelitis, orthopedics, 37(9), pp. e839-e843. 16. long, m., rack, h.j., 1998, titanium alloys in total joint replacement a materials science perspective, biomaterials, 19, pp. 1621-1639. 17. gonzález-mora, v.a., hoffmann, m., stroosnijder, r., gil, f.j., 2011, the role of hardness and roughness on the wear of different cocrmo counterfaces on uhmwpe for artificial joints, journal of biomedical science and engineering, 4, pp. 651-656. 18. ruggiero, a., d'amato, r., gómez, e., merola, m., 2016, experimental comparison on tribological pairs uhmwpe/tial6v4 alloy, uhmwpe/aisi316l austenitic stainless and uhmwpe/al2o3 ceramic, under dry and lubricated conditions, tribology international, 96, pp. 349-360. 19. al-samarai, r.a., haftirman, ahmad, k.r., al-douri, y., 2012, evaluate the effects of various surface roughness on the tribological characteristics under dry and lubricated conditions for al-si alloy, journal of surface engineered materials and advanced technology, 2, pp.167-173. 20. poggie, r.a., mishra, a.k., davidson, j.a., 1994, three-body abrasive wear behaviour of orthopaedic implant bearing surfaces from titanium debris, journal of materials science: materials in medicine, 5, pp. 387-392. 21. mundermann, a., dyrby, c.o., d’lima, d.d., colwell, c.w., andriacchi, t.p., 2008, in vivo knee loading characteristics during activities of daily living as measured by an instrumented total knee replacement, journal of orthopaedic research, 26(9), pp. 1167-1172. 22. gul, r.m., mcgarry, f.j., bragdon, c.r., muratoglu, o.k., harris, w.h., 2003, effect of consolidation on adhesive and abrasive wear of ultra high molecular weight polyethylene, biomaterials, 24, pp. 3193-3199. 13737 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 787 806 https://doi.org/10.22190/fume250525028h © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a fast insight into high-accuracy nonlinear frequency estimation of stringer-stiffened shells chun-hui he1, mostafa mohammadian2 1school of civil engineering and transportation, foshan university, foshan, china 2mechanical engineering department, esfarayen university of technology, esfarayen, north khorasan, iran orcid ids: chun-hui he https://orcid.org/0000-0003-0810-5248 mostafa mohammadian http://orcid.org/0000-0002-0714-2787 abstract. understanding the behavior of nonlinear vibrations in stringer-stiffened shell structures is crucial for enhancing the stability and efficiency of advanced aerospace and marine systems. these systems often exhibit complex responses due to geometric and material intricacies, which require analytical methods capable of effectively capturing critical dynamics. this study presents three efficient analytical methods for deriving closedform expressions for the nonlinear frequency of such systems: the adaptive location pointbased he’s formulation (alpf), the square error minimizing-based frequency formulation (semf), and the hamiltonian-based frequency-amplitude formulation (hfaf). these methods offer efficient and straightforward solutions for analyzing nonlinear oscillators without requiring complex iterative procedures. the nonlinear frequency of the stringerstiffened shell is determined using each method and validated against both exact analytical solutions and numerical results. the results show that the first two methods yield high accuracy for small amplitudes, while their accuracy decreases at higher amplitudes. in contrast, the hamiltonian-based method maintains high accuracy over a wider range of amplitudes. the original he's formulation is recognized for its simplicity and computational efficiency, making it a practical tool for rapid frequency estimation in stringer-stiffened shell systems. this comparative study offers guidance for selecting appropriate analytical tools for nonlinear vibration analysis of complex mechanical and physical systems. key words: stringer-stiffened shell, approximate analytical solution, nonlinear frequency, frequency-amplitude formulation, nonlinear vibration received: may 25, 2025 / accepted july 23, 2025 corresponding author: mostafa mohammadian mechanical engineering department, esfarayen university of technology, azadegan boulevard, corner of mader square, esfarayen, north khorasan, iran, postal code: 9661998195. e-mail: mo.mohammadyan@gmail.com, mo.mohammadyan@esfarayen.ac.ir. mailto:mo.mohammadyan@gmail.com mailto:mo.mohammadyan@esfarayen.ac.ir 788 c-h. he, m. mohammadian 1. introduction the nonlinear vibration behavior of cylindrical shells constitutes a critical topic in civil and mechanical engineering. the importance of nonlinear vibrations has grown due to their impact across various fields, including physical sciences, engineering, and mechanical systems. consequently, engineering design processes are increasingly focused on the dynamic behavior of beams and shells. in particular, the nonlinear vibration of cylindrical shells plays a vital role in mechanical engineering, especially in aerospace and marine applications. stringer-stiffened shells play a foundational role in the design of aircraft, spacecraft, and ships, which are frequently subjected to dynamic loading conditions. a deeper understanding of their nonlinear response can significantly enhance performance and structural integrity. this study aims to address the need for efficient and accurate methods to predict the nonlinear frequency of stringer-stiffened shells. such methods are essential for improving structural safety and reliability in engineering applications. the dynamic behavior and stability of shell structures have been thoroughly studied due to their critical role in modern engineering applications. gavrilenko and matsner [1] examined the buckling characteristics of stringer shells, highlighting the influence of stiffening elements on structural stability. naghsh et al. [2] utilized a meridional finite strip method to analyze free vibrations of various shell structures. golchi et al. [3] explored thermal buckling and free vibration of functionally graded (fg) truncated conical shells with stringer and ring stiffeners. bich and ninh [4] pioneered this area by investigating nonlinear vibrations in imperfect stiffened sandwich toroidal shells under thermo-mechanical loading. they later extended their work [5] to eccentrically stiffened toroidal shell segments on elastic foundations. recent research by liu et al. [6] focused on nonlinear traveling-wave vibrations of ring-stringer stiffened cylindrical shells, highlighting the complexity introduced by wave propagation. a significant body of research has been dedicated to the exploration of rotational and thermal effects on the shell’s nonlinear dynamics. liu et al. [7] analyzed nonlinear forced vibrations in rotating cylindrical shells under multi-harmonic excitations in thermal environments. vahidi et al. [8] investigated the nonlinear vibration, stability, and bifurcation of rotating axially moving conical shells, providing insight into the dynamic instabilities induced by axial motion and rotation. mohammadrezazadeh and jafari [9] proposed vibration suppression in laminated composite cylindrical shells employing magneto strictive strips. recent advancements include high-fidelity modeling and simulation techniques for stiffened cylindrical shells under various loading scenarios. bochkarev et al. [10] studied panel flutter in stiffened shallow shell structures. foroutan and dai [11] investigated nonlinear dynamics in spiralstiffened fg toroidal shells with variable thickness. azzara et al. [12] analyzed rotor dynamic responses of stiffened cylindrical structures using high-order shell models. zhang et al. [13] explored frequency responses in porous hyperplastic mooney-rivlin cylindrical shells under radial harmonic forces. du et al. [14] developed a semi-analytical method for the vibration control in cylindrical shell structures. chen et al. [15] examined the correlation between welding parameters and vibro-acoustic behavior in stiffened cylindrical shells. tan et al. [16] introduced innovative sandwich composite shell structures and established a framework for nonlinear geometric and dynamic analyses. khoddami maraghi et al. [17] investigated free vibrations of truncated conical sandwich shells with auxetic core and polymeric nanocomposite face sheets reinforced with graphene nanoplatelets. despite substantial progress, further research is needed to account for multi-physical interactions, enhance model accuracy, and develop efficient computational techniques for design and optimization. a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 789 solution methods for nonlinear differential equations are typically categorized into numerical and analytical approaches. the free vibration and stability of stringer-stiffened shells have been the subject of various numerical investigations [2, 18, 19]. in this context, developments in finite element methods for laminated shell structures [20], as well as in isogeometric finite element formulations [21], have provided highly efficient numerical tools relevant to such analyses. however, solving nonlinear problems with high accuracy presents a considerable challenge, and obtaining exact analytical solutions is often even more complex. to overcome these limitations, researchers have developed several advanced approximate analytical techniques. these methods include the harmonic balance method (hbm) [22, 23], the energy balance method (ebm) [24, 25], the homotopy perturbation method (hpm) [26], the li–he’s modified homotopy perturbation method (li-he hpm) [27], the hamiltonian approach (ha) [28, 29], the variational iteration algorithms (vim) [30, 31], the global error minimization (gem) [32, 33], the modified algebraic method (mam) [34, 35], and the variational approach (va) [36]. in recent studies, the ha [37], the hpm [38], and the global residue harmonic balance method (grhbm) [39] have been employed to determine the nonlinear frequency of stringer-stiffened shells. despite their effectiveness, each method has notable limitations when applied to complex geometries like stringer-stiffened shells. the ha often requires complex energy formulations and extensive algebraic manipulation, making it computationally demanding and less suitable for systems involving damping or external forces. the hpm, despite its flexibility, relies on iterative series expansions and the construction of a suitable homotopy. this process can be arduous and time-consuming for higher-order nonlinear systems. furthermore, the accuracy of the model tends to decrease at large amplitudes. the grhbm is recognized for its precision; however, it requires balancing multiple harmonics and computing global residuals, which leads to high algebraic complexity and necessitates the use of symbolic computation tools. despite significant progress in the field, existing analytical methodologies continue to face challenges in balancing simplicity, accuracy, and applicability to the strongly nonlinear dynamics of stringer-stiffened shells, particularly at large vibration amplitudes. the cubicquintic nonlinearities resulting from geometric and material effects demand rapid, non-iterative solutions that retain accuracy across a wide amplitude range. this study proposes three novel analytical methods: the adaptive location point-based he’s formulation (alpf), the square error minimizing-based frequency formulation (semf), and the hamiltonian-based frequency-amplitude formulation (hfaf), to derive closed-form expressions for the nonlinear frequency of stringer-stiffened shells. these approaches are optimized for simplicity and computational efficiency, effectively overcoming the complexity of earlier methods while maintaining high accuracy. their performance is validated through comparisons with exact solutions and numerical simulations. the proposed methods offer practical tools for engineers and researchers studying nonlinear vibrations in complex shell structures. 2. mathematical model as illustrated in fig. 1, a circular cylindrical shell is stiffened by uniformly distributed external stringers. the supports consist of ribs, modeled as one-dimensional elastic elements, which are evenly spaced along the shell. to determine the displacements and vibration frequencies, it is assumed that rib height is much smaller than the shell's radius of curvature. moreover, interaction between the ribs in the two directions is neglected. 790 c-h. he, m. mohammadian fig. 1 a schematic view of a stringer-stiffened shell structure to study the dynamics of the orthotropic stringer-stiffened shell under large displacements, the following governing equations are employed [40]: 2 2 4 2 1 2 2 ( , ) 0, w w r r l w x t       − + − =   (1) 4 4 4 2 2 1 1 1 1 ( , ) 0, 2 w l w b ry x r     + − =   (2) where 4 4 4 4 1 1 3 22 4 2 2 4 1 2 ,w d d d r x x y y      = + +       (3) 2 2 2 2 2 2 2 2 2 2 ( , ) 2 , w w w l w x y x yx y x y            = − +          (4) 1 1 0 1 2 , , 2 2 1 n f ne f eh h b r r       = + = + − (5) in which, w and φ denote the lateral displacement and the airy stress function, respectively. r represents the shell radius, and h is its thickness. ρ0 and ρ1 are the densities of the shell and stringer materials, respectively. n denotes the number of stringers, and f is the crosssectional area of a stringer. e and e1 represent the elastic modulus of the shell and stringer materials, respectively. additionally, v denotes poisson's ratio. the coefficients di (i = 1, 2, 3) are material-dependent parameters associated with the stiffened shell, as reported in [41]. for simply supported boundary conditions (ss bcs), the displacement w can be approximated as follows [38]: a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 791 2 2 21 1 ( ) ( , , ) ( )sin cos( ) sin , 4 f t nm m w x y t f t x ny x l r l     = +          (6) where m and n denote the wave numbers in the axial and circumferential directions, respectively. using eqs. (1) and (6), the airy function is obtained as follows: 2 22 2 1 1 1 1 1 4 2 2 1 32 2 1 2 3 5 sin cos( ) cos(2 ) 16 1 2 sin cos cos( ). 2 f m fm m b x ny ny r l nn l r fm m m x x ny l ll r −    = −            +               (7) by considering u(t) = f1(t)/r and applying the galerkin method, the governing differential equation for the dynamics of the stringer-stiffened shell is derived as: 22 2 2 3 5 1 2 3 42 2 0, d u d u du u u u u u dtdt dt        + + + + + =      (8) where the coefficients βi (i = 1, 2, 3, 4) are defined as follows: ( ) ( ) 2 2 4 4 2 4 2 2 2 2 2 1 2 1 4 1 4 3 1 4 2 62 2 4 4 4 4 2 2 2 4 3 1 4 6 4 1 0.09375 , 2 1 , 0.0625 0.5 0.75 1 , 0.25 , n l n l n n m m n n n n = = + + + − = + − − =                   (9) and the coefficients εi (i = 1, 2, …6) are defined in [41]. for doubly clamped boundary conditions (cc bcs), the displacement w can be approximated as follows: 2 2 2 21 1 3 ( ) ( , , ) ( )sin cos( ) sin . 16 f t nm m w x y t f t x ny x l r l     = +          (10) following the outlined procedure, the values βi for cc bcs are determined as follows: ( ) ( ) 4 4 2 2 2 4 1 3 1 4 6 4 2 2 4 4 2 2 2 2 2 2 2 1 4 1 4 3 1 2 62 2 4 4 4 0.140625 , 0.083 0.1875 1.5 1 , 0.1641 , 1 5.33 2.67 5.33 1 , n n n n n l n l n m m n = = + − − = = + + + −                  (11) where the coefficients εi (i = 1, 2, …6) can be found in [41]. the governing nonlinear differential equation presented in eq. (8) is highly nonlinear due to the presence of cubicquintic nonlinear restoring force terms, as well as displacement-dependent dynamic terms such as u2ü and uu̇2. these terms introduce a strong amplitude dependence into the system's response, further complicating the governing equation. in the following section, approximate analytical solutions are derived for eq. (8), considering the initial conditions: 0 (0) , 0. t du u a dt = = = (12) 792 c-h. he, m. mohammadian although eq. (8) is specifically derived for a stringer-stiffened shell structure, it also represents the governing dynamic differential equation of other structural systems. for example, by setting β4 = 0, the equation reduces to the form encountered in tapered beams [42]. similarly, setting β1 = 0 yields the well-known duffing equation, which is widely used in engineering and physics applications, such as modeling undamped fg beams [43]. therefore, the analytical methods proposed herein are not limited to the specific geometry considered but can be more broadly applied to estimate the nonlinear frequency of a wide range of conservative systems. 3. analytical solutions for nonlinear vibration of the stringer-stiffened shell this section derives closed-form expressions for the nonlinear frequency of the stringer-stiffened shell governed in eq. (8), by proposing three straightforward and noniterative approximate analytical methods. for validation purpose, the exact nonlinear frequency is also derived. each method assumes a trial solution of the form u(t) =acos(ωt), which satisfies the initial conditions specified in eq. (12) and approximates the periodic behavior of the oscillator. 3.1. adaptive location point-based he’s formulation (alpf) in this paper, we propose a new method, namely the adaptive location point-based he’s formulation (alpf), to improve he’s frequency–amplitude formulation. in the original method, a fixed location point is selected based on he’s recommendation. in contrast, alpf introduces a procedure for selecting the location point, according to the system’s amplitude and physical parameters. this adaptation improves the accuracy of the estimated frequency, especially in systems with strong nonlinearities or amplitude-dependent behavior. importantly, alpf retains the simplicity and computational efficiency of the original method. to illustrate the method, consider the differential equation of a nonlinear oscillator given by: ( ) 0, (0) , (0) 0.u f u u a u+ = = = (13) the above equation can be rewritten as ( ) 0. f u u u u   + =     (14) he [44, 45] demonstrated that the frequency of the nonlinear oscillator given in eq. (13) can be easily approximated as follows: 2 ( ) ,he u u f u u  =  (15) where ū is the location point, defined as ū = ka. this formulation has been widely applied to various types of nonlinear oscillators. the method is computationally simple, making it suitable for quick frequency estimations. however, one of the main challenges in applying he’s method is selecting an appropriate value for the location point [27]. it is more a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 793 appropriate to express the value of k as a function of both the amplitude and system parameters. to implement the proposed alpf method, eq. (15) is substituted into eq. (13), allowing the latter to be rewritten in the following form: ( ) ( ) ( ) ( ), ( ) . u ka f u u g ka u g ka u f u g ka u = + = − = (16) if the frequency obtained from eq. (15) is regarded as the exact frequency, then the right-hand side of eq. (16) reduces to zero. by applying the galerkin technique and using cos (ωt) as the weighting function, the following equation is derived: ( ) 4 0 ( ) ( ) cos( ) 0, t g ka u f u t dt− = (17) where t = 2π/ω. this equation can be solved to determine the value of k. for the present problem, eq. (8) can be rewritten as: ( )2 2 3 5 1 2 3 4 0. u u uu u u u u u u     + + + +  + =       (18) following eq. (15), the nonlinear frequency is approximated as: ( )2 2 3 5 1 2 3 42 , , .he u u u u u u u u uu u u u u = = = + + + + =       (19) when ū = ka, we have 21 ,u a k= − − and 2.u ka= − substituting these values into eq. (19) yields the approximate nonlinear frequency as follows: 2 2 4 4 2 3 4 2 2 2 1 1 , 1 2 he k a k a k a a       + + = + − (20) and the approximate solution as: 2 2 4 4 2 3 4 2 2 2 1 1 ( ) cos . 1 2 he k a k a u t a t k a a       + +  =  + −   (21) using the specific value 3 2k = recommended in [46], the approximate nonlinear frequency can be expressed as follows: 2 4 2 3 4 2 1 16 12 9 . 16 8 he a a a      + + = + (22) to obtain a more accurate frequency, eq. (8) is reformulated using the proposed alpf approach as follows: 794 c-h. he, m. mohammadian 2 2 4 42 2 3 4 2 2 2 2 1 1 22 2 4 4 2 2 3 52 3 4 1 2 3 42 2 2 2 1 1 1 2 . 1 2 k a k ad u u dt k a a k a k a d u du u u u u u u dtk a a dt                + + + =  + −    + +   − + − − −     + −      (23) if the frequency obtained from eq. (18) is the exact frequency, then the right-hand side of eq. (23) becomes zero. based on eq. (17), we obtain: 24 2 2 4 4 2 2 3 52 3 4 1 2 3 42 2 2 2 1 10 cos( ) 0. 1 2 t k a k a d u du u u u u u u t dt dtk a a dt             + +    − + − − − =    + −      (24) substituting eq. (21) into eq. (24) and solving the resulting equation yields the following expression for k: 2 2 2 1 3 1 4 , 2 k − +  −   =  (25) where 2 4 4 2 1 4 4 1 2 4 1 3 1 2 3 1 4 2 2 3 4 1 3 1 2 3 1 4 8 4 , 10 8 16 8 , 5 6 12 6 5 . a a a a a a a                    = +  = − + − −  = − + + − (26) eq. (25) provides an appropriate location point that depends on both the amplitude and the system parameters. by substituting the value of k, calculated from eq. (25), into eq. (20), the approximate frequency denoted as ωalpf is determined using the proposed alpf method. 3.2. square error minimizing-based frequency formulation (semf) the square error minimizing-based frequency formulation (semf), proposed in [47], is a mathematical refinement of he's frequency formulation. it provides a modified approach that enhances the accuracy of the approximate frequency by minimizing a square error function. the error function e(ω2), corresponding to eq. (13), is defined as follows: 2 2( ) ( ) ( ) ( ) .e f u a u t = − (27) we define the mean square error as follows: ( ) /4 2 2 2 2 0 ( ) ( ) ( ) ( ) . t e f u a u t dt = − (28) to minimize the error, it is required that ( ) /42 2 2 2 4 2 2 2 0 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) 0. ( ) ( ) t de d f u a u t f u a u t dt d d      = − + = (29) a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 795 by solving eq. (29), an approximate expression for the nonlinear oscillator frequency given in eq. (13) is obtained as [47]: /4 2 0 /4 2 0 ( ) ( ) . ( ) semf t t u t f u dt u t dt  =   (30) for the present problem, eq (30) can be formulated as follows ( )( ) /4 2 2 3 5 1 2 3 4 2 0 /4 2 0 ( ) . ( ) f t tsem u t u u uu u u u dt u t dt      + + + + =    (31) by substituting u(t)=acos(ωt) into eq. (31), the following expression is obtained: ( ) ( ) /4 3 2 3 3 2 2 1 2 0 /4 2 2 0 /4 3 3 5 5 2 3 4 0 /4 2 2 0 cos( ) cos ( ) cos( )sin ( ) cos ( ) cos( ) cos( ) cos ( ) cos ( ) . cos ( ) t t t s mf t e a t a t a t t dt a t dt a t a t a t a t dt a t dt − + = + + +                      (32) after simplification, the expression for the nonlinear frequency of the stringer-stiffened shell can be written as: 2 4 2 3 4 2 1 16 12 10 , 16 8 semf a a a      + + = + (33) and the approximate solution as 2 4 2 3 4 2 1 16 12 10 ( ) cos . 16 8 semf a a u t a t a      + +  =  +   (34) this method achieves a balance between accuracy and simplicity by optimizing the error and demonstrates robustness for amplitudes ranging from small to moderate. it should be noted that the frequency derived in eq. (33) can also be obtained using the ha [37], the hpm [38], and the first-order grhbm [39]. 796 c-h. he, m. mohammadian 3.3. hamiltonian-based frequency-amplitude formulation (hfaf) the hamiltonian-based method [48] utilizes energy conservation to estimate the frequency. for the nonlinear oscillator given in eq. (13), the hamiltonian constant h is defined as follows: 2 1 ( ), 2 du h p u dt   = +    (35) where ( ) ( ) .p u f u du=  for a conservative system, the hamiltonian remains constant during the entire oscillation process. therefore, we have: 2 1 ( ) ( ) 0. 2 du p u p a dt   + − =    (36) by substituting the approximate solution u(t)=acos(ωt) into eq. (36), the corresponding residual function is obtained as follows: ( ) ( ) ( )2 2 21 sin ( ) cos( ) . 2 r t a t p a t p a  = + − (37) by defining two average residuals 1 /4 1 1 1 1 0 4 cos( ) , t r r t dt t =  (38) 2 /4 2 2 2 2 0 4 cos( ) , t r r t dt t =  (39) the hamiltonian-based nonlinear frequency is obtained as [48]: 2 2 2 2 1 1 2 1 2 ,hfaf r r r r    − = − (40) where ω1 and ω2 are typically chosen as ω1 = 1 and ω2 = 2. the hamiltonian formulation of the present problem is expressed as 2 2 2 2 4 6 1 2 3 4 1 1 1 1 1 . 2 2 2 4 6 du du h u u u u dt dt         = + + + +        (41) considering the initial conditions given in eq. (12) and applying eq. (37), the residual function is constructed as follows: ( ) ( ) ( ) ( ) ( ) 2 22 2 2 2 1 2 4 4 6 6 3 4 1 1 1 ( ) sin( ) sin( ) cos ( ) cos ( ) 1 2 2 2 1 1 cos ( ) 1 cos ( ) 1 . 4 6 r t a t a a t t a t a t a t             = − + − + − + − + − (42) a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 797 assuming ω1 = 1 and ω2 = 2, the corresponding average residuals are obtained as follows: 4 2 4 6 2 1 1 2 3 4 2 1 7 19 1 . 15 3 30 105 3 r a a a a a         = − − − + (43) 4 2 4 6 2 2 1 2 3 4 8 1 7 19 4 . 15 3 30 105 3 r a a a a a         = − − − + (44) following eq. (40), the hamiltonian-based expression for the nonlinear frequency of the stringer-stiffened shell is obtained as: 2 4 2 3 4 2 1 70 49 38 , 70 28 hfaf a a a      + + = + (45) and the approximate solution as: 2 4 2 3 4 2 1 70 49 38 ( ) cos . 70 28 hfaf a a u t a t a      + +  =  +   (46) this method performs well at large amplitudes due to its foundation in energy principles. its energy-based formulation enables more accurate capture of nonlinear effects within this range. 3.4. exact nonlinear frequency to obtain the exact nonlinear frequency of the stringer-stiffened shell, the hamiltonian form derived in eq. (41) is employed. using the initial conditions given in eq. (12), and considering that the hamiltonian remains constant in a conservative system, the following relation is obtained: 2 2 2 2 4 6 2 4 6 1 2 3 4 2 3 4 1 1 1 1 1 1 1 1 . 2 2 2 4 6 2 4 6 du du u u u u a a a dt dt            + + + + = + +        (47) eq. (47) can be rewritten as follows: ( ) ( ) ( )2 2 4 4 6 6 2 2 3 4 2 1 1 1 2 3 . 1 a u a u a u du dt u     − + − + −   =  +  (48) solving eq. (48) for dt, we obtain: ( ) ( ) ( ) 2 1 2 2 4 4 6 6 2 3 4 1 . 1 1 2 3 u dt du a u a u a u     + =  − + − + − (49) since the period of the oscillator is four times the duration required to move from 0 to a, we have 798 c-h. he, m. mohammadian ( ) ( ) ( ) 2 1 2 2 4 4 6 60 2 3 4 1 4 . 1 1 2 3 a exact u t du a u a u a u     + = − + − + −  (50) therefore, the exact nonlinear frequency of the stringer-stiffened shell is obtained as follows: ( ) ( ) ( ) 2 1 2 2 4 4 6 60 2 3 4 2 . 1 4 1 1 2 3 exact a u du a u a u a u       = + − + − + −  (51) this integral typically requires numerical evaluation or the use of elliptic functions, serving as a benchmark for approximate methods. 4. results and discussion this section evaluates the nonlinear frequencies of a stringer-stiffened cylindrical shell using the proposed methods. the results are presented for ss bcs and are validated through comparisons with both exact and numerical solutions. tables 1-3 report the nonlinear frequencies and relative errors for non-dimensional amplitudes a ranging from 0.01 to 3. the parameter sets are as follows: table 1 (case1: β1 = 0.5, β2 = 2, β3 = 0.2, β4 = 0.1); table 2 (case 2: β1 = 0.1, β2 = 1, β3 = 0.5, β4 = 0.2); and table 3 (case 3: β1 = 0.9, β2 = 1.5, β3 = 2.5, β4 = 3.5), reflecting increasing degrees of nonlinearity. the results are compared with the exact solutions and with those reported in [40] using the hpm. to facilitate clearer comparison, the relative errors reported in tables 1-3 are also illustrated in fig. 2 as a function of amplitude. it can be observed that for small amplitudes (a ≤ 0.2), all methods exhibit high accuracy, with errors below 0.1% across all parameter sets. this behavior is expected, as the cubic and quintic terms in eq. (8) contribute minimally at low a, making the system nearly linear. the results of alpf and semf methods are identical to the exact solution for a ≤ 0.1, while the hamiltonian-based method shows slight deviations (e.g., 0.09% error at a = 0.2 in table 1), likely due to its sensitivity to higher-order nonlinear terms. at large amplitudes (a ≥ 1), the errors increase, particularly for he's method, which reaches 10.26% at a = 3 in table 3. in contrast, the alpf and semf show improved performance, with errors up to 5.89% at a = 3, benefiting from their error minimization strategies. the hamiltonian method consistently outperforms the others, maintaining errors below 3.92% at a = 3, as its energy-based framework more effectively captures the nonlinear dynamic behavior. it is noteworthy that both the alpf and semf methods accurately predict the frequencies obtained by the hpm, while offering a simpler computational procedure. furthermore, the computational time required to evaluate the frequency using the proposed methods is minimal, demonstrating their high efficiency. for instance, for the case of a = 3 in table 1, the computation times on a system with an intel core i7-7500u cpu and 12 gb ram, using matlab, are approximately 0.1 ms, 0.03 ms, and 0.006 ms for the alpf, seme, and hfaf methods, respectively. in comparison, the exact method requires about 1.3 ms due to the necessity for numerical integration. a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 799 table 1 comparison of the frequencies obtained by the proposed three methods with exact ones for β1 = 0.5, β2 = 2, β3 = 0.2, β4 = 0.1. a present ωhpm [38] ωexact ωhe ωalpf ωsemf ωhfaf 0.01 1.4142 1.4142 1.4142 1.4142 1.4142 1.4142 0.1 1.4130 1.4130 1.4130 1.4130 1.4133 1.4130 0.2 1.4093 1.4093 1.4093 1.4093 1.4106 1.4093 0.8 1.3544 1.3516 1.3524 1.3524 1.3683 1.3524 1 1.3347 1.3285 1.3304 1.3304 1.3522 1.3304 1.4 1.3162 1.2979 1.3041 1.3041 1.3356 1.3041 1.8 1.3411 1.3037 1.3176 1.3176 1.3545 1.3176 2 1.3719 1.3229 1.3416 1.3416 1.3801 1.3416 2.5 1.4985 1.4156 1.4488 1.4488 1.4900 1.4488 3 1.6820 1.5597 1.6089 1.6089 1.6537 1.6089 table 2 comparison of the frequencies obtained by the proposed three methods with exact ones for β1 = 0.1, β2 = 1, β3 = 0.5, β4 = 0.2. a present ωhpm [38] ωexact ωhe ωalpf ωsemf ωhfaf 0.01 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1 1.0016 1.0016 1.0016 1.0016 1.0016 1.0016 0.2 1.0066 1.0066 1.0066 1.0066 1.0063 1.0066 0.8 1.1175 1.1163 1.1186 1.1186 1.1121 1.1186 1 1.1926 1.1902 1.1952 1.1952 1.1843 1.1952 1.4 1.4110 1.4049 1.4204 1.4204 1.3965 1.4204 1.8 1.7215 1.7095 1.7423 1.7423 1.7024 1.7423 2 1.9093 1.8930 1.9365 1.9365 1.8885 1.9365 2.5 2.4621 2.4281 2.5036 2.5036 2.4378 2.5036 3 3.1145 3.0499 3.1623 3.1623 3.0851 3.1623 table 3 comparison of the frequencies obtained by the proposed three methods with exact ones for β1 = 0.9, β2 = 1.5, β3 = 2.5, β4 = 3.5. a present ωhpm [38] ωexact ωhe ωalpf ωsemf ωhfaf 0.01 1.2248 1.2248 1.2248 1.2248 1.2248 1.2248 0.1 1.2297 1.2297 1.2297 1.2297 1.2297 1.2297 0.2 1.2452 1.2451 1.2452 1.2452 1.2453 1.2452 0.8 1.6689 1.6500 1.6709 1.6709 1.6619 1.6709 1 1.9571 1.9197 1.9586 1.9586 1.9460 1.9586 1.4 2.7040 2.6016 2.6861 2.6861 2.6777 2.6861 1.8 3.5965 3.3897 3.5248 3.5248 3.5378 3.5248 2 4.0750 3.8032 3.9641 3.9641 3.9928 3.9641 2.5 5.3237 4.8620 5.0873 5.0873 5.1637 5.0873 3 6.6127 5.9344 6.2230 6.2230 6.3534 6.2230 800 c-h. he, m. mohammadian fig. 2 the relative error with respect to amplitude for the three proposed methods under various system parameters, (a) case 1, (b) case 2, and (c) case 3 fig. 3 illustrates the periodic solutions obtained by the proposed methods in comparison with the numerical solution. all methods demonstrate acceptable accuracy and successfully capture the periodic behavior of the stringer-stiffened shell. compared to previous studies, the proposed methods offer simpler alternatives to iterative approaches such as hpm [38] and grhbm [39], which involve complex series expansions and exhibit limitations at large amplitude values. accurate frequency prediction is essential for avoiding resonance in stiffened shells, thereby enhancing safety in applications such as aircraft and submarines. he’s method, due to its simplicity and errors below 2% for a ≤ 1, is particularly suitable for rapid design iterations during preliminary engineering analyses. the hamiltonianbased method, with errors remaining below 4% for all amplitude levels, is better suited for critical applications that demand high precision, such as spacecraft panels subjected to large dynamic loads. these methods enable engineers to optimize shell designs more efficiently, reducing costs and improving structural integrity. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10  1 = 0.5  2 = 2  3 = 0.2  4 = 0.1 amplitude, a r el at iv e er ro r (% ) he’s formula alpf & semf methods hfaf method (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 amplitude, a r e la ti v e e rr o r (% )  1 = 0.1  2 = 1  3 = 0.5  4 = 0.2 he’s formula alpf & semf methods hfaf method (b) 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10  1 = 0.9  2 = 1.5  3 = 2.5  4 = 3.5 amplitude, a r e la ti v e e rr o r (% ) he’s formula alpf & semf methods hfaf method (c) a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 801 fig. 3 comparison of the analytical solutions with the numerical solution for various system parameters and amplitudes, (a) case 1, a = 0.5, (b) case 1, a = 2.3, (c) case 2, a=0.5, (d) case 2, a = 2, (e) case 3, a = 0.5, and (f) case 3, a = 2 using the frequency-amplitude relationship derived from the hamiltonian-based frequency-amplitude method, fig. 4 illustrates the variation of the frequency ratio (ωnl/ωl) with respect to amplitude under various system parameters. in this figure, when the effect of one parameter is studied, the other three parameters are fixed at unity. it is observed that parameters β1 and β2 have a similar effect. as their values increase, the frequency ratio 0 1 2 3 4 -0.5 0 0.5  1 = 0.5  2 = 2  3 = 0.2  4 = 0.1 a = 0.5 t u (t ) numerical solution he’s formula alpf & semf methods hfaf method(a) 0 0.5 1 1.5 2 2.5 3 3.5 4 -2 -1 0 1 2  1 = 0.5  2 = 2  3 = 0.2  4 = 0.1 t u (t ) a = 2.3 numerical solution he’s formula alpf & semf methods hfaf method (b) 0 1 2 3 4 5 6 -0.5 0 0.5 t u (t )  1 = 0.1  2 = 1  3 = 0.5  4 = 0.2 a = 0.5 numerical solution he’s formula alpf & semf methods hfaf method(c) 0 0.5 1 1.5 2 2.5 3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 t u (t )  1 = 0.1  2 = 1  3 = 0.5  4 = 0.2 a = 2 numerical solution he’s formula alpf & semf methods hfaf method(d) 0 1 2 3 4 -0.5 0 0.5 t u (t )  2 = 1  3 = 0.5  4 = 0.2 a = 2  1 = 0.9  2 = 1.5  3 = 2.5  4 = 3.5 a = 0.5 numerical solution he’s formula alpf & semf methods hfaf method(e) 0 0.5 1 1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 t u (t ) a = 2  1 = 0.9  2 = 1.5  3 = 2.5  4 = 3.5 a = 0.5 a = 2 numerical solution he’s formula alpf & semf methods hfaf method(f) 802 c-h. he, m. mohammadian decreases. in contrast, parameters β3 and β4 exhibit the opposite trend. an increase in these parameters results in an increase in the nonlinear to linear frequency ratio. fig. 4 the effect of system parameters (a) β1, (b) β2, (c) β3, and (d) β4 on the variation of frequency ratio with respect to the amplitude fig. 5 presents a sensitivity analysis of the frequency ratio with respect to various system parameters under different amplitudes. as expected, at small amplitudes (here a = 0.1), the frequency ratio is equal to unity, indicating the negligible effect of nonlinear terms. figs. 5a and 5b show that, for various amplitudes, an increase in the parameters β1 and β2 results in the frequency ratio approaching unity, thereby reducing the impact of nonlinearities. specifically, fig. 5a indicates that for moderate amplitudes (here a = 0.5 and a = 1.5), the frequency ratio may be either greater or less than unity, depending on the value of β1. for instance, at a = 1.5, the frequency ratio is approximately 1.2 when β1 = 2, while it decreases to nearly 0.9 when β1 = 5. however, at higher amplitudes, the ratio exceeds for all values of β1. it is important to note that, as defined in eq. (8), the parameter β1 quantifies the contribution of displacement-dependent terms. therefore, it may be concluded that the influence of these terms on the nonlinear frequency of stringer-stiffened shell becomes significant at larger amplitudes. furthermore, figs. 5c and 5d reveal that the frequency ratio increases with higher values of β3 and β4, indicating the role of cubicquintic nonlinearities in the nonlinear frequency response of the structure. 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 a  n l / l  1 = 1  1 = 2  1 = 3  1 = 4  1 = 5 (a) 0 1 2 3 0.5 1 1.5 2 2.5 3 3.5 a  n l / l  2 = 1  2 = 2  3 = 3  4 = 4  5 = 5 (b) 0 0.5 1 1.5 2 2.5 3 1 2 3 4 a  n l / l  3 = 1  3 = 2  3 = 3  3 = 4  3 = 5 (c) 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 a  n l / l  4 = 1  4 = 2  4 = 3  4 = 4  4 = 5 (d) a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 803 using the expressions derived for the nonlinear frequency of the stringer-stiffened shell, the influence of geometric parameters can be readily examined. for instance, fig. 6 illustrates the effects of the number of stringers n and the amplitude a on the nonlinear frequency under both ss and cc bcs. the geometrical and material properties of the shell are as follows [49]: radius r = 242 mm, thickness h = 0.65 mm, length l = 609 mm, elastic modulus e = 68.95 gpa, poisson's ratio ν = 0.3, and density ρ = 2714 kg/m3. the depth and width of the stringers are taken as 7.02 mm and 2.55 mm, respectively, with the assumption that the shell and stringers are made of the same materials. as shown in fig. 6, increasing the number of stringers leads to a higher nonlinear frequency due to the corresponding increase in structural stiffness. additionally, as the amplitude increases, the frequency initially decreases and then increases. this initial drop in frequency persists up to higher amplitude levels for cc bcs compared to ss ones. fig. 7 further demonstrates that the nonlinear frequency increases with increasing shell thickness h. similarly, the effects of other geometric parameters can be systematically explored using the derived expressions. fig. 5 the variation of frequency ratio with respect to (a) β1, (b) β2, (c) β3, and (d) β4 under various amplitudes 0 1 2 3 4 5 0.5 1 1.5 2 2.5  1  n l / l a = 0.1 a = 0.5 a = 1.5 a = 2 (a) 0 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5  2  n l / l a = 0.1 a = 0.5 a = 1.5 a = 2 (b) 0 1 2 3 4 5 0.5 1 1.5 2 2.5  3  n l / l a = 0.1 a = 0.5 a = 1.5 a = 2 (c) 0 1 2 3 4 5 0.5 1 1.5 2 2.5 3  4  n l / l a = 0.1 a = 0.5 a = 1.5 a = 2 (d) 804 c-h. he, m. mohammadian fig. 6 the effects of the number of stringers n and amplitude a on the frequency of a stringer-stiffened shell when m = n = 1. (a) ss bcs, (b) cc bcs fig. 7 the effects of the number of stringers n and shell thickness h on the frequency of a stringer-stiffened shell under ss bcs. m = n = 1 5. conclusion this study developed three analytical methods: adaptive location point-based he’s formulation (alpf), square error minimizing-based frequency formulation (semf), and hamiltonian-based frequency-amplitude formulation (hfaf), to derive closed-form expressions for the nonlinear frequency of stringer-stiffened cylindrical shells. these methods were validated against both exact and numerical solutions. for small vibration amplitudes, they exhibited errors les than 0.1%, while for larger amplitudes, the hamiltonianbased method maintained an error within to 3.92%. compared to iterative techniques such as the homotopy perturbation method, the proposed methods offer efficiency and ease of application. the models effectively address the cubic-quintic nonlinearities characteristic of stiffened shells, offering practical tools for fast and accurate frequency prediction. these findings are particularly relevant to aerospace structures, spacecraft panels, and marine hulls, where avoiding resonance is essential for ensuring structural safety. it is important to note that the present study focused on the vibration response in a single dominant mode and neglected damping effects. these assumptions were adopted to facilitate the development of simple, fast, and non-iterative analytical expressions for the 0 0.5 1 1.5 2 2.5 3 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 amplitude, a f re q u en cy ( k h z) n = 10 n = 30 n = 50 n = 70 (a) ss bcs 0 0.5 1 1.5 2 2.5 3 0.8 1 1.2 1.4 1.6 1.8 2 amplitude, a f re q u en cy ( k h z) n = 10 n = 30 n = 50 n = 70 (b) cc bcs 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 shell thickness, h (mm) f re q u en cy ( k h z) n = 10 n = 30 n = 50 n = 70 a fast insight into nonlinear frequency estimation of stringer-stiffened shells with high accuracy 805 nonlinear frequency of 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liu, c., 2022, a modified frequency–amplitude formulation for fractal vibration systems, fractals, 30(03), 2250046. 48. wang, k.-j., wang, g.-d., 2021, study on the nonlinear vibration of embedded carbon nanotube via the hamiltonian-based method, journal of low frequency noise, vibration and active control, 41(1), pp. 112-117. 49. zhao, x., liew, k.m., ng, t.y., 2002, vibrations of rotating cross-ply laminated circular cylindrical shells with stringer and ring stiffeners, international journal of solids and structures, 39, pp. 529-545. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 579 593 https://doi.org/10.22190/fume200611042s © 2020 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper influence of pressure oscillations in common rail injector on fuel injection rate mikhail g. shatrov1, andrey u. dunin1, pavel v. dushkin1, andrey l. yakovenko1, leonid n. golubkov2, vladimir v. sinyavski2 1ishlinsky institute for problems in mechanics ras, moscow, russia 2energo-ecological faculty, moscow automobile and road construction state technical university (madi), moscow, russia abstract. fuel injection causes considerable oscillations of fuel pressure at the injector inlet. one of the reasons is hydraulic impact when the needle valve closes. for multiple injections, the previous injections affect the following. as both the fuel pressure in rail pac and the injection rate grow, the oscillations increase. the pressure oscillation range at the common rail injector inlet at pac=1500 bar is up to 350 bar, and at the rail pressure pac=500 bar, the amplitude decreases to 80 bar. physical properties of the fuel are also important. as the viscosity of the fuel increases, its hydraulic friction grows which results in a rapid damping of pressure oscillations. the data for an injector operating on sunflower oil is presented. as compared with diesel fuel, the oscillations range decreases from 400 to 250 bar at the same operating mode. the influence of the interval between the impulses of a double injection on the injection rate of the second fuel portion was investigated. superposition of two waves during multiple injections may result in amplification and damping of the oscillations. simulation was performed to estimate the influence of fuel type and time interval δτ between control impulses of a double injection on the injection quantity of the second portion at pressures of 2000-3000 bar. when the rail pressure pac grows, the oscillations and their impact on the injection process increase. for diesel fuel at pressure of pac=2000 bar, the variation in injection rates of the second portion is 2.36-4.62 mg, and at pac=3000 bar – 1.58-6.63 mg. key words: common rail fuel system, common rail injector, hydrodynamic effects, pressure oscillations, fuel injection rate received june 11, 2020 / accepted october 30, 2020 corresponding author: vladimir sinyavski moscow automobile and road construction state technical university, 64, leningradsky prospect, moscow, 125319, russia e-mail: sinvlad@mail.ru 580 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. 1. introduction the development of engines with perspective energetic and ecological parameters demands the development of new versions of the fuel supply system for different models of diesel engines. in this connection, important tasks are ensuring injection pressure up to 2000 bar and higher as was investigated by pflaum [1], shatrov [2], yu [3], bosch company [4], as well as the injection rate front shape control and organization of fuel distribution in the combustion chamber which was studied by shatrov [5], kamaltdinov [6], iakovenko [7], wuethrich [8]. it was demonstrated by wloka [9], grekhov [10], zhao [11], veratudela [12] that the desired fuel injection law at any operation mode of the engine is formed by variation of the control impulse duration and pressure in the common rail. it also depends on the wave phenomenon originating in the high-pressure line and having a considerable impact on the fuel injection process in the case of a multistage injection. the pressure growth in case of multiple injections makes the fuel injection process more complicated. baratta [13], beirer [14], catania [15] and experiments carried out in madi demonstrated that the pressure oscillations at the injector inlet become crucial. beirer [14] investigated the influence of hydrodynamic effects in the fuel line on pressure oscillations in the common rail injector (cri). the oscillations grow as the channel length increases and the diameter decreases. but the fuel line also influences their frequency. for example, if the fuel line length increases, the frequency decreases. this is explained by the fact that the pressure wave travel time in a long channel is higher. oscillation process increases with the growth of the fuel pressure in the common rail and control impulse duration. it was demonstrated that depending on the interval between two portions of double injection, the injection rate could vary considerably. the wave process that originates when one cri injects fuel has an impact on fuel injection process of the injectors belonging to the other diesel engine cylinders. iakovenko [7] made a conclusion about the reasons of the pressure oscillations: pressure oscillations are triggered by the so called “water-hammer” effect induced by the nozzle closure at the end of each injection. when the fuel is flowing via the nozzle holes during injection process, its kinetic energy grows which is transformed into the energy of the pressure waves when the needle valve closes and the flow stops abruptly. beirer [14] found in his research that the fuel pressure oscillations cannot be the reason of the resonance of the mechanical parts of the fuel system because their natural oscillation frequency is considerably lower than the frequency of the wave process in the hydraulic circuit. however, the authors demonstrated the possibility of the origination of the hydraulic resonance which takes place when the needle closes as soon as the compression wave reaches the nozzle due to the reflection of the injection-induced depression wave in the rail. when the injector needle valve closes on its seat, the hydraulic impact takes place. the pressure wave and the direct pressure wave (compression wave) which originate as a consequence appear to be in the same phase which causes intensive pressure oscillations in the delivery line. it is important to take into consideration this effect in the case of multiple injections and seek to avoid it. the analyses of the injector design influence on the wave process are of a high interest. for example, here it is demonstrated that in a nozzle having holes on the locking cone, the hydraulic impact when the injector is closing is not so strong as in the case of a nozzle having holes in the sack volume. influence of pressure oscillations in common rail injector on fuel injection rate 581 the results described need to be supplemented by data on the ways of prevention of oscillation effects and the influence of fuel properties on them. the present paper presents the results of solving these scientific tasks. 2. experimental setup the experimental setup (fig. 1) has a modular design making possible its adaptation for current research tasks and various designs of fuel systems. this setup consists of the following parts: ▪ asynchronous electric motor 3 (7.5 kw, 3000 rpm) with a thyristor transducer 12 enabling a smooth control of rotational speed; ▪ low-pressure fuel line which includes a low-pressure fuel pump 16, 12 v and 24 v electric power supply units 8 9, fine-mesh fuel filter 15. fig. 1 experimental setup: 1 – frame; 2 – protective casing; 3 – asynchronous electric motor; 4 – coupling; 5 – fuel tank; 6 – mounting plate; 7, 11 – fuses; 8, 9 – electric power supply units; 10 – magnetic contactor; 12 – thyristor transducer; 13 – emergency stop switch; 14 – electric motor mounting; 15 – fine fuel filter; 16 – low-pressure fuel pump; 17 – fuel pressure regulator; 19, 20 – elements mounting the electric motor on the frame the fuel system (fig. 2) of the experimental setup includes: high-pressure pump 1, fuel accumulator 2 with pressure sensor 16, fuel line 5 (length lfl=1000 mm, channel diameter dfl=2.2 mm) and electro-hydraulic injector 3. the experimental stand is equipped with a control system developed in madi. the difference is that the newly developed control system ensures the formation of any shape of electric control pulse. this feature is fundamentally important in the study of pressure oscillations in the diesel fuel system. 582 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. fig. 2 fuel system of the experimental setup: 1 – high-pressure fuel pump; 2 – fuel accumulator; 3 – electro-hydraulic injector; 4, 5 – high-pressure fuel lines; 6, 7 – fuel accumulator mounting plates; 8, 9, 10, 14 – clamping plates; 11 – fixings; 12 – fuel accumulator fixing pin; 13 – pressure sensor mounting plates; 15 – pressure sensor fixing pin; 16 – pressure sensor electromagnetic drive of the control valve is used in two electro-hydraulic injectors (fig. 3). the second electro-hydraulic injector (fuel injector nozzle hole diameter dс=0.09 mm, number of holes iс=8, fig. 3b) differs from the first injector (fuel injector dс=0.12 mm, iс=7, fig. 3a) with the presence of a fuel accumulator 6 integrated into the body 3 and with the design of the control valve 2. fig. 3 layouts of the injectors: a – injector no1, b – injector no2, 1 – solenoid, 2 – control valve, 3 – cri body, 4 – control chamber, 5 – multiplier (for version b, elements 5 and 7 are one piece); 6 (a) – channel supplying fuel to the injector nozzle; 6 (b) – fuel accumulator; 7 – injector nozzle needle; 8 – injector nozzle; 9 – needle valve spring; 10 – inlet throttle; 11 – outlet throttle; 12 – valve spring; 13 – additional throttle; 14 – channel influence of pressure oscillations in common rail injector on fuel injection rate 583 the experimental setup is complemented by two piezoelectric sensors t6000, manufactured in russia. the first sensor (no 4636 having sensibility 2.1 pc/bar and pressure measuring range 0…6000 bar) is mounted at the inlet of the cri and registers the pressure oscillations when the fuel is injected. the second sensor (no 4588 having sensibility 2.2 pc/bar and pressure measuring range 0…6000 bar) is mounted in the chamber and registers the instants of the fuel injection start and its end. the magnitude of fuel injection rate q is measured by the gravimetric method. when determining q of each point, the measurement is carried out twice, which allows us determine the random measurement error of this magnitude (q). the instrument error of high-precision scales is neglected since its value is substantially less than the random measurement error. each measurement provides q not less than ±5.0%, which is achieved by selecting a sufficient number of consequent cycles. the measurement errors of the inlet pressure in the injector and fuel accumulator are determined by the accuracy of measuring instruments. therefore, calibration of pressure sensors t6000 and dmp304 (manufactured in germany, pressure measuring range 0…4000 bar) is previously conducted. the experiment was carried out in two stages: the first stage – the injector operates in a single injection mode; the second stage – the injector operates in a multiple injection mode. 3. results and discussion the influence of some factors upon pressure oscillations at the cri inlet was investigated: fuel pressure pac, control impulse duration τimp, type of fuel used. with this, two different injectors having principally different design were studied in the experiment. two types of the fuels were used: a fossil-origin diesel fuel and renewable sunflower oil (table 1). table 1 properties of the fuels used properties diesel fuel sunflower oil density (t = 20 ос), kg/m3 820 923 kinematic viscosity (t = 20 ос), mm2/s 3.0 65.2 cetane number 45 33 low calorific value, mj/kg 42.5 37.0 fuel injection causes considerable oscillations of fuel pressure at the inlet of the injector. one of the reasons is hydraulic impact which originates when the injector nozzle needle closes. in this case, in injector no1 with pressure pac=1000 bar and control impulse duration τimp=0.6 ms (which corresponds to fuel injection rate q=16.5 mg), the injection causes the origination of pressure oscillations which have the amplitude up to 250 bar (fig. 4). evidently these oscillations have influence upon the fuel supply process in the case of multiple injections: the previous injections would influence the following ones. 584 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. fig. 4 pressure at the inlet of the cri no 1 (pac=1000 bar, τimp=0.6 ms, q=16.5 mg) as the fuel pressure and injection rate grow, the oscillation process increases. fig. 5 shows the comparison of data registered at three pressures in the fuel accumulator and constant control impulse duration τimp =0.6 ms. a single injection is used in this case. fig. 5 fuel pressure oscillations at the entry of the cri no1 at various pressures (τimp=0.6 ms): q=9.1 mg (pac=500 bar); q=16.5 mg (pac=1000 bar); q=38.3 mg (pac=1500 bar) the pressure oscillation range at the cri inlet is up to 350 bar at pac=1500 bar, and at pac=500 bar, the amplitude decreases to 80 bar. fig. 6 shows both the comparison and the variation of the first control impulse duration at constant pressure in the fuel accumulator pac=1000 bar. on the basis of this ne influence of pressure oscillations in common rail injector on fuel injection rate 585 can conclude that as the first portion of the fuel injected decreases, the fuel pressure oscillations range also decreases. fig. 6 fuel pressure oscillations at the entry of the cri no1 at various duration of the first injection (pac=1000 bar): q=2.2 mg (τimp=0.3 ms); q=16.5 mg (τimp=0.6 ms) fig. 7 shows the experimental data for modified injector no2 at the operation mode pac=1000 bar and τimp=0.6 ms. compared with the no1 version of the cri (fig. 5), the pressure oscillations are considerably lower. the pressure oscillations range for the version no1 is 400 bar, and for the version no2 – 120 bar, that is, 3.3 times lower. fig. 7 pressure at the inlet of the cri no2 (pac=1000 bar, τimp=0.6 ms, q=24.0 mg) hence, the reduced internal volume of the injector plays a considerable role and may be an efficient measure for lowering the pressure oscillations. 586 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. injector no2 has a pressure balanced valve in addition to the integrated fuel accumulator. leonard [16] proved that the balanced valve makes it possible not only to decrease the volume of fuel leaks at high pressure, but also to improve the injector working process in the case of multiple injections. physical properties of the used fuel are also important in the fuel injection process. as the fuel viscosity increases, so does the hydraulic friction which contributes to a rapid damping of the oscillations. fig. 8 shows the data obtained with the use of injector no1 when operating on more viscous sunflower oil. as compared with diesel fuel (fig. 4), the oscillations range decreases from 400 to 250 bar at the same operating mode. fig. 8 pressure at the inlet of the cri no1 (pac=1000 bar, τimp=0.6 ms, q=15.9 mg), operation on sunflower oil the influence of the interval between the impulses of a double injection on the injection rate value of the second portion was investigated. injector no1 was used. the oscillograph trace of a current passing through the electric magnet of injector no1 is shown in fig. 9. the injector control is carried out in two phases: forcing and holding. for forcing, the voltage of about 50 v is applied to the electric magnet during 0.3 ms which promotes a rapid raise of the control valve. the injector needle is kept under control by using pulse-width modulation with duty ratio of 50%. the injection rate and injection characteristic of the second portion depend on the time at which the second injection is affected related to the first injection. fig. 10 shows the results of the investigations at constant pressure pac=1000 bar with two injections each having τimp=0.6 ms when there is a variable interval δτimp between two portions of the double injection. the vertical line in the picture designates the fuel injection start instant. superposition of the waves during operation with the multiple injections may result in both the amplification and the damping of the pressure oscillations process. if the second injection is executed at the rear wave edge (pressure increase) or in the minimal pressure zone – the oscillations damping takes place. if or when the second injection is executed at the front (decreasing) wave edge or in the maximal pressure zone, the oscillations increase. influence of pressure oscillations in common rail injector on fuel injection rate 587 fig. 9 oscillograph trace of a current passing through the electric magnet of injector no1 (double injection): τimp – control impulse duration, δτimp – interval between two portions of the double injection fig. 10 pressure oscillations at the inlet of the cri no1 at various intervals between the double injection δτimp (pac=1000 bar): q=17.5 mg (δτimp=3.6 ms); q=9.5 mg (δτimp=5.5 ms) fig. 10 shows that at the interval of δτimp=3.6 ms, after injection of the second portion, the maximal pressure oscillations range is 330 bar. while at the interval of δτimp=5.5 ms, the maximal oscillation range increases (1.45 times) to 480 bar. fig. 11 shows the results of estimation of the dependence of injector rate q of the second injection on the interval between the injections. the first injection value is constant and amounts to q1=16.5 mg. the difference between the first and the second magnitudes of the injection rate is almost 2 times. 588 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. it should be mentioned that the average value of the second injection rate is considerably lower than that of the first injection rate. even if the beginning of the second injection is shifted from the first injection to the interval of δτimp =50 ms, the value of the second portion is 13.1 mg which is by 3.4 mg lower than the first one, in spite of the fact that the pressure oscillations created by the first injection are damped completely during 50 ms. the background of the revealed phenomenon has two potential explanations. first, the pressure in the fuel accumulator decreases after the first injection. the pressure deviation value is not large and according to data presented in figs. 5 and 6, it amounts to 50 bar (depending on the operation mode). fig. 11 fuel injection rate at various intervals between the injections δτimp (pac=1000 bar, τimp=0.6 ms) the second factor is the voltage slump on the injector power supply condenser. as it follows from oscillograph traces in fig. 9, the forcing current of the second injection is by 2.5 a lower than the first one which causes longer opening of the injector. this is the answer why the modern injection systems also make stringer requirements to such parameters as fuel pressure control dynamics and charging of the power supply condenser of the injectors. injection characteristic of the second fuel portion also depends on δτimp because the pressure built up in the needle valve volume is interlinked with the pressure at the cri inlet. for example, if the injection of the second fuel portion starts in the minimal zone of the pressure wave and terminates in the maximal zone of the pressure wave (fig. 10), the fuel flow velocity through the spray holes will vary with the continuing injection process from low to high value. simulation was carried out to estimate the influence of fuel type and time interval δτ (fig. 12) between the control impulses of the double injection on the value of the injection quantity of the second fuel portion at pressures 2000…3000 bar. influence of pressure oscillations in common rail injector on fuel injection rate 589 fig. 12 control impulses modeled: f – injector electromagnet force, δτ – time interval between the control impulses, τ1 – the first control impulse duration, τ2 – the second control impulse duration simulation was carried out using the software package for modeling fuel system operation developed in madi. in this software package, the fuel system is divided into the following elements: ▪ basic units: injectors, high-pressure fuel pump; ▪ connecting parts: fuel lines and fuel accumulator. each element contains equations integrated into the system and describing: ▪ the law of conservation of momentum for the mechanical moving parts of the fuel system; ▪ volume and mass balances for the internal volumes and cavities of the high-pressure line of the fuel system. the elements are connected with each other by fuel lines, in which wave phenomena takes place. they are modeled taking into account hydraulic friction in the high-pressure line. the systems of equations describing processes in the fuel system elements are the boundary conditions for calculation of the wave phenomena in the high-pressure line. the fuel flow in the high-pressure line channels is considered isothermal, and the fuel density and sound velocity are constant. the processes occurring in the volumes contained in the injector and the high-pressure fuel pump are considered equilibrium. in the computer model, yielding of the final volumes and fuel lines is not taken into account. the cri no2 was selected as a subject of the research because it provides a smaller pressure oscillations range. two equal control impulses were modeled (τ1 = τ2). we selected such duration of control impulses τ that the fuel quantity supplied during the first injection was q1 ≈ 3…4 mg. modeling results of the operation of the cri no2 on diesel fuel for two fuel pressures in the rail of pac = 2000 and 3000 bar are presented in figs. 13 and 14. as was demonstrated during experimental tests carried out in madi (fig. 11), pressure oscillations at the injector inlet were the reason of variation of the injection rates as a function of δτ. 590 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. fig. 13 fuel injection rate q2 at different intervals between injections δτ for diesel fuel (q1 =3.3 mg): pac=2000 bar fig. 14 fuel injection rate q2 at different intervals between injections δτ for diesel fuel (q1 =3.3 mg): pac=3000 bar when pressure pac grows, the oscillation phenomenon and its impact on the working process increase. when operating on diesel fuel at pressure pac=2000 bar, the spread in the injection rates of the second portion is q2 = 2.36…4.62 mg, and at pac=3000 bar, q2 = 1.58…6.63 mg. influence of pressure oscillations in common rail injector on fuel injection rate 591 the results of fuel injection rate q2 variation calculated for a higher density fuel corresponding to the sunflower oil are presented in figs. 15 and 16. fig. 15 fuel injection rate q2 at different intervals between injections δτ for a higher density fuel (q1 =3.4 mg): pac=2000 bar fig. 16 fuel injection rate q2 at different intervals between injections δτ for a higher density fuel (q1 =3.4 mg): pac=3000 bar the main difference between the test results given in figs. 13, 14 and figs. 15, 16 is a faster attenuation of the oscillations observed when passing to a more dense fuel. in the case of pac=2000 bar, the spread in the injection rates of the second portion is q2 = 2.96…4.21 mg, and at pac=3000 bar – q2 = 2.42…5.50 mg. 592 m.g. shatrov, a.u. dunin, p.v. dushkin, a.l. yakovenko, l.n. golubkov, et al. it is seen from comparison of calculated data (fig. 13 … fig. 16) that due to a higher hydraulic friction, the range of maximal pressure oscillations is lower. this will have a positive effect on the control accuracy of the second portion of a higher density fuel injected. the pressure oscillations that are described above should be taken into consideration when applying the injection rate shaping control method developed in madi in which electric control impulses are used. the method of injection rate shaping is employed in the cases when the control impulse strategy consists of the primary, main and post impulses. electric impulses are supplied from the electronic control module to the electromagnetic valves of injector no1 and injector no2 (fig. 3). duration of the first primary control impulse determines the amplitude of the front edge of the first stage of the boot-type injection rate shape. in the case of a multiple injection, one should select the proper intervals between the control impulses which could assure a boot-type injection rate shape with the desired value of oscillations of the first stage of the boottype fuel injection rate shape. 4. conclusions 1. fuel injection causes considerable pressure oscillations at the inlet of the injector. the range of these oscillations depends on injection pressure, control impulse duration, fuel physical properties and the injector design. one of the reasons of the oscillations is the hydraulic impact which takes place when the injector needle valve closes on the seat. 2. both the pressure drop in the accumulator after the pilot injection (the value of the pressure deviation is 5 mpa depending on the mode) and the voltage drop across the capacitor of the power injector (the current boost of the second injection is by 2.5 a lower than that of the first one) are responsible for the longer opening of the injectors in the case of the next injection. 3. the presence of the fuel accumulator integrated into the cri body decreases the wave phenomenon related to the fuel injection process. during recent experiments with the cri no2 modified with an integrated fuel accumulator (pac=1000 bar, τimp=0.6 ms), the impulse amplitude at the inlet to the injector was 120 bar which was for 3.3 times lower than in the case of cri no1 without the use of fuel accumulator. 4. when fuel accumulator pressure pac grows, the oscillation phenomenon and its impact on the working process increase. so, the variation range in the injection rates of the second portion is q2 = 2.36…4.62 mg, and at pac=3000 bar, q2 = 1.58…6.63 mg when operating on diesel fuel at the rail pressure pac=2000 bar. 5. after switching the cri operation to a fuel having higher viscosity, due to the growth of the hydraulic friction, there is a more rapid attenuation of the pressure oscillations caused by the pilot injection. so, when the cri no1 operates (pac=1000 bar, τimp=0.6 ms) on sunflower oil, the range of pressure oscillations decreases from 40 mpa (when the diesel engine operates on diesel fuel) to 25 mpa (when the diesel engine operates on sunflower oil). acknowledgments: this work was supported by the russian science foundation [grant number 19-19-00598]. site: https://www.rscf.ru/ https://www.rscf.ru/ influence of pressure oscillations in common rail injector on fuel injection rate 593 references 1. pflaum, s., wloka, j., wachtmeister, g., 2010, emission reduction potential of 3000 bar common rail injection and development trends, cimac congress bergen, paper no. 195. 2. shatrov, m.g., golubkov, l.n., dunin, a.u., yakovenko, a.l., dushkin, p.v., 2015, influence of high injection pressure on fuel injection perfomances and diesel engine working process, thermal science, 19(6), pp. 2245-2253. 3. yu, y., 2019, experimental study on effects of ethanol-diesel fuel blended on spray characteristics under ultra-high injection pressure up to 350 mpa, energy, 1861, 115768. 4. why higher injection pressure saves fuel and also increases performance and torque, 2013, bosch media service. 5. shatrov, m.g., malchuk, v.i., dunin, a.u., shishlov, i.g., sinyavski, v.v., 2018, a control method of fuel distribution by combustion chamber zones and its dependence on injection conditions, thermal science, 22(5), pp. 1425-1434. 6. kamaltdinov, v.g., markov, v.a., lysov, i.o., zherdev, a.a., furman, v.v., 2019, experimental studies of fuel injection in a diesel engine with an inclined injector, energies, 12(14), 2643. 7. iakovenko, a., dunin, a., dushkin, p., savastenko, e., shatrov, m., 2019, the influence of mass composition of water-fuel emulsion on ecological characteristics of a diesel engine, energies, 12(14), 2689. 8. wuethrich, d., rotz, b., herrmann, k., boulouchos, k., 2019, spray, combustion and soot of water-infuel (n-dodecane) emulsions in a constant volume combustion chamber, part ii: effects of low temperature conditions and oxygen concentrations, fuel, 248, pp. 104-116. 9. wloka, j., pötsch, c., wachtmeister, g., 2011, injection spray visualization for 3000 bar diesel injection, 24th conference of the institute for liquid atomization and spray systems. 10. grekhov, l., denisov, a., starkov, e., 2016, diesel fuel injection by pressure up to 400 mpa, international journal of pharmacy and technology, 8(4), pp. 27208-27215. 11. zhao, j., grekhov, l., yue p., 2020, limit of fuel injection rate in the common rail system under ultrahigh pressures, international journal of automotive technology, 21, pp. 649-656. 12. vera-tudela, w., haefeli, r., barro, c., schneider, b., boulouchos, k., 2020, an experimental study of a very high-pressure diesel injector (up to 5000 bar) by means of optical diagnostics, fuel, 275, pp. 1-17. 13. baratta, m., catania, a., ferrari, a., 2006, hydraulic layout effects on multijet c.r. injection system performance and design criteria to remove the pressure wave induced disturbances on sequential injection shots, 61congresso nazionale ati, perugia, 9 p. 14. beirer, p., 2007, experimental and numerical analysis of hydraulic circuit of a high pressure common rail diesel fuel injection system, tampere: tampere university of technology, finland, 198 p. 15. catania, a., ferrari, a., manno, m., spessa, e., 2008, experimental investigation of dynamics effects on multiple-injection common rail system performance, journal of engineering for gas turbines and power, asme, 130(3), 032806. 16. leonhard, r., warga, j., pauer, t., ruckle, m., schnell, m., 2010, solenoid common rail injector for 1800 bar, mtz worldwide, february, pp. 10-15. https://www.sciencedirect.com/science/article/pii/s0360544219314392 https://www.sciencedirect.com/science/article/pii/s0360544219314392 https://www.sciencedirect.com/science/journal/03605442 https://www.sciencedirect.com/science/article/pii/s0016236119304041 https://www.sciencedirect.com/science/article/pii/s0016236119304041 https://www.sciencedirect.com/science/article/pii/s0016236119304041 https://www.researchgate.net/journal/0975-766x_international_journal_of_pharmacy_and_technology https://rd.springer.com/article/10.1007/s12239-020-0062-3#auth-1 https://rd.springer.com/article/10.1007/s12239-020-0062-3#auth-2 https://rd.springer.com/article/10.1007/s12239-020-0062-3#auth-3 https://rd.springer.com/journal/12239 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 1, 2015, pp. 3 10 1adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method udc 531.2 roman pohrt 1 , valentin l. popov 1,2 1 berlin institute of technology, germany 2 national research tomsk state university, russia abstract. using the concept of stress intensity factors, we suggest a way to include adhesion into boundary elements simulation of contacts. a local criterion concerning the maximum admissible surface stresses decides whether the adhesive bonds in particular grid points fail or not. by taking into account the grid spacing, a robust methodology is found. validation is done using the theoretically derived cases of jkr adhesion. key words: boundary element method, adhesion simulation, jkr, detachment 1. introduction when two arbitrary bodies enter into contact, they exert relatively weak attracting forces, which rapidly decrease with increasing distance. this sticking is called adhesion and plays an important role in many technical applications such especially glue and adhesive tapes. in general, adhesion comes into play when one of the following conditions is met. either the surfaces are very smooth, or one of the contacting bodies is made of a very soft material. in both cases, this allows for a wide-spread intimate contact. adhesive effects are also increased when the system size is small due to the different scaling of volumeand surfaceforces. for example, in micromechanical devices (mems), engineers must design clearances and stiffnesses in such a way that the structures stay movable and do not interlock due to adhesion. the same effect is exploited in biological systems, such as the sticking pads of insects or lizards. for the understanding and prediction of such systems, an appropriate description is needed. from a theoretical point received january 09, 2015 / accepted february 12, 2015 corresponding author: pohrt, roman technische universität berlin, straße des 17. juni 135, 10623 berlin, germany e-mail: roman.pohrt@tu-berlin.de original scientific paper 4 r. pohrt, v.l. popov of view, two influential approaches have been developed in order to describe adhesive contacts for the basic case of elastic, parabolic bodies. johnson, kendal and roberts [1] (jkr), took into account the adhesion forces within the contact area. they calculated the contact radius in the equilibrium state from the minimum in the total energy, which is, in turn, obtained from the elastic deformation energy, the potential of external forces, and the surface energy of the contacting bodies. derjaguin et al. [2] (dmt theory) considered the molecular forces of attraction only within a ring outside the contact area. even though these forces contribute to the macroscopic attracting force, they are assumed to cause no deformation. the predicted maximum adhesion force coincides with the one given by bradley [3] for the contact between rigid spheres. this discrepancy between the jkr and the dmt theories was finally resolved by tabor [4] who successfully formulated two different regions of validity for both cases, based on a dimensionless parameter. accordingly, the dmt theory applies to small, rigid spheres, while the jkr theory is more appropriate for describing large, soft spheres. later, johnson and greenwood [5] pointed out the fact that the jkr theory still provides good results outside its actual area of validity. over the years, jkr theory has evolved to be the most widely used in the description of adhesion. fig. 1 qualitative presentation of an adhesive contact between a rigid, curved body and an elastic half-space 2. local criterion in bem assume two non-conforming bodies z1(x,y), z2(x,y) in contact. elastic deformations uz(x,y) inside the resulting contact zone must satisfy the condition 1 2( , ) ( , ) ( , ) 0zz x y z x y u x y   (1) in order to account for non-overlapping. the necessary surface stress can only occur inside the contact region and, for bodies with smooth geometries, it will vanish at the contact border. when the bodies keep the same contact area, but are pulled apart by distance d, then all the surface points within the contact area are deformed by that constant distance d and the deformation now reads 1 2( , ) ( , ) ( , ) 0zz x y z x y u x y d    . (2) in the case of a circular contact area, additional stress  which arises from the constant deformation by distance d, is given by adhesive contact simulation using local detachment criterion in boundary elements method 5 * 2 2 1 2( ) e d a r    . (3) here 2 2 1 1 1 2 2 * [(1 ) / (1 ) / ]e e e      is the reduced modulus of elasticity with e being the youngs modulus and  the poisson ratio of the two bodies, respectively. the stress intensity factor from eq. (3) is given by [7] * *2i e d k e a      , (4) where  is the effective adhesion energy after dupré for two surfaces. for other contact zone geometries we expect also a singularity of this type 0( )x l x  , (5) similar to the theory of fracture mechanics. here x is a coordinate in the interface plane perpendicular to the contact boundary, starting from the same and pointing into the material. l and 0 are characteristic length and stress. this singular curve applies, as long as we are sufficiently close to the contact border, which corresponds to the crack tip (see fig. 2). in fracture mechanics, as well as adhesion theory, this rise in local stresses leads to a detachment of the bodies and thereby moves the crack tip or decreases the contact zone. in a numerical representation of stresses, this singularity, eq. (5), is a problem for two reasons. first, the discretized grid is limited in its resolution and can only approach the contact border with a certain precision. second, numerical number representation is limited as well and cannot take into account very high values [6]. we will now suggest a way of overcoming these difficulties using a local, mesh dependent detachment criterion. let h be the grid spacing, the distance between two points on the numerical mesh. fig. 2 singularity (5) at the contact area boundary. by requesting points to have  < h, it is guaranteed for these points to be inside the contact zone by at least distance h (the grid spacing) 6 r. pohrt, v.l. popov assume a singularity according to eq. (5). even though the exact description of the singularity and the border is not possible using discretized grid and number format, we can safely state that any point in the vicinity of the boundary which has tensile stress below 1 2 0h l h h    (6) is at least distance h away from the singularity and should adhere in any case. any stress greater than h, the grid point is located ‘at’ the boundary, with the precision of the chosen grid. if we wish to find a local stress criterion  for detachment, it is clear that such a criterion must scale with h 1/2 just like h. concerning materials parameters e * and , the criterion must scale in the same way as stress intensity factor ki. in order to find the correct value and factor, let us consider another perspective. a rectangular grid element can only detach when its stored elastic energy uel exceeds the separation energy usurf. for a rectangular grid with dimensions x yh h the latter is simply given by surf x yu h h  . (7) the elastic energy stored in a grid element can generally be obtained from 1 2 elu uda   , (8) where u is the deformation resulting from surface stress  . with boussinesq * 2 2 1 ( , ) ( , ) ( ) ( ) x y u x y dxdy e x x y y         , (9) and the basic bem assumption that the pressure inside the element is constant, this gives 2 2 * *2 2 0 0 0 0 1 ( ) 2 ( ) ( ) y yx x h h h el h u dxdydxdy e ex x y y               . (10) after some algebra we find 3 3 2 21 ( ) 3 1 log log 2 x y x y y x x y yx xy h h h h h h h h h h h h h h h h h h                         (11) where 2 2 x yh h h  . in the common case of square grid elements x yh h h  , eq. (11) reduces to 3 32 3 2 1 1 2 log 3 2 2 1 0.473201h h               (12) adhesive contact simulation using local detachment criterion in boundary elements method 7 demanding that uel = usurf for    we can compare eq. (7) and eq. (10) to define a grid-dependent, critical stress value *x yh h e      . (13) or for square grid elements * 0.473201 e h     (14) the requirement is that no single tensile stress value solved for a local grid point must exceed  . in a numerical procedure, this can be handled in the following way. assume we have the adhesion-free normal indentation of the indenter solved (see [8]). now we wish to decrease the indentation depth from dmax to d. the procedure is done as depicted in fig. 3. in the first step, contact area a is left unchanged and the stresses are solved which are necessary for eliminating the gap between the bodies. this can be done using the techniques described in [8]. the discrete stresses are individually compared to  . all the points where this value is exceeded by tensile stress are excluded from the contact zone and the system is solved again. the loop of the algorithm is used to find the correct real contact area. fig. 3 flow chart of adhesion algorithm for every iteration, we must recalculate all the stresses inside the contact area. it is potentially a time-consuming procedure but the contact area can only decrease, so the algorithm is guaranteed to terminate. 8 r. pohrt, v.l. popov 3. numerical sample studies in this section we will show that the methodology proposed in section 2 can be effectively used to simulate the adhesive detachment of bodies in contact. several test cases exist, where analytical solutions are available. in order to illustrate the application of this rule, we will first consider the adhesive contact between a flat, cylindrical indenter with radius r and an elastic half-space. in this case physically, the whole contact zone will detach simultaneously. the total normal force required to separate the indenter from the substrate is then [9] 3 *8adhf a e   (15) the critical value for the (negative) indentation depth at the moment of detachment is given by * * 2 2 adh crit a f a d ee      (16) fig. 1 shows the resulting forces for a decreasing indentation depth. it can be seen that as d approaches dcrit, the force approaches fadh eq. (15) and then suddenly falls to zero (see arrow). at fadh, also the whole contact is suddenly lost, as depicted in fig. 1 (b). fig. 4 numerical pull-off of a flat cylindrical indenter. crosses demark points where no local detachment is found, circles show points where at least one grid point has lost contact. (a) normal force over indentation (b) normal force over contact area let us now test the classical jkr case of a parabola indented into the elastic half space. at the critical point with maximum adhesive force, the theory states [9] 1 3 1 2 2 * 2 3 * 2 3 9 3 , , 2 8 64 adh crit crit r r f r a d e e                       (17) starting from a deeper indentation, the dependencies between normalized indentation depth critd d d , normal force adhf f f , and contact radius crita a a read [9] 2 1 23 4d a a  (18) and 3 3 2 5 32 , 0.12( 1) 1f a a f d     . (19) in fig. 5 and fig. 6 these dependencies are shown with solid lines. adhesive contact simulation using local detachment criterion in boundary elements method 9 fig. 5 numerical indentation-controlled pull-off of an adhesive contact with a parabola. the grid resolution is 128128 points. crosses demark points where no local detachment is found, circles show points where at least one grid point has lost contact. (a) normal force over indentation (b) normal force over contact radius the calculation time of the overall procedure depends very much on the implementation of the stress-finding subprocedure, which usually has complexity n 2g, where ng is the number of grid points in one direction. fig. 6 numerical indentation-controlled pull-off of an adhesive contact with a parabola. the grid resolution is 512512 points. crosses demark points where no local detachment is found, circles show points where at least one grid point has lost contact. in contrast to the case shown in fig. 5, this time the inital indentation is the hertzian case with no prior pull-off. therefore, tensile stresses must first build up at the boundary, before some detachment occurs. (a) normal force over indentation (b) normal force over contact radius 10 r. pohrt, v.l. popov as points are individually excluded from the contact zone, increasing the grid resolution also increases the number of iterations used. empirically, we have found the computational time to scale with  ng 5/2 and  nd 0.55 , where nd is the number of intermediate indentation steps. in our setup, a full jkr solution as in fig 5. took 106 seconds with a grid of 256 256 (ng = 256) and nd = 45. 4. conclusion similar to crack propagation, simulating the adhesive normal contact problem is difficult because it generally has a singularity at the boundaries. when treating the problem numerically, it has been unclear how to find the correct boundary and how to deal with the finite resolution that is available. we suggest a very simple stress-based criterion to be included in bem simulations. whenever the criterion is not satisfied, the particular points are removed from the contact zone. using two examples with known analytical solutions, we show that this approach can successfully simulate the pull-off of adhesive contact. plasticity is not taken into account. the method shown here can be combined with other bem techniques, in order to simulate roughness, viscoelastic materials or frictional effects. readers who wish to implement the new methodology can do so using the instructions given here and in [8]. acknowledgements: one of the authors (r. pohrt) received funding from project po 810/22-1 by deutsche forschungsgemeinschaft. this work is supported in part by tomsk state university academic d.i. mendeleev fund program. references 1. johnson, k. l., kendall, k., roberts, a. d., 1971, surface energy and the contact of elastic solids, proc. r. soc. lond. a, 324 (1558), pp. 301-313. 2. derjaguin, b. v., muller, v. m., toporov, y. p., 1975, effect of contact deformation on the adhesion of particles, journal colloid interface sci., 55, pp. 314-326. 3. bradley, r. s., 1932, the cohesive force between solid surfaces and the surface energy of solids, philos. mag., 13, pp. 853-862. 4. tabor, d., 1977, surface forces and surface interaction, j. colloid interface sci., 58, pp. 2-13. 5. johnson, k. l., greenwood, j. a., 1997, an adhesion map for the contact of elastic spheres, j. colloid interface sci., 192, pp. 326-333. 6. kuna, m., 2008, numerische beanspruchungsanalyse von rissen, vieweg+teubner. 7. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer. 8. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17(3), pp. 1-12. 9. popov, v. l., 2010, contact mechanics and friction. physical principles and applications, springer. 12155 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume230914041f © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper evaluation of inner shear resistance of layers from mineral granular materials szabolcs fischer széchenyi istván university, department of transport infrastructure and water resources engineering, hungary abstract. the methodologies used to assess the inner shear resistance of granular layers from minerals measured in the laboratory with and without geosynthetic reinforcing layers are described in this paper. for the measurements, a multi-level shear box is applied without considering vertical loads on the top layer. in the literature and engineering practice, an accepted calculation method for determining inner shear resistance exists. it is the shear force with linear shearing speed, primarily after a peak force value. this can be accounted for in the present case by calculating the average force value for the 40-80 mm shear range using previous scientific and research achievements. the article details each possible additional method and compares it different methods. three granular materials, as well as six planar geosynthetics, were studied. for this purpose, the results of 216 measurements were considered and processed using 61 different shear function-qualification parameters. the calculations were performed using a simplified function fitting test and a selection process to maximize the allowable relative standard deviations. all three types of materials and four classification parameters were chosen as references for comparability. as a result, only one alternative parameter can be used to determine the reinforcing-weakening values with a maximum deviation of 5% while not producing insufficient results in the placement (ranking) of the individual granular material and geosynthetic pairings. this parameter is the area under the function (integral) calculated on the measurement graph of the 4080 mm shear range, in kn×mm unit; it gives correct values only if the reference granular material is the considered railway ballast, the shearing plane is the geosynthetic's plane (i.e., the so-called "0-plane"), and the reference qualification parameter is the original recommended parameter. the best relative standard deviation values were 20% and 30%. key words: planar geosynthetic, geogrid, geotextile, reinforced granular layer, inner shear resistance received: september 14, 2023 / accepted october 28, 2023 corresponding author: szabolcs fischer department of transport infrastructure and water resources engineering, faculty of architecture, civil engineering and transport sciences, széchenyi istván university, h-9026 győr, egyetem tér 1., hungary e-mail: fischersz@sze.hu 2 s. fischer 1. introduction people are traveling more and more, spending more of their lives on/with transportation, thanks to the industrial revolutions, globalization, and widespread urbanization [1,2]. this can include short trips within a residential area, but it can also be part of a longer daily commute for work, as well as work or tourist trips of several thousand kilometers. rail transportation can be a competitive option in terms of speed, allowing for travel times, mainly, up to 1000 kilometers [3,4]. railways have been an essential part of global transportation for nearly two centuries. since the invention of the steam engine in the 18th century and then the steam locomotive in the 19th century, railways have left (and continue to leave) an indelible mark on the world by significantly influencing economic and technological development, social relations and, of course, environmental sustainability [5]. the development of railways marked a turning point in transportation history. prior to the invention of steam locomotives, land transportation relied primarily on horse-drawn carriages and a primitive road network. the so-called "railway revolution," which began in 1825 with george stephenson's locomotive (locomotion), heralded a new era of public transportation. this transformation aided industrialization, urbanization, and commercial network(s) expansion [5]. when discussing railways, the concepts of dependability, punctuality, and efficiency are well-known [6]. they can transport a large amount of freight and many passengers at significantly lower ton-kilometer costs than other modes of land transport. due to wellorganized traffic and well-planned timetables, railways are less congested. moreover, rail transportation is less affected by bad weather and traffic (and accidents); it is a dependable mode of transportation [6]. some countries and railways take great pride in adhering to train timetables. according to eu statistics, estonia and latvia had the highest passenger train punctuality (around 99%) in 2018 [7]. denmark, finland, and sweden had 95% accuracy rates, while hungary had 78.41%, compared to the eu average of 90.21% [7]. one of the least polluting modes of land transportation is railway transportation. trains produce fewer greenhouse gases per kilometer than trucks and automobiles [8,9]. at the same time, the energy required to run and operate trains (including traction, acceleration, and comfort) is among the most affordable modes of transport [8-11]. electric rail systems powered by renewable energy sources can further reduce carbon emissions, assisting with global climate change efforts. in this regard, green energy is critical, and it is important to note that this is not only energy consumption but also energy production and storage. battery power is frequently used as an option in the case of electric rail systems [12]. in any case, everyone must think globally in terms of entire life cycles, i.e., people should not look only at end-users with the trendy slogan of 'zero emission'. rail transportation is one of the safest modes of transportation compared to other modes [13,14]. accidents are reduced by the isolated – albeit faster – track and strict safety standards and adherence to them [15]. to improve passenger and freight safety, modern trains employ advanced safety technologies such as automatic braking systems [16]. given the impact of risk events on infrastructure project costs, timeliness, and quality, risk management investment is required to avoid or mitigate negative consequences [17]; decision-making analysis is critical and crucial in this field [18]. evaluation of inner shear resistance of layers from mineral granular materials 3 railways can also transport large quantities of cargo such as coal, iron ore, cereals, and other raw materials (an australian example [19] shows the transport of iron ore in the northwest of australia in 2011-2012, with a transport demand of 510 million tons/year). another australian example [20] compared coal transport prices by road and rail, with an official statistic from 2005 showing a 2:1 ratio per tkm net (tkm means ton-km), or roughly 6:3 aud cents. in addition, an interesting graph highlighting these transportation costs from 1964 to 2001 was provided. it is fascinating to see how, around the mid-1980s, prices shifted from favoring road transport to favoring rail transport. as a result, transportation costs can be significantly reduced, which benefits industries that rely on bulk transport. railways can reduce road congestion by diverting a significant portion of freight traffic from highways, i.e., from general road freight traffic [21]. danielis and marcucci [21] show the total costs of congestion for road and rail transport under three scenarios, including congestion charges and political-institutional changes. this reduces not only traffic congestion but also road infrastructure maintenance costs. along with the above advantages, railways have some disadvantages, which will be discussed further below [6]. one of the significant disadvantages of railways is their limited access [22]. rail networks may not reach all remote or rural areas, posing transportation challenges in areas with insufficient rail coverage. alternative modes of transportation, such as trucks or ships, are required to cover the "last mile" in these cases. rail systems, by definition, are less adaptable than road systems. butko et al. [23] discuss railway door-to-door transport options, shkurina and maskaeva [24] discusses rail transport inflexibility and penalties for noncompliance with contracted delivery times. shkurina and maskaeva [24] also discuss identifying systemic "breaches" in the quality of the transportation process and measures to improve the quality of transportation services provided to customers. this inflexibility could harm industries that rely on "agile" supply chains or serve niche markets. construction and maintenance of rail infrastructure is expensive and is frequently financed by governments or private investors. rail may be less appealing in the short term due to the relatively long payback period for these investments, especially when compared to more cost-effective road transport options. carteni and henke [25] calculated a benefitcost ratio of at least 2.0 and a payback period of at least 15 years for the formia-gaeta tourist rail line in italy. while railways are ideal for transporting large amounts of freight, they may not be the most expedient option for time-sensitive shipments [24]. fast-moving industries, such as perishable goods [26] or time-critical manufacturing, are less well-suited to rail transportation. trains have a maximum speed limit determined by the vehicle's technical parameters and track speed. although freight train schedules can be planned to some extent, rail companies generally prefer passenger trains to run first (freight trains are "put on hold" and must wait longer in traffic jams), so delivery times are less predictable than for road transport. in most cases, this precludes them from being used for time-sensitive deliveries. a freight wagon may pass through several marshaling yards on a long journey, where it is reassigned to different trains based on its destination. all this takes time, which increases the distance traveled between the point of dispatch of the freight/wagon and the destination. freight trains are frequently run at night to avoid interfering with the heavy passenger train traffic during the day. 4 s. fischer environmental protection and awareness have been demonstrated to be increasingly important in protecting our planet, and it is worthwhile to implement as many environmentally friendly technologies as possible in all aspects of life [27]. to ensure the future of people's survival and health, we must consider reducing the high share of personal car transportation in the modal split or shifting to a significant use of green fuels, possibly even renewable energy sources. public transportation, personal (traditional or electric) bicycle transportation, e-rolling, and so on, shift the modal split in the right direction for shorter travel distances. it is easily solved by widespread adoption and preference for "carsharing" services, but public transportation should be made more appealing by introducing discount programs and other similar solutions. in this case, rail transportation should be preferred over road transportation because it is more environmentally friendly. needless to say, opportunities, such as electric vehicle transportation, must be considered. they will become an increasingly important part of land transport in the future, both for passenger and freight transport. there is some hope in the freight sector that rola trains, which have lagged for many years, will resume their essential role in freight transport. significant financial resources (foreign subsidies and domestic capital) are required to build high-quality railway infrastructure and purchase modern fleets to achieve these goals as soon as possible. if the described scenario is realized, the economic implementation of new construction, rehabilitation reconstruction, and a significant reduction in the number of maintenance interventions will be critical in the field of rail infrastructure. moreover, modern building materials and technologies will be essential. it should be noted that hungary received over one trillion huf (hungarian forint, 1 huf equals approximately 0.0025 €) in eu support for railway and urban rail development during the 2014-2020 eu cycle, which contributed to significant improvements in the country. modernization of the hungarian track design regulations (unfortunately, in parts, there are more than 30-year old regulations [28]) and widespread use of the established "e-rail system" will be required in the future because track design speeds for these are not more than 160 km/h – of course, in the relevant new harmonized european standard [29], it is possible to find design parameters that can also be applied to speeds of 300 km/h. the second most important aspect is that according to the theory of railway track degradation [30,31], someone must keep our railways in good technical condition during the current eu financing cycle (2021-2027) and in the future, taking into account that the expected lower amounts are to be spent on track rehabilitation because our high-quality reconstructed lines may face a similar fate without maintenance and maintenance funds as they did in the 1990s and 2000s. the author wanted to emphasize the development of the scientific and research discipline of civil engineering and transport sciences with the above paragraphs, especially on the socio-economic environment (for detailed information, see section 2). according to the paper, one of the most common causes of railway track geometry failures is the poor, inadequate condition of the railway substructure, low load-bearing capacity, drainage problems, and so on [32]. short and long sections can both benefit from geosynthetic reinforcement. this can be done in a variety of ways. for example, a granular layer was added on top of the substructure, i.e., beneath the crushed stone railway ballast. this layer and its lower plane may contain one or more planar geosynthetic layers supplemented by a geotextile layer. the other option is to directly place planar geosynthetic material beneath the crushed stone railway ballast to specifically and primarily geometrically stabilize it (i.e., the railway track itself). in this paper, the author examines evaluation of inner shear resistance of layers from mineral granular materials 5 how geosynthetic reinforcement affects the inner shear resistance of granular materials applied in transport infrastructure facilities. for this purpose, several granular materials and planar geosynthetics were tested in a multi-level shear box under laboratory conditions [8]. based on the previously demonstrated applicability of the multi-level shear box, the main goal of this paper is to investigate how shear functions can be used to determine classification/qualification values/parameters for each shearing plane. there were 216 measurements taken in total. the author's aim was to search for new function-qualification parameters that could replace the 2012 evaluation option [8]. the structure of the paper is the following: section 2 is a detailed literature review, section 3 is about materials and methods, section 4 is the discussion section, and section 5 is where the author summarizes the main findings of the study. 2. problem statement and further literature review the required thickness of the granular layers needs to be applied and installed, as well as the value of the (increased) load-bearing capacity available with them, is a national economic issue in the maintenance, renovation (rehabilitation), and modernization of newly constructed, as well as existing railway facilities [8,9]. this is especially important in low-load-bearing capacity subgrades. granular materials don't have enough tensile and shear strength, but they can be supplemented with geosynthetic materials (typically geogrids). this type of geogrid reinforcement property is known as interlocking effects – with some exceptions for geotextiles and geomembranes; for example, reinforcements can be achieved at least to a limited extent in the foundation of soil earthwork with high friction because they can be removed from the load with potentially significant elongation (deformation), this is known as a membrane effect [33,34]. the behavior of this interaction under relatively small displacements, when the elongation of the geogrid is no more than 1%, demonstrates the practical usefulness of the interlocking effect. in this case, the interlocking effect of the geogrid on the subsoil and the granular layer – i.e., the formation of a "composite layer" of the geogrid and granular layer – provides lateral support, thereby reducing vertical load deformation. the use of the geosynthetics (geogrid and geotextile), mentioned above, in the layer structure can also ensure the required load-bearing capacity value with a thinner granular supplementary layer. geogrids with specific parameters can be placed directly beneath the ballast bed to stabilize track geometry in railway construction. geosynthetics are also used in the construction of reinforced retaining walls, abutments, and other "ground reinforcement" (or "ground reinforcing") structures. the inner shear resistance of granular materials could be determined using a special multi-layer box, with the distance above the geosynthetic plane becoming the determining factor [8]. the behavior of geogrids on railway tracks could be evaluated by constructing experimental sections and monitoring them over time [35], it won't be covered in detail. geosynthetics are classified into the following categories: (i) geogrids, (ii) geotextiles, (iii) geocells, (iv) geonets, (v) geopipes, (vi) geofoam, (vii) geocomposites, and (viii) geomembranes. the types under consideration are (i) and (vii), where the relevant geosynthetic types are geogrids combined with geotextiles in the case of geocomposites (vii). geogrids are engineered polymer materials formed from polyethylene, polypropylene, polyester, and other polymers designed to improve the mechanical properties of soil or 6 s. fischer granular materials. they are available in various shapes and sizes, such as woven, knitted, extruded, and welded grids, each tailored to a specific application. geogrids work by confining and distributing applied loads, reducing lateral movement, and increasing the underlying soil or granular material's load-bearing capacity. dynamic loads from trains that vary in weight (axle load and train length), speed, and frequency are applied to railway tracks [36-38]. if not properly managed, these loads significantly stress the underlying ballast and subgrade layers, resulting in track deformation, settlement, and even failure. traditional ballast materials like crushed stone or gravel have limited load-bearing capacity and relatively poor long-term performance. geogrids increase the load-bearing capacity of granular layers beneath railway tracks significantly. geogrids' confinement and reinforcement distribute loads more evenly, lowering the risk of track deformation and settlement. this allows heavier trains to pass and ensures the railway infrastructure's long-term stability. geogrids also improve track stability by preventing lateral movement of ballast material. this reduces track misalignment and keeps the tracks in place, requiring less frequent realignment and maintenance. granular layers reinforced with geogrids can extend the life of railway tracks. they reduce deformation and settlement, which reduces wear and tear on track components such as ties and rails. as a result, railway infrastructure is more long-lasting and cost-effective. the improved performance and longevity of geogrid-reinforced granular layers result in lower maintenance costs. fewer disruptions, fewer track repairs, and lower operational costs over time benefit railway operators. the material and type of geogrid used in the design of reinforced granular layers for railways are critical. the selection process should be guided by factors, e.g., anticipated traffic load, soil and environmental conditions. the effectiveness of geogrid reinforcement is dependent on proper installation. to ensure a stable and uniform track foundation, the granular material and geogrid layers must be carefully placed, and the layer must be compacted. a critical design parameter is the depth at which geogrids are placed within the granular layers. when determining the depth, the load requirements, soil conditions, and desired performance goals must be considered. in railway construction, effective (adequate) drainage is critical. to prevent water accumulation and maintain track stability, geogrid-reinforced granular layers should be designed to work with drainage systems. geotextiles are synthetic or natural fabric materials commonly used inside or below the layers (ballast, supplementary/protection layers) of railway permanent way. they serve many purposes in terms of improving the performance and durability of railway tracks. geotextiles come in both woven and non-woven varieties. individual yarns are woven together to form a fabric to produce woven geotextiles. they have a high tensile strength and are commonly used in railway applications for separation and reinforcement. fibers are randomly arranged and bonded to create non-woven geotextiles. they are frequently used in railway construction for filtration and drainage. geotextiles are designed to withstand the loads and stresses imposed by trains and the surrounding materials. the second most important feature is their permeability. geotextile permeability varies and can be tailored to meet specific drainage and filtration requirements. geotextiles are designed to withstand environmental factors, such as moisture, uv radiation, and chemical exposure. geotextiles are easy to install in various railway applications due to their flexibility and adaptability. evaluation of inner shear resistance of layers from mineral granular materials 7 geotextiles act as a barrier between soil or aggregate layers, preventing intermixing and preserving each layer's integrity. this separation function is critical in preventing ballast and subgrade material contamination. they can improve subgrade soil load-bearing capacity by distributing loads more evenly. this is especially useful in areas with poor or unstable soil. owing to the fact that they allow water to pass through while retaining soil particles, non-woven geotextiles are commonly used for filtration. this helps to keep the drainage system clear and the track structure's hydraulic performance intact. water can flow through geotextiles, lowering the risk of water accumulation and track deformation. they can stabilize embankments and slopes, preventing erosion and preserving the railway alignment's integrity. since geotextiles improve track structure stability, they can reduce maintenance requirements, operational costs, and rail service disruptions. geogrids and geotextiles are used together in the category (vii) above (of course, different types of geosynthetics from the list can also be combined). the general idea is that geocomposites combine the benefits of geosynthetics. the author refers to geotextilecombined geogrid as a geocomposite for the rest of the article. using geosynthetic-reinforced granular layers in railroad construction has both positive and negative environmental consequences. on the plus side, these reinforced layers reduce the need for frequent maintenance and track repairs, potentially causing less disruption to local ecosystems and requiring less energy for maintenance activities. furthermore, the increased life of track service can help reduce material consumption and the environmental impact of track replacement. on the other hand, the environmental impact of geogrid production and disposal must be considered. the production of geogrids necessitates the use of energy and raw materials. the industry emphasizes recycling and using sustainable materials in geogrid manufacturing to counteract these effects. the received reinforcement solutions help strengthen and stabilize granular layers in transport construction when geosynthetic layers are incorporated underneath or within granular layers; these applications are discussed in this paper. several areas are mentioned in the international literature on granular materials reinforced with geosynthetics. in addition, many new findings have been published in the last 30 years. today, the most suitable research methods in this research area are (i) laboratory tests with shear box [39], triaxial tests [40], shear wave measurements [41], large-scale testing facilities [42], other laboratory tests (e.g., pull-out and/or dynamic tests, geogrid mechanical tests); (ii) field testing [33,35]; (iii) computer simulation using finite element method (fem) [43] and discrete element method (dem) [40]. the fem's other application can also be found in engineering studies [44-46]. planar geosynthetics, such as geogrid and geotextile, have become common building materials in railways. geogrids installed directly beneath railway ballast can stabilize the geometry of the railway track, extending the time interval between geometric adjustments significantly. the crushed grains are wedged (wedged) into the geogrid opening and held together by the geogrid's ribs or bands (see fig. 1 [33,34]). these 'fixed' grains connect the grains in and above this plane. this results in the formation of a quasi-strong and relatively slip-free layer. for example, laboratory and field studies have demonstrated geogrids' effectiveness in reducing subsidence (deformation) and ballast degradation [34,47]. placing a geogrid layer in the granular layer can improve the layer structure's inner shear resistance and bearing capacity [48]. however, the mechanism of grain-geogrid interaction is not fully understood. 8 s. fischer fig. 1 the zones of the interlocking effect in a granular material due to a planar geogrid layer (on the basis of [33,34]) 3. materials and methods when choosing a geogrid product, especially in the design phase, an important question is, which parameters are essential, i.e., which parameters significantly influence the interlocking effect? the quality characteristics required by the related en standard [49] are as follows:  tensile strength in the longitudinal and transverse direction, the applied unit is kn/m [50];  longitudinal and transversal elongation (approx. strain) at maximum load the applied unit is % [50,51];  stiffness at 2%, 5%, and 10% (relevant to specific conditions of use);  tensile strength of seams and joints (only relevant for geocomposites, geotextiles related to specific conditions of use);  static puncture resistance (only relevant for geocomposites, geotextiles) [52];  dynamic perforation resistance (only relevant for geocomposites, geotextiles) [53];  durability [49];  another parameter, specifically related to special conditions of use (they can be neglected in the first instance. although the reinforcement function standard specifies the five characteristics that should be included in every geogrid's performance declaration, experience has shown that other relevant parameters should be taken into account when determining a product's effectiveness. according to practical research, the grain structure composition of any granular soil has an optimal aperture size at which the interlocking effect is most significant [8,48]. according to other research, tensile stiffness at low elongations, nodal stiffness, aperture shape, geogrid thickness, fabrication technology, and other factors all influence performance [33,34,54]. the laboratory experiments were carried out as described in [8,48] in the structural testing laboratory of széchenyi istván university (győr, hungary) with 31.5/63 mm "e type" railway ballast (rb) from andesite [55], csbcg 0/56 crushed stone (andesite railway protection layer) [56] made of andesite crushed stone particles, as well as cgm1 [57] granular railway supplementary layer (made from andesite and quartz sand with evaluation of inner shear resistance of layers from mineral granular materials 9 mixing). the csbcg is the crushed stone base layers with continuous grading; csbcg 0/56 is used for the abbreviation (in hungary, the fzka 0/56 abbreviation is also applied). the cgm1 is a granular railway sub-base layer material (cgm stands for coarse grain mixture; in the original german regulation, ril. 836 [58], it is referred to as kg, i.e., "korngemisch"). cgm1 is a quasi-waterproof material and cgm2 is a quasi-waterpermeable layer; in hungary, the szk1 and szk2 [57] abbreviations are applied for cgm1 and cgm2 materials. table 1 contains the relevant details of the executed three test series. table 1 the details of the conducted three test series no. of tests details of the test test #1 the filling material was rb, without geogrid and with five different geogrid types:  geogrid type #1 (gg1);  geogrid type #2 (gg2);  geogrid type #3 (gg3);  geocomposite type #1 (gc1);  geocomposite type #2 (gc2). test #2 the filling material was csbcg 0/56, without geogrid and with four different geogrid types:  gg1, gg2, gg3, gc1. test #3 the filling material was cgm1, without geogrid and with six different geogrid types:  gg1, gg2, gg3, geogrid type #4 (gg4);  gc1, gc2. one test series consisted of the following measurements:  one type of elastic sublayer was taken into consideration (e2=7.2 mpa) [59];  layer construction on elastic sublayer: geogrid/geocomposite + 0.4 m granular material, i.e., rb or csbcg 0/56 or cgm1;  only compacted state was considered, the compaction was applied on the +0.20 m and +0.40 m height planes, the details of the compaction are described later in section 3;  no vertical pressure was applied on the top surface of the layers;  measuring of the inner shear resistance of the layer structure (horizontal pushing force) in four different planes (on the plane of geogrid, i.e., +0.00 m, +0.10 m higher, +0.20 m higher, and +0.30 m higher);  each case was measured three times. the number of the required measurements:  in test #1: 4×6×1×3=72 (i.e., 4=four shearing planes; 6=without geogrid and five geogrid types; 1=compacted filling material; 3=three times repeat);  in test #2: 4×5×1×3=60;  in test #3: 4×7×1×3=84;  altogether: 72+60+84=216. 10 s. fischer in all cases, the three-three measurements were conducted for each set-up to characterize the inner shear resistance of the considered assemblies on all the shearing planes. rb and csbcg 0/56 type protection layer materials were transported from the szob quarry of colas északkő kft., which are in accordance with all the related standards [55-57]. the cgm1 type granular protection layer material was mixed in the laboratory, except for 0/1 mm fraction (sand); every fraction was transported from the same quarry as csbcg 0/56 and rb. except for 0/1 fraction (quartz sand), all the materials were andesite. the grain size distribution curves of measured granular materials were as follows: rb [8], csbcg 0/56 [48], and cgm1 [57]. in the case of cgm1 the quarry precisely set the particle size distribution curves to the mean values between the lower and upper borderlines. the requirements and applied values are shown in tables 2 and 3. geosynthetics were chosen for the test series according to the fact that their short-term tensile strength (considering the reduction factor) would be the same, so they can be compared:  gg1: polypropylene (pp) raw material welded junction geogrid, the aperture size was 44×40 mm – biaxial geogrid that was from stiff, preloaded, extruded pp hanks, in both directions with the same design strength value, junctions were made by computer-aided laser welding.  gc1: the gg1 geogrid with 160 g/m2 mass pp non-woven geotextile.  gg2: pet (polyethylene terephthalate) raw material woven geogrid with pvc (polyvinyl chloride) coat, the aperture size was 35×35 mm – biaxial woven geogrid that were from pet piles with post-made pvc coat.  gc2: the gg2 geogrid combined with pet non-woven geotextile.  gg3: pp raw material extruded geogrid, the aperture size was 39×39 mm – extruded geogrid manufactured by stretching the punched sheet of pp in two orthogonal directions.  gg4: pp raw material extruded geogrid, the aperture size was 34×34 mm – extruded geogrid manufactured by stretching the punched sheet of pp in two orthogonal directions. table 2 particle size distribution requirements for the rb material and the applied values [55] granular materials/sieve size [mm] percentage passing by mass – rb lower boundary line percentage passing by mass – rb upper boundary line applied percentage values [8] 22.400 0.00 3.00 0.52 31.500 1.00 25.00 1.43 40.000 25.00 75.00 33.89 50.000 55.00 90.00 66.34 63.000 95.00 99.00 98.80 80.000 100.00 100.00 100.00 31.500 to 63.000 ≥50.00 >97.00 evaluation of inner shear resistance of layers from mineral granular materials 11 table 3 particle size distribution requirements for the csbcg 0/56 and cgm1 materials standards, and the applied values [55-57] granular materials/ sieve size [mm] percentage passing by mass – csbcg 0/56 lower boundary line percentage passing by mass – csbcg 0/56 upper boundary line applied percentage values – csbcg 0/56 [48] percentage passing by mass – cgm1 lower boundary line percentage passing by mass – cgm1 upper boundary line applied percentage values – cgm1 0.063 0.00 7.00 4.00 0.00 7.00 3.50 0.125 2.00 10.00 6.20 7.00 17.00 12.00 0.250 3.00 14.00 8.40 15.00 31.00 23.00 0.500 5.00 18.00 12.60 23.00 43.00 33.00 1.000 7.00 25.00 16.50 31.00 51.00 41.00 2.000 10.00 32.00 24.20 40.00 60.00 50.00 4.000 15.00 40.00 39.70 50.00 70.00 60.00 8.000 22.00 50.00 49.50 62.00 82.00 72.00 16.000 35.00 65.00 59.10 75.00 92.00 83.50 32.000 55.00 85.00 72.20 86.00 100.00 93.00 56.000 90.00 99.00 93.50 n/a 100.00 n/a 63.000 100.00 100.00 99.70 96.00 100.00 98.00 80.000 100.00 100.00 100.00 100.00 100.00 100.00 the relevant physical and mechanical characteristics of the geogrids and geocomposites can be seen in table 4. table 4 physical and mechanical characteristics of geosynthetics geosynthetic types/characteristics of the geosynthetics gg1 and gc1 gg2 gc2 gg3 gg4 raw material pp pet+pvc pet pp pp production method welded woven woven extruded extruded tensile strength at 1.0% elongation [kn/m] 7.0 n/a n/a n/a n/a tensile strength at 2.0% elongation [kn/m] 12.0 ~8.0 ~8.0 10.5 10.5 aperture size [mm×mm] 44×40 35×35 35×35 39×39 34×34 in addition, the author does not wish to go into more detailed information on the geosynthetics used because the goal of the current study is not to classify individual 12 s. fischer geosynthetics but only to provide a calculation method that could be used to substitute or replace those previously employed, and possibly to provide a means of controlling them. the multi-level shear box test was first published by fischer [8]. the area of the shear box is 1.0×1.0 m, the height is also 1.0 m, and it has a frame that consists of 10 pieces. the frames are made of steel u-profiles and are fixed to each other by m12 screws, except at the plane of shearing. the structure of the box allows for the shearing plane to be set at different depths in the box. the box part under the shearing plane moves on cylindrical rolls on the flooring of the laboratory if there is a horizontal pushing force. on the opposite side of the box, the counter force has to be actuated over the shearing plane. this force can keep the upper part of the box immobile (fig. 2). during the tests, counter and pushing forces are recorded simultaneously. these two forces should be the same because the movement of the box's lower part must be expected. if the difference between the counter and pushing force is more than 10%, the measurement must be repeated. the published results do not contain values with the mentioned error. the test starts at shearing plane no. 4 (top) and follows the planes no. 3, no. 2, and no. 1 (fig. 3). the shearing speed is 1.5 mm/s. the maximal displacement of the frames is approximately 80 mm, and it does not influence the particles' position in the planes below the shearing plane. on both sides of the box (parallel to the shearing direction) in the upper five frames are windows made of plexiglass with 200×60 mm dimensions. through these windows, the possible movement of crushed stone particles can be monitored during shear tests. it can be determined whether there is any particle movement and/or rotation of particles over and/or under the shearing plane. due to the shearing (mainly in the case of shearing in the shearing plane no. 4), the particles raise the upper frame; in this way, a vertical force is needed onto the upper frame. this force does not influence the inner shear resistance because it does not act onto and into the granular material. fig. 2 the measuring principle of the multi-level shear box [8] evaluation of inner shear resistance of layers from mineral granular materials 13 fig. 3 marking of shearing planes [8] before the tests, the friction resistance values between the frames were measured [8]; they were considered negligible values during the evaluation of the test series. the lower part of the box (under the shearing plane) should be filled with elastic material with low load-bearing capacity. material such as thermopan xps plates could be used. the e2 modulus of this layer should be determined with a static load plate test. the elasticity of the elastic supporting layer can be modified if its thickness increases or decreases. the reinforced concrete floor of the laboratory and steel plates used in the bottom of the lower frame of the box do not influence this value because of their approximately infinity elasticity. the low value of load-bearing capacity (e2 modulus 5…15 mpa) can be achieved using thermopan xps sheets with 40…50 cm thickness. the static e2 modulus, which can describe the load-bearing capacity of the support layer, was determined according to the hungarian standard msz 2509-3:1989 [59]. the bearing capacity was calculated according to [59] is e2=7.2 mpa. this means it was a significant weak support. the second layer from the bottom was sand with 10 cm thickness, laid on a geotextile layer. this layer helped the crushed stone particles penetrate into the aperture of geogrid and protects thermopan xps sheets against the sharp edges of crushed granular stone/mixture. one layer of geogrid or geocomposite (geotextile+geogrid) was laid on the top of the sand layer. ensuring quasi-same circumstances in the same measurement series was very important. it meant that in every test series, the compaction level of ballast was the same, but there was no such apparatus with which the density of the applied granular material could be measured in the shear box. in this way, the following things had to be done:  always the same compaction apparatus was used;  always the same number of compaction passes was utilized. the compaction tool's (fig. 4) main parameters were as follows:  mass: 68 kg;  power output: 1.1 kw;  nominal vibrating frequency: 3000/min;  platform: 500×500 mm. 14 s. fischer fig. 4 vibrator compaction tool and compaction processions the granular materials were compacted (fig. 4) in two layers, at 20 cm and 40 cm height. one compaction meant that on each occasion on two traces, there were three processions. one test series consists of altogether four shears: on the shearing planes no. 4, no. 3, no. 2, and no. 1. test series were conducted in the széchenyi istván university's laboratories, where granular protection layers (csbcg 0/56 and cgm1) and railway ballast (rb) were investigated. the constant parameters were the following:  e2 modulus of the support layer under the geogrid: e2=7.2 mpa;  thickness of the sand layer: 10 cm;  one layer geogrid, or one layer geocomposite;  thickness of the granular material in the shear box: 40 cm;  only compacted materials: compaction on two layers: on the +0.20 m and +0.40 m height;  number of shearing planes: 4;  no vertical pressure. the test series enabled the determination of the inner shear resistance curves in function of the distance from +0.00 m height plane. in the current paper, the aim was not to directly determine these functions; however, it was to find the definition of the possible alternative calculation methodologies – i.e., other parameters and/or characteristics of the shear graphs, etc. the instruments used for laboratory experiments are detailed in tables 5-7. the load was applied with a hi-force brand oil-pumped hydraulic work cylinder. the applied sampling frequency was 5 hz, and the shearing speed was set to approximately 1.5 mm/s. table 5 the load gauge instruments applied for the tests designation and manufacturer/type the average value of measurement error load gauge hbm w100k 40 kn – 0.10% 100 kn – 0.15% 200 kn – 0.20% 400 kn – 0.30 % evaluation of inner shear resistance of layers from mineral granular materials 15 table 6 the lvdts (linear variable differential transformer) applied for the tests manufacturer type serial no. measuring range output signal sensitivity zero shift resistance hbm w20tk 6950 ±20 mm 80 mv/v ±1.0% <±0.05% 10 mh; 56 ω hbm w20tk 6951 ±20 mm 80 mv/v ±1.0% <±0.05% 10 mh; 56 ω table 7 the applied test instruments instrument (type, ids, etc.) special characteristics amplifier: hottinger spider8, 8-channel measuring amplifier accuracy: 0.1% module: sr55 accuracy: 0.1% software: catmodul 5.0 computer: compaq armada 100s the details of the evaluation methodology are summarized in the following list:  61 different function qualification parameters were defined and calculated individually for all measured shear graphs (see fig. 5);  the 61 parameters (characteristics) were defined in such a way that the exact points of the shear functions could be clearly and easily identified in the subsequent analysis;  for each parameter, the mean, the standard deviation (sd), and the relative standard deviation (rsd) values for each shearing plane were calculated;  rsd: 10%, 20% and 30% were considered;  reference materials were: rb, csbcg 0/56, cgm1;  four types of reference rating parameters were: #1, #21, #51 and #61 from the next list (i.e., table 8);  all four shearing planes were calculated individually;  these could be calculated by defining ratios (the case of the reference material without geogrid reinforcement was 1.0 for the defined reference function parameter);  1%, 2%, and 5% allowed deviations from the reference value (i.e., from the specified ratio values, to be more precise), only the given pair of values was compared and maximized the absolute value of the difference between them for this calculation (i.e., it was not considered which value is greater when formulating and applying this condition);  only cases in which the specified conditions were met for all cases (i.e., for all 17 other measurement series in addition to the specified reference) were considered suitable results, this was necessary because the 18 different granular materialgeosynthetic pairs were relatively few for deriving global, completely general results, and only if all of them were met it could be claimed that the results obtained were suitable and acceptable. fig. 5 illustrates a typical shear curve (function) measured in the laboratory. 16 s. fischer fig. 5 a typical shear graph (curve) with the three repetitions of the measurement (rb without geosynthetic reinforcement, shearing plane #2, i.e., 10 cm above the geosynthetics' theoretical plane) in all cases, the curves were recorded up to a minimum shear length (displacement) of 80 mm, the length that the multi-level shear box used and the design of the measuring system (i.e., mainly the hydraulic cylinder) allows. the 61 qualification parameters are detailed in table 8. all the parameters can be identified in fig. 5. it is worth mentioning that all the calculations were conducted by ms excel. in the case of linear regression functions, only the tangent values were taken into consideration; however, the acceptable coefficient of determination value (r2) was reduced to 0.7. in calculating integral values, numerical integration was applied, which meant that the rule and methodology of area determination by trapezoid calculation was used (a more complicated method was not needed because all the coordinates were known for every point). the tangent of the chord (chord modulus) was computed by simply dividing. 4. results as mentioned before, the calculations were carried out from the analysis of 216 shear graphs and 61 different qualitative parameters calculated for each of them, using as reference the three granular materials and the four different reference grading parameters. the calculations were carried out in ms excel. for clarity of results, the calculations were performed systematically on separate spreadsheets and in several excel files to avoid possible calculation errors and mix-ups. all the calculations, including setting up excel worksheets, controlling them, and assessing the results, took about three working weeks. evaluation of inner shear resistance of layers from mineral granular materials 17 table 8 the details of the considered qualification parameters no. of qualif. param. meanings of qualification parameters #1 global maximum value interpreted on the full shear graph, in kn unit. #2 the calculated chord modulus (i.e., the tangent of the inclined straight line drawn between the force value for the shear length, or in other words, displacement) in kn/mm unit over the shear range 0-10 mm. #3 the tangent of the linear regression function fitted to the measured values of the shear range (displacement), 0-10 mm, in kn/mm unit. #4 the area under the shear function (integral) calculated on the measurement graph of the shear range (displacement) 0-10 mm, in kn×mm unit. #5 – #46 see points #2 – #4 above, except when they are related to the shear ranges as follows:  0-20 mm (#5 – #7); 0-30 mm (#8 – #10); 5-10 mm (#11 – #13); 5-20 mm (#14 – #16); 5-30 mm (#17 – #19); 5-40 mm (#20 – #22); 1020 mm (#23 – #25); 10-30 mm (#26 – #28); 0-40 mm (#29 – #31); 1530 mm (#32 – #34); 15-40 mm (#35 – #37); 20-30 mm (#38 – #40); 2040 mm (#41 – #43); 25-40 mm (#44 – #46). the orders in the above rows themselves are the same as for points #2 – #4. #47 the average force value calculated on the measurement graph of the 40-60 mm shear range, in kn unit. #48 see point #4, except when it is related to the 40-60 mm shear range. #49 – #60 see points #47 – #48 above, except when they are related to the shear ranges as follows:  40-70 mm (#49 – #50); 40-80 mm (#51 – #52); 45-80 mm (#53 – #54); 50-80 mm (#55 – #56); 55-80 mm (#57 – #58); 60-80 mm (#59 – #60). the orders in the above rows themselves are the same as for points #2 – #4. #61 see point #47, except when it is related to the entire shear range 0-80 mm. it should be noted that only those results were accepted for which all (i.e., relative to the reference) the 17 measurement series gave a satisfactory result. as a partial result, those cases where acceptable results were obtained for each material (rb, csbcg 0/56, and cgm1) individually could be used subsequently. however, these were not published in this paper because they are of considerable length and it is challenging to present them in a systematic way. the main results can be summarized in the following. tables 9 and 10 contain the cases and the parameters in which the first evaluation process gave an appropriate match. ref. rb_ref. 40-80 mm means that the reference granular material is the considered railway ballast (rb) and the calculated qualification parameter is #51; see table 8. ref. rb_ref. max. means that the reference granular material is considered railway ballast (rb) and the calculated qualification parameter is #1; see table 8). in i_rsdj, i refers to the shearing planes (i.e., 0 means shearing plane no. 1, and 30 means shearing plane no. 4, respectively), where the vertical distance of the planar geosynthetic layer and the shearing plane is 0-30 cm; j is the allowed maximum relative standard deviation – when calculating the three shear graphs' parameters – which differs between 10% and 30%. 18 s. fischer table 9 the results related to ref. rb_ref. 40-80 mm in the first evaluation process cases qualification parameters 0_rsd20 #1 with 5% accuracy; #47 with 5% accuracy; #49 with 2% and 5% accuracy; #50 with 5% accuracy; #52 with 5% accuracy; #53 with 1%, 2%, and 5% accuracy; #54 with 5% accuracy; #55 with 2% and 5% accuracy; #56 with 5% accuracy; #57 with 5% accuracy; #58 with 5% accuracy; #59 with 5% accuracy; #60 with 5% accuracy. 10_rsd20 #47 with 5% accuracy; #49 with 5% accuracy; #50 with 5% accuracy; #52 with 5% accuracy; #53 with 2% and 5% accuracy; #54 with 5% accuracy; #55 with 5% accuracy; #57 with 5% accuracy; #59 with 5% accuracy. 0_rsd30 all the parameters mentioned in 0_rsd20 and #58 with 5% accuracy. 10_rsd30 all the parameters mentioned in 10_rsd20. table 10 the results related to ref. rb_ref. max. in the first evaluation process cases qualification parameters 0_rsd20 #51 with 5% accuracy; #53 with 5% accuracy; #55 with 5% accuracy; #56 with 5% accuracy; #57 with 5% accuracy; #59 with 5% accuracy. 0_rsd30 all the parameters mentioned in 0_rsd30. a function fit test was carried out, which meant that:  the parameters were sorted in an increasing numerical order by reference to the above grouping, separately, i.e., ref. rb_ref. 40-80 mm (0_rsd20, 10_rsd20, 0_rsd30, 10_rsd30) and ref. rb, ref. max. (0_rsd20 and 0_rsd30);  values where any element of the data series showed an incorrect reinforcement effect (ratio to the reference values greater than 1.0), while for the reference, the ratio was less than 1.0 (i.e., a weakening effect), and for the opposite, the parameter was considered inadequate;  then a detailed mathematical, statistical fit test was performed on the remaining parameters. the above gives only one suitable parameter, as follows (ref. rb_ref. 40-80 mm; 0_rsd20, as well as 0_rsd30):  #52 with 5% accuracy. for comparison, the difference values between the values of the unique data points could be fulfilled considering the arrangement of the 18 values in a row. the differences calculated for the two data sets compared to the reference were as follows (ref. rb_ref. 40-80 mm, 0_rsd20):  in the case of #52 with 5% accuracy, the maximum difference values are +0.612% and –0.031%; the details can be seen in fig. 6. these obtained differences regarding granular materials were appropriate and acceptable because of the inhomogeneity of granular materials and the relative uncertainty of the distribution of particles within a given volume. this meant it was impossible to create a design with perfectly and precisely the same (exact) location and position within the pile twice. for this reason, a maximum relative variance of 30% and a maximum deviation of 5% were considered in the calculations. evaluation of inner shear resistance of layers from mineral granular materials 19 fig. 6 evaluation of the calculation by qualification parameter #52 in the case of ref. rb_ref. 40-80 mm and 0_rsd20 it was irrelevant considering the current paper's topic; however, the author decided to publish the meaning of the ranking in fig. 6. the data are illustrated in table 11. the first column contains the ranking based on fig. 6, while the second column is related to the original identification number of the measurements. table 11 meanings of the rankings in fig. 6 ranking based on fig. 6 the original id of the measurement meaning 1 12 cgm1 without geosynthetic reinforcement 2 10 csbcg 0/56 with gc1 reinforcement 3 6 rb with gc1 reinforcement 4 7 csbcg 0/56 without geosynthetic reinforcement 5 4 rb with gg2 reinforcement 6 18 cgm1 with gc2 reinforcement 7 3 rb with gg3 reinforcement 8 13 cgm1 with gg2 reinforcement 9 5 rb with gc2 reinforcement 10 9 csbcg 0/56 with gg1 reinforcement 11 2 rb with gg1 reinforcement 12 17 cgm1 with gc1 reinforcement 13 11 csbcg 0/56 with gg2 reinforcement 14 16 cgm1 with gg1 reinforcement 15 1 rb without geosynthetic reinforcement 16 15 cgm1 with gg4 reinforcement 17 8 csbcg 0/56 with gg3 reinforcement 18 14 cgm1 with gg3 reinforcement 20 s. fischer 5. discussion based on the results written in section 4, it should be pointed out that the results presented here are valid for the following limitations and should be considered together with them:  to answer the question of whether the inner shear resistance qualification values averaged over a 40-80 mm section given in the ph.d. thesis (see the qualitative parameter #51 above), it can be determined by a generally applicable method using a different function characteristic in such a way that the same reinforcingweakening ratios were obtained as if determined by the method developed in 2012;  the habilitation thesis in 2017 and several related papers [60] should be mentioned in which the tangent of the linear regression function fitted to the measured values of the shear range 5-40 mm, in kn/mm unit (see the qualitative parameter #21 above), noting that here the shearing plane no. 1 was considered; it was demonstrated through the detailed statistical analysis performed in this paper that the use of this parameter was not universal and cannot be accepted in all circumstances;  it was also essential to be able to determine and eliminate which granular material was measured without a geosynthetic reinforcement, i.e., which one was chosen as the reference material;  it was vital to highlight the following facts: three types of material, six types of geosynthetics (geogrid only and geocomposite) produced by three different production technologies were considered; this means that, although it was not possible to make a completely general statement for all granular materials and for all types of aperture sizes and materials, it should be noted that the other cases were likely to deteriorate the results and it was therefore necessary to verify the applicability of the method;  the presented results do not imply that a parameter other than #52 cannot be used as a rating value for inner shear resistance, but it should be kept in mind that if a different rating parameter/characteristic was chosen, the results would not be comparable with the results from the qualification with #52;  the results may vary depending on the sampling frequency and shear rate; therefore, the findings are mainly valid for the values presented in the article – a detailed analysis would need to be performed to disentangle the results from these characteristics/settings. the results presented in section 4 clearly show the following facts:  only the rb material can be chosen as a reference, and only in the "0-plane" (shearing plane no. 1) will the results be correct;  only acceptable results are obtained for relative standard deviations for 20% and 30%. it should be mentioned that in the present study, the aperture size was in a relatively small range (34-44 mm), so the effect of aperture size could not be accounted for. many geogrid types are available in the author's measurement database, so a more detailed study will be possible in the future. it should be noted that when considering a significantly larger number of samples, the need to define some acceptable maximum variance in the analyses arises. the author would like to point out here that for each calculation method, the evaluation of inner shear resistance of layers from mineral granular materials 21 permissible and acceptable calculation and prediction error (bias) to be taken into account should be specified. 6. conclusions the research focuses on the mechanics of granular materials used in transport infrastructure facilities (railways, roads, etc.), specifically the issue of load-bearing capacity for railway structures. it has been certified that achieving required load-bearing capacities with thinner granular supplementary layers is a matter of national economic importance. the study shows that geogrid reinforcements, when used in combination with granular layers in railway construction, can provide adequate stabilization. this is especially useful in low-load-bearing capacity subgrades. the study also discusses the practical utility of the interlocking effect between geogrids and granular layers in creating a "composite layer" that provides lateral support and reduces vertical load deformation. the paper assesses the use of geosynthetics in this context, specifically geogrids and geotextiles. the use of these materials in the construction of special geotechnical structures (e.g., reinforced retaining walls and other ground reinforcement structures) is emphasized. the paper describes the methodologies used in the laboratory to assess the inner shear resistance of granular layers from minerals applied for transport infrastructure facilities with and without planar geosynthetic reinforcement materials (i.e., planar reinforcing layers). a multi-level shear box was used without considering vertical loads for the measurements. in the literature and engineering practice, an accepted calculation method for determining inner shear resistance exists for the shear force with linear shearing speed, primarily after a peak force (resistance) value. in the present case, this can be accounted for by calculating the average force value for the 40-80 mm shear range using the author's ph.d. thesis from 2012. the article details each possible additional method and compares it with mathematicalstatistical methods. three granular materials (rb, csbcg 0/56, and cgm1), as well as, six planar geosynthetics (gg1, gg2. gg3, gg4, gc1, gc2) were studied. for this purpose, the results of 216 measurements were conducted and processed using 61 different shear function-qualification parameters. the calculations were performed using a simplified function fitting test and a selection process to maximize the allowable relative standard deviations. all three types of materials and four classification parameters were chosen as references for comparability. as a result, only one alternative parameter could be used to determine the reinforcing-weakening values with a maximum deviation of 5% while not producing insufficient results in the placement (order) of the individual granular material and geosynthetic pairings. this parameter was the area under the function (integral) calculated on the measurement graph of the 40-80 mm shear range, in kn×mm units; it gives correct values only if the reference granular material is the considered railway ballast, the shearing plane is the geosynthetic's plane (i.e., "0-plane"), and the reference qualification parameter is the original recommended parameter from 2012. 20% and 30% were the best relative standard deviations to consider. the study suggests using parameter #52 (i.e., the area under the shear function [integral] calculated on the measurement graph of the shear range [displacement] 40-80 mm, in 22 s. fischer kn×mm unit) to evaluate the inner shear resistance of granular materials when the reference granular material is the examined rb. the study discovered potential limitations in replacing the 2012 evaluation option [8] with other parameters, emphasizing the importance of testing the applicability of various methods. the presented results do not imply that a parameter other than #52 cannot be used as a rating value for inner shear resistance, but it should be kept in mind that if a different rating parameter/characteristic is chosen, the results are not comparable with the results from the qualification with #52, it should be noted that as a partial result, those cases where acceptable results were obtained for each material (rb, csbcg 0/56, and cgm1) individually could be used subsequently. however, these were not published in this paper because they are of considerable length, and it is challenging to present them systematically. it can be a future research possibility. further research with multi-level shear box and larger equipment may help to provide a more complete analysis of geosynthetic reinforced and unreinforced granular materials:  the application of used railway ballast material (i.e., mainly from crushed stone) that is not exclusively sharp-edged; 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engineering vol. 19, no 1, 2021, pp. 133 153 https://doi.org/10.22190/fume200920009f © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times in the context of atomic force miscroscopy maximilian forstenhäusler1, enrique a. lópez-guerra1,2, santiago d. solares1 1the george washington university, department of mechanical and aerospace engineering, washington dc, usa 2park systems inc., santa clara, ca, usa abstract. we provide guidelines for modeling linear viscoelastic materials containing an arbitrary number of characteristic times, under atomic force microscopy (afm) characterization. instructions are provided to set up the governing equations that rule the deformation of the material by the afm tip. procedures are described in detail in the spirit of providing a simple handbook, which is accompanied by open-access code and workbook (excel) sheets. these guidelines seek to complement the existing literature and reach out to a larger audience in the awareness of the interdisciplinary nature of science. examples are given in the context of force-distance curves characterization within afm, but they can be easily extrapolated to other types of contact characterization techniques at different length scales. despite the simplified approach of this document, the algorithms described herein are built upon rigorous classical linear viscoelastic theory. key words: modeling viscoelasticity, generalized maxwell model, generalized kelvin-voigt model, multiple characteristic times, numerical simulation, atomic force microscopy received september 20, 2020 / accepted december 22, 2020 corresponding author: santiago d. solares the george washington university, department of mechanical and aerospace engineering, 800 22nd street nw, suite 3000 washington, dc 20052, usa e-mail:ssolares@gwu.edu mailto:ssolares@gwu.edu 134 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares 1. introduction viscoelastic materials are those that can simultaneously store and dissipate energy when deformed [1, 2]. this occurs because they have structural mechanisms to relax some of the stresses that build up when deformed [3, 4]. these relaxations occur at different characteristic times which are often referred to as relaxation times [2, 5]. viscoelastic materials are very common in nature and diverse scientific fields focus on them. for example, biofilms are known to be viscoelastic and knowledge of their mechanical properties is crucial to understanding how they disseminate and how they could be eradicated [6-9]. similarly, human cells are viscoelastic and knowledge of their mechanical properties can, for example, help to discriminate cancerous cells from healthy ones in the early stages of the disease [10-12]. in the field of engineering, polymeric fuel cells and organic solar cells have viscoelastic components whose mechanical response in the operation of the devices is known to be ultimately linked to their performance [13-15]. all of these applications can benefit from accurate material modeling approaches to complement the experimental procedures and thus ensure reliable characterization. unfortunately, due to mathematical complexity, the viscoelastic nature of such materials is often overlooked or oversimplified in scientific studies. recent investigations [16-26] have explored more rigorous approaches to take into consideration the intricacies of real viscoelastic behaviors by combining the classical theory of viscoelasticity [1, 2, 27-29] with modern characterization techniques, such as atomic force microscopy (afm) [10, 30-39]. these efforts are especially important to close the gap between the thorough and rigorous mechanical modeling approach of early theories and the interests of scientists that deal with modern techniques. despite these efforts, we believe that there is still a long way ahead to make this viscoelastic modeling truly accessible to a broader scientific community, regardless of their mathematical background. therefore, in this study we focus on a more ‘hands-on approach’ where detailed straightforward descriptions to implement viscoelastic modeling are provided. the practical aspects of the modeling implementations are prioritized over mathematical rigor, such that only a very basic knowledge of ordinary differential equations is required to follow our presentation. the reader will be walked through the algorithms of how to solve numerically the constitutive equations that govern the deformation of viscoelastic materials, for the case of an indenter probe that deforms a viscoelastic material with multiple characteristic times. this is of special interest for afm and for nanoand micro-indentation techniques, although with few adaptations the routines can also be used for other types of characterization techniques. for the convenience of the reader, the algorithm descriptions are accompanied by an excel spreadsheet available online that contains examples with the numerical calculation methods [40]. further, users with some background in programming will benefit from the numerical implementations given in python code in an open-access repository [40]. the repository provided alongside this paper contains two libraries, afmsim and viscoelasticity, as well was implementations and examples of the functions available in these libraries. the contact-mode implementation of afm will be discussed in section 3. guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...135 2. guidelines for the simulation of linear viscoelastic behavior 2.1. model selection: generalized maxwell model & generalized kelvin-voigt model in this practical guide, generalized models are used to simulate the response of real viscoelastic materials. this provides the necessary flexibility as real viscoelastic materials have multiple characteristic times at which they accommodate and relax stresses when deformed. specifically, this document will be referring to the generalized maxwell model and the generalized kelvin-voigt model which are shown in fig. 1. these two models are mechanical analogs [2] which means that they have the same mechanical response when the appropriate equivalent parameters are chosen. the selection of one over the other obeys pure algebraic convenience, depending on whether force/stress or deformation/strain is regarded as the input. for example, if the user wishes to perform a simulation where the deformation/strain history is given or calculated ‘a priori,’ then the generalized maxwell model is the most convenient. on the other hand, if the force/stress history is known or given, then the generalized voigt model is the most convenient. fig. 1 a) generalized maxwell model and b) generalized kelvin-voigt model; mechanical model diagrams representing the linear viscoelastic relationship between stress and strain in the complex plane with multiple (n) characteristic times when strain or stress is regarded as the excitation, respectively. in a) the laplace transformed strain, (s), is regarded as the excitation and the laplace transformed stress, (s), as the response; in (b) the opposite occurs. jn and gn refer to the compliance and modulus of the nth spring, respectively. jg and ge refer to the glassy compliance and rubbery modulus, respectively. n and n refer to the fluidity and viscosity of the nth dashpot (damper), respectively. regardless of the chosen model, one must be aware that the model is only a visual representation of the material behavior described by ordinary differential equations relating the stress and strain. these are the governing equations that describe the linear viscoelastic behavior, and their most general form is as follows [20]: ∑ 𝑢𝑛 𝑑𝑛𝜎(𝑡) 𝑑𝑡𝑛 ∞ 𝑛=0 = ∑ 𝑞𝑚 𝑑𝑚𝜀(𝑡) 𝑑𝑡𝑚 ∞ 𝑛=0 (1) 136 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares where un and qm are differential coefficients, and (t)and (t)are stress and strain, respectively. this equation is model independent, where the nth and mth time derivatives are acting on the stress and strain tensors, respectively. for mathematical convenience, the differential equation can be transformed into an algebraic equation by applying the laplace transform. doing so will transform eq. (1) to, �̅�(𝑠)𝜎(𝑠) = �̅�(𝑠)𝜀(̅𝑠) (2) where �̅�(𝑠) = ∑ 𝑢𝑛𝑠𝑛 𝑛 , �̅�(𝑠) = ∑ 𝑞𝑚𝑠𝑚 𝑚 and 𝜎(𝑠), 𝜀(̅𝑠) are the transformed stress and strain, respectively. here, it is assumed zero initial conditions which should not affect generality as they can be incorporated when necessary [2]. note that �̅�(𝑠) and �̅�(𝑠) are now simply polynomials in the complex variable ‘s’. the interested reader can learn more about the laplace transform by consulting appropriate references [41-43], although this is not a strict requirement to understand the practical implementation of the algorithms described in this manuscript. we can rearrange eq. (2) in two ways, depending on whether stress or strain is regarded as the input. for example, when regarding strain as the input, eq. (2) is rearranged in the following manner: 𝜎(𝑠) = �̅�(𝑠)𝜀(̅𝑠) (3) where �̅�(𝑠) = �̅�(𝑠) 𝑢(𝑠) is the relaxance, which is the mathematical transform carrying the viscoelastic information that will rule the stress response to a given strain input. on the other hand, if the stress is regarded as the input, eq. (2) can be rearranged as: 𝜀(̅𝑠) = 𝑈(𝑠)𝜎(𝑠) (4) where the retardance, 𝑈(𝑠) = 𝑢(𝑠) �̅�(𝑠) , is introduced. here, the material transform 𝑈(𝑠), containing the mechanical information of a given material, governs the strain response when an excitation stress is imposed. as can be observed in eq. (3) and eq. (4), the retardance is the inverse of the relaxance: �̅�(𝑠) = 1/𝑈(𝑠) (5) in the case of the generalized models in fig. 1, the material transforms are ratios of polynomials in the complex variable ‘s’. these polynomial expressions, �̅�(𝑠) and �̅�(𝑠), can be expressed in terms of the spring and dashpot (damper) values in the generalized models in fig. 1. the process to find them is analogous to formulating mesh equations in electric circuit theory [2, 41]. an example for this calculation is given later in section 2.3. for now, the derivation is skipped and the relaxance and retardance for the generalized maxwell and kelvin-voigt models are given here: �̅�(𝑠) = 𝐺𝑒 + ∑ 𝐺𝑛𝜏𝑛𝑠 1+𝜏𝑛𝑠 𝑁 𝑛=0 (6) 𝑈(𝑠) = 𝐽𝑔 + ∑ 𝐽𝑛 1+𝜏𝑛𝑠 𝑁 𝑛=0 (7) where ge is the equilibrium (rubbery) modulus, which describes the elastic response of the material at long timescales, and jg is the classy compliance, which describes the elastic response of the material at short timescales [2]. both quantities are visually guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...137 defined in fig. 1. n corresponds to the total number of characteristic times in the material. n refers to the nth characteristic time. for the generalized kelvin-voigt model, 𝜏𝑛 = 𝐽𝑛/𝜙𝑛 is the retardation time related to the nth voigt unit, given by the ratio of the compliance of the n-th spring over the fluidity of the nth dashpot. for the generalized maxwell model, n = n / gn is the ratio of the viscosity of the nth dashpot over the modulus of the nth spring. parameters of this type can be found in the literature for certain materials [17, 44, 45]. the next section describes how to set up the constitutive equation to be solved in our viscoelastic simulation. here, it will become evident the importance of the mathematical concepts that were just presented in this section. 2.2. defining the governing equation for the specific case of a tip indenting a surface, a customized equation resembling eq. (1), including geometrical aspects of the physical problem under consideration, is needed. this is because there needs to be a relationship between force and indentation (instead of a relationship between stress and strain as in eq. (1)), since the afm instrument measures forces and distances, not stresses. to construct this relationship, geometrical aspects involving a boundary value problem need to be considered for which the correspondence principle may be conveniently invoked. it states that if the elastic solution for a contact problem is known, the viscoelastic solution can be directly formulated as they are analogous in the laplace domain [46]. the mathematical details are beyond the scope of this manuscript, but curious readers are directed to the literature that discusses its validity [27, 29, 46, 47]. for the case of a spherical indenter with a radius of curvature r, penetrating a viscoelastic half-space (e.g., an afm tip penetrating a flat viscoelastic surface), the relationship between force fts(t) and indentation h(t) is [27]: ∑ 𝑞𝑚 𝑑𝑚[{ℎ(𝑡)}3/2] 𝑑𝑡𝑚 ∞ 𝑚=0 = 3 16√𝑅 ∑ 𝑢𝑛 𝑑𝑛𝐹𝑡𝑠(𝑡) 𝑑𝑡𝑛 ∞ 𝑛=0 (8) notice that this equation is very similar to eq. (1), but the stress has been substituted with the force, and the strain has been substituted with the indentation. furthermore, to account for the geometry of the system, there is a coefficient on the right-hand side that depends on the radius of curvature of the indenter, and the derivatives on the left-hand side of the equation are taken on the indentation raised to the power 3/2. eq. (8) assumes that the material is incompressible with a time-independent poisson’s ratio  = 0.5) [27, 44]. with a finite number of n characteristic times, this equation can also be expressed as: 𝛼 [𝑞0𝑝 + 𝑞1 𝑑𝑝 dt + 𝑞2 𝑑2𝑝 𝑑𝑡2 +. . . +𝑞𝑁 𝑑𝑁𝑝 𝑑𝑡𝑁] = 𝑢0𝐹+𝑢1 d𝐹 dt +𝑢2 𝑑2𝐹 𝑑𝑡2 +. . . +𝑢𝑁 𝑑𝑁𝐹 𝑑𝑡𝑁 (9) where  = 16√𝑅 3 , and p = h(t)3/2. eq. (9) is our target governing equation that describes the force-indentation relationship of a parabolic tip penetrating a viscoelastic surface. the differential coefficients (u0, u1…, un, q0, q1…, qn) will depend on the numerical values of the spring moduli and dashpot coefficients in the chosen model (fig. 1). the next two sections will cover the process of finding these coefficients. once this is done, the governing equation is completely set up to be solved numerically. fig. 2 lays out the general process of viscoelastic modeling that we have discussed so far, as well as the remaining steps detailed in next sections. 138 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares as a final note, the correspondence principle is formally invalidated for the retract portion [28, 47], i.e., when the afm tip is retracting from the surface but still in contact with it. we observed that for the timescales associated with afm experiments disregarding this fact generally introduces only very small inaccuracies, so we will omit it within this manuscript. to include it, more complex contact-mechanics schemes need to be considered [28, 48], as it has been done on some recent afm studies [16, 21]. fig. 2 summary of the main steps involved in the modeling of a physical problem involving the deformation of a viscoelastic material. these steps are detailed through sections 2.1 to 2.5 of this manuscript. guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...139 2.3. obtaining the material transform for the viscoelastic model chosen in the last section we targeted the governing equation for the contact mechanics problem of a parabolic tip penetrating a viscoelastic sample, which can be solved with eq. (9). for simplicity in our illustration of the method, we will now assume that the viscoelastic model chosen has only one characteristic time: the standard linear solid (sls) model. also, it will be assumed that deformation can be regarded as input, such that the maxwell-sls model will be the most convenient model to use. specifically, the maxwell-sls model is a three-parameter model consisting of one spring arm with the equilibrium modulus ge in parallel with one maxwell arm (an arm with a spring and a dashpot in series). the maxwell-sls model can be visualized by truncating fig. 1a to contain only one maxwell arm (that is, setting n = 1). after selecting an adequate model (e.g., maxwell-sls) and selecting the appropriate constitutive equation (e.g., eq. (9)) describing the force/deformation relationship, the next step is to obtain the material transform for the selected viscoelastic model. the concept of material transform was introduced in section 2.1. as discussed there, obtaining the material transform expression for a given model involves an analog process to formulating mesh equations in electric circuit theory. details of this process can be found in the literature [2, 41]. here, for simplicity, general rules are given as a recipe, without discussing the mathematical or physical justification. these rules for obtaining the material transform are as follows: 1) relaxances are added up in parallel. for example, for the maxwell-sls model, to obtain the overall relaxance of the model we need to add up the individual relaxances of the individual elements. in this case, we need to add up the relaxance of the isolated spring (ge) with the relaxance of the maxwell arm that contains spring g1 and dashpot1. however, before we can perform this addition, we must first determine the relaxance of the maxwell arm itself, for which the 2nd rule below will be needed. 2) retardances are added up in series. for example, to obtain the relaxance of the maxwell arm (a spring in series with a dashpot), we need to add up the retardance of the spring (1/g1) with the retardance of the dashpot (1/1s). recall from section 2.1 that relaxances and retardances hold an inverse relationship. thus, for the maxwell arm the retardance is 1/1s + 1/g1 and its relaxance is the inverse of this expression, equal to 1 1/η1s+1/g1 . thus, according to the above two rules, for the example of the maxwell-sls model, the total relaxance is the addition of the isolated spring’s relaxance (ge) with the maxwell arm’s relaxance ( 1 1/η1s+1/g1 , which yields 𝐺𝑒 + 1 1/η1s+1/g1 . after some algebraic arrangements and with the substitution n = 1/g1this expression can be transformed into: 𝑄𝑆𝐿𝑆 ̅̅ ̅̅ ̅̅ (𝑠) = 𝐺𝑒 + 𝐺1𝜏1𝑠 1+𝜏1𝑠 (10) where the characteristic time 1is the ratio of dashpot coefficient to the modulus of the spring in the maxwell arm. for the generalization of a maxwell model with an arbitrary number of characteristic times (fig. 1a), the relaxances of the rest of the maxwell arms should be added. it can be easily inferred that following this process would lead us to the generalized expression in eq. (6), which governs the response of the generalized model in fig. 1a. 140 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares 2.4 calculating the coefficients for the governing equation once we have calculated the material transform (either �̅�(𝑠) or �̅�(𝑠), depending on the chosen viscoelastic model), the next step is to find the differential coefficients (u0, u1…, un, q0, q1…, qn) from this material transform. to do that, we need to perform algebraic manipulation of the material transforms �̅�(𝑠) and 𝑈(𝑠) such that they can be expressed as a ratio of polynomials in the complex variable ‘s’. for illustrative purposes we continue with the maxwell-sls model example and assign the following arbitrary values for the parameters: ge = 1.0x106 pa, g1 = 1.0x108 pa, 1 = 1.0x10-2 s. the first step is to perform the algebraic addition in eq. (10): 𝑄𝑆𝐿𝑆 ̅̅ ̅̅ ̅̅ (𝑠) = 𝐺𝑒(1+𝜏1𝑠)+𝐺1𝜏1𝑠 1+𝜏1𝑠 (11) then, we expand all terms in the numerator and denominator (the denominator will have more than one factor when the model has more than one characteristic time) and gather all the terms with common ‘s’ exponent: 𝑄𝑆𝐿𝑆 ̅̅ ̅̅ ̅̅ (𝑠) = 𝐺𝑒+(𝐺𝑒+𝐺1)𝜏1𝑠 1+𝜏1𝑠 (12) all the terms that are multiplied by s0 in the numerator are combined and assigned to q0. similarly, all the terms that are multiplied by s0 in the denominator are combined and assigned to u0. then, all terms multiplied by s1 in the numerator are combined and assigned to q1, while all terms multiplied by s1 in the denominator are combined and assigned to u1, and so on. then, it follows that all terms multiplied by sn are grouped and assigned to qn in the numerator and to un in the denominator. thus, for our specific example we have for the numerator: 𝑞0 = 𝐺𝑒 = 1.0 × 106 𝑃𝑎 (13) 𝑞1 = (𝐺𝑒 + 𝐺1)𝜏1 = 1.01 × 106 𝑃𝑎 𝑠 (14) and for the denominator we have: 𝑢0 = 1 (15) 𝑢1 = 𝜏1 = 1.0 × 10−2s (16) once we have calculated all these differential coefficients, we can plug them into eq. (9), which concludes the process of explicitly setting up the governing equation. the rest of the manuscript will now focus on explaining the steps to solve this equation numerically (i.e., through computational simulation). as a note of caution, it is worth mentioning that the algebra used to find the differential coefficients becomes quite involved when a large number of characteristic times is chosen. for convenience, the reader is invited to explore the jupyter notebook “calculation of the coefficients, (u0, u1…, un, q0, q1…, qn), for the differential equation for a 3 arm gen. maxwell, kelvin-voigt model” in an open-access github repository [40]. this notebook conveniently performs the algebra by means of the sympy library [49]. guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...141 2.5. numerical solution of the governing equation in the next two subsections, we describe the algorithms to solve the governing equation (eq. (9)) numerically. the solution is divided into two sections: the first one relates to the portion where tip and sample are in contact during the probe approach and the initial portion of the tip’s retract trajectory, before losing tip-sample contact. the second portion corresponds to the recovery of the surface after tip-sample contact is lost. in this second portion the surface remains temporarily depressed after the tip retracts, followed by surface recovery in a stress-free fashion (the rebound portion [50]). 2.5.1. numerical solution for the indentation into a viscoelastic material: contact portion of the force spectroscopy curve for the numerical procedure of eq. (9), the left-hand-side (lhs) and right-hand-side (rhs) of eq. (9) are separated according to the variable that one wishes to solve for (either force or indentation). in the case of afm simulations, the known input parameter for eq. (9) at a given time step is the surface indentation. this is because solution of the afm tip’s equation of motion (e.g., the harmonic oscillator model [30]) yields the trajectory of the tip, from which we can calculate the indentation (details of this will be provided in section 3. thus, the input for the viscoelastic governing equation (eq. (9)) is the indentation, and the output is the force (the tip-sample force). the first step in solving numerically eq. (9) is to calculate all higher order derivatives on the lhs using the indentation as the initial input (calculated from the afm tip trajectory). specifically, the tip position at the ith time step, hi, provides the information needed to calculate the zero-order derivative on the lhs of the equation, namely the term p = h(t)3/2. however, since we know that depression of the surface will lead a positive (upward) force exerted by the sample on the afm tip, we must change the sign of hi when calculating the zero-order derivative of the deformation term: 𝑝𝑖 = (−ℎ𝑖)3/2 (17) we can now calculate the rest of the derivatives on the lhs of eq. (9) using the finite difference method [42, 43]. for example, for the first order derivative ( 𝑑𝑝 dt = �̇�𝑖) the numerical approximation using backward difference [42] becomes: �̇�𝑖 = 𝑝𝑖 − 𝑝𝑖−1 δ𝑡 (18) where pi=(-hi)3/2 corresponds to the ith time step, as indicated in eq. (17), and pi-1=(-hi-1)3/2 is the analogous term corresponding to the previous time step, (i–1). the finite time step is denoted by δ𝑡. also note that at t = 0, all higher order derivatives on the lhs are initially set to zero: �̇�𝑖=0 = 0, �̈�𝑖=0 = 0, 𝑝𝑖=0 = 0, etc., and only p0=(-h0)3/2 has a non-zero value, with h0 being the initial sample deformation at time zero. the higher order derivatives at each time step are determined similarly, using eq. (19) and eq. (20): �̈�𝑖 = �̇�𝑖− �̇�𝑖−1 δ𝑡 (19) … 𝑝𝑖 𝑁 = 𝑝𝑖 𝑁−1 − 𝑝𝑖−1 𝑁−1 δ𝑡 (20) 142 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares where n is the number of characteristic times in the viscoelastic model. the key for this numerical approach is to respect the order of calculation of derivatives. when the input parameter is the deformation, as in this example, the lowest-order derivative of the deformation term must be calculated first, followed by the calculation of the higher order derivatives in ascending order. once all the derivatives of indentation have been calculated, it is time to focus on the rhs of eq. (9). when calculating the rhs of eq. (9), we use the reverse strategy described in the previous subsection. in this case, we solve first for the highest-order force derivative, and then from it we calculate the lower-order derivatives of the force. as a first step, we rearrange eq. (9) and solve for the highest force derivative at the ith time step ( 𝑑𝑁𝐹𝑖 𝑑𝑡𝑁 ) introducing two intermediate variables: 𝑑𝑁𝐹𝑖 𝑑𝑡𝑁 = 𝑠𝑢𝑚𝑎𝑄𝑖– 𝑠𝑢𝑚𝑎𝑈𝑖 (21) where 𝑠𝑢𝑚𝑎𝑄𝑖 = 𝛼 𝑢𝑁 [𝑞0𝑝𝑖 + 𝑞1 𝑑𝑝𝑖 dt + 𝑞2 𝑑2𝑝𝑖 𝑑𝑡2 +. . . +𝑞𝑁 𝑑𝑁𝑝𝑖 𝑑𝑡𝑁 ] (22) is the lhs of eq. (9), divided by the coefficient (differential constant) of the highestorder force derivative, and where 𝑠𝑢𝑚𝑎𝑄𝑖 = 𝛼 𝑢𝑁 [𝑞0𝑝𝑖 + 𝑞1 𝑑𝑝𝑖 dt + 𝑞2 𝑑2𝑝𝑖 𝑑𝑡2 +. . . +𝑞𝑁 𝑑𝑁𝑝𝑖 𝑑𝑡𝑁 ] (23) is the rhs of eq. (9), with the highest-order derivative term removed, divided by the differential constant of the highest-order force derivative. note that in calculating the variable 𝑠𝑢𝑚𝑎𝑈𝑖 we use the force derivatives of the previous time step, (𝑖 − 1), , instead of the current time step. recall also that  = 16√𝑅 3 for an incompressible material, and p(t) = h(t)3/2 (ensuring that h has a positive sign, as previously discussed – see eq. (17). once we have calculated the highest-order derivative, the lower-order force derivatives can be determined by using euler integration method [42, 43]. 𝐹𝑖 𝑁−1 = 𝐹𝑖−1 𝑁−1 + 𝐹𝑖 𝑁δ𝑡 (24) … and we can continue the process until arriving at the lowest force derivative, namely the force itself (eq. (26)): �̇�𝑖 = �̇�𝑖−1 + �̈�𝑖δ𝑡 (25) 𝐹𝑖 = 𝐹𝑖−1 + �̇�𝑖δ𝑡 (26) eq. (26) is the resulting viscoelastic force response due to indentation, which is transmitted to the afm tip. this procedure is repeated for each time step dt until the end of the afm simulation is reached, and in this manner, we obtain the force history. note that if there has been no previous indentation history (e.g., when the afm tip first approaches a pristine, undeformed sample during the simulation), the initial conditions (t = 0) for the force and for all its derivatives up to the (n–1)th derivative are equal to zero, 𝐹𝑖=0 𝑁−1 = . . . = �̇�𝑖=0 = 𝐹𝑖=0 = 0. therefore, at time zero only the highest-order derivative can have a non-zero value. an example calculation for a 3-maxwell-arm guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...143 model is provided in the next subsection. fig. 3 summarizes in a flow chart the procedure to calculate the viscoelastic surface deformation that has been discussed so far in this section. fig. 3 flow chart describing the calculation steps to obtain the force response for the contact portion of an indenter penetrating a viscoelastic surface. for each timestep of the algorithm, the indentation is regarded as the input for eq. (9). the calculation steps are detailed in section 2.5.1. this algorithm is conveniently implemented with python code in an open-access repository [40] under the afmsim library with the contact_mode function. it is also available in the excel spreadsheet within the same repository. 144 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares 2.5.2. example – 3 arm generalized maxwell model to illustrate the process described above, consider a generalized maxwell model with three characteristic times (3 maxwell arms, refer to fig. 1a). again, the procedure begins with the calculation of the deformation derivatives, since the deformation is known: 𝑝𝑖 = (−ℎ𝑖)3/2 (27) �̇�𝑖 = 𝑝𝑖 − 𝑝𝑖−1 δ𝑡 (28) �̈�𝑖 = �̇�𝑖− �̇�𝑖−1 δ𝑡 (29) 𝑝𝑖 = �̈�𝑖− �̈�𝑖−1 δ𝑡 (30) once all deformation derivatives are determined, the force response can be calculated. 𝑠𝑢𝑚𝑎𝑄𝑖 = 𝛼 𝑢3 [𝑞0𝑝𝑖 + 𝑞1�̇�𝑖 + 𝑞2�̈�𝑖 + 𝑞3𝑝𝑖] (31) 𝑠𝑢𝑚𝑎𝑈𝑖 = 1 𝑢3 [𝑢0𝐹𝑖−1 + 𝑢1�̇�𝑖−1 + 𝑢2�̈�𝑖−1] (32) eq. (21) for the 3-arm generalized maxwell model will therefore equal: 𝐹𝑖(𝑡) = 𝑠𝑢𝑚𝑎𝑄𝑖 − 𝑠𝑢𝑚𝑎𝑈𝑖 (33) using euler integration, we find the lower-order derivatives and the force itself (zeroorder derivative, eq. (36)): �̈�𝑖 = �̈�𝑖−1 + 𝐹𝑖δ𝑡 (34) �̇�𝑖 = �̇�𝑖−1 + �̈�𝑖δ𝑡 (35) 𝐹𝑖 = 𝐹𝑖−1 + �̇�𝑖δ𝑡 (36) where the initial force and force derivatives at time zero are all equal to zero, �̈�(0) = �̇�(0) = 𝐹(0) = 0. the corresponding implementation in python code is illustrated in the contact_mode function. this algorithm is also implemented in the excel spreadsheet provide in the online repository [40]. 2.5.3. numerical solution for the viscoelastic recovery after the afm tip loses contact with the surface during the retract portion of the spectroscopy curve: the rebound problem this portion of the simulation applies only for the retraction portion of the indenter trajectory, after the indenter loses contact with the viscoelastic sample. at this point the temporarily indented viscoelastic surface becomes ‘stress-free’ (since there is no longer an indenter exerting forces on it) and recovers in time according to a deformation profile that depends on its viscoelastic properties and on the previous indentation history. this process is also known as the rebound indentation portion [50]. this portion is seemingly irrelevant for the characterization of viscoelastic materials using afm force-distance curve methods because there is no way to observe from experimental observables the surface recovery. however, in the simulations it is relevant as we need to track surface guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...145 position for each time step even if the tip temporarily loses contact with the sample. this allows us to have continuity of tip-sample distance knowledge, which is relevant in the calculation of long-range noncontact forces (section 3.2). focusing now on the technical details, the onset of this portion in the simulation will be indicated by a change of sign in the force term. since forces cannot become negative (assuming the absence of van der waals forces), as this would indicate that the afm tip is grabbing and pulling the surface upwards, a flag variable in the simulation should be established to determine when the force changes from positive to negative. from this point on, the stress-free condition begins, and the calculations described in the previous section (2.5.1) are no longer applicable. specifically, the input parameter for the rebound portion will not be the indentation as in the previous section. instead, the stress-free condition, which governs the recovery, dictates that in our calculations all force values and force derivatives become zero. thus, eq. (9) reduces simply to: 𝛼 [𝑞0𝑝 + 𝑞1 𝑑𝑝 dt + 𝑞2 𝑑2𝑝 𝑑𝑡2 +. . . +𝑞𝑁 𝑑𝑁𝑝 𝑑𝑡𝑁] = 0 (37) and we solve for the highest-order derivative of the deformation (recall that we are now solving for the deformation profile): 𝑑𝑁𝑝 𝑑𝑡𝑁 = − 1 𝑞𝑁 (𝑞0𝑝 + 𝑞1 𝑑𝑝 dt + 𝑞2 𝑑2𝑝 𝑑𝑡2 +. . . +𝑞𝑁−1 𝑑𝑁−1𝑝 𝑑𝑡𝑁−1) (38) the lower-order deformation derivatives can now be determined by using euler integration: 𝑝𝑖 𝑁−1 = 𝑝𝑖−1 𝑁−1 + 𝑝𝑖 𝑁δ𝑡 (39) … �̇�𝑖 = �̇�𝑖−1 + �̈�𝑖δ𝑡 (40) 𝑝𝑖 = 𝑝𝑖−1 + �̇�𝑖δ𝑡 (41) and finally, using eq. (27), the sample surface position can be calculated: ℎ𝑖 = 𝑝𝑖 2/3 (42) it is noteworthy that in this procedure the indenter shape parameter (α) has not played a role. this is in accordance to the literature, where it has been noted that for the rebound viscoelastic problem the viscoelastic surface recovery is independent of the indenter shape [50]. further, as all the viscoelastic forces are zero, due to the stress-free condition, the only forces still present between the tip and the sample are the long-range noncontact forces between the tip and the sample surface, which correspond to dispersion forces (london or van der waals forces) and are discussed in section 3.2. the flow chart in fig. 4 illustrates the procedure to calculate the surface deformation profile as a function of time for the stress-free condition. 146 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares fig. 4 flow chart describing the calculation steps to obtain the deformation response for the viscoelastic recovery portion (rebound [50]) when the indenter loses contact with the viscoelastic surface during retract of the indenter. for each timestep the stress-free condition dictates that the force and all its derivatives in eq. (9) are zero. this allows the calculation of higher order derivatives of indentation in rhs of eq. (9) to finally obtain the resulting time-dependent surface recovery profile. the calculation steps are detailed in section 2.5.3. the algorithm has been implemented in python code and is available in an open-access repository [40] under the afmsim library and the contact_mode function. guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...147 3. case study of viscoelastic indentation with an atomic force microscope in acquisition of force distance curves 3.1. viscoelastic implementation in force-distance curve methods during an off-resonance force-distance curve acquisition, the base of the afm cantilever is brought towards the surface at a constant velocity. the governing equation that captures the dynamics of the cantilever tip interacting with the surface via the tip-sample forces is [30]: 𝑚�̈�(𝑡) + 𝑐�̇�(𝑡) + 𝑘𝑧(𝑡) = 𝑘𝑧𝑏(𝑡) + 𝐹𝑡𝑠(𝑡) (43) where t is time, k is the stiffness, c = 2𝜋𝑓𝑚 𝑄1 is the damping coefficient, f the fundamental eigenfrequency (natural frequency), m = 𝑘 (2𝜋𝑓)2 is the equivalent mass, q1the quality factor of the fundamental eigenmode, �̈�(𝑡)is the tip acceleration, �̇�(𝑡) is the tip velocity, �̇�(𝑡) is the tip position, zb(t) is the cantilever base position and fts(t) is the tip-sample force. as indicated, the dynamical variables are time-dependent. the initial position of the cantilever base, zb,initial serves as the starting point for the approach process towards the sample surface. during this experiment, the user specifies the cantilever-base approach velocity, �̇�𝑏. 𝑧𝑏(𝑡) = �̇�𝑏𝑡 (44) numerically, the new base position, zb, for each time step, i, is obtained by multiplying the approach velocity by the total simulation time, and adding the result to the initial cantilever base position (the total simulation time is simply the index i multiplied by the time step dt): 𝑧𝑏,𝑖 = 𝑧𝑏,𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + �̇�𝑏,𝑖(𝑑𝑡) (45) we now calculate the tip acceleration, �̈�(𝑡), by solving the equation of motion (eom) of the cantilever tip (eq. (46)) in introducing a subindex corresponding to the time step: �̈�𝑖 = −𝑘𝑧𝑖−1 − 2𝜋𝑓𝑚�̇�𝑖−1 𝑄1 + 𝑘𝑧𝑏,𝑖+𝐹𝑡𝑠,𝑖−1 𝑚 (46) at time zero, fts = 0, as there is no tip-sample contact yet (the experiment begins with the cantilever placed far away from the sample). z and �̇� are given initial values by the user, z0 = zb,initial and �̇�𝑖 = 0, since we assume that the cantilever was initially at rest and in equilibrium. using the tip acceleration we calculate the tip position z (eq. (47)) and velocity �̇� (eq. (48)) using the verlet integration process (a numerical method used to integrate newton’s equations of motion) and the central difference, respectively [42, 51, 52]. 𝑧𝑖+1 = 2𝑧𝑖 − 𝑧𝑖−1 + �̈�𝑖δ𝑡2 (47) �̇�𝑖 = 𝑧𝑖+1− 𝑧𝑖−1 2δ𝑡 (48) note that in order to calculate the position for time step (i + 1) with eq. (47) and the velocity for time step i with eq. (48), it is necessary to know the position at the previous two time steps i and (i 1). therefore, at the beginning of the simulation, it is necessary to define one additional initial condition of position for time step i = -1. one generally 148 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares sets this initial condition to be equal to the initial cantilever base position, namely zi=-1 = zb,initial. in our code we rename the tip position as tippos, which is given the value zi+1. if tipposis greater than the current surface position, there is no tip-sample contact. as the cantilever approaches the sample, this variable will become negative, indicating that tipsample contact has been established and that viscoelastic forces can now be calculated using eq. (9) as explained in section 2.5. fig. 5 provides examples of tip-sample interaction force curves calculated using the above procedures for different sets of parameters, using a 3-arm generalized kelvinvoigt model. note here that the spring parameters for this model are given in terms of compliances (j) instead of moduli (g). as explained in section 2.1, retardances and relaxances hold an inverse relationship (eq. (5)). specifically, the relaxance of a spring is its modulus (a measure of its stiffness) while its retardance measures how soft the spring is: its compliance. thus, the larger the value of compliance, the softer the spring is. generally, viscoelastic materials have low values of glassy compliance (jg), which means that they behave in a stiff-elastic manner when quickly deformed (i.e., at short timescales). on the other hand, viscoelastic materials have larger equilibrium (rubbery) compliances (je) when probed with very slow excitations (je = j1 + j2 + … + jn) [1, 2]. fig. 5 tip-sample interaction force curves calculated using the contact_mode function contained in the afmsim library [40] for a 3-arm generalized kelvin-voigt model. the solid line corresponds to the tip approach and the dash-dotted line corresponds to the tip retract. hysteresis is evident in all plots, as expected for a (dissipative) viscoelastic material. fig. 5a provides results for different characteristic time 1, 2x10-2 s, 1.5x10-1 s and 4.5x10-1s. the remaining viscoelastic model parameters are jg = 2.0x10-10 pa-1, j1 =9.0x10-9 pa-1, j2=7.0x10-9 pa-1, j3 = 1.0x10-10 pa-1, 2 = 0.5x10-3 s, 3 = 0.5x10-2 s. fig.5b displays results for different cantilever stiffness. the simulation parameters are jg = 2.0x10-10 pa-1, j1 = 5.0x10-9 pa-1, j2 = 7.0x10-9 pa-1, j3 = 1.0x10-10 pa-1,1 = 1.5x10-1 s, 2 = 0.5x10-3 s, 3 = 0.5x10-2 s and k = 0.5, 1 and 2 n/m. guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...149 3.2 long-range noncontact forces to accurately simulate an afm experiment, noncontact forces (van der waals or london dispersion forces) must be accounted for. as the simulation begins, if the tipposition is greater than zero, meaning there is no tip-sample contact yet, the tip and sample will experience an attractive force, which is generally modeled using the hamaker equation [30]: 𝐹𝑣𝑑𝑊 = − 𝐻∗𝑅 6 (𝑇𝑖𝑝𝑃𝑜𝑠)2 (49) where h is the hamaker constant, r is the radius of curvature of the afm tip (assumed to be nearly spherical) and tippos gives the tip-sample distance (the distance between the tip and the unperturbed surface). the negative sign indicates that the forces are attractive (i.e., the tip experiences a downward force). when there is no tip-sample contact, eq.(49) describes the only force present, so fts the tip-sample force in the cantilever tip equation of motion, eq. (43)) must be calculated using eq. (49), and there is no viscoelastic force component. once tip-sample contact is established, the tip-sample force, fts, has two contributions: the first one is the viscoelastic force described above, fi, and the second one is the attractive tip-sample force from eq. (49). when the tip is pushing onto the surface, the tip-sample distance is set to≈0.2 nm, which corresponds approximately to the diameter of a single atom of average size (that is, we assume that the atoms of the tip and the sample are touching one another, so the centers of the atoms in contact are one atomic diameter away). the total tip-sample force is then: 𝐹𝑡𝑠 = 𝐹𝑖 − 𝐻∗𝑅 6 𝑎2 (50) for simplicity, we assume that the van der waals forces are unable to deform the surface. strictly speaking this is not true, since attractive tip-sample forces are in some cases able to pull the surface upwards and cause it deform above the initial unperturbed position. this can be important when modeling the imaging of materials having very low stiffness. 3.3 time step the time step, or iteration step, is of central importance, as it has a strong effect on the accuracy of the results and significantly affects the stability of numerical methods. numerical methods are prone to instabilities when the time step is too large and can diverge very rapidly. on the other hand, extremely small time steps bring about high memory demands and lead to unnecessarily large computation time. therefore, the right balance between too small and too large a time step needs to be established. in selecting the time step, one should consider the total simulation time required, the smallest characteristic time in the material and the period of the afm cantilever (the inverse of its natural frequency). the smaller the total simulation time, the smaller the time step must be in order to ensure stability (for the case of high deformation rates). instead of recommending a specific time step that provides stability, a guideline for ranges of time steps seems more applicable. it is recommended that a time step smaller than the smallest characteristic time divided by 10 or the fundamental period t of the cantilever divided by 10,000 be used for the simulation to obtain stable results. the lower limit depends on the computational resources available. since every simulation has its own peculiarities, these general suggestions may not always be valid, so the optimum time step must in the end be determined by trial and error. 150 m. forstenhaeusler, e. a. lópez-guerra, s. d. solares 3.4 verification upon successfully running a numerical simulation, for example using the contact_mode function we have provided [40], the question remains as to whether it is actually accurate and correct. one way to verify the results is by comparing the results of the left-hand-side (lhs) with the right-hand-side (rhs) of the following equation, the boltzmann superposition integral, adjusted for a spherical tip according to lee and radok [17]. 16√𝑅 3 ℎ(𝑡)3/2 = ∫ 𝑈(𝑡 − 𝜉)𝐹(𝜉)𝑑𝜉 𝑡 0 (51) where the lhs equals 𝐿𝐻𝑆 = 𝛼 ∗ ℎ(𝑡)3/2 (52) and the rhs, the convolution of force with retardance (u*f), equals 𝑅𝐻𝑆 = 𝐽𝑔𝐹(𝑡) + ∑ ∫ 𝐽𝑛 𝜏𝑛 𝑒−(𝑡−𝜉)/𝜏𝑛𝑑𝜉 𝑡 0𝑛 (53) where 𝑈(𝑡) = 𝐽𝑔 + ∑ 𝐽𝑛 𝜏𝑛 𝑒−𝑡/𝜏𝑛 𝑛 is the compliance of the material. recall that= 16√𝑅 3 , h(t) and f(t) are the indentation and the force, respectively, as before. 𝜉is the integration variable. once the simulation data is collected, one can use the deformation data to calculate the lhs, and the force data to calculate the rhs. the rhs can be calculated with the conv function in the viscoelasticity library in our github repository [40]. calculating the rhs requires passing the force and time arrays from the simulation to the numerical convolution described in eq. (53). this can also be done using the conv function in the viscoelasticity library in our github repository [40]. when doing this, it is sometimes computationally very expensive to pass the entire force and time arrays, which could be very large if the time step of the simulation is very short. instead, one could pass a shorter (scattered) version of the arrays containing data every defined number of time steps. if the latter is decided for computational convenience, a word of caution should be given as to how scattered the version of the arrays should be. specifically, we strongly advice following the time step boundaries provided in section 3.3. an example of the consequence of passing too scattered force and time arrays is illustrated in fig. 6. this plot verifies the simulation as correct, if the convolution integral (rhs) and the results from the simulation (tip position multiplied by α constant: lhs) overlay each other. as can be seen, when the arrays are too scattered (the time step is too large), lhs and rhs do not overlay (see the blue line labeled with a time step dtconv = 9.9x10-5 s). in contrast, the yellow line (with a time step dtconv = 9.9x10-7) perfectly overlays the scattered purple star points (lhs). note that for this verification we have not considered van der waals tip-sample forces. guidelines to simulate linear viscoelastic materials with an arbitrary number of characteristic times...151 fig. 6 impact of computing the convolution of force with retardance (rhs, eq. (53)) with arrays that have different time step values. the convolutions were computed with the following time step values dtconv = 9.9x10-5 s; 9.9x10-6 s and 9.9x10-7 s and plotted with different colors as indicated in the figure’s legend. these results are compared with eq. (53) (lhs), which is plotted with purple stars. it can be observed that a time step of approximately 9.9x10-7 s has to be passed to the convolution integral (rhs, eq. (53)) in order to match the results from the simulations (lhs, eq. (52)). note how the yellow line matches the purple star markers. 𝑈(𝑡) = 𝐽𝑔 + ∑ 𝐽𝑛 𝜏𝑛 𝑒−𝑡/𝜏𝑛 𝑛 is the retardance of the material, h is the indentation, f(t) the force, the integration variable, and the coefficient is = 16√𝑅 3 . 4. conclusion we have outlined detailed guidelines for implementing linear viscoelastic simulations in the context of afm force-distance curve methods, where an indenter deforms a linear viscoelastic material. the manuscript has been written in a practical manner in an effort to make it accessible to a broad audience of researchers and students of different disciplines. in this spirit, procedural details have been prioritized over mathematical details, for which the user has been referred to relevant literature. specifically, we have provided detailed explanations for setting up the governing equation ruling the behavior of the linear viscoelastic material being indented, and the subsequent numerical solution of this equation. for the numerical solutions, we have provided conceptual explanations and given specific examples, 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generalized verlet algorithm for efficient molecular dynamics simulations with longrange interactions, molecular simulation, 6(1-3), pp. 121-142. 13240 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume241129010s © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a novel approach to predicting the cutting force in turning using dimensional analysis jelena stanojković1, miloš madić2, milan trifunović2, predrag janković2, dušan petković2 1university of priština, faculty of technical sciences, serbia 2university of niš, faculty of mechanical engineering in niš, serbia abstract. cutting forces are a critical indicator of the machining process, and their modeling is important for a variety of reasons, including tool life assessment, chatter prediction, tool condition monitoring, assessment of machining strategies and machining process optimization and control. this paper presents a new principle of modeling the main cutting force using dimensional analysis (da). dry longitudinal single-pass turning of two different steels (20mncrs5 and s235jrg2) with two different cutting inserts, was considered. taguchi's 33×21 design with 6 trials was applied to arrange seven parameters: depth of cut, feed rate, cutting speed, feed velocity, rake angle, cutting edge angle, and workpiece material parameter, i.e., tensile strength. the obtained results, including additional validation tests, showed a very good prediction capacity of the dabased model in estimating the cutting force during the turning process. the analysis includes the influence of parameters on the cutting force as well as examination of 3d surface diagrams and correlation coefficients. the chip slenderness ratio proved to be the most important dimensionless group for cutting force prediction. by performing additional experimental trials, the correction coefficient for the tool nose radius was estimated and extended models were developed. the well-known victor-kienzle model can be used to predict the cutting force if the exact values of mc and kc1.1 coefficients. the proposed da-based models proved to be valid for predicting all three cutting force components with high accuracy. key words: turning, cutting force, dimensional analysis, modeling, taguchi’s design received: november 29, 2024 / accepted march 01, 2025 corresponding author: jelena stanojković faculty of technical sciences, university of priština in kosovska mitrovica, knjaza miloša 7, 38220 kosovska mitrovica, serbia e-mail: jelena.stanojkovic@pr.ac.rs 2 j. stanojković, m. madić, m. trifunović, p. janković, d. petković 1. introduction the turning process is one of the most commonly used in the metal processing industry's production operations [1]. as one of the oldest production technologies, it still plays an important role in modern industry due to favorable cutting mechanics for a wide range of materials, and can ensure high productivity at acceptable costs [2]. nonetheless, as well as other traditional and non-traditional technologies [3-5], turning is a multi-input process characterized by the existence of multiple machinability and output performances. in addition, these performances for a particular application can be of different significance, and as a rule, are mutually opposed [5]. cutting forces are one of the most important machinability indicators for understanding the turning process. the cutting forces generated during the turning process have a direct impact on heat generation, which in turn affects tool life and machining accuracy. they also correlate with other factors, such as the quality of the machined surface, selfexcitation, and forced vibrations [6-8]. knowing the cutting forces for different cutting parameters helps the designer-manufacturer increase the machine tool's efficiency. in addition, it enables the optimization of processing parameters, which can lead to improved productivity, reduced production costs, and increased process efficiency [9-11]. in order to achieve a complete match between the predicted and real responses, it is necessary to capture all aspects of the interactions of the cutting tool, workpiece, machine tool, and environment. however, in practice, this is often difficult because the actual states of some aspects and physical phenomena are unknown or partially known in a given process or even certain parameter space. these aspects also include environmental influences, homogeneity of the workpiece material chemical composition and mechanical properties, variable lubrication and chip removal conditions, random formation of deposits and their removal, etc. [12]. therefore, general modeling approaches are based on simplifying the real process. various influencing factors, including processing parameters, workpiece material properties, the kinematics and dynamics of the cutting process, cutting tool geometry, and lubrication conditions, have led to the establishment of numerous cutting force prediction models today. various methods exist for modeling the cutting force, typically categorized as analytical, empirical, and numerical methods [13]. in practice, empirical models, with the victor-kienzle model being one of the most frequently applied approaches, are used in determining cutting forces [14]. empirical models are based on experimental determination of the model coefficients, whose form has been usually determined in advance, and typically relate cutting force as the dependent variable to different independent variables such as cutting speed, depth of cut, feed rate, material type, cutting tool geometry, etc. these empirical models are often simple to implement and provide acceptable results in a wide range of industrial applications. different researchers have previously used different approaches to predict the cutting force during the turning process, table 1. a novel aproach for predicting the cutting force in turning using dimensional analysis 3 table 1 review prediction of the cutting force during the turning process ref. tool material workpiece process parameters experimental design (trials) model type [15] coated carbide aisi 4340 steel vc, f, ap taguchi (27) linear [16] ceramic with reference aisi d3 steel vc, f, ap, ct taguchi (18) quasi-linear ann [17] coated carbide 36crnimo4 alloy steel f, ap, rε taguchi (9) linear [18] cubic boron nitride aisi h11 steel vc, f, ap, h box-behnken (29) quadratic nonlinear model with interactions [19] mixed ceramic aisi 4140 steel vc,, f, ap, rε box-behnken (29) quadratic nonlinear ann [20] polycrystalline cubic boron nitride aisi d2 steel vc, f, ap, h, rε taguchi (27) quasi-linear power [21] cubic boron nitride 7020 aisi 52100 steel vc, f, ap, h ffd (25) ann [22] tungstencoated carbide aisi 4340 alloy steel vc, f, ap, rε ffd (60) gpr svm ann [23] multi-layer coated carbide d3 steel vc, f, ap, rε taguchi (16) linear ann [24] carbide with pvd micro grain inconel 718 vc, f, ap, h taguchi (27) mechanistic [25] tialn coated carbide aisi 420 steel vc, f fd with two central points (11) quadratic ffd (full factorial design), fd (factorial design), ann (artificial neural network), grp (gaussian process regression), svm (support vector machines), gp (genetic programming), vc (cutting speed) m/min, f (feed rate) mm/rev, ap (depth of cut) mm, ct (cutting tool), h (workpiece hardness) hrc, h (chip thickness) mm. krupa et al. [26] developed a mathematical model and proposed a method for estimating the feed rate impact of stochasticity on the main cutting force in turning. they performed measurements of the main cutting force during turning on universal lathes, with the 45 steel (iso p) (carbon content 0.45%) workpiece material, a t15k6 cutting tool without coating, and feed rate values of 0.06, 0.1, 0.15 and 0.2 mm/rev. the mathematical model was developed, and dispersion characteristics (mean value, dispersion, and mean square deviation) of the main cutting force were obtained, depending on the corresponding dispersion characteristics of feed rate. fodor et al. [27] introduced a stochastic extension of the existing cutting force models. they showed that the cutting force is stochastic with a variance of the measured cutting force of around 4-9% of the average value, depending on the selected parameters. abdellaoui et al. [28] investigated the effects of tool nose radius in the turning process. they developed a thermomechanical model related to the average 4 j. stanojković, m. madić, m. trifunović, p. janković, d. petković tool/chip contact temperature, which enabled the prediction of all thermomechanical parameters when cutting with a sharp tool. as a function of the value of the minimum uncut chip thickness, two cutting situations were defined (plowing and cutting). horvath [29] presented a review of the main research results of recent years regarding the cutting force, where the cutting forces in turning depend not only on the material properties and cutting parameters, but also to a large extent on the geometry of the tool edge that determines the shape of the chip (thickness and width). he derived a new cutting force model for fine turning, which can be used to estimate the main cutting force. it can be observed from the literature review that the researchers used different methods to derive cutting force prediction models. each method makes the prediction fairly accurate, but it comes with limitations such as needing to do more experimental trials, finding the best form of the model, optimizing hyper-parameters, overfitting, and so on, which makes it not so receptive to developing and applying in real production conditions. therefore, the primary goal of this study is to propose a new cutting force prediction model, developable with a limited number of experimental trials while taking into account different parameters that influence the turning process such as the workpiece and the cutting tool properties as well as main cutting parameters. the approach is based on the principles of dimensional analysis (da), and in this study these models were developed for the prediction of cutting forces in dry single-pass medium turning of two steels, 20mncrs5 and s235jrg2. to this aim, taguchi’s orthogonal experimental design with six trials was used for the arrangement of three dimensionless groups while considering seven input process parameters. four of them are related to the machining process (depth of cut, feed rate, feed velocity, and cutting speed), two are related to the geometry of the cutting tool (cutting edge angle and rake angle), and one is the workpiece material parameter (tensile strength). the obtained experimental results of the main cutting force were compared with the predicted values to determine the percentage error. the percentage error of additional validation tests was further estimated while using parameter values outside the initial experimental hyperspace. the analysis of the obtained results included which parameter, i.e., dimensionless group, had the greatest influence on the cutting force, as well as determining the correlation coefficients. the percentage error of the experimental and validation tests of the victor-kienzle model of the main cutting force was determined with the coefficients (mc, kc1.1) taken from the literature. an extended model of the main cutting force depending on one more cutting parameter, i.e., tool nose radius, was also proposed. finally, the mutual dependences of all three components of the cutting force were analyzed with respect to the identified dimensionless group with the greatest influence. 2. dimensional analysis of the cutting force in physics and engineering, many parameters and measurements take the form of numerical quantities with corresponding dimensional units [30]. da is a powerful tool that aids in the design, ordering, performance, analysis, and synthesis of the resulting data [31]. in general, it is a mathematical tool used to simplify a problem by reducing the number of variables to the smallest number of essential parameters and a technique that enables the conversion of physical quantities from one unit of measurement to another, and the conversion of the basic physical quantities of the si system of measurements with their dimensions (length-l, time-t, mass-m, thermodynamic temperature-θ, electric current-i, a novel aproach for predicting the cutting force in turning using dimensional analysis 5 luminous intensity-j, amount of substance-n) [32]. parameters are described by dimensionless groups (π groups) that connect independent variables with dependent variables. the primary benefit of this method is its significant reduction in the number of experimental trials needed to determine the relationship between variables, thereby reducing both costs and development time. this study uses the buckingham π-theorem as the main tool for the reduction of variables. the buckingham π-theorem, as a key concept of da, helps simplify complex physical problems by reducing the number of variables involved. it is necessary to select the repeating and non-repeating variables for model development [33]. according to the buckingham π-theorem, “the number of independent dimensionless groups that may be employed to describe a phenomenon known to involve n variables is equal to the number n–m, where m is usually the number of basic dimensions needed to express the variables dimensionally” [33]. it is necessary to fulfill two basic criteria before using da [34]:  principle of dimensional homogeneity, and  all relevant variables are included in the proposed relationship. the principle of dimensional homogeneity is based on the fact that in order to establish equality, the dimensions and units in each term of an equation must be the same. the equation is dimensionally homogeneous if this condition is satisfied da [33]. in the present study, da was used to derive a model for predicting the cutting force in turning, initially for predicting the main cutting force (fc), figure 1. fig. 1 cutting force components in turning the dimensions of the cutting force and the parameters that are defined based on the basic dimensions, i.e. length (l), time (t), mass (m), which affect the main cutting force, are as follows:  main cutting force (fc): mlt-2  tensile strength (rm): ml-1t-2 6 j. stanojković, m. madić, m. trifunović, p. janković, d. petković  feed rate (f): l  cutting speed (v): lt-1  feed velocity (vf): lt-1  depth of cut (ap): l  rake angle (γ0): –  cutting edge angle (κ): – rake angle and cutting edge angle have no dimensions. the dependent parameter is the main cutting force, and the independent parameters are tensile strength, depth of cut, cutting speed, feed rate, feed velocity, rake angle and cutting edge angle. feed velocity, feed rate, and rake angle were selected as repeating parameters. by applying buckingham π–theorem, the π1 group can be represented by the following form: dcb f a mc fvrf 01   (1) by changing the dimensions, equation 1 has the following form:        c ba llttmlmlt   1212 1 (2) only those values with the same dimension are added. addition by mlt is shown in the following forms: 01:  am (3) 01:  cbal (4) 022:  bat (5) by solving the system of three equations with three variables, the values of the unknowns are obtained: a=–1, b=0 and c=–2. based on the obtained values for each dimension in the initial π1 group, equation (1), the is replaced with the obtained values and the value of the π1 group can be represented by the following form: 2 21 1 fr f frf m c mc    (6) in this case, there are four dimensionless groups, π1, π2, π3, π4, other π groups are obtained in the same way and presented in table 2. table 2 four dimensionless (π) groups π1 π2 π3 π4 2fr f m c  fv v f a p 0  if one sets: π1=f(π2, π3, π4), the main cutting force as the function of dimensionless groups can be represented by the following model: a novel aproach for predicting the cutting force in turning using dimensional analysis 7 432 0 2 1 xx p x f mc f a v v frxf                             (7) 3. experimental setup the experiments of dry longitudinal single-pass turning were performed on a “potisje” pa–c 30 universal lathe, with the power of 11 kw, spindle speed range of 20÷2000 rpm, and longitudinal feed rate range of 0.04÷9.136 mm/rev, in laboratory conditions. the cutting tool was sandvik coromant pclnr 3225p 12 tool holder (cutting edge angle of κ=95°, rake angle of γoh=–6°) with cutting inserts cnmg 120408e-sf (γoi=14.5°, tool nose radius of rε=0.8mm) and cnmg 120408e-nf (γoi=25°, tool nose radius of rε=0.8mm), manufactured by dormer pramet, of t8430 grade (pvd coated fine grained cemented carbide, suitable for higher feed rates and medium to lower cutting speeds). the workpiece materials used in this experiment were 20mncrs5 and s235jrg2 steel. 20mncrs5 is an alloy steel with excellent mechanical properties, which makes it suitable for a wide range of applications in various industries. its strength, toughness, and wear resistance ensure the durability and reliability of components made of this material. it is most widely used in the automotive and mechanical industries (for the production of machine parts that suffer heavy loads and wear, including spindles, gears, and bearings). according to dormer pramet, it belongs to the workpiece material group p3.2 (alloy steels (carbon steels with an alloying content ≤ 10%) with a hardness of 180-260 hb). s235jrg2 is a carbon construction steel that has a specific application thanks to its mechanical properties and chemical composition. it is most widely used in the construction industry, infrastructure, mechanical engineering, etc. according to dormer pramet, it belongs to the workpiece material group p2.1 (plain carbon steels (steels comprised of mainly iron and carbon) containing < 0.25% c with a hardness of < 180 hb). chemical composition of the workpiece materials 20mncrs5 and s235jrg2 are shown in table 3. the sample of cylindrical rods used in these experiments had different diameters (55 and 59 mm) and the same length of 250 mm. table 3 chemical composition of the workpiece materials chemical composition [%] 20mncrs5 c mn si p s cr ni mo al 0.018 1.11 0.24 0.016 0.026 1.07 0.08 0.018 0.026 cu as v ti b n cev. ca sn 0.19 0.007 0.007 0.024 0.0003 0.007 0.602 s235jrg2 c mn si p s cr ni mo al 0.16 0.46 0.176 0.013 0.005 0.08 0.04 0.01 0.012 cu as v ti b n cev. ca sn 0.08 0.03 <0.001 <0.001 0.0005 0.008 0.25 0.002 0.05 the hrb method was used to determine the hardness of workpiece materials. the hardness was measured on a rockwell device scale b with standard indenter and test force. tensile strength (rm) was indirectly determined by using conversion table a.1– 8 j. stanojković, m. madić, m. trifunović, p. janković, d. petković standard iso 18265:2003 metallic materials–conversion of hardness values. for 20mncrs5 hardness is 156 n/mm2 and tensile strength is 525 n/mm2, while for s235jrg2 hardness is 163 n/mm2 and tensile strength is 540 n/mm2. components of cutting force were measured using a three-component piezoelectrical force transducer, basic unit kistler type 9265a and a tool holder kistler type 9441, which was mounted on the lathe via a custom tool holder adapter creating a very rigid tooling fixture. the load signal is generated by the dynamometer and a kistler type 5007 charge amplifier. through the hbm quantumx mx840 universal measuring amplifiermeasuring data were acquired, visualized, and analyzed using the “catman” data acquisition software. the experimental setup included the machine tool, tool holder, cutting inserts, threecomponent dynamometer, charge amplifier, data acquisition unit, computer, software, input parameters, workpieces, and hardness measurement device. the experimental setup is shown in fig. 2. fig. 2 schematic representation of the experimental setup the experiment was planned according to the form of the proposed main cutting force prediction model. in the study, seven parameters were considered, four related to the machining process (depth of cut, feed rate, feed velocity, and cutting speed), two related to the geometry of the cutting tool (cutting edge angle and rake angle), and one material parameter (tensile strength). moreover, tool nose radius was also considered in additional experimental trials so as to determine the correction coefficient for the proposed cutting force prediction model. parameter ranges and levels were selected based on the tool manufacturer's recommended cutting conditions for the inserts. the experimental design used was the taguchi orthogonal design (l6) 33 ×21 [35]. parameters with their names, symbols, units, a novel aproach for predicting the cutting force in turning using dimensional analysis 9 and values are shown in table 4. the parameters were varied in accordance with the proposed main cutting force prediction model, i.e., defined by dimensionless π groups. cutting speed and cutting edge angle for both experimental designs were taken as constant values because their influence on the main force is very small, almost negligible, as discussed by stephenson and agapiou [36], for most materials, except soft metals and temperature sensitive materials. table 4 parameters with their names, symbols, units and values parameter symbol unit 20mncrs5 s235jrg2 values cutting speed v m/min 157 245 feed rate f mm/rev 0.196 0.249 0.321 depth of cut ap mm 1.2 1.8 1.9 2.0 2.9 rake angle γo ° 8.5 19 cutting edge angle κ ° 95 tool nose radius rε mm 0.8 1.2 4. results and discussion 4.1 da model of the main cutting force the main cutting force values obtained experimentally in laboratory conditions are shown in tables 5 and 6 for 20mncrs5 steel and s235jrg2 steel, respectively. based on the experimental data of the main cutting force and by minimizing the sum of squared errors using matlab, the unknown coefficients of the proposed main cutting force prediction models were determined. main cutting force prediction models in dry longitudinal single–pass turning of two workpiece materials, 20mncrs5 and s235jrg2 steels with a tool nose radius of 0.8 mm, were obtained in the following form: (20mncrs5) 01669.0 0 89362.009816.0 25254865.2                             f a v v ff p f c (8) (s235jrg2) 05254.0 0 04605.110746.0 25404551.1                              f a v v ff p f c (9) the predicted main cutting force values were compared with the main cutting force values obtained experimentally in order to determine the mean absolute percentage error (mape). the mape value of the da model for steel 20mncrs5 is about 0.38%, while for steel s235jrg2 is about 0.65%. based on these results, it can be said that the developed main cutting force prediction models obtained by applying da for both workpiece materials give quite good results from an engineering point of view. 10 j. stanojković, m. madić, m. trifunović, p. janković, d. petković table 5 experimental design 33×21 of the main cutting force of 20mncrs5 steel no. a b c v/vf π2 ap/f π3 κ/γo π4 fc [n] experimentally fc [n] da model 1 0 0 -1 693.93 7.63 5.00 969 971.2 2 0 0 1 693.93 7.63 11.18 977 984.7 3 1 -1 -1 881.57 6.12 5.00 508 506.1 4 1 1 1 881.57 9.18 11.18 742 736.8 5 -1 1 -1 538.28 9.03 5.00 1831 1830.4 6 -1 -1 1 538.28 6.23 11.18 1334 1331.4 table 6 experimental design 33×21 of the main cutting force of s235jrg2 steel no. a b c v/vf π2 ap/f π3 κ/γo π4 fc [n] experimentally fc [n] da model 1 0 0 -1 744.39 7.63 5.00 895 904.1 2 0 0 1 744.39 7.63 11.18 939 943.1 3 1 -1 -1 945.68 6.12 5.00 465 456.5 4 1 1 1 945.68 9.18 11.18 731 727.8 5 -1 1 -1 577.43 9.03 5.00 1747 1744.5 6 -1 -1 1 577.43 6.23 11.18 1235 1233.7 with both steel materials, the influence of parameters a, b and c, i.e., the influence of dimensionless groups, π2, π3 and π4, is quite the same for the main cutting force. therefore, with an increase in the value of dimensionless groups, π2, π3 and π4, the main cutting force increases, where the influence of ap/f is dominant and the most pronounced, followed by groups π4 and π2, for 20mncrs5 steel, that is, groups π2 and π4, for s235jrg2 steel. since the exponents of dimensionless groups are different, it is possible to see an increase in the main cutting force with an increase in the value of dimensionless group π4, i.e., a decrease in rake angle, more pronounced when turning s235jrg2 steel compared to 20mncrs5 steel. the same applies to dimensionless group π3. namely, the increase in the ap/f ratio (chip slenderness ratio) rapidly increases the main cutting force when turning s235jrg2 steel compared to 20mncrs5 steel. both investigated materials have approximately the same tensile strength (rm), but the calculated coefficients of the da model (c, x1, x2, x3) are different, which indicates different machinability and the fact that one da model cannot be used for accurate prediction of cutting force. if only one model was used, for example, the da model of s235jrg2 steel for 20mncrs5 steel, the mape for all experimental tests would be around 10%, and in the opposite case even 20%. the intercept values of derived models are significantly different (2.48 and 1.45) indicating that 20mncrs5 steel is more difficult to machine compared to s235jrg2 steel. parameter a (dimensionless group π2), i.e., the ratio of cutting speed and feed velocity, depends only on feed velocity because cutting speed was kept constant for given workpiece material in the experimentation. feed velocity depends on feed rate and this ratio represents only the change in the feed rate value. this parameter has a positive correlation with the main cutting force, i.e., increasing parameter a increases the main cutting force. too much increase can lead to machine tool overload and increased tool wear. in other words, a a novel aproach for predicting the cutting force in turning using dimensional analysis 11 decrease in feed rate results in a decrease in the main cutting force, since the cutting tool removes smaller layer of material. this implies that a smaller volume of material being removed requires less cutting force. furthermore, removal of smaller volume of material results in reduction of the heat generated during cutting. the reduced temperature helps to maintain the mechanical properties of the cutting tool and material, which further reduces the main cutting force [37]. parameter b (dimensionless group π3), i.e., the ratio of depth of cut to feed rate, chip slenderness ratio, has a positive correlation with the main cutting force by increasing the chip slenderness ratio the main cutting force increases sharply [37-39]. increasing the cross-sectional area of the uncut chip and the volume of the deformed material, either by increasing depth of cut or feed rate, results in an increase in the main cutting force. denkena et al. [40] have concluded the same about the influence of the undeformed cross-section of the chip and the volume of the deformed material on the main cutting force for alloy and carbon steel. parameter c (dimensionless group π4), i.e., the ratio of the cutting edge angle and rake angle, also has a positive correlation with the main cutting force, but the effect is less pronounced. thus, an increase in this parameter, which is accomplished by reducing rake angle, slightly increases the main cutting force. increasing rake angle increases shear angle and results in a decrease in chip thickness, as well as a decrease in the main cutting force and therefore a decrease in cutting power [41]. although large rake angles decrease the main cutting force, they simultaneously reduce the strength of the tool tip which ultimately may lead to its fracture. the proposed main cutting force prediction models are visualized in fig. 3. this figure shows 3d surface plots of the main cutting force (fc) for both workpiece materials, 20mncrs5 steel and s235jrg2 steel, depending on different dimensionless groups. the diagrams clearly indicate that there are no interactions between dimensionless groups, i.e., there is no qualitative change in the influence of a certain dimensionless group in relation to another dimensionless groups. the absence of significant interactions of processing parameters (depth of cut, feed rate, cutting speed, feed velocity, rake angle, cutting edge angle) on the main cutting force was also previously reported [39]. the absence of significant interactions indicates the possibility of modeling input-output dependences with simpler models, which was done in this study by creating degree models using da. adequate experimental validation of the proposed main cutting force prediction models is a necessary step to ensure that the models generalize well with new data, which is key to their application. therefore, based on the perceived significance of dimensionless groups the additional cutting regimes were tested, tables 6 and 7. 12 j. stanojković, m. madić, m. trifunović, p. janković, d. petković fig. 3 surface plot (3d plot) of the main cutting force (fc) dependance on different π groups for 20mncrs5 steel and s235jrg2 steel a novel aproach for predicting the cutting force in turning using dimensional analysis 13 the validation results for both materials show that predicted values are close to the experimentally obtained values. as can be seen from tables 7 and 8, trials 3 and 4 are within the included experimental hyperspace, where the cutting force values obtained by the da models are the closest to the values obtained experimentally. table 7 main cutting force values for different validation cutting regimes (20mncrs5) no. v/vf π2 ap/f π3 κ/γo π4 fc [n] experimentally fc [n] da model 1 538.28 3.74 5.00 739 832.4 2 484.00 3.36 11.18 810 939.3 3 807.42 7.48 5.00 715 714.9 4 606.27 7.72 11.18 1248 1285.8 5 970.72 14.61 5.00 973 915.9 6 807.42 11.21 11.18 1069 1040.9 table 8 main cutting force values for different validation cutting regimes (235jrg2) no. v/vf π2 ap/f π3 κ/γo π4 fc [n] experimentally fc [n] da model 1 577.43 3.74 5.00 735 693.1 2 519.20 3.36 11.18 794 791.1 3 866.14 7.48 5.00 658 664.4 4 650.36 7.72 11.18 1184 1232.5 5 1041.31 14.61 5.00 890 944.7 6 866.14 11.21 11.18 1005 1059.3 the mape for 20mncrs5 steel is about 1.5% and for s235jrg2 steel is about 2.5%. the other cutting regimes are outside of the experimental hyperspace, in terms of the chosen parameter values. taking into account the obtained results, the da models give fairly satisfactory extrapolation results. for all validation tests, the mape for 20mncrs5 steel is about 6.6%, which is borderline satisfactory from an engineering point of view. results for s235jrg2 steel are much more favorable and amount to a mape value of about 3.78%. based on the obtained results, it can be concluded that the da models provide the means to develop fairly accurate models for predicting the main cutting force during the turning process, using a small number of experimental trials, which result in the reduction of the cost of the experiment, savings in materials and energy, as well as reduced time prediction performances of the proposed da models in the form of regression are given in fig. 4. values from the main experiment and additional validation trials were taken into consideration. the correlation coefficient for both workpiece materials is around 0.99, confirming the considerable accuracy of the developed da models. 14 j. stanojković, m. madić, m. trifunović, p. janković, d. petković fig. 4 correlations between the predicted (da model) and the experimentally measured values of the main cutting force using the entire data set the proposed da models were further verified using residual analysis. figure 5 shows the residuals on the vertical axis and the values of the main cutting force for both steel materials (20mncrs5 and s235jrg2) on the horizontal axis by considering experimental data from the main experiment and additional validation trials. the residuals do not follow any recognizable pattern and are scattered above and below zero. fig. 5 residual plot of the main cutting force the models of the main cutting force for both steel materials yield good predictions in a wide range of chip slenderness ratio values, where significant changes in chip morphology occur. a higher chip slenderness ratio results in longer and thinner chips [4244]. figure 6 shows that the 20mncrs5 steel chips are somewhat different, and more unfavorable as their form also depends on the mechanical properties such as hardness and microstructure of the material [45]. materials with higher hardness have a lower ability to deform, so the chip breaks and is easily removed from the cutting zone, as is the case with 235jrg2 steel. a novel aproach for predicting the cutting force in turning using dimensional analysis 15 fig. 6 chip morphology as a function of chip slenderness ratio based on the initial experimental and validation tests, the main cutting force of 235jrg2 steel remains about 6% higher compared to 20mncrs5 steel. however, when the values of ap/f are less than 4, the main cutting forces are almost equal, which is not the case with the victor-kienzle model [46]. 4.2 victor-kienzle model of the main cutting force the well-known victor-kienzle model [15] is still widely used for the calculation of the main cutting force, which has the following form [46-48]: cm cc hkaf   1.11 (10) where a1 (mm2) is the cross-sectional area of the uncut chip, kc1.1 (n/mm2) is the specific cutting force for 1 mm2 cross-sectional area of the cut, h (mm) is the uncut chip thickness, and mc is the exponent of the specific cutting force for the given workpiece material. values of mc and kc1.1 can be found in the catalogs of cutting tool manufacturers and books. for 20mncrs5 steel values are mc=0.25 and kc1.1=2140 n/mm2 [49]. the mape value of the main cutting force is about 60%, while for the experimental trials and for the validation test, it is about 65%. steel s235jrg2 belongs to the universal workpiece material group, unalloyed and low-alloyed steels, c > 0.55%, not tempered, according to cutting tool manufacturer walter (mc=0.25, kc1.1=1700 n/mm2). the same values can be found in the sandvik coromant catalog [10]. for s235jrg2 steel values are mc=0.17 and kc1.1=1780 n/mm2 [49]. the mape value of the main cutting force for s235jrg2 steel is about 35%, while for the experimental trials and for the validation test, it is about 38%. the exact values of coefficients mc and kc1.1 cannot be easily found because some materials may be thermally treated, while some specific materials have properties that are between the different workpiece materials subgroups offered in catalogs of cutting tool manufacturers or available databases. 16 j. stanojković, m. madić, m. trifunović, p. janković, d. petković however, the aforementioned victor-kienzle model gives approximate values of the main cutting force, which may be in poor agreement with the cutting force values obtained experimentally, while the main advantage is fast calculation. 4.3 extended da model of the main cutting force and correction coefficient for tool nose radius the cutting forces in the turning process are dependent not only on the properties of the workpiece material and cutting parameters, but also on the geometry of the cutting tool edge, which affects the chip shape (thickness and width). tool nose radius is an important parameter that affects the cutting force. parida et al. [50] analyzed the influence of tool nose radius on the main cutting force. their study revealed that as tool nose radius increases, the main cutting force also increases due to the increased contact area. increasing tool nose radius does not only increase the main cutting force, but also affects the components of the cutting force, i.e., causing an increase in the radial and a decrease in the axial force [51-53]. the extended main cutting force prediction models with a correction coefficient are given by eqs. (11) and (12): (20mncrs5) r p f c c f a v v ff                            01669.0 0 89362.009816.0 25254865.2   (11) (s235jrg2) r p f c c f a v v ff                            05254.0 0 04605.110746.0 25404551.1   (12) where the correction coefficient was cr = 1 in the case of using the tool nose radius of 0.8 mm or cr = 1.08 in the case of using the tool nose radius of 1.2 mm. in order to determine the correction coefficient (cr), additional experimental trials were conducted for both steel materials using a cutting insert with a larger tool nose radius of 1.2 mm (tables 9 and 10). table 9 experimental trials for extended da model of 20mncrs5 steel no. v/vf π2 ap/f π3 κ/γo π4 fc [n] experimentally fc [n] extended da model 1 881.57 6.12 5.00 555 546.6 2 538.28 9.03 5.00 1972 1977.6 3 538.28 6.23 11.18 1300 1331.5 table 10 experimental trials for extended da model of s235jrg2 steel no. v/vf π2 ap/f π3 κ/γo π4 fc [n] experimentally fc [n] extended da model 1 945.68 6.12 5.00 505 493 2 577.43 9.03 5.00 1882 1884.1 3 577.43 6.23 11.18 1226 1332.4 a novel aproach for predicting the cutting force in turning using dimensional analysis 17 increasing tool nose radius increased the main cutting force and its effect was consistent, given the same cr value was obtained for both materials. 4.4 analysis of cutting force components based on the analysis of experimental results it was observed that for all three cutting force components, main cutting force (fc), passive force, i.e., radial force (fp), and feed force, i.e., axial force (ff), the most important parameter was dimensionless group π3, i.e., chip slenderness ratio [43]. the cutting force components values obtained experimentally in laboratory conditions are shown in tables 11 and 12 for 20mncrs5 steel and s235jrg2 steel, respectively. the effect of the chip slenderness ratio on cutting force components is given in fig. 7. for both workpiece materials, by increasing chip slenderness ratio, the main cutting force and feed force increase drastically, while the radial force decreases. table 11 experimental design 33×21 of the cutting force components of 20mncrs5 steel no. v/vf π2 ap/f π3 κ/γo π4 fc [n] fp [n] ff [n] experimentally 1 693.93 7.63 5.00 969 172 478 2 693.93 7.63 11.18 977 197 497 3 881.57 6.12 5.00 508 144 247 4 881.57 9.18 11.18 742 155 404 5 6 538.28 538.28 9.03 6.23 5.00 11.18 1831 1334 178 282 903 634 table 12 experimental design 33×21 of the cutting force components of s235jrg2 steel no. v/vf π2 ap/f π3 κ/γo π4 fc [n] fp [n] ff [n] experimentally 1 744.39 7.63 5.00 895 139 330 2 744.39 7.63 11.18 939 163 377 3 945.68 6.12 5.00 465 118 182 4 945.68 9.18 11.18 731 139 326 5 6 577.43 577.43 9.03 6.23 5.00 11.18 1747 1235 181 198 712 451 analysis of the ratio of cutting force components provides useful insights into the efficiency and stability of the machining process. the ratio of the cutting force components fc/ff indicates the efficiency of material removal in relation to the side load of the cutting tool. a high value of this ratio is desirable because it indicates an efficient process with reduced side load, which can improve the quality of the machined surface and extend tool life. for both materials, it ranges from 1.8 to 2.7, which indicates more efficient cutting with less feed force. the ratio of the cutting force components fc/fp varies between 3.1 and 10.2, indicating that most of the energy is directed to the removal of material, potentially reducing wear and vibration. the ratio of the cutting force components ff/fp varies around 2, which indicates a stable process with reduced lateral load and vibrations, which can improve surface quality and extend tool life. 18 j. stanojković, m. madić, m. trifunović, p. janković, d. petković fig. 7 the influence of the chip slenderness ratio (dimensionless group π3) on the cutting force components the ratios of the cutting force components for both steel materials (20mncrs5 and s235jrg2) are analogous. the ratio of the cutting force components fc/ff decreases, while the ratios fc/fp and ff/fp increase with an increase in the chip slenderness ratio. it is not possible to predict all cutting force components using only one extended da model, considering that the ratios of the cutting force components change. 5. conclusion this study proposed a novel approach to predicting the cutting force in turning using dimensional analysis. experimental investigation in dry longitudinal single-pass turning of two steel materials, 20mncrs5 steel and s235jrg2 steel, was considered using taguchi's 33×21 design with additional validation trials. based on the conducted analyses, experimental and modeling results, the following conclusions may be drawn:  the proposed models derived using da and limited experimental data proved to be able to accurately predict the main cutting force.  additional validation tests with parameter values outside the initial experimental hyper-space showed very good extrapolation capabilities of the proposed models for predicting the main cutting force.  chip slenderness ratio was the most important parameter affecting the main cutting force, followed by the ratio of cutting speed and feed velocity, while the effect of the ratio of the cutting edge angle and rake angle was the least pronounced.  the well-known victor-kienzle model can be used to predict the cutting force if the exact values of mc and kc1.1 coefficients are known for the particular material used in the cutting process.  the derived extended main cutting force prediction models showed a consistent positive correlation between tool nose radius and the main cutting force.  the proposed models proved to be valid for predicting all three cutting force components with high accuracy. it was found that by increasing the chip slenderness ratio, the main cutting force and feed force increased drastically, while the radial force decreased. in this regard, the ratios fc/fp and ff/fp are subject to change, a novel aproach for predicting the cutting force in turning using dimensional analysis 19 considering the selected values of ap and f (or chip slenderness ratio), while the ratio fc/ff is approximately constant. it can be concluded that da-based models have a very good potential for reliable and accurate estimation of the cutting force in turning. the discussed approach can take into account a large number of cutting parameters using a smaller number of experimental trials, which plays a significant role not only regarding prediction performance but also in great economy, reduced experimentation time, and saving materials and energy for conducting experiments. the limitation of the proposed da-based models is that it would be necessary to extend these models in case of significant parameter interactions. in addition, the existence of multiple dimensionless groups with a number of parameters makes it difficult 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and tool wear during dry turning of inconel 718, tribology-materials, surfaces & interfaces, 17(1), pp. 62-71. https://mac.walter-tools.com/ plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering https://doi.org/10.22190/fume231228031m © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper optimization based determination of the set of continuous-state components critical to the mechanical system state dragana makajić-nikolić1, milan stanojević1, nebojša nikolić1, petar pavlović2 1university of belgrade, faculty of organizational sciences, serbia 2academy of applied studies, šabac, serbia abstract. this paper proposes a new approach to determining the set of the most critical continuous-state components in continuous-state mechanical systems. unlike traditional importance measures, which determine the criticality of individual components, our approach provides selection of the entire set of critical components. in addition, it enables determination of the minimal budget for achieving full or any required performance level of the mechanical system. we start from the continuum structure function and the system state definition through minimal cut sets. the problem defined in this paper is formulated as a determination of a set of components whose states maximize the system performance level under the budget constraint. further, we prove that the solution of the formulated problem can be obtained based on the solution to a weighted minimal hitting set problem. the proposed approach is applied to a group of benchmark instances and the obtained results are compared to components’ rankings obtained by using traditional importance measures. key words: continuous-state, criticality, performance level, importance measures 1. introduction identification of the critical system components has an important role in the mechanical system design, improvement, and maintenance. in the system design phase, it is used in deciding on the performance levels of the particular components included in the system [1]. when the system performance should be improved, the performances of the critical components are the ones that should be improved first. in the system maintenance planning they are used to effectively allocate maintenance resources [2]. in system engineering, critical components are the components whose performance mostly influence the system received: december 28, 2023 / accepted may 19, 2024 corresponding author: dragana makajić-nikolić university of belgrade, faculty of organizational sciences, jove ilića 154, belgrade e-mail: dragana.makajic-nikolic@fon.bg.ac.rs 2 d. makajić-nikolić, m. stanojević1, n. nikolić, p. pavlović performance. performance of a system and components can be expressed using different measures: reliability, availability, maintainability, safety, utility, obsolescence, detrition, fatigue, capacity, throughput etc. [3]. regardless of the performance type, system’s and components’ performance levels are generally called system’s and components’ states. mechanical components’ and system’s states can be observed as: binary-states, where components and system can either be in a perfect functionality or a complete failure state, multi-states, with discrete degradation states between perfect functionality and complete failure, continuous-states, with continuous degradation states between perfect functionality and complete failure. in the case of binary-states, only the so called “hard failure” appears. in multi-states and continuous-states cases, components can also have “soft failure” with different levels of degradation that do not exceed the predefined failure threshold [4]. in this paper we observe continuous-state components and systems. factors that cause continuous degradation of components’ states are processes such as wear, fatigue, and erosion [5]. components and systems performance degrade continuously in many real-life mechanical systems: automobile tire whose performance degrades continuously as the tread wears; a coal-fired power plant consisting of boilers operating at fewer megavolts than their full capacity; gas turbine fueled by nozzles in which the flow of fuel can be reduces due to the nozzles blockages; internal combustion engine valves wear out after a large number of engine cycles; production systems where production capacity can deteriorate with time [6]; light emitting diode luminosity degradation [7]; the voltage degradation of membrane electrode assemblies in polymer electrolyte membrane fuel cell stack system [8] etc. the approach used in reliability and risk theory to determine the most important (critical) system components is based on the importance measures (im). the goal of im is to rank the components according to their impact to make better system design or maintenance plan. in the system design phase, im should be taken into consideration to determine the appropriate reliability of each of the components of the system being designed. in the maintenance process, im are used to identify the component whose reliability should be improved to increase the system reliability, or to order spare parts [9], and/or in the component maintenance priority approach that prioritizes the components that need to be maintained during the repair of the failed component [10]. ims can be used in diagnostics to rank components in terms of how likely they are to have caused a system failure [11]. in recent years, im are used in the optimization of system resilience, which has become an important system performance [12, 13]. the first im for binary systems was introduced by birnbaum [14]. according to birnbaum im, the importance of a component is the difference between the reliability of the system when the component is fully functional and the reliability of the system when the component fails. until now, more than sixty different binary im were introduced [15]. a considerable number of im for multi-state systems can also be found in literature [1619]. however, there are only few ims defined for continuous-state system and most of them reduce continuous-state to binary or multi-state by introducing threshold or partitioning approaches. kim and baxter [20] defined reliability importance of component j at system level α, α(0, 1), where [0, α) corresponds to the failure states, and [α, 1] corresponds to optimization based determination of the set of continuous-state components critical to the system state 3 the operating states. to formulate a criticality measure model for continuous-state system, lisnianski introduced discrete approximation [21, 22]. consequently, the continuous-state system is represented by two multi-state systems that are then used as boundary points estimation for the given continuous-state system. some of the continuous im, such as griffith im [23] and mean absolute deviation im [24], are obtained by adjustment of binary im for continuous-states. de jonge [25] reduced the continuous-state non-decreasing problem to a discrete-time discrete-state one using the probability of transition between states, and further modeled it using markov chains. chen, qiu, and zhao [26] used the monte carlo simulation method to determine optimal maintenance policies for continuousstate systems. most ims rank individual components according to the calculated values of the selected im and therefore cannot be applied to determine the simultaneous effect of a group of components on the system state [27]. this is one of the open issues on importance measures [28]. it was considered by few ims, but only for pairs of binary or multi-state components [29, 30], module of three components [31] and for preselected group of components [32]. in this paper we propose an approach for simultaneous determination of the entire set of the most important continuous-state components of a continuous-state system. we formulate this new criticality measure as an optimization problem. the similar approach was already used in [33] for binary-state components and systems. it has been shown that investing in a set of critical components obtained using optimization approach provides higher reliability of the system than the reliability provided by investing in a set of components obtained using traditional im. the optimization problem formulation in this paper starts from definition of system state function based on minimal cut sets. the resulting problem is the global optimization problem with max-min-max objective. therefore, we are about to solve them by solving the weighted minimal hitting set problem [34], also known as the minimal cost hitting set problem [35], which is complementary to the set covering problem. the paper consists of five sections. the following section introduces the continuousstates systems, definition of the problem of simultaneous determination of the critical components and its mathematical programming formulation. section 3 is devoted to solving the approach, where we prove the relationship between the solution of original max-min-max and weighted minimal hitting set problems. in section 4, computational results of experiments on a group of benchmark instances are given. we compare the results of max-min-max and weighted minimal hitting set problems. then we examine the influence of costs variances on the selection of critical components and the system state. finally, we compare the state of the systems obtained by the proposed approach with the state obtained based on two cost related im. section 5 contains the conclusions and directions for further research. 2. problem definition and mathematical model formulation let k = {1, 2, ..., n} be a set of system’s components and let xj ∈ [0, 1] be a state (performance level) of j-th component, j ∈ k. the vector x = (x1, x2, ..., xn) is called the state vector. the state of the continuous system is defined by the continuous structure function as follows [36]. 4 d. makajić-nikolić, m. stanojević1, n. nikolić, p. pavlović definition 1. a function γ: [0, 1]n[0, 1] which is nondecreasing in each coordinate of x is said to be a continuum structure function. continuum structure function expresses how the state of the coherent system depends on the components’ states. one way of determining the system state is through the minimal cut sets [37, 38]. definition 2. cut set is a set of components whose failure causes the system failure. minimum cut set (mcs) is a cut set that does not contain another cut set, that is, it is reduced to the minimum number of components whose failure causes system failure. for the binary-state components and systems, the previous definition implies that disabling the failure of some component will eliminate all minimal cut sets containing that component as a potential cause of the system failure. according to [36], continuum structure function for coherent systems can be defined as ( ) min max i j i p j c x x    (1) where p= {1, 2, ..., m} represents the set of mcs and ci is the set of components in i-th mcs. function γ is also called the system state function and it represents the state of the ”best” component in the ”worst” mcs. starting from the last interpretation of the system state, initial problem can be defined as: determining the minimal number of components whose improving of states (performance level) maximizes ( )x , i.e., the state (performance level) of the system. since the improving of the component’s performance levels requires costs and considering that the maintenance budget is almost always limited, the final definition of the problem that is observed in this paper is: allocate the available budget to the components whose increasing of performance levels maximizes the system performance level. the main assumptions of the defined problem are: 1. the continuous-state system is coherent. 2. the state space of components is [0, 1]n and the state space of the system is [0, 1], where the state 1 represents the perfect functioning, while the state 0 represents total failure. the states in (0, 1) correspond to different degradation levels, but not to complete failure. 3. all components’ failures are statistically independent. 4. the cost of providing a component j with a certain performance level linearly depends on its performance level. 5. mcss of a given system are already determined. to formulate the mathematical model of the defined problem, we introduce the set of all components (k), the set of mcs (p), and the set of components in i-th mcs (ci), such that i ∈ p, i p ic k  . for each component j ∈ k the cost of full performance level (cj) is known, as well as the total budget for achieving performance level of the system (b). decision variable that should be determined (xj) represents the performance level of jth component, j ∈ k. based on optimal values of variables, performance level of the system, pl(x), such that ( ) ( )pl x x can be obtained. the mathematical model for maximization of the system performance level (mspl) under the budget constraint, can now be formulated as follows: optimization based determination of the set of continuous-state components critical to the system state 5 j (max) ( ) min max s.t. c 0 1, i j i p j c j j k j pl x x x b x j k          the objective function represents the continuous system state defined in (1). its maximum is the value of the ”best” component in the ”worst” mcs. constraint in mspl refers to the limited budget. the formulated problem is the global optimization problem with max-min-max objective. the classes of max-min-max problems have been proven to be np-hard [39]. it is possible to reformulate this problem in such a way that the optimal solution does not depend on the available budget. in the following section, weighted minimal hitting set formulation of the problem of determining the entire set of critical components will be presented. 3. solving approach in order to formulate weighted minimal hitting set problem variant of critical components determination problem, the vector of binary variables y = (yj), j ∈ k is introduced. a variable yj is an indicator of criticality of the component j: if the component j is critical, then yj = 1, otherwise yj = 0. the remaining notation is the same as the notation used in mspl model. the mathematical model for minimization of the critical components’ cost (mccc) is formulated as follows. (min) ( ) s.t. 1, {0,1}, i j j j k j j c j fc y c y y i p y j k          mccc model corresponds to the weighted minimal hitting set problem [40]. the constraint ensures that each mcs is hit, i.e. each mcs contains at least one component j ∈ k such that yj = 1. the objective fc (y) represents the cost of the full performance level for components { | }j jy y y , since components in y hit all mcss. the following theorem determines the relationship between the mspl and mccc models, i.e., between the optimal solutions of these two problems. theorem 1. if y* = {yj}, j ∈ k is the optimal solution of mccc problem then x*={xj}, j ∈ k is the optimal solution of mspl problem, where min 1, , ( *) j j by x j k fc y        (2) 6 d. makajić-nikolić, m. stanojević1, n. nikolić, p. pavlović ̂ proof. from (2) it follows 0 ≤ xj ≤ 1, for j ∈ k and ( *) ( *) j j j j j j j k j k j k by b c x c c y b fc y fc y        therefore, x* = {xj}, j ∈ k is a feasible solution of mspl. let now prove that x∗ is the optimal solution of mspl. since for each i ∈ p there is j ∈ ci for which yj = 1, it follows that max min 1, ( *)i j j c b x fc y        . moreover, ( *) min max min 1, ( *)i j i p j c b pl x x fc y          . in case of fc(y∗) ≤b, x* is the optimal solution of mspl because pl(x*) = 1. in case of fc(y∗) > b, let us assume that x* is not an optimal solution of mspl. that means that there is a feasible solution { },jx x j k  such that ( ) ( *) ( *) b pl x pl x fc y   . further, let 1, ( ) , 0, ( ) j j j x pl x y j k x pl x      . given that ( ) min max i j i p j c pl x x    , it follows that for each ip there is ici for which ( )jx pl x , i.e. for which 1 j y  . hence, { },jy y j k  is a feasible solution of mccc. furthermore, from the definition of jy it follows that ( ) j j x y pl x  , from which we obtain 1 ( ) ( *) ( ) ( ) ( ) j jj j jj j k j k j k x b fc y c y c c x fc y pl x pl x pl x          . however, this is contrary to the fact that y∗ is the optimal solution of mccc, i.e. to the assumption that x* is not an optimal solution of mspl. hence, x* is the optimal solution of the mspl problem, which has been proved. therefore, the optimal solution of the mspl problem can be obtained from the optimal solution of the mccc problem. based on the equation (2), we conclude that there is an optimal solution x* = {xj}, j ∈ k of mspl such that all values of xj are equal to 0 (when yj = 0) or are equal to min 1, ( *) b spl fc y        (when yj = 1). if fc(y*) < b, then the system can be provided with the full performance level (spl = 1) using less founds then the provided budget b. optimization based determination of the set of continuous-state components critical to the system state 7 if fc(y*)  b, then the system can be provided with maximal performance level, in relation to the given budget b. the performance level of the entire system pl (x*) is also equal to min 1, ( *) b spl fc y        (3) the mccc model refers to the weighted minimal hitting set problem which is proven to be np-hard. however, solving mccc is much more efficient than solving mspl for the problem of determining the critical components. moreover, in mccc, the amount of available budget does not influence the optimal solution. consequently, changes in the available budget do not require a restart of the optimization process, unlike the mspl formulation. the new optimization should be done only if the cost cj of some component jk changes. in addition, mccc can be used for determining the system states that can be reached with different values of available budget. when the optimal solution is obtained, it is enough to use the formula (3) to calculate the new state (performance level) of the system or to calculate minimal budget needed to achieve a required level of system performance. 4. numerical results three sets of experiments were conducted on 13 benchmark fault trees (bft) from [41], whose characteristics are given in table 1. columns c and mcs give the total number of components and the minimal cut sets of bfts, respectively. column r shows the range of mcss, i.e. the minimal and maximal number of components in mcss. table 1 benchmark fault trees [41] no bft c mcs r 1 baobab1 61 46188 2-11 2 baobab2 32 4805 2-6 3 baobab3 80 24386 2-11 4 chineese 25 392 2-6 5 das9201 122 14217 2-7 6 das9202 49 27778 1-11 7 das9208 103 8060 2-6 8 edf9205 165 21308 1-8 9 ftr10 152 305 1-3 10 isp9603 91 3434 2-8 11 isp9605 32 5630 3-7 12 isp9606 89 1776 1-5 13 jbd9601 532 14007 1-7 all experiments were performed on a laptop computer equipped with intel i7 cpu at 2.7ghz and 16 gb of ram. problems were solved using ampl modeling environment and lgo solver for mspl problems and cplex solver for mccc problems. lgo solver 8 d. makajić-nikolić, m. stanojević1, n. nikolić, p. pavlović is intended to solve global optimization problems with potentially many local optima. cplex is considered a state-of-the-art solver for linear and mixed integer programming. random instances were generated using the library of bfts from table 1 and assigning random values to the components’ cost. the budget for improving the system performance level for each instance was determined as a constant percentage of the sum of all components’ cost. the percentage was adjusted for each bft in such a way that all instances have maximal performance level between 0.4 and 1. components’ costs were generated as random integers that have erlang’s distribution. mean and variance were adjusted differently in each set of experiments. considering that system performance level for all problems is in interval [0, 1] independently of the values of components’ cost, we were allowed to aggregate experimental results and finally find some conclusion. 4.1 comparison of mspl and mccc in the first set of experiments, we were testing the usability of models mspl and mccc. mspl is a global optimization problem that can have many local optima. mccc is an np hard binary programming problem. for the test, we used bfts and we generated 30 random instances for each of them. we solved them using both mathematical models. the average system performance levels (spl) and time in seconds needed to solve the problems are presented in table 2. table 2 mspl and mccc results mspl mccc gap [%] bft spl time [sec.] spl time [sec.] spl baobab1 0.6071 300.07∗ 0.7138 0.68 14.95 baobab2 0.6229 6.03 0.6966 0.28 10.58 baobab3 0.5438 300.03∗ 0.6621 0.55 17.87 chineese 0.5776 0.36 0.5788 0.01 0.21 das9201 0.1225 190.21 0.6584 0.13 81.39 das9202 0.5574 121.73 0.5821 0.38 4.24 das9208 0.3949 112.46 0.6313 0.06 37.45 edf9205 0.8181 300.02∗ 0.8212 0.38 0.38 ftr10 0.7258 6.96 0.7399 0.01 1.91 isp9603 0.4238 33.05 0.6055 0.05 30.01 isp9605 0.6071 8.42 0.6572 0.19 7.62 isp9606 0.6603 9.63 0.7386 0.03 10.60 jbd9601 0.7395 300.05* 0.8188 0.25 9.68 average: 0.5693 129.92** 0.6849 0.23 16.89 * optimization was interrupted after 5 minutes time limit. ** the average was calculated using the limited time. optimization based determination of the set of continuous-state components critical to the system state 9 it is obvious that lgo was not able to solve all of the problems to optimality, since the values in spl column of mspl are less than those in the spl column of mccc. the gap between obtained spls varied from 81.39% for “das9201” bft to relatively modest 0.81% for “chineese”. moreover, for “das9201” the gap is very big even the solver “concluded” that the optimal solution was found. on the other hand, optimality of solutions obtained by mccc is guaranteed. also, the difference in execution time is significantly better for mccc model. even relatively big instances like “baobab1” were solved in less than a second. after this practical confirmation of dominance of mccc model over mspl, further experiments were performed only using mccc model. 4.2 sensitivity of mccc solutions the aim of the second set of experiments was to assess the sensitivity of solutions on the dispersion of components’ cost. for each bft, we generated three sets of 10 random instances with variances 4v1 = 2v2 = v3, respectively, but with the same mean. for each value of variance, we presented the average number of components in the solution (cs), and the average system performance level of the entire system in table 3. table 3 mccc results for different variance of cost values v1 v2 v3 bft cs spl cs spl cs spl baobab1 11.0 0.6451 11.6 0.6982 11.5 0.7981 baobab2 14.2 0.5214 14.1 0.5561 14.4 0.5943 baobab3 17.7 0.6167 17.4 0.6659 18.7 0.7037 chineese 5.4 0.5611 5.6 0.5847 5.8 0.5905 das9201 9.0 0.6604 9.0 0.6631 9.0 0.6517 das9202 8.0 0.5256 8.0 0.5348 8.2 0.6858 das9208 17.2 0.6144 17.4 0.6480 17.4 0.6315 edf9205 40.0 0.8349 40.0 0.8112 40.0 0.8176 ftr10 83.6 0.7317 83.2 0.7401 83.6 0.7477 isp9603 17.2 0.5899 17.7 0.5881 17.7 0.6385 isp9605 8.0 0.6179 8.0 0.6375 8.6 0.7161 isp9606 34.2 0.7140 35.3 0.7308 35.9 0.7709 jbd9601 270.0 0.8044 269.6 0.8163 275.5 0.8356 average 41.2 0.6490 41.3 0.6673 42.0 0.7063 from the presented results, we can see that a better system performance level can be obtained if the components have more disperse costs. we explain that by the possibility that a set of cheaper components that hit all cuts can be found if there are more both cheap and expensive components, i.e., if variance is greater. 10 d. makajić-nikolić, m. stanojević1, n. nikolić, p. pavlović 4.3 comparison of mccc and traditional ims in the third set of experiments, we compared the mccc model with traditional ims. in the absence of a cost-based im for continuous states in available literature, two cost binary ims were used: cost-based component importance (cbci) and cost-effective importance measure (ceim). although both im are primarily developed for measuring the impact of the components’ reliability on the overall system reliability, the terminology below will be generalized to the state, i.e., the level of performance of components and systems. the notation in the equations will be adjusted according to this generalization. importance measure cbci was introduced in [42] and it is formulated as the birnbaum im extended by costs of components’ improvements. cbci of component j is defined as: jcbci j j c i pl     (4) where ∂cj and ∂plj represent the increase of the system cost and the system state caused by state improvement of jth component. the components with lower values of cbci ji are considered more important since they provide greater system state improvement with lower increase of costs. importance measure ceim is defined as [43]: , gi jceim j f j i i c  (5) where igi and cf,j represent the general importance (gi) and cost factor for component j, respectively. gi of component j is obtained as: jgi j pl i pl   (6) where jpl is the change in system state caused by the change in jth component state. the cost factor for jth component is calculated as: , ( ) ( ) j j k f j j e c c e c    (7) where e(cj) is the expected cost of state improvement for j-th component. components with higher value of ceim ji will be considered the most important components, since they provide higher state at lower costs. for each instance of bfts from table 1, values of cbci ji and ceim ji were calculated. then, the components were ranked by obtained values of im. for each instance the first m components that hit all mcs of a given bft were selected. the total cost of the selected components represents the budget required to achieve the state of the system equal to 1. the ratio of total cost and the given budget for an instance gives the system state which can be achieved by investing in the selected components. these values are given in table optimization based determination of the set of continuous-state components critical to the system state 11 4. the column min gives the optimal solution of minimal hitting set problem where cost cj is equal to 1 for all components j  k. the columns cc show the average number of components obtained by solving mccc and m selected component based on cbci ji and ceim ji , respectively. the columns slp contain corresponding systems’ performance levels. table 4 results of mccc and cost-based ims mccc cbci ceim bft min cc spl cc spl cc spl baobab1 11 11.6 0.8097 37.5 0.2746 40 0.1433 baobab2 14 14 0.5728 25.2 0.3582 24.6 0.2381 baobab3 17 18.7 0.5656 61.6 0.1934 60.7 0.1278 chineese 5 5.6 0.6218 16 0.2829 12.7 0.1907 das9201 9 9 0.6380 105.7 0.0724 37.4 0.1344 das9202 8 8 0.7250 31.7 0.2071 24.4 0.1568 das9208 17 17.2 0.6489 91 0.1374 59.9 0.1549 edf9205 40 40 0.8251 160.1 0.2210 136.1 0.2379 ftr10 79 83.7 0.7419 150.8 0.4184 142.5 0.4220 isp9603 17 17 0.6559 71.3 0.1737 54 0.1666 isp9605 8 8 0.7845 22.1 0.3406 18.5 0.2656 isp9606 34 34.6 0.5890 84 0.2391 77.3 0.2224 jbd9601 268 273.2 0.8183 529.5 0.4087 523.8 0.4057 average 40.5 41.6 0.6920 106.7 0.2560 93.2 0.2205 from the results in table 4 we can see that the mccc model outperforms cbci ji and ceim ji for all bfts. based on the results of mccc, the number of components that hit all mcs is the same as in the minimal hitting set case for 6 of 13 bft. the numbers of critical components obtained based on cbci ji and ceim ji are much larger, which means that they require much higher costs of selected components. consequently, system performance levels that can be achieved using available budgets are significantly lower than levels obtained by mccc model. 5. conclusions the goal of this paper was to propose a new approach for determination of the critical components, i.e., the components whose performance level has the biggest effect on the performance level of the mechanical system. criticality of the components is usually defined by means of importance measures, but most of them are not directly applicable to the determination of the group of the critical components. to overcome this lack of im, we 12 d. makajić-nikolić, m. stanojević1, n. nikolić, p. pavlović have proposed an optimization approach that enables simultaneous determination of all of the most critical components. the approach was applied on continuous-state components of continuous-state system. the problem was defined as a problem of allocation of the available budget to the components whose increasing of performance levels maximizes the system performance level. originally formulated as the global optimization problem, it was reformulated as the weighted minimal hitting set problem. the experiments that were conducted on a group of bfts showed that the proposed approach outperforms those based on the chosen cost related ims. besides determination of the critical components, the proposed approach enables calculation of the system states that can be reached with different values of available budget. since the formulated problem is np-hard, further research will be directed towards testing on large instances and developing suitable 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30(4), pp. 379–386. http://www.itu.dk/research/cla/externals/clib/ 11256 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 217 234 https://doi.org/10.22190/fume2302270018z © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper design methodology and characteristics analysis of high-order multi-segment deformed eccentric non-circular gear huacheng zhao1,2, jianneng chen1, gaohuan xu2 1faculty of mechanical engineering, zhejiang sci-tech university, china 2school of mechanical and automotive engineering, zhejiang university of water resource and electric power, china orcid ids: huacheng zhao https://orcid.org/0000-0001-7000-8452 jianneng chen https://orcid.org/0000-0002-3816-2692 gaohuan xu https://orcid.org/0000-0003-4255-5154 abstract. a novel gear mechanism named high-order multi-segment deformed eccentric non-circular gear was proposed for achieving the unification of non-circular gears with typical-form pitch curve and non-circular gears with free-form pitch curve. the transmission mechanism of the high-order multi-segment deformed eccentric non-circular gear was analyzed, and the unified mathematical expression of eccentric gears was established. the non-circular gears with free-form pitch curve could be constructed based on the proposed high-order multi-segment deformed eccentric non-circular gear by changing the parameters. moreover, the transmission characteristics were discussed, such as the transmission ratio relationship, convexity distinguished conditions, curvature radius of pitch curve. the visual design and simulation software and generation software of tooth profile for non-circular gears are compiled based on matlab, and was verified with the example. this novelty gear was applied to the drive mechanism of the metering pump. the mathematical model of non-circular gear-crank slider mechanism is established and the 3d model and virtual prototype experiment of the mechanism are accomplished. the application showed that the high-order multi-segment deformed eccentric non-circular gears were feasible in practice. key words: concavity, convexity, eccentric gear, free pitch curve, high-order, multi-segment, drive mechanism, metering pump received: february 27, 2023 / accepted may 30, 2023 corresponding author: huacheng zhao school of mechanical and automotive engineering, zhejiang university of water resource and electric power, hangzhou, 310018, china e-mail: zhaohc@zjweu.edu.cn https://orcid.org/0000-0001-7000-8452 https://orcid.org/0000-0002-3816-2692 https://orcid.org/0000-0003-4255-5154 218 h. zhao, j. chen, g. xu 1. introduction gears belong to the group of most effective mechanical power transmission components. they are characterized by very high efficiency, reliability, compact design, ease of finding a way to fit in specific applications – to name but a few properties. many researchers devoted their work to investigate different aspects of gears, such as their properties [1], health monitoring and diagnostics [2], thermal loadings [3], specific applications [4], wear resistance improvement [5], etc. this paper considers non-circular gears. non-circular gears can be accurately designed for intended movements to achieve periodic non-uniform transmission. due to their precise transmission and ease of balance, non-circular gears are widely applied in agricultural machinery, packaging machinery, pumps, and light industry machinery [6-9]. according to the basic law of gear meshing, the pitch curves of a pair of gears with variable transmission ratios are no longer circular curves. the pitch curves of non-circular gears mainly include typical-form pitch curve and free-form pitch curve. free-form pitch curve cannot be clearly expressed with a mathematical function but often as a segmental curve. a method for constructing the n-lobe non-circular gear satisfying a specified displacement law with bézier curve and b-spline curve was presented by riaza at al. [10]. medvecká-beňová [11] presented a design of pitch curves for a given set of parameters and described how to apply an eccentric elliptical gear with a continuously changing transmission gear ratio to optimize the pitch curves of non-circular gears. fanghella [12] presented a new method for designing non-circular gears under a given law of motion based on a mixed symbolic-numeric approach. li et al. [13] proposed to express non-circular gear pitch curves by means of numerical points instead of any function, for the purpose of fitting complex form pitch curves. in the attempt of analyzing the generation process of noncircular gears, vasie and andrei [14] introduced the gielis super-shape equation to fit the pitch curve. this method was focused on convex-concave gear generation. gears with typical-form pitch curve can be expressed with typical mathematical models, such as elliptical gear, eccentric gear and fourier non-circular gear. elliptical gear is the most commonly used gear and has been extensively studied. subsequently, high-order elliptical gears and deformed elliptical gears are proposed through adjusting the numbers of cycle of pitch curves [15-18]. additionally, non-circular gears with typical-form pitch curve have also been vigorously explored and investigated in the design and transmission characteristics, including eccentric noncircular gears, fourier noncircular gears, and pascal spiral gears, further to enrich gear transmission form [19-21]. eccentric noncircular gears have also been widely studied because of a simple pitch curve and convenient processing. in the studies on eccentric circular gears, the high-order eccentric gears have been obtained by keeping the polar radius of eccentric circular gear unchanged and reducing the polar angle by an integral multiple. the high-order eccentric gears can get different numbers of blade with different orders, which can realize multiple periodic transmission ratio [22]. chen et al. [23] have proposed deformed eccentric noncircular gear by changing the transmission ratio into two asymmetric segments and deduced the equation of pitch curve, which is applied to the weft insertion mechanism. the varying forms of pitch curves make it impossible to obtain non-circular gears with free pitch curve, thereby increasing great difficulty of standardized design of non-circular gears. non-circular gears with typical-form pitch curve have less flexible pitch curves and limited changes in transmission ratios and, hence, they are unable to satisfy a much broader design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 219 range of transmission requirements. to unify these two kinds of non-circular gears and standardize the design, a high-order multi-segment deformed eccentric non-circular gear is put forward to construct non-circular gears with free pitch curve. the novel gear divides the pitch curve into more than three segments in a cycle and makes the shape of the pitch curve more changeable and enriching the transmission types of non-circular gear. the advantages of periodic multistage variable transmission of high-order multi-segment deformed eccentric non-circular gear can better meet the requirements of variable transmission ratios, which can be used as a driver for the seal mechanisms of horizontal pillow packing machine, differential pump, etc. the gear pair can also be combined with other mechanisms. the high-order multi-segment deformed eccentric non-circular gear pair and crank slider mechanism can be connected to form a combined mechanism, which can greatly improve the stability of the slider speed and reduce the impact vibration. by adjusting the parameters of the combined mechanism, specific movement output can be obtained, which can be applied to metering pump, hay baler mechanism and pulsating blood flow generator. in this article, section 2 puts forward transmission mechanism of high-order multi-segment deformed eccentric non-circular gear and a mathematical model of that is built. section 3 analyzes the transmission characteristics, such as transmission ratio, curvature radius and convexity distinguished conditions. in section 4, the visual design and simulation software is developed and a design case is illustrated to verify the feasibility of the proposed gears. then, an application is given, followed by conclusions. 2. design of high-order multi-segment deformed eccentric non-circular gear 2.1. transmission mechanism of multi-segment deformed eccentric non-circular gear in general eccentric non-circular gear transmission, the driving gear 1 is an eccentric circular gear and the driven gear 2 is a non-circular gear which conjugates with the eccentric circular gear, as indicated in fig. 1. based on the analysis of equations of pitch curve of general eccentric non-circular gears, we proposed a high-order deformed eccentric non-circular gears [23]. the equations of pitch curve are as follows. 2 2 11 1 11 1 1 11 1 1 1 11 2 2 12 1 12 1 1 12 1 2 1 1 1 11 1 ( 1 sin ( ) cos( )) for (0 ) 2 2 2 ( 1 sin ( ) cos( )) for ( )  = − −         = − − − −          r r k n m k n m n m r r k n m k n m n n n m n            (1) where r11, r12 are the polar radii of the eccentric gear for two segments, r is radius of the eccentric gear, k is the eccentricity ratio (k=e/r, e is the eccentricity of eccentric gear), 1 is the polar angle of the eccentric gear, n1 is the order of the eccentric gear, m11, m12 are the deformation coefficients. the pitch curve expressed in eq. (1) for the high-order deformed eccentric non-circular gear presents two asymmetrical segments in each cycle. thus, the number of deformed segments is small, and the adjustment range of transmission ratio is limited. the pitch curve is still a typical form and cannot replace non-circular gear with free pitch curve. 220 h. zhao, j. chen, g. xu r r1 r2 1 2 x y p o 2 1w 2w 21 2o1o fig. 1 pitch curves of eccentric non-circular gears for the possibility of changing pitch curve forms and obtaining a wider range of transmission ratio, the pitch curve is divided into n (more than three) segments eccentric circle with different deformation coefficients. in this manner, a multi-segment deformed eccentric non-circular gear can be obtained. every segment is a deformed eccentric circle. eq. (2) is the equation of the pitch curve of multi-segment deformed eccentric non-circular gear. 2 2 11 11 1 11 1 1 11 2 2 12 12 1 12 1 11 11 1 11 11 12 1 2 2 1 1 1 1 1 1 1 2 ( 1 sin ( ) cos( ) for (0 ) 2 2 2 2 ( 1 sin ( ( ) ) cos( ( ) )) 2 2 2 for ( ) 1 2( 1) ( 1 sin ( ( 2 ) )) cos( ( 2 − = = − −   = − − + − − +   + − = − − + − −  j j j k k j r r k m k m nm r r k m k m nm n nm n nm nm nm j r r k m nm n k m                    1 1 1 1 1 11 1 1 1 2( 1) 2 2 ) ) for ( ) − − = = =               − +        j j j k k kk k k j nm n nm nm     (2) where r1j is the radius of the eccentric gear for segment j, j is the sequence number of the segment, while j=1,2,3…n, n is the number of segments, m1j is the deformation coefficient for segment j, 1 1 j m n and must satisfy the following equation: 1 1 1 1 n k k n m= = (3) according to eq. (2), for 1 1 1 1 1 2 j k k nm   − = =  , design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 221    2 2 1 1( 1) ( 1 sin 2( 1) cos 2( 1) ) − = = − − − − j j r r r k j n k j n  (4) for 1 0 = and 1 2 = , 1 11 (1 ) n r r r k r e= = − = − (5) hence, the pitch curve of a multi-segment deformed eccentric non-circular gear formed by eq. (2) is continuous at 1 1 1 1 1 2 j k k nm   − = =  , 1 0 = and 1 2 = . therefore, it is a continuous closed curve. 2.2. transmission mechanism of high-order multi-segment deformed eccentric non-circular gear as shown in fig. 2, combined with the formation principles of the pitch curves for a high-order eccentric gear and a multi-segment deformed eccentric gear, the pitch curve of the high-order eccentric gear is divided in each cycle into n ( 3n  ) segments to form a high-order multi-segment deformed eccentric non-circular gear. the equation of the pitch curve of the driving gear is given as follows: 2 2 1 1 1 11 1 1 11 1 1 1 11 2 2 1 2 1 12 1 1 12 1 1 11 1 11 1 1 11 1 11 1 12 1 2 2 1 1 1 1 1 1 1 2 ( 1 sin ( ) cos( )) for (0 ) 2 2 2 2 ( 1 sin ( ( ) ) cos( ( ) )) 2 2 2 for ( ) 1 ( 1 sin ( ( 2 − = = − −   = − − + − − +   + = − − i i j ij j k k r r k n m k n m nn m r r k n m k n m nn m n nn m n nn m nn m nn m r r k n m nn m                 1 1 1 1 1 1 1 1 11 1 1 1 1 1 2( 1) ) ) 1 2( 1) 2 2 cos( ( 2 ) )) for ( ) − − = = =           − +   − − − +         j j j j k k kk k k j n j k n m nn m n nn m nn m        (6) where r1ij is the radius of the eccentric gear for segment j of cycle i, while i is the cycle number of the driving gear and 11,2,3 ,i n=  . according to eq. (6), in any cycle i, for 1 1 1 1 1 1 2 j k k nn m   − = =  : 2 2 1 1 ( 1) ( 1 sin [2( 1) / ] cos[2( 1) / ]) − = = − − − − ij i j r r r k j n k j n  (7) eq. (7) shows that all the segments of the deformed pitch curve within each cycle are connected. at the junction of any adjacent cycles, for 1 0 = or 1 1 1 ( 1) 1 2 j k i k nn m   = − =  , 222 h. zhao, j. chen, g. xu 1 1 1( 1) (1 ) i i n r r r k r e − = = − = − (8) eq. (8) shows that all the segments of the deformed pitch curve ae connected. r112 o r111 r1ij 2/nnm1 2/nnm2 2/nnmj fig. 2 schematic diagram of pitch curve therefore, the pitch curve of a non-circular gear, which is given by eq. (6), is a continuous closed curve. and within a cycle of 2 , each segment of the pitch curve is connected. according to the basic requirement of gear meshing, the variation of angular displacements of the driving gear and the driven gear should be accomplished in the same period. this can be expressed by the following relationship [24]: 1 1 1 1 2 1 0 0 12 1 1 d r d i a r    = = −  (9) consequently, the closure condition of the pitch curves of a high-order multi-segment deformed eccentric non-circular gear is established as: 1 2 2 1 1 11 11 1 11 1 11 1 1 10 0 222 12 1 1 1 1 1 1 2 2 2 1 11 1 1 11 11 11 1 2 2 20 1 11 1 1 11 1 2 2 2 1 1 1 1 1 2 1 sin ( ) cos( ) ( sin ( ) cos( )) 2 sin ( ( = − = = = = + − − − − = + − − − − −       j n nn m kn nn m iji k j ji ij nn m k k nn m j k rr d d d n i a r a r r e n m e n m d a r e n m e n m r e n m nn m                 1 1 1 1 1 12 1 1 1 1 1 11 1 1 1 11 22 2 2 2 1 1 1 1 1 1 1 1 1 1 11 1 1 1 2( 1) 2 2( 1) ) ) cos( ( ) 2 2( 1) 2 2( 1) ( sin ( ( ) ) cos( ( ) )) − − = == − − − = = = = − − + − − + − − − − − + − − +        j j j jn nn m k k k kk j j j j nn m k k j j k kk k j j e n m n nn m n d j j a r e n m e n m nn m n nn m n              (10) where n2 is the order of the driven gear. design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 223 the centre distance a is solved based on eq. (10). the centre distance a is obviously controlled by the following design parameters: r, e, n1, n and mij. the process of solving the centre distance is shown below: (1) searching the unimodal interval where the centre distance is located with the advance-retreat method; (2) determining the centre distance with the golden section method. from eq. (6) and eq. (10), the equations of radius and polar angle of the driven gear are written as eq. (11) and eq. (12), respectively: 2 2 2 1 1 11 1 1 11 1 1 1 11 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 2 ( 1 sin ( ) cos( )) for (0 ) 1 2( 1) ( 1 sin ( ( 2 ) ) 1 2( 1) 2 2 cos( ( 2 ) )) for ( ) − = − − = = =  = − − −     − = − − − + − − − +        i j ij j k k j j j j k k kk k k r a r k n m k n m nn m j r a r k n m nn m n j k n m nn m n nn m nn m                  (11) 2 2 2 1 1 11 1 1 11 1 1 1 2 2 20 1 111 11 1 1 11 1 2 2 2 2 1 11 1 1 11 11 11 1 2 2 20 1 11 1 1 11 1 2 2 22 1 1 1 sin ( ) cos( ) 2 for (0 ) ( sin ( ) cos( )) sin ( ) cos( ) ( sin ( ) cos( )) 2 sin ( ( − −   − − − − − + − − − = − −   nn m j r e n m e n m d nn ma r e n m e n m r e n m e n m d a r e n m e n m r e n m n                 1 1 1 1 1 1 1 11 1 1 1 1 12 1 1 2 2 21 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 2( 1) 2 2( 1) ) ) cos( ( ) ) 2 2( 1) 2 2( 1) ( sin ( ( ) ) cos( ( ) )) 2 2 for ( − − = = − − − = = = − = − − + − − + − − − − − + − − +          j j j k kk k n j j nn m k k j j k kk k j k k j j e n m n m n nn m n d j j a r e n m e n m nn m n nn m n nn m                 1 1 1 ) =                   j k k nn m (12) clearly, the driven gear, which conjugates with the driving gear, is a non-circular gear with n2-order and n-segment. for different numbers of segments n, the following is valid: (1) if n=1, m1j=1, n1=n2=1, the driving gear 1 is a general eccentric circular gear and driven gear 2 is a non-circular gear, which conjugates with driving gear 1. (2) if n=2, the driving gear 1 is a high-order deformed eccentric non-circular gear as shown in eq. (1), and the driven gear 2 is a high-order deformed non-circular gear, which conjugates with the driving gear 1. (3) if 3n  , driving gear 1 is essentially a non-circular gear with n1-order and n-segment and the driven gear 2 is non-circular gear with n2-order and n-segment which conjugates with the driving gear 1. therefore, the high-order multi-segment deformed eccentric non-circular gears proposed in the paper are more general and unified in theory, and more extensively used in practice, as high-order eccentric non-circular gear and multi-segment deformed eccentric non-circular gear are covered. through changing the design parameters such as r, e, n1, n2, 224 h. zhao, j. chen, g. xu n and mij, the non-circular gears with a free-form pitch curve could be composed by the high-order multi-segment deformed eccentric non-circular gear. fig. 3 shows two examples of pitch curves, with different order and segment numbers, while the other parameters are the same: (a) n1=1, n2=3, n=2 and (b) n1=3, n2=2, n=3. fig. 3 cases of the pitch curves: left: n1=1, n2=3, n=3; right: n1=3, n2=3, n=3 3. analysis of transmission characteristic 3.1. transmission ratio according to the definition of the transmission ratio, the transmission ratio of high-order multi-segment deformed eccentric non-circular gear is expressed as follows. to simplify the analysis, let 1 1 1 1 1 11 2 2( 1) ( ) j j kk j n m nn m n     − = − = − + . ( 1) 2 2 1 11 2 1 12 2 2 2 1 1 1 1 ( 1 sin cos ) 2 2 for ( ) ( 1 sin cos ) − − −− = = = =   − − a r k kr a r j i r r n nr k k         (13) for   1 1 1 1 1 11 2 2( 1) ( ) 0,2 j j kk j n m nn m n     − = − − +  , when 1 1 1 1 1 11 2(1 ) / 2 j j kk j n n m nn m    − = − = + , i12 is at the minimum: 12 min ( )a r e i r e − + = + (14) for 1 1 11 1 1 1 2(1 ) / 2 j kj k j n n m nn m     − = + − = + , i12 is at the maximum: 12 max ( )a r e i r e − − = − (15) therefore, according to eq. (15) and (16), the maximum and minimum transmission ratios are designed to determine the range of i12. design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 225 3.2. concavity and convexity the pitch curves of high-order multi-segment deformed eccentric circular non-circular gear may be concave. in order to drive stably and conveniently, the concavity and convexity of the pitch curves shall be verified. the curvature radius of pitch curve of high-order multi-segment deformed eccentric non-circular gear is obtained based on the knowledge of differential geometry, hence: 3 2 2 2 2 2 2 2 = 2 dr r d dr d r r r d d            +         + −    （ ） （ ） （ ） （ ） (16) as for the driving gear, the derivations of the first segment of the pitch curve in each cycle are calculated from eq. (6). ( ) ( ) ( ) 2 1 11 1 11 11 11 1 11 1 11 1 2 2 2 1 1 1 11 1 sin 2 = sin 2 sin i e n m n mdrdr en m n m d d r e n m      = − + − (17) ( ) ( ) ( ) ( ) ( ) 2 2 2 4 2 2 222 1 11 1 11 1 1 11 1 11 12 21 11 1 11 1 11 12 2 2 2 2 2 2 2 3 1 1 1 11 1 1 11 1 cos 2 sin 2 = cos sin 4 ( sin ) i e n m n m e n m n md rd r en m n m d d r e n m r e n m        = − − − − (18) the curvature radius of the first segment curve could be solved by substituting eqs. (17) and (18) into eq. (16). when the curvature radius meets the following condition, the pitch curve is convex: 0  （ ） (19) for 1 11 1 0n m  = , the curvature radius of the first segment is at its minimum: ( ) ( ) 2 1 2 2 1 11 1 1 i r e r e n m k  − = − + −   (20) from eq. (14), it can be seen that the numerator is greater than zero. if ( )2 2 1 11 1 1 0r e n m k− + −    , the pitch curve of this segment becomes convex, so the convexity distinguished condition of this segment can be simplified as: ( )( )2 2 1 11 1 1 0n m k k− −  (21) in this case, the convexity distinguishing conditions is expressed below: 2 2 1 11 1 k n m  (22) 226 h. zhao, j. chen, g. xu for the driving gear, the convexity distinguished condition in each segment is: 2 2 1 1 1 min j k n m          (23) the curvature radius of the driven gear can also be solved using eq. (16), but the derivation is complicated. in this paper, it is calculated by euler-savery formula [24], which is expressed as: 1 1 2 1 2 1 1 1 1 sin p p r r      + = +    (24) where ( )2 2 2 1 1 11 1 1 sin sinr e n m  = − . after simplification and calculation, the convexity distinguishing condition of the pitch curve for each segment is expressed below: ( ) 2 2 1 1 2 2 1 min 1 / 0 j k n m k ae r             + −   (25) the non-circular gears can be processed by hobbing when they meet the conditions given by eqs. (23) and (25). 4. design case and discussion 4.1. compiling of design software in order to better analyze their transmission characteristics, the visual design and simulation software and the generation software of the tooth profile for non-circular gears are developed using matlab based on the equations of the pitch curve of high-order multi-segment deformed eccentric non-circular gear, which have been established in section 2, as showed in figs. 4 and 5. based on the mechanism parameters (r, e, n1, n2, n and mij), the software can determine the center distance, the transmission ratio and the curvature radius, as well as the kinematic curves of the driven gear, including the displacement, velocity and acceleration. the developed software also enables simulation of the pitch curves development. design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 227 fig. 4 software of visual design and simulation fig. 5 software of generating tooth profile 4.2. design case and discussion a gear pair meets the conditions as follows: r =100mm, e =10mm, n1 =3, n2 =4, n =3, m11 =1.5, and m11 =0.75. the visual design and simulation software is utilized to determine the following parameters: a = 233.67mm, k = 0.1 and m13 = 1. 228 h. zhao, j. chen, g. xu for the gear pair, the pitch curves and tooth profile are depicted in figs. 6 and 7 respectively. the driving gear 1 is a three-order and three-segment deformed eccentric non-circular gear, while the driven gear 2 is a four-order and three-segment non-circular gear. 111 112 113 121 122 123 131 132 133 211 212 213 222 223 221 231 232 233 241 242 243 fig. 6 pitch curves of gears 1 2 fig. 7 tooth profiles of gears the curves of transmission ratio and curvature radius are obtained using the developed visual design and simulation software, fig. 8 shows that the transmission ratio is continuous and divided into three asymmetrical segments in each cycle. as for the pitch curve of the driving gear, the convexity distinguishing conditions of each segment in the cycle are 0.049k  , 0.198k  and 0.111k  , respectively. in this example, k = 0.1, so the pitch curve is concave in first segment of each cycle for the driving gear and the curvature radius is negative, as shown in fig. 9. the curvature radius is not continuous at the connection points. similarly, the pitch curve is concave in first segment of each cycle for the driven gear. specifically, the pitch curve suddenly changes at the segmentation points, where the pitch curve transforms from convex to concave, or transforms from concave to convex. design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 229 fig. 8 curve of transmission ratio fig. 9 curvature radius of driving gear 5. application in this paper, this novel gear is applied to the drive mechanism of the metering pump. the metering pump is a special positive displacement pump with the ability to vary capacity as the process conditions require. it is capable of pumping a wide range of liquids including acids, bases, corrosives or viscous liquids. the pumping action is generated by a reciprocating piston. the function of the drive mechanism is to transform the rotary motion of the driver into reciprocating movement [25-26]. when the crank slider mechanism is used as the drive mechanism of metering pump, the curve of the output speed of slider is similar with a sine wave. the speed of the slider changes nonlinearly in the transmission process, which makes the slider gain acceleration and increases its reciprocating inertial force. it causes vibration and noise of the pump and aggravate the impact and wear between the components. 230 h. zhao, j. chen, g. xu in order to stabilize the output speed of the crank slider mechanism and reduce the impact vibration of the pump, a non-circular gear-crank slider mechanism is proposed as the drive mechanism of metering pump. figure 10 shows the schematic diagram of non-circular gear-crank slider drive mechanism of the metering pump. the motor drives the driving gear to rotate with a uniform speed and drives the driven non-circular gear to rotate at a non-uniform speed. crank o2a is fixed on the driven non-circular gear and the crank follows the driven gear at the same speed. the motion is transferred to the piston through the transmission of crank slider mechanism and realizes the reciprocating motion. φ1 φ2 ω1 ω2 o1 o2 a b 1 2 4 5 3 fig. 10 diagram of non-circular gear-crank slider mechanism: 1. driving gear, 2. driven gear, 3. crank 4. connecting rod, 5. slider 5.1. mathematical model of non-circular gear-crank slider mechanism fig.11 is the schematic diagram of the crank slider mechanism. the crank is fixed on the driven gear of the non-circular gear pair, so the angular displacement (angular velocity) of the crank is equal to the angular displacement (angular velocity) of the driven gear. the slider displacement can be expressed as: )(cossin 12212 22 1 2 2 llllls −−+−=  (26) where s is the displacement of slider, l1 is the length of crank, l2 is the length of connecting rod, 2  the angular displacement of the driven gear. φ2 ω2 o2 a b 2 3 1 b’a’ s fig. 11 diagram of crank slider: 1. crank 2. connecting rod, 3. slider design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 231 calculate the derivative of eq. (27) to obtain the velocity of the slider: ( )         − +−== 2 22 1 221 221 sin12 /)2sin( sin    l ll l dt ds v (27) the output speed of the crank can be obtained using eq. (13), so that: 12 1 2 i   == (28) from eqs. (27) and (28), it can be obtained that the velocity of the slider in the combined mechanism is: ( )         − +−= 2 22 1 221 2 12 11 sin12 /)2sin( sin     l ll i l v (29) to simplify the analysis, eq. (29) can be rewritten as: ( )         − +−= 2 22 2 2 12 1 1 sin12 )2sin( sin     i lv (30) where 21 ll= is the length ratio of the crank and connecting rod. when the driving gear rotates at a constant speed, the variation law of the output speed of the combined mechanism can be obtained from eq. (30). by adjusting the parameters of the high-order multi-segment deformed eccentric non-circular gear and the length ratio of crank and connecting rod, the speed of the stroke and return section of the combined mechanism is approximately uniform, which can reduce the reciprocating inertia force generated by the transmission system, the vibration and impact of the instantaneous flow on the pump. fig.12 shows the comparison of the velocity curves of a non-circular gear-crank slider mechanism and a crank slider mechanism. 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 1 2 3 4 5 6 7 8 v/ w 1 φ1/(rad ) crank slider mechanism non-circular gear-crank slider mechanism fig. 12 comparison of velocity curves of two transmission mechanisms 232 h. zhao, j. chen, g. xu 5.2. virtual simulation according to the mechanism parameters obtained from the analysis, the 3d model and virtual assembly of the mechanism are accomplished using solidworks software. the 3d--model is shown in fig. 13. it was imported into the adams software and constraints were added together with the drive for a virtual prototype experiment. kinematics simulation of this mechanism was conducted. the velocity curve of the slider obtained by the virtual experiment is shown in fig. 14. it can be found that the velocity curves obtained by theoretical calculations are nearly the same as that of the virtual experiment. this fact can be seen as their mutual verification. the main reason why the two curves are not completely coincident is that deformation is included in the virtual experiment as each part is considered to be deformable by adding their material properties. but some fluctuations in the velocity curve of the slider may still be observed. the reason for this is to be sought in some interstices between kinematics pairs in the transmission route, which results in the decrease of the transmission accuracy. fig. 13 3d model of the combined mechanism 0 75 150 225 300 0.1 0.2 0.3 0.5 1 v el o ci ty o f th e sl id er v/ (m m •s -1 ) time t/(s ) -300 -225 -150 -75 0.4 0.6 0.7 0.8 0.9 fig. 14 the curve of velocity of slider got by simulation design methodology and characteristics analysis of high-order multi-segment deformed eccentric… 233 6. conclusions the design method for high-order multi-segment deformed eccentric non-circular gear is proposed upon analysis of the high-order eccentric non-circular gear and the multi-segment deformed eccentric non-circular gear. unified mathematical expressions of eccentric non-circular gears are obtained, which expands their applicability. the method proposed in this paper has wide potential applicability to other non-circular gears with a typical-form pitch curve, such as elliptical gears, fourier gears and pascal spiral gears. the pitch curve shape of the high-order multi-segment deformed eccentric non-circular gear could be adjusted by changing some design parameters such as r, e, n1, n2, n and mij. hence, the pitch curves are easily adjustable (free-form). consequently, the non-circular gears with a free-form pitch curve can be obtained from high-order multi-segment deformed eccentric non-circular gear by changing the design parameters, which provides a theoretical basis for unifying these two kinds of non-circular gears. the transmission ratio and design of non-circular gears can be flexibly adjusted so that the requirements of precision transmission are met in order to improve transmission performance. the visual design and simulation software are compiled using matlab and based on the proposed mathematical expressions. the verification was done using an example. this novel gear was applied to the drive mechanism of the metering pump. comparing the results of theoretical analysis and the virtual prototype experiment, the velocity curve of the slider obtained by simulation is in a good agreement with the ideal velocity curve. this shows that the proposed non-circular gear-crank-slider mechanism can meet the functional 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design improvement for a metering pump, engineering failure analysis, 8(1), pp. 1-13. 26. borstell, d., 2005, benefits of the rotary diaphragm pump, medical device technology, 16(2), pp. 16-18. facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 29 38 https://doi.org/10.22190/fume190122014e © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  multilevel numerical model of hip joint accounting for friction in the hip resurfacing endoprosthesis galina m. eremina, alexey yu. smolin institute of strength physics and materials science sb ras, tomsk, russia, tomsk state university, tomsk, russia abstract. friction between the moving parts of the endoprosthesis has a significant impact on the endoprosthesis operation time. primarily, it concerns the endoprosthesis of hip and knee joints. to improve the tribological characteristics of the metal endoprosthesis, hardening nanostructured coatings are used. usually, titanium and titanium alloys are used as metal, and titanium nitride is used as a coating. herein, we propose an approach to multilevel modeling of the system “bone-endoprosthesis” which is based on the movable cellular automaton method and accounts for friction between the moving parts of the hip resurfacing endoprosthesis. we validated the models of the friction system materials using the instrumented scratch test simulation. then, we simulated friction at the mesolevel, explicitly considering roughness of the coating. the results obtained at the mesolevel were used as tribological characteristics of the coating in the macroscopic model of the hip resurfacing endoprosthesis. key words: hip resurfacing endoprosthesis, coatings, scratch test, friction, simulation, movable cellular automaton method 1. introduction prosthetics is an important part of the modern traumatology and orthopedics, especially for large human joints such as hips and knees. nowadays there are two options for hip joint endoprostheses: total hip replacement and hip resurfacing arthroplasty. in the first option the joint is replaced by endoprosthesis, in the second, just the surface of the femur head is replaced by the cap, which is specially shaped like a mushroom. in spite of the surgical complexity, the second approach is much better suited for young patients due received january 22, 2019 / accepted march 24, 2019 corresponding author: galina m. eremina institute of strength physics and materials science sb ras, pr. akademicheskii 2/4, tomsk, 634055, russia e-mail: anikeeva@ispms.ru 30 g.m. eremina, a.yu. smolin to their activity. mainly, both the resurfacing cup and the matching cup placed in the acetabulum are made of cobalt-chrome metal alloy. sliding friction between the cups leads to wear of the metal. the wear debris generated and released from the metal parts of the endoprosthesis accumulates in the adjacent tissue. this may lead to necrosis of the tissue cells and small phagocytes; an expressed allergic vasculitis is also observed in the literature [1]. the surface layer structure of the contacting materials has a major influence on the wear process. therefore, in the last decade, one can observe extensive research in the field of applying ultrathin hardening coatings and layers on titanium alloys used for resurfacing hip endoprosthesis. in this case, titanium alloys are used as metal and titanium nitride (tin) is used as the coating. the structure of the coating is determined by the regimes of its deposition. usually, the following technologies are used for this purpose: physical vapor deposition (pvd), chemical vapor deposition (cvd) and powder immersion reaction assisted coating (pirac) [2, 3]. the coating obtained by pvd and cvd techniques have low adhesion and may delaminate under dynamic loading, while adhesion of the coating obtained by pirac is much higher. furthermore, pirac allows depositing a wide range of compositions such as carbides, borides, and nitrides [4, 5]. that is why pirac appears to be very promising for producing hardening coating of metal implants. testing of prostheses has several stages including preclinical and clinical trials. clinical trials are carried out through the prosthesis installation in a living human body and are performed at the final stage of the testing. it is important to understand that the installation of low quality or poorly designed prosthesis may lead to problems of the patient health. that is why the preclinical study is particularly important. preclinical studies of the mechanical behavior of the prosthesis can be divided into experimental and theoretical ones. experimental studies are tests using a special technological unit that simulates the dynamic loading experienced by prosthesis in a living body. to determine mechanical properties of the material surface such standard methods as depth-sensing indentation and scratch testing as well as three-point bending are used. it is worthy to note that usually experimental studies are performed in standard atmospheric conditions and their results may significantly differ from the material behavior in the conditions of the human body. that is why part of the experiments is conducted in the condition mimicking the living body (in vitro) or using special samples installed into an animal body (in vivo). however, it is obvious that the number of in vivo tests and their capabilities is considerably restricted. theoretical studies of the mechanical behavior of the prosthesis using computer simulation make it possible to study the mechanical behavior of the prosthesis taking into account the influence of various factors. therefore, in preclinical studies, computer simulation is used to predict the mechanical behavior of prostheses. for example, computer simulation based on numerical methods of continuum mechanics allows detailed investigations of the behavior of coating and surface layer under contact loading. the influence of the coating thickness on the load-displacement curve for the instrumented indentation of ceramic coatings on the metal substrate is studied in [6-8]. the role of the surface roughness of ceramic coating in the mechanical behavior of the system “coatingsubstrate” in instrumented indentation is numerically studied in [9-12]. both instrumented indentation and scratch tests of the hardening coating are simulated in [13-17], where the authors analyze stress fields in the contact region as well as the peculiarities of the loadingdisplacement curves. multilevel numerical model of hip joint accounting for friction in hip resurfacing endoprosthesis 31 the majority of papers on simulation of the mechanical behavior of the system “boneendoprosthesis” also use numerical methods of continuum mechanics. in [18], the authors study the stress fields in the hip bone for two types of the endoprostheses (total replacement and resurfacing) and show that resurfacing leads to minimal changes of the stress distribution in the bone compared with the original “healthy” bone. dickinson et al. studied the mechanical behavior of the system “bone-endoprosthesis” for different structure and geometry of the implant [19] as well as its material [20]. in spite of a large number of the papers devoted to the modeling of the mechanical behavior of the hip endoprosthesis there is still no unified multiscale theoretical study of the system with explicit consideration of structural peculiarities of the materials. in such a model, the studies at each scale should solve their specific problems and their results should be used for modeling at the upper scale finally providing the capability to design personalized endoprosthesis. the purpose of this paper is the development of a multiscale numerical model of the mechanical behavior of the system “bone-endoprosthesis” for hip resurfacing arthroplasty that considers explicitly such geometrical parameters of the tin coating as its thickness, roughness, and mechanical properties. to reach this purpose, we solved three tasks. firstly, we validated the models of the materials used in the friction pair of the endoprosthesis. for this, we developed a numerical model for simulating instrumented indentation and scratch testing of the hardening coating on the titanium substrate. the simulation results were compared against available experimental data. secondly, we developed a model for sliding friction at the mesoscale, which considers explicitly roughness and mechanical properties of the contacting surfaces. the simulation at this scale then provided an effective coefficient of friction that can be used at macroscale modeling. the final third task was the development of a numerical macromodel of the system “bone-endoprosthesis” and studying this model. 2. description of the simulation method our numerical models are based on the method of movable cellular automata (mca), which is a representative of so-called discrete element methods (dem) and differs substantially from numerical methods used in continuum mechanics [21]. the mca considers a material as a set of particles of finite size (automata) interacting among each other and thereby simulating deformation and fracture of the real material. translational motion and rotation of the movable automata are governed by the newton-euler equations. the main advantage of the mca-method in comparison with the dem are the generalized many-body formulas for the central interaction forces acting between the particle pair of s similar to the embedded atom force filed used in molecular dynamics. it is based on computing components of the average stress and strain tensors in the bulk of automaton according to the homogenization procedure described in [21]. the use of many-body interaction forces allows, within the discrete element approach, the correct simulation of important features of the mechanical behavior of solids such as poisson effect and plastic flow. to describe the elastoplastic flow of the material, it was proposed to use the plastic flow theory (namely von mises model) by adapting the algorithm of wilkins in the mca method [21, 22]. the radial return algorithm of wilkins consists of the solution of the 32 g.m. eremina, a.yu. smolin elastic problem in increments and the subsequent “drop” of components of stress deviator tensor to von mises yield surface in the case that equivalent stress exceeds it. a pair of automata can be in one of two states: bound and unbound. thus, in the mca fracture and coupling of fragments (crack healing, microwelding etc) are simulated by the corresponding switching of the pair state. the switching criteria depend on physical mechanisms of material behavior [21, 22]. note that the knowledge of the stress and strain tensor in the bulk of an automaton allows the direct application of conventional fracture criteria written in the tensor form. herein, we used the von mises criterion based on a threshold value of the equivalent stress. 3. validation of the materials model from the literature [23], we chose the following values for the material properties of the titanium alloy ti6al4v: density ρ = 4420 kg/m 3 , shear modulus g = 41 gpa, bulk modulus k = 92 gpa, young’s modulus e = 110 gpa, yield stress σy = 0.99 gpa, ultimate strength σb = 1.07 gpa and ultimate strain εb = 0.10. geometric features of the tin coating and its mechanical properties are determined by the deposition modes at pirac [24] forming. thus, at a deposition temperature of 700º c and a treatment time of 48 hours, the coating on the titanium substrate has a thickness of 1.3 μm with an average roughness height of 0.15 μm (mode 1) and the elastic modulus, depending on the deposition regime, was equal to e1 = 258 gpa; at a deposition temperature of 800º c and a treatment time of 4 hours the coating on the titanium substrate has a thickness of 1.4 μm with an average roughness height of 0.132 μm (mode 2) and an elastic modulus of e2 = 258 gpa; at a deposition temperature of 900° c and a treatment time of 2 hours the coating on the titanium substrate has a thickness of 1.5 μm with an average roughness height of 0.265 μm (mode 3) and an elastic modulus of e3 = 321 gpa. according to this information, we chose the following values for the material properties of the tin coating: ρ = 52200 kg/m 3 , g1 = 104 gpa, g2 = 104 gpa, g3 =129 gpa, k1 = 173 gpa, k2 = 173 gpa, k3 = 205 gpa. data for yield stress and ultimate strength and strain were obtained using reverse analysis of the load-displacement curve for sensing indentation: σy = 4.50 gpa, σb = 5.50 gpa and εb = 0.075 [25]. a general view of the model for sensing indentation is shown in fig. 1a. the model specimen was a parallelepiped consisting of titanium substrate, interface and tin coating. loading was simulated by moving all the automata of the berkovich indenter with constant velocity vz = −1 m/s for loading until the required penetration depth, and vz = 1 m/s for unloading (fig. 1, b). applying the procedure of oliver and pharr [26] to the simulation results provided the dependence of the material hardness on the penetration depth (fig. 2) for three kinds of coatings obtained in different regimes of deposition. it can be seen from fig. 2 that the hardest is the coating obtained by mode 3, all the curves correspond to experimental data from [24]. thus, we may conclude that the models of the materials behavior are validated and can be used in further steps in our work. the model specimen for scratch testing was also a parallelepiped consisting of titanium substrate, interface and tin coating but elongated along axis y (fig. 3). to simulate the force acting on indenter along axis z in the experiment, we set the velocity of indenter automata to vz = −0.5 m/s (fig. 3, b) until the indenter was immersed into a multilevel numerical model of hip joint accounting for friction in hip resurfacing endoprosthesis 33 predetermined penetration depth; after that the vertical velocity was set to zero. here, we considered penetration of the indenter only up to the interface layer; the possibility of the coating to detach from the substrate was not allowed. to move the indenter along the sample surface, a constant velocity of the indenter automata along axis y was set to vy = 1 m/s. a b fig. 1 the model specimen for indentation (a) and its cross-section with loading parameters (b) represented by automata packing (the numbers indicate the model materials: 1 – titanium, 2 – interface, 3 – coating, 4 – diamond) fig. 2 dependence of the hardness on the depth of penetration (the numbers indicate the modes of coating deposition) based on the results of our simulations, images of the deformed sample were created, and the values of the critical force characteristics at certain stages of fracture were obtained. it was found that the value of the critical force for the onset of fracture and cracking is largest for the coating thickness of 1.4 μm and roughness of 0.132 μm, and smallest for the thickness of 1.5 μm and roughness of 0.265 μm. at delamination of the coating, the maximum of the critical force is typical for the specimen with a coating thickness of 1.5 μm and roughness of 0.265 μm, and the minimum for the specimen with a coating thickness of 1.3 μm and roughness of 0.15 μm (fig. 4). 34 g.m. eremina, a.yu. smolin a b fig. 3 the model specimen for scratch testing (a) and its cross-section with loading parameters (b) represented by automata packing (the numbers indicate the model materials: 1 – titanium, 2 – interface, 3 – coating, 4 – diamond) fig. 4 the force acting on the indenter from coating p versus scratching path h of the indenter (the numbers indicate the modes of coating deposition) 4. mesomodel for friction in the next step, we developed a mesomodel of sliding friction of the contacting surfaces of the endoprosthesis that explicitly accounts for surface roughness, the mean height of which is determined by the regime of the coating deposition (fig. 5). at the macroscale we merely need the effective value of the friction coefficient. therefore, the substrate and interface were not considered in this model. the scheme of loading is shown in fig. 5b and is as follows. the automata of the bottom layer were fixed, the automata of the top layer were moved with constant horizontal velocity vy = 1 m/s and subjected to pressure p = 0.75σy. multilevel numerical model of hip joint accounting for friction in hip resurfacing endoprosthesis 35 a b fig. 5 the model specimen for sliding friction at mesoscale (a) and its cross-section with loading parameters (b), the colors indicate different bodies figure 6 depicts plots of the friction coefficients versus the calculation time obtained from the simulation at mesoscale for three different tin coatings, whose parameters and roughness correspond to the different deposition regimes. average values of these friction coefficients at the final steady stage were then used in the model at the macroscale. fig. 6 plots of the friction coefficient in the model at mesoscale versus the calculation time (the numbers indicate the modes of coating deposition) 5. macromodel of hip resurfacing endoprosthesis finally, we developed a macromodel of the hip resurfacing endoprosthesis with part of the femur bone shown in fig. 7. the geometry of the model was based on the so-called 3rd generation composite femur [27], which provides geometries of cortical bone and cancellous bone as different solid bodies. for our purpose, we cut the top part of the bone 36 g.m. eremina, a.yu. smolin geometry, added resurfacing endoprosthesis (colored in cyan, and its coating colored in blue) and special loading part (colored in red) (fig. 7, b,c). a b c fig. 7 the model of the hip resurfacing endoprosthesis (a) its cross-section (b) and scheme of loading (c) a b b d fig. 8 distributions of the mean stress in the system “bone-endoprosthesis” loaded by a force of 3 kn (a,c) and 10 kn (b,c) for titanium endoprosthesis (a,b) and titanium with tin coating (c,d) multilevel numerical model of hip joint accounting for friction in hip resurfacing endoprosthesis 37 based on the developed model, we simulated compression of the part of the femur with resurfacing endoprosthesis with hardening coating and without it. the loading was applied by setting a constant velocity in both horizontal and vertical directions as shown in fig. 7c up to reaching the critical values of the resisting force of 3 kn and 10 kn [28]. the simulation results are presented in fig. 8 as distributions of mean stress. it is reported that the strength limit for the cancellous bone of healthy people reaches about 10 mpa in compression and 5 mpa in tension; it may be lower than 5 mpa and 3 mpa for some diseases correspondingly [29]. analysis of stress distribution in the model system showed that the maximum tensile stress was observed near the endoprosthesis and did not reach critical values. therefore, we tried to analyze the compression stress in the scale up to 5 mpa. from fig. 8, one can see that the maximum compression stress that can lead to fracture is observed in the femoral neck. 6. conclusions a multiscale numerical model for analyzing mechanical behavior of the hip resurfacing endoprosthesis is proposed. the material models are validated against the available experimental data using the simulation of instrumented indentation and scratching. the macroscopic coefficient of friction in the friction pair of the endoprosthesis is proposed to be found from the model for sliding friction at mesoscale where roughness of the contacting surfaces is considered explicitly. simulation at the macroscale includes a part of the femur bone with resurfacing endoprosthesis. an analysis of the stress distribution in the macromodel reveals that the critical compression stress is observed in the femoral neck. further studies could include more realistic loading mimicking not only 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optimization of the micro-channel heat sink using irreversibility analysis. the nanofluid of al2o3-water with the different nanoparticles concentration and the temperature-dependent property is chosen as a coolant. the flow is considered as fully developed, steady, and laminar in the constant cross-section of circular channels. navier-stokes and energy equations are solved for a single-phase flow with total mass flow rate and heat flow rate as constant. the objective functions related to the frictional and heat transfer irreversibilities are framed to assess the performance of the micro-channel heat sink. the optimum channel diameter corresponding to the optimum number of channels is determined at the lowest total irreversibility for both constant property solution and variable property solution. designed optimum diameter is observed maximum for 2.5% al2o3-water nanofluid with μ(t) variation followed by 1% al2o3-water nanofluid with μ(t) variation, 2.5% al2o3water nanofluid with constant property solution, and 1% al2o3-water nanofluid with constant property solution. key words: micro-channel, entropy generation, nanofluid, property variation 1. introduction in recent times, the exponential rise is experienced in the capacity and loading related to the computing/transaction/processing/storage for information and communication technology applications (such as cloud computing, artificial intelligence, 5g, etc.). the electronic devices involved in such applications have high power consumption along with high heat generation [1]. this generated high heat flux needs to be managed efficiently. the micro-channel (mc) heat sinks with liquid cooling were found more advantageous received march 08, 2021 / accepted june 18, 2021 corresponding author: rajan kumar department of mechanical engineering, dr br ambedkar national institute of technology, jalandhar, punjab 144011, india e-mail: rajank@nitj.ac.in, rajan.rana9008@gmail.com mailto:rajank@nitj.ac.in mailto:rajan.rana9008@gmail.com 504 k. kumar, p. r.chauhan, r. kumar, r. s. bharj over conventional air cooling [2]. the cooling performance of these high-performance devices can be improved either by incrementing heat transfer (ht) surface area or by using a working fluid with excellent thermophysical properties. the fluids with better thermophysical properties can be prepared by synthesizing the base fluids with particles of nanometric size (1–100 nm) and termed as nanofluid (nf) [3, 4]. the thermal performance improves with the use of nf but more pumping power is required to move the fluid [5, 6]. thus, the design and optimization of the thermal system consisting of the nf flow in the mc is an important concern for the research. the optimization of any thermal system can be assessed using entropy generation (eg), which is the measure of exergy loss of the system [7, 8]. it is always required to minimize the eg in any system for its better performance. the 2nd law analysis is important in understanding the eg and also a useful tool in designing and analyzing a thermal system with less entropy and loss of available work [9]. bejan [10, 11] used the 1st and 2nd law of thermodynamics and provided a technique for minimizing the entropy generation rate for system optimization, which is known as the entropy generation minimization approach. further, the effects of the thermophysical property variations in the case of micro-convective flows cannot be neglected because of the steeper temperature gradients [12]. rastogi and mahulikar [13, 14] implemented the theory of eg for the optimization of the mc heat sink with a laminar forced convection flow. the situation with minimum total eg rate results in the optimum channel diameter. chauhan et al. [15] numerically optimized the mc heat sink using the eg minimization principle. no change in the temperature gradient in the axial direction was analyzed but an increase in the ht eg in the axial direction occurs because of the decrease in the temperature in the denominator when micro-dimensions were approached. kumar et al. [16] performed numerical simulations to optimize the number of channels (n) corresponding to the channel diameter (dn) at the micro-scale. at the microscale, frictional eg dominates over the ht eg due to an increase in the velocity gradient and a decrease in the temperature gradient in the radial direction. heshmatian and bahiraei [17] analyzed the effects of viscosity gradient, brownian diffusion, and shear rate on the irreversibility in the titanium dioxide (tio2)-water nf flowing through the circular mc. the total eg rate decreases with the enlargement of the nanoparticles because of the decrease in the frictional eg rate. bianco et al. [18] used alumina (al2o3) nf forced convective flow in a pv/t panel. the use of nf helped in the ht improvement which reduces the thermal eg. also, the use of nf created more flow friction and resulted in a higher frictional eg. but due to the dominance of ht eg, the total eg was also less when nf was used as the working fluid. manay et al. [19] experimentally investigated the eg in tio2-water nf flow through the mc heat sink with variable nanoparticles volume fraction (vf). the reduction in the height of the channel, as well as the introduction of the nanoparticles to the base fluid, resulted in the reduction of the thermal eg. bianco et al. [20] performed a numerical study to analyze the irreversibility in the al2o3 nf flow through a three-dimensional rectangular channel. the channel top wall was considered under a constant wall heat flux of 1000 w/m2. the study found it convenient to use nfs because it helps in the reduction of the total irreversibility of the system. alfaryjat et al. [21] studied eg in the nf flowing in a hexagonal mc heat sink. the numerical investigations were performed using al2o3-water, copper oxide (cuo)-water, and silica (sio2)-water nf. the study concluded that with an increase in the reynolds number (re) and vf, a fall in the thermal eg takes place as well as a rise in the frictional eg. shashikumar et al. [22] performed eg analysis on the nf consisting of irreversibility analysis in al2o3-water nanofluid flow with variable property 505 nanoparticles of aluminum (al) and titanium (ti) alloys and water as base fluid. the eg rate was reduced with augmentation in thermal conductivity (k) because of the increase in the thermal diffusion intensity. kumar and mahulikar [23, 24] used numerical simulations to analyze various scaling effects on a single-phase water flow through the mc. the fanning friction factor for the temperature-dependent density variation was found higher than that of constant property solution (cps) due to higher radial inward velocity. whereas for the viscosity (μ) variation, the fanning friction factor was found prominently lower than the cps due to a decrease in the fluid viscosity with increasing temperature. chauhan et al. [25] executed numerical simulations for optimizing the channel geometry using the eg minimization method. it was found that the irreversibility due to conduction ht and friction got significantly affected by temperature-dependent thermal conductivity and viscosity (k(t) and μ(t)) variations, respectively. the earlier studies discussed above pay attention only to the assessment of the eg and ht parameters, along with few studies consisting of optimization of the channels. but to the best of the author’s knowledge, almost no numerical study for the channel geometry optimization has been performed with the consideration of variable nf properties. the present study consists of the following objectives: (i) to find out optimum channel diameter (d*) corresponding to an optimum number of channels (n*), (ii) to compare the eg of cps and variable property solution (vps), (iii) to examine the effect of vf of nanoparticles on eg, and, (iv) to evaluate the irreversibility ratio. 2. problem formulation and configuration in this investigation, the problem consists of channel diameter optimization corresponding to the optimum number of channels for the least irreversibility. the computational domain has a length l and a circular cross-section of radius rn. the water-based nf with alumina nanoparticles (al2o3-water) is used as a cooling liquid; it passes through the channel of the circular constant cross-section. the total heat flow rate and total mass flow rate (mtot) are 31415.926 mw and 0.180101 gm/s, respectively. both these parameters are constant for all the cases considered in the present study while the cases and range of parameters are given in table 1. table 1 range of parameters (n, dn, and heat flux) in the present study cases n dn (cm) wall heat flux (w/cm2) 1 1 0.10000 10.0000 2 2 0.07071 7.0711 3 5 0.04472 4.4720 4 25 0.02000 2.0000 5 100 0.01000 1.0000 6 400 0.00500 0.5000 7 625 0.00400 0.4000 8 2500 0.00200 0.2000 9 10000 0.00100 0.1000 506 k. kumar, p. r.chauhan, r. kumar, r. s. bharj fig. 1 is a depiction of a fully developed (hydro-dynamically as well as thermally) steadystate laminar flow in the channel of an arbitrary circular constant cross-section of 10 μm to 1000 μm diameter, subjected to the constant wall heat flux ranging from 103 w/m2 to 105 w/m2, with a fixed length of 10 cm. the fluid flow is considered as single-phase. however, the working fluid is nf, but we have taken a single-phase model for the modeling of nf which needs less computational effort and is as accurate as the correlation of thermophysical properties at various nanoparticle vf. due to the accuracy and less computational efforts, a single-phase model with thermophysical properties is the best choice for a laminar fullydeveloped flow [26]. fig. 1 schematic diagram of circular micro-channel table 2 thermophysical properties of al2o3-water nanofluid [27] properties at 1% volume fraction at 2.5% volume fraction constant variation constant variation density, ρ (kg/m3) 1016.98 1016.98 1075.18 1075.18 specific heat, cp (j/kg.k) 4050.63 4050.63 3754.93 3754.93 thermal conductivity, k (w/m.k) 0.715829 0.715829 0.743896 0.743896 dynamic viscosity, μ (pa.s) 0.00058332 µ(t) (ref eq. (1)) 0.00119432 µ(t) (ref eq. (2)) 𝜇(𝑇) = {1.43603444967752 − 1.85655480917592 × 10−2 𝑇 + 9.066346037657 × 10−5 𝑇2 − 1.978937437 × 10−7 𝑇3 + 1.6271752 × 10−10𝑇4 }(1) 𝜇(𝑇) = {45.4142966405434 − 5.80454966169067 × 10−1𝑇 + 2.782175586943 × 10−3𝑇2 − 5.9264401255 × 10−6𝑇3 + 4.733596 × 10−9𝑇4 }(2) the thermophysical properties of nf at different vf of al2o3 nanoparticles are taken as shown in table 2 [27]. the working fluid for the study is water-based nf at different vf of al2o3 nanoparticles with constant thermophysical properties and μ(t) as well. the derived expressions for μ(t) variation at 1% and 2.5% vf of al2o3 nanoparticles in the temperature range of 298-328 k are obtained as a least-square error fourth-order polynomial fitting of data by eq. (1) and eq. (2), respectively [27]. axis inlet outlet wall = constant rn l = 100000 μm irreversibility analysis in al2o3-water nanofluid flow with variable property 507 3. numerical modeling the finite volume method is used to solve the 2-dimensional governing equations for the given initial conditions and boundary conditions. the ansys workbench is used for preprocessing (geometry creation and grid generation). the governing partial differential equations are solved with the help of the fluent solver. the further subsections present the governing equations, first law-based analysis, second law-based analysis, and model validation: 3.1 governing equations the steady-state conservation equations based on the continuous, incompressible, and 2-dimensional fluid flow in cylindrical coordinates with the μ(t) variation through the mc are following [28]: continuity equation: 𝑣 𝑟 + 𝜕𝑣 𝜕𝑟 + 𝜕𝑢 𝜕𝑧 = 0 (3) momentum equation [axial direction]: 𝜌. (𝑣 · 𝜕𝑢 𝜕𝑟 + 𝑢 · 𝜕𝑢 𝜕𝑧 ) = { − 𝜕𝑝 𝜕𝑧 + [( µ 𝑟 + 𝜕µ 𝜕𝑟 ) · ( 𝜕𝑢 𝜕𝑟 + 𝜕𝑣 𝜕𝑧 )] + µ · [2 ( 𝜕2𝑢 𝜕𝑧2) + ( 𝜕2𝑢 𝜕𝑟2) + ( 𝜕2𝑣 𝜕𝑟·𝜕𝑧 )] + 2 ( 𝜕µ 𝜕𝑧 ) · ( 𝜕𝑢 𝜕𝑧 ) } (4) momentum equation [radial direction]: 𝜌. (𝑣 · 𝜕𝑣 𝜕𝑟 + 𝑢 · 𝜕𝑣 𝜕𝑧 ) = { − 𝜕𝑝 𝜕𝑟 + 2 [ µ 𝑟 + 𝜕µ 𝜕𝑟 ] · ( 𝜕𝑣 𝜕𝑟 ) – ( µ·𝑣 𝑟2) + µ · [ 𝜕2𝑣 𝜕𝑧2 + 2( 𝜕2𝑣 𝜕𝑟2) + 𝜕2𝑢 𝜕𝑟·𝜕𝑧 ] + ( 𝜕µ 𝜕𝑧 ) · [ 𝜕𝑣 𝜕𝑧 + 𝜕𝑢 𝜕𝑟 ] } (5) in the axial and radial momentum equations, the simple algorithm is used to couple the velocity and pressure fields. besides, the standard scheme is adopted to discretize the pressure gradient. energy equation: 𝜌. 𝑐𝑝 (𝑣 · 𝜕𝑇 𝜕𝑟 + 𝑢 · 𝜕𝑇 𝜕𝑧 ) = [ 𝑘 𝑟 + 𝜕𝑘 𝜕𝑟 ] . ( 𝜕𝑇 𝜕𝑟 ) + 𝑘. ( 𝜕2𝑇 𝜕𝑟2) + ( 𝜕𝑘 𝜕𝑧 ) . ( 𝜕𝑇 𝜕𝑧 ) + 𝑘. ( 𝜕2𝑇 𝜕𝑧2) + 𝜇. 𝜓 (6) the second-order upwind scheme is used for discretizing the spatial gradients in the energy equation. viscous dissipation function (ψ) for the axis-symmetry condition is expressed as: 𝜓 = {[( 𝜕𝑣 𝜕𝑧 ) + ( 𝜕𝑢 𝜕𝑟 )] 2 + 4 3 [ ( 𝜕𝑣 𝜕𝑟 ) 2 + ( 𝑣 𝑟 ) 2 + ( 𝜕𝑢 𝜕𝑧 ) 2 −( 𝜕𝑣 𝜕𝑟 ) · ( 𝑣 𝑟 ) − ( 𝜕𝑣 𝜕𝑟 ) · ( 𝜕𝑢 𝜕𝑧 ) − ( 𝑣 𝑟 ) · ( 𝜕𝑢 𝜕𝑧 ) ]} (7) in the above equations u, v,t, and p represent axial velocity, radial velocity, temperature, and pressure, respectively, whereas z and r represent the axial direction and radial direction, respectively. 508 k. kumar, p. r.chauhan, r. kumar, r. s. bharj 3.2 first law of thermodynamics based analysis the hagen-poiseuille equation for a hydro-dynamically fully developed velocity profile of the forced convective steady, laminar, incompressible, and newtonian fluid flow with mean velocity (um) inside the channel is derived as: 𝑢(𝑟) = 2𝑢𝑚 [1 − ( 𝑟 𝑅𝑁 ) 2 ] (8) the thermally fully developed temperature profile is obtained with the help of the following equation: 𝑇(𝑟 𝑧) = 𝑇0 𝑖𝑛 + 𝑞𝑤 𝑁 ′′ ∙𝑅𝑁 𝑘 [( 2𝛼 𝑅𝑁∙𝑢𝑚 ) ( 𝑧 𝑅𝑁 ) + ( 𝑟 𝑅𝑁 ) 2 − 1 4 ( 𝑟 𝑅𝑁 ) 4 ] (9) where t0,in and α are inlet temperature and thermal diffusivity, respectively. for a thermally fully developed flow with the constant wall heat flux condition, the temperature gradient along the axial direction is stated as: 𝜕𝑇 𝜕𝑧 = 𝜕𝑇𝑏 𝜕𝑧 (10) where tb is the bulk mean temperature of fluid obtained as: 𝑇𝑏 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 𝑓𝑙𝑜𝑤𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑎 𝑐𝑟𝑜𝑠𝑠−𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑓𝑙𝑜𝑤𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑎 𝑐𝑟𝑜𝑠𝑠−𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑇𝑏 = 2 𝑅𝑁 2 𝑢𝑚 ∫ 𝑢(𝑟) ∙ 𝑇(𝑟 𝑧) 𝑟𝑑𝑟 𝑅 0 (11) the governing partial differential equations (pdes) (eq. (3) – eq. (6)) for the 2d computational domain are solved numerically with the help of initial and boundary conditions. these conditions are: at the inlet (z = 0), the fully developed forced convection of hagen-poiseuille (eq. (8) & eq. (9)) is provided with temperature, t0,in= 273 k. no entrance effects, no-slip and, no temperature-jump conditions are considered in developing the profiles. the radial velocity is taken as zero (v = 0 m/s). at the wall (r = r), different constant wall heat flux boundary conditions (ref. fig. 1) are applied corresponding to the tabulated cases under consideration. the slip velocity and normal flow velocity are considered as zero (non-porous rigid wall). the wall heat flux (𝑞𝑤 𝑁 ) corresponding to the number of channels is calculated as: 𝑞𝑤 𝑁 = �̇�𝑡𝑜𝑡 𝑁.(𝜋.𝐷𝑁.𝐿) (12) at the axis (r = 0), the differentiability of flow parameters (p, t, u) at the axis of the channels in the dimensional form is, 𝜕𝑝 𝜕𝑟 = 𝜕𝑇 𝜕𝑟 = 𝜕𝑢 𝜕𝑟 = 0⁄⁄⁄ at the outlet (z = l), the pressure condition is considered as atmospheric (pexit = patm= 1.01325 bar). flow transporting variables (u, v, t) are executed using the neumann boundary condition. their normal gradients are, 𝜕𝑢 𝜕𝑧⁄ = 𝜕𝑣 𝜕𝑧 = 𝜕𝑇 𝜕𝑧⁄ = 0⁄ irreversibility analysis in al2o3-water nanofluid flow with variable property 509 3.3 second law of thermodynamic based analysis the presence of entropy in the thermal system is due to irreversibility within the system, and the eg is the measure of the magnitude of the irreversibility present in a process. the volumetric eg rate (�̇�𝑔𝑒𝑛 ) is the sum of frictional eg rate (�̇�𝑔𝑒𝑛 𝐹𝑅 ) and ht eg rate (�̇�𝑔𝑒𝑛 𝐹𝑅 ), and can be obtained as [29]: �̇�𝑔𝑒𝑛 = �̇�𝑔𝑒𝑛 𝐹𝑅 + �̇�𝑔𝑒𝑛 𝐻𝑇 (13) �̇�𝑔𝑒𝑛 = 𝜇 𝑇 𝜓 + 𝑘 𝑇2 (𝛻𝑇)2 (14) �̇�𝑔𝑒𝑛 = 𝜇 𝑇 𝜓 + 𝑘 𝑇2 (𝛻𝑇)2 (15) the formulation of the viscous dissipation function (eq. (7)) is compacted to ψ = (∂u⁄∂r)2 by taking into account only the axial velocity. in eq. (13), the first term on the right-hand side depends on the friction of the fluid whereas the second term depends on the ht along the finite temperature gradients. further, the volumetric eg rate becomes, �̇�𝑔𝑒𝑛 = 𝜇 𝑇 ( 𝜕𝑢 𝜕𝑟 ) 2 + 𝑘 𝑇2 ( 𝜕𝑇 𝜕𝑧 ) 2 + 𝑘 𝑇2 ( 𝜕𝑇 𝜕𝑟 ) 2 (16) the first-order forward difference approach is adopted to discretize the axial velocity gradient in the radial direction, the temperature gradient in the axial direction, and temperature gradient in the radial direction as: 𝜕𝑢 𝜕𝑟 | 𝑖 𝑗 = ( 𝑢𝑖 𝑗+1−𝑢𝑖 𝑗 𝑟𝑖 𝑗+1−𝑟𝑖 𝑗 ) 𝜕𝑇 𝜕𝑧 | 𝑖 𝑗 = ( 𝑇𝑖+1 𝑗−𝑇𝑖 𝑗 𝑧𝑖+1 𝑗−𝑧𝑖 𝑗 ) 𝑎𝑛𝑑 𝜕𝑇 𝜕𝑟 | 𝑖 𝑗 = ( 𝑇𝑖 𝑗+1−𝑇𝑖 𝑗 𝑟𝑖 𝑗+1−𝑟𝑖 𝑗 ) (17) on substituting these gradients (eq. (17)) in eq. (16), and multiplying it with the total volume (= number of channels (n) × volume of a channel (vn)) of channels, the total eg (sgen,tot) is obtained as: 𝑆𝑔𝑒𝑛 𝑡𝑜𝑡 = �̇�𝑔𝑒𝑛 ∙ 𝑁 ∙ 𝑉𝑁 𝑆𝑔𝑒𝑛 𝑡𝑜𝑡 = [ 𝜇 𝑇𝑖 𝑗 ( 𝑢𝑖 𝑗+1−𝑢𝑖 𝑗 𝑟𝑖 𝑗+1−𝑟𝑖 𝑗 ) 2 + 𝑘 𝑇𝑖 𝑗 2 ( 𝑇𝑖+1 𝑗−𝑇𝑖 𝑗 𝑧𝑖+1 𝑗−𝑧𝑖 𝑗 ) 2 + 𝑘 𝑇𝑖 𝑗 2 ( 𝑇𝑖+1 𝑗−𝑇𝑖 𝑗 𝑟𝑖+1 𝑗−𝑟𝑖 𝑗 ) 2 ] ∙ 𝑁 ∙ 𝑉𝑁 (18) the first, second, and third term on the right-hand side in eq. (18) gives the frictional eg (sgen,fr), ht eg in the axial direction (sgen,ht-ax), and ht eg in the radial direction (sgen,ht-rad), respectively. the sum of sgen,ht-ax and sgen,ht-rad gives the total ht eg (sgen,ht). further, bejan number (be) is calculated using, 𝐵𝑒 = 𝑆𝑔𝑒𝑛 𝐻𝑇 𝑆𝑔𝑒𝑛 𝑡𝑜𝑡 (19) in the above equations, t is taken as local temperature i.e. t=ti,j. for knowing the dominance effect of frictional irreversibility over ht irreversibility in the channel, irreversibility distribution ratio (φ) is used and is given as: 𝜑 = 𝑆𝑔𝑒𝑛 𝐹𝑅 𝑆𝑔𝑒𝑛 𝐻𝑇 = 𝑆𝑔𝑒𝑛 𝐹𝑅 𝑆𝑔𝑒𝑛 𝐻𝑇−𝑟𝑎𝑑+𝑆𝑔𝑒𝑛 𝐻𝑇−𝑎𝑥 (20) 510 k. kumar, p. r.chauhan, r. kumar, r. s. bharj when, sgen,fr = sgen,ht the diameter at intersection (dint) corresponding to the number of channels is calculated. optimum dimensions d* corresponding to n*occurs at the minimum value of sgen,tot, and is obtained by the criterion, [𝜕(𝑆𝑔𝑒𝑛 𝑡𝑜𝑡) 𝜕𝐷𝑁⁄ ] = 0 (21) 3.4 model validation the results obtained from the numerical simulation in the cps cases are validated by comparing with the results obtained from the explicit mathematical method used by genić et al. [30] and milovančević et al. [31]. comparisons are done for pure water. the present code outcomes are in good agreement with the explicit method outcomes. the comparison of the results obtained from numerical method and explicit method is available in the preceding study performed by chauhan et al. [15]. the model precision is ensured using six different grid systems that are 5000×25, 10000×50, 15000×75, 20000×100, 25000×125, and 30000×150. it is in the terms of the number of grids in the axial direction multiplied by the number of grids in the radial direction. the calculation of the poiseuille number is done for each grid system and its value increases from 5000×25 to 30000×150 grid system. the increase in the poiseuille number for the last two grids is very small whereas the computational time and effort are reasonably higher for the last grid system. therefore, the 25000×125 grid system is selected for conducting further simulations [15]. 4. results and discussion in this section of work, the irreversibility towards the micro-scale from macro-scale through a mini-scale for al2o3-water nf at different vf with and without thermophysical property variation is presented. the effects of the μ(t) variations for different vf (1% and 2.5% vf of al2o3 in base fluid water) on eg are investigated. the range of working temperature remains between 274-372 k (there is no change in the phase) in the fluid domain. so, the variations of single-phase properties are applicable. fig. 2 illustrates the variation in sgen,ht-rad along the channel diameter for cps and vps at different vfs. it is observed that the sgen,ht-rad does not play a vital role in the micro-dimension channel in comparison to the macro-dimension channel. in addition to this, the temperature gradient in the radial direction has a low value for the cases when viscosity variation is considered. at the micro-dimensions the sgen,ht-rad has approximately the same value for all the cases. the value of sgen,ht-rad experiences a sharp increase as it approaches macro-dimensions from micro-dimensions due to the lower bulk mean temperature of the fluid for bigger channel dimensions. the maximum value of sgen,ht-rad is for 2.5% al2o3 vps followed by cps water, vps water, 1% al2o3 vps, 1% al2o3 cps, and 2.5% al2o3 cps at the macro level. however, sgen,ht-rad is maximum for 2.5% al2o3 vps at the micro-level. the plot between sgen,ht-ax, and dn is shown in fig. 3. the observations showed that the value of sgen,ht-ax is insignificant in the calculation of total eg. the sgen,ht-ax is calculated by using ti,j only, as both thermal conductivity and temperature gradient in the axial direction are constant in the case of μ(t) variations. ti,j increases as channel dimensions are increased, so sgen,ht-ax decreases with an increase in the channel dimensions. although, sgen,ht-ax increases as al2o3 nanoparticles vf increases and is calculated irreversibility analysis in al2o3-water nanofluid flow with variable property 511 maximum for cps or vps at 2.5% vf of al2o3 in base fluid water at micro-scale followed by cps or vps at 1% of al2o3, and cps or vps of water. fig. 2 the sgen,ht-rad versus dn fig. 3 sgen,ht-ax versus dn 512 k. kumar, p. r.chauhan, r. kumar, r. s. bharj however, fig. 4 makes it clear that sgen,fr plays a very important role in the calculation of total eg at micro-scale. with an increase in the channel dimensions μ(ti,j) decreases due to the consideration of viscosity variation. the ti,j value for the cases with viscosity variation is almost the same as that for the cases with cps. but, the velocity gradient in the radial direction increases with a decrease in the channel dimensions; also this gradient is larger in the cases where viscosity variation is considered. in addition to this, the increase in particle concentration increases the frictional losses. an enlarged view of fig. 4 demonstrates that at a higher value of vf of nanoparticles, maximum sgen,fr is found. it is noticed that sgen,fr increases drastically towards micro-dimension for both vps and cps. however, sgen,fr is maximum for the μ(t) variation due to the presence of 2.5% vf of al2o3 nanoparticles in the fluid domain at micro-dimension. fig. 4 sgen,fr versus dn fig. 5 shows a valley (at minimum sgen,tot) in the plots of sgen,tot for each cps and vps of various fluid flows from macro to micro-scale. this valley helps to calculate d* corresponding to n*. the vf of nanoparticles increases sgen,tot at the micro-level whereas it decreases sgen,tot at the macro level. the calculation of sgen,tot includes both sgen,fr and sgen,ht. therefore, sgen,tot is maximum investigated at micro-scale because of the greatest value of sgen,fr (as from fig. 4). therefore, it is concluded that the role of vf of nanoparticles is more significant in mc as compared to macro-channels. figs. 6 and 7 illustrate the variation in be and φ along with dn , respectively. for a particular diameter of mc, sgen,ht is found to be more for the case of cps as compared to respective vps. however, in macrochannel sgen,ht is very close to sgen,tot (because the value of be equals to 1 approximately). the 0.5 value of be indicates the equal contribution of both sgen,ht and sgen,fr in the total eg. fig. 7 concludes the dominance effect of sgen,fr over sgen,ht. the maximum dominance of sgen,fr is found for 2.5% vf of al2o3-water nf among all the working fluids. the value of the irreversibility distribution irreversibility analysis in al2o3-water nanofluid flow with variable property 513 ratio approaches zero as the channel dimensions are increased. this is due to the dominance of ht eg at the macro-dimensions. fig. 5 sgen,tot versus dn fig. 6 be versus dn 514 k. kumar, p. r.chauhan, r. kumar, r. s. bharj fig. 7φ versus dn 5. optimization the problem undertaken in the present work is to optimize the channel diameter by minimizing the total eg rate, given by eq. (21) for the best possible overall performance of the mc heat sink. figs. 8, 9, and 10 illustrate the plots of variation in sgen for cps and vps along with channel diameter for water, 1% al2o3-water nf, and 2.5% al2o3-water nf, respectively. these plots depict that sgen,ht and sgen,fr intersect (at sgen,ht = sgen,fr) each other at a point. this helps to determine the values of dint for both cps and vps. fig. 8 sgen versus dn for water irreversibility analysis in al2o3-water nanofluid flow with variable property 515 for a better view of the intersection zone, an enlarged view is shown in the respective figures (figs. 8-10). the results show that as vf of al2o3 nanoparticles increase in the base fluid, dint shifts toward the right side in the plot for both cps and vps because of a faster rate of the friction change than that of ht change. additionally, it is also found that at a particular vf (1% or 2.5%) of al2o3 nanoparticles in water, dint has more value for vps as compared to cps. fig. 9 sgen versus dn for 1% al2o3 nanoparticle concentration nanofluid fig. 10 sgen versus dn for 2.5% al2o3 nanoparticle concentration nanofluid 516 k. kumar, p. r.chauhan, r. kumar, r. s. bharj table 3 shows n (actual) corresponding to dint, and d* corresponding to n* (a natural number) for various vf of al2o3 nanoparticles in water. d* is calculated maximum for 2.5% al2o3-water nf with μ(t) variation followed by 1% al2o3-water nf with μ(t) variation, 2.5% al2o3-water nf with cps, and 1% al2o3-water nf with cps. from fig. 5 and table 3 it can be interpreted that for the flow of nfs, the lowest total eg is for the case of 1% vf of al2o3 nanoparticles with cps. so, 58.72 µm is the optimum diameter of the channel under the considered condition and nf flow. table 3 optimum channel diameter corresponding to n* nanofluids n (actual) dint (µm) n* (natural number) d*(µm) 1% al2o3-water nf cps 290.18 58.70 290 58.72 1% al2o3-water nf μ(t) 155.16 80.28 155 80.32 2.5% al2o3-water nf cps 175.34 75.52 175 75.59 2.5% al2o3-water nf μ(t) 75.30 115.24 75 115.47 6. conclusions in this numerical study, the channel geometry (macro to micro-scale) is optimized for the flow of the nf with different vf using irreversibility analysis. besides, the effects of μ(t) variation on ht eg, frictional eg, and total eg are also analyzed. the study concluded that: 1. the variations in sgen,ht-ax and sgen,ht-rad are not found significant but they increase with increment in the vf of al2o3 nanoparticles. 2. the sgen,fr drastically increases as the vf of al2o3 nanoparticles increase. therefore, it has a major role in calculating sgen,tot at micro-dimension. this increment in sgen,fr is due to the increment in the flow friction with the rising vf of the nanoparticles. 3. at micro-scale, the μ(t) variation also does not have a significant impact on the value of sgen,ht-ax and sgen,ht-rad because of low temperature in the flow domain in comparison to the macro-scale. however, sgen, fr increases rapidly at micro-scale because of a larger velocity gradient in the radial direction. 4. when μ(t) variation effect is considered, sgen,tot is found maximum for 2.5% vf of al2o3 nanoparticles among all the cps and vps for various kinds of working fluids. 5. at a particular vf (1% or 2.5%) of al2o3 nanoparticles in water-based nf, dint has more value for vps comparing to cps because of a higher velocity gradient in the radial direction when viscosity variation is considered. 6. the calculated value of d* is maximum for 2.5% al2o3-water nf with μ(t) variation followed by 1% al2o3-water nf with μ(t) variation, 2.5% al2o3-water nf with cps, and 1% al2o3-water nf with cps. references 1. xie, j., amano, r.s., 2004, numerical simulation of two-phase flow in microchannel, ieee ninth intersociety conference on thermal and thermomechanical phenomena in electronic systems (ieee cat. no. 04ch37543), 2, pp. 679-686. 2. han, s., gongnan, x., chi-chuan, w., 2020, thermal performance and entropy generation of novel xstructured double layered microchannel heat sinks, journal of the taiwan institute of chemical engineers., 111, pp. 90-104. irreversibility analysis in al2o3-water nanofluid flow with variable property 517 3. oztop, h.f., abu-nada, e., 2008, numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, international journal of heat and fluid flow, 29(5), pp. 1326-36. 4. tayebi, t., öztop, h.f., 2020, entropy production during natural convection of hybrid nanofluid in an annular passage between horizontal confocal elliptic cylinders, international journal of mechanical sciences, 171, 105378. 5. zúñiga-cerroblanco, j.l., gonzalez-valle, c.u., lorenzini-gutierrez, d., hernandez-guerrero, a., cervantes de gortari, j., 2016, heat sink performance improvement by way of nanofluids, proceedings of the asme 2016 heat transfer summer conference collocated with the asme 2016 fluids engineering division summer meeting and the asme 2016 14th international conference on nanochannels, microchannels, and minichannels, washington, dc, usa, 2. 6. kumar, k., kumar, r., bharj, r.s., mondal, p.k., 2021, irreversibility analysis of the convective flow through corrugated channels: a comprehensive review, the european physical journal plus, 136(4), pp. 1-40. 7. bejan, a., 1980, second law analysis in heat transfer, energy, 5(8-9), pp. 720-732. 8. kumar, k., kumar, r, bharj, r.s., 2020, entropy generation analysis due to heat transfer and nanofluid flow through microchannels: a review, international journal of exergy, 31(1), pp. 49-86. 9. chen, k., 2004, second‐law analysis and optimization of microchannel flows subjected to different thermal boundary conditions, international journal of energy research, 29(3), pp.249-263. 10. bejan, a., 1995, entropy generation minimization: the method of thermodynamic optimization of finite-size systems and finite-time processes, crc press, boca raton. 11. bejan, a., 2002, fundamentals of exergy analysis, entropy generation minimization, and the generation of flow architecture, international journal of energy research, 26(7), pp. 545-565. 12. kumar, r., mahulikar, s.p., 2015, effect of temperature-dependent viscosity variation on fully developed laminar microconvective flow, international journal of thermal sciences, 98, pp. 179-191. 13. rastogi, p., mahulikar, s.p., 2018, entropy generation in laminar forced convective water flow due to overloading toward the microscale, asme journal of energy resources technology, 140(8), 082002. 14. rastogi, p., mahulikar, s.p., 2018, optimization of micro-heat sink based on theory of entropy generation in laminar forced convection, international journal of thermal sciences, 126, pp. 96-104. 15. chauhan, p.r., kumar, r., bharj, r.s., 2019, optimization of the circular microchannel heat sink under viscous heating effect using entropy generation minimization method, thermal science and engineering progress, 13, 100365. 16. kumar, k., kumar, r. bharj, r.s., 2020, circular microchannel heat sink optimization using entropy generation minimization method, journal of non-equilibrium thermodynamics, 45(4), pp. 333-342. 17. heshmatian, s., bahiraei, m., 2017, numerical investigation of entropy generation to predict irreversibilities in nanofluid flow within a microchannel: effects of brownian diffusion, shear rate and viscosity gradient, chemical engineering science, 172, pp. 52-65. 18. bianco, v., scarpa, f., tagliafico, l.a., 2018, numerical analysis of the al2o3-water nanofluid forced laminar convection in an asymmetric heated channel for application in flat plate pv/t collector, renewable energy, 116, pp. 9-21. 19. manay, e., akyürek, e.f., sahin, b., 2018, entropy generation of nanofluid flow in a microchannel heat sink, results in physics, 9, pp. 615-624. 20. bianco, v., marchitto, a., scarpa, f., tagliafico, l.a., 2019,numerical investigation on the forced laminar convection heat transfer of al2o3-water nanofluid within a three-dimensional asymmetric heated channel, international journal of numerical methods for heat & fluid flow, 29(3), pp. 1132-1152. 21. alfaryjat, a.a., dobrovicescu, a., stanciu, d., 2019, influence of heat flux and reynolds number on the entropy generation for different types of nanofluids in a hexagon microchannel heat sink, chinese journal of chemical engineering, 27(3), pp. 501-513. 22. shashikumar, n.s., gireesha, b.j., mahanthesh, b., prasannakumara, b.c., chamkha, a.j., 2019, entropy generation analysis of magneto-nanoliquids embedded with aluminium and titanium alloy nanoparticles in microchannel with partial slips and convective conditions, international journal of numerical methods for heat & fluid flow, 20(10), pp. 3638-3658. 23. kumar, r., mahulikar, s.p., 2018, physical effects of variable thermophysical fluid properties on flow and thermal development in micro-channel, heat transfer engineering, 39(4), pp. 374-390. 24. kumar, r., mahulikar, s.p., 2020, heat transfer characteristics of water flowing through micro-tube heat exchanger with variable fluid properties, journal of thermal analysis and calorimetry, 140, pp. 1919–1934. 25. chauhan, p.r., kumar, k., kumar, r., rahimi-gorji, m., bharj, r.s., 2020, effect of thermophysical property variation on entropy generation towards micro-scale, journal of non-equilibrium thermodynamics, 45(1), pp. 1-7. 518 k. kumar, p. r.chauhan, r. kumar, r. s. bharj 26. wen, d., ding, y., 2004, experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, international journal of heat and mass transfer, 47(24), pp. 5181-5188. 27. hussein, a.m., bakar, r.a., kadirgama, k., sharma, k.v., 2013, experimental measurement of nanofluids thermal properties,international journal of automotive and mechanical engineering, 7, pp. 850-863. 28. kumar, r., mahulikar, s.p., 2017, numerical re-examination of chilton–colburn analogy for variable thermophysical fluid properties, asme journal of heat transfer, 139(7), 071701. 29. bejan, a., 1982, entropy generation through heat and fluid flow, new york: wiley. 30. genić, s., jaćimović, b., petrovic, a., 2018, a novel method for combined entropy generation and economic optimization of counter-current and co-current heat exchangers, applied thermal engineering, 136, pp. 327-334. 31. milovančević, u.m., jaćimović, b.m., genić, s.b., el-sagier, f., otović, m.m., stevanović, s.m., 2019, thermoeconomic analysis of spiral heat exchanger with constant wall temperature, thermal science, 23(1), pp. 401-410. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 1, 2015, pp. 11 20 1new experimental investigations on the dieterich-ruina friction law udc 539.3 grzemba birthe berlin institute of technology, germany abstract. the dieterich-ruina friction law is widely used in rock friction and earthquake dynamical contexts. it has proven very powerful and versatile over the last decades, being able to reproduce stick-slip as well as foreand aftershocks and healing of faults. this paper shows that the dieterich-ruina friction law can also reproduce the accelerated creep process preceding slip event in a stick-slip system, not only for steel contact, but also for rock contact. the friction law relies on three empirical parameters, which have to be identified for each application. the most common experimental set-up is the velocity step experiment. in this paper an alternative method to identify the parameters is proposed, which uses a stick-slip experiment and a numerical fitting approach. key words: dieterich-ruina friction, stick-slip, single slider, accelerated creep, parameter fitting 1. introduction the dieterich-ruina friction law is a special version of the rate-and-state friction law introduced by j.h. dieterich in the 1970s [1, 2], which was proposed by a. ruina [3, 4]. it was originally developed to describe earthquake dynamic processes, namely rock frictional contacts experiencing frictional instabilities and it is very widely used in this area [5-12]. the popularity of the dieterich-ruina friction law (drf law) comes from its extreme versatility, being able to reproduce stick-slip, foreand aftershocks as well as healing of faults [9, 10]. the drf law states that the coefficient of friction  depends on the relative velocity v and the contact history, represented by an internal state variable . several examples for the use of internal variables in friction laws can be found in [13, 14]. a common form of the drf law [15] is given as received january 05, 2015 / accepted february 8, 2015 corresponding author: grzemba birthe technische universität berlin, strasse des 17. juni 135, 10623 berlin e-mail: birthe.grzemba@tu-berlin.de original scientific paper 12 b. grzemba 0 ln ln * * v a b v                 (1) where (.) * represents a reference value. the reference state variable value is usually given as * * cd / v  . parameters a, b and dc are the empiric parameters of the friction law. internal variable  , representing the mean lifetime of a micro-contact, is connected with the relative velocity by a kinetic equation. several forms of this kinetic equation have been proposed; the dieterich-ruina formulation is the one named ‘slip law’ in contrast e.g. to the also often used ‘aging law’, see [16] for comparison: 1 c v d     (2) the use of the drf law is not necessarily limited to earthquake dynamical processes. the drf law has shown its applicability to a wide range of materials other than rock, including steel, glass, plastic and wood [17]. the drf law is a laboratory based, empirical friction law, which means special attention has to be paid to the included parameters, which have to be found experimentally. parameter dc is a length parameter and is expected to be dependent on other system parameters and to show scaling behavior [1, 18, 19]. dimensionless parameters a and b are usually found in the order of 210 to 310 . for characteristic length parameter dc values around 0 5. to 50μm for laboratory set-ups and values around 1 to 10mm for real faults are identified [7, 18]. the most common experiment to identify the drf law parameters is a velocity step experiment as introduced by dieterich and kilgore [17] and repeated later on by others. the experiment drives a frictional contact at steady velocity before introducing a sudden step in the velocity to a value about one order faster, then holds this faster velocity and steps down again. from a measurement of the coefficient of friction over slip distance parameter dc and difference (a  b) can be identified. this method is still used for the identification of drf law parameters for specific material pairings under given load and temperatures [5 6, 20]. it is also used to decide if velocity weakening or velocity strengthening is present which can be read from the sign of (a  b). this method has a few disadvantages. firstly, the velocity step experiment can be performed with any frictional contact, independent if the configuration behaves in a drf law fashion or not. this means, even if the drf law is not a suitable model, the velocity step experiment will provide values for its parameters. this can be seen e.g. in [21,22], where the found (a  b) values depend on the sliding velocity, which should not be the case in a system obeying the drf law. secondly, the velocity step experiment does not contain stick-slip behavior. this can be advantageous if also possible velocity strengthening shall be addressed. but if the drf law shall be used for the modeling of frictional instabilities, it would be desirable to include this dynamic in the process to identify the parameters. lastly, the velocity step experiment only delivers difference (a  b) but not both parameters. this paper will present a new experimental method to deduce the drf law parameters under consideration of both macroscopic and microscopic features of the stick-slip new experimental investigations on the dieterich-ruina friction law 13 process. this will be done using a single block slider set-up. the elastically loaded single block slider has been excessively used as an analytical model to describe and investigate frictional instabilities and stick slip, mainly as a slider-on belt model for brake dynamics [23-25] and as loaded block on surface for earthquake dynamics [1, 4, 26, 27]. experimental investigations of this system, on the other hand, are rare. here, an extensive measurement series is performed using a single block slider set-up (section 2) combined with numerical modeling techniques (section 3) to show once more the eminent versatility and power of the drf law and to identify the drf law parameters of the system. 2. experiments 2.1. set-up the experimental set-up follows the concept of the single block slider model. the test stand and the measurement system are shown in fig. 1. fig. 1 experimental set-up (left) and measurement system (right) a specimen is pulled over exchangeable v-shaped base plates by a spring. the velocity of the spring’s base point is constant and realized via a linear pulling device consisting of a linear guidance and a step motor driven spindle. if the specimen does not move, the tangential load on the specimen rises linearly with time. the displacement of the specimen is measured using a laser-doppler vibrometer with 8 nm resolution. additionally, the spring force is measured via a load cell. several parameters of this set-up are varied. for the results presented here, i have used four different specimens, two made of marble, two made of steel, with different contact geometries (see fig. 2). two different sets of base plates, one made of steel and one made of sandstone, are used. the stiffness of the pulling spring is varied as well as the driving speed. for each measured configuration at least thirty 10s measurements are performed, such that a total number of 932 experiments are evaluated for the presented results. 14 b. grzemba fig. 2 used specimens. steel specimens (sharp edged and cylindric) on the left, marble specimens (regular circular and irregular) on the right 2.2. results all experimental configurations show a distinct stick-slip behavior. an example for a displacement and velocity curve over time can be seen in fig. 3. we see comparably long stick phases that alternate with quick and sudden slip events. if one zooms in the stick phases to the micro meter scale, a regular accelerated creep process can be seen (fig. 4). this preceding creep can even be used to predict the time of the next slip event [28, 29]. fig. 3 example of a typical displacement and velocity curve. the configuration of this data set is marble on sandstone with a driving velocity of 10mm/s new experimental investigations on the dieterich-ruina friction law 15 fig. 4 accelerated creep during stick phases. the shown data set is the same as in fig.3, the first slip event is very small and can hardly be seen in the macroscopic view in this paper, the experiments shall be used as a method to extract dieterich-ruina friction law parameters for the given contact. for this, first it must be shown how good the microscopic creep process and the fast slip can be described by a model using the drf law. 3. modeling and numerical fitting the experimental set-up is modeled as a single block slider with dieterich-ruina friction (see fig. 5). in this model the specimen is assumed to be rigid, which is a valid assumption because the longitudinal stiffness of the specimen is several orders of magnitude higher than the contact stiffness. 16 b. grzemba fig. 5 single block slider model with dieterich-ruina friction. the slider has mass m and the spring of stiffness c is driven with velocity 0v using the notation from fig. 5 the equation of motion is given by    0 0sgn ln ln * * c x v mx c v t x x mg a b v d                   (3) where g is the gravity constant. combined with the kinetic equation (2) a 2 2 nonlinear ordinary differential equation system is gained. this can be solved numerically, if some obstacles are overcome. i have attenuated the discontinuity of the friction law to make a numerical solution faster and more stable. to achieve this, i have approximated the signum function as a very steep hyperbolic tangent, which results in a stiff ode system, which again needs suitable solver. i have used a matlab internal solver based on a semiimplicit runge-kutta method. furthermore, for numerical stability, the arguments of the logarithms in (3) are completed with an additional 1 to avoid a sign change. this is allowed if reference velocity v * is chosen much higher than all occurring relative velocities ( * v x ). the friction law then takes the form * * 0 ln 1 ln 1 c v v a b x d                  (4) i have varied parameters a, b and dc to achieve a best fit with the experimental data. during this process it has become clear that the model underestimates the very low creep velocities in the region of 10 7 m/s. this is assumed to be a result of the rigid body assumption: the frictional contact has a finite elasticity, which is reversible. this behavior is not included in the friction law. to compensate for this shortcoming, additional very slow creep velocity v̂ is introduced, which can be estimated as 5 4 0 010 to 10v̂ v v    (5) this estimation is gained using a rough approximation of the tangential contact stiffness. for the fitting, the numerical and experimental velocity curves are compared. i have found the velocity data to be more indulgent and therefore easier to fit than the displacement data. a logarithmic error definition is used to not only consider the fast slip events for the fitting, but also make the model meet the creep curves. this is especially important because the creep contains important information on the frictional behavior. the error definition reads new experimental investigations on the dieterich-ruina friction law 17   2 num fit 1 log loglog ,i exp,i i v x x t t     (6) as fitted time interval tfit a value of approximately 16% of a characteristic stick time is used. the time resolution is 52 10 st    as in the experiments. to find the optimal parameters, a gradient method is used. all fitting procedures use the same starting values (see table 1). for the majority of all data sets the fitting works remarkably well and the experimental velocity curves are met over five to six orders of magnitude. a few examples are given in fig. 6. 22% of the data sets could not be fitted satisfyingly with this model. table 1 starting parameter values for fitting procedure a b dc 36 10 21 10 2μm from the results of these fittings, mean values for the found optimal dieterich-ruina parameters can be gained. the found mean parameters are gathered in table 2. table 2 mean values of optimal parameters for all material pairings material pairing slider specimen a in 310 b in 310 cd in μm steel-steel sharp edged 4 16. 10 66. 1 26. cylindrical 3 48. 12 16. 0 74. marble-sandstone regular circular 4 85. 9 06. 0 74. irregular circular 4 38. 8 89. 0 54. the found optimal parameters show a considerable statistical scatter, especially parameter dc. this scatter is expected if recalling the physical meaning of dc. characteristic distance dc describes the slip distance over which all micro-contacts are renewed [1]; this implies that is lies in the order of magnitude of the size of the microcontacts. since the microscopic contact configuration changes with every new contact realization, a statistical spread is anticipated. parameters a and b do not possess such interpretation, but b has been shown to be closely related to the macroscopic slip length and the peak slip velocity [28]. these macroscopic entities should be affected less by changes in the microscopic configuration and are therefore expected to scatter less. the distributions of the found values for b and dc follow the general shape of a gaussian bell curve. an example for this can be seen in fig. 7. the values found for parameter a build an exception. they are more uniformly distributed and also show a slight trend with the driving velocity, which is undesirable. i assume the reason for this in the definition of the length of the fitting interval, which depends on the characteristic stick time and therefore on the driving velocity. for future work, this definition should be revised. 18 b. grzemba fig. 6 examples for successfully fitted data sets of different material pairings and driving velocities. the inserted smaller plots show the fitting region on a logarithmic scale new experimental investigations on the dieterich-ruina friction law 19 fig. 7 histograms showing the frequency of optimal found values for the three fitted parameters (a, b, dc) for the configuration sharp edged steel specimen on steel base plates a comparison of the found mean values with values from literature shows a good agreement. the values for dc are relatively small, but this can be explained by the low normal forces and the very smooth contact surfaces in the experimental set-up. if one estimates the values for dc using the recently proposed new scaling law [19], which takes into account the normal force and the surface roughness, value around 1 m should be expected. 4. conclusions a fit of a dieterich-ruina friction law model to experimental stick-slip data is performed for different material pairings and contact geometries. most data sets can be very well reproduced by the model; the velocity curves are met over five orders of magnitude, including the slow and accelerating creep as well as the fast slip events. the ability of the drf law to reproduce the systems behavior over such a range of scales is impressive and underlines the importance and power of this friction law. additionally, the characteristic dieterich-ruina friction law parameters are gained from this fitting, which agree well with literature values. this procedure introduces a new experimental method to determine the correct empirical parameters for a given frictional contact. the experimental set-up is easier to construct and handle than the ones needed for the traditional velocity step experiments. furthermore, this method directly uses the microscopic properties of the system, because the creep process is fitted. it can be an attractive alternative especially when the focus is on systems encountering frictional instabilities. acknowledgements: i want to acknowledge the help of dr. j. starcevic with the execution of the experiments. furthermore i am very thankful to prof. v.l. popov for the helpful conversations and his advice. 20 b. grzemba references 1. dieterich, j.h. 1978, time dependent friction and the mechanics of stick—slip, pure and applied geophysics, 116, pp. 790-806. 2. dieterich, j.h., 1979, 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and state dependent friction, journal of the mechanics and physics of solids, 32, pp. 167-196. 28. popov, v.l., grzemba, b., starcevic, j., popov, m., 2012, rate and state dependent friction laws and the prediction of earthquakes: what can we learn from laboratory models? tectonophysics, 532-535, pp. 291-300. 29. grzemba, b., 2014, predictability of element models for earthquake prediction, technische universität berlin, ph.d. dissertation. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 51 63 https://doi.org/10.22190/fume171226006p © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper simulation of wear in a spherical joint with a polymeric component of the total hip replacement considering activities of daily living udc 617.582 vladimir pakhaliuk 1,2 , aleksandr poliakov 1 1 sevastopol state university, sevastopol, russian federation 2 technische universität berlin, berlin, germany abstract. the present study assesses the impact of the main typical activities of patients' daily living (adl) after total hip arthroplasty (tha) on the wear parameters of sliding couple's materials by simulating linear and volumetric wear according to the archard's law in a spherical joint with a polymeric element of the total hip replacement (thr). the mathematical wear model, built on the basis of algorithms and custom codes of the finite element analysis in ansys and matlab software systems, has been studied numerically. the activities used in the model are: level walking, stair ascending-stair descending, chair sitting-chair rising, and deep squatting. they were described by typical waveforms of the angular displacements of the thr's femoral component and the waveforms of the applied force. the results of the simulation show that for the same duration the overall wear value with adl is significantly higher than in the case of level walking according to the requirements of iso 14242-1. therefore, the evaluation of the wear value for adl is more informative for predicting the functional life time of the thr. analysis of the simulation results shows that the amount of wear calculated for all activities separately is practically the same as the overall wear value obtained at summary action of adl. this effect of the independence of contributions to the total amount of wear of each activity makes it possible to significantly simplify the solution of the problem of wear estimation for typical activities, including stochastic ones. key words: activities of daily living, total hip replacement, wear, finite element simulation, spherical joint received december 26, 2017 / accepted february 01, 2018 corresponding author: vladimir pakhaliuk sevastopol state university, universitetskaya str. 33, 299053 sevastopol, russian federation e-mail: pahaluk@sevsu.ru 52 v. pakhaliuk, a. poliakov 1. introduction activities of daily living are attracting increasing attention in the studies related to their impact on the life behavior of an ordinary patient in the post-operative period after tha since they represent more natural conditions of his life. in addition to level walking, these activities can include the most frequent ones, such as stair climbing, chair sitting and squatting [1]. in some studies, the maximum range of motion (rom) in different planes in the hip joint after tha was determined with different activities [2-4]. reference [5] estimates the effect of adl on the fatigue longevity of the implant's stem and its microscopic displacement using a test device. but the greatest number of studies is related to kinematics and kinetics of large joints of the lower limb, such as the hip, knee and ankle with various activities. for example, only kinematics during stairs up, stairs down was studied in [6-8], and kinematics and kinetics of these same activities were studied in [911]. in refs. [12, 13], the kinematics of the chair up activity (during standing up from the chair or, in other words, sit-to-stand) was studied, and the kinematics of the chair down, chair up (or stand-to-sit-to-stand) were studied in [14-16]. in [17, 18], studies are carried out of both kinematics and kinetics for the chair up of large joints of the lower limb. in refs. [19, 20], kinematics of the same joints was studied during squatting activity. in addition, for the completeness of the adl evaluation, the frequency and duration of each of the activities evaluated in [21] is of particular interest. it should be noted that in many of the above references, the results are presented in terms of the maximum values of the corresponding parameters, for example, angular displacements, loads, moments, and very rarely in the form of wave dependencies with the cycle time of the specified parameter. moreover, in spite of the fact that each of the joints has a minimum of three rotational degrees of freedom, and the vector of the applied force, as a rule, has non-zero projections in three-dimensional space, most of the data presented are given in a simplified form. for example, the angular displacement is indicated only in one plane, only the resultant vector of the applied force is indicated, the study of this motor activity is not performed during the whole time step cycle, but only on the stance phase, etc. of special interest are studies in which estimates of the adl impact on the wear parameters of artificial joint elements in terms of, for example, linear, volume or gravimetric (mass loss) wear are performed. in this connection, very few references have been found in the available literature. in particular, in ref. [22], the results of studies of wear performances on a device for wear testing a total knee replacement for various adl, which were previously taken into account by us in the synthesis of a bio like artificial knee joint [23], are presented. and ref. [1] describes similar studies of a thr's spherical joint with a polymeric element, but on a specialized device for wear testing spherical sliding couples. it is clear that for successful implementation of such tests it is necessary to know the real kinematic motion dependencies in the joint, including angular displacements in three planes for the full step cycle for all the activities used, and also their kinetics, i.e. the coordinate’s change of applied force vector during a similar step. an attempt of a mathematical simulation of wear according to the archard's law [24, 25] in a spherical joint with a polymeric element of the thr for various adl using approximating expressions depending on the load parameters, head dimensions and surface roughness was undertaken in [26]. but despite the potential possibilities of the fem used simulation of wear in a spherical joint with a polymeric component of the total hip replacement... 53 in the work, the authors made a number of serious simplifications, as a result of which it was possible only to judge the comparative evaluation of the adl impact on the volume wear of the polymeric element. the objective of this study is to assess the impact of the main typical activities of patients' daily living after tha on wear parameters and their contribution to the overall wear value by simulating linear and volumetric wear according to the archard's law in a spherical joint with a polymeric element of the thr. the study was carried out on the basis of finite element analysis methods using previously developed approaches, detailed in [27, 28]. in the computational algorithms implemented in ansys and matlab, the kinematic waveforms of the angular movements of the femoral component and the waveforms of the coordinate’s change of applied force vector, characteristic for the main typical activities, were used. a review of the available literature data suggest that the impact of the patient's daily activities in a post-operative period after tha on the wear of a friction couple has been fulfilled for the first time and can serve as additional clarifying information in selecting materials and system synthesis of innovative designs of the thr with a polymeric element [29]. the algorithms of numerical modeling of wear at the synthesis stage developed in this paper, in contrast to simulation on simulators, make it possible to significantly accelerate and reduce the cost of analyzing the set of variants of new friction couples and of choosing the best among them in accordance with the specified quality criteria. 2. materials and methods the model of the thr’s sliding couple, described in detail earlier in refs. [27, 28], contains a solid femoral ball head of cobalt-chromium alloy or ceramics (alumina or zirconia) with widely used standard diameters of 32 mm employed against a soft (uhmwpe) acetabular cup. the radial clearance between them is of 0.15 mm. such a so-called ball-in-socket mated couple can be considered as rigid-to-soft one, where only the soft cup is subjected to wear. the elasticity modulus and poisson's ratio were chosen as 1.4 gpa and 0.46, respectively, for the cup (conventional uhmwpe) and 210 gpa and 0.3, respectively, for the head. the right hip joint is defined in anatomical fixed coordinates x ' y ' z ' and shown in fig. 1. in the simulation of wear, a simplified coordinate system xyz fixed to the cup and placed in its center is used (fig. 2). the movable coordinate system used for the euler angles coinciding with the center of the cup, is placed in the center of the head and fixed to the head. the head has three rotational degrees of freedom, known as fe (flexion-extension), aa (abductionadduction), and ior (inward-outward rotation). the wear simulation for a soft cup was based on the classical archard’s law according to it, for the ideal uniformly loaded isotropic surfaces with a nominal contact pressure in the linear elastic condition, an accumulative local linear wear depth at the contact surface δh(θ,φ) in a spherical coordinate system in a discrete kind can be described as following [30] 1 ( , ) ( , , ) ( , , ) n w i i i h θ k σ θ t s θ t       (1) where σ(θ,φ,ti) is a normal contact pressure between the counter-face surfaces at the same point of time instant ti of the gait cycle; δs(θ,φ,ti) is an increment of the arc sliding 54 v. pakhaliuk, a. poliakov fig. 2 a simplified coordinate system xyz [27, 28] fig. 1 front view of the right hip joint with the specified directions of rotation (l is a resultant load vector) [27, 28] distance between the adjacent measuring points under the same conditions; kw is a wear factor which depends on the material, nature of the surface and, as was found, the nominal contact pressure. a modified formula for determining the depth of wear is presented in [28], considering the change in the wear factor as a function of the contact pressure. the finite element analysis of the wear depth model created by considering dependence (1) is performed in ansys and matlab software systems. kinematic waveforms of the angular displacements of the femoral joint component are used to determine the increment of arc sliding distance δs(θ,φ,ti), and in determining normal contact pressure σ(θ,φ,ti) by solving the contact problem, the change in magnitude of the applied force vector during the step cycle is taken into account. the calculation algorithm is presented in detail in [27, 28], where the parameters of wear are studied during the level walking in two versions: according to the demands of iso 14242-1, as well as for profiles of angular femoral positions, measured by jonhston and smidt, and for patterns of the applied force, measured by paul [31]. in fact, the patient, in addition to level walking, daily performs other motor actions, the consideration of which allows determining the wear parameters closer to the real ones. ref. [1] shows the results of the thr's wear tests on the simulator taking into account the next main typical normal activities of the patient's daily living: level walking, stair ascending and stair descending, chair sitting and rising, and deep squatting. in the same study, based on the frequency of the cycle of each activity, the relative duration of each type of activity, reduced to the same cycle duration for 1 s, is determined, and is distributed as follows: 44% level walking, 24% stair ascending-stair descending, 12% chair sitting, 12% chair rising, and 8% deep squatting. the tests during level walking presented in [1] are carried out in accordance with the iso 14242-1 demands for angular movements of the femoral joint component and the applied force. for other activities, only the resulting vector of the applied force is shown for the worst case load when the simulation of wear in a spherical joint with a polymeric component of the total hip replacement... 55 fig. 4 waveforms of three-dimensional force vector for stairs up fig. 3 waveforms of two-dimensional force vector according to iso 14242-1 patient's weight force is of 1000 n, but no kinematic profiles of the angular displacements are presented. in the current study, numerical wear simulation of the thr's spherical polymeric component is conducted, based on the activities indicated in [1] and their duration. at the same time, the wear during the level walking activity is evaluated using parameters regulated by iso 14242-1. in the design scheme of the model, the outer surface of the cup is stationary relative to the base coordinate system, so when determining the applied force, the inversion method is used, according to which the force vector is applied to the center of the head and directed toward the cup. the profiles of components of the twodimensional resulting vector of applied force fres, given in accordance with iso 14242-1 in the coordinate system shown in fig. 1, are depicted in fig. 3, and the profiles of components of the three-dimensional vector of the applied force corresponding to the load worst case with the patient's weight force in 1000 n for such motor activities as stair ascending, stair descending, chair sitting and chair rising, borrowed from ref. [32], are shown, respectively, in figs. 4-8. when converting the coordinates of the applied force vector from the coordinate system indicated in ref. [32] to the simplified coordinate system (fig. 2), the average anteversion angle equal to 12.5º is taken into account [32], and the angle between the axis of the stem's neck of the thr and the vertical component of the force vector when the position of the axis of the stem is parallel to the axis of the femur, is equal to 45°. therefore, the directional coefficients of the force vector coordinates transformation are taken as follows cos12.5 cos45 0.69xk   ; cos12.5 0.976yk   ; cos45 0.707zk   . 56 v. pakhaliuk, a. poliakov fig. 5 waveforms of three-dimensional force vector for stairs down fig. 6 waveforms of three-dimensional force vector for sitting down to chair fig. 7 waveforms of three-dimensional force vector for standing up from chair fig. 8 waveforms of two-dimensional force vector for deep squatting fig. 9 waveforms of angular movements of the femoral head according to iso 14242-1 fig. 10 waveforms of angular movements for stairs down-stairs up [10] simulation of wear in a spherical joint with a polymeric component of the total hip replacement... 57 fig. 11 waveforms of angular movements for sitting down to chair [14] fig. 12 waveforms of angular movements for standing up from chair [14] reliable profiles of the change in the coordinates of the applied force vector during deep squatting are not found in any available literature data. therefore, to estimate the wear parameters for this motor activity, the profile of the resultant force vector, given in [1], is used, which is also transformed at the transition to a simplified coordinate system in analogy with the level walking and is shown in fig. 8. the kinematic waveforms of the angular displacements of the femoral joint component required for modeling which are usually used by researchers for above adl and are available in a number of literature sources, are shown in figs. 10-13 with the corresponding references from which they are taken. fig. 13 waveforms of angular movements for deep squatting [19] numerical simulation of wear is carried out in two ways. the first one is that all adl listed above are lined up in a predetermined order with a preset duration of motor task for the one simulation cycle: 44% level walking, 24% stair ascending-stair descending, 12% chair sitting, 12% chair rising, and 8% deep squatting. in order to follow the 58 v. pakhaliuk, a. poliakov approximate number of gait steps at which the correction of the supporting surface of the cup geometry is performed by moving the nodes to the value of the obtained linear wear [28], their minimal number of 500,000 is taken. in this case, the number of replicates of each activity, which is necessary to fulfill their duration, is taken to be a multiple of 20,000 wherein they correspond to the number of steps: 220,000, 120,000, 60,000, 60,000, and 40,000, respectively. the total number of steps in the model is assumed to be 5 million, which allows 10 cycles of the geometry correction of the cup's supporting surface. given that each step is divided into 25 time slots, the solution of contact problem in ansys is carried out inside each one, and it is easy to calculate the number of such solutions that for six separate activities it makes up 150 for one cycle, while for 10 cycles it is of 1500, which significantly increases computer time. taking into account that in this work the task of obtaining absolutely accurate values of wear parameters is not being posed, and the assessment is mainly being made of the qualitative impact of each adl on wear parameters and their contribution to the overall wear value, the chosen simulation duration allows us to do this sufficiently earnestly. the second way of simulating is by evaluating the contribution of each of these adl separately to the total amount of wear. for this, only one of these activities is modeled in each cycle with its duration in the range of up to 10 cycles of the geometry correction of the cup's supporting surface, i.e. up to 5 million steps. 3. results figure 14 shows the linear wear patterns for simulation of the adl influence for the first and second ways for each of the activities, and in fig. 15 is for the same ones, but for the cumulative volume wear. fig. 14 graphs of linear wear during summarized adl and for each the adl contributed in figs. 16-17 profiles are depicted of linear and cumulative volume wear during adl simulation by the first way and in the case when the patient performs only level walking simulation of wear in a spherical joint with a polymeric component of the total hip replacement... 59 according to iso 14242-1 demands for the same cycle duration as for the total adl. fig. 18 shows the contribution of each of the above adl to the overall wear value. fig. 15 graphs of cumulative volume wear during summarized adl and for each of the adl contributed fig. 16 graphs of linear wear during summarized adl and for level walking according to iso 14242-1 at the same cycle duration as during adl 60 v. pakhaliuk, a. poliakov fig. 17 graphs of cumulative volume wear during summarized adl and for level walking according to iso 14242-1 at the same cycle duration as during adl fig. 18 bar chat on the contribution of each the adl into the overall amount of wear 4. discussion the profiles depicted in figs. 14-15 show a pronounced linear relationship observed for both linear and cumulative volume wear from the number of gait steps for all indicated adl and their total impact. their elementary analysis also makes it possible to note that the total wear calculated due to effect of each separate adl practically coincides with the overall wear value obtained at summary action of adl. this effect of the independence of contributions to the total amount of wear of each activity makes it possible to significantly simplify the solution of the problem of estimating wear for atypical activities, including stochastic ones. from the profiles shown in figs. 16-17 it follows that the total wear value for adl is significantly higher than in the case of only level walking, which is recommended by iso 14242-1 for the thr wear tests. in this case, the largest contribution to the total wear simulation of wear in a spherical joint with a polymeric component of the total hip replacement... 61 amount (about 41%) gives the stairs up-stairs down moving (see fig. 18), despite the fact that its duration is only 24% of the total cycle time. about 21% of the total wear amount occurs during sitting down-standing up and squatting, although in duration they differ 3 times. and, finally, the least contribution to total wear (about 17%) pay in level walking, despite the longest duration of the action (44% of the total cycle time). therefore, the estimation of the wear value with considering of adl is more informative for predicting the functional longevity of the thr compared to the recommended one during level walking at iso 14242-1. the linear form of the dependencies of the wear characteristics obtained in fig. 14-17 can be probably explained simply by the geometric relationships between the parameters of the figures formed from the sphere. the shape of the contact in the spherical joint on the surface of the cup as the head is deepened into it during its wear is an approximately spherical surface of the ball segment where, as the head becomes deeper, this surface and the volume of the segment increase linearly depending on the wear depth according to known mathematical relationships. this conclusion is also confirmed by the results obtained in [32] and in a number of references cited in [27, 28]. the values of wear parameters for level walking, presented in figs. 16-17, are consistent with their values in [27, 28, 30, 31], which indicates the reliability of the study results. but they are in some ways different from simulation results on the simulator specified in [1], although the used head diameter is the same and is of 32 mm. this may be due to the fact that the kinematic profiles of the angular movements of the femoral component used in [1] likely differs from those used in this study. these profiles influence the length of the slip path of points on the surface of the head over the surface of the cup and, by that, affect the depth of wear according to formula (1). therefore, the presence of such a factor could lead to serious differences in the results obtained. the results obtained in this study show that in order to extend the resource of the thr, patients after tha can be given the following recommendations: significantly restrict the activities of stair ascending-stair descending by using the elevator and deep squatting by its possible replacement for chair sitting-chair rising. 5. conclusion and outlook in this study, the effect of the main typical activities of patients' daily living after tha on wear parameters and their contribution to the overall wear value is estimated for the first time by simulating linear and volumetric wear according to the archard's law in a spherical joint with a polymeric element of the thr. the study was carried out on the basis of finite element analysis methods using previously developed approaches, detailed in [27, 28]. in the computational algorithms implemented in ansys and matlab, the kinematic waveforms of the angular movements of the femoral component and the waveforms of the coordinate's change of applied force vector, characteristic for the main typical activities such as level walking, stair ascending-stair descending, chair sitting, chair rising, and deep squatting, were used. to verify the adequacy of the mathematical model used in the study and to confirm the reliability of the results obtained, in the near future an experimental evaluation of wear will be performed on the device developed for wear testing of the thr. its design and operation principle of which is described in [33, 62 v. pakhaliuk, a. poliakov 34] and which allows to replicate with high accuracy all kinematic and kinetic parameters of movements in the joint, characteristic for adl. such confirmation is necessary for the development of well-founded methods for predicting the functional life time of the thr’s different design for arbitrary regimes of the patient's motor activity after tha. simulations in this paper have been carried out under assumption of validity of the archard’s law in its simplest form. however, it is known that it is only a very rough approximation. under certain conditions, the wear in a given tribological pair can increase dramatically or vanish almost completely [35]. it would be interesting to find out the conditions for transitions between “normal” wear, severe wear and almost wear-less conditions. further, in the present approach we have not explicitly considered the transport of the wear particles in the frictional zone, which, however, may significantly influence the wear process [36]. finally, it would be interesting to consider a possibility of similar simulations 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37. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17(4), pp. 334–340. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 203 217 https://doi.org/10.22190/fume180403017k © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper investigation of the energy recovery potentials in ventilation systems in different climates udc 697 miklos kassai 1 , laszlo poleczky 1 , laith al-hyari 1 , laszlo kajtar 1 , jozsef nyers 2,3 1 department of building service engineering and process engineering, budapest university of technology and economics, hungary 2 doctoral school of applied informatics and applied mathematics, obuda university budapest, hungary 3 doctoral school of mechanical engineering, szent istván university, hungary abstract. the aim of this research study was to investigate the energy recovery potentials in ventilation systems under different climatic conditions. the well-known heating degree day from the literature was updated using the weather data of cities with different climates from the past 40 years. as the novelty of this research with the developed procedure drawn up in this study, the energetic possibilities of heat recovery under various climate and operating conditions may be examined in more detail and more realistically than with the methods and available information of current engineering practices. to achieve this long-term and high definition the weather data of several cities are processed in order to evaluate the possibilities of heat recovery on a daily and annual basis. key words: heat recovery, energy recovery, degree day, enthalpy-hour, weather data, different climates 1. introduction the basic principle of modern engineering practice is an energy efficient design and operation [1-3]. significant energy recovery is possible from ventilation equipment in the case of proper choice of equipment and operation [4-9]. considering the selection of heat recovery equipment, the duration of economic returns is of key importance since an received april 03, 2018 / accepted june 07, 2018 corresponding author: miklos kassai budapest university of technology and economics, muegyetem rkp. 3., h-1111 budapest, hungary e-mail: kas.miklos@gmail.com 204 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers investment in integrating heat recovery equipment into the ventilation system only becomes reasonable with adequate energy savings [10-13]. therefore, it is essential to use the most accurate dimensioning data in the course of design work so that under the given climate conditions, the designer may provide the closest estimate regarding the expectable energy saving [14-18]. in the course of design works it is difficult to provide an accurate estimate with regards to the energy saving of planned heat recovery equipment since a number of parameters influence the performance of such equipment. the working point of heat recovery equipment may vary along a wide scale depending on climate conditions [19-20]. different types of equipment operate at various rates of effectiveness within such ranges, and the dependence of their exact characteristics on operating parameters is rarely known even by the manufacturers [21]. in the course of building engineering design, an energy recovery ventilation unit is selected in most cases based on the data provided in manufacturers' catalogues, or standards. the ventilation technology describes the specific energy content of moist air mass per unit with the help of total specific enthalpy, ht. the energy content of moist air mass per unit depending on physical parameters can be calculated with the help of specific enthalpy – referred to as enthalpy in the following. the total enthalpy of moist air can be described with the following equation [22]: t) c+(rx +t c=h+h=h wetpa,0dry)(pa,lst  [j/kg] (1) according to eq. (1) the total enthalpy of moist air, ht, is made up of a sensible, hs, and a latent hl, in [kj/kg] component. in eq. (1), cp represents the specific heat capacity of the dry and wet air in [kj/kg k] on constant pressure p; t represents the temperature in [k]; x represents the absolute humidity contact of the air in [kg/kg] and r0 is the phase change heat in [kj/kg]. in the course of interaction of moist air streams with different enthalpies sensible (thermal), or thermal and latent heat exchange may occur. the purpose of heat recovery units – referred to as hru in the following – is to provide thermal, or thermal and latent heat exchange between two moist air streams with different enthalpies at the highest possible rate of effectiveness. the application of hru-s in the ventilation technology enables ambient air to be preheated during the heating season, or to be pre-cooled during the cooling season using the recovered portion of heat loss from discharged air. in this way the reasonable energy consumption of auxiliary heating/cooling coils providing the indoor condition of air within the supplied space may be significantly decreased. in the following, heat recovery system components shall be interpreted according to the definitions of vdi 2071 [23], as shown in fig. 1. recirculation of air, or cooling/heating with waste heat is not considered as heat recovery according to the interpretation of [2]. the amount of recovered heat primarily depends on the heat content of ambient air conditions (depends on climate conditions) and exhaust air conditions and the effectiveness of heat recovery (given by the producer), as shown in fig. 2 as well. higher temperature difference, relative humidity difference or enthalpy difference between the ambient air and exhaust air results in higher energy recovery. the position of the time interval of heat recovery within the given day is also of key importance since the enthalpy content of ambient air varies according to the time of the day depending on the given climate. the effectiveness of heat recovery of hru-s varies depending on operating conditions (volumetric air flow, temperature, relative humidity, drum rotation, frosting, condensation, investigation of the energy recovery potentials in ventilation systems in different climates 205 etc.). effectiveness rates given by manufacturers apply to the indicated conditions of operation, and are less applicable in the case of different conditions. due to unknown equipment characteristics, dimensioning, selection and energetic classification of units are based on average annual effectiveness of heat recovery. continuously changing conditions, i.e. the effect of outdoor air conditions with known equipment characteristics could only be examined using a simulation program. in the course of this research the possibilities of heat recovery are obtained to investigate by processing high resolution weather data. the final objective is to construct tools suitable for longer term energetic estimations, with the help of which the variation of theoretically recoverable heat within ventilation units using ambient air can be demonstrated depending on the parameters of operation, such as indoor air temperature ≈ temperature of exhaust air, the time interval of operation, and weather related expectations. the purpose of evaluated weather data is to determine the theoretically recoverable sensible, latent and total energy depending on the above mentioned variables. fig. 1 diagram of the heat recovery process [23] fig. 2 components of the heat recovery process as the novelty of this research with the developed procedure drawn up in this study, the energetic possibilities of heat recovery under various climate and operating conditions may be examined in more detail and more realistically than with the methods and available information of current engineering practices. 206 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers 2. the description of the developed procedure examined cities were selected on the basis of diversity of their climates, within the possibilities offered by the wolfram alpha database. the colors of fig. 3 show the location of examined cities according to the koppen type [24] climate zones. fig. 3 location of investigated cities according to the koppen type climate zones [25] moreover, fig. 4 shows the distribution of climate regions of hungary as one of the investigated countries. fig. 4 climate regions of hungary according to the koppen type climate zones [25] in the course of this research, from among the components presented on fig. 2, the heat content of ambient air was evaluated by month and by hours of the day based on the following concept: the theoretically recoverable maximum heat is the difference of the sum of areas (integrals) under the enthalpy curve of ambient air and the enthalpy line of exhaust air for the period in question. available indicators of the condition of ambient air however, are deficient in a different degree per section. therefore, they can only be used investigation of the energy recovery potentials in ventilation systems in different climates 207 for determination of time dependent integral sums after correction or simplification/ generalization. as an initial approach the partly defective data sequences were attempted to correct by substituting in the missing values, yet this significantly distorted the original sample. as a second approach, with the omission of defective parts fewer samples were ended up; however, these were more representative of the examined phenomena due to the originality of data. by taking the average of data sequences for the same times of day with minimal correction, the change in the distribution of data, the mean values is negligible. the idea behind the evaluation method is the following: let us assume that one unit will only be operated for a one hour period each year on the 1st of january between 1:00 and 2:00 o'clock. parameter values characterizing ambient air are different every year at that time. performing the calculation for many years one by one, then adding and taking the average of the parameters characterizing ambient air, it may be determined what the expectable value of the given parameter is on average if normal distribution is assumed. with the same idea, average days can also be constructed for each month from such elemental hours as follows: all hours of an average day characteristic of the month will be the mean value of the values measured at the same time of the same day in the same month of all years. accordingly, with the method applied in the research, from these average days, such average months and an average year can be constructed, which on the basis of the examined years provides an estimate related to the given hour, or given month that can be used in terms of energetics. in the course of estimation, from the increasing sequence of values of a given hour, the limit value of any range can be selected as standard. with a division into 10 sections, decile limit values were determined. in this way 10 values were recorded for the hours of the average day of each month. the lower and upper limit values of such decile sequence represent the registered minimum and maximum values where the middle element is the mean value. in the further step of evaluation, with the use of data sequences constructed from the decile values varying from such mean value, the effect of weather values different from the average could also be taken into consideration. this is highly important since the working points of hru-s may vary within these ranges, as mentioned before. consequently, in addition to an average energetic characterization, it is possible to determine the expectable minimum and maximum operating conditions, the range of heat recovery values as well. in the course of this research the weather data of 40 years of budapest, and of 20 years of further 19 cities are processed with different climates using the downloading and evaluation programs described in wolfram mathematica 11.0. the purpose of evaluation was to determine an average daily temperature, relative humidity, pressure and total/sensible/latent enthalpy sequences projected to a month, to end up with 10 decile curves per parameter. the data sequences belonging to a given decile value are indicated as: no. i  decile value → (i1)100 [%]. accordingly, the first decile was marked as 0 [%], the last one as 100 [%]. this is to mean that according to the previous method of evaluation it can be established that of the data population available for the given time 0 [%] is located below the first decile, and 100 [%] between the first and the tenth decile. thus the resulting [%] marked decile curves mean a range constructed on the basis of data measured in the past, from among which it is practical to select the data used for dimensioning. the average expectable value is equivalent to the 50 [%] marked decile curve, while the maximum and minimum expectable values can be found on the 100 [%] and 0 [%] marked curves, respectively. the prepared energetic estimate is valid for a long term while for a short term the heat content of ambient 208 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers air varies around the 50 [%] marked decile data sequence, within the 0 and 100 [%] decile range. the long term average value of variation can also be interpreted in terms of energetics with consideration to the 50 [%] decile curves. therefore the method can only be used accurately for a long term estimation while for a short term it shows the expectable interval of the variation of values based on past events. in the course of specification of enthalpy curves, instead of the average daily sequences, the matching value triads of the original data sequence were applied. this is important since the value triads cannot be considered as independent variables in this respect. the given enthalpy sequences will accordingly contain the average joint product of the matching air temperature – relative humidity – pressure (referred to as t--p) value triads, which is not equivalent to the individually taken average values of variables at the given time since enthalpy is not a linear resultant of the given parameters. decile data sequences were derived from the enthalpy sequences for the given times using the previously described method. previously introduced decile curves will be referred to as expectations in the following since with the use of decile data sequences different from the average, we have positive or negative expectations with respect to the non-average heat content of ambient air. the theoretical process of evaluation is represented in fig. 5. in this way the operation of the enclosed program parts is more illustrative and easier to follow. for the generation of reliable monthly average (t--p) graphs from the data sequences of adequate density, certain conditions had to be defined before the evaluation of daily data sequences. in the case of examined cities, the data sequences of all days were omitted from the evaluation where the following two conditions were not met with respect to any of the given daily (t--p) lists: 1) a daily minimum expectation of 6 measurements can ensure that two adjacent pieces of data are no more than 4 hours apart on average. this is required for the interpolation in the course of enthalpy calculation. accordingly, all data sequences must contain at least 6 pieces of recorded data. 2) the first and last piece of data of the day can be no more than 4 hours apart from the beginning, or end of the given day respectively, thus any missing value for 00:00 and 24:00 hours could be reliably interpolated or substituted. the number of omitted days is represented for each city in relation to the examined time period and based on average annual distribution. a constant exhaust air temperature and humidity were assumed with a fresh-air system. the examined fresh-air ventilation system does not heat/cool, it only ensures an air supply to the premises at the prescribed temperature while all other thermal requirements are ensured by other building engineering systems. in this way the temperature of exhaust and supply air can be considered as equivalent. the effects of frosting, condensation, maintenance, etc., which decrease operating times, were also omitted. investigation of the energy recovery potentials in ventilation systems in different climates 209 fig. 5 the theoretical process of evaluation 3. presentation of the evaluation method in the following paragraphs the most significant results of weather data are presented evaluated with the developed program using the above described method, for each city. the amount of recoverable heat was summed separately for cooling and heating modes of operation since in this way the decrease in the energy requirement of the cooling calorifier and heating calorifier can be viewed separately for the given month. recoverable heat during one month of the presented graphs is the product of values of an average day characteristic of the given month, by the number of days in the given month. the expectable percentage value of total recoverable energy was calculated for each month with 50 [%] expectation for an exhaust air condition of 22 [°c] and 50 [%]. heat amounts representative of months were represented in percentage of annual heat amounts, so as to reflect the distribution of annually recoverable heat through the months of the year. 210 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers the percentage value shown on the top section of graphs shows the marking of the applied decile sequence, while the celsius value shows the temperature value of exhaust air considered as constant. the humidity of exhaust air was considered as 50 [%] in all cases. considering the limitation of the length of the paper one of the investigated cities barcelona [16] was selected to present the detailed results (figs. 6-10). in the end of the evaluation, the expected annually recoverable thermal energy of all 20 cities is compared on a single graph investigated by the developed method used for barcelona analogically. fig. 6 expected recoverable monthly total cooling enthalpy-hour in relation to the temperature of exhaust air fig. 7 expected recoverable monthly total heating enthalpy-hour in relation to the temperature of exhaust air investigation of the energy recovery potentials in ventilation systems in different climates 211 fig. 8 recoverable monthly total cooling enthalpy-hour in relation to the expectation fig. 9 recoverable monthly total heating enthalpy-hour in relation to the expectation fig. 10 monthly distribution of expected annually recoverable total enthalpy-hour based on the past 20 years with 50 [%] expectation figs. 6-10 show that a major part of cooling energy recovery occurs between the months of june and october in barcelona. the temperature of exhaust air significantly influences the expected recoverable heat amount during the intermediate period (may and 212 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers october). expectable annual total, sensible and latent enthalpy-frequency graphs show the annually expectable hours of equipment operation in heating and cooling modes at 50 [%] expectation with known values of total, sensible and latent enthalpy of exhaust air. with such time periods and the recoverable total/sensible/latent heat amounts at hand, the optimal heat/energy recovery unit type can be selected in the designing phase previously based on its estimated annual sensible and latent rates of effectiveness. the length of time periods with respect to the remaining cities shall be established in accordance with the method described here while due to the limited space, only the expectable annual total enthalpy-frequency graphs for the city of barcelona (fig. 11) shall be included in the publication. using the developed diagram (fig. 11) the expectable total heating-cooling period can be easily determined (e.g. following the black colored line, this period is around 5120 hours with 43 kj/kg total enthalpy of exhaust air and under 50 [%] expectation for barcelona). the summary graph showing the values of all cities (fig. 15) is given in the results section of this paper. fig. 11 expectable annual total enthalpy-frequency graph 4. the effects of exhaust air temperature and time interval of operation on expected theoretical maximal monthly recoverable thermal energy investigation of the effects of the time interval of operation is a complex task. the average daily time-dependant recoverable thermal energy for the given month was determined as follows. using the average daily enthalpy sequences of the given month, the recoverable heat amounting up to the given hour was calculated in each half-hour where the enthalpy of exhaust air was given as a function of air condition indicators of exhaust air. thus the algorithm shown in fig. 5 performed the computation of recoverable thermal energy assigned to every half-hour of average monthly days in the function of the expectation (decile curve), and the temperature and humidity of exhaust air. figs. 12-14 show the expected recoverable latent heat amount of barcelona in the month of december. the developed evaluation algorithm constructed the above shown graphs for all cities and all months, allows the expectable heat recovery in a month of any given city within the selected time interval of operation to be determined with a given expectation (decile). investigation of the energy recovery potentials in ventilation systems in different climates 213 fig. 12 expected monthly recoverable latent heat in the function of the expectation and the time interval of operation fig. 13 expected monthly recoverable latent heat in the function of exhaust air temperature and the time interval of operation fig. 14 expected monthly recoverable latent heat in the function of humidity of exhaust air and the time interval of operation 214 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers the resulting graph represents a given month, showing the limits of the time interval of operation on the horizontal axis, and the recoverable heat amount up to the given point of time on the vertical axis in the function of parameters of exhaust air, with the given expectation. after the time interval of operation is determined, the difference of “y” values of the respective curves will provide the heat amount recoverable during the given time interval. based on the previous figures it can be concluded that even a slight change of enthalpy of exhaust air may significantly modify the length of heating/cooling time periods relevant in terms of heat recovery. this finding is supported by the above figures entitled "expected monthly recoverable total cooling enthalpy-hour in the function of exhaust air temperature", and the enthalpy-frequency graphs of cities. fig. 12 shows that the expected monthly recoverable heat amount is quite sensitive to the changes of exhaust air, even ten-times deviation can result in the monthly summary. according to the generated graphs, this sensitivity varies according to city and periods of time, yet it is significant in all cases. using the method described for the city of barcelona, the length of expected total, sensible and latent heating/cooling periods were established for all cities. the values and their annual percentage proportions can be compared on the basis of fig. 15. total hours of operation are naturally the sum of heating and cooling hours, i.e. the empty section up to the 100 [%] value above all bars represents the percentage value of the respective cooling energy recovery hours of operation. fig. 15 shows the amount of the maximal, theoretical energy that can be recovered from the exhaust air during the ventilation operation in each investigated cities. values of fig. 15 can be used for the selection of an adequate heat/energy recovery. the method of such selection shall be the subject of future research work. 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% e x p e ct a b le p ro p o rt io n o f h e a ti n g e n e rg y re co ve ry expectable proportion of total heating energy recovery hours [h] expectable proportion of latent heating energy recovery hours [h] expectable proportion of sensible heating energy recovery hours [h] fig. 15 annual expectable proportion of heating energy recovery hours per city investigation of the energy recovery potentials in ventilation systems in different climates 215 5. results based on a comparison of the expected annually recoverable thermal energy of examined cities, a sequence of cities can be drawn up on the basis of expected total recoverable thermal energy revealing unexpected similarities which are not necessarily predictable from the climate zones of the given cities (fig. 16). 0 50000 100000 150000 200000 250000 300000 350000 e xp ec te d an nu al ly r ec ov er ab le t he rm al e ne rg y total enthalpy-hour for heating [kj h/kg year] total enthalpy-hour for cooling [kj h/kg year] ∑totalenthalpy-hour [kj h/kg year] fig. 16 expected annually recoverable total, sensible and latent thermal energy per city fig. 16 shows that budapest is in a favorable position within the sequence, and a major portion of energy recovery occurs in heating mode of operation. from among all cities, belem is outstanding, where heat recovery during the cooling period exceeds the annual amount experienced in all other cities. the least amount of energy can be saved in sydney, while the amount of savings is not negligible. the energetic similarity is shown between sydney, szeged, buenos aires, cairo, dakar and perhaps even barcelona and nanchang, with a similar magnitude of recoverable energy amounts during the heating and cooling seasons. similarly, tehran, versailles, portland, dublin, budapest, odessa, toronto and oslo can be ranked together where the expected recoverable heat amount during the heating season significantly exceeds that during the cooling season. in the case of miami, doha, mumbai and belem the opposite applies, yet expected heat recovery is significant in these cities during the cooling season. 216 m. kassai, l.poleczky, l. al-hyari, l. kajtar, j. nyers 6. conclusion the above findings apply to a constant 22 [°c] temperature and 50 [%] relative humidity condition of exhaust air. in the evaluation program these are represented in the function of the enthalpy of exhaust air; this parameter can later be substituted by any arbitrary ambience or indoor parameter. accordingly, the generated curves can be universally applied for calculations under any static or dynamically changing conditions of exhaust air. the exploitation of this opportunity shall also be the subject of future research work. using the developed methods described in this paper a large number of auxiliary tables have been generated on the basis of weather data which may be utilized in the course of 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climatology, edutus college. 25. „wikipédia,” [online], 2017, available: https://hu.wikipedia.org/wiki/barcelona#.c3.89ghajlata. 13299 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 351 375 https://doi.org/10.22190/fume241103001f © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper quantitative analysis and optimization of energy efficiency in electric multiple units szabolcs fischer1, bence hermán2, mykola sysyn3, dmytro kurhan4, szabolcs kocsis szürke5 1széchenyi istván university, faculty of civil engineering, department of transport infrastructure and water resources engineering, hungary 2széchenyi istván university, audi hungaria faculty of automotive engineering, department of vehicle manufacturing and technology, hungary 3technical university dresden, department of planning and design of railway infrastructure, germany 4ukrainian state university of science and technologies, department of transport infrastructure, ukraine 5széchenyi istván university, audi hungaria faculty of automotive engineering, department of road and railway vehicles, hungary orcid ids: szabolcs fischer https://orcid.org/0000-0001-7298-9960 bence hermán https://orcid.org/0009-0005-4620-5022 mykola sysyn https://orcid.org/0000-0001-6893-0018 dmytro kurhan https://orcid.org/0000-0002-9448-5269 szabolcs kocsis szürke https://orcid.org/0000-0001-5639-8116 abstract. the increasing urgency for sustainable transportation solutions necessitates a thorough examination of energy efficiency within railway systems. this study investigates the energy performance of siemens ventus (i.e., siemens desiro ml type) electric multiple units on austria's raaberbahn network, focusing on route-specific energy consumption and the optimization of regenerative braking. utilizing data collected from january to may 2023, the research employs a robust methodology that integrates statistical analysis, curve-fitting, and geospatial modeling to assess energy trends along routes connecting vienna, bratislava, and deutschkreutz. the findings reveal that terrain, operational practices, and external environmental factors significantly contribute to energy inefficiencies. specifically, hotspots of energy overconsumption were identified, leading to the development of tailored optimization models for each route. the analysis also produced heatmaps that illustrate critical spatial and temporal patterns, which are essential for implementing targeted interventions aimed at enhancing energy efficiency. key words: railway, electric multiple unit, siemens ventus, siemens desiro ml, energy consumption, regenerative braking energy, acceleration-deceleration, energy optimization received: november 03, 2024 / accepted january 02, 2025 corresponding author: szabolcs kocsis szürke széchenyi istván university, audi hungaria faculty of automotive engineering, department of road and railway vehicles, 9026 győr, egyetem tér 1., hungary e-mail: kocsis.szabolcs@ga.sze.hu https://orcid.org/0000-0001-7298-9960 https://orcid.org/0009-0005-4620-5022 https://orcid.org/0000-0001-6893-0018 https://orcid.org/0000-0002-9448-5269 https://orcid.org/0000-0001-5639-8116 352 s. fischer, b. hermán, m. sysyn, et al. 1. introduction in the nineteenth century, steam locomotives played an essential role in connecting societies and boosting economic growth by bridging distances and fostering economic interconnectivity like never before [1]. the steam era laid the groundwork for future transportation advancements. it should be mentioned that aviation, space exploration, and maritime navigation, which also represent significant advances in human mobility, have increased global transportation efficiency, safety, and accessibility. the most considerable improvement and development started primarily following the industrial revolution. these advancements have collectively changed how people navigate their world, making it more connected and accessible [2]. furthermore, fixed-rail systems [3-6] have played an essential role in this transportation evolution due to their inherent efficiency and dependability. they improve capacity, reliability, and energy efficiency, thereby strengthening the infrastructure of modern transportation and logistics networks [7-9]. the european union's emphasis on multimodal transportation systems, particularly railways, demonstrates a strategic approach to improving supply chain management. this focus reflects an appreciation for efficient, dependable transportation's critical role in the economy [1]. railway transportation, known for its punctuality and environmental friendliness, is highly valued in mass rapid transit systems. it emphasizes the transition to more sustainable modes of transportation, which is critical in today's environmentally conscious society [7]. furthermore, a customer satisfaction analysis in the context of railway transport reveals how passenger satisfaction significantly impacts the relevance of railway transport. this insight emphasizes the importance of understanding and meeting passenger needs to keep railway services competitive and appealing [10]. research into green freight railway operations highlights the importance of traffic management and transportation engineering. this emphasis is critical in reducing the environmental impact of freight transportation while demonstrating a commitment to sustainable logistics practices [11]. furthermore, the life cycle assessment of belgian railway freight highlights the significant environmental considerations, particularly electricity use, involved in railway operations, emphasizing the importance of environmentally friendly practices in the industry [12]. the transportation of hazardous materials by railway raises several complex safety, human, and environmental concerns. because of its complexity, it is an essential area of research for policymakers and stakeholders who must navigate these challenges to ensure safe and responsible transportation [13]. furthermore, railway transportation addresses the dual challenges of traffic congestion and environmental degradation. railways play an essential role in achieving a more sustainable future by promoting sustainable development in densely populated urban areas as well as long-distance passenger and freight transport [14]. railways, with a lower carbon footprint in the supply chain decarbonization process than trucks, emerge as a more environmentally friendly option, emphasizing the significance of rail transport in the larger context of environmental sustainability [15]. the impact of high-speed railway construction on the sustainability of urban agglomerations highlights the sector's commitment to reducing environmental impacts, focusing on the railways' role in shaping sustainable urban landscapes [16]. the role of transportation sustainability in chinese higher education demonstrates the societal impact of the railway system. it emphasizes the broad implications of transportation infrastructure for social development and well-being [14]. globally, transportation systems increasingly rely on electric traction and fixed-rail technology, indicating a shift toward quantitative analysis and optimization of energy efficiency in electric multiple units 353 more environmentally friendly and efficient modes of transportation. this shift is critical in addressing modern transportation challenges, such as increased energy efficiency and reduced environmental impact [17]. the advantages of electric traction batteries in terms of energy efficiency and performance highlight the advancements in transportation technology driving the transition to more sustainable mobility solutions [18]. addressing the challenges of power quality and energy efficiency in railway transport electric traction systems, particularly in ac-electrified railway inter-substation zones, is critical for improving their performance and sustainability [17]. the creation of theoretical frameworks for calculating rational modes of traction electrical equipment operation is critical for comprehending and mitigating the environmental effects of electric traction systems. such frameworks are crucial for developing more sustainable transportation solutions [19]. the competition between electric and combustion traction highlights the environmental benefits of electric vehicles, emphasizing the significance of sustainable transportation in the larger context of ecological stewardship [20] – emphasizing the relevance of mechanical engineering [21,22]. urban electric traction drives, with their emphasis on improving energy performance, reflect ongoing efforts to reduce the environmental impact of transportation, which aligns with global sustainability goals [23]. because of the high electrical energy losses associated with traffic, dc traction railway networks are being scrutinized for increased energy efficiency and sustainability. this necessity drives traction drive control system efficiency research, mathematical models for onboard energy storage, and power quality. these research efforts are critical to the development of more environmentally friendly and efficient railway and electric traction systems [24-26]. furthermore, strategies for reducing the environmental impact of ac traction substation electric energy quality are vital in minimizing electric traction systems' ecological footprint [27]. the study of electric motors for electric vehicles, encompassing efficiency, cost, reliability, innovation, and controllability, is crucial for comprehending the environmental impact of fixed-rail transportation and electric traction, pointing to the need for continuous innovation in this field [28]. intermittent electrification and the analysis of the annual variation in energy consumption of commuting electric vehicles are strategic approaches to decarbonizing traction energy, reflecting the importance of sustainability in the evolution of transportation systems [28,29]. enhancing transportation system sustainability through electrical traction transmission by influencing railway vehicle dynamics showcases the potential for reducing environmental impact through technological advancements [30]. the efficiency of electric vehicles is closely linked to the optimization of train schedules, utilizing the longitudinal dynamic electric vehicle model of sun et al. [31] for this purpose. this optimization, including using data to identify battery health states (if there are batteries), aims to improve energy efficiency and scheduling, showcasing the potential for technological advancements to enhance transportation efficiency [32]. driver assistance systems (dass) impact railway punctuality and energy efficiency by providing real-time driving data to operators, enabling them to make more energy-efficient decisions. this system is a prime example of how technology can be leveraged to improve the sustainability of transportation [33-36]. the implementation of railway das employs distributed computer systems and network technologies to monitor, detect, and manage energy-saving modes in power supply systems, reflecting the role of innovation in achieving energy efficiency in the railway sector [37]. tao et al. [38] introduced a new energy optimization method for traction substations that addresses time-varying parameters and environmental characteristics in mechanism modeling. a data-driven model, dynamic programming, and mimo-sofnn (multi-input multi-output 354 s. fischer, b. hermán, m. sysyn, et al. self-organizing fuzzy neural network) are used. experimental analysis validates the model's accuracy, and the proposed method reduces energy consumption by 34.8% compared to chinese freight railway company operation data. yildiz et al. [39] proposed integrated train operation optimization to reduce traction energy and increase regeneration energy. the model optimized train speed trajectory and timetable to increase braking and accelerating train group overlap. the istanbul m3 subway system is modeled and simulated using genetic and simulated annealing algorithms. the study found that the best train speed profile and timetable reduce traction energy consumption. fischer et al. [40] discussed the detection of energy loss in electric railway hauling vehicles, as well as the importance of railway energy efficiency. it examines the current situation and potential improvements for more efficient energy use. seven measurement series were conducted using scheduled railjet trains between hegyeshalom and győr railway stations in hungary. the article looks into optimizing regenerative braking energy by identifying energy-waste sources and reasons for consumption. the global energy crisis and the imperative for cleaner, "greener" energies underscore the importance of reducing electricity consumption in railways, highlighting the need for the transportation sector to adapt to changing energy landscapes [41]. all train energy consumption must be analyzed to save energy. a complex interplay of infrastructure, transportation organization, and environmental factors influences energy use in railway operations. this interplay underscores the multifaceted nature of energy use in transportation, highlighting the importance of a comprehensive approach to energy efficiency [16,42-44]. regenerative electricity is a critical technology in the quest for energy savings, illustrating the potential for innovative solutions to enhance the sustainability of transportation systems [45-47]. optimizing the timetable by considering the horizontal and vertical geometries of the railway tracks, as well as the planned (and/or actual) speed, represents a strategic approach to improving energy efficiency, underscoring the potential for planning and technological innovation to contribute to more sustainable transportation practices [45,4850]. combining these methods produces the best results, emphasizing the importance of a holistic approach to enhancing energy efficiency in the railway sector. the main focus of this paper is the analysis of energy consumption of electric multiple units. based on the energy consumption optimization, four main categories were defined [40,51]: (i) condition of the permanent way and connecting infrastructure, as well as the rolling stock, (ii) recovered energy from regenerative braking, (iii) external factors (i.e., environmental factors, temperature, etc.), finally (iv) the human factor. it plays a pivotal role in shaping energy efficiency and consumption within the railway transport sector. extensive research has been conducted to dissect the various factors influencing driving behavior and its subsequent impact on energy efficiency. notably, studies have delved into the significance of driver experience [52], highlighting its correlation with driving performance and energy consumption patterns. moreover, there exists a delicate balance between driving efficiency and service quality [53]. understanding this trade-off is essential for devising strategies that optimize energy usage without compromising operational standards. additionally, identifying and addressing barriers to energy efficiency [54] is crucial for implementing effective measures across the rail industry. further exploration into the intricacies of driver behavior reveals its profound implications for energy consumption. studies focusing on heavy-haul iron ore trains [55] shed light on the substantial influence of driver actions on energy usage. additionally, research into the quantitative analysis and optimization of energy efficiency in electric multiple units 355 physiological aspects of driving, particularly concerning monotonous tasks [56], underscores the need for strategies to mitigate fatigue and enhance driver engagement. interventions are paramount in addressing the challenge of improving energy efficiency. initiatives such as ecodriving training programs [57,58] offer promising avenues for instilling energy-conscious behaviors among drivers. fischer et al. [40] determined that the train operator's driving style and habits are identified as the primary causes for the main part of the energy losses. (it is worth mentioning that fuzzy methodology and evaluation techniques can also be applied in this field [59,60].) the original aim of this article was to analyze the driving style of train drivers and its impact on the electricity consumption of electric traction units and emus (see the (iv) point from the previous list). unfortunately, the authors were not able to do this because the received database did not contain the coding of the train drivers. the authors decided to conduct a more detailed energy consumption analysis than they did in [40] as an innovative investigation. the main focus was to define typical consumption graphs (trends) on the analyzed routes (in this case, vienna-deutschkreutz, i.e., vd; and viennabratislava, i.e., vb) between january and may 2023, which would allow for the precise localization of the consumption outliers and their location. (the opposite direction routes have the abbreviations dv and bv, respectively). only the siemens ventus (i.e., siemens desiro ml type) emus in service with raaberbahn ag were included in the analysis (emus were, of course, exclusively used for passenger transport). modern mathematical-statistical methods were used for the analysis. the structure of the current paper is as follows: section 2 deals with the applied methods, section 3 contains the results and discussion, and section 4 is the conclusions. 2. applied methods in this article, data obtained from on-board computers of the raaberbahn were provided in the following structure and detail for the vd and vb routes (see fig. 1 and table 1): 1. duration: january-may 2023 (i.e., five monthly series in total). 2. data series in bulk excel files for siemens ventus multiple units only (for the five months, five excel files were downloaded, containing all routes and all train numbers): a. 5-minute detail (sampling frequency) data with day:hour:minute accuracy, b. includes train number, current gps coordinates, cumulative electricity consumed and recuperated, c. to the best of the authors' knowledge, raaberbahn ag has provided the data of all trains operated during the given period, d. however, the data set did not include current speeds, accurate start and end times for accelerations and decelerations, nor did it include the identity of the drivers. 3. the segment numbers of stations and stops on the routes were taken from raaberbahn and öbb (austrian state railways) reports, the gps coordinates were determined accordingly, and the results were used for further calculations. 4. the speed values allowed on each section were taken from öbb, raaberbahn ag registrations and public data on the openrailwaymap.com website. the parameters that had to be calculated from the data (listed here; see section 2.2 for a detailed explanation): 356 s. fischer, b. hermán, m. sysyn, et al. 1. because of the bulk structure of the data, a train (i.e., a train with a given train number) does not run on only one section, so the first step in the classification was to associate it with a route. 2. since only gps coordinates could identify each measurement point ("data row"), the actual distance traveled between two data points was needed. to handle this, a python program was written. it was also necessary to determine the direction of traffic on the route. 3. the data thus sorted had to be filtered in the final analysis (section 2.1, section 2.2, and section 3) because a professional decision was made to consider acceleration and regenerative braking energies only in the ±2 km environment calculated by the section numbers (gps coordinates) of the general station and stop locations, considering the direction of travel. this meant that the focus was primarily on acceleration energy consumption, with the accumulation-accumulation of regenerative braking energy values being considered synchronously. 4. statistical processing was performed considering the entire data set, as detailed in section 2.2. fig. 1 segregating of sections into shorter subsections quantitative analysis and optimization of energy efficiency in electric multiple units 357 table 1 details of the considered routes (the ids are illustrated in fig. 1) 1a: vienna – bratislava (vb) 1b: bratislava – vienna (bv) id. railway station intermediate distance [m] cumulative distance [m] v vienna hauptbahnhof (vienna) 0 0 2 vienna grillgasse 2858 2858 3 gramatneusiedl 16550 19408 4 götzendorf 7340 26748 5 burck a.d. leitha 14559 41307 6 parndorf ort 6356 47663 7 parndorf 1585 49248 8 neudorf 6120 55368 9 gattendorf 3230 58598 10 pama 5850 64448 11 kittse 4980 69428 b bratislava-petrzalka (bratislava) 1920 71348 2a: vienna – deutschkreutz (vd) 2b: deutschkreutz – vienna (dv) id. railway station intermediate distance [m] cumulative distance [m] v vienna hauptbahnhof (vienna) 0 0 12 vienna meidling 3637 3637 13 ebreichsdorf 27193 30830 14 ebenfurth 10713 41543 15 neufeld a. d. leitha 2000 43543 16 müllendorf 7954 51497 17 wulkaprodersdorf 5705 57202 18 draßburg 5753 62955 19 baumgarten-schattendorf 3282 66237 20 sopron 7787 74024 d deutschkreutz 9475 83499 2.1. data ordering the distances were calculated using the python geopy module. previously, the haversine formula was also applied, but the geopy module uses a geodesic line to determine the distance, which gives a more accurate result than the haversine formula. however, due to the small distances involved, both are suitable for the task. in sections 2.1 and 2.2, the python program was employed: (i) scipy and (ii) geopy. preparations: 1. create a helper database (collect the following information in a database): a. gps coordinates of train stops: approximate, not exact. gps coordinates are centered at the stops. adding reference points for better route separation thereafter. b. gps coordinates of routes: in the sandbox editor of the gpsvisualizer.com website, add reference points for the railway track. later, calculate the cumulative distance to the reference points to obtain the length of the route and the distance to the stations from the starting point (comparison of reference points and station points.) c. collection of distances between stations: based on information recorded by öbb and raaberbahn. 358 s. fischer, b. hermán, m. sysyn, et al. 2. generating configuration files: d. to automate the database file to be created later, generate configuration files containing the variables needed to run the program. creating a database: 1. determining the nearest stops: compare the gps data in the initial excel file with the gps coordinates of the stations in the helper database. for each point, the closest station can be obtained. 2. merge waiting times for end stations: in this case, the waiting times at end stations are merged for easier data processing afterward. 3. for all data points, it is possible to know the names of the nearest stations so that it is possible to determine from what time of day to what time of day a given train has been on which route. 4. distance calculation for data points: for established route data points, the distance calculation for route reference points in the helper database is used to get the distance from the previous data point to the starting station. this operation was very time-consuming due to the distance calculation for thousands of reference points. as a solution, using the configuration files generated during the preparation, several routes and several trains were calculated in parallel. 5. other calculations: calculation of cumulative consumption and regenerative energy for specific runs (routes). 6. calculation with iqr methodology: iqr, or interquartile range, is a statistical indicator of the variance of the data. the iqr is calculated as the difference between the lower and upper quartiles. the procedure for this was as follows: a. sorting the data. 𝐶 = {𝐶1,𝐶2, … , 𝐶𝑛} (1) 𝐶1 ≤ 𝐶2 ≤ ⋯ ≤ 𝐶𝑛 (2) where {𝐶1,𝐶2, … , 𝐶𝑛} are the individual railway consumption values. b. calculating the first and third quartiles: ▪ q1: the median of the lower 25% of the sorted data. ▪ q3: the median of the upper 25% of the sorted data. c. interquartile range (iqr calculation): obtained from the difference between the first and third quartile range. 𝐼𝑄𝑅 = 𝑄3 − 𝑄1 (3) d. determination of lower and upper bound: 𝐿𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑 = 𝑄1 − 1.5 ∙ 𝐼𝑄𝑅 (4) 𝑈𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑 = 𝑄3 + 1.5 ∙ 𝐼𝑄𝑅 (5) e. filtering data: using the boundaries to filter out outliers. 𝐶𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑 = {𝑐 ∈ 𝐶|𝐿𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑 ≤ 𝑐 ≤ 𝑈𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑} (6) f. calculating the average of the filtered data (no outliers are found here) 𝐶𝑎𝑣𝑟. = ∑ 𝑐𝑖 𝑚 𝑖 𝑚 ; 𝑐𝑖 ∈ 𝐶𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑 (7) where cavr. is the average energy consumption, 𝑚 is the number of elements in the filtered data cfiltered. quantitative analysis and optimization of energy efficiency in electric multiple units 359 7. creating and exporting summary statistics tables to excel: for a given month, a given number of trains and a given route, create statistics: minimum, maximum, mean (average), standard deviation, average iqr/km calculation per station interval. the excel files also contain the data points per station in separate worksheets. there are two types of excel file generation, one where data points are filtered by station distance ±2 km and one where there is no filtering at all so that all data points can be found. fig. 2 represents the entire block scheme of the applied data ordering. fig. 2 block scheme of the applied data ordering 2.2. data processing the data processing used in the research is described in detail below. data processing is further performed on data points covering the given station distance of ±2 km. 1. generate statistical excel files for specific routes: generate multiple worksheets. calculate several parameters for given consumption values and regenerative values: minimum, maximum, mean (average), and standard deviation. on another worksheet, collect average 360 s. fischer, b. hermán, m. sysyn, et al. iqr/km calculations for kwh+ and kwh– values. values were determined for a given route, for each station and each train. 2. curve fitting: the curve fitting was done on the basis of the data from the previously generated excel files of the average iqr/km worksheet. the fitting was done using the scipy module available for python (curve.fit function). second, thirdand fourth-degree polynomial regression functions were fitted to the points, and the python module also provided a covariance matrix for each of them, based on which regression function was selected to fit the points better (the lower the covariance matrix value, the better the fit). in addition, the error bars were determined. 3. identification of point locations using a polynomial regression function fitted to the curve. for all the data points exported earlier to excel, the polynomial regression function fitted to the curve and the error bands were applied to determine the location of the points. 4. data filtering: once the position of the points has been determined, it is now possible to filter which data points were overconsumption or too low regenerative braking energy for the average and outside the range of the specified one and two standard deviations. 5. graphing: graphing the filtered data. for this purpose, the energy consumption and regenerative braking energy values measured on each route were taken into account. in this phase, only the data series calculated by the authors in kwh/km units were used. 2.3. limits of the analysis of the current research in any case, it is essential to note and highlight any factors that do not follow clearly from the chapters in section 2.1 and section 2.2 (for clarification, see section 4, too): ▪ in the analyses, neither machine manual data nor other tractive force curves were used to calculate the acceleration and regenerative braking energies of siemens ventus trains and to take the measured values into account; ▪ analyses using layout and vertical geometry data of the railway lines have not been discussed or used in this article; ▪ the authors have not used official train timetables for their research on each route and in either direction, and they have neglected to take into account the throughrolled axle-ton values; ▪ the speeds allowed on each section are given for information only in section 2.1, and are not included in the detailed analysis; ▪ the technical condition (possibly poor or good) of the railway track and the associated infrastructure, overhead contact line network, etc., and of the traction units were also neglected; ▪ the conducted and shown analyses use only the data series for january-may 2023, but the authors have not attempted to extrapolate using these to likely measurable values for other months, i.e., they have relied on factual data only; ▪ each month was treated as a unit (not merged), and the energy consumption (kwh+/km) and regenerative braking (kwh–/km) values were analyzed separately for each month. quantitative analysis and optimization of energy efficiency in electric multiple units 361 3. results and discussion based on the methodology presented in section 2, section 3 reports the results in a more detailed manner. due to the fact that two routes, and within them two directions, were considered, it was not possible to present and report a complete overview of the considerable amount of data. the first step in the evaluation of the results was to examine the different consumption data. fig. 3 shows the consumption data of four different routes (2 railway lines, considering back and forth directions) for three different trains (i.e., 4744300, 4744-301 and 4744-303). fig. 3 consumption data for different routes (bv, dv, as well as their opposite directions) considering four trains, the shown dashed lines are not real trend lines on the horizontal axis fig. 3, the serial number of the measurement is shown (it indicates the number of data series in the data set; note that the order was not relevant in this case) and on the vertical axis, the energy consumption values. the sample number shown on the horizontal axis contains about 60 values for each route. the consumption values show a significant standard deviation for all three train numbers presented, indicating that the energy consumption of the trains varies significantly between measurements. it varies between approximately 400 and 700 kwh, depending on the route and the vehicle. the vb section in subfigure a) shows medium consumption. there is also a significant standard deviation, with energy consumption peaks being less typical. the bv section is shown in subfigure b) of the figure. for all three vehicles, there is significant volatility. in subfigure c) of the figure, the vd section is shown, with the "smoothest" measured values. although there is still a significant standard deviation, the consumption trends are less hectic than in subfigure d). the highest consumption is the red line 362 s. fischer, b. hermán, m. sysyn, et al. (4744-300 vehicles) with occasional higher peaks, approaching 700 kwh. the section dv, which is the section with the highest volatility, is shown in subfigure d). the consumption data show a wider range with significant peaks for all three train numbers. the results show that there are specific energy demands on routes, which are likely to depend on the route's inclination in the vertical plane, number of stops, train speed, etc. to further examine the differences, the results are analyzed separately by month as shown in fig. 4. fig. 4 average consumption values by route in different months in 2023 fig. 4 shows energy consumption data (in kwh dimension) assigned to different routes, broken down by month, from january to may. the routes analyzed are bv: blue bars; vb: red bars; dv: grey bars; vd: yellow bars. the results show that the highest average consumption on the bv route in january was 578.54 kwh. the lowest consumption recorded on the vd route in march was 450.32 kwh. overall, the bv route has the highest fuel consumption, and the vd route has the lowest consumption. consumption is decreasing on the bv and dv routes. consumption on the vb and vd routes is relatively stable with minor fluctuations. of the routes, bv is generally the most energy-consuming, while vd consumes the least energy. considering the same route back and forth, bv vs. vb: the most significant difference is in january (90.40 kwh) and the smallest in march (22.62 kwh). this could be due to several reasons, e.g., higher loads in one direction, heavier vehicles, or more electricity needed in one direction due to different topography. when comparing the energy consumption between dv vs. vd, dv shows a higher energy consumption in all months compared to vd. the highest difference is observed in march (53.83 kwh), while the lowest difference is observed in may (30.65 kwh). all routes show a decrease in energy consumption from january to may, primarily due to better weather conditions. monthly and route-level analysis of regenerative braking energy is done by means of fig. 5. the bv route shows a particularly favorable regenerative energy use, especially in may. vd is the only route where a decrease is observed, which may indicate that traffic conditions or route characteristics do not support constant energy recovery. in the future, it may be worthwhile to investigate and optimize regenerative energy production, for example, by improving quantitative analysis and optimization of energy efficiency in electric multiple units 363 braking strategies or traffic management. when analyzing the differences between the routes, bv vs. vb, the differences between january and may are minimal, around 5-7 kwh. this suggests that the production of renewable energy is almost the same in both directions. on the other hand, the difference between dv vs. vd is more significant than the previous one, about 8-15 kwh per month. the dv route generates more regenerative energy, which may be due to higher braking or vertical inclination (e.g., more slopes in this direction; see section 2.3 and section 4). fig. 5 regenerative energy values by route in different months in 2023 fig. 6 shows the variation of energy used between different months and routes. on the bv route, energy consumption was initially very volatile (winter period), but by the spring part, it had stabilized, and the standard deviation decreased. on the dv route, it showed a low standard deviation, indicating that energy consumption is predictable and steady. the slightly increasing variation (standard deviation) on the vd line requires further investigation to understand the reasons for the variability of energy use (e.g., vehicle load, line management characteristics, weather, etc.; see section 2.3 and section 4). for further analysis of energy use, the variance of regenerative energy was also analyzed as given in fig. 7. fig. 7 shows the bv section in blue, where the variance is initially very high (54.68% in january) but decreases significantly (24.36% by may), indicating an improving trend. this is probably due to better weather conditions. vb (red line) is one of the best routes for energy recovery. dv (grey line): the variance is initially low (7.39%) but reaches a peak in april (38.52%) and then "returns" in may (10.89%). vd (yellow line) shows a similar trend to dv, with a peak in april (47.09%). by may, however, it "returns" to a lower level (10.08%). the april spikes on the dv and vd routes may indicate irregular conditions (e.g., weather, traffic, etc.; see section 2.3 and section 4) that are worth investigating. the analysis shows that the specificities of the routes have a substantial impact on the variability of energy recovery. table 2 shows the ten different trains in different months and routes. 364 s. fischer, b. hermán, m. sysyn, et al. fig. 6 spread of consumption by road for different months fig. 7 the spread of the regenerative energy generated in the different months per road table 2 shows the consumption of each train, with a monthly and route-by-route breakdown of deviations from the average consumption of the fleet. in each case, the average consumption of the fleet for the month and route were used to compare. no data is available for the grey-shaded part; the green-shaded part had a deviation of 2.5% or less. the trains marked in yellow are those where there was an average deviation of between 2.5% and 5% from the average consumption. orange indicates a deviation of 5-10%, and red indicates a deviation of 10% or more. it is important to note that individual outliers were not filtered out in the evaluation. the performance of the 4746-313 vehicles is shown to be the worst among the fleet. exceptionally high values (e.g., overconsumption in january and march) quantitative analysis and optimization of energy efficiency in electric multiple units 365 indicate significant energy efficiency problems that require urgent intervention. this vehicle should be a priority for fleet optimization. other poorly performing trains are 4746309 and 4746-315. these discrepancies may already be related to the drivers. however, no information on the drivers of each train was available during the analysis. therefore, for the reasons mentioned above, a more detailed examination of this is not evaluated. table 2 evolution of average consumption of different trains by month and route for a more accurate evaluation, each section is examined separately. for each stage, the energy consumed, the regenerative energy and the energy balance are analyzed. fig. 8 shows the consumption data associated with the route analysis. bv vb dv vd bv vb dv vd bv vb dv vd bv vb dv vd bv vb dv vd 4744-300 4744-301 4744-303 4744-304 4746-308 4746-309 4746-310 4746-312 4746-313 4746-315 march april maytrain number january february fig. 8 consumption data associated with railway stations on different routes 366 s. fischer, b. hermán, m. sysyn, et al. in fig. 8, the data points for each section are presented. in each case, the distance traveled is on the horizontal axis, and the energy used is on the vertical axis. in the subplots of fig. 8, the numbering and the additional vertical lines represent the station markers, which are related to the designations defined in table 1. for the simplicity of the presentation, the results of three measurements are shown; later calculations have been made with the entire measurement database. the values from the different trains are shown in different colors: blue for 4744-300, grey for 4744-301 and yellow for 4744-303. the red line indicates the proportional average consumption. the dot plot shows that several values are significantly above or below the average. furthermore, it can be seen that there is no clearly identifiable section where consumption is significantly above or below the average. the next step in the analysis is to analyze each station separately. the section to be analyzed is the section between bratislava and vienna (bv), which includes 9 stops between the two capitals. it is important to note that trains do not always stop at all stops, but energy consumption is presented in aggregate. unfortunately, it is not possible from the available data to identify precisely at which station a train has spent exactly how much time. the primary aim is to observe the average consumption of each section and to discover and identify the factors that influence it. in the following, the method will be illustrated by means of an example, and for ease of reference, not all the data will be used (see fig. 9). fig. 9 shows the energy consumption of train 4744-300 on the bv route as a function of distance traveled, with three different values: min (green), the lowest consumption value over a range; max (red), the highest consumption value over a range; and avr (blue), the average consumption over a range. there is a significant difference between the maximum and minimum values, indicating that the energy efficiency of some routes may vary significantly. fig. 10 shows the energy consumption of three different trains (4744-300, 4744-301, 4744-303) on the bv route as a function of distance traveled. fig. 9 consumption values for bv route – train #300 quantitative analysis and optimization of energy efficiency in electric multiple units 367 fig. 10 consumption values for bv route in kwh/km for three different trains for all trains, energy consumption starts from a higher value (around 12 kwh/km) and then decreases rapidly within the first 10-20 km. after this decrease, the consumption stabilizes and settles at around 6-7 kwh/km for the later sections. it is important to note that the increased consumption in some places may be due to the route conditions and that delays during the startup may also contribute to the higher value at the beginning of the measurement. fig. 11 shows the regenerative energy consumption (in kwh) of three different trains (4744-300, 4744-301, 4744-303) on the bv route as a function of distance traveled. fig. 11 regenerative energy values for the bv route in kwh/km for three different trains in general, there are no significant differences in the values of regenerative energy. the increase becomes faster from about the middle of the route to the end of the route (50 km), which may indicate that there are more maneuvers requiring braking on this section (e.g., more curved sections or sections with more stops; see section 2.3 and section 4). the next step in the analysis is to perform curve fitting for the average energy consumption. 368 s. fischer, b. hermán, m. sysyn, et al. curve fitting was performed for the values measured per route in the approach. fig. 12 plots the evolution of consumption (red line: kwh+), regenerative energy (regenerative energy (kwh–): green line), and total consumption (net consumption: consumption_sum (kwh+): yellow line) as a function of distance traveled (m) for a specific section after curve fitting. thus, all three polynomials are assigned to the given path, and the total length can be analyzed. all that is needed is a coordinate and a consumption (and or regenerative energy) value, and it is possible to see how the train performs compared to the average. fig. 12 curve fittings for the bv route such curve fitting has been performed for each month and each direction (fig. 13). it is important to note that the iqr method has been applied to the data used for curve fitting (see section 2.1, part 6). fig. 13 consumption curve fits for a january month on different routes (x means the distance in meter dimension) quantitative analysis and optimization of energy efficiency in electric multiple units 369 fig. 13 shows the energy consumption of trains on different routes (bv, vb, dv, vd) as a function of distance traveled in kwh/km. the curves in fig. 13 give the expected energy consumption as the distance increases. it can be seen that the different routes can be modeled with significantly different regression functions, and it is necessary to treat each of them separately. the regression functions for different months for the same route have a more minor, but not negligible, difference. fig. 14 shows the evolution of regenerative braking energy in kwh/km for four different routes as a function of distance traveled: bv (blue); dv (red); vb (green); and vd (purple). fig. 14 provides a model of how regenerative energy recovery varies with distance along each route. as before, there are significant differences between the regression functions, so it is appropriate to use a different one for each distance. the following step of the analysis was to determine the deviations from the curve fits defined for the different months and routes (figs. 15-16). fig. 14 regenerative energy curve fits for january on different routes (x means the distance in meter dimension) fig. 15 shows the energy consumption (kwh/km) for the bv route as a function of distance traveled (m). the horizontal axis shows the distance traveled (in meters), and the vertical axis indicates the energy consumption (kwh/km). the analysis is for january data and only shows points that are outside the ±2×σ (standard deviation) band. the graph shows two grey bars indicating the limits of the standard deviation (±1×σ and ±2×σ). it has to be mentioned that considering student-distribution the ±1×σ means 68.03%; hence, the ±2×σ means approx. 95% confidence interval in the case of 100 values, which is only an approximation). these can be used to examine which values are considered normal and which are outliers, respectively. points that are located outside the ±2σ band are shown as colored points on the graph. the graph shows data from different trains with different color codes. it can be observed that the outliers in energy consumption are not necessarily related to a specific train. the black curve in the graph shows the predetermined trend (as shown in the previous sections) between energy consumption and distance. the outliers indicate that energy consumption is significantly higher for certain trains. 370 s. fischer, b. hermán, m. sysyn, et al. fig. 15 energy consumption diagram for the bv route, 2023 january, only the points which are out of the 2× band ( means the standard deviation) fig. 16 shows the energy (kwh/km) from regenerative braking on the bv route in january. the figure plots energy data as a function of distance (m) and shows only points outside the ±2×σ (standard deviation) band. the regenerative energy (kwh/km) values are on a smaller scale (0.5-3.0 kwh/km) than the energy consumption data. in this case, the number of values outside the ±2×σ bands is much lower. the outliers in this case mean that for some trains, the energy recovered is significantly lower than expected. fig. 16 regenerative braking energy diagram, for the bv route, 2023 january, only the points which are out of the 2× band ( means the standard deviation) quantitative analysis and optimization of energy efficiency in electric multiple units 371 fig. 17 visualizes the overconsumption values back mapped. all months of the bv route are included in the figure. in the heat map, red markers show the most frequently overconsumed area, while yellow markers show still critical parts. areas that are not marked where overconsumption was not significant. tables 3 and 4 summarize the overconsumption values for each route and station. fig. 17 energy over consumption heat map for the bv route (the value 1.0 is related to the highest value, i.e., 17.02%, in table 3 as a reference) table 3 the overconsumption values expressed as percentages associated with the nearest stations of various routes (vb and bv) vb [%] bv [%] vienna hauptbahnhof 2.82 bratislava 5.01 vienna grillgasse 18.40 kittse 8.11 gramatneusiedl 2.15 pama 12.74 götzendorf 0.75 gattendorf 4.89 bruck-leitha 0.36 neudorf b.parndorf 17.02 parndorf ort 0.18 parndorf 1.74 parndorf 1.33 parndorf ort 3.70 neudorf b.parndorf 1.75 bruck-leitha 0.19 gattendorf 3.13 götzendorf 0.00 pama 6.17 gramatneusiedl 0.05 kittse 3.93 vienna grillgasse 0.00 vienna hauptbahnhof 0.00 372 s. fischer, b. hermán, m. sysyn, et al. table 4 the overconsumption values expressed as percentages associated with the nearest stations of various routes (vd and dv) vd [%] dv [%] vienna hauptbahnhof 4.90 deutschkreutz 7.22 vienna meidling 6.61 sopron 5.46 ebreichsdorf 0.83 baumgarten-schattendorf 2.82 ebenfurth 0.52 drassburg 0.12 neufeld-leitha 1.28 wulkaprodersdorf 0.75 müllendorf 0.35 müllendorf 3.61 wulkaprodersdorf 22.49 neufeld/leitha 0.64 drassburg 0.47 ebenfurth 6.03 baumgarten-schattendorf 0.41 ebreichsdorf 8.37 sopron 0.00 vienna meidling 0.40 deutschkreutz 0.00 vienna hauptbahnhof 0.67 in tables 3 and 4, the odd-numbered columns display the routes and their corresponding stops, while the even-numbered columns present overconsumption values expressed as percentages. these percentages represent the relative magnitude of outliers associated with a given station. for ease of identification, each overconsumption value has been linked to the nearest station. it is important to note that this does not necessarily indicate that the overconsumption occurred precisely at the station but rather somewhere along the route leading up to that station. furthermore, only values falling outside the 2×σ standard deviation range were considered in the calculations. this approach facilitates the identification of segments that appear to be critical. 4. conclusions the aim of the research was to improve railway energy efficiency by developing routespecific strategies. this study utilized a unique methodology to analyze the energy consumption of various railway routes and the utilization of regenerative braking energy. the analysis covered four routes (bv, vb, dv and vd, respectively) over a five-month period (january to may 2023). the results revealed significant variations in energy consumption driven by terrain characteristics, speed profiles, and stop frequencies. the findings also indicated substantial differences in energy consumption and regenerative feedback across the routes. understanding and decoding the causes of individual cases of overconsumption is a complex process. to address this complexity, the research analyzed not only aggregated consumption data but also processed a large volume of route-specific and direction-specific data. the study modeled the spatial (location-based) and temporal (monthly) patterns of energy use and regenerative feedback. route-specific consumption models were developed using the interquartile range (iqr) method and curve-fitting techniques to evaluate the data. these models enabled the quantification and prediction of energy consumption patterns for individual routes. deviation bands were established to identify outliers. overconsumption cases were flagged when values exceeded the upper deviation limit, while cases of low regenerative feedback were analyzed when values fell below the lower limit. this allowed the localization of outliers by route and station. quantitative analysis and optimization of energy efficiency in electric multiple units 373 the spatial distribution and temporal trends of energy consumption were also analyzed using integrated heatmap-based visualizations, facilitating the spatial identification of critical segments. to implement this approach, it is sufficient to associate consumption (or regenerative) values with coordinates along the analyzed routes. this makes it possible to evaluate how a given train performs compared to average values on the respective route. it is essential to emphasize that further investigations and analyses are required for the evaluation of consumed and regenerated electrical energy. these investigations should account for the following complex and parallel factors, which are part of future research: ▪ continuous monitoring of schedules and deviations from the timetable. ▪ 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scientific paper comparative analysis of thermal performance of two base liquids having tri-hybridized nanomaterials in a vertical channel under hall effects jafar hasnain bahria university islamabad, pakistan abstract. the use of nanoparticles with enhanced thermal properties has given nanotechnology a prominent place in modern industrial settings. as opposed to oil or water, which serve as base fluids, nanofluids with superior thermal characteristics, such as thermal diffusion, resistive nature, and increased thermal and electrical conductance, are more effective. this research aims to compare the convective heat transfer in hydromagnetic tio2-water nanofluid (nf), tio2-swcnts-water hybrid nanofluid (hnf) and tio2-swcnts-mwcnts-water tri-hybrid nanofluid (thnf) flow in a vertical channel under the influence of hall current. the flow is oscillatory due to the pressure gradient and the velocity slip is considered at the wall. the radiative heat flux is also considered along with the heat source and sink. as a result of using nondimensional variables, the fundamental equations of the flow problem are recast in a form that is dimension-free. the solution of the dimensionless equations is obtained in the exact form. the effects of involved parameters are illustrated graphically and studied concerning the flow fields, skin friction, and heat transfer rate. the results reveal that the temperature for the tri-hybridized nanofluid is greater than both hybrid and nanofluids. so, three distinct nanoparticle types can be concentrated for improved thermal performance. keywords: tri -hybridized nanofluid, hall effects, heat source/sink, beavers and joseph slip condition 1. introduction nanotubes with small structures and dimensions have attracted a lot of researchers to study their properties from a scientific and technological point of view [1]. the unique structure of nanotubes possesses excellent chemical and physical aspects, and these received: july 19, 2024 / accepted september 05, 2024 corresponding author: jafar hasnain bahria university, shangrilla road, e-8, islamabad, pakistan e-mail: jafar.buic@bahria.edu.pk javascript:; javascript:; 2 j. hasnain properties can be used in manufacturing various industrial products. cnts with pure carbon polymer chains are nano-sized cylindrical shapes with one (swcnt) or multiple layers (mwcnts) of graphene sheets. cnts with multiple layers of graphene sheet possess good stability under an argon atmosphere during the thermal phenomenon. it is because a structural transformation for catalytically grown swcnts was detected for the first time at 1200℃ [2]. titanium dioxide is significant due to its photocatalytic applications. high photocatalysis properties are obtained by manufacturing tio2 materials with a large surface area. tio2 gives high acid strength when reacted with sulfuric acid and as a result, tiso4 is produced. it gives better performance as a heat conductor due to its photocatalytic nature and hence, is useful as a coolant agent. furthermore, nanofluid becomes more stable with tio2 nanoparticles and is useful in motor vehicles to reduce the air population as well. a significant study for energy production in the photocatalysis field is given in [3]. ternary hybrid nanofluid with three different nanomaterials is an advanced category of liquid that possesses great thermal performance at energy transport when compared with normal liquids. with its unique properties, higher thermal performance, and heat rate enhancement in dynamics of fuel as well as automobile coolants, cumulative knowledge of dual particle nano liquid can be outlined. moreover, due to its advanced thermal performance it can be used as a coolant agent in radiators and to improve the flow of heat transport. some applications are magnetic resonance imaging, drug delivery and cancer removal. saleem et al. [4] analyzed the thermal analysis of cilia transport of blood flow with dispersion of three different nanomaterials in the presence of lorentz force. arif et al. [5] discussed the fractional model flow of water with heat transport using three different shaped nanomaterials in the radiator. das et al. [6] investigated the impact of an electrical double layer on mixed convective flow with three different nanomaterials in a vertical channel in the presence of lorentz force. munawar et al. [7] presented a model for hydromagnetic cilia transport of tri-hybrid nano liquid within an elastic electroosmotic pump with mixed convection phenomenon. ramesh et al. [8] investigated the behavior of base fluid due to the dispersion of three different nanomaterials in a stretchable channel with heat source and sink phenomena. some latest studies on thnf flow are listed as [912]. the heat source and sink phenomena occur due to the difference between the liquid present in the surroundings and the interface. the heat source phenomenon affects the thermal transport in the liquid, and it depends on the heat and space. some applications are aeration, reactor quality assurance, food manufacturing, combustion analysis, and dilatation. rajkumar et al. [13] investigated the system disorder due to pulsating hydromagnetic micropolar nano liquid flow in the presence of the cattaneo-christov heat flux model and heat source as well as heat sink phenomena in a channel with spongy media. shobha et al. [14] discussed the micropolar nano liquid in the presence of heat source as well as heat sink phenomena and inclined lorentz force between vertical plates. bhaskar et al. [15] discussed the heat source as well as heat sink phenomena with cross-diffusion effects of hydromagnetic nano-liquid flow within a channel in the presence of a chemical reaction. shobha et al. [16] analyzed the impact of heat source and sink phenomena in williamson nano-liquid flow with nonlinear thermal radiation within a vertical channel. jha et al. [17] scrutinized the effect of heat sink and source phenomena on free convection flow through an upright channel having suction and injection phenomena. the hall current becomes too important to ignore when magnetic forces increase in magnitude. hall current is a phenomenon that characterizes the behaviour of electrons in a https://www.sciencedirect.com/topics/engineering/photocatalysis https://www.sciencedirect.com/topics/engineering/electrical-double-layer https://www.sciencedirect.com/topics/engineering/electrical-double-layer comparative analysis of thermal performance of two base liquids having tri-hybridized... 3 conductor when they are being acted upon by electric and magnetic fields. through a second-law analysis, roja and gireesha [18] investigated the hall effect of natural convection flow in a vertical channel with a constant heat source and sink. in a mixedconvection flow driven by a shifting magnetic flux, effects of ion slip and hall currents, thermal radiative heat, and chemical reaction on a viscoelastic fluid flowing down a vertical permeable channel were investigated by singh et al. [19]. the influence of the generated magnetic field and hall current on the flow of convective viscoelastic fluid streaming down an upright channel was addressed by singh et al. [20]. using hall current and an induced magnetic field, singh and seth [21] analysed the mhd convective flow of a second-grade liquid through two alternately vertical surfaces. the hall current's impact on the movement of ionised ethylene-glycol in a vertical permeable channel with metal nanoparticles was investigated by das et al. [22]. maraj et al. [23] studied the rotating flow of an mhd (mos2sio2-h2o) hnf in a channel due to thermal periodic conditions and velocity slip at its vertical wall. motivated by the above-mentioned literature, the current study explores the effects of hall current on the flow of tri-hybridized nanofluid with two different base liquids and tio2, mwcnts and swcnts as nanoparticles. the liquids are heat absorbing and heat generating and flow through upright vertical plates under the influence of the optically thin limit of thermal radiation. exact solutions for velocities and temperature profiles are obtained and presented through graphs and tables. table 1 displays the uniqueness and research gap of the present study. the present study enhances the existing body of literature by addressing the following research inquiries.  what effect does the addition of three distinct nanoparticle types have on the heat transfer properties?  for ternary hybrid nanofluids, what role does hall current play?  how might the fluid's temperature be affected by the heat source or sink? table 1 research gap of the study ref. no. [24] [25] [26] present hall current     ternary hnf     heat source/sink     beavers and joseph velocity slip     2. mathematical development of the problem in a permeable media saturated with an incompressible tri-hybridized nanofluid, laminar flow affected by a radiative heat flux is investigated. the flow is confined to two vertical porous walls kept along the x-axis to study the effect of suction/injection (fig. 1). three types of nanoparticles are considered namely tio2, mwcnts and swcnts in the base fluid water. the two walls are electrically insulated and kept at y=0 and y=a with temperatures t0 and t1 respectively with t1>t0. the slip is also considered at the cold wall with the suction while the injection is considered at the heated wall. boussinesq's widely 4 j. hasnain recognized approximation simplifies the analysis by assuming that all fluid parameters remain constant, except for the density fluctuation with temperature in the body force part. fig. 1 schematic of the flow problem furthermore, when considering the equations for momentum and energy, density fluctuations have a negligible impact. the temperature differences that are associated with the expansion coefficient are presumed to be the same. the flow is affected by an evenly distributed magnetic field b0 applied in the y-direction. given that re is so small, the induced magnetic field is negligible. the flow is also governed by hall effects which raise the z-component for the velocity. with these assumptions, under the usual boussinesq approximation, the flow equations for the optically thin fluid subject to thermal radiation with heat source and sink are [25-27]:       2 0 * 0 02 1 , 1 thnf thnf thnf t y x yy thnf thnf thnf thnf thnf b u mw u v u p u u g t t k m                        (1)     2 0 0 2 , 1 thnf thnf thnf t y yy thnf thnf thnf b w v w w w w mu k m                (2)           *2 * 0 0 0 4 , thnf thnf t y yy thnf p p pthnf thnf thnf k q t v t t t t t t c c c                (3) with the boundary constraints 00 0 0 0 0 1 , , , 0, 0, . y yy y y y y y a y a y a k k u u w w t t u w t t                  (4) where t′ represents time, u and w are velocity components in x and z directions, v0 is uniform horizontal velocity, p is the pressure of the fluid, k is the permeability of porous comparative analysis of thermal performance of two base liquids having tri-hybridized... 5 media, b0 is magnetic flux strength, m is hall current parameter, g is gravitational acceleration, q* is a heat source, α* is mean radiative absorption coefficient, t is the fluid temperature, t0 is the temperature of reference, α is the dimensionless quantity depending on the material, a is the distance between the plates, ρthnf, μthnf, βthnf, (cp)thnf, kthnf, σthnf are density, viscosity, thermal expansion, heat capacitance, thermal conductivity and electrical conductivity of thnf, respectively. derivatives regarding x, y and t′ are indicated by the corresponding subscripts x, y and t′, respectively. the thermophysical properties of thnf are given as follows [7, 27] whereas the numerical values of these properties are given in table 2.                 2.5 3 2 1 2.52 1 1 2 3 3 1 .52 2 3 1 1 1 1 , , 1 1 thnf thnf f f f f f                                                                          3 2 1 3 1 2 2 1 1 2 3 3 2 1 3 3 2 1 1 1 1 2 1 1 1 2 2 2 , where 2 1 1 1 , 1 1 1 , p p p pthnf p p p pf f f f thnf f f f f hnf hnfthnf hnf hnf nfhnf hnf k k k k k kk k k kk c c c c c c c c k k k                                                                                                 2 2 2 2 2 3 3 3 1 1 1 3 3 3 1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 , 2 2 2 2 2 , , 2 2 2 2 2 2 where , . 2 2 nf nf nf nf f f hnf hnfnf thnf f hnff f hnf hnf nf nf f fhnf nf nf fnf nf f f k k k k k k k k k kk k k k k k                                                                      (5) in the above expressions, the term ϕ1,2,3 relates to volume fraction for nanoparticles. hybrid nanofluid, nanofluid, and fluid are indicated by the subscripts hnf, nf, and f respectively. table 2 numerical values for physical properties of nanoparticles and base fluids [7, 28, 29] properties h2o kerosene tio2 swcnts mwcnts ρ (kg/m3) 997 783 4250 2600 1600 k (wm-1k-1) 0.613 0.145 8.9583 6600 3000 cp (jkg-1k-1) 4179 2090 686.2 425 796 σ (s/m) 0.05 6*10-10 1*10-12 1400 300 β (k-1) 21*10-5 99*10-5 0.9*10-5 19*10-6 21*10-6 pr 6.2 27.6 --- 6 j. hasnain to reduce the system to a dimensionally independent form, the following nondimensional quantities are used forthcoming non-dimensional quantities are used.     0 2 1 0 * 32 0 2 2 2 2 2 *2 2 * 2 02 , , , , , , ( ) p , gr , , , , 4 , pr , , . f f f f 1 0 ff f f f pf f f f f f ff tt tx y u a wa y y f g t a a t t a g t -t a v aa p k k da s a a ca b a q a ha q k cp k                                    (6) together, the surface constraints and the flow equations can be expressed as   * * * 2 * * * * 2 1 1 , 1 t y x yy ha f s f p f f gr f mg da m                       (7)   * * * 2 * * * 2 1 , 1 t y yy ha g s g g g g mf da m                (8)     * *pr ,p t y yyc s k q            (9) with the surface constraints defined in the following manner: 0 0 00 0 1 1 1 , , 0, 0, 0, 1. y yy y yy y y y y f f g g f g                    (10) the parameters in the above expressions are hartmann number ha, viscosity ratio μ*, density ratio ρ*, electrical conductivity ratio σ*, thermal expansion ratio β*, hall parameter m, garshof number gr, electrical conductivity ratio k*, darcy parameter da, suction parameter s, prandtl number pr, radiation parameter δ, heat source/sink parameter q and slip parameter γ. 3. method of solution using f ig   the system of eqs. (7-10) can be written as   * * * 2 * * * * * 2 1 1 , 1 t y x yy ha s p gr i m da m                             (11)     * *pr ,p t y yyc s k q            (12) and comparative analysis of thermal performance of two base liquids having tri-hybridized... 7 0 00 1 1 , 0, 0, 1. yy yy y y                (13) when dealing with the pure oscillatory flow, the solutions to eqs. (11)-(13) are obtained by letting [25]        0 0, , , , .i t i t i t xp e y t y e y t y e          (14) where λ represents any positive constant and ω is oscillation frequency. using eq. (14) into eqs. (11-13) give * * 0 0 5 0 0* * ,yy yd sd q gr               (15) * 0 1 0 2 0 0,yy yk d q d q     (16) with 0 0 00 00 0 01 1 , 0, 0, 1. yy yy y y d              (17) the fluid temperature expression is produced by solving eq. (16) with restrictions (17), which leads    3 4 1 2, , q y q yi ty t e c e c e   (18) after solving eq. (15) under the constraints (17), the expression for the primary and secondary velocities of the fluid are     6 7 3 4 3 4 8 9* 5 , cos , q y q y q y q y f y t t c e c e q e q e q              (19)     6 7 3 4 3 4 8 9* 5 , sin . q y q y q y q y g y t t c e c e q e q e q              (20) at the left wall (y=0) of the channel, the non-dimensional expressions for shear stresses caused by main and secondary flows are represented by 6 7 3 4 3 4 8 9* 50 , q y q y q y q yi t x z y cf icf e c e c e q e q e y q                  (21) and the rate of heat transfer is given as  3 4 1 2 0 . q y q yi t y nu e c e c e y        (22) 8 j. hasnain the following are the values used in the aforementioned expressions as constants.         * ** * * * * 1, , , , pr, ,, t pthnf thnf thnf thnf thnf f f hnf p p f f f p f k k c k c q s c c                             2 * 2 * * 1 2 1 21 1 2 3 4 * * 2 ** * 2 * 5 5 6* * ** 2 * * 2 * * ** 5 1 7 8* * 2 * * 3 5 * * * * 4 41 1 pr, , , 2 2 41 1 i , , 21 41 , 2 p q k q q k qq q q q i c q q q sha s q mi q da k k k m q s c g q k rs q q q                                                                           * * 2 9* 2 * * * 3 4 5 4 , , c gr q q s q q q s               3 3 4 7 6 734 4 6 7 6 3 8 10 8 9 11 6 12 7 13 8 9* * 4 9 5 5 10 12 13 10 11 13 1 2 3 4 11 12 11 12 , 1 , 1 , , 1 1 , , , . q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q c c c c e e e e e e ee e e q q q qe e                                     4. results and discussion hall effects on the flow of tri-hybridized nanofluid with two different base fluids flowing through vertical parallel plates have been discussed. the fluid temperature expression is produced by solving eqs. (16) with restrictions (17), while after solving eq. (15) under constraints (17), the expressions for the primary and secondary velocities of the fluid are obtained. during the investigation, the following range of parameters is kept in mind: 0 < ϕ1=ϕ2=ϕ3≤ 0.06 (nanoparticles volume fraction), 2 < ha ≤ 3.5 (hartman number), 2 < m ≤ 3.5 (hall parameters), 0 < δ ≤ 1.5 (radiation parameter), and -1.5 < q ≤ 1.5 (heat source/sink parameter) are used. the values for the other parameters are da=1, s=1, γ=0.1, and gr=1. fig. 2a shows that the primary flow velocity is higher in the case of pure fluid, but it decreases with the addition of nanoparticles. the velocity is lowest in the case of ternary nanoparticles. as the concentration of nanomaterials in liquids increases, the liquids thicken, and the flow rates decrease. therefore, primary velocities decrease. the flow velocity is also less for kerosene-based fluid in the case of heat source and an opposite behaviour is observed for the heat sink case (see fig. 2b). the same behaviour is observed for the secondary flow velocity as can be seen in figures 2c and 2d. in comparison to pure, hybrid, and mono fluids, the ternary hybrid nanofluid has a lower temperature in both scenarios including base liquids. when the number of nanomaterials added to liquids increases, the liquids thicken significantly, slowing down the flow and reducing convection. due to this, the temperatures of the liquids decrease. the temperature for the kerosene-based ternary nanofluid is higher as compared to the water-based ternary nanofluid (see fig. 2e and 2f). comparative analysis of thermal performance of two base liquids having tri-hybridized... 9 fig. 2 comparative analysis with two base fluids (a, b) primary velocity f(y,t) (c, d) secondary velocity g(y,t) and (e, f) temperature θ(y,t) the effects of the hartmann number ha on the fluid velocities for thnf are presented in figure 3. figures 3a and 3b show the decreasing behaviour of primary fluid velocity with the increasing values of ha for both cases of heat source and sink respectively. figures 3c and 3d present the decreasing behaviour of secondary velocity with an increase in ha for both cases of heat source and sink respectively. the rise in the resistive lorentz force, which is associated with the increase in the values of the hartmann number, is the cause of the decrease in both primary and secondary velocities. the flow velocities are higher for water-based nano liquid as compared to kerosene-based nano liquid. 10 j. hasnain fig. 3 variation of (a, b) primary velocity f(y,t) (c, d) secondary velocity g(y,t) against ha for thnf the effect of hall parameter m on the fluid velocities in the cases of both heat source and heat sink are plotted in fig. 4. the reverse behaviour is observed in the velocities of thnf with the rise in m as compared to ha. when m is increased, the effectiveness of conduction decreases, which in turn reduces the magnetic damping and, as a result, enhances the velocity of the fluid. however, the variation in kerosene-based liquid and water-based liquid is like that of ha. comparative analysis of thermal performance of two base liquids having tri-hybridized... 11 fig. 4 variation of (a, b) primary velocity f(y,t) (c, d) secondary velocity g(y,t) against m for thnf the oscillatory behaviour of both primary and secondary velocities with the increasing values of ha is shown in fig. 5. both the velocities show a decreasing behaviour with escalating values of the hartmann number for both heat source and heat sink cases. the amplitude for the kerosene-based liquid is higher than water-based liquid fig. 5 oscillatory variation of (a, b) primary velocity f(y,t) (c, d) secondary velocity g(y,t) against ha for thnf the oscillatory behaviour of both primary and secondary velocities with the increasing values of m is shown in fig. 6. both the velocities show an increasing behaviour with escalating values of the hall parameter for both heat source and heat sink cases. the amplitude for the kerosene-based liquid is higher than water-based liquid. also, the amplitude is higher in the case of the heat source relative to the heat sink case. 12 j. hasnain fig. 6 oscillatory variation of (a, b) primary velocity f (y,t) (c, d) secondary velocity g(y,t) against m for thnf the surface frictions due to primary and secondary flow against ha with the variation in m are presented in fig. 7. the surface frictions due to both velocities show an increasing behaviour with escalating values of the hall parameter for both heat source and heat sink cases. however, the opposite behaviour is observed for the increasing values of ha. comparative analysis of thermal performance of two base liquids having tri-hybridized... 13 fig. 7 variation of surface friction (a, b) cfx (c, d) cfz fig. 8 is plotted to exhibit the heat transfer rate versus radiation parameter with the variation in the volume fraction of nanoparticles in thnf. the heat transmission rate increases with the radiation parameter for both liquids while it reduces with the added nanomaterials. the phenomena are the same for both cases of heat source and heat sink. fig. 8 variation of heat transfer rate nu fig. 9 presents a comparison with the results of the preceding study done by falade et al. [25]. both the figures corroborate the earlier findings with values ϕ1=ϕ2=ϕ3=q=m=0. fig. 9 comparison of results with the existing literature (a) f(y,t) and (b) θ(y,t) 14 j. hasnain 5. conclusions hall effects on the flow of tri-hybridized nanofluid with two different base fluids flowing through vertical parallel plates have been discussed. the flow is affected by thermal radiation in the presence of a heat source and sink. the following results have been concluded:  primary and secondary flow velocities of the tri-hybridized nanofluid are less than the velocities for the hnfs, nfs and pure base fluids for both the heat source and sink case.  the velocities for kerosene-based fluids are less than the velocities for the waterbased fluids for the heat source case while the opposite behavior is noticed for the heat sink case.  both the velocities decrease with an increase in ha for all kinds of kerosene and water-based fluids in both heat source and sink cases. however, the amplitudes 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creative commons license: cc by-nc-nd original scientific paper drone selection for forest surveillance and fire detection using interval valued neutrosophic edas method alize yaprak gul1, emre cakmak2, atiye ece karakas3 1department of industrial engineering, istanbul technical university, istanbul, turkey 2department of industrial engineering, istinye university, istanbul, turkey 3department of industrial engineering, piri reis university, istanbul, turkey orcid ids: alize yaprak gul https://orcid.org/0009-0007-5742-0267 emre cakmak https://orcid.org/0000-0002-3406-3144 atiye ece karakas https://orcid.org/0009-0006-8478-3031 abstract. forest fires are one of the major causes for deforestation resulting in significant economic and environmental losses. the application of drones has been extended to various areas including disaster management. since drones offer numerous advantages like real-time surveillance, task planning capabilities and autonomy, they are utilized in early detection systems for forest fires. the selection of a drone type for this purpose involves a complex system of multiple factors and conflicting information, for which the use of multi-criteria decision-making (mcdm) methods have been found to be yielding effective results. the aim of this study is to present a decision framework for drone selection problem in the context of forest fire surveillance and detection. this study contributes by (i) pointing out to the gap that the drone selection problem for forest surveillance and fire detection has been sparsely addressed, (ii) presenting an extensive literature review, (iii) extracting the relevant criteria through a literature review and interviews with the experts in field, (iv) assessing the alternatives by the proposed framework based on interval valued neutrosophic evaluation based on distance from average solution (ivn edas) method. the proposed framework is demonstrated by a case study consisting of four drone alternatives and 14 criteria. in accordance with the extant literature, the criteria related to the visual capabilities and diagnosis are evaluated as the most crucial features. a sensitivity analysis is carried out to check for the robustness by varying the criteria weights and a comparative analysis is conducted with interval valued neutrosophic technique for preference by similarity to the ideal solution (ivn topsis) and interval valued neutrosophic combinative distance-based assessment (ivn codas) methods to validate the veracity of the method. key words: drone selection, forest surveillance, fire detection, interval valued neutrosophic sets, mcdm, edas received: october 28, 2023 / accepted february 27, 2024 corresponding author: alize yaprak gul department of industrial engineering, istanbul technical university, 34367 macka, besiktas, istanbul, turkey e-mail: gul22@itu.edu.tr https://orcid.org/0009-0007-5742-0267 https://orcid.org/0000-0002-3406-3144 https://orcid.org/0009-0006-8478-3031 434 a.y. gul, e. cakmak, a. e. karakas 1. introduction forests are vital for the sustainability of the ecosystem and human life. they play a crucial role in counteracting climate change and its worsening effects. among the deforestation causes, wildfires have been responsible for about 34% of the tree cover loss since 2019 [1]. increasing wildfires and deforestation exacerbate the effects of climate change which lead to a vicious cycle of drier seasons followed by more frequent and intense fires. one of the most devastating wildfires in history occurred in the 2019-2020 australian bushfire season, the burnt area had been estimated to be between 24.3 and 33.8 million hectares [2], which was effective from june 2019 to may 2020. in total, 33 people have lost lives, 3.094 houses have been destroyed and the direct economic cost has been estimated to be $2.5 billion in addition to $4-5 billion worth of losses on the australian food and agriculture industry [3]. deforestation substantially damages the economy and the ecosystem, which is avoidable to some extent. one of the ways to reduce these damages is to take environmental precautions and extinguish the fire as soon as they occur. as time passes, the destructive effect of wildfires magnifies abruptly. a rule has been mentioned i.e., the amount of water that is required to suppress a fire burning for a minute is multiplied by ten when the burning time doubles [4]. thus, early detection and taking countermeasures on time are crucial to prevent significant damage. for this reason, various technologies have found its application for monitoring and detection of forest fire. the technologies used can be summarized as satellite-based, sensor-based, camera-based, mobile biological sensing, unmanned aerial vehicle (uav), radio-acoustic based, neural network based and fuzzy logic based technologies according to the recent classification done by chowdary et al. [5]. in practice, many of these technologies cannot be deployed as effective stand-alone systems, instead a combination of these technologies is utilized to enhance the accuracy and efficiency in detection. however, each of these technologies are associated with certain advantages and disadvantages. in comparison to the satellite-based and ground-based technologies with static topology, the use of drones brings several useful properties such greater maneuverability and real-time surveillance, especially in the areas that are difficult to access. further, drones provide autonomy, task planning capabilities and self-healing properties [6]. the radical growth in the sensor and microprocessing technologies have facilitated their usage in different areas including the fire detection systems. the sensor technologies serve with improved preciseness and fewer false alarms notably when utilizing different types of sensors. thus, the application of drones integrated with different technologies as sensors and neural networks can realize an effective and self-sufficient solution for the surveillance and detection of forest fires. the application of drones and its potential for the respective problem requires more attention for the future research. there are several factors affecting the performance of the deployed drone type in the respective context. thus, the problem of its selection contains several ambiguities, that arise from several sources such as multiple evaluation attributes, the uncertainty and bias incorporated into the human decision-making process as well as the factors and the characteristics affecting the performance of a drone. according to hristozov and zlateva [7], the performance of a drone is influenced by its internal attributes, which are the technical features, and the external attributes such as the characteristics of the area, weather and other environmental conditions. some of these attributes may be in trade-off or have interrelationships with one and other. such complex decision-making problems that involve inexplicable and unquantifiable relationships are broadly handled by the multi-criteria decision drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 435 making (mcdm) methods, since they yield appropriate results with significantly lower computation time and complexity. the vagueness and uncertainty present in information can be successfully represented by fuzzy logic introduced by zadeh [8]. turksen [9] extended fuzzy sets to interval valued fuzzy sets to enable for a range of values to be assigned in the membership information. smarandache [10] introduced the neutrosophic sets, which enable the representation of hesitancy in addition to the membership and non-membership information, as an extension to intuitionistic fuzzy sets [11]. moreover, fuzzy mcdm methods are particularly helpful in addressing the decisionmaking problems with uncertainty and ambiguity resulting from the subjective evaluations and vague relationships among criteria and alternatives. among many proposed mcdm methods, evaluation based on distance from average solution (edas) method by keshavarz ghorabaee et al. [12] processes the conflicting criteria and performs better at considering the intangibility and vagueness in the decision-making process [13]. it also handles with the biased information better since its calculation is based on the assessment by the average solution. for this reason, this study aims to handle the drone selection problem for monitoring and detection of forest fires by proposing a framework using interval valued neutrosophic (ivn) sets to present the vague and uncertain linguistic information and edas method to evaluate the alternatives based on the conflicting criteria. the findings from the case study demonstrate that “the camera accuracy” is the most significant feature for a drone used in forest surveillance and fire detection. the “ingress protection rating”, “maximum flight time” are the second and third most significant characteristics. the rest of the criteria are ranked with the following order: “wind resistance”, “camera resolution”, “zoom camera”, “maximum operation altitude”, “maximum horizontal speed”, “maximum hover time”, “operation frequency”, “obstacle sensors”, “maximum takeoff weight”, “charging time” and “hovering accuracy”. this study contributes to literature by pointing out to the gap that the drone selection problem for forest surveillance and fire detection has been sparsely addressed, by presenting an extensive literature review on the technologies deployed in forest monitoring and fire detection, and by extracting the relevant criteria through an extensive literature review and interviews with the experts in field. as a result, four drone alternatives and 14 criteria have been identified. the alternatives are then evaluated based on the proposed framework using ivn edas method. moreover, a sensitivity analysis is conducted in order to check for the robustness of the results by varying the criteria weights. the veracity of the results is validated by carrying out a comparative analysis with interval valued neutrosophic technique for preference by similarity to the ideal solution (ivn topsis) method and interval valued neutrosophic combinative distance-based assessment (ivn codas) method. the rest of this paper is structured as follows. section 2 presents an in-depth literature review on the other technologies and the application of drones in the respective context along with the examination of drone selection problem by the mcdm methods. section 3 provides the methodology of ivn edas method with the preliminaries of interval valued neutrosophic sets. then, section 4 presents the application of the method on a case study that is concluded with a sensitivity analysis and a comparative analysis with ivn topsis and ivn codas methods. lastly, section 5 gives the conclusion with the recommendations for future study. 436 a.y. gul, e. cakmak, a. e. karakas 2. literature review monitoring and early detection of a possible fire are as crucial as the fast response and suppression of wildfires to avoid irreversible damages. several studies have examined the technologies used in forest fires and investigated their benefits and drawbacks. we first summarize these technologies and then discuss the advantages of the application of drones in forest surveillance and fire detection. den breejen et al. [14] categorized the techniques used in forest fire monitoring and detection into three groups as ground-based systems, manned aerial vehicle-based systems, and satellite-based systems. in their classification, a ground-based system monitors and gathers information by utilizing equipment with static topology. manned aerial vehicle-based systems describe systems that use an aerial vehicle controlled by a human operator to patrol the area for the detection of a possible fire. in satellite-based systems, the imagery taken through the satellite is collected and processed with the image-processing techniques. recent technological advancements in the field of wireless networks, microprocessors, image-processing and artificial intelligence have enabled a variety and a combination of these methods for the monitoring and detection purposes of forest fires. alkhatib [4] generalized the detection and monitoring systems as suppression and detection techniques used by the authorities, satellite-based systems, optical sensors, and digital camera and wireless sensor networks (wsn). the suppression and detection techniques used by the authorities include watch towers, water tankers, lightning detectors, etc. among the optical sensor and digital camera technologies, the video-cameras sensitive to smoke, infrared thermal imaging cameras and light detection and ranging systems (lidar) are widely applied. in wsns, the area of interest is surveilled by multiple wireless nodes that are equipped with different types of sensors and microprocessors. depending on the movement capability of nodes, wsns are classified as wsn with static topology and wsn with dynamic topology. in wsns with dynamic topology, the nodes follow a common protocol. chowdary et al. [5] extended this classification by sensor-based, neural networks-based, drone/airborne-based, fuzzy logic, mobile biological sensing-based, and radio acoustic-based techniques. by fuzzy logic techniques, a fuzzy logic algorithm is created and applied to the visual data gathered from various sources. the proposed fuzzy logic algorithm may use instances such as temperature, smoke, light, humidity, and distance [15] and shape, size, and motion variation of the fire [16] for which membership functions are generated. then for the verification, the performance of the decision-making procedure is evaluated either by simulations or real fire datasets. in mobile biological sensing-based detection, certain animal groups may be equipped with sensor and global positioning system (gps) devices transmitting data and location information, which are examined for sudden environmental changes such as temperature or humidity or changes in animal behavior. in practice, many systems combine multiple techniques together for more precision in monitoring and detection. many forest fire surveillance systems aim to deploy a self-sufficient system that require little or no maintenance or supervision. ideally, the deployed system should monitor the forest and detect the fire as soon as possible with a low false alarm chance. also, it should inform the authorities automatically. from the studies that have dealt with these technologies’ advantages and disadvantages, it has been found that the satellite-based techniques have a low temporal and spatial resolution [17]. the quality of imagery is highly affected by the terrain, time of day, and weather [5] resulting in limited coverage, limited precision, and a lack of real-time data reporting [6]. radio acoustic-based methods are prone to fire localization errors and a high chance of false alarms [5]. in camera-based technologies, immobile cameras equipped with multiple types of sensors are positioned at specified locations drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 437 for surveillance. immobile cameras provide a limited surveillance range [18] and image and video processing techniques are associated with significant pre-processing time and effort [5]. generally used miniaturized infrared cameras have low sensitivity, which leads to a high false alarm possibility [5, 19]. neural network-based techniques require large datasets and heavy model sizes, which are also prone to various false alarms [5]. sensor-based techniques provide more accuracy in the surveillance and detection of forest fires than satellite-based techniques as these are deployable in areas that are not observable by satellites [20]. however, wsns with static topology require regular maintenance of the infrastructure and they have limited coverage and effectiveness [21]. as these systems operate as static infrastructure, they are prone to get destroyed in the event of a fire which may cause additional replacement costs. their effectiveness in terms of coverage and resolution is directly determined by the investment made into the deployed system [6]. satellite-based and ground-based techniques such as camera surveillance and wsns with static topology may be insufficient for precise real-time monitoring of large areas. drones offer greater maneuverability and real-time surveillance capabilities in areas hard to access. recent progressive improvements in micro-processing, imaging, and drone technologies facilitate lower costs and wider usage of these technologies. the type of sensors utilized in the surveillance drones can be chosen depending on the type of possible fire and topology of the area to increase precision in detection. in the equipment with different types of sensors, a similarity index of measurements helps to mitigate the chance of false alarms [22]. in addition, the major conveniences of drones are that they provide great autonomy and task-planning capabilities with efficient allocation, self-organization, and self-healing properties [6]. for the sustainability and security of the system, the selection of the drone model is as important as the system architecture, sensors, and its other fundamental components. among a great variety of drones available on the market, a drone needs to be selected that is most appropriate to the characteristics of the area and can satisfy the necessary requirements for the tasks. many factors impact the operational performance of a drone and thus its selection, these factors can be viewed as external factors and internal characteristics or attributes of a drone [7]. the external factors are the disaster type, the characteristics of the area of interest, and the weather and other environmental conditions. for instance, it has been found that strong wind and precipitation affect drones’ performance significantly [23, 24]. under the influence of the strong wind, the movement capability of a drone is significantly disrupted which may result in the deviation from its direction and poor stabilization during hovering. the stabilization of a drone is especially important for the quality of imaging. other environmental conditions such as temperature and altitude of the area must be considered since drones are not able to operate at all conditions. with the wide-ranging features and integrable nature of drones, they have a substantial role as a dynamic, deployable, and controllable component of the internet of things infrastructure [25, 26]. drones have been manufactured and used primarily for military and security purposes. their application areas are extended to natural disasters, agricultural fields and smart farming, cargo handling, delivery of medical goods, traffic management, conservation, and land monitoring, climate change and urban planning, and more. therefore, they are available with various features and attributes such as drone type, weight, speed, takeoff weight, hover time, charging time, flight time, operating altitude, etc. [27, 28]. yet some of these attributes are in trade-off with one another. as an example, drones with higher speed capability have lower payload capacity [29, 30], and typically drones with slower speed can take shorter distances and 438 a.y. gul, e. cakmak, a. e. karakas handle higher payloads [31]. as a result, the drone selection problem must consider these multiple external factors and conflicting internal attributes, which makes it a highly complex and a challenging task. this study focuses on the drone selection problem for monitoring and detection of forest fires. the application of drones in the wildfire context is studied by many authors. to our best knowledge, the studies on the drone usage in wildfires are mainly gathered on three scopes: vision-based techniques and (deep learning) image-processing techniques [19, 32, 33], the system architecture with communication, data receiving, and gis modules [34, 35], and the area coverage and coordination of multiple drones [5, 36, 37, 38]. yuan et al. [19] presented the first thorough analysis of technologies that use drones for fire detection, diagnosis, prognosis, image vibration elimination, and cooperative control of drones. bailon-ruiz and lacroix [39] reviewed a selection of studies that examine drone application in wildfire remote sensing based on an autonomy perspective using three metrics: situation awareness, decisional ability, and collaboration ability. akhloufi et al. [6] presented an extensive review with a focus on onboard sensor instruments, fire perception algorithms, and coordination strategies. the prior studies reviewed by [6, 19, 39] assumed that a set of drones are available and suited for their specific use and thus the drone selection problem has not been mentioned. regarding the drone selection problem, hristozov and zlateva [7] introduced a performance mapping model as a concept model for drone selection in specific disaster conditions to aid the drone selection decision process. to best of our knowledge, in the literature only the study by pamučar et al. [40] dealt with the drone selection problem related to forest fire uses. their study focused on determining a drone model that can be used as a physical fire suppression tool that is controlled by operators in a ground central system. our study distinguishes from the prior study in that it deals with the drone selection problem for the monitoring and detection of forest fires for its application as an early warning system that requires minimum human assistance. using a drone for monitoring and detection purposes and as a fire suppression tool require different technical characteristics. a network of drones can act as mobile wireless sensor network, where the drones are equipped with various sensors to collect data on temperature, smoke and other parameters. the collected data should be transmitted to a head cluster thereafter it is transmitted to gateway to get processed. the drones in a wsn require to follow a protocol for self-organized deployment and coverage of the area. the energy efficiency in wsn communication is of major concern for an effective communication [41, 42]. likewise, each drone has a limited payload. the higher the payload of a drone, the higher is its battery consumption and shorter is its flight time. therefore, the weight of a drone should be ideally kept low [43]. thus, the energy consumption of a drone and of the wsn communication are two fundamental issues concerning the effectiveness of the whole as a fire detection system, one contributing to the movement capability and other to the efficient data transmission. to utilize a drone additionally as a firefighting tool, it should be equipped with fire extinguishing instruments. as a result, it would increase its payload and shorten the endurance of a drone. therefore, we believe that the drone selection for monitoring and detection and for firefighting purpose should be examined separately. ideally, both drone networks shall be integrated to each other as first one would act as an early warning system signaling the latter one to counteract the starting fire timely. to address the drone selection problem, neutrosophic sets are preferred to represent the ambiguity present in the linguistic assessments of the decision makers (dms). different types of uncertainties are associated in the decision-making processes, which can be considered with the help of fuzzy logic introduced by zadeh [8]. fuzzy sets have been extended to intuitionistic fuzzy sets by atanassov [11] which describe both the membership and the non-membership drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 439 degree. to enable the representation of indeterminacy in information of which the intuitionistic fuzzy sets lack, neutrosophic sets have been developed by smarandache [10]. following that, different extensions of neutrosophic sets have been introduced such as interval-valued neutrosophic sets, type-2 neutrosophic sets, 2-tuple linguistic neutrosophic sets. numerous studies have applied edas method in neutrosophic environment of these extensions such as single-valued neutrosophic sets [44], interval valued neutrosophic sets [45], type-2 neutrosophic sets [46] and bipolar neutrosophic sets [47]. the mcdm methods have found a great variety of application areas thanks to their beneficial aspects. in the presence of complex systems which incorporate multiple criteria and when the interrelationships among the systems’ elements are vague and inexplicable by the mathematical models, mcdm methods provide a structured (group) decision making process that is easy to implement, handles the uncertainty, can deal with both quantitative and qualitative data, and provides fast and relatively reliable solutions. it is basically used to select or prioritize alternatives based by assessing on a set of criteria or attributes. mcdm methods are used in various types of decision making problems such as the selection of suppliers [48], the evaluation of potential locations for specific uses [49, 50], but also is applied for the strategy selection [51, 52], performance evaluation [53, 54], and risk evaluation [55] among many others. the edas method is utilized by various recent studies with different extensions and application areas. simic et al. [46] used a type-2 neutrosophic number based threshold-based attribute ratio analysis (itara) method integrated with edas model for route selection of petroleum transportation. menekse et al. [56] applied a pythagorean fuzzy criteria importance through intercriteria correlation (critic) method integrated with edas model for the selection of an additive manufacturing process for automotive industry. dhumras and bajaj [57] used a picture fuzzy soft dombi edas model for strategy selection for sustainable and smart (robotic) agrifarming. there are several studies that dealt with the drone selection problem using mcdm methods in other application areas of drones. to give a few examples from the recent studies, for the military purposes a fuzzy weighted average algorithm is applied for group decision-making by lin and hung [58]. hamurcu and eren [59] proposed a methodology based on analytical hierarchy process (ahp) and technique for preference by similarity to the ideal solution (topsis), and a fuzzy analytical hierarchy process-vise kriterijumska optimizacija i kompromisno resenje (ahp-vikor) hybrid model is proposed by radovanović et al. [60]. the ahp-topsis method is used by several studies for the civilian application areas [61, 62]. with the radical growth in the e-commerce, delivery by drones has gained significant attention as a delivery tool since delivery in conventional ways has become rather inconvenient due to high delivery times and increasing costs in transportation. in this context, nur et al. [28] performed an interval valued inferential fuzzy (ivif) topsis method to find the most suitable drone for package delivery purposes in urban and rural areas and banik et al. [63] proposed a graph theory and matrix approach to select a drone for the delivery of medical supplies. the examination of the literature shows that the drone selection for the usage against forest fires has been addressed only by pamučar et al. [40] in terms of selecting a drone with the capability of extinguishment. no prior study has focused on the drone selection solely for the purpose of forest surveillance and fire detection. the next section presents the proposed methodology. 440 a.y. gul, e. cakmak, a. e. karakas 3. methodology the edas method, introduced by keshavarz ghorabaee et al. [12], is a distance-based method similar to topsis method and vikor method. instead of using positive ideal and negative ideal solutions, edas method evaluates the alternatives by using the positive and negative distance to the average solution. this aspect reduces the impact of dms’ bias. it recognizes the conflicting criteria [12, 64] and represents the ambiguity in the decision-making process [13]. the presence of the conflicting criteria makes edas method more suitable for this problem. however, the mcdm methods are prone to the rank reversal phenomenon [65] and the results should be further checked against this issue. it occurs when the introduction or the exclusion of a non-optimal alternative modifies the optimal and/or the least preferred alternative. it has been found that the edas method performs better than topsis in terms of robustness to the rank reversal phenomenon [66]. the interval valued neutrosophic sets are used to represent the linguistic evaluations of the dms since this set allows for an enriched representation of information since only the lower values of the parameters (membership, hesitancy, and non-membership) must be equal to or lower than 3, and the sum of upper values can exceed 3 allowing for a flexible information representation. therefore, the edas method under the interval valued neutrosophic environment is selected as an appropriate method to consider the ambiguity in decision making and conflicting criteria. in this section, we introduce the ivn edas method and provide the details about its implementation. 3.1 preliminaries for interval valued neutrosophic sets definition 1: let 𝑋 be the universe. for each element 𝑥 in 𝑋, an ivn set is be defined by three parameters; the truth-membership 𝑇𝑁(𝑥), indeterminacy (hesitancy)-membership 𝐼𝑁(𝑥), and the falsity-membership 𝐹𝑁(𝑥), where 𝑇𝑁 = [𝑇 𝐿 𝑁(𝑥) , 𝑇 𝑈 𝑁(𝑥) ⊆ [0,1]] , 𝐼𝑁(𝑥) = [𝐼 𝐿 𝑁(𝑥) , 𝐼 𝑈 𝑁(𝑥) ⊆ [0,1]], and 𝐹𝑁(𝑥) = [𝐹 𝐿 𝑁(𝑥) , 𝐹 𝑈 𝑁(𝑥) ⊆ [0,1]]. an interval valued neutrosophic number (ivnn) satisfies the following condition that the sum of the lower values of these three elements must be lower than or equal to 3: 0 ≤ 𝑇 𝐿 𝑁(𝑥) + 𝐼 𝐿 𝑁(𝑋) + 𝐹 𝐿 𝑁(𝑥) ≤ 3. thus, an ivn set is defined as in eq. (1) [67]: n = {〈x, [t l n(x) , t u n(x)] , [i l n(x) , i u n(x)] , [f l n(x) , f u n(x)]〉 | x∈x}. (1) let 𝑎 and 𝑏 be two ivnns represented by [𝑇 𝐿 𝑎 , 𝑇 𝑈 𝑎 ], [𝐼 𝐿 𝑎 , 𝐼 𝑈 𝑎 ], [𝐹 𝐿 𝑎 , 𝐹 𝑈 𝑎 ] and [𝑇 𝐿 𝑏 , 𝑇 𝑈 𝑏 ], [𝐼 𝐿 𝑏 , 𝐼 𝑈 𝑏 ], [𝐹 𝐿 𝑏 , 𝐹 𝑈 𝑏 ], respectively. following mathematical operations are applied between two ivnns a and b as defined below [68]: 𝑎𝑐 = 〈[𝑇 𝐿 𝑎 , 𝑇 𝑈 𝑎 ], [1 − 𝐼 𝐿 𝑎 , 1 − 𝐼 𝑈 𝑎 ], [𝐹 𝐿 𝑎 , 𝐹 𝑈 𝑎 ]〉, (2) 𝑎 ⊆ 𝑏 if and only if 𝑎 ⊆ 𝑏 and only if 𝑇 𝐿 𝑎 ≤ 𝑇 𝐿 𝑏 , 𝑇 𝑈 𝑎 ≤ 𝑇 𝑈 𝑏 ; 𝐼 𝐿 𝑎 ≥ 𝐼 𝐿 𝑏 , 𝐼 𝑈 𝑎 ≥ 𝐼 𝑈 𝑏 ; 𝐹 𝐿 𝑎 ≥ 𝐹 𝐿 𝑏 , 𝐹 𝑈 𝑎 ≥ 𝐹 𝑈 𝑏 , (3) 𝑎 = 𝑏 if and only if 𝑎 ⊆ 𝑏 and 𝑏 ⊆ 𝑎 (4) drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 441 𝑎⨁𝑏 = 〈 [𝑇 𝐿 𝑎 + 𝑇 𝐿 𝑏 − 𝑇 𝐿 𝑎 𝑇 𝐿 𝑏 , 𝑇 𝑈 𝑎 + 𝑇 𝑈 𝑏 − 𝑇 𝑈 𝑎 𝑇 𝑈 𝑏 ], [𝐼 𝐿 𝑎 𝐼 𝐿 𝑏 , 𝐼 𝑈 𝑎 𝐼 𝑈 𝑏 ], [𝐹 𝐿 𝑎 𝐹 𝐿 𝑏 , 𝐹 𝑈 𝑎 𝐹 𝑈 𝑏 ] 〉, (5) 𝑎 ⊗ 𝑏 = 〈 [𝑇 𝐿 𝑎 𝑇 𝐿 𝑏 , 𝑇 𝑈 𝑎 𝑇 𝑈 𝑏 ], [𝐼 𝐿 𝑎 + 𝐼 𝐿 𝑏 − 𝐼 𝐿 𝑎 𝐼 𝐿 𝑏 , 𝐼 𝑈 𝑎 + 𝐼 𝑈 𝑏 − 𝐼 𝑈 𝑎 𝐼 𝑈 𝑏 ], [𝐹 𝐿 𝑎 + 𝐹 𝐿 𝑏 − 𝐹 𝐿 𝑎 𝐹 𝐿 𝑏 , 𝐹 𝑈 𝑎 + 𝐹 𝑈 𝑏 − 𝐹 𝑈 𝑎 𝐹 𝑈 𝑏 ] 〉. (6) definition 2: a set of ivnns is represented by the notation 𝑥𝑘 = ⟨[t 𝑳 𝑘 , t 𝑼 k ], [i 𝑳 k , i 𝑼 k ], [f 𝑳 k , f 𝑼 k ]⟩, where k is the expert (𝑘 = 1,2,3, … , 𝐾). to aggregate multiple ivnns, the interval valued neutrosophic weighted arithmetic operator (innwa) is defined as follows [68]: 𝐼𝑁𝑁𝑊𝐴(𝑥1, 𝑥2, … , 𝑥𝐾) = ∑ 𝑦𝐾 𝑘=1 𝑘 𝑥𝑘 = [1 − ∏ (1 − 𝑇 𝐿 𝑘 ) 𝑦𝑘𝐾 𝑘=1 , 1 − ∏ (1 − 𝑇 𝑈 𝑘 ) 𝑦𝑘𝐾 𝑘=1 ], 〈 [∏ (𝐼 𝐿 𝑘 ) 𝑦𝑘𝐾 𝑘=1 , ∏ (𝐼 𝑈 𝑘 ) 𝑦𝑘𝐾 𝑘=1 ], [∏ (𝐹 𝐿 𝑘 ) 𝑦𝑘𝐾 𝑘=1 , ∏ (𝐹 𝑈 𝑘 ) 𝑦𝑘𝐾 𝑘=1 ]〉, (7) where 𝑦𝑘 denotes the weight vector of the experts. definition 3: the deneutrosophication function of an ivnn 𝑎 is calculated using eq. (8) [69]: 𝐾(𝑎)= ( (𝑇𝐿 𝑎+𝑇𝑈 𝑎) 2 +(1− (𝐼𝐿 𝑎+𝐼𝑈 𝑎) 2 )(𝐼𝑈 𝑎)−( (𝐹𝐿 𝑎+𝐹𝑈 𝑎) 2 )(1−𝐹𝑈 𝑎)), (8) where 𝑎 = 〈[𝑇 𝐿 𝑎 , 𝑇 𝑈 𝑎 ], [𝐼 𝐿 𝑎 , 𝐼 𝑈 𝑎 ], [𝐹 𝐿 𝑎 , 𝐹 𝑈 𝑎 ]〉. definition 4: let 𝑎 and 𝑏 be two ivnns represented by [𝑇 𝐿 𝑎 , 𝑇 𝑈 𝑎 ], [𝐼 𝐿 𝑎 , 𝐼 𝑈 𝑎 ], [𝐹 𝐿 𝑎 , 𝐹 𝑈 𝑎 ] and [𝑇 𝐿 𝑏 , 𝑇 𝑈 𝑏 ], [𝐼 𝐿 𝑏 , 𝐼 𝑈 𝑏 ], [𝐹 𝐿 𝑏 , 𝐹 𝑈 𝑏 ], respectively. the euclidean distance between two ivnns are given by eq. (9) [70]: 𝑑𝐸(𝑎, 𝑏) = √ 1 6 (( (𝑇 𝐿 𝑎 − 𝑇 𝐿 𝑏 ) 2 + (𝑇 𝑈 𝑎 − 𝑇 𝑈 𝑏 ) 2 + (𝐼 𝐿 𝑎 − 𝐼 𝐿 𝑏 ) 2 + (𝐼 𝑈 𝑎 − 𝐼 𝑈 𝑏 ) 2 + (𝐹 𝐿 𝑎 − 𝐹 𝐿 𝑏 ) 2 + (𝐹 𝑈 𝑎 − 𝐹 𝑈 𝑏 ) 2 )). (9) definition 5: the function in eq. (10) returns the euclidean distance if the difference between two deneutrosophicated ivnns is greater than zero, otherwise it returns zero. 𝑍(𝑎, 𝑏) = { 𝑑𝐸(𝑎, 𝑏), if 𝐾(𝑎) − 𝐾(𝑏) > 0, 0, if 𝐾(𝑎) − 𝐾(𝑏) ≤ 0, (10) where 𝑎 = [𝑇 𝐿 𝑎 , 𝑇 𝑈 𝑎 ], [𝐼 𝐿 𝑎 , 𝐼 𝑈 𝑎 ], [𝐹 𝐿 𝑎 , 𝐹 𝑈 𝑎 ], 𝑏 = [𝑇 𝐿 𝑏 , 𝑇 𝑈 𝑏 ], [𝐼 𝐿 𝑏 , 𝐼 𝑈 𝑏 ], [𝐹 𝐿 𝑏 , 𝐹 𝑈 𝑏 ] . 3.2 interval valued neutrosophic edas method this section presents the steps of the proposed ivn edas method. the proposed method addresses a group decision making problem that is under the assumption that there are the set of n alternatives 𝑋 = {𝑋1, 𝑋2, … , 𝑋𝑖 , … , 𝑋𝑛} evaluated based on 𝑚 criteria 𝐶 = {𝐶1, 𝐶2, … , 𝐶𝑗 , … , 𝐶𝑚} by k dms 𝐷 = {𝐷1 , 𝐷2, … , 𝐷𝑘 , … , 𝐷𝐾}. in the rest of this section, the steps of the proposed ivn edas method are presented. 442 a.y. gul, e. cakmak, a. e. karakas step 1: the linguistic evaluations of each expert are collected regarding the alternatives’ performance with respect to the criteria. these linguistic evaluations are converted to ivnns by using the ivn scale given in table 1 to obtain the ivn matrix of each expert as presented in table 2, where 𝑥𝑖𝑗𝑘 = 〈[𝑇 𝐿 𝑖𝑗𝑘 , 𝑇 𝑈 𝑖𝑗𝑘 ] , [𝐼 𝐿 𝑖𝑗𝑘 , 𝐼 𝑈 𝑖𝑗𝑘 ] [𝐹 𝐿 𝑖𝑗𝑘 , 𝐹 𝑈 𝑖𝑗𝑘 ]〉 is an ivn evaluation of the expert k on the performance of alternative i with respect to criterion j. based on the knowledge and competency, the experts are assessed and assigned an importance weight 𝑦𝑘 . table 1 ivn scale for alternative evaluation [45] linguistic terms 〈𝑻, 𝑰, 𝑭〉 cl certainly low ⟨[0.05, 0.2], [0.6, 0.7], [0.75, 0.9]⟩ vl very low ⟨[0.15, 0.3], [0.5, 0.6], [0.65, 0.8]⟩ l low ⟨[0.25, 0.4], [0.4, 0.5], [0.55, 0.7]⟩ ba below average ⟨[0.35, 0.5], [0.3, 0.4], [0.45, 0.6]⟩ a average ⟨[0.40, 0.6], [0.1, 0.2], [0.40, 0.6]⟩ aa above average ⟨[0.45, 0.6], [0.3, 0.4], [0.35, 0.5]⟩ h high ⟨[0.55, 0.7], [0.4, 0.5], [0.25, 0.4]⟩ vh very high ⟨[0.65, 0.8], [0.5, 0.6], [0.15, 0.3]⟩ ch certainly high ⟨[0.75, 0.9], [0.6, 0.7], [0.05, 0.2]⟩ table 2 ivn decision matrix of an expert 𝑘 crit. al1 aln 𝐶1 [𝑇11𝑘 𝐿 , 𝑇11k 𝑈 ], [𝐼11k 𝐿 , 𝐼11k 𝑈 ], [𝐹11k 𝐿 , 𝐹11k 𝑈 ] [𝑇𝑛1𝑘 𝐿 , 𝑇n1k 𝑈 ], [𝐼n1k 𝐿 , 𝐼n1k 𝑈 ], [𝐹n1k 𝐿 , 𝐹n1k 𝑈 ] ⋮ ⋮ ⋮ 𝐶𝑚 [𝑇1𝑚𝑘 𝐿 , 𝑇1mk 𝑈 ], [𝐼1mk 𝐿 , 𝐼1mk 𝑈 ], [𝐹1mk 𝐿 , 𝐹1mk 𝑈 ] [𝑇𝑛𝑚𝑘 𝐿 , 𝑇nmk 𝑈 ], [𝐼nmk 𝐿 , 𝐼nmk 𝑈 ], [𝐹nmk 𝐿 , 𝐹nmk 𝑈 ] step 2: to obtain the aggregated ivn decision matrix, the ivn decision matrix of the experts are aggregated by using the innwa operator defined in eq. (7). table 3 shows the representation of the aggregated ivn decision matrix 𝑥𝐴, where 𝑥𝐴 = [𝑥𝑖𝑗𝐴] 𝑛∗𝑚 = 〈[𝑇 𝐿 𝑖𝑗𝐴 , 𝑇 𝑈 𝑖𝑗𝐴 ] , [𝐼 𝐿 𝑖𝑗𝐴 , 𝐼 𝑈 𝑖𝑗𝐴 ] [𝐹 𝐿 𝑖𝑗𝐴 , 𝐹 𝑈 𝑖𝑗𝐴 ]〉 represents the aggregated evaluation of the experts for the alternative i with respect to criterion 𝑗. step 3: each expert 𝑘 evaluates the criteria importance with linguistic terms. these evaluations are converted to ivnns using the scale given in table 4. the following criteria weights matrix is obtained for each expert 𝑘: 𝑊 = [𝑤𝑗𝑘] 1∗𝑚 (11) table 3 aggregated ivn decision matrix crit. al1 aln 𝐶1 [𝑇11𝐴 𝐿 , 𝑇11a 𝑈 ], [𝐼11a 𝐿 , 𝐼11a 𝑈 ], [𝐹11a 𝐿 , 𝐹11a 𝑈 ] [𝑇𝑛1𝐴 𝐿 , 𝑇n1a 𝑈 ], [𝐼n1a 𝐿 , 𝐼n1a 𝑈 ], [𝐹n1a 𝐿 , 𝐹n1a 𝑈 ] ⋮ ⋮ ⋮ 𝐶𝑚 [𝑇1𝑚𝐴 𝐿 , 𝑇1ma 𝑈 ], [𝐼1ma 𝐿 , 𝐼1ma 𝑈 ], [𝐹1ma 𝐿 , 𝐹1ma 𝑈 ] [𝑇𝑛𝑚𝐴 𝐿 , 𝑇nma 𝑈 ], [𝐼nma 𝐿 , 𝐼nma 𝑈 ], [𝐹nma 𝐿 , 𝐹nma 𝑈 ] drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 443 step 4: the criteria weights are aggregated by innwa in eq. (7) and normalized to obtain the aggregated criteria weights matrix 𝑊𝐴 given in eq. (12), where 𝑤𝑗 represents the aggregated weight of the criterion 𝑗. 𝑊𝐴 = [𝑤𝑗] 1∗𝑚 (12) step 5: the average solution matrix 𝐴𝑉 is obtained by taking the average of the aggregated alternative evaluations 𝑥𝑖𝑗𝐴 as given in eq. (13): 𝐴𝑉 = [𝐴𝑉𝑗] 1∗𝑚 = 1 𝑛 ∑ 𝑥𝑖𝑗𝐴 𝑛 𝑖=1 (13) table 4 ivn scale for criteria evaluation [45] linguistic terms 〈𝑻, 𝑰, 𝑭〉 cli certainly low importance ⟨[0.05, 0.25], [0.6, 0.7], [0.75, 0.95]⟩ vli very low importance ⟨[0.15, 0.35], [0.5, 0.6], [0.65, 0.85]⟩ li low importance ⟨[0.25, 0.45], [0.4, 0.5], [0.55, 0.75]⟩ bai below average importance ⟨[0.35, 0.55], [0.3, 0.4], [0.45, 0.65]⟩ ai average importance ⟨[0.40, 0.60], [0.1, 0.2], [0.40, 0.60]⟩ aa above average importance ⟨[0.45, 0.65], [0.3, 0.4], [0.35, 0.55]⟩ hi high importance ⟨[0.55, 0.75], [0.4, 0.5], [0.25, 0.45]⟩ vhi very high importance ⟨[0.65, 0.85], [0.5, 0.6], [0.15, 0.35]⟩ chi certainly high importance ⟨[0.75, 0.95], [0.6, 0.7], [0.05, 0.25]⟩ step 6: each alternative is classified as either positively positioned or negatively positioned to the average solution 𝐴𝑉𝑗 with respect to each criterion. positively positioned alternatives signify a superior performance than the average solution in terms of a criterion and contrarily, the negatively positioned alternatives indicate a worse than the average performance. the positive distance to average (pda) of a positively positioned alternative must be positive and negative distance to average (nda) must be zero. contrarily, for the negatively positioned alternatives pda must be zero and nda must be positive. the calculation of the positive and negative distance to the average matrices (pda and nda) are modified by eliminating the neutrosophic subtraction to preserve the characteristics of ivnns and to avoid negative numbers in the ivn parameters. the pda and nda are calculated as defined in eqs. (14) and (15) based on the criterion type, respectively. the euclidian distance is calculated if the difference between the deneutrosophicated aggregated alternative performance 𝐾(𝑥𝑖𝑗𝐴) and the deneutrosophicated average solution 𝐴𝑉𝑗 is positive, otherwise it assigns zero. 𝑃𝐷𝐴 = [𝑝𝑑𝑎𝑖𝑗] 𝑚∗𝑛 = { 𝑧 (𝐾(𝑥𝑖𝑗𝐴) − 𝐾(𝐴𝑉𝑗)) if 𝑗 ∈ 𝐵, 𝑧 (𝐾(𝐴𝑉𝑗) − 𝐾(𝑥𝑖𝑗𝐴)) if 𝑗 ∈ 𝐶, (14) 𝑁𝐷𝐴 = [𝑛𝑑𝑎𝑖𝑗] 𝑚∗𝑛 = { 𝑧 (𝐾(𝐴𝑉𝑗) − 𝐾(𝑥𝑖𝑗𝐴)) if 𝑗 ∈ 𝐵, 𝑧 (𝐾(𝑥𝑖𝑗𝐴) − 𝐾(𝐴𝑉𝑗)) if 𝑗 ∈ 𝐶, (15) 444 a.y. gul, e. cakmak, a. e. karakas where 𝐵 and 𝐶 denote the set of the benefit and cost criteria, respectively and 𝑧(. ) is a function defined in eq. (10). step 7: the weighted total positive distance and negative distance from the average 𝑠𝑝𝑖 and 𝑠𝑛𝑖 are calculated for each alternative 𝑖 by using eqs. (16) and (17): 𝑠𝑝𝑖 = ∑ (𝑤𝑗 ∗ 𝑝𝑑𝑎𝑖𝑗),𝑚 𝑗=1 (16) 𝑠𝑛𝑖 = ∑ (𝑤𝑗 ∗ 𝑛𝑑𝑎𝑖𝑗).𝑚 𝑗=1 (17) step 8: the weighted total positive distance and negative distance from the average values are normalized as given in eqs. (18) and (19): 𝑛𝑠𝑝𝑖 = 𝑠𝑝𝑖 𝑀𝑎𝑥(𝑠𝑝𝑖) , (18) 𝑛𝑠𝑛𝑖 = 1 − 𝑠𝑛𝑖 𝑀𝑎𝑥(𝑠𝑛𝑖) . (19) step 9: to obtain the appraisal score 𝑎𝑠𝑖 , the average of the normalized positive distance and normalized negative distance is computed as follows: 𝑎𝑠𝑖 = 1 2 (𝑛𝑠𝑝𝑖 + 𝑛𝑠𝑛i). (20) step 10: the alternatives are ranked in descending order of the appraisal score. 4. case study the methodology of this study consists of three phases, namely the preparation process, the application of the ivn edas method and the validation of results. figure 1 illustrates the flowchart of the presented methodology. this section presents the steps of the methodology implemented for the case study. 4.1. problem definition the paper aims to determine the most appropriate drone for monitoring and detection of forest fires. the drone alternatives should be evaluated with respect to multiple and conflicting criteria. although the drone characteristics and performances based on the criteria are known as its own specifications, the actual performance of a drone under mission highly depends on the environmental circumstances. this aspect brings the need for the use of fuzzy logic. the criteria were established through research of articles and books, as well as interviews with experts in the field. the authors and the experts omitted the sub-criteria that have similar or the exact opposite meaning in order to avoid overlapping in the criteria. for instance, the charging time and the battery life have rather opposing denotations as one refers to the time during which a drone cannot operate while the other signifies its up-time. as a result, 14 sub-criteria have been identified in total and presented in table 6. after determining the criteria and sub-criteria, a survey was conducted among expert dms in the field to weigh the criteria and alternatives. all dms were given different weights according to their expertise in the process of selecting a drone for detecting forest fires. the importance weights of the experts are assigned based on their experience by the authors. regarding the type of drones that have been selected for the evaluation in this study, the drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 445 rotorcraft drones are examined. it has been found that rotorcraft (multirotor) drones tend to be better off in terms of maneuverability, hovering capability, and looser requirements on takeoff and landing than fixed-wing drones [71]. therefore, four different drone models of multirotor type selected as potential alternatives for detecting forest fires, and their technical specifications are presented in table 7. the information on the experts and their weights are given as follows: fig. 1 flowchart of the methodology dm1 (40%): the first expert is a retired f16 pilot who provides drone training. he has been providing such training for about six years. he has a comprehensive knowledge on the aviation regulations, safety standards, and 10+ years of flight experience with drones. dm2 (25%): the second expert is a fire chief. although not fully experienced with drones, he had first-hand experience with forest fires for over 15 years. thus, he has an in-depth knowledge of emergency response, safety protocols and fire behavior, especially how fires spread and the factors impacting the fire behavior. he mainly contributed by assessing the importance of drone attributes. dm3 (35%): the third expert is a commercial drone user with an experience over 10 years. he has an in-depth knowledge of the technical features such as drone models, sensors, cameras, payloads, its capabilities, and the areas of drone usage. further, he has an extensive knowledge on the sensor operation. 446 a.y. gul, e. cakmak, a. e. karakas the description of each criterion is given as follows: max. takeoff weight (c1.1) (benefit): it represents the maximum weight of a drone at which it is allowed to take off. in addition to its main body, for the needs of specific usage drones can be equipped with sensors, cameras, communication tools, and components such as fire extinction equipment. max. horizontal speed (c1.2) (benefit): it expresses the maximum speed at which a drone can fly in the horizontal plane. max. hover time (c1.3) (benefit): it is the maximum time during which a drone hovers and holds its position in the air. max. flight time (c1.4) (benefit): it is the criterion that shows the total flight time of a drone during a single takeoff and landing. charging time (c1.5) (benefit): it indicates how long it takes a fully discharged battery of a drone to charge completely. max. operating altitude (c1.6) (benefit): it is the maximum altitude at which a drone can operate above sea level. table 6 criteria list for the drone selection for forest fires criteria sub-criteria references c1. technical abilities c1.1. max. takeoff weight [71, 72, 73, 74, 75] c1.2. max. horizontal speed [28, 59, 60, 73, 74, 76] c1.3. max. hover time [71] c1.4. max. flight time [59, 71, 73, 77] c1.5. charging time [28, 77] c1.6. max. operating altitude [59, 60, 72, 73, 76, 78] c1.7. wind resistance [79, 80] c1.8. ingress protection rating of drones [77, 79] c1.9. operation frequency [74, 81, 82] c1.10. hovering accuracy [77] c2. visionbased technologies c2.1. zoom camera [72, 77, 78, 83] c2.2. resolution [77, 78, 84] c2.3.thermal camera accuracy [19, 77, 83, 85] c2.4. obstacle sensors (range) [28, 77, 86] wind resistance (c1.7) (benefit): the wind resistance of a drone can be affected by its weight and size. the wind resistance level is given as a value between 0 and 12. the greater the wind resistance level, the more stable can a drone operate in high-wind circumstances [80]. ingress protection rating of drones (c1.8) (benefit): this criterion gives the ip value of a drone. ip code (also known as international protection rating) categorizes the degrees of protection given in electrical enclosures against solid object infiltration, dust, accidental touch, and liquids. the ip code consists of two digits. the first digit ranges from 1 to 6 and signifies the protection against solids, whereas the second digit ranges from 1 to 8 and indicates the resistance against liquids. operation frequency (c1.9) (benefit): it represents the frequency band that a drone uses for communication and data download. drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 447 table 7 technical specifications of drone alternatives drone alternative matrice 30 matrice 300 rtk matrice 210 rtk phantom 4 rtk max. takeoff weight (g) (c1.1) 238 930 1230 max. hor. speed (m/s) (c1.2) 23 23 20 16 max. hover time (min) (c1.3) 36 50 24 26 max. flight time (min) (c1.4) 41 55 33 30 charging time (min) (c1.5) 30– 50 70 70 30 max. operating altitude (m) (c1.6) 7000 5000 3000 6000 wind resistance (m/s) (c1.7) 15 15 12 10 ingress protection (c1.8) ip55 ip45 ip43 ip42 operation frequency (c1.9) 2.4 ghz or 5 ghz 2.4 ghz or 5 ghz 2.4 2.48 ghz; 5.7 5.8 ghz 2.4 ghz or 5 ghz hovering accuracy (c1.10) v: 0.1m h: 0.3m v: 0.1m h: 0.1m v: 0.1m h: 0.1m v: 0.1m h: 0.5m zoom camera (c2.1) 5 m 2 m 1 m 1 m resolution (c2.2) 3840x2160 1920x1080 1920x1080 3840x2160 thermal camera accuracy (c2.3) ±2°cor ±2% ±5°cor ±5% ±7°cor ±7% obstacle sensors (c2.4) f: 0.6-38m u, d, b, s: 0.5-33m f/b/l/r: 0.7-40m u/d: 0.6-30m f: 0.7-40 m u, d, b, s: 0.7-40 m 0.7-30 m hovering accuracy (c1.10) (benefit): it indicates how stable a drone stays while hovering. this criterion is given in meters (m) by how much a drone deviates from its initial position in vertical and horizontal directions. zoom camera (c2.1) (benefit): this criterion represents how far the embedded visual camera of a drone can take a clear image. resolution (c2.2) (benefit): each visual camera has a resolution value. the resolution is one of the factors that directly affect image quality. thermal camera accuracy (c2.3) (benefit): thermal cameras do not need any light source for imaging. in the context of forest fires, thermal camera imaging is especially important for early detection. obstacle sensors (range) (c2.4) (benefit): it indicates the furthest distance from which a drone can detect the obstacles. although the cost of a drone has been generally included as a criterion in the studies dealing with the drone selection problem for both military and civilian purposes, in this study, it has been excluded due to two reasons – first, the economic aspect must be of minor importance in disaster management as it is critical to human life and a cause of great economic and environmental losses. second, governments do not make public procurement decisions solely based on the economic perspective, instead sustainable and strategic options are preferred. 448 a.y. gul, e. cakmak, a. e. karakas as mentioned previously, there are interrelationships between the technical features of a drone as well as some drone attributes are highly affected by certain environmental circumstances. to give a few examples of the tradeoffs between technical internal attributes of a drone, the relationship between energy consumption and flight speed is well known. in a lower speed range, the speed does not affect the power consumption significantly. yet in a relatively higher speed range, the power consumption increases exponentially [29]. increased power consumption during operation means a drone requires to get charged more frequently and higher downtime which lowers the overall system efficiency. likewise, the payload of a drone has a significant effect on power consumption. increasing the payload of a drone rises the power consumption significantly and lowers the maximum flight time of a drone [43]. besides the interrelation among internal features of drones, their performance is also highly affected by environmental circumstances such as temperature, wind, and precipitation. drones with relatively small sizes and weights are prone to wind disturbances [87]. further, it has been found that high temperature has little effect on the flight performance of a drone, but it significantly shortens the lifetime and even damages the battery, whereas extremely low temperature worsens the flight and the battery performance [88]. it can be concluded that there is a highly complex structure (1) between the internal attributes of a drone, (2) between the environmental conditions and the internal attributes of a drone. as a result, it is hard to describe and quantify the model with clear and exact terms and relations. we conclude that the discussed problem involves uncertainties such as doubtfulness or vagueness, ambiguity, and inconsistency among its elements. the experts submit their subjective opinions based own personal experiences, background, and other individual characteristics. further, the conflicting and complex relationships between the criteria, which were explained above, contribute to the ambiguous nature of the problem. to represent the vagueness in the decision-making process and the conflicting criteria in the problem, the ivn edas method is preferred. 4.2. numerical application the application of the method is illustrated by the evaluation of four potential drone models, that are identified as suitable for forest surveillance and fire detection. the drone models are assessed based on the pre-determined 14 criteria by the three dms. the steps of the methodology are given with the obtained results as follows: step 1: in the first step, the linguistic evaluations of the experts are collected (table 8) and the ivn decision matrices are constructed by using the ivn scale in table 1. step 2: the ivn evaluations of the alternatives are aggregated using eq. (7). table 9 presents the aggregated ivn matrix for alternative 1. steps 3-4: the linguistic evaluations for criteria weights are collected, which are converted to ivnns by using table 4. the ivn matrices of the experts are then aggregated by using innwa operator. the linguistic evaluations and the average ivn weights of the criteria are presented in table 10. step 5: the average solution matrix is calculated by eq. (13). table 11 shows the average solution matrix. drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 449 table 8 linguistic evaluations of the experts dm1 dm2 dm3 criteria al1 al2 al3 al4 al1 al2 al3 al4 al1 al2 al3 al4 c1.1 h a aa l h ba aa l l ch aa ba c1.2 h h aa ch h h aa ch ch h aa ch c1.3 ch ch h aa ch h aa a h ch a a c1.4 ch h aa a ch h aa a h ch a a c1.5 h aa a a h aa a a h l ba ch c1.6 aa ch a h aa ch ba h h a l aa c1.7 aa aa aa a aa aa a ba a a ba l c1.8 aa ch a ba aa ch a ba h a a ba c1.9 aa h aa aa aa aa aa aa h h h h c1.10 aa h aa aa aa h a a aa a a a c2.1 ch h aa a h h aa a aa h a a c2.2 h ch h aa h ch h aa ch a ba ba c2.3 h ch a ba aa ch a ba h aa aa ba c2.4 h aa ba a h aa h a h h h a step 6: the pda and nda matrices are obtained and presented in table 12. the calculation of pda and nda values for alternative 1 with respect to criterion 3, which is a benefit criterion, is as follows: 𝑝𝑑𝑎13 = 𝑧(𝐾(𝑥13𝐴) − 𝐾(𝐴𝑉3)), 𝑛𝑑𝑎13 = 𝑧(𝐾(𝐴𝑉3) − 𝐾(𝑥13𝐴)), 𝐾(𝑥13𝐴) = 0.9120 ; 𝐾(𝐴𝑉3) = 0,7474; 𝑝𝑑𝑎13 = 𝑧(0.1646) = 𝑑𝐸 , (𝑥13𝐴, 𝐴𝑉3) = 0.1344; 𝑛𝑑𝑎13 = 𝑧(𝐾(𝐴𝑉3) − 𝐾(𝑥13𝐴)) = 0. table 9 aggregated ivn decision matrix of alternative 1 [(𝑻𝑳. 𝑻𝑼). (𝑰𝑳. 𝑰𝑼). (𝑭𝑳. 𝑭𝑼)] c1.1 [(0.462,0.618), (0.400,0.500), (0.329,0.487)] c1.2 [(0.634,0.796), (0.461,0.562), (0.142,0.314)] c1.3 [(0.693,0.853), (0.521,0.622), (0.088,0.255)] c1.4 [(0.693,0.853), (0.521,0.622), (0.088,0.255)] c1.5 [(0.550,0.700), (0.400,0.500), (0.250,0.400)] c1.6 [(0.487,0.638), (0.332,0.432), (0.311,0.462)] c1.7 [(0.433,0.600), (0.204,0.314), (0.367,0.533)] c1.8 [(0.487,0.638), (0.332,0.432), (0.311,0.462)] c1.9 [(0.487,0.638), (0.332,0.432), (0.311,0.462)] c1.10 [(0.450,0.600), (0.300,0.400), (0.350,0.500)] c2.1 [(0.618,0.786), (0.425,0.529), (0.148,0.328)] c2.2 [(0.634,0.796), (0.461,0.562), (0.142,0.314)] c2.3 [(0.527,0.678), (0.372,0.473), (0.272,0.423)] c2.4 [(0.550,0.700), (0.400,0.500), (0.250,0.400)] 450 a.y. gul, e. cakmak, a. e. karakas table 10 linguistic criteria evaluations and aggregated ivn criteria weights criteria dm1 dm2 dm3 aggregated criteria weights [(𝑇𝐿 , 𝑇𝑈), (𝐼𝐿 , 𝐼𝑈), (𝐹𝐿 , 𝐹𝑈)] c1.1 max. takeoff weight vl ba ba [(0.276,0.479), (0.368,0.470), (0.521,0.724)] c1.2 max. horizontal speed h a a [(0.465,0.669), (0.174,0.289), (0.331,0.535)] c1.3 max. hover time l ba a [(0.331,0.532), (0.229,0.343), (0.468,0.669)] c1.4 max. flight time h h vh [(0.588,0.791), (0.432,0.533), (0.209,0.412)] c1.5 charging time vl vl l [(0.186,0.387), (0.462,0.563), (0.613,0.814)] c1.6 max. operating altitude l vh a [(0.427,0.644), (0.260,0.380), (0.356,0.573)] c1.7 wind resistance h aa aa [(0.492,0.694), (0.337,0.437), (0.306,0.508)] c1.8 ingress protection rating of drones vh vh h [(0.618,0.821), (0.462,0.563), (0.179,0.382)] c1.9 operation frequency a ba l [(0.338,0.539), (0.214,0.328), (0.461,0.662)] c1.10 hovering accuracy vl vl l [(0.186,0.387), (0.462,0.563), (0.613,0.814)] c2.1 zoom camera a aa h [(0.469,0.672), (0.214,0.328), (0.328,0.531)] c2.2 resolution aa a h [(0.476,0.678), (0.252,0.364), (0.322,0.524)] c2.3 th. camera accuracy vh vh vh [(0.650,0.850), (0.500,0.600), (0.150,0.350)] c2.4 obstacle sensors (range) ba a vl [(0.300,0.503), (0.273,0.388), (0.497,0.700)] table 11 the average solution matrix [(𝑻𝑳, 𝑻𝑼), (𝑰𝑳, 𝑰𝑼), (𝑭𝑳, 𝑭𝑼)] c1.1 [(0.437,0.599), (0.327,0.433), (0.348,0.515)] c1.2 [(0.596,0.749), (0.440,0.541), (0.198,0.353)] c1.3 [(0.575,0.741), (0.362,0.468), (0.216,0.384)] c1.4 [(0.540,0.712), (0.321,0.425), (0.249,0.425)] c1.5 [(0.470,0.640), (0.266,0.374), (0.318,0.493)] c1.6 [(0.501,0.664), (0.307,0.419), (0.289,0.455)] c1.7 [(0.402,0.570), (0.213,0.323), (0.397,0.564)] c1.8 [(0.474,0.644), (0.263,0.371), (0.316,0.489)] c1.9 [(0.497,0.648), (0.342,0.443), (0.301,0.453)] c1.10 [(0.448,0.617), (0.214,0.323), (0.351,0.519)] c2.1 [(0.500,0.672), (0.282,0.386), (0.291,0.465)] c2.2 [(0.550,0.711), (0.361,0.469), (0.234,0.400)] c2.3 [(0.491,0.654), (0.322,0.426), (0.301,0.465)] c2.4 [(0.479,0.643), (0.297,0.397), (0.319,0.483)] table 12 the positive distance to average and negative distance to average matrices pda nda criterion al1 al2 al3 al4 al1 al2 al3 al4 c1.1 0.0446 0.1129 0 0 0 0 0.0191 0.1291 c1.2 0.0392 0 0 0.1544 0 0.0459 0.1458 0 c1.3 0.1344 0.1523 0 0 0 0 0.1111 0.1756 c1.4 0.1718 0.1141 0 0 0 0 0.1121 0.1755 c1.5 0.0973 0 0 0.0947 0 0.0797 0.1022 0 c1.6 0 0.1399 0 0.0311 0.0191 0 0.1457 0 c1.7 0.0256 0.0256 0.0089 0 0 0 0 0.0519 c1.8 0.0397 0.1659 0 0 0 0 0.1174 0.1070 c1.9 0 0.0298 0 0 0.0099 0 0.0099 0.0099 c1.10 0.0483 0.0495 0 0 0 0 0.0414 0.0414 c2.1 0.1339 0.0776 0 0 0 0 0.0722 0.1372 c2.2 0.0903 0.0989 0 0 0 0 0.0543 0.1198 c2.3 0.0390 0.1765 0 0 0 0 0.1186 0.1192 c2.4 0.0828 0.0224 0.0351 0 0 0 0 0.1329 drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 451 steps 7-10: the pda and nda values are weighted by the criteria weights and summed over the criteria to obtain the positive and negative weighted total distance to the average 𝑠𝑝𝑖 and 𝑠𝑛𝑖, respectively. then they are normalized to calculate 𝑛𝑠𝑝𝑖and 𝑛𝑠𝑛𝑖. the appraisal score of each alternative is calculated and the alternatives are ranked in descending order of the appraisal score. table 13 weighted total distances, appraisal scores and final ranking of the alternatives nspi nsni asi rank al1 0.7458 0.9785 0.8621 2 al2 1 0.9191 0.9595 1 al3 0.0299 0.1176 0.0738 4 al4 0.1967 0 0.0983 3 the rankings presented in table 13 demonstrate that matrice 300 rtk (al2) has been evaluated as the most preferable drone with an appraisal score of 0.9595 and is closely followed by matrice 30 (al1) with an as of 0.8621. the minor difference of ranking scores between al1 and al2 should be interpreted as both drone models provide relatively comparable performances. the final ranking obtained is as follows: al2≻al1≻al4≻al3. 4.3. sensitivity analysis an extensive one-at-a-time sensitivity analysis is conducted to check for the robustness of the results to the change in the criteria weights. in total 84 sets of criteria weights are built following the approach in gorcun et al. [89]. in each scenario, the initial weight of a criterion is decreased by the predetermined modification degree 𝜐 (15%, 30%, 45%, 60%, 75%, 90%), and the weights of the rest of the criteria are increased equally while satisfying that the sum of the weights equals to 1. figure 2 illustrates the criteria weights used in the scenarios from sc-57 to sc-70 which are formed by a 75%-modification degree, and figure 3 shows the resulting rankings of the alternatives in the generated 84 scenarios. fig. 2 the sets of criteria weights at 75% modification of each criterion 452 a.y. gul, e. cakmak, a. e. karakas fig. 3 rankings of the alternatives generated through 84 scenarios the result of the sensitivity analysis demonstrates that approximately a minimum modification by 30% of criterion 1.2, which corresponds to scenarios sc-16, sc-30, sc-44, sc-58 and sc-72, results in the third and fourth alternatives to change ranking positions. the same result emerges under the minimum approximate modification by 45% of criterion 1.5 and criterion 1.6. concludingly, in 100% of the scenarios, alternative 2 and alternative 1 preserved the first and second ranking positions, respectively, and in 84.5% of the scenarios (71 out of 84 scenarios) alternative 4 has been ranked at the third and alternative 3 at the fourth position. the sensitivity analysis indicates that the results regarding the first two ranking positions are robust against the weight changes in all criteria. the latter two positions are slightly sensitive to the changes in c1.2, c1.5 and c1.6. 4.4. comparative analysis in order to check for the veracity of the method, a comparative analysis is conducted by using two mcdm methods, the interval valued neutrosophic topsis, and interval valued neutrosophic codas methods. we preferred to apply the distance-based methods for the comparison in accordance with the method of application in this paper. for the application of ivn topsis, we adopted the steps given in karasan et al. [90]. the criteria weights are taken as given in table 10. the euclidian and hamming distance measures used in this section are calculated by adopting the steps presented in the study of [91] and both equations are modified by the summation over the criteria and the division by the number of criteria in the model. in the applied ivn topsis method, first, the decision matrices are aggregated by weighting with the priorities of dms to obtain the aggregated neutrosophic decision matrix. then the weighted normalized decision matrix is computed by weighting the decision matrix with criteria weights. the positive ideal and the negative ideal solution are calculated as presented in table 14. based on the determined ideal solutions, the euclidian distance of each alternative to the pis and nis is computed. drone selection for forest surveillance and fire detection using interval valued neutrosophic edas... 453 as for the second method selected for the comparative analysis, the steps of ivn codas are adopted from the study of [95] with minor changes from their method. the threshold parameter (θ) used in the threshold function is taken as 0.005 since the differences between the euclidian distances of alternatives to the negative ideal solution are significantly close. to incorporate the difference of hamming distances into the calculation, we adjusted the value of the threshold parameter. table 15 shows the results obtained by ivn topsis, ivn codas and ivn edas methods. the ivn edas method gives the same ranking with the ivn topsis and ivn codas methods indicating that the proposed ivn edas method provides veracious and consistent results with other distance based mcdm methods. table 14 ivn positive ideal solution and negative ideal solution of criteria. pis nis c1.1 [(0.027, 0.037), (0.012, 0.018), (0.01, 0.02)] [(0.014, 0.022), (0.02, 0.025), (0.025, 0.033)] c1.2 [(0.074, 0.089), (0.03, 0.039), (0.005,0.02)] [(0.044, 0.059), (0.059, 0.069), (0.035, 0.049)] c1.3 [(0.05, 0.061), (0.011, 0.018), (0.005, 0.017)] [(0.029, 0.042), (0.038, 0.045), (0.026, 0.039)] c1.4 [(0.056, 0.069), (0.008, 0.016), (0.007, 0.021)] [(0.032, 0.048), (0.042, 0.05), (0.032, 0.048)] c1.5 [(0.018, 0.024), (0.005, 0.008), (0.006, 0.013)] [(0.012, 0.017), (0.013, 0.016), (0.013, 0.019)] c1.6 [(0.054, 0.069), (0.018, 0.027), (0.009, 0.024)] [(0.028, 0.042), (0.03, 0.038), (0.038, 0.052)] c1.7 [(0.036, 0.049), (0.017, 0.026), (0.03, 0.044)] [(0.028, 0.042), (0.019, 0.028), (0.038, 0.052)] c1.8 [(0.052, 0.066), (0.008, 0.016), (0.008, 0.023)] [(0.028, 0.04), (0.026, 0.036), (0.036, 0.048)] c1.9 [(0.038, 0.049), (0.024, 0.031), (0.02, 0.031)] [(0.035, 0.046), (0.027, 0.034), (0.023, 0.033)] c1.10 [(0.016, 0.021), (0.005, 0.008), (0.009, 0.015)] [(0.013, 0.019), (0.009, 0.013), (0.012, 0.018)] c2.1 [(0.058, 0.074), (0.009, 0.019), (0.014, 0.031)] [(0.038, 0.056), (0.04, 0.05), (0.038, 0.056)] c2.2 [(0.06, 0.076), (0.027, 0.036), (0.009, 0.027)] [(0.038, 0.051), (0.042, 0.051), (0.035, 0.048)] c2.3 [(0.051, 0.064), (0.011, 0.019), (0.008, 0.021)] [(0.027,0.038), (0.036, 0.044), (0.034, 0.046)] c2.4 [(0.033,0.043), (0.006, 0.012), (0.015,0.024)] [(0.024, 0.036), (0.024, 0.03), (0.024, 0.036)] 4.5 managerial implications the criteria significant for the forest surveillance and fire detection have been established for the first time. in this respect, a framework has been proposed that can be utilized by the authorities and experts that are responsible from the safety and security of forests against wildfires. on the long run, the integration of drones to the forest surveillance systems can significantly reduce the costs, is more reliable and effective than the other techniques such as satellite based and static wsns. therefore, the adoption of the drones as the state-of-the-art technology should be considered and requires a systematic examination with real-world applications. table 15 comparison of results obtained by ivn topsis, ivn codas and ivn edas method ivn topsis ivn codas ivn edas 𝐷𝑖 𝑃𝐼𝑆 𝐷𝑖 𝑁𝐼𝑆 closeness coefficient rank relative assessment score rank appraisal score rank al1 0.0103 0.0105 0.0105 2 0.0936 2 0.8621 2 al2 0.0084 0.0124 0.0124 1 0.1107 1 0.9595 1 al3 0.0122 0.0088 0.0088 4 -0.1185 4 0.0738 4 al4 0.0128 0.0093 0.0093 3 -0.0858 3 0.0983 3 454 a.y. gul, e. cakmak, a. e. karakas 5. conclusion in order to deal with forest fires, which is a major cause of deforestation, early warning systems play a crucial role to contain wildfires under control timely. drones offer superior maneuverability and real-time surveillance capabilities in areas hard to access in comparison to other technologies. due to the aforementioned benefits, the application of drones is utilized in the early detection systems for forest fires and the type of drones has a significant impact on the effectiveness of these systems. it is noteworthy that the performance of the detection system depends on various factors such as topological characteristics of the terrain, environmental conditions and features of drones [7]. these numerous factors have interrelationships, and some may be conflicting with one another, which can be taken into account through edas method. this study is aimed at addressing the drone selection problem for forest surveillance and fire detection by using the ivn edas method thanks to its convenience at handling the conflicting criteria and enriched information representation by the ivn sets. this study contributes by pointing out to the gap that the drone selection problem for forest surveillance and fire detection has been sparsely addressed, by presenting an extensive literature review on the technologies deployed in forest monitoring and fire detection, and by extracting the relevant criteria through an extensive literature review and interviews with the experts in field. as a result, four drone alternatives and 14 criteria have been identified. the alternatives are then evaluated based on the proposed framework using ivn edas method. moreover, a sensitivity analysis is conducted in order to check for robustness by varying the criteria weights. the veracity of the results is validated by carrying out a comparative analysis with ivn topsis method and ivn codas method. the results are in compliance with the study of pamučar et al. [40] regarding the relative importance of the criteria. the criteria related to the diagnosis and monitoring are assessed as the most critical features in both studies. further, the ability of stabilization during the flight is evaluated as the second most important in this study. the flight time and the resolution for monitoring and detection are evaluated as the third and fourth most significant criteria. it is important to note that the results should be interpreted with caution. the mcdm methods are associated with a major limitation called the rank reversal phenomenon [65]. however, the methodology proposed in this study has not been examined for this limitation and should be checked for in future work. based on the findings, the set of criteria and the results should be verified with in-the-field applications and compared with the state of knowledge. in future studies, the performance of fuzzy and neutrosophic mcdm methods should be evaluated by comparing them with the software 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of niš, serbia | creative commons license: cc by-nc-nd original scientific paper an integrated swot – fuzzy piprecia model for analysis of competitiveness in order to improve logistics performances irena đalić 1 , jovo ateljević 2 , željko stević 1 , svetlana terzić 1 1 university of east sarajevo, faculty of transport and traffic engineering, bosnia and herzegovina 2 university of banja luka, faculty of economics, bosnia and herzegovina abstract. on the question: how to react in a particular situation, the management of the company must have a quick answer. in the time of fast and huge changes in production, the management must know what resources are available in the company and what kind of environment it faces. to respond promptly to the requirements of the environment, the company must define a clear strategy for its business. to define a strategy, management must know the state of the company. from these reasons, in this research it was conducted swot analysis of specific company, and after that the elements of the swot matrix were ranked using fuzzy piprecia method. this ranking shows on which element company should pay the most attention. key words: transport company, environment, swot, fuzzy piprecia 1. introduction in order to achieve the set goals of the company, it is necessary to establish a correlation between internal and external factors. management must have a relationship with the environment that allows that the company define the right goals, operate in accordance with opportunities and respond to new opportunities and dangers that occur in the environment. therefore, management must define the business strategy. based on the strategy, management decides what and how many resources need to be engaged in order to achieve the goals in the best possible way. in this research, a swot analysis of a particular company was done. based on this analysis, obtained data showing the current received march 25, 2020 / accepted july 25, 2020 corresponding author: irena đalić, university of east sarajevo, faculty of transport and traffic engineering, vojvode mišića 52, 74000 doboj, republic of srpska, bosnia and herzegovina e-mail: i.naric@yahoo.com, irena.djalic@sf.ues.rs.ba 440 i. đaliĉ, j. ateljević, ţ. stević, s. terzić state of the company. the strengths and weaknesses of the interior were examined, as well as the opportunities and threats that came from the enterprise environment. the most often used method of analyzing the state of the company is swot analysis, which can be seen from the numerous literature that deals with this topic. rothaermel [1] states in his book that swot analysis is used for strategic decisions making. swot analysis is widely used in different companies and for different decisions making in companies. a large number of researchers from different fields use swot analysis. novikov [2] formulated a standard technological model of comprehensive strategic analysis and outlined the methodology for its systematic implementation in practice. zivkovic et al. [3] demonstrated a process for quantitative swot analysis that can be performed even when there is dependence among strategic factors. bohari et al. [4] analyzed the competitiveness of the halal food business in malaysia using the ict-aided swot analysis techniques. düking et al. [5] have performed a short swot analysis of virtual reality systems for athletes. swot analysis is often used for strategic planning and strategic decision making. kolbina [6] use swot analysis as a strategic planning tool for companies in the food industry. shi [7] use swot analysis to estimate competing outlooks for energy mix in the association of southeast asian nations (asean). mondal and haque [8] map out a way to sustainable growth of the tourism industry in bangladesh using the swot analysis. gupta and mishra [9] demonstrate a swot analysis for different 19 frameworks of rcm to make a strategic decision for implementing rcm in different organizations. štěrbová et al. use swot analysis to determine the innovation strategy of contractor firms in the slovak forestry service sector in the area of further innovation activities development [10]. therefore, we can observe swot analysis as a tool that helps management of the company to make decisions. kuo et al. [11] performed smart swot strategic planning analysis in the hospitality industry. valverde et al. [12], yan et al. [13] and jasiulewiczkaczmarek [14] performed swot analysis as a method that helps management to make decisions. comino and ferretti [15] using swot analysis for strategic planning. swot analysis is used in combination with other methods to enable more precisely decision making. there is a great number of research where we can see the combination of certain methods with swot analysis. abdel-basset et al. [16] formed a model for strategic planning and decision-making combining ahp and swot methods. korableva and kalimullina [17] formed bsc-swot matrix and applied it to the optimization of organization. akhavan et al. [18] use combination of fqspm-swot for strategic alliance planning and partner selection in a holding car manufacturer company. wang et al. [19] used analysis based on anp and swot for strategic choices of china’s new energy vehicle industry. bartusková and kresta [20] using combination of ahp method and swot analysis in external strategic analysis of the selected organization. zhao et al. [21] use anp-swot approach to find a way of rare earth industry of china. pazouki et al. [22] analyze strategic management in urban environment using swot and qspm model. hatefi [23] using an integrated swot and fuzzy copras approach for strategic planning of urban transportation system. fuzzy piprecia is a recently developed method [24]. it is used in multi-criteria decisionmaking. marković et al. [25] developed a novel integrated model that involves the application of a subjective-objective model in order to achieve business sustainability and excellence. đalić et al. [26] presented a novel integrated fuzzy – rough multi-criteria decision-making (mcdm) model based on integration fuzzy and interval rough set theory. veskovićet al. [27] determined criteria significance in selecting reach stackers by applying the fuzzy piprecia method. tomašević et al. [28], vesković et al. [29] and stanković et al. [30] using fuzzy piprecia to evaluate and select criteria for decision making. an integrated swot – fuzzy piprecia model for analysis of competitiveness in order to improve... 441 after the introduction, a review of the literature was carried out, that is, research of the current papers in this field. the third part of the paper deals with the results of fuzzy piprecia method. this method defines the most important element of swot matrix and based on this management can decide on which element should pay the most attention. the last part of the paper is about concluding considerations with directions for future research. the aims of the paper are: perform a detailed analysis of the current business of the company, define corrective measures and formulate strategies for the future operations of the company. based on the results of the research, the management of the company will be able to decide what is the next best step in the business. besides, this study is the first which introducing the integration of swot analysis and fuzzy piprecia method. this is one of the main novelty of the paper. 2. methods the decision-making methodology used in this paper consists of five phases (fig. 1). in the first phase, data were collected from a particular company. at this stage, the current state of the enterprise is defined. based on the current situation, a swot matrix was formed where internal strengths and weaknesses of the enterprise were defined, as well as opportunities and threats from the external environment of the enterprise. after this, the second phase is approached, where, based on the fuzzy piprecia method, elements of the swot matrix are ranked. based on this rank the most important element has been determined. fig. 1 the decision-making methodology the explanations and steps of all the methods used in this research are presented below. analysis of the current situation in the company defining swot matrix threats strengths fuzzy piprecia: ranking elements of swot matrix weaknesses opportunities determination of the most important element 442 i. đaliĉ, j. ateljević, ţ. stević, s. terzić 2.1. operation with fuzzy numbers fuzzy number a moves at interval ( ) a x : r→[0,1] : ( ) 0 a x l l x m m l u x x m x u u m otherwise              (1) in the eq. (1), l and u are the lower and upper bounds of fuzzy number a , and m is the modal value for a . tfn (triangle fuzzy number) can be denoted by ( , , )a l m u . operations tfn 1 1 1( , , )a l m u and 2 2 2( , , )a l m u are shown as the following equations [27, 31, 32]: adding: 1 2 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )a a l m u l m u l l m m u u       (2) multiplication: 1 2 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )a a l m u l m u l l m m u u       (3) subtraction: 1 2 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )a a l m u l m u l u m m u l       (4) fraction: 1 1 1 1 1 1 1 2 2 2 2 2 22 ( , , ) , , ( , , ) a l m u l m u l m u u m la         (5) reciprocally: 1 1 1 1 1 1 1 1 1 1 1 1 ( , , ) , ,a l m u u m l           (6) 2.2. fuzzy pivot pairwise relative criteria importance assessment – fuzzy piprecia metoda the main advantage of the piprecia [33] method is that it allows criteria to be evaluated without sorting criteria by significance first, which is not the case with the swara method [34]. today, most multi-criteria decision-making problems are solved by applying group decision-making. in such cases, especially as the number of decisionmakers involved in the fuzzy piprecia model increases, achieves its benefits. the fuzzy piprecia method was developed by [24]. it consists of 11 steps shown below. step 1. forming the required benchmarking set of criteria and forming a team of decision-makers. sorting the criteria according to marks from the first to the last, which an integrated swot – fuzzy piprecia model for analysis of competitiveness in order to improve... 443 means they need to be sorted unclassified. therefore, in this step, their significance is irrelevant. step 2. in order to determine the relative importance of criteria, each decision-maker individually evaluates the pre-sorted criteria by starting from the second criterion, eq. (7). 1 1 1 1 1 1 j j r j j j j j if c c s if c c if c c             (7) r js denotes the evaluation of the criteria by a decision-maker r. in order to obtain a matrix js it is necessary to perform the averaging of matrix r js using a geometric mean. decisionmakers evaluate criteria by applying defined scales in tables 1 and 2. the second and third steps of the developed method are in close dependence and new fuzzy scales are defined to meet the second and third step of the fuzzy piprecia method. if it is taken into account that the nature of fuzzy number operations and the fact that, in the third step, the values js are subtracted from the number two, it is required to define these scales. it is important to note that by defining these scales, the appearance of number two is avoided, which in the case of calculation could cause difficulties and wrong results. therefore, no other fuzzy scales could be used that have been previously developed, but only the scales defined in this paper. table 1 scale 1-2 for the assessment of criteria scale 1-2 l m u dfv almost equal value 1 1.000 1.000 1.050 1.008 slightly more significant 2 1.100 1.150 1.200 1.150 moderately more significant 3 1.200 1.300 1.350 1.292 more significant 4 1.300 1.450 1.500 1.433 much more significant 5 1.400 1.600 1.650 1.575 dominantly more significant 6 1.500 1.750 1.800 1.717 absolutely more significant 7 1.600 1.900 1.950 1.858 when the criterion is of greater importance in relation to the previous one, assessment is made using the above scale in table 2. in order to make decision-makers easier to evaluate the criteria, the table shows the defuzzified value (dfv) for each comparison. table 2 scale 0-1 for the assessment of criteria scale 0-1 l m u dfv 0.667 1.000 1.000 0.944 weakly less significant 0.500 0.667 1.000 0.694 moderately less significant 0.400 0.500 0.667 0.511 less significant 0.333 0.400 0.500 0.406 really less significant 0.286 0.333 0.400 0.337 much less significant 0.250 0.286 0.333 0.288 dominantly less significant 0.222 0.250 0.286 0.251 absolutely less significant 444 i. đaliĉ, j. ateljević, ţ. stević, s. terzić when the criterion is of less importance compared to the previous one, assessment is made using the above-mentioned scale in table 2. step 3. determining the coefficient jk 1 1 2 1j j if j k s if j       (8) step 4. determining the fuzzy weight jq 1 1 1 1 jj j if j qq if j k          (9) step 5. determining the relative weight of the criterion jw 1 j j n j j q w q    (10) in the following steps, it is necessary to apply the inverse methodology of the fuzzy piprecia method. step 6. evaluation of the applying scale defined above, but this time starting from a penultimate criterion. 1 1 1 1 ' 1 1 j j r j j j j j if c c s if c c if c c             (11) 'r js denotes the evaluation of the criteria by a decision-maker r. it is again necessary to average the matrix 'r js by applying a geometric mean. step 7. determining the coefficient 'jk 1 ' 2 'j j if j n k s if j n       (12) n denotes a total number of criteria. specifically, in this case, it means that the value of the last criterion is equal to fuzzy number one. step 8. determining the fuzzy weight 'jq 1 1 '' ' jj j if j n qq if j n k          (13) an integrated swot – fuzzy piprecia model for analysis of competitiveness in order to improve... 445 step 9. determining the relative weight of the criterion 'jw 1 ' ' ' j j n j j q w q    (14) step 10. in order to determine the final weights of the criteria, it is first necessary to perform the defuzzification of the fuzzy values jw and 'jw . 1 '' ( ') 2 j j jw w w  (15) step 11. checking the results obtained by applying spearman and pearson correlation coefficients. 3. case study when analyzing the internal and external environment of company, it is easiest to use swot analysis. swot analysis provides data through four elements of the matrix strengths and weaknesses, as internal factors and opportunities and threats, as external factors of influence on the business of the company (table 3). internal factors are those within the company and on which, management, along with other employees in the company, can make an impact. external factors include those that cannot be influenced by management and employees. therefore, managers and employees cannot influence the occurrences that coming from the external environment of the company by their decisions and activities. table 3 elements of swot analysis internal factors strengths (+) weaknesses (-) 1. modern trucks and the ability to respond to all requests 2. worker motivation 3. professional employees and years of experience 4. offices in eu and organization and responsibility (family business) 5. recognition by brand 6. cost optimization 1. disloyalty of employee s 2. workers' omissions (information of exponents, etc.) 3. close relationship in communication between owner and worker 4. cost optimization 5. absence of test moves (employee evaluations) 6. need for one administrative worker external factors opportunities (+) threats (-) 1. expanding business 2. infrastructure growth 3. association 4. eu funds 5. training course through eco trainings 1. closing other companies 2. growth of levies 3. unexpected problems from the ground 4. unloyal competition 5. eu restrictions (cemt, etc.) 6. fluctuation of labor 446 i. đaliĉ, j. ateljević, ţ. stević, s. terzić based on the elements in table 3, it is the most easily to see what is represented as strength and what as weakness in the company, as well as what from the external environment of the company presents opportunity and what is a threat. the table shows six elements that are listed as the strength of the enterprise. all transport vehicles owned and operated by the company are the newer models. the company has enough transport and human resources to respond to all requests of service users. the company motivates employees to work and stay as long as possible in the company with rewards after a certain period spent in the company. the company has been in business for 20 years and has exclusively professional drivers trained for their business. the company operates as a very organized and responsible family-owned company that has opened offices in the eu. through years of successful business, the company has built its brand that is recognizable in the country as well as abroad. thanks to the experience and expertise of employees of the company, the conditions for satisfactory cost management have been created. after the mentioned strengths, the swot analysis also lists 6 weaknesses that the company wants to overcome. the big problem is that despite the good conditions in the company, workers often leave their jobs and look for another job. another problem is the untimely provision of information to managers by workers. the company management is intimate with the employees and cannot objectively perceive certain problems. despite good management of cost, there is space to reduce fuel consumption, which would streamline costs in the further. there is no adequate way of valuing workers in the company, what is a significant problem. the company needs one administrative worker to relieve the dispatchers of a particular administrative job. the second part of the swot analysis is related to external factors that affect the business of the company. opportunities that come from the environment the company seeks to exploit, and the threats to bypass or reduce. five opportunities are listed. the company should take the opportunity to expand its business and, based on its experience and recognizable brand, expand its business to as many users of its services as possible. in recent years, there has been a development of transport infrastructure in the domestic market, what company should use as an opportunity to develop its business in the country. today, there is the possibility of associating companies around a common interest to achieve it in a much easier way than acting individually. within the framework of the activities of the eu institutions, there are funds aimed at developing the economy at all levels and in all activities, and the company should use the opportunity to apply for them. recently, eco trainings for drivers have been organized, where drivers acquire skills of vehicle control with saving fuel consumption up to 15%, what is a very good opportunity for the company to rationalize costs in this way. further, the swot analysis also identifies six threats that the company seeks to minimize. due to the reduced volume of work or for some other reasons, there are closures of other companies in the branch of activity of a given enterprise and of course there is always a certain a dose of caution. the company is constantly faced with increasing levies both at home and abroad, what is very difficult to monitor financially. when transporting, drivers are often confronted with certain problems on the ground, such as sudden breakdowns, traffic problems, etc. as in any industry, there are those who are called unfair competition and who in some way take users from companies that legally serve the market. the eu is constantly introducing additional transport restrictions in the form of certain permits, so the company must respond to these challenges on time in order to carry out its activities. fluctuation of labor is present in the world market as well as in the domestic market, what is a significant problem that hinders a an integrated swot – fuzzy piprecia model for analysis of competitiveness in order to improve... 447 undisturbed and continuous work process. all these threats the company need to deal with in the best possible way, using its internal strengths and opportunities from the environment. for the sake of objectivity, swot analysis is carried out by management of the company together with independent external consultants. based on the swot analysis, strategies are created that minimizing threats from the environment by using the strengths of the enterprise and minimizing the weaknesses in the enterprise by taking advantage of the opportunities. the aim of swot analysis is to maximize strengths and opportunities by minimizing weaknesses and threats, and minimizing weaknesses and threats by maximizing strengths and opportunities. swot analysis provides a good analytical basis for management to get a clear picture of the company and the external environment in which it operates. in this way it provides a realistic basis for defining strategies for improving the business of the company. based on the swot analysis, the assumption is that the organization will be achieved the greatest strategic success by maximizing its own strengths and opportunities in the environment while minimizing threats and weaknesses. the analysis of the concurrence of internal and external factors, that is, determining their impact on each strategy is also very important. thus, internal strengths and weaknesses should be observe in the context of external opportunities and threats and vice versa. criteria evaluation was performed using a linguistic scale that involves quantification into fuzzy triangle numbers. table 4 shows the evaluation of the criteria for fuzzy piprecia and inverse fuzzy piprecia by the decision maker. table 4 criteria ratings for fuzzy piprecia and inverse fuzzy piprecia pipr. c1 c2 c3 c4 dm 0.500 0.667 1.000 0.400 0.500 0.667 1.100 1.150 1.200 pipr-i c4 c3 c2 c1 dm 1.000 1.000 1.050 0.286 0.333 0.400 1.200 1.300 1.350 based on the evaluation of criteria and eq. (1), a matrix sj is formed. 0.500 0.667 1.000 0.400 0.500 0.667 1.100 1.150 1.200 js             applying eq. (6), the values of the matrix kj' are obtained: ' 0.800 0.850 0.900 0,650 0.700 0.800 1.000 1.333 1.500 1.000 1.000 1.000 jk             ' 4 (1.000,1.000,1.000)k  ' 3 (2 1.000, 2 0.667, 2 0.500) (1.000,1  .333,1  .500) etc.k      448 i. đaliĉ, j. ateljević, ţ. stević, s. terzić applying eq. (7), the following values are obtained: 0.926 1.261 1.923 0.833 1.071 1.538 0.667 0.750 1.000 1.000 1.000 1.000 jq             ' 4 (1.000,1.000,1.000)q  ' 3 1.000 1.000 1.000 , , (1.667,0.750,1.000) 1.500 1.333 1.000 q        after that, it is necessary to apply eq. (8) to obtain the relative weights for the fuzzy inverse piprecia method. 4 1.000 1.000 1.000 ' , , (0.183, 0.245, 0.292) 5.461 4.082 3.426 w        the results of the applied methodology are presented in table 5. using eq. (9), the final weights of the criteria are obtained. before applying this eq., it is necessary to defuzzy the values of the criteria obtained by applying eqs. (1) (9). table 5 shows the complete previous calculation. table 5 weights of values of criteria pipr. sj kj c1 1.000 1.000 1.000 c2 0.500 0.667 1.000 1.000 1.333 1.500 c3 0.400 0.500 0.667 1.333 1.500 1.600 c4 1.100 1.150 1.200 0.800 0.850 0.900 qj wj def. wj rank c1 1.000 1.000 1.000 0.271 0.352 0.393 0.345 0.337 1 c2 0.667 0.750 1.000 0.181 0.264 0.393 0.272 0.274 2 c3 0.417 0.500 0.750 0.113 0.176 0.295 0.185 0.188 4 c4 0.463 0.588 0.938 0.126 0.207 0.368 0.220 0.231 3 pipr.–i sj kj c1 1.100 1.150 1.200 0.800 0.850 0.900 c2 1.200 1.300 1.350 0.650 0.700 0.800 c3 0.500 0.667 1.000 1.000 1.333 1.500 c4 1.000 1.000 1.000 qj wj def. c1 0.926 1.261 1.923 0.170 0.309 0.561 0.328 c2 0.833 1.071 1.538 0.153 0.262 0.449 0.275 c3 0.667 0.750 1.000 0.122 0.184 0.292 0.191 c4 1.000 1.000 1.000 0.183 0.245 0.292 0.242 an integrated swot – fuzzy piprecia model for analysis of competitiveness in order to improve... 449 spearman’s [35,36] coefficient of correlation for the obtained ranks is 1.00 what means that these ranks are in complete correlation. spearman’s coefficient is used to determine the correlation of the obtained ranks. pearson’s [24] correlation coefficient for the weights of the criteria was also calculated and it is 0.985. the pearson’s coefficient is used to determine the correlation of the values of the criteria obtained by both methods. table 6 presents the final results of weights using the fuzzy piprecia method. table 6 ranking elements by fuzzy piprecia method 0.337 strengths rank 0.274 weaknesses rank c1 0.305 0.103 1 c1 0.249 0.068 3 c2 0.141 0.048 9 c2 0.120 0.033 18 c3 0.151 0.051 8 c3 0.145 0.040 15 c4 0.172 0.058 4 c4 0.210 0.058 5 c5 0.207 0.070 2 c5 0.173 0.047 10 c6 0.121 0.041 14 c6 0.143 0.039 16 0.188 opportunities rank 0.231 threats rank c1 0.281 0.053 7 c1 0.196 0.045 12 c2 0.164 0.031 21 c2 0.199 0.046 11 c3 0.236 0.044 13 c3 0.117 0.027 23 c4 0.162 0.030 22 c4 0.142 0.033 19 c5 0.198 0.037 17 c5 0.139 0.032 20 c6 0.237 0.055 6 table 6 shows the elements ranked by importance. first, the ranking of strengths, weaknesses, opportunities and threats as a group of elements was performed. in this case, strengths have the highest value (0.337), that means that the strengths are ranked first by importance. weaknesses are in second place (0.274), while threats (0.231) and opportunities (0.188) are in third and fourth place respectively. therefore, strengths and weaknesses are more important for a company as internal factors with influence on its business than external factors, ie opportunities and threats. the total number of ranked elements is 23. the table shows that the first element a modern trucks and the ability to respond to all requests is ranked as the first element, that is, the most significant in relation to all other factors with influence on the business of the company. second by importance is brand recognition, an element that, like the first, is in the group of elements that make the strengths of the company. the lowest ranked element in this group is cost optimization, and in the overall ranking of the elements it takes 14th place. the third most important is disloyalty of employees, the element that is in the group of elements that make the weaknesses of the company. this element is also the highest ranking element in this group. the worst ranked element in this group is workers' failures, while it takes 18th place in the overall ranking of the elements. the highest ranking element from the group of elements that make opportunities from the environment of the company is business expansion, and in the overall ranking of all elements it takes 7th place. the worst ranked element in this group is eu funds, and it takes 22nd place in the overall ranking. within the group of elements that make threats from the environment, the highest ranked element is fluctuation of labor, and the worst ranked element is unexpected problems from the ground, which also takes the worst 23rd position in the overall ranking of the elements. 450 i. đaliĉ, j. ateljević, ţ. stević, s. terzić 4. conclusion this research relates to a specific company engaged in the transport of goods on the territory of bosnia and herzegovina and abroad. the company has offices in the eu. a swot analysis was conducted to get an overview of the situation in the company and based on that, data about the strengths and weaknesses in the company, as well as the opportunities and threats that come from the environment were obtained. elements from swot analysis are ranked by fuzzy piprecia method. from the results of fuzzy piprecia method it can be seen that the strengths are ranked first by the importance. the worst ranked group of elements is group of opportunities. the results shows that the first element from group of strengths, c1 a modern trucks and the ability to respond to all requests is ranked as the first element, and this is the most significant in relation to all other factors with influence on the 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theory and applications, 3(1), pp. 41-56. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 193 201 https://doi.org/10.22190/fume180526025n © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper parameters forecasting of laser welding by the artificial intelligence techniques 911.2:556 vlastimir nikolić 1 , miloš milovančević 1 , dalibor petković 2 , dejan jocić 1 , milan savić 1 1 university of niš, faculty of mechanical engineering, serbia 2 university of niš, faculty of pedagogical sciences, serbia abstract. laser welding process is used in many industrial sectors. one of the most important aspects of the laser welding quality refers to the geometrical and mechanical properties of welding joints. in order to develop optimal conditions for the laser welding process it is desirable to know in advance which machining parameters to select. though there are manuals which recommend specific parameters combinations for the desired laser welding quality it is difficult to cover all possible combinations because of the process nonlinearity. therefore, in this study the main aim is to establish an algorithm for optimal parameters forecasting of the laser welding process. the algorithm is based on an artificial intelligence approach. the main goal is to forecast the geometrical parameters of the welding joints like front width, front heights, back width and back heights of the welding joints. experimental process was performed in order to acquire training and testing data of the laser welding process. the obtained results could be of practical importance for engineers in industry. key words: laser welding, forecasting, artificial intelligence, welding joint 1. introduction the laser welding process is used in manufacturing engineering as an advanced process. one of its main merits is a high density of power, high productivity and high penetration. there is also a narrow heath-affected zone which is also a very important factor. however, before proceeding to the process itself, one needs to select optimal machining parameters in order to get the best performances of the welding joints. there are number of parameters which could have high influence on the laser welding quality. received may 26, 2018 / accepted july 15, 2018 corresponding author: vlastimir nikolić university of niš, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: vnikolic@masfak.ni.ac.rs 194 v. nikolić, m. milovanĉević, d. petković , d. jocić, m. savić the quality of the welding joints could be determined based on mechanical and geometrical parameters of the weld. laser power, welding speed, focal position and gap have high relevance for the mechanical and geometrical parameters of the weld. it is a difficult test to select optimal machining parameters in order to get the best weld quality. for such a purpose many engineers empirically select machining parameters. however, the empiric selection is prone to errors since it is affected by engineers’ knowledge of the process. there are different mechanical and geometrical parameters which are investigated by researchers. according to the results reported by schweier et al. [1] there are three mechanisms of spatter formation in the laser welding process. these spatter formations are caused by material ablation, by laser spot entry into the melting spot and by dynamics of melt spot. one of the important factors during the laser welding process with high power co2 laser is laser-induced plasma [2]. mi et al. [3] used a finite element code to predict temperature, phase fraction and stress fields during the laser welding process. taguchi based grey relational analysis was used by shanmugarajan et al. [4] for optimization of the laser welding process; it showed that the obtained results were closely correlated to the predicted values. differences in interactions between the laser and the plasma arc were investigated by chen et al. [5]. cai et al. [6] investigated the laser effect on the welding process. chen et al. [7] studied gap tolerance of the butt laser joint. using a high speed video during co2 laser-mag hybrid welding of e36 steel, huang et al. [8] studied droplet transfers at different positions. in spite of different approaches for selection of the optimal laser welding parameters there are no investigations yet which can be used universally for all different materials and process. hence in this study the main aim is to forecast optimal laser welding parameters by the algorithms which are based on the artificial intelligence (ai) approach. the ai approaches are useful since they require no internal physical knowledge of the process. there is only the need to acquire input-output data pairs for training and testing process of the algorithms. the algorithms are used for forecasting of laser welding geometrical parameters. three ai approaches are used in this study:  extreme learning machine (elm) [9, 10],  artificial neural network (ann) [11], and,  genetic programming (gp) [12]. as input parameters, laser power, welding speed, focal position and gap are used. front width, front height, back width and back height are used as output parameters. the parameters present geometrical parameters of the welding joints and represent important quality indicators of the laser welding process. 2. experimental measurement for an experimental measurement procedure low carbon steel q235 and stainless steel sus301l-ht are used. fig. 1 gives a schematic view of the laser welding process. fiber laser ipg ylr-4000 is used during the laser welding process. the laser has wavelength of 1.1 µm. lens focal length is 220 mm and diameter spot is focused to 0.2 mm. a set of input machining parameters are determined based on the previous studies in literature. table 1 shows a set of input and output parameters used in this study. parameters forecasting of laser welding by artificial intelligence 195 fig. 1 schematic view of the laser welding process table 1 main parameters used in this study inputs and outputs parameters description min max input parameter 1 laser power (w) 1500 3500 input parameter 2 welding speed (m/min) 2.5 3.5 input parameter 3 focal position (mm) -3 1 input parameter 4 gap (mm) 0 0.1 output parameter 1 weld front width (µm) 800 1600 output parameter 2 weld back width (mm) 0 1400 output parameter 3 weld front height (mm) -200 50 output parameter 4 weld back height (mm) -400 300 geometrical parameters are used as output indicators for the laser welding quality estimation and forecasting. fig. 2 shows the positions of the geometrical parameters. fig. 2 geometrical parameters of weld 196 v. nikolić, m. milovanĉević, d. petković , d. jocić, m. savić 3. extreme learning machine a fuzzy inference system in matlab software is employed in the whole process of the anfis training and evaluation. an anfis network for 2 input variables is depicted in fig. 3. fig. 3 anfis structure the fuzzy if-then rules of takagi and sugeno’s class and two inputs for the first-order sugeno are employed for the purposes of this study: if x is a and y is c then f1=p1x+q1y+r1 (1) the first layer is made up of input parameters mfs, and it provides input values to the following layer. each node here is considered as an adaptive node having a node function o=ab(x) and o=cd(x) where ab(x) and cd(x) are membership functions. bell-shaped membership functions having the maximum value (1.0) and the minimum value (0.0) are selected so that ib2 i i iiii a cx 1 1 )d,c,b,a;x(bell)x(μ            (2) where {ai, bi, ci, di} are parameter sets. the parameters of this layer are designated as premise parameters. here, x and y are inputs to the nodes. the membership layer is the second layer. it looks for the weights of every membership function. this layer gets the receiving signals from the preceding layer and then it acts as a membership function to the representation of the fuzzy sets of each input variable, respectively. second layer nodes are non-adaptive. the layer acts as a multiplier for the receiving signals and sends out the outcome in wi=ab(x)cd(y) form. every output node exhibits the firing strength of a rule. parameters forecasting of laser welding by artificial intelligence 197 the next layer, the third one, is known as the rule layer. all neurons here act as a precondition matching the fuzzy rules, i.e. each rule’s activation level is calculated whereby the number of fuzzy rules is equal to the quantity of layers [13]. every node computes the normalized weights. the nodes in the third layer are also considered non-adaptive. each of the node computes the value of the rule’s firing strength over the sum of all rules’ firing strengths in the form of wi * =wi/(w1+ w2), i=1,2. the outcomes are referred to as the normalized firing strengths. the fourth layer is responsible for providing output values as a result of the inference of rules. this layer is also known as the defuzzification one. every fourth layer node is an adaptive node having node function oi 4 = wi * xf = wi * (pix +qiy + ri). in this layer, {pi, qi, ri} is a variable set. the variable set is designated as consequent parameters. the fifth and final layer is known as the output one. it adds up all the receiving inputs from the preceding layer. thereafter, it converts the fuzzy classification outcomes into a binary (crisp). the single node of the 5th layer is considered non-adaptive. this node calculates the total output as the whole sum of all the receiving signals,     i i i i * i * i 5 i w fw xfwo (3) in the process of identification of variables in the anfis architectures, the elm or extreme learning machine is applied. the functional signals progress until the 4th layer whereby the hybrid learning algorithm passes. further, the consequent variables are found by the least squares estimation. in the backward pass, the error rates circulate backwards and the premise variables are synchronized through the gradient decline order. elm or extreme learning machine is an algorithm for training of neural networks. the type of neural networks which are trained by elm is single hidden layer feed forward networks. fig. 4 shows the structure of the single hidden layer feed forward networks which are trained by elm algorithm. the main advantage of the elm algorithm is easy application, shot training time and good generalization of results. fig. 4 elm structure 198 v. nikolić, m. milovanĉević, d. petković , d. jocić, m. savić 3. results a comprehensive research is performed using the given set of input variables. basically, an anfis model is built by the functions for each combination and then respectively trained for single epoch. subsequently, the achieved performance is reported. in the beginning only one input parameter influence is examined. from the outset, the most influential input in the prediction of the output is identified and determined. elm forecasting performances are analyzed based on root mean square error (rmse) and coefficient of determination (r 2 ). the input variable with the lowest number of errors (rmse) has the highest influence on the output parameter or the most relevance in regards to the outcome. figs. 5-8 show the forecasting of the laser weld geometrical parameters by the elm algorithm. a high forecasting accuracy based on the coefficient of determination can be observed. also, it can be noticed that the points are mostly aligned, meaning that there are no high errors. fig. 5 elm forecasting of laser weld geometrical parameters: height of weld front fig. 6 elm forecasting of laser weld geometrical parameters: width of weld front parameters forecasting of laser welding by artificial intelligence 199 fig. 7 elm forecasting of laser weld geometrical parameters: width of weld back fig. 8 elm forecasting of laser weld geometrical parameters: height of weld back tables 2-5 show the elm forecasting performances based on two indicators. also for the sake of comparison ann and gp results are also presented. ann and gp present two different approaches of artificial intelligence. based on the comparisons one can conclude that the elm has better forecasting performances than ann and gp. table 2 elm, ann and gp models for weld front height prediction elm ann gp rmse r2 rmse r2 rmse r2 5.7326 0.9952 12.1997 0.9781 17.1592 0.9567 200 v. nikolić, m. milovanĉević, d. petković , d. jocić, m. savić table 3 elm, ann and gp models for weld front width prediction elm ann gp rmse r2 rmse r2 rmse r2 9.2444 0.9975 22.5804 0.985 41.4980 0.9494 table 4 elm, ann and gp models for weld back width prediction elm ann gp rmse r2 rmse r2 rmse r2 18.7626 0.9981 73.2556 0.9708 108.4139 0.936 table 5 elm, ann and gp models for weld back height prediction elm ann gp rmse r2 rmse r2 rmse r2 9.9006 0.9973 36.6879 0.9631 44.2840 0.9462 5. conclusion forecasting of the laser weld geometrical parameters is complex due to many indicators and factors therefore, a new approach is proposed in this study in order to overcome the difficulties of the laser weld geometrical parameters forecasting by removing some unnecessary input parameters. a systematic approach is applied with the aim to select the most influential parameters for the laser weld geometrical parameters forecasting by the anfis methodology. the anfis is used to eliminate vagueness in the laser welding process and to produce the best forecasting conditions. the proposed anfis model is used to convert the complicated multiple performance characteristics into the single multi response performance index. as a result, the forecasting methodology developed in this research is useful for enhancing the multiple performances characterizing laser welding analyses. in this study the main aim is to establish a forecasting algorithm for laser weld geometrical parameters based on input machining conditions. the algorithm is based on an artificial intelligence approach. the main advantage of the approach lies in the fact that it requires no knowledge of the internal physical model of the laser welding process. there is only the need to acquire training data pairs for the ai technique. the algorithm is based on an extreme learning machine which is one type of training algorithm for artificial neural networks. based on the obtained results, the elm has shown better performances than other ai techniques. references 1. schweier, m., haubold, m.w., zaeh, m.f., 2016, analysis of spatters in laser welding with beam oscillation: a machine vision approach, cirp journal of manufacturing science and technology, 14, pp. 35-42. 2. zhao, y., zhu, k., ma, q., shang, q., huang, j., yang, d., 2016, plasma behavior and control with small diameter assisting gas nozzle during co2 laser welding, journal of materials processing technology, 237, pp. 208-215. 3. mi, g., xiong, l., wang, c., hu, x., wei, y., 2016, a thermal-metallurgical-mechanical model for laser welding q235 steel, journal of materials processing technology, 238, pp. 39-48. parameters forecasting of laser welding by artificial intelligence 201 4. shanmugarajan, b., shrivastava, r., sathiya, p., buvanashekaran, g., 2016, optimisation of laser welding parameters for welding of p92 material using taguchi based grey relational analysis, defence technology, 12(4), pp. 343-350. 5. chen, m., xu, j., xin, l., zhao, z., wu, f., 2016, comparative study on interactions between laser and arc plasma during laser-gta welding and laser-gma welding, optics and lasers in engineering, 85, pp.1-8. 6. cai, x., li, h., wei, h., yang, l., gao, y., 2014, effect of laser on the welding process of shortcircuiting transfer mig welding of aluminum alloys, the international journal of advanced manufacturing technology, 75(9-12), pp. 1829-1836. 7. chen, l., zhou, l., tang, c., huang, w., wang, c., hu, x., wang, j., yan, f., wang, x., jiang, z., shao, x., 2014, study of laser butt welding of sus301l stainless steel and welding joint analysis, the international journal of advanced manufacturing technology, 73(9-12), pp. 1695-1704. 8. chen, y.b., feng, j.c., li, l.q., li, y., chang, s., 2013, effects of welding positions on droplet transfer in co 2 laser–mag hybrid welding, the international journal of advanced manufacturing technology, 68(5-8), pp. 1351-1359. 9. huang, g.b., zhu, q.y., siew, c.k., 2006, extreme learning machine: theory and applications, neurocomputing, 70(1-3), pp. 489-501. 10. huang, g.b., chen, l., siew, c.k., 2006, universal approximation using incremental constructive feedforward networks with random hidden nodes, ieee trans. neural networks, 17(4), pp. 879-892. 11. liang, n.y., huang, g.b., saratchandran, p., sundararajan, n., 2006, a fast and accurate online sequential learning algorithm for feedforward networks, ieee transactions on neural networks, 17(6), pp. 1411-1423. 12. koza, j.r., 1992, genetic programming: on the programming of computers by natural selection, a bradford book, the mit press, cambridge, massachusetts, london. 13. pamuĉar, d., ćirović, g., 2018, vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty conditions, decision making: applications in management and engineering, 1(1), pp. 13-37. function k as a link between fuel flow velocity and fuel pressure, depending on the type of fuel facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 119 132 doi: 10.22190/fume160628003n © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper function k as a link between fuel flow velocity and fuel pressure, depending on the type of fuel udc 629.3:629.06 boban nikolić, miloš jovanović, miloš milošević, saša milanović faculty of mechanical engineering, university of niš, serbia abstract. regarding the application of vegetable oil based fuels in diesel engines, it is necessary to fully examine and understand the processes which take place in fuel delivery systems, namely, the processes of injection, mixture formation and combustion as well as emission characteristics. the paper provides an analysis of fuel flow in high pressure tubes of the fuel injection system, with the aim of determining function k as a link between fuel flow velocity and fuel pressure, and observing the influence of certain physical characteristics of the fuel upon the given function. the analysis presents the speed of sound and density, as fuel characteristics which affect the k function. the paper determines the speed of sound, density and bulk modulus for four fuels (pure rapeseed oil ro, biodiesel b100, a mixture of biodiesel and diesel b50, and diesel d), and forms appropriate k functions for each fuel in the pressure range from the atmospheric one to 1600 bar. key words: vegetable oil, rapeseed oil, biodiesel, speed of sound, density, bulk modulus 1. introduction vegetable oil based fuels represent a considerable potential as an alternative to the diesel engine fuels. the use of straight vegetable oil as fuel for diesel engines is restricted by a number of factors. vegetable oils have greater molecular mass than diesel fuel [1, 2], often accompanied by inadequate oxidation and thermal stability [1], with higher surface stress, density and viscosity [1, 2, 3]. the research performed using pure rapeseed oil in conventional direct injection diesel engines [1, 4, 5] has shown that after approximately 50 hours of operation the engine stops working due to the accumulation of coke particles in engine pistons, valves and especially injectors. simultaneously with the research of the application of straight vegetable oil, studies have also been conducted on the application received june 28, 2016 / accepted october 23, 2016 corresponding author: boban nikolić faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: nboban@masfak.ni.ac.rs 120 b. nikolić, m. jovanović, m. milošević, s. milanović of various mixtures of diesel fuels and straight vegetable oil in direct injection (di) diesel engines [1, 2, 4, 5, 6]. forson et al. [6] examine the operation of a single-cylinder, fourstroke di engine with air cooling, running on various mixtures of diesel fuel and jatropha oil. the observed exhaust emission is similar to the operation of the diesel-powered engine, regardless of the volumetric composition of mixtures. the best results are obtained when using the mixture with the volumetric composition of 97.4% diesel and 2.6% pure jatropha oil, where a small increase in effective power and degree of efficiency is noted along with a small reduction in specific consumption. however, the study does not contain any information on the operation and state of the engine after a longer period of running on these fuels. the analysis of the composition of exhaust gases from a diesel engine operating on mixtures of diesel fuel and four different vegetable oils (10–20% of volumetric share) shows a certain rise in the level of nox, co and hc emission [7]. hellier et al. [8] study the operating characteristics of a new generation di diesel engine with the injection pressure of up to 1600 bar, while operating on six different vegetable and algae oils. the oils are heated up to 60 o c and exhaust gases emission is monitored. the authors have reached the conclusion pointing to a decrease in the level of nox, but also to an increase in the levels of co, hc and pm in comparison with the engine running on diesel. what is also observed are the differences in the injection timing and the injection duration. the authors emphasize the importance of determining the physical characteristics of vegetable oils for the explanation and prediction of the behavior of injection systems and engine operation as a whole, when using alternative fuels. reviewing the research into the application of vegetable oil (and different mixtures of vegetable oil and diesel fuel and/or various additives) in diesel engines, it can be concluded that it is necessary either to adjust the engine to the vegetable oil or, vice versa, adjust the vegetable oil to the engine. the diesel engine adjustment to running on vegetable oil has usually brought about an improvement in the engine operation; however, either the results have turned out as not in agreement with the expectations or the requirements and the engine modifications have proved too complicated and costly. much better results are obtained by adjusting the vegetable oil, especially by changing it chemically with the aim of reducing the molecules by esterification using alcohol. the catalytic decomposition of the vegetable oil structure (the use of waste vegetable oil or the remains of animal fats) using alcohol (most often ethanol or methanol), or the so-called oil esterification, can yield a fuel of far superior characteristics than the basic oils (fats). this fuel is commercially known as biodiesel. rapeseed biodiesel has proved to be, in comparison to those obtained from other vegetable oils, one of the most suitable ones from the standpoint of using it as fuel for diesel engines. the majority of other biodiesels possess lower oxidation and thermal stability [9, 10, 11] with an inappropriate iodine number [9, 10, 12] or inappropriate cold filter plugging points (cfpp) [9, 13] and compressibility points (cp) [9, 11, 13-17]. exceptionally, the characteristics of the biodiesel obtained from algae position it high on the list of fuels for diesel engines [9, 18]. however, due to its being a relatively new fuel, with a specific production technology, it is much less produced and used in relation to the biodiesels based on rapeseed oil, soya and oil palms. from the standpoint of the application of biodiesel (and mixtures of biodiesel and diesel fuel) as a fuel in diesel engines, it is necessary to fully examine and understand the processes which take place in the fuel delivery systems as well as the processes of injection, mixture formation and fuel combustion, as well as emission characteristics. furthermore, it is very important that the characteristics of function k – as link between fuel flow velocity and fuel pressure, depending on the type of fuel 121 biodiesel are compatible with appropriate standards, and that those characteristics which are not prescribed by any standard (speed of sound, bulk modulus, surface stress, etc.), yet are crucial from the perspective of their influence on the processes of fuel injection, mixture formation, combustion and exhaust gases emission, are examined in detail. various researchers have recognized the importance of knowing the speed of sound, density and bulk modulus for the operation of a fuel injection system. huber et al. [19] determine by measurement the speed of sound and density of examined fuels at the atmospheric pressure, for the temperature range of 278–333 k, and starting from the molar helmholtz free energy equation, they form a model for calculating fuel characteristics for pressures of up to 500 bar and temperatures of up to 700 k. tat et al. [20, 21] present the dependence of these characteristics on the pressure of up to 350 bar at 293–313 k. ott et al. [22] determine the density and speed of sound at 0.83 bar and temperatures of 278 to 338 k. kegl [23] experimentally determines the speed of sound and density for pressures of up to 400 bar as well as the influence of a temperature change on density (from 273 to 313 k). the author also determines the bulk modulus for pressures of up to 400 bar and provides a comparative overview of results obtained by simulation. dzida et al. [24, 25] measure the speed of sound at temperatures of 293–318 k and pressures from 1 to 1010 bar, the density at the atmospheric pressure, and calculate it at pressures of up to 1010 bar. payri et al. [26] experimentally determine the speed of sound at pressures from 150 to 1800 bar and temperatures from 298 to 343 k. two pressure sensors are placed in a 12 m long “high pressure tube” (with the internal diameter of 2.5 mm) at the distance of 8.22 m from each other. the density is measured at the atmospheric pressure and calculated for higher pressures. measuring at the atmospheric pressure and temperatures from 293 to 343 k, freitas et al. [27] determine the speed of sound for different fuels and use the data to calculate and predict the acoustic characteristics of other biodiesel fuels [28]. daridon et al. [29] present the data for several different fuels where the speed of sound is measured at the atmospheric pressure and the range of temperatures from 283 to 373 k. for the same pressures and with the approach used in [24, 25], ţarska et al. [30] determine the speed of sound for the biodiesel produced from coconut and palm oil. lopes et al. [31] vary the fuel temperature from 298 to 353 k at the atmospheric pressure and determine the fuel speed of sound by measurement, with a presentation of data from the literature on the experimental values of the speed of sound for biodiesels of different origin (and some other alternative fuels). starting from the thermodynamic properties of biodiesel and mixtures, perdomo et al. [32] model a curve of change in the speed of sound with the change in the fluid temperature, at the atmospheric pressure. tat and van gerpen determine the speed of sound and bulk modulus at pressures of up to 345 bar and temperatures from 293 to 373 k, with the analysis of the change in the values of these quantities for the values of the angle of preinjection and nox emission [33]. gautam and agarwal [34] focus on the influence of fuel temperature (atmospheric pressure) on the characteristics (density, viscosity, speed of sound, bulk modulus, surface stress) of the biodiesel used in india. the determination of the fuel bulk modulus at pressures from 30 to 330 bar and a temperature of 311 k is the subject matter of the research conducted by lapuerta et al. [35]. the above research shows that the values of the speed of sound, density and bulk modulus of diesel fuel, biodiesel and their mixtures, increase with an increase in pressure, and decrease with an increase in temperature. values of the speed of sound, density and bulk modulus increase with the increasing share of biodiesel in mixtures. the complexity 122 b. nikolić, m. jovanović, m. milošević, s. milanović of the experimental determination of values of the speed of sound and density occurs with an increase in operating pressures above 600 bar, regardless of the type of fuel. here, the major problem is sealing and maintaining the absence of leaks in the apparatus elements during operation at higher pressures in standard methods which work in line with the principle of variable fluid volume. 2. analysis of fuel flow in small diameter tubes fig. 1 shows a schematic of a mechanically-controlled injection system. the main components of the injection system are a high pressure pump (hpp), a high pressure tube (hpt) and an injector. the analysis of fuel flow in hpt is based on and draws from [36, 37] with certain specificities, and with the aim of identifying and observing the importance and influence of particular physical characteristics of fuel on the operation of the fuel injection system. fig. 1 a schematic of the injection system ap, vp – hpp piston face area and velocity, vrv – relief valve casing volume, vt, at – hpt volume and cross-section area, i – active cross-section of the injector, vi – fuel velocity from the injector, vi – high-pressure volume part of the injector, vt – total volume of the high pressure system vt=vp+vt+vi the volumetric fuel flow is, at any moment of time t (or the angle of camshaft ), defined by piston velocity vk as apvp = (t) or apvp = (). if the process between the pump and the injector would take place without a delay, then there would be no “compression” of the fuel in volume vt. the link between the initial and the discharge flow (coming out of the injector) would be: iiipp vaav   , that is: paav iipp    2 (1) where  is the fuel density and p the difference in pressures before and after the injector (nozzle). in the high pressure part of the installation, fuel has to be treated as a compressible fluid, and the relative change in volume is proportional to the change in pressure: function k – as link between fuel flow velocity and fuel pressure, depending on the type of fuel 123 v v bp   (2) where b is the entropic bulk modulus of fuel compressibility. if one takes into consideration fuel compressibility and the total high pressure volume of system vt, eq. (1) changes to the form which represents the displaced amount of fuel as a sum of the injected amount of fuel and the remaining amount of fuel in the installation (due to fuel compressibility): e v dt dp ppaav tii cyliiiipp    2 (3) where pcyl is the pressure in the engine displacement and pii the pressure in front of the injector (fig. 1, cross-section ii-ii). eq. (3) neglects the influence of inertia and reflection as well as the deformation of the initial flow due to the movement of certain elements (pump, injector) during the injection process. one should bear in mind that the fuel flow velocity is 50 to 60 times smaller than the speed of sound, and that the movement of the displaced amount of fuel is 10–15 cm for a single cycle. the newly-displaced amount of fuel displaces the fuel already found in the tube in front of it, and the pressure wave travels at the speed of sound. not only does the injector react in time with the incoming signal, it also creates the return impulse (characteristic of reflection), so that the characteristic of injection in any moment of time is the consequence of the characteristics of displacement and reflection. the injector nozzle changes the flow cross-section through its movement, thus changing the ratio of the flow cross-section of the injector to the flow cross-section of the high pressure tube, i.e. the condition of reflection. to analytically solve the problem of fluid flow in small diameter tubes, it is necessary to know the velocity and pressure along the tube at any moment of time. let us observe a segment of the tube and the control volume in fig. 2: fig. 2 control volume by applying the navier-stokes equations of fluid flow to the case in fig. 2, the following is obtained:                                2 2 2 2 2 21 z w y w x w z p z z w w y w v x w u t w   (4) where u, v and w are the components of flow velocity in the direction of the x, y and z 124 b. nikolić, m. jovanović, m. milošević, s. milanović axes, z is the external forces per unit of fluid mass, (1/)(p/z) the pressure force per unit of fluid mass and  the kinematic viscosity. taking into account the axisymmetric fluid flow and the fact that the longitudinal dimensions of the tube are much larger than its diameter, the change in pressure in the radial direction is neglected, so that the system can be subjected to a one-dimensional analysis, that is, the constancy of pressure and velocity per cross-section of the tube. if the influence of friction is neglected in the injection process since the tubes are relatively short (around 1 meter), eq. (4) becomes: z p z w w t w          1 (5) during the settling process (following injection), the drop in the pressure wave peaks can be described as p = p1e -kt , where p1 is the initial value of pressure. following the completion of injection, pressure oscillates between the pump and the injector with insufficiently great amplitude to reopen the injector [36, 38]. the settling velocity is directly proportional to the size of the relief volume. on the other hand, too much relief leads to the formation of steam pockets in the high pressure volume which in turn causes the reduction of the system capacity and a delay between the beginning of displacement and the injection itself. eq. (5) can further be simplified by neglecting the second member on the left side of the equation (the convective member) due to the uniformity of fluid flow velocity w , the fact that the fluid flow velocity is much smaller than the wave propagation velocity, and the constant cross-section of the tube, which yields: z p t w       1 (6) the wave travels at the speed of sound, defined as:  b a  (7) the second equation of the link between pressure and velocity is obtained using the continuity equation. based on the applied one-dimensional analysis of the system, the continuity equation has the following form: 0           z w w zt (8) using the equation for the adiabatic change in state p- = const. and differentiating it according to t and z yields: t p pt          that is z p pz          (9) substituting eq. (9) into eq. (8) yields: function k – as link between fuel flow velocity and fuel pressure, depending on the type of fuel 125 0         z w pw z p t p  (10) taking into account the abovementioned, and adopting the fact that flow velocity w is significantly smaller than pressure wave propagation velocity a, the convective member of eq. (10) is neglected and the equation takes on the following form: 0      z w p t p   t p pz w       1 (11) eqs. (6) and (11) form a system of partial differential equations: z p t w       1 and t p pz w       1 (12) whose solution yields the dependence of pressure and velocity on z and t, i.e. it enables the knowledge of the values of fluid pressures and velocities in the desired cross-section and moment. when solving the system of eqs. (12), one has to take into account the following relations:  isontropic bulk modulus b, which is defined as: b =  (p/)  the link between the speed of sound a, bulk modulus b and density : a 2 = b /   the fact that pressure waves propagate at the speed of sound a defined as: a 2 = (p/)s=const, which when substituted by the adiabatic equation of the change in the state p- = const. yields:   p a  2 (13) it is obvious that the analytical solution of the system of eqs. (12) requires the knowledge of at least two dependencies: a = a(p,t),  = (p,t) and b = b(p,t). determining the value of the speed of sound, density and bulk modulus in the function of fuel pressure and temperature is important both for the analysis of the process occurring in fuel injection systems and the prediction of the injection system behavior when working with different fuels, particularly those operating at higher injection pressures. for the sake of a simpler solution of the system of eqs. (12), one can “conditionally” adopt that a = const.,  = const. and b = const., keeping in mind that the final form of thus obtained solutions to the system of eqs. (12) uses the values for a,  and b for appropriate working pressures and temperatures. in that case, the differentiation of the first of eqs. (12) in the z coordinate, and the second in time t yields: 2 2 2 2 2 z p a t p      (14) that is: 2 2 2 2 2 z w a t w      (15) 126 b. nikolić, m. jovanović, m. milošević, s. milanović eqs. (14) and (15) are hyperbolic, linear, partial differential equations of the second order. their solution is usually sought in the bernoulli form – which is convenient for finding the eigenfrequencies of pressure waves oscillation in the inlet and exhaust tubes of piston engines – or in the d'alambert form. for the monitoring of wave propagation, the d'alambert form of the solution is more convenient [36, 37], and it enables the definition of the current (total) pressure at every place and every moment as: drv pppppp  00 (16) where p is the current pressure, p0 the initial pressure in the tubes before the activity of the hp pump, pv the pressure from the pressure waves traveling in the direction of the fuel flow, pr the pressure from the pressure waves traveling opposite to the fuel flow, and pd the dynamic pressure pd = pv + pr. analogously, the current velocity can be presented as: rv wwww  0 (17) where w represents the current velocity, w0 the initial velocity, wv the velocity which stems from the velocity wave in the direction of the fuel flow and wr the velocity which stems from the velocity wave opposite to the fuel flow. it is necessary to find the link between the velocity and pressure waves. observe the tube segment approached by an element of velocity wave wv (fig. 3). fig. 3 elementary volume atz approached by the wave wv; at=0.25dt 2 π if positive velocity wave wv moves from the cross-section 1-1 towards 2-2, the fluid will be compressed in volume atz, since the fluid enters 1-1 in interval t at average velocity w0 + wv, and exits it at velocity w0. the time the wave takes to pass through is: a z t   (18) and the compressed volume: twav vt  (19) with the elementary volume: zav t  (20) function k – as link between fuel flow velocity and fuel pressure, depending on the type of fuel 127 the pressure wave travels at a great velocity so that the adiabatic change in the state can be assumed: p p v v     1 (21) that is, from eq. (18) to eq. (20) one can obtain: vv p p a w     (22) where pv replaces p, as the pressure wave which corresponds towv. eqs. (13) and (22) yield: vv p a w     1 (23) as an equation which shows the link between fuel flow velocity and fuel pressure. if the function of the link between fuel flow velocity and fuel pressure, k, is used to mark:   a k 1 (24) one obtains: vv pkw  (25) if the sketch in fig. 3 is used to observe the wave opposite to the direction of the flow, the following link is obtained: rr p a w     1 (26) that is: rr pkw  (27) due to the opposite direction of the velocity wave causing the attenuation in volume atz. the analysis of the influence of the displacement and reflection characteristics on the injection characteristic using wave propagation [36, 37] can present various positions of the wave for fixed observation times. on the basis of eqs. (23) and (26), the following can be concluded:  if the fluid flow direction overlaps with the direction of the velocity wave propagation, the pressure rises; and,  the velocity and pressure waves have the same sign if they travel in the direction of the fluid flow, and the opposite if they travel in the opposite direction. the movement of the injector nozzle changes the flow cross-section, thus also changing the reflection condition. during the injection process, pv changes, and when the injector nozzle travels up and down, the ratio of the flow cross-section of the injector to the flow cross-section of the high pressure tube changes as well. eqs. (23), (25), (26) and (27) show the link between the velocity and the pressure waves in the direction and opposite to the direction of fuel flow. determining the function of k (eq. 24) necessitates, depending on the type of fluid (fuel), determining a = a(p,t) and  = (p,t). 128 b. nikolić, m. jovanović, m. milošević, s. milanović 3. experimental section the tested fuels are: diesel fuel (hereinafter: d – characteristics in accordance with the en 590 standard), rapeseed oil (ro – in accordance with din en 51605), rapeseed oil methyl ester – biodiesel (b100 – in accordance with en 14214), and a mixture of diesel fuel and biodiesel (b50 – volumetric composition: 50% diesel and 50% biodiesel). the method and technique of determining the speed of sound, density and bulk modulus are based on the principle of constant volume and variable mass and density of the fluid in the high pressure part of the installation [38, 39]. the examined fluid (in a vessel with a constant volume – high pressure vessel, hpv) compresses when a controlled fluid mass is pumped. here the change in pressure is observed, the mass of the pumped fluid is determined and the time of ultrasonic wave propagation (tuwp) through the steel-fuel “sandwich structure” is measured ultrasonically. on the basis of the known path along which the ultrasonic wave travels and the known characteristics of the steel used to manufacture hpv (material, dimensions, time of wave propagation and speed of sound), the time of ultrasonic wave propagation through the working fluid is determined (the time the pressure wave takes to travel the path from the one to the other internal surface of hpv heads, propagating through the working fluid), and the velocity of ultrasonic wave propagation through the working fluid is calculated, depending on the working pressure. on the basis of the known internal volume of hpv (where the vessel dilatation is also taken into consideration [38, 39]) and the calculated fluid mass inside hpv, the density of the fluid is determined depending on the working pressure. by knowing the speed of sound and density of the tested fuels, the bulk modulus is determined (based on eq. 7), as well as auxiliary function k (based on eq. 24). an ultrasonic probe is positioned on the external surface of the hp vessel heads. an ultrasonic device ud2 – 12 was used in ultrasonic measurements. the impulse echo method with direct contact was applied – special application, with a normal combined ultrasonic probe p111-2.5-k12-002, of 2.5 mhz in working frequency. corach et al. [40] use probes with working frequencies of 1.53 mhz, 5.66 mhz and 9.43 mhz for ultrasonic measurements, stating that the results vary only slightly, maximally up to 0.05%. the schematic of the experimental line for determining the speed of sound, density and bulk modulus of the examined fuels is shown in fig. 4. fig. 4 a schematic of the experimental line a – part of the installation at the atmospheric pressure, b – part of the installation at the increased pressure, 1 – ultrasonic defectoscope, 2 – ultrasonic probe, 3 – display, 4 – high pressure vessel, 5 – high pressure tube, 6 – manometer, 7 – high pressure pump, 8 – pump tank, 9 – measuring test tube function k – as link between fuel flow velocity and fuel pressure, depending on the type of fuel 129 4. results and discussion measurements were performed for working pressures of up to 1600 bar and fuel temperature of 20 o c. measurement results are shown in diagrams in figs. 5 to 9. values of the time of ultrasonic wave propagation through the working fluid decrease with an increase in pressure for all fuels (fig. 5). tuwp is the smallest for pure rapeseed oil (ro) and the largest for diesel fuel, at the same pressure. differences in the values of tuwp for different fuels decrease with an increase in pressure. 80 85 90 95 100 105 110 115 120 0 200 400 600 800 1000 1200 1400 1600 t u w p t [ s ] t ro t b100 t d t b50 pressure (bar) 1300 1400 1500 1600 1700 1800 1900 2000 0 200 400 600 800 1000 1200 1400 1600 s p e e d o f s o u n d ( m /s ) a ro a b100 a d a b50 pressure (bar) fig. 5 time of ultrasonic wave propagation fig. 6 speed of sound analogously, the values of the speed of sound, for all fuels, increase with an increase in pressure (fig. 6). differences in the values of the speed of sound, for different fuels, decrease with an increase in pressure. moreover, there is a tendency of equalization of the speed of sound values for the fuels b100, b50 and diesel at the pressure of approximately 2000 bar [38]. density values of the tested fuels increase with an increase in pressure (fig. 7); however, this increase is smaller in percentage in comparison with the speed of sound values. for the same working pressure, rapeseed oil has the greatest density value while diesel has the lowest. differences in the values do not change significantly with an increase in pressure. bulk modulus values of the tested fuels increase with an increase in pressure (fig. 8). similar to density, differences in the values of different fuels do not change significantly with an increase in pressure. the largest values of bulk modulus can be found in rapeseed oil and the lowest in 820 840 860 880 900 920 940 960 980 1000 0 200 400 600 800 1000 1200 1400 1600 d e n s it y ( k g /m 3 )  ro  b100  d  b50 pressure (bar) 1.0e+09 1.5e+09 2.0e+09 2.5e+09 3.0e+09 3.5e+09 4.0e+09 0 200 400 600 800 1000 1200 1400 1600 b u lk m o d u lu s ( p a ) b ro b b100 b d b b50 pressure (bar) fig. 7 density of tested fuels fig. 8 bulk modulus 130 b. nikolić, m. jovanović, m. milošević, s. milanović diesel. values of the k function decrease with an increase in pressure for all tested fuels (fig. 9). this tendency is in agreement with the experimental results for the speed of sound and density of the tested fuels (figs. 6 and 7) and eq. (24). the largest values of k can be found in diesel while the lowest in rapeseed oil at the same working pressure. 5.0e-07 5.5e-07 6.0e-07 6.5e-07 7.0e-07 7.5e-07 8.0e-07 8.5e-07 9.0e-07 0 200 400 600 800 1000 1200 1400 1600 k ( m 2 s /k g ) k ro k b100 k d k b50 pressure (bar) fig. 9 function k for tested fuels 5. conclusion values of the speed of sound, density and bulk modulus increase with an increase in pressure for all fuels. in comparison with the speed of sound at the atmospheric pressure, the speed of sound of the tested fuels at 1600 bar increases for 31.3% (ro), 34.5% (b100), 36.5% (b50) and 38.6% (d). furthermore, density increases for 6.7 % (ro), 7% (b100), 7.4% (b50) and 7.9% (d), and bulk modulus for 84 % (ro), 94% (b100), 100% (b50) and 107% (d). values of the speed of sound, density and bulk modulus, at the same pressures, are different for the tested fuels (figs. 6, 7 and 8). since the k function, as a link between fuel flow velocity and fuel pressure, is reversely proportional to fuel speed of sound and density (eq. 24), the values of the k function are also different depending on the fuel and working pressure. this is particularly important if one knows that, in the mechanically controlled injection systems, the value of the fuel pressure before the injector affects the opening and closing of the injector, and coupled with the fuel flow velocity, fuel characteristics and injection system characteristics, influences the characteristics of the injected fuel spray. in the electronically controlled fuel injection systems, this effect is not significant for the injection timing and duration; yet one has to take into account the influence of the physical characteristics of different fuels on the characteristics of the injected fuel spray, thus also considering the processes of mixture formation, combustion, exhaust gases emission, and effective characteristics of a diesel engine as a whole. references 1. stefanović, a., 1999, diesel engines with fuel based on vegetable oils, (in serbian), monograph, faculty of mechanical engineering, niš. 2. mehta, a., joshi, m., patel, g., saiyad, m.j., 2012, performance of single cylinder diesel engine using jatropha oil with exhaust heat recovery system, international journal of advanced engineering technology, 3(4), pp. 1-7. function k – as link between fuel flow velocity and fuel pressure, depending on the type of fuel 131 3. pandey, r.k., rehman, a., sarviya, r.m., 2012, impact of alternative fuel properties on fuel spray behavior and atomization, renewable and sustainable energy reviews, 16, pp. 1762– 1778. 4. stefanović, a., maurer, k., 1992, some experiences in obtaining rapeseed oil and it’s use as an alternative fuel for engines, engines and motor vehicles '92, jumv, kragujevac. 5. schumacher, l.g., 1996, engine oil impact literature search and summary, final report for the national biodiesel board, columbia, usa. 6. forson, f.k., oduro, e.k., hammond-donkoh, e., 2004, performance of jatropha oil blends in a diesel engine, renewable energy, 29, pp. 1135–1145. 7. rakopoulos, d.c., rakopoulos, c.d., giakoumis, e.g., dimaratos, a.m., founti, m.a., 2011, comparative environmental behavior of bus engine operating on blends of diesel fuel with four straight vegetable oils of greek origin: sunflower, cottonseed, corn and olive, fuel, 90, pp. 3439–3446. 8. hellier, p., ladommatos, n., yusaf, t., 2015, the influence of straight vegetable oil fatty acid composition on compression ignition combustion and emissions, fuel, 143, pp. 131–143. 9. http://cdn.intechopen.com/pdfs/23666/intech-biodiesel quality standards 10. xin, j., imahara, h., saka, s., 2008, oxidation stability of biodiesel fuel as prepared by supercritical methanol, fuel, 87, pp. 1807–1813. 11. rawat, d.s., joshi, g., lamba, b.z., tiwari, a.k., mallick, s., 2014, impact of additives on storage stability of karanja (pongamia pinnata) biodiesel blends with conventional diesel sold at retail outlets, fuel, 120, pp. 30–37. 12. vujicic, dj., comic, d., zarubica, a., micic, r., boskovic, g., 2010, kinetics of biodiesel synthesis from sunflower oil over cao heterogeneous catalyst, fuel, 89, pp. 2054–2061. 13. dunn, r.o., 2011, improving the cold flow properties of biodiesel by fractionation, in ng t.b. 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determining the speed of sound, density and bulk modulus of rapeseed oil, biodiesel and diesel fuel, thermal science, 16(2), pp. s505-s514. 40. corach, j., sorichetti, p.a., romano, s.d., 2015, electrical and ultrasonic properties of vegetable oils and biodiesel, fuel, 139, pp. 466–471. http://pubs.acs.org/doi/full/10.1021/ef201608g http://pubs.acs.org/doi/full/10.1021/ef201608g plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 281 292 doi: 10.22190/fume1603281w original scientific paper the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape or concave profiles udc 539.3 emanuel willert 1 , qiang li 1 , valentin l. popov 1,2,3 1 berlin university of technology, 10623 berlin, germany 2 national research tomsk state university, 634050 tomsk, russia 3 national research tomsk polytechnic university, 634050 tomsk, russia abstract: the jkr-adhesive frictionless normal contact problem is solved for the flat annular and the conical or spherical concave rigid punch indenting an elastic half space. the adhesive solution can be derived analytically from the non-adhesive one, the latter one being calculated by the boundary element method. it is found that the annular flat punch will always start to detach at the outer boundary. the pull-off forces for both concave punch shapes almost do not depend on the pull-off boundary regime and can be significantly larger than the pull-off force for the cylindrical flat punch. key words: contact mechanics, axis-symmetry, annular contact area, adhesion, jkr-theory, boundary element method, concave rigid punch, flat annular punch 1. introduction due to the ongoing miniaturisation of indenting devices in microscopy or material testing, adhesion in those systems is getting more and more important. moreover, as the biological systems seem to have developed very efficient and powerful solutions for making use of adhesive interactions, the study of the contact mechanical interactions with their environment of insects, geckos – and other organisms relying on adhesion – has gained a lot of research interest in the past years. spolenak et al. [1] found out that toroidal or concave shapes of the contact geometry – as they are used by these organisms – lead to a much better attachment of the indenting body to the surface. the contacting received september 27, 2016 / accepted november 14, 2016 corresponding author: emanuel willert institute of mechanics, berlin institute of technology, strasse des 17. juni 135, 10623 berlin, germany e-mail: e.willert@tu-berlin.de 282 e. willert, q. li, v.l. popov bodies are usually considered to be soft, which is why the interaction range for the adhesion is small compared to the range of the elastic interaction. in this case the frictionless, adhesive normal contact problem can be solved by the theory developed by johnson, kendall and roberts (jkr), [2]. the jkr-adhesive contact problem, though, can be turned back to the boussinesq-problem, i.e. the frictionless non-adhesive normal contact problem of a rigid indenter pressed into an elastic half space. thereby many publications have dealt with the axisymmetric boussinesq problem of an annular flat punch. gubenko and mossakovskij [3] and collins [4] and [5] reduced the problem to an integral equation of the fredholm type, which can be solved iteratively. for different approximate approaches see borodachev and borodacheva [6], shibuya et al. [7] or gladwell and gupta [8]. a complete analytic however recursive solution was found by roitman and shishkanova [9]. a closed formulation could later be obtained by antipov [10] using advanced applied mathematics including riemann vector problems and mellin transforms. the boussinesq problems for a conical or spherical concave rigid punch were tackled by barber [11], gladwell and gupta [8] and shibuya [12]. barber – based on his idea presented earlier [13] that the actual contact area maximizes the normal force and can hence thereby be determined if not known a priori – gave series expansions of the solution for the ratio of the contact radii being close to zero or close to unity. the jkr-adhesive problem for axisymmetric indenters and an annular contact area was first studied by kesari and lew [14]. argatov et al. [15] demonstrated how the adhesive solution in this case can be obtained from the non-adhesive one and applied their method to conical concave (based on barber’s non-adhesive solutions) and toroidal (based on asymptotic non-adhesive solutions for narrow contact areas) indenters. in the present paper we will analyze the jkr-adhesive normal contact problem of an axisymmetric annular flat or either conical or spherical concave rigid punch. the nonadhesive solutions presented in section 2 are obtained via fast boundary element method (bem) simulations – the fundamentals of which were described by pohrt and li [16] – and will be approximated by simple analytic expressions. afterwards in section 3 the adhesive solutions are derived analytically from these non-adhesive results. section 4 will give conclusions. 2. the non-adhesive solution we consider the boussinesq problem for an annular flat or concave rigid punch pressed into an elastic half space. the normal force shall be fn, the indentation depth d and the inner and outer contact radii b and a, respectively. the hole has depth h. the concave profile shall be either conical or spherical. sketches of the problems considered and notations are shown in fig. 1, 2 and 3. in the case of the annular flat punch it is shown that the normal force can be written in the form: *2 ( ),nf e da   (1) with effective young’s modulus e * and the ratio of the contact radii: . b a   (2) the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape... 283 as the closed form analytical solution given by antipov [10] is hard to handle, we calculated it again using boundary element simulations. function γ() can be approximated by the expression: ( ) (1 ) ,m n    (3) with: 2.915, 0.147,m n  (4) obtained via a simple least-squared-error, gradient-based parameter optimization. the results of the bem calculations together with the perfectly fitting analytical approximation are shown in fig. 4. fig. 1 cross section of an annular flat rigid punch indenting an elastic half space fig. 2 cross section of a conical concave rigid punch indenting an elastic half space fig. 3 cross section of a spherically concave rigid punch indenting an elastic half space 284 e. willert, q. li, v.l. popov fig. 4 results of bem simulations (circles) and analytic approximation (3) (solid line) for normalized normal force γ = fn / 2e * da, as a function of the ratio of contact radii =b/a for the frictionless, non-adhesive indentation of an elastic half space by an annular flat rigid punch in the case of a conical or spherical concave rigid indenter the inner contact radius b is not fixed but depending on the indentation depth or the normal force. if we put  as the governing parameter for the contact problem, the solution in either case can be written in the form: 1 * 2 ( ), 2 ( ).n d h f e ha       (5) barber [11] gave solutions for dimensionless functions γ1 and γ2 in the cases of  being close to zero or close to unity. for the conical hole results of our bem calculations are shown in fig. 5. apparently it is: 1 2( ) ( )con con     (6) and hence: *2 ,nf e da (7) i.e. the indenter almost behaves like a rigid flat cylindrical punch. this is especially true for values of  close to zero. note that =0 can never be reached as this would require an infinite normal force [11]. the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape... 285 fig. 5 results of bem simulations for normalized indentation depth γ1=d/h (crosses) and normal force γ2 = fn / 2e * ha (circles), as functions of the ratio of contact radii =b/a for the frictionless, non-adhesive indentation of an elastic half space by a conical concave rigid punch in more detail, both conical solutions can be approximated in the form: 2( ) ( ln ) ( ), 1,2,incon i i i ia b c i         (8) with the fitted parameters: 1 1 1 1 2 2 2 2 0.874, 2.09, 0.54, 0.15 0.945, 1.85, 0.45, 0.12 n a b c n a b c         (9) for the spherical concave indenter the solutions can be approximated very well by the functions: ( ) (1 ) , 1,2i im nsph i ia i     (10) with the parameters: 1 1 1 2 2 2 3, 2.034, 0.91 8 / 3, 2.015, 1.03. a m n a m n       (11) note, that it can be easily proven analytically – for the spherical concave indenter – that full contact, i.e. b = 0, is established for d = 3h and fn = 16/3e * ha [11]. parameters ai in eqs. (11) therefore have not been object to optimization. the results of the bem calculations for both concave indenter profiles together with the fitted analytic approximations from eqs. (8) and (10) are shown in fig. 6. 286 e. willert, q. li, v.l. popov (a) (b) (c) (d) fig. 6 bem simulations (circles) and analytic approximations (solid lines) for normalized indentation depth γ1 = d / h and normal force γ2 = fn / 2e * ha as functions of the ratio of contact radii  = b / a for the frictionless, non-adhesive indentation of an elastic half space by an either conical or spherical concave rigid punch: (a) γ1, conical (b) γ2, conical (c) γ1, spherical (d) γ2, spherical 3. the adhesive solution in the following we will denote the non-adhesive solutions by an upper index “na”. for example, na nf shall be the normal force given above for the non-adhesive problem, whereas fn shall be the full adhesive normal force. let us again consider the annular flat punch first. the non-adhesive normal force as a function of the non-adhesive indentation depth was given in eq. (1). within the jkrtheory the adhesion is modeled via an additional energy term: ,adu a w   (12) with contact area a and effective surface energy per unit area w. hence, the total energy will be: * 2 2 2( ) (1 ) .tot el adu u u e d a a w        (13) the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape... 287 the normal force is given by the derivative: *2 ( ).tot n u f e da d       (14) as the contact radii are fixed and not connected to the indentation depth, this is the same relation as in the non-adhesive case. the contact loses its stability and detaches at the outer boundary r=a, if: * 2[ ( ) ( )] 2 0, c tot c d d u e d a w a              (15) from which we deduce the critical indentation depth: 0 , ( ) ( ) c d d        (16) and the critical adhesion force: 0 ( ) , ( ) ( ) c f f          (17) with the respective values for the flat cylindrical punch of radius a obtained by kendall [17]: 3 * 0 0* 2 , 8 . a w d f a e w e       (18) the condition for detachment starting at the inner boundary r=b is analogously given by the relation: * 2 ( ) 2 0, c tot c d d u e d b w b           (19) rom which we obtain the critical indentation depth: 0 . ( ) cd d      (20) derivative γ() is always negative so this indeed will be a real length. as it can be seen in fig. 7 the absolute value of the critical indentation depth is always smaller for the detachment at the outer boundary. thus, the contact detachment will indeed start there. 288 e. willert, q. li, v.l. popov fig. 7 critical normalized indentation depths for the detachment at the inner (r=b) and outer (r=a) boundary for the jkr-adhesive normal contact of a rigid annular flat stamp let us compare our results to analytical calculations obtained earlier. for small values of  collins [4] gives the analytical series expansion: 3 5 6 7 2 2 4 4 8 16 ( ) 1 [ ], 3 15 27 o              (21) which can be used in eqs. (16) and (17) to calculate the critical contact configuration. for narrow contact areas, i.e.  near unity, argatov et al. [15] gave the asymptotic expressions: * 3 3 * 16 ( ) 2ln , ( ) , c c a b w d e f a b e w              (22) with the transformed (small) variable: 1 . 1       (23) in fig. 8 the results of argatov et al. [15] for  > 0.85 together with the series expansion (21) used in the relations (16) and (17) are shown as dashed lines. also the results, if the approximation (3) is used for eqs. (16) and (17), are given as solid lines. obviously, the curves overlap nicely for the limiting cases. the approximation obtained by non-adhesive bem calculations also shows the transition behavior for intermediate values of , for which the two analytical but asymptotic approaches do not agree with each other very well. thereby it has to be pointed out that the sixth-order series expansion by collins gives very good results for approximately  < 0.6. the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape... 289 fig. 8 normalized critical indentation depth (red) and normal force (blue) as a function of the ratio of contact radii  = b/a. dashed: analytical results by collins [4] (together with eqs. (16) and (17)) for  < 0.85 and by argatov et al. [15] for  > 0.85. solid line: approximation (3) together with eqs. (16) and (17) to solve the adhesive problem for the concave indenters we recall the non-adhesive solution from eq. (5): na na * 1 2( ), 2 ( ).nd h f e ha      (24) thereby one can define the non-adhesive contact stiffness: na na * 2 na 1 d ( ) ( ) 2 , ( )d n n f k e a d        (25) the prime denoting a derivative. according to argatov et al. [15] the adhesive solution is given by: na na na ( ) ( ) ( ), ( ) ( ) ( ) ( ), c n n n c d d l f f k l             (26) with the length: 2 na 4 . d / dn b w l k b      (27) using the non-adhesive solution obtained above, thus inserting eq. (25) into eq. (27) we obtain: 2 1 0 1 2 2 1 ( ) ( ) . ( ) ( ) ( ) ( ) cl d                  (28) 290 e. willert, q. li, v.l. popov the critical state, for which the contact loses its stability, can be derived from the relations: * 0 2 2 1 2 2 1 d d 2 ( ) ( ) 0, d d ( ) ( ) ( ) ( ) nf d e ha h                                (29) for fixed loads, and: 0 1 1 1 2 2 1 d d ( ) ( ) 0, d d ( ) ( ) ( ) ( ) dd h h                                (30) for fixed grips. in normalized variables d/h and fn / (2e * ha) the solution obviously only depends on dimensionless surface energy d0/h and ratio . in fig. 9 the normalized relations between indentation depth and normal force for different values of d0/h are shown for the conical and the spherical concave indenter (using the approximations of the non-adhesive solutions obtained in section 2). (a) (b) fig. 9 normalized relation between indentation depth d/h and normal force fn / (2e * ha) for different values of the surface energy * 2 0 / 2 /d h a w e h   for the jkr-adhesive normal contact of a concave indenter with the profile shape being: (a) conical and (b) spherical in fig. 10 we give the results for the critical state for both indenters. the values of  and the normal force – normalized for comparison on 3 * * 0 08 2f a e w e ad    – for which the contact loses stability and detaches, are given in dependence of the dimensionless surface energy for both fixed grips and fixed loads. obviously crit is always larger for the fixedgrips-regime. interestingly the critical normal force, often called adhesive or pull-off force, almost does not depend on the regime, as the curves for the fixed grips and fixed loads overlap for both the conical or spherical concave indenter. also it is visible that for d0 >> h the concave indenters can achieve pull-off forces significantly larger (up to 50% for conical and 60% for spherical profiles) than f0, i.e. the pull-off force for a flat cylindrical punch with radius a. the jkr-adhesive normal contact problem of axisymmetric rigid punches with a flat annular shape... 291 (a) (b) (c) (d) fig. 10 dependence of critical ratio crit and normalized adhesion force fcrit / (2e * ad0) on normalized surface energy d0/h with fixed grips and fixed loads for the jkr-adhesive normal contact of a conical or spherical concave punch. (a) crit, conical, (b) fcrit / (2e * ha), con., (c) crit, spherical (d) fcrit / (2e * ha), spher. 4. conclusions based on the formalism in [15], the jkr-adhesive normal contact problem with a ringshaped contact area has been solved for the annular flat and spherical or conical concave indenter. the non-adhesive problem was solved using fast boundary element simulations and the solutions were approximated by simple analytic expressions. after that the adhesive solutions can be obtained analytically from the non-adhesive one without problems. the method used can be applied to different axisymmetric concave indenters as well. we find that the annular flat punch will always start to detach at the outer boundary. the pull-off forces for both concave punch shapes almost do not depend on the pull-off boundary regime, i.e. fixed grips or fixed loads, and can be significantly larger than the adhesion force on a cylindrical flat punch. we also gave solutions for intermediate values of the ratio of contact radii, in which case the asymptotic results obtained in the literature – with this ratio being either close to zero or unity – do not overlap each other very well. 292 e. willert, q. li, v.l. popov references 1. spolenak, r., gorb, s., gao, h., arzt, e., 2005, effects of contact shape on the scaling of biological attachments, proceedings of the royal society of london, series a, 461, pp. 305–319. 2. johnson, k.l., kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids, proceedings of the royal society of london, series a, 324, pp. 301–313. 3. gubenko, v.s., mossakovskij, v.i., 1960, pressure of an axially symmetric circular die on an elastic half space, prikladnaya matematika i mekhanika, 24, pp. 334–340. 4. collins, w.d., 1962, on some triple series equations and their applications, archive for rational mechanics and analysis, 11, pp. 122–137. 5. collins, w.d., 1963, on the solution of some axisymmetric boundary value problems by means of integral equations. viii. potential problems for a circular annulus, proceedings of the edinburgh mathematical society, series 2, 13, pp. 235–246, doi: 10.1017/s0013091500010889 6. borodachev, n.m., borodacheva, f.n., 1966, penetration of an annular stamp into an elastic halfspace, mekhanika tverdogo tela, 1(4), pp. 158–161. 7. shibuya, t., koizumi, t., nakahara, i., 1974, an elastic contact problem for a half space indented by a flat annular rigid stamp, international journal of engineering science, 12, pp. 759–771. 8. gladwell, g.m.l., gupta, o.p., 1979, on the approximate solution of elastic contact problems for a circular annulus, journal of elasticity, 9, pp. 335–348. 9. roitman, a.b., shishkanova, s.f., 1973, the solution of the annular punch problem with the aid of recursion relations, prikladnaya mekhanika, 9(7), pp. 37–42. 10. antipov, y.a., 1989, analytic solution of mixed problems of mathematical physics with a change of boundary conditions over a ring, mechanics of solids, 24(3), pp. 49–56. 11. barber, j.r., 1976, indentation of the semi-infinite elastic solid by a concave rigid punch, journal of elasticity, 6, pp. 149–159. 12. shibuya, t., 1980, indentation of an elastic half-space by a concave rigid punch, zamm zeitschrift für angewandte mathematik und mechanik, 60, pp. 421–427. 13. barber, j.r., 1974, determining the contact area in elastic indentation problems, journal of strain analysis, 9, pp. 230–232. 14. kesari, h., lew, a.j., 2012, adhesive frictionless contact between an elastic isotropic half-space and a rigid axi-symmetric punch, journal of elasticity, 106, pp. 203–224. 15. argatov, i.i., li, q., pohrt, r., popov, v.l., 2016, johnson-kendall-roberts adhesive contact for a toroidal indenter, proceedings of the royal society of london, series a, 20160218, doi: 10.1098/rspa.20160218. 16. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17(4), pp. 334–340. 17. kendall, k., 1971, the adhesion and surface energy of elastic solids, journal of physics d: applied physics, 4, pp. 1186–1195. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 289 303 modes of vibration of the beams under the influence of discontinuity in foundation udc 519.6+531 vladimir stojanović 1 , pedro ribeiro 2 1faculty of mechanical engineering, university of niš, serbia 2 demec, faculty of engineering, university of porto, portugal abstract vibrations of the timoshenko beams resting on the winkler and pasternak elastic foundation with discontinuity are investigated in this paper. a p-version finite element method that accounts for shear deformation is used. this p-element has special displacement shape functions that make it particularly appropriate for dealing with problems with discontinuities such as those introduced in the foundation. a set of ordinary differential equations is derived; geometrical non-linearity is considered in these equations for the sake of generality and for future use. natural frequencies and mode shapes of vibration (composed by transverse displacements and rotations of cross sections) of the shear deformable beam are presented for diverse sizes and location of the discontinuity in the foundation. results of the present approach are compared with the ones computed via established finite element software for various stiffness of the elastic support of the winkler and pasternak type. key words: p-version fem, mode shape, natural frequency, foundation, discontinuity, vibrations 1. introduction vibrations of the beams on elastic foundations are of a wide practical interest involving applications such as analyses of roads, rail tracks and foundations of diverse structures. there have been a large number of publications related to this problem considering different types of foundation such as winkler, pasternak, elastic or viscoelastic, linear or non-linear (refs. [1-10]). for example, mamandi et al. [4] have studied nonlinear effects of the elastic foundation in frequency domain. investigations in the nonlinear regime, with the beam actuated by moving loads are carried out in [5]. kim and cho [6] explore the vibration of a shear beam-column, resting on an elastic foundation when the system is subjected to moving loads of either constant amplitude or harmonic amplitude variation. received september 8, 2014 / accepted october 30, 2014 corresponding author: vladimir stojanović faculty of mechanical engineering, department of mechanics, a. medvedeva 14, 18000 niš, serbia e-mail: stojanovic.s.vladimir@gmail.com original scientific paper 290 v. stojanović, p. ribeiro in either case, the load moves with a constant advance velocity; a good insight of the linear dynamical behavior of the beam on elastic foundations is given. a more complex linear model [7] is also of great interest for understanding the problems of response of the beams resting on visco-elastically damped foundation under moving sdof oscillators with a number of internal variables introduced and with the aim of representing the frequency-dependent behavior of the viscoelastic foundation. with larger beam deflections, geometric nonlinearities lead to motions which are not predictable by linear formulations. chang and liu [8] have investigated deterministic and random vibrations of a nonlinear beam on an elastic foundation, subjected to a moving load. the nonlinear system of differential equations is solved by an implicit direct integration method. rotary inertia and shear deformation are neglected, while the effects of longitudinal deflections and inertia are considered, based on the bernoulli–euler hypothesis. sapountzakis and kampitsis [3] have developed a boundary element method (bem) for the geometrically nonlinear response of shear deformable beams traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation; they have shown that the bem may be advantageous for exploring nonlinear effects on this kind of problems. nonlinear foundation effects are also investigated in a different type of structures. malekzadeh and vosoughi [9] have investigated problems of composite thin beams using an efficient and accurate differential quadrature (dq) method for a large amplitude free vibration analysis. demeio and lenci [10] have used the multiple time scales (mts) method to study nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation. the finite-element method (fem) is based on approximating the solution of a problem by means of admissible functions. in the p-version of the fem, accuracy is improved by increasing the number of shape functions over the elements, without introducing more elements in the mesh; it has been often found that the p-version fem is an efficient approach to the study of mechanical systems vibrations [11-21]. recently, new shape functions have been proposed to be used when the p-version finite element method is applied to problems with steep changes in the domain [22]. in the later work, shear deformable beams with discontinuity in the cross section, due to a notch, are analyzed. the two shape functions for p-version beam elements lead to a significant improvement of the efficiency of the p-version fem in the presence of notches. this work explores the influence of discontinuity of foundations on the natural frequencies and mode shapes of beams, for various sizes and locations of the discontinuity. in the real world, one example of occurrence of large discontinuity in a foundation is provided by a railway crossing a bridge; in a simple model, the beam with discontinuity in the foundation would represent the bridge. a second type of real world example, this time with a discontinuity of small length, is an anomaly in the subsoil, which occurs very often. two types of foundation (winkler and pasternak) are considered. thefirst order shear deformation theory is used because it provides more accurate results than bernoulli-euler formulations, particularly in the case of thick beams. the equations of motion are obtained by the principle of virtual work [23]. although only linear analyses are performed in the numerical tests, the presented model includes geometrical non-linear effects for future use. the model is validated by comparison with finite element software ansys [25] in the computation of natural frequencies. modes of vibration of beams under the influence of variable discontinuity in foundation 291 2. mathematical model and equations of motion the physical model is defined by beam's length l, width b and thickness h. the discontinuity in the foundation is defined by l1 and l2 visualized in fig. 1. the beam material is assumed to be elastic, homogeneous and isotropic. the foundation is represented by an elastic layer of the winkler's and pasternak type (neglecting the shear layer reduces the foundation to winkler type). more details on the two types of foundation are given in ref. [3]. co-ordinate axis x and z are also shown in fig. 1. fig. 1 timoshenko beam supported by elastic foundation with discontinuity non-dimensional co-ordinate, ξ, represented in fig. 1 is the local coordinate typical of the finite element method. adopting the first order shear deformation theory [29], the displacement field of the model is given by: 0 0 ( ), , , ,( ) ( )u u zx z t x t x t  , (1) 0( , , ) ( , )w x z t w x t , (2) where superscript “ 0 ” indicates axis x, which crosses the cross section centroids and t represents time. letters u and w represent, respectively, the displacement components along axes x and z. the independent rotation of cross-sections about the axis parallel to y is given by θ 0 (x,t). for the sake of generality, geometrical non-linearity will be considered in the formulation. the longitudinal and shear strains are, therefore, written as: 0 0 2 0 , , ,( ) ( ) ( ) ( ) 1 , , ( , ) z , 2 x x x xx t u x t w x t x t    , (3) 0 0 ,( ) (, ,) ( ),xz xx t w x t x t   . (4) a comma in subscript, followed by a variable, represents partial derivation with respect to the latter. vector d0(ξ,t), which is formed by the components of displacement, is written as the product of shape functions by the generalized coordinates: 292 v. stojanović, p. ribeiro 0 ( ),  ( )  ( )t t d n q , (5) 0 t 0 t 0 t ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ,  0 0 ,    0 0 ,  0 0 u w u t w t t t tt                                       u w θ n q n q qn . (6) in the equation above, n u (ξ), n w (ξ) and n θ (ξ) are, respectively, the in-plane, the out-ofplane and rotation shape functions, which together form the matrix of shape functions n(ξ). time dependent generalized displacements are represented by: qu(t) (generalized longitudinal displacements); qw(t) (generalized transverse displacements); qθ(t) (generalized rotations about y axis). the shape functions associated to the discontinuity are represented by superscript “d”. consequently, the row vectors of longitudinal, transverse and rotation shape functions, are respectively, the following: t 1 2 1 2 1 2 1 2 3, ,   , ,( ) ( ) ( ) ( ) (  |          )    2 2 u d d pu l l l l f l f l f f f f                        n , (7) t 1 2 1 2 1 2 1 2 3, ,   , ,  |             2 2 ( ) ( ) ( ) ( ) ( )w d d pw l l l l f l f l f f f f                        n , (8) t 1 2 1 2 1 2 1 2 3, ,   , ,( ) ( ) ( ) ( ) ( |              ) 2 2 d d p l l l l f l f l f f f f                         n , (9) the total numbers of longitudinal, transverse and rotational shape functions employed are, respectively, pu, pw, and pθ plus two (due to functions f1 d and f2 d ). the last two functions are here associated to the discontinuity of the foundation and are explained in detail in ref. [22]. in the p-version finite element method, the number of elements is essentially defined by the geometry of the structure to analyze. for example, in ref. [14] a portal plane frame constituted by three straight beams is analyzed using three p-version beam elements. but localized steep variations, as the ones introduced by discontinuity, are thought to advocate the use of several elements, diminishing the interest of p-elements. shape functions f1 d and f2 d provide an answer to this disadvantage of the p-version approach. polynomial of types f1(ξ) f4(ξ) represent hermit cubics [18]. the constitutive equation of an isotropic beam is: 0 0 x x xz xz e g                      σ dε , (10) where e is the young modulus and g= e/[2(1+ν)] is the shear modulus of elasticity, with ν representing poisson ratio. in eq. (10) d represents matrix of elastic constants, σ and ε, respectively, the non-zero stresses and the strains in form of vectors. shear correction factor λ employed is λ=(5+5ν)/(6+5ν), as given in ref. [26], because of the good agreement with the experimentally obtained results [27]. the longitudinal strain is: 0 0 1     0 p p l x b z                              , (11) modes of vibration of beams under the influence of variable discontinuity in foundation 293 where ε0 p and zε0 b represent longitudinal and bending strains, and εl p geometrically non-linear longitudinal strain. these strains are: t t t t t t , 0 , , , ,0 2 2 2 2 2 , , , p pu b w w w xzl l l ll                         w u θ w w θ q n q n q q n n q n n q . (12) integrating the normal stress, the shear stress and the moment of the normal stress we obtain: 0 0 ,  2(1 )0 p p l x xzb t a b eh q m b d                                        , (13) where: 3 2, , (1,  ,  )  dz , 0, 12 z eh a b d z z e a eh b d     . (14) the equations of motion are achieved by the principle of virtual work, according to which: 0in v exw w w     , (15) where δwin, δwv and δwex are, in this order, virtual works done by inertia, internal and external forces due to a virtual displacement with components δu, δw and δ. these components form vector δd, as follows:   u w                 d n q . (16) the virtual work of internal forces is: t t 0 0t t 0 0 d dx ( ) 0 0 p pp p l l v b b v l a b w v b b d                                                                 ε σ q k q q , (17) and the virtual work of the inertia forces is: t din v w v    d d / 2 / 2 t / 2 / 2 ( ) dx dz h l h l b uu ww          q mq , (18) where ρ is the mass per unit volume, m the mass matrix and  d =d 2 d/dt 2 . the virtual work of external forces is given by: 2 2 ( , ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) [ ( ) ( ) ( , )] ( , ) [ ( ) ( ) ( , )] ( , ) { } j d ex w j w w p l j d j d u j u j t u w w x t w p t x x p x t k w x t w x t k w x t x p t x x p x t u x t m t x x m x t x t q q q                                               e u e w e f f m (19) 294 v. stojanović, p. ribeiro where {fu e (t), fw e (t), m e (t)} represents the vector of generalized external forces and δ(xxj) represents dirac delta function. p j (t) and m j (t) are concentrated forces or moments acting at point x=xj, p d (t) and m d (t) are distributed forces or moments. in the eq. (19) kw and kp represent, respectively, the winkler and pasternak stiffness and shear foundation moduli. using the virtual work principle, including the foundation effect, the following equations of motion are obtained:       11 12 21 22 2 3 4 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 ll b w p r b n n n                                                                              u u w w θ θ e w uu e w w w w e θ km q q m q k k k k q q qm k k k k q fq k q k q q f q m (20) matrices of type m and k are constant matrices that originate linear terms in the equations of motion, of the latter type, matrices k w and k p represent linear influences of the foundation with discontinuity and are not introduced in [22]. the matrices that depend on wq (t) kn 2 , kn 3 and matrix kn 4 – lead to non-linear terms. the superscripts l,b,r and γ represent, respectively, longitudinal, bending, cross section rotation and shear effects. adding rayleigh-type damping, with damping coefficients αr and βr, one obtains the equations of motion (in a more condensed notation) : n( ) ( ) ( ) ( ) ( ( ) )( ) ( ) ( ) ( )r rt t t t t t     mq k q m q k k q q f . (21) the mass and stiffness matrices have the following forms: 2 1 1 1t 2 2t 21 d d 2 2 c c l l l l l hbl hbl        m n n n n u u u u , (22) 2 1 1 1t 2 2t 21 d d 2 2 c c l l b l l hbl hbl        m n n n n w w w w , (23) 2 1 1 1t 2 2t 21 d d 2 2 c c l l r l l hbl hbl         m n n n n θ θ θ θ , (24) 2 11 1t 2 2t 21 2 2 d d c c l l l l l ehb d d ehb d d l d d l d d          n n n n k u u u u , (25) modes of vibration of beams under the influence of variable discontinuity in foundation 295 1 11 2 11 1t 2 2t 21 2 2 d d c c l l l l ghb d d ghb d d l d d l d d              n n n n k w w w w , (26) 12 2 11 2 1t 2t 21 d c c l l l l d d ghb ghb d d d            n n k n n w w θ θ , (27) 21 2 11t 2t 1 2 21 d c c l l l l d d ghb d ghb d d            n n k n n w w θ θ , (28) 22 2 1 1t 1 2t 2 21 2 2 d d c c l l l l ghbl ghbl         k n n n n θ θ θ θ , (29) 1 2 2 1 1 1t 2 2t 21 d d 2 2 l l w w w l l l l k k     k n n n n w w w w , (30) 1 2 2 11 1t 2 2t 21 2 2 d d l l p p p l l d d d d k k l d d l d d          n n n n k w w w w , (31) 2 t t11 1 2 2 2 0 0 2 2 21 ( ) 2 2 d d c c l l n l l dw dwebh d d ebh d d d d d d d dl l            w n n n n k q u w u w , (32) t3 22( ) ( )n nw wk q k q , (33) 2 t t12 21 1 2 2 4 0 0 3 3 21 4 ( ) 4 d d c c l l n l l dw dwebh d d ebh d d d d d d d dl l                        w n n n n k q w w w w , (34) 2 t t11 1 2 2 2 0 0 2 2 21 ( ) 2 2 d d c c l l n l l dw dwebh d d ebh d d d d d d d dl l            w n n n n k q u w u w , (35) where lc=(l1+l2)/2. 3. natural frequencies and mode shapes the p-fem will be applied to derive a model for a clamped-clamped beam with geometric and material properties based on an example from ref. [3]. used data about the 296 v. stojanović, p. ribeiro beam are: l=10m, e=210gpa, g=77gpa, i=30.55*10 -6 m 4 , a=30.55*10 -6 m 2, ρ=7850kg/m 3 , kw1=20mpa, kp1=69kn, kw2=35mpa, kp2=200kn. fig. 2 timoshenko beam supported by the elastic foundation without discontinuity (case 1) fig. 3 timoshenko beam supported by the elastic foundation with discontinuity l1=1.67m, l2=1.85m (case 2) four cases of the shear deformable beam on elastic foundation are shown on figs. 2, 3, 4 and 5. for the presented cases and two types of foundation, natural frequencies are given in tables 1-8. for the case without discontinuity in foundation, natural frequencies for simply supported beam can be easily obtained from the frequency equation (27) of ref. [24], for the timoshenko theory, by setting m=1. fig. 4 timoshenko beam supported by the elastic foundation with discontinuity l1=1.67m, l2=2.6m (case 3) modes of vibration of beams under the influence of variable discontinuity in foundation 297 fig. 5 timoshenko beam supported by the elastic foundation with discontinuity l1=l/6 (case 4) the ansys h-version element used is element beam 189, which has 3 nodes with six degrees of freedom at each node (but to analyze vibrations in a plane only 3 degrees of freedom per node are required). beam189 is an element based on the timoshenko beam theory and suitable for analyzing slender to moderately thick beam structures. the current results from ansys are obtained with 100 elements. table 1 natural frequencies [hz] of a clamped-clamped beam case 1 for kw1 and kp1 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 92.5 92.33 92.4 92.35 2 97.2 96.90 97.1 96.97 3 111.3 110.45 111.2 110.58 4 137.4 136.66 137.3 136.84 table 2 natural frequencies [hz] of a clamped-clamped beam case 2 for kw1 and kp1 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 90.7 90.55 90.7 90.57 2 95.8 95.59 95.9 95.65 3 110.5 110.27 110.6 110.40 4 136.5 136.27 136.6 136.45 298 v. stojanović, p. ribeiro table 3 natural frequencies [hz] of a clamped-clamped beam case 3 for kw1 and kp1 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 77.2 77.07 77.3 77.08 2 94.3 94.06 94.4 94.11 3 108.5 108.26 108.6 108.38 4 136.1 135.85 136.2 136.02 table 4 natural frequencies [hz] of a clamped-clamped beam case 4 for kw1 and kp1 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 35.3 35.08 35.3 35.06 2 59.6 59.30 59.6 59.30 3 77.5 77.31 77.5 77.33 4 104.5 104.33 104.5 104.42 changes in the natural frequencies have a tendency of decreasing as the discontinuity in the foundation increases. this model is important for analysis because of the easy generalization of the discontinuity in the foundation. non-linear matrices are just obtained and presented in the work and can be used for a further nonlinear analysis in the time domain. table 5 natural frequencies [hz] of a clamped-clamped beam case 1 for kw2 and kp2 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 122.1 121.74 122.2 121.78 2 126.5 125.20 126.6 125.36 3 136.1 135.91 136.2 136.22 4 158.1 157.90 158.2 158.35 table 6 natural frequencies [hz] of a clamped-clamped beam case 2 for kw2 and kp2 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 119.2 118.84 119.3 118.90 2 124.1 123.81 124.2 123.94 3 135.9 135.68 136.1 135.98 4 157.6 157.33 157.7 157.77 modes of vibration of beams under the influence of variable discontinuity in foundation 299 table 7 natural frequencies [hz] of a clamped-clamped beam case 3 for kw2 and kp2 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 97.2 97.09 97.2 97.10 2 123.1 122.88 123.1 122.96 3 133.7 133.54 133.7 133.83 4 156.9 156.66 156.9 157.10 to understand the effect of the foundation on the vibration modes, natural frequencies are not enough. it is important to present the changes in the natural mode shapes. in figs. 6-11 natural mode shapes are presented for all the cases of discontinuity in the foundation. it is evident that the effect of the discontinuity results in the change in natural transverse and rotation mode shape. it is important to underline that the asymmetrical mode shapes appear when the discontinuity exists on one side of the beam. table 8 natural frequencies [hz] of a clamped-clamped beam case 4 for kw2 and kp2 mode winkler pasternak ansys beam189 100 elements p-fem (30sf) ansys beam189 100 elements p-fem (30sf) 1 39.9 39.71 39.9 39.65 2 59.5 59.30 59.5 59.30 3 92.5 92.37 92.5 92.38 4 117.9 117.75 117.9 117.75 fig. 6 transverse components of mode shapes mode 1 300 v. stojanović, p. ribeiro fig. 7 transverse components of mode shapes mode 2 fig. 8 transverse components of mode shapes mode 3 fig. 9 cross section rotation components of mode shapes mode 1 modes of vibration of beams under the influence of variable discontinuity in foundation 301 fig. 10 cross section rotation components of mode shapes mode 2 fig. 11 cross section rotation components of mode shapes mode 3 for the cases 2 and 3 the obtained results have a significant importance in understanding the vibrations of the geometrically asymmetrical model. the advantages of the p-fem model are better approximations of the solutions in comparison with commercial software ansys (convergence occurs from above in the p-version fem, and our values are lower than the ones of ansys), with a lower number degrees of freedom. in a number of cases, the values computed via ansys for the natural frequencies of vibration considering the pasternak and winkler type foundation were the same. this rarely occurrs with the pversion model, which generally allows to detect the effect of the shear layer. 5. conclusions vibrations of shear deformable beams, elastically connected to a foundation with discontinuity, are investigated in this paper. a p-version finite element based on the timoshenko theory for bending is applied. the frequencies obtained agree with the ones computed via well know finite element software. the developed matrices are easily applicable to any size of discontinuity in the foundation, simply taking into account changes 302 v. stojanović, p. ribeiro in the boundaries of the discontinuity. various discontinuity cases (when the centre of the discontinuity is not under the middle of the beam) lead to asymmetrical vibrations. this is made evident by the mode shapes presented. the p-version finite element method developed gives the possibility 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posebno razvijenih oblika funkcija primenjenih na nosačima sa oštećenjem i upotrebljen na modelu sa osnovom koja sadrži diskontinuitet. novina ove studije predstavlja laku generalizaciju pristupa pri određivanju prirodnih frekvencija, opštih oblika oscilovanja (transverzalnih i rotacija poprečnih preseka) nosača za proizvoljno izabrane veličine i lokacije diskontinuiteta. izveden je sistem parcijalnih diferencijalnih jednačina koji omogućava dalje istraživanje u nelinearnom vremenskom domenu oscilovanja. u radu su prikazana poređenja rezultata sa različitim vrednostima krutosti nelinearne elastične osnove winkler-ovog i pasternak-ovog tipa. ključne reči: p-verzija mke, osnovni oblik oscilovanja, prirodna frekvencija, diskontinuitet, oscilacije. facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 579 600 https://doi.org/10.22190/fume210711062z © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper combining the suitability-feasibilityacceptability (sfa) strategy with the mcdm approach sarfaraz hashemkhani zolfani1, ramin bazrafshan2, parnian akaberi2, morteza yazdani3, fatih ecer4 1school of engineering, catholic university of the north, larrondo, coquimbo, chile 2department of industrial engineering and management systems, amirkabir university of technology (tehran polytechnic), tehran, iran 3esic university, madrid, spain 4department of business administration, afyon kocatepe university, afyonkarahisar, turkey abstract. suitability-feasibility-acceptability (sfa) is a fundamental tool for the development and selection of strategy. any type of decision-making problem can be resolved by multiple criteria decision making (mcdm) methods. in this research, we explore the complexity of determining the proper goal market for the chilean fish market. this study proposed a combined approach of sfa with mcdm methods in a real case study. the proposed structure helps to assign the best market for chilean export fish to west asia. three countries (saudi arabia, the united arab emirates, and oman) are selected as a target market in this region, and then related criteria are obtained from various sources. in order to develop a new market for the chilean fishery industry, five major criteria, including the potential of a target market, region's economic attractiveness, consumption of the seafood, location, cost of transportation, and country risks, were selected based on the sfa framework. calculating the criteria weights is performed by the best-worst (bwm) method, and ordering the alternatives is operated by measurement alternatives and ranking according to compromise solution (marcos) methods. the results showed that oman is the best destination (importer) for the chilean fish market (salmon fish as the case). key words: international markets, multiple criteria decision making (mcdm), suitability-feasibility-acceptability (sfa) method, export received july 11, 2021 / accepted september 23, 2021 corresponding author: morteza yazdani esic university, cam. valdenigriales, s/n, 28223 pozuelo de alarcón, madrid, spain e-mail: morteza.yazdani@esic.university mailto:morteza.yazdani@esic.university 580 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer 1. introduction in recent years, worldwide, some concepts such as globalization, fast technological changes, the appearance of new markets, and changing customer expectations cause fierce emerging competition. this phenomenon forced stakeholders to think strategically and make strategic plans for their business [1]. strategic planning is the art of creating strategies and aligning a business’s vision for the future of industries or markets. in order to select the proper strategy, many strategic options should be surveyed. the strategic planning process usually consists of three critical steps [2]: (i) strategy formulation, (ii) strategy implementation, and (iii) strategy evaluation. in strategy formulation, managers survey markets and make decisions that concentrate on their plan or generally ignore it. by choosing suitable strategies or plans, the company implements them in order to achieve the desired results. in the final step, the performance of the selected strategy is evaluated. in addition to the strategic planning process, some tools are introduced to analyze strategic possibilities. strategic analysis is a process of seeking the operating environment of a company to formulate a strategy. this analysis consists of three main factors [3]: (i) identification and evaluation of data relevant to strategy formulation, (ii) analyzing the internal and external environments, and (iii) use of the analytic method. there are various analytic methods in literature like swot analysis, pest analysis, porter’s five forces analysis, four corner’s analysis, value chain analysis, early warning scans, war gaming. these methods are used in various fields. lee et al. used swot analysis to measure the limitations and strengths of the korean space and satellite industry. sometimes researchers combined them with other methods [4]. sahani combined swot with the mcdm method like ahp and fuzzy_ahp to formulate and prioritize ecotourism strategies in western himalaya, india [5]. johnson et al. developed a matrix named sfa to evaluate and analyze the strategies [6]. this method consists of three main sections such as suitability, feasibility, and acceptability. the criteria that the managers take into consideration for comparison purposes should categorize in these three sections. the strategies are listed as options in this method, and each criterion gets scores basis on the experts' or managers' verdict. then, the options are compared by these scores. the sfa covers a varied range of criteria and items. this is an excellent characteristic that helps the managers to compare the strategies from various aspects. however, this is a plausible method but has some constraints for real problems or complicated issues. it has not a distinct method for assigning a weight of criteria. this is one of the problems that make the managers less ready to use this analytic method. the sfa method structure is very similar to the mcdm methods, both of them distinguish the priorities of alternatives by evaluating the criteria. this similarity caused the researchers to decide to utilize the mcdm methods, assign weight to criteria, and evaluate the strategies' options. the sfa strategy first selects some criteria and then evaluates them, most likely the mcdm methods. this study attempts to combine the sfa framework with the mcdm methods. showing the efficiency of this combined methodology, the researchers propose a practical case study and implement this new method on it. section 2 reviews literature about sfa and mcdm methods and explains this study's objective. section 3 describes the methodology of the sfa and mcdm methods. section 4 addresses the application of this method to exporting chilean fish and its implementation as well as the research gap and discussion. finally, section 5 provides conclusions. combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 581 2. literature review 2.1. a summary of sfa background nowadays, using mcdm methods in strategic planning is vital because these methods can be combined with the strategic method and can promote their outcome and accuracy. the mcdm methods with mathematical formulation could help the managers to select the optimum option or the best strategy with the highest degree of satisfaction in the board of managers [7]. in the following, several works that used the mcdm methods in the strategic fields are mentioned. mehrjerdi used the mcdm method for selecting the strategic system with linguistic preference and gray information. they proposed a method for selecting alternatives in the presence of uncertainty and determined optimal choice among seven possible alternatives. they compared the obtained results with quantitative strategic planning matrix (qspm), technique for order of preference by similarity to ideal solution (topsis), and simple additive weighting (saw) methods and gathered a similar ranking with topsis and saw. their results also validate that the ranking obtained by the qspm is inferior in the comparison methods [8]. selecting an appropriate vendor is often a non-trivial task, in which multiple criteria need to be carefully examined. however, many decision-makers or experts select vendors based on their experience and intuition. shyur and shih used a hybrid mcdm model for strategic vendor selection. they proposed a five-step hybrid process, which incorporates an analytic network process (anp) technique. more clearly, the anp method is used to obtain the relative weights of criteria. then, the modified topsis is adopted to rank competing products in terms of their overall performance. they reported that the proposed method is practical for ranking competing vendors regarding their overall performance concerning multiple interdependence criteria. they declared that the consideration of relationships between criteria provides the organization with a way to devise and refine adequate criteria and alleviate the risk of selecting sub-optimal solutions [9]. banihabib et al. examined the strategies to tackle water shortage for sustainable development in shahrood, iran. in their paper, a contentious plan has been proposed to transfer water from the caspian sea north of iran to this region. they used strengths, weaknesses, opportunities, and threats (swot) analysis. due to the swot model's inability to rank the alternatives, the developed strategies are ranked using mcdm models based on specified sustainable development criteria. the ranking model was programmed by using the compensatory models of saw and analytical hierarchy process (ahp) and the non-compensatory model of elimination and choice translating reality iii (electre iii). all mcdm models' results introduced water transfer as the worst strategy for a region [10]. hashemkhani zolfani et al. presented a new strategic hybrid model for international market selection based on market attractiveness and business attractiveness (maba) analysis and the edas method (one of the latest mcdm methods). they worked on a case in iran's food industry and could develop the primary model of the maba analysis in the mcdm outline. using this model leads to selecting the most suitable and profitable market, considering several quantitative and qualitative factors [11]. as noted, one essential strategic planning, which has three assessment criteria, is the sfa strategy. it has been utilized as a powerful marketing method since the researchers received valuable strategic planning results while using that in a real problem. the following paragraphs review some research projects which used sfa in their studies. in order to get incremental improvement, companies and organizations try to attain an effective strategy. georgise & mindaye examined the suitability, acceptability, and feasibility of kaizen – a japanese concept used for continuous 582 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer improvementamong smes in ethiopia. since organizations were eager to use this strategy, the researchers decided to evaluate kaizen's feasibility, suitability, and acceptability for these organizations. the study results showed that although some enterprises think it is a confusing strategy, most of them are willing to implement kaizen in their companies (acceptability). the study also found that the feasibility of kaizen practices is possible, despite being a bit challenging. as a result, the study showed that kaizen as an effective strategy is accepted in southern region, ethiopia organizations and can improve their performance. however, still, its feasibility seems challenging [12]. čirjevskis & novikova investigated the commercial viability of green energy business to make an investment choice for latvian hydropower producer and seller llc “green energy solutions”. they investigated the theoretical and practical application of such concept of the commercial viability of a strategy as an sfa, explored the latest trends of green energy business in eu and latvia, and defined strategic suitability. the research team calculated equivalent annual annuities of each alternative investment project and discussed financial feasibility to confirm disproving investments in a hydropower station or wind turbine [13]. alimardani et al. presented a new-hybrid strategic model based on the swara method and yin-yang balance theory to design products with both international and local perspectives [14]. dalic et al presented a new hybrid mcdm model applied in swot strategic tool for decision-making in a transportation company. they applied fuzzy piprecia, fucom, swot, and marcos methods in their study [15]. amoozad mehrjerdi et al. presented a hybrid mcdm model based on bwm and interval-valued intuitionistic fuzzy todim for evaluating strategies for implementing industry 4.0 [16]. ullah et al. reviewed tourism resources in ecologically sensitive coastal areas of baluchistan to assess their potential for establishing community-based ecotourism following the sfa framework. the collected information about the coastal regions was analyzed through swot analysis and fuzzy logic analysis. the results showed that the introduction of cbe within the selected localities without any investment in basic infrastructure and capacity building of communities would inevitably negatively impact the natural environment because the infrastructure and communities’ knowledge for developing the desired services were below the required standards [17]. puška et al. used multi-criteria analysis methods for ranking project management programs. they perform the marcos method for evaluating. since there are many software solutions for the project manager, selecting the best one is critical. so, the researchers choose four softwares: smart sheet, asana, microsoft project, and basecamp. they evaluate them by seven scenarios and conclude that the smart sheet is the best [18]. pamučar & savin choose the off-road vehicle for transportation activities in the serbian armed forces because selecting the proper vehicle increases the safety, quality, and efficiency of load carried out. they used the hybrid method bwm-copras for this selection. seven criteria are introduced by them with each of them having seven sub-criteria. for verification of the results, they used bwm-mabac and bwm-marca models [19]. hashemkhani zolfani et al. have proposed a vision-based weighting system (views) for the managers to consider time vision in their decision-making. they used a hybrid method edas-pmadm for this study. the three-time concept is analyzed (current, 2025, and 2030) and shows that the ranking of alternatives is changed by time. the policymakers by this method can make good decisions for the future of their company [20]. hasheminasab et al. implement the circular economic (ce) for minimizing the harmful effect of using fossil fuel. they consider three different fossil fuels (oil, gas, and coil) for selecting the most sustainable fuels. the extended-swara method is used for evaluating the ce criteria. then they used the combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 583 marcos method for ranking, and they showed that gas is the most sustainable fuel of the two others [21]. hashemkhani zolfani, et al. developed a novel integrated decision-making tool for selecting the most profitable market. they consider multiple factors like: social, political, economic, and ecological. the hybrid model of market attractiveness and business attractiveness (maba) with edas proposed and evaluated several international markets by this method [11]. behzad et al studied the waste management system. they introduced seven criteria: waste generation, composting waste, recycling waste, and landfilling waste, recycling rate, waste to the energy rate, and greenhouse gas emissions from waste. they used the hybrid method bwm-edas for weighting and evaluating the criteria and ranking them. the five countries are considered as alternatives: denmark, finland, iceland, norway, and sweden. the result showed that sweden has the best waste management profile (0.9748) [22]. hashemi et al. used the mcdm method for feature selection. they applied the topsis method for evaluating multi-label data. the ridge regression algorithm is used for constructing a decision matrix; for calculating the weight of this matrix, they implement the entropy method. they ranked the features and said the user could select a desired number of features [23]. table 1 represents some recent studies about the sfa strategy mentioned above. table 1 studies related to sfa strategy goal author/s 1 evaluating kaizen strategy usage among smes bete georgise & mindaye [12] 2 evaluating strategic options of kaizen (a business management concept) bwemelo [10] 3 assessing community-based ecotourism potentials of coastal areas of baluchistan ullah et al. [17] 4 evaluating the potential success or failure of a project abu hassan & moshdzir [24] 5 making an investment choice for corporations čirjevskis & novikova [13] according to the above research, it has been recognized that the sfa is a valuable and productive method. the scientist and stakeholder intend to use it more than before if the degree of conformity is improved. combining the mcdm method with strategic planning gives a significant result. therefore, this study tries to boost the accuracy of the sfa by using an mcdm method. in general, the research question concerns the main benefits of combining the mcdm approach in the sfa concept to improve strategy development. 2.2. research objective and novelty according to the research question, below are the main aims to reach: ▪ improve the sfa strategy for complex problems and increase its accuracy. the sfa is used just for nominal value criteria, but this combination could use the criteria with no nominal value, and, ▪ calculate the weight of criteria by a distinct method. the sfa method allocates criteria weights based on their importance. in other words, the more critical the criterion is, the more amount of weight it will be given during evaluations. however, this method has not introduced a specified way of calculating weights. section 3 explains the sfa strategy and the mcdm method, which is used in this strategy. in section 4, a case study is analyzed with these new criteria and, based on this process, concluded consequences in the last quarter. 584 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer the proposed model has novelty due to these reasons: ▪ it reveals a new perspective for strategy formulation that improves in several aspects. this enables experts and strategists to incur. ▪ there is no study in the history of strategy planning and decision-making with multiple attributes to measure the performance of strategies. ▪ application of the combined evaluation structure leads to an improved and reliable process that experts can comprehend. 3. methodology this section firstly introduces the sfa strategy processes and then describes the bestworst mcdm steps. 3.1. sfa strategy processes child was one of the significant authors who discussed strategic choice amongst organizational theorists [24]. čirjevskis and novikova claimed that the concept of strategic choice initially originated from the perception that its operational strengths and opportunities define its direction [13]. johnson et al. had a similar approach to strategic choice. they were the major contributors to the strategy choice viability by applying a clear model sfa of examining strategic opportunity through three assessment criteria: suitability, feasibility, and acceptability [25]. strategic choices involve the options for strategy in terms of both the directions in which strategy might move and the methods by which strategy might be pursued. once a set of strategic options has been established, it is time to evaluate their relative merits. the sfa framework suggests three criteria (see table 2). suitability asks whether a strategy addresses the key issues relating to the opportunities and constraints an organization faces. acceptability asks whether a strategy meets the expectations of the stakeholders. last, feasibility invites an explicit consideration of whether a strategy could work in practice. in other words, suitability is related to its strategic position and whether its strategic choice matches the external environment and company resources and capabilities. feasibility is concerned with assessing the company’s internal capabilities in terms of financial resources. finally, acceptability relates to evaluating whether the chosen strategies can meet stakeholders’ expectations in terms of outcomes. according to this model, strategic options should be evaluated before implementing them in a new context. three ‘strategic option evaluation tests’ are suggested, which helps us evaluate this nature's strategic choice before applying it to a particular environment. these are the suitability test, acceptability test, and feasibility test. the suitability test considers whether the option is the right one in given circumstances. the acceptability test considers whether the strategic option will gain crucial support from the corresponding parties or lead to opposition or criticism. further, the feasibility test considers whether a company can successfully carry out the strategic option [25]. combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 585 table 2 the saf criteria and techniques of evaluation the saf criteria scope suitability (focused on external factors) ▪ does a proposed strategy address the key opportunities and constraints an organization faces? acceptability (focused on the internal factor) ▪ does a proposed strategy meet the expectations of stakeholders? ▪ is the level of risk acceptable? ▪ is the likely return acceptable? ▪ will stakeholders accept the strategy? feasibility ▪ would a proposed strategy work in practice? ▪ can the strategy be financed? ▪ do people and their skills exist, or can they be obtained? ▪ can the required resources be obtained and integrated? 3.2. best-worst method (bwm) rezaei proposed a new mcdm method called the best-worst method (bwm). the bwm method has made substantial advancements in weight determination. according to bwm, the decision-maker identifies the best (e.g. most desirable, most important) and the worst (e.g. least desirable, least important) criteria. pairwise comparisons are then conducted between these two criteria (best and worst) and the other ones. a max-min problem is then formulated and solved to determine the weights of different criteria. the weights of the alternatives concerning different criteria are obtained using the same process. the alternatives' final scores are derived by aggregating the weights from different criteria and alternatives, based on the best alternative which is selected [26]. bwm has been successfully applied in many areas. torkayesh et al. applied it for the assessment of healthcare sectors in eastern european countries [27]. pamucar et al. addressed bwm to select the most preferred renewable energy source for a developing country [28]. ecer performed it for the sustainability evaluation of wind plants [29]. for sustainable supplier evaluation, ecer and pamucar utilized the bwm technique [30]. hashemkhani zolfani et al. handled it for selecting the best location for a newcomer in chile [31]. besides, some researchers performed it successfully in various fields [32-35]. the steps of the bwm method for calculating the weights of criteria are defined below. step 1: in this step, decision-makers determine a set of decision criteria. step 2: after selecting decision criteria, they should separate the best and the worst criteria. step 3: the preference of the best criterion over all the other criteria should be determined, for this we could use a number between 1 and 9. the resulting best-to-others vector would be: 1 2 ( , ,..., ) b b b bn a a a a= , where abj indicates the preference of best criterion b over criterion j and abb =1. step 4: the preference of all the criteria over the worst criterion is determined, and for this we could use a number between 1 and 9. the resulting others-to-worst vector would be: 1 2( , ,..., )t w w w nwa a a a= where ajw indicates the preference of criterion j over worst criterion w and aww =1. 586 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer step 5: find the optimal weights * * * 1 2( , ,..., )nw w w .the optimal weight for the criteria is the one where, for each pair of wb/wj and wj/ww, wb/wj=abj and wj/ww=ajw. to satisfy these conditions for all j should find a solution where the maximum absolute differences b bj j w a w − and j jw w w a w − for all j is minimized. considering the non-negativity and sum condition for the weights, the following problem emerges: min max , j jb bj jw j w ww a a w w    − −     (1) s.t. 1j j w = (2) 0,j for allw j (3) the above formulation could be transferred to the following formulation: min  (4) s.t. , − b bj j for w a w all j (5) , −  j jw w for w a w all j (6) 1= j j w (7) 0,j for allw j (8) by solving the above formulation, the optimal weights * * * 1 2( , ,..., )nw w w and * are obtained [26]. 3.3. measurement alternatives and ranking according to compromise solution (marcos) this method determines ideal and anti-ideal alternatives as reference values and then defines the relationship – represented as a utility function in the marcos method between them and other alternatives. though it has been introduced very recently, it attracted considerable attention from researcher communities [27], [36-41]. the following are the steps of the marcos method [42]. step 1: formation of decision-making matrix. in this step, a matrix with n criteria and m alternatives is defined. step 2: determination of ideal (ai) and anti-ideal solution (aai) and extended decision matrix. min x if j beneficial and max x if j nonbeneficial=  ij ijaai (9) combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 587 max x if j non-beneficial and min x if j beneficial=  ij ijai (10) step 3: normalization of the extended decision matrix. if j non-beneficial= ai ij ij x n x (11) if j beneficial= ij ij ai x n x (12) step 4: determination of the weighted matrix: v =n *ij ij jw (13) step 5: calculation of the utility degree of alternatives ki. − − = i i anti ideal s k s (14) + = i i ideal s k s (15) i 1 s = =  n ij i v (16) step 6: determination of the utility function of alternatives f(ki). ( ) 1 ( ) 1 ( ) 1 ( ) ( ) + − + − + − + = − − + + i i i i i i i k k f k f k f k f k f k (17) utility function in relation to the anti-ideal solution: ( ) + − + − = + i i i i k f k k k (18) utility function in relation to the anti-ideal solution: ( ) − + + − = + i i i i k f k k k (19) step 7: ranking the alternatives. all alternatives are ranked as per their values of utility functions. the advantages of the marcos method are: it considers an anti-ideal and ideal solution at the very beginning of the formation of an initial matrix, it proposes a new way of determining utility functions and their aggregation, and the possibility to consider a large set of criteria and alternatives while maintaining the stability of the method [40]. the marcos method is also used in various fields like sustainable supplier selection in the healthcare industry [40], iron and steel industry [38], assessment of battery electricity [43], and integrated to other mcdm method like fucom [40], itara [39], and used as fuzzy marcos [44]. 588 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer as mentioned, the marcos method is proper for solving real-world business problems, helping decision-makers in multifaceted problems, and contributing to the prospective multiple attribute decision making. 4. application and implementation in the last decades, the farmed atlantic salmon production was increased all over the world. chile and norway are recognized as the top producers by a 6% and 2% growth ratio in their production, respectively. for instance, during the first six months of 2020, chile has produced 246,806 tons of atlantic salmon, worth $ 1,731 million, indicating a 2.62% increase compared with the year before [45]. the greatest amount of this chilean salmon is exported to the us market. however, chile could not find an acceptable market share in the european markets because of the powerful presence of its european competitor. norway is exporting salmon not only over europe but also over asian countries like china and south korea. understandably, they would plan to increase their share of the asian markets. should chile intend to capture the asian market, it seems that the west of asia is the best target market due to the below listed reasons: first, as a major competitor, norway has not done any activity for exporting salmon in this region until now. second, the region enjoys considerable potential strategic benefits like the arabian sea and the indian ocean's availability. the target countries such as iran, saudi arabia, and turkey can also play as a hub for chile to export its salmon to other countries. considering all the above mentioned, this study's focus is on “the export of the atlantic salmon of chile to the west of asia’s region”, using the sfa strategy. the first step of this process is to define criteria for each category of the sfa. one of the essential criteria that significantly affect foreign markets' investment is our products' "potential of the target market". based on the fao report in 2011, the main aquaculture producers in the west of asia are saudi arabia and iran [46]. these countries are the major producers in this region, but they cannot supply all their demands. this provides an investment opportunity for neighboring countries like egypt to export their fishery products to the west of asia. "region's economic attractiveness" can be another factor to export. for example, the emirates have the most prominent international airline in the world. dubai international airport had 88,242,099.000 passengers in 2017 [47]. the emirates group also announced that their revenue from the first six months of 2020-21 had been us$ 3.7 billion [48].saudi arabia is one of the places where approximately 2 million muslims travel to this country for hajj. many tourists travel to turkey and iran annually because of their historical sites and cultural heritage. it’s figured out that west asia is a critical and strategic location, with the potential of millions of passengers travelling to these lands. seafood consumption is an essential issue for investors to measure and estimate people's preferences in these countries. the united arab emirates (uae) and oman are the largest seafood consumers in the region by consuming about 28.6 kg per year. the other critical criteria are the "country risks" like economic risk, business environment risk, political risk, commercial risk, and financing risk. one of the criteria that significantly affect the target country's selection is the "location and cost of transportation". as the distance between the two countries (as the exporter and the importer) increases, transportation costs are seriously growing. combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 589 hence, as shown in table 3 and fig. 2, this research considers three countries (saudi arabia, the uae, and oman) as the chilean salmon fish export destination. from the countries mentioned above, iran and turkey are omitted. due to international sanctions and unstable economic situations, iran would not be a great option. also, since maritime transportation has been one of the consideration criteria to select the target market, turkey does not seem to be an optimal option for this purpose. iran and turkey have been omitted according to the latest trend-economy site statistics. in 2018, saudi arabia, the uae, and oman imported fishery products $4, $5, and $15 million, respectively, and $19, $4, and $19 million 2019 [49]. fishery importation to saudi arabia increased for nearly 5-times in one year. it can be concluded that saudi arabia has a remarkable potential for exporting fish. economic attractiveness could be gdp growth, average inflation rate, macroeconomic stability, financial structure and development, and the target country's business environment. table 3 the sub-criteria of the region's economic attractiveness sub-criteria country saudi arabia uae oman gdp growth volatility 78.5 86 80.4 average inflation rate 100 100 100 macroeconomic stability 79 71 68.9 financial structure and development 51 46.3 36.9 business environment 81.3 88.7 68.2 source: global foreign direct investment country attractiveness [50] the annual consumption of seafood in saudi arabia, the uea, and oman is 11.3, 24.71, and 28.54 kg/person, respectively [51]. the trend of seafood consumption per capita from 1961 to 2017 is attached in the appendix. transportation cost is another critical criterion that the investors should consider because they determine the direct influence on export policy. they are transporting fishery products while noticing that the live fish should be controlled under certain conditions. a more common way of transport is via sealed containers [52]. these containers should be insulated from heat, and it is necessary to provide adequate oxygen for fish during transport. the wholesalers usually use pure bottled oxygen for oxygenating water [53]. airplanes or ships are usually preferred for intra-continental transportation. although ship freightage is less expensive than airplanes, the boat's transit time is much longer than that of the airplanes. however, as mentioned before, the fish transport system needs some other types of elements and variables. when the transition time exceeds, maintenance costs and losses of fish will increase, too. for example, the ship freighted transit time from chile to the uea is about 25 to 31 days and airplane freighted is about 1 to 3 days. in order to investigate distances, consider just the distance from the target location to chile. the shorter length is an advantage for the target location. table 4 shows these distances. table 4 distance from chile to the target location distance (miles) saudi arabia uae oman from chile to flight ship flight ship flight ship 8551 7430 9060 7873 9166 7965 source: [54] 590 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer based on euler hermes global study [55], the country risk consists of five parts (economic risk, business environment risk, political risk, commercial risk, and financing risk). this study uses five linguistic concepts as excellent, very good, good, bad, and worst for determining the value of these sub-criteria. table 5 shows these values. table 5 linguistic assessments of country risk sub-criteria economic risk business environment risk political risk commercial risk financing risk saudi arabia good good bad worst very good uae good very good good worst good oman bad good good worst bad source: [56] 4.1. research gap the first step in the sfa method is to determine the criteria. suitability is related to opportunities and constraints that an organization faces. the five criteria, the target market’s potential, and the region’s economic attractiveness, are involved in this group. the feasibility factors examine the strategy and scan its financial capability. the consumption of seafood of the target market and the cost of transportation are relevant to this group. finally, the acceptability usually surveys the risk of strategy, so the country risk is placed in this group. three countries, saudi arabia, the uae, and oman are considered option 1, option 2, and option 3. table 6 shows the sfa strategy and the criteria. table 6 the sfa strategy framework by related criteria weight suitability ▪ the potential of target market ▪ region's economic attractiveness w1 w2 feasibility ▪ consumption of the seafood ▪ location and cost of transportation w3 w4 acceptability ▪ country risks w5 all of the criteria can be measured by nominal values, except one of them that is linguistic. the sfa strategy has not proposed a procedure for transmuting this linguistic value to nominal. one of the challenges is that the deals are not balanced, and calculating these values results in the wrong answers because data should be normalized for the measurement. sfa strategy table has a column that determines the weight of criteria. the gap is to determine the weights of each criterion, the function that mcdm methods will deliver. the mcdm method normalization steps can convert linguistic concepts to nominal ones. some of these methods help the researchers to determine criteria weights. according to these benefits of the mcdm methods, combining these methods with the sfa strategy is considered in this study. combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 591 4.2. calculation with the proposed mcdm model this study uses the bwm method as an mcdm method because it requires fewer comparisons and gives more trustworthy outcomes than the other weighting tools [73]. this method works by pairwise comparison of the criteria. based on the bwm algorithm, the best and worst criteria among these five should be determined. the potential of the target market is rated as the best, and the location and transportation cost as the worst criterion. considering appendix from table 8 to 22, we obtain these weights as wpotential of target market = 0.4219, wregion's economic attractiveness = 0.1734, wconsumption of the sea-food = 0.2601, wlocation and cost of transportation = 0.0404, and wcountry risks= 0.104. the weights are achieved by the bwm excel file solver, which can be found in www.bestworstmethod.com. the ranking of options in the sfa method is realized by the marcos method. firstly, the decision matrix is defined. the decision matrix contains the values of the alternatives according to the criteria. the criteria consist of some sub-criteria. the decision matrix is given in table 7. the mcdm method provides the possibility to convert linguistic values to nominal. as country risk values are linguistic, it is possible to convert them to nominal values. risk is a negative criterion that means the lower values are better preferred. the linguistic values are excellent, good, bad, and worst transmitting to numbers 1 to 5, respectively (excellent count as 1). it has to be mentioned that commercial risk is omitted from the subcriteria of country risk because three options have the same value. the average inflation rate is also neglected from the region’s economic attractiveness for the same values. in this study, the researchers used www.mcdm.app and extracted the results. the obtained values by marcos are (saudi arabia= 0.7281, uae= 0.5281 and oman= 0.8287). it turns out that oman is the best destination for the chilean fish market, while the uae is the worst item based on our study. table 7 decision matrix table alternatives criteria sub-criteria saudi arabia uae oman potential of target market 19 4 19 region's economic attractiveness gdp growth volatility 78.5 86 80.4 macroeconomic stability 79 71 68.9 financial structure and development 51 46.3 36.9 business environment 81.3 88.7 68.2 consumption of the seafood 11.3 24.71 28.54 location and cost of transportation flight 8551 9060 9166 ship 7430 7873 7969 country risks economic risk 3 3 4 business environment risk 3 2 3 political risk 4 3 3 financing risk 2 3 4 4.3. discussion to specify which country has a good potential for the fishery products market, this paper attempts to find the answer by utilizing the sfa strategy – as a strategic choice methodthrough mcdm methods. 592 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer since the sfa strategy does not seem very efficient for the abovementioned situation, the researchers extended it by an integrated bwm-marcos methodology. this combination has also increased the capability of the sfa strategy to solve complex problems. according to the sfa framework, some related criteria and options should be defined. the evaluation criteria considered are ranked from the most significant to the least important as the potential of the target market, consumption of the seafood, region's economic attractiveness, country risks, location, and transportation cost, respectively. the selected options are the names of three countries (saudi arabia, oman, and the uae). one country should be selected among these options as the best country to export chilean fish to. then, the criteria and alternatives are evaluated and ranked. the results show that oman is the most acceptable market for the chilean fish market. put it differently, by placing in first ranking, oman best meets the criteria considered for the fish market. saudi arabia is also considered one of the top leading countries for salmon export. among the reforms that have started in saudi arabia, there are projects to encourage healthy living. they comprise the goals of increasing fish consumption. therefore, importing salmon from chile to this country is of critical importance. in the uae, the aquaculture imports are approaching $ 100 million and they are mostly imported from norway, oman, india, and turkey. therefore, the uae may have a substantial potential for chile. fig. 1 shows the structure of this combined method for the case study. fig. 1 the process and phases of the model combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 593 in order to verify the results, we have performed a sensitivity analysis by substituting the weights; we have noticed that the results are stable and confidential. table 24 shows the random tests organized for analysis and table 25 shows the ranking of the alternatives. in total, we observe that, based on 10 tests, oman is still the best option while the uae is judged to be the last choice. 5. conclusions the sfa strategy is the primary research method, which introduces some criteria and options and evaluates them. by increasing the complexity of a problem, the efficiency of this strategy decreases. sfa does not consider sub-criteria, a particular way of determining the weight, and a precise structure to prioritize the options. the deficiency of sfa bears in mind the idea of developing this method by using the mcdm methods, for instance, by applying the bwm method for determining the weight of criteria and by the marcos method for ranking alternatives or options. in addition, the combination of the mcdm methods with the sfa increases the accuracy of the selection process. a case study has been surveyed to implement the developed sfa approach. the case study was about exporting chilean fish to west asia. three countries are considered as the alternatives, including saudi arabia, the uae, and oman. the target market's potential, region's economic attractiveness, consumption of the seafood, location and cost of transportation, and country risks were five criteria selected in this study. the main challenge occurs in the process of resolving; the problem was that some criteria have nominal values and should be converted to a numeric value. this conversion in the mcdm methods is routine, but sfa does not propose a specific solution. determining the weight of criteria in sfa has no straightforward, systematic approach. however, the bwm method calculates these weights clearly. another problem with sfa was the absence of a normalization system. using mcdm methods covers all of these problems. the proposed method can be used as a great tool for managers to choose the best strategy for complex and challenging problems of their company. this principle, which suggests selecting the best strategy, can be used by different sized entities from start-up teams to holding companies. this study suggests a framework by combining the advantages of bwm and marcos methods with the sfa strategy to identify the most appropriate target market for the chilean fishery industry. the results showed that the best target market for chilean fishery industry in oman. in the future studies, the researchers can develop the sfa method with other mcdm methods like seca, edas, ahp, etc. also, it is possible to integrate various weighting methods such as fucom, lbwa, mabac, mairca, etc. it is suggested to use fuzzy logic-based methods in order to model human judgments. acknowledgements: authors of this work are very thankful to anonymous reviewers and editors for their comments and guidelines. 594 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer references 1. durmaz, y., düşün, z., 2016, importance of strategic management in business, expert journal of business and management, 4(1), pp. 38-45. 2. 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2017 [51] table 8 enter the names of the criteria (step 1) criteria criterion 1 criterion 2 criterion 3 criterion 4 criterion 5 names of criteria potential of target market region's economic attractiveness consumption of the seafood location and cost of transportation country risks table 8 select the best and the worst (step 2) best potential of target market worst location and cost of transportation table 9 enter the decision-maker's preferences (best to others: bo vector) (step 3) best to others potential of target market region's economic attractiveness consumption of the seafood location and cost of transportation country risks potential of target market 1 3 2 8 5 598 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer table 10 enter the decision-maker's preferences (others to worst: ow vector) (step 4) others to the worst location and cost of transportation potential of target market region's economic attractiveness consumption of the seafood location and cost of transportation country risks 8 6 7 1 5 table 11 the weights of criteria weights potential of target market region's economic attractiveness consumption of the seafood location and cost of transportation country risks 0.4219 0.1734 0.2601 0.0404 0.1040 calculating sub-criteria weights of the region's economic attractiveness by the bwm method: table 12 enter the names of the sub-criteria of region's economic attractiveness (step 1) criteria criterion 1 criterion 2 criterion 3 criterion 4 sub-criteria gdp growth volatility macroeconomic stability financial structure and development business environment table 13 select the best and the worst (step 2) best macroeconomic stability worst financial structure and development table 14 enter the decision-maker's preferences (best to others: bo vector) (step 3) gdp growth volatility macroeconomic stability financial structure and development business environment macroeconomic stability 4 1 8 3 table 15 enter the decision-maker's preferences (others to worst: ow vector) (step 4) others to the worst financial structure and development gdp growth volatility macroeconomic stability financial structure and development business environment 7 8 1 6 table 16 the weights of sub-criteria of region's economic attractiveness weights gdp growth volatility macroeconomic stability financial structure and development business environment 0.1755 0.5425 0.0478 0.2340 combining the suitability-feasibility-acceptability (sfa) strategy with the mcdm approach 599 table 17 sub-criteria of country risks (step 1) criterion 1 criterion 2 criterion 3 criterion 4 economic risk business environment risk political risk financing risk table 18 select the best and the worst (step 2) best financing risk worst political risk table 19 enter the decision-maker's preferences (best to others: bo vector) (step 3) best to others economic risk business environment risk political risk financing risk financing risk 2 2 6 1 table 20 enter the decision-maker's preferences (others to worst: ow vector) (step 4) others to the worst political risk economic risk business environment risk political risk financing risk 5 5 1 6 table 21 the weights of subcriteria of economy risk attractiveness weights economic risk business environment risk political risk financing risk 0.25 0.25 0.0625 0.4375 table 22 ranking of alternatives weight 0.42 0.02 0.091 0.007 0.039 0.26 0.01 0.025 0.025 0.025 0.006 0.04 beneficial (b) or nonbeneficial (nb) criteria alternative 1 alternative 2 alternative 3 b c1 19 4 19 b c2 78.5 86 80.4 b c3 79 71 68.9 b c4 51 46.3 36.9 b c5 81.3 88.7 68.2 b c6 11.3 24.7 28.54 nb c7 8551 9060 9166 nb c8 7430 7873 7969 nb c9 3 3 4 nb c10 3 2 3 nb c11 4 3 3 nb c12 2 3 4 table 23 sensitivity analysis tests c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 original weights 0.42 0.02 0.091 0.007 0.039 0.26 0.01 0.025 0.025 0.025 0.006 0.04 t1 0.42 0.02 0.091 0.006 0.039 0.26 0.01 0.025 0.025 0.025 0.007 0.04 t2 0.42 0.02 0.091 0.007 0.039 0.26 0.025 0.01 0.025 0.025 0.006 0.04 t3 0.42 0.02 0.091 0.006 0.04 0.26 0.025 0.01 0.025 0.025 0.006 0.039 t4 0.42 0.02 0.01 0.007 0.039 0.26 0.091 0.025 0.025 0.025 0.006 0.039 t5 0.42 0.02 0.091 0.007 0.039 0.26 0.01 0.025 0.025 0.025 0.04 0.006 t6 0.42 0.02 0.025 0.025 0.039 0.26 0.01 0.091 0.007 0.025 0.04 0.006 t7 0.26 0.02 0.091 0.007 0.039 0.42 0.01 0.025 0.025 0.025 0.006 0.04 t8 0.26 0.091 0.02 0.006 0.039 0.42 0.01 0.025 0.025 0.025 0.006 0.04 t9 0.26 0.02 0.091 0.006 0.039 0.42 0.01 0.04 0.006 0.025 0.025 0.025 t10 0.091 0.02 0.42 0.006 0.039 0.26 0.01 0.025 0.025 0.025 0.006 0.04 600 s. h. zolfani, r. bazrafshan, p. akaberi, m. yazdani, f. ecer table 24 ranking results of sensitivity analysis original rank score test 1 test 2 test 3 test 4 test 5 test 6 test 7 test 8 test 9 test 10 alt-1 0.7281 0.7279 0.7281 0.7279 0.7263 0.7175 0.7163 0.6308 0.6242 0.6244 0.6529 alt-2 0.5272 0.5272 0.5272 0.5272 0.5292 0.5354 0.5357 0.6144 0.6199 0.6161 0.6585 alt-3 0.8297 0.83 0.8297 0.8302 0.8327 0.8419 0.8437 0.8181 0.8213 0.8260 0.7098 r an k in g alt-1 2 2 2 2 2 2 2 2 2 2 3 alt-2 3 3 3 3 3 3 3 3 3 3 2 alt-3 1 1 1 1 1 1 1 1 1 1 1 3039 facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 331 340 doi: 10.22190/fume170602016d original scientific paper microstructure and micromechanics of shale rocks: case study of marcellus shale  udc 552.5 hui du 1 , kristen carpenter 1 , david hui 2 , mileva radonjic 1 1 louisiana state university, baton rouge, la, u.s. 2 the university of new orleans, new orleans, la, u.s. abstract. shale rocks play an essential role in petroleum exploration and production because they can occur either as source rocks or caprocks depending on their mineralogical composition and microstructures. more than 60% of effective seals for geologic hydrocarbon bearing formations as natural hydraulic barriers constitute of shale caprocks. the effectiveness of caprock depends on its ability to immobilize fluids, which include a low permeability and resilience to the in-situ formation of fractures as a result of pressurized injection. the alteration in sealing properties of shale rocks is directly related to the differences in their mineralogical composition and microstructure. failure of the shale starts with deterioration at micro/nanoscale, the structural features and properties at the micro/nanoscale can significantly impact the durability performance of these materials at the macroscale, therefore, study at micro/nanoscale becomes necessary to get better understanding of the hydraulic barriers materials to prevent failure and enhance long-term geologic storage of fluids. indentation tests were conducted at both micro and nanometer level on marcellus shale samples to get the mechanical properties of bulk and individual phase of the multiphase materials. the mechanical properties map were created based on the nano indentation results and the properties of each individual phase can be correlated with bulk response in the multiphase composite; the effect of each component on the microstructure and bulk mechanical properties can be better understood. key words: shale rock, mechanical properties, indentation received june 02, 2017 / accepted july 21, 2017 corresponding author: mileva radonjic craft & hawkins department of petroleum engineering, louisiana state university, 127 old forestry building, baton rouge, 70820, la, u.s e-mail: mileva@lsu.edu 332 h. du, k. carpenter, d. hui, m. radonjic 1. introduction shales have been a particularly interesting rock in various applications of petroleum engineering. rock-fluid interactions have been studied in the oil industry not just in the drilling phase, but also in completion, stimulation, and enhanced oil recovery projects, for over one hundred years. performing a correct evaluation of how a rock or formation reacts when under stresses involved for production is critical for the success of many operations. shales are sedimentary rocks that have distinct laminated layering characteristics and high clay and/or silt content. there are two main chemical processes responsible for these formations, with two fundamental mechanisms: 1) neoformation – precipitation from solution; and, 2) transformation – a new clay mineral inherits part of its silicate skeleton from preexisting materials such as phyllosilicate [1]. shales are typically laminated and fissile. in order for fine clay and silt particles to form, larger organic pieces must be broken down over time and deposited in environments conducive to shale formation. the processes that break down these larger pieces into clay or silt sized particles include chemical weathering in soils, formation of authigenic minerals at the sediments depositional sit, formation of diagentic minerals after deposition, and clay minerals formed by hydrothermal alteration [1]. these variations of minerals that create the shale rock make it very vulnerable to chemical reactions. shales are subject to phenomena such as hydration, swelling, shrinking, and strength reduction when exposed to water and ions [1]. most of the time, the mechanisms controlling these reactions are very complex and not completely understood. they can result in a hydrophilic nature of clay particles, which is somewhat influenced by the chemical and mechanical environments the clay materials are exposed to. the chemical effects are from the intermolecular forces between clay particles and pore fluid inside the shales, typically creating an ion exchange much like an osmotic membrane. the pore water is generally much more salty than the fresh water injected into formations during hydraulic fracturing. also, the type and amount of clay groups and subgroup in the shale play an important role in distinguishing different hydrological behaviors of the rock. this is a result of where the charge deficiency is located (silica tetrahedral or alumina octahedral sheet), as well as of a continuous charge in shale pore pressure and composition [1]. clay minerals are classified as ‘silicates’ but their chemical compositions typically have more oxygen than si, al, or mg, so many arguably consider them as (hydr)oxides of silicon, aluminium, or magnesium [2]. shale rocks predominantly composed of clay such as kaolinite, smectite, and illite. they might also have other silica and carbonate based minerals that contribute to their geomechanical strength. ian c. bourg documented different shale rock formations showing the relationship between their utility and composition. clay mineral content was identified as a very important variable that controls key material properties of these formations. shale formations with high clay content (> 35%) are utilized as seals for carbon capture and storage (ccs) and nuclear waste storage because of their low permeability and resilience to the formation of fractures [3]. 2. materials the marcellus shale is found in the appalachian basin of eastern north america. like most devonian appalachian shales with more than 2% (by volume) of organic materials, it tends to be black and classified as shales/mudrocks. these black shales can contain more https://en.wikipedia.org/wiki/kaolinite https://en.wikipedia.org/wiki/montmorillonite https://en.wikipedia.org/wiki/illite micro structure and micromechanics of shale rocks: case study of marcellus shale 333 than 20% percent (by volume) of organic material, with organic carbon totaling up to 20 weight percent of the rock [4]. this late devonian was formed in an oxygen-poor marine condition that resulted in the deposition of a dark mud and anoxic environment. it is part of the hamiliton group [5]. black shales often are enriched with redox-sensitive metals and have varying solubilities under different oxygen levels – i.e. some have higher solubility in high oxygen environments than low oxygen environments and vice versa. it has been estimated that the metal enrichment in the marcellus shale was formed roughly 400 million years ago and this created conditions of metal enrichment based on water chemistry and oxygen level [6]. the marcellus shale is made of dark-gray to black, fissile, pyritic shale. it is interbedded with dark-gray argillaceous limestone or calcareous shale [7]. some areas also contain a fossiliferous layer of limestone which is the purcell member of the marcellus shale [7], and prominent zones of calcareous concretions ranging in diameter from several centimeters to more than 1 m (3.3 ft). the clay minerals in this devonian-aged shale from the appalachian basin are illite, chlorite, kaolinite, and two types of mixed-layer clay. mixed-layer clay minerals result from the random interlayering of two or more clay minerals, including random interlayering of illite and an expandable mineral such as smectite, (called illite-smectite mixed-layer clay) and a random mixture of illite and either a degraded chlorite or a vermiculite [8]. the marcellus formation has an especially interesting shale rock because not only it is considered for carbon capture and storage (ccs) compatibility, but regarding the latest advances in hydraulic fracturing technology, it can be used in the production of natural gas. the core samples used in this experiment are from an active production well in washington county, pa, u.s. from depths of 6,300-6,450ft as shown in fig. 1. fig. 1 schematic of the well profile and the core samples id and their corresponding depths. the bulk size of each core was about 4 inch in diameter and 1inch thick 334 h. du, k. carpenter, d. hui, m. radonjic 3. methodology 3.1. sample preparation six marcellus shale samples are tested in this study including one outcrop and five core samples from an active production well. all of the samples are cut with a diamond saw into a small piece with approximately 1 x 1 inch in size and the thickness around 0.5 inch. then specimen was then grinded, polished down to 1 μm and ultrasonic cleaned. finally, the specimen was oven dried for at least 24 hours to avoid the difference caused by moisture content. 3.2. micro and nanoindentation micro indentation gives the average mechanical properties over the large area of different grains while nano-indentation could give the localized mechanical properties of a single grain. for the indentation tests, the indenter tip with a known geometry (vickers diamond) is driven into a specific site of the sample to be tested, by applying an increasing normal load. after reaching a pre-set maximum value, the normal load was paused for few seconds, then reduced until complete relaxation occurs. during the loading-unloading process, the position of the indenter relative to the sample surface is precisely monitored with an optical non-contact depth sensor. for each loading-unloading cycle, the applied load value versus position of the indenter was plotted. hardness and elastic modulus are determined through load-displacement curve using oliver & pharr’s method [9]. the schematic of indentation apparatus is shown in fig. 2 and the basic parameters used are shown in table 1. 3.3. scanning electron microscopes (sem) for sem imaging, the rock samples were first vacuumed and then sputter coated with 6nm thick carbon. the microscope used for obtaining the sem images was fei quanta 3d feg dual beam fib/sem system at 20 kv. high resolution microscopy offered an insight into sample microstructure at micro to nanometer scale. fig. 2 left: schematic of indentation apparatus; right top: single micro indentation mark, approximately 250x250 μm in area and 50 μm in depth; right bottom: nano indentation grid, each point is approximately 4x4 μm in area and 1 to 4 μm in depth and distance between two points is 20 to 25 μm micro structure and micromechanics of shale rocks: case study of marcellus shale 335 table 1 setting conditions for micro and nano indentation micro nano maximum force 10 (n) 50 (mn) loading rate 20 (n/min) 100 (mn/min) unloading rate 20 (n/min) 100 (mn/min) pause at maximum load (s) 30 10 contact load (mn) 15 0.08 poisson's ratio* 0.2 0.2 indenter type vickers vickers * poisson’s ratio was assumed to be constant at 0.2 for simplification 4. results and discussion 4.1. microstructure of the rock optical microscopy images of the outcrop sample showing a minimum fracture of all samples, as for the deep cores and the amount of fractures increased as the depths increase. the cores samples are not completely representative of the in-situ condition as they are taken into the surface condition and oven dried, release of the overburden pressure amplified the fractures, shrinkage of the swelling clays could also contribute to the development of fractures, even so, these pictures still indicated the higher stiffness at the top portion of the formation than the bottom portion, as it maintains better integrity. optical images were taken of the samples before indentation was performed. these images are shown in fig. 3. it is easy to see differences in fracture widths and basic compositional lamination differences. the sem analysis of the samples highlights major differences in textures, composition, and fracture sizes. fig. 4 shows a micrograph of the outcrop and core sample 2 at a 200μm scale. fig. 5 shows a comparison of the outcrop and core sample 7 at 100 μm. fig. 3 optical microscopy images of samples cross-sections showing the fractures along the bedding. outcrop sample has minimum amount of fractures, as the depths increase both the number and width of fractures increases 336 h. du, k. carpenter, d. hui, m. radonjic fig. 4 bse sem micrograph of outcrop and core 2 (depth 6334.1-6334.5ft) with 200μm scale; fracture width on the outcrop is slightly smaller than on the core sample 2; outcrop fracture width averages around 7μm, while sample 2 fracture width averages around 10μm fig. 5 bse sem micrograph of outcrop and core 7 (depth of 6419.25-6419.55ft) with 100μm scale; the number of fractures on core 7 is significantly higher than the outcrop, also the average fracture width is much larger (15μm compared with 7μm) from the sem micrographs, outcrop sample has a lot more iron sulfide pockets than on both of the core samples, and the average fracture width increases as the depth increases. the larger fractures in the deeper samples indicate that they most likely have lower mechanical properties as the depth increases, which is also verified in the following experiment. 4.2. micro indentation results from micro indentation (fig. 6) showed the outcrop has overall higher mechanical properties, while within the same formation, the mechanical properties have a decreasing trend as the depth increased. the significant difference in mechanical properties between top and bottom portions of the formation can result in different fracture responses because mechanical properties of the rock are the key factor for determining the likelihood of fractures initiating and propagating. micro structure and micromechanics of shale rocks: case study of marcellus shale 337 based on the results, the bottom portions of the formation are more likely to start fractures as they are less mechanically stable, but with softer grains, the fractures are likely to heal faster at subsurface condition. for the top portion, higher stress is required for fracture initiation, once fractured, and grains with higher hardness behave more rigid which will help to support the open fractures. fig. 6 mechanical properties of marcellus shale outcrop and cores measured by micro-indentation and their corresponding depths 4.3. nano indentation nano indentations were done on both outcrop and core3, and the results are plotted in fig. 7. the yellow spots represent grains with higher mechanical properties, which clearly showed more in outcrops hardness map. these rigid grains were evenly distributed which ends up an overall higher bulk hardness as shown in the result from micro indentation (fig. 6). the young’s modulus maps are relatively close comparing with the hardness maps because the calculation of hardness is based on plastic deformation of single grain, while the young’s modulus is always a composite response from all surrounding phases. from the e distribution maps shown in fig. 8, both samples have a large portion of data points laid in the range of clay minerals of kaolinite, smectite and illite while the outcrop may have higher quartz and mica content. the outcrop sample also has some high e grains, which could be chlorite or metal oxide. clay minerals have layered structures which often carry negative surface charges, which adsorb and hold cations by electrostatic force forming a double layer. the cation exchange capacity (cec) of shale is proportional to its clay content, and has been shown to be related with its geomechanical properties [10]. the existence of this double layer can also reduce effective porosity, resulting in a decrease in permeability. the thickness of the double layer is dominated by the clay mineralogy, increasing from chlorite to kaolinite to illite to smectite, it is also influenced by salt concentration of the pore fluid [11]. therefore, the type and amount of clay content are the key factors affecting shale sealing capacity, as both of them control the cec which determines the mechanical and petrophysical properties of the rock. sample id depth (ft) outcrop core 2 6334.1 core 3 6381.5 core 4 6388.6 core 6 6407.5 core 7 6519.3 338 h. du, k. carpenter, d. hui, m. radonjic fig. 7 mechanical properties maps of marcellus shale rock outcrop (left) and core3 (right) based on 100 nano-indentation test results (10x10 grid) fig. 8 young’s modulus (e) data distribution of outcrop and core3 measured by nano-indentation compared with literature e data [12, 13, 14] of common minerals found in shale rocks micro structure and micromechanics of shale rocks: case study of marcellus shale 339 the properties of clay mineral from the literature have much wider ranges due to the properties anisotropy caused by its platy microstructure. progressive burial of the sediments caused mechanical compaction during the deposition, clay platelets are forced towards a parallel bedding alignment, with a rapid reduction of porosity and permeability, created layered structured shale rock [15, 16, 17]. from the micrometer size platy grains to meso-/ marcoscale layered rock, the significant anisotropy of properties were inherited. 5. conclusions nano indentation could be an excellent two-dimensional mapping tool for examining the properties of constituent phases independently of each other in composite material microstructures. mechanical maps could be used for correlating individual phase properties with bulk response. mechanical maps could be used for correlating individual phase properties with bulk response measured by micro indentation. combing the mechanical properties map with high resolution microscopy (sem), the mineralogy/morphology can be also correlated. the mechanical properties map can be also done on other multiphase composite such as cement to study the intrinsic properties of each component, as well as the interaction and properties of the bond and interfacial regions of different phases. it might also be useful for modeling the rock/cement behavior to predict the fracture occurrence potential, as it linked the microstructural features with their mechanical properties. acknowledgements: the authors would like to thank the shared instrument facility (sif) at louisiana state university for the imaging. dr. vidic and dr. hill kindly provided samples. we would also like to thank the support from sustainable energy& environmental research (seer) lab in louisiana state university references 1. diaz-perez, a., cortes-monroy, i., roegiers, j.c., 2007, the role of water/clay interaction in the shale characterization, journal of petroleum science and engineering, 58(1-2), pp. 83-98. 2. bergaya, f. and lagaly, g. 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properties of 30 natural mudstones, journal of geophysical research, 112(b3), b03206. 11508 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 861 880 https://doi.org/10.22190/fume230521031c © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals minjia chen1, xingqi luo1, wanbo chen2, jianjun feng1, xiaohang wang3,4,5 1xi’an university of technology, shaanxi, china 2jiangxi ganneng co., ltd., nanchang, china 3state key laboratory of hydropower equipment, harbin, china 4harbin electric machinery co., ltd., harbin, china 5harbin institute of large electrical machinery, harbin, china orcid ids: minjia chen https://orcid.org/0009-0003-6057-4954 xingqi luo https://orcid.org/0009-0001-1868-7483 wanbo chen https://orcid.org/0009-0009-2424-4759 jianjun feng https://orcid.org/0000-0002-6406-1764 xiaohang wang https://orcid.org/0000-0003-2217-7037 abstract. urban water supply and drainage, the shipbuilding industry, the petrochemical industry and other professional fields rely heavily on mixed-flow pumps. enhancing the efficiency of mixed-flow pumps is crucial for achieving the ‘dual carbon’ goals and promote energy saving and emission reduction. in this study, the guide vane and impeller of the mixed-flow pump were optimised, focusing on its low head and effectiveness in power plants. the performance of the original and optimised pumps was then evaluated under various flow rate conditions, and their hydraulic performance was compared. results showed that the efficiency of the optimised mixed-flow pump was improved, resulting in effective enhancement of energy loss in the pump passage. the optimised guide vane facilitated smoother water flow into the outlet pipeline, achieving energy savings, emission reduction and contributing to the realisation of the ‘dual carbon’ goals. key words: mixed-flow pump, guide vane optimisation, impeller optimisation, numerical simulation, hydraulic performance 1. introduction a study was proposed for the first time that china aims to achieve carbon peak in 2030 and carbon neutrality in 2060 [1]. compared with developed countries, such as the united states and britain, china faces remarkable challenges in terms of its economy, energy structure and utilisation rate. at present, china’s carbon emissions account for 28.8% of received: may 21, 2023 / accepted september 17, 2023 corresponding author: xingqi luo xi’an university of technology, 710048 no. 5 jinhua south road, shaanxi, china e-mail: luoxq@xaut.edu.cn 862 m. chen, x. luo, w. chen, j. feng, x. wang the global total, with per capita emissions also amongst the highest. energy conservation and emission reduction have become the main goals in various industries to realise successfully the ‘dual carbon’ goals. mixed-flow pumps find applications in diverse fields, including irrigation, drainage, flood control and water circulation systems of power stations [2]. these pumps offer advantages, such as high head, large flow and a wide efficient range. however, the mixed-flow pump is a large-scale energy-consuming equipment that consumes a considerable amount of energy, making its operating efficiency a critical factors in achieving the ‘dual carbon’ targets [3]. the impeller, which serves as the core component of mixed-flow pumps [4], plays a vital role in determining their performance. the design theories for mixed-flow pumps include 1d, 2d and 3d approaches. [5-7]. both the 1d and 2d theories are based on the assumption that the impeller consists of infinitely thin and numerous blades, providing a foundation for solving the impeller’s internal flow under simplified assumptions. with the continuous development of computer technology, the 3d design method has been widely used in pump impeller design. ten et al. [8] proposed a 3d potential flow design based on a given circulation distribution. they derived the blade bone line equation from the blade boundary conditions and obtained the designed blade shape by solving the blade bone line. the flow control equation was solved using the spectral method to design the impeller. huang et al. [9] optimised the design of mixed-flow pump impellers based on streamline load distribution and blade trailing edge angle as design parameters, and they performed numerical simulation calculations. appropriate backloading helps to smooth the velocity gradient in the blade channel and eliminate secondary flow [10,11]. using the 3d design platform for mixed-flow pump impellers, hao and shu liang [12,13] improved the genetic algorithm and established a 3d optimisation and design system for mixed-flow pump impellers. they also explored the relationship between different blade angle deviations and the hydraulic performance of the mixed-flow pump, using both numerical simulation and model tests. the results show that when the blade rotation angle is positive, the curve moves in the direction of high flow, whereas a negative angle shifts the curve towards small flow. several scholars at home and abroad have conducted research on the internal flow analysis of mixed-flow pumps. muggli et al. [14] used a combined approach of steadystate and unsteady-state calculations to predict the performance of a mixed-flow pump. they found that unsteady flow simulations, which require more computational resources, are also feasible in an industrial environment and provide detailed information about the flow patterns and pressure fluctuations within the pump. the performance curves obtained from the two methods were consistent with the measured results. moreover, the study focused on analysing the unstable flow structure of mixed-flow pumps under certain operating conditions, providing insights into abnormal flow characteristics. the results showed that under low flow conditions, unsteady calculations were better suited for analysing the unsteady flow phenomena. saha et al. [15] created a j-groove using computational fluid dynamics (cfd) technology to investigate the causes of humps in the performance curve. they found that through experiments, it can be determined that the optimal slot shape and position have a significant effect on reducing the vortex intensity and backflow propagation in the impeller inlet area. through angular momentum absorption owing to mixing of groove reverse flow-and swirl flow, the theoretical head of the impeller is restored [16]. kim et al. [17] found that under the design working condition, the suction performance improved with increased blade thickness, incident angle and total pressure. derakhshan et al. [18] focused on centrifugal pump impellers, optimising their mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 863 geometry using optimisation algorithms and 3d navier–stokes flow solvers. the numerical results show an increase of 6.89 m in total differential pressure and a 3.59% improvement in efficiency for the centrifugal pump. kim et al. [19-21] conducted a multi-parameter optimisation design of the impeller and guide vane of mixed flow pumps using cfd technology. the hydraulic performance of the mixed-flow pump was taken as the objective function through an orthogonal test, optimising several variables that mutually affect each other to obtain the best hydraulic model for the mixed flow pump. the optimisation model exhibited improved head and efficiency compared with the original model. srivastava et al. [22] adopted experimental and numerical simulation methods to correct the setting angle of the vanes by analysing the pressure distribution in the inner flow passage of the impeller. the aim was to optimise the hydraulic performance of the mixed-flow pump. the results showed that compared with the original model, the pressure distribution on the blade surface of the impeller, designed through an inverse problem approach, became more uniform, the low-pressure area was significantly reduced and overall hydraulic performance improved. orozco murillo et al. [23] used genetic algorithm methods for single-objective and multi-objective geometrical optimisations of jet pumps for vacuum distillation of ethanol, which is an application not deeply studied in scientific literature. these devices are particularly suitable for breaking azeotropes below atmospheric pressure at ambient temperature. the optimal design considered different working pressures, five dimensionless geometric parameters that can affect the operation of the jet pump and three performance parameters. the research mentioned above highlights the challenges in optimising mixed-flow pump impellers due to numerous design parameters. the traditional method involves modifying the blade shape based on the designer’s personal experience, which is time consuming and often fails to find the global optimal solution. additionally, it lacks effective guidance on modifying the blade shape to improve performance. these limitations restrict designers from exploring a larger design space. this study presents a new design method that combines the non-dominated sorting genetic algorithm (nsga-ii) with the 3d inverse problem solving design method [24]. this method effectively optimises the wrap angle distribution of the blades and guide vanes in the mixed-flow pump. the flow patterns inside the pump before and after optimisation are compared using detailed cfd analysis. this analysis helps identify the reasons for performance improvements in each component. the intelligent optimisation design system proposed in this study overcomes the limitations of traditional methods and remarkably reduces the optimisation design period for mixed-flow pumps. the pareto solutions obtained during the optimisation process serve as references for designing high-performance mixed-flow pumps. 2. optimisation design theory in this study, the pump design theory [25] and the multi-condition intelligent optimisation design technology are used to design the wrap angle of the guide vanes and blades for the mixedflow pump [26]. in accordance with the existing data from power stations, the design parameters, such as the design flow and lift, are calculated inversely based on the requirements. the rated parameters are determined, and an initial estimation of the impeller’s basic size is made. the axial flow channel of the pump is preliminarily determined by comparing and selecting similar hydraulic models with excellent performance. the optimal ratio is determined 864 m. chen, x. luo, w. chen, j. feng, x. wang on the basis of the number of guide vanes and impeller blades, which is determined as five blades and 11 guide vanes. the overall design scheme flow of the mixed-flow pump is depicted in fig. 1. fig. 1 design scheme flow in the inverse problem design method, iterative calculations are performed for the blade shape and flow field. the blade is approximated by an infinite thin vortex, and its intensity is expressed by the circumferential average angular momentum. 2 0 2 bb rv rv d      =  (1) where r is the radius, b is the number of blades, v is a circumferential averaged swirl velocity, and θ is the angle. the pressure distribution in the interblade channel is determined by the partial derivative of the circumferential mean angular momentum along the streamline direction, as shown in eq. (2). 2 m rv p p w b m  + −  − =  (2) where p+ is the static pressure on the blade’s pressure surface, p− is the static pressure on the blade’s suction surface, ρ is the fluid density, wm is the relative shaft speed and rv m   is the blade loading. the impeller head is determined by the circumferential average angular momentum along the streamline direction as follows: ( ) ( )2 2 1 1 2 1 1 2 t n nh u v u v g g   = − =  − (3) where ht is the theoretical head of the impeller, which consists of the dynamic head and potential head. the dynamic head is represented by the specific kinetic energy. the potential head represents the increase in position head and pressure head., ω is the angular velocity, un is the tangential velocity (i.e. the circumferential velocity at which the flow rotates with the impeller), vnn is the absolute velocity component in the circumferential direction, γn is the circulation and subscripts 1 and 2 indicate the inlet and outlet sides of mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 865 the impeller, respectively. the calculation of the blade shape is the inverse problem is based on the conditions of no-slip and no-penetration of velocity on the blade surface. the blocking effect of the blade is determined by the blade thickness parameter in the continuity equation [27]. in the 3d inverse design problem, the design parameters that can be fixed are as follows: (1) projection of the blade axis surface, (2) blade thickness, (3) number of blades, (4) speed and (5) head. during the design process of a mixed-flow pump, the important target parameters are hydraulic efficiency η and the minimum pressure on the blade cpmin. the correlation between the two parameters should be considered under maximum head, design head and minimum head conditions. analysing the parameter changes separately for each working condition can lead to an increase in calculation complexity, low solution efficiency and difficulty in exploration due to the large number of objective functions. therefore, the optimisation design of the core impeller components becomes a complex multi-condition optimisation problem. the development of multi-objective multi-parameter optimisation considers the influence of algorithms, geometric parameterisation and other aspects. under multi-condition and multi-parameter solutions, the objective functions are mutually restricted, and finding the optimal solution for all the objective functions is necessary. the nsga ii [28] has shown good applicability in multi-objective, non-continuous and nonlinear optimisation problems. the optimisation data are dimensionally reduced by converting the target function of head and efficiency, as shown in eq. (4). the functional relationship is then differentiated to find the optimal solution, as shown in eq. (5). 1 2 3 min,1 min,2 min,3 ( , , ) ( , , ) ef p cp cp cp   = = (4) 2 1 2 1 2 1 2 min , min , 1 ( , ) ( ) ( , ) ( ( ) ( ) ) n ve i des j des j n vp i des i j des j j d ef ef d p p cp cp   = = = − = −   (5) where n=1,2,3 represent working conditions 1, 2 and 3. this study establishes an intelligent optimisation design system that combines cad and cfd, improving the shortcomings of traditional design methods. fig. 2 depicts a flow chart for the intelligent optimum design of overcurrent components. overall, the traditional design method is cumbersome and cannot effectively consider the simultaneous improvement of performance under multiple operating conditions. the optimisation problem has many constraints, objective functions and unknown feasible domains in the design space. the advantage of the proposed design method is the dimension reduction of optimisation problem data through objective function conversion, laying a foundation for efficient solution of the optimisation problem. it reduces human resource consumption and improves impeller design through computer self-recycling calculation. however, this method requires a high computational speed and is only applicable to the optimal design of impeller blades and guide vanes of fluid machinery, with certain limitations for the design of other overflow components (such as suction chamber, volute, etc.). the efficiency of the mixed-flow pump designed using this method is obviously improved and meets design expectations. it has been successfully implemented in production. this 866 m. chen, x. luo, w. chen, j. feng, x. wang method also reduces the effect caused by a large number of objective functions, resulting in low solution efficiency and difficulty in effective search. fig. 2 flow chart for intelligent optimum design of overcurrent components 3. physical model of a mixed-flow pump 3.1. geometric model the impeller blades, guide vanes and other overflow components of the mixed-flow pump are modelled using ug software to precisely replicate the actual shape of the pump and reduce geometric errors. fig. 3 shows a 3dl model of the pump. fig. 4 depicts the geometrical modifications of the impeller vanes and movable guide vanes. the key parameters of the mixed-flow pump are listed in table 1. in this study, numerical simulation analysis is conducted for the mixed-flow pump model before and after optimisation under low, design and large flow conditions. table 2 shows specific parameters for the three conditions. fig. 5 shows the impeller model of the pump. fig. 3 mixed-flow pump diagram mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 867 fig. 4 (a) blade and (b) guide vane table 1 design parameters of the pump parameter value rated speed n (r/min) 2325 number of impeller blades 5 diameter of the impeller (mm) 430 number of guide vane blades 11 fig. 5 model of a mixed-flow pump’s impeller 3.2. numerical calculation settings the computational domain is divided into grids using cfd icem to minimise numerical calculation errors and accurately simulate the internal flow of each overflow component in the mixed-flow pump. fig. 6 depicts the grid around impeller blades and movable guide vanes. the total number of grid nodes for the mixed-flow pump is 3.73 million. a shear stress transport turbulence model is selected as the turbulence model [29]. an automatic wall-proximity treatment is used, and the y+ value on the blade surface is kept below 30 to adapt to the turbulence model. table 2 shows the control equation and boundary conditions. the convergence criterion for calculating convergence conditions is set such that the residual value is less than 10−4. 868 m. chen, x. luo, w. chen, j. feng, x. wang fig. 6 diagram of a mixed-flow pump table 2 boundary conditions cfd solver ansys cfx inlet total pressure 5% turbulence intensity outlet flow boundary details smooth, no-slip divergent method second-order turbulence shear stress transport convergence control ≤10−4 3.3. selection of calculation conditions the primary objectives of this study are to investigate the change in the internal flow field and the performance of the mixed-flow pump with different design schemes before and after optimisation under different operating conditions. numerical simulation calculations and analysis are conducted for low, design and large flow conditions. table 3 presents specific parameters for the three conditions. table 3 parameters of experimental conditions. number condition q (m3/s) q1 low flow rate condition 9.5 q2 design flow rate condition 10.8 q3 large flow rate condition 12.5 4. analysis of optimisation results 4.1. selection of calculation conditions fig. 7 depicts a performance curve comparison between the initial and optimised models. as shown in fig. 7(left), the optimised model outperforms the initial model under the selected calculation conditions in terms of efficiency, with an increase of approximately 3%. as mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 869 depicted in fig. 7(right), the head of the optimised model is significantly higher compared with the initial model. during the transition from design to high flow conditions, neither the optimised flow-head curve [30] nor the flow-efficiency curve [31] shows any significant steep drop. under the design conditions, the optimised head and efficiency are increased by 18.5% and 2.6%, respectively, compared with the original model. these findings indicate a remarkable improvement in the external characteristics of the optimised mixed-flow pump, aligning well with the goal of ‘dual carbon’ and improving energy utilisation. fig. 7 comparison of efficiency (left) and comparison of head (right) 4.2. flow field analysis the main flow lines of the impeller and guide vanes are depicted in three dimensions in fig. 8 to investigate the flow conditions of the mixed-flow pump before and after optimisation under various operating conditions [32]. this figure provides insights into the variation rules of the internal flow field. under the small flow condition, the flow lines of the initial mixed-flow pump exhibits disorderliness near the hub at the inlet edge of the impeller components and near the pressure surface at the wheel edge. the streamline at the guide vane inlet presents a chaotic situation with certain instability and vorticity structure. the streamline at the trailing edge of the guide vane outlet disperses throughout the entire space, and the streamline inside the guide vane undergoes remarkable changes and constantly develops rearward. the whole streamline is disordered, with the presence of eddy structures. these eddies can cause flow path blockage, resulting in energy loss and performance degradation of the pump. by contrast, the results achieved after optimisation show remarkable improvements in the impeller’s streamline disorder, reduction in vortex structures and stabilisation of the flow pattern at the guide vane. this leads to reduced energy loss and increased pump efficiency. at the design condition, the streamline of the mixed-flow pump before and after optimisation tends to be smoother as flow increases. the vortex structure continues to decrease, and some vortex structures gradually disappear. the analysis shows that under the design working condition, the optimised guide vane outlet streamline is gentler compared with the initial model, the flow pattern is stable and energy loss is reduced. when operating under large flow conditions, vortex structures are generated in the guide vane channel in the initial model, affecting the water flow. however, after modification, the 870 m. chen, x. luo, w. chen, j. feng, x. wang impeller and guide vane channels can maintain stability, with a reduction in the number and size of vortex structures to some extent. the research findings indicate that, before optimisation, the flow becomes more unstable and the streamline becomes more turbulent as the mixed-flow pump operates further away from the design working condition, particularly at lower flows. however, the optimised mixed-flow pump shows a gradual stabilisation of the flow pattern under small flow conditions, effectively suppressing the occurrence of vortex structures. as the working conditions reach the design and large flow conditions, the flow in the impeller and guide vane further improves. the decrease in the dissipation rate of energy and the increase in unit efficiency are further manifestations of the improvement in internal flow. fig. 8 comparison of efficiency (a) and comparison of head (b) the internal flow pattern of the pump before and after optimisation is better compared and optimised. fig. 9 depicts segments of the streamline at the guide vane and impeller, showing the mixed flow pump’s streamline distribution under design conditions. the analysis shows that the streamline near the guide vane hub is disordered before optimisation. the streamline distribution becomes uniform after optimisation, with no vortex generation. consequently, flow separation becomes difficult, hindering the occurrence of cavitation. the operation efficiency of the mixed-flow pump has been optimised and enhanced to some extent, and adverse flow, such as turbulence, has been reduced. mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 871 fig. 9 streamline distribution of impeller and guide vane fig. 10 shows the distribution of turbulent kinetic energy at 30%, 50% and 70% of the guide vane’s cylindrical development surface to better contrast the pump’s internal flow characteristics [33]. turbulent kinetic energy indicates the development or attenuation of turbulence in time and space, and serves as a symbol for flow stability. after optimisation, the distribution of turbulent kinetic energy under small flow conditions undergoes slight changes, with a decrease in the turbulent kinetic energy in the guide vane channel. under the pre fig. 10 comparison of turbulence kinetic energy 872 m. chen, x. luo, w. chen, j. feng, x. wang optimised design flow and large flow conditions, a high-intensity turbulent concentration area exists at the back of the guide vane. however, the range of high turbulent kinetic energy distribution range in the guide vane channel narrows significantly after optimisation. fig. 11 shows the velocity distribution at the hub flow surface, intermediate flow surface and rim flow surface in the impeller after optimisation. the flow velocity increases gradually from the hub to the rim. under the designed operating conditions, no vortexes exist near the impeller blade in the flow passage. the streamlines are uniformly distributed, enabling stable flow into the guide vane. when the mixed-flow pump operates at a large flow, the flow path of the blade remains stable. however, a small high-speed area is generated at the blade’s leading edge and end with an increase in flow, leading to energy dissipation. in summary, no adverse flow phenomena, such as impingement and backflow [34], are found near the optimised impeller blade. the inner flow lines of the impeller are smooth, vortex-free structures are generated and the flow state is good, achieving the expected optimisation results. fig. 11 streamline distribution in impeller blades (a) optimised design conditions and (b) optimised large conditions fig. 12 shows the internal flow chart of the outlet pipe before and after optimisation. water flows from the guide vane to the outlet pipe, as shown by the results. the low-speed zone at the inlet end of the outlet pipe has significantly improved, with a uniform streamline (1) q1 (2) q2 (3)q3 fig. 12 streamline distribution in impeller and guide vane mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 873 distribution and reduced eddy currents. the outlet pipe’s flow rate in the mixed-flow pump is relatively slow before optimisation under design conditions, resulting in a decrease in overall efficiency. however, the outlet pipe exhibits a steady and uniform flow rate without evident speed changes after optimisation, resulting in better outlet performance. 4.3. pressure distribution figs. 13 and 14 show the static pressure distribution of the impeller blades of the mixedflow pump before and after optimisation for pressure and suction. the pressure on the pressure surface of the impeller blade increases towards the wheel rim, reaching its maximum at the rim and its lowest value at the inlet of the pressure surface. the pressure distribution around the blade’s middle region is greater compared with the rest of the wheel. this difference may be attributed to the longer wing on the wheel rim, which performs more work and generates higher pressure values. the suction surface and pressure surface exhibit distinct pressure change patterns. on the suction surface, the pressure gradually increases from the inlet to the outlet of the blade, with the maximum pressure occurring at (a) suction surface (b) pressure surface (1)q1 (2)q2 (3)q3 fig. 13 initial blade static pressure distribution (a) suction surface and (b) pressure surface (a) suction surface (b) pressure surface (1)q1 (2)q2 (3)q3 fig. 14 optimised blade static pressure distribution (a) suction surface and (b) pressure surface 874 m. chen, x. luo, w. chen, j. feng, x. wang the outlet. as the flow rate increases, the maximum pressure value gradually decreases. prior to optimisation, localised low-pressure zones appeared at the blade’s inlet edge, indicating the potential occurrence of vortexes in that region. after optimising the mixedflow pump, the extent of low pressure at the blade’s inlet edge gradually decreases. the suction surface of the blade experiences the minimum pressure value, whereas the pressure surface exhibits the maximum pressure value. the pressure gradient on the blade’s surface is small, resulting in a relatively uniform pressure distribution along the working surface. this observation indicates that the flow channel through the blade after optimisation is better distributed and yields improved performance compared with the flow channel before optimisation. figure 15 shows the pressure distribution of the impeller under design flow conditions and high flow conditions. the optimisation process eliminated low-pressure zones on the blades. the flow remains smooth without remarkable large-scale flow separation. under the designed flow conditions, the pressure distribution across the whole impeller blade is uniform, with no apparent sudden changes in pressure. this observation demonstrates that the impeller exhibits good hydraulic performance during operation after optimisation. fig. 16 illustrates the internal pressure distribution in the outlet pipe of the mixed-flow pump under a low flow rate conditions following optimisation. the analysis shows that the optimised pump’s outlet exhibits a continuous flow without any noticeable pressure variation areas, indicating a stable overall pressure distribution. this indicates that the device effectively conveys fluid, and the optimised mixed-flow pump demonstrates favourable flow characteristics under conditions of low flow. the outlet pipe should be designed to minimise pressure pulsations, which can effectively mitigate cavitation. (a) q2:design flow conditions (b) q3: high flow conditions fig. 15 optimised pressure distribution inside the impeller (a) design flow conditions and (b) high flow conditions fig. 16 pressure distribution in the outlet pipe mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 875 5. comparison of cfd numerical simulation results with actual experimental results in accordance with the high-precision cfd analysis results, the flow line distribution in the runner of the pump developed for the project is found to be reasonable, with a good flow pattern and smooth flow in the outlet pipe without any noticeable shedding. these results indicate that the unit exhibits improved hydraulic efficiency and stability, meeting the service requirements of the power plant. the designed pumps have been successfully manufactured, installed and put into operation. fig. 17 provides a schematic of the main components at the installation site. fig. 17 field component schematics: impeller (left) and painted guide vane body (right) following the installation of the optimised mixed-flow pump, several tests were conducted on both the modified and unmodified pumps under the same design conditions (i.e. design flow is 10.8 m3/s) as used in the cfd numerical simulation. a comparison revealed that the actual efficiency of the mixed-flow pump before renovation was 82.12%, whereas the actual efficiency after renovation was 89.97%. this represents a 7.85% increase in efficiency. in the numerical simulation results, the efficiency of the mixed-flow pump before modification was predicted to be 84.08%. the efficiency of the modified mixed-flow pump was estimated to be 87.1%, resulting in a 3.02% improvement in efficiency. although some errors were observed between the numerical simulation and actual experimental values, the overall trend of efficiency improvement was in good agreement. table 4 provides a comparison of the efficiencies. table 4 efficiency comparison initial efficiency optimised efficiency numerical simulation 84.08% 87.1% actual testing 82.12% 89.97% several factors contribute to the deviation between the predicted values from numerical simulation and the experimental values: 1) during the process of establishing the geometric model for the computational domain, some reasonable simplifications were made. certain parts of the mixed-flow pump were treated to promote convergence during grid division and ensure grid quality. 2) the numerical simulation only predicts the hydraulic efficiency in the fluid calculation area, considering only hydraulic losses within that region. however, the actual mixed-flow pump experiences additional losses, such as leakage loss and 876 m. chen, x. luo, w. chen, j. feng, x. wang mechanical loss, which are not accounted for in the numerical simulation. 3) differences exist between the turbulence model used in the numerical simulation and the actual turbulent flow in the actual pump. in summary, the optimisation and renovation of the water pump resulted in an overall improvement in its performance, which aligned with the actual on–site conditions and achieved the desired effect. all three remaining pumps in the project were successfully renovated and put into operation. 6. analysis of the actual operation effect operational testing on both the initial and optimised mixed-flow pumps under different test conditions, including two turbine two pump technology, two turbine three pump technology and two turbine four pump technology [35]. fig. 18 provides a schematic of the operation under these test conditions. in accordance with the operation data obtained from the dsc system, energy efficiency analysis was performed to evaluate the effect of optimisation before and after modification. statistical calculations were conducted on the operation data of the backup pump before and after modification, in combination with real-time operation data obtained from the power plant. energy-saving analysis was then performed on the circulating pump system. table 5 summarises the retrofitting effects of the circulating pump under different operation modes. the comparison results show that the head of the new pump is higher than that of the old pump when comparing the operation data before and after modification. the head of the modified pump is adjusted to match the head of the old pump based on the performance curve. at this time, the flow rate of a single pump increases by 3.05%, 2% and 0.7%. the flow rate of the modified pump is slightly higher than before. in accordance with the performance curve, the power will be reduced, further improving the energysaving rate. the efficiency improvement rates of the water pump after modification are found to be 15.3%, 11.97% and 7.19% by calculating the efficiency of water pump through head, flow and current, respectively. the energy-saving rate can reach 8% when the flow rate and pressure remain consistent with those before modification, indicating that an 8% energy-saving rate can be achieved through efficiency improvement. from this perspective, the pump’s performance meets the expected target. table 5 summary of circulating pump transformation effect under different operation modes operating mode initial optimised flow [m3/h] head [m] efficiency flow [m3/h] head [m] efficiency (a)a 47656 17.1 73.60% 41573 21.4 86.60% (b)b 44300 20.2 80.00% 41160 22.5 89.70% (c)c 38160 24.8 83.97% 36215 26.1 90.00% operating mode optimised data comparison same head flow [m3/h] energy saving rate lift rate of head flow efficiency improvement (a)a 49113 5.06% 3.05% 15.30% (b)b 45193 7.30% 2.00% 11.97% (c)c 38459 7.40% 0.70% 7.19% mixed-flow pump optimisation and flow characteristics based on the ‘dual carbon’ goals 877 a) b) c) fig. 18 test conditions: a) two turbine two pump technology; b) two turbine three pump technology; c) two turbine three pump technology 878 m. chen, x. luo, w. chen, j. feng, x. wang 7. conclusions this study focused on optimising and renovating a mixed-flow pump to align with ‘dual carbon’ goal. a novel mixed-flow pump design method was proposed, incorporating an intelligent optimisation design system that combined cad and cfd. the guide vane and blade were optimised using the3d inverse problem solving design. the effects of modification on the pump’s performance were investigated, analysing the initial and optimised mixed-flow pumps under low, design and large flow conditions. the main conclusions are as follows: (1) this study utilised the 3d inverse problem solving design method as the foundation for designing the mixed-flow pump to optimise the performance objective function under multiple working conditions and parameters. this method was supplemented by the nsga-ii and data dimensionality reduction. the combination of these techniques established an intelligent optimisation design system that effectively addressed conflicting objective functions and resulted in a better performance improvement of the optimised mixed-flow pump. (2) the overall efficiency increased by 3.2% following the optimisation and modification of the mixed-flow pump, and the head was raised by 3.5 m. this enhancement remarkably expanded the stable working range of the unit, effectively achieving energy-saving and emission reduction goals. the process reduced energy consumption and improved the energy utilisation ratio, contributing to improved efficiency and overall performance. (3) after the optimisation modification, the impeller and guide vane flow channels exhibited a significant reduction in vortex structure. the flow pattern became more stable, leading to the absence of obvious pressure pulsations in the outlet pipe. the pressure distribution became well-distributed, effectively reducing the occurrence probability of cavitation and prolonging the service life of the unit. these modifications contributed to improved reliability and performance. 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https://doi.org/10.22190/fume220822044p © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper guiding accuracy of the watt compliant cognate mechanisms nenad t. pavlović, dušan stojiljković university of niš, faculty of mechanical engineering, serbia abstract. in kinematics, cognate linkages are those that ensure the same input-output relationship or coupler curve geometry, while being dimensionally dissimilar. compliant mechanisms gain some or all of their mobility from the relative flexibility of their joints rather than from the rigid-body joints only, which is the case with classical mechanisms. in this paper the guiding accuracy of the compliant cognate watt four-bar linkages has been presented. the compliant cognate mechanisms have been developed as the counterparts of the rigid-body watt mechanism, where the coupler point can be guided on an approximate rectilinear path. the guiding accuracy of the "coupler" point on the path segment (minimal deviation between exact rectilinear and realized path) has been calculated for the new-designed compliant mechanisms. key words: compliant mechanisms, four-bar linkages, cognate linkages, watt mechanism, coupler point guiding 1. introduction compliant mechanisms gain some or all of their mobility from the relative flexibility of their joints rather than from the rigid-body joints only, which is the case with classical mechanisms. there are many advantages of using the compliant joints in the mechanism structure: a mechanism can be built in one piece, the weight can be reduced and wear, clearance, friction, noise and need for lubrication can be eliminated. therefore, they are suitable for application in micromachining. on the other hand, because of the elastic deformation, the motion range of these flexible segments is restricted [1]. there are many papers considering the structure and function of the compliant joints and compliant mechanisms. a number of authors have established basic nomenclature and classification for the components of compliant mechanisms [2, 3, 4]. a method to aid the design of a class of compliant mechanisms wherein the flexible sections (flexural pivots) are small in length compared to the relatively rigid sections has been also received august 22, 2022 / accepted october 26, 2022 corresponding author: nenad t. pavlović university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, 18106 niš, serbia e-mail: nenad.t.pavlovic@masfak.ni.ac.rs 2 n.t. pavlović, d. stojiljković suggested [2, 5]. ananthasuresh and kota [6] have presented a formal structural optimization technique called the homogenization method in order to design flexible structures (compliant mechanisms). böttcher et al. [7] have introduced new ideas of technically realizable joints from nature and their integration into elastically movable structures for motion tasks in positioning and manipulating engineering. the influence of the geometry as well as the material type of the compliant joints on guiding accuracy of some point of the compliant mechanisms has been also analyzed [8]. pavlović and pavlović [9] have presented the original designs of the compliant parallel-guiding mechanisms. there are many papers that offer information on large motion compliant mechanisms. mackay et al. [10] have introduced metrics for large-displacement linearmotion compliant mechanisms. dirksen et al. [11] have presented the topology synthesis of large displacement compliant mechanisms with specific output motion paths. the large-range compliant mechanisms have been also investigated [12–14]. several papers are dedicated to the consideration of the pantograph apparatus as a basis for developing a compliant mechanism. merriam et al. [15] have described a statically balanced concept and have demonstrated its optimization, testing, and implementation for a haptic pantograph mechanism. a number of authors have presented how compliant pantograph mechanisms can be used for linear displacement applications [16, 17]. patil et al. have dealt with a large motion displacement analyzed with an fem simulation. for the model shown in [18], the pantograph apparatus is not included but a similar double parallel mechanism is shown and used to create a specific desired motion. stojiljković et al. [19] have presented the designing of the compliant mechanism based on two symmetrical pantograph rigid-body mechanisms in order to achieve large scissors-like motion. the designing and guiding accuracy of the compliant cognate four-bar linkages developed as the counterparts of the rigid-body roberts–chebyshev mechanism, have been also presented [20, 21, 22]. the frequent method to synthesize a compliant mechanism is to design it as the counterpart of the rigid-body linkage being able to realize pre-defined function of the compliant mechanism (rigid-body replacement method). this paper deals with the design and guiding accuracy analyze of the compliant cognate four-bar linkages. the compliant cognate mechanisms have been developed as the counterparts of the rigid-body watt mechanism, where the coupler point can be guided on an approximate rectilinear path. the aim of this paper is to analyze the guiding accuracy of the "coupler" point on the path segment (minimal deviation between exact rectilinear and realized path) for the newdesigned compliant mechanisms. 2. watt rigid-body four-bar linkage the coupler point c of the watt four-bar linkage (fig. 1), located in the middle of the coupler, can be guided on an approximate rectilinear path segment [6]. guiding accuracy of the watt compliant cognate mechanisms 3 fig. 1watt four-bar linkage the links lengths of this mechanism as well as the position of the input crank, that enable realizing of horizontal displacement of coupler point c of δxc = 5 mm, with the small angular displacement of the input crank (δ ≤ 5o) and the minimal difference between realized and exact rectilinear path (δyc), has been shown as following [6]: 𝑎 = 𝐴0𝐴̅̅ ̅̅ ̅ = 61.406 𝑚𝑚 (1a) 𝑏 = 𝐵0𝐵̅̅ ̅̅ ̅ = 𝑎 (1b) c= 𝐴𝐵̅̅ ̅̅ = 1.04𝑎 (1c) d= 𝐴0𝐵0 ̅̅ ̅̅ ̅̅ ̅ = 2.1𝑎 (1d) 𝐴𝐶̅̅ ̅̅ = 𝐵𝐶̅̅ ̅̅ = 𝑐 2 (1e) 𝜑 = 313 ÷ 318 𝑜 (1f) 𝛥𝑦𝐶 = 6.4 𝜇𝑚 (1g) the cognate four-bar linkages are those that ensure the same input-output relationship or coupler curve geometry, while being dimensionally dissimilar. the roberts–chebyshev theorem states that each coupler curve can be generated by three different four-bar linkages. these four-bar linkages can be constructed using similar triangles and parallelograms. fig. 2 shows two cognate watt rigid-body four-bar linkages: a0a1cb1k0 and b0b2ca2k0, introduced in this paper, where coupler point c, located in the end of the coupler, can be also guided on an approximate rectilinear path segment. 4 n.t. pavlović, d. stojiljković fig. 2 the cognate watt four-bar linkages geometrical relations defining link positions are: 𝐴0𝐴1 ̅̅ ̅̅ ̅̅ ̅ǁ𝐴𝐶̅̅ ̅̅ ˄ 𝐴0𝐴̅̅ ̅̅ ̅ǁ𝐴1𝐶̅̅ ̅̅ ̅ (2a) 𝐵0𝐵2 ̅̅ ̅̅ ̅̅ ̅ǁ𝐵𝐶̅̅ ̅̅ ˄ 𝐵0𝐵̅̅ ̅̅ ̅ǁ𝐵2𝐶̅̅ ̅̅ ̅ (2b) 𝐴𝐶̅̅ ̅̅ : 𝐶𝐵̅̅ ̅̅ = 𝐴1𝐵1 ̅̅ ̅̅ ̅̅ : 𝐵1𝐶̅̅ ̅̅ ̅ ⇒ 𝐵1𝐶̅̅ ̅̅ ̅ = 1 2 𝐴1𝐶̅̅ ̅̅ ̅ (2c) 𝐵𝐶̅̅ ̅̅ : 𝐶𝐴̅̅ ̅̅ = 𝐵2𝐴2 ̅̅ ̅̅ ̅̅ ̅: 𝐴2𝐶̅̅ ̅̅ ̅ ⇒ 𝐴2𝐶̅̅ ̅̅ ̅ = 1 2 𝐵2𝐶̅̅ ̅̅ ̅ (2d) 𝐴𝐶̅̅ ̅̅ : 𝐶𝐵̅̅ ̅̅ = 𝐴0𝐾0 ̅̅ ̅̅ ̅̅ ̅: 𝐾0𝐵0 ̅̅ ̅̅ ̅̅ ̅ ⇒ 𝐴0𝐾0 ̅̅ ̅̅ ̅̅ ̅ = 1 2 𝐴0𝐵0 ̅̅ ̅̅ ̅̅ ̅ (2e) coordinates of the cognate mechanism joints have been calculated based on plane analytic geometry, respectively:  joint a1 𝑥𝐴1 = 𝑦𝐴0−𝑦𝐶+ 𝑦𝐴0−𝑦𝐴 𝑥𝐴0−𝑥𝐴 𝑥𝐶− 𝑦𝐶−𝑦𝐴 𝑥𝐶−𝑥𝐴 𝑥𝐴0 𝑦𝐴0−𝑦𝐴 𝑥𝐴0−𝑥𝐴 − 𝑦𝐶−𝑦𝐴 𝑥𝐶−𝑥𝐴 (3a) 𝑦𝐴1 = 𝑦𝐶 + 𝑦𝐴0−𝑦𝐴 𝑥𝐴0−𝑥𝐴 (𝑥𝐴1 − 𝑥𝐶)=𝑦𝐴0 + 𝑦𝐶−𝑦𝐴 𝑥𝐶−𝑥𝐴 (𝑥𝐴1 − 𝑥𝐴0) (3b)  joint b1 𝑥𝐵1 = 1 2 (𝑥𝐴1 + 𝑥𝐶) (3c) 𝑦𝐵1 = 1 2 (𝑦𝐴1 + 𝑦𝐶) (3d) guiding accuracy of the watt compliant cognate mechanisms 5  joint b2 𝑥𝐵2 = 𝑦𝐵0−𝑦𝐶+ 𝑦𝐵0−𝑦𝐵 𝑥𝐵0−𝑥𝐵 𝑥𝐶− 𝑦𝐶−𝑦𝐵 𝑥𝐶−𝑥𝐵 𝑥𝐵0 𝑦𝐵0−𝑦𝐵 𝑥𝐵0−𝑥𝐵 − 𝑦𝐶−𝑦𝐵 𝑥𝐶−𝑥𝐵 (3e) 𝑦𝐵2 = 𝑦𝐶 + 𝑦𝐵0−𝑦𝐵 𝑥𝐵0−𝑥𝐵 (𝑥𝐵2 − 𝑥𝐶)=𝑦𝐵0 + 𝑦𝐶−𝑦𝐵 𝑥𝐶−𝑥𝐵 (𝑥𝐵2 − 𝑥𝐵0) (3f)  joint a2 𝑥𝐴2 = 1 2 (𝑥𝐵2 + 𝑥𝐶) (3g) 𝑦𝐴2 = 1 2 (𝑦𝐵2 + 𝑦𝐶) (3h)  joint k0 𝑥𝐾0 = 1 2 (𝑥𝐴0 + 𝑥𝐵0) (3i) 𝑦𝐾0 = 1 2 (𝑦𝐴0 + 𝑦𝐵0) (3j) 3. watt compliant four-bar linkage fig. 3 shows a notch joint (circular flexure hinge) as a characteristic type of the compliant joint [6]. this compliant joint is fully determined by two parameters: the width of relatively rigid segments wr and the width of relatively elastic segments we. fig. 3 a notch compliant joint based on the rigid-body watt mechanism, a compliant watt mechanism with notch joints was developed (fig. 4a). compliant joints a0 and a have been oriented in the direction of rigid-body input crank a0a, while compliant joints b0 and b have been oriented in the direction of rigid-body follower b0b. 6 n.t. pavlović, d. stojiljković a) b) fig. 4 the basic watt compliant mechanism with notch joints (circular flexure hinges) in: a) undeformed position, b) deformed position the input force may act in the middle of the "input crank" (fa), in the middle of the "follower" (fb) or at the end of "coupler" (fc). the deformed position of the compliant watt mechanism is shown in fig. 4b [6]. the symmetry axes of the compliant notch joints cross each other at the point 1 (fig. 5) corresponding to revolute joint of the rigid-body counterpart. fig. 5 characteristic key points of a compliant notch joint the other characteristic key points of a compliant notch joint (fig. 5) can be calculated by using the set of equations: guiding accuracy of the watt compliant cognate mechanisms 7 𝑟2⃗⃗ ⃗= 𝑟1⃗⃗⃗ + 𝑤𝑅 2 𝑒𝑖(𝜙− 𝜋 2 ) (4a) 𝑟3⃗⃗ ⃗= 𝑟2⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖𝜙 (4b) 𝑟4⃗⃗⃗ = 𝑟3⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖(𝜙−𝜋) (4c) 𝑟5⃗⃗ ⃗= 𝑟4⃗⃗⃗ + 𝑤𝑅−𝑤𝐸 2 𝑒𝑖(𝜙+ 𝜋 2 ) (4d) 𝑟6⃗⃗ ⃗= 𝑟1⃗⃗⃗ + 𝑤𝑅 2 𝑒𝑖(𝜙+ 𝜋 2 ) (4e) 𝑟7⃗⃗ ⃗= 𝑟6⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖𝜙 (4f) 𝑟8⃗⃗ ⃗= 𝑟6⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖(𝜙−𝜋) (4g) 𝑟9⃗⃗⃗ = 𝑟6⃗⃗ ⃗+ 𝑤𝑅−𝑤𝐸 2 𝑒𝑖(𝜙− 𝜋 2 ) (4h) angle determines the orientation of a compliant joint in reference to the x-axis (fig. 5). fig. 6a shows the first cognate compliant watt mechanism with notch joints, introduced in this paper, that has been developed based on cognate watt rigid-body fourbar linkage a0a1cb1k0 (fig. 6b). a) b) fig. 6 the first watt cognate compliant mechanism with notch joints (circular flexure hinges) (a) and its rigid-body counterpart a0a1cb1k0 (b)  8 n.t. pavlović, d. stojiljković fig. 7a shows the second cognate compliant watt mechanism with notch joints, introduced in this paper, that has been developed based on the cognate watt rigid-body four-bar linkage b0b2ca2k0 (fig. 7b). fig. 7 the second watt cognate compliant mechanism with notch joints (circular flexure hinges) (a) and its rigid-body counterpart b0b2ca2k0 (b) the deformed position of the watt cognate compliant mechanisms with notch joints are shown in fig. 8. a) b) fig. 8 the first (a) and the second (b) watt cognate compliant mechanism with notch joints in deformed position guiding accuracy of the watt compliant cognate mechanisms 9 4. guiding accuracy of the watt compliant mechanisms the guiding accuracy of "coupler" point c (on the path segment of δxc = 5 mm) of the compliant watt mechanism as well its cognate compliant mechanisms with notch joints were analyzed. the lengths of the mechanism "links" have been defined in equations (1) with the following compliant joint parameters: the width of relatively rigid segments wr = 10 mm and the width of relatively elastic segments we = 1mm. the position analysis of the compliant mechanisms was performed using the ansys software for material piacryl (modulus of elasticity e = 3700 n/mm2, bending strength σbs = 90 n/mm2) as well as material thickness of δ = 4 mm. this material is not expensive; it is available on the market as well as suitable for further experimental research of these compliant mechanisms. the calculation was performed for the elements with a rectangular cross-sectional area using plane 183 (2-d 8-node structural solid) as a characteristic ansys element type (fig. 9). fig. 9 ansys element type plane 183 (2-d 8-node structural solid) due to the fact that the acting point of the input force does not have to be located on the "input crank" of the compliant mechanism, the guiding accuracies were compared for three different cases of location of input force acting points: in the middle of the "input crank" fa, in the middle of the "follower" fb and at the "coupler" point nearby the joint a or b– fc (fig. 4a, 6a, 7a). the results are shown in table 1. table 1 guiding accuracy δyc [μm] of the watt compliant mechanisms on the horizontal displacement of δxc = 5 mm fa fb fc 0 2.3 8.8 11.7 1 0.3 6.5 17.8 2 29.2 0.6 1.3 0 the basic watt compliant mechanism 1 the first cognate watt compliant mechanism 2 the second cognate watt compliant mechanism 10 n.t. pavlović, d. stojiljković 5. conclusion the frequent method to synthesize a compliant mechanism is to design it as the counterpart of the rigid-body linkage being able to realize pre-defined function of the compliant mechanism (rigid-body replacement method). this paper deals with the design and guiding accuracy analyze of the compliant cognate four-bar linkages. the basic compliant mechanism as well as two compliant cognate mechanisms have been developed as the counterparts of the rigid-body watt mechanism, where the coupler point can be guided on an approximate rectilinear path. the guiding accuracy, that is, the difference between realized and exact rectilinear path, have been analyzed and compared for all above mentioned compliant mechanisms being suitable to realize approximate rectilinear guiding of the coupler point. the basic compliant model with the input force acting in the middle of the "input crank", the first cognate compliant model with the input force acting in the middle of the "input crank" as well as the second cognate compliant model of the watt mechanism with the input force acting in the middle of the "follower" and in the end of "follower", realize better rectilinear guiding accuracy than the rigid-body counterpart linkage. the differences between realized and exact rectilinear path of these compliant mechanisms rate in range yc = 0.3  2.3 m on the path segment of δxc = 5 mm, comparing with the yc = 6.4 m of the rigid-body counterpart linkage. also, the producing of the rigid-body linkages is not possible without appearing of the tolerance of the links lengths and clearance in the revolute joints, which decrease the guiding accuracy. it means that the roberts–chebyshev theorem, concerning a coupler curve being generated by two different four-bar linkages, should be also applied on the compliant mechanism. the best guiding accuracy has been provided by the first cognate watt compliant mechanism (yc = 0.3 m on the path segment of δxc = 5 mm) with the input force acting point located in the middle of the "input crank". thus the cognate compliant mechanism has realized a smaller deviation between exact rectilinear and realized path than the basic compliant mechanism, and this fact confirms our suggestion that the results offered by the cognate compliant mechanisms should be taken into consideration in the synthesis procedure of the compliant mechanisms. acknowledgements: this research was financially supported by the ministry of education, science and technological development of the republic of serbia (contract no. 451-03-68/202214/ 200109). references 1. zentner, l. linβ, s., 2019, compliant systems mechanics of elastically deformable mechanisms, actuators 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(eds.), advances in asian mechanism and machine science, asian mms 2021, mechanisms and machine science, 113, springer, cham, pp. 370-378. 13142 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume241024005d © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper stability analysis of the second-order dpcm prediction filter and correlation with signal-to-quantization noise ratio nikola danković1, zoran perić2, dragan antić1, aleksandar jocić3, saša s. nikolić1, igor kocić1 1department of control systems, faculty of electronic engineering, university of niš, serbia 2department of telecommunications, faculty of electronic engineering, university of niš, serbia 3department of measurements, faculty of electronic engineering, university of niš, serbia abstract. the stability study of the differential pulse code modulation system with the special focus on a predictor is given in this paper. moreover, sufficient stability conditions for a linear prediction (recursive) filter are derived. the corresponding mathematical inequalities for the commonly used second-order predictor are derived. a method off probability estimation for the predictor coefficients is given, both deterministic and stochastic. it allows the design of the differential pulse code modulation system with the linear predictor whose coefficients meet the technical requirements. finally, the probability of stability values for the specific second-order predictor are computed and compared with the corresponding values of the signal-toquantization noise ratio (sqnr). the correlation between these values is verified for different frame lengths. this could be crucial for the optimal choice of predictor coefficients. useful conclusions are drawn regarding the stability and performances of the system. key words: normal distribution, probability density function, differential pulse code modulation, linear prediction, sqnr 1. introduction differential pulse code modulation (dpcm) is one of the most effective transmission signal techniques widely used in telecommunications and signal processing. speech [1–3], and image coding [4–8], video stream data compression [9], medical research [10–14] are some of the various dpcm applications. adaptive dpcm (adpcm) is a dpcm which received: october 24, 2024 / accepted january 28, 2025 corresponding author: nikola danković faculty of electronic engineering, university of niš, aleksandra medvedeva 4, 18000 niš, serbia e-mail: nikola.dankovic@elfak.ni.ac.rs 2 n. danković, z. perić, d. antić, a. jocić, s.s. nikolić, i. kocić includes backwards adaptive quantization and/or backwards adaptive prediction [1, 3, 9]. in recent decades, new specific types of dpcm systems have been discovered such as clustered dpcm [7] and hybrid dpcm [8]. the prediction and development of prediction models for applications in various fields has become increasingly important in recent years [15, 16]. linear prediction [1, 2, 17–20] where the prediction of the current sample is computed as a linear combination of the previous samples, is the basis of any dpcm system. a historical survey of linear prediction can be found in [21]. the dpcm transmission system is a nonlinear feedback system. due to the negative closed loop [16], although essentially a telecommunication system, dpcm is also suitable for control systems analysis. in this context, specific properties of this system, especially its linear part (recursive, prediction filter) have already been considered. a sensitivity analysis for dpcm prediction filter of arbitrary order has been performed yet. a robustness analysis for the first and second-order prediction filter is presented in [22]. beside above mentioned sensitivity and robustness, one of the most important properties of any real system is stability. some stability analyses of the dpcm transmission system have already been done for the commonly used second-order predictor [23]. it was found that the stability of the predictor, which is the linear part of dpcm system, is a sufficient condition for the stability of the whole system. in practice, every system is imperfect in some way [24]. this means that the system parameters are not deterministic but stochastic variables. in this case, the classical methods of stability analysis are not applicable, since we only know with what probability the system is stable. for these reasons, we introduced a new term: “probability of stability” [25, 26]. we have already proposed the method of stability estimation for the first-order predictor [27]. in this paper, probability of stability estimation method is presented and proven for the commonly used second-order predictor. a correlation is established between the processing quality parameter, the signal-to-quantization noise ratio (sqnr), and the probability of stability (ps). this can be very important when designing a dpcm system. the optimal choice of values for the predictor coefficients guarantees better performance of the system. the rest of this paper is structured as follows. in section 2, we explain the theoretical background of the dpcm system. in section 3, we analyse the stability of the linear part of this system (prediction filter) and give the stability conditions. numerical results for the second-order predictor are given in section 4. appropriate values for probability of stability and sqnr are determined. we analysed the results and found the correlation between these parameters in section 5. concluding remarks and ideas for future work are considered in section 6. 2. dpcm system – theoretical background dpcm is a method of converting an analog signal to a digital signal by sampling the analog signal and then quantizing the difference between the actual sample value and its predicted value. the predicted value of the actual sample is based on the previous sample or samples. the basic concept of dpcm coding a difference, is based on the fact that most source signals have significant correlation between successive consecutive samples, so the quantizer uses redundancy in sample values which in turn means a lower bit rate [1, 3, 28]. stability analysis of the second-order dpcm prediction filter and correlation ... 3 the block diagram of a dpcm/adpcm encoder is shown in fig. 1a. it consists of the quantizer, the inverse quantizer, and predictor. the linear prediction filters in encoder (fig. 1a) and decoder (fig. 1b) are of the special interest for further stability analysis. fig. 1a also shows the additional subsystem for the adaptive prediction (buffer and predictor coefficients estimator connected with dashed lines), which forms an adpcm encoder [1, 13]. the main idea of dpcm is to form the difference dn between the current sample xn and its predicted value nx̂ : ˆ n n nd x x  . (1) this difference is quantized and transmitted. let us en denote the quantization error which occurs due to quantization of the difference dn. for the linear predictor, the predicted value nx̂ is calculated as a linear combination of the previously quantized reconstructed samples yn-i. the operation of the dpcm system with the k-th order predictor is described by the following equations: ˆ n n n n n ny d e x x e     (2) 1 ˆ k n i n i i x a y    , (3) where ai, i = 1, 2,…, k are predictor coefficients. fig. 1 dpcm/adpcm system, a) encoder, b) decoder 4 n. danković, z. perić, d. antić, a. jocić, s.s. nikolić, i. kocić correlation coefficients ρi indicate the degree of correlation between i successive samples [1, 13]. in a dpcm system, the values of predictor coefficients are predetermined according to the class of signals under consideration and are known for both the encoder and decoder, but they are suboptimal. in fig. 1, the predictor coefficients ai, i=1,2,…, k are adapted according to ρi, i=1,2,…, k for each frame. the predictor coefficients could be determined from the correlation coefficients using the levinson-durbin algorithm for predictors of arbitrary order [2]. the functional dependencies between these coefficients for the second-order predictor are well known [20], and given by:   2 1 2 2 1 1 22 2 1 1 1 , 1 1 a a             . (4) dpcm systems are nonlinear feedback systems. in addition to the predictor, which is a linear element, the system contains a quantizer as a nonlinear element. the well-known parameter – sqnr, is used to evaluate the quality of the reconstructed signal at the output of the adpcm system [13, 29]:   2 1 1 adpcm 2 1 1 sqnr 10log l m jn j n l m jn jn j n x x y                   , (5) where l is the number of frames and m is a frame length. the stability analysis of the whole dpcm system is very difficult, but it is known that an element with nonlinear static characteristic with saturation stabilizes the system [30]. consideration of the stability of the prediction filter is very important in the design of the system. the basic requirement is that the predictor coefficients are located inside stability region in parametric space or very close to this region. the aim of this paper is to consider the stability of the prediction filter with the commonly used second-order predictor, although the proposed method is also applicable to higher order predictors with some alternative methods. we will also investigate in what way stability is related to the sqnr in this paper. 3. stability analysis of the linear prediction filter it is well known that the predictor is an essential part of any dpcm/adpcm system and its coefficients have a great impact on the system performances [2, 18, 20]. we will consider the stability of the prediction filter with k-th order predictor. the relation for the k-th order predictor is given by eq. (3), i.e.:     1 ˆ k i i i x z a z y z          , (6) in z-domain. transfer function of the predictor is: stability analysis of the second-order dpcm prediction filter and correlation ... 5   1 k i p i i w z a z   . (7) transfer function of the prediction (recursive) filter in the encoder has the following form:       1 1 1 1 k i i p i r k ip i i a z w z w z w z a z           . (8) transfer function of the prediction filter in the decoder is:     1 1 1 1 1 d r k ip i i w z w z a z      . (9) prediction filters are stable if all the poles of the transfer functions in eqs. (8), (9) lie inside the unit circle, i.e., if the characteristic eq. (which is the same for the both filters): 1 1 0 k i i i a z    , (10) has all its zeroes inside the unit circle. eq. (10) can be written as: 1 0 k k k i i i z a z     . (11) the stability conditions of the system described by the characteristic eq. (11) can be determined using various stability criteria (jury test, routh-hurwitz criterion, etc. [31]). we will use the routh-hurwitz stability criterion in this paper. the bilinear transformation z=(1+s)/(1-s) is a mapping unit circle inside the z-plane into the left half of the s-plane. after applying the bilinear transformation onto eq. (11), we obtain: 1 1 1 0... 0k k k kb s b s b s b      , (12) where b0, b1,…, bk are functions of the predictor coefficients a0, a1,…, ak, i.e.: b0=φ0(a1,…, ak),…,bk=φk(a1,…, ak). remark 1: other more general forms for the bilinear transformation can be found in the literature, but in its application in the analysis of system stability in the z-domain, the above-mentioned and very similar one z=(s+1)/(s-1) occurs most frequently. we have obtained absolutely the same result if we apply it. it should be noted that both are special cases of the more general form z=m(1+s)/(1-s) for m=1 and m=-1 respectively, where m is an arbitrary real number [31]. in this more general case, the region of stability in the zplane was not a unit circle, but a circle of radius m. one more general form of the bilinear transformation is z=(1+ts/2)/(1-ts/2) [32]. this form changes absolutely nothing, because sample period t has no influence on the analysis. in our case t=2. 6 n. danković, z. perić, d. antić, a. jocić, s.s. nikolić, i. kocić the prediction filter (described with characteristic eq. (11) in z-plane) is stable if all the zeroes of the characteristic eq. (12) are inside the left half of s-plane. we form hurwitz determinant dk for eq. (12): 1 3 5 2 4 1 3 2 0 0 0 0 0 . 0 0 0 0 0 k k k k k k k k k k k b b b b b b b b d b b b          (13) the necessary condition for the stability of the prediction filter in eq. (11) is that all coefficients b0, b1,…, bk are greater than zero and the sufficient condition is that all diagonal sub-determinants di, i=1,2,…, k, are greater than zero: 1 1 1 3 2 2 0, 0, 0. k k k k k k d b b b d b b d          (14) the stability region of the prediction filter in eq. (11), sk, can be determined by the above inequalities for the first-order predictor up to the k-th order predictor. in the case of the first-order predictor, the stability region s1 is given by the: 11 1a   . (15) for the second-order predictor, the stability region s2, in the parametric space a1, a2 is given by conditions: 1 2 1 2 2 1 0, 1 0, 1. a a a a a         (16) the stability region s2, is the triangle shown in fig. 2. remark 2: for the second order, the stability of the system can of course be easily determined, but because of the possible application of this method for higher-order predictors, we have given general conditions here by using the routh-hurwitz criterion [31]. we have also applied the jury stability test mentioned above and obtained the same system of inequalities. for a third-order predictor, for example, applying both stability criteria results in the following system of inequalities: a1+a2+a3<1, a1-a2+a3>-1, a1a3+1>a2+a3 2 and it defines the stability region s3. stability analysis of the second-order dpcm prediction filter and correlation ... 7 fig. 2 the stability region of the second-order prediction filter previous stability analysis was performed under the assumption that the system parameters, i.e., the predictor coefficients are deterministic. this is true if the dpcm system is perfect unlike all real systems. sometimes these imperfections have no any visible effect on the system performances, but in many cases this effect cannot be neglected. some system properties such as stability or dynamical response are directly dependent on them. mathematically, the imperfections are fluctuation of the system coefficients around the nominal values of the coefficients [25, 33]. the system can be either stable or unstable for constant parameter values. when the predictor coefficients are stochastic, they have some distribution. the most common distribution in the case of a real system design is the normal distribution. all calculations in this paper are performed for this type of distribution. we now assume that the system is stable with a certain probability. this is the reason why we need to introduce the term: probability of stability instead of the traditional stability of the system. the probability of stability is defined as follows:  1 1, , k k k s p f a a da da   , (17) where    1 1 , , k k i i i f a a f a   is the total density function and sk is the stability region. based on this, we can estimate the stability of an arbitrary order prediction filter. the proposed method can also be used for the third and higher-orders predictors, but the calculation of the integrals for the probability of stability becomes more complicated because the stability regions for which the integrals are calculated become more complex (see eq. 17). in this sense, in addition to classical integration, it is possible to use the approximate method as a monte carlo to calculate complex integrals for probability estimation. 4. numerical results – second-order predictor the most common predictors are those of the first and second order. the second order is usually an optimal order since the prediction gain often goes into saturation for higher 8 n. danković, z. perić, d. antić, a. jocić, s.s. nikolić, i. kocić order predictors [2] and this is the reason why we are focused on the second order in this paper. stability analysis in the case of the first-order predictor was studied in [27]. the most important fact we verified is that the prediction filter is always stable in the whole range of possible values for the predictor coefficient a1. when the system is not deterministic, i.e., when the parameter a1 is stochastic, we obtained the probability of stability of the prediction filter for different values of the variance. in this paper, we perform a stability analysis for the second-order predictor. we analyze a recorded speech signal of length l=10200 samples with different frame lengths, m=10, 20,…,150 samples. the adpcm system shown in fig. 1 was used to obtain the means and standard deviations of the predictor coefficients. the input signal is divided into frames of length m which are buffered. for each frame, the predictor coefficients a1 and a2 are first calculated, which are used to process the samples of that frame. in this way, we obtain round (l/m) pairs of coefficients a1 and a2 values that are adjusted for the corresponding frames. we form sets of coefficients a1 and a2 values obtained for all frames and determine their means and standard deviations, 1 2 1 2, , ,a a   , respectively. based on these values and relations, we calculate the probability of stability, ps. we set the obtained means of the predictor coefficients 1a and 2a as fixed predictor values (dpcm system), reprocess the signal and obtain the sqnr value. in this way, for the obtained values of predictor coefficients a1 and a2, we calculated the mean and standard deviation and sqnr for all values of m (table i). for illustration, the distributions of the coefficients a1 and a2 for two of eight values of the frame length (m=10 and m=150), are shown in figs. 3 and 4, respectively. normal (gaussian) distributions with the same means and standard deviations are also shown for comparison. from these figures (as well as for other values of frame length), it can be easily demonstrated that the mean (nominal) values of the predictor coefficients  1 2,a a are located into the region described by eq. (16). this means that the linear prediction filter is stable if we assume that the predictor coefficients are deterministic, i.e., their values are equal to the desired (nominal) values. since the dpcm system is imperfect in practice, the predictor coefficients cannot be perfectly matched to their projected values. fig. 3 the probability density function of predictor coefficients a1 and a2 for m=10 stability analysis of the second-order dpcm prediction filter and correlation ... 9 fig. 4 the probability density function of predictor coefficients a1 and a2 for m=150 the probability density function (pdf) for normal distribution is given as [25, 33]: 2 2 1 1 2 2 1 2 1 21 2 1 1 1 1 ( , ) exp , 2 22 2 a a a a f a a                         (18) where σ1 and σ2 are the standard deviations, while 1a and 2a are the mean values of the predictor coefficients a1 and a2, respectively. the probability of stability for the second-order system is derived from:   2 2 1 2 1 2 1 2 1 2 , ( , ) s s f a a da da p f a a da da         , (19) where s2 is the stability region of the prediction filter with the second-order predictor, while the integral in the denominator represents the total probability and is equal to 1. this gives the probability of stability form suitable for calculations. the probabilities of stability and the signal-to-quantization noise ratios for different frame lengths are given in table 1. table 1 the probability of stability (ps2) and the signal-to-quantization noise ratio (sqnr) of the second-order prediction filter for different values of frame length m [sample] 10 20 30 40 50 60 100 150 ā1 0.984 1.125 1.220 1.231 1.281 1.329 1.377 1.419 σ1 0.228 0.226 0.220 0.211 0.220 0.219 0.216 0.214 ā2 -0.228 -0.288 -0.351 -0.356 -0.399 -0.439 -0.485 -0.524 σ2 0.199 0.215 0.213 0.219 0.221 0.218 0.216 0.204 ps2 0.790 0.699 0.664 0.658 0.644 0.634 0.629 0.629 sqnr [db] 17.22 17.48 17.57 17.60 17.67 17.73 17.90 17.96 10 n. danković, z. perić, d. antić, a. jocić, s.s. nikolić, i. kocić 5. discussion as we can see, the prediction filter is stable with some probability, unlike the ideal case when the predictor coefficients are perfectly adjusted. when the deviation from projected values of predictors is increasing, the probability of stability of the prediction filter is decreasing. however, the probability of stability is also decreasing when mean values of the predictor coefficients are approaching the limits of stability region s2. thus, we see that the probability of stability is the highest for m=10 samples (about 79%), although standard deviations of the parameters are not so small, but the mean values of coefficients are deep into the stability region. for m=150 we have the similar standard deviations (0.214, 0.204), but the probability of stability of the prediction filter is the smallest (62.9%). therefore, the mean (desired) values of the predictor coefficients are closer to the boundary of the stability region more than in any other case. the stability analysis presented above is a generalization of the classical stability considered at the beginning of this section. in the deterministic case (σ=0) we obtain a probability of stability equal to 100%, corresponding to the previous analysis for perfect systems. we can see that the sqnr increases from m=10 up to m=150 while the probability of stability steadily decreases for the same range of frame lengths. we can conclude that sqnr is directly correlated with (1-ps) (“probability of instability”). this is a very interested and useful conclusion. systems where some important performance parameters (such as sqnr) are set close to the maximum may be approaching the limits of stability. fig. 5 shows the measured dependencies of the signal-to-quantization noise ratio and the probability of stability on frame length, respectively. the aim is to optimally select the predictor coefficients in order to achieve the best system performance. the values obtained for the stability probabilities as well as the corresponding values for the signal noise can help with predetermined technical conditions. fig. 5 dependencies of sqnr and ps2 on m 6. conclusions complete stability study of the prediction filter with the most commonly used secondorder predictor was presented in this paper. it means that we gave relations describing the stability analysis of the second-order dpcm prediction filter and correlation ... 11 stability region via predictor coefficients, and then verified stability of the prediction filter for a concrete signal and obtained values for predictor coefficients. finally, we generalized the stability study for a real system when the predictor coefficients are not perfectly adjusted (they are normally distributed around the projected value). probability estimation was performed for the different frame lengths. the correlation between ps and sqnr is given in the case of the second-order predictor in this paper. an increase in the value of the signal processing quality parameter, sqnr, with an increase in frame length is associated with a decrease in the probability of stability of the system, ps2. this is very important information that has not yet been utilized so to achieve the best possible system performance when designing the system and selecting the predictor coefficient values. further stability study could be probability of stability estimation of the whole dpcm system. it must include a quantizer, the nonlinear part of the dpcm system. anyway, the stability analysis of the prediction filter with predictor is the basis for this analysis. finally, the stability estimation of the prediction filter with predictor of higher order could be performed using the proposed method with classical integration as in this paper or by numerical integration using monte carlo method and find the correlation also between ps and sqnr values. this paper should form the basis for further research and application in today's popular science fields (machine learning, deep neural networks) where prediction models is very important. acknowledgement: this work has been supported by the ministry of science, technological development and innovation of the republic of serbia (grant number 451-03-66/2024-03/200102) and, in part, by the european union, within the program horizon-widera-2023-access-02 (grant agreement number 101160293). references 1. furui, s., 2001, digital speech processing, synthesis, and recognition, crc press, new york, pp. 133-212. 2. benesty, j., chen, j., huang, y.a., 2008, linear prediction, springer handbook of speech processing, springer, berlin, heidelberg, pp. 121-134. 3. sheferaw, g.k., mwangi, w., kimwele, m., a. mamuye, a., 2024, gated recurrent unit predictor modelbased adaptive differential pulse code modulation speech decoder, eurasip journal on audio, speech, music processing, 2024(6), pp. 1-17. 4. mitchell, h.b., estrakh, d.d., 1997, a modified owa 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117800-117809. 30. tarbouriech, s., garcia, g., gomes da silva jr, j.m., queinnec, i., 2011, stability and stabilization of linear systems with saturating actuators, springer, london. 31. fadali, m.s., visioli, a., 2013, digital control engineering: analysis and design, academic press, pp. 91126. 32. jiang, y., hu, x., wu, s., 2014, transformation martix for time discretization based on tustin’s method, mathematical problems in engineering, 2014(1), 905791. 33. forsyth, d., 2018, probability and statistics for computer science, springer, cham, pp. 115-137. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 329 339 https://doi.org/10.22190/fume180120016m © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper prediction of temperature distribution in the worm gear meshing aleksandar miltenović 1 , milan tica 2 , milan banić 1 , đorđe miltenović 3 1faculty of mechanical engineering, university of niš, serbia 2faculty of mechanical engineering, university of banja luka, bosnia and herzegovina 3the college of textile, leskovac, serbia abstract. worm gear transmissions have number of advantages over other types of transmission, allowing them a wide scope of applications for the transfer of power and movement. one of the important advantages of this transmission is the possibility of obtaining a large transmission ratio. the lack of worm gear transmission means a relatively low efficiency, especially for the extreme operating conditions primarily related to the high frequency of rotation. between the flanks of worm and worm gears there is considerable slippage, which results in wear at the worm gear flank and considerable significant power losses that are converted into heat. the amount of energy that is converted into heat to a large extent is determined by the friction coefficient between the flanks. it is therefore very important to take into consideration the process of tribo-system mesh of flanks and lubricant. the paper presents fem calculated distribution of transmission temperature based on the data about power losses obtained analytically. the resulting temperature distribution is compared to the experimental research. key words: worm gear transmission, temperature distribution, fem, friction 1. introduction the main function of the power transmission is to convert and manage mechanical energy from the drive to the driven machine. the most commonly used are power transmissions with gears; they are used in over 80% of mechanically transferred energy. in the division of the transmissions it is very important to distinguish transmission with pure rolling from that with helical rolling. the helical rolling transmission, besides pure rolling, has a slip along the lateral line, which results in additional power losses and received january 20, 2018 / accepted april 25, 2019 corresponding author: aleksandar miltenović university of niš, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: aleksandar.miltenovic@masfak.ni.ac.rs 330 а. miltenović, m. tica, m. banić, đ. miltenović lower efficiency. worm gears belong to the category of the helical rolling transmission. a high slip between flanks of worm and worm gear leads to high local surface pressure, which requires the use of materials pairs that are resistant to damage form like scuffing. the worm gears have a number of advantages over other types of transmissions, allowing them a wide scope of applications for the transfer of power as well as that of movement. they are used in machine tools, transportation equipment, primarily in power transfer vehicles, as well as precision devices for movement transmission. determination of heat generation using fem can be used in the cases where temperature increase is caused by friction between two materials. jain et al. [1] show the temperature distribution during the friction stir welding where he has used a thermo-mechanical model based on the lagrangian incremental technique. their fem model can predict temperature, forces and strain distribution. haddad et al. [2] present the numerical dem–fem coupling approach for determination of thermomechanical parameters of the contact interface. their numerical approach is used to estimate various parameters by giving values of compression load, sliding velocity and microscopic friction coefficient. ziegltrum et al. [3] use tehd (thermo-elastohydrodynamic lubrication) simulation to determine influences of lubricants on the gear losses that are load-dependent. the tehl simulation is based on fem; it simulates contact along the contact path of the spur gears. the thermal effect for realistic prediction of the friction coefficient is taken into consideration. milošević et al. [4] present an approach for determination of residual stress in the rail wheel during quenching process. residual stress is estimated using fem thermal simulation with relatively high precision. pech [5] uses the fem approach to calculate temperature distribution of worm gear transmission with worm gear made of plastic and worm made of steel. he has used limited input parameters; some of them he has obtained experimentally, with the intention to determine heat transfer coefficient . abukhshim et al. [6] give reviews of the temperature measurement methods and the analytical and numerical models for predicting temperature and temperature distribution in metal cutting. fem is used as a numerical approach for predicting heat generation and temperature. taburdagitan et al. [7] investigate frictional heat during meshing spur gear pair with coupled thermo-elastic finite element analysis. the numerical results obtained by the finite element analysis yield high temperatures in comparison with the experimental results. berger et al. [8] investigate standardized wear and temperature prediction for worm gears under non-steady operating conditions and develop new calculation methodology that is proposed for din 3996. the novelty of this paper lies in predicting temperature distribution for the whole transmission including lubricant, gears, shafts and housing using fem by giving analytically calculated heat according to din 3996 input, as well as thermal parameters such as thermal conductivity, specific heat capacity and coefficient of thermal expansion in realistic geometry. this approach does not need any experiment in order to calculate temperature distribution but the results reported here are confirmed by the experimentally obtained temperatures. prediction of temperature distribution in the worm gear meshing 331 2. test conditions the experiment is performed on worm gear transmission with centre distance of 30 mm. bearing of the worm shaft is done with two ring ball bearings with angular contact 7302 bep with x arrangement. position of the worm is under the worm gear. temperature of oil is measured with ni-cr-ni thermocouple, which is set just below the worm near the contact of worm gear set. bearing of the worm gear shaft is done with two ring single row ball bearings with radial contact 6007 2rs1 and 6302 2rs1. the test bench for the worm gears and the position of the measuring points is shown in fig. 1. drive of the test bench is performed via asynchronous power engine 2.5 kw. the output torque goes via magnetic coupling, which can take up to torque of 160 nm. input and output coupling of gear transmission is carried out with gear coupling. temperature of the worm on the input side of the transmission is done over an infrared thermal element. the data of the worm gear (fig. 2) are given in table 1. fig. 1 test bench fig. 2 worm gear set table 1 transmission gear parameters parameter values centre distance a [mm] 30 worm gears type zidin 3975 transmission ratio i 40 modul mx [mm] 1.2608 number of teeth z1/ z2 1/40 wheel material cusn12ni2-c-gcb worm material 16mncr5 input speed [min -1 ] 5000 torque [nm] 10-22 synthetic oil gh6-1500 40 = 1500 mm 2 /s; 100 = 232 mm 2 /s 332 а. miltenović, m. tica, m. banić, đ. miltenović 3. efficiency 3.1. overall efficiency of a transmission gear efficiency of the transmission gear is a very important parameter of transmission quality. the overall efficiency of transmission gear  can be determined with values of measured output torque values t2 and input torque values t1, i.e.: 2 2 1 1 p t p t u     (1) fig. 2 overall efficiency of the transmission for input speed n1 = 5000 min-1 and for different values of output torque t2 figure 2 presents a diagram of dependency of transmission gear overall efficiency  for input speed n1 = 5000 min-1 and for different values of output torque t2. the output torque was changed via magnetic brake within the limits of the values from t2 = 10 nm to t2 = 22 nm. at the same time, the transmission input and output torque values are measured, as well as oil and ambient temperature. the values of overall efficiency are determined according to equations (1); it is within the limits of the values:  = 0,52 – 0,71. a trend of the output torque shows that, as the load increases, so does the overall efficiency. 3.2. power losses in bearings and seals power losses in bearings and seals are determined according to the skf bearing calculator, i.e. according to the skf program module for determining power losses in bearings and seals [9]. the calculation results of a power loss in bearings for different values of output torque t2 are shown in fig. 3. bearing a of the worm shaft is loaded with both axial and radial load. as result, the power losses at this bearing are the greatest by far. the worm gear shaft has a significantly smaller number of rpm (n2 = 125 min-1). thus, the power loss in its bearings is significantly lower in comparison to the loss in the worm shaft bearings. with an increase in the load, the power losses in bearings increase as well. prediction of temperature distribution in the worm gear meshing 333 0 5 10 15 20 25 30 35 40 45 50 0 8 16 24 32 40 48 b e a ri n g p o w e r lo ss [w ] t2 [nm] bearing a bearing b bearing c bearing d fig. 3 power losses in bearings for the determined operating oil temperature the power losses increase along with the load. considering that the aforementioned power loss converts into heat, the operating oil temperature changes as well. with a change in the operating oil temperature, oil viscosity changes as well. power losses depend on oil viscosity. consequently, the calculation of the power losses in bearings is carried out for the predicted values of the operating oil temperature. thus, the obtained interdependence between power losses and load is not linear (fig. 3). one can also notice that the impact of the load on the efficiency is significantly higher that the impact of a change in the oil´s kinematic viscosity. 4. determination of working temperature – analytical approach in the gear mesh, friction causes heat generation in gear flanks of worm pair. tooth mass temperature is higher comparing to working oil temperature m: m s m    (2) increase in temperature of gear tooth m depends on power losses in worm pair pgz. according to the din 3996 [10] m can be calculated as: gz m l r p a      (3) where pgz are mesh power losses, l is heat transfer coefficient and ar cooling surface of the worm gear pair. mesh power losses pgz are experimentally determined depending on load or they can be calculated by reduction of overall power losses with power losses in bearings and seals: 1(1 )gz gldp p p   (4) according to the din 3996 [10], heat transfer coefficient l is calculated in function of input speed n1: 1(1940 15 )l kc n     (5) 334 а. miltenović, m. tica, m. banić, đ. miltenović coefficient ck has values: ck = 1 for worm gear immersed in oil; ck = 0,8 for worm immersed in oil. cooling surface of worm gear pair ar is calculated in function of worm gear width b2r and medium diameter dm2 according to [10]: 6 2 2 10r r ma b d    (6) temperature of tooth mass ϑm is calculated according to equation (2) for different values of output torque t2 is shown in fig. 9. the trend line shows that there is linear dependence between temperature of tooth mass ϑm and output torque t2. temperature of tooth mass ϑm is increasing along with increasing output torque t2 as an effect of increased power losses. friction coefficient of the worm gear pair is experimentally determined for values of output torque t2 = 10 nm to t2 = 22 nm. 5. determination of temperature distribution in the worm gear set using fem the distribution of temperature in the worm gear transmission using fem is done with the ansys workbench software. the analysis is defined as a thermal analysis in the time domain that is used in the ansys module for transient thermal analysis. worm and worm gear have complicate geometry; for generating the finite element mesh the elements of a higher order or solid 226 [11] are used. the mesh is generated with 5096742 nodes which form 3633469 elements. in the continuous operation mode of the worm gear transmission a complex temperature distribution occurs. the heat generated in the bearings and the worm gear mesh in a steady condition must be over the surface of the housing brought to the environment preferably without additional cooling. the temperature distribution between the heat source and the housing surface is determined by heat transfer. for heat conduction is authoritative thermal conductivity . in order to take into account the differential temperature in the heat transfer, heat transfer coefficient must be determined. this coefficient depends on a number of parameters. parameters and boundary conditions for fem simulations are given in table 2. simulation is carried out for n1 = 5000 min-1 and for three cases of output torque: t2 = 21.84 nm, t2 = 16.22 nm and t2 = 12.25 nm. table 2 parameters for fem simulation parameter, boundary condition unit dimension t2 = 21.84nm t2 = 16.22nm t2 = 12.25nm power losses in worm gear set pgz w 69.91 51.93 39.21 power losses in bearing a pgla w 10.05 8.69 7.49 power losses in bearing b pglb w 21.99 18.85 15.97 power losses in bearing c pglc w 0.08 0.07 0.06 power losses in bearing d pgld w 0.02 0.02 0.02 power losses in sealing of bearings pgd w 0.48 0.48 0.48 alongside the worm gear transmission model, it contains all the shafts, bearings, lubrication and housing. the fem model for the thermal simulation is shown in fig. 6. prediction of temperature distribution in the worm gear meshing 335 fig. 4 fem model of tested transmission in table 3 and fig. 4 are given thermal characteristics of the material used in analysis. the analysis takes into account thermal characteristics that are depending on temperature for all materials. bearings power losses are determined by the skf calculation [9] and introduced in the fem model in the heat transfer. in addition to the measured value of ambient temperature ϑ0 it is necessary to determine power losses in the mesh of the worm gear pair as well as those in seals. the coefficient of heat transfer to the environment based on the data available research is ok = 12-15 w/m2k [12, 13], and for simulation is used ok = 15 w/m2k. based on the generated power losses in the mesh of worm gear pair, bearings and seals, tooth mass temperature is determined by simulation and so is temperature distribution of the entire worm gearing as well as temperature of housing or some other element of transmission. table 3 thermal characteristics of material used in analysis material specific heat capacity [j/(kg k)] thermal conductivity [w/(m k)] coefficient of thermal expansion [k-1] 16mncr5 434 60.5 1.210-5 cusn12ni2 -c-gcb 20°c 376 50 17.210-6 100°c 385 56 336 а. miltenović, m. tica, m. banić, đ. miltenović fig. 5 thermal conductivity for cusn12ni2-c-gcb the generated heat in the transmission goes into the surrounding environment over a relatively large area of housing, and over a part of the foundation. the ends of the shafts have a relatively small area and over them heat goes into the environment. since the worm gear shaft rotates slowly, the share of heat that goes by convection over surface is small, so that when simulation is taken the same value of heat transfer coefficient of housing is obtained although the heat transfer from the gear on the oil and air in the transmission is, by its nature, one of the convective processes, this process is approximated as a conductive one. the same approximation is made for contact of air and oil in the transmission and in the housing. that approximation is introduced because it has a small effect on the macroscopic temperature distribution. in figs. 6 a), b) and c) are shown the results of the temperature distribution of the transmission on worm gears (figs. on the left) and the housing (figs. on the right). in fig. 6 (left) is given the distribution of temperature on the worm gear as one of the most important elements of the worm gear transmission. the figs. show that the maximum temperature is in the mesh zone and in the rest of the volume of the worm gear temperature is relatively stable and in a range of few degrees. the highest temperature distribution is for t2 = 21.84 nm and temperature is in the range of 90-96 °c. for t2 = 16.22 nm it is in the range of 80-87 °c and for t2 = 12.25 nm it is in the range of 72-77 °c. figure 6 (on the right) shows that the housing depends on the output torque. the highest temperature distribution is for t2 = 21.84 nm and temperature is in the range of 5861 °c. for t2 = 16.22 nm is in the range of 53-56 °c and for t2 = 12.25 nm is in the range of 48-51 °c. prediction of temperature distribution in the worm gear meshing 337 a) b) c) fig. 6 temperature distribution at worm gear and housing for n1 = 5000 min-1 and: a) t2 = 21.84 nm, b) t2 = 16.22 nm, c) t2 = 12.25 nm tooth mass temperature obtained by simulation is largely consistent with experimentally determinate value – fig. 7. from the above it can be concluded that it is correctly assumed that on the basis of certain analytically determinate power losses in the transmission the temperature distribution of the entire transmission in exploitation using the finite element method can be estimated. 338 а. miltenović, m. tica, m. banić, đ. miltenović fig. 7 comparison of tooth mass temperature obtain by experiment and simulation fig, 8 shows the distribution of temperature in the cross section of the transmission oil. numeric values of specified temperatures are in the range from 125.2 °c to 59.93 °c. maximum temperature of 125.2 °c is obtained in the mesh of worm gear pair and it represents the current temperature at the contact point flanks, which is consistent with similar studies in this area [15]. fig. 8 temperature distribution in the cross section of transmission oil for t2 = 21.84 nm and n1 = 5000 min-1 prediction of temperature distribution in the worm gear meshing 339 7. conclusion this study is focused on the determination of temperature distribution in the worm gear transmission. based on the generated analytically determined power losses in the mesh of worm gear pair, bearings and seals, simulation is applied to determine tooth mass temperature, temperature distribution of the entire worm gearing as well as other elements of transmission. the obtained results show a high degree of correlation results obtained experimentally and numerically. with this approach it is possible to examine thermal state of the complete transmission and identify critical points in terms of heating and successful fulfillment of the work functions of the transmission. this is of great significance for engineering practice because it offers the possibility, in the design stage of the transmission, to receive relevant information about the behavior of the transmission in the exploitation conditions and to promptly make the necessary correction structure without costly and time-consuming prototype testing. references 1. jain, r., pal, k.s., singh, b.s, 2016, a study on the variation of forces and temperature in a friction stirwelding process: a finite element approach, journal of manufacturing processes, 23, pp. 278-286. 2. haddad, h., guessasma, m., fortin, j., 2016, a dem–fem coupling based approach simulating thermomechanical behavior of frictional bodies with interface layer, international journal of solids and structures, 81, pp 1–16. 3. ziegltrum, a., lohner, t., stahl, k., 2017, tehl simulation on the influence of lubricants on loaddependent gear losses, tribology international, 113, pp. 252-261. 4. milošević, m., miltenović, a., banić, m., tomić, m., 2017, determination of residual stress in the rail wheel during quenching process by fem simulation, facta universitatis-series 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schneckengetrieben mit sich rechtwinklig kreuzenden achsen. 11. ansys inc., 2010, ansys theory manual, usa. 12. böge, a., 2011, handbuch maschinenbau: grundlagen und anwendungen der maschinenbau-technik. springer fachmedien wiesbaden,. 13. cerbe, g., wilhelms, g., 2010, technische thermodynamik: theoretische grundlagen und praktische anwendungen. gebundene ausgabe – hanser. 14. klübersynth gh 6, 2014, synthetic gear and high temperature oils based on klübercomp lube technology 15. lange, n., 2000, hoch fresstragfähige schneckengetriebe mit rädern aus sphäroguss, dissertation tu münchen. 12598 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240322021m © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dynamic vehicle routing problem for smart waste collection danijel marković1, saša marković1, dragan marinković1,2, dragan pamučar3,4,5 1faculty of mechanical engineering, university of niš, serbia 2faculty of mechanical engineering and transport systems, tu berlin, germany 3department of operations research and statistics, faculty of organizational sciences, university of belgrade, belgrade, serbia 4college of engineering, yuan ze university, taoyuan city, taiwan 5department of applied mathematical science, college of science and technology, korea university, sejong, republic of korea abstract. this paper examines waste management problems in urban areas and analyzes systems employed in waste management. the first part of the paper focuses on waste collection and transport activities in urban systems, as well as on understanding the causes behind the ineffectiveness of traditional waste collection systems. a smart waste collection container prototype is presented with the aim of achieving more efficient waste collection, coupled with the application of modern information and communication technologies in waste collection and transport systems. the waste collection problem is defined as a dynamic cvrp. to solve this problem, a methodology is proposed based on heuristics for defining optimal dynamic routes so as to minimize total costs, that is, transport costs. key words: dynamic vehicle routing problem, smart waste collection, information technologies, communication technologies 1. introduction waste management is an extremely complex process that involves a series of steps, technologies, and public participation. the success of waste management depends on the cooperation of the public and the government, where both actors bear responsibility and received: march 22, 2024 / accepted may 10, 2024 corresponding author: danijel marković, dragan pamucar faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 nis, serbia department of applied mathematical science, college of science and technology, korea university, sejong 30019, republic of korea e-mail: danijel.markovic@masfak.ni.ac.rs, dpamucar@gmail.com 2 d. marković, s. marković, d. marinković, d. pamucar should cooperate in order to achieve adequate management of this process. waste management systems are very complex. they monitor the entire cycle from generation, collection, transportation, treatment and disposal of waste and solve problems related to the construction of landfills, design and construction of recycling centers, transfer stations and various other systems for processing waste and waste products. today's waste management systems are based on the greatest possible utilization of waste, processing, recycling, reuse. municipal waste management msw is defined through four basic functions: collection, transport, processing and final disposal [1]. however, it should be emphasized that the transport function occurs between each of the mentioned functions. waste collection and transportation costs account for 60-80% of the total costs of an msw system [2]. thus, this article focuses on operational decisions at the first and second stage, collection and transportation. the function of municipal waste collection refers to the use of vehicles for collecting waste from collection points, and the function of waste transport refers to the activity of taking the collected waste to a disposal center. as a rule, the traditional method of collecting and transporting municipal waste is carried out by specialized vehicles for emptying waste containers according to a predetermined schedule. however, this conventional collection of municipal waste is very inefficient because the level of filling of the waste container very often varies from partially filled to overflowing, which leads to unnecessary consumption of resources. in relation to the traditional method of collection, there are numerous studies in which researchers treat the amount of waste at container locations as a stochastic quantity. in papers [3, 4] the authors discussed a model of waste management with stochastic demand at the locations of waste receptacles. in these papers, solving the defined problem leads to significant savings compared to the traditional way of collecting municipal waste. also, in [5] the authors presented a stochastic model of waste management where they showed the advantage of applying stochasticity compared to a deterministic model. however, even with such models, there may be an unnecessary consumption of resources, because individual containers for municipal waste may be partially filled. the amounts of municipal waste generated in containers are mostly stochastic in nature, and the accumulation of waste itself depends on several factors, the more important being seasons, holidays, sports and cultural events, etc. certain studies indicate that only about 40% of municipal waste is collected from containers that are filled more than 75% [1]. all this shows that a waste collection system should be implemented much more efficiently. one of the ways to develop the most efficient waste collection model is the application of modern information and communication technologies. the application of such technologies in the system of waste collection in urban areas contributes to the reduction of transport costs of the upc as well as a more rational use of resources [6, 7]. the goal of this paper is to present a hybrid system of municipal waste collection with the application of modern information and communication technologies. in addition to numerous studies that have been conducted with the aim of optimizing the municipal waste collection system, the development of smart cities shows that there is still a lot of room for improving waste management systems [4, 8, 9]. dynamic vehicle routing problem for smart waste collection 3 2. smart waste management a smart city is a city or settlement that uses advanced information and communication technologies ict and innovation to make its infrastructure, services, and functions more efficient, sustainable, and livable. a smart city uses sensors, data and communication networks to better manage resources, respond to the needs of citizens, improve the quality of life, and reduce the negative impact on the environment [8]. such cities aim to be smart and sustainable in all aspects, including transportation, energy management, healthcare, education. widespread and inexpensive availability of cloud computing services, rapid penetration of smart phones in urban populations, as well as the rise of the internet of things iot in the form of deployment of a variety of sensors, drive smart city technologies that offer new application domains in city planning and operations. the iot refers to the use of network protocols to form a universal network of interconnected things that are not necessarily considered as computers. it was projected that there would be 34 billion connected things in the world in 2020 [10], and the iot will dominate the smart city technologies. smart waste management systems are the answer to the challenges faced by traditional waste management practices. these systems consist of several key components that work together to optimize the waste management process, ensuring efficiency, sustainability and economic benefits. waste management as a part of a waste collection system has been a topic of interest in operations research and it has been demonstrated that optimized route planning and scheduling in waste collection can lead to significant cost reductions. consequently, route optimization has been the main motivation for developing smart waste collection systems. there are some studies of monitoring smart bins, where the authors present monitoring solutions based on radio frequency identification rfid [11]. this system utilizes load sensors to estimate the accumulated weight in the trash bins and relays the data to collectors’ pocket pcs pda. collected data is then uploaded to an online database for further processing and analysis. in [12], the authors present an iot based smart garbage system that utilizes a pay-as-you-throw payt model for food waste management. the system proposes custom-designed household garbage bins that are equipped with load sensors and cellular internet modules. the system is tested through a pilot study in seoul, korea, and the authors underline that the power consumption (battery-based operation) is the main trade-off of the system. in [13], the authors emphasize that “an efficient, cost effective and environment friendly solution for real time bin status monitoring, collection and transportation of municipal solid waste is still a major challenge to the local municipal authorities” and propose a theoretical model using rule based decision algorithms. a further review of the literature reveals many studies where researchers try to use smart technologies in the waste collection system to achieve the most optimal system possible. in the continuation of the paper, a prototype of a smart container for waste and a hybrid methodology for optimizing municipal waste collection routes will be presented. 2.1 smart container for collecting municipal waste one of the key components of smart waste collection systems is smart containers. smart containers represent a revolution in the way waste is collected and managed. the prototype of a smart container for municipal waste collection presented in this paper supports gps/gprs and includes sensors for data collection. the proposed system allows the status 4 d. marković, s. marković, d. marinković, d. pamucar of a municipal waste container to be monitored in real time. the smart vessel consists of the following elements fig. 1: standard underground container, solar panel, ultrasonic sensor, gps and gprs modem. the solar panel is used to charge the battery located under the cover, from which the system receives energy. the battery capacity needs to be defined in relation to the requirements of the case study. the ultrasonic sensor measures the filling level of the container using ultrasonic beams. these sensors can monitor any type of waste (mixed waste, paper, plastic, glass, clothing, bio-waste, liquids, electronics, metal...) in bins and containers of different types and sizes. ultrasonic technology ensures high and reliable measurement accuracy. sensors can be connected to several iot networks or gprs to provide fast data transfer. an ultrasonic sensor is placed under the lid of the waste container and measures the filling level, regardless of what is deposited inside. fig. 1 prototype of a smart container for municipal waste to measure the level of filling in the prototype of the storage tank in this study, an ultrasonic sensor or a long-range transducer type jsn-sr04t v2.0 is used. this type is a waterproof ultrasonic sensor that contains a non-contact distance measurement module and provides high accuracy. this ultrasonic sensor detects objects from 20 cm to 600 cm with an accuracy of 3 mm and a measuring angle of 20◦. also, another important feature of this sensor is its operation at temperatures from -20 to +70°c. this proposal of a smart container is designed in such a way that it gives an estimated filling of the container with waste as shown in fig. 2. in many studies, you can find systems that, in addition to the fullness of the container, also provide information about the mass of waste as well as the current temperature in the container itself. wireless sensor networks have become a key element in the sensor development process for smart waste systems. apart from the various transducers used to measure the data, one of the main purposes of the sensor is to establish a level of communication that is effective over the greatest possible distance. data that has been collected from the sensors is transmitted via gprs to the cloud where it is temporarily stored. the data is downloaded from the cloud and processed in order to obtain the real condition of the filling of containers at predefined locations. dynamic vehicle routing problem for smart waste collection 5 fig. 2 prototype of a smart container with an ultrasonic sensor figure 3 shows the communication of the smart container prototype with the master dispatcher via the cloud. after processing, the data is forwarded to the municipal waste collection vehicles. fig. 3 information flow diagram from a smart container to vehicles 6 d. marković, s. marković, d. marinković, d. pamucar 3. problem description regarding the choice of a waste collection route in a smart waste system, the search for optimal waste collection routes is considered an optimization problem that requires the reduction of collection costs and time. these costs relate to the distance traveled by each vehicle for waste and the number of vehicles required. this type of problem is known in the literature as the vehicle routing problem vrp. this name is used for a whole class of problems whose goal is to find a large number of routes for a group of vehicles located in a depot. the original vrp is also known as the capacity vehicle routing problem cvrp and includes constraints, for example, each vehicle in a fleet has a permanent and uniform load and there is one depot. the original problem has several variants [14] that add constraints to a real-life problem. with cvrp, it is assumed that the vehicles are identicalhomogeneous (equal capacity) and have a common starting point, and the only limitation that exists is the vehicle capacity. the objective function expresses the requirement to minimize the total cost (e.g. weight function of the number of routes and their total length or time) when serving all users [4]. cvrp can be expressed using basic notations from the theory of transport networks as follows. let the transport network be defined as g = (v, a), where v = {0,1, 2, ..., n} is the set of containers for collecting municipal waste, while а is a set of branches, i.e. a set of connections between municipal waste containers. the determined pair (i, j) represents a connection, i.e. a branch that connects node i ∈ v with node j ∈ v. each container location from the pool {1, ... , n} has a certain demand (in this paper, the amount of waste) that must be collected and transported to the depot. the depot demand is always 0. a set (k) of vehicles of the same capacity (q) (if the fleet has different capacities, the problem would be a different type of vrp problem) must be employed to collect waste from n container locations, m vehicles must start and finish their routes in the depot. a route is defined as a cycle of the lower cost of transport networks (g) that passes through the depot and the total demand of the vertex set must not exceed the total vehicle capacity. the purpose of the problem is to minimize the distance, time, or cost of k vehicles while meeting the following requirements: (1) the depot is the start and end point of each route; (2) each container location may be visited only once by one vehicle; and (3) the total demand of each route does not exceed the capacity (q). by introducing a smart container in the municipal waste collection route planning system, the classic cvrp becomes a dynamic cvrp. the difference between these two problems is that with classic cvrp, the vehicle receives a route plan at the beginning of the working hours and performs collection along defined routes. with dynamic cvrp, the route plan is formed in real time, that is, based on the limit of the filling of containers with waste. the transport network in which the dynamic cvrp was applied is shown in fig. 4. numerous methods can be found in the literature for solving cvrp, which belongs to the group of np-hard problems. the quality of the cvrp solution most often depends on the quality of the definition of the transport network, and considering the complexity of the problem, heuristic and metaheuristic methods are most often used to solve it. heuristic methods perform a limited exploration of the search space and usually deliver good quality results in small computing times and meta heuristic methods are generic methods of exploration in the solution space for searching and optimization problems. dynamic vehicle routing problem for smart waste collection 7 fig. 4 graphic representation of dynamic cvrp these methods provide a design line adaptable to each context, and they can generate more efficient algorithms [15, 16]. in what follows, the focus will be on the presentation of the hybrid system for the optimization of municipal waste collection routes, that is, on the collection, storage, processing and sensor data. in relation to fig. 3, which shows the information flow and communication subsystems, fig. 5 indicates which of them are part of the optimization of the municipal waste collection route. the optimization process consists of 4 steps:  container selection: the containers visited by the waste collection vehicle are selected from the defined transport network. the selection of containers is based on the fill-level criteria of each container from the transport network. a threshold will be established to select the nodes that must be collected. the criterion that is set in this case is the fill-level of 75%,  cvrp data: once the container fill data is obtained from the transport network nodes and their locations, the geographic information of each node is loaded from the geographic information subsystem to obtain the distance matrix that will be used to apply cvrp,  cvrp solver: based on the data obtained from the previous step and by defining a heuristic algorithm, problem solving is approached, that is, the design of routes for municipal waste collection,  best solution found: after the best or most favorable solution is found (during the specified time under the stopping criterion), it is forwarded to the vehicle operator. the data persistence subsystem receives information and stores the collection plan solution for that day. 8 d. marković, s. marković, d. marinković, d. pamucar fig. 5 real time routing model 4. mathematical model and method for dynamical cvrp waste collection models are most often solved as capacity-constrained vehicle routing problems – cvrp. the cvrp model assumes that all vehicles are identical and have a common starting point (depot), and the only limitation that exists is the vehicle's capacity. the objective function expresses the requirement to minimize the total transport costs when collecting municipal waste from the transport network [17, 18]. the proposed formulation is based on the following assumptions:  the fleet is homogeneous.  each route should start and end at a depot,  the vehicle is always empty when it leaves the depot,  each container location participates in only one route,  the total amount of waste collected in one route must not exceed the capacity of the vehicle on that route. dynamic vehicle routing problem for smart waste collection 9 before formulating the mathematical objective function f, certain assumptions need to be introduced. we assume that there are k vehicles at the depot, each with a known capacity 𝑄k. to ensure the feasibility of the solution, it is necessary that 𝑞 ≤ 𝑄k for each node 𝑖 (𝑖 = 1, …, 𝑛) and each vehicle 𝑘 (𝑘 = 1, …, 𝐾). this problem involves determining the minimum number of vehicles (with capacity 𝑄k) needed to serve all demands 𝑞i. the minimum number of vehicles affects traffic safety in urban areas [19]. for each 𝑖, 𝑗, 𝑘 ∈ 𝑉, the following decision variable can be defined: the variable xijk is equal to 1 if vehicle k, after visiting bin location i, proceeds to visit bin location j; otherwise, it is 0. the variable li is equal to 1 if bin level is over 75% at the bin location i; otherwise, it is 0. the mathematical formulation of the cvrp can be presented in the following manner: 1 0 0 min k n n ijk ijk ik k i j f c x l      (1) subject to: 0 0 1, 1,2, , n i k i x k k    (2) 0 0 1, 1, 2, , n jk j x k k    (3) 1 0 1, 1,2, , k n ijk k i x j n     (4) 1 , 1,2, , n i ijk k i q x q k k    (5) 1 1 , 1,2, , ; 1,2, , n n ijk jik i i x x i n k k       (6)   , 1, 2, , 1,2, ,ijk i j s x s s n k k       (7)  0,1 , , 1,2, , ; 1,2, ,ijkx i j n k k   (8)  0,1 , 1,2, , ; 1,2, ,ikl i n k k   (9) constraints given in eqs. (2) and (3) imply that every vehicle that leaves a depot returns to that same depot. the constraint shown by eq. (4) stipulates that each bin location must be visited only once by exactly one vehicle. constraint in eq. (5) implies that the total capacity of bin location requests, served by one vehicle, must not exceed the capacity of that vehicle. constraint in eq. (6) enables the preservation of vehicle flow, i.e. the fulfilment of the condition that vehicle k must leave bin location j after serving bin location 10 d. marković, s. marković, d. marinković, d. pamucar j. constraint in eq. (7) prevents the occurrence of cycles that do not represent a complete route. constraint in eq. (8) defines the passage through the branch between bin location i and j by the k vehicle and can have a value of 0 or 1. constraint in eq. (9) defines that the bin location is not visited if the bin level is less than 75% and can have a value of 0 or 1. heuristic or metaheuristic algorithms are most often used to solve dynamic cvrp. in what follows, the procedure for solving the dynamic cvrp for municipal waste collection will be presented. the procedure is based on the well-known clark-wright saving algorithm [20]. this algorithm is most often used in the literature for almost all vehicle routing problems. the pseudo code of c-w's cost-saving algorithm for solving dynamic cvrp is shown in algorithm 1. algorithm 1: c-w saving algorithm for solving dynamic cvrp start defining the distance matrix for all locations of containers whose filling is greater than 75%; defining qk; define the demand matrix by nodes for containers whose filling is greater than 75%, computei s`; sort s` in a non-increasing sequence; form a partial route; vehicle filling = qi; for for all savings from the range if operational constraints met if vehicle filling + qi ≤ qk vehicle filling = qi + vehicle filling form a route end if end if end for print routes; print vehicle fill; end the of the aforementioned methodology is reflected in the fact that after servicing each container location, a new distance matrix and a demand matrix are formed based on the criterion that the container is more than 75% empty. this means that the planning of municipal waste collection routes is done in real time. also, it is very important that a certain container that is filled less than 75% for more than 5 days is also included in the emptying system. this criterion is important for the reason that municipal waste must not remain in the container for more than five days due to the spread of unpleasant odours and negative impact on people's health in the environment. 5. optimization of the dynamical cvrp model and results the cvrp dynamic model considered in the paper is defined by the transport network shown in figure 6. the nodes of the transport network represent the locations of containers for municipal waste disposal and are defined by coordinates, i.e. latitude and longitude. at locations where there are two or more containers, their position is defined by one node or one dynamic vehicle routing problem for smart waste collection 11 coordinate. the container for collecting municipal waste, which is explained in detail in fig. 1, has a volume of 3 m3 [21]. the municipal waste collection vehicle has a capacity of 60 m3 and departs from the depot, i.e. from the node marked "1" with the assumption that it is empty, and goes around the nodes of the transport network according to the order generated based on the procedure presented in fig. 5. when servicing orders, care must be taken that the vehicle may visit each node only once. fig. 5 the transport network of container nodes in [1], this model is considered as a deterministic model where the amount of waste per node is taken as the number of containers multiplied by the volume of the container. this means that we work with the assumption that all containers are maximally filled with waste, which does not correspond to the real model. the aim of considering this model is to demonstrate the feasibility of applying the methodology presented in this paper and to obtain optimal routes for the movement of vehicles for the collection of municipal waste, that is, to keep transport costs to a minimum. the shortest distance matrix represents the real distance between all pairs of nodes within the transport network. the matrix is symmetric, that is, the distance from node i to node j is equal to the distance from node j to node i. for the purposes of solving the dynamic cvrp model, the amount of waste per node is defined based on the analysis of the periodicity of emptying the container, thus simulating the operation of the smart container. the paper emphasizes that this is a prototype of a smart container which is in the validation phase. the optimization of the model based on algorithm 1 was performed for a period of one month. the municipal waste collection period is three times a week. the period when the largest amount of waste is generated was chosen. the criterion for visiting the node was defined in the previous part of the paper, that is, that the container is 75% full and that the container must be emptied after seven days, regardless of how full it is. table 1 shows the simulation results of the dynamic cvrp model and the results of the optimization of the deterministic model considered in paper [1]. the results show that the application of 12 d. marković, s. marković, d. marinković, d. pamucar the dynamic cvrp model achieves a significant reduction in the total distance traveled by about 27%. in addition to the great savings in the traveled distance, the collection time is also reduced because the total length of the route is 16 with the deterministic cvrp, and 9 with the dynamic cvrp model. the obtained results show that the so-called idling of the vehicle is reduced, which directly affects the reduction in environmental pollution with a lower emission of harmful gases. also, table 1 shows that by about 26% fewer vehicles visited container locations with the dynamic cvrp, noting that none of the containers were overfilled. table 1 result for cvrp models the period of collection deterministic cvrp model dynamic cvrp model distance [km] visited locations distance [km] visited locations first week 32.78 80 27.96 60 second week 45.36 80 30.91 68 third week 38.44 80 26.54 56 fourth week 32.78 80 23.87 53 in total 149.36 320 109.28 237 in order to see the advantage of applying the dynamic cvrp model, fig. 6 shows the ratio of visits to locations by route, where it is clearly seen how many empty runs of municipal waste collection vehicles there are in the deterministic cvrp model. this ratio indicates that the implementation of the dynamic cvrp model in a real municipal waste collection system would significantly reduce operating costs and vehicle driving time. fig.6 graphic display of location visits by route dynamic vehicle routing problem for smart waste collection 13 6. conclusions smart waste management systems are transforming the way we manage waste, providing innovative and sustainable solutions to the challenges faced by traditional waste management practices. utilizing state-of-the-art technology and innovative design, these systems optimize the collection, transport and treatment of waste, reducing the environmental impact of waste disposal and contributing to a cleaner and healthier urban environment. in addition, smart waste management systems offer significant economic and social benefits, promoting savings, income generation and overall well-being. as cities around the world continue to expand and urbanize, the need for efficient and environmentally friendly waste management solutions will only grow. smart waste management systems offer a promising path towards achieving sustainable urban living, city by city. this paper provides an insight into the complexity of waste management problems and the need for more advanced approaches. it also indicates that the transition to smart waste management systems is justified and will contribute to more efficient collection and transport of waste. this will result in reduced costs and reduced negative environmental impacts, paving the way for a better future, which was demonstrated by the simulation of the dynamic cvrp model. the optimization of routes contributes, in addition to reducing transport costs, to the reduction of exhaust gases and fuel consumption, which is of great importance for the protection of air and the environment. optimal vehicle movement routes minimize transport costs, which significantly affects the economy and profitability of waste collection companies. this optimization also contributes to the aesthetics of the city environment itself, because the number of vehicle passes and the time spent passing through populated areas are reduced, which affects the general appearance of the city and the quality of life of citizens. by using sensors and automating the municipal waste collection process, a high degree of efficiency and precision is achieved in the planning and execution of operations. such automation significantly improves the work of waste collection companies, thus facilitating and speeding up the entire process, while reducing the so-called empty moves. the development of such a system represents the future of the development of the city of niš, which is already entering the process of implementing smart systems. the future direction of the research is the simulation of the presented methodology of dynamic routing of vehicles during the collection of municipal waste in order to obtain accurate data on the saving of transport costs. also, the future direction of research is the integration of artificial intelligence and machine learning into smart waste management systems, which have a great potential to transform the waste management industry and support sustainable living in urban environments. as these technologies become more advanced and available, their use is expected to become more widespread, leading to additional innovations and advances in waste management. however, it is important to note that the integration of these technologies also presents certain challenges, such as the need for large datasets and the potential for errors in the algorithms. it is therefore crucial that these technologies are implemented in a responsible and ethical manner, with appropriate safeguards to ensure their effectiveness and fairness. 14 d. marković, s. marković, d. marinković, d. pamucar acknowledgement: this research was financially supported by the ministry of sciencе, technological development and innovation of the republic of serbia (contract no. 451-451-0366/2024-03). references 1. marković, d., 2018, development of a logistics model for municipal waste 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problem in humanitarian aid distribution, journal of intelligent manufacturing, 34(5), pp. 2241-2261. 21. marković, d., nikolić, l., stanković, a., marković, s., 2023, hybrid waste collection system ̧ the 9th international conference transport and logistics til 2023, nis, serbia, pp. 133-137. 6941 facta universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 63 79 https://doi.org/10.22190/fume201215042h © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining using copper powder rafiqul haque, mukandar sekh, golam kibria, shamim haidar department of mechanical engineering, aliah university, kolkata, india abstract. electrical discharge machining (edm) is one of the most popular nonconventional machining processes that are being used in many high precision manufacturing industries. to increase the edm performance, a hybrid technique, namely, powder mixed electrical discharge machining (pmedm) is generally used for getting more precise requirements. in this study, an experimental investigation is carried out in order to explore the machining performance of the pmedm process on ti-6al-4v alloy using copper (cu) powder in the edm oil dielectric. taguchi’s l18 orthogonal array design has been utilized for design of experiments and the analysis of variance (anova) has been performed with the help of minitab-19 software. the optimal parametric setting of cu powder mixed edm has been found utilizing the taguchi grey relational analysis (gra) integrated approach and also validation of optimal parametric setting is done through experimentation. it is a novel approach for machining ti-6al-4v alloy by this pmedm technique in which the surface quality has been improved significantly with the addition of suitable amount of cu powder into the dielectric medium. key words: pmedm, ti-6al-4v alloy, cu powder, taguchi analysis, grey relational analysis (gra), surface roughness (sr) 1. introduction as one of the dominant non-conventional machining processes, electrical discharge machining (edm) is widely used in manufacturing industry. in edm ([1], the voltage is applied between the tool and the job material which are electrical conductors in nature. the edm process improvement may be further enhanced by incorporating the convenient modifications like mixing the electrically conductive powder materials in the dielectric fluid medium which is called powder mixed edm (pmedm). generally, pmedm received december 15, 2020 / accepted april 12, 2021 corresponding author: rafiqul haque department of mechanical engineering, aliah university, iia/27, action area – ii, newtown, kolkata, west bengal, pin – 700160, india e-mail: rh78.mech@gmail.com 64 r. haque, m. sekh, g, kibria, s. haidar exhibits better performance in terms of various response parameters like material removal rate (mrr), tool wear rate (twr), surface roughness (sr), etc. here below, fig. 1 shows the schematic diagram of a pmedm process. fig. 1 schematic view of pmedm process in last few decades, numbers of research studies have been carried out by several researchers in the domain of pmedm and they showed the enhancement of different response parameters to achieve the better performance in this machining process. the effect of mixing fine graphite powder in pmedm on tool steels with the improvement of 60% in mrr and decrement of 28% wear ratio has been carried out by jeswani [2]. mohri et al. [3] carried out their research to achieve a better surface finish by mixing of silicon powder into the dielectric medium on h-13 die steel. chow et al. [4] proposed their work by mixing of sic and aluminum powders on the edm machining of titanium alloy by showing the increment of material removal depth, sr and twr. kibria et al. [5] carried out experimental study on the performance during micro-hole machining of ti6al-4v alloy by using different dielectrics for micro-edm. in their analysis they have shown that there is a significant influence of adding powder additives in improving machining performance criteria in pmedm on ti-6al-4v alloy. ojha et al. [6] carried out their research work on the area of improvement of surface roughness by pmedm of en8 steel. here, to analyze the experiments, the response surface methodology has been adapted. the study of parametric optimization on performance characteristics of pmedm on en-8 steel has been analyzed by garg and ojha [7]. in their study, they observed various effects of some important process parameters on the performance measures. bhattacharya et al. [8] carried out their work on pmedm of die steels with the addition of silicon, graphite and tungsten powder. they observed that there is a better surface finish by edm oil compared to kerosene dielectric. also, micro-hardness of the machined surface has been significantly affected by various process parameters. unses and cogun [9] experimented on ti-6al-4v alloy by using graphite powder into the kerosene dielectric medium to enhance better performance of mrr, twr, texture properties, etc. long et al. [10] carried out their research on the machining of skd61, skd11 and skt4 die steels adding titanium powder in pmedm process by incorporating taguchi analysis. sugunakar et al. [11] showed that there is a reduction of recasting the layer thickness (rlt) by impinging the powder materials into the dielectric. improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 65 they also revealed that there is an increment of crater depth and diameter with the increase of powder from surface topography analysis. in pmedm, the maximum mrr by using al powder with copper tool and least radial over cut (roc) by using al2o3 powder with tungsten tool have been shown by ramesh et al. [12]. b. k. paul et al. [13] analyzed the performance of pmedm process in terms of material removal efficiency, sr, surface crack density and white layer thickness during the machining of inconel 718. lamichhane et al. [14] carried out their research work on pmedm on 316l stainless steel by adding hydroxyapatite (hap) nano-powder showing improvement of surface quality. singh [15] revealed his research on pmedm by using of high-speed steel t1 grade. here, gra has been introduced to optimize the input process parameters for getting a better output response in terms of twr, mrr and sr, respectively. within the selected process parameters, the comparative study of span-20 surfactant and micro-nano chromium mixed pmedm by hosni and lajis [16] stated that for machining aisi-d2 hardened steel, span-20 surfactant and nano chromium powder showed better machining performance in terms of mrr and sr. rajavel et al. [17] investigated the machinability condition of a metal matrix composite by pmedm experimentation through taguchi based gra technique. from the above studies it has been observed that there is no work carried out in pmedm on ti-6al-4v alloy using cu powder so far. hence in this work, mixing of cu powder in pmedm process on ti-6al-4v alloy material using edm oil as dielectric has been introduced for better enhancement of mrr, twr and sr. titanium alloy (ti-6al4v) has been chosen as the workpiece material in our experiment because of its high strength, low weight ratio, high corrosion resistance, high temperature resistance and low-density element and abundant in nature. also, it has a wide range of applications in the field of medical science, aerospace, automotive, chemical plant, pressure vessels, power generation, etc. in our experimentation, cu is chosen as a powder material because of its high electrical conductivity which is a very important property in edm process. it is also reported that the gra technique has been incorporated with this proposed work to find out the optimal settings of process parameters. 2. experimental methodology all the experiments have been conducted in the edm machine (make: excetek technologies, model no.: ed-30). throughout the experiment, the edm oil is used as dielectric medium. table 1 shows the properties of the edm oil. as the chamber of conventional edm machine is very large, it is difficult to do homogeneous mixing of the powder particles during the machining. hence, to perform the experiment properly and to realize homogeneous mixing of the powder particles within the dielectric medium, it is important to introduce a newly designed and developed smaller chamber (carrying capacity of 15-20 liters of dielectric fluid) into the larger chamber. all the machining operations have been carried out in a newly developed experimental setup as shown in fig. 2. 66 r. haque, m. sekh, g, kibria, s. haidar table 1 edm oil properties color colorless kinematic viscosity at 40°c, cst 3.0 – 4.0 di-electric strength, kva 45 flash point, pmcc, °c, min. 108 fig. 2 newly developed pmedm experimental set-up in this new set-up, a separate bucket is kept outside the main chamber along with a centrifugal pump having capacity of 0.25 hp which has been attached for pumping operation from the bucket to the smaller chamber. also, to achieve uniform distribution of the powder particles within the dielectric medium and to avoid settling of the powder particles at the bottom of the machining tank a motorized stirring mechanism has been incorporated in our experimentation. due to the sharing of 40% and 25% of total discharge energy by anode and cathode, respectively, in the edm process (xia et al.) [18], the whole experimentation has been performed using straight polarity, i.e. job is kept as positive and the tool electrode is negative in this study. in our experimentation, ti-6al-4v alloy is taken as job material. tables 2 and 3 show the chemical composition and physical properties of the job material, respectively. cu powder is used in the proposed experiment as powder material whose properties are shown in table 4. table 5 shows the physical properties of tungsten electrode (7.5mm diameter) which is used as tool material. improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 67 table 2 chemical compositions of ti-6al-4v alloy material elements wt.% al 5.48 c 0.369 fe 0.112 sn 0.0625 zr 0.0028 mo 0.005 cr 0.0099 si 0.0222 v 4.22 ni <0.001 cu <0.02 nb 0.0386 ti 90 table 3 physical and mechanical properties of ti-6al-4v alloy material properties typical value density (g/cm3) 4.42 melting range (°c±15°c) 1649 specific heat (j/kg °c) 560 thermal conductivity (w/m k) 7.2 tensile strength (mpa) 1000 elastic modulus (gpa) 114 hardness (rockwell c) 36 table 4 properties of copper (cu) powder material item description melting point 10830c boiling point 25620c density 8.96 g/cm3 heat of fusion 13.26 kj.mol-1 specific heat 0.39 kj/kg k thermal conductivity 401 w. m-1.k-1 thermal expansion (250c) 16.5 μm.m-1.k-1 electrical resistivity 1.673 μω-cm @200c table 5 physical and mechanical properties of tungsten tool material item description tensile strength 1725 mpa yield strength 750 mpa modulus of elasticity 400 gpa atomic weight 183.84 melting point 34220c density 19.3 gm/cm3 68 r. haque, m. sekh, g, kibria, s. haidar while analyzing several past references, it has been observed that the researcher tried to increase the performance of the edm process by applying various techniques during machining. in this proposed study, the authors carried out some preliminary investigation and after analyzing the output characteristics it is identified that the main influencing parameters which played dominancy over the performance characteristics of pmedm are peak current (ip), pulse on time (ton), pulse off time (toff) and powder concentration (cp). so, these parameters have been chosen as variable parameters to investigate the effects on performance measures, namely, mrr, twr and sr. the other process parameters are kept constant throughout the experimentation. as there are only four variables, for performing the experiments, the taguchi’s l18 orthogonal array design has been adapted for design of experiments [19] having three variables with three levels each and the remaining variable with six levels (table 6). while selecting levels of parameters, the authors have chosen the maximum range where stable machining was carried out during preliminary investigations. table 6 process parameters and their levels during pmedm technique parameters symbol level units peak current ip 4,7,10 a pulse-on-time ton 10,15,20 µs pulse-off-time toff 30,40,50 µs powder concentration cp 0,4,8,12,16,20 g/l each experiment was run for 3 times and the average value of the three was calculated for measuring mrr, twr and sr. the weight of the work piece and tool material was taken before and after the machining processes to measure mrr and twr by using an electronic weighing balance (make: aczet pvt. ltd., vasai (e); model: cy1003c). surface roughness in terms of center-line average value or simply cla value (ra) of the machined work piece surfaces have been measured by using a portable surface roughness profilometer (make: mitutoyo, japan, model:sj-410, courtesy: jadavpur university). the important process parameters which affect the performance of the pmedm process have been identified by applying the anova technique with the help of minitab-19 software. 3. results and discussion table 7 shows overall experimental results that have been carried out based on the taguchi’s experimental design (l18) by which the significant performance parameters, i.e. mrr, twr and sr (ra) were calculated. further, two important outcomes namely, mrr and sr, are recognized conflicting in nature. so, it is necessary to make a trade-off between these two conflicting response parameters for any manufacturing products or parts. this trade off can be made by any multi-objective optimization method. in this research study, the gra technique is introduced as a multi-objective optimization technique. improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 69 table 7 experimental results exp. no. ip (a) ton (µs) toff (µs) cp (g/l) mrr (gm/min) twr (gm/min) sr (μm) 1. 4 10 30 0 0.0014 0.0024 3.006 2. 7 15 40 0 0.0021 0.0043 3.457 3. 10 20 50 0 0.0029 0.0065 4.157 4. 7 10 30 4 0.0024 0.0043 2.776 5. 10 15 40 4 0.0033 0.0071 2.889 6. 4 20 50 4 0.0009 0.0015 2.938 7. 4 10 40 8 0.0009 0.0018 3.253 8. 7 15 50 8 0.0017 0.0038 3.611 9. 10 20 30 8 0.0033 0.0062 3.422 10. 10 10 50 12 0.0021 0.0057 4.156 11. 4 15 30 12 0.0013 0.0022 3.383 12. 7 20 40 12 0.0020 0.0043 3.653 13. 10 10 40 16 0.0023 0.0061 4.058 14. 4 15 50 16 0.0009 0.0016 3.547 15. 7 20 30 16 0.0026 0.0045 3.845 16. 7 10 50 20 0.0013 0.0030 5.145 17. 10 15 30 20 0.0035 0.0080 4.615 18. 4 20 40 20 0.0014 0.0019 3.643 3.1. material removal rate (mrr) mrr is defined as the weight of the material removed from the workpiece for a specified period. so, the unit of mrr is taken to be gm/min. the influence of the process parameters on the mean mrr is presented in fig. 3 and the corresponding anova analysis is drawn in table 8. it has been noticed from the response graph (fig. 3) that with the increase of peak current values from 4 to 10 a, mrr significantly increased. the available discharge energy in the inter electrode gap (ieg) is directly affected by the peak current. at a higher peak current, mrr increases significantly because of the higher current density caused by increasing spark energy with the current increase which causes overheating of the job material. at low current, the amount of energy utilized in melting and vaporizing the electrodes is not so intense due to the generation of a small quantity of heat at electrodes and its major area is absorbed by the surroundings. likewise, it is evident from the anova results that the peak current played a major role in material removing. from fig. 3 and anova results of mrr (table 8), it is clearly understood that the pulse off time is one of the influencing parameters in the case of mrr. mrr decreases with the increase of pulse off time. it happens due to the fact that with the increase in pulse off time the machining idle time increases, so no machining is happening. it is clear from the figure that mrr is inversely proportional to the pulse off time. pulse on time is another important parameter in material removing process. from fig. 3, it is noticed that mrr increases to a smaller extent with the increasing effect of pulse on time from 10 μs to 20 μs; this happens due to the supplement of energy per cycle resulting in more material melting and evaporating. further on, it is also seen that there is not such an impact of adding copper powder in material removal. it is also noticed that by adding cu powder concentration up to 8 g/l, mrr decreases and marginally increases thereafter. 70 r. haque, m. sekh, g, kibria, s. haidar this may be due to the fact that the additional powder accumulates in the gap and minimizes the discharge transitivity and as a result the mrr is reduced. fig. 3 effect of process parameters on mrr table 8 detail of anova analysis of mrr source df seq ss adj ss adj ms f p ip 2 0.000009 0.000009 0.000005 122.13 0.000 ton 2 0.000001 0.000001 0.000000 9.52 0.014 toff 2 0.000002 0.000002 0.000001 24.04 0.001 cp 5 0.000000 0.000000 0.000000 1.66 0.276 model summary error 6 0.000000 0.000000 0.000000 s r-sq r-sq(adj) total 17 0.000012 0.0001958 98.16% 94.78% 3.2. tool wear rate (twr) twr is defined as the amount of weight loss of tool material per unit time (usually minute) during machining. so, its unit is gm/min. the effect of process parameters on mean of twr is presented in fig. 4 and corresponding anova analysis is drawn in table 9. from the response graph (fig. 4) as well as from anova analysis (table 9), it is very clear that the peak current is the major significant process parameter on twr. the figure shows that with the increment of the peak current, tool wear is rapidly increased. during the current flows through the plasma column, the electrons collide with the dielectric particle due to the in which positive ions are produced and these produced positive ions flow towards the negative electrode (tool) resulting in melting and vaporizing of tool material. thus, tool wear takes place. it is also noticed that with the increment of pulse on time from 10 μs to 15 μs tool wear increases and marginally decreases thereafter. also, tool wears decreases with the improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 71 increase of pulse-off time as no machining takes place during that pulse off period. when powder is added up to 8 g/l, twr decreases a little bit and then it increases with the increase of powder concentration from 8 g/l to 12 g/l and marginally decreases thereafter. it is due to the fact that with increase of powder concentration, sparking energy decreases which lowers the tool wear rate. fig. 4 effect of process parameters on twr table 9 detail of anova analysis of twr source df seq ss adj ss adj ms f p ip 2 0.000066 0.000066 0.000033 143.78 0.000 ton 2 0.000001 0.000001 0.000001 2.48 0.164 toff 2 0.000003 0.000003 0.000001 5.56 0.043 cp 5 0.000000 0.000000 0.000000 0.42 0.817 model summary error 6 0.000001 0.000001 0.000000 s r-sq r-sq(adj) total 17 0.000072 0.0004807 98.08% 94.55% 3.3. surface roughness (sr) (ra) superior surface finish is a major requirement for all machine components. simply, surface roughness (sr) can be defined as the evaluation of the surface texture in terms of surface irregularities, waviness and flaws. surface roughness most commonly refers to the variations in the height of the surface relative to a reference plane. it is measured either along a single line profile or along a set of parallel line profiles (surface maps). it is usually characterized by: (a) ra (centre-line average, cla) value, (b) rq (root mean square, rms) value and (c) rz (average peak-to-valley height) value. in our study, we have taken ra value of surface roughness which is the most universally used roughness parameter. it is defined as the average absolute deviation of the roughness irregularities from the mean line sampling length. mathematically, it is written by the application of the following equation: 72 r. haque, m. sekh, g, kibria, s. haidar 0 1 ( )=  l ar z x dx l (1) where ‘l’ is the sampling length, ‘z’ is height of peaks and valleys of roughness profile and ‘x’ is the profile direction. the effect of process parameters on surface roughness is presented in fig. 5. powder concentration is the most crucial factor which is seen from the figure and its anova study. the anova results are shown in table 10. from the fig. 5, it is seen that with the increment of the peak current, surface roughness increases. this is because of the formation of deeper and larger craters takes place due to increase in the spark energy with the increase in the peak current. fig. 5 effect of process parameters on sr table 10 detail of anova analysis of sr (ra) source df seq ss adj ss adj ms f p ip 2 1.13766 1.13766 0.56883 8.28 0.019 ton 2 0.07565 0.07565 0.03782 0.55 0.603 toff 2 0.72550 0.72550 0.36275 5.28 0.048 cp 5 4.12681 4.12681 0.82536 12.02 0.004 model summary error 6 0.41201 0.41201 0.06867 s r-sq r-sq(adj) total 17 6.47764 0.262047 93.64% 81.98% it is also seen that when the pulse on time increases from 10 μs to 15 μs, ra value marginally decreases and thereafter increases slightly with further increment of pulse on time. from the analysis of pulse off time, it is noticed that the surface roughness increases significantly with the increment of pulse-off time. this is due to the fact that with the increased pulse off time, the debris in the machining zone is properly washed away and cleaned by the flushing of dielectric fluid. so when the next sparking takes place on the cleaned surface, the waste of the spark energy is less and full energy is utilized to create a larger and deeper crater. as a result, the surface roughness increases. it is noticed that the improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 73 surface quality gets improved with the addition of a suitable amount of the powder up to 4 g/l. with the addition of conductive powder materials, the gap between the electrodes becomes wide due to the decrement of insulating strength of the edm oil causing stable discharge. as a result, due to the reduction of electrical density, shallow craters develop on the machining surface which leads to better surface finish. oppositely, surface roughness increases with further addition of powder (up to 20 g/l) due to the fact that after removing a good amount of material results, deeper and larger craters during machining and arc formation over the workpiece takes place. this helps to generate higher surface roughness. 3.4. multi-objective optimization using the grey relational analysis (gra) it has been observed that direct application of the taguchi methodology fails in optimizing multi-response characteristics. generally, single response characteristic is optimized by the designing of taguchi. hence other methods are required with the combination of the taguchi method in order to optimize multi-response characteristics involved in the analysis. in 1982, deng proposed the grey system theory. it is being widely used for analyzing any system in which control parameters have complex characteristics. the complicated interrelationships among multiple response parameters are efficiently solved by the application of this theory. investigation of a system by means of relational coefficient, relational grade and decision may be carried out based on this grey theory. consequently, the grey relational grade will be evaluated by using gra to evaluate the multiple response parameters for getting the optimal values. several researchers like tosun and pihtili [20], kuo et al. [21], tosun [22] have done multi-objective optimization of several machining processes by applying this technique. lin and lin [23] first reported the application of gra technique for multi-objective optimization in the field of edm. later several other researchers like singh et al. [24], lin and lee [25] applied this multi-objective optimization technique to get the optimal value of the process parameters in the area of edm and wedm processes. the multiobjective optimization by using this taguchi gra integrated methodology has been performed in the following steps: a. the experimental results have been normalized first. b. grey relational coefficients have been calculated by performing grey relational generation. c. grey relational grades have been evaluated by averaging the grey relational coefficients. d. the optimal level of process parameters is selected. in this research work by considering four input parameters and three output parameters, the gra methodology is performed. the output parameters are the mrr, twr and sr. the main objective of this multi response optimization is to increase productivity while maintaining desired surface finish and geometrical accuracy. 3.4.1. normalization of the experimental results in this methodology, firstly the original data sequence has been transferred to a comparable sequence in a scale range between zero and one which is called normalization. indicating a higher value for better performance, i.e. mrr in our case, it can be normalized by the following expression: 74 r. haque, m. sekh, g, kibria, s. haidar ],....,2,1,,...2,1,[],....,2,1,,...2,1,[ ],....,2,1,,...2,1,[ mjniyminmjniymax mjniyminy x ijij ijij ij ==−== ==− = (2) where, xij is the normalized value of yij of experiment i (i=1,2,…..n) for response j (j = 1,2,….m). and, twr and sr whose lower value indicates better performance can be expressed in equation as follows: ],....,2,1,,...2,1,[],....,2,1,,...2,1,[ ],....,2,1,,...2,1,[ mjniyminmjniymax ymjniymax x ijij ijij ij ==−== −== = (3) if any response parameter requires achieving a particular value, then it can be normalized by using the following equation: 0 0 ],....,2,1,,...2,1,[ ],....,2,1,,...2,1,[ 1 xmjniymax xmjniyy x ij ijij ij −== −== −= (4) here, x0 is the desired value of the response parameter. 3.4.2. calculation of the grey relational coefficient the closeness of xij to x0j is determined by the calculation of the grey relational coefficient. the closeness of xij to x0j is indicated by a higher value of the grey relational coefficient. grey relational coefficients have been calculated by using the following equation: ],....,2,1&,....,2,1[,)( ,0 mjnixxz maxij maxmin ijj == + + =   (5) where: ijjij xx −= 0 (6) ],....,2,1&,....,2,1,[ mjnimin ijmin ===  (7) ],....,2,1&,....,2,1,[ mjnimax ijmax ===  (8) and ξ is the distinguishing coefficient, ξ ϵ (0,1). generally, fitting the practical requirements is done by adjusting the distinguishing coefficient and it is usually set at 0.5. 3.4.3. calculation of the grey relational grade the grey relational grade is defined as the average of the grey relational coefficients by considering a weight parameter for a particular experiment and it is calculated by using the following equation:  = = m j ijjji xxzw m xx 1 ,0,0 )( 1 )( (9) where wj is the weight of jth factor. by using eq. (9), the grey relational grade for all the 18 experiments has been calculated. now, the experiment which shows the highest grey relational grade is providing the optimal level of different parameters. in other words, the experiment having the maximum grey improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 75 relational grade is the best choice for getting the optimal response parameter setting among all the response parameter settings considered. 3.4.4 optimization using the gra in this research study, a conventional gray relation analysis has been proposed. the experimental data based on the taguchi’s (l18) orthogonal array design is taken for optimization. here the optimization problem is formulated as: maximize, mrr = f (ip, ton, toff, cp) subject to tool wear rate ≤ α and surface roughness ≤ β where α is the maximum allowable twr and β is the maximum allowable surface roughness (ra). as per table 7, the range of α is within 0.0015 gm/min to 0.0080 gm/min and for β is within 2.776 μm to 5.145 μm. the desired value of the tool wear rate is taken as 0.0045 gm/min and the surface roughness of the workpiece is taken as 2.8 μm. the normalized value for mrr is computed by using the eq. (2). and by using the eq. (4), the normalized value for twr and surface roughness (ra) with their desired values have been calculated. all the normalized values are shown in table 11. then from those normalized values of each of the responses, the grey relational coefficient and grade are calculated by using the eq. (5) and eq. (9), respectively, for all eighteen experiments. all these results are summarized in table 12. it is seen from the table 12 that due to the highest grey relational grade of experiment no. 4 after giving the equal weight to all the three responses, the parametric combinations corresponding to this experiment give the best multiple performance characteristics of all 18 experiments. it is found that this combination gives mrr = 0.0024 gm/min, twr = 0.0043 gm/min and sr (ra) = 2.776 μm. table 11 normalized decision matrix for mrr, twr and sr (ra) exp. no. mrr twr sr 1. 0.192307692 0.4 0.912153518 2. 0.461538462 0.942857143 0.719829424 3. 0.769230769 0.428571429 0.421321962 4. 0.576923077 0.942857143 0.989765458 5. 0.923076923 0.257142857 0.962046908 6. 0 0.142857143 0.941151386 7. 0 0.228571429 0.806823028 8. 0.307692308 0.8 0.654157783 9. 0.923076923 0.514285714 0.734754797 10. 0.461538462 0.657142857 0.421748401 11. 0.153846154 0.342857143 0.751385928 12. 0.423076923 0.942857143 0.636247335 13. 0.538461538 0.542857143 0.463539446 14. 0 0.171428571 0.681449893 15. 0.653846154 1 0.554371002 16. 0.153846154 0.571428571 0 17. 1 0 0.226012793 18. 0.192307692 0.257142857 0.640511727 76 r. haque, m. sekh, g, kibria, s. haidar table 12 grey relation coefficient, grade and rank matrix for mrr, twr and sr exp. no. grey relation coefficient grey relation grade rank mrr twr sr 1. 0.382352941 0.454545455 0.855320596 0.564072997 9 2. 0.481481481 0.897435897 0.644470861 0.674462747 5 3. 0.684210526 0.466666667 0.466123519 0.539000237 10 4. 0.541666667 0.897435897 0.985423687 0.808175417 1 5. 0.866666667 0.402298851 0.934648784 0.7345381 2 6. 0.333333333 0.368421053 0.899701977 0.533818788 11 7. 0.333333333 0.393258427 0.725351855 0.483981205 15 8. 0.419354839 0.714285714 0.594433799 0.576024784 7 9. 0.866666667 0.507246377 0.657040647 0.676984563 4 10. 0.481481481 0.593220339 0.466307867 0.513669896 12 11. 0.371428571 0.432098765 0.671637392 0.491721576 14 12. 0.464285714 0.897435897 0.582107845 0.647943152 6 13. 0.52 0.52238806 0.48510983 0.509165963 13 14. 0.333333333 0.376344086 0.614253421 0.44131028 17 15. 0.590909091 1 0.531706625 0.707538572 3 16. 0.371428571 0.538461538 0.335198135 0.415029415 18 17. 1 0.333333333 0.394664248 0.575999194 8 18. 0.382352941 0.402298851 0.584996002 0.456549264 16 here, it is clearly observed that for this optimal mrr the corresponding twr and sr (ra) are within the desired limit. hence, this experimental set up gives the optimal result. so, based on the above discussions, the optimal machining parameter combinations would be as follows: peak current (ip) = 7 a, pulse on time (ton) = 10 µs, pulse off time (toff ) = 30 µs and powder concentration (cp) = 4 g/l. fig. 6 plot of grey relational grade values on experiment numbers fig. 6 shows the plot of grey relational grade values on experiment numbers. from this plotting, it is revealed that the experiment number 4, i.e. the multi-objective parametric improvement of surface quality of ti-6al-4v alloy by powder mixed electrical discharge machining... 77 combinations setting of peak current of 7 a, pulse on time of 10 µs, pulse off time of 30 µs and powder concentration of 4 g/l gives the least surface roughness (ra) value i.e. 2.776 μm among all the experiments. 3.5. surface topography analysis fig. 7(a) and (b) shows the 3d surface topography of the best (ra = 2.776 µm) and worst (ra = 5.145 µm) surface, respectively, using cci (make: taylor & hobson) of the machined surface. from the figures, it is observed that at a higher powder concentration, the crater produced in the second case (fig. 7(b)) is larger and deeper, and more high peaks are developed due to more arcing which promotes higher surface roughness. fig. 7 3d surface stereogram of ti-6al-4v alloy after machining at (a) ip= 7 a, ton = 10 µs, toff = 30 µs and cp= 4 g/l and (b) ip= 7 a, ton = 10 µs, toff = 50 µs and cp= 20 g/l 4. conclusions in this work, the performance parameters namely mrr, twr and sr by pmedm of ti-6al-4v alloy material with cu powder are investigated. as there are response parameters of conflicting nature, the multi-objective optimization of process parameters was done by the taguchi based grey relational analysis method. this is the novel approach by using copper powder within the edm oil dielectric fluid in edm process to enhance the surface quality of ti-6al-4v alloy material. based on the experimentation and analyzing the results, the following conclusions can be drawn: (a) it has been observed that the copper powder concentration is the most significant factor in analyzing the surface quality. it is noticed that surface quality gets improved with the addition of suitable amount powder from 0 g/l to 4 g/l and thus decreases with further addition of powder up to 20 g/l. (b) the main factors which affect the mrr are peak current and pulse off time. the removal of material increases rapidly with the increase in peak current. the reverse effect observed in the case of pulse off time. there is not so severe effect of powder concentration in material removing. mrr decreases with the increase in powder. (c) the response parameter peak current has the main role in affecting the twr. with the increase in current, twr increases significantly. there is no such noticeable effect of other process parameters on twr. 78 r. haque, m. sekh, g, kibria, s. haidar (d) the combination of taguchi methodology and gra technique have been very rightly and satisfactorily chosen to analyze the multi-objective optimization parametric combinations setting and results of responses during pmedm of ti-6al4v alloy material. 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optimization of an electrical discharge machining process vidyapati kumar 1 , sunny diyaley 2 , shankar chakraborty 3 1central institute of mining and fuel research, dhanbad, jharkhand, india 2department of mechanical engineering, sikkim manipal institute of technology, sikkim manipal university, majitar, sikkim, india 3department of production engineering, jadavpur university, kolkata, west bengal, india abstract. due to several unique features, electrical discharge machining (edm) has proved itself as one of the efficient non-traditional machining processes for generating intricate shape geometries on various advanced engineering materials in order to fulfill the requirement of the present day manufacturing industries. in this paper, the machining capability of an edm process is studied during standard hole making operation on pearlitic sg iron 450/12 grade material, while considering gap voltage, peak current, cycle time and tool rotation as input parameters. on the other hand, material removal rate, surface roughness, tool wear rate, overcut and circularity error are treated as responses. based on singleand multi-objective optimization models, this process is optimized using the teaching-learning-based optimization (tlbo) algorithm, and its performance is contrasted against firefly algorithm, differential evolution algorithm and cuckoo search algorithm. it is revealed that the tlbo algorithm supersedes the others with respect to accuracy and consistency of the derived optimal solutions, and computational efforts. key words: edm process, tlbo algorithm, metaheuristics, optimization 1. introduction electrical discharge machining (edm) has already been accepted as an efficient thermo-electrical material removal process in tool and die making, aerospace and automotive industries, and also in finishing of surgical components due to its ability to maintain close tolerances and attain higher dimensional accuracy [1, 2]. in this process, a series of successive discharges between the tool (electrode) and the workpiece is responsible received february 18, 2020 / accepted june 08, 2020 corresponding author: shankar chakraborty department of production engineering, jadavpur university, kolkata, west bengal, india e-mail: s_chakraborty00@yahoo.co.in 282 v. kumar, s. diyaley, s. chakraborty for removing material in the presence of a dielectric medium (kerosene or de-ionized water). during electrical discharge, a discharge channel is developed having a temperature around 12000°c causing melting and evaporation of material from the workpiece surface. the electrode is advanced towards the workpiece until the inter-electrode gap is small enough for the higher impressed voltage to ionize the dielectric [3]. in edm process, a perfect replication of the tool shape is generated on the workpiece surface. this process is especially suitable for generating complex shape profiles on electrically conductive materials with low machinability [4]. as there is no direct contact between the tool and the workpiece, this process is free from any mechanical stress generation, chatter/burr formation and vibration problem. its machining performance is also uninfluenced by the hardness of the work material because the material removal takes place by melting due to high intensity localized heat generation. since no cutting force is generated, extremely deep narrow holes with high aspect ratio can be machined using this process with minimum tool wear. it can even generate intricate cavities in a single operation. but edm process also suffers from several drawbacks, like generation of recast layer and heat-affected zone (haz), low material removal rate (mrr), high machining time and related cost, low flexibility, capability of machining only electrically conductive materials, etc. it has been observed that the machining performance of an edm process with respect to mrr, surface roughness (sr), tool wear rate (twr), haz, radial overcut (roc) etc. is significantly affected by different electrical parameters (peak current, pulse-on time, pulseoff time, gap voltage, polarity, etc.) and non-electrical parameters (electrode material, type of the dielectric used, dielectric pressure, rotation of the electrode, etc.). thus, in order to fulfill the requirements of better response values, it is always preferred to operate an edm set-up while maintaining the settings of its different input parameters as their optimal levels. it would also lead to a higher production rate with reduced machining time. keeping in mind the requirements of finding out the optimal parametric mixes for edm processes, this paper deals with the application of teaching-learning-based optimization (tlbo) algorithm to study the influences of various input parameters of an edm process on its responses (outputs) while machining pearlitic sg iron 450/12 grade work material. for this process, gap voltage, peak current, cycle time and rotation of the tool are considered as input parameters, whereas, mrr, sr, twr, overcut (oc) and circularity error (ce) are treated as responses. both the singleand multi-objective optimization models are developed and subsequently solved using the considered algorithm. its optimization performance is also contrasted with that of firefly algorithm (fa), differential evolution (de) algorithm and cuckoo search (cs) algorithm. the tlbo algorithm supersedes the other algorithms with respect to accuracy and consistency of the derived optimal solutions, and computational effort. the results of two-tailed paired t-tests also confirm its superiority over the others. 2. review of the literature mandal et al. [5] first applied artificial neural network (ann) with back-propagation algorithm to model an edm process and non-dominating sorting genetic algorithm-ii (nsga-ii) was later adopted to optimize the said process. using controlled elitist nsga technique, bharti et al. [6] optimized different input parameters of a die-sinking edm process. the ann with back-propagation algorithm was also adopted to model the teaching-learning-based parametric optimization of an electrical discharge machining process 283 considered process. baraskar et al. [7] employed nsga-ii technique to identify the optimal settings of pulse-on time, pulse-off time and discharge current for an edm process to achieve better values of sr and mrr responses. shivakoti et al. [8] studied the effects of salt-mixed de-ionized water as a dielectric on mrr, twr, roc and taper during edm operation of d3 die steel. the taguchi method was later utilized to optimize the considered edm process parameters. aich and banerjee [9] applied weight-varying multi-objective simulated annealing technique to develop the corresponding pareto optimal front for simultaneous optimization of mrr and sr in an edm process. radhika et al. [10] considered peak current, pulse-on time and flushing pressure as the input parameters of an edm process. a hybrid optimization technique consisting of ann and genetic algorithm (ga) was later employed to minimize sr and twr, and maximize mrr. a paretooptimal front was also developed offering a set of non-dominated solutions. tiwari et al. [11] applied ga technique to simultaneously optimize mrr and sr during an edm operation. the corresponding pareto-optimal solutions were subsequently proposed. mazarbhuiya et al. [12] performed eight experimental runs in an edm set-up based on taguchi’s design plan, and applied grey relational analysis (gra) technique to determine the optimal settings of discharge current, flushing pressure, pulse-on time and polarity for achieving maximum value of mrr and minimum sr value. mohanty et al. [13] considered open circuit voltage, discharge current, pulse-on time, duty factor, flushing pressure and type of the tool material as the control parameters of a die-sinking edm process. based on a multi-objective particle swarm optimization (pso) algorithm, the optimal values of different process responses, like mrr, ewr, sr and roc were subsequently determined. while considering peak current, polarity, pulse-on time, gap voltage and spindle speed as the input parameters of an edm process, gohil and puri [14] adopted taguchi-gra technique to maximize mrr and minimize sr while machining titanium alloys. satpathy et al. [15] combined principal component analysis with technique for order of preference by similarity to ideal solution (topsis) for multi-objective optimization of an edm process, while taking into account peak current, pulse-on time, duty cycle and gap voltage as the input parameters, and mrr, twr, roc and sr as the responses. applying vikor index as a multi-objective optimization tool for an edm process, mohanty et al. [16] determined the optimal settings of current, pulse-on time and voltage for having better values of mrr, twr, sr and roc. singh et al. [17] utilized nsga-ii technique to optimize mrr and twr in an edm process while considering peak current, pulse-on time, pulses-off time and gap voltage as the input parameters. gostimirovic et al. [18] modeled the energy efficiency of an edm process with respect to mrr and sr responses. evolutionary multiobjective optimization was later performed to derive a set of optimal solutions for discharge energy taking into account discharge current and discharge duration as the input parameters. ramprabhu et al. [19] applied passing vehicle search (pvs) as a multi-objective optimization tool for optimizing various input parameters of an edm process. the performance of the adopted technique was also compared with that of other intelligent computing models. based on gra technique, tharian et al. [20] performed multi-objective optimization of mrr and sr during edm operation of al7075 alloy. huu et al. [21] proposed the application of multi-objective optimization based on ratio analysis (moora) method for having better values of mrr, sr and twr during edm operation of skd61 die steel with low-frequency vibration. analytic hierarchy process (ahp) was utilized to estimate relative weights of the considered responses. while employing response surface methodology (rsm)-based regression models, niamat et al. [22] endeavored to study the 284 v. kumar, s. diyaley, s. chakraborty influences of current, pulse-on time and pulse-off time on mrr, sr and twr in an edm process. multi-objective optimization was also performed to achieve sustainability while optimizing the conflicting responses. the above-cited review of the existing literature reveals that parametric optimization of edm processes has already caught the attention of the research community, and several optimization techniques, like ga, nsga-ii, simulated annealing, pvs, pso, etc. have been applied in this direction. those adopted algorithms have too many algorithmic parameters, which if not properly tuned, may increase the computational effort and result in local optimal solutions. similarly, numerous multi-criteria decision making approaches, such as vikor, topsis, gra, ahp, moora, etc. have also been utilized to determine the most feasible parametric mixes for edm processes. but, in most of the cases, near optimal or sub-optimal solutions have been arrived at. moreover, there is a scarcity of research works dealing with comparative analysis of the optimization performance of the available metaheuristic algorithms. in order to overcome such drawbacks, the tlbo algorithm is applied in this paper for the first time to find out the best combination of four edm process parameters while machining pearlitic sg iron 450/12 grade work material in order to simultaneously optimize the responses under consideration. the tlbo algorithm is a population-based optimization technique, requiring no algorithmic specific parameters and has already earned a broad acceptance among the researchers in the optimization domain. this algorithm is efficient, simple and capable of achieving almost global optimal solutions with less computational effort. the comparative analysis results reveal that it is more flexible, robust and reliable as compared to other mostly preferred metaheuristic algorithms, like fa, de and cs techniques. their optimization performance is compared with respect to three metrics, i.e. accuracy of the derived solutions, consistency of the solutions and convergence speed. these comparison results are also validated using the developed boxplots and paired t-test. 3. tlbo algorithm the tlbo algorithm is based on the concept of improving knowledge of the students within the classroom by the teacher first, and the knowledge is further upgraded by the mutual interaction among the students [23]. this algorithm thus consists of two phases, i.e. a) teacher phase and b) student phase. the knowledge acquired by the students from the teacher is known as the teacher’s phase. on the other hand, enrichment in knowledge through mutual interactions among the students is known as the student’s phase [24]. teacher phase in this algorithm, the teacher is supposed to be the best solution in an entire set of solutions and the learners acquire knowledge from the teacher. a teacher always attempts to improve the grades of all the students in the class by bettering the mean result of the entire class. but, from the practical point of view, it is not at all possible to uplift the mean result of the class because the learning capability of the class depends on the ability of the students to grab knowledge from the concerned teacher. let xj,k,i be any value in the solution, where j is the design variable (subject taken by the learners) (j = 1,2,...,m); k is the population member (i.e. learner) (k = 1,2,...,n) and i is the iteration number (i = 1,2,...,genmax) (genmax is the number of maximum iterations). the teacher phase begins teaching-learning-based parametric optimization of an electrical discharge machining process 285 with the identification of the teacher (best solution) from the available population, based on the objective function value. atith iteration, xk,i represents the best solution having the value of f(xk,i) being minimum among the population. this best solution is denoted as xkbest,i.. the mean result mj,i of the learners in jth subject is computed. in this algorithm, the teacher always attempts to uplift the mean result of the entire class in a particular subject. thus, the difference between the result of the teacher and mean result of the learners in each subject is represented as: difference_meanj,k,i = rj,i(xj,kbest,i – tfmj,i) (1) where rj,iis a random number between 0 and 1,xj,kbest,i is the result of the best learner in jth subject, and tf is the teaching factor which chooses the value of the mean to be modified. the value of tf can be either 1 or 2 and is decided randomly using the following equation: tf = round [1 + rand (0,1){2-1}] (2) based on the value of difference_meanj,k,i, the existing solution is upgraded using the following expression: x'j,k,i= xj,k,i + difference_meanj,k,i (3) where x'j,k,i is the updated value of xj,k,i. the x'j,k,i value is accepted if it has a better function value. at the end of this phase, all the accepted function values are retained which serve as the inputs to the learner phase. learner phase in this phase, the learners endeavor to boost their knowledge through interactions among themselves. a learner learns from other learners if they have more knowledge than him/her. for a population size of n, at ith iteration, each learner is randomly compared with other learners. for this comparison, two different learners a and b are randomly chosen so that x'a,i≠ x'b,i, where x'a,i and x'b,i are the revised values at the end of the teacher phase. x"j,a,i = x'j,a,i+rj,i (x'j,a,i – x'j,b,i), if f (x'a,i)< f (x'b,i) (4) x"j,a,i = x'j,a,i+rj,i (x'j,b,i – x'j,a,i), if f (x'b,i)< f (x'a,i) (5) if x"j,a,i has a better function value, it is accepted. at ith iteration, the learner phase is accomplished applying the following loops: fork= 1:n let the present learner be x'a,i randomly select another learner x'b,i, so that x'a,i≠ x'b,i iff (x'a,i) <f (x'b,i), forj = 1:m; x"j,a,i = x'j,a,i+ rj,i(x'j,a,i–x'j,b,i); end for else; for j = 1:m; x"j,a,i = x'j,a,i+ rj,i(x'j,b,i–x'j,a,i); end for; end if end for at the end of this phase, all the accepted function values are saved so that they become the new inputs to the teacher phase in the next generation. the flowchart for tlbo algorithm is exhibited in fig. 1. 286 v. kumar, s. diyaley, s. chakraborty fig. 1 flowchart of tlbo algorithm 4. experimental details this paper deals with the edm operation for generation of standard holes on pearlitic sg iron 450/12 grade material while considering gap voltage, peak current, cycle time and rotation of the tool (electrode) as the input parameters. this work material for edm operation is chosen due to its several favorable properties, like good wear and corrosion resistance, better castability and machinability, reasonable strength, low cost, suitability for hydraulic applications as compared to steel, malleable and grey iron castings, capability to generate intricate shapes due to better fluidity as compared to steel castings, requirement of less heat treatment resulting in better dimensional stability compared to malleable castings etc. it has found wide ranging applications in manufacturing of water pump bodies, pump housings, pump covers, pump hub for cooling system of diesel engines, manifolds for inlet teaching-learning-based parametric optimization of an electrical discharge machining process 287 and exhaust valves, castings for engine mounting arms, engine supports, fly wheels, engine couplings, drive coupling assembles, anti-vibration mountings, etc., pulleys for crankshaft assembly, castings for reduction gear boxes, bearing covers and end covers, castings for machine tool components, etc. the chemical composition and mechanical properties of sg iron 450/12 grade material are respectively provided in tables 1 and 2. table 1 chemical composition of pearlitic ductile iron element c si mn p s cr mo cu mg ti zn fe others % 3.365 2.393 0.238 0.072 <0.150 0.007 <0.010 0.37 0.085 0.032 0.027 90.75 2.661 table 2 mechanical properties of pearlitic sg iron (450/12 grade) mechanical property value tensile strength 450 mpa yield strength 310 mpa elongation 12% hardness 197 bhn density 6.95 gm/cm³ relative wear resistance excellent table 3 edm process parameters with their operating levels process parameter symbol unit level -2 -1 0 1 2 gap voltage x1 v 40 45 50 55 60 peak current x2 a 20 30 40 50 60 cycle time x3 μs 80 160 240 320 400 tool rotation x4 rpm 5 15 25 35 45 while performing edm operation on pearlitic sg iron 450/12 grade material, each of the considered edm process parameters has been varied at five different operating levels, as shown in table 3. according to the central composite design plan, for four factors with five levels, 30 experiments have been conducted in an agietron 250 c edm set-up. it has the following specifications, e.g. working area: [x 700 y 500 z 500] mm, maximum workpiece dimension: [l 1000 w 700 h 320] mm, maximum workpiece weight: 1200 kg, maximum electrode weight: 400 kg, work tank volume: 360 l, dielectric unit capacity: 1200 l and accuracy: 0.001 mm. the photograph of the edm set-up is shown in fig. 2. during the machining operation, castrol se 180 edm fluid is used as the dielectric because of its various advantageous properties, like low odor, higher stability with extended fluid life, low viscosity, high flash point, increased reliability and safe use. the specimen size has been taken as 15 × 40 mm. 288 v. kumar, s. diyaley, s. chakraborty table 4 details of experimental results exp. no. gap voltage peak current cycle time rotation mrr sr twr oc ce 1 50 20 240 25 6.00 8.572 0.589 0.8084 0.2914 2 40 40 240 25 16.60 8.581 0.671 0.8495 0.2982 3 50 40 240 45 13.07 8.092 0.623 0.8473 0.1495 4 50 60 240 25 21.07 8.532 0.698 0.8606 0.3055 5 50 40 240 25 11.98 8.712 0.612 0.8333 0.3012 6 50 40 240 5 9.28 9.612 0.599 0.8295 0.3197 7 60 40 240 25 15.98 8.902 0.663 0.8493 0.2998 8 50 40 400 25 23.26 8.742 0.726 0.9266 0.3032 9 50 40 80 25 4.10 8.622 0.531 0.7623 0.2989 10 50 40 240 25 15.21 8.531 0.659 0.8492 0.3011 11 45 30 320 35 7.41 8.235 0.590 0.8077 0.1932 12 55 30 320 15 6.10 9.092 0.586 0.8095 0.3176 13 50 40 240 25 13.07 8.626 0.627 0.8446 0.3014 14 45 30 160 15 2.25 9.207 0.522 0.7233 0.3143 15 55 50 320 15 18.14 9.367 0.677 0.8543 0.3179 16 45 30 160 35 2.16 8.265 0.518 0.7154 0.1795 17 45 50 320 15 19.29 9.247 0.696 0.8559 0.3174 18 45 50 160 35 4.98 8.635 0.545 0.7827 0.1934 19 50 40 240 25 9.95 8.732 0.597 0.8297 0.3019 20 55 30 160 35 3.59 8.475 0.529 0.7345 0.2236 21 55 30 160 15 1.97 9.212 0.511 0.7154 0.3161 22 55 50 160 35 6.25 8.685 0.593 0.8246 0.2579 23 55 30 320 35 22.13 8.345 0.719 0.9118 0.2003 24 50 40 240 25 12.84 8.826 0.618 0.8393 0.3015 25 45 50 320 35 19.78 8.436 0.692 0.8604 0.2693 26 45 50 160 15 4.96 9.232 0.548 0.7725 0.3156 27 55 50 320 35 21.24 8.176 0.702 0.8988 0.2009 28 50 40 240 25 11.06 8.696 0.605 0.8306 0.3018 29 55 50 160 15 5.65 9.172 0.552 0.7935 0.3164 30 45 30 320 15 5.80 9.497 0.567 0.8077 0.3183 it is worthwhile to mention here that all the 30 experiment runs have been performed in random order so that the machining error can be minimized. five most important responses (outputs) of the edm process are considered here, i.e. mrr (in mm3/min), sr (in μm), ewr (in mm3/min), oc (in mm) and ce (in mm). for measurement of mrr and ewr, an electronic weighing balance (a&d gr-202 type) has been employed. on the other hand, sr has been measured using hommelwerke turbo wave v7.20 roughness tester, and zeiss o-inspect 442 cmm machine (with geomet universal cmm software) has been used for measuring both oc and ce. table 4 displays the experimental design plan along with the measured values of the considered responses. in fig. 3, the photographs of the copper electrode utilized during edm operation and the machined component are provided. among these responses, mrr is the sole larger-the-better quality characteristic, and the remaining three are of smaller-the-better type. teaching-learning-based parametric optimization of an electrical discharge machining process 289 fig. 2 edm set-up fig. 3 round copper tool and machined component (5 mm depth and 20 mm dia.) 5. optimization of the edm process now, based on the experimental data of table 4 and using minitab software (r17), the following rsm-based equations are developed for the five responses, considering the main, second order and interaction effects between the considered edm process parameters. higher values of the corresponding coefficient of determination (r2) justify that these rsm-based equations are the best fit models depicting the relationships between the process parameters and responses. 1 2 3 4 1 2 2 1 4 2 3 3 4 ( ) –5.0 0.235 0.727 – 0.05 –1.436 – 0.0 ... ( 174 0.0241 0.00196 0.00149 81.5 )7 y mrr x x x x x x x x x x x x r       (6) 2 1 2 3 4( ) 9.57433 0.00305 0.00207 – 0.000105 – 0.0407 85.47( )y sr x x x x r    (7) 1 2 3 4 2 2 2 2 1 2 3 4 2 1 2 2 4 3 4 0.54 – 0.0134 0.0073 0.000319 0.00329 ... 0.000201 – 0.000008 – 0.000001 – 0.00009 – ... 0.00013 ( ) (0.000011 0.00001 84.59) y twr x x x x x x x x x x x x x x r         (8) 2 1 2 3 4 1 2 2 2 2 3 4 1 2 1 4 2 2 3 2 4 3 4 ( ) 0.147 0.0101 0.01212 0.01184 – 0.00912 – 0.000133 – ... 0.000071 – 0.0000001 – 0.000061 – 0.000022 0.000238 – ... 0.000012 – 0.000014 0.000008 84.59( ) y oc x x x x x x x x x x x x x x x x x x r        (9) 290 v. kumar, s. diyaley, s. chakraborty 1 2 3 4  2 2 2 1 2 4 1 2 1 4 2 2 3 2 4 ( ) –0.162 0.016 0.00459 0.000158 – 0.00091 – ... 0.000142 – 0.000037 – 0.000196 – 0.000069 0.000056 ... 0.000002 0.000077 90.5( )6 y ce x x x x x x x x x x x x x x x r          (10) table 5 results of single-objective optimization response method mean sd optimal value parameter gap voltage peak current cycle time tool rotation mrr fa 40.629 1.169 40.824 60 55.172 387.421 45 de 38.952 0.576 39.022 55.03 58.75 393.49 39.363 cs 41.316 1.201 41.508 56.457 57.999 392.36 44.76 tlbo 44.629 0.246 44.660 60 60 400 45 sr fa 7.938 0.001 7.904 43.482 32.00 361.18 45 de 8.009 0.001 7.987 58.05 27.39 194.14 44.23 cs 8.01 0.017 7.883 40 27.85 372.16 45 tlbo 7.874 0.0009 7.864 40 20 400 45 twr fa 0.481 0.004 0.480 50.5481 42.169 110.778 45 de 0.435 0.01 0.434 45.975 26.991 93.612 45 cs 0.437 0.009 0.436 46.963 30.184 88.668 45 tlbo 0.403 0.00069 0.402 40.024 20.001 80 45 oc fa 0.689 0.005 0.688 45.813 23.817 173.132 43.464 de 0.626 0.008 0.625 43.351 26.565 117.224 44.966 cs 0.605 0.006 0.604 43.014 28.959 86.787 45 tlbo 0.515 0.002 0.514 40.024 20.215 80 45 ce fa 0.078 0.002 0.077 45.77 26.02 116.07 45 de 0.058 0.004 0.057 45.86 20.05 94.71 44.68 cs 0.060 0.004 0.059 43.66 25.411 81.892 45 tlbo 0.021 0.001 0.020 40 20 80 45 it has already been mentioned that this paper focuses on the applications of four popular metaheuristic algorithms in the form of fa, de, cs and tlbo techniques for both singleand multi-objective optimization of the considered edm process. while solving this parametric optimization problem using the considered algorithms, the corresponding values of different algorithmic parameters are set as follows: fa: number of iterations = 500, number of fireflies = 300, light absorption coefficient = 1, initial randomness = 0.9, randomness factor = 0.91 and randomness reduction = 0.75. de algorithm: number of iterations = 500, population size = 300, lower bound of scaling factor = 0.2, upper bound of scaling factor = 0.8 and crossover probability = 0.9. cs algorithm: number of iterations = 500, population size (nests) = 300 and discovering rate of alien eggs = 0.25. tlbo algorithm: number of iterations = 500 and population size = 300. for single objective optimization, the developed equations are solved using the considered metaheuristics within the given sets of constraints as 40 ≤ x1≤ 60, 20 ≤ x2≤ 60, 80 ≤ x3≤ 400 and 5 ≤ x4≤ 45. the results of this single objective optimization are provided in table 5. it can be clearly unveiled from the table that among the four metaheuristics, the tlbo algorithm has the superiority over the others with respect to better values of the responses with higher accuracy and lower standard deviation (sd). the results of single objective optimization derive the optimal settings of the four edm process parameters to be teaching-learning-based parametric optimization of an electrical discharge machining process 291 maintained for maximization/minimization of the considered responses. the boxplots of fig. 4 prove that the optimal solutions derived using the tlbo algorithm are more consistent having the lowest variability as compared to others. on the other hand, the convergence diagrams, as exhibited in fig. 5, also demonstrate that tlbo algorithm requires less computational effort with respect to both computation speed and time. it can be interestingly noticed that the tlbo algorithm provides the optimal solutions for all the five edm responses almost within 5-10 iterations. (a) (b) (c) (d) (e) fig. 4 boxplots for the considered metaheuristic algorithms 292 v. kumar, s. diyaley, s. chakraborty (a) (b) (c) (d) (e) fig. 5 convergence diagrams for the metaheuristic algorithms table 6 t-test results with respect to the tlbo algorithm response fa de cs mrr 74.817 204.362 60.383 sr -272.183 -1423.66 -28.1014 twr -205.229 -57.451 -63.872 oc -627.432 -351.727 -436.548 ce -409.364 -183.558 -191.048 teaching-learning-based parametric optimization of an electrical discharge machining process 293 finally, in order to validate the uniqueness of the tlbo algorithm over the remaining three optimization techniques, two-tailed paired t-tests are performed with the null hypothesis and alternative hypothesis as h0(µa= µb) and hα(µa≠ µb) respectively (where α is the level of significance, and µa and µb are respectively the mean response values obtained using two algorithms being pair-wise compared). the results of the paired t-tests are provided in table 6. in this table, as the absolute values of t-statistic for all the responses for the pair-wise comparisons between tlbo and other algorithms are greater than the corresponding tabulated t-value, the null hypotheses can be rejected. it thus demonstrates the uniqueness of the optimization performance of the tlbo algorithm against the other techniques while providing the best single objective optimization solutions. based on the optimal solutions provided by the tlbo algorithm, effects of the edm process parameters under consideration on the responses are investigated using the developed response graphs of figs. 6-10. fig. 6 exhibits how the obtained mrr changes with varying values of the process parameters. it can be interestingly noticed that with the increasing values of all the edm process parameters, the mrr values also increase. higher values of gap voltage, peak current and cycle time cause the available discharge energy to increase, resulting in more melting and vaporization of material from the workpiece. the impulsive force in the spark gap also increases, which is responsible for higher mrr [25]. (a) (b) (c) (d) fig. 6 effects of edm process parameters on mrr 294 v. kumar, s. diyaley, s. chakraborty in this edm process, the tool (electrode) rotates normal to the workpiece surface and a centrifugal force is thus generated causing more debris removal from the machining zone. tool rotation in the edm process also results in formation of a thin recast layer, and the debris get easily cleared from the melt pool while exposing the workpiece to increased spark intensity. thus, less material remains in the melt cavity to be re-deposited over the workpiece surface [26]. tool rotation eases the flushing problem as encountered during the edm operation. the influences of the edm process parameters on the sr are exhibited in fig. 7. with increasing values of gap voltage and peak current, the sr of the machined components slightly increases. it almost remains unaffected due to the changes in cycle time and it shows a decreasing trend pattern with increasing values of tool rotation. the increments in gap voltage, peak current and cycle time are responsible for stronger discharge energy, creating higher temperature and formation of larger craters on the machined surface, resulting in poor surface quality [27]. it is also noticed that the tool rotation helps in a quick removal of the debris from the machining zone and as a result, with a higher tool rotation, the machined surface becomes smoother. with the tool rotation, it is expected that there would be 9-10% decrease in the average sr value. (a) (b) (c) (d) fig. 7 effects of the edm process parameters on the sr teaching-learning-based parametric optimization of an electrical discharge machining process 295 as revealed from fig. 8, higher values of gap voltage, peak current and cycle time cause increments in the twr values. at those higher parametric settings, there are micro tool wears due to availability of higher spark energy density at the machining zone. generally, lower settings of these three edm process parameters tend to enhance the possibility of carbon deposition on the tool surface, which finally helps in lowering the value of the twr. on the contrary, with increment in tool rotational speed, there is also a possibility of throwing away the carbon particles from the tool surface, which causes an increment in the twr. but, at higher tool rotation, better flushing occurs and there is a better disposition of dielectric fluid in the machining zone, leading to smaller twr [28]. (a) (b) (c) (d) fig. 8 effects of the edm process parameters on the twr during the edm operation, oc occurs due to side erosion and removal of debris. it shows increasing trend patterns with increased values of gap voltage, peak current and cycle time, as exhibited in fig. 9. at higher settings of these three edm process parameters, availability of higher gap voltage and gap width allows breakdown of dielectric at a wide gap due to higher electric field [29, 30]. at higher gap voltage and peak current, spark energy density would be more with faster machining rate, which is also responsible for higher oc. but, with increasing tool rotational speed, oc gradually increases. increased centrifugal force, resulting from a higher tool rotation, removes more debris from the machining zone. the removed debris is placed between the side wall of the electrode and 296 v. kumar, s. diyaley, s. chakraborty workpiece, causing generation of a spark between them. it leads to higher oc. at a higher tool rotational speed, there would be more turbulence taking place at the machining zone which would perhaps be responsible for removal of the debris, leading to gradual decrement in oc. the ce in the machined components occurs due to non-uniform undercut and overcut which can be effectively controlled by proper settings of different edm process parameters. as noticed in fig. 10, with increasing values of gap voltage, peak current and cycle time, ce shows an increasing trend pattern. it is also affected by tool rotation. at a higher gap voltage, peak current and cycle time, there are occurrences of secondary spark discharges caused by poor flushing, which are responsible for inferior ce. increase in ce is also occurred due to high tool wear and sporadic machining at higher voltage [31]. a higher tool rotational speed may create turbulence of the dielectric fluid at the machining zone, which would be responsible for removal of debris from the external periphery of the machined hole. thus, there may be a high chance of obtaining lower ce at higher tool rotation. (a) (b) (c) (d) fig. 9 effects of the edm process parameters on the oc teaching-learning-based parametric optimization of an electrical discharge machining process 297 (a) (b) (c) (d) fig. 10 effects of the edm process parameters on the ce it can be noticed from the results of single objective optimization of the considered edm process that separate parametric settings are obtained for different responses based on the applications of four metaheuristic algorithms. but, in a real time machining environment, it is never possible for an operator to set the edm process parameters at different operating levels in a single edm set-up. it is thus always advised to derive a unique combination of the process parameters so as to simultaneously optimize all the five responses. for this purpose, the following multi-objective optimization model is developed which is subsequently solved using the considered metaheuristics. min 5 min 4 min 3 min 2 max 1 (ce) )ce( (oc) )oc( (ewr) )ewr( (sr) )sr( (mrr) )mrr( minimize ywywywywyw z  (11) where w1, w2, w3, w4 and w5 are the weights allotted to mrr, sr, ewr, oc and ce respectively, (mrr)max is the maximum value of mrr, (sr)min, (ewr)min, (oc)min and (ce)min are the minimum values of sr, ewr, oc and ce, respectively. these values are obtained from the results of single objective optimization of the responses. in this paper, equal weights are assigned to all the responses under consideration. the solutions of this multi-objective optimization problem are provided in table 7. it can be clearly unveiled from this table that the tlbo algorithm again excels over the remaining three metaheuristics with 298 v. kumar, s. diyaley, s. chakraborty respect to accuracy and consistency of the derived optimal solutions. thus, for obtaining the most desired performance of the edm process while generating standard holes on pearlitic sg iron 450/12 grade work material, it is always recommended to set the input parameters as gap voltage = 41 v, peak current = 58.58 a, cycle time = 285 μs and tool rotation = 45 rpm. thus, based on the optimal solutions derived using tlbo algorithm, it can be concluded that lower gap voltage, higher peak current, moderate cycle time and higher tool rotational speed would concurrently optimize all the responses of the considered edm process. table 8 exhibits the percentage improvements in the response values based on the multi-objective optimization results of the tlbo algorithm against fa, de and cs techniques. it can be observed from this table that for the considered edm process, the values of mrr, sr, twr, oc and ce are significantly improved employing the tlbo algorithm as compared to other metaheuristic algorithms. table 7 results of multi-objective optimization method response mean sd optimal value z parameter gap voltage peak current cycle time tool rotation fa mrr sr twr oc ce 2.229 0.107 23.2 7.96 0.436 0.8 0.182 2.225 43.258 56.25 290 45 de mrr sr twr oc ce 2.309 0.152 21.953 7.961 0.428 0.788 0.181 2.303 42.3 57.25 278.35 45 cs mrr sr twr oc ce 2.293 0.192 22.808 7.959 0.434 0.779 0.179 2.284 41.23 58.23 283.56 45 tlbo mrr sr twr oc ce 2.223 0.09 24.012 7.959 0.409 0.708 0.178 2.218 41 58.58 285 45 table 8 percentage improvements in responses using the tlbo algorithm response fa de cs mrr 3.5 9.37 5.27 sr 0.75 0.75 0 twr 6.19 4.43 5.76 oc 11.5 10.15 9.11 ce 2.19 1.65 0.55 teaching-learning-based parametric optimization of an electrical discharge machining process 299 6. conclusions the paper studies the capability of an edm process in generating standard holes on pearlitic sg iron 450/12 grade material, which has found wide ranging applications in manufacturing of diverse mechanical components. the optimal settings of gap voltage, peak current, cycle time and tool rotation for the considered machining application are determined using the tlbo algorithm. it is observed that lower gap voltage, higher peak current, moderate cycle time and higher tool rotational speed would simultaneously optimize material removal rate, surface roughness, tool wear rate, radial overcut and circularity error of this edm process. the effects of all these edm process parameters on the responses are also investigated. the optimization performance of the tlbo algorithm is compared with three other metaheuristics, and it is concluded that the tlbo algorithm excels over the others with respect to higher accuracy of the optimal solutions with low variability and less computational effort. the results of the paired t-tests and developed boxplots also confirm this observation. this algorithm provides almost global optimal solutions for both singleand multi-objective optimization problems as it is least affected due the settings of its different tuning parameters. the applicability of this algorithm for optimization of other conventional and non-conventional machining processes can also be explored. the effects of changing weights allotted to different responses on the 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https://doi.org/10.22190/fume170602001k © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper impact of skinfold thickness on wavelet-based mechanomyographic signal  udc 616-073 eddy krueger 1,2 , eduardo m. scheeren 3 , carla daniele pacheco rinaldin³, andré e. lazzaretti 2 , eduardo borba neves 2 , guilherme nunes nogueira-neto 3 , percy nohama 2,3 1 neural engineering and rehabilitation laboratory, anatomy department, state university of londrina, londrina, brazil 2 rehabilitation engineering laboratory/cpgei/ppgeb, federal technological university of paraná (utfpr), curitiba-pr, brazil 3 rehabilitation engineering laboratory/ppgts, pontifical catholic university of paraná (pucpr), curitiba-pr, brazil abstract. surface mechanography (mmg) is a non-invasive technique that captures signs of low-frequency vibrations of skeletal muscles through the skin. however, subcutaneous structures may interfere with the acquisition of mmg signals. the objective of this study was to verify the influence of skinfold thickness (st) on the mmg wavelet-based signal in the rectus femoris muscle during maximal voluntary contraction in two groups of individuals: group i (n = 10, st <10 mm ) and group ii (n = 10, st equal to or> 20 mm). negative correlation was observed between the 19 hz, 28 hz and 39 hz frequency bands with st. there was a statistical difference in almost all frequency bands, especially in the x and y axes. all mmg axes in group ii presented higher magnitudes in frequency bands 2 and 6 hz (like low-pass filter). thus, these results can be applied to calibrate mmg responses as biofeedback systems. key words: skinfold thickness, wavelet, mechanomyography received june 02, 2017 / accepted december 22, 2017 corresponding author: carla daniele pacheco rinaldin rehabilitation engineering laboratory/ppgts, pontifical catholic university of paraná, curitiba-pr, brazil e-mail: rinaldin99@gmail.com 360 e. krueger, e. m. scheeren, c. d. p. rinaldin, a. e. lazzaretti, e. b. neves, et al. 1. introduction mechanomyography (mmg) is a noninvasive technique that can be also used for monitoring muscle biofeedback as, for instance, can be apply in the control of myoelectrical [1-3] and neural prostheses [4] and to support physical therapy sessions [5, 6]. the mmg records vibrations [7] of detectable muscle fibers on the surface of the skin. the vibrations are related: to the rate of activation of the motor units of global form; to contractile properties; to the time of contraction and relaxation [8, 9]. however, the detectable vibrations on the skin surface and the frequency of the mmg signal can be affected by the adipose tissue that attenuates the spectral and temporal characteristics of the mmg, compromising the acquisition and processing of the signals. in a previously study [10] we found a negative correlation between skinfold thickness (st) and mmg mean frequency, showing the st can be considered as a natural selective filter [11]. however, when it is applied fast fourier transform (fft) to extract the mean frequency [10] it becomes unclear which frequencies are affected by st. differing from transforms as fft that uses only basis functions as sine and cosine to process the frequency, wavelet transform presents the spectral content and temporal space in specific band of frequencies through basis functions called mother wavelet [12, 13]. peñailillo et al. [14] state that wavelet transform provides information of the frequency changes in electromyography (emg) that is not detected by fft. the cauchy wavelet [15] (caw), originally developed for the analysis of emg signals, may be used with mmg technique adjusting the processing to particularities such as time and frequency resolution requirements of the mmg signals [16]. therefore, the objective of this study is to evaluate the influence of st on wavelet-based mechanomyographic signal. based on previous studies, our hypothesis is that all frequency bands present a decrease in magnitude of energy in each frequency band with greater st. 2. methods 2.1. participants this investigation was performed according to principles of the declaration of helsinki and was approved by pontifical catholic university of paraná’s (pucpr) human research ethics committee under register n. 2416/08. twenty able-bodied volunteers (22.15±6.43 yo.) participated in this study. during the period of tests, the volunteers did not use any drug that could change their motor condition. anthropometric data such as weight, height, body mass index (bmi) and the quadriceps skinfold were collected. the volunteers were divided in two groups: group i (n= 10) with skinfold below 10 mm and group ii (n= 10) with st equal or above 20 mm. 2.2 sensors and data acquisition the developed mmg instrumentation used freescale mma7260q mems triaxial accelerometer with sensitivity equal to 800 mv/g at 1.5 g (g: gravitational acceleration). electronic circuits allowed 10 amplification. a load cell (100 kg, 2.0±0.1 mv/v) was used to acquire the quadriceps torque information. a labview™ program was coded to acquire and the display signals. the acquisition system contained a dt300 series data impact of skinfold thickness on wavelet-based mechanomyographic signal 361 translation™ board working at 1 khz sampling rate. the data were saved into european data format (edf) files. 2.3 research design the volunteers performed a standard muscular and warm-up stretching before the experimental protocol. they were seated on a bench with the hip and knee angles set to 70º [17]. after trichotomy and skin cleaning, superficial mmg sensors were positioned over the belly of rectus femoris muscle and attached with double-sided tape. the anterior ankle joint was positioned in a brace (foam coated) positioned in 60° of flexion from total knee extension (0 o ) as shows fig. 1. from three initial knee flexion repetitions, the highest torque value was obtained to measure the maximal isometric voluntary contraction. all participants were verbally instructed to provide their maximum effort and held it for 5 s. in order to minimize a possible muscular fatigue, 2 min rest was performed between contractions [18]. fig. 1 scheme of the experimental setup. the mmg sensor (triaxial accelerometer) was placed over the rectus femoris muscle belly and knee angle was positioned in 60º of flexion with a load cell 2.4 signal processing and statistical analysis inside the 5 s of maximal torque, 0.5 s before and 0.5 s after the peak torque was selected visually through the software bioproc2© version 2.4 totalizing 1 s with 1000 points (1 khz sampling rate) was computed to characterize the performance of analysis. the points were processed by the software matlab® version r2008a. ten epochs of 0.1 s [19, 20] was computed inside 1 s. a third-order butterworth filter was selected with bandpass of 5-100 362 e. krueger, e. m. scheeren, c. d. p. rinaldin, a. e. lazzaretti, e. b. neves, et al. hz [7, 21-28]. the mmg signal was processed in eleven bands (2 – 119 hz) of frequency with caw [15] to each participant. to each frequency band was computed the rms value to characterize the energy of wavelet band. spearman (ρ) correlation coefficient was applied to check the relation between mmg features and st. mann-whitney u test was applied in order to check the differences between the two groups. 3. results table 1 shows the anthropometric data of the participants. the bmi classification of the group i was normal and the group ii was considered overweight. the st between groups i (7.6±1.13 mm) and group ii (36.57±10.33 mm) was statistically significant (p < 0.01). table 1 anthropometric data bmi: body max index. i: group with skinfold below 10 mm; ii: group with skinfold equal or above 20 mm. group volunteer age (yo) weight (kg) height (m) bmi (kg/m 2 ) skinfold (mm) i 1 19 60.0 1.80 18.52 6.8 2 25 59.6 1.68 21.12 8.0 3 19 61.6 1.78 19.44 6.9 4 19 66.3 1.75 21.65 6.0 5 19 63.0 1.75 20.57 8.5 6 20 60.0 1.70 20.76 7.5 7 36 70.0 1.74 23.12 9.0 8 21 72.2 1.75 23.58 8.0 9 19 70.6 1.72 23.86 9.4 10 18 59.9 1.76 19.34 6.0 ii 11 36 90.0 1.78 28.41 36.3 12 23 110.8 1.79 34.58 52.0 13 23 106.0 1.76 34.22 51.0 14 36 88.0 1.84 25.99 30.0 15 21 90.0 1.92 24.41 21.0 16 19 82.5 1.73 27.57 43.5 17 19 90.0 1.90 24.93 20.0 18 18 79.5 1.77 25.38 40.0 19 23 76.8 1.85 22.44 37.0 20 18 80.1 1.84 23.66 35.0 table 2 shows the mmg axes mean values and standard deviation split by frequency band to each group. the frequency band to 2 hz and above 65 hz presented smallest values to all axes (z, x and y). impact of skinfold thickness on wavelet-based mechanomyographic signal 363 table 2 mean±s.d to mmg wavelet bands split by axes to each group group wavelet band (hz) axis z (mvrms) x (mvrms) y (mvrms) i 2 4.81 ± 8.10 5.93 ± 8.65 4.01 ± 5.12 6 19.20 ± 12.59 47.06 ± 33.49 20.92 ± 14.34 11 101.91 ± 58.39 225.02 ±143.48 94.09 ± 65.02 19 92.86 ± 50.74 167.52 ± 87.83 82.72 ± 45.65 28 64.31 ± 34.44 72.28 ± 36.63 52.53 ± 25.33 39 54.00 ± 31.29 55.40 ± 28.95 26.62 ± 10.96 51 29.82 ± 16.69 23.94 ± 14.41 12.33 ± 7.00 65 12.97 ± 11.58 9.82 ± 8.55 4.78 ± 4.61 81 3.93 ± 4.49 4.93 ± 5.63 2.26 ± 2.98 99 1.78 ± 3.14 2.53 ± 3.39 1.07 ± 1.82 119 1.27 ± 2.73 1.84 ± 3.34 0.73 ± 1.83 ii 2 6.03 ± 7.92 19.73 ± 30.60 8.71 ± 8.76 6 34.02 ± 25.40 108.65 ± 89.53 74.36 ± 65.76 11 93.60 ± 59.02 291.54 ±193.07 190.64 ±146.51 19 56.50 ± 32.75 121.34 ± 69.51 84.43 ± 53.05 28 52.37 ± 30.24 74.86 ± 51.81 42.19 ± 32.71 39 43.01 ± 24.37 44.26 ± 31.70 23.63 ± 13.69 51 28.06 ± 17.66 28.06 ± 24.99 14.69 ± 8.25 65 17.98 ± 14.12 16.88 ± 14.65 11.12 ± 8.98 81 8.57 ± 6.32 10.02 ± 8.08 7.07 ± 6.29 99 4.84 ± 4.23 5.34 ± 3.95 4.94 ± 4.71 119 2.73 ± 2.61 2.96 ± 3.13 3.17 ± 4.06 table 3 shows the spearman (ρ) coefficients among mmg axes and st split to frequency bands. only the frequency band to 51 hz do not show p > 0.05 to any axis. frequencies of 19 hz, 28 hz and 39 hz show a negative correlation, indicating that these frequencies are influenced by the st. the lower frequencies have positive and moderate correlations, possibly due to the attenuation of higher frequencies spreading the signal energy to lower frequency bands. table 3 spearman coefficient (ρ) between mmg features and skinfold wavelet band (hz) axis z (ρ) x (ρ) y (ρ) 2 0.289 ** 0.517 ** 0.344 ** 6 0.390 ** 0.421 ** 0.487 ** 11 -0.017 0.228 ** 0.352 ** 19 -0.257 ** -0.268 ** 0.065 28 -0.09 -0.053 -0.190 ** 39 -0.204 ** -0.220 ** -0.158 * 51 -0.135 0.007 0.126 65 0.148 * 0.217 ** 0.304 ** 81 0.249 ** 0.288 ** 0.337 ** 99 0.284 ** 0.323 ** 0.440 ** 119 0.332 ** 0.352 ** 0.386 ** *: correlation is significant at the 0.05 level (2-tailed) **: correlation is significant at the 0.01 level (2-tailed) 364 e. krueger, e. m. scheeren, c. d. p. rinaldin, a. e. lazzaretti, e. b. neves, et al. figures 2, 3 and 4 show the rms average to each frequency band across the participants. almost all bands showed statistical significance, mainly to y and x axes that present greater averages to 11 hz in group ii. regarding to 2 and 6 hz frequency bands we found a pattern in all axes where the values to group ii were always greater (p < 0.01) than group i. fig. 2 mean results (across participants) to z-axis *: p < 0.05 (2-tailed); **: p < 0.01(2-tailed) fig. 3 mean results (across participants) to x-axis *: p < 0.05 (2-tailed); **: p < 0.01 (2-tailed) impact of skinfold thickness on wavelet-based mechanomyographic signal 365 fig. 4 mean results (across participants) to y-axis *: p < 0.05 (2-tailed); **: p < 0.01 (2-tailed) 4. discussion and conclusions originally we hypothesized and found that the st attenuates the mmg frequency bands as a low-pass filter [10]. similar response was found by jaskólska et al. [29] that suggested that skinfold works like a low-pass filter, which was confirmed by a negative correlation between the st and median and peak frequencies in their study. however, we found that to lowest frequency bands (<11 hz) the magnitude is greater to group ii (st> 20 mm). cooper et al. [30] observed a negative correlation between skin fold thickness and mmg signal amplitude in gross lateral movements during muscle contractions performed by electrical stimulation in the rectus femoris, suggesting that a thick layer of adipose tissue interferes with acquisition of the amplitude of the signal of depolarization of the motor units, consequently decreasing the signal of muscular vibration. the potential of the signal amplitude of muscle activity depends on the density of muscle fibers attached to a motor neuron, which is associated with the type of fiber [31]. however, the decrease in the acquisition of muscle vibration is caused independently of the muscle fiber type, as observed by herda et al. [11]. they obtained a lower amplitude of the mmg signal of muscle strength in type i and ii fibers for the vastus lateralis muscle in isometric contractions in individuals with greater subcutaneous fat thickness, suggesting that the adipose tissue attenuates the muscular physiological signals, being a low pass filter acting in a natural way. cescon et al, [32], suggests that this effect may also be influenced, depending on the location of the accelerometer on the muscle, caused by the distance between the muscle tissue and the mmg sensor, especially in anatomical regions with a thicker layer of adipose tissue. 366 e. krueger, e. m. scheeren, c. d. p. rinaldin, a. e. lazzaretti, e. b. neves, et al. regarding to table 2, wavelet bands to 2 hz and above 65 hz presented smallest values (p < 0.05) in all axes due are located outside the frequency range of physiological contraction [31, 33]. according to maggi et al. [34], the fat density (0.95 g cm -3 ) is smaller than muscle density (1.04 g.cm -3 ) and skin density (1.20 g.cm -3 ). thus, it can be assumed that this fact leads to attenuation of high frequencies. moreover, the fat presents lower acoustic impedance than skin and muscle. it allows that an increase in the fat layer does not generate significantly change in the total energy of signal. polato et al. [35], using biaxial mmg found that the st raises and mmg values decrease significantly (r = -0.3935, p = 0.0099), therefore without the specification of frequencies that are influenced by st. figures 2, 3 and 4 show the rms average to each frequency band across the participants. axes y and x presented greater averages to 11 hz in group ii. concerning to of 2 hz and 6 hz frequencies band we found a standard in all axes where the values to group ii were always greater (p < 0.01) than group i. in general, frequencies of 19 hz, 28 hz and 39 hz present a negative correlation and frequencies of 6 hz and 11 hz have a increment in rms values due the increase in st. these results indicate that st of 36.57±10.33 mm works like low-pass filter when compared with subjects with skinfolds of the order of 7.60, to frequencies around 19 hz transferring the mmg signal energy to frequencies below this band. our results are divergent to those found by zuniga et al. [36] who investigated the effect of st at four locations over the vastus lateralis muscle during incremental cycle ergometry. they found that the st over vastus lateralis did not affect mmg temporal and spectral features. probably because the average difference between skin folds studied by zuniga et al. [36] was of the order of 8.0 mm and, in the present study, reaches the order of 29.0 mm (average group ii less average group i). in summary, individuals with a st > 20 mm obtained greater amplitudes of the mmg signal at low frequencies in all axes, suggesting that the adipose tissue is a natural lowpass filter, which can attenuate high frequencies and the higher its thickness the lower the bandwidth. for the y axis this event occurs at 11 hz of the frequency band, however, the y-axis (longitudinal), presents low influence for the recognition of muscle contraction [37]. in this sense, the amplitude of higher frequency bands (> 11 hz) is shifted to lower frequencies due to the subcutaneous tissue. our results suggest that this change in magnitude of frequency bands provides important information for the calibration of mmg systems when incorporating closed-loop control systems such as neuroprostheses. acknowledgements: we would like to thank cnpq, capes and seti-pr for important funding 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manufacturing and construction applications, forklifts are vital equipment, which are used to engage, lift and move palletized items. so, selection of the most appropriate forklift is an essential task for transportation of materials in warehouses for optimal use of the equipment. the present treatise introduces a wellknown multi-criteria decision making (mcdm) technique, namely fully consistent method (fucom) under neutrosophic environment (ne) to model and solve the problem of selecting the best forklift for warehouse. in this regard, the linguistic assessments of the criteria have been represented in terms of single valued triangular neutrosophic numbers (svtnns). a novel triangular neutrosphic score function and ranking function are also proposed. to calculate criteria weights, a novel svtn linear programming problem (svtnlpp) has been developed. the alternatives have been ranked through multi-objective optimization on the basis of ratio analysis (moora). the robustness, consistency and reliability of the proposed integrated method have been checked through comparative and sensitivity analyses. this study makes a significant contribution by developing an original integrated model which provides warehousing system managers a quantifiable analysis, based on which they may make future decisions in order to improve the overall efficiency of the organization in transport handling. key words: multi-criteria decision making, fully consistent method, neutrosophic set, forklift, warehouse, moora 1. introduction a forklift is a compact industrial vehicle with a power-operated forked platform connected to the front that can be lifted and lowered to lift or move freight. forklifts are one of the most commonly used pieces of material handling equipment. despite the emergence of a significant number of automated solutions on the market, demand for received: june 20, 2022 / accepted september 24, 2022 corresponding author: apu kumar saha department of mathematics, national institute of technology agartala, tripura-799046, india e-mail: apusaha_nita@yahoo.co.in https://orcid.org/0000-0002-2765-4296 https://orcid.org/0000-0002-3475-018x 236 s. chakraborty, a.k.saha traditional and electric forklifts is expected to remain stable. forklifts fulfill the demands of a variety of businesses, including warehouses and other big storage facilities and are thus widely utilized for carrying materials and commodities across the industry. forklifts can securely carry goods that people cannot, making them essential on every job site, when one needs to transport huge loads in a warehouse or on a construction site. special attention should be paid to storage systems, cost optimization and the enhancement of all aspects that affect efficiency and contribute to the smooth operation of logistics subsystems in the logistics sector. the issue of analyzing and selecting the most appropriate forklift in realtime industrial practice is a complex decision-making problem that should be formulated through an efficient analytical model. the efficiency of forklifts is critical to the success of any company. every day, forklifts are employed and no logistical procedures would be possible without them. as a result, it was determined to assess their efficiency, which will aid in the process optimization in this logistics subsystem. so selection of an optimal forklift for warehouse is an imperative issue. as the productivity of the forklift has an indirect impact on the productivity of logistics centers, so it is legitimate to consider the process of selecting forklifts as part of the machine tool selection (mts) [1], which can be viewed as mcdm technique. when selecting the most appropriate future machines, researchers apply a variety of decision-making processes. few of the influential factors related to forklift selection process for warehouse are its manufacturing cost, manufacturing time, load capacity, working time, lift height, etc. thus considering these parameters as selection criteria of forklift for warehouse, the problem of selecting the best forklift unit for warehouse can be modeled in terms of mcdm problem. mcdm is a powerful technique that includes individual decision-makers' value judgments to assist them and get the best option. decision makers use mcdm strategies to organize and synthesize the data they have gathered so they may feel confident and at ease with their choices [2,3,4]. the purpose of decision making techniques is to figure out and visualize the influence of an attribute on a choice. it is obvious that the impact of all criteria cannot be equal in every decision-making challenge. in comparison to another procedure that offers virtually differentiable values to indicate parameter significance, the approach that can considerably demonstrate the difference in importance for the selected parameter with regard to the decision objective is preferable. so, selection of a mcdm technique [5] to resolve any decision making problem is very vital to get satisfactory solution. as a pair wise comparison (pwc) method, the fucom reduces a large number of pwcs from similar and popular approaches such as the ahp and the bwm. for example, for n criteria, ahp [6,7] needs n(n-1)/2 pair wise comparison and bwm needs (2n 3) pair wise comparisons; whereas fucom needs only (n -1) pair wise comparison resulting it a less time-consuming technique compared to those of ahp and bwm. the following are the primary advantages of fucom over conventional mcdm methods: (i) a substantially lower number of paired comparisons, (ii) a dependable pwc of the associated criteria, and (iii) the determination of dependable values for criteria weight coefficients, all of which help to reasonable judgment [8]. badi and kridish [9] suggested an integrated framework of fucom and codas method to select the best dumping of solid waste site for landfill in the misuratacity, libya. yazdani et al. [10] proposed the fucom and the cocoso in the rough set setting to find the logistics center location in spain. ramezanzade et al. [11] proposed a hybridized decision-making technique consisting of ahp, bwm and fucom for selecting the superior hybrid renewable energy arrangement for supplying power in government organizations. feizi et al. [12] proposed two hybrid mcdms, viz., fucom selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 237 moora and fucom-moosra and applied those to solve the problem of mineral potential mapping in green field. ong et al. [13] created the fucom-vikor hybrid technique to incorporate the positive attributes as well as negative attributes of the alternatives of corresponding criteria. yousefi et al. [14] developed a combined method using z-number theoretic cocoso and fucom to study failure modes and effects analysis. further, they applied their model in an industry of automotive parts. das and sarkar [15] proposed fucom-isocov, a hybrid technique for determining performance scores of various materials while taking into account value restrictions in order to pick the optimum materials for four different applications. durmić et al. [16] used combined fucom rough saw approach for supplier selection in order to achieve sustainability, taking into account all aspects: economic, social and environmental criteria. badi et al. [17] used fucom-marcos to evaluate and prioritize aspects of green innovation, taking into account sustainability performance indicators. however, the bulk of mcdm approaches deal with discrete options that are defined by a set of criteria that may be determined using cardinal or ordinal data. furthermore, the data might be exact or ambiguous, confusing or partial. in practical mcdm contexts, fuzzy set theory and its expansions have shown to be effective tools for dealing with ambiguous and unclear data. in the context of decision theory, there are numerous multicriteria decision making (mcdm) strategies [18,19,20] that can aid in the resolution of machine tool selection difficulties under uncertainty. in 2011, taha and rostam [21] used the fuzzy ahp, moora and an artificial neural network to develop a dss for machine tool selection in a flexible manufacturing cell. for rating machine tool alternatives, ayag and gurcan [22] suggested a fuzzy mcdm model based on the modified topsis and fuzzy anp. li et al. [23] proposed a mts model using fuzzy dematel, entropy weighing and vikor. in their study of maclaurian symmetric mean using interval valued picture fuzzy (ivpf) sets, ashraf et al. [24] addressed how to use ivpf data to determine the best employee benefit plan. riaz et al. [25] introduced some picture fuzzy aggregation operators to select third party logistic provider using mcdm. in order to choose a student's career, sahu et al. [26] used a hybridized distance measure in a picture-fuzzy environment, where the assessing information about the student, subject, and student's attributes is provided in picture-fuzzy numbers. for weighing and supplier evaluation with regard to information system performance and environmental consequences, kazemitash et al. [27] employed the rough best worst technique. additionally, a case study was carried out for the supplier selection of a biofuel firm, and the outcomes suggest the efficacy of the method in tactical performance evaluation. sharma et al. [28] carried out a case study on accommodation providing hotels by developing a strategy to increase their profit margins by welcoming more and more tourists using the dominance based rough set theory (drst) on the data gathered for the variables like location, facility, value for money, etc. although fuzzy set theory is effective at dealing with issues about ambiguities or partial presence of members of a particular set, it cannot account for all of the ambiguity that arises in real-world situations, such as difficulties requiring imperfect knowledge. to overcome this challenge, atanassov [29] devised an extension, known as intuitionistic fuzzy sets (ifs), where the idea of a non-membership degree of each element was introduced in addition with the membership grade. the notion of ifs is preferable when the information provided is inadequate to determine imprecision using traditional fuzzy sets. the degree of non-membership function in a fuzzy set is always equal to one less than the degree of membership. however, this is not always the case in the real world. for 238 s. chakraborty, a.k.saha instance, if everyone in an area wants to elect one leader, the degree of membership will fall between 0 and 1 for those who will vote, but it will not be equal to one minus the membership value for those who will not vote. this is due to the possibility that a small number of members will cast illegal votes or abstain altogether. so to deal with such situation, the concepts of membership and non-membership have been further generalized by smarandache [30] and neutrosophic sets (nss) have been introduced for dealing with incomplete, abstruse and contradictory data that can be found in numerous real-world scenarios. he proposed the membership functions for truth, indeterminacy, and falsity, where the sum of these three membership values is unbounded. nss are advantageous to deal with quantitative data. karamaşa et al. [31] used neutrosophic ahp for selecting the right logistic service provider. yazdani et al. [32] used critic cocoso under interval valued fuzzy neutrosophic environment for evaluation and selection of suppliers for a dairy company in iran. 1.1. research gap one of the helpful pieces of machinery for lifting and transferring palletized things is the forklift. finding a suitable forklift unit for transport handling in a warehouse is therefore crucial. to the best of our knowledge, there isn't much research in the literature where the development of fucom under ne and consideration of the best forklift for a warehouse have been made. these research gaps have motivated us to introduce fucom under ne to model and solve the problem of optimal forklift selection for warehouse. 1.2. motivation the motivation of the present research is two-fold. ▪ first of all, a forklift is one of the most frequently used material handling instruments and is particularly helpful for industrial storage. therefore, choosing a suitable forklift for transport handling is necessary for business houses for optimal service of the forklifts. ▪ secondly, one of the most widely used mcdm strategies, fucom, is quite helpful in resolving practical decision-making issues. therefore, it makes sense to develop such a consistent mcdm technique under ne to address problems in real life. 1.3. novelty the fact that neutrosophic numbers (nns) demonstrate a broad model function for truth, indeterminacy and falsity memberships, with the sum falling between (0 -,1+), makes them useful for addressing a variety of real-world decision-making problems (dmps). as a result, nns are more able to handle a range of ambiguities that arise in dmps than other existing fuzzy numbers, such as intuitionistic and pythagorean fuzzy numbers. on the other hand, forklifts are a very valuable piece of equipment for transport handling in warehouses, thus choosing the right forklift unit is a crucial task. so, in this paper, svtnns are used as uncertain data to select the optimal forklift for a warehouse. the novelty of the present study is twofold: ▪ firstly, the problem of optimal forklift selection for a warehouse has been considered under ne and solved using the proposed integrated mcdm approach. ▪ secondly, fucom has been introduced under ne by defining a new score and ranking function for svtnns, through which the linguistic assessments of the decision makers are addressed. selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 239 1.4. contribution the main contributions of the proposed research are given below: ▪ by introducing a novel scoring and ranking function of svtnns to determine criteria weights, a novel svtnfucom has been developed. ▪ an optimal forklift selection problem for warehouse operation is modeled and solved using the proposed svtnfucom and the alternatives are ranked using moora. ▪ various comparative and sensitivity analyses have been carried out to check the robustness, reliability and consistency of the proposed integrated approach. the rest of the paper is laid out as given below: section 2 includes a brief literature review on forklift for transport handling. preliminaries on nns are covered in section 3. svtnlpp and svtnfucom with the suggested score and accuracy function have been presented in section 4. the application of the proposed approach in forklift selection presented in section 5 demonstrates the efficacy of projected integrated mcdm technique. the findings are discussed in detail in section 6 and compared with other mcdms together with sa. section 7 concludes by summarizing the proposed technique, addressing limitations and suggesting the future study direction. 2. literature review a large number of modern studies are there in literature, where mcdm techniques are used to solve the problems of transport handling. forklift is one of the useful equipment for lifting and moving palletized items. pallet load handling in warehouses, which includes receiving, transport, disposal and loading and unloading of items, are almost impossible without the employment of appropriate forklift trucks [33]. some of the popular types of forklift are warehouse forklift, side loader forklift, counterbalance forklift, heavy-duty forklift, rough terrain forklift, pallet jack forklift etc. however, the technical and operational characteristics of forklifts are largely determined by factors, such as the type of load, the type and size of the warehouse designed to handle these loads, and most importantly, the warehouse managers' ability to obtain optimal handling equipment that meets their requirements. according to a study conducted by the research firm research and markets, the forklift market will reach 2.2 million units in 2023, with a compound annual growth rate (cagr) of nearly 9% [34]. according to prnewsire [35], the forklift market is expected to rise at a 7.8% annual rate, from 2 billion in 2020 to 2.9 billion in 2035. the rising use of new technologies in the process of boosting the productivity of resources owned by businesses is the key element driving the market's dynamic growth. while the retail industry, particularly e-commerce, is driving the forklift business, the rise of automated guided vehicles is posing a significant challenge to the market. the widespread usage of forklifts is in numerous areas, such as transportation and logistics accounts for the high growth rate of this equipment. mešić et al. [36] used an integrated criticmarcos mode to compare and rank the western balkan countries' logistics performance index (bosnia and herzegovina, north macedonia, albania, serbia, and montenegro), which was calculated by the world bank for 2018. as the problem of forklift selection is also under the category of mts, so various researchers modeled this problem as mcdm problem and solved it using various integrated mcdm techniques. during the literature survey, it has been noticed that a very few researches have concentrated 240 s. chakraborty, a.k.saha on selection of optimal forklift unit for warehousing, especially using mcdm technique. in this section, a brief literature on forklift for transport handling has been discussed. ombale et al. [37] designed and developed forklift which will lift the load and then transfer the load from one place to another place. gawade et al. [38] designed electrically powered forklift for material handling in industrial warehouses and workshops. chen et al. [39] presented the power system structure of electric forklift and the battery–super capacitor hybrid energy management method of electric forklift truck. conte et al. [40] theoretically and experimentally evaluated the effective technical and economical benefits of newly commercialized electric forklift by means of a conventional electric forklift. barrois et al. [41] presented a combination of fuzzy analytic hierarchy process and fuzzy decision making trial and evaluation laboratory to select the most suitable supplier of forklift filters. in 2011, alberti et al. [1] developed a decision support system (dss) for selecting the best milling machine tool using an artificial neural network. atanasković et al. [42] applied delphy method for selection of forklift unit for warehouse. pamucar and ćirović [43] used dematel-mabac model in the process of making investment decisions on the acquisition of forklifts in logistics centers. prusa et al. [44] used topsis method to select forklift truck for logistic company. fazlollahtabar et al. [45] used fucomwaspas model to select side-loading forklift for warehouse. mahmutagić et al. [46] proposed an integrated forklift selection model using data envelopment analysis (dea), fucom) and measurement alternatives and ranking according to the compromise solution (marcos) methods. in [47], authors used an integrated critic-marcos approach for forklift selection in the warehousing system. 3. preliminaries smarandache [30] introduced neutrosophic sets (nss), a combination of fuzzy sets (fss) and ifss. they are characterized by truth (t), indeterminacy (i) and a falsity (f) membership function (mf) and they are a useful tool for dealing with incomplete, ambiguous and inconsistent information in the actual world (f). indeterminacy may be simply defined and truth, indeterminacy and falsity mfs are all independent, making this theory extremely helpful in a variety of applications. different significant concepts regarding nss are discussed in this section. 3.1. definition (neutrosophic sets) a neutrosophic sets (ns) [48] n in x , the universe of discourse is characterized by a truth, indeterminacy and falsity mfs tn, in and fn, respectively; where tn, in and fn are mappings from x to [0,1] denoted by n = {< x, (tn, in, fn >}, where x belongs to x, tn, in, fn belongs to ( 0-, 1+ ) and tn + in + fn belongs to [ 0-, 1+ ]. 3.2. definition (single-valued neutrosophic sets) let x be the universe with generic elements in  denoted by x. a single valued ns sn (svns) is characterized by truth mf, ts n (x), an indeterminacy mf, is n (x) and a falsity mf, fs n (x) and is defined as { : ( ), ( ), ( ) : }n n n n s s ss x t x i x f x x =    selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 241 3.3. definition (single-valued triangular neutrosophic number) a single valued triangular neutrosophic number (svtnn) a = <(a,b,c): ta,ia, fa> is a ns on the set of real number r , whose truth, indeterminacy and a falsity mfs are as follows:                  − −+− =  − −+− =                  − −+− =  − −+− =                  − − =  − − = otherwise cxb bc xcibx bxf bxa ab axfxb xf otherwise cxb bc xcibx bxi bxa ab axixb xi otherwise cxb bc txc bxt bxa ab tax xt a a a a a a a a a a a a ,1 , ))(( , , ))(( )(, ,1 , ))(( , , )(( )(, ,0 , )( , , )( )( where ta, ia, fa belongs to [0,1]; a, b, c belongs to r; x belongs to x. 3.4. definition (operations on single-valued triangular neutrosophic numbers) let a = < (a, b, c): ta, ia, fa>, a1 = < (a1, b1, c1): t1 a , i1 a, f1 a > and a2 = < (a2, b2, c2): t2 a , i2 a, f2 a > be three svtnns. the operations for svtnns are defined as follows: a. +++= ),max(),,max(),,min(),,,( 21212121212121 aaaaaa ffiittccbbaaaa (1) b. = ),max(),,max(),,min(),,,( 21212121212121 aaaaaa ffiittccbbaaaa (2) c. 0,,,:),,( =  aaa fitcbaa (3) 3.5. definition (score and accuracy function of svtnn) let a = < (a, b, c): ta, ia, fa> be a svtnn, then score and accuracy function of a is defined as follows: a. 12 )2)(2( )( aaa fitcba as −−+++ = (4) b. 12 )2)(2( )( aaa fitcba aa −−+++ = (5) 3.6. ranking of single-valued triangular neutrosophic number let a1 = < (a1, b1, c1): t1 a , i1 a, f1 a > and a2 = < (a2, b2, c2): t2 a , i2 a, f2 a > be two svtnns. the ranking of the svtnns is defined as follows: a. if s(a1) < s(a2), then a1 < a2. b. if s(a1) = s(a2) and if i. a(a1) < a(a2), then a1 < a2 ii. a(a1) > a(a2), then a1 > a2 neutrosophic sets (ns) n in x , the universe of discourse is characterized by a truth, indeterminacy and falsity mfs tn, in and fn, respectively; where tn, in and fn are mappings from x to [0,1] denoted by n = {< x, (tn, in, fn >}, where x belongs to x, tn, in, fn belongs to ( 0-, 1+ ) and tn + in + fn belongs to [ 0-, 1+ ]. 242 s. chakraborty, a.k.saha 3.7. multi-objective optimization method based on ratio analysis the multi-objective optimization method based on ratio analysis (moora) was presented by brauers and zavadskas [49,50]. a set of weighted targeting criteria can be incorporated into mineral potential maps using this technique. in the first stage, positive and negative signs are employed to separate targeting criteria. a set of criteria with a positive sign indicates that higher values should take precedence. negative criteria, on the other hand, prioritize lower-valued items. here moora was applied to rank alternatives. the following is an explanation of the moora method: step 1: first create the matrix below, where m and n represent the number of unit cells in the area and the number of targeting criteria, respectively [50]             = mnmm n n xxx xxx xxx x ... ............ ... ... 21 22221 11211 (6) step 2: normalize each xij in the above matrix to x* ij, with values ranging from [0, 1] for individual targeting criterion, using the following equation: nj x x x m i ij ij ij ,...,2,1, )( 1 2 1 2 * ==  = (7) step 3: determine the following optimization function to evaluate the possible values allocated to individual unit cells yi:   = += −= g j n gj ijjijji xwxwy 1 1 ** (8) where g and wj respectively denote the number of beneficial criteria and the weight allocated to the jth criterion. here rest of the criteria is considered to be non-beneficial criteria. step 4: finally rank alternatives as per values of yi in ascending order. 4. proposed method in this section, we have introduced fucom under ne for the selection of optimal forklift for warehouse, by representing the linguistic assessment of decision-maker (dm) by svtnns. to calculate criteria weights, we have used fucom and the alternatives are ranked through moora. to describe the proposed method, let us first discuss some newly introduced concepts as follows: 4.1. single valued triangular neutrosophic score and accuracy function let a = < (a, b, c): ta, ia, fa> be a svtnn. the main aim of the score function is to convert the svtnn to a crisp number. the mean of the tnn is (x + y + z)/3. the score value of mf is (2 + ta − ia − fa)/3. thus the proposed score function is defined as selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 243 9 )2)(( )( aaa fitzyx as −−+++ = (9) also the accuracy function is defined as 9 )2)(( )( aa ftzyx aa −+++ = (10) 4.2. properties of score function some of the properties of the proposed score function of svtnn are given below: i. if ta = 1, ia = 0 and fa = 0, then s(a) = 1. ii. if ta = 0, ia = 1 and fa = 1, then s(a) = 0. iii. both score and accuracy functions of svtnn are real numbers. 4.3. ranking of svtnns in respect of proposed score and accuracy functions suppose a1 = < (a1, b1, c1): t1 a , i1 a, f1 a > and a2 = < (a2, b2, c2): t2 a , i2 a, f2 a > be two svtnns. the ranking of the svtnns is defined as follows: a. if s (a1) < s (a2), then a1 < a2 b. s(a1) = s(a2) and if i. a(a1) > a(a2), then a1 > a2 ii. a(a1) < a(a2), then a1 < a2 iii. a(a1) = a(a2), then a1 = a2 4.4. ranking function of svtnn let a = < (a, b, c): ta, ia, fa> be a svtnn. the ranking function of the svtnn is defines as: 9 )maxmaxmin2)(( )( aaa fitzyx ar −−+++ = (11) 4.5. single value triangular neutrosophic linear programming problem (svtnlpp) let us take the general form of the proposed svtnlpp involving m constraints with n unknowns, where all coefficients and resources are represented by svtnns as follows: 0 x b; ax tosubject x;c (min)max t  (12) where c, a and b be the matrices of dimensions n, m×n and m respectively. they are represented by svtnns as follows: let c = < (c1, c2, c3): tc, ic, fc> be the cost vector; a = < (a, b, c): ta, ia, fa> be the coefficient matrix and b = <(b1, b2, b3): tb, ib, fb> be the vector of resources. thus the proposed svtnlpp has become 0 x >;f ,i ,t :)b ,b ,(b< x >f ,i ,t :c) b, (a, < subject to t>f ,i ,t :)c ,c ,(c < (min)max bbb321aaa xccc321  (13) 244 s. chakraborty, a.k.saha the proposed algorithm to solve system (13) is as follows: step 1: let us first consider system given by eq. (12). step 2: to convert the svtnlpp to crisp lpp, use eq. (11) and system given by eq. (13) will then become 0 x ); >f ,i ,t :)b ,b ,(b ( r x ) >f ,i ,t :c) b, (a, < ( r subject to x t)>f ,i ,t :)c ,c ,(c < ( r (min)max bbb321aaa ccc321  (14) step 3: as system (14) is a crisp lpp, use lingo 19.0 software to solve system (14) to obtain optimal solution of the proposed svtnlpp. 4.6. proposed single valued triangular neutrosophic fucom (svtnfucom) in this subsection, the procedure of the proposed svtnfucom is presented to obtain the criteria weights of any mcdm problem. suppose there are m alternatives ai, i =1, 2, …, m; n criteria cj, j = 1, 2, …, n and wj be the weight of the jth criterion. algorithm of the proposed svtnfucom is discussed below: step 1: first, rank the criteria in order of priority among the predefined set of evaluation criteria cj, j = 1, 2, …, n, starting with the criterion that is predicted to have the highest weight: (1) (2) ( ), ,...,j j j rc c c (15) where r denotes the rank of each criterion. if there is more than one criterion having same rank, the sign of ‘=’is used in place of ‘>’ between them in eq. (15). step 2: compare these criteria by determining the comparative priority (cp) k/k+1, k = 1, 2, …, n of each criterion. thus the cp vector is determined as ),...,,( )1(3 2 2 1 + = k k (16) where k/k+1 denotes priority of the criterion of the cj(k) rank to the criterion of the cj(k+1) rank. step 3: next calculate the final values of the weight vector (w1, w2, …, wn)t, where wj = <(aj, bj, cj): tw, iw, fw) >, j = 1, 2, …, n of the evaluation criteria satisfying following two conditions: 22 1 1 22 1)1( 1 1 is,that ii. i. ++ + + ++ ++ + + = = = k k k k k k k k k kkk k k k k w w w w w w w w   (17) thus the weight coefficients must satisfy the condition 2 2 + + = k k k k w w  (18) selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 245 the minimum deviation of full consistency (dfc) =  fitcba ,,:),,( is satisfied only if transitivity is satisfied, i.e. when the conditions of 1 1 + + = k k k k w w  , 2 1 2 1 + + + + = k k k k w w  and 2 1)1( 2 + ++ + = k kkk k k w w  are met so that the requirement for maximum consistency is fulfilled, i.e. dfc is χ = 0 for the obtained values of the weight vector. in order for the conditions to be met, it is necessary that the values of the weight vector (w1, w2, …, wn)t meet the condition of  − + + 1 1 k k k k w w and  − + ++ + 2 1)1( 2 k kkk k k w w , with the minimization of the value . the condition for maximum consistency is met in this way. the final model for finding the final weight vector of the criterion can be defined as below based on the following settings: min χ subject to jwjwj w w j w w j n j j k kkk kj kj k k kj kj =−−  =+ ++ + + + ,0;,1;,;, 12 1)1( )2( )( 1 )1( )(  (19) step 4: rewrite system (19) as jfitcba jfitcba jfitcba fitcba fitcba jfitcba fitcba fitcba fitcba wwwjjj n j wwwjjj k kkk kjkjkjkjkjkj kjkjkjkjkjkj k k kjkjkjkjkjkj kjkjkjkjkjkj  = −   −     = + ++ ++++++ + ++++++ ,1,1,0:)0,0,0(,,:),,( ,0,0,1:)1,1,1(,,:),,( ,,,:),,( ,,:),,( ,,:),,( ,,,:),,( ,,:),,( ,,:),,( subject to ,,:),,(min 1 2 1)1( )2()2()2()2()2()2( )()()()()()( 1 )1()1()1()1()1()1( )()()()()()(      (20) step 5: convert the svtnlpp to crisp lpp using eq. (11) in system (20) as follows: jrfitcbar jrfitcbar jfitcbar fitcba fitcba r jfitcbar fitcba fitcba r fitcbar jwwwjjj n j jwwwjjj k kkk kjkjkjkjkjkj kjkjkjkjkjkj k k kjkjkjkjkjkj kjkjkjkjkjkj  = −   −     = + ++ ++++++ + ++++++ ),1,1,0:)0,0,0((),,:),,(( ),0,0,1:)1,1,1((),,:),,(( ),,,:),,(() ,,:),,( ,,:),,( ( ),,,:),,(() ,,:),,( ,,:),,( ( subject to ),,:),,((min 1 2 1)1( )2()2()2()2()2()2( )()()()()()( 1 )1()1()1()1()1()1( )()()()()()(      (21) solve the crisp lpp (21) using lingo 19.0 software to get the final weight vector (w1, w2, …, wn)t. 246 s. chakraborty, a.k.saha 5. application of the proposed integrated model in forklift selection for warehouse transportation is an extension of the manufacturing process and one of the most significant aspects of firm logistics management. transport is required to carry raw and supporting materials to their processing locations, as well as to move commodities within the manufacturing process, so that finished or semi-finished products can be transported to market using various modes of transportation. the forklift is one of the most commonly used modes of delivery for commodities in the actual manufacturing process. the results of the research can be applied in the real-world when choosing the best forklift unit for a warehouse operation with the help of a case study. the process of selecting the best transport/handling unit (forklift) is part of the investment decision-making process. the criteria for selecting the best transport handling forklifts have been identified and illustrated in table 1. the euro-roal company possesses a number of forklifts that are more than 20 years old and ten (10) alternatives (side-loading forklifts), viz. forklift 1 to forklift 10 will be investigated to improve and refine their fleet [45]. ten (10) different alternatives are assessed with the purpose of putting the study to the test, with the best option being chosen for the investor based on the criteria. forklift models of ten distinct manufacturers have chosen and studied. all of the chosen forklift models meet the standards for warehouse operating based on their basic characteristics. one of them will be selected as a euro-roal candidate. table 1 description of evaluation criteria sl. no. criteria description cr1 purchase cost on the market, forklift prices vary depending on the manufacturer. the purchase price should not be important for the buyer when making an investment decision, but it does have a substantial impact on the final decision. once the essential parameters are completed in an ad hoc approach, the purchase price is frequently a deciding factor. cr2 age a forklift's manufacturing period is defined by its age or year of manufacture. recently made forklifts have superior specifications and alternatives for customizing them to meet individual needs. cr3 working time when choosing a forklift, one of the most crucial factors to consider is the forklift's utilization time. the lower the number of hours a forklift is used, the less likely it is to break down. cr4 maximum load capacity the maximum load capacity of a forklift is a metric that measures how much weight a forklift can lift in kg. cr5 maximum lift height the maximum lift height is a criterion that describes how high a forklift can raise anything. cr6 environmental factors environmental impact of forklift operating. cr7 supply of spare parts some representatives functioning in the republic of serbia's market do not have in stock all necessary spare parts that are subject to frequent replacements and their delivery takes weeks, leading the means to be fixed for a long time. selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 247 according to petrović et al. [51], determination of the significance of criteria is one of the most crucial steps in the dmp. the decision-maker employed a 9-level set of linguistic factors listed in table 2 to score the alternatives and criteria. at first, svtnfucom has been applied to find weights of the criteria for selecting the optimal forklift based on the linguistic assessments of expert. the linguistic assessments have been considered as svtnns. then out of 10 forklifts, the best one has been selected using moora [52]. the algorithm of the proposed method to select optimal forklift for warehouse has been discussed below: 5.1. algorithm step 1: the evaluation criteria are ranked as: cr3 > cr2 > cr4 = cr5 > cr1 > cr7 > cr6. table 2 9-level scale to rate criteria and alternatives extreme low very low low moderate low moderate moderate high high very high extreme high el vl l ml m mh h vh eh step 2: perform a pwc of the ranked criteria with respect to the first ranked criterion cr1, based on 9-level scale depicted in table 3. thus the significances of criteria are obtained for all criteria ranked in step 1. step 3: following step 3 of the proposed svtnfucom, the svtnlpp is formulated as follows: min χ jwjw w w w w w w w w w w w w w w w w w w w w w w i j i =− −− −− −− −− −−  = ,1,1,0:)0,0,0(;,0,0,1:)1,1,1(;)14,.14,.8(.:)5.1,596.1,75.1( ;)22,.14,.75(.:)76.1,004.2,5.2(;)22,.125,.75(.:)5.1,67.1,2( ;)0,11,.89(.:)33.1,5.1,2(;)1,.11,.89(.:)4,3,2( ;)14,.0,8(.:)28.1,33.1,4.1(;)14,.14,.83(.:)17.1,2.1,25.1( ;)22,.125,.75(.:)5.1,67.1,2(;)0,0,1(:)1,1,1( ;)0,11,.89(.:)33.1,5.1,2(;)1,.1,.9(.:)3,2,1( subject to 7 16 1 7 5 1 4 5 2 4 3 6 7 7 1 1 5 5 4 4 2 2 3       (22) 248 s. chakraborty, a.k.saha step 4: using eq. (11) in system (22), svtnlpp has been converted into crisp lpp as follows: min r(χ) jwjwr w w r r w w rr w w r r w w rr w w r r w w rr w w r r w w rr w w r r w w rr w w r i j i =− −− −− −− −− −−  = ,0;,1);())14,.14,.8(.:)5.1,596.1,75.1(( );())22,.14,.75(.:)76.1,004.2,5.2(();())22,.125,.75(.:)5.1,67.1,2(( )())0,11,.89(.:)33.1,5.1,2(();())1,.11,.89(.:)4,3,2(( )())14,.0,8(.:)28.1,33.1,4.1(();())14,.14,.83(.:)17.1,2.1,25.1(( );())22,.125,.75(.:)5.1,67.1,2(();())0,0,1(:)1,1,1(( );())0,11,.89(.:)33.1,5.1,2(();())1,.1,.9(.:)3,2,1(( subject to 7 16 1 7 5 1 4 5 2 4 3 6 7 7 1 1 5 5 4 4 2 2 3       (23) solving system of system (23) using lingo 19.0 version, the final weight vector of the criteria is obtained as follows: w1 = 0.09, w2 = 0.18, w3 = 0.33, w4 = 0.13, w5 = 0.13, w6 = 0.06, w7 = 0.08 and the dfc is χ ≈ 0.04. step 5: the initial decision matrix (idm) is furnished in table 3 as follows: table 3 initial decision matrix alternatives cr1 cr2 cr3 cr4 cr5 cr6 cr7 fl1 7.95 10 5012 4000 5400 5 7.67 fl2 12.9 10 7140 3000 3500 7 7.67 fl3 17.8 9 6500 5000 4500 7 5 fl4 19.3 19 4312 3000 6000 3 3.67 fl5 1.87 18 12000 3000 4000 5 3 fl6 3.4 7 4800 4000 4000 7.67 9 fl7 8.093 25 12000 4000 5900 3 5 fl8 29.8 11 3720 3000 5100 9 9 fl9 13.75 17 15350 4500 4800 3 5 fl10 18.297 13 6122 3000 4000 5 7 step 6: in the present case study, as cr1, cr2 and cr3 are non-beneficiary criteria and cr4, cr5, cr6 and cr7 are beneficiary criteria, so to make all the criteria of the same type, the idm has been normalized by using eq. (7). the normalized decision matrix (ndm) is presented in table 4. selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 249 table 4 normalized decision matrix alternatives cr1 cr2 cr3 cr4 cr5 cr6 cr7 fl1 0.135573 0.212286 0.18488 0.340195 0.356438 0.271231 0.371557 fl2 0.219987 0.212286 0.263377 0.255146 0.231024 0.379724 0.371557 fl3 0.303548 0.2 0.239769 0.425243 0.297031 0.379724 0.242215 fl4 0.329127 0.403343 0.159059 0.255146 0.396042 0.162739 0.177786 fl5 0.185369 0.382115 0.44265 0.255146 0.264028 0.271231 0.145329 fl6 0.518418 0.1486 0.17706 0.340195 0.264028 0.416069 0.435987 fl7 0.138012 0.530715 0.44265 0.340195 0.389441 0.162739 0.242215 fl8 0.508186 0.233515 0.137221 0.255146 0.336635 0.488216 0.435987 fl9 0.234482 0.360886 0.566223 0.382719 0.316833 0.162739 0.242215 fl10 0.312023 0.275972 0.225825 0.255146 0.264028 0.271231 0.339101 step 7: finally, using eq. (8) the values of yi and the ranking of alternatives are determined and furnished in table 5. table 5 ranking of alternatives forklifts yi ranking fl1 0.025137 1 fl2 -0.02921 6 fl3 -0.00639 4 fl4 -0.04607 7 fl5 -0.13615 9 fl6 -0.00532 3 fl7 -0.08772 8 fl8 0.016455 2 fl9 -0.15283 10 fl10 -0.02357 5 6. results and discussion according to the findings of this study, fl1 is identified as the optimal choice for warehouse. fl8, fl6, fl3, fl10, fl2, fl4, fl7, fl5 and fl9 have been ranked as 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th and 10th position respectively. the performance of the alternatives through the proposed approach is shown in fig. 1. from table 5, it could be observed that the values of yi are lying in between −1 and +1, that is a variation is found in the values. fig. 1 performance of alternatives through proposed integrated approach 250 s. chakraborty, a.k.saha to check the consistency of the proposed technique, a comparison has been carried in subsection 6.1 between the results obtained by the proposed method and other wellknown mcdm techniques. 6.1. comparison the results obtained through proposed integrated approach are compared with other wellknown mcdm techniques, viz. technique for the order of prioritization by similarity to ideal solution (topsis), weighted aggregated sum product assessment (waspas), complex proportional assessment (copras), additive ratio analysis (aras), multiattributive border approximation area comparison (mabac), combined compromise solution (cocoso), weighted sum model (wsm) and weighted product model (wpm). the rank of alternatives obtained through all the methods are furnished in table 6. the comparison shows that all the methods, except wsm and aras, have identified fl1 as the best alternative for warehouse. wsm and aras have identified fl8 as the best choice of forklift. a small violation could be observed in the ranking order of the comparatively lower ranked alternatives. this has been occurred as a result of the diverse methods considered for ranking the alternatives. fig. 2 shows the comparison of scores of the alternatives obtained through topsis, waspas, copras, aras, wsm, wpm, mabac, cocoso and the proposed approach. table 6 comparison between different ranking methods ranking methods ranking orders of alternatives topsis fl1 > fl8 > fl6 > fl3 > fl10 > fl4 > fl2 > fl5 > fl7 > fl9 waspas fl1 > fl8 > fl6 > fl3 > fl4 > fl2 > fl10 > fl7 > fl9 > fl5 copras fl1 > fl6 > fl8 > fl3 > fl2 > fl10 > fl4 > fl5 > fl7 > fl9 wsm fl8 > fl1 > fl6 > fl3 > fl4 > fl2 > fl10 > fl7 > fl9 > fl5 wpm fl1 > fl8 > fl6 > fl3 > fl2 > fl4 > fl10 > fl7> fl5 > fl9 aras fl8 > fl6 > fl1 > fl3 > fl4 > fl2 > fl10 > fl7 > fl9 > fl5 mabac fl1 > fl3 > fl6 > fl8 > fl4 > fl2 > fl10 > fl7 > fl9 > fl5 cocoso fl1 > fl3 > fl6 > fl8 > fl10 > fl2 > fl4 > fl7 > fl9 > fl5 integrated approach fl1 > fl8 > fl6 > fl3 > fl10 > fl2 > fl4> fl7> fl5 > fl9 as there is a variation in the ranking order of the alternatives for different mcdm approaches, so to check the correlation between the other mcdm techniques with that of the proposed one, spearman’s rank correlation coefficients have been determined as follows: 6.2. spearman’s rank correlation coefficient spearman’s rank correlation coefficient is a measure which indicates how much the mcdm techniques are correlated to each other. it lies between -1 to +1. it the correlation coefficient between the methods is +1 or nearer to +1, they are said to be strongly correlated. in this study, the correlation coefficients between the proposed method and the other existing mcdm techniques have been calculated and presented in table 7. selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 251 fig. 2 comparison between proposed approach and other mcdm methods from the table, it could be observed that the correlation coefficients between the proposed integrated approach with other existing methods are nearer to +1, which indicates the strong correlation between the mcdms with the proposed one. thus it could be concluded that the proposed method is consistent and comparable with other mcdms. for further validation of statistical consistency of the rankings obtained through the proposed integrated approach, ws coefficient between the proposed method with topsis, waspas, copras, wsm, wpm, aras, mabac and cocoso have been calculated and obtained as (0.998, 0.984, 0.941, 0.898, 0.983, 0.824, 0.901, 0.916). from the results it is clear that the proposed method is strongly correlated with the other existing mcdm techniques. 6.3. sensitivity analysis the weight coefficients of the criterion, or the proportional priority assigned to specific criteria, have a significant impact on the outcome of mcdm method. because the final selections can alter when the weight coefficients of the criteria change even little, the results of mcdm approaches should be accompanied with a study of their sensitivity to these changes. in decision making problems, the purpose of sensitivity analysis (sa) is to see how the weights of criteria affect the outcome. based on different scenarios revealed during the study, it may result in a shift in alternative precedence. it can be concluded that the obtained findings are sensitive or resilient if the achieved order of ranking changes when the importance of criterion is modified. in instances when it's difficult to determine the relative relevance of multiple aspects, sa can help decisionmakers. swapping the weights of one criterion with another is done to examine if switching the weights causes the order of precedence of alternatives to change. 252 s. chakraborty, a.k.saha table 7 spearman’s rank correlation coefficient integrated approach topsis waspas copras wsm wpm aras mabac cocoso integrated approach 1 0.975 0.95 0.958 0.933 0.983 0.9 0.883 0.917 topsis 1 0.942 0.967 0.925 0.975 0.892 0.875 0.875 waspas 1 0.892 0.983 0.967 0.95 0.933 0.9 copras 1 0.858 0.958 0.84 0.858 0.892 wsm 1 0.95 0.98 0.883 0.85 wpm 1 0.917 0.9 0.9 aras 1 0.85 0.817 mabac 1 0.967 cocoso 1 to check robustness of the proposed method, a sensitivity analysis (sa) is presented by varying the weights obtained for each criterion. in table 8, the different cases of varying the weights and their impacts on the ranking of alternatives are shown. from table 8, it could be observed that fl1 is the optimal alternative for warehouse for any choice of weights for criteria. also fl8 has obtained the second position in each of the cases; a little change could be observed in the ranking of lower ranked alternatives. so this sa verified the robustness and consistency of the proposed integrated approach. fig. 3 shows sa for the variation in weights of criteria. fig. 3 sensitivity analysis 6.4. managerial inference an effective approach for selecting the best forklift for a warehouse has been established in the suggested application. managers of related businesses can use the proposed framework to choose which forklift is best for their warehouse. this will result in significant resource and expense savings, as well as in transportation implications. all of the criteria listed will assist firms in dealing with a variety of issues and improving their attempts to find user-friendly forklifts for transportation. furthermore, the method of forklift selection criteria using industry expert responses and literature is a significant benefit of this suggested endeavor. managers will be able to test the observation stability using the implemented sensitivity analysis. selection of forklift unit for transport handling using integrated mcdm under neutrosophic... 253 the suggested model has a flaw in that subsystems linked with the criteria are not taken into account when minimizing complexity. so as future endeavor, the sub criteria may also be taken into consideration for the said purpose. table 8 sensitivity analysis cases variation in weights ranking of alternatives case 1 weights obtained through proposed method fl1 > fl8 > fl6 >fl3 > fl10 > fl2 > fl4 > fl7 > fl5 > fl9 case 2 assigning equal weights to each criterion fl1 > fl8 > fl3 >fl2 > fl6 > fl10 > fl7 > fl4 > fl9 > fl5 case 3 50% weights assigned to non-beneficial criteria 50% weights assigned to beneficial criteria fl1 > fl8 > fl3 >fl2 > fl6 > fl10 > fl7 > fl4 > fl5 > fl9 case 4 40% weights assigned to non-beneficial criteria 60% weights assigned to beneficial criteria fl1 > fl8 > fl3 >fl2 > fl6 > fl10 > fl7 > fl4 > fl9 > fl5 case 5 30% weights assigned to non-beneficial criteria 70% weights assigned to beneficial criteria fl1 > fl8 > fl3 >fl2 > fl6 > fl10 > fl7 > fl4 > fl9 > fl5 7. conclusion in this research, fucom has been introduced under ne. the linguistic assessments of the experts have been considered as svtnns and the criteria weights are calculated using the proposed novel svtnfucom. to do so, a novel svtnlpp and a new score function have been introduced. the alternatives are ranked using moora. to validate the applicability of the proposed technique, it was used to choose transport and handling methods in a warehouse system. the study was carried out in a company that primarily trades and distributes aluminum profiles. the proposed method has identified fl1 as the optimal forklift for warehouse. the model in use allows for objective consideration of input parameters that influence the final decision. the comparison, which entails use of another eight popular mcdm approaches, demonstrates consistency of previously obtained results if the model is consistently applied across all conceivable versions. to check the robustness of the proposed method, a sa is presented by varying the weights of the beneficiary and nonbeneficiary criteria. from sa also, it is seen that fl1 is the optimal choice for handling and transport in a warehouse. the unique contribution of the present study is the introduction of fucom method under ne to model and solve the problem of forklift selection for transport handling. the key advantage of the proposed integrated strategy is that it allows warehousing system managers to make more decisions in order to improve the overall efficiency of the warehousing systems. it is feasible to readily estimate the efficiency of both forklifts and other equipment by using this model, and to pay greater attention to identify and monitor input and output characteristics by using this model. in light of the findings, warehousing system managers must conduct adequate monitoring of all forklift activities and rationalize all superfluous forklift movements and operations. however, the suggested methodology has certain drawbacks, especially based on the assessment of single decision expert, final decision on forklift selection has been taken. 254 s. chakraborty, a.k.saha in future, we shall collect several assessments from different domain experts, so that the problem of optimal forklift selection for warehouse can be modeled as a mcgdm problem and based on the aggregated assessments of experts, final decision can be made. in this regard various aggregation operators under ne can be taken into consideration. from the foregoing, we may conclude that the warehousing systems of companies are extremely complicated, with several potential for cost reductions and opportunities to improve all activities and procedures inside it. future study might concentrate on ensuring that inefficient forklifts can be reusable. in this regard, to identify some recycling techniques for the damaged forklifts can also be taken into consideration, concerning circular economy. the construction of a model for estimating the efficiency of operating the selected side-loading forklift is also a future research topic related to this study. in this regard, other generalizations of fuzzy sets may also be applied to this problem. moreover, while considering the problem as mcgdm, various aggregation operators under various fuzzy environments can be considered. references 1. 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5, pp. 142-162. https://link.springer.com/article/10.1007/s10479-020-03737-y#auth-natalia-jaramillo_rueda https://link.springer.com/article/10.1007/s10479-020-03737-y#auth-giovani-jes_s_carrascal_zambrano https://link.springer.com/article/10.1007/s10479-020-03737-y#auth-giovani-jes_s_carrascal_zambrano plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 87 91 https://doi.org/10.22190/fume180108007c © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd short communication fracture mechanics simple calculations to explain small reduction of the real contact area under shear udc 539.6 michele ciavarella politecnico di bari, department of mechanics, mathematics and management, italy abstract. in a very recent paper, sahli and coauthors [12] (r. sahli et al., 2018, “evolution of real contact area under shear”, pnas, 115(3), pp. 471-476) studied the contact area evolution for macroscopic smooth spheres under shear load in presence of adhesion. it was found that contact area aa reduces quadratically with respect to shear load t, i.e. a=a0 -at2, where a0 is the contact area with no shearing, and a is the "area reduction parameter" found to be approximately proportional to a0 -3/2 across 4 orders of magnitude of a0. in this note we focus on the smooth sphere/plane contact because we believe that the case of a rough contact requires separate investigations, and we use a known model of fracture mechanics, which contains a fitting parameter  which governs the interplay between fractures modes, in order to find very good agreement between the data and the analytical predictions, developing relatively simple equations. the interaction with modes is limited. key words: adhesion, friction, fracture mechanics, area reduction, jkr model 1. introduction adhesion represents a flourishing area of tribology. several authors are focusing on this topic nowadays for its relevance in different scientific areas, which range from adhesive contact of rough surfaces [1-5] to bioinspired [6-8] and pressure sensitive adhesives [9-11]. a very important topic is the interaction between "adhesion" and "friction". recently, in a very interesting paper, sahli et al.[12] make spectacular experiments on the reduction of contact area a upon application of shear force t, suggesting it of the form a = a0 –at 2 , where a0 is the force at t=0. they suggest scaling laws for coefficient a of the form a ~ received january 08, 2018 / accepted february 02, 2018 corresponding author: michele ciavarella department of mechanics, mathematics and management, politecnico di bari, viale japigia e-mail: m.ciava@poliba.it 88 m. ciavarella a0 -3/2 for individual contacts. notably, the scaling law is shown to hold over more than 4 orders of magnitude of a0, for a plane/plane rough contact as well as for smooth spheres with radii of the order of millimeter, and for a "real" as well as "apparent" contact area. incidentally, we note that the contact area is still a fugitive concept in contact mechanics [13], due to its fractal evolution, and indeed sahli et al.[12] do not give any information on the contact radii of asperities, which is a very "resolution-dependent" quantity, as well as the contact area. therefore, we shall concentrate here on the question of macroscopic contacts, and the corresponding subset of experiments in sahli et al. [12]. we show here that qualitatively the findings of [12] are predicted by simple fracture mechanics arguments for contacts in the presence of adhesion, more in details than what is discussed in [12]. johnson, kendall and roberts (jkr-theory) firstly applied fracture mechanics concepts [14, 15] to adhesion between elastic bodies, and this was extended to the presence of tangential force by savkoor and briggs [3] who also conducted experiments between glass and rubber similar to [12] but less detailed, and clearly evidenced a reduction of the contact area when tangential load was applied, but the reduction is much less than what is expected by a "brittle model", as indeed is confirmed by [12], which in these respects is therefore not entirely surprising. johnson [17, 18] and waters and guduru [19] have proposed different models to take into account the interplay between two fracture modes, namely i and ii (mode iii is also present, but marginal) with empirical parameters and phenomenological models to generalize the "brittle" behaviour. we will refer to johnson [18] in this short communication. 2. fracture mechanics calculations we will derive here a very simple fracture model in which we retain a circular shape of the contacts, although experiments clearly show an elliptical or even more complex shape, which would make the analysis extremely complex. the energy release sum takes for mixed mode the form (after averaging over the periphery for modes ii and iii) 2 21 2 2(1 )2 i iig k k e          (1) where e * = e / (12 ) is plane strain elastic modulus, e is young's modulus and  the poisson's ratio of the soft material. here, ki, kii are irwin's stress intensity factors. if the surfaces do not slip, kii is given by [19] aπa t k ii 2  (2) in the absence of tangential force, the equilibrium dictates g=gc=w where w is the surface energy, and the standard jkr eq. (1) gives contact radius a0 for a given normal force p r ae waeπp 3 4 8 3 03 0    (3) fracture mechanics simple calculations to explain the small reduction of real contact area under shear 89 the toughness of the material in mixed-mode conditions is in many phenomenological models a function of ratio kii/ki, i.e. gc=wf(kii/ki), where f(kii/ki)=1 corresponds to the "ideally brittle" fracture, where frictional dissipation is neglected. we follow johnson 1 [18] and write gc as                  2 2 1 i ii c k k zβwg (4) i.e.         3 2 8 1 waeπ t zβwgc (5) where z=(2-)/(2(1-)) and we approximate for this step only ki 2e * w. with applied tangential force t, as we have (1), we can restate the jkr mode i problem for a reduced "effective" work of adhesion   e k zgw ii ceff 2 2 (6) i.e. 2 3 1 (1 ) 8 eff t w w z e wa            (7) where eq. (2) has been used. inserting eq. (7) into eq. (3), for the same normal load we require 32 3 3 3 0 03 44 8 (1 ) 8 3 38 e at e a e a w z e a w r re a                   (8) notice that we are conducting our analysis in load control, while displacement control would give different results, as has been shown for work of adhesion independent [20] of the shearing direction. furthermore, to keep the model simple, we are not considering any dependence of the work of adhesion on the tangential loading, even if such dependence may occur. from eq. (8), in its general form, it is not evident if this explains the findings in [12]. however, expanding in series for a small change of contact radius a=a0(1-x) leads to (being x<<1) 2 2 3 3 0 08 8 3 2 0 2 (1 ) 1 1 (1 ) 8 (1 ) 3 t t e a w e a w wr z z x e a w zt wr                         (9) for a sphere, we have a = a0 2 (1x) 2  a0 – 2xa0, thus a = (a0 –a)/t 2  2xa0/t 2 . we further expand eq. (9) for small values of t and obtain    0 0 1/ 2 3/ 4 (1 ) 4 8 3 a a a z r e e w rw                (10) 1johnson (1996) [18] assumes =0, while we keep the term z=(2-)/(2(1-)) in the calculation. 90 m. ciavarella we report in our fig. 1 the data plotted in fig. 3 of [12] for smooth contact of spheres with radius r=[7.06, 9.42, 24.81] mm. data are reported for a as a function of a0 (circles): they loosely collapse about one line with a ~ a0 -3/2 which they give as a "guide for eyes" as is here reproduced with the dashed blue line in fig.1. using our prediction (10) with material constants e = 1.6 mpa, w = 27 mj/m 2 , =0.5 and radii r = [7.06, 9.42, 24.81] mm (from [12]), we obtain various curves which asymptotically tend to a ~ a0 -5/4 and with =0.997. unfortunately, fig. 3 in [12] does not report different symbols for different radii; thus we cannot attempt to compare the dependence on the radius, nevertheless with the same  our prediction (10) covers satisfactorily the range of the experimental results. # fig. 1 comparison of results for area reduction parameter a [m 2 /n 2 ] vs. initial contact area a0 [m 2 ] in fig. 3 of [12] (only for sphere/plane contact) with our prediction (10) for three different radius of the sphere r=[7.06, 9.42, 24.81] mm (solid thick red curves, line thickness increases with radius) and =0.997. dashed line: guide for the eyes with slope –3/2 3. conclusions in this short note, we have used fracture mechanics considerations to estimate the "area reduction parameter" for a smooth macroscopic sphere following the approach of johnson [16]. we have written the energy release rate in the critical conditions as the sum of the "normal" and "tangential" contribution, where the latter is scaled by a fitting parameter . for =1 the modes i and ii do not interact, and the contact radius is independent of t, while for =0 the two modes are coupled. the adhesive problem under shear load is seen as an equivalent normal problem but with a reduced work of adhesion. with this model, and expanding for a small variation of the contact area and a small tangential load we have derived analytical predictions for the dependence of the area reduction parameter with respect to a0 which asymptotically tend to a ~ a0 -5/4 . the analytical model compared satisfactorily with the experimental data obtained by sahli and coauthors [12], showing that the interaction with the modes is extremely limited, as the best fit parameter =0.997. fracture mechanics simple calculations to explain the small reduction of real contact area under shear 91 references 1. fuller, k.n.g., tabor, d.f.r.s., 1975, the effect of surface roughness on the adhesion of elastic solids, proceedings of the royal society of london a: mathematical, physical and engineering sciences, 345, 1642. 2. pastewka, l., robbins, m.o., 2014, contact between rough surfaces and a criterion for macroscopic adhesion, proceedings of the national academy of 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tangential displacement on the adhesion strength of a contact between a parabolic profile and an elastic half-space, royal society open science, 4(8), 161010. 12801 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 329 342 https://doi.org/10.22190/fume240605035h © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a fractal-based approach to the mechanical properties of recycled aggregate concretes chun-hui he1, hua-wei liu1, chao liu1,2 1school of civil engineering, xi'an university of architecture and technology, xi’an, china 2school of science, xi'an university of architecture and technology, xi’an, china orcid ids: chun-hui he https://orcid.org/0000-0003-0810-5248 hua-wei liu https://orcid.org/0000-0003-4433-5075 chao liu https://orcid.org/0000-0002-9634-7791 abstract. the mechanical properties of porous concrete, such as strength and durability, are significantly influenced by moisture transport, particularly in the case of recycled aggregate concretes. the pore distribution and pore size of the concrete, as well as the ambient temperature in the surrounding environment, exert a significant influence on the moisture transport. this paper establishes the fractal fick’s law, the fractal darcy law, and the fractal richards equation. in conclusion, a fractal model for the diffusion and permeability in porous concrete is established. this study examines the mechanisms of moisture diffusion and water permeability in concrete. the comparison of the theoretical prediction with the experimental data indicates a high degree of congruence, thereby suggesting that the concrete’s relative humidity response can be predicted by the established model. this provides a foundation for the optimal design of concrete with required mechanical properties in special applications. key words: fractal dimension, durability of concrete, mathematical model, moisture 1. introduction concrete is the most prevalent porous material utilized globally, and it is an enduring theme in mechanical engineering [1-3], and it might be also the very promising construction material in the moon and the mars by 3-d printing technology. the humidity within concrete plays a pivotal role in determining its mechanical, chemical, and thermal properties [4-6]. consequently, the durability and reliability of concrete are significantly influenced by the transport of humidity within the concrete, especially for the recycled aggregate concretes. given the vast quantity of demolished concrete waste present globally, the utilization of aggregates derived from this waste represents an environmentally friendly and cost-effective construction material. received: june 05, 2024 / accepted july 20, 2024 corresponding author: hua-wei liu, chao liu school of civil engineering, xi'an university of architecture and technology, xi’an, china e-mails: liuhuawei@xauat.edu.cn (h. w. liu); chaoliu@xauat.edu.cn (c. liu) https://orcid.org/0000-0003-0810-5248 https://orcid.org/0000-0003-4433-5075 https://orcid.org/0000-0002-9634-7791 330 c.-h. he, h.-w. liu, c. liu in a surprising turn of events, the telegraph reported in september 2023 that reinforced autoclaved aerated concrete (raac) had failed with only a lifespan of 30 to 40 years. this presents an urgent challenge for engineers and architects. this concrete material, which has been used in construction for decades, began to corrode and produce cracks. while this has caused some unfortunate accidents in the uk and other places, it has also led to some scientific discoveries. the primary factor contributing to its deterioration is the transformation of moisture, which results in corrosion. the volume of the corroded reinforcing steel bar can be increased by up to seven times [7], which is a remarkable phenomenon. the most concerning aspect is the absence of warning signs preceding damage, which further complicates the unpredictability of the failure. although there has been considerable research into the transport of moisture in concrete using differential models [8], there has been little investigation into the underlying mechanisms in fractal spaces. the fractal diffusion theory [9,10] represents an effective tool for the analysis of moisture transfer in concrete. the intricate micro/nano-structure of the pores can be represented by a fractal network [11-13]. in this paper, we will present a mathematical model in a two-scale fractal space, as described in refs. [14-17]. 2. fractal moisture transfer the intricate porosity structure within the concrete will result in a negative pressure, which will induce the evaporation of the absorbed water. this will initiate the process of moisture diffusion in concrete, as the concrete surface has the highest relative humidity, while the center has the lowest. the diffusion process is a complex phenomenon that occurs in three dimensions [18 ]. in order to identify an analytical expression for this process, this paper examines the one-dimensional diffusion within concrete. the porous concrete is treated as a two-scale fractal space, the fractal fick’s laws can be expressed, respectively, as drh j d x    = −  (1) drh j t x       = −   (2) where j is the diffusion flux, rhd represents the relative humidity, d is the diffusion coefficient, α is the two-scale fractality for the porosity, and ∂α/∂tα is the two-scale fractal derivative. so the fractal diffusion equation can be expressed as ( )d drh rh d t x x        =    (3) when α=1, eq. (1) turns out to be the traditional richards equation, which was used to study diffusion process in soils [19]. https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-1 a fractal-based approach to the mechanical propertis of recycled aggregate concretes 331 2.1. initial surface absorption when α=1, the initial surface absorption rates of water by the concrete can be expressed in the term of the boltzmann variable [19] 1/2i t− (4) where i is the initial surface absorption rates, t is time. nevertheless, eq. (4) was found to be unsuitable for the concrete diffusion process. it should therefore be modified in accordance with the recommendation set out in [20]: ni t− (5) where n is an experimental constant. in order to investigate the physical significance of the variable n in eq. (5), it is possible to introduce a generalized boltzmann variable in a manner analogous to that for the traditional richards equation [19]: /2x t  −= (6) where λ is the generalized boltzmann variable. the initial surface absorption rate on the concrete surface can be expressed as /2i at −= (7) where a is a constant. the relationship between n and the fractality is therefore as follows 2n = (8) kumar and bhattacharjee [20] denoted n as the silting factor, now we see that it is relative to two-scale fractal dimensions. 2.2. fractal diffusion in the concrete following the initial surface absorption, the moisture diffuses from the concrete surface to the concrete interior. the diffusion process is primarily influenced by the geometry of the concrete pores. in order to study the properties of eq. (3) in more depth, the two-scale transform is introduced [21,22] s t x    =  = (9) eq. (3) becomes ( )d drh rh d s     =    (10) the two-scale transform [21,22] is regarded as a physical interpretation of the fractional complex transform [23], which was originally proposed by li & he [24]. it has been extensively employed in engineering, as evidenced by numerous references (see, for instance, refs. [25-29]). the solution of eq. (10) is https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-1 332 c.-h. he, h.-w. liu, c. liu 1/2 1/2 0 1/2 1/2 0 1 ( , ) ( ) 1 erf( ) 2 1 ( )erfc( ) 2 drh s rh rh rh d s rh rh rh d s    − −   − −     = + − −    = + − (11) where rh0=rh(0,t), rh∞=rh(∞,t), erf is the error function and erfc is the complementary error function. the solution of eq. (1) is obtained as follows. 1/2 /2 0 1/2 /2 0 1 ( , ) ( ) 1 erf( ) 2 1 ( )erfc( ) 2 drh x t rh rh rh d t x rh rh rh d t x     − −   − −     = + − −    = + − (12) the diffusion process is influenced by the pore structure of concrete, and the diffusion process in turn alters the pore structure of concrete. in order to investigate the relationship between the diffusion coefficient and the diffusion process with the pore properties of concrete, a dimensionless analysis [16,17,30] is employed in this paper. the diffusion coefficient has a dimension of m2/s. by means of dimensionless analysis, it can be assumed that 0 0( , ) ( ) ( )m nd ddrh drh d d v v dt dt =  (13) here v0 is the pore volume, m and n are dimensionless exponents. when v0 = 0, there is no diffusion; and when v0 = 1, diffusion can be studied using the traditional theory of thermodynamics. furthermore, changes in relative humidity also affect the diffusion process, which ends when the diffusion process tends to be stabilized, that is to say when drhd/dt=0. the dimensionless equation is 2 1 3[ ] [ ]m nl t l t− −= (14) that means 2 / 3, 1m n= = (15) we, therefore, obtain 2/3 0( ) ddrh d v dt  (16) the rate of change of relative humidity is related to saturation. when saturated (θ=1), drhd/dt =0. the rate of change of relative humidity is greatest when θ=0. based on the above analysis, it can be assumed that (1 )addrh dt  − (17) where a is the experimental constant. also the pore volume is proportional to porosity: 0v  (18) a fractal-based approach to the mechanical propertis of recycled aggregate concretes 333 based on the above analyses, the following equations were obtained for the diffusion coefficient versus porosity and saturation degree 2/3 (1 )ad k= − (19) where  is the concrete’s porosity, k and a are experimental constants. in literature, there was an empirical formula for soils’ diffusion [31]: 3/4 10/3(1 )d c= − (20) here c is a dimension-related parameter. 2.3. fractal permeation in the concrete in this paper, permeation is considered to occur when there is a pressure gradient, which is induced by the liquid-gas interface inside the concrete, the interface-induced force can be explained by the geometrical potential theory [32,33]. darcy's law is the most basic tool for studying permeation phenomena, but it cannot analyze the effect of concrete pore properties (pore size and distribution) on permeation properties. the fractal darcy's law can be expressed as a d p q k dx   = (21) according to the definition of the fractal derivative, eq. (21) can be approximated expressed as 1lim lim (1 ) ( ) ( )x r x r a d p a p a p a dp q k k k k r dxdx x x           − +  →  →   = = =   +   (22) where q is permeation flux, δp permeation pressure, r mean radius of pores, γ gamma function, k permeability coefficient, a pore area, μ viscosity coefficient. in the fractal space, the pore area scales as 2a r  (23) the fractal darcy's law can be expressed as 1q cr p+= −  (24) where c is a constant. the liquid-gas interface induced pressure can be described by the kelvin equation [34]: ln k rt p rh m   = − (25) where rhk is the relative humidity induced by the permeation process, r ideal gas constant (j/mol). m molar mass of water (kg/mol); t temperature (k); ρ density of water (kg/m). applying the approximate formula, ln(1+x)=x, the following approximate relationship can be obtained ln ln(1 (1 )) (1 )k k krh rh rh= − −  − − (26) https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-4 https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-18 334 c.-h. he, h.-w. liu, c. liu eq. (21) becomes 1 1ln (1 )k k rt rt q cr rh cr rh m m   + +=  − − (27) according to the law of conservation of mass in the two-scale fractal space, we obtain that ( ) 0 q t x       + =   (28) where ω is the ratio of water in liquid form to the total water in the concrete, it can be expressed as water k water moisture v rh v v  = = +  (29) where vwater and vmoisture are, respectively the volume for water and moisture. the mass conservation equation can then be written as 1( ) ( (1 )) 0k k rh rt cr rh mt x        +  − − =   (30) or ( ) ( ) 0k c krh d rh t x       + =   (31) here dc is given as follows 1 c rt d cr m +  = (32) the general solution of eq. (31) is ( )k crh f x d t = − (33) where f is a continuous function. according to the initial and final conditions, we can get  0 1 1 0 1 ( , ) ( ) exp ( ) ( ) exp ( ) k crh x t rh rh rh d x d t rt rh rh rh d x cr t m        +   = + − −   = + − −    (34) where d1 and c are experimental parameters. 2.4. coupled transport of moisture diffusion and water permeation the processes of moisture diffusion and water permeation are studied separately in the aforementioned text, although they actually occur simultaneously. the relative humidity within the concrete can be approximately expressed as a result of the coupling of diffusion and permeation. ( , ) ( , ) (1 ) ( , )d krh x t arh x t a rh x t= + − (35) https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 a fractal-based approach to the mechanical propertis of recycled aggregate concretes 335 where a is the weight function, the early stage is dominated by diffusion, the final stage is dominated by permeation, assuming that at t=t0, the diffusion and permeation contribute equally to rh, so we can choose the following weight function 0 0 t a t t = + (36) where t0 is determined according to the actual experiment. thus we obtain the coupled transport equation 0 0 0 1/2 /20 0 0 1 0 1 0 ( , ) ( , ) ( , ) 1 ( ) 1 erf( ) 2 ( )exp ( ) d k t t rh x t rh x t rh x t t t t t t rh rh rh d t x t t t rt rh rh rh d x cr t t t m      − −   +   = + + +    = + − −  +        + + − −  +    (37) 3. theoretical analysis the experimental water box was designed with dimensions of 1000mm × 680mm × 600mm. the box was maintained at a temperature of 293 k (20 ℃), concrete samples were placed within the box for the purpose of measuring relative humidity at two locations, 2 cm and 4 cm beyond the concrete surface. the relative humidity was monitored at oneminute intervals until 2880 minutes had elapsed, the samples were tested for both the recycled aggregate concrete and natural aggregate concrete. detailed experiment process was given in ref. [35]. 3.1. humidity response according to the experiment data, we obtain the following humidity response: 0.4116 293 2 0. 6 4116 293 4 293 0.35685 0.35 5 2 9 4 8 2 3 0.24160 1 erf( ) 0.24261 1 erf( ) 0.23455 1 erf( ) 0.24711 1 erf( 0.75840 98.61 0.75739 113.36 0.76545 5 6 ) 2.53 0.75289 56.5 n n r r rh rh t t t t rh rh − − − − − − − − − − − − = −  = −    +   + +  = −  = + −           (38) in the subscripts, n and r represent, respectively, natural aggregate concrete and recycled aggregate concrete. the value 293 represents temperature in kelvin, while the values 2 and 4 represent, respectively, the depth from the concrete surface in centimeters. 336 c.-h. he, h.-w. liu, c. liu fig. 1 humidity change of concrete samples with temperature of 20 ℃ (293.15 k): comparison between experimental results and calculated curves. (a)n-293-2; (b)n293-4; (c)r-293-2; (d)r-293-4. from fig. 1, it can be seen that the experimental data basically match the theoretical results. the least squares fitting method was used in the curve fitting process, and because there is a sudden change in the initial response, the calculated curve does not have a good fit with the experimental curve, but the general trend is the same. sample r-293-2 appeared obviously two times response, the beginning of the response is slow, when t=10000s, the second response occurs. this is due to the diffusion-permeability coupling. sample n-293-2 shows a similar situation, but it is not very obvious, with a faster response at the beginning, but the response becomes slower after t=100000 s. in contrast, samples n-293-4 and r-293-4 do not show the phenomenon of two responses, suggesting that diffusion dominates at x=4 cm, but it can be reasonably assumed that diffusion-permeability coupling will occur subsequently, the place near surface has a faster coupling than the center of the concrete. 3.2. space diffusion process according to the coefficients in eq. (38), the average values of x=2 cm and x=4 cm are taken, and then re-fitting based on the experimental data gives the following results. a fractal-based approach to the mechanical propertis of recycled aggregate concretes 337 0.4116 0.21 3 30 293 0.2472 29 0. 5685 3 0.757 0 0.24289 226.0301 0.75 11 1 erf( ) 0.24 3083 19 e .rf( )17 132 17 n r r t x t x h rh − − − −   +  = −  = −+       (39) according to the fitting results eq. (39), fig. 2 and fig. 3 give the diffusion process of the specimens with time and space, and the trend is basically the same, but there are errors. the fitting results eq. (39) indicate that water transport is not a simple one-dimensional transport problem, but is closer to three-dimensional transport. fig. 2 theoretical prediction for the diffusion process for the natural aggregate concrete at temperature of 20℃ (293.15k): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship fig. 2 shows the diffusion process with time and depth for natural aggregate concrete at 20 °c (293.15 k). since there are only two data sets for depth (x=2 cm and x=4 cm), there is a large deviation in the depth prediction. fig. 2(b) shows that the fitted curves at x=2 cm and x=4 cm are in general agreement with the experimental data, and the trend for x=1 cm versus 3 cm is plotted according to eq. (39). fig. 2(c) shows that the relative humidity fits better at x=4 cm and shows a large error at x=2 cm for further improvement. 338 c.-h. he, h.-w. liu, c. liu fig. 3 shows the diffusion process with time and depth for the recycled aggregate concrete at 20 °c (293.15 k). fig. 3(c) shows that the relative humidity is in good agreement at x=4 cm, with a large error at x=2 cm for further improvement. fig. 3 theoretical prediction for the diffusion process for the recycled aggregate concrete at temperature of 20 ℃ (293.15 k): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship 3.3 couple of the fractal diffusion process and the fractal permeation process in view of the above analyses, the shallower the depth, the more likely to appear the coupling phenomenon of diffusion and permeation, in the depth of the farther, mainly diffusion is dominant. considering both diffusion and permeation, the coupled transport equation for humidity is given as:     5 0.4116 0.1914 293 5 0.1914 5 0.8232 5 10 ( , ) 0.75789 0.24211 1 erf(203.2086 ) 10 0.75789 0.24211exp 4.9741( 1.1797 10 ) 10 nrh x t t x t t x t t − − −  = + − +  + + − −  + (40) a fractal-based approach to the mechanical propertis of recycled aggregate concretes 339     5 0.35685 0.2237 293 5 0.2237 5 0.7137 5 10 ( , ) 0.75917 0.2408 1 erf(153.2464 ) 10 0.75917 0.24083exp 5.0816( 5.2324 10 ) 10 rrh x t t x t t x t t − − −  = + − +  + + − −  + (41) in the case of water transport, diffusion is the dominant when t < 105. when t >105, water transport is dominated by permeation. the calculation curves of the diffusion-permeability coupled transport model for each specimen are shown in figs. 4 and 5. fig. 4 diffusion-permeability coupling for the natural aggregate concrete at temperature of 20 ℃ (293.15 k): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship fig. 4 shows the calculated curves of the diffusion-permeability coupling model for natural aggregate concrete at 20 °c (293.15 k), and the overall fitting accuracies are relatively high to meet the engineering requirements. fig. 4(b) shows that the fitted curves at x=2 cm and x=4 cm are in good agreement with the test data, so the predicted curves at x=1 cm and x=3 cm have some credibility. fig. 4(c) also shows a significant improvement in accuracy at x=4 cm, indicating that the improved model has been used for engineering predictions, it can be a better illustration of the humidity response at t=3000 s, 6000 s, 9000 s, 12000 s and 15000 s, respectively. 340 c.-h. he, h.-w. liu, c. liu fig. 5 diffusion-permeability coupling for the recycled aggregate concrete at temperature of 20 ℃ (293.15 k): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship fig. 5 shows the calculation curves of the diffusion-permeability coupling model of recycled aggregate concrete at 20 °c (293.15 k), and the overall fitting accuracy is also very good, and the accuracy basically meets the engineering requirements. fig. 5(b) shows that the fitting curves at x=2 cm and x=4 cm have some deviation from the experimental data, which is mainly due to the special pore structure of recycled aggregate concrete and the internal interfaces, how to improve its accuracy needs further theoretical and experimental research. fig. 5(c) also shows that the accuracy at x=2 cm needs to be further improved, which indicates that the closer to the surface of the concrete, the more obvious the penetration effect through the internal interface of recycled aggregate concrete. in summary, the diffusion-permeability coupling model can fully show the relative humidity changing process with time and depth, and the fitted curve is basically consistent with the experimental data, which indicates that the moisture transport model established in this paper has a certain degree of credibility. the diffusion-permeability coupling is a complex phenomenon. this paper employs weighting functions to decouple the two processes, thereby creating a framework for developing a comprehensive model of diffusion-permeability coupling in the future. furthermore, the weighting functions may be enhanced by incorporating additional factors. a fractal-based approach to the mechanical propertis of recycled aggregate concretes 341 4. conclusion in this paper, the coupling mechanism of water vapor diffusion and permeation inside concrete is investigated based on the diffusion and permeation equations in the two-scale fractal space. the relative humidity response model of concrete was established, and its theoretical results were compared with experimental data, showing a high accuracy of the established model. the conclusions are as follows: (1) this paper presents a new approach to fractal modifications of fick's law and darcy’s law in two-scale fractal space. (2) a coupled model for the moisture diffusion and water permeation is proposed. though the one-dimensional model is already doing a great job of predicting the internal humidity response characteristics of concrete, there is still plenty of room for improvement. (3) the weight function is easy to determine based on the experimental data. in order to leverage the potential of this approach, it is imperative that the mechanical field addresses the challenges of translating this theoretical analysis into practical applications. the present theory offers a promising avenue for the design of porous concretes with the specified mechanical properties. additionally, it offers a novel approach to addressing porous problems that arise in non-continuum mechanics, material science, and soil mechanics in the context of fractal space. acknowledgement: the authors would like to acknowledge the national natural science foundation of china (52178251 and 62276207), the technology innovation guidance program of shaanxi province (2023gxlh-049), the qinchuangyuan’s "scientist and engineer" team building of shaanxi province (2023kxj-242), the special research program for local service of shaanxi province (23jc047), the key industry chain core technology tackling 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by-nc-nd original scientific paper application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in multicriteria decision-making marko radovanović1, darko božanić1, duško tešić1, adis puška2, ibrahim m. hezam3, chiranjibe jana4 1military academy, university of defence in belgrade, belgrade, serbia 2department of public safety, government of brčko district of bosnia and herzegovina, brčko, bosnia and herzegovina 3statistics & operations research department, college of sciences, king saud university, riyadh, saudi arabia 4department of applied mathematics with oceanology and computer programming, vidyasagar university, midnapore, india abstract. the selection of unmanned aerial vehicles for different purposes is a frequent topic of research. this paper presents a hybrid model of an unmanned aerial vehicle (uav) selection using the defining interrelationships between ranked criteria (dibr), full consistency method (fucom), logarithm methodology of additive weights (lmaw) and grey evaluation based on distance from average solution (g-edas) methods. the above-mentioned model is tested and confirmed in a case study. first of all, in the paper are defined the criteria conditioning the selection, and then with the help of experts and by applying the dibr, fucom and lmaw methods, the weight coefficients of the criteria are determined. the final values of the weight coefficients are obtained by aggregating the values of the criteria weights from all the three methods using the bonferroni aggregator. ranking and selection of the optimal uav from twentythree defined alternatives is carried out using the g-edas method. sensitivity analysis confirmed a high degree of consistency of the solutions obtained using other mcdm methods, as well as changing the criteria weight coefficients. the proposed model has proved to be stable; its application is also possible in other areas and it is a reliable tool for decision-makers during the selection process. key words: dibr, fucom, lmaw, grey numbers, edas, unmanned aerial vehicle (uav) received: august 24, 2023 / accepted october 08, 2023 corresponding author: darko božanić military academy, university of defence in belgrade, veljka lukica kurjaka 33, 11040, belgrade, serbia e-mail: darko.bozanic@va.mod.gov.rs 388 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana 1. introduction various uavs are being used increasingly in various areas of human activity, both in civil and military sectors. successful planning, organization and execution of modern military operations are based on information about the enemy, time and space, which can be collected using various systems, including uavs. the global demand for uavs is on the rise, as they are becoming increasingly crucial in enhancing the operational effectiveness of military units, positioning them as a vital component within combat systems [1]. diverse possibilities lay the foundation for their utilization across various conditions and operational contexts. the possibilities of using uav in combat operations are diverse: mine detection, electronic reconnaissance, electronic countermeasures, hyperspectral scanning, laser target marking, radar imaging, radar jamming, use as combat stations and anti-drone systems, management and leadership of units on the battlefield in the operation zone, as logistics stations, as transport support, as support for medical teams, etc. an uav is characterized as an aerial vehicle equipped with an engine, which is either operated remotely by a controller or possesses a certain level of autonomous capability, the use of which is one-time or possiblly can be used multiple times and can carry different types of cargo. they differ in terms of purpose, construction characteristics and the source of energy used to propel them. the uav are often called drones and and if these are used for combat then are called unmanned combat air vehicle (ucav). various possibilities of uavs application make a significant contribution to defence and security, but these are also used in agriculture [2], building, transportation [3], commerce, transmission, research [4], medicine [5,6], science, architecture, archeology [7], visual recording and imaging [8,9], photogramemetry [10], geology and forestry [11], fire fighting [12], mining, oceanography and meteorology [13] ], environmental management, sports [14], mapping [15], etc. it is also possible to use them for recording, photography [16], monitoring [17], reconnaissance [18], detection [19], transport of various types of cargo [20], protection of important persons and objects, etc. the mentioned functions are significant and objective features for the implementation of unmanned aerial vehicles in various military units [21], because their utilization, there is a notable enhancement in the combat capabilities of units throughout the planning, arrangement, and execution of various military maneuvers. by selecting the optimal combat uav and equipping army units, four very important capabilities would be significantly improved and these are the abilities to: command and lead, use the information space, efficiently deploy forces and be mobile in the area of operation. for the purpose of such selection, the multi-criteria dibr-fucom-lmwabonferonni-grey edas model is used. up to now, a plethora of mcdm techniques have been formulated to assist in making specific decisions [22–25]. the very application of mcdm methods to the problem of selecting the optimal uav and the general application of these methods related to uavs have been analyzed by numerous authors in their research. therefore, ali karimoddini [26] selects the uav and evaluates its performances in order to support bridge inspection using the ahp method. by applying hybrid model interval type-2 fuzzy ahp and topsis, it is analyzed the selection of male (medium-altitude long-endurance) uav [27]. radovanović et al. [28] examine the selection of uav’s serving to the requirements of tactical army units through the application of a hybrid fuzzy ahp-vikor model. hamurcu and eren [29] perform the selection of an uav with mcdm model ahptopsis for application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in ... 389 army needs. using fuzzy anp method, liu and chan [30] analyze the importance of service quality indicators for drone-based imaging and photography. karaşan and kaya [31] employ the topsis method to identify the optimal technology for controlling unmanned aerial vehicles in terms of efficiency. the application of uavs in research related to the military sphere is presented in certain publications. milić et al. [18] investigate the utilization of uavs in particular types of operations. adamski [32] investigates the use of uavs in modern conflicts. petrovski and radovanović [15] examine the integration of uavs with command and information systems. jović [33] scrutinizes the application of uavs within counter-terrorism operations. petrovski et al. [34] show the use of uavs supported by mobile applications in crisis management. according to the performed analysis, it can be concluded that the topic of an uav/ucav selection has been discussed in the literature so far, but there is room to approach the selection/ranking issue in a different way, respectively, by applying new mcdm methods. the contribution of this paper is combined. the primary importance of this paper lies in the definition of criteria important for the selection of the optimal ucav. another, no less important contribution is the improvement of the selection methodology, that is, the formation of a decision-making model drawing upon various mcdm techniques. apart from the introductory section, the paper comprises additional five sections. in the second section it is made a brief description of the model, respectively, the applied methods. through the third section, the definition and calculation of the weight coefficients of the criteria and the ranking of alternative solutions are performed, that is, the application of the model is presented. the fourth section addresses sensitivity analysis, while in the fifth section is conducted comparative examination of the outcomes in relation to alternative mcdm approaches. at the end of the paper, the conclusion of this research is provided. 2. model description the complexity of the research issue required the application of a hybrid model of mcdm, which is formed of the methods dibr, fucom and lmaw for defining weight coefficients of criteria, as well as g-edas method for selecting the most favorable combat unmanned aerial vehicle, based on defined criteria, which is presented in the fig. 1. the first phase of the model includes the definition of criteria and the calculation of their weights using the dibr, fucom and lmwa methods, and in the second phase, the aggregation of the weight coefficients of criteria is carried out using the bonferroni aggregator. in the third phase, the alternatives are defined and the initial decision-making matrix, after which the most favorable solution is selected using the g-edas method. the fourth phase presents the sensitivity analysis, while in the fifth phase of the model a comparison of the results with other methods is performed. in the following sections it is provided a brief description of the methods used. 390 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana fig. 1 dibr-fucom-lmaw-bonferonni-grey edas model 2.1. dibr method the dibr method facilitates enhanced comprehension for decision makers regarding criteria relations by focusing solely on the connections between adjacent criteria [35]. this notably streamlines the process of deriving weight coefficients, especially in scenarios involving a substantial number of criteria. in general, with some methods that have found wide application in practice such as the ahp, in cases with an extensive array of criteria, upholding result consistency becomes more challenging; thus, this method employs a tree structure encompassing both main criteria and sub-criteria [36]. this method is still relatively underused in a significant part of the literature. tešić et al. [37] use this method with the fuzzy marcos method for the selection of a location for overcoming obstacles. in the paper [38] is shown the combination of fuzzy dibr and fuzzy-rough edas methods. lukić [39] used dibr waspas model to rank trading companies. the steps of the dibr method are presented on the fig. 2. 2.2. fucom method the fucom method is one of the newer methods. it was applied for the first in the mcmd model in 2018 in the paper [40]. the fucom method (in basic or modifed version) is used in decision-making in combination with many other methods. popović et al. [41] use gray fucom-swot model. biswas et al. [42] use a fuzzy fermatean fucom-codas in the selection of a smartphone. feizi et al. [43] present the fucommoora and fucom moosra model. kahn et al. [44] present a novel fuzzy fucomqfd approach. pamučar et al. [45] developed a mcdm model fuzzy fucomneutrosophic fuzzy marcos. stević et al. [46] present the fucom-edas model in the transport. the steps of the fucom method are presented on the fig. 3. application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in ... 391 fig. 2 steps of the dibr method [35] fig. 3 steps of the fucom method [40] 2.3. lmaw method the lmaw method can be applied to determine weight coefficients of criteria and to select the optimal alternatives from a set of offered ones. it was developed in the paper by pamučar et al. [47]. it is applied in many fields to solve various research issues. until now, the lmaw method has been used in basic and modified form [48-51]. on fig. 4 are presented the steps of the lmaw method related only to defining weight coefficients of criteria. 392 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana fig. 4 steps of the lmaw method related to calculation of weight coefficients of criteria [47] 2.4. basics of the grey theory the theory of grey numbers serves as a potent approach for addressing challenges involving partially known information. represented by ⊗x, a grey number denotes a value whose precise magnitude is not known, yet its belonging range is established. when upper �̅� and lower 𝑥 bounds are known, but specifics about the distribution for x' are uncertain, it's referred to as an interval grey number [52]. ⨂𝑥 = [𝑥, 𝑥] = [𝑥′ ∈ 𝑥|𝑥 ≤ 𝑥′ ≤ 𝑥] (1) the degree of greyness presents the distance between their limits 𝑥 − 𝑥. depending of the degree of greyness interval number can become black number, white number or white (crisp) number. further are presented the basic operations of interval grey numbers [53]: ⨂𝑥1 + ⨂𝑥2 = [𝑥1 + 𝑥2, 𝑥1 + 𝑥2] (2) ⨂𝑥1 − ⨂𝑥2 = [𝑥1 − 𝑥2, 𝑥1 − 𝑥2] (3) ⨂𝑥1 × ⨂𝑥2 = [𝑥1𝑥2, 𝑥1𝑥2] (4) ⨂𝑥1 ÷ ⨂𝑥2 = [ 𝑥1 𝑥2 , 𝑥1 𝑥2 ] (5) 𝑒⨂ 𝑥1 = 𝑒⨂[𝑥1, 𝑥1] = [𝑒𝑥1, 𝑒𝑥1] (6) to determine the crisp (whitened) value 𝑥(𝛿) for the interval grey number (⨂𝑥 = [𝑥, 𝑥]), the following expression is to be applied [48]: 𝑥(δ) = (1 − δ) 𝑥 + δ𝑥 (7) where δ0,1] and present the whitening coefficient. application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in ... 393 2.5. grey edas method the edas method was presented by keshavarz ghorabaee et al. [54]. the method has so far been applied in its basic form [55,56], as well as in fuzzy environment [57-62], grey environment [63,64] and rough environment [65]. the fig. 5 shows the general steps of the g-edas method. fig. 5 general steps of the grey edas method [64] 394 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana 3. defining the criteria and application of the mcdm model a large number of unmanned aerial vehicles with different characteristics and applications can be found on the market. the military requirements for optimal unmanned aerial vehicles for use in various operations are non-uniform in terms of their tactical, technical and economic characteristics. on the basis of the above, it is defined the goal of selecting the optimal solution (unmanned aerial vehicle) which would best meet the needs of the armed forces in terms of its tactical, technical and economic characteristics. the solution obtained by this research can be used in equipping army units with unmanned aerial vehicles. in the first phase of the application of the mcdm model are defined the criteria influencing the selection of the best ucavs, for the needs of the army's tactical units. the establishment of criteria is rooted in the examination of the existing literature and the insights provided by subject matter experts (table 1). table 1 criteria determining the selection of ucavs criterion name of the criterion type of criteria unit of measure c1 flight autonomy benefit hours (h) c2 maximum flight range benefit kilometers (km) c3 weapon features of ucav benefit linguistic c4 reliability benefit linguistic c5 maximum mass of the payload benefit linguistic c6 proce of one ucav cost us dollars ($) c7 maximum flight height benefit meters (m) c8 maximum speed benefit km/h for the linguistic-type criteria it is used the scale presented in the table 2. table 2 the scale for attribute evaluations ⨂𝑨 [66] scale ⨂𝐴 very poor (vp) [0,1] poor (p) [1,3] medium poor (mp) [3,4] fair (f) [4,5] medium good (mg) [5,6] good (g) [6,9] very good (vg) [9,10] based on the mentioned criteria, six experts defined the comparison values according to all the methods (dibr, fucom, lmaw), after which the experts' viewpoints were consolidated using the bonferroni aggregation method [67]: 𝐵𝑀𝑟,𝑠(𝑥1, 𝑥2 … , 𝑥𝑛) = ( 1 𝑛(𝑛−1) ∑ 𝑥𝑖 𝑟𝑥𝑗 𝑠𝑛 𝑖,𝑗=1 𝑖≠𝑗 ) 1 𝑟+𝑠 (18) application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in ... 395 once the criteria have been established, in the initial phase are calculated their weight coefficients using the dibr approach. implementing the dibr method's procedures, weight coefficients are computed individually for each of the six experts (table 3). in table 3 are further presented the aggregated values of criteria weight coefficients. table 3 values of the weight coefficients of the criteria dibr method criteria e1 e2 e3 e4 e5 e6 aggregated values c1 0.199 0.215 0.229 0.221 0.232 0.273 0.228 c2 0.163 0.169 0.173 0.182 0.174 0.181 0.174 c3 0.151 0.150 0.141 0.155 0.143 0.131 0.145 c4 0.128 0.113 0.111 0.117 0.112 0.103 0.114 c5 0.097 0.096 0.091 0.092 0.096 0.084 0.093 c6 0.082 0.079 0.077 0.075 0.081 0.072 0.078 c7 0.067 0.067 0.066 0.061 0.064 0.059 0.064 c8 0.060 0.057 0.058 0.052 0.052 0.052 0.055 c9 0.053 0.054 0.054 0.045 0.046 0.045 0.049 using the fucom method, the results of the weight coefficients of the criteria are obtained and presented in the table 4. the table 4 also shows the aggregated weight coefficients. table 4 values of the weight coefficients of the criteria fucom method criteria e1 e2 e3 e4 e5 e6 aggregated values c1 0.217 0.239 0.194 0.225 0.196 0.237 0.218 c2 0.168 0.159 0.151 0.150 0.168 0.140 0.156 c3 0.145 0.120 0.109 0.114 0.130 0.119 0.123 c4 0.108 0.109 0.123 0.108 0.117 0.103 0.111 c5 0.094 0.092 0.104 0.100 0.098 0.095 0.097 c6 0.080 0.080 0.091 0.090 0.087 0.085 0.086 c7 0.072 0.073 0.082 0.077 0.078 0.079 0.077 c8 0.062 0.068 0.078 0.073 0.067 0.074 0.070 c9 0.054 0.060 0.068 0.063 0.059 0.068 0.062 using the lmaw method, the results of the weight coefficients of the criteria are obtained and presented in the table 5. the table 5 also shows the aggregated weight coefficients. 396 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana table 5 values of weight coefficients of criteria lmaw method criteria e1 e2 e3 e4 e5 e6 aggregated values c1 0.141 0.143 0.130 0.142 0.127 0.149 0.139 c2 0.134 0.137 0.136 0.128 0.133 0.142 0.135 c3 0.134 0.130 0.129 0.136 0.120 0.126 0.129 c4 0.127 0.129 0.123 0.120 0.127 0.135 0.127 c5 0.119 0.121 0.115 0.111 0.112 0.116 0.116 c6 0.109 0.111 0.115 0.111 0.093 0.126 0.111 c7 0.109 0.100 0.106 0.099 0.104 0.090 0.101 c8 0.085 0.086 0.106 0.086 0.104 0.071 0.090 c9 0.042 0.043 0.040 0.067 0.080 0.045 0.052 the final criteria weight coefficients result from the aggregation of the criteria weight coefficients derived through the dibr, fucom, and lmaw approaches, as illustrated in the table 6. table 6 final values of the weight coefficients of the criteria criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 weight 0.194 0.155 0.133 0.118 0.103 0.092 0.080 0.071 0.054 upon the establishing of the final weight coefficient values of the criteria, the process of selecting the optimal alternative from the set of potential ones is done using the g-edas technique. six experts also participated in the evaluation of the alternatives. aggregation of experts' opinions is carried out using the expression [66]: ⨂𝑥𝑗 = 1 𝐸 (⨂𝑥𝑗 1 + ⨂𝑥𝑗 2 + ⋯ + ⨂𝑥𝑗 𝐸) (19) where e presents the total number of experts. the initial phase in implementing this approach involves formulating the initial decision matrix, as presented in the table 7. the values in the table present the aggregated values of the evaluations of all alternatives according to all criteria by each of the experts, obtained by applying the expression (19). table 7 initial decision-making matrix c1 c2 c3 ... c7 c8 c9 l1 u1 l2 u2 l3 u3 ... l7 u7 l8 u8 l9 u9 a1 27 40 200 300 5.50 7.50 ... 5500 7620 185 220 6.500 9.167 a2 30 42 300 500 6.83 8.83 ... 6000 8530 250 300 8.000 9.667 a3 20 28 1500 1852 4.00 5.00 ... 12000 15240 400 482 6.333 8.667 a4 40 46 900 1100 3.00 4.17 ... 6000 7620 180 217 4.000 5.000 … … … … … … … ... … … … … … … a20 52 60 750 1000 7.00 9.33 ... 7500 10000 250 280 3.333 4.333 a21 33 40 6000 7400 7.83 9.17 ... 10000 14000 350 407 4.500 5.500 a22 32 39 750 1000 6.17 8.17 ... 7500 9900 330 370 7.500 9.500 a23 40 46 20000 22800 4.50 5.50 ... 15000 18000 550 629 6.500 9.167 application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in ... 397 after applying the steps of the g-edas method, the final ranking of the alternatives was obtained, table 8. table 8 rank of the alternatives using the g-edas method alternatives φi ranking alternatives φi ranking a1 0.346 15 a13 0.230 17 a2 0.403 11 a14 0.122 23 a3 0.516 7 a15 0.127 22 a4 0.379 12 a16 0.217 18 a5 0.590 3 a17 0.194 20 a6 0.308 16 a18 0.518 6 a7 0.567 5 a19 0.486 8 a8 0.581 4 a20 0.372 13 a9 0.361 14 a21 0.680 2 a10 0.164 21 a22 0.414 10 a11 0.207 19 a23 0.690 1 a12 0.477 9 4. sensitivity analysis of the model sensitivity analysis typically encompasses the observation of shifts in alternative rankings when modifications are applied to the weight coefficients of the criteria. [68-70]. regarding the changing of criteria weight coefficients, various methodologies exist. in this study, a distinct criterion is highlighted in each scenario. the research outlines and presents ten scenarios involving changes to the weight coefficients of the criteria, as detailed in table 9. table 9 values of the weight coefficients of the criteria in relation to the scenario criteria s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 c1 0.194 0.111 0.304 0.087 0.087 0.087 0.087 0.087 0.087 0.087 0.087 c2 0.155 0.111 0.087 0.304 0.087 0.087 0.087 0.087 0.087 0.087 0.087 c3 0.133 0.111 0.087 0.087 0.304 0.087 0.087 0.087 0.087 0.087 0.087 c4 0.118 0.111 0.087 0.087 0.087 0.304 0.087 0.087 0.087 0.087 0.087 c5 0.103 0.111 0.087 0.087 0.087 0.087 0.304 0.087 0.087 0.087 0.087 c6 0.092 0.111 0.087 0.087 0.087 0.087 0.087 0.304 0.087 0.087 0.087 c7 0.08 0.111 0.087 0.087 0.087 0.087 0.087 0.087 0.304 0.087 0.087 c8 0.071 0.111 0.087 0.087 0.087 0.087 0.087 0.087 0.087 0.304 0.087 c9 0.054 0.112 0.087 0.087 0.087 0.087 0.087 0.087 0.087 0.087 0.304 the results obtained in relation to the defined scenarios are shown in the fig. 6. 398 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana fig. 6 change in the ranks of alternatives depending on the change of the weight in the fig. 7 is displayed the spearman's rank correlation coefficient for the changes in criteria weight coefficients concerning the initial ones. this figure reveals that across all scenarios, the outcomes consistently exhibit a trend towards a positive correlation, aligning closely with the ideal. notably, the most pronounced deviation arises in the scenarios s7 to s10, wherein the priority is predominantly assigned to the criteria initially possessing lower weight coefficients. this particular trend indicates a logical disarray within the system. fig. 7 the values of the spearman’s coefficient of the rank correlations. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 s-0 s-1 s-2 s-3 s-4 s-5 s-6 s-7 s-8 s-9 s-10 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s-0 s-1 s-2 s-3 s-4 s-5s-6 s-7 s-8 s-9 s-10 application of hybrid dibr-fucom-lmaw-bonferroni-grey-edas model in ... 399 5. comparative analysis in order to compare the obtained results with the results of other methods, the following methods are used: gray copras [71], mabac [72], aras [73], cocoso [74], waspas [75], mairca [76], gray topsis [77], gray marcos [78] and gray ocra [79,80]. in the fig. 8 are shown the ranks of alternatives depending on the method used to select the optimal solution. fig. 8 rank of alternatives depending on the mcdm model based on the results shown in the fig. 8, it can be clearly observed that the alternative a23 has a high degree of stability, because it took the first position in applying ten different methods of mcdm, respectively, it presents the optimal solution. the alternative a21, the second on the ranking list, occupies the same position in 9 different methods, while in only one method it took the third place, which can be considered extremely stable rating. 6. conclusion the paper proposes a new hybrid mcdm model integrating the dibr-fucomlmaw methods, the results of which are aggregated using the bonferroni aggregator with the grey edas method. the mcdm model is presented on the problem of selecting combat unmanned aerial vehicles that are in operational use by different armies of the world in order to equip tactical units of the serbian army. the dibr, fucom and lmaw methods are used to determine the weight coefficients of the criteria, previously defined by experts and by analyzing available literature. by applying grey numbers integrated in the edas method, the best alternative is selected from a set of 23 different combat unmanned aerial vehicles. to validate the outcomes generated by the proposed model, a sensitivity analysis is conducted to assess the responsiveness of the output results to variations in 0 5 10 15 20 25 g-edas g-copras mabac aras cocoso waspas mairca g-topsis g-marcos g-ocra a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 400 m. radovanović, d. božanić, d. tešić, a. puška, i. m. hezam, c. jana criteria weight coefficients. additionally, a comparison is made between the results obtained by the proposed model and those provided by other methods. the results stemming from this analysis and comparison collectively affirm the stability of the established model. one of the important limitations of the model is the fact that when defining the weight coefficients of the criteria, possible uncertainties are not considered, because the methods are applied in their basic form (with crisp values). further research should be focused on the application of all presented methods in fuzzy and rough environment, as well as on solving other research issues. acknowledgment: the fifth author, ibrahim m. hezam, was supported 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management and engineering, 6(2), pp. 581-599. 79. ulutaş, a., popović, g., stanujkić, d., karabašević, d., zavadskas, e.k., turskis, z., 2020, a new hybrid mcdm model for personnel selection based on a novel grey piprecia and grey ocra methods, mathematics, 8(10), 1698. 80. stanujkić, d., zavadskas, e.k., liu, s., karabašević, d., popović, g., 2017, improved ocra method based on the use of interval grey numbers, the journal of grey system, 29(4), pp. 49-60. https://doi.org/10.21203/rs.3.rs-1164217/v1 facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 405 414 https://doi.org/10.22190/fume190123033b © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper automatic generation of the plc programs for the sequential control of pneumatic actuators vladislav blagojević 1 , saša ranđelović 1 , vlastimir nikolić 1 , slobodan dudić 2 1 faculty of mechanical engineering, university of niš, niš, serbia 2 faculty of technical sciences, university of novi sad, novi sad, serbia abstract. nowadays, programmable logic controllers (plc) are widely used in many automated systems, especially for the control of various actuators. the most common plc programming is performed by either using a ladder diagram or a structured text. the paper presents the automatic generation of plc programs for the purpose of sequentially controlling pneumatic actuators. in this paper, the pneumatic actuators are supplied and controlled by 5/2-way as well as 5/3-way bistable pneumatic valves with electric activation. the valve type depends on the number of positions in which the actuator should come, and the position sensors are used for detecting its movement. the characteristic encoding of the movement of actuators, position sensors and control commands is performed. the advantages of the automatic generation of the plc commands and the entire program described in this paper are illustrated in a real example. key words: plc, programming, sequential control, actuator 1. introduction wherever it is necessary to automate various processes, programmable logic controllers (plc) are used. they are developed to provide flexibility and replace the old ways of controlling. there are various manufacturers of plcs, such as siemens, omron, festo, mitsubishi, abb, etc. also, there are many different plc types, from small devices in housings integrated with the processor, to large rack-mounted modular devices, with numerous inputs and outputs [1,2]. nowadays, plcs are programmed by adequate pc software. this software enables programming by using a ladder diagram and a structured text. some authors have dealt with the issue of automatic plc programs generation using adequate pc software as well as sequential control systems. dworzak and mikulczyński received january 23, 2019 / accepted july 25, 2019 corresponding author: vladislav blagojević university of niš, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: vlada@masfak.ni.ac.rs 406 v. blagojević, v. nikolić, s. ranđelović, s. dudić [3] introduced a new method for the synthesis of sequential control algorithms for pneumatic actuators controlled by monostable valves. malayappan el al. [4] used a cascading method in designing a sequential control system for an industrial robot. mroz and brol [5] developed a sequential control strategy for a pneumatic-hydraulic actuator, as well as its simulation model. they even increased the energy efficiency of the proposed system. bayoumi [6] presented a new tree-chart method to design a sequential logic controller with or without an intermediate stop of an actuator. hasdemir and kurtulan [7] and salunke et al. [8] introduced a methodology for expressing an automaton in logical domain in the matlab program that automatically generates plc commands. guttel et al. [9] focused on the safety functions, start-up and shutdown sequences of the control code and strived for a concept to automatically generate that part of the control. schumacher and fay [10] and schumacher et al. [11] used grafcet as a modeling language for automatic generation of the control code. julius et al. [12] made a systematic approach to automatically transform grafcet into the plc code while retaining the hierarchical structures. wcislik et al. [13,14] presented a new algorithm for the programming of sequential control systems in the labview software. they implemented the language of function block diagrams in plcs. adiego et al. [15] introduced a general automated methodology for formal verification of plc programs. the methodology is based on an intermediate model, used as a pivot between all the plc and formal modeling languages. qamsane et al. [16] presented a model-driven engineering method, which aims at automatically generating distributed plc-based control of automated manufacturing systems in the form of grafcet. blagojevic et al. [17] presented an algorithm for the plc programming of the sequential control systems of pneumatic actuators with a single branch and only two end positions. this paper presents a new way of automated plc programming of sequential pneumatic asynchronous automata control systems with parallel and single branches. this way of the plc programming enables a unique designation of pneumatic actuators and their movement according to certain established technological processes in steps, in order of their execution. the advantage of the algorithm is a possibility to control the stopping of actuators not only in the final but also in the middle positions. furthermore, it is possible to automatically generate plc commands and the entire program for a sequential pneumatic control system with parallel and single branches. microsoft office excel was used for the implementation of the automatic generation of plc commands. 2. pneumatic actuator system usually a pneumatic actuator system consists of a pneumatic actuator and a 5/2-way bistable direction valve, for the simple actuator movement only between end positions, fig. 1, and a 5/3-way bistable direction valve, for the actuator movement and stopping in the middle positions, fig. 2. therefore, two command signals are required for each actuator, yi+ and yi-, for the movement of the actuator in one and the opposite direction, where i=1, 2, ..., n is the actuator number. every actuator has two end position sensors and in some cases, more middle position sensors. position sensors are used to detect the movement and position of the actuator. for pneumatic actuator ci with two end positions, signal xi0 represents the initial position and xi1 the final position, fig. 1. if actuator ci has to stop somewhere in the middle automatic generation of plc programs for the sequential control of pneumatic actuators 407 positions of its stroke, there is a need for p position sensors, p>1. the signal from position sensor xi0 is for the initial position, xi1, xi2, ... xi(p-1) are for the middle positions and xip is for the final position, fig. 2. fig. 2 pneumatic system with more than two end positions with: a) symmetric actuator, b) asymmetric actuator, c) semi-rotary drive 3. sequential control pneumatic system a sequential control system is a control system with individual steps, which are processed in a predetermined order, where the progression from one step to the next depends on the matching defined conditions. if pneumatic actuators are used to perform the operations, these systems are called sequential control pneumatic systems. the pattern of a sequential control system can have only a single branch, fig. 3a, parallel branches, fig. 3b, or a combination of the previous two. fig. 1 pneumatic system with two end positions with: a) symmetric actuator, b) asymmetric actuator, c) semi-rotary drive 408 v. blagojević, v. nikolić, s. ranđelović, s. dudić fig. 3 sequential control system with: a) single branch, b) parallel branches in a pneumatic sequential control system with a single branch and with several different pneumatic actuators, every step consists of one movement of the actuator [17]. in a sequential control system with parallel branches it is possible to simultaneously execute more steps. in pneumatic sequential control systems, the designation of the movement of actuators between only two end positions is:  ci+, for the extend direction of movement for linear actuators and the right direction of movement for semi-rotary drives, and,  ci-, for the retract direction of movement for linear actuators and the left direction of movement for semi-rotary drives. if actuator ci has to stop somewhere in a middle position of its stroke, the designation of its movement is:  ci+j, for the extend direction of movement for linear actuators and the right direction of movement for semi-rotary drives, and,  ci-j, for the retract direction of movement for linear actuators and the left direction of movement for semi-rotary drives, where j=0, 1, …, p is the number of the position sensor that has to be reached by the actuator. the minimum number p=1 for only two end position sensors of the actuator, and p>1 if there are some position sensors in the middle of the actuator stroke. this paper considers the case of sequential control of the pneumatic actuators with two end positions as well as several middle positions. the pattern of the actuator movement is a combination of systems with a single and several parallel branches. 4. automatic generation of a plc program automatic plc programming of the sequential control of pneumatic actuators consists of three steps, fig. 4. the first step is to describe the actuator movement, the second step is to generate input and output addresses for sensor and command signals, and the final step is to generate plc commands and the entire program. automatic generation of plc programs for the sequential control of pneumatic actuators 409 for the automatic generation of plc commands and the entire program it is necessary to perform the exact description of the actuator movement according to sequential control, step by step. the best way is by using tables, where the number of sequential control branches is equal to the number of columns and the number of sequential control steps is equal to the number of rows, fig. 5. the movement of each actuator is entered into one cell of the table. also, it is very important to describe the actuator movement, as it was explained in chapter 3, with ci+j and ci-j, where the first two characters designate the number of actuators, the “+” or “-“ sign stands for the direction of the actuator movement and the last symbol representing the number of the position sensors that have to be reached at the end of the actuator movement. it is important that every actuator must return to the start position at the end of the whole sequential control cycle. fig. 4 block scheme of automatic plc programming of the sequential control of pneumatic actuators this way of describing the movement of actuators enables an easy generation of the output command signals, from the first three characters, and the input signals from the last character. fig. 5 table for describing actuator movement after the first step of automatic plc programming of the sequential control of pneumatic actuators, based on the given movement of the actuators and the analysis of every single character of description, in the second step it is possible to generate the necessary input and output addresses for the sensor and command signals. in this paper, the designation of input signals is ix.x and output signals is ox.x, from the festo standard for plc programming. the character or characters between character “c” and signs “+” or “-“, in the description of the actuator movement from the first step, stands for 410 v. blagojević, v. nikolić, s. ranđelović, s. dudić the number of actuators. now it is very easy to generate a list of output addresses for command signals for each actuator, for example: if “ci+p” is written then the output signal is “yi+”, or if “ci-p” is written then the output signal is “yi-”, where i is the actuator number and p is the sensor number. for example, the list of output addresses is: o0.1 y1+ o1.1 y1 o0.2 y2+ o1.2 y2 … o0.n yn+ o1.n yn the characters after signs “+” or “-“, in the description of the actuator movement, represent the number of the position sensors that have to be reached at the end of the actuator movement. after analyzing the number of actuators and the number of position sensors, it is possible to generate a list of input addresses from sensors for each actuator, for example: if “ci+p” or “ci-p” is written then the input signal is “xip”, where i is the actuator number and p is the sensor number. for example, the list of input addresses is: i0.1 x10 i1.1 x20 … in.1 xn0 i0.2 x11 i1.2 x21 … in.2 xn1 … i0.p x1p i1.p x2p … in.p xnp the final step in automatic plc programming of sequential control is generation of plc commands and the entire program. analyzing the description of the movement of actuators from the table of the first step, fig. 5, it is possible to detect the conditions for executing each step of sequential control. the condition for the execution of the next step of sequential control is that the previous step is finished. the verification of the completion of the previous step consists of detecting the existence of the signal of the sensor that has to be reached in the previous step. for example, if there are two steps, step 1: c1+1 and step 2: c1-0, then the condition for the execution of step 2 is the existence of the signal of sensor x11, and the execution command for step 2 is y1-. in a sequential control system with a single and parallel branches the condition for the execution of the next step, if there are parallel branches before it, is a logical product of all signals of the sensors that are reached in the previous step. for example, if there are two steps, where step 1 consists of three branches, step 1: c1+1 c2+2 c3+1, and step 2: c2-0, then the condition for the execution of step 2 is the existence of all the signals of sensors x11, x22 and x31, and the execution command for step 2 is y2-. after detecting all conditions from the sequential control cycle, it is possible to generate a plc program. the best way for generating a plc program is in the form of structured text by step operation, supported by the iec 61131-3 standard [2,18]. for this purpose, the syntax of the structured text of the plc program is: step k if signal from the position sensor of the previous actuator movement exists then set command for equal actuator movement in the right direction reset command for equal actuator movement in the opposite direction where k is the number of the step. https://en.wikipedia.org/wiki/iec_61131-3 automatic generation of plc programs for the sequential control of pneumatic actuators 411 in the first step of the plc program there is always a minimum of two conditions. the first condition is start or start conditions and the other condition depends on the sequential control of actuators. at the end of the plc program, after the final step operation, the jmp to step 1 command is generated. this command enables the program to return to the start of the first step. 5. automatic generation of a plc program using microsoft office excel in this paper, microsoft office excel is used for the automatic generation of plc commands. the main reason for using excel is that sequential control steps are very easy to describe by a spreadsheet, which is adjustable for text and numeric processing as well. also, there are many commands that are useful in processing the description of the movement of actuators, recognizing the number of actuators, direction of movement and number of position sensors. finally, microsoft office excel can be applied on windows, macos, android and ios platforms, and it is available on various devices, such as pcs, tablets, smartphones, etc. automatic plc programming of the sequential control of pneumatic actuators by excel consists of three parts, fig. 6. fig. 6 parts of automatic plc programming by excel the first part is the sheet for describing the actuator movement according to some sequential control. the second part consists of two sheets for generating input and output addresses for sensor and command signals. the last sheet is used for generating plc commands and program. in the “sequential control description” sheet, the operater describes the actuator movement according to a sequential control system. by recognizing the characters between “c” and signs “+” or “-”, the program detects the actuator number, and after signs “+” or “-” the sensor number. after that, the program prints the input and output addresses for sensors and command signals in the “input addresses” and “output addresses” sheets. finally, the plc commands and the entire program are generated in the “plc program” sheet. when the generation of the input and output addresses and the plc program sheet are completed by excel, it is very easy to copy and paste those sheets to adequate software for plc programming, such as fst or codesys for festo plcs, simatic for simens plcs or any other software for programming the various plcs. it is only important to use an adequate designation for input and output addresses depending on the plc type. https://en.wikipedia.org/wiki/microsoft_windows https://en.wikipedia.org/wiki/macos https://en.wikipedia.org/wiki/android_(operating_system) https://en.wikipedia.org/wiki/ios 412 v. blagojević, v. nikolić, s. ranđelović, s. dudić 6. example of the automatic generation of a plc program by microsoft office excel as an example of the automatic generation of a plc program by excel, the problem of programming the movement of four pneumatic actuators in a sequential control system with single and parallel branches is considered. actuators c1 and c4 have two end positions and actuators c2 and c3 have more middle positions to stop. the movement of the actuators is as follows: step 1 step 2 step 3 step 4 step 5 step 6 step 7 step 8 c1+1 c2+3 c2-1 c3+2 c3-0 c3+1 c3-0 c1-0 c2+2 c2-1 c2+2 c2-0 c4+1 c4-0 in this example, the four pneumatic actuators move in eight steps, where from step 1 to step 3 there is only one branch, from step 4 to step 5 there are two branches, and from step 6 to step 7 there are three sequential control branches. at the end of the sequential control system there is only one branch with step 8. the pneumatic actuator type does not influence the plc program because in this paper all of them need two command signals, due to the same type of the 5/2 or 5/3-way bistable direction valve for supplying them. the actuator type only affects the pneumatic scheme. the first step in the plc automatic programming by excel is to fill the “sequential control description” sheet, in a way that was explained in chapter 5, fig. 7. fig. 7 “sequential control description” sheet in excel by processing the characters from the “sequential control description” sheet, excel prints the input and output addresses for sensors and command signals in the “input addresses”, fig. 8, and “output addresses” sheets, fig. 9. automatic generation of plc programs for the sequential control of pneumatic actuators 413 fig. 8 “input addresses” sheet in excel fig. 9 “output addresses” sheet in excel based on the input and output addresses for sensors and command signals from the “input addresses” and “output addresses” sheets, as well as the description of the movement of actuators from the “sequential control description” sheet, and processing the condition for the execution of the next step of sequential control, the “plc program” sheet and the plc commands and program are generated, fig. 10. fig. 10 “plc program” sheet in excel 7. conclusion the automatic generation of plc commands and the entire program enables the operator to easily describe the movement of the actuators according to a certain established technological process in the way of sequential control by characteristic encoding. this way of programming plcs is especially suitable for pneumatic systems. it enables the generation of a plc program for a sequential control system with a single as well as parallel branches, by recognizing the conditions for the execution of individual steps. this paper presents implementation of the automatic generation of a plc program in microsoft office excel. this manner of plc programming by excel offers many advantages and is available on various devices such as pcs, tablets and smart mobile phones with the windows operating systems, macos, android and ios. 414 v. blagojević, v. nikolić, s. ranđelović, s. dudić references 1. laughton, m.a., warne, d.j., 2002, electrical engineer's reference book, newnes, oxford, england, 1504 p. 2. thakur, n., hooda, m., 2016, a review paper on plc & its 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science (acecs 2017), 19, pp. 4-11. 17. blagojević, v. ranđelović, s., milanović, s., 2018, automatic generation of plc programs for pneumatic actuators sequential control with two end positions, proc. xiv international saum conference on on systems, automatic control and measurements, niš, serbia, cd. 18. bryan, l.a., bryan, e.a., 1997, programmable controllerstheory and implementation, industrial text & video company, atlanta, 202 p. https://ieeexplore.ieee.org/author/37691579500 https://ieeexplore.ieee.org/author/37691579500 https://ieeexplore.ieee.org/author/37409207200 facta universitatis series: mechanical engineering vol. 16, no 3, 2018, pp. 321 335 https://doi.org/10.22190/fume180823032g © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper customization of electrospinning for tissue engineering  udc 678.3:66.095.26 nenad grujović 1 , fatima živić 1 , matthias schnabelrauch 2 , torsten walter 2 , ralf wyrwa 2 , nikola palić 1 , lazar ocokoljić 1 1 university of kragujevac, faculty of engineering, kragujevac, serbia 2 innovent e. v., biomaterials department, jena, germany abstract. this paper deals with two electrospinning technologies: the melt electrospinning with a customized jet head, adapted from the fused deposition modeling (fdm) 3d printer, in comparison with the standard solution electrospinning, aiming at fabrication of tissue engineering scaffolds. the resulting fibers are compared. the influence of the collector properties on those of the fabricated scaffold is investigated. the resulting electrospun fibers exhibit different characteristics such as morphology and thickness, depending on the technology. the micro-fibers are produced by the melt electrospinning with an inbuilt 3d printer jet head, whereas the solution electrospinning has produced nanoand micro-fibers. the scaffolds fabricated on the rotating drum collector exhibit a more ordered structure as well as thinner fibers than those produced on the stationary plate collector. further investigations should aim at fabrication of porous hollow fibers and tissue engineering scaffolds with controlled porosity and properties. key words: melt electrospinning, solution electrospinning, 3d printing, micro-fibers, nano-fibers 1. introduction the spinning process was patented by anton and formhals (1934) who described an experimental setup for the production of polymer filaments using an electrostatic force. electrospinning represents a process for fabrication of polymer nano-fibers by using an electrostatically driven jet of polymer solution or polymer melts [1]. since 1994, when this technology was put to practical use, electrospinning has been recognized for its ability received august 23, 2018 / accepted november 23, 2018 corresponding author: fatima živić university of kragujevac, faculty of engineering, sestre janjić 6, 34000 kragujevac, serbia e-mail: zivic@kg.ac.rs 322 n. grujović, f. živić, m. schnabelrauch, t. walter, r. wyrwa, n. palić, et al. to fabricate nanoand micro-fibers in different forms and of different morphologies [2]. the human extracellular matrix contains submicron fibers and the samples created by electrospinning can closely resemble that network, with a high-specific surface area, which is very important for many applications. electrospinning has attracted great attention in biomedicine mainly due to: versatility of the technology, ease of use, low cost of the custom made devices, and especially because it can fabricate diverse forms of nano and micro-fibers [3]. these fibers have been used for tissue engineering scaffolds, to provide different functions [3, 4]; possible fiber structures, morphologies and properties are widely investigated. electrospun scaffolds can provide a conductive microenvironment for the selected cells thus enhancing tissue regeneration [4, 5]. there is evidence that electrospun biomaterial samples exhibit rather good mechanical stress resistance and enhance adhesion of cells while their differentiation and proliferation due to the fiber network structure resemble human tissue [6]. the advantages of electrospinning enable its use in different biomedical applications, such as drug release and biotransformation, soft tissue implants and wound healing [7]. electrospun nanofibers as a drug delivery system show several benefits such as easy modulation of the drug release profile depending upon the properties of the used polymer or polymeric blends, such as antibiotics-loaded nanofibers as wound dressing materials [8]. two main methods are widely used (solution and melt electrospinning); the influence of the technological parameters on the properties of the final samples is studied beside input materials as well as possible fiber fillers since the complexity of intermixed influences still prevents this technology from wide practical applications [9, 10]. thakkar and misra [8] investigated the melt and the solution electrospinning methods for producing curcumin-loaded poly (pcl) fibers in order to find difference between those two methods in characterization and drug release. the results showed that the influence of the curcumin was not the same in both of these electrospinning techniques, thus indicating the necessity to consider the production parameters, beside material properties of the starting polymer. in this case [8], variation of the curcumin content did not change the morphologies of the melt electrospun fiber. however, for the solution electrospun fiber, it resulted in a different set of fiber diameters due to the influence of the curcumin content on the jet balance. a different production technology resulted in a different drug release rate with the same input polymer material. additive manufacturing has emerged as the groundbreaking technology affecting many areas due to its possibility to fabricate versatile forms of structures. fabrication of complex 3d structures also by combining several materials is especially important in tissue engineering [11] even though 3d printing is still under development regarding available materials [12]. additive biomanufacturing is an important area of research pertaining to tissue regeneration [13], control of fiber diameter and micromechanics [14] or distribution and size of pores within fabricated scaffolds for implantable medical devices [15]. furthermore, combination of 3d printing and electrospinning has been investigated in fabrication of vascular grafts [16], complex functionalized three-dimensional structures [17] and three-dimensional micro-patterns within 3d structures [18]. electrical instabilities exhibited in the solution electrospinning can be avoided by using melt materials and 3d printing. in the solution electrospinning, the produced fibers randomly fall onto the collector under the influence of the electric field due to these instabilities, thus preventing precise control of the resulting fiber mat structure. furthermore, the solvents often exhibit toxicity which is not related to melt materials. the melted material does not exhibit these electrical customisation of electrospinning for tissue engineering 323 instabilities and 3d printing additionally enables different shapes, structures, even the complex 3d structures to be easily fabricated. this is of the utmost importance in tissue engineering and mimicking of natural tissues. a significant area of application is the production of highly porous hollow fiber membranes whereas the electrospinning technology is modified by using different jet heads and collectors [19, 20]. a high porosity scaffold (83.5%) was fabricated by using a porous supporting tube as the collector for nanofibers [19]. biomimetic scaffolds were fabricated by using a complex collector with a stationary cylindrical hollow piece and a mobile internal cylinder which enabled circumferential and radial alignment of fibers [20]. some recent applications of 3d printing are for production of complex shapes of collectors in order to enable novel shapes of electrospun fibrous mats [21], multiplexed electrospinning sources [22], structuring of scaffolds [23], or planar arrays of electrospinning emitters [24]. development of different electrospinning techniques (melt and solutionelectrospinning, force-spinning, melt-blowing, flash-spinning, spinning of bicomponent fibers) has become a hot topic due to versatile applications of microand nanofibers (tissue engineering, sensors, clothing, membranes for filtration systems, fillers in composite structures, coatings, etc.) [25]. materials used in electrospinning are mainly polymers but other materials have been recently investigated such as ceramics. nanofibers made of ceramic are rather hard to produce due to many limitations of these materials but electrospinning has shown excellent results; this is especially important if wide applications of ceramic nanofibers are considered (membranes in filtration systems, fuel cells, smart textiles for wearable technologies, sensors, catalysts and many more) [26]. chemical sensors based on functional nanofibers made by electrospinning (e.g. in wearable sensors) are also a significant area of research [27]. an interesting application of this technology is related to piezoelectric polymers for selfpowered sensors [28, 29]. electrospun fibers of the piezoelectric polymers have been investigated for wearable electronics such as medical wearable active sensors for the monitoring of respiration or other health related parameters (vital signs, walking speed, gesture detection, or detection of vocal cords vibration, etc.). nanofibers of the piezoelectric polymers have been further used in fabrication of smart (functional, active) textiles for a wide range of wearable sensors, such as sensors for vital signs monitoring [30], pressure sensors [31], and tactile sensors [32]. these types of sensors are lightweight and have very high sensitivity thus representing very significant elements in many different applications. for example, these capacitive pressure sensors can detect a water droplet down to 7 mg weight, which makes them highly suitable for wearable electronics in medical applications [31]. piezoelectric smart fabrics can provide pressure and force sensing, motion and ultrasonic sensing; special yarns have been investigated [33, 34]. for example, these smart fabrics, with embedded electrospun piezoelectric nanofibers, are able to detect even small amounts of ammonia (nh3) very fast and they exhibit rapid recovery afterwards [34]. this paper elaborates upon design elements of two electrospinning technologies: the melt electrospinning with a customized jet head, adapted from fused deposition modeling (fdm) 3d printer, in comparison with the standard solution electrospinning, and resulting differences in electrospun fibers, aiming at fabrication of tissue engineering scaffolds. the influence of the collector properties on the fabricated scaffold properties was also investigated. 324 n. grujović, f. živić, m. schnabelrauch, t. walter, r. wyrwa, n. palić, et al. 2. experimental setup 2.1. materials and fabrication method three different electrospinning setups are used: the melt electrospinning with an adapted head of the fused deposition modeling (fdm) 3d printer and the standard solution electrospinning with two types of collectors (a stationary plate collector and a rotating drum collector). the nozzle tip orifice at the melt spinning with a fdm jet head has diameter of 0.3 mm. for the solution electrospinning, the conventional 1 ml syringe with a blunt tip needle (0.603 mm, inner diameter of the needle) is used. the needle temperature is considered to be the same as the ambient temperature since it is not significantly heated due to a large distance of the needle from the pump which is also not heated. biodegradable polycaprolactone (pcl) is used for the melt electrospinning and fp2 fluoropolymer, 15 wt% in 1 part dmac (dimethylacetamide) and 6 parts acetone are used for solution electrospinning with a plate collector; plga: plg8523, poly-lactide (co-glycolide), 8 wt% in hfip (hexafluoroisopropanol). parameters of different electrospinning methods are as follows:  melt electrospinning with a plate collector: voltage on the plate collector of 30 kv; distance from the nozzle tip to the collector surface of 13 cm.  solution electrospinning with a plate collector: voltage of 30 kv; polymer solution flow rate of 1.5 ml/h; distance from the needle tip to the collector surface of 200 mm; 120 mm diameter of the disk above the needle; and 25˚c temperature within the electrospinning chamber. 0.5 ml of the polymer solution is used as the starting material.  solution electrospinning with a drum collector: voltage of 30 kv, polymer solution flow rate of 1.5 ml/h; air gap at the syringe/collector distance of 250 mm and 25˚c temperature within the electrospinning chamber; 0.5 ml of the polymer solution is used as the starting material. five different drum rotation speeds are studied: 250, 500, 750, 1000 rpm. 2.2. electrospinning technology there are two basic types of electrospinning devices, namely those using: 1) the melt electrospinning technique, where the polymer wire flows through the nozzle and melts thus making the polymer fiber that falls down to the collector, and 2) the solution electrospinning technique with a liquid polymer mixture as the starting material that flows through the syringe with a needle. there is a difference between the devices where the fiber is fabricated via melting the polymer through the nozzle, and the other type where the fiber is obtained from a homogenous polymer solution. these two techniques differ in sample characteristics such as morphology, thickness of electrospun fibers, transparency, etc. selection of the appropriate technique depends on the material used. for example, some polymers worsen their properties when exposed to high temperatures hence the melting technique is not applicable. also, fiber thickness is determined by different techniques. special attention should be devoted to the adequate polymer starting form, depending on the electrospinning process that is used. in the case of the solution electrospinning, voltage is brought to the needle and the liquid polymer is put into the syringe, which is further mounted on the element that pushes the syringe plunger with precisely defined speed. the quantity of the polymer melt that is expelled through a discharge orifice at the open end of the tube and customisation of electrospinning for tissue engineering 325 into the needle is determined by the linear speed of the plunger, as setup at the beginning of the process. in the case of the melt electrospinning, voltage is brought to the collector and the polymer wire is brought to the nozzle and fed into it by the step motor with predefined speed. the polymer wire melts inside the nozzle whereas the temperature within the nozzle can be controlled and is setup according to the polymer type. in both of these methods, the polymer droplet created at the tip of the needle/nozzle is further electrified and elongated to form a thin fiber, as it falls through the electric field. the distance between the needle/nozzle tip and the collector as well as the value of electric field, determine the structure and the thickness of the electrospun fiber. the speed of the polymer passing through the needle/nozzle also has significant influence on the final fibers properties. one of the main issues in electrospinning is poor repeatability of results [3, 9]. it is not possible to predict the results with high certainty even with the same materials and process parameters; hence it is necessary to constantly monitor the process. in the case when the distance between the needle/nozzle and collector surface is too large, or the electric field is not well adjusted, the fibers can fly away from the projected collecting zone on the collector and fly randomly around the chamber. very often, different final fiber mats are produced with the same process parameters and materials. on the other hand, even small variations in fiber morphologies and structure can significantly affect final scaffold function, especially for its further nano scale responses. if the polymer solution density is not well adjusted, it can result in polymer drops directly falling on the collector without forming fibers. in the case of high viscosity polymers, such issue is not present. however, high viscosity of polymers can produce random abrupt stopping and starting of the pump which pushes the polymer solution through the syringe into the needle, thus resulting in excessive quantities of the polymer solution fed into the syringe. the issue of the possibility of such large droplets falling onto the collector and damaging the fiber mat is solved by manual removal of the droplets from the needle by using a plexiglas rod. this is one common reason for continuous monitoring of the electrospinning process. sometimes, in the case of very high viscosity polymer solutions (such as honey viscosity), the syringe can bend under the large force exerted upon it by such "thick" fluid flow. 2.2.1. melt electrospinning in the case of the melt electrospinning technique, a polymer wire flows through the nozzle and melts, thus making a polymer fiber that falls down to the collector (fig. 1). this concept with the polymer wire moving and melting through the nozzle represents the same concept as used in the 3d printer. the wire is pushed through the nozzle by a step motor while the controlled heater warms up the nozzle. 3d printer heads can be used in combination with the electrospinning chamber and such custom made devices for production of fibrous scaffolds have been recently investigated [13, 14, 15]. one such electrospinning device was designed and presented in this paper. a nozzle hole diameter through which the wire is melted is around 0.3 mm. voltage brought to the collector is of order of several 10 kv. in that way, a strong electrical field is formed, which initiate the attraction of the polymer fiber by the collector. delivering of voltage to the collector is done by an electrical transformer. the main elements of the melt electrospinning device are given in fig. 1a. 326 n. grujović, f. živić, m. schnabelrauch, t. walter, r. wyrwa, n. palić, et al. fig. 1 main elements of the melt electrospinning device: a) schematic representation of the process, b) design elements for this type of device, it is very important to adequately setup the parameters according to the specific test. if the melting temperature is too low, the material will melt but it will also stay too viscous to extract the fiber. on the other hand, if the temperature is too high, the material will burn. this is especially important in the case of different polymer mixtures (such as mixtures containing antibiotics) because improper work parameters can have a negative effect on some of the polymer properties. possible issue for this type of device is related to the wire that is used as starting material and which is made by using an extruder thus raising technology costs. one possible design of the melt spinning device is given in fig. 1b. 2.2.2. solution electrospinning the solution electrospinning technique uses a liquid polymer mixture as the starting material. the material is in the syringe connected to the needle which is located within the electrospinning device. the needle is within the bearing which is situated in the metal disk where voltage is delivered. the main elements of the solution electrospinning device are given in fig. 2a. voltage is delivered to the needle (via the metal disk) unlike the previous design where voltage is delivered to the collector. the size of the disc directly influences the size of the collector area where the sample will be created (the greater size of the disk, the greater size of the collector area). amount of the polymer mixture which goes through the needle is regulated by the module in which the syringe is installed and the flow parameters are setup. depending on the input parameters of the process, the polymer solution is pressed through the syringe. it is important that the flat top needle is used to prevent accumulation of fibers at only one side of the collector as would be in the case of the medical bevel needle. one possible design of the solution electrospinning device is given in fig. 2b. customisation of electrospinning for tissue engineering 327 fig. 2 main elements of the solution electrospinning device: a) schematic representation of the process, b) design elements 2.2.3. customization of electrospinning device an essential element of the electrospinning device is the electrospinning head (or jet head). different jet heads are used in the melt and the solution electrospinning: a nozzle (to provide wire or melt feeding) or a syringe (to provide solution flow). different designs of the jet head are investigated aiming to provide efficient material feed, including composites [22, 24]. electrospun fibers have been investigated for different biomedical applications, such as drug delivery. composite nanofibers (or core – shell systems) represent a novel direction of research in electrospinning techniques, based on the custom design of jet head and collectors. these composite nanofibers comprise the core made of one material, covered with a thin layer (shell) made of different material. special design of the element used in electrospinning device (a multi-jet electrospinning head) which introduces two different polymers into the electric field is shown in fig. 3. the needle (with one type of polymer solution – inner fluid) is located within the nozzle (with another type of polymer solution – outer fluid). altogether core – shell output droplets and subsequent fibers are fabricated within the electric field. diameter of the core zone and thickness of the shell cover are regulated and controlled by the speed of simultaneously feeding the polymers into the needle and the nozzle. different combinations of custom-design jet heads and collectors can produce versatile final fiber structures, such as hierarchically structured fibers [35], nanofibers with -shaped shell [36], or 3d nanostructured scaffold with 99.98% porosity [37]. investigation of different designs of device elements and literature review pointed out that the melt electrospinning should be used for production of micro-fibers, whereas the solution electrospinning can efficiently produce nano-fibers. one of the special designs of the jet head, as given in fig. 3, can be used to produce core-shell fiber structures. additional customization of the collector shapes, numbers and motion, can produce different final shapes of the fiber mat. depending on the final application, several different collectors can be used, some of which are shown in fig. 4. 328 n. grujović, f. živić, m. schnabelrauch, t. walter, r. wyrwa, n. palić, et al. another element of the electrospinning device, that can be custom shaped, is the collector where the fibers are collected after their forming within the electric field. a fine porous scaffold (fiber mat) is fabricated on the collector surface. the type of the collector used in the process has significant influence on the distribution of the fibers, and further on the porous scaffold sample. different types of collectors can be used for electrospinning process and new designs have been investigated, especially in combination with multiple electrospinning sources [22, 24]. high voltage brought to the collector (or the needle) creates a very strong electric field which elongates the droplet and creates fibers that are further attached to the collector. the sample is formed on the surface layer of the collector which is different from the bulk material of the collector in order to better control the final sample output. plate collectors and rotating (drum) collectors are standard forms whereas several new shapes and types of movements are currently under investigations [20, 22, 37]. plate collectors can be fixed with no motion, or they can exhibit motion, such as simple translational motion in horizontal plane (fig. 4a). translational motion produces a larger sample, whereas uniform speed results in homogenous density. speed and trajectory can be predefined and controlled. variable speed produces different thickness distribution of the sample along the sample. the spots where the collector is fixed by screws can sometimes exhibit thicker samples due to an increased quantity of metallic material in these spots thus resulting in accumulation of fibers. usually, a glass layer is installed on the plate collector in order to allow an easier control of the sample. design of the drum collector is given in fig. 4b. it rotates during electrospinning, thus creating a scaffold in the form of a round hollow pipe. rotation speed of the collector directly affects the thickness of the scaffold. aluminum foil is usually fixed on the collector's surface to enable removal of the sample afterwards. also, the collector can be designed to create tube samples, like the one shown in fig. 4c. this collector operates similarly to a drum collector. the cylinder diameter is much smaller fig. 3 multi jet electrospinning head used for composite nanofibers production fig. 4 different types of collectors: a) plate collector with directions of translational motion (variable speed), b) drum collector, c) design of the tube-form sample collector customisation of electrospinning for tissue engineering 329 than at the drum collector, thus resulting in rapid accumulation of fibers. this type of the collector has been used to produce scaffolds made of biodegradable polymers aiming at blood vessels repair. an electrospun scaffold can be used for coating a damaged vessel until its healing. 3. results and discussion a new approach to the custom design of the electrospinning device is to combine 3d printing and electrospinning techniques into one device or to use it parallel to each other in the synchronized manner [16, 17, 18]. in our work, the head of the fused deposition modeling (fdm) 3d printer was used instead of the standard melt electrospinning jet head. the main elements of such jet head are shown in fig. 5a. one fiber mat sample made at the adapted electrospinning device is shown in fig. 5b. this device setup has been used to overcome the recognized issues in fabrication of drug-impregnated fibers for drug delivery systems. drugs in the pre-fabricated drug-loaded wire can be damaged by a high melt temperature within the jet head in the case of the standard melt electrospinning device. the jet head adapted from the 3d printer also melts the wire and produces an initial polymer drop to enter the force field but it exhibits adequate temperatures thus preserving initial properties of the drug, which is further transferred into the electrospun fibers. a stepper motor is used to feed the wire into the nozzle where it melts. this device setup is well suited for drug delivery applications where larger fibers (micro to macro fibers) are needed. the wire that was fed into the adapted jet head is custom made of pcl biodegradable polymer, by using a wire extruder. fig. 5 a) main elements of the jet head in melt electrospinning device, by using adapted 3d printer head and b) sample produced at such adapted device (pcl micro-fibers) thin polymer fibers are made during electrospinning under the influence of electric force which elongates the initial polymer droplet during its falling from the needle/nozzle tip to the collector. fibers are continuous until the polymer is fed to the needle/nozzle and electric force is acting on it. in general, final fibers are randomly distributed over the collector’s surface. thickness and morphologies of the fibers are different if the meltand the solutionelectrospinning are compared. the melted polymer has higher viscosity than its solution and the nozzle orifice is larger than the needle tip. accordingly, the thickness 330 n. grujović, f. živić, m. schnabelrauch, t. walter, r. wyrwa, n. palić, et al. of the fibers made by the melt electrospinning tends to be larger (micron scale) than those made by the solution electrospinning (nano to micron scale). fp2 fluoropolymer fibers made by using the solution electrospinning with a plate collector (fig. 4a) are shown in fig. 6. fig. 6 fp2 fluoropolymer fibers produced by solution electrospinning, collected on the plate collector, two magnifications there is a clear difference between the size of the electrospun fibers produced at the solution spinning (fig. 6) and at the adapted electrospinning device (fig. 5b). the size of the nozzle tip orifice (0.3 mm) has a comparable length scale to the standard inner diameter of the syringe needle (0.603 mm). the adapted setup offers more possibilities for precise tuning of the fiber properties in the case of micro/macro fibers, and, accordingly, of the resulting fibrous scaffold. our work is fully in accordance with several authors showing that the fdm 3d printer head can be used in the melt electrospinning, especially important for composite fibers impregnated with different materials [13, 14, 15]. the 3d printer head enables a customized input material since we have produced the wire by using the extruder which enables different material combinations, including drug loading of the wire. fiber diameters and fiber density are under investigation for different applications while the combination of fdm 3d printing with electrospinning can provide good tailoring of properties at the micro/macro levels. combination of different materials, or synchronized use of additive manufacturing technologies, together with customization of elements in the electrospinning device, have been investigated in order to provide better control and repeatability of the process. beside device elements (jet head, type of collector), process parameters significantly influence the resulting electrospun fibers. the change of the collector shape (the drum instead of the plate collector) and rotation speed of the drum collector (250, 500, 750, 100 rpm), results in changes of the fiber thickness and their spatial distribution within the scaffold, as shown in fig 7. customisation of electrospinning for tissue engineering 331 fig. 7 diameter of plga polymer fibers fabricated by solution electrospinning, depending on the rotation speed: a), b) 2.4 µm, 250 rpm; c), d) 2.2 µm, 500 rpm; e), f) 2.2 µm, 750 rpm; g), h) 2 µm, 1000 rpm 332 n. grujović, f. živić, m. schnabelrauch, t. walter, r. wyrwa, n. palić, et al. the major influence of the rotation speed change is on the thickness of the produced fibers. the influence of the rotation speed of the drum is obvious, as shown in fig 7. increase of the rotation speed produces thinner fibers. fiber diameter was approx. 2.4 µm at 250 rpm, down to 2 µm at 1000 rpm. the rotating drum twists the fiber additionally, making it thinner, and continually winds the fibers onto the drum, unlike the stationary plate collector. comparison of the fiber diameter sizes for different electrospinning methods used in this work is shown in table 1. table 1 comparison of fiber diameter sizes for different electrospinning methods electrospinning method fiber diameter, µm melt electrospinning, material: pcl 20 solution electrospinning, plate collector, material: fp2 fluoropolymer 0.5 – 0.8 solution electrospinning, drum collector, 250 rpm, material: plga 2.4 solution electrospinning, drum collector, 500 rpm, material: plga 2.2 solution electrospinning, drum collector, 750 rpm, material: plga 2.2 solution electrospinning, drum collector, 1000 rpm, material: plga 2 comparison of the scaffolds produced on the plate and drum collectors indicate certain differences. the major difference between the fibers and the scaffolds produced by the solution electrospinning with the plate collector to the same technology with the drum collector is the resulting shape of the scaffold, together with different pore spatial distribution within the scaffold and the thickness of the electrospun fibers. the scaffolds produced on the plate collector comprise completely random positioning of the fibers falling upon it. the drum collector rotation influences some degree of ordered positioning of the falling fibers with simultaneous winding of the fibers onto the drum, which is similar to the spinning wheel used for textile yarn production. fibers are not completely arranged along one line in the direction of rotation, but the scaffold exhibits a much less random structure in comparison with the standard solution electrospun fiber on the plate collector. using melt materials like in 3d printing, the electrospun fibers do not exhibit electrical instabilities which are imminent to the solution electrospinning, as one of the main reasons for poor repeatability of results [3]. accordingly, better control of the resulting scaffold properties can be gained. an additional advantage of the combination of 3d printing and standard electrospinning is adaptability of the device and its low cost, thus enabling wide investigation of possible tissue engineered scaffold models. for example, combination of layers can be realized, by using two heads alternatively, one after each other, in one electrospinning process. fdm jet head can be used to fabricate one type of fiber layers, upon which a layer produced by solution electrospinning is placed, alternatively many times, thus producing scaffolds comprising two materials (e.g. drug loaded fibers throughout the scaffold made of another material). customization of the material composition of the input wire can be efficiently performed by using a wire extruder, thus enabling different composites and material combinations. hence, the drug loaded polymer wire can be prepared at the extruder and enter the fdm jet head of the custom electrospinning device. our results show that this low cost, simple solution is well suited for production of drug loaded micro-fibers and fibrous scaffolds for tissue engineering (e.g. for wound patches). however, further investigations are needed related to the precise control of all process parameters and especially of their influence on the morphology and structure of the resulting tissue scaffolds. customisation of electrospinning for tissue engineering 333 4. conclusions electrospinning has gained significant attention as efficient technology for production of nanoand micro-fibers and porous, fibrous structures for tissue engineering scaffolds. our investigation shows that customization of device elements can influence the properties of final fibers, such as: morphology or thickness, as well as the final mat properties (size, shape, thickness, porosity). it is shown that a combination of 3d printer elements (fdm head) with an electrospinning device is completely suited for fabrication of micro-fibers. a direct possibility derived from using the fdm head as the electrospinning jet head is opening up of possible input material combinations aiming towards hollow and drug loaded fibers. the 3d printer head enables customized input material since the applied wire is produced by using an extruder which enables different material combinations, including drug loading of the wire. it is shown that this is a cost-efficient solution for melt-electrospinning of micro-fibers. there is a clear difference in diameter sizes of such melt-electrospun fibers, in comparison with the standard solution electrospinning. pcl polymer fibers of micro-size diameter (approx. 20 µm) are fabricated by using melt electrospinning with an adapted 3d printer head, whereas the solution electrospinning results in thinner fibers (fp2 fluoropolymer, 0.50.8 µm and plga, 2 – 2.4 µm). fibers 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with high ultraviolet selectivity and enhanced responsivity, advanced functional materials, 27(47), article number 1704477. 37. hejazi, f., mirzadeh, h., contessi, n., tanzi, m.c., faré, s., 2017, novel class of collector in electrospinning device for the fabrication of 3d nanofibrous structure for large defect load-bearing tissue engineering application, journal of biomedical materials research part a, 105(5), pp. 1535-1548. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 345 356 https://doi.org/10.22190/fume190514042k © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper experimental investigation of the influence of train velocity and travel direction on the dynamic behavior of stiff common crossings vitalii kovalchuk 2 , mykola sysyn 1 , ulf gerber 1 , olga nabochenko 2 , jandab zarour 1 , stefan dehne 1 1 institute of railway systems and public transport, tu dresden, germany 2 department of the rolling stock and track, lviv branch of dniprovsk national university of railway transport, lviv, ukraine abstract. common crossing rails are subjected to a rapid deterioration of the rolling surface due to a dynamic loading of trains. the present study is devoted to an experimental study of the displacement and rail strain measurements in the common crossing. the experimental measurements were carried out for two stiff common crossings under the dynamic loading of high-speed train for the velocity range of 54254 km/h. the results showed 2.5 times increase of the maximal displacements within the velocity range. the absence of the difference in the displacements between the trailing and the facing travel direction is explained with the relative displacement measurements between the rail and the sleeper and the different dynamic impact loading for the wing rail. the proposed model-based analysis of the absolute measurement of rail strain enables us to estimate the dynamic factor under the impact loading. the wing rail for trailing direction is almost twice as highly loaded as the frog rail for the facing direction. the maximal dynamic factor for the trailing direction shows almost no change for the velocities of more than 200 km/h. key words: railway turnout, common crossing, experimental measurements, dynamic factor, model based analysis 1. introduction switches and crossings (s&c) are significant elements of the railway network infrastructure that enable trains to change the travel direction between tracks without delays. an average s&c density for the european railways amounts up to 1.88 units per track kilometer [1]. german railways (db ag) comprise about 70 000 turnouts including about received may 14, 2019 / accepted october 30, 2019 corresponding author: mykola sysyn technical university of dresden, hettnerstraße 3, 01069, dresden, germany e-mail: mykola.sysyn@tu-dresden.de 346 v. kovalchuk, m. sysyn, u. gerber, o. nabochenko, j. zarour, s. dehne 30 000 units at heavy-duty traffic lines [2]. 90% of the turnouts on db ag are with fixed crossing and switch angles up to 1:14. nevertheless, s&c are among the most problematical elements due to a high impact of traffic operation as well as enormous maintenance costs. because of the limitation of train velocities during the travel on s&c they are considered as a bottleneck of railway traffic in railway network. because of high maintenance efforts and short lifecycles, s&c are marked as “hungry assets” in [3]. the maintenance costs of s&c amount up to 35-50% of the overall maintenance costs of track superstructure [4, 5]. common crossing rails are the most short-living elements of s&c with lifecycle up to 5 times shorter than that of the ordinary track rails. the short lifecycles and the difficulties in predicting deterioration of rolling surface cause unplanned traffic interruptions that result in high operational hindrance costs [6]. the principal causes of the rapid deterioration of the common crossing rolling surface are an increased dynamic loading of the wheel, a lower contact area of frog and wing rail with wheel, high shear and wear loading, etc. a number of secondary causes influence the principal ones: wheel and rails geometrical profiles and mechanical properties, train velocity, lateral position of wheelset, trailing direction, stiffness of rail support, etc. the knowledge of the factors affecting deterioration mechanisms of the common crossing would potentially facilitate the development of technical solutions for the prolongation of the crossing lifecycle and the maintenance improvement. many theoretical and experimental studies appeared in the last years dealing with the dynamic interaction and deterioration mechanisms in switches and common crossings. a parametric study was carried out in [7] by means of vehicle-track simulations and it presented the impact of the train operational parameters on rail and wheel degradation. it was found that switch rail lateral loading at 1:9 type turnout is significantly influenced by the level of wheel–rail friction and less by the travel direction. the influence of vehicle speed, traction, and gauge widening and track layout is found to be small. a study of stress-strain state of stiff common crossings, optimization of its longitudinal profile and the development of rolling surface measurement system are presented in [8-10]. a detailed parametric study of the crossing geometry on the interaction influence is shown in [11]. the study indicates the longitudinal height profile of the crossing and the wheel profile as the most significant factors. a study [12] presents an experimental analysis of magnet particle images of the frog rail rolling surface during the lifecycle of the common crossing. a method for rail contact fatigue prediction using image processing and machine learning techniques is proposed. an evolution of wheel to rail contact conditions over turnout crossing is presented in the review paper [13]. experimental studies of the development of accelerations in the frog nose of common crossing during its lifecycle are presented in [14-16]. the studies show a significant increase of the acceleration components and wheel impact position for old crossings due to the growth of rail wear. a numerical modeling of dynamic vehicle-track interaction in a railway turnout is considered in [17] by means of fem and multibody models. the influence of long-term ballast settlements under the common crossing on the dynamic loading of train wheels is shown in the theoretical and experimental studies [18]. the present study is concentrated on the experimental investigation of the influence of train velocity and travel direction on the dynamic loading of stiff common crossings. the influence of wheel diameter difference on high-speed dynamic interaction of wheel and turnout is presented in the numerical study [19]. the influence of track stiffness on the dynamical response of railway switches and crossings is studied with the fem simulation in [20-21]. numerical experimental investigation of the influence of train velocity and travel direction on dynamic behavior... 347 simulations and influence analysis of the switch irregularities on the dynamic loadings are presented in the study [22]. the studies determined the form, parameters and basic patterns of the irregularities development in the rolling zone on switch frogs. almost all recent research of turnout and train interaction is based on the theoretical consideration and numerical modeling. the present paper is concentrated on the experimental investigation of turnout and train dynamic interaction. the most loaded element of turnout is considered, i.e. the common crossing. the influence of train velocity and travel direction (facing and trailing) is estimated with a conventional analysis and a model based analysis. 2. geometrical irregularity of track geometry on common crossing a turnout consists of 3 parts: switch panel, closure panel and crossing panel. the main elements of the crossing panel, namely of the stiff common crossing as shown in fig. 1, right, are: frog nose rail, wing rails, guard rails, stock rails, sleeper fastenings, etc. train vehicles are capable of traveling over a common crossing in through or diverging routes and facing or trailing directions (fig. 1, left). the current study considers a through travel direction where the trains are allowed to move at high velocities which in its turn causes high dynamic loadings and rapid deterioration. the reason of high dynamic loadings is a short geometrical irregularity that appears due to wheel rolling from the wing rail on the frog rail or vice versa. the explanation of the geometrical irregularity formation is presented in fig. 2. the contact point of the wheel and the rail shifts along the wheel conicity outside while the wheel movement from the point 1 to the point 2 (fig. 2, bottom). the wheel moves downwards owing to the radii difference. as soon as the wheel flange comes in contact with the frog rail, the contact point between the wheel and the rail jumps from point 2 on the wing rail to point 3. afterwards the wheel rolls from point 3 to 4 of the frog rail moves upwards on the primary level. in this way the vertical structural irregularity is formed. the deterioration processes that appear in the course of turnout lifecycle, being wear or plastic deformation, fig. 1 ice trailing travel on a common crossing (left) and the elements of common crossing (right) 348 v. kovalchuk, m. sysyn, u. gerber, o. nabochenko, j. zarour, s. dehne change the initial structural irregularity. the wear irregularity appears near the zone 2-3 of the wheel jump from the wing rail to the frog rail, where the highest dynamical loading is present. the irregularity increases the dynamic loading on the common crossing and accelerates the deterioration rate. fig. 2 geometrical irregularity of common crossing (top: wheel travel over the frog nose, bottom: schematic description of the geometrical irregularities formation) the form of the structural irregularity is not symmetrical; therefore, it causes different dynamic interaction in the trailing and facing directions. for an easy description of the excitation geometry function the following two demands are necessary: 1. the excitation geometry function shall be described by 2 parameters: wavelength λa which influences the excitation frequency and wave depth ẑa which can be considered as a measure of the wear, and, 2. the excitation geometry function must be asymmetric, as only in this case it can reproduce different behavior in the facing and trailing direction. the evaluation of the profile measurements for the different common crossings has found the best parametrization function in the sense of a minimal deviation between calculation and measurement. the excitation geometry function is as follows:                        2 2cos1 2 ˆ a aa a xz z   , (1) where za, xa – the local relative function and coordinate point of the excitation geometry. the function parameters are chosen according to the following assumptions: 1. the wavelength is assumed for all points with λa = 3000 mm (this value also corresponds very well to the theoretical and measured wavelength at similar points). experimental investigation of the influence of train velocity and travel direction on dynamic behavior... 349 2. a change in wave depth ẑa corresponds to an extension / compression of the excitation geometry function. 3. the magnitude of the asymmetry is set by the coefficient within the cosine argument (in case the coefficient has the value 1, the excitation geometry function is symmetric). fig. 3 shows the normalized excitation function and its variation with wave depth ẑa. fig. 3 the excitation geometry function 3. experimental measurements of train and crossing interaction the experimental measurements were carried out for 2 turnouts s1 and s2 of type ew60-500-1:12 with assembled stiff crossings. the train of type br401 used the test dynamic loadings with velocities from 54 to 254 km/h in facing and trailing directions of the through routes. the measurement equipment consisted of the strain gauge sensors and inductive displacement sensors. the sensor layout is demonstrated in fig. 4. fig. 4 sensor layout on the common crossing 1:12 (left top: schematic of the crossing, left down: sensors 16m, 16e, 36d, 36m, 36e, 36s, 56d, right: sensors layout for the differential displacement measurement) 350 v. kovalchuk, m. sysyn, u. gerber, o. nabochenko, j. zarour, s. dehne altogether 8 displacement sensors were installed on the sleepers to measure the differential wheel to sleeper displacements: 4 sensors are installed under the wing rail, 3 under the frog rail and one – under the opposite unloaded wing rail. three strain gauge sensors were installed on the wing rail foot and two on the frog rail. the most loaded areas of common crossing correspond to sensors 36d, 35m under the wing rail and 55e, 55m under the frog rail. the examples of the displacement and strain signal records within the same x-coordinate scale for the middle and high velocities are shown in figs. 5 and 6. both the strains and the displacement signals demonstrate the impact similar interaction due to the wheel jump on the wing or frog rail. outside of the impact zones the slow changes of strain and displacement owing to the track longitudinal deformation distribution can be observed that is the quasi-static deformation. the quasi-static component of deformation for the wing rail is higher than for the frog rail that could be explained with much higher bending stiffness in the frog zone. different to the wing rail, the strains in the frog rail (fig. 6, bottom) have a noticeable negative impact zone that could be only explained with the sudden loss of the contact to the wheel during its jump from the wing rail and the negative displacement frog rail. the increase of velocity causes a corresponding increase of displacement and strains for the wing rail and the frog rail. however, it is difficult to estimate the interrelations with the operational conditions only from figs. 5 and 6. to study the statistical influences of train velocity and the travel direction, fig. 7 is built that shows the displacements and strains for the facing and trailing travel directions. the increase of velocity causes a clear increase of displacement for all inductive sensors. the average displacement for all sensors is somewhat higher for the facing direction than for the trailing one. however, the maximal displacements of wing rail and frog rail for high velocities are similar. fig. 5 wing rail to sleeper vertical displacement (top) and strain in wing rail foot (bottom) experimental investigation of the influence of train velocity and travel direction on dynamic behavior... 351 fig. 6 frog rail to sleeper vertical displacement (top) and strain in frog rail foot (bottom) the analysis of strains shows a quite different relation to velocity. the strain gauge sensors 16m for facing and 55m for trailing show a significant decrease of strain with a velocity increase, while other sensors show the growth of strain. this effect can be clearly explained with the topological analysis of sensor location. these sensors are the farthest from the impact zone and record the quasi-statical strain that does not depend on the impact. the behavior of the sensors signals depending on velocity could be explained if the inertial effect of the subgrade is taken into account. the displacement sensors cannot take into account the effect due to the differential displacement measurements between the rail and sleeper. the average growth of displacements is 2.5 times for the displacements and 1.4 times for the strains within the velocity variation from 50 to 250 km/h. the analysis of strains and displacement, their mean values and variations (fig. 7), gives some relation to velocity but cannot indicate the difference between the trailing and the facing directions despite of quite different excitation geometries for both cases (fig. 3). the wheel attack angle for the trailing direction is almost 2 times higher and this should evidently cause a higher dynamic loading of stiff common crossings. nevertheless, the performed analysis of the maximal displacements and strains cannot validate this statement and, therefore, the maximal values analysis cannot be considered as a plausible way for the dynamic loading estimation. the reasons of the behavior could be found in the dynamic interaction. the travel in the trailing direction produces the impact loading and in the facing – a relatively smooth interaction. according to the study [23] where the dynamic modeling of wheel and crossing is carried out, the dynamic oscillation of the rail support in the case of impact loading produces much higher loading on the sleepers that for the smooth loading. due to the increase of the dynamic stiffness in the rail fastenings, the measured differential displacements for the trailing direction are lower than for the facing direction and, therefore, cannot be used for the dynamic factor estimation. the measured strains do not have this disadvantage due to absolute measurements. however, a lot of factors influence the strain measurements, like changing bending stiffness, the dynamic influence of subgrade, etc. to exclude the factors the model based data analysis is necessary. 352 v. kovalchuk, m. sysyn, u. gerber, o. nabochenko, j. zarour, s. dehne facing direction trailing direction fig. 7 rail to sleeper vertical displacements (top) and strain in frog rail foot (bottom) depending on train velocity and travel direction 4. model-based analysis of the measurements the basis for the calculation of the dynamic factor with model-based approach is the relation between the measured and the calculated values with the model of the beam on the elastic foundation. the values are the displacement and strain distribution along the rail. according to the analytical calculation method [23, 24], displacement z0(x) and strain distribution ε0(x) along the rail are proportional to influence functions η(x) and μ(x): 0 0( ) ( )x k f x    , (2) 0 0( ) ( )z x k f x    , (3) where kμ and kη ‒ the proportionality factors that depend on the unknown elastic properties of the track and must be determined experimentally; f0 ‒ static wheel force. moment influence function μ(x) and displacement influence function η(x) result from the superposition of the n wheels of a vehicle with their position xi relative to the position of the measurement point [25]: 1 ( ) cos sin ix xn l i i i x x x x x e l l                , (4) experimental investigation of the influence of train velocity and travel direction on dynamic behavior... 353 1 ( ) cos sin ix xn l i i i x x x x x e l l                , (5) where l ‒ characteristic length, that describes the properties of rail support and rail bending. the characteristic length is determined with the following formula: 4 4 z ei a l c   , (6) where ei – bending stiffness of rail, a – distance between sleeper axes, cz – stiffness of rail support. fig. 8 depicts the explanation of influence functions μ(x/l) and η(x/l) for the relative coordinate and their relation to the point loading. fig. 8 the beam on elastic supports (top) and influence functions μ(x/l) and η(x/l) (middle, bottom) the dynamic factor is defined as the ratio between the measured and the model values of strain or displacements. the measured strains take into account the absolute track deformations contrary to the measured differential displacements that do not take into account the subgrade dynamic deformations. therefore, only the strains are used for further analysis of the dynamic factor. it is determined by the following relation: 354 v. kovalchuk, m. sysyn, u. gerber, o. nabochenko, j. zarour, s. dehne 0 ( ) ( ) d x k x    . (7) while the vehicle parameters are included in the technical data sheets of the vehicles, the track parameters are fitted by adapting quasistatic strain curve ε0(x) to measured strain curve ε(x) in the non-dynamically affected areas as shown in fig. 9 (top). the blue line corresponds to measured strain ε(x), the red one corresponds to calculated strain curve ε0(x). the first dynamic oscillation is marked with the red and yellow markers. fig. 9 (bottom) demonstrates the calculation of dynamic factors for each wheel axle. maximal dynamic factor kdmax that corresponds to the yellow marker is taken into account for further analysis. fig. 9 determination of the experimental dynamics factor from strain measurements the results of maximal values of dynamic factor kdmax for two turnouts s1 and s2, train velocity range up to 254 km/h and facing/trailing travel direction is presented in fig. 10. the calculated set of dynamic factors is appended with prior known value 1 for the zero velocity. the diagram shows up to a twice higher dynamic factor for the trailing travel than for the facing one within the velocity range 100-160 km/h. however, the dynamic factor for the trailing travel after 200 km/h has almost no growth. the outlier points for the trailing travel of the turnout s2 can be explained with the different rail wear that makes the excitation smoother. the maximal dynamic factor for the trailing travel reaches 3.6 and for the facing travel is about 2.2. the comparison of the dynamic factor results for the model based analysis and the maximal measured value analysis (fig. 7) shows similar values for the displacement in facing direction. the factor by the maximal values for the displacement in the trailing direction is much lower than the determined with the model-based estimation due to the influence of impact loading on the differential displacement between rail and sleeper. experimental investigation of the influence of train velocity and travel direction on dynamic behavior... 355 fig. 10 dynamics factor vs. train velocity for facing and trailing directions 5. conclusions the study presents the results of experimental measurements analysis of the influence of high speeds and the travel direction on the common crossing loading. the analysis of the maximal values of differential displacement demonstrates their growth up to 2.5 times. the growth appears in all measurement points within the velocity range between 54 to 254 km/h, both for facing and trailing travel directions. the measured maximal strains are not as explicit as the displacement results. the strain gauge sensors, which are opposite to the impact zones on the wing and frog rail, show a significant decrease of the maximal values with increasing velocities of trains. the main difference between the displacement and the strain measurements is that the measured displacements do not take into account the subgrade displacements. additionally, the increase of dynamic stiffness of rail fastenings between sleeper and rail causes the dynamic displacements to be lower for trailing direction than for the facing one despite of a higher dynamic loading in trailing direction. the rail strain measurements are absolute, contrary to the differential displacement measurement that are relative. however, the ambiguity of strain measurements makes it impossible for use in a simple analysis of maximal values due to the influence of many unknown factors. the proposed model based analysis allows for excluding the factors. unlike the maximal value analysis, the model based analysis utilizes more efficiently available information provided by the measurements. the results of the model based analysis show a clear difference in dynamic loading for trailing and facing travel. the wing rail for trailing direction is almost twice as highly loaded as the frog rail for the facing direction. the maximal dynamic factor for the trailing direction shows almost no change for the velocities of more 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turnouts with stiff crossings), zevrail glasers annalen, 5, pp. 202–214. 24. heppe, a., 2010, ein beitrag zur modellierung und messtechnischen bestimmung des langzeitverhaltens von starren weichenherzstücken (a contribution to the modeling and metrological determination of the long-term behaviour of rigid common crossings), phd thesis, tu dresden, 154 pp. 25. führer, g., 1978, oberbauberechnung, veb, verlag für verkehrswesen, berlin. 151 p. https://www.scopus.com/authid/detail.uri?origin=resultslist&authorid=57194138832&zone= https://www.scopus.com/authid/detail.uri?origin=resultslist&authorid=56353229500&zone= 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https://www.scopus.com/record/display.uri?eid=2-s2.0-85067389032&origin=resultslist&sort=plf-f&src=s&sid=d7c4717af9b8b9632f17a3f4b37d8928&sot=autdocs&sdt=autdocs&sl=18&s=au-id%2857200815763%29&relpos=0&citecnt=0&searchterm= https://www.scopus.com/record/display.uri?eid=2-s2.0-85067389032&origin=resultslist&sort=plf-f&src=s&sid=d7c4717af9b8b9632f17a3f4b37d8928&sot=autdocs&sdt=autdocs&sl=18&s=au-id%2857200815763%29&relpos=0&citecnt=0&searchterm= https://www.scopus.com/sourceid/21100853663?origin=resultslist facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 153 163 https://doi.org/10.22190/fume190509032m © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  cfd modelling of formula student car intake system barhm mohamad 1 , jalics karoly 1 , andrei zelentsov 2 1faculty of mechanical engineering and informatics, university of miskolc, hungary 2piston engine department, bauman moscow state technical university, russia abstract. formula student car (fs) is an international race car design competition for students at universities of applied sciences and technical universities. the winning team is not the one that produces the fastest racing car, but the group that achieves the highest overall score in design, racing performance. the arrangement of internal components for example, predicting aerodynamics of the air intake system is crucial to optimizing car performance as speed changes. the air intake system consists of an inlet nozzle, throttle, restrictor, air box and cylinder suction pipes (runners). the paper deals with the use of cfd numerical simulations during the design and optimization of components. in this research article, two main steps are illustrated to develop carefully the design of the air box and match it with the suction pipe lengths to optimize torque over the entire range of operating speeds. also the current intake system was assessed acoustically and simulated by means of 1-d gas dynamics using the software avl-boost. in this manner, before a new prototype intake manifold is built, the designer can save a substantial amount of time and resources. the results illustrate the improvement of simulation quality using the new models compared to the previous avl-boost models. key words: internal combustion engine, intake system, linear acoustic, geometry modification 1. introduction each year the formula student (fs) hosts colligate design competition for engineering students from around the world. the competition is judged based on engineering innovation, i.e. resulting performance and cost of a formula style car designed to be produced in a small units production run. the objective of such a research work is to model and design formula student car intake system and furthermore demonstrate new techniques to determine how those systems can be optimized in order to either increase the power of the car or reduce the sound level. received may 09, 2019 / accepted october 08, 2019 corresponding author: barhm mohamad university of miskolc, faculty of mechanical engineering and informatics, h-3515 miskolc-egyetemváros, hungary e-mail: vegybm@uni-miskolc.hu 154 b. mohamad, j. karoly, a. zelentsov the intake system of an engine has three main functions. its first and most identifiable function is air filtering the air in order to ensure the engine receives clean air free of debris. two characteristics of importance to the engineers designing the intake system are its flow and acoustic performance. the flow efficiency of the intake system has a direct impact on the power the engine can deliver. melaika et al. [1] showed the effect of different air inlet restrictors on engine performance of formula student car engine using avl boost numerical simulation model and the research was carried out on restrictor diameters, which varied from 15 mm to 60 mm. the smaller intake manifold pipe diameter increased hydraulic resistance of air intake and worsened therewith cylinder volumetric efficiency that resulted in lower engine power and higher brake specific fuel consumption. mohamad et al. [2] studied the effect of ethanol-gasoline blend fuel on engine power output and emissions. their results showed great improvement in combustion process and exhaust gas characteristics. mohamad and amroune [3] used computational fluid dynamic (cfd) tools to describe the flow effects on engine exhaust chamber acoustic level and showed the transmission loss of muffler at different frequencies using 1d boost solver. abdullah et al. [4] studied the engine performance in term of fuel consumption. exhaust emission was influenced by the air intake pressure. the experimental results carried out demonstrated that the air intake pressure was influenced by the degree of opening throttle plate and venturi effect that draws the fuel to the combustion chamber in a carbureted engine. without the air filter, the combustion process is improved due to a higher air intake pressure that transforms more fuel’s chemical energy into heat energy thus raising the exhaust gas temperature above values obtained with an air filter. mohamad et al. [5] used a transfer matrix method (tmm) to perform the transmission loss of a muffler and the algorithm can also be used for other parts of the exhaust system. the result of their study of an existing muffler was compared with vehicle level test observation data. the transmission loss was optimized for new muffler design, while available literature played an important role in validation of obtained results. acquati et al. [6] used mass and momentum balance equations to model the airflow and pressure around a nominal operating values computed using mean value models for intake system of a spark ignition engine. as noticed by winterbone et al. [7], in a regular internal combustion engine (ice), during the air intake phase (the intake valve open), the cylinder volume is not filled completely, as theoretically wished. this is due to the variation in air density and pressure losses along the feeding system. as a result of this process, real volumetric efficiency of the cylinder and the engine performance as a whole cannot meet the design expectations if all the effects are not considered correctly. a component that plays an important role in this process of air supply is the intake manifold (im). its physical characteristics such as pressure loss imposed on air and lack of homogeneity of the loss between the runners (air supply unbalanced) are linked to the efficiency of fuel consumption and engine emissions (byam et al. [8]). in the current research, the intake manifold including plenum and channel ducts of honda cbr 600rr (pc 37) engine (fig. 1) are identified as components whose parameters are to be improved. cfd modelling of formula student car intake system 155 fig. 1 honda cbr 600rr (pc 37) intake system it is rather difficult to obtain the necessary input parameters for cfd analysis necessary for the process of design and optimization, because it is not possible to use a stationary type numerical analysis. for that reason, a 1d model of the intake and exhaust system of the engine was created. the 1d model of the engine was built in avl-1d boost engine simulation software. the basic parameters of the air box include the length of the intake discharge pipes (runner) and the total volume of the air box (plenum). subsequently, the values of the air pressure in the outlets of the plenum are generated based on this software. these values are used as input parameters for the cfd numerical simulation, which is furthermore combined with a geometric optimization to ensure the best final shape of the plenum. the designed process is depicted in fig. 2. 2. numerical analysis to tune the intake runners and determine the ideal runner length for the fs engine, wave theory equations were applied [9], resulting in the following equation for the runner length: fig. 2 flowchart of the design and optimization process 156 b. mohamad, j. karoly, a. zelentsov * + (1) where rpm is the targeted speed of the engine in revolutions per minute, rv (reflective value) is the reflected wave (1, 2, 3,… n), d is the diameter of the runner in mm, vw is the calculated wave velocity in millimeter per second. effective cam duration (ecd) is defined as [10]: (2) the mass flow rate through restrictor [ ] can be calculated as follows: √ ( ) (3) where ar is the area of restrictor [mm2], po is the reference pressure [pa], r is the gas constant equal to 287.04 [j/(kg k)], to is the reference temperature [k], k is the specific heat ratio equal to 1.4. the volumetric flow rate, qmax, can be calculated as follows: (4) where ρ is the gas density [kg/m3]. the analytical solution for sound pressure level (spl) of a simple plenum that is considered as an expansion chamber can be obtained by either the potential function method or the transfer matrix method as given in equation below [11]: ( ), (5) where p is the pressure [pa], while po is the reference pressure [pa]. for the 3d flow calculation, the mathematical model is based on the fundamental equations of three-dimensional nonstationary transport: the equations of momentum (navier-stokes), energy (fourier-kirchhoff) and the conservation of mass (continuity), which take the form of reynolds after the averaging procedure by the favre method: ̅ ̅̅ ̅̅ ̅ ̅ [ ( ̅̅ ̅̅ ̅̅̅ ̅̅̅ ) ̅ ̅̅ ̅̅ ̅ ̅̅̅̅ ] ( ) ( ̅̅̅̅ ) (6) ̅ ( ̅ ̅ ) where ̅i is the averaged velocity along the xi-axis [m/s], while ̅̅̅̅ i is the averaged velocity at specific instant of time along the xi-axis [m/s], t is time [s], ̅ is the averaged pressure [pa], ̅i is the component of the density vector of the volume forces along the xiaxis [n/m3], ̅ is the averaged density [kg/m3], ̅ the averaged specific energy [j/kg], μ is the dynamic viscosity [kg/(m s)], cp the heat capacity at constant pressure [j/(kg∙k)], λ is the thermal conductivity [w/(m k)], δij is the kronecker symbol, ̅ij is the averaged reynolds stress component, ̅ ̅̅ ̅̅ ̅ ̅̅̅̅ is the reynolds tensor, ̅ is the averaged temperature [k] and ̅is the averaged temperature at specific instant of time [k]. in the above equation, the einstein summation rule is used for the twice repeated i, j and k indices. cfd modelling of formula student car intake system 157 the system of transport equations in the reynolds form (6) is closed by the kε -ζ-f model of turbulence specially developed and verified for the processes of flow, combustion, and heat transfer in piston engines [12, 13]. it consists of three equations: for the k kinetic energy of turbulence, for the ε dissipation rate of this energy known from the k-ε model of turbulence, and the equations for the normalized velocity scale ζ = ̅ /k. the kε -ζ-f turbulence model proposed by hanjalic et al. [14] contains the durbin elliptical function of f, which takes into account the near-wall anisotropy of turbulence. system of eqs. (6) is used to describe, respectively, the flow velocities (navier-stokes equation), the enthalpy (energy equation) and the mass or density (continuity equation) for each control volume of the considered computational domain. the wall heat transfer is determined through the thickness of the boundary layer using hybrid wall function [15]. we also emphasize that this mathematical model is typical for cfd calculations of processes in piston engines and it is described in detail by merker et al. [16], basshuysen and schäfer [17], kavtaradze et al. [18]. 3. development of cae model the 3d model of base fs race car intake system was sketched based on specific prototype engine using advanced design software creo 4.0, including restrictor, inlet ducts, connected to outlet plenum and runner. the cross section and the dimensions of intake manifold are depicted in fig. 3. fig. 3 fs race car engine intake system dimensions 158 b. mohamad, j. karoly, a. zelentsov this study was performed using manufacturing data of a four stroke four cylinders honda cbr 600rr (pc 37) si engine. the detailed specification of the base engine selected for the simulation is given in table 1. table 1 the specification of the base engine powertrain units manufacturer / model honda cbr600 rr (pc37) cylinders & fuel cylinders: 4 fuel type: ron98 displacement & compression displacement (cc): 599 compression (_:1): 12,2:1 bore & stroke mm bore: 67 stroke: 42.50 connecting rods length mm 91 valve train chain driven, dohc valve diameters mm intake 27.5 exhaust 23 valve timing deg intake opens at 1 mm lift 22° btdc closes at 1 mm lift 43° abdc exhaust opens at 1 mm lift 40° bbdc closes at 1 mm lift 5° atdc firing order 1-2-4-3 fuel pressure bar 4.00 most of the places in the frame consisted of mating parts that had a small gap. in order to bridge that gap and make them air tight, the bridging process had to be done on each part, followed by capping the ducts. the inner volume was extracted from the solid model by means of design software solidworks. the intake restrictor and runner designs were explored using computational fluid dynamics (cfd) flow modeling software to analyze and visualize fluid flow during the design process. 4. analytical background the shape optimization of the plenum is done using cfd simulation (solver 3d fire advanced flow). for the purpose of simulation and geometrical optimization of the plenum, the intake manifold was meshed with overall 3115552 elements. some basic parameters and characteristics of the model are provided in table 2. table 2 model properties type of model 3d type of mesh mapped total number of elements 3115552 volume of model 0.004449 m3 surface area 0.346168 m2 all obtained geometric parameters of the intake manifold and other parts of the engine from the 3d scan were entered into the 1d model (such as variable diameters of the intake and exhaust pipes dependent on their length, angles in the pipe joints of the intake and exhaust manifold, materials with thermal properties, etc.). boundary condition cfd modelling of formula student car intake system 159 were set according to the engine operation, and for the air flow through restrictor to the runner k-e-ζ-f model was set as a turbulence model in order to treat the turbulence regime in the present study, including nearly random fluctuations in velocity and pressure in both space and time. a 1d model of the engine was created to obtain the input data for cfd analysis (fig. 4). fig. 4 1d model of the honda cbr 600rr (pc 37) engine values of pressures and temperatures for the individual cylinders of the engine in relation to time were subsequently generated from the 1d model. these values are used as input for the cfd analysis. the final analysis was performed as transient for complete cycles of the engine. several simulations of the engine were done to find the most appropriate length of the runner pipes for the whole operating speed range of the engine. more specifically, for the optimal length of the runner pipes was searched in the range 150-300 mm, while for the optimal total volume of the plenum in the range 2-4 liters. 5. results software calculations and multi-iterations showed that a short runner has a negative effect onto the air delivery. with the current design, the airflow would be turbulent when entering the cylinder, and thus decrease the power delivered by the engine (fig. 5). 160 b. mohamad, j. karoly, a. zelentsov fig. 5 comparison of power output from honda co., me workshop test with the base case, and in the modifications 1 and 2 from software in the version denoted as modified 1, the diameter of the inlet channel in_d was increased from 33.5 to 42 mm (see engine scheme, fig. 4), and the volume of the intake manifold pl1 was decreased from 4 to 2 liters. in the version denoted as modified 2, the length of inlet port in_l1 was increased from 100 to 250 mm, in_d diameter was reduced from 33.5 to 32 mm, and pl1 volume was reduced from 4 to 3 liters. the results show that long runners provide high torque at low rpm of the engine and short runners are advantageous at high engine speeds. in terms of the size of the plenum, a small plenum volume provides better throttle response, but a large volume plenum allows high power of the engine, fig. 6. fig. 6 comparison of torque output from honda co., me workshop test with the base case, and in the modifications 1 and 2 from software cfd modelling of formula student car intake system 161 the changes were carried out as a step-by-step search of options with different diameters, volumes and lengths. the optimality criterion was the effective performance of the engine at different operating modes (5000-11000 rpm). however, this optimization was performed under procedure in manual mode (search of optimal sizes with the use of case sets). it was not done completely automatically using boost optimizer. the analytical solution of sound pressure level (spl) of a simple plenum was performed and optimized based on intake system geometry modification. details are given in fig. 7. fig. 7 comparison of sound pressure output from honda co., me workshop test with the base case, and in the modifications 1 and 2 from software this type of analysis is rather difficult because it involves high speed compressible airflow in the intake system, pressure drops and changes of temperatures. the final distribution of the velocities, pressures and the mass flows of the air in the air box are given in fig. 8. the values are evaluated at the outlets of the pipes of the plenum for one operational cycle of the engine (rotation of the crankshaft 720°) at 8000 rpm. the initial conditions were the temperature and pressure inside the calculated volume at the initial moment of calculation (the data were taken from the calculation results in boost). fig. 8 left: pressure contour, min=73317, max=1.0041e+05 pa at 8000 rpm; right: velocity streamline min=0, max=261.05 m/s at 8000 rpm 162 b. mohamad, j. karoly, a. zelentsov the pressure drop was defined as the difference between the pressure in the exhaust manifold at the cylinder outlet and the cross section at the outlet of the computational volume. fig. 9 temperature variation at restrictor of the intake system at 2000, 5000 and 8000 rpm 6. conclusions the team from university of miskolc improved the car’s air intake system using 1davl boost within the parametric fire software workbench environment. fs regulations limit the minimum diameter of the restrictor to 20 mm, which regulates the maximum intake mass flow rate. the plenum, downstream of the restrictor, directly influences the amount of fresh air reaching the cylinders. a plenum that is too large causes the motor to react too slowly to the accelerator and, in combination with short suction pipes, triggers the engine to develop sufficient torque only at high rotation speeds. a too small plenum behaves oppositely. using the equation for the intake runner length, the length of the ideal runner was determined to be approximately 250 mm and with a diameter of 32 mm. hence, design ii of formula student racing is a better choice. acknowledgement: the authors thank formula racing miskolc for assistance with design technique, methodology, and dr. faik hamad, teesside university in middlesbrough, for comments that greatly improved the manuscript. references 1. melaika, m., rimkus, a., vipartas, t., 2017, air restrictor and turbocharger influence for the formula student engine performance, procedia engineering, 187, pp. 402-407. 2. mohamad, b., szepesi, g., bollo, b., 2018, review article: effect of ethanol-gasoline fuel blends on the exhaust emissions and characteristics of si engines, vehicle and automotive engineering, 2, pp. 29-41. 3. mohamad, b., amroune, s., 2019, the analysis and effects of flow acoustic in a commercial automotive exhaust system, advances and trends in engineering sciences and technologies iii, proceedings of the 3rd international conference on engineering sciences and technologies (esat 2018), september 12-14, 2018, high tatras mountains, tatranské matliare, slovak republic, pp. 197-202. cfd modelling of formula student car intake system 163 4. adbulleh, n.r., shahruddin, n.s., mamat, a.m.i., kasolang, s., zulkifli, a., mamat, r., 2013, effects of air intake pressure to the fuel economy and exhaust emissions on a small si engine, procedia engineering, 68, pp. 278-284. 5. mohamad, b., karoly, j., kermani, m., 2019, exhaust system muffler volume optimization of light commercial passenger car using transfer matrix method, international journal of engineering and management sciences (ijems), 4, 132-139. 6. acquati, f., battarola, l., scattolini, r., siviero, c., 1996, an intake manifold model for spark ignition engines, ifac proceedings, 29(1), pp. 7945-7950. 7. winterbone, e., pearson, r., horlock, j., 2000, theory of engine manifold design: wave action methods for ic engines, professional engineering publ. london. 8. byam, b., fsadni, j., hart, a., lanczynski, r., 2006, an experimental approach to design, build, and test a throttle body and restrictor system for formula sae racing, sae technical paper 2006-01-0748, sae world congress, detroit, mi, usa. 9. fsae mqp, 2011, retrieved from: http://moorsportsspares.com/file/2011_ttx25mkii_e.pdf (last access: 15.07.2019) 10. jawad, b., lounsbery, a., hoste, j., 2001, evolution of intake design for a small engine formula vehicle, sae technical paper, 2001-01-1211, sae world congress, detroit, mi, usa. 11. shelagowski, m., mahank, t., 2015, cfr formula sae intake restrictor design and performance, proceedings of the 2015-asee north central section conference american society for engineering education. 12. tatschl, r., schneider, j., basara, d., brohmer, a., mehring, a., hanjalic, k., 2005, progress in the 3dcfd calculation of the gas and water side heat transfer in engines, in 10. tagung der arbeitsprozess des verbrennungsmotors (proc. 10th meeting on the working process of the internal combustion engine), graz, austria. 13. tatschl, r., basara, b., schneider, j., hanjalic, k., popovac, m., brohmer, a., mehring, j., 2006, advanced turbulent heat transfer modeling for ic-engine applications using avl fire, proceedings of international multidimensional engine modeling user’s group meeting, detroit, usa. 14. hanjalic, k., popovac, m., hadziabdic, m., 2004, a robust near-wall elliptic-relaxation eddy-viscosity turbulence model for cfd, international journal of heat and fluid flow, 25(6), pp. 1047–1051. 15. popovac, m., hanjalic, k., 2005, compound wall treatment for rans computation of complex turbulent flow, proc. 3rd m.i.t. conference, boston, usa. 16. merker, g., schwarz, ch., teichmann r., 2019, grundlagen verbrennungsmotoren: funktionsweise, simulation, messtechnik, 9th ed, springer, wiesbaden. p. 1117. 17. basshuysen, r., schäfer, f., 2007, handbuch verbrennungsmotor, vieweg und sohn verlag, wiesbaden, p. 1032. 18. kavtaradze, r.z., onishchenko, d.o., zelentsov, a.a., sergeev, s.s., 2009, the influence of rotational charge motion intensity on nitric oxide formation in gas-engine cylinder, international journal of heat and mass transfer, 52(19-20), pp. 4308-4316. 7076 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 255 278 https://doi.org/10.22190/fume210129041a © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper effects of rotation on unsteady fluid flow and forced convection in the rotating curved square duct with a small curvature mohammad sanjeed hasan1, ratan kumar chanda2, rabindra nath mondal2, giulio lorenzini3 1department of mathematics, bangabandhu sheikh mujibur rahman science and technology university, gopalganj, bangladesh 2department of mathematics, jagannath university, dhaka, bangladesh 3department of engineering and architecture, university of parma, parma, italy abstract. in recent years, the analysis of flow disposition in a curved duct (cd) has greatly attracted researchers because it is broadly used in engineering devices. in the present paper, unsteady flow characteristics of energy transfer (ht) in a rotating curved square duct (csd) have been presented with the aid of spectral method. the key purpose of this study is to explore rotational effects and heat transfer (ht) of the duct. for this purpose, time-evolution calculation is performed over the taylor number (-1500 ≤ tr ≤ 1500) and other parameters are fixed; e.g., dean number (dn = 1000), curvature (δ = 0.015) and prandtl number (pr = 7.0, for water). firstly, timedependent behavior is accomplished for both clockwise and anticlockwise rotations. it is found that the flow instabilities are certainly governed by the change of tr that has been justified by sketching phase spaces (ps). to observe the flow features, velocities including axial flow (af), secondary flow (sf) and temperature profiles are disclosed for both rotations; and it is elucidated that 2to 6-vortex solutions are generated for physically realizable solutions. axial flow (af) shows that two maximum-velocity regimes are produced which induces secondary flow (sf), and, consequently, a strong bonding between the af and sf has been built up. it is observed that as the rotation is increased, the fluid is mixed considerably which boosts ht in the fluid. finally, an assessment between the numerical and experimental data has been given, and a good agreement is observed. key words: rotating curved duct, taylor number, heat-flux, phase space, chaos received january 29, 2021 / accepted may 10, 2021 corresponding author: mohammad sanjeed hasan department of mathematics, bangabandhu sheikh mujibur rahman science and technology university, gopalganj-8100, bangladesh e-mail: sanjeedlhasan@gmail.com 256 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini 1. introduction the investigation of developed fluid flows behavior and ht through a rotating curved duct (rcd) and like constrained domains is of essential interest of a great many researchers because of a multitude of its applications in the engineering fields, e.g., in transporting fluids, rotating-machinery, metallic industry, nuclear engineering, heat exchangers, gas turbines, electric generators, rocket engines and blade-to-blade passage for cooling systems. numerous investigations into different types of ducts have been studied by many authors. among them, mondal et al. [1, 2] (square duct) and yanase et al. [3] (rectangular duct), rumsey et al. [4] (u-duct), chandratilleke et al. [5] (rectangular and elliptical duct), ahmadloo et al. [6] (helical pipe) can be mentioned as outstanding analyses. one of the prominent features in analyzing curved ducts refers to examination of steady and unsteady flow characteristics. symmetric and asymmetric steady solution branches for a tightly coiled duct were gained by liu and wang [7]. they disclosed the structural displacements of bifurcation for changing the grid points and analyzed stability. besides, they calculated the dynamical responses for several pseudo-dean numbers. yanase et al. [8] used two-dimensional geometry to study transitional behavior for the cd flow with large varying aspect ratios. later, yanase et al. [9] carried out a three-dimensional coordinate system to explore the same analysis for non-rotating csd; then they differentiated their results from the previous ones. the us regarding the resistance coefficient and the fluid velocity for several reynolds numbers and gap spacing parameters across multiple staggered rows of cylinders have been represented by nazeer et al. [10]. they validated their periodic oscillation by the power spectrum analysis. zhou et al. [11] used the piv method to compare the unsteady flow characteristics in a half and fully dimpled circular cylinder for large reynolds numbers. zhang et al. [12] conducted unsteady behavior for four square-shaped cylinders where the cylinders were kept at an equal distance from each other. hashemi et al. [13] measured the fluctuation of the velocity components regarding time through the curved pipe. unsteady behaviors in a square enclosure with mixed convection have been steered out by zhang et al. [14]. wang and yang [15] numerically and experimentally reported dynamical responses of the flow through the cd for various dn. transient behaviors with thermal analysis in porous cavities for large aspect ratios have been calculated by arpino et al. [16]. mondal et al. [17, 18] computed the change of unsteady flow behavior, justifying by ps of us results, in the cooling and heating sidewalls for several dn, gr, ar and tr numbers, respectively. hasan et al. [19, 20] obtained three different types of flow oscillations such as steady-state, periodic, and chaotic flow for both rotating and non-rotating cd with small and moderate curvatures, respectively. islam et al. [21] adopted function expansion and collocation method to seek out the uss for rotating curved rectangular duct (crd) flow. kurtulmus et al. [22] observed the nusselt number effect in flow transition for converging and diverging channel. zheng et al. [23] used k ω turbulence model in a serpentine tube to obtain the temperature oscillation and the effects of centrifugal and buoyancy force in the flow transition. among the cited papers, most of the authors showed only flow transition; some of them investigated power spectrum density to justify the flow oscillation. but they did not illustrate the phase space which is an important phenomenon for regular and irregular flow transition. the present study, therefore, represents unsteady flow characteristics, and the flow transition is meticulously justified by sketching ps of the time change of flux that gives a clear view for the regular and irregular oscillation. flow visualization is another significant material for the duct because it recounts the effects of fluid and heat transfer mixing. it is known that dean [24] was the first who effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 257 derived the governing equations of curved ducts. he related that the double-vortex secondary velocity converted into the quadruple-vortex one at the limit points of the steady solution and these two-vortices are named after him: the dean vortices. he demonstrated that there are three types of potency such as centrifugal force, coriolis force and body force working in the fluid flow. here, it is noted that the centrifugal and coriolis forces are influenced by the duct curvature and rotation, respectively. ozaki and maekawa [25] showed curvature effects in the curved duct flow. numerical and experimental investigations with heat transfer through curved rectangular and elliptical ducts were illustrated by chandratilleke et al. [26]. ferdows et al. [27] explained the effects of the dean number in the helical duct for different rotational numbers. bayat et al. [28] developed numerical and experimental flow model to demonstrate the upshot of curvature and kinematic viscosity for curved and spiral micro-channel. li et al. [29] pursued the numerical study of two-dimensional flow behavior for changing the curvature and ar of the duct. li et al. [29] pursued the numerical study of two-dimensional flow behavior for changing the bend and size of the channel. they further represented the three-dimensional stream-flow velocity where the flow fields were measured by using the particle image velocimetry technique. watanabe and yanase [30] attempted to develop a 3-d model to visualize the flow structures. nowruzi et al. [31] used the energy gradient method to report flow structure through a crd with 120° inlet. considering the perturbation method as the main tool, norouzi and biglari [32] accomplished the formation of secondary flow structures. they further sketched the vector plot of the secondary flow. razavi et al. [33] performed flow structure, heat and entropy transmission through a rotating curved channel where they applied the second law of thermodynamics. the influence of centrifugal and hydrodynamic instability in the flow transition and heat conduction was attained by hasan et al. [34, 35]. sasmito et al. [36] evaluated the enhancement of heat transfer for the nano-fluid particle in coiled ducts. al-juhaishi et al. [37] elucidated the friction losses and heat generation in a curved duct. they demonstrated that heat transfer is dramatically induced by the baffles of the channel. zhang et al. [38] suggested a 3d scheme to show the impact of centrifugal force on the heat flow across the curve tube. zhang et al. [39] illustrated the chaotic flow and energy distribution under a critical buoyancy and rotation numbers. schindler et al. [40] calculated the total amount of heat transfer and the fluid mixing when the turbulent flows occurred in a heated csd. however, it is clear that the af and sf velocity and the isotherms have a close interaction, which has not been explored in literature till today. the ongoing paper, therefore, represents a strong connection between the flow velocity and the isotherms together with the transitional flow characteristics and energy transfer in the duct. 2. physical model and mathematical formulation fig. 1 presents the present study considering a two-dimensional (2d) scheme of a curved duct with constant curvature as well as a laminar flow passing through the duct. the temperature difference is kept between two horizontal duct walls in which the lower wall is heated while the upper wall is cooled. the co-ordinate system with relevant notations is shown in fig. 1. 258 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini (a) (b) fig. 1 (a) physical domain (b) cross-section of the system the set of controlling equations is given as: continuity equation: 1 0 yr r vvv v r r y r    + + + =    (1) momentum equations: 2 2 2 1 2 ( . )r r r r v vv vp v v v t r r r r        +  − = − +  − −      (2) 2 2 1 2 ( . ) r rv v v v vp v v v t r r r r            +  + = − +  − +      (3) 1 ( . ) y y r v p v v v gt t y      +  = − +  +   (4) energy equation: ( . ) t v t t t   +  =   (5) here: . r y v v v v r r y       = + +    , 2 2 2 2 2 2 2 1 1 r rr r y      = + + +    where vx, vy, and vθ are velocity components along axes r, y, and θ, respectively. to make dimensional eqs. (1-5) into dimensionless by defining the following variables, effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 259 0 0 2 0 0 0 ', ', ', ' , ', ', 2 ', = ', , = 2 , = ' r x y y z r l dx y h y l dz t t t v v u u v v u v p d d v v u w u p g z l l       = + =   = − =  = =  = =   = − =  = −  =  the proportion of width to the duct curvature radius is curvature (δ). since the present study is 2-d flow which passes uniformly in the axial direction so p ' /z ' = 0. the derived equation is as follows: continuity equation: ' ' ' 1 ' 0 ' ' ' ' ' u v u w x y z       + + + =    (6) momentum equations: 2 2 2 ' 2 2 ' 1 ' ' ' ( '. ) ' ' ' 2 ' ' u w p u v u u t x dn         +  −  = − +  −      (7) ' 2 2 0 ' ' ( '. ) ' ' ' ' ' v p g tl v v v t t y dn u     +  = − +  +   (8) 2 2 ' 2 2 ' 1 ' ' ' ( '. ) ' ' ' 2 ' ' w u w g u v w w t dn         +  +  = − +  −     (9) energy equation: ' 2 0 ' ( '. ) ' ' ' t v t t t du  +  =   (10) where, 1 ,( '. ) ' ' ' ' x v u v x y      = +  = +   and 2 ' 2 2 2 ' '' ' xx y       = + +   the vorticity vector can be written as 2 2 2 2 ' ' 1 ' ' ' ' '' ' v u x y xx y            = − = − + −       (11) and stream function as 1 ' ' ' ' u y    =  and 1 ' ' ' ' v x    = −  (12) now subtracting (9) 'y   from (8) 'x   , we obtain the equation: 2 2 ' 2 2 0 ' ' ' ' ' ' ' ' ' ' ' ' ' ' u w w u v t x y y g tl t dn xu             + + − +            =  − +      (13) 260 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini then, moderated equations for w,  and t as follows (removing () sign) are: 2 2 ( , ) , ( , ) w w w w w dn w tr t x y y x y                 + − + =  − + −      (14) 2 2 2 2 2 2 2 2 2 22 2 2 2 ( , )1 3 3 ( , ) 2 3 2 1 , 2 x t x y xx w w x y x x x y yx t w gr tr x y                                  − = − + −                      −  +  − + − +              + −  +   (15) 2 1 ( , ) 1 . ( , ) t t t t t x y pr x         + =  +      (16) where, 2 2 2 2 2x y    = +   , 1 x = + and ( , ) ( , ) p q p q p q x y x y y x      = −      the non-dimensional parameters that characterize the flow appear in eqs. (14-16): dean number, 3 2gd d l dn     =     , taylor number, 32 2 dttr      =     , grashof number, 3 2 g t dgr      =     and prandtl number, pr     =    . the boundary conditions for w and ψ are 0, 1, 0, , 1 w x y y x w x x y y      =    =   = = =  =   = = = =   (17) and for temperature t are considered as 1, , 1 1, , 1 , 1, t x x y t x x y t y x y y = = =   = − = = −  = =  =  (18) the considered fluid is water (pr = 7.0), and we run computational simulations for dn = 1000, gr = 100, δ = 0.015, and tr € [-1500, 1500]. 3. numerical analysis 3.1. numerical procedure since the method is numerically based, we have used the spectral method as a numerical technique to solve the eqs. (14-16). this is the best method for solving the navier-stokes as well as energy equations (gottlieb and orszag [41] and mondal [42]). by this method the effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 261 variables are expanded into a series of functions consisting of chebyshev polynomials. expansion functions μn(η) and ϕn(η) are stated as 2 2 2 ( ) (1 ) ( ), ( ) (1 ) ( ) n n n n f f         = −   = −  (19) where, fn(η) = cos(ncos-1(η)) is the nth order chebyshev polynomials. furthermore, w(η, γ, t), ψ(η, γ, t) and t(η, γ, t) are expanded in terms of μn(η) and ϕn(η) as: 0 0 0 0 0 0 ( , , ) ( ) ( ) ( ) ( , , ) ( ) ( ) ( ) ( , , ) ( ) ( ) ( ) m n m n m n m n m n m n m n m n m n m n m n m n w t w t t t t t t t                      = = = = = =  =    =     = −      (20) where m and n represent the truncation numbers along the x -and y –directions, respectively. in order to obtain the nonlinear algebraic equations for expansion coefficients wmn, ψmn and tmn , the expansion coefficients have been substituted into the eqs. (14-16) and the collocation method is applied. collocation points (xi, yj) are taken to be cos 1 , =1,..., +1 2 cos 1 , =1,..., +1 2 i i i x i m m j y j n n      = −    +        = −   +    (21) to obtain the time-dependent evolution of wmn, ψmn,and tmn, we substitute the required series expansion (20) into basic eqs. (14-16). 3.2. flux through the duct the flux representing total flow in a curved duct is calculated by the following formula ' ' ' ' d d d d q w dx dy vdq − − = =  (22) where, q: dimensional heat-flux; w : mean axial velocity which is defined by ' 4 q w d  = (23) converting the dimensional eq.(22) into a non-dimensional form, flux q is formulated as 1 1 1 1 q wdxdy − − =   (24) 262 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini 3.3. grid efficiency table 1 the values of q and w (0, 0) for various m and n at dn = 1000, gr = 100, tr = 500 (positive rotation) and tr = -500 (negative rotation) and δ = 0.015 m n q w (0,0) p o si ti v e r o ta ti o n 16 16 291.1740208493 382.6415407505 18 18 291.4440637777 378.9279648439 20 20 291.5043954187 378.2773776991 22 22 291.5491957619 379.0478252693 24 24 291.5511087350 379.0911636619 n eg at iv e r o ta ti o n 16 16 322.2820420248 381.0710697212 18 18 322.2817278220 381.0904822522 20 20 322.2795158289 381.3121404054 22 22 322.2790330741 381.4151170197 24 24 322.2787357133 381.5126278631 here, grid efficiency is carried out for truncation numbers m and n. as the research deals with square duct flow, so m and n are considered as equal. in this study, the values of axial flow and heat flux for both co-rotation and counter-rotation are taken for different truncation numbers where the parameters of governing equations are fixed. table 1 shows grid accuracy; it can be seen that the values do not show any substantial change for increasing or decreasing the truncation numbers. to get sufficient accuracy, m = (n) = 20 has been taken for numerical simulation. 4. results in the following discussion, transitional behavior of the fluid flow is performed for fixed grashof number, gr (= 100); curvature, δ (= 0.015) and dean number, dn (= 1000) and parameter tr (taylor number); geometric significance is duct rotation, varied from 1500 to 1500. the programming algorithm of the governing equations is developed by allocating three methods including the crank-nicolson, adam-bashforth, and function collocation method simultaneously. moreover, the characteristics of the axial velocity, secondary flow and energy distribution from contour plots for co-rotation and counterrotation have also been explored in this study. 4.1. time-dependent flow behaviors 4.1.1. case i: co-rotation (0 ≤ tr ≤ 1500) initially, we consider taylor number tr = 10. the time-dependent of the unsteady solution (us) is analyzed and found that the unsteady flow gives a multi-periodic solution in t q plane as plotted in fig. 2(a). the recognition of multi-periodicity is examined by the phase space (ps) analysis as depicted in fig. 2(b). here, the ps is calculated from the unsteady solution and sketched in the λ – γ plane, where γ is prescribed by dxdy =   . the phase space in fig. 2(b) narrates the ability of the tremble as a function of frequency at which the frequencies have potent and at which the frequencies are feeble. axial velocity, stream function and isotherms are shown in fig. 3. the observation from the af is that the particle effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 263 has pressed opposite to the duct's outer circumference. from the stream function, 2to 4vortex solution is found for tr = 10. it is found that the axial velocity and the stream function have a close bond. at t = 26.00 and t = 26.30, two high-velocity portions are produced at the outer circumference of the bend; as a result, two new vortices are generated. here, the newlyproduced two-vortices are known as dean vortices and the previous two-vortices are called ekman vortices. it is further illustrated that the two new vortices at t = 26.00 are smaller than those at t = 26.30. this has happened because of the influence of the axial flows. more explicitly, at t = 26.00, only a pair of high-velocity regions are created at the outer circumference where no dumbbells are formed. on the other hand, at t = 26.30, a pair of high velocity regions are generated with a pair of dumbbells at the left-top and left-bottom of the enclosure. as a result, newly-created two vortices at t = 26.30 are larger than the two vortices formed at t = 26.00. it is also remarked that, due to dumbbells at t = 26.30, the flow velocity is stronger than that at t = 26.00 due to the centrifugal force of the duct. temperature profiles show that heat is transferred exceedingly at t = 26.00 and t = 26.30 within the time (25.90 ≤ t ≤ 26.40). it is easily demonstrated that when the flow velocity is strong, more heat is transferred which proves that the fluid is mixed excessively when the velocity of the flows is strong. fig. 2 transient solution for tr = 10 (a) q as a function of time, (b) ps of (a) fig. 3 velocity contour (top & middle) and isotherm (bottom) for tr = 10 264 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini if the rotational speed is extended, the multi-periodic flow converts into a steady-state solution. fig. 4(a) shows steady-state solutions for tr = 200 and tr = 400. in the region 200 400tr  , as we increase the value of the taylor number, the heat-flux value decreases and the steady-state solutions are organized accordingly. axial velocity, stream function and isotherm are visualized in fig. 4(b) and found only symmetric 2vortex secondary flow patterns. fig. 4 transient solution for tr = 200 and tr=400 (a) q as a function of time, (b) velocity contour (top & middle) and isotherm (bottom) regular oscillation starts for tr > 750. here, we investigate time-development of the us for tr = 900 and the outcomes are plotted in fig. 5(a) which indicates the periodic oscillation. to ensure it more, ps is also computed as displayed in fig. 5(b). the path line of the ps, displaying multiple orbits, can show that the flow is in the initial stage of a multi-periodic oscillation instead of the periodic oscillation created at tr = 900. this has occurred because the flow characteristics are significantly affected by the coriolis force. the af, asymmetric 2-vortex stream function and isotherm are shown in fig. 6. fig. 5 transient solution for tr = 900 (a) q as a function of time, (b) ps of (a) effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 265 fig. 6 velocity contour (top & middle) and isotherm (bottom) for tr = 900 due to disorientation in periodic and multi-periodic flow, we investigate timedependent solution from tr = 950 to tr = 1300. it is elucidated that the oscillation for the required tr is certainly multi-periodic. fig. 7(a) represents the multi-periodic flow for tr = 1300. the ps drawn in fig. 7(b) shows that multiple orbits are produced and the path lines overlap with each other. it is further expressed that the area of steam function is growing for increasing tr. so, it can be said that the multi-periodic oscillation is nearly at the terminal stage. flow patterns with isotherms are drawn (see fig. 8). it is noticed that 2-vortex asymmetric secondary flow is created at tr = 1300 where the axial flow and the temperature profiles bear their usual significances. fig. 7 transient solution for tr = 1300 (a) q as a function of time, (b) ps of (a) 266 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini fig. 8 velocity contour (top & middle) and isotherm (bottom) for tr = 1300 if tr is raised more, the multi-periodic flow transforms into an irregular oscillation i.e., the flow is chaotic. the chaotic oscillation starts at tr = 1350 approximately and continues till tr = 1500. the characteristics of the unsteady flow for tr = 1450 are shown in fig. 9(a). it is observed that the flow oscillates more at the irregular (chaotic) flow than the regular (periodic/ multi-periodic) oscillation and the density of the oscillation enhances as tr increases. ps of the chaotic oscillations is also enumerated (see fig. 9(b)), which justifies the flow configuration as it was anticipated. from the phase space, the path lines pass over each other in the λ – γ plane; as a result, a critical flow characteristic has been found. it can be said that due to increase of tr, the flow oscillation is enhanced gradually. as a consequence, the vibration of fluid particle increases, for this reason; the fluid particles are amalgated which prolongs the overall heat transfer in flow. axial velocity, stream function and isotherms are depicted in fig. 10 for tr = 1450. fig. 9 transient solution for tr = 1450 (a) q as a function of time, (b) ps of (a) effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 267 fig. 10 velocity contour (top & middle) and isotherm (bottom) for tr = 1450 4.1.2. case ii: counterrotation (-1500 ≤ tr < 0) first, we realize the time-development of the us for tr = -30, and it represents the multi-periodic oscillation as fig. 11(a). ps is further plotted in fig. 11(b) for explaining the multi-periodicity properly. fig. 11 transient solution for tr = -30 (a) q as a function of time, (b) ps of (a) it is demonstrated that the path lines of the ps coincide at the starting point after completing two cycles. flow velocity (axial velocity), stream function and corresponding isotherms are exhibited in fig. 12. af patterns describe that, when two-vortex asymmetric solutions are formed, the flow velocity is towards the concave wall of the system (t = 17.70). it is obtained that the axial velocity forms a dumbbell shape at the right side of the contour (t = 17.60) and as a consequence, another vortex has appeared in the lower part of the duct. at time t = 17.80, sf consists of 4-vortex solution for generating the maximum-velocity regime. it is also observed that at t = 17.90, two maximum-velocity regimes are found where the area of upward part is smaller than the area of below part and the single dumbbell appeared at the 268 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini bottom wall; for this reason, the lower vortex is larger than the upper one, resulting in a fourvortex secondary flow. the isotherms illustrate that when the af does not produce maximumvelocity regime, the density of the isotherm stream is less but the opposite criteria are found when af generates the maximum-velocity regimes. so, it can be easily said that the af definitely affects the secondary vortex including isotherms. fig. 12 velocity contour (top & middle) and isotherm (bottom) for tr = -30 with further increase of tr in the negative direction, another flow transformation occurs at tr = -160, where the multi-periodic flow shifts to a steady-state flow. it should be mentioned here that the steady-state flow initiates at tr = 160 and terminates at tr = -390 as found from the linear stability analysis. fig. 13(a) displays the steady-state solution at tr = 230. flow characteristics including axial velocity, stream function and isotherms are visualized in fig. 13(b), where 4-vortex symmetric secondary vortices are noted for the steady-state solution. fig. 13 transient solution for try = -230 (a) q as a function of time, (b) velocity contour (top & middle) and isotherm (bottom) for try = 230 effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 269 when the values of tr are gradually increased at tr = 400, the steady-state flow turns into a multi-periodic flow (fig.14 (a)). the ps of the multi-periodic flow is represented in fig. 14(b), which shows that the streamlines have two multiple orbits before they converge at the starting point. axial velocity, stream function and isotherms are revealed in fig. 15. interesting types of flow properties are found at tr = 400. the af has pushed to the right side of the contour. the secondary vortices of the flow patterns depict that due to the coriolis force in the negative direction the dotted lines are produced at the duct's ceiling wall and the solid lines are created at the duct's bottom wall. as a consequence of these types of rotation of negatively, the ekman vortices have originated at the duct’s interior portion. here, there is also a relationship established between axial velocity, stream function, and isotherms. for example, at t = 14.45, a pair of high-velocity regions with double dumbbells of axial flow are produced at the duct's interior side; hence in a four-vortex secondary flow is generated where the two new vortices are generated at the duct's interior side. besides, the dumbbells of the bottom side of axial flow are, for this reason, smaller than those at the upper one. although the fig. 14 transient solution for try = -400 (a) q as a function of time, (b) ps of (a) fig. 15 velocity contour (top & middle) and isotherm (bottom) for try = 400 270 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini two vortices look similar, the vortex of the part below is smaller than that of the upward part. however, in brief, despite of changing direction of the axial velocity, the meaning of flow characteristics is the same as in axial and secondary flow. mainly, the flow velocity direction changes because of the duct rotation. it is noted that the fluid mixing and total heat distribution in the flow are undoubtedly affected by the change in direction of the flow rate. now, we manifest the transitional characteristics of the flow at try = -730. fig. 16(a) shows a steady-state solution. axial flow, stream function and isotherms are explored in fig. 16(b). if the carioles force is more raised at try = 1100, the steady-state solution jumps to the periodic oscillation (see fig. 17(a)). speaking more clearly about regular oscillation, ps is enumerated as drawn in fig. 17(b) which deliberates that the stream lines meet at the starting point after completing an orbit. the af and sf and isotherms are shown in fig. 18. then, we further explore unsteady characteristics for try = 1450. it is narrated that the time evolution delivers irregular oscillation as shown in fig. 19 (a). to have a better understanding of the flow movement, ps is also calculated as depicted in fig. 19(b). according to ps, it is analyzed that the direction of the stream flow moves arbitrarily in the λ γ plane, and the line spectrums oscillate continuously. fig. 20 displays af, sf and isotherms where it is seen that 2to 4-vortex solutions are found at try = 1450 where the dotted lines are seen in the top wall and the additional vortices are situated in the inner wall of the duct. at t = 19.10 for try = 1450, axial flow regions are created at the top, so that the secondary vortices are formed below. when 4-vortex solutions are found the axial flows are divided into two high-velocity regions and the fluids are mixed up more than those of the two-vortex solution as also shown by the temperature profiles. so, uss do not demonstrate only the time-dependent solutions but also disclose a connection between the liner stability, axial flows, secondary flows and the heat transfer. fig. 16 transient solution for try = -730 (a) q as a function of time, (b) velocity contour (top & middle) and isotherm (bottom) effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 271 fig. 17 transient solution for tr = -1100 (a) q as a function of time (b) ps of (a) fig. 18 velocity contour (top and middle) and isotherm (bottom) for tr = 1100 fig. 19 transient solution for tr = -1450 (a) q as a function of time (b) ps of (a) 272 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini fig. 20 velocity contour (top & middle) and isotherm (bottom) for tr = 1450 4.2. vortex generation of unsteady solutions now, the observation of the number of vortices of us for monitoring varying tr is performed by the bar diagram. fig. 21 presents the variation of θ at δ = 0.015 and tr varying from -1500 to 1500 in the (tr θ) plane. also, 2to 4-vortex solutions have been visible for varying value of tr. the maximum numbers of vortices are produced for weak rotation causing the resulting force of centrifugal, coriolis and buoyancy forces. fig. 21 schematic diagram of uss at δ = 0.015 4.3. heat transfer heat transfer (ht) through the duct for both rotation (co-rotation and counterrotation) is calculated as shown in figs. 22(a) and 22(b), respectively. here, the cooling and heating sidewalls are achieved from the time-independent (steady branch) solution which is marked by the solid lines. the symbols are determined by taking the time average of the us for constant tr. effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 273 fig. 22 nusselt number variation for varying tr: (a) positive rotation, (b) negative rotation it is clearly shown, in the above two figures, that the average nusselt number for cooled and heated side walls follows approximately linear trend with increasing tr in the negative direction while a slightly curved trend is in the positive direction. so, the heat is transferred for negative rotation more than for positive one for both cooling and heating walls. the cause of it is the duct rotation. moreover, because of the advection in the duct, time-average of the transient flow at the cooling sidewall passes less heat than that at the cooling wall. in addition, the flow transition and heat transfer throughout the duct are related to each other. the steadystate in the time-dependent flow transfers less heat than the regular and irregular oscillation. this has occurred because of fluid fixing. more precisely, in comparison to the steady-state solution, the flows vibrate more at periodic, multi-periodic, and chaotic oscillations. as a result, the fluid particles collide with each other and enhance heat transfer. 4.4 validation of the study here, a comparative study between the numerical and experimental investigations has been provided for both curved square duct (csd) flow so as to validate our current study. there are numerous authors who have disclosed the experimental outcomes of the curved square duct flow. yamamoto et al. [43] investigated experimentally the rotating curved square duct flow. they adopted the visualization technique to explore the experimental outcome with a constant curvature δ = 0.3. yamamoto et al. [43] used a water flow tank where the dye was injected continuously and this dye was adjusted by the alcohol to be the same as the weight of water. at first, they fixed the positive rotation at tr = 150 and took photographs at an angle 180o inlet for several values of dn. the same work was then conducted for negative rotation at tr = -150. in the present study, we fix our parameters according to yamamoto et al [43] experiments. then we obtain the numerical outcome of the secondary flow for several values of dn as shown in fig. 23 which shows a perfect match between our numerical computations and investigational outcomes. 274 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini fig. 23 experimental (left, yamamoto et al. [43]) vs. numerical (right, authors) results for rotating curved square duct flow; (a) tr = 150, (b) tr = -150 5. discussions there is a large interest of engineering community in studying the flow characteristics in a curved system because it has a wide range of real-life applications including heat exchangers, chemical mixing, power industry, rotating electrical machinery etc. as well as other application areas such as blood flow in human veins and arteries. the researchers are, therefore, interested in investigating different types of phenomena by changing the governing equations. here, we discuss contributions and limitations of some papers published in literature and try to explain what is new in the ongoing exploration. yang et al. [44] used cfd based fluent software and user-defined function to expose flow vibration and timedependent flow configuration through a circular cylinder. khanafer et al. [45] conducted nk5000 code to explore the effect in streamlines and isotherms with changing parameters and showed that the rotational speed expressively enhances overall heat throughout the fluid than other parameters such as ri and re number. geike [46] reported the cavitation phenomena in the viscoelastic fluid flows. dolon et al. [47] studied the flow analyses as well as the centrifugal influence in the duct size. mousavi et al. [48] conducted nanofluid and thermal radiation through a needle to narrate the velocity ratio for the change of magnetic parameters. bibin et al. [49] used vector plot to represent the axial velocity and showed temperature contour at several axial locations. umavathi and beg [50] applied finite difference method to demonstrate viscous and buoyancy effects in the secondary and isotherm contour for temperature different fluid through a vertical duct. nobari et al. [51] analyzed the behavior of af and sf pattern in annular duct at different inlets. they also calculated heat transfer in the duct but they did not visualize the isotherm variation in their exploration. riyi et al. [52] elucidated heat transport and temperature profiles in the inner and outer inlet of the curved annular duct experimentally for a large range of re. mansour et al. [53] discussed the fluid mixing in three different tubes for several dn over a large effects of rotation on unsteady fluid flow and forced convection in a rotating curved square duct… 275 range of the re but their research did not provide any discussion of flow velocity or fluid mixing contours. tanweer et al. [54] performed transitional behaviors with respect to several coefficients and observed heat transfer when the cylinder is near the moving wall. though their study discussed heat transfer in a three-dimensional coordinate system, the investigation is limited for low reynolds number. hasan et al. [55-57] adopted spectral technique to explore time-dependent flow characteristics across the circular and non-circular channels including centrifugal and body force effects. however, no study is published so far that describes hydrothermal behavior of transient flow characteristics and energy distribution with forced convection in a curved square geometry with the effects of rotation, namely, the study which is successfully accomplished in this paper. besides, a relationship among the axial velocity, stream function and isotherms is presented, and it is revealed that ht is enhanced substantially with increasing rotation, which has not been mentioned in the literature till now. the present study may improve the heat generation and transmission knowledge that constitute a new epoch in related fields to produce energy-related machinery. 6. conclusion the continuing study determines a spectral-based numerical approach on fluid flow and heat transfer through a csd at rotating stage with a small curvature considering the bottom wall heating and the top wall cooling, the sidewalls are well insulated to prevent any heat loss. a wide-range of the taylor numbers (tr) ranging from 1500 to 1500 is considered for different curvature ratio 0.015 and aspect ratio 1. the numerical findings are validated with the available experimental data. the following conclusions are drawn from the present study: ▪ time-development flow as well as phase-space of the time progression results shows that the transient flow endures in the consequence “multi-periodic → steady state → periodic → multi-periodic → chaotic”, if tr is increased in the positive direction; for negative rotation, however, the flow experiences the evolution: “multi-periodic → steady state → multi-periodic → steady state → periodic → chaotic”. ▪ flow instability at negative rotation shows more transiency than that at positive rotation, and it is found that the regular and irregular oscillation at high rotation becomes stronger than that at small rotation. ▪ 2to 6-vortex solutions are generated at the periodic, multi-periodic and chaotic flows, and the number of secondary vortices is significantly higher for the chaotic solutions while few for the steady-state solution. ▪ a strong bond between the af and sf has been established for regular and irregular oscillation which agrees well with temperature distribution. the study shows that heat transfer occurs prominently if the rotation is increased either in the positive or in the negative direction. ▪ a strong dominance among heating-induced buoyancy force and centrifugal-coriolis instability is observed that augments flow interaction and enriches heat transfer in the flow. the study reveals that chaotic solutions boost heat transfer more effectively than the steady-state or other physically realizable solutions due to formation of significant number of dean vortices at the outer concave wall. 276 m.s. hasan, r.k. chanda, r.n. mondal, g. lorenzini 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ranking udc 628.4:005.311.6 biljana milutinović 1 , gordana stefanović 2 , goran vučković 2 , mladen tomić 1 , petar đekić 1 1 college of applied technical sciences niš, serbia 2 faculty of mechanical engineering, university of niš, serbia abstract. the selection of an appropriate waste treatment scenario is a complex problem in which a set of environmental, economic, and social criteria must be taken into account. different waste treatment scenarios have different effects on the environment, which is expressed through a variety of environmental indicators. the main problem is to determine the indicators that clearly and fully express the most important influential factors. this paper presents a number of different environmental indicators and their influence on the waste treatment scenarios ranking. the study is carried out on the example of waste management in the city of niš. four scenarios are developed: the business as a usual scenario (meaning the landfilling of waste) and the three other scenarios with energy recovery and preservation of resources including composting organic waste with recycling inorganic waste, incineration of waste, and anaerobic digestion of waste. four experiments were conducted in order to assess the influence of environmental indicators: the first experiment was done using four indicators, the second by using seven indicators, the third experiment by using nine indicators, and the fourth experiment by twelve indicators. the ranking of each scenario was performed on the basis of a multi-criteria analysis, the analytic hierarchy process (ahp method). the obtained results have shown that the increasing number of environmental indicators has led to a change in the ranking of scenarios in terms of their impact on the environment. namely, it is necessary to increase the number of environmental indicators to a number which will be sufficient to carry out the relevant waste treatment scenario ranking in terms of the impact on the environment. key words: environmental indicators, waste treatment scenarios, multi-criteria analysis, ahp method received november 10, 2014 / accepted april 30, 2015 corresponding author: biljana milutinović college of applied technical sciences niš, aleksandra medvedeva 20, niš, serbia e-mail: bimilutinovic@gmail.com original scientific paper 156 b. milutinović, g. stefanović, g. vuĉković, m. tomić, p. djekić 1. introduction in assessing the sustainability of waste treatment, several aspects should be taken into account: environmental, economic, and social. each of them is expressed through specific indicators that can be identified as environmental, economic, and social. environmental indicators are essential tools for tracking environmental progress, supporting policy evaluation, and informing the public. an environmental indicator is defined as a number indicating the state and development of the environment or conditions affecting the environment. indicators can be used descriptively for a scientific purpose or normatively for a political purpose [1]. for such purposes an indicator must be placed in a context that allows for an interpretation of its values. the essential characteristic of an indicator is that it is possible to interpret or evaluate its value. an indicator may be defined as a characteristic which, when measured repeatedly, demonstrates ecological trends and a measure of current state or quality of an area [2]. some authors define an indicator as “a parameter or value that reflects the condition of an environmental (or human health) component, usually with significance that extends beyond the measurement or value itself, [and] when used alone or in combination, [the indicators] provide the means to assess progress toward one or more objectives” [3]. different waste treatments have different effects on environment, which is expressed through a variety of environmental indicators. the anaerobic digestion offers a greatly increased reduction in net greenhouse gas (ghg) production compared with other technologies. comparing incinerator facilities with energy recovery and landfill dispatching, the incinerator allows a greenhouse gas reduction of 360 kgco2eq/ t waste and in the case of a traditional landfill with no provision for landfill gas capture, the difference with respect to incinerators increases to 650 kgco2eq/t waste [4]. the greatest waste volume reduction is achieved with anaerobic digestion (95.69 %) compared with other waste treatments [5]. gasification has provided by far the best energy recovery in terms of electrical product (1083 kwh/t), while anaerobic digestion has provided a relatively poor energy recovery (151 kwh/t) [4]. to assess the environmental impact of a certain waste treatment, it is necessary to carry out an adequate analysis of all influential environmental criteria. the main problem in the analysis is to determine the number of indicators that clearly and fully express the most important influential factors. in this paper, four experiments were conducted in order to assess the influence of environmental indicators and their number, and thus four waste treatment scenarios were created. the first experiment was done using four indicators, the second by using seven indicators, the third experiment by using nine indicators and the fourth experiment by twelve indicators. the presented study was carried out on the example of waste management in the city of niš. the ranking of each scenario was performed on the basis of a multi-criteria analysis, the analytic hierarchy process (ahp method). the comparative results of the changes in scenario ranking obtained by increasing the number of different environmental indicators are presented. effect of environmental indicators to the waste treatment scenarios ranking 157 2. environmental indicators today a wide variety of environmental indicators is used. these indicators reflect trends in the environment just as they monitor the progress made in realizing environmental protection. in relation to policy-making, the environmental indicators are used for four main purposes: a) for supplying information on environmental problems in order to enable policy-makers to evaluate their seriousness; b) for supporting policy development and priority setting by identifying the key factors that cause pressure on the environment; c) for monitoring effects and effectiveness of policy responses, and d) for raising public awareness of environmental issues. since the early 1990s, the environmental indicators have gained in importance in many countries and in international forums. oecd countries have increasingly used a reduced number of indicators, so-called “key indicators”, selected from larger sets to report on major environmental issues [6]. now, there are several sets of environmental indicators, each responding to a specific purpose. during the 1990s, the environmental indicators gained significant importance and are now widely used in oecd countries. they are used in reporting, planning, clarifying policy objectives and priorities, budgeting, and assessing performance. the oecd environmental indicators are relatively small sets of indicators that have been identified for use at the international level, and should be complemented by national indicators when examining issues at the national level [7]. these key indicators have been selected from the core ones included in the core set of environmental indicators [8] and are closely related to other environmental indicators sets. these indicators are divided into two groups. the first group includes climate change, ozone layer, air quality, and waste generation. the second group that involves natural resources includes freshwater quality, freshwater resources, forest resources, fish resources, energy resources, and biodiversity. their selection has taken into account their policy relevance with respect to major challenges of the first decade of the 21st century, including pollution issues and those related to natural resources and assets, their analytical soundness and their measurability. most indicators presently used by nations are based on the dpsir-framework. they are used to characterize the main environmental issues, such as climate change, acidification, toxic contamination and wastes in relation to the geographical levels at which these issues manifest them or on which they are managed. indicators for the driving forces describe the social, demographic and economic developments in societies and the corresponding changes in lifestyles, overall levels of consumption and production patterns. primary driving forces are population growth and developments in the needs and activities of individuals. through these changes in production and consumption, the driving forces exert pressure on the environment. pressure indicators describe developments in release of substances (emissions), physical and biological agents, the use of resources, and the use of land by human activities. state indicators give a description of the quantity and quality of physical phenomena (such as temperature), biological phenomena (such as fish stocks) and chemical phenomena (such as atmospheric co2-concentrations) in a certain area. impact indicators are used to describe changes in these conditions. it is the change in the availability of species that influences human use of the environment. in terms of their strict definition, impacts can be only those parameters that directly reflect changes in environmental use functions by humans. as humans are a part of the environment, impacts also include health impacts. response indicators refer to responses by groups (and individuals) in society, as well as government attempts to prevent, compensate, ameliorate or adapt to changes in the state of the environment [9]. 158 b. milutinović, g. stefanović, g. vuĉković, m. tomić, p. djekić according to the oecd data [7], average municipal waste generation intensities per capita in 2011 for oecd countries are 530 kg per capita, which is for 30 kg less than in 2005 and 20% less waste has been disposed of in landfills since 2000. in order to evaluate the effects of different waste treatments on the environment, different environmental indicators have been used and different models have been developed. greene and tonjes [10] examine the common performance indicators to assess the environmental benefits of municipal waste systems in order to determine if there is agreement between them regarding which system performs best environmentally. the considered performance indicators are: ghg emission reductions, energy savings, landfill disposal rate, recycling rate, diversion rate, etc. the focus is placed on how indicator selection influences comparisons between municipal waste management programs and subsequent system rankings. coelho et al. [11] have proposed a new concept: the cleaner treatment based on the cleaner production principles; they have developed the cleaner treatment index (cti) to assess environmental performance of waste treatment technologies. this model focuses on five major issues: environmental liabilities, climate change, greenhouse effect, soil, water and air pollution, energy and natural resources consumption. manfredi and goralczyk [12] have developed the macrolevel waste management indicators in order to quantify and monitor the potential environmental impacts, benefits, and improvement possibilities associated with the management policies and strategies of a number of selected waste streams generated and treated in europe. the waste management indicators were developed for the eu-27. indicators were a combination of statistical waste data with emissions and resource life cycle data for the different elements of the waste treatment chain. the developed waste management indicators were: climate change midpoint, ozone depletion midpoint, human toxicity midpoint, particulate matter/respiratory inorganic midpoint, acidification midpoint, eutrophication terrestrial midpoint, eutrophication freshwater midpoint, eutrophication marine midpoint, ecotoxicity freshwater midpoint, land use midpoint, etc. josimovic et al. [13] apply the multi-criteria evaluation method to carrying out a strategic environmental assessment for the waste management plan for the city of belgrade. the method is applied to the evaluation of the impacts of the activities planned in the waste management sector on the basis of the environmental and socioeconomic indicators of sustainability. the environmental indicators taken into consideration are: water (surface water and groundwater), air and climate change, soil, biodiversity and landscape. in order to rank four different options of municipal solid waste treatment alternatives from an environmental point of view, herva and roca [14] have applied two methods: (1) the ecological footprint (ef) as a single composite indicator and (2) multi-criteria analysis integrating the ef together with other material flow indicators related to water consumption, air emissions of organic compounds, air emissions of dusts, water emissions of suspended solids, and occupied landfill volume. to choose a solid waste management system using multi-criteria decision analysis, hokkane and salminen [15] have used the following environmental criteria: global effect (co2, ch4, n2o emissions), releases with health effects (heavy metal releases to air and water: lead (pb), cadmium (cd), arsenic (as), mercury (hg), dioxin and furan), acidificative releases (so2, nox), surface water dispersed releases. in accordance with the given referential literature survey, twelve environmental indicators representing different effects of the waste treatment are selected in order to assess the impact of certain waste treatment on the environment (air, water and soil). http://www.sciencedirect.com.proxy.kobson.nb.rs:2048/science/article/pii/0377221795003258#aff1 effect of environmental indicators to the waste treatment scenarios ranking 159 3. diversity and number of indicators research table 1 presents the composition of waste generated in the city of niš. the data on generated waste are taken from the waste management plan of the city of niš in the period up to 2015 [16] and other previously published papers [5]. table 1 the quantity of waste generated in the city of niš (2010) [16] fraction production (t/year) percentage (%) food waste 24,298 33.70 yard waste 7,494 10.40 paper 11,031 15.30 plastics 12,762 17.70 glass 3,677 5.10 metals 1,370 1.90 other 11,464 15.90 total 72,100 100.00 the largest part of the waste is organic waste (food and yard waste) – 44 %, while recyclable fractions make up 40 % of total waste. 3.1. developed scenarios four scenarios were developed on the basis of the methods of waste treatment: recycling, composting, anaerobic digestion, incineration with energy recovered and waste disposal. the first scenario was the business as a usual scenario (meaning the landfilling of waste) in the city of niš. the other three scenarios were created as scenarios with energy recovery and preservation of resources. scenario 2 implied that all recyclable waste was recycled and organic waste was composted. scenario 3 implied that all amount of glass and metal was recycled and other combustible waste was incinerated. finally, scenario 4 implied that all the waste (glass, plastics and metal) which was unsuitable for anaerobic digestion, was recycled, while other types of waste were sent to an anaerobic digester. main variation factors of each scenario are given in order to evaluate environmental indicators. the annual distance driven by collection trucks was calculated according to the presumed location of the waste treatment facility. one location was presumed for the incinerator or anaerobic digester. scenario 1: most waste (68,440 t) is disposed of in the landfill, 71 % of metal and glass (3,560 t) is recycled. annual distance driven by collection trucks is 118,400 km. trucks use diesel fuel; with fuel efficiency of 2.5 km/l. energy consumed by landfill operation is 0.22 l/t of diesel fuel. energy consumption at materials recovery facility: electricity 25 kwh/t, natural gas 0.264 m 3 /t. scenario 2: inorganic waste (28,840 t) is recycled (plastic, glass, paper and metal). organic waste (31,790 t) is sent to in-vessel composting plant. other waste (11,464 t) is disposed of in the landfill. annual distance driven by collection trucks is 112,596 km. energy consumption in composting process is 21 kwh/t of electricity. scenario 3: 100 % of glass and metal (5047 t) is recycled and residual waste (67,053 t) is sent to incineration with energy recovery (cogeneration plant). annual distance driven 160 b. milutinović, g. stefanović, g. vuĉković, m. tomić, p. djekić by collection trucks is 108,149 km. energy efficiency in cogeneration plant is 75%. facility energy consumption is 70 kwh/t. scenario 4: 100 % of recyclable waste – glass, metal and plastic (17,809 t) are recycled. other waste (54,291 t) is sent to anaerobic digestion plant for the purpose of electricity generation. annual distance driven by co-mingled trucks is 108,149 km. the composition of produced biogas is co2 45%, ch4 55%. energy efficiency in anaerobic digestion process is 20%. facility energy consumption is 22% of the produced energy. 3.2. experiments description there is no unique set of indicators. whether a given set of indicators is appropriate depends on its use [7]. according to the oecd key environmental indicators [6], in order to assess the impact of certain waste treatment on the environment (air, water and soil), twelve indicators are selected. the selected indicators reflect the different impact of waste treatment on the environment in developed scenarios:  climate changes – co2 equivalents,  air quality – acid gas emissions (so2 and nox), heavy metals in air (pb), smog precursors (pm, vocs, dioxins),  freshwater quality – heavy metals in water (pb, hg) and organics in water (bod),  waste generation – waste volume reduction,  energy resources – energy consumption. ghg emissions (co2 equivalent): main concerns relate to effects of increasing atmospheric greenhouse gas (ghg) concentrations on global temperatures and the earth's climate, and consequences for ecosystems, human settlements, agriculture and other socioeconomic activities. a ghg emission (co2 and methane) occurs mostly in waste landfilling and incineration. acid gases emissions (so2, nox emission): main concerns relate to the effects of air pollution on human health, ecosystems, and buildings, and to their economic and social consequences. human exposure is particularly high in urban areas where economic activities and road traffic are concentrated. acid gas emissions (so2, nox) occur mostly in waste incineration. heavy metal in air (pb): main concerns relate to the effects of air pollution on human health, ecosystems, and to their economic and social consequences. human exposure is particularly high in urban areas where economic activities and road traffic are concentrated. a heavy metal emission in air (pb) occurs mostly in waste incineration. smog precursors (pm, vocs, dioxins): causes of growing concern are concentrations of fine particulates, vocs, dioxins, toxic air pollutants episodes in both urban and rural areas. smog precursors (pm, vocs, dioxins) occur mostly in waste incineration, recycling and composting. heavy metals in water (pb, hg) and organics in water (bod): main concerns relate to the impacts of water pollution (acidification, toxic contamination) on human health, on the cost of drinking water treatment, and on aquatic ecosystems. despite significant progress in reducing pollution loads from municipal and industrial point sources through installation of appropriate waste water treatment plants, improvements in freshwater quality are not always easy to discern, except for organic pollution. effect of environmental indicators to the waste treatment scenarios ranking 161 waste volume reduction: the main challenge is to strengthen measures for waste minimization, especially for waste prevention and recycling. except landfilling, each waste treatment involves waste volume reduction. energy consumption: main concerns relate to the effects of energy production and use on greenhouse gas emissions and on local and regional air pollution; other effects involve water quality, land use, risks related to the nuclear fuel cycle and risks related to the extraction, transport and use of fossil fuels. in order to assess the impact of specific environmental indicators on the ranking of scenarios in terms of environmental impact, four experiments were performed. table 2 presents the indicators that were monitored in the four experiments as well as the basis of evaluating the change in the results with respect to the changing number of indicators. table 2 review of selected indicators indicators first experiment second experiment third experiment fourth experiment co2 equivalent (t) x x x x nox emission (t) x x x x so2 emission (t) x x pb emission in air (kg) x smog precursors – pm (t) x x x smog precursors – vocs (t) x dioxins emission in air (teq) (g) x x x pb emission in water (kg) x x x x hg emission in water (kg) x x bod emission in water (kg) x x x waste volume reduction (%) x x x x energy consumption (gj) x the first experiment examined the effect of four indicators in ranking scenarios. the selected indicators were: ghg emission (co2 equivalent), acid gases emission (nox), heavy metals in water (pb), and waste volume reduction. in the second experiment, seven indicators were selected in order to assess the ranking of scenarios in terms of the impact on the environment. the selected indicators were: ghg emission (co2 equivalent), acid gases emission (nox), smog precursors (pm), dioxins emission in air, heavy metals in water (pb), organics in water (bod), and waste volume reduction. the third experiment examined the impact of nine indicators on scenarios ranking. the selected indicators were: ghg emission (co2 equivalent), acid gases emission (so2, nox), smog precursors (pm), dioxins emission in air, heavy metals in water (pb, hg), organics in water (bod), and waste volume reduction. in the fourth experiment, twelve indicators were selected in order to assess the ranking of scenarios in terms of their impact on the environment. the selected indicators were: ghg emission (co2 equivalent), acid gases emission (so2, nox), heavy metals in air (pb), smog precursors (pm, vocs), dioxins emission in air, heavy metals in water (pb, hg), organics in water (bod), waste volume reduction, and energy consumption. 162 b. milutinović, g. stefanović, g. vuĉković, m. tomić, p. djekić 3.3. indicators evaluation amounts of ghg, acid gases, smog precursors (pm, vocs), heavy metals in air and water (pb, hg) and organics in air (dioxins) emitted to the atmosphere and water (bod) were estimated using the data from the previous paper [5] in which the amount of gas emissions and heavy metals and organics remains in water was determined on the basis of the composition of waste using the integrated waste management model [17]. the amount of waste that remained after treatment for landfill disposal and energy consumption during waste treatment was also estimated by the integrated waste management model [17]. the integrated waste management model uses life cycle methodology to quantify the energy consumed and the emissions released from a user specified waste management system. the model has been structured so that it uses data specific to the user municipality to ensure applicability of the results and accuracy. in assessing the emissions, this model takes into account the emissions from the point at which a material is discarded into the waste stream to the point at which it is either converted into a useful material or finally disposed. emissions during transportation of waste, consumption of fossil fuels and electricity for the treatment of waste, and emissions during incineration are considered. the model does not evaluate the energy and emissions associated with the production of infrastructure (e.g. collection vehicles, waste management facilities, etc). table 3 evaluation of environmental indicators indicators scenario 1 scenario 2 scenario 3 scenario 4 co2 equivalent (t) 51,507 -85,273 17,830 -69,592 nox emission (t) -2.5 -129.1 24.2 -111.6 so2 emission (t) -3.4 -174.4 -6.5 -142.7 pb emission in air (kg) 0.0 -2.6 47.1 -3.1 smog precursors – pm (t) 17.5 -51.8 -1.7 -43.5 smog precursors – vocs (t) 7.8 -98.8 -1.2 -77.9 dioxins emission in air (teq) (g) 0.001 0.0 0.048 0.006 pb emission in water (kg) 0.81 -6.33 4.97 -42.52 hg emission in water (kg) 0.0 0.072 -0.068 0.075 bod emission in water (kg) -6.0 19.043 277.0 26.354 waste volume reduction (%) 4.65 80.59 90.48 95.69 energy consumption (gj) -11,908 -1,042,126 -684,559 -1,043,542 4. results and discussion since the environmental indicators are by nature very diverse and expressed in different units, the probability or subjective evaluations, the multi-criteria decision analysis (mcda) is the appropriate method for waste treatment scenario ranking in terms of their impact on the environment. the analytic hierarchy process (ahp) is a multi-criteria decision making technique, quite often used to solve complex decision making problems in a variety of disciplines such as manufacturing industry, environmental management, waste management, effect of environmental indicators to the waste treatment scenarios ranking 163 power and energy industry, transportation industry, construction industry, etc. the ahp hierarchical structure allows decision makers to easily comprehend problems in terms of relevant criteria and sub-criteria. additional criteria can be superimposed on the hierarchical structure. furthermore, if necessary, it is possible to compare and prioritize criteria and subcriteria in the ahp practice, and one can effectively compare optimal solutions based on this information. the decision procedure using the ahp method is made up of four steps: 1) defining the problem and determining the kind of knowledge sought; 2) structuring the decision hierarchy according to the goal of the decision – in the following order: the objectives from a broad perspective, through the intermediate levels (criteria on which subsequent elements depend) up to the lowest level (which is usually a set of the alternatives); 3) constructing a set of pair-wise comparison matrices. each element of the matrix in the upper level is used to compare elements in the level immediately below; 4) using the priorities obtained from the comparisons to weigh the priorities in the neighboring level and doing this for every element. for each element in the level below its weighed values are added and its overall or global priority is obtained. this process of weighing and adding is continued until the final priorities of the alternatives at the bottom level are obtained. the ahp software expert choice 11 is employed for the analysis. the results obtained in the four experiments after employing the procedure of multicriteria analysis using the ahp method and pair-wise criteria, performed with the expert choice 11 software, are presented in the following subsections. 4.1. first experiment the first experiment examined the effect of four indicators in ranking scenarios in terms of the impact on the environment: ghg emission (co2 equivalent), acid gases emission (nox), heavy metals in water (pb) and waste volume reduction. following the pair-wise criteria, the criteria weight with respect to the goal was obtained. fig. 1 shows the ranking of scenarios after criteria (environmental indicators) weighting. according to the obtained results in the first experiment, it can be concluded that scenario 4, which corresponds to zero waste to landfill scenario and includes recycling of waste (plastic, glass, paper and metal) and anaerobic digestion, has the best ranking in terms of environmental protection. the difference between the first and the second ranked scenario is 4%. according to the selected indicators, scenario 1, the business as a usual scenario, ranked last. fig. 1 scenario ranking for evaluated indicators weight – first experiment 4.2. second experiment in order to increase the sensitivity of analysis in the second experiment, the number of indicators was increased from four to seven. three indicators were added: smog precursors 164 b. milutinović, g. stefanović, g. vuĉković, m. tomić, p. djekić (pm), dioxins emission in air and organics in water (bod). the selected indicators were: ghg emission (co2 equivalent), acid gases emission (nox), smog precursors (pm), dioxins emission in air, heavy metals in water (pb), organics in water (bod) and waste volume reduction. fig. 2 shows the ranking of scenarios after criteria weighting. according to the obtained results in this experiment, it can be concluded that scenario 2 (composting organic and recycling inorganic waste) and scenario 4 (anaerobic digestion of waste for the purpose of electricity generation) are equally ranked in terms of their impact on the environment. the reason is that scenario 2 has a lower emission of dioxins in air and organics in water. according to the selected indicators, scenario 3, which includes incineration with energy recovery, ranked last. also, the difference between scenario 1 and scenario 3 is in the order of 1.4 %. fig. 2 scenario ranking for evaluated indicators weight – second experiment 4.3. third experiment the third experiment examines the impact of the list of indicators expanded by two new indicators: so2 emission and heavy metal in water (hg). the selected nine indicators were: ghg emission (co2 equivalent), acid gases emission (so2, nox), smog precursors (pm), dioxins emission in air, heavy metals in water (pb, hg), organics in water (bod) and waste volume reduction. following the pair-wise criteria, the criteria weight with respect to the goal was obtained. fig. 3 shows the ranking of scenarios after the nine criteria (environmental indicators) weighting. according to the obtained results in this experiment, it can be concluded that scenario 2 has the best ranking in terms of environmental protection. the reason is that scenario 2 has a lower so2 emission in the air. according to the selected indicators, scenario 1, which includes landfilling of waste, ranked last. with the increase in the number of indicators, the difference between scenarios 2 and 4 increased, but the difference between scenarios 1 and 3 decreased. this suggests that it is necessary to choose different indicators or increase their number in order to make a clearer difference between scenarios 1 and 3. effect of environmental indicators to the waste treatment scenarios ranking 165 fig. 3 scenario ranking for evaluated indicators weight – third experiment 4.3. fourth experiment in the fourth experiment the impact of twelve indicators was examined. the list of indicators was expanded by three new indicators: smog precursors (vocs) and heavy metal in air (pb) and energy consumption. the selected twelve indicators were: ghg emission (co2 equivalent), acid gases emission (so2, nox), heavy metals in air (pb), smog precursors (pm, vocs), dioxins emission in air, heavy metals in water (pb, hg), organics in water (bod), waste volume reduction and energy consumption. following the pair-wise criteria, the criteria weight with respect to the goal was obtained. fig. 4 scenario ranking for evaluated indicators weight – fourth experiment fig. 4 shows the ranking of scenarios after criteria weighting. according to the obtained results in the fourth experiment, it can be concluded that scenario 4 has the best ranking in terms of environmental protection, due to higher energy savings. according to the selected indicators, scenario 1, which includes landfilling of waste, ranked last. by increasing the number of indicators, one can notice that the difference between scenarios 2 and 4 increased by 0.8%. the difference is not entirely attributed to increasing the number of indicators. the diversification of the indicators to include other aspects, such as energy consumption, is also important, and needs to be taken into account. according to the data 166 b. milutinović, g. stefanović, g. vuĉković, m. tomić, p. djekić provided in table 3, scenario 4 has the lowest energy consumption of -1,043,542 gj, while the energy consumption in scenario 2 is -1,042,126 gj. 5. conclusion the main problem in the analysis of environmental criteria is to determine the indicators that clearly and fully express the most important influential factors. to assess the impact of waste treatment on the environment, the multi-criteria analysis ahp method is applied. the selection of environmental indicators is performed as a result of the recognized priorities in environmental criteria and according to the oecd key environmental indicators. the results obtained show that by increasing the number of environmental indicators there is a change in the ranking of scenarios in terms of impact on the environment. when selecting only four indicators, there is a small difference between scenario 2 (recycling and composting) and scenario 4 (anaerobic digestion) in favor of scenario 4, although the selected indicators reflect the impact of waste treatment on air, water and soil. in case of selecting seven indicators, additionally considering pm and dioxins emissions in air and bod in water, the difference between the waste treatment scenarios is even smaller. in this case the results show that scenario 2 and scenario 4 have the least negative impact on the environment. this result suggests that the number of indicators is insufficient to describe the complete impact of anaerobic processes on the environment; it is necessary to add to the consideration a variety of indicators that reflect the different impact of waste treatment on the environment, such as energy consumption, heavy metal (pb) emission in the air, etc. if we consider nine indicators in order to assess the waste treatment on the environment, additionally considering so2 emissions in air and hg in water, then there is a change in the scenario ranking. the best ranked scenario is scenario 2. this result suggests that it is necessary to choose different indicators or increase their number in order to make a clearer difference between the scenarios. in the case of the fourth experiment which takes into consideration twelve indicators including energy consumption, the scenario ranking changes in favor of scenario 4. it can be concluded that by increasing the number of environmental indicators and considering a variety of indicators, it is possible to make a clearer difference between the waste treatments in terms of the impact on the environment. also, it is necessary to make a careful selection of indicators and their number in order to properly assess the impact of waste treatment on the environment. based on the obtained results, the recommendations for the city of niš in the selection of environmental indicators should include the consideration of a variety of indicators (12 indicators), which reflect different effects of the waste treatment on the environment. because of the composition of waste in niš, which consists of a substantial proportion of organic waste (min 45%), the indicators that take into account the effect of anaerobic digestion on the environment should be considered. further research should turn to determining a sufficient number of indicators and selecting the indicators based on the evaluation of the most important effects of the waste treatment on the environment and human health. effect of environmental indicators to the waste treatment scenarios ranking 167 references 1. alfsen k., sabo h.v., 1993, environmental quality indicators: background, principles and examples from norway, environmental and resource economics, 3 pp. 415–435. 2. ferris r., humphrey j.w., 1999, a review of potential 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http://www.sciencedirect.com.proxy.kobson.nb.rs:2048/science/journal/03772217 http://www.sciencedirect.com.proxy.kobson.nb.rs:2048/science/journal/03772217/98/1 ../my%20documents/ecos%202015/www.ni.rs/uploads/doc/uprave/ukdes/110310lpuotpad.pdf ../my%20documents/ecos%202015/www.ni.rs/uploads/doc/uprave/ukdes/110310lpuotpad.pdf http://www.iwm-model.uwaterloo.ca/ a study of thin film lubrication at nanoscale for a ferrofluid based infinitely long rough porous slider bearing facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 89 99 original scientific paper a study of thin film lubrication at nanoscale for a ferrofluid based infinitely long rough porous slider bearing  udc 532:621.8 jimit r. patel, gunamani deheri department of mathematics, sardar patel university, india abstract. the study aims at analyzing the performance of a ferrofluid-based infinitely long rough porous slider bearing which makes use of thin film lubrication at nanoscale. the stochastic model of christensen and tonder has been employed to analyze the effect of surface roughness while the neuringer-rosensweig’s model has been adopted to study the magnetization effect. the pressure distribution in the bearing system has been obtained by solving the associated stochastically averaged reynolds type equation. the results indicate that although the transverse roughness is supposed to affect the bearing system adversely, the situation remains fairly better in the case of thin film lubrication at nanoscale. in fact, the consideration of thin film lubrication at nanoscale results in an all round improved performance, even for lower strength of the magnetic intensity. however, the couple stress adds a little more to this positive effect. key words: long bearing, ferrofluid, rough surfaces, nanoscale, load-carrying capacity 1. introduction the squeeze film phenomena develop when two surfaces approach each other at a normal velocity. due to the noticeably good behavior of the squeeze film, it is often used in different fields of real life such as machine tools, gears, rolling elements, hydraulic systems, engines, clutch plates, etc. nowadays, quite a great number of theoretical and experimental inventions are made on the bearing design systems as well as on the lubricating substances in order to enhance the efficiency of the bearing performances. one of the major inventions is the use of ferrofluid as a lubricant in the bearing systems. many authors (rosensweig [1], popa et al. received november 5, 2015 / accepted february 14, 2016 corresponding author: jimit patel department of mathematics, sardar patel university,vallabh vidyanagar, anand, gujarat india-388120 e-mail: patel.jimitphdmarch2013@gmail.com 90 jimit r. patel, g. deheri [2], nada and osman [3], urreta et al. [4], huang et al. [5], patel et al. [6]) have discussed the performance and applications of ferrofluids in different bearing systems. these investigations established that the bearing performance characteristics are enhanced due to the magnetic fluid effects. another major invention has been made in the research of thin film lubrication. since 1990s, the thin film lubrication has been well discussed as a new lubrication regime. the transition from the thin film lubrication to the boundary one, the transition from the elastohydrodynamic lubrication to the thin film lubrication, the failure of liquid film at nanoscale and the mechanism of the thin film lubrication were analyzed by luo [7], luo et al. [8], and luo and wen [9]. shen et al. [10] studied the performance of liquid crystal additives in the construction of thin film. zhang et al. [11] investigated the thin film lubrication characteristics in two phase fluid. it was found that the existence of couple stress enhanced the load carrying capacity. the study of roughness, which develops after some run-in and wear, remains of primary concern for designing the bearing system and the bearing system failure. christensen and tonder [12-14] proposed a more general method of studying the performance of both the roughness patterns (transverse as well as longitudinal). gupta and deheri [15] deployed the method of christensen and tonder to analyze the performance of a rough spherical bearing. chiang et al. [16] theoretically discussed the combined effect of couple stresses and surface roughness on the instability thresholds of a rough short journal bearing lubricated with non-newtonian fluids by adopting the theory of christensen and tonder. in their study, the couple stress escorted with longitudinal roughness to provide an increase in the stability threshold speed. on the ground of christensen and tonder’s results, patel and deheri [17] investigated the characteristics of lubrication at nanoscale on the performance of a transversely rough slider bearing. it was observed that the load carrying capacity increases due to the thin film lubrication at nanoscale and the existence of couple stress. deresse and sinha [18] dealt with a thermal and roughness effect on different characteristics of the finite rough tilted pad slider bearings. it was established that for non-parallel slider bearings the load carrying capacity due to the combined effect was less than the load capacity due to roughness effect for both the roughness models. vakis and polycarpou [19] developed a model to study the mixed nano lubrication regime expected during light contact or “surfing”, recording in magnetic storage. in fact, they investigated an advanced rough surface continuum based contact and sliding model in the presence of a molecularly thin lubricant. patel and deheri [20] dealt with the effect of various porous structures on the shliomis model based ferrofluid lubrication of the thin film squeezed between the rotating rough curved circular plates. it was established that the kozeny-carman’s model remained more suitable for bearing design as compared to the irmay’s one. further, the shliomis model based ferrofluid lubrication performed relatively better than the neuringer-rosensweig one. patel and deheri [21] analyzed the effect of transverse surface roughness on the behavior of a magnetic fluid based short bearing considering thin film lubrication at nanoscale. it was observed that the thin film lubrication at nanoscale led to a sustained improvement in bearing performance characteristics even for lower values of magnetization parameter. the aim of the present study is to analyze the performance characteristics of a magnetic fluid based rough long bearing considering the thin film lubrication at nanoscale. a study on the thin film lubrication at nano scale for a ferrofluid based infinitely long rough porous... 91 2. analysis the geometrical configuration of the bearing system is presented in fig. 1. the bearing system is infinite in z direction. the slider moves with uniform velocity u in x direction. the length of the bearing is l and the breadth b is in z direction. fig. 1 configuration of the bearing system the bearing surfaces are considered to be transversely rough. in view of the study of christensen and tonder [12-14], thickness h(x) of the lubricant film is taken as h(x) h( ) sx h  (1) where (x)h represents the mean film thickness and hs denotes the deviation from the mean film thickness characterizing the random roughness of the bearing surfaces. deviation hs is considered to be stochastic in nature and governed by the probability density function: 3 2 2 35 1 , ( ) 32 0, s s s h c h c f h c c elsewhere               where c denotes the maximum deviation from the mean film thickness. mean , standard deviation  and parameter , which is the measure of symmetry of random variable hs, are defined by the relationships as discussed by christensen and tonder [12-14]. the details are dealt with there in. the magnetic field is oblique to the stator as taken by agrawal [22]. more details of on the effect of various forms of magnitude of the magnetic field have been incorporated by prajapati [23], bhat [24] and patel and deheri [25]. according to these discussions, the magnitude of the magnetic field is considered as:        l x l x klm 1sin22 , (2) where k is a suitably chosen constant from dimensionless point of view (bhat [24]), so as to produce a magnetic field of required strength. under the usual assumptions of hydrodynamic lubrication the related stochastically averaged reynolds equation (bhat [24], prajapati [23], luo et al. [26], patel and deheri [27, 28]) takes the form: moving plate fixed plate y x l h h2 u h1 92 jimit r. patel, g. deheri 2 0 6 2 ( ) ( ) mm h hd p u dx g h               (3) where ,11 2 21 2 h hh m, l x mhh                c , ch ,   32 4412 1    and 3 2 2 2 2 3( ) 3 3( ) 3g h h h h             , while c represents the characteristic length which contributes to the couple stress effect, p is film pressure 0 denoting the magnetic susceptibility, μ is the free space permeability,  being the lubricant viscosity and  is the material constant responsible for couple stress effect. the associated boundary conditions are p = 0 at x = 0 and x = l. the dimensionless quantities are as following: 33 * 0 2 2 ( )( ) , , , , , ,m m m h h kh x h h h h h x p p h h l uul               3 3 ( ) , , , , , ( ) ( ) ( ) x l g h x l g h l h h h h h                 (4) where h is infinitesimal increment in the thickness and  being a film thickness ratio. making use of the above said boundary conditions, the dimensionless pressure distribution is obtained as: * 0 6 sin(1 ) 2 ( ) ( ) x mh h p x x dx g hl         (5) the non-dimensional load carrying capacity of the bearing system, one can find as: 1 13 * 4 0 0 ( ) 6 (1 sin(1)) 2 ( ) ( ) mh hh w w pdx dx g hul l           (6) where plbw  is the load carrying capacity. 3. results and discussions the linearity of the expression in eq. (6) with respect to the magnetization parameter suggests that an increase in the magnetization would lead to an increased load carrying capacity. it is also seen that the load carrying capacity increases by * (1  sin(1))/2 as compared to the traditional lubricant based bearing system. this is due to the fact that the viscosity of the lubricant gets increased due to magnetization. the effect of film thickness parameter hm on the load carrying capacity presented in figs. 2-6 as well as table 1 make it clear that the load carrying capacity decreases as hm increases. this decrease being more in the case of variance associated with roughness. it is interesting to note that the load carrying capacity increases with increase in the standard deviation associated with roughness. however, the effect of variance on the load carrying capacity with respect to hm is not that sharp. a study on the thin film lubrication at nano scale for a ferrofluid based infinitely long rough porous... 93 fig. 2 variation of load carrying capacity with respect to hm and  fig. 3 variation of load carrying capacity with respect to hm and  fig. 4 variation of load carrying capacity with respect to hm and  fig. 5 variation of load carrying capacity with respect to hm and  94 jimit r. patel, g. deheri fig. 6 variation of load carrying capacity with respect to mh and m table 1 variation of load carrying capacity with respect to hm and  w hm=0.95 hm=1.00 hm=1.05 hm=1.10 hm=1.15 -0.01  0.037685 0.028927 0.020168 0.011410 0.002652 -0.005 0.038416 0.029812 0.021207 0.012603 0.003998 0 0.039132 0.030678 0.022224 0.013771 0.005317 0.005 0.039833 0.031527 0.023220 0.014914 0.006607 0.01 0.040519 0.032357 0.024195 0.016033 0.007871 figs. 7-10 and table 2 dealing with the influence of characteristic length suggest that the effect of couple stress is to enhance the load carrying capacity. however, the effect of variance on the load carrying capacity with respect to characteristic length is at the best nominal. the profile of augmented load due to standard deviation is given in figs. 11-14. it is appealing to note that here even the porosity increases the load carrying capacity when thickness is considered at nanoscale for the long bearing system. fig. 7 variation of load carrying capacity with respect to  and  a study on the thin film lubrication at nano scale for a ferrofluid based infinitely long rough porous... 95 fig. 8 variation of load carrying capacity with respect to  and  fig. 9 variation of load carrying capacity with respect to  and  fig. 10 variation of load carrying capacity with respect to  and m table 2 variation of load carrying capacity with respect to  and  w  = 2.0  = 2.25  = 2.5  = 2.75  = 3 -0.01 0.053634 0.043273 0.031951 0.020592 0.010063 -0.005 0.054084 0.043906 0.032783 0.021623 0.011279 0 0.054525 0.044525 0.033597 0.022633 0.012470 0.005 0.054957 0.045132 0.034394 0.023621 0.013636 0.01 0.055380 0.045725 0.035174 0.024589 0.014777 96 jimit r. patel, g. deheri also, equally interesting is the fact that the effects of variance and skewness on the load carrying capacity run approximately opposite to the usual film thickness considered for magnetic fluid lubrication. probably, this may be happening due to the characteristics of nanoscale film thickness. the positive effect of variance gets enhanced due to the positively skewed roughness which can be seen in figs. 15-17. the fact that the aspect ratio introduces a strong impact on the load carrying capacity, when considered with respect to porosity at the nanoscale film thickness, is presented in fig. 18. fig. 11 variation of load carrying capacity with respect to  and  fig. 12 variation of load carrying capacity with respect to  and  fig. 13 variation of load carrying capacity with respect to  and  a study on the thin film lubrication at nano scale for a ferrofluid based infinitely long rough porous... 97 fig. 14 variation of load carrying capacity with respect to  and m fig. 15 variation of load carrying capacity with respect to  and  fig. 16 variation of load carrying capacity with respect to  and  fig. 17 variation of load carrying capacity with respect to  and  98 jimit r. patel, g. deheri fig. 18 variation of load carrying capacity with respect to  and m some of the figures presented above indicate that there are various parameters bound to increase the load carrying capacity and therefore, probably, this may turn out to be a very useful and effective bearing system from industry point of view it is surprising to see the effect of porosity in increasing the load carrying capacity, when the thin film lubrication is considered at nanoscale. lastly, a comparison of current investigation with the observation of patel and deheri [21] indicates that the load carrying capacity enhances by around 18-20% when thin film lubrication is considered at a nanoscale. also, it is clear that here the two roughness parameters, skewness and variance, affect the performance more sharply. 4. conclusion the results presented here suggest that this type of bearing system may be selected for a long run because here even the porosity tends to increase the load carrying capacity which is starkly different from the usual behavior of porosity on other bearing systems. in general, the situation remains better in the case of positively skewed roughness even if the couple stress effect is nominal. this investigation further reveals that the magnetization may provide a suitable measure for an enhanced performance when the thin film lubrication is considered at nanoscale. even the positive effect of standard deviation may be channelized to good use. references 1. rosensweig. r.e., 1985, ferrohydrodynamics, cambridge university press, new york. 2. popa, n.c., potencz, i., brostean, l., vekas, l., 1997, some applications of inductive transducers with magnetic fluids, sensors and actuators a, 59, pp. 197-200. 3. nada, g.s., osman, t.a., 2007, static performance of finite hydrodynamic journal bearings lubricated by magnetic fluids with couple stresses, tribol. lett., 27, pp. 261-268. 4. urreta, h., leicht, z., sanchez, a., agirre, a., kuzhir, p., magnac, g., 2009, hydrodynamic bearing lubricated with magnetic fluids, journal of physics: conference series, 149(1), article id 012113. 5. huang, w., shen, c., liao, s., wang, x., 2011, study on the ferrofluid lubrication with an external magnetic field, tribology lett., 41, pp. 145-151. 6. patel, n.s., vakharia, d.p., deheri, g.m., 2012, a study on the performance of a magnetic-fluidbased hydrodynamic short journal bearing, isrn mechanical engineering, 2012, article id 603460. a study on the thin film lubrication at nano scale for a ferrofluid based infinitely long rough porous... 99 7. luo, j.b., 1994, on experimental technique and characteristic of thin film lubrication, tsinghua university doctor dissertation. 8. luo, j.b., wen, s.z., huang, p., 1996, thin film lubrication, part 1: the transition between ehl and thin film lubrication, wear, 194, pp. 107-115. 9. luo, j.b., wen, s.z. 1997, the transition between elastohydro-dynamic lubrication and boundary lubrication, 1st world tribology conference (ed. hutchings, i. m.), london: mechanical engineering publication, pp. 9-14. 10. shen, m.w., luo, j.b., wen, s.z., 2001, nano-tribological properties and mechanisms of the liquid crystal as an additive, chinese science bulletin, 46(14), pp.1227-1232. 11. zhang, c., wan, s., and luo, j., 2002, characteristic of lubrication at nano scale in two-phase fluid system, science in china, 45(2), pp.166-172. 12. christensen, h., tonder, k.c., 1969a, tribology of rough surfaces: stochastic models of hydrodynamic lubrication, sintef, report no.10 pp. 69-18. 13. christensen, h., tonder, k.c., 1969b, tribology of rough surfaces: parametric study and comparison of lubrication models, sintef, report no.22 pp. 69-18. 14. christensen, h., tonder, k.c., 1970, the hydrodynamic lubrication of rough bearing surfaces of finite width, asme-asle lubrication conference, cincinnati. oh. paper no. 70-lub-7, october 12-15. 15. gupta, j.l., deheri, g.m., 1996, effect of roughness on the behavior of squeeze film in a spherical bearing, tribology transactions, 39, pp. 99-102. 16. chiang, h.l., hsu, c.h., lin, j.r., 2004, lubrication performance of finite journal bearings considering effects of couple stresses and surface roughness, tribology international, 37(4), pp. 297–307. 17. patel, h.c., deheri, g.m., 2009, characteristics of lubrication at nano scale on the performance of transversely rough slider bearing, mechanika, 6(80), pp. 64-71. 18. deresse, g.a., sinha, p., 2011, thd analysis for finite slider bearing with roughness: special reference to load generation in parallel sliders, acta mech., 222, pp. 1-15,. 19. vakis, a.i., polycarpou, a.a., 2013, an advanced rough surface continuum-based contact and sliding model in the presence of molecularly thin lubricant, tribology letters, 49(1), pp. 227-238. 20. patel, j.r., deheri, g.m.., 2014, effect of various porous structures on the shliomis model based ferrofluid lubrication of the film squeezed between rotating rough curved circular plates, facta universitatis, series: mechanical engineering, 12(3), pp. 305-323 21. patel, j.r., deheri, g.m., 2015, on the performance characteristics of rough short bearing considering thin film lubrication at nano scale, international journal of materials lifetime, 2(1), pp. 44-50. 22. agrawal, v.k., 1986, magnetic fluid based porous inclined slider bearing, wear, 107, pp. 133-139. 23. prajapati, b.l., 1995, on certain theoretical studies in hydrodynamic and electro-magneto hydrodynamic lubrication, phd thesis, s.p. university, vallabh vidyanagar, gujarat, india. 24. bhat, m.v., 2003, lubrication with a magnetic fluid, team spirit (india) pvt. ltd. 25. patel, j.r., deheri, g.m., 2014, slip velocity and roughness effect on magnetic fluid based infinitely long bearings, proceedings of international conference on advances in tribology and engineering systems lecture notes in mechanical engineering, springer india, pp. 97-109. 26. luo, j.b., qian, l.m., wen, s.z., 1999, the failure of fluid at nano–scale, stle. -tribology trans., 42(4), pp. 912-916. 27. patel, j.r., deheri, g.m., 2013, a comparison of porous structures on the performance of a magnetic fluid based rough short bearing , tribology in industry, 35(3), pp. 177-189. 28. patel, j.r., deheri, g.m., 2015, a comparison of different porous structures on the performance of a magnetic fluid based double porous layered rough slider bearing, international journal of materials lifetime, 1(1), pp. 29-39. 10267 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 909 920 https://doi.org/10.22190/fume220127030m © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper effect of electric pulsing on the structure, texture and hardness of cryorolled fine-grain copper mikhail markushev, irshat valeev, aygul valeeva, rafis ilyasov, elena avtokratova, stanislav krymskiy, oleg sitdikov institute for metals superplasticity problems, russian academy of sciences, ufa, russia orcid ids: mikhail markushev https://orcid.org/0000-0002-6738-1018 irshat valeev https://orcid.org/0009-0002-5162-7324 aygul valeeva https://orcid.org/0000-0003-4305-4538 rafis ilyasov https://orcid.org/0000-0003-0195-1206 elena avtokratova https://orcid.org/0009-0002-4442-3913 stanislav krymskiy https://orcid.org/0000-0002-1534-3239 oleg sitdikov https://orcid.org/0000-0001-7617-3926 abstract. a synergy effect of cryorolling and high-dense electric pulsing on the structure, texture and hardness of fine-grain cu is analyzed. more than twice cu strengthening under rolling with 90% reduction at –196°с was caused by strong rolling texture and work-hardened nanostructure with ~300 nm crystallites and ~30% fraction of high-angle boundaries. further single pulsing with current intensity kj = 3.5×104 a2s/mm4 resulted in a static recovery and slight cu softening due to the formation of a more equilibrium structure with lower dislocation density and lattice microstrain. increasing kj to 3.8×104 a2s/mm4 led to a sharp drop in the cu hardness owing to continuous recrystallization and texture randomization. at kj near 5.0×104 a2s/mm4 homogeneous ultrafine-grain structure with 1 μm defect free equiaxed grains and about 30% of twin boundaries is formed. normal grain growth to 3 µm and gradual decrease of the cu hardness were taking place at higher pulsing energies, up to 8.1×104 a2s/mm4. key words: copper, cryorolling, electric pulsing, structure, texture, hardness 1. introduction one of the promising but still quite poorly studied methods of processing ultrafinegrained (ufg) (with a grain size less than 1 µm) metals and alloys is the so-called cryogenic deformation, which is carried out at temperatures below 120k (-153°с) [1]. it is well known that lowering the deformation temperature leads to the materials strengthening received: january 27, 2022 / accepted july 15, 2022 corresponding author: mikhail markushev institute for metals superplasticity problems ras, 39 khalturin st., ufa, 450001, russia e-mail: mvmark@imsp.ru 910 m. markushev, i. valeev, a. valeeva, et al. due to the suppression of dynamic recovery and accumulation of internal stresses, mainly owing to the increase in densities of crystal defects. one of the basic results of such structure changes is intensification of dynamic and static grain refinement under straining [2-5] and subsequent thermal [2,6] or other energy action [7-9]. however, a number of data published have declared that cryogenic deformation, even using methods of the severe plastic deformation (spd) with strains of e>1, does not guarantee the ufg structure formation. for instance, it was found [10] that the spd, realized by cryorolling (cr), did not result in the expected ufg structure development and even the significant grain refinement in pure aluminum. similar behavior has been also found in copper and brass [11, 12], being conditioned by suppressing the dislocation double cross slip with temperature decrease. the latter, in turn, complicated the formation of strain-induced boundaries and grain fragmentation (subdivision) and thus restricted the development of new grains by dynamic recrystallization. nevertheless, the authors of [13-17] concluded that a complex thermomechanical treatment (tmt), involving cryorolling and subsequent high-dense electric pulsing (hdep) [18, 19], ensured the ufg sheet production. it should be noted that under such a treatment the regimes of pulsing played a dominant role in intensity and completeness of static recovery and recrystallization of a preliminary highly work-hardened material, allowing grain refinement due to the extremely fast heating and short time thermal action at hdep. another critical point is the effect of the initial (before cr) state on the structure development in the processing material. for instance, in [13] the hdep following cryorolling was shown as successful technology to process cu sheets with near homogeneous ufg structure. however, the result was obtained for the preliminary severely forged ufg billet [20]. in that case, the processing was narrowed to a quite simple task – to preserve the processed ufg structure during further cr and hdep. so, the crucial point is to analyze the ability of cu with coarser grains to form ufg structure under the similar tmt mode. thus, the aim of the present investigation was to evaluate the potential of tmt, basing on cryorolling as the only solo spd technique in combination with powerful single electric pulsing, and to study the effect of hdep energy on structuring and hardening/softening of pure copper with initial fine-grain structure. 2. material and methods the ingot out of pure copper (designated as m1 in accordance with the russian standard gost 859-2001) was preliminary isothermally forged at 850°c and further annealed at 500°c for 2 hrs to process recrystallized fine-grain structure with 10-15 µm grain size [21]. subsequent spd via rolling with reduction of about 5% per pass of mechanically cut out plates was carried out to a total strain of about 90% (e=2.3) at –196°с using six-roll mill. isothermal conditions of straining were obtained by the preliminary 1 hr holding of the billets and removable working rolls in liquid nitrogen. further hdep (see more details in [13-16]) was carried out on the dog-bone samples with 3×4 mm2 gage part, cut out from 0.4 mm thick sheets along their rolling direction. for the used integral current density (kj) from 3.5×104 to 8.1×104 a2s/mm4, the calculated temperature of gage heating (tc) was within the range of 260-900°c. with the impulse time of about 10-4 s and the current frequency of 104 hz, the depth of its penetration inside the sample was about 0.92 mm. this means the absence of the skin-effect under hdep, suggesting a uniform current distribution in the cross-section of a sample. effect of electric pulsing on structure, texture and hardness of cryorolled fine-grain copper 911 for distinct characterization of structures under tmt the scanning electron microscope "tescan mira 3 lmh" equipped with "hkl channel 5" software was used. as was recommended in [22, 23] the ebsd data and maps were obtained from 6 kikuchi-line scanning in the rolling plane with two steps of 100 and 500 nm. grain boundary misorientation angle of 15° was taken as criterion for division into lowand high-angle boundaries (labs and habs). the mean subgrain and grain sizes (d and d, consequently) were determined as equivalent diameter of not less than 900 crystallites. the average angle of misorientation (θ) and fraction of habs and twin boundaries (σ3 and σ9) (fhab and fς, consequently) were determined for boundaries with misorientation angle higher than 2°. the fraction of recrystallized grains (frec) was calculated as the ratio of the areas of recrystallized grains (with internal lattice distortion less than 2°) and the whole scanning area. pole figures and texture components were derived from 500 nm scanning of not less than 2000 crystallites [22] with an angular deviation of 15°. x-ray diffractometry (xrd) was carried out on a dron-4-07 in cu-kα radiation at a voltage of 40 kv and a current of 30 ma with a wavelength of λ = 1.54418 å, and a graphite monochromator on a diffracted beam. the scanning step of 0.1° and an exposure time of 4 s with rotation of the sample were used. the root-mean-square microstrain of the copper lattice (<ε2>1/2), as well as the coherent domain size (d), were determined by the fullprofile method in the maud software. the dislocation density (ρ) was found as ρ=2√3<ε2>1/2/db, where b is the burgers vector. strength of the copper was characterized by the vickers microhardness, estimated from not less than ten measurements per point on the mvdm 8 "affry" tester at 0.5n force and the loading time of 10 s. if not mentioned, the error of measurements of any parameter did not exceed 5%. 3. results and discussions four characteristic intervals on the cu hardness dependence vs. the regimes of the tmt studied were clearly distinguished (fig. 1). the first one (i) derived from the copper multiple strengthening due to cryorolling. in the second range (ii), lasting to hdep with low energies, hardness changed scarcely, indicating the interval of pulsing where the deformation structure was quite stable. the third (iii) stage started from some "threshold" level of current density (about 3.5×104 a2s/mm4) and occupied an extremely narrow range of pulsing energy, resulting in a sharp drop in cu hardness. and in the fourth one (iv) (at kj> 3.8×104 a2s/mm4), there was a final and less intense softening of the strained metal down to the level close to the starting fine-grain annealed condition. judging by the data in figs. 1 and 2, and tabs. 1 and 2, the hdep with the minimum energy of 3.5×104 a2s/mm4 did not cause any significant changes in the linear parameters of the structure, hardness and the lattice parameter of as-rolled copper. the structure still remained partially recrystallized deformation-like one with well-developed nanosized (sub)grains (figs. 2a-d). it should be noted that in distinction with the data published in [21], the present measurements obtained with less step of scanning, gave significantly reasonable data, regarding, first of all, the grain and subgrain sizes (tab. 1). simultaneously, hdep led to sense decrease in the coherent domain size and enhancement of angular parameters, involving fractions of habs and recrystallized grains (tabs. 1 and 2). it was also accompanied by more than 50% reductions of the micro-stresses and the dislocation 912 m. markushev, i. valeev, a. valeeva, et al. densities. obviously, the structure transformations found indicated an occurrence of static recovery and recrystallization in-situ, which, along with a noticeable increase in the structure equilibrity and homogeneity, supposed the significant cu softening. thus, the absence of a noticeable loss of hardness can only be attributed to structural strengthening due to transformations of the deformation-induced dislocation boundaries into more equilibrium (sub)grain ones. virtually, such changes have to result in cu hardening, which turned out to be sufficient to compensate softening caused by a reduction in the lattice defectiveness. fig. 1 cu hardness (hv) changes due to cryorolling (cr) and energy of high-dense electric pulsing (hdep) (bottom scale) / corresponding temperature (tc) (upper scale) when the current density reached 3.8×104 a2s/mm4, the stored deformation energy (and hardness, consequently (fig. 1)) are sharply reduced, owing to the development of a statically recrystallized ufg structure (fig. 2e,f) with a significantly higher subgrain-, grainand coherent domain size (tabs. 1 and 2). the structure was also characterized by almost complete relaxation of lattice micro-stresses due to recrystallization [24] and a drastic decrease in dislocation density to equilibrium level (tab. 2). it is also noteworthy that the almost fivefold simultaneous rise of the fʃ was found, revealing a significant role of twinning in the occurrence of static recrystallization. judging by the changes in the grain boundary spectra (figs. 2e-h and 3, and tab. 1), formation and development of annealing twins were the character for hdep structures up to completeness of recrystallization at kj = 6.8×104 a2s/mm4. with further increase in impulse capacity to 8.1×104 a2s/mm4 the weaker decreased hardness (fig. 1) was mainly conditioned by a faster grain and subgrain coarsening (figs. 2k,l and tab. 1), predominantly via normal growth, which took place even under extremely short-term heating followed by cooling to room temperature. so, the electric pulsing with high energies resulted in grain coarsening and in the loss of both the workhardening and the hallpetch grain refinement effects. from fig. 3 it also follows that some impact in cu softening at this tmt stage was attributed to the changes in grain boundary spectra. indeed, ebsd data have shown the decrease in fractions of ʃ9 twin boundaries and habs of 35-55o misorientation effect of electric pulsing on structure, texture and hardness of cryorolled fine-grain copper 913 fig. 2 grain boundary maps for cu after cr (a,b) and hdep with current density of 3.5×104 (c,d) and 3.8×104 a2s/mm4 (e,f). scanning steps are 500 (a,c,e) and 100 nm (b,d,f). hereafter red, black and blue lines indicate low-angle, high-angle and twin (σ3) boundaries, consequently 914 m. markushev, i. valeev, a. valeeva, et al. fig. 2 (continuation) grain boundary maps for cu after cr and hdep with current density of 6.8×104 (g,h) and 8.1×104 a2s/mm4 (k,l). scanning step 500 (g,k) and 100 nm (h,l) angles with increasing the impulse energy. another interesting finding was observation of a lower coherent domain size with grain coarsening (tabs. 1 and 2). virtually, it could be reasoned by the formation of new labs, dividing new grains, as was found under electric pulsing in [25]. from grain size distributions (fig. 4) three modes of dependencies, describing the processes of cu structure changes under the tmt, were evidenced. that is, the first one with a sharp peak at the nanoscale level was found in cr-ed and hdep-ed states with current density less than the threshold value, being attributed to the formation of deformation structure and its evolution under recovery. the second one was observed in hdep-ed in an intermediate power range, owing to activation of continuous static recrystallization up to its full completeness. and the third one originated from the normal grain growth at a higher energy of pulsing. therewith, the first type indicated the absence of any grain coarsening even of the finest grains, the second – the grain coalescence and/or limited grain growth. under the third one the normal grain growth and twinning changed the dependence, making it linear-like in semi-log coordinates. effect of electric pulsing on structure, texture and hardness of cryorolled fine-grain copper 915 table 1 sem-ebsd data for cu after cryorolling and electric pulsing * scanning step 100 nm, ** scanning step 500 nm [21] fig. 3 boundary spectra in cu after cr (a) and hdep with the current density of 3.5×104 (b), 3.8×104 (c), 5.0×104 (d), 6.8×104 (e) and 8.1×104 a2s/mm4 (f). scanning step is 500 nm. table 2 x-ray data for cu after cryorolling and electric pulsing condition k j , ×10 4 a 2 s/mm 4 lattice parameter å ρ 10 14 m -2 d nm <ε 1/2 > % cr 3.608±0.001 4.5 57±2 0.186±0.004 hdep 3.5 3.2 48±2 0.111±0.005 3.8 0.2 67±4 0.001+0.001 5.0 0.2 81±3 6.8 0.1 100±4 8.1 0.2 69±3 condition k j , ×10 4 a 2 s/mm 4 d μm d μm f rec % θ degree f habs % f σ % cr 0.3* 0.7* 15** 17** 33** 2** hdep 3.5 0.4 0.9 37 21 42 5 3.8 0.8 1.0 86 39 79 22 5.0 0.8 1.3 96 43 88 22 6.8 1.2 1.3 98 46 93 30 8.1 1.9 2.5 100 47 90 33 916 m. markushev, i. valeev, a. valeeva, et al. fig. 4 grain size distributions in cu after cryorolling (cr) and pulsing with different current densities (kj) all the above discussed cu structural changes resulted in appropriate texture transformations. the {100}, {110} and {111} pole figures of the cr metal (fig. 5a) showed rather strong texture with brass and s (tab. 3) dominant components, forming a texture different from the rolled copper-type one found in numerous studies, e.g. [2,26,27]. this may be explained by the fact that the materials, which form the pure metal texture when rolled at room temperature, can develop the alloy-type texture upon straining at lower temperatures [28]. transition from one type to another is allegedly the result of dislocation cross-slip behavior, which can be suppressed at cryogenic conditions. subsequent hdep with kj=3.5×104 a2s/mm4 retained the pole figures (fig. 5b) and intensities of texture components (tab. 3) almost as in the cr state. the significant changes started under pulsing with 3.8×104 a2s/mm4 (fig. 5c). however, the expected strong сubetexture, which is usually formed under static recrystallization in conventional workhardened cu and attributed to the 40° <111>-orientation relationship with the s-orientation in the rolling texture [28], has not been found in all hdep-ed conditions. instead, there were the randomization and weakening of the cr texture accompanied with a near triple decrease in brass-, twice decrease in samid retention of a fewer amount of the copper-, cubeand gosscomponents (figs. 5c-e). besides, the hdep led to increase in some other components, such as brass-r (2 3 6) <3 8 5>, dillamore (4 4 11) <11 11 8>, and especially in the component with the euler angles (50°;30/60°;0°) (tab. 3). judging by the data in [13], such texture transformations could be conditioned by the reaction of the highly deformed metal to the activation of static recrystallization, as was discussed in [12] for furnace annealing of the cr copper. and minimum texture changes with further increasing the hdep energy (fig. 5f, tab. 3) were caused by the completeness of recrystallization with the formation of equilibrium ufg structure with no sign of a drastic grain size increase. activation of normal grain growth under more powerful pulsing resulted in a slightly intense texture, owing to a higher impact of brassand scomponents. the latter could also be due to active twinning. effect of electric pulsing on structure, texture and hardness of cryorolled fine-grain copper 917 table 3 texture components in cu after cryorolling and electric pulsing condition kj, ×104 a2s/mm4 texture component, % brass (35;45;90) copper (90;35;45) s (59;37;63) brass-r (80;31;35) dillamore (50;30;0) (55;30;0) other cr 35 7 35 8 6 2 7 3.5 36 7 37 8 8 4 0 3.8 11 6 18 9 13 11 32 hdep 5.0 8 8 16 11 13 13 31 6.8 7 9 18 15 14 14 23 8.1 11 9 23 15 14 13 15 thus, the hardening/softening behavior of the copper during cryorolling and subsequent electric pulsing in the present study do not qualitatively differ from the previous ones, reported for copper [3,13], nickel [15,16] aluminum [17] and other pure metals and solid solutions [2,18,19]. therewith, similarity in their structural behavior was expectable, too. in particular, the phenomenology and nature of the structuring of the cu under cr found (figs. 2 and 3) were quite close to those observed in ni, having the same lattice and close staking fault energy. indeed, the same type of a structure a partially recrystallized deformation ufg structure was formed in both metals. the main operating mechanisms of its formation were grain fragmentation (subdivision), dynamic polygonization and recrystallization. meanwhile, due to the low deformation temperature, these processes resulted in the prevailing formation of low-energy dislocation structures, followed by their transformations into a mixed grain/subgrain structure (fig. 2 and tab. 1) with a predominance of dislocation cells misoriented to several degrees and thus, divided mainly by labs. like in other materials, the high dislocation densities (tab. 2) and the extreme refinement of grains and subgrains (tab. 1) were the main factors that ensured double strengthening of copper (fig. 1). basing on the data presented, one can also conclude that, as in the case of continuous static recrystallization, which is usually observed under furnace annealing of spd-ed metals [29,30], at the hdep with energies below the "threshold" kj value the straininduced boundaries were gradually transformed to more equilibrium ones under control of static recovery. in the kj range of 3.5×104 6.8×104 a2s/mm4 the main process of structure evolution was in-situ recrystallization with limited grain-boundary migration, accompanied with a gradual "transition" to normal grain growth. a sharp drop in hardness at the beginning of this interval, on the one hand, could be caused by fast growth of fine grains formed prior the hdep. on the other hand, new in-situ formed grains with more equilibrium boundaries acquired a similar ability, limiting grain growth. in both cases, the growth of crystallites was conditioned by a high level of the stored energy and inhomogeneity of the deformation structure. with the amount of pulse energy introduced exceeding the threshold level, the recovery intensively "released" most crystallites from defects and ensured their ability to grow (coalescence), while part of them was still "occupied" by dislocations and underwent polygonization or twinning. at the same time the grain growth was a consequence of restructuring and migration of boundaries, which absorbed defects, and reduced energy of a system and hardness of a material. it should be noted the high impact of twinning in recrystallization of cr copper and nickel under hdep. this phenomenon requires special discussion over the frame of the present paper, as some other points of effectiveness of the тмт studied. 918 m. markushev, i. valeev, a. valeeva, et al. fig. 5 pole figures for cu after cryorolling (a) and further pulsing with current density of 3.5×104 (b), 3.8×104 (c), 5.0×104 (d), 6.8×104 (e) and 8.1×104 a2s/mm4 (f) effect of electric pulsing on structure, texture and hardness of cryorolled fine-grain copper 919 4. conclusions single electric pulse processing of preliminary cryorolled fine-grain copper having highly work-hardened nanostructure with crystallite size of about 300 nm, 30% of habs fraction and intense rolling texture with brass and s dominant components, carried out with a current densities below the "threshold" value of kj = 3.5×104 a2s/mm4, led mainly to the occurrence of static recovery. this resulted in a more homogeneous and equilibrium structure with minimum changes in texture and structure parameters, as well as in hardness close to that after cryorolling. exceeding the kj “threshold” level was accompanied by a transition to in-situ continuous static recrystallization with limited grain boundary migration and a sharp softening of cu due to the loss of deformation hardening effect upon transformation of the deformation structure into a recrystallized one with a grain size about 1 µm and more random texture. increasing hdep energy in the kj interval from 4×104 to about 8×104 a2s/mm4 led to normal grain growth to 3 μm and an even higher copper softening. the nature of the processes, realized under cryorolling and electric pulsing of cu, are similar to those occurring during conventional cold straining and annealing. the obtained results testify that the combination of cryorolling and нdep may be quite advisable for processing fineand ultrafine-grain copper sheets of a wide strength range. acknowledgements: the work was supported by the russian ministry of science and higher education through the state research target and performed using the facilities of the shared 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copper during multidirectional forging at 195 k, philosophical magazine letters, 87, pp. 751-766. 30. belyakov, a., sakai, t., miura, h., kaibyshev, r., tsuzaki, k., 2002, continuous recrystallization in austenitic stainless steel after large strain deformation, acta materialia, 50, pp. 1547-1557. 11152 facta universitatis series: mechanical engineering vol. 22, no 1, 2024, pp. 45 62 https://doi.org/10.22190/fume221221011w © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper modelling and simulation of nonlinear dynamic flow field and temperature field of depyrogenation tunnel yizhi wang1,5, quanmin zhu2, tao huang3, xiaodong han4,5, min lin6 1college of intelligent science and control engineering, jinling institute of technology, china 2department of engineering, design and mathematics, university of the west of england, frenchay campus, uk 3wuhan zhongqi technology co., ltd, china 4college of computer and control engineering, minjiang university, china 5fujian provincial key laboratory of information processing and intelligent control, minjiang university, china 6school of innovation, entrepreneurship and creation, minjiang university, china orcid ids: yizhi wang https://orcid.org/0000-0002-8648-9016 quanmin zhu https://orcid.org/0000-0001-8173-1179 tao huang https://orcid.org/0009-0004-3752-7153 xiaodong han https://orcid.org/0009-0008-5185-8070 min lin https://orcid.org/0000-0002-3893-9174 abstract. nonlinear dynamics plays a crucial role particularly in equipment validation of preparation production, validation results of which will significantly influence the results and period of drug registration in drug production processes. in this research, the flow field and temperature field simulation, calculation and analysis are creatively carried out in terms of depyrogenation tunnel, a very popular preparation dryingsterilization equipment in pharmaceutical processes with strong dynamic characteristics. after construction of 3d model of this equipment using catia and mesh generation applying ansys, the computational fluid dynamic (cfd) method and verification of irrelevance method are carried out regarding 6.05 million of meshes to identify the flow velocity model and heat transfer model inside the equipment, to further provide methodology for pharmaceutical process validation and to further optimize the design of control methods. after calculation and simulation, the low-velocity vortices of different sizes inside the hood and the drying chamber are identified, which could cause vials to fall down; meanwhile, vials that are farther away from the outlet receives less heat exchange effect, which would shrink the effective sterilization area, indicating an inadequate validation methodology in pharmaceutical processes. key words: depyrogenation tunnel, flow field, temperature field, nonlinear dynamic simulation received: december 21, 2022 / accepted march 10, 2023 corresponding author: min lin school of innovation, entrepreneurship and creation, minjiang university, no. 200, xiyuangong road, shangjie town, minhou county, fuzhou 350121, china e-mail: minlin@mju.edu.cn https://orcid.org/0000-0002-8648-9016 https://orcid.org/0000-0001-8173-1179 https://orcid.org/0009-0004-3752-7153 https://orcid.org/0009-0008-5185-8070 https://orcid.org/0000-0002-3893-9174 mailto:minlin@mju.edu.cn 46 y. wang, q. zhu, t. huang, x. han, m. lin 1. introduction in pharmaceutical engineering, an extremely significant step is sterilization, especially for preparations that directly enter the human body, and its effect is directly related to the safety of people's lives. physical sterilization has little effect on the activity of drug molecules. due to this characteristic, in the pharmaceutical industry, physical sterilization is more widely used than chemical sterilization. in physical sterilization, dry heat sterilization is often used, while a dry heat sterilization depyrogenation tunnel is applied as the principal sterilization equipment. compared with the far-infrared radiation sterilization method, the implementation process of dry heat sterilization is more controllable in the control of temperature and pressure difference, and it is more in line with good manufacturing practice (gmp) requirements, resulting in higher sterilization quality and efficiency [1, 2]. consequently, under this premise, the control of the temperature and the control of differential pressure of the sterilization depyrogenation tunnel have become the key control parameters of the equipment. in this regard, wang and quan and their groups [3-5] have done research into the optimization of control systems. the optimal design was conducted using methods such as a state space method and type-2 fuzzy control. moreover, the simulation verification was carried out, and the control effect was theoretically achieved. however, in the actual application process, a series of issues were observed, namely, there was a large difference in temperature rise in the vial due to the uneven distribution of airflow in the depyrogenation tunnel; the abnormal state such as lodging of the dried object placed on the track was caused by the change of wind pressure; during the operation of the depyrogenation tunnel, problems such as energy waste were generated due to the unreasonable artificial regulation. as a result, an in-depth study of the temperature and wind speed distribution in the sterilization depyrogenation tunnel, analysis of the heating characteristics of the dried object, and reduction of energy consumption during the sterilization process of the dried object are of a great theoretical guiding value and engineering application value to improve the production efficiency of the drying of pharmaceutical packaging materials. catia software is innovatively applied to model and analyze the representative industrial complex system of the sterilization depyrogenation tunnel, filling the gap in the research on the influence of the equipment structure on the internal flow field. in addition, ampoules are usually taken as dried objects [6] in the subsequent computational process. however, vials have become popular in recent years for their high error-tolerant rate and injection efficiency [7] and the relevant computational study regarding drying and sterilization processes are yet to be fulfilled. therefore, taking the common vials as the dried objects, the flow field in the equipment is numerically simulated and studied, which provided a guiding principle for the subsequent optimization and finally the improvement of the control system. furthermore, the fluid field and thermal dynamic field are simply considered separately through a computational study for ovens. however, under the specific equipment using hot air for sterilization [3-5], the thermal dynamic field and fluid field are coupled together, which means that when the parameter of velocity is changed, it will influence the thermal dynamic field as well. however, the relevant study is left to be filled. taking the drying and depyrogenation process of vials in a depyrogenation tunnel as the research object, this research studied the heat transfer field and fluid flow field and its coupled nature in hot air circulation. specifically, it studied the problems of uneven flow field of the depyrogenation tunnel and the unreasonable control of the heating process https://dict.youdao.com/w/eng/principal/#keyfrom=dict.basic.discriminate modelling and simulation of nonlinear dynamic flow field and temperature field... 47 during the drying and depyrogontation of vials. using the computational fluid dynamic (cfd) numerical simulation method, the airflow distribution characteristics inside the tunnel and the heating characteristics of the vials were explored, which provided a theoretical basis for improving product quality and reducing production energy consumption and subsequent further modification of control system parameters. the contributions of this paper are as follows: ▪ according to the geometric parameters of the depyrogenation tunnel, using catia software, the three-dimensional physical model of the sterilization section of the depyrogenation tunnel was established, and the resistance characteristic parameters of the porous medium were determined through experiments; the 3d simulation model of the sterilization depyrogenation tunnel was simplified, and mesh-independent verification was carried out. moreover, the accuracy of the simulation model was verified by using the speed test results of the depyrogenation tunnel in no-load and cold state. ▪ the internal flow field of the depyrogenation tunnel has been simulated. through the evaluation index of flow characteristics, the uniformity of the outlet velocity of the depyrogenation tunnel was evaluated. ▪ after establishing a three-dimensional model of the vials in the depyrogenation tunnel, the transient simulation of the heating process of the vials in the depyrogenation tunnel was performed, and its energy consumption was analyzed and calculated; the influence of process parameters on the heating characteristics of vials, residence time and energy consumption of the depyrogenation tunnel was investigated, and the process parameters were identified. 2. critical review along with the rapid development of numerical simulation technology, a variety of numerical methods based on theoretical calculation, such as an iterative optimization method [8], finite element method [9], interpolation method [10] and neural network method [11] have been widely used in sterilization process research. however, realization of all these intelligent algorithms needs the support of large base of experimental data, which is to validate the drying and sterilization results of vails taken for experiments rather than to guarantee that every vial has been dried and sterilized as required. accordingly, it is particularly essential to analyze the temperature field of the heating process of the vial in the sterilization depyrogenation tunnel by using the method of numerical simulation. meanwhile, the analysis and application in drying and sterilization processing in the pharmaceutical industry using cfd regarding the airflow field and temperature field concurrently is yet to be flourished for the following reasons: ▪ the influence of thermal transfer to food flavor is under the spotlight due to the development of food industry and market demand rather than the pharmaceutical industry [12]. for example, the local convective heat transfer coefficient of oven is studied using cfd in terms of bread baking process for better flavor and less energy consumption [13-15]; a drying process and its influence to a solvent in terms of can industry is studied using a cfd method [16]; a drying and sterilization process of bottled milk is studied and the influence of the way to place the bottle inside the oven for a heating period is addressed [17]. however, the analysis of thermal process in the 48 y. wang, q. zhu, t. huang, x. han, m. lin pharmaceutical industry is keen to be studied regarding containers, such as vials and needle cylinders. as the temperature elevating process differs from material to material, the drying and sterilization process of the containers requires specific calculation. ▪ there are some studies that look into the numerical analysis of sterilization ovens. for instance, several structure improvements of sterilization ovens have been proposed with considerable simulation results after cfd regarding uniformity of airflow inside the oven [18-20]. however, from industrial perspective, the practical validation of equipment is missing which limits the structure improvement methods to be applied to practical use due to the equipment validation requirement from sterilization process validation, according to current good manufacture practices (cgmps). moreover, the oven-like sterilization equipment in these studies are usually considered as batch-based production equipment rather than continuous production equipment, also known as tunnel-like equipment. the airflow is more independent from heat distribution in ovenlike equipment and is more predictable, while the heat distribution and airflow in tunnellike equipment are mutually dependent with strong dynamics. ▪ there are other studies that look into the heat transfer process of preparation containers. vials are considered as a porous medium for overall numerical calculation, which loses the individual vial’s heating curve [21]. the temperature elevation process of an individual vial [22] and a single batch of asparaguses in a still can [23] are respectively studied to obtain their heating curve prediction. wang’s research [22] reveals the airflow distribution inside a depyrogenation tunnel using a two-dimensional model. the mentioned results are obtained via a series of simplification models, which cannot reflect the temperature and airflow three-dimensional distribution inside the depyrogenation tunnel. from the analysis above, the bottleneck issue of this area is identified as the strong dynamics from pharmaceutical industry applications compared to impaired dynamics after mathematical modelling and control, leading to solutions of desired simulation results with less reliable consequences. therefore, in order to validate the designed control system coping with the industrial applications from the purpose of cgmps, current on-site equipment, a depyrogenation tunnel, is specially adopted for computational research through geometric modelling, meshing and irrelevance validation. 3. computational modelling the hot air circulation sterilization method is normally used in the dry heat sterilization depyrogenation tunnel, as shown in fig. 1. here the typical equipment is adopted for the following geometric modelling and simulation calculation, with its detailed description found in previous work [3]. in this research, a geometric model is constructed via catia v5r20, making use of its powerful surface modelling. catia v5r20 is widely-used in design of vehicles and aerospace fields, but yet to be adopted in plant equipment design and analysis. regarding numerical simulation of fluid, it is calculated via ansys 17.0, a large-scale general-purpose finite element analysis software with abroad application internationally. modelling and simulation of nonlinear dynamic flow field and temperature field... 49 fig. 1 schematic diagram of depyrogenation tunnel 3.1. geometric modelling as shown in fig. 2, the overall structure of the depyrogenation tunnel is mainly composed of heaters, inlet air pipes, induced draft fan, fan runner, air hood, high efficiency filter, drying chamber. the part structure is specifically modeled as shown in fig. 3. fig. 2 a) overall structure of the tunnel b) heating tube area, c) induced draft fan, d) filter and drying chamber, e) air hood b) a) c) d) e) 50 y. wang, q. zhu, t. huang, x. han, m. lin fig. 3 composition structure of the tunnel impeller during the high-temperature sterilization process of the hot air circulation equipment, the hot air firstly is pumped from both sides of the heating pipe and heated from 300℃ to 350℃ by the heater, and the air flows through the hood are filtered to remove dust particles via a group of high-temperature-tolerant filters. the purified hot air enters the drying room to sterilize the vials at high temperatures. afterwards, the hot air re-enters the heater through the return air duct and is heated again, so that the process of hot air circulation is completed. meanwhile, part of the energy is lost during heat exchange between the circulating hot air and vials; while the temperature of hot air is lowered, it is still much higher than the room temperature. thus, during the reheating process, the heating tube will heat the air to the required temperature by the tunnel with less energy, so as to achieve the purpose of energy saving. the heat source structure in the heating tube area is defined as: a cylinder with a diameter of 14 mm and height of 200 mm. sixty heating rods are determined and distributed on each side, then the total number of heating rods is 120. the air inlet surface is chosen rectangular, with a size of 754 mm*247.5 mm. the rotational speed of the induced draft fan impeller is set as 3000 rpm. it is worth mentioning that these specifications defined here mainly facilitate calculation and modeling, rather than the exclusive specification for the tunnel. 3.2. mathematical modelling the sterilization process is in the middle of the continuous production line which is not a sealed container. therefore, the airflow inside the tunnel is considered as incompressible flow, and it complies with the continuity equation and momentum equation. during the flowing process of air flow, the convective heat transfer and conduction heat transfer are considered, through which energy conservation is satisfied. considering the power of heat source is rather low, heat conduction through thermal radiation plays a far less significant role and is omitted. therefore, the analysis is conducted based on continuity equation, momentum equation and energy equation respectively as follows. 3.2.1. continuity equation all fluid motion problems must comply with the law of conservation of mass, that is, within a unit of time, the mass of the in-flowing fluid and the mass of the out-flowing fluid remains the same. in other words, the net inflow fluid mass is zero. according to this principle, the mass conservation equation is obtained as: modelling and simulation of nonlinear dynamic flow field and temperature field... 51 ( ) ( ) ( w) 0 t u v x y z        + + + =     (1) 3.2.2. momentum equation the momentum equation is also known as the naiver-stokes equation, which can be described as: the rate of change of the momentum of the fluid with time was equal to the sum of various external forces acting on the fluid. the flow rate, viscosity, density, humidity and other index changes of the micro-element are included by the naiver-stokes expression, which all vary over space and time. ( ) div( ) yxxx zx x u p u f t x x y z       + = − + + + +      u (2) ( ) div( ) xy yy zy y v p v f t y x y z         + = − + + + +      u (3) ( ) div( ) yzxz zz z w p w f t z x y z       + = − + + + +      u (4) where: ρ density of air, t fluid flow time, u, v, ω velocities of the micro-element along the x, y, and z coordinate axes respectively, fx, fy, and fz are the sum of various external forces in total along the x, y and z coordinate axes respectively. 3.2.3. energy equation according to the energy equation of fluid dynamics, the growth rate of energy in the micro-element is equal to the net heat entering the micro-element and the total work done by the surface force and physical force on the micro-element. thus, the energy conservation expression is obtained as: ( ) div( ) div[ grad ] t p t h t t s t c    + = +  u (5) where: t thermodynamic temperature, cp specific heat capacity h heat transfer index of the fluid st internal heat source of the fluid. the part where the mechanical energy of the fluid is converted into heat due to the viscous effect, which is referred to as the viscous dissipation term for short. in order to construct closure equations to describe the flowing process and heat transfer equations, a state equation is introduced as eq. (6): ( ),p p t= (6) in terms of ideal gas, the equation of state is: p rt= (7) where the molar gas constant is denoted by r. 52 y. wang, q. zhu, t. huang, x. han, m. lin 3.2.4. turbulence modelling the standard k ε turbulence model is adopted for numerical simulation according to the air flow in the tunnel. ε that represents the dissipation rate in the model is defined as: ' ' i i k k u u x x        =        (8) here μt turbulent viscosity can be expressed as a function of k and ε as follows in specific: ε k ρcμ μt 2 = (9) in the standard k-ε turbulence model, k and ε represent two fundamental unknowns, and their corresponding transport equations are as follows [24]: ( ) ( ) k i t b m k i j k j k ku k g g y s t x x x             + = + + + − − +          (10) ( ) ( ) ( ) εεbkε jε t ji i s k ε ρcgcg k ε c x ε σ μ μ xx ρεu t ρε +−++                   +   =   +   2 231  (11) where: cμ, c1ε, c2ε, c3ε empirical constants, gk generation term of the turbulent dynamic energy caused by the average velocity gradient, k turbulent dynamic energy, gb generation term of turbulent dynamic energy caused by buoyancy, ym contribution of pulsation expansion in compressible turbulence, δk, δt prandtl numbers corresponding to k and ε respectively, sk, sε user-defined source items. 4. mesh generation and boundary conditions the realization of the tunnel mesh is generated by ansys icem cfd software. the mesh generated by ansys is an unstructured mesh type, with all tetrahedral elements. in order to ensure the accuracy and reliability of the calculation, the detailed structures such as the impeller of the induced draft fan and the heating pipe are subjected to mesh refinement processing, as shown in fig. 4. in order to study the influence of the number of meshes on the calculation results, a total of five different numbers of meshes were distinguished, 2.32 million, 3.45 million, 5.69 million, 6.05 million and 6.77 million of meshes, respectively. subsequently, for the above five mesh schemes, mesh independence test calculations were performed. the value of empirical constants are as follows: c1ɛ=1.44, c2ɛ=1.92, c3ɛ=1.0, cµ=0.09, σk=1.0, σɛ=1.3. modelling and simulation of nonlinear dynamic flow field and temperature field... 53 (a) tunnel integral mesh (b) mesh of heating tube and induced draft fan area (c) heating tube mesh (d) mesh of filter and drying chamber fig. 4 tunnel meshing as mentioned previously, the standard k-ε turbulence model is adopted for calculations. the inlet of the tunnel adopted the speed inlet boundary condition, with the speed of 4.2 m/s, the inlet temperature of 573 k, the impeller speed of the induced draft fan was 3000 rpm, and the rotation direction of the induced draft fan is along the positive y axis. in terms of outlet pressure, the outlet boundary condition is used, with relative pressure zero. the entire outer wall of the tunnel is set as the adiabatic boundary condition; the heating tube is set as the constant heating surface, with the heating density of 350 w/m2. the solution is based on the pressure-velocity coupled simple algorithm, and the pressure and momentum are discretized by adopting the second-order upwind scheme. in the calculation, the fluid medium is hot air and the solid material is considered as borosilicate glass. according to the theory of heat transfer, temperature has a significant influence on air density and thermal conductivity. consequently, when setting the air physical property parameters, the corresponding polynomials are fitted and obtained according to their changing laws with temperature, as shown in table 1 as below. table 1 physical parameters and expressions [25] physical parameters expression density (kg/m3) 5 2 9 32 836 0 008 1 055 10 4 841 10ρ . . t . t . t− −= − +  −  thermal conductivity (w/(m·k)) 5 8 20 003 8 314 10 2 105 10λ . . t . t− −= +  −  54 y. wang, q. zhu, t. huang, x. han, m. lin the above boundary conditions were used for the mesh independence verification of the tunnel structure, and the same boundary conditions and parameters were used for the calculation. the results of different mesh numbers were obtained by calculation, as shown in table 2. according to the comparison of the average velocity of the outlet surface, when the number of meshes increased from 2.32 million to 5.69 million, the average speed of the outlet surface decreased from 3.9795m/s to 3.697m/s. when the number of meshes reached 6.05 million, the error between the calculation result of the mesh and the calculation result of the mesh number of 5.69 million was 0.54%. continued increases in the number of meshes had less effect on the average velocity of the outlet surface. the variation trend between the number of meshes and the average velocity of the outlet surface is shown in fig. 5. taking into account a variety of cost factors, namely, the accuracy and reliability of the calculation, computing resources and computing time, the mesh with the final mesh number of 6.05 million was selected as the final calculation scheme, and this mesh scheme was adopted for subsequent calculations. table 2 calculation results of mesh independence verification number of meshes (10,000) average speed of outlet surface (m/s) relative error average temperature at the outlet surface (k) 232 3.9795 574.800 345 3.9235 1.41% 574.798 569 3.6970 5.77% 574.778 605 3.7169 0.54% 574.798 677 3.7594 1.14% 574.789 fig. 5 variation relationship between the number of meshes and the average velocity of the outlet surface modelling and simulation of nonlinear dynamic flow field and temperature field... 55 5. results and analysis 5.1. results analysis verified by mesh independence, the final mesh size of 6.05 million was deduced. the outer wall of the heating tube was fully set as a constant heating surface, with the heating density of 350 w/m2. according to this mesh, the steady-state numerical simulation calculation was carried out, and the final distribution of the flow field and temperature field inside the tunnel was obtained as follows. fig. 5 shows the overall pressure and temperature distribution of the tunnel, and it can be observed that the heating area and the outer wall of the induced draft fan show obvious high pressure areas, while the wind hood turned up in the drying room, obvious low pressure areas. there is an apparent thermal plume phenomenon on the wall near the heating tube area, mainly because the heating tube was a constant heating component, which had a relatively obvious heating effect on the heating area. driven by the inlet airflow, the heat generated a thermal plume as shown in fig. 6. according to gmp requirements to equipment with sterilization purpose, the airflow inside the tunnel for sterilization is expected as laminar. from fig. 6, considering the cross infection issue especially in pharmaceutical industries, such wind hoods are expected to be optimized in the following design. (a) pressure (b) temperature fig. 6 distribution of the overall pressure and temperature of the tunnel fig. 7 shows the streamline distribution of the airflow inside the tunnel. the airflow flowed into the impeller and the induced draft fan channel through the inlet of the tunnel, formed a relatively high-speed distribution because the rotation of the impeller part increased the airflow velocity. after coming out of the induced draft fan channel, most of the accelerated airflow was discharged from the center of the drying chamber directly, with a large discharge speed; another part of the air flow generated a low-speed vortex in the area near the wall of the drying chamber. the airflow in this part was comparatively turbulent. according to the phenomenon that vials https://fanyi.so.com/?src=onebox#comparatively 56 y. wang, q. zhu, t. huang, x. han, m. lin next to the tunnel surface would fall down occasionally and cause long-time shutdown identified in the previous study [4], this result gives a reasonable explanation. fig. 7 3d streamline distribution of the front and reverse sides of the tunnel: a) front size; b) reverse side the flow field and temperature field distribution of section x=0.683 m were shown in fig. 8. the temperature distribution inside the drying chamber was significantly higher than in other areas. however, due to the relatively low flow rate of air flow in the drying chamber, the airflow generated vortices of different sizes inside the hood and the drying chamber, which contributes to the drying effect to a certain extent. modelling and simulation of nonlinear dynamic flow field and temperature field... 57 (a) pressure (b) temperature (c) airflow velocity (d) velocity streamlines fig. 8 flow field and temperature field distribution of section x=0.683 m the flow field and temperature field distribution of section z=-0.687 m were shown in fig. 9. the temperature near the center line of the drying chamber was relatively low, while the temperature near the wall surface was relatively high. the major reason was that the airflow velocity near the centerline of the drying chamber was relatively large, and the convective heat transfer effect of the space was prominent, and the heat was carried away by the airflow, resulting in a comparatively low temperature in this area. to a certain extent, the uneven temperature distribution inside the drying chamber was caused by the low-speed vortex generated inside the drying chamber. therefore, it was necessary to optimize and improve the design of the internal structure of the drying chamber subsequently. 58 y. wang, q. zhu, t. huang, x. han, m. lin (a) pressure (b) temperature (c) airflow velocity (d) velocity streamlines fig. 9 flow field and temperature field distribution of section z=-0.687 m fig. 10 has shown the velocity and temperature distribution on the outlet surface of the tunnel under the condition that the inlet velocity was 4.2 m/s. the results suggest that the temperature and airflow velocity distribution on the outlet surface of the tunnel was not uniform. compared to the inlet temperature of 573 k, the average temperature at the outlet side was about 575 k, and the temperature rise was about 2°c. most of the air flowed out directly from the center of the tunnel, resulting in uneven distribution of velocity and temperature at the center of the outlet surface. modelling and simulation of nonlinear dynamic flow field and temperature field... 59 (a) temperature (b) airflow velocity fig. 10 velocity and temperature distribution of outlet surface by changing the inlet velocity of the tunnel, the inlet velocity was set to 6 m/s and 8 m/s respectively, while keeping the boundary conditions of other walls unchanged, and the numerical simulation calculation of the flow field and temperature field of the entire tunnel was performed. so as to explore the variation law of airflow velocity, temperature and pressure inside the drying chamber at different positions, the inner center line of the drying chamber was selected, and the variation data of the airflow velocity, temperature and pressure on the center line with the position were extracted and studied. the specific results were shown in fig. 11. (a) airflow velocity (b) pressure (c) temperature fig. 11 variation law of airflow velocity, pressure and temperature on vertical centerline of drying chamber it can be concluded from fig. 11 that under different inlet speed conditions, the farther away from the outlet, the lower the airflow velocity on the centerline, displaying a definite trend; the relative pressure changes on the entire centerline are basically the same, and the overall relative pressure is almost zero. the temperature on the centerline shows an increasing trend with the distance from the outlet, which is mainly because the farther away from the outlet, the smaller the airflow velocity, the less obvious the heat transfer effect, and the greater the temperature of the space. 60 y. wang, q. zhu, t. huang, x. han, m. lin 5.2. results validation discussion under most circumstances, the errors of computational results are caused by discretization, namely the discretization error, which is caused by mesh numbers. in this research, firstly the calculation results of mesh independence verification as shown in table 2 removes the influence of mesh numbers from computational results. secondly, the average temperature at the outlet surface regarding different mesh numbers are very close, which is 574.8k (recorded in table 2), indicating the steady state of computation results. considering the boundary conditions are measured from practical equipment, the results are proved with accuracy and reliability. 6. conclusions in this research, the tunnel is modeled geometrically using catia and 5 numbers of unstructured meshes, 2.32 million, 3.45 million, 5.69 million, 6.05 million and 6.77 million respectively, are generated using ansys. the generated meshes are verified by mesh independence, and the number of computing grids is determined to be 6.05 million finally. considering that the constant heating density of the heating tube was 350 w/m2, when the inlet velocity is 4.2 m/s, 6 m/s and 8 m/s respectively, the corresponding steady flow field and temperature field inside the tunnel are calculated, thus, the following basic conclusions are ultimately summed up as below: ▪ after entering the tunnel from the inlet, most of the airflow is discharged from the center of the drying chamber directly, with a large discharge speed; another part of the air flow generated a low-speed vortex in the area near the wall of the drying chamber. the airflow in this part was comparatively turbulent. ▪ inside the drying chamber, low-speed vortices of different intensities are generated, and the temperature in the center is relatively low, while the temperature close to the wall is relatively high. although the depyrogenation tunnel structure and its heating and sterilization process is studied using a cfd method in this research, many problems remained to be improved and solved in the future research, which include: ▪ structure improvement to reduce the low-speed vortex in this study, the low-speed vortex in the area near the wall of the drying chamber is identified, which is in line with the empirical results that vials occasionally fall down, causing an unexpected shutdown of production line. in the future work, the structure inside of the tunnel can be improved to decrease the vortex. furthermore, gmp requirements should be considered when performing structure improvement. ▪ time-variant calculation in this study, the steady-state temperature and fluid flow fields are calculated using cfd methods. however, the temperature field inside the tunnel is time-variant, taking especially the continuous production process in consideration. therefore, in the following study timevariant calculation could be conducted to express the dynamic process along time. ▪ control system optimization for energy saving in this study, the modelling and cfd calculation is carried out in terms of the depyrogenation tunnel to validate the air fluid field and temperature field results. according to the uneven distributed temperature from the wall to the central line, the reference input (temperature) of the control system could be lower. https://fanyi.so.com/?src=onebox#comparatively modelling and simulation of nonlinear dynamic flow field and temperature field... 61 acknowledgement: this work was partly supported by the national natural science foundation of china (no. 61803188), open fund project of fujian provincialkey laboratory of information processing and intelligent control (minjiang university) (grant no. mjukf-ipic202204) and research foundation of jinling institute of technology (no. jit-b-202029). references 1. huang, l., zhou, y., wang, y., lin, m., 2021, optimization of s-nitrosocaptopril monohydrate storage conditions based on response surface method, molecules, 26(24), 7533 2. krüger, o., ebner, i., kappenstein, o., roloff, a., luch, a., bruhn, t., 2021, towards a better comparability during gmp assessment–identifying the main parameters that influence the loss of volatile 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creative commons license: cc by-nc-nd original scientific paper pressure optimization in dec spray chambers: a comprehensive study on heat and mass transfer dynamics xiaoqing huang, shunyao lu, ning cai, xiantai wen college of energy and power engineering, nanjing institute of technology, nanjing, china abstract. achieving optimal contact between air and droplets is pivotal in direct evaporative cooling systems. this study presents an experimental analysis of spray cooling effectiveness in spray chambers utilizing pressure-swirl nozzles and impingement atomizing nozzle. through the analysis of experimental data related to heat and mass transfer, the study reveals that if an optimal water supply pressure is identified, increasing the water flow rate does not necessarily enhance cooling performance. furthermore, the study explores the impact of dry-bulb and wet-bulb temperatures of the air, as well as nozzle arrangement, on the determination of the optimal water supply pressure. key words: direct evaporative cooling (dec), spray cooling efficiency, heat and mass transfer, optimal water supply pressure, nozzle configuration optimization 1. introduction direct evaporative cooling (dec) systems, which utilize the latent heat of water evaporation to lower air temperature, have emerged as energy-efficient alternatives to conventional vapor-compression air conditioning [1-3]. these systems are particularly advantageous in arid and semi-arid climates, where the wet-bulb depression is sufficient to achieve significant cooling effects [4-6]. however, the core challenge lies in optimizing air-water contact efficiency to maximize thermal performance while minimizing energy and water consumption [7-9]. this requires a nuanced understanding of spray dynamics, nozzle geometry, and environmental variables, which collectively govern heat and mass transfer mechanisms in dec spray chambers [10, 11]. received: march 04, 2025 / accepted july 11, 2025 corresponding author: xiaoqing huang college of energy and power engineering, nanjing institute of technology, no. 1 hongjing avenue, jiangning district, nanjing 211167, china. e-mail: xiaoqing_huang@njit.edu.cn 2 x. huang, s. lu, n. cai, x. wen the efficiency of dec systems is fundamentally tied to the interplay between heat transfer, mass transfer, and airflow dynamics [12]. early studies focused on porous media as the primary evaporative medium, with materials such as cellulose pads and ceramic foams being widely investigated. for instance, dai and sumathy [13] modeled cross-flow dec systems using honeycomb paper, highlighting the importance of uniform water distribution for thermal homogeneity. similarly, ibrahim et al. [14] demonstrated that ceramic-based evaporators exhibit superior wettability but faced challenges in maintaining consistent water flow under variable pressures. recent advancements have shifted toward spray-based dec systems, where nozzle design and operational parameters critically influence performance [15]. pressure-swirl and impingement atomizing nozzles are commonly employed due to their ability to generate fine droplets, enhancing air-water interfacial contact [16]. however, their study did not address the nonlinear relationship between water supply pressure and cooling efficacy, particularly under fluctuating environmental conditions [17, 18]. environmental parameters, such as dry-bulb and wet-bulb temperatures, further modulate dec performance. chiesa et al. [19] demonstrated that dec efficiency in mediterranean climates correlates strongly with wet-bulb depression, achieving peak performance at relative humidity levels below 60%. saleh and talib [20] experimentally showed that an indirect evaporative cooling system's performance is influenced by operational parameters such as airflow rate and inlet conditions, achieving effective temperature reduction through optimized design and experimental validation. yang et al. [21] propose a method for summer design of evaporative cooling systems, considering lowtemperature latent heat recovery inefficiencies. system configuration, including nozzle arrangement and airflow patterns, also plays a pivotal role. vynnycky and mitchell [22] modeled the mpemba effect in evaporative cooling, showing that droplet size and evaporation rate significantly affect thermal equilibration. zhang et al. [23] conducted an experimental study on a direct evaporative cooler by using the climatic wind tunnel, focusing on its thermal performance. despite these insights, a comprehensive framework linking nozzle pressure optimization to climatic adaptability remains underdeveloped [24-26]. this study aims to investigate the relationship between water supply pressure and cooling efficiency in dec spray chambers, with a focus on identifying pressure thresholds that optimize heat and mass transfer dynamics. the work will systematically evaluate how dry-bulb and wet-bulb temperatures, nozzle configurations, and droplet atomization characteristics influence these thresholds. 2. theoretical basis for the experiment the spray cooling effect on the air side is calculated using the principles of heat and mass transfer. the reliability and accuracy of the experimental results hinge on the precision of the air side parameters, namely the dry/wet-bulb temperatures. the driving force for heat transfer is the temperature difference, while the driving force for mass transfer is the partial pressure difference of water vapor. assuming the air temperature difference dt and the air moisture content difference dm, upon contact between the air and water over an infinitesimal area da, the sensible heat transfer can be expressed as: pressure optimization in dec spray chambers: a comprehensive study on heat and mass... 3 x p bd d ( )dq gc t h t t a   (1) where, g represents the air mass flow rate in contact with water [kg/s], cp is the specific heat capacity of air [j/(kg·°c], h is the surface sensible heat transfer coefficient at the airwater interface [w/(m²·°c)] and t and tb are the mainstream air temperature and boundary layer air temperature, respectively, [°c]. moisture transfer can be quantified as follows: mp q qbd d ( )dw g m h p p a   (2) where, hmp is the moisture transfer coefficient at the air-water interface and it is determined based on the difference in water vapor partial pressure [kg/(n·s)], pq and pqb are the water vapor partial pressure in the mainstream air and the boundary layer, respectively [pa]. the following equation describes the moisture transfer, as the difference in water vapor partial pressure can be approximated by the product of the water content difference and the moisture transfer coefficient over a narrow temperature range: md bd ( )dw h m m a  (3) here, hmd is the moisture transfer coefficient at the air-water interface is determined by the difference in moisture content [kg/(m²·s], m and mb are the moisture content in the mainstream air and the boundary layer, respectively [kg/kg]. the latent heat transfer capacity can be determined using the equation below: q md bd = d ( )dq r w rh m m a  (4) where, r is the latent heat of vaporization when the temperature reaches tb [j/kg]. eq. (5) is derived based on the total heat transfer capacity z x qd d dq q q  : z b md bd =[ ( ) ( )]dq h t t rh m m a   (5) 3. experimental setup as shown in fig. 1, the experimental spray cooling system includes a high-temperature flue system, a ventilation system, and a spray system. the high-temperature flue system comprises a circulating fan, a high-intelligence hot air stove, a wind pipe, and a measuring section. the ventilation system includes an induced draft fan, an exhaust fan, a fan control device, a wind pipe, an air sampler, and a measuring section. the spray system consists of a water tank, a high-pressure water pump, nozzles, a water receiver, and a measuring section. a rectangular box measuring 2500 mm × 1450 mm × 800 mm was used. four rows of nozzles were installed on top, with a row spacing of 500 mm and a spacing of 725 mm between nozzles within the same row (see fig. 1). the water receiver was positioned at the bottom. pressure-swirl nozzles tf6 and tf8 along with impingement atomizing nozzle am4 (see fig. 2) were utilized in the experimentation. the primary parameters and instruments used for measurement are outlined in table 1. data was collected every 10 seconds using fluke equipment. 4 x. huang, s. lu, n. cai, x. wen fig. 1 schematic diagram of the spray cooling system (units: mm) a) b) fig. 2 schematic diagram of nozzle configurations a) pressure-swirl nozzles tf6 and tf8 b) impingement atomizing nozzle am4 table 1 key experimental parameters and instruments system measurement parameters instrument measuring range hightemperature flue system supply flue temperature thermal resistance temperature detector -70~500 ℃ return flue temperature thermal resistance temperature detector -70~500 ℃ dynamic pressure of flue pipeline pitot tube 2~70 m/s ventilation system inlet air temperature thermograph -50~70 ℃ outlet air temperature normal temperature thermocouple×5 -50~70 ℃ duct dynamic pressure pitot tube 2~70 m/s air sampler dry-bulb temperature thermometer×2 0~50 ℃; 50~100 ℃ air sampler wet-bulb temperature thermometer 0~50 ℃ pressure optimization in dec spray chambers: a comprehensive study on heat and mass... 5 spray system spray water temperature thermometer 0~50 ℃ spray water flow turbine flow meter lwgy-25 and digit display meter 0~10 m3/h spray system pressure pressure gauge yb150 2.5 mpa, accuracy 0.4 droplet diameter malvern spraytec particle size analyzer 0.1 μm~2000 μm 4. heat and mass transfer analysis the heat and mass transfer processes were analyzed opening tf6 nozzles in row 1 and row 2 (500 mm inter-row spacing, abbreviated as rows 1/2 tf6). initial chamber inlet conditions showed dry-bulb and wet-bulb temperatures of 80°c and 36.4°c, respectively (fig. 3). the dry-bulb temperature decreased substantially with increasing nozzle pressure (0.05-2.00 mpa, 10 increments) and water flow. thermal equilibrium occurred at 0.70 mpa, where the dry-bulb temperature approached the wet-bulb temperature. above 0.70 mpa, the cooling rate diminished progressively, exhibiting asymptotic behavior toward the wet-bulb reference state. this observed equilibrium at 0.70 mpa can be attributed to the competition between enhanced atomization and droplet coalescence. below this threshold, increased pressure reduces droplet size (fig. 5), enlarging the air-water interfacial area and improving evaporative cooling efficiency. however, beyond 0.70 mpa, the kinetic energy of droplets surpasses the energy required to overcome surface tension, leading to frequent droplet collisions and coalescence (fig. 6). larger droplets reduce the effective contact area, thereby diminishing both latent heat transfer and cooling performance. this phenomenon aligns with the findings of zhao et al. [16], who noted similar saturation effects in spray cooling systems under high-pressure conditions. fig. 4 illustrates the heat and mass transfer processes in rows 1/2 tf6. initially, the sensible heat transfer (qx) decreased with increasing water pressure, reaching equilibrium at a nozzle pressure of 0.70 mpa, exhibiting a trend similar to the dry-bulb temperature. the latent heat transfer (qq) increased with rising water content and nozzle pressure but decreased at a water supply pressure of 0.70 mpa due to air dehumidification. and the total heat transfer (qz) was in balance, but it subsequently decreased. before air saturation, evaporative cooling is the dominant driving force of spray cooling. however, after the water supply pressure reaches 0.70 mpa, heat convection between air and water becomes the dominant force. in terms of water conservation, evaporative cooling should be prioritized. furthermore, the reduction in dry-bulb temperature is not significant when the water supply pressure reaches 0.70 mpa. therefore, the optimal spray cooling pressure for tf6 is determined to be 0.7 mpa. beyond this pressure, increasing the spray quantity in the humidification chamber does not significantly reduce the dry-bulb temperature. 6 x. huang, s. lu, n. cai, x. wen fig. 3 variation in air sampler temperature in rows 1/2 tf6 fig. 4 heat and mass transfer processes in rows 1/2 tf6 the transition from evaporative cooling to convective dominance highlights a critical trade-off in dec system design. while higher pressures initially enhance moisture transfer, excessive humidity saturation (approaching 100% relative humidity) limits further evaporation, as shown by the decline in qq beyond 0.70 mpa (fig. 4). this behavior corroborates the theoretical model in eq. (4), where latent heat transfer depends on the vapor pressure gradient. once the air nears saturation, the gradient diminishes, reducing qq. practical implications suggest that maintaining pressure near 0.70 mpa optimizes both energy efficiency and water conservation, particularly in high-temperature applications. a curve depicting the variation in atomized droplet diameters (explained in table 2) of the tf6 nozzle, measured 7 cm below the nozzle, with spray pressure was plotted based on the results obtained using the malvern method, as shown in fig. 5. pressure optimization in dec spray chambers: a comprehensive study on heat and mass... 7 table 2 definitions of atomized droplet diameters symbol name meaning d[3][2] sauter mean diameter (smd) the ratio of the total volume of droplets to their total surface area. d[4][3] mass mean diameter (mmd) the ratio of the total mass of droplets to their total volume. dv(10) mass median diameter the volume of droplets with diameters smaller than dv(10) accounts for 10% of the total droplet volume. dv(50) the volume of droplets with diameters smaller than dv(50) accounts for 50% of the total droplet volume. this corresponds to the point where the areas under the volume distribution curve on both sides of dv(50) are equal. dv(90) the volume of droplets with diameters smaller than dv(90) accounts for 90% of the total droplet volume. fig. 5 variation in atomized droplet diameters of the tf6 nozzle with increased water pressure the distribution of atomized droplet diameters, as depicted in fig. 5, shows that dv[9][0] is the maximum value, followed by d[4][3], dv[5][0] and d[3][2] in descending order, with dv[1][0] being the minimum. the minimal reduction in droplet diameter beyond 0.70 mpa (fig. 5) further supports the hypothesis of energy redistribution. as pressure increases, excess kinetic energy no longer contributes to atomization but instead accelerates droplet motion, promoting collisions. this aligns with the observations of zhang et al. [23], who emphasized the role of nozzle-generated turbulence in droplet dynamics. notably, the sauter mean diameter (d[3][2]) – a critical parameter for interfacial area – plateaus beyond 0.70 mpa, indicating diminishing returns in heat transfer enhancement. 8 x. huang, s. lu, n. cai, x. wen 5. optimal water supply pressure during the experimental process, a specific water supply pressure was identified for each type of tf6, tf8, and am4 nozzles. for each type of nozzle, if the dry-bulb temperature of the air did not significantly decrease as the water pressure exceeded this specific level, it indicated the existence of an optimal water supply pressure for the spray cooling process. the optimal water supply pressure for a humid chamber is defined as the pressure at which the dry-bulb temperature of the air no longer decreases significantly and approaches the wet-bulb temperature after the nozzles' back-pressure reaches a certain value. table 3 presents the optimal water supply pressures for various types of nozzles and different nozzle arrangements. table 3 optimal water supply pressures nozzle types arrangements dry-bulb temperatures at the air inlet optimal water supply pressures tf6 725 mm×725 mm 500 mm×500 mm 500 mm×725 mm 38˚c 0.55 mpa tf6 725 mm×725 mm 500 mm×500 mm 500 mm×725 mm 80˚c 0.70 mpa tf8 725 mm×725 mm 500 mm×500 mm 500 mm×725 mm 38˚c 0.35 mpa tf8 725 mm×725 mm 500 mm×500 mm 500 mm×725 mm 80˚c 0.50 mpa am4 725 mm×725 mm 500 mm×500 mm 500 mm×725 mm 38˚c 0.25 mpa am4 725 mm×725 mm 500 mm×500 mm 500 mm×725 mm 80˚c 0.35 mpa 5.1 optimal water supply pressure variation with dry-bulb temperature at air table 3 indicates an increase in optimal water supply pressure with rising dry-bulb temperature at the air inlet. nonetheless, the increase was limited for each nozzle type. fig. 6 depicts droplet size distribution 7 cm below a tf6 nozzle across various water supply pressures. when the water supply pressure exceeded 0.8 mpa, the spraytec analyzer detected droplets larger than 1,000 µm. both the number and diameter of these droplets increased with further pressure increases. analyzing from the perspective of droplet atomization energy conservation, the change in droplet diameter was insignificant once the nozzle backpressure reached 0.7-0.8 mpa. hence, it is inferred that the work to overcome surface tension and viscous forces remained constant, with excess energy converting to droplet kinetic energy. the increased kinetic energy led to more droplet collisions, forming larger droplets detrimental to heat and mass transfer. pressure optimization in dec spray chambers: a comprehensive study on heat and mass... 9 fig. 6 droplet size distribution 7 cm below a tf6 nozzle at various water supply pressures 10 x. huang, s. lu, n. cai, x. wen 5.2 impact of wet-bulb temperature on optimal water supply pressure fig. 7 displays the variation in the air cooling effect of tf6 nozzles arranged in a 500 mm×725 mm layout, with an air inlet dry-bulb temperature of 80˚c, considering the product of the difference between dry/wet-bulb temperatures and the evaporation rate. the analysis indicates that the optimal water supply pressure remained stable at approximately 0.7 mpa within the high-temperature range. consequently, the wet-bulb temperature had an insignificant impact on the optimal water supply pressure. fig. 7 the air cooling effect in relation to the product of the difference between the drybulb and wet-bulb temperatures of the inlet air and the evaporation rate table 3 demonstrates that when using the same type of nozzle to cool air with identical parameters, variations in the tf6 nozzle arrangement had no impact on the optimal water supply pressure. 6. conclusion this study experimentally investigated the heat and mass transfer dynamics in direct evaporative cooling (dec) spray chambers, focusing on the identification of optimal water supply pressures for three nozzle types (tf6, tf8, am4). through systematic analysis of spray cooling efficacy under varying operational and environmental conditions, the work establishes critical relationships between pressure thresholds, nozzle characteristics, and cooling performance. key findings are summarized as follows: pressure optimization in dec spray chambers: a comprehensive study on heat and mass... 11  existence of optimal water supply pressure: increasing water supply pressure beyond a critical threshold (e.g., 0.70 mpa for tf6) reduced cooling efficiency due to excessive droplet coalescence and humidity saturation (fig. 4).  nozzle specific pressure thresholds: experimental data analysis identified distinct optimal pressures for tf6, tf8, and am4 nozzles under various conditions. future research should prioritize the development of real-time control algorithms that dynamically adjust water supply pressure based on meteorological forecasts, thereby enhancing climatic adaptability. additionally, integrating experimental findings with 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to, for example, widely understood transport. increasingly popular in this field are issues related to the domain of the green urban transport. in order to support the decision-making process in this area, methods of multi-criteria decision analysis (mcda) are used more and more often. however, if we solve a specific problem using different mcda methods, we get different rankings, as each method has a different methodological basis. therefore, the challenge is how to make a reliable decision. this paper presents a numerical example from the green urban transport domain, which is solved by six different mcda methods that return a complete ranking. we measure the similarity of these rankings using coefficients rw and ws, and then we propose a simple way of determining a compromise solution. the obtained compromise ranking is guaranteed to be the best match to the selected mcda methods' rankings, which is proved in the paper. finally, possible directions for further development work are identified. key words: mcda, transport selection, green urban transport, operational research 1. introduction decision-making problems appear in every aspect of our lives. many factors are involved in making a choice, which makes it much more difficult [1]. sometimes decision-making is influenced by emotions, making the choices made biased. however, making a choice should always maximize benefits and minimize possible losses. to support decision-making and to remain objective in the process, many methods have been received march 15, 2021 / accepted july 30, 2021 corresponding author: wojciech sałabun west pomeranian university of technology, szczecin; ul. żołnierska 49, 71-210 szczecin, poland e-mail: wojciech.salabun@zut.edu.pl 382 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun developed to support the decision-maker [2, 3]. the models based on these methods are designed to indicate which of the considered alternatives are better than the others. nevertheless, a more critical aspect is whether getting even different rankings can contribute to a more reliable solution. there are often used methods belonging to the multi-criteria decision analysis (mcda) methods in support systems, which work on the basis of evaluating a set of alternatives and giving them preference values thus assessing their quality concerning other options [4]. the group of mcda methods is constantly extended by newly created methods, making it difficult to choose a method to solve the selected task [5, 6]. moreover, another difficulty may be determining whether choosing a different method to solve the same problem may give different results [1]. it makes it worthwhile to lean into answering the question if the choice of mcda method matters. mcda methods can be divided into three main groups, the american school of methods, the european school of methods and rule-based methods [7, 8]. each group is based on slightly different assumptions about how the problem should be solved, but in general, each attempts to obtain a ranking that evaluates the alternatives under consideration. the effectiveness of mcda methods has been tested many times, and they have been used to solve problems of selecting industrial locations [9, 10], material suppliers [11, 12], in sports [13, 14] or medicine [15]. these methods are willingly used by experts, making them increasingly applicable to a wide range of problems. moreover, a particular decision problem can often be solved using more than one method but then we get different rankings. it is related to assumptions used in the algorithms, which can be exemplary based on the distance from the best or the best and the worst element. therefore, these solutions are not incorrect from the methodological point of view. however, it should be considered how to decide on the basis of such rankings in order to make the decision as reliable as possible. however, in the cases where mcda methods guarantee different results within one problem, it is worthwhile to determine the extent to which the obtained results are similar [16]. for this purpose, correlation coefficients can be used to assess the similarity of the analyzed rankings through numerical values. these coefficients include spearman’s weighted correlation coefficient and ws similarity coefficient [8, 17]. firstly, it is an opportunity to determine how strongly different results were returned by the methods used. secondly, it can be used to establish a compromise ranking. for this purpose, measures need to be defined to determine whether a given ranking is a better compromise. in this paper, we propose a new approach to determine the compromise ranking based on the similarity rankings. the proposed approach is extremely simple and is intended to compromise ranking based on the rankings obtained from various mcda methods. we present our methodology by using a theoretical multi-criteria problem in the form of the selection of the electric bus. for this purpose, we have selected six mcda methods that give a full ranking as a result. the obtained results from topsis, vikor, promethee ii, copras, comet, and spotis methods are then compared using the similarity coefficients to check how the rankings correspond to each other. on this basis, we define two measures indicating which of the resulting rankings the best compromise ranking is. then, we determine the compromise ranking whose result is the closest to all the rankings considered. this results in a final compromise ranking that indicates the most reliable solutions in the absence of knowledge of the reference solution. towards reliable decision-making in the green urban transport domain 383 the rest of the paper is organized as follows. in sections 2.1-2.6, the preliminaries of selected mcda methods are presented. the ranking similarity coefficients are presented in section 2.7. section 3 includes an empirical study case, in which the comparison of six given methods application, namely topsis, vikor, promethee ii, copras, comet, and spotis, is made. section 4 describes the proposed approach to obtain compromise ranking and a short discussion, and finally, the conclusions from the research are drawn in section 5. 2. preliminaries multi-criteria decision analysis (mcda) methods solve a multi-criteria problem in various areas [13, 14, 18]. with an extended trend in using these methods to solve problems, more extensions and new techniques are being developed [19, 20]. in this section, we recall the algorithms of the mcda methods (sections 2.1-2.6) and the used similarity coefficients (sections 2.7 and 2.8). it is necessary because there are many versions of these algorithms in the literature [21, 22, and 23]. 2.1. the topsis method the technique for order preference by similarity to an ideal solution (topsis) method was developed in 1992 [24]. chen and hwang proposed an approach to examining the set of alternatives based on the calculation of the distance to the ideal solution. to evaluate the alternatives’ preferences, the positive ideal solution (pis) and negative ideal solution (nis) are used [25, 26]. the method application requires an expert to define a weights vector, which describes the relevance of criteria [27]. the first step is to normalize the decision matrix in order to obtain the correct final result. next, a weighted normalized decision matrix should be calculated using the following eq. (1). , 1, , ; 1, ,=  =  = ij i ijv w r j j i n (1) where wi is the value of the i-th weighting, rij is the normalized attribute for the j-th alternative against the i-th criterion, vij is the weighted normalized attribute for the j-th alternative against the i-th criterion, n is the number of criteria, and j is the number of alternatives. positive (a*) and negative (a-) ideal solutions for a defined decision-making problem should also be identified as eq. (2): 1 1 { , , } {(max | ), (min | )} { , , } {(min | ), (max | )} p c n j ij j ij p c n j ij j ij a v v v i i v i i a v v v i i v i i    − − − =  =   =  =   (2) where ic stands for cost type criteria and ip for profit type. negative and positive distance from an ideal solution should be calculated using the n-dimensional euclidean distance. to apply such calculations, the formula presented below should be used eq. (3): 384 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun 2 1 2 1 ( ) , 1, , ( ) , 1, , n j ij i i n j ij i i d v v j j d v v j j   = − − = = − =  = − =    (3) the last step is to calculate the relative closeness to the ideal solution by eq. (4): , 1, , ( ) j j j j d c j j d d −   − = =  + (4) 2.2. the vikor method the vikor (visekriterijumska optimizacija i kompromisno resenje) method was developed by opricovic in 1998 [28]. the final preferences for alternatives are calculated using expert knowledge and the closeness of the solution to the ideal solution [29, 30]. the input data does not need to be normalized as in the topsis method [31]. each of the criteria is initially defined as a cost or profit type of criteria by eq. (5). the cost type shows that we want it to achieve the lowest possible values, while the profit type should achieve the highest possible values. , ; , i j ij i j ij i j ij i j ij f max f f min f if the i th criteria is a profi type f min f f max f if the i th criteria is a cost type  −  − = = − = = − (5) j ij j j i f f w f f   − −  − (6) for each of the criteria, the best fi * and the worst fi values are defined. then, using the formula presented in eq. (6), preference values for each criterion are calculated, taking into account the weights for criteria defined at the beginning. on this basis, the closeness to the ideal solution is calculated, considered in three different rankings calculated from eq. (7), eq. (8) and eq. (9) for s, r, and q, respectively. 1 n j ij i j j j i f f s w f f   − = − =  −  (7) j ij i j j j i f f r max w f f   −  − =   −   (8) (1 )i i i s s r r q v v s s r r   −  −  − − =  + −  − − (9) towards reliable decision-making in the green urban transport domain 385 2.3. the promethee ii method the preference ranking organization method for enrichment of evaluations (promethee) is a family of mcda methods developed by brans [32, 33]. it uses input data similar to other methods, but it optionally requires choosing preference function and some other variables. in this paper, we use the promethee ii method because the output of this method is a full ranking of the alternatives. it is the approach where a complete ranking of the actions is based on the multicriteria net flow. it includes preferences and indifferences (preorder) [34]. according to [32, 35], promethee ii is designed to solve the following multicriteria problems: 1 2{ ( ), ( ), ( ) | }nmax g a g a g a a a  (10) where a is a finite set of alternatives and gi(·) is a set of evaluation criteria either to be maximized or minimized. in other words, gi(aj) is a value of criteria i for alternative aj. with this values and weights we can define evaluation table. table 1 evaluation table a g1(·) g2(·) … gn(·) w1 w2 … wn a1 g1(a1) g2(a1) … gn(a1) a1 g1(a2) g2(a2) … gn(a2) … … … … … am g1(am) g2(am) … gn(am) step 1. after defining the problem as described above, calculate preference function values. it defined as eq. (11) for profit criteria. ( , ) [ ( , )], ,p a b f d a b a b a=   (11) where d(a,b) is the difference between two actions (pairwise comparison): ( , ) ( ) ( )d a b g a g b= − (12) and the value of preference function p is always between 0 and 1 and it is calculated for each criterion. step 2. calculate aggregated preference indices by eq. (13). 1 1 ( ) ( ) ( , ) , ( , ) , n j j j n j j j a b p a b w b a p b a w   = =  =    =     (13) where a and b are alternatives and π(a,b) shows how much alternative a is preferred to b over all of the criteria. there are some properties eq. (14) which must be true for all alternatives set a. 386 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun ( , ) 0 0 ( , ) 1 0 ( , ) 1 0 ( , ) ( , ) 1 a a a b b a a b b a      =         +  (14) step 3. next, calculate positive eq. (15) and negative eq. (16) outranking flows. 1 ( ) 1 ( , ) x a a m a x +  = −  (15) 1 ( ) 1 ( , ) x a x m a a −  = −  (16) step 4. in this paper, we use only promethee ii, which results in a complete ranking of alternatives. ranking is based on the net flow φ eq. (17). ( ) ( ) ( )a a a+ − =  − (17) a larger value of φ(a) means a higher position in the ranking. 2.4. the copras method the complex proportional assessment (copras), introduced by zavadskas [36, 37] assumes a direct and proportional relationship of the importance of the investigated variants on a system of criteria adequately describing the decision variants as well as on values and weights of the criteria [38]. this method ranks alternatives based on their relative importance (weight). final ranking is creating using the positive and negative ideal solutions [37, 39]. assuming that we have decision matrix with m alternatives and n criteria is represented as x = fij(ai)m×n, copras method is defined in the following five steps.: step 1. calculate normalized decision matrix using eq. (18). 1 ij ij m ij i x r x = =  (18) step 2. calculate difficult normalized decision matrix, which represents multiplication of the normalized decision matrix elements with the appropriate weight coefficients using eq. (19). ij ij jv r w=  (19) step 3. determine the sums of difficult normalized values which were calculated previously. eq. (20) should be used for profit criteria and eq. (21) for cost criteria. 1 k i ij j s v+ = = (20) 1 n i ij j k s v− = + =  (21) towards reliable decision-making in the green urban transport domain 387 step 4. calculate the relative significance of alternatives using eq. (22). 1 1 m min i i i i m min i i i s s q s s s s − − = + − − = −  = +          (22) step 5 final ranking is performed according ui values eq. (23). 100%i i max i q u q =  (23) where qi max stands for maximum value of the utility function. better alternatives have higher ui value, e.g. the best alternative has ui = 100. 2.5. the comet method the characteristic objects method (comet) belongs to the rule-based mcda methods [41]. the final preferences for the alternatives are calculated on the rule base, which is obtained by defining the characteristic objects and pairwise comparison made by expert [42]. moreover, it is worth noticing that comet is the first method to be completely free of the rank reversal phenomenon [42, 43]. the formal notation of this method can be shortly recalled in the following five steps [44, 45, 46, 47]. step 1. define the space of the problem – the expert determines the dimensionality of the problem by selecting number r of criteria, c1, c2, ..., cr. then, the set of fuzzy numbers for each criterion ci is selected eq. (24): 1 2 ,..., }{ , r rn r r rnc c c c= (24) where nr is a number of the fuzzy numbers for criterion r. step 2. generate characteristic objects – the characteristic objects (co) are obtained by using the cartesian product of fuzzy numbers cores for each criterion as follows eq. (25): 1 2( ) ( ) ( )rco c c c c c c=   (25) step 3. rank the characteristic objects – the expert determines the matrix of expert judgment (mej). it is a result of pairwise comparison of the cos by the problem expert. the mej matrix contains results of comparing characteristic objects by the expert, where αij is the result of comparing coi and coj by the expert. function fexp denotes the mental function of the expert. it depends solely on the knowledge of the expert and can be presented as eq. (26). afterwards, the vertical vector of the summed judgments (sj) is obtained as follows eq. (27). exp exp exp exp exp exp 0.0, ( ) ( ) 0.5, ( ) ( ) 1.0, ( ) ( ) i j ij i j i j f co f co f co f co f co f co     = =   (26) 1 t i ij j sj  = = (27) 388 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun finally, the values of preference are approximated for each characteristic object. as a result, vertical vector p is obtained, where i-th row contains the approximate value of preference for coi. step 4. the rule base – each characteristic object and value of preference is converted to a fuzzy rule as follows eq. (28): 1 2( ) ( )i i ic cif c and c and then p (28) in this way, the complete fuzzy rule base is obtained. step 5 inference and final ranking – each alternative is presented as a set of crisp numbers (e.g., ai = {a1i, a2i, ..., ari}). this set corresponds to criteria c1, c2, ..., cr. mamdani’s fuzzy inference method is used to compute preference of i-th alternative. the rule base guarantees that the obtained results are unequivocal. 2.6. the spotis method the stable preference ordering towards ideal solution (spotis) method is a recently developed method [48]. similarly to the comet method, it is declared fully resistant to the rank reversal phenomenon. this method’s main assumption is to define the data boundaries, which are used to determine ideal solution point (isp). based on it, further calculations to obtain the final preferences for alternatives are being made [49]. the definition of the data boundaries requires to select maximum sj max and minimum sj min bound for each criterion cj. ideal solution point sj * is defined as sj * = sj max for profit and as sj * = sj min for cost type of criterion. more necessary transformations during the method application are presented below. step 1. calculation of the normalized distances to ideal solution point eq. (29). ),( ij j ij i j max min j j s s d a s s s   − = − (29) step 2. calculation of weighted normalized distances d(ai, s*) ∈ [0,1], according to eq. (30). 1 ,( )( , ) n i j ij i j j d a s w d a s  = = (30) step 3. final ranking should be determined based on d(ai, s*) values. smaller values d(ai, s*) which are preferences of alternatives result in better position in general ranking. 2.7. the ranking similarity coefficients the idea of using the rankings similarity coefficients is not new and has been the subject of many works [50, 51]. particularly interesting are works related to the weighted rank measure of correlation rw [52, 53]. recently, a new coefficient is proposed by sałabun [8]. it is an asymmetric measure, where the weight of a given comparison is determined based on the significance of the position in the reference ranking. these coefficients can be presented as eq. (31) and eq. (32) respectively: towards reliable decision-making in the green urban transport domain 389 2 1 3 2 6 ( ) ( 1) ( 1) 1 ( 1) ( ) n xi yi xi yi i w r n r n r r r n n n n =  − − + + − + = −  + − −  (31) 1 | | 1 2 max{| |,| |}1 xi n xi yir i xixi r ws n r r r − = − −  = −   −   (32) where rw is a value of the weighted rank measure of correlation, ws is a value of similarity coefficient, n is a length of ranking, rxi and ryi mean the place in the ranking for i-th element in, respectively, ranking x and ranking y. 3. numerical example in green urban transport domain the problem of electric bus selection is presented as an exemplary problem from the green urban transport domain. this problem considers seven criteria and nine alternatives, where the choice of criteria was taken by the most critical elements of importance in assessing electric buses’ quality [54, 55, 56]. we assume that a reference ranking is not available, and our task is to find the best compromise solution. the defined criteria are presented in table 2, where the criterion name, its type and the unit in which the values will be given are shown. on the other hand, the decision matrix for the selected nine alternatives, which is used to calculate the final preference values in each of the mcda methods used, is included in table 3. each alternative contains the name of the electric bus model and the manufacturer’s values for the criteria considered. six vehicle manufacturers were selected, and one of the selected models was presented in four different alternatives. table 2 considered criteria ci to electric bus selection ci name type units c1 battery capacity profit kwh c2 engine power profit kw c3 range profit km c4 price cost thousands pln c5 all places profit unit c6 seating places profit unit c7 places for disabled profit unit six mcda methods were selected to solve the problem of selecting the optimal electric bus to compare the results and check whether the obtained results would be significantly different from each other. topsis method with minmax normalization, vikor method without using normalization, promethee ii method with usual type preference function, copras method in standard configuration, comet method with object evaluation by topsis method and spotis method in the standard configuration were used in the research. 390 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun table 3 decision matrix with set of alternatives ai ai name c1 c2 c3 c4 c5 c6 c7 a1 ursus city smile 18m 230 120 150 2051.74 83 24 1 a2 solaris urbino 12 electric 210 160 150 2212.86 85 32 1 a3 byd k9 324 180 250 2101.78 80 25 1 a4 volvo 7900e 250 200 200 2550.00 83 30 1 a5 proterra catalyst fc 94 190 109 2020.00 80 40 0 a6 proterra catalyst xr 220 190 264 2465.00 80 40 0 a7 proterra catalyst e2 440 190 491 2885.00 80 40 0 a8 proterra catalyst e2 max 660 190 685 3370.00 80 40 0 a9 new flyer xcelsior charge 40’s 300 159 210 3293.75 83 40 2 the obtained preference values using the mentioned methods are shown in table 4. when topsis, promethee ii, copras and comet methods are used, a higher preference value means a better-evaluated alternative. on the other hand, when preferences obtained from applying the vikor and spotis methods are analyzed, a smaller value indicates a better choice. to facilitate the interpretation of the results, the obtained rankings in the form of positional rankings are included in table 5. a histogram shows the positional rankings’ visualization for the different methods and alternatives in fig. 1. table 4 obtained preferences of alternatives for selected mcda methods ai topsis vikor prom. ii copras comet spotis a1 0.3939 1.0000 -0.1786 0.7446 0.2908 0.6589 a2 0.5139 0.5813 -0.0714 0.7831 0.5722 0.4810 a3 0.4399 0.9144 0.0179 0.8580 0.3897 0.5853 a4 0.5018 0.2783 0.1250 0.8191 0.5320 0.4977 a5 0.4535 0.9360 -0.1786 0.5988 0.4141 0.5893 a6 0.4584 0.8167 -0.0357 0.6863 0.4278 0.5661 a7 0.5008 0.6577 0.1071 0.8487 0.5304 0.4987 a8 0.5277 0.5000 0.1071 1.0000 0.5988 0.4464 a9 0.5178 0.0443 0.1071 0.9983 0.5848 0.4738 table 5 positional rankings of obtained results from application of mcda methods ai topsis vikor prom. ii copras comet spotis a1 9 9 9 7 9 9 a2 3 4 7 6 3 3 a3 8 7 5 3 8 7 a4 4 2 1 5 4 4 a5 7 8 8 9 7 8 a6 6 6 6 8 6 6 a7 5 5 4 4 5 5 a8 1 3 3 1 1 1 a9 2 1 2 2 2 2 towards reliable decision-making in the green urban transport domain 391 fig. 1 visualization of the positional rankings comparison of used mcda methods it is worth noting that for the results obtained for the promethee ii method, the preference values for the alternatives a7, a8 and a9 were the same, as they were for the pair of alternatives a1 and a5. differences in preference values were very small, which may be caused by errors in the numerical representation of floating-point numbers. the results obtained with the promethee method with ties is presented in table 6, where the preference values and the positional ranking for the set of alternatives from the decision matrix are listed. table 6 obtained preferences for promethee i and ii application ai ɸɸ+ ɸ prom. ii a1 0.5357 0.3571 -0.1786 8.5 a2 0.5000 0.4286 -0.0714 7 a3 0.4286 0.4464 0.0179 5 a4 0.3929 0.5179 0.1250 1 a5 0.4643 0.2857 -0.1786 8.5 a6 0.3929 0.3571 -0.0357 6 a7 0.3214 0.4286 0.1071 3 a8 0.3214 0.4286 0.1071 3 a9 0.3929 0.5000 0.1071 3 when the preference values are rounded to four significant decimal places, the obtained ranking correlation values may change. the rankings’ obtained similarity values when taking into account the rounding of the preference values are presented in table 7. a visualization of the newly obtained correlation relationships between the obtained rankings is shown in fig. 2, where the correlation values were represented by histograms for the analyzed alternatives and methods. table 7 correlation between promethee with ties and other methods method rw ws topsis 0.5869 0.6991 vikor 0.7971 0.7832 prom. ii 1.0000 1.0000 copras 0.6619 0.7476 comet 0.5869 0.6991 392 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun fig. 2 alternate comparison of obtained rankings from mcda methods analyzing the obtained values for the preference values derived from the application of the selected six mcda methods, it can be observed that the promethee ii method guaranteed equal preference values for the alternatives a7 a9 and for the alternatives a1 and a5. it is worth noting that the raw preference values obtained by the mcda methods included in table 4 are challenging to compare, due to the variety of values obtained. on the other hand, when analyzing the obtained positional rankings shown in table 5, which are more representative, it can be observed that some of the alternatives have different orders depending on the method used. alternative a8 can be considered the optimal choice, having been best ranked by four of the six methods. on the other hand, the second place in the rankings analyzed was most often obtained by alternative a9, which was ranked in this position five times. when the preference values obtained for the promethee method were considered and rounded to four decimal places, it is worth noting that the rankings’ similarity values differed from those when no rounding was considered. the correlation values presented in table 7 showed that only when comparing the rounded preference values with those obtained from the vikor method, the correlation increased slightly. in contrast, the correlation decreased for the other rankings. it shows that, in addition to the methods used to solve a given multi-criteria problem, the results are also affected by inaccuracies and numerical errors arising from the execution of operations and machine representation of floating-point numbers. 4. the proposed approach the next phase of our research is to determine the similarity of the rankings obtained. for this purpose, it was decided to use the weighted spearman correlation coefficient and the ws similarity coefficient. both determinants are based on the obtained ranking values from the multi-criteria decision-making methods used. table 8 shows the calculated correlations between the obtained rankings using the weighted spearman correlation coefficient. using this determinant of similarity guarantees obtaining values of the interval [-1.0, 1.0], where a value of -1.0 means a complete lack of similarity, while 1.0 means their equality. when examining the correlation of the ranking with itself, it will give a value of 1.0 in each case. in turn, the similarity results obtained using the similarity coefficient ws are included in table 9. the values obtained using this determinant guarantee the similarity defined on the interval (0.0, 1.0], where the value 0.0 means no similarity of rankings, while towards reliable decision-making in the green urban transport domain 393 the value 1.0 means the identical rankings. the similarity obtained is highly influenced by changes noted on the first positions among the analyzed orders. the most correlated rankings were obtained by the topsis, comet and spotis methods. the reason may be that these methods use the concept of reference objects. despite using the same concept by the vikor method, the obtained rankings correlations differed significantly from those mentioned above. the divergence between topsis, copras, promethee ii and vikor rankings is a frequent phenomenon appearing in the performance studies of mcda methods. meanwhile, using the ws similarity coefficient for the correlation study, it was noted that the most correlated rankings were obtained using topsis, comet and spotis methods, which shows that both coefficients equally indicate the most similar rankings. the lowest similarity of rankings was noted for the promethee ii method concerning the other rankings. table 8 rankings correlation for spearman weighted correlation coefficient rw rw topsis vikor prom. ii copras comet spotis topsis 1.0000 0.8667 0.6350 0.6650 1.0000 0.9917 vikor 0.8667 1.0000 0.8683 0.6233 0.8667 0.8750 prom. ii 0.6350 0.8683 1.0000 0.6783 0.6350 0.6650 copras 0.6650 0.6233 0.6783 1.0000 0.6650 0.7300 comet 1.0000 0.8667 0.6350 0.6650 1.0000 0.9917 spotis 0.9917 0.8750 0.6650 0.7300 0.9917 1.0000 table 9 rankings correlation for ws similarity coefficient ws topsis vikor prom. ii copras comet spotis topsis 1.0000 0.7916 0.7434 0.9051 1.0000 0.9981 vikor 0.8100 1.0000 0.8539 0.7434 0.8100 0.8119 prom. ii 0.7291 0.8698 1.0000 0.6841 0.7291 0.7375 copras 0.8619 0.7098 0.7950 1.0000 0.8619 0.8830 comet 1.0000 0.7916 0.7434 0.9051 1.0000 0.9981 spotis 0.9981 0.7935 0.7438 0.9047 0.9981 1.0000 the proposed approach is to combine the usual voting approach and the similarity coefficients of the rankings. each of the methods used takes a vote, where each alternative is given a number of points corresponding to a place in the ranking. a new ranking is then established where the highest-ranked alternative is the one that has received the least number of points. of course, the compromise solution may vary due to the number of methods and the methods chosen. table 10 shows the seven rankings that have been created based on the proposed approach, i.e., rank 1 using all six methods, rank 2 using five methods without topsis, rank 3 using five methods without vikor, rank 4 using five methods without promethee ii, rank 5 using five methods without copras, rank 6 using five methods without comet, and rank 7 using five methods without spotis. 394 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun table 10 position values for candidate compromise rankings ai rank 1 rank 2 rank 3 rank 4 rank 5 rank 6 rank 7 a1 9 9 9 9 9 9 9 a2 4 4.5 4 3.5 4 4.5 4.5 a3 6.5 6 6 7 7 6 6 a4 3 3 3 3.5 3 3 3 a5 8 8 8 8 8 8 8 a6 6.5 7 7 6 6 7 7 a7 5 4.5 5 5 5 4.5 4.5 a8 1 1.5 1 1 1.5 1.5 1.5 a9 2 1.5 2 2 1.5 1.5 1.5 in order to obtain a compromise ranking, the measure of its fit must first be defined. when the reference value is not known, one should select a ranking that best fits the selected mcda methods' rankings. thus, we use two matching measures, i.e., the minimum similarity value and the average similarity value. in table 8, the best compromise ranking from among the rankings determined by the mcda methods is the ranking obtained with the spotis method, which has the highest minimum and average similarity value of the rankings, which amount to 0.665 and 0.876, respectively. on the other hand, the worst compromise ranking would be the ranking obtained using the copras method. it has the lowest minimum and average value of ranking similarity, which are 0.623 and 0.727, respectively. an interesting situation is when comparing using the ws coefficient, where three times we obtain the same average value for the topsis, comet and spotis methods. however, taking into account the highest minimum value, the ranking determined by the spotis method is again the best (0.0004 higher than topsis and comet). tables 11 and 12 show the similarity between the candidate compromise rankings and those obtained using the mcda methods. thus rank 1 has an average rw of 0.900 and a minimum similarity value of 0.7842. this means a much higher match than was obtained from each ranking obtained using the mcda methods tested. the best fit was obtained for five methods when topsi, comet or spotis were eliminated (rank 2, 6 and 7, respectively). this is logical because in these three cases, precisely the same ranking was obtained. the matching looks different for the ws measure. different compromise ranks are determined using the average value of similarity of rankings and others in the minimum level criterion. in the first case, the best result is obtained for rank 4 because here the average value of matching is 0.9065, but the minimum value is only 0.7682. table 11 values of similarity coefficient rw for candidate compromise rankings rw topsis vikor prom. ii copras comet spotis rank 1 0.9623 0.9171 0.7992 0.7842 0.9623 0.9754 rank 2 0.9265 0.9385 0.8542 0.8302 0.9265 0.9448 rank 3 0.9483 0.9083 0.8033 0.8150 0.9483 0.9667 rank 4 0.9862 0.9027 0.7331 0.7454 0.9862 0.9946 rank 5 0.9629 0.9548 0.8108 0.7412 0.9629 0.9712 rank 6 0.9265 0.9385 0.8542 0.8302 0.9265 0.9448 rank 7 0.9265 0.9385 0.8542 0.8302 0.9265 0.9448 towards reliable decision-making in the green urban transport domain 395 table 12 values of similarity coefficient ws for candidate compromise rankings ws topsis vikor prom. ii copras comet spotis rank 1 0.9621 0.8164 0.7840 0.9144 0.9621 0.9647 rank 2 0.9043 0.8706 0.8302 0.8798 0.9043 0.9080 rank 3 0.9586 0.8140 0.7836 0.9138 0.9586 0.9622 rank 4 0.9821 0.8071 0.7682 0.9154 0.9821 0.9839 rank 5 0.9177 0.8849 0.8161 0.8659 0.9177 0.9195 rank 6 0.9043 0.8706 0.8302 0.8798 0.9043 0.9080 rank 7 0.9043 0.8706 0.8302 0.8798 0.9043 0.9080 the compromise solution, in this case, should be selected as the order indicated by rank 2, 6 or 7 (these rankings are identical), as this guarantees maximum matching for all the rankings examined. finally, we also show how strongly similar solutions have been obtained using all methods or combinations of the five elements. tables 13 and 14 show the similarity values of the rankings among themselves. as we can see, the rankings are very similar to each other, so the choice of methods is not a big problem in determining the compromise ranking, but the research presented in this paper should be extended and continued to generalize the observations of this paper. table 13 correlation matrix for rw values for candidate compromise rankings rw rank 1 rank 2 rank 3 rank 4 rank 5 rank 6 rank 7 rank 1 1.0000 0.9854 0.9971 0.9917 0.9900 0.9854 0.9854 rank 2 0.9854 1.0000 0.9883 0.9662 0.9838 1.0000 1.0000 rank 3 0.9971 0.9883 1.0000 0.9829 0.9812 0.9883 0.9883 rank 4 0.9917 0.9662 0.9829 1.0000 0.9875 0.9662 0.9662 rank 5 0.9900 0.9838 0.9812 0.9875 1.0000 0.9838 0.9838 rank 6 0.9854 1.0000 0.9883 0.9662 0.9838 1.0000 1.0000 rank 7 0.9854 1.0000 0.9883 0.9662 0.9838 1.0000 1.0000 table 14 correlation matrix for ws values for candidate compromise rankings ws rank 1 rank 2 rank 3 rank 4 rank 5 rank 6 rank 7 rank 1 1.0000 0.9387 0.9980 0.9813 0.9489 0.9387 0.9387 rank 2 0.9408 1.0000 0.9430 0.9233 0.9858 1.0000 1.0000 rank 3 0.9978 0.9407 1.0000 0.9789 0.9465 0.9407 0.9407 rank 4 0.9817 0.9185 0.9795 1.0000 0.9348 0.9185 0.9185 rank 5 0.9506 0.9854 0.9484 0.9362 1.0000 0.9854 0.9854 rank 6 0.9408 1.0000 0.9430 0.9233 0.9858 1.0000 1.0000 rank 7 0.9408 1.0000 0.9430 0.9233 0.9858 1.0000 1.0000 5. conclusions the presented study shows a simple way of building a compromise ranking, which is quite helpful when the obtained ranking results are not unambiguous. additionally, two criteria were given for the selection of the best compromise ranking. it allows obtaining a solution, which is maximally compatible with all methods involved in determining this 396 a. shekhovtsov, j. więckowski, b. kizielewicz, w. sałabun solution. the 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göhlich, d., kunith, a., ly, t., 2014, technology assessment of an electric urban bus system for berlin, wit trans. built environ, 138, pp. 137-149. 56. lajunen, a., lipman, t., 2016, lifecycle cost assessment and carbon dioxide emissions of diesel, natural gas, hybrid electric, fuel cell hybrid and electric transit buses, energy, 106, pp. 329-342. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 295 305 applicability analysis of additive manufacturing processes in the fabrication of anatomically shaped lattice scaffolds  udc 621.9 jelena milovanović, miloš stojković, miroslav trajanović university of niš, faculty of mechanical engineering abstract. manufacturing of anatomically shaped scaffolds for bonе tissue recovery as well as other similar anatomically shaped implants represents a major challenge for modern manufacturing technologies. the complexity of anatomically shaped lattice scaffolds for bone tissue recovery requires involvement of so-called additive manufacturing processes. this paper brings out the criterial matrix for the assessment of additive manufacturing processes applicability in the case of bone tissue scaffold manufacturing. moreover, this criterial matrix serves as the basis for developing calculator for the generic assessment of additive manufacturing processes applicability. in this very particular case the subject of consideration is an anatomically shaped lattice scaffold intended for the recovery of large trauma located in the upper part of proximal diaphyseal of rabbit tibia. the criterial matrix and the calculator defined for this case prove themselves as generic tools for comparative analyses of applicability of different additive manufacturing processes. furthermore, these tools can help identifying the most demanded features of some future additive manufacturing process that has to be developed for the specific case. key words: additive technologies, additive manufacturing process, scaffolds, anatomically shaped lattice scaffold, applicability, criterial matrix, calculator 1. introduction tissue engineering usually involves the use of so-called scaffolds, which are expected to provide for the necessary mechanical support to the cells seeding in the process of tissue reconstruction. the scaffolds perform the role of an artificial (highly porous) received august 12, 2015 / accepted september 30, 2015 corresponding author: jelena milovanović university of niš, faculty of mechanical engineering in niš, a. medvedeva 14, 18000 niš, serbia e-mail: jeki@masfak.ni.ac.rs original scientific paper 296 j. milovanović, m. stojković, m. trajanović extracellular matrix which ensures a proper and sufficiently rapid cell growth as well as an efficient reconstruction of the tissue that has been damaged by injury or disease. in terms of structure, the scaffolds are usually artificial lattice-like support structures of biocompatible materials. to perform their function, they should possess:  biocompatibility with native tissue,  a smooth reinnervation and revascularization of new tissue inside the volume of the graft which includes suitable communication (connection, ingrowth) related to the surrounding tissue (ensuring undisturbed transport of nutrients to cells and removal of waste products from them),  attachment of cells and grow factors to the surface of the scaffold struts (surface of scaffold struts "must be pre-processed") a process called surface functionalization [1],  appropriate mechanical properties (e.g. structural strength and elasticity or stiffness) to ensure required deformations,  high level of geometrical, i.e. anatomical consistency (congruency) of custom shaped graft during tissue recovery,  easy sterilization, and,  simplicity of fixation and implantation. another very important feature which should be added to the above list is that the scaffold design should be relatively easy to manufacture, that is, should be characterized by a high level of manufacturability. considering the complexity and uniqueness of the shape of the anatomically shaped lattice scaffolds – asls (see fig. 1).[2] and the current state of manufacturing technology [3, 4], it is clear that it is necessary to employ additive manufacturing processes – amp. fig. 1 asls design compatibility of the amp with medical imaging techniques (ct, mri ...) has opened up opportunities for the emergence of new approaches in the design of the internal architecture of the scaffolds for bone tissue reconstruction. the amp allows the creation of anatomically shaped scaffolds in addition to the control of the size and distribution of pores as well as of the entire inner architecture. however, the scaffold’s geometry complexity imposes the need to analyze applicability of different amps and to choose the most applicable one. apart from few analyses [5, 6, 7, 8] there are not many procedures and methods for analyzing the applicability of the manufacturing processes for a specific case. for example, the results of the research related to manufacturing complexity evaluation for applicability analysis of additive manufacturing processes in fabrication... 297 additive and subtractive processes in the case of hybrid modular tooling [8] have shown that the geometric parameters do not have the same influence in the cases of subtractive or additive processes, indicating the importance of generating a method (procedure) for assessing the applicability of the technological processes which could be easily adapted for a particular case. from that aspect, this paper analyzes the applicability of different amps for the manufacturing of one type of asls. at the same time, the paper discusses and proposes a method for assessing the applicability of different amps. the crucial criteria that is used for the selection of candidates between different amps is the ability of using commercial biocompatible or biodegradable materials that are applied regularly. this feature allows for the implantation of asls samples in experiments in vivo. following that criteria, the selection of the amp that can be used for the asls manufacturing is reduced to three potential amps: 1. 3d bioplotter, 2. direct metal laser sintering (dmls), and, 3. electron beam melting (ebm). 3d bioplotter is chosen as the only commercially available at developed to plot biodegradable materials and biological cells, for making temporary (biodegradable) asls of hydroxyapatite (ha) while dmls and ebm are used to make permanent asls of ti-alloys (ti6al4v and ti64). 2. manufacturability of design vs. applicability of manufacturing process in a manufacturability analysis different scaffold design variants are taken into consideration in order to identify the one which is possible to manufacture by using (given) particular manufacturing process while achieving the maximum level of required quality at minimum investment. on the other hand, the applicability analysis considers whether an amp can be applied to fabricate a reference scaffold design variant of required quality, and if so, to give further consideration to determining investment parameters (time, material, post-processing etc.) the most applicable amp among several potential candidates is the one which is featured by minimal investment. therefore, when we consider manufacturability we bring into focus design characteristics and when we consider applicability we are focused on the process’ characteristics. on the basis of the above, manufacturability and applicability can be presented using the following functions [10]: scaffold design manufacturability (m): ({ }, )i i nm m d mp (1) where i = 1, 2, 3 …, di – different design solutions of the part and mpn – manufacturing process. applicability of manufacturing processes (a): ({ }, )j j na a mp d (2) where j = 1, 2, 3 …, mpj – different manufacturing processes and dn – one (reference) design solution of the part. 298 j. milovanović, m. stojković, m. trajanović methods of assessment which are used in the determination of manufacturability can be applied for determining applicability of certain manufacturing processes. determining the applicability of certain mp is reflected in finding answers to the following [11]:  determine whether or not certain manufacturing process is applicable for the manufacture of part,  if a mp is found to be applicable, determine an applicability rating, and,  if a mp is not applicable then identify the mp attributes that cause applicability issues. 3. the criterial matrix for the assessment of amp applicability comparative analysis of the applicability of the selected amp in the case of asls manufacturing is carried out using so called abstract-quantitative evaluation [12]. this analysis begins by choosing appropriate variables for determining the applicability of the amp. these variables are chosen for a specific process of manufacturing asls and similar forms. each of these variables is assigned significance (s) which reflects the significance of variables for determining the applicability of the amp in the range of s = (0-1) (fig. 2). 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,8 0,5 0,8 0,9 0,9 0,8 0,8 0,5 0,8 1 1 0,8 series1 fig. 2 significance (s) of variables for determining the applicability of the amp values of these variables are obtained as statistical averages of summary assesments provided by multidisciplinary experts from different laboratories. applicability analysis of additive manufacturing processes in fabrication... 299 it should be kept in mind that the value of the significance (s) of some variables can be changed toward a certain (reference) scaffold design. based on the specifics of each of the amps, which is applicable to manufacture specific classes of bone scaffold (asls), a criterial matrix for the assessment of the amp applicability is defined [10]. to fill up this matrix, in addition to already defined significance value (s) for corresponding variables, each variable value (v) has to be determined (see figs. 3, 4 and 5, and table 1). it is done through abstract-quantitative and experts assessments. definitions of variables that are chosen as relevant for amp applicability assessment in the case of asls design are listed below:  complexity of operations planning takes into account the number of required operation iterations and the complexity of the same.  material quantity implies total material consumption the overall quantity of material which is necessary for manufacturing a sample (including support structures) the higher consumption of material the lower the absolute value of the variable.  material costs are determined on the basis of proportionality: bp: ebm: dmls. for this case this value is the highest for bioplotter (150 € / g) and this value is taken as maximum value of the variable: -1. consequently for the other two amps the value of this variable (considering that the price of a material is known) is determined: bp: ebm: dmls = 1: 0.0013: 0.003.  manufacturing time is also determined on the basis of proportionality: bp: ebm: dmls. the longest manufacturing process for the scaffold sample is for bioplotter (4,72h) and this value is taken as maximum value of the variable -1. for the other two amps, the value of this variable is determined: bp: ebm: dmls 1: 0.42: 0.39.  the assessment of process costs is performed according to the standard price of the machine with the amortization period of 5 years and an average interest rate of 5%. since the value for the ebm process is the highest (€ 105) in relation to the bioplotter (€ 47) and dmls (€ 89.95), the adopted value of the variable for the ebm is -1 and for the other two amp value of this variable is determined on the basis of proportionality.  complexity of the process includes: the number of process iterations (interruption of the machine to reposition, changing tool during operation, etc.) special conditions (protective atmosphere), the need for additional devices and equipment and like.  complexity of post-processing is determined in the same way as complexity of the process.  values for surface quality depend on particular case. it should be kept in mind that for the scaffold struts lower surface quality (rougher surface) is better because it increases bioadhesiveness.  geometrical accuracy takes into account the deviation of the manufactured asls sample compared to the digital model. the higher deviation of the manufactured asls sample, the lower the absolute value of the variable. 300 j. milovanović, m. stojković, m. trajanović  -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1 -0,8 -1 -1 -1 -0,448 -0,5 -0,9 0 0,5 1 0 0 series1 fig. 3 values (v) of variables for determining the applicability of 3d bioplotter in asls manufacturing -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1 -0,5 -0,5 -0,003 -0,39 -0,856 -0,4 -0,5 -0,5 1 0 1 -0,5 series1 fig. 4 values (v) of variables for determining the applicability of dmls in asls manufacturing applicability analysis of additive manufacturing processes in fabrication... 301 -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1 -0,1 -0,1 -0,0013 -0,42 -1 -0,5 -0,1 -0,2 0,8 0 1 0 series1 fig. 5 values (v) of variables for determining the applicability of ebm in asls manufacturing value of the expression: s × v is a parameter that affects the assessment of the amp applicability and  n 1 s  v is a summary assessment of the applicability of particular asls manufacturing process. bearing in mind that the value of the eliminatory coefficient possibility of manufacturing asls is equal to 0 for 3d bioplotter process because it appears as unable to manufacture such a free-form geometry, the bioplotter is not used for further comparisons. however, the values of the above mentioned variables obtained for bioplotter are shown in the criterial matrix for assessing the amp applicability (table 1, shaded columns), in order to obtain a realistic insight about advantages and disadvantages of the actual 3d bioplotter process. 302 j. milovanović, m. stojković, m. trajanović table 1 criterial matrix for the assessment of amp applicability variables for determining the applicability of the amp s(0,1-1) dmls ebm 3d bioplotter v s×v v s×v v s×v complexity of operations planning 0,8 -0.5 -0.4 -0.1 -0.08 -0.8 -0.64 material quantity 0,5 -0.5 -0.25 -0.1 -0.05 -1 -0.5 material costs 0,8 -0.003 -0.0024 -0.0013 -0.00104 -1 -0.8 manufacturing time 0,9 -0.39 -0.351 -0.42 -0.378 -1 -0.9 process costs 0,9 -0.856 -0.7704 -1 -0.9 -0.448 -0.4032 complexity of the process 0,8 -0.4 -0.32 -0.2 -0.16 -0.5 -0.40 complexity of post-processing (if there is a need for) 0,8 -0.5 -0.40 -0.1 -0.08 -0.9 -0.72 surface quality 0,5 -0.5 -0.25 -0.2 -0.10 0 0 geometrical accuracy 0,8 1 0.8 0.8 0.64 0.5 0.40 biodegradation 1 0 0 0 0 1 1 possibility of asls manufacturing 1 1 0.9 1 1 0 0 support structures necessity 0,8 -0.5 -0.40 0 0 0 0  -1.3438 -0.1184 s  significance of variables for determining the amp applicability v  values (v) of variables for determining the amp applicability comparative analysis of the value of expression s×v using dmls and ebm additive manufacturing processes for asls manufacturing is given in the diagram (fig. 6). -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1 ebm dmls fig. 6 comparison of values: s × v in dmls and ebm applicability analysis of additive manufacturing processes in fabrication... 303 the highest value of expression  n 1 s  v in the criterial matrix for assessing the applicability of amp indicates that ebm technology has the fewest features that adversely affect the applicability of the manufacturing process, i.e. ebm appears as the most suitable for manufacturing of asls and similar type of scaffolds. 4. process applicability calculator criterial matrix for the assessment of the amp applicability defined in this way serves as the basis for creating so-called calculator for generic assessment of additive manufacturing processes applicability [10]. the appropriate case (in this case manufacturing of asls), is initially entered in the calculator which is followed by the input of the appropriate processes (amp) that are used for asls manufacturing and for each of them: variables for determining the applicability of the amp, their significance -s and values -v. all these input parameters are variable and depend on the particular design and its functionality. calculator automatically calculates  n 1 s  v which is a summary assessment of the amp applicability for asls manufacturing and, according to the results, it gives the recommendation for the most appropriate technology for manufacturing of asls or similar bioforms. relational database schema that supports this calculator is given in fig. 7. database for the calculator is made in microsoft access 2007 . fig.7 relational database schema of calculator the calculator can be used to add a new amp and to change or add new variables for determining the applicability of the amp, their significance -s and values -v depending on the part being manufactured, i.e. requirements of the case. the calculator also may include a variety of cases. in this way, the calculator encompasses the ability to configure an applicability analysis to a particular case; thus, it makes it very flexible and efficient. in accordance with the above-defined approach the eliminatory coefficient in the matrix and the calculator the possibility for manufacturing of asls is calculated. for https://www.google.rs/search?q=microsoft+access+2007&rlz=1c1cafb_enrs654rs654&es_sm=93&tbm=isch&tbo=u&source=univ&sa=x&ved=0ccgqsarqfqotcowxipr1xsgcfqetfaodgfmmcg 304 j. milovanović, m. stojković, m. trajanović this specific case of design, possibility for manufacturing of 3d bioplotter is 0 (it cannot be manufactured on 3d bioplotter), i.e., this amp is shown as incapable to employ and 3d bioplotter is excluded from further comparisons. the outputs from this application are assessments of amp applicability used for manufacturing of asls and recommendation for the most applicable (quality) manufacturing process among them (fig. 8). fig. 8 result of the calculator for the assessment of selected amp applicability for asls manufacturing the criterial matrix indicates that an indisputable advantage of 3d bioplotter, compared to other technologies, is its ability to plot biodegradable materials, and even cells. however, the matrix shows that the main drawback of this technology is that it cannot be used (at the actual state of development) to make complex forms like asls. on the other hand, calculator shows that ebm is currently the optimal choice for manufacturing of asls in metal. according to comparative analysis of applicability, dmls is just behind ebm, but with slightly lower performance. however, at this point of development, ebm and dmls cannot be used to manufacture temporary asls, that is, they cannot utilize biodegradable materials. moreover, it should be noticed that the analysis of variables involved in the matrix and calculator can bring out what amp features are or could be appropriate for manufacturing of a particular design case like asls. in this way, the criterial matrix and calculator may help either improving the existing or defining new amp for a particular product design. 5. conclusions the modern society clearly shows its increasing demands for further improvements in health care. this, among other things, involves the development and manufacturing of biological implants, which, in its turn, urges a more rapid development of additive technologies, which seem to be optimal for this application. in this regard, the selection of the most applicable amp represents an essential step for determining the optimal manufacturing process for a certain biological implant. in addition, the definition of an efficient and traceable method for the assessment of applicability of certain amp to applicability analysis of additive manufacturing processes in fabrication... 305 fabricate biological implants such as, for example a bone tissue scaffold, is of great importance. however, it seems that, at present, there is no a suitable method for the purpose of an amp applicability analysis. the criterial matrix and the calculator that are described in this paper propose a kind of method and corresponding software solution for the assessment of the amp applicability for the case of anatomically shaped lattice scaffold fabrication. the proposed method appears as a very flexible one that can be changed, expanded and/or adapted according to the needs of a particular case. moreover, the application of calculator implies the ability to configure an analysis of applicability of different amp to fabricate similar bio-structures, by changing the existing and adding new variables, their values and significance as well as defining and including new assessment and elimination criteria. ultimately, this whole methodology may indicate requirements for an ideal additive manufacturing machine for bone tissue scaffold manufacturing. acknowledgements: the paper presents a case that is a result of the application of multidisciplinary research in the domain of bioengineering in real medical practice. the research project (virtual human osteoarticular system and its application in preclinical and clinical practice) is sponsored by the ministry of education, science and technological development of the republic of serbia project id iii 41017 for the period of 2011-2015. references 1. chen, q. z., rezwan, k., armitage, d., nazhat, s. n, boccaccini, a. r., 2006, the surface functionalization of 45s5 bioglass -based glass-ceramic scaffolds and its impact on bioactivity, journal of materials science: materials in medicine, 17(11), pp. 979-987. 2. stojkovic, m., korunovic, n., trajanovic, m., milovanovic, j., trifunovic, m., vitkovic, n., 2013, design study of anatomically shaped lattice scaffolds 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(jcis), pp. 194-197. 7289 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 237 253 https://doi.org/10.22190/fume201230028a © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* taguchi optimization of multiple performance characteristics in the electrical discharge machining of the tigr2 sıtkı akıncıoğlu department of machine design and construction, university of düzce, turkey abstract. electrical discharge machining (edm) provides many advantages for the shaping of metallic materials. it also provides better surface quality for ti alloys used in the defense industry. in this study, experiments were carried out with different edm parameters using the titanium (gr2) alloy. a number of novel industrial processes have been developed as a result of advances in technology. for a product to be developed, these novel approaches must be utilized to determine optimum parameters. the taguchi method was applied in the experiments with edm. the impact the test parameters had on the performance characteristics of tool wear rate, material removal rate, depth, and surface roughness were analyzed by the variance analysis (anova). quadratic regression analyses were carried out to reveal the correlation between the experimental results and the predicted values. according to the anova results for material removal rate (mrr), tool wear rate (twr), depth, and surface roughness, the most effective factor was amperage, at 99.66%, 99.56%, 87.95%, and 81.12%, respectively. the best value for average surface roughness was determined to be 3.29 µm obtained at 120 μs timeon, 8 a, and 40 μs time-off. key words: electrical discharge machining (edm), taguchi method, optimization, titanium (gr2), material removal rate 1. introduction electrical discharge machining (edm) provides many advantages in the processing of specially shaped workpieces, irrespective of material strength or hardness [1]. the effectiveness of edm is independent of the mechanical properties of the workpiece [2]. however, edm has a lower processing speed than that of traditional machining methods. furthermore, because of the cracks and micro-pits formed in the recast layer as a result of rapid cooling, less surface accuracy and a shorter cutting-tool life are obtained in molds or received december 30, 2020 / accepted march 10, 2021 corresponding author: sıtkı akıncıoğlu department of machine design and construction, university of düzce,gümüşova, düzce, turkey e-mail: sitkiakincioglu@duzce.edu.tr 238 s.akincioğlu machine parts, and thus, edm has a low mrr and poor processing precision. therefore, more detailed investigations are needed in order to overcome these disadvantages [3, 4]. the advantages of titanium alloys are recognized in many industrial sectors, including the defense, aerospace, and automotive industries, as well as for power generation, desalination, architecture, marine, and medical applications because of their high corrosion resistance, high strength, low specific gravity and high thermal resistance. titanium gr 2 is pure commercial alpha titanium. titanium grade 2 materials contain a minimum of 99% titanium elements. these alloys have very good corrosion resistance, mechanical strength and low weight, besides being extremely resistant to chemical environments such as alkali environments, organic acids and compounds, aqueous salt solutions and hot gases. titanium grade 2 materials are frequently used in the chemical industry, marine vehicles, aviation vehicles and medical parts. the selection of the parameters during machining affects surface quality and processing costs [5]. difficult-to-process titanium alloys can be processed easily with edm [2]; however, the speed of processing and the quality of the surface must be increased. these development efforts are expensive due to a high cost of the titanium alloy, the long duration of the edm process, and the need for extra testing. the taguchi design optimizes a response variable with various control factors and noise factors. it is an efficient and inexpensive experimental design method using fewer resources than full factorial design. because the taguchi method greatly reduces the number of trials necessary for the experiments, in recent years, its advantage has been increasingly recognized [6]. in many studies, researchers have successfully optimized the parameters in the edm process using the taguchi method. manisha et al. [7] machined the titanium alloy from the test material by using different parameters in the edm optimized with the taguchi method (l25). the parameters used in the experiments were determined as pulse duration (time-on), duty factor, discharge current [amperage (a)] and gap voltage (vg). the analysis of variance (anova) test showed that the most effective parameter for material removal rate (mrr) was duty (52.87%). according to the signal/noise (s/n) ratios, the optimum parameters were calculated as tone (30 μs), duty factor (9), current (50 a) and vg (6 v). vijay et al. [8] used the l9 orthogonal array taguchi method to optimize the parameters used in the processing of a titanium alloy (ti6al4v) in edm. the processing parameters were determined as three different on-time pulses, three different off-time pulses, fluid pressure, and voltage. they investigated material wear loss, tool wear loss, and surface roughness values and successfully determined the optimum parameters for mrr, tool twr, and surface roughness. sunil et al. [9] analyzed the results for mrr, twr and surface roughness obtained by processing the titanium alloy ti-6al-4v using edm via the taguchi experimental design (l18). the parameters used in the experiments were selected as duty cycle, amperage, working time, electrode, voltage, retraction distance, pulse duration, and liquid pressure. they found that the most effective parameter for mrr was the duration of the pulse (tone), the increase of the washing pressure and the increase of the surface roughness. jeavudeen et al. [10] investigated the edm machining performance of titanium and hss material of duty factor and the mixture of alumina powder to dielectric. taguchi's l9 orthogonal array was used to optimize its effect on mrr and tool wear index (twi) in the edm processing. the mrr results show that the current and dust concentration increase the material removal rate of the hss and titanium workpiece. the twi of the hss sample was also found to be higher than the titanium sample. gaikwad et al. [11] machined edm and niti alloy using a copper electrode. in their work, they discussed process parameters such as gap current, pulse on time, pulse off time. using the taguchi method, they found important process parameters affecting tool electrode taguchi optimization of multiple performance characteristics in electrical discharge machining... 239 and workpiece electrical conductivity, pulse on time, gap current, tool wear rate, and material removal rate. as a result of the study, they found the optimum tool wear rate of 0.031 mm3/min and the optimum material removal rate of 6.31 mm3/min. nagaraju et al. [12] machined 17-7 ph stainless steel using the edm method. they used current, pulse on time, discharge voltage surface roughness, overcut and clearance in electrical as processing parameters. they used taguchi l9 orthogonal array to optimize parameters. gugulothu et al. [13] investigated the effect of the ti-6al-4v alloy on the electric discharge treatment of drinking water as the dielectric liquid and graphite powder concentration. they examined the effects of discharge current, pulse on time, pulse off time and graphite powder concentration parameters on properties such as material removal rate, surface roughness and recast layer thickness. they used the taguchi method for optimization of parameters. it has been found that the discharge current is the most important parameter affecting mrr, sr and rlt. also, the dust concentration is less important. lin et al. [14] examined the performance characteristics of skd11 tool steel using edm. the machining parameters were optimized with respect to the multiple performance characteristics. the experimental results demonstrated the effectiveness of this approach. difficult materials can be easily shaped with edm. this process has a low metal removal rate and a high tool wear rate. for this reason, this study aimed to increase the metal removal rate and reduce the electrode consumption without reducing the surface quality of the titanium gr2 alloy. many superalloys have been machined with edm. however, research on edm of pure titanium alloy gr2 quality alloy is very limited. kumar et al. [15] investigated the processing of biomedical titanium alloys using edm and wedm. in their research, they examined many studies on the edm processing of titanium alloys. according to their results, it is very advantageous to process titanium alloys with edm. however, he states that the role of the edm process for surface modification of titanium biomaterials is still in the experimental phase. nimbalkar et al. [16] also reported that many studies need to be done to process titanium with edm. gupta et al. [17] also explained that there is room for improvement in processing titanium with edm. he also reported that the taguchi method can be used in the complex edm optimization of titanium. this study aims at increasing surface quality, reducing electrode consumption and shortening the processing time. thanks to the edm method, the machining of difficult materials has been made easy. although the edm machining is a time-consuming method, it can also be cost-effective. the studies have been done on titanium super alloy before surface quality, mrr and twr of the titanium gr2 the workpiece is not at the desired level. the studies on the machining of titanium super alloys with edm have been limited. comprehensive studies are required to increase the surface quality of the titanium gr2 alloy workpiece and to reduce the machining costs. in this current study, the effects of the parameters of pulse duration (timeon), waiting time (time-off), and amperage, on tool wear rate, cutting depth, and surface roughness were optimized by using the taguchi method in the machining of the titanium alloy in edm. it provided a comprehensive analysis of the edm machining of titanium gr2 using both the taguchi method and the gray relational analysis. in addition, the anova and regression analyses were performed to interpret the results more accurately. 240 s.akincioğlu 2. material and methods 2.1. electrical discharge machine the king znc k 3200 electro discharge machine (fig. 1) was used for the experimental trials in the study. fig. 1 electro-discharge machine used in the experimental work the electrical discharge machining process is a technique using sparks produced by electrical discharge to manufacture workpieces in accurate dimensions and shape [18]. the twr and mrr are calculated using standard arithmetic equations. the differences in the weights of the workpiece and the electrode per minute, before and after machining is carried out, are used in the calculation of the mrr and twr, respectively [19]. the formulae for calculation of the mmr and twr are given as (eq. 1) and (eq. 2), respectively [18]. mrr = [ (𝑊𝑖−𝑊𝑓) 𝑡 ] (1) here, mrr is the material removal rate (g/min), wi is the initial weight of workpiece (g), wf is the final weight of workpiece (g), and t is the duration of the experimental trial (min). twr = [ (𝑇𝑖−𝑇𝑓) 𝑡 ] (2) here, twr is the tool wear rate (g/min), tiis the initial weight of electrode (g), tfis the final weight of electrode (g), and t is the duration of the experimental trial (min). taguchi optimization of multiple performance characteristics in electrical discharge machining... 241 2.2. material and electrode selection in the experimental work, nine 10 × 10 mm pieces of titanium (gr2) alloys were used. table 1 presents the workpiece material chemical composition and table 2 gives its mechanical and physical characteristics. table 1 chemical composition of titanium gr2 c fe n o h titanium 0.10 0.20 0.03 0.25 0.015 balance table 2 physical and mechanical properties of titanium gr2 properties values 1 density, (g/cm3) 4.51 2 melting point, (°c) 1660 3 electrical resistivity, 10-6 (ohm•cm) 56 4 thermal conductivity, (w/m•°c) 20.8 5 specific heat, (j/kg•°c) 520 6 tensile strength, (ksi) 50 7 yield strength, (ksi) 40 8 stretching, (%) 20 9 hardness, (hrb) 82 the electrolytic copper electrode used in the edm tests was 25 mm in diameter, with a density of 8.9 g/cm3 (table 3). table 3 properties of the electrode properties values 1 melting point (°c) 1083 2 elastic modulus (n/mm2) 1.23×105 3 poisson’s ratio 0.26 4 density (g/cm3) 8.9 2.3. scanning electron microscopy and measurement of surface roughness and weight scanning electron microscopy (sem) was performed using the fei quanta feg 250 instruments to examine and measure the built-up edge (bue) occurring on the cutting tool. a mahr meter was used to measure the surface roughness values at room temperature. in order to minimize errors that may arise during the measurement, the surface roughness has been measured from five different locations. the measurement length was determined as 8 mm. a cut-off of 0.8 mm and a sampling length of 5.6 mm were set for the measurement of the surface roughness values during the machining of the workpiece. a radwag precision scale (0.001 g) was used to measure the test sample weight loss. 2.4 taguchi method the taguchi design of experiments (doe) is used extensively in the industrial sector as well as in the academic field as a method of determining the parameters that are the most effective in accordance with the experimental results. experimental design using the 242 s.akincioğlu taguchi method is a powerful statistical method for analyzing the interaction among variables [20]. the taguchi method enables manufacturers to reduce the design and production time needed to develop their products, thus lowering costs and consequently, increasing their business profits [6, 21]. moreover, the variables that are otherwise unaccountable and uncontrollable using conventional experimental design can be controlled with the taguchi doe. the objective function values are converted via the taguchi method to a signal/noise (s/n) ratio, which allows the performance characteristics of the control factor levels to be measured against these factors. the s / n ratio expresses the relation of the strength of the signal to that of the noise [22]. the taguchi test design measured each control factor combination for surface roughness (ra) and the control factors were optimized using the s/n ratios. for the calculation of the s/n ratios, the objective functions, according to the characteristic type, are expressed as “the nominal is best” (eq. (3)), “the largest is best” (eq. 4) and “the smallest is best” (eq. (5)) [23]. the nominal is best: s n = 10 log ( y̅ sy 2) (3) the largest is best: s n = −10 log 1 n ∑ 1 yi 2 n i=1 (4) the smallest is best: s n =-10 log 1 n ∑ 1 yi 2 n i=1 (5) the aim of this study was to minimize surface roughness, mrr, and twr, and to use the "the smallest is best" quality characteristic in (eq. 5). 2.5 parameters and levels table 4 shows the levels of the parameters of time-on, time-off, and amperage that were used for the experimental study. a constant cutting depth of 0.5 mm was used in the experiments. time-on, time-off and amperage values were determined by making use of the studies. these values have been tested primarily with preliminary experiments. the values obtained later were designed using the taguchi method. these parameter levels indicated that the l9 taguchi orthogonal array should be chosen as the most suitable for this study (table 5). table 4 test parameters and levels parameters symbols units level 1 level 2 level 3 time-on a (µs) 120 122 124 time-off b (µs) 40 44 48 amperage c (a) 8 12 16 table 5 taguchi orthogonal array design l9 exp. no. factor a factor b factor c 1 1 1 1 2 1 2 2 3 1 3 3 4 2 1 2 5 2 2 3 6 2 3 1 7 3 1 3 8 3 2 1 9 3 3 2 taguchi optimization of multiple performance characteristics in electrical discharge machining... 243 3. results and discussion 3.1. experimental results after a review of the relevant literature studies, the parameters were chosen accordingly for the use in the experiments. table 6 shows the results of the experimental study. according to these results, wear loss of the copper electrode and titanium workpiece material, cutting depth and average surface roughness values were evaluated. table 6 experimental results exp. no. time on (µs) time off (µs) amperage (a) total processing time (min) electrode wear loss (g) material wear loss (g) electrode height difference (mm) material height difference (mm) average surface roughness ra (µm) 1 120 40 8 511.06 0.036 0.441 0.16 0.50 3.29 2 120 44 12 207.70 0.045 0.368 0.16 0.41 4.38 3 120 48 16 126.20 0.060 0.476 0.10 0.44 5.40 4 122 40 12 278.36 0.068 0.483 0.10 0.48 4.12 5 122 44 16 118.56 0.056 0.443 0.12 0.30 5.14 6 122 48 8 705.50 0.047 0.517 0.12 0.50 4.06 7 124 40 16 118.56 0.060 0.449 0.10 0.40 5.17 8 124 44 8 666.13 0.048 0.547 0.10 0.50 4.24 9 124 48 12 313.90 0.069 0.477 0.20 0.30 4.61 since the electrical conductivity of the titanium material is low, it took a long time to remove 0.5 mm of material. after completion of the experiments, the maximum time (705.50 min) to achieve the treatment was seen in the 6th experiment, while the shortest processing time (118.56 min) was reached in the 5th and 7th experiments. it was recognized that the processing time decreased with the increase of amperage [3]. the 5th and 7th test parameters could be chosen as rough machining for this material in order to reduce the processing time. the minimum amount of electrode wear (0.036 g) was observed in the 1st experiment and the maximum electrode wear (0.069 g) in the 9th. it can be seen that the selected parameter values were not compatible with one another due to a high amount of electrode wear. the amount of electrode wear was less than that of material wear. the material wear amount and processing time were also parallel. the lowest average surface roughness was obtained under the processing conditions of the 1st experiment (120 μs time-on, 40 μs time-off, 8 a for 511.06 min). from this result, the processing time was found to be average. it appears that there was no specific relationship between the machining time and the surface roughness. lin et al. [3] achieved similar results in processing titanium using edm. surface quality decreased with the increase of amperage. 3.2. surface analysis many parameters such as pulse current, impact time, tool electrode and workpiece materials are effective on the surface integrity of the edm applied surface [24, 25]. examination of profilometry and sem images was used to determine the average surface roughness values achieved via the experiments. the best surface roughness values were achieved in the 1st experiment (8 a, 120 μs time-on, and 40 μs time-off). the highest surface roughness value (5.40 μm) was obtained in the 4th experiment (16a, 120 μs time244 s.akincioğlu on, and 48 μs time-off). it can be concluded that the increase of the amperage value had a negative effect on the surface roughness. the formation of large craters as a result of a high amperage discharge adversely affected the surface roughness. the reason for the formation of surface cracks is tensile stress caused by shrinkage of the molten material during cooling of the workpiece surface after the spark. surface cracks occur because the tensile stress of the material exceeds the final tensile stress in the white layer. increasing amperage increases pits and micro holes. thus, its amperage must increase in order to increase the mrr during electrical discharge. it creates larger and deeper craters and holes as the amperage increases [24, 26].these results are consistent with those of other studies [27]. although amperage discharge negatively affected the surface, it shortened the machining time. fig. 2 shows the average surface roughness (ra) values as 3d images obtained via profilometer, with the low ra values indicated in green and yellow. in contrast to the deeper pits, the highest ra values are indicated by the red peaks formed on the surface. with the increase of the amperage, the surface roughness value was clearly increased [7]. the surface topography varied significantly depending on the electrical parameters of timeon, time-off, and amperage. the most important factors in the deterioration of surface quality were amperage and the time-on. with the increase of the time-on and amperage, an increase in the crater size was observed in the surface area. during the short time-on, electrical sparks were found to form small craters on the surface of the workpiece [28]. fig. 2 3d surface profile of test specimens: a) 1st sample, b) 3rd sample taguchi optimization of multiple performance characteristics in electrical discharge machining... 245 3.3. microstructural analysis the very high temperatures generated at the spark point were a result of the energy discharged during the edm process, and caused the evaporation and obliteration of portions of the sample surface. with each discharge flow, spherical particles were formed in the crater, and micro fractures and crater edges of various sizes were formed on the machining surfaces (fig. 3) [29]. examination of the sem images of the machined surface showed that the first experiment yielded the lowest surface roughness. the low amperage (8 a) caused small gaps in the surface. in the 3rd experiment, the highest surface roughness value was obtained by the increase of amperage (16 a). this can be explained by the growth of the gaps in the surface profile and the heat input rates [30]. it has been reported in the literature that the craters formed on the surface grow with the increase in the amount fig. 3 sem view of test specimens and edax analysis: a) 1st sample, b) 3rd sample 246 s.akincioğlu of amperage [27]. the sem images of the machined surfaces show that the surface roughness of the1st sample (fig. 3a) is better than that of the 3rd sample (fig. 3b). the sem cross-section views (fig. 4) show the craters, micro peaks and material residue appearing on the surfaces, and a white layer (recast layer) can be seen [31]. the high amperage caused large craters to form. the white layer occurred due to sudden heating and cooling. this was the result of residual tensile stresses in the warming zone which caused surface fractures and surface defects on the workpiece [27]. the intensity and depth of surface cracks may vary depending on the machining parameters. it was found that the cracks increased with increasing amperage levels. the cause of this increase in the cracks was the surface tension which occurred in the material due to a high temperature formation during the time-on [32]. the recast layer (white layer) (fig. 4) is the outer region of the heat affected zone. it is a layer formed by the overlapping of the molten and solidified workpiece material. this heataffected solidified layer is a white layer that can be seen adjacent to the titanium [27]. these layers occurred due to a cut in the amperage. some of the particles of the adherent material were removed by the dielectric fluid. without moving away, some of them cooled down and amassed on the material to form a white layer [28]. fig. 4 sem view of test specimen cross-sections and edax analysis: a) 1.exp., b) 3. exp. the thermal under the workpiece top surface is dissipated more slowly than the top surface subjected to rapid cooling due to dielectric flow. thermal transformation depends on the thermal behavior of the workpiece. titanium alloys have lower thermal conductivity compared to steels. under the white layer formed on the surface of the workpiece, a tempered layer is formed in the heat affected area. finally, under the top two layers is the area of the workpiece that is not affected by the process. among these three layers, the white layer is the taguchi optimization of multiple performance characteristics in electrical discharge machining... 247 most important one because of its direct contact with the liquid and gap environment. the thickness of the recast layer varies according to the edm processing parameters [24, 33]. the formation of titanium carbides caused by the white layer gives rise to hardness of the recast layer. it has been reported that the average thickness of the reformed layer is increased by increasing the amperage. as the amperage increases, the surface temperature reaches the melting point faster. thus, the material removal rate increases. increasing the amount of the melted particle increases the solidified particle ratio. thus, the white layer formation also increases. as the pulse on time increases, so does the average thickness of the recast layer. thus, with the pulse on time increasing, more thermal energy is transferred to the sample surface during electrical discharge in a single pulse. when the pulse on time increases, the melting regions penetrate deeper and cause more the mrr formation [24, 34]. 3.4. analysis of signal/noise ratio the taguchi doe was used to carry out the edm tests and s/n ratios were applied to optimize the control factors [35]. table 7 shows the s/n ratios of the resulting data. table 7 input and output parameters of titanium alloy according to l9 orthogonal array exp. no input parameters output parameters time on (µs) time off (µs) ampe rage (a) mrr (g/min) s/n for mrr twr (g/min) s/n for twr depth (mm/min) s/n for depth surface rough ness ra (µs) s/n for ra 1 120 40 8 0.00086 61.2808 0.00007044 83.0435 0.00097835 60.1902 3.29 -10.3380 2 120 44 12 0.00177 55.0318 0.00021666 73.2845 0.00197400 54.0931 4.38 -12.8207 3 120 48 16 0.00377 48.4690 0.00047544 66.4582 0.00348653 49.1521 5.40 -14.6416 4 122 40 12 0.00174 55.2134 0.00024428 72.2422 0.00172434 55.2675 4.12 -12.2956 5 122 44 16 0.00374 48.5512 0.00047231 66.5155 0.00253022 51.9368 5.14 -14.2183 6 122 48 8 0.00073 62.7001 0.00006662 83.5280 0.00070872 62.9905 4.06 -12.1788 7 124 40 16 0.00379 48.4343 0.00050604 65.9162 0.00337363 49.4381 5.17 -14.2647 8 124 44 8 0.00082 61.7115 0.00007206 82.8464 0.00075060 62.4918 4.24 -12.5371 9 124 48 12 0.00152 56.3655 0.00021982 73.1588 0.00095572 60.3934 4.61 -13.2813 the s/n ratios and main effect plots for mrr, twr, depth, and surface roughness are shown in fig. 5. the lowest value for mrr (16 a, 124 μs time-on, and 40 μs time-off) was determined to be 0.00377 g/min. the lowest wear loss for twr (8 a, 120 μs time-on, and 40 μs time-off) was found to be 0.00007044 g/min. the best depth measurement (16 a, 124 μs time-on, and 40 μs time-off) was 0.00337363 mm/min. the best value for the average surface roughness was 3.29 μm (8 a, 120 μs time-on, and 40 μs time-off). according to the s/n ratios, the most suitable parameter values for mrr were a3b3c1, for twr a1b3c1, for depth a3b3c1, and for surface roughness a1b1c1. the effective control factors on the mrr, twr, depth, and surface roughness values found according to the taguchi method are shown in the graphs in fig. 5, and verify the experimental study results. the most effective parameters on the main effect graphs are indicated by the values nearest to the vertical. the most effective parameter for mrr, twr, depth, and surface roughness was determined to be amperage. the increase in amperage density leads to more energetic impulses resulting in higher material removal [36]. in the plasma channel, the accelerated ions collide with the workpiece surface. the 248 s.akincioğlu material is removed from the workpiece due to the kinetic energy of electrons and ions. as the amperage increases, the speeds of the electrons and ions increase, causing their kinetic energy to increase. in other words, as the voltage increases, the kinetic energy increases the mrr [24, 37]. fig. 5 main effect plot for s/n ratio for a) mrr, b) twr, c) depth and d) ra 3.5. anova analysis of variance (anova) is used to calculate the significance of the difference between three and more independent means in a normally distributed series. the anova compares cumulative arithmetic means of three or more groups alone. the anova result is also significant when at least one of these comparisons is significant. the effects of time-on, time-off, and amperage on mrr, twr, depth, and surface roughness were analyzed using anova, carried out at a 95% confidence level and a 5% significance level [38]. the anova results for the mrr, twr, depth, and surface roughness are shown in table 8 as 99.66%, 99.56%, 87.95% and 81.12%, respectively. therefore, amperage was found to be the most effective factor, according to the contribution percent rates. the wear loss was positively affected by the increase in amperage, while the surface roughness value was negatively affected. the increase in amperage causes the work surface temperature to reach the melting point of the material faster. thus, more material is removed from the sample surface in constant impact time. as the time on increases, the average thickness of the white layer increases. the increase in time on causes more thermal energy to be transferred to the surface of the sample during electrical discharge in a single pulse. on the other hand, as the time on increases, the melted taguchi optimization of multiple performance characteristics in electrical discharge machining... 249 isothermal regions penetrate deeper into the bulk material and the volume of the molten material increases. thus, the average thickness of the white layer increases and adversely affects the surface roughness [24]. table 8 the results obtained from anova for mmr, twr, depth and surface roughness material removal rate source df seq ss contribution adj ss adj ms f-value p-value time-on (µs) 2 0.0000000138 0.10% 0.0000000138 0.0000000069 1.66 0.376 time-off (µs) 2 0.0000000251 0.18% 0.0000000251 0.0000000126 3.02 0.249 amperage (a) 2 0.0000138805 99.66% 0.0000138805 0.0000069403 1666.44 0.001 error 2 0.0000000083 0.06% 0.0000000083 0.0000000042 total 8 0.0000139279 100.00% r-sq: 99.94% r-sq(adj): 99.76% r-sq(pred): 98.79% tool wear rate time-on (µs) 2 0.0000000002 0.08% 0.0000000002 0.0000000001 1.20 0.454 time-off (µs) 2 0.0 000000008 0.30% 0.0000000008 0.0000000004 4.47 0.183 amperage (a) 2 0.0000002632 99.56% 0.0000002632 0.0000001316 1503.96 0.001 error 2 0.0000000002 0.07% 0.0000000002 0.0000000001 total 8 0.0000002644 100.00% r-sq : 99.93% r-sq(adj) : 99.74% r-sq(pred): 98.66% depth time-on (µs) 2 0.0000004486 4.69% 0.0000004486 0.0000002243 0.84 0.542 time-off (µs) 2 0.0000001713 1.79% 0.0000001713 0.0000000857 0.32 0.756 amperage (a) 2 0.0000084095 87.95% 0.0000084095 0.0000042047 15.82 0.059 error 2 0.0000005317 5.56% 0.0000005317 0.0000002658 total 8 0.0000095611 100.00% r-sq : 94.44% r-sq(adj) : 77.76% r-sq(pred): 0.00% surface roughness time-on (µs) 2 0.16276 4.57% 0.16276 0.08138 1.74 0.365 time-off (µs) 2 0.41551 11.67% 0.41551 0.20776 4.44 0.184 amperage (a) 2 2.88725 81.12% 2.88725 1.44363 30.84 0.031 error 2 0.09363 2.63% 0.09363 0.04682 total 8 3.55916 100.00% r-sq : 97.37% r-sq(adj) : 89.48% r-sq(pred): 46.73% seq. sssequential sum of squares; adj. ssadjusted sum of squares; adj. ms adjusted mean squares; fstatistical test; pstatistical val. 3.6. regression analysis of factors the extent of the relationship among the variables was measured by regression analysis [39], which was used to calculate the formulae for mrr, twr, depth, and ra estimation. the linear models of surface roughness (ra) estimates are given as (eqs. 6), (eq. 7), (eq. 8), and (eq. 9), respectively. mrr (g/min) = 0.00114 – [0.000023ton] – [0.000015toff] + [0.000370a] (6) twr (g/min) = −0.000614 + [0.000003ton] – [0.000002toff] + [0.000052a] (7) depth (g/min) = 0.0139 – [0.000113ton] – [0.000039toff] + [0.000290a] (8) ra (m) = [−10.04 + 0.0797ton] + [0.0625 toff] + [0.1715a] (9) the regression analysis calculated the vif (variance inflation factor) = 1. this result showed that the regression model was valid. 250 s.akincioğlu 3.7. estimation of optimum parameters an evaluation was necessary to determine whether the system optimization was sufficiently accurate. the taguchi approach provided the optimum results. the estimated optimum values for mrr, twr, depth, and ra were found via (eq. 10), (eq. 11), (eq. 12), and (eq. 13), respectively. mrrp = tmrr + (a3 − tmrr) + (b3 − tmrr) + (c1 − tmrr) (10) twrp = ttwr + (a1 − ttwr) + (b3 − ttwr) + (c1 − ttwr) (11) depthp = tdepth + (a3 − tdepth) + (b3 − tdepth) + (c1 − tdepth) (12) rap = tra + (a1 − tra) + (b1 − tra) + (c1 − tra) (13) here, tmrr, ttwr, tdepth, and tra indicate the average of mrr, twr, depth, and ra values of the experiments. a comparison was made of the estimated values with the verification experiment values for determination of the confidence interval (ci). equations (eq. 14) and (eq. 15) were used to calculate the ci for mrr, twr, depth, and ra. the estimated values should be within the 95% ci limit [31]. the symbols used in the ci equations are given in table 9. ci = √𝐹𝛼;1;𝑓𝑒 𝑥𝑉𝑒𝑥 ( 1 𝑛𝑒𝑓𝑓 + 1 𝑟 ) (14) 𝑛𝑒𝑓𝑓 formula: 𝑛𝑒𝑓𝑓 = 𝑁 1+[𝑇𝑑𝑜𝑓] (15) table 9 confidence interval (ci) formulae symbols [20] no. symbol description 1 fα;1;fe f ratio at a 95% (at f table) 2 α significance level 3 fe degrees of freedom of error 4 ve error variance 5 r number of replications for confirmation experiment 6 neff effective number of replications 7 n total number of experiments 8 tdof total main factor degrees of freedom the average optimal predict (pred.) mrr, twr, depth, and ra with the ci at 95% was estimated as in (eq. 16): [(tmrr, ttwr, tdepth or tra)]– [ci] < 𝑃𝑟𝑒𝑑. < [(tmrr, ttwrt, th depthor tra)] + [ci] (16) in order to determine if the predicted experimental result values fell within the 95% ci, the quadratic regression analysis was performed to reveal the relation between the experimental results and the predicted values of the taguchi optimization. the results of the regression analysis determined that the estimated values fell within the 95% ci limit (fig. 6). taguchi optimization of multiple performance characteristics in electrical discharge machining... 251 fig. 6 comparison of predicted values and experimental results for output parameters, a) mrr, b) twr, c) depth and d) ra 4. conclusions this study investigated the parameters (time-on, time-off and a) used in the machining of a titanium alloy on an edm machine. the results obtained both experimentally and statistically are given as follows: the best value for mrr was determined to be 0.00377 g/min obtained at 124 μs timeon, 16 a, and 40 μs time-off. the best value for twr was determined to be 0.0086 g/min obtained at 124 μs time-on, 8 a, and 40 μs time-off. the best value for depth was determined to be 0.00337363 mm/min obtained at 124 μs time-on, 16 a, and 40 μs timeoff. the best value for average surface roughness was determined to be 3.29 µm obtained at 120 μs time-on, 8 a, and 40 μs time-off. ▪ according to the s/n ratios, the most suitable parameter values for mrr were a3b3c1, for twr a1b3c1, for depth a3b3c1, and for surface roughness a1b1c1. ▪ according to anova results for mrr, twr, depth, and surface roughness, the most effective factor was amperage, at 99.66%, 99.56%, 87.95%, and ty81.12%, respectively. ▪ 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method, journal of cleaner production, 19, pp. 2049-56. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 235 249 robust mixed h2/h active vibration controller in attenuation of smart beam udc 62-135:534.1, 624.9 +519.6 atta oveisi, tamara nestorović  mechanics of adaptive systems, ruhr-university bochum, germany abstract. the lack of robustness of the mechanical systems due to the unmodeled dynamics and the external disturbances withholds the performance and optimality of the structures. in this paper, this deficiency is obviated in order to reach the desired robust stability and performance on smart structures. for this purpose a multi-objective robust control strategy is proposed for vibration suppression of a clamped-free smart beam with piezoelectric actuator and vibrometer sensor in an lmi framework which is capable of handling weighted exogenous input signals and provides desired pole placement and robust performance at the same time. an accurate model of a homogeneous beam is derived by means of the finite element modal analysis. then a low order modal system is considered as the nominal model and remaining modes are left as the multiplicative unstructured uncertainty. next, a robust controller with a regional pole placement constraint is designed based on the augmented plant composed of the nominal model and its accompanied uncertainty by solving a convex optimization problem. finally, the robustness of the uncertain closed-loop model and the effect of performance index weights on the system output are investigated both in simulation and practice. key words: piezoelectric, vibration suppression, robust control, smart beam, finite element method 1. introduction the concept of adaptive materials has changed the possibilities for structure design, particularly self-diagnosis and self-controlled arrangements, namely smart structures. this perception is achieved in practice by introduction of multifunctional material based transducers which allows the structure to be sensitive towards the environmental stimuli. adaptive structures play a crucial role in challenging areas of applied science where high quality performance in extreme environments is an urgent requirement. an active structure received september 24, 2014 / accepted november 2, 2014 corresponding author: tamara nestorović universitätsstr. 150, d-44801 bochum, germany e-mail: tamara.nestorovic@rub.de original scientific paper 236 a. oveisi, t. nestorović contains elements such as sensors and actuators, which delivers data in forms of the states of the system and will affect the passive response of the structure. the evolution of mechanical and aeronautical structures requires them to be lighter and at the same time controllable. overcoming the defect of these systems, specifically their sensitivity to unwanted disturbances, has attracted many researchers over the past couple of decades in the fields of structural vibration analysis, damage detection, vibration control and noise control [1, 2]. of various suggested methods of dynamics control, the use of active control techniques in vibration suppression of the light structures is proven to be more effective, where the additional masses of stiffeners or dampers should be avoided. active techniques are also more suitable in the cases where the disturbance to be cancelled or the properties of the controlled system vary with time [3]. piezoelectric actuators are broadly employed in many practical applications due to their capability of coupling strain and electric field. in order to control structural vibrations, piezoelectric actuators can be easily bonded on the vibrating structures [4]. in terms of the dynamic performance, the high-efficient dynamic modeling and appropriate control law design are the two key points. for the purposes of dynamical modelling, the finite element method has recognized to be one of the most popular methods. reviews such as the one presented by benjeddou [5] provide a condensed overview of the development in the field of the finite element modelling (fem) modelling of active structures. the development of the fem tools has proceeded at the same rapid pace in the next decade, followed by the development of active structural control techniques, as reported in the overview by le gao et al. [6]. various types of controller design methods such as velocity feedback control [7], high gain feedback regulator [8], linear quadratic regulator (lqr) approach [9], h2 control [10], h control [11] have been studied by former scientists. in addition, some others evaluate the performance of control algorithms in vibration suppression of flexible structure experimentally [12]. the authors of this paper have made a contribution to the research field of piezoelectric adaptive structures by dedicating their work to the development of necessary finite elements for piezoelectric coupled-field problems [13], demonstrating advantages of the fem approach over other methods [14], investigating different aspects of modelling active structures [15], implementing developed tools into commercially available software packages [16], dealing with control techniques for adaptive structures [17], etc. in this work, an accurate model of a piezolaminated cantilever beam is derived by means of the finite element modal analysis. the derived formulation provides the state space model relating the actuator voltage to sensor voltage. the obtained model is capable of offering a finite order model that shall be considered as nominal system while the remaining high order states are left as multiplicative unstructured uncertainty of modeling. then, a multi-objective robust controller is designed based on the augmented plant composed of the nominal model and its accompanied uncertainty. in addition, a regional pole placement constraint is included within the linear matrix inequality (lmi) framework to improve closed-loop transient performance. the rest of the paper has the following order. in section 2 the configuration of the experimental setup is described. this will be used to verify the performance of the regulated controller in real time implementation. in section 3 the finite element based modal analysis is performed in order to calculate the eigen frequencies and mode shapes of the coupled electro-elastic system. then in section 4 the aforementioned robust controller will be introduced and finally the performance of the closed loop system will be evaluated in the next section. robust mixed h2/h active vibration controller in attenuation of smart beam 237 2. experimental setup the structure of experimental smart beam is presented in fig. 1. the piezo-laminated beam consists of a cantilever aluminum beam with young’s modulus 70 gpa and density 2.7 g/cm³. in addition since the ultimate goal is to suppress the vibration two piezoelectric actuators (duraacttm p-876.a15) are attached to the beam at the same side. (see fig. 1) fig. 1 geometry of the smart beam the feedback channel entails the measurement signal namely, the signal measured by a scanning digital laser doppler vibrometer vh-1000-d. this will provide the measurement of the velocity of the lateral vibration at a point, near the free end of the beam. schematic configuration of closed-loop vibration control system is presented in fig. 2. fig. 2 sketch of experimental setup it is worthwhile mentioning that the plant has two inputs: the control input which acts on the actuator piezo-patch and the disturbance signal which excites the system through the disturbance channel. moreover, the only output of the system is recorded using the previously mentioned vibro-meter. 238 a. oveisi, t. nestorović for implementing the controller in real time, dspace digital data acquisition and real-time control system with ds1005 digital signal process board are used. connection of the digital data acquisition system with the actuators and the computer is provided by an adc board ds2004 (analog to digital converter) and a dac board ds2102 (digital to analog converter). to increase the working range of the dac boards the control input is amplified (pi e-500). the control law, for the active vibration control of the smart beam, is then implemented on matlab platform. finally, the control system is downloaded to the dspace digital data acquisition and real-time control system. 3. system modeling one should notice that the torsional modes are not considered in controller design because they are not relevant for the bending vibration. it should be mentioned that due to the previous research the dominant mode shape of the flexible beam is the first mode shape [18]. the dynamics of the actuation is addressed by means of the fem analysis in coupled electro-mechanical domain. this leads to an ordinary differential equation which then will be converted to a linear time invariant (lti) system since it is a convenient model for the work in the computer aided control system design. it is assumed that the displacements are small enough so that the dynamics of the system remains in linear piezo-elasticity. the finite element method presents the dynamic equation of motion in matrix representation as: ,mq cq kq f   (1) with m, c and k being the mass, damping and stiffness matrices. also, q represents the nodal states of displacement ( , 1,2,...)t iu i  and electric potential ( , 1,2,...)t j j  : 1 1[ ]t t t n nq u u   (2) f shows the applied excitation which contains the external forces that is assumed to be zero because the external input disturbance is expected to affect the system from the same channel as the control input. the vector of control forces is therefore: ( ),f bu t (3) b matrix describes the position of the generalized control effort in the finite element structure with u consisting of all modal inputs. for the control design purposes the measurement signal is represented in terms of system states and plant inputs as: 0 0 ,q vy c q c q  (4) in which c0q and c0v are the output displacement and output velocity matrices, respectively; they are calculated using the fe procedure and choosing the appropriate sensor location. by applying the conventional harmonic solution of q = e it one can easily find natural frequencies j and mode shapes j (j = 1,2,...,n) solving the determinant of homogenous system of algebraic equations. the solution can be represented in matrix form as: robust mixed h2/h active vibration controller in attenuation of smart beam 239 1 2 0 0 0 0 , 0 0 n              11 21 1 12 21 2 1 2 3 4 1 2 , n n n n nn                          (5) the nodal model representation (1) can be transformed to modal coordinates by applying the following conversion: ,mq q  (6) where qm is the vector of generalized modal displacement. using the symmetricity of the mass and stiffness matrices one can easily obtain the transformed matrices as [19] : ( ),t m jm m diag m    2( ),t m j jk k diag m     (7) similarly, by using the same transformation and the orthogonality of mode shape one can find the modal damping matrices under the assumption of the proportional damping to be: ,c m k   (8) by selecting the state vector to be   t m mx q q  the state space model will be: , , x ax bu y cx du     (9) where: 00 , , , 0, 2 mq mv m a b c c c d bz                  (10) while 2 1 m mm k  and ( )jz diag  with j being the damping ratio of jth mode, bm =  t b, cmq = c0q, cmv = c0v. 4. controller design the closed-loop system by considering multiplicative uncertainty will be as shown in fig. 3. 240 a. oveisi, t. nestorović p(s) k(s) w(s)  + + u y u yw fig. 3 closed-loop system with multiplicative uncertainty where p(s) is the nominal plant, k(s) is desired controller, ∆ is a stable transfer function, where |||| < 1 and w(s) is the weighting function for multiplicative uncertainty, that satisfies following equation: ( ) 1 ( ) ( ) realp s w s p s   (11) where preal(s) is the transfer function of the real system, by considering all or some of the higher modes. note that, reduction of the order of the nominal plant will hold the designed controller’s order in a lower value, but the price will be reduced performance. for robust stability, one should have ||tyu || < 1, where tyu is the transfer function from u to y when  is removed [20]. however, to handle the stochastic aspects such as measurement noise and random disturbance, despite robust h, only h2 performance is functional. and finally, for appropriate disturbance rejection and control effort the conventional optimization problem is to minimize ||y t qy + u t ru||, where q and r are two weighting functions that indicate the relative importance of disturbance rejection and control effort, respectively. for minimizing performance index ||y t qy + u t ru||, we should minimize ||t[y u] tw||2 instead, where w is a bounded h2 norm exogenous disturbance. this will be addressed later. the transient response of a linear system is well known to be related to the locations of its closed-loop poles. this is the next issue that has to be addressed. since tyu is equivalent to tuww(s) [21], the above system can be represented in fig. 4 with all of the constraints that have to be satisfied in order to reach the predefined h2 / h performance and optimal control effort. p(s) k(s) w(s) + + 2z u z y    w q r fig. 4 desired input and outputs of augmented plant now assume that a state space representation of the open-loop system in fig. 4 (by ignoring k(s)) is: robust mixed h2/h active vibration controller in attenuation of smart beam 241 1 2 1 2 2 2 21 22 1y y x ax b w b u z c x d w d u z c x d w d u y c x d w                   (12) where u and w are control input and disturbance, respectively. our objective is to design a dynamic output-feedback controller with the state space realization: k k k k a b y u c d y         (13) where  is the state variable of the controller. therefore, the corresponding closed-loop system containing the performance and robustness channels will be: 1 1 2 2 1 cl cl cl cl cl cl cl cl cl cl x a x b w z c x d w z c x d w          (14) our three design objectives can be expressed as follows: h performance: the closed-loop rms gain from w to z does not exceed  if and only if there exists a symmetric matrix x such that [22] : 1 1 2 1 1 0 0 t t cl cl cl cl t t cl cl cl cl a x x a b x c b i d c x d i x                  (15) this lmi constraint is used to minimize ||tzw|| (closed-loop h gain from disturbance to z output channel). h2 performance: the h2 norm of the closed-loop transfer function from w to z2 does not exceed v if and only if dcl2 = 0 and there exist two symmetric matrices x2 and q such that [23]: 2 2 2 2 2 2 2 2 0 0 ( ) t cl cl cl t cl cl t cl a x x a b b i q c x x c x trace q               (16) pole placement: the closed-loop poles lie in the lmi region: { : 0}td z c l mz m z     (17) with l = l t = {ij}1i, jm and m = [ij]1i, jm if and only if there exists a symmetric matrix xpol satisfying: 242 a. oveisi, t. nestorović 1 , 0 0 t ij pol ij cl pol ji pol cl i j m pol x a x x a x            (18) for tractability in the lmi framework, we must seek a single lyapunov matrix: 2 : pol x x x x     (19) that enforces all three sets of constraints. factorizing x as: 1 1 2 1 2 0 , : , : 0t t r i s x x x x x m i n                (20) and, introducing the change of controller variables [25]: 2 2 2 : : ( ) k k k t k k k y t t k k k y k k y b nb sb d c c m d c r a na m nb c r sb c m s a b d c r             (21) the inequality constraints on x are readily turned into lmi constraints in the variables r, s, q, ak, bk, ck and dk [22], [24]. this leads to the suboptimal lmi formulation of our multi-objective synthesis problem, which is defined as: minimize 2. . ( )trace q   over variables r, s, q, ak, bk, ck, dk and 2 satisfying [26]: 2 2 2 2 2 1 2 1 1 1 2 1 2 1 2 22 2 22 2 2 0 0 t t t t k k k k y t t t k y y k k k y k y k y k y k k y k k y ij ij k ar ra b c c b a a b d c h a s sa b c c b h h c r d c c d d c b b d d h sb b d h i h d d d d i q c r d c c d d c h r i h i s ar b c a b d cr i a sai s                                                     2 2 1 , 2 0 2 2 0 21 22 1 0 ( ) ( ) 0 k y t t t t k k ij t t t t k y y k i j m k y b c ra c b a a b d c a s c b trace q d d d d                             (22) robust mixed h2/h active vibration controller in attenuation of smart beam 243 given optimal solutions  *,q* of this lmi problem, the closed-loop h and h2 performances are bounded by: * * 2 2 , ( )t t trace q     (23) 5. case study and discussion this section presents the vibration damping quality of the proposed method both in simulation and experiment. for implementation of the controller, a structure consisting of an aluminum clamped beam with two piezoelectric patches is used. the patches are attached on the same side of the beam (see fig. 1). the model of the structure for control design purposes is obtained based on the method described before. since the actuator placement plays an important role in vibration control performance the optimal placement of the actuator is addressed based on the mixed h2 / h method that is described by nestorović and trajkov [27]. firstly, two shape numbers of the clamped beam are considered as nominal model and higher order modes remain as unstructured uncertainty. in addition, a weighting function for multiplicative unstructured uncertainty that satisfies preal(s)/p(s) = wunc(s) + 1 is considered. with preal(s), p(s) and wunc(s) being the full order transfer function of the system, nominal transfer function and frequency based appropriate weighting function representing the unstructured uncertainty, respectively. fig. 5 shows the weighting functions that are considered for modeling unstructured uncertainty, disturbance and h2 / h performance. 10 -1 10 0 10 1 10 2 10 3 -80 -60 -40 -20 0 20 40 60 m a g n it u d e ( d b ) bode diagram frequency (rad/s) plant wunc wperf wdist fig. 5 the relation of weighting function to real system 244 a. oveisi, t. nestorović the desired controller design is carried out by solving convex optimization problem that is formulated in eq. (22). for obtaining an appropriate h performance, the magnitude of | | should be under unit and for increasing the performance one should minimize h2 norm from exogenous disturbance to performance index. the relative magnitudes of q and r determine the relative importance of disturbance rejection (vibration suppression) to control effort (actuator saturation). to improve transient performance, as mentioned before, one shall resort to an additional regional pole placement constraint in order to achieve a better closed-loop damping across the uncertainty range. this places the closed-loop poles into a suitable sub-region of the left-half plane that can be expressed as an additional lmi constraint. a typical example of lmi region that is commonly treated in multi-objective synthesis that guarantees h2 stability is the conic sector centered at the origin and with inner angle 2 = 2cos 1 () [22]. in this work, the closed-loop damping coefficient is assumed to be  = 0.1. the controller is designed by setting q = 10. comparison of the impulse response of the closed-loop system with this controller and the impulse response of the open-loop system (fig. 6) shows the performance of the controller in suppressing the vibration. 0 1 2 3 4 5 -40 -30 -20 -10 0 10 20 30 40 impulse response time (seconds) a m p li tu d e open-loop system closed-loop system fig. 6 impulse response of the open-loop and closed-loop system actuator voltage of this controller during the impulse response is plotted in fig. 7. as one can see, the maximum amplitude of the actuator voltage is under about 20 volts. in addition, comparison of frequency responses of closed-loop system and open-loop system is shown in fig. 8 which shows that the amplitude is reduced in the nominal model natural frequencies. robust mixed h2/h active vibration controller in attenuation of smart beam 245 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -80 -60 -40 -20 0 20 40 time (seconds) a c tu a to r v o lt a g e ( v .) fig. 7 input control for impulse response of the closed-loop system 10 1 10 2 10 3 -60 -50 -40 -30 -20 -10 0 10 20 30 m a g n it u d e ( d b ) bode diagram frequency (rad/s) open-loop system closed-loop system fig. 8 bode diagram of closed-loop system and open-loop system for investigation of the robust performance of the uncertain closed-loop system with the designed controller by structured singular value analysis fig. 9 is obtained. this plot shows upper/lower bounds of uncertain closed-loop structured singular values in frequency domain. 246 a. oveisi, t. nestorović 10 1 10 2 10 3 0 0.5 1 1.5 2 frequency (rad/sec) m u u p p e r b o u n d s fig. 9 µ bounds of uncertain closed-loop system the performance margin is the reciprocal of the structured singular value and if the magnitude of the structured singular value were under unit, in entire frequency range, the system would have robust performance. therefore, upper bounds from structured singular value become lower bounds on the performance margin and critical frequency associated with the upper bound of the structured singular value, here is critical = 87rad/sec. in addition, the system can tolerate up to 557% of the modeled uncertainty without losing desired performance. through the experimental implementation of the control law on the smart structure the possibility of the successful vibration control performance is evaluated on full order system. the vibration amplitude suppression will be demonstrated under the harmonic excitation of the piezo-beam through the control channel and the results obtained using hardware in loop system with dspace rti platform. experimental excitation is considered to be harmonic f(t) = asin(2fjt), with fj being the first bending resonant frequency of the clamped piezo-beam. the closed-loop system is implemented on the real time data acquisition platform of the dspace with sampling frequency of 10 khz. the predefined task of the controller is to guarantee the robust stability and performance in conjugation with real time vibration amplitude suppression in frequency ranges close to resonance eigenvalues. therefore, investigations are carried out in time domain by means of the experimental setup shown in fig. 10. fig. 10 experimental rig of the closed-loop system robust mixed h2/h active vibration controller in attenuation of smart beam 247 for the analysis in time domain the sinusoidal excitation signal is generated in simulink and lead out through the dspace dac. the frequency of the excitation is adjusted experimentally to reach the highest vibration amplitude representing the actual eigenfrequency. the response of the system for controlled and uncontrolled case is shown in fig. 11 based on the measurement signal generated by doppler vibro-meter. 0 1 2 3 4 5 -40 -30 -20 -10 0 10 20 30 40 time (seconds) a m p li tu d e open-loop system closed-loop system fig. 11 experimental comparison of velocity response this diagram shows the velocity magnitudes of the beam measured by dspace adc board. in addition the corresponding control effort generated for piezo-actuator patches by the dspace dac board is shown in fig. 12. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -10 -5 0 5 10 time (seconds) a c tu a to r v o lt a g e ( v .) fig. 11 control effort of the piezo-patch actuator the experimental results show the obvious performance of the robust control system in attenuating the vibration amplitude. 248 a. oveisi, t. nestorović 6. conclusion vibration control of a clamped-free beam with piezoelectric actuator and vibrometer has been achieved by using a multi-objective robust output feedback control strategy with regional pole placement constraints in an lmi framework, based on h2 / h weighting objective functions. the robustness of the closed loop smart beam with respect to external input disturbance increased to 557% of the modeled uncertainty. the regional pole placement constraints guaranteed the improvement of the transient response of the closed-loop system and the optimality of the control effort is achieved by satisfying the appropriate h2 lmi based performance index. all these constraints are presented in a lmi formulation, which is solvable in the matlab environment. finally, the performance of the approach is proven to be effective and robust on the experimental set up where the higher order modes take effect in the dynamics of the smart beam. references 1. milovančević, m., veg, a., makedonski, a., marinović, j. s., 2014, embedded systems for vibration monitoring, facta univesitatis series mechanical engineering, 12(2), pp. 171-181. 2. oveisi, a., gudarzi, m., 2013, adaptive sliding mode vibration control of a nonlinear smart beam: a comparison with self-tuning ziegler-nichols pid controller, journal of low frequency noise vibration and active control, 31(1-2), pp. 41 – 62. 3. carra, s., amabili, m., ohayon, r., hutin, p. m., 2008, active vibration control of a thin rectangular plate in air or in contact with water in presence of tonal 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piezoelectric active structures, ph.d. thesis, otto-von-guericke-universität magdeburg fakultät für maschinenbau. 14. marinkovic, d., marinkovic, z., 2011, fem and ritz method-a piezoelectric active shell case study, transactions of famena, 35(3), pp. 39-48. 15. marinkovic, d., koppe. h., gabbert. u., 2009, aspects of modeling piezoelectric active thin-walled structures, journal of intelligent material systems and structures, 20(15), pp.1835-1844. robust mixed h2/h active vibration controller in attenuation of smart beam 249 16. nestorovic, t., marinkovic, d., chandrashekar, g., marinkovic, z., trajkov, m., 2012, implementation of a user defined piezoelectric shell clement for analysis of active structures, finite elements in analysis and design, 52, pp.11-22. 17. oveisi, a., gudarzi, m., 2013, nonlinear robust vibration control of a plate integrated with piezoelectric actuator, international journal of mathematical models and methods in applied sciences, 7(6), pp. 638-646. 18. nestorović, t., durrani, n., trajkov, m., 2012, experimental model identification and vibration control of a smart cantilever beam using piezoelectric actuators and sensors, journal of electroceramics, 29(1), pp. 42-55. 19. géradin, m., 1997, mechanical vibrations theory and application to structural dynamics, 2nd edition, wiley, chichester. 20. zhou, k., doyle, j. c., 1997, essentials of robust control, prentice hall. 21. sivrioglu, s., tanaka, n., 2002, acoustic power suppression of a panel structure using h∞ output feedback control, journal of sound and vibration, 249(5), pp. 885-897. 22. chilali, m., gahinet, p., 1995, h∞ design with pole placement constraints: h∞ an lmi approach, ieee transactions on automatic control, 41(3), pp. 358-367. 23. banjerdpongchai, d., how j. p., 1998, parametric robust h2 control design with generalized multipliers via lmi synthesis, international journal of control, 70(3), pp. 481-503. 24. scherer, c., 1995, mixed h2/h∞ control, trends in control: a european perspective, volume of the special contributions to the ecc. 25. gahinet, p., 1996, explicit controller formulas for lmi-based h∞ synthesis, automatica, 32(7), pp. 1007-1014. 26. gudarzi, m., oveisi, a., mohammadi, m. m., 2012, robust active vibration control of a rectangular piezoelectric laminate flexible thin plate: an lmi-based approach, international review of mechanical engineering, 6(6), pp. 1217-1227. 27. nestorović, t., trajkov, m., 2013, optimal actuator and sensor placement based on balanced reduced models, mechanical systems and signal processing, 36(2), pp. 271-289. robustno h2/h upravljanje vibracijama u cilju prigušenja aktivne konzole ograničena robustnost u odnosu na neprecizno modeliranje dinamike i spoljašnjih poremećaja mehaničkih sistema značajno utiče na njihove performanse i optimalna svojstva aktivnih struktura. u ovom radu prikazujemo na koji način se ovaj nedostatak može prevazići u cilju postizanja željenih performansi aktivnih struktura. u tom cilju predložen je robustni upravljački zakon za redukciju vibracija aktivne konzole sa piezoelektričnim aktuatorom i laserskim senzorom u lmi okruženju, koji se uspešno može primenjivati u prisustvu eksternih ulaza sa težinskim funkcijama,a koji obezbeđuje željeno podešavanje polova i robustne performanse u isto vreme. model homogene konzole dobijen je primenom modalne analize metodom konačnih elemenata. zatim je modalni sistem redukovanog reda posmatran kao nominalni model, dok su modeli višeg reda su razmatrani kao mulitiplikativna nesigurnost. potom je na osnovu proširenog modela, koji sačinjava nominalni model sa pratećom neizvesnošću, projektovan robustni kontroler sa lokalnim podešavanjem polova, rešavanjem konveksnog optimizacionog problema. na kraju je simulacijom i eksperimantalno analizirana robustnost nizvesnog modela zatvorenog kola, kao i uticaj težinskog indeksa performansi na izlaz sistema. ključne reči: piezoelektrični materijali, redukcija vibracija, robustno upravljanje, aktivna konozla, metod konačnih elemenata. 10953 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 539 552 https://doi.org/10.22190/fume220703035f © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dynamic characteristics of mixture unified gradient elastic nanobeams seyed ali faghidian1, abdelouahed tounsi2,3 1department of mechanical engineering, science and research branch, islamic azad university, tehran, iran 2yfl (yonsei frontier lab), yonsei university, seoul, south korea 3material and hydrology laboratory, university of sidi bel abbes, faculty of technology, civil engineering department, algeria abstract. the mixture unified gradient theory of elasticity is invoked for the rigorous analysis of the dynamic characteristics of elastic nanobeams. a consistent variational framework is established and the boundary-value problem of dynamic equilibrium enriched with proper form of the extra non-standard boundary conditions is detected. as a well-established privilege of the stationary variational theorems, the constitutive laws of the resultant fields cast as differential relations. the wave dispersion response of elastic nano-sized beams is analytically addressed and the closed form solution of the phase velocity is determined. the free vibrations of the mixture unified gradient elastic beam is, furthermore, analytically studied. the dynamic characteristics of elastic nanobeams is numerically evaluated, graphically illustrated, and commented upon. the efficacy of the established augmented elasticity theory in realizing the softening and stiffening responses of nano-sized beams is evinced. new numerical benchmark is detected for dynamic analysis of elastic nanobeams. the established mixture unified gradient elasticity model provides a practical approach to tackle dynamics of nano-structures in pioneering mems/nems. key words: stationary variational principle, nanobeam, free vibrations, wave dispersion, stress gradient elasticity, strain gradient theory received: july 03, 2022 / accepted august 25, 2022 corresponding author: seyed ali faghidian department of mechanical engineering, science and research branch, islamic azad university, tehran, iran e-mail: faghidian@gmail.com; faghidian@srbiau.ac.ir 540 s. a. faghidian, a. tounsi 1. introduction micro-/nano-sensors are one of the representative elements of micro-/nanoelectromechanical systems (mems/nems) that found a variety of applications in pioneering engineering systems [1, 2]. structural analysis of vibrating modules of mems/nems is a challenging issue that stimulated a great deal of interest [3-15]. in view of inadequacy of the classical elasticity theory in appropriate prediction of the peculiar behavior of nano-scale structures, several augmented elasticity theories are introduced in literature. the constitutive model associated with the augmented elasticity theory is, accordingly, enriched via incorporation of the gradient effects or of the nonlocality. within the framework of the strain gradient theory, the material response is assumed to be a function of the classical kinematics along with the gradients of various order [16]. the strain gradient theory can efficiently predict the smaller-is-stiffer phenomenon at the ultra-small scale [17-21]. alternatively, the material response can be considered to depend on the classical kinetic variables together with the gradients of various order which yields the stress gradient theory [22, 23]. the long-range interactions can be modeled within the context of the nonlocal elasticity theory wherein the constitutive model is modified employing the nonlocal kernel function [24]. the smaller-is-softer phenomenon is, therefore, realized via implementation of the nonlocal elasticity model. as the nonlocality concept is a topic of major interest in the size-dependent mechanics [25-35], severe concern exists in its application to nano-scale continua with finite domain [36]. either the strain gradient theory of elasticity or the nonlocal elasticity model can solely predict the stiffening or the softening material behavior, and therefore, they cannot cover the wide spectrum of material characteristics at the ultra-small scale. to overcome this deficiency, dissimilar size-dependent elasticity models are integrated to introduce novel augmented theories, such as the nonlocal strain gradient model [37, 38], the nonlocal modified gradient theory [39, 40], the higher-order nonlocal gradient theory [4143], the nonlocal surface elasticity [44-46], and the mixture stress gradient theory [47]. utilization of the integrated size-dependent elasticity models for nanoscopic study of the field quantities, accordingly, received increasing interest in recent literature [48-51]. the present study provides significant insight and critical analysis of the dynamic characteristics of nano-scaled structures and offers necessary guidance for the reliable assessment of the dynamic response of nano-components in advanced mems/nems. the mixture unified gradient theory of elasticity, as a proficient alternative to the two-phase local/nonlocal gradient theory, is invoked to properly realize the size-effects. both the softening and the stiffening material behaviors can be effectively captured in view of consistent unification of the stress gradient, the strain gradient, and the classical elasticity theory. the wave dispersion response of the flexural waves along with the free vibrations of nano-sized elastic beams is rigorously examined. the dynamic characteristics of mixture unified gradient elastic nanobeams are analytically addressed and numerically demonstrated. the paper proceeds as follows; a stationary variational framework is conceived in section 2 to suitably integrate all the governing equations into a single functional. the corresponding boundary-value problem of dynamic equilibrium is determined and equipped with suitable extra non-standard boundary conditions. nanoscopic analysis of the flexural wave dispersion is performed in section 3 wherein the analytical solution of the phase velocity is derived. section 3 is, furthermore, enriched with numerical illustrations of the wave dispersion features in an elastic nanobeam. the free vibrations of dynamic characteristics of mixture unified gradient elastic nanobeams 541 nano-sized beams, associated with the framework of the mixture unified gradient theory of elasticity, is addressed in section 4. fundamental frequencies of elastic nano-scaled beams are analytically determined, numerically illustrated and commented upon. section 5 summarizes the paper and draws the conclusion. 2. mixture unified gradient elastic beams to establish the elasticity framework of the mixture unified gradient theory, an elastic homogenous beam of symmetric cross-section is considered. in the undeformed state, the beam is referred to orthogonal cartesian co-ordinates (x,y,z) with the x axis coinciding with the beam longitudinal axis and the y and z axes, respectively, coinciding with the width and the thickness directions. the beam is constrained at the ends x=0 and x=l impeding any rigid-body motion. the beam is subjected to a generalized transversal load per unit length f where the inertia force is assumed to be incorporated as the body force. the shear and warping effects are overlooked, and accordingly, the kinematics of the beam in accordance with the classical beam theory is taken in the form 1 2 3( , ), 0, ( , )xu z w x t u u w x t= −  = = (1) where 1 2 3, ,u u u denote the components of the displacement field at a generic point of the beam along with w designating the transverse displacement of the beam centroid at time t. the strain state of the beam, consequent to the assumed kinematics, can be written as ( , ) ( , )xxz w x t z x t = −  = − (2) with xxw =  denoting the curvature of the beam centroidal axis. the variational functionala consistent with the mixture unified gradient theory of elasticity is introduced as 2 2 2 0 0 0 0 2 1 12 2 2 2 12 2 1 ( , ) ( , ) ( ( , )) ( ( , )) 2 2 1 ( , ) ( , ) ( ( , )) 2 ( ) ( ( , )) ( , ) ( , ) 2 ( ) l c xx x e e xxx e c s c x e c s m x t w x t m x t m x t i i m x t w x t m x t i m x t f x t w x t dx i  =  + +   +  +  +  +  +   +  a (3) where the flexural stiffness ie is defined by the second moment of the elastic area, weighted with the scalar field of the elastic modulus e, about the flexural axis. the resultant moments m0 and m1 are, correspondingly, defined as the dual fields of the curvature  and of its first-order derivative along the beam axis x. as the mixture parameter is denoted by , the stress gradient characteristic length c and the strain gradient length-scale parameter s are introduced to address the significance of the corresponding gradient elasticity theory. within the context of the stationary variational formulation, the kinetic field variables are also treated as independent variables along with the kinematic field variables; all the field variables are, therefore, considered to be subject to variation [52]. assuming the 542 s. a. faghidian, a. tounsi virtual kinetic test fields to have compact support on the domain, the first variation of the functionala , following the integration by parts, reads ( )( ( ) ( ) ( ) 0 1 0 2 0 0 0 2 1 1 12 2 0 1 00 ( , ) ( , ) ( , ) ( , ) 1 ( , ) ( , ) ( , ) ( , ) 1 ( , ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( l xx xxx xx c xx e xxx c xx e c s x l x xx x m x t m x t f x t w x t m x t w x t m x t m x t i m x t w x t m x t m x t dx i m x t m x t w m x      = = =  −  +   +  + −       +  + −    +  −  −  + a ( )1 0 1 0 , ) ( , ) ( , ) x l x x x x l xx x t m x t w m x t w   = = = = −   +  (4) where the extra gradient-induced boundary terms are released in consideration of the virtual kinetic test fields having compact support on the domain [53]. prescribing the stationarity of the functional 0 =a , the differential and boundary conditions of dynamic equilibrium for the elastic nano-sized beam consistent with the mixture unified gradient theory cast in the form ( ) ( ) 0 1 0 1 0 0 1 0 1 0 ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( , ) 0 ( , ) ( , ) 0 ( , ) 0 xx xxx tt xxtt x l x xx x x l x x x x l x m x t m x t f x a w x t i w x t m x t m x t w m x t m x t w m x t     = = = = = =  − + =  −   − = −  = = (5) where the curvature field is considered to have arbitrary variations at the beam ends. noteworthily, the dynamic form of the differential condition of equilibrium is developed by application of the d’alembert principle. the cross-sectional mass a is defined by the cross-sectional area weighted with the scalar field of the material density . likewise, the rotatory inertia i is introduced as the second moment of area, weighted with the scalar field of the material density , about the flexural axis. to further simplify the boundary-value problem of dynamic equilibrium, the concept of the total flexural moment is utilized within the framework of the gradient elasticity theory, and is introduced as 0 1( , ) ( , ) ( , )xm x t m x t m x t= − (6) the boundary-value problem of the dynamic equilibrium of the mixture unified gradient elastic beam is, therefore, modified as ( ) 0 0 1 0 ( , ) ( , ) ( , ) ( , ) ( , ) 0 ( , ) 0 xx tt xxtt x l x l x xx x x l x m x t f x a w x t i w x t m x t w m x t w m x t     = = = = = =  + =  −   =  = = (7) dynamic characteristics of mixture unified gradient elastic nanobeams 543 as a distinguished privilege of implementing the stationary variational principle, the constitutive model of the associated elasticity theory is, furthermore, detected. the constitutive laws of the resultant fields consistent with the framework of the mixture unified gradient theory, accordingly, are cast as the subsequent ordinary differential relations ( ) 2 0 0 2 2 2 1 1 2 2 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( , ) ( ) ( , ) c xx e xx c xx e c s xxx c xx e xx c s xxxx m x t m x t i w x t m x t m x t i w x t m x t m x t i w x t w x t −  = −  −  = −  +  −  = −  −  +  (8) where the constitutive laws of the flexural resultants m0, m1 are directly determined in view of the stationarity of the functional 0 =a , and the constitutive relation of the flexural moment m is detected utilizing its definition as eq. (6). the noticeable advantage of the established stationary variational framework is to integrate all the governing equations, i.e. the differential condition of dynamic equilibrium, the classical and the extra non-standard boundary conditions along with the constitutive differential relations of the resultant fields, into the functionala . the constitutive laws of the resultant fields are of higher-order compared with the classical elasticity model, and therefore, to close the associated boundary-value problem of dynamic equilibrium on finite domains, extra non-standard boundary conditions should be imposed. realizing the proper mathematical form of the non-standard flexural resultants is of supreme importance [54]; otherwise, it yields superfluous inference vis-à-vis the size-dependent response of ultra-small structures [21, 55]. the explicit constitutive relation of the resultant moment m1 associated with the mixture unified gradient theory, subsequent to some straightforward mathematics, can be cast in the form ( ) 2 2 4 1 2 2 2 2 2 2 2 2 2 2 ( ) ( , ) ( ) ( , ) ( , ) (1 ) ( ) ( ) ( , ) ( , ) (1 ) c s c x xtt xxxtt c s c s c e c s xxx e xxxxx c s m x t q x a w x t i w x t i w x t i w x t    + = −  −  +  − −  + −  +  +  − − (9) several types of the gradient elasticity theory are, noticeably, retrieved as special cases of the established mixture unified gradient theory under ad hoc assumptions on the gradient characteristic lengths. the classical elasticity model is obtained via either setting the gradient characteristic lengths to zero or via letting the mixture parameter approach unity in the absence of the strain gradient length-scale parameter. as the vanishing of the mixture parameter yields the unified gradient elasticity theory, the strain gradient and the stress gradient theory can be, likewise, retrieved via setting the pertinent gradient lengthscale parameter to zero. moreover, the mixture stress gradient theory can be recovered via vanishing the strain gradient length-scale parameter. the peculiar size-dependent dynamic behavior of nano-scale beams can be competently captured within the framework of the mixture unified gradient theory as evinced via rigorous examination of the wave dispersion and the free vibrations of elastic inflected nanobeams. 544 s. a. faghidian, a. tounsi 3. wave dispersion characteristics the size-dependent physical characteristics of nano-structures can be effectively realized by the analysis of the wave dispersion. examination of dispersive waves can be, furthermore, exploited for inverse determination of the gradient characteristic parameters applying the inverse theory approach [56, 57]. to examine the flexural waves dispersing in the elastic nanobeam, the differential condition of dynamic equilibrium should be expressed in terms of the transverse displacement of the beam. the total flexural moment is, accordingly, obtained by imposing the dynamic equilibrium condition eq. (7)1 to the constitutive law of the flexural moment eq. (8)3, namely, 2 2 2 2 ( , ) ( , ) ( , ) ( , ) ( ) ( , ) c tt c xxtt e xx c s e xxxx m x t a w x t i w x t i w x t i w x t  =  −  −  +  +  (10) where the transverse body force is overlooked in the dynamic analysis of nano-sized beams. the governing equation on the dynamic response of the mixture unified gradient elastic beam is achieved via utilizing the detected result for the flexural moment in the differential condition of dynamic equilibrium as 2 2 2 2 ( , ) ( , ) ( ) ( , ) ( , ) ( ) ( , ) 0 c xxxxtt e xxxx c s e xxxxxx tt c xxtt i w x t i w x t i w x t a w x t i a w x t      +  −  +  +  − +  = (11) for the harmonic wave dispersion in infinitely extended homogeneous structures, vanishing the boundary conditions is tacitly fulfilled; the corresponding solution of the wave response, thus, takes the form ( , ) exp( ( ))w x t w i x vt= − (12) with i being the unit imaginary number,  and v, respectively, denote the wave number and the phase velocity along with w standing for the wave amplitude. by substituting the displacement solution eq. (12) into the governing equation on the dynamics of the elastic nanobeam eq. (11), based on the mixture unified gradient theory, the subsequent phase velocity is obtained 2 2 2 2 2 2 ( ) 1 e e c s c i i v a i      +  + = + + (13) noteworthily, the wave dispersion response of the nano-sized beam consistent with the unified gradient theory is recovered in the absence of the mixture parameter [40, 43]. the size-effects of the gradient length-scale parameters on the dispersive characteristics of waves are graphically illustrated and thoroughly discussed here. for the sake of consistency, the non-dimensional form of the gyration radius , stress gradient characteristic parameter , strain gradient characteristic parameters , wave number  , and phase velocity v are defined as 1 , , , , c s e i a l v vl l a l l i        = = = = = (14) dynamic characteristics of mixture unified gradient elastic nanobeams 545 3d variation of the non-dimensional phase velocity in terms of the stress gradient characteristic parameter  and the strain gradient characteristic parameters  is, respectively, demonstrated in figs. 1 and 2 where the logarithmic scaling of the non-dimensional wave number  is utilized [58]. as the stress gradient characteristic parameter is assumed to range in the interval [0,1] in fig. 1, two values of the mixture parameter  = 0,1/2 are prescribed for a fixed strain gradient characteristic parameter  = 1/3. likewise in fig. 2, the strain gradient characteristic parameter is ranging in the interval [0,1] while two values of the mixture parameter  = 0,1/2 are applied for a prescribed value of the stress gradient characteristic parameter  = 1/3. in all the numerical illustrations of the wave dispersion, the (logarithm of) non-dimensional wave number  is ranging in the interval [10−1,10+1]. the non-dimensional gyration radius is, moreover, prescribed as 1/ 20 = . as deducible from the numerical illustration in fig. 1, a larger value of the stress gradient parameter involves a smaller value of the phase velocity. the phase velocity of dispersive waves, therefore, reveals a softening response in terms of the stress gradient characteristic parameter . contrarily as demonstrated in fig. 2, the phase velocity associated with waves disperse in the mixture unified gradient elastic beam increases by increasing the strain gradient parameter , and accordingly, a stiffening behavior in terms of the strain gradient characteristic parameter  is observed. the effect of the stress gradient theory in the mixture unified gradient elasticity is continuously reinstated with the classical elasticity theory as the mixture parameter  varies from 0 to 1. a stiffening behavior in terms of the mixture parameter  is, thus, realized as the phase velocity of dispersive waves increases with increasing the mixture parameter . the phase velocity detected within the framework of the mixture unified gradient elasticity is observed to remain unaffected for low wave numbers. this phenomenon is in complete agreement with the fact that the dispersive behavior of flexural waves is insensitive to the physical characteristics for large wavelengths. accordingly, effects of the gradient characteristic parameters on the wave dispersion response are significantly enhanced at higher wave numbers. the classical dispersion relation of the phase velocity is, also, recovered as the gradient characteristic parameters tend to zero, or alternatively, as the mixture parameter approaches unity in the absence of the strain gradient characteristic parameter. fig. 1 wave dispersion response by the mixture unified gradient theory: v vs.  for  = 1/3 and  = 0,1/2 546 s. a. faghidian, a. tounsi fig. 2 wave dispersion response by the mixture unified gradient theory: v vs.  for  = 1/3 and  = 0,1/2 4. free vibrations characteristics the differential condition of dynamic equilibrium governing the free vibrations of a mixture unified gradient elastic beam is the same as differential equation addressed in eq. (11) and can be detected employing literally the same approach. nevertheless, the boundary-value problem of free vibrations of elastic nanobeams is defined on bounded structural domains, and accordingly, the governing differential equation of dynamic equilibrium is subject to the classical and the extra non-standard boundary conditions as expressed by eq. (7)2-3. the standard procedure of separating spatial and time variables is employed to analyze the free vibrations of elastic nanobeams, namely, ( , ) ( )exp( )w x t x i t= (15) with and , respectively, denoting the spatial mode shape and the natural frequency of free vibrations. prescribing the separation of variables eq. (15) to the differential condition of dynamic equilibrium eq. (11), the governing equation on the spatial mode shape is detected as 2 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 e c s xxxxxx e c xxxx c xx i x i i x a i x a x         −  +  + −  + +  − = (16) the analytical solution of the preceding differential equation for the spatial mode shape can be cast in the form 1 1 2 2 3 3 4 4 5 5 6 6 ( ) exp( ) exp( ) exp( ) exp( ) exp( ) exp( ) x x x x x x x =  +  +  +  +  +  (17) where unknown integration constants j (j=1..6) yet required to be determined via imposing the classical and the extra non-standard boundary conditions along with j(j=1..6) denoting the roots of the characteristic equation associated with the differential equation eq. (16). imposing the classical and the extra non-standard boundary conditions to the analytical dynamic characteristics of mixture unified gradient elastic nanobeams 547 solution addressed for the spatial mode shape yields a homogeneous sixth-order algebraic system in terms of the unknown integration constants j (j=1..6). the resulted system of algebraic equations should be singular to have the non-trivial solution, and as a result, the determinant of the coefficients of the homogeneous sixth-order algebraic system should vanish. the analytical solution approach yields a strongly nonlinear characteristic equation in terms of the fundamental frequency which requires to be numerically solved. the influence of the gradient characteristic parameters along with the mixture parameter on the fundamental frequency of mixture unified gradient elastic beams is demonstrated in figs. 3 and 4 for elastic nanobeams with kinematic constraints of interest in nano-mechanics, viz. fully-fixed and cantilever beams. the detected fundamental frequencies are, additionally, normalized utilizing the corresponding classical natural frequencies 0. the strain gradient and the stress gradient characteristic parameters are, respectively, ranging in the intervals [0.1,1] and [0,1], as two values of the mixture parameter  = 0,1/2 are implemented. the non-dimensional gyration radius of the elastic nanobeam is, also, set as 1/ 20 = . furthermore, the non-dimensional fundamental frequency is defined for the sake of consistency of the illustrations as 2 e a l i   = (18) fig. 3 normalized fundamental frequency of mixture unified gradient elastic beams with fully-fixed ends 548 s. a. faghidian, a. tounsi fig. 4 normalized fundamental frequency of mixture unified gradient elastic beams with cantilever ends as noticeably observed from the numerical illustrations in figs. 3 and 4, the stress gradient characteristic parameter  has the effect of decreasing the fundamental frequency, viz. a larger  involves a smaller natural frequency for given values of the characteristic parameters ,. the fundamental frequency of elastic nanobeams within the framework of the mixture unified gradient theory, therefore, reveals a softening response in terms of the stress gradient characteristic parameter. on the contrary, the fundamental frequency of free vibrations of a mixture unified gradient elastic beam increases by increasing either the strain gradient characteristic parameter  or the mixture parameter . a stiffening response in terms of the characteristic parameters , is, thus, confirmed for a given value of . table 1 normalized fundamental frequencies of fully-fixed nanobeams: mixture unified gradient theory ( = 0) 0/   0.1 = 0.3 = 0.5 = 0.7 = 1 = 0+ 1.19212 2.16205 3.34212 4.57294 6.45084 0.2 0.72725 1.33248 2.06215 2.82252 3.98232 0.4 0.42807 0.78774 1.21969 1.66966 2.35591 0.6 0.29615 0.54564 0.84496 1.15673 1.63220 0.8 0.22516 0.41505 0.64276 0.87994 1.24164 1.0 0.18129 0.33426 0.51767 0.70869 1.00098 table 2 normalized fundamental frequencies of fully-fixed nanobeams: mixture unified gradient theory ( = 1/2) 0/   0.1 = 0.3 = 0.5 = 0.7 = 1 = 0+ 1.19212 2.16205 3.34212 4.57294 6.45084 0.2 0.91885 1.44405 2.13588 2.87683 4.02099 0.4 0.78774 1.02674 1.38602 1.79475 2.44615 0.6 0.74694 0.87520 1.08721 1.34389 1.76976 0.8 0.73043 0.80865 0.94587 1.12062 1.42238 1.0 0.72233 0.77449 0.86948 0.99511 1.21983 dynamic characteristics of mixture unified gradient elastic nanobeams 549 as expected, the mixture unified gradient theory of elasticity reveals a stiffening response in terms of the number of kinematic boundary constraints. in the numerical illustrations of free vibrations of elastic nanobeams within the context of the mixture unified gradient theory, the influence of the gradient characteristic parameters is more noticeable in elastic nanobeams with fully-fixed ends. the fundamental frequency of the classical elastic beam is, also, retrieved as either the gradient characteristic parameters tend to zero or as the mixture parameter approaches unity in the absence of the strain gradient characteristic parameter. numerical values of normalized fundamental frequencies of fully-fixed and cantilever nano-sized beams determined within the context of the mixture unified gradient theory are collected in tables 1 through 4, correspondingly, for the mixture parameter  = 0 and  = 1/2. table 3 normalized fundamental frequencies of cantilever nanobeams: mixture unified gradient theory ( = 0) 0/   0.1 = 0.3 = 0.5 = 0.7 = 1 = 0+ 1.01961 1.10882 1.17691 1.21442 1.24183 0.2 0.94703 1.05644 1.14256 1.19185 1.22880 0.4 0.79510 0.93391 1.05452 1.13066 1.19190 0.6 0.64946 0.79839 0.94334 1.04607 1.13681 0.8 0.53583 0.67976 0.83312 0.95355 1.06970 1.0 0.45096 0.58396 0.73510 0.86370 1.00036 table 4 normalized fundamental frequencies of cantilever nanobeams: mixture unified gradient theory ( = 1/2) 0/   0.1 = 0.3 = 0.5 = 0.7 = 1 = 0+ 1.01961 1.10882 1.17691 1.21442 1.24183 0.2 0.98474 1.07291 1.14878 1.19436 1.22960 0.4 0.93391 1.00737 1.08749 1.14557 1.19711 0.6 0.90107 0.95534 1.02689 1.08913 1.15380 0.8 0.88244 0.92161 0.98013 1.03862 1.10865 1.0 0.87161 0.90034 0.94687 0.99821 1.06713 5. closing remarks a stationary variational framework is conceived to study the dynamic characteristics of elastic nanobeams within the context of the mixture unified gradient theory of elasticity. in view of consistent integration of the stress gradient theory, the strain gradient model, and the classical elasticity theory, the peculiar size-dependent response of the elastic nano-sized beams is efficiently captured. the differential condition of dynamic equilibrium, the classical and the extra non-standard boundary conditions, and the constitutive differential model of the elastic nanobeams are all incorporated to the introduced functional. to close the boundaryvalue problem associated with dynamics of elastic nanobeams, the appropriate mathematical 550 s. a. faghidian, a. tounsi form of the extra non-standard boundary conditions is determined and prescribed. various augmented elasticity theories of gradient type are retrieved as special cases of the introduced mixture unified gradient theory via appropriate assumptions on the gradient characteristic lengths. the classical elasticity model is demonstrated to be recovered via either vanishing the gradient characteristic lengths or by approaching the mixture parameter to unity in the absence of the strain gradient length-scale parameter. the wave dispersion response of nano-sized beams is rigorously examined and the analytical solution of the phase velocity of dispersive waves is addressed. the dispersive response of waves within the context of the mixture unified gradient theory is graphically demonstrated and thoroughly discussed. the free vibrations of nano-sized beams consistent with the mixture unified gradient theory of elasticity is 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original scientific paper 1an efficient co-rotational fem formulation using a projector matrix udc 531:519.6 viet anh nguyen, manfred zehn, dragan marinković structural mechanics department, berlin institute of technology, germany abstract. co-rotational finite element (fe) formulations can be seen as a very efficient approach to resolving geometrically nonlinear problems in the field of structural mechanics. a number of co-rotational fe formulations have been well documented for shell and beam structures in the available literature. the purpose of this paper is to present a co-rotational fem formulation for fast and highly efficient computation of large three-dimensional elastic deformations. on the one hand, the approach aims at a simple way of separating the element rigid-body rotation and the elastic deformational part by means of the polar decomposition of deformation gradient. on the other hand, a consistent linearization is introduced to derive the internal force vector and the tangent stiffness matrix based on the total lagrangian formulation. it results in a nonlinear projector matrix. in this way, it ensures the force equilibrium of each element and enables a relatively straightforward upgrade of the finite elements for linear analysis to the finite elements for geometrically non-linear analysis. in this work, a simple 4-node tetrahedral element is used. to demonstrate the efficiency and accuracy of the proposed formulation, nonlinear results from abaqus are used as a reference. key words: co-rotational fem, tetrahedral element, projector matrix, polar decomposition, newton-raphson-method 1. introduction beside the updated (ul) and total (tl) lagrangian formulation, the co-rotational (cr) finite element (fe) formulations represent an efficient approach to handle large non-linear deformations of flexible structures. as it allows a relatively straightforward upgrade of the finite elements for linear analysis to the finite elements for geometrically received february 15, 2016 / accepted july 16, 2016 corresponding author: viet anh nguyen tub, structural mechanics department, strasse des 17. juni 135, 10623 berlin e-mail: viet.a.nguyen@tu-berlin.de 228 v. a nguyen, m. zehn, d. marinković non-linear analysis, the co-rotational approach has attracted significant interest over the past years [1]. the main idea of the approach is to decompose the total motion of each finite element into a rigid-body rotation and a moderate deformational part, which can be measured within a local element coordinate frame. the local frame can be defined at an arbitrary point of the element. it is attached to the element and performs the same rigidbody rotation and translation as the element itself. material deformational response is determined with respect to this local frame, while geometric non-linearities are accounted for by means of the large rigid-body rotation. for sufficiently small local strains, the constant strain-displacement matrix can be applied [2]. in the past decade, many co-rotational frameworks for the geometrically non-linear analysis of flexible structures were developed. even some classic, lagrangian formulation based developments used certain aspects of the co-rotational formulation such as the element attached co-rotational frame to determine the true strains and stresses [3]. a rather thorough and systematic description of the small-strain co-rotational finite shell and beam elements is available in the literature [1, 4, 5]. in those developments, a local frame was defined at an arbitrary node of the element. the orientation change of the local frame was used to represent the element rigid-body rotation. on this basis, the rotationfree nodal displacements are determined by filtering out the rigid body rotation from the global displacements. furthermore, the internal element forces were derived using a consistent linearization of the internal element energy with respect to the global element displacements. marinkovic and zehn [2] presented a simplified co-rotational fe formulation using the linear tetrahedral element for highly efficient computation of geometrically nonlinear deformations in the field of multibody systems (mbs) and realtime simulations. this development was based on a single co-rotational frame per finite element and a linear element stiffness matrix with respect to the co-rotational frame. in the work of crisfield and moita [6, 7], a co-rotational hexahedral element has been developed that enabled computation of deformations involving large strains. for this purpose, incompatible displacement modes were used to enhance the accuracy of computing the local element displacements. in addition, the element force vector and stiffness matrix were estimated using the assumption that the spin at the element centroid was zero. espath et al. [8] used the co-rotational approach in combination with the nurbs-based isogeometric fem to solve geometrically nonlinear dynamic problems. in this work, a co-rotational formulation for the non-linear static fe analysis is given using the tetrahedral element with the volume coordinates as linear shape functions. the rigid-body rotation is estimated by the polar decomposition of the deformation gradient, which is determined based on the initial and deformed element configurations. in addition, the internal element forces and element stiffness matrix are computed using directly the consistent variations of the internal element energy with respect to the global displacements. a resulting projector matrix is used to improve the results. the paper is organized as follows. after the introduction, in the second section, the co-rotational kinematics and the method to handle the rigid-body rotation for the tetrahedral element are presented. the derivation of the internal force vector and the tangent stiffness matrix are given in the third section. the fourth section gives the numerical procedure of the co-rotational fem as a flow chart followed by a couple of numerical examples to verify the accuracy and to assess the method efficiency. finally, based on the obtained results, conclusions are drawn. an efficient co-rotational fem formulation using a projector matrix 229 2. non-linear co-rotational element kinematics in this section, important kinematical relations for the calculation of the purely deformational part are given. in addition, a method to calculate the rigid-body rotation by means of the newton-raphson iteration is described in detail. 2.1. rotation-free element displacements two coordinate systems are applied to fully describe the motion of an element. a fixed global coordinate system (xg, yg, zg) measures the global nodal positions, whereas a local co-rotational system (xl, yl, zl) is defined at a node and rigidly translated and rotated with the element. while it moves with the element, it remains an orthogonal frame. the motion of an element from the initial (undeformed) to the current (deformed) configuration is depicted in fig. 1. origin 0 of the local element system is here defined at one of the nodes. in the initial configuration, the element has an orientation matrix t0 that remains constant during the analysis. the current deformed configuration can be transformed back to the initial one by matrix (t0 t+δt r (i))t, where t+δt r (i) is the rigid-body rotation matrix determined in each iteration i within the increment at time t+δt. fig. 1 the co-rotational description of the tetrahedral element kinematics by comparing the back rotated with the initial element configuration, the pure deformational displacements of node j are computed as shown in [9]: t δt (i) t δt (i) t t δt (i) t δt (i) t j 0 j 0 0 j 0( ) ( ) ( )      u t r x x t x x , (1) where t+δt xj (i) and t+δt x0 (j) are the actual global nodal position of node j and the origin of the co-rotational system (crs), respectively. similarly, xj and x0 represent their counterparts in the initial element configuration. 2.2. rigid-body rotation in general, rigid-body rotation r at a point of a deformable 3d-continuum can be calculated by the polar decomposition of deformation gradient, f, which is given as: ruf  . (2) 230 v. a nguyen, m. zehn, d. marinković where u is the right stretch matrix that describes the non-linear deformation of an infinitesimal small volume around the considered point. for 3d solid finite elements, the deformation gradient is typically determined at the integration points of the element. however, since a linear 3d tetrahedral element is used in this paper, whose shape functions derivatives are constant over the volume of the element, there is a unique rotational matrix that describes the rigid-body rotation between two element configurations. there are various approaches to compute the deformation gradient [1, 6]. in this work, the newton-raphson method, as described by rankin [5], is applied. in the first step, the deformation gradient is estimated at the center of each element, which generally requires a choice of alternative shape functions at the center of the element. the deformation gradient matrix of an element in iteration i within the increment at time t+∆t is given as:     k 1j t(i) j tt j 0t(i)tt xvf , (3) where the summation runs over all element nodes, while constant interpolation function derivative 0vj of a node j is calculated as: t jjj10 j 0 ggg               jv , (4) and it depends only on shape functions gj and the initial element geometry [10] and finally, (i) j tt x  is the local nodal coordinate vector that can be given as: t δt (i) t t δt (i) j 0 j 0 j( )   x t x x u (5) in eq. (4) 0 j is the constant jacobian matrix of the dimension 3×3, which transforms an infinitesimal small volume element from the natural (ξ, η, ) to co-rotational cartesian frame (xl, yl, zl) in the undeformed configuration [11]. a 3d finite rotation matrix about a rotation axis r in space can be expressed as the exponential of a 3×3 skew-symmetric matrix s [12]: 2 2 2 sin sin 1 2 exp 2 2                     r s i s s (6) where skew-symmetric matrix s is defined by:               0χ 0ψ χψ0 θspin s (7) and where so-called rotational pseudo-vector rθ  , which represents a finite rotation θ along axis r, is defined by rotations ,  and  around the axes of the cartesian system, ex, ey and ez, respectively, so that: an efficient co-rotational fem formulation using a projector matrix 231 zyx eeeθ  (8) variation of eq. (6) yields: spin( )  r θ r (9) using the polar decomposition, eq. (2), and the orthogonality property of the rotation matrix, the variation of the right stretch matrix is obtained in the same way: spin( )  u θ u (10) and the following relationship can be derived [5]: 2 (axial ) ( trace )    u i u u θ (11) this equation is a basis to develop an iterative process in order to perform the polar decomposition. generally, the axial operator in eq. (11) is the inverse of the spin operator and it yields an axial vector a from the skew-symmetric part of a quadratic matrix a by:         t 2 1 axial)spin (axial aaaa (12) eq. (11) describes a way to calculate the change of pseudo vector δθ with a given right stretch matrix u. hence, the change of rotation matrix r and the actual right stretch matrix in the next iteration step can be obtained using eqs. (6) and (10). this iterative process is terminated when u becomes symmetric: 0u  axial (13) the method is performed through the following steps. in the first step, deformation gradient f is computed using eq. (3) and the right stretch matrix is initialized by: fu (0) (14) the iterative process is performed as follows: (n) (n) (n) (n 1)2(axial ) ( trace )    u i u u θ (n 1) (n 1)exp(spin( ))   r θ (15) (n)1)(n1)(n uru   the iteration is terminated, if the magnitude of vector axialu(n) is smaller than a predefined tolerance t: taxial (n) u (16) 232 v. a nguyen, m. zehn, d. marinković 3. internal element forces and tangent stiffness matrix to guarantee the quadratic convergence of the newton-raphson method, the internal force vector of each element can be formulated using variation of the internal element energy. on this basis, the tangent element stiffness matrix is estimated by a consistent linearization of the element force with respect to the global nodal displacements. 3.1. internal element force vector the incremental/iterative newton-raphson solution (nrs) for non-linear static structural mechanical problems requires a known solution at time t and estimates the next configuration at time t+δt using an iterative process. the equations for an assembled finite element structure in the iteration step i can be written as: (i) gint, δtt ext tt1)(i g δtt(i) g δtt ffuk   1)(i g δtt(i) g δtt1)(i g δtt   uuu , (17) where t+δt kg (i) is the assembled global stiffness matrix, t+δt ug (i) and t+δt∆ug (i+1) represent the global and incremental nodal displacements while t+δt fext and (i) gint, δtt f  are the global external and internal force vectors, respectively. in general, the internal force vector can be computed by the variation of the internal element energy with respect to global element displacement vector t+δt ue (i) as: (i) e tt (i) inte, tt (i) ge, δtt w u f       , (18) where t+δt ue (i) is represented as the global element displacement vector in the form: t t tt t (i) t t (i) t t (i) t t (i) t e 1 2 k[ ... ]   u u u u . (19) it should be noted here that internal element energy (i) inte, tt w is unaffected by the rigid-body motion. in the tl formulation, the 2nd piola-kirchhoff stress and the greenlagrange strain are used to compute the internal element energy. because they are invariant with respect to the rigid-body rotation, it can be written [10]: 0 e t t t (i) t t (i) t t (i) 0 e,int e v w ( ) ( ) d v     e s , (20) where (i)tt s  is the 2nd piola-kirchhoff stress matrix expressed in the actual cr frame and d0ve is an infinitesimal small volume in the initial element configuration. in addition, (i)tt e  denotes the green-lagrange strain with respect to the actual cr frame as: t t (i) t t (i) t t (i) t t (i) l nl e{ }      e b b u , (21) where (i) nl tt b  is the non-linear strain-displacement matrix and is used to describe large strain problems [11]. matrix (i) l tt b  can be decomposed in two parts: an efficient co-rotational fem formulation using a projector matrix 233 (i) l1 tt l0 (i) l tt bbb   , (22) where l0b only depends on the derivatives of the interpolation functions with respect to the natural coordinates in the initial cr system and, hence, is constant. on the other hand, (i) l1 tt b  is estimated using the local nodal displacements from the previous iteration step [11]. in this paper, only small elastic strains are considered and hence, (i) l1 tt b  and (i) nl tt b  can be neglected. in case of small strains, the 2nd piola-kirchhoff stress components are equal to the cauchy stress components expressed in the cr element frame: (i)tt(i)tt σs   [11]. hence, the global internal element forces, eq. (18), can be rewritten as: e 0(i)tt v t l0t(i) e tt t(i) e tt (i) inte, δtt vd e 0 σb u u f         . (23) the derivative in front of the integral in eq. (23) describes the change of the approximate rotation-free element displacement vector with respect to the global nodal displacements and is introduced as a non-linear projector matrix into the global system. detailed discussions about the roles and derivation of the projector matrix can be found in [1, 4, 13]. finally, the projector matrix, expressed in the co-rotational frame, is given as: t(i)tt(i) s tt(i)tt    ppip , (24) where i is the 3k×3k identity matrix and (i) s tt p  is the lever-arm matrix of the dimension 3k×3: t t (i) t δt (i) t δt (i) t δt (i) t s 1 2 k[spin( ) spin( )... spin( )]   p x x x (25) and (i)tt   p represents the non-linear rotation-projector matrix of the dimension 3×3k, which allows to find the best-fit co-rotational element configuration as [1]:                        (i) k tt (i)tt (i) 2 tt (i)tt (i) 1 tt (i)tt t(i)tt ... uuu p . (26) it can be rewritten as: s 0-1(i)ttt(i)tt gup      (27) with the 3×3 matrix: k t t (i) 0 t t δt (i) t δt (i) 0 t j j j j j 1 { ( ) ( )}      u i v x x v (28) and the constant 3×3k matrix: 0 0 0 0 s 1 2 k[spin( ) spin( )... spin( )]g v v v . (29) 234 v. a nguyen, m. zehn, d. marinković in the case of linear elastic materials, the co-rotational cauchy stress is computed using the material constitutive law: (i)tt(i)tt εhσ   , (30) with constant hooke’s matrix h and linear local strain matrix (i)tt ε  , which is obtained using the rotation-free displacements [7]: (i) e tt l0 (i)tt ubε   . (31) finally, the internal element forces for a linear tetrahedral element can be written in the closed form: (i) e tt e t(i)tt(i)δtt 0 (i) inte, δtt ukprtf   . (32) the operator a is a 3k×3k diagonal matrix of submatrices a, and ek denoting the element stiffness matrix with: el0 t l0e vbhbk  (33) where ve is the volume of a tetrahedral element. matrix ek is constant and, therefore, calculated once and saved. in this way, the computational effort is reduced. 3.2. tangent element stiffness matrix the tangent element stiffness matrix, expressed in the global coordinate system, can be calculated by the variation of the global internal element forces with respect to the global element displacements [1]: (i) e tt (i) inte, δtt (i) te, tt u f k       . (34) applying the product rule to eq. (32), one obtains: (i) e tt (i) e tt e t(i)tt(i)δtt 0 (i) e tt e(i) e tt t(i)tt (i)δtt 0 (i) e tt e t(i)tt (i) e tt (i)δtt 0(i) te, tt u u kprt uk u p rtukp u rt k                     (35) more details on eq. (35) are available in [1, 4, 5]. the variations in eq. (35) are: (i) e tt t (i)δtt 0 (i)tt(i) e tt urtpu   , (36) t δt (i) t δt (i) t t (i)spin( )     r r , (37) t(i)tt(i) s tt(i)tt(i)tt    pppp . (38) an efficient co-rotational fem formulation using a projector matrix 235 substituting eqs. (36), (37) and (38) into eq. (35), the resulting global tangent stiffness matrix is generally asymmetric and given as: t (i)δtt 0 (i) te, tt(i)δtt 0 (i) te, tt rtkrtk   , (39) where (i) te, tt k  is the local tangent element stiffness that can be expressed as: t tt t (i) t t (i) t t (i) t δt (i) t t (i) t t (i) t δt (i) t t t (i) e,t e e,int e,int[spin ] [spin ]           k p k p f p p f p (40) where t δt (i) e,int[spin ] f represents a 3k×3 matrix that contains k spin-matrices of the local nodal force vectors: t δt (i) t δt (i) t t δt (i) t t δt (i) t t e,int 1,int 2,int k,int[spin ] [spin( ) spin( ) ... spin( ) ]   f f f f , (41) with local element forces (i) inte, δtt f  computed as follows: (i) e tt e t(i)tt(i) inte, δtt ukpf   . (42) simo [14] has shown that the symmetric part of the global tangent element stiffness matrix is sufficient for a quadratic rate of convergence of the newton-raphson method, hence: t t t (i) t δt (i) t t (i) t δt (i) e,t sym 0 e,t sym 0( ) ( )   k t r k t r , (43) where the symmetric part of the local tangent element stiffness is given as: tt t (i) t t (i) t t (i) t t (i) t δt (i) t e,t sym e e,int t tt t (i) t δt (i) t t (i) e,int 1 ( ) [spin ] 2 1 [spin ] 2               k p k p p f p f p (44) a detailed discussion about the tangent stiffness matrix in eq. (44) can be found in [1]. 4. co-rotational computational procedure and numerical examples the proposed co-rotational framework for the calculation of small elastic strains but large rotation structural problems is summarized in fig. 2 and has been implemented in the commercial fe software package abaqus using the user-defined-element subroutine uel [15]. all constant variables are computed once and saved in the initialization step. the equilibrium of the structure in each increment is computed by an iterative process. the non-linear projector matrix and the element rigid-body rotation are estimated using the global nodal positions in each iteration step. furthermore, the global tangent element stiffness matrix and the global internal element forces are computed in order to assemble the global system of equations for the whole structure. the solution in the current load increment is obtained when the convergence condition is satisfied. 236 v. a nguyen, m. zehn, d. marinković fig. 2 the presented co-rotational fem with the projector matrix this paper is focused on the geometrically non-linear static analysis with linear elastic material. all the numerical examples are computed in abaqus. the results obtained by the presented cr formulation are compared to the abaqus results to verify the accuracy and assess the efficiency of the presented approach. an efficient co-rotational fem formulation using a projector matrix 237 4.1. a clamped solid block in the first example, the static geometrically non-linear analysis of a clamped solid block exposed to a single force is represented. the solid block, fig. 3, is made of steel (e=210000 n/mm2 and ν = 0.3) with the dimensions 400×200×50 mm. the single force f acts at point p in the global z-direction and has the magnitude of 1.5×107 n. the fe mesh contains 1420 linear tetrahedral elements c3d4. the resulting deformation is bending-dominated but also accompanied by torsion. fig. 3 geometry and boundary conditions of the solid block (left), fe mesh with 1420 c3d4elements (right) in fig. 4, the development of global displacements of point p with the increasing force is presented. a very good agreement of the obtained results by the presented cr formulation and those from abaqus is observable. in addition, the displacements in x-, yand z-direction of point p are given in tab. 1. the column entitled “error” gives the relative difference with respect to the abaqus solution. it should be emphasized that this example involves the maximum principal logarithmic strains of 20%. fig. 4 displacements in three directions of the solid block for the load case 1.5×107 n (left), deformation state with deformation scale factor 1 (right) 238 v. a nguyen, m. zehn, d. marinković to assess the efficiency of this formulation, the number of iteration steps used by abaqus as well as by the cr formulation is also listed in tab. 1. the geometrically nonlinear analysis was performed in abaqus once setting the automatic load increment size (initial increment size: 0.2, minimum size: 10-5, maximum size 1) and then again with the fixed load increment size of 0.4. abaqus needed 38 iterations with the automatic increment size but did not converge with the fixed increment size used. in contrast, the cr formulation has converged after 24 iterations in the case of automatic increment size and after only 12 iterations with the fixed increment size. the result is reduction of computational time of about 68%. it can be noted that the presented approach also allows for relatively large increments. table 1 results for the clamped solid block cases abaqus cr-fem error x direction -30.78 -30.70 0,3% y direction -132.12 -132.81 0.5% z direction 262.28 261.34 0.4% automatic 38 iter. 24 iter. direct fails 12 iter. max. log. strains 20% 20% 4.2. a three dimensional curved beam in the second example, a three dimensional curved beam with a single force f of 5.5×105 n at point p is considered. the geometry and the boundary conditions are given in fig. 5. the curved beam is made of steel (e = 21000 n/mm2 and ν = 0.3) with the arc length of 0.5×π×450 mm. the cross-section has dimensions of 40 mm × 40 mm. the curved beam is clamped on a one end and discretized by 3652 linear tetrahedral elements c3d4. again, the deformational behavior can be seen as a combination of bending and torsion. to evaluate the results, the displacements of point p from the geometrically non-linear calculations in abaqus and from this cr formulation are given in fig. 6. a very good agreement is obvious in this case as well. the detailed values are given in tab. 2. fig. 5 geometry and boundary conditions of the curved beam for the load case 5.5×105 n (left), mesh with 3652 c3d4 elements (right) an efficient co-rotational fem formulation using a projector matrix 239 fig. 6 displacements [mm] in three directions of the curved beam with 3652 c3d4 elements (left), deformation state with deformation scale factor 1 (right) the maximal error in the displacement components is in the y-direction and reads 1.9%. as for the efficiency, performing the computation with an automatic (initial increment size: 0.2, minimum size: 10-5, maximum size 1) and fixed load increment size of 0.3, abaqus needed 31 iterations for the both cases. in contrast, the cr-formulation required 24 iterations for the automatic increment size and only 13 iterations for the fixed one. in addition, the computed maximum principal logarithmic strain was in this case 7%. the computational time was reduced by 58%. table 2 results for the curved clamped beam cases abaqus cr-fem error x direction 198.62 201.85 1.6% y direction -142.98 -140.27 1.9% z direction 470.03 470.75 0.2% automatic 31 iter. 24 iter. direct 31 iter. 13 iter. max. log. strains 7% 7% 5. conclusions the paper presented a co-rotational fe formulation to simulate geometrically non-linear elastic behavior of 3d structures. a systematic derivation of the numerical procedure is given as well as a couple of examples. the basis of the proposed calculation framework is a co-rotational fe-formulation for 3d geometrically non-linear deformations, where the large element rigid body rotation and small elastic displacements can be separately treated in an elegant manner. the element rigid-body rotation is efficiently computed using the polar decomposition of the deformation gradient, where the newton-raphson iterative method is applied. the rotation-free nodal displacements, given with respect to the local cr element coordinate system, are assumed to be small. the internal element force vector and the tangent element stiffness matrix are 240 v. a nguyen, m. zehn, d. marinković estimated using a consistent linearization of the internal element energy with respect to the global element displacements. this results in a non-linear projector matrix, which generally improves the obtained results. finally, the presented approach allows the linear tetrahedral element to be easily upgraded into the element for the geometrically non-linear analysis, requiring rather small modifications. a very good agreement between the results from abaqus and the proposed cr formulation validates the high accuracy of the presented approach. it was also demonstrated that the computational effort could be significantly reduced by using large load increments. this fact particularly gains importance if the method is used in combination with the advantages of modern hardware tools [16]. the problems in the field of structural mechanics with large rigid-body rotation and with relatively large strains can be efficiently solved using the projector matrix. with those properties, the presented formulation could be embedded into the software packages for multi-body system simulation to provide a greater versatility compared to existing solutions in describing flexible bodies undergoing large rigid-body rotation. references 1. felippa, c. a., haugen, b., 2005, unifield formulation of small-strain corotational finite elements: i. theory, center for aerospace structures, college of engineering, university of colorado. 2. marinković, d., zehn, m., 2012, finite element formulations for effective computations of geometrically nonlinear deformations, advances in engineering software, 50, pp. 3-11. 3. marinković d., köppe, h., gabbert, u., 2008, degenerated shell element for geometrically nonlinear analysis of thin-walled piezoelectric active structures, smart materials and structures, 17(1), pp. 1-10. 4. nour-omid, b., rankin, c., 1991, finite rotation analysis and consistent linearization using projectors, computer methods in applied mechanics and engineering, 93, pp. 353-384. 5. rankin, c., 2006, application of linear finite elements to finite strain using corotation, rhombus consultants group, inc., palo alto. 6. crisfield, m., moita, f., 1996, a unified co-rotational framework for solids, shells and beams, international journal of solids and structures, 33(20-22), pp. 2969-2992. 7. moita, g.f., crisfield m.a., 1996, a finite element formulation using the corotational technique, international journal for numerical methods in engineering, 39(22), pp. 3775-3792. 8. espath, l.f.r, braun, a.l., awruch, a.l., dalcin, l.d., 2015, a nurbs-based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach, international journal for numerical methods in engineering, 102(13), pp. 1839-1868. 9. nguyen, v.a., zehn, m., marinković, d., 2014, effiziente berechnung von geometrischen und materiellen nichtlinearitäten mit einer co-rotationalen finite-elemente-formulierung, deutschsprachige nafems konferenz, bamberg, germany. 10. crisfield, m.a., 2000, non-linear finite element analysis of solids and structures, volume i, essentials, johns wiley & sons ltd, baffins lane, chichester. 11. bathe, k.j., 1996, finite element procedures, prentice-hall, inc., englewood cliffs, new jersey. 12. argyris, j.h., 1982, an excursion into large rotations, computer methods in applied mechanics and engineering, 32(1-3), pp, 85-155. 13. mostafa, m., sivaselvan, m.v., felippa, c.a., 2013, reusing linear finite elements in material and geometrically nonlinear analysis-applications to the plane stress problems, finite elements in analysis and design, 69, pp. 62-72. 14. simo, j.c., 1985, a finite strain formulation. the three-dimensional problem. part 1, computer methods in applied mechanics and engineering, 49, pp. 55-70. 15. park, k., paulino, g.h., 2012, computational implementation of ppr potential-based cohesive model in abaqus: educational perspective, engineering fracture mechanics, 93, pp. 239-262. 16. nutti, b., marinković, d., 2014, an approach to efficient fem simulations on graphics processing units using cuda, facta universitatis series: mechanical engineering, 12(1), pp. 15-25. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 223 234 1 fuzzy controller for the control of the mobile platform of the corbys robotic gait rehabilitation system udc 681.5 maria kyrarini, siniša slavnić, danijela ristić-durrant institute of automation, university of bremen, germany abstract. in this paper, an inverse kinematics based control algorithm for the joystick control of the mobile platform of the novel mobile robot-assisted gait rehabilitation system corbys is presented. the mobile platform has four independently steered and driven wheels. given the linear and angular velocities of the mobile platform, the inverse kinematics algorithm gives as its output the steering angle and the driving angular velocity of each of the four wheels. the paper is focused on the steering control of the platform for which a fuzzy logic controller is developed and implemented. the experimental results of the real-world steering of the platform are presented in the paper. key words: robot-assisted gait rehabilitation, inverse kinematics based control of the mobile robot, fuzzy logic controller 1. introduction the objective of the integrated eu fp7 project corbys [1][2] is to design and implement a generic robot control architecture that allows the integration of high-level cognitive modules to support the functioning of the robot in dynamic environments including interaction with humans. as a practical application of the control architecture, a novel cognitive mobile gait rehabilitation system is being developed during the project’s lifetime. fig. 1 shows the final cad model image of the corbys robot-assisted mobile gait rehabilitation system with an associated global coordinate system. it consists of an omnidirectional mobile platform, a powered robotic orthosis attached to the platform via a pelvis link, and a linear unit. the mobile platform provides mobility for a patient and enables overground walking training, whilst the powered robotic orthosis helps the patient to complete his/her leg movements. received october 17, 2014 / accepted november 22, 2014 corresponding author: danijela ristić-durrant university of bremen, institute of automation, otto-hahn-allee, nw1, 28359 bremen, germany e-mail: ristic@iat.uni-bremen.de original scientific paper 224 m. kyrarini, s. slavnić, d. ristić-durrant 1.1. the main components of the corbys system the mobile platform constitutes the central housing structure of the corbys mobile gait rehabilitation system. it serves as a component carrier for the linear unit and motors for the orthosis, as well as for the central power supply system. the mobile platform also houses a network module, and other system modules such as the safety module and various computers upon which the modules of the control architecture are running. four wheel hub motors are connected to the platform for driving, which provide for the system movement in the x-direction. each wheel is fitted with a steering angle motor to change its direction (rotation about y-axis) so that the platform can drive along the curves. fig. 1 cad model image of the corbys mobile gait rehabilitation system the linear unit enables both lifting and lowering of the patient’s body (patient’s pelvis movement in the y – direction) and side to side movement of the patient’s body (patient’s pelvis movement in the z – direction). it is equipped with two servo-positioning motors for the actuation of y – axis which are chosen to provide for partial body weight support. one servo motor is used for the z – axis actuation. the pelvis link is the connection between the linear unit, attached to the platform, and the powered orthosis. the pelvis link ensures that the patient’s pelvis can rotate in the frontal and transverse planes. the force/torque sensor for measurement of the interaction force between the patient and the corbys system is placed within the pelvis link. this measurement will be used for control of the mobile platform enabling it to follow walking patients while they are wearing the powered orthosis. the powered orthosis assists the patient’s lower limb joint motions. the dofs configuration of the corbys orthosis prototype follows the natural human limb kinematics. there are 6 degrees of freedom (dofs) at each leg: 3 dofs in the hip, 1 dof in the knee, and 2 dofs in the ankle joints. the hip, knee and ankle joint motions on the sagittal plane (i.e. flexion and extension) are selected as active dofs based on the biomechanical properties of human walking, while the hip dofs in the transverse and frontal planes as well fuzzy controller for the control of the mobile platform of the corbys robotic system 225 as the ankle dof in the transverse plane are passive. the movements of the orthosis joints in the sagittal plane are controlled by a push-pull control (ppc) cable-based actuation system [6]. there are three actuators per orthosis leg that actuate the hip, knee and ankle joints in the sagittal plane. the actuators are placed on the mobile platform while the ppc cables are flexible links to the joints used to transfer the rotational movement of the motors to specifically designed orthotic joints. 1.2. joystick control of the mobile platform in the fully integrated and functional corbys gait rehabilitation system, the mobile platform will be controlled in order to follow the walking patients while they are wearing the orthosis. however, in the early stages of system development a joystick control of the mobile platform had to be integrated. the joystick control should enable “manual” platform control during the system development. also, the joystick control is necessary to move the platform with attached orthosis towards the patient in the process of setting up the patient in the orthosis. in this paper, an inverse kinematics based control algorithm for joystick control of the mobile platform is presented. given the linear and angular velocities of the mobile platform using the joystick, the inverse kinematics algorithm gives as output the steering angle and the driving angular velocity of each of the four wheels of the platform. the paper is focused on the steering control of the platform for which a fuzzy logic controller is developed and implemented. the rest of the paper is organized as follows. in section 2 the inverse kinematics model of the mobile platform used for the design of the fuzzy controller is given. the design of the fuzzy controller is given in section 3. section 4 presents experimental results obtained in real-world steering of the corbys mobile platform wheels using the designed controller. 2. inverse kinematics of the four-wheel mobile platform the reference values for the control of mobile robots can be defined utilizing dynamical or kinematic model, respectively [5]. given the linear and angular velocities for the center of mass of the mobile robot, the inverse kinematics equations calculate the steering angle and the driving angular velocity of each wheel of the mobile robot. starting from the mobile platform illustration shown in fig. 2, the kinematic model of the mobile platform of corbys system is derived in the following way. the mobile platform is considered as a rigid body with four independent wheels: front right wheel (w1), rear right wheel (w2), front left wheel (w3) and rear left wheel (w4). the known variables for the inverse kinematic equations are the xand z-components of linear velocity of the mobile platform center of mass [vcx,vcz] with respect to the inertial frame {o,x,y,z}and the angular velocity of center of mass, , with respect to the inertial frame. the unknown variables are the driving angular velocity of each wheel , 1,2,3,4i i  and the steering angle of each wheel 4,3,2,1, ii . the inverse kinematic equations can be summarized as follows: 1 1 2 3 4 1 2 3 4[ ] ( , , )t cx czf v v         (1) 226 m. kyrarini, s. slavnić, d. ristić-durrant beside the inertial or reference frame {o,x,y,z}, for the description of the motion of the platform two additional frames are important as illustrated in fig. 2: the frame {c,x,y,z} which is attached to the centre of mass of the mobile platform and the frames {wi,xi',yi',zi'}, i = 1,2,3,4 that are attached to the centers of the wheels. fig. 2 top view of the mobile platform the mobile platform has the following characteristics: width w = 1026.4mm, length l = 1702mm, wheel radius r = 100mm. velocities vi, i = 1,2,3,4, of the wheels’ centre points with respect to the inertial frame {o,x,y,z}are as follows: iici ddvv   (2) where vc is the linear velocity of the mobile platform centre of mass with respect to the inertial frame whose x-component and z-component are expressed in terms of unit vectors (i, j, k) in the frame {c,x,y,z}: czcxc vvv  . (3)  is the angular velocity of the frame {c,x,y,z}, with respect to the inertial frame and di is the position vector of the point wi with respect to the frame {c,x,y,z}. as di is constant, time derivative id of the position vector in (2) is equal to zero. position vectors di, i = 1,2,3,4 of wheels’ centre points can be represented in terms of unit vectors (i, j, k) in the frame {c,x,y,z}: fuzzy controller for the control of the mobile platform of the corbys robotic system 227 1 2 3 4; ; ; 2 2 2 2 2 2 2 2 l w l w l w l w d i k d i k d i k d i k          . (4) the xand z-components of velocities vi , i = 1,2,3,4 with respect to the inertial frame, are expressed in terms of the units vectors in the frame {c,x,y,z} as:              i l vvk w vv czzcxx 2 ; 2 11  (5)              i l vvk w vv czzcxx 2 ; 2 22  (6)              i l vvk w vv czzcxx 2 ; 2 33  (7)              i l vvk w vv czzcxx 2 ; 2 44  (8) as said above, the motion of wheel wi is defined by driving angular velocity i and steering wheel angle i which are illustrated in fig. 3. fig. 3 left: driving angular velocity of the wheel (right side view). right: wheel steering angle (top view) in pure rolling conditions, driving angular velocity i and steering angle i for each wheel are calculated as: 4,3,2,1, 22    i r vv izix i , 4,3,2,1,2atan          i v v ix iz i . (9) the inverse kinematic model of the mobile platform (9) and (5-8), is used for the development of the steering control algorithm as given in the following section. 228 m. kyrarini, s. slavnić, d. ristić-durrant 3. fuzzy logic control of steering dc motor fuzzy logic provides useful methodology for a practical solution for controlling complex systems particularly in the absence of the exact model of such complex systems [3][4][7]. due to the lack of information on the steering motors and load characteristics, a fuzzy logic controller is designed and implemented for the steering angle position control of each wheel of corbys mobile platform. a block-diagram of the fuzzy logic control system is shown in fig. 4. fig. 4 block-diagram of the fuzzy logic control of steering dc motor in order to achieve desired (reference) steering angle value ref (t) as calculated from the inverse kinematics, position error e(t) = ref (t)  (t), which is different from desired and actual steering angle (t) measured by the angle sensor, and corresponding derivative de(t) are selected as two input variables. the membership functions (mfs) for two input variables are shown in figs. 5 and 6, respectively. fig. 5 membership function for the input e(t) fig. 6 membership function for the input de(t) fuzzy controller for the control of the mobile platform of the corbys robotic system 229 the output of the fuzzy logic controller is applied voltage v(t) (fig. 7). the linguistic variables are defined as nb: negative big, nm: negative medium, ns: negative small, ze: zero, ps: positive small, pm: positive medium, pb: positive big. the set of decision control rules is shown in fig. 8. the fuzzy rules contain the relationships between inputs and output. each control input has seven fuzzy sets so there are 49 fuzzy rules. the fuzzyfication, converting a numerical variable into a linguistic variable, and the set of control rules are designed experimentally. the rule base structure is mamdani fuzzy interference reasoning. the number of mfs (seven) is chosen so as to meet requirements of smooth and time efficient steering control. when the error and the change of error are big, then the applied voltage should be maximal so that the steering motor rotates with maximum angular velocity. as the steering motor rotates, the wheel is coming closer to the desired position so that the steering motor should move slowly enough to continue the wheel moving towards the desired position and avoiding big overshoot. the seven mfs enable good control on the required speed changes of the motor, which is not possible for the considered mobile platform with three mfs or five mfs. the knowledge base for the fuzzy controller is generated from the control rules of the form: if e = ei and de = dej then v = v(i,j). these rules form the “rule base” which characterizes the manner in which the steering dc motor is controlled. for example, if error is pm and error change is pm then the output (voltage) is pb, which means that the motor will rotate with high angular speed until the position error is minimized. fig. 7 membership function for output v(t) fig. 8 fuzzy rule base 230 m. kyrarini, s. slavnić, d. ristić-durrant 4. experimental results a. linear motion of the mobile platform in the experimental scenario named “linear motion of the mobile platform”, the mobile platform is moved forwards, backwards and sideways using a joystick as the input device. the system receives two signals from the joystick, as shown in fig. 9. the first signal a represents the linear velocity of the mobile platform center of mass in the xdirection vcx (forwards and backwards movements) and the second signal b represents the linear velocity in the z-direction vcz (sideways movements). from the joystick signals illustration in fig. 9, it can be seen that the movement of the mobile platform is commanded as following: after 6sec from the start of the experiment the platform should go forward for 8,5sec (motion i); 1sec after motion i the platform should move to the right for 7,5 sec (motion ii); 1sec after motion ii the platform should go backwards for 13,5 (motion iii); 0,5sec after motion iii the platform should move to the left for 10sec (motion iv). the time intervals when the platform is not moving are referred to as rest time. fig. 9 joystick signals linear velocity of the mobile platform starting from the input joystick signals, the steering angle is calculated from the inverse kinematics as shown in fig. 10a for one of the mobile platform wheels. since the desired motion of the platform is linear, all the wheels have the same steering angle. as it can be seen in fig. 10a, the output of the inverse kinematics are angles 0°, 90°, 180° and -90°for the motions i, ii, iii and iv, respectively. fig. 10b shows the adapted output of the inverse kinematics. namely, since the driving dc motors are able to drive in 2 directions only forward and backwards, the value of steering angle is limited to [-90°, 90°]. for example, in motion iii the platform should move backwards. in that case there are two options: to steer the wheel to 180° and drive forward or to steer the wheel to 0° and drive backwards. due to the limitation of steering angle, the second option is taken as the adapted inverse kinematics output. additionally, the steering angle is not changing until it fuzzy controller for the control of the mobile platform of the corbys robotic system 231 gets a new command from the joystick. so between motion ii and iii as well as between motion iii and iv the steering angle keeps the previous value. the calculated steering angle is the reference signal for the control of the steering dc motor. the reference signal and the actual steering angles, measured from the angle sensors, for each platform’s wheel are given in fig. 11. from the presented results, it is evident that the implemented fuzzy logic controller eliminates the error. fig. 10 calculated steering angle fig. 11. reference (red) and measured (blue) steering angles 232 m. kyrarini, s. slavnić, d. ristić-durrant b. rotation of the mobile platform in the second experimental scenario named “rotation of the mobile platform”, the mobile platform exhibits a rotation around the y-axis of the centre of mass frame, using the joystick as input device. the system receives one signal from the joystick as shown in fig. 12, which represents the target angular velocity of the mobile platform . from the joystick signal illustration in fig. 12, it can be seen that the movement of the mobile platform is commanded as follows: after 5sec from the start of the experiment the platform should rotate clockwise around the center of mass for 2.1sec (motion i); 1.1sec after motion i the platform should rotate counterclockwise around the center of mass for 0,4 sec (motion ii); 1.3 sec after motion ii the platform should rotate counterclockwise for 1.1sec (motion iii). the time intervals when the platform is not moving are referred to as rest time. fig. 12 joystick signal angular velocity of the mobile platform using the signal from the joystick as input, the wheels’ steering angles are calculated from the inverse kinematics as shown in fig. 13. as the driving dc motors are able to drive in 2 directions forward and backwards, the value of steering angle is limited to [90°, 90°]. additionally, the steering angle is not changing until it gets a new command from the joystick. the adapted output of the inverse kinematics to meet the above limitations can be seen in fig. 14 (red). in this scenario, the front right wheel and rear left wheel are steered at 59° and the front left wheel and rear right wheel are steered at -59°. the calculated adapted steering angle is the reference signal for the control of the steering dc motor. the reference signal and the actual steering angle, measured from the angle sensors, for each wheel are given in fig. 14 (blue). as it can be seen also in this scenario, the implemented fuzzy logic controller is efficient as eliminates the error. fuzzy controller for the control of the mobile platform of the corbys robotic system 233 fig. 13 calculated steering angle fig. 14 reference (red) and measured (blue) steering angles 4. conclusion in this paper the design and implementation of a fuzzy controller for the wheels steering control of the mobile platform of the robotic gait rehabilitation system corbys are presented. the reference control signals are calculated using inverse kinematics of the mobile platform. the results from two real-world experimental scenarios, “linear motion of the mobile platform” and “rotation of the mobile platform” that illustrate the effectiveness of the implemented fuzzy controller are given in the paper. 234 m. kyrarini, s. slavnić, d. ristić-durrant acknowledgements: this research was supported by the european commission as part of the corbys (cognitive control framework for robotic systems) project under contract fp7 ict-270219. references 1. ristić-durrant, d., slavnić, s., glackin, c., 2014, corbys project overview: approach and achieved results, in w. jensen et al. (eds), replace, repair, restore, relieve – bridging clinical and engineering solutions in neurorehabilitation, biosystems & biorobotics, 7, pp. 139-147. 2. eu fp7 project corbys, www.corbys.eu (access date: 09.11.2014.) 3. kumar, n. s., kumar, c. s., 2010, design and implementation of adaptive fuzzy controller for speed control of brushless dc motors, international journal of computer applications, 1(27), pp. 36-41. 4. rubaai, a., ricketts, d., kankam, d. m., 2002, development and implementation of an adaptive fuzzy-neural-network controller for brushless drives, ieee transactions on industry applications, 38(2), pp. 441-447. 5. rubio, j. j., aquino, v., figueroa, m., 2013, inverse kinematics of a mobile robot, neural computing & applications, 23, pp.187-194. 6. slavnić s., ristić-durrant, d., tschakarow, r., brendel, t., tüttemann, m., leu, a., gräser, a., 2014, mobile robotic gait rehabilitation system corbys – overview and first results on orthosis actuation, proceedings of the ieee/rsj international conference on intelligent robots and systems (iros 2014). 7. prema, k., kumar, n. s., dash, s. s., 2014, online control of dc motors using fuzzy logic controller for remote operated robots, journal of electrical engineering & technology, 9(1), pp. 352-362. fazi kontroler za upravlje mobilnom platformom robotskog sistema za rehabilitaciju hoda corbys ovaj rad predstavlja kontrolni algoritam na bazi inverzne kinematike za džoistik upravljanje mobilnom platformom novog mobilnog robotskog sistema za rehabilitaciju hoda corbys. mobilna platform ima četiri točka sa nezavisnim upravljanjem i pogonom. polazeći od linearne i ugaone brzine mobilne platforme inverzna kinematika definiše, kao svoj autput, ugao nupravljanja i pogonsku ugaonu brzinu svakog od četiri točka. rad je usredsređen na upravljačku kontrolu platforme za koju je razvijen i implementiran fazi kontroler. eksperimentalni rezultati upravljanja realnom platformom su prikazani u radu. ključne reči: robotski podržana rehabilitacija hoda, upravljanje mobilnim robotom na bazi inverzne kinematike, fazi kontroler http://www.corbys.eu/ fume paper 7362 facta universitatis series: mechanical engineering vol. 19, no 1, 2021, pp. 91 104 https://doi.org/10.22190/fume201225014s © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper modeling of titanium alloys plastic flow in linear friction welding vladimir a. skripnyak, kristina iokhim, evgeniya skripnyak, vladimir v. skripnyak national research tomsk state university, russia abstract. the article presents the results of the analysis of the plastic flow of titanium alloys in the process of the linear friction welding (lfw). lfw is a high-tech process for joining critical structural elements of aerospace engineering from light and hightemperature alloys. experimental studies of lfw modes of such alloys are expensive and technically difficult. numerical simulation was carried out for understanding the physics of the lfw process and the formation laws of a strong welded joint of titanium alloys. simulation by the sph method was performed using the ls dyna software package (ansys wb 15.2) and the developed module for the constitutive equation. the new coupled thermomechanical 3d model of lfw process for joining structural elements from alpha and alpha + beta titanium alloys was proposed. it was shown that the formation of a welded joint occurs in a complex and unsteady stress-strain state. in the near-surface layers of the bodies being welded, titanium alloys can be deformed in the mode of severe plastic deformation. a deviation of the symmetry plane of the plastic deformation zone from the initial position of the contact plane of the bodies being welded occurs during a process of lfw. extrusion of material from the welded joint zone in the transverse direction with respect to the movement of bodies is caused by a pressure gradient and a decrease in the alloy flow stress due to heating. the hcp-bcc phase transition of titanium alloys upon heating in the lfw process necessitates an increase in the cyclic loading time to obtain a welded joint. key words: linear friction welding, 3d numerical simulation, sph simulation, titanium alloys, mechanical response 1. introduction friction welding is a technology that allows two or more identical or dissimilar materials with different melting points to be welded using the heat generated by friction, and also the mechanical mixing of the materials at the joint surface. friction welding makes received december 25, 2020 / accepted february 03, 2021 corresponding author: vladimir v. skripnyak national research tomsk state university, 36 lenin avenue, tomsk, 634050, russia e-mail: skrp2012@yandex.ru 92 v.a. skripnyak, k. iohim, e. skripnyak, v.v. skripnyak it possible to obtain strong welded joints of various alloys containing elements such as titanium, aluminum, magnesium, etc., as well as to obtain welded joints of steels [1-4]. in modern engineering practice, two modifications of friction welding technology are used linear friction welding (lfw) and friction stir welding (fsw) [1-11]. lfw makes it possible to obtain strong joining of materials at the contact interface plane of moved bodies through the heat generation at friction and material particle transport between contacting bodies [5-7]. lfw is a manufacturing process used primarily within the aviation industry, where high integrity welds are required in aero-engine production [5]. this technology is used for welding critical parts of power equipment, such as bladed-disk (blisk) of turbines [7]. fsw is one method of solid-state joining titanium and its alloys that can prevent the mechanical properties of titanium welded joints from deteriorating by other welding processes [1, 2, 12-14]. in the last decade, experimental and theoretical studies of the physical processes underlying friction welding have been carried out. it was found that during friction welding there is a significant heating of the material in the zone of formation of the weld, which contributes to the development of large plastic deformations. in the heating zone, phase transformations can occur in metals and alloys, for example, in titanium alloys [1, 5, 6]. therefore, the selection of friction welding modes for titanium alloys with different chemical and phase compositions is a complex engineering and technological problem [12−16]. the results obtained by popov and coworkers indicate that the adhesive contact of flat bodies can clearly depend on the macroscopic shape of the contact zone [17]. therefore, the shape of the contact zone of interacting bodies formed as a result of lfw can influence the strength characteristics of the welded joint [7]. the material structure and the presence of damage in it after plastic deformation have the decisive importance to the strength of the weld seam obtained by friction welding. loss of the adhesion between the tool and the material will inevitably lead to damage nucleation and decrease in the strength of the weld. the results of experimental studies of friction welding processes have become the basis for validating the methods for modeling physical processes that underlie these technologies [1-5]. various types of numerical methods were used for numerical simulation of lfw and fsw processes [15-24]. balokhonov and coworkers numerically studied the localization of plastic deformation of an aluminum alloy in a welded joint obtained by the fsw method [18]. it was shown that different zones in welding joint demonstrate distinct deformation response to loading due to peculiar combinations of the grain geometry and crystallographic texture. smolin and coworkers showed the possibility of predicting the parameters of the friction stir welding process based on numerical simulation of the process using the movable cellular automaton method (mca) [19]. it was shown by numerical modeling, that the ratio of the rotation speed to the speed of advancement of the fsw tool significantly affects the quality of the joint and the formation of microcracks and micropores in the welded joint of plates made of duralumin alloy d−16. it was also shown that the geometric parameters of the tool pins affect the movement of the plasticized material in the zone of formation of the weld. modification of the geometric parameters and the shape of the pins make it possible to improved mechanical properties of the welded joint. modeling of titanium alloys plastic flow in linear friction welding 93 tarlakovsky and coworkers applied the numerical sph technique to study the fsw process. it has been shown that the smoothed−particle hydrodynamics (sph) method allows one to describe the process of mixing materials on a scale that depends only on the number of particles [23-24]. the sph method makes it possible to use a thermodynamic model to predict the generation of heat as a result of the interaction of the surface with the tool pin and due to the dissipation of the energy of work stresses on plastic deformations. yang and coworkers applied the numerical modeling of the fsw to predict microstructural transformations in aluminum alloy 6061 in heat-affected zones: the weld nugget zone (wnz), the thermal-mechanical-affected zone (tmaz), the heat-affected zone (haz), and the base-material zone (bz) [21]. it has been shown that the computational fsw model cannot fully simulate a real welding situation. therefore, the theoretical predictions obtained with the comsol multi-physics software differ from the experimental data. fall and coworkers showed that the size and morphology of grains observed in various zones (bz, haz and tmaz) of fsw ti-6al-4v titanium alloy are distinctly different as revealed by the microstructure analysis. the obtained experimental results confirm the significant influence of structural transformations in the grain structure of titanium alloy during friction welding on the strength characteristics of the welded joint. chauhan and coworkers showed the possibility of using the coupled euler-lagrange method and the use of the johnson-cook constitutive equation in the numerical simulation of fsw of plates made of aa 6061-t6 aluminum alloy. the results of modeling the fsw mode, taking into account the physical and mechanical response of the alloy in the process of plastic flow at high strain rate, showed that the tensile strength of the welded joint is directly proportional to the effective plastic deformation. welding efficiency 90% and 65% are found for effective plastic deformation 230 and 60, respectively. these results indicate the need for an adequate description of the mechanical behavior of friction welded materials in the ranges of temperature, strain rates, and equivalent plastic strains, which are realized during friction welding. despite efforts to create a method for modeling physical mechanisms in friction welding technologies, there is still no complete understanding of their regularities and determining factors. in this article, the processes of deformation, damage and heating of titanium alloys in the process of lfw were studies by sph method of three-dimensional computer simulation. the wide-range constitutive model developed by authors was used for describing the mechanical behavior of the alpha and alpha+beta titanium alloys [25−29]. this model is used to describe the plastic flow of titanium alloys, taking into account the effects of its localization, strain hardening, thermal softening, strain rate sensitivity of the flow stress, and patterns of damage nucleation and growth. 2. model and computational details 2.1. model of lfw in this work, the coupled 3d computational model of the lfw thermomechanical process was developed on the base of ls dyna software (ansys wb 15.2). the model was applied for lfw simulation of the friction pair deformable blocks (fig. 1). the blocks dimensions for the sph model were: a= 2.5 mm; b= 5 mm; l0= 2.5 mm; d = b/2. a system of particles was generated in blocks with an initial smoothing length equals to 0.1 mm. particles with the same initial density had the same initial volume. the sph 94 v.a. skripnyak, k. iohim, e. skripnyak, v.v. skripnyak method with variable smoothing length was used for numerical simulation by lsdyna/wb ansys 15.2 software. the lower block moved back and forth at a velocity that changed as v1(t) = v0 sin (ω t). the friction welding process was simulated at a frequency of speed change of 50 hz. the ranges of displacement amplitudes and frequencies of changing velocity were chosen in accordance with the parameters realized in the mcandrew experiments on lfw of ti-6al-4v blocks [30]. fig. 1 scheme of lfw is shown in (a), sph model of blocks in contact at lfw is presented in (b) the computational model uses the theoretical basis of continuum damage mechanics. the system of equations includes: ▪ conservation equations eq. (1) − eq. (3), ▪ kinematic relations eq. (4) − eq. (5), ▪ constitutive relations eq. (6) − eq. (7), ▪ equation of state (eos) eq. (8) − eq. (10), ▪ relaxation equation for the deviatoric stress tensor eq. (11), ▪ decomposition of strain rate into elastic and inelastic components eq. (12), ▪ equations for inelastic strain rates eq. (13) – eq. (15). i i ud dt x    =  (1) ij i j du x dt    =  (2) ij ij de dt   = (3) (1/ 2)[ / / )ij i j j iu x u x =   +   (4) (1/ 2)[ / / )ij i j j iu x u x =   −   (5) ( ) ( )m ij ij f  = (6) modeling of titanium alloys plastic flow in linear friction welding 95 ( ) ( ) ( )m m m ij ij ijp s = − + (7) ( ) ( ) ( ) ( ) x m m tp p e  = + (8) 0 ( ), , v m x t t p x x v e e e e c t e p dv= + = =  (9) ( ) 7/3 5/3 2/3 0 0 0 1 0 3 3 (( / ) ( / ) )[1 (4 ) (( / ) 1)] 2 4 m xp b b     − − −=  − − −  − (10) ( ) / 2 ( / 3)m e e ij ij ij kkds dt g= −   (11) e p ij ij ij   = + (12) / 3p p p ij ij ij kke  = + (13) /p ij ije  =   (14) / (1 )p kk growthf f = − (15) where ρ is the mass density; ui is the component of the particle velocity vector; xi are cartesian coordinates, i = 1,2,3; t is the time; σij are components of the cauchy stress tensor; p(m) is the pressure in a condensed part of material; sij components of the deviatoric stress tensor; δij is the kronecker delta; e is the specific internal energy; et is the thermal part of specific internal energy; ex is the “cold” contribution to specific internal energy; v =1/ρ is the specific volume; ε´ij, and ω´ij are components of the strain rate tensor and the bending-torsion rate tensor, respectively; superscript m means that the parameters refer to the undamaged part of the material; superscript e means the elastic part of strain rate tensor; superscript p means the inelastic part of strain rate tensor; φ(f) is a function that links the effective stresses in the damaged material with the stresses in the undamaged part of the material; γ(ρ) = γr(ρr/ρ)+0.5[1 − (ρr/ρ)]2 is the grüneisen function describing by the kerley‘s relation; ρ0 is the initial mass density of condensed phase of alloy; γr, ρr, b0, b1 are constants of undamaged part of material; cp is the specific heat capacity; t is temperature in absolute scale; d(·)/dt is the jaumann’s derivative; g is the shear modulus; growthf is the damage growth rate; f is the damage parameter corresponding to the specific volume of voids in material; λ is the plastic multiplier derived from the consistency condition dφ/dt =0; φ is the plastic potential. experimental studies of the lfw process have shown that materials in the welding zone are deformed not only at high temperatures, but also at high hydrostatic pressures [5, 6, 10]. therefore, it is important to use a nonlinear equation of state, which describes the compressibility of titanium alloys in wide ranges of temperature and pressure for obtaining adequate results of modeling the processes of lfw. the pressure was calculated by the mie-grüneisen eos model. the grüneisen function γ(ρ) was determined with the following values of the constants: γr =1.184, ρr = 4.5018 103 kg/m3, ρ0 =4.5405 103 kg/m3 for alpha titanium, and γr =1.25, ρr = 4.5018 103 kg/m3, ρ0 =4.4588 103 kg/m3 for ti-6al-4v alloy. the birch-murnaghan equation was used to describe the zero-kelvin isotherms px (m). the bulk modulus and its pressure derivative were taken equal to b0 = 111.0 gpa, b1 = 3.48 for alpha titanium and b0 = 111.0 gpa, b1 = 3.58 for ti-6al-4v, respectively. the equation of state was calibrated to pressures of 125 gpa for alpha titanium and ti-6al96 v.a. skripnyak, k. iohim, e. skripnyak, v.v. skripnyak 4v alloy. the plastic potential φ was described using the gurson-tvergaard-needleman model (gtn) [29-33]. the function φ(f) takes the form (1−f) for pressure and is implicitly defined for the deviatoric stress tensor [31, 32]. 2.2. model of mechanical behavior titanium alloys heat is released in plastically deformable layers of materials arising in the friction zone of moving blocks. the actual contact area increases due to wear and thermal softening of the materials. the temperature rise associated with energy dissipation during plastic flow can be evaluated by eq. (16) [33]: 0 0 ( / ) p eq p p eq eqt t c d     = +  (16) where β ~ 0.9 is parameter representing a fraction of plastic work converted into heat; σeq =[(3/2) σij σij − (1/2) σkk σkk]1/2 is the equivalent stress; dεp eq is the increment of the equivalent plastic strain; εp eq =[(3/2) εij εij − (1/2) εkk εkk]1/2. the specific heat capacity for alpha titanium alloys was calculated by the phenomenological relations within temperature range from 293 to 1115 k [34, 35]: 2248.389 1.53067 0.00245 [ / ] 0 1320pc t t j kgk at t t k= + −   = (17) the specific heat capacity for ti6al4v titanium was calculated using data [9]: 2 7 3375.4343 1.12061 0.00147 7.10 [ / ] 0 1373pc t t t j kgk at t t k −= + − +   = (18) the heat affected zone (haz) expands into the volume of the contacting blocks during the process of their cyclic movement. the temperature distribution in the bodies to be welded must be determined taking into account the thermal conductivity. in this work, it was assumed that the most significant is conductive heat transfer. temperature history is calculated using the heat diffusion eq. (19): 2 2 2 2 2 2 1 2 3/ ( )[ / / / ]pc t t k t t x t x t x   =   +  +  (19) where k is the thermal conductivity coefficient. the thermal conductivity coefficient was described by phenomenological eq. (20): 0 1( )k t k k t= + (20) where k0, and k1 are parameters of material. coefficients k0 = 3.48568 w/(m k), and k1 = 0.01077 w/(m k2) were determined using experimental data for ti-6al-4v [8]. the temperature dependence of the shear modulus for titanium alloys was described by eq. (21): ( ) 48.66 0.03223 [ ] ,(273 1200 )g t t gpa k t k= −   (21) the plastic flow stress σs was described by eq. (22): modeling of titanium alloys plastic flow in linear friction welding 97 0 1 2 2 3 4 0 1/2 0 1 2 3 0 exp{ (1 / )} 1 exp{ }exp{ }exp{ ln( / )} [(2 / 3) ] exp{ / } p s s m ef eq eq eq ij ij eq t p p ef eq c t t c k c t c t t dt               = − + − − − = = − + =  (22) where εp ef is the effective plastic strain, and tm is the melting temperature, σs0, c1, c2, c3, c4, k2, γ1, γ2, γ3 are coefficients of material. the value of effective plastic strain εp ef is the integral of stepwise increments of equivalent plastic strain for a period of time t. material coefficients of some titanium alloys presented in table 1. table 1 model coefficients for titanium alloys coefficients of eq. (22) 0s , gpa c1 c2, gpa c3, k-1 c4 ,k-1 k2 tm, k ti-5al-2.5sn (grade 6) 0.02 3.85 0.56 0.0016 0.00009 8.5 1875 ti-6al-4v (grade 5) 0.041 3.4 0.84 0.0026 0.00009 8.5 1933 alpha ti (grade 2) 0.04138 6.2 0.1843 0.000877 0.00004 8.5 1938 the values of coefficients γ1= 2115.08615 s-1, γ2 =38.26589 k, and γ3= 9.82388 10-5 s-1 were used to determine the normalizing parameter in eq. (22) for titanium alloys. 2.3. damage model in this paper, numerical modeling of a titanium alloys was carried out taking into account the influence of the triaxiality stress state on the evolution of damage up to the ductile fracture. the lode parameter l and the triaxiality of stress state parameter η were determined by the relations [29]: ( ) ( ) 2 /     / 3 / ii i iii i iii eq i ii l p          = − − − = − = (23) where σij are the components of the cauchy stress tensor, p = σi/3 is the pressure, σi=σkk, σii = (1/2) (σii σjj σij σij), σiii = σ11σ22σ33+2σ12σ23σ31 – σ2 12σ33 – σ2 23σ11 – σ2 31σ22 are the first, the second and the third invariants of the cauchy stress tensor, respectively. the influence of damage on the flow stress was described by gtn model [31,32]: 2 2 * * 2 1 2 3( / ) 2 cosh( / 2 ) 1 ( ) 0eq s sq f q p q f  + − − − = (24) where σs is the yield stress, q1, q2 and q3 are model parameters, f* is the parameter of material. parameter f* was calculated by relations: * * ( ) / ( ) c c u c f c c f f if f f f f f f f f if f f =  = + − −  (25) where f is the damage parameter, fu = [q1+(q1 2-q3) 1/2]/q3, q1, q2, and q3 are constants of the model, fc is the damage parameter at the beginning of nucleation damages, ff is the damage parameter value corresponding to the final fracture of the material. 98 v.a. skripnyak, k. iohim, e. skripnyak, v.v. skripnyak the model of damage process was used for determination of the damage parameter f [31,32]. 2( / )exp{ 0.5[ ) / ] } (1 ) eq eq nucl growth p p nucl n n n n p growth kk f f f f f s s f f     = + = − − = − (26) where εn and sn are the average nucleation strain and the standard deviation, respectively. the model parameters of titanium alloys were determined throughout the numerical simulation of experimental stress-strain curves. numerical values of model parameters were fitted by combined experimental and numerical techniques [38]. numerical values of model parameters are given in table 2. table 2 dimensionless parameters for the gtn model for alpha titanium alloys. parameters of eqs.(24)-(26) q1 q2 q3 f0 fn fc ff εn sn ti-5al-2.5sn (grade 6) 1.3 1 1.69 0.00 0.2 0.035 0.4 0.28 0.1 ti-6al-4v (grade 5) 1.5 1 2.25 0.00 0.02 0.023 0.145 0.35 0.1 alpha ti (grade 2) 1.5 1 1.6 0.002 0.017 0.035 0.303 0.3 0.1 the friction shear stress τfr in the contact interface between blocks is determined as [9]: 1 1   ( ) ( )fr i u t p x u  =− (27) where τfr is the friction shear stress, μ(t) is the friction coefficient, u1 is the particle velocity in sliding direction. fig. 2 shows the approximation temperature dependences of the friction coefficient and the thermal conductivity coefficient of titanium alloy. fig. 2 friction coefficient of titanium alloys versus temperature is shown in (a), the thermal conductivity coefficient versus temperature is shown in (b). experimental data [8] are shown by symbols modeling of titanium alloys plastic flow in linear friction welding 99 the falling branch of the line in fig. 2(a) reflects the decrease in the coefficient of friction of the alpha titanium alloy caused by the alpha-beta phase transition. experimental data on the coefficient of friction for titanium alloy ti-5al-2sn correlate with data for alloy ti-6al-4v [39]. 2.4. boundary conditions boundary conditions corresponding to loading condition have the form: 1 1 8 5 2 3 4 6 7 8 9 10 11 12 13 1 3 4 5 6 7 8 9 10 11 12 13 1 0 3 1 1 0 0, 1,2,3 sin( ) 0 ( ) ( ) 0 frictions i s s s fr is ij s s s s s s s s s s s s s s s s s s s s s s p p u i u v t u u t p x u t t                        = = = = = − = = = (28) where ui│sj is the components of the particle velocity vector on the surface sj, t0 was assumed equal to 295 k. the numbering of the surfaces sj of the contacting blocks is shown in fig. 1(a). initial conditions correspond to the free stress state of the material in a uniform temperature field. 3. results of numerical simulation and discussion the formation of a strong weld joint of parts when using the technology of lfw is associated with the processes of diffusion and mass transfer at the boundary of the parts. both of these processes are described using the used computational model. fig. 2(a) shows the distribution of the effective plastic deformation near the friction surface, which is formed during the cyclic movement of the lower block. obtained results indicate that the plastic flow of the near-surface layers of the block material is distributed inhomogeneously over the depth and area of the contacting blocks. the action of pressure in the contact zone of the blocks prevents the initiation of damage in the plastically deformable material, even when tensile stresses arise. due to this, titanium alloys in the near-surface layers can be deformed in the regime of severe plastic deformation. the right tab in fig. 2(b) shows the initial stage of material extrusion from the welding zone. the effect of extrusion of the titanium alloys during the lfw was recorded experimentally by lachowicz [5], li [8], turner [37]. extrusion of the alloy from the friction welding zone is caused by a pressure gradient and a significant decrease in the alloy flow stress as a result of heating. fig. 2(b) demonstrates the calculated temperature distribution in the weld zone. 100 v.a. skripnyak, k. iohim, e. skripnyak, v.v. skripnyak the simulation results indicate the deviation of the symmetry plane of the plastic deformation zone from the initial position of the contact plane of the blocks being welded (fig. 2(b) left). this effect was discovered in recent experiments by lachowicz [5]. this effect affects the flatness of welds, their mechanical properties and residual stress fields in the joints. the alpha (hcp) – beta (bcc) phase transition of titanium alloys upon heating in the lfw process lead to changing the mechanical properties of alloys. therefore, the loading time lfw and the amount of accumulated plastic deformation have to be increased to obtain a welded joint. fig. 2 the distribution calculated effective plastic strain in blocks during formation of lfw zone is shown in (a), distribution of temperature in the welded contact zone is shown in (b) fig. 3(a) shows the calculated field of equivalent stresses in the blocks at the time before the change in the direction of motion of lover block. the results explain the effect of stress on the flatness of the weld. the presence of free surfaces s3 and s4 on the contact plane of the blocks (fig. 1(a)) leads to a significant distortion of the spatial distribution of equivalent stresses, under the influence modeling of titanium alloys plastic flow in linear friction welding 101 of which severe plastic deformations develop in the zone of formation of the weld. note, that the specificity of lfw is that the distribution of equivalent stresses in the volume of the block being welded is not stationary, but depends on the frequency and amplitude of displacements. the contact area of the blocks also changes during cyclic movement. the simulation results indicate that a complex stress state is realized in lfw zone. fig. 3(b) shows the distribution of the lode parameter characterizing the complex stress state. thus, the simulation results confirm that for an adequate simulation of lfw process in metals and alloys, it is necessary to use plastic flow models in a complex stressed state. fig. 3 calculated equivalent stresses (von mises stresses) upon contact of moving blocks in the process of lfw are shown in (a), calculated values of the lode parameter (the triaxiality parameter of the stress state) at the contact of moving blocks in the process of lfw are shown in (b) the unsteady field of equivalent stresses and the resulting rotation of the symmetry plane in the plastic flow zone of the contacting blocks contribute to the occurrence of localized material flows in the transverse direction relative to the movement of the blocks. fig. 4(a) shows the calculated field of transverse displacements of the particles in the zone of formation of the weld. fig. 2-4 shows the results obtained at velocity amplitude of 1000 mm/s. fig. 4 results of material particles transfer simulation in the lfw zone is shown in (a), displacements of mesoscopic particles of the particle material into moving blocks along the normal to the plane of their contact during lfw are shown in (b), (blue particles corresponds to upper block, red particles corresponds to low block) 102 v.a. skripnyak, k. iohim, e. skripnyak, v.v. skripnyak the simulation results indicate the formation of vortices or rotational areas within the volume of plastically deformable material in the weld formation zone. fig. 4(b) shows the movement of the particles of the material into the moving blocks along the normal to the plane of their contact (blue particles correspond to the upper block, red particles to the lower block). the analysis of the evolution of the stress-strain state in the formation zone of the weld showed that the strain rates during cyclic loading vary over a wide range. under the considered loading conditions, local strain rates reached 1000 s-1. the results obtained confirmed the necessity of using wide-range constitutive equations for the numerical analysis of the process of lfw of titanium alloys. in the lfw process, the temperature in the zone of strong plastic deformation of the titanium alloy can rise above the alpha-beta phase transition temperature tαβ. note that the phase transition temperature tαβ for the titanium alloys discussed in this work can vary from 906 k to 923 k and depends on the concentration of chemical components that stabilize the beta phase. an increase in temperature is accompanied by a decrease in the plastic flow stress of the titanium alloy and the rate of heat release in the plastic deformation zone. in this regard, the question arises about the rational combination of the control parameters of lfw of titanium alloys to obtain strong joints. numerical simulation of the processes of lfw of different titanium alloys (ti-5al2.5sn (grade 6), ti-6al-4v (grade 5), alpha ti (grade 2)) under the same conditions showed that the laws of plastic flow of these alloys are qualitatively the same. note that the quantitative differences in the values of the parameters of the stress-strain state and the distributions of temperature in the welded blocks from different titanium alloys confirm the need for an individual choice of lfw modes to obtain strong welded joints. the condition for the specific power input parameter q to exceed the critical value q* can be used as a criterion for the formation of a strong weld under lfw [9]: * 2 amp cont pu q q s   =  (20) where ω is frequency of displacement, uamp is the amplitude of displacement, p is the pressure between welding parts, scont is the area of contact parts, q* is the parameter for one or a pair of materials to be welded. the critical values of q* varies in the range from 5 up to 12 for titanium alloys [9]. the results of numerical simulation make it possible to assess the correctness of the choice of the control parameters of the process of lfw of three-dimensional structural elements made of titanium alloys to obtain a strong welded joint. 4. conclusions a new coupled thermomechanical 3d model of lfw process was used for studying the plastic deformation of joining structural elements from alpha and alpha + beta titanium alloys. numerical simulation of lfw thermomechanical process was carried by the sph method. calculations were performed using the ls dyna software package (ansys wb 15.2) and the developed module for the constitutive equation. modeling of titanium alloys plastic flow in linear friction welding 103 the conclusions drawn from the numerical simulations are the following: 1) the formation of a welded joint occurs in a complex and unsteady stress-strain state. 2) in the near-surface layers of the bodies being welded, titanium alloys can be deformed in the mode of severe plastic deformation. 3) a deviation of the symmetry plane of the plastic deformation zone from the initial position of the contact plane of the bodies being welded occurs during a process of lfw. 4) extrusion of material from the welded joint zone in the transverse direction with respect to the movement of bodies is caused by a pressure gradient and a decrease in the alloy flow stress due to heating. 5) the hcp−bcc phase transition of titanium alloys upon heating in the lfw process necessitates an increase in the cyclic loading time to obtain a welded joint. acknowledgement: the work was supported by the russian science foundation (rsf), project №20-79-00102. the authors would like to thank to the rsf. references 1. heidarzadeh, a., mironov, s., kaibyshev, r., cam, g., simar, a., gerlich, a., khodabakhshi, f., mostafaei, a., field, d.p., robson, j.d., deschamps, a., withers, p. j., 2020, friction stir welding/processing of metals and alloys: 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original scientific paper alternative method for on site evaluation of thermal transmittance udc 691 aleksandar janković, biljana antunović, ljubiša preradović faculty of architecture, civil engineering and geodesy, university of banja luka, bosnia and herzegovina abstract. thermal transmittance or u-value is an indicator of the building envelope thermal properties and a key parameter for evaluation of heat losses through the building elements due to heat transmission. it can be determined by calculation based on thermal characteristics of the building element layers. however, this value does not take into account the effects of irregularities and degradation of certain elements of the envelope caused by aging, which may lead to errors in calculation of the heat losses. an effective and simple method for determination of thermal transmittance is in situ measurement, which is governed by the iso 9869-1:2014 that defines heat flow meter method. this relatively expensive method leaves marks and damages surface of the building element. furthermore, the final result is not always reliable, in particular when the building element is light or when the weather conditions are not suitable. in order to avoid the above mentioned problems and to estimate the real thermal transmittance value an alternative experimental method, here referred as the natural convection and radiation method, is proposed in this paper. for determination of thermal transmittance, this method requires only temperatures of inside and outside air, as well as the inner wall surface temperature. a detailed statistical analysis, performed by the software package spss ver. 20, shows several more advantages of this method comparing to the standard heat flow meter one, besides economic and non-destructive benefits. key words: thermal transmittance, heat flow meter method, natural convection and radiation method, software support, statistical analysis received april 19, 2017 / accepted july 03, 2017 corresponding author: aleksandar janković university of banja luka, faculty of architecture, civil engineering and geodesy, vojvode stepe stepanovića 77/3, 78000 banja luka, bosnia and herzegovina e-mail: aleksandar.jankovic@aggf.unibl.org 342 a. janković, b. antunović, lj. preradović 1. introduction building envelope represents the boundary between the inner and the outer space with the purpose to protect building occupants from different atmospheric conditions. its function is to provide a comfortable and healthy environment to the building users with its hygrothermal characteristics which are determined by the structure, adequate selection and order of used materials [1]. in modern society heat loss through the building envelope represents a significant share of the total energy consumption of the facility. therefore, the aim is to reduce heat loss, increase energy efficiency of buildings and achieve an optimal thermal comfort for its occupants. thermal performance of the building envelope is primarily determined by thermal transmittance or u-value [w∙m-2 k -1 ]. this quantity is the starting point for calculating energy consumption for heating and cooling of the building as well as its maintenance cost. for existing buildings and especially for buildings in the design phase the thermal transmittance of the envelope is determined by calculation based on the thermal characteristics of the structural elements. however, this value does not take into account the effects of irregularities and degradation of certain envelope elements caused by aging which may lead to a difference between the theoretical and the actual values [2]. therefore, it is essential to measure on site thermal transmittance in the real conditions in order to minimize errors in the estimation of heat losses through the building envelope. one simple and effective method for determining actual thermal transmittance through the building element involves measuring the heat flux density at the same time with the inside and the outside air temperature. this insitu measurement is governed by the iso 9869-1:2014 that defines the heat flow meter method or hfm method [3]. the sensor that measures the temperature of the outside air should not be exposed to direct solar radiation and precipitation. therefore, the final result is not always reliable, in particular when the building element is light (low specific heat of materials) and if multi-layered air spaces are present (even if slowly ventilated) [4]. in the last few years, new techniques have been proposed for in situ determination of thermal transmittance of the building elements. several researchers have proposed the use of quantitative thermography for a rapid in situ measurement of the overall transmittance of an envelope building in a short time [4]. albatici and tonelli [5, 6] use a technique that involves parameters measured by infrared thermography, except for the wind velocity which is measured by a hot-wire anemometer in proximity of the wall. differences with u-values determined by the hfm are of the order of 30%, while differences with calculated u-values can grow up to 80%. grinzato et al. [7] proposed a more rigorous and complex procedure involving a light metallic frame useful for accurate measurements of wall and air temperatures thus consequently getting more precise thermal transmittance evaluation. reported experimental results show a difference with u-values determined by the hfm of the order of 30-35%. for a quite long time several other methods that do not involve infrared thermography have been in use for experimental determination of thermal transmittance of the building elements, like hot box method [8, 9] or the temperature based method [10]. the hot box method is mainly used for thermal transmittance measurements of inhomogeneous components such as windows, doors and thermal bridges in laboratory and cannot be used for in situ measurements. it is a very reliable method, if the detailed and precise calibration procedure is complied [11]. the temperature based method lacks accuracy, but it is often used in practice, where the heat flux is approximated by measuring the inside alternative method for on site evaluation of thermal transmittance 343 temperature and the wall temperature assuming constant inner thermal boundary resistance. recently, a rapid in-situ measurement of the wall’s thermal resistance and u value by the excitation pulse method is developed at delft university of technology [12]. none of these techniques, except the heat box one, were standardized because they did not meet the balance between convenience and accuracy offered by the hfm method. some of them offer practicality but lack accuracy and vice versa. in accordance with the modern trends and with the aim to avoid the problems of the hfm method, a new experimental methodology based on the measuring inside and outside air temperature and the inner wall surface temperature is proposed in this paper. this new experimental method hereinafter will be referred to as the ncar method (natural convection and radiation method). 2. thermal transmittance for plane, opaque and homogenous building elements thermal transmittance u can be determined by calculation based on the thermal characteristics of the consisting materials according to the formula [13]: 1 1 1 1n k ki k e u l       (1) where lk and λk represent thickness and thermal conductivity of k-th layer of building element, respectively, while 1/αi and 1/αs denote the inner and outer resistances of the building element to heat transfer, respectively. this method refers to the dominant one dimensional heat transfer, such as the flat walls of the building envelope. in order to obtain representative values of the thermal transmittance it is necessary to comply with certain practical rules during measurement. the thermal transmittance determination by the heat flowmeter method is based on some simplifying hypotheses. in selecting the measuring point the thermal bridges and the construction joints with dominant two-dimensional and three-dimensional heat transfer have to be avoided. the sensor intended to measure the temperature of the outside air should not be exposed to the direct solar radiation and precipitation. the maximum time period between two records and the minimum test duration depends on: the nature of the element, indoor and outdoor temperatures and the method used for analysis. two methods may be used for analysis of the data in accordance with the iso 9869-1: the so-called average method, which is simple, or the dynamic method, which is more sophisticated but which gives quality criteria of the measurement and may shorten the test duration for medium to heavy elements submitted to variable indoor and outdoor temperatures. during the test the minimum difference between the inner and the outer air temperatures has to reach 10-15 °c. average method assumes that the thermal transmittance can be obtained by dividing the mean density of heat flow rate q by the mean difference between the inside and outside air temperatures, ti and te respectively, according to the formula [3]: 344 a. janković, b. antunović, lj. preradović ( ) j j i e j j q u t t     (2) where index j enumerates the individual measurements. for heavier elements, which have a specific heat per unit area of more than 20 kj∙m -2 k -1 , the analysis shall be carried out over a period which is an integer multiple of 24 h. the test shall end only when the duration of the test exceeds 72 h, the thermal resistance value obtained at the end of the test does not deviate by more than ± 5 % from the value obtained 24 h before and the change in heat stored in the wall is less than 5 % of the heat passing through the wall over the test period. also, the thermal resistance obtained by analyzing the data from the first time period during int (2 x dt / 3) shall not deviate by more than ± 5% from the values obtained from the data of the last time period of the same duration, whereby dt is the duration of the test in days and int denotes the integer part of a number. with such strict criteria and due to a great number of limitations in-situ measurement of thermal transmittance with heat flowmeter method is not always possible. the heat transfer between the inside air and the inner wall surface are affected by several transport mechanisms: heat conduction through the air adjacent to the surface, convective transport by air flows, and emission of long-wave radiation. since the conduction is negligible due to the very low thermal conductivity of the air, the heat transfer is largely controlled by the other two mechanisms that are described by quantity called the heat transfer coefficient [14]: cr   (3) where αr and αc are the coefficients that indicate the contribution to the heat transfer by radiation and convection, respectively. the convective heat transfer from the inside air to the inner wall surface happens due to the fluid motion, which is not generated by external force, but only by temperature gradients in fluid itself. this type of the heat transfer is also known as the natural convection. this assumption is valid only for the air inside the room, namely, the room which is not mechanically ventilated or air conditioned; it does not apply to the outside air, where the convective heat transfer is forced by an externally induced flow (wind). the convective heat transfer between the solid surface and the fluid in contact is described by a quantity called the convective heat transfer coefficient. there are various equations for the convective heat transfer coefficient expressed as the function of the temperature difference between the surface temperature and the temperature of the air out of the thermo-kinetic boundary layer (undisturbed air) [14]: ( )n c i isc t t    (4) where c and n represent constants and tis is the temperature of the wall surface. the equations resulting from various choices of parameters c and n are represented in table 1 together with the names of the authors who studied the natural convection and derived the corresponding values of the parameters. table 1 also includes the standard method for quantifying the convective heat transfer coefficient proposed by the american society of heating, refrigerating and air-conditioning engineers (ashrae). alternative method for on site evaluation of thermal transmittance 345 table 1 the convective heat transfer coefficients in the case of the natural convection [14] author(s) αc [w∙m-2k-1] awbi et al. [15] 1.49∙δt 0.345 khalifa et al. [16] 2.07∙δt 0.23 michejev [17] 1.55∙δt 0.33 king [18] 1.51∙δt 0.33 nusselt [19] 2.56∙δt 0.25 heilman [20] 1.67∙δt 0.27 wilkers et al. [21] 3.04∙δt 0.12 ashrae [22] 1.31∙δt 0.33 a wall surface always exchanges long-wave thermal radiation with other surfaces in its surroundings. the corresponding heat flow depends on the temperatures (to the fourth power), the material properties and the nature of the surfaces. the radiative heat transfer from the inside air to the inner wall surface is described by a quantity called the radiative heat transfer coefficient [23]: 4 4( )i is r i is t t t t        (5) where ε represents the emissivity of the wall surface and σ = 5.67∙10 -8 w∙m -2 k -4 is the stefan-boltzmann constant. if we assume stationary conditions in which the heat flux density through the wall is equal to the heat flux density that is transferred from the indoor air to the inner wall surface, then the thermal transmittance of the wall can be determined from the known values of the heat transfer coefficient and directly measured temperatures: ( ) ( ) i is i e t t u t t     (6) finally, the thermal transmittance of the wall can be determined if we consider the heat transfer from the surrounding air to the inner wall surface that occurs through the natural convection and radiation, according to the following formula: 1 4 4( ) ( ) ( ) n i is i is i e c t t t t u t t          (7) this alternative method, named the ncar method, is based on the measured temperatures of the inside and the outside air and the inside wall surface temperature, as well as emissivity of the inner wall surface. the method requires continuous monitoring of temperatures with the same recording interval as the hfm method. taking into account individual measurements (enumerated with index j), the thermal transmittance can be obtained by the following equation: 1 4 4( ) ( ) ( ) n i is j i is j j j i e j j c t t t t u t t             (8) 346 a. janković, b. antunović, lj. preradović 3. measurement methodology testo 435-2 data logger with appropriate sensors was used to measure the heat flux density and the corresponding inside and outside air temperatures necessary for determining thermal transmittance of the wall by the hfm method. on the other side, ahlborn almemo 2690 data logger with k-type thermocouples sensors for air temperatures and a film sensor for surface temperature of the wall was used for determining thermal transmittance of the wall by the ncar method. measurements were made on the northern wall of the preschool building. the structure and possible thermal characteristics of the tested wall are shown in table 2. the project documentation indicates that the wall was built from the hollow bricks plastered on both sides with the layer of lime cement mortar, but there is no specific information about thermal characteristics of the used materials. two types of hollow bricks were used (0.52 and 0.61 wm -1 k -1 ) during considered construction period, as well as various types of the lime cement mortar (0.85 0.99 wm -1 k -1 ). therefore, the range of the possible thermal transmittance values (1.241 1.404 wm -2 k -1 ) is derived. based on the type of construction, it can be concluded that it is a heavier type of building element characterized by specific heat capacity per unit area greater than 20 kj∙m -2 k -1 . table 2 thermal characteristics of the wall layers thickness [cm] thermal conductivity [wm-1k-1] thermal resistance [m2kw-1] thermal transmittance [wm-2k-1] inside air 0.130 inner lime-cement plaster 2 0.85-0.99 0.024-0.020 hollow brick 30 0.52-0.61 0.577-0.492 exterior lime-cement plaster 3 0.85-0.99 0.035-0.020 outside air 0.040 total 0.806-0.712 1.241-1.404 in order to obtain representative measurements the thermal bridges and the construction joints are avoided as well as the direct influence of heating and cooling devices. the sensor for heat flux density was mounted on the inner surface of the wall, in the area of the uniform temperature field, which was determined by the thermal imager. the thermal contact paste is applied in order to provide direct contact with the element over entire surface of the heat flowmeter. the sensors for the inner wall surface and the inside air temperatures were mounted under and in the vicinity of the heat flowmeter, while the sensor for the outside air temperature was mounted on the opposite side of the wall of the heat flowmeter. during the measurement, the sensor for the outside temperature has not been exposed to the direct solar radiation and precipitation. considering the fact that stable temperature was achieved around the heat flowmeter and inside the temperature sensors, the test duration lasted 72 h with recording interval of 15 minutes. the inside and the outside air temperatures and the heat flux density measured by the testo 435-2 instrument were used to determine thermal transmittance by the hfm method. inside air, the inner wall surface and the outside air temperatures measured by the almemo ahlborn 2690-8 instrument were used to determine the thermal transmittance by the ncar method. this method also required determining emissivity of the inner wall surface, which alternative method for on site evaluation of thermal transmittance 347 was carried out by the thermal imager. the emissivity displayed on the thermal imager had been adjusted to the real value, in such way that the temperature value shown on the thermal image was the same as the actual temperature of the inner wall surface measured by the film sensor [24]. if there is no thermal imager available during measurements, the emissivity can be assumed if the type of material is known since some materials have well known and standardized values. after the measurement is completed, the temperature sensors which served for determining u-value by the ncar method have not left marks on inner wall surface. on the contrary, because of the applied thermal paste, the plate for measuring the heat flux chopped off thin layer of plaster from the inner wall surface. 4. results the measurement period includes full days from feb., 10, 2015 to feb, 13, 2015, which is longer than the required minimum time interval demanded by the iso 9869-1:2014. the weather favored the measurements and the temperature difference between the inside and the outside air was at least 10 0 c. fig. 1 shows the temporal evolution of the weather conditions, where the indoor and the outdoor temperatures are marked with the green and purple color, respectively, and the inner wall surface temperature is marked with the blue color. for emissivity of the inner wall surface the value of 0.91 was obtained. fig. 1 the temporal evolution of inside and outside temperatures the analysis of the measurement results are performed by the average method. the thermal resistance value obtained by the hfm method at the end of the test period deviates – 4.78 % from the value obtained 24h earlier, while the same comparison for the ncar method shows deviation of 0.60 %. the thermal resistance obtained by analyzing the data for the time period covering first two days deviates – 2.52 % (for the ncar method) and -4.75 % (for the hfm method) from the values obtained for the time period covering the last two days. it is obvious that ncar method shows better agreement of the thermal resistance values obtained from different periods within the measuring interval. table 3 shows the average values of thermal transmittance obtained by the hfm and different ncar methods, according to the relations (2) and (7), respectively. the results from different ncar methods are represented depending on the numerical values of c and n constants derived from the expression for the convective heat transfer coefficient. 348 a. janković, b. antunović, lj. preradović the third value is the theoretical thermal transmittance calculated based on the thermal characteristics of the materials which constitute the building element that was analyzed. table 3 the overall thermal transmittance measured by hfm and ncar methods u [wm-2k-1] deviation from hfm [%] hfm method 1.210 ncar method ashrae [22] 1.268 4.82 awbi et al. [15] 1.322 9.29 khalifa et al. [16] 1.411 16.64 michejev [17] 1.332 10.05 king [18] 1.321 9.18 nusselt [19] 1.540 27.31 heilman [20] 1.335 10.32 wilkers [22] 1.558 28.76 theoretical 1.241-1.404 2.56-16.03 as can be seen from table 3, the mean u-value obtained using the ncar method, derived from the ashrae equations for the convective heat transfer coefficient, has the best agreement (the smallest deviation) with the mean u-value obtained by the hfm method. furthermore, it can be concluded that the hfm method and the ashrae ncar method are highly correlated. the correlation coefficient between the measurements obtained by the ashrae ncar and hfm method is very high and amounts to 0.956. table 4 detailed comparison of hfm and ncar methods hfm ncar hfm cf. ncar standard deviation [wm -2 k -1 ] 0.280 0.252 0.102 mean absolute deviation [wm -2 k -1 ] 0.234 0.213 0.083 mean absolute percentage deviation [%] 19.3 16.8 6.5 further statistical comparison performed by the software package spss ver. 2.0 shows a high agreement between two methods with a relatively small standard, mean absolute and average relative deviation (tab. 4). the ncar method indicates smaller dispersion of measurements around the mean than the hfm method, which is confirmed by the lower values of standard, mean absolute and mean relative deviation (tab. 4). this is also noticeable from fig. 2, where the changes of the thermal transmittance measured by the ncar method are more dumped, which indicates that this method is less sensitive to the measurement conditions [25]. the lower standard deviation around the mean value of the ncar method indicates a better experimental technique as well as more precise and reliable measurements than the hfm method. furthermore, the u-value obtained by the ncar method agrees with assumed interval of the theoretical u-value, unlike the thermal transmittance obtained by the hfm method. alternative method for on site evaluation of thermal transmittance 349 fig. 2 the temporal evolution of the thermal transmittance measured by the hfm (red) and the ncar (blue) method 5. conclusions the measurement of the u-value based on the ncar method has three main and meaningful advantages in comparison to the hfm method: 1) the temperature sensors do not leave marks or damage the surface on a building element, as it happens with the plates for measuring the heat flux density in the hfm method. 2) the measurement procedure can be performed with the temperature data loggers, which are considerably cheaper than the instruments for determining thermal transmittance by measuring heat flux density. 3) this method could be used to measure the thermal transmittance of the light building elements, as the sensors for measuring temperature do not significantly modify the heat flow and temperature field on the surface of a building element in contrast to the standard sensors for heat flux density. the ncar method shows fewer damped oscillations of the u-value in comparison with the hfm method, which leads to the conclusion that this method is less sensitive to the measurement conditions and that the obtained data is more reliable and precise. the main limitations of the ncar method are the same as main limitations as the hfm method, which also assumes stationary conditions and natural convection as the only type of the convective heat transfer from the indoor air to the inner wall surface: 1) this method requires calm conditions. the air conditioners and fans should not be powered on during the measurement and the room may be just slightly naturally ventilated. in this way occurrence of the forced convection is avoided. 2) since the heat flow is not constant, the minimum duration of the test should be 72h long and the minimum temperature difference between the inside and the outside air must be at least 10 0 c. the measured u-values by two given methods are in a very good agreement, which means that presented the ncar method is equally reliable as the standard hfm method. taking into account the above mentioned advantages, it can be also concluded that the ncar method is significantly simpler and cheaper than the hfm one. however, the 350 a. janković, b. antunović, lj. preradović proposed method needs further testing on different types of building elements in different conditions in order to draw more profound conclusions about the method reliability and applicability. the authors think that, if the measurement of the thermal transmittance is carried out in a proper manner following the procedure described in this paper by a technician with specific knowledge of building physics, the final result of the test can be just as reliable as the one made with the heat flow meter method. references 1. antunović, b., janković, a., preradović, lj., 2014, thermal performance of preschool education building envelope, proceedings of international conference contemporary achievements in civil engineering, subotica, serbia, pp. 545-550. 2. antunović, b., stanković, m., janković, a., gajić, d., todorović, d., 2012, measurement of thermal transmittance in the rectorate building of the university of banja luka, proceedings of international scientific conference contemporary theory and practice in civil engineering, banja luka, bosnia and herzegovina, pp. 37-46. 3. iso 9869:2014-1, thermal insulation building elements in-situ measurement of thermal resistance and thermal transmittance, international organization for standardization, geneva, switzerland. 4. nardi, i., sfarra, s., ambrosini, d., 2014, quantitative thermography for the estimation of the u-value: state of the art and a case study, journal of physics: conference series, 547(1), doi: 10.1088/1742-6596/547/1/012016. 5. albatici, r, tonelli, a.m., 2010, infrared thermovision technique for the assessment of thermal transmittance value of opaque building elements on site, energy and buildings, 42(11), pp. 2177-2183. 6. albatici, r., tonelli, a.m., chiognac, m., 2015, a comprehensive experimental approach for the validation of quantitative infrared thermography in the evaluation of building thermal transmittance, applied energy, 141, pp. 218-228 7. grinzato, e., bison, p., cadelano, g., peron, f., 2010, r-value estimation by local thermographic analysis proceedings of thermosense xxxii, orlando, usa, doi:10.1117/12.850729. 8. iso 12567-1:2010, thermal performance of windows and doors determination of thermal transmittance by the hot-box method part 1: complete windows and doors, international organization for standardization, geneva, switzerland. 9. iso 12567-2:2005, thermal performance of windows and doors determination of thermal transmittance by the hot box method part 2: roof windows and other projecting windows, international organization for standardization, geneva, switzerland. 10. cucumo, m., de rosa, a., ferraro, v., kaliakatsos, d., marinelli, v., 2006, a method for the experimental evaluation in situ of the wall conductance, energy and buildings, 38(3), pp. 238-244. 11. asdrubali, f., baldinelli, g., 2011, thermal transmittance measurements with the hot box method: calibration, experimental procedures, and uncertainty analyses of three different approaches, energy and buildings, 43(7), pp. 1618–1626. 12. rasooli, a., itard, l., ferreira, c.i., 2016, a response factor-based method for the rapid in situ determination of wall’s thermal resistance in existing buildings, energy and buildings, 119, pp. 5161. 13. schild, k., willems, v.m., 2006, building physics handbook part 1, friedrich vieweg & sohn verlag, wiesbaden, germany, 2.21. 14. min, t.c., schutrum, l.f., parmelee, g.v., vouris, j.d., 1956, natural convection and radiation in a panel heated room, heating piping and air conditioning (hpac), 62, pp. 337-358. 15. awbi, h.b., hatton, a., 1999, natural convection from heated room surfaces, energy and buildings, 30, pp. 234244. 16. khalifa, a.j.n., marshall r.h., 1990, validation of heat transfer coefficients on interior building surfaces using a real-sized indoor test cell, international journal of heat and mass transfer, 33, pp. 2219-2236. 17. michejev, m. a., 1952, základy sdílení tepla, průmyslové vydavateľství, prague, czechoslovakia, p. 387. 18. king, w., 1932, the basic laws and data of heat transmission, mechanical engineering, 54, pp. 347–353. 19. nusselt, w., 1915, das grundgesetz des wärmeüberganges, gesundheits-ingenieur, 38(42), pp. 477–482. 20. heilman, r.h., 1929, surface heat transmission, mechanical engineering, 51, p. 355. 21. wilkers, g.b., peterson, c.m.f., 1938, radiation and convection from surfaces in various positions, ashve transactions, 44, p. 513. alternative method for on site evaluation of thermal transmittance 351 22. ashrae (american society of heating, refrigerating and air-conditioning engineer), 2001, ashrae handbook, fundamentals, american society of heating, refrigerating and air-conditioning engineers, atlanta, usa. 23. hens, h., 2007, building physics – heat, air and moisture, enrst & son, berlin, germany, 73 pp. 24. binachi, f., baldinelli, g., asdrubali, f., 2014, a quantitative infrared thermography method for the assessment of windows thermal transmittance, proceedings of latest trends in applied and theoretical mechanics, salerno, italy, pp. 137 – 143. 25. antunović, b., janković, a., preradović, lj., 2015, measurement of thermal transmittance of opaque facade wall and relationship with meteorological conditions, tehnika, 70(4), pp. 593 – 598. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 175 185 https://doi.org/10.22190/fume201221005p © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper shape of a sliding capillary contact due to the hysteresis of contact angle: theory and experiment valentin l. popov1,2, iakov a. lyashenko1,3, jasminka starcevic1,2 1technische universität berlin, germany 2national research tomsk state university, russia 3sumy state university, ukraine abstract. we consider a classical problem of a capillary neck between a parabolic body and a plane with a small amount of liquid in between. in the state of thermodynamic equilibrium, the contact area between the bodies and the liquid layer has a circular shape. however, if the bodies are forced to slowly move in the tangential direction, the shape will change due to the hysteresis of the contact angle. we discuss the form of the contact area under two limiting assumptions about the friction law in the boundary line. we also present a detailed experimental study of the shape of sliding capillary contact in dependence on the roughness of the contacting surfaces. key words: capillarity, contact angle hysteresis, friction, contact area, roughness 1. introduction since coulomb, it is known that the force of dry friction may depend on humidity of the atmosphere [1, 2]. as a physical reason for this dependency, formation of capillary bridges is considered [3], because in a complete cycle of formation and destruction of a capillary bridge a certain amount of energy is dissipated. however, even if the capillary bridges are not destroyed, they can contribute to friction due to the dry friction experienced by the boundary line of a capillary bridge. the friction in the boundary line leads to the well-known effect of contact angle hysteresis [4]. as the contact angle determines the height of a capillary bridge, the contact angle hysteresis will lead to some deformation of the contact area. contact angle hysteresis is studied intensively both theoretically [5, 6] and experimentally [7-10]. heterogeneity of the surface is considered as the main physical reason for contact hysteresis. in [5], a liquid drop was considered on a porous surface, the pores of which received december 21, 2020 / accepted january 20, 2021 corresponding author: valentin l. popov affiliation: technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: v.popov@tu-berlin.de 176 v.l. popov, i.a. lyashenko, j. starcevic were filled with another liquid. this situation is observed in lubricated systems in the presence of heterogeneous liquid inclusions in the lubricating layer. in [7], the effect of roughness on the hysteresis of the contact angle and on the work performed when a drop spreads over a rough surface is studied. it was found that these quantities nonlinearly depend on the roughness amplitude. it should be noted that rms roughness was used for characterizing the roughness, which does not determine the properties of real fractal surfaces. in [8], the presence of a pronounced hysteresis of the normal force and contact angle in liquid bridges between two identical surfaces was experimentally shown when they approached and moved away in the normal direction at different speeds, from 0.02 to 2 mm/s. the review article [11] provides a brief description of the theoretical and experimental results on this topic. in [6, 10], various situations are considered in which a drop of liquid slides along an inclined plane, due to which different contact angles are observed at the leading and trailing fronts of motion. similar situation is considered in the present work, with the difference that this is not a drop but a capillary bridge between two solids which is sliding. in the present paper, following the preprint of one of the authors, [12], we present a simple theory describing the shape of the sliding capillary bridge and provide a detailed experimental study of the contact angle deformation due to contact angle hysteresis. 2. basic geometry of a capillary bridge we consider a rigid sphere with radius r in contact with a solid surface with a small amount of a liquid in between (fig.1). fig. 1 a capillary bridge between a rigid plane and a rigid sphere in equilibrium, the liquid forms a capillary bridge, which has two radii of curvature (fig. 1). the larger radius r is always positive. the sign of the smaller radius r0 depends on whether the contact angle is larger or smaller than π/2. for small contact angles, in the case of wetting of the surface, r0 is negative. there is a reduced pressure in the liquid, which leads to a force that we call capillary force. in the most cases one can assume |r0| << r, the differences between the pressure inside and outside the contact can be written as [4] 0 p r   = − , (1) shape of a sliding capillary contact due to the hysteresis of contact angle: theory and experiment 177 where γ is the surface tension of the liquid. note that in the equilibrium or at slow (quasistatic) sliding, the pressure in the liquid should be constant. this means, that the radius of curvature r0 should be constant along the whole boundary line of liquid. fig. 2 geometry of a capillary bridge fig. 2 displays the geometry of the contact angles and meniscus of a capillary bridge. it can be easily seen that the following geometrical relation is valid: 0 1 2(cos cos )r h + = . (2) the contact angles can be obtained from the equilibrium conditions for the boundary line, considering that each interface is acting on the boundary with a linear force density equal to the specific interface energy [4], see fig. 3: 1, 1, 1coss l   = + , (3) 2, 2, 2coss l   = + , (4) which gives for the contact angles 1, 1, 1cos s l    − = , (5) 2, 2, 2cos s l    − = . (6) fig. 3 scheme of tension forces acting at the boundary of a liquid bridge. γ1,l and γ1,s are the surface energies of the first solid in contact with liquid and the atmosphere, correspondingly. similar notations are valid for the second body. γ is the surface tension of liquid eqs. (5) and (6) are valid in thermodynamic equilibrium. if the boundary of the bridge is moving, it can experience a force of dry friction similar to that described for adhesive contacts in [13]. in the present paper, we will for simplicity assume that such dry friction 178 v.l. popov, i.a. lyashenko, j. starcevic force does act only on the boundary of a fluid in contact with body 1. that means that the second body is considered as being very smooth and chemically homogeneous, so that the contact angle hysteresis on its surface can be neglected. in the presence of friction force, eq. (3) will take the form 1, 1, 1coss l f   = +  , (7) where f is the linear density of the force of friction. the sign “+” should be used if the right boundary of liquid is moving to the right and “–” if it is moving to the left. for the left boundary, the use of sign is inverse to the above stated. from eq. (7), it follows 1, 1, 1cos s l f    − = . (8) substituting eqs. (1), (6) and (8) into eq. (2) gives * 1, 2, 1, 2,s s l l f f h p p     + − − = =   (9) with * 1, 2, 1, 2,s s l l    = + − − . note that eq. (7) and the following eq. (9) do not depend on whether the contact angle is larger or smaller π/2. 3. shape of the contact area for friction perpendicular to the boundary line the force of friction acting on the boundary line may depend on the nature of this force. two limiting cases can be considered. if the microscopic heterogeneity is moderate, then the movement of the liquid in the direction of the boundary line will occur without instabilities. however, the appearance of instabilities is a necessary precondition for appearance of static force of friction [13]. this means, that the movement along the boundary does not lead to friction. the movement in the direction perpendicular to the boundary, on the contrary, will always lead to appearance of instable configurations followed by rapid jumps and energy dissipation. on the macroscopic scale, this energy dissipation is perceived as being due to static force of friction. under the conditions described, the force of friction is always directed perpendicularly to the boundary, as shown in fig. 4. moreover, the magnitude of the force of friction does not depend on the velocity, so that the force line density is constant in all boundary points. however, its direction changes abruptly at the side points of the contact where the direction of movement is parallel to the boundary line. from eq. (9), it follows that the height of the gap on the front side of contact is constant and equal to * front f h p  − =  , (10) while at the back side it is * back f h p  + =  . (11) in the following, we consider a contact between a parabolic body z (r) = r2/(2r), where z is the normal coordinate and r polar radius in the contact plane. for such body, eqs (10) shape of a sliding capillary contact due to the hysteresis of contact angle: theory and experiment 179 and (11) mean that the contact area has constant (but different) radii on the front and back side of the contact, fig. 4b: * 0,front 2 f a r p  − =  , * 0,back 2 f a r p  + =  . (12) fig. 4 (a) friction forces acting on the boundary of liquid when the capillary contact is moving to the right; (b) shape of the contact area in the case of constant friction force directed perpendicular to the boundary of liquid note that the pressure difference δp can change during the movement to maintain the given volume of interfacial liquid. however, the change of volume at a constant δp is zero in the linear approximation, so that if the force of friction is small, then the pressure difference can be considered as unchanged. in the same approximation, the total force of friction can be evaluated as /2 friction 0,front 0,back 0,front 0,back 0 2 ( ) cos d 2( )f a a f a a f   = + = + . (13) definition of the angle φ is given in fig. 5. for small friction force, * 0,front 0,back 0 2 2 a a a r p +  =  (14) and the total friction force can be obtained in the first order by integrating the force density over the unperturbed shape of the contact (circle): friction 04f a f= . (15) 180 v.l. popov, i.a. lyashenko, j. starcevic 4. shape of the contact area for friction opposite to the direction of motion principally thinkable is also another law of friction for the boundary line. if the microscopic heterogeneity of the substrate becomes large, this will lead to comparable changes in the configuration of the boundary line, independently on the direction of its movement. this means, that the force of friction will be directed oppositely to the direction of motion, as shown in fig. 5a. however, for the contact angle, only the component of the friction force perpendicular to the boundary line is relevant. this component is equal to cosf f ⊥ = . (16) fig. 5 case of friction opposite to the direction of motion: (a) forces acting on the boundary line, (b) shape of the contact area (dashed line) eqs. (12) change to * 0 cos ( ) 2 f a r p    − =  . (17) this dependency is shown in fig. 5b. while the shape of the contact is asymmetrical also in this case, the asymmetry is barely visible and manifests itself mostly in the shifting the contact area from the center of the body in the direction opposite to the direction of sliding. the force of friction can be estimated as friction 02f a f= , provided that the deviation of the shape of the boundary from the circle is not too large. 5. experiment experiments have been conducted using the setup depicted in fig. 6. steel spheres having various radii of curvature were moved vertically and tangentially with precision linear stages m-403.2dg. for recording normal and tangential forces, a strain gage sensor me k3d40 was mounted between the sample and the moving stages. the contact region was illuminated from the side with 80 leds and was recorded from underneath shape of a sliding capillary contact due to the hysteresis of contact angle: theory and experiment 181 using a digital camera with resolution of 1600 x 1200 pixels. an inclination mechanism was used to ensure parallel orientation between the glass surface and tangential indenter movement. in each experiment, the following operations have been performed. first, a drop of liquid was placed on a smooth sheet of a silicate glass. then, an indenter of radius r approached the glass plate in the normal direction with a small velocity of v = 10 µm/s. the indenter was moved towards the glass as close as possible, but direct contact between the indenter and the glass was avoided. after finishing the approaching phase, indenter was moved in the tangential direction with velocity 10 µm/s by a distance of x = 10 mm. subsequently, it was moved in the opposite direction to the initial position x = 0 mm. cow's milk with 3.5% fat was used as liquid. it was chosen because milk is not transparent, which made it possible to accurately observe the evolution of contact boundary and contact form during sliding. as indenters two spherical bodies with radii r = 100 mm and r = 33 mm have been used. since the indenter was not in direct contact with the substrate, the normal force fn was negative due to capillary effects. it was approximately –15 mn for indenter with r = 33 mm and –40 mn for the indenter with radius r = 100 mm. fig. 6 experimental set-up for observing contact configuration of a capillary bridge [14, 15]. left panel – general view, right panel – contact between steel indenter and substrate, with lighting system a series of 5 experiments was performed for each indenter. in the first experiment, a smooth glass was used. in three subsequent experiments, the glass had been roughened by machining with sandpaper having numbers p180, p80 and p40. these numbers, according to the standard classification, correspond to the average sandpaper grain size of 82, 201 and 425 µm, respectively. the last (fifth) experiment was carried out after an even more 182 v.l. popov, i.a. lyashenko, j. starcevic intensive processing of the glass surface with p40 paper, which further increased the roughness value. the results obtained in the last fifth experiment will be denoted as “p40+”. results for indenter with radius of curvature r = 33 mm in this series, a stainless-steel spherical indenter with a radius of r = 33 mm was used. the surface of the indenter was polished to a mirror finish. fig. 7 shows the dependence of the experimentally measured normal and tangential contact forces on time in all five experiments of a series. in both panels of the figure, three vertical dashed lines are introduced, delimiting various stages of the experiment. before the first dashed line, the indenter approaches the glass substrate in the normal direction. displacement in the tangential direction takes place between the first and the second lines. after the second vertical line, the indenter starts moving in the opposite direction. the third line marks the moment of time when the indenter stops moving horizontally and is lifted to the initial position. from fig. 7b, it follows that the tangential force fx remains almost zero over the entire time range. in any case, it is smaller than the random background noise of the force sensor used. the normal force fn, on the contrary, can be clearly resolved and takes negative values due to capillary effects. in the experiment, the glass substrate is not perfectly flat, and the indenter cannot be shifted ideally parallel to the substrate surface. therefore, changes in the force fn during tangential motion up to the second vertical line (before changing the direction of movement of the indenter) can be associated with a change in the distance between the indenter and the glass. however, a more detailed analysis of the dependences in fig. 7a shows that the magnitude of the normal force, |fn|, is smaller after changing the direction of movement. this effect cannot be explained by the changing distance between the glass plate and the indenter since the reverse movement of the indenter in the tangential direction occurs along the same trajectory as its forward movement. to find out the reasons for this behavior, it is necessary to analyze the behavior of the observed contact area. fig. 7 dependences of the normal fn (a) and tangential fx (b) forces acting on the indenter in the experiment with an indenter with a radius of r = 33 mm on time t the contact area versus time for the case of a smooth glass substrate is shown in fig. 8, left panel. the figure shows both the full area and the area to the right and to the left from the center line of indenter (visualized by a vertical dashed line in fig. 8, right panel). the right panel of fig. 8 shows the snapshots of the contact shape at the time shape of a sliding capillary contact due to the hysteresis of contact angle: theory and experiment 183 moments marked with letters in the left panel. it is clearly seen that the area of the sliding contact is divided into regions with smaller radius (front part of the contact area) and a larger radius (back part) with a rapid transition between both parts. this strongly resembles the predicted shape shown in fig. 4. this means, that the correct assumption is that friction force is directed perpendicular to the border line and has a jump at the "turning point". however, experiment shows details which are not predicted, and which have to be mentioned and discussed. fig. 8 left panel: time dependencies of the full area of the contact, as well as the area to the left and to the right from the center line of the indenter. right panel: photographs of the contact area corresponding to the time moments marked with letters in the left panel first, note that the total area of the contact decreases with time. this is due to the fact that some portion of the liquid remains behind the actual capillary bridge. these rests are visible by the changed color of the glass. further, attentive consideration of videos [16, 17] shows that the changes in shape of the front part of the bridge occur not at once. for some time, the bridge keeps the circle shape before it starts to reshape. at the border of the initial contact area, a distinct track remains, suggesting that we have here with a sort of pinning. it looks that we have to do with a static (pinning) force which is higher than the subsequent boundary force during sliding. further, there exist a distinct difference between the shape of the capillary bridge during the initial sliding and the "back sliding". in initial sliding, the movement of the front boundary (fig. 8, right upper panel) occurs much jerkier, but at the same time, the deviation from the circle shape is smaller than by the back movement. quite obviously, this is related to the fact, that the surface properties change after the first passing of the drop. interestingly, the jerky movement is not correlated with the boundary friction force, judging by the asymmetry of the contact shape. thus, the friction force is in this case caused not by the macroscopic roughness or heterogeneity but is determined by a lower, microscopic, scale. this is confirmed by the fact, that the roughness changes substantially the well visible mesoscopic heterogeneity of the movement of the border line (which is in the case of rougher surfaces much jerkier) but practically does not influence the overall appearance of the contact shape. 184 v.l. popov, i.a. lyashenko, j. starcevic video [16] shows in detail the movement of the contact region in all 5 experiments with different roughness of the glass substrate for an indenter with radius r = 33 mm. video [17] contains similar data for an indenter with radius r = 100 mm. comparison of results of all experiments allows to conclude that the roughness and the indenter radius do not qualitatively change the behavior described above (fig. 8). the main effect of the increasing roughness is that the propagation of the contact becomes more discontinuous. the main effect of the radius of curvature is that in the case of a larger radius r = 100 mm, at the initial approach of the indenter to the glass, a drop of liquid often spreads nonuniformly. however, after some transient regime the stationary sliding resembles the situation shown in fig. 8. a detailed analysis of all details of the kinetics and properties of the stationary sliding will be published elsewhere, but the data of all experiments are already available in [16, 17]. 6. conclusions we considered the changes in the shape of the contact area of a capillary bridge when it is sliding in tangential direction. due to the friction force acting in the boundary, the contact area becomes asymmetric. two limiting laws of friction have been considered: (a) with force of friction acting perpendicular to the boundary line and (b) opposite to the direction of sliding. the predicted shapes are substantially different. this means that experimental study of the shape of capillary bridges can give information about the character of the friction law in the boundary line. experimental investigation confirms a jump-like shape for smoother indenters and more continuous for rougher indenters which mean that there may be a transition between the laws of friction (a) and (b) with increasing roughness. in the present paper, we assumed that one solid surface exhibits no friction with the liquid. this assumption can be dropped as in the general case all theoretical results remain the same with f = f1 + f2. acknowledgement: this research was partially supported by “the tomsk state university competitiveness improvement program” and by deutsche forschungsgemeinschaft (project dfg po 810-55-1). we thank one of the reviewers who made us aware that the assumption that one of the solid surfaces exhibits no friction can be dropped. references 1. bowden, f. p., tabor, d., 2001, the friction and lubrication of solids, clarendon press. 2. popova, e., popov, v.l., 2015, the research works of coulomb and amontons and generalized laws of friction, friction, 3(2), pp. 183-190. 3. persson, b.n.j., 2002, sliding friction. physical principles and applications, springer. 4. popov, v.l., 2017, contact mechanics and friction. physical principles and applications, 2nd edition, berlin: springer. 5. semprebon, c., mchaleb, g., kusumaatmaja, h., 2017, apparent contact angle and contact angle hysteresis on liquid infused surfaces, soft matter, 13(1), pp. 101-110. 6. gao, l., mccarthy, t.j., 2006, contact angle hysteresis explained, langmuir, 22, pp. 6234-6237. 7. junchao, w., wu, y., cao, y., li, g., liao, y., 2020, influence of surface roughness on contact angle hysteresis and spreading work, colloid and polymer science, 298, pp. 1107-1112. 8. shia, z., zhang, y., liu, m., hanaor, d.a.h., gan, y., 2018, dynamic contact angle hysteresis in liquid bridges, colloids and surfaces a: physicochemical and engineering aspects, 555, pp. 365-371. shape of a sliding capillary contact due to the hysteresis of contact angle: theory and experiment 185 9. heib, f., schmitt, m., 2016, statistical contact angle analyses with the high-precision drop shape analysis (hpdsa) approach: basic principles and applications, coatings, 6(4), 57. 10. wu, c.-j., li, y.-f., woon, w.-y., sheng, y.-j., tsao, h.-k., 2016, contact angle hysteresis on graphene surfaces and hysteresis-free behavior on oil-infused graphite surfaces, applied surface science, 385, pp. 153-161. 11. hubbe, m.a., gardner, d.j., shen, w., 2015, contact angles and wettability of cellulosic surfaces: a review of proposed mechanisms and test strategies, bioresources, 10(4), pp. 8657-8749. 12. popov, v.l., 2020, shape of a sliding capillary contact, arxiv:2012.13979, http://arxiv.org/abs/2012.13979 13. popov, v.l., adhesion hysteresis due to chemical heterogeneity. in: multiscale biomechanics and tribology of inorganic and organic systems, eds. ostermeyer, popov, shilko, vasiljeva, springer, 2021, pp. 473-483. 14. lyashenko, i.a., pohrt, r., 2020, adhesion between rigid indenter and soft rubber layer: influence of roughness, frontiers in mechanical engineering. section tribology, 6, 49. 15. lyashenko, i.a., popov, v.l., 2020, the effect of contact duration and indentation depth on adhesion strength: experiment and numerical simulation, technical physics, 65(10), pp. 1695-1707. 16. popov, v.l., lyashenko, i.a., starcevic, j., 2021, shape of a sliding capillary contact due to hysteresis of contact angle (experiment with indenter r=33mm), supplementary video. https://doi.org/10.13140/rg.2. 2.13032.49923 17. popov, v.l., lyashenko, i.a., starcevic, j., 2021, shape of a sliding capillary contact due to hysteresis of contact angle (experiment with indenter r=100mm), supplementary video. https://doi.org/10.13140/rg. 2.2.33165.15840 http://arxiv.org/abs/2012.13979 https://doi.org/10.13140/rg.2.2.13032.49923 https://doi.org/10.13140/rg.2.2.13032.49923 https://doi.org/10.13140/rg.2.2.33165.15840 https://doi.org/10.13140/rg.2.2.33165.15840 11600 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240228006g © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper automation of design of modular upper limb prostheses filip górski1, olga komorowska1, przemysław zawadzki1, wiesław kuczko1, magdalena żukowska1, remigiusz łabudzki1, rǎzvan pǎcurar2 1poznan university of technology, faculty of mechanical engineering, poland 2technical university of cluj-napoca, faculty of industrial engineering, robotics and production management, romania abstract. this paper presents the results of automation of the design of a modular upper limb prosthesis. the process of modernization of the existing, standard cad model in the autodesk inventor program was described, introducing structural changes and enriching the geometric form with the knowledge of the design process using the tools of the ilogic module. in addition, the cad model was combined with special design tables, thus obtaining a kbe class solution. the result is a special, intelligent generative cad model that allows for the automation of the process of designing various variants of prostheses, tailored to the patient's anthropometric characteristics. the results of the work were then integrated with the automedprint system developed at the poznan university of technology, thanks to which it was possible to test the operation of the developed solution on real data. based on the measurement results from the 3d scanning process, various variants of the modular prosthesis were automatically prepared for three patients. the final task described in the work is the process of manufacturing the selected variant of the prosthesis in the additive technique. the results of fitting the prosthesis and the opinion of the patient were also presented. key words: modular design, limb prosthesis, design automation, kbe system, generative cad model 1. introduction upper limb prosthetics are crucial in restoring function after limb loss due to injury, disease, or congenital issues [1]. amputation can occur at various levels, influencing treatment, rehabilitation, and prosthesis design [2-4]. the upper limb's complexity, with its received: february 28, 2024 / accepted january 31, 2025 corresponding author: filip górski poznan university of technology, faculty of mechanical engineering, piotrowo 3, 60-965, poznań, poland, e-mail: filip.gorski@put.poznan.pl 2 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. many degrees of freedom and ability to perform intricate movements, requires prostheses to address basic daily, professional, and social tasks [5]. depending on whether full or partial functionality is needed, prostheses can be passive (cosmetic, functional) or active (body-powered, externally powered, hybrid) [6,7]. however, a single prosthesis often cannot meet all patient needs, leading to the use of multiple types. the most effective in this area seems to be special modular passive prostheses examples are shown in fig. 1. fig. 1 examples of modular passive upper limb prosthetics modular prostheses are known worldwide but are still not widely used, and research on their design and production is limited. production of such products is still very complex, time-consuming, and expensive. it is usually done manually [8], although the use of methods and techniques known from other fields of engineering, such as intelligent cad models, reverse engineering, or additive manufacturing, is becoming more common. the effectiveness of the traditional approach to design can be increased by using the generative cad models which are enriched with a description of knowledge in the form of structured rules, relations, parameters, algorithms, and conditional instructions [10]. therefore, generative models are very often the basic element of kbe (knowledge based engineering) systems, which provide support in making design decisions, allow you to shorten the design time, and minimize construction [10-12]. issues related to the construction of generative cad models and kbe systems, both in theory and in practice, are widely described in the literature. most of these works, however, focus on the area of mechanics, which is of course justified. examples can be found in the works of [13-15]. nevertheless, works in this field have been the subject of research in the field of prosthetics and orthotics for some time [16-20]. generative cad models are also used in other sectors of medicine, such as implantology, dealing with the production of fitted cranial and craniofacial implants [21-24]. reverse engineering has become an important aspect of innovative prosthetics. these techniques enable the digitization of real objects, based on data obtained using selected imaging techniques [25, 26]. the most popular are: 3d scanning, computed tomography (e.g. when a morphological record of a body part is desired), or optical motion capture automation of design of modular upper limb prostheses 3 systems. they make it possible to visualize the surface of the stump along with its volume, omitting the complicated operations related to making casts [27-30]. the use of additive manufacturing is more and more often used in the biomedical sector, offering printouts in the field of prosthetics and orthotics, preoperative planning, or implantology. currently, prosthetic supplies are manufactured using technologies such as fdm (fused deposition modeling), sls (selective laser sintering), sla (stereolithography), and polyjet (photopolymer jetting) [31]. based on the analysis of the literature and various projects related to additively manufactured prosthetics, the tendency of fdm technology to dominate was identified. despite the relatively large thickness of cross-section layers and difficulties in the production of geometrically complex products, it is characterized by the use of low-cost equipment, simplicity, a wide range of materials, and usually no need for complicated post-processing [32-33]. this work proposes a method that integrates the above technologies to popularize modular prostheses manufacturing, addressing individual patient needs. the goal was to develop a generative cad model to automate and streamline the production of specialized prostheses. though prostheses share general geometric features, they must be tailored to individual anthropometric characteristics. key steps included developing a kbe system, parametric cad models, and integrating 3d scanning data. the innovation lies in automating design, enabling rapid production of anatomically individualized devices. this process minimizes production costs by replacing only necessary modules. 2. automation of design of modular upper limb prostheses 2.1 aim and scope of the work the approach proposed and described in this work, integrating several well-known technologies, may contribute to the popularization of the use of modular prostheses, while taking into account the individual requirements of the patient. the primary purpose of the work was to develop a generative cad model of modular upper limb prostheses to shorten and facilitate the process of manufacturing specialized prostheses by automated design tasks. variants of the prostheses are characterized by a high similarity in geometric features in general, however, due to the individual anthropometric features of the patient, they must be properly matched to them. to meet these assumptions authors assumed the following work plan:  development of the concept of the kbe system;  analysis of the structure of the modular prosthesis of the upper limb;  development of new, parametric cad models;  development of databases for processing data from the 3d scanning process;  integration and verification of the kbe system in practice. the greatest novelty of the presented prosthesis and the new contribution is the obtained level of automation of design. it has been proven that the prosthesis model is capable of quick switching between variants for different patients, which brings forth uncanny possibilities of rapid production of fully anatomically individualized devices, unknown in existing literature. 4 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. 2.2 idea of the kbe system for modular upper limb prostheses design the developed kbe system shown in fig. 2 consists of the following components: a database with design tables; intelligent, parametric cad models of prosthesis components; and vba (visual basic for applications) scripts for automated particular design tasks. all these elements are an integral part of the modular prosthesis generative cad model. the input data, in the form of mesh models obtained in the 3d scanning process, came from patients participating in the project automedprint. it should be noted that both amputation and healthy limbs were scanned, which were to be valuable sources of data to generate a variant of the new prosthesis. the generative cad model of the prosthesis was developed in the autodesk inventor professional 2022 software. the construction of such models is supported by tools from the ilogic module, in this particular case design tables and the vba scripting language. the parameterization of the designed model was developed using a file created in microsoft excel. a data structure template has been prepared in the program sheets, facilitating the interpretation of input data and performing their transformation to the form required by parametric models of prosthesis parts. excel also included some design rules that allowed for automatic processing. fig. 2 structure of developed kbe system 2.3 basic framework of a modular upper limb prosthesis the prosthesis model selected for the work consists of 3 types of main components: prosthetic socket, forearm, and hand. the structure of the model is shown in fig. 3. each of them can exist in several different variants that can be combined in various configurations. the modular prosthesis also includes auxiliary and connecting parts: pseudo-cardan coupling, acting as a movable wrist joint, an assembly adapter, and an elbow module, imitating flexion and extension in the frontal plane. the list of prosthesis parts is presented in table 1. automation of design of modular upper limb prostheses 5 fig. 3 structure of a modular upper limb prosthesis table 1 basic elements of modular upper limb prosthesis prosthetic socket prosthetic forearm prosthetic hands connecting and auxiliary elements compression and relaxation socket crs (4 variants) open forearm c-handle cross joints open funnel (2 variants) open forearm with a tip dedicated to the adapter straight fixed handle adapter (2 variants) semi-open socket (3 variants) partly closed forearm with a tip dedicated to the adapter fixed angular handle external model of the elbow crs socket for forearm amputation closed forearm with a tip dedicated to the adapter straight handle with a spring blocking part of the elbow joint the base model analyzed in this step was the result of previous work of the automedprint project team, however, its form was not suitable for carrying out design automation tasks. therefore, for all elements indicated in table 1, parametric cad models had to be prepared and linked to an excel sheet in which the results of anthropometric measurements would be placed. 2.4 development of generative cad models of prosthesis components 2.4.1 prosthetic socket modifying the base cad model was aimed at linking its relevant parameters with data stored in excel sheets. in addition, the basic prosthetic socket model (crs compression/release stabilized) had to be extended with additional variants, including an open socket, a partly open socket, and a crs type for forearm amputations. chosen modifications are presented in fig. 4. modifications had to be carried out so that, in addition to adapting to the selected values of the relevant measurement data, it could also smoothly cooperate with the forearm model. 6 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. fig. 4 chosen modifications of the prosthetic socket the most important modifications include the introduction of e.g. parameters and affecting the width of the location of the connecting elements with the forearm. this is measured from the plane of symmetry of the funnel, parallel to the xz initial plane and passing through the point determining the center of the farthest section. internal assembly of the socket relating to the forearm is usually used in the case of the youngest patients, therefore a separate parameter was introduced to determine the age of the patient. another important modification was the addition of new socket variant geometries, improving the accuracy of the stump anatomy. an intelligent sketch of the universal fixation was also made, which ensured the possibility of adapting these geometric features when changing the input basic dimensions and changing the shape of the stump. as an alternative, a type of closed fastening has been prepared. there are also slots for mounting belts. 2.4.2 prosthetic forearm the base model of the forearm is created by pulling between two sketches, the details of which are sourced from the excel table (the first one depends on the prosthetic socket, and the second on the prosthetic hand). in order to build an appropriate connection of the forearm with a part of the hand, a parameter was prepared to automatically generate a second section of this extrusion proportional to the wrist. as a result, both parts were interconnected. the forearm model was also enriched with appropriate parameters, matching and binding them to new connecting elements. the individual geometric features of the forearm model, such as the spacing of the connection or the cutout in the bottom wall have been described so that they change their size proportionally with the change in automation of design of modular upper limb prostheses 7 the length of the component. the operations of creating assembly elements from the prosthetic socket have been modified and rules controlling the dimensions of the assembly hole have been added. edge fillets were described by rules so that their values changed proportionally to the size of the entire prosthesis because constant values caused errors in the dimensional reconstruction of the model (too large radius values on small surfaces). selected changes are shown in fig. 5. appropriate changes were made one by one by making the next 3 variants of the forearm, taking into account the installation of transition elements. they were: open variant with the special ending, closed variant with the empty interior, and completely closed variant. all variants have a geometry for mounting the connecting element. fig. 5 chosen modifications of the prosthetic forearm 2.4.3 universal elbow joint the first additional element of the modular prosthesis was a new, universal elbow joint, using springs and a splined lock, as presented in fig. 6. the mechanism was placed in a closed base (external or internal part of the prosthetic socket). the base for the mechanism was made on the basis of a circle with a diameter described by parameters so that it automatically adapted to the dimensions of the selected mounting on the funnel model. the length of the element was set to the standard distribution between the fasteners. the next steps required drilling holes to attach the inner element and spring, and to connect the component to the hands and gimbals. in the execution of the mounting holes (for joints and 8 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. hands), an additional parameter describing the diameter was used, which can be manipulated in the design table depending on the designer's preferences and the size of the adapter. fig. 6 geometric features of universal elbow joint in order to enable correct position locking, appropriate features have been added to each model of forearms, enriched with rules describing the conditions of their occurrence. changing the value of specially prepared parameters in the forearm model automatically generates additional geometrical features that enable the connection of the forearm with the base of the elbow. these include, among others: lowering the surface of the upper end of the funnel, enlarging the proper mounting surface, creating a hole for the screw, a cutout for the guides, shaping extrusion for blocking, etc. the dimensions of all operations were adjusted based on the activities performed during the design of the prosthesis's basic geometry and the parameters introduced at that time. an analogous rule was also applied to prosthetic sockets, automatically changing the cutouts for assembly to the base of the mechanism. 2.4.4 prosthetic hand and universal hand holder the work consisted of introducing modifications in the structure of the models by adding the selected 4 variants of the prosthetic hand, as shown in fig. 7. the improvement of the gripping parts was made in terms of the possibility of parametric adjustment of the component width, introducing parameters related to the width of the hand. these values can be determined manually, but they can also come from measurements of the patient's healthy limb (also a 3d scan). rules have been added to hand models to automatically select mounting holes, and the connection plane has been linked by rules to the appropriate universal prosthetic hand holder plane. thanks to this, each geometric variant of the hand creates a coherent whole with it. automation of design of modular upper limb prostheses 9 fig. 7 variants of parametric 3d models of prosthetic hand another element prepared especially for the needs of the modular hand prosthesis was a special hand holder, as presented in fig. 8, allowing for the free transfer of the gripping parts between cardan joints and forearms, without the use of additional tools. the previous solution was designed "rigidly", which was a significant limitation because, in the case of different variants of the prosthesis, it usually required designing the adapter from scratch. the new model was enriched with specially described parameters so that the source of their data was measurements made on 3d scans. fig. 8 universal hand holder 2.4.5 cross joints the cross joint depicted in fig. 9 is composed of three components and serves to connect sections of the forearm and the hand. modifications of universal joints consisted of changing the features describing the base sketches and adjusting the mounting surfaces to the initial surface of the forearm and hand. the geometry of the hinge is the result of the relationship between the individual parameters of the model each of the solids is dependent on the next one to form an integral whole. this is especially visible in the case of the central body, which is closely related to the width of the other two elements, which depend on the parts of the forearm and hand cooperating with them. fig. 9 cross joint 10 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. 2.4.6 design table and design rules the design table storing and processing anthropometric data from the scanning process was prepared in excel. a partial view of this table is shown in fig. 10. the table has been linked to the relevant parameters of the modular upper limb prosthesis models. the main sheet is the "variants" sheet, which controls the selection of the prosthesis variant and thus manages the construction of cad models. the idea of the entire system assumed the preparation of a model tailored to the patient's needs without the need to use a cad program. the "variants" sheet is therefore a kind of user interface. the remaining sheets are closely related to the collection of data from the process of scanning the patient's limbs. the relationships between them are intended to calculate the final values, which are then read by the cad program and the related model parameters. fig. 10 partial view of the design table for modular upper limb prosthesis management the connection of the excel file with the cad assembly model was made using autodesk inventor ilogic tools. a number of design rules have been prepared to automate the process of rebuilding cad models. the design rules contained a series of conditional instructions on the basis of which the program made decisions on how to update a given model without the designer's participation. the rules concerned, for example, the conditions of visibility of models and features or the choice of the type of relationship between the components of the prosthesis model. 2.4.7 assembly model of modular prosthesis in the last step, the design table, prepared design rules and parametric 3d cad models of prosthesis components shown in fig. 11 were combined. as a result, an intelligent modular assembly model of the prosthesis as shown in fig. 12 was created and it is the final representation of the kbe system. generating a prosthesis variant for a new patient should start with completing the sheets storing data from the patient's measurements. however, generating another variant of the prosthesis requires only a change in the main "variants" sheet. the cad model rebuilds automatically, creating designs tailored to the individual dimensions and needs of a given patient. automation of design of modular upper limb prostheses 11 fig. 11 some of the geometrical variants of prosthesis components fig. 12 assembly model of modular upper limb prostheses 3. verification of the system for automation of prosthetic design 3.1 plan of verification process verification of the operation of the kbe system was carried out in the process of manufacturing various variants of prostheses for three patients. the research plan presented in fig. 13 included the process of 3d scanning of the patient, data entry in the design table, selection of the prosthesis variant, automatic generation of the cad model, and production of the prosthesis using incremental techniques. 12 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. fig. 13 kbe system verification process the patients were of different ages, and sex and with varying disability of upper limb, but with similar anatomy of the residual limb. for each patient, four different variants of prostheses were prepared. the description of the variants is presented in table 2. table 2 variants types for the verification process prosthesis patient #1, adult male patient #2, adult female patient #3, child female variant #1 closed crs/ open forearm / straight fixed handle open funnel/ open forearm/ c-handle open funnel/ open forearm/ c-handle variant #2 semi-open funnel/ closed forearm/ fixed angular handle semi-open funnel/ closed forearm/ straight handle with a spring semi-open funnel/ open forearm / straight handle with a spring variant #3 semi-open funnel/ closed forearm/ c-handle semi-open funnel/ open forearm/ straight fixed handle open funnel/ open forearm/ straight fixed handle variant #4 closed crs/ open forearm/ straight handle with a spring closed crs/ open forearm/ fixed angular handle closed crs/ closed forearm/ fixed angular handle automation of design of modular upper limb prostheses 13 3.2 evaluation of generative cad models the models generated for patient #1 allowed us to recognize the operation of the generative cad model in practice. the data from the 3d scanning process entered in the design table allowed for the flawless generation of individual parts of the prosthesis for variant #1. on the other hand, the reconstruction of the assembly prosthesis model resulted in irregularities in the connections between the components. the reason turned out to be errors in the prepared conditional instructions, and more precisely, the lack of certain conditions describing the joints between chosen components. these errors no longer appeared when generating variants 2-4 for patient #1. in the next step, variants 1-4 were generated for patient #2. in this case, no new problems were identified. these appeared only when generating models for the youngest patient #3. they were related to, among others, incorrect execution of the extrusion operation in the forearm model, due to a significant change in the length of the forearm part. errors also appeared when generating closed crs, open and semi-open funnel with a forearm. the problem was that the section plane at the other end of the join was too high, so the projected cut edges included areas with cuts and created an unclosed sketch. after introducing changes in operations, by adding new rules, the models were rebuilt correctly. the process of data processing in the cad system took at least a few minutes in each case. the structure of the model and its connection with the database and knowledge described by rules makes the model generation process relatively long. in practice, this is troublesome, because frequent, iterative corrections and their verification required a lot of time. 3.3 fabrication of the designed prosthesis the paper presents the effects of the fabrication of a designed prosthesis for patient #3 in variant #1, as can be seen in fig. 14, which can be considered as special bicycle prosthesis. based on the results obtained from the kbe system, the selected variant of the prosthesis was made using fdm technology and pla material. the manufactured components were assembled without major problems, and the joining surfaces were positively assessed. however, the disadvantages include the general quality of the final product irregular layers are visible (especially on the surface of the forearm) and numerous surface roughness or material shortage in the area of the connection of the funnel with the mounting element. it is assumed that the manufacturing process will require significant modifications in the further stages of work on the system. the prosthesis was tried on and tested by the patient for cycling, as shown in fig. 15. fig. 14 variant #1 of prosthesis for patient #3 14 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. fig. 15 variant #1 of prosthesis for patient #3 try-on with the patient despite the modular design and high functional capabilities of the prosthesis, its most important element is the socket, which requires proper fitting. the results of the fitting show some inaccuracies, revealing too much play around the residual limb mainly in the lower and upper segments of the components. therefore, the circumferences of the sockets should be reduced to prevent the stump from moving within their inner surface. such a change will have to be taken into account in the assumptions for converting the results of 3d scanning measurements into specific values of cad model parameters. in addition, due to the comfort of use, the patient #3 also indicated several comments regarding the improvement of the prosthesis functionality. nevertheless, in his opinion, the prototype produced is very attractive and does its job, enabling him to ride a bike. 4. discussion the result of work described in this paper is a generative cad model of the upper limb prosthesis. while the methods of generational modeling have been known in engineering for years, in the field of prosthetic design it is a certain innovation and can be an interesting alternative to the traditional approach. the main role in the project was played by the preparation of special parametric structures of the models, as well as the development of universal joints for prosthesis elements. the concept of the external elbow mechanism is an interesting alternative to the traditional connection, due to the possibility of using it in combination with an open socket. it offers the possibility of good ventilation of the residual limb, eliminating possible discomfort when using a fully enclosed socket made of impermeable plastic. however, the construction of the mechanical elbow joint should be subjected to further analysis and testing, especially in the long term of use. the developed concept of the mechanism made it possible to eliminate the screws previously used for angular adjustment. increased by the ease of use of each variant of the prosthesis. in addition, it will be necessary to intensify the research and increase the number of patients, and as a result, modify the design rules so that the adjustment of the prosthesis elements to the size related to the patient's age takes place without the need for manual corrections. the process of regeneration of new variants of the prosthesis took from several dozen seconds to several minutes. the introduction of a large number of components and relationships in the assembly model slows down the performance of autodesk inventor software, but there has never been a case in which it could not cope with the execution of the command given to it. extended model generation time can also be caused by many automation of design of modular upper limb prostheses 15 single conditional statements contained in the ilogic rule the program must go through each one separately to get the final result. therefore, future research towards modular prosthesis models should focus on improving the governing rule and presenting the program contained therein in the form of a list of loops. one of the most important elements of the built system turned out to be an extensive excel file, allowing not only to enter measurement data from the 3d scanning process but also modifying them accordingly to the form necessary for the cad program. clear sheets, filled with design tables, allow easy orientation in the working space of the program. the whole thing has been designed to give each potential user maximum freedom in managing both the dimensional and construction aspects of a given device. the data used in the research were collected in the automedprint system, however, they required manual correction to determine the correct coordinates of selected points forming sketches of sections of the residual limb. therefore, this is an aspect that should be corrected during further activities toward the automation of the design of prosthetic sockets, through the proper determination in the structure of the algorithm of the relationship describing the distance between individual points of each generated crosssection. the initial verification with the patients was successful and confirmed the possibility of smooth transition between the prosthetic sockets due to the unified assembly. despite the need to introduce corrections to the overall dimensions of the device, a properly functioning prosthesis consisting of mutually matched parts was obtained. in order to finally evaluate the designed model, however, it would be necessary to produce more prosthesis components, and then test the actual operation of the implemented solutions, especially the issue of the adapter and the elbow mechanism. it would also be necessary to re-validate with the presence of a patient who would be able to assess the ease of use of the device with one hand and the continued compliance of the design with the dimensions of the healthy limb. nevertheless, the concept of the device is a promising solution that easily allows any adjustments to the overall architecture of the prosthesis components due to the automation of the design of different variants of the prosthesis. in theory, therefore, the presented model is a useful tool to minimize human input in the general process of prosthesis generation. 5. conclusions the research presented in this paper demonstrates a significant advancement in the automation of designing modular upper limb prostheses. integration of reverse engineering, generative cad modeling and additive manufacturing, a solution was developed to facilitate the creation of modular, customized prosthetic devices. this innovative approach greatly reduces the time required for design and production but also enhances the adaptability of the prostheses to individual patient needs. the generative cad model proved to be highly effective in generating various prosthesis configurations tailored to the anthropometric data of different patients. the system's ability to automatically adjust the design parameters based on patient-specific measurements marks a considerable improvement over traditional manual methods. the use of parametric models and design rules embedded within the cad environment ensured 16 f. górski, o. komorowska, p. zawadzki, w. kuczko, m. żukowska, et al. that the prosthesis components could be easily modified and adapted, facilitating rapid prototyping and iteration. in the scope of the studies, certain challenges were identified, particularly in the fitting accuracy of the prosthetic sockets and the quality of the final printed components. these issues highlight areas for future refinement, such as improving the precision of the scanning-to-cad data translation and enhancing the resolution and consistency of additive manufacturing outputs. future work should focus on optimizing the system to minimize manual adjustments and errors in the design process. machine learning techniques could be used for that purpose, provided a sufficient number of cases to help them learn – expanding the patient pool for further testing would be crucial. additionally, exploring advanced materials and printing techniques could improve the durability and aesthetics of the prostheses. will also provide more comprehensive insights into the system's performance and areas for improvement. in conclusion, the developed kbe system for automated design of modular upper limb prostheses represents a significant step forward in prosthetic technology and in the domain of automated design systems. it offers a promising solution to the challenges of custom prosthesis design and manufacturing, contributing to improvement of the quality of life for disabled persons. continued research and development in this field hold the potential to further revolutionize prosthetic care, making it more accessible, efficient, and personalized. acknowledgement: this research was partially supported by the erasmus+ project “boosting the scientific excellence and innovation capacity of 3d printing methods in pandemic period” – bright. project reference: 2020-1-ro01-ka226-he-095517. part of the studies were realized in scope of the grant entitled “european network for 3d printing of biomimetic mechatronic systems”, supported by eea grants (project no: 21-cop-0019). this research was also partially supported by polish national center for research and development, in the scope of the “lider” program (grant agreement no. lider/14/0078/l-8/16/ ncbr/2017) and statutory activity financed by the polish ministry of science and higher education. the studies were approved by an appropriate ethical committee. references 1. murray, c.d., 2005, the social 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vol. 21, no 2, 2023, pp. 239 258 https://doi.org/10.22190/fume221220012c © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper an investigation on mean roughness depth and material erosion speed during manufacturing of stainless-steel miniature ratchet gears by wire-edm sujeet kumar chaubey1, kapil gupta1, miloš madić2 1department of mechanical and industrial engineering technology, university of johannesburg, south africa 2faculty of mechanical engineering, university of niš, serbia abstract. this paper presents the results of an investigation conducted on wire electric discharge machining (wire-edm) of miniature ratchet gears. effects of three important process parameters spark duration, ‘ton’, spark-off-duration ‘toff’, and wire tension ‘wt’ on surface quality, i.e., mean roughness depth ‘rz’ and productivity, i.e., material erosion speed ‘mes’, have been investigated by conducting seventeen experimental trials. both ton and toff have been identified as the significant parameters. further, an optimization of wire-edm parameters resulted in simultaneously best compromise values of rz 5.30 µm and mes 6.75 mm/min and is achieved the following cutting regime: ton 1.5 µs, toff 42.5 µs, and wt 1500 g. at the end, surface quality study has been conducted to evaluate the tribological fitness of the miniature ratchet gear machined at optimum combination of wire-edm parameter values. it was investigated that the generation of uniform and shallow craters on the flank surfaces of ratchet gear machined at optimum values of parameters, imparted smoother bearing area curve and lower coefficient of friction. the profile and flank surface of the ratchet gear also found free from cracks, burrs, and dirt. key words: optimization, ratchet gear, surface roughness, tribology, wire-edm 1. introduction a wheel having uniform but asymmetrical teeth at identical spaces on its circumference is widely referred to as a ratchet wheel/gear [1]. each tooth of ratchet gear has a gentle slope on one edge and a steep slope on the other edge. it receives an intermissive annular motion from an oscillating or reciprocating member. but it doesn't transfer motion and received: december 20, 2022 / accepted march 23, 2023 corresponding author: miloš madić university of niš, faculty of mechanical engineering, aleksandra medvedeva 14, 18000 niš, serbia e-mail: madic@masfak.ni.ac.rs 240 s. k. chaubey, k. gupta, m. madić power from one shaft to another shaft like a gear. a ratchet gear is a key element of mechanical devices which permits one directional rotary or linear motion during its rotation in the forward direction and fending any movement in the reverse direction. ratchet gears are widely used in power wrench, ratchet spanners, lifting devices (hoist), watches, turnstiles, ratchet winder, watch winders; cable tie, fishing reel, overrunning clutch, etc. [1, 2]. surface roughness parameters define the micro irregularities on flank surfaces of the ratchet gears. these irregularities are the indication of variation of the actual surface profile (i.e. top machined surface) from the nominal surface or means line of the surface profile [3]. in practice, surface roughness is considered as one of the most important criteria for assessing the quality of machined surfaces [4, 5]. friction and wear are two most important tribology parameters for sliding or rolling parts and largely depend on the quality of the manufactured surface. a rough surface, high coefficient of friction, crack, inclusions, burrs etc. adversely affect the tribological fitness of the mechanical components like gears [3, 6]. a compromise with the tribological fitness may lead to high consumption of energy and resources, and failure of the mechanical parts and components [7]. therefore, it becomes necessary to minimize the surface roughness for better tribological fitness of manufactured parts and components. since post finishing processes are aimed to remove surface irregularities, but are time consuming and lead to extra cost and environmental footprints, the attempt should be to generate the most favorable surface characteristics using the manufacturing process itself. the conventional processes namely extrusion, forging, die-casting, and powder particle processing, are commonly used to manufacture miniature ratchet gears from a variety of materials [8, 9]. but, these processes have some restrictions related to material selection and part, i.e., gear thickness, occurrence of tool marks, generation of a poor surface quality, etc. these limitations compel to explore alternate processes to manufacture miniature ratchet gears. in the past decade, wire electrical discharge machining (wire-edm) has drawn an attention of manufacturers worldwide due to its potential to machine difficult-to-machine materials, miniature products, machine precisely, and to run without interruption for a long period of time for big jobs [10]. this process is based upon the principle of thermoelectric erosion where the material removal takes place by erosion due to the occurrence of sparks between two electrodes as a thin wire (cathode) and workpiece (anode) and supplying a suitable dielectric between them. there has been some past attempts to use this technology for machining gears from a wide range of engineering materials. but, no record is found on machining a ratchet wheel using this process. some of the important past works are discussed here as under. wang et al. [11] fabricated ten micro spur gears (0.1 module) from x153crmov12 material by micro wire-edm and achieved 0.9 µm as the average roughness of manufactured micro gears. chaudhary et al. [12] investigated the influence of peak current, spark duration, spark-off duration, wire tension, and dielectric fluids on the maximum roughness depth of miniature gears (7mm) fabricated by wire-edm. mohapatra et al. [13] fabricated macro spur gears having a 20 mm pitch circle diameter from the copper plate by wire-edm and found 1.78 µm as the average roughness of macro gears. chaubey and jain [14] studied the influence of wire-edm parameters on surface quality of stainless steel helical and bevel gears. they successfully fabricated high manufacturing quality gears. benavides et al. [15] manufactured better surface quality miniature ratchet gears from four different materials by micro wire-edm process using ø30 µm tungsten wire. ali and mohammad an investigation on mean roughness depth and material erosion speed during manufacturing... 241 [16] achieved 1 µm average roughness and 7 µm maximum roughness depth of miniature gears by wire-edm at lower values of spark duration, voltage, and discharge current. the above review of relevant past work shows the potential of wire-edm for manufacturing miniature gears. but, after a literature review, it is identified that a detailed investigation on wire-edm of ratchet gear, is yet to be explored. with an aim to fill this gap, the present work investigates the wire-edm manufacturing of miniature ratchet wheel and its parameters’ effects on surface quality of the gears and productivity of the process. given that surface quality and process productivity are in general opposite process performances, a multi-objective optimization problem was formulated and solved in order to identify the set of process parameter values that enable achievement of best trade-off. 2. experimentation and measurements 2.1. materials and machine asymmetric miniature ratchet gears that have 8 teeth were manufactured from a 5 mm thick rectangular plate made of austenitic stainless steel of grade 304 (ss304). the specifications and material description of the ratchet gear are given in table 1. table 1 specifications of miniature ratchet gear and composition of gear material miniature ratchet gear specifications material: ss 304; profile: asymmetric, addendum diameter: 9.8 mm; no. of teeth: 8; face width: 5 mm; center-hole diameter: 3 mm composition (% wt.) of ratchet gear material (ss 304) 0.8% c; 6.8% ni; 17% cr; 1.2% mn; 1% si; 0.5% s; balance fe figure 1 presents two and three-dimensional views of a rectangular plate and miniature ratchet gear with detailed specifications. miniature ratchet gears were fabricated by a nonsubmerged type vertically traveling wire-edm machine (model: sprintcut win; brand: electronica; country of origin: india) as shown in fig. 2. a very fine soft zinc-coated brass wire of 0.25 mm diameter having 800 n/mm2 tensile strength was used as tool electrode for cutting of miniature ratchet gears from ss 304 rectangular plate in the presence of deionized water acts as a dielectric medium. the wire-edm machine used for fabricating miniature ratchet gears has computer numerical controlled (cnc) 4 axis (x, y, u, and v) with the capability to achieve wire inclination angle up to ± 300. the complex and tapered parts (i.e., gears) having dissimilar top and bottom faces can be fabricated by wire-edm using the inclination angle of wire. wire tension/rigidity is maintained between the upper and lower guide to ensure smooth machining by avoiding wire vibrations. elcam software was used to make a part program for ratchet gear cutting. 242 s. k. chaubey, k. gupta, m. madić (a) (b) (c) fig. 1 2d and 3d views of (a) gear blank in form of rectangular plate; and (b) miniature ratchet gear drawing; (c) actual pictures of ratchet gears fabricated from ss 304 plate by wire-edm an investigation on mean roughness depth and material erosion speed during manufacturing... 243 fig. 2 wire-edm machine tool used for this study and enlarged view of machining chamber during cutting of miniature ratchet gear 2.2. experiment procedure and measurements to study the effect of three variable wire-edm parameters, a total of seventeen experimental trials were planned and conducted according to the box–behnken experimental design. box-behnken is a rotatable design and requires only three levels of variation for each parameter [17], whereby parameter combinations do not coincide with factorial or fractional designs and are located at the midpoints of parameter’s ranges including several central points. this property makes this design useful in situations when one needs to avoid potential loss of experimental data which could result because of performing experimental trials in which all parameters have extreme values. each experimental trial was replicated twice in order to reduce the effect of uncontrolled variation and improve statistical accuracy of the experiment. thus, in total, thirty-four miniature ratchet gears were manufactured by wireedm. table 2 presents the values of the constant parameters, variable parameters, and their coded levels and corresponding values of the variable parameters of wire-edm. these parameters, their values and levels were selected based on literature review, some preliminary experiments, and considering machine constraints. in this study, material erosion speed (mes) and surface roughness parameter, i.e., mean roughness depth (rz), were considered as responses to study the surface quality of gears and productivity of the process. in wire-edm, material erosion speed indicates the movement of wire (in mm) per unit time (in second) according to the path of prepared gear part-program, during cutting of ratchet gears. it also indicates the productivity of wire-edm in mm/min and can be calculated by dividing the total path length (tplrg) travelled by wire by the total time (tctrg) taken for cutting miniature ratchet gear. following the geometry of miniature ratchet gear, the total path length was 71.82 mm and the total machining time was recorded from the display unit of the wire-edm machine during cutting of every ratchet gear. the following equation was used to calculate the material erosion speed.       = mincenterholegear with ratchet cuttingfor centerholegear with ratchet cuttingfor mm tct tpl mes rg rg (1) 244 s. k. chaubey, k. gupta, m. madić table 2 details of the selected wire-edm input parameters [18] variable wire-edm process parameter considered responses name, ‘symbol’ and (unit) actual (coded) levels low (-1) medium (0) high (1) spark duration ‘ton’ (µs) 0. 9 (-1) 1.3 (0) 1.7 (1) material erosion speed (mes) mean roughness depth (rz) spark-off duration ‘toff’ (µs) 40.5 (-1) 44.5 (0) 48.5 (1) wire tension ‘wt’ (g) 780 (-1) 1140 (0) 1500 (1) constant parameters wire-edm parameters: peak current ‘ip’: 12 a; servo voltage ‘sv’: 20 volts; flushing pressure ‘wp’: 15 kg/cm2; cutting speed ‘cs’: 100 % workpiece and wire: electrode material: zinc coated wire; wire diameter: 0.25 mm; wire tensile strength: 800 n/mm2; and gear materials and thickness: ss 304 and 5 mm dielectric: deionized water; dielectric conductivity: 20 mho the mean roughness depth ‘rz’ was considered as the surface roughness parameter to evaluate the flank surface quality of the miniature ratchet gears fabricated by the wireedm process. ld 130 3d-surface roughness cum contour tracing machine from mahr metrology (germany) was used for the measurements of the mean roughness depth ‘rz’. this instrument was also used to check bearing area curve (bac). to test tribological fitness for the machined ratchet gear, a linear reciprocating wear testing machine (model: cm9104 from ducom, usa) was used under dry conditions at room temperature by sliding a 6 mm diameter tungsten carbide ball (950 hv 9 hardness) for a 5 mm distance at 15 hz frequency for 15 minutes time duration under 15n normal load on a single tooth of gear sample. scanning electron microscope (sem) supra 55 from carl zeiss (germany) was used for analysis of flank surfaces of the best quality gear. 3. results and discussion table 3 presents the combination of wire-edm parameters and their corresponding values of responses obtained after conducting the experiment. the average values of the responses obtained in replicates (repl1+repl2) are considered. empirical models usually require the use of statistical tests for the assessment of significant model terms [19]. in this study analysis of variance (anova) was used for the analysis (tables 4 and 5). the following interpretations are drawn from the anova: ▪ the significance of developed models for each response was identified by using a 95% confidence interval of p-values (i.e., p-values must be less than 0.05). ▪ the developed quadratic models are significant since their p-value is less than 0.05. ▪ only ton and toff were found statistically significant for both responses. also, in the models for predicting mean roughness depth and material erosion speed, interaction of spark duration and wire tension (tonwt) and quadratic effect of the spark duration were close to statistical significance. ▪ a non-significant lack of fit, as desirable, is obtained indicating that developed models accurately fit experimental data. an investigation on mean roughness depth and material erosion speed during manufacturing... 245 table 3 experimental trials and obtained results for fabrication of ratchet gears by wire-edm exp. trials wire-edm process parameters mean roughness depth ‘rz’(µm) avg. (repl1 +repl2) material erosion speed ‘mes’ (mm/min) avg. (repl1 + repl2) machining time (min) avg. (repl1 + repl2) spark duration ‘ton’ (μs) spark-off duration ‘toff’ (μs) wire tension ‘wt’ (g) 1 0.9 (-1) 40.5 (-1) 1140 (0) 5.93 4.94 14.68 2 1.3 (0) 44.5 (0) 1140 (0) 4.76 5.28 13.60 3 1.3 (0) 48.5 (1) 780 (-1) 4.18 4.98 14.42 4 0.9 (-1) 44.5 (0) 1500 (1) 5.19 3.65 22.93 5 1.7 (1) 44.5 (0) 780 (-1) 9.34 6.1 11.40 6 1.3 (0) 44.5 (0) 1140 (0) 6.38 6.1 12.17 7 1.3 (0) 40.5 (-1) 1500 (1) 8.07 7.98 10.60 8 1.3 (0) 44.5 (0) 1140 (0) 4.8 6.53 10.67 9 1.3 (0) 40.5 (-1) 780 (-1) 9.25 7.08 10.75 10 0.9 (-1) 48.5 (1) 1140 (0) 4.19 3.61 19.87 11 1.3 (0) 44.5 (0) 1140 (0) 4.95 5.63 12.75 12 1.7 (1) 40.5 (-1) 1140 (0) 9.65 7.57 9.75 13 1.7 (1) 44.5 (0) 1500 (1) 5.22 6.46 9.95 14 1.3 (0) 48.5 (1) 1500 (1) 3.74 4.33 14.72 15 1.3 (0) 44.5 (0) 1140 (0) 6.78 4.95 12.57 16 0.9 (-1) 44.5 (0) 780 (-1) 3.69 4.06 17.67 17 1.7 (1) 48.5 (1) 1140 (0) 8.67 5.73 12.53 ▪ the values of coefficient of determination (r2) are 0.8518 and 0.9096 for mean roughness depth and material erosion speed which are close to 1. therefore, these values also confirm the adequacy of the developed response models indicating that the developed models are able to explain significant portions of the variability of the responses. ▪ the values of adequate precision greater than 4 are desirable for the developed response models. it indicates the signal-to-noise ratio, i.e., compares the range of the predictions in experimental trials to the average prediction error. the values of adequate precision are 7.458 and 10.231. therefore, it can be confirmed that developed response models indicate adequate signals. ▪ the developed response surface models for mean roughness depth ‘rz’, and material erosion speed ‘mes’ can be expressed through the following empirical equations in terms of coded values of process parameters: tofftonoffonf offontoffonz wtwtttw ttwttr 18.04.119.024.0 01.156.053.052.173.153.5 2 22 +−+− ++−−+= (2) tofftonoffonf offontoffon wtwtttw ttwttmes 39.019.012.025.0 39.063.0033.012.120.17.5 2 22 −+−− +−+−+= (3) 246 s. k. chaubey, k. gupta, m. madić table 4 results of anova for mean roughness depth measured during experiments on wire-edm of ratchet gear source sum of squares df mean square f-value p-value remarks model 58.89 9 6.54 4.47 0.0305 significant ton 24.08 1 24.08 16.46 0.0048 significant toff 18.36 1 18.36 12.55 0.0094 significant wt 2.25 1 2.25 1.54 0.2552 not significant ton toff 0.14 1 0.14 0.099 0.7626 not significant ton wt 7.90 1 7.90 5.40 0.0532 not significant toffwt 0.14 1 0.14 0.094 0.7686 not significant (ton 2) 1.33 1 1.33 0.91 0.3714 not significant (toff 2) 4.32 1 4.32 2.95 0.1294 not significant (wt 2) 0.24 1 0.24 0.16 0.6997 not significant residual 10.24 7 1.46 lack of fit 6.50 3 2.17 2.31 0.2178 not significant pure error 3.75 4 0.94 cor total 69.13 16 r-squared = 0.8518, adjusted r-squared = 0.6613, adequate precision = 7.458 table 5 results of anova for material erosion speed measured during experiments on wire-edm of ratchet gear source sum of squares df mean square f-value p-value remarks model 24.51 9 2.72 7.82 0.0064 significant ton 11.52 1 11.52 33.09 0.0007 significant toff 9.95 1 9.95 28.57 0.0011 significant wt 5.000e-003 1 5.000e-003 0.014 0.9080 not significant tontoff 0.065 1 0.065 0.19 0.6786 not significant ton wt 0.15 1 0.15 0.43 0.5349 not significant toffwt 0.60 1 0.60 1.73 0.2305 not significant (ton 2) 1.67 1 1.67 4.80 0.0645 not significant (toff 2) 0.66 1 0.66 1.88 0.2122 not significant (wt 2) 2.632e-007 1 2.632e-007 7.558e-007 0.9993 not significant residual 2.44 7 0.35 lack of fit 0.84 3 0.28 0.71 0.5961 not significant pure error 1.59 4 0.40 cor total 26.95 16 r-squared = 0.9096, adjusted r-squared = 0.7933, adequate precision = 10.231 an investigation on mean roughness depth and material erosion speed during manufacturing... 247 3.1. influence of wire-edm parameters in this section, the influence of considered input parameters of wire-edm, namely, spark duration ‘ton’, spark-off duration ‘toff’, and wire tension ‘wt’ on the considered responses (i.e., mean roughness depth ‘rz’ and material erosion speed ‘mes’) is discussed with the help of graphs, as shown in figs. 3a and 3c. in these graphs, the abscissa depicts the values of roughness parameters in µm and material erosion speed in mm/min, whereas, the ordinate shows the values of variable input parameters of wire-edm. figures 3a and 3c depict the influence of spark duration, spark-off duration, and wire tension on the selected responses, respectively. the prominent inferences drawn from figs. 3a and 3c are as follows: ▪ mean roughness depth ‘rz’ and material erosion speed ‘mes’ are continuously increasing with an increase in spark duration. their minimum and maximum values occur at 0.9µs and 1.7µs of spark duration. ▪ mean roughness depth ‘rz’ and material erosion speed ‘mes’ are continuously decreasing with an increase in spark-off duration. the minimum values of roughness parameters and maximum values of material erosion speed occur at 48.5 µs and 40.5µs of spark-off duration, respectively. ▪ wire tension has no significant influence on the mean roughness depth ‘rz’ and material erosion speed ‘mes’. though, it was observed that mean roughness depth ‘rz’ very slightly decreases with an increase in wire tension. whereas, material erosion speed slightly increases (ranging from 5.56 to 5.61 mm/min) with wire tension. spark duration is the time period of occurrence of sparks in the machining zone between wire electrode and workpiece material being machined [14, 16]. at higher values of spark duration, spark occurs for a longer period of time and consequently material removal is more due to high material erosion speed. further increase in the spark duration leads to the generation of rough surface on the workpiece by the formation of non-uniform craters and hence the surface roughness is high. spark-off duration is the time period when no spark occurs and no machining takes place. for a duty cycle having a low spark-off duration, means high frequency of spark occurrence. increase in spark-off duration leads to decrease in spark duration and material erosion speed is low for that duty cycle. uniform and shallow craters occur on the surface being machined, for such duty cycle and surface roughness decreased. wire tension is the straightness of the wire between the upper and lower guides of the wire-edm machine tool. in case of low values of wire tension, its straightness is low and due to which vibrations are high. in such case, flushing of the debris, i.e., machined material from the wire-workpiece gap, is poor which deteriorates the surface quality of the workpiece and results in rough surface. moreover, due to inefficient flushing the material erosion speed is also very low at low wire tension. the higher wire tension ensures the smooth flushing of debris from inter-electrode gaps thus ensuring the better surface quality as well as productivity of wire-edm. 248 s. k. chaubey, k. gupta, m. madić fig. 3 effect of wire-edm parameters on mean roughness depth and material erosion speed during manufacturing of ratchet gear: a) effect of spark duration; b) effect of spark-off duration; c) effect of wire tension an investigation on mean roughness depth and material erosion speed during manufacturing... 249 figures 4a and 4b present the normal probability graphs for mean roughness depth (rz) and material erosion speed (mes). these graphs show the distribution of residuals along a straight line referred to as the mean line and are helpful for residual analysis. it can be observed from figs. 4a and 4b that all seventeen residuals for the seventeen experimental data sets are located within a close proximity of the mean line, that confirms the normal distribution of the residuals for mean roughness depth and material erosion speed. in other words, it is evident from the normal probability distribution residual graphs that results data sets for mean roughness depth and material erosion speed, for all experimental runs are not much scattered with a high probability of following the same pattern, and statistically fit for establishing a relationship with the wire-edm process parameters for future predictions. a) b) fig. 4 normal probability distribution graphs of residuals for selected responses: (a) mean roughness depth ‘rz’ and (b) material erosion speed ‘mes’ 250 s. k. chaubey, k. gupta, m. madić 3.2. multi-objective optimization and experimental validation desirability function analysis (dfa) is a widely used technique for the simultaneous optimization of multiple responses to obtain the optimal set of parameters [20-22]. the idea of dfa is to transform the multi-objective into single-objective problem, by estimating overall desirability function (composite desirability) [23]. depending upon the type of the objective function, the following two equations were used to compute the desirability of each response for the ith experimental observation and using a target value equal to a minimum value of roughness response and maximum value of material erosion speed response identified from its experimental values. for minimization of the mean roughness depth (i.e., smaller-the-better type): iw i i sa ya u       − − = (4) for maximization of the material erosion speed (i.e., larger-the-better type): iw i i bs by u       − − = (5) ( )        ==  = iwn i i n n uuuuu 1 1 1 21 ,...,, (6) where ui is the desirability of all responses yi and u is the combined desirability function, s is the objective value, a is the permissible upper value of the ith response, b is the permissible lower value of the ith response, wi is the relative importance or weightage assigned to the response yi, and n is the total number of responses. a combined desirability value closer to 1 is selected to have an optimum combination. the following equations are used to compute the desirability for the ith combination of considered responses by assigning equal weightage (0.5). for minimization of the mean roughness depth (i.e., smaller-the-better type): 5.0 minmax max         − − = zz ziz r rr rr u zi (7) for maximization of the material erosion speed (i.e., larger-the-better type): 5.0 minmax min         − − = mesmes mesmes u i mesi (8) where rzmax, mesmax, rzmin, and mesmin are the maximum and minimum values of mean roughness depth and material erosion speed. the maximum values used for mean roughness depth and material erosion speed are 9.65 µm and 7.98 mm/min, respectively. while corresponding minimum values are 3.69 µm and 3.61 mm/min, respectively. an investigation on mean roughness depth and material erosion speed during manufacturing... 251 the overall desirability function, ui, for the ith combination of the selected parameters of wire-edm can be computed using eqs. (9) and (10): 1 0.5 0.5 0.5{( ) }{( ) } zi r i mes iu u u + =   (9) 0.5{( ) }{( ) } zi r i mes iu u u =   (10) the desirability predicted optimum values of wire-edm parameters are not exactly what can be set of wire-edm machine tool. therefore, the adjacent standard values of input parameters available in the wire-edm machine corresponding to their optimized values were selected for conducting the two confirmation experimental trials to validate the optimized results obtained from dfa. the average values of considered responses obtained from confirmation experimental trials were used to calculate the difference between confirmation results and optimum results obtained from dfa (table 6). table 6 results of optimization in terms of desirability predictions and experimental validation wire-edm parameter optimized value by dfa experimental validation (i.e., close standard values available in the wire-edm) avg. value error trial1 trial2 wire-edm process parameters ton (µs) 1.51 1.5 toff (µs) 42.77 42.5 wt (g) 1500 1500 responses rz (µm) 5.86 5.25 5.35 5.30 9.6% mes (mm/min) 7.06 6.72 6.77 6.75 4.4% table 6 presents optimum values of machining combination and responses obtained for the best desirability function value of 0.842. it also presents the actual values obtained after validation experimental trials. the average (of trials 1 and 2) values of mean roughness depth and material erosion speed are 5.30 µm and 6.75 mm/min, respectively. the percentage difference between confirmation results and optimum results obtained from desirability prediction is less than 10% and can be considered as the worthy agreement between predicted and actual values. 3.3. surface quality study figures 5a and 5b depict the 2d surface roughness profiles and 3d surface topographies for the best quality ratchet gear rz5.30 µm machined at optimum valued of wire-edm parameters and the ratchet gear rz9.65 µm machined using wire-edm parameter values as given in experimental trial 12, respectively. 252 s. k. chaubey, k. gupta, m. madić a) b) fig. 5 2d profiles and 3d topographies for surface roughness of (a) best quality ratchet gear, (b) poor quality ratchet gear an investigation on mean roughness depth and material erosion speed during manufacturing... 253 the deteriorated surface quality is evident for the ratchet gear having higher roughness and machined at experimental trial 12. the abbott-firestone curve referred to as the bearing area curve (bac) and describes the surface texture of manufactured parts and components [24, 25]. as shown in fig. 6a, bac represents as a graph plotted between maximum roughness and the percentage evaluation length. this graph is useful to identify the real contact area of mating surfaces and the change in that contact area, as a result of the wear of manufactured parts and components (e.g., rpk). rpk portion disappears in the running-in operation (function) of the part surface. it is also useful to determine lubricant retention for manufactured parts and components (e.g., rvk). rvk portion does not participate in any task or function during operation and serves to anchor the lubricating film. rk indicates the depth of core roughness profile area and determines the intended service life of parts. it is the actual area which comes in contact with other parts during rolling or sliding [25]. figures 6, 7 and 8 depict bearing area curves of the theoretical concept, best surface quality miniature ratchet gear rz5.30 µm and poor surface quality miniature ratchet gear rz9.65 µm (corresponding to experimental trial 12 in table 2), respectively. the bac of the best surface quality gear is smoother and bearing area ratio is higher as compared to the poor surface quality gear. it is due to the formation of uniform and shallow craters that tend to lesser irregularities on the flank surfaces of the best quality ratchet gear. it is evident from figs. 7 and 8 that the best quality ratchet gear has less material available to worn out during preliminary contact and a better lubrication capacity than poor-quality ratchet gear. the material portion ‘mr2-mr1’ (the percentage of material available during actual contact) of the best quality ratchet gear within the corresponding region is 74.98 % which is higher than the material portion 68.0% of the poor-quality gear. percentage bearing length (bl) (the total material available during tribological interaction) of the best quality ratchet gear (81%) is also higher than the poor quality gear (78.6%). these results confirm that the ratchet gear machined at optimal wire-edm parameters has the best surface quality that leads to its tribological fitness. fig. 6 theoretical concept of bearing area curve 254 s. k. chaubey, k. gupta, m. madić fig. 7 bearing area curve for the best surface quality ratchet gear fig. 8 bearing area curve for poor surface quality ratchet gear the coefficient of friction (cof) which is a ratio between friction and normal forces depends on adhesion and roughness between two surfaces. although some amount of cof is a must, but for most of the applications, a low value of cof is desirable [26, 27]. an investigation on mean roughness depth and material erosion speed during manufacturing... 255 figures 9a and 9b depict the cof curves for poor surface quality miniature ratchet gear and optimum miniature ratchet gear. these curves illustrate the variation of cof with fretting wear time in minutes. it can be seen from fig. 9a that cof increases initially and goes beyond 1. further, it becomes constant for a small duration and is then decreases for 4 minutes of wear. it starts increasing again up to 12 minutes. the maximum value of cof at 15 minutes is approximately 1.2. non-uniform and rough flank surfaces generated by wire-edm for this ratchet gear are the reasons behind this instability and variation in cof. on the contrary, as shown in fig 7b, the cof for the ratchet gear produced at optimum wire-edm parameters is lesser, i.e., not more than 0.4, due to the superior flank surface characteristic with lesser roughness. a) b) fig. 9 variation of cof with time during wear testing of wire-edm manufactured miniature ratchet gears: (a) poor surface quality miniature ratchet gear; (b) optimized miniature ratchet gear 256 s. k. chaubey, k. gupta, m. madić figure 10 illustrates scanning electron microscopy (sem) images of the miniature ratchet gear manufactured at optimum combination of wire-edm parameter values. it can be confirmed that optimum gear has an even and precise profile of teeth and center-hole, and the tooth flank shows evenly distributed shallow craters and a smooth surface free from burrs and cracks. (a) (b) (c) fig. 10 scanning electron micrographs of miniature ratchet gear manufactured at optimal parameters of wire-edm: (a) gear teeth profile; (b) and (c) magnified flank surfaces 5. conclusion an investigation on successful fabrication of asymmetric miniature ratchet gears of stainless-steel is reported. a detailed study on mean roughness depth and material erosion speed along with the surface quality of ratchet gears is carried out. some important conclusions from this research work are as follows: an investigation on mean roughness depth and material erosion speed during manufacturing... 257 ▪ based upon the analysis of variance study conducted on the experimental results, it is found that both spark duration and spark-off duration are the significant parameters, as their p values are less than 0.05. it is suggested to select both parameters carefully in wire-edm when fabricating miniature gears of stainless-steel. ▪ multi-objective optimization of wire-edm, based on the desirability function analysis, resulted in flank surface roughness 5.30 µm and 6.75 mm/min material erosion speed at optimum parameter setting 1.5 µs spark duration, 42.5 µs spark-off duration, and 1500 g wire tension. ▪ miniature ratchet gear machined at optimum wire-edm parameters was found better in tribological performances with mean roughness depth rz5.30 µm, coefficient of friction cof0.4, bearing length bl81%, material portion mr2-mr1– 74.98% values much better than the selected gear machined under non-optimal regime that is having the tribology parameter values as rz9.65 µm, cof1.2, bl78.6%, mr2-mr168%. ▪ sem study confirmed the absence of any burr, dirt, and cracks from the flank surfaces of the miniature ratchet gear machined at optimum parameters. ▪ this work is novel in the sense that it has explored and established wire-edm as an alternate of conventional manufacturing processes for fabrication of miniature ratchet gears made of stainless-steel. it is hoped that the results of this detailed investigation on surface quality of the fabricated ratchet gears of stainless-steel and productivity of the wire-edm process, would be helpful to the researchers and engineers to establish the field further. acknowledgement: we sincerely acknowledge the indian institute of technology indore (india) for providing us the measurement and testing facilities. this research is supported by the royal academy of engineering uk grant tsp1332. references 1. hatsuzawa, t., yamazaki, a., nisisako, t., yanagida, y., 2015, design and evaluation of artemia-driven micro ratchet gears, sensors and actuators a: physical, 235, pp. 182-186. 2. bralla, j.g., 1998, design for manufacturability handbook, mcgraw-hill inc., usa. 3. bhushan, b., 2000, modern tribology handbook, crc press, usa. 4. trifunović, m., madić, m., marinković, d., marinković, v., 2023. cutting parameters optimization for minimal total operation time in turning pom-c cylindrical stocks into parts with continuous profile using a pcd cutting tool, metals, 13(2), 359. 5. van, a.l., nguyen, t.t., 2023, multi-response optimization of burnishing variables for minimizing environmental impacts, technical gazette, 30(1), pp. 169-177. 6. leach, r.k., 2010, fundamental principles of engineering nanometrology, elsevier, uk. 7. holmberg, k., erdemir, a., 2017, influence of tribology on global energy consumption, costs and emissions, friction, 5(3), pp. 263-284. 8. nichols, r., 1971, miniature gears. in: bell, p.c. 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2022, desirability function analysis (dfa) in multiple responses optimization of abrasive water jet cutting process, reports in mechanical engineering, 3(1), pp. 11-19. 24. stewart, m., 1990, a new approach to the use of bearing area curve, society of manufacturing engineers, fc90229, pp. 1-13. 25. phokane, t., gupta, k., gupta, m.k., 2017, investigations on surface roughness and tribology of miniature brass gears manufactured by abrasive water jet machining, proceedings of the institution of mechanical engineers, part c: journal of mechanical engineering science, 232(22), pp. 4193-4202. 26. petare, a., jain, n.k., 2018, on simultaneous improvement of wear characteristics, surface finish and microgeometry of straight bevel gears by abrasive flow finishing process, wear, 404-405, pp. 38-49. 27. petare, a., deshwal, g., palani, i.a., jain, n.k., 2020, laser texturing of helical and straight bevel gears to enhance finishing performance of aff process, the international journal of advanced manufacturing technology, 110, pp. 2221-2238. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 99 113 https://doi.org/10.22190/fume180327013p © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. viscoelastic contacts  udc 539.3 valentin l. popov 1,2,3 , emanuel willert 1 , markus heß 1 1 technische universität berlin, berlin, germany 2 national research tomsk polytechnic university, tomsk, russia 3 national research tomsk state university, tomsk, russia abstract. until recently the analysis of contacts in tribological systems usually required the solution of complicated boundary value problems of three-dimensional elasticity and was thus mathematically and numerically costly. with the development of the so-called method of dimensionality reduction (mdr) large groups of contact problems have been, by sets of specific rules, exactly led back to the elementary systems whose study requires only simple algebraic operations and elementary calculus. the mapping rules for axisymmetric contact problems of elastic bodies have been presented and illustrated in the previously published parts of the user's manual, i and ii, in facta universitatis series mechanical engineering [5, 9]. the present paper is dedicated to axisymmetric contacts of viscoelastic materials. all the mapping rules of the method are given and illustrated by examples. key words: contact, friction, viscoelasticity, rheology, method of dimensionality reduction 1. introduction in recent years the method of dimensionality reduction (mdr) has been developed to efficiently deal with axisymmetric [1] and non-axisymmetric contacts [2]. the scope of applicability includes normal contacts with and without adhesion as well as tangential contacts [1], torsional contacts [3], contacts of functionally graded materials [4-5] and viscoelastic contacts [6-8]. in preceding papers the mapping rules of mdr have been received march 27, 2018 / accepted may 11, 2018 corresponding author: emanuel willert technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: e.willert@tu-berlin.de 100 v.l. popov, e. willert, m. heß summarised and illustrated for axisymmetric contacts with a compact contact area for homogeneous [9] and power-law graded [5] elastic materials. the present publication gives a similar “user’s manual” for the treatment of viscoelastic contacts. 2. basic assumptions let us in the beginning briefly clarify the fundamental assumptions, which define the framework of our method. the results of the set of simple rules given in this paper to solve axisymmetric contact problems of viscoelastic materials will be exactly correct, if all assumptions are met. yet the method can also be used if some of the assumptions are broken although in this case the obtained solutions might exhibit smaller or larger errors, depending on the precise circumstances. firstly, we only consider homogeneous, isotropic, linear-viscoelastic media. moreover, the deformations have to be small to ensure kinematic linearity. in this case we are also allowed to work within the half-space approximation. throughout most of the paper we will additionally demand incompressible material behaviour, i.e. poisson ratio ν shall be equal to 0.5. the treatment of compressible materials will, as far as possible, be covered in a separate section. under these assumptions the viscoelastic material can be described by a single timedependent shear relaxation function g(t), which gives the material’s stress response to a unit strain increment. stress response σ(t) to an arbitrary deformation history γ(t) is due to the superposition principle given by the sum of stresses for all past strain increments ([10], p.257), ( ) ( )d ( ) ( )d . t t g t t g t t t t            (1) as the convolution (1) in the time domain corresponds to a product in the laplace domain, eq. (1) can alternatively be written in the following way: * * *( ) ( ) ( ),s g s s s  (2) whereas a star denotes the laplace transform into the s-domain, i.e. * 0 ( ) : ( ) exp( )d ,g s g t st s    (3) and similarly for all others. also, a time-dependent shear creep function j(t), i.e. the strain response to a unit stress increment, can be defined ([11], p.215). the strain response to an arbitrary stress history is, analogously to eq. (1), given by ( ) ( ) ( )d , t t j t t t t       (4) or, equivalently, in the laplace domain method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 101 * * *( ) ( ) ( ).s j s s s  (5) hence both the material functions are coupled by the relation 2 * *( ) ( ) 1.s g s j s  (6) in case of harmonic oscillations, or put generally, in the frequency domain, for any linear viscoelastic material there is a linear proportionality between (shear) stress and (shear) strain – this directly follows from the material law in eq. (1). the coefficient of proportionality is called “complex dynamic modulus” * 0 ˆ ( ) : ( )exp( )d ( ).g i g t i t t i g s i          (7) its real part is referred to as “storage modulus” and the imaginary part as “loss modulus”. for the contact of two viscoelastic bodies with creep functions j1 and j2 both the creep functions simply have to be linearly superposed, 1 2( ) ( ) ( ).j t j t j t  (8) then a combined shear relaxation function g(t) can also be defined via eq. (6). finally, we will only consider quasi-static processes, i.e. all characteristic velocities of the contact problem must be much smaller than the smallest speed of wave propagation in the viscoelastic medium, and neglect adhesion or plasticity. 3. rheological models the viscoelastic properties of materials, as they have been briefly introduced in the previous section, are often represented in terms of rheological models. the basic elements of those models are a spring, representing ideally elastic properties, and a dashpot, representing ideally viscous properties. combining a sufficiently large set of those basic elements, any arbitrarily complex (linear-) viscoelastic behaviour can be captured. thereby only two fundamental rules of superposition exist: for two elements in parallel to each other, the respective relaxation functions have to be linearly superposed; for two elements in series the creep functions are superposed. the rheological model, which reproduces a given relaxation function g(t) shall henceforth in this paper be denoted with a simple box accompanied by the respective relaxation function (see tab. 1). we should point out that all the values of stiffness and damping in these models are to be understood as continuum variables, i.e. per unit volume, which is why we will always speak of moduli and viscosities. tab. 1 shows a compilation of the most commonly used viscoelastic material models and their rheological representations as well as the associated relaxation and creep functions. thereby δ(t) denotes the dirac δ-distribution while all the other designations are self-explanatory, based on the depicted rheological models. 102 v.l. popov, e. willert, m. heß table 1 rheological representations and material functions for the most common linear viscoelastic material models (δ denotes the dirac δ -distribution) material model material functions elastic body ( ) , 1 ( ) g t g j t g   viscous body ( ) ( ), ( ) g t t t j t     kelvinvoigt body ( ) ( ), 1 ( ) 1 exp g t g t g j t t g                   maxwell body 1 1 1 ( ) exp , 1 ( ) g g t g t t j t g            standard body 1 1 1 1 1 1 ( ) exp , 1 ( ) 1 exp ( ) g g t g g t g g g t j t g g g g g                            prony series (generalised maxwell body) 1 ( ) exp , n i i i i g g t g g t             no closed-form analytical expression for j(t) method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 103 4. two preparatory steps of the method we consider the contact of two incompressible viscoelastic bodies with combined shear relaxation function g(t) (or, equivalently, combined creep function j). the nondeformed gap between both the bodies shall be an axisymmetric function z = f(r). to solve contact problems of this configuration within the mdr two introductory steps are necessary. firstly, the axisymmetric gap has to be transformed into an equivalent (rigid) plain profile g(x) by the integral transform [9] 2 2 0 ( )d ( ) . x f r r g x x x r     (9) this relation is the same as in the elastic case and is thus explained and illustrated by several examples in the first part of this user’s manual. the inverse transform of eq. (9) reads [9] 2 2 0 2 ( )d ( ) . r g x x f r r x    (10) secondly, a one-dimensional foundation of independent, linear-viscoelastic elements, each in distance δx of each other, must be initialised, as demonstrated in fig. 1. fig. 1 one-dimensional foundation of linear-viscoelastic elements a single element of the foundation is given by the rheological model for relaxation function g(t), as shown in the previous section. for example, if the relaxation behaviour can be captured by a three-element standard solid, a single element of the viscoelastic foundation is given by a spring in series with a dashpot, the pair in parallel with another spring, as described above. the elements will have time-dependent values of normal and tangential stiffness (note that we assume incompressibility, i.e. ν = 0.5), 2 ( ) ( ) 4 ( ) , 1 4 8 ( ) ( ) ( ) , 2 3 z x k t g t x g t x k t g t x g t x               (11) or in the frequency domain, 104 v.l. popov, e. willert, m. heß ˆ ˆ( ) 4 ( ) , 8ˆ ˆ( ) ( ) . 3 z x k g x k g x           (12) 5. normal contact of axisymmetric bodies the equivalent profile defined by eq. (9) is now pressed into the viscoelastic foundation defined by eq. (11) by an indentation depth d(t). without loss of generality we will assume that the indentation starts at time t = 0, and that the viscoelastic medium previously was stress-free and non-deformed. vertical displacement w1d of an element at position x within the contact area of radius a is enforced by the indentation, 1d ( , ) ( ) ( ), .w x t d t g x x a   (13) an element comes into contact (geometrically) if the displacement of the non-contacting surface equals the displacement enforced by the indentation, i.e. n.c. 1d ( ( ), ) ( ) ( ( )).w a t t d t g a t  (14) for a monotonically increasing contact radius the non-contacting surface is not deformed. in this case the contact radius is therefore simply given by the relation ( ) ( ( )),d t g a t (15) which, for a monotonically increasing contact radius, is a universal relation independent of the material rheology, as proven by lee & radok [12]. if the contact radius has extremal values the creep behaviour of the area without direct contact must be traced and inserted into eq. (14) to give correct results [6]. the normal force in a single element of the foundation is due to the superposition principle given by 1d 0 ( , ) ( , ) 4 ( ) d , t n w x t f x t x g t t t t         (16) or in the frequency domain 1d ˆˆ ˆ( , ) 4 ( ) ( , ).nf x xg w x     (17) note that if the rheological model contains separate dashpots, like the kelvin-voigt model, stress relaxation function g(t) includes dirac-distributions, which have to be evaluated in the integral using their filter properties. instead of evaluating eq. (16), which under some circumstances may require the knowledge about the entire loading history, one can also apply the complete set of equilibrium conditions for the single element, including the inner degrees of freedom. for example, the standard element shown on the left of fig. 2 has one inner degree of freedom representing the material relaxation. the equilibrium conditions for the outer and inner degree of freedom are method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 105 1d 1 1d 1d 1d 1 1d 4 ( ), 0 ( ) . nf x g w g w w w g w        (18) for both forceor displacement-controlled conditions this gives a closed ordinary differential equation system with a unique solution. fig. 2 rheological standard element of the viscoelastic foundation under normal load (left, a) and superimposed normal and tangential load (right, b) note that under force-controlled conditions eq. (16) can be inverted to give 1d 0 ( , )1 ( , ) ( ) d . 4 t nf x t w x t j t t t x t        (19) the elements outside the contact area are, of course, free of forces, i.e. ( , ) 0, .nf x t x a   (20) an element leaves the contact (dynamically), if the upkeep of contact would require negative normal forces. the total normal force is obtained by summation of all the element normal forces, ( ) ( ) ( , ) ( ) d a t n n a t f x t f t x x     (21) and with the linear force density, ( , ) ( , ) : ,n z f x t q x t x    (22) one can also calculate pressure distribution p(r,t) in the original axisymmetric system by the relation 2 2 ( , )1 d ( , ) .z r q x t x p r t x x r        (23) we would like to stress again that all the results obtained by the described solution scheme will be exactly correct within the stated assumptions. let us now illustrate the procedure by some examples. 106 v.l. popov, e. willert, m. heß displacement-controlled indentation as the first example we consider the displacement-controlled indentation of a flat three-element standard solid by a rigid cone with slope θ and thus the profile ( ) tan .f r r  (24) the indentation depth as a function of time shall be d(t) = v0t, which corresponds to a simple indentation test to determine the (visco-)elastic properties of a material. we would like to know the total normal force as a function of time and the material properties of the standard solid. solution: the equivalent plain profile is given by ( ) tan . 2 g x x   (25) as the contact radius is monotonically increasing eq. (15) can be applied to determine the contact radius. hence 02 ( ) . tan v t a t    (26) an element at position x will therefore come into contact after a time 0 ( ) tan . 2 ( ) c x xt t x v a t    (27) the indentation velocity for all the elements in contact is equal v0. hence the application of equation (16) yields the desired normal force:     0 1 0 / 0 1 ( ) 8 exp d d 4 ( ) 2 1 exp 1 . a t t n xt a t t t f t v g g t x t v a t g t g t                                         (28) force-controlled indentation as the second example we consider the force-controlled indentation of a kelvin-voigt solid by a rigid sphere with radius r. the total normal force shall be kept constant, 0( ) const .f t f  (29) this loading situation corresponds to the ideal loading protocol of the commonly used shore hardness test for elastomers. we would like to know the indentation depth as a function of time and the material properties. method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 107 solution: the spherical profile can in the vicinity of the contact be approximated by the parabolic profile 2 2 ( ) ( ) . 2 r x f r g x r r    (30) the contact radius will be again, due to creep, a monotonically increasing function with time. hence 2 ( ) ( ) . a t d t r  (31) in a kelvin-voigt solid the stress state is a linear superposition of ideally elastic and ideally viscous stress components (this can be easily understood with the respective rheological model shown in tab. 1). the total normal force is therefore 3 0 3 3 16 ( ) ( ) 8 ( ) ( ) 3 16 d ( ) ( ( )) , : . 3 d n a t f t f g d t a t r g a t a t r t g                   (32) the solution of this equation with initial condition a(t = 0) = 0 is 3 03 ( ) 1 exp . 16 f r t a t g             (33) application of eq. (31) will then provide the indentation depth as a function of time. the solution in eq. (33) can be written in the more general form el 0 ( ( )) ˆ( ),nf a t j t f  (34) with the elastic (hertzian) normal force as a function of the contact radius, 3 el 16 ( ) : , 3 n a f a g r  (35) and the non-dimensional shear creep function for the kelvin-voigt solid, ˆ( ) : 1 exp . t j t          (36) as it turns out, the general formulation (34) is correct for arbitrary axisymmetric indenters and arbitrary linear-viscoelastic rheologies ([11], p.232) and can therefore be used to analyse general shore hardness test configurations. 108 v.l. popov, e. willert, m. heß impact test as the third example in this section and in order to complete the set of applications of the mdr to commonly used material test of elastomers, we would like to show the application of the described rules to a rebound test: a homogeneous rigid sphere with radius r, mass m, mass density ρ and initial velocity v0 impacts onto a viscoelastic half-space, whose rheology can be described by a three-element standard solid model. we would like to know the coefficient of restitution e (as a measure of energy dissipation under dynamic loading conditions) as a function of the inbound velocity and the material parameters. solution: this problem cannot be solved analytically. however, based on the mdr, a simple numerical algorithm can be implemented to give the solution of the impact problem in the quasi-static limit, i.e. assuming that the viscoelastic medium moves through a series of equilibrium states thus neglecting wave propagation in the viscoelastic material. the equation of motion of the sphere, in terms of indentation depth d, is simply given by ( ) ( ).nmd t f t (37) for all the foundation elements in contact, the displacement is enforced by the movement of the plain parabolic profile equivalent to the three-dimensional sphere (see eq. (30)), 2 1d ( , ) ( ) , . x w x t d t x a r    (38) solution of eqs. (18) will give the corresponding element forces. they can be summed up to give the total normal force, which enters the equation of motion. as stated before, an element gets into contact geometrically and leaves contact if contact upkeep would require negative values of the respective element normal force. the impact ends at time t, if all elements have left contact. any time integration scheme can be used to solve the described equation system in discrete time steps, by far the easiest one being an explicit euler method. as the required computational operations are extremely simple, the time step can be set small enough to ensure numerical stability. the solution of the impact problem, i.e. the coefficient of restitution 0 ( ) : v t t e v    (39) only depends on two non-dimensional parameters, namely [13] 1/5 0 1 1 22 3 : , , v g p p r gg           (40) and is shown in fig. 3 as a function of p1 for several different values of p2. note that the physical meaning of p1 (except for a numerical factor of the order of unity) is a ratio of two characteristic time scales: the viscoelastic relaxation time of the three-element standard solid compared to the elastic impact duration with g = g∞. method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 109 fig. 3 coefficient of restitution for the normal impact of a rigid sphere onto a flat standard solid as a function of the two governing parameters 6. tangential contact of axisymmetric bodies we now consider contacts with superimposed normal and tangential loading. thereby the application of a tangential load is completely analogous to the previous section. the elements of the viscoelastic foundation are vertically and horizontally displaced. via the superposition principle the tangential force in a single element can be calculated from tangential displacement u1d: 1d 0 ( , )8 ( , ) ( ) d , 3 t x u x t f x t x g t t t t         (41) or, vice versa, 1d 0 ( , )3 ( , ) ( ) d . 8 t xf x t u x t j t t t x t        (42) alternatively, as in the normal contact problem, the equilibrium conditions for all degrees of freedom of the elements can be evaluated. coming back to the standard element example in fig. 2 the equilibrium conditions in tangential direction (see the right side of fig. 2) read 1d 1 1d 1d 1d 1 1d 8 ( ), 3 0 ( ) . xf x g u g u u u g u        (43) note that non-contacting surface areas also relax tangentially. in the frequency domain the convolution (41) reduces to a product, 1d 8 ˆˆ ˆ( , ) ( ) ( , ). 3 xf x xg u x     (44) 110 v.l. popov, e. willert, m. heß the tangential linear force density, ( , ) ( , ) : ,x x f x t q x t x    (45) will provide the total tangential force, ( ) ( ) ( ) ( , )d , a t x x a t f t q x t x    (46) and the shear stress distribution in the original axisymmetric system, 2 2 ( , )1 d ( , ) .x xz r q x t x r t x x r          (47) to account for micro-slip in the contact we assume the validity of a local amontonscoulomb friction law in its simplest form for any contact point: if the local shear stress does not exceed the maximum value given by pressure times friction coefficient μ the surfaces are able to stick, ( , ) ( , ), stick;xz r t p r t  (48) otherwise the surface points will slip and the frictional shear stress is known, ( , ) ( , ), slip.xz r t p r t  (49) thereby it is clear that the contact area will always consist of an inner stick area with radius c and a slip area propagating inside from the edge of contact. accounting for slip in viscoelastic frictional contacts within the framework of mdr is simple (and completely analogous to the elastic case): the elements of the viscoelastic foundation simply have to obey the same local amontons-coulomb law! that is, if the indenting plain profile is moved tangentially by an increment δu (0) from a given contact configuration, the contacting elements can either stick or slip, defined by the condition (0) 1d 1d ( , ) , if ( , ) ( , ), ( , ) ( , )sgn( ), else. x n x n u x t u f x t f x t f x t f x t u            (50) radius c of the stick area is given by the condition ( , ) ( , ).x nf c t f c t   (51) tangential fretting in a viscoelastic contact as an illustrative example we consider a simple case of tangential fretting: two axisymmetric bodies with combined relaxation function g(t) are pressed together with a fixed indentation depth d0. the equivalent plain profile of non-deformed gap f(r) shall be g(x). after the normal stresses have been relaxed to their asymptotic value, small relative tangential harmonic oscillations (0) (0)( ) cos( )u t u t  (52) are enforced. we would like to know radius c of the permanent stick area. method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 111 solution: within the permanent stick area the forces in the elements of the viscoelastic foundation are known to be (0)8 ˆ ˆ( ) ( ) cos( ), arg{ ( )}. 3 xf t x u g t g          (53) here the complex dynamic modulus, introduced in eq. (7), has been used because the excitation is harmonic. the normal forces in the fully-relaxed (i.e. elastic) state are 0( ) 4 [ ( )].nf x xg d g x    (54) hence, the radius of the permanent stick area will be given by the solution of the condition (0) ˆ2 ( ) ( ) . 3 g d g c u g       (55) 7. contact of compressible materials although many (or even most) technically or biologically relevant viscoelastic media can – at least in good approximation – be considered incompressible, any linear isotropic viscoelastic material has not one but two characteristic material functions: in addition to shear relaxation g(t), already used throughout this paper, there is also bulk relaxation function k(t) giving the stress response to a unit volume strain. accounting for compressibility in viscoelastic contact problems is, in general, a rather non-trivial task [14]. however, for normal contact problems (and only for them) it is easy to show that the compressible contact problem can be traced back to an equivalent incompressible one with the effective shear creep function * eff 2 * * 3 ( ) ( ) ( ), ( ) . 3 ( ) ( ) j t j t j t j s s k s g s       (56) this obviously means that in the mdr model the rheological elements of the viscoelastic foundation simply have to be replaced by the elements shown in fig. 4. for the diagram we assume two materials with relaxation functions g1(t), g2(t), k1(t) and k2(t) respectively. a box, as in the section on rheological models, is an abbreviation for the rheological model representing the relaxation function denoted near the box. fig. 4 general rheological element representing two contacting compressible viscoelastic materials. a box is an abbreviation for the rheological model representing the relaxation function denoted near the box 112 v.l. popov, e. willert, m. heß as compressible media are not by necessity elastically similar to each other we would like to stress that this ascription to an equivalent incompressible problem is only exact for either elastically similar materials or frictionless normal contacts. displacement-controlled indentation of a compressible medium as an example let us analyse the frictionless indentation of a general kelvin-voigt solid with the relaxation functions ( ) ( ), ( ) ( ), g t g t k t k t         (57) with the shear and bulk viscosities, η and ξ, by a rigid flat cylindrical punch with radius a. the indentation depth as a function of time shall be d(t) = v0t. we would like to know the total normal force as a function of time and the material parameters. fig. 5 displacement-controlled indentation of a compressible kelvin-voigt element (based on [8]) solution: the equivalent mdr-profile of a cylindrical flat stamp is a rectangle of the same length. hence, all the elements within contact radius a are identically deformed. the equilibrium condition for the inner degree of freedom for each element reads (see fig. 5) 1d 1d 0 0 4 4 1 4 . 3 3 3 3 w k g w v k g v t                                     (58) the solution of this ordinary differential equation with initial condition  1d 0 0w t   is given by 1d 0 0( ) ( ) 1 exp , 33 4 3 : , : , : . 3 4 3 4 3 4 t w t v b c cv t k g b c k g k g                                     (59) also, for the total normal force (as the indenting body shall be a rigid flat punch, all the elements of the foundation with |x| < a are displaced in the same way), 1d 1d ( ) ( ) 2 8 [ ].n n f t f t a a g w w x       (60) method of dimensionality reduction in contact mechanics and friction: a user’s handbook. iii. 113 in the limit of fast relaxation t >> τ the total normal force will be  0( ) 8 ( ) .nf t av g b c g ct c      (61) 8. conclusions the present paper gives a concise description of the rules for the application of the method of dimensionality reduction to contacts of linear viscoelastic materials. although the given examples mostly focus on analytical solutions for accessibility, it is, of course, possible to implement the rules in a numerical scheme to efficiently simulate viscoelastic contacts with arbitrary oblique loading histories. for example, based on the method, comprehensive contact-impact solutions for viscoelastic materials have been obtained very recently and cross-checked against respective boundary-element simulations [13]. references 1. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer, berlin heidelberg. 2. argatov, i.i., heß, m., pohrt, r., popov, v.l., 2016, the extension of the method of dimensionality reduction to non-compact and non-axisymmetric contacts, zamm zeitschrift für angewandte mathematik und mechanik, 96(10), pp. 1144-1155. 3. willert, e., popov, v.l., 2017, exact one-dimensional mapping of axially symmetric elastic contacts with superimposed normal and torsional loading, zamm zeitschrift für angewandte mathematik und mechanik, 97(2), pp. 173-182. 4. heß, m., 2016, a simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, international journal of engineering science, 104, pp. 20-33. 5. heß, m., popov, v.l., 2016, method of dimensionality reduction in contact mechanics and friction: a user’s handbook. ii. power-law graded materials, facta universitatis-series mechanical engineering, 14(3), pp. 251-268. 6. argatov, i.i., popov, v.l., 2016, rebound indentation problem for a viscoelastic half-space and axisymmetric indenter – solution by the method of dimensionality reduction, zamm zeitschrift für angewandte mathematik und mechanik, 96(8), pp. 956-967. 7. kürschner, s., filippov, a.e., 2012, normal contact between a rigid surface and a viscous body: verification of the method of reduction of dimensionality for viscous media, physical mesomechanics, 15(5-6), pp. 270-274. 8. willert, e., popov, v.l., 2018, short note: method of dimensionality reduction for compressible viscoelastic media. i. frictionless normal contact of a kelvin-voigt solid, zamm zeitschrift für angewandte mathematik und mechanik, 98(2), pp. 306-311. 9. popov, v.l., heß, m., 2014, method of dimensionality reduction in contact mechanics and friction: a users handbook. i. axially-symmetric contacts, facta universitatis-series mechanical engineering, 12(1), pp. 1-14. 10. popov, v.l., 2017, contact mechanics and friction. physical principles and applications, 2nd edition, springer, berlin heidelberg. 11. popov, v.l., heß, m., willert, e., 2018, handbuch der kontaktmechanik – exakte lösungen axialsymmetrischer kontaktprobleme, springer, berlin heidelberg. 12. lee, e.h., radok, j.r.m., 1960, the contact problem for viscoelastic bodies, journal of applied mechanics, 27(3), pp. 438-444. 13. willert, e., kusche, s., popov, v.l., 2017, the influence if viscoelasticity on velocity-dependent restitutions in the oblique impact of spheres, facta universitatis-series mechanical engineering, 15(2), pp. 269-284. 14. greenwood, j.a., 2010, contact between an axisymmetric indenter and a viscoelastic half-space, international journal of mechanical sciences, 52(6), pp. 829-835. facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 121 134 https://doi.org/10.22190/fume191129013t © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  the selection of optimal reversible two-speed planetary gear trains for machine tool gearboxes sanjin troha 1 , željkovrcan 1 , dimitar karaivanov 2 , madina isametova 3 1university of rijeka, faculty of engineering, croatia 2university of chemical technology and metallurgy, sofia, bulgaria 3satbayev university, almaty, kazakhstan abstract. the application of multi-criteria optimization to two-carrier two-speed planetary gear trains is outlined in this paper. in order to determine the mathematical model of multi-criteria optimization variables, the objective functions and conditions must be determined first. two-carrier two-speed planetary gear trains with brakes on coupled shafts are analyzed in this paper. the mathematical model covers the determination of the set of the pareto optimal solutions as well as the method for selecting an optimal solution from this set. a numerical example is provided to illustrate the procedure in which the optimal two-speed planetary gear train is selected and defined by design parameters. key words: multi-criteria optimization, two-speed planetary gear trains, pareto optimal solutions, coupled shafts 1. introduction multi-criteria optimization problems are very common in many scientific and technical solutions. the optimization of gear trains as complete technical systems implies a complex mathematical model that has to describe actual system operation in actual circumstances. planetary gear trains (pgt)s are a type of geared transmission which offers many advantages in comparison to conventional gearboxes. therefore, the area of application of single-stage and multi-stage pgts in mechanical engineering is increasing. multi-stage pgts are obtained by linking the shafts of one or two single stage pgts. a special multi-stage pgt is a two-speed, two-carrier pgt consisting of two coupling shafts and four external shafts. this type of compound gear train has many important characteristics, the most notable being the ability to change the transmission ratio and the direction of rotation of the output received november 29, 2019 / accepted march 03, 2020 corresponding author: sanjin troha university of rijeka, faculty of engineering, vukovarska ul. 58, 51000, rijeka, croatia e-mail: sanjin.troha@riteh.hr 122 s. troha, ž. vrcan, d. karaivanov, m. isametova shaft under load on demand. therefore, they are particularly suited for applications as main drives in machinery, e.g. machine tools, cranes, etc. 1.1. state of the art the application of multi-criteria optimization to gear transmissions, especially planetary gear transmissions has not been the subject of many research papers. a population-based evolutionary multi-objective optimization approach, based on the concept of pareto optimality, is proposed in paper [1] for the design of helical gears. paper [2] deals with the selection of the best parameters in order to obtain the required gear quality and with the optimization of the design process itself. an analytical and computer aided procedure for the multi-criteria design optimization of multi-stage gear transmission is presented in paper [3]. the process of planetary gear transmission optimization is shown in paper [4] as a method which leads to the optimal solution.on the other hand, there were very few research efforts dealing with two-speed, two-carrier pgts until 2003 [5]. two-carrier pgts consisting of two coupled and four external shafts, which enable two-speed transmissions, have significant application as gearboxes [5]. the possible schemes of these transmissions are presented in [5-10], while possible transmission structures with convenient brake layouts which could be used as two-speed transmissions are examined in [5]. a method for investigating the transmission ratio, the internal power flows and the efficiency of complex multi-carrier gearings is presented in [7]. an optimization of the two-carrier two-speed pgts with brakes on single shafts is provided in [11]. in this example, a fishing boat transmission was chosen as input data for the numerical example of multi-criteria gearbox optimization. this paper provides an optimization of the two-carrier two-speed pgts with brakes on the coupled shafts, in continuation of the optimal selection choice methodology application. the characteristics of a machine tool transmission have been used as input data for the numerical example of multi-criteria optimization application. apart from the determination of the set of the pareto optimal solutions, the weighted coefficient method was applied in order to determine the optimal solution. 2. mathematical model for planetary gear train optimization the two-carrier two-speed pgts with brakes on coupled shafts are built from basic types of pgt. the basic type of pgt (type 2k-h, variant a) is an arrangement using a central sun gear with external gearing (1), external ring gear with internal gearing (3), planet gears with external gearing(2) and planet carrier (h), as shown in fig.1. the planets are in simultaneous mesh with the sun gear and the ring gear. also, a wolfarnaudov’s symbol can be used, indicating the torque on the main elements as a function of basic transmission ratio i0. the equations for the basic transmission ratio and the ideal torque ratio calculation are also pointed out in fig. 1. the carrier shaft is the summary element of the basic pgt, as a negative transmission ratio is obtained by stopping the planet carrier, indicating a change of the direction of rotation of the output element. the choice of optimal reversible two-speed planetary gear train for machine tools gearboxes 123 basic transmission ratio: 0130  zzi . ideal torque ratio: 3 3 0 1 1 | | 1 t z t i t z       . fig. 1 basic type of pgt and wolf-arnaudov’s symbol with torque ratios (1 – sun gear; 2 – planet; 3 – ring gear; h – planet carrier) the process of finding the optimal solution starts with the definition of a mathematical model, as stated in [12]. the complete mathematical model of the basic type of pgt was described in the aforementioned paper and a brief summary will be also presented in this section. it is necessary to define variables, objective functions and functional constraints in order to define a mathematical model. 2.1. variables the following variables are considered by this model: the number of teeth of sun gear z1, the number of teeth of planet gears z2, the number of teeth of ring gear z3, the number of planets wn , gear module nm and gear face width b. the optimization variables are of the mixed type: the gear tooth numbers are positive and negative integers, the number of planets is a discrete value, the module is a discrete standard value (acc. to iso 54), while the face width is a continuous variable. the gear tooth numbers and the number of planets are non-dimensional values while the module and the face width are given in millimeters. 2.2. objective functions the characteristics used by the model to determine the objective functions are the volume, mass, efficiency and manufacturing cost of pgts. the volume of the pgts is used as an overall dimension expression, and the gears are approximated with a cylinder volume with the diameter equal to the pitch diameter and the height equal to the face width. the fact that the planets are inside the ring gear makes it possible for the pgt volume to be expressed by eq. (1) 2 23 3 cos cos cos4            wt tn zm bv     (1) where t is the transverse pressure angle, wt23 is the working transverse pressure angle for the pair 2-3 and β is the helix angle at the pitch diameter. 124 s. troha, ž. vrcan, d. karaivanov, m. isametova since the mass of a particular gear is determined as gear volume multiplied by the density of gear material and fact that mass is determined as the sum of all gear masses in a pgt, this criterion has been expressed as eq. (2):        23 2 2 2 33 12 2 2 2 22 12 2 2 2 112 2 cos cos cos cos cos cos cos 25.0 wt t wt t w wt tn zkzknzk m bm         (2) efficiency is one of the most important criteria for the design and evaluation of the arrangement quality. the calculation of the gear transmission efficiency is generally confined to losses depending on the friction on tooth flanks in contact while neglecting the losses in bearings and losses due to oil viscosity, i.e. restricted to the calculation of contact power losses [12-14]. the model, followed by the developed computer program, is adjusted to the most commonly used variant with the sun gear as the input element, and the carrier as the output element while the ring gear remains stationary. basic pgt efficiency in this case is given by eq. (3) [12]: 0 00 1 1 i i      (3) where 0 is the efficiency with the planet carrier stationary, as expressed by eq. (4)            32113 3 0 20.035.015.0 1 zzzzz z  (4) the economic demands must be also taken into consideration in the technoeconomical optimization, as these demands are directly related to production costs. the time needed for the manufacture of gears is taken as a measure of the production costs and as an economic factor. this function is then determined as a sum of the time periods needed for the manufacturing of sun gear (tp1), planets (tp2) and ring gear (tp3), i.e. 321 ttntf wt  (5) the production times are determined according to fette, lorenc and höfler [15]. 2.3. functional constraints the functional constraints are the conditions required for the proper operation of a system. there are numerous exceptions that need to be taken into consideration for pgts to operate correctly in comparison to conventional gear transmissions. the exceptions presented in this model are related to assembly conditions, geometrical conditions and strength conditions.the assembly conditions include the conditions of coaxiality, adjacency and conjunction [16]. the geometrical conditions are related to the undercutting and profile interference, the ratio of the pressure angle to the working transverse pressure angle, the tooth thickness and the tooth space width, the transverse contact ratio value, the sliding speeds at the point of contact, the ratio of the pinion face width to the pinion reference diameter, etc. these conditions have been ensured in accordance with the actual standards (iso tc 60 list of standards 090915). the strength conditions, safety factors for bending strength and surface durability of each gear, are checked according to iso 6336-1 to iso 6336-3 [17]. the choice of optimal reversible two-speed planetary gear train for machine tools gearboxes 125 2.4. steps in the optimization process the optimization process begins by generating all solutions for the assigned input data. all 6-tuples of design parameters (z1, z2, z3, nw, mn, b) satisfying the functional constraints are generated for the given input data (transmission ratio, input number of revolution, input torque, service life in hours, application factor, accuracy grade q(din 3961), and the values of the objective functions for every 6-tuple are computed. these 6tuples form a set of feasible solutions. an optimal solution is then selected, based on the established objective functions and constraints, and determined by variables. the mathematical model of nonlinear multi-criteria problem can be formulated as follows:  1 2 max ( ), ( ), , ( ) subject to k f x f x f x x s (6) here, f1(x),..., fk(x) are objective functions, x = (x1,...,xn) is the vector of decision variables and s is the set of feasible solutions. every point x  s is mapped to the point (f1(x), f2(x),..., fk(x)) in k  dimensional objective space. therefore, one can observe the objective set: 1 2 {(( ( ), ( ), , ( ) | ) k f f x f x f x x s  (7) the notation „maxˮ determines a simultaneous maximization of all the objective functions. if any objective function has to be minimized, the minimization of function fi(x) is performed by maximization of function  fi(x). according to the structure of feasible set s, discrete multi-criteria optimization problems do exist. in this pgts problem, six decision variables exist, corresponding to the basic design parameters: x = (x1,x2,x3,x4,x5,x6) = (z1,z2,z3,nw,mn,b). furthermore, there are four objective functions: volume v(x), mass m(x), efficiency (x) and production costs t(x): 1 2 3 4 ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ) p f x v x f x m x f x x f x t x        (8) therefore, the mathematical model of nonlinear multi-criteria problem in concrete task, can be formulated as follows: 1 2 3 4 max{ ( ), ( ), ( ), ( )} subject to f x f x f x f x x s (9) as multi-criteria optimization problems are mathematically ill-defined which can be seen from the definition, a criterion for selecting the optimal solution must be defined. the most important criterion for selecting these „equally goodˮ solutions is the pareto optimality concept: the solution x  s is pareto optimal if no solution y  s exists which maintains fi(x)  fi(y) for all i = 1,...,n and maintains strict inequality, i.e. fi(x) < fi(y) for at least one index i. determination of the pareto optimal solutions set is the first stage in optimal solution finding. the optimal solution is selected in the next stage, where the weighted coefficients method is applied to select the optimal solution from the pareto solutions set. 126 s. troha, ž. vrcan, d. karaivanov, m. isametova 2.5. weighted coefficients method for this method, the following scalarized problem must be set up: 0 0 1 1 max ( ) ( ) ( ) . . m m m f x w f x w f x s t x s       (10) weighted coefficients (or weights) iw are positive real numbers and 0 0 1( ) ( ) ( ) i i i f x f f x are normalized objective functions where 0 if are normalizing coefficients [12]. all solutions obtained by using this method are pareto optimal [12]. this model may be used regardless of the existence of priority functions or not [15]. the complete optimization procedure is implemented in the plangears software. 3. two-speed two-carrier planetary gear trains 3.1. two-carrier planetary gear trains structures and labeling method in cases where two-speed transmissions are required, a mechanism obtained by connecting two basic pgts shown in fig. 1 is one of the best suited design solutions. by joining two shafts of one pgt with two shafts of another pgt a mechanism is formed with four external shafts in total, fig. 2. fig. 2 symbolic representation of a compound planetary gear train with four external shafts the two component trains can be joined in in 12 different ways, resulting in a pgt with four external shafts [18]. an alphanumerical label (s11…s56) is attached to each of the 12 structural schemes, indicating the ways of connection between the shafts of the main elements of both component trains (fig. 3). in every presented scheme it is also possible to place the brakes as well as the driving and the driven machine on external shafts in 12 different ways (v1…v12), corresponding to layout variants (fig. 4). the compound trains in consideration can be classified into three different groups according to whether the brakes are placed on the coupled shafts, on the single shafts or both on the coupled and the single shaft. the choice of optimal reversible two-speed planetary gear train for machine tools gearboxes 127 fig. 3 systematization of all schemes of two-carrier planetary gear train with four external shafts fig. 4 systematization of all layout variants (a-input shaft, b-output shaft) 3.2. operations of planetary gear trains with different layout variants by placing the brakes on two shafts, a braking system is obtained in which the alternating activation of the brakes shifts the power flow through the pgt, causing a change of the transmission ratio. some pgts of this type are described in [5,7,18,19,20]. the possible power flow paths for pgts are analyzed, and functions of the transmission ratio for some trains of this type are deduced in [5,18,19]. 15 kinematic schemes of the considered type are presented in [6], and achievable values of transmission ratios and efficiencies are given. a computer program dvobrz for the selection of an optimal variant of similar multi-speed pgts is described in [5,18,19], and charts of shifting capabilities for all possible two-speed pgts are given in [5]. each variant has its own characteristics that determine the possibilities of transmission ratio changes. some variants can be presumed to work in both transmission ratios as reducers and multipliers, while other variants work like a reducer with one ration and like a multiplier with the other ratio. also, some variants change the direction of rotation when after a transmission ratio change, while other variants keep the direction of rotation after changing the transmission ratio. the transmission ratio of each pgt stage depends only on its basic transmission ratio (ideal torque ratio). the compound train with brakes on the coupled shafts (layout variant v1 in the fig. 4) with power flows when some of the brakes are active is symbolically shown in fig. 5 by means of a wolf-arnaudov symbol. the green dotted line represents the power of relative motion. there are two possible directions: from the sun gear to the ring gear or from the ring gear to the sun gear. also, the expressions for transmission ratios and efficiency obtained by using torque method when brake br1 is activated are given on the left side, while the expressions for brake br2 activated are given on the right side [5]. 128 s. troha, ž. vrcan, d. karaivanov, m. isametova 1 (1 ) 1 i i b ii i i br i ii a ii ii t t t t t t i t t t t t                  ; 2 (1 ) (1 )(1 ) 1 b i ii i ii br i ii a t t t t t i t t t               0 0 0with losses 1 without losses 1 ( ) ( ) (1 ) i ii i ii ii iib br ib ii ii t t tt tt t t               ; with losses 0 0 2 without losses ( ) (1 )(1 ) ( ) (1 )(1 ) b i i ii ii br b i ii t t t t t t            fig. 5 power flows on the wolf-arnaudov’s symbol through the train with brakes on the coupled shafts regardless of which brake is applied, both the component trains operate actively, as seen in fig. 5. the power input and output are on the single shafts. also, the direction of the power flow is the same in both variants. when the upper brake (br1) is applied, the input element is the sun gear of the first stage. the power is transmitted through the ring gear of the first stage and the ring gear of the second stage to the output element carrier of the second stage. in the other case, when the lower brake (br2) is applied, the path from the input element (sun gear of the first stage) to the output element (carrier of the second stage) includes the carrier of the first stage and the sun gear of the second stage. as this transmission changes the direction of rotation with the transmission ratio, it is suitable for application in the machine tools which have a working motion with considerable load at low speed and a return motion to the initial position at high speed and light load. the choice of optimal reversible two-speed planetary gear train for machine tools gearboxes 129 4. results and discussion variant s15v1(figs. 6 and 7) was chosen to demonstrate the procedure of multi-criteria optimization application. a symbolic review of the transmission composition with kinematic scheme is shown in fig. 6, and the path of the power flow through the transmission is shown in fig. 7. a) b) fig. 6 symbolic review of transmission composition (a), kinematic scheme (b) a) b) fig. 7 power flow through the transmission on the conceptual scheme with brake br1 applied (a) and with brake br2 applied (b) the type of transmission is selected according to the transmission requirements of the machine tool concerned. the necessary transmission ratios are: ibr1 = 6 in one direction (with br1activated) and ibr2 = 40 in the other (with br2 activated). in this case, the ideal torque ratios for both the planetary gear stages can be defined from the shifting capabilities diagram, fig. 8 [5]. 130 s. troha, ž. vrcan, d. karaivanov, m. isametova fig. 8 shifting capabilities diagram of compound trains s15v1 the transmission ratios are defined as functions of ideal torque ratios, 1 ( / ) br i ii i t t   (1 ) ii t  and 1 (1 )(1 ) br i ii i t t   , so the ideal torque ratios are defined by using equations from fig. 8: 09.5it and 57.5iit [5]. 4.1. the first compound gear train stage (i) with brake br1 applied and brake br2 turned off, the carrier of the first stage and sun gear of the second stage are immovable. the first stage is determined in this mode as the torque of the first stage is greater in this mode. the input of the system (a) is the sun gear of the first stage (figs. 6 and 7a). the output of the system (b) is the planet carrier of the second stage (figs. 6 and 7). the power is transmitted to the ring gear of the first stage and then to the ring gear of the second stage and finally to the carrier of the second stage. the input data required for the multi-criteria optimization application is: i0 = 5.09, nin = 2850min1, tin = 33.5nm (p = 10kw), l = 8000 h, ka = 1.25, it7 for all gears, material z1/material z2/material z3= 20mocr4/20mocr4/34crnimo6, sh min = 1.1, sf min = 1.2, i = 3%, z1 = 1530. the feasible set consists of 834 solutions, from which 43 pareto solutions are selected. by application of the weighted coefficient method with weighted coefficients: w1 = 0.5, w2 = 0.0, w3 = 0.0, w4 = 0.5 the solution shown in table 1 is obtained, with a set of objective functions values shown in table 2. the weighted coefficients are chosen in accordance with techno-economic optimization requirements. the choice of optimal reversible two-speed planetary gear train for machine tools gearboxes 131 table 1 optimal solution of the first stage in the first case variable values x1 = z1 x2 = z2 x3 = z3 x4 = nw x5 = mn x6 = b 15 31 -78 3 2 16 table 2 objective functions for solution shown in table 1 f1 in mm3 f2 in kg f3 f4 in min 305815.19 1.4529 0.9865 88.523 by prioritizing other objective functions, i.e. by choosing other values for weighted coefficients, other solutions would become optimal. the differences between solutions obtained in this way are logical. for example, the efficiency can be increased only with a large number of teeth considering eq. (3) and the fact that efficiency is the only function that has to be maximized. 4.2. the second compound gear train stage (ii) with brake br2 applied and brake br1 turned off, the ring gears of both the stages are immovable. the second stage is determined in this mode as the torque at the sun gear of the second stage will be greater in this mode. as in the previous mode, the input of the system (a) is the sun gear of the first stage and the output of the system (b) is the carrier of the second stage (figs. 6 and 7b). the power is transmitted to the carrier of the first stage and then to the sun gear of the second stage and finally to the carrier of the second stage. since the ideal torque ratio of this stage is tii = 5.57, the basic transmission ratio is i0 = t = 5.57 and the transmission ratio of the second stage in this mode is i = 1  i0 = 6.57. the input parameters of the second stage are equal to the output parameters of the first stage. because of that, with brake br2 applied, the ring gear of the first stage is also immovable, and the corresponding input number of revolutions at the second stage is calculated by means of the already defined first stage: 1-min677.459 2.6 2850  i n nn outiiinii the torque on the sun gear of the second stage is calculated using the following equation: 1 (1 ) 33.5 (1 5.2) 207.7 nm ii a i t t t      the other input data required for the multi-criteria optimization application in stage is equal to the one in the first stage: l = 800 h, ka = 1.25, it7 for all gears, material z1/material z2/material z3=20mocr4/20mocr4/34crnimo6, sh min = 1.1, sf min = 1.2, i = 3%, z1 = 1530. the feasible set consists of 1778 solutions, from which 43 pareto solutions have been isolated. by application of the weighted coefficient method with weighted coefficients: w1 = 0.5, w2 = 0.0, w3 = 0.0, w4 = 0.5 the solution shown in table 3 was obtained, using a set of objective function values shown in table 4. 132 s. troha, ž. vrcan, d. karaivanov, m. isametova table 3 optimal solution of the second stage variable values x1 = z1 x2 = z2 x3 = z3 x4 = nw x5 = mn x6 = b 15 32 -81 3 2.75 27 table 4 objective function for solution shown in table 3 f1 in mm3 f2 in kg f3 f4 in min 1040497.69 4.985 0.9865 121.85 by comparing the design parameters of both the stages it can be concluded that the design parameters enable a compact transmission design that is very important for installation in a machine tool. also, the pareto optimality concept as the criterion for selecting an equally good solution can be applied to compound pgt according to these criteria. furthermore, the weighted coefficient method can be used for selecting the optimal solution from a pareto set, and it can be adjusted to varying impacts of individual criteria functions. since the numbers of teeth of all gear are known, it is now possible to determine the realized transmission ratios and efficiency with brake br1 applied and with brake br2 applied, according to equations given in fig. 5.  ideal torque ratio in the first stage: 3 1 | | | 78 | 5.2 15 i z t z     .  ideal torque ratio in the second stage: 3 1 | | | 81| 5.4 15 ii z t z     .  transmission ratio with br1 activated: 1 5.2 (1 ) (1 5.4) 6.16 5.4 i br ii ii t i t t           .  transmission ratio with br2 activated: 2 (1 ) (1 ) (1 5.2) (1 5.4) 39.68 br i ii i t t           . the deviations of the actual and required transmission ratios are in the permissible range.  basic efficiency of the first stage: 3 0 3 1 1 2 3 0.15 0.35 0.20 ( 78) 0.15 0.35 0.20 1 1 0.984 ( 78) 15 15 31 ( 78) i z z z z z z                        .  basic efficiency of the second stage: 3 0 3 1 1 2 3 0.15 0.35 0.20 ( 81) 0.15 0.35 0.20 1 1 0.98442 ( 81) 15 15 32 ( 81) ii z z z z z z                        . the choice of optimal reversible two-speed planetary gear train for machine tools gearboxes 133  total efficiency with br1 applied: 0 0 0 1 1 0.9816 (1 ) i ii i ii ii ii br i ii ii t t t t t t              .  total efficiency with br2 applied: 0 0 2 (1 )(1 ) 0.97361 (1 )(1 ) i i ii ii br i ii t t t t          . it can be noticed that both efficiencies are very high (97…98%), and that the efficiency with brake br1 on is slightly higher than the efficiency with brake br2 on. the second step can be performed by including the efficiency determined by defined gear tooth numbers. the calculation of input torque is shown here: 1 0 (1 ) 33.5 (1 0.984 5.2) 204.91 nm ii a i i t t t         . by applying the procedure of the optimal solution where only the value of the input torque has been changed, the solution shown in table 5 is obtained using the objective function shown in table 6. the feasible set consists of 1767 solutions, from which 44 pareto solutions can be deducted. the weighted coefficients have the following values: w1 = 0.5, w2 = 0.0, w3 = 0.0, w4 = 0.5. table 5 optimal solution of the second stage in the second step variable values x1 = z1 x2 = z2 x3 = z3 x4 = nw x5 = mn x6 = b 15 32 -81 3 2.75 26 table 6 objective function for solution shown in table 5 f1 in mm3 f2 in kg f3 f4 in min 1001960.7375 4.80 0.984 119.611 a difference is noticed in one variable only – face width. as the input torque is slightly lower, the expected face width will be smaller too. the distinction is negligible; therefore, it is not necessary to carry out the procedure including the efficiency determined by a defined number of teeth of the first stage. 5. conclusion an original method that combines two computer programs (dvobrz and plangears) for multi-criteria optimization of two-carrier two-speed pgts with brakes on coupled shafts has been presented in this paper. these compound gear trains consist of two basic type of pgts and have considerable application in systems which need different transmission ratios and direction changes (e.g. as machine tool gearboxes which work with 134 s. troha, ž. vrcan, d. karaivanov, m. isametova a considerably greater transmission ratio in one direction and direction changing with a smaller transmission ratio in the other). the same procedure was successfully implemented in the optimal solution choice of two-carrier two-speed pgts with brakes on single shafts, leading to a universal method of compound pgt optimization. the optimal solution is determined by considering design parameters, such as mass and production cost as objective functions, and by using multi-criteria optimization and the weight coefficient method for choosing the optimal solution from the pareto optimal solution set. this approach can be successfully used for the basic pgt type and compound gear trains assembled from basic types, as shown in this paper. the results obtained using this procedure are in accordance with the literature on technical system optimization and indicate a good choice of optimization methods. furthermore, this approach indicates a possibility for application to other pgt types. references 1. tudose, l., buiga, o., jucan, d., stefanache, c., 2008, multi-objective optimization in helical gears design, the fifth international symposium about design in mechanical engineering-kod 2008, novi sad, serbia, pp. 77-84. 2. tkachev, a., goldfarb, v., 2009, the concept of optimal design for spur and helical gears, the 3rdinternational conference power transmissions '09, october 2009, chalkidiki, greece, pp.59-62. 3. rosić, b., 2001, multicriterion optimization of multistage gear train transmission, facta universitatisseries mechanical engineering, 1(8), 2001, pp. 1107-1115. 4. brüser, p., grüschow, g., 1989, otimierung von planetengetrieben,antiebstechnik, 2, pp. 64-67. 5. troha, s., 2011, analysis of a planetary change gear train’s variants, (in croatian), phd thesis, university of rijeka, engineering faculty, rijeka, croatia. 6. kudrjavtsev, v. n., kirdyiashev, i. n., 1977, planetary gears, (in russian), handbook, mashinostroenie, leningrad. 7. arnaudow, k., karaivanov, d., 2005, systematik, eigenschaften und möglichkeiten von zusammengesetzten mehrsteg-planetengetrieben, antriebstechnik, 5, pp. 58-65. 8. ivanov, a.n., 1990, evaluation of diametric dimensions of planetary gearboxes in the design phase, (in russian),vestnikmashinostroenie, 7, pp. 16-19. 9. lechner, g., naunheimer, h., 1999, automotive transmissions, springer-verlag, heidelberg. 10. jelaska, d., 2012, gears and gear drives, university of split, croatia. 11. stefanović-marinović, j., troha, s., milovančević, m., 2017, an application of multicriteria optimization to the two-carrier two-speed planetary gear trains, facta universitatis-series mechanical engineering, 15(1), pp. 85-95. 12. stefanović-marinović, j., petković, m., stanimirović, i., milovančević, m., 2011, a model of planetary gear multicriteria optimization, transactions of famena, 35(4), pp. 21-34. 13. sriatih, a., yedukondalu, g., jagadeesh, a., 2011, mechanical efficiency of planetary gear trains: an estimate, mechanical engineering research, 1(1), pp. 97-102. 14. del castillo, j.m., 2002, theanalytical expression of the efficiency of planetary gear trains, mechanism and machine theory, 37, pp. 197-214. 15. stefanović-marinović, j., 2008, multicriterionoptimization of planetary power transmission gear pairs, (in serbian), phd thesis, university of niš, faculty of mechanical engineering, niš, serbia. 16. niemann, g., winter, h., 1989, maschinenelemente, band ii, zweite völlig neubearbeite auflage, springer-verlag berlin. 17. international organization for standardization, 2006, calculation of load capacity of spur and helical gears, international standard iso 6336-2. 18. troha, s., žigulić, r., karaivanov, d., 2014, kinematic operating modes of two-speed two-carrier planetary gear trains with four external shafts, transactions of famena, 38(1), pp. 63-76. 19. troha, s., lovrin, n., milovančević, m., 2012, selection of the two–carrier shifting planetary gear train controlled by clutches and brakes, transactions of famena, 36(3), pp. 1-12. 20. troha, s., petrov, p., karaivanov, d., 2009, regarding the optimization of coupled two carrier planetary gears with two coupled and four external shafts, machine building and electrical engineering, 1, pp. 49–56. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 115 126 https://doi.org/10.22190/fume180321014c © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper a critical assessment of kassapoglou's statistical model for composites fatigue udc 519.2:539.4 michele ciavarella 1 , giuseppe carbone 1 , vladimir vinogradov 2 1 dmmm, politecnico di bari, italy 2 school of civil engineering & geosciences, newcastle university, u.k. abstract. kassapoglou has recently proposed a model for fatigue of composite materials which seems to suggest that the fatigue sn curve can be fully predicted on the basis of the statistical distribution of static strengths. the original abstract writes expressions for the cycles to failure as a function of r ratio are derived. these expressions do not require any curve fitting and do not involve any experimentally determined parameters. the fatigue predictions do not require any fatigue tests for calibration". these surprisingly ambitious claims and attractive results deserve careful scrutiny. we contend that the result, which originates from the reliability theory where exponential distributions is sometimes used to model distribution of failures when age (or wearout) has no influence on the probability of failure, does not conform to a fatigue testing with the resulting sn curve distribution. despite kassapoglou's attempt to use a wearout law which seems to confirm this result even with wearout, we contend that a proper statistical treatment of the fatigue process should not make wear-out constants disappear, and hence the sn curves would depend on them, and not just on scatter of static data. these concerns explain the large discrepancies found by 3 independent studies which have tried to apply kassapoglou's model to composite fatigue data. key words: composite materials, fatigue, wearout models, kassapoglou model, strength-life equal rank, statistics 1. introduction the strength-life equal rank assumption wear-out models for fatigue of composite materials were first presented by hahn and kim [1], and later as a fitting approach to fatigue data by sendeckyj, which hides a derivation based on a “damage tolerance” approach [2]. a received march 21, 2018 / accepted may 12, 2018 corresponding author: michele ciavarella dmmm, politecnico di bari, italy e-mail: mciava@poliba.it 116 m. ciavarella, g. carbone, v. vinogradov significantly different model has been proposed more recently by kassapoglou [3-5] (in the following, “ref. [3-5]”), which in fact claims an extremely strong result: that of predicting sn curve of a material from just the static data. already in the end of the 1800's for metals there were attempts to relate static data to fatigue ones, and even today only crude approximations can be made on fatigue limit over static strength (the so-called fatigue ratio), which are generally based on hardness tests. the use of simplified equations for sn curve is also well known in any fatigue textbook [6], but it is always clearly shown that any such empirical equation is limited, and that it makes, in general, only a very crude estimate. this justifies the industry of fatigue machine testing, which is by no means less flourishing in composite materials although composites are known to suffer more crucially to impact than to fatigue. hence no aircraft flying today is there without having passed a very serious fatigue testing certification procedure, and for a good reason. fatigue testing is required by any certification agency to get airworthiness certificates, and the cost of testing is huge. the idea to obtain even approximate results for fatigue from just static data is therefore still obviously attractive, since experience of static strength is so much easier and cheaper to obtain. therefore, it is surprising to read in [3] that expressions for the cycles to failure as a function of r ratio are derived. these expressions do not require any curve fitting and do not involve any experimentally determined parameters. the fatigue predictions do not require any fatigue tests for calibration”. further, that comparison to several test cases found in the literature show this first simple model to be very promising”, where for several test cases”, ref. [3] intends a few references (ref. [34-41] which are [7-14] here), where the error is said to be small but which in fact is not necessarily so. take fig.6 of kassapoglou's paper [3], where the agreement is said to be very good”: ref.[3]'s curve, which should be the median value, is seen to pass close to the lowest data, and hence the error in terms of life can be easily of 2 orders of magnitude. not much better can be said regarding fig.9 (where the author admits the agreement to be “not so good”): the author prefers to measure the error based on stress, and claims a 17% error is found clearly, the error in terms of life can be of various orders of magnitude. similar problems were found in figs.10 and 11, where the data are so few that do not really permit many conclusions to be drawn. therefore, the sentence in the conclusion the approach allows analytical determination of the ratio of mean to b or a-basis life which can be used in designing certification of qualification programs” seems premature. the data show that the link with static scatter is not so strong as to make any remote estimates of the sn curve slopes. ref. [3,4,5] seem to have followed an indication in ref. [8] which at page 527 refers to the fact that many composite wohler diagram (sn curves) seem to have a life distribution with weibull shape parameter close to 1 (exponential distribution), writing “the exponential distribution is sometimes used to model distributions of failures times for the reliability of a product. it is pointed out by chatfield that the exponential distribution governs systems where age has no influence on the probability of failure”.... “this would not normally be thought applicable to the fatigue failure of reinforced plastic, for which it is known that residual performance is in fact reduced as a result of the accumulation of damage”. ref. [4, 5], however, seem to confirm this result even in the counterintuitive case of wearout, by adopting a special wearout law. we shall show that this is contrived result, as confirmed by a few recent attempts in the literature to adopt this model [15-17]. we shall draw attention in this paper to the fact that we do not expect theoretically any reason for a good predictive capability to be realistic in general. a critical assessment on kassapoglou's statistical model for composites fatigue 117 the tendency to exponential distribution of fatigue lives is not very strong in general anyway, since in many large databases a full distribution of weibull shape parameters is found [8, 18]. fleck kang and ashby [18], in an authoritative review which contains also data on composite materials, produce a large set of maps covering a huge number of references, and in particular show in fig. 5 (fig.1, here reproduced with permission) the well-known fact that endurance limit σe scales in a roughly linear way with yield strength, σy. the fatigue ratio, defined as σe/σy (but more classically for metals, σe/σfs) at load ratio r = −1, appears as a set of diagonal contours. the value of fatigue ratio, for engineering materials, usually lies between 0.3 and 1. generally speaking, it is near 1 for monolithic ceramics, about 0.5 for metals and elastomers, and about 0.3 for polymers, foams and wood; the values for composites vary more widely--from 0.1 to 0.5. naturally, for fatigue limit in composites (as well as light alloys), often intended is the value at a given fixed number of cycles. this wide variation already makes one wonder that for composites the fatigue properties depend less on static properties than on other materials. fleck kang and ashby [17] remark the wide range of fatigue ratios shown by composites relates, in part, to the wide spectrum of materials used to make them, and to the necessarily broad definition of failure: in particulate composites, failure means fracture; in fibrous composites it means major loss of stiffness. fig. 1 from fleck kang and ashby [18] (with permission). classical plot of static vs. fatigue strength for many classes of materials 118 m. ciavarella, g. carbone, v. vinogradov clearly, the exact mechanisms for fatigue limit, if there is one, are microscopic and they, however, may interact with geometry, loading conditions, etc. for composite materials and structures in general, the failure mechanism can vary. for materials for which the endurance limit depends on the formation of slip bands, it is obvious to find a correlation with yield strength, but a full microscopic model for the shape of the sn curve is more difficult. fleck et al [18] summarize about sn curve: it is the failure envelope associated with a sequence of interdependent phenomena: cyclic hardening, crack nucleation and cyclic growth, and final fast fracture. for composites, the actual nature of each phenomenon is very different but their interdependence is also clear. kassapoglou's model is based on some statistical reasoning over the distribution of static strength, and the successive application of cycles. making a certain number of (reasonable) assumptions, he seems to derive apparently simple and clear results, which he then combines with a calculation of probability of failure. we shall discuss in the present note the basic results of kassapoglou's model in details. 2. main assumptions in kassapoglou’s model in ref. [3], kassapoglou makes a certain number of assumptions, including that the probability of failure stays constant, cycle-by-cycle. since the probability of failure during the first cycle is determined by a probability distribution function for the static strength, the author concludes that it remains the same for all subsequent cycles. in the later paper [4], this was obtained rigorously. in fact, even if the probability of failure did remain constant cycle-per-cycle, this is incorrect calculation for a fatigue experiment. that reliability theory permits "failure rate" to be obtained this way does not correspond to a sn curve. as an illustrative example of these incorrect calculations let us consider the following discrete analogue: 1) there is a bucket full of balls that are numbered with 1 (which is analogous to failure of the specimen under a certain applied stress, “success”) and 0 (unsuccessful event, no failure of the specimen). assume that the probability of failure is p, so that the probability of no-success is q = 1 − p. 2) one can perform a series of experiments counting number k of trials until the picked ball has a number 1 (success) on it. this event is called first success. after each attempt, in order to implement an analogue of the main assumption of the kassapoglou's model, one must put the picked ball back into the bucket: then, and only then the probability of success is exactly the same at each single trial. the resultant distribution of number of trials until first success is described by the geometric distribution density function: probability p1 that the first success" occurs at the k th trial (cycle) is 1( ) (1 ).p k p p  (1) 3) eq. (probability first success) is simply the probability of success (failure) at last attempt (cycle) k times the probability of no-failure at previous k − 1 attempts (cycles). the average expected number n of trials until the first success” is then given by a critical assessment on kassapoglou's statistical model for composites fatigue 119 1 1 1 (1 ) ,k k n kp p p       (2) 4) as indeed should be expected. 5) we can also calculate the probability that the failure of the sample occurs during the first n experiments (cycles), i.e. that the event “first success” happens at any cycle k in between 1 and n. this probability is the cumulative distribution of 1 1 1 1 1 1 ( ) ( ) (1 ) 1 (1 ) 1 exp log . 1 n n k n k k p n p k p p p n p                    (3) 6) if now the number of trials n is treated as a continuous variable, this function represents the so-called cumulative exponential distribution with the mean value 1 1 , 1 log(1 ) log 1 n p p      (4) 7) and if p is small, the two averages (average number of trials) and (exponential average) become close. this trivial example implements the main assumption of kassapoglou's model. however, a randomly picked specimen is tested each time; each test is an independent static test with the probability of failure p being equal to the probability that the static strength of the specimen is less than applied static stress σ. obviously, this type of test is irrelevant to the fatigue phenomenon. the resulting relation between the mean number of cycles and the applied load, mistakenly claimed by the author to be a sn curve, is actually the mean number of tested specimens until failure! if the same specimen undergoes subsequent loadings, it may fail only if its static strength decreases with cycles. however, this static strength degradation should be a material specific function and is not uniquely determined by the statistics of the static strength. while dispersion in the static strength of a material reflects the possible level of initial damage observed in the material, the fatigue failure phenomenon is the result of the initial damage growth and accumulation with cyclic loading. this growth would eventually appear in any specimen independently of the strength in other specimens and the statistics that describes the static strength scatter. 3. kassapoglou’s model with strength degradation in his 2012 phd thesis [5] and in his 2011 paper [4] kassapoglou extends the model and incorporates the residual strength degradation. suppose that a constant amplitude load with maximum stress σ (r = σmin/σ = 0) is applied to a composite structure. if static failure strength σfs of this structure is less or equal to σ (i.e. σfs ≤ σ) the structure will fail at n = 0, i.e. before the first cycle is completed, whilst if σfs > σ the structure will fail at the cycle n = n. within the kassapoglou's wear-out model quantity n is treated as constant and no failure can occur for 0 < n < n. these assumptions are, as we show in the sequel, the most critical flaws of kassapoglou's model. 120 m. ciavarella, g. carbone, v. vinogradov if the test had stopped at any cycle level n < n the structure would not have failed and it would still be able to carry load. however, a strength test on the structure would show a failure strength σfs > σr > σ, where σr is the residual strength. hence, during cycling, σr decreases from the static failure strength σfs > σ at n = 0, to σ after n cycles. kassapoglou's model starts with the assumption that the change in residual strength is proportional to the current residual strength, which in the simple case of zero fatigue limit can be written in the form ,r r d a dn    (5) where a > 0 is independent of n and σr. this wear-out model assumes that under a fixed amplitude the strength of a specimen with higher residual strength stress will degrade faster than one with lower residual strength, which is physically unreasonable. that is why a typical wear-out model would define the strength degradation rate as a reciprocal to the current residual strength. the above expression in kassapoglou's model results in a residual strength / ( 1) . n n r fs fs               (6) treating σ as a constant would results, following kassapoglou's arguments, in a weibull cumulative distribution pw(σr; βr, αr) for residual strength with shape and scale parameters 1 /( 1) 1 1 ; , 1 n n n n n r r n n n               (7) which vary with n in a simple manner, implying a clear reduction in experimental scatter with a lower stress level of testing (or longer lives). in particular, for n → n − 1, αr → ∞ which means that the distribution converges to the dirac delta function, the residual strength becomes a deterministic function, and βr = σ consistently to the fact that the residual strength tends exactly to the sn curve. there are many critical inconsistencies that one can spot immediately: firstly, n from the sn curve should itself be a variety, instead of being a deterministic and constant quantity. secondly, with a constant amplitude σ, during cycling the residual strength distribution cannot approach the sn curve from below, since a specimen that has strength below σ should have failed at an earlier stage and for any n the residual strength distribution should be truncated from below by applied stress σ. one can even and easily show that, in contrast to what has been claimed by k, kassapoglou's model cannot lead to a cycle-by-cycle constant probability of failure p(n), which indeed, even within the hypothesis of kassapoglou's wear out model, would be 0( ) ( ; , ) 1 ( ; , ) ,n w nn wp n p p            (8) where δkh is the kronecker delta (see appendix ii). the discussed inconsistencies in the model development refute the claim of rigorous proof of constant cycle-by-cycle probability of failure. here, it is instructive to cite that many authors in the literature find a critical assessment on kassapoglou's statistical model for composites fatigue 121 distribution of residual strength which shows a decrease of weibull's α, rather than an increase as predicted by kassapoglou. 4. comparison with experimental data the original kassapoglou's model did not find an exact weibull distribution for the life, but over a very wide range of p values (p < 0.1 hence, unless the applied load is very high, and close to the static strength) the two ratios of mean and modal lives to b-basis life (respectively 17.86 and 8.93) are essentially constant. this suggested the author of the original kassapoglou's model that the average of the two ratios, 13.4 is very close to the value of 13.6 determined in the navy reports [6] after statistical analysis of thousands of data points. using a slightly different re-derivation (see appendix i), we show that kassapoglou's approach essentially obtains an exponential distribution of fatigue lives, for a weibull starting point in scatter of static data. hence, it is correct to say that kassapoglou's approach, in a slightly re-elaborated form (see appendix i), we easily obtain the estimate on fatigue ratio (fr, defined as the ratio between fatigue limit at 10^6 cycle, and the static value" at n = 1), as 6max,lim,10 ( 6/ ) max,1 ( ) 10 ,fr        (9) where α is a weibull shape parameter in a 2 parameters weibull distribution, of the static strength distributions. notice that in this form, kassapoglou's model includes also the rratio effect, for 0 < r < 1 ie in the cases of pure tensile loading. for α we can make use not of few sparse references like k, but the thousands of test done by the navy. the distribution of fatigue lives is not as unique as to be an exponential αl = 1. a full distribution was found, which we may define as the distribution of scatter of fatigue lives, αl. a generally accepted approximation is to take αl = 1.25, but it is clear that its distribution is relatively wide, and depends also on the method used for analysis. also, the distribution of static strength scatter α has in turn a distribution, which we can denominate αa, see fig. 2. we can use the mean value of α which results from the fig. 2 to be 26. this corresponds to a value ( 6/26)( ) 10 0.5878,meanfr    (10) which is outside the known values of ashby fatigue ratios. this suggests, as confirmed by most experimental data we shall describe, that kassapoglou's method would tend to give unconservative estimates, as too high fr. this, however, depends very much on the type of materials under examinations. if the materials are of “poor” quality, full of defects, tending to having low α, then the very steep sn curve predicted by k may neglect the possible phase in sn curve where the degradation is not so evident, resulting as extremely conservative. on the other hand, for materials having very high α like close to a metal, kassapoglou's theory fails to capture the wearout at all, and results in too optimistic sn curve. unidirectional laminates will tend to have very horizontal fatigue lines, yet their static scatter may be significant. 122 m. ciavarella, g. carbone, v. vinogradov (a) (b) fig. 2 (a) distribution of fatigue lives scatter αl and (b) of scatter of static strength α hence even if the agreement with navy experiments has some very loose qualitative agreement in terms of scatter of fatigue lives, this is an oversimplification (a single mode value instead of the full distribution) and the huge risks of using this approximation even as a crude estimate is evident. the recent investigation of kassapoglou's method by the faa (tomblin and seneviratne, 2011 [5], appendix a) finds also the sn curve predicted by the original kassapoglou's model (which we found here as the mean life curve) to have rather erratic comparison with experimental data. in particular, in 14 sets of data, kassapoglou's model was found  accurate only for 2 sets,  conservative only for 2 sets (both by 1-2 orders of magnitude),  unconservative for the vast majority of data (10 sets), of which 5 perhaps by 1-2 orders of magnitude, 3 by 2-3 orders of magnitude, and 2 by 4-5 orders of magnitude. clearly, although the statistics does not say much, it shows a tendency of kassapoglou's method to overpredict fatigue life by large factors. examples given in chapt. 6 of [3] and in [19] show that more sophisticated methods with variable p function may improve the a critical assessment on kassapoglou's statistical model for composites fatigue 123 situation, although the examples given tend to predict longer lives than the original k method --we are not able to judge if the errors and approximations from the original k theory with "constant p value" continue to manifest their negative effect here. s/n curves based on the sendeckyj analysis [2] were found generally accurate and conservative, but this is to be expected since that method is a fitting method of sn data. in a recent book, vassilopoulos & keller [16] compare 4 methods to make a statistical analysis of fatigue data, which is a problem of enormous industrial interest since aeronautical structures are designed and certified using sn curves that correspond to high reliability levels in the range above 90% and conform with design codes, but without an impossibly expensive program of fatigue testing on a population of full scale structures. the method based on the normal lifetime distribution (nld) was found as nonconservative, giving a median sn curve which is closer to the median sn curve of astm than the 95% reliability one. whitney's pooling scheme and sendeckyj's wear-out model are found to produce similar sn curves, with whitney's easier to implement, as not requiring any optimization process, and sendeckyi being also less conservative. however, this is mainly due to the need of multiple fatigue results at each stress level, and no capability to consider static data. some significant problems were found in the fitting of sendeckyi's constant process, with strange slopes of the sn curve predicted, particularly when disregarding static strength data. a discussion follows on the appropriateness of including the static data in the fitting. kassapoglou's method is not even compared to the previous four, mainly because the static data were not enough to fit weibull distributions. it is discussed, however, in its extension to describe mean stress effect, in a later chapter on constant life diagrams. however, its assumptions are negatively judged this assumption oversimplifies the reality and masks the effect of the different damage mechanisms that develop under static loading and at different stages of fatigue loading, and the restricted use of static data disregards the different damage mechanisms that develops during fatigue loading and in many cases leads to erroneous results. in the evaluation of kassapoglou's model for one database the model proved to be inaccurate for the examined material's fatigue data. 4. conclusions in the original kassapoglou's model there is confusion between what we call fatigue and statistics of the static strength of a number of specimens, which stems from an incorrect reference to the reliability theory of failure rate of products. fatigue life (number of cycles) is mistakenly replaced with the number of tested specimens to find a specimen with strength less than the applied load. this number of specimen indeed solely depends on the initial statistical distribution of the static strength, while fatigue is related to damage accumulation in a specimen and its strength degradation with cycles, which contradicts the main assumption. one can also mistakenly deduce from the proposed model that if there is no dispersion in the static strength, for instance, all the specimen have exactly the same static strength, there is no such thing as an sn curve. kassapoglou's model is an interesting attempt of using wear-out models with degradation deterministic equations to predicting sn curves from static data only for composites (something which is not easy even with metals). however, its results do not look realistic at all, and indeed 124 m. ciavarella, g. carbone, v. vinogradov we have here explained why. not surprisingly, sn curves found in many independent assessments were found to be (generally) unconservative for the vast majority of data (10 sets) considered in faa 2011 report [15], at least by 1-2 orders of magnitude. we have given reasons for this effect, both theoretically and with additional estimates from large set of results from databases of composite materials. only "fitting" models can be considered reliable, as discussed by vassilopoulos & keller [16], and it should be remarked in this respect that an additional interesting wearout model is [21-23]. references 1. hahn, h.t., kim, r.y., 1975, proof testing of composite materials. j composite materials, 9, pp. 297-311. 2. sendeckyj, g.p., 1981, fitting models to composite materials fatigue data. test methods and design allowables for fibrous composites, in: chamis, c.c. (ed), astm stp 734. philadelphia, pa: american society for testing and materials, pp. 245-260. 3. kassapoglou, c., 2007, fatigue life prediction of composite structures under constant amplitude loading, j of composite materials, 41, pp. 2737-2754. 4. kassapoglou, c., 2011, fatigue model for composites based on the cycle-by-cycle probability of failure: implications and applications, j of composite materials, 45, pp. 261-277. 5. kassapoglou, c., 2012, predicting the structural performance of composite structures under cyclic loading, phd thesis, delft univ of technology, netherlands 6. juvinall, r.c., marshek, k.m., 2011, fundamentals of machine component design, 5th ed. john wiley & sons inc, usa. 7. lee, j-w., daniel, i.m., yaniv, g., 1989, fatigue life prediction of cross-ply composite laminates. in: lagace, p.a. (ed), composite materials: fatigue and fracture, second volume. astm stp 1012, philadelphia, pa: american society for testing and materials, pp. 19-28. 8. gathercole, n., reiter, h., adam, t., harris, b., 1994, life prediction for fatigue of t800/5245 carbon-fibre composites: i. constant amplitude loading, fatigue, 16, pp. 523-532. 9. amijima, s., fujii, t., hamaguchi, m., 1991, static and fatigue tests of a woven glass fabric composite under biaxial tension-torsion, composites, 22, pp. 281-289. 10. cvitkovich, m.k., o'brien, t.k., minguet, p.j., 1998, fatigue debonding characterization in composite skin/stringer configurations, in: cucinell, r.b. (ed.), astm stp 1330, philadelphia, pa: american society for testing and materials, pp. 97--121. 11. o'brien, t.k., 1988, fatigue delamination behavior of peek thermoplastic composite laminates, j. reinforced plastics and composites, 7, pp. 341-359. 12. o'brien, t.k., rigamonti, m., zanotti, c., 1988, tension fatigue analysis and life prediction for composite laminates, hampton, va: national aeronautics and space administration. technical memorandum 100549 13. maier, g., ott, h, protzner, a., protz, b., 1986, damage development in carbon fibre-reinforced polyimides in fatigue loading as a function of stress ratio, composites, 17, pp. 111--120. 14. gerharz, j.j., rott, d., schuetz, d., 1979, schwingfestigkeitsuntersuchungen an fuegungen in faserbauweise, bmvg-fbwt, pp. 79-23. 15. tomblin, j., seneviratne, w., 2011, determining the fatigue life of composite aircraft structures using life and load-enhancement factors, report dot/faa/ar-10/6. federal aviation administration, national technical information service, springfield, usa, www.tc.faa.gov/its/worldpac/techrpt/ar10-6.pdf (last access: 12.03.2018) 16. vassilopoulos, a.p., keller, t., 2011, fatigue of fiber-reinforced composites, london: springer-verlag, uk 17. andersons, j., paramonov, yu., 2011, applicability of empirical models for evaluation of stress ratio effect on the dura.bility of fiber-reinforced creep rupture-susceptible composites, j mater sci, 46, pp. 1705-1713. 18. fleck, n.a, kang, k.j., ashby, m.f., 1994, overview no. 112: the cyclic properties of engineering materials, acta metallurgica et materialia, 42, pp. 365-381. 19. whitehead, r.s., kan, h.p., cordero, r., saether, e.s., 1986, certification testing methodology for composite structures, vol i and ii. naval air development centre report no. 87042-60 (dot/faa/ct-86-39), http://www.dtic.mil/dtic/tr/fulltext/u2/b112288.pdf (last access: 15.03.2018) 20. kassapoglou, c., kaminski, m., 2011, modeling damage and load redistribution in composites under tensiontension fatigue loading, composites: a, 42, pp. 1783-1792. a critical assessment on kassapoglou's statistical model for composites fatigue 125 21. d'amore, a., caprino, g., stupak, p., zhou, j., nicolais, l., 1996, effect of stress ratio on the flexural fatigue behaviour of continuous strand mat reinforced plastics, science and engineering of composite materials, 5, pp. 1-8. 22. caprino, g., d'amore, a., 1998, flexural fatigue behaviour of random continuous fibre reinforced thermoplastic composites, composite science and technology, 58, pp. 957-965. 23. d'amore, a., caprino, g., nicolais, l., marino, g., 1999, long-term behaviour of pei and pei-based composites subjected to physical aging. composites science and technology, 59, 1993-200. 24. post, n.l., 2005, modeling the residual strength distribution of structural gfrp composite materials subjected to constant and variable amplitude tension-tension fatigue loading, phd thesis, university of virginia. blacksburg, virginia, usa appendix i – sn curve of kassapoglou’s model in a slightly different form we have shown that kassapoglou's model is incorrect. however, a simpler form can be adapted for comparative form in a much simpler form. in particular, using this equation for a sn curve at any quantile q [ log(1 )],n q            (a1) which obviously has mean value nm = (β/σ) α and modal value nmod = (β/σ) α log2, but mode value zero (because the distribution of lives is an exponential distribution αl = 1), we obtain a closed form version of the “incorrect” k model, which can be used more easily than the original kassapoglou's model which is not in closed form, and which obtains only the mode life nc , #cn           (a2) which in the present result, coincides with the present mean value. the distribution in terms of stress for given number of cycles, pw(σ; β/n 1/α , α) 1/ 1/ [ log(1 )] ;q n             (a3) this has obviously mean value σm = (β/n 1/α )г(1+1/α), whereas median value σmed = (β/n 1/α )(log2) 1/α . in other words, in this new form, the sn derives from weibull distribution both in terms of stress and life at all levels of stress including the original static distribution pw(σ; β, α) which is obtained consistently for n 1/α = 1. appendix ii – inconsistencies in the wear out model of ref. [4, 5] suppose we want to calculate the cumulative probability distribution of failure p(n) during the first n cycles assuming kassapoglou's wear-out model / ( 1) , n n r fs fs               (a4) 126 m. ciavarella, g. carbone, v. vinogradov where the σr is the residual strength, σfs is the static strength, σ is the applied fatigue load, n is the actual cycle number, and n is the number of cycle at which the samples fails (assuming its static strength is larger than σ). equation (wear out model) simply states that: (i) all sample which have a static strength σfs larger than the fatigue stress σ will fail at the same given number of cycles n = n, and (ii) samples with static strength σfs less than σ will all fail at cycle n = 0. therefore samples may fail either at n = 0 when σsf ≤ σ or at n = n when σsf ≥ σ, no failure may occur in between i.e. for 0 < n < n. within the kassapoglou's wear-out model we have: (a) the probability that failure occurs at n = 0 is p0 = pw(σ; β, α), (b) the probability that failure occurs at n = n is pn = p(σs > σ) = pw(σ; β, α), (c) the probability that failure occurs at n satisfying the condition 0 < n < n is pn = 0. now let us calculate the cumulative probability distribution of 0 n n kk p p   with n ≥ 0, which is the probability that the sample fails within the first n ≥ 0 cycles, i.e. pn = p(0 ≤ k < n). since failure cannot occur in between 0 and n (i.e. pk = 0 for 0 < k < n) the probability that failure occurs within the firs n < n must be equal to the probability that failure occurs at cycle 0, i.e. 0 ( ; , ); 0 ,n wp p p n n      (a5) whereas considering that for n ≥ n failures has necessarily occurred one as 1; .np n n  (a6) therefore the cumulative distribution presents two steps one of amplitude pw(σ) at n = 0 ant the other of amplitude 1 − pw(σ; β, α) at n = n. in between the cumulative probability distribution is constant. we stress that, as already shown, the probability of failure cycle per cycle is not constant indeed it is: ( ; , ); , 0; 0 , 1 ( ; , ); , 0; , n w n n w n p p n n p n n p p n n p n n                 (a7) in compact notation 0( ) ( ; , ) [1 ( ; , )],n w nn wp n p p          (a8) where δjk is kronecker's delta. eq. (probability cycle per cycle) shows that the probability of failure cycle per cycle is zero for 0 < n < n, thus revealing one of the serious mistakes of the kassapoglou model where the cycle per cycle probability of failure was assumed different from zero and equal to pw(σ; β, α). indeed, eq. (probability cycle per cycle) shows that the sample life n is a discrete statistical quantity which only takes two different values n = 0 and n = n, and failure at n = 0 occurs with probability pw(σ; β, α) whereas failure at n = n occurs with probability 1 − pw(σ; β, α). this allows to calculate within the wear-out model (wear out model) the expected life of the samples as [1 ( ; , )].wn n p     (a9) which as expected differs from the value obtained by k. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 457 465 https://doi.org/10.22190/fume170927025l original scientific paper numerical computation and prediction of electricity consumption in tobacco industry udc 519.6 mirjana laković 1 , ivan pavlović 1 , miloš banjac 2 , milica jović 1 , marko mančić 1 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of mechanical engineering, university of belgrade, serbia abstract. electricity is a key energy source in each country and an important condition for economic development. it is necessary to use modern methods and tools to predict energy consumption for different types of systems and weather conditions. in every industrial plant, electricity consumption presents one of the greatest operating costs. monitoring and forecasting of this parameter provide the opportunity to rationalize the use of electricity and thus significantly reduce the costs. the paper proposes the prediction of energy consumption by a new time-series model. this involves time series models using a set of previously collected data to predict the future load. the most commonly used linear time series models are the ar (autoregressive model), ma (moving average) and arma (autoregressive moving average model). the ar model is used in this paper. using the ar (autoregressive model) model, the monte carlo simulation method is utilized for predicting and analyzing the energy consumption change in the considered tobacco industrial plant. one of the main parts of the ar model is a seasonal pattern that takes into account the climatic conditions for a given geographical area. this part of the model was delineated by the fourier transform and was used with the aim of avoiding the model complexity. as an example, the numerical results were performed for tobacco production in one industrial plant. a probabilistic range of input values is used to determine the future probabilistic level of energy consumption. key words: electricity consumption, forecasting of electricity, ar model, monte carlo simulation, seasonality pattern received september 27, 2017 / accepted november 18, 2017 corresponding author: ivan pavlović faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: pivan@masfak.ni.ac.rs 458 m. laković, i. pavlović, m. banjac, m. jović, m. manĉić 1. introduction following the deregulations that took place in many countries over the last few decades, energy markets have been rapidly evolving and bringing to the attention of practitioners and researchers the challenging problems from the modeling perspective. energy consumption has been increasing constantly due to globalization and industrialization. with an increasing need to reduce the system waste impact on the environment and an ever increasing global demand for energy, especially in developing countries, it is becoming extremely important to develop even more accurate and systematic approaches for improving the design of energy systems. industry, transportation and buildings are the three main economic sectors that consume a significant amount of energy [1]. there are two characteristics that distinguish electricity from other commodities: 1. electricity is not storable to a large extent, and, 2. transport of electricity is linked to transmission networks. these facts have consequences for the market structure, price behavior and products. on the other hand, electricity demand depends on many factors such as temperature, humidity, precipitation, wind speed, cloud cover and amount of light. the impact of temperature on both electricity demand and price has been studied in many papers. results in ref. [2] confirm that consumer responses to higher electricity prices are conditional on temperature levels, particularly during the daytime and for households with high overall levels of electricity consumption and previous experience of time-of-use tariffs. the relationship between electricity load and air temperature has an important dynamic component, and ignoring this appears to bias the estimated effects of temperature on load [3]. research [4] estimates a forecasting equation for the hourly peak electricity demand for one day in the future. everyday activities are also very important. there are different electricity demands and consumptions during weekends and working days, in winter and summer, holidays, etc. a variety of methods and ideas have been tried for electricity price forecasting (epf), with varying degrees of success. the autoregressive model (ar) is supposed to be a stochastic process that is composed of a weighted sum of the previous value and a white noise error. ref. [5] describes a price forecasting module based on neural networks and performance evaluation. an overview of modeling approaches that concentrates on practical applications of statistical methods for day-ahead forecasting, discusses interval forecasts, and moves on to quantitative stochastic models for derivatives pricing (jump-diffusion models and markov regime-switching) is presented in work [6]. in [7] zareipour begins by reviewing linear time series models and nonlinear models (regression splines, neural networks), and then uses them for forecasting hourly prices in the ontario power market. in this paper we propose a new model for the dynamics of temperature which forms the basis for pricing weather derivatives. the model generalizes the continuous-time autoregressive models suggested in benth et al. [8], to allow for stochastic volatility effects. due to transmission networks the electricity market is not global. adjacent national electricity markets start to link to one another but the capacities give natural constraints. therefore, we focus on the presentation of the serbian electricity market which is relevant for our modeling. this paper presents the electricity price forecasting for the tobacco production industry where averaged different values for a one-year period are given [9]. electricity in the factory is mainly required for the propulsion of electric motors used for machine operation, compressed air production, pumps and fans. the total installed power of the electric motor is 1.2 mw. numerical computation and prediction of electricity consumption in tobacco industry 459 2. phases of forecast and simulation probabilistic forecasting of electricity demand is an important task for all active market participants. for electricity consumption, ct at time t, fundamental model equation, given in ref. [10], has the following form tttt rgudc  )(log , (1) where dt is the deterministic seasonal pattern, u(gt) is the function of residual grid load gt. the regression method was utilized to remove the effects of trend and seasonality on hourly electricity load demands with the help of dummy variables, which presents weather conditions for the adequate region. in modeling the seasonal cycle deterministically, there are several approaches. the discrete fourier transform (dtf) is considered to be the most accurate since it removes the seasonal cycle both in the mean and in the variance. however, other researchers have proposed a novel approach in modeling the seasonal cycle, which is an extension of the dft approach. since small misspecification in a dynamic model can lead to large pricing errors, we incorporate a wavelet analysis in the modeling process in order to calibrate our model. the fundamental idea behind the wavelets is an analysis according to scale. the wavelet analysis is an extension of the fourier transform, which superposes sins and cosines to represent other functions. the wavelet analysis decomposes a general function or signal into a series of (orthogonal) basis functions, called wavelets, with different frequency and time locations. seasonal mean function dt is assumed to be a real-valued continuous function. to capture the seasonal variations due to cold and warm periods of the year, and a possible increase in temperatures due to global warming and urbanization, a possible structure of dt could be                      2425.365 )(2 cos 2425.365 )(2 sin 11 0 jj l j i i i it gtj b fti aabtad  . (2) parameter a is the coefficient of the linear trend while b is the slope. the estimated parameters of seasonal part dt are obtained using the nlinfit function in matlab. amplitude a indicates the difference between the daily winter and the daily summer temperature. for example, it is around -4.4°c in london and 9.5°c in chicago [11]. now the wavelet analysis is used to identify the seasonal part. the results of the conducted wavelet analysis [12] indicate that the seasonal part of the temperature takes the following form                          2425.365 )(2 sin 2425.365 )(2 sin 2425.365 )(2 sin 3 3 2 2 1 10 ft b ft b ft babtadt                           2425.365 )(2 sin 2425.365 )(2 sin 2425.365 )(2 sin 6 6 5 5 4 4 ft b ft b ft b  . (3) 460 m. laković, i. pavlović, m. banjac, m. jović, m. manĉić depending on the geographical area, the authors indicate different values of coefficients a and b. for example, for the new york these coefficients are presented in ref. [13]. parameter a0 is the amplitude of the sinusoid of the seasonal variance and f1f6 is the angle that refers to the maximum and minimum of the temperature in the year. all parameters are statistically significant. parameter gt in eq. (1) denotes the residual grid load, i.e. the total system load minus its deterministic part where function u(gt) could be any suitable function; a linear approach is found sufficient [10]. residual time series rt in eq. (1) is often modeled by a gaussian white noise process. the time simulation of the white noise process is given in fig. 1. fig. 1 simulation of white noise process on the basis of eqs. (1) and (3), the simulation scheme was made in matlab software and used for determining and predicting the costs for the different values of parameters from eq. (3). this scheme is initially developed in [14] and it is presented in fig. 2. numerical computation and prediction of electricity consumption in tobacco industry 461 fig. 2 simulation scheme 3. numerical results to determine the optimal values of electricity consumption the numerical monte carlo method was applied. this method finds its application in almost every field of study and presents a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. the monte carlo method can be used to predict the future economic parameters which affect the operation cost of some industry production [15]. this simulation uses time step size of δt=0.1s, while the number of simulations is n = 2000. as the example, the numerical results were performed for a tobacco production where firstly, according to the average monthly energy consumption (table 1), function u(gt) is determined and presented in fig. 3. the tobacco factory described in [9] is supplied with electricity through four transformer stations with the total leased power of 1.5mw. the consumption of electricity in the factory per month according to the data for the year 2006 is given in table 1. 462 m. laković, i. pavlović, m. banjac, m. jović, m. manĉić table 1 the electricity consumption is given by months for the year 2006 [9] 2006 consumed electricity (kwh) january 762000 february 694500 march 649500 april 504000 may 732000 june 931500 july 793500 august 655500 september 805500 october 792000 november 783000 december 744000 in total 8847000 fig. 3 approximated linear function u(gt) according to values given in table 1 many companies have an electricity demand that is, except for seasonality and some noise, quite homogeneous. they have fixed opening hours during which the electricity demand reaches a certain level, whereas at other hours the consumption drops to what we refer to as the standby level. there is a very homogeneous structure during the weekdays and the low demand on sundays. there are also special events and holidays. however, these different regimes during opening hours and closing hours are easy to predict. therefore, we can de-seasonalize the data to get something quite homogeneous. now, according to average numerical computation and prediction of electricity consumption in tobacco industry 463 time conditions in most cities in europe that correspond to weather conditions in serbia, the parameters which are used in eq. (3) are presented in table 2. these parameters are presented for working days, weekends, summer, spring, winter and autumn. table 2 the estimated parameters a0 = -0.0008; a = 1.242 b1 = 7.6994 f1 = -73.2644 b2 = 0.1317 f2 = 95.0642 b3 = 0.0469 f3 = -640.2319 b4 = -0.2743 f1 = 183.109 b5 = -0.3445 f1 = -13.1151 b6 = 0.0796 f1 = -134.5803 according to the parameters from table 2, the determination and prediction of energy costs are given in fig. 4. initial time in this figure shows the energy consumption estimated for a previous month where the further costs prediction are given for the next ten days. fig. 4 approximated energy consumption for parameters given in table 2 these results are also given for larger variations in an energy system which is taken into account using the white noise process rt in eq. (1). the results are presented in fig. 5. 464 m. laković, i. pavlović, m. banjac, m. jović, m. manĉić fig. 5 approximated energy consumption for larger variations in energy system 4. conclusion this paper introduced a time-series model for forecasting the electricity consumption of a tobacco production company. the monte carlo model was suggested for probabilistic forecasting of electricity load of an industrial company. the time-series method forecasts energy demand by the patterns and trends found in the data. when using a time-series method, the researcher uses statistical extrapolation of loads based upon historical data. electricity is one of the most important and exploited forms of energy, and it is widely used for different kinds of needs. electricity consumption varies according to the season and the time of day. energy consumption varies from season to season because more electricity is used during the winter and summer months when it is hotter or colder than in the spring and fall months when the temperatures are usually moderate. nowadays electricity is essential for economic development especially for the industrial sector. decision makers around the world widely use energy demand forecasting as one of the most important policy tools. one of the decision makers’ dilemmas is how to forecast electricity demands. the main innovation of this paper is the monte carlo model which is used to forecast electricity demands. the monte carlo simulation produces not only one answer, but rather a series of answers or a range over which the results vary as a function of probability of occurrence and also a most expected result. in other words, the monte carlo simulation generates hundreds of alternative (scenarios) for a project. the answer may fall anywhere within the range of the results produced. this paper described the electricity demand for tobacco production in a company by dummy calendar variables. the production regime process is modeled as a seasonal pattern combined with an autoregressive process, while in the standby regime we assume the demand to follow a simple white noise process. the autoregressive model (ar) is supposed numerical computation and prediction of electricity consumption in tobacco industry 465 to be a stochastic process which is composed of a weighted sum of the previous value and a white noise error. the impact of weather conditions on both electricity demand and price has been considered through the seasonal pattern. references 1. khosravani, h., castilla m., manuel berenguel m., ruano a., ferreira p., 2016, a comparison of energy consumption prediction models based on neural networks of a bioclimatic building, energies, 9(57), pp.1-24. 2. henley, a., peirson, j., 1994, residential energy demand and the interaction of price and temperature: british experimental evidence, energy economics, 20, pp.157-171. 3. peirson, j., henley, a., 1994, electricity load and temperature issues in dynamic specification, energy economics, 16, pp. 235-243. 4. engle, r.f., mustafa, c., rice, j., 1992, modelling peak electricity demand, journal of forecasting, 11, pp. 241-251. 5. shahidehpour, m., yamin, h., li, z., 2002, market operations in electric power systems: forecasting, scheduling, and risk management, john wiley and sons ltd, new york, united states. 6. weron, r., 2006, modeling and forecasting electricity loads and prices: a statistical approach, john wiley and sons ltd, chichester, united kingdom. 7. zareipour, h., 2008, price-based energy management in competitive electricity markets, vdm verlag dr. mueller e.k., germany. 8. benth, f.e., saltyte-benth, j., koekebakker, s., 2007, putting a price on temperature, scandinavian journal of statistics, 34,(4), pp. 746–767. 9. ilić, g., 2007, preliminary energy review factory-tobacco industry "vranje", annual report, faculty of mechanical engineering, university of niš, serbia. 10. berk, k., müller, a., 2016, probabilistic forecasting of medium-term electricity demand: a comparison of time series models, journal of energy markets, 9(2), pp. 1-20. 11. zapranis, a., alexandridis, a., 2011, modeling and forecasting cumulative average temperature and heating degree day indices for weather derivative pricing, neural computing and applications, 20(6), pp. 787-801. 12. benth, f.e., saltyte-benth, j., 2007, the volatility of temperature and pricing of weather derivatives quantitative finance, quantitative finance 7(5), pp. 553-561 13. göncü, a., liu, y., ökten, g., yousuff hussaini, m., 2016, uncertainty and robustness in weather derivative models, monte carlo and quasi-monte carlo methods, springer proceedings in mathematics & statistics, 163, springer, cham. 14. laković m., pavlović i., banjac m., jović m., 2017, determination and prediction of electricity consumption using the monte carlo simulation method, simterm, soko banja, serbia, isbn 978-866055-098-1. 15. laković, m., pavlović, i., jović, m., 2017, aplication of monte carlo method for large steam condenser performances determination in variable operating conditions, kodip, budva, montenegro, pp. 251-258. https://www.bookdepository.com/publishers/john-wiley-and-sons-ltd https://www.bookdepository.com/publishers/vdm-verlag-dr-mueller-e-k https://www.bookdepository.com/publishers/vdm-verlag-dr-mueller-e-k plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 413 425 https://doi.org/10.22190/fume170206029m original research paper determination of residual stress in the rail wheel during quenching process by fem simulation udc 629.4 miloš milošević, aleksandar miltenović, milan banić, miša tomić faculty of mechanical engineering, university of niš, serbia abstract. residual stresses of the rail wheels are influenced by heat treatment during the manufacturing process. the quenching process during the manufacturing results in the residual stresses within the rail wheel that may be dangerous for the rail wheel during its operation. determination of the residual stress in the rail wheel is important for understanding the damage mechanisms and their influence on the proper work of rail wheels. this paper presents a method for determining the residual stresses in the rail wheel during the quenching process by using the directly coupled thermalstructural analysis in ansys software. key words: rail wheel, residual stress, fem, thermal load 1. introduction modeling and computer simulation get an increasing importance in research projects in the field of physics, engineering, biology, medicine, etc. in engineering many technical processes can be simulated using the finite element method (fem). miltenović et al. [1] presented a method for determining the friction generated heat in a contact between the wheel and the rail during normal operation using the transient structural-thermal analysis. one of the most important issues in railway wheels is the residual stress state. the residual stress, as unavoidable during manufacturing, is an important influence factor for the damage of wheels and rails; moreover, it can be assessed by the computer simulation. the residual stress is defined as a tensile or compressive force within a material, such as steel, without application of thermal gradient or an external force. residual stresses are the product of phase transformation, plastic deformation or thermal effects such as the process of contraction upon cooling. newton’s laws require that the compressive residual received february 06, 2017 / accepted june 08, 2017 corresponding author: miloš milošević faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: mmilos@masfak.ni.ac.rs 414 m. milošević, a. miltenović, m. banić, m. tomić stresses at the surface of a material are balanced by tensile stresses within the material [2]. the residual stresses in rail wheels can be caused mechanically due to wheel/rail operation, or due to the press fitting process of a bandage wheel, as well as during the quenching process. the object of several research studies of manufacturing processes of railway wheels was to show a layer of the compressive residual stress on the surface of parts to inhibit crack propagation. the effects of the residual stress and metal removal on the contact fatigue life were estimated by seo et al. [3, 4]. furthermore, okagata [5] evaluated the fatigue strength of a railway wheel produced in japan and presented the fatigue design method of the high-speed railway wheel by considering the effect of manufacturing conditions on the fatigue strength of the material. the residual stresses of the rail wheel are primarily caused by the heat treatment process. wang [6] in his study showed the heat treatment process of a 36” (914 mm) freight car wheel manufactured by griffin wheel company. he simulated the ideal and the non-ideal heat treatment processes and the effect on the residual stress after on-tread braking. for the analysis, he used the class u wheel with the carbon content varying from 0.67% to 0.77% as specified by the association of american railroads (aar). yu [7] performed a simulation of cooling after the rail heating process by using the finite element method (fem). he obtained the result that the residual stress caused by water cooling was bigger than that of the air cooling process. handa [8] researched the influence of the wheel/rail tangential traction force on thermal cracking of railway wheels. in the fem analysis he used temperature dependent material data of the wheel steel. he concluded that the residual stress was the main cause of the tread thermal cracking and the wheel/rail tangential force. as masoudi nejad indicates in [9], most of the above mentioned authors estimated the residual stress by using numerical simulations and the finite element method in rail on simple models with coarse mesh which provided inaccurate/questionable results in the case of thermal loads. therefore, he estimated the residual stresses which were obtained during the heat treatment process of the railway mono-block wheels, by using the elasticplastic finite element model [9]. the analysis was performed by the sequential weak coupling of the thermal and structural field while neglecting dependence of the coefficient of thermal expansion on temperature and gravity effects. the noted author determined that the residual stress obtained during the heat treatment process had the significant value representing an important factor for the crack initiation and fatigue life [10]. the same author applied a similar approach [11] to accurately predict the residual stresses due to the quenching process of the uic60 rail by using the fem. this paper presents a method for residual stresses determination in the rail wheel during the quenching process by using the directly coupled thermal-structural analysis. in addition to the method used for the analysis, the novelty of the presented research is that the thermal expansion coefficient changes with temperature, as well as gravitational forces, are taken into account. the analysis is also performed for the er8 steel grade according to en 13262:2009 [12], which is common on european railways. determination of residual stress in rail wheel during quenching process by fem simulation 415 2. rail wheel manufacturing process uic code 510-2 contains the conditions relating to the design and maintenance of wheels and wheel sets for coaches and wagons used on international services. it covers wheel diameters from 330 to 1000 mm and indicates the permissible axle loads from the standpoint of stresses of the metal used for the wheel and the rail. uic code 510-2 contains detailed coordinates of the wheel rim line. it is valid for a nominal track gauge of 1435 mm and cannot be readily applied to other track gauges. fig. 1 represents the wheel profile which is used for further analysis. fig. 1 rail wheel profile (solidworks sketch) the rail wheels are manufactured by casting (in some cases by forging). after this, they are heat treated to get a specific hardness and reheated to remove the undesired residual stress that remains in the wheels after casting/forging. the heat treatment of the rail wheels is the most important step in the manufacturing process since it gives them adequate mechanical properties. the material properties depend on the cooling rates in the different parts of the wheel. a goal of the heat treatment is to homogenize the microstructure of the rim in radial and circumferential direction. the heat temperature goes from 800 to 920 o c. after the homogenization, the rims are quenched with a water spray on the tread surface by using the water spray equipment shown in fig. 2. the quenching process consists of several steps, each of which imposes different boundary conditions on the model. fig. 2 wheel quenching equipment 416 m. milošević, a. miltenović, m. banić, m. tomić the quenching process of the rail wheel increases the steel strength, improves wear resistance and induces the desirable residual stress in the rim. the water spray quenches the hot rail wheel rim which cools and shrinks. under the wheel rim, the steel that is still hot has the reduced yield strength due to high temperature. the cooling of the rim causes it to shrink compressing the plate of the wheel. in that case, the yielding occurs. after quenching the wheels are placed in a tempering furnace at approximately 500 o c for two to five hours [2], which reduces the residual stress. during this phase, there is a tension between the cooler outer rim and the hotter underneath part of rim and the plate. at the end, the rail wheel is exposed to the ambient temperature. this heat treatment process results in the beneficial residual compressive stresses in the rail wheel rim. these stresses contribute to the prevention of the formation of rim fatigue cracks in railroad service. the phases of the heat treatment are given in table 1. table 1 phases of heat treatment phase process duration film coefficient [w mm -2 c -1 ] bulk temperature o c tread other 1 pre-quench 2 min 2.837 x 10 -5 2.837 x 10 -5 ambient temperature 2 quench 4 min 0.001766 2.837 x 10 -5 ambient temperature 3 pre-reheat 2 min 2.837 x 10 -5 2.837 x 10 -5 ambient temperature 4 reheat 2 hr 2.837 x 10 -5 2.837 x 10 -5 510 5 cooling 10 h 2.837 x 10 -5 2.837 x 10 -5 ambient temperature 3. simulation for the heat treatment simulation, a direct coupled thermo-mechanical analysis was performed to estimate the residual stress, as a result of manufacturing, within the rail wheel with the wheel diameter 1250 mm. the analysis was carried out at the ambient temperature of 21.6 o c for the wheel made of the steel with carbon content of 0.56% (steel grade er8 – en 13262:2009) [12]. the young’s modulus of the rail wheel material was considered as a temperature dependent parameter, as given in table 2, with the poisson ratio value of 0.3. the bilinear kinematic hardening model was used, which also included the mentioned temperature dependence for the yield strength and tangent modulus, as given in table 3. the thermal material properties (thermal conductivity, specific heat and thermal expansion coefficient) were also counted in terms of temperatures, as given in tables 4 and 5. the fem model was made as a 2d axisymmetric model with 4894 nodes which form 1529 elements. https://www.google.rs/search?espv=2&biw=1517&bih=735&site=webhp&q=at+approximately&spell=1&sa=x&ved=0ahukewje_zhb24fsahvezrqkhanodtsqvwuifsga determination of residual stress in rail wheel during quenching process by fem simulation 417 fig. 3 finite element mesh of rail wheel table 2 mechanical material properties temperature o c young’s modulus (mpa) 25 2.06 x 10 5 100 2.02 x 10 5 200 1.96 x 10 5 300 1.88 x 10 5 400 1.8 x 10 5 500 1.7 x 10 5 600 1.6 x 10 5 700 1.49 x 10 5 870 42403 for the analysis the wheel rail was assumed to be initially at the uniform temperature of 920 o c. for the analysis phases, duration and boundary conditions (film coefficients and temperatures) were defined according to data given in table 1. thus, the heat transfer coefficient from the wheel to air was set to 28.37 w/m 2 c, as well as for the other parts of the wheel, except for the phase when the tread was exposed to the water spray during the quenching process when the heat transfer coefficient was 1766 w/m 2 c. it is necessary to mention that the convection occurred at all rail wheel surfaces during the analysis of the quenching process, while the radiation from all surfaces of the rail was omitted during the heat transfer analysis. the analysis was conducted with the influence of gravity forces in the negative direction of y axis (fig. 3). table 3 yield strength and tangent modulus bilinear kinematic hardening temperature oc yield strength (mpa) tangent modulus (pa) 22 663 20000 100 650 13300 300 543 13300 500 354 6250 600 185 2500 700 46 1200 418 m. milošević, a. miltenović, m. banić, m. tomić table 4 thermal material properties – thermal conductivity and specific heat temperature o c thermal conductivity (w/mk) specific heat (j/kgc) 21.11 49.831 457.58 37.78 49.405 465.24 93.33 47.964 490.9 148.89 46.483 516.53 204.44 45.023 542.15 260 43.401 567.77 315.56 41.8 593.44 371.11 40.161 619.06 426.67 38.481 644.69 482.22 36.763 670.3 537.78 35.002 695.97 593.33 33.203 721.6 648.89 31.365 747.26 704.44 29.588 772.89 760 27.569 1853.6 815.56 25.203 635.31 871.11 25.913 643.47 926.67 26.624 651.64 table 5 thermal material properties – coefficient of thermal expansion [3] temperature o c coefficient of thermal expansion k -1 temperature o c coefficient of thermal expansion k -1 25 1.09 x 10 -5 400 1,31 x 10 -5 100 1.09 x 10 -5 500 1,38 x 10 -5 200 1.14 x 10 -5 600 1,46 x 10 -5 300 1.24 x 10 -5 700 1,51 x 10 -5 6. results and discussion the results of the performed analysis of the heat treatment process of the rail wheel are discussed for the distribution of the temperature field as well as the residual stress. 6.1. temperature fig. 4 shows the temperature-time history of the whole heat treatment process of the rail wheel with the maximal and minimal temperature at the beginning of the analysis and during the quenching, reheating and cooling processes. determination of residual stress in rail wheel during quenching process by fem simulation 419 fig. 4 maximal and minimal temperature of rail wheel during heat treatment it can be noticed from fig. 4 that the minimal temperature during the quenching goes down to 200 o c, during the reheating this temperature raises while the maximal temperature decreases all the time. during the cooling, the maximal and minimal temperatures slowly descend from the starting temperature of 920 o c to the ambient temperature. a) after 346.17 seconds quenching b) after 4825.4 seconds reheating fig. 5 temperature distribution of rail wheel during heat treatment 420 m. milošević, a. miltenović, m. banić, m. tomić fig. 5 shows the temperature distribution through the whole wheel at the beginning of the analysis and during the quenching, reheating and cooling process in the fem model. it can be noticed that the temperature only on the thread decreases to the temperature of about 200 o c during the quenching (fig. 5a), after which during the reheating (fig. 5b) this temperature increases to around 400 o c, while during the cooling phase the temperature of the whole rail wheel slowly decreases to the ambient temperature. 6.2. residual stress the heat treatment involving a tread quenching process (fig. 2), is used to resist cracking generation at and near the tread, so the contraction of the steel allows the formation of compressive residual stresses in the outer portion of the rim, as shown in fig. 6. the stress is usually termed circumferential representing a predominant distribution of compressive stress around the rim circumference. fig. 6 rim circumferential residual stresses after tread quenching process [12] the normal stress-time histories of the whole heat treatment process of the rail wheel are shown for the maximal and minimal normal stress in the circumferential (fig. 7) and axial (fig. 8) direction at the beginning of the analysis and during the quenching, reheating and cooling process. on the diagrams, the compressive stress is indicated as negative, while the tensile stress is with positive values. fig. 7 maximal compressive and tensile stresses of rail wheel in axial direction determination of residual stress in rail wheel during quenching process by fem simulation 421 fig. 8 maximal compressive and tensile stresses of rail wheel in circumferential direction regarding maximal normal stress in the axial direction (fig. 7), it can be noticed that both the compressive and tensile stresses increase to around -450 mpa and 200 mpa during the quenching, while during the reheating and cooling these stresses reach stable values of the maximal residual stresses of -150 mpa and 100 mpa. for the normal stress in the circumferential direction (fig. 8) after some fluctuations of the stresses during the quenching, reheating and the beginning of the cooling stable values of the maximal residual stresses of around -280 mpa and 290 mpa are established at the ambient temperature. fig. 9 circumferential and axial stresses of rail wheel after 89.5 seconds – pre-quenching 422 m. milošević, a. miltenović, m. banić, m. tomić fig. 10 circumferential and axial stresses of rail wheel after 280.4 seconds quenching fig. 11 circumferential and axial stresses of rail wheel after 510.6 seconds reheating determination of residual stress in rail wheel during quenching process by fem simulation 423 fig. 12 circumferential and axial stresses of rail wheel after 43920 seconds cooling the distribution of the normal stress in the circumferential and axial directions of the rail wheel during heat treatment is illustrated in figs. 9 to 12, so that the results for the axial direction are indicated in the legend as y axis (the normal to the cross section), while for the circumferential direction as z axis. fig. 9 shows the low values of thermal stresses at the beginning of the simulation. during the quenching (fig. 10) the tensile stress grows very fast at the tread surface (157 mpa and 491 mpa) while the compressive stress prevails in the interior of the wheel (-68 mpa and -69mpa). in progress of the reheating, (fig. 11), there is a change in the distribution area of compression and tension. the compressive stress appears at the tread surface (-382 mpa and -389 mpa), while the tensile stress is high in the middle of the rim (183 and 198mpa). for the cooling phase it is characteristic that the normal stresses reduce to the circumferential and axial residual stresses declining slightly over time (fig. 12). moreover, the tensile residual stress moves from the upper middle (fig. 11) to the center middle of the rim (fig. 12) which is consistent with the fig. 6, while the compressive residual stress remains at the tread surface. from fig. 12 it is seen that the axial residual stress nominally in tension in the middle of the rim reaches 103 mpa, while the axial residual stress nominally in compression at the tread surface is -150 mpa. the same figure indicates that the circumferential residual stress nominally in tension in the middle of the rim comes to between 65 and 119 mpa, while the circumferential residual stress nominally in compression at the tread surface amounts to -289 mps. the maximal value of the circumferential residual tensile stress of 298 mpa is exposed at portions of the plate and the hub and it is not relevant. it is clear from fig. 12 that there 424 m. milošević, a. miltenović, m. banić, m. tomić are the compressive region at/near the tread and the tension region below the compressive layer. thus, it follows from fig. 12 that the separated surface is initiated at the depth at approximately 40 mm below the tread surface of the rail wheel. 7. conclusion with modern computer simulation tools it is possible to simulate residual stress occurrence as an effect of the applied technological processes of production. the results of the implemented finite element analysis in the research of the rail wheels subjected to the heat treatment process show some very significant facts about stress distribution. the results reveal that the stress field is highly sensitive to variable thermal loads. therefore, this factor significantly affects the stress field of the rail wheels during the heat treatment process and is taken into account for the determination of residual stress during the quenching process by using the directly coupled thermal-structural analysis. the results of the performed analysis are discussed for the distribution of the temperature field as well as the residual stress. the results relating to the temperature distribution on the wheel tread indicate the temperature decrease during the quenching, the temperature increase during the reheating and finally slow cooling of the whole rail wheel to the ambient temperature. the results relating to the stress distribution show the maximal and minimal normal stress in the circumferential and axial direction at the beginning of the analysis and during the quenching, reheating and cooling process. the normal stress is growing very fast during the quenching nominally in tension at the tread surface, while at the same time in compression in the interior of the wheel. redistribution of the compression and tension happens during the reheating, so that the compressive stress deploys at the tread surface and the tensile stress in the middle of the rim. during the cooling, the normal stresses decline to the residual stresses over time. the region of the compressive residual stress remains at the tread surface while the region of the tensile residual stress moves further from the upper middle to the center middle of the rim. the values of the circumferential and axial residual stresses, as well as the position of the surface separating the compressive region of the tensile one depend on the geometry, material properties and heat treatment process parameters that are for the analyzed railway mono-block wheel in a good agreement with those achieved in [9, 13] as confirmed by field measurements. acknowledgements: this study is supported by the ministry of education, science and technological development of the republic of serbia, project tr35005. determination of residual stress in rail wheel during quenching process by fem simulation 425 references 1. miltenović, a., banić, m., stamenković, d., milošević, m., tomić, m., bucha, j., 2015, determination of friction heat generation in wheel-rail contact using fem, facta univesitatis series mechanical engineering, 13(2), pp. 99-108. 2. canale, l.c.f., totten, g., mesquita, r., 2008, failure analysis of heat treated steel components, asm international, ohio, usa. 3. seo, j.w., goo, b.c., choi, j.b., kim, y.j., 2008, effect of removal and residual stress on the contact fatigue life of railway wheels, international journal of fatigue, 30(10-11), pp. 2021–2029. 4. seo, j.w., kwon, s.j., jun, h.k., lee, d.h., 2009, effects of residual stress and shape of web plate on the fatigue life of railway wheels, engineering failure analysis, 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the influence of residual stresses, engineering failure analysis, 52, pp. 75-89. 11. masoudi, n.r., shariati, m., farhangdoost, k., 2017, three-dimensional finite element simulation of residual stresses in uic60 rails during the quenching process, thermal science, 21(3), pp. 1301-1307. 12. european standard en 13262:2009, railway applications — wheelsets and bogies — wheels product requirements, afnor, 2009. 13. wang, k., 2006, the probabilistic study of heat treatment process for railroad wheels using ansys/pds, proceedings of the 13th international ansys conference, pittsburgh, pa. mu-synthesis based active robust vibration control of an mri inlet facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 37 53 original scientific paper mu-synthesis based active robust vibration control of an mri inlet  udc 681.5 atta oveisi, tamara nestorović mechanics of adaptive systems, institute of computational engineering, ruhr university bochum, germany. abstract. in this paper, a robust control technique based on μ-synthesis is employed in order to investigate the vibration control of a funnel-shaped structure that is used as the inlet of a magnetic resonance imaging (mri) device. mri devices are widely subjected to the vibration of the magnetic gradient coil which then propagates to acoustic noise and leads to a series of clinical and mechanical problems. in order to address this issue and as a part of noise cancellation study in mri devices, distributed piezo-transducers are bounded on the top surface of the funnel as functional sensor/actuator modules. then, a reduced order linear time-invariant (lti) model of the piezolaminated structure in the state-space representation is estimated by means of a predictive error minimization (pem) algorithm as a subspace identification method based on the trust-region-reflective technique. the reduced order model is expanded by the introduction of appropriate frequency-dependent weighting functions that address the unmodeled dynamics and the augmented multiplicative modeling uncertainties of the system. then, the standard d-k iteration algorithm as an output-feedback control method is used based on the nominal model with the subordinate uncertainty elements from the previous step. finally, the proposed control system implemented experimentally on the real structure is to evaluate the robust vibration attenuation performance of the closed-loop system. key words: µ-synthesis, mri scanning device, robust vibration control, subspace system identification received february 2, 2016 / accepted march 15, 2016 corresponding author: atta oveisi ruhr-universität bochum, universitätsstr. 150, d-44801 bochum, germany e-mail: atta.oveisi@rub.de 38 a. oveisi, t. nestorović 1. introduction mri is an influential method that is broadly used in medical examinations and hospital clinics. the current research in the mri technology development is mostly concentrated on improving high static magnetic field strength while keeping the speed of the current switching of the gradient coil at considerable levels. this results in a higher quality of scanning concerning the internal organs of the patient [1]. a key drawback in the development of these medical instruments is identified to be the acoustic noise that is propagated within the operational scanning [2]. one of the sources of the acoustic noise that is reported by mechefske et al. is the gradient coil [3]. the gradient coil is used to propagate measured variations in the 3d magnetic field which permits localized image slices [3, 4]. a high acoustical noise level can engender heightened anxiety, noise-induced hearing loss, and possible permanent hearing impairment [5, 6]. the vibration of the gradient coil can also affect the image quality and resolution [7]. mcjury et al. studied impair communication problems between the patient and the hospital staff due to the acoustic noise generated by mri devices and proposed an active noise control (anc) technique based on the generation of destructive anti-phase interference with mri noise [8]. cho et al. investigated the functionality of the visual and cortex motor of the ge signa 1.5-t epi commercial mri and kais 2.0-t research mri devices in the presence of the sound noise due to gradient pulsing. they identified the typical behavior of the noise signal that is induced by the scanner [9]. price et al. studied the acoustic noise of various mri devices with a magnetic field strength range of 0.2-3 t and measured the noise level between 82.5-118.4 db [10]. edelstein et al. studied the acoustic noise due to the vibrations of the gradient assembly and showed the primary effect of the eddy-current-induced vibration of cryostat inner bore and the rf-body-coil in a 1.5 t cylindrical mri scanner. they reduced the noise level that is transmitted through the vibration of the structure by blocking the signal pathways carrying the noise to the region of interest [11]. mechefske et al. proposed the design of a passive acoustic liner and the gradient coil in order to suppress the noise level for the 4-t varian/siemens unity inova whole-body mri system [12]. active vibration control as an effective solution can be employed to block the pathways that carry and transmit the noise in the whole mri structure [13]. mechefske and shao considered the acoustic sound radiated from the mri device under the lorentz forces and validated the results with the boundary element method (bem) [14]. pribil et al. compared the spectral properties of an acoustical noise which is produced by the mechanical vibration of the gradient coils during the scanning in the open-air mri equipment working up to 0.2 t [15]. the mri shell is subjected to the excitation of the uninterrupted pulse lorentz force of the coils. the study of mri gradient coil relating to the sound noise was first conducted by hurwitz et al. at the very basic level [16]. qiu and tani tackled the active vibration suppression due to the lorentz force between the pulse current applied to the coil and to the main magnetic field of the circular cylindrical shell that are equipped with mri devices [17]. yao et al. conducted a comprehensive investigation of the vibration properties of the coil using the finite element approach (fea) and experimental testing [1]. nestorović et al. implemented an optimal lq tracking controller with additional dynamics for vibration suppression of the funnel-shaped shell mri structure [18]. they designed an additional model reference adaptive controller in order to compare the performance of the lq controller experimentally [19]. jaensch and lamperth designed a micro-positioning stabilizing structure for open pre-polarization mri scanner with piezoelectric actuators by mu-synthesis based active robust vibration control of an mri inlet 39 means of pid controller [20]. roozen et al. employed seismic masses which are actuated utilizing piezo-transducers by aiming at impacting the response of the gradient coil in order to actively control the vibration [21]. the application of hybrid control system consisting of both the feedback and feedforward control channels in reducing the acoustic noise of 4-t varian unity inova mri scanner is studied by li et al. [22]. in all of the aforementioned studies, the vibration and acoustic noise cancellation are carried out by passive methods, model-free controllers, or a deterministic model-based control approach. although the model-free control methods are easy to implement, they are mostly limited in the performance especially due to a large working frequency of the mri scanners and low order dynamics of the controller [23, 24]. in addition, the active control methods that are previously carried out are based on a deterministic reduced order models that are obtained by experimental-, fem-, and bem-based identification techniques. however, due to the limitations of the identification methods [25], the obtained model is subjected to modeling uncertainties which should be addressed by the controller. to the best knowledge of the authors of this article, no research is conducted to investigate the vibration suppression excellence by means of robust control methods. the rest of the paper is organized as follows: in section 2, the experimental configuration of the multi-input/multioutput (mimo) structure that is used in this study for modeling and control design purposes is introduced. then, a short overview of the identification method which is used in this paper is presented in section 3. the interested reader can refer to [26-29]. the robust output feedback control synthesis for the reduced order model that includes the input weighting functions and uncertain elements is proposed in section 4. finally, the experimental implementation and the numerical simulation results are presented in section 5. 2. experimental setup since the ultimate goal of this research is vibration suppression, distributed piezotransducers are bounded on the top surface of the funnel as depicted in fig. 1. in order to address the actuator/sensor placement, the results are used based on the research conducted by dlr braunschweig [30]. fig. 1 distribution of the piezo-transducers on the surface of funnel 40 a. oveisi, t. nestorović the structure consists of the funnel-shaped mri inlet and six groups of sensor/actuator elements (l1, l2, l3, r1, r2, and r3). each group of transducers consists of four individual paired actuator-patches and one sensor element (pzt film sonox p53). each piezotransducer is embedded in a polymer matrix for electrical insulation and it has the overall length, width, and thickness of 50, 25, and 0.2 mm, respectively. in order to reinforce the actuation power of the function modules, the second set of actuators are bounded on the top of the first set. the base of the funnel is fixed on a vibration isolation table with a set of screws (see fig. 1). the measured output signals of the system are obtained through the six piezo-sensors that are placed on the top surface of the funnel as shown in fig. 2. fig. 2 sketch of the experimental setup dspace digital data acquisition (daq) with ds1005 ppc board is employed to compile the simulated disturbance and the designed control system in real-time analysis. the connection of the daq system with the actuators/sensors and the mainframe is provided by an analog to digital converter (adc dspace ds2004) and a digital to analog converter (dac dspace ds2102). due to the different working voltage of dspace daq (±10 v) compared to the transducers ([-60, 200] v), an amplifier (pi e500) is utilized to amplify the signal generated by daq with constant gain of 100 and similarly, the second amplifier with maximum input current up to 5 a (brüel&kjær amplifier type 2706) is utilized to amplify the generated signal by daq system for the vibration exciter (brüel&kjær shaker type 4809). the control law of the active vibration system is realized on the simulink platform and then compiled regulator law with standard fixed step explicit ode5 solver (dormand-prince method [31]) is downloaded to the dspace daq in real-time by means of experimental rig shown in fig. 3. mu-synthesis based active robust vibration control of an mri inlet 41 fig. 3 experimental rig of the closed-loop system the simulated disturbance acts on the structure by the shaker which is connected to the funnel by a force transducer (brüel&kjær deltatron type 8230-001) with the sensitivity of 22 mv/n. this transducer measures the mechanical force applied to the structure that is used as the definition of the uncertainty weighting function of input disturbance. furthermore, it should be mentioned that in the rest of the paper, due to the computational limitations, only four sets of actuators and sensors (l1, l2, r1, and r2) are activated for the modeling and control design purposes. 3. system identification in this research, predictive error minimization (pem) subspace identification algorithm [26] in the time domain for continuous lti systems is utilized to estimate the system matrices from input-output data of the mechanical configuration shown in fig. 2. the process begins with collecting data (voltage) from the input/output channels of the open-loop experimental setup. this includes the external input mechanical excitation channel that is realized by shaker (bw) and the input piezoelectrical excitation channel for applied control effort by means of functional modules (b). accordingly, the reduced order deterministic mimo lti system in the state-space framework which represents the dynamics of the funnel structure with distributed piezo-transducers in a constraint frequency range can be constructed through the trust-region-reflective newton technique as [27, 32-34] , , w n x ax bu b d y cx du w       (1) where x  r n , u  r m , d  r p , and y  r q are the state, control input, disturbance input, and output vectors, respectively. in addition, a, b, bw, c, and d correspondingly represent 42 a. oveisi, t. nestorović the matrices of states, control inputs, input disturbances, outputs, and feedforward. furthermore, wn stands for the measurement noise which is considered to be an uncorrelated zero-mean gaussian stochastic process with constant power spectral density (wn) so that { } ( ),t n n ne w w w t   in which e is the expectation operator [29]. 4. µ-synthesis the coupled piezo-laminated funnel shape structure as shown in fig. 4 has five inputs including one disturbance channel and four actuation signals. the measured output of the system consists of four channels associated with four piezo-sensors bounded on the structure. the modeling uncertainty for the input matrix and input disturbance matrix are assumed to be multiplicative unstructured uncertainty [35]. fig. 4 closed-loop system with multiplicative uncertainty in fig. 4 k(s) the controller is represented as designed on the basis of the output feedback µ-synthesis. n stands for the white gaussian output measurement noise, respectively. in addition, wdist, wrdist, wn, and wri (i = 1, 2, 3, 4) are the frequency dependent norm-bounded weighting function for external input disturbance signal, the multiplicative uncertainty of disturbance, a high pass filter representing the activation frequency of the noise signal, and weighting function of the actuation uncertainty, respectively. also, as a standard multiplicative representation, it is assumed that ||dist||  1 and ||act i||  1, i = 1,...,4 [36]. for the sake of applying the structured singular value theory to the plant under study, the signal configuration in fig. 4 should be recast into linear fractional transformation (lft) [37] representation that is shown in the standard form in fig. 5. mu-synthesis based active robust vibration control of an mri inlet 43 fig. 5 lft representation of the control system in fig. 5, the plant is an open-loop interconnection which encloses certain elements such as the nominal structure model and all the weighting functions.  -block parameterizes all of the defined uncertain elements in the previous representation of fig. 4. the inputs to the mimo plant are allocated into three standard groups in which w represents the perturbation and, similarly, the outputs are grouped into three signals with e representing the error signal. the lft representation covers the set of all the control systems fu so that {f ( ( ), ) : max [ (j ) 1]}u p s      with ω being the frequency so that for perturbation δ, stabilizing controller k(s) exists that satisfies the following equation: f [f ( ( ), ), ( )] f [f ( ( ), ( )), ] 1.l u u lp s k s p s k s       (2) for given controller k(s), constrained h∞-index function of eq. (2) with the predefined augmented uncertainty structure of lft representation in fig. 5 can be checked by examining the structured singular values of the closed-loop system if and only if k(s) satisfies the following constraint over frequency ω max [f [f ( ( ), ), ( )]] 1.l u p s k s   (3) the standard µ-synthesis minimizes the maximum structured singular value of the close-loop system over frequency: min{max[ [f [f ( ( ), ), ( )]]( )], stabilizing ( )}l u p s k s j k s   . in order to solve the µ-synthesis problem,  should be replaced with the upper bound. definition 1 [38] considering m to be a constant matrix and the uncertainty δ, the upper bound of ( )m is defined as 1inf ( )dmd  for d  d with d being the set of all matrices d that satisfy d = d. using definition 1 for the optimization function of the µ-synthesis and δ, the optimization equation is reformatted as 1 ( ) min max min [ ( ( ), ( ))( ) ],l k s d d d f p s k s j d          (4) by defining d as a frequency dependent function that satisfies d  d for an arbitrary frequency ω, and replacing max [.]   with .  , the optimization problem in eq. (4) can be rewritten as 1 ( ) , min min (p( ),k( ))( ) ,l k s d d d df s s j d      (5) then, assuming u as a block orthogonal complex matrix for complex m (definition 1) to have the same structure as d  d and satisfying u * u = uu * = i with i being the identity 44 a. oveisi, t. nestorović matrix and superscript * denoting the complex conjugate of the original matrix, it is easy to obtain 1 1[( ) ( ) ] [ ]ud m ud dmd   . this is due to the fact that the matrix multiplication with orthogonal matrix does not change  and, as a result, one can transform d into ud without changing the maximum structured singular value of the system. by the introduction of an appropriate u, it is possible to restrict dω to be real-rational, stable, and minimum-phase transfer function symbolized as ( )d s [39]. the optimization index, eq. (5), based on this transformation and due to the orthogonality of the introduced matrix is reformatted as 1 ( ) ( ) min min ( ( ), ( ))( ) ,l k s d s d df p s k s j d    (6) the transformed optimization problem of eq. (6), as shown in fig. 6, can be solved iteratively by means of d-k iteration approach [40, 41]. fig. 6 replacing µδ with the upper bound 4.1. d-k iteration algorithm 1. solve optimization problem 1 ( ) min ( ( ), ( ))( ) ,l k s df p s k s j d   with an initial guess of stable, minimum-phase, and real rational ( )d s and define pd(s) as shown in fig. 7. fig. 7 real-ration d scaling in d-k iteration 2. with known pd(s) from the previous step, solve standard h∞ problem with the optimization function of ( ) min f ( ( ), ( ))l k s p s k s  . 3. with fixed controller k(s) from the previous step, move the optimization over d(s) and look for an appropriate frequency dependent function d(s) in a certain range of frequency to satisfy the previous constraints on ( )d s . 5. experimental and numerical results in this section, the experimental investigation regarding the performance of the output feedback control system in suppressing the vibration of the mri funnel structure is studied in detail. considering the amount of variables that are involved in the modeling of the mu-synthesis based active robust vibration control of an mri inlet 45 structure and the control system and keeping in view our limitations on computation power, we shall confine our attention to a systematic procedure in evaluating the closed-loop system. for this purpose, first a reduced order model of the real structure with distributed piezoelectric sensor/actuator groups is obtained by means of system identification. in particular, a mimo lti system in state-space representation of the structure including the dynamics of the sensors/actuators is estimated in the time domain by utilizing the matlab system identification toolbox. the reduced order model is assumed to cover the dynamics of the system within the frequency range of 0-18 hz which includes two natural frequencies of the structure (9.72 hz and 15.22 hz). in order to present the identification quality, the frequency response function (frf) of the real structure is compared with bode diagram of the identified model in the nominal frequency range of fig. 8(a, b, and c). it is worth mentioning that the reduced order model is a sixth order lti object. the identified system is employed to design a robust controller on the feedback channel based on the output measurements that are collected by the four piezo-sensors (l1, l2, r1, and r2) and dspace ds2004 based on the d-k iteration approach that is explained in section 4. based on the frf of the system, appropriate weighting functions are selected on the input channels that represent the frequency dependent uncertainty bounds and activation frequency of each channel. sensor l1 sensor l2 sh a k er sensor r1 sensor r2 fig. 8a comparison of frf of the real structure with the identified model for disturbance channel 46 a. oveisi, t. nestorović actuator l1 actuator l2 se n so r l 1 se n so r l 2 se n so r r 1 se n so r r 2 fig. 8b comparison of frf of the real structure with the identified model for actuator groups l1 and l2 mu-synthesis based active robust vibration control of an mri inlet 47 actuator r1 actuator r2 se n so r l 1 se n so r l 2 se n so r r 1 se n so r r 2 fig. 8c comparison of frf of the real structure with the identified model for actuator groups r1 and r2 48 a. oveisi, t. nestorović due to the multiplicative nature of the augmented uncertainty, the weighting functions are selected as shown in fig. 9 to cover the higher order mode-shapes of the structure which are neglected in the modeling procedure. fig. 9 weighting functions associated with uncertainty presentation in control design in order to investigate the performance of the output feedback control system, the structure is excited through the disturbance channel by a sweep sine signal with a frequency range of 0-18 hz for both the open loop and closed-loop systems. the applied mechanical disturbance signal is measured by the force transducer (b&k 8230) and represented in fig. 10. accordingly, the output measurements are obtained by the piezosensors and are depicted in fig. 11. fig. 11 compares the transient response of the open loop system and the closed-loop system based on the designed controller. it can be observed that the µ-control system has successfully alleviated the vibration amplitude although the external disturbance continues to excite the structure. fig. 10 the amplitude of applied mechanical disturbance through shaker mu-synthesis based active robust vibration control of an mri inlet 49 fig. 11 the vibration suppression performance of the control system in the nominal frequency range the applied control effort for the sweep sine signal is presented in fig. 12. as expected, the control voltage for the l2 and r2 actuator groups are less than for the l1 and r1 groups which are due to the small level of strain energy that reaches to these actuators because of the distance between the disturbance source and the actuator elements (see figs. 1 and 2). fig. 12 applied control effort on the piezo-actuator groups 50 a. oveisi, t. nestorović in addition, by the introduction of the appropriate weighting functions (see fig. 9), that contain the frequency dependent information of the actuator elements, the applied control efforts have a smooth behavior with the maximum amplitude of 55 v. in order to investigate the spillover effect of the controller, the fast fourier transformation (fft) analysis was performed on the control signal to clarify the frequency content of the applied voltage [42]. the result as depicted in fig. 13 shows that although the disturbance is acting within the maximum frequency of 18 hz, the applied control effort of actuator r1 leads to excitation of higher order mode-shape. however, significant reductions in the magnitude of the applied control effort can be observed for all of the actuation elements due to the introduction of the weighting functions. in addition, in the mechanical systems, the lower eigenfrequencies have a dominant impact, which is also confirmed here by the influence of the first, second, and third eigen-frequencies. therefore, even if the high-frequency excitation would occur, it would not deteriorate the controller performance significantly due to the negligible influence of the higher modes with respect to the dominant mode-shapes. fig. 13 fft analysis on the control effort finally, by aiming at investigating the robust performance of the closed-loop system in stabilizing the unmodeled higher order dynamics of the mri funnel structure, the coupled system is excited through the disturbance channel with a sweep sine signal that charges the higher order mode-shapes of the structure in the frequency range of 20-38 hz. the applied mechanical force acts on the system within 30 sec and the experimental results for the output measurements are presented in fig. 14. in this figure, the controlled and the uncontrolled cases are brought together in four subplots to evaluate the robustness of the control system. it can be observed that the attenuation performance is reduced compared to the nominal frequency range but the control system mostly suppressed the vibration. furthermore, as it can be seen in fig. 15, the applied control effort on actuator r1 saturates for a short period between 6.8 and 7.7 sec which leads to performance loss that is mostly detected in sensor elements l1 and r1. this time period stands for the higher order unmodeled dynamics with the natural frequency around 27 hz. mu-synthesis based active robust vibration control of an mri inlet 51 fig. 14 robust vibration attenuation performance for unmodeled dynamics by comparing the control effort in the nominal frequency range (fig. 12) and the highfrequency area (fig. 15), it can be seen that the control system loses its optimality since by using more actuation energy less performance is achieved [42]. this emphasizes the importance of the nominal system order and the weighting 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numerical study and design optimization on energy absorption of thin-walled tube with a yoshimura pattern yudong fang1,2, weijian han2,3, zhenfei zhan4, li huang2,3 1college of mechanical and vehicle engineering, chongqing university, chongqing, china 2college of materials science and engineering, nanjing tech university, nanjing, china 3yangtze delta region institute of advanced materials, suzhou, china 4school of mechatronics and vehicle engineering, chongqing jiaotong university, chongqing, china abstract. this paper presents a numerical investigation and design optimization of the energy absorption of an origami tube. the arrangement of the tube's surface is inspired by a yoshimura pattern. a numerical simulation model is developed in abaqus/explicit to simulate the axial crushing of the origami tube. the simulation model is validated by comparing its deformation with the deformation observed in physical tests. considering the limitations of existing single parametric analysis in the study of the origami tube with a yoshimura pattern, the paper emphasizes a comprehensive parametric study, which includes a full factor analysis that considers the structural topological parameters, as well as a global sensitivity analysis that considers all the structural parameters. furthermore, there has been limited reporting on structural optimization for thin-walled tubes with yoshimura pattern. the paper also highlights multi-objective optimization for origami tubes with a yoshimura pattern. the optimization aims to enhance energy absorption efficiency and minimize the peak crushing force. to address the computational demands, a surrogate model-based optimization approach is employed for the optimization problem. the optimization process results in a set of pareto front solutions, offering a range of peak crushing force values from 151.5 to 288.69 kn and specific energy absorption values from 16.06 to 19.52 kj/kg. key words: origami tube, parametric study, design optimization, surrogate model received: october 03, 2023 / accepted march 07, 2024 corresponding author: zhenfei zhan school of mechatronics and vehicle engineering, chongqing jiaotong university, chongqing 400044, china e-mail: zhenfeizhan@cqjtu.edu.cn 2 y. fang, w. han, z. zhan, l. huang 1. introduction thin-walled tubes have a wide range of applications in the automotive and aerospace industries as energy absorption devices [1]. in particular, in the design of passive vehicle safety systems, crash boxes are commonly employed as typical thin-walled structural components to ensure the crashworthiness of vehicle bodies. the crash box is designed to minimize damage to other key components, such as the front rail of the vehicle, in lowspeed crashes, thereby improving maintenance costs and providing occupant safety in highspeed crashes. consequently, when designing a thin-walled tube as the crash box, it needs to absorb more energy and exert a smaller crushing force on the front rail during dynamic impacts. research indicates that the energy absorption characteristics of a thin-walled structure are closely related to the mode of crush deformation. among various deformation modes, such as radial bending and tangential torsion, the desired mode in a crash scenario is axial progressive folding deformation, as it corresponds to the optimal energy absorption characteristics of a thin-walled tube [2]. adding inducing grooves or inducing holes is an effective approach to achieve the desired deformation mode and decrease the crushing force [3-5]. in vehicle passive safety design, inducing grooves and holes are commonly arranged on the crash box, which is made of steel, while crash boxes made of aluminum alloy typically use inducing holes. conical tubes are another typical type of tube that has been numerically and experimentally proven to perform well under axial and oblique impacts [6]. novel designs based on conical tubes have been developed by combining design elements such as taper, variable thickness, and foam filler [7-9]. according to the research work of najibi et al. [10], thin-wall tubes filled with lightweight foam take high specific energy absorption when the foam material is well chosen. multi-cell thin-walled tubes have also attracted significant attention in the development of energy absorption devices. these tubes are formed by specific structures, such as holes or cross-shaped structures arranged in the cavity of the tube, extending from the top surface along the tube axis to the bottom surface. for example, a typical multi-cell tube is a quad tube with a centered cross connecting the four center points of the tube edges. chen et al. [11] and kim et al. [12] found that multi-cell tubes exhibit more stable deformation and higher specific energy absorption compared to traditional single-cell thinwalled tubes. combining the characteristics of multi-cell thin-walled tubes and conical tubes, the multi-cell tapered thin-walled tube was developed. googarchin et al. [13] studied the crashworthiness of multi-cell square tubes and multi-cell tapered tubes through theoretical derivation and numerical simulation, revealing that the tapered multi-cell tube exhibited better crashworthiness than the square tube under the same conditions. origami tube is considered a promising choice for designing energy absorption devices. similar to origami, where the folding process of plastic hinges formed by two folding faces can be observed, the axial collapse deformation of a thin-walled tube can be approximated to an origami process. numerical and experimental studies indicate that thin-walled tubes with origami patterns exhibit outstanding energy absorption, stable axial collapse modes, more uniform crushing load distribution, and low initial buckling force during the crushing process [14-16]. aghamirzaie et al. [17] proposed a novel multi-layered origami thin-wall tube. compared with the conventional octagonal tube, the proposed origami tube could achieve a 55% decrease in the initial peak force while maintaining the specific energy absorption. zhai et al. [18] developed an origami honeycomb, which, when compared to numerical study and design optimization on energy absorption of thin-walled tube... 3 traditional honeycomb structures, exhibits a more stable folding process during out-ofplane crushing. gaining insightful understanding of the effects of structural parameters on the performance of an origami tube is helpful in designing desired origami tubes with suitable layouts of origami patterns and appropriate parameters. based on numerical simulation and physical test, wickeler et al. [19] carried out a two-level and three-factor full factorial design of experiment to investigate the shape deformation mechanism of thin-walled tube with miura-ori pattern. zhou et al. [20] conducted a parametric study based on single-factor analysis to investigate the effects of geometric parameters on the energy absorption of an origami tube with diamond patterns. the parameters studied included the length and width of the tube, tube thickness, length of the unfolded origami pattern unit, and width of the folded lobe of the origami pattern unit. yang et al. [21] examined the effects of thickness on the energy absorption of circular tubes, origami tubes with diamond patterns, and origami tubes with yoshimura pattern. the specific energy absorption increases with increasing tube thickness. when the tubes have the same thickness, the origami tube with yoshimura pattern exhibits the highest specific energy absorption among the three. ye et al. [22] conducted a parameter analysis to investigate the effects of tube geometries and ply stacking sequence on the energy absorption behavior of an origami tube made of carbon-reinforced plastics. the existing parametric study on origami tubes mainly using single factor analysis, the coupling effects of parameters are still unknown. furthermore, conducting a global analysis that covers a reasonable parameter range is necessary. structural optimization is a powerful approach to enhance the performance of designed tubes [23,24]. baykasoğlu et al. [25] optimized a hybrid structure consisting of an aluminum square tube filled with a functionally graded lattice structure to improve the crashworthiness of the hybrid structure under multiple impact loading conditions. surrogate models based on least square regression were used to reduce computational burden in the optimization process. based on polynomial neural networks and evolutionary algorithms, najibi et al. [26] performed a multi-objective optimization to efficiently synthesize a thin-walled structure with a combination of circular and square sections. a set of pareto solutions was yielded by minimizing peak crushing force and maximizing specific energy absorption. yang et al. [27] performed multiple-objective optimization for a multi-cell origami tube by reducing the initial peak force and the fluctuation of the loading-displacement curve under dynamic impact. quadratic response surfaces were trained as surrogates of numerical simulations to reduce computational burden. fiumarella et al. [28] presented a multiple-objective study on an origami tube with diamond patterns formed by carbon fiber/epoxy prepreg woven. the initial peak force and total energy absorption were optimized using a genetic algorithm. surrogate models were employed in the optimization process to avoid extensive time-consuming simulations. more research on the optimization of energy absorption characteristics of origami tubes can be found in references [29-32]. as an extension of previous research work [21], this paper presents a numerical study and design optimization for the energy absorption of an origami tube with yoshimura pattern. different from the previous research work, which involved a simple single-factor analysis to investigate the effects of thickness, the current study involves a comprehensive parametric study. the parametric study includes a full factor analysis that considers the structural topological parameters, as well as a global sensitivity analysis that considers all the structural parameters. as there is a lack of reported research on structural optimization 4 y. fang, w. han, z. zhan, l. huang for thin-walled tubes with the yoshimura pattern, a multi-objective optimization is then performed to maximize the specific energy absorption of the tube and minimize the peak crushing force. surrogate models are employed to ensure optimization efficiency. the remainder of this paper is organized as follows: section 2 presents an origami tube with yoshimura pattern. section 3 describes the construction of a finite element analysis (fea) model used to simulate the structural performance of the tube under quasi-static crushing. section 4 presents a parametric study to investigate the effects of structural parameters on the performance of the origami tube. section 5 describes a multipleobjective optimization based on surrogate models and a genetic algorithm to optimize the origami tube. the conclusions are provided in the last section. 2. origami tube the specific origami pattern examined in this study is the full-diamond pattern, also known as the yoshimura pattern. as depicted in fig. 1, the origami tube consists of isosceles triangular facets. the length of the tube is denoted as l. and it can be uniformly divided into multiple cells along the axial direction. the number of cells is represented as nc. the cross-sectional shape of the tube, when viewed from the top, forms a regular polygon. the diameter of the origami tube, denoted as d, is defined as the diameter of the circumscribed circle of the regular polygon. the number of the edges of the regular polygon is denoted as ne. the structural topology primarily depends on the design parameters nc and ne. in the case shown, nc and ne are equal to 4 and 6, respectively. a view of the intersection angle of two neighbored isosceles triangular facets along the axial direction, θ, and the base angle of the isosceles triangular facet, α is shown in fig. 1(d). with the given l, d, ne and nc, intersection angle of two neighbored isosceles triangular facets along the axial direction, θ and the base angle of the isosceles triangular facet, α could be derived by eqs. (1) and (2), respectively. the prefolding angle of the origami tube is the supplementary angle of θ. fig. 1 thin-walled tube with a yoshimura pattern: (a) front view; (b) left view; (c) top view; (d) a view of intersection angle of two neighboured isosceles triangular facets along the axial direction, θ, and the base angle of the isosceles triangular facet, α numerical study and design optimization on energy absorption of thin-walled tube... 5 = 2arctan [1 cos( )]c e l dn n    (1) = arctan sin( )sin( ) 2 c e l dn n    (2) the generation of the origami tube could be summarized as: 1) along the axis direction, 2nc+1 regular polygons with ne edges are distributed as shown in fig. 2(a). all regular polygons have the same circumscribed circle. the distance between two neighbored regular polygon in the axis direction is l/(2ne). the projections of two neighbored regular polygons in the x-y plane are shown in fig. 2(b). if rotating one projection with an angle of , where  = π/ne, the two projections will coincide. 2) with the given l, d, ne and nc, the coordinates of all the endpoints in fig. 2(a) could be derived when the coordinate origin is coinciding with the center of the bottom edges of the origami tube. 3) creating isosceles triangular facets with three endpoints, among whom two endpoints belongs to an edge of the ith regular polygon, while the other one is the closest endpoints of the neighbored regular polygon. the creation of isosceles triangular facets is shown in fig. 2(c). in this paper, the base value of l, d, ne and nc are set to 100 mm, 90 mm, 6, and 4 respectively. the thickness of the tube, t, is set to 2 mm. the setting values of l and d are based on the geometry measurement of a vehicle crash box [33]. fig. 2 a view of endpoint distribution and the creating of isosceles triangular facets: (a) a distribution of the endpoints of the origami tube; (b) projection of the endpoints on the xy plane; (c) creation of two neighbored isosceles triangular facets 3. finite element simulation numerical simulation is an economical and reliable tool for studying the behavior of thin-walled structures [34,35]. in order to investigate the energy absorption and crushing force of the origami tube with yoshimura pattern in an axial quasi-static crushing process, a fea model is created using abaqus/explicit. as depicted in fig. 3(a), the bottom 6 y. fang, w. han, z. zhan, l. huang edges of the origami tube are fixed, while a constraint is applied to the top edges of the origami to ensure that the top edges of the origami tube undergo compression from top to bottom. referencing to [21], the tube is compressed by a rigid body for a distance of 80 mm, with a speed of 8 m/s. to avoid initial penetration, the distance between the rigid body and the tube in the z-axis direction is set to 0.5 mm. as shown in figs. 3(b) and (c), the origami tube is discretized using doubly curved thin or thick shell elements with hourglass control, finite membrane strains, and reduced integration shell elements (s4r/s3r in abaqus). the effects of mesh size on the energy absorption of the origami tube is analyzed as shown in fig. 4(a). as the mesh size decreases, the energy absorption also decreases. when the mesh size is reduced from 2 mm to 1 mm, the energy absorption curves tend to overlap. the effects of mesh size on the displacement vs. force curve are analyzed, as shown in fig. 4(b). it can also be observed that the two corresponding displacement vs. force curves have a high degree of overlap when the mesh sizes are set to 1.5 mm and 1 mm, respectively. therefore, the mesh size is set to 1.5 mm. self-contact is applied to the tube, while surface-to-surface contact is used to model the contact between the origami tube and the rigid plate. referencing to [21], the friction coefficients for the contacts are set to 0.25. to avoid calculation error caused by the mesh size being smaller than the thickness of the surface, as recommended by abaqus user’s guide, parameter ‘scalethick=0.25’ has been set on *surface to allow a rescaling of the contact thickness where necessary for this surface. fig. 3 simulation model created in the abaqus: (a) boundary condition and loading; (b) a view of the mesh; (c) an enlarged view of the meshing of a single cell since traditional manufacturing techniques are not applicable to the origami tube, additive manufacturing is a feasible option for fabricating it. in reference [21], raw brass was used to form the origami tube, and the material properties were determined through uniaxial tensile tests on samples fabricated by 3d printing [36]. the properties of the material applied to the simulation model are presented in table 1. table 1 material properties applied to simulation model [21] density poisson’s ratio young’s modulus yield strength ultimate tensile strength ultimate tensile strain 8494 kg/m3 0.35 110 gpa 165 mpa 385 mpa 0.2 numerical study and design optimization on energy absorption of thin-walled tube... 7 fig. 4 mesh size affects on the energy absorption and force-displacement curve of the origami tube: (a) internal energy-time curves; (b) force-displacement curves the energy principle is always used to analyze the rationality of the model. the energy change along with the displacement of the rigid plate is presented in fig. 5. as shown in the figure, the total energy introduced by the external work equals the sum of the internal energy and kinetic energy of the tube, which fits the principle of conservation of energy. the ratio of artificial strain energy (allae), an indicator of hourglass occurrence in hexahedral and shell meshes in abaqus, to internal strain energy is below 5%, indicating that the effects of hourglass are within an acceptable level. fig. 5 energy vs. the displacement of the rigid wall in the simulation the crushing force and von mises stress distribution versus displacement of the rigid plate are presented in fig. 6. as shown in the figure, the initial peak force is introduced when the rigid plate starts contacting the origami tube. the stress is then transferred from the top edges of the tube to the bottom edges. the crushing force appears to be very stable as the displacement of the wall increases from 0 to 60 mm. however, when the tube is close to full deformation, the crushing force increases dramatically. the von mises stress distribution is approximately symmetrical along the centerline of the origami tube. when the origami tube is fully deformed, there are 4 folding units, corresponding to 4 cells of the origami tube. the origami tube exhibits a very regular axial collapse in axial quasi-static crushing. 8 y. fang, w. han, z. zhan, l. huang fig. 6 contour of the von mises stress distribution and the crushing force vs. the displacement of the rigid wall yang et al. [21] performed a physical test to investigate the deformation behavior of the origami tube with a yoshimura pattern. the simulation results could be compared to the physical test presented in yang’s work when the parameters t, d, ne and nc are set to 0.81 mm, 65.6 mm, 8, and 4, respectively. fig. 7 displays the comparison of experimental and simulated deformation in the axial quasi-static crushing of an origami tube. both the simulation and physical test indicate that buckling of the origami tube starts from the middle cell as shown in fig. 7(a) and ends at the cell close to the rigid plate as shown in fig. 7(c). the fe simulation and physical test exhibit consistency in terms of the deformation location and degree of deformation. therefore, the fe model demonstrates good fidelity. fig 7 simulation results vs. physical test corresponding to different displacement of the plate numerical study and design optimization on energy absorption of thin-walled tube... 9 4. parametric study the parametric study is a process used to investigate the effects of a set of parameters on system responses or functions. it helps to gain insight into complex systems or processes and contributes to the development of new systems or designs [37,38]. in order to investigate the effects of design parameters on the energy absorption of the origami tube, a full factorial analysis considering ne and nc, as well as a global sensitivity analysis considering ne, nc, d, and t are conducted based on the data collected from fea. the range of variables used in the study is listed in table 2. table 2 summary of the design parameters parameters lower bound upper bound type step ne 4 8 discrete 1 nc 3 7 discrete 1 d 80 mm 90 mm continuous t 1 mm 3 mm continuous the specific energy absorption (sea) and peak force (pf) are commonly used measurements to describe the energy absorption and crushing force of a thin-walled structure. the sea is calculated using the following equation: 0 = ( ) , = s e e f s ds sea m (3) where s represents the displacement of the rigid plate in the z-axial direction. f(s) is the crushing force of the origami tube on the rigid plate. e is total energy absorbed by the origami tube. m is the mass of the origami tube. a full factor analysis is performed to investigate the effect of ne and nc on sea and pf. fig. 8 presents contour plots of sea and pf as functions of ne and nc, considering different diameters and tube thicknesses of the origami tube. the gradient changes in color along the x-axis and y-axis reflect the effects of ne and nc on these measurements. regardless of the combinations of d and t being considered, it is evident that for sea, the color gradient along the x-axis is more pronounced, while the gradient along the y-axis is relatively subtle. this suggests that ne has a greater influence on sea compared to nc. increasing ne leads to an increase in sea, whereas increasing nc slightly decreases sea. this is because the larger value of ne, the cross-section of the origami tube is more close to a circle. a circular crosssection, in geometric terms, achieves the maximum area with the minimum perimeter, making it perform better in terms of energy absorption capability. 10 y. fang, w. han, z. zhan, l. huang fig. 8 contour plots of the sea and pf in different terms of ne and nc: (a) d=80 mm, t=2 mm; (b) d=90 mm, t=2 mm; (c) d=90 mm, t=1 mm for pf, the color gradient changes are evident both along the x-axis and the y-axis, indicating that both ne and nc have noticeable effects on pf. the influence law is complex because of the interaction between ne and nc. this is because both ne and nc affects the prefolding angle of the origami tube, according to eq. (1). the larger the value of nc or the smaller the value of ne, the larger the prefolding angle of the origami tube. the larger the prefolding angle, the easier of the origami tube to be crushed, resulting in a smaller peak crushing force. there is an interaction term of ne and nc in eq. (1), indicating the effects of ne and nc could be complex. when nc equals 6 or 7, the pf increases along with the increase of ne. but when nc is below 5, the pf fluctuates along with the change of ne. furthermore, comparing figs. 8(a) and (b), it can be concluded that the effects of d on sea and pf are not significant. although the value of d has been changed from 80 mm to 90 mm, the color distributions in the two figures are highly similar. in contrast, the color distributions in figs. 8(b) and (c) differ significantly, indicating that the thickness of the tube, t, can have noticeable effects on both sea and pf. the change of color distribution results from the thickness of the tube being changed from 2 mm to 1 mm. the thicker the origami tube, the earlier it will be fully compressed, resulting in a higher peak crushing force. sobol sensitivity analysis is a widely used global analysis method for investigating the effects of parameters on system responses. herein, a global sensitivity analysis-based parametric study is conducted using the given parameter ranges listed in table 2. since sobol sensitivity analysis requires a large number of samples, performing the analysis directly based on the fe simulations would result in a computationally expensive process. to overcome this challenge, surrogate models are recommended as an alternative approach to improve efficiency [39,40]. in this paper, instead of using the time-consuming fea model, kriging models with gaussian correlation function and second-order polynomial function as global numerical study and design optimization on energy absorption of thin-walled tube... 11 approximations are trained to predict sea and pf. a total of 200 training samples are prepared by using optimal latin hypercube sampling method and fe simulations. the accuracy of the surrogate models is evaluated using 10-fold cross-validation, with the r square metric providing an objective and intuitive evaluation of global accuracy. the r square values for the surrogate models built to predict sea and pf are 0.986 and 0.943, respectively. fig. 9 presents the validation results of the surrogate models. as shown in fig. 9, the red lines are perfect fitting lines. the scatters are predictions of the surrogate model vs. predictions of the fe model. the scatters located on the perfect fitting lines denote the surrogate model has the same prediction values as the fe model. the more scatters located on or near the perfect fitting line, the higher accuracy of the surrogate model. the r square value is closer to 1, the higher accuracy of the model. according to the distribution of scatters and r square values, all the surrogate models are acceptable to provide data support in a sensitivity analysis. fig. 9 surrogate model (sm) predictions vs. fea predictions referring to [41], sobol sensitivity analysis under the support of surrogate models is performed. in this global sensitivity analysis, the total number of sample is set to 104. fig. 10 shows the main effect of a single parameter (first-order indice) and the interaction effect of two parameters (second-order indice). the values on the main diagonal represent the main effects of the parameters, while the other values represent the interaction effects of two parameters. a threshold of 0.01 is set to judge the parameter is sensitive or not, while a threshold of 0.1 is set to judge a sensitive parameter is highly sensitive or not [42]. according to the sensitivity analysis results, sea, is highly sensitive to ne and t. in addition, sea is sensitive to d and nc. pf is only highly sensitive to t. ne and nc are sensitive parameters in determining pf. ne and t have the most significant effects on sea and pf, respectively. the sea and pf are also sensitive to the interaction effect of ne and t. in addition, pf is sensitive to the interaction effect of ne and nc. the ranking of the first four parameters that affect sea, from small to large, is as follows: ne, t, nc and d, while the ranking of the first four parameters that affect pf, from small to large, is as follows: t, ne, interaction between ne and nc, and nc. 12 y. fang, w. han, z. zhan, l. huang fig. 10 first-order effects and interaction effects 5. optimization in this section, multi-objective optimization is performed to determine the preferred ne, nc, d and t. the optimization objectives are to maximize sea and minimize the pf, while satisfying the constraint on the energy absorption (e ≥9 kj). the optimization formulation is summarized as follows: find: , , , and , 1 to minimize: and , subject to: 9kj e cn n d t pf sea e  (4) the range of the parameters that need to be optimized is provided in table 2. the nondominated sorting genetic algorithm (nsga-ii) has been proven to be an effective method for solving multi-objective optimization problems in various domains, including engineering, finance, and data science [43]. in this paper, nsga-ii is employed to search for optimum designs that address the specific optimization problem in eq. (4). the population size and the number of generation are set to 40 and 100, respectively. in the optimization process, instead of calling fe simulations, the surrogate models built in section 4 are used to improve optimization efficiency. to address the errors introduced by surrogate models, an adaptive optimization strategy is introduced as: using fe simulation to validate the pareto solutions obtained from the ith round optimization. if the relative error of the responses of all pareto solutions are below 0.01 and the solutions meet constraint conditions, the adaptive optimization is converged. otherwise, the pareto solutions will be added to the training sample set to update the surrogate models. the updated surrogate model will be used in the i+1th round of optimization. totally 15 rounds of optimization have been performed to get the adaptive optimization converged and yield a set of pareto front solutions as shown in fig. 11. due to the inclusion of discrete variables ne and nc in the optimization process, the pareto front is not smooth. in fig. 11, three typical pareto front solutions are marked as a, c, and b, corresponding to the solution with minimum peak force, the solution with the maximum specific energy absorption, and the compromised solution, respectively. the compromise solution is selected by adapting a utopia point-based decision strategy from the pareto solution set. the decision strategy based on the utopian point can be referred to in [37,44]. numerical study and design optimization on energy absorption of thin-walled tube... 13 fig. 11 pareto front solutions and three typical pareto front solutions marked marked as a, b, and c table 3 shows the summary of the three typical pareto solutions and the corresponding responses obtained by fea. all the pareto solutions meet the constraint condition. the pf range of the pareto solutions is from 151.5 kn to 288.69 kn, while the sea range is from 16.06 kj/kg to 19.52 kj/kg. all three typical pareto solutions prefer an origami tube with a high value of ne , and a small value of nc. in addition, all three typical pareto solutions prefer to have a thick thickness and a small diameter for the designed origami tube. table 3 summary of the pareto front solutions parameters solution a solution b solution c ne 8 8 7 nc 4 3 3 d (mm) 80 80 80 t (mm) 2.71 2.67 2.87 e (kj) 9.33 10.24 11.89 pf (kn) 151.50 196.73 288.69 sea (kj/kg) 16.06 18.15 19.52 fig. 12 displays the von mises stress distribution of the pareto solutions vs. the deformation of the rigid plate. as all three origami tubes are symmetrical, the stress distribution is also symmetrical. the origami tube a starts its collapse from the bottom area of the tube, while the origami tubes b and c start their collapse from the top area and middle area of the tube, respectively. the final configurations of the three origami tubes are quite different. the obvious difference between origami tube a and origami tube b is the number of cell of the origami tube, nc, indicating nc could have effect on the sequence of deformation and the finial configuration of the origami tube. observed from the final configuration, origami tube c has three fully folded units, which correspond to the three cells of the origami tube, indicating that this origami tube exhibits good axial progressive folding. however, we cannot find the same number of folded units as the number of cells in origami tubes a and b from their final configurations. from the view of deformation mode, the origami tube c could be the best solution among the three typical pareto. 14 y. fang, w. han, z. zhan, l. huang solutions. fig. 12 von mises stress distribution of origami tubes corresponding to the three typical pareto solutions vs. the deformation of the rigid plate fig. 13 displays the crushing force of the three typical origami tube vs. the displacement of the rigid plate. the three curves have similar trending when the displacement of the rigid plate is below 0.02 m. when the displacement of the rigid plate is close to 0.08 m, the crushing force of the tube c increase dramatically, while the crushing force of the origami tubes a and b on the rigid plate decrease and then slightly increase. this because the tube c is close to fully compressed as it has thicker thickness, while tubes a and b still have space to undertake the axial compression. when the origami tubes a and b trend to be fully compressed, their crushing forces start to increase slightly. from the view of crushing force control, the origami tubes a and b outperform origami tube c as the crushing force curves of the two origami tubes are more gentle. fig. 13 the displacement of the plate vs. the crushing force corresponding to the three typical pareto front solutions numerical study and design optimization on energy absorption of thin-walled tube... 15 6. conclusion this paper presents a numerical study and design optimization of an origami tube with yoshimura pattern under axial compression. a fea model was developed in abaqus to simulate the axial quasi-static crushing process of the origami tube. the effects of structural parameters that govern the topological shape of the tube, such as the number of edges in the tube section and the number of cells in the axial direction, on energy absorption were investigated through a comprehensive factor analysis. additionally, a global sensitivity analysis based on surrogate models was conducted to explore the impact of these parameters on energy absorption and peak crushing force from a global perspective. finally, an adaptive multiple-objective optimization, employing surrogate models and a genetic algorithm, was applied to optimize the origami tube. the following findings were observed: 1. high-accuracy surrogate models could be built to describe the numerical relationship between the structural parameters of the origami tube and the peak force and energy absorption obtained from fe simulation. 2. the energy 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42. tang, y., reed, p., van werkhoven, k., wagener, t., 2007, advancing the identification and evaluation of distributed rainfall‐ runoff models using global sensitivity analysis, water resources research, 43(6), pp. 1-14. 43. ding, f., liu, s., li, x., 2022, pareto optimality of centralized procurement based on genetic algorithm, tehnicki vjesnik/technical gazette, 29(6), pp. 2058-2066. 44. gunantara, n., 2018, a review of multi-objective optimization: methods and its applications, cogent engineering, 5(1), 1502242. https://ascelibrary.org/journal/ajrua6 https://ascelibrary.org/journal/ajrua6 facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 399 418 https://doi.org/10.22190/fume200528033b © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a hybrid lbwa ir-mairca multi-criteria decision-making model for determination of constructive elements of weapons darko božanić 1 , aca ranđelović 1 , marko radovanović 2 , duško tešić 1 1 university of defense in belgrade, military academy, belgrade, serbia 2 1st army brigade, serbian armed forces, novi sad, serbia abstract. the paper demonstrates a model whose goal is to define the construction elements of weapons necessary to meet minimum requirements of users. the complexity of the problem, conditioned by different construction elements of weapons and specific situations of their use, is the reason for using methods of multi-criteria decisionmaking. in the paper we used the hybrid lbwa – ir-mairca model. with this model, one can conduct an analysis of characteristics of the existing weapons, based on which we define the construction elements for modifying the existing and manufacturing new weapons. regarding a large number of different types of weapons, the paper is limited to the analysis of close-quarters combat weapons. the lbwa method was used to calculate weight coefficients of the criteria. the mairca method, which was modified by interval rough numbers, was used to select the best close-quarters combat weapon that has the best characteristics in accordance with the requests of the users. based on the analysis, the users have the option to clearly and precisely define requests for improvement of the existing, and manufacturing new weapons. key words: multi-criteria decision-making, lbwa, mairca, interval rough numbers, constructive elements 1. introduction defining construction elements of different types of weapons is the process that must be carried out by both the constructor and the user. the user’s role is to define the requests that the constructor’s role is to implement. often those requests are not aligned received may 28, 2020 / accepted july 18, 2020 corresponding author: darko boţanić university of defence in belgrade, military academy, pavla jurišica šturma 33, 11000 belgrade, serbia e-mail: dbozanic@yahoo.com 400 d. boţanić, a. ranđelović, m. radovanović, d. tešić with the constructors’ abilities. in order to avoid misunderstandings, we have developed a model that would be used to define requests, make improvements of the existing, or develop new weapons based on the existing ones. due to complexity of research problems, we have used close-quarters combat weapons as an example because they take an important place while conducting modern military and police operations. modern military operations are conducted in a variety of operational theaters. a number of factors that follow and affect modern military operations have had an effect on the development of firearms in order to maximize the effects they have on the objective, that is, in order to accomplish the end state easier. besides pistols, revolvers, rifles, and machine guns, close-quarters combat weapons take a significant place in modern militaries. they have been developed as a necessity to provide a high rate of fire at close distances, especially in urban environments which are very different from other environments where combat operations are conducted (close-quarters, a high number of objects, small shooting distances, horizontal and vertical sectors of fire, a high number of targets, etc.). close-quarters combat weapons are individual, light weapons, designed for engaging combatants at distances of up to 200 meters. due to their practical rate of fire, they accomplish a high density of fire. most often, they are of small sizes and weigh less than traditional rifles, and use pistol ammunition [1], which is one of the biggest differences from the traditional assault rifles. these types of weapons are most commonly used by military and police special forces as well as crews of armored vehicles, helicopters or airplanes [2]. there are a variety of close-quarters combat weapons on the market with different characteristics. most of the armed forces have different types of close-quarters combat weapons. design elements, quality of the material and construction reveal significant differences between these weapons in terms of their precision, rate of fire, number of malfunctions, etc. on the other hand, the requests from the military and police forces when conducting different kinds of operations are undefined. consequently, an objective was set to develop a model able to determine those close-quarters combat weapons that are best suited for the needs of the serbian army. the research results are useful for acquiring new weapons as well as for determining the most suitable close-quarters combat weapons for use in the serbian army in the current state of affairs. the research results then serve as the base for defining construction tasks of new weapons, that is, modifications of the existing ones. also, these research results were used for purchasing close-quarters combat weapons for the serbian army besides determining those close-quarters combat weapons which are best suited for the army units for conducting their combat tasks. previous research studies of this problem can be primarily connected to different analyses of characteristics of weapons as well as different approaches to selecting the best types of weapons. according to the resources available to the authors, the selection of the close-quarters combat weapons has not been performed by means of the multi-criteria decision-making method yet; therefore, we have considered the selection of other types of weapons. dağdeviren et al. [3] show the selection of optimal weapons using the ahp, topsis and fuzzy topsis methods. ashari and parsaei [4] select the infantry rifle using the electra iii method. radovanović et al. [5] select the best anti-armor system of the second and third generation, using the ahp method. jokić et al. [6] compare different calibers for automatic rifles using the vikor method. brady and goethals [7] analyze efficiency of different types of 155 mm projectiles using monte carlo simulations. a number of authors conducted comparative analyses of weapons using their characteristics. a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 401 jenkins and lowrey [8] conducted a comparative analysis of the weapons in use in the united states army as well as weapons recommended for replacement. comparison was conducted using a quantitative analysis of the weapons characteristics “head to head“. gordon et al. [9] conducted a comparative analysis of the weapons in use in the united states army as well as in a number of armies in the world by comparing basic combat characteristics. radovanović et al. [10] select the most suitable anti-armor rocket system using numerical analysis of tactical and technical combat characteristics. the complexity of the selection process and the process of defining the most desirable characteristics are the reasons behind the decision to use multi-criteria decision-making methods. a large number of methods were analyzed and due to specific problems of research, we defined a hybrid model that consists of two methods: lbwa (level based weight assessment) and mairca (multi attributive ideal-real comparative analysis method) method, modified by interval rough numbers (ir-mairca). 2. lbwa – ir-mairca model a lbwa – ir-mairca hybrid model is defined through four phases as shown in fig. 1. phase 1 defining criteria that affect the selection phase 2 weight coefficients calculation phase 4 sensitivity analysis phase 3 selection of the best alternative experts’ evaluation experts’ evaluation, lbwa method ir-mairca change of weight coefficients of criteria fig. 1 lbwa – ir-mairca model in the first phase of the model, the criteria are defined using experts’ evaluations that the selection of the close-quarters combat weapon depends on. in the second phase, the initial matrix was defined using expert evaluations and lbwa method to calculate weight coefficients of the criteria. in the third phase, the selection of the best alternative was conducted using the ir-mairca method. in the last phase, we conducted sensitivity analysis by altering weight coefficients of the criteria. based on the obtained results, the user 402 d. boţanić, a. ranđelović, m. radovanović, d. tešić can realistically define the requests that the new close-quarters combat weapon needs to satisfy. further on in the paper, we show the lbwa and ir-mairca methods in detail. 2.1. lbwa method lbwa method is one of the newer methods for determining weight coefficients of the criteria. the model was first demonstrated in the ţiţović and pamučar paper [11]. a big advantage of this method is a relatively simple mathematical calculation, whose simplicity does not depend on the number of criteria. also, this method can be used in both individual and group decision-making. at the very beginning of the lbwa method, just like in many other methods, the first thing we do is to define criteria. if n is the number of criteria, then we have a set s = {c1, c2,..., cn}. after defining the set of criteria (s), we start using the lbwa method that goes through following steps [11]. step 1 determining the most significant criterion from the set of defined criteria s = {c1, c2,..., cn}. the most significant criterion is the one which has the biggest effect on the decision, i.e. it has the biggest weight coefficient. step 2 grouping criteria by the significance level. if we define the most significant criterion as c1, in reference to it, we define which level the rest of the criteria belong to based on the following:  level s1: on level s1 we group criteria from set s whose significance is equal to the significance to criterion 1c or up to two times less than c1;  level s2: on level s2 we group criteria from set s whose significance is exactly two times less than c1 or is up to three times less than c1;  …  level sk: on level sk we group criteria from set s whose significance is exactly k times less than significance of c1 or up to k + 1 times less than significance of criterion c1. by using the above mentioned rules, the decision-maker makes a rough classification of the observed criteria. if the significance of a criterion cj is denoted by s(cj), where j  {1, 2, ..., n}, then we have 1 2 ks s s s    , where for each level i  {1, 2, ..., k}, it is true that it is 1 2 ,{ , , } { : ( ) 1} si i i i j js c c c c s i s c i      (1) also, for each p, q  {1, 2, ..., k} such that p q holds p qs s  . thus, in this way is well defined partition of the set of criteria s. step 3 within the formed subsets (levels) of criteria influence, we compare criteria based on their significance. each criterion pi ic s in the subset 1 2 ,{ , , } si i i is c c c is assigned with an integer {0,1, , } pi i r so that the most important criterion c1 is assigned with i1 = 0, and if pi c is more significant than qi c then ip < iq, and if pi c is equivalent to qi c then ip = iq. maximum value of the comparison scale is defined using the expression (2)  1 2max , , , kr s s s (2) a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 403 step 4 based on the defined maximum value of criteria comparison scale (r), expression (2), we define elasticity coefficient r0  n (where n represents a set of real numbers) that needs to meet the condition that r0 > r,  1 2max , , , kr s s s . method creators recommend that initial values of weight coefficients should be defined based on elasticity coefficients r0 = r + 1. since parameter r0 affects smaller changes of weight coefficient changes, taking another value of elasticity coefficients is recommended for additional adjustments of the weight coefficients in accordance with personal preferences of decision-makers. step 5 criteria influence function calculation. influence function :f s r is defined in the following way. for each criterion pi ic s we define a function of influence 0 0 ( ) p p i i r f c i r i    (3) where i represents the number of levels/subsets to which the criterion is assigned, r0 represents the elasticity coefficients, while {0,1, , } pi i r represents the value assigned to criterion pi c in the scope of the observed level. step 6 calculation of optimal values of weight coefficients of criteria. using expression (4) we calculate the weight coefficients of the most influential criterion: 1 2 1 1 ( ) ( )n w f c f c     (4) the weight coefficient values of the rest of the criteria are obtained using expression (5) 1( )j jw f c w  (5) where 2,3, ,j n , and n represents the total number of criteria. 2.2. ir-mairca method due to a high level of uncertainty following the decision-making processes, evident is an increase in the number of researchers who modify classical methods of multi-criteria decision-making in their papers, using different areas that address these issues in appropriate ways. since the selection process of the close-quarters combat weapon involves uncertainty, in this paper we used the modification of the mairca method using interval rough numbers. interval rough numbers take a significant place when addressing uncertainty and there are numerous papers about them [12, 13, 14, 15]. interval rough number irn(a), is defined as [13]:  ' ' '( ) ( ), ( ) , , ,l u l u l uirn a rn a rn a a a a a            (6) where the value rn(a l ) represents the lower class of the irn(a) object, which is defined by a lower a l and upper boundary a u , where a l  a u , and value rn(a 'u ) represents the upper class of the irn (a) object, defined by the lower a 'l and upper boundary a 'u , where a 'l  a 'u . 404 d. boţanić, a. ranđelović, m. radovanović, d. tešić using the interval rough numbers in the mairca method modification requires knowledge of the basic arithmetic operations that are specific for interval rough numbers. if we assume that there are two interval rough numbers 1 2 3 4( ) ([ , ],[ , ])irn a a a a a and 1 2 3 4( ) ([ , ],[ , ])irn b b b b b , then the basic arithmetic operations with them are performed as follows [12]: (1) addition “+”: 1 2 3 4 1 2 3 4 1 1 2 2 3 3 4 4 ( ) ( ) ([ , ],[ , ]) ([ , ],[ , ]) ([ , ],[ , ]) irn a irn b a a a a b b b b a b a b a b a b         (7) (2) subtraction “-“ 1 2 3 4 1 2 3 4 1 4 2 3 3 2 4 1 ( ) ( ) ([ , ],[ , ]) ([ , ],[ , ]) ([ , ],[ , ]) irn a irn b a a a a b b b b a b a b a b a b         (8) (3) multiplication “×” 1 2 3 4 1 2 3 4 1 1 2 2 3 3 4 4 ( ) ( ) ([ , ],[ , ]) ([ , ],[ , ]) ([ , ],[ , ]) irn a irn b a a a a b b b b a b a b a b a b         (9) (4) division “/” 1 2 3 4 1 2 3 4 1 4 2 3 3 2 4 1 ( ) / ( ) ([ , ],[ , ]) /([ , ],[ , ]) ([ / , / ],[ / , / ]) irn a irn b a a a a b b b b a b a b a b a b   (10) (5) scalar multiplication where 0k  1 2 3 4 1 2 3 4( ) ([ , ],[ , ]) ([ , ],[ , ])k irn a k a a a a k a k a k a k a        (11) the mairca method was first published in papers [16, 17]. since then it has been used in a large number of papers in its initial version [18, 19, 20, 21, 22, 23] or as a modified mairca method in fuzzy and rough environments [13, 24, 25, 26, 27, 28, 29, 30]. modified ir-mairca method has seven steps [13, 27]. step 1 forming initial decision-making matrix ( y ). as with similar methods of multicriteria decision making, the first step is to form an initial decision-making matrix, where l number of alternatives is being evaluated based on n number of criteria: 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ... ( ) ( ) ... ( ) n n n l l l ln l n c c c a irn y irn y irn y a irn y irn y irn y y a irn y irn y irn y              (12) where n represents the total number of criteria, and l represents the total number of alternatives. a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 405 interval rough vector 1 2( ( ), ( ),..., ( ))i i i ina irn y irn y irn y , where ( )ijirn y  ' ' '[ ( ), ( )] ([ , ],[ , ])l u l u l u ij ij ij ij ij ijrn y rn y y y y y , represents the value of the i-th alternative by jth criterion ( 1,2,..., ;i l 1,2,...,j n ). step 2 determining preferences based on the choice of alternatives pai. in the largest number of cases, the decision-makers are neutral towards the choice of alternatives. however, the mairca method offers a possibility to the decision-maker to have a preference towards some of the offered alternatives and to express it through the use of the method. if the decision-maker is neutral towards the choice of the alternative, preference based on one of the l alternatives is 1 1 ; 1, 1,2,..., i i l a a i p p i l l     (13) where l represents the total number of alternatives that are selected. step 3 calculation of matrix elements of theoretical estimations (tp). theoretical estimations matrix elements (irn(tpij)) are interval rough numbers calculated using the following expression: ( ) ( ) [ ( ), ( )]l u pij ai i ai i iirn t p irn w p rn w rn w    (14) where pai represents preferences towards the choice of alternatives, irn(wi) the weight coefficients of the evaluation, and irn(tpij) theoretical estimation of the alternative for the given criterion of evaluation. after the calculation, we get the matrix of theoretical estimations: 11 12 1 21 22 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ( ) ( ) ... ( ) p p p n p p p n p pl pl pln l n irn t irn t irn t irn t irn t irn t t irn t irn t irn t               (15) step 4 selecting elements of real estimations matrix (tr). calculation of elements (tr) is performed using the expression:    ' ' ' '( ) ( ) ( ) , , , , , ,l u l u l u l u rij pij nij pij pij pij pij ij ij ij ijirn t irn t irn x t t t t x x x x                  (16) where irn(tpij) represents elements of the theoretical estimations matrix, and irn(xij) represents elements of the normalized initial matrix of decision-making (x): 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ... ( ) ( ) ... ( ) n n n l l l ln l n c c c a irn x irn x irn x a irn x irn x irn x x a irn x irn x irn x              (17) 406 d. boţanić, a. ranđelović, m. radovanović, d. tešić elements of matrix (x), that is, normalization of elements of the initial decisionmaking matrix is performed using the following expressions: a) for criteria of the benefit type (criteria where the larger value is more desirable)   ' ' ' '( ) , , , , , , l u l u ij ij ij ij ij ij ij ijl u l u ij ij ij ij ij ij ij ij ij ij ij ij ij y y y y y y y y irn x x x x x y y y y y y y y                                               (18) b) for criteria of the „cost“ type (criteria where the smaller value is more desirable)   ' ' ' '( ) , , , , , , u l u l ij ij ij ij ij ij ij ijl u l u ij ij ij ij ij ij ij ij ij ij ij ij ij y y y y y y y y irn x x x x x y y y y y y y y                                               (19) where iy and iy represent minimum and maximum values of the border intervals of the observed criterion, respectively: 'min{ , }l l ij ij ij j y y y  (20) 'max{ , }u u ij ij ij j y y y  (21) step 5 calculation of the matrix of total gap (g). gap gij represents interval rough number obtained using expression:    ' ' ' '( ) ( ) ( ) , , , , , , ij l u l u l u l u ij pij r pij pij pij pij rij rij rij rijirn g irn t irn t t t t t t t t t                  (22) where irn(tpij) represents elements of the theoretical estimations matrix, and irn(trij) elements of real estimations matrix (trij). from the calculation, we get the matrix of total gap (g): 11 12 1 21 22 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ... ( ) ( ) ... ( ) n n l l ln l n irn g irn g irn g irn g irn g irn g g irn g irn g irn g              (23) where n represents the total number of criteria, l represents the total number of alternatives that are being selected, and gij represents the obtained gap of alternative i according to criterion j. step 6 calculation of criteria functions values (qi) by alternatives. values of criteria functions are calculated by adding the gap elements of matrix (g) by columns: 1 ( ) ( ), 1,2,..., n i ij j irn q irn g i m    (24) where n represents the total number of criteria, m represents the total number of alternatives that are being selected. a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 407 ranking the alternatives is done by converting interval rough numbers to real numbers. the conversion of interval rough number ' ' ( ) ([ , ],[ , ]) l u l u i i i i iirn q q q q q into real number qi is done using the expression: ' '( ) ; ( ) ; ( ) ( ) ( ) u l u lui i ui i i li i i ui li rb q rb q q q rb q q q rb q rb q        (25) '(1 )l u i i i i iq q q      (26) step 7 determining the dominance index of the first-ranked alternative (ad,1 j) and the final ranking of the alternatives. the dominance index of the first-ranked alternative represents the element that defines its advantage over other alternatives and that is why it is necessary mostly due to subjectivity during the decision-making process. by defining the dominance index, we can see the difference between the first-ranked and other alternatives more clearly. dominance index is determined using the expression: 1 ,1 , 2,3,.., j d j n q q a j m q     (27) where q1 represents the criteria function of the first-ranked alternative, qn represents criteria function that is ranked last, qj represents criteria function of the alternative used to compare the first-ranked alternative to, m represents the total number of alternatives. besides the dominance index, in order to finish defining the first-ranked alternative, it is necessary to determine dominance threshold id using the following expression: 2 1 d m i m   (28) where m represents the total number of alternatives. if the dominance index ad,1 j is greater than, or equal to the threshold di (ad,1 j  id), we keep the obtained ranking. if dominance index ad,1 j is smaller than threshold id (ad,1 j < id), we cannot conclude with certainty that the first-ranked alternative has enough of an advantage over the observed alternative. 4. description of criteria and calculation of weight coefficients throughout the first phase of using the model, we defined the criteria that influence the selection of the best alternatives, that is, the best close-quarters combat weapon. defining criteria and their weight coefficients represents a significant phase for decisionmaking models [31]. complexity and specificity of the research problem has forced us to rely on experts in order to define criteria used to make the selection. for the selection of the best alternative, we defined eight criteria shown in the next part of the paper from the most significant (c1) to the least significant (c8). initial velocity of the bullet (c1) is the velocity that the bullet reaches at the moment when it leaves the muzzle; it represents the distance in the unit of time (m/s). larger initial 408 d. boţanić, a. ranđelović, m. radovanović, d. tešić velocity means a larger fire power of the weapon; therefore, it increases the kinetic energy of the bullet, and with that the effect (the degree of materialization) it has on the target [8]. weapons of a smaller caliber reach a higher initial velocity of the bullet than those with a bigger caliber and, therefore, accomplish better results. a higher initial velocity gives higher accuracy and efficiency of the weapon. reliability (c2) is one of the most significant exploitational characteristics of the weapon that is expressed as the number of malfunctions proportional to the number of fired bullets (the number of malfunctions for 6000 fired bullets). it is also important for the weapon to function in different environments, in high and low temperatures, with dirty parts, in different positions of firing, etc. experience so far shows that close-quarters combat weapons are reliable weapons; however, after a longer use, it is possible to start malfunctioning. the most common reasons for malfunctions are: wear and tear of parts, bad ammunition, bad maintenance and careless and unprofessional handling [32]. practical rate of fire (c3) represents the number of bullets fired in one minute. there is a difference between theoretical and practical rate of fire. practical rate of fire has a significant effect when conducting combat tasks. this characteristic is important for every weapon type due to its close correlation to fire density which causes greater effects on the objective. by increasing fire density, we increase the probability of hitting the target. it is expressed as the number of bullets in a minute (bullets/min). a higher rate of fire directly affects the efficiency of the weapon. practical rate of fire is determined through experiments or calculations using the template according to malinovski [2]: 60 pn c n tt t s e    (29) where tn represents time, tp time of loading the weapon, tc time of one cycle of automatic work, e number of bullets in a magazine and s number of bullets in a burst. efficient range (c4) represents distance (in meters) up to which we can expect to hit the target with enough kinetic energy to neutralize it [33]. a higher efficient range means engaging targets at higher distances which provides greater security and protection for the shooter [10], that is, it increases the efficiency of the weapon. mass of the weapon (c5) represents the unavoidable characteristic of a close-quarters combat weapon because modern warfare demands using weapons of smaller mass. mass of the weapon affects mobility and ability to shift fire [5]. in order to produce a weapon of small mass, manufacturers use new kinds of materials, mostly polymers. close-quarters combat weapons made of these materials are light; however, their characteristics are the same as those made of metal. the mass is in kilograms. smaller mass increases mobility, it is easier to handle increasing efficiency when conducting combat tasks. length of the weapon (c6) distance (in millimeters) from the tip of the muzzle to the stock. it represents the characteristic that most affects handling and carrying the weapon. the longer the weapon, the less mobile it is and handling is more difficult in smaller space. because of this, and in order to use the weapon more efficiently, we are leaning towards weapons smaller in size. in this way we increase mobility and provide better handling indoors [5]. modern close-quarters combat weapons usually have collapsible stock, or the telescope type stock, that significantly reduce the length. this can significantly increase efficiency, but also reduce accuracy. a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 409 lifetime of the barrel (c7) is a characteristic defined as the number of fired bullets without affecting the characteristics of the barrel and maintaining given specifications. manufacturing method and material are crucial for the lifetime of the barrel as well as the pressure and temperature when firing the weapon. most common tolerable standards are 0,07 mm change in caliber, and damage to the barrel of less than 50%. length of the barrel (c8) barrel is one of the main parts of the weapon; therefore, its length affects the accuracy and precision of the rifle. longer barrels enable longer firing distances, achieving more precise and accurate results, but, at the same time, they increase mass and length of the entire weapon. when firing, the bullet rotates in the barrel for a longer time which provides a more stable travel of the bullet [34]. also, in the longer barrels, gases have a longer effect on the bottom of the bullet and provide higher initial speeds. unlike other automatic weapons (rifles, snipers, machine guns), close-quarters combat weapons are technically designed with shorter barrels, which means reduced precision when firing over longer distances. therefore, they are used for accomplishing combat tasks at shorter distances. also, the length of the barrel affects the degree of efficiency as well as materialization on the objective. all of the listed criteria are of a numeric character and can be divided in two subsets:  set of benefit type criteria 1 3 4 7 8{ , , , , }c c c c c c  ,  set of cost type criteria 2 5 6{ , , }c c c c  . after determining the criteria, in the second phase of the research, we conducted the calculation of weight coefficients using the lbwa method, as described in the previous part of the paper: step 1 for the most significant criterion we chose 1c . step 2 the experts sorted the criteria in roughly 6 levels: 1 1 2 3 2 4 5 3 6 4 7 5 6 8 { , , }, { , }, { }, { }, { } { }. s c c c s c c s c s c s s c        step 3 using expression (2) we obtained the maximum value of the criteria comparison scale  1 2 3 4 5 6max , , , , , 3r s s s s s s  therefore, criteria comparison scale is in range iip  {0,...1,...3}. in this step, we once again relied on the experts who conducted comparison on each level. for the final value of comparison of two criteria, we took the middle value of comparisons of all experts and we obtained following values: level 1s : 1 0i  , 2 0.6i  , 3 3i  . level 2s : 4 1.2i  , 5 2i  . level 3s : 6 0.7i  . level 4s : 7 1.1i  . level 6s : 8 1.5i  . 410 d. boţanić, a. ranđelović, m. radovanović, d. tešić step 4 elasticity coefficient needs to be r0  4, in this particular case, we defined r0 = 4. step 5 defining criteria influence function using eq. (3): 1 2 8 4 4 4 ( ) 1; ( ) 0.869;... ( ) 0.157 1 4 0 1 4 0.6 6 4 1.5 f c f c f c            step 6 using expression (4) we calculated the weight coefficient of the most influential criterion: 1 0.869 .. 5 7 1 0 . 0. .2 1 15 w     values of weight coefficients of the rest of criteria are obtained using eq. (5). 2 8 0.869 0.251 0.218; ... 0.157 0.251 0.039. w w       we obtained following weight coefficients: 0.25,0.22,0.14,0.11,0.1,0.08,0.06,0 4( ).0jw  . after calculations of weight coefficients, we can move on to the next phase of the model. 4. selection of the best alternative using the ir-mairca method the third phase of using the model implies using the ir-mairca method, according to the steps described in the second part of this paper. a part of data about alternatives was taken from the existing literature, whereas a part of it was obtained through different kinds of measurements and estimations. step 1 in the first step of using the ir-mairca method, we formed the initial decisionmaking matrix (y), table 1, where all the alternatives have been evaluated based on all criteria. table 1 initial decision-making matrix (y) alternatives criteria c1 c2 c3 c8 a1 [(640,715),(777,930)] [(76,88),(95,98)] [(92,115),(145,195)] ... [(264,376),(406,406)] a2 [(714,800),(825,870)] [(85,101),(122,127)] [(58,72),(91,121)] ... [(58,72),(91,121)] a3 [(285,315),(375,400)] [(9,11),(12,14)] [(48,72),(95,129)] ... [(48,72),(95,129)] a4 [(255,285),(370,410)] [(8,9),(10,11)] [(56,74),(93,125)] ... [(56,74),(93,125)] a5 [(620,660),(680,700)] [(3,3),(4,5)] [(59,84),(96,148)] ... [(59,84),(96,148)] a6 [(400,465),(485,500)] [(17,24),(26,29)] [(51,63),(132,175)] ... [(51,63),(132,175)] a7 [(270,330),(378,422)] [(14,17),(21,22)] [(39,58),(91,132)] ... [(130,150),(175,175)] a8 [(250,260),(267,290)] [(33,38),(40,43)] [(39,49),(78,108)] ... [(140,170),(410,470)] a9 [(315,370),(377,405)] [(113,167),(194,211)] [(60,74),(94,126)] ... [(238,255),(305,367)] a10 [(320,344),(370,380)] [(83,92),(106,115)] [(97,119),(172,222)] ... [(195,225),(230,230)] a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 411 step 2 decision-makers did not have different preferences towards the choice of alternatives, therefore, pai was determined using eq. (13): 1 0.1 10iap   step 3 elements of theoretical estimations matrix (tp) were calculated using expression (14), table 2 table 2 theoretical estimations matrix (tp) alternatives criteria c1 c2 c3 c8 a1-10 [(0.025,0.025), (0.025,0.025)] [(0.022,0.022), (0.022,0.022)] [(0.014,0.014), (0.014, 0.014)] ... [(0.004,0.004), (0.004,0.004)] step 4 using eqs. (18)-(21) we conducted normalization of elements of the initial decision-making matrix. normalized initial matrix (x) is shown in table 3. table 3 normalized initial decision-making matrix (x) alternatives criteria c1 c2 c8 a1 [(0.574,0.684),(0.775,1)] [(0.543,0.558),(0.591,0.649)] ... [(0.42,0.735),(0.82,0.82)] a2 [(0.682,0.809),(0.846,0.912)] [(0.404,0.428),(0.529,0.606)] ... [(0.213,0.32),(0.392,0.606)] a3 [(0.051,0.096),(0.184,0.221)] [(0.947,0.957),(0.962,0.971)] ... [(0,0.087),(0.093,0.31)] a4 [(0.007,0.051),(0.176,0.235)] [(0.962,0.966),(0.971,0.976)] ... [(0.239,0.239),(0.239,0.239)] a5 [(0.544,0.603),(0.632,0.662)] [(0.99,0.995),(1,1)] ... [(0.183,0.183),(0.183,0.183)] a6 [(0.221,0.316),(0.346,0.368)] [(0.875,0.889),(0.899,0.933)] ... [(0.189,0.189),(0.211,0.211)] a7 [(0.029,0.118),(0.188,0.253)] [(0.909,0.913),(0.933,0.947)] ... [(0.042,0.099),(0.169,0.169)] a8 [(0,0.015),(0.025,0.059)] [(0.808,0.822),(0.832,0.856)] ... [(0.07,0.155),(0.831,1)] a9 [(0.096,0.176),(0.187,0.228)] [(0,0.082),(0.212,0.471)] ... [(0.346,0.394),(0.535,0.71)] a10 [(0.103,0.138),(0.176,0.191)] [(0.462,0.505),(0.572,0.615)] ... [(0.225,0.31),(0.324,0.324)] after normalization of the initial decision-making matrix we satisfied the conditions to calculate elements of real estimations matrix (tr). we calculate elements of the real estimations matrix using eq. (16), table 4. table 4 real estimations matrix ( rt ) alternatives criteria c1 c2 c8 a1 [(0.014,0.017),(0.019,0.025)] [(0.012,0.012),(0.013,0.014)] ... [(0.002,0.003),(0.003,0.003)] a2 [(0.017,0.02),(0.021,0.023)] [(0.009,0.009),(0.012,0.013)] ... [(0.001,0.001),(0.002,0.002)] a3 [(0.001,0.002),(0.005,0.006)] [(0.021,0.021),(0.021,0.021)] ... [(0,0),(0,0.001)] a4 [(0,0.001),(0.004,0.006)] [(0.021,0.021),(0.021,0.021)] ... [(0.001,0.001),(0.001,0.001)] a5 [(0.014,0.015),(0.016,0.017)] [(0.022,0.022),(0.022,0.022)] ... [(0.001,0.001),(0.001,0.001)] a6 [(0.006,0.008),(0.009,0.009)] [(0.019,0.02),(0.02,0.021)] ... [(0.001,0.001),(0.001,0.001)] a7 [(0.001,0.003),(0.005,0.006)] [(0.02,0.02),(0.021,0.021)] ... [(0,0),(0.001,0.001)] a8 [(0,0),(0.001,0.001)] [(0.018,0.018),(0.018,0.019)] ... [(0,0.001),(0.003,0.004)] a9 [(0.002,0.004),(0.005,0.006)] [(0,0.002),(0.005,0.01)] ... [(0.001,0.002),(0.002,0.003)] a10 [(0.003,0.003),(0.004,0.005)] [(0.01,0.011),(0.013,0.014)] ... [(0.001,0.001),(0.001,0.001)] 412 d. boţanić, a. ranđelović, m. radovanović, d. tešić step 5 in this step, using expression (22), we calculated gap matrix ( g ), table 5. table 5 gap matrix (g) alternatives criteria c1 c2 c8 a1 [(0,0.006),(0.008,0.011)] [(0.008,0.009),(0.01,0.01)] ... [(0.001,0.001),(0.001,0.002)] a2 [(0.002,0.004),(0.005,0.008)] [(0.009,0.01),(0.013,0.013)] ... [(0.002,0.002),(0.003,0.003)] a3 [(0.019,0.02),(0.023,0.024)] [(0.001,0.001),(0.001,0.001)] ... [(0.003,0.004),(0.004,0.004)] a4 [(0.019,0.021),(0.024,0.025)] [(0.001,0.001),(0.001,0.001)] ... [(0.003,0.003),(0.003,0.003)] a5 [(0.008,0.009),(0.01,0.011)] [(0,0),(0,0)] ... [(0.003,0.003),(0.003,0.003)] a6 [(0.016,0.016),(0.017,0.019)] [(0.001,0.002),(0.002,0.003)] ... [(0.003,0.003),(0.003,0.003)] a7 [(0.019,0.02),(0.022,0.024)] [(0.001,0.001),(0.002,0.002)] ... [(0.003,0.003),(0.004,0.004)] a8 [(0.024,0.024),(0.025,0.025)] [(0.003,0.004),(0.004,0.004)] ... [(0,0.001),(0.003,0.004)] a9 [(0.019,0.02),(0.021,0.023)] [(0.012,0.017),(0.02,0.022)] ... [(0.001,0.002),(0.002,0.003)] a10 [(0.02,0.021),(0.022,0.022)] [(0.008,0.009),(0.011,0.012)] ... [(0.003,0.003),(0.003,0.003)] step 6 using expression (24) we calculated values of criteria functions (qi) by alternatives, that is calculation of total gap, table 6. table 6 matrix of total gap alternatives alternatives gap irn(qi) a1 [(0.025,0.038),(0,0.058)] a2 [(0.031,0.042),(0,0.061)] a3 [(0.046,0.056),(0,0.071)] a4 [(0.047,0.053),(0,0.067)] a5 [(0.032,0.038),(0,0.049)] a6 [(0.036,0.046),(0,0.061)] a7 [(0.041,0.05),(0,0.066)] a8 [(0.056,0.065),(0,0.08)] a9 [(0.059,0.074),(0,0.091)] a10 [(0.047,0.058),(0,0.074)] further on, using eqs. (25) and (26) we converted interval rough numbers to real numbers; based on this, we defined the initial ranking of alternatives, table 7. table 7 initial ranking of alternatives alternatives alternatives gap qi initial ranking of alternatives a1 0.0445 2 a2 0.0467 3 a3 0.0599 7 a4 0.059 6 a5 0.0415 1 a6 0.049 4 a7 0.0538 5 a8 0.0692 9 a9 0.0796 10 a10 0.062 8 a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 413 step 7 in the last step of the ir-mairca method, the dominance index of the firstranked alternative is determined using eq. (27), as well as the exact final ranking of alternatives, table 8. since eq. (28) yielded dominance threshold id =0.09, we notice that the advantage of initially first-ranked alternative (a5) is not significant enough compared to second-ranking (a1) and third-ranking alternative (a2). therefore, we can conclude that the decision-maker can choose any of the listed alternatives as the first-ranking one. this is a significant characteristic of the mairca method considering to a certain extent the omnipresent subjectivity of decision-makers while defining entrance parameters for weight coefficient criteria calculations. in the practical sense, during the last step of the mairca (ir -mairca) method, we take into consideration errors while defining criteria of weight coefficients, regardless of the methods used to determine the weight coefficients. table 8 final ranking of alternatives alternatives dominance index (ad,1-j) final ranking of alternatives a1 0.037 1* a2 0.065 1** a3 0.230 7 a4 0.219 6 a5 0.000 1 a6 0.094 4 a7 0.154 5 a8 0.348 9 a9 0.478 10 a10 0.257 8 from obtained rankings, we can conclude that the construction elements of the new type of the close-quarters combat weapon, or modifications to the existing ones, have to be based on the characteristics of first-ranked alternatives a1, a2 and a5. the initial request of the users gives the best characteristics of these three types of weapons, which would be adjusted to realistic possibilities for construction. through the developed model of multi-criteria decision-making, we can constantly compare requests for the new close-quarters combat weapon with the existing ones, and by doing that, we can conduct checks and corrections. this would lead to maximization of the weapon’s characteristics by the user; at the same time, the constructor would have the ability to constantly check the quality of his work in practice. 5. sensitivity analysis sensitivity analysis is the last step that needs to be applied. weak results of sensitivity analysis take the whole research process to the beginning [35]. there are different approaches to the sensitivity analysis of models; most often authors in their papers use sensitivity analysis by changing weight coefficients of criteria [36]. this analysis implies evaluation of alternatives based on different weight coefficients of criteria, that is favoring one criterion in each scenario. in this research we defined eight scenarios, table 9. 414 d. boţanić, a. ranđelović, m. radovanović, d. tešić table 9 weight coefficients of criteria in different scenarios criteria s-0 s-1 s-2 s-3 s-4 s-5 s-6 s-7 s-8 c1 0.25 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 c2 0.22 0.1 0.3 0.1 0.1 0.1 0.1 0.1 0.1 c3 0.14 0.1 0.1 0.3 0.1 0.1 0.1 0.1 0.1 c4 0.11 0.1 0.1 0.1 0.3 0.1 0.1 0.1 0.1 c5 0.10 0.1 0.1 0.1 0.1 0.3 0.1 0.1 0.1 c6 0.08 0.1 0.1 0.1 0.1 0.1 0.3 0.1 0.1 c7 0.06 0.1 0.1 0.1 0.1 0.1 0.1 0.3 0.1 c8 0.04 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3 rankings of alternatives obtained using different scenarios are shown in table 10. table 10 rankings of alternatives obtained using different scenarios criteria s-0 s-1 s-2 s-3 s-4 s-5 s-6 s-7 s-8 a1 1(2) 1 2 1 2 4 1 1 1 a2 1(3) 2 7 3 1 5 5 2 2 a3 7 8 6 8 8 8 8 5 9 a4 6 7 5 7 7 6 7 4 7 a5 1 3 1 2 3 2 3 3 3 a6 4 4 3 4 4 1 2 6 4 a7 5 5 4 6 5 3 4 7 5 a8 9 9 9 9 9 9 9 10 8 a9 10 10 10 10 10 10 10 9 10 a10 8 6 8 5 6 7 6 8 6 obtained rankings, shown in table 10, imply that favoring certain criteria affects the differences in rankings; this further implies that the developed model is sensitive to the changes of weight coefficients. rankings of alternatives by different scenarios are visible in the graph below, fig. 2. fig. 2 graph of alternatives rankings by scenarios a hybrid lbwa ir-mairca multi-criteria decision-making model for determination... 415 worst-ranked alternatives (a8, a9) in a large number of scenarios kept their rankings, as well as best-ranked ones (a5, a1, a2). however, even though the correlation between rankings seems pretty obvious, a serious analysis demands quantitative indicators. in that sense, we checked rankings correlation using the spearman’s rank coefficient: 2 1 2 6 1 ( 1) n i i d s n n     (30) where: s the value of the spirman coefficient; di the difference in the rank of the given element in vector w and the rank of the correspondent element in the reference vector; n number of ranked elements. the values of the spearman’s coefficients range from -1 ("ideal negative correlation") up to 1 ("ideal positive correlation"). in table 11, one can see values of the spearman’s coefficients by comparing all scenarios to each other. in the first row of table 11, when comparing scenario s-0 (values of weight coefficients obtained through research) to others we got values compared to the final ranking – values outside of the parentheses and compared to the initial ranking (values inside of parentheses). table 11 rankings of alternatives obtained using different scenarios scenarios s-0 s-1 s-2 s-3 s-4 s-5 s-6 s-7 s-8 s-0 1 0.93(0.93) 0.75(0.88) 0.90(0.92) 0.93(0.92) 0.75(0.85) 0.81(0.88) 0.86(0.85) 0.91(0.90) s-1 1 0.73 0.98 0.99 0.79 0.92 0.81 0.99 s-2 1 0.76 0.67 0.88 0.87 0.68 0.70 s-3 1 0.95 0.78 0.91 0.78 0.96 s-4 1 0.78 0.87 0.79 0.98 s-5 1 0.91 0.54 0.78 s-6 1 0.65 0.90 s-7 1 0.75 s-8 1 from table 11 we can see that the correlation of rankings by scenarios is very high. certainly the most important correlation on rankings is between scenarios s-0 and others, where the value of spearman’s coefficient does not go below 0.75; it is a satisfactory value. the lowest correlation of rankings is between scenarios s-5 and s-7 (0.54); however, it is expected for lower correlations to exist in situations where the weight coefficient of criteria increases significantly. essentially, there are no scenarios where the correlation is absent; neither is there a scenario whose correlation approaches ideally uncorrelated rankings. this implies that using this model, that is the ir-mairca method, we can reach good solutions, even in the cases when weight coefficients deviate from realistic requests. 416 d. boţanić, a. ranđelović, m. radovanović, d. tešić 6. conclusion in this paper, we have demonstrated the use of lbwa – ir-mairca model when selecting the close-quarters combat weapon that is most suitable for tasks executed by the members of the serbian army equipped with this kind of weapons. based on the selected weapon, one with the best characteristics, we can plan constructing a new weapon, or modifying an existing one; this we can do on the basis of realistic requests and realistic capacities of the constructor. throughout the paper, we have demonstrated all phases of developing and using a multi-criteria decision-making model. we have defined the selection-affecting criteria and calculated their weight coefficients using the lbwa method. this method has proved to be very applicable and simple in the process of collecting data from the experts. the choice of the best alternative was done using the mairca method, which was improved using interval rough numbers which significantly improved the decision-making process since it opened possibilities for observing characteristics of each weapon. a significant step of this method, determining the first-ranked alternative in relation to others, made three alternatives first-ranking. this is significant since the mairca method additionally eliminates subjectivity when making decisions. we have also conducted sensitivity analysis of the model. results obtained from sensitivity analysis show that output values (rankings of alternatives) change depending on weight coefficients. on the other hand, changes in rankings while changing weight coefficients of criteria, demonstrated clearly the dominance of the first-ranked alternatives. everything listed above implies that the model provides the same or similar results, regardless of possible minor errors that can occur in the process of defining weight coefficients of the criteria, as a consequence of subjectivity of 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by university of niš, serbia | creative commons license: cc by-nc-nd review article peakforce quantitative nanomechanical mapping for surface energy characterization on the nanoscale: a mini-review heebo ha1, sebastian müller2, roelf-peter baumann2, byungil hwang1 1school of integrative engineering, chung-ang university, republic of korea 2material physics, structures and surfaces, raa/os, basf se, germany orcid ids: heebo ha https://orcid.org/0000-0003-3047-2263 sebastian müller https://orcid.org/0009-0009-4843-1575 roelf-peter baumann https://orcid.org/0009-0002-5781-9650 byungil hwang https://orcid.org/0000-0001-9270-9014 abstract. surface energy characterization is important to design the fabrication process of reliable electronic devices. surface energy is influenced by various factors such as surface functionality and morphology. owing to the high surface-to-volume ratio, surface energy at the nanoscale can be different from that of the bulk. however, the conventional methods for characterization of surface energy such as a sessile drop or washburn methods cannot be used for nanoscale samples, owing to the limited volume for characterization. recently, surface energy characterization on the nanoscale using atomic force microscopy (afm) with peak force-quantitative nanomechanical mapping (pf-qnm) imaging mode has been proposed. the nanoscale afm tips measure the adhesion forces at the nanoscale, which are converted into surface energy through pre-calibrated curves. successful surface energy characterization of nanoscale metal samples using afm with the pf-qnm method has been reported previously. this minireview discusses the recent progress on surface energy characterization at the nanoscale using afm with the pf-qnm method. the fundamentals of the pf-qnm mode are introduced, and the results of surface energy characterization are summarized. consequently, the future research direction for surface energy characterization at the nanoscale is discussed. key words: surface energy, peak force-quantitative nanomechanical napping, nanoscale, metal received: november 26, 2022 / accepted: january 11, 2023 corresponding author: byungil hwang school of integrative engineering, chung-ang university, 84 heukseok-ro, dongjak-gu, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr https://orcid.org/0000-0003-3047-2263 https://orcid.org/0009-0009-4843-1575 https://orcid.org/0009-0002-5781-9650 https://orcid.org/0000-0001-9270-9014 2 h. ha, s. müller, r.-p. baumann, b. hwang 1. introduction with the development of low-dimensional devices with unique characteristics, building a heterostructure of two-dimensional (2d) layered materials on three-dimensional (3d) materials and thin film hetero epitaxy have been widely investigated [1-3]. integrating 2d materials with 3d materials and layer-by-layer stacking of thin films result in unprecedented physical and chemical properties, which can be used in various applications such as electronic skin [4-6], health care [7-9], diagnostic devices [10-12], and power generators [13-16]. nevertheless, building continuous heterostructure and direct transfer of 2d materials or thin films on different substrates is challenging. therefore, to develop reproducible and reliable exfoliation and transfer 2d materials and thins film stacking on different thin films or 3d structures, surface characterization of these materials is needed. despite the importance of surface characterization of materials, the conventionally adopted method to determine the surface energy involves measuring the contact angle between the liquid droplet and the solid surface interface [17-22]. this liquid droplet contact angle measurement can only measure at bulk scale or microscale owing to the inert size of the liquid droplet, thereby limiting nanoscale measurements. moreover, unlike bulk or microscale materials, nanoscale materials show different properties owing to the high surface area to volume ratio, which implies that small changes in the surface energy at a few nanometers scale can significantly alter the property of nanomaterials. thus, characterizing the surface energy of materials at the nanoscale is important. there are different methods to determine surface energy by measuring the contact angle. typically, the sessile drop method is used, which measures the contact angle of a liquid droplet on a solid surface using the young equation [23-25], 𝛾𝑠𝑔 − 𝛾𝑠𝑙 − 𝛾𝑙𝑔 ∗ 𝑐𝑜𝑠𝜃 = 0 (1) where 𝜃, 𝛾𝑠𝑔 , 𝛾𝑠𝑙 , and 𝛾𝑙𝑔 are the contact angle, interfacial energy at the substrate-gas, substrate-liquid, and liquid-gas interfaces, respectively [18, 19]. in the case of solid powder, the washburn adsorption method is used, which calculates the contact angle by measuring the weight of the solution adsorbed by the capillary phenomenon when the powder-filled cylinder is dipped into the solution; this is done using the washburn equation, 𝑐𝑜𝑠𝜃 = 𝜂 𝐶𝜌2𝛾𝐿 ∗ 𝑚2 𝑡 (2) where 𝜃, 𝜂, 𝐶, 𝜌, 𝛾𝐿, 𝑡, 𝑚, and 𝑡 are the contact angle, liquid viscosity, material constant, density, surface tension, weight, and time, respectively [26, 27]. although these methods are effective in bulk solid materials, with a decrease in material size, methods for evaluating surface energy through contact angle become complicated and are not able to achieve the necessary resolution, especially at the nanoscale. in addition, the washburn method cannot measure the surface energy of a single powder. to address this challenge, recently, a new technology was proposed to characterize surface energy at the nanoscale using atomic force microscopy (afm) with peak forcequantitative nanomechanical mapping (pf-qnm) imaging mode [17, 28, 29]. this method obtains a calibration curve comprising a plot of adhesion forces versus contact angle/surface energy for macroscale samples; then, by coupling the adhesive force of nanoscale material measured using afm pf-qnm with calibration curve, the surface peakforce quantitative nanomechanical mapping for surface energy characterization... 3 energy of the nanomaterials is estimated. although several attempts have been made for surface energy characterization of nanoscale metal strips using afm pf-qnm, further research is required for various forms of samples, such as powder, 2-d materials, and porous systems. for this, it is important to summarize as well as discuss the recent progress in the surface energy characterization technology using afm pf-qnm. in this minireview, the recently reported results of surface energy characterization using afm pfqnm are summarized. the fundamentals of the afm pf-qnm method are introduced, where the theoretical background and the process to evaluate the surface energy on the nanoscale are presented. our perspective on the afm pf-qnm method is shared, which provides ideas for future research on surface energy characterization at the nanoscale using the afm pf-qnm method. 2. surface energy characterization using afm-pf-qnm method 2.1. principles of afm pf-qnm imaging mode fig. 1a shows the schematic of the surface energy measurement process of nanoscale metals. in the first step of measurement, a calibration curve is acquired, where adhesion forces versus contact angle/surface energy of macroscale metals are plotted [28]. to achieve different hydrophobicity values, the macroscale metals were treated with several thiol sams. after the sam treatment, the contact angle/surface energy and adhesion force of the macroscale metals are measured using the sessile drop method and afm-qnm, respectively. the calibration curve is obtained by correlating the macroscale contact angle/surface energy and adhesion force. afterward, the adhesion force of nanoscale metals is measured using afm-qnm. finally, the contact angle and the surface energy of nanoscale metals were extracted by substituting the adhesion force of nanoscale metals into the obtained calibration curves. pf-qnm is a novel functioning mode of afm that can record real-time force versus separation curves by periodical tapping of peak force on the desired area of the sample using a low frequency. the maximum vertical force that presses down the sample surface is recorded as the peak force. in the pf-qnm mode, the peak force shows a constant value since the software calculates the force between the afm tip and the surface of samples based on the feedback reflecting the surface morphology and mechanical properties. fig. 1b shows a plot of force versus time during one tapping cycle of the pf-qnm operation (fig. 1d): (a) positioning the afm tip on the surface of the sample surface at a given height and ready to touch the surface, (b) touching the sample surface with the tip and then applying pressure until amounting to the peak force, (c) withdrawing the tip from the surface of the sample, (d) entirely detaching of the tip from the surface of the sample after reaching the maximum attractive force point, which represents the adhesion force, (e) returning of the tip to its original height position. after one tapping cycle (a-e) is completed, the next tapping cycle starts all over again. 4 h. ha, s. müller, r.-p. baumann, b. hwang fig. 1 (a) schematic of the ca and se measurement procedure for the nanoscale materials via nca-afm (b) illustration of afm pf-qnm imaging plot of force and zpiezo/tip movement as a function of time, and (c) force curve for analyzing the material-specific properties. (d) schematic of the afm pf tapping technique; cantilever oscillated below resonance, resulting in a continuous series of forcedistance curves [28]. reproduced with permission from ref. [28]. the force versus separation curve quantitatively denotes the various mechanical properties of the surface at each point, such as adhesion, derjaguin-muller-toporov (dmt) modulus, and energy dissipation (fig. 1c) [30-32]. adhesion force is the required force to detach the afm tip from the surface. in the force versus separation curve, the adhesion force is the force difference between the maximum attractive force point and the noncontact point. young’s modulus can be calculated with the reduced modulus, which is derived using the dmt model expressed as follows: 𝐹𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 = 4 3 𝐸∗√𝑅(𝑑 − 𝑑0)3 + 𝐹𝑎𝑑ℎ (3) where 𝐹𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 is the tip-sample force, 𝐸∗ is the reduced elastic modulus of the tip and the sample, r is the tip radius, 𝑑0 is the surface rest position, (𝑑 − 𝑑0) is the sample deformation, and 𝐹𝑎𝑑ℎ is the adhesion force during the contact [31, 33, 34]. the young’s modulus of the sample is calculated with the reduced modulus values using the following equation: peakforce quantitative nanomechanical mapping for surface energy characterization... 5 𝐸∗ = ( 1−𝑣𝑠 2 𝐸𝑠 + 1−𝑣𝑡𝑖𝑝 2 𝐸𝑡𝑖𝑝 )−1 (4) where 𝑣𝑠 is the poisson’s ratio of the sample, 𝑣𝑡𝑖𝑝 is the poisson’s ratio of the tip, 𝐸𝑠 is the young’s modulus of the sample, and 𝐸𝑡𝑖𝑝 is young’s modulus of the tip [31, 33, 34]. the areal gap in the curves between the approaching and withdrawing of one cycle of tapping represents the energy dissipation released during the pf-qnm imaging mode. 2.2. surface energy of nanoscale metal strips park et al. characterized the surface energy of au nano ribbons with a width of 430 nm and a thickness of 100 nm [20]. fig. 2b shows the sem image of au nano ribbon tested in the study. to make a calibration curve, thiols such as 1-undecanethiol (ch3(ch2)9sh), 11mercapto-1-undecanol (oh(ch2)11sh), 11-mercaptoundecanoic acid (cooh(ch2)9ch2sh), and 1h,1h,2h,2h-perfluorodecanethiol (cf3(cf2)7ch2ch2sh), which have end groups of cf3, ch3, cooh, and oh, respectively, were used to modify the surface of the reference substrates. the macro contact angle and surface energy of thiolmodified substrates were characterized using the sessile drop method, which were then plotted with the adhesion forces measured for the corresponding substrates using the pfqnm method. the calibration curves were made for the flattened au thin film surface fabricated through the process in fig. 2a. fig. 2c and d shows the plots of contact angle versus adhesion forces and surface energy versus adhesion forces, respectively. the linear changes in the macro contact angle (fig. 2c) and adhesion forces (fig. 2d) as a function of the adhesion forces are shown with slopes of -27.53 and 16.26, respectively. then, the adhesion forces of the au nano ribbons were measured using the pf-qnm method, which yielded a value of ~2.31 nn. thus, the contact angle and surface energy were estimated as 78.83° and 38.57 mn/m, respectively, by extracting the values corresponding to the adhesion force of the au nano ribbons from the calibration curves (blue dot lines in figs. 2c, d). these values were similar to those of bulk au, which has a contact angle of ~74.38° and a surface energy of ~38.59 mn/m. although this study was the first to report on the surface energy characterization for nanoscale samples using the pf-qnm method, they provided results only for a single sample, which is insufficient to understand the mechanism of surface energy change on the nanoscale. ha et al. investigated the effect of grain size on the surface energy of nanoscale au thin films using the pf-qnm method [28]. to form different grain sizes, au thin films with different thicknesses of 10, 50, and 100 nm were deposited on the sio2 substrates using the thermal evaporation method (figs. 3a-c). the change in the thickness from 10 nm to 100 nm caused the increase in rms roughness by ~0.5 nm (figs. 3d-f), which was negligible compared to the film thickness and afm tip radius. all the deposited film showed the identical crystallographic orientation as confirmed by x-ray diffraction (xrd) as shown in fig. 3g. the grain sizes estimated through xrd data were 10.0, 29.0, and 41.2 nm for thicknesses of 10, 50, and 100 nm, respectively (fig. 3h). surface morphology and lattice constants for all the samples as shown in the high-resolution transmission electron microscopy (hrtem) images (figs. 3j-o), which confirmed that there was no significant distortion of microstructure during further deposition to increase the thickness of au thin films. in addition, xps peaks of all the samples showed the same binding energies of 87.87 and 84.20 ev as confirmed by x-ray photoelectron spectroscopy (xps) results in fig. 3i. 6 h. ha, s. müller, r.-p. baumann, b. hwang therefore, the au thin films deposited with different thicknesses was a suitable material system to investigate the effect of grain boundary densities on the surface energy. fig. 2 (a) sem image of au nano ribbon. the calibration curves plotted by using the data from different thiols for evaluation of (b) contact angle and (c) surface energy of nanoscale au ribbon [17]. reproduced with permission from ref. [17]. to make the calibration curve, the macro contact angle and surface energy for the substrates treated with the different silanes such as n-[3-(trimethoxysilyl)propyl]ethylenediamine, (3aminopropyl)triethoxysilane (aptes), and octyltrichlorosilane (ots) were measured using the sessile drop method. then, the adhesion forces for each substrate were measured using the pf-qnm method (figs. 4e-g). figs. 4a and b shows the calibration plots for contact angle and surface energy, respectively. the macro contact angle values linearly decreased as the measured adhesion force (𝐹𝑎𝑑ℎ) increased, as shown in fig. 2c, whereas the surface energy was proportional to 𝐹𝑎𝑑ℎ. these plots provide the equation to calculate the nano contact angle and surface energy, which are as follows: nca (contact angle) = −1.19 × 𝐹𝑎𝑑ℎ + 153.77 (5) and se (surface energy) = 0.51 × 𝐹𝑎𝑑ℎ + 1.86 (6) once the 𝐹𝑎𝑑ℎ values for the nanoscale samples are obtained, the values of contact angle and surface energy can be calculated using the above-mentioned equations. the adhesion forces of the au thin films with different thicknesses, that is, grain sizes, were measured using the pf-qnm method; then, the measured values were used as input for the derived equations to evaluate the nanoscale contact angle and surface energy. figs. 3c and d are the calculated contact angle and surface energy as a function of grain boundary density (𝐷𝐺𝐵), which is calculated by dividing the diameter of the afm tip with a radius of ~125 nm with the average grain size. the adhesion force was smaller for the au thin films with lower 𝐷𝐺𝐵 than those with higher 𝐷𝐺𝐵 , which resulted in a linear increase in the peakforce quantitative nanomechanical mapping for surface energy characterization... 7 contact angle as a function of 𝐷𝐺𝐵 (fig. 3c). the surface energy is inversely proportional to 𝐷𝐺𝐵 , where a higher 𝐷𝐺𝐵 showed higher surface energy (fig. 3d). consequently, the rational equations, 𝐹𝑎𝑑ℎ = 40 𝐷𝐺𝐵 + 28.8 (7) and 𝑆𝐸𝑛𝑐 (surface energy of the nanocrystalline surface) = 21.2 𝐷𝐺𝐵 + 16.0 (8) were derived, which indicated that the surface energy of the nanoscale metals is inversely proportional to the grain size of the samples. fig. 3 (a-c) afm topography images and profile sections on the edge of the evaporated au layers on the sio2 substrate. afm topography images of (d) 10 nm, (e) 50 nm, and (f) 100 nm thick au layer; corresponding rms surface roughness values are also indicated. (g) xrd pattern and (h) variation in full width at half maximum (fwhm) and gs. (i) xps spectra results and (j-l) hrtem images of au films and (m-o) atom lattice with different thicknesses [28]. reproduced with permission from ref. [28]. 8 h. ha, s. müller, r.-p. baumann, b. hwang fig. 4 calibration plot consisting of adhesion forces from different alkyl silane selfassembled monolayers measured using the afm pf-qnm mode versus (a) macro ca and (b) se of sams evaluated using a krüss dsa100 drop shape analyzer. variation of nanoscale adhesion forces measured on top of 10, 50, and 100 nm thick au layers and converted using the calibration curve into an equivalent nanoscopic (c) ca and (d) se. adhesion force mapping images of (e) 10 nm, (f) 50 nm, and (g) 100 nm thick au layer [28]. reproduced with permission from ref. [28]. in the other work by ha et al., the effect of pattern size in nanoscale on the surface energy au strips were investigated [29]. the thermally evaporated au strips were manufactured on si substrate to have thicknesses (t) of 50 nm and 100nm (fig. 5i). the length (l) of the strips was fixed as 1000 nm while the width (w) was varied as 200 nm, 300 nm, 400 nm and 500 nm (figs. 5a-h). the vacuum-deposition process has a great influence on the nucleation and growth of au crystal on the si substrate. the morphology of au strips that corresponds to the grain size of au and the roughness of surface was largely governed by the processing temperature, deposition rate (ǻ/s) and status of vacuum during the thermal evaporation step, which consequently influenced the nanoscale contact angle and surface energy. the increase in grain size of au resulted in the higher rms values as shown in fig. 5j. the thinner film might cause a slightly higher density of nucleation that promotes the formation of small au grains on the substrate. on the other peakforce quantitative nanomechanical mapping for surface energy characterization... 9 hand, the deposition time for thicker au thin films was longer than those of thinner films that provides sufficient time for atoms to be arranged step-by-step, thereby forming larger and uniformly oriented grains. fig. 5 afm topography images of au strips with thicknesses of (a-d) 50 nm and (e-f) 100 nm and widths of (a, e) 200 nm, (b, f) 300 nm, (c, g) 400 nm, and (d, h) 500 nm. (i) thickness and (j) rms values of au strips with the different widths measured by afm mapping [29]. reproduced with permission from ref. [29]. the nano contact angle (nca) and surface energy (se) of the au strips were characterized by using a afm-pf qnm method. the calibration curves were prepared for the substrates treated with the different alkyl silanes such as methoxy(polyethyleneoxy)propyl]-trimethoxysilane (mpt), aptes, and ots (figs. 6a and b). based on the calibration plots, the standard equations can be derived for nca and se as follow. nca (contact angle) = −1.29 × 𝐹𝑎𝑑ℎ + 158.22 (9) se (surface energy) = 0.53 × 𝐹𝑎𝑑ℎ + 0.75 (10) figs. 6c and d show the converted nca and se values for the measured adhesion forces of au strips with different widths using the standard equation (9) and (10), respectively. the nca and se values showed no clear relation with the widths of au strips in the 10 h. ha, s. müller, r.-p. baumann, b. hwang identical thickness, which was attributed to the similar adhesion forces. on the other hand, the change in film thickness resulted in the large difference in nca and se values, which was due to the increased adhesion force by the increased thickness. therefore, it was concluded that the grain size was governing factor to determine se of au strips while the change in dimension in few hundreds of nanometer scale of nanoscale led to no significant change in se of au thin films. fig. 6 calibration plot consisting of the adhesion force from different alkyl silane selfassembled monolayers measured by afm in pf-qnm mode versus the (a) macroscopic contact angle or (b) surface energy. (c) translated nca and (c) surface energy of au strips as a function of the widths of au strips [29]. reproduced with permission from ref. [29]. 3. conclusions this mini-review discusses the recent trends in the novel technology for surface energy characterization at the nanoscale using afm pf-qnm. pf-qnm is a special function of afm that can be used to obtain the adhesion profile throughout the scanned area. the adhesion force is created by surface characteristics such as functionalities or morphology; thus, the surface energy is correlated to the adhesion forces. by deriving the calibration curves that present the relationship between the adhesion force versus and contact angle/surface energy, the contact angle or surface energy at the nanoscale is calculated. this method can overcome the limitations of the previous surface energy characterization peakforce quantitative nanomechanical mapping for surface energy characterization... 11 methods using the sessile drop method or washburn method. several reports demonstrated the potential of pf-qnm for surface energy characterization at the nanoscale by successfully measuring the surface energy of au nanoribbons and nanocrystalline au thin films. although these studies revealed the important relationship between the surface energy of nanoscale samples and morphology such as grain size, other factors such as roughness or defect densities are still unclear. moreover, once the sample size further decreases to the atomic scale, even defect densities on the surface significantly influence the surface energy. therefore, further research is required on the surface energy characterization on the atomic scale using the pf-qnm method. furthermore, atomic simulation should be carried out simultaneously to investigate pf-qnm-based surface energy characterization. pf-qnm opened a new research area on surface energy change at the nanoscale. we hope that this mini-review encourages further studies on surface energy at the nanoscale, especially by correlating experimental characterization results and theoretical simulation evidence. acknowledgement: this work was supported 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research presented in this paper is the development of the optimal micro configuration for vibration monitoring of pumping aggregate, based on microchip’s microcontroller (mc). hardware used is 10-bit mc, upgraded with 12/bit a/d converter. software for acquisition and data analysis is optimized for testing turbo pumps with rotation speed up to 2000 rpm. this software limitation is set for automatic diagnostics and for individual and manual vibro-diagnostic; the only limitation is set by accelerometer performance. the authors have performed numerous measurements on a wide range of turbo aggregates for establishing the operational condition of pumping aggregates. key words: micro configuration, vibration monitoring, microcontrollers 1. introduction the development of non-invasive methods of monitoring has enabled the transition from preventive to predictable maintenance. there are various indicators of machine condition (temperature, pressure…); however, the method of using vibrations for determining machine operating conditions has proved the best. a mc-based monitoring system is developed because vibrations-based monitoring has been used in a large number of cases in which the machine condition has been determined. this system is created to meet certain demands: monitoring platform based on the use of common pc, a low cost of device, mobility, 12-bit resolution and appliance on rotational machines. the microcontrollers application in industry is a new research topic, based on fundamental research [13] as well as on industrial application [14, 15]. received may 06, 2014 / accepted july 3, 2014 corresponding author: miloš milovanĉević faculty for mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: milovancevic@masfak.ni.ac.rs original scientific paper 172 m. milovanĉević, a. veg, a. makedonski, j. stefanović 2. use of mc in monitoring microcontrollers (mc) are electronic components designed for developing electronic systems for digital command and control. with the use of such systems it is possible to control several electronic devices and systems and to collect and process various electric and non-electric quantities. digital systems based on microcontrollers are programmable to perform distinct processing depending on the condition of control circuit, and to perform command in circuit by using the results obtained from the processing. the main difference between microprocessors and microcontrollers is that the latter are designed to incorporate a complete digital computer in one chip, because, besides the processor, they also contain memory and peripheral units. this results in the desired system having a minimal number of components as well as realizing savings in space and the time needed for device designing. there are several microcontroller manufacturers at present, having very diverse mc families in their production range. most important are intel, motorola and microchip. there is a very wide range of microcontroller usages because they can be programmed, depending on the desired usage, to obtain the type of behavior of the digital device we design [1]. with the intention to determinate an analysis of pumping aggregate operating condition, the authors of this paper have designed special mc developing environment, whose description is here given in brief. the optimal micro configuration for vibration monitoring is designed based on axiomatic design regarding electronic components selection. the basic idea in the optimal micro configuration design is to meet the frame conditions that are required for pump aggregates vibration monitoring. in order to define all components of the micro configuration it is necessary to define functional requirements of the system and the conditions in which the system will be tested. considering micro systems for vibration monitoring the signal characteristics determine system accuracy and vibration monitoring quality in general. selection of the microcontroller that serves as a base for micro system has been done primarily by taking into account economic aspects of new-developed system that is pc (personal computer) dependant [2]. microchip, a company that has been producing microcontrollers for more than a decade, and microcontroller pic16f877a have been developed as symbioses of microprocessor (cpu), memory and periphery, with pic as acronym for (peripheral interface controller) [3]. this microcontroller is based on cmos technology with risc architecture and implemented flash and eeprom memories. thus, pic16f877a represents the best compromise between price and technology. the main signal characteristics that have been chosen as the main requirements for design of optimal micro system are: resolution, stability and repeatability of signal. in order to meet the requirement of resolution an a/d converter is added, since the microcontroller has 10-bit resolution, a mcp3204 12-bit a/d converter is connected via spi connection protocol [4]. 3. signal repeatability analyses the analyses of the main selected signal characteristics have significant influence in an optimal micro system selection. a significant signal resolution has been obtained by introducing mcp3204 12-bit a/d converter and stability is ensured by selecting leading enbedded systems for vibration monitoring 173 electronic components manufactures and implementing their components. thus, testing signal repeatability in laboratory conditions, on signal generators tektronix 3102 and tektronix dpo 4034, is a method for selecting an optimal micro configuration. the following diagrams represent part of intensive signal repeatability testing of the selected optimal micro configuration. fig. 1 diagrams of signal repeatability testing by exponential signal signal repeatability has been tested by two types of complex signals: exponential and triangle. exponential signal in repeatability testing is presented in fig. 1 in the following order: a) original signal screen shot of referent testing signal, b) fast furrier transformation (fft) diagram of collected signal from tested configuration, c) collected signal from tested configuration without transformation. 174 m. milovanĉević, a. veg, a. makedonski, j. stefanović fig. 2 diagrams of signal repeatability testing by triangle signal triangle signal in repeatability testing is presented in fig. 2 in the following order, a) original signal screen shot of referent testing signal, b) fast furrier transformation (fft) diagram of collected signal from tested configuration, c) collected signal from tested configuration without transformation. this testing has proved that the selected optimal micro configuration for vibration monitoring based on microcontroller has suitable performance regarding signal characteristics. enbedded systems for vibration monitoring 175 4. identification of pumping aggregate vibration parameters it is necessary to provide several measures of supervision to assure the correctness of turbo pumps operating conditions. the control of vibrations and their measuring via electric means is considered the very basis for supervision. the primary objective of all the supervision measures is timely recognition of critical operating conditions. the operation of centrifugal pumps is accompanied by two undesirable side effects: vibrations and noise. the intensity level of vibrations and noise characterize the perfection of pump operation, its construction and pump condition during exploitation period, as well as cavitation phenomenon in the pump. more about all these effects of the emitted noise as a side effect of centrifugal pumps is given in [5, 6, 7]. the source of centrifugal pumps vibrations are mechanical, hydraulic and electrical processes caused by the pump construction, operating regime, exploitation and manufacturing technologies used. due to the blade passage frequency (bpf) with frequency fz = z / 2 = zn, where z is the number of impeller blades and n is the rotational speed in rps. unbalance of rotational masses of rotor is caused by oscillation with frequency j1 =  / 2. vibrations from the collisions of parts in the contact are produced in bearings, gear box, couplings and connected shafts of pump and driver. rolling bearings may produce vibration with frequency often lower than 30 khz [8]. disturbance force is generated by connecting of pump and driver shaft to the geared coupling, with frequency j2 = zs   / 2, (zs  number of coupling teeth). electromotor vibrations are caused by disturbance forces generated by variations of electromagnetic field, with frequency for this case: je =   zw / 2, (zw i number of motor poles). mechanical vibrations of pumps are the subject of numerous research projects [9]. the analysis of the obtained results leads to the conclusion that the level of vibration can be lowered by respecting certain instructions and recommendations for balancing of rotational masses, selection of bearings, couplings, eccentricity between the shaft axis of pump and driver, etc. 5. results horizontal pumps have a significant role in water transportation. it also defines the importance of providing flawless work. the electro motors of horizontal pumps are extremely burdened from the aspect of continuous exploitation for maintaining a permanent operation. an adequate choice of measuring point at pump aggregate of horizontal pump can indicate the condition of operation for electromotor bearings and rotor, the pumping aggregate bearings and coupling, and complete aggregate construction likewise. the following measuring points are chosen:  first measuring point is chosen for diagnosing the operational condition of the first bearing at electromotor.  second measuring point is defined to diagnose the condition of the driving electromotor second bearing 176 m. milovanĉević, a. veg, a. makedonski, j. stefanović  third measuring point is determined in such a manner that it is possible to diagnose both the condition of the pump first bearing and the elastic coupling.  fourth measuring place is defined to diagnose the condition of the pump second bearing. fig. 3 measuring point at horizontal pump aggregate cvnr 5-3, no.1 the measured result analysis is generated by means of frequency spectra. the presented diagrams are created from fft algorithm, adapted for pump aggregate diagnostics. measuring point 1, fig 4. a), horizontal and vertical acceleration not passing the 1 m/s² can be observed, indicating the electro motor (em) bearing proper operating condition. likewise, there are no vibrations in frequency range 700-900hz which indicates that the motor fan is installed correctly. measuring point 2, fig 4. b), high acceleration amplitude at frequency at 310hz is manifestation of an incorrect coupling working condition; the second electromotor bearing is in a good operating condition. measuring point 3, fig 4. c), based on the acceleration, the correct bearing operation can be concluded, while, an incorrect coupling operating condition can be based on analyzing previous diagrams. measuring point 4, fig 4. d), a satisfactory operating condition of second bearing of pump can be concluded. diagram presented in fig. 4 (a, b) points to the following facts: for electro motor it is possible to determine bearing malfunction as well as other mechanical defects as an incorrect coupling operating condition. diagram presented in fig. 4 (c, d) present pump bearing malfunction but also a high frequency range is appearing as a result of hydrodynamic processes in a pump.  first measuring point is chosen for diagnosing the operating condition of the bearing at the upper part of electromotor  second measuring point is defined to diagnose the condition of the driving electromotor lower bearing  third measuring point is determined in such a manner that it is possible to diagnose both the condition of the pump bearing and the elastic coupling  fourth measuring place is defined to enable a diagnosis of the vibrations caused by nonlinear oscillations of the complete pump aggregate. enbedded systems for vibration monitoring 177 fig. 4 fft diagrams for horizontal pump aggregate cvnr 5-3, no.1 at measuring points (mp), a) mp 1, b) mp 2, c) mp 3, d) mp 4. 178 m. milovanĉević, a. veg, a. makedonski, j. stefanović 1 4 2 3 fig. 5 measuring points at well pump aggregate bp 350-3g, no. 1 3.2.1. result analysis of pump aggregate measurement measuring point 1, fig. 6. а) based on diagram, the correct em bearing operation can be concluded but there are some vibrations on high frequencies which are product of fan vibrations. measuring point 2, fig. 6. b) based on diagram, the correct em bearing operation is concluded; there are vibrations on frequencies in the range of 400-700hz that are caused by an incorrect coupling connection between the shafts. measuring point 3 fig. 6. c) based on diagram, the correct bearing operation can be concluded; coupling connection vibrations are increased by the pump aggregate body vibration induced by a loose connection between aggregate and ground. measuring point 4 fig. 6. d) based on diagram, intensive vibrations in the rage of 300500hz can be concluded by a loose connection between aggregate and ground. in order to understand the results previously presented, it is necessary to emphasize that the frequency amplitude diagrams that are presented, are the result of several years of work in order to determine the operating condition for most of the pump aggregates used in industry. the data presented is a small segment of the research. over 230 pump aggregates have been analyzed in order to improve the mathematical apparatus and software for vibration analysis. the table gives absolute values of acceleration; bold values are marked as analyzed in the previous diagrams. the table is part of the project report and the starting point is in the analysis of operating condition. enbedded systems for vibration monitoring 179 fig. 6 fft diagrams for well pump aggregate bp 350-3g, no. 1 at measuring points (mp), a) mp 1, b) mp 2, c) mp 3, d) mp 4. 180 m. milovanĉević, a. veg, a. makedonski, j. stefanović 6. conclusions the examination of pumps vibration phenomenon provides the data about the vibration magnitude and its frequency components as well as their change with respect to operating parameters. on the basis of the obtained results the safety level for the pump and the whole plant is evaluated. besides the mentioned ones, in most cases it is necessary to determine the cause of non-stationary occurrences. the operating ranges that should be avoided are determined in many cases. the primary sources of vibrations at centrifugal pumps are mechanical, hydraulic and electric processes caused by the design of a pump, its manufacturing technology, operating regime and exploitation condition. it is possible to eliminate mechanical and electrical sources, partially or completely, thus lowering the level of vibrations. however, hydraulic vibrations are hard or almost impossible to avoid [12]. hydraulic processes which happen in pumps are complex and non-stationary as a rule. for description of such processes it is possible to form mathematical models whose evaluation is performed after very comprehensive, expensive and long-lasting research projects. for these reasons such models have not been taken into consideration in this paper – given are the experimental results obtained by new-developed embedded system, based on the new generation of microcontrollers. in the diagnosis of pumping aggregate malfunctions, frequency spectrums have a crucial role in defining the causes of failure. the created monitoring system has significant results in frequency vibration analyses regarding mechanical defects detection of the pumping aggregate. references 1. matić, n., andrić, d., 2000, pic mikrokontroleri, mikroelektronika beograd. 2. milovanĉević, m., cvetković, m., 2009, application of new microcontroller generation for pump aggregate working condition analyses, journal research and design in commerce & industry, 23/24, pp. 35-41. 3. milovanĉević, m., cvetković, m., 2009, applicative approach to vibro-diagnostic model optimization for turbo pumps, journal research and design in commerce & industry, 25, pp. 41-48. 4. milovanĉević, m., veg, a., 2009, application of axiomatic design on vibro-diagnostic system, 9th international conference ″research and development in mechanical industry″ radmi, pp. 295-301, serbia. 5. ĉudina, m., 2003, detection of cavitation phenomenon in a centrifugal pump using audible sound , mechanical system and signal processing, 17(6), pp. 1335-1347. 6. ĉudina, m., prezelj, j., 2008, use of audible sound for safe operation of kinetic pumps, international journal of mechanical science, 50(9), pp. 1335-1343. 7. ĉudina, m., prezelj, j., 2009, detection of cavitation in operation of kinetic pumps. use of discrete frequency tone in audible spectra, applied acoustics, 70(4), pp. 540-546. 8. milovanĉević, m., stefanović marinović, j., anċelković, b., veg a., 2010, embedded condition monitoring of power transmission of a pellet mill, transactions of famena 34(2), pp. 71-79 9. grjanko, l. p., papir, a. n., 1975, lopastine nososi, mašinostroine leningrad. 10. milenković, d., 1988, nestabilno strujanje kroz kola turbomašina izazvano globalnim gubitkom stabilnosti, 18. jugoslovenski kongres teorijske i primenjene mehanike, vrnjaĉka banja, pp. 320-326. 11. gost 13731—68 (state all union standard), mechanical vibration, vibration characteristics control 12.1.003-76. 12. milovanĉević, m., milenković, d., troha, s., 2009, the optimization of the vibrodiagnostic method applied on turbo machines, transactions of famena, 33(3), pp. 63-71. 13. danković, d., vraĉar, lj., prijić, a., prijić, z. 2013, an electromechanical approach to a printed circuit board design course, ieee transactions on education, 56(4), pp. 470-477. enbedded systems for vibration monitoring 181 14. prijić, a., danković, d., vraĉar, lj., manić, i., prijić, z., stojadinović, n., 2012, a method for negative bias temperature instability (nbti) measurements on power vdmos transistors, measurement science & technology, 23(8), 085003. 15. vraĉar, lj., prijić, a., vuĉković, d., prijić, z., 2012, capacitive pressure sensing based key in pcb technology for industrial applications, 5th ieee sensors journal, 12(5), pp. 1496-1503. embedded sistemi za monitoring vibracija cilj istraživanja prikazanog u radu je razvoj optimalne mikrokonfiguracije za monitoring vibracija pumpnih agregata zasnovanom na mikročipovom mikrokontroleru (mc). mikrokontrolerski 10-bitni hardver unapređen je 12-bitnim a/d konvertorom. softver za akviziciju i analizu podataka optimizovan je za testiranje turbo pumpi, čiji je broj obrtaja rotora do 2000o/min. ovo softversko ograničenje odnosi se na automatski režim rada, ako se testiranje izvodi manuelno, jedino ograničenje potiče od ograničenje akcelerometra. autori su izveli veliki broj merenja na širokom spektru turboagregata u cilju utvrđivanja stanja radne ispravnosti pumpi. kljuĉne reĉi: mikrokonfiguracija, monitoring vibracija, mikrokontroleri 12892 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 827 860 https://doi.org/10.22190/fume240920015m © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper porosity effects on nonlinear static performances of functionally graded shells considering thickness stretching hanen mallek1,2, hana mellouli2,3, lotfi ben said4, mondher wali2,5, fakhreddine dammak2, muapper alhadri4 1higher institute of applied sciences and technology of sousse, issat sousse, university of sousse, tunisia 2laboratory of electrochemistry and environment (lee), national engineering school of sfax, (enis), university of sfax, tunisia 3gafsa preparatory engineering institute, ipei gafsa, university of gafsa, tunisia 4department of mechanical engineering, college of engineering, university of ha’il, ha’il city, saudi arabia 5higher school of sciences and technology of hammam sousse, essths, university of sousse, tunisia orcid ids: hanen mallek https://orcid.org/0000-0003-1681-9558 hana mellouli https://orcid.org/0000-0001-6739-1550 lotfi ben said https://orcid.org/0000-0001-7652-3068 mondher wali https://orcid.org/0000-0001-8384-8936 fakhreddine dammak https://orcid.org/0000-0002-9629-671x muapper alhadri https://orcid.org/0000-0001-5848-5803 abstract. this study delves into the intriguing realm of nonlinear responses exhibited by porous functionally graded (fg) 3d shell structure. a power law approach is formulated to simulate the mechanical behavior of fg materials, considering two distinct porosity distributions. this approach provides a comprehensive exploration of porosity phenomena in fg materials. the finite element (fe) formulation is based on an improved first-order shear deformation theory (fsdt) with the inclusion of thickness stretching parameters. the theoretical framework improved in this manner provides a more accurate representation of the transverse shear stress distribution within the structure, capturing the complexities of its behavior under loading conditions. this research enriches understanding of fg materials by integrating porosity that may be distributed evenly or unevenly, thus contributing to the advancements in the field. key words: porous fgm, thickness stretching, fe analyses, porosity, shell structure received: september 20, 2024 / accepted april 05, 2025 corresponding author: hanen mallek laboratory of electrochemistry and environment (lee), national engineering school of sfax, (enis), university of sfax, route de soukra km 4, 3038 sfax, tunisia e-mail: hanen.mallek@enis.tn 828 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri 1. introduction functionally graded materials (fgms) have garnered increasing interest in modern technologies over the last decades due to their unique composition, synthesized from a blend of two distinct material phases in specific volume ratios. these materials exhibit gradual variations in properties along specified directions, typically through-thickness, thereby mitigating abrupt changes in thermal or mechanical properties and reducing interlaminar stresses. leveraging the remarkable advantages of fgms in industrial and engineering applications, numerous researchers have focused on analyzing the behaviors of fgm plates and shell structures using finite element (fe) methods [1]. recent studies have also explored the integration of fgm materials with piezoelectric materials to create smart structural components. for instance, mallek et al. [2] investigated the piezoelastic response of fgm shells embedded with intelligent materials under mechanical and electrical loads, highlighting the potential for advanced multifunctional applications in structural engineering. the materials industry has witnessed a significant evolution, with porous materials gaining prevalence owing to their distinct characteristics, including lightweight nature, excellent energy absorption, and heat resistance [3]. these attributes have motivated the emergence of a novel category of lightweight materials known as functionally graded porous (fgp) materials, characterized by a graded distribution of internal pores within the microstructural section [4]. fgp materials, offering unique mechanical properties, have recently found applications in various domains, including static, free vibration, and dynamic analysis. numerous studies have investigated both linear and nonlinear behaviors of fgp beams [5-7]. for instance, wang and wu [8] conducted free vibration analysis of cylindrical porous fg shells, combining sinusoidal shear deformation theory with the rayleigh-ritz method to discern the effect of graded porosity on structural characteristics. additionally, nguyen et al. [9] proposed an efficient approach based on the polygonal finite element (fe) method to predict static and free vibration responses of fgp plates reinforced with graphene platelets (gpls), considering various porosity distributions. subsequent works by other authors [10-12] have explored the bending, buckling, and vibration behaviors of fgp structures, highlighting the influence of porosity distributions and mechanical properties. notably, zghal and dammak [13, 14] investigated buckling responses and vibrational behavior of fgp plates and shells with varying mechanical properties using a mixed fe formulation, emphasizing the critical role of porosity percentages and distributions. the literature review underscores the importance of considering porosity effects and distributions in predicting the mechanical behavior of fgp materials. indeed, wattanasakulpong et al. [5] demonstrated the emergence of porosity and micro-voids during the multi-step infiltration technique in layered fgms, attributed to solidification temperature differences among material phases. two models for studying the mechanical behaviors of fgp materials have been proposed: one based on mixture rules, assuming continuous variation of effective material properties and even or uneven dispersion of porosities [15], and another employing a specific continuous function to determine porous material properties throughout the structure's thickness [16]. while considerable attention has been devoted to analyzing the mechanical behavior of fgp beams and plates, there is a notable dearth of investigations concerning porous fg shell structures, as noted by wang and wu [8]. thus, it is imperative to explore the behaviors of porous fg shells to unveil the effects of porosity. shell structures find widespread applications [17] in various industrial fields, including pressure vessels, rollforming, and deep drawing, often undergoing large deformations [18] and finite rotations porosity effects on nonlinear static performances of fg shells considering thickness stretching 829 where linear models prove inefficient. consequently, accounting for geometric nonlinearities becomes crucial in shell modeling, given the significant influence of nonlinear parameters in numerous analysis fields, particularly in porous fg structural design. within this framework, abid et al. [19] examined the geometrically nonlinear static responses of porous fg ring and cylindrical panels, considering even and uneven porosity distributions along the thickness direction. the high-order shear deformation theory (hsdt) employing double directors' vectors was utilized with an isoparametric concept, highlighting the disparities between linear and nonlinear deflections of fgp shell structures. it was deduced that under high loading conditions, linear shell modeling may yield inaccurate predictions, emphasizing the importance of geometric nonlinearity in fe formulations. moreover, when modeling moderate or thick shell structures under the plane stress assumption (psa), neglecting the thickness stretching effect of normal deformation may lead to inaccuracies, especially for thick structures. in this regard, evolving an efficient fe formulation capable of accommodating arbitrary three-dimensional constitutive laws without condensation is crucial. to achieve this objective, the standard fsdt shell formulation with five parameters is extended to a 6parameter theory, including the full set of stress and strain fields. this extension involves considering an extensible director vector, acknowledging the sixth degree of freedom as a scalar stretching component. the exploration of mechanical behaviors using the 3d theory of elasticity was pioneered by simo and rifai [20] employing a 6-parameter shell model, which avoids displacement formulation errors and thickness locking phenomena. various methods have been proposed to address thickness locking, including quadratic displacement distribution through the thickness direction and enriching the thickness strain via methods like the enhanced assumed strain (eas) method. recent efforts by beheshti and ramezani [21] introduced a nonlinear extensible shell model to investigate the large deformation of fg shell structures, incorporating seven parameters to account for thickness stretch, while employing the eas concept to mitigate numerical instability issues and enhance strain components. triply periodic minimal surface (tpms) structures are currently renowned for their exceptional energy absorption capabilities, which make them highly suitable for applications requiring efficient mechanical performance. in this regard, several recent studies have explored advanced approaches such as nonlocal strain gradient-based isogeometric analysis for functionally graded tpms nanoplates reinforced with graphene platelets. these studies aim to improve the accuracy of mechanical behavior predictions for such complex materials. furthermore, new computational frameworks [22, 23] have been proposed for modeling 3dprinted graphene-reinforced functionally graded tpms plates, emphasizing multiscale techniques to better capture the mechanical properties and performance in additive manufacturing contexts. these advancements offer a deeper understanding of the behavior of graphene-based composites and present innovative computational strategies for their analysis. from the literature review, it is evident that previous studies primarily focused on analyzing the responses of fg plates and shell structures with internal pores, employing the plane stress assumption (psa) model, which neglects thickness stretch. however, a comprehensive investigation into the large elastic deformations and finite rotations of fg shell structures with internal pores, considering three-dimensional constitutive equations and through-the-thickness stretching, remains unexplored. this paper aims to address this research gap. our work introduces a 3d extensible shell model utilizing six independent kinematic parameters, including thickness stretching, and incorporates the eas method. considering the limitations of conventional theories like fsdt and hsdt, an efficient finite element shell model is proposed based on an improved fsdt theory, ensuring a realistic parabolic distribution of transverse shear strains (tss) across the structure's 830 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri thickness. additionally, the importance of thickness stretching is highlighted by incorporating linear variation through the thickness via the eas concept to analyze large elastic deformations. moreover, locking problems inherent in extensible models are addressed by applying the assumed natural strain (ans) method to improve membrane behavior. the subsequent sections of this paper delve into the materials modeling and geometrically nonlinear analysis methods considered herein. the proposed finite element method necessitates an effective elastic part, including offsetting the neutral surface, accounting for geometric nonlinearity in stiffness matrices, enhancing quantities for bending and membrane deformations using the eas method, and improving tss using a modified fsdt theory and ans method. 2. generalized formulation and fe modeling of fgp shells the study of shell structures is significant in several fields of structural mechanics. with the expansion of fe methods, various theories have been proposed to examine shell issues. in this work, the extensible director shell approach is used to develop the governing mathematical expressions for the elastic case to evaluate the large deflection performance of porous functionally graded structures. three essential concepts underpin the extensible fsdt shell theory. the first concept illustrates the position of the director dispersion in the deformed state. the second concept, called parabolic tss, assumes that the distribution of tss across the structure's thickness is formulated as polynomial of degree two. the inclusion of the thickness stretching parameter in shell elements is established in accordance with the third concept. 2.1. fgp properties’ estimation the increasing application of functionally graded materials in thin-walled structures necessitates the development of accurate predictive models to characterize their material behavior. in this investigation, the rule of mixture (rom) is employed to analyze the performance of these advanced materials. fgms can be categorized into two classes: perfect and imperfect. perfect fgms are composed of two phases: ceramic and metal where the composition and properties change gradually within the material. imperfect fgms, on the other hand, are characterized by the presence of internal pores within their microstructure (see fig.1). fig. 1 perfect and imperfect fgm porosity effects on nonlinear static performances of fg shells considering thickness stretching 831 2.1.1. perfect fgm material properties fgms are composite materials composed of distinct phases, typically metals and ceramics, which vary gradually in volumetric fraction along the thickness direction. this gradual variation results in material properties that smoothly transition from one constituent to another. typically, the top surface consists of ceramic while the bottom surface is made of metal. in this study, the mechanical properties of the material evolve according to a power law distribution (where n denotes the power low index), considering the specific material properties of the ceramic and metal phases (as illustrated in fig. 2 with the porosity volume fraction α set to 0). it is worth mentioning that when n equals zero, the material is entirely ceramic, and as n tends to infinity, the material transitions to being entirely metal. in situations where there are asymmetrical variations in material properties, particularly in terms of e, the offset 𝑧̅ of the neutral surface from the discretized mid-plane is defined as detailed in reference [24]. /2 /21 /2 1 /2 ( ) ; ( ) h h h h e e z dze z e e e z z dz − −  =  =   =    (1) where �̅� and �̅�1 are the effective moduli for membrane and bending, respectively. subsequently, the analytical representation of 𝑧̅ /h is approximated as [24]: ( 1)( 1) / ; 2( 1)( 2)( ) c m en k n z h k n n n k e − + = = + + + (2) in this study, the modified rule of mixture (mrom) refers to an enhancement of the traditional rule of mixture used in composite materials analysis. the rule of mixture (fig. 2), also entitled the fgm homogenization schema, is used to predict the overall properties of an fgm based on the properties of its individual constituents and their volume fractions. in the context of wattanasakulpong and ungbhakorn's work [25], the modified rule of mixture extends this concept by incorporating additional factors or adjustments to more accurately predict the behavior of functionally graded materials (fgms). these factors may firstly refer to the porosity volume fraction α that defines the proportion of the fgm's volume that is occupied by voids or pores. these porosities can be distributed either evenly or unevenly (as shown in fig. 2) across the thickness of the structure. it is assumed that the top surface is predominantly ceramic, while the bottom surface is primarily metal, with a continuous variation of the material composition throughout the thickness. the material properties of the porous fgm follow the rule of mixtures, where the porosities are distributed uniformly across the thickness, as described by: ( ) ( ) ( ) 1 2 2 c m c m c m z p z p p v p p p b h    = − + − + −    (3) where α (0≤ α ≤1) represents the porosity volume fraction, 𝑃𝑚 and 𝑃𝑐 denote the material properties of the metal and ceramic respectively such as the young’s modulus e, the poisson’s ratio υ and the density ρ. further, 𝑉𝑚 and 𝑉𝑐 indicate the volume fractions of metal and ceramic respectively assuming the condition of 𝑉𝑚 + 𝑉𝑐 = 1 and h denotes the structure’s thickness. 832 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri fig. 2 fgm homogenization scheme when α is zero, it indicates that the fgm is completely dense without any voids or pores. increasing this factor results in more voids along the fgm’s volume. secondly, the factor b represents the uneven parameter. setting b to zero means that the porosity is evenly distributed throughout the material. conversely, setting b to unity indicates an uneven distribution of porosity within the fgm (fig. 2). not forgetting the power law index n of the fgm that determines the rate at which the material composition and properties change along the material's gradient. in fact, the volume fraction of the ceramic phase varies along the structure's thickness, denoted as:   1 ( ) ; / 2, / 2 2 n c z v z z h h h   = +  −    (4) as the power law index n increases, the fgm structure becomes richer in metal phase, while lower values of n result in a structure that is predominantly ceramic. when n is set to zero, the structure is entirely ceramic, and as n increases, the structure shifts towards a higher metal content. 2.1.2. imperfect fgm material properties porosities arise as a consequence of the manufacturing process, manifesting as defects within the microstructure. it is imperative to account for these porosities when studying the mechanical behavior of fgms. in this study, two distinct porosity distributions along the thickness direction are examined: even and uneven, as depicted in fig. 1. in contrast to the even distribution of porosity, where the distribution is uniform across the thickness, the uneven distribution reveals a concentration of porosities primarily within the middle zone. fig. 3 illustrates the evaluation of effective fgp material properties along the thickness, considering both even and uneven distributions of porosities. the exponent n represents the material power-law index describing the material profile gradation. when n is set to zero, a single-phase ceramic material is obtained. the porosity volume fraction α varies from zero to 1, setting α to zero implies a perfect fgm shell, while increasing introduces porosity and micro-voids, rendering the fgm imperfect. further, b represents the uneven parameter. when it is set to zero, an evenly distributed porosity is achieved, whereas setting it to unity indicates an uneven porosity distribution. porosity effects on nonlinear static performances of fg shells considering thickness stretching 833 it is important to acknowledge that the variations of porous fg material properties along the thickness direction are asymmetric. in mixed finite element formulations, this asymmetry can pose challenges during simulation due to the coupled nature of bending and membrane effects. to address this issue, the concept of a neutral surface can be employed. all effective quantities are then evaluated relative to this neutral surface. the analytical expression for the offset of the neutral surface from the discretized mid-plane is given as follows [26]: ( 1)( 1) / ; ( 1)( 2)(2( ) ( 1)( 1)(1 0.5 )) c m en k n z h k n n n k k n b e − + = = + + + − + + − (5) the distribution of the offset 𝑧̅/ℎ along with the variation of the material exponent n and the porosity volume fraction α is plotted in fig. 3. a) b) fig. 3 variation of the offset 𝑧̅/ℎ versus different values of the material exponent n and porosity volume fraction α for the (a) even; (b) uneven configuration 834 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri 2.2. extensible shell kinematics and fundamental equations this section provides a brief overview of the kinematics underlying the nonlinear extensible director shell model. the initial and deformed configurations are denoted as c0 and ct, respectively. the present research utilizes curvilinear dimensions (𝜉1, 𝜉2, 𝜉3 = 𝑧) to characterize the location and placement of physical parameters. 2.2.1. extensible director shell kinematics in the following, the fsdt model, also sometimes denoted as 5-parameter model, is here enhanced with an extensible director, thus being able to depict the shell thickness stretch during its deformation. according to the cited theory, the displacement field of an arbitrary point (q) on the shell, in both its initial state and its current configuration can be expressed using curvilinear coordinates:   1 2 1 2 1 2 0 01 2 1 2 1 2 ˆ( , , ) ( , ) ( , ) ˆ; ; / 2, / 2 ˆ( , , ) ( , ) ( , ) q p q p z z z z z z h h z z              = + = −  − = + x x d x x d (6) in which the in-plane curvilinear coordinates (𝜉1, 𝜉2) are tangent to the mid-surface, whereas the thickness coordinate 𝜉3 is perpendicular to this mid-surface, as depicted in fig. 4. xp and xp represent the position vector located on the mid-surface in c0 and ct, respectively. ho refers to the initial thickness of the structure and d designates the director shell vectors in c0. fig. 4 geometry and kinematics of the shell element the initial improvement to the shell's kinematic properties involves incorporating an extensible director vector using an extra variable known as the shell thickness stretch. consequently, the deformation of the director within ct is characterized by both an inextensible unit vector t and a thickness stretch parameter represented as λ: porosity effects on nonlinear static performances of fg shells considering thickness stretching 835 1 2 1 2 1 2 1 2( , ) 1 ( , ) ( , ); ( , ) 1         = − = d t t (7) according to eq. (4), the vectors xq and d are treated as independent quantities, with xq expressed by three components, while d is described by only two. the shell thickness can be determined in relation to the thickness stretching parameter λ (𝜉1, 𝜉2) as follows [27]: 0 01 / , (1 )h h h h − = = − (8) it is worth mentioning that for the reference configuration c0 where there is no thickness stretch, the scalar function λ is initially defined as zero. the derivative of the director vector d is expressed as below: , , ,(1 ) , 1,2k k k k = − − =d t t (9) for the ease of defining the shell deformation gradient, the covariant basis vectors are determined in the initial and actual configurations as: , 3 , 3 ; , 1,2 ˆ ; k k k k k k z k z = + = = = + = g a d g d g a d g d (10) where (𝑨1, 𝑨2, 𝑨3) denote the tangent basis vectors of the mid-surface in the undeformed reference configuration and (𝒂1, 𝒂2, 𝒂3) are their image at the actual configuration. the riemannian metric at the initial configuration c0 is designated by g, that of the current configuration ct by g, having the following expressions: . , , 1,2,3 . i j i j i j   =   =  =   g g g g g g (11) introducing the expressions of the covariant basis vectors and the riemannian metric (eqs. (10) and (11)) yields the covariant components of the 3d metric tensor in ct as: 2 0 1 ij i j ˆ ( ) ˆ. , + , , 1,2,3, , 1,2 . kβ kβ k k3 k k 33 g a zb o z g g c zc i j k g    = + +  = = = =  = g g d d (12) where 𝑎𝑘𝛽, 𝑏𝑘𝛽, 𝑐𝑘 0 and 𝑐𝑘 1 represent the kinematic variables, which are expressed as: , , 0 1 , . , . . , , 1,2 . , . kβ k β k k β β k k k k k a b k c c    = = + = = = a a a d a d a d d d (13) for later use, geometrical variables are also added in the initial state c0 as: 1 2 1 2d d , , d d dξ ξa a a a a ξ ξ= =  =a a (14) 836 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri 2.2.2. green compatible strain tensor a suitable strain measure, accommodating arbitrarily large rotations and displacements, is provided by the green compatible strain tensor 𝑬𝑐, given as: ( ) 2 c 0 1 ij ij 3 c 2 33 33 ˆ ( ) 1 1 ˆ( ), ( ), 2 +z , , 1,2 2 2 (1 ) 1 / 2 c kβ kβ k c c ij k k k e e zχ o z e g g e k e e       = + +  = − = − = =  = = − − e g g (15) the components of strain field, including membrane 𝑒𝑘𝛽, transverse normal 𝑒33, bending 𝜒𝑘𝛽 and shear (𝛾𝛼 0, 𝛾𝛼 1 ) strains, are defined as energetically conjugate parts to the kinematic variables. 0 0 0 1 , ( ) / 2 , ( ) / 2 , , 1,2 , ( 1) kβ kβ kβ k k k k k k k k e a a χ b b k γ c c γ        = − = − = = − = − (16) by utilizing these relations, the variation of the corresponding strains can be evaluated as: , , 33 , , , , , , 0 1 , , , ( . . ) / 2 , ( 1) ( . . . . ) / 2, , 1,2 . . , ( 1) kβ k β β k k k β β k k β β k k k k k k k δe δ δ δe δχ δ δ k δγ δ δ δγ            = + = −  = + + + =  = + = + − a x a x a d a d x d x d a d x d (17) whereby, the variation of the extensible shell director is computed as follows: , , , , ,(1 ) , (1 )k k k k k          = − − = − − − −d t t d t t t t (18) the cited quantities are condensed obtaining the following matrix form: 11 11 0 1 22 22 0 11 1 0 1 33 2 2 12 12 , , , 0 2 2 e e e e                       = = = =                    e χ γ γ (19) next, these strain vectors are organized into a generalized strain 𝞢𝑐 as follows: 0 1 12 1 c       =       e χ σ γ γ (20) 2.2.3. improved fsdt theory the mindlin's first-order shear deformation theory (fsdt) is based on the assumption of a linear shear strain distribution 𝐸𝑘3 𝑐 across the thickness of the shell. however, it is well porosity effects on nonlinear static performances of fg shells considering thickness stretching 837 established that shear strain follows a parabolic distribution across the thickness, vanishing at the top and bottom surfaces of the shell. to correct this discrepancy, several studies have introduced constant shear correction coefficients. additionally, tanov and tabiei [28] incorporated a shear function in the fsdt formulation, resulting in a parabolic distribution of transverse shear stresses. building on the work of shi [29], the present study improves the fsdt theory by assuming a parabolic shear strain distribution through the thickness, using a quadratic function f(z) as detailed in the following: 2 0 1 3 2 5 ˆ2 ( )( ), ( ) 1 4 , 1,2 4 c k z z e f z z f z k h   = + = − =    γ γ (21) 2.3. shell variational formulation the enhanced assumed strain (eas) method is employed for defining the strain tensor, aiming to prevent numerical locking problems and enhance the deformation gradient. according to this technique, the green strain field e comprises the compatible strain field 𝑬𝐶 and the enhanced incompatible strain field �̃�: c= + e e e (22) the eas formulation is employed to represent the three-dimensional equilibrium equations in terms of three independent variables: the green strain tensor e, the 2nd piolakirchhoff stress tensor s, and the deformation field φ. as demonstrated in the work of simo and rifai [20], the following variational formulation is articulated: int int0 ; : ( )c ext v w w w w dv = − = = + s e e (23) where 𝑊𝑒𝑥𝑡 represents the virtual work of the external loading and dv expresses the shell initial volume element. note that the stress tensor s is considered to be orthogonal to the enhanced strain tensor �̃� over the element as outlined by simo et al. [30]. this assumption of orthogonality is expressed as: 0 v dv = e s (24) moreover, no additional shear enhancement components are included in �̃�. thus, the internal virtual work, upon integration across the thickness of the shell, can be expressed as: ( )0 1 int 0 1( ) ( ) z a w da     = + + + + + n e e m χ χ t γ t γ (25) where the terms (n, m, 𝑻𝑘(k=0,1)) represent the membrane, bending, and shear stress resultants expressed in vector form as: 11 11 1 122 22 0 1 0 12 233 33 0 1 12 12 n m t tn m , , t tn m n m               = = = =                     n m t t, (26) 838 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri where its constituents are defined as following: h /2 h /2 33 33 -h /2 -h /2 h /2 h /2 33 33 -h /2 -h /2 h /2 h /2 0 1 -h /2 -h /2 / d , / d ˆˆ / d , z d , , 1,2 ˆ( ) / d , / d kβ kβ k kβ k k3 k3 n s g a z n s g a z m zs g a z m s z k t f z s g a z t zs g a z     = =   = = =   = =        (27) to simplify the expression of the internal work as shown in eq. (23), the stress and strain resultants are organized in a generalized vector form, yielding: 0 0 1 1 12 1 , ,c                           = = =                     n e e m χ χ r σ σ t γ 0 t γ 0 (28) thus, eq. (25) can be expressed as follows: ( )c t int a w da = + σ σ r (29) 2.4. governing equations of fgp shell a linear relationship is assumed between the generalized stress resultant r and the generalized strain 𝞢 as follows: t .=r h σ (30) where 𝑯𝑇 represents the material tangent modulus, described in terms of the linear elastic in-plane h and out-of-plane 𝑯𝜏 sub-matrices: ( ) 11 22 33 34 44 2 2 11 12 22 2 2 2 2 2 2 33 34 44 2 ( ) (1 ) ( ) ( ) ( ) ( ) t h/ h / h / h / sym ˆ ˆ, , ,z,z dz ˆ ˆ, , f z ,zf z ,z f z dz −  −      =        =   =    h 0 0 0 h 0 0 h h h h h h h h h h h h (31) for isotropic materials, the matrices h and 𝑯𝜏 are specified as follows: 1 0 1 0 1 0( ) 1 0 0 1(1 )(1 2 ) 2(1 ) 0 0 0 1 2 ) 2 e z e , /  −       −     = =     −+  −  +      −   h h (32) porosity effects on nonlinear static performances of fg shells considering thickness stretching 839 2.5. fe modelling of fgp shell in this section, the fe implementation of the current theoretical formulation based on a nonlinear four-node shell element is established according to the extensible director theory to predict the static fgp responses of plate and shell structures. thus, the development of an isoparametric fe, which involves rotation-displacement-thickness-stretch, can be achieved. this extensible shell element according to the director model encompasses a total of six degrees of freedom at each node i: including 3 displacements, 2 rotational degrees and the stretching parameter presented in the generalized displacement vector as 𝜱𝑰 = (𝒖𝑰, 𝒕𝑰, 𝜆𝑰). the displacement vector, written in the form u=x-x and its variation are approximated using the standard finite element bilinear shape functions 𝑁𝐼, as: 4 1 ( ) ( )i i i i i , , n , , =  = x u u x u u (33) the unit director vector t and the λ parameter are discretized using the same functions: 4 1 ( ) ( )i i i i i , , n , , =   =  t t t t (34) as the approximation of the different fields is assessed in the local cartesian system, eq. (35) defines the jacobian matrix j to facilitate the transformation from curvilinear coordinate vectors to the local cartesian system {𝒏1 0, 𝒏2 0, 𝒏𝟎}. here, 𝒏𝟎 represents the normal field to the mid-surface in the initial configuration, computed as 𝒏𝟎 = 𝑨𝟏 ∧ 𝑨𝟐/‖𝑨𝟏 ∧ 𝑨𝟐‖: 0 0 1 1 2 1 0 0 1 2 2 2 . . . .   =      n a n a j n a n a (35) subsequently, the derivatives of the cartesian standard shape functions defined at node i are converted to the global cartesian basis as follows:   1,1 ,1 ,2 ,2 i i i i n n n n −       =           j (36) 2.5.1. compatible membrane and transverse normal strain as detailed in the study by bischoff and ramm [31], to mitigate the occurrence of parasitic strain distribution, the assumed natural strain (ans) method is employed to interpolate the compatible thickness strain defined at the sampling points instead of the gauss points (refer to fig. 5). 4 33 33 33 1 , ( 1)i i i i i i e n e e     = = = − (37) 840 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri the interpolated compatible membrane strains (𝑒11, 𝑒22 and 𝑒33) and transverse normal strain 𝑒33 are connected to the node’s kinematic variables 𝛿𝜱𝑛 through the discrete operator matrix 𝑩𝑚, as outlined in appendix a .m n =e b φ (38) fig. 5 presentation of transverse normal and shear strain interpolation points 2.5.2. compatible bending strain the linearized compatible bending strain is written in term of the node’s kinematic variables 𝛿𝜱𝑛 and the discrete bending operator matrix 𝑩𝑏 (detailed in appendix a), as follows: .b n =χ b φ (39) 2.5.3. compatible first term of the transverse shear strain it is hypothesized that the shear locking issue, which hampers the effectiveness of the displacement finite element (fe) formulation, particularly for thin shells, can be effectively mitigated by the assumed natural strain (ans) method [32, 33]. the compatible first term of the tss 𝜸0 is computed within the element at sampling points (a, b, c and d) as follows: ( ) ( ) ( ) ( ) 2 0 2 00 1 10 1 1 0 1 00 2 22 (1 ) (1 )1 (1 ) (1 )2 b d a c              − + + = =    − + +    γ (40) here, (a, b, c and d) represent a set of mid-points along the boundaries of the element, as illustrated in fig. 5. this interpolation technique safeguards against unreasonable modes of compatible shear strains, enabling deformations without parasitic shear strains during pure bending. next, the variation of the constant tss is interpolated within the element as: 0 0 .s n =γ b φ (41) porosity effects on nonlinear static performances of fg shells considering thickness stretching 841 in which, 𝑩𝑠 0 is the discrete first shear strain operator matrix, illustrated in appendix a. 2.5.4. compatible second term of the transverse shear strain the calculation of the compatible second tss follows the same procedure as the first shear strain, utilizing the ans method. 1 2 1 2 1 1 1 1 1 1 1 1 1 1 2 2 2 (1 ) ( ) (1 ) ( )1 2 (1 ) ( ) (1 ) ( ) b d a d                − + + = =    − + +    γ (42) the variation of the second tss strain is presented in the matrix form as follows: 1 1.s n =γ b φ (43) here, the discrete operator matrix for the second shear strain-displacement 𝑩𝑠 1 is depicted in appendix a. ultimately, the virtual compatible strain tensor is obtained using eqs. (39), (40), (42), and (43) as follows: 0 1 . t c n      = = σ e χ γ γ b φ (44) 2.5.5. enhanced transverse normal strains approximation the enhanced strain field �̃� according to the eas method is expressed as following: 0 0 e g g =e t mα (45) where α refers to the internal variable vector. �̃� denotes the interpolation function matrix and the index ‘0’ represents its evaluation at the center of the element. four configurations of enhancement for the �̃� matrix are developed with one, four, five, or eight parameters. in eq. (18), a linear distribution is assumed along the shell thickness for the in-plane normal strains 𝐸𝛼𝛽 𝑐 and 𝐸𝛼3 𝑐 , while the transverse normal strain �̃�33 is presumed constant with respect to the thickness coordinate z. this constant distribution may result in artificial stresses in the thickness direction. to address this issue, the transverse normal strain 𝐸33 𝑐 should be augmented by an additional linear strain �̃�33. this enhancement component is introduced discretely using the enhanced assumed strain (eas) concept. according to the eas approach, the interpolation of �̃�33 is treated independently for each element and is introduced via additional parameters, which are condensed within the element. the following enhancement is achieved using either one or four additional parameters. the expression of the interpolation matrix when considering one additional parameter, �̃�1 is introduced as follows:  1 2 0 0 0 0 0 t z , h  = =m (46) next, the variation of the generalized incompatible strain 𝛅�̃� is determined using the interpolation matrix �̃�1 defined in the curvilinear basis (refer to appendix b), as follows: 842 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri 1 =σ b  (47) if four strain-supplemented parameters are taken into account, the expression of the interpolation matrix �̃�4 can be described as outlined in appendix b. 2.5.6. enhanced membrane strains approximation to tackle the membrane locking, the eas method is employed. within this approach, the membrane strains are enhanced through incompatible strains. hence, selecting the additional enhanced strain modes is crucial. four bilinear enhanced strain components can be incorporated to form a complete polynomial, resulting in: 1 11 2 22 1 2 12 0 0 0 span 0 0 0 2 0 0 e e e                                                           (48) this technique was initially explored by bischoff and ramm [31] to improve the compatible membrane strains. the interpolation for the enhanced strains can now be articulated as follows: 1 1 1 2 0 2 4 4 4 4 2 0 1 2 3 3 0 0 0 , , 0 0 0 0 0 t a a     −            = = =               g g g e g α g g t 0 g g  (49) where 𝑇𝟎 is the following matrix evaluated at the center of the element as: 2 2 2 2 11 21 11 21 2 2 2 2 12 22 22 12 2 2 2 2 2 2 2 2 11 12 22 21 11 22 12 21 2 2 j j j j j j j j j j j j j j j j     =    +  t (50) here, 𝐽𝛼𝛽 represent the jacobian components expressed in eq. (36). the fifth and eighth strain parameters are then derived, considering not only the enhanced thickness strain component but also the membrane strains, allowing for the matrices �̃�5 and �̃�8, as illustrated in appendix b. it is now appropriate to summarize the relations between strain and kinematic variables at the node for the compatible and enhanced strains in matrix notation as follows: . , .n   = =σ b φ σ b α (51) 2.5.7. linearization eq. (24) delineates the nonlinear shell model, which is solved using the newton iterative algorithm. the consistent tangent operator for the newton procedure can be derived from the directional derivative of the weak form in the direction of the increment δφ: porosity effects on nonlinear static performances of fg shells considering thickness stretching 843   0 t int ext t n nw dw         −  + = + =             φk l f f φ α αl h h (52) where k, l and h are given by: , , t d g d t a t t t t a a da da da = + = = =    k k k k b h b l b h .b h b h b (53) 𝑲𝐷 and 𝑲𝐺 are the material and geometric tangent matrices. the components of the geometric tangent matrix 𝑲𝐺 are given in the appendix c. in eq. (50), 𝒇𝑖𝑛𝑡 and h, are given by the following expressions ,int t t a a da da= =  f b r h b r (54) removing the internal variable vector α from eq. (53) at the element level yields the element tangent matrix 𝑲𝑇 , expressed as: 1t t d g −= + −k k k l h l (55) and the residual vector r defined as: 1t ext int−= −r l h h+ f f (56) at the element level, transitioning from the generalized displacement vector 𝜱𝑛 = (𝒖, 𝒕, 𝜆)𝑛 to the nodal variable vector 𝜞𝑛 = (𝒖, 𝜣, 𝜆)𝑛, where 𝜣 represents the two rotations vector in material description, is significant. at node i, the relationship between the generalized displacement vector 𝜱𝑛 = (𝒖, 𝒕, 𝜆)𝑛 and the nodal variable vector 𝜞𝑛 = (𝒖, 𝜣, 𝜆)𝑛 is elucidated by: 7 6 1 4i i i i i 0 , i ,.., , 0 0      0 1         = = =       _ i 0 φ 0 λ   (57) for all element nodes, the transformation matrix π is expressed as: ( )1 2 3 4diag , , ,    = (58) for further insights into nodal transformation, refer to mallek et al. [34]. 3. numerical tests and discussion this work investigates geometrically nonlinear performances of fgp shells via the formulated extensible director shell approach. the 3d shell structure is examined for its nonlinear responses taking into account the thickness-stretch. existing literature lacks such results, necessitating model verification. through numerical examples, the approach 844 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri proves versatile for analyzing nonlinear structural responses of 3d shell structures, including porous functionally graded materials. verification tests and simulations validate the methodology's reliability and accuracy. new numerical results are presented showing the influence of porosity dispersion on the large deformation responses of the semicylindrical shell structure made by fgp. this research provides crucial insights into the geometrically nonlinear behavior of 3d shell structures with the phenomenon of the thickness-stretch included. 3.1. linear behavior of simply supported rectangular plate the first example is considered as dimensionless. it deals with static bending of a simply supported rectangular plate with the edges a = 7.5 and b = 5 and with the thickness of h = 0.2. the material properties are taken as follows: the elastic modulus e = 1 and the poisson’s ratio υ = 0.3. the plate is subjected to a uniformly distributed transverse load q = 1, as shown in fig. 6. the symmetry allows to model only one quarter of the plate. refinement of the fe mesh was done in order to study the convergence of the proposed model. fig. 6 simply supported rectangular plate table 1 gives a comparison of the central transverse deflection w obtained with the current element and the same results reported in [35]. in [35], a mixed enhanced finite plate element was used. the reference solution used for comparison is 6624.4 which represents an exact 3-d solution given by piltner [36]. as can be seen, by increasing the number of elements, the results converges and the results by the 32×32 mesh is taken as representative. table 1 central deflection results of simply supported plate mesh 2x2 4x4 8x8 16x16 32 x 32 piltner and joseph [35] 6503.6 6591.2 6612.3 6617.9 6619.4 present 6488.8 6587.7 6613.6 6620.1 6621.8 3.2. nonlinear behavior of fgp square plate under uniformly distributed load the second example examines deformation of a simply supported square plate under a uniform load. the plate has a width of l=100 mm and a thickness of h=1mm. the material properties include the poisson’s ratio υ = 0.3 and the young modulus of aluminum and porosity effects on nonlinear static performances of fg shells considering thickness stretching 845 zr/o2: 𝐸1= 70 gpa and 𝐸2= 151 gpa, respectively. to simplify analysis, the symmetry is used and one quarter of the plate is discretized using 16×16 enhanced four-node elements. the plate surface is subjected to a uniform load q. the parameters considered in this test are the normalized load p= 𝑞𝐿4 𝐸1ℎ4 and the normalized central deflection �̅� = 𝑤/ℎ. fig. 7 illustrates the normalized load central deflection curves for various porosity volume fractions, considering a material exponent of n=0.1. the response of the fgp plate closely aligns with findings from nguyen et al. [37]. notably, the configurations with evenly distributed pores exhibit higher deflections compared to the uneven configurations due to reduced bending stiffness. additionally, an increase in porosity volume fraction, particularly with even distribution, results in higher deflections, whereas this effect is less pronounced with uneven distribution due to pores vanishing at the plate's surfaces. a) b) fig. 7 results of normalized central deflections of the porous fg plate versus the load parameter using the material exponent n=0.1 at the (a): even; (b) uneven configuration 3.3. nonlinear behavior of fgp cantilever plate under uniformly distributed load the next test investigates tip deformation of a cantilever plate subjected to a shear line load at the free end. this example shows the significance of the enhanced thickness -5 -4 -3 -2 -1 0 -0.25 -0.2 -0.15 -0.1 -0.05 0 load parameter n o rm a liz e d c e n tr a l d e fl e c ti o n present: perfect present: even (=0.2) present: even (=0.6) nguyen et al.[37]: perfect nguyen et al.[37]: even (=0.2) nguyen et al.[37]: even (=0.6) -5 -4 -3 -2 -1 0 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 load parameter n o rm a liz e d c e n tr a l d e fl e c ti o n present: perfect present: uneven (=0.2) present: uneven (=0.6) nguyen et al.[37]: perfect nguyen et al.[37]: uneven (=0.2) nguyen et al.[37]: uneven (=0.6) 846 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri strains �̃�33. the plate with dimensions l=10 mm, b=1 mm, and h=0.1mm is clamped at one end and subjected to a distributed transverse line load of the magnitude -1n/mm at the free end, as shown in fig. 8. a) b) fig. 8 plate’s geometry: a) underformed; b) deformed in this problem, the enhanced four-node element is employed for modeling. the functionally graded material plate is fabricated using transition alloys of metals and ceramics to enhance the ductility of the structure. it is assumed that the top surface is rich in ceramics, while the bottom surface is rich in metals. the material properties of the fgm vary continuously in the thickness direction following a power law distribution. the poisson’s ratio of the fg material is υ = 0.3 and the young’s modulus of the ceramic and metal phases are 𝐸𝑐= 3.8 × 105 gpa and 𝐸𝑚= 2.1 × 105 gpa, respectively, as reported by beheshti and ramezani [21]. in fig. 9, the nonlinear tip displacement of the fgm plate is presented using the extensible director shell model, with enhancements of membrane and transverse strains, across various values of the material exponent. it is observed that as the material exponent fig. 9 load-deflection curves of the fgm cantilever plate for various values of the material exponent porosity effects on nonlinear static performances of fg shells considering thickness stretching 847 n transitions from ceramic to metal, the tip deflection increases, with metallic plates exhibiting greater deflection due to the higher bending stiffness of ceramic plates. furthermore, the nonlinear response of the fgm cantilever plate closely aligns with the findings reported by beheshti and ramezani [21], who employed a seven-parameter shell model to account for the thickness change and utilized the enhanced assumed strain (eas) method to mitigate the locking phenomena. hence, the current nonlinear model demonstrates the capability to yield convergent results and avoid locking. fig. 10 depicts a comparison between the linear and nonlinear deflections of cantilever plates made of ceramic and metal. it is evident that the displacement magnitudes are consistently overestimated in linear analysis. this discrepancy arises because the result of the linear analysis is dominated by the bending stiffness, while the nonlinear behavior correctly accounts for the influence of the membrane stiffness. furthermore, the nonlinear effect is more pronounced for metallic plates, which exhibit lower stiffness compared to ceramic plates. fig. 10 linear and nonlinear deflection curves of cantilever plate made of ceramic and metal 3.4. nonlinear behavior of hinged fgm cylindrical panel the next example deals with the cylindrical panel, often referred to as the roof-like panel, depicted in fig. 11. the example aims to assess the accuracy of the presented enhanced element in nonlinear analysis. the geometrical characteristics of the panel include a length of l=508 mm, a radius of r=2540 mm, a thickness of h=12.7 mm and a span angle of θ = 0.1 rad. the shell is free at its straight edges, hinged over the two curved edges, and subjected to a concentrated normal force f acting in the middle of the panel. the panel is made of a mixture of two material phases with the following mechanical properties: elastic modulus 𝐸𝑚=3102.75 mpa and 𝐸𝑐=3615.77 mpa, for metal and ceramic, respectively, and a poisson’s ratio υ =0.3. due to the structure's symmetry, one quarter of the panel is discretized by 4×4 enhanced elements. 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 displacement [mm] l o a d q [ n /m m ] nonlinear deflection: ceramic linear deflection: ceramic nonlinear deflection: metal linear deflection: metal 848 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri a) b) fig. 11 (a) undeformed and (b) deformed shapes of the fgm hinged cylinder the nonlinear deflection at the loaded point, obtained using the developed extensible 3d director shell model, is depicted in fig. 12 for various values of the material exponent n, and are compared with the results by the models available in the literature [21]. it is evident that the obtained results closely align with the numerical solutions presented by beheshti and ramezani [21], particularly in the case of the fully metallic shell, displaying good agreement across different material exponent n. this underscores the high accuracy of the developed fe shell model in predicting large displacements. furthermore, as illustrated in fig. 11, the load required to deform a metallic panel is significantly lower than that required for a ceramic panel due to the lower rigidity of the metal. fig. 12 load-deflection diagram of the hinged fgm cylindrical panel for various values of the material exponent at loaded point porosity effects on nonlinear static performances of fg shells considering thickness stretching 849 3.5. nonlinear behavior of fgm ring shell under shear force at the end the slit annular plate serves as a sensitive benchmark to assess the capability and accuracy of the finite element formulation in handling large deformations. as illustrated in fig. 13 (a), the structure is subjected to a distributed vertical shear force with a magnitude of q=2120 n/m over the free edge, while a clamped boundary condition is applied at the other edge. the geometric parameters of the plate include the inner radius of r = 6 mm, the outer radius of r =10 m, and the thickness of h = 0.03 m. regarding material properties, the young's modulus of the metal and ceramic phases are 𝐸𝑚= 21 × 103mpa and 𝐸𝑐= 38 × 103 mpa, respectively, with a poisson coefficient ν assumed to be zero. the ring is discretized by 8×46 enhanced elements. upon application of predefined line load over the free edge, the initially planar structure undergoes arge displacements and rotations, thus deforming into a doubly crooked structure, as depicted in fig. 13 (b). a) b) fig. 13 geometry of the ring plate: a) undeformed shape; b) deformed shape the nonlinear deflection at point a on the outer circle of the functionally graded (fg) ring obtained with the developed model is compared to the results of the models developed by beheshti and ramezani [21]. the first model is a displacement-based four-node model, while the second model is an enhanced model incorporating the enhanced assumed strain formulation and accounting for the thickness change. fig. 14 displays the load–deflection curves at point a fig. 14 load-deflection diagram of the fgm ring shell for various values of the material exponent at point a 850 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri for different material exponents n. it is evident from the plotted curves that the present nonlinear deflection of the studied structure closely aligns with the results by the enhanced model of beheshti and ramezani [21], although the authors employed a completely different numerical approach based on the strain enhancements and thickness stretch. hence, this test serves as a highly sensitive benchmark for assessing the accuracy of finite rotation analysis. 3.6. parametric study of a porous semi-cylinder in this example, a semi-cylindrical shell structure (fig. 15 (a)) is considered to analyze its large deflection response subjected to an end pinching force at point a equal to f=2000. due to the symmetry of the structure, a half of the cylinder is modelled using 40×32 finite elements. the geometrical characteristics of the cylinder are given dimensionless: length l=3.048, radius r=1.016 and thickness h=0.03. in the first step is to validate the nonlinear response of the isotropic semi-cylinder by comparing the result with the solutions by mallek et al. [2]. the isotropic material considered for the validation has the following material properties: young’s modulus e =2.068×107 and poisson’s coefficient υ=0.3. fig. 16 depicts results of large deflections at the point a of the isotropic semi-cylinder versus the load parameter. as may be seen in this figure, a good accuracy is associated for the present model where present results coincide with the referenced solutions. a) b) fig. 15 (a) undeformed and (b) deformed shapes of the porous fgm semi-cylinder once the nonlinear response of the isotropic semi-cylinder is validated, the subsequent phase involves applying porous functionally graded material (fgm) to the analyzed structure. this fgm comprises a blend of two materials characterized by mechanical properties, 𝐸𝑚= 21 × 109 pa, 𝜐𝑚=0.3 and 𝐸𝑐= 38 × 109 pa, 𝜐𝑐=0.3. fig. 17 illustrates deflection outcomes at point a of the porous semi-cylinder, considering an fgm material exponent (n=0.1) for various porosity distributions: perfect, even, and uneven configurations. this figure demonstrates that the increasing porosity volume fraction α results in significant deflections of the semi-cylinder for both even and uneven configurations. this is due to the reduced bending stiffness caused by a higher density of pores within the structure. moreover, the even distribution of porosity exhibits greater deflections compared to other configurations with the same porosity volume fraction, especially at higher values of α due to its impact on the bending stiffness of the structure. in an even distribution, pores are uniformly spread throughout the material, affecting its mechanical properties consistently across its volume. this results in a more pronounced reduction in bending stiffness compared to uneven distributions where pores may cluster or be concentrated in specific areas. when the porosity porosity effects on nonlinear static performances of fg shells considering thickness stretching 851 is evenly distributed, a larger portion of the material experiences a reduction in stiffness, leading to more significant deformations under loading conditions. essentially, the even distribution creates more consistent reduction of the structure's bending stiffness, allowing for greater overall deflections. conversely, in uneven distributions, the concentration of pores in certain regions may lead to localized reduction in stiffness, but other areas may remain relatively unaffected. this can result in a less uniform reduction in bending stiffness throughout the structure, leading to comparatively lower deflections than those observed with even distributions for the same overall porosity volume fraction. fig. 16 load-deflection diagram of the isotropic semi-cylinder at point a furthermore, the influence of the fgm material exponent n on the large deflection response at point a, considering the porosity volume fraction α, is highlighted in fig. 18 for even and uneven configurations of pores. this figure indicates that the increasing fgm material exponent n leads to higher deflections due to a decrease in bending stiffness resulting from a greater proportion of the metal phase in the structure. additionally, fig. 19 depicts the impact of the ratio r/h on the nonlinear bending response at point a of the cylindrical structure, considering the fgm material exponent (n=0.1) for various pore configurations. high values of the ratio r/h correspond to increased deflection of the porous semi-cylinder. this is expected from the geometric influence, flexibility, and reduced bending stiffness associated with the higher r/h ratios. with a larger radius relative to thickness, the shell becomes geometrically less stiff, allowing for greater deformation under applied loads. additionally, the thinner profile in relation to the curvature enhances flexibility, thus making the structure more susceptible to bending. consequently, the shell experiences reduced resistance to bending moments, resulting in more significant deflections under the same loading conditions. 0 0.5 1 1.5 2 0 500 1000 1500 2000 deflection w a l o a d present mallek et al. [2] 852 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri a) b) fig. 17 results of deflections at point a of the porous semi-cylinder for the material exponent of the fgm n=0.1 for the (a): even; (b) uneven configuration 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 100 200 300 400 500 deflection w a [m] lo ad ( × 10 3 [ n ]) perfect even (=0.2) even (=0.6) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 100 200 300 400 500 deflection w a [m] l o a d ( × 1 0 3 [ n ]) perfect uneven (=0.2) uneven (=0.6) porosity effects on nonlinear static performances of fg shells considering thickness stretching 853 a) b) fig. 18 results of deflections at point a versus different material exponent n of the porous semi-cylinder for the porosity volume fraction at the (a): even; (b) uneven configuration 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 100 200 300 400 500 deflection w a [m] l o a d ( × 1 0 3 [ n ]) n=0 n=1 n=3 n=6 0 0.1 0.2 0.3 0.4 0.5 0 100 200 300 400 500 deflection w a [m] l o a d ( × 1 0 3 [ n ]) n=0 n=1 n=3 n=6 854 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri fig. 19 influence of the ration r/h on the large deflections at point a of the porous semicylinder considering the material exponent n=0.1 of the fgm for the different configurations of porosity with various values of α finally, fig. 20 illustrates the distributions of normal 𝜎𝑥𝑥 and transverse shear stresses 𝜎𝑥𝑧 at point a for different configurations of the porous fgm, considering the material exponent (n=0.1). these stress distributions underscore the influence of porosity volume fractions, as an increase in pore density reduces the structure's stiffness and leads to higher stress values. as evident from fig. 20 (b), a notable parabolic variation of transverse shear stress throughout the thickness of the structure is observed, owing to the utilization of an enhanced fsdt. this improved theoretical framework provides a more accurate representation of the transverse shear stress distribution within the structure, capturing the complexities of its behavior under loading conditions. 0 10 20 30 40 50 60 0 0.5 1 1.5 2 d e fl e c ti o n w a [ m ] r/h perfect even(=0.2) even(=0.6) uneven(=0.2) uneven(=0.6)) porosity effects on nonlinear static performances of fg shells considering thickness stretching 855 a) b) fig. 20 stress distributions at point a along the thickness of the porous semi-cylinder with a material exponent n=0.1 for (a) 𝜎𝑥𝑥; (b) 𝜎𝑥𝑧 -2 -1 0 1 2 3 -0.5 0 0.5 stress  xx [×109pa] z /h perfect even(=0.2) even(=0.6) uneven(=0.2) uneven(=0.6) 0 0.5 1 1.5 2 2.5 3 3.5 -0.5 0 0.5 stress  xz [×108pa] z /h perfect even(=0.2) even(=0.6) uneven(=0.2) uneven(=0.6) 856 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri 4. conclusions this study introduces an innovative finite element-based model, incorporating enhancements to the standard fsdt, to investigate large deformation behaviors in porositycontaining functionally graded (fg) shell structures. emphasizing thickness stretching through the enhancement of the third bending strain, with additional parameters facilitated by the eas method, and augmentation of the membrane strains with enriched parameters, a comprehensive six-degree-of-freedom shell model is developed. the investigation evaluates the model's performance through simulations of various boundary conditions and different types of perfect fgms, demonstrating convergent and accurate results compared to existing literature. a parametric study explores the impact of porosities on fgm characteristics, particularly focusing on cylindrical shell structures. the presence of pores reduces the structure's stiffness, resulting in significant deformations, particularly notable in unevenly distributed porosity configurations. moreover, an increase in porosity volume fraction exacerbates the deflection of porous fgm shell structures, underscoring the structural implications of porosity in fgm materials. this research significantly 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n.v., nguyen, h.x., lee, s., nguyen-xuan, h., 2018, geometrically nonlinear polygonal finite element analysis of functionally graded porous plates, advances in engineering software, 126, pp. 110-126. 858 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri appendix a: strain-displacement interpolation membrane : .m n =e b φ i i i m mu m  =  b b 0 b t 1 ,1 t 2 ,2 t t 1 ,2 2 ,1 i i i mu i i n n n n+      =        n n b 0 n n 0 0 ( 1) 0 i m i i n        =  −     b bending : .b n =χ b φ i i i i b bu bt b  =  b b b b ,1 ,1 ,2 ,2 ,1 ,2 ,2 ,1 t i t i i bu t i t i n n n n+      =        d d b 0 d d 1 1 2 2 2 1 1 2 i t i t i bt i t i t l l l l+      =        n n b 0 n n 1 1 2 2 1 2 2 1 . . 0 . . t i t i i b t i t i  +      =        n m n m b n m n m , , ,(1 )   = − −d t t , ,(1 )i i il n n   = − − , ,( )i i in n  = − +m t t transverse shear, first terme : 0 0 .s n =γ b φ 0 1 0 s  −=b j b 1 2 2 2 2 2 ,1 1 1 ,1 1 ,1 1 1 ,1 10 1 4 4 2 3 3 ,2 2 2 ,2 2 ,2 2 2 ,2 2 . . . . t t t t t t b b b b b b b b b b t t t t t t a a a a a c c c c c n p n n p n n p n n p n   − − =  − − d a a t d a a t b d a a t d a a t 3 3 3 4 3 3 ,1 1 1 ,1 1 ,1 1 1 ,1 1 3 3 3 4 4 4 ,2 2 2 ,2 2 ,2 2 2 ,2 2 . . . . t t t t t t d d d d d d d d d d t t t t t t c c c c c a a a a a n p n n p n n p n n p n − −  − −  d a a t d a a t d a a t d a a t , (1 )i i m mp n  = − transverse shear, second terme : 1 1.s n =γ b φ 1 1 1 s  −=b j b 1 1i i    =  b 0 0 b 2 2 3 3 ,1 ,1 ,1 ,1 ,1 ,1 ,1 ,11 4 3 3 4 ,2 ,2 ,2 ,2 ,2 ,2 ,2 ,2 b b d d a c c a n n n n n n n n            =      b 1 2 3 4 ,1 ,1 ,1 ,1 1 2 3 4 ,2 ,2 ,2 ,2 ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) b b d d a c c a n n n n n n n n          − − − − +   − − − −   porosity effects on nonlinear static performances of fg shells considering thickness stretching 859 appendix b: the interpolation matrix for enhancement of membrane and transverse normal strain transverse normal strain with one additional parameter : 6 1 0 5 12 1 / 1a a      =       0 b 0 transverse normal strain with four additional parameters : 6 4 4 1 4 5 4 12 4 t         =       0 b v 0 1 1 4 0 2 1 2 1 /a a           =       v enhanced membrane and transverse normal strain with one additional parameter : 4 4 5 1 4 4 4 4 12 5         =         g b b 0 0 enhanced membrane and transverse normal strain with four additional parameters : 4 4 8 4 4 4 4 4 12 8         =         g b b 0 0 appendix c: geometric tangent matrix components 6 6 ( ) ( ) ( ) ij ij ij ij ij gij ji ji ij ij ij ij t t t t ji ji ji ji ij uum ubf ubc ubf ubc bbf bbc ll  + +    = + + +    + +  i i ulf ulc k i i blf blc ulf ulc blf blc components of gijk : ( )11 12 12 22 ,1 ,1 ,2 ,2 ,1 ,2( ) ( )i j j i j j ij a uum n n n n n n n n n n da= + + + ( )11 12 12 22 ,1 1 2 ,2 1 2( ) ( )i j j i j j ij a ubf n m l m l n m l m l da= + + + ( )11 12 12 22 ,1 1 2 ,2 1 2 0 ( ) ( ) ij i i ii i a bbf if i j bbf l m m l m m da =   = − + + +  n n n n t ( )11 12 12 22 ,1 1 2 ,2 1 2( ) ( )i j j i j j ij a n m m n m m da= + + +ulf m m m m 860 h. mallek, h. mellouli, l. ben said, m. wali, f. dammak, m. alhadri ( )11 12 12 22 1 1 2 2 1 1 2 2( ) ( )ij ij ij ij ij a m n m n m n m n da= − + + +blf n n n n 33 0ij i ii a ll if i j ll n n da =   =  , ,(1 )i i il n n   = − − , ,( )i i in n  = − +m t t , , ij i j i jn n n n n  = + 0 01 08 0 a b b a b b c c c c d d a d a d − − − −    − −  =  +   − −  ubc 1 8 − − − −    − −  =  +   − −  a b b 0 a b b c c 0 ulc 0 c c d d a 0 d a d 1 8 +    +  = −  +   +  α β β 0 α β β γ γ 0 blc 0 γ γ δ δ α 0 δ α δ  1 1 2 2 3 3 4 4 1 diag . . . . 4 = −bbc g d i g d i g d i g d i 2 0(1 ) a a t da= − 1 0(1 ) a b t da= − 2 0(1 ) a c t da= + 1 0(1 ) a d t da= + (1 )aa a= − (1 )bb b= − (1 )cc c= − (1 )dd d= − aa=a t bb=b t cc=c t dd=d t 2(1 )a a a a = −g a 1(1 )b b b b = −g a 2(1 )c c c c = −g a 1(1 )d d d d = −g a 1 a b= +g g g 2 b c= +g g g 3 c d= +g g g 4 d a= +g g g 2 aa=α a 1 bb=β a 2 cc=γ a 1 dd=δ a plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 269 280 doi: 10.22190/fume1603269t original scientific paper adhesion effects within the hard matter – soft matter interface: molecular dynamics udc 539.8 alexey tsukanov 1,2 , sergey psakhie 1,2 1 institute of strength physics and materials science, siberian branch of russian academy of sciences, tomsk, russian federation 2 tomsk polytechnic university, tomsk, russian federation abstract. in the present study three soft matter – hard matter systems consisting of different nanomaterials and organic molecules were studied using the steered molecular dynamics approach in order to reveal regularities in the formation of organic-inorganic hybrids and the stability of multimolecular complexes, as well as to analyze the energy aspects of adhesion between bio-molecules and layered ceramics. the combined process free energy estimation (copfee) procedure was used for quantitative and qualitative assessment of the considered heterogeneous systems. interaction of anionic and cationic amino acids with the surface of a [mg4al2(oh)12 2+ 2cl–] layered double hydroxide (ldh) nanosheet was considered. in both cases, strong adhesion was observed despite the opposite signs of electric charge. the free energy of the aspartic amino acid anion, which has two deprotonated carboxylic groups, was determined to be –45 kj/mol for adsorption on the ldh surface. for the cationic arginine, with only one carboxylic group and a positive net charge, the energy of adsorption was –26 kj/mol, which is twice higher than that of chloride anion adsorption on the same cationic nanosheet. this fact clearly demonstrates the capability of “soft matter” species to adjust themselves and fit into the surface, minimizing energy of the system. the adsorption of protonated histamine, having no carboxylic groups, on a boehmite nanosheet is also energetically favorable, but the depth of free energy well is quite small at 3.6 kj/mol. in the adsorbed state the protonated amino-group of histamine plays the role of proton donor, while the hydroxyl oxygens of the layered hydroxide have the role of proton acceptor, which is unusual. the obtained results represent a small step towards further understanding of the adhesion effects within the hard matter – soft matter contact zone. key words: adhesion, interface, soft matter, layered hydroxide, steered molecular dynamics received september 15, 2016 / accepted november 07, 2016 corresponding author: alexey a.tsukanov institute of strength physics and materials science sb ras, tomsk, 634055, russia e-mail: a.a.tsukanov@yandex.ru 270 a. tsukanov, s. psakhie 1. introduction in the last decades soft matter engineering has a very rapid development. the importance of the soft matter extends to a wide range of applications in such fields as materials science [1], energetics [2, 3], catalysis [4, 5], and pharmaceutics. it especially stands for biomedicine since biological nano-objects (bno) such as polypeptides, proteins, bio-membranes as well as almost all biologically-active compounds (drugs, genes, viruses, etc.) are soft-matter systems with flexible, non-uniform and multifunctional surfaces [6, 7]. one of the main characteristics of the soft matter is its ability to form high-level complicated functional nano-objects from comparatively simple building blocks [8]. on the other hand, certain condensed materials such as naturally occurring layered ceramics (lc), in particular cationic clays, layered double hydroxides (ldh) and metal oxyhydroxides possess special properties such as large surface charge (which allows them to act as host nanoparticles for ionic molecules), large specific surface area, low toxicity, chemical inertness and biocompatibility, which makes them extremely promising in such biomedical/nanomedical applications as drug and gene delivery [9-13]. in addition, lowdimensional aluminum oxyhydroxide nanoparticles are amphoteric compounds with high proton and hydroxyl buffer capacity [14], which can affect the ionic balance in the cellular environment. the use of lc in the role of hosting “nanocontainers” is explained by the fact that almost all biologically-active molecules (including modified ones) may be intercalated in between hydroxide nanolayers. there are two particularly important aspects in this context: first, interaction effects within the hard matter – soft matter interface (hsi) between the biologically-active compound (“guest”) and the layered hydroxide nanosheets (“host”) determine the formation of (conditionally) stable organic-inorganic nanohybrids (nh); second, the interaction between the inorganic outer surface of the nanohybrid and the cell membrane determine the mechanism and efficiency of the cellular uptake of the nh. predicting behavior and interaction of hard matter and soft matter subsystems is a challenge. to shed light on several effects which can rise within the hard matter – soft matter interfacial region a series of steered molecular dynamics (smd) simulations utilizing all-atom models was conducted with different pairs of organic and inorganic nanomaterials. 2. computational technique 2.1. potential energy functional to accurately compute any energies and forces of the studied thermodynamic system, suitable potentials need to be selected. in the case of soft matter systems and systems having covalent bonds, the most typical terms in the potential energy functional of the bonded atoms are the bond stretching term, angle bending, urey-bradly 1-3 stretching term, dihedral or torsional angle bending and improper rotation or inversions term (fig. 1). eq. (1) describes a typical form of potential energy functional with harmonic type of the terms as it implemented in charmm [15]: adhesion effects within hard matter – soft matter interface: molecular dynamics 271 2 2 2 0 0 0 2 2 0 ( ) ( ) ( ) (1 cos( )) ( ) b ub bonds angles unbonded dihedrals impropers u k b b k k s s k n k u                             (1) where the unbounded terms such as lennard-jones potential and electrostatic interactions are also included: 6 12 2 ij ij i j unbonded ij i j i jij ij ij q q u c r r r                            . (2) the most famous all-atom force fields for soft matter systems are gromacs [16], amber [17], dreiding [18], charmm, etc. the potentials for hard matter systems are, e.g., eam for solid metals [19, 20] and liquid metals [21, 22], clayff for ceramics [23], airebo and tersoff for carbon nanostructures and other compounds [24-26] and others. there is also a hybrid force field interface, which is suitable for heterogeneous system modeling [27]. fig. 1 typical terms included in the potential energy functional of force fields for bonded atoms for bound atoms, special coefficients are often used reducing the lennard-jones and coulombic terms (so-called 1–2, 1–3, 1–4 special bonds). in addition, to facilitate computations, a cutoff for pairwise interactions with a smoothly decreasing envelopefunction is used. so-called “full electrostatics” (long-range) is usually calculated with the particle-particle particle-mesh (pppm) [28, 29] or particle-mesh ewald (pme) methods [30, 31]. in the present study, the cutoff distance of 12 å for pairwise interaction, with a smooth decrease starting at 10 å was utilized, as well as the pppm method for “longrange” coulombic interactions with a relative accuracy of 0.001. molecular dynamics (md) modeling of the layered ceramics nanostructures requires not only unbounded terms but also an explicit treatment of the covalent bonds in hydroxyl groups between oxygen and hydrogen atoms [23]. 272 a. tsukanov, s. psakhie to parameterize all molecular systems considered in the present study, the following force fields are used: charmm [15] for organic molecules, tip3p [32] for water molecules, as well as clayff [23] for layered ceramics, with a simple modification according to [33]. 2.2. free energy of adsorption to quantify interactions between different parts of the system or the energy change between two states of the system, free energy analysis is often very useful. in the present study of interaction effects within the hsi region, the combined process free energy estimation (copfee) procedure [34] was utilized, which is based on the potential of mean force (pmf) analysis [35] for constant velocity steered md. the central idea of the copfee approach is a combination of two oppositely directed processes, which allows us to reduce the dependence on the velocity of the pulling procedure and to estimate the level of irreversible energy dissipation in comparison with the free energy change. briefly, to estimate the free energy of adsorption (of some molecule) consider the following combination of two processes: a forward process – forced adsorption, in which an external force is performing the work of translocating an adsorbate from “infinity” (actually some point in the solvent, where the adsorbate is fully hydrated) onto the surface of the adsorbent (fig. 2); and then a reverse process – forced desorption, when the external force acts to remove the adsorbed molecule/nanoparticle away from the adsorbent surface to any point in the solvent that is equidistant with the initial point (fig. 2). during both the processes the work done by the external force is integrated, and two free energy profiles (pmf profiles along reaction coordinate) are obtained as functions of z, in the direction perpendicular to the adsorbent surface. fig. 2 two stages of combined process free energy estimation procedure (copfee) for an organic anion interacting with an ldh nanosheet: left – forced adsorption of adsorbate on the adsorbent surface under the action of an external force, right – forced desorption, the reverse process. during the steered md simulation, the free end of the abstract spring is moving with constant velocity in the respective direction. adsorbate colors: yellow – c, white – h, blue – n, red – carboxylic o. adsorbent colors: purple – al, black – mg, gray – hydroxylic o, white – h adhesion effects within hard matter – soft matter interface: molecular dynamics 273 as is known from thermodynamics, work aext of the external force is spent on change of gibbs free energy g of the system (in case of an isothermal-isobaric ensemble) and on entropy generation (if the process is not reversible): exta g t s   . (3a) writing this for both the forward and reverse processes gives: fwd fwd ext ads rvs rvs ext ads a g t s a g t s         (3b) where t is known (and constant), both a fwd and a rvs are can be estimated from the smd simulations, and ∆g, δs fwd and δs rvs are unknown variables. thus, there are three unknown variables in the system (3b) and only 2 equations. to overcome this problem, the following assumption can be made [34]: if the pulling velocities in both forward and reverse processes are equal and sufficiently small, the entropy generation in both processes should also be approximately equal: fwd rvss s  . (3c) the system (3b) becomes solvable with this assumption: 1 1 ( ) ( ) ( ) 2 2 2 fwd rvs fwd rvs fwd rvs ads ext ext ext ext t g a a s s a a        (4a) if the adsorption is energetically favorable, the external work in forward process a fwd is negative (with a sufficiently small pulling velocity), while a rvs is positive and fwd ext rvs ext aa  . thus, the free energy of adsorption can be simply estimated as minus the arithmetic mean of the absolute values of the external work for the forward and reverse processes [34]: 2 fwd rvs ext extads copfee a a g     . (4b) 3. results and discussion 3.1. organic anion adsorption on mg4/al2-ldh nanosheet first we consider a combined adsorption-desorption constant velocity smd process for the aspartic amino acid anion (in zwitterionic state) on a [mg4al2(oh)12 2+ 2cl – ] layered double hydroxide nanosheet, which has a strong positive surface charge (about 0.7 c/m 2 ) and exposes polar hydroxyl groups on its surface. the aspartic acid anion (asp) has two carboxylic groups with local negative charges on oxygen atoms. the total charge of the asp molecule is -1 e. it is convenient for further discourse to put the origin of the coordinate system in the center of mass of the mg4/al2-ldh fragment and to orient the z-axis perpendicular to the nanosheet plane. the pmf profile (the cumulative work as a function of the distance between mg4/al2ldh nanosheet central plane and asp center of mass) for the forward process obtained with smd with a constant pulling velocity v = 0.1 å/ns is represented by the blue curve in 274 a. tsukanov, s. psakhie fig. 3. comparing the reverse process (fig. 3, green line) it is immediately obvious that the obtained curves are not equal, because of entropy generation during the irreversible part of the process (fig. 3, 5.6 < z < 6.4 å). it is necessary to note that pmf profiles are relative functions, which is why a zero level must be chosen for both the dependences. since we are interested in the free energy of adsorption, which is the difference of g between some distant point in the bulk water solution and the point corresponding to the nearest local minimum of pmf, it seems most convenient to choose the zero level at the initial point (in the water) for both the curves. the too fast perturbation (in comparison with thermal motions) of a single-moleculethick water layer on the adsorbent surface is the most probable reason for the observed entropy generation .this could probably be mitigated with a lower velocity of the pulling process during smd simulation, while greatly increasing the computational cost of the numerical experiment. fig. 3 free energy of asp amino acid anion adsorption on ldh using copfee method. the profile has three local minima: m3 (6.5-7.0 å) – adsorbate is separated from ldh surface by a single-molecule-thick water layer, m2 (5.0-5.4 å) – first carboxylic group of asp forms h-bonds with surface oh-groups, m1 (4.2-4.5 å) – both carboxylic groups of asp contact with hydroxylic ldh surface – completely adsorbed state. color code is similar to the colors of fig.2, except that carbon atoms of the adsorbate are in cyan, and water (several molecules in contact zone) in light blue. the rest of the water and other ions are not shown for clarity following the procedure described with the assumption eq. (3c), we obtain an estimate for the free energy of asp acid anion adsorption on the mg4/al2-ldh surface of –45 kj/mol (minimum m1 in fig.3). this is a very high value, indicating strong adhesion within the hard matter. such strong interaction energy within hsi allows the formation of hybrid organic-inorganic multimolecular complexes, as was demonstrated in unbiased (non-steered) molecular dynamics simulations [33]. adhesion effects within hard matter – soft matter interface: molecular dynamics 275 the half-width of the “corridor” between forward and reverse pmf profiles is ∆ = ±6 kj/mol, which provides a rough estimate of entropy generation during the forced processes. 3.2. arginine and chloride adsorption on cationic nanosheet free energy of adsorption of anionic molecules on an ldh nanosheet was considered in the previous section as well as in other works [33, 34]. the behavior of cationic amino acid residues such as arginine (arg), lysine and protonated histidine at a cationic nanosheet is also an important question since these also are typical building blocks of proteins and polypeptides, and the possibility of adsorption is not obvious due to electrostatic repulsion. here the copfee procedure was applied to characterize the interaction of arginine amino acid with a [mg4al2(oh)12 2+ 2cl – ] nanosheet. the all-atom 3d structure of mg4/al2-ldh nanosheet was built based on crystallographic data from [36] as in the previous case, wherein the interlayer co3 2– anion was replaced by dissolved cl – . fig. 4 amino acid cation (arginine) adsorption on a cationic nanosheet of ldh in comparison with an inorganic anion (chloride ion), using copfee method. the gray curve corresponds to copfee result for chloride anion adsorption onto the ldh nanosheet. colors of atoms is similar to the colors of fig.3, except for carbon atoms of arginine, which are reddish, and chlorine, which is yellow the obtained profile for arg adsorption (fig. 4, black curve) has a minimum m1 (67 å), in which the carboxylic group of arg forms several hydrogen bonds with ldh hydroxyl groups, while the positively charged amino group prefers to be located away from the ldh surface (fig.4). carboxylic groups of arg as well as those of asp amino acid are proton acceptors, while the surface hydroxyl-groups of ldh are proton donors. there is also a plateau m2 (8-10 å) on the free energy profile, where the arginine carboxylic group is separated from the adsorbent by a one-molecule-thick water layer (fig. 4, m2 inset). 276 a. tsukanov, s. psakhie using the copfee procedure the surprisingly high estimation for free energy of adsorption of –26 kj/mol was obtained with the half-width of the forward-reverse pmf corridor ∆ = ±3.5 kj/mol. the result means that, despite the fact that arg is a cation, its adsorption onto the positively charged mg4/al2-ldh nanosheet is favorable. this result is quite non-obvious, especially considering that arginine adsorption is even more favorable than adsorption of the chloride anion (fig.4, grey curve), which has a free energy of –12 kj/mol (using copfee as well). 3.3. adsorption of carboxyl-less cation on aluminum oxyhydroxide nanosheet in both the previous cases, the considered organic ions had one (arg) or two (asp) carboxylic groups, which are good terminals (especially in deprotonated state) for hbonding with the hydroxide surface of ldh. thus, to understand the behavior of carboxyl-less cationic molecules, a smd simulation of protonated histamine interacting with an aluminum oxyhydroxide (boehmite) nanosheet was additionally conducted. the full-atom model of alooh was made using structural data from [37]. oxyhydroxide model parametrization was performed in accordance with the clayff force field, but null lennard-jones parameters were replaced by r0 = 0.449 å, ε = 0.046 kcal/mol [15, 33] to allow proper interactions with the charmm subsystem of the model. the net charge of the alooh nanosheet is zero. however, its surface has positive charge due to oriented hydroxylic groups with exposed protons outside. in the all-atom histamine model, charmm-compatible parametrization was utilized, using the swissparam [38] web-service (http://www.swissparam.ch). partial atomic charges were obtained using self-consistent field (scf) calculation in the nwchem package [39] with hartree-fock basis hf/6-31g** [40, 41]. the histamine molecule in the protonated state has a charge of +1 e, which is concentrated in the amino-group region, where the partial atomic charge of hydrogen atoms in the amino-group is +0.390 e, while the nitrogen contributes –0.642 e. using the copfee procedure we obtain a value of –3.6 kj/mol for the free energy of adsorption of protonated histamine (phst) on the alooh nanosheet (fig. 5, black curve). the depth of the free energy well is comparable to the kt level, which is about 2.5 kj/mol at model temperature t = 310 k (p = 0.101 mpa.). the obtained result shows that, despite electrostatic repulsion, the soft molecule can assume a suitable conformation to allow hydrogen bonding and thus make the adsorbed state energetically favorable, but not as stable as in previous cases. as can be seen in fig.5, in the adsorbed state phst is oriented with its nh3 + -group towards the boehmite nanosheet. furthermore, the hydrogen atoms of the boehmite oh-groups that are closest to phst are pushed apart due to coulombic repulsion, and one of the nh3 + -group protons found contact with hydroxyl oxygens between the hydroxyl hydrogens of the alooh. using the copfee procedure we obtain a value of –3.6 kj/mol for the free energy of adsorption of protonated histamine (phst) on the alooh nanosheet (fig. 5, black curve). the depth of the free energy well is comparable to the kt level, which is about 2.5 kj/mol at model temperature t = 310 k (p = 0.101 mpa.). the obtained result shows that, despite electrostatic repulsion, the soft molecule can assume a suitable conformation to allow hydrogen bonding and thus make the adsorbed state energetically favorable, but not as stable as in previous cases. as can be seen in fig.5, in the adsorbed state phst is http://www.swissparam.ch/ adhesion effects within hard matter – soft matter interface: molecular dynamics 277 oriented with its nh3 + -group towards the boehmite nanosheet. furthermore, the hydrogen atoms of the boehmite oh-groups that are closest to phst are pushed apart due to coulombic repulsion, and one of the nh3 + -group protons found contact with hydroxyl oxygens between the hydroxyl hydrogens of the alooh. fig. 5 combined process for histamine (in protonated state) adsorption-desorption on an aluminum oxyhydroxide nanosheet. colors for atoms: aluminum – purple, bridging oxygen – black, hydroxyl oxygen – grey, hydrogen – white, nitrogen – blue, carbon – orange, chlorine – yellow. water is not shown thus, the current case is unusual in that the adsorbed molecule is a proton donor, while the hydroxyl oxygens of the nanosheet are proton acceptors, unlike the previous cases. 4. conclusion in the present work, three different soft matter – hard matter couples of nanomaterials are studied to reveal regularities in the nanohybrid formation and the stability of supermolecular complexes, as well as to analyze the energy aspects of adhesion between bio-molecules and layered metal hydroxides, which holds great promise in a wide range of biomedical applications. the conducted smd study shows that anionic and cationic bio-molecules can form (conditionally) stable nanocomplexes with positively charged layered hydroxide nanosheets, even despite the electrostatic repulsion in the case of a cationic adsorbate. since both anionic and cationic amino acids (in zwitterionic state) are capable of being adsorbed on the [mg4al2(oh)12 2+ 2cl – ] surface, all 20 amino acids in zwitterionic state could exhibit the same behavior. comparing the free energy of adsorption of asp, having two carboxylic groups, and arg with one carboxylic group onto [mg4al2(oh)12 2+ 2cl – ] surface, it may be concluded that each deprotonated carboxylic group adds about 2025 kj/mol to the depth of the free energy well; however, the exact mechanism can be very 278 a. tsukanov, s. psakhie complicated, and many different aspects such as charge, shape, size, hydrophilicity and so on must be taken into consideration. comparing the free energy profiles obtained for arg and chlorine adsorption, it seems that the flexibility of the molecule and non-uniform spatial distribution of the electric charge on the molecule allow it to “sneak” in between the hydroxylic surface and the surrounding water molecules, thus minimizing the energy of the system more efficiently than the simple chlorine anion. looking at the interaction of phst with the boehmite nanosheet, it can be concluded that layered metal hydroxides can act not only as proton donors in hydrogen bonding but also as proton acceptors. in this case, the depth of free energy well is much lower, however. moreover, in the case of a deprotonated surface hydroxyl group, the remaining oxygen atom can be a stronger proton acceptor, which may produce strong adsorption sites in defective zones, edges and cleavages. as shown in the considered cases, the interactions within the interface of hard and soft matter may be quite nontrivial, while playing a crucial role in the formation of hybrid multimolecular nanocomplexes and in the modification of cellular environments via selective adsorption of bio-molecules and ions, both of which is important in modern nanomedical and biomedical applications. despite first steps in this direction, the interface between living and nonliving matter remains a rich object for multidisciplinary investigation, including contact mechanics, chemistry, biology, medicine and computational methods, especially molecular simulations. all md simulations are performed using the lammps package (sandia national laboratory, usa) [42] on the lomonosov-1 cluster 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usa, crc press, pp. 283–307. 44. humphrey, w., dalke, a. schulten, k., 1996, vmd visual molecular dynamics, j. molec. graphics 14, pp. 33–38. 45. hanwell, m.d., curtis, d.e., lonie, d.c., vandermeersch, t., zurek, e., hutchison, g.r., 2012, avogadro: an advanced semantic chemical editor, visualization, and analysis platform, journal of cheminformatics, 4(1), pp. 1. facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 135 151 https://doi.org/10.22190/fume190210004p © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  fast user activity phase recognition for the safety of transfemoral prosthesis control aleksandr poliakov, vladimir pakhaliuk sevastopol state university, sevastopol, russian federation abstract. in the process of creating powered transfemoral prostheses, one of the most important tasks is the provision of the user safety while walking. experience shows that security depends not only on the mechanical strength of such devices, but also on the quality of their control systems, which, among other things, must ensure that latency and error rates of recognition are acceptable for each of the possible changes in gait. incorrect or late recognition of the activity mode at best can lead to suboptimal assistance from the auxiliary device, and at worst to loss of stability of the user with a subsequent fall. loss of stability can also occur due to exceeding the critical time or critical errors of the activity phase recognition and the associated incorrect commands generated by the control system. in this paper, a method for quickly recognizing the phase of the user's activity based on the properties of hu’s moment invariants is substantiated. its use in the intelligent control systems will minimize the critical errors that contribute to the loss of the user's equilibrium with the powered transfemoral prosthesis. key words: powered transfemoral prosthesis, safety, activity mode, activity phase, recognition, moment invariant 1. introduction modern transfemoral prostheses (tfp) are sophisticated high-tech devices that enable people with disabilities to overcome the severe psychological consequences associated with amputation of the lower limb above the knee, to lead an active life and to be involved in a community of healthy people. thanks to tfp, people with disabilities can carry out the usual motor actions typical for a healthy person, including: walking on flat and rough terrain, ascending / descending the stairs, sitting down / getting up from the chair, standing still, riding a bicycle, etc. this was made possible after the appearance on the market of available powered tfp, operating under the influence of commands generated by the intelligent control system (ics). one of the key problems of ics is in recognition of the user's intentions to carry out a specific activity mode and generate commands to tfp received february 10, 2019 / accepted november 25, 2019 corresponding author: vladimir pakhaliuk sevastopol state university, universitetskaya str. 33, 299053 sevastopol, russian federation e-mail: pahaluk@sevsu.ru 136 a. poliakov, v. pakhaliuk actuators, facilitate the implementation of this regime. currently, many algorithms are known to solve this problem, but none of them allows obtaining an absolutely exact solution. in this regard, work to improve the icss for powered tfps and to improve the reliability and performance of algorithms used in them is still relevant. reference [1] presents general considerations for the synthesis of ics for powered tfp in the framework of the intellectual-synergetic concept (isc). as in most modern icss, the isc-based control system provides three-tier architecture, including the high level controllers (hlc), middle level controllers (mlc) and low level controllers (llc) [2]. the basic idea of isc is that the ics created on its basis consists of two subsystems: intellectual (is = hlc + mlc) and synergetic (ss = mlc + llc). at the same time, is is responsible for recognizing the locomotor and non-locomotor intentions of the user in the short and medium term, and ss is for their bio-like implementation [1, 3]. in principle, as noted in the review [4], controllers of such ics can perform their functions using different algorithms, each of which can be optimal from the point of view of different criteria and conditions. but an unconditional requirement for such algorithms is the provision of the user safety in the process of locomotion with tfp. despite the fact that is is not directly involved in the control of tfp actuators, it indirectly plays an important role in the controlling as well as developing a strategy and control tactics that are justified as a result of recognizing the intentions of the user to implement a certain movement in the near or medium term. objective errors in the recognition of intentions and their classification depend on many factors and, in general, are random, and therefore inevitable when using any recognition algorithms. moreover, the tfp user can change his initial intentions within a relatively short period of time, which leads to subjective errors. therefore, the exact solution of the problem of recognizing intentions is impossible. to describe the time during which a classification decision must be made that provides an opportunity for the ics of tfp to make the appropriate transition between the activity modes, the term "critical time" [5] was introduced, while to describe any errors that lead to subjective perception by the user state of unstable equilibrium, the term "critical error" was introduced [6]. these terms allowed the division of all possible errors with respect to the recognition and classification of intentions, into non-critical ones, i.e. such that the worst case scenario is the lack of assistance to the user on the part of the auxiliary device, and critical ones in which the user can not only lose balance with a subsequent fall, but can even feel unsure when using an auxiliary device. further studies in this direction have enabled zhang et al. to conclude that not all errors lead to instability of equilibrium [7]. in addition, the authors found that such characteristics as the accuracy of activity mode recognition and error rate may not be sufficient for a real estimation of the recognition algorithm potential. in this regard, they proposed to evaluate the quality of recognition algorithms on the time duration and quality of identifying critical errors, which seems more reasonable from a functional point of view [7]. after comparing the identified critical errors with their effect on the functioning of the auxiliary devices, it was found that they depended more on the phases in which the error occurred and on the changes in the mechanical work value in the artificial joints associated with these errors. in this paper, a reliable method for fast and qualitative recognition of the phase of tfp user activity is grounded based on the features of hu’s moment invariants [8]. its use in the ics of powered tfp, equipped with wearable imus and pressure sensors, will minimize the time of recognition of intentions and critical errors contributing to the loss of user balance during walking, thus increasing its safety. fast user activity phase recognition for the safety of transfemoral prosthesis control 137 in addition to providing security, the qualitative recognition of the activity phase in is is important for the reliable determination of the boundary conditions of the problem of planning the bio-like movements of tfp elements in ss [1]. it is known that most ics to achieve these goals use information about the detected activity in hlc mode and the current state of the device determined using sensors installed in the device or directly in the user [4]. but such an approach does not allow direct solving of the problem of planning synergetically optimal (bio-like) motions, even under condition of qualitative recognition of the activity mode because it does not provide information about the time remaining before the end of this activity mode [1]. consequently, the use of the fast phase activity recognition algorithm in is will allow increasing the quality of planning the movements of tfp elements not only for realization of typical periodic modes of activity but also in the process of volitional control of powered tfp. 2. materials and methods the "ideal" powered tfp should help the user make many possible moves in different modes of activity. such movements as normal walking can be typical for a person, and walking on railway sleepers, overcoming an unknown obstacle, etc. can be atypical. the ability to perform the full range of possible movements is not available to every healthy person and, especially, not to every disabled person. but, as is known, through systematic training, a healthy person can always acquire certain skills to perform the necessary movements [9]. theoretically, this is available to every disabled person using tfp with advanced ics. in ics, the powered tfp should be perceived as one of the subsystems of an integral biomechanical system controlled by the central nervous system (cns). because cns commands are not directly available for tfp actuators, the ics must perform a number of cns functions. one such function is to generate and send commands to the tfp actuators needed to implement the required motion. but the ics must first recognize the intent of the disabled person, which in fact corresponds to the recognition of cns signals transmitted to the musculoskeletal system. namely, we can assume that the task of recognizing intentions is the key for is as a part of ics. there are many methods and algorithms for recognizing intentions, modes and activity phases that are used in control systems of various auxiliary devices and are described in detail in the review [4]. but the work on their improvement continues at the present time. we offer a fairly simple and very fast method for recognizing the phase of activity, based on the use of reference patterns of activity regimes presented in is as matrices of hu’s moment invariants. 2.1. images of activity modes each of the activity modes of the tfp user can be investigated experimentally and presented in the is database as a set of informative parameters, which can be classified as: φh – hip joint angle; φk – knee joint angle; φa – ankle joint angle; yh – position of the hip joint relatively ground support; th, tk, ta – torques in the hip, knee and ankle joints, respectively; vgrf – vertical ground reaction force; hgrf – horizontal ground reaction force etc. but at the moment the question of which of these parameters are independent remains open. at the same time, it can be argued that the relationships between the above and other parameters of the state are in most cases nonlinear. therefore, the experimental 138 a. poliakov, v. pakhaliuk set of all possible parameters, measured at some phase in an arbitrary mode of activity, can be considered unique. in other words, informative parameters, which can be dependent on a certain mode of activity, can be considered independent of the set of possible modes of activity. consequently, a certain point ω in the n-dimensional parametric activity space ω can be represented by a set of coordinates pijk, k=1,…,n: ωij ={pij1, pij2, …,pijn}, where n is the number of received informative state parameters; i is the phase number corresponding to the activity mode with the number of j; p is a parameter identifier. in order to uniquely represent the point ω in n-dimensional space, in addition to the independence of n parameters, it is necessary that the space itself satisfies the completeness condition [10]. but in practice such a representation is impossible because up to the present time, a complete set of independent parameters that uniquely determine human activity is unknown. this indicates that a certain point ω can theoretically represent an infinite set of states (phases) corresponding to an infinite set of activity modes in space ω. one approach that makes it possible to reduce the degree of uncertainty in the activity phase is to localize the space of interest ω. in other words, if only one, for example, level normal walking is tested, instead of all possible modes of activity, then the probability that two or more phases will be represented by the same set of parameters will be extremely low. but an increase in the area under study, by including other modes of activity, will inevitably lead to an increase in the probability of the emergence of additional states corresponding to the same set of informative parameters if their number does not correspond to the condition of completeness of the activity space. given the conditions noted above, suppose that the activity phase of the user's tfp is uniquely determined by a set of four informative parameters: φh, φk, φa, vgrf, which are assumed to be conditionally independent. then an arbitrary point ω in a 4-dimensional parametric space is represented as follows: ωij={φijh, φijk, φija, vgrfij}. the set of all admissible points ωij defines a bounded activity space, which is the union of subspaces of activity modes , which is a union of subspaces activity modes j, j=1,…,s, i.е.:  = 1  2  ...s. having located informative parameters in the lexicographic sequence: φh  φk  φa  vgrf, and normalizing them so that max * max * max * max * ,,, ij ij ij ija ija ijh ijk ijk ijk ijh ijh ijh vgrf vgrf vgrf           , where * ijh, * ijk, * ija, vgrf* ij, are the real values of parameters, and max ijh  , max ijk  , max ija  , max ij vgrf , are the modules of their maximum possible values, we can obtain the image of point ωij. then the set of such images arranged in a given order so that 0 1 2 ... n, will represent the image of activity mode j. finally, the combining the s images obtained in this way gives a complete image of the limited activity space  of a specific tfp user. the images of the activity phases, activity modes and space of potentially possible human activity can be obtained experimentally and are presented graphically for clarity. as an example, fig. 1 shows charts of the change in the normalized informative parameters for the three typical activity modes: level normal walking (1), ascending the stairs (2) and descending the stairs (3), which we obtained as a result of processing information from imu-sensors installed on the thigh, shin and foot and pressure sensors installed in the sole of the shoe of three adult healthy men aged 22, 23 and 24 years. elementary analysis shows that these charts are in general similar to those that are currently available in many literary sources. fast user activity phase recognition for the safety of transfemoral prosthesis control 139 level normal walking ascending the stairs descending the stairs a) b) c) d) fig. 1 parameters changing at different activity modes: a) hip joint angle; b) knee joint angle; c) ankle joint angle; d) vgrf figs. 2, 3 and 4 show graphical images of the activity modes formed for 20 phases in each of the above modes of activity: ω1, ω2 and ω3, respectively. it is easy to see that the graphical image of each phase (ωij) is practically different from all others, which confirms its uniqueness in studied activity space  = 1  2  3. therefore, the problem of recognizing the phase of the activity of the tfp user can be reduced to the problem of pattern recognition, choosing the proper method, which provides high recognition quality 140 a. poliakov, v. pakhaliuk in combination with high speed. from this point of view, the methods of pattern recognition, based on the use of hu’s moment invariants [8, 10-13], are of interest. 0) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) fig. 2 image of the level normal walking fast user activity phase recognition for the safety of transfemoral prosthesis control 141 0) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) fig. 3 image of ascending the stairs 142 a. poliakov, v. pakhaliuk 0) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) fig. 4 image of descending the stairs fast user activity phase recognition for the safety of transfemoral prosthesis control 143 2.2. hu’s moment invariants suppose that there is a reference activity space  and available for research, which represents all the potentially possible activity modes of the tfp user. but even in this case, it is possible to estimate the belonging of the real activity phase to a certain region of space  only approximately. this is due to the variability of the user's and environmental conditions, as well as the systematic and random errors of the real sensory systems. namely, in real conditions, the image of the activity phase the tfp user, determined in accordance with the information coming from the sensors, each time will differ from the reference image. in the general case, the relationship between reference f(ω) and observed g(ω) image can be described, the so-called function of degradation, widely used in the field of image recognition [10]: ( )g d f , where d is some degradation operator. because operator d is unknown or described by a parametric model with unknown parameters, the main problem solved in the recognition process is the comparison of unknown image f(ω), observed by measuring image g(ω), based on a priori information on degradation. its solution can be obtained on the basis of the hu's moment invariants [8], which are very effectively used to recognize images of objects specified as contours in the plane by k points bi={𝑥bi, 𝑦bi} [11]. in the problem of recognizing the activity phase of the tfp user, invariant i can be considered as a functional defined in activity space ω on the set of admissible images of phases ωij that does not change its value under the action of degradation operator d, that is, it satisfies the condition ( ) [ ( )]i f i d f . (1) taking into account the above discrepancies and limitations, in practice the condition (1) can be considered fulfilled if i(f) does not differ significantly from i[d(f)] and each of these invariants belongs to the same class. in order to successfully solve the recognition problem, it is necessary that the values of i for the phase images belonging to different classes differ significantly from each other. as a rule, such discrimination of classes with the help of a single invariant cannot be performed. to fulfill this condition, we must use the set of invariants il, l=1,2,…, which can be obtained at the conditions of the problem under consideration. thus, for example, for each image of phase ωij, the seven hu's moments that are invariant to the full group of affine transformations can be calculated [8]: hri 1 , 4 2 2 r m i  , 6 3 3 r m i  , 6 4 4 r m i  , 12 5 5 r m i  , 8 6 6 r m i  , 12 7 7 r m i  , (2) where h is a constant (in the given problem it can be chosen arbitrarily, for example, h = 1);    k i bi x k x 1 1 and    k i bi y k y 1 1 are the mean coordinates values of the image contour points; 1 1 ( ) ( ) i i k p q pq b b i x x y y k      are the central moments of order 𝑝 + 𝑞 ≤ 3; 0220  r ; ms, s = 1,…,7 are the moments invariant to the operations of rotation, transfer, and mirror mapping, determined by expressions: 144 a. poliakov, v. pakhaliuk 02201  m , 2 2 2 20 02 11 ( ) 4m      , 2 2 3 30 12 21 03 ( 3 ) (3 )m        , 2 2 4 30 12 21 03 ( ) ( )m        , 2 2 5 30 12 30 12 30 12 21 30 2 2 21 03 21 30 30 12 21 03 ( 3 )( )[( ) 3( ) ] (3 )( )[3( ) ( ) ], m                              2 2 6 20 02 30 12 21 30 11 30 12 21 30 ( )[( ) ( ) ] 4 ( )( )m                   , 2 2 7 21 03 30 12 30 12 21 03 2 2 30 12 21 03 30 12 21 03 (3 )( )[( ) 3( ) ] ( 3 )( )[3( ) ( ) ]. m                              as an example, table 1 gives the matrix of hu's moment invariants (2) calculated for level normal walking. matrices characterizing other modes of activity can also be represented in a similar way. the combination of such matrices provides another way of representing the image of the limited space of the potential activity of the tfp user. this space can be obtained as a result of experimental studies of the motor activity of healthy people, therefore, in a certain sense it can be considered a reference one. but in the process of training the user to walk with tfp, the activity space can be modified to suit its individual characteristics. 3. results to ensure safety and obtain initial data for planning the optimal motions of the elements of the powered tfp, it is sufficient to relate with a high degree of reliability the current state of the user to a certain discrete image of phase ωij in the activity mode space ωj, so that ωi-1j  ωij  ωi+1j, given the errors of the state evaluation with the help of a sensory system. suppose that each of the sensors used to identify the informative parameters of the user state gives an approximate value of the parameter for the current state: maxmin cskcskcsk ppp  . in addition, taking into account the fact that the typical activity modes ωj of different users are similar, we will assume that the rationing of informative parameters allows us to evaluate their values regardless of anthropometric data and the user's physical states. this allows us to calculate the vectors of the hu's moment invariants for possible mean and boundary values of the parameters, i.e. for mid ijkp , min ijkp , max ijkp , which are determined as a result of the experiments. the components of such vectors are generalized coordinates of the space of invariants j7 defining in it some point hij = {iijs, s = 1,...,7}. points mid ijh , min ijh , max ijh generated by parameters mid ijkp , min ijkp and max ijkp respectively, characterize the same image of activity phase ωij and can be considered as homogeneous elements of space j7 forming cluster clij  j7. the volume and location of clij in j7 are implicitly determined by the boundary values of the informative parameters of phase image ωij. therefore, we assume that all the points hij generated by the values of informative parameters maxmin ijkijkijk ppp  belong to cluster clij whose center is point mid ijh . fast user activity phase recognition for the safety of transfemoral prosthesis control 145 the space of invariants j7 is metrical, therefore in it the metric can be given 2( , ) ( )a b s s d h h d i  , (3) where d(is) is the difference of the coordinates with number s of points hb and ha. table 1 hu's moment invariants for 20 phases of level normal walking phase no. i1 i2 i3 i4 i5 i6 i7 0 4.519 0.929 0.782e-2 0.101e-2 -1.396e-6 -0.739e-3 2.609e-6 1 4.543 0.882 0.373e-2 0.495e-3 -6.637e-7 -0.457e-3 1.105e-7 2 4.611 0.776 0.169e-2 0.340e-3 8.553e-8 -0.207e-3 -2.914e-7 3 4.598 0.794 0.954e-3 0.182e-3 7.231e-8 -0.128e-4 -2.409e-8 4 4.548 0.871 0.944e-2 0.566e-4 -9.382e-9 0.529e-4 -4.076e-9 5 4.526 0.906 0.167e-2 0.152e-3 4.394e-9 -0.600e-4 8.116e-8 6 4.529 0.902 0.465e-2 0.463e-3 -2.843e-7 -0.372e-3 6.473e-7 7 4.548 0.871 0.944e-2 0.951e-3 -2.090e-6 -0.845e-3 1.974e-6 8 4.585 0.817 0.158e-1 0.157e-2 -6.600e-6 -0.140e-2 4.298e-7 9 4.627 0.758 0.224e-1 0.216e-2 -1.328e-5 -0.189e-2 7.082e-6 10 4.599 0.797 0.218e-1 0.226e-2 -1.532e-5 -0.203e-2 4.157e-6 11 4.508 0.937 0.102e-1 0.116e-2 -3.964e-6 -0.111e-2 -5.757e-7 12 4.506 0.941 0.105e-2 0.959e-4 -1.202e-8 -0.801e-4 -2.931e-8 13 4.594 0.806 0.125e-2 0.173e-3 4.013e-8 -0.622e-4 8.811e-8 14 4.682 0.691 0.510e-2 0.640e-3 2.039e-7 -0.239e-3 1.300e-6 15 4.645 0.748 0.383e-2 0.676e-3 2.501e-7 -0.239e-3 1.230e-6 16 4.586 0.841 0.146e-2 0.293e-3 4.517e-8 -0.107e-3 2.173e-7 17 4.559 0.884 0.135e-2 0.206e-3 -1.891e-8 -0.105e-3 1.161e-7 18 4.541 0.908 0.282e-2 0.376e-3 -1.564e-7 -0.233e-3 3.723e-7 19 4.526 0.925 0.508e-2 0.668e-3 -5.598e-7 -0.452e-3 1.146e-6 the metric (3) makes it possible to measure distance 7 : ( , )b a b a r h h j  including distance ij csr between an arbitrary point hcs, characterizing the current state of the user, and cluster center ij mid ij clh  . thus, this distance can serve as a measure that allows us to judge whether point hcs belongs to cluster clij. in fact, this means that the current state of the user described by point hcs corresponds to activity phase ωij determined by cluster clij if ij csr has a minimum value among all distances from this point to the cluster centers included in j7. if it is not possible to determine the minimum distance ij csr , it can be assumed that the user makes a move that is not represented in is as the typical mode of activity ωj. in this case, the hlc takes the mlc to volitional control. thus, to recognize the activity phase of the tfp user, you must perform the following steps, i. e. you must:  generate a database of reference images of i phases for each of j possible modes of activity in the form of a set of vectors hij = {iijs, s = 1,...,7} representing cluster centers clij  j7;  determine the vector of informative parameters of the current state of the user pcs = {csh, csk, csa, vgrfcs} and calculate the corresponding values of the generalized coordinates of invariant space j7 : hcs = {icss, s = 1,...,7}; 146 a. poliakov, v. pakhaliuk  calculate distances ij csr between points hcs and hij : clij  j7 and determine the smallest among them; and,  decide on whether the current state belongs to cluster clij and compare this state to activity phase ωij. however, one should take into account the fact that space j7 is extremely inhomogeneous and the generalized coordinates of the clusters centers characterizing the phases of activity differ significantly from one another (see the data in table 1). therefore, the generalized coordinates of vectors hij and hcs, which have significantly smaller values compared to the others, will practically have no effect on distance ij csr between the points in j7. at the same time, such generalized coordinates are very sensitive to possible errors of the sensor system [11] and, therefore, are effective for clarifying the belonging of point hcs to a certain cluster clij. given the above features, for the preliminary estimation of the belonging point hcs to cluster clij mapping space j4 described by generalized coordinates {i1,i2,i3,i4} can be chosen. in the case of obtaining obvious minimum distance ij csr in j4, at this stage it is possible to decide on the belonging point hcs to cluster clij and, consequently, about the correspondence of the current user state to activity phase ωij. if in the study of this space several small distances close to each other are obtained:, ji csr )1(  , …, )1( ji csr ,…, estimating the belonging of point hcs to cluster clij should be clarified by performing an analysis of mapping space j3 described using generalized coordinates {i5,i6,i7}. in cases where it is impossible to determine explicit minimum distance ij csr , a decision is made to switch the mlc to volitional control. as an example, fig. 5a shows clusters in hyperplane 3 of the space of invariants 4j , which is described by generalized coordinates {i2,i3,i4}. each cluster in p3 is represented by three points the coordinates of which are calculated using parameters mid ijkp , min ijkp and max ijkp at the activity phases of 0, 4, 9, 14, 19 corresponding to activity level ω1 (level normal walking). the visual analysis in fig. 5a shows that clusters cl01 and cl191 arranged in p3 hyperplane are relatively close to each other. therefore, distances 01 csr and 191 csr calculated from the centers of these clusters to some point hcs may differ slightly. in such cases, it is expedient to refine the decision on the classification of point hcs and its correspondence to a certain phase of activity from the analysis of mapping space j3, described by generalized coordinates {i5,i6,i7}. a fragment of this space, including clusters cl01 and cl191, is shown in fig. 5b. we will illustrate the process of recognizing the activity phase using a simple example. suppose that as a result of the experiments a database of reference patterns of activity modes ω1, ω2 and ω3 is formed and the current state of the tfp user is measured, characterized by the following normalized informative parameters: h = 0.0739557, k = 0.106382, a = 0.414937, vgrf = 0.623085. consequently, the image of the current phase of activity cs is a vector: {0.0739557,0.106382,0.414937,0.623085}, cs   to which point hcs corresponds in space j7: fast user activity phase recognition for the safety of transfemoral prosthesis control 147 { , 1,2,...,7} {4.534,0.895,0.772 2,0.786 3, 0.162 6, 0.728 3,0.108 7} cs csk h i k e e e e e            . a) b) fig. 5 clusters in the activity mode 1 (level normal walking): a) in hyperplane p3 with generalized coordinates {i2,i3,i4} at the activity phases 0, 4, 9, 14, 19; b) in space j3 with generalized coordinates {i5,i6,i7} at the activity phases 0, 19 fig. 6 shows distances diagrams ij csr from point hcs to cluster centers clij, (i = 0,2,...20; j = 1,2,3) in hyperplane p3. a) b) c) fig. 6 distances ij csr from point hcs to centers of clusters clij in hyperplane p3 for i = 0,2,…,20: a) 1i csr ; b) 2i csr ; c) 3i csr elementary analysis of the charts in fig. 6 suggests that in the example considered, it is impossible to uniquely determine the smallest distance minij csr in hyperplane p3, if we consider that: the base of reference patterns of activity regimes is formed on the basis of the average data obtained for discrete phases of activity; the current phase of the activity 148 a. poliakov, v. pakhaliuk of the tfp user may differ from the discrete phase represented in the base of the reference images of the activity modes; the sensor system approximately evaluates the current state of the tfp user. therefore, to make a final decision on whether the current phase of activity belongs to a cluster, it is advisable to perform an analysis of space j3, choosing for this a number of clusters, the distances from the centers of which to point hcs in hyperplane p3 are relatively small. in this example, 5 clusters were selected for each activity mode clij : j = 1: i = 1, 5, 6, 7, 18; j = 2: i = 5, 6, 8, 9, 13; j = 3: i = 5, 6, 7, 8, 15. the charts of the distances from point hcs to centers clij in space j3 are shown in fig. 7. their analysis allows us to conclude that the vector of informative parameters used in this example does not allow one to uniquely match point hcs to a single cluster since the distances from this point to the centers of the nearest clusters in space j3 are practically equal to each other: 6371 cscs rr  . such a result can be obtained only in the cases when the image of the current activity phase and the images of the activity phases closest to it differ slightly from each other. comparing the images of activity phases 6371,, cs , it is easy to see that this is the case under the conditions of the example in question (see fig. 8). a) b) c) fig. 7 distances from point hcs to centers of clusters in space j3: a) ;18,7,6,5,1,1 iri cs b) ;13,9,8,6,5,2 iri cs c) 15,8,7,6,5,3 iri cs a) b) c) fig. 8 images of user activity phases: a) 71; b) cs; c) 63 fast user activity phase recognition for the safety of transfemoral prosthesis control 149 4. discussion the method of recognition, grounded in this work, is based on the representation of the activity phase of the tfp user in the form of an artificially created image. nevertheless, such an artificial image roughly describes some user state in the space of all possible states and can be used to compare it to a certain phase of activity. it is assumed that the image of each phase is unique and can be represented in space j7 in the form of a vector, the components of which are hu's moment invariants. if we take into account that the artificial images of the activity phases are not subject to displacements, reflections and rotations in space and that they change their shape only by changing the vertical component of the image, then the set of all hu's moment invariants turns out to be redundant for solving the problem of their recognition. but this makes it possible to perform the stratification of space j7 into two subspaces j4 and j3, in each of which the recognition of the phase images can be performed independently. in this case, the recognition results in space j4 and, in particular, in its hyperplane p3, are more significant, because invariants i2,i3,i4 are less sensitive to measurement errors of informative parameters than invariants i5,i6,i7 [11]. in this connection, subspace j3 in this method is used only if it is necessary to make an accurate decision about the correspondence of the current state of the tfp user to a certain activity phase. the most important advantage of the recognition algorithm proposed in this paper in comparison with others is the time of its realization, which, due to the use of the minimum number of arithmetic and logical operations, is significantly smaller compared to the "critical time" of recognition. this makes it possible to increase the time necessary for making a reasonable decision in is for a further mode of traffic, which in turn increases the safety of the tfp user. it should also be noted that, in comparison with neuronet methods successfully used both in the practice of pattern recognition and in the practice of recognizing the activity modes, the proposed recognition algorithm does not require a learning procedure. for its practical implementation, only a lot of coordinates of artificially created images are required, which turn out to be similar for the majority of healthy people belonging to a certain group of physical activity. moreover, even if the current user state is determined by the tfp sensor system with some errors, the image of this state is close in shape to the reference image. in fact, this makes it possible to determine with a high degree of reliability the belonging of the current state to a certain area of the user-implemented activity mode and reduces the probability of generating in ss driving moments in tfp hinges that can lead to "critical errors". the method was tested on a set of vectors { , 1,...,4} { , , , } k h k a k vgrf       , whose coordinates were chosen randomly within intervals maxmin k mid kk   at a given phase of a certain reference activity mode. in this case, the given phase could differ from the discrete phase list in the activity mode, i.e.: * * ( 1) : , 0,2,...,20; 0 1.0 ij i ji j i i i i              . in most cases, the classification of the "current state" given in this way when using this method was correctly executed. in some test cases, one of which was described earlier, ambiguous solutions of the recognition problem were obtained, which did not allow attributing the "current state" to a certain phase of activity. it can be assumed that for some values of the coordinates of user state vectors ω, even if the conditions 150 a. poliakov, v. pakhaliuk maxmin k mid kk   are satisfied, an incorrect classification of the "current state" is possible, but in our test studies such solutions were not obtained. it should be noted that in many cases, in which the boundary values of the informative parameters were chosen as the coordinates of the state vectors, it was impossible to make an unambiguous decision about the classification of the "current state". the results of the tests showed that the proposed method is not inferior to neural network and neural fuzzy methods in efficiency, but it requires significantly less computer time to solve the problems of phase activity recognition. its effectiveness increases significantly when recognizing the phases of specific activity modes implemented by the tfp user at a given time since the difference in phase images within a particular mode is more noticeable than in the entire activity space available to the user. 5. conclusion and outlook in this paper, the method for fast recognition of the activity phase of the tfp user is based on the properties of the hu's moment invariants. its use in isc will significantly reduce the time of phase recognition and minimize critical errors that contribute to the loss of equilibrium the user with tfp in different modes of activity. the evaluation of the probability of successful phase recognition based on this method is the subject of a separate study, so this issue in the study was not considered. at the same time, it should be noted that we performed a series of numerical experiments in which it was assumed that the data registered using sensors characterizing the value of a particular parameter keep normal distribution laws with mathematical expectations corresponding to the reference values of the parameters accepted as informative for different dispersion values. as it turned out, in most cases, the activity mode corresponding to the experimental data was recognized correctly. however, there were also incorrectly recognized regimes, especially in the cases where the variances of the distribution laws were chosen as large enough. this suggests that the probability of correct recognition of the activity mode using the proposed method increases with increasing accuracy of the recorded parameters. the method can be used to recognize the phase in the entire activity space that is accessible to a specific user. at the same time, the activity mode realized by the user at a given time is also recognized. however, in such an option, the possibility of an incorrect classification of the current state turns out the greatest. this method is more effective when determining the phase at a given mode of activity, because the images of the phases in this case are more different from each other. in this connection, in ics algorithms, it is advisable to use activity mode recognition subsystems based, for example, on long short-term memory (lstm) networks, which allow using time dependences of data streams of sensors [14]. however, it can be assumed that the modification of our approach to the construction of artificial images of phases can lead to the appearance of more noticeable differences in them, which in turn will significantly increase the reliability of their recognition, including in the entire activity space. in conclusion, it should be noted that this paper presents only the mathematical side of the method of recognizing the modes and phases of patients' physical activity when walking with the tfp. at the same time, we assumed that the sensors, controllers, and software in the ics of a real prosthesis will be similar to those commonly used in prosthetic control systems and described in detail in the scientific literature [4]. that is, the time required to register, fast user activity phase recognition for the safety of transfemoral prosthesis control 151 filter and classify the values of informative parameters will correspond to that observed in existing control systems. this suggests that the proposed method is characterized by high speed in relation to generally accepted ones at present, because the number of mathematical operations necessary for its implementation is much smaller than in known methods. acknowledgements: the reported study was funded by the internal grant of sevastopol state university, project number 512/06-31. references 1. poliakov, a., pakhaliuk, v., kolesova, m., shtanko, p., ovchinnikova, m., 2017, transfemoral prostheses control in a frame of intellectual-synergetic concept, proc. of the 2017 2nd int. conf. on autom., mech. and elect. eng. (amee 2017), 87, pp. 245-253. 2. varol, h.a., sup, f., goldfarb, m., 2010, multiclass real-time intent recognition of a powered lower limb prosthesis, ieee transactions on biomedical engineering, 57(3), pp. 542–551. 3. poliakov, a., pakhaliuk, v., lozinskiy, n., kolesova, m., bugayov, p., shtanko, p., 2016, biosimilar artificial knee for transfemoral prostheses and exoskeletons, facta universitatis-series mechanical engineering, 14(3), pp. 321-328. 4. tucker, m.r., olivier, j., pagel, a., bleuler, h., bouri, m., lambercy, o., millan, j.-r., riener, r., vallery, h., gassert, r., 2015, control strategies for active lower extremity prosthetics and orthotics: a review, journal of neuroengineering and rehabilitation, 12(1), pp. 1-29. 5. huang, h., zhang, f., hargrove, l.j., dou, z., rogers, d.r., englehart, k.b., 2011, continuous locomotionmode identification for prosthetic legs based on neuromuscular mechanical fusion, ieee transactions on biomedical engineering, 58(10), pp. 2867-2875. 6. zhang, f., liu, m., huang, h., 2012, preliminary study of the effect of user intent recognition errors on volitional control of powered lower limb prostheses, conf. proc. ieee eng. med. biol. soc., pp. 2768-2771. 7. zhang, f., liu, m., huang, h., 2014, effects of locomotion mode recognition errors on volitional control of powered above-knee prostheses, ieee transactions on neural systems and rehabiliation engineering, 23(1), pp. 64-72. 8. hu, m.k., 1962, visual pattern recognition by moment invariants, ire transactions on information theory, 8(2), pp. 179-187. 9. bernstein, n.a., 1991, on dexterity and its development, physical education and sport, moscow, 287 p. 10. flusser, j., suk, t., zitova, b., 2009, moments and moment invariants in pattern recognition, john wiley and sons ltd pub., chichester, 294 p. 11. abramov, n.s., khachumov, v.m., 2014, object recognition based on invariant moments, bulletin of the peoples' friendship university of russia series mathematics, computer science, physics, 2, pp. 142-149. 12. arafah, m., moghli, q.a., 2017, efficient image recognition technique using invariant moments and principle component analysis, journal of data analysis and information processing, 5, pp. 1-10. 13. al-azzo, f., taqi, a.m., milanova, m., 2017, 3d human action recognition using hu moment invariants and euclidean distance classifier, international journal of advanced computer science and applications, 8(4), pp. 12-21. 14. hochreiter, s., schmidhuber, j., 1997, long short-term memory, neural computation, 9(8), pp. 1735-1780. 11904 facta universitatis series: mechanical engineering vol. 22, no 1, 2024, pp. 77 100 https://doi.org/10.22190/fume230615028m © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient flow shop scheduling solutions toufik mzili1*, ilyass mzili2, mohammed essaid riffi1, dragan pamucar3,4, vladimir simic5, laith abualigah6,7,8,9, bandar almohsen10 1department of computer science, faculty of science, chouaib doukkali university, ei jadida, morocco 2department of management, laboratory of research in management and development, faculty of economics and management, hassan first university, settat, morocco 3department of operations research and statistics, faculty of organizational sciences, university of belgrade, belgrade, serbia 4college of engineering, yuan ze university, taiwan 5faculty of transport and traffic engineering, university of belgrade, belgrade, serbia 6hourani center for applied scientific research, al-ahliyya amman university, jordan 7meu research unit, middle east university, amman, jordan. 8applied science research center, applied science private university, amman, jordan. 9department of electrical and computer engineering, lebanese american university, byblos, lebanon 10mathematics department, college of science, king saud university, saudi arabia orcid ids: toufik mzili https://orcid.org/0000-0002-5733-3119 ilyass mzili https://orcid.org/0000-0002-9096-8271 mohammed essaid riffi https://orcid.org/0000-0003-1576-2174 dragan pamucar https://orcid.org/0000-0001-8522-1942 vladimir simic https://orcid.org/0000-0001-5709-3744 laith abualigah https://orcid.org/0000-0002-2203-4549 bandar almohsen https://orcid.org/0000-0002-2160-4159 abstract. this paper presents a novel hybrid approach, fusing genetic algorithms (ga) and penguin search optimization (pseoa), to address the flow shop scheduling problem (fssp). ga utilizes selection, crossover, and mutation inspired by natural selection, while pseoa emulates penguin foraging behavior for efficient exploration. the approach integrates ga's genetic diversity and solution space exploration with pseoa's rapid convergence, further improved with fssp-specific modifications. extensive experiments validate its efficacy, outperforming pure ga, pseoa, and other metaheuristics. key words: hybrid metaheuristics, pseoa, scheduling problem, combinatorial optimization, artificial intelligence, swarm intelligence. received: june 15, 2023 / accepted august 12, 2023 corresponding author: toufik mzili department of computer science, faculty of science, chouaib doukkali university, avenue jabran khalil jabran, b.p 299-24000, el jadida,morocco e-mail : mzili.t@ucd.ac.ma https://orcid.org/0000-0002-5733-3119 https://orcid.org/0000-0002-9096-8271 https://orcid.org/0000-0003-1576-2174 https://orcid.org/0000-0001-8522-1942 https://orcid.org/0000-0001-5709-3744 https://orcid.org/0000-0002-2203-4549 https://orcid.org/0000-0002-2160-4159 78 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. 1. introduction combinatorial optimization problems [1] are a class of complex and diverse problems that encompass many application domains, such as logistics, production planning, communication network design, and many others. among these problems is the multistage flow shop problem (fssp), a classical scheduling problem that aims at optimizing the sequence of tasks to be performed on a set of machines to minimize a given criterion, such as the total task completion time (makespan) or the sum of the completion times of all tasks (total flow time). metaheuristics are high-level stochastic optimization approaches that aim at guiding the search process in the solution space by drawing on principles from nature, physics, or social systems. they have been widely used to solve large and complex combinatorial optimization problems because of their flexibility, adaptability, and ability to escape local optima. among the most commonly used metaheuristics are genetic algorithms (ga) [2], particle swarm optimization (pso) [3], rat swarm optimization [4], and ant colony optimization (aco) [5]. metaheuristics and hybrid metaheuristics offer several advantages for solving combinatorial optimization problems in various domains, including the flow shop problem. some of the main advantages are: flexibility: metaheuristics are generic optimization methods that can be applied to a wide variety of combinatorial optimization problems without requiring specific knowledge of the problem structure. this makes them particularly useful for solving problems from different domains. adaptability: metaheuristics can be easily adapted to take into account the specificities and constraints of a particular problem. dynamic adaptation mechanisms can be integrated to adjust the parameters of the algorithm during execution, thus improving the overall performance of the method. balanced exploration and exploitation: metaheuristics are designed to balance exploration of the solution space (searching for new regions of the space) and exploitation of promising solutions (improving existing solutions). this balance allows metaheuristics to converge to high-quality solutions while avoiding local optima. robustness: metaheuristics are often robust to perturbations and variations in the problem parameters. they can generally provide solutions of acceptable quality even in the presence of noise or uncertainty. efficiency: hybrid metaheuristics, which combine the strengths of different metaheuristics, can offer better performance than individual metaheuristics. hybridization exploits the advantages of each method, such as the genetic diversity of genetic algorithms, the fast convergence of particle swarm optimization algorithms, and the adaptability of ant colony optimization algorithms. this can lead to solutions and faster convergence. in the field of optimization, numerous methods have been developed to solve various problems. these techniques are used in many fields, including engineering, economics and machine learning. these methods can be classified according to their principles and approaches. the classification presented highlights the distinct characteristics of each method, as well as their respective advantages and limitations. fig.1 provides a visual summary of this classification, offering an overview that enables the different optimization methodologies to be explored in greater detail. hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 79 fig. 1 classification of methods for solving optimization problems in this paper, we focus on the hybridization of two popular and efficient metaheuristics, namely genetic algorithms and penguin swarm optimization (pseoa), to solve the fssp. genetic algorithms inspired by the process of natural selection and evolution of species, use operators such as selection, crossover, and mutation to evolve progressively towards optimal or near-optimal solutions. on the other hand, penguin swarm optimization (pseoa) is a recent and promising metaheuristic based on the foraging behavior of penguins. pseoa combines individual and collective foraging behavior to explore the solution space and converges to optimal solutions through a dynamic adaptation mechanism. the main contributions of this work can be summarized as follows: proposal of an innovative hybrid approach combining genetic algorithms (ga) and penguin swarm optimization (pseoa) to solve the multistage flow shop problem (fssp). exploring the hybridization of gas and pseoa, leveraging the strengths of each metaheuristic, including the genetic diversity and exploration of the solution space offered by gas, and the rapid convergence and adaptability of pseoa. modifications and improvements to the basic mechanisms of gas and pseoa to strengthen their performance in the context of the fssp, including the adaptation of selection, crossover, and mutation operators for ga, as well as the introduction of dynamic adaptation mechanisms for pseoa. extensive experimental evaluation of the proposed hybrid approach on a diverse set of multistage flow shop problems of different sizes and complexities, demonstrating its effectiveness and competitiveness over pure gas and pseoas as well as other state-ofthe-art metaheuristics. discussion of the findings and implications of this study for future research on metaheuristics applied to combinatorial optimization and, more specifically, to the fssp, including the limitations of the hybrid approach and the possibilities for improvement and extension to address other combinatorial optimization problems and new challenges in the scheduling domain. the rest of this paper is structured as follows. section 2 presents a review of the literature on metaheuristics applied to the fssp, focusing on genetic algorithms, pseoa, and their hybridizations. section 3 provides a detailed description of the flow shop problem, as well as the main mathematical formulations and evaluation criteria used. section 4 presents the genetic 80 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. algorithms and pseoa, explaining the key concepts, mechanisms, and parameters involved in each. section 5 describes the proposed hybrid approach, highlighting aspects of hybridization between genetic algorithms and pseoa, as well as modifications and improvements made to improve the overall performance of the hybrid approach in the context of fssp. section 6 presents the experimental results obtained by applying the hybrid approach to a set of flow shop problems of different sizes and complexities. a comparison of the performance of the hybrid approach with that of pure gas and pseoas as well as other state-of-the-art metaheuristics is also provided, demonstrating the effectiveness and competitiveness of the proposed hybrid approach. section 7 discusses the conclusions drawn from this study, as well as the implications for future research in the field of metaheuristics applied to combinatorial optimization and, more specifically, to the fssp. we also discuss the limitations of the hybrid approach and the possibilities for improvement and extension to address other combinatorial optimization problems and new challenges in scheduling. 2. related work the field of multi-objective optimization has become an important area of research, especially for planning problems in manufacturing. various algorithms for multi-objective planning problems have been developed over the years, such as genetic algorithms (ga), particle swarm optimization (pso), and ant colony optimization (aco). this paper focuses on multi-objective planning in hybrid flow shop systems where the allocation of production resources considers multiple objectives. shao et al. [6] propose an original multi-objective evolutionary algorithm (moea-ls) for the multi-objective distributed hybrid flow shop scheduling problem (mdhfsp) with multiple neighborhood-based local searches, where each mdhfsp contains a set of factories. they solve the mdhfsp phase in a hybrid flow shop-scheduling problem using parallel machines. they start the solution with a sophisticated weighting mechanism and generate subsequent solutions using multiple local search operators. an adaptive weight update mechanism is also used to avoid stalling at the local optimum. their results confirm the efficiency and effectiveness of the proposed algorithm in processing mdhfsp. han et al. [7] considered a hybrid flow shop scheduling problem with worker constraints (hfsspw) and constructed a mixed integer linear programming model. they proposed seven multi-objective evolutionary algorithms with heuristic decoding (moeahs) to solve the problem. the results showed that the proposed moeahs performed excellently in terms of the makespan objective and demonstrated highly effective performance compared to other methods. amirteimoori et al [8] presented a parallel hybrid pso-ga method for task and carrier organization in a flexible flow shop environment. the researchers used the gurobi solver, parallel genetic algorithm (pga), parallel particle swarm optimization (ppso), and hybrid parallel pso-ga approach (ppsoga) to address the problem instance. the results show that the ppsoga approach outperforms the other algorithms in terms of solution quality and computational efficiency. qiao et al. [9] proposed a genetic algorithm to adapt the flow to a warehouse system and planned a two-stage hybrid with a large setup time in the first stage and stop requirements in hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 81 the second stage. the researchers identified feasible planning conditions and created a local search method to improve the accuracy of the solution. the results showed that the algorithm could find a satisfactory solution within 1% of the lower bound in three minutes, proving its effectiveness. authors vali et al. [10] introduced a flexible job shop scheduling problem with an aim to optimize patient flow and minimize the total carbon footprint. the authors proposed a metaheuristic optimization algorithm, named chaotic salp swarm algorithm enhanced with opposition-based learning and sine cosine (cssaos). the method was executed in a real-world scenario study and showed superior performance in contrast to other established metaheuristic algorithms. miyata and nagano [11], in their paper "an iterated greedy algorithm for distributed blocking flow shop with setup times and maintenance operations to minimize makespan," present the variable search neighborhood (vns) algorithm, referred to as ig_nm. this algorithm is designed to tackle the distributed blocking flow shop scheduling problem, taking into account sequence-dependent setup times and maintenance operations. a comparative study between the ig_nm algorithm and mixed integer linear programming (milp), along with recent methodologies from the literature, was conducted through computational experiments. the outcomes indicate that ig_nm surpasses other literature-based metaheuristics. cui et al. [12] introduced an improved multi-population genetic algorithm (impga) for tackling the distributed heterogeneous flow shop-scheduling problem (dhhfsp), a complex, np-hard problem. the proposed algorithm features a strategic inter-factory neighborhood structure based on greedy job insertion and a novel move evaluation method for efficient neighborhood movements. the impga also incorporates a guided sub-population information interaction and a re-initialization procedure with an individual resurrection strategy to enhance convergence speed and robustness. the results demonstrated that the impga outperformed state-of-the-art algorithms for dhhfsp, proving its effectiveness in finding optimal solutions. furthermore, hou et al. [13] investigated the distributed blocking flow-shop sequencedependent scheduling problem (dbfsdsp) with the objectives of minimizing makespan, total tardiness, and total energy consumption. they proposed a cooperative whale optimization algorithm (cwoa) to solve the problem. computational experimentation on an extensive benchmark demonstrated that cwoa outperforms state-of-the-art algorithms in terms of efficiency and significance in solving dbfsdsp. 3. flow shop scheduling problem the flow shop scheduling problem (fssp) [14] is a crucial component of organizing tasks over a specific period while taking into account temporal constraints (like deadlines and chaining constraints) and limitations on resource utilization and availability. a scheduling problem can be defined as follows: given a set of tasks that need to be executed and a collection of machines (resources) available for performing these tasks, the goal is to identify the most efficient sequence of tasks on the machines, thereby minimizing the total processing time for all tasks across all machines. the majority of scheduling problems are classified as np-hard optimization problems. 82 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. scheduling problems are typically defined by four main elements: tasks, resources, constraints, and objectives [15]. hence, it is crucial to discuss these four fundamental elements before delving into scheduling problems ▪ tasks the manufacturing of products in a production workshop requires the execution of a set of elementary operations or tasks. a task is located in time by a start date 𝑡𝑖 and/or by an end date 𝑐𝑖 and an execution duration pi = ci ti some technical or economic constraints can associate with the task’s earliest start dates ri or latest end dates di. ▪ resources a resource is anything that is used to perform a task. they are available in a limited quantity and time. in the manufacturing context, resources can be machines, workers, equipment, facilities, energy, etc. in the case of the flow shop problem, a machine represents a resource. ▪ constraint constraints are the conditions that must be met when constructing a schedule for it to be feasible. ▪ objective the objectives of companies are diversified and the scheduling process has become more and more multi-criteria. the criteria that must be satisfied by a scheduling process are varied. there are several classes of scheduling objectives: time-related objectives: for example, minimization of total execution time (cmax), average completion time, total set-up time or delay to delivery dates (lmax or tmax) resource-related objectives: maximize the load on a resource or minimize the number of resources needed to complete a set of tasks. cost-related objectives: minimize launch, production, storage, transportation, etc. costs. energy or flow-related objectives. let j={1,...,n} be a set of n jobs. each job must be executed on 𝑚 machines in m={1,...,m} in the same order. each job 𝑗 ∈ 𝐽 consists of 𝑚 ordered operations oj1,...,ojm, and each operation must be executed on one of the m machines. let o={oji, j∈ {1,...,n} and i ∈ {1,...,m}} be the set of all operations to be ordered. each operation 𝑂ji ∈ 𝑂 is associated with a fixed execution time pji the operations are interdependent by two types of constraints. each machine can only process one job at a time and each job must pass through each machine only once. the flow shop problem (fssp) consists in finding a feasible order that minimizes the makespan cmax, i.e., the time needed to complete all the jobs. an order can be represented by a vector of completion times of all operations (cji,i ∈ {1,...,m} and j ∈ {1,...,n}). by adopting the following notations: ▪ oji: operation i of job j, ▪ pji: execution time of oji, ▪ cji: completion time of oji. we can propose the objective function of fssp, which is: minimize cmax = cn×m (1) hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 83 4. methodology 4.1. inspiration penguins are seabirds that are adapted to aquatic life and cannot fly. their wings are modified to function as flippers that help them swim, and they are highly efficient in the water, being able to dive more than 520 meters to search for food. while swimming underwater is less tiring for them, penguins still need to return to the surface regularly to breathe. they can breathe by swimming rapidly at speeds of 7 to 10 kilometers per hour. during dives, their heart rate slows down, and their hunting eyes remain open, with a protective nictitating membrane covering their cornea. their retina enables them to distinguish shapes and colors. penguins primarily feed on fish and squid, and to do so, they hunt in groups and coordinate their dives to maximize their search for food. penguins use vocalizations to communicate with one another, and each penguin's vocalizations are unique, like fingerprints in humans. this uniqueness allows for the identification and recognition of individual penguins within large colonies, where they can otherwise look quite similar. the amount of food required by penguins varies depending on their species, age, the variety of available food, and the amount of food available in their specific area. studies have shown that a colony of 5 million penguins can consume up to 8 million pounds of krill and small fish daily. 4.2. description of the penguin search optimization algorithm (pesoa) penguin search optimization [16] introduces a novel metaheuristic inspired by the collaborative hunting strategies employed by penguins. this concept is particularly intriguing as penguins exhibit the remarkable ability to synchronize their diving patterns to enhance overall energy efficiency. the underlying foundation of optimizing feeding behavior typically revolves around an economic premise: if the energy acquired outweighs the energy expended to secure it, the pursuit becomes a fruitful endeavor. penguins, as living organisms, draw from this concept to gather insights into foraging time, costs, and prey energy content. these factors guide their decisions on whether to forage in a specific area, contingent upon the available resources and travel distances. the algorithm's core principles are encapsulated in the following rules: rule 1:a penguin population is organized into multiple groups. rule 2: each group accommodates a variable number of penguins, adaptable based on localized food availability. rule 3: penguins collectively hunt and wander randomly, ceaselessly seeking sustenance while oxygen reserves permit. rule 4: simultaneous dives to the same depth are possible. rule 5: penguins within a group initiate search from a designated position ("hole i") and various depths ("j1, j2, ..., jn"). rule 6: individual penguins within a group randomly forage and share findings with peers after several dives, fostering intra-group communication. rule 7: each level can accommodate varying numbers of penguins based on food availability. rule 8: inadequate food prompts a group, or even its entirety, to migrate to a different hole, ensuring inter-group communication. 84 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. rule 9: the group with the highest fish consumption discloses the location of the prosperous food source, denoted by the hole and level. in this algorithm, the penguins are characterised by their position in the holes and levels, and by the number of fish they have eaten. the distribution of penguins is based on the probability of fish being present in the various holes and levels. the penguins are grouped together and start their quests from random starting positions. the penguins use a communication system to share information about the food they have found during their dives. groups that have found little food follow those that have found a lot of food on their next dive. the penguins are divided into groups and search for food in holes at different levels. their position is adjusted according to the previous solutions. after several dives, the penguins determine the best solution in terms of food found. the distribution probabilities of the holes and levels are then updated according to this best solution. using eq. (2), an updated solution is calculated for each penguin in each group. dnew = dlastlast + rand ∗ |xlocalbest − xlocallast| (2) rand() is used to generate a random number for the distribution. there are three options: the best local solution, the most recent solution, and the new solution. the calculations of the updated solution equation (eq. (2)) are repeated for every penguin in each group. after several dives, penguins communicate which solution worked best, represented by the number of fish consumed. lastly, the probability of the new distribution of holes and levels is calculated. 4.3. motivations the effectiveness of a metaheuristic is associated with its ability to strike a balance between exploiting promising areas and exploring the global search space, as well as achieving an ideal balance between intensification and diversification. pseoa demonstrates better control over local intensive search strategies and more efficient exploration of the entire search space. additionally, the fewer parameters make pseoa less complex and potentially more versatile. comparisons between pseoa, pso, and ga reveal that pseoa is more robust and efficient than the other algorithms due to its group-based search strategy rather than solely relying on updating the next best-found position. simulations also indicate that the distribution of penguins in the final step is well-balanced between the global and other local minima. given a sufficiently large number of penguins, the algorithm can detect all local minima and the global minimum in the search space. 5. proposed hybrid and discrete pesoa pesoa is a metaheuristic defined for solving continuous optimization problems. the application of standard pesoa to solve discrete (combinatorial) optimization problems is impossible, since there is a difference in the type of variable and also in the techniques for finding the solution. consider the objective function of optimization problem 𝑓(𝑥). the variable 𝑥 for a continuous optimization problem represents a real value but for a combinatorial optimization problem represents a scheduling or configuration. if we consider the traveling salesman problem, the variable 𝑥 represents the hamiltonian cycle. hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 85 the search space in the continuous case is an interval, while for discrete optimization, is the set of solutions meeting the constraints of the problem. the work in this paper will focus on the design of the adaptation of pesoa to combinatorial optimization problems. the result is a discrete version of pesoa adaptable to various pocs. the adaptation process requires a decomposition of our problem into the set of key elements: group, penguin, position, objective function, and search space. a) group ▪ a group is a set of penguins ▪ the number of penguins in all groups is equal to the population size ▪ the number of groups in a population is not fixed. ▪ individuals in a group are not stable. ▪ in a combinatorial optimization problem, a group represents a subset of the problem solutions. b) penguin ▪ a penguin is an individual of the population, the number of penguins equals the population size. a penguin is characterized by the position and quantity of food. c) position ▪ a position is a solution adopted by an individual in the population. ▪ moving from one position to another is a movement in the given problem solution space. d) objective function in pesoa, the amount of food consumed by the penguin is used as the objective function. in our problem, the objective function, or fitness, is a numerical value assigned to every solution in the search space. the most optimal solution is the one with the highest objective function value. e) search space the search space represents the potential positions of the penguins. generally, the positions are coordinates (x,y) in r2 .to move to another position, it is enough simply, to modify the current position by adding a real value to (d') a coordinate (or two coordinates). the search space in a combinatorial optimization problem can be represented by the given problem solutions and the neighboring solutions. ▪ coordinates: the coordinates are the elements that directly affect the quality of the solution through the objective function. this function must be well defined in order to simplify the representation of the coordinates. ▪ displacement: in the combinatorial case, the coordinates of a solution in the search space are changed through the properties of the problem being treated. in general, the change of position in the combinatorial space is done by a change in the order of the elements of the solution, by a combination, a permutation, or a set of methods or operators called perturbations or movement (purple). 86 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. ▪ neighborhood: the notion of neighborhood requires that the neighboring solution of a given solution must be generated by the smallest possible perturbation. this perturbation must make the minimum of changes to the current solution. it is, therefore, necessary to specify subsets of solutions called regions, according to the metric of the search space. ▪ step: the step is the distance between two solutions. it is based on the topology of the space and the notion of neighborhood. in this work, we have classified the steps according to their length, which is the nature and the number of successive perturbations. 5.1. adaptation of ga-pesoa to fssp the objective of solving the flow shop scheduling problem (fssp) is to determine a feasible task sequence for a set of machines that optimizes an objective function. the hybrid genetic and penguin search optimization algorithm (ga-pesoa) adopts a search procedure to obtain optimal solutions when solving fssp. ga-pesoa has been utilized to solve fssp to illustrate its superior effectiveness when compared to other metaheuristic algorithms. this section will demonstrate how to represent a solution in the search space and how to transition from the current solution to a new one. 5.2. fssp solution in the ag-pesoa context, the location of a penguin in a population is considered as a potential solution in the search space. take, for example, table 1, which illustrates a 4-task, 3-machine scheduling problem. each operation is linked to a particular machine with its corresponding processing time. table 1 example for 4 × 3 fssp instance m1 m2 m3 6 2 4 3 6 2 1 3 1 2 1 5 in the context of the fssp, a sequence of tasks representing a solution consists of a permutation of tasks, which is essentially a sequence of n integers denoted by s = {1, 2, 3, ..., n}. this sequence is symbolically represented by s = {1, 2, 3, ..., n}. each integer "i" in the sequence represents a task index, which is responsible for organizing the order of tasks on the designated machines according to their structural arrangement within "s". for example, let's consider a hypothetical initial solution: 2-1-4-3, which specifies the sequential execution of tasks on each machine. the information in table 1, together with the initial solution provided, forms the basis for creating a visual representation similar to a gantt chart (as shown in figure 2). this visual representation is essential for calculating the objective function (cmax) corresponding to the specific sequence (2-1-4-3). this analytical process results in an important measure, which is a significant reflection of the solution's efficiency. in this context, the determined value of 20 represents the efficiency of the solution. hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 87 fig. 2 an example of a gantt chart for the initial 2-1-4-3 solution 5.3. moving in space moving from one solution to another in the search space involves an operation based on concepts of length and topology. length is represented by the cost of the solution, and techniques to move through topology are performed by the operators of equation 3. 𝑆𝑛𝑒𝑤 = 𝑆𝑖𝑑 + 𝑟 ∗ (𝑆𝑏𝑒𝑠𝑡 − 𝑆𝑖𝑑) (3) 5.4. operation subtraction (-) this operation involves two solutions, s1 and s2, and yields a displacement vector q. it also extracts the permutations that were applied to s2 to obtain s1: let s1 = {j1,j2,j3,j4,… jn-1,jn} and s2 = {j2,j3,j1,j4,… jn,jn-1} q=s1-s2 then q= {(j1,j2),(j2,j3),…,(jn,jn-1)} in other words, s1-s2=q’ s1+q=s2 5.5. operation addition ⊕ the operation involves solution s2 and vector q, resulting in a new solution snew. this involves applying permutations of vector q to solution s2 to obtain a new solution snew, as exemplified by: either s2 = {j1,j2,j3,j4,j5,… ,jn} and q = {(3,1),(4,3),(2,5)} snew =s2⊕ q then snew = {j4,j5,j1,j3,j2,...,jn} 5.6. operation multiplication ⊗ the operation involves a multiplication between a real random number r (rϵ[0,1]) and a displacement vector q. its role is to decrease the number of permutations of the vector q based on the value of r. we consider a displacement vector q with n permutations. ▪ q={(c1,c2),(c3,c4),(c5,c6)…,(cn-1,cn)} and real number 0≤r≤1. ▪ q’=r⊗ q then m= n×r ≤n , q’={(c1,c2),(c3,c4),…,(cm-1,cm)} ▪ the fig. 3 describe the mathematical operators. 88 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. fig. 3 illustration of the function of the ga-pesoa operators at the fssp 5.7. genetic algorithm operators in ga-pesoa, the generation of new solutions for solving the fssp is performed by combining the operators of the genetic algorithm and the pesoa mechanism. the genetic algorithm operators (selection, crossover, and mutation) are used to create a new population. two different solutions are chosen from the population to represent the parents of the new population, and crossover is applied to produce two new offspring (solutions). crossover is used to recombine the genetic makeup of two solutions (parents) to produce two solutions that inherit the characteristics of their parent solutions. after the crossover operation, a mutation is applied to the solutions to maintain diversity within the population. the structures of the genetic algorithm operations (crossover, mutation) are illustrated in fig. 4. in the example in fig. 4, we have two parent solutions (parent1, parent2). suppose that the two randomly chosen positions for the crossover are 4 and 7; we then obtain the children child1 and child2, but child1 and child2 are not legitimate. we consider the duplicated tasks in the child 1 as superfluous tasks, which are 2 and 10, and deprived tasks, which are 3 and 9. in child 2 the duplicated tasks are 3 and 9, and the deprived tasks are 2 and 10. then we exchange them to create the new legitimate children child1' and child2'. the mutation operator creates random changes in the order of the tasks. to do this, we randomly select a solution and two positions, then swap the first position of one task with the second position of another task. fig. 4 shows an example of mutating a child1 solution with positions 6 and 10. to integrate the pesoa mechanism with the genetic algorithm, we can use the pesoa search process to guide the exploration of the solution space. for instance, after applying the genetic algorithm operators and obtaining new offspring, we can employ pesoa to refine the offspring's solutions further. this can be achieved by simulating the penguins' hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 89 collaborative hunting strategy and synchronizing their dives to optimize global energy. by combining both approaches, ga-pesoa can potentially find better solutions for the fssp than using either method alone. fig. 4 example of application of ga operators (crossover mutation) 5.8. the 2-opt local search technique operator the 2-opt technique is a local search optimization method commonly used to solve the traveling salesman problem (tsp). however, it can also be adapted to the shop floor scheduling problem (fssp) to improve the solutions generated by the ga-pesoa algorithm. in the fssp, the objective is to find an optimal schedule of tasks to be processed on a set of machines. suppose we have a current schedule (solution) represented by a permutation of tasks and we want to improve it using the 2-opt technique. here is a step-by-step example of how to apply the 2-opt technique to update a penguin's solution for fssp: 1. begin with the current schedule (solution) for a penguin: [1, 2, 3, 4, 5, 6] 2. select two random positions in the permutation (excluding the first and the last positions). for example, positions 2 and 4 (tasks 2 and 4). 3. reverse the order of the tasks between the two selected positions, resulting in a new schedule: [1, 4, 3, 2, 5, 6] 4. calculate the objective function value (e.g., makespan) for both the original and the new schedules. 5. if the new schedule has a better objective function value, accept it as the new solution for the penguin. otherwise, keep the original schedule. 6. repeat steps 2-5 for a predefined number of iterations or until a termination criterion is met (e.g., no improvement in the objective function value for a certain number of consecutive iterations). 90 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. by applying the 2-opt technique to each penguin's solution for the fssp, we can potentially find better schedules by exploring local neighborhoods of the current solutions. this can be a helpful addition to the ga-pesoa algorithm, allowing it to further refine and improve the generated solutions. the final algorithm incorporates all the mechanisms described as follows: algorithm 3: discrete pesoa 1: generate a random population of 𝑃 solutions x={x1,x2,x3,...,xp} ; 2: objective function f(xi),xi ϵ{x1,x2,x3,...,xp} ; 3: select the best solution from p (xbest and xgbest). 4: while (t < maxceneration) or (the stop criterion) do 5: for i=0 to p do 6: while (reserve_oxygene > 0) do 7: calculate xi(t+1) using equation xi(t+1) = xi(t+1) + rand*(xbest -xi(t)) 8: end while 9: if (f(xi(t+1))< f(xbest(t+1))) then 10: xi(t+1)= xi(t) 11: end if 12: else then 13: cross-over the current solution 𝑥𝑖 𝑡 with xbest(t+1))); 14: update the xi(t+1) by the best child of crossing xi(t) and xbest(t+1). 15: end else 16: xi(t+1)= 2-opt(xi(t)) 17: end for 18: select the best xi(t+1))) 19: updating population solutions x(t) by x(t+1) 20: if (f(xbest(t+1))< f(xbest)) then 21: xbest=xbest(t+1) 22: end if 23: if (f(xbest)< f(xgbest)) then 24: xgbest=xbest 25: end if 26: end while. 27: post-processing of results and visualization hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 91 6. experiments and calculation results in this section, an evaluation was carried out on the ga-pesoa using a set of test cases taken from the or library and executed in the c++ programming language. data for these experiments was collected on a macbook air equipped with an intel m1 processor and 8.00 gb ram. the results of this experiment are shown in tables 4, 5, 6 and 7, revealing the numerical results as follows: the initial column serves to identify the instance under examination, called "instance", while the "n × m" column elucidates the specific configuration involving n tasks spread over m machines. the "bks" column gives an overview of the most optimal results, as indicated by the alternative algorithms. the "best" columns effectively summarize the outstanding results obtained by applying the ga-pesoa approach to the particular instances in question. the ga-pesoa parameter values are shown in table 2 : table 2 values of the parameters used parameters meaning values maxceneration maximum number of iterations 1000 p population size 60 ro2 oxygenated reserve 10 to evaluate the performance of the ga-pseoa, we will compare its results with those of various existing algorithms, as listed in table 3. in table 4, 5, 6, and 7, we present a comparison between ga-pseoa and other methods. additionally, figs. 5-8 illustrate the graphical representation of the corresponding results. table 3 nomenclature used abbreviation full name source isdh improved std dev heuristic [17] isdh-ls improved std dev heuristic with local search ibh improved best-so-far heuristic ga genetic algorithm sso social spider optimization (sso) algorithm [18] ns-sgde self-guided differential evolution with neighborhood search [19] hwa hybrid whale optimization algorithm [20] masc memetic algorithm with novel semi-constructive evolution operators [21] neh nawaz-enscore-ham [22] sga standard genetic algorithm neh-nga nawaz-enscore-ham with neighborhood generation algorithm ihsa improved harmony search algorithm [23] pso particle swarm optimization [17] 92 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. table 4 comparison in small instances instance (nxm) bks neh sga neh-nga ihsa pso ga-pseoa car01 11 × 5 7038 7038 7038 7038 n/a n/a 7 038 car02 13 × 4 7166 7376 7166 7166 n/a n/a 7 166 car03 12 × 5 7312 7399 7312 7312 n/a n/a 7 312 car04 14 × 4 8003 8129 8003 8003 n/a n/a 8 003 car05 10 × 6 7720 7835 7720 7720 n/a n/a 7 720 car06 8 × 9 8505 8773 8505 8505 n/a n/a 8 505 car07 7 × 7 6590 6590 6590 6590 n/a n/a 6 590 car08 8 × 8 8366 8564 8366 8366 n/a n/a 8 366 rec01 20 × 5 1247 1320 1249 1247 17 874 19 556 1 247 rec03 20 × 5 1109 1116 1111 1109 15 098 17 417 1 109 rec05 20 × 5 1242 1296 1245 1242 17 793 19 210 1 242 rec07 20 × 10 1566 1626 1584 1566 25 647 28 407 1 566 rec09 20 × 10 1537 1583 1561 1537 24 347 26 796 1 537 rec11 20 × 10 1431 1550 1473 1438 22 706 25 362 1 431 rec13 20 × 15 1930 2002 1956 1935 33 136 36 669 1 930 rec15 20 × 15 1950 2025 1982 1950 33 066 35 905 1 950 rec17 20 × 15 1902 2019 1959 1907 31 901 35 215 1 902 rec19 30 × 10 2093 2185 2175 2098 51 080 59 231 2 093 rec21 30 × 10 2017 2131 2076 2022 48 935 57 782 2 017 rec23 30 × 10 2011 2110 2070 2016 47 921 56 316 2 011 rec25 30 × 15 2513 2644 2636 2518 65 926 76 201 2 513 rec27 30 × 15 2373 2505 2470 2378 63 788 73 432 2 373 rec29 30 × 15 2287 2391 2415 2292 59 655 n/a 2 287 rec31 50 × 10 3045 3171 3249 3150 118 184 n/a 3 045 rec33 50 × 10 3114 3241 3189 3149 125 914 n/a 3 114 rec35 50 × 10 3277 3313 3327 3282 124 035 n/a 3 277 fig 5. comparison of the best-obtained solution for small instances hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 93 table 5 comparison in complex instances (50x5) from [24] instance bks isdh isdh-ls ibh ga sso ns-sgde hwa masc ga-pseoa tail31 2724 82 183 79 471 79 562 80 701 2724 2724 2724 2724 2724 tail32 2834 90 846 88 454 86395 86 105 2839 2834 2834 2834 2834 tail33 2621 2 738 82 218 81 122 80 561 2621 2621 2621 2621 2621 tail34 2751 86 173 84 250 83 257 84 991 2753 2751 2751 2751 2751 tail35 2863 87 367 85 680 85 763 86 789 2863 2863 2863 2863 2863 tail36 2829 89 192 85 739 86 354 84 781 2832 2829 2829 2829 2829 tail37 2725 85 884 82 335 83 010 81 998 2725 2725 2725 2725 2725 tail38 2683 85 103 83 365 84 082 81 934 2703 2683 2683 2683 2683 tail39 2552 80 444 79 978 77 992 77 916 2561 2552 2552 2552 2552 tail40 2782 88 675 85 946 84 142 85 670 2782 2782 2782 2782 2782 fig. 6 comparison of the best-obtained solution for complex instances (50x5) table 6 comparison in complex instances (50x10) from [24] instance bks isdh isdh-ls ibh ga sso ns-sgde hwa masc ga-pseoa tail41 2991 12 0090 11 6969 11 6122 11 7654 3053 3021 3021 3024 2991 tail42 2867 11 8203 11 3873 11 2619 11 7445 2938 2896 2891 2882 2867 tail43 2839 11 7403 11 4235 11 4880 11 0999 2890 2888 2869 2852 2839 tail44 3063 12 2769 11 8586 11 6836 11 7599 3071 3075 3063 3063 3063 tail45 2976 12 0773 12 0242 11 9499 12 0528 3024 3027 3001 2982 2976 tail46 3006 12 0201 11 6570 11 8467 11 6090 3050 3029 3006 3006 3006 tail47 3093 12 2457 11 9751 12 0075 12 2151 3133 3124 3126 3099 3093 tail48 3037 11 6975 11 6003 11 5720 11 8636 3046 3055 3046 3038 3037 tail49 2897 11 8063 11 6323 11 6140 11 5648 2927 2928 2897 2902 2897 tail50 3065 12 1804 11 8031 11 6781 11 8053 3131 3092 3078 3077 3065 94 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. fig. 7 comparison of the best-obtained solution for complex instances (50x10) table 7 comparison for complex instances (50x20) from [24] instance bks isdh isdh-ls ibh ga sso ns-sgde hwa masc ga-pseoa tail51 3850 178954 173683 172254 178630 3974 3916 3876 3889 3850 tail52 3704 169880 166390 166792 166887 3808 3744 3715 3720 3704 tail53 3640 175244 172739 170554 167089 3772 3702 3653 3667 3640 tail54 3720 172895 168989 171055 167904 3849 3793 3755 3754 3720 tail55 3610 172514 166783 169655 173415 3746 3677 3649 3644 3610 tail56 3681 172492 169714 170539 168755 3795 3743 3703 3708 3681 tail57 3704 177382 171602 174218 173165 3835 3784 3723 3754 3704 tail58 3691 169268 165782 165601 173020 3829 3757 3704 3711 3691 tail59 3743 174213 169524 171078 172826 3870 3795 3763 3772 3743 tail60 3756 178270 173446 173635 175483 3875 n/a 3767 3778 3756 fig. 8 comparison of the best-obtained solution for complex instances (50x20) hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 95 6.1. discussion the following section presents a comprehensive comparison of various algorithms, including the hybrid genetic algorithm and penguin search algorithm (ga-pseoa), on distinct instances and complexities of scheduling problems. the best-known solutions (bks) are provided as a benchmark for each instance. 6.1.1. comparison of distinct instances of the scheduling problem table 4 compares the performance of ga-pseoa with other algorithms (neh, sga, neh-nga, ihsa, and pso) on distinct instances of the scheduling problem. ga-pseoa demonstrates exceptional performance, often matching the best-known solutions and consistently outperforming ihsa and pso when results are available. fig. 5 further illustrates ga-pseoa's consistent capability to achieve high-quality solutions through a low curve representing the best solution values discovered across all instances. 6.1.2. comparison of complex instances (50x5) table 5 presents a comparison of ga-pseoa with other algorithms (isdh, isdh-ls, ibh, ga, sso, ns-sgde, hwa, and masc) on complex instances (50x5) from taillard (1993). ga-pseoa consistently matches or closely approaches the best-known solutions for each instance. fig. 6 displays ga-pseoa's consistent ability to obtain high-quality solutions through a small curve representing the best solution values found across all instances. 6.1.3. comparison of complex instances (50x10) table 6 compares ga-pseoa with other algorithms (isdh, isdh-ls, ibh, ga, sso, ns-sgde, hwa, and masc) on complex instances (50x10) from taillard [24]. gapseoa again demonstrates exceptional performance, consistently matching or closely approaching the best-known solutions for each instance. fig. 7 further emphasizes ga-pseoa's consistent ability to obtain high-quality solutions through a small curve representing the best solution values found across all instances. 6.1.4. comparison of complex instances (50x20) table 7 compares ga-pseoa with other algorithms (isdh, isdh-ls, ibh, ga, sso, ns-sgde, hwa, and masc) on complex instances (50x20) from taillard [24]. gapseoa consistently matches the best-known solutions, underlining its ability to efficiently tackle complex scheduling problems. fig. 8 highlights ga-pseoa's consistent capability to obtain high-quality solutions through a small curve representing the best solution values found across all instances. 6.2. validation of ga-pseoa performance using the anova test the performance of the ga-pseoa method is validated using the anova test, followed by dunnett's multiple comparison tests. these tests compare the performance of ga-pseoa with other algorithms by examining the average differences. the following information is provided for each comparison: 96 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. ▪ dunnett's multiple comparison tests: algorithm pairs being compared (ga-pseoa vs. another algorithm). ▪ mean diff: the average difference between the two compared algorithms. ▪ 95.00% ci of difference: the 95% confidence interval for the difference between the mean values. ▪ below threshold? indicates if the difference is below the threshold of significance (yes or no). ▪ adjusted p value: the adjusted p-value for comparison. 6.2.1. cases of lower complexity in table 8, the anova test with dunnett's multiple comparison tests is used to examine the mean differences in performance between ga-pseoa and four other algorithms (neh, sga, neh-nga, ihsa, and pso). the test yields a p-value, which indicates the statistical significance of the differences between the means. table 8 anova test comparison for less complex instances dunnett's multiple test mean diff, 95,00% ci of diff, below threshold? adjusted p value ga-pseoa vs. neh -99,54 -132,5 to -66,55 yes <0,0001 ga-pseoa vs. sga -41,65 -69,07 to -14,24 yes 0,0017 ga-pseoa vs. neh-nga -7,385 -18,56 to 3,789 no 0,2950 ga-pseoa vs. ihsa -47756 -69484 to -26029 yes <0,0001 ga-pseoa vs. pso -36792 -50266 to -23317 yes <0,0001 the test results reveal that, except for neh-nga, there are significant average differences between ga-pseoa and all other algorithms. the average difference between ga-pseoa and neh is -99.54, with a 95% confidence range spanning from -132.5 to 66.55. the average difference between ga-pseoa and sga is -41.65, with a 95% confidence range spanning from -69.07 to -14.24. the average difference between gapseoa and ihsa is -47,756, with a 95% confidence range of -69,484 to -26,029. the average difference between ga-pseoa and pso is -36,792, with a 95% confidence range between -50,266 and -23,317. the test also indicates that there is no statistically significant difference between the mean values of ga-pseoa and neh-nga. this implies that there is no evidence to suggest that ga-pseoa is more or less effective than neh-nga. in conclusion, the anova test with dunnett's multiple comparisons demonstrates that ga-pseoa is significantly more effective than neh, sga, ihsa, and pso, but not significantly different from neh-nga. the results suggest that ga-pseoa is a more competitive approach than other algorithms for the given task. the qq-plot in fig. 9 further illustrates the distribution of average differences between ga-pseoa and the other algorithms, highlighting the distinct behavior of ga-pseoa compared to other researched algorithms. hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 97 fig. 9 qq-plot distribution displaying the distinct performance of ga-pseoa compared to other algorithms for less complex instances 6.2.2. cases of higher complexity the differences in mean performance between ga-pseoa and the other seven algorithms (isdh, isdh-ls, ibh, ga, sso, ns-sgde, hwa, and masc) are compared using dunnett's multiple comparison test. this test gives a p-value that indicates the statistical significance of the differences between the means. table 9 anova test comparison for the more complex instances dunnett's multiple test mean diff, 95,00% ci of diff, below threshold? adjusted p value ga-pseoa vs. isdh -120877 -143540 to -98213 yes <0,0001 ga-pseoa vs. isdh-ls -120600 -139703 to -101498 yes <0,0001 ga-pseoa vs. ibh -120558 -139943 to -101174 yes <0,0001 ga-pseoa vs. ga -121177 -140849 to -101505 yes <0,0001 ga-pseoa vs. sso -57,10 -85,64 to -28,57 yes <0,0001 ga-pseoa vs. ns-sgde -8,283 -27,30 to 10,73 no 0,7390 ga-pseoa vs. hwa -12,41 -19,40 to -5,428 yes 0,0002 ga-pseoa vs. masc -13,00 -20,98 to -5,024 yes 0,0006 the test results reveal that, except for ns-sgde, there are significant average differences between ga-pseoa and all other algorithms. the average difference between ga-pseoa and isdh is -120,877, with a 95% confidence interval ranging from -143,540 to -98,213. the average difference between ga-pseoa and isdh-ls is -120,600, with a 95% confidence interval spanning from -139,703 to -101,498. the average difference between ga-pseoa and ibh is -120,558, with a 95% confidence interval between 139,943 and -101,174. the average difference between ga-pseoa and ga is -121,177, with a 95% confidence interval covering -140,849 to -101,505. the average difference between gapseoa and sso is -57.10, with a 95% confidence interval ranging from -85.64 to -28.57. the average difference between ga-pseoa and hwa is -12.41, with a 95% confidence 98 t. mzili, i. mzili, m.e. riffi, d. pamucar, v. simic, g.dhiman, et al. interval between -19.40 and -5.428. the average difference between ga-pseoa and masc is 13.00, with a 95% confidence interval ranging from 20.98 to 5.024. the test also indicates that the mean difference between ga-pseoa and ns-sgde is not significant, meaning that there is no evidence to suggest that ga-pseoa is more or less effective than ns-sgde. in conclusion, dunnett's multiple comparison test demonstrates that ga-pseoa performs significantly better than isdh, isdh-ls, ibh, ga, sso, hwa, and masc, but is not significantly different from ns-sgde. these results suggest that ga-pseoa is a competitive method in comparison to other algorithms for the given task. in fig. 10 the qq plot displays a bimodal distribution of performance differences between ga-pseoa and the other algorithms, indicating that ga-pseoa performs considerably better than other algorithms in some cases, while its results are significantly worse in others. the points furthest from the edges of the line reveal that these performance differences can be substantial. fig. 10 qq-plot distribution showing the distinct performance of ga-pseoa compared to other algorithms for more complex instances 7. conclusions in conclusion, the innovative hybrid approach presented in this paper, which combines the strengths of genetic algorithms (ga) and penguin search optimization (pseoa), has proven to be highly effective in solving the multistage flow shop scheduling problem (fssp). by integrating the genetic diversity and solution space exploration capabilities of ga with the rapid convergence and adaptability of pseoa, this hybrid method significantly outperforms pure ga, pseoa, and other state-of-the-art metaheuristics in comprehensive experimental evaluations. the promising results of this study pave the way for further research in metaheuristics applied to combinatorial optimization, particularly in the context of fssp, and emphasize the need to address the limitations of the hybrid approach while exploring avenues for improvement and extension to tackle other combinatorial optimization problems and challenges in the scheduling domain. hybrid genetic and penguin search optimization algorithm (ga-pseoa) for efficient... 99 the successful application of the hybrid ga-pseoa approach in the context of fssp highlights its potential for enhancing efficiency and productivity in manufacturing systems. the implementation of this method can contribute to substantial advancements in manufacturing processes, leading to more streamlined and cost-effective operations. future research will focus on several key aspects to advance the current state of knowledge. firstly, refining the performance of the hybrid ga-pseoa approach for fssp will be prioritized to improve its efficiency and impact. additionally, the exploration of potential applications for this hybrid optimization technique in other optimization tasks will expand its scope and influence across various domains. furthermore, the development of hybrid optimization algorithms that blend the strengths of ga-pseoa with other optimization techniques has the potential to yield significant improvements in the overall efficiency and effectiveness of the method. acknowledgement: the authors extend their appreciation to the deputyship for research & innovation, ministry of education in saudi arabia, for funding this research work through the project no. 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taillard, e., 1993, benchmarks for basic scheduling problems, european journal of operational research, 64(2), pp. 278–285. an application of multicriteria optimization to the two-carrier two-speed planetary gear trains facta universitatis series: mechanical engineering vol. 15, no 1, 2017, pp. 85 95 doi: 10.22190/fume160307002s © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper an application of multicriteria optimization to the two-carrier two-speed planetary gear trains udc 621.833.65-027.231 jelena stefanović-marinović 1 , sanjin troha 2 , miloš milovančević 1 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of engineering, university of rijeka, croatia abstract. the objective of this study is the application of multi-criteria optimization to the two-carrier two-speed planetary gear trains. in order to determine mathematical model of multi-criteria optimization, variables, objective functions and conditions should be determined. the subject of the paper is two-carrier two-speed planetary gears with brakes on single shafts. apart from the determination of the set of the pareto optimal solutions, the weighted coefficient method for choosing an optimal solution from this set is also included in the mathematical model. key words: multi-criteria optimization, planetary gear train, two-speed planetary gear trains, pareto optimal solutions 1. introduction multi-criteria optimization problems are very common in many scientific and technical solutions. the optimization of gear trains as concrete technical systems implies complex mathematical model that has to describe real system operation in real circumstances. planetary gear trains (pgt) are a type of gear trains with many advantages. since that their application is increasing in mechanical engineering and conveyor systems as single stage and multi-stage. multi-stage planetary gear trains are obtained from single stage gear trains by linking one or two planetary units shafts. adequate design of pgt could be used as (multiple speed) gearboxes. significant application as gearboxes has two-carrier planetary gear trains which consist of two coupled and four external shafts and enable two-speed transmissions [1]. there is a significant number of possible schemes of these transmissions [1-6]. some transmissions structures could be used as two-speed transmissions by applying convenient brakes layout [1]. received march 07, 2016 / accepted september 20, 2016 corresponding author: jelena stefanović-marinović faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: jelenas@masfak.ni.ac.rs 86 j. stefanović-marinović, s. troha, m. milovanĉević there are not many papers focus on the application of multi-criteria optimization to gear transmissions, especially planetary gear transmissions. some papers are specified below. the multi-objective optimization approach, based on the concept of pareto optimality, is used in order to design helical gears [7]. the choice of the best parameters optimization is the important step in the process of obtaining the required gear quality and the optimization of the designing process itself [8]. planetary gear transmissions are the subject of some papers in which the research relevant to the optimizations given. arnaudow and karaivanov [3] present a simple, descriptive and easy-to-handle method for investigating the transmission ratio, the internal power flows and the efficiency of complex multi-planetary gearings. this paper provides an optimization of the two-carrier two-speed planetary gears with brakes on single shafts. for the numerical example of multi-criteria optimization application, the input data suitable for the usage at the fishing boat gearbox is chosen. apart from the determination of the set of the pareto optimal solutions, the weighted coefficient method is applied for choosing the optimal solution. 2. mathematical model for planetary gear train optimization in this paper multi-criteria optimization is applied to the two-carrier two-speed planetary gear trains with brakes on single shafts. these compound planetary gear trains consist of the basic type of planetary gear train. the basic type of a planetary gear train (pgt) is a design which has a central sun gear (external gearing 1), central ring gear (internal gearing 3), planets (satellites 2) and carrier (h), shown in fig.1. planets are in simultaneous contact with the sun gear and the ring gear. the multi-criteria optimization is limited to geared pairs. fig. 1 basic type of planetary gear train (1sun gear; 2 – planet; 3 – ring gear; h – carrier) the process of finding the optimal solution starts by defining a mathematical model [9]. the complete mathematical model of the basic type of a pgt is described in the aforementioned paper and will also be briefly presented in this section. the mathematical model is defined by the variables, objective functions and conditions required for the proper functioning of a system expressed by the functional constraints. an application of multicriteria optimization to the two-carrier two-speed planetary gear train 87 2.1 variables under the mathematical model definition, it is necessary to determine the variables since each objective function is the function of several parameters. in this paper, the following variables are considered: the number of teeth of the central sun gear z1, the number of teeth of planets z2, the number of teeth of the ring gear z3, the number of planets nw, the gear module mn and the face width b. the optimization variables are of the mixed type: the numbers of gear teeth (z1, z2, z3) are integers, positive and negative, the number of planets (nw) is a discrete value, the module (mn) is a discrete standard value (acc. to iso 54 (din 780)), while the face width (b) is a continual variable. the numbers of gear teeth and the number of planets are non-dimensional values, while the module and the facewidth are given in millimeters. 2.2 objective functions in this model, the following characteristics are chosen for objective functions of a planetary gear train: volume, mass, efficiency and production cost of gear pairs. the volume of gear pairs is used as an overall dimension expression and the approximation of the gear by the cylinder volume with the diameter equal to the pitch diameter and the height equal to the face width. the fact that planets are inside the ring gear makes it possible for the gear volume to be expressed by: 2 23 3 cos cos cos4            wt tn zm bv     (1) where αt is the transverse pressure angle, αwt23 is the working transverse pressure angle for the pair 2-3 and β is the helix angle at the pitch diameter. mass is determined as the sum of all gear masses in a gear train. since the mass of a particular gear is determined as gear volume multiplied by the density of gear material, this criterion receives the final expression, given by eq. (2):          23 2 2 2 33 12 2 2 2 22 12 2 2 2 112 2 cos cos cos cos cos cos cos 25.0 wt t wt t w wt tn zkzknzk m bm         (2) efficiency is one of the most important criteria for the design and evaluation of the construction quality. power losses in planetary transmissions consist of losses in the gear contact, losses in bearings and losses due to oil viscosity. the calculation of the gear transmission efficiency is generally confined to losses depending on friction on tooth sides, i.e. on the calculation of contact power losses [9-11]. then, the following expression for efficiency is considered: 0 00 1 1       i where η0 is the efficiency when the carrier is immovable            32113 3 0 20.035.015.0 1 zzzzz z  (3) and i0 is basic transmission ratio i0=z3/z1. 88 j. stefanović-marinović, s. troha, m. milovanĉević economic demands must also be taken into consideration in the techno-economical optimization. first, these demands are related to production costs. these costs consist of the production material and the production process costs. the time needed for the production of gears is taken as a measure of the production costs and as an economic factor. this function is then determined as a sum of time periods needed for the production of the central sun gear (tp1), the planets (tp2) and the ring gear (tp3), i.e. 321 ppwpt ttntf  (4) production times are determined according to the technologies of fette, lorenc and höfler [12]. 2.3 functional constraints planetary gear trains represent a specific group of gear trains. therefore, there are numerous exceptions that need to be taken into account for these transmissions to function correctly compared with classical gear transmissions. the exceptions presented in this article are related to mounting conditions, geometrical conditions and strength conditions. the mounting conditions comprise the condition of coaxiality, the condition of adjacency and the condition of conjunction [13]. geometrical conditions relate to the undercutting and profile interference, the ratio of the pressure angle to the working transverse pressure angle, the tooth thickness and the space width, the transverse contact ratio value, the sliding speeds, the ratio of the pinion face width to the pinion reference diameter, etc. these conditions are ensured in accordance with the actual standards (iso tc 60 list of standards 090915). the strength conditions, safety factors for bending strength and surface durability of each gear, are provided according to iso 6336-1 to iso 6336-3 [14]. 2.4 optimization procedure the base of the optimization process of pgt presented in this paper is the comparison of solutions with different parameters in the same conditions and the selection of the best variant. the optimization process begins with generating all solutions for the assigned input data. for the given input data (transmission ratio, input number of revolution, input torque, service life in hours, application factor, accuracy grade (q-din3961)), all 6-tuples of design parameters (z1, z2, z3, nw, mn, b) satisfying the functional constraints are generated and the values of the objective functions for every 6-tuple are computed. these 6-tuples form a set of feasible solutions. based on the established objective functions and constraints, an optimal solution is selected, determined by variables. the mathematical model of nonlinear multicriteria problem can be formulated as follows: 1 2max{ ( ), ( ),....., ( )} subject to kf x f x f x x s (5) here, f1(x),...., fk(x) are objective functions and x=(x1,......,xn) is the vector of decision variables and s is the set of feasible solutions. every point xs is mapped to the point (f1(x), f2(x), …,fk(x)) in k-dimensional objective space. therefore, one can observe the objective set: 1 2{(( ( ), ( ),......, ( )) }kf f x f x f x x s  (6) an application of multicriteria optimization to the two-carrier two-speed planetary gear train 89 the notation "max" determines simultaneous maximization of all the objective functions. if any objective function is to be minimized, minimization of the function fi(x) is equivalent to the maximization of the function –fi(x). according to the structure of the feasible set s, there exist discrete multi-criteria optimization problems. in our planetary gears problem, six decision variables exist, corresponding to the basic design parameters: x = (x1, x2, x3, x4, x5, x6) = (z1, z2, z3, nw, mn, b). furthermore, there are four objective functions: volume v(x), mass m(x), efficiency η(x) and production costs t(x): 1 2 3 4 ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ) f x v x f x m x f x x f x t x        (7) then, mathematical model of nonlinear multi-criteria problem in concrete task, can be formulated as follows: 1 2 3 4max{ ( ), ( ), ( ), ( )} subject to f x f x f x f x x s (8) it is often useful to know the best possible values for each objective function. these values form a so-called ideal point f * =(f1 * ,...., fk * ) in the objective space. its components are computed as: * max ( ), 1,....i if f x i k x s    (9) as it can be seen from the definition, multi-criteria optimization problems are mathematically ill-defined. the most important criterion for selecting these "equally good" solutions is the pareto optimality concept: solution xs is pareto optimal if there is no solution ys such that holds fi(x)≤ fi(y) for all i=1,......, n and for at least one index i holds strict inequality, i.e. fi(x)< fi(y). determination of the pareto optimal solutions set is the first step in optimal solution finding. next step is optimal solution choice from pareto solutions set. in this model weighted coefficients method is applied for choosing optimal solution from pareto solutions. 2.5 weighted coefficients method in this method the following scalarized problem is constructed: 0 0 1 1max ( ) ( ) ...... ( ) . . m m mf x w f x w f x s t x s       (10) here, the weighted coefficients (or weights) wi are positive real numbers and fi 0 (x)= (fi 0 ) -1 fi(x) are normalized objective functions where fi 0 are normalizing coefficients. in this approach, the components of the ideal point f * =(f1 * , f2 * , f3 * , f4 * ) are used as normalizing coefficients, i.e. fi 0 = fi * for i=1, 2, 3, 4. therefore, absolute values of all objective functions are between 0 and 1, which simplifies the choice of the weighted coefficients. all solutions obtained by this method are pareto optimal [9]. the weighted coefficients method has very clear physical meaning and experience in application on technical systems optimization. 90 j. stefanović-marinović, s. troha, m. milovanĉević this model is suitable in the case of priority functions existence, also in the case of equal priority functions [12]. a shortened algorithm for the complete optimization procedure is shown in fig. 2. the complete optimization procedure is implemented in the plangears software. fig. 2 shortened algorithm for the optimization procedure 3 two-speed planetary gear trains speed change under load is advantage, and in some occasions request of mechanical system (e.g. machine tools, cranes etc.). two-speed two-carrier planetary gears which consist of two coupled and four external shafts and have brakes on single shafts could be use in these systems. a special type of mechanism is obtained by settings the brakes on two external shafts that allow energy flow managing throw the transmission and transmission ratio changing. the layout of brakes at these compound gear transmissions has many possibilities [15, 16]. in fig. 3 ideal torque ratios and torque ratios, as well as wolf-arnaudov’s symbol, of the basic type of pgt are pointed. the carrier shaft is summary element, since by carrier stopping negative transmission ratio is obtained. an application of multicriteria optimization to the two-carrier two-speed planetary gear train 91 prerequisite: 0 13(h) 31(h) 1     ideal torque ratio: 3 3 0 1 1 1 t z t i t z       torques: 1 3 h: :t t t  0 01: ( ) : ( 1)i i   fig. 3 torque ratios and wolf-arnandov’s symbol of the basic type of pgt 3.1 structures of compound two-carrier pgt and their labeling a planetary gear train with four external shafts is shown in fig. 4. two component trains can be joined in a whole in 12 different ways, called the planetary gear train with four external shafts [16]. to each of 12 structural schemes an alphanumerical label (s11…s56) is attached, which indicates the ways of connecting the shafts of the main elements of both component trains (fig. 5). in every presented scheme it is possible to put brakes as well as the driving and the operating machine on external shafts in 12 different ways (v1…v12), which will be here called layout variants (fig. 6). fig. 5 systematization of all schemes of two-carrier planetary gear train with four external shafts fig. 6 systematization of all layout variants fig. 4 planetary gear train with four external shafts (compound train) 1t 3 0 1t i t   0 11ht i t   1t 3t 1 3 h 2 ht t 0 1t 3 0 1t i t    h 0 11t i t   92 j. stefanović-marinović, s. troha, m. milovanĉević 3.2 analysis of the operations of compound trains with different layout variants by situating the brakes on two shafts a braking system is obtained in which the alternating activation of the brakes shifts the power flow through the planetary gear train, which causes a change in the transmission ratio [1, 3, 6, 15, 16, 17]. kudrjavtsev and kirduashev [2] present 15 kinematic schemes of the considered type and achievable values of transmission ratios and efficiencies of both gears. troha et al. [1, 15, 17] present the computer program for the selection of an optimal variant of similar multi-speed planetary gear trains including shifting capabilities charts for all possible two-speed planetary gear trains. the considered compound trains can be divided into three different groups depending on the layout of brakes on the shafts. the first group consists of the compound trains with brakes on the coupled shafts. the second group consists of the compound trains with brakes on the single shafts. the third group consists of the compound trains with brakes on the coupled and the single shaft. all compound trains within separate groups have some specific common characteristics [16]. the compound train with brakes on the single shafts (layout variants v6 and v12) is symbolically shown in fig. 7. when the left brake is turned on, power is transmitted through the left component train (component train i), and when the right brake is turned on, power is transmitted through the right component train (component train ii). input and output of power are on the coupled shafts. in this case, whatever the brake is turned on, only one component train operates actively while the other operates idly. therefore, the transmission ratios of the compound train are equal to the transmission ratios that the component trains accomplish. br1 br2 br1 br2 v6 v12 fig. 7 possible power flows through the train with brakes on the single shafts each variant has its own characteristics that determine the possibilities of converting. it could be presumed that some variants work in both speeds lake reducers and multipliers, while some other variants work like reducers with one speed and like multiplier with other speed. also, some variants change direction of rotation when change speed, while some other variants keep the direction of rotation during speed changing. transmission ratio of each planetary gear train unit depends only on basic transmission ratio (ideal torque ratio). some of these variants have very interesting kinematic characteristics from the point of the practice. for example, transmission variant s36v6 which changes the direction of rotation during speed changing is suitable for application in machine tools where exist one working motion with considerable resistances and low speed and other working motion during returning to the starting position with great speed. transmissions with brakes on single shafts have their own constraints. considerable constraint is transmission ratio range: between 0.0769 and 13. if transmission ratio besides this range is need, usage of transmissions with brakes on coupled shafts or on coupled and single shaft is recommendation. two-shaft operating mode unloaded work an application of multicriteria optimization to the two-carrier two-speed planetary gear train 93 4. results and discussion for the example of multi-criteria optimization application variant s36v6 is chosen (figs. 8 and 9). this type of transmission has considerable application at the systems which need transmission ratio changes under load. for instance, it is applicable as a drive of fishing boats propellers. in that case the transmission works with transmission ratio i=4 in one way and with transmission ratio i=-4 in the other way. the transmission is situated on the propeller shaft between engine and propeller. these data is adopted for the input data for optimal solution choice. a) b) fig. 8 symbolic review of transmission composition (a), kinematic scheme (b) a) b) fig. 9 power flow throws the transmission when the brake br1 is activated (a) and when the brake br2 is activated (b) 4.1 the first stage of compound gear train (i) when the brake br is activated and the brake br2 inactivated the ring gear in first stage is immovable. the input of the system is a, and sun gear of the first stage and sun gear of the second stage have the same number of revolution. since the ring gear in the first stage (3i) is reactive, the power is transmitted to the carrier of the first stage (hi) and than to the output b. the second stage works at the three-shaft mode, but no resistance, i.e. at idle. the first stage is determined in this mode. the next input data is chosen for the multi-criteria optimization application: i0=-3.0, nin=1800 min -1 , tin=3119.34 nm (p=588 kw), l=8000 h, ka=1.25, it7,for all gears, material z1/material z2/material z3=20mocr4/20mocr4/34crnimo6, shmin=1.1, sfmin=1.2, δi=3%, z1=15÷30. 94 j. stefanović-marinović, s. troha, m. milovanĉević the feasible set consists of 7713 solutions. the number of pareto solutions is 20. by application weighted coefficient method with weighted coefficient: w1=0.5, w2=0.0, w3=0.0, w4=0.5 the solution shown in table 1 is obtained, with a set of objective functions values shown in table 2. table 1 optimal solution obtained by weighted coefficient method variable values x1=z1 x2=z2 x3=z3 x4=nw x5=mn x6=b 23 23 -67 5 5 34 table 2 objective function for solution shown in table 2 f1 [mm3] f2 [kg] f3 f4 [min] 3218716.637 18.476 0.986 257.25 4.2 the second stage of compound gear train (ii) when the brake br2 is activated and the brake br1 inactivated, sun gear of the first stage and sun gear of the second stage rotate, but the carrier of the second stage is reactive and power transmitted through the sun gear of second stage and ring gear of second stage to the output b. the second stage is determined in this mode. if the carrier is immovable, the required transmission ratio is equal to the basic transmission ratio (ideal torque ratio). taking into account the dependence between the basic transmission ratio and the transmission ratio with the immovable transmission ratio, the next input data is chosen for the multi-criteria optimization application in this stage: i0=-4.0, nin=1800 min -1 , tin=3119.34 nm, l=8000 h, ka=1.25, it7 for all gears, material z1/material z2/material z3=20mocr4/20mocr4/34crnimo6, shmin=1.1, sfmin=1.2, δi=3%, z1=15÷30. the feasible set consists of 10466 solutions. the number of pareto solutions is 73. by application weighted coefficient method with weighted coefficient: w1=0.5, w2=0.0, w3=0.0, w4=0.5 the solution shown in table 3 is obtained, with a set of objective functions values shown in table 4. table 3 optimal solution obtained by weighted coefficient method variable values x1=z1 x2=z2 x3=z3 x4=nw x5=mn x6=b 20 29 -79 3 5.5 44 table 4 objective function for solution shown in table 4 f1 [mm3] f2 [kg] f3 f4 [min] 6476760.79 31.04 0.986 274.597 it can be concluded that the pareto optimality concept as the criterion for selecting equally good solution makes sense to apply to compound pgt according to these criteria. an application of multicriteria optimization to the two-carrier two-speed planetary gear train 95 5. conclusions in this paper, an original model for multi-criteria optimization of two-carrier two-speed planetary gear trains with brakes on single shafts has been presented. these compound gear trains consist of two basic type of planetary gear trains and have considerable application at the systems which need transmission ratio changes under load. the optimal solutions of chosen variant of these gear trains are obtained for the both transmission ratios. the weight coefficient method is used for choosing the optimal solution from the pareto optimal set. this approach is original in the planetary gear train optimization and can be successfully used for the basic planetary gear train type and compound gear trains consist of basic type, as is shown in this application. the results obtained in this way are in accordance with the literature on technical system optimization and indicate a good choice of the applied methods. furthermore, this approach indicates a possibility for application to other planetary gear train types. references 1. troha, s., 2011, analysis of a planetary change gear train’s variants, (in croatian), phd thesis, university of rijeka, engineering faculty, rijeka, croatia., 395 p. 2. kudrjavtsev, v. n., kirdyiashev, i. n., 1977, planetary gears, (in russian), handbook, mashinostroenie, leningrad., 535 p. 3. arnaudow, k., karaivanov, d., 2005, systematik, eigenschaften und möglichkeiten von zusammengesetzten mehrsteg-planetengetrieben, antriebstechnik, 5, pp. 58-65. 4. ivanov, a., n., 1990, evaluation of diametric dimensions of planetary gearboxes in the design phase, (in russian),vestnik mashinostroenie, 7, pp. 16-19. 5. lechner, g., naunheimer, h.,1999, automotive transmissions, springer-verlag, heidelberg, 438 p. 6. jelaska, d., 2012, gears and gear drives, university of split, croatia, 444 p. 7. tudose, l., buiga, o., jucan, d., stefanache, c., 2008, multi-objective optimization in helical gears design, proc. the fifth international symposium about design in mechanical engineering kod 2008, pp. 77-84, novi sad, serbia. 8. tkachev, a., goldfarb, v., 2009, the concept of optimal design for spur and helical gears, proc. the 3rd international conference power transmissions 2009, pp.59-62, kallithea, greece. 9. stefanović-marinović, j., petković, m., stanimirović i., milovanĉević, m., 2011, a model of planetary gear multicriteria optimization, transactions of famena, 35(4), pp. 21-34. 10. sriatih, a., yedukondalu g., jagadeesh, a., 2011, mechanical efficiency of planetary gear trains: an estimate, mechanical engineering research, 1(1), pp. 97-102. 11. del castillo, j.m., 2002, the analytical expression of the efficiency of planetary gear trains, mechanism and machine theory, 37, pp. 197-214. 12. stefanović-marinović, j., 2008, multicriterion optimization of planetary power transmission gear pairs, (in serbian), phd thesis, university of niš, faculty of mechanical engineering, niš, serbia, 291 p. 13. niemann g., winter h., 1989, maschinenelemente, band ii, springer-verlag berlin, 376 p. 14. international organization for standardization, 2006, international standard iso 6336-2, calculation of load capacity of spur and helical gears. 15. troha, s., lovrin , n., milovanĉević, m., 2012, selection of the two–carrier shifting planetary gear train controlled by clutches and brakes, transactions of famena, 36(3), pp. 1-12. 16. troha, s., žigulić, r., karaivanov, d., 2014, kinematic operating modes of two-speed two-carrier planetary gear trains with four external shafts, transactions of famena, 38(1), pp. 63-76. 17. troha s., petrov p., karaivanov, d., 2009, regarding the optimization of coupled two carrier planetary gears with two coupled and four external shafts, machinebuilding and electrical engineering, 1, pp. 49–56. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 141 148 https://doi.org/10.22190/fume190131019k © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper wedging of frictional elastic systems sangkyu kim 1 , yong hoon jang 1 , james r. barber 2 1 school of mechanical engineering, yonsei university, korea. 2 department of mechanical engineering, university of michigan, u.s.a. abstract. we consider discrete two-dimensional elastic systems with coulomb friction contacts, and investigate the conditions that must be satisfied if these are to be capable of becoming ‘wedged’ --i.e. of remaining with non-zero elastic deformations when all external loads have been removed. the condition for wedging is reduced to the requirement that a prescribed set of constraint vectors should fail to positively span the n-dimensional vector space of nodal displacements. we also show that the range of admissible wedged states increases monotonically with the coefficient of friction f and that there exists a unique critical coefficient fw such that wedging is impossible for f < fw and possible for f > fw. key words: wedging, coulomb friction, positive span, contact mechanics 1. introduction if a system of contacting elastic bodies with frictional interfaces is subjected to timevarying loads, it can become wedged, meaning that it remains in a state of deformation with non-zero contact forces and tangential displacements, even when all external loads have been removed [1]. this concept is illustrated by the simple system of fig. 1, comprising two rigid blocks, the upper block being supported by a spring of stiffness k. if the coefficient of friction between the blocks and/or between the lower block and the supporting plane surface is sufficiently high, the system will remain in the loaded configuration even when force f is removed. received january 31, 2019 / accepted may 14, 2019 corresponding author: yong hoon jang school of mechanical engineering, yonsei university, 50 yonsei-ro, seodaemun-gu, seoul, 120-749, republic of korea. e-mail: jyh@yonsei.ac.kr 142 s. kim, y-h. jang, j.r. barber fig. 1 a simple system susceptible to wedging wedging is important in many practical applications. for example, it may lead to incorrect configurations of assembled components in automated processes, such as pinin-hole assembly [2,3]. but also, systems such as screwed fasteners [4] and hip replacements [5] depend crucially on the occurrence of wedging for their very functionality. clearly we would like to be able to predict the frictional conditions under which wedging is possible for a given system. if coulomb friction is assumed with coefficient of friction f, the simple system of fig. 1 leads us to expect that there should exist a critical value fw, such that wedging is possible for f > fw and is not possible for f < fw. one strategy for finding fw is to postulate incipient slip throughout the contact area and explore the conditions under which a nontrivial solution exists with zero external loads. if such a solution can be found for a particular coefficient of friction f0 , and if the corresponding normal contact tractions are everywhere compressive, the same deformation pattern would define a wedged state for f > f0 and hence f0 would define an upper bound for fw. this formulation leads to an eigenvalue problem for f0 which was explored in the discrete formulation by hassani and hild [6, 7] and in the continuum formulation by hild [8], principally from the perspective of the implied non-uniqueness of the quasi-static solution in frictional contact problems [9, 10]. a modified algorithm, tailored specifically to the wedging problem, was proposed by barber and hild [11], who used a finite element model to incrementally reduce the coefficient of friction from a wedged configuration in an attempt to discover the value of f at which the system finally relaxes back to the undeformed state. in this paper, we shall explore the conditions for wedging in the restricted class of discrete two-dimensional problems with coulomb friction that are `stationary' in the terminology of dundurs and stippes [12]. in other words, all contact nodes in the undeformed state remain in contact during deformation, and no additional contact nodes are established. 2. problem description we first discretize the elastic contacting bodies [e.g. by the finite-element method] and use static reduction [13] to obtain the contact stiffness matrix        cb ba k t , (1) wedging of frictional elastic systems 143 such that the normal and tangential nodal forces pi, qi and the corresponding relative nodal displacements wi, vi are related by an equation of the form ( ) ( ) w w t t                       tq vq a b p wp b c . (2) we adopt the convention that the normal forces pi are positive when compressive, and the normal displacements wi are positive when the nodes separate. in eq. (2), pi w (t), qi w (t) are the nodal forces that would be produced by the external loads if the nodes were all welded in contact at v = w = 0. the matrix b is a measure of the coupling between the normal and tangential contact problems [14] and plays a crucial role in the historydependence of the frictional evolution problem. since the system is two-dimensional, each contact node i must, at any given time t, be in one of the four states ,separation 0 0 0 slip backward 0 0 slip forward 0 0 stick 0 0     iii iiii iiii iiii wpq vwfpq vw-fpq vwfpq    (3) where the dot denotes a time derivative. 2.1. the eigenvalue problem for a wedged state, there are no external loads and we are restricting attention to problems where all the nodes remain in contact (w = 0), so eq. (2) reduces to bvpavq  ; . (4) for hild’s eigenvalue problem [7], each node must be in a state of stick, but with incipient slip |qi|=fpi. clearly the direction of slip may be different at each node, but we can accommodate this by introducing a diagonal matrix λ such that λi = sgn(qi). in other words, λi =1 for incipient backward slip and for incipient forward slip. we then have bvavpq λλ ff  or , (5) using eq. (4). for any given λ, this defines a generalized linear eigenvalue problem for f, so the possibility of wedging can be explored by (i) solving the eigenvalue problem for all possible values of λ [i.e. all possible diagonal matrices whose non-zero elements are all either +1 or −1] and then (ii) checking the resulting eigenfunctions to determine which permit a solution in which the normal nodal forces are all non-tensile. 2.2. the p-matrix condition the perceptive reader will recognize a relation between the eigenvalue equation (5) and klarbring’s ‘p-matrix’ condition [15] for the frictional ‘rate’ problem [the statement of the coulomb friction evolution problem in terms of time derivatives] to be well posed. for two-dimensional problems, klarbring’s condition is satisfied if and only if all matrices of the form a + fλb are p-matrices [16] — i.e. they have positive determinants 144 s. kim, y-h. jang, j.r. barber as have all their principal minors. the matrix a is a stiffness matrix and hence is also a pmatrix, so klarbring’s condition is always satisfied in the absence of friction [f = 0]. also, a + fλb varies continuously as f is increased, so if the condition is to be violated, it must correspond to a condition where this matrix or one of its principal minors has a determinant equal to zero. cases where the full matrix is singular correspond to eigenvalues of (5), but there appears to be no similar link between principal minors of the matrix and hild’s eigenvalue problem. 3. the displacement vector space v ahn et al. [17] introduced the idea of tracking the evolution of a frictional state in the n-dimensional vector space v = {v1, v2, …, vn} and showed that the instantaneous state is represented by a point in this space that is ‘pushed’ by the frictional constraints during periods of slip. the inequalities qi ≤ fpi and -qi ≤ fpi governing stick at node i take the form . tqtfpvfba tqtfpvfba w i n j w ijijij w i n j w ijijij )()()( and)()()( 11    (6) for the simple 2-node case, the instantaneous state is defined by the point p(v1, v2) and each inequality excludes the shaded region on one side of a straight line as shown in fig. 2. here, the lines i, ii govern backward and forward slip respectively at node 1 and lines iii, iv govern slip at node 2. if the external loads pi w (t), qi w (t) change, the constraint lines move whilst retaining the same slope. fig. 2 illustrates the resulting motion of p if constraint iv first advances to the dotted line and then recedes, after which i advances. fig. 2 evolution of the frictional state for a two node system, as governed by the four constraints (inequalities) (6), here labeled i,ii,iii,iv, after ahn et al. [17] 3.1. constraint vectors when there are no external loads, the inequalities (6) take the form vfba vfba n j jijij n j jijij    11 0)( and0)( (7) wedging of frictional elastic systems 145 using eq. (4). it is convenient to write these in the compact form k ,   0vc (8) where ck, k  (1, 2n) comprise a set of unit constraint vectors defined by 2 1 2 ( ) ( ) ; | ( ) | | ( ) | t t i i iit t i i f f f f      a b e a b e a b e a b e c c , (9) and ei is the unit vector in direction vi. each of these constraints excludes the region on one side of a hyperplane through the origin, and the corresponding constraint vector is defined so as to point perpendicularly into the excluded region. 3.2. a necessary and sufficient condition for wedging for the purpose of this section, a wedged state is defined as one in which all nodes i  (1, n) are in contact [wi =0], and satisfy the unilateral inequality pi ≥ 0. theorem 1. a wedged state is possible if and only if there exists a non-null displacement vector v that satisfies all of the inequalities (7). in other words, a necessary and sufficient condition for wedging is that there should exist at least one non-null n-vector u such that ck · u ≤ 0 for all k  (1, n). if this condition is satisfied by a given vector u, it is clear that it will also be satisfied by λu, where λ is any positive scalar multiplier. thus, the admissible wedging space, if it exists, will comprise a cone with vertex at the origin ofv . this case is illustrated for the 2-node case in fig. 3(a), where wedging is possible for vectors v in the unshaded region between constraints ii and iii. (a) (b) fig. 3 (a) a two-node case where wedging is possible between constraints ii and iii. (b) a case where klarbring’s p-matrix condition is not satisfied, but where wedging is not possible the arrows on the constraint boundaries in fig. 3(a) indicate the direction of slip implied if the constraint were to move so as to exclude more space due to the imposition of external loads. in particular, if constraint ii were to move so as to reduce the extent of the unshaded region, the point p(v1,v2) would eventually become ‘trapped’ between ii and iii with no admissible slip motion being possible. this represents a special case 146 s. kim, y-h. jang, j.r. barber where klarbring’s p-matrix criterion is violated and hence the rate problem is not wellposed. more general investigation of the two-node system shows that any set of constraints allowing a wedged region leads to a situation where the p-matrix criterion is violated. however, the converse is not always true. fig. 3(b) shows a case where p can become trapped at a between ii and iii, but where the slopes of the remaining constraints iii, iv are such as to preclude wedging. we conclude that for the two-node system, violation of klarbring’s condition is a necessary but not sufficient condition for wedging. however, we are not aware of a proof of this result for the more general nnode case. 3.3. positive span theorem 1 implies that wedging is impossible if and only if for every non-null nvector v, at least one of the 2n constraint inequalities (8) is violated — i.e. every point in v is excluded by at least one of the constraints. an alternative statement of this is condition is that the set of 2n vectors ck positively spans the n-dimensional vector space v . algorithms for testing whether a given set of vectors spans a vector space are discussed by regis [18]. if the coefficient of friction f = 0 , the constraint vectors (9) for node i reduce to 2 1 2; | | | | t t i i iit t i i    a e a e a e a e c c , (10) and hence are equal and opposite. the same result is obtained for all values of f, for any node i for which i t 0eb . (11) the pair of constraints (10) spans all vectors in v except those in the common orthogonal hyperplane. thus, if (10) is satisfied at a subset of m nodes, the admissible region is reduced to the intersection of m such hyperplanes, which comprises a vector space v* of dimension (n-m). in particular, if m=n, this reduces to a single point [the origin] and wedging is impossible. this arises (i) if f = 0, or (ii) if (11) is satisfied for all nodes i  (1, n), in which case b=0 and the system is ‘uncoupled’ [14]. at the other extreme, as f → ∞, if eq. (11) is not satisfied, we obtain |||| i t i t i i t i t -i eb eb eb eb  212 ; cc , (12) and the two constraints for each node become identical. in this case, the complete set of 2n constraint vectors comprises only n independent vectors, but a minimum of n+1 independent vectors is needed to span an n-dimensional vector space [18, 19]. more generally, if (11) is satisfied at m (< n) nodes, there will be (n-m) constraints of the form (12), but these are insufficient to span the reduced vector space v* of dimension (n-m) . we conclude that if b ≠ 0, the system must be capable of wedging at sufficiently large f. as f is increased from f = 0 + , the admissible region due to the pair of vectors c2i-1 , c2i increases monotonically from half of the hyperplane orthogonal to a t ei to the half-space wedging of frictional elastic systems 147 bounded by the hyperplane b t ei. it follows that the admissible region due to the entire set of constraint vectors also increases monotonically [once it is non-null], so in general there must exist a unique critical coefficient of friction fw such that wedging is possible for f > fw and impossible for f < fw . 3.4. tensile nodal forces and separation solutions of the system of equations (2) are physically meaningful only if all the normal nodal forces pi are non-negative. however, if any pi < 0 it follows that |qi| > fpi, so at least one of the two constraints c2i-1·v ≤ 0 and c2i·v ≤ 0 must be violated. thus if a wedged state v for the system is identified by this criterion, it is not necessary to check the signs of the normal tractions, since these will necessarily be all positive. 4. conclusions the principal conclusion of the paper is that wedging for a discrete two-dimensional system with coulomb friction is possible if and only if the set of 2n constraint vectors ck defined by eq. (9) fails to positively span the nodal displacement vector space v . we also show that any states satisfying this condition automatically satisfy the condition that all nodes remain in contact with non-tensile nodal forces, and that for any given system, there exists a unique fw such that wedging is possible for f > fw and not for f < fw . acknowledgements: y. h. jang and s. kim are pleased to acknowledge support from the national research foundation of korea (nrf) funded by the korea government (msip) (grant no. 2018r1a2b6008891). we are also grateful to the reviewers for identifying significant mathematical errors in an earlier version of the paper. references 1. zhechev, m.m., khramova, m.v., 2008, geometrical conditions for wedging in mechanical systems with coulomb friction, journal of mechanical engineering science, 223, pp. 1171–1179. 2. mosemann, m., wahl, f.m., 2001, automatic decomposition of planned assembly sequences into skill primitives, ieee transactions on robotics and automation, 17(5), pp. 709–718. 3. sturges, r.h., laowattana, s., 1996, virtual wedging in three-dimensional peg insertion tasks, journal of mechanical design, 118(1), pp. 99–105. 4. bickford, j.h., 2007, introduction to the design and behavior of bolted joints: non-gasketed joints, crc press, 4. ed. boca raton. 5. falkenberg, a., drummen, p., morlock, m.m., huber, g., 2019, determination of local micromotion at the stem-neck taper junction of a bi-modular total hip prosthesis design, medical engineering and physics, 65, pp. 31–38. 6. hassani, r., hild, p., ionescu, i.r., sakki, n-d., 2003, a mixed finite element method and solution multiplicity for coulomb frictional contact, computer methods in applied mechanics and engineering, 192, pp. 4517–4531. 7. hild, p., 2002, on finite element uniqueness studies for coulomb’s frictional contact model, international journal of applied mathematics and computer science, 12, pp. 41–50. 8. hild, p., 2004, non-unique slipping in the coulomb friction model in two-dimensional linear elasticity, quarterly journal of mechanics and applied mathematics, 57, pp. 235–245. 9. eck, c., jarušek, j., 1998, existence results for the static contact problem with coulomb friction, mathematical models and methods in applied sciences, 8(3), pp. 445–468. 148 s. kim, y-h. jang, j.r. barber 10. haslinger, j., nedlec, j.c., 1983, approximation of the signorini problem with friction, obeying the coulomb law, mathematical methods in the applied sciences, 5(1), pp. 422–437. 11. barber, j.r., hild, p., 2006, on wedged configurations with coulomb friction, in: wriggers p., nackenhorst u., (eds.), analysis and simulation of contact problems, springer-verlag, berlin, pp. 205–213. 12. dundurs, j., stippes, m., 1970, role of elastic constants in certain contact problems, asme journal of applied mechanics, 37(4), pp. 965–970. 13. thaitirarot, a., flicek, r.c., hills, d.a., barber, j.r., 2014, the use of static reduction in the finite element solution of two-dimensional frictional contact problems, journal of mechanical engineering science, 228, pp. 1474–1487. 14. flicek, r.c., brake, m.r.w., hills, d.a., 2017, predicting a contact’s sensitivity to initial conditions using metrics of frictional coupling, tribology international, 108, pp. 95-110. 15. klarbring, a., 1999, contact, friction, discrete mechanical structures and discrete frictional systems and mathematical programming, in: wriggers p., panagiotopoulos p. (eds.) new developments in contact problems, springer, wien, pp. 55–100. 16. andersson, l-e., barber, j.r., ahn y-j, 2013, attractors in frictional systems subjected to periodic loads, siam journal of applied mathematics, 73, pp.1097–1116. 17. ahn, y-j., bertocchi, e., barber, j.r., 2008, shakedown of coupled two-dimensional discrete frictional systems, journal of the mechanics and physics of solids, 56(12), pp. 3433–3440. 18. regis, r.g., 2016, on the properties of positive spanning sets and positive bases, optimization and engineering, 17(1), pp. 229–262. 19. davis, c., 1954, theory of positive linear dependence, american journal of mathematics, 76, pp. 733–746. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 243 254 https://doi.org/10.22190/fume190403028r © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper microstructure-based simulations of quasistatic deformation using an explicit dynamic approach varvara romanova 1 , ruslan balokhonov 1 , evgeniya emelianova 1 , olga zinovieva 2 , aleksandr zinoviev 2 1 institute of strength physics and materials science, sb ras, tomsk, russia 2 airbus endowed chair for integrative simulation and engineering of materials and processes, university of bremen, bremen, germany abstract. microstructure-based simulations of the deformation processes require substantial computational resources due to the necessity of using detailed meshes with a large number of elements. an approach that considerably reduces the computational costs implies simulation of quasistatic deformation within a dynamic approach involving a solution of the motion equations rather than the equilibrium equations. it enables a transition from implicit to explicit time integration providing a significant gain in the computational capacity. in this paper, we show that the explicit dynamic approach can be successfully used in the microstructure-based simulations of quasistatic deformation, considerably reducing the computational costs without losing the information and solution accuracy. the following conditions have to be met to ensure a close agreement between the dynamic and static solutions: (i) the load velocity in the dynamic calculations must be smoothly increased to its amplitude value and then kept constant to minimize the acceleration term appearing in the equation of motion and (ii) the constitutive model employed must describe a quasi-rateindependent response. an examination of the mesh convergence and the strain-rate dependence for a polycrystalline aluminum model has supported this conclusion. key words: microstructure-based simulations, quasistatic deformation, explicit dynamic approach, crystal plasticity received april 03, 2019 / accepted june 28, 2019 corresponding author: varvara romanova institute of strength physics and materials science sb ras akademicheskii prospect 2/4, 634055 tomsk, russia e-mail: varvara@ispms.tsc.ru 244 v. romanova, r. balokhonov, e. emelianova, o. zinovieva, a. zinoviev 1. introduction the microscale deformation mechanisms are known to be strongly affected by the material microstructure. the knowledge of deformation and fracture mechanisms operating in loaded materials at different scales is of critical importance since gradual accumulation of irreversible microdeformation and damage is commonly followed by macroscopic failure of the engineering structure. along with the experimental methods, numerical simulations explicitly taking the material microstructure into account appear to be useful tools for studying the multiscale deformation processes. while considerable progress in this field has been made in the recent few decades, the microstructure-based numerical analysis in a 3d case remains to be a challenge for the researchers due to mathematical complexity of the 3d problem, difficulties in its numerical implementation and high computational demands on numerical solving the boundary-value problem (bvp). on the one hand, the microstructure model has to contain a sufficient number of structural elements for the microand mesoscale processes to be simulated as realistically as possible; the microstructure constituents and interface regions have to be approximated in sufficient detail to ensure a reasonable accuracy of the solution. this necessitates the use of detailed meshes with a large number of elements. in some cases, the high-resolution meshes require the memory and computational time so large that the microstructure-based numerical analyses become impractical. it is, therefore, challenging to reduce the computational costs without losing the information and solution accuracy. an approach that considerably reduces the memory, disk space and computational time requirements implies simulation of quasistatic deformation processes in a dynamic formulation where the equations of motion are solved instead of the equations of equilibrium [1-3]. this enables a transition from implicit to explicit time integration, which provides significant advantages from the viewpoint of computational capacity. the benefit of the dynamic calculations becomes much more significant for any kind of nonlinearity, e.g., nonlinear constitutive behavior, microstructural inhomogeneity, nonlinear loading history, and the like. among the numerical problems, namely, those where the explicit dynamic approach has a distinct advantage over the static one, there are contact problems in which it is quite difficult to make the implicit solution converge [4]. the drawback of the dynamic simulations is the conditional stability of the numerical scheme, which places strong restrictions on the time step value so that too many computational steps would be necessary to achieve a reasonable degree of straining at quasistatic loading rates. to overcome this trouble the loading is artificially sped up, which under certain conditions would result in wave dynamics untypical for quasistatic deformation. moreover, the material free surface and interfaces of different kinds (e.g., grain boundaries, matrix–particle or substrate–coating interfaces, etc.) are the sources of wave reflection, refraction and dissipation and as such they would affect the numerical solution to a greater or lesser extent. for this reason, the dynamic approach has limited applications in the microstructure-based simulations, where the material interfaces are treated explicitly. in this paper we discuss the dynamic approach applicability in microstructure-based simulations of quasistatic deformation phenomena. the conditions under which the static and dynamic solutions converge to a high degree of accuracy are analyzed for an aluminum polycrystal as an example. microstructure-based simulations of quasistatic deformation using an explicit dynamic approach 245 2. boundary-value problems in statics and dynamics the mechanical boundary-value problems (bvps) discussed in this section are formulated in a rectangular cartesian coordinate system in the absence of body forces. in the dynamic formulation the elastic-plastic bvp includes the equations of motion ,i ij ju  , (1) the strain rate-displacement relations , , 1 ( ) 2 ij i j j iu u   (2) and the constitutive equations in the rate form of the generalized hooke’s law ( )e p ij ijkl kl ijkl kl klc c      . (3) here ρ is the current density, ui are the components of the displacement vector, σij are the stress tensor components, cijkl is the fourth-order tensor of elastic moduli, εij, ε e ij and ε p ij are the components of the total, elastic and plastic strain tensors; the upper dot denotes the time derivative. the kinematic boundary conditions take the form i is u   , (4) and the traction boundary conditions are ij j is n t    , (5) where υi are the velocity vector components prescribed on the sυ surface, and ti and ni are the force and normal vector components on the sσ surface. the governing equations of a static problem are the equations of equilibrium , 0ij j  (6) complemented by the strain-displacement relations , , 1 ( ) 2 ij i j j iu u   . (7) and the constitutive equations (3). the traction boundary conditions are given by eq. (5) while the kinematic boundary conditions are given by surface displacements ui prescribed on the su surface u i is u u . (8) among the numerical methods for solving partial differential equations (pdes) the finite-element (fe) method is considered to be the most universal and widely used one. it implies a transition from the strong formulation of the bvps to a weak integral form using, e.g., the virtual work principle. the integral equations are approximated by a system of algebraic equations determined on a finite-element mesh. the finite-element 246 v. romanova, r. balokhonov, e. emelianova, o. zinovieva, a. zinoviev method is discussed in detail in a large number of papers (e.g., [5]). let us briefly address the main features of the fe implementation using explicit and implicit approaches. both in dynamic and static formulations the deformation process is simulated in a stepwise manner: the load applied to the boundaries is incremented and the stress, strain and displacement fields are updated at the end of each time increment to achieve a new state of the static or dynamic balance. the time integration methods in current use are based on implicit or explicit approaches. with an implicit fe solver the unknown quantities to be calculated at each time increment are expressed through the parameters that are also unknown at the beginning of this increment. therefore, iteration algorithms are necessary to achieve a numerical solution. the static equilibrium eq. (6) can be solved by implicit numerical methods alone. the general form of the global fe equations of the static bvp is    [ ] 0 k u f , (9) where {f} and {u} are the global vectors of the nodal forces and displacements, and [k] is the global stiffness matrix relating the forces and displacement vectors. in the implicit computational procedure the stiffness matrix has to be inverted, which requires substantial computational resources. while the implicit solution is assumed to be attained at relatively large time increments, the iteration procedure and calculations of the inverse stiffness matrix make the computations rather expensive in terms of memory, disk space, and computational time. any kind of nonlinearity (e.g., nonlinear constitutive behavior, irregularly-shaped interfaces, complex geometry of the computational domain, nonlinear loading path, etc.) additionally increases the number of iterations necessary to distinguish the features of nonlinear phenomena to a proper accuracy. in contrast to the static equilibrium problem, hyperbolic equations of motion can be solved using an explicit scheme. the unknown quantities appearing in the pdes are expressed through the parameters known from the previous time step. the components of acceleration and velocity vectors at the (n+1)-th time increment are expressed through the values known from the n-th and (n+1/2)-th time steps 1 1 1/2n n n n i i iu u t u    , (10) 1 1/2 1/2 2 n n n n n i i i t t u u u        . (11) the accelerations are calculated at the beginning of the time increment from the equation of motion written in the matrix form as follows       [ ]  m u f k u , (12) where [m] is the lumped mass matrix. the explicit solution requires neither iterations nor inversion of the stiffness matrix, which substantially reduces the computational costs for each time increment. the drawback of the explicit schemes is their conditional stability. the stability condition is associated with the velocity of the fastest process described by the pdes. for the mechanical problem the time step has to be proportional to the smallest element size and inversely proportional to the velocity of mechanical wave propagation in the material (i.e., the sound velocity) with the coefficient <1. the stability condition microstructure-based simulations of quasistatic deformation using an explicit dynamic approach 247 ensures that the stress wave across each time increment does not cover the distance exceeding the smallest mesh step. while each increment of the explicit time integration is much less computationally consuming than that of the implicit calculations, the time step providing the stability of the explicit scheme is too small for the simulations of long-time processes to be practical. particularly, too many computational steps would be necessary to achieve a reasonable degree of straining at quasistatic loading rates. to overcome these troubles, the load velocities in the explicit simulations of quasistatic processes are artificially increased. another method for reducing the computational time in the dynamic calculations involves scaling the material density to speed up the stress wave propagation. 3. microstructure-based simulations 3.1 generation of polycrystalline geometrical models the construction of a geometrical microstructure model, which is the starting point in the microstructure-based numerical analysis, is a sophisticated problem in a 3d case. a rigorous approach is based on processing a series of experimental microstructural images obtained by means of specimen sectioning, x-ray tomography and other time and money consuming techniques. alternative methods rely on the computer-aided design of synthetic models with microstructural features similar to those of real materials. earlier [6] we have proposed a semi-analytical method of step-by-step packing (ssp), which enables generating 3d microstructures with a wide variety of geometrical features. in the general case, the spp-procedure includes the following steps. a 3d volume is discretized with a regular or irregular mesh. certain mesh elements are selected to be seeds of the microstructure elements. each kind of seeds is associated with a certain analytical function according to which the volumes surrounding the seeds are grown in a stepwise manner. the mesh elements, whose coordinates fall within any of the incremented seed volumes, are added to this microstructure phase. the ssp-procedure is repeated until the growing phases reach the preset volume content. the main parameters controlling the resulting microstructure geometry are the number of seeds, their types and spatial distributions, and the laws of their growth. the equations of ellipsoids, spheres, cylinders and the like are the basic analytical functions enabling us to construct microstructures typical of many materials. in this paper the ssp-method has been employed to generate three-dimensional polycrystalline models on regular meshes with different resolutions. in order to obtain the same polycrystalline structure on different meshes a set of grain seeds was once randomly selected and then applied in all ssp-simulations. all grains were grown by the equation of a sphere at the same growth rate. the ssp generation was terminated when the entire computational domain was packed with grains. this algorithm provides polycrystalline aggregates with convex polyhedral grains characterized by plane faces and straight-line edges much similar to those constructed analytically by a voronoi tessellation. an advantage of the ssp-models generated on a mesh by default is that they can be directly imported into the finite-element or finite-difference computations. 248 v. romanova, r. balokhonov, e. emelianova, o. zinovieva, a. zinoviev two polycrystalline models generated on the coarsest and finest meshes are shown in fig. 1 together with a selected grain presented for 100×20×100, 160×32×160, 200×40×200, 250×50×250, 320×64×320 and 400×80×400 meshes. the mesh resolution was found to have but a minor effect on the generated polycrystalline structures. while finer meshes describe smoother grain interfaces, the shape, size and spatial arrangement of the grains are the same as those for coarser meshes. a) b) fig. 1 polycrystalline models generated on 100×20×100 (a) and 400×80×400 meshes (b) and a selected grain meshed with a step of 20, 12.5, 10, 6.25 and 5 µm (left-to-right) 3.2 constitutive behavior of aluminum grains the constitutive response of aluminum grains is described in the framework of the crystal plasticity theory [5]. a polycrystal is treated as an aggregate of single crystals of varying crystallographic orientations with respect to the specimen coordinate system (xyz-axes in fig. 1). the generalized hooke’s law (3) is written with regard to a local coordinate system associated with the crystal axes. the plastic strains appearing in eq. (3) are thought to result from dislocation gliding in the active slip systems ( ) ( )p ij ij       , with ( ) ( )1 ( ) 2 ij i j j is m s m    (13) where si (α) and mi (α) are the components of slip direction and slip plane normal vectors for a slip system (ss) α. equations (13) provide a relation between specimen strains and dislocation slip in the directions prescribed by the orientation tensor θij. the shear strain rate ( ) in eq. (13) is the unknown quantity and has to be determined by a constitutive dependence on the resolved shear stress τ (α) =σijθij (α) acting on the slip system α. rateindependent models provide an ambiguous definition for a set of active slip systems [7]. in order to overcome this ambiguity, the rate-dependent models are commonly used in crystal plasticity simulations microstructure-based simulations of quasistatic deformation using an explicit dynamic approach 249 ( ) ( ) * ( ) ( ) sgn crss            , (14) where * and ν are the parameters controlling the strain rate sensitivity. of primary importance is the description of critical resolved shear stress (crss) τ (α) crss with proper account of the strengthening mechanisms for particular materials. in this paper, a simple phenomenological equation for τ (α) crss is used to describe the grain boundary strengthening and strain hardening ( ) ( ) 0 ( )poly p crss eqf       , (15) where τ0 (α) is the crss of a single crystal, τ poly takes into account the crss value increasing due to the presence of grain boundaries. the third term of the sum describes the strain hardening as a function of the accumulated equivalent plastic strain ε p eq. in what follows, calculations are presented for an aluminum alloy which is characterized by face-centered cubic (fcc) crystalline structure. twelve <111>{110} sss are potentially active in fcc metals, all of them are activated at the same value of τ (α) crss. the strain hardening function for the aluminum alloy is determined as 2 2/ / 1 1( ) (1 ) (1 ) p p eq eqa bp eqf a e b e          , (16) where a1, a2, b1 and b2 are chosen to fit the experimental data [8]. the model parameters and material constants used in the simulations are c1111=108 gpa, c1122=61 gpa, c2323=28 gpa, τ0 (α) =2 mpa, τ poly =18 mpa, a1=73 mpa, a2=0.07, b1=16 mpa, b2=0.002. 3.3 numerical implementation the polycrystalline constitutive models were imported into the finite-element software package abaqus/standard and abaqus/explicit through umat and vumat user subroutines, respectively. the crystal plasticity fe-implementation in abaqus/standard is reported in [5]. let us consider briefly the explicit numerical procedure. the simultaneous solution to eqs. (3), (13) and (14) at each time increment calls for an iterative procedure. we employed the method of simple iterations which provided a fast solution convergence with a reasonable accuracy. the solution is shown to converge for one or two iterations provided that the parameters of eq. (14) are well-defined, with a purely elastic state being chosen as the initial approximation. in the abaqus/explicit solver, the constitutive equations are formulated with respect to local orientations given by θij. the tensors and vectors used in the calculations of constitutive behavior are automatically rotated with respect to the local coordinates before they are imported to vumat. thus, we do not have to reformulate the constitutive eqs. (3), (13)-(17) for each local coordinate system. the boundary conditions, formulated with respect to the specimen coordinate system, simulate uniaxial tension along the x-axis. the specimen top surface in all simulations is free of external forces, while the bottom surface is taken as a symmetry plane. the lateral surfaces parallel to the tensile axis are assumed to be free of external forces. 250 v. romanova, r. balokhonov, e. emelianova, o. zinovieva, a. zinoviev 4. computational results 4.1 mesh convergence the mesh convergence of the numerical solution has been checked for six polycrystalline models of 0.2×0.04×0.2 cm consisting of 1600 grains generated on different meshes. two models generated on the coarsest and finest meshes are shown in fig. 1. the set of grain orientations was identical in all simulations. a) b) c) d) e) f) fig. 2 equivalent stress (a-c) and plastic strain fields (d-f) in the polycrystalline structure meshed with 100×20×100 (a, d), 200×40×200 (b, e) and 400×80×400 resolution (c, f) explicit calculations of uniaxial tension have been performed for the models loaded at the same strain rate of 10 2 s -1 . all components of the stress and plastic strain tensors obtained for different meshes have been compared to analyze the effect of the mesh resolution on the numerical solution accuracy. for the sake of illustration, the equivalent stress and plastic strain fields are presented in fig. 2 for three meshes, with the general conclusion being supported by the whole set of numerical data. the respective stress-strain curves and those of mesh dependence of the maximum stress and strain values are plotted in fig. 3. for all mesh approximations the stress and strain distributions are found to be reproduced with a reasonable accuracy. even the stress and strain fields calculated for the coarsest mesh (fig. 2a, d) qualitatively reproduce the main features which become more detailed on the finer meshes (fig. 2b-c, e-f). for the meshes finer than 160×32×160, qualitative differences between the stress-strain patterns become hardly distinguishable. microstructure-based simulations of quasistatic deformation using an explicit dynamic approach 251 100 200 300 400 340 360 380 400  eq fitting curve m a x . e q u iv a le n t s tr e s s , m p a no. of mesh elements along x-axis 0,08 0,10 0,12 0,14  p eq m a x . e q u iv a le n t p la s tic s tra in 0,00 0,03 0,06 0,09 0 100 200 300 s tr e s s , m p a strain mesh size 100x20x100 160x32x160 200x40x200 320x64x320 400x80x400 × a) b) fig. 3 mesh dependence of the maximum values of equivalent stresses and plastic strains (ε = 0.029) (a) and averaged stress-strain curves for different mesh densities (b) the stress-strain curves for all mesh approximations mostly coincide except the curve for the coarsest mesh which lies somewhat lower (fig. 3b). it is worth noting that the maximum values of stresses and strains also tend to converge upon mesh refinement (fig. 3a), though at a slower rate than the averaged values do. this supports the conclusion made by harewood and mchugh [1] that the rate-dependent models enable eliminating mesh sensitivity of the solution when plastic strain localizes in shear bands. 4.2 explicit dynamic simulations of quasi-static deformation in quasi-static simulations using the dynamic approach it is critical to ensure that the inertia effects are insignificant. apparently, the dynamic and static solutions converge if the inertia term appearing in the left-hand side of eq. (1) vanishes. the acceleration value is non-zero in the cases where the velocity is changed and is neglected when it is kept constant. thus, the load velocity in the initial deformation stage has to be increased smoothly to minimize the acceleration and, thus, to eliminate the dynamic effects involved. we have found that the wave effects become negligible if the time of the velocity increase up to the amplitude value is 3-4 times larger than that necessary for the elastic wave to propagate through the computational domain. taking into account the ratio of the model longest length to the speed of elastic wave propagation in aluminum, the time of load increasing in the explicit simulations was chosen to be 3 µs. the solutions to the implicit static and explicit dynamic problems have been compared for the polycrystalline structure approximated by a mesh consisting of 1,600,000 elements. note that the maximum number of elements, which might be calculated with abaqus/explicit, is an order of magnitude larger than that in the implicit static simulations run in the same computer. the element-by-element comparison between static and dynamic stress and strain fields showed a coincidence to within 0.1% for the most part of the elements, with only few elements belonging to interface regions demonstrating a disagreement of 3-5% (fig. 4a). 252 v. romanova, r. balokhonov, e. emelianova, o. zinovieva, a. zinoviev a) 0,0 0,5 3,5 4,0 0 20 40 60 n o f fi n it e e le m e n ts , % discordance between dynamic and static solutions, % b) 0,00 0,04 0,08 0,12 0 100 200 300 strain s tr e s s , m p a strain rate, 1/s 10 2 10 3 10 4 fig. 4 the element-by-element comparison between plastic strain fields obtained in dynamic and static calculations (a) and the stress-strain curves calculated using an explicit dynamic approach at different load velocities (b) in the dynamic simulations of quasi-static deformation, where load velocities are artificially increased, it is important to use rate-independent constitutive models. thus, the strain-rate sensitivity parameters, appearing in the rate-dependent crystal plasticity eq. (14), have to be chosen in order to eliminate the rate dependence. harewood and mchugh [1] reported that a material becomes quasi-rate-independent for large ν values, which may have an adverse effect on the iteration convergence. in our simulations the ν value was equal to 10, which reduced the strain-rate sensitivity of the average material response for the strain rates up to 10 3 s -1 . the grain scale stress and strain fields, however, demonstrated conspicuous differences. with increasing the load velocity, the regions of plastic strain localization became wider, while the strain values in the shear bands decreased (cf., e.g. fig. 5a and b). this is due to the fact that the local strain rates in the strain localization regions can be several orders of magnitude higher than the strain rate prescribed by the boundary conditions. if so, the plastic strain rates calculated by eqs. (13)-(14) might not be high enough to achieve a static balance between the load and the material response. to overcome this drawback, we suggest substituting constant * in eq. (14) by the relationship * eqk  , (17) where eq is the equivalent total strain rate and k is the constant value <1. in our simulations k was chosen to be 0.8. larger values of k had an adverse effect on iteration convergence, while smaller values led to rate-dependent effects. the calculation results for * given by eq. (17) are plotted in fig. 4b and 5 for the tensile strain rates varied by three orders of magnitude. a close agreement of the averaged stress-strain curves indicates the rate-independence of the material model at the macroscale. the corresponding plastic strain fields compared in fig. 5 at two most different strain rates support this conclusion for the microscale as well. the differences in the plastic strain patterns are almost undistinguishable either qualitatively or quantitatively. microstructure-based simulations of quasistatic deformation using an explicit dynamic approach 253 a) b) fig. 5 equivalent plastic strain fields calculated with abaqus/explicit for the strain rates 10 2 (a) and 10 4 s -1 (b). tensile strain is 8% finally, let us compare the computational costs needed for solving the same 3d quasistatic problem using the abaqus implicit and explicit solvers. the estimations of the element number, which can be accommodated in the memory, were performed only for c3d8r finite elements for the initial static and explicit dynamic steps. the maximum number of elements in the explicit calculations was found to be 10-12 times larger than that in the implicit computations. table 1 normalized computational time in static and dynamic calculations no. of cpus 1 2 3 4 abaqus/implicit runtime 1 0.64 0.57 0.52 abaqus/explicit runtime 0.16 0.08 0.06 0.04 in order to compare the runtime values of the implicit and explicit calculations, quasistatic uniaxial tension up to the same strain value was solved with abaqus/standard and explicit. the tension velocity in the dynamic calculations was chosen to provide a close agreement between the quasistatic and dynamic solutions. the calculation time values normalized to those required for nonparallel static calculations are given in table 1 for different numbers of cpus. the time needed for the explicit calculations is much shorter than that for the implicit ones. the dynamic calculations become even more advantageous in the case of parallel computations. 5. conclusion the numerical solution to the mechanical boundary-value problem with an explicit account of the material microstructure requires substantial computational resources due to a necessity of using detailed meshes with a large number of elements. an approach that considerably reduces the requirements for computer memory, disk space, and computational time implies the solution of quasistatic problems in a dynamic formulation, where the equation of motion is solved instead of static equilibrium equation. this enables a transition 254 v. romanova, r. balokhonov, e. emelianova, o. zinovieva, a. zinoviev from implicit to explicit calculations providing a significant improvement of computational capacity. in this paper we have shown that the explicit dynamic approach can be successfully used in the microstructure-based simulations of quasistatic deformation, substantially reducing the computational costs without losing the information and solution accuracy. the following conditions have to be met to ensure a close agreement between the dynamic and static solutions: (i) the load velocity in the dynamic calculations must be smoothly increased to its amplitude value and then kept constant to minimize the acceleration term in the initial loading stage and (ii) the constitutive model used must be quasi-rate-independent. an examination of the mesh convergence and strain-rate dependence of the polycrystalline aluminum model has supported this conclusion. acknowledgements: this work is supported by the russian academy of sciences within the fundamental research program for 2013–2020. the constitutive model presented in section 3 has been developed within the joint research project of deutsche forschungsgemeinschaft (grant no. pl 584/4-1) and the russian foundation for basic research (grant no. 18-501-12020). references 1. harewood, f.j., mchugh, p.e., 2007, comparison of the implicit and explicit finite element methods using crystal plasticity, 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polycrystals, physical mesomechanics, 12(3-4), pp. 130-140. 7. busso, e.p., cailletaud, g., 2005, on the selection of active slip systems in crystal plasticity, international journal of plasticity, 21(11), pp. 2212-2231. 8. teplyakova, l.a., bespalova, i.v., lychagin, d.v., 2009, spatial organization of deformation in aluminum [1 ī 2] single crystals in compression, physical mesomechanics, 3(12), pp. 166-174. 12969 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 673 688 https://doi.org/10.22190/fume240904048n © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper analysis of nanobeam-based microstructure in n/mems system using van der waals forces muhammad nadeem1, qura-tul ain2, naif almakayeel3, yabin shao4, shuqiang wang5, meshal shutaywi6 1school of mathematics and statistics, qujing normal university, qujing china 2institute of computing science and technology, guangzhou university, guangzhou, china 3department of industrial engineering, college of engineering, king khalid university, abha, saudi arabia 4research institute of microscale optoelectronics, school of jia yang, zhejiang shuren university, shaoxing china 5shenzhen institutes of advanced technology, chinese academy of sciences 6department of mathematics, college of science & arts, king abdul aziz university, rabigh, saudi arabia orcid ids: muhammad nadeem https://orcid.org/0000-0002-9349-4729 qura-tul ain https://orcid.org/0000-0003-4442-4756 naif almakayeel https://orcid.org/0000-0001-9461-5935 yabin shao https://orcid.org/0009-0003-1925-9231 shuqiang wang https://orcid.org/0000-0003-1119-320x meshal shutaywi https://orcid.org/0000-0003-1454-2962 abstract. nano/microelectromechanical system (n/mems) have garnered significant global attention for their transformative applications across various cutting-edge fields, including wearable devices, 5g service, collecting power, and space travel. these systems face challenges when exposed to excessive forces and leading to pull-in instability that can undermine a reliable performance. therefore, it is essential to quickly and precisely comprehend the periodic motion of the system to minimize the possibility of pull-in instability and maintain consistent operation. this article explores the application of the homotopy transformation scheme (hts) for forecasting dynamic behavior of n/mems. the most advantage of hts is its ability to deliver results without relying on hypotheses, assumptions, or constraints on variables. to illustrate its effectiveness, we use a generalized n/mems oscillator system, presenting an accurate analysis of analytic solution. specifically, we focus on a microstructure of nanobeambased system actuated by van der waals (vdw) forces to highlight its periodic characteristics. the results reveal that hts provides a robust comparison and excellent agreement with the exact solution, underscoring its reliability and precision. key words: sumudu transform, n/mems, homotopy perturbation method, approximate solution received: september 04, 2024 / accepted november 17, 2024 corresponding author: yabin shao zhejiang shuren university, shaoxing 312028 china e-mail: shaoyabin@zjsru.edu.cn (shao) mailto:shaoyabin@zjsru.edu.cn 674 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi 1. introduction nano/microelectromechanical systems (n/memss) represents a cutting-edge technology characterized by their exceptional geometrical precision, reliable performance, and high sensitivity. these systems achieved various advanced properties through meticulous control of applied voltage and involved both fixed and dynamic elements; whenever an effort is exerted for activation [1, 2]. the n/mems have a variety of potential areas, consisting of wearable instruments, 5g service, and collecting power [3-6]. the transformative impact of these systems on various technologies has prompted extensive research into their operational properties and potential applications. recent advancements in nanotechnology have brought significant attention to the dynamic issues of nanotubes within nanostructures and systems. size-dependent behavior in clamped-clamped microbeam systems has important mechanical and structural responses due to the change in beam dimensions at the microscale level. small-scale clamped-clamped beams may display nonlinear behavior, particularly at significant deflections or when subjected to electrostatic forces in mems systems. nonlinear components emerge as importance in dealing with problems, influencing frequency and equilibrium, and providing size-dependent nonlinearity [7, 8]. researchers have found that oscillators in n/mems exhibit complex dynamics, such as pull-in instability, phase diagrams, and hysteresis being crucial to their dynamic analysis [9, 10]. the basic mode of oscillation coincides with the main stretching of the beam. the ends are fixed for a doubly clamped beam, thus its initial frequency is higher than that of simply supported or free ends. this fixed structure enhances elasticity, which implies that the beam needs more energy to oscillate, and raises the fundamental frequency. nonlinear oscillatory dynamics are often utilized in energy-harvesting instances, in which the oscillations of the beam transform mechanical power to electrical power. subsequently, damping and harmonic stability are essential for performance. the criteria for the existence of periodic solutions and their associated properties are crucial for the efficiency and aesthetics of nanotube-based n/mems and resonators [11-13]. the casimir effect arises from quantum fluctuations and can be viewed as an attractive force between conducting plates; nevertheless, it comes from the same fundamental principles that govern vdw interactions. the vdw force originates from the relationships across temporary dipoles induced by electron density fluctuations within molecules. this force is applicable to all molecules, irrespective of their features. this indicates that even though the plates are formed of nonpolar objects, the vdw force remains present. this force becomes crucial in influencing the system's behavior when the plates are close together. in recent years, various researchers have particularly focused to overcome the difficulty of obtaining analytical solution for n/mems system. these solutions provide fruitful insights that are vital for comprehensive understanding of the behavior and characteristics of these advanced systems. most of the numerical methods often fail to explicitly capture the nonlinear frequency amplitude relationship. as a result, various analytical methods have been developed in the literature to address n/mems oscillators. these methods include the variational iteration approach [14], energy balance technique [15], iteration perturbation scheme [16], residual harmonic balance strategy [17], parameter expansion approach [18], adomian decomposition technique [19], power series [20], he-transform [21], and frequency formulation [22]. the homotopy perturbation method (hpm) is an extremely successful tool for the resolution of a diverse array of nonlinear problems. this method involves expressing the outcomes as a sequence that swiftly turns to precise results. a standout feature of hpm is analysis of nanobeam-based microstructure in n/mems system using van der waals forces 675 its ability to achieve high accuracy with just a single iteration. this characteristic significantly enhances its appeal and practicality in numerous fields of technology and scientific research. pull-in instability exhibits significant nonlinearity as a result of the inverse-square relationship of electrostatic forces relative to the gap distance. nonlinear behaviors demand iterative and perturbation techniques to identify the crucial points related to instability. consequently, hpm provides a feasible analytical method for deriving approximation solutions [23]. qayyum and oscar [24] proposed the least square homotopy perturbation approach to analyze multiple order problems related to boundary values. anjum and he [25] studied the nonlinear vibration properties of nanotube-based n/mems oscillators using the hpm and the laplace transform. hpm has been widely regarded as an auspicious and innovative approach for addressing nonlinear problems [26-28]. on the other hand, the sumudu transform [29] is a mathematical tool used to simplify the analysis of differential equations. it simplifies the solution of differential problems by converting them into an algebraic formula. it is highly effective in handling with initial value problems and certain types of nonlinear problems [30, 31]. in this work, we design the homotopy transformation scheme (hts) through the association of sumudu transform and the homotopy perturbation method. this technique is straightforward to utilize for nonlinear differential problems. using this scheme, we can obtain the results of the n/mems system in the form of a convergent series. we focus on the solution of nonlinear behavior of dynamic oscillatory for doubly clamped nanobeam that is controlled and governed by vdw forces. we evaluate the outcomes of hts by comparing them to the derived results via some mathematical software by utilizing the rk method of 4th order and other established schemes. 2. basic idea of sumudu transformation let a be the set of function, which is defined as [32]: 1 2( ) : m, , ( ) , ( 1) [0, ).j t r ja t r r t me if t =      −   (1) for every function in the set a, it is necessary for the constant m to be a finite number, whereas the constants r1 and r2 may be either finite or infinite. furthermore, the sumudu transform is defined as an integral equation such that = 0 0 1 ( ) ( ) ( ) ( ) , 0, t tt r t e dt t e dt t       − −    = =   (2) where θ is an independent of t parameter that might be real or complex. the inversion expression for st is as follows   1 1 ( ) . 2 a i t a i d r e r i      +  −    =       (3) the subsequent characteristics are beneficial in the calculations of sumudu space. 676 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi   ( )   ( ) ( ) ( ) ( ) 2 2 1 ! , , 0( ) ( ) , 0 ' 0( ) ( ) , 0 0( ) ( ) ... . n n n n n n t n n r t r t r t                −   =    = −  = − −   = − − −  (4) we introduce these properties during the development of hts so that our proposed scheme can provide accurate results. 3. development of homotopy transformation scheme here, we propose the development of hts for a general problem. we explain the stepby-step formulation of proposed scheme. we show the process of handling the nonlinear terms and derived a series result by using the application of hpm. this obtained series can easily converge to the exact results only after a few steps of series. let us examine the process of hts by starting a nonlinear fractional differential problem such as, 2 2 ( ) ( ), d f r t dt  + = (5) with following constraints (0) , (0) 0.b = = (6) eq. (5) can be modified in the following manner: 2 2 2 ( ) ( ), d g r t dt    + + = (7) where g(ϑ)=f(ϑ)-ω2ϑ. let us define the term g(ϑ) such as 2 2 1 22 ( ) ( ) ( ), d l g l g r t dt     + + + = (8) in which l1 be linear and l2 is a nonlinear operator with a source parameter r(t). operating the linearity of st to eq. (8), we can write it as follows:    2 1 22 2 ( ) (0) (0) ( ) ( ) ( ) ( ) . r r l g l g r t           − − + + + = (9) using condition of eq. (6), we can simplify it to obtain r(θ) such as 2 1 2 2 2 2 2 2 22( .( ) [ ] [( ( )] (1 ) (1 ) ( ) ) 1 )r l g l g b r t           = − + + + + + (10) analysis of nanobeam-based microstructure in n/mems system using van der waals forces 677 now, using the property of inverse st on eq. (10), we get 1 1 2 2 2 1 2 2 2 2 ( ) ( ) ,( ) cos [ ] [ ( )] (1 ) (1 ) t b t l g l g r t           − −  = − +  + +     +   (11) we can also write it as follows 2 12 2 2 1 (( ) ( ) [ ] (1 ) ) ( ) ,l g lt g t g      −   = −   + +  (12) where  1 2 2 2 ( ) (1 .( ) ) cosg rb t tt    −   = +    +  (13) let the general expression of eq. (12) is 0 ( ) ( ).n n n t p t   = =  (14) we introduce the concept of hpm to decompose the terms of nonlinear operator l2 in eq. (12) such as 2 0 ( ) ( ),n n n l g p h   = =  (15) with p∈ [0,1] as a small term of perturbation and hn(ϑ) represents the he's polynomials that are produced by the following formula 0 1 2 0 0 1 ( ) ( , , , ) , 0,1,2, ! n n n n n in i p h h l g p n n p       = =     = = =        (16) on utilizing eqs. (14) and (15) to eq. (12), we can obtain such as 2 1 12 2 0 0 0 ( ) ( ) [ ( ) ] . (1 ) n n n n n n n n n p t g t p l p g p h         − = = =    = − +   +      (17) after some successive repetitions along with the analysis of components p, we construct the following results:       0 0 2 1 1 1 1 0 02 2 2 2 1 2 1 1 12 2 2 1 1 1 12 2 ( ) ( ), ( ) ( ) ( ) , 1 ( ) ( ) ( ) , 1 ( ) ( ) ( ) . 1 n n n n p t g t p t l g h p t l g h p t l g h                    − − − − − = =   = = − +  +    = = − +  +    = = − +  +  (18) 678 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi finally, the above series can be expressed as follows: 0 1 2 0 ( ) lim ( ).i p i t t      → = = + + + =  (19) this approach efficiently utilizes the principle of continuous iteration and mathematical investigation to reveal an extensive solution for the proposed model, and effectively demonstrates the ability of hts procedures to address intricate problems. 4. problem formulation nanobeams serve as the primary structural components in n/mems systems and play a crucial role in their functionality. to effectively analyze the behavior of these beams, various beam theories are employed. these theories not only facilitate the analysis of beam behavior but also enhance the design process that ensures the development of robust and high-performing n/mems system. consider a nanobeam with clamped-clamped boundary conditions as shown in fig. 1, where l is the length between the two clamped ends of the beam, b is the width of the beam in the plane perpendicular to its length, h shows the thickness of the beam in the plane perpendicular to its width and length, and ρ is the mass per unit volume of the material of the beam. fig. 1 schematics of a double-sided driven clamped-clamped microbeam this study delves into the dynamic behavior and properties of such a nanobeam structure that focusing on its transverse vibrations and the resulting mode shapes and natural frequencies. by leveraging the euler-bernoulli beam theory and applying the appropriate boundary conditions, our purpose is to demonstrate a comprehensive analysis of the mechanical response of the nanobeam under various conditions. this investigation is essential for applications where precise control and understanding of nanoscale mechanical elements are required. consider the von-karman nonlinearity for mid-plane stretching, depending on the euler-bernoulli beam idea as follows: 24 2 2 4 2 2 0 ( , ) 0, 2 l q q es q q ei s n d k l              + − + − =          (20) where q is a function of ψ and η that indicates the spatial location and time respectively, e denotes the effective young's modulus of the beam, while i indicate the moment of inertia analysis of nanobeam-based microstructure in n/mems system using van der waals forces 679 about the cross-section of y-axis. additionally, ñ signifies the residual axial load. the function k(ψ, η) describes the vdw forces among nanobeam and the substrate and is stated as follows [33]: 3 ( , ) , 6 ( ) ha b k d q   =  − (21) where ah denotes the hamaker coefficient with a value of 30 × 10-19 j < ah < 50 × 10-19 j, s=bh, whereas d shows the distance among nanobeam and the substrate. we can define the nondimensional parameters as follows: , , , q t l d t    = = = (22) where 4 . hbl t ei  = (23) employing the value of eq. (22) into eq. (20), we can obtain the following form ( ) 214 4 2 4 2 2 3 0 0, 1 n d t                 + − + − =      −     (24) where n is the load of axial, α shows the aspect ratio, whereas λ is the van der waals force such that: 2 42 3 12 , 6 , .hanl d l n ei h deh       = = =        (25) moreover, we have the dimensionless boundary conditions, which are stated as below ( ) ( )0, 1, (0, ) (1, ) 0, 0. t t t t         = = = =   (26) since the singularity in eq. (24) renders it challenging to resolve when δ=1. therefore, we consider the separation of variables approach to minimize eq. (24) to an ordinary differential equation, such that δ(φ,t) be the result of two distinct functions as follows: ( , ) ( ) ( ),t t   =  (27) here ϑ(t) is a time-dependent function and ς(φ) denotes the first mode shape function that fulfills the boundary criteria. putting the value of eq. (27) into eq. (24), and then multiplying the result with ς(φ)(1-δ)3 and taking integration over the dimensionless domain, we derive 1 14 2 3 2 3 4 2 0 0 21 1 12 3 2 0 0 0 (1 ) (1 ) (1 ) 0, d d t n d d d                      −  +  −  −         −  + −  =             (28) 680 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi which may turn to the following form: 2 2 3 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 102 ( ) 0, d q q q q q q q q q q q dt          + + + + + + + + + + = (29) here, the expression of qj(j=0,1,2,3,…,10) is defined in the appendix. the eq. (29) has the following initial conditions such that (0) , (0) 0.tb = = (30) since eq. (29) is extremely nonlinear, thus it was difficult to solve using several wellknown analytical techniques [34-36]. thus, we construct hts to solve eq. (29) with the initial conditions of eq. (30). 5. application of homotopy transformation scheme for n/mems in the following part, we construct the formulation of hts for the nanobeam-based microstructure system actuated by van der waals forces. therefore: 2 2 3 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 102 ( ) 0. d q q q q q q q q q q q dt          + + + + + + + + + + = (31) dividing eq. (31) by q0, we get 2 2 2 ( ) 0, d g dt    + + = (32) where 2 2 3 2 2 3 4 5 6 1 2 3 4 5 6 7 8 9 102 ( ) ( ) ( ) . d g d d d d d d d d d d dt            = + + + + − + + + + + (33) utilizing the idea of st on both sides of eq. (32), we have  2 2 2 ( ) ( ) 0. r b r      − + +  = (34) where 2 2 3 1 2 3 42 2 2 3 4 5 6 5 6 7 8 9 10 ( ) ( ) . d d d d d dt d d d d d d             = + + + + − + + + + + (35) we can simplify it to obtain r(θ) such as   2 2 2 2 2 ( ) . 1 1 b r       = −  + + (36) now, using the property of inverse st, we get   2 1 2 2 ( ) cos . 1 t b t      −   = −   +  (37) analysis of nanobeam-based microstructure in n/mems system using van der waals forces 681 let the general expression of eq. (32) is 0 ( ) ( ).n n n t p t   = =  (38) using the idea of hpm, we obtain eq. (37) as follows: 2 1 2 52 2 0 0 0 ( ) cos ( ) . 1 n n n n n n n n n p t b t p d p p h           − = = =     = − − +    +        (39) on comparing the components p, we construct the following results:   0 0 2 1 1 1 2 2 cos , , 1 p b t p       − = =   = = −   +  (40) the above equation can also be written as     2 1 1 0 1 2 32 2 2 1 4 5 62 2 cos cos 2 cos3 1 cos 4 cos5 cos6 1 t t t t t t              − −   = −  +  +  +   +     +  +   +  (41) where the coefficients γn (n=0,1,2, 3,...,6) can be evaluated as 3 2 4 6 2 3 6 8 101 0 4 3 52 2 7 92 1 5 3 2 4 6 2 3 6 8 101 2 3 52 2 7 92 3 3 4 6 2 3 8 10 4 3 3 5 , 2 8 2 8 16 3 53 1 , 4 4 8 15 , 2 2 2 2 32 5 , 4 4 16 3 8 8 d b d b d b d bd b b d d b d bd b b bd d b d b d b d bd b b d b d bd b b d b d b d b b         = − + + + + +       = − + + + +       = − + + + +       = − + +       = − + +    5 9 5 6 10 6 , 16 , 16 . 32 d b d b  =  = (42) 682 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi after adopting inverse st to resolve eq. (41), we obtain the following: 0 1 2 1 2 2 3 4 2 2 5 6 2 2 (cos 1) sin (cos cos 2 ) 2 3 (cos cos3 ) (cos cos 4 ) 8 15 (cos cos5 ) (cos cos 6 ). 24 35 t t t t t t t t t t t t t                       = − − − −   − − − −   − − − − (43) in order to preserve its periodicity for the system, the components of sint t must be zero. consequently, 1 0, 2  = (44) or 3 52 2 7 92 5 3 53 1 0. 4 4 8 d b d bd b b bd   − + + + + =    (45) it produces the following result: 2 4 5 7 5 2 2 8 6 5 , 8 6 d d b d b d b  + + = + (46) subsequently, the estimated first-order analytical result is as follows: 0 1 2 3 4 5 6 ( ) cos cos 2 cos3 cos 4 cos5 cos 6 , hts a a b t a t a t a t a t a t        = + + + + + + + (47) therefore, using eqs. (46) and (47), we can derive the periodic motion of doubly clamped n/mems system built on nanobeams with vdw forces, and where: 0 3 5 6 72 4 2 0 1 0 22 2 2 3 5 64 3 4 5 62 2 2 2 1 , , , 3 8 15 24 35 48 3 , , , . 8 15 24 35 a a a a a a a                = − =  − − − − − − =       = = = = (48) 6. numerical findings and analysis this section compares the analysis of hts results with the numerical results of literature study in [17]. figure 2 illustrates the variation in displacement obtained for n/mems parameters under vdw forces, in which we demonstrate mathematical software (dotted line), the hts (black square with straight line), and srhbm [17] (arrow head with red line). this shows an analysis of the mid-plane deviation of nanobeams obtained numerically using rk4 and qualitatively with hts and srhbm. analysis of nanobeam-based microstructure in n/mems system using van der waals forces 683 fig. 2 comparison of hts solution with srhbm and rk4 solution fig. 3 presents the error analysis of hts and srhbm results over time with various components of identical parameters, in which we demonstrate the hts results (arrow head with red line), and srhbm [17] (circle with red line). the parametric values are described to show the efficiency of model. fig. 3 error analysis between hts and srhbm solutions 684 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi fig. 4 examines the variation in parameters by using our purposed scheme, whereas the other three parameters remain unchanged. the results of the hts and by using the mathematical software are in agreement at various values of the amplitude parameters. fig. 4 effects of amplitude on deflection of nanobeam fig. 5 shows the impact of hts for the significant results of n/mems system. only one parameter is changed, while the other three parameters remain unchanged at α=0, 50, 100, 150. fig. 5 effects of gap parameter on deflection of nanobeam analysis of nanobeam-based microstructure in n/mems system using van der waals forces 685 fig. 6 illustrates the various elements of the problem influenced by hts solution. the other three remain steady parameters and only one factor is changed at λ=0, 0.3, 0.6, 0.9. fig. 6 effects of parameters on deflection of nanobeam fig. 7 highlights the variation of the problem affected by the hts solution. the other three factors endure constant; only one is changed. the results from hts and using software are in perfect alignment with the parameter's value n=0, 5, 10, 15. in our graphical fig. 7 effects of axial load parameter on deflection of nanobeam 686 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi structure, we can conclude that hts results have strong agreement for n/mems system based on microstructure using vdw forces. in the fundamental oscillation concept, the fact is recognized that the closed orbits illustrated in these phase visuals refer to oscillatory behavior. it is anticipated that significant values exist within the range of parameters that lead to the existence of pull-in solutions. 7. conclusion this study examined a general oscillation in a highly nonlinear n/mems system, which describes the motion of a microbeam under the influence of vdw forces. our proposed method rapidly converges to results and demonstrates high accuracy, even with the firstorder approximation step. the derived results confirm the efficiency of the suggested approach and provide better performance than other techniques. micro-sensors used for monitoring and controlling vibrations in machinery are a practical application of mems in mechanical engineering. this research contributes to our comprehension and improvement of mems performance by examining signal stability, energy efficiency, and resilience in severe conditions, which are crucial for dependable, long-term functioning in mechanical domains. in the future research, we intend to expand this scheme to address other nonlinear challenges in n/mems 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journal of low frequency noise, vibration and active control, 41(2), pp. 572-603. 35. he, c.h., amer, t.s., tian, d., abolila, a.f., galal, a.a., 2022, controlling the kinematics of a spring-pendulum system using an energy harvesting device, journal of low frequency noise, vibration and active control, 41(3), pp. 1234-1257. 36. he, c.h., tian, d., moatimid, g.m., salman, h.f., zekry, m.h., 2022, hybrid rayleigh-van der pol--duffing oscillator: stability analysis and controller, journal of low frequency noise, vibration and active control, 41(1), pp. 244-268. https://doi.org/10.1002/mma.6583 688 m. nadeem, q.-t. ain, n. almakayeel, y. shao, s. wang, m. shutaywi appendix: the components defined by eq. (29) are obtained as follows: 1 1 1 1 1 2 3 4 5 0 1 2 3 4 0 0 0 0 0 1 1 2 2 5 6 0 0 1 1 1 3 3 2 7 0 0 0 1 1 4 4 2 8 0 0 , 3 , 3 , , , ( ) , ( 3 3 ) , (3 3 ) , ( ) 3 =  = −  =  = −  = −     =  −  = −   +      =   −   −     = −  +   +               q d q d q d q d q d q n d q n d q n d d d q n d d                1 2 0 1 1 1 1 3 2 4 2 9 10 0 0 0 0 , 3 , .     = −    =         d q d d q d d        where ς(φ)=sin(πφ). facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 229 239 https://doi.org/10.22190/fume201203001h © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper passive atmospheric water harvesting utilizing an ancient chinese ink slab chun-hui he1, chao liu1, ji-huan he1,2,3, ali heidari shirazi4, hamid mohammad-sedighi4,5 1school of civil engineering, xi’an university of architecture & technology, xi’an, china 2school of mathematics and information science, henan polytechnic university, jiaozuo, china 3national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china 4mechanical engineering department, faculty of engineering, shahid chamran university of ahvaz, ahvaz, iran 5drilling center of excellence and research center, shahid chamran university of ahvaz, ahvaz, iran abstract. extraction of atmospheric water using a passive mechanism instead of a complex and advanced equipment has become an emerging subject. there is a clear record in mengxibitan by shen kuo(1031~1095) that an ink slab has the ability to collect water from the air. its mechanism is exactly similar to the fangzhu [1], a recently investigated device for atmospheric water harvesting (awh). based on the fangzhu device, a mathematical model for the awh mechanism in ink slab-like materials is suggested. using he’s frequency formulation and two-scale fractal derivatives the possible working mechanism of ink slab-like materials is investigated. the potential applications of ink slab-like structures for awh in interior and exterior architecture are also presented and discussed. it is revealed that efficiency of the slabs highly depends on velocity and temperature of the flowing air and also its low-frequency characteristics. key words: nanotechnology, chinese civilization, mengxibitan, fangzhu, fractal oscillator, two-scale fractal derivative received december 03, 2020 / accepted december 31, 2020 corresponding author: chao liu affiliation, postal address : school of civil engineering, xi’an university of architecture & technology, xi’an 710055, pr china e-mail: chaoliu@xauat.edu.cn mailto:chaoliu@xauat.edu.cn(c 230 c.-h. he, c. liu, j.-h. he, a.h. shirazi,, h. mohammad-sedighi 1. introduction the freshwater inadequacy has become an alarming issue in the contemporary lives of humankind as well as flora and fauna. this issue escalates especially in the lands with low or none natural water accessibility. furthermore, responding to an increasing demand for consumable water due to climate change and population growth has turned into a difficult task based on polluted water sources. several water purification methods employed to overcome this matter, filtration [2], solar water purification [3], and reverse osmosis [4], are examples of wastewater and seawater refinement solutions. however, the mentioned strategies highly depend, firstly, on natural water sources such as lakes and rivers [5], and, secondly, on advanced equipment. as a promising alternative, researchers turned toward atmospheric water harvesting (awh) methods to overcome this problem in arid regions. earth's atmosphere contains approximately 50,000 cubic kilometers of water [6], this phenomenon introduced awh as a sustainable water source. fog/dew collection [7] mechanisms accelerate the growth of droplets, based on wettability engineering. however, dependency on saturated local humidity significantly restricts its applications in areas with dry environment. moisture harvesting, on the other hand, has become a suitable alternative in ahw because of its independence of climate conditions since moisture is water vapor and is widespread across the earth [5]. the goal of this method is to liquefy the trapped moisture in the atmosphere by cooling the air below its dew point and subsequently condense the air [3]. however, this cooling procedure is intense energy costly. based on the interaction between nanostructure and water molecules, passive awh materials emerged as a novel alternative to address this problem. the capability of harvesting water in low and high relative humidity and low energy demand are the main advantages of these materials. zhao and et al. [6] improved these materials by suggesting a super moisture-absorbent gel. however, the fundamental design principles of these materials are still mostly unknown [6]. the nanostructure of these materials can be obtained by carefully studying the water harvesting mechanism among desert animals and plants. gurera and bhushan [8] studied passive water harvesting mechanisms among desert species. comanns [9] studied passive water collection in animal's integuments. on the other hand, with the development of solar panels, off-grid and self-sufficient architecture has become an attractive subject to engineers. however, few studies focused on freshwater problems in rural and arid areas and potential solutions in architecture. passive awh materials can provide novel ideas in architectural engineering. this paper introduces an ancient ink slab, which was recorded in mengxibitan (梦溪笔谈 ) by shen kuo(沈 括)(1031~1095), for water collection from air. shen wrote: 孙之翰，人尝与一砚，直三十千。孙曰：“砚有何异而如此之价也？”客曰：“砚以 石润为贤, 此石呵之则水流。”孙曰：“一日呵得一担水才直三钱，买此何用？”竟不受。 literally, someone gave sun zhihan an ink slab as a gift, which was worth 30,000 dollars. “has this ink slab anything special? why is this gift so expensive?” sun asked. “the ink slab is special for its stone moist; when you blow it, flowing water can be collected.” the giver replied. “though the ink slab can collect a pail of water per day, it is only worthy of 3 dollars, so it is not worthy buying it.” sun said, and rejected it. passive atmospheric water harvesting utilizing an ancient chinese ink slab 231 mengxibitan was a collection of the most advanced technology and natural phenomena at his times; it is still perhaps the most prestigious and influential book on ancient science and technology, and natural phenomena in china. sun zhihan (998 ~ 1057) was an official in huazhou, which is adjacent to xi’an city, china. the ink slab is a traditional tool for painting and writing, which plays an important role in chinese civilization. the recorded ink slab was special for its water collection by blowing its surface. the fabrication technology has been lost for a long time; no one can re-build the ink slab so far. there is a similar ancient device for water collection from the air; it is called as fangzhu, which was widely recorded in many ancient chinese classics. a detailed discussion is given in ref. [1] and refs. [10-12]. wang found that the nanoscale surface morphology of fangzhu plays an important role in its water collection [10]. this paper shows that the material of fangzhu has some similar properties as the modern metamaterials, which are famous for attenuation of sound and vibration [13]. the fangzhu material has the low frequency property favorable for water transmission [14, 15]. first we suggested a working mechanism for ink slab material by explaining fangzhu working principles. in the second and third sections, a mathematical model for water molecule vibration behavior is suggested and solved. finally, potential applications of ink slab-like material in modern architecture are discussed in the last section. 2. ink slab and fangzhu material fangzhu was an ancient device famous for its water-harvesting ability. if it is the fact that the recorded ink slab is actually a fangzhu-like device, its water collection property can be easily revealed. according to ref. [1], fangzhu is easy to collect water from the air when it is placed against a moving air flow. fig. 1-a, illustrates a traditional chinese ink slab under the warm air flow. a key factor in attracting water molecules in these materials is a hydrophilic-hydrophobic surface. the convex surface (super hydrophobic) generates surface potential duo to its curvature, thus easily attracting water molecules in the flowing air to its surface; the attracted water molecule then integrates into water droplets (fig.1-b). on the contrary, the concave surface (super hydrophilic) merges water droplet and stores the extracted water. the recorded ink slab might be fangzhu in an ink slab form (see fig. 1-c). the record emphasizes the water collection property by blowing to its surface. according to ref. [1], water harvesting of fangzhu highly depends on the flowing air velocity and temperature; higher temperature results in higher efficiency for water collection. when we blow on the surface of the ink slab, a higher velocity and a higher temperature enable the ink slab to work much more efficiently. 232 c.-h. he, c. liu, j.-h. he, a.h. shirazi,, h. mohammad-sedighi (c) fig. 1 (a) traditional chinese ink slab and its potential nanostructure, (b) nanoscale unit cell of the slab(c) a real ink slab in the modern time and ancient chinese characters for fangzhu (方诸 ) passive atmospheric water harvesting utilizing an ancient chinese ink slab 233 3. fangzhu oscillator and its low frequency property fangzhu represents a long lost device characterized by some incomparable properties which cannot be found in both natural and artificial materials. the main property of fangzhu material is to absorb water molecules which vibrate with an extremely low frequency [1]. it was reported that the low frequency property is beneficial for a long transmission of air/moist permeability through a microscale capillary tube [14, 15]. this property is used for sound and vibration attenuation in modern metamaterials [13]. the fangzhu oscillator, which describes the motion of a water molecule in fangzhu’s surface, can be written in the form [1]: 2 1 2 1 22 2 1 2 1 0, 0, 0 ( + )p q d x dt x x h     + + + − =   (1) with the initial conditions: (0) , (0) 0x a x = = (2) where 1 and 2 are surface factors for the nanoscale concave and convex areas in fangzhu’s surface, respectively. likewise, p and q are also surface-dependent constants while h is the distance between the concave and the convex, x is the distance between an absorbed water molecule and the convex area. eq. (1) with the initial conditions of eq. (2) is difficult to be solved analytically; the homotopy perturbation method [11, 16], the variational iteration method [17], the taylor series method [18], and the reproducing kernel method [12] can effectively solve fangzhu's oscillator. in this section, we will apply he’s frequency formulation [19, 20] to reveal the frequency property of the fangzhu oscillator. the variational principle can be established by the semi-inverse method [21], which is: 2 2 21 2 0 1 ( ) ( ) ( + ) 2 2 2 t p qdx j x x x h dt dt p q  − − = + −        (3) the hamiltonian invariant is: 2 2 21 2 1 ( ) ( + ) 2 2 2 p qdx x x h h dt p q  − − − + = (4) where h is the hamilton constant. by the initial conditions of eq. (4), the hamilton constant can be identified, and eq. (4) becomes 2 2 2 2 21 2 1 2 1 ( ) ( + ) ( + ) 2 2 2 2 2 p q p qdx x x h a a h dt p q p q    − − − − − + = − + (5) differentiating eq. (5) with respect to t results in: 2 2 -1 2 -1 1 22 + ( + ) =0 p qdx d x dx dx x x h dt dt dt dt   − − − (6) 234 c.-h. he, c. liu, j.-h. he, a.h. shirazi,, h. mohammad-sedighi which is equivalent to eq. (1). eq. (4) implies that the total energy during the oscillation keeps unchanged. we use he’s frequency formulation to elucidate the frequency property of the fangzhu oscillator. consider a nonlinear oscillator in the form 2 2 ( ) 0, ( ) / 0 d x f x f x x dt + =  (7) he’s frequency formulation is [19, 20] 2 ( ) = f a a    (8) where θ is a constant, satisfying 0˂θ˂1. as an example, we consider the following oscillator [22]: 2 2 1 0 d x dt x + = , (0) , (0) 0x a x = = (9) we choose θ=0.8, and eq. (8) leads to the following result: 1 1.25 = = 0.8a a  (10) the exact frequency can be obtained as [22] exact 1.2533 = a (11) the relative error is as small as 0.26%. now he’s frequency formulation and its various modifications have been widely applied to various nonlinear oscillators [23-29]. we predict the fangzhu’s frequency property by eq. (8) as follows: 1 2 2 2 2 2 = (0.8 ) (0.8 + )    p q a a h + + − (12) a water molecule is absorbed on the convex and then is transferred to the concave; it requires that the frequency is as low as practically possible: 1 2 2 2 2 2 0 (0.8 ) (0.8 + )   p q a a h + + −  (13) and 1 2 2 2 2 2 0 (0.8 ) (0.8 + ) p q a a h   + + − → (14) for the sake of practical design of these materials eq. (14) can be used to optimize the surface geometrical properties and materials to achieve water harvesting effects. passive atmospheric water harvesting utilizing an ancient chinese ink slab 235 4. fractal oscillator abro et al. revealed that nano fluid can be modeled by fractional differential equations [30]. wang et al. pointed out that the water molecule’s vibration along the fangzhu’s surface should consider the unsmooth morphology, and a fractal modification is suggested, which is [10]: 2 1 2 2 2 1 2 1 0 ( + ) p q d x dt x x h    + + + − = (15) with the initial conditions (0 ) , (0 ) 0x a x  = = (16) where /dx dt  is a two-scale fractal derivative [31-33]. its approximate solution can be expressed as ( )cos( )x a a a t  = + − (17) where a is a positive constant,  is defined by eq. (12). the frequency property of the fractal oscillator is important. we consider a special case of eq. (17): 1 0.5cos( )x t  = + (18) fig. 2 shows a low frequency property of the fractal oscillator when time tends to infinity. based on capillary effect in transporting water molecule mass, the low frequency can effectively guarantee both safe water transmission and an extremely low loss of water [14]. the equivalent frequency can be approximately written as 1 eq t    − = (19) when 1  , we have the low equivalent frequency when t>>1. the frequency-amplitude relationship given in eq. (12) reveals that a lower frequency results in larger amplitude, which implies that the vibrating water molecule can be transmitted to a longer distance. a) 236 c.-h. he, c. liu, j.-h. he, a.h. shirazi,, h. mohammad-sedighi b) c) (d) (e) fig. 2 frequency property of a fractal oscillator at the initial stage for different values of  . (a) 1 = ; (b) 100 = ; (c) 1000 = ; three-dimensional illustrations for ω=1 (d) and ω=100 (e) passive atmospheric water harvesting utilizing an ancient chinese ink slab 237 4. potential applications of ink slab-like materials many architects have been recently attracted to the idea of self-reliance houses and complexes. in the case of water harvesting, these novel materials can play an influential role in the development of this idea. in this manner, the ink slab-like materials can be modified into tile panels for the exterior of houses (fig. 3-a). these panels can be used to cover building facades or rooftops. the extracted water will be collected through pipelines and ducts toward the water tank. the functionality of this system relies on gravitational force; therefore, very low external energy is needed. in interior design, a similar mechanism can be used to collect water into air condition vents. extracted water can be stored via outlet piping toward water tank. (fig. 3-b) can be used to collect water into air condition vents. extracted water can be stored via outlet piping toward water tank. (fig. 3-b). fig. 3 (a) exterior roof platform made of ink slab-like materials, (b) interior ink slab-like material usage in air condition vent few pieces of equipment and a simple collecting system allow these materials to be used in various scenarios. fig. 4 shows the flexibility of these materials by maintaining the visual beauty in modern architectural elements and also harvesting fresh atmospheric water at the same time. fig. 4 using ink slab-like material in architectural elements and maintaining visual beauty 238 c.-h. he, c. liu, j.-h. he, a.h. shirazi,, h. mohammad-sedighi 5. conclusion according to the record in the mengxibitan, the device was extremely rare, and the ink slab implies that fangzhu could still be seen about 1,000 years ago. when we blow on the fangzhu-like ink slab, the hot air, high air velocity and high humidity are all favorable for water collection as discussed in ref. [1]. it is shown how these materials can significantly improve the idea of self-reliance houses through various examples. the mechanism of the atmospheric water harvesting (awh) revealed in this paper is extremely helpful for re-designing the long-lost device for water collection from the air, and there are new promises and future challenges for modern applications in architectural engineering and other fields. acknowledgement: h.m. sedighi is grateful to the research council of shahid chamran university of ahvaz for its financial support (grant no. scu.em99.98). references 1. he, c.-h., he, j.-h., sedighi, h.m., 2020, fangzhu (方诸): an ancient chinese nanotechnology for water collection from air: history, mathematical insight, promises, and challenges, mathematical methods in the applied sciences, https://doi.org/10.1002/mma.6384. 2. jiao, m., yao, y., chen, c., jiang, b., pastel, g., lin, z., wu, q., cui, m., he, s., hu, l., 2020, highly efficient water treatment via a wood-based and 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paper in silico analysis of an articular cartilage regenerative rehabilitation under conditions of mesenchymal stem cells implantation and their mechanical stimulation aleksandr poliakov, vladimir pakhaliuk sevastopol state university, sevastopol, russia orcid ids: aleksandr poliakov https://orcid.org/0000-0002-2940-8945 vladimir pakhaliuk https://orcid.org/0000-0002-9992-1189 abstract. one of the most important tasks of modern medicine is the development of effective technologies for the treatment of joint diseases caused by damage to the articular cartilage. the results of experimental studies and a number of successful clinical practices indicate that its solution can be found within the framework of a new medical direction regenerative rehabilitation, which synergistically combines the methods of regenerative and rehabilitation medicine. in particular, regenerative rehabilitation of articular cartilage defects involves the use of cellular technologies, the effectiveness of which is enhanced by mechanical stimulation of chondrogenic cells, which accelerates their proliferation, differentiation, and formation of an extracellular matrix. the simulation results indicate that its outcome depends not only on a set of parameters determined by the state of the tissue in the defect aria, but also on their combination. one of the main goals of this work is to find the best combination of parameter values that are practically achievable in the process of articular cartilage regenerative rehabilitation using cellular technologies and mechanical stimulation of cells. its solution is based on the study of a regenerative tissue rehabilitation mathematical model, the state parameters set of which is determined by the sobol-statnikov method, based on a systematic study of the parameter space uniformly distributed in a multidimensional cube. the practical significance of the results of the work lies in the fact that they can be used to evaluate the effectiveness of mechanical stimulation various methods of articular cartilage defects in the process of regenerative rehabilitation. key words: articular cartilage, osteoarthritis, stem cell implantation, mechanical cell stimulation, regenerative rehabilitation, mathematical model received: september 19, 2023 / accepted december 25, 2023 corresponding author: aleksandr poliakov sevastopol state university, 299053, universitetskaya, 33, sevastopol, russian federation e-mail: a.m.poljakov@sevsu.ru https://orcid.org/0000-0002-2940-8945 https://orcid.org/0000-0002-9992-1189 400 a. poliakov, v. pakhaliuk 1. introduction it is known that the current mechanical and biophysical properties of the cellular tissue microenvironment depend on the mechanical impact experienced by cells in the past and largely determine their future fate [1]. in particular, mechanical impacts on mesenchymal stem cells (msc), which have a high regenerative potential, affect their ability to synthesize elements of the extracellular matrix (ecm), including fibronectin, laminin, collagen and proteoglycans, and form the microenvironment necessary to increase their own viability, proliferation and differentiation [2,3]. taking into consideration these facts, a hypothesis was formulated according to which, by artificially creating a certain cellular microenvironment, it is possible to control the life cycle of mscs implanted in the area of articular cartilage defect, stimulate their proliferation and differentiation in the desired direction in order to ensure the regeneration of damaged tissue. this hypothesis is confirmed by the results of a number of studies in silico [4–7], in vitro [8– 10], and in vivo [11–13]. there is an opinion that the regenerative rehabilitation methods of cartilage tissue defects, which have been intensively developed in recent years, have great potential for its successful practical implementation [14]. however, these methods still need to be improved. one of the main issues to be resolved on the way to their widespread introduction into medical practice is how exactly it is necessary to influence msc implanted in the area of articular cartilage defect in order to achieve the best effect of its restoration. this is an extremely complex task, the solution of which depends on many factors, the influence of which on the achievement of the final results is ambiguous and is characterized by a high degree of uncertainty. at the same time, it is known that the mechanical and biophysical state of articular cartilage is characterized by many measurable parameters that change as a result of mechanical stimulation of the tissue. these include: the rates of diffusion, proliferation, differentiation and death of cells, the effect of growth factors and nutrients on the deposition and degradation of ecm, the degree of reproduction and degradation of growth factors, the rate of consumption of nutrients by cells, etc. the question arises: what are the optimal values of the above, as well as other parameters that ensure a stable, irreversible and final process of tissue regeneration? in addition, no less important (and perhaps even more important) is the question of how to ensure the creation of a cellular microenvironment corresponding to optimal parameters. in other words: how to carry out mechanical stimulation of cells, providing the best conditions for the regeneration of damaged tissue? the answer to this question, if it exists, can be obtained by comparing the results of in vivo experiments with the results of in silico studies performed on the basis of adequate mathematical models for tissue regenerative rehabilitation. it is clear that this is an extensive way, but, unfortunately, currently there are no other ways apparently to find the optimal solution to the multiparametric problem of repairing articular cartilage defects. known methods of its treatment cannot be considered not only optimal, but also sufficiently reliable, because in most cases they do not lead to positive results. this is largely due to the peculiarities of the cartilage tissue, which is practically devoid of a vascular network, as a result of which the nutrition of its cells is carried out mainly due to the diffusion of nutrients. therefore, unlike, for example, bone tissue, hyaline articular cartilage after destruction in the vast majority of cases is not able to fully recover. after reparative regeneration, at best, it is transformed into hyaline-like cartilage [15]. at the same time, it should be noted that cases of complete restoration of local articular cartilage defects as a result of msc cell therapy followed by physiological procedures are known. it can be assumed that such outcomes are possible as a result of achieving an optimal cellular microenvironment in the tissue defect area as a result of in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 401 its optimal mechanical stimulation. with this in mind, the most important task, the solution of which is necessary for organizing effective processes of articular cartilage defects regenerative rehabilitation, is to determine the optimal parameters of the cellular microenvironment, which must be achieved by mechanical stimulation of cells. articular cartilage has a very complex structure, represented by solid and liquid phases, and its rigidity is determined to the greatest extent by the fibrous structure of the extracellular matrix and the incompressibility of the interstitial fluid [16]. when loaded, the stiffness of cartilage changes along with a change in the ratio of solid and liquid phases. at the same time, at the initial stage of loading, the degree of hydration of the matrix does not change, since osmotic pressure prevents the release of interstitial fluid. its extrusion from the matrix begins only when the load acting on the joint will lead to an intra-articular pressure greater than the osmotic pressure. when the load is removed, the reverse process occurs the absorption of interstitial fluid by cartilage. the viscous component of the synovial fluid, consisting of molecules larger than the pores of the matrix, always remains in the joint space and the low-viscosity component moves from the joint space to the cartilage and back during loading/unloading of the joint. this promotes the transfer of cells and nutrients into the cartilage and the removal of metabolic products from the cartilage into the joint cavity. the state of articular cartilage under conditions of mechanical loading (stimulation) is characterized by a number of parameters, estimates of which can be obtained as a result of in vivo experiments, thanks to modern methods and analysis tools. at the same time, the measured values of the key parameters αi, which have the greatest impact on the course of cellular processes, are within some segments min max i i i    . when modeling the processes for regenerative rehabilitation of articular cartilage, taking into account the personalized characteristics of the tissue, the boundary values of the parameters * min m i i = and ** max m i i = may be deliberately chosen to be larger than min i and max i , respectively, to be able to evaluate as many methods of mechanical stimulation of cells as possible. obviously, the simulation results will depend on which point ak=(αk1, αk2,…,αkn) in the parameter space a=(α1, α2,…,αn) will be investigated. theoretically, it is possible to simulate an infinite number of variants for regenerative rehabilitation processes, if we assume that the parameters αi are continuously distributed on the segments * **[ , ]i i  . if the parameters αi are represented as discrete sets, then a finite number of models can be investigated. but in any case, in order to obtain the most complete information about the processes of regenerative rehabilitation, the parameter space a should be systematically investigated. one of the main objectives of this work is to search for the values of the cellular microenvironment key parameters, which are practically achievable using cellular technologies and mechanical stimulation of cells, under which the best conditions for regenerative rehabilitation of articular cartilage defects are achieved. to solve it, a mathematical model of regenerative tissue rehabilitation is used, the combination for state parameters of which is determined by the sobol-statnikov method, based on a systematic study of the parameter space a uniformly distributed in a multidimensional cube [17,18]. 402 a. poliakov, v. pakhaliuk 2. materials and methods 2.1. mathematical model for regenerative rehabilitation of an articular cartilage defect fig.1 shows the layout of a local articular cartilage defect on the articular surface of the knee joint, as well as its element on an enlarged scale and the associated coordinate system. fig. 1 representation scheme of the elementary area for the articular cartilage defect and the associated coordinate system if to assume that the development of the process for articular cartilage regenerative rehabilitation begins with the subchondral bone, and the properties of the tissue or tissueengineered structure located in the defect area change slightly in planes parallel to the subchondral bone, then, for the purpose of simplification, this problem can be approximately considered as one-dimensional. we will place the origin of the coordinate system on the surface of the subchondral bone in the conditional center of the defect, and direct the x coordinate axis along the outer normal vector ν to the articular surface. we will assume that the articular cartilage regenerative rehabilitation is performed under conditions of implantation in the area of the msc defect, i.e., based on the asi articular stem cell implantation cellular technology that meets the criteria established by the international society for cellular therapy's mesenchymal stromal cell committee (isct msc) [19]. to solve this work problems, a mathematical model of the “diffusion-reaction” type is used, the advantages of which were confirmed when modeling various types of biological processes. the history of this type model development is briefly presented in [20]. initially, starting in the mid-1980s, it was used to simulate the dynamics of tumor-induced angiogenesis. in 1990, it was modified to predict epidermal wound healing by taking into account the interaction between in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 403 endothelial cells and a chemoattractant. this took into account the chaotic movement of cells, their proliferation and death, while the chemical substance produced by the cells diffused and underwent natural breakdown. the results of this model were in very good agreement with the results of experimental studies. in 1993, chaplain and stuart developed a model for the chemotactic response of endothelial cells to tumor angiogenesis factor [21], which was modified in 1995 by byrne and chaplain [22] to account for the development of blood vessels in response to tumor angiogenesis factor. in addition, in 1996, the same authors established an important link between mathematical models of tumor-induced angiogenesis and wound healing angiogenesis [23], which led to the emergence of numerous mathematical models of wound angiogenesis, which were based on the concept of the “diffusion-reaction” model. the most important among them, according to many researchers, are the models developed in 1996 by pettet, byrne, mcelwain and chaplain, which inspired many researchers to study wound angiogenesis [24]. in 2001, bailón-plaza and meulen [25] used a “diffusion-reaction” model to study fracture healing processes taking into account the influence of growth factors, the equations of which had the following structure: 1) msc density = migration + mitosis + differentiation; 2) chondrocytes density = mitosis + differentiation endochondral replacement; 3) osteoblasts density = mitosis + differentiation – removal; 4) connective/cartilage ecm density = synthesis – degradation; 5) bone ecm density = synthesis – degradation; 6) osteogenic growth factor concentration = diffusion + production – decay; 7) chondrogenic growth factor concentration = diffusion + production – decay. the migration of msc in the model was simulated based on the experimentally observed behavior of randomly populated cells, considering the effects of ecm density and environmental heterogeneity; to account for cell proliferation, the logistic growth law was used, according to which the rate of cell division decreases linearly with increasing cell density due to space and nutrient limitations. the size of the cell population, in accordance with experimental data, stabilized around the maximum density, and cell death due to apoptosis was balanced by the mitosis of new cells. in addition, the enhancing and inhibitory effects observed at low and high ecm densities, respectively, were taken into account. the differentiation of mesenchymal cells into osteoblasts or chondrocytes was regulated by the concentrations of osteogenic and chondrogenic growth factors, respectively. in this case, the dependence of the differentiation rate on the concentration of growth factors was determined considering experimental dose-dependent curves. other processes occurring in the fracture area during its healing were similarly modeled. in 2011, adopting the approach proposed by bailón-plaza and meulen, lutianov, naire, roberts and kuiper presented a “diffusion-reaction” model to predict articular cartilage regeneration using cell therapy [4], which was studied in a modified form by campbell, naire and kuiper in 2019 [5,6]. this model, which previously studied by popov, poliakov and pakhaliuk under other conditions [7], is also used in this work. its equations are: 404 a. poliakov, v. pakhaliuk 01 1 2 3 1 0 [ ] ( ) ( ) ( )s s s s s s s s diffusion deathdifferentiation proliferation с n d c p c h n n p c h c c p c h n n t n n  =   + − − − − −  + , (1) 04 1 2 5 1 0 [ ] ( ) ( ) ( )c c c c s s s c diffusion deathdifferentiation proliferation с n d c p c h n n p c h c c p c h n n t n n  =   + − + − − −  + , (2) 6 7 0 { } ( )n s c diffusion reaction n n d n p c p c t n n  =   + +  + , (3) 8 0 [ ]m c diffusion reaction m n d m p c t n n  =   +  + , (4) 9 11[ ]g s reactiondiffusion g d g p c p g t  =   + −  , (5) 12 13[ ]b c diffusion reaction b d b p c p g t  =   + −  , (6) where i j k x y z     = + +    ; ( , , )x y z i j k grad x y z          = + + =    ; 2 2 2 2 2 2 grad i j k divgrad x y z          = + + =    ; where φ(x,y,z) is a scalar function. a detailed description of the model, represented as system of eqs. (1-6) and the results of its study are presented in [7]. the numerical values of the parameters used in this case corresponded to the values given in the literature [4–6]. their brief description is presented in appendix a. previously, these values were established experimentally or justified theoretically and used in a number of studies on bone fracture healing, implant osseointegration, and articular cartilage regeneration [26,27]. boundary conditions of the problem [7]: ▪ at the point x=0: ( ), 0, 0, 0, 0, 0s c s c g b n m c c g b n m d f t d d d d d x x x x x x       − = = = = = =       ; (7) ▪ at the point x=d: 00, 0, , , , 0s c s c g b m c c g b m d d d g d b n n d x x x x x      = = = − = − = =        . (8) in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 405 initial conditions at t=0 [7]: (0) 0 3( ), 0, 0, 0, ,s s c sc c h x g b c n n m m m= = = = = = + , (9) where (0) sc is the initial msc density; h(x) is the seeding profile of msc by defect depth; m3 is the initial ecm density; ms is the scaffold density. 2.2. methodology for the systematic study of a mathematical model by the sobol-statnikov method 2.2.1. research objective it should be noted that the problem solved in the framework of this work is essentially a multicriteria one. this is due to the fact that it is rather difficult to formulate its global criterion, despite the fact that its qualitative content is interpreted quite unambiguously: complete regeneration of the articular cartilage defect. but, as practice shows, even if it is possible to achieve this result, the ways to achieve it are most likely ambiguous. in addition, there is a possibility that during the regeneration of a tissue defect, undesirable complications may occur or other quality criteria will deteriorate. another problem is that the global criterion, the qualitative content of which is formulated above, is quite difficult to give a quantitative form. therefore, its evaluation should be carried out indirectly along with the evaluation of other criteria. for example, the mathematical model represented as system of eqs. (1-6) makes it possible to evaluate the rate of ecm synthesis and interstitial cartilage growth by increasing the density of chondrocytes. therefore, ecm density (m) and chondrocytes density (cc) on the articular surface can be taken as quality criteria for the process of an articular cartilage defect regenerative rehabilitation, assuming that their increase promotes the regeneration of cartilage tissue. in addition, msc, which can differentiate into chondrocytes, play an important role in the processes of articular cartilage regeneration. therefore, the density of these cells in the defect area or on its surface can also be attributed to the quality criterion of the cartilage tissue regeneration process. thus, the task is to determine the set of model parameters at which the highest density of ecm (m), chondrocytes (cc) and msc (cs) on the articular surface in the area of the defect is achieved over a certain period of time. 2.2.2. parametric constraints the solution of the problem is sought taking into account the set of variable parameters of the αi model, which, depending on the state of the cellular microenvironment, can take different values belonging to some segments * **[ , ]i i  . their description is given in table 1. the minimum * i and maximum ** i values of the parameters are accepted to be smaller or larger, respectively, than in [4–6]. this was done in order to evaluate as many options as possible for potentially possible processes of regenerative rehabilitation, in which the parameters of the cellular environment αi can take extreme values. 406 a. poliakov, v. pakhaliuk table 1 variable parameters of the mathematical model parameter functional dependencies and parameter values parameter dimension dimensionless parameter values, * ** i i − ds0 – msc diffusion constant 0 * 12s sd m d (mm3/hour)·(g/mm3) 0.001 – 0.01 dc0 – chondrocyte diffusion constant 0 * 12c cd m d (mm3/hour)·(g/mm3) 0.0005 – 0.0015 dn – nutrient diffusion coefficient dn ≈ 4.6 mm2/hour 100 – 300 dm – ecm diffusion coefficient dm ≈ 2.5·10-5 mm2/hour 0.001 – 0.01 dg – fgf-1 diffusion coefficient dg ≈ 2.0·10-3 mm2/hour 0.5 – 1.5 db – bmp-2 diffusion coefficient db ≈ 2.0·10-3 mm2/hour 0.5 – 1.5 p2 – msc differentiation rate p2 ≈ 3.75·10-3 1/hour 0.5 – 1.5 p3 – msc death rate p3 ≈ 3.75·10-3 1/hour 0.5 – 1.5 p40 – chondrocyte proliferation constant 0 9 4 4 10p −  g/(hour·mm3) 0.006 – 0.018 p400 – chondrocyte proliferation rate 00 4 4 2 10p −  1/hour 0.006 – 0.018 p5 – chondrocyte death rate p5 ≈ 3.75·10-3 1/hour 0.5 – 1.5 p80 – ecm production constant 0 13 8 3.75 10p −  (g/mm3)/((cells/mm3)·(1/hour)) 0.2 – 1.2 p800 – fgf-1 ecm deposition rate 008 0 1p =  0.2 – 1.2 p81 – ecm degradation constant 1 9 8 3.75 10p −  (1/hour)·(1/(cells/mm3)) 0.5 – 1.5 p9 – fgf-1 production constant p9 ≈ 10-17 (g/mm3)/((cells/mm3)·(1/hour)) 22.0 – 28.0 p11 – fgf-1 degradation rate p11 ≈ 5.8·10-2 1/ hour 13.0 – 16.0 p12 – bmp-2 production constant p12 ≈ 10-17 (g/mm3)/((cells/mm3)·(1/hour)) 22.0 – 28.0 p13 – bmp-2 degradation rate p13 ≈ 5.8·10-2 1/hour 13.0 – 16.0 2.2.3. functional limitations the functional limitations of the regenerative rehabilitation process are set directly in the equations of the mathematical model. the following effects are taken into account. proliferation of msc and chondrocytes can occur as long as the concentration of nutrients n in the defect area is greater than the critical value n1, which is ensured by introducing the in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 407 heaviside function into the model equations of the form 1 1 1 0, ( ) 1, n n h n n n n  − =   . on the contrary, as soon as n becomes less than n1 due to a lack of nutrients, the cells gradually begin to die, which is ensured by the heaviside function of the form 1 1 1 0, ( ) 1, n n h n n n n  − =   . the process of msc differentiation into chondrocytes is explained similarly. it occurs as long as the msc density (cs) is greater than the threshold value cs0 . as soon as cs becomes smaller cs0 , differentiation stops. these effects in the model equations are provided by the heaviside function of the form 0 0 0 0, ( ) 1, s s s s s s c c h c c c c  − =   , where 0 0max 0min ( ) ab s s sc c c e−= − ; 0maxsc , 0minsc are the maximum/minimum threshold msc density, respectively, a is the threshold msc density reduction factor. in addition, the functional limitations that must be taken into consideration in the simulation process can include the initial values of the key parameters of the model given in table 2 [4]. table 2 initial values of the mathematical model key parameters parameter functional dependencies and parameter value parameter dimension dimensionless parameter value (0) sc – initial msc density (0) 52.5 10sc   cells/mm3 0.25 (0) cc – initial chondrocytes density (0) 210 %cc − of total cell density cells/mm3 0.0001 m3 – initial ecm density assumed mmax/104 g/mm3 0.0001 ginit – initial fgf-1 concentration ginit ≈ 10-12 g/mm3 0.01 binit – initial bmp-2 concentration binit ≈ 10-12 g/mm3 0.01 it should be noted that in practice these values significantly depend on the state of a particular tissue, as a result of which the processes of regenerative rehabilitation are highly personalized. 2.2.4. criteria constraints it is known that after reaching a certain critical value, the reproduction of ecm stops. therefore, in order to exclude obtaining impossible results of the model study, a criterion constraint of the form m≤mmax=10-4 g/mm3 is introduced. the same applies to the differentiation of msc into chondrocytes. those, after reaching the critical density of chondrocytes, their reproduction in the defect area stops, which is taken into consideration in the model by introducing a criterion constraint of the form cc≤ccmax0=106 moles/mm3. finally, it is assumed that the density of msc in the defect area should also be limited. as a first approximation, one can take cs≤csmax0=106 moles/mm3. 408 a. poliakov, v. pakhaliuk taking into consideration the results of studies presented in [19-21], these criteria restrictions in dimensionless form are presented as follows: m≤mmax=1; cc≤ccmax0=0.4; cs≤csmax0=0.6. (10) constraints in eq. (10) assume that the maximum density of chondrogenic cells in the ctotal,max0 articular cartilage defect should not exceed 106 cells/mm3 at zero ecm (m=0) density 1 6 3 ,max 0 max 0 max 0 max 1 ( ) 10 /total s c m c c c cells mm m −   = − +     . in the dimensionless form ctotal,max0 ≤ 1, therefore, cs + cc ≤ 1. 2.2.5. selection of trial points according to the sobol-statnikov method, to select test points ak = (αk1, αk2,…, αkr), it is advisable to use sequences q1, q2,…,qk,… uniformly distributed in the parameter space with fairly good uniformity characteristics and, if possible, simple algorithms for calculating coordinates [18]. then, by the cartesian coordinates of the next point qk = (qk1, qk2,…, qkr), you can find the generalized coordinates of the point ak using the following relationship * ** *( ), 1,ki i ki i iq i r   = + − = , (11) where r is the number of mathematical model variable parameters. to determine qk, lпτ-sequences of points are used, which are based on the concepts of a binary segment, a binary parallelepiped, a multidimensional cube, and a пτ-grid. definition 1. any segment that can be obtained by dividing segment [0,1] into 2m equal parts (m=0,1,2,…) is called binary. all binary segments can be numbered and denoted as ls, s=1,2,… let k=(k1,k2,…,kr), where some or even all of kj, j=1,2,…,r may coincide. definition 2. a set of points with coordinates (x1,x2,…,xr) such that , 1,2,...,i kix l i r = is called a binary parallelepiped пk. any binary box belongs to the unit dimensional cube kr. definition 3 [17]. a grid consisting of n=2ν points of an r-dimensional cube kr is called a пτgrid, if each binary parallelepiped пk with volume 2 k v n   = has 2τ points of the grid at ν > τ. definition 4 [17]. a sequence of points p0,p1,…,pi,… of an r-dimensional cube kr is called an lпτ-sequence if any of its binary segments containing at least 2τ+1 points is a пτgrid. it is proved that all lпτ-sequences are uniformly distributed in kr [17]. moreover, all lпτ-sequences are perfectly distributed [28]. the main idea of the sobol-statnikov method is that for a systematic study of the parameter space of a model, perfectly distributed lпτsequences in kr can be successfully used. let i = em…e2e1 be the number of the trial point in the binary system. then its cartesian coordinates qk = (qk1,qk2,…,qkr) can be calculated by the formula (1) (2) ( ) 1 2 ... m ki i i m iq ev e v e v=    , (12) where the symbol (*) denotes the "exclusive or" operation. in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 409 eq. (12) can also be calculated in the usual arithmetic way. to do this, considering the given k, first calculate the number m = 1 + [ln k / ln 2], (13) and then for i=1,2,…,r 1 ( ) 1 1 1 2 2{ 2 } 2{ 2 } 2 m m l s s i s ki i l s l q k r− + − − − = =               , (14) where ( )s ir are the numerators of guiding numbers [17]. in formulas (13) and (14), [*] is the integer part of the number *, and {*} is the fractional part of the number indicated as *. thus, the set of cartesian coordinates qki, calculated by eq. (14) makes it possible to determine the set of generalized coordinates of test points given in eq. (11), uniformly distributed in the parameter space of the model for the process of an articular cartilage defect regenerative rehabilitation, which makes it possible to systematically study this process in silico. 3. results of the study for a systematic study of the influence the parameters of the mathematical model on the course of the processes for an articular cartilage defect regenerative rehabilitation, according to eq. (13) and eq. (14), the cartesian coordinates lпτ-sequences of points uniformly distributed in an 18-dimensional cube were calculated, which were used to calculate by the eq. (11) generalized coordinates of sample points uniformly distributed in the 18-dimensional parameter space, taking into consideration the parametric constraints given in table 1. as an illustration, fig.2 shows the distribution of sample points ak = (αk1, αk2,…, αkr) in the 3-dimensional parameter space. fig. 2 illustration of the test points ak={αk1, αk2, αk3} distribution in a 3-dimensional parameter space: a) 10 test points (k=1..10); b) 100 trial points (k=1..100) in this work, the study of the mathematical model represented by the system of eqs. (16) was carried out taking into consideration the generalized coordinates of 56 test points uniformly distributed in the 18-dimensional parameter space. the numerical values of the 410 a. poliakov, v. pakhaliuk generalized coordinates of the test points are presented in appendix b; the results of the mathematical model study are in appendix c. the first stage of the mathematical model study was carried out taking into consideration the time of the regenerative rehabilitation process of about 110 days (in a dimensionless form t=10). at the same time, at 6 test points (nos. 20, 22, 25, 27, 36, 56), unstable solutions of the system of eqs. (1-6) were obtained, and therefore they were excluded from consideration. the analysis of the results for the mathematical model study was carried out under the assumption that all test points are independent and uniformly distributed in the 18dimensional parameter space. in principle, another 56 sample points evenly distributed in this space could be chosen for the study. formally, this would have no effect on the results of the numerical experiment. to confirm this fact, in fig. 3 shows different sets of 10 sample points in 3d parameter space. their generalized coordinates were determined using lпτ-sequences, which means that from a mathematical point of view they are uniformly distributed in the parameter space, which, in general, is also confirmed visually. therefore, we can assume that the test points used to study the mathematical model were randomly selected, which means that the results of the study are also random. fig. 4 shows the distributions of the quality criteria values {m, cc, cs} in 3-dimensional and 2-dimensional criteria spaces, and table 3 presents the correlation coefficients of these criteria. the results of the mathematical model study indicate that there is a strong negative relationship between the ecm density and the msc density in the tissue defect area, which is explained by the essence of the tissue regeneration process. it is known that msc under certain conditions can differentiate into chondroblasts, which, in turn, provide the secretion of ecm, producing type ii collagen and sulfated glycosaminoglycans associated with noncollagen proteins proteoglycans. ultimately, chondroblasts clog in the matrix cavities (lacunae) and gradually turn into mature cells with lower synthetic activity chondrocytes. thus, msc produced during proliferation differentiate into chondroblasts, which contribute to the secretion of ecm and increase its density. at the same time, the msc density decreases. fig. 3 illustration of the test points ak={αk1, αk2, αk3} distribution in a 3-dimensional parameter space: а) 10 test points (k = 11…20); b) 10 test points (k = 31…40) in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 411 table 3 quality criteria correlation coefficients {m, cc, cs} m cc cs m 1 -0.4186 -0.9977 cc -0.4186 1 0.4341 cs -0.9977 0.4341 1 at the same time, a moderate negative relationship is observed between the ecm density and the density of chondrocytes, indicating that the dynamics of changes in these parameters are interrelated, but expressed not brightly. that is, there is a tendency for a decrease in the density of ecm with an increase in the density of chondrocytes and vice versa. fig. 4 distribution of quality criteria values in criterion spaces calculated at 50 trial points: a) 3-dimensional space of criteria {m, cc, cs}; b) 2-dimensional space of criteria {m, cs}; c) 2-dimensional space of criteria {m, cc }; d) 2-dimensional space of criteria {cc, cs}. regression lines are shown in blue. and, finally, from the data of table 3 and the nature of the location of points in the 2dimensional criterion space {cs, cc}, shown in fig. 4d, it follows that there is a moderate positive relationship between the density of msc and the density of chondrocytes, which 412 a. poliakov, v. pakhaliuk is also explained by the essence of the regenerative process: an increase in the density of msc in the defect area contributes to an increase in the density of chondrocytes and, consequently, the regeneration of cartilage tissue. table 4 presents the values of the correlation coefficients between the quality criteria of the problem under study and the mathematical model variable parameters. it is easy to see that p80 – ecm production constant parameter, which determines the ecm synthesis rate, has the most significant impact on the quality criteria values 0 18 8 8p p p m= − . (15) table 4 correlation coefficients between quality criteria {m, cc, cs} and the mathematical model variable parameters m cc cs ds0 -0.1146 0.1763 0.1311 dc0 -0.0127 -0.0283 0.0113 dn 0.1612 0.0400 -0.1738 dm 0.0132 0.1602 -0.0145 dg 0.1795 -0.1186 -0.1786 db -0.1130 -0.3195 0.1078 p2 0.1132 0.2691 -0.1351 p3 0.1091 -0.1554 -0.0998 p40 0.0661 -0.0452 -0.0773 p400 0.1151 0.0201 -0.1216 p5 -0.0966 -0.0354 0.0881 p80 0.8771 -0.5113 -0.8724 p800 -0.0936 0.0031 -0.0916 p81 -0.1924 0.3921 0.1985 p9 0.0274 -0.1558 -0.0415 p11 -0.1151 0.0277 -0.1076 p12 -0.1548 0.2757 0.1574 p13 0.0187 0.1269 -0.0072 this parameter does not depend on the density of the ecm, but has a direct impact on its dynamics. in addition, as follows from table 3, it has a moderate effect on the change in the density of chondrocytes and a strong effect on the change in the density of msc. another parameter: p81 – ecm degradation constant, has a moderate effect on the change in the density of chondrocytes. it follows from eq. (15) that its increase contributes to a decrease in the ecm synthesis rate with an increase in the ecm density. the results of the mathematical model study indicate that other parameters that are important in terms of the implementation of a sustainable process of articular cartilage regeneration have an insignificant effect on its dynamics. at the same time, some of their combination can lead to a violation of the process stability or its atypical development. by typical is meant a process of regenerative rehabilitation in which an increase in matrix density is constantly observed in the area of the articular cartilage defect, which ultimately ends in the formation of native tissue characterized by a physiologically limited density. atypical processes are multidirectional during regenerative rehabilitation and, as a rule, do not lead to the formation of tissue with physiological density. in addition, processes that cannot be in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 413 implemented within the framework of the mathematical model under study should be considered atypical. that is, in such cases, solving the system of eqs. (16) is impossible. an illustration of a typical increase in ecm density and the corresponding pattern of changes in nutrient density in space and time are shown in fig. 5, and an example of an atypical process is shown in fig. 6. fig. 5 illustration of changes in state variables of a mathematical model corresponding to a typical regenerative rehabilitation process (test point no. 51): a) change in ecm density (m); b) change in nutrient concentration (n) fig. 6 illustration of changes in state variables of a mathematical model corresponding to a typical regenerative rehabilitation process (test point no. 56): a) change in ecm density (m); b) change in nutrient concentration (n) in typical cases, the growth of ecm density from the very beginning of the process is purposeful. at the same time, the process of nutrient consumption quickly becomes established. in atypical cases, at the initial stage, the process of ecm density growth is multidirectional, and the process of nutrient consumption only after a certain period of time tends to take a stable form. in some cases, the process of an articular cartilage defect regenerative rehabilitation becomes unstable and in most of them cannot be assessed. 414 a. poliakov, v. pakhaliuk 4. conclusion in this paper, a systematic analysis of the regenerative rehabilitation processes of a local articular cartilage defect was performed using a mathematical model represented by a system of non-linear differential equations in partial derivatives (1-6) of the "diffusionreaction" type. this model is quite plausible and makes it possible to judge the dynamics of the process for given parameters values, which are determined mainly on the basis of the experimental studies results in vitro. at the same time, it is known that many parameters of the cellular environment depend on the mechanical stimulation of cells, which largely determines their future fate. taking into consideration this fact, the concept of regenerative rehabilitation of articular cartilage defects was developed, based on the simultaneous use of cellular and rehabilitation technologies. however, for its successful practical implementation, it is necessary to find answers to a number of questions, including: ▪ what should be the parameters values of the cellular environment in vivo, at which a stable, irreversible and high-quality regeneration process of damaged tissue is achieved? ▪ how should mechanical stimulation of cells implanted in the defect area be organized in order to ensure the condition of the damaged tissue with parameters that ensure a stable, irreversible and high-quality regeneration process? the results of the mathematical model study performed in this work allow us to note that there are parameters of the cellular microenvironment of the tissue, in which the regenerative rehabilitation process of an articular cartilage defect is stable, irreversible and can result in complete regeneration of the damaged tissue. the problem of achieving the values of these parameters under in vivo conditions remains topical. one way to find its solution is to compare the parameters of the cellular microenvironment observed after physiotherapeutic procedures with the optimal parameters determined as a result of a mathematical model systematic study. if with the help of the available physiotherapeutic methods and means it will not be possible to ensure the compliance of the parameters, then there will be a need to develop new approaches to the mechanical stimulation of tissues and cells. it is necessary to note a number of limitations adopted in this study. 1. first of all, this concerns the rationale for choosing a one-dimensional model for studying, generally speaking, the spatial process of regenerative rehabilitation. this was done primarily to simplify solutions to the system of nonlinear partial differential equations representing the model. it should be noted that even in the one-dimensional case, obtaining a set of such solutions for each set of state parameters a trial point is a non-trivial task. due to the impossibility obtaining analytical solutions, in this work, to study the model at 56 test points, uniformly distributed (in the sense of i.m. sobol) in an 18-dimensional parameter space, has been used the finite element method, implemented in matlab using the built-in pdepe function on supercomputer at the afalina shared use center of sevastopol state university. 2. in the mathematical model, the size of the defect was neglected and the change in the parameters for the tissue stress-strain state was not taken into account, which did not allow assessing the value of the chondrogenic index and the probability of achieving the desired result the formation of native hyaline cartilage at the end of the regenerative rehabilitation process for the defect. 3. the mathematical model did not consider the possibility of msc differentiation into bone and fibrous tissue cells, because of which the information obtained from the results of its study was incomplete and could potentially contribute to the formulation of incorrect conclusions. in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 415 4. the mathematical model did not take into consideration the possibility of using technologies for implanting autologous chondrocytes or their combinations with msc into the defect area, which limited the obtaining more complete results achieved by methods of regenerative tissue rehabilitation. 5. when determining the density of the formed ecm, the biodegradation for the tissueengineered structure scaffold populated with cells and implanted into the defect area at the initial stage was not taken into account. 6. the boundaries for the mathematical model variable parameters have been chosen, in general, arbitrarily, considering only a limited number of experimental studies in which they were assessed. however, the results obtained in this study indicate a high potential for articular cartilage regeneration under conditions for parallel use of cellular and rehabilitation technologies, at least in the initial stages of osteoarthritis. in the context of this work, the use of neural networks and machine learning methods to predict and control the processes of regenerative tissue rehabilitation is of interest. in further studies devoted to this topic, the mathematical model for regenerative rehabilitation will be refined, which will make it possible to obtain more accurate information about the course of this process in vivo and plan more reliable protocols for the treatment of osteoarthritis. acknowledgement: the paper is a part of the research done at the laboratory of biomechanics at sevastopol state university without external funding. authors are grateful to the management of the “afalina” shared use center at sevastopol state university for the opportunity to perform numerical experiments on a hybrid supercomputer cluster 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b., meldon, j., freed, l., vunjak-novakovic, g., 2000, glycosaminoglycan deposition in engineered cartilage: experiments and mathematical model, aiche journal, 46, pp. 1860-1871. 26. zhou, s., cui, z., urban, j.p., 2004, factors influencing the oxygen concentration gradient from the synovial surface of articular cartilage to the cartilage-bone interface: a modeling study, arthritis rheum, 50(12), pp. 39153924. 27. sobol, i.m., 2020, increasing the efficiency of statistical sampling with the aid of infinite-dimensional uniformly distributed sequences, proc of the 1st world cong of the bernoulli soc, 2020, pp. 743-746. in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 417 appendix a parameters of the mathematical model for regenerative rehabilitation articular cartilage defect parameter functional dependencies and parameter value parameter dimension dimensionless parameter value cs – msc density cs=cs(x,t) cells/mm3 cc – chondrocytes density cc=cc(x,t) cells/mm3 m – ecm density m=m(x,t) g/mm3 n – nutrient concentration n=n(x,t) moles/mm3 g – fgf-1 concentration g=g(x,t) g/mm3 b – bmp-2 concentration b=b(x,t) g/mm3 ds – msc diffusion coefficient 0 2 2 1 s s m d d m m = + mm2/hour d* s – maximum msc diffusion coefficient * 4 33.6 (10 10 )sd − −   mm2/hour dc – chondrocytes diffusion coefficient 0 2 2 1 c c m d d m m = + mm2/hour d* c – maximum chondrocytes diffusion coefficient * 4 33.6 (10 10 )cd − −   mm2/hour p1 – msc proliferation rate max 1 1 s m s c p a c   = −      cells/hour 01 2 2 2 m m a p m m = + max max 0 max 1s s m c c m   = −    p10 – msc proliferation constant 0 * 6 1 2 12 4 10p m p −= =  g/(mm3·hour) 12 p* 1 – maximum msc proliferation rate * 1 0.2p = cells/hour m1 – reference ecm density m1=10-5 g/mm3 0.1 m2 – reference ecm density m2=10-5 g/mm3 0.1 p4 – chondrocyte proliferation rate max 4 1 c m c c p b c   = −      1/hour 0 004 42 2 2 0 m m g b p p m m g g = + + + max max 0 max 1c c m c c m   = −    418 a. poliakov, v. pakhaliuk g0 – fgf-1 reference concentration g0=10-10 g/mm3 p* 4 – maximum chondrocyte proliferation rate * 4 4 2 10p −=  1/hour n1 – critical nutrient concentration n1=9.5·10-12 moles/mm3 0.1 n0 – threshold nutrient concentration n0=2.3·10-11 moles/mm3 0.24 – 0.81 cs0 – msc threshold density 0 0 0 0max min min( ) ab s s s sc c c e c−= − + cells/mm3 cs0 max – maximum threshold msc density 0 ,max 0 max 2 total s c c = cells/mm3 0.35 cs0 min – minimum threshold msc density 0 0min max0.9s sc c= cells/mm3 0.315 p6 – nutrient uptake constant by msc p6=1.5·10-14 moles/(nc·hour) 10000 p7 – nutrient uptake constant by chondrocytes p7=1.5·10-14 moles/(cells·hour) 10000 p8 – ecm synthesis rate 0 18 8 8p p p m= − (g/mm3)/((cells/mm3)· (1/hour)) a – threshold msc density reduction factor a =1010 1/(g/mm3) 100 γ – fgf-1 flux coefficient γ=10-2 mm/hour 0.01 χ – bmp-2 flux coefficient χ=10-2 mm/hour 1 b0 – bmp-2 reference concentration b0=10-10 g/mm3 n0 – initial nutrient concentration n0 = (2.85 – 9.5)·10-11 moles/mm3 in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 419 appendix b general coordinates of test points 1 2 3 4 5 6 7 8 ds0 0.0055 0.0034 0.0078 0.0021 0.0066 0.0044 0.0089 0.0016 dc0 0.001 0.0013 0.0008 0.0011 0.0006 0.0009 0.0014 0.0014 dn 200.0 150.0 250.0 275.0 175.0 225.0 125.0 237.5 dm 0.0055 0.0078 0.0033 0.0089 0.0044 0.0021 0.0066 0.0038 dg 1.0 0.75 1.25 1.125 0.625 1.375 0.875 0.6875 db 1.0 1.25 0.75 0.625 1.125 1.375 0.875 0.5625 p2 1.0 0.75 1.25 0.875 1.375 0.625 1.125 0.9375 p3 1.0 1.25 0.75 0.875 1.375 1.125 0.625 1.0625 p40 0.012 0.0150 0.009 0.0165 0.0105 0.0075 0.0135 0.0158 p400 0.012 0.009 0.015 0.0135 0.0075 0.0165 0.0105 0.0143 p5 1.0 1.25 0.75 1.125 0.625 0.875 1.375 0.5625 p80 0.7 0.45 0.95 1.075 0.575 0.825 0.325 0.3875 p800 0.7 0.95 0.4500 1.075 0.575 0.325 0.825 0.6375 p81 1.0 0.75 1.2500 0.625 1.125 0.875 1.375 1.0625 p9 25.0 26.5 23.5 24.25 27.25 25.75 22.75 23.875 p11 14.5 13.75 15.25 14.125 15.625 13.375 14.875 15.4375 p12 25.0 23.5 26.5 27.25 24.25 25.75 22.75 26.875 p13 14.5 15.25 13.75 14.875 13.375 14.125 15.625 15.0625 9 10 11 12 13 14 15 16 ds0 0.0061 0.0038 0.0083 0.0027 0.0072 0.0049 0.0094 0.0013 dc0 0.0009 0.0007 0.0012 0.0008 0.0013 0.0011 0.0006 0.0010 dn 137.5 287.5 187.5 162.5 262.5 112.5 212.5 181.25 dm 0.0083 0.0061 0.0016 0.0072 0.0027 0.0049 0.0094 0.0030 dg 1.1875 0.9375 1.4375 1.0625 0.5625 1.3125 0.8125 0.9688 db 1.0625 1.3125 0.8125 0.6875 1.1875 1.4375 0.9375 0.7813 p2 1.4375 0.6875 1.1875 0.5625 1.0625 0.8125 1.3125 1.4688 p3 0.5625 0.8125 1.3125 1.4375 0.9375 0.6875 1.1875 0.7813 p40 0.0098 0.0068 0.0128 0.0083 0.0143 0.0173 0.0113 0.0071 p400 0.0083 0.0173 0.0113 0.0068 0.0128 0.0098 0.0158 0.0161 p5 1.0625 1.3125 0.8125 1.1875 0.6875 0.9375 1.4375 0.9688 p80 0.8875 0.6375 1.1375 1.0125 0.5125 0.7625 0.2625 1.1063 p800 1.1375 0.8875 0.3875 0.7625 0.2625 0.5125 1.0125 0.8563 p81 0.5625 1.3125 0.8125 1.1875 0.6875 1.4375 0.9375 1.2188 p9 26.875 25.375 22.375 23.125 26.125 27.625 24.625 25.5625 p11 13.9375 14.6875 13.1875 15.0625 13.5625 15.8125 14.3125 14.0313 p12 23.875 25.375 22.375 23.125 26.125 24.625 27.625 26.6875 p13 13.5625 14.3125 15.8125 13.1875 14.6875 15.4375 13.9375 13.6563 17 18 19 20 21 22 23 24 ds0 0.0058 0.0035 0.0080 0.0024 0.0069 0.0047 0.0092 0.0018 dc0 0.0005 0.0008 0.0013 0.0007 0.0012 0.0015 0.0009 0.0010 dn 281.25 131.25 231.25 206.25 106.25 256.25 156.25 268.75 dm 0.0075 0.0097 0.0052 0.0086 0.0041 0.0018 0.0063 0.0047 dg 1.4688 0.7188 1.2188 1.3438 0.8438 1.0938 0.5938 0.7813 db 1.2813 1.0313 0.5313 0.9063 1.4063 1.1563 0.6563 0.8438 420 a. poliakov, v. pakhaliuk p2 0.9688 1.2188 0.7188 1.0938 0.5938 1.3438 0.8438 1.0313 p3 1.2813 1.0313 0.5313 0.6563 1.1563 1.4063 0.9063 1.3438 p40 0.0131 0.0161 0.0101 0.0176 0.0116 0.0086 0.0146 0.0154 p400 0.0101 0.0131 0.0071 0.0116 0.0176 0.0086 0.0146 0.0109 p5 1.4688 1.2188 0.7188 1.3438 0.8438 0.5938 1.0938 0.9063 p80 0.6063 0.8563 0.3563 0.2313 0.7313 0.4813 0.9813 1.0438 p800 0.3563 0.6063 1.1063 0.4813 0.9813 0.7313 0.2313 1.0438 p81 0.7188 1.4688 0.9688 1.0938 0.5938 1.3438 0.8438 0.6563 p9 22.5625 24.0625 27.0625 27.8125 24.8125 23.3125 26.3125 26.6875 p11 15.5313 13.2813 14.7813 13.6563 15.1563 14.4063 15.9063 14.5938 p12 23.6875 25.1875 22.1875 22.9375 25.9375 24.4375 27.4375 22.5625 p13 15.1563 15.9063 14.4063 14.7813 13.2813 14.0313 15.5313 14.5938 25 26 27 28 29 30 31 32 ds0 0.0063 0.0041 0.0086 0.0030 0.0075 0.0052 0.0097 0.0011 dc0 0.0015 0.0012 0.00072 0.0013 0.0008 0.00059 0.0011 0.0013 dn 168.75 218.75 118.75 143.75 243.75 193.75 293.75 290.625 dm 0.0092 0.0069 0.0024 0.0058 0.0013 0.0035 0.00803 0.0070 dg 1.2813 0.5313 1.0313 1.4063 0.9063 1.1563 0.6563 1.2969 db 1.3438 1.0938 0.5938 0.9688 1.4688 1.2188 0.7188 1.4219 p2 0.5313 1.2813 0.7813 1.4063 0.9063 1.1563 0.6563 1.2344 p3 0.8438 0.5938 1.0938 1.2188 0.7188 0.9688 1.4688 1.3906 p40 0.0094 0.0064 0.0124 0.0079 0.0139 0.0169 0.0109 0.0126 p400 0.0169 0.0079 0.0139 0.0154 0.0094 0.0124 0.0064 0.0159 p5 1.4063 1.1563 0.6563 1.2813 0.7813 0.5313 1.0313 0.7969 p80 0.5438 0.7938 0.2938 0.4188 0.9188 0.6688 1.1688 0.9969 p800 0.5438 0.2938 0.7938 0.4188 0.9188 1.1688 0.6688 1.1531 p81 1.1563 0.9063 1.4063 0.5313 1.0313 0.7813 1.2813 1.0781 p9 23.6875 22.1875 25.1875 25.9375 22.9375 24.4375 27.4375 25.0938 p11 13.0938 15.3438 13.8437 15.7188 14.2188 14.9688 13.4688 13.3281 p12 25.5625 24.0625 27.0625 27.8125 24.8125 26.3125 23.3125 22.4688 p13 13.0938 13.8438 15.3438 13.4688 14.9688 15.7189 14.2188 13.5156 33 34 35 36 37 38 39 40 ds0 0.0056 0.0034 0.0079 0.0023 0.0068 0.0045 0.0090 0.0017 dc0 0.0008 0.00054 0.0010 0.0009 0.0014 0.0012 0.00067 0.00073 dn 190.625 240.625 140.625 115.625 215.625 165.625 265.625 153.125 dm 0.0025 0.0048 0.0093 0.0037 0.0082 0.0059 0.0014 0.0099 dg 0.7969 1.0469 0.5469 0.9219 1.4219 0.6719 1.1719 1.4844 db 0.9219 0.6719 1.1719 1.2969 0.7969 0.5469 1.0469 1.4844 p2 0.7344 1.4844 0.9844 1.3594 0.8594 1.1094 0.6094 1.2969 p3 0.8906 0.6406 1.1406 1.0156 0.5156 0.7656 1.2656 0.9531 p40 0.0066 0.0096 0.0156 0.0111 0.0171 0.0141 0.0081 0.0103 p400 0.0099 0.0129 0.0069 0.0114 0.0174 0.0084 0.0144 0.0107 p5 1.2969 1.0469 0.5469 1.4219 0.9219 0.6719 1.1719 0.8594 p80 0.4969 0.7469 0.2469 0.3719 0.8719 0.6219 1.1219 1.1844 p800 0.6531 0.4031 0.9031 0.2781 0.7781 1.0281 0.5281 0.7156 p81 0.5781 1.3281 0.8281 1.2031 0.7031 1.4531 0.9531 0.5156 p9 22.0938 23.5938 26.5937 27.3438 24.3438 22.8438 25.8438 26.9688 in silico analysis of an articular cartilage regenerative rehabilitation under conditions of ... 421 p11 14.8281 14.0781 15.5781 14.4531 15.9531 13.7031 15.2031 15.3906 p12 25.4688 23.9688 26.9688 27.7188 24.7188 26.2188 23.2188 26.5938 p13 15.0156 15.7656 14.2656 14.6406 13.1406 13.8906 15.3906 14.8281 41 42 43 44 45 46 47 48 ds0 0.0062 0.00395 0.0085 0.0028 0.0073 0.00508 0.0096 0.0014 dc0 0.0012 0.00148 0.00098 0.0011 0.0006 0.00086 0.00136 0.00077 dn 253.125 103.125 203.125 228.125 128.125 278.125 178.125 221.875 dm 0.0054 0.0031 0.0076 0.00198 0.0065 0.0087 0.0042 0.0062 dg 0.9844 1.2344 0.7344 0.8594 1.3594 0.6094 1.1094 1.2031 db 0.9844 0.7344 1.2344 1.3594 0.8594 0.6094 1.1094 1.1406 p2 0.7969 1.0469 0.5469 1.1719 0.6719 1.4219 0.9219 0.7656 p3 1.4531 1.2031 0.7031 0.5781 1.0781 1.3281 0.8281 1.1719 p40 0.0163 0.01331 0.0073 0.0148 0.0088 0.0118 0.0178 0.0129 p400 0.0167 0.0077 0.0137 0.0152 0.0092 0.0122 0.0062 0.0066 p5 1.3594 1.1094 0.6094 1.4844 0.9844 0.7344 1.2344 0.7031 p80 0.6844 0.9344 0.4344 0.3094 0.8094 0.5594 1.0594 0.3406 p800 0.2156 0.4656 0.9656 0.5906 1.0906 0.8406 0.3406 0.5594 p81 1.0156 0.7656 1.2656 0.6406 1.1406 0.8906 1.3906 0.6719 p9 23.9688 22.4688 25.4688 26.2188 23.2188 24.7188 27.7188 22.6563 p11 13.8906 14.6406 13.1406 15.0156 13.5156 15.7656 14.2656 13.7969 p12 23.5938 25.0938 22.0938 22.8438 25.8438 24.3438 27.3438 27.1563 p13 13.3281 14.0781 15.5781 13.7031 15.2031 15.9531 14.4531 13.2344 49 50 51 52 53 54 55 56 ds0 0.0059 0.00367 0.00817 0.00255 0.00705 0.0048 0.0093 0.00198 dc0 0.00127 0.00102 0.00052 0.00139 0.00089 0.00064 0.00114 0.0012 dn 121.875 271.875 171.875 196.875 296.875 146.875 246.875 134.375 dm 0.0017 0.00395 0.00845 0.00508 0.0096 0.0073 0.0028 0.0079 dg 0.7031 1.4531 0.9531 0.5781 1.0781 0.8281 1.3281 1.0156 db 0.6406 0.8906 1.3906 1.0156 0.5156 0.7656 1.2656 1.2031 p2 1.2656 0.5156 1.0156 0.6406 1.1406 0.8906 1.3906 0.7031 p3 0.67188 0.9219 1.4219 1.2969 0.7969 0.5469 1.0469 0.7344 p40 0.0069 0.0099 0.0159 0.0114 0.0174 0.0144 0.0084 0.0092 p400 0.0126 0.00956 0.0156 0.01406 0.00806 0.0170 0.01106 0.0148 p5 1.2031 1.4531 0.9531 1.0781 0.5781 0.8281 1.3281 0.6406 p80 0.8406 0.5906 1.0906 0.9656 0.4656 0.7156 0.2156 0.2781 p800 1.0594 0.8094 0.3094 0.9344 0.4344 0.6844 1.1844 0.3719 p81 1.1719 0.9219 1.4219 0.5469 1.04688 0.7969 1.2969 1.2344 p9 25.6563 27.1563 24.1563 24.9063 27.9063 26.4063 23.4063 23.7813 p11 15.2969 13.0469 14.5469 13.4219 14.9219 14.1719 15.6719 14.7344 p12 24.1563 25.6563 22.6563 23.4063 26.4063 24.9063 27.9063 22.2813 p13 14.7344 15.4844 13.9844 15.1094 13.6094 14.3594 15.8594 14.9219 422 a. poliakov, v. pakhaliuk appendix c results of the mathematical model investigation 1 2 3 4 5 6 7 8 m 0.5982 0.215 0.7321 1.0776 0.3499 0.5466 0.1400 0.3152 cc 1.0376 0.7795 0.8062 0.3101 0.4375 0.7856 0.9931 1.316 cs 0.2779 0.476 0.2032 0.0148 0.3888 0.3077 0.5108 0.4146 fgf1 3.6621 3.0659 4.0465 4.4377 1.5657 6.9173 3.1281 1.8207 bmp2 3.8201 2.3315 5.3087 1.7329 6.9741 4.6249 2.2303 3.577 9 10 11 12 13 14 15 16 m 0.9373 0.4199 1.0681 0.5450 0.6240 0.3824 0.2195 0.8316 cc 0.7614 1.0803 0.5413 0.7406 1.1207 0.9292 1.6943 0.8042 cs 0.0979 0.3612 0.0273 0.3076 0.2650 0.3829 0.4629 0.1486 fgf1 5.0074 4.5112 4.8063 3.1708 2.2640 5.9213 2.4906 3.0667 bmp2 5.3259 3.8617 2.6619 1.4368 7.4866 5.185 3.4894 3.9660 17 18 19 20 21 22 23 24 m 0.6515 0.5012 0.2882 0.2686 0.8423 1.0990 cc 0.9789 1.3741 1.1998 0.5790 0.6557 0.2010 cs 0.2493 0.3152 0.4348 0.4516 0.1506 0.0019 fgf1 3.8033 2.4157 7.9171 3.1099 1.9360 1.9905 bmp2 2.7256 2.8415 2.8215 4.8565 2.0348 2.3157 25 26 27 28 29 30 31 32 m 0.7490 0.3722 0.7919 0.6564 0.8198 0.8320 cc 0.9692 1.3878 0.7556 0.9994 0.6076 0.6377 cs 0.1949 0.3794 0.1730 0.2488 0.1567 0.1472 fgf1 1.3102 3.3675 3.8432 3.6281 3.0314 3.9435 bmp2 4.9400 3.8044 7.3184 8.3667 1.9080 6.3305 33 34 35 36 37 38 39 40 m 0.2013 0.5419 0.0856 0.8856 0.3978 0.8963 1.0964 cc 0.6829 1.3217 0.7386 0.6010 1.4700 0.5758 0.1069 cs 0.4827 0.3055 0.5416 0.1229 0.3653 0.1193 0.0023 fgf1 2.9481 3.5707 1.6303 6.5989 2.4895 4.3165 4.1863 bmp2 2.2565 2.6339 7.6867 5.0906 3.8176 1.9505 6.6589 41 42 43 44 45 46 47 48 m 0.5745 0.7012 0.2586 0.0167 0.4678 0.5642 0.7082 0.1572 cc 0.9367 0.8136 0.9458 0.1744 0.9135 1.3071 0.7062 0.7271 cs 0.2904 0.2307 0.4538 0.5183 0.3407 0.2943 0.2187 0.5069 fgf1 3.4220 3.4029 5.5431 3.4370 6.0843 1.2618 5.6291 4.4711 bmp2 2.4714 2.1959 5.4392 4.1516 2.3561 2.6471 4.0417 5.9224 49 50 51 52 53 54 55 56 m 0.6195 0.3160 0.6739 0.9191 0.4149 0.5855 0.1199 cc 1.070 0.6936 0.7177 0.5920 1.6777 0.9843 1.3427 cs 0.2705 0.4232 0.2367 0.1075 0.3547 0.2877 0.5180 fgf1 2.1485 9.9384 2.8559 1.9884 4.6092 4.4418 3.3682 bmp2 2.1416 1.6862 4.2940 2.0489 5.0816 3.9575 4.0476 facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 143 153 application of smart mobile phones in vibration monitoring 1 udc 62-135:534.1 ljubomir vračar 1 , miloš milovančević 2 , petra karanikić 3 1 faculty of electronic engineering, university of niš, serbia 2 faculty of mechanical engineering, university of niš, serbia 3 technology transfer office, university of rijeka, croatia abstract. the purpose of the research presented in this paper is the development of the smart mobile phone application for vibration monitoring of pumping aggregate, based on microchip’s microcontroller (mc). hardware used is based on bluetooth connection between smart sensor and smart mobile phone. software for acquisition and data analysis is optimized for imbedded application in smart sensors. smart acceleration sensor in conjunction with bluetooth connection to smart mobile phone creates one touch mobile vibration monitoring system. the authors have performed numerous measurements on a wide range of aggregates for establishing the operating functionality of the newly created system. the possibility of system application i rail vehicle vibration monitoring is also analyzed. key words: smart phone, vibration monitoring, smart sensors, bluetooth connection 1. introduction vibration monitoring is one among many methods of technical diagnostics for continuous monitoring of the technical state of a device by observing the level of mechanical oscillation in a real time. the mechanical oscillation is the device manifestation during its operation. certain parts become vibration exciters, others, depending on excitation, react specifically. therefore, the vibration monitoring is one of the most important methods used in technical diagnostics for identifying a technical state. with the use of vibration diagnostics we are able to detect an incipient failure, locate its place and predict the length of time during which a device is going to work before the failure occurs or a preventive action is performed [1, 3, 4, 5, 7]. according to the iso 10816 standard, all machines are sorted into four classes and for each of them references are given for allowed level of vibrations, points for measuring received february 25, 2015 / accepted may 5, 2015 corresponding author: miloš milovanĉević faculty for mechanical engineering, university of niš, a. medvedeva 14, 18000 niš , serbia e-mail: milovancevic@masfak.ni.ac.rs professional paper 144 lj. vraĉar , m. milovanĉević, p. karanikić vibrations, way of measuring and selection of measuring parameters. it is possible to carry out further fusion of certain machine classes, regarding the possibility of causing prospective dynamic problems, into two general groups: group 1 (g1) class i machines according to iso 10816-1, small machines, typically electric motors up to 15 kw. class ii machines according to iso 10816-1, medium machines, electro motors with power from 15 kw to 75 kw without special foundation, rotational machines with special foundation and power up to 300 kw. the primary characteristic of this machine group is embedment of rolling bearings. the most common causes of dynamic problems with this group of machines are: imbalance, defects at rolling bearings, misalignment and support errors. less frequent are errors of electrical origin, damages at gears, errors caused by the effects of aerodynamic and hydraulic force, defects in belt transmitters, defects caused by pulsation and between rotor and stator [2, 3, 4, 6, 9, 10]. group 2 (g2) class iii machines according to iso 10816-1, large machines on rigid foundations class iv machines according to iso 10816-1, large machines on soft foundations the machines of this group may have sleeve or rolling element bearing. the most common causes of dynamic problems with this group of machines are: imbalance, misalignment, malfunctions on roller and sleeve bearings, support errors and resonance phenomenon. less frequent are errors of electrical origin, errors caused by the effects of aerodynamic and hydraulic force, contacts (friction) between rotor and stator, and anisotropy of rotor [1, 8]. based on our experimental research (a long-term one with over a thousand turbo aggregates examined) and using other sources from the referential literature [2, 5, 6, 7], frequency spectra components are statistically analyzed regarding causal dynamic problems in real exploitation conditions for these two groups of machines, as shown in the following tables. they contain data obtained in a scientific research project of turbo pumps exploitation optimization, conducted at the city of niš waterworks system “naissus”: table 1 possibility of certain defect appearance at machines (g1) damage type misalignment 17.5% errors at roller and sleeve bearings 27.5% support errors 7.5% imbalance 40.0% other defects 7.5% table 2 possibility of certain defect appearance at machines (g2) damage type misalignment 15% errors at roller bearings 15% support errors 15% imbalance 25% other defects 15 % application of smart mobile phones in vibration monitoring 145 2. vibration characteristics vibrations occur as a result of rotating or straight-line moving bodies. the course of vibrations is mainly under the influence of technical states of single machine components such as shafts, gear boxes, crank mechanisms, cam mechanisms, antifriction bearings, as well as by imbalances of rotating parts, backlash in friction bearings, wear, material fatigue, occurring cracks, corrosion and other parameters affecting a smooth machine run. the vibration itself is then defined as a dynamic phenomenon when particles or solid bodies move around a zero equilibrium position. they are given by a combination of six movements, namely by a shift in orthogonal coordinate system x, y, z and rotation about these axes. vibrations can be described by amplitude and a phase at a certain period of time. depending on the time variations of values, the vibrations are of a periodical, nonperiodical or random character. as for periodical vibrations, a time course of vibration measured values repeats. a harmonic vibration which has a sinusoidal waveform is based on these vibrations. for harmonic vibrations we need to set only one determining value and the other ones can be calculated. the basic way of describing oscillations is to determine their displacement x, velocity v, acceleration a, maximum amplitude xmax, a root of mean square xrms, and an absolute value xave (fig. 1). the measurement of displacement x is convenient for low-frequency events such as measuring backlashes, etc. which might be calculated the following way: max sinx x t (1) 2 f  (2) where xmax maximum amplitude (maximum displacement), ω angular frequency, f frequency (oscillation), t time. velocity can be expressed as the characteristics of motion which informs us about the way of changing the position of a body (particle) in time. velocity is a vector physical value because it defines both the change magnitude and its direction. velocity might be determined as the time derivation of trajectory (displacement) using the equation given below: max cos dx v x t dt    (3) acceleration can be expressed as the characteristic of motion which shows the way the velocity of a body (particle) changes in time. the acceleration is a vector physical value since it gives both the change magnitude and its direction. it is possible to calculate momentary acceleration and average acceleration. the acceleration might be also determined as the time derivation of velocity using the formula below [11].  2 2 max maxsin sin dv a x t x t dt          (4) if the acceleration has the orientation opposite of motion, it is called deceleration and has a negative sign. 146 lj. vraĉar , m. milovanĉević, p. karanikić fig. 1 harmonic oscillation with the illustration of maximum amplitude xmax, a root of mean square xrms, and an absolute value xave mean absolute value xave can be expressed as follows: 0 1 ave t x x dt t   (5) where t – a period expressed by the formula. the root of mean square can be calculated by the equation below [12, 13]: 2 max 0 1 1 2 t rmsx x dt x t   (6) in order to interpret the measured values correctly, it is advisable to transform the oscillation time course into a frequency domain, i.e. vibrations are to be replaced by a sequence of its oscillation components. it can be said that the time signal contains the information about when a certain event occurred, but the frequency spectrum contains the information about how often the same event occurs in an observed signal. the procedure during which complex signals are subdivided into their frequency components is called a frequency analysis which applies either selective band-pass filters or more often a fast fourier transformation (fft). along with the fft also a wavelet, cosine or walshhadamard transform can be used for expressing a signal by orthogonal basis functions. in the paper we have applied a fast fourier transformation; therefore, we introduce the formula expressing its transformation which is as follows [12]: 2( ) ( ) j ftf f x t e dt      (7) where f – frequency, j – imaginary unit, x(t) – continuous signal. application of smart mobile phones in vibration monitoring 147 3. realized embedded system although measuring systems used for diagnostics and vibration measurement already exist at the market, the system that has been described in this paper has certain advantages. first, this system enables the user to measure and monitor vibration from a distance, more precisely, the user does not have to be near the place where the measuring is taking place. this is important, primarily, at potentially dangerous sites, big industrial systems, where some unpredictable malfunctioning may occur. it is also important in a situation when the measuring has to be made in the environment that can be harmful for human health, that is, the environment containing some poisonous substances, or the like. another advantage of this system is the fact that monitoring can be done by widely available devices, such as laptop computers and smart phones, via the embedded bluetooth connection. all diagnostic measurements and calculations (dfft transformations) are done within the embedded microcontroller, so the diagnostic accuracy is not dependent on the performance or the state of the operative system (software) of the user’s device. the system that has been realized contains three major units: the sensor with the signal acquisition circuit, the microcontroller with the proper software and the bluetooth module (fig. 2). the vibration sensor used is of the analogue type, and in order to achieve the satisfactory measuring accuracy, the external 12 bit a/d converter has been used. the microcontroller is attached to the a/d converter by the spi connection, and to the bluetooth module by the rs232 port [14, 15]. fig. 2 block diagram of the realized embedded system vibration sensors are used for measuring physical values (displacement, velocity and acceleration) and their transformation into an electric signal which is later worked with. there is a great variety of different vibration sensors which are used for achieving results as accurate as possible. we have the sensors of acceleration, velocity and displacement. the principle of the vibration sensor function is the motion of seismic matter of mass m towards an object of mass m whose vibrations are measured. for the calculation the formula below is used: 0hma bv ky ma   (8) 148 lj. vraĉar , m. milovanĉević, p. karanikić where y displacement, v velocity, ah acceleration of seismic matters motion, ao object acceleration, m seismic matters mass, k spring stiffness, b damping coefficient [2]. for the system realized the accelerometer adxl 311 by analog devices company is used [1, 13, 14]. it is a dual-axis sensor realized in the imems technology with the analogue voltage output. the sensor can measure dynamic and static acceleration ranging ±2 g, which is sufficient for the realized application (fig. 3). capacitors c1 and c2 together with the inner resistance of the sensor make a low pass filter and define the maximum frequency detected by the sensor: 3 1 5 2 32 db f f k c c        (9) therefore, with the standard defined valued for c1 and c2 of 4.7 nf, maximum frequency is slightly higher than 1 khz. fig. 3 detailed scheme of the part of the system with the sensor the operational amplifier mcp6022 12 has been used in the configuration unity gain and it presents an impedance buffer. a/d converter mcp3202 13 has two channels with 12 bit resolution and it communicates with the microcontroller via spi connection. the software enables the switching of the measuring channels and reading of the measured voltage values. the chosen microcontroller is the pic18f45k22 14, by microchip company. it has been chosen for its stability with the interior oscillator at even 64 mhz with the 3.3 v voltage as well as for its containing sufficient software and operational memory necessary for the complicated calculation of the dfft analysis, fig. 4. all measuring and dfft analysis are done inside the microcontroller, so that the final results are independent of the device and the software that user applies. application of smart mobile phones in vibration monitoring 149 fig. 4 the scheme of connecting the bluetooth module and the microcontroller in order to connect the system to the laptop computer or the smartphone and in order to read out the measured results the bluetooth connection is used, via the rn41 module by microchip. this module has the embedded ceramic chip antenna, which makes the dimensions of the system even smaller. it is connected to the microcontroller by the standard rs232 connection. 4. application of wi-fi system in industry 4.1. vibration monitoring of water pumps horizontal pumps play a significant role in water transportation which also defines the importance of providing for flawless work. electro motors of horizontal pumps are extremely burdened regarding their continuous exploitation in terms of maintaining permanent operation. an adequate choice of measuring point at the pump aggregate of the horizontal pump can indicate the operational condition of electromotor bearings and rotor, the pumping aggregate bearings and coupling, as well as complete aggregate construction. following measuring points are chosen (fig. 5):  first measuring point is chosen for diagnosing the operational condition of the first bearing at electromotor.  second measuring point is intended to diagnose the driving electromotor second bearing condition.  third measuring point is determined in such a manner that it is possible to diagnose both the condition of pump first bearing and of the elastic coupling.  fourth measuring place is defined to diagnose the pump second bearing condition. 150 lj. vraĉar , m. milovanĉević, p. karanikić fig. 5 measuring point at horizontal pump aggregate cvnr 5-3, no.1 measuring result analysis is generated by using frequency spectra. the presented diagrams are created from fft algorithm, adapted for pump aggregate diagnostics. measuring point 1, fig. 6. a), horizontal and vertical acceleration not passing 1 m/s² can be observed, indicating the proper electromotor (em) bearing operating condition. also there is no vibration in frequency range 700-900hz which indicates that the motor fan is installed correctly. measuring point 2, fig. 6. b), high acceleration amplitude at frequency at 310hz is manifestation of an incorrect coupling working condition, the second electromotor bearing is in a good operating condition. measuring point 3, fig. 6. c), based on the acceleration, the correct bearing operation can be determined as well as an incorrect coupling operating condition based on the analyses of previous diagrams. measuring point 4, fig. 6. d), a satisfactory operating condition of the second bearing of pump can be determined. diagram presented in fig. 6 (a, b) points to the following facts: for electromotor it is possible to determine bearing malfunction as well as other mechanical defects such as an incorrect coupling operating condition. diagram presented on a figure 6 (c, d) presents pump bearing malfunction but also a high frequency range is appearing as result of the hydro-dynamic processes in a pump. in order to understand the results previously presented, it is necessary to emphasize that the above presented frequency amplitude diagrams are the result of several years of work aimed at determining operating condition for most pump aggregates used in industry. the data presented is a small segment of the research. over 230 pump aggregates have been analyzed in order to improve the mathematical apparatus and software for vibration analysis. application of smart mobile phones in vibration monitoring 151 fig. 6 fft diagrams for horizontal pump aggregate cvnr 5-3, no.1 at measuring points (mp), a) mp 1, b) mp 2, c) mp 3, d) mp 4. 152 lj. vraĉar , m. milovanĉević, p. karanikić 4.2. vibration monitoring of railway vehicles complex conditions of exploitation of railway vehicles as well as the conditionality of the high level of safety and security require a high reliability operation of all vehicle components. to ensure a failure-free rail vehicles operation it is necessary to carry out diagnostics of working correctness based on the given parameters. the complexity of identifying the parameters on the basis of which the vehicle diagnosis is done is worked out in detail in the uic regulations. value of dynamic characteristics and their measurement technique, recording and analysis are given in uic 518 standard. wi-fi system for vibration monitoring is easy to customize according to regulations of standard uic 518. the granting of the exploitation permit for a railway vehicle, regarding its dynamic behavior, should be based on an experimental examination (rather than on simulations). it is important to comply with the provisions of the uic 518 concerning the conditions under which testing is performed as follows: track geometry, the characteristics of the vehicle and driving conditions and characteristics of the zone for testing (on the right tracks, in large radius curves, small radius curves, state vehicles (empty, loaded ...)). thus the application of wi-fi system for vibration monitoring can be useful since system is compact and mobile. vibrations of mobile mechanical systems such as railway vehicles are more complex and more difficult to determine. thus, vibrations are consequences of a large number of relative and absolute motions of elements of rail vehicles and therefore the wi-fi system can be used for conducting complex condition monitoring. 5. conclusions examination of the pumps vibration phenomenon provides the data about the vibration magnitude and its frequency components as well as their change with respect to operating parameters. on the basis of obtained results the safety level for pump and the whole plant is evaluated. to this we should add that, in most cases, it is necessary to determine the cause of non-stationary occurrences. operating ranges that should be avoided are determined in many cases. primary sources of vibrations at centrifugal pumps are mechanical, hydraulic and electric processes caused by the design of a pump, its manufacturing technology, operating regime and exploitation condition. it is possible to eliminate the mechanical and electrical sources, partially or completely, lowering the vibrations level in that way. however, hydraulic vibrations are hard or almost impossible to avoid. hydraulic processes which happen in pumps are complex and non-stationary as a rule. for description of such processes it is possible to form mathematical models whose evaluation is performed after very comprehensive, expensive and long-lasting research projects. for these reasons such models are not taken into consideration in this paper – given are the experimental results obtained by newly developed embedded system, based on the new generation of microcontrollers. in the diagnosis of pumping aggregate malfunctions, frequency spectrums have a crucial role in defining the causes of failure. the created monitoring system has significant results in frequency vibration analyses regarding mechanical defects detection of the pumping aggregate. application of smart mobile phones in vibration monitoring 153 acknowledgements: this research was supported by the ministry of education, science and technological development, republic of serbia, project number tr-32026. references 1. matić 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http://ww1.microchip.com/downloads/en/devicedoc/41412f.pdf http://ww1.microchip.com/downloads/en/devicedoc/rn-41-ds-v3.42r.pdf http://ww1.microchip.com/downloads/en/devicedoc/rn-41-ds-v3.42r.pdf plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 249 257 finite-element modelling of the tympanic membrane retraction pocket under negative pressure in the tympanic cavity  udc 531/534:[57+61] gennadi i. mikhasev 1 , sergei m. bosiakov 1 , lyudmila g. petrova 2 , marina m. maisyuk 2 1 belarusian state university, belarus 2 state enterprise "national centre of otorhinolaryngology", belarus abstract. the finite-element calculation of the static stress-strain state of the middle ear was made in this paper. the malleus, incus and stapes models were constructed on the basis of tomographic data. the tympanic membrane model was obtained using the equations of elliptic hyperboloids. the tympanic membrane consists of the pars tensa and pars flaccida, which have different thicknesses and elasticity moduli. absolute deformations of the tympanic membrane were defined at different values of negative pressure in the tympanic cavity. the critical values of elastic modulus for the pars tensa posterosuperior quadrant were found for the point at which the tympanic membrane touches the auditory ossicles. obtained results can be used to predict the thickness of a cartilaginous graft which is overlaid on the posterosuperior quadrant of the pars tensa in order to eliminate the retraction pocket. key words: middle ear, tympanic membrane, retraction pocket, finite element analysis, strain-state state, negative pressure 1. introduction the retraction pocket of the tympanic membrane (tm) is a clinical manifestation of atelectatic otitis media. for many reasons this pathology of the middle ear deserves special attention. in particular, it is necessary to correctly assess the nature of the pathogenesis and best formulate a treatment strategy to repair retraction pockets that have formed as a result of the cholesteatoma growth in the middle ear [1, 2]. two factors may lead to the formation of the tm retraction: а) pressure imbalance in the middle ear cavity received april 1, 2015 / accepted july 2, 2015 corresponding author: sergei mihailovich bosiakov belarusian state university, 220030, minsk, nezavisimosti ave., 4, belarus e-mail: bosiakov@bsu.by original scientific paper 250 g.i. mikhasev , s.m. bosiakov , l.g. petrova , m.m. maisyuk (long persisting negative pressure in the middle ear cavity); b) changes in the structure of tm itself (thinning and changing physical properties) [3]. the tm consists of two parts [4]: pars tensa (which is mainly responsible for sound transmission) and pars flaccida (for the detection of pressure differentials). the elasticity of the pars tensa is associated with its fibrous layer, which is absent in the pars flaccida. the posterosuperior quadrant of the pars tensa has a weak network of only circular collagen fibers, which is why this part of the tm has a thickness of about 60 µm, while the thickness of the rest of pars tensa is approximately 90 µm [5, 6]. therefore, the pars flaccida and the posterosuperior quadrant of the pars tensa are the most anatomically vulnerable areas of the tm for the formation of retraction pockets. the process of retraction is accompanied with irreversible changes in the tm structure and consequently, with a significant reduction of its elastic characteristics and thickness. for this reason, the weakened parts of pars tensa come into contact with the underlying auditory ossicles (the long crus of incus, the incudo-stapedial articulation). this inevitably leads to their erosion [7] and epidermal ingrowth into the middle ear cavity with cholesteatoma formation. retraction pockets are therefore dangerous because they are asymptomatic for a long time but ultimately lead to a permanent irreversible hearing loss. an «early» surgical treatment of the retraction pocket is a preventive measure against middle ear cholesteatoma development. surgery typically aims at mounting a cartilaginous graft in the area of the posterosuperior quadrant of the pars tensa in order to increase the rigidity of this part of the tm and preserve its auditory function [8]. therefore, it is useful to calculate the magnitude of the tm deformation at different levels of negative pressure in the tympanic cavity (tc) in order to determine a suitable thickness for the cartilaginous graft. this paper is dedicated to finite-element modeling of the stress-strain state (sss) of the middle ear tm at different values of negative pressure as well as those of elasticity moduli of the pars tensa. two cases are taken into consideration: when the elastic characteristics of pars tensa are within normal values; and when the elasticity modulus values are low. 2. the geometric and the finite-element middle ear models a solid 3d model of the tm was obtained by gluing different parts of the pars tensa and pars flaccida. the thickness of the pars flaccida was 30 µm. the posterosuperior quadrant of the pars tensa consisted of two layers, and the final part consisted of three layers. the thickness of the posterosuperior quadrant of the pars tensa was 60 µm, and the final part was 90 µm. the dimensions of the tm are indicated in fig. 1 (all dimensions are in mm). solid 3d models of the malleus, incus and stapes were generated in solidworks 2010 (solidworks corporation, usa) using a computed tomography of the middle ear. the characteristic dimensions of some middle ear models are shown in fig. 2. the finite-element middle ear model is shown in fig. 3. contact between the tm, handle of the malleus and middle ear ossicles is described by means of the ‘bonded–type’ (owing to the absence of slip and penetration). all the gaps between contacting surfaces are closed in this type of contact, i.e. the contacting surfaces are bonded. the finite finite-element modelling of the tympanic membrane retraction pocket... 251 element segmentation was carried out in a semi-automatic mode. maximal dimensions of the finite-element rib for the middle ear ossicles and tm were 0.3 mm and 0.1 mm, respectively. the number of nodes and elements used for the middle ear model components is shown in table 1. the transverse dimension of the finite element rib for each layer of the tm was 30 µm – both for the pars tensa and pars flaccida. the tm is rigidly fixed in an external circuit in the tympanic ring (tr) [8]. the variance of anchorage corresponding to ossification of the tm and tr contact area was considered. negative pressure in the tc, leading to tm retraction and retraction pocket formation was modeled by applying a static, evenly distributed load on the outer side of the tm. fig. 1 geometrical dimensions of the tm: a) external view, b) plane section of membrane x = 0 (0a1a2a3 is the posterosuperior quadrant of the pars tensa, a2a4a5 is the pars flaccida) fig. 2 geometric model of the middle ear and its characteristic dimensions: a) internal view; b) side view; c) external view a) b) c) 252 g.i. mikhasev , s.m. bosiakov , l.g. petrova , m.m. maisyuk table 1 number of nodes and elements in the middle ear model middle ear model component number of nodes number of elements tm 274 341 135 852 pars flaccida 3 984 1 849 malleus 12 097 7 631 incus 16 308 10 424 stapes 4 600 2 449 fig. 3 discrete model of the middle ear: a) finite element models of the tm, malleus, incus and stapes; b) finite-element segmentation of tm layers in the area of pars tensa the elastic modulus of the malleus, incus and stapes is 13.7 gpa, poisson’s ratio was 0.3 (this corresponds to compact bone tissue [9]). the elastic moduli of the pars tensa (excluding the posterosuperior quadrant 0a1a2a3) and pars flaccida are 33 mpa and 11 mpa, respectively. the poisson’s ratio for these two areas is 0.4 [10, 11]. the elastic modulus of the posterosuperior quadrant 0a1a2a3 has different values when modeling middle ear sss for the normal and pathologically affected states of the pars tensa. 3. tympanic membrane deformations in the normal state the finite-element calculation of the middle ear sss was carried out using ansys workbench 14 (ansys inc., usa) for different values of pressure on the tm. fig. 4 shows simulated deformations of the tm and middle ear ossicles for the standard pressure, ps = 20 pa (pspl = 120 db), which is caused by the static pressure difference between the ear canal and the tc. critical sound pressure in decibels (db) is defined based on the weberfechner law [12], as pspl = 20 log (ps/prs), prs = 210 5 pa. the elastic modulus of the posterosuperior quadrant 0a1a2a3 is 33 mpa. fig. 4 shows that maximal absolute deformations (approximately 3.060 µm) are observed with negative pressure of 20 pa in the central part of the pars flaccida (see fig. 4, a, b). a large deformation also occurs in the pars tensa with maximal values of approximately 1.7 µm (see fig. 4, b). other parts of the tm also undergo deformation besides the pars flaccida and the pars tensa (the range of absolute deformations is from 1.0 µm to 1.7 µm). a) b) finite-element modelling of the tympanic membrane retraction pocket... 253 fig. 5 shows the middle ear sss under negative pressure of 34.0 kpa (184.6 db). this corresponds to the pressure that develops when the incus touches the tm (i. e. when the minimal distance between them is 0.1 nm). fig. 4 absolute deformations of middle ear under negative pressure of 20 pa: a) internal view; b) external view; c) side view fig. 5 absolute deformations of middle ear under negative pressure of 34,0 kpa: a) internal view; b) external view; c) side view fig. 5 shows that the maximal value of pars tensa displacements is approximately 5.20 mm. the direction of maximal deformation is that in which the tm makes contact with the side part of incus. this corresponds to a clinical manifestation of the retraction pocket in the posterosuperior quadrant of the tm (pre-cholesteatoma) [13, 14]. the obtained values of displacements correspond to the documented results [12], according to which the distance between the posterior point on the tm and head of the stapes is approximately 3.4521 mm. maximal deformation of the pars flaccida is also relatively high at approximately 3.468 mm. values for the pars flaccida and pars tensa maximal deformation and distribution (see fig. 5, с) show that significant sagging of these parts may be observed in overpressure. the maximum displacement of other tm parts does not exceed 2.31 mm. 4. tm deformations in pathological changes of pars tensa elastic properties negative pressures are defined as the point where the incus touches the tm, as in the case when recurrent inflammatory diseases in the middle ear cause a decrease in the elastic modulus of the pars tensa [15]. the condition of the incus and tm contact a) b) c) a) b) c) 254 g.i. mikhasev , s.m. bosiakov , l.g. petrova , m.m. maisyuk remained unchanged. tables 2 and 3 show values for the maximal displacements of the tm in the posterosuperior quadrant with an elastic modulus between 9 mpa and 11 mpa. absolute middle ear displacements under pressures between 6.7 kpa and 8.3 kpa are shown in figs. 6 and 7. posterosuperior quadrant elasticity moduli are 9 mpa and 11 mpa, (the elastic moduli of other parts of the pars tensa and pars flaccida are 33 mpa and 11 mpa, respectively). fig. 6 absolute deformations of the middle ear under negative pressure of 6,7 kpa: a) internal view; b) external view; c) side view table 2 maximum deformations of the tm, when the elastic modulus of the posterosuperior quadrant of the pars tensa is 9 mpa, for different values of negative pressure (the pressure level in the last column corresponds to contact between the incus and tm). pressure ps, kpa 0.5 1.0 2.0 4.0 6.7 pressure pspl, db 148.0 154.0 160.0 166.0 170.5 maximum deformation of the tm, mm 0.2443 0.4886 0.9771 1.9542 3.2733 table 3 maximum deformations of the tm, when the elastic modulus of the posterosuperior quadrant of the pars tensa is 11 mpa, for different values of negative pressure (the pressure level in the last column corresponds to contact between the incus and tm) pressure ps, kpa 0.5 1.0 2.0 4.0 6.7 8.3 pressure pspl, db 148.0 154.0 160.0 166.0 170.5 172.4 maximum deformation of tm, mm 0.2023 0.4046 0.8093 1.6186 2.7111 3.3585 figs. 6 and 7 show that the qualitative distribution of the absolute deformations in the posterosuperior quadrant is virtually identical for models with different elastic moduli of the pars tensa. the quantitative difference lies in absolute deformations of tm. practical interest lies in the assessment of the elastic modulus of the posterosuperior quadrant of the pars tensa at which contact occurs between the tm and incus at pressure of 20 pa. finite-element calculations show that contact between the middle ear ossicles and the posterosuperior part of the tm is observed below an elastic modulus of 32.5 mpa. the a) b) c) finite-element modelling of the tympanic membrane retraction pocket... 255 distribution of absolute deformations in the middle ear under negative pressure of 20 mpa, and with a posterosuperior quadrant elastic modulus of 32.5 mpa, is shown in fig. 8. fig. 7 absolute deformations of the middle ear under negative pressure of 8,3 kpa: a) internal view; b) external view; c) side view fig. 8 shows the deformation of only the posterosuperior quadrant of the pars tensa for negative pressure of 20 pa and a posterosuperior elastic modulus of 32.5 mpa. there are no absolute deformations of other middle ear parts. fig. 8 absolute deformations of the middle ear (with an elastic modulus of 32.5 mpa for the posterosuperior quadrant of tm): a) internal view; b) external view; c) side view 5. mesh convergence the grid convergence was studied for the tm finite element models with the maximum thickness of the finite element rib equal to 0.2 mm, 0.1 mm and 0.05 mm. the increase in the membrane maximal deformation in the second segmentation variant was 5% when compared to the first one. the increase in the maximal deformation in the third variant of segmentation was 0.3% compared to the second one. the second variant of the tm sampling was accepted on this basis in the middle ear finite-element calculation. the decrease in the element rib thickness for the middle ear ossicles models (less than 0.3 mm) did not lead to the tm maximal deformation change. a) b) c) a) b) c) 256 g.i. mikhasev , s.m. bosiakov , l.g. petrova , m.m. maisyuk 5. conclusion the following results were obtained during the finite-element simulation of a middle ear model with negative pressure in the tc:  maximal deformation of the tm is observed in the posterosuperior quadrant of the pars tensa for all values of negative pressure.  the largest deformation observed for the standard negative pressure (ps = 20 pa) was 3.060 µm.  the negative pressure corresponding to contact between the tm and incus was 34 kpa; the maximal deformation of the tm in this case was 5.2 mm (normal elastic properties).  the negative pressure corresponding to contact between the tm and incus, with an elastic modulus between 9 mpa and 11 mpa for the posterosuperior quadrant of pars tensa, was 6.7 kpa and 8.3 kpa, respectively (the elastic moduli for the final part of pars tensa and pars flaccida were 33 mpa and 11 mpa, respectively).  the elastic modulus of the posterosuperior quadrant of the pars tensa corresponding to contact between the tm and middle ear ossicles under the standard pressure of 20 pa was 32.5 mpa.  the distribution of absolute deformations of the tm with different and identical elasticity moduli for the pars flaccida and pars tensa under various negative pressures differ from each other. this shows the need to describe the elastic characteristics of the pars flaccida and pars tensa with different elasticity moduli when calculating the stress-strain state in the middle ear. the qualitative analysis of the above results allows us to make the following conclusions which could assist both researchers and clinicians:  it is the reduction of the tm elastic properties in the posterosuperior quadrant of the pars tensa that may result in the retraction formation.  to increase the rigidity of a weakened segment of the eardrum it is recommended to apply an early surgical approach involving the installation of a cartilage transplant.  one needs to perform further finite-element simulations in order to study the effect of the cartilage transplant and estimate an optimal thickness of a cartilaginous graft. acknowledgements: this paper is the result of project implementation: «trans-atlantic micromechanics evolving research: materials containing inhomogeneities of diverse physical properties, shapes and orientations» supported by fp7-people-2013-irses marie curie action «international research staff exchange scheme». references 1. ars, b., decraemer, w., 1989, tympanic membrane lamina propria and middle ear cholesteatoma, proc. third international conference on cholesteatoma and mastoid surgery 1989, amsterdam / new york, kugler publications, pp. 429-432. 2. ohinishi, t., shirahata, y., fukami, m., hongo s., 1985, the atelectatic ear and its classification, auris-nasus-larynx, 12(1) s211-s213. 3. esteve, d., dubreuil, ch., delia vedova, cl., normand, b., lavieille, j. p., martin, ch., 2001, physiologic et physiopathologie de la fonctiond'ouverture de la trompe auditive apports de la finite-element modelling of the tympanic membrane retraction pocket... 257 tubomanometrie, journal français d'oto-rhino-laryngologie, audiophonologie, chirurgie maxillofaciale, 50(5) pp. 233-241. 4. schrapnell, h., 1832, on the form and structure of the membrane tympani, london medical gazette, 120 pp. 120-124. 5. ars, b. ars-firel, n., 1991, dynamics of the organogenesis of the middle ear structures. anatomical variants, middle ear structures, organogenesis and congenital defects, в. ars and p. van camvenberge eds., amsterdam / new york, kugler publications, pp. 11-25. 6. ars, b, ars-pirct, n., 1997, morpho-functional partition of the middle ear cleft, acta oto-rhinolaryngologica belgica, 51 pp. 181-184. 7. borqstein, j., gerritsma, t., bruce, i., 2008, erosion of the incus in pediatric posterior tympanic membrane retraction pockets without cholesteatoma, international journal of pediatric otorhinolaryngologie, 72(9) pp. 1419-1423. 8. neumann, a., jahnke, k., 2005, die trommelfellrkonstruktion mit knorpel: indikationea. techniken und ergebnisse, hno, 53 pp. 573-586. 9. tanne, k., nagataki, t., inoue, y., sakuda, m., burstone, c. j., 1991, patterns of initial tooth displacements associated with various root lengths and alveolar bone heights, american journal of orthodontics and dentofacial orthopedics, 100 pp. 66-71. 10. wada, h., koike, t., kobayashi, t.,1997, three-dimensional finite-element method (fem) analysis of the human middle ear, middle ear mechanics in research and otosurgery (k.-b. hüttenbrinked.), dresden: department of otorhinolaringology, university of technology, pp. 76-81. 11. kirikae, j., 1960, the middle ear, tokyo, the university of tokyo press, 196 pp. 12. mikhasev, g., ermochenko, s., bornitz, m., 2010, on the strain–stress state of the reconstructed middle ear after inserting a malleus–incus prosthesis, mathematical medicine and biology, 27, pp. 289-312. 13. sudhoff, h., tos, m., 2000, pathogenesis of attic cholesteatoma: clinical and immunohistochemical support for combination of retraction theory and proliferation theory, american journal of otolaryngology, 21(6), pp. 786-792. 14. hours, a. f., decat, m., gersdorf, m., 1998, our classification of tympanic retraction pockets, acta oto-rhino-laryngologica belgica, 52, pp. 25-28. 15. sade, j., berco, e., 1976, atelectasis and secretory otitis media, annales oto-rhino-laryngologie, 85(25), pp. 66-72. 16. koike, t., wada, h., kobayashi t., 2000, analysis of the finite-element method of transfer function of reconstructed middle ears and their postoperative changes, the function and mechanics of normal, diseased and reconstructed middle ear, j. j. rosowski and s. n. merchant eds., the hague, the netherlands: kugler publications, pp. 309-320. 11384 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 265 286 https://doi.org/10.22190/fume230215017z © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper modeling and simulation of industrial robot arms using simscape multibody cheng zhang1, 2, songxiao li3, zhuo zhang4 1faculty of innovation engineering, macau university of science and technology, china 2school of electronic and information engineering, zhuhai college of science and technology, china 3department of mathematics, shantou university, china 4college of robotics, guangdong polytechnic of science and technology, china orcid ids: cheng zhang https://orcid.org/0009-0009-4892-2374 songxiao li https://orcid.org/0000-0001-7911-2758 zhuo zhang https://orcid.org/0009-0009-6240-0046 abstract. the dynamic simulation modeling problem of industrial robot arm is solved, and the trajectory planning dynamic simulation is performed in this paper. in response to the lack of trajectory planning and motion controller interfaces in the robotic modelling study, including the lack of dynamic simulation visualization, a simscape multibodybased method for building a dynamic model of industrial robot arm is proposed and the effectiveness of the model is verified through dynamic simulation. the simulation model integrates the robotic arm trajectory planning, motion controller and data acquisition module. it has a clear structure and the parameters are easy to modify. it can reasonably simulate the structure and parameters of the research object and facilitate the subsequent research of related algorithms. it provides an innovative and open-source research and development platform for the dynamic simulation study of the robot arm. key words: robotic dynamic model, industrial robot arms, robot simulation system, simscape multibody 1. introduction robots are known as the "pearl at the crown of manufacturing". at present, the world is experiencing the wave of "machine replacement", and artificial intelligence has become a strategic technology leading a new round of scientific and technological revolution and industrial transformation. as a representative of artificial intelligence, industrial robots integrate advanced technologies in the fields of machinery, electronics, sensors, wireless communication, sound recognition, image processing and artificial intelligence, involving received: february 15, 2023 / accepted may 28, 2023 corresponding author: songxiao li department of mathematics, shantou university, 515063, shantou, guangdong, china e-mail: jyulsx@163.com https://orcid.org/0009-0009-4892-2374 https://orcid.org/0000-0001-7911-2758 https://orcid.org/0009-0009-6240-0046 266 c. zhang, s. li, z. zhang multiple disciplines. it is a country's level of science and technology development and an important symbol of national economy modernization and informatization. at present, various industries have integrated industrial robots into the production line, and made use of industrial robot technology to meet the actual needs of the development of circular economy and improve the development of time efficiency, making the unique advantages of higher production efficiency and lower cost. compared with traditional industrial processing auxiliary equipment, industrial robots have lower maintenance costs, higher degree of intelligence and better flexibility to meet the requirements of modern production. it can better adapt to a variety of varieties, small batch, on-site processing requirements. on the whole, industrial robots have become a wide range of tools for various tasks in modern society, forming a new generation of "intelligent robot manufacturing"[1]. under these circumstances, it is particularly important to further research on industrial robot related technology. many scholars have carried out research and achieved many achievements, including but not limited to the problems of industrial robot grasping, assembly, process control and industrial man-machine cooperation in typical operations. a group of authors [2-4] studied the trajectory planning strategy of industrial robot arm and optimized it. özyer [5], yilmaz et al. [6] and gharib [7] carried out studies based on algorithms such as adaptive neural network control, adaptive fuzzy control and robust control for industrial robot arm. rojas-garcia et al. [8] and gutierrez-giles et al. [9] also proposed optimization methods for industrial robot arm control strategy from the perspective of force control. the premise of the various robot algorithms is how to build a precise, flexible model. the effectiveness and accuracy of the dynamic model of the industrial robot arm are crucial for its practical application. usually, robot manufacturers only provide kinematic data, but this can only guarantee a certain level of localizer accuracy, without than insight into its dynamic features. therefore, how to establish the modeling of industrial robot arm is a very important research content in industrial robot technology. it is the primary premise of all follow-up research, and also one of the hot spots and key points in the field of robot research. in recent years, scholars have studied the modeling of industrial robot arm. in general, the common methods for dynamic modeling of industrial robot arm include newton-euler method, lagrange method, robot modeling with matlab / simulink or robot toolbox, robot modeling with multi-body system, and parameter identification method based on acceleration, speed, position and voltage, etc. urrea and pascal [10] introduced the acquisition process of the dynamic equation of the 6r articulated industrial robot, and obtains the expression of the generalized dynamic force of the robot to represent the equation of the inverse dynamic model. fernández et al. [11] used the euler-lagrange equation to model the robot. bugnar et al. [12] studied the transfer moments of the industrial robot arm joints, developed a mathematical model of the 6-dof robot, and determined the kinematic and kinetic equations. urrea and pascal [13] derived a direct kinematic mathematical model, an inverse kinematic equation, and a dynamic model of the effector torque in the joint. šuleki ć et al. [14] studied the forward and reverse kinematics of the dobot kinematic robot and simulated it in the matlab environment. ho et al. [15] proposed a fast inverse kinematics algorithm, which has a closed solution for a specific 6-dof arm. bien [16] proposed the solution algorithm for a closed inverse kinematic model of a human two-arm robot, and verified the effectiveness of the algorithm through simulation. wang et al. [17, 18] proposed dynamic model for articulated industrial robots suitable for machining applications, and determined the link and rotor inertia, joint stiffness and damping parameters. alam et al. [19] and nguyen et al. [20] respectively designed the parameter identification model of industrial robots, which is formulated based on acceleration, speed, position and voltage. modeling and simulation of industrial robot arms using simscape multibody 267 harfoush et al. [21] proposed an algorithm based on the position of a set of measured end effectors to identify the sources of error included in the kinematic model. huynh et al. [22] describes the data-driven modeling approach and its application in controlling a new set of series elastic actuators. segota et al. [23] used the multilayer perceptron (mlp) algorithm to model the robot to obtain the best combination of output parameters. truc et al. [24] defined an implementation method for solving the optimal geometric structure model of the plane z-type mechanism, and verified the method by using the geometric modeling software. barrientos-diez et al. [25] presented a fast and low-cost fencing robotic arm modeling system. ogbemhe et al. [26] provides a workspace determination and analysis based on spatial and plane parallel manipulator graphics technology, and completes its implementation in the catia workbench using cad software. aboulissane et al. [27] introduced a kinematic model of robot in drive and configuration space. after the comparison and analysis of the above modeling methods, the following remarks can be made: (1) some modeling methods require special measuring of the physical parameters such as the center of mass and mass of the robot. especially when the model parameters change, the measurement is cumbersome and the accuracy is not always sufficiently good. (2) some studied robot models are not intuitive and convenient for subsequent trajectory planning and control strategy data. (3) some modeling methods only discuss the simple axial range of axis motion, not including the case of physical interference between joints, while the coupling between joints is a relatively common case. the main contribution of this paper is to establish a dynamic simulation model of an industrial robot arm, and to simulate the motion trajectory planning of the robot involving all the above mentioned challenging factors. the parameters of the center and mass of the model are automatically calculated by the multi-body system, which is accurate and convenient, and can satisfy the mathematical description of various parameters of the robot. at the same time, the parameters coupling is solved and realized for the structured robot. the model is set up with sufficiently rich data interface to carry out the research and verification of trajectory planning and control algorithm. the study protocol of this paper is shown in fig. 1. it includes the fig. 1 research scheme for the dynamic modeling of the industrial robot arm 268 c. zhang, s. li, z. zhang following work: (1) analyzing the kinematic structure and kinematic model of industrial robots; (2) reconstructing the cad model of the robot; (3) establishing the dynamic model of the robot with mbs and setting up external interfaces such as controller data; (4) conducting the motion trajectory planning simulation to prove the validity and rationality of the model. 2. kinematic analysis 2.1. kinematic characteristics dobot mg400 (hereinafter referred to as mg400) is the subject of this paper, which is an industrial super small desktop manipulator developed and released by dobot in december 2020. it combines the characteristics of the series mechanism and also the parallel mechanism. the quadrilateral link of the robot arm restricts its own partial working space, it has good stiffness, precision and speed. mg400 can be widely used in loading and unloading, stacking, visual sorting, distribution, viscose, automatic testing and other industrial scenarios [28-30]. the paper deals with modeling and simulation of mg400. the method used is also applicable to other similarly structured robotic arms. the motion mechanism of mg400 can be modeled as a planar mechanism with four degrees of freedom, as shown in fig. 2. the definitions of joints j1 to j4 refer to mg400 hardware user manual (hereinafter referred to as the manual) officially published by dobot. the definition of each component is shown in fig. 3. the degree of freedom f can be calculated by formula, so that n is the number of active components, pl is the number of low pairs, and ph is the number of high pairs. then the degree of freedom f of the mechanism is given as: 3 2 3 10 2 13 0 4l hf n p p= − − =  −  − = (1) fig. 2 joints of mg400 fig. 3 components of mg400 in fig. 3, the red part represents the moving component driven by j1, the blue part represents the moving component driven by j2, the yellow part represents the moving modeling and simulation of industrial robot arms using simscape multibody 269 component driven by j3, and the green part is the moving component driven by j4. j1 and j4 are parallel to each other in space, and j2 and j3 are parallel to each other in space. since there are two parallelogram mechanisms composed of l2, l2a, l2b, l33, l33a and l33b, the assembly l4 remains perpendicular to the horizontal plane, independent of the joint angles of j2 and j3. j1, j2 and j3 are orthogonal in space. j1 only affects the rotation angle between the axz1 vertical plane (xoz plane of the j1 reference coordinate system) and the axz0 vertical plane (xoz plane of the base coordinate system). j2 and j3 only affect the position of the execution point at the end of the robot arm on the axz1 vertical plane. the manual does not give the dimensional parameters of each component of the robot arm, and it is challenging to obtain the dimensional parameters through actual measurement. the official support center of dobot released the cad model of mg400 in november 2021. there is a certain error between the default coordinate system of l0, l1 and l4 part models and the parameters given in the manual. this error can be eliminated by creating a reference coordinate system in solidworks. for the convenience of analysis, the parameters used in this paper are obtained by measuring the default coordinate system of the cad model. the definition of each component reference coordinate system is shown in fig. 4, where ai represents the reference coordinate system of component li during modeling (base is l0). when the joint angle from j1 to j4 is 0, the conversion parameters of each coordinate system under a0 are shown in tab. 1. the euler angles (rx, ry, rz) under the a0 reference are obtained by specifying the order of the three basic rotations. the order is x-y-z. table 1 the conversion parameters of each coordinate system based on a0 coordinate system coordinate system tx (mm) ty (mm) tz (mm) rx (deg) ry (deg) rz (deg) a0 0 0 0 0 0 0 a1 -5 0 123 0 0 0 a2 38.5 0 228 -90 0 -90 a2a -0.47 0 250.5 -90 0 -90 a2b 38.5 0 403 -90 0 0 a31 38.5 0 228 -90 0 -90 a32 -4.5 0 228 -90 0 -90 a33 -4.5 0 403 -90 0 0 a33a 67.43 0 437.47 -90 0 0 a33b 213.5 0 403 -90 0 0 a4 279.5 0 367 0 0 0 fig. 4 the definition of reference coordinate system for each component of mg400 270 c. zhang, s. li, z. zhang 2.2. forward kinematics forward kinematics is the process of obtaining the position and pose of a point at the end of a robot arm through the known angle values of each joint, which is given as: 1 2 33 4[ , , , ] [ , , , ]x y z zp p p r f    = (2) in order to simplify the analysis, the motion mechanism of the robot arm can be equivalent to the series mechanism as shown in fig. 5, in which the key parameters such as d-h joint reference coordinate system have been marked. fig. 5 the equivalent d-h model of mg400 according to fig. 5 and tab. 1, the robot arm standard d-h parameter tab (sdh) is shown in tab. 2, where {i} represents the number of the movable connecting rod corresponding to the d-h joint, and link{i} represents the connecting rod corresponding to the {i} joint. the range of motion of θ2 and θ3 are mutually influenced, that is 2 3( ) ( 25,105) +  − . the joint of l33b is a driven joint, so the joint speed has no upper limit. angle θ4 is structurally connected to l33b. the initial angle represents the angle value of θi in the initial state of the robot arm. table 2 mg400 d-h parameter table {i} link {i} θi (deg) di (mm) ai (mm) αi (deg) 1 l1 θ1 105 43.5 -90 2 l2 θ2 0 175 0 33 l33 θ33 0 175 0 33b l33b -θ2-θ33 0 66 90 4 l33b θ4 -36 0 0 modeling and simulation of industrial robot arms using simscape multibody 271 according to tab. 2 and sdh homogeneous transformation matrix formula [31-33], the homogeneous transformation matrix between adjacent connecting rods is easily obtained. in order to express homogeneous transformation matrix conveniently, let the following equation hold: cos( ) sin( ) cos( ) sin( ) i i i i i j i j i j i j c s c s       + + =  =  = +   = + (3) the coordinates of the base center of the robot arm are defined as the base coordinate origin of the robot arm. based on the parameters a0 and a1 in table 1, the conversion equation from a0 to a1 is obtained: 0 1 0 0 5 0 1 0 0 0 0 1 123 0 0 0 1 bt −     =       (4) therefore, the homogeneous transformation matrix of the robot arm is given as: 1 4 1 4 1 2 33 2 1 4 1 4 1 2 33 20 1 2 33 33 4 0 1 2 33 33 4 2 2 33 0 ( (350 350 219)) / 2 0 ( (350 350 219)) / 2 0 0 1 175 175 192 0 0 0 1 + + + + + + + − + +    + +  = =  − +     b b b b c s c c s s c s c s t t t t t t t c s (5) columns 1 to 3 in 4 b t are the direction vectors of the robot arm end execution point under the reference of the base coordinate system xb (x0), yb (y0) and zb (z0) axes respectively, and column 4 is the position vector of the robot arm end execution point under the reference of the base coordinate system. from the above derivation, it can be seen that the position of the end execution point of the robot arm is only related to the three joints j1, j2 and j3, but not j4. the attitude of the end execution point of the robot arm (the attitude after rotating θ1+θ4 around zb) is only related to j1 and j4, but not j2 and j3. assuming that the coordinates of the end execution point of the robot arm under the reference of the base coordinate system are [px, py, pz] and the attitude angle is rz, the forward kinematics equation of the robot arm is given as follows: 1 2 33 2 1 2 33 2 2 2 33 1 4 cos (175cos( ) 175sin 109.5) sin (175cos( ) 175sin 109.5) 175cos 175sin( ) 192 x y z z p p p r              = + + +  = + + +  = − + +  = + (6) 272 c. zhang, s. li, z. zhang 2.3. inverse kinematics inverse kinematics is opposite to forward kinematics. that is to calculate the angle of each joint through the known pose, which is shown as eq. (7). 1 2 33 4[ , , , ] [ , , , ]x y z zf p p p r    = (7) (1) calculate θ1 let the robot arm end execution point homogeneous transformation matrix be shown as eq. (8). 4 1 4 1 4 1 2 33 2 1 4 1 4 1 2 33 2 2 2 33 0 0 0 1 0 ( (350 350 219)) / 2 0 ( (350 350 219)) / 2 0 0 1 175 175 192 0 0 0 1 x x x x y y y yb z z z z n o a p n o a p t n o a p c s c c s s c s c s c s + + + + + + +      =       − + +    + +  =  − +     (8) multiply eq. (8) left and right by 0 1 1 1 0 bt t− − . then we get eq. (9). 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 2 33 2 2 33 2 4 4 43.5 228 0 0 0 1 0 175 175 66 0 0 1 175 175 36 0 0 0 0 0 1 x y x y x y x y z z z z y x y x y x y x n c n s o c o s a c a s p c p s n o a p n c n s o c o s a c a s p c p s c s c s s c s c + + + + + + −    − − − −    − − − −     − + +    − − +  =       (9) since eq. (9) holds, the elements on the left and right of the equation must be equal. according to the position elements of matrix (3, 4) in eq. (9), we get eq. (10). 1 1 0y xp c p s− = (10) therefore, 1tan y x p p  = (11) eq. (11) can be further written as: 1 atan2( , )y xp p = (12) (2) calculate θ3 according to the matrix (1, 4) and (2, 4) position elements in eq. (9), we get eq. (13). modeling and simulation of industrial robot arms using simscape multibody 273 1 1 2 33 2 2 33 2 43.5 175 175 66 228 175 175 36 x y z p c p s c s p s c + + + − = + +  − = − + (13) therefore, we have eq. (14). 2 2 2 1 1 33 2 2 175 ( 109.5) (192 ) sin 2 175 x y zp c p s p   − + − − − =  (14) according to the initial angle of θ33 and the height relationship between the level of a33b and the level of a2, we have eq. (15). 33 33 1 33 33 33 1 1 sin (3,4) (3,4) cos 1 sin (3,4) (3,4) b b b b t t t t     −  =  − −  (15) in eq. (15), 33 b t is the homogeneous transformation matrix from a0 to a33b, 1 b t is the homogeneous transformation matrix from a0 to a2, then we have eq. (16). 33 33 33atan2(sin ,cos )  = (16) (3) calculate θ2 transforming eq. (13), we obtain eqs. (17-19): 2 2 1 1 33 2 33 33 2 33( 109.5)( 1) 175 ( 1) (192 ) 175+ − − + − = − −x y zp c p s s s s p c s c (17) 2 2 2 33 2 33 33 1 1 33175 ( 1) 175 (192 ) ( 109.5)( 1)− + = − − + − −z x ys s s c p c p c p s s (18) 2 2 2 33 33 33 1 1 33(175( 1) 175 ) (192 ) ( 109.5)( 1)− + = − − + − −z x ys s c p c p c p s s (19) by further transformation, the following equations are obtained: 33 1 1 33 2 2 2 33 33 (192 ) ( 109.5)( 1) sin 175( 1) 175 z x yp c p c p s s s c  − − + − − = − + (20) 2 2cos 1 sin = − (21) then we have eq. (22). 2 2 2atan2(sin ,cos )  = (22) (4) calculate θ4 according to the matrix (1,1) and (2,1) position elements in eq. (8), we get the following equation: 1 4 1 4 x y n c n s + + =  = (23) since atan2( , )z y xr n n= (24) according to eq. (6) and eq. (24), eq. (25) is obtained. 4 1atan2( , )y xn n = − (25) 274 c. zhang, s. li, z. zhang the influence of the initial state of the robot arm joint and the motion trajectory of the corresponding initial state is not considered, because it solves the minimum angle value of θ4 when the robot arm is in a certain pose, and we have eq. (26). 1 4 4 1 4 atan2( , ) 360 180 atan2( , ) 360 180 y x y x n n n n      − +  −  =  − −   (26) according to the above calculation, the inverse kinematics equation of the robot arm is shown as eq. (27). 1 33 2 2 2 1 1 2 33 2 33 1 1 33 2 2 33 33 2 4 1 atan2( , ) atan2 2 175 ( 109.5) (192 ) ( 2 175 , 1 sin ) atan2 (192 ) ( 109.5)( 1) ( 175( 1) 175 , 1 sin ) atan2( , ) 360 y x x y z z x y y x p p p c p s p p c p c p s s s c n n        =  =   − + − − −      −  =  − − + − −  − +  −  = −  (27) 2.4. kinematic verification in order to verify the correctness of eq. (6) and eq. (27), the kinematics solution function is written by matlab. ten groups of joint angles that conform to the motion range of the joints of the robot arm are randomly generated, and eq. (6) is used to calculate the forward kinematics solution. then, according to the obtained position and posture of the terminal execution point, the inverse kinematics solution is obtained by using eq. (27). the correctness of eq. (6) and eq. (27) is verified by comparing the results of the two calculations. the specific results of forward and inverse kinematics solutions are shown in tab. 3 and tab. 4. table 3 forward kinematics conditions and calculation results of mg400 no. θ1(deg) θ2(deg) θ3(deg) θ4(deg) px(mm) py(mm) pz(mm) rz(deg) 1 -1.86 60.70 7.26 290.68 327.62 -10.66 115.45 -71.19 2 125.10 11.76 48.32 -217.58 -133.65 190.19 211.66 -92.48 3 -150.23 56.85 -16.85 -14.46 -338.58 -193.69 175.21 -164.68 4 129.51 42.09 13.21 258.80 -207.68 251.84 178.01 28.31 5 97.76 38.44 -39.66 -187.25 -53.08 389.66 332.80 -89.49 6 123.68 -21.85 40.03 -239.09 -116.82 175.27 299.81 -115.41 7 153.18 53.40 -13.34 -20.82 -342.62 173.24 183.72 132.36 8 -140.92 50.02 -69.50 -308.56 -317.17 -257.55 362.82 -89.48 9 6.93 -14.36 67.61 228.63 169.56 20.60 221.32 -124.44 10 71.18 -8.52 50.46 13.39 68.96 202.33 248.11 84.57 modeling and simulation of industrial robot arms using simscape multibody 275 table 4 inverse kinematics conditions and calculation results of mg400 no. px(mm) py(mm) pz(mm) rz(deg) θ1(deg) θ2(deg) θ3(deg) θ4(deg) 1 327.62 -10.66 115.45 -71.19 -1.86 60.70 7.26 -69.32 2 -133.65 190.19 211.65 -92.48 125.10 11.76 48.32 142.42 3 -338.58 -193.69 175.21 -164.68 -150.23 56.85 -16.85 -14.46 4 -207.68 251.84 178.01 28.31 129.51 42.09 13.21 -101.20 5 -53.08 389.66 332.80 -89.49 97.76 38.44 -39.66 172.75 6 -116.82 175.27 299.81 -115.41 123.68 -21.85 40.03 120.91 7 -342.62 173.24 183.72 132.36 153.18 53.40 -13.34 -20.82 8 -317.17 -257.55 362.82 -89.48 -140.92 50.02 -69.50 51.44 9 169.56 20.60 221.37 -124.44 6.93 -14.36 67.61 -131.37 10 68.96 202.33 248.11 84.57 71.18 -8.51 50.46 13.39 when solving the inverse solution of the robot arm, the minimum value of the rotation angle of each joint of the robot arm is obtained. when the rotation angle of θ4 exceeds the range of [pi, pi], the algorithm will let θ4 map to the range of [-pi, pi], and select a smaller inverse solution under the condition of ensuring the attitude of the robot arm. according to tab. 3 and tab. 4, it can be concluded that the kinematic analysis represented by eq. (6) and eq. (27) is correct. 3. construction of mbs model 3.1. build solid module mechanism of mbs model to establish the solid module mechanism of the mbs model of the robot arm, it requires understanding the parameters of each part of the robot arm, but it is difficult to directly obtain the specific parameters of each part of the robotic arm. therefore, the relevant parameters in this article are generally obtained through cad model analysis. the solid module mechanism of the established mbs model is shown in fig. 6 (a). solidworks is used to establish the reference coordinate system shown in tab. 1 in the cad model, and (a) (b) fig. 6 a) solid module mechanism of mbs model and b) its workspace 276 c. zhang, s. li, z. zhang all solidworks related entities are output as step format files under the corresponding reference coordinate system. the solid module model is used for the dynamic simulation of the system, and the working space of each joint needs to be accurately consistent with the real robot arm, so that the various parameters of the model can be directly applied to the real robot arm. the workspace of the robot arm model system is obtained in joint space as shown in fig. 6 (b). comparing with the mg400 manual, it is obtained that the model established can accurately simulate the working state and parameters of the actual robot, and is reliable and feasible. 3.2. build robot arm mbs model a robot arm mbs system has been established, mainly including trajectory planning, controller, robot mbs model, and simulation data module [34-36]. the mbs model is composed of joint module and environment module. the joint module is divided into rotation and translation according to the solid module motion mode, as shown in fig. 7 which includes solid module coordinate system transformation, joints and part solid modules. the joint module is used to simulate the motion state of each joint of the robot arm and the components connected on the joint, and can also simulate the motion state of the outer shaft of the robot arm. the environment module is shown in fig. 8. fig. 7 joint module of the mbs model fig. 8 environment module of the mbs model in order to make the model consistent with the equivalent model, the position of l33 rotating joint is coincident with a2b. this causes l2a, l2b, l31, l32, l33a and l33b in the robot arm model to be driven components. according to tab. 1 and tab. 2, the solid transform parameters of each joint module can be obtained, as shown in tab. 5. modeling and simulation of industrial robot arms using simscape multibody 277 table 5 the solid transform parameters of joint module coordinate system conversion module tx (mm) ty (mm) tz (mm) rx (deg) ry (deg) rz (deg) rt 0_1 -5 0 123 0 0 0 rt 1_2 43.5 0 105 -90 0 -90 rt 1_2a -0.47 0 250.5 -90 0 -90 rt 2_2b 175 0 0 0 0 90 rt 1_31 43.5 0 105 -90 0 -90 rt 31_32 0 -43 0 0 0 0 rt 2_33 175 0 0 0 0 90 rt 2b_33a 28.93 -34.47 0 0 0 0 rt 33_33b 175 0 0 0 0 0 rt 33b_4 66 36 0 90 0 0 in tab. 5, rt {i}_{j} represents the conversion parameters from a{i} to a{j}, and {i} and {j} represent the number of the robot arm assembly and the corresponding coordinate system. in this model, the corresponding joint modules of l1, l2, l33 and l4 are used as the active components of the robot arm, and the input can be the target angle (j{i}) or torque (t{i}). the visualization of the robot arm is obtained by importing the parts of models shown in fig. 6 into the solid module. as the mg400 is made of aluminum alloy, the model parameter density is set to 2700 kg/m3, and the physical properties of the components such as mass, center of mass coordinates and moment of inertia will be automatically calculated according to the geometric characteristics of the model, as shown in tab. 6. table 6 the physical properties of solid module modules solid module mass (kg) centroid coordinates (cm) moment of inertia (g·cm2) l1 4.781 [0.462, -0.006, 5.699] [163602, 156446, 200109] l2 0.234 [5.963, -0.286, -0.049] [1049.56, 9144.35, 8310.61] l2a 0.179 [7.378, -1.967, 0.001] [1009.9, 5914.17, 5415.16] l2b 0.099 [-0.511486, -2.702, 0.002] [408.52, 663.483, 634.498] l31 0.012 [-0, -1.728, -0.003] [43.717, 3.788, 45.756] l32 0.025 [8.934, 0.204, -0.453] [3.752, 966.085, 968.624] l33 0.101 [8.21, 0.487, 0.011] [68.215, 4436.23, 4433.04] l33a 0.122 [8.78, -0.668, -0.001] [541.489, 3825.9, 3563.67] l33b 0.431 [5.526, 0.337, 0.003] [2676.58, 2873.23, 4239.39] l4 0.004 [0, 0, 0] [3.824, 3.824, 0.223] connecting each module which are shown in fig. 3 to get the mbs model of the robot arm, as shown in fig. 9. the physical quantities in mbs model are simulated by its basic equations, which have original physical effects and characteristics. the motion state of the system is solved by the basic motion equation at each sampling time, and the visual images are synchronously generated in mechanics explorers. 278 c. zhang, s. li, z. zhang fig. 9 the mbs model of robotic arm 3.3. mbs dynamic simulation modeling the mbs dynamic simulation model after adding the trajectory planning module, control algorithm module and data acquisition module is shown in fig. 10. these interfaces all greatly facilitate the follow-up research and development of the robot. in addition, the mbs model retains the interface between the robot arm and the robot outer axis and the tool, enabling the simulation system to be used repeatedly and effectively in subsequent studies of the robot arm, in this way to facilitate subsequent ongoing development. fig. 10 the dynamic simulation model of mg400 modeling and simulation of industrial robot arms using simscape multibody 279 4. simulation results and analysis the dynamic simulation of the mbs model system by the above. in this paper, the palletizing operation task of the industrial mechanical arm is selected as the trajectory planning target. according to the operation requirements, given the coordinates of target points such as initial point (a), lift hand point (b1, b2) , drop hand point (c1, c2) and task point (d1, d2). the schematic diagram of the movement of the robotic arm operation is shown in fig. 11. after setting the target points for j1, j2, j33 and j4 joints, trajectory planning simulations were performed using three methods: the quintic polynomial trajectory, the b-spline and the minimum jerk polynomial trajectory. generate the target point parameters for the robotic arm, and the driving signal of each joint module is obtained by solving the inverse solution. then the signal is directly transmitted to the mbs model to obtain the real-time torque or force. all simulation data is available via the rotating joints and transformation sensor modules. similarly, a control algorithm module, added to the system, using torque as a drive signal, can be transmitted to the motion controller module to verify the effect of the control algorithm. the paths and their process dynamic maps planned by the above three trajectory planning algorithms are obtained, as shown in figs. 12-14. after the path planning, 3d dynamic animation can also be generated in the mechanical explorer, which provides a nice and straightforward option to observe the trajectory. a few screenshots of the obtained trajectories are shown in fig. 15. (a) (b) fig. 12 path (a) and its dynamic map (b) planned by quintic polynomial trajectory fig. 11 the schematic diagram of the movement of the robotic arm operation 280 c. zhang, s. li, z. zhang (a) (b) fig. 13 path (a) and its dynamic map (a) planned by b-spline (a) (b) fig. 14 path (a) and its dynamic map (b) planned by minimum jerk polynomial trajectory modeling and simulation of industrial robot arms using simscape multibody 281 fig. 15 3d simulation visualization of robot arm the obtained joint angles, angular velocities and angular accelerations are shown in fig. 16, fig. 17 and fig. 18, respectively, and the output torques are shown in fig. 19, including quintic polynomial trajectory, b-spline, and minimum jerk polynomial trajectory. as can be observed from these figure, in the motion data obtained from the model, there is an obvious proportional relationship between the torque and acceleration of j1. j2 and j33, which are greatly affected by gravity, and j4 is not equipped with any fixtures, so its torque is almost zero. it can be obtained by comprehensive comparison that the quintic polynomial trajectory is an algorithm involving both the starting point and the end point, yielding the smoothest path between the two points. the b-spline can be inserted into multiple pathway points, those on the path between the start and the end. the path generated by the b-spline algorithm does not pass through the path points. it has the advantages of a smooth curve and short path, but it will cause discontinuity in acceleration and torque. the minimum jerk polynomial trajectory can insert several pathway points between the starting point and the end point. unlike the b-spline, the path planned by this method is through the pathway point. the advantage is that the path is the smoothest, while ensuring the complete continuity of angle, angular velocity, angular acceleration and angular acceleration. the above data well simulate the actual operation of the robot arm. 282 c. zhang, s. li, z. zhang fig. 16 movement angle of each joint of mbs model fig. 17 angular velocity of each joint of mbs model modeling and simulation of industrial robot arms using simscape multibody 283 fig. 18 angular acceleration of each joint of mbs model fig. 19 torque of each joint of mbs model 284 c. zhang, s. li, z. zhang the error between the actual simulated trajectory and the ideal trajectory at the end execution point of the robot arm model is shown in fig. 20 and the error statistics are shown in tab. 7. the average error of simulation is as low as 10-11 mm, and the minimum error is 0, which is a very good accuracy. to sum up, the developed robot arm model can simulate well the actual kinematics and dynamics of the robot arm. fig. 20 error between simulation trajectory and ideal trajectory of robot arm table 7 error between simulation trajectory and ideal trajectory of the end execution point of robot arm mbs model method maximum error (mm) minimum error (mm) average error (mm) quintic polynomial trajectory 3.8730e-09 0 9.0446e-11 b-spline 2.1104e-09 0 4.4756e-11 minimum jerk polynomial trajectory 4.4408e-09 0 1.2206e-10 5. conclusion the mbs modeling and simulation of dobot robot arm are carried out by using simscape multibody in this paper. first, the kinematic properties of the robot arm are analyzed and the forward and inverse kinematic equations of the robot arm are established. secondly, simulink is used to establish the mbs model logic framework of the robot arm, and then the solid model is imported into the corresponding solid module in the model, calculate the corresponding physical properties of the corresponding components, and obtain the visualization of the robot arm. third, an offline dynamic simulation model of the robot arm was established by adding a trajectory planning module, a control algorithm module, and a data acquisition module to the mbs model. fourth, the motion trajectory of modeling and simulation of industrial robot arms using simscape multibody 285 the simulation model is planned and simulated in the joint space. the simulation results show that the simulation model can accurately simulate the motion state of the robot arm, the visual simulation animation is easily obtained to intuitively observe the motion process of the robot arm, and the output simulation data is convenient for the study and analysis of the subsequent control algorithm. the peripheral mechanisms can also be extended by adding joint modules or related follow-up studies by embedding the code into the trajectory planning module or control algorithm module. based on the dynamic simulation function of the simulation model, it is more economical and easier to study the robot arm with the same structure as mg400. acknowledgement: the paper is supported by the 'three levels' talent construction project of zhuhai college of science and technology, the higher education teaching reform project of zhuhai college of science and technology 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https://doi.org/10.22190/fume220730040h © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper bending fatigue behavior of ag nanowire/cu thin-film hybrid interconnects for wearable electronics byungil hwang1, yurim han1, paolo matteini2 1school of integrative engineering, chung-ang university, republic of korea 2institute of applied physics “nello carrara”, national research council, italy abstract. enhancing the mechanical reliability of metal interconnects is important for achieving highly reliable flexible/wearable electronic devices. in this study, ag nanowire and cu thin-film hybrid interconnects were explored as a novel concept to enhance mechanical reliability under bending fatigue. bending fatigue tests were conducted on the cu thin films and cu/ag nanowire/polyimide (cap) interconnects. the increase in resistance was larger for the cu thin films than for the cap. the single-component cu electrodes showed multiple crack initiation and propagation due to bending strain, which degraded the electrical conductivity. in cap, however, no long-range cracks were observed, even after 300,000 cycles of bending, although a wavy structure was observed, probably due to the delamination of the ag nanowires under repeated bending. our study confirms that flexible ag nanowire and metal thin-film hybrids can enhance the mechanical reliability of metal thin-film interconnects under bending fatigue. key words: thin film, stability, bending, ag nanowire, flexibility 1. introduction with the progress in the field of flexible/wearable electronics, there are increasing demands for flexible interconnects that are reliable under repeated bending deformation [17]. metal thin films are the most commonly used materials for flexible interconnects due to their high electrical conductivity and compliant nature [8, 9]. metal electrodes can work as flexible electrodes for devices that undergo strain under the elastic limit. however, actual electronic devices can undergo more than 300,000 cycles of bending deformation in their lifetime of operation [10]. this causes fatigue damage on the metal thin film even though the strain is under the elastic limit. schwaiger et al. reported that the dislocation received: july 30, 2022 / accepted september 28, 2022 corresponding author: byungil hwang school of integrative engineering, chung-ang university, 84 heukseok-ro, dongjak-gu, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr 554 b. hwang, y. han, p. matteini accumulation at the interface between a metal thin film and polymer substrate occurs under many cycles of bending, which leads to the formation of voids at the interfaces [11, 12]. with repeated bending, such accumulated voids create protrusions at the surface of the metal thin film, where the imposed stress is localized, resulting in the failure of the electrodes. the cracks block the electrical current flow, which can cause malfunctioning of the devices using the metal thin-film electrodes. to enhance mechanical reliability, several alternatives have been studied, including graphene, carbon nanotube, metal nanowire, metal/graphene hybrid system, or holey metal thin-film electrodes [9, 13, 14]. graphene, carbon nanotube, and metal nanowire are highly flexible and transparent in the visible range, but their conductivity is insufficient for interconnects. metal-graphene hybrid systems have shown better reliability under repeated bending deformation, but the scale-up of the electrode is limited due to the size of graphene [15]. in addition, the fabrication process using multiple transfers is complex, which significantly increases the processing cost. the mechanical reliability of metal interconnects is enhanced by forming holes on the metal thin films [16]. crack propagation is blocked by the holes, which reduces long-range crack densities on metal thin films during repeated deformations. however, a complex patterning process is required to form the holes, which increases the fabrication costs. to overcome the limitations of the previously reported technologies, hybrid systems of flexible materials and metal thin films are a promising option. by integrating a flexible material (which can be simply formed on the polymer substrates) with a metal thin film, both high conductivity and flexibility can be achieved. considering this, in this study, ag nanowire and cu thin-film hybrid interconnects were investigated. ag nanowires are highly flexible because of their nanoscale dimensions and network structure; in addition, their mass production is possible via the polyol reduction process [17-23]. the coating process is a very simple solution-based process using a spray or doctor blade system. in this study, ag nanowires were coated on polyimide (pi) substrates using a doctor blade. cu thin films were deposited on the ag nanowire/pi samples via a magnetron sputtering system. the reliability of the cu/ag nanowire/pi (cap) interconnects was examined using a bending fatigue tester. our study showed that the cap had better reliability than cu thinfilm interconnects. 2. materials and methods 2.1. sample preparation and characterization for this study, 0.1 wt% ag nanowire suspension in ethanol was purchased from sg flexio (south korea) and used as received. fig. 1 shows the schematic illustration of fabrication process of cap interconnects. first, a 125-m-thick pi substrate (dupont, kepton e) was sequentially cleaned with ipa, ethanol, and di water in an ultrasonic bath. each cleaning step was carried out for 5 min. next, ag nanowires were coated on the pi substrate using a mayer bar (#rd 6). third, the ag nanowire networks were air-dried at 80 °c for 10 min using a hot plate to evaporate a remaining solvent. lastly, a cu thin film was deposited on the pi substrate using dc magnetron sputtering at 100 w at room temperature for 15 min. thickness of cu thin film was ~400 nm. the microstructures of cap and cu thin film were analyzed using field-emission scanning electron microscopy (fe-sem, sigma, carl zeiss). x-ray diffraction (new d8-advance, bruker-axs, usa) bending fatigue behavior of ag nanowire/cu thin-film hybrid interconnects for wearable electronics 555 was used to analyze the crystallographic orientation. cu κα (λ = 0.154 nm) radiation was used for xrd analysis in the 2θ range between 30º and 100º. fig. 1 schematic illustration of a fabrication process of cap film 2.2. cyclic bending fatigue test bending fatigue tests were conducted using a cyclic bending fatigue tester (south korea, ck trading co. ltd.), which can measure electrical resistance in situ with a resolution of ~0.003 ω. fig. 2 presents the mechanism of a bending fatigue tester. the samples were mounted between the two horizontal plates, and both ends of the samples were fixed to the plates with screw bolts. the resistance was measured in situ across the two ends of the samples that were in contact with the metal pads. the value of the imposed strain was calculated using ε = y/2r, where ε, y, and r are the imposed strain value, the distance from the neutral axis, and the radius, respectively. by adjusting the distance between the two plates, the value of r varies, which results in the change of bending strain imposed on the samples. during bending tests, the lower plates move forward and backward repeatedly in the horizontal direction, thereby imposing many cycles of bending. by measuring the resistance change in situ during the bending tests, the crack evolution in the samples can be evaluated. fig. 2 schematic of a bending fatigue testing system 3. results 3.1. crystallographic analysis using xrd fig. 3 shows xrd patterns of single cu and cap films. similar xrd patterns were observed for both cu and cap films, where typical (111) peaks at 43.71º and 43.69º were strongly indicated with weak (200) texture at 50.44º and 50.28º, respectively as shown in fig. 3a,b. the observed peaks at diffraction angle below 40º was noise as can be seen in xrd pattern of pi in fig. 3c. there were no peak shifts at each peak, which confirmed cu 556 b. hwang, y. han, p. matteini and cap films had the same texture. in addition, the grain size is calculated according to the scherrer equation: cos    = k d (1) where d, κ, λ, β and θ are the grain size, the shape factor, the wavelength, the full width at half maximum (fwhm), and the diffraction angle, respectively. here, κ is 0.9 for typical cu and λ is 0.154 nm. the fwhm was evaluated for the (111) peak of cu and cap films, which were 0.56 and 0.55, respectively. with those parameters, the grain sizes are calculated as 15.3 nm and 15.5 nm for cu and cap films, respectively. therefore, the crystallinity of cu and cap films was confirmed to be similar; artifacts that can be resulted from the different crystallinity is minuscule, especially for understanding of deformation behavior under bending fatigue. (a) (b) (c) fig. 3 xrd analysis results of (a) cu thin film, (b) cap film, and (c) polyimide substrate 3.2. cyclic bending fatigue test results fig. 4a shows the resistance change of cap and cu thin film as a function of bending cycles under bending radius of 3 mm. the y-axis represents the normalized resistance change calculated by dividing the resistance gap between the measured resistance and the initial resistance value with the initial resistance value, and the x-axis represents the number of bending cycles. to confirm the reliability under severe conditions, up to 300,000 cycles bending cycles were imposed on the samples. for the cu thin film, the resistance increased rapidly up to 40,000 cycles of bending and then showed a transient point, where a slower increase in resistance was observed until the end of the bending cycles. at the end of the bending tests, the resistance increased by 300% compared to the initial resistance. meanwhile, the resistance of cap increased faster than that of the cu thin films in the initial period of the bending cycles. the transient point, where the trend of resistance increase changed from rapid increase to gradual increase, was observed at 20,000 cycles of bending, which is smaller than that observed for cu thin films (fig. 2b). however, the amount of resistance increase at the transient point was smaller for the cap than for the cu thin films. a ~113% increase in resistance was observed for the cap film at the transient point, whereas a ~151% increase in resistance was observed for the cu thin films. in addition, the increase in resistance at the end of the bending cycles was smaller for the cap than for the cu thin films. these results confirm that cap is more reliable under large numbers of bending cycles compared to single cu thin films. bending fatigue behavior of ag nanowire/cu thin-film hybrid interconnects for wearable electronics 557 (a) (b) fig. 4 (a) normalized resistance changes in cap and cu thin films as a function of the number of bending cycles, and (b) replotted results in the early stage of bending cycles, 30,000 cycles to investigate the mechanism of enhanced reliability of cap under bending fatigue, sem images were taken for the cap and cu thin films after imposing 300,000 cycles of bending, as shown in fig. 3. the cu thin films showed multiple cracks on the entire film area. the enlarged image of the cracks shows extrusion formation (fig. 3a). on the contrary, there was no clear crack formation in the cap, as seen in fig. 3b. wavy traces were observed in the cap after imposing 300,000 cycles of bending (the lower inset of fig. 3b). fig. 5 sem images of (a) cu thin films and (b) cap after 300,000 bending cycles 4. discussion the enhanced mechanical reliability for cap film can be attributed to two factors: (1) stress relaxation by forming wavy structure and (2) reduced strain by the relocation of neutral plain due to the addition of ag nanowires. as shown in the upper inset of fig. 3a, cu thin films showed no cracks before bending. under repeated bending, dislocation accumulation occurred at the interface between the thin film and substrate (fig. 6a), which resulted in extrusion formation (the lower inset of fig. 3a). with the progression of bending fatigue, the stress localized on the extrusion, which led to the generation of cracks (fig. 6a). the crack formation degraded the electrical conductivity of the metal thin films. as shown in the upper inset of fig. 3b, the cap showed no wavy structure before bending. the wavy structure could be attributed to the delamination of the thin film from the 558 b. hwang, y. han, p. matteini substrate owing to the weak adhesion of ag nanowires with the pi substrate (fig. 6b). in the cap, cu was attached to the substrate with relatively high bonding forces, whereas the ag nanowires were weakly attached by physical bonding forces, such as van der waals or electrostatic forces. under the cyclic bending, the weak interface resulted from ag nanowires can cause delamination or interfacial sliding of thin films, forming the wavy structure. such wavy structure might have induced micro-cracks, which were not observed in the sem images because the cracks were closed in the no-strain state (fig. 6b). because of the micro-cracks, an increase in resistance was observed in the cap. however, there were no long-range cracks in the cap, unlike in the cu thin films; thus, the increase in resistance was smaller for the cap than for the cu thin films. in the initial stage of bending, however, the cap underwent the deformation by forming wavy structure, which was earlier than the point where the crack formation and propagation of single cu thin films occurred. during which, short range of micro cracks can be formed at the wavy structure of cap; thus, the resistance can be slightly increased. the micro cracks are difficult to observe in sem or tem because the cracks are closed. thus, the resistance increase of cap in the initial stage was higher than those of single cu thin films. (a) (b) fig. 6 illustration of failure mechanism of (a) cu and (b) cap films under cyclic bending second, the relocation of neutral plane position by the addition of ag nanowires can vary the strain imposed on the metal thin film as shown in fig. 7. the imposed bending strain is calculated by the equation: 2 2 1 0 1 0 ( ) 2 ( ) n i i i i n i i i i b y y s b y y + = + = − = −   (2) where s, b and y(i+1) − y1 (δy) are the neutral plane position, the biaxial modulus and the thickness of each layer, respectively [14, 24]. the biaxial modulus, b, is calculated by the equation: bending fatigue behavior of ag nanowire/cu thin-film hybrid interconnects for wearable electronics 559 ν1− = e b (3) where e is the elastic modulus and ν is poisson’s ratio. table 1 shows parameters used for the calculation of neutral plane position. with those parameters, the calculated neutral positions were 69.3 and 71.2 µm for the cu and cap films, respectively. therefore, the imposed strains on the cu and cap films are calculated as 2.8% and 2.7%, respectively. the difference between the calculated imposed strain values was ~0.1%. although the strain value itself is not significant; this small value can make a reasonable change in the results of cyclic bending tests where more than 300,000 cycles of bending is imposed. fig. 7 illustration of change in imposed strain on metal thin film by the relocation of neutral plane position 5. conclusion in this study, cu/ag nanowire/pi (cap) interconnects were explored as a novel concept to improve the mechanical reliability of single-component cu electrodes. bending fatigue tests were conducted for the cap and cu thin films while monitoring the resistance in situ. the resistance change reflected the crack or failure evolution in the thin films during the bending tests. the results of the bending fatigue test showed that the cap exhibited better mechanical reliability compared to the cu thin films, with a lower increase in resistance after 300,000 cycles of bending. microstructural analysis through sem showed a high density of long-range cracks in cu thin films after imposing 300,000 cycles of bending; cap showed no such long-range cracks but exhibited a wavy structure, probably due to the delamination of ag nanowires by the bending strain. our study confirms that flexible ag nanowires and metal thin-films hybrids can enhance the mechanical reliability of metal thin-film interconnects under bending fatigue. acknowledgement: this work was supported by a national research foundation of korea (nrf) grant funded by the korean government (msit) (no. nrf-2019k1a3a1a25000230). this research was supported by the chung-ang university graduate research scholarship in 2022. references 1. hwang, b., an, y., lee, h., lee, e., becker, s., kim, y.-h., kim, h., 2017, highly flexible and transparent ag nanowire electrode encapsulated with ultra-thin al2o3: thermal, ambient, and mechanical stabilities, scientific reports, 7(1), 41336. 2. kim, d., 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measurements for comparison of microprocessor controlled knee joints  udc 621:617.3 julius thiele, simon gallinger, peter seufert, marc kraft berlin institute of technology, department of medical engineering abstract. a test device for lower limb exoprostheses has been developed at the department of medical engineering of the tu berlin which is able to apply realistic loads to prostheses. hence, the gait simulator meets the increasing demands on functional and fatigue testing of microprocessor controlled knee joints (mpk). an exemplary comparison of two mpk was performed to prove that known differences in the functional quality of the mpk can also be demonstrated in simulator tests. significant differences between the mpk could be found. the mpk could not be tested in their full range of function though. to enable comprehensive functional and fatigue testing, the gait simulator has to be modified to achieve higher walking velocities and step lengths. key words: amputee, transfemoral, knee joint, motion analysis, gait simulator 1. introduction the gait simulator, a test device for lower limb exoprostheses, is developed at berlin institute of technology to fulfill the requirements for comprehensive functional and fatigue testing of modern prosthetic knee joints, especially microprocessor controlled knees (mpk). technological advances in exoprosthetics lead to a steady increase in the mobility of amputees. the resulting and partially complementary demands on the prostheses concerning low weight, high functional capability, and improved strength result in complex designs which are strongly optimized to their field of application. due to that, it is important to test the prostheses under realistic conditions. the current standards din en iso 10328 and din en iso 22675 are not able to address these requirements as the amount of loading and the number of cycles does not comply with field studies [1]. in contrast, the simulator is able to apply real time series of multi-axis received october 12, 2015 / accepted november 5, 2015 corresponding author: julius thiele berlin institute of technology, department of medical engineering, dovestr. 6, 10587 berlin e-mail: julius.thiele@tu-berlin.de preliminary report 194 j. thiele, s. gallinger, p. seufert, m. kraft loads. furthermore, it is possible to study prostheses with a high reproducibility from an objective point of view. especially the influence of subjects concerning inter-individual and day-to-day variability as well as accommodation time on different prostheses set ups can be eliminated. particularly, the simulator is suited for measurements which are usually not possible due to ethical restrictions or excessive stress for the subjects, such as simulations of falls and stumbles. in this study, the simulator was used to examine differences between c-leg 3 1 and plié 2.0 2 to prove its abilities for functional testing of mpk. 2. methods and materials the gait simulator is based on two kinematic chains (hip module a and foot module b, fig. 1), which are linked by the leg prosthesis during stance phase and separated during swing phase [2, 3]. the simulator is designed to apply loads in a realistic manner using five servo-hydraulic actuators with volume-controlled servo valves for precise load application. the hip module consists of three serially coupled actuators for the movements of flexion/extension (a1), of adduction/abduction (a2) and of inversion/ eversion (a3). for all drives, moments are captured by integrated strain gauge based fig. 1 gait simulator with integrated oktapod measuring system, a: hip module, b: foot module, c: oktapod 1 provider: otto bock healthcare gmbh, max-näder-str. 15, 37115 duderstadt, germany 2 provider: freedom innovations, 30 fairbanks, ste. 114, irvine, ca 92618, usa the gait simulator for lower limb exoprostheses 195 measuring devices. the foot module consists of two serially coupled actuators for vertical (b1) and horizontal movement (b2) of the instrumented foot plate which records ground reaction forces. the kinematic chains are coupled within the stance phase by the legprostheses. an additional six dof force and moment sensor [1] (oktapod measuring system c with data logger c1 and li-ion battery c2) was integrated in the prosthesis to obtain reference data for comparison of actual and nominal values of the simulator control. 2.2. control of the simulator the control of the gait simulator has a robust design and enables a quick setup of new testing scenarios. the required test data are either directly collected during measurements with subjects or calculated from test records. nine independent closed loop controls with pd controllers are used to control the five actuators of the simulator. during the swing phase all the actuators are position controlled. during the stance, a force/moment control is used. only the actuator for extension and flexion movement is always position-controlled because of the need for correct knee angles for the reproduction of physiological loads during gait. the sequence control ensures correct switching between the force/moment-controlled stance phase and the position-controlled swing phase. for the correct transition between the two phases the sequence control is designed as a finite state machine with three states, which are processed in a defined order during simulation of a gait cycle. an overview of the states is given in fig 2. fig. 2 overview of the states of the finite state machine 196 j. thiele, s. gallinger, p. seufert, m. kraft the first state “swing phase” starts when the foot detaches from the base plate. in this state, a real-time analysis is performed to estimate the position of heel strike. if the calculation of the estimated position finishes, the next state “contact” is entered, where the heel and the base plate are brought into contact. after the initial contact, the ground reaction force rises. when a defined force is reached, the control switches to the state “stance phase” and back to the force/moment-controlled movement. this phase ends when the plate gets to the position “toe off” and the swing phase starts again. due to the coupling of the actuators during the stance phase, a movement of one actor also changes the measured forces and moments of the other control loops. that effect depends on the individual characteristics of the prostheses like geometry, flexibility and damping. therefore, a decoupling is implemented in the two most affected control loops. 2.2.1. decoupling of the force control of the vertical foot actuator from the flexion/extension angle for decoupling of the vertical force from the flexion/extension angle, the compliance of the prostheses is needed. it is acquired by experimental examination and recording of so called foot-compliance-curves (fig. 3). to that end, the position of the foot during a constant vertical force and variable shank angle is determined. the resulting field of curves is used to calculate the variable compliance of the system during a single step (fig. 3, purple trace). fig. 3 resulting field of curves after examination of the foot compliance the gait simulator for lower limb exoprostheses 197 2.2.2. decoupling of the moment control of horizontal foot actuator from flexion/extension angle the compliance of the prosthesis in horizontal direction is smaller than in vertical direction. hence, only the geometry of the leg prostheses is needed for decoupling of the sagittal moment from the flexion/extension angle. additionally, the control loop is supported by an iterative learning pre-control (fig. 4), which adjusts manipulated variable y after a few runs of target values wt to minimize the control deviation. fig. 4 scheme of the closed-loop control with the self-learning pre-control 2.3 prosthesis to determine whether the current development level allows the use of the gait simulator to identify differences in the functional quality of mpk joints, measurements were first performed with the c-leg 3 (ottobock healthcare) and the plié 2.0 (freedom innovations) only. these joints were selected on the basis of the results of a gait lab analysis, which demonstrated that the quality of the swing phase damping between these two joints differed significantly [4]. 2.4. test procedure for controlling of the simulator, kinematic and kinetic reference data were used, which were recorded by the oktapod six dof force and moment sensor system [4]. via the servo-hydraulic drives of the simulator, movements were induced and forces and moments were applied simultaneously. thereby, the measured data were also recorded by means of the oktapod measuring system, which was integrated in the prosthesis (fig. 1). a significant limitation of the set up was the maximum step length of the simulator. due to constructive restrictions, it is limited to 0.8 meter. acceleration and deceleration of the foot plate before the heel strike and after toe off additionally consume useable length of the foot module. hence, the subjects were asked to walk at medium velocity with short steps, and at slow velocity with normal steps to obtain reference data for the simulator (table 1). thereby, the c-leg 3 and the plié2.0 knee joints were used in combination with the prosthetic foot variflex with evo 3 . prosthetic alignments and setup of the prostheses were conducted by a certified prosthetist. the same prostheses with identical knee joint settings were used in the gait lab and during simulator measurements. 3 provider: össur hf., grjothals 1-5, 110 reykjavik, iceland 198 j. thiele, s. gallinger, p. seufert, m. kraft table 1 walking velocity and double step length of the reference values velocity [m/s] double step length [m] walking at slow velocity 0.69 1.11 walking at medium velocity 0.97 1.23 3. results of gait simulator measurements 3.1. optimizations after the first measurements the first measurement data acquired by the simulator show two main deviations from the target values. first, the movement of the shank generates a moment when the knee reaches its extended position at the end of the swing phase. the impact results in a high stop moment which leads to oscillations of the sagittal moment (fig. 5, red line). also, the shank is reflected at the extension stop resulting in an oscillation of the knee flexion angle. second, the knee angle maximum produced by the simulator deviates from the reference values as seen in fig. 6. those two problems were improved by an optimization of the thigh angle. based on the original angle curve, a fifth degree polynomial was fitted through four defined points, while the transitions between fitted and original curves were continuously differentiable (fig. 7). within those borders, the angular speed and acceleration could easily be manipulated. the results are shown in fig. 5 and fig. 6. the stop moment was decreased by half, the knee-angle maximum was raised by 12° and the reflection at the extension stop was nearly eliminated. fig. 5 sagittal hip moment (my) before and after optimization of thigh angle the gait simulator for lower limb exoprostheses 199 fig. 6 knee flexion angle before and after optimization of thigh angle fig. 7 thigh angle before (red) and after optimization (green) 3.2. reproducibility of measurement successive steps of the gait simulator were recorded with the oktapod measuring system and compared to each other. the mean correlation factor over all signals was calculated from the pairwise comparison of each step with a reference step. the results 200 j. thiele, s. gallinger, p. seufert, m. kraft are presented in table 2. the variation is slightly increased at higher velocities. however, the minimum of the correlation factor of 0.982 is still very high, showing very high reproducibility. table 2 mean deviation between six successive steps of the gait simulator; mean correlation factor over all signals mean correlation factor, slow velocity mean correlation factor, medium velocity c-leg 0.997 0.982 plié2.0 0.994 0.986 3.3. deviation between reference values and measured values at first, a comparison of the nominal values for simulator control and the measured actual values was conducted to assess the performance of the simulator control. exemplary results for plié2.0 are presented in fig. 8. in general, the curve characteristics of the forces and moments applied by the gait simulator show a high correspondence to the gait analysis data. however, in swing phase, the highest deviations could be found for the frontal and the sagittal hip moment. also the movements recorded for the thigh and the shank differ significantly from the target values in swing phase, which causes the low maximum knee flexion angle. remaining optimization potentials are described in the discussion section. 3.4. comparison of the swing phase control of the two knee joints previous gait lab tests have shown that significant differences could be found for the swing phase control of different mpk [4]. for the investigated slow, comfortable and fast walking velocity, c-leg 3 and plié 2.0 showed the largest differences regarding to the maximum knee flexion angle. with plié 2.0 a strong correlation between maximum and walking velocity with a high slope of the regression line of 28.1°/m/s was identified for level walking. in contrast, with c-leg 3 the slope of 3.5°/m/s was very small. with able bodied subjects as well as with amputees on the contralateral side constant maximum knee flexion angles with different speeds of level walking ensure low energy consumption and sufficient toe clearance to avoid falls [4, 6-10]. during functional testing of the two mpk with the gait simulator, similar tendencies to the results of the gait lab tests could be found. the increase of the maximum knee flexion angle with rising walking velocity measured with the simulator was 37.8°/m/s with the plié2.0 and 21.4°/m/s with the c-leg 3 (fig. 9). the higher slopes resulting from simulator tests could be related to the smaller velocities simulated on the testing machine compared to the gait lab tests. the gait simulator for lower limb exoprostheses 201 fig. 8 comparison of target and actual values measured with the simulator for plié2.0, coordinate system located in the hip center with shown orientation fig. 9 knee flexion angle [°] over walking velocity [m/s] measured with the simulator 202 j. thiele, s. gallinger, p. seufert, m. kraft 4. discussion the first measurements with the simulator showed a high reproducibility comparing successive gait cycles. in some parameters, there was a good concordance between the reference data used for control and the measured data. this could be observed for the values of the ground reaction forces and the hip moment in the sagittal plane, which represents the main functional plane of prosthetic knee joints. thigh and shank angles and the resulting knee flexion angle showed higher deviations. the reason for the differences could be found in the missing translatory movement of the hip module during swing phase which also accelerates the shank. due to these restrictions, the prosthetic knee joints could not be tested within the optimum operation range of their swing phase control. this is also ascribed to the tested walking velocities and step lengths that were below the comfortable values measured in the gait lab tests. both the tested knee joints are subject to this system-dependent influence as they are controlled by the gait simulator with the same reference values. for that reason, the differences between the knee joints in the shank angle and the resulting knee flexion angle can be attributed to different joint characteristics. comparing the maximum of the knee flexion angles in the swing phase, the plié2.0 has shown a higher dependence on the walking velocity than the c-leg. comparable results could also be found in the previous gait lab tests [4, 5, 11]. the results of this study suggest that the functional differences between knee joints can be shown with the gait simulator too. however, further research is needed to enable realistic test conditions. 5. outlook the high reproducibility of the simulator tests may enable the examination of the influence of changes in the setup of the prosthesis. thus, the interaction of different prosthetic components may be investigated, like the influence of different prosthetic feet used with the same mpk. the measured foot-compliance-curves already enable characterizations of prosthetic feet. to examine the interaction with the mpk, further measurements with different prosthetic components are necessary. furthermore, the optimization of the gait simulator should focus on increasing the achievable walking velocities and step lengths to allow testing of the prosthetic knee joints under conditions that correspond to everyday usage. to extend the scope of the investigations, the simulation of other gait situations in addition to level walking is planned. especially the tests providing information on the potential of a prosthetic knee joint to prevent falls could be safely conducted with the simulator. in the future an optimized gait simulator could provide the basis for comprehensive and reproducible functional testing of prosthetic components. acknowledgements: the paper is a part of the research done within the project bemobil (fkz 16sv7069k), founded by the federal ministry of education and research germany. the author want to thank wulf wulff for designing and developing the gait simulator during his phd project at the tu berlin. the knee joints were provided by courtesy by otto bock healthcare gmbh. the gait simulator for lower limb exoprostheses 203 references 1. oehler, s., 2015, mobilitätsuntersuchungen und belastungsmessungen an oberschenkelamputierten , degruyter, berlin. (isbn: 978-3-11-026786-0) 2. wulff, w., and kraft, m., 2009, entwicklung eines gangsimulators für beinprothesen, orthopädietechnik, 5, pp. 310–318. 3. wulff, w., and kraft, m., 2010, konzeption und entwicklung eines gangsimulators für beinprothesen, medizinisch-orthopädische technik, 3, pp. 26–28. 4. thiele, j., westebbe, b., bellmann, m., kraft, m., 2014, designs and performance of microprocessorcontrolled knee joints, biomedizinische technik/biomedical engineering, 59(1), pp. 65–77. (doi: 10.1515/bmt-2013-0069) 5. bellmann, m., schmalz, t., and blumentritt, s., 2010, comparative biomechanical analysis of current microprocessor-controlled prosthetic knee joints, archives of physical medicine and rehabilitation, 91, pp. 644–652. 6. blumentritt, s., schmalz, t., and jarasch, r., 2009, the safety of c-leg: biomechanical tests, journal of prosthetics and orthotics, 21, pp .2–15. 7. hafner, b. j., smith, d. g., 2009, differences in function and safety between medicare functional classification level-2 and -3 transfemoral amputees and influence of prosthetic knee joint control, the journal of rehabilitation research and development, 2009, 46, pp. 417–433. 8. kahle, j. t., highsmith, m. j., hubbard, s. l., 2008, comparison of nonmicroprocessor knee mechanism versus c-leg on prosthesis evaluation questionnaire, stumbles, falls, walking tests, stair descent, and knee preference, the journal of rehabilitation research and development, 45, pp. 1–14. 9. winter, d. a., 1983, biomechanical motor patterns in normal walking, journal of motor behavior, 15, pp 302–330. 10. hanlon, m., anderson, r., 2006, prediction methods to account for the effect of gait speed on lower limb angular kinematics, gait posture, 24, pp. 280–287. 11. bellmann, m., 2009, funktionsprinzipien aktueller mikroprozessor gesteuerter prothesenkniegelenke, orthopädie-technik, 60, pp. 297–303. facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 87 93 https://doi.org/10.22190/fume190103012l © 2019 by university of niš, serbia | creative commons licence: cc by-nc-nd short communication numerical implementation of fretting wear in the framework of the mdr qiang li, fabian forsbach, justus benad berlin university of technology, berlin, germany abstract. two numerical methods are proposed to improve accuracy of the numerical calculation of fretting wear in the framework of the method of dimensionality reduction (mdr). due to the singularity of the transformation equations, instabilities appear at the border between the stick and slip regions after many transformations from the one-dimensional to the three-dimensional contact and back. in these two methods, the transformation equations are reformulated to weaken the singularity of the integrals and a stable simulation of fretting wear is realized even with the wear models which go beyond the classical archard law. with an example of dual-oscillation, we show the change in the worn profile of a parabolic indenter as well as the stress distribution on the contacting surface during the oscillating cycles under the archard’s law of wear and coulomb’s law of friction. key words: fretting wear, method of the dimensionality reduction, singularity, numerical simulation 1. introduction in recent years, the method of the dimensionality reduction has been applied to various contact problems. the classic contacts of rotationally symmetric indenters like the hertzian normal contact, partial sliding, or jkr (johnson-kendall-roberts)-type adhesive contact, etc. can be easily understood and resolved very quickly in the framework of the mdr [1]. a similar approach, the method of memory diagrams (mmd), provides semi-analytical solutions for axisymmetric contact problems, for example, friction-induced energy loss [2]. the paper [3] provides a series of guidelines for using the mdr in the applications of homogeneous media or graded material, in elastic and viscoelastic contacts. if the contacting bodies and the material properties do not change in the whole contact case, one can obtain the results by simulation merely in the framework of the one-dimensional contact, for example, relaxation damping [4] received january 03, 2019 / accepted march 01, 2019 corresponding author: qiang, li affiliation: berlin university of technology, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: qiang.li@tu-berlin.de 88 q. li, f. forsbach, j. benad or adhesive pull-off [1]. however, in some cases, for example, in the wear contact, the surface profile of the indenter changes due to wear so that one has to come back to the three-dimensional contact by using the transformation equations to calculate a new surface profile. then the 3d profile is transformed into the corresponding new 1d profile for the solution of the contact problem. the procedure is then repeated for the whole contact. fig.1 shows this simulation procedure as presented in the papers [5, 6]. fig. 1 procedure of numerical calculation of wear using the mdr these transformation equations are listed in the following for the profile, from 3d f(r) to 1d g(x), and for the displacement and stress, from 1d u(x), q(x) to 3d w(r), p(r): 2 20 ( ) ( ) d x f r g x x r x r     , (1) 2 20 2 ( ) ( ) d r u x w r x r x    , (2) 2 2 1 ( ) ( ) d r q x p r x x r       . (3) these abel equations have a singularity at point x r . numerically, the integral can be calculated in different ways, for example, using simpson’s 1/3 rd rule, or via a semianalytical technique with piecewise approximation in a segment by a constant or linear profile [7, 8]. from fig.1, we can see that in the case of numerical description of wear, the transformation from 1d to 3d and the back transformation have to be carried many times, which will lead to instability if the integral is numerically approximated with a low accuracy. in the recent paper [9], the transformations (1)-(3) are rewritten by using the integration by parts to avoid the singularity, and the corresponding numerical implementation is given in detail. with an example of gross slip wear, this method shows very accurate results, where, however, only transformations (1) and (3) are necessary. in the fretting wear, we observe instability using the same method where the eq. (2) is also involved. in this short communication, we will give two further approaches to improve the integral precision and stability in the fretting wear simulation. numerical implementation of fretting wear in the framework of the mdr 89 2. methods 2.1. method a as suggested in [10], the singularity of the integrand can be weakened by splitting it into two parts to improve accuracy of an approximate integration. focusing on the eq. (2), it can be rewritten as 1 2 2 2 2 20 2 ( ) ( ) ( ) d r u x u x w r x r x r x          , (4) with 1( ) ( )u x xu x , 2( ) ( ) ( )u x u x xu x  . (5) eq. (4) can further be written with a derivative:   2 2 1 0 2 d dsin ( ) ( ) ( ) ( ) ( ) d d d r r x x r w r u x u x x u x x x x                . (6) this integral can be numerically calculated with greater accuracy than eq. (2). in a discrete form, eq. (6) is  2 2 2 2 1 1 1 111 2 ( ) sin sin k j k j k j j k kk k k k j j u r x r x w x x u u x r r                                           . (7) first derivative u΄k is obtained via forward difference. for eqs. (1) and (3), however, we use the method from the recent paper [9] which gives highly accurate results. 2.2. method b here we rewrite all the three transformations (1)-(3) simply in the following form 1 0 dsin ( / ) ( ) ( ) d d x r x g x x f r r r    , (8) 1 0 2 dsin ( / ) ( ) ( ) d d r x r w r u x x x    , (9) 11 dcosh ( / ) ( ) ( ) d dr x r p r q x x x      . (10) using the trapezoidal method, they can be written in the discrete form 1 11 1 2 sin ( / ) sin ( / ) 2 k j k k j k k j k j k f f g x r x r x               , (11) 90 q. li, f. forsbach, j. benad 1 1 1 1 2 2 sin ( / ) sin ( / ) 2 k j j k j k j k k k w x r u x r u               , (12) 1 11 1 1 1 cosh ( / ) cosh 2 ( / ) k n k k j k j k j k j q q p x r x r                 . (13) the first derivatives in eqs. (11)-(13) can be obtained via central differences. different from the method in [9] by using the technique of integration by parts, the second derivatives are not necessary in the above method b. fig. 2 shows the numerical errors of 1d profile, normal stress and deformation at each discrete point in the contact area in comparison with the hertzian theory. the error is estimated as the absolute value of (results a or b – theory)/theory. it is noted that the adjustment as suggested in [9] is not applied for stress in the method a. it is seen that for the profile, there is no difference for the two methods, but for the deformation, the method a is better. a) b) c) fig. 2 error estimation with an example of hertzian contact for: (a) 1d profile; (b) normal stress; (c) normal deformation 3. implementation of fretting wear now we numerically simulate fretting wear with the approaches given in section 2. the indenter with initial parabolic profile f=r 2 /(2r) is pressed into the elastic half space with elastic modulus e * and poisson’s ratio . the indenter oscillates with displacement-controlled periodic functions in both vertical and horizontal directions: (0) 1 (0) 2 ( ) sin ( ) sin( ) z z z x x x u t u u t u t u u t         . (14) the numerical algorithm is the same as in fig.1, and we use the local formulation of archard’s law of wear for the change in 3d surface profile: (0) ,3d( ) ( )( ( ))wear x xf r k r u u r    , (15) where kwear is the wear coefficient, τ(r) is the tangential stress on the surface, ux (0) is the tangential movement of the indenter and ux,3d is the change of tangential displacement of the elastic half space in a time increment. numerical implementation of fretting wear in the framework of the mdr 91 a) b) c) d) fig. 3 simulation of fretting wear: (a) 3d profiles and (b) corresponding 1d profiles at three states; (c) normal and tangential stresses at state 1 and (d) state 3 it is known that if the tangential oscillation amplitude is small, there will be a stick region in the center and a slip region at the boundary of the contact area with coulomb’s law of friction. the contact problem can be solved in the 1d contact with winkler’s spring foundation [3]. the obtained 1d tangential stress qx and tangential movement ux,1d are then transformed into three-dimension τ(r) and ux,3d for substitution into the wear law (5) to get the new profile. in the mdr, the stick and slip regions of ‘springs’ are determined according to the following rule: (0)( ) , in the stick region,if ( ) ( ) ( ) ( ) , in the slip region x x x x z z x x u x u k u x f x f x u x k         . (15) in the numerical simulation, we first assume that all the springs in the contact region move with the same value ux (0) as the indenter and then the tangential forces of these springs are calculated kxux,pre. subsequently, we check the condition (15): the springs that meet condition kxux,pre <µfz are located in the stick region, where the relative tangential movement is zero, ux =ux (0) , as described in (15). the other springs having the relation 92 q. li, f. forsbach, j. benad kxux,pre>µfz are in the slip region, where the tangential displacement must be smaller than assumed value ux<ux,pre, and it will be calculated with the rule in (15). in fretting wear, the indenter moves also backward, so the sign in (15) should be the same as ux,pre. numerically, we simulated various cases of the fretting, including variation of phase, different oscillation frequencies in normal and tangential directions, amplitude of oscillation, coefficient of friction, and also other laws of wear, for example the case when the surface change has a power-law function of stress or relative tangential movement. both methods show stable results. here we give only one example as shown in fig.3. the main parameters are set as ux (0) = uz (0) =0.1 uz (0) , ux (0) =0, φ=/4, 1=2=100, µ=0.1 and discrete points n=100. the states 1, 2, 3 in figs.3a and b indicate 50.72, 100 and 199.24 cycles of oscillation (the large value of wear coefficient is used to observe the wear behavior quickly). the 3d worn profile and the corresponding 1d profile are shown in figs.3a and b. figs.3c and d show the 3d normal and tangential stresses at state 1 and 3, where the indenter is located shortly before 3/4 and 1/4 period, respectively. the difference of the 3d profile between these two methods is very small. only when the surface is strongly worn, the stress singularity at the border of stick region can be observed. 4. conclusion we provide two approaches to improving the accuracy of the numerical implementation of the transformations for the fretting wear. both the methods provide a fast and stable simulation in the fretting contact. besides the example presented in section 4, we have tried a few simulations with other laws of wear, for example f~τ 1.5 , and using not only the methods a and b but also their combinations. it is found that method b gives sometimes unstable simulations for a relatively large wear coefficient. generally, for the wear or other contact problems, we suggest the users of the mdr to use the numerical implementation of transformations in the following consequents: a) method a; b) combination of method a for eqs. (1) and (3) and method b for eq. (2); c) method b. in the tests there is almost no difference observed between suggestions (a) and (b). acknowledgements: the authors thank dimaki a.v. for providing the numerical code for fretting contact studied in [5] and thank popov v.l. for the discussion. references 1. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer, berlin. 2. kevin truyaert, k., aleshin, v., koen van den abeele, delrue, s., 2019, theoretical calculation of the instantaneous friction-induced energy losses in arbitrarily excited axisymmetric mechanical contact systems, int. j. solids struct., 158, pp. 268-276. 3. popov, v.l., heß, m., 2014, method of dimensionality reduction in contact mechanics and friction: a user’s handbook. i. axially-symmetric contacts, facta universitatis-series mechanical engineering, 12, pp. 1–14. 4. popov, m., popov, v.l., pohrt, r., 2015, relaxation damping in oscillating contacts, sci. r., 5, 16189. 5. dimaki, a.v., dmitriev, a.i., chai, y.s., popov, v.l., 2014, rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction, int. j. solids struct., 51, pp. 4215-4220. numerical implementation of fretting wear in the framework of the mdr 93 6. dimaki, a.v., dmitriev, a.i., menga, n., papangelo, a., ciavarella, m., popov, v.l., 2016, fast high-resolution simulation of the gross slip wear of axially symmetric contacts, tribol. trans., 59, pp. 189-194. 7. dasch, c.j., 1992, one-dimensional tomography: a comparison of abel, onion-peeling, and filtered backprojection methods, appl. opt., 31, pp. 1146-1152. 8. kolhe, p.s., agrawal, a.k., 2009, abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing techniques, appl. opt., 48, pp. 3894-3902. 9. benad, j., 2018, fast numerical implementation of the mdr transformations, facta universitatis-series mechanical engineering, 16, pp. 127–138. 10. krylov, v.i., 1962, approximate calculation of integrals, the macmillan company, now york. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 125 139 https://doi.org/10.22190/fume190301031n © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper on the discretization of a bistable cantilever beam with application to energy harvesting max-uwe noll, lukas lentz, utz von wagner chair of mechatronics and machine dynamics, technische universität berlin abstract. a typical setup for energy harvesting is that of a cantilever beam with piezoceramics excited by ambient base vibrations. in order to get higher energy output for a wide range of excitation frequencies, often a nonlinearity is introduced by intention in that way, that two magnets are fixed close to the free tip of the beam. depending on strength and position of the magnets, this can either result in a mono-, bior tristable system. in our study, we focus on a bistable system. such systems have been investigated thoroughly in literature while in almost all cases the beam has been discretized by a single shape function, in general the first eigenshape of the linear beam with undeflected stable equilibrium position. there can be some doubts about the suitability of a discretization by a single shape function mainly due to two reasons. first: in case of stochastic broadband excitations a discretization, taking into consideration just the first vibration shape seems not to be reasonable. second: as the undeflected position of the considered system is unstable and the system significantly nonlinear, the question arises, if using just one eigenshape of the linear beam is a suitable approximation of the operation shapes during excited oscillations even in the case of harmonic excitation. are there other, e.g. amplitude dependent, possibilities and/or should multiple ansatz functions be considered instead? in this paper, we focus mainly on the second point. therefore, a bistable cantilever beam with harmonic base excitation is considered and experimental investigations of operation shapes are performed using a high-speed camera. the observed operation shapes are expanded in a similar way as it is done in a theoretical analysis by a corresponding mixed ritz ansatz. the results show the existence of distinct superharmonics (as one can expect for a nonlinear system) but additionally the necessity to use more than one shape function in the discretization, covering also the amplitude dependence of the observed operation shapes. key words: bistable beam, energy harvesting, discretization, vibration shape analysis via high speed camera received march 01, 2019 / accepted june 20, 2019 corresponding author: max-uwe noll chair of mechatronics and machine dynamics, technische universität berlin, einsteinufer 5, 10587 berlin e-mail: max-uwe.noll@tu-berlin.de 126 m.-u. noll, l. lentz, u.v. wagner 1. introduction harvesting of energy from ambient vibrations has attracted much interest and corresponding research in the past decades. a common method for transferring the mechanical energy into electric one is the usage of piezoceramics fixed on corresponding vibration structures. a survey of such systems gives, for example, the paper [1]. the classical setup in that case is a cantilever beam with piezoceramics bonded close to the clamping excited by ambient base vibrations. these systems perform well when they are designed as a linear resonator with its eigenfrequency tuned to the frequency of the excitation. a higher energy output, especially for broadband or stochastic excitations, can be realized in such systems, if nonlinearities are introduced by intention resulting in a broadband characteristic for large responses of excited vibrations compared to sharp resonance peaks of the linear system [2]. a common setup is to fix two magnets symmetrically close to the free end of the beam, as shown in fig. 1. fig. 1 vibrational energy harvesting system [3] depending on strength and position of the magnets, this can either result in a mono-, bi or tristable (e.g. [4, 5, 6] respectively) system. in this paper we investigate the bistable configuration and focus on the occurring vibration shapes during non-chaotic operation. in that case, there are two main types of solutions, namely so-called intrawell solutions around one of the two stable equilibrium positions and so-called interwell solution with large displacements covering both stable equilibrium positions. for the modeling of the system, knowledge is necessary for the discretization of the beam with respect to its longitudinal coordinate x, introducing a mixed (dependence on both x and time t) ritz ansatz 1 ( , ) ( ) ( ) n i i i w x t w x p t   (1) into the partial differential equation describing the continuum vibrations of the beam. herein w(x,t) is the lateral displacement of the beam relatively to the moving frame and n the chosen ansatz order. in this ansatz shape functions wi(x) shall be given (they must fulfill all the boundary conditions) while functions pi(t) can be calculated from the thereby discretized model equations e.g. by harmonic balance. this let arise the question of suitable shape functions wi(x) and ansatz orders n. for the sake of simplicity and in order to concentrate on the question of how to discretize a bistable beam, the piezoceramics are neglected in the following. only the on the discretization of a bistable cantilever beam with application to energy harvesting 127 setup of the cantilever beam with magnets and base excitation is considered. the classical paper describing a corresponding modeling is that of moon [7] discretizing the beam with the first eigenshape of the linear euler-bernoulli beam and modeling the magnetic forces by a third order polynomial. this discretization results in a duffing-oscillator with negative linear and positive cubic restoring term. most publications follow this model when adding the piezoceramics for the energy harvesting system, e.g. [8] where it was shown, that the nonlinear system performs better than its linear counterpart for a non-resonant excitation and [9] where the system was analyzed for the case of a stochastic excitation. publications using more than one ansatz function are very rare. in [10] multiple ansatz functions are applied for a buckled beam system, nevertheless ending finally up again with a onedegree of freedom model. there can be some doubts about the suitability of this discretization by a single shape function mainly due to two reasons. first: in the case of stochastic broadband excitation a discretization, taking into consideration just the first vibration shape, seems not to be reasonable. therefore, the authors have in prior investigations [11, 12] added a second ansatz function in that way, that not just the first, but also the second eigenfunction of the linear beam is considered in the ritz ansatz. second: as the undeflected position of the considered system is unstable and the system significantly nonlinear, the question arises if using just the first eigenshape of the linear beam is a suitable approximation of the operation shapes during excited oscillations even in the case of harmonic excitation. are there other, e.g. amplitude dependent, possibilities and should multiple ansatz functions be considered instead? in the work [13] and [14] this topic was already addressed, and it is shown how the usage of linear shape functions leads to erroneous results. furthermore, a purely theoretical method to compute nonlinear, i.e. amplitude dependent shape functions, is presented. later the concept of nonlinear normal-modes was transferred to the analysis of discrete nonlinear systems, see e.g. the review [15] and the references therein. on the other hand in [16] it is shown, that for a bistable system, in that case a buckled beam, the two first modes coexist during the snapping process. in [17] exact solutions of postbuckling configurations of beams are calculated, but also the interaction between vibration modes is shown. nevertheless, as already mentioned, it is state of the art in considering energy harvesting systems to use just one mode shape for discretization. an example for this is [18], where both the buckled beam as in the two papers mentioned before as well as the bistable cantilever beam, that is considered in the actual paper, is discretized by a single degree of freedom. in this paper, the questions, if the mixed ritz ansatz, eq. (1), gives a suitable modeling as well as how many and which ansatz functions should be used, is discussed in the case of harmonic base excitation by analyzing experimental results. therefore, operation shapes have been captured by a high-speed camera. the observed operation shapes are expanded into their frequency content and then the vibration shape corresponding to each frequency is analyzed. 2. experimental setup the chosen setup for the experimental investigations is shown in fig. 2. the steel cantilever beam is placed in an aluminum frame excited by a shaker and the bistability is realized by two magnets placed approximately symmetrical to the undeflected position of 128 m.-u. noll, l. lentz, u.v. wagner the beam, therefore becoming unstable. the beam’s static deflection, when there is no base excitation, as well as the dynamic displacement of the beam tip are measured using a laser triangulation sensor which is attached to the moving frame, hence directly providing the relative displacement of the beam tip from its undeflected position without the superposed movement of the supporting frame. the base excitation of the system is captured by a (nonmoving) laser vibrometer. laptop i is used to process the data delivered by these two measurement devices with the software package vanalyzer. fig. 2 experimental setup with bistable cantilever beam excited by a shaker (left) and beam with magnets in detail (right) [19] laptop ii is utilized to control the high-speed camera photron fastcam mini ax100 by the software package phontron fastcam viewer for high speed digital imaging. the dimensions of the cantilever beam together with its first natural frequency (without magnets) are given in table 1. table 1 properties of the cantilever beam property value beam length beam width beam thickness 1st eigenfrequency 250 mm 20 mm 1 mm 13.1 hz on the discretization of a bistable cantilever beam with application to energy harvesting 129 the magnets are intended to be placed symmetrically with respect to the undeflected position of the beam in order to realize an approximately equal distance of the two equilibrium positions from the undeflected beam position. the magnets are glued to a magnet carrier, also made of aluminum, which ensures a constant and reproducible distance between the magnets of approximately 14 mm. the exact placement of the magnet carrier with respect to the beam is however limited by the manual adjustment of the carrier with finite accuracy and reproducibility. the resulting distances of two realized experiments together with the frequencies of small free vibrations around the two stable equilibrium positions (intrawell solutions) are given in table 2. table 2 properties of bistable beam configuration (magnet distance approx. 14 mm) left right setup i (static experiment in fig. 6) equilibrium position frequency setup ii (dynamic experiments) equilibrium position -6.91 mm 15.1 hz -7.01 mm 7.13 mm 14.5 hz 7.40 mm frequency 16.0 hz 16.0 hz a paper stripe with geometry given in fig. 3 is attached to the front side of the beam. the beam itself has a thickness of one millimeter, but for a higher detection rate of the markers their size is chosen to be two millimeters in diameter on an all-black background of the three millimeters wide paper stripe, which therefore exceeds the thickness of the beam. fig. 3 geometry of marker stripe the camera was positioned manually in front of the beam setup with three goals regarding its arrangement. first: the camera was orientated orthogonal to the beam’s plane of motion. second: the camera was placed as far as possible from the beam to reduce errors due to perspective influences. third: the camera was placed as close as needed so that the beam took up almost the full height of the pictures in order to use the full resolution to capture each marker. the camera is taking monochrome black and white videos with a frame rate of 4000 frames per second and each frame has a resolution of 384×944 pixels. each video has a duration of at least one second. to further reduce the amount of data and processing time only every second frame was considered for further evaluation. the authors are aware that taking a picture is a sampling like process of the time continuous movement of the markers. that means that not only the sampling rate needs to be at least two times the highest frequency that is expected to be occurring in the measured signal (nyquist criteria), but 130 m.-u. noll, l. lentz, u.v. wagner also needs to be filtered for even higher frequencies to prevent aliasing effects. since there is no possible way to filter these analog signals, we have checked the results from the digital image analysis to the frequencies determined by the laser triangulation sensor, for which the signal has been aliasing filtered properly before sampling. both results are in good accordance with each other. the very first frame of each video is replaced by a frame showing the static, undeflected beam without magnets (fig. 4 (left)), which was taken before the magnets were applied. this first frame is taken as the reference to determine the relative displacement of each marker on each frame from its position when the beam is undeflected. the software gom correlate is used to analyze each frame of the video and to detect the relative distance from its position on the reference frame. the calibration of the distance measurement from the frames is done using the known distance of the markers on the maker strip, which results in a resolution of roughly 3.5 pixel/mm. fig. 4 reference frame of undeflected beam without magnets (left) and marker detection and evaluation of marker displacement with gom correlate (right) further it is necessary to eliminate the relative movement of the supporting frame to the nonmoving camera in order to find the relative displacement of each marker. this is done by subtracting the current displacement of the marker that is the nearest to the beam clamping from all other marker displacements of that current frame. 3. results of experimental investigations and their analysis the aim of the following investigations is to decide if the observed operational vibration shapes of the harmonically excited bistable beam can be expanded in the eigenshapes of the euler-bernoulli beam as well as how many ansatz functions are required according to the on the discretization of a bistable cantilever beam with application to energy harvesting 131 mixed ritz ansatz (eq. (1)). fig. 5 shows the first two eigenshapes 1, 2 of the beam together with the static bending line caused by a constant lateral tip force. fig. 5 beam eigenshapes and static bending line according to the linear euler-bernoulli beam theory in fig. 6 the measured static bending line resulting in the “right” stable equilibrium (positive displacement according to table 2) resulting from the magnets from the setup i is displayed. this measured bending line shows a high agreement with the theoretical static bending line. fig. 6 static bending of the beam in “right” equilibrium position (table 2, setup i) now the operation shapes are measured with a high-speed camera. fig. 7 shows typical examples of the two main types of solutions, namely the intrawell solution (left) and the interwell solution (right). the phase diagram of the last marker at the tip of the beam is shown, where the velocity has been determined by the time derivative of the displacement after filtering the signal by a butterworth low pass filter. both solution types do not show any period multiplication in these tests, which restricts in the stationary case the occurring response frequencies to the excitation frequency and corresponding superharmonics while subharmonics do not occur. 132 m.-u. noll, l. lentz, u.v. wagner fig. 7 phase diagram of beam tip: intrawell solution (left) and interwell solution (right) (setup ii) for distinct points jx̂ time series for the displacement can be derived. a corresponding result is shown in fig. 8 in the case of harmonic excitation with f0=14 hz. fig. 8 time series of three points xj, j = 32,42,52 of the beam in case of an intrawell (left) and interwell solution (right) resulting from harmonic excitation with f0=14 hz (setup ii) again in fig. 8 on the left an intrawell solution is shown while an interwell solution is displayed on the right. an fft (fast fourier transformation) analysis for each of the time series ˆ ( )j ix t is performed. a corresponding result is shown in fig. 9. fig. 9 fft analysis of the time series in fig. 8 (setup ii) on the discretization of a bistable cantilever beam with application to energy harvesting 133 according to the nonlinear character of the system, distinct superharmonics of the excitation frequency in the response can be expected. due to the symmetric characteristic of the interwell solutions, odd superharmonics are expected while even and odd superharmonics can be expected in the intrawell case due to the asymmetric restoring characteristic around the stable equilibrium positions originating from the magnets. this is confirmed by the experimental results. subharmonics do not occur, as there are no period multiplications. having considered these initial results, we will first reconsider the mixed ritz ansatz (eq. (1)) and the corresponding theoretical analysis. using e.g. harmonic balance as solution method for the discretized system equations and considering an excitation being proportional to cost with =2f0 where  is the circular excitation frequency, one can expect, that time function pi(t) in the mixed ritz ansatz can be expanded as follows 0 ( ) cos( ) m i ik ik k p t a k t      (2) where k=0,1,2,3, … in the case of the intrawell solution and k=1,3,5,7, … in the interwell case. the intrawell case contains both a constant solution part (k=0) due to the deflected stable equilibrium position as well as even superharmonics due to the non-symmetric magnet force characteristic. on the other hand, the interwell solution has in theory a zero mean value and is symmetric, which limits k to odd numbers. these considerations are almost fully confirmed by the results shown in fig. 8 and 9. only a small constant part of the interwell solution is possibly due to the non-perfect symmetry of the magnets (table 2). inserting the expansion (2) in the mixed ritz ansatz results after sorting in 1 0 ( , ) ( ) cos( ) ik n m i ik i k w x t w x a k t        , (3) as already mentioned in the introduction, we are interested in the question which and how many shape functions wi(x) are necessary for a good representation of intraand interwell solutions in our setup. therefore, we will now do the same expansion with the experimental results. for the j-th distinct point jx̂ (position on the beam) the time series of its movement is expanded by a fourier expansion 0 ˆ( , ) cos( ) m j jk jk k w x t a k t      (4) with coefficients jka~ and a phase shift jk~ . while ik in eq. (2) describe the phase shift compared to the excitation, jk~ depend on the time sequence of the video to be analyzed, and are triggered by the starting of the measurement. as we are interested purely in vibration shapes this is not a restriction. for each considered multiple k of excitation circular frequency , from jka~ a shape ˆ ( )kw x shall be formed, which is then expanded in considered shape functions wi(x). to do so, the following circumstances must be taken into consideration. jka~ are taken from the absolute values of the fourier expansion (which is performed in a complex notation), therefore they have positive values only. this means that in the cases where the beam vibration has vibration nodes it is to be considered that there is a phase shift of  between 134 m.-u. noll, l. lentz, u.v. wagner jk~ , even for one fixed mode k. therefore, applicability of eq. (4) requires, that for each k all jk~ are constant for all j with the exception that a jump with size  is possible in the case of nodes. if this is the case, subscript j can be neglected in the following and the final phase shift k̂ is described by kk 1 ~ˆ   (5) while jka~ are replaced by jkâ following the scheme        .~~if~ 0~~if~ ˆ 1 1   kjkjk kjkjk jk a a a (6) from jkâ the shape ˆ ( )kw x is formed, which is expanded as follows 1 ˆˆ ( ) ( ) n k ik i i w x a w x   (7) with coefficients ikâ . in general, displacement w(x,t) from the experiments is therefore given by 0 ˆˆ( , ) ( )cos( ) m k k k w x t w x k t      (8) inserting eq. (7) in (8) results in 0 1 ˆ ˆ( , ) ( )cos( ) m n ik i k k i w x t a w x k t       (9) and after sorting in 1 0 ˆ ˆ( , ) ( ) cos( ) n m i ik k i k w x t w x a k t        (10) which is almost equal to the theoretical result (3). the only differences are that subscript i is missing in k̂ , i.e. if the expansion is possible as described, the phase shift for the k-th harmonic does not depend on the number i of the mode wi(x), and reference times for the phase shifts may differ as described above. in the following, the steps (4)–(7) are performed with the experimental results. fig. 10 shows a more detailed analysis of the frequency contents of the intrawell solution (left) and interwell solution (right) of the beam tip with a logarithmic scale. constant parts (zero frequency) are not considered in the following as they are due to the static bending line, which is geometric almost similar to the first eigenshape 1 (fig. 4) and therefore anyway covered by the following expansion in 1 and 2. in the case of the intrawell solution we limit our result in accordance with fig. 10 (left) to k=1, 2 and 3, while we will take in the case of the interwell solution k=1, 3, 5 and 7 into consideration. on the discretization of a bistable cantilever beam with application to energy harvesting 135 fig. 10 fft of beam tip displacement of intrawell (left) and interwell (right) oscillation (setup ii) fig. 11 shows 1 ˆ ja for the intrawell and interwell solution respectively, forming the corresponding shapes 1 ˆ ( )w x . in the following these shapes shall be expanded according to eq. (7), where coefficients ikâ are found from the experimental data using a least square approach. as shape functions wi(x) we take the first two eigenfunctions 1, 2 of the beam as sketched in fig. 5, i.e. we limit n by 2, which means i = 1, 2. the corresponding expansion shows, that shapes 1 ˆ ( )w x are almost identical to 111 ˆ a , and 0ˆ 21 a (red lines in fig. 11 (top)). fig. 11 (bottom) shows the corresponding phases 1 ~ j , which should be equal, or only have a jump of  at a vibration node, for all j in order to allow the expansion (4). in fact, it can be seen that there are only small deviations occurring mainly close to the clamping, i.e. at points with only small displacements. fig. 11 vibration shape 1 ˆ ja of intrawell (top left) and interwell (top right) for frequency f=f0 and corresponding phases j~ beneath (setup ii) 136 m.-u. noll, l. lentz, u.v. wagner for the intrawell solution fig. 12 shows the 2 ˆ ja and 3 ˆ ja forming the shapes 2 ˆ ( )w x and 3 ˆ ( )w x corresponding to twice and three times excitation frequency 2f0 and 3f0. fig. 12 vibration shape of intrawell oscillation for frequency f = 2f0 (left) and f = 3f0 (right) (setup ii) while 2 ˆ ( )w x gives a somewhat smooth curve there are several small deviations in 3 ˆ ( )w x but the shapes of both functions can be expanded with the chosen ansatz functions 1 and 2. fig. 13 shows this step in the case of the interwell solutions for shapes 3 ˆ ( )w x , 5 ˆ ( )w x and 7 ˆ ( )w x . all shapes can be expanded in good approximation by 1 and 2. fig. 13 vibration shape for interwell oscillation for frequency f = 3f0 (left), f = 5f0 (middle) and f = 7f0 (right) (setup ii) from these results it can be concluded that expansions (3) and (10), respectively, can give a good approximation of the experimentally observed shapes and that the beam eigenshapes are suitable functions for the expansion, which is in agreement to most used models in literature (cf. section 1). on the other hand, the shapes corresponding to the superharmonics k with k>1 can only be suitably expanded, when using at least two shape functions in the ritz ansatz (1), while most publications restrict to a single one! on the discretization of a bistable cantilever beam with application to energy harvesting 137 finally, the amplitude dependence shall be discussed. fig. 14 shows 1 ˆ ja forming shape 3 ˆ ( )w x in the case of the intrawell (left) and the interwell solution (right) in the case of different excitation and, therefore, response amplitudes as well. the shapes are almost proportional and are represented for all excitation amplitudes in good approximation by 1. this changes for the higher harmonic shapes. in fig. 15 and 16 the shapes for twice the excitation frequency in the case of the intrawell and three times excitation frequency in the case of the interwell solution are displayed. it can be seen especially in fig. 15 that the corresponding shapes change with excitation amplitude, but all shapes can be very well approximated by 1 and 2. from this it follows that the ansatz (1), (2) with w1,2 = 1,2 can represent all the observed behavior in good approximation, but at least two ansatz functions are necessary while there is no need for amplitude dependent (nonlinear) ansatz functions in that case. fig. 14 vibration shape for frequency f=f0 for different amplitudes of the excitation (intrawell left, interwell right) fig. 15 vibration shape for frequency f=2f0 (intrawell) for different amplitudes of the excitation fig. 16 vibration shape for frequency f=3f0 (interwell) for different amplitudes of the excitation 138 m.-u. noll, l. lentz, u.v. wagner 4. conclusions energy harvesting performed by a bistable cantilever beam has attracted much attention. in general, the beam is in corresponding modeling discretized by the first eigenshape of the linear beam in a corresponding mixed ritz ansatz. in this paper, this common assumption has been proofed for suitability. therefore, the beam has been harmonically excited and corresponding response vibrations have been captured by a high-speed camera. distinct markers have been applied to the beam and their positions were tracked over a series of frames. in accordance with the ritz ansatz in theory the experimental results have been analyzed for both intrawell and interwell solution performing the following steps: first a fourier expansion of the responses has been performed. as there was no period multiplication only the excitation frequency and superharmonics exist in the responses. for these harmonics, corresponding shapes could be identified which are afterwards expanded in the two first eigenfunctions of the undeflected cantilever beam. the results show that the general ansatz to separate the solution of the beam vibration into a product of functions depending on t and x respectively is possible and sufficient. on the other hand, a good approximation of the experimentally observed shapes can only be reached if at least two ansatz functions are applied. only for considering the excitation frequency in the response a single ansatz function is sufficient while for superharmonics a second ansatz function is needed to sufficiently approximate the observed vibration shapes. further, the existing amplitude-dependence of the shapes due to the nonlinearities, can also be covered by the two ansatz functions. acknowledgements: this work has been funded by deutsche forschungsgemeinschaft (dfg) by grant wa 1427/23,2. references 1. priya, s., 2007, advances in energy harvesting using low profile piezoelectric transducers, journal of electroceramics, 19, pp. 165-182. 2. erturk, a., inman, d., 2009, a piezomagnetoelastic structure for broadband vibration energy harvesting, applied physics letters, 94, 254102. 3. noll, m.-u., 2018, energy harvesting system, doi:10.6084/m9.figshare.7492208.v1. 4. mann, b. p., owens, b., 2010, investigations of a nonlinear energy harvester with a bistable potential well, 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g., 2017, a simplified lumped model for the optimization of postbuckled beam architecture wideband generator, journal of sound and vibration, 409, pp. 165-179. 11. lentz, l., nguyen, h. t., von wagner, u., 2017, energy harvesting from bistable systems under random excitation, machine dynamics research, 41(1), pp. 5-16. on the discretization of a bistable cantilever beam with application to energy harvesting 139 12. lentz, l., 2018, on the modelling and analysis of a bistable energy-harvesting system, phd thesis, tuberlin, germany, 130 p. 13. shaw, s., pierre, c., 1994, normal modes of vibration for non-linear continuous systems, journal of sound and vibration, 169(3), pp. 319-347. 14. szemplinska-stupnicka, w., 1983, non-linear normal modes and the generalized ritz method in the problems of vibrations of non-linear elastic continuous systems, international journal of non-linear mechanics, 18(2), pp. 149-165. 15. mikhlin, y., avramov, k., 2011, nonlinear normal modes for vibrating mechanical systems, review of theoretical developments, applied mechanics reviews, 63(6), 060802. 16. cazottes, p., fernandes, a., pouget, j., hafez, m., 2009, bistable buckled beam: modeling of actuating force and experimental validations, journal of mechanical design, 131(10), 101001. 17. nayfeh, a. h., emam, s. a., 2008, exact solution and stability of postbuckling configurations of beams, nonlinear dynamics, 54, pp. 395-408. 18. vocca, h., cottone, f., neri, i., gammaitoni, l., 2013, a comparison between nonlinear cantilever and buckled beam for energy harvesting, eur. phys. j. special topics, 222(7), pp. 1699-1705. 19. noll,m.-u., 2019, experimental setup of an energy harvesting system ii, doi:10.6084/m9.figshare.7764851.v1. 8029 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 341 361 https://doi.org/10.22190/fume211028008y © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dynamic analysis of the rigid-flexible excavator mechanism based on virtual prototype yongliang yuan1, jianji ren2, zhenxi wang2, xiaokai mu3 1school of mechanical and power engineering, henan polytechnic university, jiaozuo, china 2school of computer science and technology, henan polytechnic university, jiaozuo, china 3school of mechanical engineering, dalian university of technology, dalian, china abstract. in this paper, the excavator’s dynamic performance is considered together with the study of its trajectory, stress distribution and vibration. many researchers have focused their study on the kinematics principle while a few others focused their work on dynamic performance, especially the vibration analysis. previous studies of dynamic performance analysis have ignored the vibration effects. to address these challenges, the rigid-flexible coupling model of the excavator attachment is established and carried out based on virtual prototyping in this study. the dipper handle, the boom and the hoist rope are modeled as a flexible multi-body system for structural strength. the other components are modeled as a rigid multi-body system to catch the dynamic characteristics. the results show that the number of flexible bodies has little effect on the excavation trajectory. the maximum stress determined for the dipper handle and the boom are 96.45 mpa and 212.24 mpa, respectively. the dynamic performance of the excavator is greatly influenced by the clearance and is characterized by two phases: as the clearance decreases, the dynamic response decreases at first and then increases. key words: excavator, dynamics, virtual prototype, rigid-flexible coupling 1. introduction excavators are widely used as primary production equipment for surface mining operations. in practical engineering, the dynamic response of an excavator reflects its performance. the conventional design process uses a laboratory test to simulate the physical prototype in order to evaluate the excavator performance, which is very costly and time-consuming for investigating the dynamic performance of the physical prototype [1]. received: october 28, 2021 / accepted february 22, 2022 corresponding author: jianji ren school of computer science and technology, henan polytechnic university, jiaozuo 454003, china e-mail: renjianji@hpu.edu.cn 342 y. yuan, j. ren, z. wang, x. mu recently, virtual prototype simulators have been widely used to simulate surface mining equipment [2]. in order to improve the dynamic performance of the excavator, it is necessary to build the excavator model based on the virtual prototype technology just as to build a physical prototype. for this purpose, the excavator could be analyzed based on a dynamics model. many kinematic and dynamics models of excavators were studied in the last decades [3,4], for example, li et al. [5] conducted an excavator model study and confirmed its feasibility with the virtual prototype software; they obtained its kinematics principle. frimpon et al. [6] used newton-euler to establish the dynamics model and obtained the displacement, the speed and the acceleration of an excavator in adams. šalinić et al. [7] established the lagrange equation model to obtain the kinematics principle of the excavator. awuah-offei et al. [8] proposed the numerical method and obtained the lifting speed and thickness of cutting relational curves. mitrev et al. [9] developed a plane multi-body mechanical model of a hydraulic excavator, and proved its applicability. ding et al. [10] proposed a new dynamic mathematical model, which broadened the scope of excavation and improved the excavating force. though the kinematics principle of excavators has been well studied, most of the previous work only dealt with a rigid multi-body system of the excavator rather than a more realistic rigid-flexible system. the excavator is a typical rigid-flexible coupling system, which could influence the dynamic performance. however, there are a few studies on the rigid-flexible coupling of excavators. ma used matlab/simulink to establish the model and carry out the simulation [11]. he et al. [12] used the rigid rod to replace the steel wire rope to establish the virtual prototype model, and carried out the simulation. he et al. [13] established the flexible body of the rope and obtained the stress change trend of the wire during the digging stage. jiang et al. [14] established the rigid-flexible coupling model of hydraulic excavators and obtained the vibration performance. strzalka et al. [15,16] offered an approach to a priori estimation with areas of the structure exposed to high stresses, which is decisive for a highly efficient fatigue analysis. zhao et al. [17] established a rigid-flexible coupling model, and obtained the stress of the key points in the cracked area. the above researchers focused on stress of the components based on the rigid-flexible coupling model and have not yet conducted a vibration analysis, especially the effect of the assembly clearance on the vibration of the excavator attachment. in order to investigate the dynamic performance of excavator attachments, this paper proposes a rigid-flexible coupling model based on virtual prototyping. the reminder of the paper is organized as follows: section 2 contains the structural components of the excavator and kinematic sketch of attachment. establishment of rigid-flexible coupling model is described in section 3. section 4 presents the numerical simulation settings. section 5 discusses the simulation results, especially the effect of different clearances on the boom and the dipper handle. section 6 gives the concluding remarks and future work. 2. the structural components of the excavator and kinematic analysis 2.1. structural components of the excavator the excavator is complex major equipment, which is mainly utilized in construction, mining, etc. the excavator has many advantages, including simple operation, low strength of the workers and high efficiency. the excavator consists of three major components, including the upper body, the lower body and the attachment. the whole system and main components of the excavator are shown in fig. 1. dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 343 fig. 1 the system composition of a typical large-scale excavator table 1 main parameters of the excavator technical parameters of the excavator total weight (ton) 1460 theoretical yield (m3/h) 6600 hoisting speed (m/s) 1.58 main motor powers (kw) 21000 thrusting speed (m/s) 0.75 maximum digging radius (m) 23.85 hoisting force (n) 2890 maximum digging height (m) 18.10 thrusting force (n) 2227 maximum permissible gradient (°) 13 2.2. the kinematic principle of excavator attachment for an excavator, the excavator attachment is one of the most important factors influencing its performance. therefore, the establishment of the kinematic sketch of excavator attachment is quite necessary before the dynamics simulation. the mathematical model of the attachment is shown in fig. 2. it can be seen from the fig. 2 that ∠dfe is the angle between the hoist rope and the center line of the dipper handle. it directly affects the magnitude of this force, which is along the dipper handle. as a result of the lifting force and the pushing force in the opposite direction, the pushing force will decrease and reduce power. thus, in order to reduce the loss of energy, the angle of ∠dfe could be increased as much as possible. according to fig. 2, the angle of ∠dfe can be divided into two components: 1 2dfe   = + (1) where ∠dfe is the angle between the hoist rope and the dipper handle center line. θ1 is the angle between the end of hoist rope to the center of pulley and the center line of the dipper handle. θ2 is the angle between the end of the hoist rope to the center of pulley and the hoist rope. furthermore: 2 2 2 2o f o c fc= + (2) 2 2 2 1 2 1 2 1 2 arccos 2 o f o f o o dfe o f o f + −  =   (3) where o1f is the distance from the end of the hoist rope to the center of pulley. o2f is the distance from the center of gear to the end of the hoist rope. o1o2 is the distance from the 344 y. yuan, j. ren, z. wang, x. mu center of the gear to the center of the pulley. o2c is the equivalent distance to df, which is the perpendicular distance from the center of the dipper handle (f) to o1o2. fc is the distance from the end of the hoist rope to o2c. fig. 2 the kinematic sketch of the excavator attachment the bucket trajectory can be decomposed along the vertical and horizontal direction; the equations are given as: 2 2 sin( ) cos( ) x o a y h o a       =  + +  = −  + + (4) where α is the angle between o2c and the vertical direction. β is the angle between o2c and the center line of the dipper handle. γ is the rotation angle of the dipper handle. h is the distance from the center of the gear to the ground. the distance is defined in eq. (5), which is from the pulley to the end of the hoist rope. 1 2tan o e ef  = (5) where, o1e is the distance from the center of the pulley to the center of the hoist rope. 2.3. the differential equations of excavator attachment the dynamics of mechanical systems is usually described by high order differential equations. however, many parameters are involved in this higher-order differential equation. therefore, in order to obtain an explicit expression between the parameters and dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 345 the system dynamic performance, the system dynamics model is often simplified. the establishment of system differential equations needs to calculate the kinetic energy, potential energy and generalized force of the system. the kinetic energy of the lifting mechanism is given as: 2 2 2 2 2 2 2 1 0 3 1 4 2 1 4 3 5 4 5 1 6 1 1 1 1 1 1 1 2 ( ) 2 2 2 2 2 2 t j j j i j j m l   = + + + + +          (6) where ji (i = 1,2, … 4) are: the moment of inertia of the lifting motor, the driving gears of the lifting reducer, the driven gears of the lifting reducer and the roller, respectively; i is the transmission ratio of the gear; θi (i =3,4,5) are the rotation angle of the lifting motor, the driving gears of the lifting reducer and the roller, respectively; θ6 is the rotation angle between the boom and the platform; m1 is the mass of the pulley; l1 is the distance between the pulley and the hinge of the boom. the kinetic energy of the dipper handle components is given as: 2 2 2 2 2 2 2 5 7 6 8 7 9 2 6 2 3 6 3 4 6 4 1 1 1 1 1 ( ) ( ) ( ) 2 2 2 2 2 t j j j m l m l m l   = + + + + +          (7) where ji (i = 5,6,7) is the moment of inertia of the crowd motor, the driving gears of the pushing reducer and the driven gears of the pushing reducer, respectively; θi (i =7,8,9) are the rotation angles of the crowd motor, the driven gears of the lifting reducer and the crowd gear, respectively; mi (i = 2,3,4) are the mass of the pushing motor, driving gears and driven gears; l2 is the distance between the pushing motor and the hinge of the boom; l3 is the distance between the center of the belt and the hinge of the boom; l4 is the distance between the pushing gear and the hinge of the boom. the kinetic energy of the execution components is given as: 2 2 2 3 8 6 b d 10 b d 1 1 0 1 0 1 1 1 ( ) ( )( 2 sin ) 2 2 2 t j j j m m s s s s= + + + + +  +   (8) where j8, jb and jd are the moment of inertia of the boom, the dipper handle and the dipper, respectively; θ10 is the rotation angle of the dipper handle; mb and md are the mass of the dipper handle and the dipper, respectively; α1 angle between the hoist rope and the boom; s0 is the dipper handle with the displacement of the boom; s1 is the pushing gear with the displacement of the pushing motor. the total kinetic energy is given as: 1 2 3 2 2 2 2 2 2 2 0 3 1 4 2 1 4 3 5 4 5 1 6 1 2 2 2 2 2 2 5 7 6 8 7 9 2 7 2 3 7 3 4 7 4 2 2 8 6 b d 10 b d 1 1 0 1 1 1 1 1 1 1 2 ( ) 2 2 2 2 2 2 1 1 1 1 1 ( ) ( ) ( ) 2 2 2 2 2 1 1 1 ( ) ( )( 2 sin 2 2 2 t t t t j j j i j j m l j j j m l m l m l j j j m m s s s = + +   = + + + + +      + + + + + +    + + + + + +                 2 0 )s+ (9) potential energy has many components, including the potential of the hoist rope components, the dipper handle and the boom. it is given as: 346 y. yuan, j. ren, z. wang, x. mu 2 2 25 11 1 4 1 2 2 10 1 1 6 1 1 4 3 2 5 5 1 1 1 1 1 sin sin ( ) 2 2 l fe a v r o c l k k l e a i     = + +  − + − + −                 (10) where a1 is the sum of cross-sectional areas of the hoist rope; l5 is the length of deformation of the hoist rope; f1 is the hoist rope lifting force; ki (i = 1,2) is torsional rigidity of the lifting motor and roller, respectively; e1 is the elastic modulus of the hoist rope; r is the diameter of the drum. the potential energy of the dipper handle is given by: 2 2 2 3 22 2 1 1 2 2 9 7 6 1 5 2 3 3 9 3 2 2 2 3 7 1 6 2 4 8 7 1 1 cos sin 2 2 2 3 1 1 ( ) ( ) 2 2 e a e a b o cf df v r l r df e a l e i o a k r r k        = + + − +          + − + −           (11) where ai (i =2,3) is cross-sectional areas of the guy rope and the dipper handle, respectively; ei (i =2,3) is elastic modulus of the guy rope and the dipper handle, respectively; ki (i = 4,5) is torsional rigidity of the belt and between the belt and the belt pulley, respectively; ri (i =1,2) is the diameter of the driving belt pulley and the driven belt pulley, respectively; f is thrust of the dipper handle; r is radius of the gear for the dipper handle; l6 is total length of the hoist rope; l7 is the width of the boom connection; i is the moment of inertia of the dipper handle. the potential energy of the boom is given as: 2 1 2 1 3 4 4 ( ) 2 f f l v e a  + = (12) where f1 ’ is the hoist rope force acting on the boom; f2 ’ is the guy rope force acting on the boom; a4 is cross-sectional areas of the boom; e4 is elastic modulus of the boom. the total potential energy is given as: 1 2 3 2 2 25 11 1 4 5 2 10 1 1 6 1 1 4 3 2 5 5 1 1 1 2 2 22 2 9 7 6 1 3 7 1 6 2 4 8 7 3 3 2 2 3 21 1 2 5 9 1 1 sin sin ( ) 2 2 1 1 1 cos ( ) ( ) 2 2 2 2 1 sin 2 3 v v v v l fe a r o c l k k l e a i e a f df r l k r r k df e a e a b o c r l = + +     = + +  − + − + −            + + + − + − + −         +                        2 1 2 1 2 4 43 2 ( ) 2 f f l e ae i o a  + +  (13) according to the lagrange generalized coordinates, the equation of motion can be given as: ( 1,2, )i i i i i d t t v d q i n dt q q q q      − + + = =       (14) file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); file:///e:/users/admin/appdata/local/youdao/dict/6.3.69.8341/resultui/frame/javascript:void(0); dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 347 by substituting eqs. (12) and (13) into eq. (14), the differential equation of motion reads: 0 3 1 4 3 2 4 1 4 21 4 1 4 3 2 5 1 2 2 3 2 24 1 1 2 3 5 4 5 2 5 5 2 1 9 3 2 2 5 11 1 1 1 6 8 6 5 2 10 1 1 6 1 5 1 1 2 2 ( ) 2 2 2 2 ( ) 0 sin 0 3 sin sin sj k q j j k k i e a b o c j j k r i l e i o a l fe a m l j r o c l l e a + − =   + + − + − =        + + − + =         + + + +  −                        1 1 2 2 9 7 6 1 7 1 3 3 5 7 2 7 2 2 3 7 3 3 4 7 4 4 3 1 7 1 6 2 4 8 7 6 8 4 8 7 2 2 7 9 9 7 6 1 3 3 sin cos cos 2 2 ( ) ( ) ( ) ( ) ( ) 0 ( ) 0 2 cos t l e a f df r l l q df e a j m l l m l l m l l k r r r k j k e a f df j r r l df e a         + + + − − =          + + + + − + − = + − =  + + +                       5 11 1 b d 10 5 2 10 1 1 6 1 2 1 5 1 1 0 2 ( ) sin sin sin 0 l fe a j j r o c l o c l e a                         − =           + + + +  −  =               (15) where qs is the torque of the lifting motor; qt is the torque of the crowd motor. 3. setting up the dynamic model the objective of this paper is to obtain the dynamics performance of the excavator attachment during one digging process. therefore, the lower body and its associated effect have been ignored in the current work. the 3d dynamic model is established and imported into adams. as the boom and the dipper handle make a great effect on the excavator, it is better to define them as flexible bodies in the simulation, which will generate more realistic results compared with the conventional full rigid simulation. 3.1. grid-independent descriptors in order to obtain an accurate model and reduce computing time, a grid independent analysis has been carried out. this work compares the first order natural frequency of the boom and the dipper handle with different grids. as shown in figs. 3 and 4, the frequency tends to be stable as the number of grids increases. it can be found that the frequency is independent of the number of grids when the number of grids is greater than 2.8×106. this paper proposes a method for the components which allow for local grid refinement. the rigid regions of the boom and the dipper handle 348 y. yuan, j. ren, z. wang, x. mu are set in accordance with the actual situation. it can not only guarantee the accuracy of the results, but it can also reduce the computing time. (a) boom (b) dipper handle fig. 3 the kinematic sketch of the excavator attachment 3.2. modal calculation the flexible body model has been established in the ansys environment, as described in section 3, which can be used for calculating its eigenfrequencies and its eigenmodes [18,19]. the eigenfrequencies and eigenmodes are analyzed theoretically in this study. in order to resolve these important dynamic parameters of the system, the free vibration equation of the system is obtained by ignoring the damping factors in the system:        y y+ =m k 0 (16) where m and k are the mass matrix and stiffness matrices of the component, respectively, y are displacements. suppose the differential eq. (1) has the following form of results:     ( )1siny t= + u (17) where {u} is the eigenmode vector;φ1 is the frequency of simple harmonic motion; λ is the initial phase angle. {u} is given as:    1 2 3 t nu u u u=u (18) by substituting eq. (17) into eq. (16), a new equation is obtained:    ( )   2 1− =k m u 0 (19) the eigenfrequencies and the eigenmodes are then defined by the following equation:        1 2 − =m k u u (20) dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 349 figs. 4 and 5 depict the eigenfrequencies and eigenmodes of the boom and the dipper handle. in order to avoid resonance and improve the excavator performance, the researchers modify the structure of the boom and the dipper handle, which can achieve the objective. first mode and natural frequency: 3.12hz second mode and natural frequency:3.81hz third mode and natural frequency: 10.36hz fourth mode and natural frequency:18.02hz fig. 4 dipper handle: eigenfrequencies and eigenmodes computation first mode and natural frequency: 24.52hz second mode and natural frequency: 28.50hz third mode and natural frequency: 32.87hz fourth mode and natural frequency:43.46hz fig. 5 boom: eigenfrequencies and eigenmodes computation the boom and the dipper handle used in this case have the same general characteristics as those studied previously, except that the boom and the dipper handle are now taken as a flexible body. the boom and the dipper handle are saved as the neutral file (.mnf) and imported into the adams to obtain a rigid-flexible coupling model of the attachment. the rigid-flexible coupling model is shown in fig. 6. 350 y. yuan, j. ren, z. wang, x. mu fig. 6 rigid-flexible coupled model 4. numerical simulation settings 4.1. basic assumptions of the model the kinematics principle of excavators is extremely complex and the main reason for this is that vibration exists in the process; it is very important to simplify the model and give the basic assumptions. therefore, this paper makes following basic assumptions: (i) this paper will not consider the dimension tolerance of the model and various errors; (ii) in addition to the boom, the dipper handle and the hoist rope, the other parts of the system are treated as rigid bodies. the initial condition of the rigid-flexible model is shown in table 2. table 2 the initial conditions of the excavator components position boom the boom is fixed; the distance is 12.1 m, which is between the gear shaft and the ground. dipper handle the angle is 23.8°, which is between the axis of the dipper handle and the vertical direction. 4.2. constraints and motions in order to ensure the reliability of the excavator model, the constraints are set according to the actual situation, which are shown in table 3. dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 351 table 3 the constraints of the excavator boom dipper handle pulley bucket saddle guy rope hoist rope platform boom — — r — r r — r dipper handle — — r r c — — — pulley r r — r — — c — bucket — r r — — — r — saddle r c — — — — — — guy rope r — — — — — — — hoist rope — — c r — — — r platform r — — — — — r — where r represents rotation; c represents contact. to give the resistance force applied to the bucket, edem is used to simulate one digging cycle; figs. 7-9 show the mass change, the resistance force change, and the control velocities of the dipper handle and the hoist rope, respectively. fig. 7 mass change during the mining process fig. 8 resistance force change during the process fig. 9 control velocity of the hoist rope and dipper handle 352 y. yuan, j. ren, z. wang, x. mu 4.3. solver setting adams/solver uses multistep integration methods that contain a predictor and a corrector [20,21]. adams has many integrators and formulations to meet different requirements. in this paper, the researcher chooses the gear stiff integrator (gstiff) and the integrator formulation (si1), which takes into account the constraint derivatives when solving the equations of motion just as it monitors the integration error on the impulse of the lagrange multipliers in the system [21]. this work selects the integration tolerance as 0.001, the simulation time is 12s and the step size is 0.02s. 5. results in order to catch the dynamic characteristics of the excavator, it can be divided into two cases in order to carry out a rigid-flexible coupling study. one case is that the dipper handle is regarded as a rigid body and the boom is regarded as a flexible body while the other case is that the dipper handle and the boom are both regarded as a flexible body. the trajectory of the excavator is shown in fig. 10. fig. 10 trajectory of the excavator where tc is the theoretical curve trajectory; plane is the plane before the material is excavated; double, single and none are the trajectory of different number of flexible bodies, respectively. as shown in fig.10, the value of the (tc) is significantly lower than the simulation value; the main reason is that there is vibration during the whole process. in conjunction with the results of the simulation, the number of the flexible body has little effect on the trajectory, and the rigid body is closer to the theoretical trajectory. the study concludes that the excavator in the process is not steady in work and that certain vibration exists. in this study, the stress distribution is obtained according to the post-processing module. the top ten maximum stress nodes as well as their occurrence times are when the boom and the dipper handle are in the state of maximum stress. the results of the boom are shown in figs. 11 and 12 and table 4. dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 353 fig. 11 illustrates that the maximum stress is located at node 294. the value of the stress is 212.24 mpa and it occurs at time 5.72 s. the node is located at the boom corner. this is the position exposed to a high risk of failure. the objective is to obtain the stress curve which is 294th node in the entire process. firstly, this study obtains the load spectrum from adams and imports it into the ansys. then, the load spectrum is loaded into the model, which calculates the modality information. the von mises stress curve of the 294th node can be obtained in the post-processing module of ansys. the stress mainly is concentrated in 70-175 mpa of the 294th node in the whole process. hence, the boom meets the strength requirements. table 4 the maximum stress node number and position of the boom hot spot stress node time location (mm) # mpa id (sec) x y z 1 212.27 294 5.72 -2117.33 2418.46 6291.37 2 189.02 826 5.72 -1959.91 2307.07 4309.16 3 184.75 187077 6.39 -2135 2418.98 6291.37 4 192.43 146014 5.72 -2424.58 1972.99 4351.58 5 181.08 296 5.72 -2134.49 2436.64 6291.37 6 178.99 496 5.72 -2411.37 1976.95 4366.2 7 174.10 146015 5.72 -2425.18 1972.71 4338.11 8 173.53 19612 5.72 -3616.54 1037.46 4391.37 9 172.55 58 5.72 -2415.69 1976.09 4325.26 10 169.48 29260 5.72 -2135.51 2401.3 6291.37 fig. 11 stress distribution of the boom 354 y. yuan, j. ren, z. wang, x. mu fig. 12 the stress curve of node 294 the dipper handle is a core component of the attachment. the results of the dipper handle are shown in figs. 13 and 14 as well as table 5. from fig. 13, it can be seen that the maximum stress is located at node 70. its value is 96.45 mpa, and it occurs at time 3.27 s. the maximum stress occurs in the holes that enable connection with the bucket. this position is at the highest risk of failure. fig. 13 stress distribution of the dipper handle dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 355 fig. 14 the stress curve of 70th node table 5 the maximum stress node number and position of the dipper handle hot spot stress node time location (mm) # mpa id (sec) x y z 1 96.45 70 3.27 -6906.14 -3265.95 3822.05 2 95.23 69 3.30 -6906.14 -3265.94 4115.05 3 92.53 3256 3.27 -6930.42 -3245.89 3822.05 4 86.71 3173 3.30 -6930.42 -3245.88 4115.05 5 83.41 49626 3.30 -6902.92 -3234.28 4115.05 6 82.02 49358 3.30 -6871.51 -3269.42 4146.55 7 79.75 49865 3.27 -6904.16 -3233.27 3822.05 8 78.90 2838 3.30 -6882.79 -3287.08 4146.55 9 78.10 2399 3.27 -6293.56 -2350.94 3746.04 10 77.51 39203 3.27 -6917.16 -3256.16 3805.85 the main reasons for this can be broadly classified into two types. one refers to the clearances between the dipper handle and the saddle while the other is the uneven distribution of the wire rope in the mining process. in order to investigate the influence of the clearances on the boom, this paper studies the clearances of 0-14 mm while the results are shown in figs. 15-18. fig. 15 illustrates that the vibration of the boom is most noticeable in the y direction, followed by the x direction and z direction. for example, the amplitude vibration range of the boom is mainly between 6349 mm and 6357 mm in the y direction. this paper has analyzed the clearance effect of different directions, and obtained the probability density distribution. this study obtains the von mises stress curve of the dipper handle by using the same method as above. the stress mainly is concentrated in 20-60 mpa at node 70 in the whole process. hence, the dipper handle meets the strength requirements. as shown by the above analysis, it can be clearly seen that it is easier for the boom to fail than for the dipper handle. 356 y. yuan, j. ren, z. wang, x. mu fig. 15 the influence of different clearances on the boom in the 3d space fig. 16 the influence of different clearances on the boom in the x direction fig. 17 the influence of different clearances on the boom in the y direction dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 357 fig. 18 the influences of different clearances on the boom in the z direction it can be observed from figs. 16-18 that the boom vibration is similar to the normal distribution in different directions. in the x direction, with the clearance decreasing, the vibratory magnitude of the boom decreases at first, and then increases. the y direction and z direction are similar to the x direction. the boom vibration is most noticeable in the y direction and x direction; the optimal value is near 3.5 mm. the variances of the different clearances are shown in table 6. table 6 the variance of the different clearances in different directions x y z 0mm 0.71 1.28 0.11 3.5mm 0.69 1.22 0.07 7mm 0.77 1.32 0.16 10.5mm 0.72 1.26 0.06 14mm 0.72 1.25 0.08 in order to investigate the influence of clearances on the dipper handle, this study examines the clearances of 0-14 mm; the results are shown in figs. 19-22. fig. 19 the influences of different clearances on the dipper handle in the 3d space 358 y. yuan, j. ren, z. wang, x. mu fig. 20 the influences of different clearances on the dipper handle in the x direction fig. 21 the influences of different clearances on the dipper handle in the y direction fig. 22 the influences of different clearances on the dipper handle in the z direction fig. 19 illustrates that the range of motion is relatively large in the 3d space. the main reason for this is that the dipper handle is moving. the clearance effects of different directions are analyzed in this paper and the probability density distribution is obtained. figs. 22-25 illustrate the influences of different clearances on the dipper handle in different directions. obviously, the effect in the z direction is smaller than in the x direction and y direction. the vibration amplitude of the dipper handle is the smallest when the clearance value is 3.5 mm. therefore, it can be concluded that the optimal value is near 3.5 mm. in addition, the variances of the different clearances in different directions are shown in table 7. dynamic analysis of a rigid-flexible excavator mechanism based on virtual prototype 359 table 7 the variance of the different clearances in different directions x y z 0mm 1446.32 930.54 5.96 3.5mm 1449.42 904.89 5.87 7mm 1484.51 925.97 8.48 10.5mm 1480.90 908.08 8.24 14mm 1469.57 925.63 6.18 in order to study the influence of the clearances on the lifting force, this study examines the clearances of 0-14 mm and the results are shown in fig. 23. fig. 23 the influences of different clearances on the force of the hoisting ropes fig. 23 shows that the value of the theoretical curve (tc) is significantly lower than the simulation value. the main reason is that the dipper handle as well as the bucket has the vibration in the whole process. when the clearance value is 3.5 mm, the force of the hoisting ropes is the smallest. the deviation is about 25% between the simulation and the theoretical value in the acceleration phase. in the stationary phase, the deviation value is about 5.6%. when the bucket is out of the material, the value of simulation is lower than the value of tc, mainly resulting from the fact that friction becomes ineffective. 6. conclusion dynamic performance of the intelligent excavating process is developed in this paper. firstly, the structure and primary performance parameters are introduced. then, the rigid-flexible coupling virtual prototyping model of the excavator attachment is established. in addition, the stress distribution is obtained, including the top ten maximum stress nodes and the stress curves. to explore the laws of the influence of flexible body and clearance, the comparisons between different excavating scenarios with respect to different clearances are conducted. the results show that with the clearance decreasing, the vibratory magnitude of the boom decreases firstly, and then increases. the actual model with different clearances of macroscopic fluctuations is taken into account in the numerical experiments for the analysis of the excavator’s dynamic characteristic. it can be seen that the analysis of the dynamic characteristic model is flexible and available. 360 y. yuan, j. ren, z. wang, x. mu the excavator is complex equipment, which consists of three major components, including the upper body, the lower body and the attachment. all the corresponding structures exert great influences on the dynamic performance. for the future 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yifei lv7, kaiyang zhong8 1hainan vocational university of science and technology, haikou, hainan, china 2universiti sains malaysia, penang, malaysia 3department of mobile machinery and railway transport, vilnius gediminas technical university, vilnius, lithuania 4department of industrial management engineering, korea university, seoul, korea 5department of mechanical engineering, libyan academy, misurata, libya 6institute of mechanical science, vilnius gediminas technical university, vilnius, lithuania 7faculty of economics and management, alfa university college, selangor, malaysia 8college of information science & electronic engineering, zhejiang university, hangzhou, china orcid ids: meijing song https://orcid.org/0009-0009-0355-239x željko stević https://orcid.org/0000-0003-4452-5768 ibrahim badi https://orcid.org/0000-0002-1193-1578 dragan marinković https://orcid.org/0000-0002-3583-9434 yifei lv https://orcid.org/0009-0002-3593-2820 kaiyang zhong https://orcid.org/0000-0002-5575-0306 abstract. autonomous vehicles (avs) have become a tangible presence on roads, indicating the emergence of a promising transportation technology for the future, possibly arriving sooner than anticipated. nevertheless, the extensive integration of this technology is contingent on various factors, with the foremost being the level of public acceptance and adjustment to this advanced technology. several factors, including safety, privacy, and cost, play crucial roles in fostering acceptance. consequently, this research delves into the key determinants shaping individuals' willingness to embrace avs. in this paper, a novel model, which consists of two methods: piprecia and aroman with interval rough numbers (irns) has been developed. the irn piprecia serves to define criterion weights, while the most significant contribution of the paper is the extension of the aroman method with irns for evaluating the public acceptance of autonomous vehicles and adapting all the necessary conditions for their use. the results show that a rapid implementation with extensive testing strategy represents the best solution. key words: autonomous vehicles, irn piprecia, irn aroman, mcdm received: july 29, 2024 / accepted september 21, 2024 corresponding author: željko stević department of mobile machinery and railway transport, vilnius gediminas technical university, vilnius, lithuania and department of industrial management engineering, korea university, seoul, korea e-mail: zeljko.stevic@vilnius.lt; 172317@korea.ac.kr https://orcid.org/0009-0009-0355-239x https://orcid.org/0000-0003-4452-5768 https://orcid.org/0000-0002-1193-1578 https://orcid.org/0000-0002-3583-9434 https://orcid.org/0009-0002-3593-2820 https://orcid.org/0000-0002-5575-0306 128 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong 1. introduction in recent years, interest in autonomous vehicles (avs) has begun to grow [1,2]. we can therefore expect them to spread and expand further and faster, as their popularity increases globally [3]. consequently, research is increasing to understand the extent to which people accept this technology and their willingness to make it an important transportation option [4,5]. a thorough examination of prior studies on popular acceptance levels unmistakably reveals this divergence in individuals' perceptions worldwide regarding the concept of avs and, consequently, their openness to embracing this technology [6,7]. in arab countries, the experimentation with avs remains constrained, with limited initiatives in place [8,9]. these experiments specifically focus on users of this technology, as indicated by previous research [10]. beyond potential safety implications, legal and financial considerations are also taken into account [11-13]. other aspects that are of additional concern can also be identified. the exploration of these considerations involves soliciting individuals' opinions to gauge their willingness and capacity to adopt this technology [14,15]. for instance, despite avs exhibiting elevated safety levels in comparison to human drivers, safety remains a predominant apprehension for many individuals [16,17]. the heightened safety levels are attributed to the capacity of these vehicles to make intelligent decisions in anticipated traffic scenarios [18]. the heightened focus on safety may stem from the potential impacts on both individuals and transportation infrastructure [19,20]. the capability to make intelligent decisions arises from the advanced technologies integrated into these cars, empowering them with intelligent behavior. these technologies encompass image processing tools that enhance their perceptual capabilities. this provides them with a notable advantage in terms of heightened confidence [21]. however, the utilization of intelligent sensing technologies may have adverse implications for the car owner in case of an accident. furthermore, the restricted coverage of intelligent sensors could result in a failure to comprehensively grasp the entire environment, potentially resulting in erroneous decisions. this challenge can be effectively addressed through the incorporation of thermal imaging cameras capable of recognizing humans and animals, especially during nighttime driving [22]. users also voice concerns regarding privacy protection. these worries revolve around the type of data that avs can store and the potential for unauthorized access or hacking. there is a fear of losing control over their vehicles due to security breaches and the illicit use of the vehicles [23]. this could lead to data loss and misuse [24,25]. apart from safety and privacy concerns, the financial viability of adopting avs significantly influences the decision-making process. the substantial expenses associated with these vehicles may lead individuals to hesitate in embracing them [12,26]. the factors outlined above will inevitably impact public trust and approval of avs to varying degrees. anticipated resistance is understandable, given the novelty of this technology and the associated information gap. addressing this issue requires additional research and inquiry. consequently, this study seeks to delve into the determinants influencing the acceptance of avs within libyan society. its significance lies in providing policymakers, researchers, and specialists with insights to formulate strategies and plans for the future. numerous prior studies have aimed to comprehend the public acceptance of avs. these studies, primarily relying on stated preference surveys and employing descriptive analysis [27], have served various objectives. some focus on forecasting the future adoption of avs technology [28], while others investigate its potential repercussions on public health [29]. moreover, numerous investigations have delved into the advantages of avs, encompassing reduced logistics costs [30], diminished accident rates [7, 31], lowered fuel consumption [31], assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 129 and simplified parking [32]. these advantages contribute to the increasing public acceptance of avs. conversely, various studies have underscored the existence of potential obstacles that should be addressed to expedite the implementation of avs [27]. many prior investigations have examined individuals' attitudes toward the incorporation of avs. these studies have encompassed assessments of driver confidence, perceptions of avs capabilities, and confidence in the reliability of the systems, considering their likeness to human drivers. alongside the potential advantages of avs, apprehensions about privacy, identity, and societal norms have also been raised [33]. these vehicles have the potential to decrease travel durations and exhibit high fuel and parking efficiency. nonetheless, several obstacles may impede widespread acceptance of avs as a global alternative. these barriers encompass substantial initial expenses, irregularities in licensing standards, poorly defined liability, security vulnerabilities, and apprehensions regarding privacy [21]. in this paper, a novel model which consists of two methods: piprecia (pivot pairwise relative criteria importance assessment method) and aroman (alternative ranking order method accounting for two-step normalization) and interval rough numbers (irns) has been developed. the emphasis is on the novelty of the work through the development of the novel interval rough aroman method. the previously developed irn piprecia serves to define criterion weights, while the most significant contribution of the paper is the extension of the aroman method with irns for evaluating the public acceptance of autonomous vehicles and adapting all the necessary conditions for their use. 2. methods in this section, the work methodology is presented with a focus on the algorithms of the applied, i.e. developed method. the emphasis is on the novelty of the work through the development of the novel interval rough aroman method. the flow of the research and the steps of the approaches used are presented in more detail below. 2.1. interval rough piprecia method piprecia is a method frequently used to determine the significance of criteria, which can be seen by its application in various areas [34-36] and by numerous extensions of the method [37-39]. the paper [40] introduces the extension of the method with irns and its steps are given below. in order to obtain the value functions of the criteria, two linguistic scales converted into irns are used. the scales differ depending on whether criteria have greater significance (table 1) or less significance (table 2). table 1 scale for assessing the criteria with greater significance linguistic term abbr. scale 1-2 irn almost equal value ae 1 [1.00, 1.05] [1.10, 1.10] slightly more significant sm 2 [1.10, 1.20] [1.20, 1.25] moderately more significant mms 3 [1.20, 1.35] [1.30, 1.40] more significant m 4 [1.30, 1.50] [1.40, 1.55] much more significant mm 5 [1.40, 1.65] [1.50, 1.70] dominantly more significant dm 6 [1.50, 1.80] [1.60, 1.85] absolutely more significant am 7 [1.60, 1.90] [1.70, 1.95] 130 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong table 2 scale for assessing the criteria with less significance linguistic term abbr. scale 0-1 irn weakly less significant wl 1 [0.80, 0.90] [0.85, 0.95] moderately less significant mls 1/2 [0.70, 0.80] [0.75, 0.85] less significant l 1/3 [0.60, 0.70] [0.65, 0.75] really less significant rl 1/4 [0.50, 0.60] [0.55, 0.65] much less significant ml 1/5 [0.40, 0.50] [0.45, 0.55] dominantly less significant dl 1/6 [0.30, 0.40] [0.35, 0.45] absolutely less significant al 1/7 [0.20, 0.30] [0.25, 0.35] step 1. assess the criteria by experts using the scales given above. with the irn piprecia method, experts first assess the significance of the second criterion in relation to the previous criterion.             1 1 1 1,1 , 1,1 if 1,1 , 1,1 if 1,1 , 1,1 if j j r j j j j j c c irn s c c c c − − −      = = =    (1) irn [sr j] denotes the assessment of the criteria by each expert r. step 2. since it is a group decision-making, the initial irn matrix is obtained by aggregating experts' estimates using one of available aggregators.   1 1 1/ 1/ 11 1 1 , 1 ( ) 1 ( ) 11 ( )( ) 1 2 1 ( ) 1 ( ) , ( ), ( ),.., ( ) n n ljljj j nn ljlj jj ljjj lj n ujj n uj j j uj f f ww ff n f w f irndwga irn irn irn                 = = == = =       − −     ++                  −  +                   =      1 1/ 1/ 1 , 1 ( ) 1 ( ) n ujj n uj j ujj f w f       = =    −   +                                                      (2) here, the aggregation is conducted using the irn dombi weighted geometric averaging aggregator, which is an operator frequently used for averaging values [41-44]. step 3. calculate the coefficient irn [kj].  [1,1],[1,1] if 1 [ ] 2 [ ] if 1j j j irn k s j = = = −  (3) step 4. calculate the interval rough weight irn [qj]. 1 [1,1],[1,1] if 1 [ ][ ] if 1 [ ] j j j j qrn q j k − = =  =     (4) assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 131 step 5. calculate the relative interval rough weight irn [wj]. 1 [ ] [ ] [ ] j j n j j q irn w q = =  (5) the inverse irn piprecia method is used in the following steps. step 6. reassess the criteria by experts, starting from the penultimate criterion. 1 ' 1 1 [1,1],[1,1] if [ ] [1,1],[1,1] if [1,1],[1,1] if j j r j j j j j c c irn s c c c c + + +    = = =   (6) irn [sr j ’] denotes the assessment of criteria by expert r. then, it is required to aggregate all experts’ estimates. step 7. calculate the coefficient irn [kj ’].  [1,1],[1,1] if [ '] 2 [ '] ifj j j n irn k s j n = = = −  (7) step 8. calculate the interval rough weight irn [qj ’]. 1 [1,1],[1,1] if '' if ' jj j j n qq j n k + = =  =     (8) step 9. calculate the relative interval rough weight irn [wj ’]. 1 [ '] [ '] [ '] j j n j j q irn w q = =  (9) step 10. calculate the final values. ( ) 1 [ ''] [ ] [ '] 2 j j jw irn w irn w= + (10) step 11. test the results by applying spearman and pearson correlation coefficients. 2.2. a novel interval rough aroman method aroman is a method created in [45] for evaluating and ranking alternative solutions. in this section, for the first time in the literature, the extension of the aroman method with irn has been presented, and it is explained in detail throughout the following several steps. 132 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong step 1. define the initial interval rough matrix (ш). 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ... ( ) ( ) ... ( ) n n n m m m mn m n c c c a irn ш irn ш irn ш a irn ш irn ш irn ш ш a irn ш irn ш irn ш       =       (11) where alternatives are denoted by m, and criteria by n. step 2. normalize the initial interval rough group matrix. 1 2 1 11 12 1 2 21 22 2 1 2 ... ( *) ( *) ... ( *) ( *) ( *) ( *) ... ... ... ... ... ( *) ( *) ... ( *) n n n m m m mn m n c c c a irn н irn н irn н a irn н irn н irn н н a irn н irn н irn н       =       (12) where irn (нij) denotes the values of the interval rough normalized matrix (н). step 2.1. a1) for “benefit type” criteria (linear) ( ) ' ' ' '( ) [ , ],[ , ] , , , l u l u ij ij ij ij ij ij ij ijl u l u ij ij ij ij ij ij ij ij ij ij ij ij ij ш ш ш ш ш ш ш ш irn н н н н н ш ш ш ш ш ш ш ш − − − − + − + − + − + −     − − − −  = =      − − − −        (13) b1) for “cost type” criteria (linear) ( ) ' ' ' '( ) [ , ],[ , ] , , , u l u l ij ij ij ij ij ij ij ijl u l u ij ij ij ij ij ij ij ij ij ij ij ij ij ш ш ш ш ш ш ш ш irn н н н н н ш ш ш ш ш ш ш ш + + + + − + − + − + − +     − − − −  = =      − − − −        (14) where шij and шij + denotes minimum and maximum values of the rough boundary interval of the criteria, respectively: 'min{ , }l l iј ij ij i ш ш ш− = (15) 'max{ , }u u iј ij ij i ш ш ш+ = (16) step 2.2. a2) for “benefit type” criteria (vector) ( ) ' ' ' ' ' ' ( ) [ , ],[ , ] , , , ij ij l u l u ij ij ij ijl u l u ij ij ij ij ij u l u l ij ij ш ш ш ш irn в в в в в г г г г       = =              (17) b2) for “cost type” criteria (vector) ( ) ' ' ' ' ' ' ( ) [ , ],[ , ] 1 , , , ij ij l u l u ij ij ij ijl u l u ij ij ij ij ij u l u l ij ij ш ш ш ш irn в в в в в г г г г       = = −              (18) where ( )' ' 2 1 ( ) [ , ],[ , ] ( ) m l u l u j j j j j иј i irn г г г г г ш = = =  (19) assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 133 step 2.3. perform aggregated averaged normalization the aggregated averaged normalization is performed by the following equation: ( ) ( ) ( *) 2 ij ij ij н в irn н   +  = (20) where нij * is the aggregated averaged normalization and α is a weighting factor varying from 0 to 1. in this specific case, α is 0.5. step 3. compute the weighted matrix: ( ) ( )' ' * * ' * ' ' * '( ) [ , , , ] [ w , w , w , w ]l u l u l l u u l l u u ij ij ij ij ij ij ij ij ij ij ij ij ijm n irn б б б б б n n n n  = =     (21) irn (wj) denotes criteria weights. step 4. summarize the normalized weighted values of the criteria type min (мi) and the normalized weighted values of the max type (џi) individually. it can be done using eqs. (22) and (23): ( )' ' 1 1 ( ) [ , , , ] ( ) for n l u l u i i i i i ijm j irn м м м м м б c  = = = (22) ( )' ' 1 1 ( ) [ , , , ] ( ) for n l u l u i i i i i ijm j irn џ џ џ џ џ б b  = = = (23) step 5. compute the final ranking of the alternatives. the final ranking of the alternatives (чi) is obtained by eq. (24): ( )' ' (1 ) 1 ( ) [ , , , ] ( ) ( )l u l u i i i i i i im irn ч ч ч ч ч м џ −  = = + (24) where γ is the coefficient in interval 0.1-0.9. 3. case study this case study aims to analyze the level of acceptability of avs in libya. with the ongoing development of the automotive industry, the potential for avs to be implemented on libyan roads generates interesting inquiries regarding their reception and integration within the local environment. avs possess the capacity to fundamentally transform transportation, boost road safety [46], and optimize mobility efficiency. yet, the effective incorporation of avs into the transportation system of libya hinges on comprehending the attitudes, views, and apprehensions of the local populace towards this nascent technology . libya's transportation infrastructure [47], like to those in many developing nations, mainly depends on private car usage and lacks a significant presence of public transportation. libya has one of the highest rates of road traffic fatalities globally [48]. this is a hurdle in terms of public adoption of autonomous mobility. hence, the objective of this study is to investigate the pivotal factors that impact the public's willingness to embrace avs in libya. consequently, this will offer vital knowledge to steer decision-making procedures about the implementation of avs. it will establish the foundation for future efforts to encourage popular approval and facilitate the shift towards a more autonomous transportation system. 134 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong 3.1. forming the mcdm model in this section, the formation of the mcdm model is shown, that is, the description of the strategies of public acceptance of avs (table 3) and nine evaluation criteria that have been used for evaluation (table 4). table 3 description of suitable strategies for evaluation strategy description s1 gradual adoption with limited use cases in this scenario, avs are phased in for limited use cases like controlled settings (e.g., dedicated lanes, closed campuses) or specific applications. before widespread adoption, av technology must gain public trust. s2 geographically targeted deployment deploying avs in certain cities is another option. deployment in certain places allows localised testing, infrastructure development, and public participation. this method allows for context-specific public acceptance assessments. s3 piloted av programs piloted av programs involve av developers and local transportation authorities conducting small-scale real-world experiments. this option permits controlled testing and data collecting with public input. it seeks transparency, communication, and engagement to address public problems. s4 rapid deployment with extensive testing this scenario aggressively deploys avs into the transport system after significant testing and validation. this scenario targets efficiency, congestion, and transportation system improvements with better technical readiness and public acceptance. table 4 description of used criteria criterion description c1 safety safety is a key public acceptance factor. compare av safety to human-driven vehicles. this covers accident rates, reliability, and av technology robustness. c2 trust and reliability assess public trust in av technology. take into account system transparency, av performance, and autonomous system reliability. c3 privacy and data security examine av privacy and data security issues. assess public opinion on data collecting, storage, and unauthorized access. consider how privacy measures affect public approval. c4 job displacement and economic impact assess public opinion on av deployment's effects on jobs and the economy. assess job displacement problems in transportation and delivery. assess the perceived economic benefits and drawbacks of avs. c5 environmental impact assess public opinion of avs' environmental impact. consider energy efficiency, greenhouse gas reduction, and avs' potential for sustainable transportation. assess how environmental benefits or concerns affect public acceptance. c6 accessibility and inclusivity assess public opinion on av accessibility and inclusion. consider how avs can carry disabled or limited-mobility people. evaluate public opinion on avs' transportation equity and accessibility potential. c7 legal and regulatory framework assess public opinion on av law and regulation. assess if people think current safety and liability regulations are enough. assess public trust in av deployment laws and regulations. c8 social and cultural acceptance consider the social and cultural adoption of avs. examine cultural norms, technology attitudes, and public acceptability of avs on roads. c9 user experience and comfort assess public opinion on av comfort and user experience. consider ride quality, convenience, and usability. assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 135 3.2. determining criteria weights using the irn piprecia method determining the criteria weights is of crucial importance in mcdm [48]. in this approach, their importance has been evaluated by four experts according to the irn piprecia and inverse irn piprecia methodology, which is shown in table 5. four experts in the areas of transportation engineering, artificial intelligence, and the psychology of accidents were contacted. all the experts had more than 15 years’ worth of expertise in their line of specialization. this provided a comprehensive insight into the various factors that contribute to the assessment of the public perception and safety issues arising from the use of self-driving cars. then, the experts' estimates are aggregated by the irn dombi weighted geometric averaging operator to obtain a matrix irn [sr j]. the following example represents the aggregation procedure for c4: ( ) ( ) ( ) ( ) 1 2 4 4 3 4 4 2 ( ) [ ] , ( ) [ ] , ] 0.80,0.90 ) ,[0.85,0.95] 0.70,0.80 ,[0.75,0.85] 0.70,0.80 ,[0.75,0.85] 0.60,0.70 ,( [ ] , ( [0.65,0.] 7) [ 5 e e e e irn c irn c irn c rn c = = = = in the aggregation process, the weights of experts are wdm=(0.250,0.250,0.250,0.250). the computation procedure is given below: table 5 evaluation of criteria by four experts according to required scales p c1 c2 c3 c4 … c8 c9 e1 [1,1.05], [1.1,1.1] [0.7,0.8], [0.75,0.85] [0.8,0.9], [0.85,0.95] … [1.1,1.2] [1.2,1.25] [0.6,0.7], [0.65,0.75] e2 [0.7,0.8], [0.75,0.85] [0.7,0.8], [0.75,0.85] [0.7,0.8], [0.75,0.85] [1.2,1.35] [1.3,1.4] [0.7,0.8] [0.75,0.85] e3 [0.7,0.8], [0.75,0.85] [0.7,0.8], [0.75,0.85] [0.7,0.8], [0.75,0.85] [1.2,1.35] [1.3, 1.4] [0.7,0.8] [0.75,0.85] e4 [0.7,0.8], [0.75,0.85] [0.7,0.8], [0.75,0.85] [0.6,0.7], [0.65,0.75] [1,1.2] [1.2,1.25] [0.8,0.9] [0.85,0.95] p-i c1 c2 c3 c4 … c8 c9 e1 [1.1,1.2], [1.2,1.25] [1.1,1.2], [1.2,1.25] [1,1.05], [1.1,1.1] [1,1.05], [1.1,1.1] … [1.2,1.35] [1.3,1.4] e2 [1.1,1.2], [1.2,1.25] [1.1,1.2], [1.2,1.25] [1.1,1.2], [1.2,1.25] [1,1.05], [1.1,1.1] [1.1,1.2] [1.2,1.25] e3 [1.1,1.2], [1.2,1.25] [1.1,1.2], [1.2,1.25] [1.1,1.2], [1.2,1.25] [1,1.05], [1.1,1.1] [1.1,1.2] [1.2,1.25] e4 [1.1,1.2], [1.2,1.25] [1.1,1.2], [1.2,1.25] [1.2,1.35], [1.3,1.4] [0.7,0.8], [0.75,0.85] [1,1.05] [1.1,1.1] then, experts' estimates are aggregated by the irn dombi weighted geometric averaging operator to obtain a matrix. irn [sr j]. the following example represents the aggregation procedure for c4: ( ) ( ) ( ) ( ) 1 2 4 4 3 4 4 2 ( ) [ ] , ( ) [ ] , ] 0.80,0.90 ) ,[0.85,0.95] 0.70,0.80 ,[0.75,0.85] 0.70,0.80 ,[0.75,0.85] 0.60,0.70 ,( [ ] , ( [0.65,0.] 7) [ 5 e e e e irn c irn c irn c rn c = = = = in the aggregation process, the weights of experts are wdm=(0.250,0.250,0.250,0.250). the computation procedure is given below: 136 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong ( ) 4 1 1/ 4 1 4 1 1/ 4 1 3.30 0.7974 1 0.242 1 0.333 1 0.250 ... 0.2501 ( ) 0.242 0.3331 ( ) ( ) 3. 4 1 ( ) 1 ( ) 4 ljj lj j j lj jj lj j ljj l f w f lim u f w f c c irnd c         = = = = = = − −       +  + + −            +       =   −   +        = = =     4 1 1/ 4 1 70 0.900 1 0.243 1 0.324 1 0.250 ... 0.250 0.243 0.324 3.55 0.8554' 1 0.239 1 0.338 1 0.250 .... 0.2501 ( ) 0.239 0.3261 ( ) ljj lj j j lj l f w f c     = = = − −     +  + +           = = − −       +  + + −            +       =   ( ) 4 1 1/ 4 1 ( ) 3.90 0.9514' 1 0.244 1 0.320 1 0.250 ... 0.2501 ( ) 0.244 0.3201 ( ) [0.797,0.900],[0.855,0.951] jj lj j ljj lim u f w f c     = = = = − −       +  + + −            +                           =    =   where f(irn φ4) is calculated: 1 4 4 1 4 4 1 1 4 1 4 4 1 4 4 1 1 ( ) 0.80 ( ( )) 0.242 3.30 ( ) ( ) 0.90 ( ( )) 0.243 3.70 ( ) ( ( )) ( ) 0.85 ( '( )) 0.239 3.55 ( ) ( ) 0.95 ( '( )) 0.244 3.90 ( ) i i i i i i i i lim f lim lim lim f lim lim f irn lim f lim lim lim f lim lim              = = = =  = = =     = = = =  = = = = = =                   after the entire procedure, including the inverse irn pirecia steps, has been conducted, the final weights (table 6) are obtained. table 6 ranking of the criteria after applying the irn piprecia method irn piprecia inverse irn piprecia final wj rank c1 [0.119,0.160],[0.145,0.197] [0.057,0.167],[0.142,0.346] [0.088,0.164],[0.143,0.272] 1 c2 [0.093,0.136],[0.118,0.175] [0.052,0.134],[0.113,0.260] [0.072,0.135],[0.116,0.217] 3 c3 [0.072,0.113],[0.095,0.152] [0.046,0.107],[0.091,0.195] [0.059,0.110],[0.093,0.173] 6 c4 [0.060,0.103],[0.083,0.145] [0.042,0.086],[0.073,0.148] [0.051,0.095],[0.078,0.146] 8 c5 [0.052,0.099],[0.076,0.146] [0.046,0.089],[0.074,0.144] [0.049,0.094],[0.075,0.145] 9 c6 [0.053,0.108],[0.087,0.169] [0.058,0.102],[0.089,0.159] [0.056,0.105],[0.088,0.164] 7 c7 [0.058,0.129],[0.105,0.214] [0.075,0.123],[0.111,0.183] [0.066,0.126],[0.108,0.198] 4 c8 [0.054,0.134],[0.106,0.237] [0.095,0.143],[0.135,0.204] [0.075,0.139],[0.121,0.220] 2 c9 [0.044,0.120],[0.091,0.222] [0.086,0.116],[0.109,0.155] [0.065,0.118],[0.100,0.188] 5 assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 137 based on the results obtained and the significance of the criteria, the final ranking is as follows: c1>c8>c2>c7>c9>c3>c6>c4>c5. 3.3. assessing strategies using the irn aroman method experts evaluated potential strategies using interval numbers. since it is a group decision-making when applying the rules for operations with interval numbers, an initial decision matrix, shown in table 7, is obtained. the following calculation involves a three-phase normalization procedure depending on the type of criteria. since the experts use linguistic terms for evaluation, all criteria have been modelled as benefit, so eqs. (13) and (17) are applied. for example, in order to perform linear normalization for the first alternative according to the first criterion, it is necessary to do the following: ( )11 5.580 5.500 6.530 5.500 , , 7.380 5.500 7.380 5.500 ( ) [0.043,0.548],[0.117,0.617] 5.720 5.500 6.660 5.500 , 7.380 5.500 7.380 5.500 irn н  − −     − −  = =  − −    − −   ( )11 5.580 6.530 5.720 6.660 ( ) [0.399,0.533],[0.427,0.570] , , , 13.989 12.247 13.405 11.679 irn в      = =           while: ( ) 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 (5.58) (5.88) (5.50) (6.36) , , (6.53) (6.97) (6.25) (7.03) ( ) [11.678,13.405],[12.247,13.989] (5.72) (6.29) (5.67) (6.75) , (6.66) (7.32) (6.58) (7.38) irn г   + + +      + + +  = =   + + +     + + +   table 7 initial interval rough matrix c1 c2 c3 c4 … c8 c9 a1 [5.58,6.53] [5.72,6.66] [6.11,6.79] [6.29,7.04] [3.14,3.86] [3.63,4.47] [3.59,5.14] [3.91,5.55] ... [4.32,4.82] [4.42,4.88] [2.88,3.97] [3.24,4.35] a2 [5.88,6.97] [6.29,7.32] [5.70,6.79] [6.11,7.07] [4.67,6.15] [4.93,6.34] [2.88,3.97] [3.50,4.47] [2.42,3.94] [3.00,4.73] [4.11,4.79] [4.35,5.29] a3 [5.50,6.25] [5.67,6.58] [4.77,6.12] [5.04,6.24] [3.32,4.60] [3.42,4.75] [3.54,4.82] [3.76,4.96] [4.83,6.14] [5.25,6.47] [2.40,3.63] [2.95,4.33] a4 [6.36,7.03] [6.75,7.38] [6.62,7.25] [6.93,7.53] [5.70,6.79] [6.11,7.07] [4.70,5.79] [5.36,6.50] [1.73,3.31] [2.22,3.92] [4.40,5.63] [4.95,6.33] after that, it is necessary to apply eq. (20): ( )11 (0.043 0.500) (0.399 0.500) ( *) [0.110,0.270],[0.136,0.297] 2 irn н  +  = = . in this way, the final normalized interval rough matrix shown in table 8 is obtained. 138 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong table 8 normalized interval rough matrix c1 c2 c3 c4 ... c8 c9 a1 [0.11,0.27] [0.14,0.30] [0.23,0.32] [0.25,0.36] [0.07,0.15] [0.11,0.21] [0.13,0.31] [0.17,0.37] ... [0.24,0.32] [0.26,0.34] [0.10,0.23] [0.14,0.28] a2 [0.16,0.34] [0.22,0.40] [0.19,0.32] [0.23,0.36] [0.20,0.36] [0.23,0.39] [0.07,0.19] [0.13,0.26] [0.10,0.24] [0.15,0.32] [0.21,0.30] [0.24,0.37] a3 [0.10,0.23] [0.13,0.28] [0.09,0.25] [0.12,0.27] [0.08,0.22] [0.10,0.24] [0.13,0.28] [0.16,0.31] [0.28,0.43] [0.33,0.48] [0.06,0.19] [0.12,0.28] a4 [0.23,0.35] [0.29,0.41] [0.29,0.37] [0.32,0.41] [0.29,0.41] [0.33,0.45] [0.23,0.37] [0.31,0.47] [0.04,0.19] [0.09,0.25] [0.23,0.38] [0.30,0.47] after that, the procedure of weighting the normalized matrix with the criteria weights obtained with the irn piprecia method is performed. further, the calculation is performed using eqs. (23) and (24), so the final results obtained are presented in table 9. table 9 final results obtained applying the irn piprecia – irn aroman model 1 ( ) ( ) n i ij j irn џ б = = (1 )( ) ( ) ( )i i iirn ч м џ −= + av rank [0.081,0.286],[0.159,0.532] [0.729,0.854],[0.924,0.961] 0.867 3 a1 [0.092,0.336],[0.195,0.636] [0.797,0.893],[0.945,0.972] 0.902 2 a2 [0.062,0.258],[0.134,0.483] [0.695,0.834],[0.913,0.956] 0.849 4 a3 [0.129,0.388],[0.255,0.729] [0.854,0.924],[0.961,0.980] 0.930 1 a4 since decision-making is reduced to only benefit criteria in this case, the matrix irn (mi) is not calculated, i.e. it is equal to zero, while the calculation example for benefit criteria is as follows: ( )1 0.010 0.017 0.044 ... 0.007, 0.044 0.043 0.016 ... 0.027 ( ) [0.081,0.286],[0.159,0.532] 0.020 0.029 0.011 ... 0.014, 0.081 0.077 0.037 ... 0.052 irn џ  + + + +     + + + +   = =  + + + +     + + + +   in the last step, the matrix is calculated as follows: ( ) 0.5 (1 0.5) 1( ) [0.729,0.854,0.924,0.961] (0) (0.081)irn ч −= = + with a value γ = 0.5. the findings indicate that safety stands out as the foremost criterion, especially in a country with one of the highest road accident death rates globally. following closely in second place is the cultural acceptance of avs, underscoring the presence of diverse cultural challenges in this context. trust and reliability secured the third position in the rankings, primarily attributed to the scarcity of information available about these cars, relying heavily on personal observations and social media. in contrast, environmental factors were deemed the least important, stemming from a lack of environmental awareness among many individuals and the perception that their impact is not considered significant. job displacement and economic impact were assigned the lowest ranking, primarily because of the significant personal dependence on cars for mobility, which diminishes the anticipated impact of this technology on employment. assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 139 the outcomes indicate that the most effective strategy involves rapid implementation coupled with extensive testing. executing this strategy poses a significant challenge for decision-makers, as they may encounter difficulties in adequately preparing and allocating the required resources. additionally, there is a need for comprehensive and targeted public awareness campaigns. the strategy of focusing on specific geographical areas, particularly in large urban cities, ranks second in effectiveness. this approach enables the gathering of data and the accumulation of knowledge before broader implementation. it also aids in pinpointing weaknesses and identifying necessary improvements before expanding the application to larger geographical areas. 4. verification tests in this section of the study, several verification tests which should demonstrate the usability of the developed irn aroman method have been created. first, a sensitivity analysis has been carried out with changes in the weights of nine criteria. the second test involves a comparative analysis with four other methods, and the third includes a rank reversal analysis. 4.1. sensitivity analysis determining the influence of the criteria weights on the final ranking of alternative solutions is practically an indispensable step, which is also confirmed by the following studies [50-54]. it is very important to determine if and how the new simulated values affect the changes in the final values of the alternatives. figure 1 shows the values of the new criteria weights. fig. 1 criteria weights in new 90 scenarios in this case, a total of 90 scenarios in which new values are simulated for all criteria in the percentage values of 5-95 have been created, so that each criterion in any of the scenarios tends 140 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong to zero. after the new criteria values have been set, it is necessary to perform calculations in 90 models using the irn aroman method. the results are presented in fig. 2. although there is a large number of scenarios in which the new weights of the nine criteria have been defined, it is important to note that there are no changes in the ranks of the strategies, so the initial results remain the same throughout the entire sensitivity analysis. fig. 2 results of sa 4.2. comparative analysis a comparative analysis has been conducted with four other mcdm methods: aras (additive ratio assessment) [55], copras (complex proportional assessment) [56], saw (simple additive weighting), [57], cocoso (combined compromise solution) [58] with irns in order to verify the stability of the developed model, which is shown in figs. 3 and 4. in fig. 3, we can notice that there are no changes in the ranks of alternative solutions, regardless of which method is applied. it should be noted that this is because of a small number of alternatives considered in this paper, but there would be certain differences in the ranks if the number of alternative solutions increases, which is understandable in one way. fig. 3 ranks in ca assessing public acceptance of autonomous vehicles using a novel irn piprecia irn aroman... 141 figure 4 shows the values of the alternatives in the comparative analysis conducted in order to test if some alternatives are similar to each other. fig. 4 values in ca 4.3. rank reversal analysis the third verification test involves changing the size of the initial decision matrix, the results of which are shown in fig. 5. fig. 5 results of rra a total of eight sets have been formed in this analysis, where there is the elimination of the worst strategy in the first three sets and recalculation is done with the irn piprecia – irn aroman model. the fourth set includes adding the strategy with the worst characteristics, so 142 m. song, y. lv, ž. stević, i. badi, d. mari̇nkovi̇ć, k. zhong that there are five alternatives in total. the fifth set is formed in such a way that the worst alternative is replaced by the second worst alternative. the remaining three sets imply the size of the initial matrix with four alternatives, but with a smaller number of criteria, because one of the criteria is eliminated in each of these three sets, so that there is a 4x6 matrix in the last set. regardless of the diversity in defining sets in rank reversal analysis, there is no change in ranks, that is, the initial results remain the same. 5. conclusion in this paper, the evaluation of strategies for the introduction of autonomous vehicles has been carried out creating a novel approach which includes the extension of the aroman method with irns. in this way, the contribution from the scientific and methodological aspects is presented since the developed model can be applied in different areas. transportation holds a crucial role in shaping individuals' engagement in various life activities. this study offers insights into the public acceptance of avs and the policies aimed at integrating them into diverse transportation modes, marking a new era in the field of transportation. the choice of a specific mode of transport is intertwined with numerous influencing factors. these factors can exhibit variations not only from one country to another but even within different urban areas within the same country. public acceptance stands out as a pivotal element in the seamless integration of avs into established transport networks. decision-makers engaged in the integration of this technology should, therefore, consider these factors diligently to arrive at the most suitable decisions regarding their incorporation into respective communities. to comprehend the implications of these factors, the research employed a mcdm method, which revealed that security, privacy, and trust are the foremost considerations, emphasizing the necessity for a strategy development that incorporates these aspects. these 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commons license: cc by-nc-nd original scientific paper the ceramic modular head improvement in the design of a total hip replacement vladimir pakhaliuk1, aleksandr poliakov1, ivan fedotov2 1sevastopol state university, sevastopol, russian federation 2national research nuclear university mephi, moscow, russian federation abstract. for the first time, a design of a modular ceramic ball head of a total hip joint replacement (thr) friction pair has been developed, which has the properties of a metal in conjunction with the stem neck and the properties of a ceramics on bearing surface of the pair. this is achieved by creating a low-cost, low-toxic, durable fixed connection of the head made of alumina or zirconia ceramics and the titanium-based alloy sleeve to obtain a brazed joint that is efficient in human synovial fluid. with the help of finite element analysis, a quantitative assessment of the strength and rigidity of the proposed head design was performed and its use in modern hip arthroplasty was indicated. the approbation of the proposed design solutions for creating a thr ceramic head with a titanium-based alloy sleeve brazed was carried out. key words: total hip replacement, ceramic friction couple, modular ceramic ball head, brazing 1. introduction in the designs of modular total hip joint replacements (thr), which have found wide application in medical practice in the treatment of diseases and injuries of the hip joint, spherical friction pairs are most often used with heads, made of metal alloys (cobaltchromium, titanium-based), ceramics (alumina al2o3, zirconia zro2) and their mixtures. at the same time, both types of materials, metal and ceramics, have fairly good tribological properties in terms of minimizing wear of a spherical sliding friction pair due to their high hardness. but the ions of cobalt and chromium, which are released from the all-metal head, are toxic in nature, have the property of spreading throughout the body and accumulating in its vital tissues, which in some cases can lead to very severe allergic complications [1]. in addition, in [2], it was studded that one of the lowest friction coefficients is possessed by a pair of ceramic head-on-polyethylene liner made of ultrahigh molecular weight polyethylene received december 12, 2020 / accepted january 29, 2020 corresponding author: vladimir pakhaliuk sevastopol state university, universitetskaya str. 33, 299053 sevastopol, russian federation e-mail: pahaluk@sevsu.ru 68 v. pakhaliuk, a. poliakov, i. fedotov (uhmwpe). the simulation results we performed earlier indicate that the amount of wear products of such a friction pair is significantly less than in a metal-on-uhmwpe or metal-onmetal pair [3-4]. in addition, it can be reasonably assumed that when using a friction pair of a partially regular micro texture in the form of dimples, applied, for example, to the thr head surface, a ceramic-on-uhmwpe pair will be more effective from a tribological point of view. the presence of such an artificial lubricant pockets, as a rule, prevents the bonding of articulated surfaces of the tribological pair, promotes removal of the wear process products into dimples from a contact area, and feeds the frictional contact with a portion of the lubricant during its operation [5]. thus, the above performances of thr with a ceramic-on-uhmwpe friction pair indicate their competitive advantages compared with analogues. however, in order to achieve a global advantage, it is necessary at the design and technological levels to ensure the necessary strength of the connection between the ceramic head and the thr metal stem, as well as its resistance to shock loads. both metal and ceramic heads are mated with thr stem neck made of titanium-based alloy through the presence of a blind taper hole in them. in this case, the acetabular part of the thr (liner into the cup) mated to the head can be made of both ceramics and uhmwpe. but ceramics has one of the essential disadvantages fragility. therefore, in the place of the specified conical connection in case of extreme load, for example, when a patient jumps from a height, there is a high probability of the ceramic head destruction [6-7]. at the same time, in the case of revision prosthetics, it becomes necessary to install a thr with only a ceramic pair, since it is impossible to extract from the tissues all small sharp fragments of destroyed ceramics, which, if they get into a friction pair, especially with the use of uhmwpe, would lead to a catastrophic avalanche-like wear of a new pair [6]. in view of the foregoing, it becomes obvious that the reliability of the thr, along with ensuring the best tribological properties, can be significantly increased if the design connection of the tapered stem neck with the ceramic head is provided, which would exclude the destruction of ceramics under any heavy loads (including extreme ones), arising from the vital activity of the patient. known thr head [8] consists of external and internal elements, of which the internal is a metal base, for instance of tantalum, with a tapered hole. the hole has a various shape of the outer surface, on which an outer element made of ceramics or, for example, sapphire single crystal, the outer surface of which has a spherical shape. this design has a number of obvious disadvantages. first, it is unclear what technology should be used to deposit ceramics on the inner metal element made of tantalum in order to provide a strong and reliable connection between them. most likely, such a technological process must take place at a very high temperature. it is known that the most widely used types of ceramics for such heads are alumina and zirconium ceramics or their mixture [9]. but the temperature coefficient of linear expansion of tantalum and, for instant, alumina ceramics, differs on average by 25%, which, when the head cools down, in the case of a presumptive deposition of such a layer of ceramics, can lead to the probable destruction of this brittle layer due to its different deformation as compared to the metal tantalum base. similar consequences will take place in the case of using sapphire, which is characterized by pronounced anisotropic properties. the most probable technology for its deposition also requires growing a single crystal on tantalum. consequently, when developing it, it is necessary to take into account differences up to 30% in the temperature coefficient of linear expansion of sapphire in two mutually perpendicular directions with the temperature coefficient of linear expansion of tantalum. the ceramic modular head improvement in the design of a total hip replacement 69 secondary, processing the outer surface of a spherical sapphire head is a very laborious operation, provided that the required accuracy of its dimensions and shape is achieved. and, finally, the tantalum material itself and the indicated probable technologies for creating the outer ceramic layer on it are very expensive and will not be competitive with existing analogues on the market. another solution was proposed and promoted by smith & nephew, which made the head from an all-metal zirconium alloy (97.5% zirconium and 2.5% niobium), and on its surface formed a layer of zirconia ceramics (zirconia oxide), called oxinium, through a special process of blowing the head oxygen and its diffusion into the crystal structure of the metal alloy [10]. according to the company's advertising, such a head has the property of metal and ceramics at the same time, which should prevent the damage of ceramics and increase the wear resistance of the thr. but the thickness of the ceramic layer at the specified head does not exceed 7 microns, at the border of which there is a sharp decrease in hardness. in addition, despite the company's advertising materials, the degree of adhesion of the ceramic layer to the base metal at such a thickness remains debatable. at the same time, there is a high probability of damage to the ceramic layer when foreign abrasive bodies enter the friction pair, since there is still not a sufficient amount of long-term reliable information on the performance of thrs with such materials in vivo. in addition, the head has an increased mass due to the high specific gravity of the zirconium alloy, which shifts the center of mass of the entire thr from the position of the center of mass of the natural joint, increasing the shoulder of the dynamic component of the load at the thr and stimulating the weakening of its fixation. a number of thr designs have been proposed, where adapter sleeves are used to interface the ceramic head with the stem neck. for instant, in the design of the ceramic head, a cylindrical hole is made for mating with the stem neck, where a plastic sleeve is inserted [11]. since the head, as a rule, is fixed from axial movement in the thr acetabular component during its installation, then in the presence of certain dynamic loads during the operation of the thr, this will contribute to the creation of axial cyclic forces, leading to a relative displacement of the head and stem in their cylindrical connection. the presence of a soft plastic sleeve aggravates the process, leading to loosening of the non-strength connection between the head and neck, instability in the elements of the thr, disruption of the joint biomechanics and manifestation of pain syndromes, and, consequently, the need for revision prosthetics. in another known design of the thr, taper mates of the adapter metal sleeve made of titanium-based alloy are made both with the head and with the stem neck. but the use of such a sleeve is mainly recommended for revision prosthetics for fixing the head on the stem, which is well-fixed in the femur and does not require its extraction, and the neck of which has mechanical damage [12-15]. in [16], a numerical analysis of a sleeve connection with a ceramic head was carried out, which shows an improvement in the stress-strain state of ceramics in the case of using a sleeve. but on the other hand, it was shown in [17] that during the operation of the thr there are micro displacements of the sleeve, both rotational and translational, in the hole of the head and on the stem neck. therefore, this phenomenon does not exclude the possibility of ceramics destruction when exposed to extreme loads on the thr. there is also a thr ball head [18], which consists of external and internal elements, the external of which contains a blind cylindrical hole and is a part made entirely of ceramics or, for instance, a single crystal of sapphire or ruby, with an external spherical surface. in the specified hole there is an internal head element a cylindrical metal sleeve with an internal taper through hole and an external shoulder from the side of the inlet of the taper hole. for fixed mating of the two head elements, an adhesive or soldered joint is used. 70 v. pakhaliuk, a. poliakov, i. fedotov but this head has a number of significant disadvantages. a constructive drawback in the form of the presence of an external shoulder on the internal element of the head (sleeve) leads to a serious decrease in the range of angular motion in the artificial joint of the patient. in this case, to a probable impingement of the cup liner edge with the specified shoulder during the operation of the joint and, thereby, to a possible violation of the edge integrity and the ingress of products of its destruction into the tissues surrounding the thr. the technological disadvantage lies in the fact that the soldering of ceramics or single crystals to the metal sleeve, indicated in the reference, must be carried out with biocompatible solders. but in the available literature, the soldering as a low temperature process with biocompatible solders practically does not occur and is probably hardly possible, because only active metals can "wet" the ceramics and among them only 3 are biocompatible titanium, zirconia and hafnium, and brazing alloys based on titanium, zirconia and hafnium are high-temperature. in this case, when using brazing (a high-temperature process) for anisotropic sapphire single crystals and ruby to a metal sleeve with different temperature coefficients of linear expansion, cracks are likely to form in ceramics, which is unacceptable in products of this type. thus, the main aim of this work is to develop the design of a thr modular ceramic head, which has the properties of a metal in joint with the stem neck and the properties of a ceramics on the bearing surface of a friction pair by creating a low-cost, low-toxic, strong connection of alumina or zirconia ceramics and a titanium-based alloy to obtain a brazed joint that is efficient in human synovial fluid. 2. head design development another technological disadvantage relates to the creation of an adhesive bond, for example, specified in [18]. if the size of the gap between the elements to be brazed can be on the order of a proportion of a millimeter, then in order to achieve a strong adhesive bond, this gap should already be several millimeters. this statement is confirmed by the image in fig. 1, which shows a section of the adhesive joint between the stem neck of the titanium-based alloy and the head made of alumina ceramics with an outer diameter of 32 mm of the existing nonmodular thr design. here, component 1 is an alumina ceramic head, 2 is a titanium-based alloy stem neck, and 3 is an adhesive layer connecting these two components of the thr. the section shows that the thickness of the glue at this diameter of the head is about 4.5 mm and, in this case, there is no sufficiently strong thickness of the head at the inlet of its tapered hole to organize modularity when the adapter sleeve fastens by the glue. therefore, in the above thr design, only the stem neck can be motionlessly connected to the head. 2.1. design description to solve the problem, a thr head model was developed, shown in fig. 2, which consists of an external ceramic element 4 with a spherical outer surface 5 and an internal blind hole, in which the internal head element is placed an adapter sleeve 6 made of titanium-based alloy, for example, vt1-0 (pure titanium) or ti-6al-4v. the sleeve is made, for instant, with a cylindrical shape of the outer surface 7 (but it can also be conical) without an outer shoulder, and its inner end 8 is made of any shape, and it contains an axial blind tapered hole 9. the surface of the hole in ceramic element 4, for example, of a cylindrical shape, mating with the surface 7 contains a specially made macro-texture for improving the adhesion of the brazing alloy with the ceramics while brazing. the sleeve 6 in the specified hole of the element 4 the ceramic modular head improvement in the design of a total hip replacement 71 fig. 1 sectional view of the adhesive joint between the stem neck of the titaniumbased alloy and the alumina ceramic head fig. 2 model of the proposed thr head design with a quarter of its cut along the axis of symmetry along its surface 7 is installed with a gap of about 50-200 micrometers at the radius and with a similar gap along the surface 8 to fill the gap with biocompatible brazing alloy, and is fixedly connected to the element 4 by means of brazing. at the moment, the design of this head is undergoing the stage of obtaining a patent for an invention. the proposed design of the thr head provides the following. reducing the intensity of the occurrence of fretting-corrosion in the conical connection of the thr stem neck and the head sleeve, made of a homogeneous material, which excludes the occurrence of an electrolytic couple between them in an aggressive environment of the body, as well as its exclusion in the joint of the sleeve and the ceramic head due to the creation of a fixed connection between them. the absence of an external shoulder at the sleeve provides an expansion of the angular range of motion in the hip joint, which is very important for the patient, and significantly reduces the likelihood of impingement of the thr cup liner edge with the stem neck, thereby increasing thr resource. in addition, to create a strong brazed joint with ceramics, it is necessary to make a special macro-texture on the ceramic surface of the joint to increase its active area and improve the adhesion of the brazing alloy to the ceramics. for this purpose, a specially made macro-texture is applied to the ceramic element of the head on the cylindrical or conical surface of its hole. for instance, it can be provided by a milling method and have the form of concentric grooves with any pitch and depth (for example, a pitch of 1 mm and a depth of 500 μm), and such a texture is necessary to ensure high strength of the brazed joint. the brazing alloy wetting of the ceramic element with the applied macro-texture is provided by the preliminary application of a titanium coating on it by the method of gas-phase or magnetron deposition. and at the blind end of the sleeve and the inner end of the holes in the ceramic element, the surface shape can be any, since brazing alloy is placed in this area, which can rise into the gap, both due to capillary forces, and by creating possible additional axial pressure on the sleeve. the specified head design combines all the advantages of both metal and ceramic heads. among them, the presence of a metal sleeve with a traditionally tapered axial hole provides a fixed connection of the ceramic head with the sleeve due to the absence of micro displacements along the outer surface of the sleeve, thereby significantly increasing the stability of the thr and protecting the head from possible splitting in the tapered conjunction. 72 v. pakhaliuk, a. poliakov, i. fedotov all of these factors create conditions under which the service life of the entire thr increases and thereby the patient's quality of life improves. the design of the spherical joint of the thr with the indicated head is shown in fig. 3. here, 10 is the cup liner of the thr acetabular component, which is articulated with its inner spherical surface with the corresponding surface 4 of the head, thereby providing the formation of a spherical friction pair. component 11 is a part of the thr tapered stem neck, on which the head is placed during implantation of the thr. fig. 3 spherical joint design of thr with head developed 2.2. finite element simulation to confirm the operability of the thr composition shown in fig. 3, its finite element analysis was performed for a standard outer head diameter of 32 mm and a tapered hole with a standard cone 14/16 at a length of 1 inch. in the designs of thr, three standard external head diameters are mainly used: 22, 28 and 32 mm. moreover, the larger its diameter, the higher is the reliability of the thr, since it is more resistant to dislocation. from these positions, a head with a diameter of 32 mm was adopted for research. during the research, the following materials were accepted for the corresponding thr elements. for ceramic head and liner: alumina ceramic (al2o3) with young's modulus e=380 gpa, poisson's ratio is μ=0.22, ultimate strength in bending is σb=325 mpa. for the internal element of the head (sleeve) and for the stem, titanium-based alloy vt1-0 was used: young's modulus is e=110 gpa, poisson's ratio is μ=0.32, conditional yield stress σ0.2 =265 mpa. in the ansys workbench software, the 2-d axially symmetric problem of interaction of a ceramic head with an adapter sleeve and with no sleeve was solved. since the aim of the study was to compare stresses, strains and deformations in the heads of two designs, the type of thr static analysis with a mesh element size of 0.5 mm, obtained by curvature size function, was chosen. meshing method was quadrilateral dominant, which known to give good accuracy and acceptable solution time. in this case, for the presence of a sleeve, the use of brazing between the outer surface of the sleeve 6 and the inner surface of the hole in the ceramic head 4 was simulated by the bonding option. the contact between the stem neck 11 and the sleeve 6 in the presence of the sleeve or between the ceramic head 4 and the stem neck 11 in the no sleeve case was simulated with the rough option. the boundary conditions were set as a complete fixation the liner 10 along its outer surface from all displacements. the load f in the form of a concentrated force of 2.6 kn applied to the stem neck, acts along its longitudinal axis and is the ceramic modular head improvement in the design of a total hip replacement 73 the projection of the reaction on the head longitudinal axis of the concentrated maximum force of 3.0 kn, acting vertically on the head and induced by the acetabular component of the thr in accordance with iso 14242-1 standard [19]. the load profile specified in the above standard has a variable shape with a maximum value of 3.0 kn, a minimum value of 0.3 kn and a period of 1 hz. when designing the loading mechanism for the hip joint wear simulator according to the shape of this profile, the values of dynamic loads additionally acting on the hip joint were investigated [20]. it turned out that they do not exceed 3.5% and are negligible compared to the load indicated in this study, and, based on this, a static analysis was performed here. 3. results and discussion as a result of the solution, patterns of the distribution of contact pressures and deformations on the contact surfaces, as well as stresses and strains and total deformations in all components of the finite element model over the thickness of their section, are obtained. but from results obtained, we are mainly interested in solutions in both cases for the ceramic element of the head, since the excessive loads can lead to failure of this element with high probability. fig. 4 shows the model in the form of a head half-section of the thr with no sleeve, and fig. 5 shows the results of calculating the indicated head with patterns of stress, strains and total deformation distributions. fig. 6 shows the model in the form of a head halfsection with a sleeve brazed, and fig. 7 shows similar calculation results for the said head. the sleeve is placed in the head hole by reducing the thickness of its wall and has a thickness of 2 mm at the inlet to the hole, while maintaining the fitting size for the stem neck. fig. 4 the model of the thr head without a sleeve from the analysis of fig. 5, which shows the results of calculating the thr ceramic head without a sleeve, it can be seen that the most dangerous area manifests itself at the inlet of the 74 v. pakhaliuk, a. poliakov, i. fedotov stem neck into the tapered opening of the head. here, the maximum equivalent von mises stresses in the ceramic head are 19.3 mpa at the hole inlet, equivalent strains is 5.1·10-5 and total deformation (displacement) reaches 3.3·10-4 mm. fig. 5 the results of calculating the thr ceramic head without a sleeve, shown in figure 4: on left is the distribution of equivalent von mises stresses and strains, and on right is the distribution of total deformations fig. 6 the model of the thr head with a sleeve brazed made of vt1-0 alloy while maintaining the fitting size for the stem neck the ceramic modular head improvement in the design of a total hip replacement 75 fig. 7 the results of calculating the thr ceramic head with the sleeve shown in figure 6: on left is the distribution of equivalent von mises stresses and strains, and on right is the distribution of total deformations under excessive shock loads, the formation and growth of cracks begins from here. the analysis of fig. 7, which shows the results of calculating the thr ceramic head with a sleeve, shows that the most dangerous zone has shifted to the area where the ceramics has practically the greatest thickness. in this case, the maximum equivalent von mises stresses in the ceramic head decreased by more than 2 times to 8.0 mpa, equivalent strains to 2.1·10-5 and total deformation to 2.4·10-4 mm. the visualization of the magnitude and distribution of total deformation is quite informative, since if it is present, it becomes obvious how the deformation is redistributed between interacting bodies, and its reduction in the presence of a titanium-based sleeve indicates that the latter "takes" the deformation onto itself, since it is softer than ceramics. this confirms the hypothesis that the presence of the sleeve improves the stress-strain state of the ceramic head [16]. but in the investigated case of the head design, the problem is posed and considered not only the above improvement of the stress-strain state, but also, mainly, the eliminating of possible destruction the ceramics under excessive loads. this is achieved by creating a fixed connection of the metal sleeve with the ceramics by brazing, which excludes the axial micro displacement of the sleeve in the head hole, which takes place in its existing designs [17], and thereby eliminates the occurrence of the splitting effect in ceramics. at the same time, the hole in the developed ceramic head can be either cylindrical or conical with a similar shape of the outer surface of the sleeve. in existing designs of a head with a sleeve, the hole is made only of a conical shape, and when the sleeve is axially displaced from the action of axial force, a splitting effect is created in the ceramics, which increases the likelihood of its destruction in extreme cases. to demonstrate the feasibility of the proposed design solutions, two standard ceramic heads with an outer diameter of 32 mm with titanium-based alloy sleeves brazed were manufactured. since the design of the head is new and has no analogues in the world, it 76 v. pakhaliuk, a. poliakov, i. fedotov was important to check the possibility of obtaining it by using high-temperature brazing with titanium-based alloy. the internal tapered opening of the head was machined to create grooves and thereby increase the contact area of the brazing alloy with the ceramics. the biocompatible powder brazing alloy of the tizrco system was laid on the bottom of the head hole. brazing of the two heads was carried out in a xerion xvac 1600 vacuum oven to prevent oxidation of the brazing alloy and titanium-based alloy sleeve. the every sleeve is made with inner thread hole to secure the grip of the testing machine. an alumina ceramic brazed head with an indicated sleeve is shown in fig. 8. fig. 8 alumina ceramic head with titanium sleeve brazed mechanical tests of the brazed heads were carried out on an instron lx150 testing machine. during the tests, the separation of the sleeve from the alumina ceramic head was simulated by using special assembly. the deformation diagram for two brazed heads is shown in fig. 9. the results obtained show that the brazed sleeve can withstand a pull-off load of over 20000 n. the test was carried out either to the destruction of the head, or to the detachment of the gripping pin. the diagram 9 shows that for sample 1 there was a breakdown of the ceramic element of the head, and for sample 2 a rupture of the pin. fig. 9 deformation diagram of the brazed joint of the thr ceramic head with a titaniumbased alloy sleeve the ceramic modular head improvement in the design of a total hip replacement 77 preliminary mechanical tests on two specimens conducted to evaluate the possibility of a reliable fixation of the sleeve inside the head. in both cases, the failure was not along the brazed joint, and, therefore, the structural strength of the joint is higher or equal to the strength of the head itself. in addition, the resulting load values will never be achieved in the brazed joint. to determine the average bond strength, additional experimental studies should be carried out on a larger number of samples. thus, preliminary tests indicate the possibility of obtaining a strong brazed joint of a ceramic head with a titanium-based alloy sleeve for the manufacture of a thr with increased reliability. 4. conclusion and outlook for the first time, a design of a modular ceramic ball head of a thr friction pair has been developed, which has the properties of a metal in conjunction with the stem neck and the properties of a ceramics on the bearing surface of the friction pair. this is achieved by creating a low-cost, low-toxic, durable fixed connection of a head made of alumina or zirconia ceramics and a titanium-based alloy sleeve to obtain a brazed joint that is efficient in human synovial fluid. with the help of finite element analysis, a quantitative assessment of the strength and rigidity of the proposed head design was performed and its use in modern hip arthroplasty was indicated. the approbation of the proposed design solutions for creating a thr ceramic head with a titanium-based alloy sleeve brazed was carried out. further investigations can be directed towards performing a similar analysis for the standard diameter of other head sizes. further research will be directed towards performing additional experimental studies on a larger number of samples to obtain more accurate statistical indicators of the average strength of the specified brazing joint, as well as to perform a similar analysis for the standard diameter of other head sizes. in addition, it is planned to simulate the impact test of the developed head in the ls dyna software to take into account the severe dynamic process of its loading, leading to the destruction of ceramics, as well as its wear tests in accordance with iso 14242-1 at the simulator of sevastopol state university [21]. thus, the studies performed make it possible to move forward the creation of a thr design with high tribological performances along with an increased service life compared to existing designs, which will help to reduce the likelihood of patient revision prosthetics and improve their quality of life in the postoperative period. acknowledgement: the study was carried out with the financial support of the russian foundation for basic research (grant no. 20-03-00046a). references 1. poliakov, a., pakhaliuk, v., popov, v.l., 2020, current trends in improving of artificial joints design and technologies for their arthroplasty, frontiers in mechanical engineering 6, art. 4, 16 p. 2. tankut o.v., 2010, substantiation of hip arthroplasty using single crystal sapphire in the joint of hip prosthesis, phd thesis, sytenko institute of spine and joint pathology, ukraine, 356 p. 3. pakhaliuk, v., polyakov, a., kalinin, m., kramar, v., 2015, improving the finite element simulation of wear of total hip prosthesis' spherical joint with the polymeric component, procedia engineering, 100, pp. 539-548. 78 v. pakhaliuk, a. poliakov, i. fedotov 4. pakhaliuk, v., polyakov, a., 2018, simulation of wear in a spherical joint with a polymeric component of the total hip replacement considering activities of daily living, facta universitatis-series mechanical engineering, 16(1), pp. 51-63. 5. pakhaliuk, v., polyakov, a., kalinin, m., bratan, s., 2016, evaluating the impact and norming the parameters of partially regular texture on the surface of the articulating ball head in a total hip joint prosthesis, tribology online, 11(4), pp. 527-539. 6. kumar, n., arrora, gen n.s., datta, b., 2014, bearing surfaces in hip replacement – evolution and likely future, med j armed forces india, 70(4), pp. 371-376. 7. gallo, j., goodman, s.b., lostak, j., 2012, advantages and disadvantages of ceramic on ceramic total hip arthroplasty: a review article, biomed pap med fac univ palacky olomouc czech repub, 156(3), pp. 204-212. 8. volkov, v.v., kovalenko, o.v., kalinin, m.i., pakhaliuk, v.i., poliakov, a.m., brekhov, a.n., 2011, hip replacement head, ua patent no 95,382. kharkiv: ukrainian institute of intellectual property. 9. piconi, c., maccauro, g., muratori, f., branch del prever, e., 2003, alumina and zirconia ceramics in joint replacements, j appl biomater biomech, 1(1), pp. 19-32. 10. hunter, g., dickinson, j., herb, b., graham, r., 2005, creation of oxidized zirconium orthopaedic implants, journal of astm international, 2(7), pp. 1-14. 11. korzh, m.o., filipenko, v.a., radchenko, v.o., litvinov, l.a., voloshin, o.v., sliunin, y.v., timchenko, i.b., golukhova, a.g., tankut, v.o., tankut, o.v., 2007, hip endoprosthesis, ua patent no 79,551. kharkiv: ukrainian institute of intellectual property. 12. massin, p., lopes, r., masson, b., mainard, d., 2014, does biolox® delta ceramic reduce the rate of component fractures in total hip replacement?, orthopaedics & traumatology: surgery & research, 100(6), supplement, pp. s317-s321. 13. dickinson, e.c., sellenschloh, k., morlock, m.m., 2019, impact of stem taper damage on the fracture strength of ceramic heads with adapter sleeves, clinical biomechanics, 63, pp. 193-200. 14. leibiger, t., mcgrory, b.g., 2015, custom titanium sleeve for surgical treatment of mechanically assisted crevice corrosion in the well-fixed, noncontemporary stem in total hip arthroplasty, arthroplasty today, 1, pp. 107-110. 15. güttler t., 2006, experience with biolox® option revision heads. in: benazzo f., falez f., dietrich m. (eds) bioceramics and alternative bearings in joint arthroplasty. ceramics in orthopaedics, steinkopff, pp. 149-154. 16. falkenberg, a., dickinson, e.c., morlock, m.m., 2020, adapter sleeves are essential for ceramic heads in hip revision surgery, clinical biomechanics, 71, pp. 1-4. 17. koch, c.n., figgie, m.j., figgie, m.p., elpers, m.e., wright, t.m., padgett, d.e., 2017, ceramic bearings with titanium adapter sleeves implanted during revision hip arthroplasty show minimal fretting or corrosion: a retrieval analysis, hss j., 13(3), pp. 241-247. 18. bouvet, j.c., 1991, prosthesis ball joint, ep patent no 0,406,040. arpajon: european patent office. 19. implants for surgery – wear of total hip-joint prostheses – part 1: loading and displacement parameters for wear-testing machines and corresponding environmental conditions for test. – iso 14242-1:2002(e). 20. pakhaliuk, v.i., poliakov, a.m., desyatov, i.b., kalinin, m.i., stupko, m.g., 2011, the kinematic and dynamic performances of the loading mechanism of the hip joint wear simulator, annals of daaam & proceedings, pp. 595-596. 21. poliakov, o., pakhaliuk, v., lazarev, v., shtanko, p., ivanov, y., 2013, stand and control system for wear testing of the spherical joints of vehicle suspension at complex loading conditions, ifac proceedings volumes, 46(25), pp. 106-111. https://www.x-mol.com/paperredirect/1213064663885090836 https://www.x-mol.com/paperredirect/1213064663885090836 http://www.iso.org/iso/en/cataloguedetailpage.cataloguedetail?csnumber=23257&scopelist=all plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 99 108 1determination of friction heat generation in wheel-rail contact using fem udc 629.4 aleksandar miltenović 1 , milan banić 1 , dušan stamenković 1 , miloš milošević 1 , miša tomić 1 , jozef bucha 2 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of mechanical engineering, slovak technical university, slovakia abstract. the modeling of the friction heat generation has become increasingly important in product design process including areas such as electronics, automotive, aerospace, railway (e. g. wheel and rail rolling contact, braking systems, and so on), medical industries, etc. determination of generated friction heat in the contact of wheel and rail is important for understanding the damage mechanisms on these two bodies such as wear. this paper presents a method to determine the friction generated heat in contact of wheel and rail during normal operation using transient structural-thermal analysis in ansys software. key words: wheel-rail, friction, heat generation, fem, computer simulation 1. introduction today, computer simulation has allowed engineers and researchers to optimize product design process efficiency and explore new designs, while at the same time reduce costly experimental trials. generated friction heat in some physical processes like contact of rail and wheel during operation is an influential factor for damage forms and other processes. there are a great number of studies and research papers dealing with a rail/wheel contact problem. ertz and knothe [1] have concluded that the contact of wheel and rail can be investigated very efficiently with hertzian contact with polynomial approximation. they have also presented methods for the calculation of contact temperatures using blok’s flash temperature formula. the bulk temperature of the wheel increases with time by continuous friction heat. he has shown that the wheel temperature during normal operating condition cannot be more than double the average temperature of the cold wheel. lyu et al. [2] have received april 10, 2015 / accepted june 20, 2015 corresponding author: aleksandar miltenović faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, niš, serbia e-mail: aleksandar.miltenovic@masfak.ni.ac.rs original scientific paper 100 a. miltenović, m. banić, d. stamenković, m. milošević, m. tomić, j. bucha analyzed wheel/rail contact including the influence of temperature and humidity on the friction coefficient and wear. gallardo-hernandez et al. [3] have compared temperature in a twin-disc wheel/rail contact in simulation and in experiment using a thermal camera. they have used two methods to calculate temperatures and he has carried out an experiment with 0.5 – 5 % slip. they have also given a diagram of the friction coefficient in the function of slip where the friction coefficient varies up to 0.6. wu et al. [4] analyzed wheel/rail contact using thermal-elastic-plastic deformation and residual stress after wheel sliding on a rail. they have simulated a sliding contact process by translating the normal contact pressure and the tangential traction across the rail surface. the results indicate that the friction thermal load of contact between wheel and rail has a significant influence on the residual deformation, plastic strain and residual stress at the rail surface. since the friction coefficient plays an important role in study of wheel/rail contact, there is a large number of papers which describe methods for friction measurement. tomeoka et al. [5] have performed friction control between wheel and rail. firstly, fundamental tests with two-roller-rigs were carried out to evaluate the friction performances of several types of friction modifiers. then, for the purpose of realizing the friction control with them, authors developed an on-board system, which sprayed friction modifier from a bogie to the top of rail accurately. areiza et al. [6] did experimental measurement of coefficient of friction in rails using a hand-pushed tribometer. numerical simulations can be used to check the old and to develop new and more efficient designs [7]. milošević et al. [8] have presented the procedure of modeling thermal effects in braking systems of railway vehicles. miltenović et al. [9] have presented the basic procedure for determination of friction heat generation in the wheel/rail contact using fem. 2. wheel-rail geometry 2.1 rail profile there are 23 rail profiles specified in the standard en 13674-1:2011 [10]. this european standard specifies vignole railway rails of 46 kg/m and a greater linear mass for conventional and high speed railway track usage. the two classes of the rail straightness are specified, differing in requirements for straightness, surface flatness and crown profile. moreover, the two classes of profile tolerances are specified. fig. 1 represents rail profile 50e2 which is used in further analyses. 2.2 wheel profile uic code 510-2 [11] contains the conditions relating to the design and maintenance of wheels and wheel sets for coaches and wagons used on international services. it covers wheel diameters from 330 to 1000 mm and indicates the permissible axle loads from the standpoint of stresses of the metal used for the wheel and the rail. uic code 510-2 contains detail coordinates of the wheel rim line. it is valid for a nominal track gauge of 1435 mm and cannot be readily transposed to apply to other track gauges. fig. 2 represents the wheel profile which is used in further analyses. determination of friction heat generation in contact of wheel-rail using fem 101 fig. 1 rail profile 50e2 [12] fig. 2 wheel profile 3. analysis setup for analysis purposes the simplest case is taken where the wheel speed is constant and when there is only one contact point between wheel and rail. in a case like this slip and friction are also constant. to estimate the generated heat amount is a challenging task that requires a multidisciplinary approach as well as analysis of many influencing factors [12] and the authors of this article offer an approach to simplify the case study. the basic model of the wheel/rail contact was made in solidworks before being exported in ansys. for the fem analysis only the upper part of the rail 10 meters long and the outer ring of the wheel were used. in the following analyses the ansys elements solid 226 (3-d 20-node coupledfield solid, fig. 3) and solid 227 (3-d 10-node coupled-field solid) were used that support the thermoplastic effect which calculate temperature based on plastic deformation by partial conversion of work into heat. fig. 3 element solid 226 used in analyses 102 a. miltenović, m. banić, d. stamenković, m. milošević, m. tomić, j. bucha for meshing of the rail the body solid 226 was used. the complete geometry of the wheel body was meshed with solid 227 instead of 226 (fig. 4). table 1 shows numbers of elements and nodes in the meshed wheel/rail model. fig. 4 mesh of wheel and rail table 1 number of elements and nodes wheel rail sum elements 76502 62400 138902 nodes 114735 94820 209555 the properties of the material used in the analysis are listed in table 2. table 2 material properties used in this study parameter unit value density [kg/m 3 ] 7850 coefficient of thermal expansion [c -1 ] 1.2∙10 -5 young’s modulus [pa] 2∙10 11 poisson’s ratio [-] 0.3 tensile yield strength [pa] 2.5∙10 8 tensile ultimate strength [pa] 4.6∙10 8 isotopic thermal conductivity [w/mc] 60.5 specific heat [j/kgc] 434 the analysis of the friction heat generation in the wheel/rail contact was defined as a direct coupled transient structural-thermal analysis. the rate of frictional dissipation of contact elements in ansys is evaluated using the frictional heating factor and is given by: determination of friction heat generation in contact of wheel-rail using fem 103 vfghtq   (1) where τ is the equivalent frictional stress,  is sliding rate and fgth is fraction of frictional dissipated energy converted into heat (the default value of 1 was used).[13] fig. 5 analysis setup of wheel and rail for case with sliding 1% and speed 30 km/h due to limited computer resources, analyses were limited on 0.6 sec which is enough for wheel to make more than one whole rotation. during normal operation there is sliding between the wheel and the rail. in order to take into account the sliding effect, the rail was translated during an analysis in the same time with the wheel in the same direction 1%, 2% or 3%, which represent some of slip ratios [14]. this moving of the rail represents the sliding that is expected during the normal train operation. for analyses purposes, the wheel was analyzed with the speeds of 30 km/h and 60 km/h. however, because of the slip ratio, the exact speeds of the wheel during the simulation were 29.7 km/h and 59.4 km/h in the case when the sliding was 1%, 29.4 km/h and 58.8 km/h for 2% sliding and 29.1 km/h and 58.2 km/h for 3% sliding. at the same time, the pressure load on the wheel was 5 t and 10 t representing the weight of the wagon. the friction coefficient between the wheel and the rail was 0.1 or 0.3. 5. results tables 3 and 4 present the results for generated temperatures of the analyzed friction heat generation of the wheel and the rail in the contact area. these results represent relative increase of the temperature, since, for the simulation circumstances, the atmosphere temperature was 22 °c and weight of 5 t. in table 3 temperatures for speed 30 and 60 km/h and the friction coefficient 0.1 are given, while table 4 gives temperatures for 30 and 60 km/h and the friction coefficient 0.3. table 5 gives temperatures on wheel and rail for the speed 30 km/h, the friction coefficient 0.3 and with different weights (5 and 20 t). temperatures in the tables are average values of the highest temperatures of wheel or rail area. 104 a. miltenović, m. banić, d. stamenković, m. milošević, m. tomić, j. bucha table 3 relative temperatures increase [k] on wheel and rail for speed 30 and 60 km/h and friction coefficient 0.1 wheel rail 30 km/h 60 km/h 30 km/h 60 km/h sliding 1% 6.4 8.1 4.4 5.7 2% 13.1 16.1 8.8 10.4 3% 19.7 21.5 13.3 16.2 table 4 relative temperatures increase [k] on wheel and rail for speed 30 and 60 km/h and friction coefficient 0.3 wheel rail 30 km/h 60 km/h 30 km/h 60 km/h sliding 1% 7.5 8.9 4.9 6.3 2% 15.2 16.6 9.6 11.6 3% 20.8 22.9 15.7 17.8 table 5 relative temperatures increase [k] on wheel and rail for speed 30 km/h, friction coefficient 0.3 and different weights wheel rail 5 t 10 t 5 t 10 t sliding 1% 7.5 10.8 4.9 7.11 2% 15.2 20.2 9.6 13.3 3% 20.8 26.3 15.7 19.6 figs. 6 to 12 show comparisons of temperatures of the wheel/ rail contact areas for varying: speed 30 and 60 km/h, friction coefficient 0.1 and 0.3, sliding 1, 2 and 3 % and weight 5 and 10 t. fig. 6 gives comparisons of the temperature increase for speeds of 30 and 60 km/h and the friction coefficient 0.1. compared with temperatures on the rail, temperatures on the wheel are approximately 66 – 68 % higher for the speed of 30 km/h, also, for the speed of 60 km/h temperatures on the wheel are approximately 70 – 75 % higher. determination of friction heat generation in contact of wheel-rail using fem 105 fig. 6 comparisons of the temperature increase for speeds of 30 km/h and 60 km/h and friction coefficient 0.1 fig. 7 comparisons of the temperature increase for speeds of 30 km/h and 60 km/h and friction coefficient 0.3 fig. 7 represents comparisons of the temperature increase for speeds of 30 and 60 km/h and the friction coefficient 0.3. compared with temperatures on the rail, temperatures on the wheel are approximately 63 – 74 % higher for the speed of 30 km/h, also, for the speed of 60 km/h temperatures on wheel are approximately 69 – 77 % higher. fig. 8 comparisons of the temperature increase on the wheel and rail for speeds of 30 km/h and 60 km/h and friction coefficient 0.1 106 a. miltenović, m. banić, d. stamenković, m. milošević, m. tomić, j. bucha fig. 8 shows comparisons of the temperature increase on the wheel and rail for speeds of 30 and 60 km/h and the friction coefficient 0.1. compared for the cases of the speed of 30 km/h, temperatures on the wheel are approximately 16 % higher for the speed of 60 km/h, also, for the rail, temperatures are approximately 19 % higher for the speed of 60 km/h. fig. 9 comparisons of the temperature increase on the wheel and rail for speeds of 30 km/h and 60 km/h and friction coefficient 0.3 fig. 9 gives comparisons of the temperature increase on the wheel and rail for speeds of 30 and 60 km/h and the friction coefficient 0.3. compared for the cases of the speed of 30 km/h, temperatures on the wheel are approximately 12 % higher for the speed of 60 km/h, also, for the rail, temperatures are approximately 17 % higher for the speed of 60 km/h. similarly, fig. 10 represents comparisons of the temperature increase on the wheel for the same speed and friction coefficients 0.1 and 0.3. compared with the temperatures for the case with the friction coefficient 0.1, temperatures on the wheel for the speed of 30 km/h are approximately 10 % higher for the friction coefficient 0.3, also, for the speed of 60 km/h temperatures on the wheel are some 6 % higher for the friction coefficient 0.3. a) 30 km/h b) 60 km/h fig. 10 comparisons of the temperature increase on the wheel for the same speed and friction coefficients 0.1 and 0.3 determination of friction heat generation in contact of wheel-rail using fem 107 a) 30 km/h b) 60 km/h fig. 11 comparisons of the temperature increase on the rail for the same speed and friction coefficients 0.1 and 0.3 a) 5 t b) 10t fig. 12 comparisons of the temperature increase on rail and wheel for speed 30km/h, friction coefficient 0.1 and different weights fig. 11 gives comparisons of the temperature increase on the rail for the same speed and friction coefficients 0.1 and 0.3. compared with the temperatures for the case with the friction coefficient 0.1, temperatures on the rail for the speed of 30 km/h are some 14 % higher for the friction coefficient 0.3, also, for the speed of 60 km/h temperatures on the wheel are approximately 10 % higher for the friction coefficient 0.3. finally, fig. 12 depicts comparisons of the temperature increase on the rail and the wheel for the speed of 30 km/h, the friction coefficient 0.1 and for different weights. compared with the temperatures for the case with the weight of 5 t, temperatures on rail are approximately 31 % higher for the weight of 10 t and for the wheel the temperatures are some 32 % higher. 6. conclusion the paper presents an approach to determine the friction generated heat in the wheel/rail contact by using of the fem transient structural-thermal analysis in ansys software. the performed research shows that temperatures on the wheel are greater than temperatures on the rail (66-75%) in all the cases of slip ratios, speeds and friction coefficients. moreover, the increase of the sliding leads to increase of friction generated temperature on the wheel/rail contact surface. 108 a. miltenović, m. banić, d. stamenković, m. milošević, m. tomić, j. bucha the increase of the speed (30 to 60 km/h) leads to 12 – 16 % increase of the temperature on the wheel and 17 – 19 % on the rail. the increase of the friction coefficient (0.1 to 0.3) leads to 6 – 10 % increase of the temperature on the wheel and 10 – 14 % on the rail contact surface. the increase of the weight (5 t to 10 t) leads to 31 – 32 % higher temperatures on the rail and wheel. further research should compare temperatures obtained by means of simulation with real temperatures in the exploitation. also, it would be very interesting to calculate friction generated heat in the wheel rail contact in the cases of acceleration and braking of trains, as well as in the cases when there are two contact points between the wheel and rail. references 1. ertz m., knothe k., 2002, a comparison of analytical and numerical methods for the calculation of temperatures in wheel/rail contact, wear, 253, pp. 498–508. 2. lyu y., zhu y., olofsson u., 2015, wear between wheel and rail: a pin-on-disc study of environmental conditions and iron oxides, wear, 328-329, pp. 277–285. 3. gallardo-hernandez e.a., lewis r., dwyer-joyce r.s., 2006, temperature in a twin-disc wheel/rail contact simulation, tribology international, 39(12), pp. 1653–1663. 4. wu l., wen z., li w., jin x., 2011, thermo-elastic–plastic finite element analysis of wheel/rail sliding contact, wear, 271(1–2), pp. 437–443. 5. tomeoka m., kabea n., tanimoto m., miyauchi e., nakata m., 2002, friction control between wheel and rail by means of on-board lubrication, wear, 253, pp. 124–129. 6. areiza y.a, garcés s.i., santa j.f., vargas g., toro a., 2015, field measurement of coefficient of friction in rails using a hand-pushed tribometer, tribology international, 82(b), pp. 274-279. 7. vukić m., vučković g., ţivković p., stevanović ţ., tomić m., 2013, 3d numerical simulations of the thermal processes in the shell and tube heat exchanger, facta universitatis, series mechanical engineering, 11(2), pp. 169-180. 8. milošević m., stamenković d., milojević a., tomić m., 2012, modeling thermal effects in braking systems of railway vehicles, thermal science, 16(2), pp. 515-526. 9. miltenović a., banić m., stamenković d., milošević m., tomić m., 2014, determination of friction heat generation in contact of wheel-rail set using fem, proc. xvi international scientific-expert conference on railway railcon‘1+-4, niš, pp. 21-24. 10. bs en 13674-1:2011, 2011, railway applications track rail part 1: vignole railway rails 46 kg/m and above, bsi, england. 11. uic code 510-2, 2004, trailing stock: wheels and wheelsets. conditions concerning the use of wheels of various diameters, union internationale des chemins de fer, france. 12. mijajlović m., 2013, numerical simulation of the material flow influence upon heat generation during friction stir welding, facta universitatis, series mechanical engineering, 11(1), pp. 19-28. 13. miltenović v., banić m., miltenović a., 2013, prediction of heat generation in meshing of crossed helical gears, proc. 5th international conference on gears, munich, germany pp. 586-597. 14. arias-cuevas o., li z., lewis r., gallardo-hernandez e.a., 2008, rolling–sliding laboratory tests of friction modifiers in leaf contaminated wheel–rail contacts, proc. stle/asme international joint tribology conference ijtc 2008, miami, florida, usa. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 153 168 software for synthesis of compliant mechanisms without intersecting elements  udc 621+004.42; 515.1; 62-4 andrija milojević, nenad d. pavlović university of niš, faculty of mechanical engineering, niš, serbia abstract. a compliant mechanism is defined as a single-piece structure that transfers motion or force through elastic deformation. the synthesis of this kind of mechanisms represents a challenging task, especially because their flexible segments usually must undergo large, nonlinear deflections which include a difficult nonlinear analysis. in this paper the new software for synthesis of compliant mechanisms is described. the software uses an improved topology optimization technique that is especially useful when the designer does not have a particular compliant mechanism already in mind. the intersection between the elements in the compliant mechanisms obtained by using the existing topology optimization technique often increases stiffness of the structure which needs to be flexible. the topology optimization technique is improved in the software so that compliant mechanisms without intersecting elements are obtained. the methodology that the software uses and its capability will be shown on the examples of the synthesis of a compliant gripper and a compliant displacement inverter. key words: compliant mechanism, synthesis, topology optimization, compliant gripper, compliant displacement inverter 1. introduction a compliant mechanism can be defined as a single-piece flexible structure which uses elastic deformation to achieve force and motion transmission [1]. compliant mechanisms differ from conventional rigid-link mechanisms in that they gain their mobility from the deflection of the flexible members, contain no pin joints and are intentionally flexible. a compliant mechanism is a combination of a structure and a mechanism, since the jointless feature resembles a structure, while the function of the structure resembles a mechanism. compliant mechanisms are designed to be flexible enough to transmit motions, yet stiff enough to withstand external loads. the transition from the conventional mechanism to the compliant  received december 2, 2013 corresponding author: andrija milojević faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: andrijamilojevic87@gmail.com 154 a. milojević, n. d. pavlović one is shown in fig. 1. the function of both the grippers is to transfer the input force to the output and to grasp an object. while the rigid-link gripper gains functionality by a rigid-body mechanism, the compliant one undergoes elastic deformations due to an input actuation. a) b) fig. 1 a) rigid-link gripper; b) compliant gripper [2] the main advantages of compliant mechanisms over classical ones are: simplified manufacturing, reduced assembly costs, no wear, no backlash, reduced noise, easier maintenance, no need for lubrication, built-in restoring force, better scalability and accuracy. because of these advantages compliant mechanisms are used in many applications including manufacturing, aerospace, robotics, biomedical devices, mems, grippers, motion amplifiers, positioning devices, adaptive structures, shape morphing structures, surgical tools, etc. (see e.g. [2], [3], [4]). the compliant mechanism synthesis represents a challenging task and has been well studied in the past [5], [6]. the compliant mechanism synthesis is not so straightforward because the flexible segments usually must undergo large, nonlinear deflections, introducing high stresses and a difficult nonlinear analysis. a nonlinear analysis usually includes solving nonlinear differential equations for accurate prediction of flexible segments motion. to include the solutions to these equations in the design process would be tedious and overly complex, thus it is efficient and easy to use tools for compliant mechanism synthesis. different methods are used for the optimal synthesis of compliant mechanisms [7], [8], [9], but more appropriate and more robust tools for the synthesis of compliant mechanisms for real-world applications are missing. this paper presents a new software and a novel approach to the compliant mechanisms synthesis. the software uses a topology optimization technique that is especially useful when the designer does not have a particular compliant mechanism already in mind. the intersection between the elements in compliant mechanisms obtained by using the existing topology optimization technique often increases complexity and stiffness of the structure which can significantly lower the mechanism functionality. the topology optimization technique is improved in the novel developed software, so that compliant mechanisms without intersecting elements are obtained. the software, its efficiency, functionality, capability and methodology will be shown on the example of the synthesis of a complaint gripper and a compliant displacement inverter which are common benchmarking problems that have been broadly studied [8], software for synthesis of compliant mechanisms without intersecting elements 155 [9]. the synthesis of the mechanisms (gripper and displacement inverter) with and without intersecting elements will be shown for comparison. the developed software intends to provide an easy-to-use tool for compliant mechanism design, which, we hope, will lead to novel solutions for today's engineering problems. 2. synthesis of compliant mechanisms with distributed compliance compliant mechanisms can be subdivided into two groups: mechanisms with lumped compliance and mechanisms with distributed compliance [10], [11], [13]. mechanisms with distributed compliance (fig. 7, fig. 9, fig. 15) make use of longer and thicker bending elements. they gain their mobility through elastic deformation of flexible members and compliance is distributed more or less equally in the entire mechanism (i.e. they deform as a whole). the continuum synthesis approach is used for the design of mechanisms with distributed compliance [5], [7], [9], [10]. the synthesis methodology used in the continuum based approach involves two stages: generation of the mechanism topology and determination of optimum size, geometry, and shape of various constituent elements of the mechanism (dimensional synthesis). in this paper more attention is paid to topology optimization, because this is a more difficult and "creative" part of the design process. the allowable space for the design in a topology optimization problem is called the design domain (fig. 3). the topology is defined by the distribution of material and void within the design domain or as the pattern of connectivity of elements in a structure (fig. 7, fig. 9, fig. 15). the continuum based approach focuses on the determination of the optimal topology (the best material connectivity in a compliant structure). the designer only needs to define the size of the design domain in which the mechanism should fit, location of the supports, input and output ports, size of applied loads (fig. 3) as well as properties of the material from which the mechanism should be produced. then, through the topology optimization, the optimal structural form (optimal topology) of a compliant mechanism for a specified input force and output deflection requirements is automatically generated (fig. 7, fig. 9, fig. 15). 3. new software for optimal synthesis of compliant mechanisms in the past many computer-coded algorithms and software for synthesis of compliant mechanisms have been developed [7], [9], [12], but most of them require some knowledge from the field of topology optimization. this has motivated us to develop a new tool for the synthesis of compliant mechanisms, that can be used by experts and designers who are not necessarily familiar with the details of topology optimization techniques. the software provides an interactive and automated design route to synthesize compliant mechanisms for any desired input-output force/displacement characteristics while meeting given constraints. the algorithm on which the software is built is shown in fig. 2. 156 a. milojević, n. d. pavlović fig. 2 computational procedure in the first step the problem specification is given by the user, including the size of the design domain, the input force (intensity, point and direction), the desired output deflection (size, location and direction), the material property (young modulus) from which the mechanism should be built and other constraints (such as location of the fixed nodes). in the second step the design domain is parameterized. the physical design space must be broken down so as to be represented by a set of variables that an optimizer can act on. since mechanisms with distributed compliance consist of individual flexible members, the grounded structure approach (gsa) [7] is used for the parameterization. since a compliant mechanism relies on elastic deformation (bending) of constituent elements, beam elements are used to generate the topology. therefore, the prescribed design domain is divided into a number of nodes, and a network of beam elements connecting these nodes (fig. 4) serves as an initial guess. the design variable is the thickness of each element. a thickness value of zero deactivates the element, removing it from the structure; other values represent thickness values. in the software, the thickness of each element can be one of three predetermined discrete values defined by the user. in the software the designer has the option of specifying the degree of nodal connectivity, that is, every node can be connected by an element to every other node in the grid, or only to nearby neighbors. after the parameterization is done, in the third step, the optimization starts. because of the broad design space and number of elements, topology synthesis problems are solved with optimization methods. when the parameterization is discrete (i.e. elements are either on or off), the discrete optimization methods are used. these methods include the field of evolutionary optimization, of which genetic algorithms [8] are applied here. genetic algorithms are a commonly used method in topology optimization of compliant mechanisms [2], [4], [7], and they have been well studied in the past. after the optimization procedure converges, some elements will be removed from the original exhaustive set. the remaining elements will define the topology for the compliant mechanism, which represents the optimal solution for the given problem; the optimized compliant mechanism is a network of a subset of beam elements (fig. 6). software for synthesis of compliant mechanisms without intersecting elements 157 the capabilities of the developed software and its efficiency will be shown on the example of the synthesis of a compliant gripper and a compliant displacement inverter, which are common benchmarking problems in the topology optimization field [5], [12]. 4. synthesis of the compliant gripper and the displacement inverter the goal is to design two compliant mechanisms: a compliant gripper and a compliant displacement inverter, by using the newly-developed software with the topology optimization technique that is commonly used for the synthesis of compliant mechanisms. in the first case we want to design a compliant gripper that can realize desired output motion δ at the indicated location, when input load f is applied (fig. 3a). this will allow the device to grip some object at the output point. in the case of the displacement inverter we want to design a compliant structure that can realize desired output displacement δ in the opposite direction as input force f (fig. 3b). the design domain for both cases in fig. 3 represents the upper-half view, since this is assumed to be a symmetric problem without any loss of generality in the solution procedure. a) b) fig. 3 design domain and problem setup for synthesis of the compliant gripper (a) and the compliant displacement inverter (b) 4.1. problem specification and parameterization in the software, the size of the design domain, in which the mechanism should fit, must be defined first (only the rectangular design domain can be defined). then, the user needs to input the desired number of nodes, creating nl x nh grid of nodes. this defines the resolution with which the design domain will be parameterized. after this, the software automatically parameterizes the design domain using the gsa. the design domain is broken down so that the grid of nodes is interconnected by beam elements (fig. 4). a) b) fig. 4 initial ground structure: a) compliant gripper; b) compliant displacement inverter 158 a. milojević, n. d. pavlović in the obtained ground structure all the nodes are initially interconnected i.e. the ground structure is "fully connected". a fully connected structure can lead to overlapping elements that are difficult to produce. thus, certain filters are implemented in the software that eliminates the overlapping elements. this partially connected structure results in much fewer design variables and greatly reduced computation time, while still effectively representing the design domain. the user can also define the degree of nodal connectivity, that is, the number of nodes that one node can be interconnected maximally in the directions of the width and the height of the design domain (how many fields one beam element can cross). this can significantly reduce the grid resolution of the ground structure which can speed up the process of optimization. for the case of the compliant gripper and the compliant displacement inverter, design specifications of the design domain, size of grid nodes and degree of nodal connectivity are given in table 1. the design domain in fig.4a needs to have a void, thus some of the fields in the ground structure must be eliminated. in the software this is done by typing the mark of the right corner node of the field which must be eliminated. in the case of the compliant gripper the node mark 16 is selected and after this the new ground structure for the compliant gripper is obtained (fig. 5). fig. 5 ground structure with a void next, input force f and supports need to be defined. this is done by typing the node mark in which the force and supports are to be put. the same stands for defining output displacement δ. the user also needs to input the size and the orientation of the force and the displacement. for the specified cases, parameters for defining force, supports and output displacement are given in table 1. table 1 design specifications for compliant gripper and displacement inverter problems design parameters compliant gripper compliant displacement inverter element modulus e=2.1 gpa e=2.1 gpa design domain 60 mm  30 mm 50 mm  25 mm grid size 4  4 3  2 force node = 13; α = 180°; f = 30 n node = 4; α = 180°; f = 2 n supports node = 1 nodes = 1, 3 displacement node = 12; direction: y; δmin = 5 mm; node = 6; direction: x; δmin = 2 mm; element thickness 0.5 mm, 1 mm, 1.5 mm 0.5 mm, 1 mm, 1.5 mm degree of nodal connectivity 3 2 software for synthesis of compliant mechanisms without intersecting elements 159 when the problem is set, and the design specifications are defined, the software starts with the optimization procedure and the optimal solution of the given problem is searched. 4.2. objective function an objective function is needed to drive the optimizer toward a desired goal. the main goal in the synthesis of the compliant gripper and the compliant displacement inverter is to maximize the output displacement while meeting the given constrains. this is done by maximizing the ratio of output displacement δout to input displacement δin which is known as the geometric advantage. all objective function terms are calculated from the results of the linear finite element analysis (fea), implemented in the software, with only one beam element placed between each of the nodes in the topology being analyzed. 4.3. constraints two constraint penalties are added to the objective function to monitor deflection. the first is a constraint that requires the output deflection in the desired direction be greater than minimum value δmin. the second constraint requires that the output deflection perpendicular to the desired direction, should be less than maximum value min  . also, total element length (lt) constraint is used; lt is the sum of the lengths of all the elements in a given design, and primary intention is to reduce complexity, not necessarily weight. the final form of the objective function that is used in the software for the synthesis of the compliant gripper and compliant displacement inverter is:         ||| )δ|δ()δδ( δ δ 321 targettoutmaxminout in out llwwwmaximize (1) where w1, w2, and w3 are relative weighting constants and ltarget represents the desired total element length. in the formulation, the penalties are applied only when δout<δmin or out max     . the values for constrains and for weighting constants are given in table 1 and table 2. 4.4. discrete nonlinear topology optimization for a given model of the design space, the variables can be optimized via a genetic algorithm to yield a mechanism topology. genetic algorithms are a form of nonlinear optimization that seeks global optima, and are especially suited for discrete, nonlinear problems. every design is evaluated with fea, which is used to assign the above fitness function, eq. (1). in the software, to initiate the genetic search, the user must define the following parameters [14]: size of initial population which represents the number of randomly generated designs; number of generations i.e. the number of iterations which will be used in the optimization process; crossover probability – the probability that crossover will be performed between a pair of parent designs (random number of element variables are swapped between two parent designs in the process of crossover); mutation probability – the probability that the design will mutate (elements in the design mutate to different thickness). the algorithm parameters used in the synthesis of the compliant gripper and the displacement inverter are: 160 a. milojević, n. d. pavlović initial population of 200 designs, total number of 500 generations, crossover probability of 80% and mutation probability of 9%. table 2 constraints used in the synthesis of the compliant gripper and the compliant displacement inverter constraints compliant gripper compliant displacement inverter weighting constants w1=0.6; w2=0.2; w3=0.2 w1=0.6; w2=0.2; w3=0.2 max  5 mm 0.1mm ltarget 354 mm 214 mm 4.5. results after the optimization is finished, the software displays the obtained solution of the given task. the solution to the problem contains the compliant structure in which some of the elements are eliminated in the process of optimization. the remaining elements define the optimal topology of the compliant mechanism that can realize the given task. the obtained results for the compliant gripper problem and compliant displacement inverter problem, by using the developed software, are shown in fig. 6 [15]. a) b) fig. 6 optimal topology of the compliant gripper (a) and the displacement inverter (b), only the top-halves of the mechanisms are shown (deformed position is shown with dashed lines) [15] in both cases, the topology of the compliant mechanisms is optimized. when the force is applied at the input port the compliant gripper deforms and realizes the output deflection of: δout =10.66 mm, and in the case of the displacement inverter the output deflection is δout =17.93 mm. the geometrical advantage of the compliant gripper is δout/δin=1.42, and in the case of displacement inverter δout/δin=0.994. finally, the results are verified via solid model finite element analysis [15]. physical prototypes of the mechanisms in fig. 6, made from plastic, are shown in fig. 7. another optimization is run for the case of the displacement inverter but now with a higher grid resolution of the initial ground structure (fig. 8a). here the grid size 3x3 is used, the input node is 7 and the output node is 9 (all other design and algorithm parameters are the same as in the case of the displacement inverter in fig.3b). the obtained solution is shown in fig. 8b. software for synthesis of compliant mechanisms without intersecting elements 161 a) b) fig. 7 physical prototypes, made of plastic, of the compliant gripper (a) and the compliant displacement inverter (b) a) b) fig. 8 a) initial ground structure; b) optimal topology of the displacement inverter, only the top-half of the mechanism is shown (deformed position is shown with dashed lines) due to the applied force at the input port (node 7) the compliant displacement inverter deforms and realizes the output deflection (node 9) of: δout=2.0229 mm, while the displacement of the input port is δout=1.0891 mm. the geometrical advantage of the obtained mechanism is δout/δin=1.8574. this means that a displacement inverter that has the geometrical advantage greater than 1 (thus acting as an amplifier) can be obtained if higher grid resolution is used (in our case the output displacement is nearly doubled). the results are verified via solid model finite element analysis (fig. 9a). the analysis shows linear dependence between the input and the output displacement (the results are given in table 3). the manufactured model of the compliant displacement inverter, made of plastic, is shown in fig. 9b. table 3 results of the finite element analysis for the compliant displacement inverter input displacement (mm) output displacement (mm) geometrical advantage 1 1.7427 1.7427 2 3.4854 1.7427 3 5.2281 1.7427 162 a. milojević, n. d. pavlović a) b) fig. 9 a) finite element analysis of the compliant displacement inverter; b) mechanism prototype made from plastic (the whole mechanism is shown in both cases) 5. synthesis of compliant mechanisms without intersecting elements in the topology optimization of compliant mechanisms the choice of initial ground structure is very important as the ground structure represents the space of design variables in which the optimal solution (for a given problem) is searched for. in the literature, the reduced initial ground structure is commonly used [12] with the degree of nodal connectivity 1 (fig. 10). because the ground structure is reduced, nodes are placed at the intersection point between two elements i.e. in this ground structure intersections do not exist (fig. 10). although this kind of ground structure greatly reduces computation time and compliant mechanisms without intersecting elements are obtained, it cannot always efficiently represent the design domain; because of the reduction many of the good designs are lost at the beginning. partially connected ground structure (fig. 4) is also used in the literature [2], [7]. although this kind of ground structure can more effectively represent the design domain (higher degree of nodal connectivity), the solutions with intersecting elements are obtained (as in fig. 6a and fig. 8b or see [2], [7]) where at the place of intersection the nodes are not defined. there are some differences between the results obtained by fem analysis and by using the novel software in mechanisms in fig. 6a and fig. 8b, as at the places of intersection the nodes were not defined. defining the nodes at the intersecting points between elements would greatly increase the computational time because the ground structure becomes highly complex with many intersecting elements (fig. 4a). a designer must keep in mind that when it comes to manufacturing of compliant mechanisms obtained with this kind of ground structure, elements that intersect themselves must be manufactured in separate planes. this can be sometimes very difficult to achieve. other solution is to produce elements in the same plane with intersections (like in fig. 7a and fig. 9b), but then the stiffness of the system will increase and the functionality of the mechanism will decrease regarding the originally obtained mechanism. this deficiency has motivated us to improve the existing topology optimization technique so that the intersections between elements would be eliminated in the process of optimization. this represents a novel approach to the synthesis of compliant mechanisms. software for synthesis of compliant mechanisms without intersecting elements 163 fig. 10 reduced initial ground structure to eliminate intersections between elements in the topology optimization process, a computer-coded algorithm as a search filter is created. in every iteration during the optimization process, the algorithm searches for intersecting points between elements in the structure (fig. 11). then the total number of intersections (for every structure) is calculated. the goal is to eliminate all the intersections in the structure thus minimizing their total number represents the objective function term. here the total number of intersections (nint) is used as a constraint rather than an objective function term. the final form of the objective function that is used in the software for the synthesis of compliant mechanisms without intersecting elements is:         intntargettnn in out nwllwnstraintsvarious conwmaximize 21 ||) ( δ δ (2) the synthesis of compliant mechanisms without intersecting elements will be shown on the example of the synthesis of the compliant gripper and the compliant displacement inverter. fig. 11 finding the intersections between elements in different steps of optimization 6. synthesis of the compliant gripper and the displacement inverter without intersecting elements for the synthesis of the compliant gripper and the compliant displacement inverter without intersecting elements the same design domain is used as in fig. 3. the only 164 a. milojević, n. d. pavlović difference is that instead of the force, now the displacement is applied at the input port. design specifications for both compliant mechanisms are given in table 4. table 4 design specifications for the synthesis of the compliant gripper and the compliant displacement inverter without intersecting elements design parameters compliant gripper compliant displacement inverter element modulus e=2.1 gpa e=2.1 gpa design domain 50 mm  25 mm 50 mm  25 mm grid size 3  4 3  2 input displacement node = 10; direction: x; δin = 2 mm node = 4; direction: x; δin = 1 mm desired output displacement node = 9; direction: y; δmin = 1 mm node = 6; direction: x; δmin = 0.5 mm supports nodes=1, 4, 7 nodes=1, 3 element thickness 0.5 mm, 1 mm, 1.5 mm 0.5 mm, 1 mm, 1.5 mm degree of nodal connectivity 3 2 initial ground structures with an initial number of intersections for the compliant gripper and the compliant displacement inverter problem are shown in fig. 12 (intersections are shown with red circles). a) b) fig. 12 initial ground structure with an initial number of intersections: a) compliant gripper (number of intersections 81); b) displacement inverter (number of intersections 9) the objective function used in the synthesis of the compliant gripper and the compliant displacement inverter is similar as in eq. (1), where the constraint of the total number of intersections is now added (constrains are given in table 4 and table 5):         inttargettoutmaxminout in out nwllwwwmaximize 4321 ||| )δ|δ()δδ( δ δ (3) the algorithm parameters used in the synthesis of both compliant mechanisms are: initial population of 200 designs, total number of 500 generations, crossover probability of 80% and mutation probability of 9%. software for synthesis of compliant mechanisms without intersecting elements 165 table 5 constraints used in the synthesis of the compliant gripper and the displacement inverter without intersecting elements constraints compliant gripper compliant displacement inverter weighting constants w1=0.3; w2=0.3; w3=0.1; w4=1 w1=0.3; w2=0.3; w3=0.1; w4=1 max  0.5 mm 0.5mm ltarget 167 mm 111 mm after defining all the design and algorithm parameters, the optimization procedure is started and the optimal topology of compliant mechanisms is searched for. the obtained optimal topologies of the compliant gripper and the compliant displacement inverter are shown in fig. 13. a) b) fig. 13 optimal topology of the compliant gripper (a) and the displacement inverter (b) without intersections, only the top-halves of the mechanisms are shown (deformed position is shown with dashed lines) unlike solutions in fig. 6a and fig. 8b here the compliant mechanisms without intersections are obtained. the results for both mechanisms are shown in table 6. table 6 results for the obtained optimal topologies of the compliant gripper and the compliant displacement inverter without intersections results compliant gripper compliant displacement inverter with grid size 32 compliant displacement inverter with grid size 33 output displacement 8.53 mm 0.996 mm 3.9023 mm realized max  0.40 mm 0 mm 0 mm geometrical advantage: δout/δin 4.2693 0.996 1.9512 lt 171 mm 120 mm 222 mm the compliant gripper mechanism analysis shows that the mechanism expresses a better performance than the one with intersections (fig. 6a). geometrical advantage of the gripper mechanism without intersections is much greater than the one with intersections (table 6). this is due to the fact that intersections between elements increase stiffness of the structure thus lowering the functionality of the mechanism. because the geometrical 166 a. milojević, n. d. pavlović advantage of the compliant gripper without intersections is greater than 1 (δout/δin=4.2693), the compliant gripper also acts as an amplifier of the output displacement in respect to the input displacement. it is important to note that the topology of the compliant gripper in fig. 13a is mainly formed of elements that would not exist if a simplified initial ground structure was used (fig. 10). in the case of the compliant displacement inverter, both the compliant mechanisms in fig. 6b and fig.13b are without intersections. this is because the initial ground structure with a small grid size is used in both the cases and thus the number of possible solutions is small. although the topology of the mechanisms is slightly different, their geometrical advantage is nearly the same (table 6). to confirm the elimination of the intersections in the case of the compliant displacement inverter, another optimization is run but now with higher grid resolution: 3x3 (fig. 14a). all the other design and algorithm parameters are the same as for the displacement inverter in fig.13b; here ltarget=223 mm, the input and the output port are nodes 7 and 9, respectively. the optimal topology of the compliant displacement inverter is shown in fig.14b. again the compliant mechanism without intersections is obtained, unlike mechanism in fig. 8b. the previously made conclusion is confirmed here, the compliant displacement inverter that has the geometrical advantage greater than 1 (thus acting as an amplifier) can be obtained with higher grid resolution (table 6). compared to the compliant mechanism in fig. 8b, the geometrical advantage of the compliant displacement inverter without intersection is better (table 6). similar to the case of the compliant gripper, the optimal topology of the displacement inverter contains beam elements that would not exist if the simplified ground structure was used. a) b) fig. 14 a) initial ground structure with initial number of intersections (44); b) optimal topology of the compliant displacement inverter, only the top-half of the mechanism is shown (deformed position is shown with dashed lines) all the results are verified via commercially available fem analysis software (fig. 15). a) b) c) fig. 15 finite element analysis of: a) compliant gripper; b) displacement inverter (grid size 32); c) displacement inverter (grid size 33) software for synthesis of compliant mechanisms without intersecting elements 167 7. conclusions this paper presents original software for an optimal synthesis of compliant mechanisms without intersecting elements. the software uses the topology optimization technique that is especially useful when the designer does not have a particular compliant mechanism already in mind. in the software the user only needs to input the problem specifications and the software then (through the topology optimization) automatically generates the optimal structural form (optimal topology) of a compliant mechanism which represents the solution for a given task. the methodology as well as the algorithm on which the software is built are also described. the existing topology optimization technique is improved so that compliant mechanisms without intersecting elements are obtained. this represents a novel approach to the synthesis of compliant mechanisms. the capability of the software with the newly introduced approach is shown on the example of the synthesis of two compliant mechanisms: a compliant gripper and a compliant displacement inverter, with and without intersecting elements for comparison. the optimal topologies of both mechanisms (with and without intersections) are obtained. the results show that compliant mechanisms without intersecting elements have better performances. it is also shown that the objective function in eq. 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software for synthesis of compliant mechanisms, mechanical engineering in xxi century masing 2013, faculty of mechanical engineering niš, niš, 273-278 pp. softver za sintezu gipkih mehanizama bez presečnih elemenata gipki mehanizam se definiše kao pokretna, materijalno koherentna struktura koja može da prenese sile i transformiše kretanje samo zahvaljujući elastičnoj deformaciji odgovarajučih segmenata strukture. sinteza ove vrste mehanizama predstavlja izazovan zadatak, posebno zato što su njihovi elastični segmenti često izloženi velikim, nelinearnim deformacijama što zahteva znatno težu, nelinearnu analizu. kada projektant nema unapred osmišljenu formu gipkog mehanizma za sintezu se uobičajeno koriste postupci optimizacije topologije. preseci elemenata, koji postoje u gipkim mehanizmima dobijenim primenom postojećeg postupka optimizacije topologije, često povećavaju krutost strukture koja mora biti dovoljno elastična. u ovom radu je predstavljen novi softver za sintezu gipkih mehanizama koji unapređuje aktuelne postupke optimizacije topologije time što eliminiše presečne elemente odgovarajućih segmenata gipkog mehanizma. metodologija koju softver koristi i njegove mogućnosti predstavljeni su na primeru sinteze gipkog hvatača i gipkog invertora pomeranja. ključne reči: gipki mehanizam, sinteza, optimizacija topologije, gipki hvatač, gipki invertor pomeranja plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering https://doi.org/10.22190/fume240110026s © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper ann and hybrid modelling of eccentrically patch loaded steel i-girders biljana šćepanović1, snežana rutešić1, olga mijušković1, luisa maría gil-martín2, enrique hernández-montes2 1university of montenegro, podgorica, montenegro 2university of granada, granada, spain abstract. majority of eccentrically patch loaded girders collapse at lower load level than their geometrical equivalents loaded in the web plane, due to different collapse mechanism. however, some eccentrically patch loaded girders, even with significant load eccentricity, behave as if loaded in the web plane, having the same collapse mode and ultimate load as in the case of centric load. hence, the ultimate strength of eccentrically patch loaded steel i-girders should be found out in two phases: firstly, girder collapse mode should be estimated; secondly, depending on expected collapse mode, ultimate load should be calculated appropriately. both tasks are demanding, with plenty of mutually dependent influential parameters. artificial neural network (ann) modelling, being suitable for multi-parameter analysis, is quite convenient method for studying eccentrically patch loaded steel i-girders in both mentioned tasks, as confirmed through the examples elaborated in the paper. not only that it is valuable on its own, as a standalone technique, but also in combination with and as support to other methods (hybrid modelling). in comparison of five procedures for ultimate load determination (empirical expression, mechanical model, two versions of refined mechanical model, one of them by ann, and standalone ann forecast model), ann modelling, providing high quality results, qualified in top-two procedures for ultimate load, regardless of collapse mode type. it is also proved that certain shortages of mechanical model may be overcome by its coupling with ann modelling. such hybrid modelling provided remarkably more accurate results than original mechanical model. study confirmed need of revision of mechanical model. as the first step, new, more demanding constraints of mechanical model validity are proposed in this paper. key words: eccentric patch load, collapse mode, ultimate load, artificial neural network, hybrid modelling received: january 10, 2024 / accepted march 15, 2024 corresponding author: biljana šćepanović university of montenegro, faculty of civ. eng, bul. džordža vašingtona bb, 81000 podgorica, montenegro e-mail: biljanas@ucg.ac.me, biljazs38@gmail.com 2 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes 1. introduction the term “patch load” designates the load acting over a small area or length of a structural element, as a concentrated or locally distributed load, fig. 1. a typical situation in structural engineering practice is when compressive patch load affects the flange of iprofile so that the web is locally pressed in the zone under the loading. local compression might provoke local instability that may result in element carrying capacity loss and, consequently, collapse of the whole structure. patch load examples are numerous and present in different types and segments of structures, such as crane girders loaded by crane wheels, bridge girders during erection by launching, rafters supporting secondary girders, column-beam joints, ship structures subjected to truck wheel or steel coil, etc. [1-3]. a comprehensive research work has been done worldwide, tackling different aspects of patch loaded steel i-girders [4,5]. the main research topic is the ideal case of patch load acting in the web plane, i.e. centric patch load, fig. 1a. numerous experimental studies provide solid amount of data as a base for further statistical, numerical (primarily by finite element method (fem)) and analytical modelling, enabling proposals of different plastic collapse mechanisms, their mathematical formulations and adjoining expressions for ultimate load calculation. increase in web (buckling) stability and girder resistance to patch load may be achieved by: (a) increase in web thickness; (b) longitudinal (web) stiffeners [6]; (c) transverse stiffeners [7]; (d) simultaneous transversal and longitudinal stiffening [8]; (e) delta stiffeners, i.e. triangular cell loaded flange [9,10]; (f) use of corrugated instead of flat web plates [11,12]. not all these measures are fully efficient in case of load that changes its position (e.g. cranes, bridges, ships, etc.). furthermore, production of such girders may be cost/time demanding. hence, the rationalisation of patch loaded i-girders with unstiffened, flat web plate is still in focus of research. additionally, investigation attention in this domain is still devoted to specific influential parameters and circumstances, such as: initial conditions (geometry imperfections and residual stresses) [13,14]; patch load length [14,15]; interaction of patch loading with bending and/or shear [16], etc. in line with needs of environmental protection and cost-effectiveness, hybrid steel i-girders (with different mechanical properties of flange/web steel) [17], as well as options of substitution for mild steel (aluminium [18], stainless steel [19], high-strength steel igirders [20]) or its upgrade by cladding (i-girders having mild steel web, cladded by titanium [21] or stainless steel [22]) have been studied recently. a girder behaviour under patch loading at elevated temperatures or in fire conditions is analysed [23]. a piece of research findings concerning centric patch loading is implemented in european design standards, eurocodes – en1993-1-5 [24]. however, review and further adjustments of standards are needed, so to avoid too low underestimation of ultimate load, as well as to cover more diverse cases regarding girder geometry and load arrangement. in the engineering reality, a certain eccentricity of patch load relative to the web plane is unavoidable, i.e. eccentric patch load is more realistic case, fig. 1b. this case is among issues which are not currently incorporated in eurocodes, with the appropriate calculation procedure. in some circumstances, the load eccentricity may be treated as an imperfection that will not cause significant change in girder’s behaviour and carrying capacity compared to the centric load case. however, in other situations the load eccentricity should not be neglected, since it may be the reason for dangerous decrease in ultimate load (even up to 75%), as a result of girder’s behaviour being completely different from the one of centrically loaded girder. ann and hybrid modelling of eccentrically patch loaded steel i-girders 3 fig. 1 patch loaded steel i-girder: (a) centric patch load; (b) eccentric patch load [25] compared with the case of centric patch load, rather modest amount of research has been devoted to the case of eccentric patch load. while over 40 experimental researches (with more than 800 tested samples) dealt with centrically patch loaded i-girders, influence of load eccentricity was analysed in only eight experimental studies (with less than 200 tested samples). majority of tests (135 specimens) were done at the university of montenegro, in three series of experiments, in: 1998, reported by lučić [26]; 2001, reported by lučić and šćepanović [27]; and 2007, reported by šćepanović et al. [28]. the latest testing, with only four tested girders, was done at the university of navarra, upon the initiative of the university of granada, in 2009, reported by gil-martín et al. [29]. experimental work was followed by non-linear finite element method (fem) modelling [28,29]. while over 30 mathematical expressions (mostly based on collapse mechanism) for centric ultimate load might be found in scientific literature and design norms, only one mathematical/mechanical model for eccentric ultimate load calculation, based on the plastic collapse mechanism, is published by graciano and uribe-henao [30]. prior to the development of this mathematical/mechanical model, a few empirical expressions for eccentric ultimate load have been proposed, last modifications by šćepanović et al. [28] and gil-martín et al. [29]. design norms still do not encompass eccentric patch load. arabzadeh and varmazyari [9] analysed influence of delta stiffeners in eccentrically patch loaded girders, by means of fem modelling. inaam and upadhyay [11], as well as maiorana, poh’sié and emechebe [12] touched on an issue of eccentric patch load in girders with corrugated web, in their fem based analyses. experimental research demonstrated that majority of eccentrically patch loaded girders collapse at lower load level than their geometrical equivalents loaded in the web plane. lover ultimate load is a consequence of different girder’s behaviour, i.e. different collapse mechanism or collapse mode, in case of eccentric load (flange warping/twisting, followed by web bending) compared with the case of centric load (web buckling, followed by flange “sinking”), fig. 2. 4 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes fig. 2 collapse modes typical for centric and eccentric patch load: (a) centric collapse mode; (b) eccentric collapse mode [25] however, some eccentrically patch loaded girders, even with significant load eccentricity, behave as if loaded in the web plane, having the same collapse mode and ultimate load as in the case of centric load. hence, the ultimate strength of eccentrically patch loaded steel i-girders should be found out in two steps: at first, girder collapse mode should be estimated; secondly, depending on expected collapse mode, ultimate load should be determined appropriately. obviously, it is rather complicated task to formulate unique mechanical model (i.e. collapse mechanism and ultimate load expression based on it) that may be applied to all eccentrically patch loaded girders. numerous parameters and their combinations determine behaviour, i.e. collapse mode and ultimate load of eccentrically patch loaded girders. the most prominent one is the load eccentricity, e, fig. 1b. girder dimensions, fig. 1, primarily plate thicknesses, tw and tf, as well as dimensionless geometry parameters, such as tf /tw, hw /tw, etc. are also of a great significance. a change of only one girder dimension results in the variation of several influential parameters, what makes the analysis rather complex. various artificial intelligence (ai) methods and techniques are highly suitable for multi-parameter analysis, particularly for solving compound stochastic problems, characterised with plenty of uncertainties and inter-dependencies. artificial neural networks (ann), machine learning (ml), deep learning (dl), etc. are nowadays fairly present in different areas of structural engineering, with high potential to complement, alternate or even replace traditionally used empirical modelling, based on statistics methods, and fem based numerical modelling, as summarised in comprehensive reviews of salehi and burgueño [31], sun et al. [32], wang et al. [33], tapeh and naser [34]. widening field and growing number of examples of successful ann applying in design, construction, protection, monitoring and maintenance of diverse structures, such as reported by chojaczyk et al. [35], markovic et al. [36], zarezadeh et al. [37], include the topic of patch loading, too. ann modelling/forecasting, as a sophisticated ai calculation and modelling method, is quite convenient approach for studying centrically patch loaded i-girders, as presented by kurtoglu et al. [18], mai et al. [38], kumar et al. [39]. in case of eccentric patch load, with all its complexity and intricacy, ann modelling is even more welcome and valuable, ann and hybrid modelling of eccentrically patch loaded steel i-girders 5 as demonstrated by šćepanović et al. [25]. however, this technique has not been used to its full potential, which is evident in both phases of eccentrically patch loaded steel igirders analysis: collapse mode estimation and ultimate load determination. ann modelling may be successfully used on its own, but also as an advantageous tool coupled with other methods (i.e. hybrid modelling), e.g. with the only one existing mechanical model of collapse mechanism or with some of only few existing empirical expressions for ultimate load calculation. benefits of ann modelling, as well as hybrid modelling are elaborated in the paper, through the examples. 2. collapse modes of eccentrically patch loaded steel i-girders eccentrically patch loaded steel i-girders mostly lose carrying capacity due to local elastic-plastic bending, with emphasised flange torsion and almost undeformed web, as in fig. 3, but not necessarily. as already mentioned, under certain circumstances, eccentrically loaded girders may lose carrying capacity the same way as if there is no load eccentricity. three different types of collapse mode, each with several variants, are observed in experimentally tested eccentrically patch loaded steel i-girders: eccentric (e, fig. 3), centric (c, fig. 4) and mixed (m, fig. 5) collapse mode. flange torsion, web deformation, position and shape of yielding lines (primarily one in web, but also one in flange), as well as their development level, in case of eccentric collapse mode, vary, depending on girder geometry and load eccentricity, as displayed in fig. 3. fig. 3 eccentric collapse mode of eccentrically patch loaded i-girders (e) – variants, depending on girder geometry: (a) tw < tf & bf /tf ≥ 12.5; (b) tw = tf & bf /tf ≥ 12.5; (c) bf /tf = 10 (thick flange) 6 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes fig. 4 centric collapse mode of eccentrically patch loaded i-girders (c) – variants: (a) with two yielding lines (and one buckle) in web; (b) with three yielding lines (and two buckles) in web fig. 5 mixed collapse mode of eccentrically patch loaded i-girders (m) – variants: (a) centric-mixed collapse mode (gentle torsion of flange, centric collapse of web); (b) eccentric-mixed collapse mode (pronounced torsion of flange, undefined web collapse) based on the experimental data, collapse mode identification criteria summarised in table 1 have been established. these criteria correlate dimensionless geometry parameters tf /tw and a/tw = hw /tw with the dimensionless load eccentricity e/bf , e/tf or e/tw. such correlations provide high level of reliability in collapse mode identification. combining (checking) several of these criteria increases the reliability level. it has to be kept in mind that data from experiments done at the university of montenegro [26-28] were used for this analysis, i.e. girders dimensions, load eccentricity and load length in range: (0) 5 mm  e  30 mm, 3 mm  tw  10 mm, 3 mm  tf  15 mm, a = hw = 700 mm, bf = 150 mm, c = 50 or 150 mm; and dimensionless geometry parameters in range: (0) 1/30  e/bf  1/5, (0) 0.33  e/tf  8.33, (0) 0.50  e/tw  8.33, 1  tf /tw  5, 10  bf /tf  50, 70  a/tw = hw /tw  233, a/hw = 1, c/a = 0.071 or 0.214. hence, these are the limitations of high-level validity area for criteria from table 1. girders fulfilling criteria in dark grey fields of table 1 have eccentric collapse mode (e), with the reduced ultimate load. girders fulfilling criteria in light grey fields of table 1 ann and hybrid modelling of eccentrically patch loaded steel i-girders 7 might have any collapse mode (e, m, c) and these criteria imply that such girders should be carefully analysed each separately, with its specific characteristics, considering several influential parameters and their combinations, as well as paying particular attention to the initial deformation. girders fulfilling criteria in white (i.e. no highlighted) fields of table 1 will behave as if there is no load eccentricity (c collapse mode), having same ultimate load as in case of centric patch load. primary influential parameter tf /tw enables high quality identification of collapse mode in correlation with each one of the three dimensionless eccentricities. however, web slenderness a/tw = hw /tw, not encompassing flange thickness tf, correlated with any of the three eccentricities does not provide comprehensive, exact collapse mode determination. as a complement to criteria from table 1, but also in case of its standalone use, ann modelling proved to be confidential method for collapse mode identification in case of girder geometry when all collapse modes are possible, as illustrated in figs. 6 and 7. several networks of various architecture were created and trained, by means of ann software developed for the needs of research at the university of montenegro. two ann forecast models for collapse mode are chosen for representation herein. both ann models displayed at figs. 6 and 7 have good match with experimental data, particularly considering “pure” eccentric and centric collapse modes (e, c). tolerable discrepancy may be observed in domain of mixed collapse modes. trend of ann models in that domain is completely in accordance with experimental findings. better fitting of individual points may be achieved by further ann modelling, which is generally endless activity. another software or selection of training/verification dataset, as well as fine tuning of ann models (e.g. different choice of numerals for collapse modes in this specific software etc.) will provide different forecasts. note: in order to apply available ann software, collapse mode types needed to be assigned numerals and they are chosen in interval [-1, 1], as follows: eccentric collapse mode e = -1; mixed collapse modes -1 < m <1; centric collapse mode c = 1. table 1 criteria for identification of collapse mode in eccentrically patch loaded i-girders dimensionless load eccentricity correlated dimensionless parameter criterion collapse mode* e/bf hw /tw hw /tw < 1050 · e/bf + 35 e hw /tw ≥ 1050 · e/bf + 35 e, m, c tf /tw tf /tw < 15 · e/bf + 0.5 e 15 · e/bf + 0.5 ≤ tf /tw ≤ 15 · e/bf + 1.5 e, m, c tf /tw > 15 · e/bf + 1.5 c e/tf hw /tw hw /tw ≤ 85 · e/tf + 60 e 85 · e/tf + 25 ≤ hw /tw ≤ 85 · e/tf + 105 m** hw /tw ≥ 85 · e/tf + 70 c tf /tw tf /tw < e/tf + 0.5 e e/tf + 0.5 ≤ tf /tw ≤ e/tf + 1.7 e, m, c tf /tw > e/tf + 1.7 c e/tw tf /tw tf /tw < 0.3 · e/tw + 0.8 e 0.3 · e/tw + 0.8 ≤ tf /tw ≤ 0.3 · e/tw + 1.8 e, m, c tf /tw > 0.3 · e/tw + 1.8 c * e = eccentric, m = mixed, c = centric collapse mode ** criterion is not sufficiently precise, detailed analysis of each girder, separately, is necessary 8 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes fig. 6 illustration of collapse mode identification by means of ann modelling and according to criteria from table 1, compared to experimental data, for girders geometry: tw = 5 mm, tf = 12 mm, a = hw = 700 mm, bf = 150 mm, c = 50 mm fig. 7 illustration of collapse mode identification by means of ann modelling and according to criteria from table 1, compared to experimental data, for girders geometry: tw = 10 mm, tf = 10 mm, a = hw = 700 mm, bf = 150 mm, c = 50 mm ann and hybrid modelling of eccentrically patch loaded steel i-girders 9 figs. 6 and 7 exhibit very good match of ann modelling of collapse mode with criteria from table 1, not only in outer zones of exclusively centric or eccentric collapse modes (c or e), but also in the middle zone, where all types of collapse mode are possible (c, m, e). some girder dimensions or dimensionless parameters, analysed separately, without correlating them with other parameter(s), may be used as a rough identifiers of collapse mode. girders having thin flange (tf ≤ 6 mm) or thick web (tw ≥ 8 mm, e.g. fig. 7) or low web slenderness (hw /tw < 100, e.g. fig. 7) or ratio tf /tw < 2 (e.g. fig. 7) are affected even with the lowest eccentricity, as confirmed in fig. 7. on the other side, girders with the ratio tf /tw > 3 have centric collapse mode even at the highest load eccentricity. similarly to these conclusions, drawn from the experimental testing, the accuracy limitations of mechanical model, by graciano and uribe-henao [30], are set up by means of load eccentricity, e, and tf /tw ratio: the mechanical model describing eccentric collapse mode should be used when e ≥ 10 mm and tf /tw ≤ 2.5. it may be concluded that two sets of limits for collapse modes are in a proper agreement, as confirmed in figs. 6 and 7. obviously, apart from the load eccentricity, e, collapse mode depends on both plate thicknesses, tf and tw, as well as on their relation, so that parameter tf /tw has the key-role in collapse mode determination, regardless of being analysed in co-relation with the other parameters or separately, only by itself. 3. ultimate load of eccentrically patch loaded steel i-girders from the engineering practice aspect, the most significant characteristic of eccentric and mixed collapse modes is decrease in ultimate load with the increase in load eccentricity. experiments revealed that the ultimate load decrease of over 40% is possible even in cases of such a small load eccentricity of only 5 mm (i.e. e/bf = 1/30), while larger eccentricity, of 25 mm (i.e. e/bf = 1/6), may decrease ultimate load up to 75%. fem modelling, based on non-linear analysis, may provide confident results for ultimate load, but it is not always comfortable to be used by engineers. for common situations in practice, simple calculation procedures are appreciated. two types of such expressions for ultimate strength of eccentrically patch loaded girders may be found in literature: empirical expressions [28,29], obtained by statistical processing of primarily experimental, but also fem results, and mathematical expression based on plastic collapse mechanism [30]. ann modelling may be of great help in determination of ultimate load, whether ann forecast models are used directly for ultimate load [25], or this tool is coupled with some of previously mentioned methods/expressions, for determination of certain parameters figuring in those expressions for ultimate load. 3.1 expressions for ultimate load 3.1.1 empirical procedure – expression for reduction coefficient empirical procedure relates the ultimate load of eccentric collapse mode to the ultimate load of centric collapse mode, in girders of same geometry with and without load eccentricity, by introducing reduction coefficient, r, as: r = ultimate load of eccentrically loaded girder ultimate load of centrically loaded girder . (1) 10 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes the ultimate load of centrically loaded girder might be determined by some of numerous expressions proposed in literature, including design norms. reliable reduction coefficient, r, will then provide ultimate load of eccentrically loaded girder with acceptable accuracy. for the further analysis in this paper, the following expression for reduction coefficient, proposed by šćepanović et al. [28], will be used: 𝑅 = 𝑚 ⋅ ( 𝑒 𝑏𝑓 ) 2 + 𝑛 ⋅ ( 𝑒 𝑏𝑓 ) + 1.01 ≤ 1   𝑚 = −0.864 ⋅ ( 𝑡𝑓 𝑡𝑤 ) 2 − 14.40 ⋅ ( 𝑡𝑓 𝑡𝑤 ) + 38.00   𝑛 = −12.30 + 4.22 ⋅ ( 𝑡𝑓 𝑡𝑤 ) } . (2) eq. (2) provides valid results for input in range of data used for its derivation (experimental data reported in [26-28] and fem data reported in [28]), as follows: 1  tf /tw  5, (0) 1/30  e/bf  1/5, 45  a/tw  233, 6.25  bf /tf  50, 1  a/hw  2, 0.036  c/a  0.071 or c/a = 0.214. 3.1.2 expression based on mechanical model of collapse only one mathematical/analytical expression based on plastic collapse mechanism has been proposed so far, by graciano and uribe-henao [30]: 𝑃𝑢 = 𝑀𝑓 ∙ (2𝛽ξ + 𝑐ξ + 𝑏𝑓−2𝑒 𝛽 ) + 𝑀𝑤 ∙ 2𝛽+𝑐 𝛼𝑐𝑜𝑠𝜃 𝑀𝑓 = 𝑓𝑦𝑓𝑡𝑓 2 4⁄ 𝑀𝑤 = 𝑓𝑦𝑤𝑡𝑤 2 4 ⁄ ξ = 𝑏𝑓 (𝑒(𝑏𝑓 − 2𝑒))⁄ 𝑐𝑜𝑠𝜃 = 3 8 𝑓𝑦𝑓𝑎(𝑐+0.7𝑎) 𝐺𝑏𝑓𝑡𝑓 𝛽 = √ 𝑀𝑓(𝑏𝑓 2⁄ −𝑒)∙𝛼𝑐𝑜𝑠𝜃 𝑀𝑤+𝑀𝑓ξ∙𝛼𝑐𝑜𝑠𝜃 �̃� = 1.9447𝑡𝑤 ( 𝑒 𝑏𝑓 ∙ ( 𝑡𝑤 𝑡𝑓 ) 3 ) −0.451 } , (3) where: mf and mw are plastic moments per unit length of yielding line in the flange and web, respectively; fyf and fyw are flange and web yield stresses, respectively; g = 81000 mpa is the shear modulus for steel; α, β, θ and ξ are geometrical parameters describing yielding lines and deformed girder geometry in the collapse mechanism. they differ in character, i.e. in method of obtaining. while ξ is used only as a symbol to shorten writing, α, β and θ have clear geometrical meaning explained in [30]. while β and θ are defined analytically, based on mechanical models, by mathematical calculation, α (= distance of the web yielding line from the loaded flange, fig. 3) may be determined as ᾶ, using the expression from eq. (3), obtained by regression analysis, based on the experimental data [30]. to provide accuracy of eq. (3), graciano and uribe-henao [30] set up following limits: e ≥ 10 mm and tf /tw ≤ 2.5. these limits and need of more precise definition of eq. (3) validity area are going to be discussed herein. ann and hybrid modelling of eccentrically patch loaded steel i-girders 11 3.2 ann modelling as support for ultimate load calculation in addition to its successful standalone application for direct determination of ultimate load [25], ann modelling may be used as a very convenient and welcome tool combined with other methods in order to improve them and provide more precise values of ultimate load. such coupling of ann modelling with expressions for ultimate load may be characterised as hybrid modelling. it is going to be presented herein, at the example of refining eq. (3) by means of ann modelling. experimental database from 1998 [26], 2001 [27] and 2007 [28], containing 135 test samples, has been used for various ann training and ann forecast models creation. also, comparison of different methods for determination of ultimate load and/or necessary parameters has been done using these test data. tables 2 and 3 comprise tested girders geometry data, experimental collapse mode type, α and pu values obtained by different methods (all calculated and ann determined pu values are related to respective experimental values, so to provide easier data comparison): experimental position of web yielding lines αexp, (table 2); experimental collapse loads pu,exp, (table 2); ultimate loads pu,ann obtained by ann forecast models (as pu,ann /pu,exp), (table 3); ultimate loads pu,eqs(1-2) obtained by eqs. (1) and (2) when pu,centr = pu,exp (e = 0) (as pu,eqs(1-2) /pu,exp = req(2) /rexp), (table 3); ᾶ values, obtained by expression from eq. (3), (table 2); ultimate loads pu,eq(3) obtained by eq. (3), including expression for ᾶ (as pu,eq(3) /pu,exp), (table 2); ultimate loads pu,eq(3)-exp obtained by expression for pu from eq. (3), but with αexp values (as pu,eq(3)-exp /pu,exp) – experimentally refined eq. (3), (table 2); αann values, obtained by ann modelling, (table 2); ultimate loads pu,eq(3)-ann obtained by expression for pu from eq. (3), but with αann values (as pu,eq(3)-ann /pu,exp) – hybrid modelling of pu, (table 2). complete calculation, as well as creation of ann models were done with real values of yield stresses fyf and fyw, determined experimentally (table 3), and g = 81 gpa. table 2 summary of α values (αexp, ᾶ, αann), experimental collapse loads (pu,exp) and ultimate loads determined by original and refined eq. (3) (pu,eq(3), pu,eq(3)-exp, pu,eq(3)-ann), all related to pu,exp, for experimental database from 1998, 2001 and 2007 tests [26-28] no. girder tw tf e co ll a p se m o d e αexp pu,exp ᾶ pu,eq(3) pu,exp pu,eq(3)-exp pu,exp αann pu,eq(3)-ann pu,exp [mm] [mm] [mm] [mm] [kn] [mm] [mm] 1 eb i – 1 3 15 0 c 133 2 eb i – 2 3 15 5 c 128 238.7 3.10 55.5 3.14 3 eb i – 3 3 15 10 c 127 174.6 2.01 55.4 2.04 4 eb i – 4 3 15 15 c 135 145.4 1.49 55.2 1.52 5 eb i – 5 3 15 20 c 134 127.7 1.28 54.9 1.31 6 eb i – 6 3 15 25 c 124 115.5 1.24 54.6 1.27 7 eb ii/iii 1 6 15 0 c 341 8 eb ii/iii 2 6 15 5 c 321 186.9 1.25 53.2 1.30 9 eb ii/iii 3 6 15 10 c 313 136.7 0.84 52.9 0.88 10 eb ii/iii 4 6 15 15 e 50 282 113.9 0.75 0.80 52.5 0.79 11 eb ii/iii 5 6 15 20 e 50 235 100.0 0.78 0.83 52.1 0.83 12 eb ii/iii 6 6 15 25 e 55 192 90.4 0.86 0.91 51.6 0.92 13 eb iv 1 8 15 0 c 401 12 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes 14 eb iv 2 8 15 5 c 418 168.8 0.97 52.7 1.03 15 eb iv 3 8 15 10 e 45 394 123.5 0.68 0.75 52.3 0.73 16 eb iv 4 8 15 15 e 60 301 102.9 0.72 0.77 51.9 0.79 17 eb iv 5 8 15 20 e 50 245 90.4 0.78 0.86 51.5 0.85 18 eb iv 6 8 15 25 e 45 209 81.7 0.84 0.94 51.0 0.91 19 eb v 1 5 10 0 c 229 20 eb v 2 5 10 5 cm 212 115.2 0.97 43.5 1.00 21 eb v 3 5 10 10 e 40 197 84.2 0.68 0.71 42.8 0.71 22 eb v 4 5 10 15 e 43 175 70.2 0.62 0.64 42.0 0.64 23 eb v 5 5 10 20 e 47 153 61.6 0.62 0.63 41.1 0.64 24 eb v 6 5 10 25 e 50 129 55.7 0.66 0.67 40.2 0.69 25 eb vi 1 10 10 0 c 720 26 eb vi 2 10 10 5 em 575 90.2 0.39 24.0 0.48 27 eb vi 3 10 10 10 e 22.5 365 66.0 0.44 0.59 23.3 0.58 28 eb vi 4 10 10 15 e 22.5 313 54.9 0.45 0.60 22.6 0.60 29 eb vi 5 10 10 20 e 22.5 275 48.3 0.47 0.63 22.0 0.64 30 eb vi 6 10 10 25 e 22.5 220 43.6 0.57 0.75 21.5 0.76 31 eb vii 1 5 12 0 c 230 32 eb vii 2 5 12 5 c 225 147.4 1.23 50.1 1.26 33 eb vii 3 5 12 10 cm 212 107.8 0.85 49.7 0.88 34 eb vii 4 5 12 15 em 180 89.8 0.80 49.2 0.83 35 eb vii 5 5 12 20 e 25 170 78.9 0.73 0.83 48.7 0.76 36 eb vii 6 5 12 25 e 25 149 71.3 0.75 0.86 48.0 0.78 37 eb viii 1 3 3 0 e/em 79 38 eb viii 2 3 3 5 e 22.5 44 27.0 0.41 0.42 22.5 0.42 39 eb viii 3 3 3 10 e 22.5 37 19.8 0.35 0.34 22.0 0.35 40 eb viii 4 3 3 15 e 22.5 29 16.5 0.40 0.37 21.5 0.37 41 eb viii 5 3 3 20 e 22.5 23 14.5 0.47 0.41 21.1 0.42 42 eb viii 6 3 3 25 e 22.5 20 13.1 0.52 0.44 20.7 0.45 43 eb ix 1 3 6 0 c 95 44 eb ix 2 3 6 5 em 80 69.1 0.86 33.1 0.88 45 eb ix 3 3 6 10 e 40 69 50.5 0.66 0.67 32.2 0.67 46 eb ix 4 3 6 15 e 25 57 42.1 0.64 0.67 31.3 0.66 47 eb ix 5 3 6 20 e 25 47 37.0 0.68 0.71 30.4 0.69 48 eb ix 6 3 6 25 e 25 39 33.4 0.75 0.78 29.5 0.76 49 eb x 1 3 9 0 c 102 50 eb x 2 3 9 5 c 105 119.6 1.44 25.5 1.48 51 eb x 3 3 9 10 c 107 87.5 0.91 44.9 0.93 52 eb x 4 3 9 15 cm 90 72.9 0.86 44.2 0.88 53 eb x 5 3 9 20 e 50 85 64.0 0.78 0.79 43.4 0.80 54 eb x 6 3 9 25 e 50 70 57.9 0.85 0.86 42.5 0.87 55 eb xi 1 3 12 0 c 116 56 eb xi 2 3 12 5 c 113 176.5 2.42 53.0 2.44 57 eb xi 3 3 12 10 c 115 129.1 1.53 52.7 1.55 58 eb xi 4 3 12 15 c 110 107.5 1.26 52.4 1.28 59 eb xi 5 3 12 20 cm 105 94.5 1.13 52.1 1.15 60 eb xi 6 3 12 25 cm 115 85.4 0.92 51.7 0.94 61 eb xii 1 4 4 0 c/cm 120 62 eb xii 2 4 4 5 e 25 70 36.1 0.47 0.49 22.4 0.50 63 eb xii 3 4 4 10 e 25 50 26.4 0.48 0.48 21.9 0.50 64 eb xii 4 4 4 15 e 25 45 22.0 0.47 0.45 21.4 0.47 65 eb xii 5 4 4 20 e 25 40 19.3 0.49 0.45 21.0 0.48 66 eb xii 6 4 4 25 e 25 35 17.5 0.54 0.48 20.6 0.51 67 eb xiii 1 4 6 0 c 125 68 eb xiii 2 4 6 5 em 110 62.4 0.64 28.7 0.67 69 eb xiii 3 4 6 10 e 25 86 45.7 0.55 0.59 27.8 0.58 70 eb xiii 4 4 6 15 e 25 68 38.0 0.58 0.62 27.0 0.61 71 eb xiii 5 4 6 20 e 25 50 33.4 0.70 0.74 26.3 0.73 72 eb xiii 6 4 6 25 e 15 45 30.2 0.72 0.87 25.5 0.75 73 eb xiv 1 4 8 0 c 140 74 eb xiv 2 4 8 5 c 129 92.1 0.99 37.2 1.02 75 eb xiv 3 4 8 10 c 130 67.4 0.65 36.3 0.67 76 eb xiv 4 4 8 15 e 50 100 56.1 0.68 0.68 35.3 0.70 77 eb xiv 5 4 8 20 e 37.5 86 49.3 0.69 0.71 34.3 0.71 ann and hybrid modelling of eccentrically patch loaded steel i-girders 13 78 eb xiv 6 4 8 25 e 37.5 75 44.6 0.72 0.73 33.3 0.74 79 eb xv 1 4 10 0 c 155 80 eb xv 2 4 10 5 c 148 124.6 1.39 44.8 1.41 81 eb xv 3 4 10 10 c 140 91.1 0.95 44.2 0.97 82 eb xv 4 4 10 15 cm 138 75.9 0.77 43.4 0.79 83 eb xv 5 4 10 20 em 128 66.7 0.71 42.7 0.74 84 eb xv 6 4 10 25 em 115 60.3 0.72 41.8 0.74 85 eb xvi 1 5 6 0 c 187 86 eb xvi 2 5 6 5 e 25 130 57.7 0.56 0.60 27.2 0.59 87 eb xvi 3 5 6 10 e 25 105 42.2 0.48 0.52 26.5 0.51 88 eb xvi 4 5 6 15 e 25 74 35.2 0.57 0.62 25.7 0.61 89 eb xvi 5 5 6 20 e 25 59 30.9 0.66 0.69 25.0 0.69 90 eb xvi 6 5 6 25 e 25 55 27.9 0.66 0.69 24.3 0.69 91 eb xvii 1 5 8 0 c 209 92 eb xvii 2 5 8 5 cm 200 85.1 0.65 35.5 0.68 93 eb xvii 3 5 8 10 e 35 145 62.3 0.60 0.63 34.6 0.63 94 eb xvii 4 5 8 15 e 32 130 51.9 0.54 0.58 33.6 0.57 95 eb xvii 5 5 8 20 e 28 98 45.6 0.64 0.69 32.6 0.67 96 eb xvii 6 5 8 25 e 25 83 41.2 0.69 0.76 31.6 0.73 97 eb xviii 1 6 6 0 c/cm 208 98 eb xviii 2 6 6 5 e 25 170 54.1 0.44 0.49 23.3 0.50 99 eb xviii 3 6 6 10 e 20 130 39.6 0.46 0.58 16.8 0.63 100 eb xviii 4 6 6 15 e 15 104 33.0 0.53 0.74 16.7 0.70 101 eb xviii 5 6 6 20 e 88 29.0 0.60 16.6 0.78 102 eb xviii 6 6 6 25 e 69 26.2 0.75 16.6 0.95 103 eb xix 1 6 9 0 c/cm 330 104 eb xix 2 6 9 5 e 20 285 93.6 0.56 0.69 19.6 0.69 105 eb xix – 3 6 9 10 e 20 217 68.5 0.51 0.67 19.3 0.68 106 eb xix – 4 6 9 15 e 20 155 57.0 0.61 0.82 19.0 0.84 107 eb xix – 5 6 9 20 e 20 125 50.1 0.69 0.94 18.7 0.96 108 eb xix – 6 6 9 25 e 107 45.3 0.76 18.5 1.06 109 eb xx – 1 6 12 0 c 300 110 eb xx – 2 6 12 5 em 265 138.2 1.05 46.6 1.09 111 eb xx – 3 6 12 10 e 25 311 101.1 0.60 0.73 26.2 0.73 112 eb xx – 4 6 12 15 e 25 235 84.2 0.66 0.82 25.5 0.82 113 eb xx – 5 6 12 20 e 25 202 74.0 0.68 0.86 24.8 0.86 114 eb xx – 6 6 12 25 e 165 66.9 0.76 24.2 0.99 115 b i/ii 5 5 10 0 c 259 116 eb xxi – 2 5 10 5 c 250 115.2 1.50 43.5 1.54 117 eb xxi – 1 5 10 10 cm 240 84.2 0.95 42.8 0.99 118 b iii 1/1 5 10 15 e 75 202 70.2 0.80 0.79 40.7 0.84 119 b iii ½ 5 10 20 e 70 171 61.6 0.81 0.80 39.8 0.86 120 b iii 1/3 5 10 25 e 65 141 55.7 0.90 0.88 38.9 0.95 121 b iii ¼ 5 10 30 e 60 122 51.3 0.98 0.96 37.9 1.03 122 b i/ii 11 10 10 0 c 801 123 eb xxi – 4 10 10 5 e 22.5 790 90.2 0.51 0.65 24.0 0.64 124 eb xxi – 3 10 10 10 e 50 640 66.0 0.43 0.45 23.3 0.57 125 b iii 2/1 10 10 15 e 50 387 54.9 0.55 0.56 25.5 0.73 126 b iii 2/2 10 10 20 e 47.5 297 48.3 0.67 0.67 24.9 0.89 127 b iii 2/3 10 10 25 e 45 254 43.6 0.75 0.74 24.2 1.00 128 b iii 2/4 10 10 30 e 42.5 219 40.2 0.86 0.84 23.6 1.13 129 b i/ii 6 5 12 0 c 266 130 eb xxi – 6 5 12 5 c 255 147.4 1.97 50.1 2.02 131 eb xxi – 5 5 12 10 c 255 107.8 1.19 49.7 1.23 132 b iii 3/1 5 12 15 e 85 228 89.8 0.96 0.97 48.4 1.01 133 b iii 3/2 5 12 20 e 70 177 78.9 1.06 1.07 47.8 1.12 134 b iii 3/3 5 12 25 e 65 162 71.3 1.05 1.06 47.0 1.10 135 b iii ¾ 5 12 30 e 60 147 65.7 1.08 1.09 46.3 1.13 for all girders: a = hw = 700 mm, bf = 150 mm for samples nos. 1-114: c = 50 mm for samples nos. 115-135: c = 150 mm series b i, b ii, b iii – 1998 tests [26] series eb i eb iv – 2001 tests [27] series eb v eb xxi – 2007 tests [28] 14 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes table 3 summary of material properties (fyf, fyw) and ultimate loads determined by eqs. (1) and (2) and forecasted by ann models (pu,eqs(1-2), pu,ann), both related to pu,exp, for experimental database from 1998, 2001 and 2007 tests [26-28] no. series fyw fyf e [mm] [mpa] [mpa] 0 5 10 15 20 25 30 1-6 eb i 327.3 268.7 1.00 1.04 1.05 0.99 0.99 1.00 pu,eqs(1-2)/pu,exp 1.02 1.01 1.03 0.99 1.00 1.01 pu,ann/pu,exp 7-12 eb ii/iii 309.3 268.7 1.00 0.39 0.37 0.38 0.41 0.43 pu,eqs(1-2)/pu,exp 0.99 1.01 1.00 1.00 1.01 1.00 pu,ann/pu,exp 13-18 eb iv 262.4 268.7 1.00 0.84 0.77 0.87 0.93 0.96 pu,eqs(1-2)/pu,exp 1.23 1.00 0.92 1.01 1.00 0.93 pu,ann/pu,exp 19-24 eb v 270.9 309.7 1.00 0.96 0.90 0.89 0.89 0.93 pu,eqs(1-2)/pu,exp 1.01 1.00 0.99 1.01 1.01 1.04 pu,ann/pu,exp 25=30 eb vi 309.7 309.7 1.00 0.96 1.13 0.99 0.88 0.97 pu,eqs(1-2)/pu,exp 0.97 1.01 1.18 1.09 1.00 0.99 pu,ann/pu,exp 31-36 eb vii 270.9 288.8 1.00 0.96 0.93 0.99 0.94 0.93 pu,eqs(1-2)/pu,exp 0.99 0.98 1.03 1.09 0.98 0.97 pu,ann/pu,exp 37-42 eb viii 274.4 274.4 1.00 1.38 1.22 1.17 1.16 1.16 pu,eqs(1-2)/pu,exp 0.91 1.18 1.08 1.17 1.35 1.45 pu,ann/pu,exp 43-48 eb ix 274.4 287.3 1.00 1.05 1.07 1.14 1.21 1.28 pu,eqs(1-2)/pu,exp 1.11 1.10 1.04 1.00 1.04 1.05 pu,ann/pu,exp 49-54 eb x 274.4 282.0 1.00 0.97 0.93 1.04 0.99 1.03 pu,eqs(1-2)/pu,exp 1.08 0.99 0.91 0.98 0.89 0.94 pu,ann/pu,exp 55-60 eb xi 274.4 288.8 1.00 1.03 1.01 1.05 1.10 0.85 pu,eqs(1-2)/pu,exp 0.97 0.96 0.99 1.04 1.05 0.91 pu,ann/pu,exp 61-66 eb xii 285.0 285.0 1.00 1.31 1.37 1.14 1.01 1.01 pu,eqs(1-2)/pu,exp 0.92 1.14 1.20 1.02 0.95 0.97 pu,ann/pu,exp 67-72 eb xiii 285.0 287.3 1.00 0.94 0.98 1.02 1.18 1.16 pu,eqs(1-2)/pu,exp 1.06 0.97 0.99 0.97 1.08 0.98 pu,ann/pu,exp 73-78 eb xiv 285.0 300.3 1.00 0.96 0.84 0.95 0.97 0.98 pu,eqs(1-2)/pu,exp 1.07 1.05 0.96 1.10 1.07 1.07 pu,ann/pu,exp 79-84 eb xv 285.0 309.7 1.00 0.99 0.97 0.90 0.87 0.84 pu,eqs(1-2)/pu,exp 0.99 0.99 1.02 0.99 0.96 0.97 pu,ann/pu,exp 85-90 eb xvi 270.9 287.3 1.00 1.14 1.09 1.22 1.24 1.17 pu,eqs(1-2)/pu,exp 0.92 1.04 0.98 1.07 1.03 0.89 pu,ann/pu,exp 91-96 eb xvii 270.9 300.3 1.00 0.88 1.00 0.94 1.06 1.11 pu,eqs(1-2)/pu,exp 1.01 0.93 1.10 1.05 1.14 1.12 pu,ann/pu,exp 97-98 eb xviii 287.3 287.3 1.00 0.94 0.92 0.86 0.80 0.89 pu,eqs(1-2)/pu,exp 99-102 458.3 287.3 1.04 0.98 0.99 0.95 0.93 1.01 pu,ann/pu,exp 103-108 eb xix 458.3 282.0 1.00 0.96 1.03 1.19 1.24 1.28 pu,eqs(1-2)/pu,exp 1.00 0.91 0.94 1.02 1.00 0.97 pu,ann/pu,exp 109-110 eb xx 287.3 288.8 1.00 1.00 0.75 0.87 0.89 0.96 pu,eqs(1-2)/pu,exp 111-114 458.3 288.8 1.01 1.05 1.00 1.08 1.01 0.99 pu,ann/pu,exp 115, 118-121 b i/ii, iii 1 286.5 274.7 1.00 0.92 0.84 0.87 0.90 0.97 0.99 pu,eqs(1-2)/pu,exp 116-117 eb xxi 270.9 309.7 122, 125-128 b i/ii, iii 2 274.7 274.7 1.00 0.78 0.72 0.89 0.91 0.93 1.11 pu,eqs(1-2)/pu,exp 123-124 eb xxi 309.7 309.7 129, 132-135 b i/ii, iii 3 286.5 267.4 1.00 0.98 0.90 0.91 1.04 0.99 0.93 pu,eqs(1-2)/pu,exp 130-131 eb xxi 270.9 288.8 ann and hybrid modelling of eccentrically patch loaded steel i-girders 15 graphical presentation of data from tables 2 and 3 is given separately for girders with eccentric or mixed collapse mode (e, m – 92 samples, nos. 10-12, 15-18, 20-24, 26-30, 33-36, 38-42, 44-48, 52-54, 59, 60, 62-66, 68-72, 76-78, 82-84, 86-90, 92-96, 98-102, 104108, 110-114, 117-121, 123-128, 132-135, figs. 8a and 9a) and for girders centric collapse mode (c – 43 samples, nos. 1-9, 13, 14, 19, 25, 31, 32, 37, 43, 49-51, 55-58, 61, 67, 7375, 79-81, 85, 91, 97, 103, 109, 115, 116, 122, 129-131, figs. 8b and 9b), having in mind sense(lessness) and/or (in)appropriateness of certain methods implementation, depending on collapse mode type of eccentrically patch loaded girder. fig. 8 depicts α values: αexp, ᾶ and αann. it is obvious that ᾶ values are excessively overestimated for the majority of test samples having eccentric or mixed collapse mode (e, m), fig. 8a. on the other hand, αann values are mostly in range of αexp. this is valid for samples with load length c = 50 mm (i.e. samples nos. 1-114(117)). situation is different for samples with load length c = 150 mm (i.e. samples nos. 114(117)-135). a satisfying agreement of ᾶ and αexp may be noted in this zone, while certain discrepancy between αann and αexp occurs in these samples. an acceptable match between ᾶ and αexp in girders with c = 150 mm, fig. 8a, also stated by graciano and uribe-henao [30], is expected, having in mind that expression for ᾶ is fitted to these experimental data (1998 [26] tests data were used in the regression analysis that provided the expression for ᾶ [30]). however, strong disagreement of ᾶ values with experimental data for c = 50 mm (2001 [27] and 2007 [28] tests), indicates that conditions for application of eq. (3) should be revised. this mismatch is also expected and easily explainable. experimental samples from 1998 [26], 2001 [27] and 2007 [28] have same dimensions a = hw = 700 mm, bf = 150 mm, but different plate thicknesses tf and tw, as well as their ratios, which immensely influence girder behaviour, so that three experimental campaigns complement each other in constituting comprehensive database. considering this simultaneously with the fact of difference in load length, it is clear that the expression for ᾶ in eq. (3) may not be assumed as universal and should be applied only for girders with geometry in range of data used for the regression analysis. although parameter α from eq. (3) does not have (the same) realistic meaning in girders with centric collapse mode (c), i.e. eq. (3) is not intended for girders with centric collapse mode, illustration fig. 8b is kept herein, not only to display significant difference between ᾶ and αann values, but also to point out non-realistically high values of ᾶ. actually, several samples in fig. 8b have parameters e and tf /tw at the very border of domain suitable for use of eq. (3), according to graciano and uribe-henao [30]. majority of samples from fig. 8b confirm that eq. (3) does not provide reliable results for girders with e < 10 mm or tf /tw > 2.5, in addition to the fact that it should not be applied for these girders at all, because they have completely different collapse mechanism. consequently, it may be concluded that complete set of eq. (3) may provide valid results for input in the following range (i.e. range of experimental data reported by lučić [26], which were used for statistical data procession during eq. (3) derivation [30]): 1  tf /tw  2.4, 1/10  e/bf  1/5, 70  a/tw  140, 12.5  bf /tf  15, a/hw = 1, c/a = 0.214. furthermore, eq. (3), describing eccentric collapse mechanism, should be applied only in girders with eccentric collapse mode, regardless of the existence and level of load eccentricity, and this is, actually, the first precondition for application of eq. (3). these constraints are more demanding and more comprehensive than those set up by graciano and uribe-henao [30] (e/bf ≥ 1/15, in the context of available experimental data, and tf /tw ≤ 2.5), which are close to, but do not completely fit within newly proposed 16 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes conditions and do not cover all influential parameters. hence, newly proposed constraints (in the previous paragraph) will provide higher reliability of eq. (3) results. discrepancy between αann and αexp in samples nos. 114(117)-135, fig. 8a, is easily explained by the fact that ann for αann models were trained on experimental data in which c = 50 mm prevailed. hence, those models may not provide perfect forecast for input with c = 150 mm. (a) (b) fig. 8 α values in girders with: (a) eccentric or mixed collapse mode (e, m); (b) centric collapse mode (c) in fig. 9, pu values are compared: experimental values pu,exp, values pu,eq(3) obtained by original eq. (3), and values pu,eq(3)-exp and pu,eq(3)-ann obtained by refined eq. (3). refinement of eq. (3) is achieved by using αexp or αann instead of ᾶ. ann and hybrid modelling of eccentrically patch loaded steel i-girders 17 (a) (b) fig. 9 pu values in girders with: (a) eccentric or mixed collapse mode (e, m); (b) centric collapse mode (c) it is clearly noticeable that almost all values of ultimate load obtained by eq. (3), in its original as well as refined form, are immensely underestimated in girders with eccentric or mixed collapse mode (e, m), fig. 9a. calculated values (pu,eq(3), pu,eq(3)-exp and pu,eq(3)-ann) are in average around 71% of experimental values (pu,exp), while some of them are as low as only 34% pu,exp (table 4, middle segment). both percentages are too low and may not recommend application of eq. (3) in the whole range of available experimental data. if only samples with load length c = 50 mm are analysed (i.e. samples nos. 1-114(117), with e or m collapse mode), level of underestimation is slightly worse: calculated ultimate loads are in average around 68% pu,exp, while minimal calculated value remains 34% pu,exp. concerning only samples with load length c = 150 mm (i.e. samples nos. 114(117)-135, 18 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes with e or m collapse mode), both percentages of underestimation are more favourable, i.e. underestimation is less emphasized: calculated values of ultimate loads are in average around 86% pu,exp, while minimal calculated value is 48% pu,exp. certain differentiation of these percentages, for c = 150 mm, in comparison to those reported by graciano and uribehenao [30] are generally due to differences in yield stresses fyf and fyw (real values are used herein, while graciano and uribe-henao [30] used nominal values), but may also be due to the fact that more samples from this category are analysed herein (15 samples) than by graciano and uribe-henao [30] (12 samples). values pu,eq(3)-ann are closest to pu,exp. in average, they are 75% pu,exp for the whole dataset from fig. 9a (table 4, middle segment), 71% pu,exp for c = 50 mm and even 93% pu,exp for c = 150 mm. this is significantly better match than for pu,eq(3) values, obtained by original eq. (3), which are the most distant from pu,exp: in average 68% pu,exp for the whole dataset from fig. 9a (table 4, middle segment), 65% pu,exp for c = 50 mm and 82% pu,exp for c = 150 mm. the beneficial difference of even 11% (c = 150 mm) speaks in favour of eq. (3) refinement, achieved by its coupling with ann α modelling, fig. 10. 92 samples, with eccentric or mixed collapse mode, are presented in fig. 9a. only two of them have ratio tf /tw > 2.5 and other twelve have load eccentricity e < 10 mm. hence, even 85% (78/92) of samples satisfy preconditions for eq. (3) defined by graciano and uribe-henao [30]. still, results of eq. (3) are not appropriate for engineering practice. however, only 29% (27/92) of samples satisfy more realistic (and stricter) preconditions for eq. (3), suggested herein. this is the main reason for described unacceptable disagreement of original eq. (3) and experimental results. thorough analysis of data from table 2 reveals that exactly these 27 samples (15 samples with c = 50 mm, nos: 22-24, 2830, 34-36, 106-108, 112-114; 12 samples with c = 150 mm, nos: 118-121, 125-128, 132135, that were used for the derivation of eq. (3)) have the best match of calculated and experimental values of ultimate load, with average calculated value of 81% pu,exp (table 4, lower segment), i.e. for 10% better than for the whole dataset from fig. 9a (table 4, middle segment). again values pu,eq(3)-ann have smallest deviation from pu,exp. in average, they are 88% pu,exp (table 4, lower segment), i.e. for 13% better than for the whole dataset from fig. 9a (table 4, middle segment). statistics enhancement is well pictured in fig. 11. when samples with different load length are considered separately, results are also significantly better, for both categories. as assumed, samples with c = 150 mm have the best match of refined/hybrid calculated and experimental results: in average pu,eq(3)-ann = 98% pu,exp. obvious shortage of eq. (3), i.e. its expression for ᾶ, may be overcome by ann modelling of α. with αann values, instead of ᾶ, pu values are remarkably more precise. although calculated pu values (by any version of eq. (3)) might not be valid ultimate load of eccentrically loaded girders with centric collapse mode (no ultimate load reduction due to load eccentricity, therefore no need and no sense to calculate ultimate load by eq. (3) or any other procedure related to eccentric load mode), fig. 9b is displayed herein. it is interesting that, unlike the case of eccentric or mixed collapse mode from fig. 9a, these “calculated ultimate loads” in fig. 9b are mostly overestimated, significantly higher than pu,exp, which confirms senselessness of these data. again, even if it is not known that these samples have centric collapse mode, they are almost all out of validity domain of eq. (3), according to graciano and uribe-henao [30], so that use of eq. (3) would be inadequate. however, it should be pointed out that fig. 9b has neither practical purpose nor meaning, primarily because these samples have centric collapse mode, which is not described by eq. (3), regardless of the boundaries for load eccentricity and girder geometry. ann and hybrid modelling of eccentrically patch loaded steel i-girders 19 3.3 comparison of methods for ultimate load determination based on summaries from tables 2 and 3, general comparison of following methods and procedures for ultimate load determination has been presented: empirical procedure, defined by eqs. (1) and (2) pu,eqs(1-2); mechanical model, mathematically defined by eq. (3) in its original form pu,eq(3); mechanical model coupled with experimental data αexp, i.e. experimentally refined eq. (3) pu,eq(3)-exp; mechanical model coupled with ann modelling data αann, i.e. ann refined eq. (3) (hybrid model) pu,eq(3)-ann; and ann forecast model pu,ann. for the sake of comparison simplicity, all calculated (by eqs. (1-3)) and forecasted (by ann model) values of ultimate load (pu,calc) are related to respective experimental collapse loads (pu,exp). statistical parameters (max, min and mean value, standard deviation and coefficient of variation) for ratios pu,calc/pu,exp are determined and summarised in table 4. statistics has been done initially for the complete database (135 samples – upper segment of table 4), then only for samples with eccentric or mixed collapse mode (92 samples – middle segment of table 4), and finally only for samples with eccentric or mixed collapse mode, satisfying preconditions for applying eq. (3), as suggested in chapter 3.2 (27 samples – lower segment of table 4). graphical presentation is enclosed for later two cases, figs. 10 and 11. table 4 statistics parameters for five procedures of ultimate load determination pu,eqs(1-2) pu,exp pu,eq(3) pu,exp pu,eq(3)-exp pu,exp pu,eq(3)-ann pu,exp pu,ann pu,exp all girders (135 samples) max 1.38# 3.10 ca lc . o n ly f o r g ir d er s w it h ec c. a n d m ix ed co ll ap se m o d e 3.14 1.45 min 0.37 0.35 0.35 0.89 mean value 0.98 0.82 0.88 1.02 stand. dev. (s) 0.16 0.40 0.40 0.08* coeff. of var. (v) 0.17 0.49 0.45 0.08* eccentric or mixed collapse mode – e, m (92 samples) max 1.38# 1.13 1.09 1.15 1.45 min 0.38 0.35 0.34 0.35 0.89 mean value 0.99 0.68 0.70 0.75 1.03 stand. dev. (s) 0.18 0.17 0.17 0.19 0.09* coeff. of var. (v) 0.18 0.25 0.24 0.25 0.09* validity domain for eq. (3), chapter 3.2 (27 samples) max 1.28 1.08 1.09 1.13 1.09 min 0.87 0.45 0.56 0.60 0.97 mean value 0.97 0.75 0.81 0.88 1.02 stand. dev. (s) 0.11 0.17 0.15 0.16 0.04 coeff. of var. (v) 0.11 0.22 0.18 0.18 0.04 although displayed herein, data from dark grey fields in table 4 should be exempted from further discussion, since they include ultimate loads obtained by eq. (3) for girders for which this method is not intended – girders with centric collapse mode. mean values are above 0.80 (much better than in the middle segment of table 4), but these are fake information, as a consequence of improper use of eq. (3), and it should be recognised. 20 b. šćepanović, s. rutešić, o. mijušković, l.m. gil-martín, e. hernández-montes fig. 10 ratios of calculated and experimental ultimate load (pu,calc/pu,exp) for girders with eccentric or mixed collapse mode (e, m) – set of 92 samples fig. 11 ratios of calculated and experimental ultimate load (pu,calc/pu,exp) in domain of eq. (3) validity – set of 27 samples ann and hybrid modelling of eccentrically patch loaded steel i-girders 21 statistical parameters of ultimate load obtained by empirical procedure (pu,eqs(1-2)) or forecasted by ann models (pu,ann) do not depend on whether girders with centric collapse mode are included in sample set or not – statistical values are same for set of 135 samples, having various collapse modes (upper segment of table 4), and for set of 92 samples with eccentric or mixed collapse mode only (middle segment of table 4). this is due to the fact that both methods are derived from the complete experimental database (135 samples), by statistical “best fitting” techniques. mean values of both methods are very close to 1, i.e. almost perfect (bolded numbers in table 4; fig. 10). somewhat lower standard deviation and coefficient of variation may bring certain advantage to ann modelling (* marked numbers in table 4), judging only by statistics. in reality, the fact that empirical procedure approximates experimental values from the safe side (fig. 10), also having lower max unsafe discrepancy (# marked numbers in table 4), stands in favour of this method. regarding the variants of eq. (3), its underestimation of ultimate load (italic underlined numbers in table 4) is not tolerable in engineering practice, despite its evident refinement by coupling with ann modelling (bold italic underlined number in table 4). this method is on the safe side (figs. 10 and 11), but not simultaneously economical. statistical parameters for eq. (3) variants are notably enhanced when it is applied in a real domain of its validity (light grey fields, lower segment of table 4; fig. 11). however, even in this narrow domain, i.e. on dataset of only 27 samples, other methods (empirical procedure with reduction coefficient and ann modelling) provide better statistics and are more favourable, which implies necessity of further revision of eq. (3), considering complete existing experimental database, exactly as graciano and uribe-henao [30] concluded. 4. conclusion ann modelling proved to be valuable in the analysis of eccentrically patch loaded steel i-girders. well-structured and properly trained artificial neural networks provide reliable forecast models for different needs. they may be used as a self-contained tool for resolving both key-issues: identification of collapse mode type and determination of collapse load. in addition, ann forecasting may be successfully combined with other methods, in order to complement and/or to refine them, by providing confident determination of certain parameters needed for the other method. such procedure is denoted as hybrid modelling. regarding collapse mode identification, whether used standalone or complemented with empirical criteria, ann forecast models have very good results. as concerns ultimate load determination, pure ann modelling provides nearly perfect results, regardless of collapse mode type, which positions this method in top-two procedures for ultimate load. this is of particular importance when girder geometry enables any (or more) collapse mode(s). applying ann modelling within mathematical model, as its support to determine specific influential parameter necessary for ultimate load calculation, is highly beneficial, fixes insufficiencies and significantly contributes to the enhancement mechanical model. although coupling with ann modelling considerably improved mechanical model for ultimate load and new, more demanding constraints of its validity are proposed, further revision of mechanical model should be done, based on all available experimental data. having in mind that options for improvement of ann models could hardly be exhausted, future investigation will reveal new advantages and benefits, as well as new fields of use of this ai technique in studying eccentric patch loading in steel i-girders. 22 b. šćepanović, s. rutešić, o. mijušković, 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niš, serbia abstract. although business process management systems (bpm) have been used over the years, their performance in unpredicted situations has not been adequately solved. in these cases, it is common to request user assistance or invoke predefined procedures. in this paper, we propose using the active semantic model (asm) to detect and handle exceptions. this is a specifically developed semantic network model for modeling of semantic features of the business processes. asm is capable of classifying new situations based on their similarities with existing ones. within bpm systems this is then used to classify new situations as exceptions and to handle the exceptions by changing the process based on asm’s previous experience. this enables automatic detection and handling of exceptions which significantly improves the performance of bpm systems. key words: business process management systems, exception detection, exception handling, active semantic model, analogy-based reasoning 1. introduction business process management systems (bpms) are software systems which manage business processes. so far, these systems have been proven useful for management of the processes with a solid structure, in which changes do not occur often. on the other hand, bpms are also used in environments in which there is a constant need for deviation from predefined process (e.g., logistics, healthcare). in the terminology used for bpms, deviations from the predefined model are called exceptions. exceptions can be anticipated in which case the issue is handled by incorporating it in the model at the process modeling time. this approach leads to the creation of very received: november 15, 2021 / accepted may 17, 2022 corresponding author: dragan misic faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18 000 niš, serbia e-mail: dragan.misic@masfak.ni.ac.rs 2 d. misic, m. stojkovic, m. trifunovic., n. vitkovic complex models with many branches describing alternative pathways in the case an exception occurs. nowadays, this approach is almost completely abandoned. unanticipated exceptions are handled at the process execution time. in that case, it is necessary to have a way of changing the business process and adjusting it to the new circumstances. this approach is used in most modern bpms [1]. exception handling in bpm systems is of great importance for the successful process execution. research shows that the management of process exceptions requires a lot of resources, it costs a lot [2], its success is critical [3] and it is time-consuming. one of the possible ways of making a system capable of recognizing and categorizing the unknown situation as an exception is by enabling recognition of similarities and differences between the semantic features of an unpredicted situation and the known ones previously semantically interpreted and categorized as an exception. that is exactly how the active semantic model (asm) functions, and we use it in this paper for detection and handling of the exceptions. in the early stage of its use, this system is able to assist people to solve problems. with time, it collects knowledge and it becomes capable of offering intelligent advice and proposing solutions independently. the main contribution of our work is the use of asm for presenting the knowledge and making conclusions based on that knowledge. asm may independently offer a solution to the problem that occurred and recommend the process adaptation accordingly. in that way we reduce the need for direct human involvement in handling exceptions, because after the training, asm is capable of handling challenging situations independently. asm is more flexible than other methods in artificial intelligence such as case-based reasoning or ontologies, because asm is able to reuse the events from other domains to make decisions in a new domain. additionally, the formalisms do not need to be defined in advance, unlike with the ontologies. this paper represents a sequel to our work on this subject which is presented in [4] and [5]. in [4], we describe how we expanded the process model defined in the xpdl (xml process definition language) language by adding constructions which refer to assignment of resource to activities. in [5] we describe how asm is used for detecting exceptions. in this paper, we show that asm is now able to solve problems (it offers a solution to a certain situation), and this we applied to the problems which occur due to inadequate resources. its ability to reach conclusions is also improved since we have significantly improved the algorithm based on which asm recognizes the topological similarity it uses for handling exceptions. asm is now capable of making meaningful judgments, conclusions and decisions in new and unexpected situations. compared to other approaches to semantic modeling (e.g. ontologies), asm has an autonomous, flexible and significantly more analytical mechanism of semantic categorization of data. the rest of this paper is organized as follows: in section 2 we present some of the relevant papers in this area. in section 3 we show the asm structure and its reasoning methods. section 4 explains how asm is connected to bpms and how it can assist with detection and handling of exceptions. the results of the asm conclusions are explained in detail in section 5. application of asm in the bpm systems is discussed in section 6. the paper is concluded in section 6. detection and handling exceptions in business process management systems using active semantic ... 3 2. related work as mentioned earlier, the exceptions during the execution of business processes are frequent in practice. handling these exceptions is therefore significant for organizations employing bpms systems. for instance, in [6] the authors analyzed the relations between the occurrence of exceptions and operational performance. their research indicates that the exceptions lead to poorer operational performance: the processes where the exceptions occur take longer to complete than the processes with no exceptions, underlying the importance of the bpms systems that can adapt to changes. in [7], the authors performed the analysis of existing process management systems with respect to their support for flexible, emergent and collaborative processes. they conclude that the contemporary systems do not adequately support these three process characteristics. in [8], the authors work on supporting users at the inflection point. the inflection point is a place in the process execution where an unforeseeable eventuality arises. at that moment, the mechanism that gives recommendations about adaptation to new circumstances is launched. this is done based on the search of the existing workflow specifications, which are located in the repository. examination of the previous cases is also used by the authors who apply the casebased reasoning [9] techniques in order to identify and solve exceptions. for example, in [10], authors use so-called adaptation cases. the adaptation cases describe situationspecific adaptation traces, which can be transferred to another, similar situation and replayed there. based on defined adaptation cases, it is possible to apply the adaptation which was used earlier to a new situation, too. in [11], the authors suggest using the conversation cbr techniques [12] to update the process. the authors extend the basic cbr mechanism with automatic question creation technique that leads the user through describing the new process. the questions are created based on the analysis of existing processes. the model used in the adept system [13] uses an ad-hoc approach. the processes are described via specific language in which there are operators for dynamic insertion, deletion or transfer of activities during the process execution. the disadvantage is primarily the fact that there are no algorithms that will automatically determine the circumstances under which it will apply a specific workflow adaptation. in the paper [14], the authors state that the existing mechanism for handling exceptions in business process execution language is not completely satisfactory. the main disadvantage lies in the fact that the behavior of the system in the cases when the exception occurs must be defined in design time. in [15] and [16], the authors describe the application of ontologies to enhance the flexibility of bpm systems. they allow for defining ad-hoc activities followed by the decision which process to run based on ontologies [15]. in [16], the ontologies help define advice for users when creating new processes. the processes are offered to the user only if they are in accordance with the rules which are defined within the ontologies. in agile bpm [1], the reactive rule model is utilized to recognize exceptional circumstances automatically and to determine the necessary process instance flow adaptations. for this purpose, failures trigger new obligations, which are the principal 4 d. misic, m. stojkovic, m. trifunovic., n. vitkovic motivators for agents to act. based on obligations, agents can dynamically replace/re-plan the failed goal, trigger a repair action, or abort/roll-back the execution. for process monitoring, detection of unanticipated exceptions and automated resolution, the cognitive process management system can be used [17]. this system relies on the technologies from the field of knowledge representation and reasoning. for modeling the primary domain where the processes are run, situation calculus is used; for structure specification and control flow of the process, it uses the indigolog (agent programming language) while for process adaptation it uses automatic planning. as can be seen, there are many different approaches to handling exceptions in bpms. some of these approaches only help people with solving problems. others attempt to offer a certain level of automation, i.e. the use of previously defined solutions. our opinion is that the level of automation in handling exceptions may be raised if the knowledge about the problem and the process is presented in a way computer can easily use it. as mentioned before, asm may offer a solution to the problem that occurred and recommend the process adaptation accordingly. this is not the case with the approaches which represent the tools that help humans with the system adaptation [13]. asm is an analogy-based reasoning technique (abr), and so are the aforementioned case-based reasoning techniques [10, 11]. in comparison with these techniques, asm offers better solutions because it is not limited exclusively to the solutions which come from the same domain. when compared with the approaches that achieve flexibility by using the ontologies [15,16], it can be said that our system overcomes some of the main problems which exist in ontologies, such as that the semantic reasoners designed to work with dl-founded ontologies showed themselves weak in making relevant entailments beyond the predefined and embedded logical formalism of deduction. similar is also true for reasoning flexibility – ability to make relevant, but quite different entailments about the same concept for semantically distant or different contexts (vocabularies) with a single set of logical inference rules and axioms on disposal. finally, having analytical ability to autonomously dissolve a portion of knowledge about one concept or group of concepts from one context and apply it to a quite different (semantically distant) concept or a group of concepts that are inherent to equally different context is something which appears not as a strong side of the richly axiomatized ontologies which rely on first-principles reasoning approach. 3. active semantic model asm is a specific model of the semantic network that was originally developed inhouse and is described in more detail in previous papers [18, 19, 20]. the specificity of this model originates from its feature that the meaning (semantics) of a certain concept (which is usually represented as a node in a semantic network) is defined not in its attributes, but in the attributes of its relations to other concepts (the term association is favored instead of relation since this structure helps the asm algorithm to associate or point out correspondent inference i.e., semantic categorization). every single association is featured by eleven attributes where two of them are tags (names) of concepts this relation associates: cpti, cptj. since these attributes – tags of concepts, can exist in more detection and handling exceptions in business process management systems using active semantic ... 5 than one relation, they are a kind of junction of these associations, and in this way, they can be considered as virtual nodes of semantic network. beside two different concepts, every single association is defined by additional three groups of attributes: topological (roles (ri, rj), type (t), direction (d), character (c)), weight (accuracy (h), significance (s)) and affiliation (context id, instructor id or user id). this kind of associations' structure enables application of an original algorithm for efficient recognition of similarities between network's sub-graphs or network fragments. (the term plexus of associations is preferred instead of associations’ sub-graph due to its feature to connect the concepts from different contexts, hence, not just in one layer, i.e., not just in a graph-plane). this algorithm drives analogy-based reasoning process in the core of the model’s inferring engine. the ability to determine the type and degree of similarity between sub-graphs of the network makes the inferring engine extraordinary autonomous, flexible and analytical in data semantics interpretation. these features are especially important in the cases of unpredicted inputs and small or incomplete networks [18, 19, 20]. 3.1 communication between the user and asm the most usual case of communication between the user and asm is being performed by entering the new concepts into the semantic network. this is performed by forming the new associations that include the new concept. by associating the new concept with other concepts that exist already in the asm semantic network, the inferring engine of asm is being triggered immediately. this results in proposing (creating) additional new associations between the new concept and other concepts and contexts in the network by asm itself autonomously. each new association is an elementary piece of knowledge that enables further semantic categorization of a new concept. the user can correct the attributes of the associations that are proposed by asm or remove the proposed associations; thus the user corrects its semantic categorization i.e., the way how asm infers. in addition, by correcting it, the user keeps improving asm for future autonomous analogy-based reasoning. hence, while associating a new concept into its network, asm enlarges its semantic network gaining a new piece of knowledge in addition to improving, at the same time, the algorithms for analogy-based reasoning. also, by proposing the new associations autonomously by employing abr, asm provides new semantic categorizations of a new concept or a new context, which are actually a kind of intelligent responses that the user expects from asm. so, asm always works in both regimes acquiring the knowledge and providing the intelligent inferences at the same time. fig. 1 shows an example of how asm learns and infers simultaneously. after the user forms a few new associations that connect one or several new concepts with the rest of asm’s semantic network, i.e., to the other several concepts, which exist already in the asm’s semantic network, building an input association plexus plxx in this way, asm, firstly, scans the network looking for a set of association plexuses {plxn} which are topologically analogous to the input association plexus plxx in which the new concept cpt1 appears: plxx ≍ (plxn) (fig. 1). actually, asm recognizes fragments (subgraphs or plexuses of associations) of more complex structures that exist in the semantic network of asm (e.g., plxnfrg(ctxn)) which are topologically analogous to the input plexus. once it recognizes the topological analogy between the input association plexus plxx (which is new to asm) and existing association plexuses {plxn}, a 6 d. misic, m. stojkovic, m. trifunovic., n. vitkovic procedure for upgrading the input association plexus plxx is triggered and performed according to the model of the existing association plexuses {plxn} that are topologically analogous to plxx. the upgrading of plxx is performed by creating new associations between the “known” concepts that exist in asm’s semantic network, and “unknown” concepts (that are included in plxx). these new associations are being created by asm autonomously. for example, asm reacts by proposing creation of a new association ax 1,4 between concepts cpt1 and cpt4, emulating association an 11,14 (between concepts cpt11 and cpt14) from topologically analogous plexus plxn which is a fragment of context ctxn. the new association will have the same topological parameters as association an 11,14. in that way asm categorizes the new concepts semantically, in other words, forms their meaning in the new (current) context; these new associations are practically the resulting conclusions about them. various algorithms which asm uses for reasoning are explained in more details in [18, 19, 20]. cpt1 cpt3 cpt2 cpt16 аx 1,3 аn 11,13 аx 1,2 аn 11,12 plxx ≍ plxn r1 r2 r3 r4 r1 r2 r3 r4 plxx ctxn plxn аn 12,14 аn 14,15 cpt15 cpt11 аn 11,14 аn 11,15 аn 13,16 cpt14 cpt4 cpt5 cpt6 ctxx cpt12 cpt13 fig. 1 topologically analogous association plexuses: plxx ≍ plxn 4. connecting asm and bpm systems the process model consists of activities which are executed in a specific order. this model is later used for the creation of specific instances, which correspond to the real processes. for process management we used the md system, developed at the faculty of mechanical engineering in niš [4]. it is based on the enhydra shark engine [21]. for the model definition and process execution in the enhydra shark system, the xpdl is used. xpdl is a standard xml based format defined by the workflow management coalition (wfmc). its aim is to enable the exchange of process definitions between various tools used to create processes. enhydra shark uses xpdl not only for data exchange with other systems, but also as the main way of representing processes within the system. enhydra shark is not able to handle process’ exceptions and respond appropriately in the situations when the process deviates from the model that is predefined. such situations detection and handling exceptions in business process management systems using active semantic ... 7 are resolved by asm and the expert system [5]. solving problems can lead to changing individual activities, but also to the changing of the entire process and its definition. illustration of the connection between the md system and asm is shown in fig. 2. in the first version of the md system for handling exceptions, an expert system was used with the expert shell jess. the core of the md system was written in java that matched the expert shell jess and made it relatively simple to connect the expert system with the process management system. the process in the md system is described by one definition and multiple instances made based on that definition. the task of the expert system is to help with detecting exceptions that may arise during the process execution as well as to propose a solution. the solution often consists of a proposed change to the process. the changes can refer only to the current process instance, but also to all other instances created from the same process definition. this is defined within the rules of the expert system that update the process upon their execution. the rules are defined for specific processes. if a new process should be monitored, then it is necessary to define suitable rules that will handle the exceptions that may arise in the new process. the procedure that performs this task is defined by a set of functions written in java and jess script language that are later invoked from the action part of the rule [4]. bpm system (md) asm domain asm knowledge base expert system (exception handling) expert rules fig. 2 md system with asm exception detection and handling using an expert system are limited by the rules that are defined in the system. in an attempt to overcome this limitation, the md system is connected to asm that is capable of making conclusions based on analogies. this connection improves the quality of the md system by significantly enhancing the system’s capabilities for automatic detection and handling of exceptions. without asm, it was possible to detect exceptions from signaling that a resource is missing based on the values of control parameters and whether the execution time was over the time limit of a certain activity. asm enables the system to consider the big picture and connect the situations that were previously labeled as exceptions and took places in completely different processes, with the current situations. asm is also able to offer a solution based on the analogy with some of the previous situations. the application of asm for detecting and handling exceptions will be explained in more detail using the orthopedic implant design and manufacturing process as an example. the outputs of this process are the orthopedic implants adapted for the patient. the process is managed by the previously mentioned md system. 8 d. misic, m. stojkovic, m. trifunovic., n. vitkovic the preparation of an orthopedic implant includes the preparation of osteofixational material comprised of the scaffold and the fixator. the scaffold is a piece of the bone implant assembly (entitled ossification material in figures), whose main functions are to substitute the missing part of the bone tissue and to hold the bone graft inside the volume of the scaffold during the tissue recovery process. this allows the communication with neighboring tissues. fixator is another piece of the bone implant assembly, which should fix (fasten) traumatized parts of the bone into regular anatomical position and transfer a part of the load that bone bears while organism is trying to heal the bone, that is, generate a missing piece of the tissue. the proto-tissue or bone graft (entitled “bone part” in figures) is the third piece of the compound bone implant assembly that usually consists of fat tissue, stem and/or progenitor cells and other soft and liquid substances. in order to enable asm to perform the process analysis, it is necessary to semantically describe the process and its elements. that means that it is necessary to present all the essential elements of the process by using the structures from asm. the semantic description of the process elements is done by the system administrator at the time of initiation of the first process instances, and it is based on the process definition. defining of concepts and associations between them is done by using a graphical editor. initially, the concepts which represent data from xpdl process definition are displayed in the editor. it is up to the administrator to accurately describe the process, by which he improves the quality of later asm’s conclusions. that means the administrator has a role of asm’s instructor. an example of an asm context for the process which manages the process of preparing and manufacturing the osteofixation material (om) is shown in fig. 3. om manufacturing ostefixation material manufacturing fixator bone part compound activity object focal concept type subtype type scaffold type fig. 3 asm context for osteofixation material preparing and manufacturing each process activity is represented by a particular plexus of associations in the asm. these plexuses also contain descriptions of the data required for the execution of these activities as well as the data made while executing the activity. the data which describes the activity is both defined by the administrator and taken from xpdl process definition. there is also information about the resources that are required for the normal execution of the activities. most importantly, this may include the material resources because people who perform these activities are represented by separate xpdl elements (participant element). an example of defining the activity within asm (scaffold modeling from fig. 6) is shown in fig. 4. detection and handling exceptions in business process management systems using active semantic ... 9 design process quality om modeling and manufacturing subject part assembly scafold model product concept attribute activity object activity similar concept activity object subject subject 3d bone model object activity similar concept associate cause consequence similar concept short deadline scaffold modeling helps out expert 1 similar concept activity number of revisions attribute concept ct image associate help engagement assembly part part fig. 4 association plexus (context) for the model of the activity scaffold modeling it should be mentioned that the first step is to give the description of the activity model to asm. the contexts of specific activity instances represent subtypes of the initial context. the association between these contexts is shown in fig. 5. ctx: scaffold modeling ctxn: instance of scaffold modeling type subtype fig. 5 connection between the context of activity model (ctx) and the context of specific instances (ctxn) if an exception occurs during the process execution, the first issue is to recognize the new situation as an exception. as we mentioned before, activities and their execution environment are described by the semantic network which also contains data that the activity uses. the new situation is usually manifested through the data that characterize a process. what usually happens in such situations is that certain new data occur or that the existing data receive some special values. in the system’s learning phase, the instructor should characterize the new situation as an exception and offer a solution to it by introducing the association between the concept exception and that solution. in the application phase, asm should recognize an analogy between the new situation and what it has learned, and independently propose the qualification of a new situation as an exception at first besides offering the solution. 10 d. misic, m. stojkovic, m. trifunovic., n. vitkovic 5. process of implants design and manufacturing as an example of the asm based reasoning we will show how asm draws conclusions on the example of managing processes that may occur in the same company. let us suppose that there is a process of designing and manufacturing osteofixation material (scaffold and fixator) which is adapted to the patient. the process begins in a hospital, when a patient comes to the doctor with a fracture which is to be treated. the first thing the doctor should do is to define the type of the treatment which the patient will undergo. he makes that decision based on radiology image. if there are no parts of bone missing, a fixator will be set and it will allow the bone to heal properly. if a small part of bone is missing, it is needed to design and manufacture a scaffold, which is filled with cellular material that will allow the missing bone part to regenerate. it may happen that it is needed to set a fixator, in addition to the scaffold. the third possibility occurs when a large part of bone is missing; in this case it is needed to make a fixator as well as the missing bone part. after making the decision about the treatment, the process is continued, part in the hospital, part in the company which manufactures osteofixation material. if the patient needs a scaffold, the first step is to create a parametric model of the bone and the scaffold from the parameters determined by the doctor. using that model, the manufacturer creates the scaffold and designs and constructs the fixator. the scaffold and the fixator are then sent to the hospital where the surgeon will use them for the operation. the process diagram defined inside the md system is shown in fig. 6. 5.1 training an asm the activity of this process which we are interested in is scaffold modeling. the scaffold is geometrically very complex, so its modeling is not simple (it is a kind of armature needed for the bone recovery of a specific patient). the proper modeling of the scaffold requires time as well as the extensive experience of the engineers. in the operation of the scaffold modeling the engineer creates so-called scaffold struts connecting nodes one by one and puts them on the surface. this structure differs for each patient. cross-section, intersection angle and density of the scaffold struts may change depending on desired mechanical strength of the scaffold. the process consists of iterative sequences. the accelerated work of the engineers may easily be the cause of the relocation of the connecting points of the scaffold struts or some other mistakes while modeling, which leads to problems with the structure of the scaffold modeled in such manner. this activity is followed by control activity (activity model control from fig. 6). in the case that there is something wrong with the design, the model is returned for revision. if the number of revisions is excessively increased (e.g. more than five), we conclude this is a sign that there is something wrong with the modeling, and that certain steps must be taken. the reaction to such situation is anticipated and embedded in the system (as an ifthen procedure, which is a part of the process). also, there are defined deadlines for scaffold modeling, which depend on the patient’s injury and the urgency of the surgery. detection and handling exceptions in business process management systems using active semantic ... 11 fig. 6 process for ostefixation material preparing and manufacturing (process diagram created within the md system) 12 d. misic, m. stojkovic, m. trifunovic., n. vitkovic occasionally it happens that the deadline for manufacturing of osteofixation material is very short. therefore, the deadline for scaffold modeling is also very short. asm will be notified of that by adding a new concept – short deadline (fig. 4). short deadline will make an engineer hurry up with the model design, which may cause an increased number of mistakes. in order to preserve the quality of the model and prevent the bottleneck from occurring in this activity, the first predefined reaction is to engage an additional expert. asm association plexus for this activity at model level is modeled as shown in fig. 4. however, it sometimes happens that the deadline is missed, despite the engagement of the additional expert. this is the case when the number of revisions stayed below the specified limit (e.g. 5), so the embedded procedure for the case of an excessively increased number of revisions was not launched. asm is notified of this by adding the concept small number of revisions which is an attribute of the concept number of revisions. this refers to a specific activity instance (fig. 5). missing a deadline is an exception for bpms. in cooperation with the engineers involved in the process, the system administrator is documenting that a short deadline may cause the operation to fail, despite the engagement of an additional expert. in such cases, the number of revisions stayed small. therefore, the situation which is characterized both by a small number of revisions and short deadline may lead to missing the deadline. there are two ways of solving this problem. the first one is to embed an if-then procedure in bpms, and that procedure would be executed in the case when such (numerically expressed) short deadline and the number of model revisions occur. it should be noted that this procedure can be applied only if the same situation occurs again in the same context. such formalized knowledge, however, is impossible to apply to a case from a different domain. it is also impossible to apply it to a case from the same domain if the conditional parts do not completely match. in order to enable the acquired experience to be applied to other domains, the administrator can provide asm with new information. thus, new associations are manually added to asm by the administrator (fig. 7). these associations are added to the context which refers to a specific instance of the activity scaffold modeling (ctxn). the first association connects the concept short deadline with the concept unsuccessful operation and, therefore, defines it as a cause of operation failure. the second cause of an unsuccessful operation is a small number of revisions. this is represented by the association between the concept small number of revisions and the concept unsuccessful operation. based on experience and conversations with the engineers, the system administrator reached the decision to set the accuracy of the first association to 50%. by doing this, he wanted to highlight that there is a 50% probability that a short deadline will cause the operation to fail. it is also estimated that in the given context, this association is very significant, so the association significance is set to 75% or 100%. the same parameters are set for the associations which are related to the concept small number of revisions and unsuccessful operation. detection and handling exceptions in business process management systems using active semantic ... 13 design process quality om modeling and manufacturing subject part assembly concept attribute scafold model product concept attribute activity object activity similar concept activity object subject subject 3d bone model object activity similar concept associate cause consequence similar concept short deadline scaffold modeling helps out expert 1 similar concept activity number of revisions attribute concept ct image unsuccessful operation small number of revisions subtype type associate help engagement assembl y part part cause consequence consequence cause fig. 7 association plexus (context) for the activity scaffold modeling with added associations (at instance level) the altered context of the activity scaffold modeling represents an exception. asm is notified of that by adding a new association between the context and the concept exception (fig. 8). scaffold modeling activity plexus in fig. 8 represents the whole context shown in fig. 7 (it refers to a specific activity instance – ctxn). ctxn: scaffold modeling activity plexus exception attribute concept fig. 8 categorizing the situation as an exception 14 d. misic, m. stojkovic, m. trifunovic., n. vitkovic in addition to documenting the new situation by adding new associations, the system administrator, in cooperation with the engineers, has considered the ways of overcoming such situations in the future. for such cases, it could be useful to consider the application of a specific designing method characterized by applying so-called udfs (user defined features). that approach, which involves usage of partially pre-defined geometric forms, can accelerate the process of modeling, and simultaneously decrease the number of model revisions. in the case of bone scaffold design, udfs could be pre-defined forms of scaffold’s struts and connecting nodes. this conclusion leads to the decomposition of the scaffold modeling activity into two activities. during the first activity, the udfs would be prepared, and in the second activity the scaffold would be designed using the prepared elements (udfs). the second activity is now performed much faster because it is needed only to define positioning references and dimension parameters for each udf. using of udfs for designing complex geometric forms could be semantically interpreted as a subtype of some more general activity, which can be e.g. entitled as sequential job decomposition. this relation should also be taken into account that is embedded into the asm network. asm is notified of the conclusions made by the administrator and the engineers by adding the new associations to the system, which connect the concept exception with the concept alternate, which is further connected with the concept udf based scaffold modeling which is a subtype of the sequential job decomposition (fig. 9). the accuracy of the association which indicates a possible solution is 50%. this describes the fact that the sequential job decomposition is not the only possible solution. the offered solution may be permanently applied to this process, so the asm administrator will teach asm by associating this solution with the general context of the scaffold modeling activity – ctx (fig. 9), thus signalizing that the process should be permanently changed. 5.2 semantic categorization of a new process in some new situations that are more or less similar to the previous ones, asm can now apply what it has learned from these previous situations. the process of recognizing an unpredicted exception and its categorization is performed by comparing the similarities of the context describing the current situation (activity) with the already existing contexts. in this case, the contexts of new activities are compared to the context of the scaffold modeling activity. the comparison of the context similarity according to the content is based on the similarity of plexus topology (a kind of subgraph isomorphism) [4, 18, 19]. in accordance with the topological similarity (difference) which it recognizes between these contexts, asm will semantically categorize the new situation in regard to the existing situations. if a new situation is similar to the situation categorized as an exception, then it is suggested that the new situation should also be categorized as an exception, with the calculated/assessed magnitude of the assertion accuracy. detection and handling exceptions in business process management systems using active semantic ... 15 fig. 9 the associations which define the problem solution after an exception is detected, asm will try to offer a solution for the occurring problem. the procedure is similar to the one which is used when an exception is being detected. asm compares the similarity of the plexus of associations which describe an exception, to the plexuses which exist in the asm network and which are described an exception. if the asm discovers that there was a solution to the problem in any of the predefined plexuses, it will offer such a solution to the new situation as well. the process which we used as an example of the application of the knowledge implemented in the asm refers to the manufacturing of customized hip endoprosthesis. the process model is shown in fig. 10 (due to complexity of presenting the whole process, only the part of the process relevant for the paper theme is shown). hip endoprostheses consists of three elements. those are femoral head, femoral neck and femoral body insert. during the process execution, the adaptation of parametric model of all three elements to a specific patient is done based on ct image, after which those elements are manufactured. after manufacturing, the elements are put together and inserted into the patient. fig. 10 part of process for hip endoprosthesis designing and manufacturing 16 d. misic, m. stojkovic, m. trifunovic., n. vitkovic at the beginning of the process, the administrator defines the asm model for the hip endoprosthesis manufacturing process (fig. 11). endoprosthesis manufacturing hip endoprosthis manufacturing femoral head femoral body insert compound activity object focal concept part assembly part femoral neck part fig. 11 asm context for process hip endoprosthesis manufacturing the activity of the process we are interested in is the activity endoprosthesis assembling. in this activity, the operator initially puts the elements that are completed in the clamping tools, after which the elements are being attached to form one unit. the operator uses a specially designed jig to position the parts accurately. this custom-made positioning mechanism is considered as the main production means for this operation, though there is also an additional tool (means), which may also be used for assembling if needed. the geometric accuracy of the assembly is determined regarding the angular deviation of so-called femoral neck shaft angle (nsa) from predefined value. the neck shaft angle is an angle between the femoral neck axis and the femoral corpus axis. since the value of this angle differs for each patient, this deviation is expressed as a percentage, and must not be larger than, for example, 3%. if the deviation is larger than 3%, the process is stopped. after that, the engineer will search for the problem causes and try to eliminate them. this is a formalism implemented in the process as an if-then procedure. ordinarily, the geometric accuracy of the manufactured assembly is below the required one due to fast manipulation, but it may occur for some other reasons as well. the context, i.e., plexus of associations that semantically describes the assembling activity is shown in fig. 12. this context models the general concept of this activity (ctx). the previous remark that the context which describes the instance of activity is a subtype of the context which describes the model of activity is valid in this case, too. a sudden requirement for a larger than usual production batch may lead to the acceleration of the assembling process, which, in its turn, may further lead to an increased deviation from the required geometric accuracy of the produced assemblies. the case when this deviation exceeds the allowed limit is covered by a specific if-then procedure. if the angular deviation remains below the allowed value, this procedure is not being launched. in the following section we will explain in detail the manner in which asm makes conclusions about a new situation using the knowledge about a familiar situation which once occurred. detection and handling exceptions in business process management systems using active semantic ... 17 assembly process quality endoprosthesis manufacturing subject part assembly concept hip assembly product concept attribute activity object activity similar concept activity object subject subject femoral neck object activity similar concept parts positioning mechanism 2 cause similar concept large batch consequence assembling supplement parts positioning mechanism 1 similar concept activity increment of nsa deviation attribute concept femoral head femoral body insert object similar concept similar concept mechanism 2 engagement part assembly part fig. 12 associations' plexus (context) that models the context of endoprosthesis assembling activity of assembly according to the plexus that models generic assembling activity 5.3 applying learned associations in a new situation by performing the topological analysis of the network, asm can determine that there are topological analogies between the current situation (described by the input context instance) and the plexuses which are already in the network. within the same process, asm determines the degree and the quality of the similarities between certain association plexuses. in this particular case, asm recognizes that the context of the activity scaffold modeling is topologically similar to the context of the activity assembling in the current process (at instance level). following the procedure of upgrading the current context based on the topologically analogous one, asm initially proposes adding the associations between the concepts unsuccessful operation and large batch and small increment of nsa deviation, which are already introduced (embedded) in the network (fig. 13). the next step is to categorize the new context with additional concepts as a possible exception. asm suggests making an association between the context which describes the activity instance and the concept exception. in the process of further upgrade, asm proposes making a connection between the concept exception and the context which describes the activity model with the concept alternate. in the end, the udf based scaffold modeling and the sequential decomposition of that activity are offered as a way 18 d. misic, m. stojkovic, m. trifunovic., n. vitkovic of solving the problem (fig. 14). these conclusions refer to the context which describes the activity model, and, therefore, they should be applied to all instances of that process. at that moment, the associations with the concept udf based scaffold design and the concept sequential job decomposition are offered. the administrator will refuse the former one because that connection is not applicable in this context, but he should accept the latter one and consider what this decomposition might refer to in a particular case. the process of context upgrade is shown in figs. 13 and 14. asm is currently connected with the md system so as to provide recommendations for solving the problem. these recommendations are then implemented by the rules in the expert systems. these rules are used to change the process in accordance with the recommendations by asm. if asm connects a situation with the term exception and if the concept alternate also appears, the md system will send a signal to the expert system that a change in the process is required. the change will be performed by calling an appropriate rule. assembly process quality endoprosthesis manufacturing subject part assembly concept hip assembly product concept attribute activity object activity similar concept activity object subject subject femoral neck object activity similar concept parts positioning mechanism 2 cause similar concept large batch consequence assembling supplement parts positioning mechanism 1 similar concept activity increment of nsa deviation attribute concept femoral head femoral body insert object similar concept similar concept mechanism 2 engagement part assembly part unsuccessful operation small increment of nsa deviation cause consequence cause consequence subtype type fig. 13 the context of the activity assembling with added associations based on the similarity with the context of the activity scaffold modeling (marked in red) detection and handling exceptions in business process management systems using active semantic ... 19 fig. 14 the upgrade of the piece of network related to the assembling activity instance which is featured as an exception according to the partially analogous model of scaffold modeling activity instance that is featured as an exception also 6. discussion our research results related to the application of asm for detecting and handling exceptions show that asm brings significant improvements in this domain. in this work, we have shown how a new process (prosthesis implanting) can benefit from the knowledge collected in a difference process (designing a scaffold), where these two processes are not closely connected. the processes used here as examples for managing exceptions come from close fields (mechanical engineering), but the application of asm is not limited to such situations. asm can draw conclusions even in completely different contexts. for example, the conclusion that it is necessary to decompose work could also be drawn from the analogy with construction engineering in the case where a building could not be finished on time, so it was required to split the work. 20 d. misic, m. stojkovic, m. trifunovic., n. vitkovic in our previous work [4], we used the rules of the expert system to represent the process knowledge. the drawback of that approach is that for solving a problem it is necessary to define the rules (knowledge) specifically related to that problem. collected knowledge can be represented in other ways. nowadays, ontologies are widely used for this purpose. instead of relying on ontologies, we have decided to use asm as a mechanism for knowledge representation and reasoning. the advantage of asm over ontologies derives from the fact that asm imitates human way of thinking, i.e. it is capable of drawing conclusions even on the basis of incomplete information. asm is thus able to draw conclusions in a new area, on the basis of analogy with an area that is not directly connected to the first one. since asm bases its reasoning on the knowledge that it has previously built into the network, there is a risk of the so-called indoctrination. negative indoctrination of asm, which results in the production of incorrect conclusions, can occur in two cases. in the first, the user/teacher can transfer their misconceptions (ignorance) by incorporating their knowledge about a certain domain. in the second case, when the asm is "taught" about a certain domain by several users/teachers, there is a danger that the semantic content will be inconsistent. this second case is particularly interesting because the growth of knowledge and the semantic network can be significantly accelerated by providing access to asm via the web. so, the asm concept cannot guarantee that the expert/teacher did his job rightly just as no one can guarantee to have got completely correct inferences from any kind of an ai method. unique and completely correct inference is possible just in the case of strictly defined corpus of knowledge, like in formal logic or mathematics. however, for a great majority of situations in the real world, this is not possible. in the md system, asm can be used in two complementary ways: for exception detection and exception handling. exception detection is explained in a greater detail in [5], while in this work, we made a step further and used asm for solving challenging situations. asm in this case manages to recognize a situation as an exception and offers advice based on the analogy with the knowledge previously incorporated into the asm model. in addition to offering an intelligent advice, the system which handles exceptions should enable the realization of the offered solution. sometimes it is possible to do that without human participation, and sometimes it is not. in the example we have presented, the human is left to try to modify the process on the basis of the asm’s recommendations. the propagation of changes suggested by asm is currently done via expert rules developed previously [4]. these rules enable the process update that can be applied to new or already defined process instances, or a combination of the two. the changes are implemented via the java methods invoked from the action parts of these rules. 7. conclusion and future work exception handling is one of the problems that are not solved adequately in the existing business process management systems. the process of solving this problem can be divided into two stages. the most important step is that the system detects an exception in the first stage in order to be able to solve the exception in the second one. detection and handling exceptions in business process management systems using active semantic ... 21 in this paper, we described the md system that uses asm and expert rules to handle exceptions. the expert rules are used as a mechanism for handling exceptions that can be predicted in advance. when an exception is detected, a sequence of rules is initiated that modify the process according to the new situation. for detection and handling of exceptions that cannot be predicted in advance, we used asm. asm makes conclusions based on the analogy between current and some of the previous situations. for asm to be able to make conclusions, the business process and all activities involved in it are represented as a semantic network. when a new concept and association are added to the network, a mechanism is triggered to find if that new situation is an exception and if so, to potentially propose a solution based on the similarity with some previous situation. the solution is then forwarded to the system via the expert rules that adapt the process. in our previous work, we began to use asm for handling of the exceptions. at first, we only used it for detection of the exceptions. one of the reasons for the relatively limited use of asm at that moment (a few years ago) is the fact that algorithms for recognition of the topological analogies were not fully developed at the time. in the meantime, we have significantly improved these algorithms, and we wanted to show how they can be put to the best use. in this paper we took a step forward and also used asm for solving problems. the examples we used for presenting the capabilities of drawing conclusions come from similar processes, but the parts of the processes which are used in analogies are very distant. the situation which is used for asm’s learning is from the area of designing and modeling while the situation in which the collected knowledge is used is from the area of manufacturing. the examples could have been from the processes which are used in a completely different area but we rather wanted to describe a situation which is likely to happen in the same company. currently, the asm’s conclusions may only be applied by using the rules from the expert system, but our plan for the future is to enable asm to independently apply its proposals. concept bodies may include formalized knowledge structures. this is so-called firmly structured knowledge contained in unambiguous mathematical and logical formalisms transformed into procedural or object-oriented programs. these programs can be called for execution when appropriate. when it comes to the development of asm, we plan to further develop procedures for creation of heuristics and knowledge crystallization. we also plan to develop structural elements for semantic categorization of events, i.e. the contexts which come one after another in a certain timeline. asm could thus be used in the systems where the time dimension has a semantic value. in addition to the use of asm for adapting processes to new circumstances, we plan to enable the use of asm as a support tool in the adaptive case management systems in further work. initially the system will function in a manner which will let the user define the steps that the process should include; but with the spreading of its knowledge base, asm will increasingly become able to advise the user about further doings. acknowledgements: this research was financially supported by the ministry of education, science and technological development of the republic of serbia (contract no. 451-03-9/202114/200109). 22 d. misic, m. stojkovic, m. trifunovic., n. vitkovic references 1. kir, h., erdogan, n., 2021, a knowledge-intensive adaptive business process management framework, information systems., 95, 101639. 2. sadiq, s., orlowska, m., 2000, on capturing exceptions in workflow process models, in: abramowicz, w., orlowska, m. (eds.) bis 2000, pp. 3–19. 3. casati, f., ceri, s., paraboschi, s., pozzi, g., 1999, specification and implementation of exceptions in workflow management systems, acm transactions on database systems 3, pp. 405–451. 4. mišić, d., domazet, d., trajanović, m., manić, m., zdravković, m., 2010, concept of the exception handling system for manufacturing business processes, computer science and information systems, 7(3), pp. 489-509. 5. mišić, d., stojković, m., domazet, d., trajanović, m., manić, m., trifunović, m., 2010, exception detection in business process management systems, jsir, 69(03), pp. 1038-1042. 6. dijkman, r., turetken, o., van ijzendoorn, g.r., de vries, m., 2019, business processes exceptions in relation to operational performance, business process management journal, 25(5), pp. 908-922. 7. ariouat, h., andonoff, e., hanachi, c., 2016, do process-based systems support emergent, collaborative and flexible processes? comparative analysis of current systems, procedia computer science, elsevier, 96(c), pp. 511-520. 8. allen, d., chapman, a., blaustein, b., mak, l., 2015, what do we do now? workflows for an unpredictable world, future generation computer systems, 42, pp. 1–10. 9. richter, m., weber, r., 2013, case-based reasoning, springer, berlin heidelberg. 10. minor, m., bergmann, r., görg, s., 2014, case based adaptation of workflows, information systems, 40, pp. 142-152. 11. zeyen, c., müller, g., bergmann, r., 2018, a conversational approach to process-oriented case-based reasoning, ijcai'18: proceedings of the 27th international joint conference on artificial intelligence, pp. 5404 – 5408. 12. aha d.w., breslow, l.a., munoz-avila, h., 2001, conversational case-based reasoning, applied intelligence, 14, pp. 9-32. 13. reichert, m., dadam, p., 2009, enabling adaptive process-aware information systems with adept2, in: cardoso, j., van der aalst w., (eds.), handbook of research on business process modeling, information science reference, new york, pp. 173-203. 14. laznik, j, juric, m., 2013, context aware exception handling in business process execution language, information and software technology, 55(10), pp. 1751-1766. 15. yao, w., kumar, a., 2013, conflexflow: integrating flexible clinical pathways into clinical decision support systems using context and rules, decision support systems, 55(2), pp. 499-515. 16. dang, j., hedayati, a., hampel, k., toklu, c., 2008, an ontological knowledge framework for adaptive medical workflow, journal of biomedical informatics 41, pp. 829-836. 17. marrella, a., mecella, m., 2018, cognitive business process management for adaptive cyber-physical processes, in: teniente, e., weidlich, m., (eds), business process management workshops, springer: berlin/heidelberg, germany, pp. 429–439. 18. stojkovic, m., trifunovic, m., misic, d., manic, m., 2015, towards analogy-based reasoning in semantic network, computer science and information systems, 12(3), pp. 979-1008. 19. trifunovic, m., stojkovic, m., misic, d., trajanovic, m., manic, m., 2014, recognizing topological analogy in semantic network, international journal on artificial intelligence tools, 24(03), pp. 1550006-1 – 1550006-25. 20. trifunović m., stojković m., trajanović m., mišić d., manić m., 2013, interpreting the meaning of geometric features based on the similarities between associations of semantic network, facta universitatis, series: mechanical engineering, 11(2), pp. 181-192. 21. https://sourceforge.net/projects/sharkwf/ (last access: 15.12.2021) https://www.emerald.com/insight/search?q=remco%20dijkman https://www.emerald.com/insight/search?q=oktay%20turetken https://www.emerald.com/insight/search?q=geoffrey%20robert%20van%20ijzendoorn https://www.emerald.com/insight/search?q=meint%20de%20vries https://www.emerald.com/insight/publication/issn/1463-7154 javascript:void(0); https://dl.acm.org/doi/proceedings/10.5555/3304652 http://www.sciencedirect.com/science/article/pii/s0950584913000840 http://www.sciencedirect.com/science/article/pii/s0950584913000840 http://www.sciencedirect.com/science/journal/09505849 https://sourceforge.net/projects/sharkwf/ microsoft word rev 13939-72201-1-ce_li xu_w12 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume250731034x © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fault-tolerant control of low-altitude unmanned aerial vehicle swarms using a reduced-order sliding mode observer li xu school of electronic and information engineering, ningbo university of technology, ningbo, china abstract. the operational demands placed on unmanned aerial vehicles (uavs) have increased significantly, rendering single low-altitude uavs insufficient for addressing complex missions and navigating dynamically evolving information environments. to overcome these limitations, the deployment of low-altitude uav swarms was adopted, offering enhanced agility, adaptability, and scalability. however, the expansion in swarm size leads to a higher probability of actuator faults and increased complexity in fault diagnosis and mitigation. in this study, a fault-tolerant control framework was proposed for low-altitude uav swarms, incorporating a reduced-order sliding mode observer for real-time fault detection and isolation. a dynamic model of the uav swarm was first established based on newtonian mechanics and coordinate transformation techniques, accurately capturing inter-agent dynamics and external disturbances. building upon this model, a reduced-order sliding mode observer was constructed to enable prompt estimation of actuator faults while ensuring robustness against uncertainties and model inaccuracies. fault estimates obtained in real time were subsequently utilized in the design of a fault compensation controller to maintain swarm stability and performance in the presence of partial actuator failures. the stability and reachability of the proposed observer were rigorously validated through lyapunov-based analysis, ensuring convergence of the fault estimation error. to evaluate the proposed methodology, an experimental uav swarm platform operating at low altitude was developed. experimental results demonstrate the high accuracy, resilience, and computational efficiency of the fault-tolerant control system under various fault scenarios, highlighting its practical applicability to real-world uav swarm deployments. the proposed framework offers a scalable and robust solution for maintaining mission continuity in safety-critical uav swarm operations. key words: unmanned aerial vehicle swarm, mathematical model, sliding mode observer, fault-tolerant control, lyapunov stability theorem received: july 31, 2025 / accepted december 21, 2025 corresponding author: li xu school of electronic and information engineering, ningbo university of technology, ningbo 315211, china e-mail: xuli@nbut.edu.cn 2 l. xu 1. introduction in recent years, uavs have been widely used in wide-area searches, low-altitude exploration, forest fire monitoring and other fields. a single uav can hardly meet the needs of the increasingly complex information environment, and uav swarms are becoming a research hotspot in various countries [1-5]. uav swarms replace the individual units in a collaborative manner, which can significantly improve mission coverage and search efficiency without increasing costs [6, 7]. on the other hand, the uav itself is a nonlinear, multiple-input multiple-output (mimo), strongly coupled underactuated system, and the design of a single uav control algorithm has been very complicated. due to the geometric increase in the number of components, the fault incidence of multiple uavs will be greatly increased. therefore, it is a meaningful issue to consider the coordination and fault tolerance of multiple uavs after partial faults. however, the coordination of uavs under partial faults and their formation control design are still one of the challenges in this field [8]. in recent years, many researchers have developed many practical strategies for the collaborative formation control of uav swarms. dong et al. [9] presented an improved distributed control strategy for formation maneuverability and verified its effectiveness through experiments. liao et al. [10] put forward a control scheme to produce a collision avoidance mechanism so that the uav swarms can complete formation and reconstruction without internal collision. kang et al. [11] developed a distance-based continuous formation control method. belkadi et al. [12] proposed a distributed path planning strategy under the framework of multi-agent systems. yang et al. [13] presented a uav swarm power control method and trajectory planning derived from a depth deterministic strategy gradient. liu et al. [14] came up with a multi-objective optimization model of uav swarm trajectory using the decomposition method and trajectory search. however, the control schemes raised in some studies [9-14] stem from a necessary assumption, i.e., a single uav in a uav swarm can be regarded as an ideal particle or there are no uncertain factors such as faults and interference in each single uav system, which is difficult to guarantee in practical applications. in practice, one or more individuals in the low-altitude uav swarm system inevitably receive environmental disturbance or may have actuator failures. these disturbances may spread to other individuals through the uav network, and in severe cases, a chain reaction will be triggered, resulting in collision or misidentification between each other, where the advantages of the uav swarm may become the cause of a serious accident [15-17]. however, as the number of the low-altitude uavs in the formation increases, the possibility of actuator failure will be greatly improved. the uav swarm is essentially a strongly coupled nonlinear multi-agent system, which makes its fault diagnosis and faulttolerant control a significant challenge. in recent years, the fault-tolerant control of a single uav has seen great progress by using passive or active fault-tolerant technology. aiming at the actuator fault problem of a uav, hasanshahi et al. [18] designed a robust fault-tolerant controller by modeling the nonlinear dynamics as an lpv system and incorporating adaptive fault estimation, thereby achieving trajectory tracking under faulty conditions. yang et al. [19] combined the model reference adaptive control method with the fractional order method to offset the impact of actuator failure and mass change on the uav. liu et al. [20] developed a fault-tolerant tracking robust controller of the uav with uncertain parameters based on a fixed-time fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 3 interference observer, and the proposed control strategy is feasible and effective in terms of disturbance suppression, fault tolerance, buffeting suppression and non-heteronormality. wang et al. [21] developed a robust fault-tolerant controller according to the hierarchical distributed comprehensive theory, aiming at the possible external disturbance, actuator failure and wireless network-associated failure during uav operation. liu et al. [22] presented a fault-tolerant control scheme based on neural networks by taking advantage of the strong approximation of radial basis function neural networks to compensate for the effects of parameter uncertainties, external disturbances, and actuator faults. however, the above studies are all aimed at the failure of a single uav, which is insufficient for the overall risk control of modern large-scale uav formations. in recent years, some scholars have conducted a preliminary exploration on the fault-tolerant control of uav swarms. yu et al. [23] used sliding mode control and model reference adaptive control to compensate for the impact of actuator faults on uav swarms, but it belongs to passive fault-tolerant control and lacks fault diagnosis. reppa et al. [24] established the diagnosis and isolation of uav swarm faults through multi-sensor fusion and applied a global decision logic, but failed to compensate for the faults. from the above analysis, it can be learned that the research on fault diagnosis and faulttolerant control of a single uav has led to many important achievements, but there is still an open challenge for the research of low-altitude uav swarms. the fault-tolerant control difference between a single uav and uav swarms can be summarized as follows: a) the controller of the uav swarm consists of an outer-loop group controller and an inner-loop individual uav controller. the output of the inner-loop individual uav controller serves as the input for the outer-loop group controller. faults can propagate through the inner and outer control loops; hence, the fault diagnosis and fault-tolerant control of the uav swarm must consider not only the impact of faults on the individual uav but also their effects on the entire group. b) in a uav swarm, in addition to individual uav malfunctions, there can also be communication failures between uavs. moreover, these communication failures can be amplified as information is passed between uavs. therefore, fault diagnosis and fault-tolerant control require the establishment of a global error system for the uav swarm to estimate and compensate for these faults. to address the two key research gaps for low-altitude drone swarms mentioned above, the novelty of our work lies in: 1) the fault-tolerant control method with active compensation that enhances formation stability in the event of actuator failure. 2) the designed reduced-order sliding mode observer for fault estimation that enables more efficient fault detection and thus enhances system stability. 3) the framework capable of handling effectively multiple actuator failures and maintaining swarm formation stability under disturbances. the rest of this study is organized as follows. in section 2, the position relationship of adjacent wingmen in the low-altitude uav swarm was analyzed, and the formation model of the inner and outer-loop uav swarm under the wingman strategy was established. in section 3, a sliding mode observer with reduced order was designed for fault diagnosis, and its reachability and stability were proved. in section 4, the estimated faults and disturbances were solved by using the sliding mode observer, and active fault-tolerant control was carried out to realize the trajectory tracking in the case of actuator faults. in section 5, the effectiveness of the developed method was verified by experiments. section 6 summarizes the content of the study. 4 l. xu 2. construction of the uav cluster model 2.1 construction of a single uav motion model assuming the earth's curvature is neglected and the ground coordinate system is adopted as the inertial coordinate system, the definitions of the ground reference frame and the body reference frame are illustrated in fig. 1, with specific details as follows: (1) ground coordinate system sg-oxyz: the origin o is a selected point on the ground. the x-axis lies in the horizontal plane and points in a specific direction; the z-axis is perpendicular to the ground and points toward the earth's center; the y-axis also lies in the horizontal plane, is perpendicular to the x-axis, and is determined in accordance with the right-hand rule. (2) body coordinate system sb-obxbybzb: the origin ob is located at the center of mass of the unmanned aerial vehicle (uav) and is fixed to the aircraft. the xb-axis lies in the aircraft's plane of symmetry, is parallel to the aircraft's design axis, and points toward the nose. the yb-axis is perpendicular to the aircraft's plane of symmetry and points toward the right side of the fuselage. the zb-axis lies in the aircraft's plane of symmetry, is perpendicular to the xb-axis, and points toward the bottom of the fuselage. (3) aircraft attitude angles are determined by the relationship between the body coordinate system and the ground coordinate system, which are known as euler angles. specifically:  the pitch angle θ is the angle between the body axis xb and the horizontal plane, with a positive value corresponding to the nose-up attitude.  the yaw angle ψ is the angle between the projection of the body axis xb onto the horizontal plane and the ground axis x, with a positive value corresponding to the nose yawing to the right.  the roll angle ϕ is the angle between the body axis zb and the vertical plane passing through the body axis xb, with a positive value corresponding to the aircraft rolling to the right. fig. 1 the relationship between the body coordinate system sb and the ground coordinate system sg fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 5 newton's second law can be applied to establish the linear motion and angular motion dynamics equations of a single uav under the action of the external force f and the external torque m. the linear motion dynamics equation of a uav with the external force f can be expressed as follows: d d m t  v f (1) where v denotes the velocity vector of the uav's center of mass, t denotes the time and m denotes the mass of the uav. the angular motion dynamics equation of a uav under the action of the external torque m can be expressed as: d dt  l m (2) where l is the moment of momentum. to establish the relative motion relationship of the uav relative to the ground coordinate system, the body coordinate system was selected to establish the dynamic coordinate system. assuming that the velocity of the body coordinate system relative to the inertial coordinate system is v and the angular velocity vector is ω, then it can be written: d dt t     v v v i v (3) d dt t     l l l i l (4) where iv and il are the unit vectors along the flight speed v and the moment of momentum l, respectively, t   v and t   l represent the relative derivatives in the dynamic coordinate system, and ω is the total angular velocity vector in the dynamic coordinate system relative to the inertial coordinate system. v and ω can be represented via their components, as follows:   u v wv i j k (5) p q r  i j k (6) where i, j and k are the unit vectors in the directions of the x-axis, y-axis and z-axis, respectively; u, v and w represent the velocity v components along the xb-axis, yb-axis and zb-axis, respectively, and p, q, and r represent the components of the rotational angular velocity ω of the aircraft coordinate system relative to the ground coordinate system on the xb-axis, yb-axis and zb-axis of the aircraft coordinate system, respectively. in what follows, eq. (3) can be adequately rewritten. the first term on the right-hand side of eq. (3) reads: 6 l. xu u v w u v w t t t t              v v i i j k i j k   (7) the second item on the right-hand side of eq. (3) is: ( ) ( ) ( )p q r wq vr ur wp vp uq u v w         i j k v i j k (8) the combined force can also be represented via its components: b b bx y zf f f  f i j k (9) where bxf , byf and bzf are the resultant components along each axis. from eqs. (7)-(9) and eq. (1), one may write: ( ) ( ) ( ) b b b x y z f m u vr + wq f m v wp+ur f m w uq+vp            (10) unless otherwise specified, a dot above a variable in the text indicates the derivative of that variable with respect to time. in the uav system, considering the combined forces of the total aerodynamic force and engine thrust on various coordinates bxf , byf and bzf , gravity g is represented via its components with respect to the dynamic coordinate system. eq. (10) can be then rewritten as follows [25]: sin sin cos cos cos b b b x y z f u vr g wq m f v wq ur g m f w uq vp g m                              (11) the resulting eq. (11) represents the linear motion dynamics model of the uav. similarly, considering the angular motion dynamics model of the uav, eq. (4) can be suitably rewritten as it will be demonstrated below. the first term on the right-hand side of eq. (4) is: b b bx y zl l l t t t t          l l i i j k (12) where , b bx yl l and bzl denote the components of the moment of momentum with respect to the xb-axis, yb-axis and zb-axis, respectively. the right hand-side of the equation can be decomposed into: fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 7 b b b b b b b b b b x x x z y y z z x z l pi ri t l qi t l ri pi t                          (13) where bxi , byi and bzi are the moments of inertia of the uav with respect to the xb-axis, yb-axis and zb-axis, respectively, and b bx zi is the product of inertia of the uav with respect to the xb-axis and zb -axis. the second term on the right-hand side of eq. (4) can be rewritten as follows: ( ) ( ) ( ) b b b b b b b b b z y x z y x x y z p q r ql rl rl pl pl ql l l l         i j k l i j k (14) by decomposing the combined torque into each axis of the dynamic coordinate system and synthesizing eq. (2) and eqs. (12-14), the angular motion dynamics equation of the uav can be obtained: 2 2 2 2 2 2 2 2 [ ( ) ( )] / ( ) [ ( ) ( )] / [ ( ) ( ) ] / ( ) b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b z x x z z z x z y z x z x y z x z x z x z y x z y x z x x z x x z x y x z x y z x z x z p i m i m i i i i qr i i i i pq i i i q pr i i m i p r i r i m i m i i i i pq i i i i qr i i i                                (15) where bxm , bym and bzm are the torque components about the xb-, yb-, and zb-axis, respectively. eqs. (11)-(15) constitute the dynamic model of a uav. to further describe the motion of the uav, a kinematic model needs to be constructed. in the inertial system, it can be obtained from geometric relationships: tan sin tan cos sin cos ( cos sin ) / cos q r p r q r q                           (16) eqs. (11), (15) and (16) establish the complete motion model (including dynamics and kinematics) of a single uav, which is the six-degree-of-freedom nonlinear equation of the uav. using [u, v, w, p, q, r, θ, ψ, ϕ] as a state variable, the model can be written in the form of a state space. due to the nonlinear relationship in the motion model and the close correlation between force and torque and motion parameters, it is difficult to obtain analytical solutions, and numerical solutions can only be obtained with the help of computers. therefore, the model obtained after partial linearization using the small perturbation principle is as follows [26, 27]: 8 l. xu ( ) ( ) ( ( ), ( )) ( ) ( ) t t t t t t     x ax g x u y cx  (17) where x(t) = [u, v, w, p, q, r, θ, ψ, ϕ]t is the state variable of the system, u(t) = [δe, δt, δα, δr]t = [u1,u2,u3,u4]t is the input of the system, being the elevator, throttle, flaps, and ailerons; y(t) is the output vector of the system; g(x(t),u(t)) is the nonlinear vector function; a and c are the coefficient matrices after linearization. assuming that the nonlinear part of g(x(t),u(t)) is independent of the control input u(t), eq. (17) can be decoupled into the form of bu(t)  g(x(t)), where g(x(t)) is a continuous nonlinear function that satisfies the lipschitz condition. 2.2 construction of the uav cluster motion model the motion model construction of a single uav was completed in section 2.1. now the uav cluster model under the lead wingman strategy in the planar coordinate system is considered, as shown in fig. 2. fig.2 schematic diagram of the uav formation under the lead wingman strategy according to the geometric relationship in the figure, the motion equation of the i-th uav can be obtained as follows: cos sin i i i i i i i i x v y v             (18) where the planar coordinate system is composed of the x-axis and y-axis defined in section 2.1 of the ground coordinate system, xi, yi represent the position of the i-th uav, vi, ωi represent the forward and angular velocities of the i-th uav, and ψi denotes the angle between the forward velocity of the i-th uav and the x-axis. fig. 1 shows that the relative forward distance fij and the lateral distance lij between the lead aircraft and the wingman could be expressed as: fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 9 ( )sin ( )cos cos( ) ( )cos ( )sin sin( ) ij i j j i j j i j ij i j j i j j i j f y y x x d l y y x x d                      (19) where d denotes the distance between the control and actual centers [28, 29]. the forward error ijfe and the lateral error ijle between the forward and lateral distances, as well as the expected forward and lateral distances, can be written as a state space expression to obtain: cos( ) sin( ) sin( ) cos( ) ij ij f j ij j i j i j i ij j i j i j il e v l d v f de                                             (20) comparing eqs. (18) and (20), it can be seen that the output of the kinematic model of a single uav is the input of the kinematic model of the uav cluster. 2.3 construction of the uav motion model in the presence of faults to construct a fault model for the uav cluster, it can be assumed that one or more uavs have malfunctioned, i.e., the i-th uav has malfunctioned. the motion model of a single uav given in eq. (17) can be rewritten as: ( ) ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) i i i i i i i i i i i i i t t t t t t t t        x a x b u g x d e f y c x φ (21) where φi(t) represents the unknown nonlinear disturbance and parameter uncertainty of the system, and fi(t) denotes the unknown fault term of the system that occurred on the i-th uav, di and ei are the time-invariant coefficient matrices. 3. sliding mode observer and fault-tolerant controller design the outer-loop uav group control system needs to be designed. the reconstruction can be obtained as follows: * * * * cos( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) cos( ) ij ij f j ij j i j i j i ij j i j i jl i i j i j i i i j i j i i e v l d v f de d v v d                                                                         (22) where * iv and * i are the expected forward and angular velocities of the i-th uav. to make the system stable, the introduction of a proportional-integral (pi) control rate can be derived as follows: * 1 1 * 2 2 ( ) cos( ) sin( ) sin( ) cos( )( ) ij ij ij ij ij ij f p f i f i j i j i i i j i jl i ip l i l e k e k e dt d v v de k e k e dt                                              (23) where kpi, kii, i=1, 2 respectively represent the proportional gain and integral gain of the 10 l. xu pi control law. when * *, iiv  and vi, ωi are equal, i.e., when the uav tracks the desired signal, the following can be obtained: 1 1 2 2 ( ) ( ) ij ij ij ij ij ij f p f i f l p l i l e k e k e dt e k e k e dt                   (24) obviously, eq. (24) is asymptotically stable, which means that when each uav in the uav group is stable under normal or fault conditions, the stability of the uav group can be guaranteed. to ensure that the state variables of the inner-loop uav can track their reference values, a sliding mode tracking controller for the inner-loop uav was designed. assumption 1. for the nonlinear term in the nonlinear model of a single uav in eq. (17), there is a continuous bounded function (xi(t)) > 0 such that it can satisfy the following inequality: ( ( )) ( ( )) i t t i i i i it ts l cg x λ x s l cb (25) where si is the sliding surface. the sliding surface of the sliding mode tracking controller of the inner-loop uav was designed as follows: 0 0 t i i i idt  y ys e k e (26) where eyi = yri liyi represents the output tracking error of the i-th uav, yri denotes the expected output of the i-th uav, yi denotes the actual output of the i-th uav, and li represents the dimension adaptation matrix. the following derivation of eq. (26) can be obtained: i i i i ri i i i i    y y ys e k e y lcx k e    (27) its stability was proved below. a lyapunov function vi was considered as follows: 1 v 2 t i i i s s (28) the derivation of eq. (28) can be obtained: v ( )t t i i i i ri i i i yi   s s s y lcx k e   (29) to guarantee that the system can be stable on the sliding surface, i.e., vi < 0, the input ui = ui1 + ui2 of the controller was designed as: 1 1 1 2 ( ) (( ) (( )( ) ) ( sign( )) ) ( ( ))( ) t t t t i i i i ri i yi i i i t t i i i i t i i t           u b bb l c l c l c y k e s ax λ x s l cb u s l cb (30) fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 11 where ρi > 0 is the designed gain, and sign(si) is the designed sign function vector. substituting eq. (30) into eq. (29), the following can be obtained from eq. (25): 2 1 ( ( ))( ) v sign( ) ( ( )) ( ( )) sign( ) ( ( )) ( ( )) 0 t t t t ti i i i i i i i i i i it i i t t t i i i i i i i i i i i n t t t t                 λ x s l cb s s s l c b s l cg x s l cb s s λ x s l cb s l cg x s (31) this indicates that the system can stabilize on the sliding surface after reaching the sliding surface. after designing the sliding mode controller, it is necessary to design a sliding mode observer to observe the fault. the reconstruction of eq. (21) can be obtained as follows: 1 11 1 12 2 1 1 1 2 21 1 22 2 2 2 2 2 ( ) ( ) ( ) ( ( ), ( )) ( ) ( ) ( ) ( ) ( ) ( ( ), ( )) ( ) ( ) ( ) ( ) i i i i i i i i i i i i i i i i t t t t t t t t t t t t t t t t                x a x a x g x u d e f x a x a x g x u d y φ e f x φ (32) where xi1, xi2  rp are the upper and lower parts obtained by decomposing the system state vector xi, g1(xi(t),ui(t)) and g2(xi(t),ui(t)) are the upper and lower parts obtained by decomposing g(xi(t),ui(t)), fi(t) denotes the unknown fault term of the system and has a known upper bound γ2, φi(t) represents the unknown nonlinear disturbance and parameter uncertainty of the system, and d1 ,d2 , e1 and e2 are the time-invariant coefficient matrices. assumption 2: in eq. (21), the matrix pair (a, c) is detectable. from assumption 2, there exists a matrix l such that the matrix a-lc is stable and satisfies the following lyapunov equation: ( ) ( )t    a lc p p a lc q (33) where q > 0, and there exists a unique solution p > 0 for eq. (33). the linear transformation zi  txi was introduced, where: 1 1 2n p p         0 i p p t i (34) 1 2 1 2 2 3 2 3 ,t tq              p p q q p p p q q (35) substituting eq. (32) leads to: 1 1 1 21 11 1 11 21 12 22 2 1 1 2 21 1 21 22 2 2 2 2 2 ( ) (( ) ) ( , ) ( , ) ( ) ( , ) ( ) ( ) i i i i i i i i i i i i i i i t t                       iz ka a z a ka k a ka y kg t z u g t z u z a z a k a z g t z u e f d y z φ (36) where k = p1 -1p2 , z = col (z1,z2 ) and z1  rn-p. 12 l. xu for eq. (36), after linear transformation, a sliding mode-based observer was designed, and its stability was proved. theorem 1: for the uav nonlinear system after linear transformation, if λmin(q1) > 2lg ‖[p1 p2]‖min is satisfied, the reduced-order observer eq. (37) will be asymptotically stable on the sliding mode surface. 1 21 11 1 11 21 12 22 1 1 2 1 1 2 21 1 22 21 2 2 2 ˆ ˆ( ) (( ) ) ˆ ˆ ( , ) ( , ) ˆ ˆ ˆ ˆ( ) ( , ) ˆ ˆ i i i i i i i i i i i i i i i                        z ka a z a ka k a ka y kg t z u g t z u z a z a a k z g t z u v y z (37) where 1 1 1 1ˆ ˆ( , ) ( , )i i i i g i il     g t z u g t z u t z t z (38) was obtained by the lipschitz condition satisfied by the nonlinear term and the sliding mode switching term vi as follows: 1 22 21 2 1 2 2 ˆ ˆˆ( ( ) )sign( )i i i i i iw       v a a k y y d t z e y y (39) where w is a positive real number. proof. suppose 1 1 1ˆi i z ze , and ˆy i i y ye , then the error equation can be expressed as: 1 11 21 1 1 2 21 1 22 21 2 2 2 ( ) ( ) ( ) ( )y y i i it t                 e a ka e g kg e a e a a k e f vφe g d (40) where 1 1 ˆ( , ) ( 2, , ) 1,l l i i l i i l   g g t z u g t z u (41) the designed sliding surface is as follows:  1( , ) | 0y y s e e e (42) consider the following lyapunov function: 1 1 1 1v t e p e (43) the derivative of eq. (43) can be obtained: 1 1 1 11 21 11 21 1 1 1 1 1 2v [ ( ) ( ) ] 2 [ ][ ]t t t t n p    e p a ka a ka p e e p i k g g  (44) eq. (33) is decomposed in the form of eq. (35), which can be expressed as: 1 11 21 11 21 1 1( ) ( )t    p a ka a ka p q (45) fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 13 combining eq. (43) and eq. (44) leads to:          1 21 1 1 1 1 1 2 1 1 1 1 1 1 2 2 1 min 1 1 2 v 2 2 ( ) ( ) 2 0 t t t t t g g l l                e q e e p p g g e q e e p p t z z e q p p   (46) thus, the system can be stabilized on the sliding surface. via the designed observer, the fault was estimated. when the system reaches the sliding mode surface for sliding mode motion, 0, 0y y e e (47) then eq. (40) can be changed to: 21 1 2 2 20 ( ) ( )i i eqt t    a e g d e f vφ (48) because of 10 lim 0 t e it can be derived as follows:  2 2 ( ) ( ) i eq i t t       f φ v d e (49) suppose there exists  1 2 t p p d r  m m m satisfying the following:   1 21 2 2 32           0 h hm d e hm (50) further, combining eq. (49) and eq. (50) can yield the following: 1 2 3 ( ) ( ) i eq i t t              h h m φ v h f0 (51) thus, the fault and disturbance can be estimated as: 1 3 2 ˆ ( )i eqt f h m v (52) 1 1 1 1 2 3 2 ˆ ( ) ( )i eqt   h mφ m h h v (53) where 1 22 21 2 2 1 2 ˆ ˆ ˆ( ( ) ) ˆ i i eq i i i i w           y y v y y a a k e t z d y y (54) 14 l. xu where  is a very small real number. a compensation control scheme is presented. from the previous analysis, it is known that the estimation errors of faults and disturbances are bounded and stable. therefore, a compensation control term was added to the controller in the absence of faults to offset the adverse impact of actuator faults on the uavs. the fault-tolerant control method can be rewritten as follows: * i i fi u u u (55) where ( 1) ˆˆ( ) ( ( ) ( ))t t f i i it t  u b bb dφ ef (56) 4. experimental results due to the limitations in the number of experimental equipment and venues, three uavs, including one leader uav and two wingmen, were subjected to physical verification experiments to verify the effectiveness of the fault-tolerant control method for low-altitude uav clusters via the reduced-order sliding mode observer and the stability of the entire formation. to ensure the online operation of various complex algorithms, after using the flight control unit for low-altitude control, all uavs in the experiment were equipped with an arm-based microcomputer, odroid-xu4. this computer runs a 64bit operating system and has an ros development environment built on it. most of the communication and data interaction in the project is based on the communication protocol of the ros. the sensor system used in this experiment is a sensor model more suitable for coverage path planning, which better describes the coverage effect of the sensor for different location areas around it, by quantitatively describing the coverage effect on a certain range of area surrounding the flight path, rather than at a fixed point. when the uav moves along a certain path, the sensor system will continuously collect the surrounding information, during which the same point will be observed several times. in addition, it is straightforward to optimize the coverage path of the uav, i.e., a segment of the curve, to calculate the coverage path that can collect the most accurate and reliable position information. the spatial localization system used in the experiments is the optitrack system, which consists of a number of infrared cameras distributed on the top around the room to achieve the tracking of uavs within a defined range. its infrared cameras are connected to a pc serving as the host computer through an optihub integrator, which provides positional information for the indoor target by recognizing the calibration sphere attached to the target. a dual-frequency wireless router was used in this study to set up a local area network to connect all experimental devices. the entire network environment is shown in fig. 3, which includes both a wired and a wireless local area network. the experimental setup included a leader uav (qbot2-l) and two follower uavs (qball-x4 and qbot2-f). fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 15 fig. 3 low-altitude uav swarm experimental platform in considering the failure of a single actuator, the four motors were named separately to distinguish the four motors of the quadrotor. the specific naming method was to take the nose as the forward. the motor inputs were designated as follows: the rear motor as u1, the front as u2, the left as u3, and the right as u4. the fault selection in this compensate is a 10% failure of the rear motor, occurring 35 seconds after the start of the mission. the control parameters were selected as: [1, 2.682,0,5]t d (57) 1 [ 1.753, 0.0612,0.1356]  f (58) 2 0.1287 1.4021 3.1981 0.0112 4.1372 0.0498          f (59) 2.682 1.093 0.2534 0 0.416 4.211 0 143.234 0.4532           h (60) 4.1 single actuator failure it can be seen from fig. 4 that when the fault occurs, because the rear motor fails by 10%, the rear motor is facing the positive direction of the z-axis. therefore, the positive direction of the z-axis decreases, resulting in a decrease in the pitch angle. the observer will compensate for the estimated fault size in the controller to return the uav to the original path. the pitch angle is maintained at the original angle and the pitch angle control command increases rapidly. after about 8 seconds, the uav returns to the proper path, and the pitch angle control command returns to zero. it can be seen from the optitrack curve that the actual pitch angle is not greatly affected. in addition, it can be seen that the effectiveness 16 l. xu of the reduced-order sliding mode observer and the fault-tolerant controller is evident and will be more intuitively reflected in the subsequent trajectory diagram. fig. 4 pitch angle tracking when a single actuator fails remark 1: too many transitions can be seen in the results. some transitions are inevitable in the dynamic adjustment of uav systems, while others are caused by external interference noise. they can be eliminated by increasing the hardware's anti-interference ability. as it can be seen in fig. 5, the roll angle is almost not affected by the failure of the rear motor. the roll angle command only changes for a split second at 35 seconds, but the whole is not affected, which is consistent with the theoretical situation. fig. 5 roll angle tracking when a single actuator fails fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 17 fig. 6 shows that, at 35 seconds, due to the failure of the rear motor by 10%, the lift provided is reduced. therefore, the height is affected and slightly reduced. but within two or three seconds, the height is adjusted back to the set 0.7 meters, which shows the effectiveness of the height control method. fig. 6 height tracking when a single actuator fails diagrams in figs. 7-8 demonstrate that the reduced-order sliding mode observer can basically estimate the size of the fault. but due to the large fluctuation of noise, the observer estimates the fault quickly and can quickly compensate for the estimated value into the motor control instruction of the corresponding channel, where the motor command ordinate is the percentage of the full speed of the motor rotor. it should also be pointed out that in figs. 4 to 8, approximately the first 15 seconds are used by the uav to take off to the departure site. therefore, all angles are still adjusted. fig. 7 fault estimation when a single actuator fails 18 l. xu a) b) c) d) fig. 8 actuator command when a single actuator fails: a) u1; b) u2; c) u3; d) u4 qbot2-l and qbot2-f are two ground-based two-degree-of-freedom unmanned robots, with qball-x4 being a quadrotor uav. qbot2-l acts as the leader, while qbot2-f and qball-x4 serve as wingmen, building a formation. fig. 9 presents top-down views, which are the projection views of the formation on the ground. the objective is for the wingmen to always follow the leader, even if a malfunction occurs. in fig. 9, qbot2-l starts from (0,0) with a route of (0.75, -0.75), (-0.75, -0.75), (0.75, 0.75), and (0.75, 0.75). meanwhile, qbot2-f begins from (0.1, 1) and qball-x4 from (0.8, 0.45), both following qbot2-l. when qball-x4, the uav, reaches (0.3, -0.9), its motor u1 malfunctions, causing it to stray from the formation. however, after approximately 0.4 meters, around the location (-0.1, -0.9), it resumes its tracking of the leader, qbot2-l, to the designated position. this scenario demonstrates the effectiveness of the proposed fault diagnosis and fault-tolerant control scheme. the fault has little impact on the uavs. after about 0.4 meters, uavs track the long aircraft to the designated position again, and the entire formation is still stable, which shows the effectiveness and practical application value of the fault diagnosis and the fault-tolerant control scheme designed in this topic. fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 19 fig. 9 top view trajectory when a single actuator fails 4.2 failure of two actuators on the basis of the original rear motor failure, the right motor in the asymmetrical position also fails at 49 seconds, and the fault form is u4 f  105%*u4, and the specific experimental results are shown below. as can be seen from figs. 10-11, the pitch angle at 35 seconds is similar to that of a single fault in fig. 5, and the pitch angle control command changes only briefly at 49 seconds. at 49 seconds, the roll angle decreases because the output of the right motor suddenly becomes larger and the right motor is in the negative direction of the x-axis. therefore, the roll angle decreases and the roll angle control command increases rapidly. fig. 10 pitch angle tracking when two actuators fail 20 l. xu fig. 11 roll angle tracking when two actuators fail in fig. 12, the height decreases slightly at 35 seconds and increases slightly at 49 seconds, but both quickly adjust back to 0.7 meters, showing that the control is still ideal in the event of actuator failure of the motor in two asymmetrical positions. fig. 12 height tracking when two actuators fail it can be seen from figs. 13-14 that the adaptive sliding mode observer can do better fault estimation for both faults. however, compared with the situation of a single motor failure, the effect of fault estimation is slightly reduced, and the control instructions of the corresponding two motors are slightly slowed down when compensating for the fault, but this is also related to factors such as battery voltage during the experiment. overall, the fault-tolerant controller and the adaptive sliding mode observer reach the pre-target, which shows that the fault diagnosis scheme is still ideal in dealing with multiple faults. fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 21 a) b) fig. 13 fault estimation when two actuators fail: a) u2; b) u4 a) b) c) d) fig. 14 actuator commands when two actuators fail: a) u1; b) u2; c) u3; d) u4 qbot2-l acts as the leader, while qbot2-f and qball-x4 serve as wingmen, forming a formation. fig. 15 presents top-down views, which are the projection views of the formation on the ground. the objective is for the wingmen to always follow the leader, 22 l. xu even if a malfunction occurs. in fig. 15, qbot2-l starts from (0, 0) with the same route as in fig. 9. meanwhile, qbot2-f begins from (-0.5, 0.7) and qball-x4 from (-0.4, 0), both following qbot2-l. it can be observed that the u1 failure occurs at point (0.2, -1), and the u4 failure occurs at point (-0.7, -0.6). furthermore, both failures occur simultaneously. it is evident that the flight quality of the uavs decreases. due to the adoption of the directed graph communication topology in this formation, the leader uav (qbot2-l) does not receive information from the follower uav (qball-x4) or wait for adjustments from it. to catch up with the leader, the follower directly flows in a curve at point (-0.75, -0.75), instead of hovering in the air waiting for the qbot2-l to change direction, as it would have done without failure. overall, the entire formation is still able to maintain a good formation even in the latter half, where both failures occur simultaneously. the failure of the uav qball-x4 does not affect the tracking of the qbot2-f on the other side. this demonstrates that the formation scheme studied in this research, along with the fault diagnosis and the fault-tolerant control scheme for uav swarms, has a certain engineering application value. fig. 15 top view trajectory when two actuators fail 5. conclusion this study proposed a fault-tolerant control strategy tailored for low-altitude uav swarms subject to actuator faults. a reduced-order sliding mode observer was designed to enable real-time fault diagnosis, with the estimated fault information actively fed into a sliding mode tracking controller to achieve timely compensation. this structure enhances both the robustness and real-time responsiveness of the system under actuator failure conditions. without the need for additional fault-specific sensors, actuator faults were reconstructed online using only the available system state information. in particular, the proposed reduced-order observer significantly improved computational efficiency while maintaining diagnostic accuracy, which is especially beneficial for embedded platforms with limited computational resources. the overall stability of the closed-loop system, incorporating the observer and controller, was rigorously established using the lyapunov fault-tolerant control of low-altitude unmanned aerial vehicle swarms using... 23 based methods. the proposed control scheme was further validated through experimental tests on a custom-built low-altitude uav 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s., aloboud, e., aldawood, a.s., youcef-toumi, k., 2019, adaptive task allocation for multi-uav systems based on bacteria foraging behaviour, applied soft computing, 83, pp. 105643-105661. 28. aljalaud, f., kurdi, h., youcef-toumi, k., 2023, autonomous multi-uav path planning in pipe inspection missions based on booby behavior, mathematics, 11(9), 2092. 29. liu, z.x., yu, x., yuan, c., zhang, y.m., 2015, leader-follower formation control of unmanned aerial vehicles with fault tolerant and collision avoidance capabilities, international conference on unmanned aircraft systems, denver, usa, pp. 1025-1030. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 231 244 doi: 10.22190/fume170505011p © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper automotive applications of evolving takagi-sugeno-kang fuzzy models  udc 681.5 radu-emil precup, stefan preitl, claudia-adina bojan-dragos, mircea-bogdan radac, alexandra-iulia szedlak-stinean, elena-lorena hedrea, raul-cristian roman politehnica university of timisoara, dept. automation and applied informatics, romania abstract. this paper presents theoretical and application results concerning the development of evolving takagi-sugeno-kang fuzzy models for two dynamic systems, which will be viewed as controlled processes, in the field of automotive applications. the two dynamic systems models are nonlinear dynamics of the longitudinal slip in the antilock braking systems (abs) and the vehicle speed in vehicles with the continuously variable transmission (cvt) systems. the evolving takagi-sugeno-kang fuzzy models are obtained as discrete-time fuzzy models by incremental online identification algorithms. the fuzzy models are validated against experimental results in the case of the abs and the first principles simulation results in the case of the vehicle with the cvt. key words: automotive applications, anti-lock braking systems, continuously variable transmission systems, dynamics, evolving takagi-sugenokang fuzzy models 1. introduction the main property of the evolving takagi-sugeno-kang fuzzy models, which gives them advantages over other fuzzy ones, consists in computing the rule bases by a learning process, that is, by continuous online rule base learning as shown in the classical and recent papers exemplified by [1–10]. the takagi-sugeno-kang fuzzy models are obtained by evolving the model structure and parameters in terms of online identification algorithms. the adding mechanism in the structure of online identification algorithms plays an  received may 05, 2017 / accepted june 20, 2017 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania e-mail: radu.precup@upt.ro 232 r.-e. precup, s. preitl, c.-a. bojan-dragos et al. important role because it adds or removes new local models; the evolving structure and parameters are, therefore, ensured. a well-acknowledged classification of the online identification algorithms oriented on the evolving takagi-sugeno-kang fuzzy models is presented in [11]. this classification highlights three categories of online identification algorithms, i.e., ii and iii, briefly pointed out as follows. category i of the adaptive algorithms starts with the initial structure of the takagi-sugeno-kang fuzzy model, given by other algorithms or by the experience of the specialist in the modeling or operation of the nonlinear dynamic system being modeled. the number of space partitions/clusters does not change over time, and these algorithms adapt just the parameters of the membership functions and the local models. category ii is represented by the incremental algorithms, which are applied in this paper, with examples referred to as the widely applied algorithms ran [12, 13], sonfin [14, 15], neurofast [16, 17], denfis [18, 19], scfnn [20, 21], ets [22], [23], flexfis [24, 25], and panfis [26]. these algorithms implement just adding mechanisms. category iii consists of evolving algorithms. besides the adding mechanism, category iii also implements removing and some of these algorithms merging and splitting mechanisms as well. this paper presents a part of the recent results obtained by the process control group of the politehnica university of timisoara, romania, in the development of evolving takagi-sugeno-kang fuzzy models obtained by online incremental algorithms. the paper continues the work carried out in [27] concerning the presentation of real-world applications of the evolving takagi-sugeno-kang fuzzy models that describe the dynamics of nonlinear systems in crane systems [28, 29], pendulum systems [30, 31], prosthetic hand fingers [32] and twin rotor aerodynamic systems [33]. the main difference with respect to [27] is that this paper applies incremental online identification algorithms to the derivation of evolving takagi-sugeno-kang fuzzy models for other process applications in order to characterize their dynamics. two automotive applications are treated in this paper: one concerns modeling of the longitudinal slip dynamics in the anti-lock braking systems (abss) laboratory [34], while the other one models the vehicle speed dynamics in the vehicles with the continuously variable transmission (cvt) systems. the evolving fuzzy models presented in this paper and the online identification algorithms are important because they are developed with the intention to be used in the process control. relevant process and control applications are presented in [35–43], with both crisp and fuzzy models. however, the online identification algorithms must be adapted accordingly in order to cope with the specific nonlinear elements and operating conditions of these processes [44–50]. the paper is organized as follows: an overview of incremental online identification algorithms is presented in the next section. several results related to the derivation of the evolving takagi-sugeno-kang fuzzy models for the two automotive applications are given in section 3. the examples of fuzzy models are validated against experimental results in case of the abs and the first principles simulation results in the case of vehicle with the cvt. the conclusions are outlined in section 4. automotive applications of evolving takagi-sugeno-kang fuzzy models 233 2. overview on incremental online identification algorithms the basic version of incremental online identification algorithm is implemented using the theoretical aspects described in [27] and [33] in terms of the software support of efs lab presented in [51] and [52]. the flowchart of the basic version of incremental online identification algorithm is presented in fig. 1, where tsk is the abbreviation of takagisugeno-kang. this algorithm proceeds in accordance with the following steps described as follows and also given in [27] and [33]: fig. 1 flowchart of basic version of incremental online identification algorithm [27] step 1. the rule base structure is initialized by setting all the parameters of rule antecedents so as to initially contain just one rule, namely nr = 1, where nr is the number of rules. the subtractive clustering is next applied to compute the parameters of the evolving takagi-sugeno-kang fuzzy models using first data point p1, with the general expression [22] of data point p in the input-output data set at discrete time step k, with notation pk: ,] ... [] ... [] [ ,] ... [ 1121 21 121     ntnnt n tt tn kkkk ppppyzzzy ppp zp p (1) where t stands for matrix transposition. 234 r.-e. precup, s. preitl, c.-a. bojan-dragos et al. the expression of the input-output data set is: ,}...1|{ 1 n k dkp (2) where d is the number of input-output data points or data points or data samples or samples. the rule base of takagi-sugeno-kang fuzzy models with affine rule consequents, also called the first-order sugeno fuzzy inference systems in some software programs and toolboxes implementations, is: ,...1 ,... then is and ... and is if: rule 110 11 rnniiiinini nizazaayltzltzi  (3) where zj, j = 1 … n, are input variables, n is the number of input variables, ltij, i =1 … nr, j = 1 … n, are input linguistic terms, yi is the output of the local model in the rule consequent of the rule with index i, i =1 … nr, and ail, i =1 … nr, l = 0 … n, are the parameters in the rule consequents. a more flexible input-output map of the takagi-sugeno-kang fuzzy models can be ensured if other expressions are included in the rule consequents. however, this complicates the parameter estimation specific to the rule consequents. the takagi-sugeno-kang fuzzy model structure considered in this paper includes the algebraic product t-norm as an and operator and the weighted average defuzzification method. this leads to the expression of output y of the takagi-sugeno-kang fuzzy model: ,...1 ,/ ,]1[ ,/ 1 111 r n i iii i t i n i ii n i i n i ii ni yyyy r rrr                         πz (4) where the firing degree of rule i and the normalized firing degree of rule i are i(z) and i, respectively, and the parameter vector of rule i is i, i =1 … nr. the expression of the firing degree is: ,...1 ),(...)()())(),...,(),((and)( 22112211 rniniininiii nizzzzzz  z (5) and the expression of the parameter vector is: ....1 ,]...[ 10 r t iniii niaaa π (6) the other parameters specific to the incremental online identification algorithm are initialized as follows using [22]: ,1)( , ,1 ,1 ,4.0 , ,]0 ... 0 0[])( ... )( )[(ˆ * 11 * 1 1112111   pzz icπππθ pnk r kr s ttt n tt r (7) where )1()1(   nnnn k rrc is the fuzzy covariance matrix related to the clusters, i is the nr(n+1) th order identity matrix,  = const,  > 0, is a large number, kθ̂ is an estimation of the parameter vector in the rule consequents at discrete time step k, and rs, rs > 0, is the automotive applications of evolving takagi-sugeno-kang fuzzy models 235 spread of all gaussian input membership functions ij, i =1 … nr, j = 1 … n, of the fuzzy sets of input linguistic terms ltij: ,...1 ,...1 ],))(/4(exp[)( 2* 2 njnizzrz rjijsjji  (8) * jiz i =1 … nr, j = 1 … n, are the membership function centers, * 1p in (7) is the first cluster center, * 1z is the center of rule 1 and also the projection of * 1p on axis z in terms of (1), and )( * 11 pp is the potential of * 1p . step 2. data sample index k is incremented, viz. replaced with 1k , and next data sample pk that belongs to the input-output data set defined in (2) is read. step 3. the potential of each new data sample pk(pk) and the potentials of the centers )( * lkp p of existing rules (clusters) with index l are recursively updated as: ].)()()(2/[)()1()( ,)( ,)( ,)( ],2)1)(1/[()1()( 1 1 2 )1( * 1 * 1 * 1 * 1 1 1 1 1 1 1 1 2 1 1 2                    n j j kklklklklk n j k l j l j kk n j k l j lk n j j kk kkkkk dppkpkp pppp kkp pppp p (9) step 4. the possible modification or upgrade of the rule base structure is carried out by means of the potential of the new data compared to the potential of the existing rules’ centers. the rule base structure is modified if certain conditions mentioned in [22], [2734] are fulfilled. step 5. the parameters in the rule consequents are updated using either the recursive least squares (rls) algorithm or the weighted recursive least squares (wrls) algorithm. these updates result in updated vectors kθ̂ and ck, k = 2 … d. step 6. the output of evolving takagi-sugeno-kang fuzzy model at next discrete time step k+1 is predicted as 1 ˆ ky : ,ˆˆ 1 k t kky θψ (10) where the general expression of (10) and the expressions of the vectors are: ].] 1[...] 1[] 1[[ ,]...[ , 21 21 t n ttt tt n ttt r r y zzzψ πππθθψ   (11) step 7. the algorithm continues with step 2 until all data points of the input-output data set presented in expression (2) are read. as emphasized in [31] and exemplified for a popular nature-inspired evolutionarybased optimization algorithm, rls and wrls in step 5 can be replaced with other optimization algorithms. some classical and nature-inspired evolutionary-based algorithms and various applications subjected to optimization problems are presented in [53–64], with focus on charged system search and gravitational search algorithms. along with limiting model complexity, i.e., number of rules and parameters, to allow for model generalization, there is a need to limit minimal model complexity to avoid 236 r.-e. precup, s. preitl, c.-a. bojan-dragos et al. trivial solutions. a way of doing it is to insert the constraints related to the optimization problem solved in step 5 because providing adequate learning data is not sufficient. however, not only upper constraints but lower ones as well should be considered as, for example, using a simple model with just one rule could mean that the evolving takagisugeno-kang fuzzy model is not a meaningful way of modeling the system. 3. two automotive applications 3.1. modeling the longitudinal slip in the anti-lock braking systems the continuous-time nonlinear state-space model of the abs process is derived on the basis of [34] and [65]: ),)(( ,)( ,)( 1311 2022222 11011111 mubcm mxdrfxj mmxdrfxj n n       (12) where  is longitudinal slip, j1 and j2 are inertia moments of the wheels shown in fig. 2, x1 and x2 are angular velocities, d1 and d2 are friction coefficients in the axes of the wheels, m10 and m20 are the static friction torques that oppose the normal rotation, 1m is the brake torque, r1 and r2 are the wheels radii, fn is the normal force that the upper wheel pushes upon the lower wheel, () is the friction coefficient, 1x and 2x are angular accelerations of the wheels, u is the control signal applied to the actuator, namely the direct current (dc) motor which drives the upper wheel, and the actuator’s nonlinear model is reflected in the nonlinear map b(u). fig. 2 abs experimental setup in the intelligent control systems laboratory of the politehnica university of timisoara, romania [65] automotive applications of evolving takagi-sugeno-kang fuzzy models 237 longitudinal slip  is defined as: ,0 ),/()( 2221122  xxrxrxr (13) the controlled output of the abs process is  if the longitudinal slip control is targeted, and notation y =  is employed in this sub-section in relation with the takagi-sugenokang fuzzy models presented in section 2. setting the sampling period to 0.01 s, several values of u have been generated in order to cover different ranges of magnitudes and frequencies. output y =  has been measured from the abs equipment. the evolution of the system input versus time is presented in fig. 3, which includes the input data for both training and validation (testing). fig. 3 abs process inputs versus time expressed as training data and validation (testing) data [34] the input signal illustrated in fig. 3 has been applied to the laboratory equipment to generate input-output data points (zk, yk), k = 1 … d, needed to be applied to the algorithm. fig. 3 illustrates the inputs that correspond to the set of d = 240 data points of the training data and the inputs of the other set of d = 60 data points of the testing data. the output values computed by the takagi-sugeno-kang fuzzy models and measured from the equipment will be next presented. a part of the real-time experimental results is exemplified in fig. 4 as the time responses of y versus time of the takagi-sugeno-kang fuzzy model with the input vector: ,] [ 211 t kkkkk yyuu z (14) with wrls applied in step 5 of the incremental online identification algorithm, and the real-world abs. the response of the real-world abs shown in fig. 4 is one of the results of the real-time experiments on the laboratory equipment. 238 r.-e. precup, s. preitl, c.-a. bojan-dragos et al. fig. 4 longitudinal slip position y =  versus time of the takagi-sugeno-kang fuzzy model (red) and real-world abs (blue) on the validation (testing) data set [34] as shown in [34], this fuzzy model has evolved to 9 rules and has 117 identified parameters. the performance is acceptable; it can be improved but this is constrained by the number of data samples considered for this process. a reduced number of data samples have been used, so the mode complexity has an upper bound in order to allow for model generalization. 3.2. modeling of the vehicle speed in the vehicles with continuously variable transmission systems the nonlinear system represented by the vehicle with the cvt system is presented in [66] as a vehicular power train system, which consists of other sub-systems, i.e., the internal combustion engine, the torque converter, the cvt and the vehicle. the main equations which model these sub-systems are given as follows. the internal combustion engine corresponds to a honda civic 1.6i sr 1598 cc car, which produces 113 ps din at 6300 rpm and 143 nm of torque at 4800 rpm. the engine is also characterized by the moment of inertia jeng = 0.284 kg m 2 and the engine speed is constrained. the engine speed dynamics is modeled as an inertia system: %,100%0 ,)( )( ),( , 2 2 max max1             throttle throttle tt tthrottlet ttj meng mp p engeng cengengeng  (15) which indicates the nonlinear torque characteristic. the torque converter (which consists of a pump, a turbine and a stator) is usually modeled by using the capacity factor/torque ratio versus speed ratio steady-state curves 2 and 3 given in the form of look-up tables. data on such tables is given in [66]. automotive applications of evolving takagi-sugeno-kang fuzzy models 239 the metallic v-belt driven cvt is dedicated to low-torque engine up to 200 nm. this cvt is characterized by gear ratio icvt. the overall transmission equations are: , , , ),(4 wcvtc trfrgw ctqcvttr vcvt i tit tiit vi     (16) where nonlinear map 4 is given as a look-up table with the parameters presented in [66]. the vehicle is a compact hatchback weighting about 1200 kg and an equivalent rotational moment of inertia jveh = 150 kg m 2 , and the common size of the wheels is 15” with 185/85 tires. the equations that characterize the vehicle dynamics are: .6.3 ,5.0100 ),sgn( , 2 wwv wroll wwdrag rolldragwwveh rv t ct tttj      (17) summing up the equations of the sub-systems using the structure, connection and parameters given in [66], the first principles model of the vehicle with cvt system is: ,9.0 , )),9.0(( )),9.0(()9.0(03.0 , )),9.0(( 5.34.37225.01047.32 2 1242 2 2 2 1 12423242 1242 2 2 2 1 1 2 1 6 1 xy xxx x xxxxx xxx x uuxuxx         (18) where the characteristic variables are input variable u = throttle(%), state variables x1 = eng and x1 = w, and the output variable represented by the vehicle speed y = vv (km/h). the nonlinear state-space model has been linearized in [66] around several operating points in order to allow for relatively simple controller designs. setting the sampling period to 0.1 s, several values of u have been generated in order to cover relatively wide ranges of magnitudes and frequencies. output y = vv has been obtained as the response of the nonlinear state-space model, which is the first principles model given in (18). the evolution of the system input versus time is presented in fig. 5, which includes the input data for both training and validation (testing). the input signal illustrated in fig. 5 has been applied to the system model given in (18) in order to generate input-output data points (zk, yk), k = 1 … d, needed to be applied to the algorithm. fig. 5 shows the inputs that correspond to the set of d = 3000 data points of the training data and the inputs of the other set of d = 3000 data points of the testing data. the output values computed by the takagi-sugeno-kang fuzzy models and obtained as the response of the system model given in (18) will be next presented. 240 r.-e. precup, s. preitl, c.-a. bojan-dragos et al. fig. 5 vehicle with cvt system inputs versus time expressed as training data and validation (testing) data a part of the real-time experimental results is exemplified in fig. 6 as the time responses of y versus time of the takagi-sugeno-kang fuzzy model with the input vector considered in (14), and the first principles model given in (18). the same input vector has been used as in the previous process model for the sake of simplicity. but the rls has been applied for this automotive process application in step 5 of the incremental online identification algorithm. fig. 6 vehicle speed y = vv versus time of the takagi-sugeno-kang fuzzy model (red) and real output (blue, i.e., output of first principles model) on the validation (testing) data set this fuzzy model has evolved to 5 rules and has 65 identified parameters. the performance is very good although the number of parameters is rather small. since the responses presented in fig. 6 are very close, the zoomed responses are illustrated in fig. 7. automotive applications of evolving takagi-sugeno-kang fuzzy models 241 fig. 7 zoomed vehicle speed y = vv versus time of the takagi-sugeno-kang fuzzy model (red) and real output (blue, i.e., output of first principles model) on a part of the validation (testing) data set 4. conclusions this paper has presented some results obtained by the process control group of the politehnica university of timisoara, romania, in the application of evolving takagisugeno-kang fuzzy models to two automotive process applications. a relatively simple incremental online identification algorithm, previously used by the authors in other nonlinear systems applications, has been applied to obtain the structure and parameters of the takagi-sugeno-kang fuzzy models. the main limitation of the models and the algorithm is the performance dependence on the parameters of the incremental online identification algorithm. this leads to parametric sensitivity and robustness problems, which can be solved by the proper definition and solving of optimization problems that include parametric sensitivity and robustness models related to the algorithm itself. the model-based fuzzy control of these automotive processes on the basis of evolving takagi-sugeno-kang fuzzy models will be treated as a future research direction. acknowledgements: the research was supported by the grants of the partnerships in priority 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by-nc-nd original research article normalized weighted geometric bonferroni mean operator of interval rough numbers – application in interval rough dematel-copras model udc 519.8:656 dragan pamučar, darko božanić, vesko lukovac, nenad komazec university of defence in belgrade, military academy, serbia abstract. this paper presents a new approach to the treatment of uncertainty and imprecision in the multi-criteria decision-making based on interval rough numbers (irn). the irn-based approach provides decision-making using only internal knowledge for the data and operational information of the decision-maker. a new normalized weighted geometric bonferroni mean operator is developed on the basis of the irn for the aggregation of the irn (irnwgbm). testing of the irnwgbm operator is performed through the application in a hybrid ir-dematel-copras multi-criteria model which is tested on the real case of selecting an optimal direction for the creation of a temporary military route. the first part of the hybrid model is the irn dematel model, which provides objective expert evaluation of criteria under the conditions of uncertainty and imprecision. in the second part of the model, the evaluation is carried out by using the new interval rough copras technique. key words: interval rough numbers, dematel, copras, bonferroni mean operator 1. introduction the decision-making theory comprises many multi-criteria decision-making models (mcdm) that support solving of various problems such as those in management science, urban planning issues, problems in natural sciences and military affairs, etc. according to triantaphyllou and mann [1], mcdm plays an important role in real-life problems, considering that there are many everyday decisions to be taken which include a number of criteria, while according to chen et al. [2], the multi-criteria decision making is an received may 03, 2018 / accepted june 07, 2018 corresponding author: dragan pamuĉar university of defence in belgrade, military academy, pavla jurišica šturma 33, 11000 belgrade, serbia e-mail: dpamucar@gmail.com 172 d. pamuĉar, d. boţanić, n. komazec, v. lukovac efficient systematic and quantitative manner of solving vital real-life problems in the presence of a large number of alternatives and several (opposing) criteria. the mcdm area is an area that has experienced remarkable advances in the last two decades, as demonstrated by numerous models developed in this area: the ahp (analytical hierarchical process) method [3, 4], the topsis (technique for order of preference by similarity to the ideal solution method) method [5], the vikor (vlsekriterijumska optimizacija i kompromisno resenje) method [6], the dematel (decision making trial and evaluation laboratory) method [7], the electre (elimination and choice expressing reality) method [8], the copras (complex proportional assessment) method [9], the mabac (multi-attributive border approximation area comparison) [10, 11], the edas (evaluation based on distance from average solution) method [12,13], the codas (combinative distance-based assessment) method [14, 15], mairca (multiattributive ideal-real comparative analysis) method [16,17]. as already mentioned, the mcdm models are used to solve many problems. in complex mcdm models, a large number of experts participate in order to find the most objective solution [18]. such models require the application of mathematical aggregators to obtain an aggregated initial decision-making matrix. there are many traditional aggregators used in group mcdm models, such as dombi aggregators [19], bonferroni aggregators [20], einstein and hamacher operators [21], heronian aggregation operators [22]. these aggregation operators have been widely used in theories of uncertainty such as fuzzy mcdm models [23-26], single-valued neutrosophic mcdm models [27-29], linguistic neutrosophic models [30, 31], etc. in this paper, a new approach in the theory of rough sets is applied to the treatment of uncertainty and imprecision contained in the data in group decision-making, namely, an approach based on interval rough numbers (irn). since this is a new approach, only traditional arithmetic aggregators have been used so far in the mcdm models based on rough numbers [34-36]. this paper presents the application and development of a new normalized weighted geometric bonferroni mean operator for the irn aggregation (irnwgbm). the application of the new irnwgbm operator is shown in hybrid irdematel-copras model. in the literature, there are numerous examples of using the dematel model for determining weight coefficients [17, 37], as well as the copras model for evaluating alternatives [9]. however, so far in the literature the dematel and copras models based on interval rough numbers are not familiar. to the best of this author’s knowledge, there is no hybrid ir-dematel-copras model in the field of mcdm, which in this way takes into consideration mutual dependence of criteria, evaluates alternatives and treats imprecision and uncertainty with the irn. one of the goals of this paper is the development of a new irnwgbm operator for the irn aggregation. the second goal of this paper is the improvement of the mcdm area through the development of a new hybrid ir-dematel-copras model based on the irn. the rest of the paper is organized as follows. the second chapter presents a mathematical analysis of interval rough numbers and the development of new irnwgbm operator. the third chapter presents the algorithm of hybrid ir-dematel-copras model, which is later tested in the fourth chapter using a real example of selecting an optimal direction for the creation of a temporary military route. in the fifth chapter, the concluding observations are presented with a special emphasis on the directions for future research. normalized weighted geometric bonferroni mean operator of interval rough numbers... 173 2. interval rough numbers and normalized weighted geometric bonferroni mean operator if we suppose that there is a set of k classes which present the preferences of a dm, r=(j1,j2,...,jk), provided that these belong to the series which meets the condition where j1<j2<,...,<jk and another set of m classes which also present the preferences of a dm, r * =(i1,i2,...,ik). all the objects are defined in the universe and related to the preferences of a dm. in r * every object class is presented in the interval ii={ili,iui}, meeting the condition where ili≤iui (1≤i≤m), as well as the condition where ili,iuir. then, ili presents the lower limit of the interval, while iui presents the upper limit of the interval of the i-th class of objects. if both upper and lower limits of the class of objects are sorted so that i * l1< i * l2<…< i * lj, i * u1<i * 2u<…<i * uk (1≤j,k≤m), respectively, then we can define the two new sets containing the lower class of objects rl * =(i * l1,i * l2,…,i * lj) and upper class of objects ru * =(i * u1,i * u2,…,i * uk), respectively. then, for any class of the objects i * lir (1≤i≤j) and i * ui  r (1≤i≤k) we can define the lower approximation i * li and i * ui as follows [38]:  * * *( ) / ( )  li l liapr i y u r y i (1)  * * *( ) / ( )  ui u uiapr i y u r y i (2) the upper approximations i * li and i * ui are defined by applying the following expressions:  * * *( ) / ( )li l liapr i y u r y i   (3)  * * *( ) / ( )ui u uiapr i y u r y i   (4) both classes of objects (upper and lower class of the objects i * li and i * ui) are defined by their lower limits *( )lilim i and *( )uilim i and upper limits *( )lilim i and *( )uilim i , respectively * * *1 ( ) ( ) ( )li l li l lim i r y y apr i m   (5) * * * * 1 ( ) ( ) ( )ui u ui l lim i r y y apr i m   (6) where ml and m * l present the sum of objects contained in the lower approximation of the classes of objects i * li and i * ui, respectively. the upper limits *( )lilim i and *( )uilim i are defined by eqs. (7) and (8) * * *1 ( ) ( ) ( )li l li u lim i r y y apr i m   (7) * * * * 1 ( ) ( ) ( )ui u ui u lim i r y y apr i m   (8) 174 d. pamuĉar, d. boţanić, n. komazec, v. lukovac where mu and m * u present the sum of objects contained in the upper approximation of the classes of objects i * li and i * ui, respectively. then, the uncertain class of objects i * li and i * ui can be shown with their lower and upper limit * * *( ) ( ), ( )li li lirn i lim i lim i    (9) * * *( ) ( ), ( )ui ui uirn i lim i lim i    (10) as can be seen, every class of objects is defined by its upper and lower limits, which consist of interval rough numbers, defined as follows * * *( ) ( ), ( )i li uiirn i rn i rn i    (11) interval rough numbers are characterized by specific arithmetic operations differing from the arithmetic operations with classic rough numbers. detailed arithmetic operations with the irn and mutual comparison of the irn are presented in pamuĉar et al. [34]. definition 1 [20]. let (a1,a2,…,an) be a set of non-negative numbers, the function nwgbm: r n →r, wi (i=1,2,…,n) be the relative weight of ai (i=1,2,…,n), wi  [0,1] and 1 1 n ii w   . if p,q≥0 and normalized weighted geometric bonferroni mean operator satisfies:  , 1 1 2 , 1 1 ( , ,..., ) i j i w wn p q w n i j i j nwgbm a a a pa qa p q       (12) then nwgbm p,q is called a normalized weighted geometric bonferroni mean (nwgbm) operator. definition 2 set irn(ξi)=[rn(ξi l ),rn(ξi u’ )]=([ξi l ,ξi u ],[ξi ’l ,ξi ’u ]) (i=1,2,..,n) as a collection of interval rough numbers (irns) in ψ, then the irnwbm can be defined as follows  , 1 1 2 , 1 1 ( ( ), ( ),..., ( )) ( ) ( ) i j i w wn p q w n i j i j i j irnwgbm irn irn irn pirn qirn p q             (13) where iw is the relative weight of irn(ξi), wi  [0,1] and 1 1 n ii w   , wj is the relative weight of irn(ξj), wj  [0,1] and 1 1 n jj w   . according to the arithmetic operations applied in interval numbers and definition 2, we can obtain the following theorems: theorem 1 set irn(ξi)=[rn(ξi l ),rn(ξi u’ )]=([ξi l ,ξi u ],[ξi ’l ,ξi ’u ]) (i=1,2,..,n) as a collection of irns in  , then according to eq. (12), aggregation results obtained is still rn, and we can get the following aggregation formula normalized weighted geometric bonferroni mean operator of interval rough numbers... 175         , 1 1 2 , 1 ' ' 11 , 1 , 1 1 , 1 1 ( ( ), ( ),..., ( )) ( ) ( ) 1 1 ( ) ( ) , ( ) ( ) 1                                            i j i i ji j ii i j i w wn p q w n i j i j i j w ww wn n l l u u ww i j i j i j i j i j i j w w l l w i j i j irnwgbm irn irn irn pirn qirn p q prn qrn prn qlim p q p q p q p q       1 , 1 ' ' ' '1 1 , 1 , 1 1 , , 1 1 ,                                                  i j i i j i j i i w wn n u u w i j i j i j i j w w w wn n l l u uw w i j i j i j i j i j i j p q p q p q p q p q p q (14) proof.  ' ' '( ) ( ), ( ) , , ,l u l u l u i i i i i i iirn rn rn                  ;  ' ' '( ) ( ), ( ) , , ,l u l u l u i i i i i i ipirn prn prn p p p p                  ;  ' ' '( ) ( ), ( ) , , ,l u l u l u i i i i i i iqirn qrn qrn q q q q                  ;  ' ' ' '( ) ( ) , , ,l l u u l l u u i j i j i j i j i jpirn qirn p q p q p q p q                       ' ' ' '1 11( ) ( ) , , , i j i ji j i ii w w w ww w l l u u l l u uw ww i j i j i j i j i jpirn qirn p q p q p q p q                                       1 , 1 1 1 , 1 , 1 ' ' ' '1 1 , 1 , 1 1 ( ) ( ) 1 1 , , 1 1 ,                                                              i j i i j i j i i i j i j i i w wn w i j i j i j w w w wn n l l u uw w i j i j i j i j i j i j w w w wn n l l u uw w i j i j i j i j i j i j pirn qirn p q p q p q p q p q p q p q p q p q            so, theorem 1 is true. theorem 2 (idempotency). set irn(ξi)=[rn(ξi l ),rn(ξi u’ )]=([ξi l ,ξi u ],[ξi ’l ,ξi ’u ]) (i=1,2,..,n) as a collection of irns in ψ, if irn(ξi)= irn(ξ), then , , 1 2( ( ), ( ),.., ( )) ( ( ), ( ),.., ( )).     p q p q nirnwgbm irn irn irn irnwgbm irn irn irn 176 d. pamuĉar, d. boţanić, n. komazec, v. lukovac proof. since irn(ξi)= irn(ξ), i.e. ξi l =ξ l , ξi u =ξ u , ξi ’l =ξ ’l and ξi ’u =ξ ’u for i=1,2,..,n, then         , 1 1 2 , 1 ' ' '1 1 1 , 1 , 1 , 1 1 ( ( ), ( ),..., ( )) ( ) ( ) 1 1 1 1 , , , i j i i j i j i j i i i w wn p q w n i j i j i j w w w w w wn n n l l u u l l uw w w i j i j i j i i j i j i j i j i j i j irnwgbm irn irn irn pirn qirn p q p q p q p q p q p q p q p q p q                                                           ' 1 , 1 ' ' ' '1 1 1 1 , 1 , 1 , 1 , 1 1 1 1 1 , , , 1 i j i i j i j i j i j i i i i w wn u w j i j i j w w w w w w w wn n n n l l u u l l u uw w w w i j i j i j i j i j i j i j i j p q p q p q p q p q p q p q p q p                                                                                       ' '1 1 1 1 , 1 , 1 , 1 , 1 1 1 , 1 , 1 1 1 1 ( ) , ( ) , ( ) , ( ) , i j i j i j i j i i i i i j i j i i w w w w w w w wn n n n l u l uw w w w i j i j i j i j i j i j i j i j w w w wn n l uw w i j i j i j i j p q p q p q p q q p q p q p q                                                                         ' '1 1 , 1 , 1 ' ' , , , , , ( ), ( ) ( ) i j i j i i w w w wn n l uw w i j i j i j i j l u l u l urn rn irn                                             the proof of theorem 2 is completed. theorem 3 (boundedness). let irn(ξi)=[rn(ξi l ),rn(ξi u’ )]=([ξi l ,ξi u ],[ξi ’l ,ξi ’u ]) (i=1,2,..,n) as a collection of irns in ψ, let  ' '( ) min ,min , min ,minl u l uirn              and  ' '( ) max ,max , max ,maxl u l uirn              , then , 1 2( ) ( ( ), ( ),.., ( ) ( )p q nirn irnwgbm irn irn irn irn       . proof. let irn(ξ)=min(irn(ξ1),irn(ξ2),…,irn(ξn))=([minξi l ,minξi u ],[minξi ’l ,minξi ’u ]) and irn(ξ + )=max(irn(ξ1),irn(ξ2),…,irn(ξn))=([maxξi l ,maxξi u ],[maxξi ’l ,maxξi ’u ]). then we have minξi l =min(ξi l ), minξi u =min(ξi u ), minξi ’l =min(ξi ’l ), minξi ’u =min(ξi ’u ), maxξi l =max(ξi l ), maxξi u =max(ξi u ), maxξi ’l =max(ξi ’l ) and maxξi ’u =max(ξi ’u ). based on that we have                 ' ' ' ' ' ' ( ) ( ) ( ); min max ; min max ; min max ; min max ; i l l l i i i u u u i i i l l l i i i u u u i i i irn irn irn                          according to the inequalities showed above, we can conclude that irn(ξ ) ≤ irnwgbm p,q (irn(ξ1),irn(ξ2),…,irn(ξn)) ≤ irn(ξ + ) holds. normalized weighted geometric bonferroni mean operator of interval rough numbers... 177 theorem 4 (commutativity). let rough set (irn(ξ1 ’ ),irn(ξ2 ’ ),…,irn(ξn ’ )) be any permutation of (irn(ξ1),irn(ξ2),…,irn(ξn)). then there is irnwgbm p,q (irn(ξ1),irn(ξ2),…,irn(ξn)) =irnwgbm p,q (irn(ξ1 ’ ),irn(ξ2 ’ ),…,irn(ξn ’ )). proof. the property is obvious. 3. irn dematel-copras model 3.1. extension of the dematel method based on interval rough numbers using the dematel method, the dependent factors are considered and the degree of dependency between them is determined [32]. the method is based on the graph theory and enables visual planning and problem solving. this method allows better understanding of the relationship between factors, the relationship between the level of structure and the strength of factor influence [33, 34]. as the result of the method application, total direct and indirect effects of every factor upon other factors as well as those received from other factors are obtained. in order to comprehensively consider imprecision and uncertainty existing in group decision-making, in this paper the modification of the dematel method is performed by using interval rough numbers. their use eliminates the need for additional information in order to determine uncertain number intervals [34]. so far, in the literature the modification of the dematel method by applying interval rough numbers (ir'dematel) for determining interval rough coefficients of weight criteria has not been considered. in the following part, the steps of the ir'dematel method are elaborated: step 1: expert analysis of factors. assuming that there are m experts and n factors which are considered, every expert should determine the degree of influence of factor i on factor j. a comparative analysis of the pair of i-th and j-th factor by e-th expert is marked with xij e , where: i=1,...,n; j=1,...,n. the value of every pair xij e has one whole number value with the following meaning: 0 – no influence; 1 – low influence; 2 – middle influence; 3 – high influence; 4 – very high influence. the response of the e-th expert is shown with nonnegative matrix of the range n×n, and every element of the e-th matrix in the expression x e =[x e ij]n×n marks the whole nonnegative number x e ij, where 1 ≤ e ≤ b. ' ' 12 12 1 1 ' ' 21 21 2 2 ' ' 1 1 2 2 0 ; ; ; 0 ; ; 1 , ; 1 ; ; 0 e e e e n n e e e e e n n e e e e n n n n nxn x x x x x x x x x i j n e b x x x x                  (15) where xij e and xij e’ present linguistic expressions from the predefined linguistic scale by which expert e presents his comparison in the pairs of criteria. therefore, matrices x 1 , x 2 , …, x m are those of the response of every of b experts. the diagonal matrix elements of the responses of all experts have the value zero because the same factors have no influence. if expert e has uncertainty during a pair comparison of criteria (i,j), that is, if expert e cannot decide between two values from the linguistic scale, then both values from the 178 d. pamuĉar, d. boţanić, n. komazec, v. lukovac scale are stated in matrix x e . then, at the position (i,j) in matrix x e we have different values of x e ij, that is, x e ij ≠ x e’ ij. if there is no uncertainty, expert e unequivocally selects one value. then, in the matrix of the comparison of criteria (x e ) is entered the same value at the position ( ,i j ), that is x e ij= x e’ ij. step 2: determination of the matrix of average responses of experts. based on the matrices of the responses x e =[x e ij]n×n of all m experts, by applying eqs. (1-11) are determined classes of objects and defined interval rough numbers in matrices x 1 , x 2 , …, x b . that is how interval rough matrices x e (i=1,2,...,b) are obtained and presented in the form 0 ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 e e ij ij e e ij ij e e e ij ij irn x irn x irn x irn x x irn x irn x              (16) where e (e=1,2,...,b) presents the mark (number) of the expert, and irn(x e ij) presents the interval rough number presented in the form irn(x e ij)=[rn(xij el ),rn(xij e’u )]=([xij el , xij eu ], [xij e’l ,xij e’u ]). by applying the irnwgbm operator, eq. (14), we obtain averaged interval rough number irn(x e ij)=[rn(xij el ),rn(xij e’u )]. in this way we obtain averaged rough matrix of average responses z 12 1 21 2 1 2 0 ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 n n n n irn z irn z irn z irn z z irn z irn z             (17) step 3: based on matrix z initial direct-relation matrix d=[irn(dij)]n×n is calculated, see eq. (18). by normalization every element of matrix d takes the value between zero and one. matrix d is obtained when every element irn(zij) of matrix z is divided by rough number irn(s), see eqs. (18)-(20) 12 1 21 2 1 2 0 ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 n n n n irn d irn d irn d irn d d irn d irn d             (18) where irn(dij) is obtained by applying eq. (18) ' ' ' ' ( ) ( ) , , , ( ) l u l u ij ij ij ij ij ij u l u l ij ij ij ij irn z z z z z irn d irn irn s s s s z                      (19) the value of interval rough number irn(s) is obtained by applying eq. (20) normalized weighted geometric bonferroni mean operator of interval rough numbers... 179         ' ' 1 1 1 1 ( ) max max ,max , max ,max n n n nl u l u ij ij ij ijj j j j irn s x x x x                   (20) step 4: the total relation matrix (t=[irn(tij)]n×n) of the range n×n is calculated, according to eq. (21). element irn(tij) presents direct influence of factor i on factor j, and matrix t shows total relations between every pair of factors. 2 1 lim( ... )m i mm t d d d d        (21) where 2 1 1 2 1 1 1 2 1 1 1 ... ( ... ) ( ) ( )( ... ) ( ) ( ) ( ) i m m m m m d d d d d i d d d d i d i d i d d d d i d i d d i d                               (22) where i is the unit matrix of the range n×n. based on eqs. (21) and (22), the total relation matrix is obtained: 11 12 1 21 22 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n nn irn t irn t irn t irn t irn t irn t t irn t irn t irn t             (23) where irn(tij)=[rn(tij l ),rn(tij ’u )] is interval rough number by which indirect effects of factor i on factor j are expressed. then, matrix t shows mutual dependence of every pair of factors. step 5: calculation of the sum of rows and columns of total relation matrix t. in total relation matrix t the sum of rows and sum of columns is presented by vectors r and c with the range n×1:  ' ' 1 1 1 1 11 1 ( ) ( ) , , , n n n n nl u l u i ij ij ij ij ijj j j j nj n irn r irn t t t t t                            (24)  ' ' 1 1 1 1 11 1 ( ) ( ) , , , n n n n nl u l u i ij ij ij ij iji i i i ni n irn c irn t t t t t                            (25) value ri presents the sum of the i-th row of matrix t, and shows total direct and indirect effects which criterion i provided to other criteria. value ci presents the sum of the j-th column of matrix t, and shows total direct and indirect effects that criterion j received from other criteria [10]. step 6: determination of weight coefficients of criteria (wi). the calculation of weight coefficients of criteria is performed based on the values obtained in step 5, see eq. (26) 180 d. pamuĉar, d. boţanić, n. komazec, v. lukovac 2 2 2 2 ' ' ' ' 2 ' ' 2 ' ' ' 2 ' ' 2 ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) l l l l l j i i i i u u u u u j i i i il u j j j l l l l l j i i i i u u u u u j i i i i w r c r c w r c r c irn w rn w rn w w r c r c w r c r c                           (26) where the values ri + ci and ri  ci are obtained by applying eqs. (27) and (28) 1 1 1 1 ' ' ' ' 1 1 1 1 , , ( ) ( ) ,                                  n n n nl l u u ij ij ij ijj i j i i i n n n nl l u u ij ij ij ijj i j i t t t t irn r irn c t t t t (27) ' ' 1 1 1 1 ' ' 1 1 1 1 , , ( ) ( ) ,                                  n n n nl u u l ij ij ij ijj i j i i i n n n nl u u l ij ij ij ijj i j i t t t t irn r irn c t t t t (28) the normalization of weight coefficients is performed by applying eq. (29) ' ' ' ' 1 1 1 1 1 ( ) ( ) , , , ( ) l u l u j j j j j j n n n n nu l u l j j j j jj j j j j irn w w w w w irn w irn w w w w w                                (29) where n denotes the number of the evaluation criteria, irn(wi) final values of weight coefficients which are used in the decision-making process. 3.2. extension of the copras method based on interval rough numbers every mcdm method is characterized by specific mathematical apparatus. the copras method is characterized by a somewhat more complex aggregation process of the values of criteria functions and a simplified procedure for data normalization (the character of criteria is not considered min/max). in the following part, the mathematical apparatus of the copras method is briefly presented. step 1: a group of experts (e=1,2,…,b) is formed where b presents the number of experts who select the criteria and define the elements of the initial decision-making matrix. the problem is formally presented by the selection of one of m options (alternatives), ai, i=1,2,…,m, which are evaluated and compared mutually based on n criteria (xj, j=1,2,…,n) whose values we know. the alternatives are shown with vectors x e ij; x e* ij, where x e ij; x e* ij presents the value of the i-th alternative by j-th criterion. based on eqs. (1)-(11), the evaluations of experts by vectors xij are transformed into interval rough vectors ai=(irn(xi1), irn(xi2),…, irn(xin)), where irn(xij)= [rn(xij l ),rn(xij ’u )]=([xij l ,xij u ],[xij ’l ,xij ’u ]) presents the value of the i-th alternative by j-th criterion (i=1,2,…,m ; j=1,2,…,n). since the criteria affect differently final values of alternatives, to every criterion is attributed weight coefficient wj, j=1,2,…,n which reflects its relative significance in the evaluation of alternatives. normalized weighted geometric bonferroni mean operator of interval rough numbers... 181 in that way matrices xe=x1,x2,...,xb (e=1,2,…,b) are obtained in which b experts performed the comparison in pairs of criteria. 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ... ( ) ( ) ... ( ) n n n e m m m mn m n c c c a irn x irn x irn x a irn x irn x irn x x a irn x irn x irn x              (30) where m is the number of alternatives, and n is total number of criteria. step 2: normalization of the initial decision-making matrix (xe). basic objective of the normalization of criteria values is the transformation of different values of criteria (benefit or cost) into the values allowing mutual comparison. by applying the irnwgbm operator, from eq. (14), averaged interval rough number irn(xij)=[rn(xij l ),rn(xij ’u )] is obtained which is further normalized in matrix d. normalized values are shown in matrix d 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ... ( ) ( ) ... ( ) n n n m m m mn m n c c c a irn d irn d irn d a irn d irn d irn d d a irn d irn d irn d              (31) the elements of normalized matrix irn(dij) are obtained by applying the expression    ' ' ' ' ' ' 1 1 1 1 1 , , ,( ) ( ) , , , ( ) , , , l u l u ij ij ij ijijl u l u ij ij ij ij ij m m m m m l u l u ij ij ij ij ij i i i i i x x x xirn x irn d d d d d irn x x x x x                                          (32) where irn(xij) presents the elements of the initial decision-making matrix (xe), irn(dij) presents normalized values of the elements of the initial decision-making matrix, m presents total number of alternatives. step 3: in the third step the weighted normalized matrix (z) is formed in which the elements of normalized matrix (d) are multiplied by weights of criteria (irn(wj)) 11 12 1 21 22 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ( ) ( ) ... ( ) n n m m mn irn z irn z irn z irn z irn z irn z z irn z irn z irn z             (33) where the elements of matrix z are obtained by multiplying normalized elements of the matrix given in eq. (31) by weight coefficients of criteria irn(wj), respectively, irn(zij)= irn(dij)* irn(wj). 182 d. pamuĉar, d. boţanić, n. komazec, v. lukovac step 4: in the following or fourth step, are summed the values of matrix z by columns. the values are summed up depending on the group of criteria they belong to (min or max). the values of max criteria (higher values of criteria are desirable) are obtained by the application of eq. (34), respectively, eq. (35) ( ) ( ) i i ij z irn s irn z   (34) where zi=+ presents the set of max criteria, respectively, 1 ( ) ( ) ( ) i k ij j i irn s irn d irn w    (35) where k presents the total number of max criteria. the values of min criteria (lower values of criteria are desirable) are obtained by applying eq. (36), respectively, eq. (37) ( ) ( ) i i ij z irn s irn z   (36) where zi=  presents the set of min criteria, respectively, 1 ( ) ( ) ( ) i p ij j i irn s irn d irn w    (37) where p presents total number of min criteria. step 5: in the fifth step by applying eq. (38) the relevance (influence) of every observed alternative from the set of alternatives being compared is determined. min 1 min 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( )                           m ii i i m i i i m ii i m i i i irn s irn s irn q irn s irn s irn s irn s irn s irn s irn s irn s (38) step 6: in the last or sixth step, the alternatives are ranged based on the values of criteria functions which are assigned to every alternative, where as the most desirable alternative is selected the one with the highest value of criteria function. 4. application of the nwgbm operator in the ir’dematel-copras model the ir'dematel-copras model with the nwgbm operator was tested on the problem of selecting an optimal direction for making a temporary military route. the temporary military route represents a type of route with limited duration [39]. these routes are mostly used for a short time, usually during combat operations, sometimes for disposable use. they are built on the directions of the movement of units in situations normalized weighted geometric bonferroni mean operator of interval rough numbers... 183 where the existing roads are insufficient or highly damaged [39]. such roads are built for taking position, supply, when the existing roads or road network is to be circumvented or because of certain objects settlements located on the existing roads, etc. the existing literature elaborates the methodology for defining direction of the temporary military route taking into account primarily the length of route and the scope of works. other segments, which have a significant influence, are usually not elaborated. for this reason, the criteria that influence the selection of the temporary route direction are further elaborated in table 1. table 1 criteria for selecting the temporary route direction name of criterion description of criterion scope of works (c1) this criterion defines the scope of works necessary for the construction of a particular road section. the scope of works depends on the type of soil and its carrying capacity, in relation to the maximum type of load planned for transport via the route considered. the criterion is presented through qualitative parameters and belongs to the group of min criteria. critical points (c2) through this criterion a number of potential regions is defined, where it is possible for an enemy with significant prospects of success to set an ambush. the criterion is of quantitative character and belongs to the group of min criteria. length of route (c3) this criterion defines the length of route, which further affects the time when the units are retained on it. this increases or decreases the security of the people and means using a temporary military route. the criterion is of quantitative character and belongs to the group of min criteria. masking the movement (c4) in this criterion, through linguistic descriptions are defined the possibilities of masking the movement of units while moving on a temporary military route. the criterion is described by linguistic values and belongs to the group of max criteria. capacities for reparation and reconstruction of route (c5) capacities necessary for reparation and reconstruction of the route. for the purpose of quantification of this criterion, a working group of components is defined including: grader, dozer, roller, loader and two self-loaders. the evaluation of the criteria is based on the required number of working groups it and belongs to the group of min criteria. capacities for providing supply, respectively, the movement of units on the route made (c6) these units monitor the movement of own forces, as well as the activities of the enemy. with their presence, they should prevent attacks on the vehicles moving along the way. the basic unit that quantifies this criterion is the shooting unit. the evaluation of the criteria is based on the required number of shooting units and it belongs to the group of min criteria. 184 d. pamuĉar, d. boţanić, n. komazec, v. lukovac the application of the hybrid ir'dematel-copras model with the nwgbm operator is shown on the example of the evaluation of six routes for the construction of a temporary military route in southern serbia. the routes considered are marked with a1 to a6. in the first phase of the ir'dematel-copras model, the weight coefficients of the criteria are calculated using the ir'dematel model.in the first step of the ir'dematel model, an expert analysis of the factors is performed. in this research, three experts took part in the evaluation of the criteria using the scale: 0 – no influence; 1 – low influence; 2 – middle influence; 3 – high influence; 4 – very high influence; 5 – extremely high influence. the weight coefficients of the experts were determined (0.337, 0.314, 0.349) t . after the expert evaluation, three matrices of comparisons were obtained in pairs of criteria with the dimension 66, (table 2). table 2 expert evaluation of criteria expert 1 c1 c2 c3 c4 c5 c6 c1 0;0 3;4 4;5 5;5 3;4 4;4 c2 4;5 0;0 3;4 4;4 5;5 5;5 c3 2;3 1;2 0;0 4;4 3;4 4;5 c4 2;2 3;3 4;5 0;0 4;4 3;4 c5 2;3 2;2 2;3 1;2 0;0 5;5 c6 3;4 2;3 2;3 3;4 2;3 0;0 expert 2 c1 c2 c3 c4 c5 c6 c1 0;0 4;5 5;5 5;5 5;5 5;5 c2 5;5 0;0 4;5 4;4 4;4 5;5 c3 2;3 3;4 0;0 4;5 4;5 4;4 c4 3;4 3;4 3;4 0;0 4;5 4;5 c5 2;3 1;2 3;4 2;3 0;0 4;5 c6 4;5 3;4 2;2 2;3 3;4 0;0 expert 3 c1 c2 c3 c4 c5 c6 c1 0;0 5;5 4;5 3;4 5;5 4;4 c2 5;5 0;0 3;4 4;5 5;5 4;5 c3 3;4 2;3 0;0 5;5 4;5 5;5 c4 3;4 3;4 3;4 0;0 3;3 4;4 c5 2;3 1;2 2;3 1;2 0;0 4;5 c6 3;4 4;5 3;4 3;4 3;4 0;0 in accordance with the procedure for implementing the ir-dematel model, the initial matrices of comparison in pairs of criteria are transformed into the interval rough matrices by means of eqs. (1-11). thus, we obtain three interval rough matrices of the criteria, table 3. normalized weighted geometric bonferroni mean operator of interval rough numbers... 185 table 3 interval rough matrices of comparisons in criteria pairs expert 1 c1 c2 ... c6 c1 [(0, 0),(0, 0)] [(3, 3.67),(4, 4.67)] ... [(4, 4.33),(4, 4.33)] c2 [(4, 4.67),(5, 5)] [(0, 0),(0, 0)] ... [(4.67, 5),(5, 5)] c3 [(2, 2.33),(3, 3.33)] [(1, 2.33),(2, 3.33)] ... [(4, 4.33),(4.67, 5)] c4 [(2, 2.67),(2, 3.33)] [(3, 3),(3, 3.67)] ... [(3, 3.67),(4, 4.33)] c5 [(2, 2),(3, 3)] [(1.67, 2),(2, 2.33)] ... [(4.33, 5),(5, 5)] c6 [(3, 3.33),(4, 4.33)] [(2, 2.67),(3, 3.67)] ... [(0, 0),(0, 0)] expert 2 c1 c2 ... c6 c1 [(0, 0),(0, 0)] [(3.67, 4),(4.67, 5)] ... [(4.33, 5),(4.33, 5)] c2 [(4.67, 5),(5, 5)] [(0, 0),(0, 0)] ... [(4.67, 5),(5, 5)] c3 [(2, 2.33),(3, 3.33)] [(2.33, 3),(3.33, 4)] ... [(4, 4.33),(4, 4.67)] c4 [(2.67, 3),(3.33, 4)] [(3, 3),(3.67, 4)] ... [(3.67, 4),(4.33, 5)] c5 [(2, 2),(3, 3)] [(1, 1.67),(2, 2.33)] ... [(4, 4.33),(5, 5)] c6 [(3.33, 4),(4.33, 5)] [(2.67, 3),(3.67, 4)] ... [(0, 0),(0, 0)] expert 3 c1 c2 ... c6 c1 [(0, 0),(0, 0)] [(3.67, 4),(4.67, 5)] ... [(4, 4.33),(4, 4.33)] c2 [(4.67, 5),(5, 5)] [(0, 0),(0, 0)] ... [(4, 4.67),(5, 5)] c3 [(2.33, 3),(3.33, 4)] [(2.33, 3),(3.33, 4)] ... [(4.33, 5),(4.67, 5)] c4 [(2.67, 3),(3.33, 4)] [(3, 3),(3.67, 4)] ... [(3.67, 4),(4, 4.33)] c5 [(2, 2),(3, 3)] [(1.67, 2),(2.33, 3)] ... [(4, 4.33),(5, 5)] c6 [(3, 3.33),(4, 4.33)] [(2.67, 3),(3.67, 4)] ... [(0, 0),(0, 0)] in the second step of the ir-dematel model, using nwgbm, eq. (14), the aggregation (averaging) of the interval rough matrices of the experts’ responses is carried out. thus, we obtain a centralized interval rough matrix of the average responses of the criteria experts, table 4. table 4 averaged interval rough matrix of criteria c1 c2 ... c6 c1 [(0.0,0.0),(0.0,0.0)] [(3.4,3.9),(4.4,4.9)] ... [(4.1,4.5),(4.1,4.5)] c2 [(4.4,4.9),(5.0,5.0)] [(0.0,0.0),(0.0,0.0)] ... [(4.4,4.9),(5.0,5.0)] c3 [(2.1,2.6),(3.1,3.6)] [(1.9,2.8),(2.9,3.8)] ... [(4.1,4.6),(4.5,4.9)] c4 [(2.4,2.9),(2.9,3.8)] [(3.0,3.0),(3.4,3.9)] ... [(3.4,3.9),(4.1,4.5)] c5 [(2.0,2.0),(3.0,3.0)] [(1.4,1.9),(2.1,2.6)] ... [(4.1,4.6),(5.0,5.0)] c6 [(3.1,3.5),(4.1,4.5)] [(2.4,2.9),(3.4,3.9)] ... [(0.0,0.0),(0.0,0.0)] 186 d. pamuĉar, d. boţanić, n. komazec, v. lukovac averaging of the elements of interval rough matrix of comparison in pairs of criteria at the c3-c1 position is performed by applying eq. (14):                        1, 1 0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349 1 0.337 1 0.337 1 0.314 1 0.314 0.349 0.337 1 0.349 2, 2.33 , 3, 3.33 , 2, 2.33 , 3, 3.33 , 2.33, 3 , 3.33, 4 2 2 2 2.33 2 2 2 2.331 2 2.33 2 2.33                                   p qirnwgbm               0.349 0.314 1 0.349 0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349 1 0.337 1 0.337 1 0.314 1 0.314 0.349 0.337 0.349 0.314 1 0.349 1 0.349 2.11 2 2.3 2.3 2.3 3 2.3 2.3 2.3 31 2.55 2 3 2.3 3 2.3 1                                                          0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349 1 0.337 1 0.337 1 0.314 1 0.314 0.349 0.337 0.349 0.314 1 0.349 1 0.349 0.337 0.314 0.337 0.349 1 0.337 1 3 3 3 3.3 3 3 3 3.3 3.11 2 3.3 3 3.3 3 3.3 3.3 3.3 41 2                                                 0.314 0.337 0.314 0.349 0.337 1 0.314 1 0.314 0.349 0.337 0.349 0.314 1 0.349 1 0.349 3.3 3.3 3.3 4 3.56 4 3.3 4 3.3 2.11,2.55 , 3.11,3.56                                              after determining the averaged matrix of criteria (table 4), using eqs. (18-20), the third step of the ir-dematel model is carried out, which assumes the determination of the initial direct-relation matrix. in the next step, using eqs. (21-23), the initial directrelation matrix is transformed into total relation matrix of the criteria. on the basis of total relation matrix, direct and indirect effects are determined (table 5), which criterion i provided to other criteria and received from other criteria, see eqs. (24) and (25). table 5 direct and indirect effects of the criteria criteria irn(ri) irn(ci) irn(wj) rank c1 [(1.7,2.6),(2.9,5.0)] [(1.3,1.9),(2.4,4.2)] [(0.056,0.153),(0.205,0.548)] 2 c2 [(1.8,2.8),(2.9,5.4)] [(1.0,1.8),(2.0,4.3)] [(0.054,0.158),(0.193,0.577)] 1 c3 [(1.3,2.0),(2.3,4.4)] [(1.3,2.0),(2.5,4.4)] [(0.047,0.135),(0.182,0.522)] 5 c4 [(1.4,2.0),(2.4,4.5)] [(1.4,2.1),(2.4,4.4)] [(0.051,0.138),(0.186,0.521)] 6 c5 [(1.0,1.5),(2.1,3.6)] [(1.6,2.4),(2.6,4.8)] [(0.048,0.133),(0.183,0.500)] 4 c6 [(1.2,1.8),(2.3,4.2)] [(1.8,2.6),(2.9,5.0)] [(0.055,0.151),(0.203,0.545)] 3 in the last step, using eqs. (26-29), we obtain final interval rough weight coefficients of the criteria, table 5. after determining the weight coefficients of the criteria, the evaluation of the alternatives using the ir-copras method is carried out. as with the ir-dematel model, three experts evaluated six ways for the construction of a temporary military route. experts evaluated the alternatives by assigning a certain value from the scale 1-9: 1  very low influence; 2  medium low influence; 3  low influence; ...; 8  high influence; 9  very high influence. the results of the expert evaluation of the alternatives are shown in table 6. normalized weighted geometric bonferroni mean operator of interval rough numbers... 187 table 6 expert evaluation of the alternative expert 1 alt./crit. c1 c2 c3 c4 c5 c6 a1 (7;8) (7;8) (3;4) (1;2) (5;6) (3;4) a2 (9;9) (7;8) (5;6) (9;9) (7;8) (9;9) a3 (7;8) (3;4) (5;6) (5;6) (7;8) (8;9) a4 (9;9) (7;8) (7;7) (9;9) (9;9) (8;9) a5 (7;8) (5;6) (5;5) (7;8) (5;6) (5;6) a6 (7;8) (7;8) (7;8) (5;6) (7;8) (7;8) expert 2 alt./crit. c1 c2 c3 c4 c5 c6 a1 (3;4) (5;6) (3;4) (5;6) (3;4) (5;5) a2 (9;9) (8;9) (5;5) (7;8) (7;8) (9;9) a3 (8;9) (7;8) (5;6) (5;6) (5;6) (8;9) a4 (9;9) (9;9) (7;8) (9;9) (8;9) (8;9) a5 (7;8) (8;9) (5;6) (7;8) (7;8) (7;8) a6 (6;7) (7;8) (7;8) (5;6) (7;8) (5;6) expert 3 alt./crit. c1 c2 c3 c4 c5 c6 a1 (5;6) (9;9) (7;8) (7;8) (1;2) (3;3) a2 (9;9) (9;9) (9;9) (8;9) (9;9) (9;9) a3 (7;8) (3;4) (5;6) (7;8) (8;9) (8;9) a4 (9;9) (8;9) (8;9) (9;9) (8;9) (8;9) a5 (7;8) (7;8) (5;6) (7;8) (8;9) (5;6) a6 (5;5) (3;4) (5;5) (7;8) (8;9) (5;5) using eqs. (1-11), the elements from table 6 are transformed into interval rough numbers, which using eq. (14) are aggregated into the initial decision-making matrix, table 7. table 7 initial decision-making matrix alt. c1 c2 ... c6 a1 [(3.99,6),(5,7)] [(6.01,8.01),(6.9,8.4)] ... [(3.21,4.08),(3.48,4.48)] a2 [(9,9),(9,9)] [(7.5,8.5),(8.44,8.89)] ... [(9,9),(9,9)] a3 [(7.11,7.54),(8.11,8.54)] [(3.41,5.15),(4.42,6.15)] ... [(8,8),(9,9)] a4 [(9,9),(9,9)] [(7.49,8.49),(8.44,8.89)] ... [(8,8),(9,9)] a5 [(7,7),(8,8)] [(5.87,7.38),(6.87,8.38)] ... [(5.21,6.09),(6.21,7.09)] a6 [(5.49,6.49),(5.87,7.38)] [(4.72,6.54),(5.72,7.54)] ... [(5.22,6.11),(5.6,7.1)] averaging of the elements of the evaluation matrices of alternatives at the a1-c2 position is carried out using eq. (14): 188 d. pamuĉar, d. boţanić, n. komazec, v. lukovac                          1, 1 0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349 1 0.337 1 0.337 1 0.314 1 0.314 0.349 0.337 0.349 0.314 1 0.349 1 0.349 6,8 , 7,8.5 , 5,7 , 6,7.67 , 7,9 , 7.67,9 6 5 6 7 5 6 5 71 2 7 6 7 5                                            p qrnwgbm                 0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349 1 0.337 1 0.337 1 0.314 1 0.314 0.349 0.337 0.349 0.314 1 0.349 1 0.349 0.337 0.314 0.337 0.349 1 0.337 1 0.3 6.01 8 7 8 9 7 8 7 91 8.01 2 9 8 9 7 7 6 7 7.671 2                                                        0.314 0.337 0.314 0.349 37 1 0.314 1 0.314 0.349 0.337 0.349 0.314 1 0.349 1 0.349 0.337 0.314 0.337 0.349 0.314 0.337 1 0.337 1 0.337 1 0.314 0 6 7 6 7.67 6.9 7.67 7 7.67 6 8.5 7.67 8.5 9 7.67 8.51 2 7.67 9                                             .314 0.349 0.349 0.337 0.349 0.314 1 0.314 1 0.349 1 0.349 8.4 9 8.5 9 7.67 6.01,8.01 , 6.90,8.40                                        after normalizing the initial matrix using eq. (32) and summing the elements of the normalized matrix using eqs. (34-38), we obtain final rank of the alternative shown in table 9. table 9 values of criteria functions of alternatives and their ranking alt. irn(qi) irn(pi) rank a1 [(0.06,0.17),(0.24,0.79)] [(100.0,100.0),(100.0,100.0)] 1 a2 [(0.03,0.13),(0.17,0.63)] [(60.98,75.12),(67.99,80.04)] 5 a3 [(0.04,0.15),(0.18,0.68)] [(70.18,86.84),(72.43,86.29)] 4 a4 [(0.03,0.13),(0.16,0.62)] [(62.31,76.44),(65.62,78.41)] 6 a5 [(0.04,0.15),(0.19,0.70)] [(75.09,88.55),(78.12,87.79)] 2 a6 [(0.04,0.15),(0.19,0.67)] [(73.14,85.35),(78.57,85.00)] 3 in the following section, the analysis of the influence of parameters p and q from the irnwbm operator to final ranges from the table 9 was performed. the analysis assumes taking different values of parameters p and q and their impact on final values of the irn weight coefficient of criteria, as well as the influence on the averaging of the value of the initial decision-making matrix or ranks from the table 9. the considered values of parameters p and q and their influence on changing the alternatives rank are shown in table 10. normalized weighted geometric bonferroni mean operator of interval rough numbers... 189 table 10 ranking order by different parameters p and q parameters p and q ranking order parameters p and q ranking order p=1 q=1 a1>a5>a6>a3>a2>a4 p=5 q=0 a1>a5>a6>a3>a2>a4 p=0 q=1 a1>a5>a6>a3>a2>a4 p=10 q=10 a1>a5>a6>a3>a2>a4 p=1 q=0 a1>a5>a6>a3>a2>a4 p=0 q=10 a1>a5>a6>a3>a2>a4 p=2 q=2 a1>a5>a6>a3>a2>a4 p=10 q=0 a1>a5>a6>a3>a2>a4 p=0 q=2 a1>a5>a6>a3>a2>a4 p=50 q=10 a1>a5>a6>a3>a2>a4 p=2 q=0 a1>a5>a6>a3>a2>a4 p=10 q=50 a1>a5>a6>a3>a2>a4 p=5 q=5 a1>a5>a6>a3>a2>a4 p=50 q=50 a1>a5>a6>a3>a2>a4 p=0 q=5 a1>a5>a6>a3>a2>a4 p=100 q=100 a1>a5>a6>a3>a2>a4 changes in the values of parameters p and q lead to certain changes of the values of the criteria functions of alternatives. however, the values of the criteria functions are such that they do not lead to changes in final ranges of alternatives, as shown in table 10. table 10 shows the influence of randomly selected values of parameters p and q on final ranges of alternatives in the ir-dematel-copras model. on the basis of the obtained results we can conclude that in the considered multi-criterion problem, changes of parameters p and q have no influence on the final rank of alternatives. 5. conclusion the recognition of imprecision and uncertainty in the multi-criteria decision-making is a very important aspect of an objective and impartial decision-making. there are often difficulties in presenting information about decision attributes by accurate (precise) numerical values. these difficulties are the result of doubts in the decision-making process just as they are due to the complexity and uncertainty of many real indicators. this paper presents a new approach to the exploitation of imprecision and uncertainty in group decision-making, which is based on interval rough numbers. the application of interval rough numbers in the multi-criteria decision-making is presented through a hybrid model consisting of the ir-dematel model and the ir-copras method. in addition to the modification of the dematel and the copras models, the irnwgbm operator for interval rough numbers is developed in this paper. the application of the irdematel-copras model and the irnnwgbm operator is presented through a case study in which the evaluation of alternatives for the construction of a temporary military route is performed. this study shows that the irnnwgbm operator can be effectively applied in 190 d. pamuĉar, d. boţanić, n. komazec, v. lukovac group decision-making models, respecting imprecision and uncertainty. since this is a new irn aggregator, which has not been applied as yet in the mcdm, the direction of future research should focus on the application of the irnnwgbm in other models based on the irn approach. references 1. triantaphyllou, e., mann, s.h., 1995, using the analytic hierarchy process for decision making in engineering applications: some 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(only in serbian). 12018 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 335 349 https://doi.org/10.22190/fume230730050b © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a new criteria importance assessment (cimas) method in multi-criteria group decision-making: criteria evaluation for supplier selection sara bošković1, stefan jovčić1, vladimir simic2,3, libor švadlenka1, momčilo dobrodolac2, nebojša bacanin4,5 1university of pardubice, faculty of transport engineering, department of transport management, marketing, and logistics, czech republic 2university of belgrade, faculty of transport and traffic engineering, serbia 3yuan ze university, college of engineering, department of industrial engineering and management, taiwan 4singidunum university, faculty of informatics and computing, serbia 5meu research unit, middle east university, amman, jordan orcid ids: sara bošković https://orcid.org/0000-0002-0331-4262 stefan jovčić https://orcid.org/0000-0002-9162-2133 vladimir simic https://orcid.org/0000-0001-5709-3744 libor švadlenka https://orcid.org/0000-0001-6484-6660 momčilo dobrodolac https://orcid.org/0000-0001-8112-8329 nebojša bacanin https://orcid.org/0000-0002-2062-924x abstract. decision-making is a challenging task for logistics managers when solving the supplier selection problem. it is usually affected by numerous conflicting criteria that are not equally important to all decision-makers. criteria evaluation is one of the crucial parts here. the primary purpose of this paper is to propose a novel integrated criteria importance assessment method based on objective judgment and group decision-making. the developed method is applied to the criteria importance evaluation for supplier selection. first, we proposed the criteria importance assessment (cimas) method based on the expert’s opinion, where the years of experts’ experience were given in the form of an expert’s weight. second, the obtained criteria weights are further integrated within the well-known critic method, and the hybrid criteria weights are determined. the input data matrix is based on the experts’ criteria evaluation on the one-to-ten-point scale. the data were further analyzed by the novel cimas method and were utilized within the critic (objective) method. the paper's main contribution is the proposal of the novel cimas method. another contribution is coupling the subjective (cimas) and the objective (critic) methods. the results reveal that the most important criterion for supplier selection is on-time distribution, followed by distribution cost, external image and appearance in public, social responsibility rate, and air pollution, respectively. the sensitivity and comparative analysis were also performed, and the technique confirmed a high level of stability. key words: decision-making, experts, supplier evaluation, cimas, critic received: july 30, 2023 / accepted december 18, 2023 corresponding author: stefan jovčić the university of pardubice, faculty of transport engineering, department of transport management, marketing, and logistics, studentská 95, 532 10 pardubice, czech republic. e-mail: stefan.jovcic@upce.cz https://orcid.org/0000-0002-0331-4262 https://orcid.org/0000-0002-9162-2133 https://orcid.org/0000-0001-5709-3744 https://orcid.org/0000-0001-6484-6660 https://orcid.org/0000-0001-8112-8329 https://orcid.org/0000-0002-2062-924x mailto:stefan.jovcic@upce.cz 336 s. bošković, s. jovčić, v. simić, l. švadlenka, m. dobrodolac, n. bacanin 1. introduction the multi-criteria decision-making (mcdm) process has always been an intriguing issue for both researchers and managers all around the world. each decision-making problem in life is affected by many criteria that interact with each other. only some of the requirements are equally important to each decision-maker. for that reason, it is necessary to assign the proper importance to them. many methods, that are used to evaluate the criteria, have been suggested by various authors in the mcdm field [1,2]. two sorts of methods in the criteria weighting exist, the subjective weighting methods (direct rating (dr), simple multi-attribute rating technique (smart), delphi, full consistency method (fucom), best-worst method (bwm), analytic hierarchy process (ahp), stepwise weight assessment ratio analysis (swara), etc.) and the objective weighting methods (entropy, critic, standard deviation, etc.). in this paper, the authors conducted research on the subjective methods that were proposed in the literature. the web of science database was this survey’s primary source of information. it is noticed from the literature that there are a bunch of publications regarding the criteria evaluation methods. saaty [3] proposed the ahp method. diakoulaki [1] developed the criteria importance through intercriteria correlation (critic) method. gabus and fontela [4] developed the decisionmaking trial and evaluation laboratory (dematel) method. rezaei [5] developed the bwm. keršuliene et al. [6] proposed the swara method. pamučar et al. [2] developed the full consistency method (fucom). all the proposed methods found their application in various fields. for instance, xie et al. [7] applied dematel for the site selection of supermarkets. ortega et al. [8] used the bwm to find a sustainable park-and-ride location. kant and gupta [9] evaluated urban freight strategies using the bwm method. regarding the ahp method, it is applied in various contexts such, as forest fire hazard modeling [10], risk assessment of metro systems [11], project post-evaluation [12], etc. the swara method has applications to various problems such as supplier evaluation [13], road sections ranking [14], risk assessment in coal supply chain management [15], glasshouse location selection [16]. when it comes to the fucom method, it has been applied to various branches such as in the manufacturing environment [17], human resources [18], landfill site selection [19], forklift efficiency analysis [20], etc. besides mcdm methods for criteria importance assessment, in recent years, the field of mcdm has been growing rapidly. for instance, bošković et al. [21] developed the aroman method to select the best electric vehicle alternative for last-mile delivery. this method has been further developed and applied to cargo bike delivery concept selection by bošković et al. [22]. in addition, yalçın et al. [23] developed an intuitionistic fuzzy-based aroman model for performance evaluation of ecoports. švadlenka et al. [24] developed a picture-fuzzy decision-making approach for sustainable last-mile delivery. simić et al. [25] developed a neutrosophic lopcow-aras model for prioritizing industry 4.0-based material handling technologies in smart and sustainable warehouse management systems. tirkolaee et al. [26] proposed an integrated decision-support framework for resilient supply chain network design. they used the bwm and waspas methods that were further integrated within the type-2 neutrosophic fuzzy numbers (t2nn). čubranić-dobrodolac et al. [27] proposed a fuzzy-aroman-fuller technique to select a professional driver. on the other hand, nikolić et al. [28] applied an interval type-2 fuzzy aroman decision-making method to improve the sustainability of the postal network in rural areas. pamucar et al. [29] a new criteria importance assessment (cimas) method in multi-criteria group decision-making... 337 prioritized sustainable mobility-sharing systems using integrated fuzzy dibr and fuzzy-rough edas models. in this paper, a novel integrated criteria importance assessment method based on objective judgment and group decision-making is developed. first, we proposed the criteria importance assessment (cimas) method based on the expert’s opinion, where the years of experience of each expert are given in the form of an expert’s weight. thus, the obtained criteria weights are further integrated within the well-known critic method, and the hybrid criteria weights are determined. the authors of this paper followed the logic that the more years of experience one expert has, the more critical his criteria assessment decision is. to assess the criteria, the experts use a one-to-ten-point scale and thus formulate an input data matrix. the proposed integrated approach is to help decision-makers determine the criteria weights in multi-criteria decision-making problems, including a team of experts. the first part of the proposed method belongs to a group of subjective methods since a team of experts assesses the criteria in a decision-making process. the second part integrates the well-known critic method, which belongs to the objective methods. the critic method belongs to the objective methods developed and verified by diakoulaki et al. in 1995 [1]. the correlation analysis obtains the contrasts among criteria [1]. this paper's main contribution lies in coupling subjective and objective methods to get more precise and accurate criteria assessments that are further used in decision-making. in addition, this paper emphasizes its further contributions: i) the original subjective method (cimas) to assess the criteria importance is proposed; ii) the proposed cimas method appears for the first time in the literature; iii) the method is general (includes the experts’ judgment) and is applicable to assess any criteria in real-life problems. iv) the proposed method is applied to evaluate the criteria for sustainable supplier selection. (v) coupling the subjective (cimas) with the well-known objective (critic) method, an integrated approach gives more precise and confident results. the rest of the paper is structured as follows: section 2 thoroughly explains the proposed cimas method. in section 3, the cimas method is applied to assess the criteria that affect the supplier selection problem. sub-section 3.1 is the sensitivity analysis on the cimas method, where three scenarios are performed. sub-section 3.2 is the comparative analysis with the critic method. sub-section 3.3 describes the hybrid criteria weights based on combining the cimas and the critic methods. the main contribution should be found in this sub-section. section 3.4 gives some managerial recommendations regarding the newly proposed method. section 4 concludes the paper and gives some future research directions. 2. criteria importance assessment method this paper proposes a new approach, the criteria importance assessment (cimas), in decision-making. this method should be used to identify the criteria’ importance in a decision-making process using experts’ assessment. one important fact regarding the expert knowledge and experience is that this method considers the experts’ experience in the years spent in the considered field. as mentioned above, the authors of this paper followed the logic that the more years of experience one expert has, the more critical the criteria assessment decision is. the experts’ weights are further used in the criteria assessment. the calculated expert weights should be significant later in the decision-making process. in addition, the experts use a one-to-ten-point scale to formulate an input data matrix where 338 s. bošković, s. jovčić, v. simić, l. švadlenka, m. dobrodolac, n. bacanin the most significant importance of one criterion is denoted by 10. in contrast, the lowest one is denoted by 1. a flowchart of the proposed method is depicted in fig. 1. fig. 1 a flowchart of the proposed method the cimas method is described through the following steps: step 1. define a set of evaluation criteria: the first step in a decision-making process is to define the criteria by which importance should be assessed. the criteria may be identified either from the literature or by consulting experts. step 2. identify the number of experts: decision-maker identifies the number of experts involved in the field and collects information regarding their years of experience. step 3. establish the experts’ experience (years): the experts clarify their experience in the number of years spent in the field, and the decision-maker defines the importance of each expert. step 4. calculate the importance of each expert: the importance of each expert is calculated by following this logic: let us suppose that “q” experts participate in the criteria assessment procedure. expert 1 has five years of experience in the considered field, expert 2 has three years of experience, while expert q participates with seven years of experience. the importance of experts’ evaluations (the expert weights) should be determined as 5/15 for expert 1, 3/15 for expert 2, and 7/15 for expert q. in the general case, the expert’s importance can be calculated by eq. (1). 𝑊𝐸𝑖 = 𝐸𝑖 ∑ 𝐸𝑖 𝑞 𝑖=1 , 𝑖 = 1, 2, … , 𝑞 (1) a new criteria importance assessment (cimas) method in multi-criteria group decision-making... 339 step 5. formulate an input data matrix based on the experts’ assessment (1-10 scale): in this step, an input decision-making matrix is defined. the experts use a one-to-tenpoint scale to formulate an input data matrix where the most significant importance of one criterion is denoted by 10, while the lowest one is denoted by 1. the structure of the input data matrix is presented in table 1. 𝐸𝑖 represents one of the q experts involved in a decisionmaking, 𝐶𝑗 is one of the p considered evaluation criteria, 𝑥𝑖𝑗 is the expert’s number i assessment of criteria number j importance (on a scale of 1-10), and finally, the values from 𝑊𝐸1 to 𝑊𝐸𝑞 are the experts’ weights. table 1 input data matrix experts/criteria 𝐶1 𝐶2 … 𝐶𝑗 … 𝐶𝑝 experts’ weights 𝐸1 𝑥11 𝑥12 … … … 𝑥1𝑝 𝑊𝐸1 𝐸2 𝑥21 𝑥22 … … … 𝑥2𝑝 𝑊𝐸2 𝐸𝑖 … … … 𝑥𝑖𝑗 … … … 𝐸𝑞 𝑥𝑞1 𝑥𝑞2 … … … 𝑥𝑞𝑝 𝑊𝐸𝑞 step 6. normalize the input-data matrix: after the input data matrix is formulated, step 6 is data normalization. it means that the input data are structured in intervals 0 and 1. it further facilitates the decision-making process. in this method, the normalization technique is applied by eq. (2) and presented in table 2. 𝑥𝑖𝑗 ∗ = 𝑥𝑖𝑗 ∑ 𝑥𝑖𝑗 𝑞 𝑖=1 , 𝑖 = 1, 2, … , 𝑞; 𝑗 = 1, 2, … , 𝑝 (2) table 2 normalized input-data matrix experts/criteria 𝐶1 𝐶2 𝐶𝑗 𝐶𝑃 𝐸1 𝑥11 ∗ 𝑥12 ∗ … 𝑥1𝑝 ∗ 𝐸2 𝑥21 ∗ 𝑥22 ∗ … 𝑥2𝑝 ∗ 𝐸𝑖 … … 𝑥𝑖𝑗 ∗ … 𝐸𝑞 𝑥𝑞1 ∗ 𝑥𝑞2 ∗ … 𝑥𝑞𝑝 ∗ step 7. multiply each value of the normalized input data matrix by the importance of each expert (expert-weighted matrix): in this step, the normalized input data are multiplied by the experts’ weights obtained in step 4. it is calculated by eq. (3) and presented in table 3. 𝑥 ∗̂ 𝑖𝑗 = 𝑥𝑖𝑗 ∗ · 𝑊𝐸𝑖 , 𝑖 = 1,2, … 𝑞; 𝑗 = 1, 2, … , 𝑝 (3) table 3 expert-weighted matrix experts/criteria 𝐶1 𝐶2 … 𝐶𝑗 … 𝐶𝑃 𝐸1 𝑥 ∗̂ 11 𝑥 ∗̂ 12 … … … 𝑥 ∗̂ 1𝑝 𝐸2 𝑥 ∗̂ 21 𝑥 ∗̂ 22 … … … 𝑥 ∗̂ 2𝑝 𝐸𝑖 … … … 𝑥 ∗̂ 𝑖𝑗 … … 𝐸𝑞 𝑥 ∗̂ 𝑞1 𝑥 ∗̂ 𝑞2 … … … 𝑥 ∗̂ 𝑞𝑝 340 s. bošković, s. jovčić, v. simić, l. švadlenka, m. dobrodolac, n. bacanin step 8. identify the maximum and minimum value of each criterion in the expertweighted matrix: the primary purpose of this step is to identify the maximum (𝑅𝑗 𝑚𝑎𝑥) and minimum value (𝑅𝑗 𝑚𝑖𝑛) of each criterion by columns. it is calculated by eqs. (4) and (5) and presented in table 4. 𝑅𝑗 𝑚𝑎𝑥 = max 𝑖 𝑥 ∗̂ 𝑖𝑗, 𝑗 = 1, 2, … , 𝑝, (4) 𝑅𝑗 𝑚𝑖𝑛 = min 𝑖 𝑥 ∗̂ 𝑖𝑗 , 𝑗 = 1, 2, … , 𝑝. (5) table 4 maximum and minimum value of each criterion in the expert-weighted matrix experts/criteria 𝐶1 𝐶2 … 𝐶𝑗 … 𝐶𝑃 𝐸1 𝑥 ∗̂ 11 𝑥 ∗̂ 12 … … … 𝑥 ∗̂ 1𝑝 𝐸2 𝑥 ∗̂ 21 𝑥 ∗̂ 22 … … … 𝑥 ∗̂ 2𝑝 𝐸𝑖 … … … 𝑥 ∗̂ 𝑖𝑗 … … 𝐸𝑞 𝑥 ∗̂ 𝑞1 𝑥 ∗̂ 𝑞2 … … … 𝑥 ∗̂ 𝑞𝑝 𝑅𝑗 𝑚𝑎𝑥 𝑅1 𝑚𝑎𝑥 𝑅2 𝑚𝑎𝑥 … 𝑅𝑗 𝑚𝑎𝑥 … 𝑅 𝑝 𝑚𝑎𝑥 𝑅𝑗 𝑚𝑖𝑛 𝑅1 𝑚𝑖𝑛 𝑅2 𝑚𝑖𝑛 … 𝑅𝑗 𝑚𝑖𝑛 … 𝑅𝑝 𝑚𝑖𝑛 step 9. calculate the difference between minimum and maximum values: this step calculates the difference (𝐵𝑗) between the minimum and maximum values from the previous step by applying eq. (6). 𝐵𝑗 = 𝑅𝑗 𝑚𝑎𝑥 − 𝑅𝑗 𝑚𝑖𝑛 , 𝑗 = 1, 2, … , 𝑝 (6) step 10. apply the final ranking formula: this step obtains the criteria importance (𝐿𝑗) by using the final ranking equation, i.e. eq. (7). 𝐿𝑗 = 𝐵𝑗 ∑ 𝐵𝑗 𝑝 𝑗=1 , 𝑗 = 1, 2, … , 𝑝 (7) 2.1. checking reliability index (ri) when it comes to subjective methods, which in this case is the cimas method, an important issue is measuring the reliability of the collected answers by the experts. in the ahp method, the authors proposed a well-known approach to measure the inconsistency rate. in that case, the inconsistency rate should be less than 0.1 to consider the results reliable. in the case of cimas, this approach cannot be utilized. in the case of cimas, the authors applied another approach [27] for measuring the reliability index (ri). the procedure implies a second round of interviewing the experts, where they do not know the results of the first round regarding the criteria assessed. in the second round, the experts are asked to consider the criteria on a scale from 0 to 100 % and thus determine the percentage importance of each criterion. the experts should be careful that the evaluation for all “n” criteria must give the sum of 100 %. finally, the results from both rounds are compared and they lead to the conclusion regarding the reliability of the results. if we a new criteria importance assessment (cimas) method in multi-criteria group decision-making... 341 denote the answers from the second round of the interview by 𝑃𝑗 and previously obtained cimas weights by 𝐿𝑗, then the reliability index (ri) can be calculated by eq. (8). 𝑅𝐼 = ∑ |𝐿𝑗∗100−𝑃𝑗|𝑛 𝑗=1 100 (8) if the ri is less than 0.1, the results should be considered reliable. contrary, the experts should repeat the criteria assessment procedure. 3. results and discussion this section demonstrates the applicability of a newly proposed cimas method in the case of the criteria assessment for the supplier selection process. five criteria have been identified from the literature: two economic (distribution costs [30-33], and on-time distribution [30, 31,34, 35]), one environmental (air pollution [30, 33, 34, 36, 37]), and two social (external image and appearance in public [30, 31, 33, 34] and social responsibility rate [30, 31, 33, 34]). the criteria assessment problem is depicted in fig. 2. fig. 2 illustration of the problem the first economic criterion is distribution cost. it is the cost that possible suppliers would request from the company for the distribution process. the second economic criterion is ontime distribution, which means whether the supplier respects the promised timeframes. the third criterion is air pollution which belongs to the environmental pillar. the fourth social criterion is the external image and appearance in public. it presents the general impression of possible suppliers in public, such as their building, transport fleet, logo, behavior toward customers, etc. the fifth and last social criterion is the social responsibility rate. five experts participated in the criteria assessment. all these experts belong to the logistics field with various years of experience (table 5). experts 1 to 3 are the ceos of the logistics companies in the czech republic. experts 4 and 5 are the ceos of the middlesized logistics companies in slovakia. the names of the companies are not mentioned because of their internal policy. the experts were interviewed by telephone due to the ongoing covid-19 crisis. they were asked to give the importance of each criterion by following the one-to-ten-point scale, where the highest priority of one criterion is denoted by 10. at the same time, the lowest one is represented by 1. in addition, the experts’ weights 342 s. bošković, s. jovčić, v. simić, l. švadlenka, m. dobrodolac, n. bacanin are determined based on their experience in the field. therefore, the data were generated from the experts’ criteria assessments for this case study. according to the cimas method, the following results are obtained (table 5 table 7). the criteria rank is depicted in fig. 3. table 5 input data matrix distribution cost on-time distribution air pollution external image and appearance in public social responsibility rate expert – years of experience expertweights e1 9 6 6 6 4 2 𝑊𝐸1 =0.0909 e2 8 8 7 9 6 4 𝑊𝐸2 =0.1818 e3 10 9 6 7 7 1 𝑊𝐸3 =0.0455 e4 9 6 8 9 8 5 𝑊𝐸4 =0.2273 e5 10 9 6 8 6 10 𝑊𝐸5 =0.4545 ∑=22 table 6 normalized input data matrix distribution cost on-time distribution air pollution external image and appearance in public social responsibility rate e1 0.1957 0.1579 0.1818 0.1538 0.1290 e2 0.1739 0.2105 0.2121 0.2308 0.1935 e3 0.2174 0.2368 0.1818 0.1795 0.2258 e4 0.1957 0.1579 0.2424 0.2308 0.2581 e5 0.2174 0.2368 0.1818 0.2051 0.1935 the results of the criteria assessment for supplier selection are given in the following ranking order: the highest importance was assigned to the distribution on time, followed by the cost of distribution, external image, and appearance in public, social responsibility rate, and air pollution, respectively. to check the reliability of the obtained results, we interviewed the experts in the second round to collect information about the percentage distribution of criteria importance. the results are shown in table 8. as it can be concluded, the rate of inconsistency is below 0.1 (ri = 0.06445), which means that reliability is satisfactory. table 7 the maximum and minimum value of each criterion, its difference obtained final weights. distribution cost on-time distribution air pollution external image and appearance in public social responsibility rate e1 0.0178 0.0144 0.0165 0.0140 0.0117 e2 0.0316 0.0383 0.0386 0.0420 0.0352 e3 0.0099 0.0108 0.0083 0.0082 0.0103 e4 0.0445 0.0359 0.0551 0.0524 0.0587 e5 0.0988 0.1077 0.0826 0.0932 0.0880 𝑅𝑗 𝑚𝑎𝑥 0.0988 0.1077 0.0826 0.0932 0.0880 𝑅𝑗 𝑚𝑖𝑛 0.0099 0.0108 0.0083 0.0082 0.0103 𝐵𝑗 0.0889 0.0969 0.0744 0.0851 0.0777 ∑=0.4230 𝐿𝑗 0.2102 0.2291 0.1758 0.2011 0.1837 a new criteria importance assessment (cimas) method in multi-criteria group decision-making... 343 fig. 3 criteria ranking table 8 calculation of the reliability index (ri) 𝐿𝑗 e1 (%) e2 (%) e3 (%) e4 (%) e5 (%) average 𝑃𝑗 |𝑊𝑗*100-𝑃𝑗| 𝑅𝐼𝑗 distribution cost 0.2102 25 30 15 15 15 20 1.0244473 0.010244 on-time distribution 0.2291 30 25 20 20 35 26 3.0944179 0.030944 air pollution 0.1758 25 5 15 30 5 16 1.5840832 0.015841 external image and appearance in public 0.2011 15 15 35 25 10 20 0.1139869 0.00114 social responsibility rate 0.1837 15 25 15 5 35 19 0.6280995 0.006281 0.06445 3.1. sensitivity analysis in this section, the sensitivity analysis is performed to check the deviations of the cimas method. the sensitivity analysis is performed in a way that the most experienced experts were excluded from the model to observe how the rank of criteria will change. in our case, we examined three scenarios. scenario 1, where the high-experienced expert (expert 5) was excluded. scenario 2, where the second most experienced expert (expert 4) was excluded, and scenario 3, where an inexperienced expert (expert 1) was excluded. the results of the sensitivity analysis are presented in fig. 4. fig. 4 sensitivity analysis the sensitivity analysis results revealed that when the most experienced expert (expert 5) was excluded from decision-making, there was a significant change in criteria ranking. the other two scenarios had better compatibility with the original state. however, when 0.2102 0.2291 0.1758 0.2011 0.1837 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 distribution cost on-time distribution air pollution external image and appearance in public social responsibility rate 0 5 10 c1 c2 c3 c4 c5 original scenario 1 expert 5 excluded scenario 2 expert 4 excluded scenario 3 expert 1 excluded 344 s. bošković, s. jovčić, v. simić, l. švadlenka, m. dobrodolac, n. bacanin the least experienced expert (expert 1) was excluded, this did not significantly affect the ranking order. it may be concluded that the more experienced experts significantly influence the criteria to rank more than those inexperienced. 3.2. comparative analysis with the critic method in this sub-section, the newly proposed cimas method was compared to the well-known critic method proposed by diakoulaki et al. [1]. the critic method, as mentioned at the beginning of the paper, belongs to the objective methods used to obtain the criteria weights. the authors of this paper utilized the same input data (in the critic method) obtained from the five experts, and the results of the comparative analysis are depicted in fig. 5. it may be noticed from fig. 5 that both the critic and the cimas methods ranked the on-time distribution (c2) criterion with the highest importance. on the other hand, the critic method ranked the distribution cost (c1) as the second best, while the cimas ranked the c1 as the third best. in the next section, given the noticeable similarity between the two methods, the paper’s authors coupled the newly proposed method with the critic one to obtain hybrid criteria weights that should be utilized further in decision-making. fig. 5 comparative analysis between the critic and the cimas methods 3.3. hybrid weights based on the integration of the critic and cimas methods in this sub-section, the obtained criteria weights from the cimas and critic methods are effectively coupled, and the hybrid criteria weights are obtained. the hybrid criteria weights are calculated by applying the equation of an arithmetic mean: 𝑊𝑗 ℎ𝑦𝑏𝑟𝑖𝑑 = 𝑊𝑗 𝐶𝑅+𝐿𝑗 2 , 𝑗 = 1, 2, … , 𝑝 (9) where: 𝑊𝑗 𝐶𝑅 are the criteria importance obtained by the critic method, while 𝐿𝑗 are the criteria obtained by the cimas method. the obtained hybrid criteria weights are presented in table 9 and fig. 6. 0.2407 0.2571 0.2171 0.1678 0.1173 0.2102 0.2291 0.1758 0.2011 0.1837 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 distribution cost on-time distribution air pollution external image and appearance in public social responsibility rate lj wcr a new criteria importance assessment (cimas) method in multi-criteria group decision-making... 345 table 9 hybrid criteria weights 𝑊𝑗 𝐶𝑅 𝐿𝑗 𝑊𝑗 ℎ𝑦𝑏𝑟𝑖𝑑 distribution cost 0.2407 0.2102 0.2255 on-time distribution 0.2571 0.2291 0.2431 air pollution 0.2171 0.1758 0.1965 external image and appearance in public 0.1678 0.2011 0.1844 social responsibility rate 0.1173 0.1837 0.1505 fig. 6 shows that the obtained hybrid criteria weights (the red line) are in between the critic and newly proposed cimas method, which may indicate more accurate results in the decision-making process. this should be seen as a powerful technique that combines objective and subjective criteria weights for better decisions. fig. 6 hybrid criteria weights 3.3.1. sensitivity analysis of the integrated model the sensitivity analysis is proposed to prove the robustness of the newly proposed framework with the hybrid criteria weights. the authors introduced the parameter λ (between the intervals 0 and 1) and tested the robustness of the hybrid critic-cimas model. the robustness of the model was tested by applying eq. (10). the results are presented in fig. 7. 𝑊𝑗 𝐻𝑦𝑏𝑟𝑖𝑑 (𝜆) = 𝜆𝑊𝑗 𝐶𝑅+(1−𝜆)𝐿𝑗 2 , 𝑗 = 1, 2, … , 𝑝 (10) 346 s. bošković, s. jovčić, v. simić, l. švadlenka, m. dobrodolac, n. bacanin fig. 7 sensitivity analysis on the hybrid criteria weights applying eq. 10 allows two scenarios where different methods were used. in the first one, when λ=0, only the criteria weights obtained by the cimas method ( 𝐿𝑗 2 ) are considered. in the second one, when λ=1, only the criteria weights obtained by the critic method ( 𝑊𝑗 𝐶𝑅 2 ) are considered. both criteria weights are considered in all other cases, λ = (0.1-0.9). it may be noticed from fig. 7 that the criteria weights are stable when λ is between 0.4 and 1. in the rest, minimal overlaps are noticed. it can be concluded that the criteria ranks are very stable to variations in the operational parameter λ. 3.4. managerial implications the evaluating criteria weights for an effective supplier selection process is an initial phase and crucial to all logistics decision-makers. the criteria weights may be determined differently since many multi-criteria decision-making methods were developed. sometimes, the criteria weights may be obtained either by objective or subjective methods. to reduce the level of uncertainty and boost more precise final ranking decisions, we developed an integrated criteria importance assessment method based on objective judgment and group decision-making. this integrated approach may serve managers to assess the criteria efficiently and effectively. one of the possibilities of this approach is that the managers may choose either to apply the cimas method or the critic one to find the criteria's importance. the recommendation to managers may be to use an integrated approach due to its high level of robustness. apart from the logistics industry, the integrated approach may be applied by managers to assess the criteria in other branches of industry such as supply chain, medicine, education, automotive, construction, etc. one of the advantages of the integrated approach is its simplicity of implementation, which leads to the expectation that managers worldwide would be able to apply it to solve their decision-making problems. a new criteria importance assessment (cimas) method in multi-criteria group decision-making... 347 4. conclusions the primary aim of this paper was to propose a novel criteria importance assessment (cimas) method to help decision-makers assess the criteria when dealing with complex mcdm problems. as its name suggests, the novel method should be suitable in cases when multiple criteria affect the decision on the best alternative. the main advantage of the proposed method is to help managers and decision-makers identify the weights of each criterion and rank them in descending order. the experts' knowledge and experience are essential to the newly proposed method. the final rank of the criteria exclusively depends on the experts' knowledge and experience, which sometimes could be a disadvantage if the decision is made by an inexperienced expert. the paper’s authors performed the sensitivity analysis to observe how the criteria to rank would change if some experts were excluded from the criteria evaluation (fig. 4). the sensitivity analysis results revealed that the most experienced expert significantly impacted decision-making if his knowledge and experience had been omitted from decision-making. on the contrary, if the least experienced expert had been excluded, that did not significantly affect the criteria ranking order. the secondary aim of this paper was to integrate the newly proposed cimas method within objective one (critic) and thus obtain more accurate hybrid criteria weights. the proposed integrated approach was applied to evaluate the hybrid criteria weights affecting supplier selection. the obtained results within the integrated approach presented the following hybrid ranking order (table 10, fig. 6): the highest importance was assigned to the on-time distribution, followed by the cost of distribution, air pollution, external image and appearance in public, and social responsibility, respectively. in addition, the sensitivity analysis was performed to check the stability of the newly integrated approach where the criteria rank presented a high level of robustness (fig. 7). this paper highlights its significant contributions: i) the original mcdm method, named the criteria importance assessment (cimas), is proposed; ii) the cimas method belongs to the subjective methods and is presented for the first time in the literature; iii) the method is general (includes the expert-knowledge and experience) and applies to any other multi-criteria decision-making problem in many branches of industry; iv) the application of the cimas method is demonstrated in evaluating the criteria which are further used for the supplier selection process. v) coupling the subjective (cimas) with the well-known objective (critic) method, an integrated approach gives a more precise and confident criteria rank which was confirmed by the sensitivity analysis. besides contributions, there is a limitation of the proposed cimas method. the method uses only the crisp values, and its application is not suitable in an uncertain environment. the possible future research directions of this paper should be: 1) to extend the methodology on fuzzy sets; 2) to apply the methodology to many different fields; 3) to couple the cimas method with some of the existing mcdm methods to obtain criteria weights; 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series: mechanical engineering vol. 17, no 3, 2019, pp. 397 404 https://doi.org/10.22190/fume190422037t © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper the role of digital information models for horizontal and vertical interaction in intelligent production pancho tomov, lubomir dimitrov technical university of sofia, faculty of mechanical engineering, bulgaria abstract. intelligent production is the future of industrial production. it is the leading way to a new industrial era and it best defines the concept of the fourth industrial revolution. getting the real-time data on quality, resources and costs it provides significant advantages over classical production systems. intelligent production must be built on sustainable and service-oriented technological and business practices. they are characterized by flexibility, adaptability and self-learning, resilience to failures, and risk management. the high levels of automation, on the other hand, become a mandatory standard for them, which is possible thanks to a flexible network of production-based systems that automatically monitor the production processes. flexible systems and models that are capable of responding in real time allow internal processes to be radically optimized. production benefits are not limited to one-off production conditions, and the capabilities include optimization through a global network of adaptive and self-regulating manufacturing components belonging to more than one operator. key words: fourth industrial revolution, industrial information models, automation, production based systems 1. introduction the introduction of intelligent production is a production revolution in terms of cost and time savings. intelligent production brings many advantages over conventional production, or it is the transition to future smart production. the interaction between embedded systems based on highly specialized software and dedicated user interfaces that are integrated into digital networks create a whole new world of system functions. a well-established production and engineering system is one of the prerequisites for the normal operation of intelligent production systems. this system shall fully satisfy, under the existing conditions, the requirements of operation ability and functionality of industrial received april 22, 2019 / accepted august 10, 2019 corresponding author: pancho tomov technical university of sofia, faculty of mechanical engineering, 8 “kliment ohridski” blvd, sofia, bulgaria e-mail: pkt@tu-sofia.bg 398 p. tomov, l. dimitrov processes in an industrial plant. this part includes services and features for operational activity that are executed as web-based software components and can be used continuously in contact with both internal and external objects. at&t’s bell laboratories stands as an exemplar of this model, with many notable research achievements but a notoriously inwardly focused culture. other celebrated twentieth-century examples of this model include ibm’s tj watson research center, xerox parc, ge’s schenectady laboratories, merck, and microsoft research. in other countries, such as japan, the closed model remains quite popular to this day [1, 2]. this traditional innovation process is closed because projects can only enter it in one way, at the beginning from the company’s internal base, and can only exit in one way, by going into the market. traditionally, new business development processes and the marketing of new products took place within the firm boundaries. in future, intelligent embedded systems will contribute to more efficient manufacturing and more manageable technological processes, with the principles of network econometrics undergoing full communicative change [3]. this is also related to the impact of web technologies on production technologies, which is shown in fig. 1. fig. 1 web impact of web technologies on production technologies the connections of information technologies with people, machines and products have been rapidly realized thanks to the rapid development of technology transfer standards and a comprehensive information infrastructure. the current development and future development as a change in the requirements in information and communication technologies on the way to the fourth industrial revolution is given in fig. 2. 2. horizontal and vertical integration there are conditions and opportunities, based on models and trends in the generated data, to make decisions in real time. it is also possible to build digital information integration models for horizontal and vertical action. horizontal integration means the integration of various information technology systems in the production and automation equipment at different stages of the production and planning process in order to find a the role of digital information models for horizontal and vertical interaction in intelligent production 399 constantly optimal solution [4]. vertical integration means the integration of information technology in it systems at different hierarchical levels in the production and automation equipment in order to find a consistently optimal solution. the vertical and horizontal machine-internet, machine-human and machine-machine collaboration along the value chain, in real time, is the basis of the intelligent production system. fig. 2 change in the requirements to information and communication technologies horizontal compatibility means integration of different technology and information systems into different stages of production and product planning. therefore, the automated system is formed by a single material, energy and information flow that makes connections both inside the company and with external companies. in doing so, it maintains optimal and continuous horizontal compatibility at the level of intelligent solutions. vertical compatibility means the trouble-free integration of different information technologies into information systems at different hierarchical levels in the vertical production structure. the goal is to maintain an optimal management solution. the interaction between embedded systems based on highly specialized software and dedicated user interfaces that are integrated into digital networks create a whole new world of system functions. thus, in intelligent production, the ability to communicate and decentralize data processing, as well as optimization, is done through embedded systems 400 p. tomov, l. dimitrov equipped with dedicated software and hardware. these embedded systems are connected wirelessly (partially) to the information net-works of other systems of stakeholders, companies, and others with a view to exchanging data and accessing web-based services. all this requires interoperable communication interfaces and standardized protocols, continuously integrated it systems, control and fast, real-time communication. fig. 3 shows the development of the it architecture of intelligent systems [5, 6]. erp is a software architecture that facilitates the flow of information among the different functions within an enterprise. similarly, erp facilitates information sharing across organizational units and geographical locations. it enables decision-makers to have an enterprise-wide view of the information they need in a timely, reliable and consistent fashion. erp provides the backbone for an enterprise-wide information system. at the core of this enterprise software is a central database which draws data from and feeds data into modular applications that operate on a common computing platform, thus standardizing business processes and data definitions into a unified environment. with an erp system, data needs to be entered only once. the system provides consistency and visibility or transparency across the entire enterprise. a primary benefit of erp is easier access to reliable, integrated information. a related benefit is the elimination of redundant data and the rationalization of processes, which result in substantial cost savings. fig. 3 development of the it architecture of intelligent system as a result, data and services can be used in real time, creating great flexibility and ability to meet customer requirements. it is also important to follow the development of the information technology architecture shown in fig. 4. fig. 4 development of the information technology architecture the role of digital information models for horizontal and vertical interaction in intelligent production 401 3. turning innovation into product concept the concept formation aims at further development of the innovation idea and its transformation into a real innovative product designated for the market, as well as at turning consumer wishes into a manufacturing-wise feasible product. usually the company has at least several innovative ideas about a product [7]. we are aware of different approaches towards the formation of product concept, but those that got established in practice are less. fig. 5 shows the concepts that got established as approaches for solving the problem. fig. 5 approaches towards the product concept formation in the case of the adaptable approach the most important thing is to satisfy consumer requirements and the market requirements. the own development provides greater capacities, as well as better freedom of usage and creates competitive advantage for the company but is related to the maintenance of higher innovative potential. the acceptance of ready solution (open innovation) is related to the need of particular prerequisites to realize it and the need of preliminary research about its purposefulness. nevertheless, in all the cases the concept should be directly related to market and it should define certain market positions [8]. after the formation of the product concept we proceed to the next phases, design and production. in view of the smooth running of these processes they should be planned in advance and we should choose the instrumentation for time and resources management. 4. design and implementation of innovative products when it comes to designing innovative products and processes we take the technicaleconomic indicators as input data that include: production order, including the necessary number of items to be produced, technical economic product indicators (type, mass, size etc.), requirements towards the construction and technology of the product manufacturing, accounting value with stages and performance terms, economic and social effectiveness, ecological requirements, specific requirements towards the production, storage and realization of the product [9]. the design and implementation of innovative products pass through several stages to be reviewed in short. the process of designing innovative products is graphically shown in fig. 6. 402 p. tomov, l. dimitrov in most cases the suggestion for the most effective technological concept for product manufacturing includes [10]:  basic information about the technical terms and conditions of manufacturing the product,  managerial information about the types of technological processes, standards and work organization,  inquiry information that includes description of progressive methods of processing, catalogues, reports, technical norms and operational production regimes. the design takes place during a single stage that combines draft and working project. the draft solution includes the selection of version of the innovative product. usually it is being developed in several versions thus resulting in certain complexity because of the versatility of the factors impacting the choice of the optimal version. in principle, one of the indicators is perceived as the basic optimization criterion (objective function) [11]. hence the draft project should be perceived as a systematic solution in order to clarify the cause and effect of the basic parameters and distribute their impact onto the end result, including producing the item. fig. 6 process of innovative products’ design the draft solution of the innovative product usually includes: draft constructivetechnological documentation of the product (draft drawings and documentation), description of the main technical-economic product indicators, bill of quantities that contains the costs under the individual stages, design, aesthetics, market requirements, effectiveness or expected profit from the production and realization of the product, other specific requirements, ecology etc. 5. software packages in terms of horizontal integration, companies shall be able to use next-generation erp systems that are suitable for use in an integration (hybrid) environment. these are intelligent erp systems using service oriented architecture (soa). this allows the use of functions and services by other software providers through standardized interfaces [12]. the role of digital information models for horizontal and vertical interaction in intelligent production 403 these erp systems are integrated with technological processes that are compatible with intelligent production systems. the internet of things allows direct communication of an erp system with cps (cyber physical systems) and intelligent products at the production level. by using in-memory databases and large amounts of data, cps sensors can process the information in real-time. thus, in the case of production changes, simulation is performed using in-memory technology in real time. direct access to production data from the erp system ensures transparency of technological and business processes for all individual orders. these solutions are easier to perform because the simulations and forecasts (created by the erp system) are presented in a handy way on mobile devices such as tablets or smartphones. the new erp system uses cloud computing for services access to internet (ios) capabilities. this part of the internet includes services and features that are executed as web-based software components and can also be used in contact with external companies and users. in building cyber physical systems (cps), depending on their intended complexity and type, different types of software are used. firstly, these are the most used software tools included in: product development (cad/cae), process planning (cap, cam), order management (crm, erp, /pps, scm), operational management (mes, bde, qm) and service (ips) [13]. secondly, these are the software packages built in recent years by industrial companies, which enable them to track their entire value chain, bringing them closer to the requirements of the (cps). thirdly, the trend of software development towards web services and platform connectivity is outlined. many companies, technology centers and research institutes work in this direction. this may also be the case for digital business software. a portfolio of software based systems is developed, which is based on a backbone data platform and includes the modules; product lifecycle management (plm) module. it allows you to virtually completely create and optimize new, unproduced products. it enables you to effectively manage the product lifecycle from the idea and its design to its production, maintenance and recycling [14, 15]. manufacturing execution system/ manufacturing operations management (mes/mom) module. this module is highly saleable, offers a variety of functions and allows production to be combined with quality and transparency as well as to speed up the production process. this complete solution maintains the entire value chain of product development, planning, production, growth and operation. totally integrated automation (tia) module. it is an open system architecture that covers the entire production process and provides effective interaction of all automation components. this comprehensive approach to totally integrated automation includes:  industrial communication  industrial security  integrated engineering  industrial data management  integrated security. technological building blocks of the digital single market the most important technological building blocks of the digital single market defined by the european union are:  new generation networks (5g)  computer cloud services.  internet of things.  technologies for processing large information arrays.  cybersecurity. 404 p. tomov, l. dimitrov these are areas of extreme priority in terms of rapid development of the necessary directions for the fourth industrial revolution. 6. conclusions on the basis of the above, the following conclusions can be drawn: the connections of information technologies with people, machines and products have been rapidly realized thanks to the rapid development of technology transfer standards and a comprehensive information infrastructure. the vertical and horizontal machine-internet, machine-human and machine-machine collaboration along the value chain, in real time, is the basis of the intelligent production system. a portfolio of software based systems, based on a backbone data platform, is proposed and includes the modules;  product lifecycle management (plm).  manufacturing execution system/manufacturing operations management (mes/ mom).  totally integrated automation (tia) flexible systems and models that are capable of responding in real time allow internal processes to be radically optimized. references 1. berger, s., 2013, making in america: from innovation to market, the mit press, cambridge, ma, usa. 2. chesbrough, h., 2003, open innovation: the new imperative for creating and profiting from technology, harvard business school press. 3. kagermann, h., 2014, chancen von industrie 4.0 nutzen, springer vieweg, wiesbaden. 4. tomov, p., 2017, increasing the efficiency of automation of production processes by reporting the parameters of the parts' flow, tem journal, 6(3), pp. 484-487. 5. mandl, c., hauser, m., mandl, h., 2013, the co-creative 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29 https://doi.org/10.22190/fume191024009j © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy; dry machining and alternative cooling techniques sonja jozić, ivana dumanić, dražen bajić faculty of electrical engineering, mechanical engineering and naval architecture, university of split, split, croatia abstract. the latest trends in machining research show that great efforts are being made to understand the impact of different cooling and lubrication techniques as well as cutting parameters on machining performances. this paper presents the investigation results of different cutting parameters and different cutting environments such as dry machining, minimum quantity lubrication (mql) and minimum quantity lubrication with compressed cold air (mql+cca) on average surface roughness, cutting force and material removal rate. the experiments were designed based on three input parameters and three different cutting environments when turning of en aw2011 alloy. taguchi-based grey relational analysis was used to identify the optimal process parameters by which minimum values of surface roughness, minimum value of cutting force and maximum value of material removal rate will be achieved. the results showed that minimum quantity lubrication in the stream of compressed cold air, in comparison to dry and minimum quantity lubrication machining, gives the best machining performances. therefore, the use of mql + cca method, which reduces the amount of lubricant may represent in the described extent of turning operations an alternative to turning processes most often carried out by wet method that causes considerable costs for purchasing, maintaining and using cutting fluids. key words: turning, dry machining, minimum quantity of lubrication, compressed cold air cooling, taguchi design, grey relational analysis received october 24, 2019 / accepted december 19, 2019 corresponding author: sonja jozić university of split, faculty of electrical engineering, mechanical engineering and naval architecture, r. boškovica 32, 21 000 split, croatia e-mail: sjozic@fesb.hr 14 s. jozić, i. dumanić, d. bajić 1. introduction nowadays, modern mechanical industries are constantly trying to design products and processes that can run faster, last longer and operate more precisely. contemporary highperformance machines that undergo higher loads and increase moving speed of the moving parts are requiring that bearings, seals, shafts, machine guides, gears and other mechanical elements have to be dimensionally and geometrically accurate or the surface texture of the manufactured parts must be precise [1]. therefore, the objective of machining operations is to produce mechanical elements with specified quality as productive as possible. to achieve the highest possible efficiencies in machining processes, understanding the relationships between process responses and the process controlling factors must be attained. the machining processes generate heat that is distributed into tool, workpiece, chips and environment. the most efficient way to reduce the temperature during machining is to apply a cutting fluid. therefore, the cooling process becomes an integral part of every machining process. the most popular method of cooling the cutting zone is a conventional one, by flooding. the main disadvantage of this cooling method is a high flow rate of refrigerants and the fact that only a small part reaches the contact area between the cutting edges and the material being machined. as a result of the economic (high cost of cutting fluids) and ecological (negative effects on humans and the environment) pressures, the industry seeks for newer methods to minimize the consumption of harmful lubricants, [2]. in recent years, several technologies have been developed in order to increase the overall effectiveness of the process like flood cooling, cryogenic cooling, solid coolants/lubricants, high pressure coolants (hpc), minimum quantity lubrication (mql)/near dry machining (ndm), internal tool cooling and cooling with compressed air/gases [3]. the concept of dry machining, which has been suggested by many contemporary researchers, has many advantages such as: the absence of adverse effects on humans and the environment, the reduction of variable machining costs due to the lack of cutting fluid, easier recycling of a chip that is not contaminated with the cutting fluids and the possibility of applying high cutting speeds and reducing the cutting forces and thus the longer tool life [4]. the loss of its positive effects, namely lubrication, cooling and chip flushing, is imposed by the elimination of the coolant. also, during the dry machining, the mechanical and thermal loads of the cutting tool are increased [5]. minimum quantity lubrication is a cooling technique during which a very small amount of lubricant agent is applied during the machining process. mql results in a significant reduction in the cutting temperatures along with favorable work-chip and worktool interactions. this leads to a reduction in the surface roughness as well as cutting forces. dhar et al. [6] investigated the mql technique when turning aisi-1040 steel and their results of the study indicated that the mql machining system is more efficient than a conventional flood coolant system. priarone et al. [7] performed milling experiments of titanium aluminides with different types of cooling conditions; dry, wet and mql. they have shown that mql gave the best results in terms of surface roughness even when compared to wet machining [7, 8]. sharma et al. [9] concluded that mql leads to decreasing of the cutting temperature which results in improving the tool life as well as the surface roughness. maruda et. al [10] and maruda et al. [11] concluded that the use of the mql cooling technique reduces the machined surface roughness parameters and cutting force values when turning the carbon steel compared with compressed air cooling and dry experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 15 machining. the mql research in the literature has shown so far that this technique has proved its efficiency, e. g.: allowing reduction in lubricant use from 50 % to 90 % [12] and lower energy consumption, better performances and environment protection [13]. the extension of the mql system is a system where minimal quantity of lubricant is combined with compressed cold air (mql + cca). in such systems, the lubricant is used to reduce friction while compressed cold air enhances the cooling and flushing action [14]. pervaiz et al. [14] and pervaiz et al. [15] explored the machinability of ti6al4v using a vegetable oil-based mql system mixed with sub-zero temperature air. it was observed that the vegetable oil-based cooling strategy had promising potential to replace conventional flood cooling methods. yuan et al. [16] reported that mql with compressed cold air significantly reduced cutting force, tool wear and surface roughness when machining titanium alloy (ti6al4v). singh and sharma [17] concluded that using a ranque-hilsch vortex tube in addition to the mql cooling process during turning of commercially pure titanium led to an improvement reducing surface roughness values and cutting forces when compared to mql process. multiple characteristic optimization methods have been the focus of the recent research directed at improving product quality and reducing the costs of the machining process. among various optimization methods are grey relational analysis (gra), technique for order preferences by similarity to ideal solution (topsis), genetic algorithms (ga), desirability analysis (da), metaheuristic algorithms and other methods that allow multiple performance characteristics to be optimized simultaneously. diyaley and chakraborty [18] used six most popular metaheuristic algorithms to determine optimal values of the cutting parameters during roughing and finishing milling operations in order to minimize total production time and total production cost. gopal and prakash [19] used gra and topsis to optimize milling parameters in order to minimize cutting force, surface roughness and temperature during end milling process, and both methods gave the similar optimal cutting parameters. fratila and caizar [20] conducted an experimental investigation on milling of almg3 alloy under conventional flood lubrication, minimal quantity lubrication and dry milling. by using the taguchi optimization methodology, they found optimal cutting parameters for minimal cutting power and surface roughness. yan and li [21] used a multiobjective optimization method based on weighted grey relational analysis and response surface methodology (rsm) to optimize the cutting parameters in milling process. they tried to evaluate trade-offs between sustainability, production rate and cutting quality. lin [22] used the taguchi method together with gra to deal with optimization of the turning operations, for the machining of s45c steel, with the multiple performance characteristics (tool life, cutting force, surface roughness). s. tripathy and d.k. tripathy [23] performed multi-response optimization of the cutting parameters using grey relational analysis together with the topsis method. to obtain favorable process output values such as radial thrust force, cutting power and coefficient of friction when dry turning of ti-6al-4v. li et al. [24] combined the taguchi-based grey relational analysis and the kernel principal component analysis (kpca) to optimize machining parameters such as type of inserts, feed rate and depth of cut. mia et al. [25] optimized cutting forces, surface roughness, cutting temperature, and chip reduction coefficient when turning of ti-6al-4v in dry and high-pressure coolant condition using gra combined with the taguchi method. in their next paper [26] they experimentally investigated surface roughness, cutting force, and feed force when turning the same 16 s. jozić, i. dumanić, d. bajić workpiece material under cryogenic (liquid nitrogen) condition. they performed multiresponse optimization according to the models of responses by response surface methodology (rsm) and artificial neural network. aman et al. [27] utilized the taguchi method and the response surface method (rsm) to find out the influence of depth of cut, cutting speed, cutting environment, cutting tool geometry and feed rate on power consumption. it can be observed that whereas different studies have found mql and mql+cca as an improvement and benefits over dry machining, wet machining and flood lubrication, fuller understanding of mql and mql+cca process is still required. also, few research projects have been performed for which the cutting environments (dry, mql and mql+cca) were variables. it is evident from the review of available literature that no studies have been found on the use of mql in turning of the aluminum alloy en aw-2011. this alloy is often called a free machining alloy and it is well suited to the use in automatic lathes. boswell et al. [28] provided an overview of the effectiveness of mql during conventional machining; turning, milling, grinding, drilling for different workpiece materials such as steel, aluminum alloys (a7175, a6061, a1050) and difficult-to-cut materials (ti48al-2cr-2nb intermetallic alloy, ti-6al-4v titanium alloy, nickel-based alloys and ironbased alloys). as noted earlier, the objective of this study is to research the effects of cutting environments (dry machining, mql, mql + cca) on surface roughness, cutting force component in the z direction and on the material removal rate in the longitudinal turning at varying cutting parameters (cutting speed, feed rate, depth of cut). cutting force components in x and y direction were not taken into consideration because of the negligible small values of fx and fy in comparison to fz. in this study, the use of the taguchi method to design the tests and grey relational analysis to find an optimum cutting environment from the machining response, such as cutting force, surface roughness and material removal rate was reported. confirmation test was established after optimizing machining parameters. 2. experimental work the aim of the experimental study was to evaluate the influence of cutting parameters and cutting environments on the surface roughness, cutting force and the rate of material removal. the experiments were carried out on the universal lathe machine with a spindle motor of 11 kw power and the spindle speeds in the range from 11.2 rpm to 2240 rpm. turning inserts, the vcgt 160404-as produced by iscar were mounted on the tool holder svjcr 2020k-16. the workpiece material was en aw-2011, aluminum alloy. the chemical composition of the material in mass fraction was al (91.4%), cu (4%), fe (0.7%), pb (0.6%), bi (0.6%), si (0.4%) and zn (0.3%). physical and mechanical properties of the alloy are presented in table 1. the workpieces were shafts with dimensions of 300 mm length and 75 mm in diameter with machined slots to perform experiments and measurements much easier, fig. 1. the workpiece was clamped into the chuck, while the other side was supported by a tailstock to prevent vibration during machining. the surface roughness was measured using the mitutoyo surfest sj 301 profilometer. sampling length and the cut-off length for the measurements of surface roughness were experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 17 selected to be 5.6 mm and mm 0.8, respectively. every measurement was repeated at three different locations and the average value was considered. the system for measuring the cutting force consists of dynamometer kistler 9257a mounted between the cutting tool and the tool support as well as multi-channel charge amplifier kistler 5007a. the charge amplifier serves to amplify and convert the electrical charge delivered from the dynamometer into corresponding voltage and then forwards the signals to the a/d interface board (bmc usb-ad16f). the nextview 4 software was used for analyzing the cutting forces. all the measuring instruments were calibrated before the measurements were carried out. measurements and analysis equipment are shown in fig. 2. fig. 1 dimension of the workpiece table 1 physical and mechanical properties of used material in experimental part alloy en aw-2011 density g/cm³ 2.82 coefficient of thermal expansion (20-100 °) 10-6/k 23.4 modulus of elasticity mpa 72500 tensile strength (rm) mpa 320 yield strength (rp0,2) mpa 270 elongation a50 (%) 8 brinell hardness 90 fig. 2 measurement and analysis equipment minimum quantity lubrication (mql) is an intermediate solution for switching from the conventional use of cutting fluids to dry machining. as such it belongs to semi dry 18 s. jozić, i. dumanić, d. bajić methods of cooling and lubrication. mql technique shows the great potential and advantages over conventional cutting fluids systems, especially if the economic and environmental aspect of application of chosen techniques of cooling and lubrication in machining is taken into consideration. fig. 3 schematic view of the experimental set up the minimum quantity lubrication system vectolub was applied in this work for the turning process. in this system, every micropump, whose working frequency is set by a pneumatic pulse generator, delivers the lubricant through the coaxial line into a bifluid projection nozzle. the compressed air is provided by an external compressor and delivered into the mql. the air and the lubricant are parallel conducted in a coaxial line. in the nozzle, the lubricant is broken down and transported into microdroplets by the air. thus, a homogeneous lubricant film at the friction point is formed. in this study, cold air guns which use a vortex tube were used to improve the performance of the mql process as shown in fig. 3. a high-pressure air stream enters the vortex tube tangentially and after passing through the chamber, the compressed inlet air accelerates at a high rate of speed. the hot gas stream leaves the tube through the control valve, while the cold gas stream passes through the cold end, near the entrance nozzle. experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 19 2.1. controllable parameters and process responses cutting speed vc, feed rate f, the depth of cut ap and cutting environments were considered as controllable variables. the values of machining parameters were selected from the manufacturer’s handbook recommended for the tested material together with the machine tool capabilities. all factors had three levels as shown in table 2. cutting environment as the fourth controllable variable were dry, mql and mql with compressed cold air (mql + cca). table 2 machining parameters and their levels machining parameters level 1 level 2 level 3 cutting speed, vc [m/min] 200 300 400 feed rate, f [mm/rev] 0.05 0.15 0.25 depth of cut, ap [mm] 1 1.75 2.5 cutting environment, ce dry mql mql+cca the process responses that have been considered are surface roughness, cutting force and material removal rate. in this study, surface roughness average (ra) is the surface parameter used to evaluate surface roughness. the required surface roughness cannot be achieved as easily as physical dimensions due to the influence of many factors. the most important of them are: workpiece and tool material, tool geometry, tool wear, machining parameters, dynamic behavior of machining system, application of cooling and lubrication agents. cutting forces are important physical variables that provide relevant information about machining processes such as machinability, tool conditions, power consumption, etc. cutting parameters, type of cooling methods, and cutting tool geometry are the most important parameters that affect cutting forces. the material removal rate (mrr) indicates the amount of material removed from the workpiece per unit of time. the higher the values of the cutting parameters, the greater value of the material removal rate is provided, and thus the less cycle time. 2.2. design of experiments and experimental results the influence of the controllable factors on the process performances was examined using the taguchi approach. the taguchi method generally requires a small number of experiments and l9 orthogonal array was chosen to investigate the influence of machining parameters (vc, f, ap), on cutting output variables (fz, ra, mrr). the influence of cutting environment on mentioned response variables was examined by performing three sets of experiments: dry turning, mql turning and mql+cca turning. therefore, it is carried out a total of 27 experiments. highly efficient production can be achieved by increasing the volume of the removed material per time unit. in this study, the material removal rate in the turning process is calculated using the equation: c p mrr v a f (1) the design of experiments together with experimental results is given in table 3. 20 s. jozić, i. dumanić, d. bajić table 3 design of experiments based on l9 orthogonal array and experimental results cutting speed [m/min] feed rate [mm/rev] depth of cut [mm] cutting condition ra [µm] fz [n] mrr [mm3/s] 1 200 0.05 1 dry mql mql+cca 0.53 0.41 0.47 46.55 42.30 39.88 166.67 2 200 0.15 1.75 dry mql mql+cca 1.43 1.28 1.34 176.18 160.35 144.94 875 3 200 0.25 2.5 dry mql mql+cca 4.91 4.68 4.77 427.06 343.38 337.39 2083.33 4 300 0.05 1.75 dry mql mql+cca 0.35 0.22 0.26 82.94 73.71 69.94 437.50 5 300 0.15 2.5 dry mql mql+cca 1.45 1.31 1.36 267.77 230.26 231.73 1875 6 300 0.25 1 dry mql mql+cca 4.58 4.42 4.44 223.77 164.66 179.26 1250 7 400 0.05 2.5 dry mql mql+cca 0.48 0.42 0.46 114.17 107.94 97.19 833.33 8 400 0.15 1 dry mql mql+cca 1.49 1.42 1.42 117.64 109.82 100.04 1000 9 400 0.25 1.75 dry mql mql+cca 4.61 4.39 4.44 270.74 272.21 229.13 2916.67 3. results and discussion the different machining aspects of en aw-2011 studied by using dry, mql, mql + cca cutting environments are presented below. 3.1. surface roughness and material removal rate the three machining parameters, namely, the cutting speed, the feed rate and the depth of cut are found to influence surface roughness but as mentioned in the introduction the feed rate is the parameter that has the highest influence on surface roughness. fig. 4 represents the value of surface roughness with the increasing of feed rate at varied cutting environments for cutting speed of 200 m/min. from fig. 4 it can be observed that the surface roughness increases with the increase in the feed rate for all cutting environments. this is because at large feed rates the distance between peaks and valleys of the feed marks generated by the tool grooves is much more important. experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 21 fig. 4 influence of feed rate to surface roughness experimental runs with the results are given in table 3 where it can be clearly seen that mql resulted in lower surface roughness values when compared to mql + cca and dry cutting process for all the combinations of parameters. this is because in mql process, the formation of defects (built up edge, debris, adhesion of micro particles, etc.) is reduced due to better lubrication in comparison with mql + cca where compressed cold air carries off a small amount of lubricant from the cutting zone. 3.2. cutting forces fig. 5 shows cutting force fz under different cutting environment. the cutting force has minimal value during the 1st trial by using mql + cca environment. the cutting force increases with the increase of the depth of cut and the feed rate due to an increase of the cutting action area. also, the increase of these two parameters causes the material removal to become difficult due to increasing shear force. mql + cca resulted in lower cutting forces values when compared to mql and dry cutting process. fig. 5 effect of cutting parameters on cutting force under different cutting environments the reduced temperature at the cutting zone accomplished by compressed cold air and lubricate effect of mql leads to avoiding the adhesion and welding of chips, reduction in friction produced and hence a reduction in the cutting force. the maximum value of cutting force (third trial) at the same parameters came out to be 27 % lower for mql + cca compared to dry processing, which is a quite significant reduction. 22 s. jozić, i. dumanić, d. bajić 4. optimization methodology grey relation analysis the taguchi's method uses a signal-to-noise ratio (s/n) as the core criterion for analysis of experimental data. the purpose of the s/n ratio is to identify a control factor that reduces process variability by reducing the impact uncontrollable factors i.e. factors of noise. a positive effect on an output variable denotes the term "signal" while the term "noise" refers to an undesirable effect on an output variable. therefore, estimating the deviation of the output variable from the desired value is equal to the s/n ratio. the high value of s/n ratio indicates the optimal level of control factors and it is determined by using three types of quality characteristic: "higher the better", "smaller the better" and "nominal the best." for machining in general the small values of surface roughness ra and cutting force fc are desirable as well as a large value of the material removal rate. the following equations are used to calculate the s/n ratio, for smaller the better characteristic, equation (2), and higher the better characteristic, equation (3):          n i iy n ns 1 21 log10/ (2)           n i iy n ns 1 21 log10/ (3) where: i – experiment number, yi – measured value of output variable, n –replicates number. based on the calculation of s/n ratio and the mean effect plot analyses, three sets of optimal input parameters, within the offered levels, will be obtained, one set per cutting environment (dry, mql, mql+cca) and furthermore, each set has optimal input parameters for every output variable. this will allow the determination of optimal parameters for each process response separately. the following is explained determining of optimum parameters when dry turning, for minimum surface roughness. the s/n ratio analysis was used to generate the response table for the s/n ratio of surface roughness, ra, (table 4). delta value is the difference between the highest and lowest average value of each input variable and ranks are assigned according to the delta value. the factor with the largest delta has the greatest influence on ra. main effects plot for s/n rations for surface roughness, based on the data in table 4, is shown in fig. 6. table 4 response table of s/n ratio ra for dry turning level vc f ap 1 -3.80 7.00 -3.72 2 -2.44 -3.27 -2.42 3 -3.45 -13.44 -3.56 delta 1.36 20.44 1.30 rank 2 1 3 experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 23 fig. 6 main effect plots of s/n ratio ra for dry turning the trend of the plot indicates that ra is greatly influenced by variations in feed rate, (rank 1). from the known theory and also from the experimental result (table 3) it can be seen that surface roughness increases as feed rate increase. minimum surface roughness is achieved with levels of the control parameters for which the s/n ratio has a maximum value, a2 (cutting speed of 300 m/min), b1 (feed of 0,05 mm/rev) c2 (depth of cut of 1,75 mm). ultimately, it is more effective and beneficial if it is possible to identify only one set of optimal parameters for all cutting environments and for all the responses. the grey relational analysis, gra, was adopted in the study to find optimal controllable variables for multi-objective function. furthermore, the statistical analysis of variance was performed to find the influence of the controllable process factors on the multi-objective function. the first step of the grey relation analysis, gra is the grey relation generation performed in order to make experimental data comparable. during this step, the cutting force, surface roughness and mrr are normalized between zero and one. depending on characteristics of a data sequence, various methodologies of carrying out grey generation are available. in this study, a linear data pre-processing method for the cutting force and surface roughness is the lower-the-better and is expressed as: ijij ijij ij yy yy x minmax max    (4) where xij is the value after the grey relation generation, min yij is the smallest value of original data yij for the jth response of the experiment i, and max yij is the largest value of yij for the jth response. the normalized value of the original sequence for material removal rate which is larger-the-better performance characteristic can be expressed as: ijij ijij ij yy yy x minmax min    (5) table 5 shows the normalized values of surface roughness ra, cutting force fz and material removal rate mrr. 24 s. jozić, i. dumanić, d. bajić table 5 normalized experimental results cutting environment ra [µm] fz [n] mrr [mm3/s] 1 dry mql mql+cca 0.9346 0.9602 0.9474 0.9828 0.9938 0.3999 0 2 dry mql mql+cca 0.7435 0.7746 0.7616 0.6480 0.6889 0.7287 0.2576 3 dry mql mql+cca 0 0.0491 0.0299 0 0.2161 0.2316 0.6970 4 dry mql mql+cca 0.9735 1 0.9929 0.8888 0.9126 0.9224 0.0985 5 dry mql mql+cca 0.7378 0.7692 0.7579 0.4114 0.5083 0.5045 0.6212 6 dry mql mql+cca 0.0708 0.1038 0.1002 0.5251 0.6778 0.8520 0.3939 7 dry mql mql+cca 0.9453 0.9581 0.9496 0.8081 0.8242 0.8520 0.2424 8 dry mql mql+cca 0.7288 0.7450 0.747 0.7992 0.8194 0.8450 0.3030 9 dry mql mql+cca 0.0633 0.1108 0.1011 0.4038 0.3999 0.5112 1 the second step of the gra is the determination of referent sequence x0j for jth response. the performance of experiment i is considered as the referent for response j if normalized value xij is equal to 1 or nearer to 1 than the value for any other experiment. in the next step, the grey relation coefficient is used for determining relation degree between xij and x0j. the larger the grey relation coefficient, the closer xij and x0j. grey relation coefficient ξij can be calculated as: maxij maxmin       ij (6) where  is the index of distinguishability called distinguishing coefficient   (0,1]. in this work, the value of coefficient  is assumed as 0.5. 0 | | ij j ij x x   (7) min min{ , 1,2,..., ; 1,2,..., } ij i m j n     (8) max max{ , 1,2,..., ; 1,2,..., } ij i m j n     (9) experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 25 the last step is the determination of the grey relation grade which enables multiple optimizations and is calculated using equation:    n j iji n y 1 1  (10) where n is a number of experiment. the larger value of grey relation grade, the closer the corresponding controllable parameter combination to optimal. therefore, the experiment with the highest grey relation grade would be the best choice. table 6 shows the grey relation coefficients and grade for each experiment. the highest grey relational grade is the order to 1. the nearest optimum controllable parameters combination is in the experiment 1 with mql cutting environment (a1b1c1d2). the optimal levels, levels with the highest grey relation grade of the process parameters are mql + cca cutting environment, high cutting speed (400 m/min), low feed rate (0.05 mm/rev) and intermediate depth of cut (1.75 mm). these optimal levels are shown in bold in table 7. optimal combination of parameters is a3b1c2d3. table 6 grey relational coefficients and grey relational grade cutting environment ra [µm] fz [n] mrr [mm3/s] grade grey order 1 dry mql mql+cca 0.8844 0.9263 0.9049 0.9667 0.9876 1 0.3333 0.7281 0.7491 0.7461 5 1 2 2 dry mql mql+cca 0.6609 0.6893 0.6771 0.5868 0.6164 0.6482 0.4024 0.5501 0.5694 0.5759 21 19 18 3 dry mql mql+cca 0.3333 0.3446 0.3401 0.3333 0.3894 0.3942 0.6226 0.4298 0.4522 0.4523 27 25 24 4 dry mql mql+cca 0.9496 1 0.9860 0.8181 0.8513 0.8656 0.3568 0.7081 0.7360 0.7361 6 4 3 5 dry mql mql+cca 0.6560 0.6842 0.6738 0.4593 0.5042 0.5023 0.5690 0.5614 0.5858 0.5817 20 16 17 6 dry mql mql+cca 0.3498 0.3581 0.3572 0.5129 0.6081 0.5814 0.4521 0.4383 0.4727 0.4636 26 22 23 7 dry mql mql+cca 0.9014 0.9227 0.9084 0.7227 0.7400 0.7716 0.3976 0.6739 0.6867 0.6925 9 8 7 8 dry mql mql+cca 0.6484 0.6622 0.6640 0.7134 0.7346 0.7629 0.4177 0.5932 0.6049 0.6149 15 12 11 9 dry mql mql+cca 0.3480 0.3599 0.3574 0.4561 0.4545 0.5057 1 0.6014 0.6048 0.6210 14 13 10 26 s. jozić, i. dumanić, d. bajić table 7 means for grey relational grades parameter level 1 level 2 level 3 rank (max-min) vc [m/min] 0.5837 0.5871 0.6326 3 (0.0489) f [mm/rev] 0.7174 0.5819 0.5040 1 (0.2134) ap [mm] 0.6012 0.6337 0.5685 2 (0.0652) ce 0.5871 0.6068 0.6093 4 (0.0222) 5. analysis of variance for grey relation grade the percentage contribution by the sum of squares each of the process parameters to the total sum of squared deviations was used to evaluate the importance of the controllable parameter on the performance characteristic. the sum of squared deviations ss for the considered factor can be defined as:    k t t m y y m k ss 1 2 2 (11) where k is the number of levels, m is the total number of experiments, yt is the total sum of the grey relation grade at tth level and y is the total sum of the grey relation grade. the total sum of the squared deviations sst can be defined as: m y yss m i n j ijt 2 1 1 2    (12) where yij are individual observations. finally, percentage contribution p can be calculated as: tss ss p  (13) it can be observed from table 8 that the feed rate had the greatest influence on the grey relation grade. for multiple performance characteristics, the cutting environment did not have an impact on the experiments. the cutting environment might have an effect on some performance characteristics individually. table 8 percentage contribution of input parameters on response variables ss percentage contribution vc 0.0134 5% f 0.2099 78% ap 0.0191 7% ce 0.0027 1% sst 0.2706 experimental analysis and optimization of the controllable parameters in turning of en aw-2011 alloy... 27 6. confirmation test once the optimal level of controllable parameters is selected, the final step is to predict and verify improvement of the performance characteristic using the optimal level of the controllable parameters. based on the previous discussion, the most significant parameters with optimal level are cutting speed at level 3, feed rate at level 1 and depth of cut at level 3. the estimated grey relation grade can be calculated as: 1 ˆ ( ) p t i t i y y y y     (14) where yt is the total mean of the grey relation grade, p is the number of the controllable parameter that significantly affects the multiple performance characteristic and yi is the mean of the grey relation grade at the optimal level. table 7 shows the estimated grey relational grade for optimal controllable parameters calculated using eq. (12). also, table 8 presents the results of the confirmation experiment. in this study, a confirmation experiment was conducted by utilizing the levels of the optimal parameters combination of the turning process, a3b1c2d3. as shown in table 9, the surface roughness was improved from 0.41 µm to 0.3 µm, the cutting force was slightly reduced from 42.30 n to 36.28 n and the material removal rate increased from 166.67 mm3/s to 583.33 mm3/s. also, a good agreement between the actual and predicted grey relational grade was obtained and multiple characteristics in turning operations are improved. table 9 results of the confirmation experiment initial controllable parameters optimal controllable parameters prediction experiment setting level a1b1c1d2 a3b1c2d3 a3b1c2d3 ra [µm] 0.41 0.3 fz [n] 42.30 36.28 mrr [mm3/s] 166.67 583.33 grg 0.7491 0.7814 0.7737 improvement in grg 0.0246 7. conclusion this paper presents a comprehensive experimental evaluation of an alternative cooling method when turning en aw-2011. the use of the taguchi method and the grey relational analysis to optimize the turning operations with the multiple performance characteristics has been reported. optimum values of cutting parameters and optimum machining condition were found out to minimize two response variables: the surface roughness and cutting force and also to maximize the material removal rate. the conclusions are summarized as follows: i. using mql and mql+cca lead to improvement in surface roughness and cutting force when compared to dry turning. ii. the maximum percentage reduction in cutting force during mql+cca turning is iii. 27 % and 19 % in comparison to dry and mql turning, respectively. 28 s. jozić, i. dumanić, d. bajić iv. using the optimal levels of the controllable parameters (a3b1c2d3) in the machining process, it is possible to decrease surface roughness and cutting force as well as to maximize the material removal rate. v. feed rate is the dominant factor affecting the main 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by-nc-nd original scientific paper novel smart design of rear half shaft of large urban transport vehicle fabio alberti1, danilo d’andrea2, alarico macor3, giacomo risitano2, antonio rossetti4, aleksandar sedmak5,6 1department of mechanical and aerospace engineering, politecnico di torino, italy 2department of engineering, university of messina, italy 3department of engineering and management, university of padua, italy 4istituto per la tecnologia della costruzione, consiglio nazionale delle ricerche, italy 5faculty of mechanical engineering, university of belgrade, serbia 6university of brasov, faculty of civil engineering, brasov, romania orcid ids: fabio alberti https://orcid.org/0000-0003-2457-8344 danilo d’andrea https://orcid.org/0000-0002-9809-8434 alarico macor https://orcid.org/0000-0003-2244-6524 giacomo risitano https://orcid.org/0000-0002-0506-8720 antonio rossetti https://orcid.org/0000-0003-0635-6748 aleksandar sedmak https://orcid.org/0000-0002-5438-1895 abstract. new design method of rear half shaft of a large urban transport vehicle with a conventional (power shift) and hydro-mechanical (power split) transmission is presented. the so-called smart design is based on the automatic cumulative fatigue damage determination combining the rainflow method and palmgren-miner rule. previous study, comparing the power shift transmission with other alternatives, clearly demonstrated that the power split transmission offers significant advantages in terms of pollutant emissions and fuel consumption, making it a better choice for the ecological transition. anyhow, from a mechanical point of view, the results of this work show that the load condition of the hydro-mechanical power split transmission reduce significantly the fatigue life of a component initially designed for a conventional transmission. key words: large urban transport vehicle, power split/shift transmission, smart design, cumulative fatigue damage 1. introduction problems of air pollution and road traffic noise are two important environmental troubles that could be harmful to the health and well-being of urban populations [1–3]. many studies have been conducted focusing on the design of new type of transmission for high power received: february 03, 2024 / accepted may 15, 2024 corresponding author: aleksandar sedmak faculty of mechanical engineering, university of belgrade, kraljice marije 16, 11000 belgrade, serbia asedmak@mas.bg.ac.rs https://orcid.org/0000-0003-2457-8344 https://orcid.org/0000-0002-9809-8434 https://orcid.org/0000-0003-2244-6524 https://orcid.org/0000-0002-0506-8720 https://orcid.org/0000-0003-0635-6748 https://orcid.org/0000-0002-5438-1895 584 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak density vehicles such as wheel loader [4] buses for urban transport [5], taxies [6,7] and other urban vehicles [8], to improve the efficiency of the system and reduce consumptions and emissions. in recent years, electric vehicles have been introduced, which have the potential to significantly reduce carbon emissions and improve air quality in urban areas, compared to conventional vehicles [9]. in addition, electric vehicles are generally quieter than internal combustion engine vehicles, which can help reduce noise pollution. this can be particularly beneficial for densely populated areas where noise pollution can have a significant impact on quality of life [10]. despite the hybrid electric or full electric transmission is becoming more and more common in the automotive field, the costs and the reliability related to this technology make it unsuitable for all types of applications [11,12]. in this sense, the use of a hydro-mechanical transmission could be a valid solution attending for a more reliable and less expensive technology in the electric field. d’andrea et al. [13] have compared a hydrostatic transmission with a hydro-mechanical transmission designed for an urban street sweeper, highlighting the advantage of using the hydro-mechanical transmission from the fuel consumption point of view. the same authors, have been investigated the fatigue damage of a front half shaft of a road car comparing three different fatigue damage criteria [14]. related to a similar mechanical component, the drive axle housing of a truck was analyzed using static, fatigue, modal, harmonic response, and random vibration analyses [15]. two lightweight optimization schemes were proposed: topology optimization, which reduced mass by 17.4%, and a multi-objective optimization approach. the latter approach achieved a reduction in mass of 4.35%, decreased equivalent stress by 21.05%, increased the minimum life of the component by 72.28%, and improved the natural frequencies. asi [16] presents a failure analysis of a rear axle shaft from a car involved in an accident. the axle shaft was found to have fractured into two pieces. the purpose of the investigation was to determine whether the failure was the cause or a consequence of the accident. various evaluation techniques were employed, including visual examination, photo documentation, chemical analysis, micro-hardness measurement, tensile testing, and metallographic examination. the results indicate that the axle shaft failed due to reversed bending fatigue caused by improper welding. decker et al. [17] investigate the mechanics and interactions of wheel forces, suspension component forces, and axle casing stress in a multiaxial-randomly loaded airsuspension rear-axle for commercial vehicles. fatigue, frequency, and correlation analyses are performed using load-time and stress-time data from a prototype vehicle tested on a track. the study identifies critical load situations affecting the axle's fatigue behavior under operational conditions, leading to optimized design solutions and experimental validation. wu et al. [18] have investigated a hydro-mechanical continuously variable transmission for the all-terrain vehicle, evaluating the vehicle dynamic performance and the speed ratio control by simulation and experiment on a transmission prototype. bayrakceken et al. [19] have been focused their study on the fracture analysis of a universal joint yoke and a drive shaft in the power transmission system of an automobile. the failures in these components can be attributed to various factors such as manufacturing and design faults, maintenance issues, raw material defects, material processing flaws, and user-related errors. investigation involves spectroscopic and metallographic analyses, as well as hardness measurements for each part. additionally, stress analyses using the finite element method are conducted to determine the stress conditions at the failed sections. barone et al. [20] establish the t-n torsion fatigue curve at r = -1 for the front half-shaft mechanical system in an existing car. experimental fatigue tests were conducted on 15 front half-shafts. crack propagation evaluations and novel smart design of rear half shaft of large urban transport vehicle 585 failure analysis were performed to identify the causes of the fractures. as a result, the t-n fatigue curve for the front half-shaft mechanical system was obtained. by developing a multidisciplinary approach, it is possible to investigate the fatigue damage in mechanical components of the same vehicle equipped with different powertrain solutions, ultimately aiming for an improved design. in this work, a comparison between the conventional transmission (power shift) and hydro mechanical transmission (power split) [21] of an urban bus has been carried out from both energetic and mechanical point of view. from the energetic point of view, a comparison between power shift and power split transmission has been already carried out by the authors in a previous work, highlighting the advantage of using power spilt transmission for the reduction of consumptions and emissions in urban buses [21]. in addition, the workflow adopted in this work has been already developed by the authors in a previous work [22] and it is indicated as “smart design”. smart design consists of an automated and multi-object algorithm aimed at evaluate the fatigue damage of a mechanical component under specific load condition (e.g., mission test) for the best possible design, taking into account others design needs (e.g., environmental sustainability. the target component chosen for this work is the rear half shaft of an urban bus related to the traction. through a cooperation between the university of messina and the urban transport company atm messina s.p.a., it was possible to acquire different driving cycles in order to test both transmissions. the experimental setup consists of a portable control unit model scads xs compact provided by siemens company, together with a gps and a tablet for monitoring the signals in real-time. both power shift and power split transmission model have been developed in simcenter amesimtm environment. the models output of interest is the torque evaluated at the traction wheel, which represents the input load condition of the finite element model (fem) of the target component, developed in ansys workbench r2 2020tm. the stress distribution evaluated from the fem, has been used for evaluating the fatigue damage of the component, starting from the s-n curve of the material which the half shaft is made [23]. the automated algorithm developed in matlabtm is based on the rainflow method [24] and goodman criterion [25]; the fatigue damage has been assessed by the palmgren-miner rule [26,27]. a key limitation in current methodologies is their inability to accurately model complex interactions in new powertrain technologies, especially under variable amplitude loading conditions. previous studies have often failed to account for the nuanced effects of modern transmission systems on component stress and fatigue life, relying only on the energy efficiency of the vehicle without considering any mechanical aspects. this study aims to address these gaps by introducing an enhanced analytical approach that combines advanced fem analysis with the goodman criterion for stress adjustment. by integrating these methodologies, a more accurate prediction of fatigue life under realistic operational conditions has been provided. despite it was demonstrated that, with a proper management of the internal combustion engine (ice) in terms of torque and angular speed, the power split transmission turns out to be more energy efficient with respect to a traditional stepped gear transmission [5], it was remarked that the transmission change leads to a different fatigue damage, as shown in [22] by the comparison between a conventional and hybrid electric transmission. also in this work, the results show that the power split transmission reduces the half shaft fatigue life in comparison to the power shift transmission, leading to the component failure. this approach allows for a more refined assessment of fatigue damage, particularly in automotive components subjected to complex load profiles. by surveying the drawbacks and limitations of previous studies, this research not only advances the theoretical framework but also significantly enhances practical applications in fatigue assessment. 586 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak 2. materials and methods 2.1. reference vehicle the reference vehicle is a bus for urban transport. the transmissions topologies adopted in this work are the same already investigated by the authors in a previous work [21]. in particular, it was assessed that the adoption of a hydro mechanical power split transmission in urban buses leads to a reduction of fuel consumption with respect to conventional power shift transmission. the main geometric parameters of the bus are shown in table 1. 2.2. driving cycle acquisition different driving cycle of a typical route of the urban bus have been acquired in order to test both transmissions on a real mission test. the driving cycles have been acquired in the city of messina and the instrumentation used consisted of a portable control unit scads xs compact model, a gps and a tablet, as already used in a previous work by the authors [22]. the gps position of the vehicle extracted from google earth tm are shown in fig. 1, whereas fig. 2 shows the acquired driving cycles. table 1 geometrical and engine parameters of the reference vehicle parameter value vehicle model tector 7 diesel euro vi driving layout rear wheel drive vehicle mass 19000 kg length 12 m height 3.066 m wheelbase 6.120 m maximum speed limited to 80 km/h engine tector 7 euro vi type 6 cylinders in line, common rail engine volume 6.7 l maximum torque 1000 n at 1250-1700 rpm maximum power 210 kw at 2500 rpm a) b) c) d) fig. 1 driving cycles acquisition, gps positions: (a) cycle a, (b) cycle b, (c) cycle c, (d) cycle d novel smart design of rear half shaft of large urban transport vehicle 587 a) b) c) d) fig. 2 driving cycles acquisition, speed profile: a) cycle a, b) cycle b, c) cycle c, d) cycle d 2.3. rear half shaft finite element model (fem) the target component chosen for the cumulative fatigue damage calculation is the rear half shaft of the bus, which geometrical dimensions are shown in fig. 3. fig. 3 geometrical dimensions of the rear half shaft of the reference vehicle 588 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak the provided cad of the shaft has been imported in spaceclaim 2020 r2tm, cad extension of ansys workbench 2020 r2tm software used for the fe simulations. in order to save computational time, it was possible to take advantage of the symmetry of the shaft simulating only one sector instead of the full shaft. hence, three planes have been created: one orthogonal to the axis of symmetry of the shaft and two along the flank of the teeth. then, through the boolean operations, it was possible to cut the shaft along the planes direction as shown in fig. 4a and fig. 4b respectively. a) b) fig. 4 cad geometry imported in spaceclaim 2020 r2tm (a) creation of the planes (b) cutting of the geometry along the planes 2.4 mesh convergence before starting the fe simulations, it is necessary to perform a convergence analysis of the mesh through a static simulation. it is useful for choosing the right elements type and dimension in order to save computational time and maintain at the same time a good accuracy of the results. at the first, the cyclic region symmetry has been implemented, setting as low boundary one flank of the teeth and high boundary the other. in addition, a local cylindrical reference system has been created on the geometry using the same rotation direction of the shaft. the mesh element type used in the fem is a tetra 3d element, which consist of a high order 20-node solid186 element with an initial dimension of 10 mm (fig. 5a). in order to simulate the transmitted torque by the shaft, a spider beam (rigid connection) has been created on one flank of the teeth. this methodology allows simulating the force exchanged between the crown and the pinion, connecting all the nodes of the face to a remote point called master node (fig. 5b). then, all the boundary conditions have been settled. a fixed constrain has been inserted on one side of the shaft, locking all the displacements and rotations along the three axes. a cylindrical constrain has been inserted to the superior face of the shaft simulating a bearing support, locking the radial displacement and maintaining the axial and tangential displacement free. a static load of 1000 n (ramped) along x-axis, which is the exchanging force direction, has been associated to the remote point previously created (fig. 5c). the simulation time has been settled to one second. novel smart design of rear half shaft of large urban transport vehicle 589 a) b) c) fig. 5 mesh convergence analysis through static simulation (a) remote point creation (b) mesh creation with an initial size of 10 mm (c) load and constrain conditions the results of the static simulation (total deformation) with the initial size of the mesh are shown in fig. 6. the main advantage of the symmetry is that only one tooth can be modeled (fig. 6a) instead of the full shaft (fig. 6b), reducing the computational time. starting from the initial dimension of the mesh, different static simulations were performed reducing the mesh size, until the convergence of the total deformation value has been achieved (fig. 7a). the optimal compromise between the accuracy of the results and the computational time has been obtained with a mesh size of 3 mm, confirmed also by a 590 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak sensitivity analysis conducted varying the mesh size and recording the corresponding deformation variation (fig. 7b). the size of 3 mm was also adopted for the following fatigue analysis. the fem model consist of 90429 nodes and 57515 elements. a a) b) fig. 6 results of the static simulation total deformation (a) one sector (b) full shaft a) b) fig. 7 results of mesh convergence, (a) total deformation value (b) sensitivity analysis novel smart design of rear half shaft of large urban transport vehicle 591 2.5. transient simulation with respect to the static simulation, in order to consider a variable load, it is necessary to perform a transient simulation. in this case, the fe software package does not allows taking advantage of the symmetry of the shaft using the symmetry cyclic region. hence, the entire shaft has been investigated, using as load condition the torque values evaluated from amesimtm software for both power shift and power split transmission (fig. 8). among the acquired driving cycles shown in fig. 2, cycle a has been chosen as case study for testing the design method, since: i) it more accurately represents the typical or most demanding operating conditions that the vehicle or component experiences, including factors like average speed and typical load conditions; ii) the quality of the data collected during cycle a have been higher or more consistent compared to the other cycles, probably attributed to fewer data anomalies or more stable testing conditions during the data collection; iii) cycle a covers a broader range of operating scenarios like urban and extra-urban run, providing a more comprehensive assessment of the component's performance across different stress levels and operational environments. as a consequence, the choice of cycle a align more closely with the specific research objectives or hypotheses being tested. investigating both the speed profile and gps position in fig. 1 and fig. 2, it is possible to observe that the driving cycle consist of a high-speed zone related to the run in extra urban areas and a mid-speed zone related to urban areas. the high-speed values are therefore reflected in high demanded torque. considering that the material has been modelled as linear-elastic, the torque values are proportional to the maximum stress values computed from the fem analysis. hence, the low torque values do not influence the maximum stress neither the fatigue life of the component, if the stress values are below the fatigue limit of the material. taking into account these considerations, in order to save computational time only the first 300 s of the driving cycle have been considered. the boundary conditions are the same of the static simulation (fig. 5c) as well as the mesh size, with the difference that the torque (moment) is directly applied to all the flank of the teeth as shown in fig. 9. fig. 8 torque at the rear half shaft computed from amesimtm software 592 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak fig. 9 load and constrain conditions applied to the transient simulation 2.5. smart design algorithm the smart design algorithm represents a novel, rapid, and automated methodology proposed by the authors to assess fatigue damage in mechanical components and to optimize their design through multi-objective analysis. with respect to the previous work by the authors [22], the input data of the algorithms is the stress distribution evaluated from the fem (fatigue analysis). at the first, it is necessary the knowledge of the fatigue behavior of the material through experimental tests. the material of which the half shaft is made is the aisi 4130 steel (25crmo4 according to iso 683-18, en 10083-1 standard), typically used for the production of shaft in heavy vehicles [28]. the mechanical properties, as well as the chemical composition of the employed material are shown in table 2 [20]. table 2 mechanical properties and chemical composition of aisi 4130 steel [20] parameter value tensile strength 670 mpa yield strength 435 mpa carbon (c) % 0.28 ÷ 0.33 chromium (cr) % 0.80 ÷ 1.10 molybdenum (mo) % 0.15 ÷ 0.25 manganese (mn) % 0.40 ÷ 0.60 silicon (si) % 0.15 ÷ 0.35 phosphorus (p) % 0.035 sulfur (s) % < 0.04 the s-n curve of the material has been obtained from a literature study [23] in which the authors have been conducted tensile fatigue test on regulated specimen with a load ratio r = -1. the experimental results have been then fitted with the basquin equation [29,30] according to eq. (1): 𝜎𝑎 = a𝑁𝑓 b (1) where 𝜎𝑎 is the stress amplitude, nf are the failure cycles, a and b are two regression constants. fig. 10a and fig. 10b show the specimen geometry and the s-n curve with basquin equation obtained from experimental fatigue tests respectively [23]. novel smart design of rear half shaft of large urban transport vehicle 593 a) b) fig. 10 fatigue tensile tests, [23], (a) geometry of the specimens (b) s-n curve the stress distribution evaluated from the numerical simulation derives from a random load in the time domain. therefore, it is necessary to extrapolate an equivalent stress distribution referred at r = -1, in order to evaluate the fatigue damage from the s-n curve of the material (fig. 10b). at the first, the driving cycle has been repeated until the design target is reached, in this work, 800000 km. then, the rainflow method [24] has been applied. the rainflow method is a fatigue analysis technique that decomposes complex variable amplitude load histories into simple, reversible stress cycles. it identifies and counts these cycles by analyzing peaks, troughs, and reversals in the load spectrum, which are critical for assessing the fatigue life of a component. this method allows for the precise calculation of fatigue damage by correlating these cycles with the material's s-n curve. hence, the obtained load histories are characterized by a certain stress range σa, stress mean σm, number of working cycle n and load ratio r which, in the vast majority of cases, turns out to be different to r = -1. in order to convert all the load histories into an equivalent one at r = -1, the goodman criterion [25] has been applied, as in eq. (2) 𝜎𝑎 𝜎𝑎𝑟 + 𝜎𝑚 𝜎𝑢 = 1 (2) where σa is the stress range computed from the rainflow method, σar is the equivalent stress range referred to r = -1, σm is the stress mean and σu is the ultimate strength. following the application of the goodman criterion, the load histories are characterized by specific numbers of working cycles at r = -1. these adjusted load histories are then intersected with the fitting line of the experimental data, evaluated according to eq. (1), to derive the number of failure cycles, which are subsequently used in calculating fatigue damage. the fatigue damage has been measured applying the palmgren – miner rule [26,27], defined by eq. (3): ∑ 𝑛𝑖 𝑁𝑖 𝑖 (3) where ni is the number of working cycles and ni the number of failure cycles. 594 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak 3. results and discussions the transmission models developed in amesimtm environment have been validated comparing the speed profile of both transmissions with the imposed one (mission test). the control system acts as follow: ▪ power shift transmission. the transmission in controlled by a rotary friction placed after the differential gear (final drive). the rotary friction controls the amount of torque necessary to follow the speed profile acting as a brake (friction torque) when it is open or partially open. ▪ power split transmission. the transmission is controlled by the swash plate angle of the hydraulic pump. more is the swash plate angle more is the oil flow rate inside the hydraulic circuit used to accelerate the vehicle. the quality of the control system is reflected in a near-perfect adhesion between the mission test and the speed profiles of both transmissions as shown in fig. 11. fig. 11 comparison between the mission test and the speed profiles of power shift and power split transmission the results of the transient simulation are shown in fig. 12a and fig. 12b in which the equivalent stress (von-mises) has been compared for both power shift and power split transmission. from the investigation of fig. 12 is possible to assert that, despite the torque levels are quite similar for both transmission (fig. 8), the power split transmission presents a higher stress distribution, especially in the first part of the driving cycle, as also shown in fig. 13. the maximum stress values are recorded in the restricted sections and junctions, as expected from the fatigue theory about the stress concentration, with a peak value of 2000 mpa for the power split transmission and 1300 mpa for the power shift transmission. the rainflow method has been applied to both stress distributions. the rainflow histograms novel smart design of rear half shaft of large urban transport vehicle 595 regarding the power shift and power split transmissions are shown in fig. 14a and fig. 14b, respectively. a) b) fig. 12 results of the transient simulation in terms of equivalent stress (von mises) power shift (b) power split fig. 13 von mises equivalent stress comparison between power shift and power split at the end, all the stress cycles have been referred to a load ratio r = -1 through the goodman criterion expressed in eq. (2). through the smart design algorithm, it was possible to calculate the fatigue damage on the component by applying the miner rule expressed in (3). stress cycles below the fatigue limit of the material does not contribute to the fatigue damage calculation while, for every stress cycle above the fatigue limit, the 596 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak working cycles ni and the failure cycles ni have been calculated respectively. the stress cycles for the power shift and power split transmission are shown in fig. 15. a) b) c) d) fig. 14 rainflow method results: a) cycle amplitude power shift, b) cycle mean power shift, c) cycle amplitude power split, b) cycle mean power split novel smart design of rear half shaft of large urban transport vehicle 597 fig. 15 equivalent stress cycles referred to a load ratio r = -1. comparison between power split (green) and power shift (red) transmission the measured fatigue damage for power shift transmission is: ∑ 𝑛𝑖 𝑃𝑆ℎ 𝑁𝑖 𝑃𝑆ℎ = 36000 3.5∙106 ≈ 0.01 ≈ 1%𝑖 (4) as can be observed from fig. 15, one load history related to the power split transmission (labeled as psp and green square in the figure) exceeds the s-n curve. this indicates that this load history causes the failure of the component, particularly occurring at ~300 mpa and 72000 cycles. 4. conclusions in this study, a multidisciplinary approach has been adopted to assess the fatigue damage of the rear half shaft in an urban bus, focusing on two transmission types: power shift and power split. the findings of the work reveal that, although the power split transmission offers substantial environmental benefits by reducing fuel consumption and emissions, it introduces significant mechanical challenges due to the integration of the hydraulic branch. specifically, the power split transmission was found to exacerbate fatigue damage compared to the power shift transmission, thereby reducing the component's lifespan. this critical insight necessitates the redesign or adaptation of components to withstand the load profiles imposed to the power split transmission, while maintaining fatigue resistance. consequently, the research provides a vital foundation for further studies and design optimizations, which are essential to advance sustainable propulsion systems without compromising mechanical reliability. future work should focus on developing and testing components that can endure the demands of power split transmissions. additionally, collaborations with vehicle manufacturers to apply these findings in real-world settings would be instrumental in refining the design of sustainable, robust mechanical systems for 598 f. alberti, d. d’andrea, a. macor, g. risitano, a. rossetti, a. sedmak urban buses. in conclusion, while the environmental advantages of the power split transmission are clear, ensuring the mechanical integrity and longevity of components in such systems is paramount. the study underscores the need for a 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license: cc by-nc-nd original scientific paper contact interaction of an underground pipeline with soil under dynamic impacts karim sultanov institute of mechanics and seismic stability of structures named after m.t. urazbaev uzbekistan academy of sciences, tashkent, uzbekistan orcid id: karim sultanov https://orcid.org/0000-0002-5526-9862 abstract. the interaction between a semi-infinite underground pipeline and the surrounding soil during longitudinal seismic wave propagation is examined in the article. the wave travels through the soil with its front perpendicular to the pipeline's axis, and the nonlinear laws of interaction (friction) on the surface of their contact, including the amonton-coulomb law, are used to solve related wave problems for the pipeline and soil. the method of characteristics and the finite difference method are sequentially used to solve these problems numerically. the implementation of the law of interaction (friction) processes is shown, depending on the parameters of the wave in soil and the mechanical characteristics of soil. it was determined that a wave in soil that involves the pipeline generates a powerful wave in the pipeline. the amplitude of this wave is many times greater than the amplitude of the wave in soil, and it propagates through the pipeline without attenuation. the dependences of the amplitude of this wave on the parameters of the law of friction, waves in soil, and the mechanical characteristics of soil are determined. key words: pipeline, soil, underground, law of interaction, friction, wave, numerical solution 1. introduction as noted by popov and heß [1], contact and friction are significant phenomena in various fields of science and technology, including underground utilities. underground pipelines serve as essential engineering life support lines for the population and industry, making them crucial for the economy of any country. hence, ensuring the strength and durability of underground pipelines is a critical and relevant problem. when an underground pipeline is subjected to external dynamic impacts, primarily seismic, it interacts with the surrounding soil, resulting in a frictional force on the surface of their contact. the complexity of these friction forces is evident from experimental results obtained by sultanov [2]. received: december 27, 2023 / accepted march 28, 2024 corresponding author: karim sultanov institute of mechanics and seismic stability of structures named after m.t. urazbaev, academy of sciences of the republic of uzbekistan, durmon yuli street 33, 100125 tashkent, uzbekistan e-mail: sultanov.karim@mail.ru https://orcid.org/0000-0002-5526-9862 486 k. sultanov the specific characteristics of the contact and friction between underground pipelines and soil are notable due to the significant difference in strength between a steel pipeline and loamy soil. this specificity manifests in the fact that the strength of contact between soil particles and the outer surface of the pipeline is greater than the contact between the particles of soil [2]. this phenomenon occurs because, over time, soil particles stick to the pipeline due to cohesion processes and the corrosion properties of the steel pipeline, as noted by sultanov [2]. when an underground pipeline moves or shifts relatively to the soil, it involves the soil particles in a specific thickness called the contact layer [2, 3]. this interaction between the pipeline and soil results in a complex friction process. the law of interaction between the underground pipeline and soil is quite complex and has been developed based on the results of fundamental experimental research obtained by sultanov [2] and sultanov and vatin [3]. the interaction of an underground pipeline with the surrounding soil, under the impact of dynamic (seismic) loads, considering the free (daytime) earth surface is a complex threedimensional spatial wave problem. this problem can be solved using numerical modeling methods. however, great difficulties arise when implementing numerical modeling in a three-dimensional solution domain. this problem, with reasonable simplifications, is reduced to two one-dimensional wave problems [3]. thus, when seismic waves propagate in the “pipeline-soil” system, we accept a nonlinear law on the surface of their contact, which includes the amonton-coulomb law [2]. it is known that when solid bodies interact by the amonton-coulomb law, the friction force instantly reaches its limiting value, that is, the rate of change of the friction force is equal to infinity. at the beginning of the interaction curve, there is a small-width section where the friction force increases to a limiting value, depending on the relative displacement. when the pipeline interacts with soil, which is an essentially “soft” body, this pre-limit section significantly expands and becomes complex and curvilinear, as noted by sultanov [2] and sultanov and vatin [3]. in the limiting section, the amonton–coulomb law is satisfied. in the pipeline-soil system, during the propagation of high-frequency seismic waves in soil with a frequency f=50 s-1 and an amplitude of 0.5 mpa, by the interaction law, 50 and more times increase in the amplitude of longitudinal stresses was observed in the pipeline compared to the stress amplitude in soil [3]. here, the problem is considered for the case of propagation of a lowfrequency seismic wave in soil and changes in longitudinal stresses in the pipeline. underground pipelines, as life support lines for the population and industry, are currently of significant importance in the global economy. as the means of conveying liquid and gaseous products, water, etc., underground pipelines are the most profitable and economical structures. however, the destruction of operating underground pipelines under external influences can lead to huge environmental and economic disasters. one of these external influences is seismic loads. an analysis conducted by kwong et al. [4] clearly show the aftermath of strong earthquakes. one of the main issues considered by musa [5] is the condition of the contact of various structures with media. in the problem of interaction of an underground pipeline with the surrounding soil, the main issue is the proper selection of interaction law, adequate to the interaction process. the stress-strain state of the pipeline was determined, and, based on this, its seismic resistance was assessed by fan et al. [6] and popov [7]. zhang et al. [8], xue et al. [9], yiğit et al. [10], and demirci et al. [11] considered the mechanical behavior of underground pipelines crossing a fault. numerous authors [12-17] discussed issues of interaction between an underground pipeline and soil. dai et al. [18] and huang et al. [19] showed how important it is to correctly model the underground pipeline contact interaction of an underground pipeline with soil under dynamic impacts 487 interaction with soil under the impact of a seismic wave. it was proven that goodman's model, consisting of spring deformation and frictional contact, agrees with reality. sarvanis et al. [20] developed a model of the longitudinal interaction of an underground pipeline with soil, considering the dilatancy of the contact layer of soil. soil dilatancy determines the increase in soil pressure on the pipeline. sarvanis et al. [20], based on in-depth experimental and theoretical studies, determined that soil strain could cause significant stresses in the pipeline body, leading to rupture. psyrras et al. [21] and bakhodirov et al. [22] made an in-depth analysis of extensive literature sources to show that correct modeling of the interaction process of an underground pipeline with soil is important in the reliable determination of stresses in the pipe. issues of modeling the viscoelastic properties of materials, including soils, numerical modeling of dynamic processes, and the behavior of underground pipelines under seismic impacts were examined in references [2327]. the study aims to determine the longitudinal stresses in underground main pipelines under the impact of low-frequency seismic waves, based on the nonlinear laws of interaction (friction) of an underground pipeline with soil. 2. law of longitudinal interaction of an underground pipeline with soil summarizing the results obtained by sultanov [2] and sultanov and vatin [3], a schematic representation of the pipeline-soil interaction is shown in fig.1, where  is the shear stress arising in the process of the pipeline interaction with soil, u=ug-uc is the relative displacement, ug is the absolute longitudinal displacement of soil in the pipeline’s axis direction, and uc is the absolute longitudinal displacement of the pipeline. the research conducted by sultanov [2], sultanov and vatin [3], and bakhodirov et al. [22] show that shear stress  appears in a certain contact layer of soil surrounding the pipeline. relative displacement does not occur since due to cohesion processes, directly on the pipe-soil contact surface, soil particles strongly cohere to the outer surface of the pipe. relative displacement occurs in the thickness of the contact layer  [2]. curve 1 (0ace) in fig. 1 corresponds to the process of the complete first cycle of interaction when the soil contact layer is not damaged. this is the first cycle of the interaction of an embedded (for a long time) pipeline. if this cycle passes completely along the 0ac curve, then at point c, the contact layer is damaged and the pipeline interacts with the destroyed soil layer along the ce line. subsequent cycles of interaction between the underground pipeline and soil follow the dashed curve 2 (0c). fig. 1 diagrams of the interaction of an underground pipeline with soil. 1 for undamaged contact layer of soil, 2 for damaged contact layer of soil 488 k. sultanov in the first case, the peak value of the shear stress = appears at the value of the relative displacement u=u. it can be assumed that in the first cycle, up to 70% of the relative displacement u, the interaction occurs elastically, i.e. elastic bonds between particles in the contact layer of soil are kept. the law of interaction is called elastic, based on this process of deformation of the contact layer at the beginning of this stage of interaction. in the case of damaged soil of the contact layer, there are no elastic forces between soil particles (dashed curve 2). in this case, in the interaction process, the soil particles are repacked and the soil structure changes. in section ac of curve 1 (fig. 1), intensive destruction of the structure of the soil contact layer occurs; this process ends at u=u*, where u* is the critical value of the relative displacement, at which the structural bonds of the soil contact layer are completely destroyed. in section ce, a further increase in the relative displacement does not affect the value of , and the amontonscoulomb law is fulfilled. it is known that under shear strain of soil, diagram () for all soil types with low moisture content ( is the shear strain of soil) also has the form of curve 1 or 2 (fig. 1). the repeated cycle of interaction of the pipeline with soil, when the contact layer is completely damaged, passes along curve 2 (fig. 1). in section 0uu*, dependence (u) is nonlinear in all cases due to changes in the soil structure under the pipeline-soil interaction. the nonlinearity of the pipeline-soil interaction curves 1, 2 in fig. 1 is related to the destruction of the structural bonds between the soil particles and their re-packing. on the outer contact surface of the steel pipe and soil (loess, loamy soil), soil particles do not separate from the pipeline [2]. since in section ce of the interaction process, the value of  depends on n, determined by the amontons-coulomb law, then in section 0ac (curve 1) or 0c (curve 2) the value of  should depend on the value of the normal pressure (stress) n acting on the outer surface of the pipeline. otherwise, there is a discontinuity in the value of  at point c. as noted above, under the longitudinal interaction of an underground pipeline with the surrounding soil, the soil contact layer of a certain thickness is subjected to shear strain. this soil contact layer on one side is under the shear load of the soil mass, and on the other side is under the pipeline's load. therefore, the mechanism of shear strain of this layer is rather complicated. considering the above, the law of longitudinal interaction of an underground pipeline with soil has the following form [3]: for * n n  , *0 u u  : ( , , ) ( , , ) ( , ) ( , ) s n s s n s xd n s xs n s d du i u i u u k i dt k i dt + = +         (1) for * n n  , u>u*: u nc f = + (2) for * n n  : 0 = , (3) where kxd is the variable dynamic coefficient of soil stiffness (as u → ), kxs is the variable static coefficient of soil stiffness (as 0u → ), s is the variable parameter of soil shear viscosity, contact interaction of an underground pipeline with soil under dynamic impacts 489 u du dt= is the velocity of displacement of the pipeline relative to soil, is=u/u* is the parameter characterizing the structural destruction of the soil contact layer, 0is1, for is=0, the soil contact layer or the contact bonds between the outer surface of the pipeline with soil are not damaged, and for is=1, these bonds are completely destroyed, fu is the internal friction coefficient of soil, * n is the soil tensile strength (here compressive stresses are assumed positive). note that a change in is from zero to 1 occurs in each interaction cycle and it is irreversible. in the case of a structurally intact or not completely destroyed soil contact layer, changes in is occur along curve 1, and, in the case of a destroyed soil contact layer along curve 2 (fig. 1). in both cases, when the underground pipeline interacts with soil, it is considered that there is a change in the structure (destruction, in the case of curve 1 or repacking, in the case of curve 2) of the soil contact layer. parameter s and the shear viscosity coefficient s are related by:  ( )s xd xs xd xs sk k k k= −  (4) the functions of stiffness coefficients kxd, kxs are determined from the results of experimental studies conducted by sultanov [2] and have the following form:  *( , ) ( )exp (1 )xd n s xd n sk i k i= −   (5)  *( , ) ( )exp (1 )xs n s xs n sk i k i= −   (6) where k* xd and k* xs are the secant coefficients of dynamic and static stiffness of soil for u=u*, the damaged contact layer of soil. for an undamaged soil contact layer (is=0), from eqs. (5) and (6) we obtain: * *exp( ), exp( )xdn xd xsn xsk k k k= =  (7) where kxdn and kxsn are the initial values of the stiffness coefficients,  and  are the dimensionless coefficients characterizing the degree of change in kxdn and kxsn. from eq. (7), it follows that: *ln( / )n= +    (8) where *= k* xd / k* xs, n= kxdn / kxsn. it has been already determined that *  n [2], therefore, according to eq. (8),   . following experimental results obtained by sultanov [2], the values of , for curves 1 and 2 (fig. 1) differ. based on the experimental results, the following was obtained [2]: * *( ) ; ( )xs n ns n xd n nd nk k k k= =    (9) where kns, knd are static and dynamic interaction coefficients. sultanov [2] and sultanov and vatin [3] determined that * *( )( )m n n su c= + −     (10) where cs is the velocity of shear wave propagation in soil. 490 k. sultanov the value of s under the change in the soil structure is determined from the following equation [3]:  *( , , ) exp (1 )s n s s si u i= −    (11) where  is the coefficient characterizing the range of change in the parameter of shear viscosity of soil, * s is the viscosity parameter of the structurally damaged soil. to determine the approximate values of * s and * s, the following formulas are obtained, with some assumptions from eqs. (1) and (4): * * * * ; ( 1) n s s ns s s f c k f c = = −      (12) the value of the coefficient of static interaction kns is determined from the results of experiments by the method given by sultanov [2] or proposed by sultanov and vatin [3], this coefficient and the thickness of the contact layer can be determined through the strength characteristics of soil according to the following relationships: * * * * ns n f c k u u = +    (13) * * n gu f c  = +   (14) where g is the soil shear modulus. as can be seen, the interaction model eq. (1) describing curves 1 and 2 (fig. 1) is rather complicated. however, an account for the defining factors, such as the velocity of interaction, the destruction of the soil contact layer, the cohesion force, and the force of internal friction of soil, leads to such complex equations. in the simplest cases, without considering these factors, according to bakhodirov et al. [22], eq. (1) can be replaced by eq. (15): xk u= (15) where kx for curves 1 and 2 is determined experimentally or from eqs. (5-6), respectively. in this case, the law of interaction is determined by eqs. (15), (2), and (3) considering eqs. (7-13). as noted by bakhodirov et al. [22], coefficient kx is a constant equal to the value of k* xs and the interaction curve 2 in fig. 1, according to eq. (15), will turn into a straight line, which is not consistent with the experimental results obtained by sultanov [2]. experimental studies have established the dependence of the interaction force  on the depth of the underground pipeline [2]. the static stress (pressure) on the pipe associated with the soil’s weight above it, n=ns is normal to the outer surface of the pipeline. it is known that during the seismic wave propagation, even in cases when the wave front is perpendicular to the axis of the pipeline, there is the dynamic normal stress n=nd acting on the pipe, which is comparable to ns, and, sometimes, it can be greater than ns. when an arbitrary seismic wave interacts with an underground pipeline, the following formula holds: n ns nd= +   (16) where ns is determined by the depth of the pipeline in soil, and nd is determined by the pressure of a seismic wave. contact interaction of an underground pipeline with soil under dynamic impacts 491 when a longitudinal seismic wave propagates parallel to the pipeline’s axis, the normal stress nd is approximately determined by the following formula: nd gk=   (17) where k is the lateral pressure coefficient of soil, and g is the longitudinal seismic stress in soil under longitudinal wave propagation. in the case of strong earthquakes in loess soils, the approximate maximum value of the longitudinal stress is max p  0.7 mpa [3]. then for k = 0.3, we obtain max nd  0.21mpa according to eq. (17), which conventionally corresponds to a substantially deep embedment of the pipeline (h  10.5 m). therefore, the value of max nd can even exceed ns. this shows that calculations conducted according to eq. (15), for kx = const, for the seismic stability of underground pipelines, without considering nd, are approximate. eqs. (1-3) with the corresponding defining relations, eqs. (4-13), and eqs. (16, 17) are non-linear interaction laws between an underground pipeline and the surrounding soil. 3. statement of the problem, basic equations, and methods for their solution the embedment of the underground pipeline in soil is shown schematically in fig. 2a. at a depth of h from the free earth surface there lays a pipeline with an outer diameter dh and internal diameter db, x is the pipeline axis of circular section (fig. 2a). due to the cylindrical pipeline, the problem is three-dimensional, which significantly complicates the numerical modeling of the problem under consideration. at the state of rest, the pipeline is subject to static pressure from the overlying soil ns. strictly speaking, the values of ns along the perimeter of the outer surface of the pipeline are different, especially for pipes with large external diameters dh. in the case of seismic wave propagation in soil, for example along the x-axis, additional pressure nd is initiated in the pipeline determined by eq. (17). as a result, the total pressure on the pipeline is n=ns+nd, as was said above. under the influence of a seismic wave, the soil and the pipeline are strained along the xaxis in different ways and relative displacement is formed in the x-axis direction. this leads to the formation of friction force  between the pipeline and soil. for f, the longitudinal displacement of soil is ug and normal dynamic pressure on the pipeline-soil contact surface is nd. the values of ug and nd must be determined from the solution to the threedimensional wave problem. this is a labor-intensive and complex problem. however, this problem can be simplified and reduced to two one-dimensional wave problems: an external one for soil and an internal one for the pipeline. these simplifications are as follows: ▪ the soil, in which the pipeline is embedded is considered an external cylindrical body with external diameter dg=2h+dh, internal diameter dh, and length l, the axis of which coincides with the axis of the pipeline. ▪ section x=0 is taken as the initial section for the pipeline and soil. ▪ longitudinal seismic wave acts in the initial section x=0 along the x-axis only in soil. ▪ the initial section x=0 of the pipeline is load-free and the pipeline is involved in motion under the influence of friction on its outer surface. ▪ the underground pipeline and the surrounding soil are considered a coaxial pipe-inpipe system. 492 k. sultanov fig. 2 а) pipeline location in soil and conditional wave fronts, b) wave front location on the characteristic plane the wave pattern for the pipeline and soil on the characteristic plane x, t is shown in fig. 2b. a plane wave begins to propagate in soil from the initial section x=0. the front of this wave fg remains perpendicular to the x-axis of the pipeline. at section x=0 of the pipeline, a frictional force occurs from the soil motion and a plane wave with front fc begins propagating through the pipeline (figs. 2a, b). since the propagation velocities of longitudinal waves in soil and steel pipelines are different, the tilts of these fronts to the x-axis are different. these waves propagate along two characteristic planes x, t. in fig. 2b, they are combined and shown on the same plane. numerical calculations are conducted for any time layer t2, and in the t1 layer, the wave parameters are considered known. in this case, the time step dt for both problems is equal, and the space step dx differs. wave fronts reflected from the pipeline into soil fgc (figs. 2a, b) and from soil into the pipeline are ignored. the equations of motion of the pipeline and soil along the x-axis coinciding with the axis of the pipeline, considering the friction force on the surface of their contact, have the following form: contact interaction of an underground pipeline with soil under dynamic impacts 493 0 / / 0 / / 0 i i i i i i i v t x v x t   −   + =   −   =      (18) where vi is the particle velocity (mass velocity), i, i are longitudinal stresses and strains, 0i is the initial density, i=sign(v) for the pipeline, i =−sign(v) for soil, v=v2=vg is the soil particles velocity,  is the reduced friction force acting per unit length of the pipeline and t denotes the time. the values of  for the pipeline and soil are determined from the following relation: 2 24 / ( )i hi hi bid d d= −  (19) where  is the friction force (shear stress) determined from eqs. (1-3), dhi are the outer diameters of the pipeline and cylinder soil, dbi are the inner diameters of the pipeline and soil. the equations of state of the pipeline and soil are assumed linear-viscoelastic (eyring’s model or a model of a standard linear body): / ( ) i i i i i di si i di si di si i d d d dt e dt ie e e e e + = + = −         (20) in eqs. (18-20), i =1, 2. for i = 1, the parameter values refer to the pipeline, and for i = 2, to soil. it is known that eq. (20) for esi = edi describes an elastic medium. the parameters introduced in the calculations are c= ed1/es1 = edc /esc, g = ed2/es2 = edg /esg , where edc and esc are the dynamic and static strain moduli of pipeline material, respectively, and edg and esg are the dynamic and static strain moduli of soil. taking in computer calculations the values of c and g close to 1 or greater than 1, we can examine elastic or viscoelastic pipelines and soils. the solution to the problem is reduced to integrating the nonlinear system of eqs. (18) closed by eq. (20) separately for the pipeline (i=1, an inner problem) and soil (i=2, an outer problem). these systems are connected by non-linear conditions on the contact surface of the pipeline and soil, which are the laws of change in the interaction force (friction)  determined by eqs. (1-3). boundary conditions are: for x=0, the seismic wave that changes according to the following ratios is set as: max sin( ), 0 0, t t t t =   =        (21) where t is the half-period of the wave,  is the wave duration, max is the wave amplitude,  is the longitudinal stress of the propagating along the x-axis. the conditions at the wave fronts in soil and the pipeline are zero, and the initial conditions of the problems are zero. eqs. (18-20) are hyperbolic, they have characteristic equations and characteristic relations on these equations. by applying the method of characteristics, partial differential eqs. (1820) are reduced to ordinary differential equations, solved using the finite difference method in an implicit scheme. using the described scheme, an algorithm was compiled, and a program for solving the problem in the fortran-2005 language implemented on a 494 k. sultanov computer was developed. special approaches to the numerical solution of nonlinear systems of equations were developed. numerical results were obtained as changes in the parameters of waves in soil and the pipeline over time in fixed sections and along the coordinate at fixed time points. the stability of the solution algorithm and the reliability of numerical solutions obtained were substantiated by sultanov and vatin [3]. obtaining numerical solutions for low-frequency waves requires large computer resources. this is due to the discretization steps of the ever-expanding domain of the solution of the wave problem on the characteristic plane x, t. at large discretization steps, the numerical solution instability arises and, at small steps, large resources are required. with the use of powerful modern computers, this problem was overcome and stable numerical solutions to the problem were obtained. 4. results obtained and discussion a steel pipeline surrounded by soil is considered. its parameters c=edc/esc and g=edg/esg characterize the stiffness of the pipe material and soil. in numerical calculations, an underground steel pipeline is assumed elastic (c=1.02), and the soil is assumed viscoelastic (g=2). in this case, a pipeline surrounded by soil has a length of l=1000000 m so that in the calculation the wave in the pipeline does not reach the final cross-section x=l. then, for greater convenience, the parameters with index c (i=1) refer to the pipeline, and with index g (i=2), they refer to soil. the following data is used as an initial set for the calculations. cases obtained under various deviations from this set are discussed separately: ▪ for underground pipeline: dh1=0.15 m, db1=0.14 m, gc=78 kn/m3, c=104 s-1, c0c=5000 m/s, c=1.02, edc=c2 0c0c, esc=eds/c, ▪ for soil: gg=18 kn/m3, c0g=1000 m/s, k=0.3, g=1000 s-1, g =2.0, edg= c2 0c0g, esg=edg/g, dh2 =2.0 m, db2 =0.15 m, ▪ seismic wave parameters are: t=1 s, =50 s, max=0.35 мpа, ▪ interaction parameters are: h=2m, kn=100 m-1, =2.5, fu=0.5, u*=0.003 m. the program for the numerical solution to the problem allows us to determine the wave parameters or the variables of the problem separately for soil and the pipeline, and jointly for both cases. consider the calculation results. calculations were conducted for the first arrival (first half-cycle) of the wave only. fig. 3 shows changes in the longitudinal stresses in soil. contact interaction of an underground pipeline with soil under dynamic impacts 495 fig. 3 longitudinal stresses in soil at distances x = 0, 5, 10, 15, 20, 25, 30 m curves 1-7 curves 1-7 in fig. 3 refer to distances x=0, 5, 10, 15, 20, 25, 30 m from the initial section x=0 of the pipeline and soil. the stress amplitudes in the considered distances are the same gmax=0.35 mpa. the calculation results showed that the wave in soil propagates with this given amplitude and practically does not attenuate. soil in this case is viscoelastic (g=2.0). as seen from fig. 3, the friction force  in eq. (18) does not lead to wave attenuation in soil due to the significant thickness of the soil layer surrounding the underground pipeline (h=2.0 m). in all the options of calculations considered below for g=2.0, the pattern of changes in longitudinal stresses in soil has the form shown in fig. 3. changes in longitudinal stresses in the pipeline in its sections x=0, 5, 10, 15, 20, 25, 30 m (curves 1-7) are shown in fig. 4. the wave (eq. (21)) does not act in the initial section of the pipeline, this section of the underground pipeline is free from the wave load, and accordingly, the stress is zero (straight line 1, fig. 4). in subsequent sections of the pipeline, a significant increase in the stress amplitude is observed. in the cross-section at x=5 m, the stress amplitude reaches cmax=75.7 mpa (curve 2), at x=10 m 125 mpa (curve 3), at x=15 m 138 mpa (curve 4), at x=20 m 140 mpa (curve 5), at x=25 m 140 mpa (curve 6), and at x=30 m -140 mpa. fig. 4 longitudinal stresses at max=0.35 mpa in sections of the underground pipeline x=0, 5, 10, 15, 20, 25, 30 m curves 1-7 496 k. sultanov the stress amplitude in the pipeline increases with distance from the initial section (curves 1-7, fig. 4). at distance x=20 m, it reaches 140 mpa and remains unchanged. the stress wave propagates with this amplitude through the pipeline without damping. the maximum stress values in the pipeline (140 mpa) are 400 times greater than the stress amplitude in soil 0.35 mpa. this is due to the active force of the underground pipeline interaction (friction) with soil. this force acting on the outer surface of the underground pipeline, involving the pipeline into motion, increases stresses. let us consider the change in the interaction force  corresponding to this case caused by the relative displacement u (fig. 5), where curves 1 – 3 refer to distances x=0, 5, and 10 m. in this option of calculations, when the relative displacement reaches the value of u*=0.003 m, the interaction process should occur by the amontons-coulomb law. fig. 5 changes in the force of interaction (friction) between the underground pipeline and soil at x=0, 5, 10 m curves 1-3 as seen in fig. 5, the interaction process does not pass to the stage of the amontonscoulomb dry friction (curves 1-3). the process of interaction occurs completely according to the law given by eq. (1). the maximum values of the friction forces significantly decrease with distance and then become negligible (fig. 6). in fig. 6, curves 1-4 refer to pipeline sections at x=15, 20, 25, 30 m, respectively. as can be seen, the maximum values of the friction force decrease by an order of magnitude compared to the value in the initial sections of the pipeline. this leads to the propagation of a stress wave through the pipeline without attenuation. straight lines 10–70 in fig. 5 refer to the change in friction force when eq. (15) is used instead of eq. (1). in this case, as seen in fig. 5, the process of interaction occurs in all sections of the pipeline along the same straight line. the values of the friction force are approximately two times less than in the case of eq. (1). the interaction process passes to the stage of dry friction (u>u*=0.003 m) and is not considered. the significant difference between interaction curves 1-3 in fig. 5 and curves 1-4 in fig. 6 from curves 10-70 in fig. 5 is the result of considering in eq. (1) dynamic pressure normal to the outer surface of the pipeline, structural changes in the contact layer of soil, and interaction rates. these parameters are not considered contact interaction of an underground pipeline with soil under dynamic impacts 497 in the case of the law given by eq. (15) (curves 10–70). an account for these parameters leads to a completely different pattern of the interaction process between the pipeline and soil. in this case, the law of interaction becomes essentially non-linear. fig. 6 changes in the force of interaction (friction) between the underground pipeline and soil at x=15, 20, 25, 30 m curves 1-4 an increase in the amplitude of the propagating wave in section x=0 to max = 0.5 mpa, leads to a greater increase in the stress amplitude in the sections of the pipeline (fig. 7). fig. 7 longitudinal stresses at max=0.5 mpa in sections of the underground pipeline x=0, 5, 10, 15, 20, 25, 30 m curves 1-7 in this option, the steady-state stress amplitude reaches cmax = 200 mpa (curves 1-7, fig. 7). curves 1-7 in fig. 7 refer to the same sections of the pipeline as curves 1-7 in fig. 4. however, the stress amplitude in the pipeline in this option is 400 times greater than the 498 k. sultanov amplitude of the wave in soil (max = 0.5 mpa). the changes in the interaction force corresponding to this case are shown in fig. 8. fig. 8 changes in the interaction force in sections of the underground pipeline at x=0, 5, 10 m curves 1-3 an increase in the wave amplitude in soil also leads to an increase in the value of the friction forces in the outer surface of the underground pipeline. compared to the previous case, the value of  increases by 1.3 times at the initial section of the pipeline. however, the process of interaction in this case does not pass to the stage of dry friction. the maximum value of , at the distance from the initial section of the pipeline x=10 m (curve 3, fig. 8), is approximately two times less compared to the initial section x=0 (curve 1, fig. 7). as the calculation results show, at a distance of x=30 m, the values of =0.08 mpa are 11 times less than in the initial section. a decrease in the value of  with distance allows for the longitudinal waves initiated in the initial section (up to 15-20 m) of the pipeline, to propagate in the pipeline without attenuation. an increase in the maximum value of the wave stress in the initial section up to max=0.7 mpa, which corresponds to destructive earthquakes, results in the stress increase in the pipeline to cmax = 280 mpa, the value of  increases to 0.13 mpa. in this case, the pipeline experiences plastic strain under such stresses, which contradicts the initial assumption that the pipeline is elastic. in cases where the longitudinal stress in the pipeline exceeds the elastic limit of the material, we will assume that the pipeline is damaged. more reliable results can be obtained when considering the elastic-plastic properties of the pipeline material, which is a task for future studies. in this case, the interaction process passes to the stage of the amontons-coulomb dry friction only in the initial section of the pipeline up to x = 5 m. beyond this distance, the interaction process occurs according to eq. (1). contact interaction of an underground pipeline with soil under dynamic impacts 499 in the case of initial data max = 0.35 mpa, like the case shown in fig. 4, the values are k = 0.1, kn = 500 m-1 and fu = 0.3, other parameters remain unchanged. the patterns of interaction are shown in fig. 9. fig. 9 changes in the interaction force in sections of the underground pipeline at x=0, 5, 10, 15, 20, 25, 30 m curves 1-7 in this case, when the values are u*=0.0006 m, and in the initial section of the pipeline up to x=15 m, the interaction process occurs mainly in the second stage – the amontonscoulomb stage (curves 1-4). at distance x=20 m, the process of interaction partially occurs by the law of dry friction (eq. (2)). in the section of the pipeline x = 20 m (curve 5), the relative displacement reaches the value of u=0.6510-3 m, which slightly exceeds the values of u*. farther along the contact surface of the pipeline, the law of interaction eq. (1) is fulfilled. despite the same wave amplitude values in soil max=0.35 mpa, as in fig. 4, the maximum value of the friction force  is 0.023 mpa at x=0, which is approximately three times less than in fig. 5. here, the value of the stress amplitude is cmax=130 mpa and it is only 7% less than in fig. 4. these results show that the longitudinal stresses in the pipeline under seismic impacts mainly depend on the longitudinal seismic stresses in soil, and, therefore, on the seismic pressure on the pipeline. thus, in the contact interaction of an underground pipeline with soil under seismic effects, the laws of interaction (eqs. (1-3)) adequately describe the interaction process. the numerical results obtained above show that when determining the stresses in the pipeline, the correct choice of the law of interaction between the pipeline and soil is of paramount importance. sultanov and vatin [3] showed that under strong ground motion, longitudinal stresses in underground pipelines may reach up to cmax = 85 mpa and more. the results obtained confirm this statement. 500 k. sultanov 5. conclusions the proposed two-stage nonlinear law of interaction of an underground pipeline with the surrounding soil describes the interaction process and is applicable in calculations to determine longitudinal seismic stresses in the pipeline. the law of interaction considered in the study, in contrast to the laws used nowadays when solving problems of the underground pipeline strength under seismic impacts, accounts for the transition of the process of the pipeline-soil interaction to the stage of amontonscoulomb friction and the dependence of the friction forces caused by the wave pressure in soil on the pipeline during both stages. this law also considers the destruction of the soil contact layer around the pipeline when it interacts with soil and the rate of interaction. the numerical solution to the wave problems considered for the pipeline-soil system showed that seismic stresses in the pipeline increase manifold compared to the longitudinal stresses in soil. this is explained by the lower stiffness of the soil surrounding the pipeline compared to the steel pipeline rigidity and the nonlinear properties of the interaction law. in this case, the dynamic soil pressure on the pipeline is determinant. as a result of the greater deformability of soil, the interaction force (friction) is active and leads to a significant increase in stresses in the pipeline. interaction force (friction), actively acting in the initial section of the pipeline, creates a powerful wave with an amplitude many times greater than the wave in soil. farther the value of the friction force between the pipeline and soil decreases, and this wave propagates through the pipeline with insignificant attenuation (for a low-frequency excitation wave and viscoelastic soil). as the calculation results showed, longitudinal stress waves and their amplitudes in underground pipelines strongly depend on the law describing the process of interaction of the underground pipeline with soil. this article discusses a more realistic soil-pipeline interaction law. acknowledgement: the research was conducted at the expense of basic budget funding from the academy of sciences of the republic of uzbekistan. references 1. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, edition: 1st publisher: springer, 253 p. 2. sultanov, k.s., 1993, laws governing the interaction of underground structures with soil during their relative displacement, international applied mechanics, 29(3), pp. 217–223. 3. 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mechanical engineering vol. 15, no 1, 2017, pp. 163 182 doi: 10.22190/fume161010009k © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd review article from art to engineering: a technical review on the problem of vibrating canvas part i: excitation and efforts of vibration reduction udc 534.1 kerstin kracht, thomas kletschkowski haw hamburg, department of automotive and aeronautical engineering abstract. cultural assets are witnesses of past times with versatile worth. the irreplaceability of those treasures of art makes their protection our major task. this article reflects the commitment and results of 40 years of conservators’ research to protect canvas objects of cultural heritage particularly from mechanical loads. it gives a classification of mechanical loads that act upon canvas during transport, exhibition and storing in depot. furthermore, it gives an overview of different approaches which were used over years to protect canvas from various mechanical loads. this article tends to bridge the gap between restorers’ knowledge and methods and concepts known from engineering dynamics. restorers’ first steps using engineers’ methods are brought up and the necessity of theoretical modeling which has not started so far are pointed out. key words: canvas, paintings, transport, shock, vibrations, elastic isolation 1. introduction studying shock and vibration hazards in view of art and cultural objects has been an important topic during the last 40 years in conservation science. in spite of the long research period restorers and conservators are facing numerous unsolved problems. current research in the field of 'preventive conservation' is summarized by the term 'green museum', which '[...] brings together conservation science with the sustainable development for the preservation of art and cultural heritage [...]' [1]. environmental influences like (a) temperature, (b) humidity, (c) uv radiation, (d) dust and (e) mechanical loads are discussed in different articles, papers and forums. references for the state-of-art are [2, 3] and the workshop [4]. in these, effects of the first four influences (a) – (d) are examined extensively. more recently, the effect of mechanical loads has been intensively discussed [5, 6]. received october 10, 2016 / accepted march 21, 2017 corresponding author: thomas kletschkowski haw hamburg, department of automotive and aeronautical engineering, berliner tor 9, 20099 hamburg, germany e-mail: thomas.kletschkowski@haw-hamburg.de 164 k. kracht, t. kletschkowski cultural assets are exposed to mechanical loads during (i) transport, (ii) exhibition and (iii) storing in depots. however, it is just not clear yet which load level is critical and how art objects could be protected to avoid damages [7, 8]. the main contribution of this paper is a literature review. restorers‟ questions are summarized and transferred to engineering problems. because of the huge range, this paper focuses on a particular class of cultural assets: paintings on textile carriers. the article begins with the motivation of this topic (section 2), followed by the description of common excitation mechanisms (section 3). relevant frequency ranges as well as amplitude levels are discussed. afterwards in section 4, conservators‟ efforts concerning shock and vibration reduction during transport, exhibition and storing in depots are discussed. bridging the gap between restorers‟ knowledge and the art of engineering is subject of section 5. the paper is closed by a summary, conclusion and outlook in section 6. 2. motivation nowadays the presentation of one‟s own collection is insufficient for museums to keep pace with national and international competition [9]. the evolution of the number of attendees in german museums in the period from 1996 until 2014 is shown in fig. 1. as can be observed the total number is strongly influenced by the number of special exhibitions visitors. fig. 1 total attendees and special exhibition visitors‟ numbers [10] the evolution of the special exhibition numbers in german museums in the period from 1996 until 2014 is shown in fig. 2. with reference to 'staatliche museen zu berlin' (smb) [10], the german museums stress an increased number of special exhibitions as one reason for the increased attendance. among these, 'blockbusters' like 'the moma in berlin' (berlin, 2004, 1.2 million visitors), 'the most beautiful french come from new york' (berlin, 2007, 667.000 visitors) and 'friedrich the great' (potsdam, 2012, 350.000 visitors) play a special role [10]. from art to engineering: a technical review on the problem of vibrating canvas  part i 165 the number of special exhibitions seems to have reached a saturation point of approximately 9000 per year in germany (fig. 2). so, the attendance could be maximized by extended exhibitions with spectacular works of art, which are usually not loaned. fig. 2 numbers of special exhibitions [10] summing up, the risk of damages during transport increases because of a constantly high number of special exhibitions with an increasing number of presented works of art especially in view of a keen demand for cultural objects with an endangered condition. furthermore, in-house and museum to museum/depot transports are caused by reorganization processes and building renovation. this should be acknowledged; the whole museum collections are moved [12]. besides transportation, cultural objects are exposed to mechanical loads during exhibition and depot storage by visitors, construction works, concerts, public transportation systems and other kinds of environmental excitation. the different kind of excitation scenarios are discussed in the following section. 3. excitation scenarios of canvas the diversity of mechanisms requires a detailed consideration. the mechanical load during in-house and loan traffic transport is discussed in section 3.1. the examination of excitations during exhibition is discussed in section 3.2. mechanical loads during depot storage are characterized in section 3.3. 3.1. transportation several scenarios entail the transportation of art works. examples are loan traffic, reorganization and renovation, as well as cleaning and restoration of objects. in principle, in-house movement and art in transit must be discussed in a different way. 166 k. kracht, t. kletschkowski within the museum, objects are usually moved by hand or by using hard rubber tired trolleys. there are a lot of online tutorials for restorers on how to care for and handle art objects [13]. for a better understanding of the excitation mechanisms, a transport scenario is described for a special exhibition [14]: the transportation route of cultural assets loan starts in the home museum. the canvas is taken by the restorer from the installation site and usually wrapped with wrapping film. next the canvas is fixed in the transport case. the packed canvas is moved, usually by means of a hard rubber tired trolley, into a special truck, which is equipped with air springs. in a few publications canvas transportation by truck is discussed considering (i) the paintings‟ orientation [19] and (ii) its position on the platform (especially the closeness to the sideboard) [20]. depending on the destination, transportation is continued by airplane – sometimes by ship and train. the last section of the journey requires the transportation by truck again. the canvas journey is completed by its unpacking and positioning at the new installation point. after the end of the special exhibition, the canvas travels back in the same way. according to marcon [15], measurements are the only option to characterize the shipping environment. hence, vibration and shock measurements during transport are dealt with in a couple of papers and articles: important sources of past measurements are by caldicott [16], marcon [17] and saunders [18, 19]. recent investigations are done by palmbach [20] and läuchli [21] as well as braun [22] and kracht [23]. palmbachs and läuchlis data are acquired, analyzed and documented based on the german standard specification (din en 30786). they investigate several different packaging systems during handling, trucking, shipping etc., considering route quality and the position as well as the orientation of the transport case. läuchli and bäschlin have pointed out that “shock immissions” occur mostly during handling in museums. continuous vibrations, on the other hand, are rather generated during trucking and flight/shipping. fig. 3 presents measurement results according to transportation processes by averaging the results of palmbach [20], läuchli and bäschlin [21], braun [22] and kracht [23]. fig. 3 emission levels and frequency ranges of different vehicles [21] figure 3 represents emission levels from vehicle to packages and relevant frequency ranges of the excitation without specification of the measurement direction. the dimensions of the amplitudes and frequency ranges are similar in every direction, and the out-of from art to engineering: a technical review on the problem of vibrating canvas  part i 167 plane levels are greater than the levels in the in-plane directions. precise values should be obtained from palmbach [20], läuchli and bäschlin [21], braun [22] and kracht [23]. the increasing trend of measuring vibrations and shocks during transportation leads to the development of monitoring systems. by now there are several companies which sell or lend data loggers [22]. these small devices are frequently integrated on the bottom or at the side walls of transport cases. some models enable their installation at the rear side of the canvas [24]. with such devices temperature, humidity, shocks and vibrations are recorded. note that shocks are time discrete and dynamic events. in contrast, the changing of temperature and humidity are assumed to be quasi-static processes. as a consequence, the recording of mechanical vibrations is still a challenge due to a large memory space, permanent power consumption during permanent recording and a dynamic range of embedded sensors, mostly accelerometers [25]. as mentioned before, vibration and shocks of canvas during transportation are in most cases measured with embedded lightweight accelerometers, which are mounted at the frame or in the middle of the canvas. non-contact measurements of the canvas using laser technology are also known [26]. although the laser signal is proportional to the distance (triangulation) or to the relative velocity (vibrometer) between the laser and the canvas, the laser movement during the measurements has not been considered so far. 3.2. exhibition for most restorers and conservators, the „tight‟ installation of cultural assets during an exhibition is the best preservation situation: the climate is controlled, uv and dust contamination is observed and the importance of shock and vibration immissions is much smaller compared to the transport problem. but the question remains whether the mechanical stress exposition is negligible. in 2014, gmach [12] introduces building dynamics in the context of damage scenario and risk estimation. she states that mechanical stress of canvas during exhibition is produced by vibrations of buildings, which are mostly excited by car traffic, railway dynamics, visitors‟ footsteps (depending on footwear), construction works and earthquakes at certain locations. gmach investigates shock and vibration emissions in several museums in munich with different car traffic, railway and industry concentration. four different construction types with three different materials (reinforced concrete, wood and brick) and subsoil are considered. gmach applied two different methods during her investigations. for executing the first method, the heel impact test, a person is standing at the center of the floor on his/her tiptoe and lets himself/herself tumble onto the heels. the resulting impact excitation allows us to draw conclusions related to the natural frequencies of the floors. the results of this test type are shown in fig. 4. it has to be noted that gmach measures the vibration of the floor with low-frequency-sensitive triaxial accelerometers (deltatron type 8340, type 4506 b003 and type 4507 b005 – all manufactured by brüel & kjaer). the velocities declared by gmach are converted according to din 4150-4. the main vibration response direction of the floor is an out-of-plane direction. the eigenfrequencies can be clustered into two groups. the natural frequencies of wooden floors with wooden parquet (8.5 – 40 hz) are lower than the eigenfrequencies of steelconcrete bond with stone flooring (60 – 80 hz). 168 k. kracht, t. kletschkowski fig. 4 emission levels and frequency ranges of heel impact test in different exhibition halls of different museums [12] gmach‟s second method comprises the measurement of the vibration response of the floors caused by different excitation scenarios (construction work, rail traffic, visitors). the results are shown in fig. 5. fig. 5 emission levels and frequency ranges of excitation scenarios (construction works, rail traffic, visitors) in different exhibition halls of different museums [12] the results in fig. 5 can be clustered into 3 groups [12]: 1. vibration at low frequencies (5 – 40 hz) and highest velocity amplitudes (6 mm/s – max. 13 mm/s), caused by visitors‟ footsteps on wooden truss floor. 2. vibration at midrange frequencies (50 – 100 hz) and velocity amplitudes (up to 2 mm/s), caused by railway traffic. 3. vibration at high frequencies (150 – 200 hz) and low velocity amplitudes (up to 0.064 mm/s), caused by construction work (especially drifter drills). note: referring to din 4150-3, the velocity amplitudes of historic buildings in the frequency range up to 150 hz should be permanently less than 3 mm/s. the allowable mechanical load level of canvas depends without doubt on its condition and on the properties of the load signal. however, thickett [27] investigates the vibration and shock levels which cause damage(s) to museum objects. from art to engineering: a technical review on the problem of vibrating canvas  part i 169 kracht [23] investigates the excitation of canvas during exhibition for two different suspension systems: steel hook and nylon rope. fig 6 shows the setup for the measurement at the steel hook. fig. 6 measurement setup for the investigations of the vibration excitation of the suspension system: steel hooks (oil painting “the holy family” by luciano podormo, bode-museum, berlin) [23] the measurement results are shown in fig. 7. the clustering of gmach‟s results [12] are confirmed. fig. 7 auto spectra of the signals of the measurements at the “holy family” excited by rail traffic and visitors [28] 170 k. kracht, t. kletschkowski bakker [29] and weihser [30] highlighted another excitation mechanism. the attractive surroundings of museums and the special acoustics of museum buildings make them favored venues of different kinds of concerts. 3.3. depots nearly 70% of a museum‟s collection is usually stored in depots [31]. depots are excited by the same mechanisms as the exhibition halls – traffic, railway, etc. however, the initiation of mechanical vibrations inside the building depends on the in-house transfer paths. mostly, depots are installed at the basement of the building, exhibitions usually at the higher floors. furthermore, in contrast to exhibitions, canvasses are stored space-saving at moving grid walls. kracht [23] investigates three different kinds of moving grid walls and five different canvas mounting systems. an example of a usual grid wall, the sensor position and the measurement direction are shown in fig 8. fig. 8 storage of canvas at typical grid walls in the paintings‟ depot, alte nationalgalerie, staatliche museen zu berlin [23] the top edges of a grid wall are moveable since mounted in rails with ball-bearings. the rails of 20 up to 50 or more grid walls are parts of a huge steel rack. furthermore, the front corners of the grid walls are often supported by a guiding hard rubber wheel on rough concrete or sometimes on tile flooring. as a consequence, if one grid wall is moved, the excited steel rack will hence induce vibrations to all the other grid walls. these design details are responsible for high excitation levels. the peak and rms values of three different types of moving grid walls are documented in fig. 9. the two grid walls in depots 1 and 2 have a hard rubber guide wheel in the front edge. the grid walls in depot 3 are mounted in a pending position. the stopping-systems are also different. the grid walls in depot 1 can be stopped by arm power or by a grid wallto-rack-shock. the grid walls in depot 2 have a spring, which prevent shocks between grid wall and rack. the grid walls in depot 3 have a dashpot instead of a spring. finally, only the ball-bearings of the depot 3-grid walls are free-moving. from art to engineering: a technical review on the problem of vibrating canvas  part i 171 fig. 9 rmsand peak-values of three different moving grid walls [32] these characteristics cause different vibration levels (rms between 0.1 m/s² and 2 m/s²) and shock levels (peak values between 1.2 m/s² and 12.5 m/s²) levels. kracht detects vibrations with a continuous excited frequency range between 3 and 150 hz at all moving grid walls. the highest measured amplitudes are between 7 and 48 hz. 4. efforts to reduce shock and vibration – state-of-the-art in most cases, damages caused by continuous vibrations are neglected in the literature. the low amplitude of mechanical vibrations (compared to shock levels) makes a high cumulative number of vibration cycles necessary to make damages visible [7, 8]. for this reason, efforts to protect canvas during transport are focused on shock absorbing treatments. moreover, restorers and conservators usually assume that shocks do not occur to canvas during storing in depots as well as during exhibition. hence, the efforts of vibration and shock reduction by conservators and restorers are focused on the field of transportation. in the following three subsections, the state-of-the-art of: (a) transport utility design, (b) mounting systems during exhibition and (c) storing for depots are discussed. 4.1. transport utility design since the 70's '[...] the technologies of packing and transporting paintings have been the subject of extensive investigations. [...].' [11]. a report is given by hackney and green [34]. among other findings they state: '[...] none of the respondents incorporate any specific vibration protection in their case [...]' because '[...] vibration specifically has not been shown to cause damage [...]' (ref. p. 74). this position still influences the design of transport utilities in such a way that mainly shocks are considered during the design process, and continuous vibrations are neglected [15]. based on the properties of cushioning materials, the fragility rating of objects [42] and expected shock loads, a „circular slide rule‟ [43] and a computer program 'padcad' is developed by the canadian conservation institute for supporting package designer [15, 17]. a description of the cushion design procedures with examples using measurement curves and notes about cushion material behavior like buckling as well as creep is given by richards [11]. last but not least chemical and bio-chemical material properties are discussed in [35]. 172 k. kracht, t. kletschkowski green, one of the few engineers in the field of heritage preservation, presents in [44] the idea of a vibration isolation transport utility. herein he suggests: '[...] the low natural frequencies of canvas make the successful application of vibration isolating materials difficult. [...]'. building on this, green develops 'performance criteria for packing' in general [36]. green also develops a transit frame for paintings with cushioning material which isolates canvas from most shocks during handling and transit [37]. various packages from simple slides to expensive box-in-box systems are used. the art of packaging differs from museum to museum and from country to country. restorers [21] consider the actual best solution for vibration reduction and shock absorbing is the box-in-box transport case. canvasses shipped in such crate are damaged least [45]. figs. 10 and 11 show a typical case and a close-up of the elastic support between inner and outer case. fig. 10 box-in-box transport case [38] fig. 11 support between inner and outer case [38] in their project, bäschlin, läuchli and palmbach [39] have investigated five different package systems in a number of transports over years. the key motivation of the measurements is the understanding of the damage potential for assessing risks and developing preventive strategies [13]. they establish that most efforts of vibration reduction amplify the input instead of reducing the incoming noise in a frequency range up to 40 hz. the research group names the stiffness of the elastomer support as a reason for amplification [21]. in 2016 kracht analyzes the vibration behavior of a transport crate and the interaction between transport crate and canvas [40]. as an example, fig. 12 shows the absolute value of a transfer function between the response of the midpoint of a canvas and the excitation force signal (chirp 0 to 500 hz, average 10 n) at the outer case measured near the right lower corner. fig. 12 amplitude of transfer function between midpoint of canvas and outer case [40] from art to engineering: a technical review on the problem of vibrating canvas  part i 173 in fig. 13 four absolute values of the transfer functions between the corners of a test canvas and the excitation force signal (chirp) at the outer case are shown. the magnitude of the two most dominant modes of this particular canvas during this test is approximately 4 mm at the midpoint. the corresponding resonance frequency modes are 5 hz (tilting of the rigid case) and 45 hz (torsion mode of the case). these measurement results are fundamental for vibration isolation design and optimization. fig. 13 amplitude of transfer function between canvas frame and outer case [40] 4.2. mounting and support systems in exhibition halls during exhibition, canvasses are presented in an upright position. they are mounted at steel hooks or with nylon/steel ropes or standing at pedestals. in fig. 14, left, a steel hook as a coupling element between wall and canvas frame is shown. fig. 14, right, presents the mount system based on nylon rope. in comparison with fig. 7, the results of the measurements at the nylon rope fixation in fig. 15 show that the steel hook causes no vibration isolation – in contrast to the nylon rope support. it must be also noted that the fig. 14 attachment of the work “the holy family” by luciano podormo (fig. 6) with a steel hook (left) and with a nylon rope (right) [23, 28] 174 k. kracht, t. kletschkowski auto spectra of the measurement at nylon rope support shows one peak in x-direction at 20 hz with a magnitude of 0.75 (m/s²)². in contrast, the auto spectra of the measurements at the steel hook support show a broad band with a maximum magnitude of 610 -5 (m/s²)² at 5 hz in x-direction. fig. 15 auto spectra in x-direction of the vibration measurements at nylon rope fixation excitation, which is excited by visitors (magnitude of the auto spectra in the other directions are approx. 10 -6 (m/s²) ²) [28] canvas painted from both sides is often presented on pedestals (fig. 20 left) or in a free hanging position (fig. 20 right). the pedestal of the canvas in fig. 20 is a wooden box. in [41] the operational vibration behavior of the system, shown in fig. 16, left, is analyzed. tab. 1 contains the measurement results: peak values and relevant frequency ranges. for these measurements 6 visitors excited the floor around the system. fig. 16 presentation solutions of canvas painted from both sides: on pedestals (left) [41] and hanging at four steel ropes (right) [38] from art to engineering: a technical review on the problem of vibrating canvas  part i 175 tab. 1 vibration measurement results of the system shown in fig. 16 (left) excited by 6 walking visitors [41] sensor direction (fig. 16) peak in [mm/s] frequency range in [hz] x 0.03 4 to 65 y 0.035 10 to 35 z 0.13 4 to 25 yfloor 0.04 n. s. zcanvas 0.2 3 to 10 4.3. mounting and support systems in depots in contrast to restorers‟ and conservators‟ assumptions, the results of vibration measurements at moving grid walls prove that vibrations and shocks (dependent on the moving grid walls design) can occur in practice. thus, the mounting system of the canvas should be able to absorb and to neutralize shocks as well as to isolate the system from incoming vibrations. in fig. 17 common suspension systems, which have no elastic or energy absorbing elements, are shown. fig. 17 suspension systems at moving grid walls [32] the vibrations of five different steel hook-grid wall-combinations are analyzed by [32]. rms and peak values are shown in figs. 18 and 19. fig. 18 rms-values of different moving grid walls and mounting systems [32] 176 k. kracht, t. kletschkowski fig. 19 peak-values of different moving grid walls and mounting systems [32] hanging systems 1, 2 and 3 in figs. 18 and 19 are mounted in a hanging position at grid walls in depot 1. hanging system 4 in figs. 18 and 19 is mounted in a hanging position at a grid wall in depot 2 and hanging system 5 in figs. 18 and 19 is mounted in depot 3. compared to the results shown in fig. 9 it can be observed that the excitation of the moving grid walls in depot 1 is the most significant one, but the peak values of the canvas‟ mounting points in depot 2 are larger than in depot 1. it can be concluded that the low-level excitation of the moving grid walls in depot 3 causes a low-level excitation of the canvas‟ mounting points. 5. bridging the gap: roads to the art of engineering this section aims to arbitrate between restorers‟ mind and engineers‟ abstract conceptions. on the one hand, restorers‟ first steps using engineers‟ methods are brought up and, on the other hand, the advantages and the necessity of theoretical modeling which has not started yet are pointed out. the challenges of saving canvasses from shocks and vibrations will be shown by means of a minimal model of the canvas-in-crate-system in fig. 10. furthermore, an excursus to environmental testing and the fact that canvasses are complex fiber-matrix-composites with a very complex dynamic behavior will show the wide research potential: observation and monitoring are the basis of conservators‟ and restorers‟ work. as shown in sections 3 and 4, principles of canvas‟ excitation as well as effects of elastomer mounts [46] are analyzed phenomenological by case studies during real transportation and storage scenarios. a fine step forward using engineers‟ methods is checking the capability of transport cases especially cushioning materials based on environmental testing [48]. state-of-the-art is the simulation of an aircraft flight or the conditions during a truck ride by electrodynamic or hydraulic/pneumatic shakers. therefore, the vibration and shock transmission has to be considered: from art to engineering: a technical review on the problem of vibrating canvas  part i 177 based on the engineers‟ input-output-assumption [33], fig. 20 shows the vibrations and shocks transmission exemplary from the rolling truck wheels by road roughness through the wheels, platform and package to the shipped canvas considering the reaction of each subsystem. fig. 20 path of shock and vibration transmission with considered reactions the determination of load profiles for shaker tests are common due to din en 61373 and din en 60721-2-9. recorded data for example at the package are clustered and smoothed with filter function. the results in terms of power-spectral-densities (psd) serve the target value of a control loop. exemplary the psds (target value, actual value and control limits) of a mid-range air spring truck transportation simulation are shown in fig 21. fig. 21 desired and actual values of power-spectral-density (psd) within the break-off limits referring to air spring truck level ii [49] the practical approach of case studies and environmental testing is limited in terms of finding feasible solutions by the uniqueness of each canvas, the great number of canvasses as well as the complexity of vibration isolation together with shock absorption. the challenge of vibration and shock reduction of canvasses during transportation can be shown with an engineers‟ basic: 1-degree-of-freedom minimal model: thinking of the vertical movement of the box-in-box transport crate shown in figs. 10 and 11, the dynamic of the canvas-frame-system can be modeled as shown in fig. 22 – abstracting the canvas and frame as rigid body (approximation zero order) and introducing parameters k (spring stiffness), d (damper constant), m (mass) and the coordinates of movement x(t) and u(t). 178 k. kracht, t. kletschkowski fig. 22 simplified model of the base support of the box-in-box transport crate in vertical direction the complex transfer function is given by 2 { ( )} 2 1 ( ) { ( )} 2 1 f x t j dη h jη f u t η j dη       (1) with introducing mk / 0  (natural frequency), (2 )d b/ km (damping ratio) and  = /0 (frequency ratio between excitation frequency and natural frequency) of the system in fig. 22 [47]. mostly, the excitation of canvas can differ regarding continuous random vibration and discrete shock events. the plot of absolute value of h(jη) in fig. 23 shows the remarkable characteristics of vibration isolation: i. the higher damping factor d, the lower the amplitude during resonance (η=1), ii.the higher damping factor d, the lower the effect of vibration isolation (η>√2). fig. 23 absolute value of transfer function h(jη) this means that a high damping factor is necessary to avoid resonance, but a high damping factor reduces the vibration isolation effect. furthermore, the springs of the isolation system must be weak, because the first natural frequency of paintings on textile carrier is generally very low (< 4 hz) [23] and from art to engineering: a technical review on the problem of vibrating canvas  part i 179 various excitations occur from also very low frequencies, e.g. truck excitation from 1 hz (fig. 3). the consequence of weak springs becomes obvious in fig. 24. fig. 24 base excitation and responses of the system in fig. 22 fig. 24 shows an excitation time signal, which includes random vibration and an abort, as well as the responses of the dynamic system in fig. 22 with different damping factors d in the time domain, which is calculated by numeric integration of the normalized (with tω:τt 0 and 0x ξ : x/xx  ) differential equation second order (dynamic force equilibrium of the system in fig. 21): ( ) ( ) ( ) ( ) ( )ξ τ 2dξ τ ξ τ 2dξ τ ξ τx x x u u      (2) regarding the first maximum absolute amplitude of the diagram in fig. 24 dependent on the damping factor, the third characteristic of the vibration isolation shock absorption problem can be figured out: iii. the lower damping factor d, the higher the deflections of the canvas. thus, fact iii. requires also a high damping factor to avoid high deflection of canvas. the question about the prevention of resonance and about shock absorption seems to have been answered. the simultaneous reduction of vibration excitation of canvas is up to now an unresolved problem. 6. summary, conclusion and outlook this paper gives a summary of the 40 years of conservators‟ research on canvas vibration caused by various excitation scenarios and restorers‟ efforts to reduce mechanical load. to this end, this paper contains a summarized data collection, which compresses the huge data volume in restorers‟ literature. 180 k. kracht, t. kletschkowski it is acknowledged that the customary solutions for transporting, presenting and storing canvas in matters of vibration reduction are improvable. these utilities are not designed in consideration of modeling and analyzing the dynamic behavior of each subsystem. until nowadays restorers themselves establish most of vibration reduction solution. it is figured out that most of these solutions amplify the input instead of reducing the incoming noise. in this context, the practice and methods of environmental testing are suggested. the difficulties of reducing the mechanical loads of canvas during transportation are shown by means of a minimal model. generally very low natural frequencies of paintings on textile carriers (first natural frequencies often < 4 hz) and the excitation of canvas from 1 hz in many cases require a vibration isolation system with weak springs. as a consequence, a high damping factor to avoid the resulting high deflections of canvas after shock loads and during resonance is necessary. in contrast to this, the high damping inhibits the reduction of the excitation caused by continuous vibrations. furthermore, in the context of the very low natural frequencies of canvas, it is noteworthy that transport crates tend to rigid body tilting modes at frequencies lower than 10 hz (ref. fig. 17). scopes to solve the problem of vibrating canvas are 1. the reduction of load levels, 2. the development of passive or active adjustable spring damper elements and 3. the stiffening of canvas. therefore, the systematic investigation and modeling of each subsystem are required. engineering literature provides several approaches in terms of excitation reduction and spring damper elements design. the transfer to the preservation of paintings poses the challenge. contrary to this, 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series: mechanical engineering vol. 15, no 3, 2017, pp. 439 456 https://doi.org/10.22190/fume170508024c © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper a novel hybrid method for non-traditional machining process selection using factor relationship and multi-attributive border approximation method udc 658.5 prasenjit chatterjee 1 , supraksh mondal 2 , soumava boral 3 , arnab banerjee 1 , shankar chakraborty 4 1 department of mechanical engineering, mckv institute of engineering, india 2 department of mechanical engineering, mallabhum institute of technology, india 3 subir chodhury school of quality and reliability, indian institute of technology, kharagpur, india 4 department of production engineering, jadavpur university, india abstract. selection of the most appropriate non-traditional machining process (ntmp) for a definite machining requirement can be observed as a multi-criteria decision-making (mcdm) problem with conflicting criteria. this paper proposes a novel hybrid method encompassing factor relationship (fare) and multi-attributive border approximation area comparison (mabac) methods for selection and evaluation of ntmps. the application of fare method is pioneered in ntmp assessment domain to estimate criteria weights. it significantly condenses the problem of pairwise comparisons for estimating criteria weights in mcdm environment. in order to analyze and rank different ntmps in accordance with their performance and technical properties, mabac method is applied. computational procedure of fare-mabac hybrid model is demonstrated while solving an ntmp selection problem for drilling cylindrical through holes on non-conductive ceramic materials. the results achieved by fare-mabac method exactly corroborate with those obtained by the past researchers which validate the usefulness of this method while solving complex ntmp selection problems. key words: non-traditional machining processes, mcdm, factor relationship, mabac received may 08, 2017 / accepted november 08, 2017 corresponding author: prasenjit chatterjee affiliation: department of mechanical engineering, mckv institute of engineering, howrah711204, india e-mail: prasenjit2007@gmail.com 440 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty 1. introduction increasing global competition and rapid progress in manufacturing expertise are the major state of things in today’s commercial environment. they have forced the manufacturing organizations to reallocate the business priorities in the direction of quality, cost optimization and responsiveness to market changes. the manufacturing scenario of the 21 st century is budding out as the integration of fragmented consumer market and rapidly varying production technologies. these changes are motivating the manufacturing organizations to struggle along several product dimensions including design, manufacturing and others. although manufacturing has never been utilized as a viable weapon historically, still, the market place of the 21 st century demands manufacturing technologies to presume an imperative role in the new competitive arena. at present, customers aspire for a huge variety of products. the growing need for generating and machining complex and precise shapes in newer materials, like glass, titanium, ceramics, high strength temperature resistant (hstr) alloys, fiber-reinforced composites, stainless steel, refractory materials and other difficult-to-machine alloys in nonexistence of adequately hard and strong cutting tool materials has resulted in the advancements of a number of new machining processes, commonly known as non-traditional machining processes (ntmps). now-a-days, ntm is considered as one of the major strategic resources for operational enhancement and for maintaining competitive position of an organization in the global marketplace. this competitive environment has refurbished interest with respect to research on economic analysis and ntmp validation methods, which can be utilized to aid organizations in selecting suitable ntmps to meet their operational and business objectives. the conventional machining processes mostly remove materials in the form of chips by applying forces on the work material with a wedge-shaped cutting tool that is harder than the work material. these forces stimulate plastic deformation in the workpiece which leads to shear deformation next to the shear plane and also lay the foundation for chip formation. as compared to the conventional machining processes, the ntmps use variety of mechanical, thermoelectrical, electrochemical and chemical energies to provide machining or removing materials in the shape of chips or atoms to get the preferred accuracy and burr-free machined surface [1,2]. material removal perhaps occurs by means of chip formation or without chip formation. for example, in abrasive jet machining (ajm), chips are in infinitesimal dimension and for electrochemical machining (ecm), material removal takes place because of electrochemical dissolution at atomic level. thus, an exhaustive knowledge about various machining characteristics is very important for efficient exploitation of the capabilities of different ntmps. comparing to the conventional machining processes, ntmps possess superior process capabilities whose application domain may go on increasing in diverse ranges. as ntmps can provide new ways of satisfying the demands of modern technological advances in many areas, including data transmission and miniaturization, the designers and nowadays manufacturing engineers are venturing towards the applications of different ntmps to fulfill the machining and surface quality requirements. latest possibility of choices from a group of available ntmps has been unlocked for designing and machining extremely intricate products. existence of a huge number of ntmps along with multifarious uniqueness and capabilities, and lack of proficiency in ntmp selection domain have a novel hybrid method for non-traditional machining process selection using factor relationship… 441 forced the manufacturing and design engineers to develop structured approaches for ntmp selection for assorted machining applications. the uncertainties regarding material requirements, shape applications, technical capabilities and other process attributes with the availability of many alternatives make ntmp selection for a meticulous application a very difficult and risky task. to properly select the predominant ntmp for a specific application, the process engineer must understand the limitations as well as strengths of each ntmp with specific functionalities and applicability [3]. 2. review of literature on ntmp selection to the utmost level of information, there are not so many published works on selection of ntmps in multi-criteria decision-making (mcdm) environment. before probing for an appropriate ntmp, it is obligatory to be acquainted with the nature of application where the ntmp would be implemented. even though an ntmp can be employed very efficiently for a particular application, changes in the application type, material and other requirements can reduce its efficiency significantly. therefore, the selection approach must start with a clear identification of the application domain. cogun [3,4] developed a computer-aided ntmp selection approach for some given industrial applications using an interactively generated 16-digit classification code. the developed system was used to categorize and rank the viable ntmps. yurdakul and cogun [5] proposed a combined analytical hierarchy process (ahp) and technique for order preference by similarity to ideal solution (topsis)-based procedure for selecting suitable ntmps for the given industrial application viewpoints and characterized the alternative ntmps using a number of criteria, including workpiece material suitability, shape applications, process capability and cost considerations. chakraborty and dey [6] developed an ahp-based expert system to help the decision maker (dm) for selecting the most appropriate ntmps for some given applications. the expert system was based on the priority assessments for different criteria and sub-criteria as associated to the explicit ntmp selection problems. chakraborty and dey [7] designed a quality function deployment (qfd)-based expert system for ntmp selection, considering various product and process characteristics. the weights of ntmp selection criteria were used to calculate overall scores for the possible ntmps. das chakladar and chakraborty [8] utilized a combined ahp-topsis-based methodology for selection of the best ntmps for some given machining applications. edison chandrasselan et al. [9] proposed a web-based knowledge base system for identifying the most appropriate ntmp, while considering material requirements, shape applications, process economy and process capabilities parameters. edison chandrasselan et al. [10] illustrated a knowledge-based system for recognizing the most suitable ntmp from 20 alternatives of engineering significance. the developed knowledge-based system considered material variety and some process capability constraints, including tolerance, corner radii, surface damage, taper, width of cut, surface finish, hole diameter, depth-to-diameter ratio (for cylindrical holes) and depth-towidth ratio (for blind cavities) to select the best ntmp for a particular machining application. das chakladar et al. [11] applied a digraph-based decision-making approach to select the most appropriate ntmps for some real time manufacturing applications. sadhu 442 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty and chakraborty [12] applied a two-phase decision-making model, based on data envelopment analysis (dea) and weighted overall efficiency ranking method, to select and rank feasible ntmps for some certain shape characteristics and work material combinations. das and chakraborty [13] developed an analytic network process (anp)-based graphical user interface model to select the most appropriate ntmps, captivating interdependency and feedback relationships among different criteria, affecting the ntmp selection decision. chakraborty [14] applied multi-objective optimization by ratio analysis (moora) method to choose ntmps for various engineering applications. temuçin et al. [15] provided some methodical approaches and a decision support model in fuzzy and crisp situations to deal with ntmp selection problems. karande and chakraborty [16] applied an integrated preference ranking organization method for enrichment evaluation (promethee) as well as a geometrical analysis for an interactive aid (gaia) method for solving ntmp selection problems. the suggested approach would act as a visual decision support to the process engineers. temucin et al. [17] proposed a fuzzy decision support system for selecting ntmps considering the vagueness of several inter-related decision criteria. choudhury et al. [18] developed a topsis-ahp based expert system for ntmp selection and considered several paradoxical criteria. prasad and chakraborty [19] developed a quality function deployment (qfd) based expert system module in visual basic 6.0 to automate the ntmp selection. temuçin et al. [20] proposed a fuzzy decision model for ntmps selection. a graphical user interface (gui), built in seted 1.0 software, was also developed to validate the potentiality of the proposed approach. roy et al. [21] suggested a fuzzy ahp and qfd technique for the purpose of ntmp selection. madic et al. [22] again explored the applicability of moora and ahp methods while solving ntmp selection problems and compared the results with topsis method. khandekar and chakraborty [23] proposed the application of fuzzy axiomatic design (ad) principles to select best ntmp for generating cavities on ceramics and micro-holes on hardened tool steel and titanium materials, based on their practical/industrial importance. boral & chakraborty [24] applied case-based reasoning (cbr) approach for ntmps selection using a gui, developed in visual basic 6.0. roy et al. [25] proposed a novel approach combining fuzzy ahp with qfd for ntmps selection and ranked the alternatives by applying grey relational analysis (gra) methodology. regarding the above survey of referential literature, it has been observed that in most of the ntmp selection papers, the past researchers have mainly applied ahp, topsis, qfd and dea models. very few applications are related to promethee, moora and evamix etc. methods. ahp is an expedient method of breaking down an intricate and unstructured problem into its various constituent parts to amalgamate the judgments in order to establish the highest priority variables which may influence the outcome of the situation. however, the computational requirement of ahp is tremendous even for a small problem. it suffers from inconsistencies between judgment and ranking criteria. rank reversal may occur due to the changes of the order of the alternatives when a new alternative is added to the problem. when the number of the levels in the hierarchy increases, the number of pairwise comparisons also increases which is a very time consuming effort [26]. the fundamental theory of the topsis method is based on the concept that the best alternative has the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. the topsis method instigates two reference points using a novel hybrid method for non-traditional machining process selection using factor relationship… 443 vector normalization; however, it does not contemplate the relative significance of the distances from these reference points. it means that the best alternative in the topsis method may not always mean that it is the closest to the ideal solution [27]. as a typical formulation of dea builds a separate linear program for every alternative, so it becomes computationally intensive. also as dea is an extreme point method, measurement error can root considerable problems. dea does not provide estimates which can effortlessly be validated with conventional statistical procedures and it does not tender the ranking of the alternatives [28, 29]. a major objective of qfd is to transform customer requirements (crs) into engineering characteristics (ecs) of a product. qfd is a very time consuming approach. numerous complexities may be countered while prioritizing crs and ec using ordinal ratings. differentiating between diverse and contradictory crs is very difficult [30,31]. promethee method does not provide the possibility to really structure a decision problem. in the case of many criteria and alternatives, it may become difficult for the dm to obtain a clear view of the problem and to evaluate the results due to the involvement of different preferential parameters like preference functions which may be very difficult to define in real time scenarios [32]. the literature survey also indicates that criteria weights for ntmp selection problems are generally determined by expert opinion-based pair-wise comparisons using ahp method. the criteria weights being one of the vital decisive phases in the selection process, the accuracy of expert evaluation essentially depends on the number of criteria. when this number is too large, an expert may no longer be proficient to evaluate the criteria to determine their relative importance. thus, it is evident that the past researchers have adopted different decision-making tools for evaluating, justifying and selecting ntmps, but all those methods are either very complicated or require lengthy computations and sometimes need the help of linear programming tools to solve the developed models. also, for the decision-making problems with large number of criteria and smaller number of alternatives, those approaches may occasionally give poor results. to overcome such difficulties, the present paper proposes a novel hybrid method encompassing a new criteria weighting technique, namely factor relationship (fare) and multi-attributive border approximation area comparison (mabac) approaches, as shown in fig. 1. the proposed model allows criteria weight calculation based on the relationship between one criterion with the others to reduce the amount of expert assessment, while the precision of evaluation augments and also provides a more precise and accurate rankings of the feasible ntmp alternatives. 444 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty fig. 1 proposed fare-mabac hybrid method f a r e criteria ranking and interrelationship determination of potential impact of each criterion determination of potential equilibrium of the attributes calculation of total impact or dependence determination of weight of the attribute normalization of decision matrix weighted normalized matrix calculation of the distance matrix estimation of the baa matrix calculation of criteria function determination of ranking alternatives m a b a c a novel hybrid method for non-traditional machining process selection using factor relationship… 445 3. methods 3.1. fare method ginevicius [33] developed fare method for estimating criteria weights in mcdm background. the procedural steps to apply fare method to determine criteria weights are described as follows [33-35]: step 1: determination of potential impact of the attributes: initially, the potential impact of the criteria is found out using: )1(  nsp (1) where p is the potential of the system’s criteria impact and s is the maximum value of the evaluation scale used, as given in table1. step 2: ranking criteria and assessment of their interrelationship: criteria are now ranked based on their importance while the relationship among the criteria is assessed using table 2. any criterion with a lower rank has less significant impact on other criteria having higher ranks and consequently it ought to transmit a larger part of its potential impact to others. table 1 scale of quantitative evaluation of interrelationship between the system’s attributes type of the effect produced rating of the effect produced by interrelationship (in points) almost none 1 very weak 2 weak 3 lower than average 4 average 5 higher than average 6 strong 7 very strong 8 almost absolute 9 absolute 10 table 2 measurement scale for pair wise comparison verbal judgment or preference numerical rating extremely preferred 9 very strongly to extremely preferred 8 very strongly preferred 7 strongly to very strongly preferred 6 strongly preferred 5 moderately to strongly preferred 4 moderately preferred 3 equally to moderately preferred 2 equally preferred 1 446 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty step 3: determination of impact of the attributes on the main attribute: the impact of criterion aj on the main criterion is computed and then, this impact is transformed as follows: 1 1j ja s ã  (2) where, a1j is the impact of the j th criterion on the first main criterion and ã1j is the part of the j th attribute’s potential impact transmitted to the main criterion. step 4: determination of total impact: the total impact and consistency of any criterion is calculated using eq. (3). the subset considered is reliable, consistent and steady if the total impact of its criteria with a positive sign is equal to the total impact with a negative sign, i.e. their sum is always equal to zero. ijn j ijajp   ,1 (3) the total impact can also be estimated using eq. (4). the total impact or dependence of a criterion exemplifies its dominance over the others. therefore, the most significant criterion in the matrix presented should be the first one with maximum total dominance. 1 1.j jp p n a  (4) where pj is the total impact (dependence) of the j th criterion and n is the total number of criteria. step 5: computation of attribute weights: lastly, the criteria weights are derived using: 1 1 ( 1) ( 1) f j j j s p p na s n w p ns n       (5) where ps is the total potential of a set of criteria, calculated using eq. (6) and pj f is the actual total impact of the j th criterion of the system, calculated using eq. (7): . . ( 1)sp n p n s n   (6) 1 1 ( 1)f j j jp p na s n p p      (7) where pj is the total impact produced by the j th criterion of the system signifying its total dependence on the other criteria and p is the potential impact of the criteria. 3.2. mabac method the mabac method was developed at the research centre of university of defense in belgrade. once the weight importances of the criteria are assessed using fare method, the provisions are all laid to instigate the mathematical formulation of mabac method. the elementary concept of this method can be realized in the explanation of the a novel hybrid method for non-traditional machining process selection using factor relationship… 447 distance of criterion function of each alternative from the border approximation area. in this section, six easy steps to execute mabac method are presented as follows [36-39]: step 1: formation of the preliminary decision matrix (x): the primary step is to assess m alternatives according to a set of n predefined criteria, describing the alternatives. this decision matrix is presented in the formation of vectors ai=(xi1,xi2,…,xin), where xij is the value of the i th alternative according to the j th criterion (i = 1,2,…,m; j = 1,2,…,n). 1 11 12 1 2 21 22 2 1 2 ... ... ... ... ... ... ... ... n n m m m mn a x x x a x x x x a x x x              (8) step 2: normalization of initial decision matrix (x): normalize the initial decision matrix (x) using linear normalization method. the reason of normalization is to attain dimensionless assessments of different criteria to make them comparable with each other. 1 11 12 1 2 21 22 2 1 2 ... ... ... ... ... ... ... ... n n m m m mn a r r r a r r r r a r r r              (9) the components of the normalized decision matrix (r) are determined using the following equations: a) for beneficial criteria (for which higher values are always desirable)      jj jij ij xx xx r (10) b) for non-beneficial or cost criteria (for which lower values are always preferable)      jj jij ij xx xx r (11) where xj + and xj are the maximum and minimum values of j th criterion according to the alternatives. step 3: determination of the weighted normalized decision matrix (v): the elements from the weighted matrix (v) are calculated according to ( 1)ij j ijv w r   (12) where rij are the elements of the normalized matrix (r), wj are the weight coefficients of the criteria. 448 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty 11 12 1 1 11 2 12 1 21 22 2 1 21 2 22 2 1 2 1 1 2 2 ... ( 1) ( 1) ... ( 1) ... ( 1) ( 1) ... ( 1) ... ... ... ... ... ... ... ... ... ( 1) ( 1) ... ( 1) n n n n n n m m mn m m n mn v v v w r w r w r v v v w r w r w r v v v v w r w r w r                                             where n is the total number of criteria, m is the total number of alternatives. step 4: estimation of the border approximation area (baa) matrix (b): the elements of matrix (b) for each criterion are determined according to: 1/ 1 m m j ij i b v            (13) where vij are the elements of weighted matrix (v), and m is the total number of alternatives. after calculating value gj for each criterion, a border approximation area matrix (b) is formed with format n  1. 321 c...cc [ ... ]1 2b b b bn (14) step 5: calculation of the distance matrix of alternatives ( q ) from the baa: 11 12 1 21 22 2 1 2 ... ... ... ... ... ... ... n n m m mn q q q q q q q q q q              (15) the distance of the alternatives from the baa is determined as the difference between the elements in weighted matrix (v) and the value of border approximation area (b): 11 12 1 1 2 21 22 2 1 2 1 2 1 2 ... ... ... ... ... ... ... ... ... ... ... ... ... ... n n n n m m mn n v v v b b b v v v b b b q v b v v v b b b                               (16) 11 1 12 2 1 11 12 1 21 1 22 2 2 21 22 2 1 21 1 2 2 ... ... ... ... ... ... ... ... ... ... ... ... ...... n n n n n n m m mnm m mn n v b v b v b q q q v b v b v b q q q q q q qv b v b v b                                  (17) where bj is the baa for j th criterion and vij is the weighted matrix of elements (v), n is the number of criteria, m is the number of alternatives. a novel hybrid method for non-traditional machining process selection using factor relationship… 449 alternative ai may belong to baa (b), upper approximation area (b + ) or lower approximation area (b ), i.e. ai  {b  b +  b }. upper approximation area (b + ) is the area which contains ideal alternative (a + ), while lower approximation area (b ) is the area which contains anti-ideal alternative (a ), as shown in fig. 2. fig. 2 presentation of the upper (b + ), lower (b ) and border (b) approximation areas [36] the belonging of alternative ai to approximation area (b, b + or b ) is determined on the basis of eq. (18): if > 0 if 0 if < 0 b qij a b qi ij b qij           (18) in order to select alternative ai as the best in the set, it is necessary for it to belong to upper approximate area (b + ) for as many criteria as possible. if value qij>0, that is qijb + , then alternative ai is near or equal to the ideal alternative. if value qij<0, that is qijb , it indicates that alternative ai is near or equal to the anti-ideal alternative. step 6: calculation of criteria function (si) values and ranking the alternatives: calculation of the criteria function values for the alternatives is obtained as the sum of the distance of the alternative from the border approximation area (qi), as indicated by eq. (19). by adding together the rows of elements of matrix q, the final values of the criterion functions for the alternatives can be obtained. finally, alternatives are arranged in the descending order of si values and the alternative with the highest si value is ranked as the best one: 1 , 1,2,..., , 1,2,..., n i ij j s q j n i m     (19) 450 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty 4. illustrative example to reveal the computational flexibility and expediency of the proposed hybrid faremabac method, an ntmp selection problem for drilling cylindrical through holes on nonconductive ceramic materials is considered here. the diameter (d) and slenderness ratio (l/d) of the hole are given as 0.64 mm and 5.7 respectively, where l designates hole depth. yurdakul and cogun [5] developed an mcdm approach for deciding appropriate ntmps for different industrial applications. the authors characterized different ntmps using a number of attributes, including shape applications, workpiece material suitability, process capability and cost considerations to provide a way of evaluating the levels of achievement of ntmps with respect to their capabilities and output quality. in the manufacturing domain, ntmps are used generally for machining different shapes, including blind cavities, producing through profiles or holes, cutting operations and surface finishing operations. work material suitability mainly indicates the easiness of an ntmp to machine a particular material. based on the shape application and workpiece material suitability constraints, yurdakul and cogun [5] eradicated a number of ntmps, including ecm, ecg, ech, edm, wedm, pac and wjm from further deliberations for this cylindrical through hole drilling operation. yurdakul and cogun [5] observed ajm, usm, chm, ebm and lbm processes as the most feasible alternatives to be judged and developed the corresponding decision matrix, as shown in table 3. tolerance (tl), surface finish (sf), surface damage (sd), taper (t), material removal rate (mrr), work material (m) and cost (c) were chosen as the most pertinent attributes, affecting this ntm process selection decision. tl and sf are the two most important product capability attributes, used to measure the performance of an ntmp for a particular machining application. tolerance can be defined as the difference between the maximum and minimum dimensions of a component. depending on the type of application, the permissible variation of dimension is set according to the available standard grades. it also relates to the capability of an ntmp stating how closely the process can achieve the required surface finish on the given work material. material removal rate is one of the most important criteria leading to the fact that higher mrr leads to lower machining time. the effectiveness of an ntmp is usually measured in terms of its mrr. machining cost is an important criterion for ntmp selection. it generally comprises tooling and fixture, power consumption and tool wear costs. tooling and fixture costs include workpiece holding and adjustment costs. power consumption includes costs associated with electricity used for material removal or in driving pumps, compressors, motors, heating units, beam generators, cost of electrolytes, dielectrics, chemicals, acid solutions or gases consumed during the machining operation. tool wear cost includes entire tool and tool replacement costs. yurdakul and cogun [5] developed a generalized overall machining cost formula to provide an approximated cost score for ntmps, based on tooling and fixture, power consumption and tool wear cost elements. among these seven considered criteria, tl (mm), sf (cla) (µm), sd (µm), t (mm/mm) and c are non-beneficial in nature, whereas, mrr (mm 3 /mm) and m are beneficial. also, among these attributes, tl, sf, sd, t and mrr are expressed quantitatively, having absolute numerical values, whereas m and c have qualitative measures for which ranked value judgments on some qualitative scales are used, as shown in tables 4-6 [5]. table 7 shows the transformed decision matrix for this ntmp selection problem. the criteria weights were determined by yurdakul and cogun [5] as wtl = 0.32, wsf = 0.19, wsd = 0.04, wt = 0.04, wmrr = 0.19, wm = 0.11 and wc = 0.11 using ahp a novel hybrid method for non-traditional machining process selection using factor relationship… 451 method. yurdakul and cogun [5] solved this ntmp selection problem using a combined ahp-topsis method and observed the ranking of the alternative ntm processes as usm > lbm > ebm > chm > ajm. now, this ntmp problem is solved using the proposed fare-mabac method and the results are then compared. table 3 ntmp selection matrix for drilling cylindrical through holes on ceramics [5] ntm process tl sf sd t mrr m c ajm (a1) 0.05 0.6 2.5 0.005 50 good 4 usm (a2) 0.013 0.5 25 0.005 500 good 5 chm (a3) 0.03 2 5 0.3 40 poor 2 ebm (a4) 0.02 3 100 0.02 2 good 1 lbm (a5) 0.02 1 100 0.05 2 good 1 table 4 qualitative scale for workpiece material suitability level [5] linguistic variable poor fair good scale value 1 2 3 4.1. calculation of criteria weights using fare method in this phase, the calculation of criteria weights are performed using the steps as explained in section 3.2. first of all, the potential impact of the attributes is computed as 60. next the initial priority of the ntmp selection criteria (tl > sf > mrr > m > c > sd > t) are decided according to their weight values as determined by yurdakul and cogun [5] and an extensive literature review as presented earlier. the priority order indicates that tl is the most preferred criterion, while t is the least important criterion for the considered ntmp selection problem. since tl is the most significant criterion, the impact of other criteria will be transferred through it and, therefore, their direct impact on this ntmp selection problem will be decreased, as exhibited through table 8. as shown in this table, the matrix of potential equilibrium has a particular structure, i.e. aij = aji. for example, priority of pl to sf is 3, then priority of sf to pl is automatically settled as -3. in this case, a row or column of the matrix demonstrates the total (summed up) effect or dependence of a particular criterion on other criteria compared with it. table 5 cost levels of different ntmps [5] ntmp tooling and fixture cost rating power consumption rating tool wear cost rating overall machining cost score a1 low (3) low (3) medium (5) 4 a2 low (3) low (3) high (7) 5 a3 low (3) very low (1) very low (1) 1.8 (2) a4 very low (1) low (3) very low (1) 1.2 (1) a5 very low (1) very low (1) very low (1) 1.2 (1) 452 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty table 6 qualitative scale used to gauge machining cost elements [5] linguistic variable very low low medium high very high scale value 1 2 4 3 5 6 7 8 9 table 7 transformed decision matrix ntmp tl sf sd t mrr m c a1 0.05 0.6 2.5 0.005 50 3 4 a2 0.013 0.5 25 0.005 500 3 5 a3 0.03 2 5 0.3 40 1 2 a4 0.02 3 100 0.02 2 3 1 a5 0.02 1 100 0.05 2 3 1 table 8 summary matrix of potential equilibrium of the ntmp selection criteria criteria tl sf sd t mrr m c total effect (dependence) tl 3 3 4 4 3 5 22 sf -3 2 3 3 4 4 13 sd -3 -2 2 -4 -3 -3 -13 t -4 -3 -2 -2 -2 -2 -15 mrr -4 -3 4 2 4 4 7 m -3 -4 3 2 -4 4 -2 c -5 -4 3 2 -4 -4 -12 total effect -22 -13 13 15 -7 2 12 in order to calculate the total impact (pj), summation of each row is computed using and also exhibited in table 8. this table reveals that the total impact of the criteria set with a positive sign is equal to the total impact with a negative sign. consequently, the set is totally consistent. total potential (ps) value of 420, required for determining the weights, is now computed using eq. (6). for this example, n = 7 and p = 60. once the ps value is calculated, the individual potentials of all the considered ntmp selection criteria are determined and the criteria weights are estimated, as given in table 10. these weights are subsequently used for mabac-based analysis. table 9 total impact of ntmp selection criteria criteria tl sf sd t mrr m c sum tl 7 7 6 6 7 5 38 sf -7 8 7 7 6 6 27 sd -7 -8 8 -6 -7 -7 -27 t -6 -7 8 -8 -8 -8 -29 mrr -6 -7 6 8 6 6 13 m -7 -6 7 8 -6 6 2 c -5 -6 7 8 -6 -6 -8 a novel hybrid method for non-traditional machining process selection using factor relationship… 453 table 10 individual potential and criteria weight values criteria pj f ps = n.p = n.s(n1) wj = pj f /ps tl 98 7 x 60 = 420 98/420 = 0.23 sf 87 87/420 = 0.21 sd 33 33/420 = 0.08 t 31 31/420 = 0.07 mrr 73 73/420 = 0.17 m 62 62/420 = 0.15 c 52 52/420 = 0.12 4.2. application of mabac method for selection of ntmps through the second phase, the evaluation and ranking of ntmp alternatives is now performed by applying the mabac method. after forming the transformed decision matrix of table 7, normalization of its elements is carried out respectively for beneficial and non-beneficial type criteria, as shown in table 11. table 11 normalized decision matrix for ntmp selection problem ntmp tl sf sd t mrr m c a1 0 0.9600 1.0000 1.0000 0.2500 0.0964 1.0000 a2 1.0000 1.0000 0.7692 1.0000 0 1.0000 1.0000 a3 0.5405 0.4000 0.9744 0 0.7500 0.0763 0 a4 0.8108 0 0 0.9492 1.0000 0 1.0000 a5 0.8108 0.8000 0 0.8475 1.0000 0 1.0000 table 12 weighted normalized matrix ntmp tl sf sd t c mrr m a1 0.2333 0.4060 0.1571 0.1476 0.1500 0.1906 0.2952 a2 0.4667 0.4143 0.1390 0.1476 0.1200 0.3476 0.2952 a3 0.3595 0.2900 0.1551 0.0738 0.2100 0.1871 0.1476 a4 0.4225 0.2071 0.0786 0.1439 0.2400 0.1738 0.2952 a5 0.4225 0.3729 0.0786 0.1364 0.2400 0.1738 0.2952 the elements of weighted normalized matrix (v) are then estimated by multiplying the weight coefficients of the criteria with the elements of the normalized matrix using eq. (12), as given in table 11. for example, element v12 of weighted matrix (v) is obtained as follows: v12 = w2 x (r12 + 1) = (0.96+1) x 0.21 = 0.4060 where r12 is an element of normalized matrix (r), w2 is the weight coefficient of criterion sf. now, border approximation area matrix (b) is obtained by taking geometrical average of the values, as given in table 13. for example, baa for criterion sf is obtained as follows: 1/7 7 1/7 2 1 (0.4060 0.4143 0.2900 0.2071 0.3729) 0.3275ij i b v x x x x             454 p. chatterjee, s. mondal, s. boral, a. banerjee, s. chakraborty table 13 border approximation area (b) matrix baa tl sf sd t c mrr m bj 0.3706 0.3275 0.1159 0.1258 0.1852 0.2064 0.2570 the next step is the estimation of distance matrix (q) elements of ntmp alternatives from border approximate area matrix (b).the distance of the alternative ntmps from matrix b is determined using eq. (17) as the difference between the elements in weighted matrix (v) and the values from the elements of matrix b, as exhibited in table 14. for example, element q11 is calculated as follows: q11 = v11  b1 = 0.2333  0.3706 = 0.1372 table 14 distance of ntmp alternatives from border approximation area (b) ntmp tl sf sd t c mrr m a1 -0.1372 0.0785 0.0412 0.0218 -0.0352 -0.0158 0.0382 a2 0.0961 0.0868 0.0231 0.0218 -0.0652 0.1412 0.0382 a3 -0.0111 -0.0375 0.0392 -0.0520 0.0248 -0.0193 -0.1094 a4 0.0520 -0.1204 -0.0373 0.0180 0.0548 -0.0326 0.0382 a5 0.0520 0.0454 -0.0373 0.0105 0.0548 -0.0326 0.0382 the last step is the calculation of criteria function (si) values for each ntmp alternative. the si values of the alternative ntmps along with their ranks are presented in table 15. for example, si value of a1 (ajm) is computed as below: si (a1) = -0.1372 + 0.0785 + 0.0412 + 0.0218  0.0352  0.0158 + 0.0382 = 0.0085 the results of table 15 imply that alternative a2 (usm) is ranked as the first one, and alternative a3 (chm) as the worst and the least favorable ntmp. the results show that the rankings of the ntmps are exactly the same as those derived by yurdakul and cogun [5] using a combined ahp-topsis method, which leads to the confirmation that the proposed fare-mabac model can be an effective, efficient and simple method for solving complex ntmp selection problems. table 15 ntmps with si values and corresponding ranks ntmp si value rank ahp-topsis [5] a1 -0.0085 3 3 a2 0.3421 1 1 a3 -0.1653 5 5 a4 -0.0272 4 4 a5 0.1310 2 2 a novel hybrid method for non-traditional machining process selection using factor relationship… 455 5. conclusions this paper presents a new fare-mabac model for ntmp selection problems in manufacturing domain. this combined application is based on an uncomplicated weight calculation tool which involves least amount of mathematical calculations, followed by a simple ranking methodology. the analytical results show that the ranking preorder 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5department of mathematics and computer science, faculty of science, beni-suef university, beni-suef, egypt abstract. a critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. this paper establishes a fractal-differential model for this purpose, and a fractal duffing-van der pol oscillator (dvdp) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. utilizing the two-scale transforms and he-laplace method, an analytic approximate solution may be attained. unfortunately, this solution is not physically preferred. it has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. on the other hand, the linearized stability theory is employed in the autonomous arrangement. consequently, the phase portraits around the equilibrium points are sketched. for the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. it is revealed that the exciting external force parameter plays a destabilizing role. furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture. key words: fractal vibration theory, duffing-van der pol oscillator, homotopy perturbation method, expanded frequency analysis, linearized stability theory, multiple timescales technique received january 18, 2022 / accepted february 12, 2022 corresponding author: ji-huan he school of civil engineering, xi’an university of architecture & technology, xi’an 710055, china e-mail: hejihuan@suda.edu.cn 2 j.-h. he, g. m. moatimid, m. h. zekry 1. introduction the fractal vibration theory has become a significant topic in both mathematics and mechanical engineering. the smart receptor system of a three-dimensional printing processing can be exactly controlled by the gecko-like fractal vibration system [1]. the fractal toda oscillator exhibits some special properties, which the traditional vibration theory cannot account for [2]. the pull-in instability can be completely eliminated for an electrostatic nano/micro-electromechanical (n/mems) system by suitably controlling the fractal vibration system [3-5]. the nonlinear wave travelling in a fractal space can be converted into a fractal oscillator, and some fascinating properties are revealed [6]. the fractal vibration theory provides an effective tool for studying various unusual phenomena under the micro-gravity condition [7-9]. the fractal vibration can effectively model the vibration arising in porous media, and the low frequency property plays an important role in the anti-vibration design [10, 11]. furthermore, the fractal fangzhu oscillator explores the mechanism of an ancient technology for the collection of water from air [12]. much literature focused on simple fractal oscillators [13-15], this paper considers a more generalized fractal vibration system in the following form: 2 2 3 2 ( ) cos , d z dz b cz az dz f d d       − − + + = (1) where dz/dτα is the two-scale fractal derivative [16 and 17]; z,τ,b,c,a,d,f and ω are displacement, time, the coefficient of the damping term, the coefficient of the nonlinear damping term, the root of natural frequencies, the cubic stiffness parameter, the amplitude of the excitation force, and frequency of this excitation, respectively. eq. (1) displays a nonlinear vibration system in a fractal space as discussed by tian et al. [3-5], where the air is considered as a porous medium and the fractal dimension α reflects the air distribution. when it vibrates in a vacuum, one has α=1. when f=b=c=0, eq. (1) becomes the fractal duffing oscillator [1 and 13-15], which can model the nonlinear vibration arising in a three-dimensional printing technology, concrete vibration and other applications. for more convenience, the graph x() = cos α will be plotted for various values of the parameter α in fig. 1. it is obvious that the periodic solution becomes an asymptotic one. when α=1, eq. (1) reflects the forced duffing -van der pol driven oscillator. in view of the two-scale transforms [16], one gets z=x√b/c, τα=t/√a, b=µ√a and ω=ω√a to convert eq. (1) into the following form: 2 3(1 ) cosx x x x x f t − − + + =  , (2) where β=bd/ac, f=(f/a)√c/b, , and the dot is the derivative with respect to t. the two-scale fractal theory is now widely used to model various phenomena arising in porous media or un-smooth boundaries. the fractal rheological model gives a quick physical insight into the rheological property of a non-newtonian fluid [18 and 19]. the fractal approach provides a simple and good tool for mechanical and electrical properties of composite materials [20]. the fractal two-phase flow model can figure out the hidden properties in a polymer filling process, which cannot be revealed by the different models [21-23]. now, the fractal variational principle has become a predominant topic in the forced nonlinear oscillator in a fractal space 3 establishment of governing equations in a fractal space [24 and 25]. it is worthy to observe that when µ=0, eq. (2) yields the duffing equation. simultaneously, the cancellation of the parameter β results in the vdp oscillator. in 1920, van der pol performed experimental studies on the dynamics of an oscillator electrical circuit. it was formulated by a second-order nonlinear differential equation, which was later known as the van der pol oscillator. alghassab et al. [26] examined the duffing oscillator together with the van der pol oscillator with a nonlinear damping term (dvdp). apart from the linearized analysis, they exploited the lyapunov theory in their analysis. zhihong and shaopu [27] presented a novel weak signal detection approach grounded on the dvdp oscillator. they found that the planned method was stronger than the duffing oscillator. under certain parametric conditions, gao and feng [28] deduced the first integrals without complicated computations of the dvdp. their methodology was constructed of a series of variable transformations with the preller-singer method. motsa and sibanda [29] presented a novel application of the successive linearized method to the classical dvdp. they obtained accurate values of frequency and amplitude. utilizing the homotopy analysis method, cui et al. [30] explored a high accuracy curve and solutions of the dvdp. their stability analysis adopted the floquet theory. additionally, their results were validated in light of the spectra analysis and bifurcation theory. moatimid [31] examined the stability criteria of the parametric duffing oscillator. the numerical calculations showed that the damper and the stiffness parameters have a destabilizing influence. the study of the parametric excited systems is actually imperative in light of external excited fields. wei et al. [32] examined the major response of the dvdp to collective deterministic and arbitrary parametric excitations. they employed the method of multiple time scales to govern the equations of amplitude and phase. dao et al. [33] considered the dvdp under the parametric exciting force. they evaluated the cases of a small parameter quasi-linear as well as the general case. han et al. [34] scrutinized the effects of a gradually changing parametric excitation of the dvdp equation. they examined the periodic bifurcation delay behaviors when there are no spinning points. their study showed that, at the first delay behavior, the delay time is decided by the initial time. the phenomenon of the entertaining of a generalized vdp was considered by kovacic [35]. by regulating the method of averaging, the approximate amplitude-frequency response was obtained. the inspiration of the controls of the reinstating and damping force in the fig. 1 basic property of the fractal vibration, x() = cosα 4 j.-h. he, g. m. moatimid, m. h. zekry existence of this occurrence was examined. shen et al. [36] investigated the primary and sub-harmonic simultaneous resonance of the duffing oscillator with a fractional order derivative. the approximate analytical solution of the resonance was obtained. furthermore, the rightness and acceptable exactness of the analytical solution were verified by a numerical simulation. recently, wu et al. [37] have examined periodic explanations of a harmonic forced duffing oscillator with a time-delay state feedback by using the incremental harmonic balance method. fractal differential equations are convenient in describing the numerous occurrences in porous media or on un-smooth boundaries. it is to be noted that the exact solutions are generally unreachable. therefore, these nonlinear equations should be solved by expanding approximate methods. many efforts were exerted to analytically treat these phenomena, ranging from the traditional perturbation methods to the multiple time scale technique. as known, the straightforwardness of the perturbation methods renders the nonlinear equations into linear ones. certainly, all these perturbation methods are primarily contingent on a small parameter. therefore, the presence of such a parameter is necessary to attain an approximate solution. frequently, the solution of such a problem might be limited by the presence of this parameter. fundamentally, the exact solution of this equation is difficult and a perturbation technique is required. all classical perturbation methods ranging from the traditional methods to the multiple-scales method depend mainly on a small parameter. consequently, the problem cannot be solved without the occurrence of such a parameter. in recent years, many promising and powerful methods have been introduced to construct analytical solutions for such equations. one of the pioneer methods is the homotopy perturbation method (hpm). this promising approach was first proposed by the chinese mathematician he [38]. homotopy is constructed in terms of a small embedded parameter. the hpm is an amalgamation of the traditional perturbation performance and the homotopy method, which has removed the dominant restrictions of the perturbation methods. this hpm brings about an actual quick convergence of the solution series in most cases. therefore, only after a few iterations, one reaches accurate solutions. hpm is a universal technique which can solve various kinds of nonlinear equations. this technique surpasses the traditional perturbation techniques. gómez-aguilar et al. [39] introduced the fractional equations of the mass-spring-damper system with caputo and caputo–fabrizio derivatives. their results indicated that the mechanical components display visco-elastic performances manufacturing temporal fractal diverse scales. yang et al. [40] considered boundary-value problems in a fractal heat transfer. the exact solutions of nondifferentiable type were attained by the local fractional differential transform method. recently, elías-zúñiga et al. [41] introduced a new methodology for descending the careful, steady-state solutions of fractal damped and forced differential equations. motivated by the aforementioned aspects, the aim of this examination is to investigate the dvdp since it is widely used in different fields. as previously shown, the dynamics of this equation have been widely investigated in recent years. therefore, the task now is to investigate an analytical approximate solution together with the stability profile. to develop the existing problem, the rest of the paper is organized as follows. a periodic solution, based on an adaptation of the expanded natural frequency analysis, is given in section 2. the stability analysis of the autonomous system, based on the linearized theory, is realized in section 3. additionally, the phase portraits are drawn in this section. in section 4, the stability analysis of the resonance as well as the non-resonance cases are accomplished by utilizing a combination of the homotopy concept and the multiple time forced nonlinear oscillator in a fractal space 5 scale analysis. finally, concluding remarks grounded on the consequences are summarized in section 5. 2. an approximate bounded solution through resources of the nonlinear frequency analysis as known, the classical hpm fails to solve the damping nonlinear oscillator, especially for the van der pol oscillator because it contains linear and nonlinear damping forces. therefore, the main purpose of this section is to attain an analytical bounded approximate solution for the dvdp equation. the the classical hpm does not enable us to satisfy this end properly. consequently, one necessarily seeks another new technique to realize a bounded solution of the considered equation. to this end, consider the homotopy equation that is given by eq. (3), where the natural frequency of the current model is the unity. 2 3( ( 1) cos ) 0; [0,1]x x x x x f t   + + − + −  =  (3) along with this approach, the time dependent function may be determined as follows:   = = 0 )();( n n n txtx  (4) the following stability examination will be constructed on the nonlinear frequency analysis given by moatimid [31]. in accordance with this approach, an expanded artificial frequency σ2 may be formulated as follows:   = += 1 2 1 j j j (5) where the parameters σj (j=1,2,…) will be determined from the invalidation of the sources of the secular terms. by combining eqs. (3) and (4), the homotopy equation may be rewritten as: 2 2 2 3 1 2 3( ( 1) ( ) cos ) 0x x x x x x f t       + + − − + + + −  = (6) it is worth mentioning that if the standard initial conditions are considered as approached by moatimid [36], the stiffness parameter µ will disappear. therefore, to preserve the presence of this parameter in the following analysis, an adjustment of the initial conditions must take place. consequently, a modification of the initial conditions may be written as follows: ,0)0( =x and   = = 0 )0( j j jx  (7) where the constants λi will be determined later as a function of the characteristics of the given equation. 6 j.-h. he, g. m. moatimid, m. h. zekry this will be done, together with the parameters σj as given in eq. (5), in light of the cancellation of the secular terms. taking the laplace transforms to eq. (6), while considering the initial conditions given in eq. (7), the result becomes 2 30 31 2 2 2 2 2 2 2 2 2 2 2 3 1 2 32 2 { ( ; )} .... 1 { ( ( 1) ( ) cos )}. t t l x t s s s s l x x x x f t s                    = + + + + + + + + − − − + + + −  + (8) employing the inverse transform of both sides of eq. (8), one gets: 2 30 31 2 1 2 2 3 1 2 32 2 ( ; ) ...... sin 1 { ( ( 1) ( ) cos )}t t x t t l l x x x x f t s                     −   = + + + + −      − − + + + −   +  (9) by utilizing the expansion of the dependent parameter x(t;ρ) as given in eq. (4), and by equating the coefficients of like powers ρ on both sides, one gets ,sin)(: 0 0 0 ttx     = (10) 1 2 31 1 0 0 1 0 02 2 1 : ( ) sin( ) { ( 1) cos }t tx t t l l x x x x f t s         −   = − − − + −   +  (11) characteristically, the uniform effective expansion emerges from the invalidation of the secular terms. for this purpose, one gets the following results: ▪ the cancellation of the secular terms, i.e. excluding the coefficients of the functions sin σt and cos σt gives  20 = (12) and  31 = (13) the second-order of eq. (9) yields 2 1 2 22 2 1 0 0 0 1 1 1 2 0 0 12 2 1 : ( ) sin { ( ( 1) 2 ) 3 }t tx t t l l x x x x x x x x x s          −   = − − + − − +  +  (14) ▪ once more, the cancellation of the secular terms needs  4/51 −= (15) and 2222 2 8/)9(  +−= (16) the third-order of eq. (9) results in 2 2 2 0 0 1 0 2 1 0 13 13 3 2 2 2 2 1 2 2 1 3 0 0 1 2 0 ( ( 1) ( 2 ) 2 )1 : ( ) sin 3 ( ) t t x x x x x x x x x x t t l l s x x x x x x x          −   − + + + −   = −    + − − + +     (17) forced nonlinear oscillator in a fractal space 7 ▪ once more, the cancellation of the secular terms necessitates 2 2 2 2 2 2 2 2 2 4 2 2 3 4 2 2 4 2 3 2 2 2 2 2 (81 5 )( ) (9 ) 72 (9 ) 24 (69 6 ) , 96 ( ) (9 ) f f              + − − − − + −  + = − − (18) and 4222 3 64/)9(9  += (19) it follows that the bounded solution x1(t) then becomes 2 2 2 2 21 1 2 2 2 2 2 2 2 1 ( ) sin ( ( 4 ( ))cos 3 ( )sin 4 cos 4 ( ) ( )cos3 ( )sin 3 ). x t t f t t f t t t                     = + − + − + − +  − − − − − (20) as mentioned before, for an easy follow-up of the analysis, the function x2(t) will be given in the appendix. therefore, the periodic analytic approximate solution of the governing equation of motion given in eq. (2) may be written as follows: 2 0 1 2 1 ( ) lim( ( ) ( ) ( ))x t x t x t x t    → = + + (21) by the inverse of two-scale transforms, t=√aτα, we can obtain the following solution of the fractal dvdp equation along with x0(√aτα)+ x1(√aτα)+ x2(√aτα). in fact, the bounded approximate solution as given in eq. (21) necessitates that the arguments of the trigonometric functions are real. for this purpose, combining eqs. (13), (16) and (19) into eq. (5), it follows that the synthetic frequency satisfies a certain characteristic equation. the calculation showed that this equation represents a polynomial of sixth order. once more, by means of hpm, one may obtain the approximate solution as follows: 2 4 6( ) , [0, 1]a b c    = + −  (22) where the constants a, b and c, may be listed as: )8/(9 22  −=a , ,)8(9/)819264( 22  −−+=b .)8(9/64 2  −=c (23) following a similar procedure as given above, an approximate solution up to the second-order, of the simulated frequency, may be written as: 2 2 10 sss  ++ (24) where the constants si are given as: 2/)(, 2/3 10 acbasas +== (25) and )11187( 8 1 2222/5 2 cacbabas ++= (26) in accordance with the real value of the expanded frequency, the approximate bounded solution requires the following criterion: .08 2 − (27) 8 j.-h. he, g. m. moatimid, m. h. zekry eq. (27) reads that the analytic approximate bounded solution cannot be realized where the stiffness parameter becomes negative. for more opportunities, the attained analytic approximate analytic solution must validate the numerical solution. for this objective, the fourth-order runge–kutta method (rk4) it utilized to confirm the approximate solution. therefore, fig. 2 is designed for demonstrating the analytic bounded solution given from the expanded frequency approach together with the numerical solution that is given by rk4. through this figure, the usage of the sample chosen system showed that the approximate value of the expanded frequency equals σ≌ 3.29. additionally, the calculations revealed that the initial condition becomes ẋ(0)=5.821. as shown from this figure, the amplitudes of the solution in the two cases are nearly coincident. this indicates a good agreement between the two methodologies. fig. 2 comparison of the approximate solution that is given in eq. (21) with the numerical one for the case: µ=0.2, β=2, f=0.05 and ω=0.4 additionally, the bounded solution of the considered dvdp will be graphed. this solution is given by eq. (21). simultaneously, it depends mainly on the value of the expanded frequency (σ) that is given in eq. (24). it is obvious that the validation of the value σ requires that 8β >µ2 as specified in the inequality (27). to prove the accuracy of the method, we depict in fig. 3, the percent relative error e(t)=x2(t)/(x0(t)+x1(t) +x2(t)), i.e., the stepwise error, from the results depicted in this figure, we assure the accuracy as well as the stability of raising the order of the approximation. fig. 3 the error function e(t) for a system having the particulars: µ=0.2, β=2, f=0.05 and ω=0.4 forced nonlinear oscillator in a fractal space 9 along with these restrictions, in what follows two graphs are plotted for some chosen systems to indicate the bounded solutions, as shown in figs. 4, and 5. fig. 3 displays the analytical bounded approximate solution for a sample chosen system. the inspection of this figure indicates the behavior of the solution, which is a bounded one. throughout this figure, the calculations showed that the approximate value of the expanded frequency equals σ≌ 8.716. additionally, the calculations illustrated that the initial condition developed as ẋ(0)=13.017. fig. 4 the approximate solution that is given in eq. (21) for a system having the particulars: µ=0.2, β=2, f=50 and ω=4 by contrast, fig. 4 represents a system whose particulars are held fixed, except for the values of the parameter f. as shown from this figure, the amplitude of the wave solution increases as f also increases. this shows a destabilizing influence of this parameter. physically, it is a convenient evidence because of the excitation of the external force. it should be noticed that the domain of the solution function is much shorter than the previous one. this occurs only to clearly indicate the influence. fig. 5 the approximate solution that is given in eq. (21), for a system having the particulars given in fig. 2 but for the variation of the parameter f. 10 j.-h. he, g. m. moatimid, m. h. zekry 3. linearized stability of the autonomous system the analysis here is concerned with the governing dvdp equation at which the external force is absent as (f→ 0). for this purpose, consider the following transformations: ,yx = and .yx  = (28) it follows that the governing equation that is given by eq. (2) may be represented by the following first -order nonlinear ordinary differential equations: ),( yxfx = and ,),( yxhy = (29) where f(x,y)=y and h(x,y)=-x+µ(1−x2)y−β x3. the details of the current approach are given in our previous recent work [42]. to restrict the length of the paper, they will be omitted. for more opportunities, some numerical calculations are given here to indicate the nature of the characteristic eigenvalues and consequences. therefore, the phase portraits are plotted. table 1 illustration of some sample values of the eigenvalues and the corresponding stability/instability sample chosen system fixed point roots of the eigenvalues stability/instability 1 0,1 ==  (0,0) pure imaginary .2,1 i= a stable center see fig. 6 2 1,1 =−=  ( 1,0) real, equal, and of different signs .414.1,414.1 21 −== saddle point (unstable) see fig. 7 3 5.0,2 =−=  1 ,0 2       real, distinct, and of different signs .29473.1 ,54473.1 2 1 −= = saddle point (unstable) see fig. 8 4 1,1 −=−=  (0,0) complex conjugates with negative real part )31( 2 1 1 i−= stable focus see fig. 9 unfortunately, the previous criterion ignores the contribution of the external excitation force. therefore, the following analysis will be included to indicate this influence throughout the multiple scale technique. forced nonlinear oscillator in a fractal space 11 fig. 6 the phase portrait of a system having the particular: β =1 and µ=0. fig. 7 the phase portrait of a system having the particular: β =−1 and µ=1. fig. 8 the phase portrait of a system having the particular: β =−2 and µ=0.5. fig. 9 the phase portrait of a system having the particular: β =−1 and µ=−1. 4. the stability analysis along the multiple-time scale technique as known the poincare-lindstedt method is an approach to postulate an asymptotic approximation of the periodic solution. on the other hand, it cannot be used to attain solutions that develop periodically on a slow timescale. the method of multiple scales is a more general approach that involves two key tricks. therefore, the method of multiple scales encompasses a technique that is used to establish a uniform valid approximation to the general solution of perturbation problems in which the solution depends on extensively diverse scales. nayfeh [43] divided the time into several time scales which will be considered to be independent variables. based on this conception, nayfeh formulated his theory. in agreement with the transformation of the derivatives, the original differential equation is divided into a set of partial differential equations. subsequently, it seems that 12 j.-h. he, g. m. moatimid, m. h. zekry the problem has been complicated. the results obtained from this approach have shown that there are disadvantages of this complexity, which are far overbalanced by more compensations. this technique does not only offer a uniform expansion, but it is also still valid to be applied to numerous extremely nonlinear problems in extensive several areas of practical applications and engineering. now, one seeks a uniform approximate solution eq. (2) in the following form: )(),(),();( 2 101100  ottxttxtx ++= (30) it should be noted that the number of the independent time scales generates the appropriate order of expansion. specifically, if the expansion is carried out up to the second-order, then only three scales t0, t1 and t2 are required. replacing eq. (30) as well as the transformation of the derivatives given below in the homotopy equation eq. (5), and then comparing the coefficient of like powers ρ, one finds the following equations: 0 2 0 0: ( 1) 0d x + = (31) and 0 02 2 3 0 1 0 1 0 0 0 0 0: ( 1) 2 ( 1) ( ) 2 i t i tf d x d d x x d x x e e    −  + = − − − − + + (32) where ....... 2 2 10 2 2 1 1 0 0 +++=+   +   +    ddd tdt dt tdt dt tdt dt dt d  and 2 2 2 2 0 0 1 1 0 22 2 ( 2 ) ..., n n d d d d d d d d tdt     + + + +   therefore, it is convenient to write the solution of eq. (32) in the following form: .,.)(),( 0 1100 ccetattx it += (33) where a(t1) is an unknown complex function of the time scale t1, and ..cc characterizes the complex conjugate of the foregoing terms. the governing equation of the complex function a(t1) is obtained in such a way that the time-dependent functions x1 must be of an oscillatory nature. substituting eq. (33) into eq. (32), one finds 0 0 032 2 3 0 1 1 1 ( 1) ( 2 ( 3 ) ) ( ) . ., 2 it it i t d x id a i a i a a e i a e fe c c      + = − + − + − + + + (34) where the function ā represents the complex conjugate of the function a. the uniform valid expansion of the function x1 requires a cancellation of the secular term. accordingly, in order to attain a uniform valid expansion, the secular terms must be omitted. the invalidation of these secular terms arises from the coefficients of the exponentials exp(±it0). therefore, the uniform valid expansion requires 0)3(2 2 1 =+−+− aaiaiaid  (35) forced nonlinear oscillator in a fractal space 13 eq. (35) is a nonlinear first-order differential equation with complex coefficients. it is known as the amplitude equation. occasionally, it is called the solvability condition. it follows that uniform solution of the particular solution of eq. (34) may be formulated as .. )1(2 1 )( 8 1 ),( 00 2 33 101 ccfeeaittx tiit + − ++=  (36) 4.1. stability analysis in the non-resonance case to complete the stability profile, along with the non-resonance case, one goes back to the amplitude eq. (35). essentially, this equation might be utilized to regulate the indefinite function a in terms of the time-independent variable t1. additionally, the stability presentation is contingent principally on the nature of this function. for this determination, one may apply the polar form solution as: )( 11 1)( 2 1 )( ti etta = (37) where the functions ζ (t1) and 𝛾(t1) are two real functions on the time scale 1t . substituting eq. (37) into eq. (35), and then associating the real and imaginary parts on the two sides, one gets the following differential equations: 3 1 82     −= dt d (38) and 2 1 8 3   = dt d (39) eqs. (38) and (39) are coupled first-order differential equations with real coefficients. as ρ→1, the parameter t1 will be transformed into t. certainly, the stability conditions need that both of the functions ζ(t) and 𝛾(t) become bounded functions. for straightforwardness, these equations may be solved by using a perturbation approach as follows: consider that the function ζ (t) may be perturbed around a steady-state response as: ζ (t)= ζ 0+ ζ 1(t). in this case, the zero-order solution yields 20 = (40) it follows that the first-order solution becomes tet  −=)(1 (41) in order to obtain a finite solution, the coefficient of the damped parameter (µ) must be positive. combining equations (40), (41) and (39), the first-order solution of the function 𝛾1(t) may be written as: 1 3 ( ) (1 ), 2 tt e    −= − (42) where the initial condition: 𝛾1(0)=0 is used. 14 j.-h. he, g. m. moatimid, m. h. zekry it is worthy to notice that the stability occurs regardless of the sign of the parameter β, provided that 𝜇 >0. as a final result in the case of non-resonant, the approximate periodic solution of the dvdp given in eq. (2) may be written as follows: 0 1 1 ( ) lim( ( ) ( ))x t x t x t   → = + (43) where x0(t) and x1(t) are the time-dependent functions obtained from eqs. (33) and (36), respectively. because of the presence of the non-homogeneous oscillatory term through the nonresonance case, the analysis will examine the resonance cases. therefore, the following subsection discusses the resonance case. 4.2. stability analysis of the resonance case as seen from the previous subsection, the non-resonance case fails to contain all parameters of the problem at hand during the stability criteria. consequently, the following discussion is devoted to presenting resonance cases. typically, only the resonance case is obtained. this case is definitely harmonic resonance. it is defined here as a sharp resonance. this benefit helps in introducing extra secular terms. accordingly, one may write += 1 (44) where 𝛿 is some detuning small parameter. it follows that the governing equation x1 has an extra secular term. inserting eq. (44) into eq. (34), one gets 0 01 32 2 3 0 1 1 1 ( 1) ( 2 ( 3 ) ) ( ) . ., 2 it iti t d x id a i a i a a fe e i a e c c     + = − + − + + − + + (45) the removal of the secular term from eq. (45) requires 02/)3(2 12 1 =++−+− ti feaaiaiaid  (46) really, the particular integral of eq. (45) is different from the previous corresponding equation as given in eq. (34). subsequently, the periodic approximate solution of the firstorder will be transformed. the particular solution, in case of the harmonic resonance, becomes ..)( 8 1 ),( 033 101 cceaittx it ++=  (47) eq. (46) is a first-order nonlinear differential equation with complex coefficients. as previously mentioned, its solution may be considered as: 1)()( 11 ti etta = , (48) where 𝜓 (t1) is a real and time-dependent function of the independent parameter t1. once more, as ρ→1, the parameter t1 will be converted to t. substituting eq. (48) into eq. (45), and then separating the real and imaginary parts, one gets: , 22 3     −= dt d (49) .0 2 32 3 =+− f  (50) forced nonlinear oscillator in a fractal space 15 the combination of eqs. (49) and (50) yields (2 3 ) . 6 12 d f dt         + − = − (51) eq. (51) is a linear differential equation in the function 𝜓. its solution may be written as ,)32( 6 1 32 )(               −+− − = texp f t      (52) where the initial condition: 𝜓 (0)=0 is used. it follows that the stability occurs in such a way that (2 2 3 ) / 0,  − −  (53) where eq. (44) is used. the instability criterion shown in the inequality (53) is independent of the parameter f. this happens because the parameter f lies away from the argument of the exponential function, as seen in eq. (52). on the other hand, as ω=1, the frequency of the exciting force has no implication in the stability criterion. this occurs because the natural frequency of the given dvdp is the unity. in the latter case, regardless of the sign of the parameter β, the stability criterion yields µ>0, which gives the same condition given in the non-resonance case. it is appropriate to corroborate this stability criterion numerically. therefore, in what follows, two stability profiles are plotted. in the following figures, the white areas symbolize the unstable regions, whereas the dark areas stand for the stable regions. fig. 10 represents the stability diagram, at which the parameter µ is plotted versus the parameter ω, to show the influence of the parameter β on the stability picture. as seen, the plane is partitioned into white and dark regions. this figure is interpreted as follows: for positive values of the parameter µ, the increase of the parameter β matches larger stable fig. 10 the stability criterion as given in the inequality (53), with variation of β 16 j.-h. he, g. m. moatimid, m. h. zekry regions. on the other hand, at the negative values of the parameter µ, the increase of the parameter β meets smaller stable regions. therefore, the parameter β plays a dual role in the stability configuration. it should be noticed that this influence does not appear in the preceding analysis. fig. 11 plots the stability profile, at which the parameter µ is plotted versus the parameter β to indicate the influence of the parameter ω on the stability picture. as seen, the plane is partitioned into white and dark regions. this figure illustrates that for positive values of the parameter µ, the increase of the parameter β meets larger stable regions. on the other hand, at negative values of the parameter µ, one finds that the increase of the parameter β matches smaller stable regions. once more, the parameter ω plays a dual role in the stability diagram. this is the first mechanism for the parameter ω in the entire analysis of the current paper. fig. 11 the stability criterion as given in the inequality (53), with variation of ω. 4. discussion and concluding remarks generally, a conservative vibration system must be free of the damping forces, and its amplitude will be fixed as t tends to infinity. as pointed out in ref. [44], the inertia force in a fractal space can be approximately expressed as 2 2 2 2 = +(1 ) d z d z dz p p dd d   − (54) where p is a function of the fractal dimension. in our paper, a weak damping effect is considered. its amplitude changes extremely slow, and we assume its value not to change for simplicity. forced nonlinear oscillator in a fractal space 17 by the inverse two-scale transform [45-47], x=z√c/b, t=√a τα, the above results are converted to the fractal cases. because of the great importance of the motivation of the dvdp in extensive applications in several areas in the sciences, many researchers have examined this problem. really, the dvdp is one of the significant mathematical models for demonstrating a dynamical system having a single unstable fixed point and a single stable limit cycle. the forced dvdp does not have global analytical first integrals. this classical technique of he-laplace does not have any authority to avoid the presence of the secular terms that have been assimilated during obtaining the approximate solution. accordingly, the resulting approximate solution has an increased amplitude with the increase of time. on the other hand, the bounded solution is accomplished by means of nonlinear frequency analysis. actually, this approach is easy, straightforward, powerful, and promising. it may be applied to analyze different types of highly nonlinear problems. through this method, a dispersion relation that guarantees the periodic nature of the solution is reached. once more, the hpm is utilized to realize an approximate solution of this characteristic equation. simultaneously, stability restrictions are addressed. the periodic solution, in light of the second method, is graphed. by means of the linearized stability analysis, the stability criterion of the autonomous system is carried out. finally, the multiple time scales together with the homotopy concept are used to govern those stability criteria of the whole system. the latter analyses include the resonance as well as the non-resonance cases. it is concluded that the multiple scales method is one the most powerful mathematical tool in analyzing the nonlinear oscillation system that arises in physical applications and engineering. overall, the concluding remarks of the present work may be summarized as follows: ▪ from the mathematical analysis of the nonlinear expanded frequency, one finds ▪ to maintain the presence of the cubic stiffness parameter µ in the stability criterion, a modification of the initial conditions has been made. ▪ the criterion of the analytical bounded approximate solution is given as 36β−µ2>0. ▪ in accordance with the multiple scales method, the following outcomes are reached: ▪ the stability criterion in the non-resonance case, regardless of the sign of the parameter β, requires that µ>0. ▪ the stability criterion, in the resonance case, needs µ(2ω−2−3β) β <0. ▪ because the natural frequency of the considered system is the unity, the 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(16) may be written as follows: 2 2 2 21 2 2 2 222 2 2 2 2 2 2 2 8 1536 ( )sin 36 3 ( ) sin3 (3 5 )sin5 sin 1 ( 1) 768 (2 ) 768 ( 2 ) sin( (2 )) sin( ( 2 )) ( )cos ( 1)( 1) (3 ) ( 1)( 1) ( 3 )1 ( ) 3072 8 36 3 ( ) ( f f x t t t t t f f t t x t t x x f t               + + + + − −  −   −  −  +  − + + − − + + +  −  − +  −  − − + = − + + −  − 2 2 2 2 2 2 , 4608 2304 ) cos3 cos cos( (2 )) 1 ( 1) ( 1) ( 3 2 ) 2304 cos( ( 2 )) ( 1) ( 3 2 ) f f t t t f t                    −  + + +   −  + − +  +      − +    − − +  +  2 4 2 2 2 21 2 2 2 2 24 96 (3 78 11 ) ( ) (813 216 ) 48 (13 ) (73 72 ) 1 ( 9)( 1) f f x t t t t      −  +  = − − + + + +    −  −  −  , and 2 4 2 4 2 2 2 22 2 2 2 288(( 183 10 ) 4 (9 10 ) ) ( ) 44 864 48 (24 ) ( 9)( 1) t x t t t f       − −  + + −  + = − + − + +  −  − . the function x2(t) that appears in eq. (33) may be written as follows: 2 2 2 21 22 23 24 4 2 2 2 2 2 2 2 2 1 cos sin cos3 sin3 (8 cos5 (3 5 )sin5 ) 1 96 ( 3 cos sin ) 96 ( 3 cos( ( 2 )) ( 2 )sin( ( 2 ))) ( ) sin 96 ( ) (3 )( ) 96 ( 3 cos( ( 2 )) ( 2 x t x t x t x t t t f t t f t t x t t f t                            + + + + + − + −  +   − − +  − − = + + + − − − − + +  + 2 2 2 , )sin( ( 2 ))) (3 )( ) t                   +    + −  2 2 2 2 2 4 2 2 4 21 4 2 2 2 2 2 ( (9 )( ) 36(111 18 ) ) (9 )( ) x f            = − − + +  −  − − ,                         −+− −−− +−−− −− = )18111(12 ))(9(29 ))(9(42 ))(9( 1 4224 22222 2222222 22222422     f x 2 2 23 4 2 2 2 9 (4 ( )) ( ) x f       = − − − , and 2 2 2 2 2 2 24 4 2 2 2 9 ( ( ) 2 ( 2 ( ))) ( ) x f        = − + − + − − . 10461 facta universitatis series:mechanical engineering vol. 20, no 2, 2022, pp. 279 306 https://doi.org/10.22190/fume220201009s © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper effect of static and harmonic loading on the honeycomb sandwich beam by using finite element method babak safaei1,2, emmanuel chukwueloka onyibo1, dogus hurdoganoglu1 1department of mechanical engineering, eastern mediterranean university, famagusta, turkey 2department of mechanical engineering science, university of johannesburg, gauteng, south africa abstract. the aim of this paper is to present a proposed honeycomb core shape and compare it with a normal hexagonal shape core in a sandwich beam. the sandwich cores are simulated in finite element with different materials; aluminum and epoxy-carbon with six layers are used as face sheet and the results are compared to those obtained theoretically. simulation of 3-point bending test is performed in commercial software ansys to verify the analytical results with the numerical ones. hence, for simplicity one layer of the skin is used on the equivalent model of sandwich for lesser computational time and more accurate evaluation. simulation of harmonic analysis of hexagonal core and proposed core shape is carried out in frequency domain to identify the core with less deformation under high frequency and it can withstand harmful effects. the proposed core shape model having the same cell numbers and material as the normal hexagonal model is compared with experimental results; it is observed that the proposed core shape model has good flexural stiffness, resonance, fatigue, and stress resistance at a higher frequency. key words: honeycomb sandwich, static analysis, harmonic analysis, finite element method received: february 01, 2022 / accepted march 09, 2022 corresponding author: babak safaei department of mechanical engineering, eastern mediterranean university, famagusta, north cyprus via mersin 10, turkey department of mechanical engineering science, university of johannesburg, gauteng 2006, south africa e-mail: babak.safaei@emu.edu.tr mailto:babak.safaei@emu.edu.tr 280 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu 1. introduction the honeycomb structure is a natural or man-made structure that has a honeycomb geometry to reduce the volume of material to attain minimum weight and minimal cost [1]. the geometry of the honeycomb structures can vary greatly, but the typical characteristic of such structures is the sequence of hollow cells created between the thin vertical walls. the honeycomb structures are widely used in almost every part of the manufacturing sector, because of their advantages, including extremely low weight/force ratios, which leads to lower weight, lower fuel usage. composite sandwich panels are used in aerospace and civil infrastructure applications because they have a higher flexural/transverse shear stiffness and, as a result, a higher corrosion resistance [2]. the cell arrangement is mostly hexagonal in section. researchers have experimented with a lot of shapes on sandwich structures, circular [3], triangular [4], square or rhombic [5], and pyramid lattice structures [6]. honeycomb normally has a regular hexagonal geometry (the sides are equal, the angles are all 120º and the cell walls are of the same thickness). due to this, their deformations can be easily analyzed and equations of orthotropic properties are obtained [7]. sandwich panel behaviors depend mainly on the geometric arrangement of core and facing materials [8]. honeycomb sandwich panels normally consist of two thin face sheets or skins and a lightweight thicker core. moreover, the core is made of different materials which depend on the desired mechanical properties needed. in some cases, sheet metal is often used as a skin material. the core is bound to the skin by brazing along with the glue or metal elements. burlayenko and sadowski [9] filled the honeycomb cores with foam to enhance the damage resistance and equally change the structural response of the sandwich structure, which is preferably used in waterproof, sound, and heat insulation. in addition, the sandwich core is known for low density, high compression, stiffness, and shear properties. manufacturing of honeycomb sandwich is mainly by corrugation, expansion, molding and 3d printing while the most adopted manufacturing method is expansion and corrugation [10,11]. skin materials are laminates of glass or carbon fiber reinforced thermoplastics or mainly thermoset polymers (unsaturated polyesters, epoxies), while commonly used composite is fiber-glass, carbon fiber reinforced plastic, and aluminum [12]. alhijazi et al. [13] presented and analyzed the elastic properties of luffa and palm natural fiber composites (nfc) with epoxy and ecoepoxy matrixes, with the influence of fiber volume fractions taken into account. however, honeycombs are known to have four common types, aluminum honeycomb, thermoplastic honeycomb, nomex honeycomb and stainless steel honeycomb; moreover, aluminum possesses the highest strength to ratio[14]. zaid et al. [15] concluded that the corrugated core sandwich structure has a better strength to weight ratio. the experimental eigenvalue responses of the epoxy-filled skew sandwich construction are computed for the first time by katariya et al. [16] to demonstrate the appropriateness of equivalent type single-layer higher-order theory for the analysis (including through-thickness stretching term effect). the strength increases exponentially relative to the core thickness while the increase in weight is negligible, and honeycomb is easily milled, routed, cut, edged, fastened and bonded, making it the first choice to reinforce any component structural area. mechanical property and energy absorption capability of aluminum honeycomb structures vary with impact velocity [17]. liu et al. [18] investigated the flexural characteristics of a stiffened foam core sandwich structure. however, the honeycomb core is known for its great stiffness. li et al. [19] stated that sandwich composites with truss effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 281 core materials have the best flexural stiffness and strength in bending deformation, which is ideal in structural parts. wang et al. [20] proposed a novel multilevel modeling approach for calculating young's modulus of polymers reinforced with graphene nano-platelets. chemami et al. [21] added orthogrid to improve the stiffness of soft honeycomb and thereby reduce the interfacial mismatch in the sandwich structure. barbaros et al. [22] analyzed how functionally graded materials (fgms) are prepared, manufactured, used, as well as their elastic properties. li et al. [23] studied the behavior of composite sandwich beams with glass fiber reinforced polymer (gfrp) face sheets and a balsa wood core in terms of flexural creep. ghanati and safaei [24] investigated the elastic buckling of regular hexagonal thin sheets made of homogeneous and isotropic materials under in-plane hydrostatic and uniaxial compression, with internal supports, translational and rotational elastic edge supports, and a mix of free, simple-support, and clamped boundary conditions. ribeiro faria et al. [25] investigated the dynamic behavior of sandwich beams with honeycomb cores loaded with magnetorheological gels and composite material skins under the effect of vibrations. hence, under various forms of mechanical loads, the static deflection, frequency, and transient responses of the multilayer sandwich shell (flat/curved) structure were estimated by katariya and panda [26]. fazilati et al. [27] showed that honeycomb structures are commonly used as energy shock absorbers due to their strong crashworthiness characteristics of high energy absorption potential and high strength-toweight ratio. ha et al. [28] presented the nap-core sandwich's fabrication, characteristics, and uses, with a focus on the sandwich's sensitivity and a special focus on the effect of symmetry in nap cores [29]. lu et al. [30] compared nomex and aluminum with sandwiches made from carbon fiber/epoxy and concluded that the bending strength of carbon fiber/epoxy honeycomb is greater. choosing the best honeycomb core for your application can be a very major factor. hence, not only mechanical property strengths and stiffness, but also the problem of the environment must also be considered. in order to study and observe the low velocity impact energy response of sandwich structures used for railways, sakly et al. [31] introduced finite element method (fem), to simulate ballast impacts, a high-speed and low weight test bench was designed. the graded sandwich shell structure's thermal eigenvalue responses are numerically assessed under varied thermal loadings while considering temperature-dependent characteristics as investigated by sahoo et al. [32]. he et al. [33] carried out experimentally and numerically the low velocity effect of carbon fiber face sheets and aluminum alloy cores. however, many researchers used carbon fiber as skin [34]. it comes in two weaves, unidirectional (all fibers are parallel) and bidirectional (fibers cross at a 90-degree angle). furthermore, recently many researchers carried out experiments on 3d printed cores [19,35–37]. moreover, studying the nonlinear behavior of the sandwich is necessary to truly understand the system's reaction. on an elastic substrate, the nonlinear forced vibrations of fgm sandwich cylindrical shells with porosities are investigated by liu et al. [38]. li et al. [39] studied nonlinear vibrations of fiber-reinforced composite cylindrical shells (frccss) with bolted joint boundary conditions, both theoretically and empirically, where the nonlinear amplitude-dependent material properties of fiber-reinforced composites (frcs) and partial bolt loosening boundary conditions are considered. a coupled nonlinear modeling for composite cylindrical shells is constructed using the modified donnell nonlinear shell theory and maxwell static electricity/magnetism equations to analyze nonlinear forced vibrations in multi-physics domains as investigated by liu et al. [40]. liu et al. [41] created a new method for solving nonlinear forced vibrations of functionally 282 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu graded (fg) piezoelectric shells in multi-physics fields. yang et al. [42] analyzed the geometrically nonlinear harmonically soft stimulated oscillation of size-dependent composite truncated conical microshells (ctcms) formed of fgms combined with magnetostrictive layers. according to yang et al. [43] the proposed research examines the geometrical nonlinear flexural response of arbitrary-shaped microplates with variable thickness composed of fg composites at small scales. the microstructural-dependent nonlinear stability behavior of micropanels under axial compression was studied by sahmani and safaei [44] combining moving kriging meshfree formulations with the thirdorder shear flexible shell model and modified strain gradient continuum mechanics. liu et al. [45] investigated a size-dependent numerical solution approach to assess nonlinear buckling and post-buckling of cylindrical microsized shells constructed of checkerboard randomly reinforced nanocomposites subjected to axial and lateral compressions. hence, a smart multifunctional sandwich plate with one core lightweight porous layer, two intermediate polymer/graphene nanocomposite layers, and two active faces constructed of piezoceramic material is presented by moradi-dastjerdi and behdinan [46]. in addition, they used an innovative and trustworthy method, to examine the free vibration behaviors of a multifunctional smart sandwich plate [47]. the mechanically excited layered skew sandwich shell panels' time-dependent deflection responses are estimated by katariya et al. [48] numerically using a general model built theoretically utilizing higher-order shear deformation theory, which includes the effects of large displacement. the frequency responses of a free vibrated composite sandwich panel structure are numerically studied by katariya et al. [49] by taking geometrical nonlinearity into account by using generalized green–lagrange strain kinematics. however, the mathematical models of static and fatigue response of honeycomb structures panel were developed by several researchers, due to the multiple numbers of coefficients and parameters based on their experiments, it is difficult to implement. due to this, ansys and abaqus were introduced by a few researchers to study the response and investigate the behavior of sandwich structures. upreti et al. [50] investigated the normal frequency of a honeycomb sandwich composite structure consisting of unidirectional carbon/glass fibers. the dynamic stiffness matrix approach is used to calculate the natural frequencies for out-of-plane free vibration of three-layer symmetric sandwich beams by gholami et al. [51]. jin et al. [52] studied a low-frequency vibrating lightweight cylindrical honeycomb sandwich structure. safaei [53] examines the damped vibrational behavior of a light sandwich plate subjected to a periodic force over a short period of time. kim and hwang [54] investigated the natural frequencies of honeycomb sandwich beams with embedded debonding or delamination between the face-layer laminates and the honeycomb core. abbadi et al. [55] stated that the presence of defects has no effect on the static behavior of the material. herranen et al. [56] investigated the strength of gfrp sandwich panels with different core materials through experimentation and simulation using the ansys apdl program. ansys is one of the favorite among designers, alhijazi et al. [57] found out that the ansys workbench is highly programmed and incredibly versatile, enabling users to customize it to their particular application or analysis. fatigue experiments were carried out to investigate the behavior of honeycombs with and without an artificial defect, therefore concluding that the drilling type defect influenced the fatigue life significantly compared to brinell. the fatigue life in w-direction is lesser than the l configuration. sandwich panels with defects (brinell and drilling hole) have a lower face cracking failure mode for both configurations. herranen et al. [58] studied finite element effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 283 analysis (fea) subjected to 4-point bending was conducted with ansys apdl v12.1 software for glass fiber plastic (gfrp) to investigate the strength. selvaraj et al. [59] used the ansys program to model the laminated composite single-core and double-core sandwich beams numerically; hence, the study findings explicitly demonstrated that their stiffness is higher than that of single-core composite sandwich beams, due to the greater stiffness provided by double-core composite sandwich beams. safaei et al. [60] developed a galerkin-based fem model to analyze the elastic stress field in a platelet reinforced composite subjected to axial load. this paper aims to investigate the static loading effect on the sandwich beam. its flexural stresses were clearly analyzed and harmonic analysis of the two core shapes was investigated; the amplitude of the response, phase angle and corresponding stresses were detailed. epoxy carbon ud (230gpa) was used as face skin, aluminum, honeycomb and san foam materials used as core. in addition to that, the aluminum alloy was later used as face skin while maintaining honeycomb as the core for validation. the ansys fem simulation results were validated analytically and it is in accordance with the stiffness of the sandwich beams equation. the natural frequency of honeycomb core and of mode shapes is illustrated. in order to gain a better insight into certain systems' deformation mechanisms, ansys is used as the finite element simulation and comparisons between both honeycomb cores materials deformation, stress resistance and mode shapes. lastly, improved flexural stiffness, resonance resistance and optimization check between the normal hexagonal honeycomb core and proposed shape core are confirmed. 2. problem modeling 2.1. problem objective the response of honeycomb structures under static loading with different cores, aluminum, honeycomb, and san foam with the same face sheet will be investigated in this study, in view of modal analysis and static structure. in order to find the most influenced output like equivalent stress, stiffness and natural frequency, the design of the experiment and parameter correlation is used in the ansys tool for understanding the parametric response. lastly, harmonic analysis of two different shaped geometry cores are investigated and stresses reviewed at their peak frequencies. 2.2. geometry and material properties the honeycomb core model is created using commercial solidworks software (igs file format) fig. 1(a) and imported to the ansys workbench for assembling in static structural. ansys composite preppost (acp) is used for the design of the face sheet, combining laminate layup from a composite material as shown in fig. 1(b). in addition, the honeycomb core is bounded on both sides by two face sheet (epoxy carbon 230 ud) as depicted in fig. 1(c). lastly, fig. 1(d) depicts the proposed honeycomb model design. 284 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu (a) (b) (c) (d) fig. 1 schematic of honeycomb sandwich structure; (a) hexagonal honeycomb; (b) laminate layup composite; (c) bounded surface of honeycomb core and face sheet; (d) proposed core honeycomb inclined edge model effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 285 2.3. material properties epoxy provides a solid, durable framework with good carbon fiber adhesion. pre-preg is a polymer consisting of "pre-impregnated" fibers. unidirectional (ud) carbon fiber has high bending power in a 0-degree orientation associated with the fibers against the progression of forces. the same configuration of the core is used for an isotropic solid sandwich made of epoxy carbon (thickness = 0.2 x 5mm). the lay ups used for orthotropic face sheets are [0/0/0/0/0]; therefore, the laminate thickness is 1mm on each skin. table 1 lists the material properties that were used in the sandwich beam structure analysis. aluminum, honeycomb, and san foam are used as the core in bending stress analysis, epoxy carbon ud (230 gpa) prepreg is used for the face sheet. lastly aluminum alloy and honeycomb are used as skin and core, respectively, for analytical validation. table 1 material properties used for skin and core material young’s modulus e (mpa) poisson’s ratio ν shear modulus g (mpa) density ρ (kg/m3) san foam 85 0.3 32.692 103 aluminum 7100 0.33 2669 2770 epoxy carbon 121000 0.27 4700 1490 honeycomb 1 0.49 1e-06 80 resin epoxy 3780 0.35 1400 1160 2.4. meshing of fea meshing is a method of transforming geometrical bodies into finite element entities. in this design of the sandwich structure, 8 nodded brick elements are used. in order to divide the domain into discrete elements, and to solve the equilibrium equation for each nodal location, the solid168 element type was used; this is a high order element that gives sufficient bending properties for this work. solid168 is an explicit dynamical element in a higher order, 3-d 10-node. it is ideal to model irregular meshes, for example those created from different cad/cam systems. the element is defined with a total of 10 nodes, each with three degrees of freedom: node x, y and z. finite meshing is performed to take account of the transition of tension between the core material and the face sheet, as seen in fig. 2(a). pham used different types of meshing methods and sizing [61]. moreover, meshing can affect the output results in order to be close to the analytical solution; fine meshing is adopted for the face sheet and program-controlled mesh for the honeycomb core in fig. 2(b). the number of nodes and elements used for both face skin and honeycomb core are shown in table 2. fig. 2(c) shows the meshing of a solid model of sandwich beam (all components and materials of the sandwich assigned by acp module in ansys). furthermore, figs. 2(d) and 2(e) show the meshing of the hexagonal core and proposed core model. however, the higher the nodes and elements, the longer computational time, but better the results, when compared to the exact solution. 286 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu table 2 number of nodes and elements name material nodes elements thickness (mm) face-sheet epoxy carbon 868 803 1 hexagonal honeycomb core aluminum alloy 42469 5680 10 proposed core aluminum alloy 45787 6750 10 (a) (b) (c) (d) (e) fig. 2 finite element meshing: (a) meshing of sandwich structure (hexagonal honeycomb core); (b) meshing of sandwich structure (proposed honeycomb core); (c) solid model meshing; (d) meshing of honeycomb core; (e) meshing of proposed honeycomb core 2.5. boundary conditions and loads the honeycomb sandwich structure has been assigned a fixed support at one end and displacement at the other end in a line of z-axis (horizontal to the sandwich); hence in order to mimic simply supported beam conditions, the top face skin is loaded with uniformly force in the static structural analysis of ansys workbench as shown in fig. 3(a). figs. 3(c) and (d) analysis are carried out to investigate the modal frequencies and harmonic effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 287 response, therefore treating the honeycomb core as a cantilever beam (one end fixed). lastly, as shown in fig. 3(e), concentrated loading of 3-point bending test of equivalent shell model of the sandwich structure is performed to validate the analytical calculation with the simulation method. (a) (b) (c) (d) (e) fig. 3 boundary conditions: (a) loads and boundary condition of static analysis of normal hexagonal honeycomb sandwich beam; (b) loads and boundary condition of static analysis of proposed honeycomb core sandwich beam; (c) harmonic response boundary condition of honeycomb hexagonal core; (d) harmonic response boundary condition of proposed honeycomb core; (e) 3-point bending test of sandwich and boundary condition of equivalent shell model of honeycomb sandwich structure 3. evaluation of behavior of sandwich beam in this section, approaches are identified to evaluate the stress in the core and skin of sandwich beams when they are subjected to flexural or shear loads. 3.1. discrete sandwich modeling the honeycomb sandwich beam is constructed with the complex hierarchical description of the core using a discrete modeling technique in the commercial software ansys. this method is made less preferable because it will take high computing time due to the cell details and a large number of cells involved in a full-scale honeycomb shell structure. 288 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu 3.1.1. equivalent sandwich model in order to equate the bending rigidity of the honeycomb sandwich structure and the equivalent model, the honeycomb sandwich structure is now replaced by an equivalent model in ansys acp, where all layers are identified and materials assigned. 3.2. theoretical analysis 3.2.1. sandwich beam the sandwich beam is made up of two thin skins with a thickness of t and a core with a thickness of c as shown in fig. 4. the overall depth of the beam is h and the width is b. the distance between the center lines of the upper and lower faces is d. d is the flexural stiffness of sandwich beam calculated by the following equation: 𝐷 = 𝐸𝑓 · 𝑏𝑡3 6 + 𝐸𝑓 · 𝑏𝑡𝑑2 2 + 𝐸𝑐 · 𝑏𝑐3 12 (1) where ef and ec are elastic modulus of the face and core, respectively. fig. 4 sandwich beam schematic model in three-point bending the honeycomb core's elastic modulus is significantly lower than that of the face sheet. the face is only 0.5 mm thick, significantly thinner than the core, which measures 4 mm in height. the following equations are met: 3 ( 𝑑 𝑡 ) 2 > 100,6 𝐸𝑓 𝐸𝑐 𝑡 𝑐 ( 𝑑 𝑐 ) 2 > 100 (2) as a result, the first and third terms of eq. (1) can be omitted [62]. therefore, the total flexural stiffness supplied by the faces around the sandwich's centroid axis is determined as 𝐷 = 𝐸𝑓 · 𝑏𝑡𝑑2 2 (3) when the sandwich beam is loaded by central point force p, according to the standard sandwich beam theory [62] , the usual stress on the face is at x = l/2, 𝜎𝑓 = p𝐿𝑧 4𝐷 𝐸𝑓 ( 𝑐 2 ⩽ 𝑧 ⩽ ℎ 2 ,ℎ 2 ⩽ 𝑧 ⩽ 𝑐 2 ) (4) effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 289 the shear stress in the honeycomb core is 𝜏𝑐 = p 2𝑏𝑑 (𝑐 2 ⩽ 𝑧 ⩽ 𝑐 2 ) (5) in the three-point bending, central deflection w of a sandwich beam is the sum of flexural deflection of the face sheets and shear deflection of the core. 𝑤 = 𝑇1 + 𝑇2 = − p𝐿3 48𝐷 − p𝐿 4𝐴𝐺𝑐 (6) where a = bd2/c,gc is the honeycomb core's shear modulus, and t1 and t2 are flexural and shear deflections at mid-span, respectively. 3.3. analytical validation in the literature, there are standard methods for calculating deflection and stresses. ashby [63] introduced an equation of sandwich beams that can also be used to validate the aluminum sandwich and honeycomb core of the equivalent model of the finite element. table 3 shows the validation and closeness between the simulation and analytical results of the sandwich flexural strength; hence, fem can be used for validation. analytical results were numerically obtained for the stress and strain of sandwich beam in the commercial ansys software shown in tables 3 and 4. as a result of that, their characterization and prediction of most mechanical properties are known; similarly, this analysis was performed by gibson [64]. table 3 comparison between simulation results and analytical calculated results stress (n/mm) strain deformation (mm) fem 37.703 0.00053102 0.4297 analytical 37.939 0.00053434 0.4254 3.4. fem the fea is carried out in order to estimate the maximum stress and deflection under the three-point bending test of the sandwich beam. the values of the parameter used, face thickness t = 0.5 mm, core thickness c = 4mm, span l = 108 mm, beam width b = 35 mm, axial load in strut p = 100 n, moduli of elasticity of core ec= 1 n/mm, moduli of elasticity of face sheet ef = 71000 n/mm, the overall thickness of sandwich h = 5 mm, the distance between center-lines of opposite faces d = 4.5mm and shear modulus of honeycomb gc = 70 n/mm. as shown in table 3, the nearness of both results are in agreement, fem still remains a valid method in analyzing and predicting the performance, strength and behavior of a component. a force of 100 n was used to keep the deformation of the sandwich beam in a linear state. 290 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu 4. results and observation 4.1. design of experiments 4.1.1. face sheet an optimization was carried out to determine the best laminate angle ply orientation with the same thickness to reduce the total deformation of the sandwich based on the loading criteria considered. three layer of the laminate each with two levels (0/90), a full factorial method is generated by the matlab code and imported to the ansys parameter set workbench for evaluation while there is a 45% reduction in deformation when laminate layup ply angle is at (90,90,90,90,90) degree. this is in accordance with the fact that the ud carbon fiber has its highest strength at 90 degrees along its fiber axis. moreover, this is due to the direction of load applied; therefore, (0, 90, 0, 90, 45, -45) degree is used; hence it gives reasonable stiffness irrespective of the direction of the applied load. 4.2. static analysis 4.2.1. deformation and stresses the sandwich beam is aligned in 3 point bending test setup, subjected to uniformly distributed load, and point load. moreover, when a sandwich beam is loaded, internal forces develop within it to maintain equilibrium. in this case the internal forces have 3 components, compression, tension and shear force. hence, the sandwich beam sagged and became shorter; therefore, the internal forces in the top became compressive, as a result, the bottom of the sandwich beam structure got longer and the normal forces in the bottom became tensile. the deformation and the stresses were observed as illustrated in fig. 5. fig. 5(a) shows that the sandwich beam is in compression on the top skin and tension on the bottom skin due to the applied load; the deflection occurs throughout the length of the sandwich beam, starting from the boundary conditions. the stresses keep increasing to the middle of the sandwich beam where the maximum bending stress and deformation occurred in the mid as shown in fig. 5(b). it is observed that the maximum deflection occurred in the middle of the sandwich beam. hence, fig. 5(c) shows the rate of deformation of the sandwich beam along its length, starting gradually from both supports and reaching its maximum at the mid. fig. 6 shows a stress plot along the top surface of the hexagonal honeycomb sandwich beam. fig. 6(a) shows the geometry path of the hexagonal sandwich beam created along the top surface. it was investigated that the maximum stress equally occurred at the center of the honeycomb sandwich beam, the hollow nature of the hexagonal made the unstable nature of the line graph to be in a zigzag manner as depicted in fig. 6(b). it is as a result of errors; moreover, the force is either tension or compression along the line. lastly, fig. 7 depicts the stress plot along the top surface of the proposed honeycomb core model. similarly, fig. 7(a) shows the geometry path of the proposed sandwich created along the neutral axis while fig. 7(a) depicts the stresses along the neutral axis. the simulation results plot is shown in fig. 8. where the hexagonal honeycomb was assigned 3 different materials and simulated in 3 point bending test, the same number of forces was applied and the stresses recorded for each amount of load. this was repeated for 3 materials, namely honeycomb, aluminum alloy and san foam. in fig. 9 the same simulation setup was carried out but the hexagonal shaped core was replaced with the proposed core model. all the results were evaluated and plotted. it effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 291 was observed that there was significant stress resistance in the proposed model and it is also important to note that aluminum alloy became more elastic than the honeycomb material used in the hexagonal shaped core. to investigate the rate of deformation on sandwich structure for hexagonal shaped core, the materials were assigned and total deformation was recorded and plotted as shown in fig. 10. the same was done for the proposed model in fig. 11. it was observed that the proposed model has reasonable resistance to deformation irrespective of the material used. finally, fig. 12 shows the effect of core thickness to equivalent von mises stress and shear stress. it was noted that when the core thickness is increased, both stresses decrease. the findings determined by means of the ansys workbench have been compared to the results calculated from the spreadsheet. the analysis of results reveals that the results of the fem method and the analytical approach have less than 1% difference. (a) (b) (c) fig. 5 deflection along the surface of sandwich structure under 3-point bending loading (a) total deformation of the sandwich beam; (b) maximum deflection at the middle of the sandwich beam; (c) directional deformation along the length of the sandwich beam 292 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu a) b) fig. 6 stress plot along the top surface of the hexagonal honeycomb structure (a) the geometry path on the top surface of the hexagonal honeycomb structure; (b) stress plot along the length of the hexagonal honeycomb structure a) b) fig. 7 stress plot along the top surface of the proposed honeycomb structure (a) the geometry path on the top surface of the proposed honeycomb structure; (b) stress plot along the length of the proposed honeycomb structure effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 293 fig. 8 stresses comparison of hexagonal core materials to loading fig. 9 comparison of proposed core materials to loading fig. 10 comparison of sandwich core deformation of hexagonal honeycomb core 294 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu table 4 shows fem and analytical results of the rate of change in length of the sandwich beam when load of 100 n is applied. the estimated stress of the sandwich beam can be obtained from the eq. (7), combining the stress on the skin and the core. table 5 depicts the comparison of stresses and validation analysis of the sandwich beam structure under 3-point bending test in analytical and fem. 𝜎𝑠 = 𝑀(ℎ/2) (𝐸𝐼)𝑒𝑞 𝐸𝑠 , 𝜎𝑐 = 𝑀(𝑐/2) (𝐸𝐼)𝑒𝑞 𝐸𝑐 (7) where h, c, (𝐸𝐼)𝑒𝑞 , ec and es are sandwich height, core height, flexural stiffness, core young’s modulus and skin young’s modulus, respectively. table 4 analytical and fem results at force of 100 n compared method strain fem 0.00053102 analytical 0.00053434 fig. 11 comparison of sandwich core deformation of proposed geometry core similarly, in order to investigate the proposed core model, the same parameter effect of applied load is maintained as used in hexagonal cores simulations. hence, the behavior is similar to a hexagonal shaped model with improved stress resistance about 10.9%, 12.3%, 42.8% for honeycomb, san foam and aluminum alloy, respectively. in this case of the proposed shaped geometry simulation, aluminum became more elastic than honeycomb and foam materials compared to the hexagonal shaped core model. response of the sandwich cores deformation in respect to the applied load. it is observed that aluminum core resists deformation more; it has very high compressive strength compared to foam honeycomb cores. this is similar to the work done by aslan et al. [65]. he has found that the structure made of carbon fiber and aluminum honeycomb has the highest edgewise compression strength, but poor bending strength, and the structure made of polyurethane foam and carbon fiber has the greatest flat-wise compression strength and bending strength. moreover, the deformation response of the proposed honeycomb model is similar to the effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 295 hexagonal honeycomb shaped core but deformation resistance improved by 32.9%, 12.9%, 34.4% for honeycomb, aluminum alloy and san foam core materials, respectively. stress comparison of sandwich cores materials, aluminum, san foam and honeycomb of the same parameter effect of the applied load on the hexagonal shaped core. the stress is more on the foam core and lesser in honeycomb, irrespective of its lowest young’s modulus and density. hence, honeycomb is more elastic than aluminum and foam (fig. 12). effect of core thickness to shear stress and equivalent von mises stress. accordingly, as the core thickness increases both stresses decrease, this is in agreement with potluri et al. [66] work. table 5 equivalent stress comparison of analytical and fem force [n] equivalent stress analytical [mpa] equivalent stress fem [mpa] 10 3.77 3.79 20 7.54 7.59 30 11.31 11.38 40 15.08 15.18 50 18.85 18.97 60 22.62 22.76 70 26.39 26.56 80 30.16 30.35 90 33.93 34.14 100 37.70 37.94 fig. 12 core thickness vs. stress 4.2.2. proposed honeycomb core inclined edge model the hexagonal honeycomb was modified in order to change its geometry to improve its strength and stiffness. fig 1(d) is designed from the traditional hexagonal honeycomb structure and modified, with no addition in cell number, both having the same thickness and material. hence, simulation was carried out in an ansys workbench with the same face sheet thickness and material properties; what changed was the core. it was discovered that the new shape has 79.9% deformation resistance; as a result, it is stiffer than a normal hexagonal structure. in general, for stiffer components honeycomb core inclined edge model should be used. fig. 13 illustrates the strain resistance comparison between the 296 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu hexagonal core sandwich and the proposed core model of the sandwich beam. hence, in order to investigate the extension of the hexagonal shaped core model and the proposed shaped core model in the ansys workbench mechanical, the sandwich beam was setup and assigned boundary condition in order to mimic the simply supported beam. the force ranging from 10 n to100 n was applied as a distributed load on the top surface of the sandwich structure and the corresponding strain recorded and plotted. the simulation was carried out with the same materials, parameters and loading; what changed was the shape of the core. it was observed that when the proposed core shape was used there was a significant reduction in the sandwich beam strain of the proposed shape core model. fig. 13 comparison between the hexagonal honeycomb sandwich beam and proposed shape core model sandwich beam 4.2.3. analysis and comparison table 6 compares the results of the fea and experimental data approaches done by boudjemai [67] employed for the modal analysis of the hexagonal honeycomb sandwich beam under clamped-free boundary conditions. for comparison, the first three frequencies have been chosen and results recorded; hence a good agreement between the results is obtained. furthermore, the proposed core sandwich structure was used with the same dimensions and materials as the experimental data provided by boudjemai [67] in ansys and the results were compared as depicted in table 6. moreover, the first and third mode of the sandwich are in bending mode; the proposed core model shows great improvement in frequencies that will cause deformation on the sandwich beam, having 12.8% and 30.7% increase for the first and third mode, respectively. effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 297 table 6 comparison of results frequency fea proposed model experimental[67] 1 133.45 151.75 134.5 2 309.69 308.21 311 3 818.51 929.69 711 4.3. modal analysis and natural frequencies determination of the vibration characteristics assumes determination of natural frequency and mode shapes of the honeycomb sandwich structure. the natural frequency of the sandwich beam system comprises the frequencies at which it vibrates with increasing amplitude. hence, the phenomenon is referred to as resonance. fig. 14 shows the mode shapes of the first 6 natural frequencies on the sandwich beam structure with core thickness of 10mm and face thickness of 1mm. fig. 15 depicts the mode shapes of natural frequency of honeycomb core and lastly fig. 16 shows the first 6 modes shapes of natural frequency of proposed core. in addition, it has a significant effect on the shear stresses. this is due to the differences in mass and stiffness which is caused by variation in core and skin thickness of the honeycomb sandwich beam. additionally this is in accordance with upreti et al. [50] and potluri et al. [66] work. table 7 shows natural frequencies of the honeycomb sandwich beam structure, hexagonal honeycomb core and proposed core model. mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 fig. 14 first 6 mode shapes contours of the sandwich structure 298 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 fig. 15 first 6 mode shapes contours of honeycomb core mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 fig. 16 first 6 mode shapes contours of proposed model effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 299 table 7 natural frequency mode natural frequency of sandwich beam (hz) total deformation of sandwich beam (mm) natural frequency of honeycomb core model (hz) total deform. of honeycomb core model (mm) natural frequency of proposed core model (hz) total deformation of proposed core model (mm) 1 196.53 480.87 113.11 759.24 322.50 342.64 2 622.53 525.62 227.38 704.13 359.64 283.82 3 661.36 795.44 618.00 911.05 1242.30 317.00 4 990.69 680.32 702.54 819.35 1317.30 425.78 5 1148.40 583.92 1078.10 685.84 1868.20 382.03 6 1547.20 794.27 1327.30 507.74 2489.20 366.01 4.4. honeycomb solid and equivalent model natural frequency acp in the ansys workbench was used to develop two types of honeycomb sandwich beam that give approximately the same results. a similar study was performed by rahman et al. [68]. the models are shown in fig.17. fig. 17(a) shows the equivalent model designed from acp where all the necessary material parameters are assigned. the solid model shown in fig. 17(b) is also designed from the acp and added solid layer; hence, using both is effective when considering the kind of boundary condition needed to apply. in addition, table 8 depicts the difference in their nodes and elements and table 9 shows the closeness of the equivalent model and solid model of sandwich beam mode results. the difference in their results is negligible; the equivalent model takes lesser time due to lower numbers of nodes and elements. a similar test was carried for the loading and deformation; it was found out that equivalent models run faster than the solid model. in addition, the solid model was added resin epoxy which made it stiffer than the equivalent model. in general, both models can be used for any analysis. this is in accord with rahman et al. [68] study on the efficiency of continuum, discrete and equivalent models. the complex cellular geometry is converted into a set of effective continuum properties that can be employed in most sandwich calculations using these models. (a) (b) fig. 17 honeycomb sandwich beam structure (a) equivalent model; (b) solid model 300 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu table 8 nodes and elements of both models model nodes elements solid 11928 10153 equivalent shell 994 923 table 9 natural frequency comparison of results of equivalent model and solid model mode equivalent frequency (hz) solid frequency (hz) 1 457.76 457.68 2 2166.1 2181.5 3 2362 2364.3 4 2817 2819.2 5 6858.5 6915.8 6 7692.5 7710.8 4.4.1. dynamic analysis and harmonic analysis in order to effectively understand the significance of hexagonal honeycomb and the proposed honeycomb model designs, harmonic analysis is carried out. additionally, the linear dynamics of forced frequency response analysis in the frequency domain of both models is observed. 4.4.2. hexagonal honeycomb model what is investigated is the behavior of the honeycomb structure under the steady-state sinusoidal (harmonic) loading at a given frequency. moreover, the time-history response of the honeycomb will be ignored in this analysis. therefore, what is taken into consideration is the honeycomb structure dynamic behavior in the frequency domain instead of the time domain. the two modes are spread over a frequency sweep from 0 to 900 hz, modes while higher frequencies were ignored for this analysis. hence, a single harmonic force of 100 n was applied in z-axis orientation as shown in fig 3(b). in order to obtain a reasonable resolution, the solution interval was made 50 and damping control ratio of 2%. fig 18(a) depicts the frequency response plot of the honeycomb core, the peak frequencies were observed but for more accurate results in capturing the peak, a cluster was added for more resolution. in general, the frequency at which peak deformation occurs is 108 hz, where the honeycomb core experiences the most deformation. however, with respect to this, contour results were created and evaluated fig 18(b), while fig 18(c) depicts the phase response plot. lastly, considering the equivalent stresses, the model is set to the peak frequency of 108 hz and the corresponding sweeping angle is the same as the directional deformation plot and was evaluated. fig 18(d), in conclusion, based on the design criteria more material or fgm should be added in the region of a yellowish part to bring the stresses within a preferred reach in order to enable the honeycomb structure to overcome fatigue, and other harmful effects of forced vibration. the input and output of harmonic analysis are both sinusoidal acting at the same excitation frequency. effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 301 (a) (b) (c) (d) fig. 18 harmonic response of honeycomb core: (a) frequency (hz) vs. amplitude (mm) plot (maximum displacement in amplitude is dictated); (b) 108 hz the maximum deformation occurred at the corresponding peak response and sweep angle 0f 149.79 °; (c) phase response plot; (d) equivalent stresses acting on the honeycomb core at the peak response 4.4.3. proposed shape honeycomb core model similarly, the same steps were carried out as in hexagonal harmonic analysis for the proposed model. fig. 19(a) shows the frequency response plot, fig. 19(b) depicts the contour plot on the core, fig. 20(a) illustrates the phase response plot of the proposed model and lastly, fig. 20(b) shows the point of maximum bending. hence, a damping ratio for the core is defined as 2%, under the harmonic system, a force of 100 n was applied in the z direction, the one with the frequency sweep from 0 to 900 hz. moreover, the frequency at which peak deformation occurs is 324 hz. the local results were recorded in table 10. 302 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu however, as a result of that, the peak displacement occurs at the first natural frequency from the modal analysis of 324hz while maximum spatial resolution is used. it is observed that the maximum value reported from the frequency response plot fig 19(a) corresponds to the peak value on the contour plot as shown in fig 19(b). table 10 analysis and results model maximum frequency (hz) phase angle (°) directional deformation (mm) equivalent stress (mpa) frequency sweep (hz) harmonic force (n) hexagonal core 108 149.79 535.70 29892.00 0 900 100 proposed core 324 93.614 69.76 21297.00 0 900 100 (a) (b) fig. 19 harmonic analysis of proposed honeycomb shape core: (a) frequency response plot (maximum displacement in amplitude is dictated); (b) directional deformation on the core at 324 hz (contour plot) effect of static and harmonic loading on honeycomb sandwich beam by using finite element... 303 (a) (b) fig. 20 harmonic analysis of proposed honeycomb shape core: (a) 93.61° reflect a phase shift between the sinusoidal input loads and corresponding response due to 2% damping assumed in the proposed honeycomb model; (b) equivalent stresses acting on the proposed honeycomb core at the peak response (point of maximum bending) 5. conclusion both a comprehensive analysis and simulation were carried out and the results are depicted and plotted, and essential observation was noted. the stresses reviewed from the peak frequencies of both cores are analyzed. it is observed that the proposed core model has fewer stresses and deformation at a much higher frequency than the traditional hexagonal honeycomb core. it is clear that the distribution of mass and the stiffness in the proposed core model are better than the hexagonal shaped core as compared with the experimental results. the core thickness and the face sheet thickness have a significant effect on shear stress, deformation and equivalent maximum stress. hence, increasing the thickness of the face sheets is the least efficient way to accomplish a rigid sandwich beam (not including the cost involved, especially when producing in large quantities) since it would increase the weight exponentially. nevertheless, it was well observed that the type of core material affects its elastic behavior and strength of the sandwich beam. the skin thickness is directly proportional to mass and inversely proportional to the sandwich beam 304 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu structure's elastic strain. as a result of the simulation compared, the proposed shape core has 79.9% extension resistance more than the hexagonal shape core. moreover, the equivalent sandwich beam model is the fastest to solve computationally, while the discrete sandwich beam model takes a longer time. in general, we can conclude that sandwich beam structure with carbon fiber skin layers is stiffer than metal skins, lighter and of weight lesser. in addition, it was investigated that ply angle orientation has a significant effect on strength of the sandwich beam, same with the number of layers, but increasing the layers will equally increase the overall weight, instead the core thickness should be increased. in the situation of modal analysis, it was found that an 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study on vibration response of a non-uniform beam with nonlinear boundary condition peng wang1, nan wu2, haitao luo3, zhili sun1 1school of mechanical engineering & automation, northeastern university, shenyang, china 2school of mechanical engineering, university of manitoba, winnipeg, canada 3shenyang institute of automation, chinese academy of sciences, shenyang, china abstract. forced vibration of non-uniform beam with nonlinear boundary condition is studied in this paper by proposing an iterative model combining adomian decomposition method and modal analysis. an exponentially tapered beam with a hardening nonlinearity spring boundary is simulated as a case study. the model accuracy is proved by comparing iteration results and analysis solutions with linear and weakly nonlinear boundary conditions. sin-weep nonlinear frequency spectrum is then obtained by the proposed model. the influence of boundary nonlinearity on the vibration response of non-uniform beam is analyzed. and the effect of different excitation amplitudes on nonlinearity in the vibration response is studied. the mathematical model and numerical solutions proposed in this paper can be used to solve and analysis broad vibration problems on general non-uniform beams with different nonlinear boundary conditions under various excitations. key words: nonlinear boundary, non-uniform beam, iterative method, adomian decomposition method, duhamel integral, vibration characteristics 1. introduction in practical engineering applications, non-uniform beam structures, including functionally graded material (fgm) beam structures are widely used because they can optimize weight and change strength by changing cross-sectional area and material properties. over the years, many experts and scholars have studied the dynamics of non-uniform beams [1-4], including vibration characteristic analysis (natural frequency and modal shape solution) and vibration utilization (energy harvester and stability analysis) [5-9]. at the same time, the dynamics of structures with nonlinear boundary (including multi segment linear boundary condition) is a hot topic that has been studied in recent years. in real life, spring [10,11], rubber bearing [12,13], received march 24, 2021 / accepted may 16, 2021 corresponding author: zhili sun, nan wu northeastern university, 110819, university of manitoba, r3t 2n2 e-mail: zhlsun@mail.neu.edu.cn, nan.wu@umanitoba.ca mailto:zhlsun@mail.neu.edu.cn 782 p. wang, n. wu, h. luo, z. sun concrete and elastic material and foundation [14] and soil [15,16] all have nonlinear characteristics, which have been taken into account in the static analysis of structures, but the nonlinearity of boundary is usually ignored in the dynamic analysis of structures. for the vibration governing equation of beam structure, the boundary condition determines the result of solution. in order to obtain more accurate structural dynamic characteristics, it is necessary to consider the nonlinearity of boundary condition, so it is meaningful to study the dynamic characteristics of non-uniform beam structure with nonlinear boundary. at present, there are many methods to solve the dynamic problems of uniform beam structure with nonlinear boundary condition. iterative method is one of the effective methods. ma and silva solved the fourth order differential vibration equation of uniform beam with nonlinear boundary by iterative method [17]. sun and wang studied the existence of monotone positive solutions for a class of elastic beam equations with nonlinear boundary conditions by monotone iterative method [18]. dang and huong reduced the nonlinear fourth-order problem to a series of second-order linear problems with linear boundary conditions, and proposed an iterative method for solving the fourth-order nonlinear equations of beam structures with nonlinear boundary conditions [19]. liu and li proposed a fast iterative method to transform ordinary differential equations into integral equations to solve nonlinear beam equations with nonlinear moment boundary conditions [20]. alves et al. studied the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions by monotone iterative method [21]. at present, the iterative method to solve the dynamic problem of uniform beam structure with nonlinear boundary condition is to solve the nonlinear equations about deflection. the deflection here has no practical physical significance, and it is more about solving a mathematical problem. in addition to the iterative method, the reproducing kernel method and the expansion method can be used to solve the nonlinear boundary problems of beam structures. in [22] and [23], the analytic approximate solutions of a class of fourth order differential equations with nonlinear boundary conditions are studied by using the iterative reproducing kernel method and reproducing kernel hilbert space method, respectively. geng and cui obtained a series solution to solve the singular nonlinear second-order periodic boundary value problem in the reproducing kernel space [24]. sedighi et al. redefined the preloading nonlinearity as the boundary condition of cantilever beam with a new exact equivalent function (ef) of preloading nonlinearity, and obtained the corresponding analytical solution by using the parameter-expansion method (pem) [25]. li and zhang studied the existence and uniqueness of monotone positive solutions for a class of elastic beam equations with nonlinear boundary conditions based on a new fixed point theorem of generalized concave operators [26]. sedighi et al. used the newly introduced equivalent function to model the preloaded nonlinear boundary conditions of the beam, and obtained the analytical solution of the nonlinear vibration equation of the beam by he’s parameter expanding method [27]. wang et al. considered the nonlinear fourth-order two-point boundary value problem (bvp) of elastic beam equation, and studied the existence, nonexistence and uniqueness of convex monotone positive solution of elastic beam equation with parameter  by using the fixed point theorem of cone expansion [28]. in addition to the above methods, there are also a series of methods to solve the dynamic problems of uniform beam with nonlinear boundary. song uses the theorem of infinitely many critical points to study the existence of infinitely many solutions to the boundary value problem of elastic beam deflection on a fourth-order nonlinear elastic foundation that depends on two real parameters [29]. mao et al. studied the nonlinear response of flexible structures with nonlinear general support conditions by using the modal revision method [30]. rahman et al. studied the forced nonlinear study on vibration response of a non-uniform beam with nonlinear boundary condition 783 vibration of euler bernoulli beam on nonlinear elastic foundation by using the improved multilevel residual harmonic balance method [31]. li and xu proposed an accurate fourier series method for vibration analysis of multi span beam systems under arbitrary boundary conditions [32]. bai and wang discussed the existence of positive solutions for a class of nonlinear fourth order beam equations by using fixed point theorem and degree theory [33]. wu proposed an iterative numerical method to solve the exact forced vibration of a cracked beam by considering the multiple modes and bilinear characteristics of the cracked beam [34]. for the non-uniform beam structure with linear boundary, especially for the general non-uniform beam with both cross-section width and thickness changing along the beam length, adomian decomposition method (adm) is a very effective method to solve its natural frequencies and mode shape functions [5]. the specific solving process of adm is to decompose the solution of the equation and express it in the form of infinite series sum [35]. it is not necessary to use linearization, perturbation, iteration, model simplification, difference method and finite element method to solve the vibration differential equation with nonlinear term. keshmiri et al. has used adm to do the research work on solving the nature (mode shape functions and nature frequencies) and energy harvesting of non-uniform beam with introducing taylor series [5-7]. but it is hard to obtain the general vibration response solution of non-uniform beam with nonlinear boundary because the boundary varies with time during the vibration process for the nonlinear vibration system. furthermore, the mode function and natural frequency also change with the boundary. even though, there are also some literatures on the dynamic problem of non-uniform beam with nonlinear boundary condition. based on the hamiltonian principle, lin derived the governing differential equations for the non-uniform time-varying elastic boundary conditions of the pre-twisted non-uniform beam with coupled bending vibration and solved them by the method of separation of variables [36]. kuo and lee et al. transformed the governing differential equations into a set of self-adjoint linear fourth-order ordinary differential equations with variable coefficients by using the perturbation method, and studied the static deflection of a general elastic end-constrained non-uniform beam on nonlinear elastic foundation bearing axial and transverse forces [37]. lee et al. extended the mindlin-goodman method and used the exact solution of the general elastic constraint non-uniform beam given by lee and kuo to study the dynamic and static responses of the non-uniform beam with non-uniform elastic boundary conditions [38]. tsiatas proposed a boundary integral equation method for solving nonlinear problems of non-uniform beams on nonlinear three parameter elastic foundation [39]. jang proposed an analysis method of moderately large deflections, which effectively considered the geometric nonlinearity caused by the moderately large deflections and the non-uniformity of the beam, and successfully and completely solved the moderately large deflections of the infinitely large non-uniform beam on the base of nonlinear elasticity question [40]. lohar et al. assumed that the beam is on an elastic foundation and bears uniformly distributed loads. considering different boundary conditions, static and dynamic parts are used to solve the large-amplitude free vibration behavior of axially functionally graded beams with different tapers [41]. these methods are used to transform or approximate the vibration governing equations of non-uniform beam with nonlinear boundary. the transformation process is complex, and some transformed equations cannot be solved directly, so numerical approximation method have to be used to solve them anyway. from the literature review, different models and methods were proposed to study the vibration characteristics of uniform and non-uniform beam structure with nonlinear boundary, but to the best of authors’ knowledge, the solution without transforming the vibration 784 p. wang, n. wu, h. luo, z. sun governing equation to solve the vibration response of general non-uniform beam (linearly or nonlinearly tapered along both width and thickness of beam cross-section) with a nonlinear taper function describing the variation of both width and thickness along the length direction (including non-uniform cylinder beam) and nonlinear boundary has not been proposed. in this paper, the adm method and an iteration process are introduced to solve, simulate and study the vibration response of non-uniform beam with nonlinear boundary condition. under solid spring properties, the vibration response of the non-uniform beam can be solved by adm and duhamel integral. in the numerical example section, the influences of different excitation amplitudes and frequencies on boundary nonlinearity are studied. when the properties of the spring are determined, the proposed iterative method can be used to solve the vibration response of the non-uniform beam with nonlinear boundary. 2. theoretical model in this section, general mathematical model and numerical progress describing and solving the vibration of a non-uniform beam sitting on a non-linear boundary are presented. the nonlinear boundary condition is considered to be with multi-linear elastic properties, while adm method is used to solve the natures of the non-uniformed beam with different linear elastic foundation, and the iteration numerical method is applied to solve the vibration response considering the linear boundary condition in each short time iteration step. 2.1. vibration model of non-uniformed beam sitting on elastic foundation the equation of free vibration of a non-uniform beam without considering damping is given below, 2 2 2 2 2 2 ( , ) ( , ) ( ) ( ) 0 w x t w x t a x ei x t x x      + =      , (1) where x is the position variable along the beam length, ρ is the mass density, a(x) is the cross-sectional area, w(x, t) is the deflection function, t is time, e is the modulus of elasticity, i(x) is the second moment of area. the general boundary condition of the beam is defined as, 2 1 2 0 2 1 2 0 ( , ) ( , ) ( ) 0 ( , ) ( , ) + ( ) 0 l x t x w x t k w x t ei x x x w x t w x t k ei x x x = =     − =           =     , (2) 2 22 2 2 2 ( , ) ( ) ( , ) 0 ( , ) ( , ) ( ) 0 l x l t x l w x t ei x k w x t x x w x t w x t k ei x x x = =     + =           − =     , (3) study on vibration response of a non-uniform beam with nonlinear boundary condition 785 where 1lk and 2lk are the tension spring constants on the left end, 1tk and 2tk are the torsional spring constants on the right end. in this study, a sample non-uniform beam structure with exponentially varied circular cross section sitting on a nonlinear tension spring foundation is analyzed and shown in fig. 1. at the left fixed position, kl is a tension spring, which has a nonlinear property. other non-uniform beam with general nonlinear boundary conditions can also be solved by the progress described below. with the multi-linear assumption of the non-linear spring stiffness, the vibration in each linear domain of kl can be solved by treating the system as a linear one. the whole nonlinear vibration response can be solved by iteration process with small iteration time step, while in each time step, the system is considered as linear with non-change spring stiffness constant. d left end free end d l 0 l lk fig. 1 main and right views of an exponentially increasing tapered beam with nonlinear boundary condition for the above sample structure, its boundary condition can be expressed as below, 2 2 0 0 ( , ) ( , ) ( ) 0 ( , ) 0 l x x w x t k w x t ei x x x w x t x = =     − =           =   , (4) 2 2 2 2 ( , ) [ ( ) ] 0 ( , ) ( ( ) ) =0 x l x l w x t ei x x w x t ei x x x = =   =           . (5) by using mode superposition method, w(x, t) in eq. (1) can be decomposed into two parts, 1 ( , ) ( ) ( )i i i w x t w x q t  = = , (6) where wi(x) is the i-th mode shape and qi(t) is the i-th corresponding generalized coordinate of the free vibration response or external force. substituting eq. (6) into eq. (1), one ordinary differential equation is obtained as, 786 p. wang, n. wu, h. luo, z. sun 2 4 3 22 2 4 3 2 ( )( ) ( ) ( ) ( ) ( ) ( ) +2 + =0 ( ) ( ) ( ) i i i i i d i xdi x d w x d w x d w x a x w xdx dx i x i x ei xdx dx dx  − , (7) where ωi is the i-th natural frequency. in order to solve the above equation, adm is applied [5]. the operator form of the eq. (7) is rewritten as,   2 ( ) ( )'( ) ''( ) ( ) 2 '''( ) ''( ) ( ) ( ) ( ) i x i i i i a x w xi x i x l w x w x w x i x i x ei x  = − − + , (8) where lx is the fourth order differential operator. lx -1 is applied on the both sides of eq. (8) at the same time, where lx -1 is the fourth-order integral operator. 2 3 1 2 3 4 1 2 ( ) 2! 3! ( ) ( )'( ) ''( ) 2 '''( ) ''( ) ( ) ( ) ( ) i i x i i i x x w x c c x c c a x w xi x i x l w x w x i x i x ei x  − = + + +   − + −    , (9) where c1 to c4 are constants that can be determined by boundary conditions, the detailed progress defining c1-c4 can be found in [5]. wi(x) is written in series form, 0 ( )= ( ) ii k k w x w x  =  , (10) where k is the number of terms in series form. the larger the value of k is, the more accurate the solution is. a precise solution is often obtained with very small values of k [35]. substituting eq. (10) into eq. (9), we have 2 3 1 2 3 4 0 1 2 0 0 0 ( ) 2! 3! ( ) ( ) '( ) ''( ) 2 '''( ) ''( ) ( ) ( ) ( ) i i i i k k k k x k k i k k x x w x c c x c c a x w x i x i x l w x w x i x i x ei x    =    − = = = = + + +      − + −          . (11) for each term in the series, we can have 2 3 0 1 2 3 4( ) 2! 3!i x x w x c c x c c= + + + , (12) 1 2 1 ( ) ( )'( ) ''( ) ( ) 2 '''( ) ''( ) 0 ( ) ( ) ( ) i i i i k k x k k i a x w xi x i x w x l w x w x k i x i x ei x  − +   = − + −     . (13) substituting eq. (12) and eq. (13) into eq. (11), the i-th mode shape function, wi(x), can be obtained. with different boundary conditions and structural design, the mode shape function from adm can be different and hence not given here. the natural frequency can be obtained by introducing the mode function into the boundary condition and solving the study on vibration response of a non-uniform beam with nonlinear boundary condition 787 eigenvalue problem. the detailed solving process of the mode shape functions and nature frequencies of the non-uniform beam with general linear boundary conditions can be found in references [5-7] and is hence not provided here. after the natures (natural frequencies and modes shapes) of the structure vibration are solved by free vibration analysis and adm, the forced vibration response can be solved using the modal analysis. the forced vibration governing equation of non-uniform beam structure considering damping under action of f(x, t), which is from a base motion, y sin(  t), is given below, 2 2 2 3 2 2 2 2 2 ( , ) ( , ) ( , ) ( ) ( ) + ( ) ( ) ( , ) =( ) sin( ) w x t w x t w x t a x ei x c x i x f x t t x x x t a x y t          + =        , (14) where c(x) is the strain rate damping coefficient. y is the amplitude of the base displacement and  is the angular frequency of the base vibration. substituting eq. (6) into eq. (14), the following equation can be obtained, 2 2 1 2 22 2 2 2 2 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) sin( ). i i i i i i i i i d q t a x w x dt d w x dq t d w xd ei x q t c x i x a x y t dtdx dx dx  =   = = +   =           (15) based on the understanding from the free vibration governing equation, for the i-th mode of free vibration, we have 22 2 2 2 ( ) ( ) ( ) ( )i i i d w xd ei x a x w x dx dx     =    , (16) with the understanding from the relationship given in the above equation, both sides of eq. (15) multiplied by wj(x) (i=j) and integrated from 0 to l in space domain leads to, 2 20 1 2 22 2 2 20 1 1 2 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) sin( ) ( ) ( ) l i i j i l i i i i j i i l j d q t a x w x w x dx dt d w x dq t d w xd ei x q t c x i x w x dx dtdx dx dx y t a x w x dx     =   = =   +     =     . (17) since in general, the damping function does not possess the orthogonality property, it is assumed that the structural damping is in the form of c(x)=αe where α is a constant. according to the orthogonality of normal vibration modes, eq. (18) can be obtained, while αωi 2=2ξiωi is defined, 22 2 0 2 2 0 sin( ) ( ) ( )( ) ( ) 2 ( ) ( ) ( ) ( ) l ii i i i i i i l i y t a x w x dxd q t dq t q t f t dtdt a x w x dx       + + = =   , (18) where ξi is the modal damping ratio of the corresponding i-th order natural mode. 788 p. wang, n. wu, h. luo, z. sun eq. (18) can be solved by duhamel integral, and the final time domain solution for i-th mode is, ( ) 0 2 0 5 2 2 2 3 2 2 2 2 2 0 2 2 2 2 2 2 2 2 1 ( ) ( ) sin( ( )) ( ) ( ) = ( (2 2 ) ( ) ) ( ) ( ) ( (( )sin( ) 2 cos( )) (( )si i i i i i i i i i i i i i i i t t i i d d l i l d i i d i i d i t i i d d d i i d d i i d q t f e t d y a x w x dx a x w x dx e t t − − − =   − −  + − + + − − − + − + − +                                       n( ) 2 cos( )))i it t−    , (19) where id is the damped frequency corresponding to the i-th vibration mode, 2= 1id i i   , fi() is the force coefficient corresponding to the i-th vibration mode. according to eq. (12), eq. (13) and eq. (19), the response of the beam structure is, 1 ( ) 0 1 ( , ) ( ) ( ) ( ) ( ) sin( ( ))i i i i i i i t ti i d i d w x t w x q t w x f e t d         =  − − = = =   −    . (20) 2.2. iteration process considering nonlinear boundary condition an iteration process is proposed in this section with multi-linear assumption for the nonlinear foundation of non-uniform beam. it is assumed that the elastic foundation in each short period iteration step is linear with constant stiffness constant, while the stiffness constant changes between different iteration steps. the premise of iterative method is that at t=0 s, the initial condition of vibration is known, and there is no force applied to the beam. at this time, the beam is considered to be at rest, and the deflection and velocity at any position of the beam are, ( , )=0 ( , ) =0 w x t dw x t dt      . (21) the step length of each iteration is δt=tn+1-tn, 1≤n<∞, and n is the number of iteration steps. the initial condition for time t1 (t1=0 s) is, 1 1 = ( , )=0 ( , ) =0 t t w x t dw x t dt      . (22) because some variables, like time, vibration natural frequencies, mode shape functions, in each iteration step are different in the process of iteration of the nonlinear system, they are defined by iteration step subscripts. the subscript 1 represents these variables in the first time period/iteration step in period t1-t2. the subscript 2 represents these variables in the second time period/iteration step in period t2-t3 and so on. the subscript n represents study on vibration response of a non-uniform beam with nonlinear boundary condition 789 these variables in the n-th time period/iteration step in period tn-tn+1. detailed specific presentation method of variables in the iteration section is shown in table 1. i and j in table 1 represents the i-th and j-th vibration modes, respectively. table 1 representation of variables in the iterative process time period variable (1≤i<∞, 1≤j<∞) t1≤t≤t2 1( , )w x t total vibration response 1 ( ) i w x i-th mode shape function 1 ( ) i q t , 2 1 ( ) i t t dq t dt = i-th corresponding generalized coordinate of the external force and its derivatives 1i  , 1i d i-th natural frequency and damping frequency 1 1( ) i f t + force coefficient corresponding to the i-th vibration mode tn≤t≤tn+1 ( , )nw x t total vibration response ( ) inw x , ( ) jnw x i-th and j-th mode shape functions ( , ) nfreew x t free vibration response in , ni d i-th natural frequency and damping frequency ina , inb coefficients in the i-th corresponding generalized coordinate of free vibration ( ) in nf t + force coefficient corresponding to the i-th vibration mode ( ) ni free nq t , ( ) ni n free t t dq t dt = i-th corresponding generalized coordinate of free vibration and its derivatives at time tn 1( ) in nq t + , 1 ( ) i n n t t dq t dt += i-th corresponding generalized coordinate of the external force and its derivatives at time tn+1 ( ) ni freeq t , ( ) ni forceq t i-th corresponding generalized coordinate of free vibration and external force during tn-tn+1 when the time is in t1-t2, there is no free vibration and the response of the forced vibration is, 1 1 1 1 1 1 1 1 1 1 1 ( ) 1 1 10 1 ( , ) ( ) ( ) ( ) ( ) sin( ( )) i i i i i i i i i i d t t t t d w x t w x q t w x f t e t t d           = = − − − = = +    − −    ( 1 2t t t  ). (23) to simulate the nonlinearity of spring stiffness, kl is divided into n sections by different deflection intervals at the spring boundary location. it is considered that the value of kl corresponding to each deflection interval is the same, and kl, the mode shape functions and 790 p. wang, n. wu, h. luo, z. sun natural frequencies in different deflection intervals are different. by bringing each kl into the boundary condition, following the progress described in section 2.1, mode shape functions and natural frequencies can be obtained for the n-th time step (tn-tn+1) during iteration. the value of kl in the boundary condition of solving wni(x) at time tn-tn+1 is the value of kl corresponding to the deflection at the spring boundary location, wn(x,t), x=0 for the sample structure shown in fig. 1, at time tn. to clarify the iteration progress, we start the derivation from the second time step, t2-t3, while the first step vibration solutions is only from the duhamel integral assuming the structure is at rest before excitation as described in eq. (23). during the period, t2-t3, when the value of kl does not change from the one in the previous period, t1-t2, we have w1i(x) = w2i(x), 1i(x) = 2i(x), the free vibration response from the initial condition at t2 is, 2 2 2 22 2 2 1 ( , ) ( ) ( cos sin )i i i i ii i t free d d i w x t w x e a t b t      − = =  + ( 2 3t t t  ), (24) where 2i a and 2i b are determined by 1 2( ) i q t and 1 ( ) i dq t dt at t=t2. 2 2 2 22 2 2 2 1 2( cos sin ) ( )i i i i ii i t d de a t b t q t     − + = ; (25) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 ( cos sin ) ( ) ( sin cos ) i i i ii i i i ii i ii i i t d d i d t d d i d t t t t e b dq t t t e a dt               − − = − − + = ; (26) 2i a and 2i b can be obtained by the above formula, 2 2 2 2 2 2 2 1 2 1 2 2 1 2 2 2 ( ) (( ( ) ) cos( ) ( )sin( )) = i ii i i ii i i i i t i d d d t t d dq t e q t t q t t dt b         =  + + ; (27) 2 2 2 2 1 2 2 2 2 2 ( ) sin( ) = cos( ) i i i i i i i t d d q t e b t a t      − ; (28) during the period, t2-t3, the total vibration response is, 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 ( ) 2 2 20 ( , ) ( ) ( )= ( )( ( )+ ( )) = ( )( ( cos sin ) 1 ( ) sin( ( )) ) i i i i i i i i i ii i i i i i i free force i i t d d i t t t t d d w x t w x q t w x q t q t w x e a t b t f t e t t d               = =  − = − − − =  + + +    − −     ( 2 3t t t  ). (29) where qfree2i (t) is i-th corresponding generalized coordinate of free vibration during t2-t3, qforce2i (t) is i-th corresponding generalized coordinate of the external force induced vibration during t2-t3. study on vibration response of a non-uniform beam with nonlinear boundary condition 791 on the other hand, when the value of kl at time t2-t3 is different from that of t1-t2 with relative large boundary deflection, w(0, t), at t=t2 reaching to a different deflection interval, the corresponding vibration mode shape functions and generalized coordinates of free vibration at time t2-t3 also change. in order to calculate the generalized coordinates of free vibration at time t2-t3, the deflection function at time t2 need to be reassigned, 2 21 2 2 2 2 1 ( , ) ( , )= ( ) ( ) i i free free i w x t w x t w x q t  = =  (30) where w1(x,t2) is the deflection at time t2 from the previous iteration period, t1-t2.. wfree2(x,t2) is the response of the free vibration at time t2, 2 ( ) i w x is the mode shape function of the structure starting at time t2, 2 1( ) ( ) i i w x w x , qfree2i (t2) is the i-th generalized coordinate of free vibration at time t2. both sides of eq. (30) are multiplied by 2 ( ) j w x and integrated on 0-l giving 21 2 2 2 2 20 0 1 ( , ) ( ) ( ) ( ) ( ) j i ji l l free i w x t w x dx w x q t w x dx  = =   . (31) considering the significant vibration natural frequencies and mode shapes changes due to the different boundary conditions with varying kl and the inevitable error in the mode shape functions derived by adm method, when 2 2( ) i freeq t (i=1,2,3) are solved, mode shape functions 2 ( ) j w x (j=1,2,3) are introduced in eq. (31), respectively. through eq. (31), if only the generalized coordinates of first three orders are taken consideration as an example, we can then obtain, 2 3 13 1 1 3 1 2 1 3 2 3 1 3 1 1 2 1 3 2 2 2 1 2 20 0 2 1 2 2 20 0 2 2 2 2 2 2 2 20 0 0 0 2 2 2 2 2 2 2 20 ( ) ( ( ) ( , ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) / ( ( ) ( ) ( ) ( ) ( ) ( ) l l free l l l l l l q t w x w x t dx w x dx w x w x t dx w x w x dx g w x w x dx w x w x dx g w x w x dx w x dx w x dx w x dx h w x w x dx w x w x dx = − + − −         1 3 2 3 1 0 0 0 2 2 2 2 2 2 20 0 0 ( ( ) ( ) ) ( ) ( ) ( ) ) l l l l l l l w x w x dx h w x w x dx w x dx− +        , (32) 2 22 3 2 2( ) ( )free freeq t g h q t= +  , (33) 2 1 2 1 21 2 2 1 33 1 2 2 1 2 2 2 20 0 2 2 20 2 20 ( ) ( ( ) ( , ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ) ( ) l l free free l free l q t w x w x t q t w x w x dx q t w x w x dx w x dx = − −      , (34) where 792 p. wang, n. wu, h. luo, z. sun 1 2 1 2 1 2 1 1 2 2 2 2 1 2 2 2 2 1 20 0 0 0 2 2 2 2 2 2 20 0 0 ( ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ( ) ( ) ) l l l l l l l w x dx w x w x t dx w x w x dx w x w x t dx g w x dx w x dx w x w x dx − = −        , (35) 1 2 1 3 2 3 1 2 1 1 2 2 2 2 2 2 2 2 20 0 0 0 2 2 2 2 2 2 20 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) l l l l l l l w x w x dx w x w x dx w x w x dx w x dx h w x dx w x dx w x w x dx − = −        , (36) 21 2( )freeq t , 22 2( )freeq t and 23 2( )freeq t are the first 3-th generalized coordinates of free vibration at time t2. we assume the free vibration response function during time t2-t3 to be, 2 2 2 2 1 2 2 2 2 2 1 ( , ) ( ) ( ) = ( ) ( cos sin ) i i i i i i i i i free free i t i w x t w x q t w x e a t b t      =  − = =   +   ( 2 3t t t  ), (37) where a2i and b2i here are determined by the initial conditions of the current iteration step at t=t2, 2 2( ) i freeq t and 2 2 ( ) i free t t dq t dt = , which can be obtained from eqs. (32-36). 2 2 2 2 22 2 2 2 2( cos sin ) ( )i i i ii i i t d d freee a t b t q t     − + = ; (38) 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 ( cos sin ) ( ) ( sin cos ) i i i ii i i i ii i ii i i t d d i d freet d d i d t t t t e b dq t t t e a dt               − − = − − + = ; (39) a2i and b2i can hence be obtained by the above formula, 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) (( ( ) )cos( ) ( )sin( )) = i ii i i i i i i i i freet i free d d free d t t d q t e q t t q t t dt b         =  + + ; (40) 2 2 2 2 2 2 2 2 2 2 ( ) sin( ) = cos( ) i i ii i i i t free d d q t e b t a t      − ; (41) when kl changes from the previous iteration, during the second time step, t2-t3, the total vibration response is hence, study on vibration response of a non-uniform beam with nonlinear boundary condition 793 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 ( ) 2 2 20 ( , ) ( ) ( )= ( )( ( )+ ( )) = ( )( ( cos sin ) 1 ( ) sin( ( )) ) i i i i i i i i i ii i i i i i i free force i i t d d i t t t t d d w x t w x q t w x q t q t w x e a t b t f t e t t d               = =  − = − − − =  + + +    − −     ( 2 3t t t  ). (42) for general vibration solution during the iteration, when the time is tn-tn+1, if the value of kl is equal to the one in previous period tn-1-tn, the total vibration response is hence, 1 1 1 ( ) 0 ( , ) ( ) ( )= ( )( ( ) ( )) = ( ) ( ( cos sin ) 1 ( ) sin( ( )) ) i i i n ni i i ni i i n i ni i n i n ni i ni ni n n n n free force i i t n n d n d i t t t t n n d n d w x t w x q t w x q t q t w x e a t b t f t e t t d               = =  − = − − − = +  + + +    − −     ( 1n nt t t +  ). (43) where ( ) ni freeq t is i-th corresponding generalized coordinate of free vibration during tn-tn+1, ( ) ni forceq t is i-th corresponding generalized coordinate of the external force during tn-tn+1. ina and inb are given as follow with known 1 ( ) in nq t− from the previous iteration step, 1 1 1 ( ) (( ( ) )cos( ) ( )sin( )) = i d nn ii i i n n i ni i i n i ni t n i n n n d n d n n d n t t n d dq t e q t t q t t dt b         − − − =  + + ;(44) 1 ( ) sin( ) = cos( ) i n ni i i ni i ni t n n n d n n d n q t e b t a t     −  − ; (45) on the other hand, if the value of kl during the time period tn-tn+1 changes from that of tn-1-tn, the total vibration response is, 1 1 1 ( ) 0 ( , ) ( ) ( )= ( )( ( ) ( )) = ( ) ( ( cos sin ) 1 ( ) sin( ( )) ) i i i n ni i i ni i i n i ni i n i n ni ni ni n n n n free force i i t n n d n d i t t t t n n d n d w x t w x q t w x q t q t w x e a t b t f t e t t d               = =  − = − − − = +  + + +    − −     ( 1n nt t t +  ). (46) ani and bni can be obtained by the above formula with the known vibration response, -1( , )n nw x t , at t=tn from the previous step, 794 p. wang, n. wu, h. luo, z. sun ( ) (( ( ) )cos( ) ( )sin( )) = i d n nn ii i n n n n ni i i i i n i ni t free i n free n d n d free n d n t t n d dq t e q t t q t t dt b         =  + + ; (47) ( ) sin( ) = cos( ) i n ni n i ni i i ni t free n n d n n d n q t e b t a t      − ; (48) where the first 3-th generalized coordinates of free vibration at time tn, ( ) ni free nq t (i=1,2,3), are given as below, 3 13 1 1 3 1 2 1 3 2 3 1 3 1 1 2 1 3 2 10 0 10 0 2 0 0 0 0 2 2 ( ) ( ( ) ( , ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) / ( ( ) ( ) ( ) ( ) ( ) ( n l l free n n n n n l l n n n n n l l l l n n n n n n n n n n n n n q t w x w x t dx w x dx w x w x t dx w x w x dx m w x w x dx w x w x dx m w x w x dx w x dx w x dx w x dx n w x w x dx w x w x − − = − + − −         1 3 2 3 1 0 0 0 0 2 2 0 0 0 ) ( ( ) ( ) ) ( ) ( ) ( ) ) l l l l l l l n n n n n dx w x w x dx n w x w x dx w x dx− +        , (49) 2 3 ( ) ( ) n nfree n free nq t m nq t= + , (50) 1 1 21 2 1 33 1 10 0 0 2 0 ( ) ( ( ) ( , ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ) ( ) n n n l l free n n n n free n n n l free n n n l n q t w x w x t q t w x w x dx q t w x w x dx w x dx −= − −      , (51) where we have, 1 2 1 2 1 2 1 1 2 2 1 10 0 0 0 2 2 2 0 0 0 ( ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ( ) ( ) ) l l l l n n n n n n n n n l l l n n n n w x dx w x w x t dx w x w x dx w x w x t dx m w x dx w x dx w x w x dx − −− = −        ,(52) 1 2 1 3 2 3 1 2 1 1 2 2 0 0 0 0 2 2 2 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) l l l l n n n n n n n l l l n n n n w x w x dx w x w x dx w x w x dx w x dx n w x dx w x dx w x w x dx − = −        . (53) 3. numerical studies, results and discussion as shown in fig. 1, a non-uniform cylindrical beam with positive exponential cross-section variation function is chosen for numerical case studies. the length of the beam is l=0.2 m and the diameter of the left end of the non-uniform beam is d=0.01 m. the diameter at x position along the length of the beam is ( ) n xd x d e =  , with the taper ratio as n=2.0. the material of study on vibration response of a non-uniform beam with nonlinear boundary condition 795 the beam is 6061 aluminum alloy with the elastic modulus as e =70×109 n/m2 and the density as ρ=2.7×103 kg/m3. the cross-section area and bending moment of inertia are 2( ) ( / 2)n xa x d e =  and 4( ) 1/ 64 ( )n xi x d e =   , respectively. the sampling interval is 0.0001 s. the relationship between spring force and deflection is shown by a diagram in fig. 2. in this paper, the continuous nonlinear boundary is treated as a multisegment linear approximation. k1, k2, ..., in a diagram shown by fig. 2 represent the kl values after the multi-linear segmentation. it is seen that the multi-linear assumption cannot present the exact nonlinear behavior/property of the reality nonlinear material with certain error leading to ‘slower’ hardening process with displacement increment. considering certain number of linear segments, more obvious error can be noticed while the nonlinearity is more significant. however, such error can be further limited by introducing more refined multi-linear segments leading to more precise spring force considered during the calculation. in this work, we consider 10 segments for the case study to present the theory. the deflection intervals are determined according to the deflection range calculated by all kl values, and the left bound of deflection interval is defined to be smaller than the amplitude of steady state deflection. and it is more reasonable to define the interval according to the steady state amplitude to ensure that all kl values can be obtained during the vibration under harmonic excitation. when the deflection of the left end of the beam is greater than the maximum value in the interval of deflection, the elastic stiffness coefficient kl will change. there are many kinds of relationships between the tension (torsion) coefficient and deflection for different nonlinear springs, we just study one of them here. deflection s p ri n g fo rc e real deflection and spring force curve deflection and spring force curve in calculation k1 k2 k3 k4 k5 fig. 2 relationship of deflection and spring force 3.1. verification of mathematical model in this section, the accuracy of the proposed mathematical model is verified by using the iterative and non-iterative method to calculate the vibration response of non-uniform beam with linear boundary, and the iterative method to calculate the vibration response of beam with weakly nonlinear boundary defined by that kl changes within a small range 796 p. wang, n. wu, h. luo, z. sun (kl=20000-21800 n/m). the excitation angular frequency of  =672 rad/s and the base motion amplitude of y=0.005 m are chosen for the verification. for the kl values with weak nonlinearity, the amplitudes of steady state deflections and their corresponding sectional intervals are shown in table. 2. when values of kl are different, the mode functions and natural frequencies of the beam are different, and the amplitudes of steady state deflections at left end are also different. the amplitudes of steady state deflections at left end shown in table. 2 will be larger closer to resonance. since the variation range of kl is very small, it is considered that the values of kl change linearly with deflections variation at the left end of the beam. when the deflection at the left end is larger than the corresponding range of sectional intervals, the value of kl will change. vibrations of the non-uniform beam at its free end in time domain are shown in fig. 3. table 2 kl values, amplitudes of steady state deflection and corresponding sectional intervals kl values (n/m) amplitudes of steady state deflections at left end (m) sectional intervals of deflection at left end (m) 20000 0.008154417514000 0-0.000855 20200 0.008205248588000 0.000855-0.00171 20400 0.008256681170000 0.00171-0.002565 20600 0.008308729902000 0.002565-0.00343 20800 0.008361411807000 0.00343-0.004275 21000 0.008414724816000 0.004275-0.00513 21200 0.008468682431000 0.00513-0.005985 21400 0.008523295153000 0.005985-0.00684 21600 0.008578582863000 0.0684-0.007695 21800 0.008634553316000 0.007695 fig. 3 deflections of non-uniform beam solved by iterative and non-iterative method in time domain study on vibration response of a non-uniform beam with nonlinear boundary condition 797 it can be seen from fig. 3 that the results of vibration response of non-uniform beam with linear boundary obtained by the iterative and non-iterative method are consistent with each other which preliminarily proves the accuracy of the iterative process. the difference between the vibration response of the beam with weakly nonlinear boundary and that with linear boundary solved by the iterative method is very small, which further proves the correct consideration of the nonlinearity with the iterative method. 3.2. sin-sweep curves the vibration of the non-uniform beam with nonlinear boundary condition under different excitation frequencies are studied in this section. for the varying frequency of the base-motion excitation from 200 to 850 rad/s, the magnitude of the base-motion induced inertia force is kept equal. the kl values, the amplitudes of steady state deflections and their corresponding sectional intervals are the same as the values given in table. 3. the value of kl is determined by the deflections at the left end and varies nonlinearly. it only represents one of the cases that the spring coefficient changes with the deflection. maximum steady state vibration deflection at free end of non-uniform beam under a harmonic sweep test with different excitation frequencies are shown in fig. 4. table 3 kl values, amplitudes of steady state deflections and corresponding sectional intervals kl values (n/m) amplitudes of steady state deflections at left end (m) sectional intervals of deflection at left end (m) 20000 0.008154417514000 0-0.002 22759 0.008912748913000 0.002-0.004 31594 0.012528617870000 0.004-0.006 41120 0.020219170700000 0.006-0.008 50863 0.026634596390000 0.008-0.01 60702 0.017120586210000 0.01-0.012 70591 0.010843838260000 0.012-0.014 80511 0.007735876899000 0.014-0.016 90450 0.005971986427000 0.016-0.018 100402 0.004850948731000 0.018 it can be seen from fig. 4 that for a beam with hardening nonlinear spring boundary (kl value changes following the deflection interval given in table 3) under the increase of excitation frequency, the maximum steady state of the deflection will continue to increase and then suddenly drop. as a common nonlinear vibration phenomenon, this is because the natural frequencies of the beam with nonlinear boundary is varying during the vibration at different excitation frequencies and vibration amplitudes. through the sin-sweep frequency progress, the deflection will become larger when the excitation frequency is close or equal to the higher natural frequency of the structure with hardening nonlinear spring boundary. when the excitation frequency is passing the range of the natural frequencies, the deflection will decrease significantly. at the same time, with the larger the excitation amplitude (0.0005-0.01 m), the stronger the level of the boundary nonlinearity can be noticed leading to wider frequency range with high vibration amplitude in the spectrum as shown in fig. 4 (200-550 rad/s-200-765 rad/s). 798 p. wang, n. wu, h. luo, z. sun fig. 4 steady state vibration amplitude at free end of non-uniform beam under different excitation frequencies 3.3. influence of the degree of boundary nonlinearity on non-uniform beam vibration based on the proposed iterative method, further studies on the vibration response of non-uniform beam under nonlinear boundary condition are carried on. in order to study the influence of boundary nonlinearity on the vibration characteristics of non-uniform beam, the vibration response at the free end of the beam in time and frequency domain with fixed spring stiffness kl of 20000 n/m (linear boundary condition), 20000-31594 n/m, 20000-60702 n/m and 20000-90450 n/m are simulated. the excitation angular frequency is  =672 rad/s and the base motion amplitude is y=0.005 m. under this certain excitation, the multi-linear kl values, the amplitudes of steady state deflections at the left end of the beam and their corresponding sectional intervals are shown in table. 3. the relationship between deflection and kl in table. 3 is used for different degrees of nonlinear boundary. for kl in the ranges of 20000-31594 n/m, 20000-60702 n/m and 20000-90450 n/m, when the deflection is larger than the maximum value of deflection intervals (0.006 m, 0.012 m and 0.018 m) during vibration, the kl keeps unchanged. the vibration response at free end of the beam is shown in fig. 5 (normalization was done in fig. 5 (b)). the normalization process is to divide the magnitude by the maximum magnitude value to make the magnitude between 0-1. study on vibration response of a non-uniform beam with nonlinear boundary condition 799 (a) (b) fig. 5 vibration response of non-uniform beam at free end with different degree of boundary nonlinearity ((a) time domain, (b) frequency domain) from fig. 5, while the spring constant at the boundary is changing during vibration, compared with the linear boundary, with the increment of time, the deflection of the beam with nonlinear boundary will only approach a semi-steady state with variable amplitude in the time domain signal and clear wide bandwidth in frequency domain. this phenomenon is more obvious for stronger nonlinearity case with larger spring constant variation range (such as kl=20000-80511 n/m as given in table 3). from the fig. 5(b) it can be seen that for the 800 p. wang, n. wu, h. luo, z. sun beam with kl=20000 n/m, the frequency corresponding to the first peak and second peak are the natural frequency and excitation frequency. for the beam with kl=20000-31594 n/m, 20000-60702 n/m and 20000-80511 n/m, there are only one obvious peak. this is because the beam with linear boundary just has one natural frequency close to the excitation frequency. the natural frequencies of the beam with nonlinear boundary will change with the boundary conditions. the range of natural frequencies is relatively wide and includes excitation frequency, so from the fig. 5(b), there will only be one peak at excitation frequency, and the peaks at natural frequencies are not very obvious. fig. 6 time periods of the system vibration staying at different kl values with different levels of boundary spring nonlinearity it is also interesting to see that when ranges of kl are 20000-60702 n/m, with the chosen base motion excitation constants,  =672 rad/s and y=0.005 m, the beam experiences relatively larger vibration amplitude. but when the range of kl becomes 20000-31594 n/m or 20000-80511 n/m, the vibration amplitude decreases. this phenomenon is mainly due to the longer period of vibration close to resonance when the kl is between 20000 and 60702 n/m. from table 3, it is noted that the beam steady state vibration amplitude reaches to relatively higher value, when the spring support stiffness, kl, is between 41120-60702 n/m leading to the natural frequency of the beam close to the excitation frequency,  =672 rad/s. fig. 6 shows the times periods of the beam vibration staying at different spring support constants in different nonlinearity cases (in total 0.6 s). from fig. 6, it can be seen that the beam experiences longer period of vibration at kl =60702 n/m closer to resonance, when kl is between 20000 and 60702 n/m compared with other two cases. although the above values are just obtained from one specific case study, it can be concluded that if higher boundary of the hardening nonlinear spring stiffness leads to the resonance or close to resonance vibration, the vibration amplitude of the whole system can be larger, while the exact responses will differ with different system constants especially different nonlinear spring intervals. study on vibration response of a non-uniform beam with nonlinear boundary condition 801 3.4. influence of excitation amplitudes on boundary nonlinearity it is noticed that different excitation amplitudes may introduce different level of linearity (different nonlinearity spring stiffness variation range) in the vibration of the structures with nonlinear boundary. in this section, the excitation angular frequency is fixed as =672 rad/s. the kl values, the amplitudes of steady state deflections and their corresponding sectional intervals are shown in table. 3. when the excitation amplitudes are 0.0005 m, 0.0025 m, 0.005 m and 0.01 m, respectively, the times periods of the beam vibration staying at different kl values (in total 1 s) are shown in fig. 7. fig. 7 time periods of the system vibration staying at different kl values under different excitation amplitudes it can be seen from fig. 7 that when the excitation amplitude is 0.0005 m, the beam vibration at the spring boundary is with extremely low amplitude (<0.002 m), and the spring stiffness does not change during the vibration with the linear boundary. with the increment of excitation amplitude, the range of the spring stiffness variation and level of the nonlinearity increases. when the amplitude becomes 0.01 m, the level of the boundary nonlinearity is the strongest covering all the kl variation range. at the same time, with different amplitudes, the vibration period of the beam staying at each kl is also different. following the given the deflection intervals in table 3, with different excitation amplitudes, the beam deflections at the spring boundary falling into each interval are different, and the vibration period on each kl is naturally not the same. when the spring properties are known, the vibration period of the beam staying at a certain kl can be controlled by changing the excitation amplitude, and then the vibration state of the beam can be adjusted. on the other hand, while the excitation frequency and amplitude are known, by adjusting the nonlinearity spring property (spring stiffness variation intervals), the vibration status can be controlled as well. 802 p. wang, n. wu, h. luo, z. sun 4. conclusions in this paper, an iterative method is proposed to accurately solve the force vibration response of non-uniform beam with nonlinear boundary. taking time node as iteration step, considering the change of natural frequencies and mode shape functions in the iteration process, the vibration response of beam in each short time period/iteration step is calculated by adm and duhamel integral considering both the transient and steady response. the influences of different excitation amplitudes and frequencies on boundary nonlinearity of non-uniform beam are studied. the results of numerical examples reveal the following conclusions: (1) comparing iteration numerical results and analysis solutions with linear boundary condition, adm method is found to be accurately combined with iteration progress solving non-uniform beam structural vibration with the nonlinear boundary condition, although inevitable minor error can be noticed in the calculated vibration mode shape functions from adm leading to imperfect vibration modes’ orthogonality. (2) through sin-sweep simulations, clear nonlinear spectrum with the hardening nonlinear boundary support can be noticed. with the increase of excitation frequency, the maximum steady state of the deflection will continue to increase and then suddenly drop because of non-resonance. checking spectrums with different excitation amplitudes, the maximum steady state beam deflection will be larger and the frequency range with high vibration amplitude in the spectrum will be wider with the increase of the excitation amplitude. (3) under the same base motion excitation, the beam vibration amplitude and the vibration period staying at a certain boundary stiffness range varies with different level of boundary nonlinearity. while the excitation frequency and amplitude are known, by adjusting the nonlinear spring property, the vibration status can be controlled. 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sahoo, s., 2016, geometric nonlinear free vibration of axially functionally graded non-uniform beams supported on elastic foundation, curved and layered structures, 3(1), doi: 10.1515/cls-2016-0018. 10243 facta universitatis series: mechanical engineering vol. 21, no 2, 2023, pp. 273 292 https://doi.org/10.22190/fume220108017w © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper design and optimization of the variable-density lattice structure based on load paths fenghe wu1, hui lian1, guobin pei1, baosu guo1, zhaohua wang2 1school of mechanical engineering, yanshan university, qinhuangdao, china 2school of mechanical engineering, taiyuan university of science and technology, taiyuan, china abstract. lattice structure is more and more widely used in engineering by replacing solid structure. but its mechanical performances are constrained by the external shape if the unit cells are directly filled in the design domain, and the traditional topology optimization methods are difficult to give the explicitly mechanical guidance for the distribution of internal unit cells. in this paper, a novel design and optimization method of variable-density lattice structure is proposed in order to simultaneously optimize the external shape and the internal unit cells. first of all, the envelope model of any given structure should be established, and the load paths need to be visualized by the theory of load path. then, the design criteria of external shape are established based on the principle of smoother load paths in the structure. an index of load flow capacity is defined to indicate the load paths density and to control the density distribution of unit cells, and a detailed optimization strategy is given. finally, three examples of a cantilever plate, an l-shaped bracket and a classical three-point bending beam are used to verify the method. the results show that the models designed by the proposed method have better mechanical performances, lower material usage and less printing time. key words: lattice structure, load paths, additive manufacturing, optimization 1. introduction lattice structure (ls) is a kind of cellular structure composed of a large number of unit cells arranged in a specific manner, which has some excellent performances including high strength relative to low mass, good energy absorption and high thermal and acoustic insulation [1-3]. some multifunctional usages of cellular structures in all scales ranging from nanometers to centimeters or even larger have been reported in recent literature [4-5]. additive manufacturing (am) technology further promotes the application of ls in received: january 08, 2022 / accepted april 02, 2022 corresponding author: zhaohua wang school of mechanical engineering, taiyuan university of science and technology, no. 66, waliu road, wanbailin district, taiyuan, china, 030024 e-mail: wangzhaohua@tyust.edu.cn 274 f. wu, h. lian, g. pei, b. guo, z wang engineering by building engineering components in a bottom-up, layer-by-layer manner as opposed to traditional subtractive manufacturing such as milling, turning, forging, and so on [6, 7]. for example, an aero-bracket with the ls features was designed and manufactured using different types of am processes and the weight of initial model is reduced by 50% [8]. therefore, it is a trend that the solid structure is replaced by the corresponding ls to improve the mechanical performances and achieve a lightweight design in the future. generally, the design domain of the solid structure is directly filled with uniform-density unit cells [9], and the mechanical performances of the ls model are verified by simulation and experiment. but, in fact, different regions in a given structure usually bear different magnitude of loads [10, 11], while the mechanical performances of uniform-density lattice structures still have the potential to be improved, which depends on the topology, shape, size and orientation of the unit cell [1]. by controlling these variables, ls can be modified in space to improve the performances such as stiffness or strength [12-15]. in the following section, some existing research studies about the design, analysis and optimization of variable-density ls are reviewed and discussed. a global relative density mapping method for the design of non-uniform ls is proposed in references [16-17], which introduces the topology optimization method into the ls design. the diameter of unit cells is changed through the linear interpolation results of continuum topology optimization to reduce the weight and improve the performances. zhang et al. [18] proposed an efficient design-optimization of variable-density hexagonal cellular structure based on three key techniques: homogenization, optimization, and construction (hoc). then, the variable-density topology optimization of ls considering stress constraints, structural dynamic characteristics and heat conduction problems are studied in references [19-21]. huang et al. [22] proposed a bidirectional evolutionary structural optimization (beso) method to optimize the distribution of porous materials and composites with periodic micro-structures and maximize the stiffness of macro-structure. coelho et al. [23] proposed a topology optimization method for three-dimensional (3d) ls, in which the optimization scheme is composed of two main loops, namely, the outer loop solves the macro design of materials while the inner loop uses the homogenization method to optimize the topology of cells. above all, most of the methods rely on current topology optimization (to) methods, including beso [24], moving morphable component (mmc) [25], and level set-based method (lsm) [26], and many excellent research results have been obtained. however, stress constraint optimization as one of the classical problems in the to method still faces many challenges, such as singularity phenomenon and local optimal solution [19]. on the other hand, the to methods based on the black box model are difficult to give the explicitly mechanical guidance for the structural optimization [27]. in addition, the traditional solid structures usually contain some material redundancy regions and complex features such as corners and lightening holes. the mechanical performances of ls, designed by directly filling the unit cells in the design domain of the solid structures, are constrained by the external shape, which can be improved. load path [28, 29] is a concept for tracking transferred load within a structure starting from loading points and ending at supporting points, which has been studied in structural design in recent years [30-32]. kelly et al. [33-34] inspired by the balance characteristics of fluid stream tube, proposed a load path method based on the equilibrium principle of stress components. the method has been extended to load paths visualization of 3d models and applied to the design of an aircraft wing spoiler and fiber steered composite [35-37]. the advantage of this method is that it can give clear load paths in a given structure and provide design and optimization of variable-density lattice structure based on load paths 275 guidance for the performance analysis of the local region. in this study, a new design and optimization method based on the theory of load paths for the variable-density ls is proposed, which can optimize the external shape and internal unit cells simultaneously. the outstanding contributions of this paper include the following: (1) the design criteria of external shape are established based on the principle of smoother load paths in the structure. (2) an index of load flow capacity s is defined to indicate the load paths density and control the density distribution of unit cells for the ls. (3) the detailed optimization strategy of internal unit cells including mathematical model and algorithm is given to realize the structural lightweight. this paper is structured as follows: section 2 describes the proposed method including four steps, and the detailed process of each step is described. in section 3, three examples of a cantilever plate, an l-shaped bracket as well as a classical three-point bending beam are used to verify the proposed method by numerical simulation or experiment. section 4 concludes our work. 2. methodology the proposed method consists of four steps, as shown in fig. 1. first, the envelope model of a given structure is established according to the working conditions and functional features (step 1). then, the finite element model (fem) is established and the load paths are visualized by the load path theory (step 2). next, the design criteria of external shape are established by analyzing the load transferred law (step 3). finally, an index of load flow capacity s is constructed and the optimization strategy and algorithm of variable-density unit cells are given (step 4). the detailed design process of each step is described in this section, and three examples are used to verify the proposed method in next section. 2.1. establishment of envelope model the traditional structures usually contain features such as corners and lightening holes, which may interfere with the direction of load paths. it is necessary to abandon these features designed by traditional design methods. so, the envelope model of a given structure is firstly established in this method by analyzing the working conditions and functional features. the basic criteria [38] are as follows: (1) the working conditions of the parts are analyzed to ensure that the constrained and loaded regions remain unchanged. (2) the internal regions and functions are analyzed and the internal holes that do not affect the normal work are filled. (3) the outer boundary of the parts is analyzed and the complex boundary is filled into the regular boundary. 2.2. load paths visualization the theory of load path visualization is described in [34]. a hypothetical force stream tube is introduced as shown in fig. 2; the load paths are defined as curves that surround stream tube where the force remains constant. the tube inherits properties similar to that of a fluid stream tube in that no force (fluid) crosses the boundaries and the magnitude of the forces at the ends are such that equilibrium is satisfied over the tube, i.e. ba ff = . (1) 276 f. wu, h. lian, g. pei, b. guo, z wang fig. 1 design and optimization method based on load paths fig. 2 schematic diagram of the force stream tube for arbitrary point p on the pipe wall, there are a normal stress σn and a shear stress τt. according to the principle of force equilibrium, the stress components at point p satisfy eq. (2).    =− =− direction)-( 0cossin direction)-( 0sincos y x tn tn   . (2) angle α and θ is taken as the direction of load path and the normal line at point p. for a 3d structure, the stress components at a point form a second order tensor; it can be design and optimization of variable-density lattice structure based on load paths 277 represented in a [3×3] matrix. if each row gives three stresses acting on a plane that the normal is aligned with one of the coordinate axes, then             = zzzyzx yzyyyx xzxyxx     , (3) where, σxy is the shear stress acting on the plane that the normal is in the x direction and pointing y direction. eigenvalues and eigenvectors of this matrix provide the principal stresses and principal stress vectors. load paths can be defined by plotting contours aligned with total stress pointing vectors given by the columns of the stress matrix. each column of the matrix gives the stress component in the corresponding coordinate direction on the three planes that form the sides of the corner element depicted in fig. 3. fig. 3 construct for force component the pointing stress vectors are thus defined at every point in the domain by      ++= ++= ++= kjiv kjiv kjiv zyzxzz zyyxyy zxyxxx    . (4) using the pointing stress vector, the force acting on the infinitesimal element can be described as the vector dot product:        = = =    danvf danvf danvf zz yy xx    . (5) normal n = nxi+nyj+nzk. since the force in x-direction along the curves is zero, it is required that normal n is perpendicular to pointing stress vector vx along the curves. therefore, the load paths are described by the sequential connection of the pointing stress vectors. three pointing stress vectors (vx, vy and vz) are defined, and they describe the load paths in the corresponding direction. 278 f. wu, h. lian, g. pei, b. guo, z wang the stress components can be solved based on the fem and the direction of load paths in each element can be calculated using eq. (1) and eq. (5). the nodes and elements information, including node number, coordinates and stress components, are extracted in ansys 18.0. the load paths can be fitted in the post-processing software tecplot by importing the stress information. 2.3. design criteria of external shape the external shape of the envelope model is usually designed as regular shape, resulting in some material redundancy regions or stress concentration regions in the structure. generally, the features of chamfer, fillet or hole are used to improve the stress state of these regions, but there is a lack of mechanical theory guidance. the load paths can directly reflect the load transferred law and provide guidance for shape design. waldman et al. [34, 39] found that the stresses are low and the normalized pointing vectors show re-circulating structures or “eddies”, as shown in fig. 4. the force flow patterns arise because of the requirements of equilibrium, and it is suggested that the regions occupied by these eddies can be removed in the process of design. fig. 4 formation of eddies in low stress regions on the other hand, the discontinuous load paths are easily found (region 1 in fig. 5). the corresponding regions are considered that the load transfer is blocked by the structural shape and more materials should be added. in addition, there are some regions where the load does not pass through at the corner of the structure (shown in region 2 in fig. 5), it is suggested to remove the supplementary materials or replaced by other features, such as fillet, chamfer. based on the above analysis, the design criteria of external shape are established as follows: the regions with eddies or where the force does not pass should be removed, or replaced by the features of filled, chamfer, as well as hole. more materials should be added in the regions with discontinuous load paths. the functional requirements of the part should be considered in the process of design and the general principle is to arrange the materials along the load paths as much as possible. design and optimization of variable-density lattice structure based on load paths 279 fig. 5 other features of load paths in an l-shaped plate 2.4. optimization strategy of internal unit cells 2.4.1. load flow capacity s the density of load paths reflects the magnitude of loads in the local regions, which can be used to guide the distribution law of internal unit cells of lattice structure and improve the mechanical performances. there are several load flow fields in the structure with balanced force, and any load stream tube with different cross sections can be randomly selected in the fields. as shown in fig. 6, the cross-sectional areas of two ends of a force stream tube are a1 and a2, respectively. then, a small load flow channel is selected inside the tube, and the cross-sectional areas of the two ends are assumed to be da1 and da2. according to the theory of load paths and the principle of force equilibrium, the internal forces of two small cross-sections of load flow channel are equal in the x-direction. so, 2211 dapdap xx = , (6) where px1 and px2 are the stress components of regions da1 and da2 along x-direction. fig. 6 schematic diagram of internal load flow the entire force stream tube is a collection of load flow channels, and the internal forces of force stream tube along x-direction can be integrated as shown in eq. (7). 280 f. wu, h. lian, g. pei, b. guo, z wang  = 21 2211 a xa x dapdap . (7) thus, the internal forces through each cross section is equal to that in any other cross section in a force-balanced structure, and the cross-sectional area is inversely proportional to the stress components distributed on the surface. it is assumed that the transferred loads at the unit cross section bc of load flow channel is p, and the length l=1, as shown in fig. 7. the components along the coordinate axis are represented by px and py. based on the equilibrium principle, we have     += +=   cossin sincos 1 1 xyyy yxxx p p . (8) fig. 7 internal force decomposition at unit cross section by substituting eq. (8) into eq. (7), we get   +=+ 1 2 2222211111 )sincos()sincos( a a yxxyxx dada  . (9) according to the geometric relationship in fig. 7, eq. (9) can be simplified as   +=+ 1 2 2 2 2 2 21 2 1 2 1a a yxxyxx dada  . (10) similarly, the entire force flow tube in the y-direction can be written as follows:   +=+ 1 2 2 2 2 2 21 2 1 2 1a a yxyyxy dada  . (11) based on the principle of internal force continuity, the internal force values are constant for all load paths and can be explicitly expressed. the component of the transferred loads in x-direction is represented by σx and τxy, while the component in y-direction is represented by σy and τxy. the vector sum is defined as load flow capacity s, which can be expressed as 222 2 yxyxs  ++= . (12) design and optimization of variable-density lattice structure based on load paths 281 load flow capacity s reflects the characteristics of load paths distribution for each direction in a given structure. the magnitude of s for any point in the structure indicates the load paths density. for a large s value, the load paths are relatively dense in the point, otherwise the opposite. so, load flow capacity s can be used to control the density distribution of unit cells for the ls. 2.4.2. optimization algorithm the load paths objectively describe the load transferred law in the structure. the load paths in an ideal load-bearing structure will be evenly distributed, and the uneven load distribution is likely to cause stress concentration. so, the optimization criterion is that if the s value in some regions is large, the corresponding density of internal unit cells needs to be increased in order to improve the structural load-bearing performance. a cross-cell structure is selected and the length of four sides is taken as the design variables to update the dimensions of unit cells, as illustrated in fig 8. di,j is the length of unit cell at the i-th row and j-th column, and the shared edge of two adjacent unit cells is determined by average side length, which can effectively avoid the step phenomenon, also known as stress concentration caused by the dimension difference between the adjacent unit cells. fig. 8 variable-density strategy of internal unit cells the detailed optimization algorithm of variable-density lattice structure is as follows: (1) side length di,j of the unit cells is used as the design variable, and the variation range is set that has an upper limit of dmax and a lower limit of dmin. the initial value of side length is set as dmax. (2) the stress values at the node of fem are extracted, and the stress components at the element centroid are calculated by interpolation, and the value of load flow capacity si for all elements is obtained by eq. (12). 282 f. wu, h. lian, g. pei, b. guo, z wang (3) the average value of load flow capacity for internal unit cells of a given lattice structure is calculated using n s s n m i ji  == 1 , , (13) where n is the number of elements for unit cell. (4) side length di,j of the unit cell is calculated by eq. (14). the maximum value of load flow capacity in all unit cells is denoted as smax. f is the volume fraction, which is used to control the global density of unit cells. if di,j > dmax, then di,j = dmax; if di,j < dmin, then di,j = dmin. , , max max i j i j s d d f s =   . (14) (5) the density value obtained in step (4) is then assigned to the corresponding unit cell, and the loop iteration is performed until convergence is reached. if the change rate of volume is less than r, it is regarded as convergence, as shown in eq. (15). r v vv k kk  −+ )1( , (15) where k represents the number of iterations, vk is the volume of the k-th iteration step. r represents the change rate of volume, which is set to 0.01 in this study. finally, the optimization model is established as shown in eq. (16).              = ++= = = −  = + kuf s nss sfsdd rvvv d yixyixii n m iji jiji kkk ji 2 /)( )/( subject to )/( objective find 222 1 , max,max, )1( ,  , (16) where, u, f, k are the displacement vector, force vector, and stiffness matrix. 3. examples in this section, three examples, including a cantilever plate, an l-shaped bracket as well as a classical three-point bending beam are designed and optimized by the proposed method. 3.1. cantilever plate the problem setting of a square cantilever plate is illustrated in fig. 9 with an area of 60×60 mm2 and a thickness of 1 mm. the left top and left bottom edge corners are constrained design and optimization of variable-density lattice structure based on load paths 283 and a vertical load f is applied at the middle of the right edge. the vertical load is distributed over 5 nodes to avoid stress concentration. fig. 9 a square cantilever plate the fem is established and the load paths are calculated by the section 2.2; the results are shown in fig. 10(a). region 1 belongs to the region where the loads do not pass, and region 2 forms the phenomenon of eddies, where the materials can be removed. according to the design criteria in section 2.3, the materials are arranged along the outermost load path and a new shape with load paths is given in fig 10(b). fig. 10 load paths visualization of the cantilever plate (a) load paths of initial shape (b) load paths of new shape in order to further verify the rationality of the proposed design criteria of external shape, the stiffness and strength of two models in fig. 10 are compared by fea. the boundary conditions and the selected material properties are exactly the same. the element type is solid 185 and the element size is 0.2mm. the fea results including displacement and stress distribution of initial shape and new shape are shown in figs. 11 and12. the results show that the volume of new shape decreases by 22.44%, the maximum displacement decreases from 0.0086 mm to 0.0064 mm, accounting for 34.50%, the maximum stress decreases from 88.83 mpa to 61.08 mpa, accounting for 37.50%. it shows that the mechanical performances of the cantilever plate are improved obviously by retaining the regions with effective load paths and removing the regions with invalid load paths. 284 f. wu, h. lian, g. pei, b. guo, z wang fig. 11 the fea results of initial shape of the cantilever plate (a) displacement distribution (b) stress distribution fig. 12 the fea results of a new shape of the cantilever plate (a) displacement distribution (b) stress distribution then, a unit cell of 5×5 mm2 is selected to fill the inner region of the new shape in fig. 10b. considering the accuracy requirements of additive manufacturing technology, dmin is set to 0.4 mm and dmax is set to 3.0 mm. volume fraction f is set to 0.5. the unit cells are filled according to the optimization algorithm in section 2.4 and the model of variable-density lattice structure is established and shown in fig. 13. fig. 13 variable-density lattice structure of the cantilever plate design and optimization of variable-density lattice structure based on load paths 285 the fea with same loads and constraints is performed to compare the mechanical performance of the variable-density model based on the proposed method (model a1), the variable-density model based on von mises stress (model a2), and the uniform-density model (model a3), as shown in fig. 14. the volume, maximum displacement and maximum von mises stress are listed in table 1. the results show that the maximum displacement and stress of model a1 are smaller than the other two models, and the weight reduction effect is the best, which shows that the proposed design and optimization method has good applicability. fig. 14 the comparison of three different cantilever plates (a) model a1 (b) model a2 (c) model a3 table 1 results for different cantilever plates parameter model a1 model a2 model a3 volume (mm3) 2404 2517 2673 maximum displacement (mm) 1.00296 1.97867 2.11406 maximum von mises stress (mpa) 76.7985 86.5097 84.0265 3.2. l-shaped bracket the second example is an l-shaped bracket as shown in fig.15; the top edge is constrained and a vertical load f is applied at the middle of the right edge. the load paths 286 f. wu, h. lian, g. pei, b. guo, z wang are calculated by the section 2.2 as shown in fig. 16. region 1 belongs to the region where the loads do not pass, and region 2 forms the phenomenon of eddies, where the materials can be removed. the discontinuous load paths are found in region 3, and more materials should be added. fig. 15 l-shaped bracket fig. 16 load paths of l-shaped bracket on this basis, the same size of unit cell, value of dmax, dmin, and volume fraction f are set with the example of cantilever plate, the optimization of unit cells density is carried out. the new model is established as shown in fig. 17. fig. 17 the lattice structure of l-shaped bracket similarity, the fea with same loads and constraints is performed to compare the performance of variable-density model based on the proposed method (model b1), the variable-density model based on von mises stress (model b2), and the uniform-density design and optimization of variable-density lattice structure based on load paths 287 model (model b3), as shown in fig. 18. the volume, maximum displacement and maximum stress are listed in table 2. the results show that the maximum displacement and stress of model b1 are smaller than the other two models, and the weight reduction effect is the best. fig. 18 the comparison of three different l-shaped brackets (a) model b1 (b) model b2 (c) model b3 table 2 results for different l-shaped brackets parameter model b1 model b2 model b3 volume (mm3) 1391 1466 1498 maximum displacement (mm) 2.1073 2.3959 2.2725 maximum von mises stress (mpa) 28.71 30.15 31.74 3.3. three-point bending beam the third example is a three-point bending beam and the length, width and height are 100×20×25 mm, as shown in fig. 19. the upper platform has a width of 20 mm, and each lower platform has a width of 15 mm. a load is applied to the upper surface, and two fixed constraints are applied to the lower surface. 288 f. wu, h. lian, g. pei, b. guo, z wang fig. 19 the structure of three-point bending beam the load paths of cross-section in front view are calculated by section 2.2 as shown in fig. 20. the regions with eddies and where the loads do not pass are removed. the same size of unit cell, value of dmax, dmin, and volume fraction f are set with the example of cantilever plate, the optimization of unit cells density is carried out. a new three-point bending beam is established as shown in fig. 21. fig. 20 load paths visualization of cross-section in front view fig. 21 a new three-point bending beam with unit cells in order to further verify the proposed method, a group of physical models was fabricated by a 3d pro x3 printer using poly-lactic acid (pla). the variable-density bending beams based on stress (model c2) and the uniform-density bending beams (model c3) are designed and fabricated using the same device, as shown in fig. 22. fig. 22 physical models of different three-point bending beam (a) model c1 (b) model c2 (c) model c3. design and optimization of variable-density lattice structure based on load paths 289 the experiment was carried out using a microcomputer-controlled electronic universal testing machine. the linear displacement with 1 mm/min was applied; the displacement and corresponding loads were recorded by the data acquisition system. three groups of experiments were carried out on each model to assure that the experiments are fulfilling the repeatability conditions and the results are not under the influence of exceptional characteristics or flaws of a single sample. the experimental results are shown in fig. 23. fig. 23 experimental results of three groups of physical models the comparison of volumes, manufacturing times, ultimate loads is listed in table 3. two variable-density bending beams (model c1 and model c2) show the significant advantages compared to the uniform-density bending beam (model c3). compared with the variable-density model c2 based on stress, the volume of the new model c1 is further reduced by 6.13%, the printing time is reduced by 11.65%, but the average ultimate load is increased by 7.56%. these results show that the mechanical performances of variable-density lattice structure designed by the proposed method are significantly improved. and the material usage and printing time are reduced. table 3 performance comparison of three models parameter model c1 model c2 model c3 volume (mm3) 33540 35730 39080 fabrication time (min) 364 412 443 average ultimate load (n) 6922.9 6436.2 5463.4 4. conclusion a novel design and optimization method of variable-density ls is proposed to optimize the external shape and internal unit cells simultaneously in this paper. the method is divided into four steps, as follows: step1, the envelope model of any given structure is established according to the working conditions and functional features; step2, the load paths are visualized by the load path theory; step3, the external shape is re-designed by 290 f. wu, h. lian, g. pei, b. guo, z wang analyzing the load transferred law; step4, the distribution of internal unit cells is optimized according to the dense of load paths. three examples of a cantilever plate, an l-shaped bracket and a classical three-point bending beam are used to verify the proposed method by numerical simulation or experiment. the results show that the models designed by the proposed method have better mechanical performances, lower material usage and less printing time. the major contributions of this work can be summarized as follows: (1) the design criteria of external shape are established based on the principle of smoother load paths in the structure, which give the designers a straightforward optimization suggestion. (2) a new index of load flow capacity s is defined to indicate the load paths density and control the density distribution of unit cells. (3) the detailed optimization strategy of internal unit cells including mathematical model and algorithm is given to realize the structural lightweight. 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https://doi.org/10.22190/fume180912034b © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper efficient calculation of the bem integrals on arbitrary shapes with the fft udc 519.6:539.3 justus benad berlin university of technology, berlin, germany abstract. this paper builds upon the results of a recent study which illustrates how the fast fourier transformation (fft) can be used to accelerate the boundary element method (bem) for arbitrary shapes. in the present work, we further deepen this understanding and focus especially on implementation details in order to calculate the boundary integrals with the fft. different numerical techniques are compared for an exemplary shape. also, additions to the concept are mentioned such as the introduction of a high-resolution grid close to the boundary and a low-resolution grid farther away. key words: laplace equation, navier equation, boundary element method, fft 1. introduction over the past years, the boundary element method (bem) has become a highly efficient tool in contact mechanics for the calculation of the deformations and stresses with the half-space approximation [1, 2]. the boundary integral equations are simple convolutions over the surface of a half-space which can be evaluated rapidly with the fast fourier transformation (fft) [3, 4]. for many practical applications, however, the complex geometry of the contacting bodies exceeds the limitations of the half-space approximation. one of these fields is the modeling of deformations and stresses in bio mechanics, where the complex geometries of joints such as the hip and the knee joint (see for example [5, 6]), clearly exceed the range of application of the half-space theory. another field is given by a wide range of aerospace applications, such as the simulation of deformations and stresses in highly stressed parts of aircraft engines. turbine blades and discs are among these critical components [7, 8]. highly undulating surfaces, cooling holes, and other complex geometrical elements in close proximity to the contact regions of turbine blade fir-tree connections cause difficulties when using the half-space theory to model deformations and stresses [9]. therefore, other modeling techniques received september 12, 2018 / accepted november 10, 2018 corresponding author: justus benad tu berlin, institut für mechanik, fg systemdynamik und reibungsphysik, str. d.17. juni 135, 10623 berlin, de e-mail: mail@jbenad.com 406 j. benad have to be used for these kinds of problems, such as the finite element method (fem) or the classical bem for arbitrary shapes [10]. generally, this comes at the cost of a much higher computational complexity. many valuable methods have been proposed in the past to reduce the computational complexity of both the fem and the classical bem for arbitrary shapes [10-14]. a recent study [15] illustrates how the classical bem for arbitrary shapes can be accelerated with the fft in a similar way as it is done with great success in contact mechanics within the framework of the half-space approximation. the study highlights that although the boundary integrals of the bem are not simple convolutions over the boundary of arbitrary shapes, they can still be regarded as convolutions over the space which is one dimension higher than that of the boundary. this makes the evaluation of the boundary integrals with the fft on the surface of an arbitrary shape possible if the dimension of the fft is increased. for a three-dimensional shape with n 2 surface points, the computational complexity is o(n 4 ) to invert the classical bem matrix. when the boundary integrals are evaluated with the fft, this complexity is reduced to o(n 3 log n 1.5 ). this reduction along with highly efficient implementations which are available for the fft make this technique appealing. in the present work, we will build upon the results found in [15] and further deepen the understanding of how the technique can be implemented. in an exemplary simulation, different numerical techniques are compared for a curved shape. also, we mention additions to the method which further decrease the computational complexity. the overall goal of this work and future works will be to strive towards a complexity comparable to that of the fft on the half-space which is o(n 2 log n). 2. interpretation of the boundary integral formulations as convolutions the boundary integral formulation of the laplace equation ( ) 0u x  (1) is * * 0 0 0 0( ) ( ) ( ) ( ) ( ) ( ) s s c x u x q x u x x ds u x q x x ds     . (2) it relates the values for ( )u x and ( ) ( ) ( )q x n x u x  on boundary s of a region with outward normal vector n to the value of u at a certain point 0x in the region. it is c = 1 when 0x lies inside the region, and c = 1/2 when point 0x lies on the smooth boundary. term * 0( )u x x is the fundamental solution of eq. (1). for instance, it is  * 0 0 1 ( ) ln 2 u x x x x      (3) for the two-dimensional case of x yx xe ye  . for this case boundary s in eq. (2) is a line, whereas for the three-dimensional case with x y zx xe ye ze   boundary s is a surface. the boundary integral formulation of navier’s equation 1 1 ( ( )) ( ) ( ) 1 2 u x u x b x          (4) is efficient calculation of the bem integrals on arbitrary shapes with the fft 407 * * 0 0 0 0( ) ( ) ( ) ( ) ( ) ( ) , s s c x u x t x u x x ds u x t x x ds       (5) where c is defined as above, u and t n  are the values on boundary s and * t is known from the fundamental solution 0 0 3 0* 0 0 3 4 ( ) ( ) ( ) 16 (1 ) x x x x i x x x x u x x              (6) with the material law. both of the boundary integral formulations (2) and (5) from above can be interpreted as convolutions over the space which is one dimension higher than the boundary. recall the two-dimensional convolution one obtains for the half-space and which can be solved rapidly with the fft. it is 0 0 0 0( , ) ( , )a ab b s u x y k x x y y dx dy   , (7) where u is the deformation,  is the load, a is the direction of the deformation and b is the direction of the load, (a,b)  {x, y, z}. here, two coordinates x and y lie in the plane of surface s over which the integration is performed. this characteristic makes it possible to perfectly align a uniform grid on which the fft is performed to calculate the convolution in (7) with the half space surface (see fig. 1a). for arbitrary shapes, one has to enclose the entire surface with a uniform three-dimensional grid in order to apply the fft (see fig 1b). integral formulations (2) and (5) clearly represent convolutions over this space. one only has to pay attention to appropriately set zeros at the grid points where there is no surface so as not to distort the results. in [15] an exemplary implementation of this technique is presented which illustrates the feasibility of the concept. using the fft on the three-dimensional space to obtain the integrals on the surface lowers the computational complexity to o(n 3 log n 1.5 ) from the complexity of o(n 4 ) which is needed to invert the classical bem matrix. a) b) fig. 1 a uniform grid aligned with the even surface of a half-space (a), and an arbitrary shape fully enclosed with a uniform three-dimensional grid (b), [15] 408 j. benad 3. implementation details we will now highlight a few important details one has to consider when the technique described above is implemented. surface normal vectors the components of the normal surface vectors in integral formulations (2) and (5) depend on x . this characteristic does not occur for the half-space because the surface is even and stretches to infinity. for the arbitrary case, however, the components have to be considered in detail. this can be illustrated with an example for the two-dimensional laplace equation. with 2 2 0 0( ) ( )r x x y y    , (8) the fundamental solution (3) can be written as * 0 0 1 ( , ) ln( ) 2 u x x y y r      . (9) with eq. (9) and q n u  , one obtains 0 0* 2 (( ) ( ) ) 2 x yn x x e y y e q r       . (10) equations (9) and (10) can now be inserted into eq. (2) which yields the boundary integral formulation 0 0 0 0 0 0 2 ( ) ( )1 1 ( , ) ( , ) ln( ) + . 2 2 x y s s x x n y y n c x y u x y r qds u ds r         (11) now we can turn our attention to the components of outward normal vector n . as stated above, they depend on the vector’s position on the border: nx = nx(x,y), ny = ny(x,y). in order to reveal the convolution terms, one has to split the integral formulation according to the normal vector components: 0 0 0 0 0 0 0 2 0 0 0 2 0 0 1 ( , ) ( , ) ln( ( , )) ( , ) 2 ( )1 ( , ) ( , ) 2 ( , ) ( )1 ( , ) ( , ) . 2 ( , ) s x s y s c x y u x y r x x y y q x y ds x x u x y n x y ds r x x y y y y u x y n x y ds r x x y y                   (12) (note that in eq. (12) we have also switched the variables in the kernel functions to highlight the convolution.) the procedure above has to be applied in the same style for the boundary element formulation of the navier equation in order to reveal the convolution terms. efficient calculation of the bem integrals on arbitrary shapes with the fft 409 boundary shape and interpolation when the boundary of the arbitrary shape is not aligned with the uniform grid on which the fft is performed one simply obtains the values on the boundary where it cuts through the grid (see fig. 2). subsequently, the data points on the grid can be interpolated to a certain desired spacing on the surface. fig. 2 an arbitrary boundary which cuts through a uniform grid. the intersection points are marked with black dots conjugate gradient method note that although the fft gives the results of the boundary integrals, the final results for the potential and the flow, or the deformations and the stresses which we seek still reside within these integral formulations (see eqs. (2) and (5)). therefore, an iterative method such as the conjugate gradient method has to be used to obtain them. note that with the fft technique used for the half space, only the surface stresses occur within the integral and require a conjugate gradient method, and not the surface deflections (see eq. (7)). further reduction of the computational complexity the kernel functions change very little far away from evaluation point 0x . in close proximity to the point they change rapidly (see fig. 3). fig. 3 kernel functions of the integral formulation (12) therefore, one may obtain a further reduction in computational complexity when the uniform grid has a finer resolution close to the evaluation point than it has farther away from it. this technique is of particular interest if the values of only very few points of the 410 j. benad domain are of interest, which is the case for the method described here. in contrast to the fft for the half-space where the results of the fft are of interest on all discretization points in the domain, we seek here only the values which lie directly on or very near a certain boundary shape. with a finer uniform grid in close vicinity of the surface, one may save the effort of computing non-required values at many discretization points which lie within the arbitrary surface or which surround it. in order to realize this concept, one can perform a rough calculation on the entire uniform grid which fully encloses the surface but which has a low resolution. subsequently, the influence of the rough discretization points close to the evaluation points is deleted and replaced with the influence of many discretization points from a finer local grid. several of these finer local grids are placed along the surface but not in the areas where we have no interest in the results (see fig. 4). note that the convolutions over these finer grids can be obtained rapidly parallel to each other since their results do not depend on each other. fig. 4 fine local uniform grids placed along the surface of an arbitrary shape which is fully enclosed by a rough uniform grid 4. simulation example in this section we will present a small example to illustrate how the fft can be used to obtain the boundary integrals on arbitrary shapes. contrary to the example given in [15] where a simple rectangular boundary was used, we will here extend the calculation method to more sophisticated curved shapes. we will show different implementation techniques which all have the same order of computational complexity. however, their accuracy varies significantly. technique i: fft, nearest-panel boundary consider the boundary of an arbitrary two-dimensional shape as displayed in fig. 5 and let it be fully enclosed by a uniform grid. each panel through which the boundary cuts is highlighted gray in fig. 5. the two intersection points of the boundary with the edges of a panel shall define the beginning and end of a constant boundary element. this leads to an irregular size distribution of the elements along the boundary. for this first preliminary study, we will accept this. efficient calculation of the bem integrals on arbitrary shapes with the fft 411 fig. 5 an arbitrary two-dimensional shape fully enclosed with a uniform grid. the intersection points of its boundary (gray thin line) with the grid (black dots) define start and end points of the boundary elements (bold black straight lines). in the left graph the grid has n = 10 panels on each side. the right graph displays a finer resolution with n = 56 we now seek to calculate the boundary integrals (12) of the laplace equation (1) on the boundary of the chosen shape. in order to use the fft for the operation, each active panel (marked gray in fig. 5) is loaded with the potential and flow of the constant boundary element which cuts through it. we can then perform the convolutions with the kernel functions from eq. (12) using the fft in the same way as in [15]. in order to test the method, we use a simple analytical solution for the laplace equation. here, we use the exemplary solution a 1 / 2u x  (13) of eq. (1). with the known geometry of the chosen shape and eq. (13), we then have the exemplary boundary values for the potential and the flow. the boundary integrals (12) are now obtained with the fft and the resulting numerical values ui of the potential are compared with the analytical solution ua on the boundary. the results are displayed in the first graph in fig. 9. it becomes apparent that this first technique of loading only the nearest boundary panels and taking the results directly from the convolution delivers only poor results. the numerical values oscillate with a great error around the analytical solution. also, it shall be mentioned that for a finer resolution the numerical values on the boundary obtained with this first method do not converge to the analytical solution. technique ii: fft, weighted nearest-panel boundary in order to obtain more accurate results than with technique i, we distribute the potential and flow of one boundary element not only to one panel but over three panels which are weighted so that their center corresponds precisely to the midpoint of the boundary element. the resulting active panels are shown in fig. 6. to each main panel which is directly cut by the boundary (marked with a square), there is one horizontal 412 j. benad neighbor (marked with a cross) and one vertical neighbor (marked with a circle). depending on the position of the panel midpoint we must choose either the left or the right neighbor for the additional horizontal panel, and either the upper or the lower neighbor for the vertical panel, so that by weighing the elements their center can lie directly at the midpoint of the boundary element. fig. 6 the boundary elements (solid black lines) of the chosen shape shown together with their active panels (gray) from technique ii: to each main panel which is directly cut by the boundary (marked with a square), there is one horizontal neighbor (marked with a cross) and one vertical neighbor (marked with a circle). the left or the right neighbor for the additional horizontal panel and the upper or the lower neighbor for the additional vertical panel is determined depending on which one is closest to the midpoint of the boundary element. in the left graph the grid has n = 10 panels on each side. the right graph displays a finer resolution with n = 56 after the potential and the flow have been distributed to the active panels, we can proceed just as in technique i. we perform the convolutions with the kernel functions from (12) with the fft and resulting numerical values ui of the potential are compared with analytical solution ua on the boundary. the results can be seen in the second diagram in fig. 8. there is a definite improvement when compared to the first technique; however, there is still a high oscillating error. technique iii: fft, direct center integral, weighted nearest-panel boundary in order to further improve the results, we now calculate the boundary integrals (12) directly in a region close to a desired boundary value (see fig. 7). outside this small region, we continue using technique ii. efficient calculation of the bem integrals on arbitrary shapes with the fft 413 fig. 7 exemplary midpoint of a boundary element (cross) around which a square is placed in which the integral is calculated directly. the grid has a resolution of n = 56 panels on each side and the small square has r = 7 panels in the horizontal and vertical direction note that the direct integration in a small region around the boundary element does not increase the order of the computational complexity of the method. however, the accuracy is further improved by the technique as can be seen in the third graph in fig. 8. here, a small square with r = 7 panels on each side is used. it shall be mentioned that for a finer resolution of the grid the numerical values converge to the analytical solution. therein, the number of panels r for the direct integration does not increase when the grid is refined. technique iv: fft, direct center integral, weighted nearest-panel boundary, interpolation the raw data from technique iii can still be improved. a large portion of the error which remains is due to the alternating boundary element sizes caused by the uniform grid which cuts through the arbitrary boundary. thus, the data is linearly interpolated to a constant spacing along the boundary and each resulting value is averaged with its two neighbors to eliminate the influence of the alternations. this technique further improves the accuracy of the results as can be seen in the last graph in fig. 8. the order of the computational complexity remains the same. the last point to make in the paper is that, in addition to the boundary values, also inner values of the problem are calculated with each method above. these inner values are not needed in order to obtain the boundary values but they are simply available after the boundary integral is obtained. this is due to the application of the fft on the entire domain. at a certain distance to the boundary these values are highly accurate. fig. 9a displays the inner values as they are obtained with the convolution in technique i. close to the boundary the error starts to increase rapidly. fig. 9b displays the inner values as obtained in technique iii and technique iv. here, the graphs show only the portion of the results from the convolutions. the additional portion which leads to the numerical values for u on the boundary (compare fig. 8) comes from the direct integration which is not shown in fig. 9b. the last two graphs (see fig. 9c) show again the inner values as obtained in the technique iii and technique iv, only now the resolution n of the grid is increased ten 414 j. benad times. as the number of panels for the direct integration remains the same (r = 7) while the overall resolution is increased, the values which represent the potential within the shape now stretch all the way to its boundary. fig. 8 numerical results for u (black dots) displayed along boundary s of the shape and compared with the analytical solution for u on the boundary (red line). the graphs are displayed for a grid with n = 56 panels on each side. for the chosen shape this corresponds to n = 214 surface points. (see the red line in fig. 9 for a view of the analytical solution for u displayed over the x-y plane.) efficient calculation of the bem integrals on arbitrary shapes with the fft 415 fig. 9 exemplary results for the inner values available after the convolutions in a) technique i, n = 56, r = 7, b) technique iii & iv, n = 56, r = 7, c) technique iii & iv, n = 560, r = 7. the red line shows the analytical solution for u on the boundary. (note that the grid lines in c) are not displayed only for a better visual illustration.) 416 j. benad 5. conclusions in this study we have further deepened the understanding of how the boundary integrals of the bem can be obtained on arbitrary shapes with the fft. we have built upon the results in [15] and presented an exemplary numerical calculation of the boundary integrals of the laplace equation for a chosen curved shape. the accuracy of different implementation techniques is compared. we have also drawn attention to several general aspects of the concept: the importance of the normal surface vectors is highlighted, that is, the one with which one can split the boundary integral formulations on arbitrary shapes to obtain simple convolution integrals. we have further illustrated how the arbitrary boundary cuts through the uniform grid. then it is highlighted that with the boundary integral formulations of the laplace and navier equation there is always a conjugant gradient method necessary to obtain the desired values on the boundary. furthermore, a concept is mentioned to further reduce the computational complexity of the method by introducing a high-resolution grid close to the boundary and a low-resolution grid farther away. it can be concluded that the use of the fft to obtain the boundary integrals of the bem is appealing and worthy of further investigation. this is due to the lower computational complexity of the method o(n 3 log n 1.5 ) than the inversion of the full bem matrix o(n 4 ), and to the potential for further decreasing this complexity. furthermore, very efficient implementations are available for the fft. also, it should be mentioned that the method discussed in this work is very similar to the technique used with great success for the half space in contact mechanics. thus, many proven techniques from this field could be adopted in the future. among them is the rapid calculation of the contact area, the acquisition of stick and slip zones, and 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aluminum alina y. morkina1,2, danila v. tarov2, gulnara r. khalikova1,3, alexander s. semenov4, pavel s. tatarinov4, ilya a. yakushev4, sergey v. dmitriev1,3 1institute for metals superplasticity problems of russian academy of sciences, ufa, russia 2research laboratory for metals and alloys under extreme impacts, ufa university of science and technology, ufa, russia 3ufa state petroleum technological university, ufa, russia 4polytechnical institute (branch) in mirny, north-eastern federal university, mirny, russia orcid ids: alina y. morkina https://orcid.org/0000-0002-3989-0376 danila v. tarov https://orcid.org/0009-0009-4539-1248 gulnara r. khalikova https://orcid.org/0000-0002-6712-8469 alexander s. semenov https://orcid.org/0000-0001-9940-3915 pavel s. tatarinov https://orcid.org/0000-0001-7430-8591 ilya a. yakushev https://orcid.org/0000-0003-2539-7334 sergey v. dmitriev https://orcid.org/0000-0002-6744-4445 abstract. the effect of electroplasticity (ee) is used to increase the deformability of metals and alloys, save the energy used to deform them, and reduce tool wear. despite the practical importance of ee, there is no consensus in the literature on its physical nature, and further research is needed to elucidate it. in this work, the tensile deformation of aluminum and copper wires stimulated by the repetitive electric pulses is experimentally investigated. copper and aluminum have low and high stacking fault energy, respectively, and it is interesting to compare the behavior of these metals during electrically assisted plastic deformation. joule heat release is precisely controlled in the experiments. both metals show qualitatively similar behavior, with the increment in plastic deformation decreasing with pulse number at fixed tensile stress. this result supports the dislocation mechanism of ee in these metals. key words: effect of electroplasticity, tensile deformation, copper, aluminum, creep, current pulses received: september 20, 2024 / accepted november 20, 2024 corresponding author: sergey v. dmitriev institute for metals superplasticity problems of ras, khalturin st. 39, 450001 ufa, russia e-mail: dmitriev.sergey.v@gmail.com https://orcid.org/0000-0002-3989-0376 https://orcid.org/0009-0009-4539-1248 https://orcid.org/0000-0002-6712-8469 https://orcid.org/0000-0001-9940-3915 https://orcid.org/0000-0001-7430-8591 https://orcid.org/0000-0003-2539-7334 https://orcid.org/0000-0002-6744-4445 616 a.y. morkina, d.v.tarov, g.r. khalikova, et al. 1. introduction troitskii first identified the effect of electroplasticity (ee) in metals in 1969 [1], and subsequent research has extensively explored this phenomenon through numerous studies [2]. the effect is the reduction in yield stress when high-density electric current pulses are applied to metallic materials. the temperature increase in the deformed material can be relatively small, and joule heating alone is insufficient to account for the significant increase in plasticity observed. as a result, the energy-saving approach to metal forming can be developed with the help of ee. the ability to form parts with minimal heating allows for more energy efficient manufacturing processes, reducing overall energy consumption. in recent years, studies on ee have gained considerable attention due to its usefulness in a wide range of industrial sectors [3-5]. due to the skin effect [6,7], ee can be observed only in relatively thin workpieces during rolling [8,9], drawing [10-13], stamping [14,15], cutting [16], forging [17,18], and forming [5,6,19-23]. at the same time, the fundamental principles and mechanisms underlying ee remain not fully understood. researchers have consistently pointed to two key mechanisms responsible for ee: non-uniform joule heat release and electron wind action [24-26]. although the heat released at defects such as dislocations and grain boundaries increases their mobility and enhances ductility, the overall increase in average temperature remains negligible. in addition, the electron wind exerts a force on the dislocations, which also contributes to the increase in their mobility. experimental studies have provided evidence supporting the electron wind theory as an explanation for ee [26]. however, one study [27] observed non-directional migration of twin boundaries in response to electropulsation, which was attributed to the interaction between electrons and dislocations rather than the electron wind force. the mechanisms of ee have been discussed in several research and review articles [28-31]. in recent years, research has been conducted to gain a deeper understanding of ee and to explore its potential applications in the processing of various metallic materials. a method for self-healing of metallic materials using pulsed electric current to restore the microstructure has been developed [32]. the research by kim et al. [33] showed that the application of electric current during tensile testing of an al-mg-si alloy can induce structure relaxation and void formation processes. lahiri et al. selected various parameters that affect the reduction of flow stress during electropulsation to explain the mechanism of electroplasticity in terms of crystal plasticity [34]. ee can improve the ductility of a mgnd-zn-zr alloy [35] and hcp metals [36,37]. the effect of strain rate on plasticity and microstructure was investigated in the context of electrically assisted tensile testing of the aluminum alloy 5754-h111 [38]. the influence of microstructure on ee has been investigated in an aluminum alloy [39]. pulsed electric current has been used to improve the formability of the 5052-h32 aluminum alloy [40]. high-density pulsed electric current is able to induce precipitation hardening of steel [41], assist the deformation of the haynes 230 alloy [42] and the invar 36 alloy [43]. the ee was studied in a high entropy alloy under uniaxial tension, where the study [44] reported a decrease in yield stress and an increase in strain with increasing current density. in the study [45], the effect of the angle between the direction of the applied load and the electric pulse current on the ee in the ti6al-4v alloy was investigated. increased plasticity in monatomic metallic glasses was observed when electric pulses were applied, as reported in [46]. the in-situ study of comparison of the effect of electroplasticity in copper and aluminum 617 individual dislocation motion was carried out in single crystal nickel and copper under combined electromechanical actuation as reported in [47] and [48], respectively. the loading scheme can strongly influence the development of plastic flow in metals [49,50] and thus the manifestation of ee [51-56]. in the present study, ee is studied for copper and aluminum wires under tensile loading. there are two main goals of this work. first is to assess the effect of multiple electric pulses at fixed tensile stress. joule heat release does not depend on the pulse number and if the deformation increment will also be same for each current pulse, this would suggest that the observed plastic deformation is related to the temperature increase of the wire. on the other hand, if ee is realized through the depinning of dislocations, then the increment of plastic deformation is expected to reduce from pulse to pulse, as dislocations will eventually pinned by stronger obstacles. the observation that the deformation increment decreases with increasing pulse number under constant stress was reported in [57] for nanoindentation and compression. the second goal is to compare ee in two metals with significantly different stacking fault energy. copper and aluminum have considerably different values of the stacking fault energy, about 0.04 and 0.2 j/m2, respectively [58-60]. as the results, twins are easily formed in copper but not in aluminum. if a difference in the behavior of copper and aluminum will be observed, this would reveal the contribution of twins to ee, otherwise, in both metals ee is realized through the activation of dislocation motion. in our previous study [61] the effect of several high-density electric current pulses on the deformation of copper wires in different structural states under tension was investigated. in this work, a comparison between the uniaxial tension of copper and aluminum wires under the action of repetitive electric pulses is carried out. additionally, the cooling time of the samples after several pulses at different applied voltages was examined. 2. materials and methodology 2.1. microstructure characterization in this work, copper and aluminum wires of m1 and kas 8176 grades, respectively, are used as samples. the chemical composition of the metals studied is given in tables 1 and 2, respectively. the choice of these materials is based on their use in the production of electrical wires by drawing, and this process can be improved by ee. it is also interesting to compare these metals because, as mentioned in the introduction, there is a significant difference in the values of the stacking fault energy in copper and aluminum. samples with a length of 360 mm were obtained from copper (aluminum) wire with a cross-sectional diameter of 1 mm (1.79 mm). the wires were examined in their as-received condition and after annealing to a larger grain size. copper wire was annealed at 500°c for 2 hours and oven cooled, and aluminum wire was annealed at 450°c for 2 hours and oven cooled. table 1 chemical composition of grade m1 copper mass fraction of element cu no less impurities, no more bi fe ni zn sn sb as pb s o 99.90 0.001 0.005 0.002 0.004 0.002 0.002 0.002 0.005 0.004 0.05 618 a.y. morkina, d.v.tarov, g.r. khalikova, et al. table 2 chemical composition of kas 8176 aluminum mass fraction of element al no less impurities, no more si fe cu mn mg cr ni zn ti ga 99.35 0.03-0.15 0.40-0.15 0.10 the surface of specimens for ebsd analysis was prepared as follows: material blanks were cut into plates by electrospark method, then these plates were mechanically thinned on abrasive paper with gradual reduction of abrasive value, then polishing was carried out on diamond paste also with gradual transition to finer abrasive. the final polishing was carried out by jet electropolishing on the "tenupol 3" machine using electrolyte of the following composition: for copper wire 20% aqueous solution of h3po4 (at room temperature), for aluminum wire 750 ml methanol, 250 ml hno3 (at 28°c). the polishing time was 20 sec. ebsd (electron back skatter diffraction) analysis was performed on a tescanmira 3lmh scanning electron microscope using the built-in "channel 5.0" software. imaging was performed at an accelerating voltage of 20 kv. for all states, the imaging step was set to 0.5 μm for copper wire and 0.3 μm for aluminum wire, with a maximum allowable misorientation error of 2°. the number of kikuchi lines used to index the diffraction patterns was at least 5. to increase the reliability of the ebsd data obtained in the analysis of crystallographic grain misorientation, regions consisting of five points or less were removed from the ebsd maps as unreliable [62]. the resulting misorientation maps show the different crystallographic orientations in different colors. boundaries with misorientations between 2 and 15° were considered low-angle boundaries (labs), and those with misorientations greater than 15° were considered high-angle boundaries (habs), with twin boundaries labeled ʃ3. structures were analyzed quantitatively according to astme11210 recommendations. the equivalent diameter was used as the grain size. the ebsd inverse pole figure (ipf) map and corresponding misorientation angle distributions for copper and aluminum wires in the as-received condition are shown in figs. 1 and 2, while similar plots for the annealed copper and aluminum wires are shown in figs. 3 and 4. the left (right) columns in these figures show the results for the cross sections along (normal to) the wire axis. it can be seen from figs. 1 and 2 that the structure in the as-received wires is recrystallized with random orientation of the grains in the longitudinal direction and preferential <111> orientation in the transverse direction for copper and <111> + <100> for aluminum. the ratio of twin boundaries for copper wire reaches 0.53 for as-received and 0.81 for annealed wires (see figs. 1 and 3). the average grain size for as-received copper wire, excluding twin boundaries, is 9.1 ± 0.4 µm and 8.2 ± 0.2 µm in the longitudinal and transverse directions, respectively (fig. 1). for as-received aluminum wire, the grain sizes are 7.6 ± 0.3 µm and 6.2 ± 0.2 µm in the longitudinal and normal directions, respectively (fig. 2). the grain size in copper increases significantly after annealing, up to 113.8 µm for the longitudinal and 50.4 µm for the normal cross section (fig. 3). in contrast, the grain size in annealed aluminum increases only slightly to 12.3 ± 0.3 µm and 6.2 ± 0.2 µm for the longitudinal and normal cross sections, respectively (fig. 4). comparison of the effect of electroplasticity in copper and aluminum 619 fig. 1 the microstructure of the as-received copper wire: (a,c) grain orientation distribution maps, (b,d) grain boundary distribution according to misorientation angle. results for section (a,b) along the wire axis and (c,d) normal to the wire axis fig. 2 the microstructure of the as-received aluminum wire: (a,c) grain orientation distribution maps, (b,d) grain boundary distribution according to misorientation angle. results for section (a,b) along the wire axis and (c,d) normal to the wire axis 620 a.y. morkina, d.v.tarov, g.r. khalikova, et al. fig. 3 the microstructure of the annealed copper wire: (a,c) grain orientation distribution maps, (b,d) grain boundary distribution according to misorientation angle. results for section (a,b) along the wire axis and (c,d) normal to the wire axis fig. 4 the microstructure of the annealed aluminum wire: (a,c) grain orientation distribution maps, (b,d) grain boundary distribution according to misorientation angle. results for section (a,b) along the wire axis and (c,d) normal to the wire axis comparison of the effect of electroplasticity in copper and aluminum 621 in summary, copper contains a large number of twins, while twins are almost absent in aluminum, which is a consequence of the large difference in stacking fault energy between these metals. it is also important to note that the grain size of the annealed copper is an order of magnitude larger than that of the as-received copper, whereas the grain size of the aluminum wires is only slightly increased by annealing. fig. 5 stress-strain curves for as-received (black) and annealed (red) specimens deformed at a strain rate of 1.1×10-3 s-1: (a) copper and (b) aluminum 2.2. stress-strain curves tensile tests were performed on long copper and aluminum wires (360 mm) at room temperature. the tests were performed using a schenck trebel rmc100 universal electromechanical machine with a force of 100 kn. figure 5 shows the stress-strain curves for copper and aluminum, with the black curves representing the initial condition and the red curves representing the annealed condition. it can be seen that for both metals the stress levels required for deformation are lower in the annealed condition. this is particularly noticeable in the case of copper, where there is a large difference in grain size between the annealed and as-received specimens, while for aluminum this difference is small. according to the results obtained, the yield stresses of copper and aluminum in both structural states were determined. the yield stress of copper (aluminum) in the initial state is 125 mpa (55 mpa), and after annealing it is 95 mpa (45 mpa). based on these values, the tensile stress in the ee experiments was selected. 3. results 3.1. joule heat release a detailed description of the experimental setup and methodology is given in our previous study [61]. 622 a.y. morkina, d.v.tarov, g.r. khalikova, et al. first, the joule heat release on the samples due to a current pulse from a battery of capacitors charged to different voltages is measured. the dead load is chosen to be 65 n, which corresponds to a tensile stress of 83 mpa for copper and 26 mpa for aluminum. in both cases, the specimens are loaded in the elastic region. the voltage range of the capacitor bank is varied from 45 to 100 v in 5 v steps. after each current pulse, the tensile strain of the specimen associated with thermal expansion, ∆𝜀j, is measured and the increase in temperature is calculated using the formula ∆𝑇 = ∆𝜀j 𝛼 , (1) where the coefficient of thermal expansion α at room temperature is 16.9×10-6 k-1 and 22.2×10-6 k-1 for copper and aluminum, respectively. figure 6 shows the function of temperature increase with voltage on the capacitors for copper and aluminum in two states, as-received and annealed. the results for the samples in the as-received state are indicated by black curves and for the annealed state by red curves. for better accuracy, the experiments were done twice and the results obtained were averaged. note that the results presented in double logarithmic scale are well approximated by straight lines, implying the power dependence between ∆t and v in the form ∆𝑇~𝑉𝑠. (2) the blue dashed line in fig. 6 has a slope of s=2.4, which is very close to the slope of the experimental curves for both metals regardless of the structural state. fig. 6 temperature increase of samples due to joule heat release as a function of voltage across the capacitor battery for copper (solid) and aluminum (dashed) lines. results for as-received (annealed) samples are shown in black (red). the blue dashed line shows the power dependence of ∆t on u with the exponent s=2.4. each curve is the average of the results from two experiments comparison of the effect of electroplasticity in copper and aluminum 623 the energy stored in the capacitor is proportional to the square of the voltage, e=cu2/2, where c is the capacitance of the capacitor. one could expect that the joule heat release would also be proportional to the square of u, but the experiment showed a slightly faster growth. this can be attributed to the increased resistance of the wires as the temperature rises. from figure 6 it can be seen that the temperature increase of the annealed specimen is greater than that of the as-delivered specimen. this can be explained by the decrease in grain boundary density in the annealed specimens and thus the dissolution of grain boundary segregations. it is well known that elements in solid solution significantly increase the electrical resistivity of metals, resulting in an increase in the heat dissipation due to the passing current. note that for copper, the difference in heat dissipation between annealed and as-received samples is greater than for aluminum. this is because the grain size difference between annealed and as-received copper wires is greater than for aluminum wires. in the previous experiment for copper samples [61], the battery voltage was chosen to be 60 v, for which the wire temperature increase is about 25 k, see fig. 6. in this study, this voltage across the capacitor is used for copper, but a different voltage is chosen for aluminum because the aluminum wire has a different cross-sectional area and different physical properties. the voltage is chosen so that the homologous temperature increase of aluminum wires is equal to that of copper wires. the melting point of m1 copper is equal to tm1=1358 k and a temperature increase of 25 k corresponds to 0.0184tm1. the melting point of aluminum is tm2=933 k, and 0.0184tm2=17.2 k. approximately this heating occurs when a current pulse is applied with a battery voltage of 75 v, and this voltage is chosen in experiments with aluminum wires. in the previous experiment for copper specimens [61], the battery voltage was chosen to be 60 v, for which the increase in the wire temperature is about 25 k, see fig. 6. in this study this voltage across capacitor is used for copper but a different voltage is chosen for aluminum since the aluminum wire has different cross section area and different physical properties. the voltage is chosen such that the increase in the homological temperature of aluminum wires is equal to that of copper wires. the melting point of m1 copper is equal to tm1=1358 k and an increase in temperature by 25 k corresponds to 0.0184tm1. the melting point of aluminum is tm2=933 k, and 0.0184tm2=17.2 k. approximately this heating occurs when a current pulse with a battery voltage of 75 v is applied and this voltage is chosen in the experiments with aluminum wires. in summary, experiments are performed with copper wires at a voltage across the capacitor of 60 v and with aluminum wires at a voltage of 75 v. the temperature increase t in both types of wires is 0.0184 of their homologous melting temperatures, which is approximately 25 k for copper and 17 k for aluminum. 3.2. cooling and creep of wires after the current pulse in the course of experiments at sufficiently high tensile stresses, the phenomenon of creep of samples after passing a current pulse was observed. it is known that the creep rate of metals depends significantly on temperature, so the question arose about the nature of the observed creep and its relation to joule heating. to answer this question, the cooling and creep rates of the samples were studied. 624 a.y. morkina, d.v.tarov, g.r. khalikova, et al. to study the cooling rate of the specimens, the same tensile stresses were applied as in the experiment described in section 3.1, namely, 83 mpa for copper and 26 mpa for aluminum; in both cases, the tensile stresses lie below the yield stress. the voltages across the capacitor bank were 60, 80, and 100 v for the copper specimens and 75, 110, and 140 v for the aluminum specimens. to improve accuracy, three experiments were made for each capacitor voltage and the data were averaged. in the experiments, the decrease in length of the wires due to their cooling after the current pulse was recorded and recalculated into the temperature decrease using eq. (1). fig. 7 shows the results obtained for copper and aluminum wires, where the abscissa shows the time in seconds and the ordinate shows the temperature change of the samples in kelvin. the black, red and blue curves correspond to the voltage on the capacitor batteries of 60 (75), 80 (110), and 100 (140) v used for the experiment with copper (aluminum) wires. the results for copper in the as-received state and after annealing are plotted in fig. 7(a). exponential decrease of temperature in time can be seen as the logarithmic scale is used for the ordinate. the structural state does not noticeably affect the cooling of the samples as the difference is within the scattering range of the experimental data. it can be seen that the temperature decreases down to t=10 k in 20, 30, and 45 s for the current pulse from the capacitor charged to 60, 80, and 100 v, respectively. fig. 7 cooling rate of samples after an electric current pulse from the capacitor charge to different voltages. (a) results for as-received (solid lines) and annealed (dashed lines) copper. (b) results for annealed aluminum. the plots show the average result of three experiments the cooling of the aluminum wire was measured only for the annealed sample because, as mentioned above, the grain size in the annealed and as-received aluminum samples differs not significantly, see figs. 2 and 4. in fig. 7(b) it can be seen that aluminum cools comparison of the effect of electroplasticity in copper and aluminum 625 down longer compared to copper and this is due to the fact that the diameter of the aluminum wire is 1.79 times larger. but even at the capacitor voltage of 150 v cooling down to t=10 k takes only 50 s. to study the creep rate, high tensile stresses of 191 mpa were chosen for the copper specimens in both structural states, and 80 and 70 mpa for the as-received and annealed aluminum specimens, respectively. the capacitor bank voltages were chosen to be 60 v for copper and 75 v for aluminum. figure 8 shows the results for copper (aluminum) wires with solid (dashed) curves. the results for the samples in the as-received condition are shown as black curves and the results for the annealed samples are shown as red curves. it can be seen that creep is much faster as the wire heats up after the current pulse. however, the deformation does not stop when the sample is completely cooled and continues at a slower rate. it can also be observed that the creep of the annealed specimens stops earlier compared to the original specimens. this is particularly noticeable for copper specimens. fig. 8 creep strain as the function of time for specimens after single electric current pulse at a fixed tensile stress for copper (solid) and aluminium (dashed) lines. results for as-received (annealed) specimens are shown in black (red) 3.3 wire elongation under the action of repetitive electrical pulses the next step was to make a series of 15 current pulses with fixed tensile stress in the range of 95.5-171.9 mpa and 34-76 mpa for copper and aluminum wires, respectively. these load ranges capture the elastic and inelastic regions. figure 9 shows the results for the specimens in the initial state, where the abscissa stands for the pulse number and the ordinate denotes the strain induced by a current pulse. figure 9(a) shows the results for copper and (b) for aluminum. 626 a.y. morkina, d.v.tarov, g.r. khalikova, et al. fig. 9 the increase in plastic deformation as a function of the electric current pulse number at various values of the fixed tensile stress. fifteen current pulses are applied for each tensile stress, after which the load is increased. results for the as-received specimens of (a) copper and (b) aluminum. each curve is the average of two experiments. the capacitor bank voltage is 60 v for copper and 75 v for aluminum fig. 10 the increase in plastic deformation as a function of the electric current pulse number at various values of the fixed tensile stress. fifteen current pulses are applied for each tensile stress, after which the load is increased. results for the annealed specimens of (a) copper and (b) aluminum. the capacitor bank voltage is 60 v for copper and 75 v for aluminum. the graphs show the average result of two experiments comparison of the effect of electroplasticity in copper and aluminum 627 it can be seen that qualitatively the results do not differ for copper and for aluminum. at tensile stresses below 121 mpa (52 mpa) the effect of electroplasticity is practically not manifested in copper (aluminum), since this stress range is below the flow stress. but with increasing tensile stresses beyond the elastic limit, the effect of electroplasticity is enhanced. the first current pulse gives the largest deformation of the specimen, and with each subsequent pulse this effect becomes weaker until saturation occurs. this can be attributed to the fact that dislocations, whose motion is activated by the current pulse, are gradually fixed on the obstacles or reach the grain boundaries or come to the surface. the fraction of mobile dislocations decreases from pulse to pulse, which leads to a decrease in the plastic strain increment with pulse number. figure 10 shows the results of the same experiment but for copper and aluminum wires after annealing. it can be seen that the effect of electroplasticity for the annealed specimens is manifested at lower tensile stresses because their yield strength is lower compared to the specimens in the as-received condition. for the annealed copper wire, the strain due to current pulses increased compared to the as-received wire, see fig. 9(a) and fig. 10(a). this is because the metal became more ductile due to the increase in grain size during annealing. however, the same cannot be said for the aluminum specimens. this is due to the fact that aluminum already had rather large grains in as-received state, which grew only slightly as a result of annealing. fig. 11 increment of plastic deformation as a function of fixed tensile stress for four successive electric current pulses. after every four pulses, the tensile stress increased by about 5%. results for the as-received specimens of (a) copper and (b) aluminum. the capacitor bank voltage is 60 v for copper and 75 v for aluminum. the graphs show the average result for two experiments in the next experiment, 4 current pulses were applied at a fixed tensile stress, after which the latter was increased by about 5%. the results for samples in the as-received (annealed) state are shown in fig. 11 (fig. 12), where the dependence of strain increment 628 a.y. morkina, d.v.tarov, g.r. khalikova, et al. on tensile stress is plotted in (a) for copper and (b) for aluminum. the black, red, blue and magenta curves denote the first, second, third and fourth current pulses, respectively. it can be seen that regardless of the microstructure of the samples, the dependence revealed in the previous experiment is maintained the first pulse gives the largest increase in relative elongation, and with each subsequent pulse the strain increment decreases. this is most pronounced for samples after annealing. fig. 12 increment of plastic deformation as a function of fixed tensile stress for four successive electric current pulses. after every four pulses, the tensile stress increased by about 5%. results for the annealed specimens of (a) copper and (b) aluminum. the capacitor bank voltage is 60 v for copper and 75 v for aluminum. the graphs show the average result for two experiments 4. conclusions the influence of electric current pulses on plastic deformation of wire samples from m1 grade copper and kas 8176 grade aluminum in two structural states as-received and after annealing has been investigated. the heat release on the specimen during pulsed current with different voltages on the capacitor batteries has been evaluated. at capacitor voltages of 60 v for copper and 75 v for aluminum, an increase in sample temperature of 25 k and 17 k was observed, which is 1.84% of the melting point of these metals. the results presented in figs. 9 through 12 are obtained at these capacitor voltages and, therefore, the observed strain increments during the current pulses of the samples occurred at such a small joule heating, less than two percent of the melting point. thus, the classical effect of electroplasticity was observed in this work, when plastic deformation occurs under the action of pulsed current practically without heating the metal. figures 9 through 12 show that each current pulse causes an increase in strain of the order of 10-3, and it increases with increasing tensile stress and comparison of the effect of electroplasticity in copper and aluminum 629 decreases from pulse to pulse at a fixed tensile stress. if the tensile stress does not exceed the yield stress, the current pulses do not cause elongation of the specimens. the results for copper and aluminum are qualitatively similar, although the stacking fault energy is significantly different in these two metals. this supports the dislocation mechanism of ee and the presence of numerous twins in copper does not affect the electroplastic deformation. the decrease in wire elongation with pulse number at fixed tensile stress can be explained as follows. joule heat released mainly on dislocations increases their mobility. while moving, the dislocations reach obstacles or grain boundaries or the specimen surface. the number of mobile dislocations decreases and so does the increment of plastic deformation. however, an increase in tensile stress leads to a resumption of the effect of electroplasticity. the volume-averaged joule heat release is small because heat is released mainly at defects, particularly dislocations, which are effective scatterers for electrons. it can be seen from figs. 9 through 12 that in annealed specimens the plastic deformation caused by electric pulses is somewhat greater than in as-received ones, but the effect of electroplasticity decays faster with the number of pulses due to a smaller number of dislocations. this is clearly seen in the case of copper. in aluminum, however, this effect is barely noticeable due to the insignificant difference in grain size between the as-received and annealed states. the cooling and creep rates at a fixed tensile stress were also investigated. it was shown that the cooling rate is almost independent of the structural state of the specimen and is 2050 s for copper wire and 10-50 s for aluminum wire, depending on the capacitor battery voltage. in addition, it was found that creep continues at a slower rate after the specimen has cooled completely. our results confirm that the effect of electroplasticity can be used as an energy saving way of metal forming, because the heat is mainly released at dislocations, increasing their mobility and stimulating plastic deformation, while the net joule heat is small. acknowledgement: this research was supported by the ministry of science and higher education of the russian federation within the framework of the state task of the ufa university of science and technologies (no. 07503-2024-123/1) of the youth research laboratory “metals and alloys under extreme impacts”. kh.g.r. and s.v.d. thank the state task of russian ministry of science and education. the electron backscatter diffraction (ebsd) analysis was supported by the program of fundamental researches and it was performed on the equipment of the center for shared access “structural and physicomechanical investigations of materials” at imsp ras. references 1. troitskii, o.a., 1969, electromechanical effect in metals, journal of experimental and theoretical physics letters, 1(10), pp. 18-22. 2. okazaki, k., kagawa, m., conrad, h., 1978, a study of the electroplastic effect in metals, scripta metallurgica, 12(11), pp. 1063-1068. 3. stolyarov, v., misochenko, a., 2023, a pulsed current application to the deformation processing of materials, materials, 16(18), 6270. 4. dong, h.-r., li, x.-q., li, y., wang, y.-h., 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spectrum of mechanical engineering and operational research, 1, pp. 27-43. 62. humphreys, f.j., 2004, characterisation of fine-scale microstructures by electron backscatter diffraction (ebsd), scripta materialia, 51(8), pp. 771-776. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 623 637 https://doi.org/10.22190/fume200116018a © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper 1 experimental investigation of tool wear and induced vibration in turning high hardness aisi52100 steel using cutting parameters and tool acceleration nitin ambhore1,2, dinesh kamble2 1department of mechanical engineering, sinhgad college of engineering, sp pune university, india 2department of mechanical engineering, vishwakarma institute of information technology, sp pune university, india abstract. in machining of high hardness steel, vibration of cutting tool increases tool wear which reduces its life. tool wear is catastrophic in nature and hence investigation of its assessment is important. this study investigates experimentally induced vibration during turning of hardened aisi52100 steel of hardness 54±2 hrc using coated carbide insert. in this context, cutting tool acceleration is measured and used to develop a novel mathematical model based on acquired real time acceleration signals of cutting tool. the obtained model is validated as r2= 0.93 while its residuals values closely follow the straight line. the predictions are confirmed by conducting conformity test which revealed a close degree of agreement with respect to the experimental values. the artificial neural network (ann) examination is performed to determine the model regression value. the study shows that the examined reports forecasts of ann are more exact than regression analysis. the future directon of this investigation is towards developing a low-cost microcontroller-based hardware unit for in-process tool wear monitoring which could be beneficial for small scale industries. key words: tool wear, vibration, regression, artificial neural network 1. introduction in recent years dry machining for hard materials proved to be one of the promising and eco-friendly alternatives. turning of high hardness material with hardness range 4570 hrc is carried out by a single point cutting tool and is referred to as hard turning. it is widely used in aviation, automotive industries for manufacturing components such as received january 16, 2020 / accepted april 06, 2020 corresponding author: nitin ambhore mechanical engineering department, sinhgad college of engineering, sp pune university, india e-mail: nitin.ambhore@gmail.com 624 n. ambhore, d. kamble shafts, bearings, camshaft gears and landing gear, engine attachment fittings and constant velocity joints, and so on [1]. the hardened steels are favored because of their special mechanical properties like high hardness, high wear opposition etc. among hardened steel, aisi52100 steel is widely used for the production of bearing as it offers the advantages of high wear resistance and rolling fatigue strength [2-3]. hard turning is made feasible because of cutting edge advancement in tool materials such as cubic boron nitride (cbn), ceramic and coated carbide tools. as of late, carbide tools with different coatings are being utilized as a cheap substitution to expensive pcbn and ceramics tools. several research studies conducted by aurich et al. [4], suresh et al. [5], chinchanikar and choudhury [6], jiang et al. [7] have reported that the low-cost carbide cutting tools with different coatings can achieve the same performance as that of ceramic and cbn/pcbn. but in actual, tool wear is exposed to enormous mechanical burdens and in this way creates vibration all through the process. in hard turning, the cutting tool is subjected to massive mechanical loads and therefore produces vibration throughout the process. vibration influences the machining performance and specifically tool wear, surface finish and tool life; it also creates unsavory noise in the workplace [8-9]. therefore, the effect of cutting tool vibrations during the machining process must be studied. many researchers have tried to contemplate and investigate the vibrations in metal cutting. several research studies presented diverse mathematical/statistical predictive models for cutting force, surface roughness, tool vibrations, and tool wear , etc. models dependent on cutting parameters give a specific estimation of tool wear regardless of tool condition and thus can only help in the selection of the process parameters. to obtain real-time value of tool wear during turning, the model should include a signal that could represent the condition of the tool. dimla [8] presented tool wear analysis using vibration signals in the machining of en24 steel. the vibration characteristics showed that the measured wear values correlated well with certain resonant peak frequencies. salgado et al. [10] reported a significant relationship between surface roughness and tool vibration utilizing soft computing techniques. abouelatta and madl [11] reasoned that the thought of hardware vibration alongside cutting parameters expands the precision of a model. chen et al. [12] pointed out that the relative vibrations between cutting tool and workpiece cause the poor machined surface quality and unusual tool wear which drops down the profitability. suresh at al. [5] presented regression model experimental results showed that the cutting speed has higher influence on the tool wear than feed rate depth of cut. upadhyay et al. [13] developed regression models and reported that feed is the main factor that influences surface roughness followed by acceleration in a radial direction. hessainia et al. [14] inferred that the feed is the overwhelming component impacting the surface harshness, while vibrations on both radial and tangential have discovered an irrelevant impact on surface unpleasantness. ghorbani et al. [15] reported tool life predictive models based on fatigue strength of tool material and parameters of tool vibrations for different combination of workpiece and cutting tool. dmello et al. [16] performed high speed turning experiments on ti-6al-4v material using uncoated carbide insert. it is seen that tool vibration in speed direction has a major influence on surface roughness parameter and, feed rate showed a significant effect on surface roughness with more than 70% contribution. prasad et al. [17] developed multiple linear regression models for the displacement amplitude of the tool. the anova result demonstrates that the displacement of the cutting tool is affected by the workpiece hardness and cutting speed. experimental investigation of tool wear and induced vibration in turning high hardness aisi52100... 625 mir and wani [18] reported regression model for tool wear and surface roughness during hard turning of aisi d2 steel using pcbn, mixed ceramic and coated carbide tools. the results show that the tool cutting speed has the highest influence on tool wear. zeqin et al [19] proposed surface roughness model considering the influences of toolwork vibration components in feeding, cutting and in feed cutting directions as inputs. the developed model with three direction vibrations makes a better prediction for the single diamond turned surface. the majority of the studies focused on predicting surface roughness using vibration signals. some of the research projects tried to predict tool wear for using vibration signals for workpiece hardness less than 45 hrc. however, less research work has been reported which takes the actual vibration acceleration for monitoring tool wear during hard turning. thus, the objective of the present work is develop a new mathematical model to predict real-time tool wear based on real-time acceleration of cutting tool in dry turning of hardened aisi52100 steel. 2. experimental procedure and method 2.1. materials and machining conditions hard turning experiments were performed on simpleturn5076 cnc lathe equipped with 7.5 kw spindle power. the workpiece material utilized in this examination was aisi52100 steel. the workpiece rod was heated at 8500c, then quenched in oil and then being tempered around at 2000c for two hours, thus producing a tempered martensitic microstructure with a hardness of 54±2 hrc. the workpiece was held in three jaws and supported by a center in the tailstock and all experiments were carried under dry conditions. the hardened steel rods have been trued, centered, and cleaned at a moderate machining speed and feed before conducting experiments. the chemical composition of the workpiece material is 1.03% c, 1.38% cr, 0.35% mn, 0.002% p, 0.16% si, and 0.005% s and remaining fe. the machining condition, namely, cutting speed, feed and depth of cut are selected on the basis of preliminary experiments, work-piece hardness, literature review and the tool manufacturer’s recommendation. the cutting parameters ranges are cutting speed 60-180 m/min, feed 0.1-0.5 mm/rev, and depth of cut 0.1-0.5 mm. 2.2. measurement setup the setup used to measure vibration in feed, radial and, tangential directions, is schematically shown in fig. 1. a bruel & kjaer 4535b001 type-30859 tri-axial piezoelectric accelerometer with sensitivity 9.8mv/g was placed on tool holder (pclnr 2525m12) close to the insert. the coated carbide tool insert was selected of iso designation cnmg120408mf5 with th1000 grade. the tool has a rhombic shape with an included angle of 800, 4.8mm thickness and nose radius 0.8mm with the following tool geometry: including angles = 800, back rake angle = 60, clearance angle = 50, approach angle = 950 and nose radius =0.8 mm. dino-lite digital microscope model: ad4113zta with magnification rate 200x was employed to capture images of flank wear after each pass. 626 n. ambhore, d. kamble fig. 1 experimental setup 2.3. design of experiments (doe) in this work, the central composite rotatable design (ccrd) technique was implemented for planning trial runs. the design suggested 20 experimental runs which include 8 factorials, 6 axial and 6 replications of center points. in ccrd, a central run was repeated six times to check the repeatability of the output variables. in order to maintain rotatability, the value of α depends upon number of factors in design and it varies in between -1.682 to +1.682 for five levels [20]. the cutting parameters and levels are illustrated in table 1. table 1 machining parameter levels levels cutting speed v (m/min) feed f (mm/rev) depth of cut d (mm) -1.682 60 0.1 0.1 -1 90 0.2 0.2 0 120 0.3 0.3 1 150 0.4 0.4 1.682 180 0.5 0.5 experimental investigation of tool wear and induced vibration in turning high hardness aisi52100... 627 3. results and discussion 3.1. vibration analysis the ccrd recommended 20 experimental runs to be conducted while accelerations in feed vx, radial vy and, tangential vz directions are recorded and tool wear vb is measured. a new cutting edge is used for each cutting condition. vibration signals are captured at three locations, at the start, middle, and end of the process. the tool is removed and its wear is measured with the help of a microscope after every pass. this process is repeated until the tool wear reached 0.2 mm. the tools wear images for some cutting parameters are presented in figs. 2-4. fig. 2 tool wear at v= 120 m/min, f = 0.5 mm/rev, d = 0.3 mm fig. 3 tool wear at v=90 m/min, f=0.4 mm/rev, d= 0.4 mm fig. 4 tool wear at v=180 m/min, f=0.3 mm/rev, d= 0.3 mm 628 n. ambhore, d. kamble while conducting experiments, continuous chip formation was observed at cutting condition v = 150 m/min, f = 0.4 mm/ rev and d = 0.2 mm. the continuous types of chips came in contact with the accelerometer mounted near the insert. therefore, the sudden rise and fall in acceleration values are observed as shown in fig. 5. such values are neglected in developing a mathematical model. the frequency response from fft analyzer revealed fluctuation in vibration frequency in feed, radial, and tangential directions observed from 16 hz to 15 khz. the frequency response of cutting tool in tangential direction at cutting condition v = 120 m/min, f = 0.3 mm/ rev, d = 0.3 and v = 120 m/min, f = 0.5 mm/ rev, d = 0.3 mm is shown in figs. 6 and 7, respectively. it is observed that frequency started increasing onwards 5000hz. the components of the tool vibration reflect various occurrences during turning in the frequency domain, including the tool holder vibration and machine self-vibration. fig. 8 represents acceleration signals for a cutting condition at which no chip formations take place and hence the no sudden rise and fall in acceleration values are observed. the tool vibration frequency for different cutting conditions is shown in table 2. the acceleration amplitude signals without cutting are also captured for a better understanding of the machine vibration level and the response without cutting is illustrated in fig. 9. it is observed from the acceleration signals that the vibrations of cutting tool during cutting are higher than the vibrations without cutting. signals acquired do not represent different concurrences of turning. this only shows the vibration of the machine; it is helpful in finding the natural frequency of a tool holder. table 2 tool frequency at various conditions cutting parameter frequency range, hz v (m/min) f (mm/rev) d (mm) vx vy vz ---14-745 19-600 15-1500 150 0.2 0.2 44-9520 26-10695 65-10750 150 0.4 0.2 121-8646 76-11180 96-12810 180 0.3 0.3 16-11160 45-11940 16-10880 90 0.4 0.4 96-9562 35-14130 11-14260 120 0.5 0.3 397-6453 353-6512 107-8342 120 0.3 0.3 59-6550 76-6652 172-8970 60 0.3 0.3 42-8260 23-8760 32-9320 120 0.3 0.3 397-6453 353-6512 107-8342 experimental investigation of tool wear and induced vibration in turning high hardness aisi52100... 629 fig. 5 acceleration response for v = 150 m/min, f = 0.4 mm/ rev and d = 0.2 mm fig. 6 acceleration response for v = 120 m/min, f = 0.3 mm/ rev and d = 0.3 mm fig. 7 acceleration response for v = 120 m/min, f = 0.5 mm/ rev and d = 0.3 mm high-frequency zone high-frequency zone 630 n. ambhore, d. kamble fig. 8 acceleration response for v = 120 m/min, f = 0.3 mm/ rev and d = 0.1 mm fig. 9 acceleration vs frequency graph for without cutting fig. 10 acceleration vs. flank wear at v=60m/min, f=0.3mm/rev, d=0.3mm no sudden change in acceleration without cutting experimental investigation of tool wear and induced vibration in turning high hardness aisi52100... 631 the variation of tool acceleration for different values of tool wear at cutting speed 60 m/min, feed f=0.3 mm/rev and depth of cut d=0.3 mm is shown in fig. 10. the acceleration of the cutting tool in the tangential direction is observed as higher than that in the feed and radial directions. a similar trend is also observed for other cutting conditions. at the start of the cutting process, the acceleration signals increase for tool wear 0.05-0.08 mm. this is because of the sharp edge of the flank rapidly wears out due to a high initial pressure; it is accurately detected by an increase in acceleration amplitude in the feed, radial and, tangential directions. for tool wear 0.085-1.35 mm, the acceleration signals increases with the uniform rate. it is additionally observed that when the device wear is more than 1.35 mm, the tool wear rate increases because of the increase in the interface temperature and the normal pressure on the flank. this ultimately results in a sub-surface plastic flow and sometimes leads to catastrophic tool failure. the tool vibration shows quick response with a higher rate of tool wear. vibrations in the radial directions are observed as high when contrasted with those in the feed and radial directions. fig. 11 shows the acceleration of the cutting tool at different stages of tool wear. fig. 11 acceleration vs tool wear at v=120 m/min, f=0.3 mm/rev, d=0.3 mm fig. 12 (a-c) shows the trend of acceleration amplitude with varying cutting speed, feed and, depth of cut. the vibration signals have an increasing pattern with an increase in cutting speed increase as appeared in fig. 12(a). the cutting speed has a noteworthy effect on vibration in each of the three directions because frequency depends upon the rotational speed of the workpiece. vibration amplitude in the tangential direction is found higher than in the feed and radial directions. fig. 12(b) enlisted the feed rate effect at 100 mm/min cutting speed and at 0.5 mm depth of cut. from the figure it is seen that the vibration signals are observed as high for low values of feed and decrease further with an increase in feed rate. fig. 13(c) reports variation in acceleration with the varying depth of cut and for constant feed 0.3 mm/rev and cutting speed 100 mm/min. the tool acceleration observes an increase in the tangential direction with the increase of depth of cut followed by the radial and feed directions. this is in good agreement with suresh et al [5]. 632 n. ambhore, d. kamble fig. 12 variation in acceleration for varying (a) cutting speed (b) feed (c) depth of cut 3.2 regression analysis (ra) a new multiple regression model is proposed as a function of cutting parameters and tool acceleration in three directions; it is described below, 1 2 3 4 5bv ax bx cx dx ex g= + + + + + (1) where x1 is the cutting speed, x2 the feed, x3 the depth of cut, x4 the acceleration in feed direction(vx), x5 the acceleration in radial direction (vy), x6 the acceleration in tangential direction (vz) and a, b, c, d, e, f, and g are constants. the statistical analysis treatment is performed on the obtained results using the datafit statistical customize tool. in the analysis, a confidence level of 95% is chosen. the analysis of variance (anova) results shows that the statistical significance of the fitted model is evaluated by p-value (prob>f) and f-value. all p-values less than 0.5 indicate the corresponding term is highly significant. terms with a p-value higher than 0.05, are considered as insignificant for the model. the regression equation is obtained: 1 2 3 4 5 61.4055 0.1015 0.1348 0.3341 0.4192 0.1958 0.1277  bv x x x x x x= + + + + − − (2) experimental investigation of tool wear and induced vibration in turning high hardness aisi52100... 633 the goodness of the model is checked by regression coefficient (r2) value. r2 value close to 1 is desirable. r2 value for the tool wear model is found as 0.93 which is fairly enough and which concludes that the factor cutting speed, feed and depth of cut, accelerations vx, vy and vz have a significant effect on tool wear and can provide reliable estimates. the diagnostics checking of the model has been carried by examining the residuals. from the normal probability plot, it is observed that the residuals lie close to a straight line with maximum error 11% which illustrates that the error is normally distributed; the model does not indicate any inadequacy and it provides reliable prediction. some experiments have been conducted for different cutting parameters which are not the part of a designed experimental set. the machining parameter used for the selected test and the corresponding output is presented in table 3. tool wear comparison between experimental and ra model is presented in table 4. fig. 13 residual plot for tool wear table 3 confirmation test-cutting condition and acceleration values run conditions vx mm/sec2 vy mm/sec2 vz mm/sec2 1 v=75m/min, f=0.15mm/rev, d=0.25mm 0.0214 0.0325 0.0412 2 v=135m/min, f=0.45mm/rev, d=0.35mm 0.01862 0.0235 0.0256 3 v=165m/min, f=0.15mm/rev, d=0.45mm 0.0835 0.0723 0.07345 634 n. ambhore, d. kamble table 4 tool wear comparison between experimental and ra model run conditions vb-experiment (mm) vb-model (mm) error % 1 v=75m/min, f=0.15mm/rev, d=0.25mm 0.186 0.177 4.83 2 v=135m/min, f=0.45mm/rev, d=0.35mm 0.195 0.199 -2.05 3 v=165m/min, f=0.15mm/rev, d=0.45mm 0.201 0.194 3.48 3.3. artificial neural network artificial neural networks (anns) are computation models intended to reproduce the way in which the human mind forms data. artificial neural network modeling is found very useful in solving nonlinear and complex problems in the field of engineering. typically an ann network is comprised of three layers, namely, input layer, hidden layer, and output layer. ann requires sufficient input and output data instead of a mathematical equation [20]. anns can combine and incorporate both literature-based and experimental data to solve problems. the conduct of a neural system is controlled by the exchange elements of its neurons, by the learning rule, and by the structure itself. in the ann model, many input and target sets are utilized to set up a network. the network is re-adjusted on the basis of a comparison between output and target until the network output yields the target [21-22]. the neural network is created in matlab software of version r2012. the training data used during training of the neural network is collected from 20 experiments and back-propagation algorithm based on levenbergmarquardt back is used. to train the ann model, v, f, d, vx, vy and vz are considered as input data whereas tool wear vb is taken as an output parameter. the basic layout of the ann model is as shown in fig. 14. fig. 14 ann network experimental investigation of tool wear and induced vibration in turning high hardness aisi52100... 635 table 5 tool wear comparison run no. vb-experiment (mm) vb-ann (mm) error % 3 0.184 0.168 8.69 7 0.193 0.189 2.07 10 0.199 0.194 2.51 18 0.186 0.179 3.76 fig. 15 regression plot for tool wear fig. 16 tool wear comparison between experimental, ra and ann approach 636 n. ambhore, d. kamble the optimal performance of the network is evaluated based on performance parameter correlation coefficient value (r) for both training and testing data for tool wear prediction in an ann model. the correlation coefficient for the tool wear model is observed as 0.98. the closeness of the ann model predictions to the experimental results is high; the correlation coefficient between the ann model predictions and the experimental results are close to 1. fig. 15 shows regression curves ann training, testing, validation, and the overall data set for vb. from tables 4 and 5, it can be seen that the predicted values by ra and ann approach, for tool wear (vb) are closer to each other with an acceptable margin of error. the maximum error found between ann model predictions and experimental results are found 9.74%, and between ann model predictions and experimental results is observed as 8.69% and comparison is shown in fig. 16. therefore, the proposed tool wear model can be effectively used for predicting tool wear. 4. conclusion in this paper, the attempt has been made to utilize vibration signals in order to evaluate tool wear in dry turning of hardened aisi52100 steel using pvd coated carbide insert cnmg120480 of coating layers tisin-tialn. this investigation proposes a new tool wear prediction model based on real-time acceleration signals which will provide real time tool wear. the advantage of the proposed model is examined by r2 value and is found as 0.89 which is close to one. also, the diagnostics checking of the model has been carried by examining the residuals. it is observed that the residuals lie close to a straight line which illustrates that the error is normally distributed; the model does not indicate any inadequacy and it provides reliable prediction. further, the ann model is developed and the regression value for the model is found as 0.98. both models anticipated the tool wear within reasonable accuracy making them suitable for real-time prediction. the vibration frequency is observed in the range 16-15khz. the vibration in the tangential direction is found higher for the variable cutting conditions. the future work of this investigation is to develop a hardware unit for in-process tool wear monitoring suited for small scale factories. references 1. shihab, s.k., khan, z.a., mohammad, a. siddiquee, a.r., 2014, a review of turning of hard steels used in bearing and automotive applications, production & manufacturing research: an open access journal, 2(1), pp. 24-49. 2. chinchanikar, s., choudhury, s.k., 2015, machining of hardened steel – experimental investigations, performance modeling and cooling techniques: a review, international journal of 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experiments, 8th edition. wiley, pp. 288-292. 21. albu, a., precup, r., teban, t., 2019, results and challenges of artificial neural networks used for decision-making and control in medical applications, facta universitatis-series mechanical engineering, 17(3), pp. 285 – 308. 22. yilmaz, m., kayabasi, e., akbaba, m., 2019, determination of the effects of operating conditions on the output power of the inverter and the power quality using an artificial neural network, engineering science and technology, an international journal, 22, pp. 1068–1076. 11963 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 251 263 https://doi.org/10.22190/fume230728040a © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper analysis of the mechanical characteristics of date seed powder-based composite carbon fiber reinforced polymers abdelsalam abdelhussien1, guolai yang1, emad kamil hussein2, lei li1, hamid al-abboodi3, barhm mohamad4 1school of mechanical engineering, nanjing university of science and technology, nanjing, jiangsu, china 2mechanical power engineering department, mussaib technical college, al furat al awsat technical university, iraq 3state key laboratory of solidification processing, school of materials science and engineering, northwestern polytechnical university, china 4department of petroleum technology, koya technical institute, erbil polytechnic university, iraq orcid ids: abdelsalam abdelhussien https://orcid.org/0000-0003-4676-3292 guolai yang https://orcid.org/0000-0003-2499-6407 emad kamil hussein https://orcid.org/0000-0003-3831-1659 lei li https://orcid.org/0009-0007-0121-2426 hamid al-abboodi https://orcid.org/0000-0003-0642-9396 barhm mohamad https://orcid.org/0000-0001-8107-6127 abstract. composites with fiber reinforcement are often a popular lightweight option. due to their unique qualities, fiber-reinforced composites are the best alternative to traditional materials. mechanical parameters of a carbon fiber-reinforced epoxy resin (cfre) composite reinforced with date seed granulated powder (dsgp) were examined at the room temperature. the mechanical qualities included tensile, flexural, and impact strength. enhanced mechanical properties were noticed compared to carbon fiber reinforced epoxy composite produced using the hand lay-up method and vacuum bag. the dsgp-reinforced cfre with weight ratios of 0%, 15%, 25%, and 35% were considered. the test results revealed the maximum values of breaking force, maximum bending force, energy absorption capacity, and hardness number for 15%, 35%, 25%, 25% wt dsgp samples. the results show that the optimal composition for carbon fiber reinforced epoxy with date seed granules is in the range from 15% to 25% wt. gds. this study reveals that carbon fiber-date seed-reinforced composites are excellent substitutes for carbon fiber composites since they offer better mechanical properties at a lower cost. key words: mechanical properties, maximum bending force, energy absorption capacity, reinforced epoxy, seed-reinforced composites received: july 28, 2023 / accepted october 28, 2023 corresponding author: guolai yang school of mechanical engineering, nanjing university of science and technology, 200 xiaolingwei street, xuanwu district, nanjing, jiangsu, 210094 china e-mail: yangglnjust@126.com https://orcid.org/0000-0003-4676-3292 https://orcid.org/0000-0003-2499-6407 https://orcid.org/0000-0003-3831-1659 https://orcid.org/0009-0007-0121-2426 https://orcid.org/0000-0003-0642-9396 https://orcid.org/0000-0001-8107-6127 252 a. abdelhussien, g. yang, e. k. hussein, l. li, h. al-abboodi, b. mohamad 1. introduction due to their superior qualities, ease of manufacture, environmental friendliness, resistance to chemicals and erosion, high quality and solidity, and economically beneficial qualities, composite materials find extensive use in commercial enterprises. researchers are primarily focused on improving the thermal and mechanical qualities of the newly evolved composites that have excellent properties, as mentioned above. composites are also characterized by further remarkable properties including high levels of strength, stiffness, and elasticity. the performance of the composites is improved by the filler and fiber mix. the most common material used for matrix of composites is epoxy [1]. a typical composite material is a system of materials made up of two or more materials that have been combined and joined [2]. if the composition is done on a microscopic scale, the new material is known as an alloy for metals or a polymer for plastics [3]. frequently, composites are used in a metal matrix or a polymer matrix. both synthetic and natural reinforcements are applied [4]. glass, carbon, and aramid fibers are examples of synthetic reinforcements [5]. most commonly, composite materials consist of a continuous matrix phase in the bulk and reinforcement as a single distributed non-continuous phase. the fiber type significantly affects the fiber-reinforced polymer matrix composites used. the importance of the matrix type is secondary, and the fabrication process is of the third importance compared with the effect of the fiber material [6-8]. hence, mechanical characteristics mostly depend on the fiber content. the fiber-matrix bonding, fiber's characteristics, the micro-voids, and the selected fabrication techniques affect the resulting strength [9]. delamination and flexure failure are the two failure modes regarded to be significant in designing polymer composites, and their effects on the properties of the composite panel for similar fiber/resin systems were investigated [10]. the adhesion bond of the resin-matrix increases the mechanical properties of cf/ep composites [11]. micrographs of cf/ep composites samples in tensile tests show intrinsic manufacturing defects and cracks propagation, which leads the carbon fiber to get pulled out without breaking. cracks propagation in the interfacial zones affects the shearing force between the reinforcement and matrix and, therewith, the mechanical properties [12]. fiber content has a positive effect on mechanical properties in the matrix. carbon fibers improve strength, yield strength and peak load in composites [9]. the vacuum bagging process provides high quality, defect-free, non-porous composites when through-thickness pathways and prepared formats are used [13]. both experimental and sem analysis showed that there was good adhesion between the matrix and fibers [14]. cfrp composites offer advantages over metals and other polymers when used for spring applications. the tensile modulus was improved by using reinforced carbon fibers [9]. due to the fact that the tensile strength of composites is greater than the compressive strength, failure in these materials typically manifests as a crack under the roller and delamination in the specimen [15]. more efficient stress transfer from the matrix to the reinforcement increases the ultimate strength, which is due to good interfacial adhesion. choosing a proper reinforcement and matrix combination is one of the most effective ways to improve the capacity of composites. changes to the interface may have an impact on the unidirectional (ud) cfrps' fracture modes and mechanical properties [16]. when ud cfrps undergo longitudinal tensile failure, those with weak interfacial bonds exhibit splitting or broom fracture behavior, whereas those with strong interfacial bonds exhibit step-like or brittle fracture behavior [17]. in essence, fiber reinforced polymer (frp) failure mechanisms heavily depend on the matrix parameters. analysis of the mechanical characteristics of date seed powder-based composite... 253 due to its simple processing, high strength, and chemical resistance, epoxy resin is the most widely used polymer matrix in the creation of composite for structural and automotive purposes. researchers used inorganic and synthetic particles made of aluminum, titanium oxide, silica, and different types of carbon nanotubes (cnt) as toughening materials to enhance the mechanical properties of frps in order to overcome the drawbacks of frps [18-20]. for instance, a study on the possible improvement of interfacial compressive strengths of cfrp composites employing scrolling carbon nanotube (cnt) sheets around individual carbon fibers was reported [21]. natural fiber-reinforced composites have become more prevalent in a variety of applications. glass fibers were supplemented with carbon nanotubes and sawdust (additional reinforcements) to enhance the mechanical properties, which helped to some extent to solve the aforementioned issues. raw sawdust can be used as a material to improve biodegradable polymers [22]. natural fiber-reinforced composites have become more prevalent in a variety of applications, this is because more people are becoming aware of how synthetic materials affect the environment. various plant parts are used to produce natural fibers [23-25]. nevertheless, given the abundance of date plants, particularly in tropical areas, date seeds are frequently viewed as a waste of biomass [26-28]. after eating the fruit's flesh, the majority of date seeds produced in most third-world nations are still thrown away [29-31]. in this work, different plain fabric structures were considered to improve the mechanical properties of composites. a composite material of carbon fiber-reinforced epoxy (cfre) and date-seed granulated powder (dsgp) were investigated and used as a filler reinforcement material. this investigation was to ascertain the impact of date seed granules on the mechanical characteristics like hardness, strength, impact, tensile, and bending of carbon fiber-reinforced epoxy composite. additionally, the ideal cfre to dsgp ratio was established in this work in terms of strength, hardness, and toughness. 2. materials and methods 2.1. materials the polymer matrix material used was epoxy resin epolam 2040 and hardener 2047 under the trade name sika sikaaxson shanghai under the trade name sika sikaaxson shanghai (china). it is shown in fig. 1(a). the mechanical properties of epoxy and carbon fiber are given in tables 1 and 2, respectively. the plain weave carbon fiber fabric has a weight of 92 g/m². the density of the carbon fibers used was 1.79 g/cm³, fig. 1(b). the carbon fibers were supplied by hangzhou impact new material co., china. a hammer mill was used to grind the date seeds into a powder. a date fruit's seed (pit) and meaty pericarp make up the entire fruit. while dietary fiber, protein, fat, ash, and polyphenols are present in tiny levels in the fleshy pericarp, carbohydrates make up the majority of it. the date seed contains a tiny embryo and is a hard-coated seed that is normally rectangular and ventrally grooved. it makes up between 6 and 20% of the fruit's overall weight and weighs between 0.5 and 4 g. fruit maturity, grade, and variety affect the weight [32-35]. dates have the following chemical compositions: 3.1-7.1% moisture, 2.3-6.4% protein, 5.0-13.2% fat, and 0.9-1.8% ash. additionally, seeds have the highest concentrations of antioxidants (580-929 m trolox equivalent/g), phenolics (3,102-4,430 mg gallic acid equivalent/100 g), and dietary fiber (78-80 g/100 g) [16]. 254 a. abdelhussien, g. yang, e. k. hussein, l. li, h. al-abboodi, b. mohamad a) b) fig. 1 a) epoxy and b) carbon fiber table 1 mechanical properties of epoxy property flexural modulus [mpa] tensile strength [mpa] mix ratio by weight at 25°c color density [g/cm³] viscosity [mpas] resin 2900 75 100:30 light amber 1.1 1100 table 2 mechanical properties of carbon fiber property young’s modulus [gpa] tensile strength (mpa) elongation to failure [%] density [g/cm³] thickness [mm] weight [g/m²] carbon fiber 238 3845 1.6 1.79 0.15 92 2.1.1. extraction and preparation of date seeds using a knife and a firm brush, the date seed (ds) surface layers were removed locally. the ds was next cleaned with distilled water and left to air dry for 24 hours. an oven set to 70°c for 8 hours was used to dry out the ds's moisture content. the ds was then broken up and put through a hammer mill to create a 200 µm spherical powder as shown in fig 2. in order to ensure the correct fiber-to-matrix orientation between the granulated date seed and the carbon fiber-reinforced epoxy resin composite, lignocellulose characterization of the date seed powder was performed. fig. 2 date seed after hammer milling in form of 200 µm spherical powder analysis of the mechanical characteristics of date seed powder-based composite... 255 2.2 composite preparation composite samples were fabricated using two different methods: the hand lay-up technique and vacuum bagging. the vacuum bagging process involved several components, including vacuum bagging film, vacuum pump, vacuum chamber, molding material, sealing tape, resin feed connector, pipe for resin infusion, epoxy resin, peel-ply material, and resin infusion mesh. these components were crucial for the successful implementation of the vacuum bagging process. these methods were used because of their simplicity and availability of the items. the laminating mixture of the epoxy resin and the granulated date seed powder was applied by pouring in the prepared mold and using a paint roller to properly dress the surface to fit in. the carbon fiber was then applied to the poured resin using the woven-proven method. in consolidating the laminate, paint rollers were used, thoroughly wetting the reinforcement and removing the entrapped air. subsequent layers of carbon fiber reinforcements laminate mixture of epoxy resin and granulated date seeds were added to build the required composite thickness (table 3). both cobalt amine-based accelerator and hardener (epolam 2042 hardener) were employed as curing agents for epoxy resin and were later vacuum bag consolidated for approximately an hour. the constructed samples were kept at room temperature overnight to finish the curing process. table 3 weight variation of each test piece no cfr epoxy [%] ds powder [%] weight of cfre [kg] weight of ds [kg] total weight of composite [kg] 1 100 0 0.297 0.285 2 85 15 0.264 0.016 0.280 3 75 25 0.297 0.031 0.328 4 65 35 0.297 0.083 0.380 various compositions of the composite have been considered in this work: 1. 100 % carbon fiber epoxy and 0 % date seeds powder 2. 85 % carbon fiber epoxy and 15 % date seeds powder 3. 75 % carbon fiber epoxy and 25 % date seeds powder 4. 65 % carbon fiber epoxy and 35 % date seeds powder 3. mechanical tests tensile, three-point test (bending), impact and hardness tests were performed using the generated composite samples. for each test, ten samples were used, and the mean values of results are given. an electronic universal testing machine from sinotest equipment co., td. with a 100 kn load frame (in line with astm d3039 standards), was used for the tensile test. to prepare the rectangular composite laminate for mechanical testing, the finished product, measuring 300 mm × 250 mm and with a thickness of 3 mm, underwent precise slicing into the required sizes. for the tensile testing, we followed astm d 3039 guidelines and cut rectangular coupons measuring 250 mm × 25 mm from the cured composites. the testing was carried out at a controlled displacement rate of 2 mm/min. rectangular samples are preferred over dogbone samples in evaluating the tensile properties of composites because the latter often experience splitting at regions where width changes. to ensure consistency, the rectangular samples were designed with a gauge length of 175 mm. 256 a. abdelhussien, g. yang, e. k. hussein, l. li, h. al-abboodi, b. mohamad for the bending strength test, as per astm d 790 specifications, each specimen was 4 mm thick and 13 mm wide. the test involved placing the specimen on two supports and loading it in the middle. the flexural testing was conducted to assess the composite's resistance to deflection, and it was carried out at a loading rate of 2 mm/min. to determine the izod impact strength, we employed the izod impact device in accordance with astm 7136 standard. each sample had dimensions of 55 mm × 10 mm × 5 mm, as illustrated in fig. 3. a) b) c) fig. 3 sample dimensions in: a) tensile; b) bending; and c) impact test additionally, the hardness test was performed using the hardness shore d method, following astm 2240 guidelines, as depicted in fig. 4. fig. 4 from left to right: tensile device, binding device, impact device, and hardness device analysis of the mechanical characteristics of date seed powder-based composite... 257 4. results and discussion 4.1. tensile characteristics the highest force a material can endure when being pulled apart and the amount of stretching it can withstand before failing are both referred to as tensile strength. the mean results of the experimental runs of tensile tests on four different ds powder-reinforced cfr epoxy ratios are given in fig. 5. fig. 5 maximal breaking force for different composites it can be seen that the cfr15 %wt from ds has the highest ultimate force breaking value (58.12 kn). the maximum breaking force of the reference sample was measured to be 48 kn. the incorporation of ds to the epoxy obviously increased the tensile strength it should be emphasized that this result is in agreement with the observations made in the available literature [15-22] that blending an amount of ds to epoxies considerably increased the tensile strength. this is achieved by the variable interfacial adhesion of ds and epoxies in various systems. for active reinforcing potential, adequate load transmission between the epoxy matrix and ds is necessary. since efficient load transfer during the elastic deformation stage under loading is possible due to the interfacial adhesion, a 10% increase in the breaking force was seen in the current study, and, consequently, it was possible to obtain efficient reinforcement in terms of both strength and tensile modulus. it is crucial to point out that the mechanical characteristics of ds modified epoxies rely on the ds content as well. the samples used in the tensile test are depicted in fig. 6. 258 a. abdelhussien, g. yang, e. k. hussein, l. li, h. al-abboodi, b. mohamad fig. 6 sample tensile test: a) and b) 0% ds before and after the test; c) and d) 15% ds before and after test; e) and f) 25% ds before and after the test; g) and h) 35% ds before and after the test 4.2. flexural test to evaluate the stiffness, the bending strength and modulus of the composite materials were evaluated using a three-point bending test. the experimental results are shown in fig. 7, which displays the outcomes of the flexural test performed on the carbon fiber reinforced epoxy resin composite and various compositions of powder date seed. obviously, the maximum bending force is exhibited by cfr-% 35 wt. ds and it is 1.010 kn. this indicates that the composition will be able to withstand the most bending caused by flexural loads at this composite combination ratio. it is inevitable that fractures in flexural tests start on the tension side (the side opposing the loading direction) and steadily advance to the compression side until the composite fails completely. beginning of a fracture on the tension side means that the material's strength has been loosened, which will cause the curve to depart from linearity. because of this, the maximum stress at failure on the tension side is regarded as a material's flexural strength. flexural strengths are improved when the ds loading is raised to 35 wt% percent, which boosted both flexural force by 24% in comparison to the reference sample. similarly to this, 15 wt%, 25 wt% percent of ds loading added to gfrp improved flexural strength, when measured at lab temperature, was 7% and 9 %, respectively. this increase in strength and modulus might be explained by the effective load transfer from the polymer matrix to the ds via the delicate matrix/ds contact. analysis of the mechanical characteristics of date seed powder-based composite... 259 fig. 7 maximum bending force in different composites fig. 8 shows the samples used for bending, before and after testing. a) b) c) d) e) fig. 8 sample bending test: a) 0% ds after the test, b) 15% ds during the test, c) 25% ds before the test, d) 35% ds under the test, and e) 35% ds after the test 260 a. abdelhussien, g. yang, e. k. hussein, l. li, h. al-abboodi, b. mohamad 4.3. impact test the impact resistance ability of the materials was also tested. the hybrid biocomposite (carbon fiber + (epoxy + 25% ds)) added more strength to the material compared to the reference cfrp. it has increased by 17% with addition of ds, as shown in fig. 9, and it is about 4.7 j of energy absorbing. the fracture behavior for laminate structure design combinations is depicted in a closeup image, fig. 10. the behavior of composite materials might change depending on the laminate structure. before they fail, composite materials may show a variety of damage, including matrix cracking, fiber fracture, fiber pullout, fiber rupture, and fiber debonding. typically, when fibers break, the entire structure might also fail. according to damage behavior seen upon impact, fiber rupture is the primary cause of the total damage, as shown in fig. 10c. this suggests that in order to assure the capacity of the composite lamination structure, the choice of reinforcing type is a very crucial issue to be considered. additionally, the impacts of various matrix types revealed varying capacities for withstanding impact, with epoxy resin with a 25 wt% ds having a greater capacity than the standard. the behavior of the hybrid and architecture-oriented structures under impact also differed dramatically. the impact energy absorption of the various composite lamination structures revealed that, when compared to other composite structures, the carbon fiber + epoxy with 25wt%ds specimen had a greater energy absorbing performance, reaching the value of 4.7 j. overall, the findings demonstrate the impacts of various composite material reinforcement, matrix, and structural types. in the analysis of contrasts between the experimental. fig. 9 maximum strength impact analysis of the mechanical characteristics of date seed powder-based composite... 261 a) b) c) d) fig. 10 sample impact: a) 0% ds before the test, b) 15% ds after the test, (c) 25% ds after the test, and d) 35% ds before the test 4.4. hardness test hardness of the composites was determined by using shore d hardness testing machine. the machine measures the ability of the sample to resist penetration by measuring the depth of indention). the four samples: carbon fiber epoxy resin, reinforced carbon fiber epoxy resin with 35 %wt. gd, 25 %wt. gd, 15 %wt. gd and 0 %wt. gd were tested for hardness by determining their various shore d hardness number, which is based on the depth of the indent. the results are presented in fig. 11, which shows that the granulated date seed reinforced carbon fiber epoxy resin with 25 % wt. gd has the highest hardness value of 84hrb. the chart also shows that the hardness of the reinforcement increased as the percentage composition of the date seed granule reduced, with the exception at the 15-25 %wt. ratio of the reinforcement. fig. 11 shore hardness number hrb in different composition 262 a. abdelhussien, g. yang, e. k. hussein, l. li, h. al-abboodi, b. mohamad 5. conclusions the focus of this study was on evaluating the mechanical properties of carbon fiberreinforced polymers using date-seed micro fillers (ds) in combination with epoxy. the aim was to analyze the effects of different design configurations of laminated structures. for this purpose, composite specimens were fabricated using both vacuum bagging and hand layup techniques. upon analyzing the results, the study yielded the following findings: 1. during the mechanical testing and loading of the specimens, in comparison to gfr epoxy, deformation was greater with the carbon fiber reinforced epoxy resin gds composite. 2. as the content of gds increased, the composite's impact strength also increased. 3. the specimen's ability to absorb energy is greater in the cfr 25 %wt gds composite and declined as gds's content increased. 4. the best variation of the reinforcement of carbon fiber reinforced epoxy with the date seed granules is the gfr-15-25% wt. gds composition, according to observations made regarding the mechanical properties of carbon fiber reinforced epoxy resin reinforced with 35, 25, and 15 and 0% wt. gds. it is important to note that the carbon fiber reinforced epoxy composite performs best at 15 %wt. gds, having the highest breaking force, 35%wt. gds the highest bending force, 25% wt. gds the highest energy absorption capacity, and 25 %wt. gds the highest hardness. acknowledgement: the authors would like to acknowledge the financial supports 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no 1, 2018, pp. 65 75 https://doi.org/10.22190/fume180109011c © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper the “sport” of rough contacts and the fractal paradox in wear laws udc 539.6 michele ciavarella 1 , antonio papangelo 1,2 1 department of mechanics, mathematics and management, politecnico di bari, italy 2 department of mechanical engineering, hamburg university of technology, germany abstract. in a recent paper in science, namely, “the contact sport of rough surfaces”, carpick summarizes recent efforts in a “contact challenge” to predict in detail an elastic contact between the mathematically defined fractal rough surfaces under (very little) adhesion. he also suggests the next steps that are needed to “fulfill da vinci’s dream of understanding what causes friction”. however, this is disappointing as friction has been studied since the times of leonardo and in 500 years, no predictive model has emerged, nor any significant improvement from rough contact models. similarly, a very large effort we have spent on the “sport” of studying rough surfaces has not made us any closer to being able to predict the coefficient of proportionality between wear loss and friction dissipation which was already observed by reye in 1860. recent nice simulations by aghababaei, warner and molinari have confirmed the criterion for the formation of debris of a single particle, proposed in 1958 by rabinowicz, as well as reye’s assumption for the proportionality with frictional loss, which is very close to archard anyway. more recent investigations under variable loads suggest that reye’s assumption is probably much more general than archard’s law. the attempts to obtain exact coefficients with rough surfaces models are very far from predictive, essentially because for fractals most authors fail to recognize that resolution-dependence of the contact area makes the models very ill-defined. we also suggest that in the models of wear, rough contacts should be considered “plastic” and “adhesive” and introduce a new length scale in the problem. key words: rough contact, fractals, adhesive wear, reye’s law, archard’s law, rabinowicz’ criterion  received january 09, 2018 / accepted february 07, 2018 corresponding author: michele ciavarella department of mechanics, mathematics and management, politecnico di bari, viale japigia 182, 70126 bari, italy e-mail: mciava@poliba.it 66 m. ciavarella, a. papangelo 1. introduction of all the basic tribological phenomena of contact, adhesion, friction, lubrication and wear, very few have really benefited from the detailed studies many academics are putting into exploring rough contacts. in a recent comment in science, namely, ―the contact sport of rough surfaces‖ [1], carpick has summarized a recent competition where ―approximate models of interacting surfaces competed against a supercomputer solution‖, and concluded that ―the multiscale approaches had the winning score‖. also that, although multiscale models were already invented by archard [2], ―the time has come to standardize multi-scale descriptions with elastic coupling‖, providing, however, no experimental evidence that this effect of elastic coupling is really so important in any tribological model. one of the basic laws of friction dictates that ―force of friction is independent of the apparent area of contact‖ but was already noticed by leonardo da vinci although not properly published at a time. do we have any model today that can predict the friction coefficient, based on the rough surface details? on the contrary, despite very large attempts to correlate friction coefficient, including roughness, no clear formula is known. perhaps some effects of roughness on friction in viscoelastic materials are understood qualitatively, but basically they correspond to a single scale of roughness, and they are extremely sensitive to the so-called large wavevector cutoff, which remains rather arbitrary [3]. archard’s model did predict the linearity of the contact force with the real contact area already in 1953, and at that time, this was a real innovation. in fact, he even anticipated ―fractals‖ which were much later used in a more refined form in the ―contact sport‖ of rough surfaces. yet it was only much later that ciavarella et al. [4] remarked that linearity involves a linear coefficient which is scale-dependent, and in particular it goes to zero for realistic fractal geometry. this was done with the not very popular choice of a weierstrass series as a fractal, but later gaussian model theories [5] did not change the basic conclusion that ―no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension — the contact area consists of a set of fractal character for all values of the geometric and loading parameters‖. hence, the contact area was found to have a (limiting) fractal dimension of (2−d), where d is the fractal dimension of the surface profile. the idea [5] of a ―magnification‖-dependent solution is rather a mathematical trick for avoiding the discussion of an ―ill-defined‖ nature of the contact area. wear remains the least scientifically understood tribological process. the most common approach in wear refers to archard [2] as wear volume v is proportional to sliding length l, normal force w, and inverse with hardness h of the material [6]. hence, wear rate v is proportional to pressure p via a wear coefficient k: h kp v  (1) this proportionality is satisfied for many pure metals, see ref. [7], but more in general, there is a more complex dependence (see fig.1), in some cases even contradicting the archard wear law inverse dependence on hardness, especially for very high hardness. the latter result may be connected to kragelsky’s [8] observation that catastrophic wear occurs when surface layers are harder than substrate, whereas the opposite is suggested for minimizing wear (see also popov et al. [9]). therefore, even just looking at fig.1 reveals the ―sport‖ of rough contacts and the fractal paradox in wear laws 67 that there is more in k than what ―wear coefficient‖ constant suggests. it probably depends on other factors, and indeed, [7] suggests there are two ranges: in range i the wear coefficient is constant and low, and the wear resistance increases proportionally to hardness. instead, in range ii, the wear rate can either increase (but less than in range i), or be independent of, or decrease with hardness depending on the particular toughness. fig. 1 some example dependencies of wear resistance for various materials (from [7]) hence, while the influence of hardness is not rigorous in the archard law, the proportionality with normal load is more robust and indeed it was proposed much earlier in europe by reye [10] in a rather unfortunately unknown paper (it has 50 citations in google scholar, which in the days of bibliometric assessments of research impact, should mean it is a rather bad paper!). in italy, for example, reye's assumption is universally suggested in undergraduate engineering course of brakes design at least after the ii world war, and it is usually put in terms of wear rate being ―proportional to frictional work expenditure". before continuing our discussion, an interesting paper to cite is a very recent one by akbarzadeh & khonsari [11]. they investigated pin-on-disk experiments with a variable loading sequence. the experimental results show that the loading sequence affects weight loss (contradicting another old known law in fatigue, palmgren-miner), which correlates well with the dissipated power, regardless of the loading sequence, while archard’s law was shown to be inaccurate. hence, reye’s assumption is probably more general than archard’s. the general picture remains confused, and there is no alternative to measurements. meng and ludema [12] confirm, after a detailed analysis, that no single predictive equation or group of limited equations could be found for general and practical use in wear, and suggest that the ―reasons include the perpetuation of erroneous and subjective expressions for the mechanisms of wear‖, and ―the paucity of good experiments to verify proposed models‖. we suspect that this situation will continue for many more attempts to propose miraculous predictive equations for wear. 68 m. ciavarella, a. papangelo 2. the rabinowicz critical length scale one of the interesting ideas for adhesive wear (one of the most prevalent types of wear) was suggested in 1958 by rabinowicz [13] in wear and was later forgotten (the paper has 12 citations in google scholar, even worse than reye’s paper [10]), distinguishing the regimes of plastic smoothing and the formation of wear particles in a contact of homogeneous sliding bodies. its summary says that "whether a fragment transferred during sliding comes off as a loose wear particle is shown to depend on whether its elastic energy exceeds the surface energy at its point of attachment. calculations show that, for an equiaxed fragment, there is a critical size, such that smaller fragments remain adherent while larger fragments come off in loose form. experimental support for this critical size concept is cited. the material parameters involved are such that smaller fragments are formed with hard metals than with soft metals, thus explaining the better surface finish during sliding observed with the former". there was considerable experimental evidence in support of the criterion, as for example, in the tests involving steel sliding on steel,with only the largest fragment formed (diameter 120 µm) "could be loosened from the surface to which it adhered by gentle mechanical treatment". in a recent paper, aghababaei et al. [14] have confirmed by explicit mesoparticle simulation, that the transition does exist so strongly supporting to general rabinowicz’s competition of plastic deformation and fracture. the rabinowicz-aghababaei length scale is more in details 2 jσ w gλa  (2) a function of shear modulus g, the fracture energy per unit area w and junction strength σj with a shape factor λ. therefore, the basic ideas for wear return regularly but were formed already a long time ago: the novelty is that with new simulation techniques, it is tempting to think that one could be perhaps able to "predict" wear coefficients, which are otherwise experimentally widely differing by more than 7 orders of magnitude. to explain the concept of wear coefficient we need to mention archard who proposed that "we postulate: worn volume ~a 3 and effective sliding distance ~a, therefore, the contribution of this contact to the wear per cm of sliding ~a 2 ; also load supported by contact ~a 2 . therefore, for this contact, the contribution to the wear rate is proportional to the load supported by it. a similar argument applies to all other contacts, and the total wear rate is proportional to the load". he had in mind, of course, the plastic junctions of bowden and tabor [15], for which the real contact area is obviously proportional to load, via hardness. hence, the proportionality of wear rate with normal load is known almost universally as the "archard wear law", because of the hardness dependence, which we have seen is not so universal. the linearity with load also occurs under elastic regime (as proved also by archard in his multiscale model), with an important difference that in this case there is no reference to hardness. however, archard's explanation for his law is unconvincing in that he assumes a constant density of asperities with height, which is universally known to be not a reasonable assumption of a real surface and more importantly, all asperities are wearing in his model, in contrast with the rabinowicz-aghababaei critical size concept. instead, reye's assumption, proportionality of wear with frictional work, has also been confirmed again by very nice modern simulations in small systems by aghababaei et al. [16]. the ―sport‖ of rough contacts and the fractal paradox in wear laws 69 3. multiscale contact attempts to derive wear coefficient and the fractal paradox despite our concern that the archard law is inevitably connected to the concept of hardness, and the latter is strongly connected to plastic deformations, recent attempts are emerging to infer wear coefficients by upscaling the concept of the rabinowicz length scale in multiscale contact. frérot et al. [17] for example inserted the rabinowiczaghababaei critical scale in the archard model, i.e. assumed k is the probability that contact area a is larger than a * (where a * is the area of critical length scale a * ) for a given load w ( ) ( , ) a k p a a p a w da       (3) and found common characteristics with experimental observations. more precisely, starting from analytical predictions, using the quasi-realistic greenwood-williamson model with exponential distribution of asperities =(c/σs)exp(-zs/σs) for zs>0, and σs being a scale parameter of the order of rms amplitude, p(a,w) turns out a negative exponential independent of applied load w            ss σb a exp σb w,ap 1 (4) where b=πr for elastic, and b=2πr for plastic model. therefore, (i) wear coefficient k is also independent of the load,           sσb a expk (5) also, in this model (ii) there is a linear relationship between the contact area and the applied load in both elastic and plastic cases. both (i) and (ii) observations are consistent with experimental observations. k values reported in experiments are in the range 10 -8 10 -1 , and there are too many constants in the model to judge if the prediction is realistic or not. notice the dependence on the rabinowicz length scale but also that on roughness amplitude as for lower roughness lower wear rate is predicted. one aspect that is not discussed, however, in [17] is the dependence on radius. also a lower radius shows a lower wear rate, and given that the sensitivity to radius to resolution of the instrument (or to truncation of the fractal process) is extremely high, this would mean that the elastic model predicts always infinitesimally small wear! we can add this to many rough contact paradoxes, which are nevertheless often forgotten, when attempting to model precisely the trend towards meaningless results. notice that this is not a consequence of the simple asperity model. lately, the academic desire to improve models has led to a large effort all over the world to improve elastic models with fractal surfaces, as we have discussed in the opening of the paper, so that carpick calls this ―contact sport‖ [1, and see refs. therein]. this is due to the fact that it is much more difficult to include plasticity in models rather than refining asperity models. in fact, more refined models in [17] with more "exact" bem numerical calculations show that the probability density function of the cluster areas follows a negative power-law with exponent 1.5 in a certain interval of a, where it is independent 70 m. ciavarella, a. papangelo of w, up to a upper bound value am which increases with the load. this results in small differences with respect to the simple and analytical archard-gw exponential model: namely, the wear coefficient transitions from zero (i.e. no observable wear) to a constant value. this constant value decreases if a * /λs 2 increases, where λs is the smallest wavelength in the roughness spectrum. this dependence (which is the counterpart of the dependence on the radius of the gw model) is not fully discussed in [9] but obviously means that the truncation of the fractal process is a very sensitive parameter also in the ―refined‖ bem models, as in the fractal limit, λs→0 or r→0 and there is no longer predicted wear. the returning problem as we have discussed recently in [18] is that the ―real contact area‖ is often an ill-defined, ―magnification‖ dependent quantity. in the bowden-tabor view, which was a plastic model, later used by archard to derive his hardness dependent law, this problem obviously does not exist. of course, macroscopic quantities like stiffness or load vs. separation, are not so magnification dependent, and on these properties, elastic models can be used. finally, notice that another (more ambitious) interpretation of the wear coefficient in [17] based on an alternative assumption, namely that wear coefficient k would be equal to the ratio between the area of contact satisfying the rabinowicz-aghababaei critical scale criterion, to the total real contact area, ie. not making the archard connection with the probability of wear particle formation shows features in contrast with experiments, and here it is disregarded. summarizing, the analytical archard-gw exponential model with the rabinowiczaghababaei critical scale concept has qualitatively correct features, and could be further considered for other investigations, provided we knew the scale for truncation of the fractal process. one can obviously decide this scale is universal and the reason in this regard. but it may well be quite a strong assumption. 4. plasticity models the discussion in the paper so far, and the very nature of the archard law, with its hardness in the denominator, highly suggest that a plastic model should be used in any attempt to extrapolate wear coefficients. in an attempt to solve the paradoxes of ciavarella et al. [4], gao et al. [19] added plasticity in the weierstrass model. while the elastic model [4] showed that the ―fractal limit‖ consists of an infinite number of contact spots, with zero size, subjected to infinite contact pressure, gao et al [19]’s model found that while the total contact area and contact pressure are well defined, it remains impossible to predict the actual number of contacts or their size. they also suggest to add adhesion (but the problem with both plasticity and adhesion has not even been attempted yet) or truncating the fractal process where the fractal description breaks down. this is indeed typically done in numerical simulations such as those described in ―contact challenge‖ exercises described in [1], but in that case the truncation is rather arbitrary, and not many authors make even an attempt to discuss how they make this truncation. on the other hand, this poses some questions about the use of elastic simulations. gao et al. [19]’s model shows that, for coarse resolutions, the elastic model is a good description of the full solution. hence, it becomes important to look at the 3 dimensionless parameters defined by gao et al. [19], which in practice essentially depend on wavelength scale parameter and amplitude parameter in the power spectrum density (continuous approximation) of the the ―sport‖ of rough contacts and the fractal paradox in wear laws 71 weierstrass profile, the elastic modulus and the yield strength. therefore, the results of the previous paragraph, where radius and amplitude of roughness entered directly into the prediction of the k wear coefficients, would need to be modified only if the resolution at which the problem is looked is increased enough to see plasticity effects, and this depends on an additional set of parameters. if the plastic regime is relevant, then the constants involved are the same, but in a different mix. notice also that the results of gao et al. [19] have been compared with much more sophisticated predictions with detailed finite element simulations of pei et al. [20], and show very similar features, for example the predicted true contact area as a function of the ratio of yield stress to elastic modulus. in gao’s calculations, the elastic–plastic transition (the elastic regime) disappears. pei et al. [20], however, report more in details that plasticity produces qualitative changes in the distributions of the connected contact regions, with large clusters becoming more likely. however, as seen in fig. 2, the elastoplastic model shows that the distribution of contact clusters is much closer to the very simple overlap model than to the elastic one. once again, decades of discussion about elastic models (the ―contact sport‖ of rough surfaces) seem largely unjustified. we hope at least that in the future, simplicity will guide our efforts rather than competitions over precise results of mathematical problems of little real tribological interest. a) b) c) fig. 2 an example of contact area prediction for: a) elastic; b) elasto-plastic and c) rigid overlap model (adapted from [20]) 5. adhesion models in the paper so far, and in the review of previous models, an important ingredient for "adhesive wear" is surprisingly missing. they do not consider that at the contact interface the average size of the micro contacts will strongly depend on the strength of adhesion, which is a contradiction to the idea that plasticity junction should be formed in the first place, that is before possibly breaking in a wear particle. the rabinowicz criterion is based on consideration of competition of adhesion and plasticity. analysis carried out in [21] illustrates that the rabinowicz criterion is valid even if this competition occurs directly in the same interface. in most of the most previous works, the size of asperity is assumed as given and independent of adhesion. however, in an adhesive contact of curved surfaces, the contact size does depend on adhesion, too. in this section we discuss this aspect of the problem. let us note that there 72 m. ciavarella, a. papangelo also exist other approaches where the poorly defined notion of an asperity is avoided [9, 22]. however, even within such a concept, the effective work of adhesion (which is proportional to the real contact area, which is also an ill-defined quantity [18]) does play a central role, and this area is also dependent on adhesion. for the limited scope of the present paper, we shall attempt an elementary step towards a more complete picture, adding adhesion at the interface. in the presence of adhesion, the standard jkr equation [23, 24] gives the contact radius a for a given normal force p, r ae waeπp 3 4 8 3 3    (6) where 1/e * =1/[e1/(1-ν1 2 )]+1/[e2/(1-ν2 2 )] and ei, i=1,2, are young's moduli and νi, i=1,2, poisson's ratios of each material. in other words, contacts cannot be smaller than these sizes, respectively upon approach, and before detachment. moving, therefore, to a multiasperity contact, we disregard the archard's uniform multi-asperity contact model as unjustified geometrically, despite its leading to reasonable results. indeed, surfaces are today considered to be approximately gaussian. an exponential tail is, however, a reasonable assumption, for qualitative purposes. consider a density of asperities n and f, g the functions relating area and load for each asperity, to its compression δ, which is here given by the height of the asperity minus the separation between mean planes (zs-u). the exponential distribution of heights permits to elucidate some results very clearly in the present context, obtaining the number of asperities in contact, total area a, and load p, as                                                                                  sss δ sss δ ss δ s σ δ d σ δ exp σ δ gnp σ δ d σ δ exp σ δ fna σ δ d σ δ exp σ u expcnn 1 1 1 (7) where δ1=0 in the phase of loading, because jkr by assumption takes contact only when there is intimate contact, but δ1= δ0 upon unloading. irrespective of whether we are in the phase of loading or unloading, it appears clear that area a and load p are proportional to each other, as in the classical repulsive case, and regardless of the jkr assumption. notice in fact that these results hold also for plastic and any other constitutive law within the approximation of asperities, of course. hence, if we do consider the jkr solution (see appendix for details), if ain > a * 1 1 2 3 3 3 1/ 2 * 2 * ˆ (2 )in in j r w gw a a r ee r w                 (8) the ―sport‖ of rough contacts and the fractal paradox in wear laws 73 then the entire distribution of asperities will wear, and this will be proportional (or nearly proportional) to load. therefore, we retain the main experimental observations known as the archard law. for simplicity, we are using, of course, the same surface energy which enters in the interface problem also in the detachment of wear particles -as a reasonable first guess - and otherwise it is trivial to generalize the model keeping two separate material properties. also, to make qualitative assessment, we assume that only one of the bodies is wearing out, the softer one which has also a smaller elastic modulus so that we can consider e * ≈ e. therefore, we can consider g = e / [2(1ν)] 3/ 2 2 3 1/ 32(2 ) (1 )j e r w            (9) as σj/e=310 -3 for many materials, assuming λ=1 e w *r 6104 (10) for a crystalline material, w/e=0.05a0 where a0 is atomic spacing, of the order of less than a nanometer; therefore, we can estimate very crudely w/e≈10 -11 m, and the condition becomes μm40r (11) naturally, under this condition, all the asperities will wear out, and the k coefficient will take an extremely unlikely value of one. it is, therefore, unlikely that this condition provides anything very useful in practice. it is more likely to be another of the possible paradoxical results of applying multiscale contact models to archard and rabinowicz’s ideas. however, notice that the paradoxical behavior occurs here not in the fractal limit, but rather on the contrary, if the resolution is not increased enough for asperity radii to decrease below this value. obviously, if the condition (11) is not fulfilled, one would have to integrate the asperity radii greater than the rabinowicz aghababaei length scale, similarly to what is done in [17]. the conclusion would simply be that for small a * , wear coefficient k will remain unity until the condition (11) becomes violated, and then it would start to decrease, vaguely similarly to the adhesionless counterpart of [17]. actually, we can predict that for large a * , wear coefficient k will tend asymptotically exactly to the adhesionless one, since for large contact areas the load is also large and the adhesive correction is likely to be small. 6. conclusions in the present paper, we have discussed that the contact ―sport‖ of simulating an elastic multiscale contact with fractal accurate models that has dominated part of the specialized literature, is still very remote from solving any actual real tribological problem as prediction of friction coefficient or, even worse, wear coefficient. we have shown that many classical ideas that return periodically have been also confirmed in modern ways by numerical simulations. however, the most promising form of wear law 74 m. ciavarella, a. papangelo is reye’s assumption rather than archard’s, at least according to some recent experiments under a variable amplitude load. archard’s models are often surprisingly discussed assuming no plasticity, clearly despite the hardness term in the denominator; it calls for an elasto-plastic analysis, and in this respect we have not made much progress since the qualitative ideas of reye, archard, bowden-tabor. also, the ―adhesive‖ wear law calls strongly for adhesion, and yet no model so far has ever discussed the effect of adhesion. we have attempted some qualitative discussion of these two aspects, and, concerning adhesion, we have effectively introduced another critical length scale in the problem, which is the minimum size of adhesive contacts based on jkr theory [23, 24]. the effect of adhesion is as intuitively expected, that of increasing the contact areas, and hence the wear coefficient with respect to the non-adhesive case. however, at least within the exponential approximation, the normal load and the area remain proportional. also for some geometries, and combination of material properties, the effect would seem to obtain wear coefficients of unity; in particular, we write this in terms of an upper bound to the radius of asperities, above which this very high wear is predicted. as we estimate this upper bound to be quite high (hundreds of microns) but not incompatible with worn particle size measurements, we find reason for possible further discussion. acknowledgements: the authors thank r. aghababaei and v. popov for useful discussions and for sharing preprints of their submitted papers. a.p. is thankful to the dfg (german research foundation) for funding the project ho 3852/11-1. references 1. carpick, r.w., 2018, the contact sport of rough surfaces, science, 359(6371), pp. 38-38. 2. archard, j.f., 1953, contact and rubbing of flat surfaces, journal of applied physics, 24, pp. 981-988. 3. ciavarella, m., 2018, a simplified version of persson's multiscale theory for rubber friction due to viscoelastic losses, journal of tribology, 140(1), 011403. 4. ciavarella, m., demelio, g., barber, j.r., jang, y.h., 2000, linear elastic contact of the weierstrass profile, in proceedings of the royal society of london a: mathematical, physical and engineering sciences, vol. 456, no. 1994, pp. 387-405. 5. persson, b.n., 2001, theory of rubber friction and contact mechanics, the journal of chemical physics, 115(8), 3840-3861. 6. dimaki a.v., dmitriev, a.i., menga, n., papangelo, a., ciavarella, m., popov, v.l., 2016, fast highresolution simulation of the gross slip wear of axially symmetric contacts, tribol. trans., 59, pp. 189-94. 7. hornbogen, e., 1975, the role of fracture toughness in the wear of metals, wear, 33(2), pp. 251-259. 8. kragelski, i.v., 1965, friction and wear, butter worth. 9. popov, v.l., gervé, a., kehrwald, b., smolin, i.y., 2000, simulation of wear in combustion engines, computational materials science, 19(1), pp. 285–291. 10. reye, th., 1860, zur theorie der zapfenreibung, j. der civilingenieur., 4, pp. 235-255. 11. akbarzadeh, s., khonsari, m.m., 2016, on the applicability of miner’s rule to adhesive wear, tribology letters, 63(2), pp. 1-10. 12. meng, h.c., ludema, k.c., 1995, wear models and predictive equations: their form and content, wear, 181, pp. 443-457. 13. rabinowicz, e., 1958, the effect of size on the looseness of wear fragments, wear, 2, pp. 4–8. 14. aghababaei, r., warner, d.h., molinari, j.-f., 2016, critical length scale controls adhesive wear mechanisms, nature communications, 7, 11816. 15. bowden, f.p., tabor, d., 2001, the friction and lubrication of solids, clarendon press. 16. aghababaei, r., warner, d.h., molinari, j.-f., 2017, on the debris-level origins of adhesive wear, proceedings of the national academy of sciences, 114(30), pp. 7935–7940. 17. frérot, l., aghababaei,r. molinari, j.f., 2018, on the understanding of the wear coefficient: from single to multiple asperities contact, submitted, personal communication. the ―sport‖ of rough contacts and the fractal paradox in wear laws 75 18. ciavarella, m., papangelo, a., 2017, discussion of “measuring and understanding contact area at the nanoscale: a review‖ (jacobs, tdb, and ashlie martini, a., 2017, asme appl. mech. rev., 69(6), p. 060802), applied mechanics reviews, 69(6), 065502. 19. gao, y.f., bower, a.f., 2006, elastic–plastic contact of a rough surface with weierstrass profile, proceedings of the royal society of london a: mathematical, physical and engineering sciences, 462(2065), pp. 319-348. 20. pei, l., hyun, s., molinari, j.f., robbins, m.o., 2005, finite element modeling of elasto-plastic contact between rough surfaces, journal of the mechanics and physics of solids, 53(11), 2385-2409. 21. popov, v.l., 2017, generalized rabinowicz’criterion for adhesive wear for elliptic micro contacts, aip conference proceedings, 1909 (1), 020178. 22. li, q., popov, v.l., 2018, on the possibility of frictional damping with reduced wear: a note on the applicability of archard’s law of adhesive wear under conditions of fretting, physical mesomechanics, 21(1), pp. 94-99. 23. johnson, k.l., kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids, proceedings of the royal society of london, series a, 324, pp. 301-313. 24. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer, berlin heidelberg. appendix – on jkr equations we introduce the following dimensionless notation for the approach, and the load in spherical contact rw p p̂; r ˆ; r a â      2 (12) where  = (e * r/w) 1/3 . the jkr theory gives 332 8 3 4 2 ââp̂;â∠  (13) the jkr theory can be presented in a curve fitted form indistinguishable from the actual jkr curve, in order to be easily used in terms of ̂ 1/ 2 3/ 2 0 0 0 ˆ ˆ ˆ ˆˆ ˆ 1.1( ) 0.43( )p p         (14) where 2 / 3 0 ˆ ˆ(3/ 4) , 5/ 6p     are the jkr values at pull-off in displacement control. to obtain the contact radius as a function of displacement which is easier to use in asperity models, we find a good best-fit as 6 2 1/ 2 3 4 3/ 2 0 0 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ 4.594 10 0.9( ) 8.698 10 ( ) 3.342 10 ( )a                   (15) these equations show an important feature. there is a jump-in to contact situation, where the surfaces approach each other, and jump-out of contact which this depends on the control, but we shall assume that for a multiasperity contact, displacement control is more realistic given we assign the displacement 1/ 3 1/ 3ˆ ˆ(2 ) ; 2 in outa a    (16) plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 305 323 1effect of various porous structures on the shliomis model based ferrofluid lubrication of the film squeezed between rotating rough curved circular plates udc532:621.8 jimit r. patel, gunamani deheri department of mathematics, sardar patel university abstract: efforts have been made to analyze the shliomis model based ferrofluid lubrication of a squeeze film between rotating rough curved circular plates where the upper plate has a porous facing. different models of porosity are treated. the stochastic modeling of christensen and tonder has been employed to evaluate the effect of surface roughness. the related stochastically averaged reynolds type equation is numerically solved to obtain the pressure distribution, leading to the calculation of load carrying capacity. the results presented in graphical form establish that the kozeny-carman model is more favorable as compared to the irmay one from the design point of view. it is observed that the shliomis model based ferrofluid lubrication performs relatively better than the neuringer-rosensweig one. although the bearing suffers due to transverse surface roughness, with a suitable choice of curvature parameters and rotational ratio, the negative effect of porosity and standard deviation can be minimized by the ferrofluid lubrication at least in the case of negatively skewed roughness. key words: squeeze film, curved circular plates, roughness, magnetic fluid, rotation, porous structures 1. introduction wu [1] has analyzed the effect of rotation on the squeeze film performance in porous circular disks. the criteria for neglecting the inertia effects are laid down in this study. murti [2] has discussed the squeeze film behavior between the porous circular disks when the upper disk had a porous facing. by means of the fourier bessel’s series the equation for pressure distribution is derived. it is shown that the enhanced value of permeability parameter has diminished pressure over the entire disk. vora and bhat [3] received september 3, 2014 / accepted october 18, 2014 corresponding author: jimit r patel department of mathematics, sardar patel university, vallabh vidyanagar, anand,gujarat india-388120. e-mail: patel.jimitphdmarch2013@gmail.com original scientific paper 306 j. r. patel, g. deheri have obtained the load carrying capacity of a squeeze film between the curved porous rotating circular plates. the effect of the rotating fluid inertia reduces the load carrying capacity significantly. paras ram and vikaskumar [4] have examined the effects of magnetic field dependent viscosity on the steady axi-symmetric ferrofluid flow due to the rotating disk in a porous medium. verma [5] has investigated the magnetic fluid lubrication effect of a squeeze film between two approaching surfaces in the presence of an externally applied magnetic field, oblique to the lower surfaces. the magnetization effects induce enhanced load carrying capacity and longer squeeze time. lu et al. [6] have evaluated the effects of fluid inertia in magneto hydrodynamic annular squeeze films. it is observed that the inertia correction factor in the magneto-hydrodynamic load carrying capacity is more pronounced with large hartmann numbers. hsu et al. [7] have studied the magnetohydrodynamic squeeze film behavior between circular disks by taking rotational inertial effects into consideration. it is established that the overall squeeze film performance characteristics of rotating circular disks are improved when using an electrically conducting fluid in the presence of a transverse magnetic field. zueco and beg [8] have numerically analyzed the hydromagnetic squeeze film flow between two parallel rotating circular disks with induced magnetic field. this study finds applications in hydro magnetic lubrication of breaking devices and rotating machinery. huang et al. [9] have experimentally studied the ferrofluid lubrication with an external magnetic field. with an appropriate magnetic field the load carrying capacity is significantly enhanced. patel and deheri [10] have analyzed the effect of various porous structures on the performance of a shliomis model based ferrofluid lubrication of the film squeezed in rotating rough porous curved circular plates. it is found that by suitably choosing curvature parameters and rotational inertia, the adverse effect of transverse roughness could be overcome by the positive effect of ferrofluid lubrication in the case of negatively skewed roughness when the kozeny-carman model is deployed. ting [11] has considered the fluid inertia effects to develop a mathematical analog of determination of porous annular disk squeeze film behavior. it is concluded that the inertia effect results in decreased load carrying capacity and reduced squeeze time. a review of the performance of porous squeeze films between lubricated plates is presented by wu [12]. bhat and deheri [13] have dealt with the squeeze film behavior in porous annular disks lubricated with a magnetic fluid. the load carrying capacity rose sharply with increasing magnetization. bhat and deheri [14] have theoretically analyzed magnetic fluid based squeeze film in curved porous circular disks. the pressure, load carrying capacity and the response time are found to increase with increasing magnetization. however, the effects due to magnetization are independent of the upper plate’s curvature. elsharkawy and nassar [15] have discussed the hydrodynamic lubrication of porous squeeze film bearing. the increase in the permeability parameter causes decreased load carrying capacity. gupta and deheri [16] investigate the effect of surface roughness on the behavior of squeeze film in a spherical bearing. it is shown that the performance of the bearing system is adversely affected by the composite roughness of the surfaces. deheri et al. [17] have analyzed the magnetic fluid lubrication of a squeeze film between transversally rough curved plates. in this case it is established that the combined effect of magnetization and curvature parameters leads to an increased load carrying capacity although the effect of standard deviation is adverse in general. basti [18] has studied the combined effect of couple stress and surface roughness on the performance of the film effect of various porous structures on the shliomis model based ferrofluid lubrication... 307 squeezed between curved annular plates. performance of the squeeze film suffer due to radial roughness pattern. however, circumferential roughness pattern improves the squeeze film characteristics for couple stress lubrication. prakash and tiwari [19] consider the surface roughness effect on the response of a squeeze film between two circular plates one with porous facing. deheri et al. [20] analyze the performance of a magnetic fluid based squeeze film between porous circular plates considering porous matrix of variable film thickness. a significant observation is that with a proper selection of thickness ratio parameter, a magnetic fluid based squeeze film bearing with variable thickness could be made to perform considerably better than that of a conventional porous bearing with a uniform porous matrix thickness working with a conventional lubricant. vadher et al. [21] have discussed the behavior of hydro magnetic squeeze films between conducting rough porous circular plates. with a suitable choice of the conductivity parameter the negatively skewed roughness induces load carrying capacity further enhanced significantly owing to magnetization. lastly, patel and deheri [22] analyze the performance of a ferrofluid based squeeze film in rotating rough curved circular plates resorting to shliomis model. recently, patel and deheri [23] have investigated the effect of different porous structures on the performance of a shliomis model based magnetic squeeze film in rotating rough porous curved circular plates. the results presented in graphical form show that the adverse effect of transverse roughness could be compensated by a positive effect of magnetization in the case of negatively skewed roughness, a suitable choice of the rotation ratio and of curvature parameters. it is also found that this compensation appears to be more in the case of the kozeny-carman model as compared to that of irmay one. this paper tends to analyze the effect of various porous structures on the shliomis model based ferrofluid lubrication of a squeeze film between the curved rough rotating circular plates with the upper one having a porous facing. 2. analysis the configuration of the curved circular bearing is given in fig. 1. here the bearing consists of two plates, each of radiuses a. the lower and upper plates rotate with angular velocities l and u respectively, about the z-axis. the upper disk moves normally towards the lower disk with uniform velocity 0 . h . according to the discussion of christensen and tonder [24-26] the stochastic modeling of transverse roughness, film thickness h of the lubricant film is considered to be: shhh  (1) where h is the mean film thickness and hs is the deviation from the mean film thickness characterizing the random roughness of the bearing surfaces. deviation hs is derived by the probability density function: 2 2 3 7 35 ( ) , ( ) 32 0, s s s c h c h c f h c elsewhere          (2) 308 j. r. patel, g. deheri where c is the maximum deviation from the mean film thickness. the details regarding mean , standard deviation  and parameter  which is the measure of symmetry of random variable hs, are taken from the christensen and tonder’s study [24-26]. fig. 1 configuration of the bearing system in view of the discussions of bhat [27] and abhangi and deheri [28], it is assumed that the rotating upper plate lying along the surface determined by the relation: 2 0 exp( );0uz h r r a    (3) approaches with normal velocity 0 . h to the rotating lower plate lying along the surface given by: 2 0[exp( ) 1];0lz h r r a     (4) where  and  are curvature parameters of the corresponding plates and h0 is the central film thickness. film thickness h(r) is defined by (bhat [27], abhangi and deheri [28]): 2 2 0( ) [exp( ) exp( ) 1];0h r h r r r a        (5) shliomis [29] has established that magnetic particles of a magnetic fluid can relax in two ways when the applied magnetic field changes. the first is by the magnetic particles’ rotation in the fluid and the second by rotation of the magnetic moment in the particles. brownian relaxation time parameter b gives particle rotation while relaxation time parameter s describes the intrinsic rotational process. assuming a steady flow and neglecting inertial and second derivatives of s , the equations governing the flow become: effect of various porous structures on the shliomis model based ferrofluid lubrication... 309 0)( 2 1 ).(0 2    ishmq s p (6) )(0 hmis s  (7) )(0 ms ih h mm b    (8) where s is the internal angular momentum, i is the sum of moments of inertia of the particles per unit volume, q 2 1 , together with: 0).(,0,0.  mhhq (9) (bhat [27]), q is the fluid viscosity in the film region, h is external magnetic field,  is magnetic susceptibility of the magnetic field, p is the film pressure,  is the fluid viscosity and μ0 is the permeability of the free space. by making use of eq. (7), in eqs. (6) and (8), one finds that: 0)( 2 1 ).( 00 2  hmhmqp (10) and )(0 m h h mm b  (11) neglecting sb terms, substitution of m in above equation, leads to: 20 0 0 ( . ) ( . ) 4 1 [ ( . ) ( .. ). ( . ) ] 0 2 p b b m h q m h h m h m m h                       (12) from eq. (11), it is easily observed that an initial approximation to m is: h h mm 0 (13) substituting the value of m on the right side of eq. (11), a second approximation to m is found to be: )(0 0 m h m h h mm b  (14) again, substituting this value of m on the right side of eq. (11), third approximation to m is availed as: ]).[(. 22220 0 hh h m hmhm b  (15) 310 j. r. patel, g. deheri in view of the equations of shliomis model in cylindrical polar coordinates with a uniform magnetic field, when both surfaces are solid and the upper one rotates, the governing equation for the film pressure is obtained as (bhat [27]): 3. 3 2 2 2 3 1 0 1 3 1 3 1 3 1 ( 12 ) 12 24 ( ) 10 320 b a u u a nd dp d d h l r h l r h r dr dr r dr r dr                                                4222 2 633 3 5 756 251 14 13 616 27 uuu r dr dp r dr dp r dr dp rh (16) with l1 being layer thickness. neglecting  3 b term, following the discussions of christensen and tonder [24-26] regarding the modeling of roughness and under the usual assumptions of hydro-magnetic lubrication (bhat [27], prajapati [30], deheri et al. [31]) the modified reynolds equation when both plates rotate, takes the form: . 2 1 0 1 2 2 2 1 ( ( ) 12 ) 12 (1 ) 24 3 1 ( ( )) 10 u r r l l d dp g h l r h l r dr dr d r g h r dr                          (17) where. 3 2 2 2 2 3( ) 3 3( ) 3g h h h h             the following non-dimensional quantities are considered: 3 2 2 20 . 20 0 [exp( ) exp( ) 1], , , , h ph r h br cr r p b a h a a h             2 3 2 0 3 . 0 0 0 0 , , , , (1 ), , ,u a r u l h c a s h h h h                         2 * 3 2 2 * 3 * *1 1 ' 3 3 2 0 0 1 (1 ) , , , , , (1 )6(1 ) l c s f u e d l d ll l e l a d el h h e                       (18) the related boundary conditions are: 0 (1) 0, 0 r dp p dr         (19) effect of various porous structures on the shliomis model based ferrofluid lubrication... 311 2.1 a globular sphere model a porous material is filled with globular spherical particles (mean particle size dc) as shown in fig. 2. fig. 2 structure model of porous sheets given by kozeny-carman the kozeny-carman equation is well known in fluid dynamics. relatively better results for pressure drop are obtained when this model is applied to laminar flow. liu [32] and patel and deheri [33] have suggested that the use of the kozeny-carman formula turns in the relation: 2 3 2 '72(1 ) cd e l e l    (20) where e is the porosity and l/l' is the length ratio. from experimental investigations the length ratio is proposed to be around 2.5 under suitable situations. thus, the kozenycarman formula takes the form: 2 3 2180(1 ) cd e e    (21) resorting to the boundary conditions (19), the non-dimensional form of the pressure distribution in the case of kozeny-carman, is derived as: 2 1 1 ( ) ( 6 6 ) (3 4 3) 10( ) ( ) r r f f r s rg h p sa dr dr g h a g h a             (22) where: 3 2 2 2 2 3 ( ) 3 3( ) 3g h h h h             the non-dimensional load carrying capacity of the bearing system is determined from:    1 0 0 . 4 3 0 2 rpdrw ha h w 1 13 3 2 0 0 1 ( ) 3 3 (3 4 3) 2 20( ) ( ) f f r s r g h w sa dr dr g h a g h a                 (23) 312 j. r. patel, g. deheri 2.2 a capillary fissures model fig. 3 structure model of porous sheets given by irmay in fig. 3, the model of porous sheets given by irmay consists of three sets of mutually orthogonal fissures (a mean solid size ds). assuming no loss of hydraulic gradient at the junctions, irmay [34] and patel and deheri [33] obtain permeability: 2 2 31 (1 ) 12(1 ) sd e e             (24) where e is porosity. in view of the boundary conditions (19), the dimensionless pressure distribution for the irmay model is found to be: 2 1 1 ( ) ( 6 6 ) (3 4 3) 10( ) ( ) r r f f r s rg h p sd dr dr g h d g h d             (25) the non-dimensional load carrying capacity is calculated as: 13 0 . 4 002 h w w rpdr a h     1 13 3 2 0 0 1 ( ) 3 3 (3 4 3) 2 20( ) ( ) f f r s r g h w sd dr dr g h d g h d                 (26) 3. results and discussions the dimensionless pressure distribution is calculated from eq. (22) and eq. (25) while the non-dimensional load carrying capacity is determined from eq. (23) and eq. (26) for the kozeny-carman model and the irmay one, respectively. it is obvious that the load carrying capacity increases by: 1 3 0 3 ( ) r dr g h a   effect of various porous structures on the shliomis model based ferrofluid lubrication... 313 and: 1 3 0 3 ( ) r dr g h d   as compared to the case of conventional lubricants. it is observed from eqs. (23) and (26) that the bearing can support a certain amount of load for both the models even when there is no flow. setting the roughness parameters to be zero this discussion reduces to the effect of various porous structures on the shliomis model based ferrofluid lubrication of a squeeze film between rotating curved circular plates. also, considering the magnetization parameter to be zero this discussion turns out to the performance of different porous structures on a squeeze film in rotating curved circular plates (bhat [27]). further, in the absence of porosity this study reduces to the performance of a squeeze film in rotating curved circular plates. lastly, taking curvature parameter to be zero, this is the effect of rotation on squeeze film performance as discussed in wu [1]. it is clearly seen from eqs. (23) and (26) that these expressions for non-dimensional load carrying capacity are linear with respect to magnetization parameter . accordingly, increasing values of magnetization could lead to an increased load carrying capacity in the case of both the models. the graphical representation of the results corresponding to the kozeny-carman model is shown in figs. 4-18 while figs. 19-33 account for the case of the irmay model. from the graphical representation the following appears to be true, 1. a close glance suggests that the load carrying capacity is larger in the kozeny carman model as compared to the irmay one. hence the former is more suitable for the shliomis model based ferrofluid lubrication (figs. 4, 5, 19, 20). 2. in increasing the load carrying capacity, the upper plate’s curvature parameter has a dominant role to play. the increase in the load carrying capacity due to the upper plate’s curvature parameter is bigger in the case of the kozeny-carman model as compared to the irmay one (figs. 5, 7, 8, 13, 14, 15, 20, 22, 23, 28, 29, 30). 3. the effect of  _ and * on the load carrying capacity with respect to  is not that significant. the decrease in the load carrying capacity due to * is bigger in the case of the irmay model. it is observed that there is a heavy reduction in the load carrying capacity due to * in the case of the irmay model (figs. 4, 6-12, 19, 21-27). 4. the reduction in the load carrying capacity due to porosity is higher in the case of irmay (figs. 6, 13, 21, 28). 5. surprisingly, the decrease in the load carrying capacity due to the standard deviation is higher in the case of the kozeny-carman model (figs. 9, 14, 16, 24, 29, 31). 6. regarding the irmay model the effect of skewness on the load carrying capacity with respect to * is not that significant as seen in fig. 25. it can be noted that the positive skewness reduces the load carrying capacity while the load carrying capacity increases due to the negatively skewed roughness. it is easily observed that the trends of load carrying capacity with respect to variance is quite similar to that of skewness as can be seen in figs. 18 and 33 (figs. 10, 11, 15-18, 25, 30-33). 314 j. r. patel, g. deheri fig. 4 variation of load carrying capacity with respect to  and ψ fig. 5 variation of load carrying capacity with respect to  and b fig. 6 variation of load carrying capacity with respect to ψ and e fig. 7 variation of load carrying capacity with respect to ψ and b effect of various porous structures on the shliomis model based ferrofluid lubrication... 315 fig. 8 variation of load carrying capacity with respect to ψ and c fig. 9 variation of load carrying capacity with respect to ψ and σ fig. 10 variation of load carrying capacity with respect to ψ and ε fig. 11 variation of load carrying capacity with respect to ψ and  _ 316 j. r. patel, g. deheri fig. 12 variation of load carrying capacity with respect to ψ and s fig. 13 variation of load carrying capacity with respect to e and b fig. 14 variation of load carrying capacity with respect to b and σ fig. 15 variation of load carrying capacity with respect to c and ε effect of various porous structures on the shliomis model based ferrofluid lubrication... 317 fig. 16 variation of load carrying capacity with respect to σ and ε fig. 17 variation of load carrying capacity with respect to ε and s fig. 18 variation of load carrying capacity with respect to  _ and s fig. 19 variation of load carrying capacity with respect to  and * 318 j. r. patel, g. deheri fig. 20 variation of load carrying capacity with respect to  and b fig. 21 variation of load carrying capacity with respect to * and e fig. 22 variation of load carrying capacity with respect to * and b fig. 23 variation of load carrying capacity with respect to * and c effect of various porous structures on the shliomis model based ferrofluid lubrication... 319 fig. 24 variation of load carrying capacity with respect to * and σ fig. 25 variation of load carrying capacity with respect to * and ε fig. 26 variation of load carrying capacity with respect to * and  _ fig. 27 variation of load carrying capacity with respect to * and s 320 j. r. patel, g. deheri fig. 28 variation of load carrying capacity with respect to e and b fig. 29 variation of load carrying capacity with respect to b and σ fig. 30 variation of load carrying capacity with respect to c and ε fig. 31 variation of load carrying capacity with respect to σ and ε effect of various porous structures on the shliomis model based ferrofluid lubrication... 321 fig. 32 variation of load carrying capacity with respect to ε and s fig. 33 variation of load carrying capacity with respect to  _ and s a close look at the graphs tends to reveal that the adverse effect induced by standard deviation and porosity can be compensated, to a large extent, by the positive effect of magnetization and this compensation is relatively larger in the case of the kozeny-carman model. further, the combined effect of magnetization and negatively skewed roughness goes a long way in minimizing the adverse effect of porosity, standard deviation and positive variance. the negative effect of rotation and porosity can also be reduced by a judicious choice of magnetization and curvature parameters when variance (-ve) is involved. it is found that the negatively skewed roughness causes an increase in the load-carrying capacity and gets raised by 5.88% more in the case of the kozenycarman model. lastly, it is revealed that there is at least 2.6% of increase in the load carrying capacity due to shliomis model as compared to the neuringer-rosensweig one (bhat [27]). 4. conclusion this investigation suggests that the roughness aspect must be taken into consideration while designing this type of bearing systems. the kozeny-carman model may be preferred over the irmay one. in fact, magnetization plays a little role in improving the overall performance of the bearing system in the case of the irmay model when the roughness is even moderate. although there are several factors reducing the load carrying capacity, the bearing can support a good amount of load even in the absence of flow, which is far away from being true in the case of traditional lubricants. a suitable combination of magnetization, rotation ratio and curvature parameter may be more fruitful in the case of the kozeny-carman model. 322 j. r. patel, g. deheri references 1. wu, hai, 1971, the squeeze film between rotating porous annular disks, wear, 18(6) pp. 461-470. 2. murti p.r.k., 1974, squeeze-film behavior in porous circular disks, journal of lubrication technololgy, 96(2), pp. 206-209. 3. vora, k.h., bhat, m.v., 1980, the load capacity of a squeeze film between curved porous rotating circular plates, wear, 65, pp. 39-46. 4. ram, paras, kumar, vikas, 2012, ferrofluid flow with magnetic field-dependent viscosity due to rotating disk in porous medium, international journal of applied mechanics, 4(4), pp. 1250041-58. 5. verma, p.d.s., 1986, magnetic fluid-based squeeze film, international journal of engineering science, 24(3), pp. 395-401. 6. lu, r.f., chien, r.d., lin, j.r., 2006, effects of fluid inertia in magneto-hydrodynamic annular squeeze films, tribology international, 39(3), pp. 221-226. 7. hsu, c.h., lai, c., hung, c.r., lin, j.r., 2008, magneto-hydrodynamic squeeze film characteristics between circular discs including rotational inertial effects, proceedings of the institution of mechanical engineers, part j: journal of engineering tribology, 222(2), pp. 157-164. 8. zueco, joaquín, bég, o. anwar, 2010, network numerical analysis of hydromagnetic squeeze film flow dynamics between two parallel rotating disks with induced magnetic field, tribology international, 43(3), pp. 532-543. 9. huang, w., cong, s., liao, s., 2011, study of the ferrofluid lubrication with an external magnetic field, tribology letters, 41, pp. 145-151. 10. patel, j.r., deheri, g.m., 2013, shliomis model based ferrofluid lubrication of squeeze film in rotating rough curved circular disks with assorted porous structures, american journal of industrial engineering, 1(3), pp. 51-61. 11. ting, l.l., 1972, a mathematical analog for determination of porous annular disk squeeze film behavior including the fluid inertia effect, j. basic eng., 94(2), pp. 417-421. 12. wu, hai, 1978, a review of porous squeeze films, wear, 47(2), pp. 371-385. 13. bhat, m.v., deheri, g.m., 1991, squeeze film behaviour in porous annular discs lubricated with magnetic fluid, wear, 151(1), pp. 123-128. 14. bhat, m.v., deheri, g.m., 1993, magnetic-fluid-based squeeze film in curved porous circular discs, journal of magnetism and magnetic materials, 127(1–2), pp. 159-162. 15. elsharkawy, a.a., nassar, m.m., 1996, hydrodynamic lubrication of squeeze-film porous bearings, acta mechanica, 118(1-4), pp. 121-134. 16. gupta j.l., deheri, g.m. 1996, effect of roughness on the behavior of squeeze film in a spherical bearing, tribology transaction, 39, pp. 99-102. 17. deheri, g.m., patel, r.m., abhangi, n.d., 2010, magnetic fluid based squeeze film between transversely rough curved plates, advanced tribology(proceedings of cist2008 & its-iftomm2008), pp. 54-55. 18. basti, d.p., 2013, effect of surface roughness and couple stress on squeeze films between curved annular plates, isrn tribology, 2013, article id 640178. 19. prakash, j., tiwari, k., 1983, roughness effects in porous circular squeeze-plates with arbitrary wall thickness, journal of lubrication technology, 105, pp. 90-95. 20. deheri, g.m., patel, h.c., patel, r.m., 2007, behaviour of magnetic fluid based squeeze film between porous circular plates with porous matrix of variable thickness, international journal of fluid mechanics, 34(6), pp. 506-514. 21. vadher, p.a., vinodchandra, p.c., deheri, g.m., patel, r.m., 2008, behaviour of hydromagnetic squeeze film between two conducting rough porous circular plates, journal of engineering tribology, 222 (part j), pp. 569-579. 22. patel, j.r., deheri, g.m., 2013, shliomis model based magnetic fluid lubrication of a squeeze film in rotating rough curved circular plates, caribbean journal of science and technology, 1, pp. 138-150. 23. patel, j.r., deheri, g., 2014, shliomis model-based magnetic squeeze film in rotating rough curved circular plates: a comparison of two different porous structures, int. j. computational materials science and surface engineering, 6(1), pp. 29–49. 24. christensen, h., tonder, k.c., 1969, tribology of rough surfaces: stochastic models of hydrodynamic lubrication, sintef, report no. 10, pp. 69-18. 25. christensen, h., tonder, k.c., 1969, tribology of rough surfaces: parametric study and comparison of lubrication models, sintef, report no. 22, pp. 69-18. http://www.sciencedirect.com/science/article/pii/0020722586900959 effect of various porous structures on the shliomis model based ferrofluid lubrication... 323 26. christensen, h., tonder, k.c., 1970, the hydrodynamic lubrication of rough bearing surfaces of finite width, asme-asle lubrication conference, cincinnati. oh. paper no. 70-lub-7. 27. bhat, m.v., 2003, lubrication with a magnetic fluid, india: team spirit (india) pvt. ltd. 28. abhangi, n.d., deheri, g.m., 2012, numerical modeling of squeeze film performance between rotating transversely rough curved circular plates under the presence of a magnetic fluid lubricant , isrn mechanical engineering, 2012, article id 873481. 29. shliomis, m. i., 1972, effective viscosity of magnetic suspensions, sov. physics jetp., 34, pp. 1291-1294. 30. prajapati, b.l., 1995, on certain theoretical studies in hydrodynamic and electro-magneto hydrodynamic lubrication, phd thesis, s.p. university, vallabh vidyanagar, gujarat, india. 31. deheri, g.m., andharia, p.i., patel, r.m., 2005, transversely rough slider bearings with squeeze film formed by a magnetic fluid, international journal of applied mechanics and engineering, 10(1), pp. 53-76. 32. liu, j., 2009, analysis of a porous elastic sheet damper with a magnetic fluid, journal of tribology, 131, pp. 0218011-15. 33. patel, j.r., deheri, g.m., 2013, a comparison of porous structures on the performance of a magnetic fluid based rough short bearing, tribology in industry, 35(3), pp. 177-189. 34. irmay, s., 1955, flow of liquid through cracked media, bull. res. counc. isr, 5a(1), pp. 84. dejstvo raznih poroznih struktura na podmazivanje gvozdenim fluidom, zasnovano na šliomisovom modelu, filma stisnutog između obrtnih hrapavih zakrivljenih prstenastih ploča u radu su prikazana nastojanja da se analizira podmazivanje gvozdenim fluidom, zasnovano na šliomisovom modelu, filma stisnutog između obrtnih hrapavih zakrivljenih ploča pri čemu gornja ploča ima poroznu površinu. obrađeni su različiti modeli poroznosti. primenjen je stohastički model kristensena i tondera za procenu dejstva površinske hrapavosti. numerički je rešena s tim povezana stohastički usrednjena jednačina rejnoldsovog tipa za dobijanje distribucije pritiska što dovodi do izračunavanja nosivosti. grafički prikaz rezultata pokazuje da je kozeni-karmanov model mnogo povoljniji u poređenju sa irmejevim sa projektantskog stanovišta. takođe se pokazalo da podmazivanje gvozdenim fluidom, po šliomisovom modelu, relativno bolje deluje od nojringerrosensvajgovog. iako ležaj trpi zbog poprečne površinske hrapavosti, odgovarajućim izborom parametra zakrivljenosti i koeficijenta obrtanja, negativno dejstvo poroznosti i standardna devijacija se mogu smanjiti na minimum podmazivanjem gvozdenim fluidom, barem u slučaju negativno zakošene hrapavosti. ključne reči: stisnuti film, zakrivljene prstenaste ploče, hrapavost, magnetni fluid, obrtanje, porozne strukture plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 9 18 https://doi.org/10.22190/fume171121003w © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper dugdale-maugis adhesive normal contact of axisymmetric power-law graded elastic bodies udc 539.3 emanuel willert berlin university of technology, berlin, germany abstract. a closed-form general analytic solution is presented for the adhesive normal contact of convex axisymmetric power-law graded elastic bodies using a dugdalemaugis model for the adhesive stress. the case of spherical contacting bodies is studied in detail. the known jkrand dmt-limits can be derived from the general solution, whereas the transition between both can be captured introducing a generalized tabor parameter depending on the material grading. the influence of the tabor parameter and the material grading is studied. key words: adhesive contact, power-law elastic grading, dugdale-maugis model, axial symmetry, method of dimensionality reduction 1. introduction propelled by the technological demand for versatile high-performance materials and the study of biological materials and contact solutions, living nature developed in several circumstances, functionally graded materials (fgm), i.e. media with continuously inhomogeneous mechanical properties, have encountered a lot of scientific interest and research in the past years. the use of fgm is proven to be possibly beneficial in physical [1] and biological [2] applications. whereas rigorous solutions for non-adhesive contact problems of fgm, at least for some special forms of inhomogeneity, have been available for quite a long time [3-5], the adhesive contact of fgm is still in the focus of current research [6-9]. these latter studies, nonetheless, only concern the limiting case of a negligible range of the adhesive interaction, established by johnson, kendall and roberts (jkr, [10]) in 1971. after derjaguin, muller and toporov (dmt, [11]) a few years later presented a different theory of long-range adhesive interactions giving a different result received november 21, 2017 / accepted january 11, 2018 corresponding author: emanuel willert technische universität berlin, sekr. c8-4, straße des 17. juni 135, d-10623 berlin e-mail: e.willert@tu-berlin.de 10 e. willert for the critical pull-off force in a parabolic contact, a discussion started, which was only finally resolved by maugis [12], who – based on a model of the adhesive stress first introduced by dugdale [13] – was able to show the transition between what was proven by tabor [14] to be correct descriptions of limiting cases. the present paper generalizes maugis’ solution for the adhesive contact of homogeneous spheres to arbitrary axisymmetric bodies with elastic-grading in form of a power-law. as the contact problem of interest can be ascribed to the frictionless, non-adhesive normal contact of power-law graded elastic materials a solution procedure based on the method of dimensionality reduction (mdr) can be applied. 2. general axisymmetric solution we consider elastic grading of the young modulus e with depth z in form of a powerlaw: 0 0 ( ) , 1 1. k z e z e k z          (1) thereby constants e0 and z0 as well as poisson ratio ν may be different for the contacting bodies. exponent k, however, needs to be the same for both of them. as the exponent may take positive or negative values, both soft surfaces with a hard core and hard surfaces with a soft core can be studied. it has been shown that the frictionless normal contact of axisymmetric power-law graded elastic bodies can be exactly mapped onto a plain contact of a rigid profile g with a one-dimensional foundation of independent linear springs, each in distant δx from each other [15,16]. thereby the equivalent plain profile g = g(x) within this mapping procedure called method of dimensionality reduction (mdr) can be calculated from the axisymmetric gap f = f (r) between the non-deformed three-dimensional bodies by the integral transform: 1 2 2 1 0 ( ) ( ) d . ( ) x k k f r g x x r x r       (2) stiffness δkz of a single spring at position x is given by the expression: 0 ( ) , k z n x k x c x z         (3) with: 1 2 2 1 2 1 01 2 02 1 1 : . ( , ) ( , ) nc h k e h k e              (4) dimensionless auxiliary function h can be determined from exponent k and poisson’s ratio according to: dugdale-maugis adhesive normal contact of axisymmetric power-law graded elastic bodies 11 2(1 )cos( / 2) (1 / 2) ( , ) : , ( , ) ( , )sin[ ( , ) / 2] [(1 ) / 2] k k k h k c k k k k                (5) with: 12 [(3 ( , )) / 2] [(3 ( , )) / 2] ( , ) : (2 ) k k k k k c k k                 (6) and ( , ) : (1 ) 1 1 k k k             (7) and gamma function γ. note that the spatial distribution of the spring stiffness in eq. (3) obeys the same power-law as the elastic grading. if equivalent profile g is pressed into the foundation of springs by an indentation depth d the vertical spring displacement w1d(x) in the area of direct contact is elementarily given by: 1d ( ) ( ), ,w x d g x x a   (8) with contact radius a. normal force fn as well as the local distributions of pressure p and relative displacement w in the original three-dimensional system can be calculated from w1d(x) according to: 1d 0 1d 2 2 1 0 1d 2 2 1 0 ( ) d , ( )d *( ) , ( ) ( )d2cos( / 2) ( ) . ( ) kn n k n k k r r k k c f w x x x z c w x x p r z x r x w x xk w r r x                   (9) the second of these latter eqs. (9) can be inverted to give: 0 1d 2 2 1 2 cos( / 2) ( )d ( ) . ( ) k k n x z k rp r r w x c r x       (10) if we now assume a dugdale model of a constant adhesive stress σ0 within the adhesive zone with radius b: adh 0( ) , ,p r r b   (11) the corresponding displacements in the mdr model are due to eq. (10) given by: 2 2 10 0 1d,adh 2 cos( / 2) ( ) ( ) , . (1 ) k k n z k w x b x x b c k        (12) hence, the one-dimensional displacements in the dugdale-maugis adhesive contact are: 12 e. willert 1d 2 2 10 0 ( ), , ( ) 2 cos( / 2) ( ) , . (1 ) k k n d g x x a w x z k b x a x b c k               (13) for the three-dimensional stresses to be finite at the edge of direct contact these displacements must be continuous at x = a, which results in: 2 2 10 02 cos( / 2) ( ) ( ) . (1 ) k k n z k d g a b a c k       (14) the total external normal force is due to the first of eqs. (9) given by: adh 00 1 2 2 adh 0 2 12 2 2 [ ( )] d , 4cos( / 2) 1 1 3 : f , ; ; , 2 2 21 a kn n k k c f d g x x x f z k a k k k a f b bk b                          (15) with the hypergeometric function: 2 1 0 ( ) ( ) f ( , ; ; ) : , 1, ( ) ! γ( ) ( ) : . γ( ) n n n n n n a b z a b c z z c n x n x x        (16) radius b of the adhesive zone is not known a priori but can be determined from the condition that the gap between the deformed surfaces at r = b has to equal the range h of the adhesive stresses. as the gap between the deformed surfaces can be easily calculated from three-dimensional relative displacement w, indentation depth d and axisymmetric non-deformed gap f, we obtain the additional relation ( ) ( ).h w r b d f r b     (17) to close the equation system. evaluating eq. (17) with the help of the third of eqs. (9) and using the identity: 2 2 1 0 2cos( / 2) [ ( )]d ( ) ( ) b k k k x d g x x d f b b x         (18) one obtains: 2 2 1 2 11 2 0 0 2 12 2 2cos( / 2) [ ( )]d ( ) 3 cos (1 ) 4 1 32 2 f , ; ; . 3 2 21 2 b k k a kk k n k x d g x x h b x k k k z b a k k a k kc bk b                                                 (19) dugdale-maugis adhesive normal contact of axisymmetric power-law graded elastic bodies 13 equations (14), (15) and (19) completely solve the given contact problem. in the homogeneous case k = 0 they are reduced to the axisymmetric generalization of maugis’ results given very recently by popov et al. [17]. the stresses in the area of direct contact could theoretically be calculated inserting eq. (13) into the second of eqs. (9). 3. the jkr limit it is of course possible to retrieve the known solution in the jkr limit of adhesion from the relations derived in the previous section. for this purpose we study the limit of negligible adhesion range h → 0, whereas the surface energy per unit area, δγ = σ0h, is kept constant. in this case the radius of the adhesive zone can be written in the form: (1 ),b a   (20) with a small parameter ε. using the linearization: ( ) ( ) ( ) ,g a x g a g a x    (21) performing the integration and neglecting all terms of higher than first order in ε leads to: 1 10 0 12 1 10 0 2 2cos( / 2) ( ) (2 ) , 1 2 ( ) 2 cos ( / 2) cos (2 ) (2 ) . 2 1 (1 ) k k k n k k k k n z ak d g a k c z ak d g a k h k ck                               (22) hence, 12 10 0 2 0 2 cos ( / 2) (2 ) (1 ) k k k n z ak h ck           (23) and therefore: 102 ( ) , k k n z d g a a c      (24) which perfectly coincides with the known solution in the jkr limit [8]. the normal force via the same mechanism is also reduced to the known relation: 00 2 [ ( )] d . a kn n k c f d g x x x z   (25) note that eq. (23) is actually independent of the profiles of the contacting bodies. 4. parabolic contact let us now consider the specific case of parabolic contact with the radius of curvature r, i.e.: 2 ( ) . 2 r f r r  (26) 14 e. willert the equivalent profile is accordingly: 2 ( ) . (1 ) x g x r k   (27) thus, evaluating the general solution derived above, the solution of the dugdale-maugis adhesive normal contact problem in case of power-law elastic grading is given by: 2 2 2 10 0 13 2 02 2 0 1 2 2 2 12 2 2 cos( / 2) ( ) , (1 ) (1 ) 4 4cos( / 2) (1 ) (3 ) 1 1 1 3 1 f , ; ; . 2 2 2 k k n kk n n k k z ka d b a r k c k c a k a f b brz k k k a k k k a b b                                                 (28) radius b of the adhesive zone can be determined from the condition:   1 2 2 1 2 32 2 2 1 2 2 1 0 0 2 2 cos( / 2) 1 1 3 f , ; ; 1 1 2 2 2 4 cos( / 2) 1 3 5 f , ; ; 1 2 (1 )(3 ) 2 2 2 3 cos 1 4 2 2 31 2 k k k k n k a k k k a h d k b b b k a k k k a r k k b b k k k z b kck                                                                   1 2 2 1 2 1 3 f , ; ; . 2 2 k a k k a k b b                      (29) introducing the normalized variables : , : , : , : ,n c c c c fd a b d a f m d a f a     (30) with the critical values in the jkr limit under force-controlled boundary conditions [6]: 2 2 2 2 3 0 1 2 2 2 3 0 (1 ) (3 )1 , (1 )(3 ) 8 (1 ) (3 ) , 8 3 , 2 k k c n k k c n c k k r zk d k k r c k k r z a c k f r                                 (31) dugdale-maugis adhesive normal contact of axisymmetric power-law graded elastic bodies 15 and the generalized tabor parameter for power-law elastic grading, i.e. the ratio of the characteristic height of the adhesive neck and the adhesion range: : , cd h   (32) equations (28) can be written in the form: 2 1 2 1 3 3 2 2 1 1 2 12 2 3 16cos( / 2) ( 1) , 1 (1 ) 2 1 4cos( / 2) 1 ( ) 1 1 1 1 1 3 1 1 f , ; ; . 2 2 2 k k k k k k k d a a m k k k k f a ma k k m k k k m m                                               (33) the compatibility condition (29) in dimensionless variables reads:     2 11 2 2 2 13 2 12 2 3 cos / 21 2 1 1 1 3 1 f , ; ; 1 1 2 2 2 ( ) 4 1 1 3 5 1 cos f , ; ; (1 )(3 ) 2(1 ) 2 2 2 2 3 cos (1 ) 32 2 2 3(1 ) (1 ) 2 k k k k k k k d k m m ma k k k k k k k m m k k ma k kk k                                                                      2 11 2 1 1 3 1 f , ; ; , 2 2k k k k m m                 (34) which in the homogeneous case coincides with maugis’ solution [12] (maugis uses a slightly different scaling for normalization). the jkr limit is given by the known relations [17]: 1 jkr 2 2 3 jkr 3 2 3 4 , 1 1 2 . k k k k d a a k k f a a           (35) as the adhesive force in the dmt limit, dmt adh 2 ,f r   (36) is independent of the elastic contact properties (it is actually the force for the adhesive contact of rigid spheres derived by bradley [18]), the dmt limit of eqs. (33) reads: 16 e. willert dmt 2 dmt 3 3 , 1 4 . 3 k k d a k f a k        (37) to illustrate above findings and the influence of material grading figs. 1 and 2 show the implicitly defined force-indentation relations as well as the respective jkrand dmt limits for two different values of the power-law exponent k. fig. 1 force-indentation-curves for the dugdale-maugis adhesive normal contact of powerlaw graded elastic spheres for k = -0.5 and several values of the tabor parameter λ fig. 2 force-indentation-curves for the dugdale-maugis adhesive normal contact of powerlaw graded elastic spheres for k = 0.5 and several values of the tabor parameter λ dugdale-maugis adhesive normal contact of axisymmetric power-law graded elastic bodies 17 note that the dmt limit is only well-defined for positive indentation depths although the branch without direct contact (and therefore negative indentation depths) can be seen as its “natural” continuation. to denote this slight distinction a small gap is left between the dmt limit and the curve without direct contact in fig. 1. obviously the convergence for higher values of the tabor parameter towards the jkr limit is much faster for larger values of k. for k = 0.5 there is already no noticeable difference between the solution for λ = 1 and the jkr limit. also the normalized indentation depths are getting much higher for larger values of k. interestingly, the critical pull-off forces in the jkrand dmt limit are the same for k → 1 (as it was pointed out already in [6]). in this case the left branch of the jkr curve and the curve without direct contact will be practically indistinguishable. 5. conclusions based on the mdr a closed-form analytic solution has been obtained for the dugdalemaugis adhesive normal contact of arbitrary convex axisymmetric, power-law graded elastic bodies. as the most common and probably most relevant special case the contact of spherical or parabolic bodies has been studied in detail. the common limits for very large (jkr) or very small (dmt) values of the tabor parameter are derived from the general solution. in dimensionless variables the relations between indentation depth, contact radii and normal force only depend on the tabor parameter and exponent k of the elastic grading. thereby the convergence for larger values of the tabor parameter towards the jkr limit is faster for higher values of k. the presented solution is of course based on strong contact-mechanical assumptions (half-space hypothesis, absence of friction or roughness) and quantitatively problematic physical models (power-law grading with either infinitely stiff or infinitely soft surfaces, dugdale model for the adhesive stress); it is, however, to the author’s best knowledge, the only tool, to rigorously study the influence of both material grading and adhesion range in a closed form, for example in microor nano-applications, for which the range of the (adhesive) molecular forces becomes relevant. and although other models might seem physically more appropriate, they will probably neither allow for analytic treatment nor show a qualitatively different behavior. for future work it would be interesting to compare the obtained analytical results with numerical or experimental findings. references 1. suresh, s., 2001, graded materials for resistance to contact deformation and damage, science, 292, pp. 2447-2451. 2. scherge, m., gorb, s., 2001, biological microand nano-tribology – nature’s solutions, springer, berlin heidelberg. 3. booker, j.r., balaam, n.p., davis, e.h., 1985, the behaviour of an elastic non-homogeneous halfspace. part i–line and point loads, international journal for numerical and analytical methods in geomechanics, 9(4), pp. 353-367. 4. giannakopoulos, a.e., suresh, s., 1997, indentation of solids with gradients in elastic properties: part i. point forces, international journal of solids and structures, 34(19), pp. 2357-2392. 5. giannakopoulos, a.e., suresh, s., 1997, indentation of solids with gradients in elastic properties: part ii. axisymmetric indentors, international journal of solids and structures, 34(19), pp. 2393-2428. 18 e. willert 6. chen, s., yan, c., zhang, p., gao, h., 2009, mechanics of adhesive contact on a power-law graded elastic half-space, journal of the mechanics and physics of solids, 57(9), pp. 1437-1448. 7. guo, x., jin, f., gao, h., 2011, mechanics of non-slipping adhesive contact on a power-law graded elastic half-space. international journal of solids and structures, 48(18), pp. 2565-2575. 8. jin, f., guo, x., zhang, w., 2013, a unified treatment of axisymmetric adhesive contact on a powerlaw graded elastic half-space. journal of applied mechanics, 80(6), 061024. 9. liu, z., meyers, m.a., zhang, z., ritchie, r.o., 2017, functional gradients and heterogeneities in biological materials: design principles, functions, and bioinspired applications, progress in materials science, 88, pp. 467-498. 10. johnson, k.l., kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids. proceedings of the royal society of london, series a, 324, pp. 301-313. 11. derjaguin, b.v., muller, v.m., toporov, y.p., 1975, effect of contact deformations on the adhesion of particle, journal of colloid and interface science, 53(2), pp. 314-326. 12. maugis, d., 1992, adhesion of spheres: the jkr-dmt-transition using a dugdale model. journal of colloid and interface science, 150(1), pp. 243-269. 13. dugdale, d.s., 1960, yielding of steel sheets containing slits, journal of the mechanics and physics of solids, 8(2), pp. 100-104. 14. tabor, d., 1977, surface forces and surface interactions, journal of colloid and interface science, 58(1), pp. 2-13. 15. heß, m., 2016, a simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, international journal of engineering science, 104, pp. 20-33. 16. heß, m., popov, v.l., 2016, method of dimensionality reduction in contact mechanics and friction: a user’s handbook. ii. power-law graded materials, facta universitatis-series mechanical engineering, 14(3), pp. 251-268. 17. popov, v.l., heß, m., willert, e., 2018, handbuch der kontaktmechanik – exakte lösungen axialsymmetrischer kontaktprobleme, springer, berlin heidelberg. 18. bradley, m.a., 1932, the cohesive force between solid surfaces and the surface energy of solids, the london, edinburgh, and dublin philosophical magazine and journal of science, 13(86), pp. 853-862. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 229 244 https://doi.org/10.22190/fume200603024r © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics elias randjbaran1, dayang l. majid 1 , rizal zahari 1,2 , mohamed t. h. sultan 1 , norkhairunnisa mazlan 1 1 department of aerospace engineering, faculty of engineering, serdang, selangor, malaysia 2 systems engineering department, military technological college, muscat, sultanate of oman abstract. investigations of the angled ballistic impact behavior on carbon kevlar® hybrid fabrics with assorted volumes of carbon nanotubes (cnts) into epoxy are presented. the ballistic impact behavior of the epoxy composites with/without cnts is compared. individual impact studies are conducted on the composite plate made-up of carbon kevlar hybrid fabrics with diverse volumes of cnts. the plate was fabricated with eight layers of equal thickness arranged in different percentages of cnts. a conical steel projectile is considered for a high velocity impact. the projectile is placed very close to the plate, at the centre and impacted with sundry speeds. the variation of the kinetic energy, the increase in the internal energy of the laminate and the decrease in the velocity of the projectile with disparate angles are also studied. based on the results, the percentage of cnts for the ballistic impact of each angle is suggested. the solution is based on the target material properties at high ballistic impact resistance, the inclined impact and the cnt volumes. using the ballistic limit velocity, contact duration at ballistic limit, surface thickness of target and the size of the damaged zone are predicted for fabric composites. key words: carbon nanotubes, oblique impact, ballistic impact, kevlar fiber, carbon fiber received june 03, 2020 / accepted july 18, 2020 corresponding author: elias randjbaran department of aerospace engineering, faculty of engineering 43400 upm, serdang, selangor, malaysia e-mail: elias@gmx.co.uk 230 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan 1. introduction reinforcing composite panels affects their mechanical properties as well as their weight, which are two crucial considerations for a wide range of applications such as the process of responsibility of naval vessels designers, armored vehicle designers and manufacturers, aircraft designers, etc. [1-6]. due to effective behavior and safe performance, the use of cnt/epoxy composites as materials for structural design of vehicles and other constructions has led to the requirements of sustaining them. a special attention ought to be paid to the commercial aircraft structures that must be lightweight and divulge exceptional impact resistance properties in order to meet growing demands in this industry. peculiarly, a body of commercial aircrafts is mostly made of composite materials and metals [5-9]. due to weight issue as well as to an increasing role of saving energy in the modern life, weight deduction can be one of the major challenges of the commercial aircraft designer despite retaining the quality and guarantee for the safety [6-8 &10]. therefore, some parts of the body can be replaced with laminate composite materials [8-10]. moreover, the most economical and conventional fabric is glass fiber, despite the fact that the glass fiber use has a lot of drawbacks, e.g. consistently triggered physical problems, health issues and respiratory irritations, allowing for air flow, trap allergens, while moisture can get trapped inside the fiberglass insulation material and thus become a bedding material for pests [10-12]. consequently, modern attempts at circumventing the drawbacks of using glass fiber, which can be inevitable, and replacing it with kevlar and carbon fiber as well as adding cnt into resin epoxy can be recommended. numerous studies regarding the ways of revitalizing the use of natural fibers have been published and the trend of applying natural fibers is increasing, but technically, in industrial scales, none of manufacturers has used or even recommended them [13-15]. eventually safe, durable materials, but expensive kevlar/ carbon fiber hybrid fabrics with cnt epoxy resin are studied in the current research to normal and oblique impact resistance. carbon fibers, despite their high price, are predominantly used for ballistic protection under rifle bullet threats [15-18]. although commercial monolithic carbon fiber plates could be manufactured to a large thickness, they may not meet the design needs for certain thickness and multiple thinner plates, which are used to meet design specifications. in addition, multi-layer plates can provide flexibility to users in terms of mobility and maintenance [17-19]. for this reason, most experimental studies as well as those in the published works relating to the ballistic performance of carbon fiber plates focus on replacing monolithic plates with layered plates of the same thickness. the chief goal of these experiments is to reduce the overall weights of armor plates while maintaining the same ballistic protection as epoxy-cnt carbon-kevlar composite plates [18-21]. although some other fibers including asbestos or wood or paper have been sporadically used, the fiber-reinforced polymer (frp) is made of a resin matrix reinforced with fibers that can be conventionally glass, carbon, or kevlar. [22-24]. since the fibers are used to support the load by providing excellent strength to the structure, the resin serves as matrix material to transfer the load to the reinforcement and then to hold and protect the fibers [25-28]. the frp composites have a variety of superior properties such as a high strength-to-weight ratio; they could provide high dimensional stability, excellent fatigue, light-weight engineering solutions, corrosion, wear resistance, etc. [29-33]. because of these attractive properties, the frp composites have been widely used for numerous applications such as aerospace and effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics 231 commercial aircrafts, automobile, sports goods, robot arms, chemical containers, and fishing rods [34-36]. especially in commercial aircraft industry, the frp composites have been in a remarkably increasing demand. for example, the innovative frp composite fuselage has been widely applied in airbus a350 aircraft, and about 53% of the materials in a350 airframe are made of composites. the frp composite part fabrication with a shorter cycle time is desired to meet a high demand for frp composites, although they have a higher production cost [37-39]. on another note, the commercial aircraft structures are exposed to bird strike events causing serious damages, which usually occur in flight or on take-off and include bird strikes or impact by hail, tyre rubber, engine fragments or runway debris. ballistic impact scenarios are low mass with often ballistic impact speeds transpiring from ammunitions with implementation to security protection structures. a few papers were published on this expensive and sometimes-lethal flying risk, though annual damages from bird and bat collisions with aircraft were estimated at and up to $1.2 billion in commercial aviation worldwide; that is why so many councils are dedicated to the problem at numerous conferences. by providing a comprehensive guide to preventing and minimizing damage caused by a bird strike on the aircraft, the researchers attempt to bridge the current gap. the fact sheet of the federal aviation administration’s (faa) about wildlife hazard mitigation programme on 9th january, 2019, declared, from 1990 to 2017, the reported costs for civil aircraft in usa totalled $765 million for the 28-year period as well as costs are adjusted to the reported strikes in which costs were not provided; regarding the estimated number of strikes that were not reported, losses could be as high as $500 million per year. moreover, 311 human injuries are attributed to wildlife strikes with us civil aircraft. in addition, 287 human fatalities are attributed to wildlife strikes globally (fig. 1). hence, advance material development and analyses pertaining to the ballistic impact strikes present a continuous challenge to the composites’ community. in this work, the ballistic impacts are simulated for so-called bird strike impacts. fig. 1 locations of bird-strike damage (boeing report) [38] 232 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan 2. methodology in this paper, the carbon/kevlar reinforced epoxy composite with cnt fillers of a varying loading ratio were fabricated; they underwent ballistic impact testing and their responses were analyzed. 2.1. fabrication for the absorbing nanostructured composites processing, carbon nanotubes were dispersed in an epoxy resin matrix. the carbon nanotubes (type cm-95/mwcnt) were acquired from iljin nanotech co. ltd. (seoul, south korea). according to the manufacturer, this mwcnt has size of 0.01-0.015 µm of diameter and 10-20 µm of length. as shown in fig. 2, cnt-epoxy composites sample were fabricated using cnt, a dispersing agent byk 9077, and araldite 2016. cnts and byk 9077 with ethanol were blended for five minutes; soon after it there followed sonication for an hour at fixed output power of 25 watts. the mixture was added to the resin and stirred for 10 minutes with a high shear blending homogenizer at 3500 rpm, followed by 45 minutes sonication treatment. the blend was then placed in a vacuum degassing oven and remained at room temperature (25°c) for 48 hours under 1 bar, followed by 24 hours under 1 bar at 80°c for degassing process. an open mould process in which components or successive plies of reinforcing material or resin-impregnated reinforcements are applied to the mould and the composite is built up and worked by hand. curing is normally conducted at ambient temperatures, but may be accelerated by heating, if desired. in addition, the hand lay-up process produces the largest amount of reinforced composite products. fig. 2 schematic diagram of the fabrication protocol for cnt-epoxy composites 2.2. ballistic impact tests fig. 3 displays a schematic diagram that is in accordance to the nij 0108.01 standard produced by the national institute of justice (nij) and the detailed procedures for ballistic protection certification and testing [41]. the gas gun tunnel facility utilized in this work was developed at the department of aerospace engineering, university putra malaysia (upm) as reported here [44]. the noted standard is focused on shields, blankets, and visors. compressed (pressurized) nitrogen gas is used as a working fluid. the reason for this is that nitrogen gas has the advantages of being colorless, odorless, effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics 233 tasteless, and non-toxic. moreover, it is commercially available and as a chemical element it contains a high thermal conductivity. its lightness as well as its inferior density in comparison to atmospheric (due to a lower mass of the nitrogen particles) air is another criterion for its selection to carry out the experiment. another advantage of nitrogen gas lies in its low density which facilitates the movement of the bullet as the sound waves travel faster in a less dense material than in a denser one [40-44]. fig. 3 schematic of ballistic test setup following the nij standard [44] fig. 4 illustrates inclined target, which can be rotated from 90 to 140 degrees with an angle indicator, all made of stainless steel. this bullet trap features a spent bullet depository and ballistic rubber front panel, 12.7 mm thick with self-healing properties. the ballistic front panel can slow down the rounds, minimize the impact noise, and lead dust generation [43-46]. fig. 4 set-up for inclined target 234 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan the energy that must be dissipated in the collision is approximately relative kinetic energy ek of the bird, defined by: ek= 0.5mv 2 , (1) where m is the mass of the bird and v is the relative velocity (the difference of the velocities of the bird and the plane, resulting in a lower absolute value if they are flying in the same direction and a higher absolute value if they are flying in opposite directions) [46-48]. the ballistic limit formula for armor tm5-855-1 by [43]: v50 = 3922 wt -0.5 d 0.7 (eh sec (θ)) 0.8 , (2) where v50 is the ballistic limit velocity in fps, d is the caliber of the projectile, in inches, eh is the thickness of the homogeneous armor, θ is the angle of obliquity, wt is the weight of the projectile (in lbs). in eq. (2), the theoretical ballistic limit is investigated but the coefficient of materials is not considered as a correlation. moreover, in this equation one velocity is only pointed out due to the fact that the initial and final speeds are the ballistic limit. v50 ballistic limit is the means of comparing the ultimate performance of armor materials. v50 ballistic limit is the velocity at which a specific projectile (bullet) is expected to penetrate the armor half of the time. the ballistic limit of armor is most frequently conducted using the procedures of tm5-855-1. in other words, a given projectile cannot generally pierce a given target when the projectile velocity is lower than the ballistic limit [41-43]. fig. 5 clearly illustrate that the thicknesses of plates a and b are 30mm and 23mm. on one hand, it means plate a is approximately 30% thicker than b. on the other hand, by increasing the angle up to 45 degrees, the effective thickness is 32.5mm. it means that b is not only definitely lighter but it is also about 8.34% thicker than a. triangular profile inserts are enforced to have uniform area density from angled plate (in b) and the thickness of normal plate (in b) must be diminished to compensate for the weight of these impacts. furthermore, an illustration of proof of angled plate proposes less weight benefit while protecting a certain width, which is a comparison of a vertical slab of plate on the left and a section of plate sloped at an angle of 45 degrees on the right and the horizontal distance fig. 5 thicknesses of right and inclined angle [12] effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics 235 via the plate is the same. however, the normal thickness of the sloped plate is lower. therefore, the actual cross-sectional area of the armor, and hence its mass, is the same in each case. due to mass, the normal will have to decrease if the angles are elevated. ti = tn  sec(α) (3) where α is the angle of the sloped armor plate from the vertical, tn is the normal thickness, ti is the inclined thickness. for instance, as the second of 60° is 2, the armor sloped sixty degrees back from the vertical presents to a projectile travelling horizontally a line-of-sight thickness (ti) twice armor’s normal thickness (tn). 3. results and discussion 3.1. morphology fig. 6 shows pure multi-walled carbon nanotubes (mwcnts). furthermore, figs. 7a and 7b show the surface morphologies captured from specimens coated with the solutions of the active layer (fig. 7a) and not well-dispersed (fig. 7b). the sem images clearly illustrate that the dispersion of the epoxy-cnts in the matrix is not well for the nanocomposite formed, which favors cnt agglomeration and entanglement that form various cnt clusters due to an excessive amount of cnt (more than 0.3%). on the other hand, the well-dispersed epoxy-cnts is observed for the composite prepared (fig. 6): this can be ascribed to the monomers polymerized on the surface of the epoxy-cnts; this inhibits π–π interaction of the nanotubes and thereby decreases agglomeration. moreover, the well-dispersed epoxy-cnt means there is a continuous percolation path for free charge carriers and eventual collection at the impact resistance thereby improving cell performance in mechanics of impact. fig. 6 mwcnts magnified by sem (500nm and 1.00µm) 236 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan fig. 7 shows the images obtained by sem of the materials processed observing in fig. 6 that the nanotubes are imperfectly distributed and the presence of clusters in island (0.3% cnt/epoxy composite) in composite. however, within the island the cnts are dispersed indicating the formation of an interconnected network. according to the sem fracture surface images showing tested composite laminates in fig. 7, it can be seen that the fibers are covered with epoxy, stipulating magnificent adhesion between the resin and the fibers with the appearance of cnts, which could be observed equally on both the fiber surfaces and the connected irrefutably interfacial regions between the matrix and the fibers. furthermore, it can contribute immensely to enhanced stress transfer compared to the composites based on neat epoxy resins. the excellently-dispersed cnt network within these fabric laminates can be used for damage sensing presently by monitoring how mechanical resistivity transforms from progressive cracking showing the crosssectional areas of composite reference specimen and hybrid cnt/epoxy specimen after impact tests, respectively, where very clear inter-laminar impact failures are observed. the type and amount of aggregation of cnt structures are evidently determined by rigidity of nanotubes and whether their diameters are thin enough to allow indubitably their buckling and self-aggregation into low-density, particle-like, intertwined, and coiled assemblages. moreover, the production technique can unquestionably impact the sorts of cluster structure and whether it is stabilized incontrovertibly by adequate agglomeration or some sort of inter-tubular aggregation (physical entanglement). consequently, contingent on these characteristics, agglomerate structures of nanotubes can appreciably differ from thin-walled cnt to thick-walled, rigid cnt; examined, they exhibit a strong tendency to aggregate into microscopic bundles (clusters) demonstrably and in turn agglomerate loosely into small clusters that are substantially visible; forming cnt agglomeration exists inevitably in all cnt-epoxy composites. conceptually, presented as spherical assemblages of intertwined tubes verified by image interpretation from the samples were purposefully assumed as agglomerates (in fig. 7). fig. 7 damaged samples a) dispersed 0.3% cnt on surface and b) 1.5% cnt agglomeration effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics 237 fig. 8 illustrates ballistic limit speeds of seven specimens, after attempting six successful ballistic impacts, at various targets’ angles with different volumes of cnt added into the epoxy resin including 0, 0.1, 0.3, 0.5, 0.7, 1, and 1.5 percent. in these single line graphs, x-axis indicates percentages of cnt while y-axis indicates ballistic limit in meter per second. in addition, fig. 5 shows how various percentages of cnt increased fairly rapidly until 0.3% cnt, and then remained constant at the same level in 1% and 1.5% cnt. fig. 8 ballistic limit at angles with different amount of cnt ballistic limit speeds at 120 degrees of target differ from 168m/s to 196m/s in different percentages of cnt from 0 to 1.5%. the ballistic limit speed in the neat epoxy resin composite is 152m/s. by adding cnt into the epoxy resin the speed resistance can be improved. in addition, the positive effect of the cnt inclusion into the epoxy resin is obviously illustrated that by adding 0.1% the speed is 176m/s while by adding 0.3% the speed is at maximum level from 196m/s to 223m/s; it shows at least 18m/s improvement of reinforcement technique. by increasing percentages of cnt from 0.3% to 0.5%, which is the maximum point, the increasing of the speed can happen. by adding more than 0.3% cnt, the ballistic limit is gradually dropped from 196m/s to 185m/s at 0.7% of cnt. by increasing the amount of cnt the ballistic limit speed is decreased; by adding 1% of cnt, the speed rises from 166m/s to 199m/s and then the speed drops from 166m/s to 199m/s. however, the speeds were still more than the neat epoxy by adding more than 0.3% of cnt. moreover, there was a plateau of the ballistic limit between 154-207m/s by adding cnt from 1 to 1.5%. in conclusion, the line graph displays the optimum percentage of cnt is 0.3% and by adding the cnt ballistic limit increasingly dropped. nevertheless, it is apparent that the ballistic limit velocities increase with inclination and that the saturation limit for cnt content is 0.3 %. 238 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan fig. 9 shows ballistic limit speeds of seven specimens at 140 degrees of target with different volumes of cnt added into the epoxy resin including 0, 0.1, 0.3, 0.5, 0.7, 1, 1.5 %, and theoretical graph for the ballistic limit velocity. in this seven-line graphs, x-axis indicates angles degree while y-axis indicates ballistic limit in meter per second. increasing the angles can improve the body resistance against the impact, i.e. the highest ballistic limit velocity is for 0.3% cnt at 140 degree and the lowest is for neat epoxy at 90 degree. the slanting protective layer can fabricate confirmation by an instrument, thickness of the plate disengaged by the cosecant of the point from impact course. along these lines, the case for the occasion of a 100mm piece of plate at a 40° inclination from vertical with the shell course being parallel to the ground; according to 100 x sec (40) = 100 x 1.56 = 156. in the hindrance, by increasing the angles, the thickness is growing. therefore, by saving the same mass due to increased thickness, the ballistic energy absorption can rise without increasing the mass. it means that, for example, at specimen thickness at 100mm, at 40 degrees the thickness can be achieved at 156. in other words, the energy absorption can be increased at 56%. however, the lowest ballistic limit of neat epoxy is higher than the theoretical one; this resistance happens due to the use of carbonkevlar fabric. the energy absorption of the ballistic impacts is directly proportional to their angles. based on equation 3, in table 1, the greater the energy, the larger the angle (up to 90 degrees). fig. 9 various specimens versus ballistic limit velocities table 1 ballistic limit by using tm5-855-1 degree speed (m/s) 90 135 100 137 110 142 120 151 130 166 140 192 effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics 239 fig. 9 illustrates and compares the amount of the ballistic limit energy absorption (in joules) of seven carbon-kevlar hybrid fabrics with various percentages of cnt added to the epoxy (from 0% to 1.5%) impacted on six different angles (from 90 to 140 degrees). fig. 9 ballistic energy absorption of specimens at different angles on the whole, the composite with inclusion of 0.3% cnt spent more energy on 140 degrees than 0.3% in the angles. all the composites with 0.3% cnt spent most of their energy absorption on different angles; whereas, the least amount of energy was spent on the neat epoxy in the 90 degrees compared to the neat epoxy in the 100 degrees also 1% and 1.5% cnt in the 90 and 100 degrees. furthermore, the most significant difference in the ballistic energy absorption between the two composites was in cnt percentage. besides, in terms of angles, samples in the 140 degree spent about 250 j to more than 300 j on this as opposed to the 1% and 1.5% cnt in 90, 100, and 110 degrees at about 150 j. similarly, the 130 degree ballistic energy absorption was higher on the 1% and 1.5% cnt in 130 degrees than the neat epoxy and 1% or 1.5% cnt in 90, 100, and 110 degrees (around 200 j and 210 j, respectively). in the 120 degrees ballistic energy absorption on 0.3 % cnt (just over 350 j) was over 60% that of neat epoxy, 1% and 1.5% cnt in 90, 100, and 110 degrees, which was only 150 j. on the other hand, the amount of the energy absorption depleted on the remaining samples was higher in with cnt epoxy. above 250 j was depleted by the 120, and 130 degrees on most samples, which was slightly more than the 100 degrees deplete about 225 j. neither of the 140 degrees depleted much on 0.3% cnt, which accounted for over 300 j of energy absorption also in 0.3% cnt but approximately 220 j in the 90 degrees. high shear mixing to disperse homogeneously, the entanglement of cnts produced by the agglomerates of cnts caused by the intermolecular van der waals force must be broken for homogenization, which can construct further fillers on obtainable surface area. a further intensification of similarity to composite material could be attained by a chemical functionalization of the carbon nanotube surface, through covalent or ionic bonds to the resin epoxy matrix. the bonds enable a stress transfer between the resin epoxy and cnts, which leads to enhancing of the interfacial interactions. furthermore, the covalent bonding is stronger as physical interactions anticipate a negligible influence on the ballistic 240 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan impact performance of cnts. therefore, the reinforcement surface functionalization of cnts accompanying noticeable dispersion of nanotubes in the matrix is the vital issue in developing cnt/resin epoxy composites. cnts may result in tube breakage. the results illustrate some breakages transpire which signifies cnts decrease by accumulating energy input. the rate at which cnts is reduced abates as the material is dispersed and the tube separation increases. consequently, tube breakage is not a serious problem and the aspect ratio of the tubes remains very high. therefore, satisfactory dispersion can be achieved at the expense of an acceptable reduction in cnts. fig. 10 illustrates the intact composite specimens (a) and impact c-scan proceeds before (b) and after (c) the ballistic impact test. soon after that, the ultrasonic inspection shows that there are no types of delamination and voids on the samples before the ballistic impact occurrence in fig. 10 (b). the contrast in tints and shades tone prevails because of the depth of the signal or deviation of the amplitude emitted under the sample, and can be helped with surface irregularities, which are inherent to the process. however, after conducting an oblique ballistic impact event, the extent of damage is clearly revealed in fig. 10 (c). fig. 10 c-scan inspections at 0.3% cnt fig. 11 displays in macroscopic scale the behavior of the composite plates in relation to the reinforced laminates at 0.3% cnt; it also shows total amount of the absorbed ballistic impact energy the specimen absorbed during the entire oblique impact test at 140 degrees. therefore, the deformation of the laminates is observed as a repercussion of the beneficial combination of carbon fiber reinforcement with the cnt epoxy, which presents higher capability to disfigure without fracture. moreover, it presents the types of damages occurring in the tested laminates on back. the observation called a small damage describes the initiating of delamination partially relating to fracture and it describes the delamination of various layers and the delamination of some layers. however, the specimen is completely penetrated by a high speed impact without complete fracture and the fractured specimens were impacted with adequate energy to clinch a complete failure. accordingly, measuring the total ballistic impact energy absorbing capability is named penetration resistance. correspondingly, the amount of the energy preliminary to the total penetration is related to the highest load, namely, the load the structure can bear before breaking utterly. hence, effects of volume of carbon nanotubes on the angled ballistic impact for carbon kevlar hybrid fabrics 241 maximum amount of the tolerable force by the material without even whole perforation is saturation ballistic impact energy [48-51]. to sum up, it can be clearly seen that resistance of reinforce matrix caused less fracture, which occurred despite the fact that the tearing of the fabrics matrix is still strong enough to trap the bullet. fig. 11 back face’s specimens after impact at 140 degree with 0.3% cnt 4. conclusions investigational studies were conducted experimentally on the ballistic impact behavior of the nanoparticle dispersed unidirectional laminate specimens carbon kevlar® hybrid fabrics/epoxy composites with and without cnt. for comparison, studies were carried out on unidirectional laminate specimens carbon kevlar® hybrid fabrics/epoxy with and without cnt. damage and energy absorbing oblique ballistic impacts at different angles (90, 100, 110, 120, 130, and 140 degrees) for the different materials investigated are presented. experimental data are given for the ballistic impact behavior of the seven types of specimens with cnts (0, 0.1, 0.3, 0.5, 0.7, 1, and 1.5%) examined. the specific observations are: firstly, the ballistic limit speed (v50) can state comprehensibly that incorporating moderate loading of an optimized weight fraction of cnts at 0.3% into carbon fiber / kevlar® hybrid fabric due to interlocking and van der waals forces improved the mechanical properties of the resultant nano-composite, accordingly were attributed to the cnts being well dispersed within the resin epoxy; secondly, a higher amount of the cnts (0.5-1.5 wt. %) provided insufficient improvements in the impact energy absorbing properties, consequently at more than 0.3% of the cnts significant reductions were technically recorded; thirdly, improvements in impact mechanical properties can be analytically achieved by increasing the angles of targets about 26%, which is providing the most significant improvements in impact resistance due to rising the applied thickness of target analytically; lastly, cnts reinforced composites exhibit more advantageously impact mechanical properties than neat epoxy. moreover, cnts normally exhibit elevated stiffness and extraordinary ballistic impact resistance, whose superior properties are predominantly 242 e. randjbaran, d.l. majid, r. zahari , m.t.h. sultan , n. mazlan attributed to the presence of robust sp 2 -bonded carbon-carbon lattice in their external shells; contrary to this, considering increasing the volume of cnts, agglomeration can be occurred. acknowledgements: the paper is a part of the research done within the project code frgs/1/2019/ stg07/upm/02/10 and vot no. 5540209. the authors would like to thank the department of 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implications for reducing bird strikes, landscape and urban planning, 179, pp. 38-45. 52. rocha, r.j.b., pina, l.m.p., gomes, m., sousa, j.p., 2020, evaluation of impact resistant composites for aircraft cockpit, materiales compuestos, 4(1), pp. 1-9. facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 735 750 https://doi.org/10.22190/fume210125033h © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper the enhanced homotopy perturbation method for axial vibration of strings ji-huan he1,2,3, yusry o. el-dib4 1school of science, xi'an university of architecture and technology, xi’an, china 2school of mathematics and information science, henan polytechnic university, jiaozuo, china 3national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, china 4department of mathematics, faculty of education, ain shams university, roxy, cairo, egypt abstract. a governing equation is established for string axial vibrations with temporal and spatial damping forces by the hamilton principle. it is an extension of the well-known klein-gordon equation. the classical homotopy perturbation method (hpm) fails to analyze this equation, and a modification with an exponential decay parameter is suggested. the analysis shows that the amplitude behaves as an exponential decay by the damping parameter. furthermore, the frequency equation is established and the stability condition is performed. the modified homotopy perturbation method yields a more effective result for the nonlinear oscillators and helps to overcome the shortcoming of the classical approach. the comparison between the analytical solution and the numerical solution shows an excellent agreement. key words: homotopy perturbations method, exponential decay parameter, damping duffing equation, damping nonlinear klein-gordon equation 1. introduction string axial vibrations arise in various fields, e.g., suspension bridge, drill string, and ring spinning; the vibration property will significantly affect the safety and the life span, and it can be described by nonlinear differential equations with possible two-fractal derivatives, which are difficult to solve exactly. the scientists have made great efforts to provide information about the approximate solutions of these equations [1-6]. of the nonlinear vibration systems that have been fascinating to engineers and scientists, by far received january 25, 2021 / accepted march 11, 2021 corresponding author: yusry o. el-dib department of mathematics, faculty of education, ain shams university, roxy, cairo, egypt e-mail: yusryeldib52@hotmail.com 736 j.-h. he, y. o. el-dib the most perplexing is the frequency-amplitude relationship. a simple but accurate approach to the relationship is much needed in practical applications [7-9]. it was revealed that nanofiber’s surface morphology can be controlled by the vibration’s frequency-amplitude relationship [10], and a successful finding of the fangzhu’s vibration property has revealed the hidden nanotechnology in ancient china [11-13]. the variational principle for nonlinear differential equations is another effective analytical method [14,15]. the string axial vibration can be modeled by the klein-gordon equation (kg), which is a significant group of partial differential equations; it is also found in relativistic quantum mechanics and field theory. el-dib [16] introduced the multiple-scale homotopy perturbation method (he’s multiple-scale method) as an outer perturbation of the nonlinear kg equation. a highly accurate periodic unsteady solution was derived from three order perturbations. furthermore, a complete approximate solution consisting of a temporal solution, which functions as an outer expansion along with the spatial solution serving as an inner expansion of the nonlinear kg equation, was achieved. el-dib et al. [17,18] recently proposed a new approach to the study of the nonlinear instability analysis. their analysis has revealed a nonlinear pde that controls the surface deflection of the interface. they have transferred the characteristic equation to a kg equation. through a traveling–wave solution, they have examined the stability profile. it is important to note that the temporal damped kg equation is not naturally dissipative. so, the introduction of the dissipative mechanisms is necessary to force the energy to decay to zero when time goes to infinity. indeed, the appearance of the damping terms given in the damped kg equation is not required by itself to gauge the energy e(t) associated with the problem, but it is a non-increasing function of the parameter t. uniform decay rate estimates to the problem have been considered by cavalcanti et al. [19-21]. the maximum principles of the optimal control problems, governed by a damped kg equation with state constraints, were examined by parka and jeong [22]. lin and cui [23] investigated a damped nonlinear kg equation, taking into consideration a kernel space. they assumed a new technique in solving this equation to emphasize the feasibility and the rigor of the method. the local solution of the initial-boundary value problem for a class of fourth-order wave equations with a high damping term was obtained by using fixed point theory [24]. he and el-dib [25] used the reducing rank method to solve a third-order damped nonlinear equation produced from the strong damped nonlinear kg equation one of the most traditional techniques to find an approximate solution is the homotopy perturbation method (hpm). the major property of the hpm is its ability and flexibility to examine a wide class of nonlinear differential equations conveniently and accurately [26]. it has been developed and improved by scientists and engineers. the hpm with two expanding parameters was suggested by el-dib [27]. a combination of the hpm and the laplace transforms were used by el-dib and moatimid [25]. el-dib [29] suggested a modified version of the hpm by the multiple scales technique. this new modification works well, particularly for nonlinear oscillators. ren et al. [30] developed the multiple scale by combining it with the hpm to improve it by incorporating some known technologies. it provides solutions to nonlinear equations for which the classical perturbation method proved unsuccessful. a modification of the hpm with the so-called reducing-rank technique has been introduced by shen and el-dib [31] and el-dib and elgazery [32]. in this paper, we present an efficient method for the approximation of the damped nonlinear klein–gordon equation, with the aim to solve and study the stability of the periodic solution. the enhanced homotopy perturbation method for axial vibrations of strings 737 the key features of our methods are based on the coupling of the exponential decay parameter to the homotopy perturbation method to overcome the shortcomings of the classical method. 2. mathematical model of the damping nonlinear klein-gordon equation the governing equation can be obtained from the following variational principle [14,15]: 2 1 { } 0 t t k p w dt− + =    (1) where k, p, and w represent, respectively, the kinetic energy, the potential energy, and the work done by the external force; k. p and w can be expressed, respectively, as 2 0 0 1 ( ) 2 l ly y k dx ydx t t        = −    (2) 2 0 0 0 0 1 ( ) 2 l l l ly y y p t dx ydx t dx ydx x x x           = − = −       (3) 0 l w f udx =  (4) here y is the axial displacement,  is the density, t is the surface tension of the string, f is the external force,  represents the temporal damped coefficient,  stands for the spatial damped coefficient, + 0tt t xx xy y ty y f  − + − = (5) eq. (5) can be re-written as +2 2 0tt xx t xy py y y f − + − = (6) where p=t/, f=f/, /  = and /  = , while y(x,t) is the displacement at location x and time t. when ,0==  eq. (6), the klein-gordon equation becomes: 0tt xxy py f− − = (7) when f y= − (8) we have a linear damped klein-gordon equation +2 2 + 0tt xx t xy py y y y  − + = (9) in this paper, we consider a cubic nonlinear case 3f y qy= − + (10) we have the following one-dimensional nonlinear wave equation with linear damping parameters: ).,(;22 3 txyyqyyyypyy xtxxtt ==+++−  (11) where  refers to the natural frequency, and q stands for the cubic-stiffness parameter. 738 j.-h. he, y. o. el-dib eq. (11) becomes the classical nonlinear klein–gordon equation that appears in various branches of science, including quantum mechanics, nonlinear optics fluid mechanics, and solid physics. in the case of  > 0 and  = 0, with some specific nonlinearities, eq. (11) becomes a one-dimensional temporal damped nonlinear klein-gordon equation [33]. el-dib et al. [34] discussed the time fraction of the damped kg equation to overcome the presence of the non-zero damping coefficients. besides, they applied the modified multiple scales method for the same aims. if the cubic term in eq. (11) becomes vanishing because it is small compared to the linear term, then the right-hand side term contributes to the order of y3 and thus can be neglected. in the linear limit, eq. (11) has the well-known exact temporal solution for the damped system given by: ( ) cos( ), t x py t ae t − − = +     (12) where a and  are real constants determined by the initial conditions. linear frequency is given by . 3 2 22 p   −−= (13) this procedure cannot be successfully used when the nonlinear terms are involved. the solution of the damping nonlinear klein-gordon eq. (11), by applying the classical analysis of the homotopy perturbation method [26], leads to the disappearance of the damping coefficient. in what follows the homotopy perturbation with the exponential decay of the slow variable is elaborated [3,35]: 2.1. solution with the modified hpm by exponential decay description to build the homotopy equation, we select the linear partial differential operator and the nonlinear operator as: yyyl ttt += and .22 3qyyypyny xtxx −++−=  (14) the corresponding temporal homotopy equation may be constructed as 3( ; ) ( ) ( 2 2 ) 0; [0,1],tt xx t xh y q y y q py y y qy q= + + − + + − =    (15) where function y(x,t) becomes ( , ; ) ( , ; ) .q ty x t q u x t q e− = );( qyh is the homotopy function. as the primary solution can be obtained by making parameter q tend to zero, we find ).;,(lim),( 0 0 qtxutxy q→ = (16) the final form of the approximate solution of the original eq. (11) has the form ).;,(lim),( 1 qtxuetxy q t → −=  (17) assuming that eq. (15) admits the solution [3] .),,();,( tqeqtxuqtxy −= (18) the enhanced homotopy perturbation method for axial vibrations of strings 739 function u(x,t,q) is expanded as ...),(),(),(),,( 2 2 10 +++= txuqtxqutxuqtxu (19) also, one can use the expanded frequency as ...,2 2 1 2 +++= qq (20) where i is unknown in total frequency 2 and determined by the disappearance of the secular term. employing eq. (18) with eq. (15), and using expansions (19) and (20), we obtain homotopy function h(u;q) arranged in the form .0...),,(),()();( 0122 2 01100 =+++= uuuhquuqhuhquh (21) the estimation of ; 0,1,2,...nh n = is as follows: 0 0 0 ( ) lim ( ; ), q h u h y q → = (22) 1 1 0 0 ( , ) lim ( ; ), q h u u h y q q→  =  (23) 1 0 0 1 (, ,..., , ) lim ( ; ). ! r r r rq h u u u h y q r q→  =  (24) according to eqs. (12-14) and equating each hi to zero, we obtain 2 0 0 0( ) ( ) ( , ) 0,tth u u x t=  + = (25) 2 2 2 1 1 0 1 0 1 0 0( , ) ( ) ( , ) ( ) ( 2 ) 0,tt tt xx xh u u u x t t u p qu u=  + −  + + −  +  − − =  (26) 2 2 2 2 2 1 0 2 1 1 1 0 1 2 2 2 2 21 0 0 2 0 1 0 0 02 ( , , ) ( ) ( , ) ( ) (2 3 ) ( ) (2 3 ) 0. tt tt xx x tt xx x h u u u u x t t u p u qu u t u p u u t qu u u =  + −  + −  +  − − +  + +  − −  − − − =       (27) handling eq. (25) with eq. (26) yields 2 3 1 0 0 1 0 0( ) ( , ) 2 .tt xx xu x t p u u u qu + =  −  + + (28) eq. (27) will be simplified by using eqs. (25) and (28), to become 2 2 2 3 2 1 0 1 2 0 0 0( ) ( , ) ( 2 3 ) ( ) 2 .tt xx x xxu x t p qu u p u u tqu + =  −  + + +  + +  −   (29) suppose the solution of eq. (25) has the form 0 ( , ) cos( ), x pu x t ae t − =  +   (30) where unknown arbitrary function b(x) is chosen to be ( ) , x pb x ae  − = with constant a. by making use of (20) into (18), we transformed the right-hand side into a polynomial in exp( ).x p  − 740 j.-h. he, y. o. el-dib ( ) ( ) ( ) 22 2 2 1 1 3 3 3 3 ( , ) cos 4 1 cos 3 3 4 x x p p tt x p u x t a qe e a t p e a q t       − − −    + =  + +  +    +  + (31) avoiding the secular term that appeared in eq. (31) gives . 4 33 2 2 2 1 x pqea p   − −−= (32) consequently, the total solution of eq. (31) becomes 33 1 2 ( , ) cos(3 3 ). 32 x p qa u x t e t − = −  +    (33) employing eqs. (30) and (33) into eq. (29), and using eq. (32), the result reduces to 42 4 2 2 2 2 2 2 2 22 3 3 2 2 2 52 5 2 3 3 ( ) ( , ) cos( ) 2 128 1 3 3 cos(3 3 ) 2 8 128 3 cos(5 5 ). 128 x x x p p p tt x x p p x p q a u x t tqa e e ae t qa t a qe e t p q a e t − − − − − −    + =  + − −  +      + − − −  +     −  +               (34) avoiding the secular terms from eq. (34) requires that .0 128 3 2 3 4 2 422 22 2 =  −−+ −− x p x p e aq eqat   (35) according to the solvability condition of eq. (35), the solution of eq. (34) becomes 22 3 3 2 2 2 4 53 2 3 5 2 4 1 1 3 3 ( , ) cos(3 3 ) 1616 256 3 sin(3 3 ) cos(5 5 ). 32 1024 x x p p x x p p u x t qa t a qe e t p qa q e t a e t − − − −    = + +  +        +  + +  +             (36) the approximate second-order solution can be performed by inserting eqs. (30), (33), and (36) into eq. (18), and taking q = 1 yields: 3 3 2 33 2 323 2 2 4 52 5 4 ( , ) cos( ) cos(3 3 ) 32 3 sin(3 3 ) 32 3 3 cos(3 3 ) 16256 cos(5 5 ). 1024 t x t x p p t x p t xx pp t x p qa y x t ae t e t qa e t qa a qe e t p q a e t     − + − +          − +      − +−       − +    =  + −  +  +  +    + +  +     +  +                   (37) the enhanced homotopy perturbation method for axial vibrations of strings 741 if the higher orders of the perturbation are done, the compact form of (1 – 2t + …) becomes 2 te − [3]. this solution still contains unknown frequency . its determination is the subject of the following section. 2.2. the nonlinear frequency in the perturbed form to get stability condition the two solvability conditions (32) and (35) can be used to establish the frequency formula by inserting them into the expansion (20), and making q = 1 yields: . 128 3 4 3 4 2 422 222 x p x p t e aq qea    −      +−  −−= (38) without upgrading the power of the frequency, we need to obtain an approximate solution of the above frequency equation. to accomplish this solution, we put eq. (38) in the perturbed form by introducing an artificial small parameter  [0, 1]. thus, we have 42 42 2 2 2 2 3 3 ( ) . 4 128 ( ) t x x p p q a a qe e   − + −    = − −           (39) consider the following expansion: ...2 2 1 22 +++=  (40) inserting eq. (30) into eq. (29) and equating the identical power of  to zero, we obtain . 128 3 4 3 2 42 2 1   aq qa −−= (41) to perform the first-order approximate solution of eq. (39), we employ eqs. (13) and (41) in eq. (40), which yields . 128 3 4 3 2 42 222   aq qa −−= (42) the stability requires  to be real, which needs the following condition: .0 128 3 4 3 2 42 22 −−   aq qa (43) employing the linear frequency as defined by eq. (13), we obtain .0 128 33 4 33 422 22 2 2 2 −         −−−         −− aq p qa p     (44) 2.3. numerical illustrations the numerical solution of the damped nonlinear kg-equation (11) is obtained by using a mathematical software for the system of p=1, =3, q=0.3, a=1, =0.1, =0.1. the results are displayed in fig. 1. it is noted that the small values in damping coefficients  742 j.-h. he, y. o. el-dib and  lead to the gradual damping in the periodic wave solution. the analytical solution as given by eq. (37) is also calculated numerically for the same system as used in the numerical solution, assuming that =0. the results are located in fig. 2. fig. 1 the numerical solution of eq. (11) for a system having p=1, =3, q=0.3, a=1, =0.1, =0.1 fig. 2 the analytical approximate solution of eq. (11) as given by eq. (37) using the full frequency as given by eq. (39). the system is considered as given in fig. (1) for comparison, the two kinds of solutions of eq. (11) are collected together in one graph as shown in fig. 3. it is observed that there is an excellent agreement between the numerical and analytical solutions along the time. the enhanced homotopy perturbation method for axial vibrations of strings 743 fig. 3 the comparison between the numerical and analytical solutions of eq. (11) for the same system as given in fig. 1 the examination of the impact of temporal damping coefficient  on the picture of the periodic wave solution of the modified analytical solution of eq. (37) is displayed in fig. 4. the numerical values are the same as those used in fig. (2), except that the spatial variable is fixed to value x=1. it is observed that as =0 the pure periodic solution is found. by a slight change in  from the zero value, the damping influence is observed. the damping influence of the increase of spatial damping coefficient  is shown in fig. 4. the calculation is made for the same system as illustrated in fig. 4 by fixing  at the zero value. it is noted that increase in  decreases the amplitude of the wave solution. fig. 4 the variation of temporal damping coefficient  for the same system as given in fig. 1 except that the spatial variable is considered fixed at the value x=1 744 j.-h. he, y. o. el-dib fig. 5 the variation of spatial damping coefficient  for the same system as given in fig. 4 except that temporal damping coefficient  is considered fixed at the value =0 3. alternative process across the traveling wave description traveling waves arise naturally in many physical systems, usually qualified by partial differential equations. therefore, an alternative homotopy equation can be formulated by introducing traveling wave variable (x,t) defined as ( , ) 2 2 .x t x p t= +   (45) according to the above new independent variable, we have 2 2 22 ( ), 2 ( ), 4 ( ), 4 ( ),t x tt xxy p y y y y p y y y          = = = = (46) where the prime denotes the total derivative with respect to variable . employing eq. (45) with eq. (11), it will handle the following damping duffing equation: 2 3 0 1 ( ) ( ) ( ) ( ),y y y ry p  + + =     (47) where 2 0 and r are given by the following notations: 2 0 2 2 2 2 , 4 ( ) . 4 ( ) p p q r p p = + = +       (48) accordingly, the homotopy equation can be built in the form 2 3 0 1 ( ; ) 0; [0,1]h y q y y q y ry q p    = + + − =      (49) the enhanced homotopy perturbation method for axial vibrations of strings 745 the homotopy expansion is applied associated with the modification by the exponential decay parameter [3] as follows: 1 22 0 1 2( ; ) ( ( ) ( ) ( ) ...), q py q e y qy q y − = + + +      (50) introducing modified frequency  which is given by ,...2 2 1 2 0 2 +++=  qq (51) where i is unknown, determined through the perturbation analysis. substituting eqs. (50) and (51) into eq. (49), and proceeding as requested by the homotopy perturbation method, the above unknowns can be determined and have the following results: 0 ( ) cos( ),y a= +   (52) where a and  are two arbitrary constants determined by the initial conditions. besides, we have 3 1 2 ( ) cos(3 3 ), 32 ra y = − +     under the condition 2 1 3 . 4 ra = − (53) moreover, we get ( ) 3 2 3 2 2 2 2 2 5 4 3 3 cos(3 3 ) sin(3 3 ) 32 32 128 cos(5 5 ), 1024 ra ra ra y p p r a               = + + + +    + + (54) under the condition: 2 4 2 2 2 2 1 3 3 . 44 128 r a ra pp    = − + + (55) as known, the approximate second-order solution of eq. (47) is given by 1 22 0 1 2 1 ( ) lim ( ( ) ( ) ( ) ...) , q p q y e y qy q y      − →   = + + +    (56) also, the approximate frequency is formulated by 2 2 2 0 1 2 1 lim( ...). q q q → = + + +    (57) inserting y0(),y1() and y2() into eq. (50) yields 12 2 1 2 2 2 2 2 4 2 4 ( ) 3 cos( ) cos(3 3 ) 32 32 3 sin(3 3 ) cos(5 5 ). 128 1024 p p y ra ra e ae ra r a p                − −   = + + − + +    + + + + (58) 746 j.-h. he, y. o. el-dib where the frequency equation is determined [35] as . 128 3 4 3 4 1 2 42 2 2 2 0 2    ar era p p +−−= − (59) it is worth observing that through performing the final solution (58), we replace the terms (1-/p+…) which refer to the first two terms of exponential exp(-/p); therefore, this compact form is used in eqs. (58) and (59). without upgrading the power of frequency 2, we proceed to obtain an approximate solution of eq. (49). this approximate solution is formulated in the form . 4 3 4 1 128 3 4 3 4 1 2 2 2 0 42 2 2 2 0 2       −− +−−= ra p ar ra p   (60) the necessary stability requirements need to be positive 2. this requires .0 4 3 4 1 128 3 4 3 4 1 2 2 2 0 42 2 2 2 0        −− +−− ra p ar ra p   (61) the above stability is arranged in terms of the definition given in eq. (48), which leads to the same stability condition as in eq. (44). 3.1. numerical estimations the comparison between the numerical solution of the duffing eq. (47) and the modified hpm solution given by (58), with the full frequency as given by (59), is shown in fig. 6. the numerical solution is made by the use of mathematical software. the calculation is made for the same system as given in fig. 1. the numerical solution is located in red, while the analytical solution is plotted in blue. this graph indicates that there is an excellent agreement between the numerical solution and the analytical modified hpm solution. fig. 6 the comparison between the numerical solution and the analytical solution to the duffing eq. (47), for the same system of fig. 1 the enhanced homotopy perturbation method for axial vibrations of strings 747 fig. 7 the modified hpm solution in eq. (58) for the same system as given in fig. 2 with the variation of parameter  in fig. 7, we plot the modified hpm solution given by eq. (58) with the full frequency as given by eq. (59) for variation of temporal damping coefficient =0.1, 0.2, 0.3, 0.4. inspection of the graph shows that as  increases, the amplitude of the wave solution decreases gradually until vanishing. this indicates the loss of energy as  increases. the variation of spatial damping coefficient  with fixation of =0.1 is displayed in fig. (8). the conclusion turns out to be similar to that of . the examination of the impact of coefficient p on the picture of the wave solution is indicated in fig. 9. it is observed that as p increases, so does the amplitude of the wave solution. this observation is expected because parameter p represents the inverse of the damping coefficient of the duffing eq. (47). fig. 8 the same system of fig. 2 with the variation of  fig. 9 the same system of fig. 2 with the variation of p 748 j.-h. he, y. o. el-dib 4. results and discussion if damped parameters  and  tend to zero in eq. (11), the result is the classical klein-gordon equation: ).,(;3 txyyqyypyy xxtt ==++  (62) further, let →0 and →0 in the frequency formula (38), it reduces to . 128 3 4 3 2 42 22  −−= aq qa (63) this is equivalent to those obtained before in [16] by using the classical hpm. the classical hpm has successfully solved this equation. el-dib [16] applied the he’s-multiple scale technique to find the periodic solution and to obtain the stability conditions. further, el-dib [27] investigated eq. (62) by applying the method of variable separation to variable y(x,t), and investigated the periodic solution using the technique of the multi-homotopy perturbations. these previous studies failed to analyze the nonlinear damped klein-gordon eq. (11). in ref. [35], the authors tried to overcome the shortcomings of the classical hpm. they converted the integer second-time partial derivative to the fractional nonlinear partial differential of order (+1; 0<1), and applied the properties of the riemann–liouville fraction derivatives, through the hpm. this technique allowed the authors to obtain the periodic solution of eq. (11) and the stability conditions in terms of parameter  and letting →1 in the final results. the new modification to hpm discussed in the present study depends on the idea of the normal form technique. it is known that this technique is successfully used in the linear damped equation. in the current work, we succeeded to adapt the technique of the normal form when coupled with the hpm, in which the exponential decay parameters are established. the analysis shows that the amplitude behaves as an exponential decay by the damping parameter. an excellent agreement with the numerical solution is observed in these calculations. this scheme yielded a more effective result for the nonlinear oscillators and helped to overcome the shortcoming of the classical approach. 5. conclusion the periodic solution of nonlinear models has much significance and has drawn a great deal of interest. in the current work, we have investigated the solution of the damping nonlinear wave equation. the normal form transformation is used to solve the damping linear klein-gordon equation. further, the frequency equation is estimated as displayed in eqs. (12) and (13). although the classical hpm is not successful in analyzing the damping nonlinear oscillator when it is associated with the exponential decay parameter as defined in eq. (18), the solution is obtained. this modification has been successful in suppressing the shortcoming of the classical hpm. the first-order approximate solution is derived in eq. (37), the frequency-amplitude equation is established in eq. (42) and the stability condition is performed in eq. (44). furthermore, the traveling-wave description is used to transform the klein-gordon equation in the damping duffing equation into a new dependent variable. the same procedure of homotopy perturbation with the exponential the enhanced homotopy perturbation method for axial vibrations of strings 749 decay parameter is used, and the solution and the frequency formula are obtained. numerical calculations are done and some graphs are displayed. the periodic damping analytical solution is found to have an excellent agreement with the numerical solution. this conclusion is reached in the case of the partial differential eq. (11), and, also, in the case of the ordinary nonlinear duffing eq. 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https://arxiv.org/search/math?searchtype=author&query=martel%2c+y https://arxiv.org/search/math?searchtype=author&query=yuan%2c+x https://doi.org/10.1016/j.rinp.2020.103345 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 479 493 https://doi.org/10.22190/fume170911026b © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper  determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal pump using numerical simulation results udc 681.5 jasmina bogdanović-jovanović, dragica milenković, živojin stamenković, živan spasić university of niš, faculty of mechanical engineering, serbia abstract. one of the most important aims in the turbo pump design is to achieve an optimal design of the pump impeller. the basic assumption in the design procedure of the impeller is that of the axisymmetric fluid flow. it can be confirmed or disputed by using the method presented in the paper, which uses the results of numerical simulation of fluid flow in the pump impeller. the method is actually a procedure for determining averaged axisymmetric flow surfaces and meridian streamlines. furthermore, according to the obtained streamlines, a correction of the impeller blade geometry can be made (if the streamlines deviate significantly from the assumed axisymmetric ones). it is also possible to calculate the specific works of the elementary stages and compare them with the previous assumptions. the pump impeller torque can be calculated as well. key words: centrifugal pump, averaged flow surface, meridian streamlines, numerical simulations 1. introduction in view of the vast number of applications of centrifugal pumps in various systems, such as water supply, irrigation, drainage, water cooling systems, etc., it is crucial to ensure they are working in the most effective way possible. turbo machine designing has always been a complex task, which is largely based on the designer’s experience, due to many assumptions made in the process of calculating the impeller geometry. thus, the impeller calculation has to be followed by the prototype(s) testing and later, if needed, received september 11, 2017 / accepted november 27, 2017 corresponding author: jasmina bogdanović-jovanović university of nis, faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: bminja@masfak.ni.ac.r 480 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić impeller blade correction (which is usually necessary for obtaining a more efficient impeller and higher pump efficiency). a special attention in the pump designing must be paid to the pump impeller. the basic assumption is that the fluid flow in the pump impeller is axisymmetric, which is the case of profile cascades with an infinite number of infinitely thin blades [1-4]. the continuity of the flow must be maintained, and the fluid flow energy must be increased by the required design value [5, 6]. by averaging the flow velocities according to the circular coordinate, the fluid flow in the real pump impeller can be derived into that of the fictive impeller with an infinite number of infinitely thin blades, which creates the equal flow declination as the real pump impeller [7, 8]. an accurate determination of meridian streamlines in the centrifugal pump impeller is very important since they represent meridian traces of the axisymmetric flow surfaces in the pump impeller. on the other hand, numerical simulations of the flow in the radial pump impeller provide for obtaining all flow parameters in the flow domain (pump impeller). there are many attempts to use the cfd results of flow in turbo pump impellers in the process of developing their optimal design [9-12]. values of the numerically obtained flow parameters, especially of all the velocity components could be very useful, as will be presented in the method for determining averaged axisymmetric flow surfaces. this method for averaging flow parameters and calculating averaged flow surfaces according to the circular coordinate, in order to obtain a model of the fictive impeller with an infinite number of infinitely thin blades, is presented in the paper. meridian streamlines are calculated using the integral continuity equation for the previously averaged flow parameters [8]. the comparison of the obtained meridian streamlines and the streamlines defined in the process of blade designing can be made. furthermore, the specific works of elementary stages in the meridian cross-section can be calculated and compared with the predefined specific works of elementary stages. the pump impeller torque can be calculated as well, and also compared with the numerical simulation results. 2. equations for averaging flow parameters over circular coordinate and flow equations observing the universal curvilinear orthogonal coordinate system (q1, q2, q3), there are following coordinate surfaces [7, 8]: meridian planes q3=const., axisymmetric surfaces q2=const. and axisymmetric flow surface q1=const. perpendicular to q3=const. (fig. 1). lame's coefficients are also defined as a function of curvilinear coordinates, l1(q1,q2), l2(q1,q2) and l3=r/ro, where r=r(q1,q2) is a radial distance from the origin of cylindrical coordinate system (r, , z) [7, 8]. direction of circumferential coordinate q3( 3 oe ) does not have to be the same as that of the impeller rotation (  ); also, the right (positive) coordinate system is used (dashed blade lines in fig.1 correspond to the left (negative) coordinate system). using cylindrical coordinate system (r, , z): q1=z, q2=r and q3=ro (l1=1, l2=1, l3=r/ro), where ro is the radial distance in the pump inlet. determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 481 fig. 1 cross-section of the centrifugal pump impeller in curvilinear coordinate system if circumferential velocity u=-r<0 and absolute velocity c3=-cu<0, where  is the angular velocity and cu is the circumferential component of the absolute velocity, then relative velocity w3=c3+r=-cu+r>0 for unit vectors 3 o oe u  , also, if circumferential velocity u=r>0 and c3=cu>0, then w3=c3-r=cu-r<0 for 3 o oe u . the suction side of the impeller blade is denoted by “l” and the pressure side is denoted by “g” (fig. 1), indexes “a” and “b” are: a=l, b=g for 3 o oe u  , respectively a=g, b=l for 3 o oe u  [8]. absolute (c) and relative (w) flow velocities are related, taking into account   o ze (in the direction of the rotation axis),  ,    c w u w r , therefore, rot rot 2  c w . (1) according to the cosines theorem, the velocity triangle exists: c 2 =w 2 + 2 r 2 -2cu, therefore, w 2 = c 2 - 2 r 2 +2cu. scalar and vector functions can be considered as an averaged value over the circular coordinate and pulsation component. the averaged pulsation component is equal to zero [5]. 2.1. averaging of flow parameters in turbo pumps any scalar function f(q1,q2,q3) can be averaged over circumferential coordinate q3: 3 1 2 3 1 2 ( ) 1 2 1 2 3 3 3 ( ) 1 ( ) ( )d δ b a q q ,q q q ,q f q ,q f q ,q ,q q q   , (2) where the circumferential coordinate can be written as q3=rφ (fig. 1). 482 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić point m(q1,q2) in the meridian cross-section of the centrifugal pump impeller belongs to radial coordinate r=r(q1,q2). all the flow parameters, such as pressure p or vector components (relative wj and absolute cj flow velocity), can be averaged over circumferential coordinate q3. the averaging interval is q3=q3b(q1,q2)q3b(q1,q2), where q3b(q1,q2) and q3a(q1,q2) are equations of two neighboring blade surfaces in the blade passage. formula (2) becomes [7, 8]: 3 3 ( ) ( ) 1 2 1 2 3 3 1 2 3 ( ) ( ) 1 1 ( , ) ( , , ( ))d ( ) ( , , ( ))d ( ) δ δ ( ) b b a a q r r q r r f q q f q q q r q r f q q r r q r         , (3) where, dq3=dq3(r)/l3 and q3=q3(r)/l3=r(r)/l3, for (r)=b(r)–a(r) [deg]. since coordinate q3 is circular, the averaging is obtained over a circular arc in the blade passage area. depending on the number of numerically calculated flow parameters the integral in eq. (3) can be resolved numerically using the trapezoidal rule. in the ansys cfx software, even only one blade with half of the blade passages on each side can be numerically simulated (in order to reduce the computational time due to impeller symmetry). the same strategy applies to the fluent software [10]. anyway, regardless of whether the whole impeller or just a single blade is numerically simulated, only one blade with half of the blade passages around it will be enough for analyzing numerical simulation results, as shown in fig. 2. fig. 2 averaging over circular coordinate q3 (around the blade) averaging of the scalar function for the values as given in fig. 2 can be written:       1 1 2 ( ) ( ) 1 2 1 2 1 2 ( ) ( ) 1 1 , , , ( ) d ( ) , , ( ) d ( ) δ ( ) δ ( ) m n m r r r r f q q f q q r r f q q r r r r                , (3a) where, 1 1δ ( ) ( ) ( )mr r r     , 2 δ ( ) ( ) ( )n mr r r     [deg]. the application of the trapezoidal rule for eq. (4) yields: 1 2 1 2 1 1 1 1 2 1 1 1 ( , ) ( )( ) ( )( ) 2δ ( ) 2δ ( ) m n j j j j j j j j j j m f q q f f f f r r                                , (4) determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 483 where: 1 1δ ( ) ( ) ( )mr r r     , 2 δ ( ) ( ) ( )n mr r r     ,  [deg], fj=fj(r), j=1,2,...m1,m2,...,n. 2.2. averaging of derivatives of flow parameters, gradient, divergence and curl the corresponding derivatives of the flow parameters can be averaged as follows [7]: 1 2 1 2 3 3 1 δ , where δ ( , ) ( , ) δ b a f f f f q q f q q q q         and (5)  3 3 3 3 3 1,2 3 1,2 3 1,2 1,2 1,2 1,2 δ1 1 δ , where: δ δ δ b a q f q q q qf f f f f q q q q q q q q                                                  (6) inclination angles of blade profiles, a,b and a,b, are measured in the negative direction of the circumferential flow velocity, as shown in fig. 3. also, for a radial pump impeller, the vectors perpendicular to blade surfaces, 1 , 2 , 3 ,, ,a b a b a bn n n , are given in fig. 3. thus, the averaged derivative (6) can be transformed into the next equation: 1 2 1 23 1 2 3 1 2 3 3 3 (δ )1 1 δ δ δ , , , , l nq ff f q q q q l n                , for 1,2 1,2 1,2 3 3 3 δ b a n n n f f f n n n                    (7) where, 3 1 1 1 , 1 3 3 3, , a b a b a b q l l n ctg q l l n                    and 3 2 2 2 , 2 3 3 3, , a b a b a b q l l n ctg q l l n                    for a point m(q1, q2). fig. 3 cross-sections of the centrifugal pump impeller where 3 o oe u  (a=l, b=g) 484 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić gradient of a scalar function f, divergence and curl of any vector v become [7, 8]: 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 grad grad(δ ) δ grad δ δ δ δ 1 1 1 div div(δ , ) δ , =div δ , δ δ δ 1 1 1 rot rot(δ , ) δ , rot δ δ δ δ n n f q f f f f q l q n l q n n n v q v v v v q l q n l q n n n v q v v v q l q n l q n                                         3 ,v                (8) also, the following relations are obtained: 3 3 3 3 3 3 3 3 3 1 1 1 grad δ div δ , rot δ δ δ δ                          n n n f f , , v ,v v ,v l q n l q n l q n . (9) 2.3. averaging of fluid flow equations there are two equations used for describing a turbulent incompressible fluid flow for the stationary operation of turbo machine, and the first one is continuity equation: div 0w . (10) neglecting gravity, the flow equation is given by the formulation [13]: [ , rot ] gradw c e , (11) where e is the energy per unit mass of the fluid of density ρ, according to the bernoulli’s integral for relative fluid flow: 2 2 1 2 3 ( ω) , ( , , ) 2 2 p w r e e e q q q      . (12) averaging the continuity eq. (10) over the circular coordinate, it becomes: 3 3 3 1 div(δ , ) δ , =0 n q w w l n        . (13) equation (13) can be transformed due to the requirement of perpendicularity of vectors n and w , then the second term of eq. (13) is equal to zero. the remaining part of the equation has been developed and then multiplied with the number of blades zl = 2/, where:  = 2/zl is an angular blade pitch, k = φ/τ = zlφ/2 is the blockage factor, which represents the cross-section reduction due to the real thickness of the impeller blades, and q3 = ro [7, 8]. thus, eq. (13) becomes: 3 2 1 3 1 2 1 2 (2 ) (2 )o or kl l w r kl l w q q         . (14) determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 485 equation (14) is a necessary and sufficient condition for the existence of meridian stream function, 1 2( , )m m q q  , where the meridian velocity is 1 1 2 2 o o mw w e w e  , thus 3 2 2 3 1 1 1 2 2 and 2m m o or kl l w r kl l w q q            . (15) equation (15) can be used to calculate the averaged meridian stream function (meridian streamlines) m , for known averaged velocities and blockage factor. it determines volume flow rate (q) through axisymmetric cross-sections, d dm q  second term in eq. (13) can be neglected only in the case of a very thin boundary layer (a fully developed primary turbulent flow with very high reynolds numbers). in practice, the fluid flow in turbo pumps is highly turbulent but often showing separations of the boundary layers, when the second term in eq. (13) should not be neglected and the meridian streamlines should be determined by using the requirement of equal flow rate passing between the axisymmetric flow surface and the hub surface (not using eq. (15)). averaging of the flow eq. (11), where the flow values can be treated as the sum of mean and fluctuation values, w w w  and rot rot (rot )c c c   , it becomes [7, 8]:  rot rot grad       w, c w ,( c ) e . (16) equation (16) can be finally written in the next form [8]: (1) (2) (3), rot gradw c f f f e       , (17)  (1) (2) (3) 3 3 3 3 3 3 2 2 1 1 , δ , , δ , , , rot where, δ δ ( ω) and 2 2 n n f w w f e f w w l q n l q n p w r e                          (18) equation (18) represents the forces obtained as the result of averaging, which should be treated as mass forces of the blades acting on the averaged fluid flow. since    3 1 f f , especially if w is changing linearly over circumferential coordinate q3,  3 f can be neglected [13]. also, for axisymmetric flow and model of inviscid fluid,  2 0f  . but if the fluid flow is not axisymmetric,  2 f has to be taken into account. thus,    1 2  f f f :       (1) 1 2 2 3 3 2 1 3 1 3 3 (1) 2 3 3 1 1 1 2 3 2 3 3 (1) 1 3 3 2 3 1 1 2 2 3 3 1 δ( )ctg δ( ctg ) δ δ 1 δ( )ctg δ( ctg ) δ δ δ( ctg ) δ( ctg ) δ δ δ f w w w w w w w w l q f w w w w w w w w l q w f w w w w w w w l q                                    (19) 486 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić and      2 2 2 1 2 3 3 3 3 3 3 3 1 1 1 δ( ,ctg ), δ( ,ctg ), δ , δ δ δ f e f e f e l q l q l q          (20) where, 1 3 1 3 2 3 2 3/ / ctg and / / ctgn n f f n n f f      . assuming axisymmetric flow surfaces (when  2 0f  ) the general equation of the averaged axisymmetric flow in centrifugal pump can be obtained. the surfaces are perpendicular to  1 f and vector )1(n  is in the direction of  1 f ( 0],[ )1()1( fn  ), then 0),( )1( wn  . since  (1) and  (1) are corresponding blade angles and considering 1 1c w , 2 2c w , 3 3c w r  , for 3 o oe u  and 3 3c w r  , for 3 o oe u , eq. (11), i.e. eq. (17) transforms into the general equation of the averaged fluid flow in pump impellers: (1) (1)3 3 2 2 1 1 2 2 1 1 1 2 1 2 1 2 2 ( ) ( ) ( ) ( )1 1 1 1 1 0 rc rc l c l c e ctg ctg rl q rl q l l q q c l q                     (21) (1) (1)3 3 2 2 2 1 1 1 2 2 1 1 1 2 1 2 1 2 2 ( ) ( )1 1 or, 2 2 ( ) ( )1 1 1 0. rw rwr r ctg r ctg r rl q q rl q q l w l w e l l q q w l q                                    (22) real and viscous fluids differ from non-viscous ones; and, according to the energy losses, e , the averaged bernoulli’s integral for relative and absolute flow can be written [8]: ( ) ( ) δ ( ) ( ) ( ) ( ) δ ( )m o m g m o m u o m g me e e g rc e           (23) where ( )o me  is averaged mechanical energy of the relative fluid flow in inlet control surface “o”, δ ( )g me  is averaged flow energy loss from the inlet control surface to the arbitrary circular cross-section, when the meridian trace is line const.m  in eq. 23, ( )og  relates to averaged absolute flow, 2( ) / / 2o o og p c   , where 3uc c  for 3 o oe u  (for “+”) and 3uc c for 3 o oe u (for “‒“). in vaneless parts of the pump, the differential equation of the averaged flow is obtained for 0f  . 3. numerical simulation of fluid flow in radial pump impeller the given methodology for obtaining averaged flow surfaces and averaged streamlines in the radial pump impeller is illustrated in the case of a centrifugal pump, which is constructed and tested as the part of the previous project of the department of hydroenergetics, faculty of mechanical engineering niš (“improving the constructive solution of a centrifugal pump in order to increase its operating area and to improve cavitation characteristics”). it is a centrifugal norm pump (fig. 4), used mainly for water supply or irrigation purposes, which operates with the following operating parameters: flow rate q=30 l/s, pump head h=14,9 m, number of revolutions n=1490 min -1 , efficiency =0,78, inlet determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 487 diameter is d1=110 mm, outlet diameter d2=250 mm (cut to 220 mm) and 6 impeller blades. specific speed is nq=nq1/2 h -3/4 =33,3. detailed impeller geometry is given in [8]. fig. 4 geometric model of the centrifugal pump impeller and a detail of the discretization mesh used in cfd a discretization mesh is generated using the ansys workbench bladegen and it consists of 166998 nodes and 700506 elements, mostly tetrahedral, 589332, pyramidal 463 and wedges 110711, (fig. 4). spiral casing is deliberately omitted since in some studies [14] it has been shown that for such models of centrifugal pumps the spiral does not affect too much pump performance characteristics. numerical simulations of the centrifugal pump are performed using the ansys cfx 14.0, which solves rans equations, with standard k- turbulent model [15, 16]. specified conditions are: mass flow at the impeller inlet and static pressure at the outlet. high resolution advection scheme is used. numerical solving of governing equations continues until the root mean square values of the equation residuals become smaller than 10 -5 . 488 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić fig. 5 numerical and experimental results of the pump head characteristic the volume flow rate is changed from 18 l/s to 36 l/s, giving the following operating characteristic chart. validation of obtained results is presented in fig. 5; the numerical and experimental curves show a very small difference in the values obtained (1,7%). 4. determination of meridian streamlines of averaged flow using the integral continuity equation first of all, it is necessary to determine a series of control cross-sections across meridian lines (lk, k=1,2,…), as shown in fig. 6 [8]. fig. 6 control cross-sections determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 489 each of these lines contains an arbitrary number of calculation points (n) and for every point an averaged flow parameter (such as wr, wz, w, cr, cz, c, w 2 , c 2 , p and pt , eq. (4)) is calculated, using the values obtained by numerical simulations. using the integral continuity equation, for each meridian trace represented with a line l, the volume flow rate through an axisymmetric flow surface, for k=zlδφ/2π is: o ( ) ( ) ( ) ( ) 2 d d n n n o o r l z r l z r r l z r q kc r r kc r z           . (24) in the centrifugal pump impeller, 6 different cross-sections are selected, two of which are outside of the blade area (a-a in front of the blades and f-f behind the blades) and four cross-sections are in the impeller blade area (b-b, c-c, d-d and e-e), as shown in fig. 7. these streamlines represent the initial (or the first) approximation. there are nine points on each meridian control line, which are used for averaging all the flow parameters. along every circular arc passing through these points, there are ten evenly distributed points in which all the flow parameters are obtained by the cfd. fig. 7 designed meridian cross-sections the following values are needed: the blockage factor due to blade thickness, k = zl/2 number of blades zl = 6, and angle δφ  60 o . note that in blade area k < 1, and in vaneless space k=1. volume flow rates in j cross-section of the pump impeller can be calculated: 1 1 1 1 1( )( ) ( ( )( )), for 1,2,...,j j j j j j j j j jq q f f r r f f z z j n              (25) where, qo = 0, and f and f are functions calculated knowing the values of averaged axial ( zw ) and radial relative velocity ( rw ) in the pump impeller: . . ( ) ( , ) ( , ) ( , ) ( , ), for 1,2,..., ( ) ( , ) ( , ) ( , ) ( , ), for 1,2,..., k j k j j z l j j j j j j z j j j r l j j j j j j r j j f krw f z r k z r r z r w z r j n f krw f z r k z r r z r w z r j n             (26) 490 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić flow parameters are obtained by numerical simulation of the flow in the pump impeller, for the nominal operating pump regime. the volume flow rates are calculated for all cross-sections (a, b, c, d, e, f) and the average value is q=0,0282 m 3 /s=28,2 l/s, obtaining an error of 6% mostly due to blade thickness and flow separation from the blade surfaces and reverse fluid flow. in cross-sections a-a and f-f, which are outside the blade area (in front and behind the impeller blades), the volume flow rates are close to the designed nominal value. thus, the values obtained are 29,436 l/s and 29,945 l/s, respectively. 4.1. correction of meridian streamlines in radial pump impeller considering that in the observed control cross-sections the flow rates between the hub and these surfaces are not equal, it is clear that the assumed axisymmetric flow surfaces in the initial approximation are not averaged meridian streamlines. therefore, the next step is the correction of streamlines. fig. 8 shows the initial meridian line (j), the corrected meridian line (j’), and the distances between these lines and the next (or neighboring) line, where k-k is the flow cross-section perpendicular to the meridian streamlines. fig. 8 an illustration for the correction methodology volume flow rates between the streamlines can be calculated using the mean values of meridian velocity ( )( . j avgmc ): ( ) ( ) 1 . . . 1 . ( ) ( ) 1 . . . 1 . 1 2 , for ( ) 2 1 2 , for ( ) 2 j j j j j m avg m avg m j m j j j x j x m avg m avg m j m x q r n c c c c q r n c c c c                   (27) assuming a linear change of the meridian velocity along meridian traces nj and nx, the relation can be obtained: 2 . . 1 2( / ) a( / ) , where a 1 (2 a) x j x j m jx j m j n n n n cq q c            . (28) since j jx j x jxjx q qq q q qqqq       1 then , . (29) and the quadratic equation is obtained:                                 j jx j x j x q qq n n n n 1)a2(bfor ,0b2a 2 . (30) determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 491 there are two possible solutions of eq. (30): . . 1 1 if a 0, , then ( 1 1 ab) a x m j m j j n c c n          and (31) j jx j x jmjm q qq n n cc        1 then , 0,a if 1.. . (32) note that as a result of eq. (31) one should take only a real and positive solution. in fig. 9 the initial approximation of the meridian streamlines (dashed lines) and the corrected meridian streamlines (full lines) are presented. a noticeable difference is clearly visible but in some parts of the impeller there is a greater difference of the corrected streamlines, compared to the initially assumed. fig. 9 meridian streamlines of averaged flow and urc distribution in the impeller outlet 4.2. calculation of specific work of pump elementary stages and torque the specific work of elementary stages in the pump impeller can be calculated: ( ) ω[( ) ( ) ]k j u f u ay s rc rc  (33) in fig. 9 a change of the specific work in an outlet cross-section (f-f), calculated for the given example of centrifugal pump is presented. it clearly shows an unequal distribution of urc from the hub to the shroud of the pump impeller. first three cross-sections have similar values of specific work (15% smaller than design value). in the next (fourth) cross-section 492 j. bogdanović-jovanović, d. milenković, ž. stamenković, ž. spasić this value increases, obtaining the design value in the next 3 cross-sections. in the last two cross-sections the specific work exceeds design value up to even 35%. the common practice in the design of all types of turbomachinery is the assumption of equality of the specific work of elementary stages. there are some earlier suggestions proposing the use of an unequal distribution of the specific work of elementary stages in turbomachinery designing [17, 18]. the justification of such an assumption (unequal distribution of urc ) is shown in fig. 9. the pump impeller torque can be calculated using the formula: 2 1 (1,2) ω ( ,d ) ω ( ,d ) (2) (1)(p) k k u m u m k k a a m m r c c a r c c a m m       , (34) where a1 and a2 are control cross-sections on inlet (1) and outlet (2) of the pump impeller. therefore, 1,2 1,2 2 2 1,2 1,2(1,2) 2 2 , ( ( , ))k u z u r l l m r c c dr r c c dz l l z r     (35) if the calculation points are uniformly and densely distributed along control lines l1 and l2 (n1,n210), then the integrals of eq. (34) can be determined, with good accuracy, using the trapesoidal rule. using the notation, for j=0,1,2,...n1,2: (1,2) (1,2) 1,2. 1,2. (1,2) 2 (1,2) (1,2) (1,2) (1,2) 2 (1,2) (1,2) (1,2) , , ( , ), ( , ) j j j j j u z j j j j u r j j j z z r r g r c c g z r g r c c g z r       (36) equation (34), i.e. eq. (35), for inlet and outlet cross-sections, becomes: 1,2 1,2 (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) 1 1 1 1 1 1 (1,2) ( )( ) ( )( ) n n k j j j j j j j j j j m g g r r g g z z              (37) in the given example of a centrifugal pump, it is obtained mk=29,785 nm, which differs for only 5% from the value obtained by numerical simulation. 5. conclusion the principle and method of averaging flow parameters and flow equations, using the results of numerical simulations of flow in the centrifugal pump impeller, are presented in the paper. meridian streamlines of the averaged flow are obtained, as well as the distribution of specific work in the impeller. the specific work distribution in the impeller outlet shows unequal distribution, which indicates it deviates more or less from the pre-designed values. the impeller torque was estimated as well. such information is of the greatest importance in the design and calculation of the pump impeller. the meridian streamlines in the centrifugal pump impeller should be corrected during the design process. using the numerical simulation data, according to the procedure described in the paper, much more relevant information is available to the pump designer. at the same time, this procedure shortens the time required for pump model developing and testing. determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal… 493 acknowledgements: the paper is a part of the research done within the project tr33040, “revitalization of existing and designing new micro and mini hydropower plants (from 100 kw to 1000 kw) in the territory of south and southeast serbia”, supported by the ministry of science and1 technological development of the republic of serbia. references 1. obradović, n., 1973, basics of turbomachinery, gradjevinska knjiga, belgrade, serbia (in serbian). 2. babić, m., stojković, s., 1990, basics of turbomachinery: operating principles and mathematical modeling, naučna knjiga, belgrade, serbia (in serbian). 3. ristić, b., 1987, pumps and fans, naučna knjiga, belgrade, serbia (in serbian). 4. krsmanović, lj., gajić, a., 2005, turbomachinery theoretical basics, university of belgrade, faculty of mechanical engineering belgrade, belgrade, serbia (in serbian). 5. voronjec, k., obradović, n., 1973, fluid mechanic, gradjevinska knjiga, belgrade, serbia (in serbian) 6. lakshminarayana, b., 2002, fluid dynamics and heat transfer of turbomachinery, john wiley & sons, inc., new york, usa. 7. bogdanović-jovanović, j.b, bogdanović, b.p, milenković, d.r., 2012, determination of averaged axisymmetric flow surfaces according to results obtained by numerical simulation of flow in turbomachinery, thermal science, 16 (suppl.2), pp. 647-662. 8. bogdanović-jovanović, j., 2014, determination of averaged axisymmetric flow in hydraulic turbomachinery runner, phd thesis, university of niš, faculty of mechanical engineering, serbia, 263p 9. yedidiah, s., 2008, a study in the use of cfd in the design of centrifugal pumps, engineering applications of computational fluid mechanics, 2(3), pp. 331-343. 10. stuparu, a., resiga, r., muntean, s, 2011, a new approach in numerical assessment of the cavitation behaviour of centrifugal pumps, international journal of fluid machinery and systems, 4(1), pp. 104-113. 11. tan, m.g., he, x.h., liu, h.l., dong, l., wu, x.f., 2016, design and analysis of a radial diffuser in a single stage centrifugal pump, engineering application of computational fluid mechanics, 10(1), pp. 500-511. 12. an, z., zhounian, l., peng, w., linlin, c., dazhuan, w., 2015, multi-objective optimization of a low specific speed centrifugal pump using an evolutionary algorithm, engineering optimization, 48(7), pp. 1251-1274. 13. etinberg, i.e., rauhman, b.s., 1978, hydrodynamics of hydraulic turbines, sent petersburg, russia. 14. stamenković, ž., bogdanović-jovanović, j., manojlović, j., 2013, determination of centrifugal pump operating parameters in turbine operating regime, proceedings, 16. symposium on thermal science and engineering of serbia, sokobanja, serbia, pp. 846-855. 15. ferziger, j.h., peric, m., 2002, computational methods for fluid dynamics, springer, berlin, heidelberg. 16. casey, m., 2004, best practice advice for turbomachinery internal flows, qnet-cfd network newsletter (thematic network for quality and trust in the industrial application of cfd), 2(4), pp. 40-46. 17. bogdanović-jovanović, j., bogdanović, b., božić, i., 2014, design of small bulb turbines with unequal specific work distribution of the reunner's elementary stages, facta universitatis, series: mechanical engineering, 12(1), pp. 73-84. 18. bogdanović, b., spasić, ž., bogdanović-jovanović, j., 2012, low-pressure reversible axial fan designed with different specific work of elementary stages, thermal science, 16(suppl.2), pp. s605-s615. facta universitatis series: mechanical engineering vol. 18, no 2, 2020, pp. 205 217 https://doi.org/10.22190/fume200611025p © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper finite element modeling of temperature fields on the cutting edge in the dry high-speed turning of aisi 1045 steel roberto pérez 1 , luis hernández 1 , ana quesada 1 , julio pino 2 , enrique zayas 3 1 cad/cam study center, engineering faculty, university of holguín, holguín, cuba 2 civil engineering, faculty of technical sciences, universidad estatal del sur de manabí, ecuador 3 departament of mechanical engineering, school of engineering of barcelona (etseib), universitat politècnica de catalunya, barcelona, spain abstract. high-speed turning is an advanced and emerging machining technique that, in contrast to the conventional machining, allows the manufacture of the workpiece with high accuracy, efficiency and quality, with lower production costs and with a considerable reduction in the machining times. the cutting tools used for the conventional machining cannot be employed for high-speed machining due to a high temperature induced in machining and a lower tool life. therefore, it is necessary to study the influence of high cutting speeds on the temperature distribution in different typologies of cutting tools, with the aim of evaluating their behavior. in this paper, a finite element method modeling approach with arbitrary lagrangian-eulerian fully coupled thermal-stress analysis is employed. the research presents the results of different cutting tools (two coated carbide tools and uncoated cermet) effects on average surface temperature fields on the cutting edge in the dry high-speed turning of aisi 1045 steel. the numerical experiments were designed based on different cutting tools like input parameters and different temperature field zones like dependent variables in the dry high-speed turning of aisi 1045 steel. the results indicate that the dry high-speed turning of aisi 1045 steel does not influence significantly the temperature field zones when p10, p15 or p25 inserts are used. therefore, the use of a dry high-speed turning method, which reduces the amount of lubricant and increases productivity, may represent an alternative to turning to the extent here described. key words: high-speed turning, fem modeling, temperature fields, tool-chip interface friction, round edge received june 11, 2020 / accepted july 15, 2020 corresponding author: pérez, roberto affiliation, address: cad/cam study center, university of holguín, cuba e-mail: roberto.perez@uho.edu.cu 206 r. pérez, l. hernández, a. quesada, j. pino, e. zayas 1. introduction high-speed machining (hsm) constitutes one of the new technologies that, in contrast to the conventional machining, allows the manufacture of parts with high accuracy, efficiency and quality, with lower production costs and with a significant reduction in the machining times. besides, the hsm offers other advantages such as excellent dimensional accuracy and a better surface finish just as it facilitates the machining of materials with high surface hardness [1]. the metal cutting process can be considered as a deformation process where the deformation is highly concentrated in a small area. therefore, the cutting can be conducted as a chip formation process and may be simulated using the finite element method (fem). the classic orthogonal cutting model for continuous chip formation adopts continuous plane-strain deformation conditions. primary representation of the cutting model is illustrated in fig. 1. in the conventional machining cutting at low speeds, the friction mechanism is more effective at the tool flank face. however, in the hsm due to the considerable cutting speed, there exists an increase in the chip-tool contact friction, and it is much more significant at the tool rake face [2]. fig. 1 primary representation of the machining process, adapted from [2, 3] the distribution of normal stress (σn) and shear stress (f) describe the cutting temperature and tool wear. these distributions are generally represented as expressed in fig. 2 in dry conditions [4, 5]. over length lp, normal stress is very high and the metal adheres to the rake face; therefore, a plastic flow occurs in the work material. in this region, the shearing stress is independent of the normal load which is recognized as the sticking region of friction. on length ls, smaller normal stresses occur and the typical condition of sliding friction applies, where coulomb´s friction law can be functional. as a result of the metal cutting process, heat generation occurs in several regions or interfaces [7, 8, 9]. the primary sources of heat generation in machining are the plastic deformation of the layer machined and adjacent at the cutting surface as well as the friction that occurs between the tool and the workpiece being machined. finite element modeling of temperature felds on the cutting edge in the dry high-speed turning... 207 fig. 2 normal and frictional stress distributions on the tool rake face, adapted from [6] fig. 3 shows an idealized orthogonal machining process using a sharp tool. in this idealization, there are two heat sources; the primary shear plane source (q1) and the rake face source (q2), with the thermal constriction resistance (r1 and r2). the latter accounts for the secondary plastic deformation area in the chip, and the tool-chip friction. the inset in fig. 1 shows that point-to-point contact is encountered in the sliding zone. the heat generated in the process comes into the tool through a limited number of small contacts [10]. fig. 3 contact configuration and heat sources in the tool-chip-workpiece system, adapted from [10] it has been perceived that 60 to 80 percentage of the heat generated in the cutting process is evacuated through the chip. the percentage increases to the same magnitude as the cutting speed does. in the technological process of turning on average it has been observed that 50 to 86 percentage of the heat is evacuated through the chip, 10 to 40 percentage moves the tool, from 3 to 9 percentage moves into the workpiece and about one percentage radiates to the environment. heat distribution is affected by several factors, the most important of them being the cutting speed and the cut thickness of the cutting process [7, 8, 9, 11, 12]. 208 r. pérez, l. hernández, a. quesada, j. pino, e. zayas the cutting temperature is not constant throughout the cutting tool, the chip or the workpiece. it can be observed that the maximum temperature is developed not on the very cutting edge, but at the tool rake, some distance away from the cutting edge [13]. in recent years, the quantity of research projects related to the evaluation of the useful life of cutting tools, using different cooling technologies, has increased; however, there are few studies on the operation of dry turning at high cutting speeds in carbon construction steels, used in the manufacture of machine parts, as shown below. authors like özel et al. [3], leopold [14], and chinchanikar et al. [15] developed finite element studies for the orthogonal cutting in hsm. continuous re-meshing and adaptive meshing are the principal tools employed for avoiding the difficulties associated with deformation-induced element distortion, and for resolving fine-scale features in the solution. the model accounts for dynamic effects, heat conduction, mesh-on-mesh contact with friction and full thermos-mechanical coupling. this primary approach did not look at the scope, the temperature distribution by specific areas of the cutting tools. özel and altan [16] develop a methodology for simulating the cutting process in flat end milling operation and predicting chip flow, cutting forces, tool stresses and temperatures using fem. as an application, machining of p-20 mold steel at 30 hrc hardness using uncoated carbide tooling was investigated. the highest tool temperatures were predicted at the primary cutting edge of the flat end mill insert regardless of cutting conditions, using the commercially available software deform-2d. these temperatures increase wear development at the primary cutting edge. however, a predictive curve analysis adjusting the temperature distribution was not performed. özel [17] investigates the influence of edge preparation in cubic boron nitrite (cbn) cutting tools on process parameters and tool performance by utilizing fem simulations and high-speed orthogonal cutting tests on aisi h-13 hot work steel. distribution of temperatures in the workpiece, chip and the cbn tool was obtained from fem simulations. the temperature generated at the chip-tool interface was found substantially higher than the other temperatures. the author does not make a predictive analysis of fitting curves for the temperature distribution by regions of the cutting tool. in the work of fang et al. [18] the thermal impact in cutting are considered. the effects of land length and second rake angle of the grooved tool on chip formation, cracking and temperature are discussed. some simulation results are compared with other published analytical and experimental results. several authors [19-26] illustrate a dynamic explicit arbitrary lagrangian-eulerian (ale) based on fem modeling using commercially available software’s. fem techniques such as adaptive meshing, explicit dynamics and fully coupled thermal-stress analysis are combined to simulate hsm with an orthogonal cutting model. finite element modeling of temperature distribution induced by round edge cutting tools is performed in numerous materials. the authors do not obtain adjustment curves of the temperature distribution in every surface of the cutting tools. a structural model of a soft/hard composite-coated textured tool was proposed by yun et al. [27] and validated through a three-dimensional numerical simulation. its dry turning performance as applied to aisi1045 steel was analyzed via orthogonal experiments for different coating parameters. the authors do not differentiate the analyses by heat transfer zones in the cutting tool. a similar analysis was carried out by akbar et al. [28] on aisi / sae 4140 steel with uncoated and tin-coated tools. finite element modeling of temperature felds on the cutting edge in the dry high-speed turning... 209 an improved understanding of heat partition between the tool and the chip is required to obtain more accurate finite element models of machining processes. akbar et al. [29] performed an orthogonal cutting of aisi/sae 4140 steel with tungsten-based cemented carbide cutting inserts at cutting speeds ranging between 100 and 628 m/min. chip formation was simulated using a fully coupled thermo-mechanical fem. the results show that over a wide range of cutting speeds, the accuracy of fem output such as tool– chip interface temperature, are significantly dependent on the specified value of heat partition. the analysis discretized by thermal transfer areas was not performed. from another perspective, heisel et al. [30] describe the procedure for creating and verifying a thermos-mechanical fem model of orthogonal cutting. the temperature was measured in the secondary shear zone and at the exterior surface of the chip, which was analytically estimated as well and compared with the experimental values checking the proposed model. finally, the values for temperature, among other variables are related to those calculated with the fem cutting model. no detailed analysis of heat behavior by cutting tool zones was obtained. the objective of the present study is to increase a better understanding of the influence of the dry high-speed turning on temperature distribution in the different zones of the cutting tools. based on 2-d thermo-viscoplastic fem cutting simulations, the predicted distributions of temperature within the tool coating and substrate have been investigated. orthogonal turning fem simulation tests with different cutting tool materials and machining conditions have been carried out in order to achieve the abovestated objective. during and after each test, several numerical parameters have been quantified to calibrate the finite element model. 2. materials and methods 2.1. sample characteristics in the present research, the aisi 1045 carbon steel was selected as work material, in hot rolled conditions, as widely used in the manufacturing of parts for metal mechanics and the automobile industry. this steel is considered a critic standard for manufacturing machine parts. the composition of aisi 1045 steel is perlite–ferrite at 50%, with a medium carbon percentage, which is the maximum percentage for its group, leading to improved mechanical properties at the cost of making machinability more difficult. the chemical composition and mechanical properties of aisi 1045 are given in table 1 [31]. the microstructure and grain size were examined on the whole transversal section of the workpiece, using an optical microscope nikon epiphot. the grain size was 8 µm. the hardness of the sample was measured on the complete cross-section, using a microhardness tester shimadzu. the average hardness was 258 hb. 2.2. characteristics of inserts and experimental arrangement of the research two coated carbide tools (gc4215-p15 and gc4225-p25) were used during cutting tests as well as the uncoated cermet ct5015-p10, the tested tools were manufactured by sandvikcoromant. the cutting tool characteristics are given in table 2. 210 r. pérez, l. hernández, a. quesada, j. pino, e. zayas table 1 chemical composition of aisi 1045 steel elements chemical composition, % proof strength (0,2% yield), mpa tensile strength, mpa elongation, % hrc hardness c si mn p s cr ni al 0,450 0,150 0,710 0,036 0,007 0,122 0,024 0,035 300 570 14 26 table 2 characteristics of inserts insert coatings substrates first layer second layer third layer width (μm) w ti co nb al2o3 ct5015-p10 20,68 47,2 17,71 8,71 5,7 gc4215-p15 tin al2o3 al2o3 15 96,19 1,44 2,38 gc4225-p25 ti(c,n) al2o3 10 94,77 2,1 3,13 this research is focused on the study of the dry high-speed turning of the aisi 1045 steel, using three inserts with three levels of cutting speeds and five levels of machining times. also, two replicas for the acquisition of information have been made (90 tests). the experiments have been conducted on ten solid cylindrical workpieces with an initial diameter of 80 mm and a length of 300 mm. the depth of cut (a = 0,5 mm) and feed rate (f = 0,1 mm/r) were kept constant. 2.3. finite element analysis modeling 2.3.1. principles of finite element analysis modeling two-dimensional cutting simulations were carried out to explore the effect of the dry high-speed turning in three different cutting tools, using deform-2d software developed for significant plastic deformation problems. the simulation of chip flow was achieved by adaptively remeshing the workpiece nearby the tool tip. the workpiece was modeled as rigid-plastic and the oxley law was implemented to describe material flow as a function of strain, strain rate and temperature [3]. 1838 isoparametric quadrilateral elements were utilized for the workpiece mesh. on the other hand, the tool was modeled as rigid and meshed with 2709 elements. materials thermal parameters were set as a function of the actual temperature, based-on software database [3]. 2.3.2. flow stress data of the workpiece and substrate properties of the coatings this study considers johnson-cook (jc) work material models for aisi 1045 steel [6]. the jc work material model describes the flow stress of the material by considering strain, strain rate, and temperature effects as given in eq. (1): ̅ [ ( )̅ ] * ( ̇̅ ̇̅ )+ * ( ) + (1) finite element modeling of temperature felds on the cutting edge in the dry high-speed turning... 211 constants a (plastic equivalent strain, mpa), b (strain related constant, mpa), c (strain-rate sensitivity, mpa), n (strain-hardening parameter), and m (thermal softening parameter) of the jc constitutive model are obtained by the split hopkinson pressure bar shpb tests conducted at strain ranges of 0,05 to 0,2, strain-rate of 7500 l/s, and temperature ranges of 35°c to 625°c for aisi 1045 steel as given in table 3[6]. table 3 johnson-cook material constants material tm (°c) a (mpa) b (mpa) c n m aisi 1045 1460 553,1 600,8 0,0134 0,234 1 two coated carbide tools (cvd) gc4215-p15 and gc4225-p25 were compared during cutting tests as well as the uncoated cermet ct5015-p10. all the tested tools were manufactured by sandvik and the tool holder having the designation sclcr/l 2020k 12. after the tests, the cutting tools were analyzed using the jeol scanning electron microscope. every cutting edge of the cutting tools was previously inspected using the nikon epiphot optical microscope [31]. the simulation considered a uniform coating thickness of 15 μm on the rake face, cutting edge, and flank face for the p15 insert and 10μm for the p25 insert. the interface between the coating and the substrate was assumed to be perfectly bonded together, so that an interfacial slip during loading was neglected. the thermal properties for the workpiece material and the cutting tools were obtained from the literature. to model the thermal contact without significant resistance, a heat transfer coefficient of 45 n/sec/c was utilized. a shear friction factor of 0,6 and a mesh cutting tool of 2500 were used. the tool wear usui model was applied. 2.3.3. boundary conditions fig. 4 briefly shows the concept of this finite element model. to model the chip formation, the coupled eulerian-lagrangian finite element method was employed using the commercial software package deform-2d. with this model, the chip formation is simulated similarly to a fluid flow model. the fluid in this case is the workpiece material which continuously enters the eulerian domain at the top and then flows against the tool which is fixed in space. the software is accomplished of controlling overlock based on the boundary conditions defined for the workpiece; therefore, the nodes on the left side boundary are fixed in the x axis direction. at the same time, the nodes at the bottom remain fixed in the y direction. thermal boundary conditions are defined to facilitate thermo-mechanical modeling of solutions. the leading boundary conditions are: (a) heat is generated due to the plastic deformation of the part while the tool penetrates the material, assuming that 90% of all plastic deformation work is converted to heat; (b) the heat generated is due to the friction between the tool and the chip; and (c) the workpiece emits heat to the environment. the results will specifically check the influence of cutting speed on the twodimensional model of the dry high-speed machining of aisi 1045 steel for different cutting tools (p10, p15, p25). the overlock has been carried out in such a way that when there is a penetration of one element over another with a value greater than 0,0014 mm, the software will perform meshing. 212 r. pérez, l. hernández, a. quesada, j. pino, e. zayas fig. 4 modeling of orthogonal cutting with coupled eulerian-lagrangian method table 4 shows the numerical variables considered for simulation in this research. the arrangement of each run of the numerical simulation was: rake angle; cutting edge radius; incidence angle; cutting speed of 600 m/min; and p10, p15 and p25 inserts. table 4 variables considered in the numerical simulation cutting speed (m/min) zone p10 p15 p25 600 rake angle a-600.p10 a-600.2 a-600.3 cutting edge radius r-600.p10 r-600.2 r-600.3 incidence angle i-600.p10 i-600.2 i-600.3 the temperature of the cutting tool will be analyzed according to three zones: the zone defined by the rake angle, the zone defined by the cutting edge radius and the zone defined by the incidence angle. fig. 5 shows the discretization of the cutting tool and its zoning based on the aforementioned criteria. points from p1 to p20 correspond to the rake angle surface of the cutting tool, points p21 to p28 correspond to the cutting edge radius and from p29 to p32 to the incidence angle surface. fig. 5 discretization of the cutting tool by zones finite element modeling of temperature felds on the cutting edge in the dry high-speed turning... 213 3. results and discussion the results related to the finite element simulation of the temperature distribution in the three evaluated cutting tools in the dry high-speed machining of aisi 1045 steel are presented below. 3.1. effect of high speed manufacturing and coatings on tool temperature fig. 6 shows the isothermal lines in the y-z plane, restricted for clarity to the region closer to the cutting tool zones. fig. 6 (a) represents the case of no coating (the uncoated cermet ct5015-p10) and perfect contact over the full contact length. fig. 7(b) represents similar condition, but with a coated carbide tool gc4215-p15. in fig. 7(c), the tool was a coated carbide tool of type gc4225-p25. the figure shows that for the same high cutting speed of 600 m/min, the presence of coating causes the field temperatures to shift from the cutting edge to the other extreme point of the contact length (lc in fig. 3), as showed in figs. 6 (b) and 6 (c). in the case of coated inserts, a similar behavior is observed in terms of temperature distribution, and in their distribution in the contact length. the highest temperature point is reached in practically the same position as the contact length for the three cutting tools studied. for the studied conditions, it indicates that high speed machining does not have a decisive influence on the location of the point of maximum temperature in the contact length. a) p10 b) p15 c) p25 fig. 6 effect of high-speed manufacturing and coating on temperature distribution in the tools 214 r. pérez, l. hernández, a. quesada, j. pino, e. zayas 3.2. temperature distribution in the considered contact zones in order to evaluate the ways in which temperature behaves in the three modeled contact zones, their behaviors are presented below for the three cutting tools under study. fig. 7 behavior of temperature in the zone of the cutting edge radius fig. 7 shows the behavior of the temperature in the zone of the cutting edge radius (cer), as a function of time for the three tools. it is observed that in the case of the tool with three coatings (p15), its behavior is slightly superior compared with the other two tools. in general, it is observed that the use of a high cutting speed does not imply an appreciable manifestation of temperature differences in this zone. fig. 8 represents the behavior of the temperature in the contact zone of the incidence angle (ia), as a function of time for the three tools. in this case, it can be seen that the tool with three coatings (p15) has the best behavior compared to the other two tools, which are similar. fig. 8 behavior of the temperature in the zone of the incidence angle finite element modeling of temperature felds on the cutting edge in the dry high-speed turning... 215 the behavior of the temperature distribution in the area of the rake attack surface (ra) is presented in fig. 9. it shows that the insert with three coatings (p15) heats up slightly more than the other two inserts. like the aforementioned regarding the cutting edge radius, the difference is not significant. fig. 9 behavior of the temperature in the zone of the rake angle in general, it is observed that the behavior of the three inserts is similar in the case of the rake attack surface and the cutting edge radius but not so in the incidence angle. it is concluded from the thermal point of view that the dry high speed machining in aisi 1045 steel, for the analyzed conditions, does not produce significant changes when varying the inserts analyzed. 7. conclusion this paper presents a comprehensive experimental numerical evaluation of the influence of high cutting speeds on the temperature distribution in different typologies of cutting tools, with the aim of evaluating their behavior on the dry high-speed turning of aisi 1045 steel. the results indicate that the dry high-speed turning of aisi 1045 steel does not influence significantly the temperature field zones, when we use the p10, p15 or p25 inserts. therefore, the use of the dry high-speed turning method, which reduces the amount of lubricant and increases productivity, may represent an alternative to turning to the extent here described. acknowledgements: the paper is a part of the research done within the project ―experimental study and numerical simulation to improve the performance of cutting tools in high-speed machining on machine tools‖ of the university of holguin. the authors would like to thank to the cad/cam study center at the engineering faculty in university of holguin (cuba), the civil engineering career at the faculty of technical sciences in the universidad estatal del sur de manabí (ecuador), and the mechanical engineering department at the polytechnic university of catalonia (barcelona, spain) for the support provided. 216 r. pérez, l. hernández, a. quesada, j. pino, e. zayas references 1. sulaiman, s., roshana, a., ariffin, m.k.a., 2013, finite element modeling of the effect of tool rake angle on tool temperature and cutting force during high speed machining of aisi 4340 steel, proc. 2nd international conference on mechanical engineering research, 1–4 july, kuantan, pahang, malaysia. 2. özel, t., altan, t., 2000, determination of workpiece flow stress and friction at the chip–tool contact for highspeed cutting, international journal of machine tools & manufacture, 40, pp. 133–152. 3. özel, 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journal of manufacturing and materials processing, 2(4), 66, doi: 10.3390/ jmmp2040066 12402 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 515 527 https://doi.org/10.22190/fume240104009b © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper interlayer effect on deformation and fracture of dendritic structure formed during wire-feed electron-beam additive manufacturing of al-si alloy ruslan balokhonov1, veronika utyaganova1, diana gatiyatullina1,2, aleksandr zemlianov1, varvara romanova1 1institute of strength physics and materials science sb ras, russia 2national research tomsk state university, russia orcid ids: ruslan balokhonov https://orcid.org/0000-0001-9994-5685 veronika utyaganova https://orcid.org/0000-0002-2303-8015 diana gatiyatullina https://orcid.org/0000-0003-4697-1817 aleksandr zemlianov https://orcid.org/0000-0001-9763-5712 varvara romanova https://orcid.org/0000-0001-8693-3810 abstract. interfaces and surfaces play an important role in tribology, mechanics and materials science, causing plastic strain localization and stress concentration of different spatial scales. the interfacial inhomogeneity is highly pronounced in 3d printed materials due to thermo-cycling and layer-by-layer building. in this paper, the inlayer and interlayer structure of a eutectic al-si alloy fabricated by wire-feed electron-beam additive manufacturing is investigated by optical and electron microscopy. model structures inheriting the experimental morphology are created, and their deformation and fracture are simulated using abacus/explicit, with the user-defined subroutines being developed to describe the constitutive behavior of aluminum dendrite, silicon and eutectic materials. a two-scale computational approach is implemented to study the influence of the interlayer formed in the heat-affected zone on the dendritic structure strength. key words: additive manufacturing, aluminum alloys, microstructure, multiscale numerical simulation, plastic deformation, fracture 1. introduction tribology is a multidisciplinary science that requires considering solid as a multiscale system, where deformation and fracture develop at different time and scale levels [1]. for instance, the macroscopic contact mechanics has to consider, at least phenomenologically, the atomic-scale interactions, the adhesion, the influence of plasticity and internal stresses, received: january 04, 2024 / accepted march 02, 2024 corresponding author: ruslan balokhonov institute of strength physics and materials science sb ras 634055, tomsk, pr. akademicheskii, 2/4, russia e-mail: rusy@ispms.ru https://orcid.org/0000-0001-9994-5685 https://orcid.org/0000-0002-2303-8015 https://orcid.org/0000-0003-4697-1817 https://orcid.org/0000-0001-9763-5712 https://orcid.org/0000-0001-8693-3810 516 r. balokhonov, v. utyaganova, d. gatiyatullina, a zemlianov, v romanova etc. [2]. in the interplay of tribology, mechanics and materials science, the key role belongs to rough surfaces and curvilinear interfaces between the microstructural components: matrix and particles, coating and substrate, polycrystalline grains, layers, interlayers, etc. the interfacial and surface asperities are the sources of stress concentration and plastic strain localization of different spatial scales, which may result in earlier crack nucleation and growth causing premature material failure [1, 3]. the multiscale nature of deformation is most pronounced in additively manufactured materials with distinct interfaces formed due to the layer-by-layer fabrication of the sample and the material thermo-cycling in between the layers during melting. additive manufacturing has been ever increasingly used in a variety of industries due to its capacity of fabricating products of intricate shapes with an optimal consumption of materials [4, 5]. an application of additive technologies allows reducing the weight and accelerating the fabrication of components critical for the production process [4, 6], as well as minimizing the material and energy consumption via a recovery of used rebuildable components [7]. the aluminum alloys used in additive manufacturing are attractive materials due to their high specific strengths and cost-effective recycling [8]. the aluminum-silicon alloys possess low thermal expansion coefficients and high corrosionand wear resistance [9, 10]. an addition of si also improves the characteristics of casting and mechanical treatment of these alloys [11]. the material microstructure depends on the technological process parameters, especially on the alloy solidification rate [12, 13]. on the whole, the eutectics is a strengthening phase, but its coarse-grained and agglomerated structures may decrease the product strength and plasticity [14, 15]. among the most popular printing processes are the laserand electron-beam melting [16, 17]. the selective laser melting (slm) of eutectic silumines selectively melts the powder by a laser beam along the scanning trajectory [18]. the wire-feed technologies have acquired an ever increasing application; they are efficient in printing of industrial prototypes and assembly components. the use of wires as the initial stock in laser melting processes allows saving more energy than in slm [19]. the processing conditions necessary for a stable deposition of an al 5087 wire were studied by a laser beam forming singleand multiple-layer strips [20]. the electron beam wire-feed additive manufacturing (ebam) has clear advantages over the laser beam process, since it occurs in vacuum, which rules out an oxidation of aluminum alloys, and the energy density and the depth of thermal energy penetration are much higher than those of a laser [21, 22]. the additively manufactured alloys differ from the conventional cast alloys of the same chemical compositions by their complex hierarchical structure with characteristic features at different scale levels [23]. earlier we demonstrated that the material inside individual layers of ebam al-si12 alloy represent a two-level composite [24]. on a scale of tens of microns there are aluminum dendrites surrounded by a eutectic network that at a lower level represents an aluminum matrix with volumetric silicon particles measuring hundreds of nanometers [25]. the purpose of this work is to study experimentally the ebam al-si12 alloy material structure in the heat affected zone between the layers and the common features of deformation and fracture of composites at different scale levels using the methods of numerical modeling. interlayer effect on deformation and fracture of dendritic structure formed during wire-feed... 517 2. wire-feed additive manufacturing of al-si12 alloy and microstructure analysis using an ad hoc setup designed and built at the institute of strength physics and materials science sb ras (tomsk, russia), we fabricated a bulk sample shaped as a thinwalled product (fig. 1a). the material used for printing was a 1.2 mm wire of a eutectic alloy, alsi12 esab оk autrod 4047, which was deposited on a 5 mm-thick substrate of an аlmg5 alloy. the chemical compositions of both alloys (alsi12 and аlmg5) are listed in table 1. the manufacturing followed an optimal regime of an exponential decrease of the heat input, which allows fabricating the samples with a minimal content of defects, such as porosity and discontinuities, and prevents the spread of the top layers. the heat input was calculated by the following formula: hi=(u⋅i⋅60)/(v⋅1000), where u – accelerating voltage (v), i – current (a), v – table motion velocity (mm/min). in this case, the heat input values were changed due to the exponentially decreasing current on every layer during printing of the sample. the heat input dependence of a layer-by-layer printing of alsi12 alloy is given in fig. 1b. fig. 1 thin-walled alsi12 alloy sample fabricated by wire-feed ebam (a) heat input parameters during layer-by-layer printing (b) table 1 compositions of аlmg5 substrate and alsi12 wire alloy al, % mg, % si, % mn, % fe, % cr, % ti, % zn, % v, % zr, % almg5 balance 5.21 0.41 0.48 0.43 0.23 0.11 0.16 alsi12 balance 13.3 0.11 0.08 0.03 0.035 0.005 the ebam sample structure was examined by metallography and scanning and transmission electron microscopy (sem, tem). the metallographic tests were performed using a standard technique. the samples with a size of 10х10 mm were cut from the middle part of the ebam workpiece, ground on an abrasive paper with a grit of 200 to 2000, and polished on diamond paste and suspension. the surfaces of metallographic sections were examined in an olympus lext ols4100 laser scanning confocal microscope. the sample microstructure was studied in the tescan vega ii lmu and leo evo-50 scanning electron microscopes (sem); the latter was equipped with a system for the energy dispersive microanalysis (eds). thin foils for tem examinations were prepared by thinning the samples cut from the thin-walled workpiece fragments to a thickness of ~ 160 µm via mechanical grinding on abrasive paper and using an em-09100is ion slicer at 7 kv for 7–8 hrs. the 518 r. balokhonov, v. utyaganova, d. gatiyatullina, a zemlianov, v romanova fine structure was studied in a jem-2100 transmission electron microscope (tem, tokyo boeki ltd., japan) at an accelerating voltage of 200 kv. the studies performed in this work have demonstrated that the alsi12 ebam alloy has a complex hierarchical microstructure at different scale levels with characteristic features of the layer-by-layer printing, which distinctly differs from a typical microstructure of the cast alsi12 alloy in terms of its morphology and phase distribution (fig. 2). inside the printed layers, there are large dendrites in a network (fig. 2a). in-between the layers, immediately in the electron beam affected zone, a 30–50 µm-thick interlayer is formed, containing large particles (fig. 2b). fig. 2 optical images of alsi12 ebam alloy at a scale of 100 (а) and 20 µm (b) fig. 3 sem images inside alsi12 layer: in se beams (a) and eds mapping of this area (b). tem eds mapping of fine eutectic structure (c) the dendritic structure inside the layer represents a two-level composite material (fig. 3a). at a scale of tens of microns (level 1), it consists of plastic inclusions in the form of dendrites of a primary solid solution of α-aluminum in a harder matrix representing a eutectic network (fig. 3b). at a lower micron scale (level 2), the eutectics, in its turn, is revealed by tem as volumetric submicron silicon particles in an aluminum matrix (fig. 3c) and, therefore, represents a metal-matrix composite material. silicon particles in the eutectics have a variety of shapes (rectangular, oval, rounded and elongated tooth-like interlayer effect on deformation and fracture of dendritic structure formed during wire-feed... 519 shapes). the particle size in the dendritic interlayer is several times larger than that of silicon particles in the eutectics (cf. figs. 4c and 3с). to sum up, the interlayer material, similarly to the eutectic material, is a composite with micron particles forming a cellular substructure rather than a structure with uniformly distributed silicon submicron-sized particles. it was also shown in the works of some authors studying the microstructure of am materials, manufactured by slm (e.g. [22]), that the structure morphology in the heat affected zone (haz) in eutectic aluminum-silicon alloys differs from that inside the layer. they [22] attribute the precipitation of coarse silicon particles to high temperature gradients in the haz. fig. 4 sem eds mapping of an area in alsi12 alloy in-between layers (а) and aluminum (b) and silicon (c) distribution patterns 3. computational analysis of deformation and fracture in dendritic structures using the experimental data presented in figs. 2b and 3c, we designed the model structures at level 1 (fig. 5a and b) and the structures of an aluminum matrix – silicon particle composite at level 2 (fig. 5c). an explicit three-phase model structure (fig. 5a), which is formed in-between the layers in the haz, contains an interlayer of large silicon particles. a two-phase dendrites–eutectics model structure (fig. 5b), typical for the in-layer material, was designed relying on two considerations necessary for a caeteris paribus comparative analysis. firstly, the dendrite morphology both in the interlayer area and outside it has to be consistent with that in the three-phase structure. secondly, the total volume fraction of aluminum in the dendrites and the eutectic network has to be the same for the twoand three-phase structures and be equal to 86% (typical for alsi12). henceforward the term “phase” has no conventional physical meaning and is used only to distinguish between “dendrites-eutectic” and “dendrites-eutectic-interlayer” structures. the dendritic structures (fig. 5а, b) are of the sizes on the order of 100х100 µm and are discretized by a regular grid containing 1000×1000 square finite elements. the 2.5x2.5 µm model of the eutectic material (fig. 5c) is approximated by a 480х480 mesh. the dynamic boundary-value problems on loading the composites are solved in a three-dimensional formulation simulating a plane strain state using abaqus/explicit in a two-level approach. firstly, at level 2 we solve the problem of an elastic-plastic deformation of a metal-matrix composite and then by averaging over the volume we identify the effective properties of the eutectic material. then these properties are used at level 1 in the numerical study of the plastic strain localization and crack propagation in the dendritic structures under tension. 520 r. balokhonov, v. utyaganova, d. gatiyatullina, a zemlianov, v romanova fig. 5 model structures: dendritic structures in haz (a) and inside the layer (b) and composite material in the eutectic network (c) in order to describe the elastic-plastic behavior of aluminum, we used the hooke’s law: 2 ( / 3 )p ij ij ij kk ij ij kk ij ijp s k= − + = + − −         , (1) where σij and sij are the stress tensor and deviatoric tensor component, p is the pressure, εij and p ij are the total and plastic deformation components, δij is the kronecker delta, k and µ are the bulk and shear moduli; the upper dot indicate the time derivatives. the plastic strain rate is found according to the associated plastic flow rule. the yielding condition includes the isotropic strain hardening into consideration: ( ) ( )exp( )p p p u u y rf = − − −      , (2) where σu and σy are the ultimate stress and the yield point, respectively, and p r characterizes the strain hardening degree. silicon particle in the eutectic network and in the interlayer are assumed to be elastic and do not fracture. cracks nucleate both in the aluminum and in the eutectics as soon as the accumulated equivalent plastic strain reaches its respective critical value: p p сal=  or p p сeut=  . (3) taking into account the models given by (1–3), a vumat user subroutine was developed for abaqus/explicit, which was used to simulate a kinematic tension of the structures. the elastic moduli of aluminum and silicon were taken from a reference book. the plasticity characteristics in the dendrites and the eutectics were assumed to be different; they were selected based on the concept of a dislocation mean free path to their pinning sites, which is determined either by the average size of dendrites (about 5–10 µm) or the average spacing between the silicon particles in the eutectic network (about 0.5–1 µm). in a manner similar to the hall–petch law, the yield stress and plasticity of aluminum in the dendrites or eutectics were selected in accordance with the experimentally obtained flow curves for the aluminum samples with the corresponding grain sizes [26, 27]. the elastic modulus values for the eutectic material in the dendritic structures at level 1 (fig. 5a, b) were calculated based on the aluminum volume fraction in the composite of φ=74% at level 2 (fig. 5c): interlayer effect on deformation and fracture of dendritic structure formed during wire-feed... 521 (1 )eut al sik k k= + −  , (1 )eut al si= + −     . (4) the parameters of the hardening function (2) for the eutectic material at level 1 were obtained by calculating the composite tension at level 2 via averaging of the equivalent stresses <> and accumulated equivalent plastic strains <p> over the domain shown in fig.5c, followed by an approximation of the resulting relationship ( )( ) ( )exp( )eut p p eut eut eut p peut u u y rf = = − − −        . (5) the results of modeling the composite tension at level 2 are given in fig. 6. due to a difference between the elastic moduli of aluminum and silicon, a stress concentration in the vicinity of the matrix–aluminum interface is observed as early as in the elastic deformation stage. upon further loading, plastic shears develop in the matrix in the sites of the highest stress concentrations, which propagate throughout the composite volume inbetween the particles and form plastic shear localization bands at an angle 45 degrees to the axis of tension (fig. 6c). within the stages of a developed plastic flow, the level of stresses in the particles is about 2–3 times higher than that in the matrix (cf. fig. 6a and b). fig. 6 equivalent stress distribution in aluminum matrix (a) and silicon particles (b), accumulated equivalent plastic strain (c) at 3 % homogenized deformation of eutectic composite (d) the averaged flow curve of the eutectic composite is presented in fig. 6d. by approximating this curve, we obtained the yield stress and strength values and those of the hardening coefficient 522 r. balokhonov, v. utyaganova, d. gatiyatullina, a zemlianov, v romanova in (5). these properties are used as the characteristics of plasticity and isotropic hardening of the eutectic network material during deformation of dendritic structures at level 1. the values of the model constants are listed in table 2. the results of modeling the dendritic structure deformation are presented in figs. 7–10. figure 7 illustrates the stressed state in the developed plastic flow stage of the composites. the eutectic network withstands a more than twice higher stress level than do the aluminum dendrites (compare pairwise fig. 7a, c and b, d). this is accounted for by two factors. firstly, aluminum in the eutectic network, due to a nearly an order of magnitude smaller dislocation mean free path, demonstrates higher strength characteristics, corresponding to its submicrocrystalline state, than aluminum in the micron-sized dendrites. secondly, the submicron silicon particles in the eutectic network cause an additional strengthening effect. table 2 model parameters material ρ, g/cm3 k, gpa µ, gpa σy, mpa σu, mpa εr p , % εc p , % al in eutectics 2.7 66 26 125 150 20 al in dendrites 2.7 66 26 60 100 20 90 eutectics 2.6 74 36 130 157 5 50 si 2.33 98 60 by performing a pairwise comparison of fig. 7a, b and c, d, we can reveal the following interlayer effect. on the whole, the stressed state of the eutectics outside the interlayer area hardly depends on its presence (cf. fig. 7a and c). the situation in the dendrites due to the fig. 7 equivalent stress distributions in eutectic network (a, c) and aluminum dendrites in matrix (b, d) in pre-fracture stage of dendritic structures with (a, b) and without (c, d) interlayer taken into consideration interlayer effect on deformation and fracture of dendritic structure formed during wire-feed... 523 presence of an interlayer is somewhat different. owing to the availability of comparatively large silicon particles, a specific inhomogeneous stressed state pattern is formed: thin lengthy stress concentration regions are found in-between the particles, forming a characteristic netstructure (fig. 7b). because of the interlayer, there are quantitative differences in the regions delineated by dashed circles: the stress concentration in these regions is higher in the three-phase structure than it is in the two-phase structure (cf. fig. 7b and d). the fine net-structure of the stressed state in the interlayer with characteristic narrower stress concentration regions results from the plastic strain localization in-between the silicon particles (fig. 8). high stress concentrations near the silicon particles/aluminum interface, which are generated as early as in the stage of elastic deformation of the threephase dendritic structure, cause a nucleation of the local sites of plastic deformation in the region of interfacial asperities. as the loading increases, the entire interlayer is covered by a system of localized shear bands, with the regions of the highest localization being observed along the interface between the interlayer and the eutectic network (dashed squares in fig. 8b). such an intensive deformation of the interlayer in the three-phase structure results in a redistribution of the stress-strain state and a much stronger strain localization in some regions of the dendritic structure surrounding the interlayer compared to the two-phase structure case (compare pairwise the areas delineated by dashed circles in fig. 8a, b and c, d). fig. 8 equivalent plastic strain distributions in the entire volume (a, c) and in separate aluminum dendrites (b, d) in pre-fracture stage of dendritic structures with interlayer (a, b) taken into consideration and without it (c, d) 524 r. balokhonov, v. utyaganova, d. gatiyatullina, a zemlianov, v romanova figure 9a presents the probability densities of plastic strain distribution over the volume in the two-phase and three-phase structures, calculated on the data of figure 8, separately for the dendrites and the eutectic network. we have found out that the eutectic network has been deformed rather weakly: the major part of its volume experienced small plastic strains of fractions of a percent (peaks in the curves with square symbols in fig. 9a). note that the strain localization patterns are quantitatively identical for the twoand three-phase structures (blue and purple curves with square symbols in fig. 9a coincide). the main differences are observed in the aluminum dendrites straining with a higher intensity than the eutectic network. a large volume fraction of dendrites, both in the threeand two-phase structures, experiences a strain of 5–8 % (major peaks in curves with triangular symbols in fig. 9a). it is noteworthy that, similarly to the eutectic network, there are peaks at the plastic strain of parts of a percent, but their volume fraction is much smaller. in the three-phase structure, there are more regions with a plastic deformation of 8–20 % and a smaller number of those where the deformation is in the range of 1–5 %, than in the two-phase structure (cf. blue and red curves with triangular symbols in fig. 9a). the regions with pronounced plastic strain localization occupy equal volume fractions in the twoand three-phase structures: blue and red curves with triangular symbols in fig. 9 coincide for deformations of more than 20%. fig. 9 probability densities constructed for distributions in fig. 8 (a) and flow curves of dendritic structures with and without interlayer taken into consideration (b). states c and l are shown in fig. 10 c and l the macroscopic response of the composite dendritic structures under tension is given in fig. 9b, where strain is the relative elongation of the computational domains. two main conclusions can be made at this point. firstly, the two-phase structure withstands a higher stress level than the three-phase structure. this occurs due to the following circumstance. despite the fact that the aluminum and silicon volume fractions in the twoand three-phase structures are the same (86 % al and 14% si), the volume fraction of the eutectics is larger in the two-phase structure. therefore, the latter contains a larger volume fraction of the high-strength aluminum, wherein the dislocation free mean path is an order of magnitude shorter than that in low-strength aluminum in the dendrites. secondly, the strength of the two-phase structure without an interlayer is higher than that of the structure with an interlayer effect on deformation and fracture of dendritic structure formed during wire-feed... 525 interlayer. this is related to the above-described character of plastic strain localization in the interlayer and its effect on the material stress-strain state beyond the interlayer. thus, a higher stress concentration and plastic strain localization in the area of the interlayer– eutectic network interface causes an earlier crack nucleation (fig. 10а) than this occurs in the two-phase structure (fig. 10j). the crack propagation character in both structures is similar. the first fractured regions appear in the aluminum dendrites. however, these fracture events are few; further on all secondary cracks nucleate in the eutectics in the vicinity of the interface with the dendrites in the cases where the equivalent plastic strain reaches its critical value εceut p (fig. 10d, m). primarily, these cracks propagate in the inter-dendritic space from one interface to another and stop (fig. 10e, n). a high stress concentration in the crack tip results in an intensive plastic strain localization in the dendrites, which is accumulated up to its critical value εcal p and the cracks grow through the dendrites (fig. 10 f-i and o-r). fig. 10 equivalent stress during fracture of dendritic structures with (a–i) and without (j–r) interlayer 526 r. balokhonov, v. utyaganova, d. gatiyatullina, a zemlianov, v romanova 4. conclusions using the methods of optical and electron microscopy, we have experimentally studied the microstructure of an al-si12 eutectic alloy fabricated by the additive layer-by-layer electron-beam wire melting. the results demonstrated that inside every layer the aluminum dendrites measuring tens of microns are formed, which are surrounded by a eutectic network. the eutectic network is identified by transmission microscopy as an aluminum matrix with volumetric homogeneously distributed primary-silicon submicron-sized particles. it was found out that in-between the layers in the heat affected zone an interlayer is formed from aluminum inherited from the dendrites. the interlayer contains agglomerated micro-sized silicon particles that are distributed inhomogeneously in the form of a cellular structure. based on the experimentally obtained images, a number of model structures were designed: dendritic structures with and without interlayer and a composite eutectic material structure. in order to describe the mechanical response of silicon and aluminum, we used the isotropic constitutive models of elasticity and elasticity-plasticity including fracture, respectively. the structures and the models were integrated into abaqus/explicit. the calculations of the composite structure tension were performed. a two-level approach was implemented, involving a primarily solution of the problem of a deformed aluminum matrix – silicon particles composite at the micro level and, by averaging over the volume, the effective properties of the eutectic network were identified. these properties were then used at the level of hundreds of microns in order to study the deformation and fracture of the dendritic structures. the results of modeling demonstrate that in the interlayer in the region of its interface with the eutectic network a high stress concentration is formed, which results in a plastic flow localization, an early 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of piezoelectric nanoresonator sayyid h. hashemi kachapi1, sayyideh gh. hashemi kachapi2 1department of mechanical engineering, faculty of engineering and technology, university of mazandaran, babolsar, iran 2department of physics, university of kashan, kashan, iran orcid ids: sayyid h. hashemi kachapi https://orcid.org/0000-0002-5384-9724 sayyideh gh. hashemi kachapi https://orcid.org/0009-0004-8518-4386 abstract. this study explores the influence of advanced continuum mechanics frameworks, specifically strain gradient theory (sgt), nonlocal theory (nlt), and gurtin-murdoch surface/interface theory (gmsit), on the nonlinear vibrational characteristics of a piezoelectric nanoresonator (penr). these findings are then juxtaposed with those derived from conventional continuum mechanics. the penr is driven by a nonlinear electrostatic force, incorporating both static and dynamic voltage components, and is situated within a viscoelastic foundation of the pasternak type. to dissect the nonlinear frequency response and stability of the penr, a methodology combining hamilton’s principle, galerkin’s method, complex averaging techniques, and arc-length continuation is utilized. the analysis reveals that disregarding nanoscale and interfacial effects results in significant deviations from the actual vibrational response of the penr. notably, under diverse boundary constraints, the material length scale parameter (associated with sgt) induces a reduction in the penr's rigidity, whereas the nonlocal scale parameter (associated with nlt) yields the opposite effect, enhancing stiffness. moreover, the oscillation amplitude and the extent of instability predicted by the advanced theories (nlt and sgt) surpass those obtained via classical analysis. fluctuations in the surface/interface parameters are demonstrated to exert a considerable influence on the penr's normalized natural frequency, resonance frequency, resonance amplitude, nonlinear dynamics, and system stability, manifesting as either augmentations or diminutions contingent upon the specific parameter under consideration. key words: piezoelectric nanoresonator, nonlocal strain gradient theory, gurtin–murdoch surface/interface, nonlinear frequency response, complex averaging received: september 03, 2024 / accepted march 10, 2025 corresponding author: sayyid h. hashemi kachapi department of mechanical engineering, faculty of engineering and technology, university of mazandaran, babolsar, iran e-mail: shhashemi.kachapi@umz.ac.ir , sha.hashemi.kachapi@gmail.com mailto:shhashemi.kachapi@umz.ac.ir mailto:sha.hashemi.kachapi@gmail.com 312 s.h. hashemi kachapi, s. gh. hashemi kachapi 1. introduction the relentless march of technological innovation has spurred the proliferation of intelligent materials and adaptable structural systems [1, 2]. within this domain, piezoelectric nanostructures, endowed with exceptional attributes, have garnered substantial interest, promising transformative applications across diverse sectors [3, 4]. notably, piezoelectric shell architectures are extensively deployed in sensing and actuation technologies [5, 6]. precise dynamic analysis and mathematical modeling of these nanoscale systems necessitate the incorporation of scale-dependent parameters. consequently, advanced continuum theories, encompassing nonlocal [7], strain gradient [8], and gurtin-murdoch surface/interface [9] frameworks, are employed to probe their nonlinear vibrational and dynamic behaviors. drawing upon nonlocal elasticity principles, feng et al. explored the axial vibrational characteristics of single-walled carbon nanotube-based mass sensors [10]. soltani et al. [11] leveraged nonlocal elasticity to analyze the lateral-torsional buckling behavior of axially functionally graded nonlocal beams. furthermore, arefi demonstrated that an increase in the nonlocal parameter amplifies rotations, in-plane displacements, and transverse deflections within a piezoelectric nanoshell exhibiting double curvature [12]. employing nonlocal strain gradient theory, boyina et al. [13] investigated wave propagation in viscoelastic timoshenko nanobeams under the influence of surface and magnetic field effects. similarly, liang et al. explored wave dispersion characteristics in lipid tubules, modeling them as shells, within the framework of nonlocal strain gradient theory [14]. nonlinear vibration responses of piezoelectric nanosensors and nanoresonators have been addressed by hashemi kachapi et al. [15, 16] considering nonlocal and surface/interface effects. adopting the gurtin-murdoch approach to surface elasticity, fang et al. [17, 18] investigated the nonlinear buckling, post-buckling, and vibration characteristics of piezoelectric nanostructures. hashemi kachapi et al. [19, 20] employed surface/interface and nonlocal strain gradient effects to investigate the effects of the small scale on the natural frequencies, nonlinear vibration, and stability analysis of piezoelectric nanostructures subjected to electrostatic and harmonic excitations. it is noteworthy that, beyond the work of hashemi kachapi et al. [20], studies that consider simultaneously both surface and small-scale effects in piezoelectric materials remain scarce. farrokhian et al. [21] investigated the vibration and damping behavior of smart sandwich nanotubes using nonlocal and surface piezoelasticity theories. additionally, ghorbani et al. [22] revealed that the material length scale (mls) parameter augments natural frequency, while the nonlocal (nl) parameter diminishes it. lovisi [23] explored the bending response of functionally graded cracked nanobeams, incorporating surface and nonlocal effects. sun et al. [24] employed nonlocal and surface effects to analyze the buckling of piezoelectric nanoshells under external voltages and compressive loads. it is worth noting that in other works by hashemi kachapi et al. [25, 26], nonlinear vibration and stability analyses considering nonclassical effects were investigated for double-walled cases of piezoelectric nanostructures. while prior research has extensively explored the vibrational and stability characteristics of piezoelectric nanostructures, a notable gap persists in studies that concurrently incorporate strain gradient, gurtin-murdoch surface/interface, and nonlocal effects. the present investigation extends the research trajectory initiated by hashemi kachapi et al [20]. however, it distinguishes itself through a distinct subject matter, particularly regarding the excitation forces and resulting findings, which are entirely independent of our previous comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 313 work. unlike the earlier study, this analysis examines a nanostructure subjected to a complex nonlinear electrostatic excitation, encompassing both direct (dc) and alternating (ac) voltage components, while also considering the influence of a visco-pasternak medium. the primary objective of this study is to elucidate the impact of these combined effects on the nonlinear vibration and stability behavior of a piezoelectric nanoresonator, in comparison to predictions derived from classical continuum theory. to achieve this, a comprehensive analytical framework is employed, integrating hamilton’s principle, the galerkin method, and a hybrid complex averaging technique coupled with arc-length continuation. this methodology facilitates a direct comparison of three nonclassical theoriesstrain gradient theory (sgt), nonlocal theory (nlt), and gurtin-murdoch surface/interface theory (gmsit)with classical continuum theory (ct). consequently, the study investigates the influence of nanoscale and surface/interface effects, the elastic medium, electrostatic and piezoelectric voltages, and other relevant parameters on the dimensionless natural frequency (dnf), pull-in voltage, nonlinear frequency response, and stability of the piezoelectric nanoresonator. 2. mathematical formulation fig. 1 depicts the configuration of a cylindrical nanoshell-based piezoelectric nanoresonator subjected to a visco-pasternak medium and nonlinear electrostatic excitation. fig. 1 a piezoelectric nanoresonator subjected to nonlinear electrostatic excitation the nanoresonator is actuated by a superimposed electrostatic field, consisting of an alternating voltage component, 𝑉𝐴𝐶 , and a direct voltage component, 𝑉𝐷𝐶. the geometric configuration of the piezoelectric nanoresonator is defined by the following parameters: mid-surface radius, 𝑅; axial length, 𝐿; piezoelectric material layer thickness, 2ℎ𝑝; and cylindrical shell thickness, 2ℎ𝑁. a cartesian coordinate system is established with its origin positioned at the mid-surface of the nanoshell, where 𝑥, 𝜃, and 𝑧 denote the coordinates in the axial, circumferential, and radial directions, respectively. the supporting medium is characterized by a visco-pasternak foundation, with 𝐾𝑤 representing the winkler foundation 314 s.h. hashemi kachapi, s. gh. hashemi kachapi stiffness coefficient, 𝐾𝑝 denoting the shear layer stiffness of the pasternak foundation, and 𝐶𝑤 signifying the damping coefficient of the viscous medium for transverse motion. additional physical and geometric properties of the nanostructure are detailed in hashemi kachapi et al. [19]. 2.1. governing equations this section delineates the derivation of the governing equations of motion and associated boundary conditions for the piezoelectric shell, utilizing hamilton's principle: ∫ (𝛿𝑇 − 𝛿𝜋 + 𝛿𝑤𝑣𝑓 + 𝛿𝑤𝑒)𝑑𝑡 = 0, 𝑡 0 (1) in this expression, 𝛿𝑇, 𝛿𝜋, 𝛿𝑤𝑣𝑓 and 𝛿𝑤𝑒 represent the first variations of strain energy, kinetic energy, work done by the viscoelastic foundation, and work done by the nonlinear electrostatic excitation, respectively. it is crucial to note that all constitutive relations, coefficients, and expressions related to nonlocal strain gradient theory, surface/interface theory, nonlocal strain gradient surface/interface theory, nanoscale stress-strain relationships, and the physical and geometric properties employed herein are comprehensively detailed in hashemi kachapi et al [20]. the first variation of strain energy, 𝛿𝜋, is expressed as: 𝛿𝜋 = ∫ ∫ { 𝑁𝑥𝑥 ( 𝜕𝛿𝑢 𝜕𝑥 + 𝜕𝑤 𝜕𝑥 𝜕𝛿𝑤 𝜕𝑥 ) −𝑀𝑥𝑥 ( 𝜕2𝛿𝑤 𝜕𝑥2 ) + 𝑁𝜃𝜃 ( 1 𝑅 ( 𝜕𝛿𝑣 𝜕𝜃 + 𝛿𝑤) + 1 𝑅2 𝜕𝑤 𝜕𝜃 𝜕𝛿𝑤 𝜕𝜃 ) −𝑀𝜃𝜃 ( 1 𝑅2 𝜕2𝛿𝑤 𝜕𝜃2 ) + 𝑁𝑥𝜃 ( 1 𝑅 𝜕𝛿𝑢 𝜕𝜃 + 𝜕𝛿𝑣 𝜕𝑥 + 1 𝑅 𝜕𝛿𝑤 𝜕𝑥 𝜕𝑤 𝜕𝜃 + 1 𝑅 𝜕𝑤 𝜕𝑥 𝜕𝛿𝑤 𝜕𝜃 ) −𝑀𝑥𝜃 ( 2 𝑅 𝜕2𝛿𝑤 𝜕𝑥𝜕𝜃 ) } 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥 (2) the first variation of kinetic energy can then be expressed as: in eq. (2), the force resultants, 𝑁, and moment resultants, 𝑀, are defined as specified in hashemi kachapi et al. [19]. the first variation of kinetic energy is given by: 𝛿 ∫ 𝑇𝑑𝑡 𝑡2 𝑡1 = −∫ ∬{𝐼 (( 𝜕2𝑢 𝜕𝑡2 ) 𝛿𝑢 + ( 𝜕2𝑣 𝜕𝑡2 ) 𝛿𝑣 + ( 𝜕2𝑤 𝜕𝑡2 ) 𝛿𝑤)}𝑅𝑑𝜃𝑑𝑥𝑑𝑡 𝑡2 𝑡1 (3) where 𝐼 = ∫ 𝜌𝑁 ℎ𝑁 −ℎ𝑁 𝑑𝑧 + ∫ 𝜌𝑝 −ℎ𝑁 −ℎ𝑁−ℎ𝑝 𝑑𝑧 + ∫ 𝜌𝑝 ℎ𝑁+ℎ𝑝 ℎ𝑁 𝑑𝑧 + 𝜌𝑆,𝐼 = 2( 𝜌𝑁ℎ𝑁 + 𝜌𝑝ℎ𝑝 +𝜌𝑆 + 𝜌𝐼 ) (4) the first variations of the work done by the viscoelastic foundation and the nonlinear electrostatic excitation, are expressed as [19, 20, 27]: 𝛿𝑊𝑣𝑓 = −∫ ∫ ∫ (𝐾𝑤𝑤 − 𝐾𝑝𝛻 2𝑤 + 𝐶𝑤 𝜕𝑤 𝜕𝑡 ) 𝛿𝑤 𝑤 0 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥, (5) 𝛿𝑊𝑒 = ∫ ∫ ∫ 𝜋𝛶(𝑉𝐷𝐶+𝑉𝐴𝐶 cos(𝜔𝑡)) 2 (√(𝑏−𝑤)(2𝑅+𝑏−𝑤) [cosh−1( 1+ 𝑏−𝑤 𝑅 )] 2 ) 𝛿𝑤 𝑤 0 2𝜋 0 𝐿 0 𝑅𝑑𝜃𝑑𝑥 (6) comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 315 the explicit definitions of all coefficients and expressions presented in eqs. (5) and (6) are provided in hashemi kachapi et al. [19, 20]. substituting eqs. (2-6) into eq. (1) results in the derivation of the governing equations of motion and associated boundary conditions for the piezoelectric nanoresonator (penr), as comprehensively documented in hashemi kachapi et al. [19, 20]. by employing the dimensionless parameters defined in hashemi kachapi et al. [19], and subsequently incorporating nonlocal and material length scale effects, along with nonlocal strain gradient surface/interface effects (as detailed in hashemi kachapi et al [20]), the dimensionless governing equations are obtained. (1 − �̅� �̅�2) ( 𝛼1𝑢 𝜕2𝑢 𝜕𝜉2 + 𝛼2𝑢 𝜕2𝑢 𝜕𝜃2 + 𝛼3𝑢 𝜕2�̅� 𝜕𝜉𝜕𝜃 + 𝛼4𝑢 𝜕�̅� 𝜕𝜉 +𝛼5𝑢 𝜕�̅� 𝜕𝜉 𝜕2�̅� 𝜕𝜉2 + 𝛼6𝑢 𝜕�̅� 𝜕𝜉 𝜕2�̅� 𝜕𝜃2 + 𝛼7𝑢 𝜕2�̅� 𝜕𝜉𝜕𝜃 𝜕�̅� 𝜕𝜃 ) = (1 − �̅� �̅�2) 𝜕2𝑢 𝜕𝜏2 , (7) (1 − �̅� �̅�2) ( 𝛼1𝑣 𝜕2𝑢 𝜕𝜉𝜕𝜃 + 𝛼2𝑣 𝜕2�̅� 𝜕𝜉2 + 𝛼3𝑣 𝜕2�̅� 𝜕𝜃2 + 𝛼4𝑣 𝜕�̅� 𝜕𝜉 𝜕2�̅� 𝜕𝜉𝜕𝜃 +𝛼5𝑣 𝜕2�̅� 𝜕𝜉2 𝜕�̅� 𝜕𝜃 + 𝛼6𝑣 𝜕�̅� 𝜕𝜃 + 𝛼7𝑣 𝜕�̅� 𝜕𝜃 𝜕2�̅� 𝜕𝜃2 ) = (1 − �̅� �̅�2) 𝜕2�̅� 𝜕𝜏2 , (8) (1 − �̅� �̅�2) ( 𝛼1𝑤 𝜕𝑢 𝜕𝜉 + 𝛼2𝑤 𝜕𝑢 𝜕𝜉 𝜕2�̅� 𝜕𝜉2 + 𝛼3𝑤 𝜕𝑢 𝜕𝜉 𝜕2�̅� 𝜕𝜃2 + 𝛼4𝑤 𝜕2𝑢 𝜕𝜉2 𝜕�̅� 𝜕𝜉 + 𝛼5𝑤 𝜕2𝑢 𝜕𝜉𝜕𝜃 𝜕�̅� 𝜕𝜃 +𝛼6𝑤 𝜕2𝑢 𝜕𝜉𝜕𝜃 𝜕�̅� 𝜕𝜃 + 𝛼7𝑤 𝜕𝑢 𝜕𝜃 𝜕2�̅� 𝜕𝜉𝜕𝜃 + 𝛼8𝑤 𝜕2𝑢 𝜕𝜃2 𝜕�̅� 𝜕𝜉 + 𝛼9𝑤 𝜕�̅� 𝜕𝜉 𝜕2�̅� 𝜕𝜉𝜕𝜃 +𝛼10𝑤 𝜕2�̅� 𝜕𝜉2 𝜕�̅� 𝜕𝜃 + 𝛼11𝑤 𝜕2�̅� 𝜕𝜉𝜕𝜃 𝜕�̅� 𝜕𝜉 + 𝛼12𝑤 𝜕�̅� 𝜕𝜃 + 𝛼13𝑤 𝜕�̅� 𝜕𝜃 𝜕2�̅� 𝜕𝜉2 +𝛼14𝑤 𝜕�̅� 𝜕𝜃 𝜕2�̅� 𝜕𝜃2 + 𝛼15𝑤 𝜕2�̅� 𝜕𝜃2 𝜕�̅� 𝜕𝜃 + 𝛼16𝑤�̅� + 𝛼17𝑤�̅� 𝜕2�̅� 𝜕𝜉2 + 𝛼18𝑤�̅� 𝜕2�̅� 𝜕𝜃2 +𝛼19𝑤 𝜕2�̅� 𝜕𝜉2 + 𝛼20𝑤 𝜕�̅� 𝜕𝜉 𝜕2�̅� 𝜕𝜉𝜕𝜃 𝜕�̅� 𝜕𝜃 + 𝛼21𝑤 𝜕2�̅� 𝜕𝜉2 ( 𝜕�̅� 𝜕𝜉 ) 2 + 𝛼22𝑤 𝜕2�̅� 𝜕𝜉2 ( 𝜕�̅� 𝜕𝜃 ) 2 +𝛼23𝑤 𝜕4�̅� 𝜕𝜉4 + 𝛼24𝑤 ( 𝜕�̅� 𝜕𝜉 ) 2 + 𝛼25𝑤 𝜕4�̅� 𝜕𝜉2𝜕𝜃2 + 𝛼26𝑤 ( 𝜕�̅� 𝜕𝜉 ) 2 𝜕2�̅� 𝜕𝜃2 +𝛼27𝑤 𝜕2�̅� 𝜕𝜃2 + 𝛼28𝑤 𝜕4�̅� 𝜕𝜃4 + 𝛼29𝑤 𝜕2�̅� 𝜕𝜃2 ( 𝜕�̅� 𝜕𝜃 ) 2 + 𝛼30𝑤 ( 𝜕�̅� 𝜕𝜃 ) 2 +𝛼31𝑤 𝜕4�̅� 𝜕𝜉2𝜕𝜏2 + 𝛼32𝑤 𝜕4�̅� 𝜕𝜃2𝜕𝜏2 + 𝛼33𝑤 ) (9) = (1 − �̅� �̅�2) ( 𝜕2�̅� 𝜕𝜏2 + 𝐶�̅� 𝜕�̅� 𝜕𝜏 + 𝐾𝑤�̅� − 𝐾𝑝�̅� 2�̅� − �̅�𝑒(�̅�𝐷𝐶 + �̅�𝐴𝐶 𝑐𝑜𝑠(�̅�𝜏)) 2 √(𝑚1�̅� − �̅�)(2𝑚4�̅� + 𝑚1�̅� − �̅�) × [𝑐𝑜𝑠ℎ−1 (1 + 𝑚1�̅� − �̅� 𝑚4�̅� )] 2 ) , where �̅�2 = 𝜕2 𝜕𝜉2 +𝑚0 2 𝜕2 𝜕𝜃2 and the coefficients of 𝛼𝑖𝑢(𝑖 = 1. .7), 𝛼𝑗𝑣(𝑗 = 1. .7) and 𝛼𝑘𝑤(𝑘 = 1. .33) are defined in hashemi kachapi et al [20]. the corresponding boundary conditions, expressed in dimensionless form, are given by: 316 s.h. hashemi kachapi, s. gh. hashemi kachapi �̅� = 0: ( 𝛼1𝑢 𝑏𝑐 𝜕𝑢 𝜕𝜉 + 𝛼2𝑢 𝑏𝑐 𝜕�̅� 𝜕𝜃 + 𝛼3𝑢 𝑏𝑐�̅� +𝛼4𝑢 𝑏𝑐 ( 𝜕�̅� 𝜕𝜉 ) 2 + 𝛼5𝑢 𝑏𝑐 ( 𝜕�̅� 𝜕𝜃 ) 2 +𝛼6𝑢 𝑏𝑐 ) 𝛿�̅�𝜉| | 0 1 + ( 𝛼7𝑢 𝑏𝑐 𝜕𝑢 𝜕𝜃 + 𝛼8𝑢 𝑏𝑐 𝜕�̅� 𝜕𝜉 +𝛼9𝑢 𝑏𝑐 𝜕�̅� 𝜕𝜉 𝜕�̅� 𝜕𝜃 )𝛿�̅�𝜃| 0 2𝜋 = 0, (10) 𝛿�̅� = 0: ( 𝛼1𝑣 𝑏𝑐 𝜕𝑢 𝜕𝜃 + 𝛼2𝑣 𝑏𝑐 𝜕�̅� 𝜕𝜉 +𝛼3𝑣 𝑏𝑐 𝜕�̅� 𝜕𝜉 𝜕�̅� 𝜕𝜃 )𝛿�̅�𝜉| 0 1 + ( 𝛼4𝑣 𝑏𝑐 𝜕𝑢 𝜕𝜉 + 𝛼5𝑣 𝑏𝑐 𝜕�̅� 𝜕𝜃 + 𝛼6𝑣 𝑏𝑐�̅� +𝛼7𝑣 𝑏𝑐 ( 𝜕�̅� 𝜕𝜉 ) 2 + 𝛼8𝑣 𝑏𝑐 ( 𝜕�̅� 𝜕𝜃 ) 2 +𝛼9𝑣 𝑏𝑐 ) 𝛿�̅�𝜃| | 0 2𝜋 = 0, (11) 𝛿�̅� = 0: ( +𝛼1𝑤 𝑏𝑐 𝜕𝑢 𝜕𝜉 𝜕�̅� 𝜕𝜉 + 𝛼2𝑤 𝑏𝑐 𝜕𝑢 𝜕𝜃 𝜕�̅� 𝜕𝜃 + 𝛼3𝑤 𝑏𝑐 𝜕�̅� 𝜕𝜉 𝜕�̅� 𝜕𝜃 + 𝛼4𝑤 𝑏𝑐 𝜕�̅� 𝜕𝜃 𝜕�̅� 𝜕𝜉 +𝛼5𝑤 𝑏𝑐 �̅� 𝜕�̅� 𝜕𝜉 + 𝛼6𝑤 𝑏𝑐 𝜕�̅� 𝜕𝜉 + 𝛼7𝑤 𝑏𝑐 𝜕3�̅� 𝜕𝜉3 + 𝛼8𝑤 𝑏𝑐 𝜕3�̅� 𝜕𝜉𝜕𝜃2 +𝛼9𝑤 𝑏𝑐 𝜕�̅� 𝜕𝜉 ( 𝜕�̅� 𝜕𝜉 ) 2 + 𝛼10𝑤 𝑏𝑐 𝜕�̅� 𝜕𝜉 ( 𝜕�̅� 𝜕𝜃 ) 2 + 𝛼11𝑤 𝑏𝑐 𝜕3�̅� 𝜕𝜉𝜕𝜏2 ) 𝛿�̅�𝜉 | | 0 1 = 0, (12) 𝜕�̅� 𝜕𝜉 = 0: ( 𝛼23𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜉2 + 𝛼24𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜃2 +𝛼25𝑤 𝑏𝑐 + 𝛼26𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜉2 +𝛼27𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜃2 + 𝛼28𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜏2) 𝛿 ( 𝜕�̅�𝜉 𝜕𝜉 ) | | 0 1 + (𝛼29𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜉𝜕𝜃 ) 𝛿 ( 𝜕�̅�𝜃 𝜕𝜉 )| 0 2𝜋 = 0, (13) 𝜕�̅� 𝜕𝜃 = 0: (𝛼30𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜉𝜕𝜃 ) 𝛿 ( 𝜕�̅�𝜉 𝜕𝜃 )| 0 1 + ( 𝛼31𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜉2 + 𝛼32𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜃2 +𝛼33𝑤 𝑏𝑐 + 𝛼34𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜉2 +𝛼35𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜃2 + 𝛼36𝑤 𝑏𝑐 𝜕2�̅� 𝜕𝜏2) 𝛿 ( 𝜕�̅�𝜃 𝜕𝜃 ) | | 0 2𝜋 = 0. (14) the coefficients of 𝛼𝑙𝑢 𝑏𝑐(𝑙 = 1. .9), 𝛼𝑚𝑣 𝑏𝑐 (𝑚 = 1. .9) and 𝛼𝑛𝑤 𝑏𝑐 (𝑛 = 1. .36) are defined in hashemi kachapi et al [20]. utilizing the lsqcurvefit function within the matlab toolbox, a nonlinear leastsquares curve-fitting technique, the electrostatic force in eq. (6) is approximated as a polynomial. consequently, the dimensionless work performed by the electrostatic force can be represented as: 𝛿𝑊𝑒 = ∫ ∫ {∫ ( �̅�𝑒(�̅�𝐷𝐶 + �̅�𝐴𝐶 cos(ω𝜏)) 2 × (𝐶1̅ + 𝐶2̅�̅� + 𝐶3̅�̅� 2 +⋯+ 𝐶�̅��̅� 𝑛−1) ) �̅� 0 𝛿�̅�} 2𝜋 0 𝐿 0 𝛿𝜃𝛿𝜉 (15) which 𝐶1̅ − 𝐶�̅� are constant. 2.2. solution procedure to transform the governing partial differential equations into a set of ordinary differential equations, the galerkin method is employed. the displacement field is approximated using a series expansion of generalized coordinates and mode functions, as follows [28]: comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 317 [ 𝑢(𝑥, 𝜃, 𝑡) 𝑣(𝑥, 𝜃, 𝑡) 𝑤(𝑥, 𝜃, 𝑡) ] = ∑ ∑ [ [ 𝑢𝑚,𝑗,𝑐(𝜏) cos(𝑗𝜃) +𝑢𝑚,𝑗,𝑠(𝜏) sin(𝑗𝜃) ] 𝜒𝑚𝑗(𝜉) [ 𝑣𝑚,𝑗,𝑐(𝜏) sin(𝑗𝜃) +𝑣𝑚,𝑗,𝑠(𝜏) cos(𝑗𝜃) ] 𝜙𝑚𝑗(𝜉) [ 𝑤𝑚,𝑗,𝑐(𝜏) cos(𝑗𝜃) +𝑤𝑚,𝑗,𝑠(𝜏) sin(𝑗𝜃) ] 𝛽𝑚𝑗(𝜉) ] 𝑁 𝑗=1 𝑀1 𝑚=1 + ∑ [ 𝑢𝑚,0(𝜏)𝜒𝑚0(𝜉) 𝑣𝑚,0(𝜏)𝜙𝑚0(𝜉) 𝑤𝑚,0(𝜏)𝛽𝑚0(𝜉) ] 𝑀2 𝑚=1 = ∑ [ 𝑢𝑖(𝜏)𝜒𝑖(𝜉)𝜗𝑖(𝜃) 𝑣𝑟(𝜏)𝜙𝑟(𝜉)𝛼𝑟(𝜃) 𝑤𝑠(𝜏)𝛽𝑠(𝜉)𝜓𝑠(𝜃) ] 𝑀2+𝑀1×𝑁 (𝑖,𝑟,𝑠)=1 (16) in the galerkin method, the trial functions, 𝜒𝑖(𝜉), 𝜙𝑟(𝜉) and 𝛽𝑠(𝜉), must satisfy all geometric and natural boundary conditions. substituting eq. (16) into eqs. (7-14) and applying the galerkin procedure yields the reduced-order equation of motion: [(𝐾)𝑢 𝑢 + (𝐾𝑏𝑐)𝑢 𝑢]{�̅�} + [(𝐾)𝑢 𝑣 + (𝐾𝑏𝑐)𝑢 𝑣 ]{�̅�} + [(𝐾)𝑢 𝑤 + (𝐾𝑏𝑐)𝑢 𝑤]{�̅�} + [(𝑁𝐿)𝑢 𝑤 + (𝑁𝐿𝑏𝑐)𝑢 𝑤]{�̅�2} = [(𝑀)𝑢 𝑢]{�̈̅�} + �̅�𝑢𝑝 𝑏𝑐, (17) [(𝐾)𝑣 𝑢 + (𝐾𝑏𝑐)𝑣 𝑢]{�̅�} + [(𝐾)𝑣 𝑣 + (𝐾𝑏𝑐)𝑣 𝑣]{𝑣} + [(𝐾)𝑣 𝑤 + (𝐾𝑏𝑐)𝑣 𝑤]{�̅�} + [(𝑁𝐿)𝑣 𝑤 + (𝑁𝐿𝑏𝑐)𝑣 𝑤]{�̅�2} = [(𝑀)𝑣 𝑣]{�̈̅�} + �̅�𝑣𝑝 𝑏𝑐 , (18) [(𝐾)𝑤 𝑢 ]{�̅�} + [(𝐾)𝑤 𝑣 ]{�̅�} + [(𝐾)𝑤 𝑤 + (𝐾𝑏𝑐)𝑤 𝑤 − (𝐾𝑣𝑝)𝑤 𝑤 − (𝐾𝑒2)𝑤 𝑤] {�̅�} + [(𝑁𝐿)𝑤 𝑢 + (𝑁𝐿𝑏𝑐)𝑤 𝑢 ]{�̅��̅�} + [(𝑁𝐿)𝑤 𝑣 + (𝑁𝐿𝑏𝑐)𝑤 𝑣 ]{�̅��̅�} + [(𝑁𝐿)𝑤2 𝑤 + (𝑁𝐿𝑏𝑐)𝑤2 𝑤 − (𝑁𝐿2𝑒)𝑤2 𝑤 ]{�̅�2} + [(𝑁𝐿)𝑤3 𝑤 + (𝑁𝐿𝑏𝑐)𝑤3 𝑤 − (𝑁𝐿3𝑒)𝑤3 𝑤 ]{�̅�3} = ([(𝑀)𝑤 𝑤 + (𝑀𝑏𝑐)𝑤 𝑤]){�̈̅�} + ([(𝐶)𝑤 𝑤] + [(𝐶𝑏𝑐)𝑤 𝑤]){�̇̅�}) + �̅�𝑤𝑝 + �̅�𝑤𝑝 𝑏𝑐 − �̅�𝑤𝑒 − �̅�𝑒 { ((�̅�𝐴𝐶cos�̅�𝜏) 2 + 2�̅�𝐴𝐶�̅�𝐷𝐶cos�̅�𝜏) × (𝐶4̅(𝑁𝐿𝑒)𝑤3 𝑤 + 𝐶3̅(𝑁𝐿𝑒)𝑤2 𝑤 + 𝐶2̅(𝐾𝑒)𝑤 𝑤 + 𝐶1̅�̅�1) } (19) where all coefficients and phrases of eqs. (17-19) are defined by hashemi kachapi [20] and only coefficients of visco-pasternak medium , (𝐾𝑣𝑝)𝑤 𝑤 , are presented in following eq. (20) of current work: (𝐾𝑣𝑝)𝑤 𝑤 = ∬ ( �̅�𝑤 (𝛽𝑟𝛽𝑜𝜓𝑠𝜓𝑝 − �̅�(𝛽𝑟𝛽𝑜 ′′𝜓𝑠𝜓𝑝 +𝑚0 2𝛽𝑟𝛽𝑜𝜓𝑠𝜓𝑝 ′′)) −�̅�𝑝(𝛽𝑟𝛽0 ′′𝜓𝑠𝜓𝑝 +𝑚0 2𝛽𝑟𝛽𝑜𝜓𝑠𝜓𝑝 ′′) +�̅�𝑝�̅� ( 𝛽𝑟𝛽0 ′′′′𝜓𝑠𝜓𝑝 + 2𝑚0 2𝛽𝑟𝛽0 ′′𝜓𝑠𝜓𝑝 ′′ +𝑚0 4𝛽𝑟𝛽𝑜𝜓𝑠𝜓𝑝 ′′′′ ) ) 𝑑𝜉 𝑑𝜃, (20) to solve the nonlinear equations of the system (eqs. 17-19), the complex averaging method combined with arc-length continuation is employed [19, 20, 29]. 318 s.h. hashemi kachapi, s. gh. hashemi kachapi 3. results and discussions validation studies for piezoelectric nanostructures are documented in hashemi kachapi et al. [19, 20]. this section examines the influence of diverse material and geometric parameters, both with and without the inclusion of nonlocal, strain gradient, and surface/interface effects, on the dimensionless natural frequency, stability characteristics, and frequency response. analyses are performed for clamped-clamped (cc), simply supported-simply supported (ss), clamped-simply supported (cs), and clamped-free (cf) boundary conditions. the surface and bulk material properties of the aluminum (al) nanoshell and pzt piezoelectric layer are tabulated in tables 1 and 2, respectively [19, 20]. table 1 surface and bulk properties of al 𝑬𝑵[𝑮𝑷𝒂] 𝝊𝑵 𝝆𝑵[𝒌𝒈 𝒎𝟑⁄ ] 𝝀𝑰[𝑵 𝒎⁄ ] 𝝁𝑰[𝑵 𝒎⁄ ] 𝝉𝟎 𝑰 [𝑵 𝒎⁄ ] 𝝆𝑰[𝒌𝒈 𝒎𝟐⁄ ] 𝟕𝟎 0.33 2700 3.786 1.95 0.9108 5.46 × 10−7 table 2 surface and bulk properties of pzt-4 𝑪𝟏𝟏𝒑[𝑮𝑷𝒂] 𝑪𝟐𝟐𝒑[𝑮𝑷𝒂] 𝑪𝟏𝟐𝒑[𝑮𝑷𝒂] 𝑪𝟐𝟏𝒑[𝑮𝑷𝒂] 𝑪𝟔𝟔𝒑[𝑮𝑷𝒂] 𝑬𝒑[𝑮𝑷𝒂] 𝟏𝟑𝟗 139 77.8 77.8 30.5 95 𝝊𝒑 𝜌𝑝[𝑘𝑔 𝑚 −3] 𝜂33𝑝[10 −8 𝐹 𝑚⁄ ] 𝜆𝑆[𝑁 𝑚⁄ ] 𝜇𝑆[𝑁 𝑚⁄ ] 𝜏0 𝑆[𝑁 𝑚⁄ ] 𝟎. 𝟑 7500 8.91 4.488 2.774 0.6048 𝒆𝟑𝟏𝒑[𝑪 𝒎𝟐⁄ ] 𝑒32𝑝[𝐶 𝑚2⁄ ] 𝑒31𝑝 𝑆 [𝐶 𝑚⁄ ] 𝑒32𝑝 𝑆 [𝐶 𝑚⁄ ] 𝜌𝑆[𝑘𝑔 𝑚2⁄ ] −𝟓.𝟐 −5.2 −3 × 10−8 −3 × 10−8 5.61 × 10−6 the remaining physical and geometrical parameters of the penr used in the following results are listed in table 3 [19, 20, 22]. table 3 the material and geometrical parameters 𝑹[𝒎] 𝑳 𝑹⁄ 𝒉𝑵 𝑹⁄ 𝒉𝒑 𝑹⁄ 𝒃 𝑹⁄ 𝑪𝒘[𝑵. 𝑺/𝒎] 𝟏 × 𝟏𝟎−𝟗 10 0.01 0.005 0.1 1 × 10−3 𝑲𝒘[𝑵 𝒎𝟑⁄ ] 𝐾𝑝[𝑁 𝑚⁄ ] 𝑉𝑝[𝑉] 𝑉0 𝑉𝐷𝐶[𝑉] 𝑉𝐴𝐶[𝑉] 𝟗 × 𝟏𝟎𝟏𝟕 2.07 1 × 10−5 1 1.5 0.5 𝝁[𝒎𝟐] 𝜂[𝑚2] (𝟏 × 𝟏𝟎−𝟏𝟎) 𝟐 (1 × 10−11)2 3.1. nonlinear frequency response and stability analysis this section compares the predictions of three nonclassical continuum theories-nonlocal theory (nlt), strain gradient theory (sgt), and gurtin-murdoch surface/interface theory (gmsit)with those of classical continuum theory (ct). the analysis investigates the impact of various geometric and material parameters, incorporating and excluding strain gradient, nonlocal, and surface/interface effects, on the dimensionless natural frequency (dnf), stability, and nonlinear frequency response of the piezoelectric nanoresonator (penr). arc-length continuation is employed as the numerical solution method, utilizing the penr specifications provided in tables 1-3. specifically, the nlt analysis considers the nonlocal parameter, �̅�; the sgt analysis considers the material length scale parameters, the comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 319 �̅� and �̅� parameters; and the gmsit analysis incorporates surface/interface (s/i) effects. recognizing the substantial influence of surface/interface densities on nanostructure vibration, two distinct surface density cases, as defined in table 4, are presented. table 4 two case of surface density case 1 case 2 𝝆𝑰[𝒌𝒈 𝒎𝟐⁄ ] 𝜌𝑆[𝑘𝑔 𝑚2⁄ ] 𝜌𝐼[𝑘𝑔 𝑚2⁄ ] 𝜌𝑆[𝑘𝑔 𝑚2⁄ ] 𝟓. 𝟒𝟔 × 𝟏𝟎−𝟕 5.61 × 10−6 5.46 × 10−8 5.61 × 10−7 figs. 2 and 3 illustrate the dimensionless natural frequencies of the piezoelectric nanoresonator (penr) as a function of the dimensionless nonlocal scale parameter �̅�and the dimensionless material length scale parameter �̅�, respectively, both with and without surface/interface effects. as depicted in fig. 2, surface/interface densities corresponding to case 1 result in a reduction of penr stiffness, leading to a lower dimensionless natural frequency compared to the absence of surface/interface effects. conversely, surface/interface densities associated with case 2 induce an increase in penr stiffness, consequently yielding a higher dimensionless natural frequency. fig. 2 natural frequency vs. dimensionless nonlocal scale parameter �̅� with �̅� = 0.01: influence of surface/interface effects for different boundary conditions in all cases, the dimensionless natural frequency (dnf) decreases with increasing nonlocal scale parameter �̅� with �̅� = 0.01, attributable to the reduced stiffness of the porous elastic nanorod (penr). as shown in fig. 3, the dnf increases with the dimensionless material length scale parameter �̅� with �̅� = 0.1 across all scenarios, owing to the enhanced penr stiffness. these results align with the previously discussed effects of surface/interface 320 s.h. hashemi kachapi, s. gh. hashemi kachapi densities. the trends in figs. 2 and 3 demonstrate that the nonlocal scale parameter �̅� and material length scale parameter �̅� inversely influence penr stiffness: �̅� reduces stiffness, thereby lowering the dnf, while �̅� enhances stiffness, increasing the dnf. additionally, the dimensionless natural frequency associated with the cc boundary condition consistently exceeds those of cs, ss, and cf configurations. this disparity arises from the greater structural rigidity inherent to the cc boundary condition compared to others. fig. 3 natural frequency vs. dimensionless material length scale parameter �̅� with �̅� = 0.1: influence of surface/interface effects for different boundary conditions fig. 4 illustrates the influence of the length to radius ratio (𝑳 𝑹⁄ ) on the dimensionless natural frequency (dnf) as a function of the dimensionless nonlocal scale parameter (�̅�) for ss penr. the results incorporate all surface/interface effects, with �̅� = 𝟎. 𝟎𝟏. as evident from fig. 4, the dimensionless natural frequency (dnf) decreases with increasing values of both the length to radius ratio (𝑳/𝑹) and the nonlocal scale parameter (�̅�). this trend is attributed to the reduction in stiffness of the penr. fig. 5 depicts the variation of the dnf as a function of the dimensionless material length scale parameter (�̅�) for ss penr under different 𝑳/𝑹 ratios. the analysis incorporates all surface/interface effects, with �̅� = 𝟎. 𝟏. as shown in the figure, the dimensionless natural frequency increases with the nonlocal parameter but decreases with the 𝑳/𝑹 ratio. comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 321 fig. 4 natural frequency vs. dimensionless nonlocal scale parameter �̅�: influence of 𝐿 𝑅⁄ ratio with all surface/interface effects and �̅� = 0.01 fig. 5 natural frequency vs. dimensionless material length scale parameter �̅�: influence of 𝐿 𝑅⁄ ratio with all surface/ interface effects and μ̅ = 0.1 fig. 6 compares the dimensionless natural frequency predicted by three nonclassical theories, nlt, sgt, and gmsit, against the classical theory ct for ss penr under varying piezoelectric thickness to radius ratios (𝒉𝒑 𝑹⁄ ). as evident from the figure, the 322 s.h. hashemi kachapi, s. gh. hashemi kachapi gmsit (case 2) yields the highest dnf, indicating greater structural rigidity in this configuration compared to other cases. the lowest natural frequency is observed when gmsit (case 1) is considered in combination with sgt, i.e. gmsit+sgt (�̅� = 𝟎. 𝟏, �̅� = 𝟎. 𝟎𝟏), indicating that this combination results in the lowest system stiffness. additionally, the natural frequency predicted by classical theory is higher than those predicted by nlt and sgt, demonstrating a reduction in system rigidity when these nonclassical theories are considered. nonlocal theory predicts a higher natural frequency than strain gradient theory due to its greater rigidity. surface/interface effects, depending on the surface/interface densities, can result in natural frequencies higher or lower than those predicted by classical theory. however, the combined application of nlt and sgt consistently yields a lower natural frequency than classical theory. furthermore, the 𝒉𝒑 𝑹⁄ ratio significantly influences the dimensionless natural frequency (dnf). an increase in this ratio leads to a decrease in penr stiffness and, consequently, a decrease in the natural frequency. fig. 6 dimensionless natural frequency vs. piezoelectric thickness to radius ratio ℎ𝑝 𝑅⁄ : a comparison of non-classical and classical theories fig. 7 compares the dimensionless natural frequency predicted by three nonclassical theories, nlt, sgt, and gmsit, with classical theory (ct) at various piezoelectric actuation voltages (�̅�𝒑) for ss penr. it reiterates the effects of the various theories on natural frequencies, confirming the previous findings. the impact of piezoelectric voltage on the natural frequency varies across the different theories. gmsit cases 1 and 2, as well as the combined gmsit+sgt, exhibit the most significant changes, particularly at low surface density. in these instances, increasing voltage leads to increased system rigidity and, consequently, an initial rise in frequency, followed by a plateau with a slight slope. in contrast, other theories show minimal sensitivity of the penr's natural frequency to piezoelectric voltage variations. comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 323 fig. 7 dimensionless natural frequency vs. piezoelectric voltage v̅p: comparison of nonclassical and classical theories fig. 8 presents a comparison of the dimensionless natural frequency predicted by nlt, sgt, gmsit, and ct as a function of direct pull-in voltage (dc) for an ss nanoshell. the inclusion of surface/interface effects is known to enhance system rigidity, thereby necessitating a greater dc voltage to achieve pull-in. in contrast, the reduced stiffness associated with nlt, sgt, and ct leads to a lower pull-in voltage, with these theories predicting pull-in at roughly the same voltage level. fig. 8 dimensionless natural frequency vs. direct pull-in voltage (dc) of ss penr: a comparison of non-classical and classical theories 324 s.h. hashemi kachapi, s. gh. hashemi kachapi the nonlinear frequency response of ss and cc penrs, as obtained from nlt, sgt, gmsit, and ct, is compared in figs. 9 and 10, respectively, for �̅�𝐷𝐶 = 1.7 and �̅�𝐴𝐶 = 0.5. as depicted in fig. 9, the frequency analysis indicates that the incorporation of surface/interface effects in gmsit case 2 (case 1) leads to a hardening (softening) of the ss penr. this, in turn, results in an increase (decrease) in the resonant frequency and a corresponding decrease in the resonance amplitude. fig. 9 nonlinear vibration and stability analysis of ss penr: comparison of non-classical and classical theories the gmsit model does not predict instability under the applied voltage conditions. conversely, ct, nlt, and sgt exhibit instability, manifested as saddle-node bifurcations and nonlinear hardening behavior. the nonclassical theories, nlt and sgt, predict a larger oscillation amplitude and a wider instability range than the classical theory (ct). moreover, an increase in 𝜏̅ and �̅� results in an amplified oscillation amplitude and an expanded instability range, accompanied by a reduction in resonance frequency. as depicted in fig. 10, an increase in the dimensionless nonlocal parameter results in a decrease in the resonance amplitude. this suggests that small-scale effects, as captured by the nonlocal model, enhance the flexibility of the cc penr. generally, the cc boundary condition yields result comparable to those of the ss boundary condition. however, a notable exception is observed in the sgt model, where an increase in 𝜏̅ and �̅� induces comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 325 nonlinear softening instability in the cc boundary condition, while the ss boundary condition exhibits nonlinear hardening instability. fig. 10 nonlinear vibration and stability analysis of cc penr: comparison of nonclassical and classical theories the frequency response and stability characteristics of the penr, subjected to different direct (�̅�𝐷𝐶) and alternating (�̅�𝐴𝐶) voltages, are illustrated in figs. 11-16. these figures compare the predictions of ct, nlt, sgt, gmsit, and the combined theories gmsit+nlt and gmsit+sgt for ss boundary conditions. 326 s.h. hashemi kachapi, s. gh. hashemi kachapi fig. 11 frequency response and stability analysis of ss penr (ct) under varying dc and ac voltages fig. 12 frequency response and stability analysis of ss penr (nlt) under varying dc and ac voltages comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 327 fig. 13 frequency response and stability analysis of ss penr (sgt) under varying dc and ac voltages fig. 14 frequency response and stability analysis of ss penr (gmsit) under varying dc and ac voltages 328 s.h. hashemi kachapi, s. gh. hashemi kachapi fig. 15 frequency response and stability analysis of ss penr (gmsit+nlt) under varying dc and ac voltages fig. 16 frequency response and stability analysis of ss penr (gmsit+sgt) under varying dc and ac voltages comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 329 it can be concluded from figs. 11-16 that the lowest resonance frequency for the unstable amplitude is related to fig. 12 for the nlt theory (of course, in the absence of gmsit case1, which has the lowest resonance frequency for the unstable amplitude) and the maximum value is related to fig. 14 for the gmsit (case2). as already mentioned, the three theories of ct, nlt, and sgt (respectively for figs. 11-13), due to the softening of the system in these cases reach the unstable amplitude at lower voltages from dc and ac and in order to reach the unstable amplitude in the gmsit theory, more voltages of dc and ac are needed. for example, according to the data in this work, ct, nlt and sgt theories in �̅�𝑨𝑪 = 𝟎. 𝟓 and �̅�𝑫𝑪 = 𝟏. 𝟒𝟓 reach unstable amplitude, while gmsit theory and the simultaneous use of this theory with two non-classical theories, gmsit + nlt (fig. 15) and gmsit + sgt (fig. 16), are required to reach the unstable amplitude to �̅�𝑨𝑪 = 𝟐 and �̅�𝑫𝑪 = 𝟒 voltages . furthermore, the results show that in all cases, with increasing of ac and dc voltages, the frequency amplitude is increased and the nanostructure shows nonlinear hardening behavior with saddle-node bifurcations. also, figs. 17-22 present frequency response and stability analysis penr for different values of direct (�̅�𝑫𝑪) and alternating (�̅�𝑨𝑪) voltages based on different theories of ct, nlt, sgt, gmsit, gmsit combined with nlt and sgt for cc boundary conditions. the results of figs. 17-22 for cc penr approximately are similar to figs. 11-16 for ss penr. it is clear from figs. 11-22 that boundary condition ss than condition cc, in some approaches and under equal loading conditions, earlier reaches the instability region. fig. 17 frequency response and stability analysis of cc penr (ct) under varying dc and ac voltages 330 s.h. hashemi kachapi, s. gh. hashemi kachapi fig. 18 frequency response and stability analysis of cc penr (nlt) under varying dc and ac voltages fig. 19 frequency response and stability analysis of cc penr (sgt) under varying dc and ac voltages comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 331 fig. 20 frequency response and stability analysis of cc penr (gmsit) under varying dc and ac voltages fig. 21 frequency response and stability analysis of cc penr (gmsit+nlt) under varying dc and ac voltages 332 s.h. hashemi kachapi, s. gh. hashemi kachapi fig. 22 frequency response and stability analysis of cc penr (gmsit+ sgt) under varying dc and ac voltages 4. conclusion this study investigates the influence of surface/interface parameters, specifically, lamé's constants (𝜆𝐼,𝑆, 𝜇𝐼,𝑆), residual stress (𝜏0 𝐼,𝑆), piezoelectric constants (𝑒31𝑝 𝑠𝑘 , 𝑒32𝑝 𝑠𝑘 ), and mass density (𝜌𝐼,𝑆) on the pull-in instability, dimensionless natural frequency, and nonlinear dynamic response of a piezoelectric nanosensor. the nanosensor is subjected to two piezoelectric layers, a nonlinear electrostatic force, harmonic excitations, and structural damping. to achieve this, hamilton's principle, assumed mode and lagrange-euler theories, along with arclength continuation and complex averaging methods, are employed. the following conclusions are drawn from this investigation: ▪ under conditions of lower surface/interface densities, an augmentation of stiffness is observed, resulting in an elevated natural frequency compared to scenarios lacking surface/interface effects. conversely, higher surface/interface densities lead to a reduction in stiffness and a corresponding decrease in natural frequency. ▪ cf (cc) boundary condition with s/i case 1 has the lowest (highest) natural frequency: this indicates that when one end of the nanoresonator is clamped and the other is free, and the surface/interface properties correspond to case 1, the nanoresonator will vibrate at its slowest natural frequency. ▪ an increase in surface/interface lamé's constants 𝜆𝐼,𝑆 and 𝜇𝐼,𝑆, surface/interface residual stress 𝜏0 𝐼,𝑆 , and the magnitude of negative surface piezoelectricity constants 𝑒31𝑝 𝑠𝑘 and 𝑒32𝑝 𝑠𝑘 , results in an augmentation of the piezoelectric nanosensor (pens) comparison of nonclassical controllers on nonlinear vibrations analysis of piezoelectric... 333 stiffness. consequently, this stiffness increase leads to a higher pull-in voltage in both the dimensionless natural frequency (dnf) and nonlinear dynamic response (ndr) analyses. ▪ an increase in surface/interface mass density 𝜌𝐼,𝑆 leads to a reduction in the piezoelectric nanosensor (pens) stiffness, resulting in a significant decrease in the dimensionless natural frequency (dnf). however, this increase in mass density does not significantly impact the 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https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=2ahukewjuvfurgzrpahwyqhuihysmaeoqfjabegqiebac&url=https%3a%2f%2fonlinelibrary.wiley.com%2fjournal%2f15214001&usg=aovvaw2dqpwnctfmikcxbs1xlo57 12403 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 459 471 https://doi.org/10.22190/fume231215004k © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper selection of optimal technological parameters for forming nominally flat surfaces with lubricating microcavities viktor p. kuznetsov1, igor v. tatarintsev2, vladimir v. voropaev2, andrei v. korelin1 1ural federal university, russia 2kurgan state university, russia orcid ids: viktor p. kuznetsov https://orcid.org/0000-0001-8949-6345 igor v. tatarintsev https://orcid.org/0009-0008-3303-0851 vladimir v. voropaev https://orcid.org/0000-0001-8508-073x andrei v. korelin https://orcid.org/0000-0003-0650-2805 abstract. this article demonstrates a multi-step process for forming a nominally flat surface with circumferential lubricating microcavities on a tribological assembly of an х38crsi steel shaft. the process includes finish turning, preliminary strengthening burnishing, deformation profiling of microcavities by a honing stone and smoothing of microprotrusions. the study determines the optimal technological parameters for each of the transitions based on microhardness and oil absorption power maximization, as well as roughness and periodic impact minimization. the process optimization is conducted using the tagichi experiment design method. the optimal combination of technological parameters for hardening burnishing was discovered to be: normal force f = 200 n; feed rate f = 0.025 mm/rev and three tool passes. these burnishing parameters practically eliminate the influence of periodic impacts at spatial frequencies determined by the tool feed rate and the number of spindle rotations during turning. we determined suitable honing stone grit parameters and application force that yield microcavities 3.8 to 8.1 μm in depth, as well as the optimal parameters for smoothing of microprotrusions, resulting in a bearing area roughness of sa = 0.15 µm and oil absorption power of 13.7410–5 mm3/mm2. key words: nominally flat surface, lubricating microcavities, burnishing, deformation profiling, microhardness, roughness received: december 15, 2023 / accepted february 09, 2024 corresponding author: vladimir voropaev kurgan state university, 640020, sovetskaya st. 63, building 4, kurgan, russia e-mail: sen_vvv@mail.ru https://orcid.org/0000-0001-8949-6345 https://orcid.org/0009-0008-3303-0851 https://orcid.org/0000-0001-8508-073x https://orcid.org/0000-0003-0650-2805 460 v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin 1. introduction increasing the wear resistance of precision surfaces of tribological assembly parts remains an important technological challenge. one possible solution is controlling mechanical machining process modes to reduce the surface's roughness and increase its microhardness. additional machining operations can be applied to increase the oil absorption power of the surface, methods for the formation of nominally flat surfaces with lubricating microcavities. such methods are implemented by mechanical and physical machining. in a review article, pawlus et al. [1] analyzed methods for producing nominally flat surfaces with microcavities for liquids. koszela et al. [2] describe a technology for the formation of oil-retaining microtopography on flat surfaces of tribological assemblies by finish milling and subsequent eccentric burnishing. lubricating microcavities (oil pockets) are created in the form of dimples by steel and ceramic balls. the depth of cavities is 5 to 13 µm, their diameter is 0.15 to 0.33 mm. mezghani et al. [3] proposed a method for obtaining a nominally flat surface with microcavities 4 to 6 µm in depth and 50 to 100 µm in width, by using a cylinder tool with honing stones. zhang et al. [4] studied the formation of a surface with lubricating microcavities up to 5 µm in depth, obtained by using laser ablation. both particular parameter values and a multitude of surface topography parameters can be used to assess the quality of nominally flat surfaces and their tribological properties. molnar [5] demonstrated that tribological properties of a surface can be characterized by the following 3d roughness parameters: maximum peak height – sp, maximum valley depth – sv, peak material volume – vmp, valley void volume – vvv, skewness – ssk and kurtosis – sku. it was found that low values of the maximum peak height (sp) and the functional-volumetric parameter (vmp) provide for a higher wear-resistance of a surface, and a higher vvv value indicates the increase of the surface's oil absorption power. moreover, molnar [6] uses a comprehensive analysis of height and functional parameters to determine the impact of topography on wearresistance. it was established that a simultaneous reduction of spk and ssk parameters to negative values leads to a better wear-resistance of the sample. nadolny [7] studied the flatness parameters fltt, fltp, fltv, fltq and roughness parameters sa, st, sq, sp, sv after a threestep mechanical machining by milling, grinding, and burnishing. this study showed a stable correlation between flatness and roughness parameters. kuznetsov et al. [8] proposed a technology for highly productive formation of nominally flat oil-retaining microtopographies on surfaces of shafts by using multi-step machining within one setup on a cnc machine, consisting of, consequently: finish turning, preliminary hardening burnishing, production of lubricating microcavities by a special tool equipped with a selfaligning honing stone and finish smoothing burnishing. preliminary hardening burnishing allows one to simultaneously attain submicronic roughness and maximum load bearing capability. thereafter, a special tool equipped with a self-aligning honing stone is used to produce lubricating microcavities of needed depth and oil absorption power by selecting the honing stone grit and its application force. the finish smoothing burnishing by a spherical indenter with a ≥6 mm radius removes protrusions created by the deformation profiling of lubricating microcavities. to effectively implement a multi-step machining procedure of this kind, careful optimization of the operating parameters is necessary to achieve the desired end result and reduce harmful interference between steps. the optimization is carried out based on orthogonal arrays proposed by g. taguchi, which has seen extensive use as an optimization method. this technique allows for discovering systematic, simple and efficient solutions, minimizing the effort needed to achieve the desired result [9]. selection of optimal technological parameters for forming nominally flat surfaces with... 461 for example, this method is used for optimizing drilling [10-11] and turning [12] parameters. this method was also successfully used for studying and predicting characteristics of a surface layer of construction steels after nanostructuring burnishing [13-14]. for evaluating surface quality, power spectral density (psd) analysis is a flexible mathematical tool, that is especially useful for identifying the influence of periodic impacts with different spatial frequencies during surface machining. this parameter is used to analyze functional surface properties in problems of contact mechanics, adhesion and thermal conductivity [15]. use of psd-analysis in contact mechanics was studied by peresadko et al [16]. surface roughness may have essential influence on the contact stiffness [17] and wear properties of contacting materials [18]. popov and pohrt [19] presented an evaluation of the psd function spectra after ball milling, burnishing and finishing using a honing stone. it was discovered that a psd function spectrum has five main constituents, varying in amplitude and wavelength. the appearance of said constituents correlates with periodical phenomena that occur in the mechanical machining system during the milling process. the dominating peak has a wavelength of 0.05 mm, corresponding to the milling feed value of 0.04 mm/tooth. the emergence of a peak with a wavelength of 0.26 mm, which is the first harmonic of the dominating peak, can be attributed to the formation of overlap traces. the psd function spectrum also includes periodical components, whose wavelength does not correlate with the kinematics of the process. these components can be attributed to dynamic impacts in the machining system, removal mechanisms, or chip segmentation. only one dominant component of the psd function that influences the forming of the surface texture was discovered during the burnishing process. periodic impact has a significantly lower stepover than the burnishing feed of 0.02 mm. it is presumed that this impact is connected with the traces of plastic deformation of the material. the frequencies of psd function peaks for surfaces after grinding are also different from those achieved during the milling process. it was discovered that milling traces do not get covered by burnishing traces, thus confirming the optimal and correct choice of technological parameters for burnishing of alcu4mgsi alloy curved surfaces. martins et al. [20] used psd analysis to investigate the influence of periodic feed rate impacts during turning and subsequent deep rolling at different tool contact pressures. it was found that at the lowest deep rolling feed rate of 0.06 mm/rev, the roughness peaks after the previous turning have a larger amplitude than at higher feed rates. increasing the deep rolling feed rate to 0.09 and 0.12 mm/rev reduces the influence of prior turning. it was determined that at the lowest deep rolling feed rate, peaks with a wavelength of 0.06 mm (spatial frequency 16.667 mm–1) are present on the psd function, which correspond to the deep rolling feed rate. it was demonstrated that in case the deep rolling pressure increases, the turning feed peaks presented on the power spectral density function become almost invisible. increasing the pressure and consequently the intensity of surface plastic deformation favors the concentration of peaks associated with the feed rate of turning, eliminating the peaks formed during the previous turning process. hardening with a higher feed rate of 0.15 mm/rev further increases the amplitude of the peaks in the psd function plot. using the lowest feed rate during deep rolling results in more complex surface deformation and a series of peaks with a large spatial distribution. the aim of this study is to determine the optimal combination of technological parameters for hardening burnishing (normal force, feed rate and number of tool passes) according to the criteria of microhardness hv, mean absolute error of microprofile sa and the minimization of periodic impacts’ influence that appear after the turning step, the optimization of grit and 462 v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin application force of the honing stone during deformation profiling of lubricating microcavities by the criterion of surface oil absorption power, and normal force during smoothing burnishing by the criterion of minimum plateau roughness and maximum surface oil absorption power vvv. 2. materials and research methods as a sample tribological assembly part, we used a cylindrical workpiece 23 mm in diameter and 150 mm long, made of х38сrsi (с 0.37%; cr 1.34%; si 1.1%; mn 0.47%; cu 0.21%; ni 0.05%; s 0.024%; p 0.019%) steel with hardness of 440 hb after oil quenching at a heating temperature of 900°c and tempering at 320°c with air cooling. we selected this material because of our efforts to increase the wear resistance of the shaft surface that is in contact with the needle bearing in the burlak snow and swamp-going vehicle's transmission. the multi-step technology for the formation of nominally flat microtopography with lubricating microcavities on a cnc turning machining center okuma/genos l300m (japan) under consideration consists of four consecutive processing step: finish turning and pre-burnishing, deforming profiling of microcavities and smoothing burnishing. finish turning was carried out using a zcc wnmg080408 cutting plate at a feed rate of ft=0.05 mm/rev and the spindle speed of 1,650 rpm. after finish turning the microhardness value was 4969 hv0.05. hardening burnishing was performed using a diamond tool of spherical shape with a radius of r=2 mm. deforming profiling of lubricating microcavities was done with a special tool with a honing stone 20 mm in length and 4 mm in width. the work surface grit was 160/125 and 200/250. the final smoothing burnishing was carried out using a tool with a diamond spherical indenter with a 6 mm radius. examples of workpiece machining by burnishing and deforming profiling of lubrication microcavities are shown in fig. 1. to analyze the topography of the processed surfaces, we used a wyko-nt1100 optical profilometer and wyko vision, matlab, gwyddion and mountainmap software. to measure microhardness, we used the ahotec eco hard xm1270c microhardness tester. fig. 1 examples of workpiece surface machining by burnishing (a) and deforming profiling of microcavities using a tool with a honing stone (b) selection of optimal technological parameters for forming nominally flat surfaces with... 463 3. experimental research 3.1. finish turning after finish turning, the microprofile's arithmetic mean deviation (average roughness parameter) was measured at sa=443 nm. surface morphology and profilogram after turning are shown in fig. 2. using an optical profilometer with the wyko vision software, we determined the surface’s oil absorption power to be vvv=6.9510–5 mm3/mm2 after turning. fig. 2 surface morphology (a) and profilogram (b) after finish turning the surface spectrum after turning is influenced by two periodic peaks, as determined by power spectral density (psd) estimation (fig. 3). the first peak has a spatial frequency of ωe=1.147 mm–1, a period of te=0.8 mm, and is related to spindle rotation speed of n=1,650 rev/min. the second peak has a spatial frequency of ωt=19.463 mm–1 and corresponds to the turning feed rate ft=0.05 mm/rev. fig. 3 surface psd function after finish turning periodical frequency psd function components at frequencies higher than ωt are believed to be connected to the geometry of the turning plate [18]. 464 v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin 3.2. determination of optimal pre-burnishing parameters using taguchi method further, to determine the optimal combination of preliminary strengthening burnishing parameters, we utilized the taguchi method with an orthogonal array l9 and three levels of process parameters: burnishing force f, feed rate f and number of tool passes np. based on the experimental studies of the process, we determined the optimal parameters for hardening burnishing to maximize microhardness hv0.05. the study determined the correlation between pre-burnishing parameters and the arithmetic mean deviation of the microprofile of the surface area sa. table 1 shows the mean microhardness values hv0.05 and arithmetic mean deviation of the sa profile over five measurements. table 1 machining modes and results exp. no altered parameter hv0.05 sa (nm) 1 f = 150 n f = 0.01 mm/rev np = 1 623 147.6 2 f = 150 n f = 0.025 mm/rev np = 2 602.8 82.74 3 f = 150 n f = 0.04 mm/rev np = 3 595 68.03 4 f = 200 n f = 0.01 mm/rev np = 2 642.6 161.33 5 f = 200 n f = 0.025 mm/rev np = 3 657.4 73.35 6 f = 200 n f = 0.04 mm/rev np = 1 592.6 96.46 7 f = 250 n f = 0.01 mm/rev np = 3 584.2 440.06 8 f = 250 n f = 0.025 mm/rev np = 1 630.4 109.96 9 f = 250 n f = 0.04 mm/rev np = 2 615.8 90.53 we used minitab 19 software package to analyze the results and rank the technological parameters according to their degree of influence on the output characteristic of the process. the evaluation of how process parameters influence the microhardness is presented in fig. 4. fig. 4 evaluation of how process parameters influence the output characteristic hv0.05 selection of optimal technological parameters for forming nominally flat surfaces with... 465 we discovered that the tool feed rate has the most significant impact on the hardening of the surface layer. the optimal combination of technological parameters for hardening burnishing can be determined using the highest values of each parameter. plots in fig. 4 demonstrate that the optimal combination of the parameters based on microhardness hv0.05 is a normal force of f=200 n, feed rate of f=0.025 mm/rev and number of passes np=3 (experiment 5). the microhardness value in this experiment was 657.4 hv0.05. simultaneously, the average value of sa=73.35 nm was slightly lower than the best value (sa=68.03 nm) achieved in experiment 3. however, the microhardness of the surface area after burnishing with parameters f=150 n, f=0.04 mm/rev and np=3 (experiment 3) was only 595 hv0.05. using the optimal combination of technological parameters for burnishing established in experiment 5 yielded the results closest to the calculated values shown in fig. 4. the psd analysis was used to evaluate the choice of burnishing parameter combination for reducing the influence of periodic impacts of the turning step. the assessment of psd function was done for surfaces formed after hardening burnishing that used technological parameters according to experiments 3, 5, 7. for these studies, we used the technological parameter combinations with feed rates of 0.04, 0.025 and 0.01 mm/rev, and normal force values of 150, 200 and 250 n in three tool passes. at the feed rate of f=0.04 mm/rev and the normal burnishing force of f=150 n, the psd function value at spatial frequency ωt=19.463 mm–1 was 176 nm2mm, and at frequency ωe psd=1129 nm2mm (fig. 5a). after reducing the feed rate to f=0.025 mm/rev and increasing the normal force to f=200 n in experiment 5, the psd function had significantly lower values at both spatial frequency ωt=19.463 mm–1 where psd=30.6 nm2mm and at frequency ωe=1.147 mm–1 where psd=991 nm2mm (fig. 5c). this indicates that the burnishing parameters adopted in experiment 5 minimize the influence of periodic impacts on the microprofile formation during the finish turning. after reducing the feed rate to f=0.01 mm/rev and increasing the burnishing force to f=250 n (experiment 7), the psd function amplitude significantly increases at frequencies ωe и ωt to 31523 nm2mm and 3293 nm2mm (fig. 5e). this mode significantly deteriorates the surface roughness, including to the point of destruction. reducing the feed rate f and increasing the normal force f during hardening burnishing increased the mean square deviation of surface irregularity height sq from 86.05 nm after machining with parameters of experiment 3 (fig. 5b) to 94.38 nm (experiment 5, fig. 5d) and 610.24 nm (experiment 7, fig. 5f). asymmetry of the ssk profile, which is the measure of spatial deviations from the middle plane, increased from 0.36 (experiment 3) by about two times to 0.65 (experiment 5) and spikes quite significantly to 1.21 after the machining with technological parameters of experiment 7. thus, technological parameters of burnishing used in experiment 5 are optimal and essentially nullify the influence of periodic impacts seen during the turning of sample surfaces. surfaces so obtained are best prepared for further technological transitions. so, the following machining transitions of deformation profiling of lubricating microcavities and smoothing burnishing were performed on samples after hardening burnishing with the following combination of technological parameters: f=200 n, f=0.025 mm/rev and np=3. 466 v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin fig. 5 psd functions and sample surface topography after burnishing with parameter combinations used in experiments 3 (a, b), 5 (c, d) and 7 (e, f) selection of optimal technological parameters for forming nominally flat surfaces with... 467 3.3. deformation profiling with the honing stone optimal honing stone grit parameters and application force were determined during the technological step of lubricating microcavity formation. the value of surface oil absorption power vvv was maximized for this purpose. the deformation profiling of microcavities was carried out using honing stones with grit values of 160/125 and 200/250, with application force values fp amounting to 200 and 250 n, providing for application pressure of 12.5 to 15.5 bar respectively for a 4×20 mm honing stone. workpiece linear speed was set at 30 m/min. similar parameters of nominally flat honing were used in [2]. fig. 6a and b presents a 3d-topography and a profilogram of the surface after deformation profiling with the honing stone application force of fp=200 n. the depth of microcavities obtained at fp=200 n and the grit value of 160/125 was 1.7 to 5 µm. the height of formed microprotrusions was 1.8 µm to 4.5 µm. the oil absorption power of the surface after deformation profiling with the process parameters presented above vvv=18.4710–5 mm3/mm2 was determined using the mountainsmap software. fig. 6 3d-topography (a) and profilogram (b) of the surface after deformation profiling with the honing stone application force of fp=200 n and the grit value of 160/125 the optimal parameters for deformation profiling were ultimately determined to be honing stone application force fp=250 n and grit value of 200/250, which resulted in the microcavities with a depth from 3.8 to 8.1 µm. in this case, the oil absorption power of the surface reaches its maximal value vvv=59.0710–5 mm3/mm2 (fig. 7a, b). 468 v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin fig. 7 3d-topography (a) and profilogram (b) of the surface after deformation profiling with the honing stone application force of fp=250 n and the grit value of 200/250 3.4. smoothing burnishing in the final step, we studied burnishing by a diamond indenter with the radius r=6 mm at two force values, 275 n and 300 n, seeking to remove microprotrusions and form a nominally flat surface while retaining the lubricating microcavities. minimizing the parameter sa and maximum microprofile peak deviation from the flatness fltp [21] and maximizing the oil absorption power of the formed surface vvv were taken as optimization criteria. sa, fltp and vvv parameter values after smoothing burnishing are presented in table 2. table 2. flatness fltp and oil absorption power vvv values after smoothing burnishing surface parameters flatness and oil absorption power fb=275 n fb=300 n sa (nm) 244.8 149.7 fltp (nm) 122.5 82.67 vvv (mm3/mm2) 10.7610–5 13.7410–5 3d-topography and profilogram of the surface after smoothing with the normal force of fb=300 n are shown in fig. 8. selection of optimal technological parameters for forming nominally flat surfaces with... 469 fig. 8 3d-topography (a) and profilogram (b) of the surface after smoothing with the normal force of fb=300 n as can be seen from the above, the optimal smoothing burnishing force fb=300 n yields the maximum surface oil absorption power vvv=13.7410–5 mm3/mm2 and minimum fltp parameter: fltp=82.67 nm. the mean absolute error of the microprofile amounted to sa=0.15 μm. 4. conclusions we carried out a series of studies relating to the formation of nominally flat surfaces with circumferential microcavities (grooves) on a shaft tribological assembly made of x38crsi steel during multi-step machining on a turning center. this method of machining had been previously suggested by nadolny and kapłonek [7]. the initial data of the surface after finish turning were: mean absolute error of microprofile sa=440 nm, microhardness 496 hv0.05, oil absorption power vvv=6.9510–5 mm3/mm2. also, psd analysis of the surface topography revealed the influence of periodic frequencies e and t associated respectively with spindle speed n=1650 rev/min and turning feed ft=0.05 mm/rev. the dominant influence of spatial frequency ωe=1.147 mm–1 is due to periodic impacts of the machine tool mechanical system in the region of low frequencies, multiple of the spindle speed. further study was aimed at determining the optimal parameters of technological steps of burnishing and deformation profiling of lubricating microcavities. using the taguchi l9 method of experiment planning, we established the relationship between the technological parameters of hardening burnishing (normal force, feed rate and number of tool passes) and microhardness hv0.05 and the mean absolute error of the surface microprofile sa. we discovered that normal burnishing force f=200 n, feed rate f=0.025 mm/rev and three tool passes are optimal and provide maximum hardening of the surface layer up to 657.4 hv0.05 and nearly the minimum value of the mean absolute error of the microprofile sa=73.4 nm. the choice of the optimal combination of hardening burnishing parameters was assessed by the criterion of minimizing the influence of periodic impacts occurring at the preceding transition of finish turning by psd-analysis. the psd function amplitude was evaluated for surfaces formed after hardening burnishing with the best and worst parameters of microhardness hv0.05 and roughness sa. these surface topography parameters corresponded to burnishing feed rates of 0.04, 0.025, 0.01 mm/rev and normal forces of 150, 200 and 250 n at three tool strokes. 470 v.p. kuznetsov, i.v. tatarintsev, v.v. voropaev, a.v. korelin the best surface roughness sa=68 nm was achieved at normal burnishing force f=150 n and feed rate f=0.04 mm/rev. the magnitude of psd function amplitude at periodic frequency e is 1035.5 nm2mm and at frequency t the psd function amplitude is 11.2 nm2mm. however, the microhardness did not exceed 595 hv0.05 in that case. it was discovered that when the feed rate is reduced to 0.01 mm/rev and the burnishing force is increased to 250 n, there is a significant increase in the amplitude of the psd function to 3293 nm2mm at periodic frequency t and to 33182 nm2mm at frequency e. the mean absolute error of the microprofile increases to sa=440.1 nm. we suspect that increasing normal force and decreasing feed rate aggravates the effect of machine tool mechanical system oscillations. the established optimal parameters of hardening burnishing were used for surface preparation for subsequent transitions of deformation profiling of lubricating microcavities and final smoothing burnishing of microprotrusions. deformation profiling of lubricating microcavities was carried out by a special tool with a self-aligning honing stone. we revealed the relationship between the parameters of the honing stone grit, the force and pressure of its application against the surface with the depth of microgrooves and oil absorption power of the processed surface area. it was established that the optimum parameters of deformation profiling are the honing stone grit of 200/250 and the pressure of its application to the surface of 15.5 bar at the force 250 n. machining with these parameters allowed us to obtain microcavities with depths from 3.8 to 8.1 μm and the oil absorption power of vvv=59.0710–5 mm3/mm2. a similar level of microcavity depth and oil absorption power of tribological assembly surfaces had been obtained by other researchers [2-3]. smoothing burnishing with a diamond indenter 6 mm in radius with normal force fb=300 n and feed rate of f=0.2 mm/rev enabled us to practically eliminate microprotrusions after deformation profiling of lubricating microcavities. the mean absolute error of the microprofile amounted to sa=0.15 μm. simultaneously, due to reduction of the depth of lubricating microcavities to 1.7...4.1 µm the oil absorption power of the surface sank to 13.7410–5 mm3/mm2. after the deformation profiling and flat-top burnishing steps, the microhardness of the bearing area decreased to 605 hv0.05. this is due to the surface de-hardening after honing. at the optimum mode of preliminary strengthening burnishing at force f=200 n and feed rate of f=0.025 mm/rev, the parameters of the surface psd amplitude function increased insignificantly and amounted to 30.6 nm2mm at t and 1386.9 nm2mm at e, respectively. for further optimization, assistance of burnishing by ultrasonic oscillations may be considered [22], as well as application of coatings [23]. acknowledgement: funding from the ministry of science and higher education of the russian federation (ural federal university, the state assigment № 075-03-2023-006 of 16.01.2023 (feuz-2024-0020)) is acknowledged. selection of optimal technological parameters for forming nominally flat surfaces with... 471 references 1. pawlus, p., reizer, r., wieczorowski, m., 2021, analysis of surface texture of plateau-honed cylinder liner a review, precision engineering, 72, pp. 807-822. 2. koszela, w., pawlus, p., rejwer, e., ochwat, s., 2013, possibilities of oil pocket creation by the burnishing technique, archives of civil and mechanical engineering, 13(4), pp. 465-471. 3. mezghani, s., demirci, i., yousfi, m., mansori, e.m., 2013, running-in wear modeling of honed 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coating, mechanics of advanced materials and structures, 29(16), pp. 2390-2403. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 307 323 toward an integrated information system for the design, manufacturing and application of customized implants  udc 004.8:004.42 dragan mišić, miodrag manić, nikola vitković, nikola korunović university of niš, faculty of mechanical engineering in niš abstract. the adjustment of products to the needs of customers has been present in various industries for many years. personalized medicine is a field that has been rapidly developing recently. this kind of medical help mainly implies the use of medications which are adjusted to each patient individually. in this paper, we describe an information system which manages the process of designing and manufacturing personalized products in the area of orthopaedics. the system output comprises patient-adjusted orthopaedic implants. in addition to the process management, the information system ought to enable the process to be adjusted to unexpected situations which may occur in different stages of designing and manufacturing. the information system should also assist doctors and engineers in the decision making process. this aid is realized in the form of the expert system which provides doctors and engineers with advice about defining an appropriate treatment for the patient. key words: wfms, exceptions, expert system, personalized medicine 1. introduction modern products and services currently on the market must develop in accordance with the ubiquitous globalization. nowadays, a company cannot expect the customers to buy their products only because there is no alternative. in order to keep the customers, the company must establish a close relationship with them, whereby they cannot rely solely on the price lower than that of the competition. the use of the internet technologies and online business leads to the possibility of easily comparing prices, and the customer may thus choose to buy a product from some other company, if he finds the price or some other characteristic more fit to his needs. received april 20, 2015 / accepted july 30, 2015 corresponding author: dragan mišić mašinski fakultet niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail: misicdr@gmail.com original scientific paper 308 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović one of the possible reactions to such environment is to attempt to establish a closer relationship with the customers, which would contribute to customer satisfaction and their loyalty. in this context one can speak of personalization as a business model. in [1] the author defines personalization in the following manner: “personalization is about building customer loyalty by building meaningful one-to-one relationship; by understanding the needs of each individual and helping satisfy a goal that efficiently and knowledgeably addresses each individual need in a given context“. in the domain of industrial products and services, the personalization falls under marketing, but it must also be connected to the production which must be able to respond to the customers‟ needs. in comparison with the personalization in industry, personalization in medicine has just recently begun to gain importance. personalized medicine derives from the belief that the same illnesses that afflict different patients cannot be treated in the same manner [2]. the assumption is that every individual responds differently to specific medications in accordance with his unique genome. introducing personalized medicine into practice has brought up different problems which should be regarded not only from the medical point of view, but from the technical and, also, legal aspect [3, 4]. the term „personalization‟ in medicine mainly refers to the use of medications and treatments which are adjusted to a specific patient, and thus there are not many examples of applying this principle when manufacturing orthopaedic implants. in this paper, we describe the system which enables the designing and manufacturing of the orthopaedic implants which are adjusted to specific patient (customized implants). another aspect of this paper which has not been well researched is the connection between medical institutions and production organizations which produce for these institutions. in our paper, we discussed the connection between an orthopaedic clinic and an implant manufacturer. usually, the clinic purchases a certain number of implants via public procurement process and then uses them for all patients for a period of time. the only contact between the company and the clinic is tender by which the clinic makes a purchase. instead of this usual way of working, we propose the integration of the processes in clinics and production organizations by using the information system which would manage the whole process, starting from the doctor‟s diagnosis, and ending with the manufacturing of the implant which is adjusted to a specific patient, if the adjustment is required. when talking about use of information systems in medicine, it is mainly referred to health is. the health information system (his) is dealing with processing of data, information and knowledge in health care environments [5]. these systems deal with information flow in medical facilities; however, they are rarely connected to information systems of companies that make equipment used in hospitals. in [6], for example, is described integration in medicine manufacturing enterprises. they recommend use of soa and rfid integration technologies. some authors are trying to apply the supply chain management technologies to health care [7]. those authors emphasize the fact that the supply chain management in a health care setting is characterized by some unique features, which makes it difficult to transfer knowledge from the industrial sector to the health care one in a direct way. toward an integrated information system for design, manufacturing and application of customized implants... 309 we propose the integration of information systems of clinics and companies via the workflow management system (wfms). the role of this system is to manage the process which is carried out in two different organizations, namely, the orthopaedic clinic, on one hand, and the enterprise which designs and manufactures implants, on the other. the idea behind the use of such system is to enable flexibility when selecting and embedding implants. this flexibility is achieved through a process which deals with the manufacturing of customized implants, on one side, and through the possibility to respond to exceptions which may occur during the process, on the other. exception handling in workflow systems is a problem which does not have a real solution yet. nowadays, there a lot of different approaches connected to it [8, 9, 10, 11]. in this paper, we use an expert system for problem solving. this expert system is integrated with the basic process management system and is in charge of dealing with unexpected situations and possible modifications of the process according to the new conditions [12]. the knowledge base of the expert system is created by purchasers (in this case, doctors) as well as the implant manufacturers. besides being used for dealing with exceptions, the expert system also provides support for doctors and engineers in decision making process. in orthopaedics, one of the most important decisions which the doctor makes is the choice of appropriate osteofixation material. choosing the right treatment requires a certain amount of experience and knowledge which experienced doctors possess. this knowledge is heuristic, so the easiest way to present it in computer form is in the form of rules which are part of some expert system. during the expert system exploitation, less experienced doctors may use knowledge and experience of their colleagues to help them with making decisions related to the patient treatment. some research projects [13] show that, even in the environments where there are developed clinical decision support systems (cds), the implementation faces various problems. probably the most important one is that these systems are not integrated with the clinic workflows. in this paper, we propose integration of cds systems with the workflow management systems. the integration is performed by automatically calling the cds system when executing certain activities, which then offers to doctors and engineers intelligent advice related to the choice of right treatment. the system is capable of doing so due to all the data that exist within the workflow and the data which it collects through a series of questions that the users give answers to. the rest of this paper is organized as follows: in section 2 we show the process of designing and manufacturing osteofixation material and its application. in third section we describe the manner of handling exceptions which may occur during the execution of this process. the fourth section explains the expert system which is used as a support for choosing osteofixation material. the paper is concluded in section 5. 2. managing the process of creating osteofixation material within the project vihos (virtual human osteoarticular system and its application in preclinical and clinical practice) [14] which is carried out at the faculty of mechanical engineering and the faculty of medicine at the university of niš, the 3d parameterized geometrical and bone models for fea (finite element analysis) are being developed. also, a method for design of patient specific internal and external fixators and scaffolds is 310 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović developed. the mentioned models are developed by the use of the method of anatomical features which is presented in [15]. the maf consists of several procedures which enable the creation of geometrically precise and anatomically correct geometrical models of the human bones. based on such bone models, additional models of fixators, scaffolds and other elements can be constructed. in general, the maf is a reverse engineering (modeling) method and its main procedures are:  ct scanning of the human bone data, importing the data into cad and adequate filtering of the input point cloud.  creation of anatomical model [15] morphologically and anatomically defined descriptive model of human bone.  definition and creation of referential geometrical entities (rges) [15] geometrical entities which are created on polygonal model of the human bone in relation to anatomical features. they can be: planes, views, lines, points, curves, etc.  definition and creation of the basic geometrical elements in this procedure, the basic geometrical elements are created, namely, spline curves, planes, surface patches, lines, etc. these elements can be used for the creation of different types of geometrical models like surface, volume or parametric.  creation of parametric models this procedure enables creation of the parametric model which consists of anatomical points whose coordinates are defined as parametric functions. arguments of these functions are morphometric parameters defined in medicine literature, as described in [9]. this model enables the construction of complete bone models in the cases of incomplete data about bone geometry. this can be the case when part of the bone is missing because of some bone disease (e.g. osteoporosis), or for example, when only one x-ray image is available. on the basis of the obtained models it is possible to achieve another goal of the project, and that is to develop osteofixation material which could be adapted to a specific patient. osteofixation material is a term that usually refers to implants for bone fixation and support, and they can be: fixator, scaffold and/or the graft, etc. in order to make appropriate osteofixation material models, which will be used in surgery that a given patient will later undergo, the cooperation between doctors and engineers is required. within the proposed system, this cooperation is carried out by information system, the work of doctors and engineers is considered to be a unity and a part of a common process, the process begins in orthopaedic clinic, continues in the company which designs and produces customized osteofixation material, and it ends also in clinic, where the planning of the surgery and the surgery itself are being conducted. since the work on designing and implementing of osteofixation material is considered a process, it is only natural that this process is managed via computer, i.e. via appropriate software workflow management system. since the said process is carried out partly in hospital, partly in company, the best option is to implant everything by using two different processes which communicate via messages. due to the lack of the tool which is used here (enhydra shark), it is not possible to carry out the process in that manner. that is why everything stays within one process which is carried out in both hospital and company. in order to successfully manage the process, it is necessary to know who the owner of the process is. since the responsibility is divided in this case, our option is that the owner is neither the company nor the hospital. instead, a specific interdisciplinary organization toward an integrated information system for design, manufacturing and application of customized implants... 311 is formed, in order to govern, monitor and execute the overall process. the members of that organization are doctors and hospital staff, on the one hand, and engineers and company staff, on the other. depending on the type of the patient‟s injury, there is a possibility of using standardized osteofixation material (doctor can select only those osteofixation materials that are already in stock). it is also possible that the injury is somehow specific, in which case it is needed to adjust the products to the patient. orthopaedic fixators are made in the specific dimension range (sizes), in order to enable the application of fixators to bones of different patients. application of predefined internal fixators to the specific patient may be problematic because of the difference in the size and shape of the particular bone and the fixator. one of the solutions for this problem is the application of the so-called customized fixators. the geometry and topology of those fixators are adapted to the anatomy and morphology of the bone belonging to the specific patient. application of customized fixators has a positive effect on patients, but, on the other hand, it requires additional time for preoperative planning and fixator manufacturing. therefore, these fixators are used in situations where the application of predefined fixators can lead to complications in the surgical interventions or in the process of recovery. the process of designing and manufacturing of customized implants may be schematically presented, as in fig. 1 [16]. fig. 1 automated design process phases and manufacture of customized implants the process in fig. 1 may be physically realized by defining activities which would be monitored by wfms. the process diagram realized in jawe editor [17] is shown in fig. 2. 312 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović the above mentioned process may be carried out in a few different ways. the first possibility is that the use of customized osteofixation material is not required. in that case, after defining the treatment, the standard material is immediately sterilized and the operation may begin. in the case when it is recommended to use the customized osteofixation material, the first task is to make appropriate 3d models of fracture and fractured bones. when it comes to the fracture itself, it may be such that all parts of a bone are present, but it may also happen that a part of the bone is missing. if the missing part is small, a scaffold may be used as a support for fracture healing, while in cases when the missing bone part is large, it is required to make the missing part using the methods of reverse engineering. the four described cases are at the same time the four possible paths which the process may follow. after the activities related to designing and manufacturing of the missing bone part or scaffold comes designing and manufacturing of customized fixator. the complete osteofixation material is sent to sterilization at the end of the process, and later used in surgery. fig. 2 the process of creating customized osteofixation material toward an integrated information system for design, manufacturing and application of customized implants... 313 a precondition for execution of branches in which the customized osteofixation material is designed and manufactured is the existence of corresponding models. bone models may be obtained from radiology images, if the images are of good quality (fig. 3). some activities of the process which is monitored are in charge of definition of these models. fig. 3 3d model of bone and fracture it may happen with some patients that some parts of the bone are missing, or the image quality is not satisfying. in such cases, the required models are obtained by using the methods of reverse engineering, which are described at the beginning of this chapter. the models of bone and fracture created in the first stages of the monitored process are also used for the manufacturing of customized fixator (fig. 4). in this case, the models are used for adjusting the shape and the dimensions of fixator. fig. 4 fixator adapted to the specific bone model for the creation of the geometrical models of customized fixators a new design method has been developed and presented in paper [18]. geometrical models of internal fixator by mitkovic for tibia bone created by this method could be applied for the preparation of preliminary models for fea, for the preoperative planning, for the production of customized internal fixators, etc. when the production of all elements of osteofixation material is finished, the process is continued in the hospital which receives these elements. after their sterilization comes the surgery, when the osteofixation material becomes functional. the process activities which we are describing are shown in fig. 2. a detailed description of particular activities may be found in [19]. 314 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović 3. exception handling in the process of creating osteofixation material business process management systems have proved themselves to work best in environments where the processes are well scheduled, whereby those processes do not change often. the problem occurs in the situations when the process must deviate from predefined scenario. nowadays, these deviations occur rather frequently. they also occur in production companies, which are forced to quickly react to changes in the environment due to the constant battle with the competition, but they occur in medical facilities as well. the processes in medicine are variable by their nature, since it is difficult to predict everything that will be required for a specific patient treatment. the process we describe in this paper is carried out partly in a production organization, partly in a hospital, so it is natural to expect that the situation in which it is impossible to bring results should occur during its execution, if the process is not modified. one of the possibilities is to embed all possible process deviations in the model. this solution is not adequate for two reasons. firstly, it is difficult to predict all possible scenarios, and secondly, the process model would become more complex, difficult to read and monitor. the process shown in fig. 4 is complex enough on its own, so it is necessary to avoid its further complexity. we have attempted to find the solution in integration of the system for process management with the expert system, which is in charge of dealing with problems and possible modifications of the process according to new circumstances. the expert system functions on the basis of rules, which are defined independently of the process model. these rules are activated in the situations when the system or human notice that the regular process execution is not possible. the rules may offer advice or modify the process, thus solving the problem. since they are independent of the process model, they do not burden it. the proposed solution is realized by the wfms md, developed at the faculty of mechanical engineering in niš [12]. this system uses open source system enhydra shark as a workflow engine, which is modified and adapted to specific needs. for execution of the rules, we use the expert shell jess [20]. the connection between these systems is shown schematically in fig. 5. exception system core process editor xpdl expert system process definitions process instances business data business environment md system users administrator knowledge engineer knowledge acquisiton database rules meta rules process manager fig. 5 the connection between wfms and expert system toward an integrated information system for design, manufacturing and application of customized implants... 315 we will provide an example of exception handling, in order to explain it thoroughly. the example is the process of creating fixator for tibia. one of the activities which are executed in the hospital at the beginning of the process is the preparation of images which will later be used for creation of 3d osteofixation material model. in some cases it is not possible to obtain an adequate image. for example, such situation occurs when there is a part of bone missing, or when it is not allowed to expose the patient to radiation. without the image, it is impossible to continue the process, because all further activities are based on the existence of those images. the solution may be found in process modification. for example, if the bone whose part is missing is tibia, possible solutions are to scan the other sound leg or to start the process of reverse engineering of tibia based on available data, if it is not possible to scan that other leg, either. at the moment when such problem occurs, the process is stopped. md system may be notified of it by entering specific values of parameter which is related to the process continuation, but in some situations the system itself is able to independently reach the conclusion that something is wrong. in this case, after the execution of the activity images acquisition, it checks whether there is an image of the fracture. if the image does not exist, that is a signal that the exception occurred. another possibility is that the user notifies the system of the exception. the processes which are defined and monitored via wfms systems may refer to various areas. for that reason, there are so-called meta rules, which ought to determine whether there is domain knowledge (defined rules) for the problem which has occurred. these meta rules enable modularization and easier management of knowledge. in particular case, the meta rule which determines whether the system possesses knowledge about ways of solving the problem is not necessary. its role is overtaken by the rule which checks whether the image exists. if it does not exist, and the process is that of creating the customized implants and the activity is image acquisition, the rules which are related to solving this problem are loaded. depending on the manner of detection, and on the specific process which is monitored, it may occur that the expert system requires additional information in order to conduct the concluding procedure. additional information refers to that which is not in the process itself, because all data from process are transferred to the expert system. in accordance with the aforementioned, we have developed rules which ask certain questions concerning the given problem to the user. in the expert system, the questions are presented as rules. the user‟s responses enter the fact base and present additional facts on the basis of which the expert system makes conclusions. interface by which the questions are asked to the user is completely coordinated with the whole system, thus the user is not aware that he is actually communicating with the expert system. the questions are usually multiple choice or yes/no questions. as mentioned before, if there is not an image of damaged bone, it is possible to use the image of the other sound leg, or measure the distance between certain points on the sound leg. the first option is to record the sound leg. the system cannot automatically determine whether it is possible to make image of the sound leg; instead, it asks the user (doctor). the rule which defines asking this question is: 316 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović (defrule recording (answer (ident images) (text ?d)) => (if (eq ?d yes) then (assert (question (type multi) (text "is it possible to make a ct scan of a sound leg?") (ident possible_recording) (valid yes no))) else (assert (answer (ident no-solution) (text no-solution))) ) ) if the user responds to the question which is asked based on this rule with yes, the rules which modify the existing process are loaded. the rule which loads these rules is: (defrule load-image-rules (answer (ident images) (text yes)) (answer (ident possible_recording) (text yes)) => (clear) (batch image_rules.clp) ) in file image_rules.clp, there is only one rule, which ought to modify the process according to new circumstances. that rule is: (defrule recording_sound_leg (exception (id ?id) (type ?t) (currentactivityid ?currentactivityid) (resactivityid ?resactivityid) (sourceprocessid ?processdefid)) (answer (ident possible_recording) (text yes) (place before current) => (bind ?packagedefinition (fetch packagedefinition_1)) (bind ?process (get-process-from-definition ?processdefid ?packagedefinition )) ; (bind ?prevresactivities (call org.enhydra.jawe.misicproba findpreviousactivities ?currentactivityid ?process)); (bind ?transitions (call ?process get "transitions")) ; (bind ?activities (call ?process get "activities"));(bind ?currentactivity (call ?process getactivity ?currentactivityid)); (bind ?prevresactivity (call ?prevresactivities get 0)); (bind ?nextresactivity (call ?nextresactivities get 0)); (bind ?b1 (call org.enhydra.jawe.misicproba createactivity "recording sound leg" "recording_sound_leg" ?activities ?process "description" 1 "" “doctor" 1 1 "" "" "" nil)) (bind ?tnew1 (call org.enhydra.jawe.misicproba createtransition ?transitions ?prevresactivity ?b1 nil)) (bind ?tnew2 (call org.enhydra.jawe.misicproba createtransition ?transitions ?b1 ?nextresactivity nil)) (call org.enhydra.jawe.misicproba removeactivity ?currentactivity ?nextresactivity ?transitions) ; (bind ?strategy (new org.enhydra.shark.strategy nil nil ?processdefid nil "current_instance_only" nil ?processdefid ?resactivityid "change" nil "packagedefinition_1")) (bind ?strategies (fetch strategies)) (?strategies add ?strategy)) within this rule, the process definition is modified by deleting the activity images acquisition, and the activity recording sound leg is inserted instead. the process toward an integrated information system for design, manufacturing and application of customized implants... 317 modification may generally refer only to the current process instance, but it may also refer to all future instances. in the specific case, the recording of the sound leg is added only for that particular patient, so the modification of definition refers only to current instance (value current instance only). this is defined by list "strategies", in which the conclusions of the expert system related to application of the solution are inserted. this is a list because the solutions may be applied in several ways. for example, if the new process definition ought to be applied to the current instance as well as all future instances, then in addition to the option current instance only there would be the option all_new_instances in this list. after the execution of this rule, the new process definition is returned to the wfms. further execution of this process instance continues on the basis of the new process definition. specifically, this means that the activity recording sound leg appears in the doctor‟s work list. rules for other exceptions which may occur during the execution of the process may be defined in a similar manner. the rules are defined independently of the system operation, thus it is not required to modify the process model. by adding a new rule in the rule base, the rule immediately becomes available for execution during the processing of exceptions. 4. decision support system it is already mentioned that the system which we describe in this paper may function as a decision support system as well. since the process is carried out partly in clinic, partly in production organization, this system ought to support the work of doctors as well as that of engineers. the advantage of this system over other similar solutions stems from the fact that the system is embedded in the wfms, and it may thus help to execute the process in the best possible way. we have also created a decision support system via expert system, i.e. the expert system shell jess. what distinguishes it from the mechanism for exception handling is the fact that this system is called as a part of activity execution but not in the case when the activity is impossible to execute. this means that the activities are predefined in a manner which enables the expert system to be called during their execution. the expert system offers recommendations to a doctor/engineer about executing the activity in the best possible way. for making successful conclusions, data and knowledge are required. when it comes to the process of manufacturing osteofixation material, it may be said that data and knowledge are placed within the knowledge model and the topological and geometrical models of product. the topological model describes its position and orientation within the fracture and bone models. the geometrical model contains geometrical data about points and surfaces which it is composed of. it is a surface model which describes the geometrical shape of osteofixation material. within the knowledge model, there are all necessary data which are related to physical, biological and mechanical characteristics, as well as characteristics of implementation and other things necessary for its manufacturing, ways of sterilization, implementation, removal, disposal, etc. 318 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović the knowledge model contains a lot of different information. for example, it contains information about the type of implant whose purpose may be bone reconstruction, softtissue reconstruction, augmentation, cosmetic reconstruction. it also contains data about whether the vascularization is necessary, whether the biodegradability is required, etc. the cds may use all the information in decision making. we will explain the manner of functioning of clinical decision support system by using the activity selection of fixator material as an example. within this activity, the appropriate material for fixator manufacturing is being chosen. the right choice depends on many factors which should be taken into account. in addition, there is a variety of materials which are appropriate under some, and less appropriate under other circumstances. in the literature, there is a large number of different materials which are used for the manufacturing of implants. thus, in [21] the following materials are mentioned:  metals o stainless steel o vitallium (cobalt-chromium) o titanium o gold  calcium ceramics o hydroxyapatite o tricalcium phosphate o hydroxyapatite cement o bioactive glass  polymers o silicone o polymethylmethacrylate o hard tissue replacement (htr) polymer o polyesters (dacron, mersilene) o biodegradable polyesters (polyglycolic acid, o poly-l-lactic acid) o polyamides (supramid, nylamid) o polyethylene (medpor) o polypropylene (prolene, marlex) o cyanoacrylates o polytetrafluoroethylene (teflon, gore-text)  biologic materials o collagen o alloderm  discontinued materials o polyurethane o proplast as mentioned before, the choice of materials depends on various conditions. in most cases there is not only one possibility, instead, a number of different materials may be used. we have illustrated the process of choosing materials with a few rules, which give recommendations about the right choice of materials. they can be further developed in the case that the material is not available (but this case is not presented in the paper). toward an integrated information system for design, manufacturing and application of customized implants... 319 various factors may have influenced the choice of materials. some of them are:  the place where the implant is prepared (whether it should be prepared in the operating room immediately prior to the implementation or not)  whether the implant is going to be high load-bearing  what amount of money can be given for the implant  the complexity of shape  the required weight of the implant  the purpose of the implant (cosmetic reconstruction or other) based on this information, a few rules which illustrate the process of choosing material are defined. those rules are shown in table 1. table 1 the rules for choosing material conditions material  the implant should be prepared in the operating room immediately prior to the implementation calcium ceramics – hydroxyapatite  load-bearing implant  low price  not compex shape metal – stainless steel  high load-bearing implant  very complex shape  need low weight metal – titanium (additive technology)  implant for cosmetic reconstruction  very complex shape  need low weight biologic materials collagen the system does not have an access to the information which is required for the activation of the rules from table 1 so that the users must enter them. in this case, a doctor and an engineer should do this together. the system of questions and answers is also used in this case. the questions are defined in the form of rules whose action part enters certain facts in the fact base and enables new questions to be asked or to reach an appropriate conclusion. the first question which is asked is whether the implant should be prepared in the operating room immediately prior to the implementation. the rule containing this question looks like this: (defrule implant-preparing (initial-fact) => (assert (question (type yes-no) (text "is the implant prepared in the operating room?") (ident operating_room) (valid yes no))) ) if the user confirms this, the rule which recommends hydroxyapatite as the material type is executed. 320 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović (defrule def_1 "material is hydroxyapatite" (answer (ident operating_room) (text yes)) => (store material hydroxyapatite) ) if this question is answered negatively, the system continues with questions, i.e. it continues entering the required data. the questions related to load, price, shape, etc. are asked. there is a defined question for each of the parameters which may affect the decision making process. the system asks questions as long as there are some that the user has not answered yet. some of the rules which are used for defining the questions posed to the user are: (defrule load_question (answer (ident operating_room) (text no)) => (assert (question (type yes-no) (text " is the implant high load-bearing?") (ident high-load ) (valid yes no))) ) (defrule price_question (answer (ident operating_room) (text no)) => (assert (question (type multi) (text "the price of implant") (ident price) (valid low high))) ) (defrule shape_question (answer (ident operating_room (text no)) => (assert (question (type multi) (text " is the shape of the implant complex?") (ident shape) (valid complex not_complex))) ) (defrule weight_question (answer (ident operating_room) (text no)) => (assert (question (type multi) (text "is the implant heavy or lightweight?") (ident weight) (valid high low))) ) (defrule reconstruction_question (answer (ident sala ) (text no)) => (assert (question (type yes-no) (text "is it a cosmetic reconstruction?) (ident cosmetic_reconstruction) (valid yes no))) ) after the user has answered these questions, the expert system should have enough data to reach a conclusion about the type of fixator material which should be used. some of the rules that do that are: (defrule solution_metal_stainless_steel (answer (ident sala) (text no)) (answer (ident high-load) (text yes)) (answer (ident price) (text low)) (answer (ident shape) (text not_complex)) toward an integrated information system for design, manufacturing and application of customized implants... 321 => (store material stainless_steel) (printout t "material is stainless steel" crlf) ) (defrule solution_metal_titanium (answer (ident sala) (text no)) (answer (ident high-load) (text yes)) (answer (ident weight) (text low)) (answer (ident shape) (text complex)) => (store material titanium) (printout t "material is titanium" crlf) ) (defrule solution_biologic_material_collagen (answer (ident sala) (text no)) (answer (ident cosmetic_reconstruction) (text yes)) (answer (ident weight) (text low)) (answer (ident shape) (text complex)) => (store material biologic_materials_collagen) (printout t "material is collagen" crlf) ) these rules choose the most appropriate fixator material. the chosen material is put in the variable material, which is later accessible from the system for process management. the value of the variable which is read is used for definition of the value of the variable defined at the process level and the activity selection of fixator material. the user can access that variable, defined within the system, in subsequent activities in which the process of making fixators is defined. if the expert system does not have any solution, the value of variable remains empty. in that case, the user chooses and enters the appropriate value on his own. decision support system which is modeled in this manner is open for expansion, since it is not necessary to compile a program code or to change the activity in some other way. in case that some new materials become available, it is enough to enter new rules which are related to this case in the rule base, and the expert system will take that into account when defining its recommendations. 5. conclusions in this paper, we have described the system for management of the process of designing and manufacturing customized osteofixation material. the process of designing and manufacturing osteofixation material is carried out partly in hospital, partly in production organization which produces this material. in this paper, we discuss customized osteofixation material, and this process is started when the patient needs a specific type of the service. in the cases when the standard osteofixation material can be used, the process is carried out by the usual procedure, which we did not discuss in this paper. the connection between activities which are carried out in the hospital and those which take place in the production organization is achieved by the workflow management 322 d.mišić, m.manić, n. vitković, m. trajanović, n.korunović system. the novelty of this work is that the processes in the hospital and in the production organization are not considered individually. instead, they are considered to be a unity, managed by one computer system, thus integrating businesses which are carried out in two different environments. processes of designing and manufacturing osteofixation material are modified frequently, with the goal to provide a patient with the best treatment possible. instead of changing process definition and adapting it to the new conditions every time a change occurs, we enhanced the system by connecting it to the expert system which deals with unpredicted situations. the main workflow management system is integrated with the expert system which solves problems on its own, in case that unpredicted problems occur. the conclusions which expert system makes may lead to a change of the process definition, at the level of one or more instances of the process. those modifications, however, are carried out automatically, thus allowing the user to continue with working on the new process, without having to manually modify the model. the expert system is used also as a support for making decisions in activities whose execution requires experience. in this paper, we describe this process by using the activity of choosing fixator material as an example. that choice is a result of experience, not estimation. the experience and knowledge of both doctors and engineers are presented by rules, which attempt to recommend the material which is the most appropriate for the conditions given. in this case, the integration with the expert system is realized by embedding the calling of the system in the activity. the user is not aware of the fact that he is communicating with another system. the main advantage of this work is the fact that it is easier to present experiential knowledge by rules, and that it is not necessary to change the process activity in case that the knowledge is upgraded or changed. in such cases it is enough to change the rules, and the expert system (and the activity in which it is embedded) will automatically offer new conclusions. the rules which are used for solving problems are placed in the file which is not related to the system, thus the rules can be easily modified. acknowledgements: the paper is part of the project iii41017 virtual human osteoarticular system and its application in preclinical and clinical practice, sponsored by republic of serbia for the period of 2011-2014. references 1. ricken, d., 2000, personalized views of personalization, communications of acm, 43(8), pp.27-28. 2. annas, g.j., 2014, personalized medicine or public health? bioethics, human rights, and choice, revista portuguesa de saúde pública, 32(2), pp. 158-163. 3. isa, a., 2015, 10 years of personalizing medicine: how the incorporation of genomic information is changing practice 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milovanović j., 2014, the concept of the information system for managing business processes of designing and manufacturing of osteofixation material, icist 2014, 4th international conference on information society and technology, kopaonik, srbija 20. jess, the rule engine for the javatm platform, sandia national laboratories, http://herzberg.ca.sandia.gov/, last access 30.01.2015. 21. thorne, c., 2007, grabb and smith's plastic surgery, 6th edition, wolters kluwer, philadelphia, p. 976. http://www.sciencedirect.com/science/article/pii/s0950584913000840 http://www.sciencedirect.com/science/article/pii/s0950584913000840 http://www.sciencedirect.com/science/journal/09505849 http://vihos.masfak.ni.ac.rs/ http://www.together.at/prod/workflow/twe facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 705 718 https://doi.org/10.22190/fume190225020j © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper analysis of the influence of the digging position on the loading of the slewing platform bearing in hydraulic excavators vesna jovanović, dragoslav janošević, jovan pavlović university of niš, faculty of mechanical engineering, serbia abstract. the paper defines the general mathematical model of hydraulic excavators for determining the boundary and possible digging resistances and equivalent loads of the axial bearing of the slewing platform drive mechanism throughout the excavator’s working area. using a developed mathematical model and program, in the case of a 100000 kg hydraulic excavator with a shovel manipulator bucket volume of 6,5 m3, a detailed analysis was performed to examine the influence of the position and digging resistance on the loading of the axial bearing of the excavator slewing platform drive mechanism. the results of the performed analysis show that the equivalent loads, relevant for the selection of axial bearing of the excavator slewing platform, occur in the zone of the working area when the kinematic chain of the excavator has positions in which the manipulator mechanisms have coordinated interaction when they can overcome the greatest resistance forces in the stable operation of the excavator. key words: hydraulic excavator, slewing platform, loading of axial bearing 1. introduction in the synthesis of the slewing platform drive mechanism of hydraulic excavators, one first chooses the axial bearing, which attaches the slewing platform to the support and movement mechanism of the excavator. the choice of this bearing is very complex because the excavator performs various functions in its working area with a variety of different positions and in very different working conditions. the recent research on the slewing platform drive mechanisms of excavators includes: development of general mathematical models of excavator [1-4], analysis of loading of large diameter axial bearings [5-7], the study of the influence factors on the bearing loads [8-11], the synthesis of the slewing platform drive mechanism [12] and the development of hybrid drives of the slewing platform drive mechanism which enables the energy recovery that occurs received february 25, 2019 / accepted may 05, 2019 corresponding author: vesna jovanović university of niš, faculty of mechanical engineering, serbia, aleksandra medvedeva 14, 18000 niš, serbia e-mail: vesna.jovanovic@masfak.ni.ac.rs 706 v. jovanović, d. janošević, j. pavlović when the platform slews [13-15]. within the research conducted by jovanović [12], general mathematical models and programs have been developed for determining the spectra of equivalent loads of the axial bearing of the slewing platform drive mechanisms in the entire working area of hydraulic excavators. by comparing the load spectra with the allowed loads, the size of the axial bearing can be reliably chosen. this paper presents the research results that show in which working area i.e. the position of the kinematic chain of the excavator, and under which conditions of operation the resistance of digging, the equivalent loads occur in the load bearing spectrum, for the choice of the size of the bearing of the slewing platform drive mechanism. during the research, a new method of analyzing the loading of the axial bearing of the slewing platform of hydraulic excavators was used, using the load spectra determined for the entire working area of the excavator according to the boundary forces that can overcome the drive mechanisms of the excavator and to the boundary forces that allow the stability of the excavator. 2. mathematical model the research uses the general mathematical model of the excavator developed according to the physical model of the excavator with the support and movement mechanism l1 (fig. 1a), slewing platform l2 and the manipulator with a boom l3, stick l4 and shovel bucket l5. the support and movement mechanism and the slewing platform of the excavator are linked with a slewing joint in the form of an axial bearing. the slewing platform drive mechanism consists of a hydraulic motor with a planetary gearbox with a gear on the output shaft coupled with a toothed axial bearing. the actuators of the drive mechanisms of the excavator manipulator are two-way hydraulic cylinders of boom c3, stick c4 and bucket c5. the numerical and experimental research [12] of the load of the axial bearing of the slewing platform drive mechanism have shown that: a) the greatest axial bearing loads in the digging operation of the excavator and b) the dynamic loads of the bearing, caused by the moving of the kinematic chain of the excavator in relation to the static loads, are relatively small bearing in mind that the digging process takes place slowly. taking into account the results of the previous research on the loading of excavator axial bearing, a mathematical model of the excavator was developed with the following assumptions: 1) the support surface and the members of the kinematic chain of the excavator are modeled as rigid bodies; 2) during the manipulative task the excavator is subjected to external (technological) forces digging resistance w and the gravitational force (weight) of: the members of the kinematic chain, the members of the drive system and the land grabbed by the bucket of the excavator; 3) during the digging operation, the kinematic chain of the excavator with a planar configuration is viewed as an open configuration chain whose last member – the bucket, is subjected to the possible digging resistance at the cutting edge of the bucket [16]. in analyzing the loading of the axial bearing of the slewing platform drive mechanism, the parameters of the kinematic chain members and the parameters of the drive mechanisms of the excavator are known. 2.1. the space of the excavator model the space of the excavator model is determined by absolute coordinate system oxyz with unit vectors k,j,i  along coordinate axes ox, oy and oz. the excavator support analysis of the influence of the digging position on the loading of the slewing platform bearing … 707 surface lies in the horizontal axis of the oxz absolute coordinate system, while vertical axis oy of the same system falls on the axis of the support and movement member-slewing member kinematic pair when the excavator is positioned on the horizontal surface. the internal (generalized) coordinates of the mathematical model of the kinematic chain of the excavator are angles θi (fig. 1a) of the relative position of member li in relation to previous member li-1 at rotation around the axis of joint oi. fig. 1 axial bearing load: a) mathematical model of excavator with a loading manipulator, b) load of slewing platform drive mechanism by changing length ci of the hydraulic cylinders of the drive mechanisms in the interval of limit values ci = [cip, cck], generalized coordinates θi are changed in interval θi = [θip, θik], where θip is the initial and θik is the final angle of the relative position of member li to previous member li-1. the relative angle of movement θio of member li in relation to previous member li-1 is expressed by the difference: a) b) 708 v. jovanović, d. janošević, j. pavlović ipikio  −= (1) the position of member li in relation to the horizontal oxz plane of the absolute coordinate system is determined by the angle: 5,4,3i i 3i ii == =  (2) vectors: ir  the center of joint oi, tir  the center of masses of the kinematic chain members and wr  the center of the cutting edge of the bucket, are defined, in the absolute coordinate system, by the equations: 5,4,3,2ia 1i 1i iioi == − = sr  (3) where: aio – the transfer matrix used to transfer the vector quantities from local coordinate system oi xi yi zi of member li to absolute coordinate system oxyz 2.2. axial bearing load according to the defined mathematical model, force f2 and moment m2 of the load of the axial bearing of the excavator slewing platform drive mechanism are determined by the equations [12, 16]: m2c 5 3i ci 5 2i i02 mm wfggf +++=  == (4) ( )( ) ( )( ) ( ) m2w 5 3i 2ctici 5 2i 2tii02 mm wrrgrrgrrm −+−+−=  == (5) where: mi member mass, mci mass of the actuators of the manipulator drive mechanisms, fc2 reaction force of the slewing platform, and wm possible force of digging resistance. the reaction force of the slewing platform drive mechanism with one slewing drive is determined by the equation (fig. 1b): ( ) ( ) kifc2 2222 2425 y2 cossin dd m2  +−+ − = (6) where: m2y moment of the slewing platform drive, α2 position angle of the platform drive, θ2 platform slewing angle, d25 diameter of the toothed ring of the axial bearing of the platform, d24 gear diameter on the drive output shaft. the resulting force of possible digging resistance wm (fig. 1a) acts on the cutting edge of the bucket and comprises component wxym normal to the cutting edge and lateral component wzm collinear with the cutting edge of the bucket: 2 zm 2 xymm ww +=w (7) whose direction is determined by the unit vector: analysis of the influence of the digging position on the loading of the slewing platform bearing … 709 ( ) ( )k jiw 2wm2wmw wwm2w2wwmm cossinsincos sinsinsincoscoscosort   ++ ++−= cos cos (8) where: φwm angle between the direction of the action of the possible force of digging resistance wm and the force of the digging resistance normal to the cutting edge of bucket wxym, determined by the equation: xym zm wm w w arctg= (9) the intensity of the possible force of the digging resistance applied to the cutting edge of the bucket is   m5m4m3pmsmxym ,w,w,w,wwminw = (10) where: wsm, wsm boundary forces of the digging resistance that limit the stability of the excavator, w3m, w4m, w5m boundary forces of the digging resistance that can be overcome by the drive mechanisms of the manipulator boom, stick, and bucket. the direction of the normal component of the possible force of the digging resistance wxym in relation to the horizontal oxz plane of the absolute coordinate system is defined by the angle: w5w  += (11) where: θw angle of the digging resistance in relation to the positive o5x5 axis of the local coordinate system of bucket l5. the direction of the normal component of digging resistance wxym in the absolute coordinate system is determined by the unit vector: kjiw 2wwxym sinsincosort  cos 2 ++= (12) the boundary force of digging resistance wsm that is limited by the static stability of the excavator is determined, depending on the position of the kinematic chain of the excavator and оrtwxym, from the equilibrium conditions for one of the possible rotary joints о11, о12 between the support and movement member and the support surface, whose axes represent the potential excavator rollover lines (fig. 1a) [17]:           − + − − =    − + − − = = 12w11w 11w12w 1xym12w 12o 12sm 12w11w 11w12w 1xym11w 11o 11sm sm π, 0y π, 0y , )ort)(( m w π, 0y π, 0y , )ort)(( m w w     ewrr ewrr (13) where: msm11, msm12 gravitational moments for longitudinal x-x or transverse z-z excavator rollover line of the excavator, r11,r12 vectors of the position of the center of the appropriate first rotary joint о11,о12, e1 unit vector of the first rotary joint о11,о12, φ11, φ12 angles of the position center of the cutting edge of the bucket in relation to the corresponding rotary joints defined by the equations:         − − = 11w 11w 11 )( arccos rr irr          − − = 12w 12w 12 )( arccos rr irr  (14) 710 v. jovanović, d. janošević, j. pavlović the gravity moments of the kinematic chain of the excavator for calculating possible longitudinal x-x or transverse z-z excavator rollover line are: ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )        −−−−−−= −−−−−−= =    = = = = = = = = 5k 3k 1125tz112ctkck 5k 1k 112tkk12o 5k 1k 5k 3k 1115tz111ctkck111tkk11o 1o gmmg mgm gmmg mgm m ejrrejrrejrr ejrrejrrejrr (15) where: rctk vector of the position of the center mass of the hydraulic cylinder, mz mass of the material scooped with the bucket. depending on the bucket position, the weight of material is: 270 0 270 c v 5 o o 5z       = o o 5 z 90 90os m   (16) where: ρz density of the material, v volume of the bucket. from the non-sliding conditions of the excavator, i.e. the support and movement mechanism, in relation to the support surface during the digging operation and due to the action of the digging resistance force, the boundary of the digging resistance force wpm was determined according to the size of the adhesion force occurring between the support and movement mechanism and the support surface: cos μmg w w p pm   = (17) where: m mass of the excavator, μp coefficient of adherence of the excavator movement mechanism to the support surface. the boundary forces of digging resistance wim that can be overcome by the drive mechanisms of manipulator boom (i=3) and bucket (i=5), for the known оrtwm and the position of the kinematic chain of the excavator, during the operation of the maximum torques of mechanism мpimax, are determined from the equilibrium conditions for axial joints оi (i=3,5) of the manipulator: ( )( ) 5,3i ort mm w ixymiw oimaxpi im = − −− = ewrr (18) where: moi total moment of the gravitational forces of the kinematic chain members and the drive mechanisms of the excavator and the weight of the material scooped with the bucket for the individual axes of joints оi. the total moment of the excavator gravitational forces is determined by a set of moments: 5,4,3 i mmmm oizoicoigoi =++= (19) where: moig moment of the gravitational forces of the members of the excavator kinematic chain for the individual axes of the joints oi, moic moment of the gravitational forces of the members of the excavator drive mechanisms, moiz moment of the weight of the material for the individual axes of joints oi . the moment of the gravitational analysis of the influence of the digging position on the loading of the slewing platform bearing … 711 forces of the members of the excavator kinematic chain for individual axes of joints oi, is determined by the equation: ,54,3 i ))((mgm i 5k 1k itkkoig =−−=  = = ejrr (20) where: ei unit vector of joint axis oi. the moments of the gravitational forces of the members of the excavator drive mechanisms for individual axes of joints oi are determined by the following equations [18, 19]: ( )( ) ( )( ) ( )( ) ( )( ) ( )( )         =−−= =−−−−= =−−−−= =  = = 5 i 2 m gm 4 i ejrr 2 mn g 2 mn gm 3 i ejrrmng 2 mn gm m 555a 5c 5oc 445a 5c5c 444a 4c4c 4oc 5k 4k 33ctkckck333a 3c3c 3oc oic ejrr ejrr ejrr (21) where: ra3, ra4, ra5 coordinates of the joints in which the hydraulic cylinders are linked to the members of the kinematic chain (fig. 2). the moment of the material weight for the individual axes of joints oi is determined by the equation: ( )( ) 5,4,3 i gmm ir5tzoiz =−−= ejrr (22) where: mz mass of the material scooped with the bucket, assuming that the center of mass coincides with the bucket mass center. the maximum drive moments of manipulator mechanisms for both directions in operation (upon piston pushing and piston pulling in the hydraulic cylinder): fig. 2 models of drive mechanisms of the excavator with a loading manipulator 712 v. jovanović, d. janošević, j. pavlović ( ) ( ) ( )        = − = == = 0θ,0θ,0θ 5,4,3i p 4 πdd nrθsignm 0θ,0θ,0θ 5,4,3i p 4 πd nrθsignm m 543max 2 2i 2 1i ciciimax2pi 543max 2 1i ciciimax1pi maxpi   (23) where: rci transmission function of the drive mechanism of member li, i  angular velocity of the active member of the kinematic pair mechanism around the axis of joint оi, pmаx maximum pressure of the hydrostatic excavator system. the transmission function of the drive moment of the boom, stick and bucket mechanisms for the axis of joint о3 , о4 , о5 (fig. 2): ) ba2 cab sin(arccos c b asin c b ar 33 2 3 2 3 2 3 3 3 33 3 3 33c −+ ==  , (24) ) ab2 cab arccossin( c b asin c b ar 44 2 4 2 4 2 4 4 4 44 4 4 44c  −+ −==  , (25) ) ca2 sca sin(arccosasinar 55 2 54 2 5 2 5 5555c −+ ==  (26) where: )arccoscos(sb2sbs 4 44 445 2 4 2 554 a ia   −−+=  ai,bi,ci (i=3,4,5) intensity of the position vector joints in which the hydraulic cylinder mechanisms are linked to the kinematic chain members and the length of the hydraulic cylinders, s4 kinematic length of the stick. the boundary force of digging resistance w4m that can be overcome by the drive mechanism of the manipulator stick, for the known оrtwxym and the position of the excavator kinematic chain, during the operation of maximum drive moment мp4max, is determined from the condition of equilibrium for the axis of joints о4 and о5 of the manipulator because the stick and bucket mechanisms are dependent mechanisms (fig. 2): ( )( ) 04 =+++− max4p4o54c54cxym4wm4 mmrfortw ewrr (27) where: rc54 transmission function of the drive moment of the bucket mechanism for the axis of joint о4, fc54 force in the hydraulic cylinder of the bucket subjected to boundary resistance w4m. the transmission function of the drive moment of the bucket mechanism for the axis of joint о4 is determined by equation [20]: 54554c m 54c sinbri == (28) where: 554 2 5 2 5 2 54 554 2 4 2 5 2 54 54 cs2 acs arccos bs2 sbs arccos −+ + −+ = (29) from the condition of equilibrium for joint о5 when the boundary force of digging resistance w4m acts: ( )( ) 05 =++− 5o5c54cxym5wm4 mrfortw ewrr (30) analysis of the influence of the digging position on the loading of the slewing platform bearing … 713 the force in the bucket hydraulic cylinder is determined: ( )( ) 5c 5oxym5wm4 54c r mortw f −−− = 5ewrr (31) by substituting force fc54 in eq. (30), boundary digging resistance w4m was obtained: ( )( ) ( )( ) 4 5xym5w54cxym4w5c 5o54c4omax4p5c m4 ortrortr mr)mm(r w ewrrewrr −−− ++− = (32) depending on the moment of rotation of the tracked support and movement mechanism in relation to the surface and the operation of the horizontal component of the normal force of the possible digging resistance, the component of the possible force of the digging resistance, collinear with the cutting edge of the bucket, is determined by the following equation: w wxmozm x 1 zw 4 lmg μw       +  = (33) where: μo coefficient of the turning resistance of the tracks against the excavator support surface, l length of the continuous tracks footprint, wxm component of the possible digging resistance, xw, zw coordinates of the action of the possible digging resistance on the cutting edge of the bucket. the size of axial bearings of the excavator slewing platform is determined on the basis of the equivalent loads defined by the bearing manufacturers in the form (fig. 1b) [21]: ▪ for equivalent force: sr2a2es f)fbfa(f += (34) ▪ for equivalent moment: sr2es fmcm = (35) where: f2a, f2r axial and radial force of the axial bearing loading, m2r moment of the axial bearing loading, a, b, c coefficients depending on the type of bearing, the type and size of the machine and its working conditions, fs factor of static safety. 3. analysis the analysis of the influence of the position and digging resistance in the entire working area on the change in the equivalent load of the axial bearing of the excavator slewing platform was carried out on an excavator of the weight of m =100000 kg and with the loading manipulator bucket of v = 6,5m3 in volume. the analysis used the hodographs of boundary and possible digging forces and the spectra of axial bearing loads of the slewing platform drive mechanism defined for certain segments of the entire working area of the excavator. 3.1. analysis of the impact of possible forces of digging resistance using the developed program based on the defined mathematical model of the excavator, the hwi (fig.3) and possible (hwm) resistance forces for two different positions of the excavator kinematic chain are defined: φ3=21,51°, φ4=-99,74°, φ5=-91,83° (fig. 3a), φ3=9,58°, φ4=-54,52°, φ5=-61,83° (fig. 3b). 714 v. jovanović, d. janošević, j. pavlović for a specified position of the kinematic chain, the hodograph of the boundary resistance (hwim) presents, in relation to the center of the cutting edge of bucket ow, a line perpendicular to the vector of the boundary resistance (wim min), of minimal value determined from the operation of the driving mechanism in both directions [22]. the hodograph of the potential digging resistance, in relation to the center of the cutting edge of bucket ow, is a polygon bounded by the hodographs of the boundary digging resistances (forces) limited by the excavator stability and the hodographs of the boundary digging resistances (forces) limited by the manipulator drive mechanisms. normal n-n drawn on the current speed of digging v, divides the hodograph of the possible digging resistance (force) into: a) the zone with a possible natural direction of action corresponding to the digging technology, and b) the zone with a direction of action inappropriate to the digging technology. for the two different positions of the excavator kinematic chain, the comparative analysis shows (fig. 3a, b and table 1) that the hodographs of the digging resistance forces vary considerably. for different boom and stick positions, the same bucket position (φ5 =-61.83°) and the same direction of the digging resistance force (φw =238.17°), which corresponds to the excavator digging technology with the shovel bucket, the intensity of the possible digging resistance in the first position is equal to the boundary force of the digging resistance that can be overcome by the bucket mechanism (wm = w5m). in the second position, the possible digging resistance is considerably lower in relation to the first position and is limited by the stability of the excavator (wm = wsm). fig. 3 hodographs of possible digging forces (resistances) for two different positions of the excavator kinematic chain there is also a difference in the size of the equivalent loads (fes, mes) of the axial bearing due to the difference in the position of the excavator kinematic chain and the difference in the possible forces of digging resistance. the obtained results point to a complex problem of determining the load of the axial bearing of the slewing platform drive mechanism since it has many different positions of the kinematic chain during operation throughout the entire working area; moreover, there are many different possible forces of digging resistance in every position. analysis of the influence of the digging position on the loading of the slewing platform bearing … 715 table 1 the position and load parameters of the excavator valid for the choice of axial bearing name and symbol of quantities dimension value position i position ii horizontal reach, xw m 5,30 8,09 vertical reach, yw m -0,64 -0,99 position angle of boom, φ3 º 21,51 9,58 position angle of stick, φ4 º -99,74 -54,52 position angle of bucket, φ5 º -61,83 -61,83 angle of action force of digging resistance, φw º 238,17 238,17 possible force of digging resistance, wm kn 823,24 326.889 boundary digging resistance force allowed by stability, wsm kn 841,22 326.889 boundary digging resistance force of boom mechanism, w3m kn 835,99 497,46 boundary digging resistance force of stick mechanism, w4m kn 928,42 538,30 boundary digging resistance force of bucket mechanism, w5m kn 823,24 641,32 equivalent force, fes kn 3325,76 1934,83 equivalent moment, m es knm 6815,27 3916,97 3.2. analysis of the impact of the digging position in order to analyze the influence of the change in the position of the kinematic chain of the excavator, i.e. the change in the digging position throughout the entire working area, on the change in the equivalent loads of the slewing platform axial bearing, the axial bearing spectrum load was used. the spectrum of the axial bearing equivalent loads was determined using the developed program by spatial simulation by changing the position of each member of the manipulator kinematic chain, in its range of movement for 60000 different positions of the manipulator kinematic chain, i.e. possible digging positions in the entire working area of the excavator of the weight of 100000 kg and a loading manipulator with the bucket of 6,5m3 in volume. the obtained spectrum of axial bearing loads is shown in the diagram (fig. 4a) (al3,…,al6), which presents the dependence of the allowed values of equivalent moments of mes and equivalent forces fes of axial bearing loads. in order to find the position of the kinematic chain in which the equivalent loads arise for the choice of the axial bearing size, the entire working area of the excavator is divided into four segment fields (fig.4b) and the equivalent loads of the axial bearing of the slewing platform mechanism are determined for each segment field. the spectra of the equivalent load of axial bearings for each segment field of operation are shown, in appropriate colors, in the bearing diagrams of the available axial bearings. a comparison of the spectra shows that the bearing loads are very different in segment fields. according to the equivalent loads in the segment fields, the slewing platform mechanism corresponds to the different sizes of the available axial bearings. the position of the kinematic chain of the excavator in which the corresponding equivalent loads for the choice of the axial bearing size are found, is determined using the developed program, on the condition that the point, from the constellation of the spectra of the equivalent bearing load, is the least removed from the allowed load bearing capacity of selected bearing size al6. in the isolated position of the excavator kinematic chain, when the corresponding bearing loads are applied, according to the hodograph of the possible forces of digging resistance (fig. 3), the boundary forces of digging resistance have approximately the same values (table 1), and the force of the possible digging resistance has the direction which belongs to the zone with the possible natural action directions that correspond to the digging technology. 716 v. jovanović, d. janošević, j. pavlović the analysis shows that the set of equivalent loads, which are close to the equivalent bearing loads, corresponds to the positions (fig. 4c) of the excavator kinematic chain in the zone of the working area when the manipulator drive mechanisms have such positions and coordinated interaction that they can overcome the largest digging resistance forces allowed by the excavator stability. fig. 4 analysis of axial bearing load of slewing platform drive depending on the position of the loading manipulator of excavator: a) equivalent loads in the entire working area, b) segmental fields of the working area of the excavator, and c) area of relevant loads analysis of the influence of the digging position on the loading of the slewing platform bearing … 717 4. conclusion the choice of the axial bearing size in the slewing platform drive mechanism in hydraulic excavators is very complex due to a large number of different impacts on the bearing load. one of the more important impacts is the digging position in the excavator’s working area. hydraulic excavators are distinguished by the ability to perform various functions with various tools (bucket, grapple, hook, grabber). it is characteristic that in carrying out any possible functions in their working area, hydraulic excavators have many different positions and working conditions. using the developed mathematical model and program, the paper analyzes the effects of the digging position on the load of the axial bearing of the slewing platform drive mechanism in an excavator equipped with a shovel bucket. the analysis uses the hodographs of the boundary forces of digging resistance and the axial bearing load spectra. the hodographs of the digging resistance forces show that the possible forces of digging resistance, as a primary load of the axial bearing, are very variable and depend on the digging position or the position of the excavator kinematic chain. using the axial bearing load spectrum, the working area fields are determined – i.e. the position of the excavator kinematic chain, and the vectors and the hodographs of the digging resistance forces, as well as the possible digging resistance vectors, which yield the equivalent loads relevant to the choice of the axial bearing of the slewing platform drive mechanism in hydraulic excavators. acknowledgements: this paper presents the results of the research conducted within the project "research and development of new generation machine systems in the function of the technological development of serbia" funded by the faculty of mechanical engineering, university of niš, serbia. references 1. jiaqi, x., hwan-sik, y., 2016, a review on mechanical and hydraulic system modeling of excavator manipulator system, journal of construction engineering, 2016, pp. 1-11. 2. mitrev, r., janošević, d., marinković, d., 2017, dynamical modelling of hydraulic excavator considered as a multibody system, tehnički vjesnik, 24, pp. 831-836. 3. vujić, d., lazarević, o., batinić, v., 2017, development of dynamic-mathematical model of hydraulic excavator, journal of central south university, 24, pp. 2010-2018. 4. mitrev, r., marinkovic, d., 2019, numerical study of the hydraulic excavator overturning stability during performing lifting operations, advances in mechanical engineering, 11(5), doi: 10.1177/1687814019841779. 5. hua, w., peiyu, h., bitao, p., xuehai, g., 2017, a new computational model of large three-row roller slewing bearings using nonlinear springs, proc. institution of 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cu content and austempering parameters on the hardness and corrosion properties of austempered ductile iron (adi) ladislav vrsalović1, nikša čatipović2, senka gudić1, stjepan kožuh3 1university of split, faculty of chemistry and technology, split, croatia 2university of split, faculty of electrical engineering, mechanical engineering and naval architecture, split, croatia 3university of zagreb, faculty of metallurgy, sisak, croatia orcid ids: ladislav vrsalović https://orcid.org/0000-0001-6111-5475 nikša čatipović https://orcid.org/0000-0002-5031-0766 senka gudić https://orcid.org/0000-0001-5390-5215 stjepan kožuh https://orcid.org/0000-0003-2536-2713 abstract. the effect of copper content (0.031 wt.% cu, 0.32 wt.% cu, 0.51 wt.% cu and 0.91 wt.% cu) on the hardness and corrosion properties of adi was investigated. samples austenitization were carried out at 850°c for 60 min followed by its austempering at temperatures from 250°c to 420°c for different time (30 to 60 min) in 50% (kno3 + nano3) salt bath. it was concluded that hardness rises with copper content but decreases with higher austempering temperatures and times. the corrosion properties of the samples with minimum and maximum cu content were investigated by electrochemical methods in 0.5 m nacl solution. samples with a higher copper content have shown higher values of polarization resistance (rp) and lower values of corrosion current (icorr). after polarization measurements, corroded surfaces were analyzed with sem/eds analysis. key words: copper content, adi, hardness, corrosion properties, sem, eds 1. introduction ductile iron (di) is a high carbon cast ferrous material with a composition similar to grey iron, in which carbon is present in the form of spheroids graphite instead of conventional flake or plate form, which is characteristics of grey iron [1, 2]. it is significantly used in different industrial applications, including construction, mining, agriculture, automotive parts and machines, tubes, etc. [3-5]. the reason for its common use lies in its advantages over steel castings and grey iron castings. appropriate heat treatment and alloying leads to improve received: january 06, 2022 / accepted april 03, 2022 corresponding author: nikša čatipović faculty of electrical engineering, mechanical engineering and naval architecture, ruđera boškovića 32, 21100 split, croatia e-mail:ncatipov@fesb.hr https://orcid.org/0000-0001-6111-5475 https://orcid.org/0000-0002-5031-0766 https://orcid.org/0000-0001-5390-5215 https://orcid.org/0000-0003-2536-2713 34 l. vrsalović, n. čatipović, s. gudić, s. kožuh the strength, hardness, and wear resistance of di. austempering treatment can achieve the transformation of ductile iron into alloy with the microstructure of acicular ferrite and retained austenite [6]. austempered ductile iron (adi) was commercialized during the 1970s. its application steadily grew due to its high tensile strength, good fatigue resistance under dynamic loading conditions, abrasion resistance and toughness [5-9]. the benefits of using adi material are reflected in its good mechanical properties, such as good vibration properties, lighter weight than other structural materials, and uniform castability. investigations of mechanical properties of di and adi have been in the focus of many published studies, but the related studies on its corrosion properties are significantly less published. corrosion susceptibility of di lies in the fact that graphite is nobler than the iron matrix, and it acts as a cathode when the di surface is exposed to the corrosive environment [2]. the presence of alloying elements, like copper, nickel and molybdenum, leads to microstructural changes in di, which affect its mechanical and electrochemical behavior [6, 10-12]. nickel addition increased pearlite content, decreased the nodule count, and reduced the graphitic corrosion of ductile iron [10]. an increase in molybdenum content leads to a rise in the quantity of retained austenite and carbide. at the same time, the acicular ferrite becomes finer, which leads to increased wear and corrosion resistance [6]. higher silicon content in this material increased its susceptibility to localized corrosion in chloride media [13]. the increase in austenite temperature and processing time decreases its corrosion rate [13,14]. also, surface treatments, like laser surface alloying, plasma oxidation treatment and plasma nitriding increase the corrosion resistance of ductile iron [15,16]. this paper deals with investigating the influence of cu content on hardness and corrosion properties of the di and adi in 0.5 m nacl solution. the as-cast specimens of di were first austenitized at 850°c for 60 min and then austempered in 50% nano3 50% kno3 salt bath at temperatures from 250°c to 420°c for 30 to 60 min to produce adi. 2. experimental procedure the chemical composition of the ductile iron samples is shown in table 1. table 1 chemical composition of ductile iron samples alloy c(%) si(%) mn(%) cu(%) s(%) p(%) cr(%) v(%) 1. 3.63 2.61 0.135 0.031 0.0035 0.022 0.005 0.004 ni(%) mo(%) al(%) ti(%) sn(%) w(%) mg(%) 0.085 0.003 0.017 0.013 0.033 0.017 0.041 alloy c(%) si(%) mn(%) cu(%) s(%) p(%) cr(%) v(%) 2. 3.63 2.61 0.135 0.32 0.0035 0.022 0.005 0.004 ni(%) mo(%) al(%) ti(%) sn(%) w(%) mg(%) 0.085 0.003 0.017 0.013 0.033 0.017 0.041 alloy c(%) si(%) mn(%) cu(%) s(%) p(%) cr(%) v(%) 3. 3.63 2.61 0.135 0.51 0.0035 0.022 0.005 0.004 ni(%) mo(%) al(%) ti(%) sn(%) w(%) mg(%) 0.085 0.003 0.017 0.013 0.033 0.017 0.041 alloy c(%) si(%) mn(%) cu(%) s(%) p(%) cr(%) v(%) 4. 3.63 2.61 0.135 0.91 0.0035 0.022 0.005 0.004 ni(%) mo(%) al(%) ti(%) sn(%) w(%) mg(%) 0.085 0.003 0.017 0.013 0.033 0.017 0.041 beneficial effect of cu contentand austempering parameters on hardness and corrosion properties... 35 the first step was to heat treat ductile iron samples to produce adisamples. all samples were austenitized at the same temperature of 850°c for the same time of 60 min. after austenitization, austempering was performed according to the experimental plan, table 2. the austempering temperature was changed between 250°c and 420°c, and austempering time was varied between 30 min and 60 min. planning of experiments in respect to input parameters was performed by design expert software. the cubic model was selected for response surface method (rsm) study type with d-optimal initial design [17]. table 2 experimental plan according to design expert software sample id wt. % cu austempering temperature, [°c] austempering time, [min] 10 0.32 % 335 45 11 420 47 12 250 42 13 320 60 14 320 60 15 349 30 16 349 30 20 0.91 % 420 60 21 420 30 22 420 60 23 299 30 24 250 51 25 250 51 30 0.031 % 420 30 31 335 60 32 335 45 33 255 36 34 420 60 35 250 60 40 0.51 % 420 60 41 335 39 42 420 60 43 250 30 44 250 60 45 420 30 vickers hardness hv10 was measured at an applied load of 98 n using a universal hardness machine according to iso standard 6507-1:2005. the electrodes for the electrochemical measurements were made from di and adi cylindrical samples φ = 10 x 8 mm, soldered on insulated copper wire and then protected by two-component epoxy resin, leaving a surface area of 0.5 cm2 to contact the solution. investigations have been performed by open circuit measurements (eoc) in time of 60 min, followed by electrochemical impedance measurements (eis) at eoc, from 50 khz to 30 mhz frequency range, with 5 points per decade and a.c. sinusoidal potential perturbation (±10 mv). after eis measurements, the linear polarization method was performed in the potential region of ±20 mv around eoc, with the scanning rate (s.r.) of 0.2 mv s−1, afterwards potentiodynamic polarization method in the potential region of −250 mv to 600 mv vs eoc, with the s.r. = 0.5 mv s−1. all electrochemical investigations were performed in triplicate. a 36 l. vrsalović, n. čatipović, s. gudić, s. kožuh potentiostat/galvanostat (eg&g model 273a, usa) with eg&g m5210 lock-in amplifier were used. measurements were performed in 0.5 m nacl electrolyte with pt-sheet auxiliary electrode and saturated calomel electrode (sce) as the reference electrode. metkonforcipol 1v grinder-polisher was used for mechanical treatment of electrodes with wet sic emery papers (from 400 to 1500) and polishing with al2o3 polishing suspension (0.05 m) up to mirror finish. the electrode was then degreased in ethanol, washed in deionized water using asonic ultrasonic cleaner, and then immersed in the electrolyte solution. detailed surface analysis was performed with scanning electron microscope tescan vega 5136 mm paired with energy dispersive spectroscopy microscopy (sem/eds). 3. results and discussion 3.1. hardness measurement average hardness values were calculated from three different measurements. results are shown in table 3. table 3 hardness measurements of adi samples wt. % cu sample id hardness hv10 measurement 1 measurement 2 measurement 3 average 0.32 % 10 279 268 272 273 11 266 258 254 259 12 455 421 433 436 13 330 360 317 336 14 294 294 330 306 15 264 256 258 259 16 279 232 228 246 0.91 % 20 270 306 312 296 21 266 268 274 269 22 266 274 287 276 23 397 429 401 409 24 327 351 351 343 25 493 498 493 495 0.031 % 30 245 240 254 246 31 333 330 325 329 32 317 299 297 304 33 238 232 233 234 34 272 274 268 271 35 330 333 325 329 0.51 % 40 272 272 266 270 41 339 351 339 343 42 249 251 247 249 43 446 488 483 472 44 446 442 455 448 45 264 274 260 266 fig. 1 shows the dependence of hardness on austempering temperature and time for each adi alloy. the hardness decreases with increasing austempering temperature and time. higher hardness at lower austempering temperatures and times relates to smaller amounts of carbon beneficial effect of cu contentand austempering parameters on hardness and corrosion properties... 37 enriched retained austenite and a larger amount of carbides, mainly silicon carbides that form simultaneously as ausferrite. increasing of austempering temperatures and times, increases the volume fraction of retained austenite, which leads to a decrease in hardness, as seen in fig. 1. also, it is clear that hardness rises with the increase of copper content in the specimens. this is because, with larger amounts of copper in ductile iron samples, there is more perlite in starting microstructure which promotes carbide formation. fig. 1 dependence of vickers hardness hv10 on austempering temperature and time for each adi alloy the rsm analysis of measured hardness values for used experimental parameters yields an equation for each adi alloy: 0.031 % cu: hv = 534.8 − 0.81154 ∙ 𝑇𝑎 + 0.48636 ∙ 𝜏𝑎 (1) 0.32 % cu: hv = 552.02917 − 0.81154 ∙ 𝑇𝑎 + 0.48636 ∙ 𝜏𝑎 (2) 0.51 % cu: hv = 602.07953 − 0.81154 ∙ 𝑇𝑎 + 0.48636 ∙ 𝜏𝑎 (3) 0.91 % cu: hv = 603.64003 − 0.81154 ∙ 𝑇𝑎 + 0.48636 ∙ 𝜏𝑎 (4) the correlation coefficient of this fit is r2 = 0.6670. 38 l. vrsalović, n. čatipović, s. gudić, s. kožuh 3.2. electrochemical measurements fig. 2 a) illustrates the eoc decay for di samples in 0.5 mnacl solution while fig.2 b) showed changes of eoc for adi with the lowest and the highest amount of cu. the electrode potentials for all samples changed to negative values after immersion into the electrolyte solution due to the adsorption of chloride ions on the electrode surface, which trigger the corrosion processes on the surface of the alloy. similar behavior for di and adi exposed to the chloride solution was established in the literature [18,19]. this change is most noticeable in the first 20 minutes of electrode immersion, after which potential changes with time become less pronounced and the stabilization of the open circuit potential values occurred. the final values of eoc of all investigated samples do not differ significantly (potential differences are less than 20 mv). fig. 2 open circuit potential changes for a) ductile iron samples and b) austempered ductile iron samples in 0.5m nacl solution at 20°c fig. 2 b) showed eoc measurements of adi samples with the highest and lowest cu content (0.91% and 0.031%), which have a similar trend of lowering values of eoc with time like di samples. slightly more negative values of eoc have been observed for the adi sample with the lowest percentage of cu, and the final potential difference was around 20 mv. adi samples showed more positive eoc values than ductile iron samples with the same cu composition, implying higher corrosion resistance than di samples. to obtain a physical image of the observed systems and explain the influence of cu content on corrosion of di and adi at eoc, impedance measurements have been performed, and the results are shown in figs. 3 and 4 in the form of the nyquist plot. only one capacitive loop was observed, which described the dielectric properties of corrosion product film on di surface 14,20,21. the capacitive loop diameter increases with the cu content of in alloy. beneficial effect of cu contentand austempering parameters on hardness and corrosion properties... 39 fig. 3 nyquist plots for ductile iron in 0.5 mol dm-3 nacl solution fig. 4 nyquist plots for austempered ductile iron in 0.5 mol dm-3 nacl solution since the analysis of impedance measurements showed that the capacitive loops are flattened, a constant phase element (cpe) was used instead of an "ideal" capacitor. the impedance of cpe (zcpe) is given by the expression 22: 𝑍𝐶𝑃𝐸 = [𝑄(𝑗𝜔)𝑛]−1 (5) where constant q accounts for a combination of properties related to both the surface and the electroactive species, jω is the complex variable for sinusoidal perturbation with ω = 2f and n is the exponent of cpe. exponential term n can assume values from −1 to +1. when the value of n is close to 1, the cpe behaves like an ideal capacitance, while n values relative to 0.5 indicate diffusion processes. consequently, the cpe represents a warburg diffusion component. furthermore, for n values close to 0, the cpe represents the resistance and inductance for n close to −1. the obtained spectra are accompanied by a simple equivalent circuit, shown in fig. 5, which consist of rel (electrolyte resistance), r (resistance) and q (capacitance) of the surface corrosion product film, and the calculated values are presented in table 4. according to the fitting results, the n values for q are about 0.77-0.87. hence, in the proposed 40 l. vrsalović, n. čatipović, s. gudić, s. kožuh equivalent circuit, q is a constant phase element (cpe) representing surface layer capacitances combined with diffusion processes. fig. 5 proposed equivalent circuits for modeling the impedance data the addition of cu increases the resistance and reduces the capacity which is connected to the better protective properties of the surface corrosion layer. exponential term n also increases, indicating a reduced diffusion through the surface layer due to the increase in its compactness. the same effect was observed in the heat treatment of specific di samples, i.e., samples with the lowest and highest cu content (fig. 4 and table 4). table 4 impedance parameters for di and adi in 0.5 m nacl solution sample di adi % cu q1 × 103 (ω-1 sncm-2) n1 r1 (kω cm2) q2 × 103 (ω-1 sncm-2) n1 r1 (kω cm2) 0.031 1.23 0.77 0.95 1.01 0.82 1.59 0.32 1.15 0.79 1.42 0.51 1.11 0.81 1.54 0.91 0.93 0.84 2.11 0.76 0.87 2.84 according to parallel plate capacitor theory, the surface film capacity, c, is inversely proportional to its thickness, d. hence, the decrease of q, with the increase of cu content and heat treatment, leads to the corresponding increase in the thickness of the surface layer and additionally prevents iron corrosion at eoc. alloying elements have a significant impact on the properties of the surface films. thus, a better oxide film (with a more compact structure, higher resistance and thickness) will be formed on the alloy surface, which contains alloying elements that facilitate passivation, and this will be manifested with the increase in the corrosion resistance of the alloy, i.e., lower corrosion current and greater polarization resistance. according to literature, the 1 wt. % of copper addition effectively increased the retained austenite content, as copper can slow down stage ii of bainitic transformation to offer a wider processing window, i.e., copper addition creates an increased range of austempering durations to obtain additional content of the retained austenite [23,24]. furthermore, the addition of 1 wt. % cu leads to a homogeneous distribution in microstructure regardless of as-cast and adi, i.e., there was no copper-rich phase found in both the irons with copper addition [23]. the beneficial effect of cu in di and austempering heat treatment of sample on the corrosion resistance of the alloy (table 1 and 2) could be explained as the copper addition reduced the number of the graphite nodules and thus alleviated graphite corrosion, while the austempering heat treatment allowed the formation of the retained austenite, which has an action like a corrosion inhibitor [23].graphite corrosion is a form of selective corrosion in beneficial effect of cu contentand austempering parameters on hardness and corrosion properties... 41 which graphite act as a strong cathode to iron and electrolytically accelerated the attack on the surrounding matrix [10,23]. thus, experimental results indicated that the corrosive resistance of di dependent on the number of graphite nodules in the microstructure and the retained austenite content in its microstructure. therefore, it may be said that less nodule account and more retained austenite content could provide better corrosion resistance. this is also in accordance with the investigations of seikh and associates [20], which observed that corrosion rate decrease linearly with the increasing volume fraction of retained austenite. after eis investigations, linear and potentiodynamic polarization measurements were conducted on the samples to determinate corrosion parameters such as values of polarization resistance (rp), corrosion potentials (ecorr) and corrosion current densities (icorr). fig. 6 presents results of linear polarization measurements for the di samples a) and potentiodynamic polarization curves b) in 0.5 m nacl solution, while fig. 7 presents results for adi. fig. 6 results of linear polarization measurements a) and potentiodynamic polarization measurements b) for ductile iron samples in 0.5m nacl solution at 20°c fig. 7 results of linear polarization measurements a) and potentiodynamic polarization measurements b) for austempered ductile iron samples in 0.5m nacl solution at 20°c a) b) 42 l. vrsalović, n. čatipović, s. gudić, s. kožuh in figs. 6 a) and 7 a) it can be seen that the slope of the linear parts of the curves rises with the increase in the mass percentage of cu in di and adi samples. the values of the polarization resistance, determined from the slopes of these linear parts, according to equation (5), are showed in table 3. as rp value is inversely proportional to the material corrosion rate, a higher rp value corresponds to the higher corrosion resistance of alloy [23,26]. 𝑅𝑝 = ∆𝐸 ∆𝑖 (6) the results of potentiodynamic polarization measurements for di in 0.5 m nacl solution (fig. 6 b)) show that increasing the percentage of cu in the alloy leads to a decrease in anodic and cathodic current densities, which results in a lower value of corrosion current density. therefore, it can be concluded that increasing cu content in the alloy increases its corrosion resistance. adi sample with higher cu in the alloy shows lower anodic current densities and higher positive ecorr value than lower cu adi sample. the corrosion current density values for adi samples have a significantly smaller difference with the alloy's copper content than ductile iron (table 5). table 5 corrosion parameters for di and adi in 0.5 m nacl solution sample di adi % cu rp (kω cm2) icorr (a cm-2) ecorr (v) rp (kω cm2) icorr (a cm-2) ecorr (v) 0.031 0.771 21.78 -0.693 1.275 6.96 -0.715 0.32 1.042 15.18 -0.676 0.51 1.115 11.67 -0.717 0.91 1.351 9.27 -0.670 1.710 6.34 -0.672 the higher corrosion resistance of adi with a higher percentage of cu could be attributed to the combinatory effect of copper addition which reduced the nodule count to mitigate graphite corrosion and austempering high treatment, which led to the formation of the retained austenite, which has a protective role in adi structure [24]. 3.3. surface analysis to get more insight into corrosion processes on di and adi samples in nacl solution, sem analysis were performed on corroded surfaces. also, eds analysis was performed on selected di and adi samples. after polarization investigation and prior surface analysis, electrodes were cleaned ultrasonically in deionized water and dried in a desiccator. fig. 8 shows sem micrographs of the surface. there are notable differences in sem micrographs between corrode di and adi. the cathodic graphite nodules are clearly seen in figs. a) and b) and also severe corrosion of matrix around the nodules according to the reaction (7): fe2+ + 2cl− → fecl2 (7) the intensity of corrosion leads to graphite nodules peeled off, and the round cavities remained on the sample surfaces. along with graphite (galvanic corrosion), uniform corrosion is also present. in adi, graphite corrosion still occurs to a lesser extent, while uniform corrosion was minimal due to the occurrence of retained austenite in the matrix [7,14]. beneficial effect of cu contentand austempering parameters on hardness and corrosion properties... 43 fig. 8 sem micrographs of di with 0.031% cu a) and 0.91% cu b) and adi with 0.031% cu c) and 0.91% cu d) after polarization measurements in 0.5 mol dm-3 nacl solution eds analysis of the corroded di sample with 0.91% cu is presented in fig. 9 and table 6. 44 l. vrsalović, n. čatipović, s. gudić, s. kožuh fig. 9 eds analysis of di sample (3 different position): a) at the edge of graphite nodule, b) at the center of the graphite nodule, c) in the surrounding matrix table 6 eds analysis of corroded di sample for positions 2 and 3 position 2 position 3 element wt. % at. % wt. % at. % c 57.39 67.77 1.33 4.24 o 33.86 30.01 16.50 39.60 fe 6.42 1.63 72.69 49.97 cu 1.21 0.27 7.57 4.57 ni 0.69 0.17 0.67 0.44 si 0.17 0.09 0.47 0.64 mn 0.27 0.07 0.78 0.54 eds analysis was performed on the three positions on corroded di surfaces. position 1 was on the edge of the graphite nodule where dominant compounds were fe and cu oxides along with a small percentage of c, si, mn and ni. the different composition was found at the center of the graphite nodule (position 2), where dominant elements were c and o with the minor percentage of fe, cu, si, ni and mn. position 3 in the matrix around the nodule showed the dominant percentage of fe and o with a relatively high percentage of cu and c and a small percentage of other elements (si, mn and ni). eds analysis of the corroded adi sample with 0.91% cu is presented in fig. 10. and table 7. beneficial effect of cu contentand austempering parameters on hardness and corrosion properties... 45 fig. 10 eds analysis of adi sample (3 different positions): a) at the edge of graphite nodule, b) at the center of the graphite nodule, c) in the surrounding matrix table 7 eds analysis of corroded adi sample for positions 2 and 3 position 2 position 3 element wt. % at. % wt. % at. % fe 83.72 63.48 85.09 65.74 o 7.72 20.44 5.54 14.93 c 2.94 10.38 4.07 14.63 si 2.22 3.35 2.82 1.91 cu 2.13 1.42 1.15 1.77 ni 0.66 0.48 0.79 0.58 mn 0.60 0.46 0.55 0.43 eds analysis was performed on the three positions on corroded adi surfaces. position 1 was at the center of the nodule where the amount of c is dominant. all other elements are presented in low percentages. position 2 was in the matrix, where a dominant element is fe along with o and c, while the other elements are present in low percentage. the percentage of oxygen present in these two positions on the adi surface is significantly smaller than a similar position on the di surface. a similar composition in position 2 has been found in position 3, on the boundary between matrix and another partially deformed nodule. it should be noted that the content of carbon measurement with eds is not accurate due to a low atomic number of carbon and hydrocarbon molecules on the surface. 46 l. vrsalović, n. čatipović, s. gudić, s. kožuh 4. conclusion in the present study, the influence of cu content on hardness and corrosion properties of the di and adi in 0.5 m nacl solution was investigated. based on the experimentation and analyzing the results, the following conclusions can be drawn: ▪ the hardness decreases with increasing austempering temperature and time and rises with the increased copper content in the specimens. ▪ the beneficial effect of higher cu content in di and adi towards corrosion resistance has been proved by polarization and eis measurements. higher polarization resistance and lower corrosion current density values for a higher percentage of cu in di and adi and also increase in alloy surface resistance (r) and decrease in 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3, 2017, pp. 353 366 https://doi.org/10.22190/fume170928019b original scientific paper bicycle helmet design and the virtual validation of the impact, aerodynamics and production process udc 539.3+533.6+621.7 bojan boshevski, ile mircheski ss. cyril and methodius university in skopje (ukim), faculty of mechanical engineering, republic of macedonia abstract. this paper presents the development process of a bicycle helmet through individual research, creation, presentation and analysis of the results of the most important product development stages. the quality of the development and manufacturing process of the protective equipment for extreme sports is an imperative for a successful product and its flawless function. the design of the bicycle helmet is made following the rules of the design in order to create a well-founded and functional product. after creating design sketches, a virtual prototype was developed in "solidworks" using the required ergonomic dimensions. 3d printed model of the human head with adapted ergonomic dimensions and the designed bicycle helmet was developed in order to verify the applied ergonomic measures. the virtual model will be used as an input in the finite element analysis of the helmet impact test based on the en1078 standard and the aerodynamic simulations executed in "solidworks simulation and flow simulation", for verification of the impact and aerodynamic properties. virtual testing of aerodynamic features and the ability of the bicycle helmet to allow ventilation of the user's head indicate that the helmet performs its function in the desired way. also, the virtual prototype will be used for the production process simulation in "solidworks plastics" in order to analyze the production of the bicycle helmet. the polycarbonate helmet outer shell is subject to a number of simulations for the sake of analyzing the production process in order to obtain the desired characteristics of the polycarbonate outer shell and to avoid the disadvantages that occur in the manufacturing process. the main goal of this paper is to develop a safety bicycle helmet with improved ergonomic, validation of impact, aerodynamic characteristics and production process in order to produce a high quality product for mass use. key words: design, bicycle helmet, ergonomics, 3d printing, virtual testing received september 28, 2017 / accepted november 15, 2017 corresponding author: ile mircheski ss. cyril and methodius university, faculty of mechanical engineering, karpos ii bb, 1000 skopje, macedonia e-mail: ile.mircheski@mf.edu.mk 354 b. boshevski, i. mircheski 1. introduction both proper design and protective equipment production for extreme sports form the basis for the given purpose of this type of sports equipment, that is, the user’s protection and avoidance of injury with or without lasting consequences. the most important difference, if it comes to the point when the supplementary protective equipment needs to fulfill its purpose, is whether the user of the equipment benefits from its use in a given situation or the equipment does not fully and properly perform its function. the second most important factor in the use of protective equipment is the possibility for the user to make the most of the functions that are foreseen in the design and production. at the same time, the additional equipment should not cause a decrease in the physical performance of the user himself; on the contrary, it should enable an uninterrupted activity performance in addition to increasing the possibility to achieve a better desired performance. this paper synthesizes the designing process and the virtual production testing of this product in order to enable construction and optimization of those product’s key features that lead to a successful product. using the designers’ approach and sketching, the final version of the bike helmet is created and converted to a 3d model using software for virtual prototyping. simulations for testing and verification of the product’s impact and aerodynamic functioning in use are made in a virtual environment. after verification of the product’s aerodynamic features, another simulation is carried out; only this time the outcome is used for optimization of the helmet’s first layer, the outer shell, the surface that receives the first impact and is responsible for the overall reaction of the bicycle helmet after the impact. 2. literature review the most common injuries in cycling and sport activities are head ones. the protective safety equipment is a bicycle helmet. safety is one of the most important factors when it comes to protective helmets used in sports, industry, etc. the paper [1] aims at improving the protective characteristics of the bicycle helmets, namely, to improve especially energy absorption of the liner foam in the bicycle helmet. the study is focused on finding a replacement material for eps liner foam with improved energy absorbing characteristics. the impaxx energy absorbing foam is used; it presents a strong potential in overcoming such problems of eps foam. this study has carried out the finite element analysis of the helmet impact test using ls-dyna software. the studies [2, 3, 4] presented assessment of a helmet with dual layer liner based on the shock absorbing test. the fem is used in analyzing of helmet impact using ls-dyna software. the simulation of the helmeted head form drop test is implemented in order to determine a structure of the liner that reaches high effective protection and to design helmet models with single and dual liners. the paper [5] compares the results of three virtual drop tests of the protective equipment produced of three completely different types of plastic and the results of the stress and deformation values. the inspiration for writing the paper [5] comes from a large number of injuries induced in rugby matches. one of the most common injuries in this sport is the brain concussion, which occurs as a result of collision between two or more rugby players. the purpose of this research is to determine the influence of the velocity against the player’s protective helmet. in order to check the helmet’s performance, testing with a pneumatic linear impact system, rugby helmet and all the necessary parameters that make this testing bicycle helmet design and virtual validation of impact, aerodynamics and production process 355 reliable is conducted. the experiment has eighteen phases and in each new phase, a new rugby helmet is used and tested three times. the results obtained from the experiment are used as inputs in injury simulation of a finite elements human brain model. after the test, it was concluded that the collision velocity between players is a significant factor and that it is necessary to design a rugby helmet with a higher impact speed tolerance. the human head is one of the most vulnerable parts of our body and because of this, it is necessary to carry a protective helmet while riding a motorcycle. the purpose of the paper [6] is to create a more functional and protective design of a motorcycle helmet, with all the necessary characteristics of a supreme helmet with top quality, including proper head ventilation as one of the most important features. in order to properly construct this type of protective equipment, a database that contains information on the effect of the overall design on all the necessary features that the helmet should have, has been created. by using this database and determining the most significant subsystems and their separation to the smallest details which are linked and associated with the subsystems, few sketches for motorcycle designs have been created. using this method with separation of subsystems, then joining them together again, allows a creation of a design that meets all the necessary requirements. in order to improve the homologation process of protective sports helmets the paper [7] develops a new method for angled impact testing. besides the development of this testing method, the complete testing process, and an opportunity for making cheap, reliable, easy to use test equipment and the criteria by which one helmet would be given a certificate that approves its use are explained in this paper. the criteria that are necessary for optimizing the protection of the head with the help of helmet are created and improved. a head model created from finite elements is subject to virtual testing of impact in order to obtain the impact values in kpa and mj, that cause minimal or severe head injury. after the impact a testing is done, helmets are placed on the tested model of the human head and through a virtual simulation, an assessment of the ability to protect the head in relation to the previously set criteria for head injury is made, resulting in the possibility of creating a better protective equipment for the head [8]. there are many ways of discovering the tendency of a model towards turbulences. because of this, the paper [9] presents a comparative analysis for detection of the most appropriate model for the helmet turbulence analysis. in order to determine the air flow characteristics of a helmet, 2d virtual simulation with a simplified model of a human head wearing a helmet is conducted. the results of this simulation do not match with the numerical results obtained from the turbulence model. to find out where the problem arises the 3d simulation of a similar simplified model is made. with the obtained results from the 3d simulation, the turbulence model which gives the most reliable results regarding the experimentally obtained results is determined. in the papers [10, 11] are investigated the aerodynamic improvements of applying a truncated airfoil shape with a trailing edge modification to a helmet design. solidworks flow simulation is used to evaluate the aerodynamic forces. a common production helmet design is progressively truncated to determine the optimal truncation length and the effect of multiple trailing edge modification is tested. scale models of the final improved design and the production helmet were tested in the wind tunnel to verify the computational results. 356 b. boshevski, i. mircheski the goals in this paper are the design process, ergonomic analysis, the virtual validation of impact test, aerodynamic verification and production testing of the bicycle helmet in order to enable construction and optimization of the product’s key features which lead to a successful product. 3. conceptualization of the bicycle helmet the creation of a functional and well-founded product starts with design development through sketches shown in the next page, figs. 1, 2 and 3. this phase is very important for the product because it includes an elaborative design as well as functional characteristics which need to be compatible with the product’s aesthetics, i.e. to ensure that the proper product’s functioning is synthesized with the aesthetical value. it is vital to ensure balance between the design of the product and its functionality in this type of products. in fig. 1 is shown a sketch in perspective for the bicycle helmet and in fig. 2 is shown a sketch in multiple views. fig. 3 illustrates the placement of the ventilation openings on the helmet’s polycarbonate shell and the airflow in and around the helmet. in order to analyze the airflow inside and around the bike helmet, section 6.2 presents airflow verification. for this type of product, there should be a balance between the functional characteristics of the product and its design. the functional characteristics should not be neglected because of the design and vice versa. however, not focusing on the design could lead to an unreliable and poor quality product; therefore, creating a safe product to use is the most important task. 3d cad model and ergonomic verification of the helmet are shown on fig. 4. this prototype is made based on the previously created design sketches, made by using software for parametric modeling „solidworks“. 3d scanned model of a male head in real ergonomic dimensions, which meets the designing standards for this type of protective equipment is used as a base for 3d modeling of the designed bicycle helmet for mountain biking. the virtual 3d model is created by using an advanced surface modeling technique available in the 3d modeling software since this design cannot be created by means of simple modeling tools. fig. 1 product design sketch in perspective fig. 2 product design sketch in multiple views bicycle helmet design and virtual validation of impact, aerodynamics and production process 357 fig. 3 bicycle helmet air circulation sketch fig. 4 virtual model of 3d scanned head and designed bicycle helmet 4. applied ergonomics in the design and validation of the ergonomic properties with rapid prototyping on 3d printer the ergonomics is an important issue in the product designing process. the body sizes the smallest and biggest representative of the selected users’ population direct influence on the product's dimensions. with the use of necessary and exact ergonomic measures shown in fig. 5, the development process would result in a potentially safe product. the creation of the product which is safe to use depends not only on the ergonomics itself, but also on the other design and developmental processes. the ergonomic measures used for designing this protective bicycle helmet are the key measures of the men’s head in the range of 95 th percentile. this percentile is chosen because the product will be more universal in terms of covering a bigger group of users without changing the product’s safety qualities. as a starting point for this product design a 3d scanned male head in real scale, shown in fig. 6 is used. verification of the designed virtual model ergonomic properties is accomplished by creating a 3d printed model of the bike helmet, positioned on the previously 3d scanned male head, scaled down together to allow a flawless printing process. the completed 3d printed prototype of the bicycle helmet positioned on the male’s head, shown in fig. 7, represents the applied ergonomics and its accuracy. 5. the design of bicycle helmet and 3d modeling the aesthetic values of the bicycle helmet design are created and developed by using guidelines enclosed in functionalism. in this case, it is essential that the form follows the function. the basic inspiration for the design shape comes from the aerodynamic qualities of the rain drop, which represents a conjunction between naturalism and functionalism. perforations on the bicycle helmet are minimized, and because of their correct placement, they allow proper air circulation and reduce the risk of helmet’s shell weakening because of their position. 358 b. boshevski, i. mircheski fig. 5 applied ergonomic measures [12] fig. 6 3d scanned human head fig. 7 3d printed model bicycle helmet design and virtual validation of impact, aerodynamics and production process 359 the elements for fixing the straps, the plastic parts for joining the straps together and the parts of the strap buckle are minimalistic and in tone with the colors and the shape used for the design of the helmet’s shell. figs. 8, 9, 10 and 11 show the bicycle helmet design and represent the style that this design belongs to. fig. 8 bicycle helmet front view fig. 9 bicycle helmet side view fig. 10 bicycle helmet rear view fig. 11 bicycle helmet on the 3d scanned head model 6. virtual testing of the bicycle helmet during the product development process, the phase in which the product’s qualities are simulated and verified is unavoidable. this part of the development process is very important for the product and its progress. in this process phase it is possible to realize the characteristics of the product which are going to be kept, and those that will have to be modified in order to optimize the helmet’s performance and its manufacturing in terms of efficiency, effectiveness and economic justification. this paper contains examination and verification of the impact and aerodynamic features of the designed bicycle helmet, when the bicycle helmet is in an active (riding) position, a passive (riding) position and in a simulation of an optimized manufacturing process of the bicycle helmet shell made of polycarbonate. 360 b. boshevski, i. mircheski 6.1. finite element analysis of the helmet impact test the impact tests should be performed in order to confirm the bicycle helmet protection and satisfy the helmet safety standards. the basic bicycle helmet is composed of outer shell and liner foam. the outer shell protects the head from impact and injures. the outer shell is usually made of polycarbonate (pc) or acrylonitrile butadiene styrene (abs). in the case study, for outer shell is used pc material with the following material properties: young's modulus of 2440 mpa, density of 1400 kg/m3 and poisson's ratio of 0,4. expanded polystyrene (eps) foam is material with a good energy absorption characteristic which is used for bicycle helmet. the protective liner foam is made of eps foam. the thickness of the pc outer shell is 2,5 mm and the liner thickness in this case study is 25 mm. the material properties used in this case study for eps liner foam are: young's modulus of 40 mpa, density of 1100 kg/m3 and poisson's ratio of 0,1. the flat anvil is a rigid body. the distance between the helmet and the anvil in the beginning is 1,5 m as initial condition. the meshed model with tetrahedron elements is shown in fig. 12 and prepared to be used in the fea. the number of total finite elements is 86964. the bike helmet fe model drops onto a flat anvil plate under gravity and load from body weight from 120 kg. the virtual testing is performed with the finite element analysis (fea) using solidworks simulation software package. the fea is used to obtain stress field and results of von mises stress field for outer shell from the bike helmet. fig. 13 shows the results from the von mises stress field for the bike helmet. the maximum von mises stress for the pc outer shell is 44,5 mpa. the yield strength for pc material is 60 mpa and the outer shell is in elastic range. in this case study the cyclist will be protected from head injuries since the bicycle helmet materials will be in elastic range. fig. 12 the meshed model of bicycle helmet fig. 13 fea of the bicycle helmet under impact test 6.2. examination and verification of aerodynamic properties the most important feature of bicycle helmets is their aerodynamic; therefore, it is important to examine and verify its functioning. testing of aerodynamic properties is bicycle helmet design and virtual validation of impact, aerodynamics and production process 361 usually done before the serial production begins in order to determine the correctness of the design and its proper functioning. virtual testing for the bicycle helmet design is performed using “solidworks flow simulation“. the input values for initial and boundary conditions used for the virtual and experimental testing of the aerodynamic properties are: input velocity of 25 km/h, atmospheric pressure of 101325 pa, earth acceleration of 9.81 m/s 2 , and ideal walls of the air tunnel. in fig. 14 for the first virtual testing are presented the results for the velocity of air around of helmet in every point and airflow lines. the helmet is positioned on a human head in a vertical position in order to properly display the velocity, airflow lines and the ventilation of the head outside the bike helmet. the results from the presented flow charts for the first virtual testing reveal a large turbulence after contact with helmet. the airflow in the first virtual testing is more complex and requires a longer time to stabilize. in the second virtual testing, shown in fig. 15, the helmet is also positioned on the human head and the input values are identical to the first virtual testing, but this time the head is set at an angle of 30° in relation to the lower plane, representing an active bicycle riding position, where the helmet aerodynamics should fully emphasize its performance. the difference between the passive position (first virtual testing) of the helmet and the active position (second virtual testing) of the helmet can be seen in fig. 15 and the airflow diagram. the diagram in the second virtual testing shows faster stabilization of the velocity, lower turbulence after contact with helmet and improved ventilation and air circulation in the helmet. fig. 14 airflow lines and velocity diagram for a vertically placed bicycle helmet fig. 15 air flow and velocity diagram for inclination of human head at an angle of 30° in addition to the aerodynamics of the bicycle helmet, the above mentioned entrance of the air into the helmet or the ventilation of the user's head, which is made possible by positioning the ventilation openings in the precisely determined places are very important in order to enable constant air circulation. the air circulation in the bicycle helmet is shown in fig. 16. 362 b. boshevski, i. mircheski fig. 16 air circulation inside the bicycle helmet in order to verify the results from the virtual testing of aerodynamics the bicycle helmet was tested in a wind tunnel and experimentally verified with the equipment shown in fig. 17. the wind tunnel at the faculty of mechanical engineering in skopje and the measurement equipment were used to perform verification. the result for the airflow lines between the virtual testing and the experimental one with the same initial and boundary conditions are compared and the airflow lines are in correspondence as presented in fig. 18. fig. 17 wind tunnel used for experimental testing bicycle helmet design and virtual validation of impact, aerodynamics and production process 363 a) b) fig. 18 the airflow lines for a vertically placed bicycle helmet obtained by: a) virtual testing and b) experimental testing 6.3. virtual testing of the production process of the bicycle helmet outer shell virtual testing of a plastic part injection molding process is an economical and effective option when it comes to larger and more complex products. instead of doing physical production testing with certain values of the inputs, the testing in a virtual environment allows combining values in a faster and simpler way in order to obtain mutually compatible input values that will result in a high quality and feasible product. virtual testing of the bicycle helmet production process was made using solidworks plastics. in order to obtain results that are credible and applicable in reality, more virtual testing was done by changing the input values and taking into account several parameters that the quality and endurance of the bicycle helmet depend on. the parameters that were analyzed are: the position of the injection gates, sink marks, the residual stress in the part, the total displacement of the part after the process and the weld lines. those parameters have a significant impact on the 364 b. boshevski, i. mircheski design and aesthetic of the bike helmet. the input values that have been changed in order to keep the residual stress within their safe limit are: the number of injection gates (four in the last virtual test) symmetrically positioned with a 3mm diameter nozzle and shown in fig. 19; the temperature of the mold is increased to 95 °c and the packing time is set to 18 seconds. with the change of the input values, the desired values of the analyzed parameters are obtained. sink marks are shown in fig. 20, which have maximum value of 0.0601 mm; they occur in the safe areas, that is, in the helmet areas that are not critical for the user’s safety. the residual stress shown in fig. 21 appears in its proportional limit with a maximum value of 17.89 mpa and because the stress is lower than the maximum limit value of the material’s yield strength, the bicycle helmet shell would not undergo major plastic deformations. the total deformation of the plastic part, shown in fig. 22, is 0.8337 mm. due to the size of the part, is acceptable and negligible. the weld lines which are shown in fig. 23 are critical points where the material collides and where the part is the weakest; they are in the areas that would not be affected under the influence of mechanical forces on the helmet, while the helmet is on the user’s head. with the obtained results, the virtual testing of the production process is fully optimized and the virtual testing is successful. fig. 19 positions of the injection nozzle fig. 20 sink marks of the pc shell fig. 21 residual stress at post-filling end fig. 22 total stress displacement of the pc shell of pc shell bicycle helmet design and virtual validation of impact, aerodynamics and production process 365 fig. 23 weld lines of the pc shell 7. conclusion this paper shows the basic developmental process of a product bicycle helmet. the main objective of this paper is to present the process of a protective bicycle helmet development which will not end up with just an idea and a virtual 3d model, but will continue with the rest of the developmental phases. the ergonomic characteristics of the designed bicycle helmet are validated with a 3d printed prototype in real dimensions. the bicycle helmet impact test model used solidworks simulation while the design was based on virtual testing. this study indicates that the design and the virtual validation of the helmet impact test are satisfied; it also predicts safety of helmet during impacts. example for this type of limitation is the airflow around and in the designed bicycle helmet, which in this case represents the imperative for a successful product. because of this, it is very important to position the ventilation openings correctly and this is only possible with previously conducted virtual air flow testing. besides the helmets’ design, the virtual testing for the helmet’s performance and its outer shell manufacture are presented in order to point out that it is very important to realize and overcome the limitations which appear and affect the development process and the helmet’s design. with the use of modern computer technology and the right parameters we can get reliable results as well as a realistic representation of the function of the developed and tested product. using the computer technology and software for modeling, virtual testing and the result analysis, the time required for designing and preparation of the manufacturing process is significantly reduced, which, in its turn, reduces the costs indirectly. 366 b. boshevski, i. mircheski references 1. tso-liang, t., cho-chung, l., chien-jong, s., van-hai n., 2013, design and analysis of bicycle helmet with impaxx foam liner, advanced materials research, 706-708, pp. 1778-1781. 2. tso-liang, t., cho-chung, l., van-hai, n., 2013, design of dual layer liner with deformable semi-spherical convex for bicycle helmet, applied mechanics and materials, 284-287, pp. 681-686. 3. tso-liang, t., cho-chung, l., van-hai, n., 2016, assessment of a bicycle helmet liner with semispherical cones, proc imeche part l: j materials: design and applications, 230(1), pp. 344-352, doi: 10.1177/1464420715569290 4. tso-liang, t., cho-chung, l., van-hai, n., 2014, innovative design of bicycle helmet liners, j materials: design and applications, 228(4), pp. 341–351. 5. post, a., oeur, a., hoshizaki, t. b., gilchrist, d. m., 2013, the influence of velocity on the performance range of american football helmets, report, human kinetics university of ottawa, ottawa, canada, school of mechanical & materials engineering, university college dublin. 6. zhihua, w., peter, r. n., childs, d., kak, l., 2013, application of an effects database in idea generation approach for helmet design, report, department of mechanical engineering, imperial college london, london, uk, department of innovation design engineering, royal college of art, london, uk. 7. halldin, p., kleiven, s., 2013, the development of next generation test standards for helmets, report, royal institute of technology, stockholm, sweden. 8. deck, c., 2013, model based head injury criteria for head protection optimization, report, university strasbourg & cnrs, strasbourg, france. 9. shishodia, b. s, sanghi, s., mahajan, p. a., 2013, comparative study of turbulence models performance for the study of air flow in helmets, report, applied mechanics department indian institute of technology delhi, new delhi, india. 10. bradford, w. s., jenkins, e. p., 2011, aerodynamic bicycle helmet design using a truncated airfoil with trailing edge modifications, proceedings of the asme 2011 international mechanical engineering congress, denver, colorado, usa. 11. rodrigue-millan, m., tan, l. b., tse, k. m., lee, h. p., miguelez, m. h., 2017, effect of full helmet systems on human head responses under blast loading, material & design, vol. 117, pp. 58-71, https://doi.org/10.1016/j.matdes.2016.12.081. 12. human factors standardization subtag, 2000, human engineering design data digest, washington, pp. 79-82. 12583 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 643 671 https://doi.org/10.22190/fume240305023s © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* selection of data conversion technique via sensitivity-performance matching: ranking of small e-vans with probid method željko stević1,2, mahmut baydaş3, mustafa kavacık3, emrah ayhan4, dragan marinković5 1department of mobile machinery and railway transport, vilnius gediminas technical university, vilnius, lithuania 2school of industrial management engineering, korea university, seoul, korea 3necmettin erbakan university, faculty of applied sciences, konya, turkey 4anadolu university, faculty of economics and administrative sciences, eskişehir, turkey 5institute of mechanical science, vilnius gediminas technical university, vilnius, lithuania orcid ids: željko stević https://orcid.org/0000-0003-4452-5768 mahmut baydaş https://orcid.org/0000-0001-6195-667x mustafa kavacık https://orcid.org/0000-0003-3782-8154 emrah ayhan https://orcid.org/0000-0001-6152-7509 dragan marinković https://orcid.org/0000-0002-3583-9434 abstract. sensitivity analyses are frequently performed to determine the robustness of mcdm methods, of which there are more than 200 types. in the past, rankings were compared to each other rather than to an external ranking. thus, the direction and meaning of sensitivity can become unclear and complex. in addition, sensitivity analysis is usually performed only based on weight coefficients, but the effect of the normalization type is neglected. in this study, the most appropriate data conversion technique was investigated through an innovative sensitivity procedure to select the small electric van, which is an environmentally friendly logistics and transportation vehicle. seven different normalization types based on the probid method (and two additional alternative mcdm methods) were used as parameters, resulting in 105 different mcdm rankings. according to the findings, mcdm rankings, which have low sensitivity, were also the performing methods that produced the highest correlation with price. what is striking is that careless choice of normalization type can be so effective as to manipulate the results. although the most appropriate technique may vary depending on the data type, the fixed gold standard we recommend offers a flexible solution for all applications. a suitable data converter will result in the choice of a reliable electric vehicle. key words: normalization technique, mcdm, sensitivity analysis, electric vehicles, logistics, transport received: march 05, 2024 / accepted may 11, 2024 corresponding author: mahmut baydaş necmettin erbakan university, faculty of applied sciences, konya, türkiye e-mail: mbaydas@erbakan.edu.tr https://orcid.org/0000-0003-4452-5768 https://orcid.org/0000-0001-6195-667x https://orcid.org/0000-0003-3782-8154 https://orcid.org/0000-0001-6152-7509 https://orcid.org/0000-0002-3583-9434 mailto:mbaydas@erbakan.edu.tr 644 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć 1. introduction the transportation sector has been a sector that has especially triggered commercial growth since the beginning of human history. transport has played a key role in determining the location of industries and cities and shaping the prosperity of regions. moreover, in many industrialized countries, transportation has established itself as an important industry and has therefore become a complex sector. the belief that the increasing travel demand in the world's major cities can be met by building more highways has had to change due to the resulting congestion. humanity quickly embraced the freedom of movement opened up by motor vehicles. but it was only realized over time that this also led to urban sprawl, air pollution, traffic noise and accidents. it was a known fact that the use of oil would not continue forever due to limited fossil fuel resources and the need to reduce greenhouse gas emissions. the fact that the current situation in transportation is not sustainable has led to the idea that fundamental changes are required in the technology of transportation systems [1]. this led to the production of more environmentally friendly electric vehicles, whose development continues to this day. however, in a short time, performance types of these (like gasoline-powered ones) were produced under very different brands. moreover, choosing a suitable electric vehicle has become an important selection and ranking problem due to the abundance of different alternatives, performance criteria, weight methods, and normalization techniques. in daily life, we encounter different problems that require choosing the most suitable one among alternatives. it is not easy for the average person to immediately visualize, calculate and conclude a decision matrix to solve a multi-criteria and multi-alternative decision problem. therefore, in such problems, people may make regrettable decisions based on only a few criteria. in addition, the multi-criteria decision making (mcdm) methodology is frequently used in solving such problems where the weight importance of the criteria is different and the approach to them can be focused on maximum benefit and minimum cost [2-8]. mcdm methods also normalize different units of different criteria and are successfully used in solving problems in many application areas. moreover, the original calculation algorithm of each mcdm method is unique. this also means that different mcdm methods may produce different rankings for a given problem [9]. when the input variables or parameters of mcdm methods change, the ranking results may also change. for example, when weight coefficients and normalization techniques change, the ranking results produced by some mcdm methods often change depending on the situation. this situation is expressed by sensitivity analysis in the literature [10-14]. the mcdm holistic algorithm consists of some additional input parameters. and these may include data type, number of alternatives, number of criteria, problem type, normalization technique, weighting method, even threshold values and preference function in the initial decision matrix. the final rankings produced by the mcdm integrity are affected by different inputs at different rates. of course, since there are so many parameters or input types in the mcdm architecture, understanding how they affect the final sensitivity of the mcdm method is a very complex issue and it is very difficult to solve the problem in one go. for example, normalization, weight coefficients and the mcdm basic equation interact simultaneously and affect each other to certain extent, which in turn affects the final result. there is no consensus on how to determine the quality of an mcdm method or the reliability of the results it produces, and moreover, there is no consensus on exactly what selection of data conversion technique via sensitivity-performance matching: ranking... 645 the limits of sensitivity analysis are and how to determine them [15]. a classical criterion emphasized for sensitivity analysis in mcdm methods is that the best ranked alternative expresses the stability conditions [16]. in fact, we can think that this determination is a partial solution, that is, it is not sufficient. because what is important is not just the change in the position of one alternative, but the degree to which the ranking of all alternatives changes, which can be easily measured with any rank correlation analysis. as it is known, the number of inputs of a decision-making problem is very high, and this further increases the complexity. thus, it becomes difficult to verify precise model recommendations. for example, when it comes to alternatives with close values, changing the criterion weight coefficients as a parameter can easily change the position of the alternatives, which is natural. thus, focusing on the stability of the ranking of all alternatives rather than the stability of the alternatives may provide a more holistic and robust view. although there is no objective guarantee, many authors have recently described the comparison of the sensitivities of the results obtained using many different mcdm methods as "accuracy" and "robustness" analysis [17]. it is a controversial issue whether the concept of "accuracy" is actually measured here. on the other hand, what is meant by robustness is low sensitivity, that is, minimal fluctuation of the ranking results. however, in our opinion, this is a judgment or inference that is open to criticism in terms of the semantics of the methodology. identification of classical sensitivity analysis with the concepts of "robustness" or "accuracy" in a direct approach without being subjected to a reasonable cost-benefit analysis requires evidence. of course, excessive sensitivity can be harmful in the sense of "unnecessarily affecting the rankings", but in our opinion, it is a controversial claim that all types of sensitivity are harmful. according to many research results, sensitivity is actually affected by the chosen mcdm method, the characteristics of the problems, the data type, and many components, parts or parameters in the mcdm integrity such as normalization and weight coefficient. because of this dependence on input variables, the results of each study are often empirical (i.e., relevant only to the observed sample). and it has not been possible to generalize them with a framework. performing a sensitivity analysis beforehand for the selection of an mcdm method, which is a complex problem, of course gives an idea, which is actually useful. but first of all, it may be useful to objectively evaluate what the results of classical sensitivity analysis for mcdm methods mean from a different perspective. for example, it is important to consider which direction sensitivity goes and to what extent this involves robustness or verification. of course, it is an expected result that an mcdm method will be affected by the change in input parameters. however, in order to understand change and give meaning to change, we need a fixed base or a solid reference point. this can be difficult to formulate clearly in mcdm methods. however, by taking advantage of data analytics and with a rational approach, we need to create a static, somewhat similar and natural external order against the dynamic mcdm results. thus, we compare the dynamic and refreshed mcdm results with a static sequence and observe the sensitivity. this approach has previously been successfully applied in the comparative capacity assessment of mcdm methods by taking advantage of the natural relationship of financial performance and stock return [8,9,18]. moreover, there are authors who suggest that this may mean a sensitivity analysis approach from a different perspective [19]. essentially, when considered well, this type of approach can be applied to all commercial products with a price-performance relationship. 646 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć the performance of commercial products can already be calculated with mcdm using different criteria and alternatives. it is already known that there is a reasonable level of relationship between performance and price, which companies fine-tune. we can take advantage of the relationships that static price produces with dynamic mcdm-based performance. thus, it can be easily observed how and to what extent the change in mcdm input parameters affects the statistical relationship with price, provided that other factors are kept constant. the mcdm components (criteria, alternatives, weight coefficient, normalization, mcdm basic algorithm, threshold values, preference function, etc.) essentially have the same purpose in all mcdm methods (in terms of each task type allocated). this means that if an mcdm component reduces or increases an existing relationship with the other party, it can be described as a different indicator of sensitivity. in this study, we measure sensitivity through the relationship with a static and external factor and strive to discover the degree, direction and nature of sensitivity. in our study, we examine normalization types as the determining parameter in sensitivity, which is rare. not only that, we also measure how sensitivity is determined when we change the weight coefficient and mcdm basic algorithms. in this study, we measure how changing the input parameter of an mcdm method specifically for a new method, probid, affects the final ranking sensitivity produced, roughly using the "price-performance relationship". we apply the selection of the small commercial transportation and logistics electric vehicle to our case. the probid original equation or method may have a special empirical situation that prevents justified generalization. and avoiding this risk, the well-known saw and codas methods will also be tested in comparative analyses. it is known how normalization affects sensitivity in mcdm studies, but this is an issue that has not been adequately conceptualized. generally, to understand sensitivity, criterion weighting coefficients are changed and the results are monitored. in fact, the sensitivity of the normalization method used may be a more determining parameter. in this study, the selection of an appropriate normalization method based on the sensitivity-mcdm performance special relationship will be evaluated through the small electric van selection. on the other hand, we think that the findings obtained from our study can be a solution to the data conversion problem not only for the mcdm family but in every field (for example in machinery, automotive, energy and industry engineering, artificial intelligence, machine learning or finance, etc.) and will be remarkable. the data preprocessing phase is an advancement where information can be parsed more easily. normalization is the best technique for pre-processing data before the data training phase. normalization is a process where data within a model is categorized to increase attribute connectivity [20]. the interesting point is that only a few techniques, especially min-max and z-score, are widely used in artificial intelligence or machine learning studies. [20-24]. we think that the adequacy of this choice may be a research gap open to question. according to the expertise and insights of mcdm methods, which are known to focus more on the problem of normalization selection than many other disciplines, using a random or static normalization technique is an objectionable choice. it can greatly influence, change, and even manipulate the final results. for example, a statistical relationship or causality investigated between two factors such as x and y may increase, decrease, or become meaningless depending on normalization [25]. this situation will question the reliability of the analysis results. as a matter of fact, in this study, we will show that the normalization technique can directly affect or mislead the relationship between price and mcdm final rankings that measure performance. in other words, the normalization method can reach a level that can manipulate the price selection of data conversion technique via sensitivity-performance matching: ranking... 647 performance relationship. on the other hand, for example in machine learning, the subject is generally viewed in terms of prediction verification performance. however, we think that the results found (even if they are low or high) may be misleading because the wrongly chosen normalization type transforms the data incorrectly. working with an arbitrary and incorrectly chosen type of normalization can be the same as working with a different data set. in this study, we focus on the literature in the first part. the literature consists of two parts. first, we focus on the importance of electric transport vehicles in the transportation and logistics industry and investigate the importance of choosing a commercial logistics transport vehicle with a good price-performance ratio. in the second part, we focus on the literature on normalization and sensitivity analysis. in the third part, the application part, we conduct an innovative sensitivity analysis for probid, vikor and codas methods in order to choose the most suitable/best small e-van among 51 alternatives. here, we also try different weighting methods as in the literature. thus, we observe how three mcdm basic algorithms, seven normalization techniques, and five weighting method parameters individually affect the sensitivity of mcdm final results. mcdm sensitivity analysis in this study was analyzed with a total of 105 different alternative methodological scenarios, based mainly on the performance-price relationship. finally, we evaluate our findings in the discussion section and make final inferences and recommendations based on the available evidence in the conclusion section. 2. literature overview in this study, we select small electric van, commercial vehicle category, with mcdm methodology. to understand the nature of mcdm methods and, according to some, "sensitivity analysis", which means validation and robustness, has become popular and widespread in recent years. it is known that in recent years, sensitivity analysis has been used as a criterion in choosing a good mcdm method. however, sensitivity analysis is generally done by changing the weighting coefficient. normalization methods are also a strong sensitivity indicator. normalization methods can change the ranking results just like the weight coefficients. the individual effects of the original mcdm equation, normalization and weighting methods on the final result of the mcdm methods are significant. a comprehensive sensitivity framework for this, through a fixed external factor (e.g. price), has not been addressed before to such a scope as in this study. this study approaches classical sensitivity analysis with critical thought, improves it, and proposes an alternative sensitivity approach. in this study, we investigate whether an mcdm has low sensitivity but also shows high performance (by observing the relationship between price and mcdm). therefore, we examine the literature in three parts. first, we emphasize the undeniable importance of electric vehicles in terms of transportation, logistics, environment, and sustainability, based on the literature. secondly, we review past studies on electric vehicles and mcdm methods. third, we provide a framework for the reader by summarizing the existing literature on sensitivity analysis and normalization, an important topic within the methodology of evaluating mcdm methods. 2.1. sustainable transportation the transportation sector is seen as the sector that contributes the most to carbon dioxide emissions. because the operating system of the vehicles used on the roads is mostly 648 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć powered by fossil fuels. the rapid development of technology significantly affects all countries in the world. technological development and environmental awareness have given birth to electric and alternative energy vehicles in the global transportation system. it is estimated by the international energy agency that, with the more intensive use of such vehicles, the emission rate will drop below 50% by the 2050s, especially in developed countries such as the usa [26]. in the world energy outlook special report published by the agency in december 2023, it was stated that if countries fully implement their national energy and climate commitments, oil and gas demand will be below the current level, that is, 45%, before 2050. it has been stated that if emissions from the sector reach net zero, oil and gas use will decrease by 75% by 2050 [27]. sustainable transportation, also often referred to as green transportation due to its emphasis on sustainable transportation or environmental aspects, stems from sustainable development. according to the oecd's definition, sustainable transportation is defined as transportation that does not harm human health and the ecosystem and includes the use of renewable and non-renewable energy resources at an acceptable level. sustainable transportation narrowly focuses only on resource depletion or air pollution problems. social and economic welfare in a broad sense is also included in the content [28]. black [29] claimed in his study that the transportation system is not sustainable due to reasons such as limited oil reserves, air quality problems, global atmospheric problems, excessive deaths, traffic congestion and urban sprawl. nakamura, hayashi [30] discussed different strategies used to reduce carbon emissions in the transportation sector. he compared the policies followed in different cities with the double classification they created. it has been revealed that each city has its own solutions to reduce carbon emissions. within the strategy expressed as improve, electric vehicles are mentioned as the next generation of the lowemission vehicle class. sustainable transportation policy covers many interrelated issues such as climate, air quality, safety, traffic safety and health [31]. some countries in the eu tax vehicles annually based on the pollution they emit. for this purpose, germany has created annual taxes for different automobile classes and has especially exempted electric vehicles from tax for 5 years after they enter the traffic. tax rates are offered at a significant discount for environmentally friendly and energy efficient vehicles. on the other hand, the tax rates applied to large vehicles that do not save energy and cause more harm to the environment are quite high. in order to encourage the purchase of more efficient and cleaner vehicles, the uk taxes vehicles annually according to sixband carbon emission figures [32]. additionally, free charging stations have been created in london to encourage people to purchase electric vehicles [33]. 2.2. electric vehicles and mcdm methods since batteries are used as the power source in electric vehicles, they do not create exhaust emissions and effectively reduce dependence on fossil fuels and air pollution. since renewable energy can be used when charging electric vehicles, energy consumption decreases. thus, it prevents the decrease in energy resources. however, today, efforts are being made to develop the technology of high-performance electric vehicles and improve their costs [34]. the use of electric vehicles in traffic is accepted as a part of the sustainable transportation system planned in cities. electric vehicles have many advantages compared to motor vehicles. it reduces operating costs, users feel less noise and vibration inside the selection of data conversion technique via sensitivity-performance matching: ranking... 649 car, it accelerates better at low speeds, has appropriate charging systems, and has zero exhaust emissions [35]. 2022 has been a difficult year for the automobile industry due to different factors (wars, the ongoing impact of the coronavirus pandemic, increasing energy-fuel costs, increase in inflation). 14% of new cars sold in 2022 are electric vehicles. this rate is expected to reach 18% by the end of 2023. 60% of world electric car sales occur in china. while the eu ranks second, it is reflected in the data as the usa third [36]. according to the statement of the european automobile manufacturers association, alternative energy vehicles had a share of more than half of the eu automobile market in 2022. while petroleum-powered cars have a share of 36.4% and diesel-powered cars have a share of 16.4%, battery electric vehicles (bev) have a share of 12.1%, hybrid electric vehicles 22.6% and plug-in hybrid vehicles (phev) 9.4% [37]. this situation was realized as 6.7% for battery electric vehicles, 7.2% for hybrid electric vehicles and 1.7% for plug-in hybrid vehicles in the usa [38]. as of the third quarter of 2023, the share of these vehicles has increased to 18% in total. the share of other internal combustion engine vehicles in the usa is around 82% [39]. small electric vans are the subject of this study. small vans have a small and economical structure and are the vehicles preferred by smes and tradesmen for product deliveries within the city. it moves easily through the narrow streets and bends of cities, is easily parked, and provides convenience by reducing costs. the biggest reason for choosing small vans is the need to carry loads of a certain size and weight. for this, the powerful engine and torque it will have and the speed and acceleration it will use when carrying loads in the city are important [40]. the sales share of electric light commercial vehicles worldwide was 3.6%, accounting for one quarter of passenger cars. battery electric vehicles (bev) constitute 98% of electric light commercial vehicle sales. there are reasons for this situation, such as intensive use, driving range, wide geographical coverage, lower maintenance and service costs, and the promotion of bevs. among light commercial vehicles sold globally, korea had the largest share of electric vehicles with 27%. this situation was realized as 15% in china. in the eu and us markets, the share of electric light commercial vehicles in sales remained below 10%. countries regarding light commercial vehicles have made changes to their incentive plans and reduced their support per vehicle. despite this, world light commercial vehicle sales have continued to increase [36]. choosing the most suitable electric transportation vehicle is important, especially from the perspective of cost-benefit analysis and price-performance. electric vehicles have different performance characteristics based on multiple criteria. a significant number of multi-criteria decision making (mcdm) methods have been used in the past. moreover, we come across thousands of examples of modified versions of methods with the authors' intention of improvement. kijewska et al. [41] evaluated the electric freight vehicles used in city logistics within the framework of their technical specifications using the promethee method. it was evaluated according to engine, battery, price and performance and sub-factors. price and engine were the most influential factors. evi walk-in van and mt-ev wiv vehicles according to engine factor features; according to the price factor, ewolf omega 0.7, motorcars suv and zero truck vehicles were chosen as the best. ecer [42] used mcdm techniques (seca, marcos, mairca, cocoso, aras and copras) to evaluate battery electric vehicles (bev) such as battery, acceleration, range, price, etc. it tried to identify the best one by sorting it according to its technical features such as. the most important features were found to be price, allowable load and energy consumption, and as a result of the sensitivity analysis, tesla model s was chosen as the best bev. 650 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć in their study to rank electric vehicles with the mcdm method and select the best among them, sonar, kulkarni [43] used the ahp method to determine the weights of the criteria and the mabac method to select electric vehicles. as a result, hyundai kona was chosen as having the best performance among all the selected alternatives. tata tigor model is also suitable for those looking for a low-priced vehicle. chawla et al. [44] selected the best electric scooter according to five criteria: price, engine power, driving range, maximum speed and battery charging time, using the integrated mcdm model. fuzzy ahp was used to calculate the weights, and the topsis method was used to rank the alternatives. according to the results, battery charging time was determined to be the feature with the most weight, and engine power was found to be the feature with the least weight. if all criteria are weighted equally, ola s1 pro is found to be the best model, and according to ahp's weighting, simple one is found to be the best model. pradhan et al. [45] introduced a model that allows consumers to choose the most suitable electric vehicle based on two sets of criteria using the mcdm method. the quality function distribution (qfd) model was used to weight the criteria. hyundai kona electric was chosen as the most suitable model with the highest score. puška et al. [46] evaluated 9 small vans according to 12 criteria using a mixed method consisting of entropy, critic and merec weight determination methods. charging time and cargo volume were determined as the most important criteria. according to most conditions, the toyota proace city verso electric l2 model was chosen as the most suitable model. in the evaluation made with the merec method, the renault kangoo etech model came to the fore. saxena, yadav [47] used the fuzzy analytic hierarchy process (fahp) to determine the importance levels of factors for the adoption of electric light commercial vehicles in india. a telephone survey was administered to 32 logistics operators and their opinions on the impact of the selected factors were obtained. high capital costs, driving range concerns and long charging times have been found to be the most important negatives in the adoption of electric light commercial vehicles. tian et al. [48] determined the weights and importance levels of the criteria that are important in the selection of electric vehicles with the mcdm method based on sensitivity analysis. it made weighting by combining the best-worst method and the deviation maximization method. the ranking of electric vehicles was carried out with the extended oreste based on hifs. in the evaluation made based on comfort, cost performance, appearance, interior space, fuel consumption, space, handling and power criteria and consumer comments, toyota's levin hev e+ was determined as the most superior model among the alternatives. while consumers were very satisfied with the appearance criterion, they expressed more negative statements about fuel consumption. wang et al. [49] tried to identify electric distribution vehicles with the best features in urban logistics by using fuzzy-coarse swara and marcos methods. according to the fuzzy-rough swara method, the most important features in choosing an electric distribution vehicle were found to be the range and price of the vehicle. according to the fuzzy-coarse marcos method, in line with the opinions of experts, kangoo e-tech electric was determined as the vehicle with the best features and was confirmed by sensitivity analysis. 2.3. normalization, mcdm and sensitivity analysis the word ‘sensitivity’ is derived from the latin ‘sensibilitas’ meaning ‘capacity to feel’. over time, the term has evolved in both meaning and usage. for example, this concept also selection of data conversion technique via sensitivity-performance matching: ranking... 651 includes some meanings related to awareness, reaction, and response. as in many scientific fields, sensitivity analysis in the mcdm methodology is more concerned with the degree to which the change of a certain input affects the final results [10,50]. so, in general terms, sensitivity analysis is about the relationship between the influencer and the affected. it is a comparative study of why the affected are affected too much, too little, or at average levels, rather than the influencer. the general judgment or acceptance on this issue is that high sensitivity for mcdm methods is considered to be a negative situation because it means being easily affected by an impact factor. it is claimed that an overly sensitive mcdm sequence also reduces quality because it deviates from the ideal more easily and is damaged more quickly. although this determination seems reasonable and correct in general, sometimes it is useful to determine whether the sensitivity of an mcdm can be positive or not. a positive sensitivity cannot be considered negative. to give real-life examples, it is sometimes desirable for a sensor device to be extremely sensitive, and this is beneficial. therefore, when you change the weight coefficients of an mcdm method (perhaps because it is the right thing to do), it may be positive that the rankings change a lot. especially if sensitivity is measured with the help of a specific reference point, the results can be interpreted more meaningfully. as explained above, sensitivity analysis for mcdm methods generally refers to the degree to which a final ranking is affected by one or more inputs [51]. frequently, attempts have been made to measure the effect of changing weight coefficients on the sensitivity of the ranking [52,53]. however, for example, normalization may also be a determinant of sensitivity. it can be said that sensitivity has not been examined systematically and indepth. in studies where robustness evaluation is made, the effect of the change in weights on sensitivity is meant. however, the effect of other components on the change of mcdm results seems to be generally neglected [54]. the concept of sensitivity should not be understood as just changing a few mcdm components. it can be easily observed how changing any mcdm input component affects the results, provided that other factors are kept constant. these mcdm components include data type, criteria, alternatives, weight coefficient, normalization techniques, mcdm basic algorithm and even threshold values, preference function, etc. it might even happen. moreover, changing the mcdm basic equation itself and evaluating the ranking results is a common sensitivity analysis. of course, not every impact factor changes the rankings in the same direction and degree. for example, in promethee, a threshold value or preference function parameter may not affect mcdm results as much as changing the weight coefficients. the intention and purpose behind performing the sensitivity analysis of mcdm is related to the choice of the best of the alternatives for the decision-maker. in other words, decision-makers perform sensitivity analysis to make a sound decision or avoid a wrong choice. in this context, choosing a reliable and robust mcdm tool is useful in choosing the most appropriate alternative for a problem. [54,11-14]. therefore, the health of a measurement can also ensure the reliability of the decision. on the other hand, the sensitivity level of an mcdm method may vary depending on the conditions, especially the data and the problem. many influencing factors such as what the problem is, data type, number of alternatives and criteria, weight coefficients, normalization, and the basic mcdm equation can explain this situation. therefore, to keep the work more robust, authors have used more than one method simultaneously for a given problem in the past. instead of choosing the most appropriate alternative using a single method, it is suggested to compare the procedures applied and the results obtained with the results obtained by other methods [55]. according 652 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć to the highlighted definition, sensitivity analysis can be performed depending on any parameter or input change. as a different example, triantaphyllou and mann [16] also pointed out the normalization technique and "rank reversal", which means that the change of alternatives disrupts consistency. on the other hand, among the sensitivity analysis techniques for mcdm methods, even the monte carlo simulation method has been used on initial data [56]. on the other hand, if you look carefully, the fact that normalization techniques have different calculation procedures and, as a result, produce different ranking results (provided that other mcdm components are kept constant) is a source of concern for the decision-maker. these procedures mean that very different sequences or alternative solutions are produced even when tested in a fixed mcdm method. there is almost a consensus in the literature that normalization or data transformation techniques directly affect the results. moreover, the best method for all scenarios cannot be mentioned for now [24,57-63]. the findings in the literature are quite controversial in terms of whether the answer sought regarding the choice of normalization is correct or appropriate. although some normalization techniques are rejected and some are highlighted depending on the data structure or statistical tools such as standard deviation and entropy, it can be said that there is not yet a solid, consistent, and reliable objective criterion required for normalization selection [24]. according to the general opinion in past studies, a low-sensitivity mcdm method that maintains the order of all alternatives by changing the weight coefficients and prevents drift is consistent and acceptable [54]. even if this is generally true, it is problematic to understand sensitivity only by changing the best alternative. because hypothetically, when you choose the wrong weighting method, there is the possibility of choosing the wrong alternative, and when you choose the right weighting method, there is a possibility of choosing a correct and different alternative. therefore, the best alternative may change as the weight inputs change. we think that it is useful to observe the ranking of all alternatives statistically, rather than observing one alternative as in the study here. 3. methods and materials in this study, we provide the selection of the best small electric van alternative, one of the environmentally friendly mobile vehicles, with mcdm, based on an innovative sensitivity model analysis based on the selection of an appropriate normalization. and we show methodologically whether this choice is a reasonable choice. as it is known, there are at least 200 mcdm methods available, and moreover, there is no common trend regarding which one to choose. recently, many authors have been trying to legitimize their chosen or newly produced mcdm algorithms with sensitivity analysis. we would like to demonstrate the degree of verifiability of these efforts with the model we propose here. additionally, in this study, we showed that classical sensitivity analysis can be objectively criticized, and then we focused on developing an alternative sensitivity analysis. some information about the methodological framework we used in the selection of an small electric van is included in table 1 below. selection of data conversion technique via sensitivity-performance matching: ranking... 653 table 1 normalization methods, mcdm methods, performance criteria and weighting technique used in this study normalization method weighting method mcdm methods performance criteria sum, vector, min-max, max, rank based, z-score, decimal entropy, equal, critic, sd, subjective probid, saw and codas acceleration, electric range, total power, fastcharge time, efficiency, useable battery, weight unladen, cargo volume, total torque, top speed the phases shown in fig. 1 were used during the implementation of this research. fig. 1 flow diagram of the study we aim to provide a gold standard insight into which data transformation/normalization technique should be chosen for the best small e-van selection (and other mcdm problems) through the performance-sensitivity pattern matching proposed in figure 1. we measure how the sensitivity of an mcdm method is affected by changing the weight coefficients and normalization techniques individually each time. 654 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć 3.1. performance criteria although small e-van have many technical features, detailed explanations based on the literature of the technical criteria used in the analysis part of the study are mentioned below. acceleration (c1 criteria): it shows how many seconds the vehicle can reach 100 kilometers. as it is known, in current technology, "acceleration" is a factor that increases energy consumption. a small amount of acceleration increases energy consumption by several hundred percent [64]. in their study, sovacool et al. [65] found that, as an interesting finding, male participants gave more importance to "acceleration" than women. electric range (c2): the range criterion is an element that shows how many kilometers the electric vehicle can travel with a full battery. electric vehicles find a place in the market mostly due to their high level of acceleration. however, this reduces energy efficiency and shortens the range distance [64]. the issue of range is expressed as the biggest problem in electric vehicles. it is estimated that the range limitation problem will be solved by developing battery technology [66]. total power (c3): total power is the motive power of the vehicle, which expresses the horsepower in internal combustion engine vehicles [66]. as the weight of the battery in electric vehicles increases, it increases the total power demand and decreases its efficiency [65,64]. fastcharge time (c4): as a feature that increases the preference for electric vehicles, "fastcharge time" ensures that the range barrier is ignored in the purchasing choice. [67], who examined the relationship between daily driving distance and standard and fast charging mechanisms, found that fast charging is more effective. nilsson, nykvist [68] stated that charging technology should be improved, long-range vehicles should be charged very quickly, and this will reduce the current charging time by half. efficiency (c5): it refers to the fuel efficiency of electric vehicles, that is, the efficiency of the battery system. therefore, the performance of batteries is very important for electric vehicles. the efficiency of the battery is also expressed by power density. it shows the amount of power the battery can transmit without being damaged [69]. in the study by egbue, long [66], participants stated that electric vehicles save fuel and are more efficient than motor vehicles. useable battery (c6): the battery technology of electric vehicles continues to be developed today. the usable battery is the capacity that indicates that 95%-99% of it can be used even though it shows 100% on the vehicle's dashboard. vehicle manufacturing companies use software that prevents the battery from charging up to 100% to keep it healthy [70]. agrawal et al. [71] described the useable battery as “the duration within which the maximum ev battery capacity gradients below a certain threshold of its original capacity and needs to be replaced for regular use”. weight unladen (c7): it refers to the weight of the vehicle without a load. curb weight is the weight of a vehicle that includes all its equipment, tires, other in-vehicle accessories, and at least 90% energy filling required during production [72]. the share of batteries in the weight of electric vehicles is large. as electric vehicle technology develops, the weight of the battery should be reduced without reducing the performance of the vehicle [42]. cargo volume max. (c8): it refers to the load that the vehicle can carry if the rear seats are folded or completely removed. the highest cargo volume affects the vehicle's energy consumption. therefore, the range that the vehicle can travel may vary depending on the selection of data conversion technique via sensitivity-performance matching: ranking... 655 battery capacity [73]. aiello et al. [74] stated that the maximum carrying capacity should be taken into account when comparing vehicles that will perform urban distribution activities. total torque (c9): it refers to the rotational power of the vehicle's engine. directcurrent electric motors are used for the movement of electric vehicles. although these engines have different features, the torque required in the vehicles is the same. torque constitutes part of the power that enables the wheels and also the vehicle to move [75]. top speed (c10): it refers to the highest speed level reached by the electric vehicle. jensen et al. [76] found that participants' opinions about top speed changed before and after using an electric vehicle. electric vehicles with a maximum speed below 120 km/h were not accepted by the participants. 3.2. normalization methods the sensitivity of an mcdm method to the type of data normalization is not an issue that is often addressed in sensitivity analyses. we propose a rare use case in sensitivity analysis. below is information about normalization types and equations [77,78,61,25]. for sum normalization: fij= fij ∑ fkj m k=1 i∈{1,2,…,m}; j∈{1,2,…,n} (1) for vector normalization: fij= fij √∑ fkj 2m k=1 i∈{1,2,…,m}; j∈{1,2,…,n} (2) for minimum-maximum normalization: fij= fij-mini∈mfij maxi∈mfij-mini∈mfij i∈{1,2,…,m}; j∈{1,2,…,n} for benefit objectives fij= maxi∈mfij-fij maxi∈mfij-mini∈mfij i∈{1,2,…,m}; j∈{1,2,…,n} for cost objectives (3) for maximum normalization: fij= fij maxi∈mfij i∈{1,2,…,m}; j∈{1,2,…,n} for benefit objectives fij= mini∈mfij fij i∈{1,2,…,m}; j∈{1,2,…,n} for cost objectives (4) for ranking based data conversion: for each criterion, the best value is assigned first rank, while the worst value is assigned n rank. thus, the weighted preference function for the unit cell in each criterion column is calculated as follows: fij= rij×wj (5) where rij is the rank of solution i for criteria j. this data conversion method is recommended as an alternative to the normalization method. this method is used instead of normalization techniques in the fuca method. 656 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć for z-score normalization: nij= xijμj σj = xij ∑ xij m i=1 m √ ∑ (xijμj) 2m i=1 m nij= xijμj σj (6) z-score refers to the measurement of the standard deviation of a value from the mean of a given distribution. for decimal normalization: this method moves the decimal point of series values. the movement of decimal points depends on the number of digits of the maximum value in the series. decimal scaling produces a normalized series with values in the range 0 to 1; where d is the number of digits of the maximum value. fij=fij/10d i∈{1,2,…,m}; j∈{1,2,…,n} (7) 3.3. mcdm methods 3.3.1. preference ranking on the basis of ideal-average distance (probid) method in this study, a new mcdm method, probid, (and codas as an alternative) were used. in 2021, wang et al. [77] developed the probid method, which follows a similar methodology to the distance-based topsis and vikor methods. the mathematical stages of the probid method can be followed below. the basic idea of the probid method is that it covers ideal solutions from the most positive ideal solution (pis) to the most negative ideal solution (nis). in this respect, probid is actually similar to methods such as topsis and vikor. the probid calculation comprises six steps [77]: phase 1. by applying vector transformation, an initial decision matrix containing m rows and n columns is obtained. fij= fij √∑ fkj 2m k=1 i∈{1,2,…,m}; j∈{1,2,…,n} (8) phase 2. a weighted decision matrix is obtained by multiplying each column by a determined weight coefficient: 𝑣𝑖𝑗 = 𝐹𝑖𝑗 × 𝑤𝑗 𝑖 ∈ {1,2, … ,𝑚}; 𝑗 ∈ {1,2, … , 𝑛} (9) phase 3. the highest value pis is determined (a(1)), 2nd pis (a(2)), 3rd pis (a(3)), …, and mth pis (a(m)) (i.e., the most nis). a(k)={ (large(vj,k)|j∈j), (small(vj,k)|j∈j') }={v(k)1, v(k)2,v(k)3, …,v(k)j,…, v(k)n} (10) where, k ∈ {1,2,...,m}, j= set of benefit objectives from {1, 2, 3, 4, …, n}, j’ = set of cost objectives from {1, 2, 3, 4, …, n}, large (vj,k ) means the kth largest value in the jth weighted normalized objective column (i.e., vj) and small (vj,k) means the kth smallest value in the jth weighted normalized objective column (i.e., vj). then, find the average value of each objective column. v̅j= ∑ v(k)j m k=1 m for j∈{1,2,…,n} (11) selection of data conversion technique via sensitivity-performance matching: ranking... 657 the average solution is then: a̅={v̅1, v̅2, v̅3,…,v̅j, …, v̅n } (12) phase 4. calculate the euclidean distance of each solution to each of the m ideal solutions as well as to the average solution: si(k)=√∑ (vij-v(k)j) 2n j=1 i∈{1,2,…,m}; k∈{1,2,…, m} (13) then, the distance to average solution is found as: si(avg)=√∑ (vij-v̅j) 2n j=1 i∈{1,2,…,m} (14) phase 5. at this stage, the overall positive-ideal distance, which is the weighted total distance of a solution to the first half of the ideal solutions, is determined: si(pis)= { ∑ 1 k s i(k) m+1 2 k=1 i∈{1,2,…,m} when m is an odd number ∑ 1 k s i(k) m 2 k=1 i∈{1,2,…,m} when m is an even number (15) and, determine the overall negative-ideal distance, which is essentially the weighted sum distance of one solution to the second half of ideal solutions. si(nis)={ ∑ 1 m-k+1 si(k) m k= m+1 2 i∈{1,2,…,m} when m is an odd number ∑ 1 m-k+1 si(k) m k= m 2 +1 i∈{1,2,…,m} when m is an even number (16) here, weight is increasing with the ideal solution number (i.e., k increasing to m). overall positive-ideal and negative-ideal distances of each solution (i = 1, 2, ..., m) are thus calculated by equations 8 and 9 respectively. to better visualize the calculations of ideal and non-ideal distances, a small dataset with 4 pareto-optimal solutions (s1, s2, s3, s4) and 2 objectives (f1 and f2) is plotted in fig. 2. as shown, the green (continuous) arrowed line s4(3), for example, represents the euclidean distance between optimal solution s4 to the 3rd pis (a(3)). following equations (8) and (9), s4(pos-ideal) = s4(1) + (1/2) s4(2) and s4(neg-ideal) = (1/2) s4(3) + s4(4). phase 6. calculate the pis/nis ratio (ri) and then performance score (pi) of each solution as follows: ri= si(pos-ideal) si(neg-ideal) i∈{1,2,…,m} (17) pi= 1 1+ri 2 +si(avg) i∈{1,2,…,m} (18) on the other hand, the mcdm method called sprobid is a simple variation of probid. the first 4 steps of sprobid are the same as those of probid. in stage five, instead of using the first half of ideal solutions to find si(pis) and the second half of ideal solutions to find si(nis), sprobid considers only the top and bottom quarters of ideal solutions for finding si(pis) and si(nis), respectively. 658 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć si(pos-ideal)= { ∑ 1 k s i(k) m\4 k=1 i∈{1,2,…,m} when m≥4 si(1) i∈{1,2,…,m} when 0<m<4 (19) here, m\4 is the integer quotient of m divided by 4, which discards the remainder and retains only the integer portion. in case the number of pareto-optimal solutions is smaller than 4, only the euclidean distance between optimal solution si and the most pis is taken. si(neg-ideal)= { ∑ 1 m-k+1 si(k) m k=m+1-(m\4) i∈{1,2,…,m} when m≥4 si(m) i∈{1,2,…,m} when 0<m<4 (20) where, m+1-(m\4) gives the starting position of calculating negative-ideal distance. if fewer than 4 non-dominated solutions exist, only the euclidean distance between optimal solution si and the mth pis (i.e., most nis) is taken. in step six of sprobid, the final score is simplified to the ratio of negative-ideal distance over positive-ideal distance. pi= si(nis) si(pis) i∈{1,2,…,m} (21) the farther a solution is from nis and the closer it is from pis, the higher the performance score 𝑃𝑖. the solution with the highest 𝑃𝑖 is recommended to the decision maker. 3.3.2. combinative distance-based assessment (codas) in the codas method, which has become popular in the last five years, the ranking performance of an alternative is measured by its distance from the negative ideal point [77]. each pair of alternatives is compared according to their distance from this ideal value. here, the superiority of the alternatives over each other can be determined by two criteria. the priority criterion is the euclidean distance of the considered alternatives to the negative ideal (in cases where the euclidean distance cannot be used, taxi distance is preferred as an alternative). this distance-based method, somewhat similar to topsis, is actually preferred in cases where the best alternative has the farthest distance from the negative ideal [79]. 3.3.3. simple additive weighting (saw) the score for each alternative is calculated as follows: the final score is the sum of the normalized value multiplied by the predetermined weight coefficient across the entire row for each criterion. in other words, it represents the weighted total score for each alternative. among the selected alternatives, the one with the highest score is considered the best solution [80]. 3.3.4. weighted methods the most common use of sensitivity analysis is the sensitivity of an mcdm to weight coefficients. below is information about five different weight coefficient assignment methods [77,80,81]. the entropy approach involves the subsequent three stages. selection of data conversion technique via sensitivity-performance matching: ranking... 659 stage 1. normalize the first decision matrix: fij= fij ∑ fkj m k=1 i∈{1,2,…,m}; j∈{1,2,…,n} (22) stage 2. calculate the entropy of values of each criterion: ej=1 ln(m) ∑ (fij ln fij) m i=1 j∈{1,2,…,n} (23) stage 3. determine the weight for each criterion: wj= 1-ej ∑ (1-ej) n j=1 j∈{1,2,…,n} (24) for benefit and cost criteria fij= fij-mini∈mfij maxi∈mfij-mini∈mfij , fij= maxi∈mfij-fij maxi∈mfij-mini∈mfij (25) according to that, calculate the standard deviation method of values of each criterion: σj= √∑ (fij-fj m i=1 ) 2 m (26) critic weighted method (criteria importance through intercriteria correlation) has three phases. phases 1: “m” is the number of rows and “n” is the number of columns; fij= fij-mini∈mfij maxi∈mfij-mini∈mfij i∈{1,2,…,m}; j∈{1,2,…,n} if it is beneficial (27) fij= maxi∈mfij-fij maxi∈mfij-mini∈mfij i∈{1,2,…,m}; j∈{1,2,…,n} if it is cost-oriented (28) phases 2: a binary correlation matrix is created to measure the dependency/correlation between two criteria. ρ jk = ∑ (fij-fj m i=1 )(fik-fk) √∑ (fij-fj m i=1 ) 2 √∑ (fik-fk m i=1 ) 2 j,k∈{1,2,…,n} (29) phases 3: the standard deviation of the criteria is calculated. σj= √∑ (fij-fj m i=1 ) 2 m j∈{1,2,…,n} (30) here, 𝐹𝑗 = 1 𝑚 ∑ 𝐹𝑖𝑗 𝑚 𝑖=1 . it is the arithmetic mean of the jth normalized objective values. finally, the weight coefficients for each criterion are determined as follows. cj=σj∑ (1-n k=1 ρ jk ) j∈{1,2,…,n}, wj= cj ∑ ck n k=1 j∈{1,2,…,n} (31) mean/equal weighting method: the equal weighting method is the simplest way to create weights for each criterion. it is based on the assumption that all n criteria are of equal 660 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć importance. it is based on the assumption that all n criteria are of equal importance and therefore equal weight coefficients are assigned to each: wj= 1/n j = ∈{ 1, 2, .....n} finally, subjective weighting was also used in practice. weight coefficients are assigned high by the author, especially for some criteria, to better understand the sensitivity of mcdm and to show it to the reader. 3.3.5. statistical method used in this study, the relationship between mcdm performance results and price was obtained by spearman rank correlation. we should emphasize that our aim here is part of sensitivity analysis. we would like to remind you that as mcdm results change, the existing relationships with the fixed reference point also change. this change means sensitivity. the spearman rank correlation coefficient measures the statistical dependence between two ranking-based variables [82,83]: rs=1 6 ∑di 2 n (n2 -1) here, rs is the symbol for spearman's rho coefficient. di is the symbol for the difference between pairwise rankings. and n represents the number of alternatives in the formula. 4. application when purchasing an small e-van in transportation and logistics, people also pay attention to the technical features of the vehicles. this situation becomes more prominent when purchasing an electric vehicle because it is difficult to choose the best alternative for matrix problems where the alternatives and criteria are numerous. moreover, evaluation methodologies such as weighting coefficient, normalization selection and sensitivity analysis are also used for this selection. we rather help them by guiding them to make a reasonable decision. there are 51 small e-van categories for our study. the data regarding the brands, models, 10 different criteria and price amounts of the vehicles in this category were obtained from the open access web address ‘https://ev-database.org/’ [84]. the price of each small electric van is in euro currency. it is a known fact that normalization affects mcdm results [85,86]. we show how this affects sensitivity in the following application from an innovative gold standard perspective. the matrix in table 2 shows the alternatives (small electric van models and brands), criteria (performance dimensions), and price information. the small e-van used in the study are all brands and models between 2020-2023. tables 3-5 show the spearman correlation (rho) results between price and mcdm rankings for a total of 105 different scenarios with probid, saw, and codas methods. while reading the table, we can read from two different perspectives: row and column. if we read along the lines, this means that we must first assume that the normalization method will be constant. on the other hand, when we read from the column, this means that we act with the assumption that the weight assignment method will be constant. thus, we can see from the table that the relationship levels with price always change. in other words, provided that all components remain constant, either the weight coefficient or the normalization type is changed and the correlation results are monitored. selection of data conversion technique via sensitivity-performance matching: ranking... 661 table 2 alternatives (a), performance criteria (c) and price information for small electric van selection c1 c2 c3 c4 c5 c6 c7 c8 c8 c10 price a1 135 11.7 205 100 26 226 46.3 1739 2126 260 32790 a2 135 11.7 205 100 26 226 46.3 1813 2693 260 43640 a3 130 12.1 185 100 26 250 46.3 1969 3061 260 51940 a4 130 13.3 265 100 38 257 68 2131 3061 260 57940 a5 130 12.1 180 100 26 257 46.3 1989 3497 260 52730 a6 130 13.3 260 100 38 262 68 2161 3497 260 58730 a7 130 12.1 185 100 26 250 46.3 1969 3061 260 53640 a8 130 13.3 265 100 38 257 68 2131 3061 260 59640 a9 130 12.1 180 100 26 257 46.3 1989 3497 260 54430 a10 130 13.3 260 100 38 262 68 2161 3497 260 60430 a11 130 12.1 185 100 26 250 46.3 1969 3061 260 55990 a12 130 13.3 265 100 38 257 68 2140 3061 260 61990 a13 130 12.1 180 100 26 257 46.3 1989 3497 260 56990 a14 130 13.3 260 100 38 262 68 2167 3497 260 62990 a15 180 9.2 345 180 50 261 90 2385 2179 350 68990 a16 132 12.6 225 90 40 200 45 1874 1979 245 49445 a17 160 12 210 150 44 286 60 2531 5010 362 68949 a18 160 12 210 150 44 286 60 2501 4630 362 68056 a19 160 12.1 310 150 41 290 90 2660 5010 362 72519 a20 160 12.1 310 150 41 290 90 2635 4630 362 71626 a21 160 12 205 150 28 293 60 2405 4990 366 61571 a22 160 12 305 150 41 295 90 2555 4990 366 65140 a23 160 12 210 150 28 286 60 2380 4630 366 60678 a24 160 12 310 150 41 290 90 2530 4630 366 64248 a25 132 12.6 225 90 40 200 45 1872 1730 245 39990 a26 135 11.7 205 100 26 226 46.3 1739 2126 260 43050 a27 135 11.7 200 100 26 232 46.3 1813 2693 260 44750 a28 130 12.1 180 100 26 257 46.3 1989 3497 260 51825 a29 130 13.3 260 100 38 262 68 2161 3497 260 57775 a30 130 12.1 185 100 26 250 46.3 1969 3061 260 50992 a31 130 13.3 265 100 38 257 68 2131 3061 260 56942 a32 130 12.1 180 100 26 257 46.3 1989 4900 260 64075 a33 130 13.3 260 100 38 262 68 2161 4900 260 70075 a34 130 12.1 185 100 26 250 46.3 1969 4900 260 63250 a35 130 13.3 265 100 38 257 68 2131 4900 260 69250 a36 130 13.1 185 100 26 250 46.3 1989 3497 260 52730 a37 130 14.3 260 100 38 262 68 2161 3497 260 58730 a38 130 13.1 185 100 26 250 46.3 1969 3061 260 51940 a39 130 14.3 265 100 38 257 68 2131 3061 260 57940 a40 135 11.7 200 100 26 232 46.3 1841 2693 260 42440 a41 135 11.7 205 100 26 226 46.3 1765 2126 260 41240 a42 130 13.1 185 100 26 250 46.3 1989 3497 260 54430 a43 130 14.3 260 100 38 262 68 2161 3497 260 60430 a44 130 13.1 185 100 26 250 46.3 1969 3061 260 53640 a45 130 14.3 265 100 38 257 68 2131 3061 260 59640 a46 132 12.6 220 90 40 205 45 1870 2500 245 39300 a47 135 11.2 210 100 26 220 46.3 1739 2126 260 37800 a48 135 11.2 205 100 26 226 46.3 1813 2693 260 40150 a49 130 13.1 260 100 38 262 68 2161 3497 260 65385 a50 130 13.1 265 100 38 257 68 2131 3061 260 64530 a51 145 10.2 345 150 30 223 77 2459 2123 310 64581 662 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć if the correlation results are very volatile or have high variability, it means that the sensitivity of that mcdm ranking will be high. so, the table below primarily shows that each of the 7 different normalization types and 5 weighting methods are used in different scenarios together with codas, probid, and saw. in addition, the degree of relationship between 105 different mcdm rankings and price was obtained. the variation of all correlations in the table (on a methodological basis) can be measured by “standard deviation”, thus providing an innovative measure of sensitivity. what we mean by the concept of average here is the average correlations between price and a method or technique. standard deviation is the measure of the degree of change in the correlations produced by the same mcdm sequence, that is, it expresses the degree of sensitivity of the probid, codas, and saw methods. table 3 shows the spearman correlation (rho) and standard deviation results between price and probid rankings for a total of 35 different scenarios. table 3 sensitivities of the probid method to weights and normalization types entropy sd critic equal subjective stdv average sum 0.918 0.93 0.24 0.92 0.329 0.314 0.667 vector 0.917 0.929 0.24 0.92 0.329 0.314 0.667 min max 0.897 0.877 0.863 0.878 0.762 0.048 0.855 max 0.889 0.875 0.847 0.876 0.843 0.018 0.866 rank based 0.765 0.73 0.742 0.159 0.21 0.276 0.521 decimal 0.917 0.891 0.877 0.909 0.892 0.014 0.897 z-score 0.762 0.318 0.021 0.207 0.277 0.245 0.317 stdv 0.066 0.204 0.338 0.325 0.275 average 0.866 0.793 0.547 0.696 0.520 table 3 clearly shows that in general, low sensitivity (standard deviation) in an mcdm ranking is matched to produce high correlation (rho) with price. for example, when we change the weight method, provided that the other components are constant, we see that the methods that provide the lowest standard deviation and the highest relationship with the price are the decimal and maximum methods. on the contrary, it is clear that sum, vector, rank based and z-score methods produce high sensitivity and low correlation with price. on the other hand, when we take a reading from the column, that is, when we keep the weight method constant and change the normalization method, we see that the entropy method clearly produces the lowest sensitivity and the highest correlations with price. on the other hand, it can also be seen that the critic method produces high sensitivity and low correlation with price. in other words, decimal achieved the highest correlation average and the lowest sensitivity degree in normalization and entropy in weighting. additionally, when we create a combination of successful techniques, we notice that the decimal/entropy/probid combination produces one of the highest correlations with 92 %. while this is the case for the probid method, it is a matter of curiosity what the findings are for other methods. because the mcdm basic algorithm may have special cases that prevent us from generalizing, we wanted to test the pattern matching (low sensitivity-high mcdm performance) we obtained here in other methods. for this purpose, we added the distance-based codas and the aggregation-based saw method to this study. as can be clearly seen in the tables below, low sensitivity-high performance (or vice versa) patterns match here. although the mcdm basic equation changes the results slightly, the general trend is the same. table 4 shows the rho and standard deviation results between price and codas rankings for a total of 35 different scenarios. also, shows that for codas, selection of data conversion technique via sensitivity-performance matching: ranking... 663 in general, the low sensitivity (standard deviation) in an mcdm ranking is matched by producing a high relationship (rho) with price. accordingly, it can be said that entropy techniques are more successful in weighting and sum and vector techniques are more successful in normalization. on the other hand, it can also be seen that the critic method and the subjective weight coefficient assignment by the authors produce high sensitivity and low correlation with price. additionally, when we create a combination of the most successful techniques, we note that it is not a coincidence that the sum&vector /entropy/codas combination produces the highest correlation with 92.5%. table 4 sensitivities of the codas method to weights and normalization types entropy sd critic equal subjective stdv mean sum 0.925 0.916 0.881 0.919 0.845 0.030 0.897 vector 0.925 0.916 0.881 0.919 0.837 0.033 0.896 min max 0.773 0.384 0.289 0.359 0.105 0.219 0.382 max 0.832 0.569 0.098 0.429 0.273 0.251 0.440 rank based 0.732 0.55 0.186 0.519 0.163 0.221 0.43 decimal 0.924 0.905 0.884 0.912 0.898 0.013 0.905 z-score 0.911 0.907 0.887 0.907 0.82 0.034 0.886 stdv 0.075 0.210 0.346 0.241 0.335 mean 0.860 0.735 0.587 0.709 0.563 table 5 shows the rho and standard deviation results between price and saw rankings for a total of 35 different scenarios. the same seems to be true for the saw method. table 5 clearly shows for the saw method that in general the low sensitivity (standard deviation) of an mcdm ranking is matched by producing a high correlation (rho) with price. accordingly, it can be said that entropy techniques are more successful in weighting and sum and vector techniques are more successful in normalization. on the other hand, it can also be seen that the critic method and the subjective weight coefficient assignment by the authors produce high sensitivity and low correlation with price. table 5 sensitivities of the saw method to weights and normalization types entropy sd critic equal subjective stdv mean sum 0.913 0.908 0.886 0.911 0.89 0.011 0.902 vector 0.913 0.907 0.886 0.911 0.89 0.011 0.901 min max 0.932 0.867 0.363 0.754 0.254 0.274 0.634 max 0.935 0.896 0.378 0.82 0.328 0.263 0.671 rank b 0.726 0.546 0.284 0.507 0.226 0.182 0.458 decimal 0.909 0.914 0.889 0.912 0.899 0.009 0.905 z-score 0.909 0.908 0.893 0.907 0.835 0.028 0.890 stdv 0.068 0.125 0.272 0.139 0.303 mean 0.891 0.849 0.654 0.817 0.617 when we create a combination of successful techniques, we notice that the sum&vector/ entropy/codas combination produces one of the highest correlations with 91.3%. although the general trend points to this combination, the actual record was produced by a specific slightly different combination: the max/entropy/codas combination produced the best correlation at 93.5%. although entropy is effective, the interaction seems to extract the first one from a different combination. 664 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć 5. results and discussion in this study, low sensitivity and high correlation production with price (and vice versa) seem to be a kind of “pattern matching”, which we encounter in the three methods used. in fact, we can see and interpret the above findings better by means of fig. 2 (for probid). fig. 2 normalization, sensitivity and performance relationship for probid method the better mcdm rankings have lower sensitivity and develop a good relationship with price. this situation emerges as an infallible feature in all possible scenario sets, as can be observed in fig. 2 and fig. 3 (for codas). fig. 3 normalization, sensitivity and performance relationship for codas method an mcdm method with low sensitivity has higher performance. in this study, we adjusted the sensitivity by the weight coefficient and by changing the normalization techniques. so, as a result, decimal-based mcdm methods gave better results. this interesting aspect can be easily observed in the figs. 2 and 3, and in fig. 4 (for saw). selection of data conversion technique via sensitivity-performance matching: ranking... 665 fig. 4 normalization, sensitivity and performance relationship for saw method as can be clearly seen from the graphs above, one of the determinants of sensitivity, regardless of the mcdm method, is the normalization techniques used. we can summarize other findings as follows: ▪ the lowest sensitivity and highest performing data transformation tool for all three mcdm methods is the decimal scaling technique. ▪ on the other hand, in the opposite case, the data conversion technique that improves the highest sensitivity and lowest performance for an mcdm is z-score for probid, min-max for codas and rank based conversion technique for saw. this shows that the interaction of the basic mcdm equation with normalization techniques also affects the result to some extent. ▪ max, rank based or min-max techniques appear to reduce the performance of the saw method, while others result in high performance. a similar situation also applies to the codas method. for both methods, decimal, sum and vector seem to be the best techniques. ▪ it is noteworthy that the min-max method, which is most commonly used in the data pre-processing stage in artificial intelligence and machine learning algorithms, fails in the codas and saw methods. this shows that the interaction of a calculation format with the type of normalization is important. in addition, although the other commonly used z-score technique performs relatively better, it is not the data conversion technique that produces the best performance. in fact, the worst case scenario for probid is produced with z-score. this shows that we need to question the widespread use of min-max and z-score techniques, especially in ai and ml applications. sum and vector techniques, which are two successful methods even though they are not used much in ai and ml studies, produced almost the same result, which is interesting. ▪ undoubtedly, the best method for weighting was determined to be entropy. because entropy produced the lowest sensitivity and highest correlation with price. ▪ when we cluster among 105 matrices, it can be seen that the most efficient results are as above. however, when we only search for the combination that provides high correlation without classifying or clustering the 105 matrices, we reach an interesting result. the max/entropy/saw combination was able to produce a record correlation 666 ž. stević, m. baydaş, m. kavacik, e. ayhan, d. mari̇nkovi̇ć of 93.5%. let us remind here that the entropy method generally provides the lowest sensitivity and highest correlation average. thus, we can say that general trends and record results may differ to a certain extent. ▪ as a result, it can be said that the decimal method is the best data converter in terms of average relationship generation performance and low sensitivity, and on the other hand, the entropy method is the best weighting method. ▪ it is clear that the normalization method can bring the price-performance relationship to a level that can be manipulated. for example, when you choose the decimal technique, it can be seen that all three mcdm methods produce a statistical relationship of over 90% with the price. on the other hand, tragically, when you use z-score for probid, you may not even produce a relationship that is significant on average. when you use min-max converter for codas or rank based converter for saw, low correlations such as 38.2% and 45.78% are produced on average. ▪ a much more tragic situation is this: if the critic/z-score/probid combination is used, almost no relationship with the price can be produced (0.021%). in other words, if the normalization technique is chosen correctly, you will get a 90% relationship with the price, and if it is chosen incorrectly, you will get a 0% relationship. this shows how normalization and weighting coefficient combined manipulate the results. ▪ another meaning of this is the following: if we look at the situation in the example of ml or ai studies, to look for a relationship between two variables such as x and y, let us assume that the relationship is detected in the first case. in another normalization case, there may be no relationship at all. this strange situation will also affect causality analyses. ▪ the sensitivity rating-performance level coupling for mcdm in this study appears to be a unique pattern matching discovery. ▪ we propose an alternative framework on how to determine the best alternative for probid, saw and codas: if we look at the overall performance results, it is a fact that the three methods jointly produce the most efficient results with decimal scaling & entropy weight coefficient. in this case, the best alternative for all three mcdms is common: the “mercedes eqv 300 extra-long”, which is the a19 alternative. ▪ when we look at the ranking of other alternatives for probid, the second alternative is the a22 model and the third alternative is the a24 model. the a46 alternative is in last place. ▪ when we look at the ranking of other alternatives for codas, the second alternative is a22 and the third alternative is a20. the a26 alternative is in last place. ▪ when we look at the ranking of other alternatives for the saw method, the second alternative is a8 and the third alternative is a31. a1 and a26 are in the last place. ▪ the second option we recommend for the selection of the best alternatives is this: the mcdm rankings that provide the highest correlation among 105 matrices, regardless of sensitivity, is the sum&vector/sd combination for probid. the best alternative here is “mercedes evito tourer extra-long 90 kwh” (a22). for codas, sum&vector/entropy is the most performant. and the best alternative here is the “mercedes eqv 300 extralong” (a19). for saw, sum&vector/entropy is the most performant. the best alternative here is “mercedes eqv 300 extra-long” (a19). the common best alternative for codas and saw is the same. we obtained the same results in the evaluation above. but it is noteworthy that the result changed for probid here. selection of data conversion technique via sensitivity-performance matching: ranking... 667 ▪ the most striking finding of this research is that we recommend that decision-makers be careful when choosing an electric vehicle, as choosing an incorrect or random normalization/data transformation technique may distort the ranking results. ▪ wrong selection of the most suitable small e-van may mean increased costs for the decision-maker. a good and low-risk choice can be possible with a good methodology selection. it should be noted that the choice of a good normalization technique is highly dependent on the data, mcdm method, weight coefficient and conditions. ▪ the choice of normalization technique is dynamic. a suitable mcdm configuration with all its components will result in a good electric vehicle choice. 6. conclusion in this study, a sensitive mcdm evaluation methodology was made to find the best alternative among small electric van types, which is an electric environmentally friendly transportation and transport vehicle. as it is known, sensitivity analysis in terms of the selection and robustness of mcdm methods is a subject that has been studied extra in recent years. the research gap in these studies is multifold. first, precision is often measured by weighting methods. however, the results of this study also showed that normalization also affects sensitivity. second, there is no external reference point for sensitivity in other studies. a controversial benchmarking methodology is used as there is no reference point, meaning that the sentiment measurement in the mcdm rankings produced is novel. thirdly, sensitivity analysis is generally done by only looking at whether the best alternative has changed. however, the entire ranking needs to be evaluated from a holistic perspective. on the other hand, sensitivity can also affect an mcdm's external relationships. in other words, normalization or weighted method sensitivity is decisive. normalization-based sensitivity can distort or even manipulate correlations between mcdm performance and prices of small electric van alternatives when we approach the issue in this study example. at this point, choosing the appropriate weight coefficient and normalization technique is extremely critical. according to the findings obtained in this study, the relationship between price and performance can vary between ninety percent and zero percent, depending on the choice of normalization technique. in other words, while there is a very strong relationship between two factors, the relationship disappears with an incorrect normalization choice. this interesting result clearly showed decision-makers that the type of normalization can have a manipulative aspect if not chosen carefully in all data processing procedures. the selection of normalization technique is dynamic. a suitable data converter with all its components will result in the choice of a reliable electric vehicle. finally, so far the selection of optimal components for e-vehicles was mainly based on established engineering criteria, such as demonstrated by kalmaganbetov et al. 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force. this force is dominated by the frequency-dependent impedance behavior of the relevant electrical layers. however, many existing models are only valid at frequency extremes and use very simplified contact mechanical approaches. in the present paper, a rc impedance model is adopted to characterize the behavior in the relevant range of frequencies of the ac excitation voltage. it serves as an extension to the macroscopic model for electrovibration recently developed by the authors, which is based on several wellfounded approaches from contact mechanics. the predictions of the extended model are compared to recent experimental results and the most influential electrical and mechanical parameters are identified and discussed. finally, the time responses to different wave forms of the excitation voltage are presented. key words: friction, adhesion, electrovibration, surface haptics, tactile display 1. introduction electrovibration is a powerful technology of surface haptics that enables effective tactile feedback on touch screen surfaces of smartphones, tablets, navigation systems and similar devices of consumer electronics. it uses electrostatic attraction to enhance sliding friction between a fingerpad and touch surface. this can be done by applying an ac voltage to the conductive layer of the screen, which causes polarization of the fingerpad, resulting in an electroadhesive contribution to the normal contact force that increases the frictional force. the latter is controlled by changing the amplitude, shape and frequency of the voltage to create a variety of different tactile effects [1, 2]. although the magnification of the perceived frictional force caused by an alternating voltage, was known far earlier, before grimnes [3], named this effect "electrovibration", received january 05, 2021 / accepted february 04, 2021 corresponding author: fabian forsbach department of system dynamics and friction physics, berlin university of technology, sekr. c8-4, straße des 17. juni 135, 10623 berlin, germany e-mail: fabian.forsbach@tu-berlin.de 40 f. forsbach, m. heß it is still not fully understood today. one major problem represents the complex coupling between contact mechanics and electrodynamics, which is why highly simplifying assumptions are often made in theoretical modeling. even without considering electrostatic interactions, understanding the tribological behavior of human skin including its effect on tactile perception is a challenging task because of the layered structure and non-linear viscoelastic material behavior of the skin. in addition, the skin surface of the fingerpad has a very specific topography and its ridges are far away from being smooth. they are punctuated by a variety of concave shaped sweat pores which allow to lubricate the skin and hence change its tribological properties [4, 5]. the induced electrostatic interaction between the fingerpad and conductive layer of the screen in the state of full slip further increases the difficulty in finding a suitable model. in the last decade, intensive research has been carried out on the study of the electrodynamics of contact, both experimentally and theoretically, and several interesting models have emerged [6, 7, 8]. however, experiments are often carried out under restrictive contact mechanical conditions, such as a fixed apparent contact area, although it is obvious that the contact area of the real, electroadhesive frictional contact depends on both the normal force and the applied voltage. if the corresponding measurement results serve as the basis for theoretical modeling, effects like the reduction of the (apparent and ridge) contact area during transition from stick to slip caused by an increasing tangential force cannot be considered. moreover, the mechanical part of the contact is often represented only in the simplest way via a combination of spring and damper [9]. recently, the authors proposed a new macroscopic model for electrovibration that is based on several well-founded approaches from contact mechanics [10]. the model provides plausible results for all contact mechanical quantities, particularly, it adequately predicts the friction force and the friction coefficient over the entire range of relevant voltages and applied normal forces, which is supported by a comparison with experimental results. however, in the electrodynamic part of the model, the outer layer of skin, the interfacial air gap and the coating of the conductive layer of the screen are assumed to be purely capacitive. therefore, the applicability of the model is limited to high-frequency excitations. in the present paper, the electrodynamic part is improved by considering the resistive properties of all three components to represent the frequency dependence of the frictional force. for this purpose, the rc impedance model proposed by shultz et al. [8] is adopted. the manuscript is structured as follows: at the beginning of chapter 2, we first briefly summarize the main approaches of the model recently proposed by the authors to map electrovibration. subsequently, the improvement of its electrodynamic part is explained in detail which allows to study the frequency dependence of the frictional force. chapter 3 is addressed to the relevant electrical and mechanical parameters for the coating of the touchscreen, the stratum corneum, the interfacial air gap and the ridge contact area as a function of the applied normal force used in our simulations. since no experimental study provides a complete set of measured relevant parameters, they had to be taken from various works. chapter 4 contains the results emerging from the simulations with the new electromechanical model. the influence of numerous parameters on the electrostatic contribution to the frictional force as a function of frequency is analyzed and compared with known experimental data. the time response is studied for both a pure sinusoidal and a square-wave excitation. some conclusive remarks in chapter 5 close the manuscript. a rigorous model for frequency-dependent fingerpad friction under electroadhesion 41 2. macroscopic model for electroadhesion in a recent work the authors propose a macroscopic model for the frictional contact between a fingerpad and an ac voltage supplied coated touch surface in a state of full slip (fig. 1). in contrast to highly simplified models, it is based on sound contact mechanics approaches, which are briefly summarized below. for a detailed description of the model, the reader is referred to the original paper [10]. pressure-controlled friction is assumed, i.e., the frictional force obeys the generalized amontons-coulomb law [11], which includes an additional contribution to the normal contact force to account for electroadhesive interaction: t 0 n el 0 n el r( ) ( )f f f f a  = + = + . (1) it should be noted that eq. (1) emerges as a limiting case of the adhesive tangential contact in a coulomb-dugdale approximation as well [12]. µ0 and fn denote the friction coefficient and the applied normal force, respectively. 𝜎el represents the electrodynamic contribution to the normal contact force per unit area which can be expressed in terms of the air gap thickness da and voltage ua 0 r,a a 2 2el a . 2 u d    = (2) ar is the ridge contact area in a state of full slip, which is significantly smaller than the ridge contact area ar,0 under pure normal loading. supported by experiments [13] and fe simulations [10], the decrease of the ridge contact area during the transition from stick to slip can be described by fig. 1 electroadhesive frictional contact with the relevant layers of the finger-touchscreen interface and their representation in the rc impedance model 2 r t r,0 t r,0 2( ) . c a f a f a = − (3) the empirical parameter c2 must be determined by experiments, fe simulations or similar. the ridge contact area under pure normal loading, but taking into account adhesion, must be determined from the following equation that results from shull’s compliance method 42 f. forsbach, m. heß ( ) 2 1/ 1/ 1/ 1/ r,0 r,0 r,0 1 1/ 1/ n 2 , m m n m m nwn f a a a n m    − − + − = − − (4) where 𝛼, 𝛽, m and n are parameters of non-adhesive powerlaw-relationships between ridge contact area ar,0 and normal force f1 as well as indentation depth 𝛿1, namely r,0 1 1 ,( ) ma ff = (5) r,0 1 1 ,( ) na  = (6) and w denotes the work of electroadhesion per unit area (see [10] for details). in the present paper the parameters in eqs. (5) and (6) are taken from experimental data (see section 3.4 and table 1 for details). according to heß and popov [14], the work of electroadhesion per unit area can be calculated by a el a a( ) .d d w d d  =  (7) by using eqs. (1) to (7), the authors calculated the friction coefficient and friction force which both agree well with experimental measured data over the entire range of relevant voltages and applied normal forces [10]. however, since the outer layer of skin, the interfacial air gap and the coating of the conductive layer of the screen are assumed to be purely capacitive, the applicability of the model is restricted to the high-frequency regime. here, we improve the electrodynamic part of the model by mapping the impedance of each electrical layer as a resistor in parallel with a capacitor (fig. 1). now, suppose a sinusoidal excitation voltage ( )0 sin .( ) u tu t =  (8) the impedance of each layer is then given by  , i,a,sc , 1 x x x x r z x i r c =  +  (9) with the capacitances cx and resistances rx of the respective layer. for the rc circuit depicted in fig. 1, the amplitude ua,0 of the voltage across the interfacial air gap, ( )a a,0 sin ,( )u t u t =  + (10) relates to the amplitude of the excitation u0 as i a,0 a 0 a sc . u z u z z z = + + (11) with eq. (9), this equation can be written as a 2 2 aa,0 0 a sc a a s 2 c sci i i 2 2 2 2 2 2 2 2 2 2 a i sc i i s 2 2 c 2 1 1 1 1 1 1 1 , r u u r r r rr r                  + = + + + + + + + + + + +             (12) a rigorous model for frequency-dependent fingerpad friction under electroadhesion 43 where 𝜏 i = rici, 𝜏 a = raca and 𝜏 sc = rsccsc. the phase angle of the gap voltage ua(t) is given by sc a i a sc i2 2 2 2 sc i 2 2 a sc a i a sc i2 2 2 2 sc i ( ) ( ) arctan . 1 1 1 1 1 1 r r r r r              − − + + + + +  +    +   =    + +     (13) in the limiting case of very small frequencies, the electrical circuit behaves purely resistive, yielding a,0 a 0 i a sc 0 l ,im u r u r r r→ =  +  +     (14) whereas for high frequencies the behavior is purely capacitive, yielding a,0 i sc 0 i a a sc i sc .lim u c c u c c c c c c→ = + +       (15) the latter equation leads to the original model of [10]. 3. electromechanical parameters the electrical and mechanical parameters used for the model are listed in table 1. to the authors knowledge, there is unfortunately no experimental investigation where the complete set of relevant parameters is measured. the capacitances for the three different layers can be determined via  0 r, r , i,a,sc , x x x a c x d   =  (16) where 휀0 is the permittivity of free space, 휀r,x is the relative permittivity of the respective layer and dx is the thickness of the layer. fig. 2 resistivity 𝜌sc and relative permittivity 휀r,sc of stratum corneum taken from [17] 44 f. forsbach, m. heß 3.1. 3m coating the commonly used 3m coating of the touchscreen is 1µm thick and the relative permittivity 휀r,i is usually given with 3.9 [2]. due to the high resistivity of silica (sio2), the coating is often described as an insulator [6, 7]. however, in [15], a resistive behavior is measured for low frequencies which may be the result of imperfections in the 3m coating. an adequate approximation of the impedance measured in [15] in the relevant frequency range can be achieved by the circuit in fig. 1, a capacitator with the properties of the silica layer parallel to a resistor with ri=0.2m𝛺. 3.2. stratum corneum the resistance of the stratum corneum is determined via sc sc sc r , d r a  = (17) with the resistivity and thickness of stratum corneum 𝜌sc and dsc, respectively. the thickness varies in the range of 200-450 µm [16]. its electrical properties are highly frequencydependent. the resistivity 𝜌sc and relative permittivity 휀r,sc in the relevant range of frequencies was measured by [17]. fig. 2 shows the experimental data as well as the fits 3 3 20.023 0.0795 0.2133 4.922 sc 9 10 mz z z − + − + =  (18) and 5 4 3 20.00286 0.03564 0.15785 0.33963 0.6235 1 r, c 4 8 s . 59 10 ,z z z z z − + − + − + = (19) where z=log10(f / 1hz). these fits were used in the model to account for the frequencydependent behavior. table 1 parameters for the model of sliding friction symbol parameter name value and unit µ0 friction coefficient 0.3 fn applied normal force fn = 0.5 n 휀r,i relative permittivity of 3m coating 3.9 휀r,a relative permittivity of the interfacial gap (air) 1 휀0 permittivity of free space 8.854 ⋅ 10−12 as/vm di thickness of the 3m coating 1 µm da thickness of the interfacial gap 3.5 µm dsc thickness of stratum corneum 350 µm ra resistance of the interfacial gap 1 ⋅ 106 𝛺 ri resistance of the 3m coating 2 ⋅ 105 𝛺 m, n, 𝛼, 𝛽 parameters of power-law expressions for ridge contact area m=0.52, n=1.41, 𝛼=54.4mm2/nm, 𝛽=32.0 mm2-n c2 empirical parameter for area reduction c2 = 5000 mm4/n2 a rigorous model for frequency-dependent fingerpad friction under electroadhesion 45 3.3. interfacial gap the properties of the interfacial air gap are a subject of current research. however, recent studies [8, 9] indicate an equivalent gap thickness da of 1-5µm between the finger pad ridges and the comparatively smooth display surface. this is further supported by topography measurements of the microstructures on the ridge [18]. the experimentally observed resistance of the interfacial gap is mainly a constriction resistance of the rough contact. the charges can only pass the interfacial gap at the microcontacts or at fluid filled spots and can thus not flow freely. this gap resistance is therefore highly dependent on the interface properties such as the amount of sweat produced by the sweat ducts or contamination. to the authors knowledge there is no direct measurements of the gap resistance, but in [8] a value of 7 m𝛺 is determined indirectly. 3.4. ridge contact area the ridge contact area ar is a further highly subject dependent quantity. the parameters for the power-law relations of the non-adhesive ridge contact area (eqs. (5) and (6)) in table 1 are taken from [19]. thus, at a normal force of 0.5n non-adhesive ridge contact area is measured to approximately 37mm². however, this value can vary significantly depending on the finger size and its angle to the contacting surface. in the following section, the influence of the ridge contact area is investigated by keeping the exponents m and n in the power-law relations in eqs. (5) and (6) constant but scaling the parameters 𝛼 and 𝛽 accordingly. the parameter c2 controlling the area reduction due to the tangential loading is taken from [10]. 4. frequency dependence of electroadhesion in this section, the model prediction of the electroadhesive contribution to the normal force fel is investigated and compared to experimental data by [6]. furthermore, the influences of the uncertain equivalent thickness and resistance of the interfacial gap as well as the subject dependent thickness of the stratum corneum and ridge contact area are investigated. fig. 3 shows the model predictions of the average inferred electroadhesive force in terms of the excitation frequency and the experimental data found by [6] for different subjects. the black curves correspond to the parameters in table 1. furthermore, a variation in the experimentally found ranges is shown for the parameters listed above. in all cases, the model predictions show the same characteristics: for low frequencies, the electrical circuit behaves mainly resistive and the voltage across the interfacial gap approaches the limit in eq. (14). the inferred electroadhesive force is proportional to the square of that voltage, fel ∝ ua 2 (see eq. (2)), and therefore also approaches a limit value. similarly, for frequencies higher than 2000 hz the circuit is mainly capacitive (see eq. (15)) and the electroadhesive force approaches another, larger limit. in agreement with the experimental data, the forces increase by up to 80% from 100 hz to 10000 hz. for decreasing thickness of the stratum corneum and increasing ridge contact area (on the upper left and right of fig. 3, respectively), the electroadhesive force increases approximately linearly in the whole frequency spectrum. however, compared to the variations in the thickness of the stratum corneum, small differences in the ridge contact area have far greater influence on the force. in [20], using another, much simpler modeling approach, the wide46 f. forsbach, m. heß spread experimental results were explained by the variability in the thickness of the stratum corneum. with the mechanically and electrically much more comprehensive model presented in this paper, this result cannot be confirmed. fig. 3 inferred electroadhesive force fel in terms of excitation frequency f for an excitation voltage amplitude of u0 = 140v and an external force of fn=0.5n predicted by the proposed model and, in grey, experimentally found by [6] the influence of the interfacial gap parameters is shown on the lower left and right of fig. 3. if the resistance of the interfacial gap is increased, the force in the resistancedominated low frequency range is increased as well. for the proposed model, values in the range of 1-2m𝛺 appear appropriate. this range is significantly lower than the value calculated in [15] (7m). however, the interface resistance is expected to vary extensively depending on multiple interface and environmental parameters as described in section 3.3. the equivalent interfacial gap thickness influences the whole frequency range significantly (see lower right of fig. 3). the range of 2-5µm appears appropriate and is, as described in section 3.3, in agreement with the values found in the literature. in light of eq. (12) and ca≪csc as well as ca≪ci, the electroadhesive force is roughly proportional to (da)-2 and, thus, increases rapidly for small interfacial gap thicknesses. a rigorous model for frequency-dependent fingerpad friction under electroadhesion 47 the experimental data points for the different subjects in fig. 3 vary extensively: some show convergence for high frequencies, while other do not (yet) converge and in addition there is a significant quantitative scatter. the reason for this is unclear and requires further experimental investigation. particularly, measurements of (ridge) contact area and skin hydration level of the different subjects are of interest. unfortunately, to the authors’ knowledge, this is the only experimental investigation of the frequency dependence. thus, while adequate agreement with some individual subjects is possible by fitting of the parameters in the discussed ranges (see for example the black curve in fig. 3), the model should be validated with a more complete data set of a future experimental study. however, the parameter variations discussed above offer some possible explanations for the observed experimental behavior. fig. 4shows the modeled time response of the gap voltage ua and the inferred force fel for one high frequency case with capacitive behavior (5000 hz) on the right and one in the transition range between resistive and capacitive behavior (100 hz) on the left. the model parameters are chosen as in table 1. for the 100 hz case, the gap voltage leads the excitation voltage by some degrees and the amplitude is approximately 60% of the excitation amplitude. contrary, in the high frequency case the gap voltage is in phase with the excitation voltage and the amplitude transfers to more than 80%. since the electroadhesive force is proportional to the square of the gap voltage, its frequency is twice the excitation frequency and the amplitude is significantly higher for the high frequency case. fig. 4 modeled time response of the voltage ua and the inferred electroadhesive force fel for a sinusoidal excitation voltage with amplitude of u0 = 200v at frequencies of 100hz (left) and 5000 hz (right), both at an external force of fn = 0.5n fig. 5 shows the time responses for a square wave excitation for the frequencies of, again, 100 hz on the left and 5000 hz on the right. for each sudden change in excitation voltage, a maximum or minimum in the gap voltage follows. during the phases of constant excitation voltage, the gap voltage drops due to leakage through the resistors. this voltage drop is significant for the 100 hz case, causing a very volatile time response of the inferred force. for the high frequency case, the gap voltage drop is much less significant and the resulting model prediction of the force is almost constant in time. 48 f. forsbach, m. heß fig. 5 modeled time response of the voltage ua and the inferred electroadhesive force fel for a square wave excitation voltage with amplitude of u0 = 200v at frequencies of 100hz (left) and 5000 hz (right), both at an external force of fn=0.5n 5. conclusions to model the observed frequency dependence in the electroadhesive frictional contact of a finger pad with a touchscreen, an extension of the macroscopic model recently developed by the authors has been proposed. the impedance behavior observed in experiments, resistive for low frequencies and capacitive for high frequencies, was successfully modeled by three parallel circuits of resistor and capacitator in series, one for each electrical layer. the electromechanical parameters were chosen according to recent measurements and the model predictions of the inferred electroadhesive force were compared to a recent experimental study [6]. the model predictions agree qualitatively and quantitatively reasonably well with experimental results. the present model further shows that the significant scattering of the experimental data for different subjects may be due to the variability of crucial parameters such as the ridge contact area, the equivalent interfacial gap thickness and the electrical resistance of the interfacial gap. however, further validation with a more complete experimental data set is needed. finally, the time response of the developed model to different wave forms and frequencies of the excitation voltage is presented and discussed. references 1. bau, o., poupyrev, i., israr, a., harrison, c., 2010, teslatouch: electrovibration for touch surfaces, uist 2010 23rd acm symposium on user interface software and technology, pp. 283-292. 2. vardar, y., güçlü, b., basdogan, c., 2017, effect of waveform on tactile perception by electrovibration displayed on touch screens, ieee transactions on haptics, 10, pp. 488-499. 3. grimnes, s., 1983, electrovibration, cutaneous sensation of microampere current, acta physiologica scandinavica, 118, pp. 19-25. 4. liu, x., lu, z., lewis, r., carré, m.j., matcher, s.j., 2013, feasibility of using optical coherence tomography to study the influence of skin structure on finger friction, tribology international, 63, pp. 34-44. a rigorous model for frequency-dependent fingerpad friction under electroadhesion 49 5. van kuilenburg, j.v., masen, m.a., van der heide, e., 2013, a review of fingerpad contact mechanics and friction and how this affects tactile perception, proceedings of the institution of mechanical engineers, part j: journal of engineering tribology, 229, pp. 243–258. 6. meyer, d.j., peshkin, m.a., colgate, j.e., 2013, fingertip friction modulation due to electrostatic attraction, 2013 world haptics conference (whc), pp. 43-18. 7. vezzoli, e., amberg, m., giraud, f., lemaire-semail, b., 2014, electrovibration modeling analysis, haptics: neuroscience, devices, modeling, and applications lecture notes in computer science, pp. 369–376. 8. shultz, c.d., peshkin, m.a., colgate, j.e., 2015, surface haptics via electroadhesion: expanding electrovibration with johnsen and rahbek. 2015 ieee world 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america, 115, pp. 471-476. 14. heß, m., popov, v.l., 2019, voltage-induced friction with application to electrovibration, lubricants, 7(12), 102. 15. shultz, c.d., peshkin, m.a., colgate, j.e., 2018, on the electrical characterization of electroadhesive displays and the prominent interfacial gap impedance associated with sliding fingertips. 2018 ieee haptics symposium (haptics), pp. 151-157. 16. fruhstorfer, h., abel, u., garthe, c.-d., knüttel, a., 2000, thickness of the stratum corneum of the volar fingertips, clinical anatomy, 13, pp. 429–433. 17. yamamoto, t., yamamoto, y., 1976, dielectric constant and resistivity of epidermal stratum corneum, medical & biological engineering, 14, pp. 494–500. 18. choi, c., ma, y., li, x., ma, x., hipwell, mary, 2021, finger pad topography beyond fingerprints: understanding the heterogeneity effect of finger topography for human–machine interface modeling, acs applied materials & interfaces, 13, pp. 3303–3310. 19. dzidek, b.m., adams, m.j., andrews, j.w., zhang, z., johnson, s.a., 2017, contact mechanics of the human finger pad under compressive loads, journal of the royal society interface, 14, 20160935. 20. vezzoli, e., ben messaoud, w., nadal, c., frédéric, g., amberg, m., lemaire-semail, b., bueno, m.a., 2015, coupling of ultrasonic vibration and electrovibration for tactile stimulation, european journal of electrical engineering, 17, pp. 377-395. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 321 332 https://doi.org/10.22190/fume180110005b © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper advanced morphological approach in aerospace design during conceptual stage andreas bardenhagen 1 , dmitry rakov 2 1 aircraft design and aerostructures at the institute of aeronautics and astronautics, technische universität berlin, germany 2 a. a. blagonravov mechanical engineering institute, russian academy of sciences, russia abstract. this paper presents an advanced morphological approach supporting designers and developers in their search for as well as synthesis and analysis of new engineering solutions during the conceptual design stage. the proposed method is based on the cluster analysis and the set theory, the set of rules and engineering implementations maximizing the gain of the products potential. the number of possible combinations using the standard morphological technology is extremely large. we present a mathematical framework that handles this problem. the method was evaluated with case studies of new engineering solutions in aerospace, ecology and adaptive soundproofing system. the case studies verified the significant potential of the proposed approach in comparison with the methods presently in use. key words: conceptual design support, solution space, morphological matrix, design modeling, new engineering solutions, unmanned aircraft systems (uas) 1. introduction in engineering practice it is usual to strive for a new optimum engineering solution (es) within the constraints directly at the beginning of the project. the search for a new es makes it necessary to start with the main tasks of the system analysis, namely, decomposition, analysis and synthesis of an es. an analysis is carried out to determine the properties of the es and its operation, while the synthesis task consists in the development of an es model, the definition of its structure (structural synthesis) and the received january 30, 2018 / accepted december 21, 2018 corresponding author: rakov dmitry affiliation, address: mechanical engineering institute, 101990 moscow, m.harinonevsky per, 4. russia e-mail: rdl@mail.ru 322 a. bardenhagen, d. rakov corresponding parameters (parametric synthesis or parametric optimization) which are necessary for an efficient functioning of the es and the achievement of the stated goal. the structure represents the most concrete way of representing a system, and the most concrete stage in generating systems during the conceptual stage of design process [1, 2]. polovinkin has examined three conceptual design levels of an es [3]. the characteristics of the synthesized systems on the third level of an optimization can be improved (statistically) in average by 10-15% (fig. 1). at levels 1 and 2 the characteristics can be improved in average by 30-35 % and sometimes more. fig. 1 improving system performance depending on conceptual design levels the es structure is composed of a set of elements (constructive, functional or technological) and a set of their relationships. the search of the rational structure of an es is the attainment of compromise levels for a number of criteria. the target function for optimum search does not correspond to the main requirements of the theoretical method for optimization because it is discontinuous or cannot always be determined; it exists in operator notation; it is not based on analytical expressions; it is not differentiable, nor unimodal; neither is it separable nor additive. it is impossible to build a hyper surface of the target function and to predict its change on an increment of variables [4]. during the stage of structural synthesis of a new es intuitive (brainstorming, mind mapping, triz, synectics etc.) and discursive (morphological analysis, cause-and-effect diagram, osborn-checklists, etc.) techniques can be used. the most common method among the discursive techniques is a morphological analysis (ma). the morphology analysis was developed by fritz zwicky – a swiss astrophysicist based at the california institute of technology (caltech) [5]. researchers applied the morphological analysis first to astronomical studies and the development of rocket propulsion systems. as a problem-structuring and problem-solving technique, the ma was designed for multi-dimensional and non-quantifiable problems where causal modeling and simulation do not function well, or not at all [6]. the morphological design will be structured into a set of functional and characteristic attributes [7]. against each of these attributes, the designer will have to select a advanced morphological approach in aerospace design during conceptual stage 323 conceptual solution. the combination of all these solutions then generates the final design concept [8]. the ma has been applied by a number of researchers in the fields of engineering science [9, 10]. this is considered as the hierarchical morphological multicritieria design – providing conceptual lens. this approach is based on ordinal estimates of design alternatives for system parts/components, and on new interval multiset estimates for design alternatives with special attention to the aggregation of modular solutions [11]. the morphological approach is widely used in germany. the society of german engineers has developed two sets of rules for engineers: vdi 2222, “design methods: methodical development of engineering principles” [12] and vdi 2221, “design methods of technical systems and products” [13], in which it is recommended to use the morphological approaches to find a new es. the disadvantage of morphological methods is the impossibility to analyze all possible variants – potentially the number of generated variants in the morphological array can be enormous (up to a hundred thousand and millions possible variants). 2. methodological background to reduce the dimensionality of a morphological array an advanced morphological approach is developed. applying the means of this approach makes it possible to solve two groups of tasks:  direct task: after creating a morphological array the search for engineering solutions occurs with the help of clustering.  inverse task: during the search for es the nearest vicinities of the more effective variants based on known es are screened. the proposed approach can be represented by the following set [14, 15]: {t, z, w, v, o, l, m, n, k, c, p} in the realized approach and computer program the following options can be selected in table 1. to illustrate this approach two es have been synthesized and studied. 3. advanced morphological approach for the adaptive noise insulation system current r&d efforts to develop quiet aircraft configurations feature, amongst other solutions, engines relocated to the positions above the wing or at the rear fuselage as well as fully or semi fuselage integrated engines. all of these solutions imply higher noise levels in the cabin. effective insulation technologies are therefore required to reduce cabin noise. the developed methodology was applied using the inverse task approach to search for perspective noise insulation systems based on 4s-technology (steerable sound suppression system). 324 a. bardenhagen, d. rakov table 1 sequence of tasks no task task definition 1 t formulation of the problem t1 synthesize and choose the best es t2 reverse es search 2 z solution level z1 choice of the best function z2 choose of the best structure 3 w criteria w1vector criterion w2 scalar criterion 4 v additional information v1 no v2 well-known or existing solutions v3 cross-consistency assessment matrix 5 o measurement method o1 point scale 6 l system investigations l1 integrated system l2 the study of the subsystems 7 m variants assess m1 variants assessment in general, after the synthesis of the parts m2 evaluation of individual subsystems before the synthesis 8 n variants generating n1 loop through all variants n2 loop through all variants with choice n3 random selection n4 random selection with choice 9 k clustering method k1 hamming distance k2 l1-norm 10 c target function c1 additive c2 multiplicative 11 p number of levels of the system under consideration p1 one p2 two and more the investigations demonstrated the possibility of reducing significantly the cabin noise level by using thin-walled porous materials [16, 17]. this effect has been achieved by the new 4s-technology due to the creation of deformation zones on the surface and along the thickness of an insulating material as well as the interaction of that material with elastic membranes. a variable level of deformation and, hence, density of this sound advanced morphological approach in aerospace design during conceptual stage 325 insulation material enables an adaptation to a desired degree of sound suppression and also to a desired frequency band. this ultimately allows the creation of active and adaptive systems for sound insulation alongside passive ones. the synthesis process for this problem goes through the following steps. 1. for the problem solving the following set of subtasks is selected from table 1: {t2, z2, w1, v2, o1, l1, m1, n2, k1, c1, p1 } 2. the morphological matrix is created and reduced to the key characteristics (attributes) shown in table 2. the total space of the morphological matrix contains 1152 potential variants of engineering solutions. table 2 morphological matrix with possible 4s system variants attribute (descriptors) option 1 option 2 option 3 option 4 1 stress increased pressure reduced pressure combined 2 source of stress pneumatic mechanical electric hydraulic 3 membrane flexible hard 4 control no controlled adaptive 5 combination of volume 1 2 3 n 6 combination of layers homogeneous material heterogeneous material 7 kind of a material porous cellular 3. the reference variant (green cells in the matrix) consists of the following structure: {p1 2 , p2 1 , p3 1 , p4 1 , p5 1 , p6 1 , p7 1 } pi n where i-subscript is an attribute, n superscript is a particular option of the i-th attribute. 4. reference variants are entered into a morphological matrix, and then a set of permuted variants is generated. 5. in the next stage of the variant generation their estimated value (estimation) is calculated, the initial selection is carried out and an array of rational variants for the subsequent analysis is formed. 6. henceforth, the clustering of the variants by measuring a predefined level of similarity based on the hamming distance is carried out. for final analysis 60 generated rational variants, grouped in 8 clusters, were chosen (fig. 2). 326 a. bardenhagen, d. rakov fig. 2 clusters and variants in the solutions space 7. after clustering and choosing variants, the final choice set derived from the morphological matrix was analyzed for optimization and experimental investigations (figs. 3 and 4). {p1 2 , p2 1 , p3 1 , p4 1 λ p4 2 λ p4 3 , p5 1 , p6 1 λp6 2 , p7 1 λ p7 1 } fig. 3 expanded set of variants with 4s technology advanced morphological approach in aerospace design during conceptual stage 327 fig. 4 sound-insulating experimental panel enclosed in an airproof package, vacuum pump and indicator in the rear during the experiments the type of materials, the degree of deformation, porosity, rigidity, etc. were varied. the tension in the investigated samples was created mechanically and pneumatically. the tests were carried out in a frequency band from 63 up to 5000 hz. in the appropriate low-pressure (stress range) the experimental panels exhibit different sound insulation properties. pressure change is less than 1% from atmospheric pressure. the average sound insulation achieved was 12 db on the basis of the 4s-effect (fig. 5). fig. 5 sound reduction index of foamed 30-mm polyurethane panel under low-pressure (decrease from atmospheric pressure less as 1%) 328 a. bardenhagen, d. rakov 4. advanced morphological approach for stratospheric unmanned aircraft systems the number of potential roles for unmanned aircraft systems (uas) systems is legion, especially in the civil field. the demands defined by the customer lead to the system requirements which determine principally shape, size, performance and costs of the air vehicle, but also of the overall uas operating system. some of the more important parameters involved, beginning with the air vehicle, are briefly discussed below [19]. the studied uas shall have performance potential to fulfill the following mission:  uas with civil mission (e.g. observation, research, communication node, etc.)  flight altitude: 12-20 km (above jetstream)  long flight in the stratosphere (as long as possible on station; ideal: >1 week)  no range requirement – defined position be hold within area of 4 km 2  max. 10 m/s wind during climb – no jetstream – max. 10 m/s wind on position for 4h/day flight time  payload 1kg, constant 50w electric energy consumption. for the problem solving the following set of subtasks was selected from table 1: {t1, z1, w1, v2, o1, l1, m1, n2, k1, c1, p1} the morphological matrix and the criterion table are given in tables 3 and 4. the complete morphological matrix contains in total 248,832 potential uas variants. first, 12,000 variants are generated using random selection. then 256 best variants identified by experts (~2% of the 12,000) are selected for analysis and grouped into 16 clusters (fig. 5). for the clustering a geometric approach based on the compactness hypothesis is used. the solution space also contains 15 reference variants (fig. 6). table 3 criterion table criteria comments 1 uas system cost estimated cost of complete system (ground support & uas) 2 cost per mission cost per mission/flight incl. cost for fuel/energy, operators, etc. 3 total weight /mission flight time this is a technical key performance indicator for a long endurance mission 4 emissions emissions like co2, etc. and noise 5 reliability 6 energy efficiency 7 speed (wind and time for climb) capability of uas to reach mission altitude and endure wind 8 flight duration time of uas to stay on the predefined position (time for climb/descent excluded) 9 safety (flight in the stratosphere) safe operation including hazards from fuel, tethers, electromagnetic waves, etc. advanced morphological approach in aerospace design during conceptual stage 329 table 4 morphological matrix category px attribute (descriptors) option px 1 option px 2 option px 3 option px 4 lift 1 lift aerodynamic thrust aerostatic thrust 2 thrust coupled to lift generation independent from lift generation energy storage 3 internal energy storage non chemical, reversible (e.g. lipo battery) chemical, irreversible (e.g. fuel tank) mechanic (e.g. fly-wheel) energy supply 4 external energy supply non continuous (e.g. solar, microwave) interrupted, discontinuous (e.g. tank) power generation 5 engines electric internal combustion (e.g. diesel engine) gas turbine reaction engine, e.g. rocket motor 6 engines single engine twin engines more than 2 engines flight control 7 flight height control aerodynamic (e.g. elevators) changing of thrust aerostatic 8 flight directional control aerodynamic (e.g. rudder) thrust imbalance (e.g. two engine geometry fuselage 9 fuselage no one fuselage twin-boom geometric characteris tics wing 10 increasing the wing area no yes 11 wing area control no yes, e.g. max. solar radiation use flight guidance 12 trajectory constant height changing height 13 guidance remote controlled autonomous 330 a. bardenhagen, d. rakov fig. 5 solutions space of uas with 16 clusters fig. 6 solutions space of uas with 15 reference variants based on the cluster analysis and after expert analysis, the following conclusions can be drawn [20]: 1. many variants have incompatible options or, based on today's knowledge, are impossible to implement (fig. 5: clusters 7, 8, 13, 15 and 16) 2. among the reference variants (fig. 6), the configuration sharp, pat.us 8307922, [21] and the configurations solareagle, helios, solar impulse have the highest relative value, advanced morphological approach in aerospace design during conceptual stage 331 i.e. “estimation” in fig. 6 has the highest value. the reference configuration showing the lowest value for the required mission is the stratosphere rotor platform [22] 3. better reference variant no. 32 is in cluster 4. improved reference variables are located in cluster 4. without external energy supply, uas solutions correspond to the aerodynamic configuration (cluster 4) or helicopter configuration (cluster 5) with power supply by cable. 4. many generated and selected variants as well as complete clusters have hybrid properties (table 3: attributes p1 and p2). 5. cluster 14 contains aerodynamic electrical uas with energy storage on board with external power supply with aerodynamic or thrust control. for further examinations, the following four areas are of interest: 1. aerodynamic configurations with energy storage on board as well as external power supply and with aerodynamic or thrust vector flight control. 2. investigations of hybrid uas. 3. aerodynamic configurations or helicopter configurations with power supply by cable. 5. conclusions the major aim of the presented approach is the expansion of the number of potential variants, their clustering and efficient selection during the solution space synthesis, in order to increase the number of innovative solutions in engineering design. in two case studies the technique demonstrates the power of the approach for generating design concepts. in addition to the above, the proposed approach clarifies and arranges the structuration of the decision task. the validity of decision-making increased while the multitude of variants among which the selection is carried out is broadened. this enables quality improvement of the developed engineering systems. references 1. hubka, v., eder, w.e., 1998, theory of technical systems: a total concept theory for engineering design, new york:springer-verlag, 278 p. 2. milčić, d., miltenović, v., 1999, application of artificial intelligence methods in gear transmitters conceptual design, facta universitatis-series mechanical engineering, 1(6), pp.721-734. 3. polovinkin, a., 1991, design theory of new engineering solutions: technology patterns and their applications. edn. informelectro, 104 p. 4. mishin, v., osin, m., 1978, introduction to aircrafts design, moscow: edn. mashinostroenie, 128 p. 5. zwicky, f., 1966, entdecken, erfinden, forschen im morphologischen weltbild, droemer/knaur, münchen, zürich, 206 p. 6. ritchey, t., 2006, problem structuring using computer-aided morphological analysis, journal of the operational research society, 57(7), pp. 1-12. 7. pahl, g., beitz, w., 1996, engineering design: a systematic approach, 2nd edn. springer, london, 617 p. 8. matthews, p., 2011, challenges to bayesian decision support using morphological matrices for design: empirical evidence, research in engineering design, 22(1), pp. 29-42. 9. grote, k.h., antonsson, e. (eds.), 2011, springer handbook of mechanical engineering, springer, 1520 p. 332 a. bardenhagen, d. rakov 10. fargnoli, m., troisi, r., rovida, e., 2006, the morphological matrix: tool for the development of innovative design solutions, 4th international conference on axiomatic design, icad, june 13-16, 2006, florence, italy, pp. 1-7. 11. levin, m.s., 2015, modular system design and evaluation, springer, 413 p. 12. vdi standard: vdi 2221, 1993, systematic approach to the development and design of technical systems and products, beuth verlag. 13. vdi standard: vdi 2222 ,1997, part 1. methodic development of solution principles. beuth verlag. 14. andreychikov, a., andreichikova, o., 2014, system analysis and synthesis of strategic decisions in innovation: conceptual design of innovative systems, moscow, edn. urss, 424 p. 15. rakov, d., 1996, morphological synthesis method of the search for promising technical systems, ieee aerospace and electronic systems magazine, 12, pp.3-8. 16. rakov, d., thorbeck, j., 2008, ein beitrag zur beeinflussung des schalldämmungsverhaltens von akustischen dämmelementen, lärmbekämpfung zeitschrift für akustik, schallschutz und schwingungstechnik, 1/2008, pp. 41-43. 17. rakov, d., thorbeck, j., pecheykina, m., 2018, design and modeling of adaptive noise suppression systems with morphological approach. in: hu, z., petoukhov, s., he, m. (eds), advances in artificial systems for medicine and education aimee 2017, advances in intelligent systems and computing, vol. 658. springer, cham, pp. 266-272. 18. rakov, d., timoshina, a., 2010, structure synthesis of prospective technical systems, ieee aerospace and electronic systems magazine, 25(2), pp. 4-10. 19. austin, r., 2010, unmanned aircraft systems: uavs design, development and deployment, wiley, 372 p. 20. bardenhagen, a., gavrilina, l.v., klimenko, b.m., pecheykina, m.a., rakov, d.l., statnikov, i.n., 2017, a comprehensive approach to the structural synthesis and evaluation of engineering solutions in the design of transportation and technological systems, journal of machinery manufacture and reliability, 46(5), pp. 453–462. 21. delaurier, j., gagnon, b., wong, j., williams, r., hayball, c.,1985, research on the technology of an airplane concept for a stationary high altitude relay platform (sharp), 32nd annual general meeting of the canadian aeronautics and space institute, montreal, may 27, pp. 5-22. 22. wilfred, p.s., petrides, t., 1961, supersonic rotary wing platform, patent usa 3116040, date of priority 26. june 1961. facta universitatis series: mechanical engineering vol. 18, no 4, 2020, pp. 525 536 https://doi.org/10.22190/fume201005043r © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper evolution of the carbon nanotube bundle structure under biaxial and shear strains leysan kh. rysaeva1, dmitry v. bachurin2, ramil t. murzaev1, dina u. abdullina1, elena a. korznikova1,3, radik r. mulyukov1,3, sergey v. dmitriev3,4 1institute for metals superplasticity problems of the russian academy of sciences, russia 2institute for applied materials, karlsruhe institute of technology, germany 3ufa state petroleum technological university, russia 4national research tomsk state university, russia abstract. close packed carbon nanotube bundles are materials with highly deformable elements, for which unusual deformation mechanisms are expected. structural evolution of the zigzag carbon nanotube bundle subjected to biaxial lateral compression with the subsequent shear straining is studied under plane strain conditions using the chain model with a reduced number of degrees of freedom. biaxial compression results in bending of carbon nanotubes walls and formation of the characteristic pattern, when nanotube crosssections are inclined in the opposite directions alternatively in the parallel close-packed rows. subsequent shearing up to a certain shear strain leads to an appearance of shear bands and vortex-like displacements. stress components and potential energy as the functions of shear strain for different values of the biaxial volumetric strain are analyzed in detail. a new mechanism of carbon nanotube bundle shear deformation through cooperative, vortex-like displacements of nanotube cross sections is reported. key words: carbon nanotube bundle, plane strain conditions, lateral compression, shear deformation, deformation mechanisms received october 05, 2020 / accepted november 11, 2020 corresponding author: sergey v. dmitriev ufa state petroleum technological university, kosmonavtov st. 1, ufa 450062, russia. national research tomsk state university, lenin ave. 36, tomsk 634050, russia e-mail: dmitriev.sergey.v@gmail.com 526 l.kh. rysaeva, d.v. bachurin, r.t. murzaev, d.u. abdullina,... 1. introduction in nature, there are many allotropic modifications of carbon with different physical properties. one of such modifications is carbon nanotube (cnt), which is but a rolled sheet of graphene. the peculiarity of cnts is that they interact rather weakly with each other and can form cnt bundles [1-4]. the interest in cnts is primarily due to their unique mechanical properties such as very high tensile strength, high young's modulus, and ultimate fracture strain [5-8]. cnts are flexible and lightweight and have good thermal and electric conductivity. mechanical applications of cnts include high-strength ropes [2,9], fibers [10-14], polymerand metal-matrix composites [15-17], solid lubricants [17-20], and many others. tensile [2,9-14] and compressive [21-28] behavior of vertically aligned cnt brushes and forests have been most extensively studied experimentally. straining of vertically aligned cnts implies deformation along their axis, while straining of horizontally aligned cnts means deformation normal to their axis. lateral compression of isolated cnts or cnt bundles has not been widely studied yet [29-33]. it should be noted that the methods of winding, drawing, micromechanical rolling, and shear pressing [34-37] can be applied to obtaining horizontally aligned cnt bundles from the vertically aligned ones. it has been revealed that cnt bundles can deform elastically up to hydrostatic pressure of 1.5 gpa, and the hydrostatic deformation of cnt lattice is reversible up to 4 gpa [38]. karmakar and co-authors have demonstrated that deformation of cnt bundles under non-hydrostatic pressure is reversible (elastic) at stresses below 5 gpa [39]. in the last two decades, considerable attention has been paid to computer modeling of the mechanical properties of cnt bundles for a better understanding of their properties and deformation mechanisms. since a nanotube bundle can be represented as a material with highly deformable elements, new deformation mechanisms different from those typical for conventional materials can be expected. in particular, transformation of vertically aligned cnt forest into a horizontally aligned one under pressure has been studied via mesoscopic modeling [40,41]. different morphological patterns of cnts subjected to large deformations have been revealed by means of continuum shell model [42]. the rigidity of cnt crystal does not decrease with increasing cnt diameter [1]. authors [43-45] have found that cnts can exist in two stable configurations (circular and collapsed ones) depending on their diameter. nonlinear coarse-grained potentials specially developed for cnts have allowed studying the mechanical response and failure of cnt bundles [46]. the chain model developed in ref. [47] has been applied successfully to the investigation of structure and properties of carbon nanoribbons [47-51] and dynamics of surface ripplocations [52]. recently, the chain model has been used for simulation of evolution of cnt bundle structure under both lateral uniaxial and biaxial compression [53-55]. the present work is devoted to a study of structural evolution of zigzag cnt bundle under biaxial compression with the subsequent shear straining using the chain model [47,53]. the latter uses a reduced number of degrees of freedom but at the same time gives at least one order of magnitude acceleration of computations without loss of accuracy, in comparison with full atomistic modeling under plane strain conditions. despite the fact that the chain model [47,53] can be applied for various cnts, here we restrict ourselves to the consideration of only single-walled cnts of equal diameter. evolution of carbon nanotube bundle structure under biaxial and shear strains 527 2. model and computational details the computational cell of the modeled bundle has a parallelogram shape and contains 1080 zigzag cnts (30 horizontal layers and 36 vertical layers) of equal diameter aligned along the z-axis. for clarity, fig. 1 represents only a part of the computational cell (two horizontal layers and two vertical layers). each cnt consists of 60 carbon atoms. total number of atoms in the cell is n=60×30×36=64800. periodic boundary conditions are applied along the two (x,y) cartesian directions, i.e., infinitely elongated along the z-axis cnts are considered. lateral compression of the cnt bundle is applied using plane strain conditions, namely when each carbon atom stays rigidly in “its own” atomic row along the z-axis, but can move freely along the (x,y) plane. thus, each atom has only two degrees of freedom, which allows us to reduce the dimensionality of the problem from three-dimensional to two-dimensional one. fig. 1 geometry of the computational cell. only two horizontal layers and two vertical layers are presented for clarity, while in simulation 30 horizontal layers and 36 vertical layers are used. carbon atoms in the upper row are indicated by large green circles, and those in the lower row are indicated by small circles. each cnt contains 60 carbon atoms. the cnt cross-sections represent a triangular lattice the following geometrical parameters for the model of cnt bundle are chosen. the valence bond length in graphene is =1.418 å. the distance between neighboring atomic rows in the zigzag cnt is a=1.228 å. the cnt diameter is d=23.46 å and the equilibrium distance between the neighboring cnt walls is d=3.30 å. thus, the centers of neighboring cnts are at the distance of a=d+d=26.76 å. the hamiltonian describing the interaction between carbon atoms includes the four terms, namely h=k+ub+ua+uvdw. (1) here the first term, k, gives the kinetic energy of the carbon atoms, ub stands for the energy of valence bonds, ua is the energy of valence angles, and uvdw is the energy of van der waals interactions between cnts. the model parameters were calculated based 528 l.kh. rysaeva, d.v. bachurin, r.t. murzaev, d.u. abdullina,... on the interatomic potential developed for sp2-carbon by savin et al. [56] and further applied for investigation of various phenomena [56-61]. the initially constructed cnt bundle is subjected to relaxation in order to obtain minimum energy configuration. a biaxial compression (with xx<0 and yy<0) is applied by a strain increment of xx=yy= −0.0025, which is followed by giving to carbon atoms small random displacements in the range from −10-6 to 10-6 å along the xand yaxes and further minimization of potential energy u=ub+ua+uvdw. the relaxation of the system is stopped, when the absolute value of the maximal force acting on atoms becomes less than 10-10 ev/å. this biaxially strained structure is deformed again by applying subsequently a shear strain up to =0.30 with the step of =0.01. the strain state of deformed system is thus characterized by two parameters: the absolute value of the volumetric strain, | |=|xx+yy|, (2) and shear strain, . in the present study, no thermal effects are taken into consideration. for further details related to the simulation setup and construction of the hamiltonian of the chain model, we refer the reader to our previous publications [53-55]. 3. simulation results and discussion firstly, structural evolution of the cnt bundle under simple biaxial compression is considered, and, secondly, biaxial compression followed by a shear strain is analyzed. fig. 2 structural evolution of cnt bundle at different values of biaxial volumetric strain,  (presented under each structure). translation cells of the structures are shown by red lines and includes four cnts marked for clarity with red, blue, black and light green colors. for a more detailed visualization, only a part (lower left corner) of the computational cell is shown evolution of carbon nanotube bundle structure under biaxial and shear strains 529 3.1. biaxial compression fig. 2 demonstrates the evolution of cnt bundle structure subjected to biaxial volumetric strain, , along the xand y-axes. at strains of |θ | < 0.02, cnts preserve mainly their circular cross-sections and only small distortions of cnts, barely visible to the naked eye in the figure, are observed. an increase of biaxial strain up to |θ | = 0.03 results in an appearance of elliptical cross-sections. at that, as clearly seen in fig. 2, in the horizontal close-packed rows, the elliptic cross-sections are inclined alternatively in the opposite directions. the latter is the necessary condition to maintain an equilibrium state within the bundle. the straining up to |θ | = 0.1 leads to a further compression of cnts within the bundle and no collapsed cross-sections are observed. note that the deformation is performed up to the compressive biaxial volumetric strain |θ | ≤ 0.1 to avoid the first-order phase transition related to a formation of the collapsed cnts, as revealed in ref. [55]. fig. 3 structural evolution of cnt bundle at different values of biaxial volumetric strain, |θ | (horizontally) and subsequent shear strain, γ (vertically). for a more detailed visualization, only a part (lower left corner) of the computational cell is shown 3.2. shearing of biaxially pre-strained cnt bundle the structural evolution of the cnt bundle under biaxial volumetric strain, , followed by shear straining is presented in fig. 3. at low biaxial pre-strains in the range of |θ | ≤ 0.02, the deformation is almost uniform, and only small distortions (will be discussed later in more detail) in the form of shear bands in a triangular lattice of cnt bundle are observed at  = 0.3. it should be noted that all nanotubes in the computational cell are deformed approximately in the same way, and their cross-sections have the elliptical non-collapsed shape inclined towards the direction of the applied shear strain. an increase of the biaxial pre-strain up to |θ |=0.03 results in an appearance of longitudinal-transverse shear bands and the nuclei of vortex-like displacements. 530 l.kh. rysaeva, d.v. bachurin, r.t. murzaev, d.u. abdullina,... =0.03 =0.15 =0.30 | |=0.02 | |=0.03 | |=0.05 | |=0.07 | |=0.10 fig. 4 displacement field of the centers of mass for each cnt in the computational cell at different values of biaxial volumetric strain, |θ | (vertically), and shear strain, γ (horizontally) evolution of carbon nanotube bundle structure under biaxial and shear strains 531 moreover, the compression of the horizontal cnt layers becomes inhomogeneous: some layers turn out to be less deformed in contrast to the elements of the neighboring layers, where the deformation is much more pronounced (best seen for |θ |=0.03 and  = 0.3). the alternation of such layers with different compression values is to maintain an equilibrium state, which is seen in fig. 3 for |θ |=0.03. further increase in biaxial deformation from |θ |=0.05 to 0.1 and subsequent shear result in disordering of the cnt bundle structure. the fraction of collapsed cnts gradually increases with an increase in biaxial deformation and subsequent shear strain and is maximal at |θ | = 0.1 and  = 0.3. fig. 4 illustrates the true displacement of the centers of mass of cnts with respect to the uniformly deformed state. at that, the uniform biaxial compression and shear deformations were subtracted for clarity. at pure shear without preliminary biaxial deformation (| | = 0.0), no formation of shear bands or vortices in the structure occurs (not shown in fig. 4). the same can be said about the case of | | = 0.02, as seen, at low biaxial strains, no significant distortions in the structure of cnt bundle are observed. with an increase in the preliminary biaxial deformation, shear bands and vortices begin to appear, which become wider with an increase in the shear strain. the vortices are best seen in fig. 4 in the structure at | | = 0.1 and  = 0.30. since the average size of vortices is significantly smaller than the size of the computational cell, it can be argued that in this work we consider a representative volume of a cnt bundle. fig. 5 dependence of stress components σxx (a) and σxy (b) on shear strain, γ at different values of biaxial volumetric strain, | |, as indicated in the legends the structural evolution can be also well seen on the stress-strain curves shown in fig. 5. the behavior of stress components xx and yy is very similar, which is expected due to the hexagonal symmetry of the structure, that is why only xx is shown in (a). an increase in biaxial compression (| | > 0.03) leads to the first increase of both xx and yy components and saturation at  = 0.03, so that xx and yy practically do not change with increasing shear strain. for all values of | |, pressure p=-(xx + yy)/2 decreases with increasing . note that for | | =0 for  > 0 pressure becomes negative. this means that the cnt bungle with no pre-stress shrinks volumetrically under shear deformation. the behavior of xy component is shown in fig. 5(b). for the structures with | | > 0.03, the stresses linearly decrease with 532 l.kh. rysaeva, d.v. bachurin, r.t. murzaev, d.u. abdullina,... increasing strain as shown in fig. 5(c). this occurs because the structure of nanotubes initially has a rearranged structure that noticeably differs from the initial one, and due to this, the nucleation of vortex segments occurs already at the compressive state, and the following shear deformation leads to their growth and increase in number. shear stress for | | =0 has the opposite sign as compared to the other values of volumetric strain. fig. 6 demonstrates potential energy of the cnt bundle per atom as the function of shear strain for different values of the compressive biaxial volumetric strain (indicated for each curve). in figs. 6(b) to 6(d), the three components of the potential energy u=ub+ua+uvdw are given separately, i.e., the energy of van-der-waals interactions (uvdw), the energy of valence bonds (ub) and the energy of valence angles (ua), respectively. fig. 6 potential energy of the cnt bundle per atom as the function of shear strain, γ, calculated for different values of the compressive volumetric strain, | | (indicated for each curve). total potential energy, u=ub+ua+uvdw, is shown in (a). its three components are presented separately: the energy of van-der-waals interactions, uvdw (b), the energy of valence bonds, ub (c), and the energy of valence angles, ua (d) the energy of the van-der-waals interactions is negative since it is described by the lennard-jones potential with the zero energy level corresponding to well-separated atoms. the creation of the van-der-waals bonds leads to the reduction of energy below zero. energies ub and ua are positive because for them the zero energy level corresponds to the unstrained flat graphene so that the bending deformation to create a cnt results in an increase of potential energy above zero. evolution of carbon nanotube bundle structure under biaxial and shear strains 533 as seen in fig. 6(b), at | | ≤ 0.03 the energy of van-der-waals interactions, uvdw, shows the tendency of a very slow reduction with increasing shear strain. this can be explained by elliptization of cnts and, as a consequence, an increase in the contact area between adjacent cnt walls, which effectively results in the formation of new van-der-waals bonds and, therefore, to the reduction of the corresponding energy. at | |=0.02, in the range of  < 0.1, uvdw slightly increases. this can be a result of the interplay between cnt elliptization, which leads to a decrease of uvdw and pore opening between cnts due to a better packing under not very high compressive strain. at | | ≥ 0.05, uvdw decreases with increasing shear strain first very rapidly and then more slowly. this is obviously due to the collapse of cnts, which leads to the formation of new van-der-waals bonds between the inner walls of collapsed cnts. fig. 6(c) reveals that the energy stored by valence bonds, ub, is for two orders of magnitude smaller than the corresponding values of both uvdw and ua. this is due to very high rigidity of the valence bonds with respect to their tension/compression. one can conclude that the deformation of the cnt bundle occurs mainly due to the bending of the cnt walls and van-der-waals interactions between cnts with a negligible contribution from the cnt walls compression or tension. fig. 6(d) clearly shows that the energy stored by valence angles, ua, increases with , which is due to elliptization of cnts growing with shear strain. at | | ≥ 0.05, a reduction of ua is seen, when  < 0.05. in this initial range of shear strain, collapse of cnts takes place which frees up the space for non-collapsed cnts and their cross-sections become closer to circles leading to the net reduction of ua. 4. conclusions in summary, for the first time, structural evolution of zigzag cnt bundle with the nanotubes of the same diameter under biaxial lateral compression with the subsequent shearing was investigated in the frames of the chain model. nanotube bundle is a material with highly deformable elements, and, therefore, the deformation mechanisms in them can differ significantly from those typical for conventional polycrystalline or amorphous materials. simulations revealed that simple biaxial compression results in elliptization of cnt cross sections and formation of the characteristic pattern where cnts in the horizontal close-packed rows are inclined in the opposite directions. formation of shear bands and vortex-like displacements in cnt bundle subjected to biaxial pre-straining with subsequent shearing were found. the analysis of the potential energy during shear showed that the energy stored by the valence bonds is for two orders of magnitude smaller than both the energy of van-der-waals interactions and the energy stored by the valence angles. thus, the deformation of cnt bundle occurs mainly due to the bending of the cnt walls and van-der-waals interactions between cnts play a negligible role. it should be emphasized that the formation of shear bands is very common for conventional crystalline and amorphous materials, while the formation of vortex-type displacement patterns observed in the present study can be regarded as a new deformation mechanism which can be realized in the material with highly deformable structural units. investigation of the effect of the cnt diameter and multi-walled cnts nanotubes on the deformation behavior and structural evolution may become a natural continuation of the present work. in addition, investigation of cnt bundle with armchair orientation (in 534 l.kh. rysaeva, d.v. bachurin, r.t. murzaev, d.u. abdullina,... contrast to zigzag orientation studied here) are also of interest for future research. test calculations show that the chain model used in this work reproduces well the stiffness of the cnt wall in bending and compression, as well as van der waals interactions; nevertheless, it is important to compare the results reported here with the results of fullatomic modeling, which is planned to be done in one of the future works. acknowledgements: the work of e.a.k. 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improving bending rigidity of graphene nanoribbons by twisting, mech. mater., 137, 103123. 3503 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 127 143 https://doi.org/10.22190/fume200220034s © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper user defined geometric feature for the creation of the femoral neck enveloping surface miloš stojković1, milan trifunović1, jelena milovanović1, stojanka arsić2 1university of nis, faculty of mechanical engineering, nis, serbia 2university of niš, faculty of medicine, nis, serbia abstract. there is a growing demand for application of personalized bone implants (endoprostheses or macro-scaffolds, and fixators) which conform to the anatomy of the patient. hence the need for a cad procedure that enables fast and sufficiently accurate digital reconstruction of the traumatized bone geometry. research presented in this paper addresses digital reconstruction of the femoral neck fracture. the results point out that the user-defined (geometric) feature (udf) concept is the most convenient to use in digital reconstruction of numerous variants of the same topology, such as in this kind of bone region. udf, named femoneck, is developed to demonstrate capability of the chosen concept. its geometry, controlled by a dozen of parameters, can be easily shaped according to the femoral neck region anatomy of a particular patient. that kind of the cad procedure should use a minimally required set of geometric (anatomical) parameters, which can be easily captured from x-ray or computed tomography (ct) images. for the statistical analysis of geometry and udf development we used ct scans of proximal femur of 24 caucasian female and male adults. the validation of the proposed method was done by applying it for remodeling of four femoral necks of four different proximal femurs and by comparing the geometrical congruency between the raw polygonal models gained directly from ct scan and reconstructed models. key words: femoral neck, proximal femur, femur, bio-shape, user-defined feature, cad received february 20, 2020 / accepted march 18, 2021 corresponding author: miloš stojković university of niš, faculty of mechanical engineering, 18000 niš, aleksandra medvedeva 14, niš, serbia e-mail: milos.stojkovic@masfak.ni.ac.rs mailto:milos.stojkovic@masfak.ni.ac.rs 128 m.stojković, m.trifunović, j. milovanović, s. arsić 1. introduction the personalized medicine (pm) market encompasses tailor-made medical products segmented into pm diagnostics, pm therapeutics, pm care, and nutrition & wellness. considering a remarkable growth as well as volume increasing trends, two market research studies [1, 2] indicate the global personalized medicine market as the greatest single business opportunity of our lifetime. these two market observations, and a series of similar ones, highlight “personalized medical care” as the keyword that will be unavoidable in the terminology of health care in near future. in the relevant market niche of bone implants (endoprostheses and macro-scaffolds) and fixators, personalization becomes an undoubtable trend, too. parthasarathy [3] observes that personalized implants for reconstruction of the bone defects (craniomaxillofacial) show better performance over their generic counterparts. due to precise adaptation to the region of implantation, personalized bone implants enable faster and fuller reinnervation and revascularization of the traumatized region [4] and, consequently, better and more efficient recovery of the bone and neighboring tissue. in addition, application of the personalized implants usually requires less invasive surgical intervention and less time [3]. new manufacturing technologies (especially additive ones) eliminate most of the constraints regarding shape, material, size and internal structure design of the implants [3], allowing the designers to optimize them in accordance with the required mechanical and physiological properties of the region of implantation. hence, it is obvious that personalized implants and fixators aimed for bone tissue recovery are already in a queue for extensively developing forthcoming products [5]. however, even though it may seem that all prerequisites for easy production of personalized implants are met, there is still a long way ahead to achieve a commercially efficient, standardized production procedure. 1.1. approaches in the personalized implants design despite the common opinion that the design method is the smallest challenge in this case, in real life a great difficulty arises right from the lack of an optimal design procedure for personalized implants or fixators. the ideal scenario would be if an orthopedic surgeon is able to redesign the personalized implant during the analysis of the radiologic images of the patient’s traumatized region, that is, without external help of the cad designer (fig. 1). there are two general approaches that can be applied for this kind of automatic cad procedure. the first one is to use cloud of points generated from radiographic images (computed tomography (ct) scans) as anchor points for facet tessellation of outer or even internal surfaces (e.g. trabecular structure). however, to the design corresponding personalized implant, that is, endoprosthesis and scaffold, the creation of a bone geometric model native to the geometric kernel of a cad program is almost unavoidable. any further changes in design, i.e. in geometry, are much easier for the cases where the cad program manipulates with native geometric model. user defined geometric creature for the creation of the femoral neck enveloping surface 129 fig. 1 workflow diagram explaining the approach: a) acquiring and healing radiographic image (x-ray, ct scans), b) digital reconstruction of the bone region geometry, b) capturing anatomic parameters, d) modeling the implant – complementary geometry another, and probably better, variant of this approach is to use cloud of points acquired from radiographic images (ct scans) as referential points, not for facet tessellation, but for initial shape modeling based on subdivision surfaces: subd or t-splines [6, 7] (fig. 2). the geometry created in this way would be native to the geometric modeling kernel of a cad program and could also be parametrically controlled [8]. having in mind that these controlling parameters correspond to specific anatomical and morphometric measurements captured from x-ray images or ct scans by the orthopedic surgeon, there is a need for a skeleton model of the subd model. fig. 2 developing subd model within the polyhedron of control points 130 m.stojković, m.trifunović, j. milovanović, s. arsić the second approach involves the use of user-defined (geometric) features (udf) – a compound of basic geometric features of the cad program, but mutually harnessed and controlled by a set of geometric (anatomical) parameters. in brief, this approach uses a kind of previously prepared generic shape that fits into the region of the bio-shape of interest (e.g. femoral neck or trochanteric region [9], knee [10], or sternum [4]). the generic shape is constructed by means of standard geometric features. their mutual geometric relations, which keep these features in a consistent topology, are driven by the imposed geometric constraints (e.g. tangency, perpendicularity, etc.), logical (e.g. if-then rules) and mathematical relations. at the top of the design structure there are several parameters, which are the driving variables for all these relations, directly or indirectly. when these constraints directly correlate to the distinctive morphometric measurements that can be captured from radiologic images, the generic shape can be easily created and personalized by the surgeon. yet, the main advantage that comes out from using udfs in digital reconstruction of bio-shapes is a built-in association between the digital model of reconstructed bio-shape (part of the bone) and the corresponding and complementary model of the implant or fixator. in this approach a collection (set or base) of models and corresponding udfs must be prepared for every single bone region of interest, in advance. though it may seem as a rather extensive task, it is limited in scope, and the existence of this collection can bring remarkable benefits to the patients’ health care. 1.2. femoral neck fracture case one of the most frequent cases of bone fracture is a femoral neck fracture. there are several approaches in classification of femoral neck fractures (lat. fracturaecollifemoris), as shown in [11]. these fractures are highly complex, and their treatment is a challenging clinical problem, especially in the situations where fixation elements should be customized for the specific patient. therefore, the existence of an appropriate and accurate cad model of the femoral neck could bring significant improvement to the surgical (orthopedic) treatment of the femoral neck fracture. the goal of the research reported in this paper was to explore the femoral neck geometry, looking for the most efficient cad procedure for digital reconstruction of the femoral neck volume enveloping surface. the decision was to use udfs that consist of regular cad features for modeling the generic solid shape or surface that matches the femoral neck region with maximal possible geometric congruency with the real one. 2. related work in [12, 13] the authors present the approach which is based on reshaping (scaling) the standard sample of the human bone 3d generic model to match x-ray image of a particular patient bone. the model created with this approach does not have precisely defined geometric entities (points, planes, spline curves). it can be very hard to control the accuracy of the 3d model, without precisely defined geometry. in [14] the authors propose the process of creating contour curves based on cross-sections of bone obtained from ct slices. this method may not give satisfactory results since there is no information on cross-sections other than from ct slices. the approach presented in [15] uses the curves obtained from different cross-sections for creating femur 3d model. user defined geometric creature for the creation of the femoral neck enveloping surface 131 some methods for digital reconstruction of patient-specific surface models are based on deformation of 3d model relative to the measurements or geometry captured from x-ray images. non-stereo corresponding contours (nscc) method [16] uses 2d contours identified semi-automatically on bi-planar patient-specific radiographs. the nscc algorithm [17] is used to perform first a rigid matching, and then a non-linear deformation of the generic object, by kriging, as a method of interpolation, in order to minimize the distance between its 3d retroprojected contours and the corresponding region contours identified on both radiographs. the main limitation of this method is reliance on one generic surface object of the considered anatomy, i.e. not taking into consideration shape variations. the method presented by galibarov et al. [18] uses a library of generic proximal femur models instead. the contour extracted from the radiograph is used for selection of a closest matching 3d model from a library. the selected generic model is then warped to improve correlation with the extracted contour. problems can occur for the femur shapes which are not covered by an assumed size distribution. according to the authors, error values doubled when there was not a relatively close match in the library of generic models. one general limitation is the fact that planar pelvic radiographs do not capture very well three-dimensional morphology of the regions with complex geometry (e.g. greater trochanter region). another method that uses a set of whole femur sample models was presented by wu et al. [19]. the authors claim that, in the situations when major adjustment is needed in some local region, the model must be overall deformed to maintain the correlation between parameters. a well-processed femur model was selected as a template for guiding other sample models to achieve quick compatible segmentation. based on mesh segmentation, complete morphological parameters of all femur sample models were calculated. then, according to partially known parameters, the best matching sample and (group) average model was selected to perform global interpolation resulting in a rough femur model. the rough model regions are then further deformed locally. only longitudinal parameters were well controlled during the deformation process, while some complicated parameters, such as shaft curvature radius, needed to be more delicately controlled. a statistical shape analysis provides an important and increasingly popular means for generating patient-specific surface models. statistical shape models (ssm) aim at describing the natural variability of a shape, e.g. the morphological variation of the same bone from different subjects [20]. the general idea behind ssm is to perform a linear decomposition of the shape variability from a set of training data by defining a mean shape and modes of deformations under some mathematical criteria. the power of this approach depends on the variations contained in the given training database, which is one of its disadvantages [21]. the reconstruction technique presented by zheng and schumann [21] uses the point distribution model (pdm) constructed from a training database consisting of 30 ct scans of patient hips without pathology. it requires two x-ray radiographs (ap and ax view) as the input. the user needs to interactively define one outer contour from the ap view of the proximal femur and one to two contours from the ax view. three anatomical landmarks (the center of the femoral head, a point on the axis of the femoral neck, and the apex of the greater trochanter) are used for pdm initialization. the locations of these landmarks on the mean model of the pdm are extracted before the reconstruction while their locations in the reference coordinate system of the input radiographs are defined interactively from the input radiographs. the initial scale and the initial rigid transformation are obtained by performing a paired point scaled rigid registration. subdivision surfaces allow the design of efficient, hierarchical, local, and adaptive algorithms for modeling, rendering, and manipulating free-form objects of arbitrary topology. 132 m.stojković, m.trifunović, j. milovanović, s. arsić the basic idea of subdivision is to define a smooth surface as the limit surface of a subdivision process in which an initial control mesh is repeatedly refined with newly inserted vertices [22]. the subdivision based modeling can be dated back to chaikin’s corner cutting algorithm for defining free-form curves starting from an initial control polygon through recursive refinement. the scheme was later extended by doo and sabin and catmull and clark for defining free-form surfaces starting from an initial control mesh of arbitrary topology. the most important advantage of subdivision surfaces is the ability to handle control meshes of arbitrary topology. another advantage is that the continuity conditions along all patch boundaries are automatically maintained with subdivision surfaces. application of subdivision surfaces for a piece of human femur bone is presented in [23]. general surface representation that combines b-spline and catmull-clark subdivision surfaces for modeling objects with arbitrary topology and that provides an algorithm for simultaneously fitting smoothly connected multiple surfaces from unorganized measured data was proposed. 3. methods 3.1. creation of the user defined geometric feature: femoneck if one adopts the concept of trochanteric wedge as a specific wedge-shaped morphological bony structure between the femoral head and the body (shaft) [9], consideration of femoral neck as a transition structure that connects the femoral head and the trochanteric wedge is imposed. multi-sections surface appears as the most appropriate basic cad feature to be used for udf creation, considering the main shape of the femoral neck region (fig. 3). besides the main shape, geometry of the femoral neck udf should include smooth transition surfaces to both ends of the femoral neck, that is, to the femoral head and the trochanteric wedge. fig. 3 femoral neck defined as a smooth transition structure between the femoral head and the body. concept of using shell for reconstruction of the human femur neck the most challenging task is to define the guiding line, or a curve, after which the sections should be lined up, as well as to identify how the complex sections should be designed. the following activities were carried out to develop the proper udf: 1. collecting the raw material (ct scans) for analysis of femoral neck geometry 2. identifying referential geometric entities (rge) for digital reconstruction: a. including definition of the femoral neck guiding line, or a curve, for the multisections surface user defined geometric creature for the creation of the femoral neck enveloping surface 133 3. analysis of the geometry of cross sections: a. identifying the minimally sufficient set of cross-sections b. identifying the simplest, but sufficiently geometrically congruent, basic 2d sections that fit chosen cross-sections c. identifying the most robust dimension schema which can drive the designed 2d sections, and correspondent 2d section parameters d. identifying additional parameters e. identifying the parameter’s tree and their relations 4. identifying referential entities for udf placement 3.2. material research included both geometric and anatomical analyses conducted over twenty ct scans of femur proximal part, made by 64-slice ct (msct, aquillion 64, toshiba) with the resolution of 0.5 mm. all 24 samples came from caucasian adults, of different gender and age: ▪ 6 x 2 (both left and right femur) female samples, aged between 25 and 67 ▪ 6 x 2 (both left and right femur) male samples, aged between 22 and 72 ct data were transformed from clouds of points to initial polygonal models, as presented in [4, 9, 24, 25]. 3.3. identifying referential geometric entities the next step in the reverse modeling process, following the generation of initial polygonal model, is recognition and definition of rges [9,24]. for this task, and further geometry creation activities, we used cad software catia v5. in the case of the proximal femur region, the geometric entities that we identify as referential are: ▪ point of center of the femoral head – p_cfh (lat. caput femoris)(fig. 4) ▪ inferior margin of the trochanter wedge – imtw(fig. 4) ▪ tkeel plane, normal to the bottom line of the trochanter wedge (fig. 4) ▪ femoral neck axis – fna(fig. 5) ▪ angle between fna and femur body(fig. 5) ▪ femoral neck curve – fnc(fig. 5) creation of fnc starts with construction of fna. according to the procedure for definition of proximal femur rges [24, 25], the femoral neck axis starts from p_chf and ends perpendicularly to the inferior margin of trochanteric wedge (in anterior-posterior plane and view). 134 m.stojković, m.trifunović, j. milovanović, s. arsić a) b) fig. 4 proximal femur rges [9] lateral-medial aspect of the axis is needed for determination of fna spatial location. in this view, the projection of a small trochanter boundary is used as a reference which fna touches tangentially (fig. 5). following the fna direction, a series of cross sections of the femoral neck volume is being created. the centers of gravity of these cross sections are used as control points for spline generation, i.e. fnc approximation (the main reference is p_cfh, through which fnc passes). a) b) fig. 5 fna spatial location; a) fna in tkeel projection, medial aspect; and b) fnc and fna in a-p (anterior-posterior) projection 3.4. analyzing the geometry of femoral neck contour curves once fnc is defined, it becomes a guiding curve for a series of the planes normal to the fnc, which cut the polygonal model in the femoral neck region creating a series of cross-sections of the femoral neck enveloping surface (fig. 6). these intersection contour curves are being used to analyze the femoral neck geometry, trying to identify the minimal set of the regular geometric features that could combine in a robust udf which will enable an easy and accurate remodeling of the particular femoral neck geometry. user defined geometric creature for the creation of the femoral neck enveloping surface 135 a) b) c) fig. 6 femoral neck curve creation and corresponding sections fig. 7 series of cross sections representing contour curves of the femoral enveloping surface 136 m.stojković, m.trifunović, j. milovanović, s. arsić a series of cross-section curves (fig. 7) are used as a set of underlying 2d patterns to sketch approximate contour curves. approximation should be done by combining minimal number of basic geometric elements which can accurately describe contour curves and be applicable to all contour curves. therefore, the analysis is being focused just on structurally similar cross-sections which are above the trochanteric region and below the femoral head (sub-capital cross-sections). after all, only these cross-sections are real representatives of the enveloping surface of the femoral neck geometry. the most representative cross-section is the mid-cervical one. the analysis shows that each cross-section can be approximated with sufficient accuracy by the sketch made of two partial ellipses connected with tangent lines. it turns out that each 2d sketch, i.e. contour curve of the femoral neck enveloping surface, can be designed by combining four basic geometric elements: two ellipses and two lines. additionally, each contour curve can be surrounded by a trapezoid, made of auxiliary constructional lines, whose function is to control position and shape of basic geometric elements of the contour curve (fig. 8). a) b) fig. 8 intersection approximated with 2d basic geometric elements 3.5. identifying the parameters of femoneck udf the shape of a femoral neck contour curve reconstructed by the user-defined 2d sketch that consists of two ellipses and two straight lines inscribed in the control trapezoid can be managed easily (fig. 9) by changing four parameters of ellipses (major axis and minor axis lengths: cal, cpl, cas, cps), and three variables of trapezoid (height: h, and base angles: al, am). the position and orientation (rotation) of the trapezoid and the inscribed contour curve are controlled by additional three parameters: two offsets of the midsegment mid-point from p_cfh projection (lm_shift, ap_shift), and torsion angle (tacs) (the angle between trapezoid base and projection of trochanteric wedge axis: imtw) (fig. 10). the rotation angle of the trapezoid base corresponds to the femoral neck cross-sections torsion angle related to the trochanter region cross-section and is directly related to the parameter of specific contour curve identification (id_cs) [9]. user defined geometric creature for the creation of the femoral neck enveloping surface 137 a) b) fig. 9 approximate reconstructing of a contour curve by udf2d sketch – black compound curve (parameters shown in the figure can be read in table 1) fig. 10 angular orientation of trapezoid surrounding 2d sketch as explained herein before, each contour curve corresponds to the specific crosssection of the femoral neck enveloping surface, and each cross-section corresponds to the specific plane which is normal to the femoral neck curve (fnc) at the specific point located in a specific distance from p_cfh. the length of fnc arc from the point that identifies specific cross-section plane to the p_cfh is the last parameter (cross-section distance – csd) that is directly related to the specific contour curve (id_cs). hence, the femoral neck contour curve defined in this way, driven by this set of parameters, composes 2d geometric udf that is named femoneck_section (fig. 11). 138 m.stojković, m.trifunović, j. milovanović, s. arsić fig. 11 controlling the shape of femoneck_section udf instances by changing the variable parameters of the femoneck_section (most of numerical parameters), the contour curve is being shaped to match the corresponding bone contour (table 1, and fig. 11). table 1 parameters of femoneck udf the user defined feature used for remodeling of the femoral neck geometry (we named it femoneck) includes one multi-sections surface element and two variable-radius fillet elements (oval transition surfaces) towards the femoral head and the trochanteric wedge. considering that the main shape of the femoral neck will be formed by using basic geometric feature type of user defined geometric creature for the creation of the femoral neck enveloping surface 139 multi-sections surface, the contour curves (femoneck_sections) will take the role of the main components which shape the enveloping surface (fig. 12). in this way, femoneck_section udf becomes a part of supreme femoneck udf enabling an easy and fine adjustment of the enveloping surface shape in order to achieve maximal geometric congruency with the real bone surface (fig. 13). moreover, by employing statistical analysis of parameters [26, 27], it is possible to fully automate an adjustment procedure, i.e. searching for optimal values of the variable parameters. fig. 12 user defined cad feature: femoneck a) b) fig. 13 femoneck udf in use: reconstruction of human femur neck outer surface 3.6. referential entities for udf placement the user should identify the center of femoral head (p_cfh) and the femoral neck axis (fna) as positioning references for femoneck udf placement. they can both be defined by the surgeon while analyzing x-ray digital images (ap and lm projections). all necessary input elements for p_cfh and fna reconstruction are generated by sketching the circles around the femoral head and small trochanter contours in ap and lm projections, as well as underlining the inferior margin of the trochanteric wedge in ap view. finally, the user should define distance between cross-sections to control the smoothness of multi-sections surface and fineness of its details. 140 m.stojković, m.trifunović, j. milovanović, s. arsić 4. results the application of femoneck udf was tested in the femoral neck reconstruction of four new cases (different from the learning base of 24 models). two input models are made from the ct scans of proximal femurs of two women, and the other two models from the ct scans of proximal femurs of two men. for each geometry input, we applied the femoneck udf to reconstruct the geometry of each femoral neck. afterwards, the comparison between the geometry of raw polygonal model and the reconstructed model was done. the threshold for acceptable deviation in congruency between cross-sections and reconstructed contour curves is set to 0.5 mm. each femoral neck was sliced by 13 planes at distance of 1.5 – 2.2 mm depending on size of the bone. for the presentation of the results, five representative cross-sections and corresponding femoneck_section instances were chosen: 1st – close to trochanteric wedge, 4th – between trochanteric and mid-cervical cross-section, 7th – mid-cervical, 9th – between mid-cervical and sub-capital, and 13th – sub-capital. fig. 14 percent of the reconstructed contour curve length: femoneck_section instance that deviates more than 0.5 mm from corresponding cross-section curve after semi-automatic adjustment (without using statistical analysis) the average value of the portion of femoneck_sections that deviated from their corresponding cross-sections of femoral neck for more than 0.5 mm was 6.35 % of their length. the results, presented in fig. 16, indicate that the greatest deviation was encountered near the trochanteric region, which is expected due to a very irregular transition between the neck and the trochanteric wedge. in the sub-capital region, again, an abrupt change of shape around the femoral head rim has led user defined geometric creature for the creation of the femoral neck enveloping surface 141 to a slight increase in deviation. considering the achieved accuracy in the context of orthopedic interventions, femoneck has been showed as a usable and sufficiently accurate solution for digital reconstruction of the femoral neck geometry and its geometrically complementary parts – implants. 4. discussion before the beginning of the discussion, it should be emphasized that digital reconstruction of the bone surfaces is not an aim by itself, but just a first step in a process of modeling personalized bone implant geometry and its manufacturing process. the efficiency and accuracy of digital reconstruction of the bone geometry by udf substantially depend on the way its structure is prepared. selection of basic geometric features that constitute an udf, as well as definition of their mutual relations, which includes topologic and dimensional interdependences, makes a difference between robust and applicable udfs and those who are not. the main shortcoming of udf application for the cases like specific bone region geometry reconstruction comes from necessity to invest a considerable effort in udf structure preparation. however, once well structured, it becomes a powerful cad tool for creation of “families” of shapes that are topologically congruent, like endless variations of the femoral neck shape from one patient to another. the most common current alternative in remodeling complex bio-shapes, like bone surfaces, is tessellation of elementary facets over the cloud of points. despite its capacity to remodel complex surfaces very precisely and quickly, the greatest issue with this approach is geometric inertness for further modification. application of subdivision surfaces seems an even more attractive cad approach due to fascinating easiness of shaping the digital geometry of any complexity. another great advantage of this approach is that the geometry is native to the geometric kernel of cad software. the shortcoming is that it requires not just a very skillful, but also talented cad expert for digital sculpturing to efficiently and accurately shape a very complex geometry of the bone, and later corresponding implant. regarding personalized implants design procedure, this approach seems as more convenient than manipulation with tessellated models; yet it still shows very similar limitations – each subsequent modification of the same bone topology (e.g. femoral neck) requires no small intervention of the cad designer on geometry free forming. the best solution for designing personalized bone implants and fixators could be to create a collection of specific udfs that are made by combining subd surface features (and corresponding volumes) and regular geometric features. the biggest challenge regarding this kind of udfs is managing the topology of subd model within the udf. it should be controlled by the parameters that directly correlate to distinctive morphometric measurements. one solution that may enable this kind of direct control could be usage of user-defined polyhedron [28], whose vertices are coupled with control points of subd surfaces by specific topologic and dimensional relations and constraints. the user defined subd model for digital reconstruction of a particular bone region, as femoral neck, can take a role of a base model geometry for creating a set of corresponding geometric complements – implants and fixators. 142 m.stojković, m.trifunović, j. milovanović, s. arsić 5. conclusions the paper presents research regarding cad methodology that should be employed for digital reconstruction of complex topologies, like human bone enveloping surfaces. the focus of the research was on geometric remodeling of the femoral neck region (lat. collumfemoris). even though the femoral neck is not characterized by very complex topology, like some other human bone regions, it was chosen as the most appropriate to easily present the research results. moreover, in real life, this part of thigh bone is very often being fractured and, consequently, there are many diverse demands for implant solutions regarding this region. the results point out that the udf concept is the most convenient to use in digital reconstruction of numerous variants of the same topology, even the very complex one. specific udf – femoneck, was created to test validity of that finding. testing the application of femoneck has shown that it is very robust and sufficiently accurate in digital reconstruction of femoral neck geometry. acknowledgements: the paper represents a summary about a part of the research that is conducted within the project “virtual human osteoarticular system and its 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piezoelectric shell element for linear and geometrically nonlinear dynamic analysis of smart structures udc 624.01+531;519.6 gil rama berlin institute of technology (tu berlin), department of structural and computational mechanics, germany abstract. composite laminates consisting of passive and multi-functional materials represent a powerful material system. passive layers could be made of isotropic materials or fiber-reinforced composites, while piezoelectric ceramics are considered here as a multifunctional material. the paper is focused on linear and geometrically nonlinear dynamic analysis of smart structures made of such a material system. for this purpose, a linear 3node shell element is used. it employs the mindlin-reissner kinematics and the discrete shear gap (dsg) technique to alleviate the transverse shear locking effects. the electric potential is assumed to vary linearly through the thickness for each piezoelectric layer. a co-rotational formulation is used to handle the geometrically nonlinear effects. a number of examples involving actuator and sensor application of piezoelectric layers are considered. for the validation purposes, the results available in the literature and those computed in abaqus are used as a reference. key words: shell element, piezoelectricity, active laminates, co-rotational fem, actuator, sensor, geometrically nonlinear dynamic 1. introduction laminated thin-walled structures made of isotropic or orthotropic materials are widely used in engineering practice. this is a consequence of the optimization strategy to reduce the structural dead-load whereby the structural carrying capacity is kept at a very high level. besides numerous advantages offered by thin-walled structures, they also tend to suffer from structural stability issues and are rather sensitive to vibrations. the use of multi-functional materials offers a great potential to cope up with those challenges. received: february 25, 2017 / accepted march 21, 2017 corresponding author: gil rama tu berlin, fachgebiet für strukturmechanik und –berechnung, str. des 17. juni 135, 10623 berlin, germany e-mail: gil.rama@tu-berlin.de 32 g. rama piezoelectric materials are characterized by a sufficiently strong coupling between the mechanical and the electric fields, so that they can be employed for an adequately designed actuator as well as sensor devices. in the actuator case, the inverse piezoelectric effect is used to affect the mechanical field through a purposeful change of the electric field. oppositely, the direct piezoelectric effect is used for sensors to gain information on the induced deformation, i.e. strain-field in the material. the finite element method (fem) has established itself as the method of choice in the field of structural analysis including coupled-field problems, such as the piezoelectric effect. over the last couple of decades, numerous elements have been developed for static and dynamic analyses of piezoelectric thin-walled structures. a survey of piezoelectric solids, beams, plates and shells developed in the 90’s is given by benjeddou [1]. solid elements, such as those proposed by lee et al. [2] and willberg and gabbert [3], provide high fidelity fe modeling but at the price of a high numerical effort when applied to thin-walled composites. therefore, shell type finite elements are usually addressed as numerically more efficient for this type of structures when the global structural behavior is aimed at. most of the composite shell fe formulations are based on the equivalent single-layer approach and mainly rely on the kirchhoff-love or mindlin-reissner kinematics. the kirchhoff-love kinematics leads to zero transverse shear strains/stresses and is therefore applicable to rather thin shells. the mindlin-reissner kinematics takes the transverse shear strains into account, so that the resulting theory is referred to as the first-order shear deformation theory (fsdt). the mindlin-reissner plate and shell elements are notorious for shear locking when rather thin structures are modeled. various techniques have been developed to eliminate the effect, such as the assumed natural strain (ans) [4], enhanced assumed strain (eas) [5], reduced integration schemes and the discrete shear gap (dsg) method [6]. all of them were also used in the development of piezoelectric shell elements. marinković et al. [7] developed a full biquadratic degenerated shell element with a choice between the full and uniformly reduced integration scheme. the element was used to check the convergence of fe results for the coupled electro-mechanical field [8] and it was also implemented in abaqus [9] for the users’ convenience. zemčík et al. [10] developed a linear 4-node element with the dsg method implemented to resolve shear locking effects and eas to handle the membrane locking effects. yang et al. [11] presented a linear quadrilateral piezoelectric shallow shell element with the ans technique, while nguyen et al. [12] proposed a linear triangular shell element based on the dsg approach. besides the equivalent single-layer theories, layer-wise theories were also addressed in modeling of smart laminated structures. a number of those approaches rely on the carrera unified formulation (cuf) for multilayered plates and shells [13]. cinefra et al. [14] proposed a 9-node plate element that implements mixed interpolation of tensorial components (mitc) approach and variable through-the-thickness layer-wise kinematics to perform linear static analyses. this development was extended to free-vibration analyses of piezoelectric plates [15]. milazzo [16] used both equivalent single-layer and layer-wise approaches for piezoelectric laminated plates whereby the coupled-field problem was reduced to mechanical one. the theoretical contributions of tzou [17] and numerical developments by rabinovitch [18], kulkarni and bajoria [19], lentzen et al. [20], klinkel and wagner [21] addressed the geometrically nonlinear effects in the behavior of smart thin-walled smart structures. however, a 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis... 33 so far this aspect was much less in the focus of the researchers compared to the developments for linear analysis; thus, further contributions would be worthwhile. in the present work a recently developed linear 3-node shell element [22] is applied to resolve a linear and geometrically nonlinear dynamic response of piezoelectric laminated shells. the basic features of the element are briefly described and several dynamic linear and nonlinear sensor and actuator cases are considered to verify the applicability of the developed element formulation by comparing the obtained results with the solutions from the available literature. 2. features of the linear piezoelectric shell element only the most important features of the triangular piezoelectric shell element, which is used in this work, are presented here. a detailed element formulation can be found in [22]. the element uses five mechanical degrees of freedom, three translations and two rotations, per node and, in addition, as many electrical degrees of freedom as piezolayers. the electrical degrees of freedom are the differences of electric potentials between the electrodes of a piezolayer. the mechanical field of the element is enhanced by the cell smoothed – discrete shear gap (cs-dsg) formulation. the mindlin-reissner kinematical assumptions are implemented and, hence, the transverse shear effects are included. the discrete shear gap technique proposed by bletzinger [6] is implemented to alleviate the transverse shear locking. the strain smoothing technique suggested by nguyen et al. [12] is applied to improve the accuracy and stability of the element, and, furthermore, to render the element formulation independent of the node numbering sequence. two different coordinate systems presented in fig. 1 are used within the formulation: global (x, y, z) and local (x, y, z). the structural displacement field is given with respect to the global coordinate system that is fixed in space. fig. 1 geometry and coordinate systems of the 3-node shell element the local element coordinate system (x, y, z) is used to derive the mechanical strain and stress fields as well as the electro-mechanical coupling. regarding the piezoelectric layers, they are assumed to operate by using the piezoelectric e31-effect, which implies that the in-plane strain field is coupled to the electric field acting in the thickness direction. electric field e within the piezoelectric layers is assumed to be 34 g. rama constant, which leads to a linear distribution of electric potential across thickness  so that the following relations hold: k k k h e z e      (1) where k is the difference of electric potentials between the electrodes and hk is the layer thickness (k in the subscript pertains to the layer number in the sequence of layers). the element formulation is also extended to the geometrically nonlinear analysis. for this purpose the element-based co-rotational (cr) fe formulation [22, 23] is used, thus covering structural deformations characterized by the finite local rotations, whereby the strains remain small. 3. finite element equations the coupled electro-mechanical dynamic fe equations may be derived using the hamilton’s principle for a piezoelectric continuum [24]. the fe system of equations for a geometrically nonlinear dynamic analysis by means of an implicit time integration scheme reads: t tt t (k) t t t t t (k) uu uuu uu t t t (k) u t t t t (k 1) ext in t t t t (k 1) ext in [k ] [k ][m ] [0] {u} [c ] [0] {u} { u} [k ] [k ][0] [0] {0} [0] [0] {0} { } {f } {f } {q } {q }                                                        (2) where [muu] is the mass matrix, [cuu] the damping matrix, [kuu], [ku], [ku] and [k] are mechanical stiffness, piezoelectric direct and inverse coupling, and dielectric stiffness matrices, respectively, while vectors {∆}, {∆u}, { u }, { u } comprise the incremental differences of electric potentials of the piezolayers, incremental displacements, nodal velocities and accelerations, respectively. vectors {fext}, {fin}, {qext} and {qin} on the right hand-side of the fe equations are external and internal mechanical forces and electric charges, respectively. index k in the superscript denotes the iteration number. rayleigh damping is used to introduce the dissipative effects in the fe equations. it consists of stiffness and mass proportional terms: uu uu uu[c ] [k ] [m ]   (3) where α and β are the rayleigh damping coefficients [25]. 4. numerical examples in what follows, a set of examples is studied to demonstrate the applicability of the element for linear and geometrically nonlinear dynamic analysis of smart thin-walled structures. the considered structures are made of composite laminates with various combinations of fiber-reinforced, isotropic and piezoelectric layers. the properties of all a 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis... 35 the used materials are given in table 1, where y denotes the young’s modulus and  the poisson’s ratio (with indices referring to the material orientation). the values in empty cells of table 1 are considered to be equal to zero in the studied examples. the thickness and stacking sequence of layers vary in the examples and will be specified for each example separately. table 1 layers material properties (given in principal material directions) t300/976 aluminum steel ptz-4 gr/ep pic 151 ptz y11 [gpa] 150.0 70.3 210 81.3 132.28 61.0 63.0 y22 [gpa] 9.0 70.3 210 81.3 10.76 61.0 63.0 y33 [gpa] 9.0 70.3 210 64.5 10.76 48.4 63.0 υ12 [-] 0.3 0.345 0.3 0.33 0.24 0.3 υ13 [-] 0.3 0.345 0.3 0.43 0.24 0.3 υ23 [-] 0.3 0.345 0.3 0.43 0.49 0.3 density [kg/m³] 1600 2690 7800 7600 1578 7760 7600 piezoelectric constants e31 = e32 [cm-2] -14.8 9.6 -22.87 dielectric constant [f m-1] d31 (× 10-8) 1.1505 1.710 1.5 the examples include both actuator and sensor cases. in the actuator case the piezopatches are subjected to a predefined electric voltage, thus causing mechanical excitation due to the inverse piezoelectric effect. in the linear analysis the computation of induced mechanical loads t {f,e} is performed on the element level as follows: t t ,e u ,e a,e{f } [k ] { }   (4) where the matrices and vectors are defined on the element level. in the nonlinear analysis the system matrices, including the piezoelectric coupling terms, have to be updated first. in the framework of the cr-formulation, the element piezoelectric coupling matrix is updated using element rotation matrix t [re]: t t 0 u ,e e u ,e[k ] [r ] [k ]  (5) in the sensor case, the direct piezoelectric effect is used to induce electric voltage t {s,e} (again, computed on the element level) due to the external mechanical loads, whereby the external electric charges are equal to zero: t 1 t s,e ,e u,e e{ } [k ] [k ] {u }     (6) where, again, all the vectors and matrices are defined on the element level (index ‘e’). in the linear analysis the above equation is used directly, whereas in the geometrically nonlinear analysis the rotation-free (i.e. purely deformational) displacements are computed first. as a sensor patch/layer is discretized by a number of finite elements, a constraint is introduced that the induced electric voltages in the sensor layer are equal in all those elements. in this manner, the obtained sensor voltage reflects the average value of the in-plane strains caused in the sensor layer by the action of external mechanical loads. 36 g. rama 4.1. modal analysis of a simply supported piezoelectric plate in the first case a modal analysis of a composite piezoelectric plate is performed. in order to verify the cs-dsg3 formulation and to illustrate the influence of the electromechanical coupling on the dynamic properties two cases with different electric boundary conditions are investigated. in the first one, the electrodes of the piezolayers are shortcircuited (sc). hence, the electric potential {} is equal to zero. this leads to the purely mechanical eigenvalue problem, i.e. the natural frequencies and modes are the same as if only purely mechanical field was considered. in the second case, the electrodes are assumed to be open (o) which implies zero electric charge as a boundary condition. from eq. (2) follows: 1 s u{ } [k ] [k ]{u}     (7) hence, an electric potential difference is generated in the sensor layer if the shell is deformed. due to the open electrodes, the electric voltage induces mechanical stresses through an inverse piezoelectric effect. in the modal analysis these stresses are taken into account by a modified stiffness matrix obtained by substituting {s} into eq. (2): * 1 uu u u[k ] [k ] [k ][k ] [k ]     (8) the electro-mechanical coupled eigenvalue problem reads then: * uu[k ] ²[m ] {u} 0    (9) it is obvious that the natural frequencies and mode shapes are in this case influenced by the properties of the piezoelectric material. the natural frequencies are increased in the open electrodes case compared to the short-circuited case because of the additional stiffness term. both the cases are studied on the same structure, at all edges simply supported square laminated piezoelectric plate (dimensions aa = 0.20.2m, see fig. 2). fig. 2 geometry of the simply supported plate with different electric boundary conditions (sc) and (o) the composite ply layup is [p/0°/90°0°/p]. the outer layers are made of piezoelectric ptz-4 ceramics and the composite layers of graphite epoxy (gr/ep). the thickness of each piezoelectric layer is 0.0004 m and each composite layer is 0.001068 m thick. saravanos [26] computed the first natural frequency for this structure using different a 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis... 37 meshes. these results are used for the comparison with the current formulation. in order to make the results comparable to [26] the value of density of all layers is set to one kg/m³. table 2 shows the result convergence for the first natural frequency determined by using three different meshes (32, 128 and 288 elements). for an easier comparison these results are normalized with respect to a reference solution. for the sc-case the reference solution is obtained with abaqus using a 24×24 elements mesh and the biquadratic s8 element while the reference solution of case (o) is analytical and presented in [26]. the difference to the reference solutions is in both cases less than 0.5 % for the mesh with 288 elements. table 2 the normalized first eigenfrequency – convergence analysis closed circuit f1,ref =22915 hz (abaqus s8 24×24 mesh) open electrodes f1,ref =24594 hz [26] mesh present abaqus [26] present [26] 32 1.203 1.220 1.090 1.193 1.109 128 1.045 1.050 1.034 1.040 1.054 288 1.003 1.024 1.023 1.001 1.044 4.2. transient analysis of an active beam structure (linear dynamic actuator case) the undamped dynamic behavior of a clamped beam with two pairs of piezopatches bonded onto its outer surfaces is studied in this example. the beam geometry is depicted in fig. 3. it is made of aluminum, while the piezopatches are made of pic 151 (table 1). fig. 3 geometry of the active beam structure with two pairs of piezopatches the oppositely polarized piezopatches are subjected to a time-varying voltage. the voltage is a sinusoidal function with amplitude of 100 v and frequency of 100 hz. this induces time-varying bending moments with respect to the structure’s mid-surface, which are uniformly distributed over the patch edges. the resulting transverse beam tip deflection (w) is observed in a time interval of 0.1 s with constant time-step of 0.0001 s (1000 steps) using the newmark time integration scheme [25]. the first three eigenfrequencies of the beam considered as a purely mechanical structure are 31.1 hz, 131.8 hz and 349.9 hz. hence the answer of the structure subjected to an excitation with the frequency of 100 hz is dominated by the first 38 g. rama two natural mode shapes. the transient analysis is carried out using a fe mesh with 320 elements, which yielded a converged solution for the first three eigenfrequencies and mode shapes. for the purpose of verification, a transient analysis of the same structure was computed in abaqus using the s3 shell element, the same mesh and time-step, whereby the equivalent mechanical nodal excitations were pre-computed and directly applied. the obtained time histories of the tip deflection are shown in fig. 4. the results of the present formulation are in a very good agreement with those from abaqus. fig. 3 linear dynamic behavior of the active beam (1000 steps) 4.3. nonlinear dynamic analysis of a two-edge-simply-supported laminate the previous example was computed using the assumption of linearity. hence, the structural stiffness and induced piezoelectric loads were calculated using the initial configuration as a reference configuration. this example will be calculated using both the assumption of linearity and a geometrically nonlinear approach. the geometry of the laminate composite plate simply supported over two shorter parallel edges is shown in fig. 5. the laminate consists of three layers. the aluminum mid-layer is 0.5 mm thick and each outer ptz-4 layer has a thickness of 0.25 mm. fig. 5 geometry of the two-edge-simply-supported structure the same type of excitation from the previous case is used here as well. it is the timevariable electric voltage with amplitude of 100 v and frequency of 100 hz. the response of the structure is computed for a time interval of 0.1 s with a constant time-step of 0.0001 s using the newmark scheme. the comparison between linear and nonlinear dynamic response is obvious in fig. 6. it shows that, even in the range of relatively small deformations, the linear and the geometrically nonlinear response could differ significantly. such a result emphasizes the a 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis... 39 necessity of taking into account nonlinear effects. the geometrically nonlinear computation is verified by means of abaqus. as already mentioned in the previous example, the equivalent mechanical excitation is first pre-computed and then directly applied in abaqus. it should be emphasized that the induced bending moments are of the follower type as their orientation depends on the current structural configuration, and this is how they are defined in abaqus (the option ‘follow nodal rotation’ was used). again, observing the structure’s mid-point deflection, practically congruent geometrically nonlinear results obtained by means of the developed element and in abaqus can be seen in fig. 6. fig. 6 two-edge-simply supported structure under harmonic excitation 4.4. clamped piezoelectric plate (linear dynamic sensor case) a composite plate clamped over one edge is considered next. the plate geometry is shown in fig 7. the composite consists of six layers. the outer two are oppositely polarized ptz layers and the remaining four are t300/976 plies. each t300/976 layer has a thickness of 0.25 mm and each ptz layer is 0.1 mm thick. the antisymmetric composite stacking sequence is [p/-45°/45°/-45°/45°/p] with respect to the global x-axis. this structure has been already considered in the available literature [27, 28] as a static linear actuator case. for this reason, the exact same static case will be computed here first. after that, a dynamic sensor case will be addressed. in the linear static case a uniform surface load p = 100 n/m² acts upon the plate. both ptz layers are used as actuators subjected to three different voltages: 0 v, 30 v and 50 v. fig. 7 clamped piezoelectric plate subjected to uniform surface load 40 g. rama the shape of the plate centerline is computed using a mesh of 200 elements. the comparison between the obtained results and the solutions of lam et al. [27] and zhang [28] shows a rather good agreement. for the sake of better readability, only the results of lam et al. [27] and those obtained by the present formulation are presented in fig. 8. fig. 8 centerline deflection subjected to uniform load and different input voltages in the linear dynamic analysis, the piezolayers are used as sensors and the composite plate is subjected to harmonic varying concentrated force. the force acts at point a (see fig. 9) with an amplitude of 0.2 n and frequency of 1 hz. the induced sensor voltage of the lower layer is observed in a period of 4 s with a time-step of 0.005 s using the newmark scheme and the same mesh as in the previous static analysis. fig. 9 clamped piezoelectric plate subjected to harmonic varying concentrated force zhang et al. [29] studied this example using the sh851uri biquadratic shell element (uniformly reduced integration) along with the modal superposition method using the first 12 modes. fig. 10 shows a good agreement in the amplitude and frequency of the sensor potential response between the current formulation and zhang et al. [29]. the minor local differences are attributed to a different time-step (not specified in [29]) and a different damping definition. a 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis... 41 fig. 10 dynamic sensor response of piezoelectric plate under harmonic concentrate force 4.5. simply supported piezoelectric plate (nonlinear dynamic sensor case) the next example illustrates the influence of geometrically nonlinear effects on the dynamic sensor response when piezoelectric plate structures are considered. fig. 11 shows the plate geometry together with the boundary conditions. the laminate consists of three layers. the outer two are oppositely polarized piezoelectric ptz layers with a thickness of 0.1 mm, while the mid-layer is 0.5 mm thick and made of steel. the plate is discretized so that the fe mesh consists of 512 elements. the plate is subjected to a concentrated force with periodic time dependent amplitude (see fig. 11). the vertical displacement of point b (see fig. 10) and sensor response of the upper (1) and lower (2) layers is observed in a time period of 0.2 s using a time-step of 0.005 s. fig. 11 simply supported piezoelectric plate geometry in the first step the dynamic response of the structure is determined with abaqus using the same mesh and time-step. fig. 12 shows the vertical deflection of point b computed as linear and geometrically nonlinear dynamic response. again, a good agreement between the results from abaqus and the present formulation can be noticed. 42 g. rama fig. 12 vertical deflection of a simply supported piezoelectric plate under harmonic excitation in the second step, the linear and nonlinear sensor voltage response is computed for the same test case. the obtained results for the upper (1) and lower (2) piezoelectric layers are presented in fig. 13. in the linear analysis, the stiffness matrix is determined for the initial configuration and the mechanical excitation leads to bending deformation. as a result of the opposite polarization of the ptz layers, the computed sensor voltages of the layer (1) and (2) are equal (see fig. 13). in the nonlinear analysis the stiffness matrix changes continuously with the structural deformation. the deformation involves membrane and bending effects and, consequently, the sensor voltages of the upper and lower layer differ. the difference between the linear and the nonlinear results depends on the boundary and loading conditions. in this case, one can easily notice the differences in periods and amplitudes of the linear and nonlinear sensor response, demonstrating the importance to account for the geometrically nonlinear effects for adequate accuracy. fig. 13 sensor voltage response of a simply supported piezoelectric plate under harmonic excitation a 3-node piezoelectric shell element for linear and geometrically nonlinear dynamic analysis... 43 5. conclusions the dynamics is of particular importance for smart structures as their advantages are quite often used to achieve active vibration suppression, radiated noise attenuation, etc. simulation of the smart structures dynamic behavior is a significant prerequisite for their successful design, including control laws, i.e. algorithms that define the control strategy. the recently developed linear triangular shell type finite element [22] was used in this paper to perform dynamic analyses of thin-walled piezoelectric laminated structures. both linear and geometrically nonlinear computations were performed. actuator and sensor cases were considered. for the nonlinear computations, the co-rotational fe formulation was used. the verification of the results was done using either results available in the literature or the results from abaqus by properly prepared equivalent mechanical models. a high agreement of the results validates the developed element. furthermore, the geometrically nonlinear examples demonstrate that, depending on the boundary and loading 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license: cc by-nc-nd original scientific paper metaheuristic-based tuning of proportional-derivative learning rules for proportional-integral fuzzy controllers in tower crane system payload position control radu-emil precup1,2, raul-cristian roman1, elena-lorena hedrea1, emil m. petriu3, claudia-adina bojan-dragos1, alexandra-iulia szedlak-stinean1 1politehnica university of timisoara, department of automation and applied informatics, timisoara, romania 2romanian academy – timisoara branch, center for fundamental and advanced technical research, timisoara, romania 3university of ottawa, school of electrical engineering and computer science, ottawa, canada orcid ids: radu-emil precup https://orcid.org/0000-0002-2060-7403 raul-cristian roman https://orcid.org/0000-0003-2551-7842 elena-lorena hedrea https://orcid.org/0000-0003-3277-6340 emil m. petriu https://orcid.org/0000-0002-0274-1035 claudia-adina bojan-dragos https://orcid.org/0000-0003-2465-9283 alexandra-iulia szedlak-stinean https://orcid.org/0000-0002-1572-2060 abstract. this paper proposes the use of metaheuristic optimization algorithms to tune the proportional-derivative (pd) learning rules within the framework of iterative learning control applied to low-cost takagi-sugeno proportional-integral (pi)-fuzzy controllers for tower crane system payload position control. four pd learning rules are considered: direct rule with current (in the iteration domain) control error, direct rule with previous control error, indirect rule, and open-closed-loop rule. the fuzzy controllers are tuned by the extended symmetrical optimum method applied to the linear pi controllers, and then by the modal equivalence principle. set-point filters are included for overshoot reduction. a unified design approach is formulated for all four pd learning rules in terms of optimally computing the gains in the iteration domain using metaheuristic optimization algorithms that solve optimization problems with objective functions expressed as the sum of the squared control error multiplied by time, where the two variables are the parameters of the pd learning rules. seven popular metaheuristic optimization algorithms are implemented. real-time experimental results from ten received: september 14, 2024 / accepted november 01, 2024 corresponding author: radu-emil precup politehnica university of timisoara, department of automation and applied informatics, bd. v. parvan 2, 300223 timisoara, romania romanian academy – timisoara branch, center for fundamental and advanced technical research, bd. mihai viteazu 24, 300223 timisoara, romania e-mail: radu.precup@aut.upt.ro https://orcid.org/0000-0002-2060-7403 https://orcid.org/0000-0003-2551-7842 https://orcid.org/0000-0003-3277-6340 https://orcid.org/0000-0002-0274-1035 https://orcid.org/0000-0003-2465-9283 https://orcid.org/0000-0002-1572-2060 568 r.-e. precup, r.-c. roman, e.-l. hedrea et al. iterations of these optimization algorithms support the performance comparison of the fuzzy control systems. key words: iterative learning control, metaheuristic algorithms, proportional-derivative learning rules, proportional-integral-fuzzy controllers, tower crane systems 1. introduction iterative learning control (ilc) performance improvement of the control system by performing repeated experiments (iterations or trials or cycles). according to [1], the collected data from previous experiments is used in an iterative way. a systematic way to design ilc structures is the optimal tuning [2]. the proportional-derivative (ρd) learning rule is a representative component in several ilc structures, with recent applications given in [3] and [4]. for better readability, the nomenclature of all symbols used in this paper is included at the beginning of the supplementary appendix given in [5] and organized in table 1, which describes the abbreviations, variables, and parameters. a classification of optimal ιlc is given in [6], several data-driven ilc techniques are reviewed in [7], and recent applications to wafer stages, ball screw feed-drive systems, turntable systems and a combination with immune deep reinforcement learning are reported in [8], [9], [10] and [11], respectively. general classifications of ilc are given in [10] and [12] as direct ilc (i) and indirect ilc (ii), which combine ilc with feedback control loops: (i) ilc computes the control signals or improves their computation strategy; (ii) the controllers generate the control signals and ilc updates some parameter gains and/or variables of the control loops including reference inputs (or set-points) and controller tuning parameters. the fusion of fuzzy control and ιlc was coined by the authors back in 2006 [13] and continued in [14] to exploit the advantages of both control strategies. this has resulted in several low-cost proportional-integral (ρι)-fuzzy controllers, and fuzzy logic has been incorporated into the learning rules specific to ilc. another way of mixing fuzzy control with ιlc is the approximation of the unknown dynamics of the controlled processes, considering fuzzy models to cope with the nonlinear mechanisms of the processes, and further the inclusion in adaptive ιlc structures. the most recent applications of adaptive structures deal with high-speed trains [15], permanent magnet synchronous motors [16], multi-agent systems [17], [18], and treating the sampling period as a parameter to compensate for its influence on the control system performance using the optimizationbased design [19]. building upon authors’ recent approaches to the performance improvement of fuzzy control systems for tower crane (tc) systems, using ilc with (a) direct pd learning rule with current (in the iteration domain) control error and the two parameters (i.e. the gains) optimally tuned using the metaheuristic grey wolf optimizer (gwo) given in [20], (b) direct pd learning rule with previous control error using the metaheuristic slime mould algorithm (sma) [21] and (c) indirect pd learning rule using sma [22], the new contributions of this paper are: metaheuristic-based tuning of proportional-derivative learning rules… 569 ▪ a new ilc-based fuzzy control system structure, with the notation (d), derived from the open-closed-loop ilc with direct pd learning rule formulated in [23] for linear systems and a continuous-time derivative term. ▪ the systematic performance improvement of ilc-based fuzzy control systems for tc systems focusing on payload position control is proposed using the four pd learning rules (a), (b), (c) and (d) specified above, with seven popular metaheuristic optimization algorithms involved in tuning the two pd gains: gwo, sma, particle swarm optimization (pso) [24], gravitational search algorithm (gsa) [25], charged system search (css) [26], whale optimization algorithm (woa) [27], and african vultures optimization algorithm (avoa) [28]. these learning rules are combined with takagi-sugeno pi-fuzzy controllers leading to a new generation of low-cost fuzzy controllers. ▪ a novel unified design approach for all four pd learning rules. this approach optimally computes the gains of each learning rule in the iteration (or the experiment) domain using metaheuristic optimization algorithms that solve optimization problems with objective functions expressed as the sums of squared control errors multiplied by time and the variables represented by the two gains of the pd learning rules. ▪ the control system performance comparison based on real-time experimental results after a few iterations (namely, ten ones) of the metaheuristic optimization algorithms on the tc system laboratory equipment. the main advantage of these new contributions with respect to the state-of-the-art is the fact that the novel design approach does not require any complicated computations for the optimal ilc in order to guarantee convergence. these computations become even more complicated as pi-fuzzy controllers are involved, whose models must be considered in the design, and they are avoided in the proposed design approach. the takagi-sugeno pi-fuzzy controllers are designed by the fuzzification of the linear pi controllers tuned by the extended symmetrical optimum (eso) method [29], [30], applied to a simplified process model using the modal equivalence principle [31]. the eso method is advantageous as it offers a tradeoff to the empirical performance indices of the control systems using only one design parameter, and set-point filters are included for overshoot reduction. the stability of the fuzzy control system is ensured by using modeling based on fuzzy basis functions. although not shown in the paper, the proposed controllers reject constant load type disturbances. the presence of the integral block in the fuzzy controller structure helps in this regard. however, after rejecting the disturbances, the ranking of the ilc-based fuzzy controllers will not remain similar. the paper treats the following topics: the control system structures and the design approach are presented in the next section. the τc system and its model are briefly described in section 3. the experimental results are given in section 4, and the conclusions are highlighted in section 5. 2. ilc-based fuzzy control system structures and unified design approach 2.1. control system structures the single input-single output (siso) fuzzy control system structure with direct pd learning rule with current control error is shown in fig. 1 (a), where: r – the reference input 570 r.-e. precup, r.-c. roman, e.-l. hedrea et al. or the set-point, assumed to be repetitive, u – the control signal, d – the disturbance input also assumed to be repetitive, e – the control error, y – the controlled output, f – the setpoint filter, c – the fuzzy controller, p – the controlled process (namely, a siso sub-system of the tc system), m – the memory block, kp and kd – the proportional gain and the derivative gain of the pd learning rule, k – the index of the current sampling interval or the discrete time index, q–1 – the backward shift operator in the iteration domain, and j – the subscript that indicates the current iteration (or cycle or trial or experiment). the expression of the ρd learning rule used in fig. 1 (a) is expressed as follows after adapting the pd learning rules given in [13] and [20]: )],()([)()()( ]1[][][]1[][ kekekkekkuku jjdjpjj −− −++= (1) where the control error is ).()()( ][][ kykrke jj −= (2) fig. 1 structures of fuzzy control systems with: direct pd learning rule with current control error (a) (adapted from [20]), direct pd learning rule with previous control error (b) (adapted from [21]), indirect pd learning rule (c) (adapted from [22]), and openclosed-loop pd learning rule (d) the structure of sιsο fuzzy control system with direct pd learning rule with previous control error is shown in fig. 1 (b), with the following rule obtained by adapting the pd learning rules given in [13] and [21]: )].()([)()()( ]1[][]1[]1[][ kekekkekkuku jjdjpjj −−− −++= (3) metaheuristic-based tuning of proportional-derivative learning rules… 571 the structure of sιsο fuzzy control system with indirect pd learning rule is shown in fig. 1 (c), with the following rule obtained by adapting the pd learning rule given in [22]: )],()([)()()( ]1[][][]1[][ kekekkekkrkr jjdjpjj −− −++= (4) and it outlines a second notation for the set-point, namely r[j](k) indicating the set-point produced by the ilc algorithm and applied to the control loop, different to r(k), which indicates the set-point of the control system. the structure of sιsο fuzzy control system with open-closed-loop pd learning rule is shown in fig. 1 (d), where the following new pd learning rule is obtained by adapting the pd learning rule given in [23] to a discrete time formulation: )].()([)()()( ]2[]1[][]1[][ kekekkekkuku jjdjpjj −−− −++= (5) as specified in the previous section, c is a low-cost takagi-sugeno pi-fuzzy controller, and its design and tuning start with the tuning of the linear pi controller. a simplified model of the controlled process (p) with the following transfer function with respect to the control signal is used: )],1(/[)( stsksp p += (6) with the parameters kp (the process gain) and t (the small or parasitic time constant) partly known in the process model, and partly obtained, as in [22], by least-squares identification making use of input-output data obtained after real-time experiments conducted on the real process. using these control system structures and models, the unified design approach is described in the next subsection. 2.2. unified design approach pi controllers are recommended in [29] and [30] to control the processes with transfer functions of type (6). the transfer function of the (linear) pi controller is )],/(11[)( stksc ic += (7) where kc is the controller gain and ti is the controller integral time constant. as mentioned in the previous section, the eso method [29], [30] is successful in tuning the pi controller parameters because it ensures a convenient tradeoff to a set of empirical control system performance indices. the tradeoff is achieved using a single design parameter, , within the range 1 <   20, as shown in fig. 2, which illustrates the empirical performance indices σ1 – overshoot, = ttt rr /ˆ – normalized rise time, = ttt ss /ˆ – normalized settling time, both times defined in the unit step modification of r, and φm – phase margin. the pi tuning conditions specific to the eso method are [30] . ), /(1  == tttkk ipc (8) and the transfer function of one simple version of set-point filter f, which improves the performance by the cancellation of a zero in the closed-loop transfer function with respect to r, is [30] ). 1/(1)( stsf += (9) 572 r.-e. precup, r.-c. roman, e.-l. hedrea et al. fig. 2 empirical performance indices versus design parameter β [30] the control system performance is next improved by replacing the linear pi controller with the takagi-sugeno pi-fuzzy controller illustrated in fig. 3 (structure and input membership functions), with: z–1 − the backward shift operator in the discrete time domain, tiso-fb − the two inputs-single output fuzzy block, e(k) – the control error, u(k) – the control signal, e(k) − the increment of the control error, and u(k) − the increment of the control signal. the subscript [j] is dropped out in fig. 3 for the sake of simplicity. fig. 3 low-cost takagi-sugeno pi-fuzzy controller structure and input membership functions (adapted from [22]) the recurrent expression of the incremental discrete-time pi controller is obtained using tustin’s discretization method )],( )([)( kekekku p += (10) with the parameters [14], [32] ),2/( 2 ),/5.01( sisiscp tttttkk −=−=  (11) where ts > 0 is the sampling period, which is set to meet the requirements of quasicontinuous digital control. tiso-fb uses the weighted sum defuzzification method, and the inference engine uses the max and min operators. the tiso-fb rule base consists of two rules, r1 and r2, which are defined as follows to enable the low-cost design and implementation of the fuzzy controller [22], [32]: metaheuristic-based tuning of proportional-derivative learning rules… 573 )],( )([)( then )n is )( and p is )(( or ) zeis )( and p is )(( or )p is )( and n is )(( or ) zeis )( and n is )(( or ) zeis )(( if :r2 )],( )([ )( then )p is )( and p is )(( or )n is )( and n is )(( if :r1 kekekku kekekeke kekekekeke kekekku kekekeke p p   +=   +=  (12) where the parameter , 0.25    0.75, is inserted to further diminish the overshoot. the modal equivalence principle [31] is next applied leading to the following tuning condition [32]: . ee bb = (13) to summarize, the parameters of the low-cost pi-fuzzy controller are , which is set to ensure the trade-off to the empirical performance indices of the linear control system, , which is set in relation to the overshoot alleviation, and be, which is set in relation to fig. 3 to ensure the firing of both rules, possibly taking into account stability constraints [14], or optimally tuned [32]. after showing in section 2 in [5] that it is not possible to guarantee the stability of the fuzzy control system due to the saturation and dead zone static nonlinearities placed on the process inputs, a lyapunov-based stability analysis is offered, using the takagi-sugeno fuzzy model of the process and fuzzy basis functions. this analysis is important because it avoids the popular parallel distributed compensation based on solving linear matrix inequalities. the stability analysis, supported by [33] and [34], is easy for the user and is included in one of the steps of the design approach. as pointed out in section 1, the gains kp and kd of the ρd learning rules in the four ilcbased fuzzy control system structures are obtained as the solutions to the following optimization problem, which is processed using [20], [21] and [22]: ,),( 1 )( ),(minarg 1 2 ][ *  =  == sn k j d kke n jj ρρρρ ρρ (14) where d2 is the feasible domain of the parameter vector , * is the optimal parameter vector, i.e. the vector solution to the optimization problem, with the expressions ,][ ,][ *** t dp t dp kkkk == ρρ (15) t indicates matrix transposition, j() is the objective function, and ns is the number of data samples, which sets the width of the time horizon, nsts. the optimization problem expressed in (14) is solved in this paper using seven metaheuristic optimization algorithms. an important issue in this regard is to map the optimization algorithms onto the optimization problem. these algorithms operate with a total number of n agents, and each agent is assigned to a position vector xi[j] ,...1 ,] ... ... [ ][][ 1 ][][ nidxxx qtq ji f jijiji == ρx (16) where: f jix ][ – the position of ith agent in fth dimension, f=1…q, q=2 in this particular case, j – the index of the current iteration in both ilc (fig. 1) and the optimization algorithms, j=1…jmax, and jmax – the maximum number of iterations. pso is mapped onto (14) in terms of 574 r.-e. precup, r.-c. roman, e.-l. hedrea et al. , ,...1 ),( , * ,][][ ρpρρx ==== bestgjiji nijs (17) where si[j] is the fitness function specific to pso, and pg,best is the best swarm position vector. gsa, css, gwo, woa and sma are mapped onto (14) in terms of ,)(minarg ,...1 ),( , * ][ ...1 ][][ max ρxρρx ==== = ji ni jiji jnijs (18) where si[j] is the fitness function specific to gsa, css, gwo, woa and sma. avoa is mapped onto (14) in terms of , ,...1 ),( , *1 ][][][ max ρxρρx ==== bv jjiji nijs (19) where si[j] is the fitness function specific to avoa, and 1 ][ max bv jx is the first best vector solution obtained at the iteration jmax. in all algorithms, the search domain is the feasible domain d2 in (14). the unified design approach applied to the ilc-based fuzzy control systems with the four pd learning rules consists of the following steps, which are obtained after organizing the steps given in [20], [21] and [22]: step 1. the sampling period ts is set. the performance specifications imposed to the fuzzy control system are expressed in terms of the optimization problem expressed in (14). the associated dynamic regime is defined in order to assess the objective function values through experiments conducted on the real-world process. step 2. the value of jmax is set. step 3. the values of the parameters ,  and be of the takagi-sugeno pi-fuzzy controller are set. the tuning condition (13) is applied to obtain the value of be. an experiment is conducted to check the stability condition of the fuzzy control system given in theorem 1 in [5], and only those controller parameters that satisfy the stability condition are used next. step 4. the parameters specific to the metaheuristic optimization algorithms are set, and the algorithms are applied to compute the values of the gains * pk and * dk . the above steps are actually a quick and useful guide to easily tuning the controllers. steps 1 to 4 are applied to the tc system in order to validate the proposed design approach in a payload position control example. details of the tc system equipment are given in the next section. 3. tower crane system the laboratory equipment involved in conducting the experiments and assessing the efficiency of the fuzzy controllers and design approach proposed in this paper is built around the tc system with technical information given in [35], and it is presented in detail in section 1 of the supplementary appendix given in [5]. a photo of the laboratory equipment is illustrated in fig. 4. the tc system equipment is a multi input-multi output system focused on controlling the cart position, the arm angular position, and the payload position. a relatively simple control approach is to separately design three siso control systems to control each position (output). to keep the paper to a reasonable length, only the results for the payload position control are presented; moreover, the payload position control is the most challenging out metaheuristic-based tuning of proportional-derivative learning rules… 575 of the three siso ones. a diagram with details of the equipment and its communication with a computer are presented in fig. 1 given in [5]. fig. 4 photo of tower crane system laboratory equipment [36] the description of the tc system equipment and its modeling carried out in [36], with hardware and software details given in [37], is supported by [38], which was used in the derivation of the state equation of payload position dynamics. using the information given in this section, in [5], and the theoretical results proposed in section 2, the experimental validation is carried out in the next section. 4. experimental results and comparison the real-time experiments were conducted on the tc system experimental setup described in the previous section and [5] to apply the novel unified design approach formulated in section 3 to the four ilc-based fuzzy control systems with pd learning rules. as mentioned in the previous sections, the payload position control is considered in this paper, therefore the controlled output y in section 2 is actually the payload position y3 (m) = x9 (m) of the tc system as described in [5]. 4.1. application of design approach aspects concerning the steps of the design approach are highlighted as follows. the dynamic regime used to evaluate the value of the objective function defined in (14) is characterized by: zero initial conditions, no disturbances applied, but arbitrary disturbances can appear, and the reference input is next given as a reference trajectory along with the tc system responses. the objective function is evaluated using the sampling period set to ts = 0.01 s, the width of the time horizon is set to 70 s, corresponding to a number of ns = 70/0.01 + 1 = 7001 data samples. applying least-squares identification, the parameters of the process transfer function in (7) obtain the values kp = 0.2505 and t = 0.0533 s. setting the design parameter =9, the tuning conditions specific to the eso method given in (8) lead to the parameters of the linear pi controller kc = 24.99 and ti = 0.48 s. applying (11), the parameters of the discrete576 r.-e. precup, r.-c. roman, e.-l. hedrea et al. time pi controller are kp = 11.8531 and  = 0.0211. setting the parameter of the pi-fuzzy controller be = 0.04, the other parameter results from (13), be = 8.410–4. the third parameter of the pi-fuzzy controller is set to  = 0.2550. these parameter values guarantee, via the stability analysis in [5], the stability of the fuzzy control system. the feasible domain of  is set by modifying the domains used in [20-22] to obtain the unified domain }.1.01.0|{}1.01.0|{ −−= ddpp kkkkdρ (20) the initial parameters of the four pd learning rules are different to those used in [20-22]: ,0 ,0 ]0[]0[ == dp kk (21) where the subscript 0 is forced to capture the situation without ilc, namely without the pd learning rule in fig. 1. this ensures a fair comparison of the control system structures, the pd learning rules, and the optimization algorithms on actually 1 + jmax iterations. the maximum number of iterations is set to jmax = 10 to ensure the low-cost implementation, and for the same reason, the number of agents for all seven metaheuristic optimization algorithms is set to n = 10. in the context of ensuring the full transparency of the experimental validation and comparison results, the non-random parameters specific to the seven optimization algorithms are set as follows: the weight parameters of pso: c1 = c2 = 0.3, a linear decrease of the inertia weight parameter w[j] of pso with the bounds wmax = 0.9 and wmin = 0.4: ,)( maxminmaxmax][ jwwjww j −−= (22) the exponential decrease law of the gravitational constant g[j] of gsa: ,max/ 0][ jj j egg − = (23) with the initial value g0 = 100, the exponent parameter  = 8.5 and the parameter at the denominator of the force gsa = 10–4 to avoid possible divisions by zero, each agent in css is considered as a charged sphere of uniform volume charge density with radius a = 1, the css acceleration and velocity parameters ka[j] and kv[j], respectively: ),/1(2),/1(2 max][max][ jjkjjk jvja +=−= (24) and the parameter at the denominator of the separation distance css = 10–4 to avoid possible divisions by zero, the threshold in the position update equation specific to sma z = 0.03, and the parameter in the fitness weight of each slime mould specific to sma sma = 10–3 to avoid possible divisions by zero, and the controlling parameters specific to avoa set to: p1 = 0.6, p2 = 0.4, p3 = 0.6, avoa = 0.8, avoa = 0.2, and avoa = 2.5. the results of the comparison may be different for parameter settings other than the values specified above. additional details on the implementation of the optimization algorithms are given in [32] and [39-41], where it is also highlighted that these parameter settings ensure a good trade-off to exploration and exploitation. in the same context of ensuring full transparency, the data processed in the statistical analysis conducted to compare the performance of all four control system structures and seven optimization algorithms is freely available in the data_fume_2.m matlab file [42]. metaheuristic-based tuning of proportional-derivative learning rules… 577 4.2. experimental results, comparison and discussion the optimization algorithms were run 30 times. all results are presented in averaged values. the results obtained after conducting the variance (anova) test of the minimum objective function jmin evaluated after running these seven optimization algorithms on the four ilc-based fuzzy control system structures are presented in fig. 5, with the general notation m-q, where m indicates the pd learning rule, namely m  {c, p, i, o}, c – direct rule with current control error (fig. 1 (a)), p – direct rule with previous control error (fig. 1 (b)), i – indirect rule (fig. 1 (c)), and o – open-closed-loop rule (fig. 1 (d)), and q indicates the optimization algorithm, namely q  {pso, gsa, css, gwo, woa, sma, avoa}. fig. 5 and table 2 given in [5] show that the best jmin is achieved by the indirect pd learning rule and five optimization algorithms for this process and this fuzzy controller, in the following descending order, with small differences: i-woa, i-gsa, i-css, i-gwo and i-pso. the average values of the optimal parameters of the four pd learning rules obtained after running the seven optimization algorithms are presented in table 2 given in [5]. typical responses of the fuzzy control system before and after the application of the four pd learning rules and an optimization algorithm are shown in figs. 2 to 5 in [5]. the performance improvement offered by ilc combined with metaheuristic optimization algorithms is clearly highlighted. the main results of the non-parametric statistical tests conducted on the 28 cases (four ilc-based fuzzy control system structures multiplied by seven optimization algorithms) are presented in tables 3 and 4 given in [5]. these results show that the best performance is exhibited by the indirect pd learning rule and three optimization algorithms, namely icss, i-gwo and i-woa, followed by the same rule and other two optimization algorithms, namely i-gsa and i-pso, again with small differences. the following performance indices were considered for comparison: rise time, settling time, percent overshoot, steady-state error, and mean sum of squared control signals. accordingly, experiments were conducted to compare the performance of the control system structures and the results are summarized in table 5 given in [5] both initially and after 10 iterations. the results show significant improvement of the rise time, settling time, overshoot and mean sum of squared control signals after ten iterations over the same indices calculated initially. a numerical comparison and a discussion of the experimental results are given in section 3 of [5]. the comparison and the discussion provide suggestive evidence of the advantages and disadvantages of the four controllers whose pd learning rules are tuned using different metaheuristic optimization algorithms; they also show which one performs better or best. the results of the comparison carried out in [5] cannot be generalized to any controlled process. however, experience in controlling certain processes can be used and incorporated into the fuzzy control rules. such representative processes include the finite element formulation for piezoelectric active laminated shells [43, 44], biomedical systems including image processing [45], natural language processing [46], spiking neural p systems [47], bed combustion processes [48], human well-being and resilience [49], complex planetary gearboxes [50], bin packing [51], telesurgical robotic systems [52], vanets [53], robust evolving cloud-based controllers [54], cloud computing systems for traffic management [55], cognition processes [56], and friction compensation in haptic interfaces [57]. 578 r.-e. precup, r.-c. roman, e.-l. hedrea et al. fig. 5 anova test of minimum objective function j for all four ilc-based fuzzy control system structures and seven optimization algorithms metaheuristic-based tuning of proportional-derivative learning rules… 579 5. conclusions this paper proposed the performance improvement of low-cost fuzzy control systems with four proportional-derivative (pd) learning rules in the context of iterative learning control (ilc) in terms of optimally tuning the two pd parameters (or gains) using seven metaheuristic algorithms. a unified design approach is formulated, and the performance improvement is proved and compared by real-time experimental results that correspond to payload position control of tower crane systems. as specified in section 1, the first advantage of the design approach proposed in this paper is the avoidance of complicated computations specific to the classical optimal ilc approaches in relation to the need to guarantee convergence. as shown in fig. 3 given in [5], the average objective function changes quickly as the iterations go by (fast decrease of the objective function j). this is especially true for the ilc-based fuzzy control system structure with an indirect pd learning rule, optimized by five algorithms, which ensure the average jmin = 0.0062. the value of j remains practically constant after iteration 1. although the use of metaheuristic algorithms usually requires many evaluations of j and thus is generally a limitation, these results clearly show the fast decrease of j after only iteration 1, which is the second advantage of the design approach. the same conclusion is drawn after examining the evolution of the average values of the parameters kp and kd of the pd learning rules in fig. 4 [5]. the stability analysis of the four control systems performed in [5] can also be performed in a different approach in the context of data-driven control, since ilc can also be considered as a data-driven control technique. this can be seen as related to the optimization problem solved by the authors for optimal reference input tracking, and the results given in the book [36] can be used as a starting point for different approaches to stability analysis. future research will be focused on the further reduction of the number of evaluations of the objective function. incorporating gradient information into data-driven control is expected to be a viable solution and suggests another avenue for future study. acknowledgement: this work was supported by a grant of the ministry of research, innovation and digitization, cncs/cccdi uefiscdi, project 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faculty of economics, administration, and social sciences, kadir has university, türkiye 3international trade department, faculty of applied sciences, istanbul gelişim university, türkiye abstract. selecting an independent quality control, test, and inspection firm (iqctif) as a strategic partner in railway wagons and locomotive manufacturing is critical for safety, compliance, and efficiency, as errors can lead to accidents, high maintenance costs, and reputational damage. recent railway accidents in germany, spain, china, and turkey highlight the risks of inadequate quality control. to mitigate these risks, multicriteria decision-making (mcdm) approaches help balance security tests, certifications, cost, impartiality, and technological infrastructure. despite its importance, optimizing these processes through outsourcing remains an underexplored research area. this study proposes an ivq-rofs-based critic-waspas model to identify the most suitable strategic partners in railway manufacturing. the proposed model was preferred in this study due to its advantages, such as having an extremely flexible structure, providing results at a high level of accuracy, and allowing effective and successful management of uncertainties. according to the results, "railway certifications" (c2) is the most critical criterion, and tüv süd is the most suitable partner for the railway manufacturing industry. a comprehensive robustness check confirmed the model's reliability. key words: railway manufacturing industry, independent audit & quality control, test, and inspection firm (iqctif), critic, waspas, interval-valued q-rung orthopair fuzzy set received: march 11, 2025 / accepted july 14, 2025 corresponding author: ömer faruk görçün logistics department, faculty of applied science, kadir has university, kadir has street, no:1, b-203 cibali, fatih – i̇stanbul 34083, türkiye e-mail: omer.gorcun@khas.edu.tr 2 ö.f. görçün, c.n. durmuş, m. çanakçioglu 1. introduction failures in railway vehicle manufacturing quality control processes have recently caused fatal accidents, injuries, financial losses, and damage to trust and reputation. in 1998, 101 people were killed, and 88 passengers were injured because of a high-speed train derailment in germany [1]. post-accident investigations showed that factors such as defects in the materials used in the production of the high-speed train (such as metal fatigue) [2] and poor management of quality control processes were primarily responsible for the accident. the wenzhou train accident in china in 2011 resulted in the death of more than 40 people due to a lack of quality control in the signaling system [3]. official reports on this accident have cited serious design flaws and a lack of supervision and control as the main culprits [4]. then, in the train accident that occurred in spain in 2013, in which eighty people lost their lives [5], the railway authority claimed that the fault was with the train personnel, while technical reports cited inadequate testing of the brake systems as the most crucial cause of the accident [6]. recently, railway accidents have occurred in some rail system projects in turkey due to a lack of certification, resulting in significant losses [7]. while collision-type accidents in railway transportation generally occur due to human errors, the factors that cause derailment can be listed as deterioration in the wheels of the railway vehicle, deterioration on the rail surface, deterioration of the track geometry, etc. [8]. therefore, derailment-type accidents are caused mainly by systemic problems. cannon et al. [9] highlighted that the real cause of such accidents is improperly structured quality control and monitoring systems for rolling stock. according to the statistical data published by uic, considering the accidents that occurred only in turkey, more than three times as many derailment-type accidents took place as collision-type accidents. a well-structured quality control and monitoring system enables the production of reliable locomotives and components that fully comply with international standards. this situation is critical in reducing risks and accidents in railway transportation and creating a safe, reliable, comfortable, and efficient transportation system [10]. the integration of preventive maintenance and production planning, particularly within the framework of industry 4.0 and smart manufacturing, has been shown to significantly improve production efficiency and reduce operational disruptions, as emphasized by babaeimorad et al. [11]. in this direction, accidents and externalities caused by production processes that do not comply with the standards, regulations, and certifications determined by international railway authorities or cannot meet the requirements specified in these regulations, and the defects of the wagons, locomotives, and bogies produced cause severe economic losses in addition to loss of trust and reputation in the eyes of all stakeholders of the industry [12]. a properly configured quality control and assurance system improves supply chain efficiency and effectiveness in the rolling stock manufacturing industries [13], optimizing the waste of materials used in production and maintenance and repair costs [14]. however, quality control, monitoring, auditing, and sustainability practices have expanded international and national legal regulations regarding the number and content. in this context, they have become more complex and sophisticated due to international organizations' and governments' increasing sanctions and pressures on safety, security, and sustainability [15]. in addition to the railway industry, it has become almost impossible for the departments and units to be established within the enterprise to undertake the responsibilities regarding these applications in nearly every sector and successfully manage these processes [16]. assessment of the quality inspectors for the railway manufacturing industry in... 3 therefore, manufacturers must meet these requirements, especially in the railway sector, through outsourcing and strategic partnerships with professional quality service providers [17]. in this context, selecting a strategic partner for quality control and process management of production enterprises in the railway industry is a critical and strategic decision regarding railway transportation safety, security, and sustainability [18]. this methodical technique improves motor selection and can be used as a template for similar assessments in other technical contexts, including strategic partner evaluation and process optimization in highly regulated industries [19]. on the other hand, it may be more rational to outsource these services to internationally professionalized sources, providing them with higher quality, more convenient, and continuous access [20]. first, professional organizations continuously and in real-time follow international standards related to quality control, sustainability, and safety associated with rolling stock production. they have deep and extensive expertise in this field. accordingly, manufacturers adapt more quickly to changing and differentiating regulations in different markets [21] while increasing their competitiveness tremendously. a professional iqctif conducts objective evaluations and identifies issues impartially, thanks to its independence [22]. in addition, professional businesses also have relationships with different industries and provide services for these industries as well. accordingly, it can also use other industry experiences and practices for companies that are railway vehicle manufacturers, thus providing critical advantages. in addition to the accelerating development of technology, the diversification and complexity of legal regulations have incredibly complicated the quality control and inspection processes, and accordingly, it has become challenging for production enterprises to produce these services by establishing departments within their bodies [23]. considering the costs to be incurred for the establishment of the department in the first place, the employment of expert personnel to work in these departments and the costs of continuing education for these specialists, the provision of technologically advanced and highly sophisticated equipment and tools used in quality control processes, as well as the costs to carry out certification processes, such an approach can lead to excessively high and unbearable expenses [24]. on the other hand, professional businesses only demand payment from production enterprises for the services they provide, as they cover installation costs, etc. therefore, production enterprises can provide higher-quality control and inspection services at very reasonable costs. in addition, professional independent quality control, test, and inspection firms (iqctifs) produce and complete these services quickly and effectively as they standardize and simplify their quality control and inspection activities with the advantage of the experience they have gained over many years [25]. in the railway production industry, enterprises can obtain a significant competitive advantage by supplying their products to the market more easily and quickly, depending on the inspection processes that are completed quickly. therefore, the services produced by such enterprises also increase the trust and reputation of the manufacturer regarding the products they make at an extraordinary level. more importantly, professional organizations produce more precise, detailed, and high-quality testing services by continuously investing in advanced technologies required for tests and inspections, as they have determined quality control and inspection activities as their main activities [26]. establishing in-house infrastructure is costly, and equipment may quickly become outdated. 4 ö.f. görçün, c.n. durmuş, m. çanakçioglu regarding production enterprises, selecting the appropriate quality control and inspection company involves critical challenges and risks. focusing solely on selecting iqctifs based on economic criteria and determining the most suitable option through static assessment models often reflects a tendency to address the short-term needs of businesses. for instance, practitioners may base their choices on subjective assessments and prioritize a criterion of limited relevance for longstanding objectives. thereby, they risk overlooking critical factors and key criteria for sustainable success. such an approach significantly hinders manufacturing enterprises from effectively achieving their long-term strategic goals. the lack of a structured roadmap for selecting iqctifs can lead to critical challenges in modeling and managing existing and potential risks associated with the selection process. failing to evaluate audit firms based on essential factors identified within a strategic framework increases the likelihood of making deficient choices. such missteps can significantly increase the business's operational, financial, and reputational risks and have long-term consequences. evaluating iqctifs without a structured roadmap poses serious challenges and risks. such an approach may expose businesses to inconsistency and indecisiveness in decisionmaking processes. additionally, decisions made without a well-structured framework are prone to subjective biases, which can undermine the credibility and reliability of the outcomes. moreover, this lack of transparency can tarnish the company's reputation among stakeholders, significantly erasing trust and credibility. this inefficiency can lead to wasted time and other resources that decision-makers could have utilized more effectively. consequently, such mismanagement results in substantial time and labor losses and imposes significant financial costs on the organization. in addition, determining whether stakeholders should outsource activities related to appropriate quality control and process management for production enterprises is a decision-making problem that is affected by highly complex uncertainties and needs to be balanced by considering contradictory criteria. similarly, the presence of diverse and distinct service features in various application domains — such as e-services of brokerage firms — has resulted in multi-criteria decision-making dilemmas for users, as emphasized by jana et al. [27]. moreover, when addressing this decision-making problem, linear and nonlinear interactions and relationships between criteria also need to be considered. for example, c4 (technological infrastructure and equipment) directly affects c3 (safety testing capability). the development of technological infrastructure increases the accuracy of security tests. likewise, there can be a nonlinear relationship between c7 (cost and pricing) and c8 (service timeliness and flexibility). services with higher costs can often be faster, but after a certain level of cost, the increase in speed may become limited. in this context, ivq-rof clusters precisely reflect nonlinear threshold effects and complex interactions by defining membership, anti-membership, and degrees of uncertainty over a wide range. thanks to its capacity to manage ambiguities, it can analyze linear and nonlinear dependencies between criteria flexibly and dynamically. hence, it produces more accurate and reliable results in multi-criteria decision-making processes. in this context, selecting the most appropriate stakeholder requires using a robust, consistent, and reliable decision-making tool extended based on an advanced fuzzy set to model and manage uncertainties at a perfect level. at the same time, this model should handle the interactions between the selection criteria and be able to perform multidimensional and comprehensive analysis. assessment of the quality inspectors for the railway manufacturing industry in... 5 according to the authors' knowledge, no study has focused on selecting quality control and process management companies using multi-criteria decision-making approaches in the relevant literature. in recent literature, hybrid mcdm models that combine weight determination and ranking phases, such as the dibr ii–rough mabac method, have proven effective in evaluating organizational and technical strategies under uncertain and resource-constrained environments [28]. generally, researchers have focused on issues such as inspection procedures in production processes [29], inspection and quality costs in the manufacturing sector, and the effects of quality management practices on performance and competitive advantage [30]. in conclusion, even though the selection of iqctif is a significant and critical research problem, it is an issue that has been largely neglected and ignored by members of the research community. accordingly, considering the above requirements and critical research gaps, this study aimed to develop and recommend a reliable, robust, and consistent decision-making tool for decision-makers of all manufacturing enterprises, especially the railway vehicle manufacturing industry. this proposed decision-making framework should also include an advanced fuzzy set to manage uncertainties effectively. following these requirements, this study developed an interval-valued q-rof (q-rung orthopair fuzzy) set based on criteria importance through intercriteria correlation (critic) developed by diakoulaki et al. [31] and weighted aggregated sum product assessment (waspas) approaches introduced by zavadskas et al. [32] to address this decision-making problem. cronbach's alpha was calculated to verify the validity and relevance of the selected criteria, ensuring that the evaluation was based on relevant and influential factors. in this context, each of the nine experts was asked 14 questions to evaluate the criteria, and for each question, the experts were asked to rate between 1 and 10. here, the total variance was calculated as 21.6944, while the variance value of the total scores was determined as 91.5. then, using the formulation of the cronbach alpha technique, the cronbach alpha value was calculated as 0.82159. when the results are between 0.80 and 1.00, they are highly reliable [33]. the subjective weight coefficients derived from the mathematical framework of the fuzzy set were combined with the weights obtained by the critic method. a range-valued q-rof-based waspas method was used to rank and prioritize strategy alternatives. the suggested model has a practical and straightforward procedure that allows practitioners to employ it effectively without requiring advanced mathematical knowledge. it is also highly trustworthy and robust for modeling ambiguities, structuring decision issues, and delivering consistent and dependable conclusions. these features make the model well-suited for applications that require precision and adaptability in complex decision-making scenarios. the proposed model has identified ten alternatives and fourteen evaluation criteria for selecting an iqctif. the findings show that risk management practices emerged as the most critical and practical criterion, while the independence and impartiality strategy was determined as the most appropriate strategic approach. the contributions of this study are essential and multifaceted. first, the integrated model proposed in this research improves the decision-making processes of enterprises by providing a more structured and practical framework for selecting iqctifs. furthermore, the versatile model can be adapted to address similar decision-making challenges in other industries, demonstrating its broader applicability. furthermore, the proposed model provides a well-structured, effective, and reliable strategy-based 6 ö.f. görçün, c.n. durmuş, m. çanakçioglu framework, ensuring that resources are appropriately allocated and that evaluation processes are carried out systematically. it supports better decision-making and improves organizations' strategic alignment and operational efficiency. in addition, the ivq-rof-based subjective & critic & waspas integrated model proposed in the study has not been used in the relevant literature before, but has been developed for the first time to solve the decision problem addressed in this study. as such, it represents a comprehensive hybrid decision-making approach that is rare in the available literature. this study has some critical managerial and political implications. considering the findings and conclusions obtained in this study, decision-makers can strategically structure their decision-making processes rationally and systematically, especially regarding the selection of iqctifs. in addition, the study provides a comprehensive roadmap for decision-makers, emphasizing the need to focus on sustainable growth and long-term alliances, not just short-term operational goals. in addition, the results obtained in the study can further increase the trust environment in the sector by making the evaluation processes more transparent based on business models and determined strategies. first, it underlines the need for businesses to adopt a more holistic framework for their decision-making processes while at the same time identifying the critical and practical criteria that affect the choice of iqctif and ensuring that the decision-making problem is structured more accurately and appropriately. given the identified research gap and objectives, this study aims to address the following key research questions: rq1. what are the most critical evaluation strategies for selecting iqctifs in an interval-valued q-rung orthopair fuzzy (ivq-rof) environment? rq2. how do the critic and waspas methods improve decision-making when selecting iqctifs? rq3. how does the proposed model compare to other mcdm approaches regarding robustness, reliability, and accuracy? after a comprehensive introduction, the rest of the study is structured as follows. part 2 presents the results of a detailed literature review. chapter 3 introduces the proposed integrated decision-making model and illustrates the implementation steps and mathematical operations. chapter 4 describes the implications of the proposed decisionmaking model for addressing and resolving the decision-making problem related to evaluating and prioritizing iqctifs' choices. chapter 5 summarizes and discusses the results obtained. chapter 6 concludes the study and shows the limitations of the study and research guidelines for future studies. 2. suggested framework this section outlines the mathematical formulation and implementation steps of the proposed ivq-rofns-based decision-making model. the model was chosen to identify strategies for iqctif selection, the core of the decision-making problem, and was designed to address certain structural limitations. the following subsections present the proposed approach's structural features, fundamental algorithms, and step-by-step implementation. 2.1 preliminary information about the ivq-rofns yager [34] introduced q-rofss, defined by the pair (q, ν), to manage complex uncertainties in decision-making. however, due to structural limitations, they may not fully capture or process uncertainties [35]. to overcome these issues, joshi et al. [36] proposed assessment of the quality inspectors for the railway manufacturing industry in... 7 ivq-rofss as an improved version. their flexible structure offers a broader and more accurate approach to handling uncertainty. the following section introduces the basics of ivq-rofss and their primary algebraic operations. a q-rofs on the universal set x is given below [35]:            0,1 , 0,1 ,0 1 q q f f f fx x x x        (1)       , , q q f ff x x x x x   (2) here, ϛf(x) is the membership degree and υf(x) depicts the non-membership degree of x ∈ x. on the other hand, the degree of indeterminacy is calculated as:    1 q qq f f fx x     (3) an iv-q-rofs on the universal set x is signified below [34, 35]:                   , 0,1 , , 0,1 ,0 1 h h h h h h q q u u u ul lx x x x x x                (4)         , , , , h h h h q q q q u ul lh x x x x x x x            (5) in eq. (5), the membership degree's lower and upper bound's lower and upper bounds of the non-membership degree are exhibited, respectively. moreover, the indeterminacy membership degree is defined as follows:            , 1 , 1 h h h h h h q q q qq qu u ul l l f x x x x x x                  (6) the iv-q-rof number (iv-q-rofn) can be written as φ= ([α, β], [γ, δ]) for simplicity. moreover, the conditions given below are satisfied [34].        , 0,1 , , 0,1 ,0 1q q          (7) let φ1= ([α1, β1], [γ1, δ1]) and φ2= ([α2, β2], [γ2, δ2]) be two iv-q-rofns. next, some operations, score, and accuracy functions on iv-q-rofss are outlined below: ζ > 0, q ≥1 [34–36].                  1 2 1 2 1 2 1 22 21 22 1 2 1 2, , , q q q q q q q qq q                          (8)                  1 2 1 2 1 2 1 2 1 2 1 2 1 2, , , q q q q q q q qq q                          (9)          1 1 1 1 11 1 , 1 1 , , q qq q                          (10) 8 ö.f. görçün, c.n. durmuş, m. çanakçioglu          1 1 1 1 1, , 1 1 , 1 1 , 0, q qq q                           (11)       1 1 1 1 1, , , , c      (12)               1 1 1 1 1 1 1 1 ; 0,1 4 q q q q s s              (13)               1 1 1 1 1 1; 0,2 2 q q q q a a           (14) the ivq-rof weighted arithmetic average (ivq-rofwaa) operator is denoted below: here, φi = φ1,φ2,…, φm, 0 ≤ ζi ≤ 1. the sum of ζi should be equal to 1.             1 2 1 1 1 1 1 1 1 1 , ,... 1 1 , 1 1 , , i i i i m m m m m q q q q i i i i ivq rofwaa                                        (15) the ivq-rof weighted geometric average (ivq-rofwga) operator is described below: here, φi = φ1,φ2,…, φm, 0 ≤ ζi ≤ 1. the sum of ζi should equal 1 [34, 35].             1 2 1 1 1 1 1 1 1 1 , ,... , , 1 1 , 1 1 i i i i m m m m m q q q q i i i i ivq rofwga                                      (16) eq. (17) gives the minkowski distance between φ1 and φ2 are indeterminacy membership degrees of φ1 and φ2, respectively [34-36]:                           1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 , 4 p p p q q q q q q p p p pp q q q qq q u ul l m                                (17) when p =1, eq. (17) is employed to compute the hamming distance, while with p =2, it determines the euclidean distance. 2.2 identifying the experts' reputations at this stage, each expert's weight values (reputation) are calculated by considering various characteristics of the experts, such as age, title, industry in which the expert works, field of expertise, graduation degree, graduated department, and industry experience of the participant. it is recommended that professionals with higher expertise and knowledge, like the esteemed members of our audience, evaluate the experts proposed by the researchers to obtain more logical and rational results in this study. step 1: evaluation of the characteristics of the experts: to include the knowledge, experience, and other characteristics of the members of the expert committee that are assessment of the quality inspectors for the railway manufacturing industry in... 9 expected to affect the results, the determined professionals evaluate each feature of the experts as well as the impact of these characteristics separately for each expert. for this, the linguistic assessment scale given in table 1 is used. table 1 the linguistic appraisal scale for ivq-rofns system abbr. μl μu vl vu system abbr. μl μu vl vu extremely low el 0.05 0.1 0.85 0.9 medium-high mh 0.55 0.65 0.3 0.35 very low vl 0.1 0.2 0.7 0.75 high h 0.65 0.75 0.2 0.25 low l 0.2 0.3 0.55 0.65 very high vh 0.75 0.8 0.15 0.2 medium-low ml 0.3 0.4 0.45 0.55 very very high vvh 0.8 0.85 0.1 0.15 medium m 0.45 0.55 0.35 0.4 extremely high eh 0.85 0.9 0.05 0.1 step 2: conversion of assessments into ivq-rof numbers: researchers convert linguistic assessments of traits and experts into numerical equivalents corresponding to table 1 in the second round. then, matrices for characteristics and specialists are prepared. step-3: aggregating the ivq-rof numbers for the assessments and calculating the score values and weights: with the help of eq. (15), first, the ivq-rof number values are combined in matrices for the features, and then the score values are calculated using eq. (13) for each feature. then, with the help of eq. (18), the final weight values of the features are calculated. the same operations are carried out to determine the characteristics of specialists. in the process of combining ivq-rof numbers, the relative weight coefficients of the features are included in the calculations.     1 kg g k z kg g g s q x w s q x w     (18) 2.3 ivq-rof-subjective & critic & waspas this chapter shows the mathematical notion of the extended form of the hybrid model consisting of a combination of subjective & critic, and waspas methods with the help of ivq-rofns and the basic implementation steps. in the first stage, subjective weights of the criteria are identified with the help of the structure of the ivq-rofns. step-4: identify the subjective weights of the criteria: at this stage, it uses the mathematical notion of ivq-rof sets to evaluate the weight values and impact levels of the criteria that affect iqctif selection. below are the implementation steps for this. step-4.1: perform the linguistic evaluations for the criteria by the experts: in this step, each expert considers the criteria individually and evaluates them separately, considering the linguistic evaluation scale given in table 1. the researchers then collect these assessments to create linguistic matrices converted into the corresponding q-rof numbers on the rating scale. step-4.2: aggregating the experts' evaluations: after the linguistic assessments are converted into q-rof numbers, the researchers combine these numbers using eq. (10). step-4.3: calculation of score function values of criteria: eq. (13) is used to determine the score function values of the criteria. afterward, the final subjective weights of the criteria are identified with the help of eq. (19). 10 ö.f. görçün, c.n. durmuş, m. çanakçioglu 1 sub kjsub j n sub kj j x w x        (19) step-5: assessment of the alternatives by the experts: in this process, r number of experts evaluate m number of alternatives, considering n number of criteria. while making these assessments, experts consider the linguistic assessment scale shown in table 1. next, the linguistics appraisals are converted to the ivq-rofns corresponding to the linguistics scale, and the initial ivq-rof matrices involving ivq-rofns are formed. after the linguistic evaluations are completed, the initial ivq-rof decision matrices x(k) including matrix elements as follows:         , , , k k k kk ij ij ij ij ijx            (20) step 6: aggregation of the ivq-rof matrices: in this step, the researchers aggregate these matrices with the help of eq. (15), generating the initial ivq-rof decision matrix. in this process, the reputation degrees of the experts are considered when implementing eq. (15). step 7: computation of the score function values of the alternatives: in this step, the score function of the ith alternative, considering the jth criterion, is computed with the help of eq. (13), and the score matrix is constructed. step 8: criterion weighting with the ivq-rof-critic approach: in this step, the application steps of the critic method [28] are followed. the score matrix determined in the previous steps is the initial decision matrix when applying the critic method. step-8.1: normalization of the elements of the initial decision-making matrix: in this step, the initial decision matrix elements are normalized by employing eq. (21). then, the normalized matrix is constructed [35].             ; ij ij ij ij ij ij ij ij ij ij ij x min x ; x c max x min x r max x x x b max x min x            (21) here, c denotes the cost criteria and b denotes the benefit criteria. besides, rij denotes normalized values, and min(xij) is the minimum value of jth criterion concerning alternatives, max(xij) is the maximum value of jth criterion concerning alternatives. step-8.2: calculate the correlation and sum of information for each criterion: the researchers compute correlation coefficients between each criterion pair in the normalized matrix. for this purpose, standard deviations for each criterion are calculated with the help of eq. (22).   2 1 r ijij i j r r n      (22) here, rij is computed as ∑rij/n. next, each criterion's sum of information (kj) is identified by employing eq. (23). assessment of the quality inspectors for the railway manufacturing industry in... 11   1 1 ; 1,2,..., n j j jk k k j n      (23) step-8.3: identification of the final weights of the criteria: in the final step of the procedure, the weight coefficients of the criteria are calculated using eq. (24). next, the criteria are sorted considering their relative weight coefficients. 1 jobj j n j k k w k    (24) step 9: combine the subjective and objective weights: in this step, the criteria's subjective and objective weight values are combined with the help of the β coefficient. it takes a value between 0 and 1. both weight coefficients are combined, as follows:  . 1 . objsub j j jw w w    (25) step 10: ranking the performance of alternatives: the ivq-rof-based waspas method was used to compute and rank the relative importance of each alternative. the following outlines the method's application steps and calculation procedure. step-10.1: construction of the initial decision matrix: like the critic approach, the same initial decision matrix (score matrix) is used. therefore, to avoid duplication, the operations in the critic method's first step are again not shown at this stage. step-10.2: normalization of the initial decision matrix elements: in the second step, the matrix elements are normalized by employing eq. (26), generating the normalized matrix.     ; ij ij ij ij ij ij ij x ; x b max x min x x c x          (26) where ρij symbolizes the normalized value of each element. step-10.3: calculate the measures of wsm and wpm: in this step, the weighted sum model (wsm) and weighted product model (wpm) measures for each option are computed with the help of eqs. (27) and (28).  1 ij ij j x w  (27)    2 1 j ij n w ij m     (28) step-10.4: compute the aggregated measures of the waspas method: by implementing eq. (29), the combined measures for each option are identified as follows:      1 2 . 1 . ij ijij      (29) here, α denotes the parameter of the waspas technique taking a value between 0 and 1. the combined measures are calculated, and decision alternatives ranked based on these values. 12 ö.f. görçün, c.n. durmuş, m. çanakçioglu 3. numerical illustration this section applies to the proposed ivq-rofns-based decision-making model to identify strategies for selecting iqctifs and presents the results. the researchers assembled an expert panel to ensure the model's effective implementation and produce realistic results. the panel, comprising three professionals and nine experienced managers, was selected through a rigorous screening process. the panel included a professor affiliated with two universities' rail systems departments and an expert consultant from the industry. other members included a professional with 35 years of experience who served twice as general manager of i.e.t.t. and once as director general of tulomsaş, a 30-year veteran who led another railway vehicle manufacturing company for two terms, and an electrical engineer who served as r&d manager at tuvasaş. the researchers and the professionals identified the key characteristics and skills required for expert participation in the evaluation process. as a result, seven essential factors were identified: age (f1), title (f2), industry (f3), field of expertise (f4), degree (f5), department (f6), and industry experience (f7). detailed expert information is provided in table 2. table 2 profile of the experts dms f1 f2 f3 f4 f5 f6 f7 dm1 62 former gen. man. railway manufacturing ph. d mechanical eng. 35 dm2 60 former gen. man. railway manufacturing bachelor's mechanical eng. 25 dm3 57 professor university rail systems ph. d mechanical eng. 20 dm4 54 professor university rail systems ph. d electrical eng. 17 dm5 47 r&d manager railway technics master's electric eng. 20 then, together with researchers, professionals, and experts, they defined the problem of making decisions. accordingly, the determination and prioritization of iqctif alternatives have been determined as the primary research problem. then, the application steps of the proposed model were started. after identifying the expert panel, the researchers discussed and determined the alternatives to meet the requirements with professionals and experts. alternatives were determined according to the fact that each option can be matched with two or more factors. below are the alternatives chosen at the end of the negotiations and the factors with which each is matched (table 3). initially, ten alternatives were identified. in the pre-selection process, considering the requirement, it was discussed whether the other options could respond to the determined factors (i.e., e1certifications, e2 rolling stock expertise, e3 presence in turkey, e4 digital capabilities, and e5 major rail clients, and after the preselection process, experts and researchers decided to continue with five alternatives. after the iqctif selection strategies were determined, the experts and professionals were asked to choose the criteria that affect the evaluation of these alternatives, and they were given a blank questionnaire form; it was stated that they could write as many criteria as they wished in this form. then, the forms were collected and combined to prepare a list of criteria. then, with the help of experts, the researchers eliminated the same repetitive criteria covered by another criterion and updated the list. in the next stage, each criterion was discussed separately with the experts. the criteria on which an entire agreement and consensus were reached were left on the list, and the others were eliminated. the table below shows the criteria set. the identified criteria are presented in table 4. assessment of the quality inspectors for the railway manufacturing industry in... 13 table 3 alternatives and eligibility for pre-selection criteria dms candidates e1 e2 e3 e4 e5 result ac1 applus+   x x  x ac2 bureau veritas       ac3 dnv gl  x    x ac4 ensco x  x   x ac5 intertek x x   x x ac6 lloyd's register       ac7 mermec x  x   x ac8 ricardo rail   x    ac9 sgs       ac10 tüv süd   x    table 4 the criteria to assess the iqctif alternatives code criteria definitions c1 experience and references the firm's past projects, references, and customer feedback. c2 railway certifications certifications like iso 9001, iris, and ecm are required for compliance. c3 safety testing capability performing crash, braking, emc, and fire safety tests effectively. c4 technological infrastructure and equipment accuracy of equipment and use of advanced testing technologies. c5 international recognition global recognition and compliance with tsi and fra standards. c6 independence and objectivity no conflict of interest between quality firms and manufacturers. c7 cost and pricing competitive service fees, budget-friendly, and long-term agreements. c8 service timeliness and flexibility completing inspections promptly and responding to urgent testing. c9 railway material testing mechanical and chemical testing of components. c10 digital reporting and data analytics digital test result sharing and data analytics for optimization. c11 local and regional offices offices or labs in turkey or europe for on-site services. c12 regulatory compliance compliance with tcdd, uic, era, and local regulatory standards. c13 training and consultancy services providing training and consultancy for the company's engineers. c14 sustainability and environmental policies reducing carbon footprint, green production, and eco-friendly testing. after the alternatives and criteria were determined, the practices related to the determination of the reputation ratings of the experts, which is the next stage of the proposed model, were implemented. 3.1 computing the experts' weights (reputation) steps 13: the professionals first evaluated the characteristics they identified with the help of the linguistic terms in table 1. then, he assessed the characteristics of the experts involved in the evaluation process. first, the relative weight values of the determined characteristics were calculated to marginalize the analysis process in which the reputation of the experts was determined. table 5 shows the weight values for the characteristics and the results obtained. table 5 the weight values of the features pr1 pr2 pr3 μl μu vl vu score v. weights f1 mh m h 0.5655 0.6661 0.2965 0.3449 0.5190 0.1292 f2 vvh h vvh 0.7693 0.8264 0.1713 0.2154 0.6056 0.1508 f3 h mh vvh 0.7149 0.7801 0.2585 0.3032 0.5677 0.1414 f4 vh vh vh 0.7500 0.8000 0.1500 0.2000 0.5858 0.1459 f5 h vvh mh 0.7149 0.7801 0.2585 0.3032 0.5677 0.1414 f6 vh h vh 0.7259 0.7858 0.1768 0.2247 0.5728 0.1426 f7 vvh mh vvh 0.7623 0.8156 0.2565 0.2994 0.5973 0.1487 14 ö.f. görçün, c.n. durmuş, m. çanakçioglu the table shows that f2 and f7 were identified as the most influential expert characteristics. the remaining characteristics were ranked with varying values. the researchers used linguistic variables based on this information to assess the experts' characteristics. these evaluations are presented in appendix table a. the linguistic assessments were then converted into ivq-rof number values according to table 2 and aggregated using eq. (10). next, the score function for each expert was calculated using eq. (13), and their reputation was determined with eq. (18). the calculated score function values and weight coefficients are presented in table 8 (see also table 6). table 6 the weight values of the experts (reputation degrees) weights 0.1292 0.1508 0.1414 0.1459 0.1414 0.1426 0.1487 f1 f2 f3 f4 f5 f6 f7 sv w rank dm1 0.7483 0.7714 0.7707 0.7546 0.8006 0.7483 0.7865 0.7690 0.2138 1 dm2 0.6904 0.7105 0.7300 0.7105 0.6682 0.7483 0.7112 0.7102 0.1975 3 dm3 0.7546 0.7546 0.7105 0.7714 0.8189 0.7649 0.6682 0.7485 0.2082 2 dm4 0.6891 0.6891 0.6440 0.7105 0.8189 0.7483 0.5928 0.6983 0.1942 4 dm5 0.6023 0.6193 0.6682 0.6678 0.7306 0.7300 0.6682 0.6699 0.1863 5 as shown in the table, after determining the reputation of each expert, the other implementation steps of the proposed ivq-rof-subjective & critic & waspas model were implemented. 3.2 implementation of the suggested model at this stage, the ivq-rof-subjective and critic methods were used to determine the criteria weights. in contrast, the ivq-rof-waspas methods evaluated and ranked the strategy alternatives. the applied procedure is presented below. step 4: the experts evaluated the criteria using the linguistic terms provided in table 1. these evaluations were then aggregated by the researchers and converted into ivq-rof numbers. table 7 presents the experts' linguistic assessment of the criteria. table 7 the linguistic evaluations of the experts for the criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 dm1 vh vvh eh vvh vh h h h vvh h h vvh h m dm2 vvh eh vvh vvh vvh vvh vvh vvh vvh vvh vh eh vh vh dm3 eh eh eh eh vvh vvh vh vh eh vvh vh eh vh vh dm4 vvh vvh eh vvh vvh vvh vh vh vvh vh vh eh vh vh dm5 vh vvh vvh vh vh vh h vh vvh vh h vvh mh mh steps 4.14.3: the researchers then combined these values with the help of eq. (15). the obtained combined ivq-rof values were converted into score function values using eq. (13) and then weighed to calculate the final relative weighting values (table 8). assessment of the quality inspectors for the railway manufacturing industry in... 15 table 8 the weight values of the experts (reputation degrees) μl μu vl vu sub j sub jw rank c1 0.79511 0.84632 0.12094 0.16792 0.77559 0.07341 6 c2 0.82246 0.87314 0.08641 0.13411 0.80474 0.07617 3 c3 0.83287 0.88350 0.07725 0.12412 0.81625 0.07726 1 c4 0.80434 0.85520 0.10803 0.15529 0.78521 0.07432 5 c5 0.78175 0.83209 0.12495 0.17356 0.76167 0.07209 7 c6 0.76733 0.82449 0.14331 0.19009 0.75061 0.07105 8 c7 0.72922 0.79445 0.16822 0.21651 0.71857 0.06801 11 c8 0.74477 0.80255 0.15738 0.20587 0.72935 0.06904 10 c9 0.81195 0.86244 0.09352 0.14229 0.79327 0.07509 4 c10 0.75680 0.81431 0.15043 0.19802 0.74056 0.07010 9 c11 0.71591 0.78175 0.17356 0.22279 0.70710 0.06693 12 c12 0.83209 0.88273 0.07804 0.12495 0.81538 0.07718 2 c13 0.70494 0.76926 0.20650 0.25312 0.69512 0.06580 13 c14 0.68271 0.74341 0.25387 0.29911 0.67149 0.06356 14 step 5: in the second step of the model, the researchers converted the expert evaluations of the strategy alternatives into the corresponding ivq-rofns in table 1, and these numerical values were combined with the help of eq. (15). then, the ivq-rof decision matrix shown in appendix, table c, was created. steps 67: in this step, they combined the matrices created according to each expert's opinion with the help of eq. (15) and included the degree of prestige of the experts in this process. step 7: then, with the help of eq. (13), the score function value of each alternative was calculated according to each criterion. table 9 shows the options' score function values. table 9 the score matrix c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 a1 0.7768 0.6086 0.6086 0.6826 0.6948 0.7421 0.7421 0.7421 0.6394 0.7421 0.7081 0.6086 0.7076 0.7206 a2 0.7626 0.6815 0.6672 0.7306 0.6682 0.7804 0.7306 0.7804 0.6672 0.7804 0.6948 0.7075 0.7064 0.7419 a3 0.7736 0.5610 0.6063 0.7082 0.8148 0.7804 0.8259 0.7542 0.7070 0.7509 0.8259 0.5610 0.8148 0.8259 a4 0.6550 0.7706 0.7706 0.7500 0.6531 0.7306 0.6682 0.7071 0.7706 0.6682 0.5928 0.7706 0.5928 0.6550 a5 0.7982 0.6115 0.6115 0.7111 0.7445 0.7445 0.7599 0.7599 0.6115 0.7599 0.7339 0.6115 0.7339 0.7599 step 8: after the score matrix was created, the application steps of the ivq-rofcritic approach were employed to determine the criteria's relative weight coefficients. step-8.1: the initial decision matrix elements were normalized using eq. (21). next, the normalized decision matrix is constructed as shown in table 10. then, the standard deviation for each vector that makes up the matrix is calculated using eq. (22). table 10 the normalized matrix c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 a1 0.8508 0.2274 0.0140 0.0000 0.2581 0.2307 0.4688 0.4775 0.1755 0.6585 0.4947 0.2274 0.5173 0.3840 a2 0.7518 0.5751 0.3707 0.7121 0.0937 1.0000 0.3959 1.0000 0.3503 1.0000 0.4377 0.6992 0.5119 0.5084 a3 0.8286 0.0000 0.0000 0.3798 1.0000 1.0000 1.0000 0.6429 0.6004 0.7369 1.0000 0.0000 1.0000 1.0000 a4 0.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 a5 1.0000 0.2412 0.0315 0.4221 0.5652 0.2779 0.5813 0.7194 0.0000 0.8166 0.6055 0.2412 0.6357 0.6136 sd 0.3940 0.3890 0.4294 0.3760 0.4062 0.4669 0.3598 0.3694 0.3905 0.3808 0.3587 0.4059 0.3583 0.3628 16 ö.f. görçün, c.n. durmuş, m. çanakçioglu step 8.2: a correlation matrix is created by calculating the correlation between each pair of criteria. table 11 shows the correlation matrix. then, after the correlation matrix is constructed, the total amount of information for each criterion is determined using eq. (23). to do this, the difference (1-τik) between each element of the correlation matrix is calculated, and the sum of each vector (∑(1-τjk)) is multiplied by the standard deviation value (σj). the resulting value exhibits the total amount of information (kj) for a criterion. table 11 the correlation matrix c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c1 1.000 -0.873 -0.960 -0.765 0.575 0.453 0.754 0.792 -0.892 0.891 0.783 -0.824 0.816 0.755 c2 -0.873 1.000 0.966 0.827 -0.846 -0.455 -0.938 -0.528 0.613 -0.660 -0.944 0.991 -0.940 -0.880 c3 -0.960 0.966 1.000 0.875 -0.698 -0.409 -0.843 -0.637 0.790 -0.771 -0.861 0.941 -0.874 -0.797 c4 -0.765 0.827 0.875 1.000 -0.456 -0.074 -0.596 -0.302 0.695 -0.486 -0.610 0.835 -0.607 -0.475 c5 0.575 -0.846 -0.698 -0.456 1.000 0.460 0.939 0.296 -0.215 0.373 0.923 -0.865 0.892 0.904 c6 0.453 -0.455 -0.409 -0.074 0.460 1.000 0.652 0.766 -0.104 0.709 0.667 -0.355 0.704 0.753 c7 0.754 -0.938 -0.843 -0.596 0.939 0.652 1.000 0.548 -0.389 0.629 0.999 -0.919 0.991 0.981 c8 0.792 -0.528 -0.637 -0.302 0.296 0.766 0.548 1.000 -0.672 0.980 0.587 -0.417 0.653 0.658 c9 -0.892 0.613 0.790 0.695 -0.215 -0.104 -0.389 -0.672 1.000 -0.769 -0.426 0.573 -0.469 -0.388 c10 0.891 -0.660 -0.771 -0.486 0.373 0.709 0.629 0.980 -0.769 1.000 0.667 -0.561 0.727 0.705 c11 0.783 -0.944 -0.861 -0.610 0.923 0.667 0.999 0.587 -0.426 0.667 1.000 -0.920 0.996 0.984 c12 -0.824 0.991 0.941 0.835 -0.865 -0.355 -0.919 -0.417 0.573 -0.561 -0.920 1.000 -0.905 -0.842 c13 0.816 -0.940 -0.874 -0.607 0.892 0.704 0.991 0.653 -0.469 0.727 0.996 -0.905 1.000 0.987 c14 0.755 -0.880 -0.797 -0.475 0.904 0.753 0.981 0.658 -0.388 0.705 0.984 -0.842 0.987 1.000 step-8.3: in this step, the total amount of information for each criterion is divided by the total amount of information of the criteria to determine the relative weight values. eq. (24) is used for this. table 12 shows the values obtained at the end of the implementation. table 12 the absolute value matrix and weights of the criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c1 0.000 1.873 1.960 1.765 0.425 0.547 0.246 0.208 1.892 0.109 0.217 1.824 0.184 0.245 c2 1.873 0.000 0.034 0.173 1.846 1.455 1.938 1.528 0.387 1.660 1.944 0.009 1.940 1.880 c3 1.960 0.034 0.000 0.125 1.698 1.409 1.843 1.637 0.210 1.771 1.861 0.059 1.874 1.797 c4 1.765 0.173 0.125 0.000 1.456 1.074 1.596 1.302 0.305 1.486 1.610 0.165 1.607 1.475 c5 0.425 1.846 1.698 1.456 0.000 0.540 0.061 0.704 1.215 0.627 0.077 1.865 0.108 0.096 c6 0.547 1.455 1.409 1.074 0.540 0.000 0.348 0.234 1.104 0.291 0.333 1.355 0.296 0.247 c7 0.246 1.938 1.843 1.596 0.061 0.348 0.000 0.452 1.389 0.371 0.001 1.919 0.009 0.019 c8 0.208 1.528 1.637 1.302 0.704 0.234 0.452 0.000 1.672 0.020 0.413 1.417 0.347 0.342 c9 1.892 0.387 0.210 0.305 1.215 1.104 1.389 1.672 0.000 1.769 1.426 0.427 1.469 1.388 c10 0.109 1.660 1.771 1.486 0.627 0.291 0.371 0.020 1.769 0.000 0.333 1.561 0.273 0.295 c11 0.217 1.944 1.861 1.610 0.077 0.333 0.001 0.413 1.426 0.333 0.000 1.920 0.004 0.016 c12 1.824 0.009 0.059 0.165 1.865 1.355 1.919 1.417 0.427 1.561 1.920 0.000 1.905 1.842 c13 0.184 1.940 1.874 1.607 0.108 0.296 0.009 0.347 1.469 0.273 0.004 1.905 0.000 0.013 c14 0.245 1.880 1.797 1.475 0.096 0.247 0.019 0.342 1.388 0.295 0.016 1.842 0.013 0.000 sum 11.495 16.668 16.278 14.139 10.720 9.234 10.192 10.276 14.653 10.566 10.156 16.268 10.030 9.655 k 4.529 6.484 6.989 5.316 4.354 4.311 3.667 3.796 5.722 4.024 3.643 6.603 3.594 3.503 wij 0.0681 0.0975 0.1050 0.0799 0.0654 0.0648 0.0551 0.0571 0.0860 0.0605 0.0548 0.0992 0.0540 0.0526 step 9: in this step, the subjective and objective weights of the criteria were combined by employing eq. (25). table 13 demonstrates the subjective, objective, and final weights of the criteria. in this work, the β parameter value is accepted as 0.5. assessment of the quality inspectors for the railway manufacturing industry in... 17 table 13 the subjective, objective, and final weights of the criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 sub jw 0.0734 0.0762 0.0773 0.0743 0.0721 0.0710 0.0680 0.0690 0.0751 0.0701 0.0669 0.0772 0.0658 0.0636 obj jw 0.0681 0.0975 0.1050 0.0799 0.0654 0.0648 0.0551 0.0571 0.0860 0.0605 0.0548 0.0992 0.0540 0.0526 jw 0.0707 0.0868 0.0912 0.0771 0.0688 0.0679 0.0616 0.0630 0.0805 0.0653 0.0608 0.0882 0.0599 0.0581 according to the results regarding the relative significance of the criteria, safety testing capability (c3) is determined as the most effective criterion with a weight value of 0.09116, followed by regulatory compliance (c12) with a weight value of 0.08821 and railway certifications (c2) with a weight value of 0.08681. other criteria are ranked as (c9) railway material testing (0.08054) > c4 technological infrastructure and equipment (0.07711) > c1 experience and references (0.07074) > c5 international recognition (0.06877) > c6 independence and objectivity (0.06792) > c10 digital reporting and data analytics (0.06529) > c8 service timeliness and flexibility (0.06305) > c7 cost and pricing (0.06157) > c11 local and regional offices (0.06084) > c13 training and consultancy services (0.05990) > c14 sustainability and environmental policies (0.05810). then, the second stage of the proposed model started, and the ranking performances of the iqctifs were determined using the ivq-rof set-based waspas approach. step 10: at this stage, the application steps of the ivq-rof-waspas method, which is the second part of the proposed model and used to determine the ranking performance of the alternatives, were started. step 10.1: the decision matrix used in the first application step of the ivq-rofcritic method was also used. therefore, these processes for creating the first and second decision matrix (table 12) are not shown here again to avoid repetition. step-10.2: the elements of the first decision matrix were normalized using eq. (26), and the normalized decision matrix shown in table 14 was generated. table 14 the normalized matrix c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 a1 0.9732 0.7898 0.7898 0.8858 0.8527 0.9509 0.9004 0.9509 0.8298 0.9509 0.8574 0.7898 0.8685 0.8726 a2 0.9555 0.8844 0.8659 0.9481 0.8201 1.0000 0.9146 1.0000 0.8659 1.0000 0.8413 0.9182 0.8670 0.8983 a3 0.9693 0.7280 0.7869 0.9191 1.0000 1.0000 0.8091 0.9664 0.9175 0.9622 1.0000 0.7280 1.0000 1.0000 a4 0.8207 1.0000 1.0000 0.9733 0.8015 0.9362 1.0000 0.9060 1.0000 0.8562 0.7178 1.0000 0.7275 0.7932 a5 1.0000 0.7936 0.7936 0.9228 0.9137 0.9539 0.8794 0.9736 0.7936 0.9736 0.8887 0.7936 0.9007 0.9201 step 10.3-10.4 at this stage, each alternative's weighted sum model (wsm) and weighted product model (wpm) measurements were calculated using eqs. (27) and (28). then, using these values, the total distance of each alternative was determined with the help of eq. (29), and the other options were ranked according to this value. table 15 shows the results for ranking the alternatives with wsm, wpm, and the total distance of each alternative. 18 ö.f. görçün, c.n. durmuş, m. çanakçioglu table 15 the normalized matrix wsm wpm ij rank a1 0.8698 0.8675 0.8687 5 a2 0.9114 0.9097 0.9105 1 a3 0.9016 0.8951 0.8983 3 a4 0.9065 0.9005 0.9035 2 a5 0.8852 0.8821 0.8837 4 as shown in table 15, tuv-sud (a2) has been identified as the most valuable and priority option to be a collaborator of the turkish railway vehicle manufacturing industry. it is followed by sgs (a4) and ricardo rail (a3), respectively. other alternatives have been ranked as lloyd's register (a5) and bureau veritas (a1). 4. robustness check a comprehensive robustness test assessed the proposed model's validity and reliability. in the first stage of the process, the effect of changes in the criteria weights on the final ranking results was examined. each criterion's weight was gradually reduced by 10% to zero, while the reduced value was equally redistributed to the others to maintain a total weight sum of 1. in this context, a total of 140 scenarios were created. the impact of varying expert weights on the rankings was also examined. as shown in figure 1, the place of the worst alternative (a1) in the ranking did not change in any scenario, while the a5 alternative could not maintain its place in the ranking in only one scenario. the best alternative, a2, maintained its place in the ranking, except for eleven scenarios. the similarity between the 140 scenarios and the original ranking results was calculated as 0.8643. the expert weights changed similarly in the second stage, and the results obtained were evaluated. the similarity between the original ranking results and the rankings that occurred when the expert weights were changed was determined as 1.0. in sources such as saaty [37] and belton and stewart [38], it is recommended that consistency and similarity values should be above 85-90%, and it is emphasized that such ratios provide reliable results in the decision processes of mcdm methods. in this study, the average similarity rates were computed as high as 86.43% and 100.00%. it shows that the model's results exceed the reliability standards accepted in the literature. in the second step, we tested the resistance of the proposed model against the rank reversal problem, an essential limitation in decision-making approaches. in n-1 scenarios, the lowest-ranked alternative was excluded, and rankings were recalculated to test for rank reversal. we meticulously compared the alternatives' ranking results with the previous scenario's ranking performance in each scenario. the results of this thorough comparison process are presented in table 16. table 16 the results of the test for the rank reversal problem scenarios ranking results in the scenarios original a2 > a4 > a3 > a5 > a1 sc1 a2 > a4 > a3 > a5 sc2 a2 > a4 > a3 sc3 a2 > a4 assessment of the quality inspectors for the railway manufacturing industry in... 19 sc4 a2 fig. 1 new ranking results based on modified criterion weights in the third stage, the results of the proposed decision-making model were compared with the ivq-rofns-based decision-making approaches used in the relevant literature. in this context, aras [39], topsis [40], copras [41], vikor [42], and todim [43] methods were applied based on ivq-rof. the results are shown in fig. 2. as seen in this figure, when only the ivq-rof-based topsis method was used, there was a displacement between the first two alternatives, and the other options remained in place. on the other hand, the ranking results obtained for all other methods remained the same as those obtained using the proposed model, and all alternatives showed the same ranking performance. 20 ö.f. görçün, c.n. durmuş, m. çanakçioglu fig. 2 comparative analysis results the parameter value was changed in the last stage, and the ranking results were examined. figure 3 shows the new ranking results obtained according to the scenarios. when the results were examined, it was observed that there was no change in the ranking performance of a1 and a5 for all scenarios. at the same time, the best alternative, a2, lost its place only in the 11th scenario, where the objective weight value was zero, and maintained its place in the other ten scenarios. a3 ranked second when the subjective weight value was 1 to 0.70, falling to third place when the objective weights increased. a4, on the other hand, shows a similar trend, but when the subjective weight values are accepted as zero, they rise to the top of the ranking. to solve this highly complex decision-making problem, which is highly affected by uncertainties, the critic & waspas procedure based on range value q-rung orthopair fuzzy (ivq-rof) sets was developed in this study. the proposed model effectively manages uncertainties in the dynamic structure of the sector. to solve the decision-making problem reasonably and logically, these requirements discussed above highlight the necessity of decision models that can accommodate subjective and ambiguous criterion evaluations in a decision-making environment where complex uncertainties and ambiguities are effective, as sudha et al. [44] stated in their study. the proposed decision-making model allows the uncertainties and contradictory information that arise in the decision-making processes and affect the results to be handled more effectively and sensitively. in addition, the critic approach, a part of the model used to assess criterion weights, deals with the interactions and relationships between criteria in more depth, enabling the relative weights of the criteria to be determined more objectively, effectively, and with high accuracy. the ivq-rof-waspas approach produces more balanced, reliable, and comprehensive results when considering both the additive and multiplicative properties of the strategy alternatives. this framework makes it possible to evaluate strategies multidimensionally and prioritize them according to the importance of the scores they assessment of the quality inspectors for the railway manufacturing industry in... 21 achieve. in addition, the decision-making model developed in this study has an advantageous and practical basic algorithm and can be easily applied by almost any decision-maker without the need for advanced mathematical knowledge. moreover, despite the highly reliable and robust structure of the model, the fact that it can be easily applied to this extent systematizes the evaluation processes, reduces the time to be spent on the evaluation processes, and makes significant contributions to the reputation of the company because it produces results at a high level of accuracy. fig. 3 new ranking results based on modified  parameters 5. results and discussion the results and findings obtained in this study show that criteria such as safety test capability (c3), regulatory compliance (c12), and railway certificates (c2) are the most effective and critical criteria in the evaluation and selection of iqctifs for industries producing products such as railway wagons, locomotives, and railroads. these results are consistent with the fact that the railway industry is an industry that is subject to the most legal regulations and standards at national and international levels, and that safety is the most critical and vital factor. when the findings obtained for each criterion are evaluated separately, the requirement for producing railway vehicles by safety and security standards explains the safety test capability (c3) criterion in the first place. it is critical and vital for production enterprises to be able to perform essential tests such as crash tests, braking tests, electromagnetic compatibility, and fire safety tests to prevent a railway accident that can cause fatal disasters. in addition, businesses that carry out such security tests and 22 ö.f. görçün, c.n. durmuş, m. çanakçioglu practices in a very robust and impartial manner gain customers' trust and significantly increase their reputation. one of the most crucial elements in railway transportation is the concept of interoperability. while this ensures that transportation is uninterrupted, it provides significant advantages in terms of safety and costs. the most essential condition is that the tools and equipment comply with international standards. therefore, compliance with legal regulations and standards determined by international railway authorities is extraordinarily critical and vitally important. deficiencies in the compliance of the producer enterprises with these obligations may lead to significant problems such as certificate cancellations, penalties, and sanctions to be applied, the inability of wagons to cross the border in crossborder transportation, and the obligation to transfer to other wagons. accordingly, the extent to which these firms comply with regulatory requirements and standards is critical in selecting iqctifs. to achieve this, the fact that the companies that will be stakeholders have internationally accepted certificates such as iso 9001, iris (international railway industry standard), and ecm (maintenance responsible entity) regarding testing, control, and inspection can be considered as an essential proof that the determined quality standards are met, and reliable service is provided. in addition, criteria such as railway material testing (c9) and technological infrastructure and equipment (c4) are also directly related to safety, security, and compliance with regulations in the railway industry. in this context, high-quality and technological elements guarantee the durability of railway vehicle components while increasing the accuracy and reliability of the results obtained. although experience and references (c1), which are listed as an essential criterion at a medium level, are decisive in terms of reliability in the evaluation and selection processes of the companies in question, considering the technical criteria, it is not enough for the partner company to have experience and references. in addition, according to the study's findings, the cost and pricing (c7) criterion is of low importance. the main reason is that a cost-oriented choice is unreliable enough to meet safety and technical competence requirements. given the consequences of accidents, especially in the railway industry, reducing costs at the expense of safety and regulatory compliance is unacceptable. these findings and conclusions have critically important policy and managerial implications. first, safety, certification, and regulatory compliance should be prioritized when selecting partners to receive quality control, testing, and inspection services. the evaluation process should include businesses that can fully meet these requirements. in addition to regulations and legal regulations, the enterprises' certificates should be considered in the selection processes. besides, the technological saturation levels of businesses and their investment in technological requirements are another essential focus. in addition, the findings provide valuable insights for all stakeholders. first, companies to be included in the selection processes should consider strategies to increase their security testing capabilities and competitiveness regarding their certifications. processors that meet these requirements long-term and sustainably can establish strategic alliances with businesses in the railway industry. at the same time, businesses should constantly review and update their compliance with regulations regarding testing, control, and verification processes. otherwise, these businesses may significantly lose their competitiveness in the market. at the same time, data analytics, artificial intelligence, and digital reporting systems can dramatically improve the quality of service by making testing, control, and assessment of the quality inspectors for the railway manufacturing industry in... 23 inspection processes faster and more accurate, and such technological elements can improve quality control processes. the findings highlight tuv-sud (a2) as the top alternative due to its leadership in rail safety testing and regulatory compliance. the fact that the company takes an active role in many railway projects in europe and other major markets and fully complies with the standards set by regulatory organizations such as tcdd, era, and uic, as well as the company's advanced test laboratories and technology infrastructure, which offers a great advantage for companies in the sector, is in line with the findings obtained in this study. it was followed by sgs (a4) and ricardo rail (a3), while other alternatives were lloyd's register (a5) and bureau veritas (a1). in this context, sgs is a global business that provides a wide range of industrial testing and certification services, specializing in safety, material durability testing, and conformity assessments in the railway sector. ricardo rail is an engineering-oriented company with in-depth know-how in railway technology and system safety. it has a strong structure and knowledge, especially in vehicle dynamics, signaling systems, and infrastructure engineering. however, his specialization in a more specific area caused him to take third place. lloyd's register (a5) and bureau veritas (a1) focus mainly on the maritime and energy sectors. their experience in the railway field is relatively small, and their expertise in railway vehicles and systems is limited. in addition to having deficiencies, especially in technical and railway safety certificates, their market experience is minimal. the study's findings clarify that the most critical components of the railway industry's quality control and inspection processes are safety testing capability, regulatory compliance, and railway certifications. these criteria are essential because rail transport requires high safety standards and strict adherence to international regulations. this company stands out as one of the most reliable options in the sector with its advanced test infrastructure, competence in international certification processes, and many years of experience in the railway sector. sgs (a4) and ricardo rail (a3) were ranked second and third due to their wide range of testing and certification services, strong engineering infrastructure, and industry expertise. on the other hand, lloyd's register (a5) and bureau veritas (a1) focus more on the maritime and energy sectors, lagging due to their limited expertise in the railway sector. these results provide essential policy and managerial implications for manufacturers, authorities, and other stakeholders. it is emphasized that safety, regulatory compliance, and technical competence should be prioritized instead of cost-oriented approaches when selecting iqctifs. in addition, railway manufacturers need to increase their technological investments and continuously improve their inspection processes to comply fully with safety standards. the digitalization of testing, control, and certification processes, the use of artificial intelligence-supported analysis, and cooperation with internationally certified enterprises are among the main strategies that will increase the competitiveness of the railway sector. although this study presents essential findings on evaluating and selecting quality control and inspection firms in the railway sector, it contains some limitations. the alternative companies assessed in the study were chosen according to specific criteria and do not cover all global quality control and inspection firms. therefore, more comprehensive analyses can be made by considering a wider pool of firms in future research. 24 ö.f. görçün, c.n. durmuş, m. çanakçioglu regulatory requirements in the railway sector may vary from country to country. although the study is based on international standards, examining local regulations and practices in different countries in more detail will be helpful. in addition, although the methods used in the study were based on objective criteria, the evaluation of some factors was carried out based on expert opinions. in future research, subjective biases can be minimized by taking the views of more industry experts and company representatives. given the increasing use of technologies such as artificial intelligence, machine learning, and automation in quality control and inspection processes, it will be helpful to evaluate the impact of these technologies on the performance of iqctifs in future research. some topics are proposed for future research. comparison of the method used in the study with different multi-criteria decision making (mcdf) methods may help test the consistency and reliability of the results obtained. the effects of regulatory requirements in the european, us, and asian markets on different iqctif selection processes can be analyzed. in addition, the effectiveness of artificial intelligence, big data analytics, and digital certification processes in quality control and inspection processes can be examined. in addition, creating a comprehensive strategic roadmap on how the quality control and inspection processes in the railway sector will be shaped in the future can also be an essential research topic. such research will help rail manufacturers, regulatory authorities, and iqctifs make more informed decisions and contribute to improving safety and efficiency in the industry. 6. conclusion this investigation proposes a comprehensive and robust decision-making model in which ivq-rof (interval-valued q-rung orthopair fuzzy set) based critic and waspas methods are integrated to address the highly complex decision-making problem affected by uncertainties and ambiguities, such as the evaluation and selection of independent quality control, testing, and inspection firms (iqctif) in the rolling stock manufacturing industry, and to produce reasonable solutions. this decision model developed in this study can manage multidimensional criteria with remarkable interactions between them by integrating subjective expert evaluations and analyses based on objective data. according to the findings obtained in this study, safety test competence (c3), regulatory compliance (c12), and railway certificates (c2) criteria stand out in the selection of iqctif in terms of their significance and influence. these outcomes indicate the need for the railway industry's high safety requirements and strict compliance with international regulations. evaluations of alternative iqctifs show that tüv süd (a2) is the most suitable strategic partner regarding safety testing, compliance with regulatory standards, and advanced technological infrastructure. the proposed decision-making framework for evaluating iqctifs has been confirmed by robustness checks to be ten-degree consistent, stable, and reliable. the proposed integrated decision-making approach stands out as a robust evaluation and decisionmaking tool for real-world applications by overcoming structural problems that point to the most fundamental limitation of traditional decision-making approaches, such as the rank reversal problem. assessment of the quality inspectors for the railway manufacturing industry in... 25 this research makes significant contributions in terms of theoretical and industrial applications. a new hybrid framework has been developed in the theoretical context, considering uncertainties and inter-criteria relationships. it offers a transparent, systematic, and easily applicable method to decision-makers regarding managerial and policy implications. it provides a model for selecting strategic partners for the railway production sector and other sectors that require high regulation. in addition, the importance of decision processes based on long-term sustainability, compliance, and technical competence instead of short-term cost-oriented approaches was emphasized. this model can be developed in future studies with sectoral risk indicators, dynamic assessment frameworks, and artificial intelligence-supported decision support systems, and tested with wider applications in different 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vvh h vh eh vh eh eh vvh vvh vvh vh h h h 4 mh vvh h h vvh mh mh vh mh h vvh h eh eh vvh vh vvh vh mh mh mh 5 m h mh mh h mh h h h mh h vh vh vh vh h vh vvh h h h appendix table b: linguistics assessments of the experts for the alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 a1 dm1 eh mh mh vh h vvh vvh vvh mh vvh vh mh vh vvh a1 dm2 vvh h h h h vh vh vh h vh h h h vh a1 dm3 vvh mh mh h vh vh vh vh h vh vh mh vh vh a1 dm4 vvh mh mh h vh vh vh vh h vh vh mh h vh a1 dm5 h mh mh h h vh vh vh mh vh h mh vh mh a2 dm1 vvh h h vh h vvh vh vvh h vvh h h h vh a2 dm2 vvh vh vh vh h vvh vh vvh vh vvh h vh h vh a2 dm3 vvh h mh vh h vvh vh vvh mh vvh vh vvh vh vvh a2 dm4 vvh h h vh h vvh vh vvh h vvh vh h vh vvh a2 dm5 h h h vh h vvh vh vvh h vvh h h vh h a3 dm1 vh mh mh vvh vvh vvh eh vh mh vh eh mh vvh eh a3 dm2 vh mh mh vh vvh vvh vvh vh mh h vvh mh vvh vvh a3 dm3 vvh m vh h eh vvh eh vh eh vvh eh m eh eh a3 dm4 vvh m m h eh vvh eh vh eh vvh eh m eh eh a3 dm5 eh m m h eh vvh eh eh m vvh eh m eh eh a4 dm1 h vh vh h mh vh h h vh h mh vh mh h a4 dm2 h vvh vvh vh h vh h vh vvh h mh vvh mh h a4 dm3 h vvh vvh vvh h vh h vh vvh h mh vvh mh h a4 dm4 h vvh vvh vvh h vh h vh vvh h mh vvh mh h a4 dm5 mh vvh vvh vvh h vh h h vvh h mh vvh mh mh a5 dm1 eh h h vvh vvh vvh eh eh h eh vvh h vvh eh a5 dm2 eh h h vh vvh vvh vvh vvh h vvh vh h vh vvh a5 dm3 eh mh mh vh vvh vvh vvh vvh mh vvh vvh mh vvh vvh a5 dm4 eh mh mh vh vvh vvh vvh vvh mh vvh vvh mh vvh vvh a5 dm5 vvh mh mh vh vvh vvh vvh vvh mh vvh vvh mh vvh vvh plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 11, no 2, 2013, pp. 169 180 3d numerical simulations of the thermal processes in the shell and tube heat exchanger  udc 66.04:536.2 mića vukić 1 , goran vučković 1 , predrag živković 1 , žarko stevanović 2 , mladen tomić 3 1 university of niš, faculty of mechanical engineering, niš, serbia 2 institute "vinĉa", belgrade, serbia 3 school of higher technical professional education niš, serbia abstract: a shell and tube heat exchanger (sthe) is one of the most often used apparatuses in chemical industry. one of the main goals of the sthe manufacturers is to improve their exploitation reliability and efficiency. two approaches to the sthe design improvement are possible: experimental investigation, which is very expensive and time-consuming because of the shell side complex geometry, and numerical investigations. numerical simulations can be used to check the old and to develop new and more efficient sthe designs. in this paper, the results of the numerical investigations of fluid flow and heat transfer in the laboratory experimental sthe are presented. numerical simulation has been performed by using the phoenics code. the tube bundle has been modeled by using the concept of porous media. standard k- turbulence model is used. key words: heat exchanger, local heat transfer, numerical investigation 1. introduction shell and tube heat exchangers are apparatuses in which heat exchange between hotter and colder fluid is performed. the fluid flowing through the tubes is called a tube fluid, and the fluid flowing around the tube bundle is called a shell side fluid. the baffles, placed in the shell side space, are providing a cross flow direction of the shell side fluid; hence, a more intensive heat exchange between fluids could be realized. besides, the baffles are tube bundle carriers, which helps them decrease deflection in the horizontal as well as vibrations in the horizontal and vertical apparatuses.  received november 20, 2013 corresponding author: mića vukić faculty of mechanical engineering, a. medvedeva 14, 18000 niš, serbia e-mail: micav@masfak.ni.ac.rs 170 m.vukić, g. vuĉković, p. živković, ž. stevanović, m. tomić sthes usually have a combined fluid flow, which means that there is parallel in one, and a counterflow in the other part of the exchanger. if the sthe is with a so-called „full tube bundle“, the shell side fluid flows through the baffle cuts along the tubes. on the shellside, there is not just one stream. in addition to the main cross-flow one, four leakage or bypass streams exist as a result of design type baffle to tubes, baffle to shell and tube bundle to shell gaps (tube–to–baffle hole leakage stream, bundle bypass stream, pass–partition bypass stream and baffle–to–shell leakage stream). basically, one can conclude that heat transfer between fluids in the sthes is highly influenced not only by thermal and flow quantities, such as inlet temperatures and velocities, but also by baffle cut size, baffle spacing, size of inlet and outlet zones and number of baffles 5, 7, 8, 9, 10, 11. considering the fact that the flow visualization and a detailed measurement of the shellside turbulent characteristics in the heat exchanger are very difficult, calculated pressure, velocity and temperature fields, as well as turbulent characteristics fields (e.g. field of turbulent kinetic energy dissipation), can be of great significance in solving this, very complex, turbulent thermal – flow processes in the shell and tube heat exchangers, when it comes to further investigation and designing of reliable and efficient shell and tube heat exchangers 7, 8. 2. mathematical model numerical solution of heat transfer, fluid flow and other processes can begin when the laws that govern these processes are mathematically expressed, mainly using partial differential equations. partial differential equation describing some of the mass, momentum or energy transport phenomena, is actually expressing the principle of conservation for a certain transport property. turbulent heat transfer mathematical model in the sthe (fig. 1) where heat is exchanged between the shell side and the tube side fluid, is formulated with the following assumptions: the flow fields are stationary, three-dimensional, incompressible, viscous, turbulent with large re numbers in the defined boundaries. the thermo-physical properties of the process fluids, in the concrete case are considered to be constant. the heat loss outside the exchanger, and the radiation are not taken into consideration. fig. 1 laboratory sthe with cross segmental baffles the resistance to heat transfer through the tube wall as well as the fouling resistance are also neglected. numerical investigation of thermal processes in shell and tube heat exchanger 171 the flow characteristics of both fluids have a direct influence on the efficient heat exchange and pressure drops in both the shell and tube bundle. the approaches of the distributed resistance and the porosity concept have been used in modeling both the process of heat transfer and the sthe geometry. in this paper, the time averaged conservation equations: of mass, momentum and energy, on the shell side of sthe, in the referent curvilinear coordinates are given with the following simple form 1: exj m jmj j ssu x gu x                     )( )( )()( )( '' δ δ (1) the averaged conservation equations are formally equal to the “instant” ones only by replacement of instant values with the average ones; as a result of averaging they have extra terms, namely, so-called reynolds turbulent stresses in the momentum conservation equation and turbulent flux of scalar in the equation for scalar, of the form  j u  . there is a set of turbulent models connecting the reynolds turbulent stresses and the turbulent scalar fluxes with the mean flow characteristics closing the set of equations. one of the frequently used turbulent models is a two-equation k-ε turbulent model. it is based on introducing effective diffusivities for transport quantities under consideration, which are the sum of molecular and turbulent diffusivities. the terms and coefficients that form expression (1) depend on the conservation equation and should be specified for each conservation equation. diffusion coefficient γ and standard source term s, depending on the nature of dependent variable , are given in table 1. table 1 diffusion coefficient and standard source term  γ s mass 1 0 0 momentum u (i) eff ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )] i ji j i jm j m j m j p g u u u u x                 energy cp,oto eff h μ σ 0 turbulent kinetic energy k eff k μ σ  p dissipation rate for k  effμ σ 2 1 2c c ρ k k   p ( ) ( ) ( ) ( ) ( ) ( ) ( )[ 2 /3 ( )]ij ij m k ik tur tur m jg g u u       p k at the right hand side of equation (1) s ex is an extra source term. in this paper, special attention will be paid to s ex . the extra source term expresses: flow resistance, heat and mass exchange between the fluids. the extra source term in the momentum equation takes into account the flow resistance caused by the tube bundle in the sthe 172 m.vukić, g. vuĉković, p. živković, ž. stevanović, m. tomić shell (distributed resistance). expressions of local hydraulic resistance are taken from ref. 2, 8. extra source term in the energy equation for the shell side fluid can be given as 7: )( co ex t ttks o   (2) volume overall heat transfer coefficient kco w/m 3 k, reduced to the volume unit that is enveloped by the heat transfer surface, is defined as:             oc sv o v c co hh d hh k 11 4 11 1 (3) where hc and ho w/m 2 k are the heat transfer coefficients scaled to the heat exchange surface unit. in this paper, in numerical experiments for determination of the volume overall heat transfer coefficient, two approaches are used. firstly, the average volume overall heat transfer coefficient is determined by generalizing our experimental results, and, secondly, the procedure for determination of the local volume overall heat transfer coefficient is developed and built in the cfd-code, which can thus build all the needed fields. in this procedure, the local re numbers for the fluid on the shell side are defined in correlation to the external tube diameter and the resulting fluid velocity. in order to determine the thermal-flow fields in the sthe shell as well as the temperature field for the tube fluid, a set of differential equations has to be solved together with the algebraic (constitutive) relations and the corresponding unambiguity conditions. 3. numerical investigation solving the set of partial differential equations with the phoenics software package is based on using the finite volume method. effective flow side surfaces acf and cell volume v are proportional to geometrical quantities acf * and v * , where the proportion coefficients are the porosity factors that are denoted as  with corresponding subscripts for sides and volumes. this principle is known as the porous media concept where the porosity factor takes values  = 0.0  1.0. the chosen face or whole cell is completely blocked for  = 0 (when the cell is at the wall), and completely free for  = 1. the porosity factors can be different for each cell; therefore, with complex geometrical configurations additional dependent variables can be introduced. for solving the general discretization equation simple procedure is used of patankar and spalding 3 which uses staggered grid and combines the continuity equations with momentum ones for pressure determination. all the numerical experiments have been made for the same initial fluid parameters as in real experiments 7. first discretization of the integration domain in bfc coordinates has been made by grid generator in the satelite mode using q1 file. by testing a large number of different density grids, with and without matching the frames, a unique nonuniform grid numerical investigation of thermal processes in shell and tube heat exchanger 173 for all numerical experiments is used, with number of cells in x 1  x, x 2  y and x 3  z directions, respectively, 24  24  128, where z coordinate is pointed in direction of the shell axis (fig. 2). the grid uniqueness is represented in the aspect that for all the numerical experiments the same grid lines in the cross section are used (grid density in xy plane is adjusted to different sizes of the baffle cut), while the grid density, in z direction, is adjusted to the number, position and thickness of segmental baffles in the shell. in the cells that are on the segmental baffles, the fluid flow is blocked unlike the heat transfer which is free. fig. 2 segment of numerical grid (plane x = 12, y = 15, z = 128) then by using the porosity concept, in q1 file and ground file the sthe geometry modeling is carried out. considering the position and symmetry of the tube bundle in the shell, and the prechosen numerical grid, eight constant porosity factor fields have been defined in the cross section. determination of the extra source terms of the distributed resistance and volume heat transfer coefficients has been made by linearization technique of the special source terms. the corresponding subroutines are written in ground and q1 file 5. for turbulence modeling on the shell side, a standard k-ε model has been used 8. firstly, a series of numerical experiments with the average volume heat transfer coefficient is made for verifying both mathematical and numerical models. then a series of experiments with local heat transfer coefficient on the shell side is performed. naturally we have to emphasize that in all the numerical experiments the average value of the heat transfer coefficient on the tube side has been used, determined on the bases of the real experiments. this is absolutely justifiable by the fact that neither the volume flow rate nor the fluid temperature at the tube bundle inlet varied. in order to verify mathematical and numerical models, the numerical results are compared to the experimental ones 7. we have compared the values of the fluid tempera174 m.vukić, g. vuĉković, p. živković, ž. stevanović, m. tomić tures at the outlet for both fluids as well as the fluid temperatures inside the shell (in the shell axis). fig. 3 shows the comparison between the experimental and the numerical values of the heating fluid temperature by using the average and the local volume overall heat transfer coefficient in the shell axis for the case of one segmental baffle np = 1, with baffle cut of bc = 22%, without leakage flow between the baffle and the shell, for the heating fluid flow rate of vt = 4m 3 /h and the initial temperature of t1t = 60°c. fig. 3 variation of heating fluid temperature along the sthe’s axis fig. 4 variation of heating fluid temperature along the sthe’s axis fig. 4 shows the comparison between the experimental and the numerical values of the heating fluid temperature by using the local volume overall heat transfer coefficient in the shell axis for the case of three segmental baffle np = 3, with baffle cut of bc = 22%, with leakage flow between the baffle and the shell, for the heating fluid flow rate of vt = 3m 3 /h and initial temperature of t1t = 60°c. fig. 5 shows the comparison between the experimental and the numerical values of the heating fluid temperature by using the average and the local volume overall heat transfer numerical investigation of thermal processes in shell and tube heat exchanger 175 coefficient in the shell axis for the case of five segmental baffle np = 5, with baffle cut of bc = 22%, with leakage flow between the baffle and the shell, for the heating fluid flow rate of vt = 3m 3 /h and the initial temperature of t1t = 60°c. fig. 6 shows the comparison between the experimental and the numerical values of the heating fluid temperature by using the local volume overall heat transfer coefficient in the shell axis for the case of five segmental baffle np = 5, with baffle cut of bc = 32%, with leakage flow between the baffle and the shell, for the heating fluid flow rate of vt = 3m 3 /h and the initial temperature of t1t = 60°c. fig. 5 variation of heating fluid temperature along the sthe’s axis fig. 6 variation of heating fluid temperature along the sthe’s axis on the basis of the shown results we conclude that there is satisfactory agreement between the experimental and the numerical results and that the employed numerical model can be successfully applied to the simulation of the thermal-flow processes in the shell and tube heat exchangers. the maximum deviation between the measured and the numerical values of certain temperature for all experiments was 5%. in the figs. 8-10 we have the results of numerical simulations for the experiments with five segmental baffles np = 5, with the baffle cut of bc = 22%, for the heated fluid 176 m.vukić, g. vuĉković, p. živković, ž. stevanović, m. tomić parameters: flow rate of vh = 9m 3 /h and the initial temperature of t1h = 15°c, heating fluid parameters: flow rate of vt = 3m 3 /h and the initial temperature of t1t = 60°c. fig. 8 shows velocity field on shellside fluid for case without leakage flow between the baffle and the shell. fig. 9 shows velocity field on shellside fluid for case with leakage flow between the baffle and the shell. fig. 8 velocity field (plane y = 13)  no leakage fig. 9 velocity field (plane y = 13)  with leakage numerical investigation of thermal processes in shell and tube heat exchanger 177 fig. 10 shows velocity field on shellside fluid for case with leakage flow between the baffle and the shell. fig. 10 velocity field (plane x = 8)  with leakage figs. 11-15 present the results of numerical simulations for the experiments with one segmental baffles np = 1, with the baffle cut of bc = 22%, for the heated fluid parameters: flow rate of vh = 9m 3 /h and the initial temperature of t1h = 15°c, heating fluid parameters: flow rate of vt = 3m 3 /h and the initial temperature of t1t = 60°c. fig. 11 shows velocity field around segmental baffle, for case with leakage flow between the baffle and the shell. fig. 11 recirculation zone behind baffle  with leakage fig. 12 shows temperature field for tubeside fluid, for the case without leakage flow between the baffle and the shell. 178 m.vukić, g. vuĉković, p. živković, ž. stevanović, m. tomić fig. 13 shows temperature field for shellside fluid, for the case without leakage flow between the baffle and the shell. fig. 14 shows temperature field for shellside fluid, for the case with leakage flow between the baffle and the shell. fig. 15 shows heat transfer coefficient field around segmental baffle for shellside fluid, for the case with leakage flow between the baffle and the shell (plane z-x). fig. 12 temperature field for tubeside fluid (plane y = 12)  no leakage fig. 13 temperature field for shellside fluid (plane y = 12)  no leakage numerical investigation of thermal processes in shell and tube heat exchanger 179 fig. 14 temperature field for shellside fluid (plane y = 12)  with leakage fig. 15 heat transfer coefficient field (plane y = 12, z = 45  70)  with leakage 4. conclusions in this paper the results of numerical experiments carried out on the basis of phoenics software package applied on thermal-flow processes in a shell and tube heat exchanger, are shown. for the illustrative purposes the cross correlations between the experimental and the numerical results are shown. satisfactory agreement is observed. on the basis of the results shown above we conclude that there is satisfactory agreement between the experimental and the numerical results and that the employed numerical model can be successfully applied to the simulation of the thermal-flow processes in the heat exchangers. the results of the performed numerical simulations show that the sthe’s heat exchange strongly depends on the shell side geometry. 180 m.vukić, g. vuĉković, p. živković, ž. stevanović, m. tomić references 1. i. demirdžić, metod konačnih zapremina za proračun transportnih procesa u turbulentnim tokovima proizvoljne konfiguracije, akademija nauka i umjetnosti bosne i hercegovine, posebna izdanja: knjiga lxvi, odeljenje tehniĉkih nauka: knjiga 13, sarajevo, 1984. 2. l. huang, j. wen, t. karayiannis, 1996, cfd modeling of fluid flow and heat transfer in shell and tube heat exchanger, phoenics journal, vol.9, no.2, pp.181-209. 3. s. patankar, 1980, numerical heat transfer and fluid flow, hemisphere, washington. 4. h. rosten, b. spalding, 1987, the phoenics equations, cham tr/99. 5. b. spalding, 2005, a simplified cfd method for the design of heat exchangers, proceedings of ht2005, asme summer heat transfer conference, san francisco, california, usa, july 17-22 2005. 6. w. shyy, s. tong, s. correa, 1985, numerical recirculating flow calculation using a body-fitted coordinate system, numerical heat transfer, vol. 8, pp.99-113. 7. m. vukić, 2004, experimental and numerical investigations thermal and fluid flow processes in shell and tube heat exchangers, (in serbian) phd thesis, niš, serbia. 8. stevanović, ž., ilić, g., radojković, n., vukić, m., stefanović, v., vuĉković, g., 2001, design of shell and tube heat exchangers by using cfd technique – part one: thermo-hydraulic calculation, facta univesitatis series mechanical engineering, 1(8) pp. 1091-1105. 9. m. prithiviraj, m. j. andrews, three-dimensional numerical simulation of shell-and-tube heat exchangers. part ii: heat transfer, numerical heat transfer, part a: applications: an international journal of computation and methodology, volume 33, issue 8, 1998. 10. yong-gang lei, ya-ling he, rui li, ya-fu gao, effects of baffle inclination angle on flow and heat transfer of a heat exchanger with helical baffles, chemical engineering and processing: process intensification, volume 47, issue 12, 2008, pp 2336–2345. 11. jian-fei zhang, ya-ling he, wen-quan tao, 3d numerical simulation on shell-and-tube heat exchangers with middle-overlapped helical baffles and continuous baffles – part i: numerical model and results of whole heat exchanger with middle-overlapped helical baffles, international journal of heat and mass transfer, volume 52, issues 23–24, 2009, pp 5371–5380. 3d numeričke simulacije prenosa toplote u dobošastom izmenjivaču toplote dobošasti izmenjivač toplote je jedan od najčešće korišćenih aparata u hemijskoj industriji. glavni ciljevi proizvođača ovih aparata su povećanje njihove pouzdanosti i efikasnsoti u eksploataciji. moguća su dva pristupa u cilju poboljšanja dizajna ovih aparata: eksperimentalno istraživanje koje je veoma skupo i dugotrajno, zbog složene geometrije međucevnog prostora aparata i numeričko istraživanje. numeričkim simulacijama može se proveriti stari dizajn i usavršiti novi, efikasniji dizajn aparata. u ovom radu prikazani su rezultati numeričkog istraživanja termo-strujnih procesa u laboratorijskom dobošastom izmenjivaču toplote. trodimenzionalne numeričke simulacije turbulentnih termo-strujnih procesa obavljene su korišćenjem softverskog paketa phoenics. za modeliranje geometrije izmenjivača toplote korišćen je koncept poroznosti. za modeliranje turbulentnih napona korišćen je standardni k- turbulentni model. kljuĉne reĉi: izmenjivač toplote, lokalni prenos toplote, numeričko istraživanje 12351 facta universitatis series: mechanical engineering vol. 22, no 3, special issue, 2024, pp. 423 433 https://doi.org/10.22190/fume231208013n © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper triboinformatics: machine learning methods for frictional instabilities michael nosonovsky1,2, aleksandr s. aglikov2 1university of wisconsin-milwaukee, mechanical engineering department, usa 2itmo university, infochemistry scientific center, russia orcid ids: michael nosonovsky https://orcid.org/0000-0003-0980-3670 aleksandr s. aglikov https://orcid.org/0000-0001-7135-1156 abstract. the study of friction is traditionally a data-driven area with many experimental data and phenomenological models governing structure-property relationships. triboinformatics is a new area combining tribology with machine learning (ml) and artificial intelligence (ai) methods, which can help to establish correlations in data on friction and wear. this is particularly relevant to unstable motion, where deterministic models are difficult to build. there are several types of friction-induced instabilities including those caused by the velocity dependency of dry friction, coupling of friction with another process (wear, heat generation, etc.), the elastic adams instabilities, and others. the onset of sliding is also an unstable process. ml/ai methods, such as topological data analysis and various ml algorithms, which have been already used for various aspects of data analysis on friction, can be applied also to the frictional instabilities. key words: frictional instabilities, painlevé paradoxes, triboinformatics, machine learning 1. introduction due to the extremely interdisciplinary and multiscale character of tribological processes, new mathematical and computational methods rapidly evolve [1]. many current problems of contact mechanics and tribology are complex dynamic problems, which often lead to instabilities and require new mathematical methods, models, and algorithms. thus, ostermeyer and co-workers proposed a boundary layer machine approach for the modeling of wear [2]. popov and co-workers applied several new methods including non-linear diffusion-reaction models of cellular processes [3], the burridge-knopoff model combined with voronoi tessellations to study the stick-slip instability for the state-and-rate friction [4], the effects of dynamic menisci on adhesion [5], a method to solve viscoelastic contact problems with arbitrary loading histories [6], and an energetic criterion for adhesion in viscoelastic contacts received: december 08, 2023 / accepted march 09, 2024 corresponding author: michael nosonovsky university of wisconsin-milwaukee, 3200 n cramer st., milwaukee wi 53217 usa itmo university, 9 lomonosova st., st. petersburg, russia e-mail: nosonovs@uwm.edu https://orcid.org/0000-0003-0980-3670 https://orcid.org/0000-0001-7135-1156 424 m. nosonovsky, a. s. aglikov with non-entropic surface interactions [7]. forsbach and co-authors [8] investigated a two-scale fem-bam approach for fingerpad friction under electroadhesion. many of new multi-scale and dynamic applications are related to biomimetic and biocompatible applications [3, 9-10]. various machine learning (ml) algorithms have been used successfully for similar engineering problems including the k-nearest neighbor (knn), decision trees (dt), and random forests (rf) [11, 12]. stability analysis has been the subject of intensive studies in mechanics and the theory of dynamical systems since at least the middle of the 19th century. the theory of stability addresses the stability of equilibria states and stability of solutions of differential equations of dynamical systems, under small perturbations of initial conditions [13]. typically, a dynamical system is represented by a point in that system’s configuration space (x1, x2,… xn), where xi are coordinates and n is the total number of degrees of freedom, and by a governing differential equation which establishes the evolution of the system in the time domain. different approaches to the stability analysis have been developed to provide stability criteria: lyapunov stability, algebraic stability (the routh–hurwitz stability criterion), frequency stability analysis (the nyquist stability criterion), poincare diagrams, structural stability, etc. [13] traditionally physicists and engineers are trying to ensure that their systems are stable and concentrate on stable behavior. however, the analysis of the behavior of unstable systems is much more difficult. the stable motion is deterministic and predictable since the state of the system at any instance of time is defined by the initial conditions. contrary to that, unstable motion is virtually unpredictable since any small fluctuation or error tends to grow exponentially, and after a short time, it becomes impossible to predict the position of a system in the configurational space. thus, turbulence remains a very difficult problem to handle [14]. for the analysis of unstable behavior, statistical methods can be used, and novel machine learning (ml) and data science algorithms can be applied [15]. the application of ml methods in tribology is a new area of research referred to as triboinformatics [16]. there are different types of instabilities in fluid mechanics including the rayleigh– taylor instability, plateau–rayleigh instability, rayleigh–bénard instability, kelvin– helmholtz instability, saffman–taylor instability and many others. no comprehensive classification of fluid instabilities has been suggested so far. while fluid mechanics involves many types of instability, solid mechanics also can lead to unstable motion. in particular, dry friction can lead to the so-called frictional instabilities [17]. dry friction is governed by the coulomb friction law, f=w, which relates the friction force, f, to the normal load force, w, through the coefficient of friction (cof),  [18]. friction is usually thought of as a stabilizing factor, however, can lead to destabilization under several circumstances. this includes the situations when the cof depends upon the sliding velocity (the so-called “frictional weakening” when cof decreases with increasing velocity) or upon other parameters, such as the temperature at the frictional interface, or when friction is combined with the constitutive law of the material, such as the linear elastic dependency between stresses and strains of the material [17]. there are two types of dry friction: the static and kinetic friction, which correspond to the stick and slip states or to the states of rest and motion of the dynamical systems (table 1). typically, the static friction force is greater than the kinetic friction, which can be viewed as another manifestation of the frictional weakening, and it can lead to the so-called stick-slip selfexited vibrations. in terms of rheological properties of materials, friction corresponds to the plasticity. the transition to the plastic flow is sometimes viewed as a phase transition [19]. there are situations when it is useful to introduce the third state which is in a sense in between the static and kinetic friction. this is the ultraslow motion, when the rate of deformation is comparable with relaxation rates of the involved materials. the unstable triboinformatics: machine-learning methods for frictional instabilities 425 motion can also be viewed as the third state, in addition to the states of rest and of deterministic stable motion. some solutions, which result in instabilities, may originate due to ill-posedness of the dynamic problem. nosonovsky and breki [20] suggested that a three-valued logic can be applied to such cases. three-valued logic is an algebraic approach in which a logical variable, p, can have a third value p = undefined, in addition to p = true and p = false of the boolean logic (table 1). moreover, it is suggested that ml approaches can be applied to the unstable motion to establish correlations in data, because traditional deterministic methods fail to predict the state of the system due to exponentially growing fluctuations. in the present paper we will review main types of frictional instabilities and will expand the ternary logic approach for the frictional instabilities. table 1 instability and the non-binary character of motion and friction initial state opposite state third state motion rest motion unstable motion friction stick slip ultraslow motion material elastic plastic relaxation / creep logical predicate, p(x) p(x)=true p(x)=false p(x)=undefined modeling approach statics dynamics machine learning 2. friction-induced vibrations and instabilities in this section we will discuss different types of frictional instabilities. since some of these instabilities are related to the ill-posedness of the dynamic problems with dry friction, we will start from the discussion of the frictional paradoxes. 2.1. painlevé paradoxes we start the discussion of friction-induced instabilities with the painlevé paradoxes, which are not instabilities per se; however, they are related to some frictional instabilities and may provide an interesting perspective to understand the latter [20-24]. painlevé paradoxes can be found in simple (1dof) mechanical systems with dry friction consisting of rigid undeformable bodies (sliders and connecting rods) fig. 1 (a) the setup for the painlevé paradox (two sliders connected by a coupler) and (b) introduction of an elastically compliant link with the spring coefficient k 426 m. nosonovsky, a. s. aglikov a simple example of the painlevé paradox is found in the system consisting of two sliders connected by a rigid coupler link of length l (under angle φ) and sliding in parallel grooves (fig. 1a). both sliders have the same mass m; however, friction in the upper groove is negligible (=0), while the lower slider is subject to finite friction with the cof =0. an external force p is applied to the upper slider parallel to the groove. the normal reaction force r acts at the upper slider, so that the tension force in the link is t = r/sin(φ), and the friction force is f=μ|r|sign(v); the direction of the friction force depends on the sign of the sliding velocity 𝑉 = �̇�. the equations of motion of the system are given by 2𝑚�̈� = 𝑃 − 𝜇|𝑅|sign(�̇�) (1) 𝑚�̈� = 𝑃 − 𝑅/ tan 𝜑 (2) when cof is small so that for μ tan(φ) < 2 assuming motion in the positive direction (v > 0, p > 0), the equations of motion have a solution 𝑚�̈� = 𝑃 1−𝜇 tan 𝜑 2−𝜇 tan 𝜑 (3) and assuming motion in the negative direction (v < 0, p < 0), 𝑚�̈� = 𝑃 1+𝜇 tan 𝜑 2+𝜇 tan 𝜑 (4) however, if v < 0 and μ tan(φ) > 2, an additional solution exists 𝑚�̈� = 𝑃 1−𝜇 tan 𝜑 2−𝜇 tan 𝜑 (5) in case of v > 0 and μ tan(φ) < 2, no solution exists at all [20]. the coulomb-amontons’ law of dry friction is not always logically compatible with the laws of newtonian mechanics. this is why the paradoxes, i.e., the situations when the solution is not unique or non-existent, arise in some case. in mathematics, the logically inconsistent problem is called ill-posed, and various ways of regularizing such problems exist. one possibility is the use of the inconsistent logic, such as the ternary (three-valued) logic. nosonovsky and breki [20] suggested regularizing the paradox by using the ternary logic with the logical variable 𝑃𝑖 ≡ (�̇�𝑖 = 0) characterizing the state of the system as “rest” (𝑃𝑖 = 𝑡𝑟𝑢𝑒), “motion” (𝑃𝑖 = 𝑓𝑎𝑙𝑠𝑒)), and “paradox” (𝑃𝑖 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑). the undefined state is a paradoxical situation when no solution or non-unique solutions exist. the standard logical operations of the three-valued logic can be defined, thus, if at least one part of the dynamical system is in the paradox state, the entire system is in the paradox state, and so on. the overall state of the system is given by the conjunction of the degrees of freedom 𝑃 = 𝑃1⋀𝑃2⋀ … ⋀𝑃𝑛. thus, the system is at rest if all velocities are defined and equal to zero, the system is moving if at least one velocity is defined and non-zero, and the system is undefined otherwise. similarly, the stability can be defined with the logical variable 𝑆𝑖 ≡ 𝑃𝑖 ∨ �̅�𝑖, so that the entire system is stable when and only when all its parts are stable, 𝑆 = 𝑆1⋀𝑆2⋀ … ⋀𝑆𝑛. if the link is compliant (elastically deformable), the paradox is resolved [21-22]; however, friction-induced instabilities can originate in a compliant system (fig. 1b). in the next section we will discuss the ill-posedness of the problems with frictional instabilities. triboinformatics: machine-learning methods for frictional instabilities 427 2.2. frictional elastic instabilities in the previous section, we established the relation of the frictional paradoxes to frictional instabilities. the most common case of frictional elastic instabilities are the socalled adams instabilities, which arise during frictional sliding of two elastic half-spaces with different elastic properties (i.e., with the elastic moduli e1 and e2, densities 1 and 2, and poisson’s ratios 1 and 2) pressed together by a normal force per area, w, applied at infinity (fig. 2a). the contact is modeled in the 2d plain-strain elasticity. the sliding is driven by a constant external shear force per area, p, applied at the infinity, so that the system is not conservative (work is done against friction). during the frictionless contact of two elastic half-spaces, the interfacial waves, also referred to as slip waves or generalized rayleigh waves (grw), can propagate along the interface. the grw is a generalization of the elastic surface wave (rayleigh wave) propagating along the surface of a single elastic half-space. the amplitude of the surface wave decreases exponentially with the distance from the surface. similarly, the amplitude of the grw decreases with the distance from the interface in both half-spaces. depending on the elastic properties of the sliding bodies and the cof, a self-excited dynamic instability can rise, in the form of destabilized interfacial elastic waves (grw) whose amplitude grows exponentially with time [25]. frictional elastic instabilities emerge also during sliding of a periodic wavy elastic surface against a flat elastic half-space [26]. the degree of instability (the rate of growth of the wave amplitude) is proportional to the cof and sliding velocity. in other words, the displacements during the vibrations in the x and y-directions have the form 𝑢(𝑥, 𝑦, 𝑡) = 𝑈0(𝑥, 𝑦)e(±𝜆𝑖+𝜇𝑉𝐶)𝑡 (6) 𝑤(𝑥, 𝑦, 𝑡) = 𝑊0(𝑥, 𝑦)e(±𝜆𝑖+𝜇𝑉𝐶)𝑡 (7) where 𝑈0 and 𝑊0 are the modes, 𝜆 is the frequency, and c is a constant. the real part of the term under the exponent 𝜇𝑉𝐶 is the rate of growth of the instability (fig. 2b). fig. 2 (a) two elastic half-spaces sliding relative to one another. (b) position of complex eigenvalues characterizing instabilities of dynamic solutions renardy [27] established ill-posedness of the frictional sliding of an elastic surface in contact with a rigid surface. ranjith and rice [28] showed that the dynamic sliding of dissimilar half-spaces is ill-posed in the sense that in the limit of short wavelength of the 428 m. nosonovsky, a. s. aglikov interfacial waves the degree of instability is inversely proportional to the wavelength and unbounded. in other words, for the wavenumber k, 𝑈0(𝑥, 𝑦) = 𝑢0(𝑦) cos ( 𝑘𝑥 2𝜋 ) (8) 𝑊0(𝑥, 𝑦) = 𝑤0(𝑦) cos ( 𝑘𝑥 2𝜋 ) (9) the degree of instability is proportional to the wavenumber, 𝐶~𝑘, in the short wavelength limit 1/𝑘 → 0. ranjith and rice also suggested a way of regularizing the ill-posedness by modifying the friction law and introducing the so-called rate-and-state friction with internal variables [28]. generally speaking, dynamic instabilities and ill-posedness in the short wavelength limit are closely related to one another. 2.3. instabilities of velocity-dependent friction another type of friction-induced instability is due to frictional weakening. as the force of friction decreases with the increasing velocity, a small fluctuation of sliding velocity can trigger a positive feedback loop. a higher velocity leads to lower friction, lower resistance, acceleration, and further uncontrolled increasing velocity. on the other hand, a lower velocity leads to higher frictional resistance, consequent deceleration and further decreasing velocity. the sliding velocity cannot grow forever, and the growth continues until the frictional weakening stops. in that case the sliding velocity can drop due to increased friction. this leads to friction-induced self-excited non-linear vibrations, such as the stick-slip [17, 29]. it has been suggested in the literature on friction-induced self-organization, that a thermodynamic stability criterion can be applied to the frictional sliding [29-31]. in particular, the entropy production rate, �̇�, controls the stability of a tribological system at a steady state. the entropy production rate for such a system is often at a minimum. the stability is governed by the sign of the second variation of the entropy production rate �̇� = 𝜇𝑊𝑉/𝑇, where w is the normal load, 𝜇𝑊 is the friction force, v is the sliding velocity and t is temperature [1719]. the second variation is then given by 𝛿2�̇� = 𝑊 𝑇 𝛿𝑊𝛿𝜇 = 𝑊 𝑇 𝑑𝜇 𝑑𝑉 (𝛿𝑉)2 > 0 (10) and the stability is dependent on the sign of 𝑑𝜇 𝑑𝑉 . in a more complex case, the cof may depend on a structural parameter of the material, 𝜑 (e.g., the thickness of a film at the surface of the material, fig. 3). the stability criterion can then be written as 𝛿2�̇� = 𝑊𝑉 𝑇 𝑑2𝜇 𝑑𝜑2 (𝛿𝜑)2 (11) if the stability condition is violated for a certain value of 𝜑, then further growth of the film will result in decreasing friction and wear, which will facilitate the further growth of the film [17, 29]. such an in situ formed tribofilm can often play a protective role as it minimizes friction and wear. the standard example is a cu film formed at the bronze-steel frictional interface due to migration of cu+ ions from the bulk of bronze to the interfaces caused by the temperature gradient near the interface. triboinformatics: machine-learning methods for frictional instabilities 429 fig. 3 schematic showing in situ formed tribofilm at a bi-material unstable interface similarly, the parameter 𝜑 can mean the surface roughness or the rate of wear. eq. (10) provides a very general criterion of frictional stability [17, 29]. 2.4. instabilities of the onset of sliding the onset of frictional sliding is another process which has many features of destabilization. it has been suggested that the initiation of sliding has many features similar to fracture along the interface, in particular, to the shear (mode-ii) crack initiation (fig. 4) due to rapid dynamic effects [17, 20] during the transition from static to kinetic friction. stress distribution during the onset of sliding is consistent with crack propagation model based on the linear elastic fracture mechanics [32], while frictional dynamic instabilities at a bimaterial interface are of the same type as the adams instabilities [33]. fig. 4 the similarity of mechanical systems with sliding friction (left) and mode ii (shear) crack (right) 3. machine-learning methods since dynamically unstable motion often cannot be simulated by traditional computational methods due to the divergence, novel statistical ml methods can be applied. the difference between traditional and ml methods is that the dependency between the input and output parameters in the latter is expressed in the form of an algorithm, instead of a formula or equation, which usually governs traditional systems. 430 m. nosonovsky, a. s. aglikov 3.1. topological data analysis the method of topological data analysis (tda) including the persistence analysis shows how these topological features, such as holes and voids, depend on the spatial resolution [34]. the idea of this method is that experimental data points are presented as points in an abstract multi-dimensional data space. following that, the above-mentioned topological features are sought. an example of the input may be a visual image (a photograph) presented as a matrix of 256 × 256 pixels or a map representing a surface roughness surface zij(xi,yj) with 1 ≤ i, j ≤ 256. such matrix can be divided into 3 × 3 square pixel patches, with each patch forming a data point in a 9d space thus providing a large number of 254 × 254 = 64516 patches / data points per image. these data points have a certain distribution in the 9d data space, which depends on various features of the image or a random rough surface. thus, due to the anisotropy of the images (e.g., horizontal and vertical features dominating over those inclined under an arbitrary angle) the points would form certain structures that can be detected by the tda algorithms. for surface roughness data, tda provides “persistence diagrams” and “barcodes” which represent the dependency of topological invariants on the length scale resolution. topological invariants are connected components (h0), 1d holes (h1), and higher dimensional voids (h2, h3, etc.). the intervals of the feature appearance and disappearance are plotted as bars in the barcode diagram or as a point characterizing their appearance and disappearance in the persistence diagram (see ref. [34] for more technical details). these features show the types of anisotropic features at a certain scale level. thus, the output of the algorithm is a dependency of anisotropic features on scale. 3.2. the onset of sliding one area where tad can be used is the analysis of the onset of sliding. the frictional sliding of two elastic rough surfaces may involve three local states: stick, slip, and separation. the growth of local slip zones and their merger in a one single-connected slip zone during the increase of shear loading facilitates the onset of global sliding. here we suggest a simplified procedure of studying this transition to a single-connected slip zone by a similar transition of stick-separation to a single-connected stick zone during the increase of normal loading. when a rough random elastic surface z(x,y) is pressed against a flat surface by the normal load force w, the asperities of a rough surface deform. the exact determination of the stress distribution would require a solution of the contact elasticity problem. however, it is appropriate to assume that stresses at asperity contacts are roughly proportional to the height of the asperity, σ(x,y) ~ z(x,y). with increasing load w, stresses grow, however the static friction inequality can hold, τ(x,y) < μσ(x,y). the transition to the slip state is given locally by the relationship τ(x,y) = μσ(x,y). the stress state at every point of contact depends both on the shear load at that point and, due to the elastic relaxation, on the states of neighboring points thus forming slip zones of correlated neighboring points. the “islands” of the slip zones grow with the shear load until they merge forming a network of slip zones (fig. 5). this makes the transition to the global sliding of the two bodies in contact similar to classical problems of percolation. in such problems, the slip zone diameters are usually related to the distance from the critical load, 𝑊𝑐𝑟 , by a power law 𝑑~(𝑊 − 𝑊𝑐𝑟)𝛾, where 𝛾 is the critical exponent [17]. triboinformatics: machine-learning methods for frictional instabilities 431 while it is difficult to evaluate the map of the slip zones at a frictional interface, one can obtain an experimental height profile of a surface, z(x,y). given σ(x,y)~z(x,y), one can argue that the condition of the local transition to slip, τ(x,y)=μσ(x,y), could be replaced by a similar condition τf = kz(x,y), where τf is the frictional strength of the interface, and k is a coefficient. therefore, the study of the transition can be replaced by the study of the separation zones. fig. 5 shows the cross-sections of rough bronze surface 5 × 5 m sample measured by an atomic force microscope (afm) corresponding to layers at different heights from 1% (when 0% corresponds to the highest asperity) to 87% (when 100% corresponds to deepest valley). the surface shown in dark color changes from separated islands to a single-connected area between 37% and 47%. it is anticipated that at the transition point critical phenomena can occur, which are typical to the phase transition. these phenomena would also correspond to the onset of sliding, which is a rapid dynamic event that is difficult to study by traditional methods. in this case, the rough surface serves as an input of the algorithm, while quantitative characteristics of the transition (e.g., critical exponents relating the dependency of the slip zone diameters on the distance from the transition point) could be the output. fig. 5 the transition from separation to contact with increasing normal load at the interface between a rough deformable and flat rigid surface 4. conclusions many modern problems of contact mechanics and tribology involve unstable motion or unstable solutions, which make it difficult to solve them by traditional approaches. dry friction can cause dynamic instabilities, which are often related to the ill-posedness of mechanical models of the coulombian friction with unlimited rate of instability in the short-wavelength limit of the solution. various ways of regularization of the ill-posedness have been suggested in the literature. unstable motion is effectively unpredictable from the initial conditions, since any small fluctuation grows exponentially with time. we have suggested a threefold classification of dynamic possible solutions: (i) static problems, (ii) dynamic problems with stable solutions, and (iii) dynamic problems with unstable solutions. for unstable or close to instability solutions, new methods of analysis are suggested. 432 m. nosonovsky, a. s. aglikov one way to handle the unstable motion is to apply statistical and machine learning tools. for such methods, structure-property dependencies are often expressed in the form of an algorithm, rather than as an equation. the tda method can provide scale dependency of characteristic features of a random rough surface, such as directions of anisotropy, if present. the onset of sliding can be studied with the tda tools and with methods of phase transition analysis, such as critical exponents. in general, the unstable nature of many frictional processes leads to the novel mathematical and computational methods of tribological data analysis. acknowledgement: this work was presented at the international workshop “interplay of mechanics, tribology, and materials science” at samarkand state university (uzbekistan), 4-6 september, 2023. support from the priority-2030 program (itmo university) is acknowledged. the authors would like to thank prof. valentin popov (tu berlin) and ekaterina v. skorb (itmo university). m. n. acknowledges sabbatical support from uwm. references 1. popov, v.l., 2018, is tribology approaching its golden age? grand challenges in engineering education and tribological research, frontiers in mechanical engineering, 4, 16. 2. ostermeyer, g.-p., müller, m., 2006, dynamic interaction of friction and surface topography in brake systems, tribology international, 39(5), pp. 370-380. 3. popov, 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and supershear ruptures driving slip in bimaterial friction, nature communications, 7, 11787. 34. zhukov, m., hasan m.s., nesterov, p., sabbouh, m., burdulenko, o., skorb, e.v., nosonovsky, m., 2021, topological data analysis of nanoscale roughness in brass samples, acs applied materials & interfaces, 14(1), pp. 2351-2359. modeling of defects detection by analyzing thermal images facta universitatis series: mechanical engineering vol. 12, no 2, 2014, pp. 123 136 1modeling of defects detection by analyzing thermal images udc 004.932 boban r. anđelković, biljana r. đorđević, nataša r. jovanović faculty of mechanical engineering, university of niš, serbia abstract. in this paper a contactless method for the detection of an object’s defect by applying the thermovision camera is proposed and implemented. the thermal image processing is realised in two perpendicular directions and the analysis of the results is presented. an advantage of the proposed method is a simple and efficient application of the standard techniques, which are not specially designed for thermal image processing. by comparing the analysis of results obtained by this method with those obtained by already known methods, the applicability of this method could be confirmed. using the powerful software packages and applications the thermovision images are being transformed into digital records that can be processed on personal computers. the conducted experiments show that this contactless method is very efficient in the detection of the defects in forms of cavities and damages caused by overload or by material fatigue. key words: thermovision image, kernel operation, rangefilt method, thresholding, defect detection 1. introduction in the 1980s, vavilov and taylor took into consideration the non-destructive thermal methods for material testing (tndt methods) and pointed out their ability to provide quantitative information about the hidden defects in the material [1]. there are several significant thermal and spectral features of the materials which need to be considered during the testing of the compactness of their structure and the detection of a possible defect. if a solid material possesses a defect in its structure in the form of cavity, aperture or damage of any shape, the density of the material changes at that very place. the density change affects the process of thermal transmission and thermal conductivity of the material. these changes cause surface discontinuity, which is possible to record with a thermovision camera. the functional dependence of the specified values is described by equation (1): received january 09, 2014 / accepted may 16, 2014  corresponding author: boban r. anđelković university of nis, departmentd of mechanical design, development and engineering, a. medvedeva 14, 18000 nis, serbia e-mail: bandjel@gmail.com original scientific paper 124 b. anđelković, b. đorđević, n. jovanović / pk c   (1) where: α heat diffusion [m 2 /s], k thermal conduction [w m -1 k -1 ],  specific density of the material [kg m -3 ], cp specific thermal capacity [j kg k -1 ]. thermal resistance e [w s 1/2 m -2 k -1 ] is the parameter that represents the characteristics of the material and it can be calculated with equation (2): 1/ 2( )pe k c   (2) the materials of the lower thermal resistance value will heat up more and have higher temperatures, compared to the materials with higher thermal resistance values. the thermal difference is defined with equation (3): t q e t    (3) where: ∆t  temperature increase of the surface [k], q  amount of energy that has been brought through the material [j], t  time [s]. the inner and outer defects of the material are capable of easily transmitting heat because of the lower thermal resistance values. by heating, a higher thermal difference compared to the environment can be achieved in these areas. the result of that effect is that defects on the thermovision image will be ’warmer’ surfaces. the temperature increase depends on the amount of the accumulated energy and the thermal features of the material. by using a special thermovision camera, it is possible to record thermal radiation from the surface of an object to determine its conditions and features. by detecting energy anomalies it is possible to determine and define the defects that can be noticed during the production process [2]. thermovision is a method of preventive maintenance that can be successfully applied to monitoring and diagnostics of the conditions of industrial equipment, facilities, systems and products. thermovision cameras of the new generations have an increased resolution and sensitivity comparing to the previous generations, which makes the detection of the anomalies more accurate. for the display and analysis of the thermal images in colour personal computers are used. the images are being formed by means of three basic colours: red, blue and green. a thermogram can display only 65536 different temperatures, which means that by using the combinations of the shades of three basic colours, it is possible to display a thermogram in an adequate manner (one temperature one colour). most often the colour scales in which higher temperature is represented by a lighter colour nuance are used. the relative spectral characteristics of a huge amount of thermal sources can be approximated by the spectral characteristics of the black object heated to a certain temperature. when it is heated, the black object transmits radiations, whose spectrum depends only on the radiation temperature. max planck showed that this dependence is defined by eq. (4): 25 1 1 c t em c e      (4) where t is temperature, c1 and c2 are constants. for each temperature it is possible to calculate rgb coordinates of the colour lights. these colours are approximate to those radiated by steel during heating and melting and can be formed by applying planck’s formula (4). modeling of defects detection by analyzing thermal images 125 that is how the defect analysis process is reduced to that of defect detection, i.e. the detection of a characteristic shape in a thermovision image. the defect detection process is based on the process of image segmentation. an image, by segmentation, gets divided into meaningful regions. the aim is to separate the pixels which form the objects of interest from all other pixels in the image. image segmentation, usually, comprises three subsystems [3]. the first subsystem is binary classification (binarization) of pixels which is applied to each pixel and which assigns one of the two possible values to it: 0 – not an object or 1 – an object. these values are joined by a black and white pixel, respectively. classification is always a specific application. the basic tendency is to realise homogeneous regions related to a certain pixel characteristic. this phase is imperfect and pixels may end up unclassified. the second subsystem is the determination of the availability of neighbouring pixels of the same class and their interconnectedness. they define the spatial adjustment. these groups can be represented by defining a certain property of each pixel individually or by a list of pixel coordinates which define the boundary of a homogeneous region. the third subsystem is the description of the considered characteristic (size, position or shape) in terms of a scalar or vector quantity. 2. the method and materials the control of the product is a significant phase of the serial production. the human visual control, which is represented in the greater part of industry, presents the roughest form of elimination of badly produced pieces. the need for creating new and more precise forms of control and safety of products, considering the development of science and technology, has evolved and the demands for accuracy have increased. in the last decades, the method of non-destructive testing has stood out as the primary method of examination and testing of defects in materials [4]. the thermovision methods are contactless measuring methods and, as such, they have become the basic, necessary methods for controlling most of the production and exploitation phases [5]. they enable the detection of defects in a material on a very big surface in a very short period of time. this type of testing has become a standard during the testing in high-tech systems, aerospace and aeronautics. the requirement to use the thermovision methods is the existence of a specific thermal difference between the object which is being tested and its environment. the thermovision methods are defined as passive or active, depending on the existence of an additional source of infrared radiation. passive methods enable the recording of the material surfaces and structures, whose temperatures are significantly different from those of the environment, and that do not require an additional source of infrared radiation [6], [7], [8]. the applications of the passive methods are multiple; the most important areas where they can be applied are: control of industrial processes [9], maintenance of machines and equipment, traffic monitoring, detection of forest fires [10], applications in medicine, biology and agriculture, detection of gases and non-destructive testing of electronic printed circuit boards, etc. active methods of the thermovision examining require the use of additional sources of infrared radiation in order to achieve a specific thermal contrast. based on the knowledge of the parameters of the outer impulse, the quantitative estimation of the temperature of the tested object is accomplished [11]. in ref. [12], the implementation of a measurement system for thermovision examining of solar cells and modules during the period of their exploitation is shown, which is included in the methods of active thermovision. 126 b. anđelković, b. đorđević, n. jovanović in the experiment presented in this paper, the thermovision system is applied in combination with a personal computer. the research is conducted in order to diagnose the state of the product in the glass industry. typical glass structure defects come in the form of bubbles, little bubbles, drops and defects in the form of stone, fibre or railroad (fig. 1). fig. 1 typical defects of glass panels in the interior of the product, unknown objects and dirt can be detected or the discontinuity of the density can be spotted as a result of disturbance of the production process. the external defects come in the form of physical damages of various shapes. 3. the experiment the goal of the research is to confirm the possibility of applying non-destructive methods of examining and detecting material defects. the conducted experiment employs a thermovision system in combination with a personal computer. the research is performed with the aim to diagnose the state of glass industry products. the material of the used samples is glass. such a manner of testing allows for the detection of defects in the material on a big surface and in a relatively short period of time. the system is conceived in such a way so as to perform the process of product control without human presence. the used glass samples are rectangular in shape and positioned on an assembly line. the assembly line oscillations and external influences are neglected, thus the speed could be considered constant. the applied method belongs to the group of active methods because it requires an additional source of energy in order to achieve the needed thermal difference between the object and its environment. the infrared lamp is used for that purpose. it provides the needed amount of heat for the samples, so the thermovision camera can record them. an ic lamp is positioned above the assembly line. samples are heated up by its effect. the uneven distribution of temperature is the consequence of the assembly line motion. a thermovision camera is set up next to the ic lamp so that their visible areas do not overlap. the process of recording the object and the formation of the thermogram is shown in the scheme in fig. 2. fig. 2 block diagram of the process of thermography recording modeling of defects detection by analyzing thermal images 127 once the object is recorded, the signal is transmitted from the thermovision camera to the computer where it gets adequately transformed. the transformed signal is then being processed by means of appropriate software and a thermogram is formed. the obtained results can be displayed on the screen in the form of a 3d presentation or a 2d diagram. with the aim of detecting possible defects in the material, first the recording of the reference sample without defects is performed in the state of inaction. the analysis of the reference model thermogram yields the values of temperature dilatations which are caused by surface roughness. after that, the mobile reference sample without defects is recorded in the process of motion and its thermogram is formed. the analysis of this thermogram provids for the distribution of temperature dilatations which are the consequences of the object surface roughness and its motion. by recording the unexamined object, forming its thermogram and analyzing it in comparison to the previously obtained thermograms of the reference model, one can observe the existence of a defect in the material and determine its shape and position. the process of examining can be divided into two phases: the recording and the analysis of the thermogram. 3.1. the capture of the thermogram the heat that an object radiates goes through the atmosphere, passes through the camera lens and gets to the detector. the detector transforms the received signal into an electrical one. the transformed signal is then transferred to the computer system and shown on the screen in the form of a colour image–thermogram. the camera registers the simultaneous radiation from the surface of the object and the radiation of the atmosphere. it is necessary to perform compensation for atmospheric effects, as the only thing that is relevant for the research is the energy that is transmitted from the object. as it passes through the atmosphere, the intensity of the infrared light is reduced due to the absorption from the unknown particles that are in the air. the compensation is accomplished with the help of the correction parameters, which include:  the distance between the object and the camera. the greater distance causes higher atmospheric absorption and greater reduction of radiation intensity.  the relative environment moisture. a higher percentage of moisture leads to a higher collapse of radiation intensity.  the environment temperature. with the increase of the environment temperature, the atmospheric radiation increases as well. in order to minimize the atmospheric effects, the experiment is carried out in the laboratory conditions. 3.2.the analysis of the thermogram the matlab software is used for the analysis of the thermogram. filtering is performed in two mutually perpendicular directions. the obtained results show the presence or absence of defects. a defect can be detected if its shape and direction of distribution are at a random angle or match the direction of the filter kernel movement. if the defect is in the direction perpendicular to the direction of the filter kernel movement, there is no possibility for it to be detected. with the aim of solving this problem, filtering is conducted in both directions simultaneously. this enables the detection of defects regardless of their 128 b. anđelković, b. đorđević, n. jovanović shape and position. the height (width) of the filter kernel is in line with the dimensions of the processed image and it is defined using equation (7) (equation (8)). the value of the other dimension of the kernel is minimum possible value (3) with regard to achieving the most accurate results. if the adopted value is higher than the minimum, the area encompassed by the filter kernel would be too large, thus reducing the possibility of detecting smaller dimension defects. the process of filtering is done by applying several different filters and the comparative analysis of the obtained results is performed. max-min method filtering (rangefilt) is based on the assessment of local extreme values in the region defined by a filter kernel. the assessment of minimal and maximal values is carried out using the morphological functions of dilatation and erosion. in the boundary regions, the padding behaviour of morphological functions is applied. to be realised, these functions imply a certain basic environment. if necessary, it is possible to create structural elements and use the extraction method to generate the environment of a desired shape and dimensions. the result of such filtering is image j, in which the value of each output pixel corresponds to the difference of the estimated maximal and minimal value within the previously defined environment of the appropriate input image i pixel. the dimensions of the output image are the same as those of input image i. entropy filtering (entropy filtering) is based on the calculation of the value of the entropy of the current pixel neighbouring environment. the one-dimensional entropy of the grey image (h) represents the measure of the average amount of contained information generated by the source which sends n independent messages, each with possibility pi. in the sense of image analysis, entropy represents the sharpness of the image. accordingly, the images which have blurred edges have lower values of entropy than those with sharp and clearly defined edges. in this case, the homogeneity of the material is analysed, along with the absence of defects and unknown objects contributing to the temperature discontinuity. thus, it is more favourable if the values of entropy are lower, since they indicate the absence of great temperature dilatations and the absence of defects in the tested material. the mathematical relation used to describe image entropy is: 1 2 0 log l i i i h p p      (5) the processing of edge pixels is performed by adding pixels symmetrically. the values of added pixels are symmetrically mapped around the edge pixel of input image i. apart from the default neighbouring environment, it is possible to form an environment of a desired shape and size. the application of appropriate functions leads to the creation of a structural element, upon which the desired environment is generated by extraction. the result of such a type of filtering is image j, in which the value of each of the input pixels corresponds to that of entropy within the previously defined environment of the appropriate input image i pixel. the dimensions of the output image are the same as those of input image i. statistical filtering (stdfilt) implies the determination of standard deviation (σ) in the previously defined local environment around the appropriate input image i pixel. the mathematical relation used to describe the standard deviation is: 1 2 0 ( ) l i i i m p     (6) modeling of defects detection by analyzing thermal images 129 where pi represents the probability of appearance of the i–th shade of grey colour, and m is the mean value of the image and the level of saturation. images are commonly in the 8 bit format, thus the number of possible shades of grey is l=256. standard deviation represents the measure of the average image contrast and it points to the dispersion of the pixel intensity values in the defined environment. a higher value of standard deviation implies the existence of contrast, and thus the existence of temperature discontinuity and defect in the material. the calculation of the standard deviation of edge pixels is carried out by adding pixels symmetrically. the values of the added pixels are mapped symmetrically onto the image i edge pixel. the default neighbouring environment can be altered if necessary. the application of appropriate functions defines the structural element which is used for generating the desired neighbouring environment by the extraction method. the result of such a type of filtering is image j in which the value of each output pixel corresponds to that of the standard deviation within the previously defined environment of the appropriate input image i pixel. the above functional dependences differ significantly in the speed of realization. if time interval δt is needed to perform a single basic mathematical operation, to calculate the difference between extreme values and carry out filtering which is based upon it time interval δt is also needed. the mathematical operation (equation (5)) for calculating requires time interval 5 δt, while the mathematical operation (equation (6)) for calculating standard deviation requires time interval 7 δt. a comparative analysis of the mentioned ways of filtering points to the fact that filtering based on the assessment of extreme values and their differences can be conducted most easily and quickly; thus, it is applied in the analysis of the recorded thermovision images. filtering is performed using the rangefilt function and the differences of the extreme values of intensity within the filter kernel are defined. filter kernels are, with their size, customised to the image dimensions. the process of filtering is realised in the direction of x-axis, y-axis and in both directions simultaneously. low values indicate the homogeneity of the structure. by analysing the obtained results their maximum value is determined and approved as the reference parameter. the reference and examined samples are exposed to the same working conditions. after the capture of the non-examined object, the analysis of its thermogram is made. the existence of the defect could be established by comparing the obtained results with the reference parameters. 3.3. the system proposed for defect detection the system that is proposed (fig. 3) consists of an infrared lamp, a glass panel on the assembly line, thermovision camera, and the computing system for data processing. the position of the infrared lamp and the thermovision camera is determined in such a manner that the areas of their effect do not overlap. the moving of the assembly line causes uneven heating of the glass panel. when one end of the glass panel comes out of the range of the infrared lamp effect, the other end is still under its thermal influence. when the glass panel finds itself completely in the position of the recording of the thermovision camera, the front end which left the heating area earlier has lower temperature than the rear end. this thermal difference cannot be excluded and it has to be considered when it comes to larger objects. when dealing with smaller objects, the difference between colder and warmer end is negligible. 130 b. anđelković, b. đorđević, n. jovanović after the capture, the signal goes from the thermovision camera to the computing system, being transformed in an appropriate manner and shown as a video on the display. the analysis of the thermogram is conducted by using the matlab software and it consists of the next steps:  image input of the unheated sample without damages and defects in the state of inaction,  pre-processing and adjusting the image to the demands of the applied filters,  filtering,  the analysis of the temperature dilatations, fig. 3 computer controlled thermovision system  the estimation of the parameters in the state of inaction,  image input of the heated and moving sample with damages and defects the reference sample,  pre-processing and adjusting the image to the demands of the applied filters,  filtering,  the analysis of the temperature dilatations,  the estimation of the extreme temperature values of the reference sample on the moving assembly line,  image input of the examined object the image is in colour and in .bmp format,  pre-processing of the input image means the conversion of the data to a suitable format. a colour image is being converted into a grey 8-bit image (uint8),  the analysis of the image parameters the size of the image in pixels (mxn),  forming filter kernel h that is necessary for the analysis and the process of filtering and which defines the local environment of the pixels,  local filtering over the previously defined environment. the filtering process is conducted in a way similar to the convolution. the filter kernel is moving linearly on the image. the start position of the filtering kernel is the upper left corner, and the process ends when the kernel of the filtering is in the lower right corner. in comparison with the convolution, the mathematical operation that is conducted over the selected pixels is simpler and requires less time for realization, so the process of filtering goes a lot faster. the local filtering is made with the rangefilt function, which can be used to calculate the absolute difference of the maximal and minimal intensity value of pixels of kernel h. the calculated difference is approved as a criterion for testing homogeneity of the surface texture. if the intensity difference is equal to zero, or negligible, the analysed environment can be considered as a homogeneous one. otherwise, when the results show a significant modeling of defects detection by analyzing thermal images 131 difference of the pixel intensity, the surface cannot be considered as homogeneous, which is the result of the existence of the defects. the result of applying the rangerfilt function on the input image is the forming of matrix j which is made of calculated differences of the local environment intensity. by experimenting, two filterings are realised in two perpendicular directions.  the first filtering is realised in the direction of x-axis, using kernel h1, dimensions 3×m1h. size m1h is determined from the conditions of the filter kernel centre with the initial (1,1) and last pixel (m,n) matching equation (7). the applied kernel is:   1 1 2 1 3, 561 hm m h ones    (7) the obtained results form matrix j1.  the second filtering is realised in the direction of y-axis, using kernel h2, dimensions (n1h×3), defined by equation (8):   1 2 2 1 783, 3 hn n h ones    (8) the obtained results form matrix j2.  the third filtering is realised in both directions simultaneously, using kernel h3. the dimensions of this kernel are (3×3). the obtained results form matrix j3.  the determination of the size and the position of the global maximum is being achieved by applying the max function in the matlab program software. the function analyses each column separately and detects maximum values of each column. as a result there is a row composed of local maximums of all columns. by applying the same function to the obtained row, the obtained result is the global maximum. the defining of the column, row and size of the global maximum is achieved by applying the functions: [ , ] max (max ( , [ ], 1))maxj col j (9) [ , ] max (max ( , [ ], 2))maxj row j (10) 4. results and discussion the analysis of thermal images enables continuous monitoring of the thermal changes of the surface. since it is not possible to present all thermograms and all the measuring results, this paper presents only characteristic samples. fig. 4a shows a sample without defects in the state on inaction. the thermal distribution is shown in fig. 4b. what can be seen is relatively even distribution as a result of the homogeneity of the structure. the deviations that can be seen are caused by surface roughness. figs. 4c and 4d present graphical representation of all thermal changes in the directions of x-axes and y-axes. the difference of extreme values is on a relatively small scale. fig. 5a presents the thermal layout of the samples on the moving assembly line. the darker area corresponds to the lower, and the lighter area to the higher temperature. the three-dimensional presentation of the temperature distribution is given in fig. 5b, and the layout on x-axis in fig. 5c. with respect to the previously obtained results, the changes in 132 b. anđelković, b. đorđević, n. jovanović the temperature distribution can also be seen. a thermal difference on the edges of the object is in the direction of the assembly line movement. the thermal values oscillations are caused by surface roughness and motion. the thermal layout in the direction perpendicular to the direction of moving is approximately constant. a) b) c) d) fig. 4 temperature distribution of the reference model in the state of inaction a) b) c) fig. 5 temperature distribution of the reference model in the moving process for the estimation of the parameters of the reference model the filtering in two perpendicular directions is conducted. the applied coordinate system is adjusted to the experiment conditions. the direction of the moving object on the assembly line is in the direction of x-axis. a three-dimensional presentation of the results obtained by filtering in the direction of x-axis is given in fig. 6a. the filtering kernel is formed in such a manner, so that it registers only the changes on x-axis. the detection of the defect in the direction of y-axis is not possible. fig. 6a shows the changes of the thermal differences on x-z plane. in the direction of y-axis the values of the changes are constant. the detection of the defects in the direction of y-axis is achieved by using the same process, but in the direction perpendicular to the previous one. the three-dimensional presentation of the results obtained by filtering in the direction of y-axis is given in fig. 6c and the thermal changes in y-z plane in fig. 6d. the detection of the defects in the direction of x-axis is not possible by using this filter. in x-axis direction the values of these changes are constant. a) b) c) d) fig. 6 filtered thermograms reference model in the direction of x-axis and y-axis modeling of defects detection by analyzing thermal images 133 a reliable detection of defects, regardless of their shape and size, can be realised by using the simultaneous filtering in two perpendicular directions. the most reliable results are obtained by applying the filtering kernel with dimensions (3×3). the three-dimensional presentation of this filtering is shown in fig. 7a. the distribution of the thermal changes in the plane x-z is given in fig. 7b, and the distribution of these changes in plane y-z is given in fig. 7c. a) b) c) fig. 7 two-dimensional filtering thermograms reference model the change in the numerical values of the two-dimensional filtering results is in the dependence of the current filtering kernel position. this change is the result of surface roughness of the reference sample. the criterion in the process of defect detection is the maximal value of thermal deviation. this value is considered to be the threshold value. by analysing the thermogram of the untested object and comparing its parameters with the approved threshold value, the data about the existence of defects on the object can be obtained. if thermal deviations are greater than the threshold value in some points, it means that the distribution of the heat in those spots is significantly increased. discontinuity of the thermal distribution is the direct result of the existence of defects. fig. 8a shows a sample with the defect in the process of moving. the heat distribution is shown in fig. 8b. a significant deviation can be seen in certain domain, which is the result of the defect in the structure. figs. 8c and 8d give graphical presentations of all thermal changes in the directions of x and y-axes. a) b) c) d) fig. 8 temperature distribution of the tested object the three-dimensional presentation of the thermogram filtering results of the tested object moving in the direction of the moving assembly line (x-axis) is given in fig. 9a, and the layout of the thermal differences in x-z plane in fig. 9b. comparing with the results obtained by analysing the thermogram of the reference sample, a significant deviation from the threshold value can be seen in several positions. this proves the existence of the defect in the direction of y-axis. 134 b. anđelković, b. đorđević, n. jovanović a) b) c) d) fig. 9 filtering the thermogram of sample in x-axis and y-axis direction by filtering the same thermogram in the direction perpendicular to the previously considered direction, the given results show the existence of the defect on x-axis. these three-dimensional results are given in fig. 9c, and the layout of the thermal differences on y-z plane in fig. 9d. by two-dimensional filtering of the thermogram in x-y plane, the defects in both directions are detected. a comparison of the obtained results with the reference values is enabled by applying the same filtering kernel which was used for the filtering of the reference model. the results of the analysis indicate the existence of defects. the three-dimensional presentation of the obtained results is shown in fig. 10a. the distribution of the thermal changes in x-z plane is given in fig. 10b and the distribution of these changes in y-z plane is given in fig. 10c. by using appropriate algorithms, the position of the largest detected defect on the tested object is determined. the numerical extreme values and their positions are given in table 1. a) b) c) fig. 10 two-dimensional filtering of the thermogram of a sample table 1 numerical extreme values and their positions object without defect in sleep mode moving object without defect moving object with defect max. temp. difference 0.0510 0.0510 0.5647 rowmax (pix) 128 128 275 colmax (pix) 99 99 150 min. temp. difference 0 0 0 rowmin (pix) 1 1 1 colmin (pix) 1 1 1 modeling of defects detection by analyzing thermal images 135 4. conclusions the presented principle of thermography, the procedure for testing, and the obtained results show that this method of indirect testing is very useful when it comes to assessing and discovering defects with the possibility of monitoring the development of defects. due to the described values, the testing by thermography and the method for estimation of the usage suitability are accepted. determination of the existence and the location of defects in products after a certain exploitation period is the subject of much experimental research. there are various diagnostic methods based on the usage of thermovision systems. the presence of defects causes undesirable changes in product features. defects most often occur during the production or exploitation processes. the implemented system presents an original solution which increases the possibilities of testing and product quality control in exploitation. based on the presented research results it is possible to achieve an increase in efficiency and expand the basic possibilities of the thermovision system application. with the research presented in this paper, the conditions are created for the contactless determination of defects or the presence of unknown objects in the structure of a product. acknowledgements: this research has been funded by the serbian ministry for science under the projects tr-33035 and tr-35005. references 1. maldague, x.p.v., 2001, theory and practice of infrared technology for nondestructive testing, willey j., son inc, new york. 2. russia heriansyah, r., abu-bakar, s.a.r., 2007, defect detection in thermal image using thresholding technique, 6th wseas international conference on circuits, systems, electronics, control & signal processing, cairo, egypt, pp. 29-31. 3. corke, p., 2011, robotics, vision and control, springer-verlag berlin heidelberg. 4. petrušić, z., mančić, d., veljkovi,ć m., radmanović, m., 2009, active thermovision method for inspection of solar cells, infoteh, jahorina, 8, pp. 449-455. 5. mančić, d., radmanović, m., petrušić, z., lazić, m., stajić, d., 2004, termička analiza jednog rešenja pfc kola, v simpozijum indel 2004, banja luka, pp. 69-74. 6. blagojević, m., petrušić, z., mančić, d., radmanović, m., 2005, thermal analysis of current probe based on sensor csa-1v, 13th international symposium on power electronics, novi sad, paper no. t4-4.8, pp. 1-5. 7. prijić, a., prijić, z., pešić, b., pantić, d., ristić, s., mančić, d., petrušić, z., 2008, design and optimization of s-type thermal cutoffs, ieee trans. on components and packaging technologies, 31(4), pp. 904-912. 8. rao, m.r.p.d., 2007, review of nondestructive evaluation techniques for frp composite structural components, measurement science and technology, college of engineering and mineral resources, west virginia. 9. choi, c.j., kim, j.y., yang, d.j., song, k.s., kwac, l.k., 2007, study on the application of infrared thermovision camera for efficiency analysis of solar cell, material science forum, vols. 544-545, pp. 467470. 10. stević, z., rajčić vujasinović, m., antić, d., 2006, computer based infrared thermography system for monitoring and diagnostics in electroenergetic plants, infoteh, jahorina, vol. 5, pp. 225-229. 11. wierzbicki, l., stabik, j., wróbel, g., szczepa, m., 2010, efficiency of two non-destructive testing methods to detect defects in polymeric materials, journal of achievements in materials and manufacturing engineering, 38(2), pp. 163-170. 12. wierzbicki, l., stabik, j., wróbel, g., szczepa, m., 2008, detecting of defect in polymeric materials using pulsed infrared thermografy, archives of materials science and engineering, 30(1), pp. 29-32. 136 b. anđelković, b. đorđević, n. jovanović modeliranje detekcije defekta analizom termovizijskih slika u ovom radu predložen je metod detekcije defekta primenom termovizijske kamere. obrada slike je ostvarena u dva perpendikularna pravca i obavljena je analiza dobijenih rezultata. prednost predložne metode ogleda se u jednostavnoj i efikasnoj primeni standardnih tehnika koje nisu specijalno namenjene za obradu termalnih slika. uporednom analizom rezultata ostvarenih ovom metodom sa rezultatima dobijenim već poznatim metodama može se potvrditi njena primenljivost. moćnim softverskim paketima i aplikacijama termovizijske slike se transformišu u digitalne zapise koje je moguće obrađivati i na personalnim računarima. realizovani eksperimenti su pokazali da je ova beskontaktna metoda vrlo efikasna u detekciji nedostataka u obliku šupljina, oštećenja usled preopterećenja ili zamora materijala. ključne reči: termovizijska slika, kernel operacija, rangefilt metoda, metoda praga, detekcija defekta facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 719 734 https://doi.org/10.22190/fume201210046v © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical simulation of single point incremental forming for asymmetric parts george-christopher vosniakos, gabriel pipinis, protesilaos kostazos national technical university of athens, school of mechanical engineering, manufacturing technology laboratory, athens, greece abstract. single point incremental forming (spif) that will produce non-symmetric sheet metal parts has been rarely dealt with so far. spif of a francis hydro-turbine vane made of aluminum alloy is studied as a typical example in this work. at first, a concave geometry, encompassing the desired vane shape is designed, from which the formed part will be ultimately cut out. the necessary spif toolpaths are created by using the cam software normally used for milling processes. based on these toolpaths, a finite element simulation is setup using shell elements with a particular emphasis on substantial time scaling and due care on tool-sheet contact parameters. for validation purposes the part was manufactured and digitized by a white light scanner. it exhibited tolerable deviation from the targeted nominal geometry. simulation predicted a significant part of this deviation, proving its indispensability in checking out toolpaths and process parameters for non-symmetric parts, yet at non-negligible computational time. key words: single point incremental sheet forming, non-symmetry, toolpath, finite elements, time scaling 1. introduction the single point incremental forming (spif) process for sheet metal parts does not require a die but only a blank holding fixture and a simple forming tool to operate on a computer numerically controlled (cnc) milling machine. therefore, it is suited to small batch manufacturing and prototyping [1]. spif process is associated with better formability compared to conventional forming processes. the exact reason for this as well as the principal forming mechanism seems to be unclear. the latter has been related to the stretching of the sheet, which has a lot in common with conventional drawing; this has been adopted by many researchers, due to received december 10, 2020 / accepted april 26, 2021 corresponding author: george-christopher vosniakos affiliation: national technical university of athens, school of mechanical engineering, heroon politehneiou 9, 15780 athens, greece e-mail: vosniak@central.ntua.gr 720 g.-c. vosniakos, g. pipinis, p. kostazos its simplicity. the area in which yielding manifests itself is small and it is constantly changing; thus neck formation is suppressed leading to enhanced formability [1]. on the contrary, many researchers have acknowledged the importance of thru thickness shear [2] and bending [3] in the ‘radial’ direction. the isotropic stress also seems to have an important role in the forming process [4]. as a result, the use of fracture forming limit (ffl) is recommended [2]. spif tools are usually made of tool steels or carbides and are classified according to end shape: round-ball, which is the most common one, cylindrical and roller-ball. the rolling ball seems to benefit surface quality and formability due to reduction of friction, but it is more complicated to produce and limits the maximum sheet-wall angle so that contact of the sheet is only made to the rolling ball [5]. tool size has been mostly studied for round-end tools. in principle, a tool with a smaller radius has a positive effect in formability [6]. however, it has been also pointed out [7] that as the tool radius gets smaller a “squeezed-out wall” defect appears in the surface. step down and material properties also contribute to the presence and intensity of this defect. thus, tool selection should be based on tool size to sheet thickness ratio [8]. formability is also related to the maximum achievable wall angle [9]. the toolpath most often starts from the edge of the cavity that is to be constructed and ends at its bottom. when multiple trajectories are used, some of them may start from the bottom of the cavity. indeed, multiple trajectories can lead to substantial increase in formability [10]. feed speed and the step down speed may be constant so as to create a helical toolpath. discrete step down have also been used to create a “z-level” toolpath, but the helical toolpath results in more even strain distribution [11]. step down (and stepdown speed for helical toolpaths) is selected in dependence of the other process parameters and it heavily influences surface quality as well as formability of the part [3]. spindle speed is selected so as to achieve favorable friction conditions. for tools with a round-end one of the following two seemingly contradictory strategies is used: (a) minimizing the average sliding speed in the contact area between the part and tool (b) increasing the temperature in the contact area due to a high relative speed between the sheet and the tool, thus increasing formability. the optimum spindle speed varies in dependence of the rest of the process parameters [12]. for the manufacture of some complex parts a “featured based trajectory” has been proposed. the toolpath has a constantly variable stepdown speed as to adapt to the distinct features of the part shape (e.g. bottom edges with variable depth) [13]. in the overwhelming majority of spif applications reported in literature axisymmetric parts have been processed. however, lack of symmetry is most interesting since it constitutes the general case of engineering part shapes encountered. this work is devoted to exploring spif of non-symmetric parts by example of a francis hydro-turbine vane, aiming to point out the use of numerical modeling and simulation in such cases. section 2 reviews numerical modeling techniques pertaining to spif. section 3 presents the case study. section 4 outlines simulation setup and section 5 the results obtained. section 6 describes the validation experiment. a discussion of results is provided in section 7. the conclusions drawn are summarized in section 8. numerical simulation of single point incremental forming for asymmetric parts 721 2. literature review on finite element analysis of spif the explicit fea method seems to be preferred in forming process simulations, including the spif case because it is faster [14,15], even though the implicit method may lead to better accuracy [16]. there is a limitation in the maximum time step (courant time step). to further decrease the time step two methods can be implemented: time scaling and mass scaling. these methods may cause a significant (artificial) increase in the total kinematic energy, with a negative effect in the overall accuracy of the results. note that in spif the total kinematic energy is typically a small fraction of the total energy [17]. solid elements may be thought to represent the sheet in a more accurate way than shell elements. however, many issues arise if there is no sufficient through-thickness discretization, typically less than 4 elements: shear locking, hourglass modes, poor nonlinear bending. such issues are dealt with at the expense of computational time [18] or by novel element types [19]. a 3d shell, the solid-shell element, offers a better representation of the problem, and can resolve solid-element issues. it has been also used in spif simulations for high accuracy. however, this still needs usually 2-3 through thickness elements; it seems to fail in patch tests and it has limited adaptive re-meshing capability [20]. shell elements, despite their being 3d, can successfully deal with bending, through thickness shear, stretching normal to the surface and others with suitable formulation [18]. they are faster than solid and solid-shell elements and they achieve very good results in forming processes if they are used with 5 to 9 through thickness integration points. the latter are necessary for simulating plastic bending [21, 22]. a better compromise between accuracy and computation time in many cases including spif [14] is struck by adaptive re-meshing in areas with significant concavity or stress gradient, e.g. near the spif tool. however, adaptive re-meshing is neither standard nor robust in most fea programs [21]. as far as boundary conditions are concerned, fixed end support was applied in the area where the sheet is clamped. however, very small, in-plane translation of the fixed nodes, due to elasticity or slippage, may heavily influence the results of the simulation [23]. as far as material property modeling is concerned, the use of an anisotropic yield criterion, such as hill’s, is important, especially for cold rolled sheets [21]. in-plane anisotropy can be assumed. thus, the yield locus is calculated based on the yield stress and lankford coefficient (usually r0 ,r45,r90) that are easily determined [24, 25]. the exponential hardening law (swift power law) is a popular choice for the simulation of many materials [3, 21]. the mixed work-kinematic model takes into consideration the bauschinger effect as well, which seems to have a substantial impact in spif process [26]. however, the mixed model is based on two parameters that are difficult to be determined as they are strain dependent. recent literature summarizes finite element modeling issues as mentioned above and suggests possible solutions [27]. 3. part and toolpath geometry the part to be manufactured is a francis hydro-turbine vane with a 132 mm chord length, see fig. 1(a). the part was manufactured from aa6082o annealed according to the material provider’s recommendations (leichtmetall). manufacturing of a cavity (cup-shape) starting from a flat sheet is necessary from which the vane will be finally cut-out, e.g. by laser. the cavity was created in a 3d cad environment, starting from the vane’s convex surface 722 g.-c. vosniakos, g. pipinis, p. kostazos geometry, i.e. neglecting its varying thickness and taking into account the following considerations: (a) the wall angle has to be restricted so as to minimize the probability of fracture during spif. of particular importance was the cavity area near the vane in order to avoid thinning of the formed part (b) geometry of the cavity corresponding to the roughing phase was modified so as to minimize curvature and avoid features that could increase dimensional deviation (c) size of the flat sheet was restricted so as to reduce manufacturing time and cost (d) the vane had to be positioned at a sufficient distance away from the edge of the cavity, where excessive dimensional deviation is expected. spring-back was not taken into consideration when designing the cavity. a b c d fig. 1 (a) vane ideal shape (b) cavity contour lines for roughing (a) and finishing (b) passes (c) helical toolpath for roughing (d) zigzag toolpath for finishing two toolpaths corresponding to roughing and finishing were created using solidcamtm. sample contours corresponding to the roughing and finishing cavity shape that was considered as a guide for constructing the toolpaths are compared in fig. 1(b). roughing was assigned a helical toolpath and finishing was assigned a zig-zag toolpath see fig. 1(c-d). the numerical simulation of single point incremental forming for asymmetric parts 723 parameters used in both phases were consistent with literature recommendations for each material used. a round ball-end tool with diameter 7 mm was chosen for formability and surface quality [28]. spindle speed was 50 rpm in order to keep relative speed between the sheet and the tool at low levels. feed was set to f=1000 mm/min and stepdown was set to d=0.445mm for formability and surface quality [28]. 4. numerical simulation model setup to simulate spif for the non-symmetric cavity presented above, the finite element software ls dyna r8.1 was employed with the default explicit integration method. the main issues regarding model setup are presented next. 4.1. meshing and re-meshing the mesh consists of two parts, the sheet (slave surface) and the tool (master surface). the tool was simulated with a hollow sphere meshed with hexa elements. the sheet was assumed as a surface discretized mainly by square shell elements (belytscko-tsay) with five through-thickness integration nodes. this element formulation offered sufficient accuracy with high robustness and low computational cost. re-meshing and fusion were based on the total angle change relative to the surrounding elements in order to fulfill three criteria, namely to: (a) sufficiently represent the sheet curvature, especially at the edge of the cavity bottom (cup) (b) minimize the number of elements, especially in areas with minor interest in the simulation, and (c) keep contact constant between the remeshed elements and the tool. note that contact between the original mesh and the tool may cause sudden re-meshing and excessive strains in the contact area. a very aggressive fusion strategy seems to also cause stability issues. the value of the angle based on which re-meshing takes place has to be reset three times in order to satisfy the above requirements. this was necessary in order to fulfill the third criterion in the beginning of the process, where the angles and the deformations were relatively small. resetting the value in fusion was unexpectedly not possible, reducing the positive impact of the whole re-meshing-fusion algorithm. areas near the fixed support were initially remeshed and excluded from adaptive remeshing for two reasons: (a) deformation in these areas was significant making a fine mesh important (b) adaptive re-meshing causes a sudden change in geometry leading to oscillation of the sheet. 4.2. tool-sheet contact a penalty based segment to segment search algorithm was utilized. node forces are calculated based on the distance among surfaces or edges instead of the classic nodesurface distance. although computationally more demanding, this method was selected as contact simulation was a particularly challenging task. in particular, the number of elements in contact was low. in addition, the size of the master elements had to be optimized in order to minimize the impact of the acute edges and vertices in the master surface, and the poor master/slave element size ratio. therefore, discretization problems were caused as well as noise in the contact. thus, the segment to segment algorithm led to a more gradual transition of the contact between neighboring elements. 724 g.-c. vosniakos, g. pipinis, p. kostazos 4.3. material model the yield locus was calculated by the barlat’s yld2000 model, drawing on the equivalent shear yield energy. it captures a plane anisotropic behavior and depends on initial yield strength and lankford coefficients in the 0o, 45o, 90o directions with respect to rolling direction. the parameters needed for the stress-strain model and the yield locus were defined from tensile tests according to ε8_m and e517_m astm international standards performed on an instrontm model 4482 testing machine. the ‘dogbone’ specimens were created by cnc milling using mild cutting conditions and cutting fluid. the rest of the parameters were adopted from the respective alloy manufacturers (leichtmetall). hardness was assessed by a vickers hardness tester. material parameter values are shown in table 1. analysis (regression-extrapolation) of the tensile test concluded that stress-strain dependence was best represented by the exponential model. table 1 aa6082o properties used property value density (gr/mm3) 0.0027 young modulus (gp) 69 poisson ratio 0.33 yield strength (mpa) 83.5 strength coefficient -k (mpa) 242 strain hardening –n 0.21 hardness (hv) 76 elongation at break 0.18 4.4. boundary conditions sheet clamping was simulated as fixed support. 3d rigid body motion is imposed on the spherical tool, according to the g-code created during toolpath generation stage, see section 3, and a time-displacement file resulting from g-code processing by a matlab custom-written script. the feeding speed of the tool was increased by 300 times compared to the actual speed (time scaling). the spindle speed was neglected. 4.5. stability enhancement due to the reduced integration formulation of the belytchko tsay elements, kinematic hourglass control had to be used. in addition, damping was implemented in the nodes, especially in the contact area, to reduce the impact of time scaling. selective mass scaling was used in some elements near the fixture because they possessed shorter edges. 5. simulation results after termination of the spif simulation, a spring-back simulation was carried out using an implicit integration method. during this stage, the sheet was set free from the boundary conditions and the final sheet shape was obtained as a 3d solid body. a full simulation run lasts about 290 hours on a 6-core amd ryzer 7 cpu memory being of lesser importance. numerical simulation of single point incremental forming for asymmetric parts 725 5.1. plastic strain and thinning thinning estimation is very important for the prediction of fracture-cracking. no excessive thinning was observed in the simulated case. at the 2nd stage of the process (zigzag toolpath), a small increment in strain appears, see fig. 2, albeit much less confined to the tool contact zone in comparison to deformation at the 1st stage (helical toolpath). strain magnitude as well as thinning, see figs. 2 and 3, are correlated to the wall angle. fig. 2 simulated effective plastic strain after 1st stage (left) and 2nd stage (right) fig. 3 simulated shell thickness distribution 5.2. force on the tool spif force can be broken down into three components: one along the tool axis (z direction), constituting the axial component, which is the largest, and two on the horizontal plane (x and y directions) constituting the radial component. fig. 4(a) and (b) depicts the variation of these forces for the roughing and finishing phases, respectively. the maximum axial force reached 1750 n whilst maximum radial force reached 700 n. note that equivalent tensile yield stress was calculated at 200 mpa, the average equivalent strain acquiring a value of about 0.45. 726 g.-c. vosniakos, g. pipinis, p. kostazos a) b) fig. 4 tool force envelope (a) helical roughing toolpath (b) zig-zag finishing toolpath during spif with a helical toolpath, see fig. 1(c), the force constantly increases in the first half of the process duration. this is due to the material hardening and the lower wall angle. then, the force remains stable for most of the second half of the process duration, whilst at the end it decreases due to the lower wall angles locally. the variation pattern for the finishing phase is different, forces hardly reaching half the magnitude of the roughing phase, see fig. 4(b). 6. experimental validation 6.1. fixture and tool the blank holding fixture was designed on solidworkstm and tested for strength and deformation on ansystm employing the worst-case forces that resulted from spif simulation. the required clamping force was calculated on ansys, taking in consideration the cyclic nature of spif loading, resulting in total necessary pre-tension of 400 kn numerical simulation of single point incremental forming for asymmetric parts 727 distributed over 20 m12 bolts. maximum deformation of the fixture resulted to 0.17 mm and was deemed acceptable, see fig. 5. fig. 5 deformation of the work-holding fixture the fixture was manufactured from arc welded square steel tubes (40x40mm crosssection and 2mm wall thickness), see fig. 6(a). a ball-end tool with a radius of 7 mm was employed, see fig. 6(b). it was manufactured from stainless steel (304l) on a haas tl-1 cnc lathe. its hardness was measured at 270 hv30, which was deemed sufficient for spifing of aluminum, whose hardness was 36 hv30. fig. 6 equipment used (a) blank holding fixture with formed sheet (b) spif tool the spif process was carried out on an okuma mx45vae machining center possessing exceptional rigidity. the spindle motor’s maximum power was 14 kw. 6.2. part quality the manufactured part is shown in fig. 7(a). as far as surface quality is concerned, the formation of engravements or ‘squeezed out walls’ is conspicuous in some places. 728 g.-c. vosniakos, g. pipinis, p. kostazos a b fig. 7 manufactured part (a) convex surface (b) overlaid on simulated part of fig. 3 shell thickness measurements were taken at 45 points marked on the convex surface of the manufactured part and at the corresponding points of the simulated part (see fig. 3) as retrieved by figure overlaying, see fig. 7(b). comparison is shown in fig. 8. fig. 8 shell thickness comparison at 45 points between real and simulated parts note that simulated thickness was measured within a range of ±0.033 mm due to fea postprocessor granularity. thickness measurements on the manufactured part were taken by a teledictor 2000tm ultrasonic gauge. the deviation results that are shown in table 2 exhibit a mean of 0,052 mm and a standard deviation of 0,036 mm, which is practically equal to the accuracy range of simulated thickness measurements. the mean relative deviation between measured and simulated thickness is only 2,22%. numerical simulation of single point incremental forming for asymmetric parts 729 in addition, the manufactured part was digitized using an imetricstm model icam m300 white light scanner and associated software with a nominal accuracy of 70 μm. the concave surface was used as reference for alignment purposes between nominal and real shape. a dimensional comparison of the manufactured part to the simulation prediction on one hand and to the designed nominal shape on the other hand is presented in fig. 9. fig. 9 part dimensional comparison between real and (a) simulated (b) nominal in fig. 9(a) rms deviation between real and simulated parts is 0.234 mm, whereas according to fig. 9(b) deviation between real and nominal pats is somewhat larger, i.e. 0.290 mm. simulation offers good prediction of the final geometry near the “edge” of the cavity, and, in many cases, satisfactory prediction of the deviations in regions with intense curvature, e.g. near vane edge. note that final production of the vane requires a metal cutting (finishing) process, typically laser cutting that was not performed in this case. 7. discussion several factors related to the setup of the numerical simulation model may have affected its accuracy. these are briefly discussed next. heat dissipation related to friction between the tool and the sheet surface has been neglected in the model; this may change the yield characteristics of the material locally. to some extent, this is overcome by ample use of lubricant, yet its effect has not been quantified. on a related note, spindle speed was not taken into account in modeling either. finally, the squeezed out wall effect cannot be captured by simulation in the current formulation of mesh discretization and material behavior. simplifications have been adopted in the simulation model to alleviate computation load. regions near the contact and the ‘edge’ between the cavity bottom and walls have a 730 g.-c. vosniakos, g. pipinis, p. kostazos substantial curvature compared to shell thickness. moreover, stress derivative is considerable, especially in the through-thickness direction near the contact. due to the shell formulation, these aspects of the problem may not be simulated accurately enough. in fact, throughthickness shear force distribution cannot be assessed for validity, although it is comparable to the tensile stress distribution. contact stiffness selection affects forces on the contact node. an increased value leads to smaller penetration and increased node speed and its determination was based on experience and experimentation. according to the boundary conditions employed, the tool is rigid and the sheet is fully clamped on the rigid fixture. this simplification may have a significant impact due to the increased spif forces. indeed, substantial elastic deflections are predicted from supplementary simulations reaching 0.5 mm for the fixture and 0.7 mm for the tool. focusing on the impact of element orientation, it is noteworthy that the toolpath is not symmetric with respect to the mesh. thus, orientation of shell edges in relation to the toolpath varies from 0o to 45o. it is known that belytchko-tsay elements are prone to warping [29], especially at increased relative angles between the toolpath and the edges. spif of a fully symmetric conical shape was simulated to check such problems, see fig. 10. sheet shape representation near the contact with the tool depends on the size of the elements in the direction normal to the toolpath. stress and strain derivatives as well as curvature are intense there. finally, a hypothesis is outlined next, regarding the way in which the shape is deformed away from the contact area, especially in the radial direction and in the presence of substantial curvature. fig. 10 impact of finite element orientation in symmetrical part (a) presence of warping (b) deviation of plastic strain intensity in the tangential direction (c) detail of (b) referring to fig. 11(a), 18 nodes are monitored. displacement along z and x axis is followed for the whole of simulation duration, see fig. 11(b) and (d). numerical simulation of single point incremental forming for asymmetric parts 731 fig. 11 node translation (a) node positions (b) nodes displacement in z direction (c) difference between z-displacement of adjacent nodes (d) displacement normal to the toolpath (a: nodes close to edges, b: nodes distant from edges c: final cross-section) 732 g.-c. vosniakos, g. pipinis, p. kostazos note that in fig. 11 ‘a’ denotes nodes that are near the edge of the cavity in a convex shape and tend to deform for a longer period, ‘b’ denotes nodes that are in a convex area and tend to rebound finally ending up with lower final deformation and ‘c’ denotes the final cross section of the region to which the nodes belong. when the tool is located at a lower position with respect to a region with high curvature, the tensile stresses caused will lead to deformation and decrease of curvature there. the difference of the displacement (translation) along z-axis between adjacent nodes depicts the resulting deformation, see fig. 11(c). simulation seems to be able to predict this kind of deformation sufficiently. radial movement of the nodes verifies the hypothesis. an abrupt change in displacement occurs when the tool is in the nodes’ region, see fig. 11(c). then, nodes in concave regions continue being displaced in the negative direction for a while, whereas nodes in convex regions are displaced in the positive direction. 8. conclusions usefulness of simulation in planning spif for non-symmetric parts has become obvious, as far as the toolpath and process parameter selection is concerned. for the particular choices made in the framework of the case study presented, shape difference between the nominal and real formed parts were predicted to some extent, but what is most important, an insight into the deformation mechanism was gained. such deviations range within a few tens of a mm (rms value) which, taking into account the simplifications adopted in numerical modeling of the spif process are acceptable. simulation duration on a normal personal computer, taking several days, despite the explicit solver being used and the simplifications in modeling, is considered high. it is certainly prohibitive, if a number of alternative scenarios need to be studied, but it can be manageably reduced if high-end computers, gpu / parallel programming techniques are used. the main difficulty stems from the sheer length of the toolpath to be simulated resulting in a very large number of discretized positions of the tool relative to the part. future work may follow different directions: (a) based on the current model (possibly improved through adaptive formulation at the tool contact regions) a systematic study is required so as to determine the optimal way to design the toolpath. (b) augmentation of the simulation model is desirable, e.g. to incorporate the effect of spindle speed, thermal effects due to friction and lubrication employed, along with a change in constitutive equations of the material due to temperature. acknowledgement: nikos melissas and kostas kerasiotis of ntua’s manufacturing technology laboratory are gratefully acknowledged for constructing the jigs and fixtures used in experiments and for collaborating in various aspects of experimental measurements, respectively. numerical simulation of single point incremental forming for asymmetric parts 733 references 1. tera, m., breaz, r.-e., racz, s.-g., girjob, c.-e., 2019, processing strategies for single point incremental forming—a cam approach, the international journal of advanced manufacturing technology, 102(5-8), pp. 1761-1777. 2. jackson, k., allwood, j., 2009, the mechanics of incremental sheet forming, journal of materials processing technology, 209(3), pp. 1158-1174. 3. centeno, g., bagudanch, i., martínez-donaire, a.j., garcía-romeu, m.l., vallellano, c., 2014, critical analysis of necking and fracture limit strains and forming forces in single-point incremental forming, materials & design, 63, pp. 20-29. 4. fang, y., lu, b., chen, j., xu, d.k., ou, h., 2014, analytical and experimental investigations on deformation mechanism and fracture behavior in single point incremental forming, journal of materials processing technology, 214(8), pp.1503-1515. 5. lu, b., fang, y., xu, d. k., chen, j., ou, h., moser, n. h., cao, j., 2014, mechanism investigation of frictionrelated effects in single point incremental forming using a developed oblique roller-ball tool, international journal of machine tools and manufacture, 85, pp. 14-29. 6. ham, m, jeswiet, j., 2006, single point incremental forming and the forming criteria for aa3003, cirp annals, 55(1), pp. 241-244. 7. hussain, g., khan, h.r., gao, l., hayat, n., 2013, guidelines for tool-size selection for single-point incremental forming of an aerospace alloy, materials and manufacturing processes, 28(3), pp. 324-329. 8. hussain, g., gao, l., hayat, n., 2011, forming parameters and forming defects in incremental forming of an aluminum sheet: correlation, empirical modeling, and optimization: part a, materials and manufacturing processes, 26(12), pp. 1546-1553. 9. kopac, j., kampus, z., 2005, incremental sheet metal forming on cnc milling machine-tool, journal of materials processing technology, 162-163, pp. 622-628. 10. skjoedt, m., bay, n., endelt, b., ingarao, g., 2008, multi stage strategies for single point incremental forming of a cup, international journal of material forming, 1, pp. 1199-1202. 11. blaga, a., bologa, o., oleksik, v., breaz, r. 2011, influence of tool path on main strains, thickness reduction and forces in single point incremental forming process, proceedings in manufacturing systems , 6(4), pp. 2-7. 12. mcanulty, t., jeswiet, j., doolan, m., 2017, formability in single point incremental forming: a comparative analysis of the state of the art, cirp journal of manufacturing science and technology, 16, pp. 43-54. 13. lu, b., chen, j., ou, h., cao, j. 2013, feature-based tool path generation approach for incremental sheet forming process, journal of materials processing technology, 213(7), pp. 1221-1233. 14. 14. suresh, k., regalla, s.p., 2014, effect of mesh parameters in finite element simulation of single point incremental sheet forming process, procedia materials science, 6, pp. 376-382. 15. dejardin, s., thibaud, s., gelin, j.c., michel, g., 2010, experimental investigations and numerical analysis for improving knowledge of incremental sheet forming process for sheet metal parts, journal of materials processing technology, 210(2), pp. 363-369. 16. naranjo, j., miguel, v., martínez-martínez, a., gómez-lópez, l.m., manjabacas, m.c., coello, j., 2015, analysis and simulation of single point incremental forming by ansys®, procedia engineering, 132, pp. 1104-1111. 17. ambrogio, g., filice, l., gagliardi, f., micari, f., 2005, sheet thinning prediction in single point incremental forming, advanced materials research, 6-8, pp. 479-486. 18. ansys inc., 2006, thin wall structure simulation. ansys manual. 19. marinković, d., rama, g., zehn, m., 2019, abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree of freedom, facta universitatis series mechanical engineering, 17(2), pp. 269-283. 20. bambach, m., 2005, performance assessment of element formulations and constitutive laws for the simulation of incremental sheet forming (isf), in: o’nate e., owen d.r.j. 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forming process: part b, materials and manufacturing processes, 29(4), pp. 454-460. 29. haufe, a., schweizerhof, k., dubois, p., 2013, properties & limits : review of shell element formulations motivation – from shells to solids, in: ls-dyna developer forum 2013, filderstadt, germany, pp. 1-35. facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 613 632 https://doi.org/10.22190/fume210106020k © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper on the influence of multiple equilibrium positions on brake noise sebastian koch, emil köppen, nils gräbner, utz von wagner chair of mechatronics and machine dynamics, technische universität berlin, germany abstract. brake noise, especially brake squeal, has been a subject of intensive research both in industry and academia for several decades. nevertheless, the state of the art simulations does not provide a predictive tool, and extensive experimental investigations are still necessary to find an appropriate design. actual investigations focus on the consideration of nonlinearities which are in fact essential for this phenomenon. unfortunately, by far not all relevant effects caused by nonlinearities are known. one of these nonlinear effects that the actual research focuses on is the limit cycle behavior representing squeal. in contrast to this, the actual paper considers the influence of the equilibrium position established while applying the brake pressure. the elements of the brake, namely, the carrier, caliper and pad, are highly nonlinear and elastically coupled and allow for multiple equilibrium positions depending e.g. on the initial conditions and transient application of the brake pressure while the frictional contact between the pads and the disk may excite small amplitude self-excited vibrations around this equilibrium, i.e. squeal. the current paper establishes a method and corresponding setup, to measure the position engaged by the brake components using an optical 3d-measuring system. subsequently, it is demonstrated that in fact different equilibrium positions can be engaged for the same operation parameters and that the engaged position can be decisive for the occurrence of squeal. in fact, certain positions result in squeal while others do not for the same operation parameters. taking this effect into consideration may have significant consequences for the design of brakes as well as simulation and experimental investigation of brake squeal. key words: brake noise, nonlinearities, equilibrium positions, digital image correlation received january 06, 2021 / accepted february 15, 2021 corresponding author: sebastian koch chair of mechatronics and machine dynamics, technische universität berlin, einsteinufer 5, 10587 berlin e-mail: koch@tu-berlin.de 614 s. koch, e. köppen, n. gräbner, u.v. wagner 1. introduction brake squeal and other brake noises are typical examples for nvh (noise, vibration, harshness) problems in the automotive industry. these phenomena in general do not represent safety risks but are merely comfort issues. their avoidance nevertheless requires a considerable amount of development and testing, making brake noise a topic of numerous scientific and technical publications. several review papers, e.g. [1] and [2], provide overviews on the topic. the brake squeal simulation is still a tool with only a limited predictive character which almost always requires, in addition, experimental investigations. it is well known that the alteration of operating parameters such as brake pressure, brake torque, speed or brake disk temperature during such experiments has a strong influence on the squealing behavior, see e.g. [3]. therefore, the necessary number of tests to classify the brake in such manner is extensive, e.g. [4]. the situation becomes more complicated by the fact that there is a possibility for squealing to sometimes occur and sometimes not even during the tests with the identical operation parameters [5-7]. ref. [6] mainly considers thereby changes in the direction in which the brake components obviously capture significantly different positions as well as the corresponding influence on the squealing behavior. the state of the art procedure to experimentally classify the squealing behavior of the passenger car brakes is sae j2521. during this test, the entire brake and the wheel suspension are mounted on a dynamo test bench. a huge amount of braking operations is performed on different parameters, while the noise level is recorded to categorize whether the brake is squealing or not. on the other hand, the industrial state of the art for the brake squeal simulation is based on the consideration of simulation data gathered from finite element (fe) models. the analysis is divided into multiple steps [8]. in the first step, the brake pressure is applied quasi statically to determine the equilibrium position and the contact forces. therefore, a nonlinear contact analysis is used. then the model is linearized with respect to the found equilibrium position. obviously, the equations of motion of this linearized model depend on this equilibrium position. the linear model is finally used for a complex eigenvalue analysis (cea). since these models contain, if correctly set up, the mechanism of selfexcitation due to the friction forces between the disk and the pad, there is a possibility of instability of the equilibrium position and the mode shapes with eigenvalues with a positive real part are considered as modes potentially associated with squealing [9]. however, linear instability does not represent the observed behavior of a squealing brake, as unstable solutions in linear models show an increasing amplitude above all boundaries with time while the real brake squealing is represented by a more or less stationary behavior with distinct frequencies and finite amplitudes. therefore, the behavior observed in the brake squealing can only be represented by nonlinear models [10]. hereby, nonlinearities limit the increasing vibration caused by the self-excitation finally ending in a limit cycle [10, 11]. several attempts have been made in the past years to describe these effects and to investigate the resulting effects, e.g. [12-19]. these attempts are essential for several reasons. one reason is that the nonlinearity limiting the limit cycle could be a key for avoiding squealing, i.e. if this nonlinearity could be designed in a way that the amplitudes of the limit cycles are irrelevant for noise, the problem of brake squeal would be solved. another reason is that nonlinearities can play a key role, when it comes to the desired predictive character of the simulation methods. in [11] it is shown that the mode shape belonging to the largest positive eigenvalue real part is not necessarily the one occurring on the influence of multiple equilibrium positions on brake noise 615 in the limit cycle, as nonlinearities may limit that mode much earlier than another mode shape with a smaller positive real part. as a conclusion, nonlinearities are essential for the description of brake squeal and may play a key role for its suppression. nevertheless, as the description above makes obvious, nonlinearities, if actually considered, are so far considered in detail only at one place of the simulation process, namely in limiting the increasing vibrations. another obvious point, where nonlinearities play a role, is in determining the equilibrium position. here, in general, only one equilibrium position and its dependency on parameters such as brake pressure are considered, while the immanent nonlinearities could also result in multiple equilibrium positions with the same parameters due to different initial conditions. this fact is completely ignored in state of the art simulations. most people having done experiments with squealing brakes may have experienced that a squealing brake can be brought to silence or vice versa, if the positions of some parts of the brake are manipulated e.g. by pressing temporarily a screwdriver on them. sometimes it is visible with the naked eye that the position of the brake parts is not the same after this manipulation as it was before, so that a new equilibrium position has been reached, possibly with decisive influence on the noise behavior. for both types of above mentioned influences of nonlinearities, the initial conditions play an essential role in determining which solution appears. with respect to the limit cycle, the initial conditions decide in the case of coexistent stable limit cycle and stable trivial solution about the appearance of squealing [20], but the same happens, if different equilibrium positions are possible due to nonlinearities, where the stability behaviors may differ from each other. in reality, nonlinearities affect the noise behavior in both cases and the dependence on the initial conditions is the explanation why brake squeal sometimes occurs and sometimes not for the same operation conditions. the present paper aims to investigate experimentally the influence of the actually engaged equilibrium position on the noise behavior and, therefore, to investigate another influence of nonlinearities on the brake squeal. in reality, the engaged equilibrium position is determined during the process of applying the brake pressure. this process is highly nonlinear especially due to the new contacts occurring therein. therefore, it is highly probable that different equilibrium positions can be established depending on the initial conditions, respectively the state of the brake before the brake pressure is applied. additionally, there might be some influence of external excitation on the engaged equilibrium position. as experiments show, the concept of one engaged stationary equilibrium position is an idealization in simulations, as these equilibrium positions may also change periodically during the turning of the disk due to disk wobbling or other imperfections. nevertheless, the following investigations show that there is significant dependence of the occurrence of squealing on the (medium) absolute and relative positions of the brake parts; they also show that significant different positions are possible even for constant operation parameters. in the metrological investigation it is necessary to consider that the changes in the positions are slow (quasi static) and with respect to the displacement in the order of 0.1 millimeters or more. squealing, however, has usually a much smaller displacement amplitude (µm range or smaller) and it occurs at frequencies in the range of approximately 1 to 16 khz. while conventional set-ups regarding the investigation of brake squeal are focused on measuring these high frequency vibrations by using accelerometers or laser vibrometers, other methods must be established for measuring the equilibrium positions. 616 s. koch, e. köppen, n. gräbner, u.v. wagner the paper is structured as follows. first, the developed test bench is presented. the test setup comprises an optical 3d measuring system filming the brake. the position data of respective parts can be determined using digital image correlation (dic). this method allows the determination of the position of many points on the structure for low frequencies (quasi static) and comparatively large displacements. based on this, two test series are presented. test series number one is used to investigate whether changes in equilibrium position can occur under almost identical parameters for braking torque, speed and temperature and whether these changes have a significant influence on the squealing behavior. in a second test series, what is investigated is the extent to which an increasing braking torque influences the equilibrium position. the results are collected and discussed in a manner which illustrates the essential influence of the equilibrium position on the squealing behavior. 2. experimental setup the main task of the experimental setup is to detect differences in the equilibrium position of the examined floating caliper disk brake while the disk is rotating at a specific speed and a specific brake pressure is applied. especially the brake components carrier, caliper and pad are considered. furthermore, the test bench must be able to detect squealing events and to record the corresponding parameters, namely rotational speed, brake torque, respectively, brake pressure and temperature. fig. 1 left: main components, i.e. the carrier, caliper and pad of the investigated industrial floating caliper brake [21]. right: overview of the test bench with mounted brake and optical 3d measuring system [22] fig. 1 shows one of the brake test benches at mmd tu berlin, which was already used in several prior works, e.g. [11]. it includes an industrial floating caliper disk brake driven by an electric motor via the original drive shaft from the inner side. to allow for a wider range of equilibrium positions, the clamp connecting the carrier and the caliper (fig. 16 left) in the serial setup was omitted in these two first test series. nevertheless, on the influence of multiple equilibrium positions on brake noise 617 similar effects are also observed when the clamp is mounted, as demonstrated in a third test series. the entire control of the brake pressure and rotational speed is done manually. compared to industrial test benches this set up does not allow the investigation of high rotational speeds or high torques and, therefore, it is not capable of analyzing the brake performance. nevertheless, brake squeal generally requires only low rotational speeds and torques. the main advantage of this set-up is that most parts of the brake are perfectly accessible for optical measurements. here, this accessibility is used to observe one side of the brake using a gom aramis optical 3d measuring system, determining the position of the brake parts with a high spatial resolution. this 3d measuring system essentially consists of two cameras and a lighting system. it enables the capturing of grayscale images with 25 frames per second (fps) and a resolution of 2752 by 2200 pixel. self-adhesive point markers with an inner diameter of 1.5 mm are attached to the components carrier, pad and caliper (see fig. 2 right). the image sequence recorded by the 3d-system makes it possible to determine the spatial movement of these point markers by using dic for homologous point tracking in all three dimensions. the equipment used is state of the art. fig. 2 details of the test bench [23]. left: 3d-system, single camera, angle sensor and mounting for the coordinate system. right: accelerometers, temperature sensor and point markers nevertheless, the computation algorithm as described in [24] for the determination of displacement data shall be briefly sketched. the glued-on point markers are identified and tracked via image recognition techniques. the center of the point determines the position of the measurement point. therefore, the tracking resolution is in subpixel accuracy and for this setup in the order of approximately 5µm [25, 26]. the subpixel resolution is possible since the points consist of several pixels and, therefore, the center can be interpolated. each camera records a separate image sequence, and the point tracking is also done separately. according to [27] the data from both cameras can be combined to determine the 3d position of the point markers. 618 s. koch, e. köppen, n. gräbner, u.v. wagner beside the 3d-system there is also a single camera, photron fastcam mini ax100 with an applied resolution of 896 by 768 pixel, for possible 2d measurements. this camera is utilized to display a live image of the brake to adapt or record certain initial conditions before the brake pressure is applied. by overlaying the live image and a previously stored reference image it is possible to (approximately) reproduce spatial initial conditions. fig. 3 shows two examples of this live image. in both cases this is overlaid with a reference image. on the left hand side, the live image is almost the same as the reference. therefore, it is hard to recognize that two images are overlaid. this also means that in this case the conditions stored in the reference image are almost identical to the conditions shown in the live view. on the right hand side, it can be seen that there are two images overlaid (especially in the region highlighted with the red circle). this indicates that the actual conditions differ from the reference conditions. fig. 3 exemplary field of vision of the single camera live image overlaid with a previously stored reference image. left: high agreement between live and reference image. right: relatively high divergence of the two positions visible by naked eye especially in downside part marked by the red circle furthermore, four triaxial accelerometers are attached to the carrier and to the pad to investigate the high frequency oscillations of the brake components and to detect whether the brake is squealing or not. the alignment takes place tangentially, radially and normal to the disk plane, respectively. the effective braking torque is determined by the strain gauges attached to the drive shaft. the values of temperature, rotational speed and rotational angle of the brake disk are measured by additional sensors. all signals are recorded simultaneously and synchronized with the information from the image sequence. 3. measurements to determine experimentally whether different initial conditions result in different equilibrium positions and if those changes have a significant influence on the squealing behavior, two test constellations are conducted. in the first constellation (test series 1), multiple tests are performed where the operation parameters like braking torque, rotational speed and temperature are equal for all tests and constant during the actual test phase. when on the influence of multiple equilibrium positions on brake noise 619 assuming a constant friction coefficient, the brake pressure is proportional to the braking torque. since the braking torque can be measured more accurately than the brake pressure, the brake torque is considered in the following. since all operating parameters are kept constant, only different initial conditions or the positions of the brake components before the brake pressure is applied are possible. therefore, test series 1 is used to determine whether these initial conditions have a significant influence on the positions respectively the equilibrium positions of the brake components after the brake pressure is applied and whether these differences have a significant influence on the squealing behavior. parts of the measurement setup and evaluation procedure were developed in the master’s thesis of the second author. each test run includes three steps. first, an initial position is adjusted manually before the brake pressure and the rotational speed are applied. the live view of the single camera and the overlaid reference image (fig. 3) are used for this purpose. then the rotational speed of the disk is adjusted, and the brake pressure is slowly increased to a specific level. to ensure that during all tests the temperature of the disk is the same, it is measured by an infrared sensor. in the case of a temperature lower than the target one, a warm-up cycle is done until the target temperature is reached. finally, the actual test starts, where all data are recorded in a time interval of 31.24 s while the brake pressure and rotational speed are constant. the time interval of 31.24 s results from the maximum number of images which can be stored uncompressed by the camera. in the second constellation (test series 2), the main procedure is equal except that the braking pressure is continuously increased during the recoding of the data. in total, 146 measurements were carried out, 70 for test series 1 and 76 for test series 2. 4. evaluation procedure the evaluation procedure is shown by two example measurements from test series 1. the two chosen measurements are denoted by m1 with audible squealing and m2 without audible squealing, respectively. during a single measurement, 782 individual images are recorded with a frame rate of 25 fps whereby the positions of all point markers are determined for each image. all calculated position data refer to the coordinate system shown in fig. 4, where the directions are oriented in tangential ex, radial ey and out of plane ez direction. the position of the i-th point pi can be written as �⃑�𝑖 = ( 𝑥𝑖 𝑦𝑖 𝑧𝑖 ) . (1) the position and orientation of the coordinate system are defined by additional point markers which are attached to a frame mounted to the base of the test bench. therefore, slight changes in the camera position do not influence the origin of the reference coordinate system and the position of the points on the brake is always measured relatively to this inertial coordinate system. fig. 5 shows the position results for the two considered measurements m1 and m2 for p1, p2 and p3. 620 s. koch, e. köppen, n. gräbner, u.v. wagner point position on the brake p1 carrier top p2 pad top p3 caliper top p4 carrier bottom p5 pad bottom p6 caliper bottom fig. 4 example of an image recorded by the optical 3d measuring system. applied coordinate system and measurement points p1to p6 (green dots) according to the table on the right. the brake disk rotates counterclockwise it should be emphasized that the visible fluctuations in the positions in fig. 5 are not related to brake squeal, as the displacement amplitudes here are much higher and the frequencies are much lower compared with the squeal. squeal only takes place in m1, while m2 is silent. instead of this, as will be shown later, the fluctuations visible in fig. 5 are related to the actual rotational angle of the disk and its origin can, therefore, be related to the brake disk wobbling or other out-of-roundness imperfections of the brake. focusing on squealing, which takes place at much higher frequencies than the turning of the disk, it can even be said that the equilibrium position might change periodically during the turning of the disk. the effect observed very often during our measurements and in general is that the brake does not squeal permanently while turning, but only at certain ranges of rotational angles; or in the case of permanent squealing the intensity varies with the rotational angle. as a result, the concept of one stationary equilibrium position applied in simulations can hardly be found in measurements. instead, brake squeal is a low amplitude high frequency vibration around an actual position. this position is varying itself in amplitudes of higher order of magnitudes with the frequency of disk turning. additionally, as fig. 5 demonstrates, there are significant differences in the positions possible even for the same operation parameters, and these differences in position, as will be shown later on in several measurements, can make a difference between squeal (m1) and non-squeal (m2)! this is not only true for the absolute, but also for relative positions of the brake parts as will be shown in the following. based on the positions considered so far, also the absolute value of the relative position, i.e. distances ∆𝑖𝑗= |�⃗�𝑖 − �⃗�𝑗| (2) between two points pi and pj can be determined. on the influence of multiple equilibrium positions on brake noise 621 fig. 5 positions of p1 (top), p2 (middle) and p3 (bottom). measurement m1 (squealing) in blue and measurement m2 (no squealing) in orange. the medium positions in case of audible squeal (m1) and without squeal (m2) differ significantly assuming that the parts of the brake are approximately rigid with respect to the displacement scale considered in the positions, the actual absolute position can be determined by the positions of the measured points. nevertheless, the relative positions or their absolute value of the connected brake parts are considered in the following. the following results, in fact, show that significant differences in relative positions can be observed in squealing and non-squealing cases, so that even the condensed information from eq. (2) seems to be sufficiently significant. corresponding results for the two cases m1 and m2 are shown in fig 6. besides the positions, brake torque m and rotational angle φ of the brake disk are also measured during the test period. compared to the position measurements that are recorded at every 0.04 s, the measurement set-up allowed much higher sampling rates for these signals. therefore, the mean value of these signals is calculated at every 0.04 s. 622 s. koch, e. köppen, n. gräbner, u.v. wagner (a) ∆12 (b) ∆13 (c) ∆23 fig. 6 distances ∆23 according to eq. (2) for m1 (squeal, blue) and m2 (no squeal, orange). significant differences of the distances can be observed between the squealing (m1) and the non-squealing case (m2) in addition to these parameters, it is necessary to determine whether the brake is squealing or not. therefore, the data recorded by the accelerometer at the bottom of the carrier (see fig. 2 right) are used while the data of other accelerometers would also have been suitable for this task. in fig.7 the time series of this signal are shown for test m1 (left) with audible squealing and test m2 (right) without squealing. it can be clearly seen that the amplitude is higher while the brake is squealing. however, several previously performed tests have shown that a simple consideration of the amplitude is not sufficient to determine a squealing event since the amplitude varied also due to other parameters like brake pressure or rotational speed. therefore, it is hard to define a specific threshold for the amplitude which indicates whether squealing occurs or not. to overcome this issue the almost mono-frequent characteristics of the brake squeal are taken into account. to integrate this in the evaluation process a specific band acceleration level la is defined according to [28] as 𝐿𝑎 = 20 ∙ log10 ∑ 10𝐿𝑎,𝑖0.2 𝑖 db with: 𝑎0 = 5 ∙ 10−5 m s2 𝐿𝑎,𝑖 = 20 ∙ log ( 𝑎𝑖 𝑎0 ) db, (3) where ai is the value of the i-th measuring point in the power spectrum in a range of ± 20 hz around the previously determined squealing frequency of 2.65 khz. reference acceleration a0 is chosen in the way that the minimum band acceleration level in a series of measurements results in 0 db. this specific band acceleration level is very sensitive to the frequencies close to the squealing one; hence a better squealing indicator than the overall amplitude or intensity of the signal. from a practical point of view, it should be mentioned that a significantly different squealing frequency that could have occurred during the measurements would have been audible and would therefore have been taken into account. however, it must be considered that this indicator will fail if the brake squeals at a significantly different frequency. nevertheless, as the test bench needs permanent supervision this would probably be recognized. on the influence of multiple equilibrium positions on brake noise 623 fig. 7 time signals of the accelerometer at carrier bottom used for the detection of squeal. left m1 with audible squealing events and right m2 without the time series of the accelerometers are recorded with a sample rate of 30 khz during the test. since the optical measurement system has a frame rate of 25 fps the position information is recorded at every 0.04 s. to calculate the corresponding band acceleration level also within this time interval, the time series of the acceleration data is divided into synchronous time sequences of 0.04 s. then the results in each interval are transferred to the frequency domain by using the fast fourier transformation. as a result, the power spectrum of the acceleration related to each image taken by the camera with 25 fps is available. two plots of exemplary frequency bands for the band acceleration level are shown in fig. 8.the red colored area (± 20 hz around 2.65 khz) indicates the values used for the calculation of the band acceleration level. fig. 9 shows the band acceleration level for the complete measurement m1 (left) and m2 (right). it should be noticed that in the case of squealing (m1) la is much higher but varies with a constant period. this period correlates with the rotational speed of the brake. by defining fig. 8 examples of a 0.04 s time sequence transferred to the frequency domain for m1(left, with squealing event) and m2 (right, without squealing event). the power spectrum is shown in blue and the range of ± 20 hz around the squealing frequency considered for calculating la according to eq. (3) is shown in red. the black dot marks the previously determined potential squealing frequency of 2.65 khz. [29] 624 s. koch, e. köppen, n. gräbner, u.v. wagner fig. 9 band acceleration level la according to eq. (3) for the measurement m1 (left) and m2 (right) a threshold value of 60 db for the band acceleration level, which indicates that the brake squeals; when this value is exceeded, the figure shows that the squeal is not constant but appears repetitively with each revolution of the brake disk. this was also audible during the tests, where the squealing event was not continuous but repeated with every rotation of the disk. to investigate this phenomenon in more details fig. 10 shows the band acceleration level and the braking torque as a function of the rotational angle of the disk. it can be seen that both values are strongly related to the rotational angle, but the torque does not differ significantly between the squealing and the non-squealing cases. this indicates that the disk includes some imperfections, such as wobbling, which has an influence on the brake torque and on the squealing condition. since the aim of the first test series is to identify the influence of the actual relative position on the squealing behavior and this relative position is changing with the rotational angle, the rotational angle is also considered for the entire test duration. fig. 10 band acceleration level la as squealing indicator (left) and braking torque (right) as function of rotational angle φ for the measurements m1 (blue) and m2 (orange) to visualize the influence of the position changes on the squealing behavior, the band acceleration level as the squeal indicator is plotted in color as a function of the rotational angle and relative position. fig. 11 shows this for the two measurements m1 and m2. each point in the plot represents the data recorded every 0.04 s. on the influence of multiple equilibrium positions on brake noise 625 each vertical path corresponds to one test series. it is noticeable that the variation of the relative position during each individual measurement is small compared to the differences between both measurements. one series of relative positions is related to squeal while the others are related to non-squeal. the only differences are varied initial conditions, while all the operation parameters are kept constant. since only two tests are considered in the plot, large white areas are present indicating that these relative positions were not reached. in the following, the complete test series will be discussed, while using the way of representation as in fig. 11. as written earlier, the initial position is fixed manually with the help of a single camera which only allows a rough and not very precise determination of the initial position. it should also be mentioned that, in general, the initial positions are far away from the actual equilibrium positions, as the brake components are elastically hinged and, therefore, move with displacement amplitudes of several millimeters, if the brake torque is applied. nevertheless, the following results show that a large range of positions can be reached but only some of them make the brake susceptible to noise. (a) ∆12 (b) ∆13 (c) ∆23 fig. 11 band acceleration level la (squealing indicator) according to eq. (3) as function of the distances for both measurements m1 (with squealing event) and m2 (without squealing event) 626 s. koch, e. köppen, n. gräbner, u.v. wagner 5. results the results for the complete test series 1 are shown in figs. 12 and 13. following the description of fig. 11, each plot shows the band acceleration level according to eq. (3), which is visualized by color as a function of the rotational angle of the disk and the relative position between two specific points on different parts of the brake. in test series 1 all investigated parameters are identical for all measurements and constant during each test session. hereby the braking torque is m ≈ 53 nm, the temperature t = 31 °c and the rotational speed v = 28.8 rpm. only the initial conditions are varied manually by adjusting them to the reference image, so that finally varying position series are kept. this shows that the relative positions of the parts carrier, caliper and pad vary, and that multiple equilibrium positions are possible due to different initial conditions. (a) ∆12 (b) ∆13 (c) ∆23 fig. 12 band acceleration level la as function of distances ∆12, ∆13and ∆23 according to eq. (2) and rotational angel φ for constant braking torque m, rotation speed v and temperature t for the upper 3 point markers p1, p2 and p3. red areas indicate squealing. all 70 measurements of test series 1 are included in each image. on the influence of multiple equilibrium positions on brake noise 627 (a) ∆45 (b) ∆46 (c) ∆56 fig. 13 band acceleration level la as function of distances ∆45, ∆46 and ∆56 according to eq. (2) and rotational angel φ for constant braking torque m, rotation speed v and temperature t for the bottom 3 point markers p4, p5 and p6. red areas indicate squealing. all 70 measurements of test series 1 are included in each image [30]. it is noticeable that for a certain rotational position of the disk the squealing tendency depends only on the relative position. there are distinct areas (blue color only) where the brake never squealed and significant areas (red color) where the brake always squealed. these distinct areas, where the squealing indicator is high, show the strong dependence of the squealing behavior on the relative positions. in test series 2 the braking torque respective the brake pressure is increased slowly by hand (see fig. 14) during the test procedure. temperature and rotational speed remain unchanged. again, at the beginning of each measurement, the initial conditions were varied relative to the reference image as in test series 1. the potential squealing frequency is again 2.65 khz. due to the limited recording time, less data per braking torque are available. nevertheless, the data is sorted by braking torque at every 5 nm1. 1the data points are divided into braking torque interval of 0.5 nm. 628 s. koch, e. köppen, n. gräbner, u.v. wagner fig. 14 exemplary time profiles for quasi-static increase of the braking torque (right) with corresponding band acceleration level la (left) for test series 2 fig. 15 shows again the band acceleration level as a function of the rotational angle and the distance. when a braking torque of approx. 50 nm is reached, a red-colored area with a high band acceleration level is visible and so is an area of the positions which have led to squealing. there are also areas that are permanently without squeal. this result is consistent with the one determined from test series 1 (fig. 13b) as the torque is m ≈ 53 nm in that case. if a braking torque level of approx. 65 nm is exceeded, the squealing disappears completely for all investigated initial conditions. test series 2 shows the well-known influence of the braking torque on the squealing behavior. in connection with equilibrium positions, however, it shows that only the combination of specific relative positions, which is dependent on the respective initial condition and the transient process (e. g. applying the corresponding braking pressure), leads to squealing. for example, distances ∆46 between 28.35 µm and 28.45 µm did not cause any squeal until a certain brake pressure was reached. due to the elastic connecting elements between the components of a brake, it is obvious that the brake components shift significantly when a braking torque is applied. it should be noted that during the measurement test series 1 and 2 and, as already mentioned in the introduction of the experimental setup, the clamp (see fig. 16 left) connecting the carrier and the caliper was removed. in the passenger car brakes, such a part is sometimes used for functional reasons, but not specifically to avoid noise. based on the previous results such a clamp might restrict the range of possible equilibrium positions and, therefore, be capable of avoiding squealing. to investigate whether the mounting of the clamp has a corresponding effect on the investigated brake a third test series was conducted. in this test the clamp was installed while the test procedure was otherwise similar to test series 1. the results in fig. 16(right) show in fact that the clamp fixes the components in such a way that only smaller changes, e. g. in distance ∆46, are possible. nevertheless, the main properties that different positions are possible for constant operation parameters while some of them are resulting in squealing and some not, are still valid. on the influence of multiple equilibrium positions on brake noise 629 (a) m ≈ 40 nm (b) m ≈ 45 nm (c) m ≈ 50 nm (d) m ≈ 55 nm (e) m ≈ 60 nm (f) m ≈ 65 nm fig. 15 band acceleration level la as function of distance ∆46 and rotational angel φ in the case of increasing braking torque m at constant rotational speed v and temperature t (test series 2). red areas indicate squealing. all 76 measurements of test series 2 are included in each image 630 s. koch, e. köppen, n. gräbner, u.v. wagner fig. 16 left: clamp mounted for the 3rd test series. right: band acceleration la as function of distances ∆46 according to eq. (2) and rotational angel φ for constant braking torque m, rotation speed v and temperature t with mounted clamp (test series 3). red areas indicate squealing 6. summary and outlook state of the art simulations of the brake squeal do not provide a predictive tool and extensive experimental investigations are still necessary to find appropriate designs. actual investigations on this topic focus on the consideration of nonlinearities but do so in most cases by trying to model and simulate the limit cycle behavior representing squeal. in contrast to this, the actual paper considers the influence of the equilibrium position engaged during stationary braking due to the transient process. in fact, the elements of the brake, namely the carrier, caliper and pad, are highly nonlinear elastically coupled and allow for multiple equilibrium positions as is demonstrated in the present paper. it is also demonstrated that the engaged position may have a decisive influence on the occurrence of squeal, i. e., depending on which equilibrium position is engaged, the brake squeals or not for the same operation parameters. for the purpose of these investigations an experimental setup and a corresponding analyzing method are established to measure the position engaged by the brake components using an optical 3d-measuring system. these experimental results indicate observations that most people doing experimental work with brake noise probably have made: squealing can be stopped or initiated e. g. by pressing on brake parts with a screw-driver (i.e. possibly manipulating the equilibrium position) and for same operation conditions brake squeal sometimes occurs and sometimes not. the (relative) positions of brake parts are hereby in general not considered. assuming that the results are representative, the conclusions are that essential effects for brake squeal are actually not considered in state of the art industrial and actual scientific investigations, which is a possible explanation for the poor predictive character of actual simulation tools. consequences for simulations should be that the possibility of multiple on the influence of multiple equilibrium positions on brake noise 631 equilibria has to be considered. for stability analysis, the equations of motion then must be linearized with respect to these multiple possible equilibria with the ultimate possibility, that for some equilibria significant instability may occur, and for others not, even in the case of constant operation parameters. on the other hand, consequences for the design of a silent brake could be that the enforcement of specific equilibrium positions could be helpful in avoiding squealing. finally, a consequence for experimental investigations of the nvh behavior of brakes could be that stationary positions of brake parts should be measured by default. acknowledgements: the authors thank the extrusion research and development center of the tu berlin, in particular sören müller and rené nitschke 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beranek, l.l., ver, i.l., 1992, noise and vibration control engineering: principles and applications, john wiley & sons, 966 p. 29. koch, s., 2021, power spectrum with squealing frequency, dataset: doi:10.6084/m9.figshare.13663694.v1 (https://figshare.com/articles/figure/power_spectrum_with_squealing_frequency/13663694/1, last access: 11.02.2021) 30. koch, s., 2021, equilibrium positions, dataset: doi:10.6084/m9.figshare.13673059.v2 (https://figshare. com/articles/figure/equilibrium_positions/13673059/2, last access: 11.02.2021) 12572 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 633 642 https://doi.org/10.22190/fume240314022z © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper gecko-inspired fractal buffer for passenger elevator yu-ting zuo1,2, fu-fang luo3, shi-long zeng3 1college of mechanical engineering, baoji university of arts and sciences, baoji, china 2shaanxi key laboratory of advanced manufacturing and evaluation of robot key components, baoji university of arts and sciences, baoji, china 3gansu province special equipment inspection and testing institute, lanzhou, china orcid ids: yu-ting zuo https://orcid.org/0000-0001-5260-6926 fu-fang luo https://orcid.org/0009-0001-6030-3279 shi-long zeng https://orcid.org/0009-0000-6718-2150 abstract. elevator accidents are common occurrences and can cause serious injuries and property damages. if an elevator falls from a tall building and hits the ground at high speed, passengers in the car have little chance of survival. we designed a fractal buffer with hierarchical structure, which is inspired by the gecko’s pad system, to minimize the damage. a fractal-fractional oscillator is established to show the frequencyamplitude relationship of the fractal buffer using he’s frequency formulation. the fractal-structured vibration-absorbing metamaterial in low frequency contributes to the elevator safety. this paper opens a new window for designing safe and reliable buffers, and provides entrepreneurs with new ideas for the next generation of elevators. key words: metamaterials, passenger elevator, elevator crash, energy absorption, fractal theory, hierarchical structure, duffing oscillator, he’s frequency formulation 1. introduction the first safety elevator in history was exhibited at the 1854 new york world's fair by an american mechanical engineer elisha graves otis. subsequently, as the technology developing, the traction drive passenger elevator was used. due to the defects during elevator construction, or lacking regular maintenance or faulty operation, accidents happen easily. elevator accidents include failure of the elevator’s door which may trap passengers, failure of the brake system which may lead to elevator crash such as rushing to the top or falling to the ground. among all of these, elevator crash is the most serious case. assuming that an elevator car falls from the 10th floor (35 meters) and if other systems fail, it will reach the speed of 94.29 km/h when it hits the ground. in a traction drive elevator system, received: march 14, 2024 / accepted may 10, 2024 corresponding author: yu-ting zuo baoji university of arts and sciences, baoji 721016, china e-mail: zuomath@163.com 634 y.t. zuo, f.f. luo, s.l. zeng a buffer is installed at the bottom of the elevator shaft. there are two types of buffers, one is for energy accumulation, such as polyurethane buffers and steel spring buffers. the other is for energy dissipation, such as oil buffers [1]. a buffer is used to absorb the enormous kinetic energy to prevent the car from falling. usually, the buffer is made of polyurethane, alloy or other materials, which have the advantage of high elasticity and effective energy absorption. to avoid elevator accidents, yao et al. proposed an elevator fault monitoring system based on the internet of things technology [2]. tome et al. designed an automatic speed measurement system applied to elevator overspeed governors [3]. lozzi et al. investigated the failure models of an inclined elevator [4]. zheng et al. recommended a soft-landing system for a full-load elevator [5]. however, there are still many things need to be improved in an elevator buffer system. at present, elevator buffers are mainly made of a single material with simple structure, which cannot fully absorb the energy within a short period of time. if the kinetic energy of the falling car cannot be fully absorbed by the buffer, passengers will get injured easily. in this paper, inspired by the unique structure of the gecko’s pad [6,7], we propose a fractal buffer with a hierarchical structure which can drastically absorb the kinetic energy brought by the elevator car in a short distance, so that passengers are better protected and have greater chance of survival. the two-scale fractal theory [8,9,10] is used in the design of the elevator buffer. this study provides a theoretical insight into the innovative elevator design and sheds light on advanced elevator manufacturing. 2. gecko’s pad and its fractal structure fractal geometry is a valuable mathematical tool for the description of surface microstructures [11,12] and the quantification of complex networks [13,14]. the surface fractality and the network fractality provide a straightforward yet effective approach to predicting the complexity, simplifying theoretical analysis and enhancing its reliability. the gecko is famous for its ability of fast movement, strong adhesiveness, and smooth landing. when a gecko falls from a high place to the ground, it hardly gets hurt. many attempts were made to discover the mechanisms of the gecko's pad and many geckoinspired systems appeared. for example, gecko-like dry adhesive surfaces [15], geckoinspired adhesives [16,17,18], gecko-inspired gripping pads [19], switchable dry/wet adhesives [20], soft robotic grippers [21] and other bio-inspired materials [22]. mathematically, the microstructure of a gecko’s pad can be approximated by fractal geometry, which is characterized by self-similarity. the microstructure of a gecko’s pad is shown in fig. 1. there is a hierarchical structure in the gecko’s pad system. when a gecko falls onto the ground, the kinetic energy can be completely absorbed by the pad in a very short time, therefore the structure of the gecko’s pad has a super energy absorption capability. to understand better the relationship between the structure of a gecko’s pad and the energy absorption ability, we use fractal theory as a mathematical tool. fractal theory is good at describing unsmooth boundaries, porous media, discontinuous problems, etc. in the 1960s, mandelbrot raised a question about measuring the length of the coastline of britain, and he found that as the measurement scale decreased to a very small value, the length of the coastline tended to infinite large. this work, published in science, ushered in the era of fractals [23]. gecko-inspired fractal buffer for passenger elevator 635 fig. 1 gecko’s pad system currently, hausdorff fractal dimension is widely used in engineering, and it requires self-similarity at all scales. a hierarchical structure with three or four cascades is often used, but an additional cascade with self-similarity requires high cost. therefore, in this paper we allow the two-scale fractal dimensions to mimic the fractal nature of the gecko’s pad system. a gecko’s pad has multiple cascades from centimeters to nanometers, a soft landing induces only the first nanoscale cascade, and a hard landing puts the whole pad system into work in a very short time. this property is very useful for designing a fractal buffer. 3. fractal buffer system inspired by the gecko’s pad, a fractal buffer can be made of some metamaterials with a hierarchical structure. the main idea is to build a gecko-inspired system that can completely absorb the kinetic energy of the fast-falling elevator car, so that the car ends up with soft landing in a short time. the speed of the elevator car should reduce gradually so as to avoid any injury to the passengers. xu et al. proposed a fractal cushioning using the hilbert fractal structure and investigated its quasi-static property [24]. traditionally, the buffer is made of alloy or polyurethane, which can absorb energy to some extent. however, in an elevator crash, it is of little help to the passengers. to improve the property of the buffer, we start by changing its material. by using metamaterials [25], we can obtain a wider range of elastic coefficients. fractal metamaterials have been employed extensively in noise and sound absorption [26] and wave absorption [27]. they exhibit distinctive mechanical properties [28] and 636 y.t. zuo, f.f. luo, s.l. zeng damping characteristics [29]. a fractal buffer made of metamaterials is proposed, as shown in fig. 2. the top of the metamaterials has the smallest elastic coefficient, while the bottom has the largest elastic coefficient. several layers of different materials are placed according to different elasticity coefficients. by taking into account the weight of the elevator car (including passengers) and the fall distance, the choice of buffer metamaterials and the corresponding thickness can be determined. metamaterials can be fabricated by 3d printing technology [30,31]. generally, 3d printed mechanical metamaterials have negative poisson's ratio, steady-state structures, and origami structures [32], these properties offer a total new road to design smart programmable mechanical materials, e.g., metamaterial bones [33] and smart fabrics [34]. fig. 2 hierarchical metamaterial with different properties at different cascades the gecko's pad is made of proteins, a type of soft and high molecular weight material. similarly, man-made soft materials such as metamaterials [25] and auxetic materials [35,36] with hierarchical structure can be applied to the design of fractal buffers. the best candidate for metamaterials or auxetic materials with hierarchical structure is the fractal-like porous structure, such as menger's sponge. different porosity leads to different property, and the fractal dimensions are the key factor influencing the property. the different porosities for different cascades in a fractal buffer result in different elastic coefficients and can be adjusted during the manufacturing process. 4. theoretical analysis the fractal buffer system consists of a cascade of rigidities. the top layer has the lowest stiffness, while the bottom layer has the highest stiffness. the equivalent coefficient of elasticity can be calculated as 0 0 0 0 1 1 ... n n n an a n k k k k = + + + , (1) where k is the equivalent elastic coefficient, ki (i = 0 ~ n) is the elastic coefficient of the buffer in the i-th layer, and n0 is number of the first layer, a is the self-similarity index, generally, we choose a=2 or a=3. the equivalent elastic coefficient is a function of the displacement of the fractal buffer: ( ) exp( )k k h b ch= = , (2) gecko-inspired fractal buffer for passenger elevator 637 where b and c are positive constants, h is the displacement. eq. (2) implies that when the fractal buffer is hit, it can adjust its elastic coefficient according to the displacement, so that the elevator car can land smoothly and passengers can be prevented from injury in elevator accidents. if we view the elevator car as a mass point, its total potential energy is e mgh= , (3) where m is the mass of the car, g the gravity acceleration, h is height. the initial velocity is: 2u gh= . (4) the governing equation of the fractal buffer can be expressed as 2 3 2 0, (0) 0, (0) 2 d u du m ku u u gh dt dt  + + = = = , (5) where εu3 is the nonlinear term, and ε can be negative for metamaterials, du/dtα is the twoscale fractal derivative [10], α is the two-scale fractal dimension, which can be calculated by he-liu fractal dimension formulation [8]. furthermore, the genetic programming methodology [13] offers an alternative approach to the accurate determination of fractality. as demonstrated in ref. [13], the accuracy of this approach is as high as 98%. the approximate solution of eq. (5) reads [37] sinu a t= , (6) where a is the amplitude and ω is the frequency. according to he’s frequency formulation [38], we have 2 /2 ( ) u a df u du  = = . (7) here f(u) is given as 3 ( ) ku u f u m + = . (8) so we obtain 2 2 3 4 k a m m   = + . (9) according to eq. (6), we have cos du a t dt    = . (10) that means (0) du a dt = . (11) according to the initial conditions, we, therefore, have 2a gh = . (12) https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 638 y.t. zuo, f.f. luo, s.l. zeng combining eqs. (9) and (12), we have 2 2 3 2 k gh m m    = + . (13) solving ω from eq. (13) yields 22 4 24 4 k k mgh m   + + = . (14) from eq. (12), the amplitude can be solved as 2 8 2 4 24 mgh a k k mgh = + + , (15) and the period reads 2 2 4 2 2 4 24 m t k k mgh     = = + + . (16) we assume that the total energy is absorbed by the fractal buffer system within one quarter period, and the landing velocity is zero. the falling time is t v  = , (17) where v is average velocity of the falling car, δ is the maximal amplitude. it requires that a  . (18) according to eqs. (2) and (15), the amplitude can be solved from the following transcendental equation 2 8 2 exp( ) 4 exp(2 ) 24 mgh a b ca b ca mgh = + + . (19) the average velocity can be calculated as 2 2 gh v = . (20) so the falling time is 2 t v gh   = = . (21) it should meet the following requirement 4 t t  . (22) https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-25 gecko-inspired fractal buffer for passenger elevator 639 that is 2 2 2 4 24 m gh k k mgh     + + . (23) in view of eq. (2), we write eq. (23) in the form 2 2 2 2 exp( ) 4 exp(2 ) 24 m gh b c b c mgh      +  + . (24) eq. (24) is the criterion for designing the needed fractal buffer. 5. fractal buffer’s maximal displacement we have obtained the criterion given in eq. (24) for smooth landing during elevator accidents, but it is an inexplicit form. an explicit form for the maximal displacement of the fractal buffer’s motion is much needed. we consider the following equation 2 2 2 ( ) 0 2 exp( ) 4 exp(2 ) 24 m r gh b c b c mgh     = − =  +  + . (25) in order to solve δ from eq. (25), we apply chun-hui he’s iteration method [39-41]. during experiment, its value can be approximately estimated, for example, 1 / 2 = , (26) 2 / 4 = , (27) where λ is the height of the fractal buffer. according to ref. [41], we have 2 1 1 2 1 1 2 2 exp( ) 4 exp(2 ) 24 m r gh b c b c mgh    = −  +  + , (28) and 2 2 2 2 2 2 2 2 exp( ) 4 exp(2 ) 24 m r gh b c b c mgh    = −  +  + . (29) so an explicit form for the maximal displacement of the fractal buffer’s motion is obtained 1 2 2 1 1 2 r r r r  −   = − . (30) the iteration can be continued for a better explicit form. when the buffer reaches its maximum displacement, a much longer time is needed, which can be guaranteed by the fractal vibration system. fig. 3 shows the dynamic property of the fractal buffer for different fractal dimensions. it is obvious that a smaller value of the fractal dimension leads to a larger half period. when the half period tends to infinity, the fractal buffer becomes https://baike.baidu.com/item/%e5%b8%8c%e8%85%8a%e5%ad%97%e6%af%8d/4428067#3-12 640 y.t. zuo, f.f. luo, s.l. zeng extremely stable and the passengers inside the car can be protected. this extended period property is also called the low frequency property [42,43]. fig. 3 dynamical property of the fractal buffer with different fractal dimensions 6. conclusion in order to prevent injury to passengers in an elevator crash, a fractal buffer system, which can completely absorb the kinetic energy of the falling elevator, is proposed, and the criterion for designing the needed fractal buffer is discussed. the system exhibits a low equivalent frequency, which can be adjusted by the fractality, and the maximal displacement should be at the quarter period. these two properties guarantee the system’s safety and reliability during an accident. the metamaterials or auxetic materials with special property, as given by eq. (2), can be fabricated with ease by 3d printing technology [32,33] and by spinning technology [44,45]. the 3d-printed metamaterials exhibit a negative poisson's ratio, which can be adjusted during the printing process. furthermore, the fractality can be precisely determined, allowing for the smart fabrication and ai optimization of the fractal buffer system. the amplitude-frequency of the fractal buffer’s vibration can be readily and efficaciously revealed by he’s frequency formulation [38,46]. in a forthcoming article, we will discuss the effect of the negative poisson's ratio on the periodic property of the fractal buffer system. acknowledgement: the paper is supported by science and technology program of gansu administration for market regulation (grant no. sscjg-ts-b202202) and national natural science foundation of china. gecko-inspired fractal buffer for passenger elevator 641 references 1. miralbes, r., cuartero, j., castejon, l., 2013, biomechanical response and behavior of 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property, fractals, 29(4), 2150105. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 2, 2014, pp. 183 193 1dfcl: dynamic fuzzy logic controller for intrusion detection udc [004.77+004.8]:681.5 abdulrahim haroun ali1, shahaboddin shamsirband1, nor badrul anuar1, dalibor petković2 1department of computer system & technology, university of malaya, 2university of niš, faculty of mechanical engineering, department for mechatronics abstract: intrusions are a problem with the deployment of networks which give misuse and abnormal behavior in running reliable network operations and services. in this work, a dynamic fuzzy logic controller (dflc) is proposed for an anomaly detection problem, with the aim of solving the problem of attack detection rate and faster response process. data is collected by pinger project. pinger project actively measures the worldwide internet’s end-to-end performance. it covers over 168 countries around the world. pinger uses simple ubiquitous internet ping facility to calculate number of useful performance parameters. from each set of 10 pings between a monitoring host and a remote host, the features being calculated include minimum round trip time (rtt), jitter, packet loss, mean opinion score (mos), directness of connection (alpha), throughput, ping unpredictability and ping reachability. a set of 10 pings is being sent from the monitoring node to the remote node every 30 minutes. the received data shows the current characteristic and behavior of the networks. any changes in the received data signify the existence of potential threat or abnormal behavior. dflc uses the combination of parameters as an input to detect the existence of any abnormal behavior of the network. the proposed system is simulated in matlab simulink environment. simulations results show that the system managed to catch 95% of the anomalies with the ability to distinguish normal and abnormal behavior of the network. key words: intrusion detection, fuzzy system, pinger, round trip time (rtt), packet loss received june 21, 2014 / accepted july 11, 2014 corresponding author: abdulrahim haroun ali university of malaya, department of computer system & technology, kuala lumpur, malaysia e-mail: haroun@gmail.com original scientific paper 184 a. h. ali, s. shamsirband, n.b. annuar, d. petković 1. introduction an anomaly is a deviation of normal behavior or common order. in computer networks, the behavior tends to follow a specific pattern. the followed pattern is due to the network operation time, user’s activities or network usage policies. network base lining is a common technique for identifying the network behavior. in the case of attacks or network faults, the network behavior tends to be anything but normal. hence it’s a worthwhile approach to indicate the presence of attacks or faults. ref. [1] suggests that intrusion detection system (ids) uses an anomaly detection approach to estimate the normal behavior. any deviation that exceeds predefined threshold is considered malicious. unlike signature-based intrusion detection system, the anomaly detection approach is capable of detecting new attacks. network anomalies is experienced due to internal and external factors. internal factors include failure of network nodes or overload of traffic in network nodes. network node such as router which is under pressure of handling traffic beyond its capabilities automatically delays the response time of networks or drop the extra packets. external factors are commonly described as attacks such as denial of service (dos) attacks. denial of service (dos) attacks floods the network with unwanted packets that fill up the host memory buffer to make the network unable to process any request. in dos attacks, the response time of the network and packet loss are rapidly rising and it results in an abnormal behavior of the network. in this paper, dynamic fuzzy logic controller (dflc) mechanism is proposed to monitor and detect network abnormalities. basically, fuzzy logic is a precise logic of imprecision and approximate reasoning [3, 8, 9, 10]. it is widely used in different research disciplines as potential solution [11, 12, 13, 14]. ref. [2] suggests the use of fuzzy logic in controlling traffic in broadband communication networks. ref. [4] shows how fuzzy logic technique is used for correcting climatological ionosphere models. the proposed dynamic fuzzy logic controller consists of a number of fuzzy logic rules that precisely detect any network abnormalities. in order to evaluate the proposed system, this study uses pinger data. pinger project is ping end-to-end reporting network monitoring infrastructure. it uses a ping computer program to measure the performance of several networks around the world. a ping involves sending an internet control messages protocol (icmp) echo request [1] to a specified remote node which responds with an icmp echo reply [5]. pinger performance parameters include round trip time (rtt), packet loss, jitter, mean opinion score (mos), ping predictability, ping reachability, zero packet loss frequency, directivity and tcp throughput. pinger is currently active in over 168 countries around the world. malaysia alone is having three monitoring host and more than 30 remote hosts. the objective of this study is to detect a network anomaly using dynamic fuzzy logic controller (dflc). all pinger matrices contribute to the normal behavior of the network. unconditional changes in any of the matrices indicate abnormal behavior and hence it needs attention to prevent the network from potential threat of faults. dflc is used for deciding whether the received matrices indicate a threat or safety. this paper is organized as follows. section 2 discusses literature review. section 3 explains the architecture of the proposed dflc system. section 4 shows the results of the experiment and comparisons. section 5 concludes the paper with discussion and conclusion. dfcl: dynamic fuzzy logic controller for intrusion detection 185 2. related work pinger is a network monitoring infrastructure which sends a set of 11 pings of 100 byte packet size, followed by 10 pings of 1000 byte packet size from a monitoring host to a remote host at an interval of 1 second [5]. from the ping, icmp packets returns two vital network performance parameters; round trip time and packet loss. round trip time (rtt) is the time taken for the packet to be accepted by the router interface, any delays caused by queuing, and the time taken for the packet to be transmitted from the interface [5]. round trip time needs to be very low for application; as such it requires high level of interactivity such as telnet, voice or video communications. is it impossible to reduce rtt to less than the time taken by the medium to transmit data? in fiber networks, rtt is impossible to be less than the time taken for light to travel the distance along the fiber [5]. packet loss is a parameter that gives a clear picture of congestions in network nodes. if a network node is congested (buffer is full), it discards all extra packets [6]. ping program used in pinger, allows a number of other useful matrices to be calculated. it includes ping unpredictability which is derived from a calculation based on the variability of packet loss and round trip time [7]. ping unreachability which measures the extent of unreachability of the monitored host, if all 10 pings did not receive a reply the remote host would be declared unreachable. the opposite parameter of ping unreachability is quiescence, if all 10 pings have received a reply the node or network is considered quiescence (non busy). among other parameters are tcp throughput, mean opinion score (mos), directivity which calculates how direct the link between monitoring host and the remote host is. pinger project methodology needs to take into consideration two major limitations; periodic sampling and the use of icmp packets [5]. periodic sampling allows ping to be sent at regular interval in order to understand the network performance. hence, all network activities happening at a time outside the period of sampling are not going to be recorded. also rare activities that can only occur during the period of sampling may result in making the network to be marked as poorer than it really is. the use of icmp packets results in another limitation to pinger methodology due to its features in a network. in the networks which implements quality of service (qos), icmp packets are given low priority compared to other ones like tcp and udp in the network. some network blocks icmp packets prevent the network from network attacks such as smurf attack uses icmp packets [5]. 3. methodology the fuzzy logic-risk analysis model is built on the following constituent parts and logic: fuzzy linguistic variables and fuzzy expression for input and output parameters are shown in table 1. for each variable, three membership functions are used which are low, medium, and high for inputs. the output variable (output) uses only two membership functions, normal and abnormal. the characteristics of the inputs and output variables are shown in table 1. 186 a. h. ali, s. shamsirband, n.b. annuar, d. petković table 1 fuzzy linguistic and abbreviation of variables for each parameter inputs range parameters linguistic variables maximum rtt 0−100 minimum rtt 0−100 average rtt 0−100 packet loss low, medium, high 0−100 outputs output abnormal, normal 0−1 the four inputs will be defined as maximum rtt (which is the maximum round trip time received after sending the 10 ping request), minimum rtt (which is the minimum round trip time received after sending the 10 ping request), average rtt (which is the average round trip time of the received 10 ping’s reply) and packet loss (which is the percentage of packet lost during the 10 ping request and reply). a valid range of input is considered and divided into three classes, or fuzzy sets for all the inputs. table 6 shows the range is low, medium and high for each input. membership functions for input and output fuzzy model. in choosing the membership functions for fuzzification, the event and type of membership functions are mainly dependent upon the relevant event. in this model, gaussian-shaped membership function is employed to describe the fuzzy sets for input; for output variable triangular-shaped membership function is used [8, 9, 10]. the input variables have been partitioned according to the experiment parameter ranges. the degree of belongingness of the values of the variables to any selected class is called the degree of membership as shown in fig. 3. the output defined in fuzzy sets “normal” means the input from four features received indicate no anomaly, hence it is safe. while “abnormal” meaning that there input received indicates the detection of abnormal behavior of the network. fig. 1 membership function minimum rtt (inputs mf) dfcl: dynamic fuzzy logic controller for intrusion detection 187 fig. 2 membership function maximum rtt (inputs mf) fig. 3 membership function average rtt (inputs mf) fig. 4 membership function packet loss (inputs mf) 188 a. h. ali, s. shamsirband, n.b. annuar, d. petković fig. 5 membership function represents countermeasure (output mf) expert knowledge is used to characterize inputs and outputs and connect the inputs and outputs by a set of inference rules using if/then statements; according to the number of the fuzzy sets of the inputs. the system will have sixteen possible combinations (inference rules). the fuzzy output set is the indication of the existence of an anomaly from the input received by the system. fuzzy inference system (fis) for input and output parameters is shown in fig. 6. fig. 6 fuzzy inference system 5. structure of fuzzy rules the type of the response of the anomaly detection system will be based on the calculated average rtt, minimum rtt, maximum rtt and the packet loss. for example, if the packet loss is low while any of the remaining inputs is high, then the system will indicate an anomaly. if all four inputs are low, then the system will indicate it as a normal behavior of the network, hence no anomaly. the opposite of that (previous example) means that definitely the system will indicate anomaly. this methodology will catch any abnormal behavior of the network. dfcl: dynamic fuzzy logic controller for intrusion detection 189 a set of 16 rules have been constructed based on the actual experimental qualitative analysis shown in table 2 and the characteristics of the inputs and output variable are shown in table 2. experimental results are simulated in the matlab using matlab simulink. table 2 the pseudo code for dynamic fuzzy logic anomaly detection (dflad) 1. if (artt is low_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is normal) 2. if (artt is high_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is abnormal) 3. if (artt is med_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is normal) 4. if (artt is low_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is med_pl) then (output1 is abnormal) 5. if (artt is low_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is high_pl) then (output1 is abnormal) 6. if (artt is high_artt) or (max_rtt is high_max) or (min_rtt is high_min) or (p.l is high_pl) then (output1 is abnormal) 7. if (artt is med_artt) and (max_rtt is med_max) and (min_rtt is med_min) and (p.l is low_pl) then (output1 is normal) 8. if (artt is low_artt) and (max_rtt is low_max) and (min_rtt is med_min) and (p.l is low_pl) then (output1 is normal) 9. if (artt is low_artt) and (max_rtt is med_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is normal) 10. if (artt is low_artt) and (max_rtt is med_max) and (min_rtt is med_min) and (p.l is low_pl) then (output1 is normal) 11. if (artt is med_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is normal) 12. if (artt is med_artt) and (max_rtt is low_max) and (min_rtt is med_min) and (p.l is low_pl) then (output1 is normal) 13. if (artt is med_artt) and (max_rtt is med_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is normal) 14. if (artt is med_artt) and (max_rtt is med_max) and (min_rtt is med_min) and (p.l is low_pl) then (output1 is normal) 15. if (artt is high_artt) and (max_rtt is low_max) and (min_rtt is low_min) and (p.l is low_pl) then (output1 is abnormal) 6. defuzzification defuzzification is the conversion of a fuzzy quantity to a precise value, just as fuzzification is the conversion of a precise value to a fuzzy quantity. in this method, the resultant membership functions are developed by considering the union of the output of each rule, which means that the overlapping area of fuzzy output set is counted as one, providing more results. fig. 9 is an example to demonstrate the appropriate assent between input parameters change and output values predicted by fuzzy based model. the close assent of output values obviously displays that fuzzy logic model can be used to predict output values under consideration thus, the proposed fuzzy logic model gives a promising solution to predict output value in the specific range of parameter. 190 a. h. ali, s. shamsirband, n.b. annuar, d. petković fig. 7 output value in relation to change of average rtt and packet loss 7. simulation the experiment is simulated in simulink environment using the required components to achieve the objectives. the components used include multiplexers, fuzzy logic controller and a scope that relates to the output. fig. 10 illustrates the design of the simulink environment used in this experiment. fig. 8 simulink design fig. 8 shows our four inputs being multiplexed to the fuzzy logic controller. it is the fuzzy logic controller where we install our fuzzy logic instance which we have created using the fuzzy rules shown in table 8 and the membership functions. after the fuzzy system makes the decision, it will push the results to the scope and stored in the system. dfcl: dynamic fuzzy logic controller for intrusion detection 191 fig. 9 shows the simulation in the fuzzy logic block. the results obtained from this simulation are highly accurate as the all predicted anomalies are caught by the system leaving the remaining inputs as normal behavior of the network. fig. 9 fuzzy simulation on progress 4. results the proposed system is simulated in matlab simulink using pinger data set. in order to compare the results, the same data set is simulated in weka environment using naivebayes and decision tree (j48) machine learning techniques. the performance of each simulation is evaluated using accuracy rate and misclassification rate. the following formulas are used: instancesofnumber total instances classifiedcorrectly ofnumber ratesaccuracy = (1) 192 a. h. ali, s. shamsirband, n.b. annuar, d. petković number of correctly classified instancesmisclassification rate 1 total number of instances = − (2) table [3] presents the evaluation of the proposed system, naivebayes and decision tree results in terms of accuracy and misclassification rates. in table 3 very high accuracy for decision tree (j48) should be noted. table 3 experimental results naive bayes decision tree (j48) proposed system (dflc) data split training set (%) 30 30 0 accuracy rate (%) 92.163 99.0593 95.614 misclassification rate (%) 7.837 0.9404 4.386 5. discussion and conclusion conclusively, the proposed dynamic fuzzy logic controller has proved to be an optimal approach to detecting anomalies in networks. with pinger dataset, simulation results prefer dflc compared to naivebayes but not to decision tree in terms of accuracy and misclassification rates. the study shows that processing time is not a setback, as it is only a single system integrated with the monitoring host. yet it is recommended that the monitoring host should have enough computing resources and power as huge datasets need to be processed by the system. references 1. feizolah, a., anuar, n.b., salleh, r., mat kiah, k.l., 2013, anomaly detection using cooperative fuzzy logic controller, intelligent robotics systems: inspiring the next, 376, pp. 220-231. 2. lim, h.h., qiu, b., 2001, fuzzy logic traffic control in broadband communication networks, the 10th ieee international conference on fuzzy systems, 1(5), pp. 99-102. 3. zadeh, l., 2008, is there a need for fuzzy logic?, annual meeting of the north american fuzzy information processing society – nafips, 8(9), pp. 1-3. 4. giannini, j.a., kilgus, c., 1997, a fuzzy logic technique for correcting climatological ionospheric models, ieee transactions on geoscience and remote sensing, 35(2), pp. 470-474. 5. cottrell, w.m., matthews, w., 2000, the pinger project: active internet performance monitoring for the henp community, ieee communications magazine, 38(5), pp. 130-136. 6. postel, j., 1981, internet control message protocol, rfc editor, united states. 7. mathis m., 1997, the macroscopic behavior of the tcp congestion avoidance algorithm, computer communication review, 27(3), pp. 67-82. 8. petković, d., issa, m., pavlović, d. n., zentner l., 2013, intelligent rotational direction control of passive robotic joint with embedded sensors, expert systems with applications, 40(4), pp. 1265-1273. 9. shamshirband, s., petković, d., ćojbašić, ž., nikolić, v., anuar, n.b., mohd shuib, n.l., mat kiah, m.l., akib, s., 2014, adaptive neuro-fuzzy optimization of wind farm project net profit, energy conversion and management, 80(4), pp. 229–237. 10. zakaria, r., sheng, o.y., wern, k., shamshirband, s., petković, d., pavlović, t.n., 2014, adaptive neurofuzzy evaluation of the tapered plastic multimode fiber based sensor performance with and without silver thin film for different concentrations of calcium hypochlorite, ieee sensors journal, doi: 10.1109/jsen.2014.2329333. dfcl: dynamic fuzzy logic controller for intrusion detection 193 11. shamshirband, s., petković, d., anuar, n.b., mat kiah, m.l., akib, s., gani, a., ćojbašić, ž., nikolić, v., 2014, adaptive neuro-fuzzy generalization of wind turbine wake added turbulence models, electrical power and energy systems, 62, pp. 490–495. 12. petković, d., shamshirband, s., iqbal, j., anuar, n.b., pavlović, d.n., mat kiah, m.l., 2014, adaptive neuro-fuzzy prediction of grasping object weight for passively compliant gripper, applied soft computing, 22, pp. 424–431. 13. shamshirband, s., petković, d., anuar, n.b., gani, a., 2014, adaptive neuro-fuzzy generalization of wind turbine wake added turbulence models, renewable and sustainable energy reviews, 36, pp. 270–276. 14. petković, d., shamshirband, s., ćojbašić, ž., nikolić, v., anuar, n.b., md sabri, a.q., akib s., 2014, adaptive neuro-fuzzy estimation of building augmentation of wind turbine power, computers & fluids, 97, pp. 188–194. dfcl: dinamički fazi kontroler za detekciju intruzije intruzije su problemi pri mrežnom prenosu koje dovode do pogrešne primene i abnormalnog ponašanja u pouzdanim opercijama mreže i servisa. u ovom radu, dinamički fazi kontroler (dfk) je predložen za detekciju anomalije u mreži, sa ciljem da se reši problem detekcije brzine napada i proces bržeg reagovanja. podaci su skupljeni u okviru projekta pinger. reč je o projektu koji aktivno meri performanse do krajnih korisnika svetske internet mreže. pokriva više od 168 zemalja u celom svetu. pinger koristi univerzalan internet ping kako bi izračunao parametar korisnih performansi. pri svakom setu od 10 pinga između praćenog hosta i udaljenog hosta, karakteristike koje se računaju uključuju minimum round trip time (rtt), jitter, packet loss, mean opinion score (mos), directness of connection (alpha), propusna moć, ping nepredvidljivost i ping dosezanja. set od 10 pinga se šalje od praćenog čvora do udaljenog čvora svakih 30 minuta. primljeni podaci prikazuju trenutnu karakteristiku i ponašanje mreže. svaka promena u primljenim podacima ukazuje na primenu potencijalne pretnje ili abnormalnog ponašanja. dfk koristi kombinaciju parametara kao ulaz da detektuje bilo koje abnormalno ponašanje mreže. predložen sistem je simuliran u matlab-u. rezultati simulacije pokazuju da sistem može da uhvati 95% anomalija sa mogućnošću da odvoji normalno i abnormalno ponašanje mreže. ključne reči: detekcija intruzije, fazi sistemi, pinger, vreme cirkulacije (rtt), gubitak podataka. dfcl: dynamic fuzzy logic controller for intrusion detection abdulrahim haroun ali1, shahaboddin shamsirband1, nor badrul anuar1, dalibor petković2 1. introduction 2. related work 3. methodology 5. structure of fuzzy rules 6. defuzzification 7. simulation 4. results 5. discussion and conclusion references 12108 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 667 686 https://doi.org/10.22190/fume231102003s © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper multi-objective optimization and performance analysis of bimorph magneto-electro-elastic energy harvesters meisam moory shirbani, sayed ehsan alavi faculty of engineering, shohadaye hoveizeh campus of technology, shahid chamran university of ahvaz, dashte azadegan, iran orcid ids: meisam moory shirbani https://orcid.org/0000-0001-5121-4138 sayed ehsan alavi https://orcid.org/0000-0002-5669-7498 abstract. this study thoroughly investigates the multi-objective optimization of a magneto-electro-elastic (mee) harvester in bimorph configurations and by the new method of harris hawk’s optimization (hho). the harvesters are configured in both series and parallel connections and under harmonic excitation to explore the effects of various parameters on the performance of the harvesting system. the primary objective is to maximize the total harvested power. optimization involves various parameters, including dimensions, relative displacement changes, voltage, and current values. the pareto fronts from the hho method reveal optimal points in different configurations and scenarios. notably, the optimal points are selected based on the criterion of maximum total power. the results reveal distinct optimal points for each objective function, demonstrating trade-offs between performance metrics. these findings provide valuable insights into the design and operation of efficient energy harvesters in mee systems. the parallel configuration outperforms the series connection in terms of the current generation. moreover, the evaluation of the overall performance of the energy harvesters in terms of total harvested power indicated that both series and parallel connections could lead to promising outcomes. however, the series connection exhibited a more dominant effect on maximizing the total harvested power, proving its relevance in pursuing the highest possible power output. key words: multi-objective optimization, mee harvester, harris hawk’s optimization (hho), series and parallel connections received: november 02, 2023 / accepted january 14, 2024 corresponding author: meisam moory shirbani faculty of engineering, shohadaye hoveizeh campus of technology, shahid chamran university of ahvaz, dashte azadegan, b.p 78986-64418, iran e-mail: m.mooryshirbani@scu.ac.ir https://orcid.org/0000-0001-5121-4138 https://orcid.org/0000-0002-5669-7498 668 m.m. shirbani, s.e. alavi 1. introduction energy harvesting involves converting environmental sources such as wind, heat, and vibrations into electrical energy. mechanical vibrations are present in almost every industrial device as a reliable energy source. energy harvesting from ambient vibrations has gained substantial popularity due to its accessibility in various places and times [1-3]. this field involves multiple aspects of engineering disciplines, including mechanics, materials science, and electronics. smart materials that can couple at least two fields, such as piezoelectric materials [4-6], mee materials [7,8], and others [9,10], are employed in this energy conversion process. the potential applications of this technology were extensive, with one of its most efficient domains being wireless sensor networks. piezoelectric energy harvesters have been used, often as cantilever beams and usually made of two layers [11-13]. among notable initial works, the design and fabrication of a generator by roundy et al. [14] in 2003 is noteworthy. this generator can produce 375 μw of power per 1 cm3 with a density of 5.8 mw/gcm3, sufficient output power for many lowpower microelectronic devices. among the most significant works conducted in the analytical and numerical modeling of energy harvesting from piezoelectric materials, the contributions of erturk and inman are noteworthy [15,16]. they delved into the analytical modeling of piezoelectric energy harvesters by considering electrical and mechanical parameters, subsequently comparing and validating their results with experimental data. utilizing a simple cantilever beam structure and operating in a frequency range of 450 hz, they successfully achieved a power density of 6.8 mw/gcm3. researchers are employing various techniques, such as changing the type of piezoelectric material, altering electrode patterns, and layering energy harvesters to enhance the harvested power [17,18]. the mee composites exhibit sensitivity to a magnetic field in addition to their sensitivity to an electric field [19-21]. with these dual sensitivities, these materials can interconvert mechanical, electrical, and magnetic energies, positioning them as superior and more versatile than piezoelectric materials. in 2018, shirbani et al. [22] proposed using mee composites in energy harvesters to enhance the electrical power density. they provided analytical modeling of the coupled mechanical, electrical, and magnetic dynamic responses of the suggested energy harvesters. an important outcome of their work was the positive impact of the new structure and circuit design used for the external coils around the composite layers, resulting in a 43% increase in the total harvested power. they also explored the influence of temperature differences between the different layers of mee energy harvesters [23]. the optimization technique holds significant importance in mechanics, as it strives to discover the most optimal solutions to intricate problems while weighing various constraints and objectives [24-26]. mechanics employ optimization techniques to design and analyze structures, systems, and processes that can efficiently withstand forces, stresses, and dynamic conditions. optimization algorithms help engineers find the best configuration or parameters that lead to better performance, lower costs, and better safety. this paper aims to provide a comprehensive overview of the bimorph mee energy harvester research, mainly focusing on applying multi-objective optimization using the hho method. the optimization process involves the simultaneous consideration of various objective functions, enabling the identification of optimal solutions that balance competing design criteria. multi-objective optimization offers a holistic approach to achieving the multi-objective optimization and performance analysis of bimorph... 669 highest levels of efficiency and performance, catering to different operational scenarios and requirements. the hho is a relatively recent optimization algorithm inspired by the hunting behavior of harris’s hawks [27], a type of raptor bird. the algorithm was proposed to solve complex optimization problems by simulating the social hunting behaviors of these birds. hho aims to balance exploration and exploitation, leveraging the collaborative behavior of hawks during hunting. the hho algorithm was introduced in a research paper by heidari et al. [27] in 2019. the algorithm was designed to address optimization problems by imitating the cooperative hunting strategies of harris’s hawks. the hho algorithm has gained attention due to its unique optimization approach inspired by nature. however, like any optimization algorithm, its effectiveness depends on the problem and its characteristics. researchers continue to explore its potential and refine its techniques for various applications [28,29]. 2. energy harvester equations and optimization approach in the present work, energy harvesters are analyzed as cantilever beams under the assumptions of the euler-bernoulli beam theory, coupled with layers possessing active mee properties. in this context, energy harvesting refers to the extraction of vibrational energy derived from the vibration of the base of the beam. with transverse vibration, the beam undergoes strain. as a result of electrical and magnetic polarization, an electric and magnetic potential difference is established along the polar axis of the beam, enabling the generation of electrical power. the beam features embedded electrically conductive electrodes (for utilizing the electric field generated and extracting electrical energy) with minimal thickness, spanning the entire beam length. additionally, external coils (for utilizing the induced magnetic field and extracting magnetic energy) are present. other assumptions include the linear behavior of magnetic, electric, and elastomer properties concerning strain, electric, and magnetic fields. figures 1a and 1b illustrate the analyzed configurations. these configurations feature symmetric arrangements with layer connections in series (fig. 1a) and parallel (fig. 1b). in fig. 1, ie, re, vm, im, and rm represent the voltage, current generated, and external electrical resistance across the electrodes and the voltage, induced current and external electrical resistance across the external coils. furthermore, wbt denotes the transverse base excitation. the fundamental equations that characterize mechanical, electrical, and magnetic behaviors of materials with active mee properties are expressed as follows [22]: i ik k ik k ik kc e e f h = + + (1a) k ik i kj j kj jd e h e g h= + + (1b) k ik i kj j kj jb f g e h = + + (1c) in these equations, σi, di, and bi represent the stress tensor, electric displacement, and magnetic flux density. also, cij, hij, and μij denote the tensor of elastic stiffness, and electric and magnetic permeability coefficients, respectively. the distinguishing feature of this class of materials from piezoelectric materials lies in the simultaneous influence of the strain fields εk, electric field ek, and magnetic field hk on stress, electric displacement, and 670 m.m. shirbani, s.e. alavi magnetic flux density, characterized by the piezoelectric coefficients eij, piezomagnetic coefficients fij, and magnetoelectric coefficients gij. the strain component along the beam ε1 according to the relative displacement wrel is expressed as: 2 1 ,2 ( , )rel rel xx w x t z zw x   = − = −  (2) fig. 1 symmetric configurations of the mee energy harvester; (a) series connection, (b) parallel connection of harvester layers 2.1. equations governing mechanical motion coupling for bimorph mee energy harvesters in this section, the euler-bernoulli cantilever beam with the configurations depicted in fig. 1 is considered the energy harvester. by applying a base excitation in the transverse direction wbt, the overall displacement change of each point along the beam w(x,t) is as follows: ( , ) ( ) ( , )bt relw x t w t w x t= + (3) the governing differential equation for the forced vibrations of the euler-bernoulli cantilever beam under base excitation is expressed by eq. (4) [7]: 2 22 2 2 2 ( , ) ( , ) ( , )( , ) [ ]rel rel bt m w x t w x t w x tm x t c m m tx t t    + + = −    (4) the bending moment m(x,t) is the area moment of inertia, cm is the damping coefficient, m is the mass per unit length l, and t is the time. the structural equations for expressing the coupling relationship in the mee layers (denoted by the superscript m) in the axial direction using eqs. (5a) to (5c), are presented as follows: 1 11 1 31 3 31 3 m m m m m m mc e e f h = − − (5a) multi-objective optimization and performance analysis of bimorph... 671 3 31 1 33 3 33 3 m m m m m m md e h e g h= + + (5b) 3 31 1 33 3 33 3 m m m m m m mb f g e h = + + (5c) also, the stress-strain constitutive equation for the sublayer with homogeneous properties (marked with superscript h) is expressed as follows: 1 11 1 h h hc = (6) it should be noted that based on the polarities indicated in fig. 1, for the series connection (denoted by the superscript s), the electric field e3 in both upper and lower layers is the same. for the parallel connection (denoted by the superscript p), the electric fields e3 in both the upper and lower layers are equal in magnitude but opposite in direction [7]. 3 ( ) ,0.5 0.5 , 0.5 0.5 2 s s e h m h m h h m v e t h z h h h h z h h = −   + − −   − (7a) 3 3 ( ) ,0.5 0.5 2 ( ) , 0.5 0.5 2 p p e h m h m p p e m h h m v e t h z h h h v e t h h z h h  = −   +   = + − −   −  (7b) furthermore, the magnetic field h3 generated in the mee layers is calculated using the induced voltage across the terminal of the external wire coils vm and ampere’s law, along with the number of wire coil turns n, as given by eq. (8) [7]: 3 ( ) ( ) ( ) m m m m m ni t nv t h t h r h = = (8) the bending moments for each connection are calculated and expressed using the structural equations, electric and magnetic fields, and euler-bernoulli assumptions: 2 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( )[ ( ) ( )] ( )[ ( ) ( )] elec mag elas rel eq em e mm m m x t m x t m x t w x t m x t ei v t h x h x l v t h x h x l x  = + − − + − −    (9) where melas, melec, and mmag are the components of the bending moments that comprise elastic, electric, and magnetic components. the electric-mechanical and magneticmechanical coupling coefficients θem and θmm are calculated for each connection. 312 ( )p s m em em m hbe h h =  = + (10a) 31 ( ) m p s mm mm m h m nf b h h r  =  = + (10b) the equivalent flexural rigidity of the composite section eieq and mass per unit length are calculated for the bimorph configuration. 672 m.m. shirbani, s.e. alavi , 3 3 311 11( ) ( 0.5 ) (0.5 ) 12 3 h m s p eq h m h h c b c b ei h h h h = + + −  (11a) , (2 )s p m m h hm b h h = + (11b) the densities of the mee and homogeneous layers are represented by ρm and ρh. furthermore, b, hm, and hh represent the beam’s width, the mee layer’s thickness, and the homogeneous layer’s thickness, respectively. the derived motion equations can be solved using traditional modal analysis methods. by using variable separation techniques, the displacement variation of each point on the beam with respect to its base can be expressed through eq. 12 [15]: 0 ( , ) ( ) ( )rel r r r w x t w x t t  = = (12) the time response of the beam and the shape of the r-th mode one-dimensional vibration of the cantilever beam are determined by tr(t) and wr(x). the coupled mechanical equations of motion in modal coordinates are derived after substituting eq. (12) into eq. (9) and then applying the orthogonality conditions of the mode shapes. 2 2 2 ( ) ( ) 2 ( ) ( ) ( ) ( )r r r r r r emr e mmr m r d t t dt t t t v t v t f t dtdt     + + + + = (13) the modal electric-mechanical and magnetic-mechanical coupling coefficients αemr, αmmr are defined here and are calculated for each connection as follows: 31 ( ) 2 ( )p s m r emr emr m h x l dw x be h h dx   =   = = +     (14a) 31 ( ) ( ) m p s r mmr mmr m h m x l f n dw x b h h r dx   =   = = − +     (14b) the natural frequency of the r-th mode is calculated using eq. (15) [7]: 2 2 4 eq r r ei ml  = (15) the damping ratio ζr and the mechanical force function fr(t) in modal coordinates are also defined as follows: 2 m r r c m   = (16a) 2 2 0 ( ) ( ) ( ) l bt r r d w t f t m w x dx dt = −  (16b) multi-objective optimization and performance analysis of bimorph... 673 2.2. governing electrical coupling equations of electrodes in bimorph mee energy harvesters the differential equation governing the electrical circuit involving the two electrodes around the mee layers is extracted in this section. the two electrode terminals and the external wire coils are connected to external electrical resistances re and rm to use the generated voltages ve and vm. the electric charge created at the two electrode terminals qe(t) can be determined through gauss’s law, which stipulates that the electric charge equals the integral of the electric displacement d3 m over the electrode surfaces ae. furthermore, the electric current ie(t) equals the time derivative of the electric charge produced [7]. 3 ( ) ( ) ( , ) e me e e a dq t d i t d x t da dt dt = =  (17) consequently, the following equation is obtained for each connection using eqs. (5b) and (17) and applying ohm’s law (v = ri). 1 ( ) ( ) ( ) ( )e m re r m c m e mer me re dv t dt t dv t c v t dt r dt dt      = + = + (18) this equation defines cm, υmer, and χme as the internal capacitance, modal mechanicalelectrical, and magnetic-electrical coupling coefficients of the mee layers, respectively. in addition to these parameters, the constants γcm and γre for each connection are obtained as follows: 33 m m m h bl c h = (19a) , 31 ( ) 0.5 ( )s p m r mer m h x l dw x e h h b dx  = = − + (19b) , 31 ( ) 0.5 ( )s p m r mer m h x l dw x e h h b dx  = = − + (19c) 2 1 m m p s c c = = (19d) 2 2 e e p s r r = = (19e) 2.3. governing electrical coupling equations of external coils in bimorph mee energy harvesters during vibrational excitation, the relative motion between the mee layers and the external wire coils causes a change in the magnetic flux passing through the coils, resulting in an electric voltage. for the present energy harvester beam, the induced voltage in the wire coils results from changes in the magnetic flux density b3 m. consequently, vm can be expressed as follows [7]: 674 m.m. shirbani, s.e. alavi 3( ) ( , ) i m m a d v t b x t da dt =  (20) by substituting eqs. (12) and (20) into eq. (5c), the differential equation for the electrical coupling in the circuit of the two wire coils is obtained: 1 ( ) ( ) ( ) ( )m r e c m m mmr em r dv t dt t dv t l r v t dt dt dt    = + = + (21) in these equations, lc, υmmr, and χem represent the self-inductance coefficients of the wire coils, the modal mechanical-magnetic and electric-magnetic coupling coefficients, respectively. these coefficients are derived for each connection as follows: 2 33 m c m bl l n h = (22a) , 31 ( ) 0.5 ( )s p m r mer m h x l dw x e h h b dx  = = − + (22b) , 33 m s p me m m g n bl r h  = (22c) 2.4. frequency response of bimorph mee energy harvesters under harmonic base excitation the base excitation of the beam in the transverse direction at the excitation frequency ω is given as wbt(t) = wat ejωt, where wat represents the amplitude of the base transverse displacement. in this case, the expressions for the modal coordinates response of the beam, the voltages ve(t) and vm(t) are assumed as follows: ( ) , ( ) , ( )j t j t j t r ar e ae m amt t t e v t v e v t v e  = = = (23) where tar, vae, and vam represent the amplitudes of the modal coordinate response of the beam, the voltage across the two electrodes, and the voltage across the external coils, respectively. also, the modal mechanical force function is rewritten as follows: ( )2 0 ( ) ( ) l j t j t r ar at rf t f e m w w x dx e = = −  (24) by substituting eq. (23) into eqs. (13), (18), and (21), the following expressions for ve(t), vm(t), and wrel(x,t) are obtained: 2 2 1 2 2 2 2 1 1 ( ) (2 ) ( ) 1 ( ) ( ) (2 ) ( ) (2 ) m e j tmer ar r r r r e mer emr mer mmr c m me me r e r rr r r r r r j f e j v t j j j c n r j j                          =   = =  − +  =        + + + −       − + − +           (25) multi-objective optimization and performance analysis of bimorph... 675 2 2 1 2 2 2 2 1 1 ( ) (2 ) ( ) ( ) ( ) (2 ) ( ) (2 ) j tmmr ar r r r r m mmr mmr mmr emr m c em em r rr r r r r r j f e j v t j j r j l n j j                        =   = =  − +  =        + + + −       − + − +           (26) 2( ) ( ) ( ) ( , ) ( ) c m em e rel r mm r r j l v t v t w x t w x j    + − = (27) where the parameters nme and nem represent the ratios of vm(t) to ve(t) and vice versa, respectively, have been obtained as follows: 31 31 31 31 1 ( ) ( ) 1 ( ) ( ) m e m m m c m em r em me m m em e m me m c f nr j c j e rv n n v j f nr e r j l           + −     = = =  − +  (28) finally, by having the expressions for ve(t) and vm(t), the analytical relationships for the harvested electrical power pe(t), the harvested magnetic power pm(t), and the total harvested magneto-electric power pme(t) by the mee energy harvesting beam can be obtained. these equations provide a quantitative understanding of the energy harvesting performance of the mee beam. they can be used to optimize their design and operation for various applications in wireless sensor networks and biomedical implants. 2( ) ( ) 2 e e e v t p t r = (29) 2( ) ( ) 2 m m m v t p t r = (30) ( ) ( ) ( )me e mp t p t p t= + (31) 2.5. multi-objective optimization of bimorph mee harvesters by new method of hho algorithm optimization techniques are mathematical tools used to find the best solution for a given problem within a set of constraints. these techniques are widely used in various domains, such as engineering, economics, biology, and artificial intelligence. they help solve complex problems by finding the optimal solution that maximizes or minimizes a specific objective function. optimization techniques have become increasingly popular in industries and academia due to their ability to provide efficient and effective solutions to problems. one such innovative approach is the hho algorithm, a population-based method inspired by the hunting behavior of harris’s hawks. it can be used for various optimization problems by utilizing appropriate formulas. the hho simulates the behaviors of harris’s hawks in hunting scenarios to optimize solutions. the algorithm involves exploration and exploitation stages based on the prey’s energy levels. different strategies are modeled to mimic the prey’s escape patterns and the hawk’s pursuit behaviors [27]. the principles of the hho algorithm are summarized and illustrated in fig. 2. 676 m.m. shirbani, s.e. alavi fig. 2 flowchart of the hho algorithm for the determination of the best design variables in the following, the bimorph mee harvesters have been optimized using the novel algorithm of hho and multi-objective optimizations. five different multi-objective optimization cases, as introduced in the chart displayed in fig. 3, are specified on the plots to determine the best performance of the energy harvesting device. fig. 3 five different multi-objective cases and corresponding functions optimized by the hho method multi-objective optimization and performance analysis of bimorph... 677 also, the geometric and electrical design variables and their limits in terms of optimization are listed as follows: 6 100 10 & 1 & 10 10 h m e m l mm b mm h h mm r r n       (32) 3. results and discussion this section presents the numerical results of the optimal connections for the series and parallel configurations of the energy harvesting beams. table 1 provides the values of the properties of the harvesting layers. it should be noted that a composite material consisting of batio3-cofe2o4 with a volumetric percentage of 50% is selected as the active layer, while aluminum is chosen as the homogenous layer. batio3-cofe2o4 is a composite material with potential applications in energy harvesting due to its piezoelectric and magnetic properties. table 1 properties of the mee energy harvester layers 11 hc (nm-2) 75×109 g33 (nsv-1c-1) 2600×10-12 11 mc (nm-2) 225×109 μ33 (ns2c-2) 90×10-6 e31 (cm-2) -3.5 ρh (kgm-3) 2707 f31 (na-1) 350 ρm (kgm-3) 5550 h33 (c2n-1m-2) 6.3×10-9 ζ 0.01 during the continuation of the analysis, the results of multi-objective optimization of the mee energy harvester in series and parallel connections using the hho algorithm were extracted and presented in tables 2 and 3, which will help analyze the performance of the energy harvesting device and make informed decisions to optimize its efficiency. also, as an example, in figs. 4a and 4b, the multi-objective optimization results and the pareto front for the first optimization case are shown. in these figures, points a, b, and c are optimal in this research phase. the power of pm is maximized for the series connection at point a, while the power of pe is minimal. neither of the two harvested powers is maximized at point b, but their values are in the desired optimal state. at point c, the power of pe is maximized, while the power of pm is almost minimized. point c was selected as the optimal point based on the maximum total power pme criteria. in fig. 4b, point a was selected for parallel connection as the result of optimization due to the maximum of both electrical powers. other optimized variables have been obtained with another similar method and are listed in tables 2 and 3. 678 m.m. shirbani, s.e. alavi fig. 4 multi-objective optimization results and the pareto front for the first case of optimization case; (a) series connection, (b) parallel connection table 2 optimal multi-objective optimization variables for series connection b (mm) l (mm) hh (mm) hm (mm) n rm (ω) re (ω) series layers 5.61 90.50 0.857 0.143 6 0.008733 107.992 case1 8.80 92.72 0.878 0.122 10 0.0156 95.568 case2 8.92 90.01 0.872 0.128 9 0.0212 126.744 case3 8.63 89.12 0.888 0.112 7 876.35 31681.40 case4 9.91 98.81 0.857 0.143 4 51.48 34750.10 case5 table 3 optimal multi-objective optimization variables for parallel connection after obtaining the values of the optimized variables with the help of eq. (25), the frequency response of the voltage ve generated by each energy harvesting beam connection is plotted against the dimensionless excitation frequency η, considering the vicinity of their respective first natural frequencies. the results are depicted in fig. 5, where fig. 5a corresponds to the series connection, and fig. 5b corresponds to the parallel connection. the parameters of the electrical circuits of the two-terminal electrodes and external coils, as well as the geometric values of the harvesting beams, are selected based on the values obtained from tables 1 and 2 to plot these results. these figures show that the maximum voltage ve is generated near the first natural frequency of the beam harvester. considering fig. 5a, the maximum value of ve for the series connection in the second optimization case is measured as 0.808 vs2/m. this value is notably higher than the minimum ve generated in the third case, indicating that the second optimization case can b (mm) l (mm) hh (mm) hm (mm) n rm (ω) re (ω) parallel layers 8.61 89.74 0.898 0.102 5 18.17 15940.02 case1 9.53 91.11 0.857 0.143 7 0.0156 95.568 case2 10 95.04 0.896 0.104 8 0.0212 126.744 case3 10 92.01 0.900 0.100 6 651.52 10011.50 case4 6.91 96.41 0.900 0.100 8 10.13 10020.31 case5 multi-objective optimization and performance analysis of bimorph... 679 create a significantly higher ve. similarly, fig. 5b demonstrates that the maximum optimal value of ve for the parallel connection in the second optimization case is measured as 0.5771 vs2/m. this configuration generates up to 44.67% higher voltage ve compared to the minimum optimization case, which is the third one. furthermore, it is observed that the maximum voltage occurs for excitation ratios less than η = 1. the optimal excitation ratios for the series and parallel connections are measured as 0.99904 and 0.99944, respectively. this phenomenon implies that the electrical circuits connected to the energy harvesting beams increase the damping ratio of the beams. this excitation ratio is defined as the damping ratio of the harvesting beams in the presence of electrical coupling, denoted as ηdam. utilizing the relation ηdam = (1-ζelec 2)0.5, where ζelec represents the electrical damping ratio of the harvesting beams, the values of ζelec are determined as 0.044 and 0.033 for the series and parallel connections, respectively. the series connection, characterized by a lower electrical damping ratio, exhibits a higher maximum voltage ve, indicating the advantageous utilization of coupled electrical circuits for this configuration. fig. 5 frequency response of the generated voltage ve over the electrodes in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers by using eq. (26), the frequency response of the voltage vm is plotted for both the series (fig. 6a) and the parallel connections (fig. 6b). the maximum values of vm are calculated as 0.025 vs2/m and 0.0212 vs2/m for the series and parallel connections, respectively. these values represent a 23% difference in the better performance of the series connection. both of these maximum values occur in the third case. furthermore, the maximum values of vm are generated at the excitation ratios of 0.99986 and 0.99957 for the series and parallel connections, respectively. by utilizing the relation ηdam = (1-ζmag 2)0.5 and the excitation ratios obtained from the results of fig. 6, the magnetic damping ratio of the harvesting beams, ζmag, can be determined. the values of ζmag for the series and parallel connections are measured as 0.017 and 0.029, respectively. the series connection, characterized by a higher magnetic damping ratio, exhibits a lower maximum voltage vm. in the following, the frequency responses of currents ie and im are obtained using ohm’s law and eqs. (25) and (26). then, the frequency response of ie is plotted for both series and parallel connections in figs. 7a and 7b, respectively. the optimal values of ie are obtained 680 m.m. shirbani, s.e. alavi for the series and parallel connections in the fifth and third optimization cases, respectively, as 22.321 μas2/m and 37.328 μas2/m. these values signify a 71.72% better parallel connection performance than the series connection. as a result, the parallel connection is essential for designing a new harvesting device with the highest possible amount of current, ie. fig. 6 frequency response of the generated voltage vm over the external coil in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers fig. 7 frequency response of the generated current ie over the electrodes in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers according to fig. 8a, the maximum value of im for the series connection in the fifth optimization case is 200.903 μas2/m. this value indicates an increase of about 21.55 times compared to the first case. also, fig. 8b illustrates that the maximum amount of im in the parallel connection occurs in the first optimization case, measuring 480.71 μas2/m. this value is 20.83 times higher than the minimum value in the fourth case. also, the values multi-objective optimization and performance analysis of bimorph... 681 obtained for the maximum values of im show that the parallel connection produces 1.39 times more electric current im . fig. 8 frequency response of the generated current im over the external coil in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers the comparison of different connection’s performance examines how the arrangement of mee layers, geometric dimensions, and electrical circuit parameters affect the harvested power by the electrodes pe and external coils pm. these results aid in selecting a more suitable and improved structure for these harvesters. the frequency response of pe is plotted for both series and parallel connections in figs. 9a and 9b. the optimal amount of pe for the series connection occurs when the fifth optimization case is implemented. it has been observed that the maximum value of pe will be attained in the third case for the parallel connection. the power values achieved are 8.66 μws4/m2 and 7.453 μws4/m2, respectively, indicating that the series connection produces around 16.19% more power pe than the parallel connection. as a result, the series connection is preferred for achieving maximum generated power pe. the results related to the frequency response of the harvested power by the external coils pm are extracted in the next step. the displayed results in fig. 10a for the series connection still emphasize the significance of the fifth optimization case, which exhibits a remarkably more significant difference than other cases. this difference is so substantial that it indicates an 8.43 times difference with the fourth optimization case. the results for the parallel connection in fig. 10b depict the superior performance of the first optimization case and a 77.07% difference with the second optimization case. the comparison between both connections demonstrates that the parallel connection performs better in generating 102.31% more harvested power pm. 682 m.m. shirbani, s.e. alavi fig. 9 frequency response of the harvested power pe by the electrodes in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers fig.10 frequency response of the harvested power pm by the external coil in five cases of optimization; ) a) series connection, (b) parallel connection of bimorph mee layers as mentioned, the ultimate goal in the design of harvesters is to achieve the maximum total harvested power. to this end, the total mee harvested power pme results for both connections are depicted in figs. 11a and 11b, respectively. the maximum values of pme have been measured for the series and parallel connections as 9.701 μws4/m2 and 7.833 μws4/m2, respectively. the highest value of pme in the series connection, achieved in the fifth optimization case, is 23.85% higher than the value obtained in the first case for the parallel connection. these results show the importance of series connection to achieve the highest total harvested power pme, which is the most crucial goal of a harvester. multi-objective optimization and performance analysis of bimorph... 683 fig. 11 frequency response of the total mee harvested power pme in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers in the final step, using eq. (27), the tip relative displacement of the beam wrel(l) is calculated for both connections. the frequency responses of wrel(l) are then depicted in figs. 12a and 12b, respectively. one of the reasons for examining this parameter is its fundamental role in the design of the harvester’s volume. in other words, besides the dimensions of the beam, a specific range should be left empty for the beam’s deflection so that the existing vibrations in the environment can be fully utilized. the results in figs. 12a and 12b reveal that both connections exhibit similar maximum relative displacement changes in the fifth case. the maximum wrel(l) values for the series and parallel connections are 132.617 μms2/m and 131.457 μms2/m, respectively. the choice of different objective functions in different optimization cases has led to 47.95% and 37.55% greater deflection wrel(l) in the series and parallel connections, respectively. fig. 12 frequency response of the relative tip displacement wrel(l) in five cases of optimization; (a) series connection, (b) parallel connection of bimorph mee layers 684 m.m. shirbani, s.e. alavi for a better comparison, the maximum values extracted from the plotted graphs in tables 4 and 5 are provided for both connections, respectively. in the fifth optimization case, where the maximum power pme occurs, the contribution of pe to the total power pme is 89.33% for the series connection. in this case, the optimal maximum value of the current im is 9.01 times the generated current, ie. in the first optimization case of the parallel connection, the contribution of electrical power pe to the maximum total harvested power pme is 73.16%, indicating a larger share of the circuit connected to the external coils in power generation compared to the series case. the optimal maximum value of im is 17.93 times the amount of the ie, which is a significant amount for producing electrical current in a harvesting system. table 4 optimal values of the objective functions in the series connection case5 case4 case3 case2 case1 series layers 9.701 5 2.9258 6.922 3.902 pme (µws4/m2) 8.666 4.88 2.619 6.605 3.866 pe (µws4/m2) 1.038 0.123 0.031 0.0312 0.003561 pm (µws4/m2) 22.321 17.56 19.967 16.354 10.874 ie (µas2/m) 200.903 16.812 30.277 27.99 8.906 im (µas2/m) 0.7759 0.5561 0.2622 0.808 0.7104 ve (vs2/m) 0.0103 0.0146 0.025 0.0223 0.007978 vm (vs2/m) 132.617 93.969 89.639 108.862 91.859 wrel (µms2/m) table 5 optimal values of the objective functions in the parallel connection 4. conclusions this study comprehensively investigates the multi-objective optimization of a mee energy harvester operating in both series and parallel connections. the main goal is to maximize the total harvested power by considering various objective functions, such as harvested power, voltage, and current. the study uses the hho algorithm and looks at points on the pareto fronts for each connection and objective function. the best candidate in each iteration is considered the target or a solution close to the optimum in the hho algorithm. the algorithm randomly places hawks in different places based on specific identification strategies. the resulting equations describing the beam’s mechanical displacement response, the harvested powers, voltages, and currents were formulated and presented according to the specified parameters. the frequency response of the bimorph configuration of the mee energy harvesting system under harmonic excitation was case5 case4 case3 case2 case1 parallel layers 7.833 6.578 7.716 1.258 2.455 pme (µws4/m2) 5.731 5.392 7.453 1.099 1.436 pe (µws4/m2) 2.1 1.186 0.2606 0.157 1.017 pm (µws4/m2) 26.814 18.684 37.328 4.688 5.365 ie (µas2/m) 480.71 151.318 24.489 22.024 451.033 im (µas2/m) 0.4273 0.5771 0.3989 0.4688 0.5367 ve (vs2/m) 0.00873 0.0156 0.0212 0.0143 0.00431 vm (vs2/m) 107.992 95.568 126.744 104.142 131.457 wrel (µms2/m) multi-objective optimization and performance analysis of bimorph... 685 investigated after finding and using the 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profile of axisymmetric indenter due to the initially displaced dual-motion fretting wear facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 55 61 original scientific paper limiting profile of axisymmetric indenter due to the initially displaced dual-motion fretting wear  udc 539.3 qiang li berlin institute of technology, department of dynamics and tribology, germany abstract. recently the final worn shape of elastic indenters due to fretting wear was analytically solved using the method of dimensionality reduction. in this paper we extend this model to dual-motion fretting wear and take into account that the indenter is initially pressed with constant indentation depth and moved horizontally with constant displacement. two key parameters, the maximal indentation depth during oscillation and the stick area radius in the final state as well as the liming shape of indenter are analytically calculated. it is shown that the oscillation amplitudes and the initially indented or moved displacements have an influence on the final shaking-down shape. key words: fretting wear, dual-motion, tangential force, oscillation 1. introduction fretting wear is a surface destruction process in the frictional contacts subjected to oscillating load with small amplitude [1]. this phenomenon occurs very often in the vibrating connections of mechanical elements, such as clamping devices, interference fit joints, gear or bearing contacts and electrical connectors, etc. [2-4]. fretting leads to material loss, crack formation as well as fatigue failure [5]. in the last few decades many experimental and theoretical investigations have been intensively carried out to understand this process, for example, by using the finite element method or that of the boundary element [6, 7]. however, there are still some unsolved basic problems, especially under complicated loading [8]. recently, a new method known as that of dimensionality reduction (mdr) was applied to analyzing the process of fretting wear as well as its final ‘shake down’ state for arbitrary axisymmetric shape of elastic or viscoelastic indenter [9-11]. in received january 4, 2016 / accepted march 1, 2016 corresponding author: qiang li technical university of berlin, 10623 berlin, germany e-mail: qiang.li@tu-berlin.de 56 q. li paper [9] a general theoretical solution of the limiting profile due to fretting wear was given for an arbitrary axisymmetric indenter. for the case of an elastic indenter under the tangential oscillation [10], a rapid numerical procedure based on the mdr was later developed to simulate the wear process, and its results for the final state of wear also verified the solution in [9]. furthermore, a similar mdr-based procedure was suggested for a gross-slip wear problem and the results are exactly same to the solution obtained by the full fem formulation, and it is for several orders of magnitude faster than the fem [12]. fretting wear of viscoelastic indenters was analyzed in papers [13] and [14], where the analytical solution of limiting profile due to dual-motion oscillation was presented in [13], and the numerical simulation of fretting wear under the tangential oscillation was carried out in [14]. these final worn shapes for spherical indenters under multiple-mode fretting conditions have been validated by experimental investigation [15]. till now most work focuses on fretting wear only under the tangential oscillation and less on dual-mode fretting. in this paper, we consider the fretting wear of elastic indenter oscillating in both tangential and normal directions, and take into account the factor that the indenter has initially constant displacements in both normal and tangential direction. 2. wear criterion in fretting contact this paper is an extension of the solutions in [9]; therefore, firstly we give a very brief discussion of wear condition in [9]. we consider a contact between a rigid axis-symmetrical body and an elastic half space. under the normal load the indenter is pressed into the half space and then oscillates tangentially. it is known that, if the oscillation amplitude is small enough, there will be an annular slip-zone generated at the boundary of contact area and a circular stick-zone at the inner area, as illustrated in fig.1. fig. 1 schematic representation of the stick-slip area in fretting contact the stick and slip condition can be determined by the classic amontons’ law: if tangential stress  is smaller than normal pressure p multiplied by a constant coefficient of friction µ, τ < µp, the surfaces of contact bodies stick together, and in the slip region the tangential stress remains constant and equal to product µp: , in stick area , in slip area    p p     (1) according to the reye-archard-khrushchov wear law [16, 17], the wear volume is proportional to the normal force (or pressure), the relative tangential displacement and of contacting bodies and reversely proportional to the hardness. from this law, the wear in the limiting profile of axisymmetric indenter due to initially displaced dual-motion fretting wear 57 local contact area vanishes when the normal pressure becomes zero or there is no relative displacement between two bodies. as described in [9], this no-wear condition can be written as: either 0 no wear condition: or 0x p u     (2) in the process of fretting wear, the surfaces in the stick area have no relative displacement, so that no wear occurs in this contact area during the whole process. due to slip at the boundary wear occurs in this area, but the normal pressure will reduce to zero finally; therefore, there is no wear any more in this local contact area in the final state. in this paper we analyze this limiting profile of indenter. 3. solution for pre-stressed dual-motion problem the analytical solution of limiting profile in [9] was obtained based on the method of dimensionality reduction (mdr). using this method the three-dimensional normal and tangential contact problems for axis-symmetric bodies can be mapped into one-dimensional contact with a properly defined foundation [18-21]. according to the rules of the mdr, three-dimensional pressure distribution p(r) can be calculated from the profile of one-dimensional indenter g(x): * 2 2 ( ) ( ) d      r e g x p r x x r . (3) from no-wear conditions, eq. (2), it follows that there are two parts in the contact areas in the final state: in the inner contact area with radius c no wear occurs because of no relative displacement δux=0, so that the final profile keeps its initial form g∞(x) = g0(x) for r < c; at boundary r > c the pressure in the final state reduces to zero, p(r) = 0. from eq. (3), p(r) = 0 means that g′(x) = 0 and g(x) = const for c < x < a and the value of const is equal to maximum indentation depth dmax achieved during the whole oscillation process. thus, the one-dimensional mdr-transformed profile in the final shakedown state has the form 0 max ( ), for 0 ( ) , for g x x c g x d c x a        (4) according to the reverse transformation in the mdr, the three-dimensional limiting shape can be calculated as [9] 0 0 max 2 2 2 2 0 ( ), for 0 ( ) ( )2 2 1 , for c r c f r r c f r g x dx d dx c r a r x r x               . (5) eq. (5) gives the solution for limiting shake-down-state shape of the indenter. given an initial three-dimensional profile of indenter, its limiting shape can be determined if the two parameters are known: radius c of the stick area in the limiting state and maximum indentation depth dmax. in the following we discuss how these two governing parameters can be determined in our pre-stressed dual-motion problem. 58 q. li such a contact is taken into consideration. the indenter is pressed into an elastic half-space with an indentation depth d0, and moved horizontally with a distance x0, then oscillates harmonically according to (0) 0 0 sin( )z z zd d u d u t      (6) in vertical direction and (0) 0 sin( ) x x xu x u t (7) in horizontal direction.  is phase shift between normal and tangential oscillations. this movement is illustrated in fig. 2. here we consider small amplitude of oscillations under the assumption that uz (0) < d0 and ux (0) < x0, and all these four parameters are positive. now we calculate the two important parameters. (1) the maximum indentation depth. this one can be easily obtained by eq. (6): (0) (0) max 0 0 ( ) max{ sin( )}z z z t d d u t d u      . (8) fig. 2 illustration of dual-motion of the indenter (2) radius c of the stick area. according to eq.(1), radius c can be determined by the condition that tangential force kxux(c) of springs at each time moment is smaller than or equal to coefficient of friction µ multiplied by normal force kzuz(x): * (0) * (0) 0 0( sin( )) ( sin( ) ( ))x x z zg x x u t e x d u t g c            (9) solving this inequality with respect to g(c) gives * (0) (0) 0 0* ( ) sin( ) ( sin( ))z z x x g g c d u t x u t e          (10) or * (0) (0) 0 0*( ) ( ) min sin( ) ( sin( ))z z x x t g g c d u t x u t e               (11) from eq.(11), it can be seen that the value of g(c) is dependent on phase shift . if it is not fixed, that means phase shift  is not constant but changes all the time, then g(c) has a very simple and general form * (0) (0) 0 0* ( ) ( )z x g g c d u x u e     . (12) limiting profile of axisymmetric indenter due to initially displaced dual-motion fretting wear 59 however, if the phase shift is constant, the solution of eq. (11) is not easy to calculate. here we consider only a special case of same oscillation frequencies: ωx = ωz = ω. solving the eq. (11) gives 2 * * * 2(0) (0) (0) (0) min 0 0* * * ( ) 2 cosz z x x g g g g c d x u u u u e e e                . (13) now the two parameters, radius c of the stick area in the limiting state and maximum indentation depth dmax are obtained. substitute eqs. (8) and (13) to the limiting profile eq. (5), then the three-dimensional limiting shape of the indenter can be calculated. it is seen that the two parameters as well as the limiting shape depend on oscillation amplitudes uz (0) and ux (0) , phase shift  between normal and tangential movements, and also the initial pre-indented and –displaced distance d0 and x0. radius c of the stick area is briefly discussed here. from eq.(13), the smallest stick radius is given when phase shift  = π, and the value are the same to (12) in the case of no-fixed phase. the maximum stick radius (minimum wear volume) is realized at  = 0: * * (0) (0) 0 0* * ( ) z x g g g c d x u u e e      . (14) if the phase  = ±π/2, the stick area is given by 2 * * (0)2 (0)2 0 0* * ( )           z x g g g c d x u u e e  . (15) it is noted that these results of the stick area as well as the related limiting profile are independent of the frequencies of normal and tangential oscillations. with an example of parabolic indenter we show how the limiting profile can be calculated in the case of pre-displaced dual-motion. the one-dimensional profile of spherical indenter with radius r is given by g(x) = x 2 /r [18]. from eq. (8) the maximal indentation depth is equal to dmax = d0+uz (0) . if the phase shift between normal and tangential oscillations is  = π, then the stick radius is calculated by eq. (12) as c 2 /r = d0–uz (0) – (x0+ux (0) )g * /(µe * ). substituting these two parameters dmax and c into basic solution, eq. (5), the final profile in this case is then obtained. an example of this final shape is shown in fig. 3. the worn shape of the indenter can be clearly seen, that is, this part lies in the slip area and the strongest wear is almost in the middle of the slip region. -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 r f( r) initial profile limiting profile fig. 3 an example of limiting shape of parabolic indenter 60 q. li 4. conclusion we extended the basic solution of limiting shape of axis-symmetric profiles due to fretting wear in paper [9] to the case of pre-stressed dual-motion fretting wear. it means that the indenter is pressed into the half space with initial indentation depth and initial tangential displacement; it oscillates in both vertical and horizontal directions. the emphasis of the analysis is placed on two parameters – the maximum indentation depth during the oscillation process and the radius of the stick area in the final state, which determine the limiting shape of worn profile according to basic analytical solution in [9]. for the particular case in this paper we derived and obtained the relation of these two parameters, and it is shown that they depend on the oscillation amplitudes, the phase shift between normal and tangential movements, as well as on the initially indented and displaced distance. especially the different phase shift between normal and tangential oscillations for the same frequency will result in a different size of the stick area as well as a different limiting profile. with an example of parabolic indenter oscillating on a half space, we present its final worn shape. the worn area is clearly observed and the volume of material loss can be further calculated by comparison with the initial shape of profile. acknowledgements: the author thanks v.l. popov for valuable discussions. references 1. kennedy, p., peterson, m.b., stallings, l., 1982, an evaluation of fretting at small slip amplitudes. in: materials evaluation under fretting 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dreidimensionaler kontakte auf systeme mit niedrigerer räumlicher dimension, cuvillier-verlag, göttingen. 21. heß, m., 2012, on the reduction method of dimensionality: the exact mapping of axisymmetric contact problems with and without adhesion, phys. mesomech., 15, pp. 264–269. facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 249 259 https://doi.org/10.22190/fume180424019n © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dynamical behavior of a heat pump coaxial evaporator considering the phase border’s impact on convergence udc 621.1 arpad nyers 1 , zoltan pek 2 , jozsef nyers 2,3 1 university of pécs, faculty of engineering and information technology, hungary 2 obuda university, doctoral school of applied informatics and applied mathematics, hungary 3 szent istván university, mechanical engineering phd school, hungary abstract. using a dynamical mathematical model, we investigated transient behavior of a water-water heat pump’s evaporator. the model consists of time and space dependent partial differential equations of water, pipe wall and refrigerant. mathematically the thermal expansion valve (tev) and compressor are described with lumped parameters and represent the boundary conditions. during the numerical solution of this system of equations the problem emerged of divergence of solutions. it was determined that the cause of the divergence solution was in the location of phase change of the refrigerant. the aim of this paper is, firstly, to display and propose a new approach to the elimination of divergence. in addition, it examines dynamic behavior of the heat pumps’ coaxial evaporator. key words: heat pump, coaxial evaporator, dynamical behavior, phase border, convergence, numerical simulation 1. introduction the evaporator is probably the most important part of the heat pump since it represents the place where heat is recovered; this, to a great extent, influences energy efficiency of the system. the process efficiency is significantly influenced by the amount and quality of injected refrigerant into the evaporator. the injection process must be controlled and for the design of the controller the transient behavior of the evaporator must be known. we tried to take into account all those physical phenomena which significantly influence the process. received april 24, 2018 / accepted june 10, 2018 corresponding author: jozsef nyers obuda university, doctoral school of applied informatics and applied mathematics, budapest, hungary e-mail: jnyers1@gmail.com 250 a. nyers, z. pek, j. nyers several dynamic models of heat pumps have been developed but few of them dealt with the heat pump’s coaxial evaporator. macarthur and grald [1] presented a model of vaporcompression heat pumps. the heat exchangers were modeled with detailed distributed formulations while the expansion device was modeled as a simple fixed orifice. pavkovic et al. [2] dealt with a fully distributed dynamic numerical model of a shell and a tube type of the refrigerant evaporator, suitable for control and design purposes. avoidance of the fraction model is used to account for a slip effect. the solution is achieved by the finite volume method. fu et al. [3] presented a dynamic model of air-to-water dual-mode heat pump with screw compressor having four-step capacities. the dynamic responses of adding additional compressor capacity in a step-wise manner were studied. kima et al. [4] presented a dynamic model of the water heater system driven by the heat pump; the finite volume method was applied to describe the heat exchangers while the lumped parameter models were used to analyze the compressor and the hot water reservoir where dynamic simulations were carried out for various reservoir sizes. rasmussen and alleyne [5] developed an air to air heat pump system using a moving boundary-lumped parameter approach to the heat exchangers. đorđević et al. [6] experimentally studied the heat transfer of the archimedean spiral coil made of a transversely corrugated tube as a heat absorber of the parabolic dish solar concentrator. the results showed enhancement of the heat transfer rate in comparison with the spirally-coil smooth tubes, up to 240%. nyers et al. presented in their earlier works [7-10] the structure and functioning of the heat pump’s evaporator, its dynamic and discrete mathematical model as well as the numerical method for solving the model as well. szamarszkij et al. [11] described in detail the methods for solution of gas dynamics equations. kajtar and kassai [12-15] dealt with computerized simulation of the energy consumption. some further works [16-21] dealt with the same topics as in this paper, including mathematical modeling, numerical procedures, heat pump and heat pump heating-cooling system. in this study we propose a new numerical method for solving the divergence of solution at the phase change border which can also be applied to the most complicated model. the obtained results presented by graphs and diagrams visibly prove the solution improvement. the improvement is achieved by means of the proposed method for calculating the time step. 2. physical model of evaporator constructively, the considered evaporator is a coaxial tube heat exchanger. from thermal aspect, in the evaporator the heat transfer is performed between the refrigerant and the well water. in the observed case, the refrigerant freon r134a flows inside the evaporator pipes while the cooled mass, the well water, flows inside the evaporator’s shell. in the evaporator, the parallel pipes are connected with the baffles. the baffles are placed perpendicularly to the pipes. the pipes bundle at a distance of about 150 mm in the direction of the tube axis. water flows between the baffles approximately as a sinusoid. the evaporator works most efficiently when the full length of parallel pipes is filled with vapor-liquid mix phase of the refrigerant. the quantity of the evaporated refrigerant depends on the compressor capacity, the temperature and the mass flow rate of well water, respectively. the thermo-expansion (tex) valve doses an adequate quantity and quality of the refrigerant into the evaporator. this valve uses the sensors for monitoring dynamical behavior of a heat pump coaxial evaporator considering the phase border’s impact ... 251 temperature and pressure of the outlet refrigerant from the evaporator. based on the measured temperature and pressure, the valve realizes optimal dosing of the refrigerant. in case of a refrigerant quantity deficit, the liquid flows for a shorter length of the evaporator’s pipes while in the remaining part of the evaporator flows the superheated vapor. the heat transfer of the refrigerant's vapor is multiple times lower than of the vapor-liquid mix phase. the refrigerant vapor in the dry superheated section superheats. the refrigerant over-dosing means that the liquid phase cannot evaporate and some quantity of the liquid leaves the evaporator. this phenomenon most reduces the refrigeration effect, i.e. the performance of the heat pumps' evaporator. the non-evaporated liquid evaporates in the compressor and the consequences of the hydro hummer can happen in the compressor. in the correct operation mode the superheated, dry vapor flows out of the evaporator. the level of the superheating is up to 4 °c. the evaporator is connected to the compressor. the compressor sucks out the superheated vapor from the evaporator. the compressor using mechanical work compresses the vapor which primarily increases the temperature on the adequate level. the compression is unfortunately accompanied with an intensive increasing of the vapor pressure as well. fig. 1 the physical model of heat pump's evaporator with the system parameters fig. 2 photo of the heat pump with coaxial evaporator and shell-tube condenser 3. dynamic mathematic model of evaporator in this paper, both the mathematical model and its numerical solution represent an improvement over those reported in our earlier work [7, 10]. the mathematical model is built up with following assumptions:  evaporator consists of horizontal bundled pipes, the refrigerant is distributed uniformly between them, the heat resistance of the pipe wall neglected,  refrigerating flow inside the pipes is homogeneous and one-dimensional,  heat resistance of the pipe wall is neglected, and,  axial heat conduction is neglected everywhere. 252 a. nyers, z. pek, j. nyers the mathematical model built on these assumptions contains the equations of the refrigerant, water, pipe wall, thermal expansion valve tev and compressor. the mathematical description of the refrigerant’s dynamic behavior is the most complicated part of the model. it contains the conservation equations of mass, impulse and energy which are valid in the evaporation region, at the phase change border and in the superheated region as well. conservation equations of mass ( ) 0 , p ρ w 1 , ρ t z v         (1) where p is the pressure, w the velocity, ρ the density, v the specific volume, z the space increment while t denotes the time. conservation equations of impulse ( ) ( ) ( ) 0, 2ρ w ρ w p f x t z           (2) where f(x) is the friction factor in flowing refrigerant as a function of vapor quality, x. the equation of energy conservation yields: ( / 2 ) ( ( / 2 )) ( ) 0 2 2ρ w ρ i p w ρ w ρ i q x , t z                (3) where i is the refrigerant enthalpy and q(x) is the heat flux as a function of vapor quality. equation (3) is written in a divergent form since this is a solid basis for the construction of a reasonable discretization of the equations: it assures the validity of discrete conservation laws [11]. the above equations are complemented by equations of mixture and vapor state, equations of heat transfer, friction coefficients and heat flux. the equations of state differ in the evaporator and the superheated region (the phase change point will be determined by the vapor quality): evaporation region: xo <= x <= xmax vapor-liquid equilibrium equation as a function of pressure, p, and temperature, t: 1( ) 0f p,t , (4) specific enthalpy of refrigerant mixture: ( )t g ti i x i i ,    (5) where the subscripting t and g denote that the quantity is related to liquid and vapor, respectively. specific volume of refrigerant mixture: ( )t g tv v x v v ,    (6) dynamical behavior of a heat pump coaxial evaporator considering the phase border’s impact ... 253 the heat flux between the pipe wall (index c) and the refrigerant (index f) as a function of vapor quality: ( ) ( ) ( )u c fq x α x 4/d t t    (7) where du is the inner diameter. the coefficients of heat transfer and friction as a function of vapor quality are denoted by (x) and f(x), respectively. phase change border x = xmax =1 vapor-liquid equilibrium equation: 1( ) 0f p,t , (8) equation of the superheated vapor state: 2 ( ) 0f p,t, v , (9) fictive equation, (due to the number of equations): 3( ) 0maxf x,x , (10) again, the coefficients of heat transfer and friction as a function of vapor quality are denoted by (x) and f(x), respectively. superheated region: x = xmax =1 equation of superheated vapor state: 2 ( ) 0f p,t, v , (11) equation of superheated vapor enthalpy: 4 ( ) 0f p,t, v,i , (12) fictive equation (due to the number of equations): 5( 1) 0f x, , (13) also, in this case, the coefficients of heat transfer and friction as a function of vapor quality are denoted by (x) and f(x), respectively. the water through the pipe wall transfers heat to the refrigerant. the dynamical heat transient behavior of the water and the wall are described by the energy conservation law: ( ) 0v v v v v c t t w c t t t z           (14) where the subscripting v means that the quantity is related to water, while the sign (+) is used in the case of concurrent flow and the sign (-) in the case of countercurrent flow. ( ) ( ) 0c cv v c cf c f t c t t c t t t          (15) where ccv and ccf are the coefficients of interaction between the tube and water and the tube and the refrigerant, respectively. 254 a. nyers, z. pek, j. nyers the refrigerant flows through the thermal expansion valve tev and the compressor. these components of the system are described using algebraic equations with lumped parameters, as follows: thermal expansion valve’s equation, tev based on darcy – weissbach formula: 2( ) ( ) ( ) 0conp p c t w     (16) where the subscripting con means that the quantity is related to the compressor and c(t) is the throttle coefficient of the tev valve as a function of time. the compressor`s steady-state equation for the piston mechanism reads: 1 /( ) 0 n con c h c c p c a w v n p c            (17) where n is the exponent of polytrophic, 1≤ n ≤ 1.26, cc is the coefficient of clearance volume, a is the surface area of piston, vh the work volume of compressor and nc the number of revolutions of compressor. in practice, the piston speed is high so that the vapor heat loss to the environment is low; therefore, the process of compression can be approximately regarded as an adiabatic, consequently n=1.26. 4. numerical solution method the above mathematical model can be solved only approximately, using a numerical approach. to obtain the corresponding formula, the conservation laws are integrated over small sub regions (cells) of the z, t-domain in which the solution is to be determined. the area integrals can be transformed to line integrals using green’s integral theorem; the line integrals are approximated by simple formula like the trapezium and rectangle rule. weighting parameters are introduced for stability reasons. in this way a system of coupled nonlinear, algebraic equations arises for the solution of which the newton method is applied. the linear algebraic system to be solved in every step of the newton method has a distinct structure; it is of a block-tri diagonal form. this is a result of the discretization described above and of the fact that we write the mass conservation equation not for the density but for the specific volume and then discretize it over the cell {xi ≤ x ≤ xi+1, ti ≤ t ≤ ti+1}, whereas the remaining equations are discretized over the cell {xi-1 ≤ x ≤ xi, ti ≤ t ≤ ti+1}. the block-tri diagonal linear equations are solved by block-gauss elimination. special treatment needs the point of phase change where the vapor quality reaches the value x=1. at this point the approximation is written down not on a rectangular cell but on a triangular one. by appropriate choice of the time step we manage the phase change point to move from time level to time level by at most one spatial interval only (a shift of this point by more than one interval has an adverse influence). the solution of this problem is obtained as follows: in the first iteration on a given time level, using old time step ∆to, the number m of ∆z-intervals is found by which the phase change point moves. dynamical behavior of a heat pump coaxial evaporator considering the phase border’s impact ... 255 fig. 3 phase boundary displacement after ot time the solution of this problem is obtained as follows: in the first iteration on a given time level, using old time step ∆to, the number m of ∆z-intervals is found by which the phase change point moves. the number of space intervals m, zm z1 t t ,k-km o t        (18) m t t o   (19) where k is the number of increments and kt is the number of increment at phase boundary when the vapor quality x=1. ∆t is taken as a new time step for the next iteration. when the iteration on a fixed time level has converged, ∆to is taken as the first time step on the next time level since in our problem, after a period of considerable changes in the solution, convergence to a new stationary state follows. in that latter r time interval, the phase change point settles down and time step (much smaller than ∆t is not necessary for reaching sufficient accuracy) but would mean a waste of computing time. 5. results and discussion using real-life physical data we simulated the evaporation in a pipe, taking into account the different processes listed above. the evaporator consists of a 13x10/8 mm copper pipe of 10 m length. the compressor supplies 10.m f  kg/s r134a refrigerant whereas the mass flow of water is 50.mv  kg/s. the system’s transient behavior is affected by a jump in the water temperature for 4°c (from 14°c to 6°c) or for +6°c (from 14°c to 20°c). the transient processes induced by these temperature changes are well depicted by our computations, see figs. 4-7. also visible is the advantage of the described time step choice over an arbitrarily specified time step, especially if looking at the computed refrigerant temperatures. in the case of an arbitrary time step the temperature at the phase change point attains too low or high values and oscillates in the superheated region, significantly differing from reasonable values; see the figs. 4 (-4°c) and 6 (+6°c) displaying the tf values. this non-physical effect practically disappears if using the proposed time step choice; see figs. 5 (-4°c) and 7 (+6°c). 256 a. nyers, z. pek, j. nyers fig. 4 refrigerant temperature changes in time and pipe length affected by water temperature jump of -4 °c (from 14 °c to 10 °c). in the superheated section no natural oscillations are visible because of the wrong time step. fig. 5 refrigerant temperature changes in time and the pipe length is affected by a water temperature jump of -4 °c (from 14 °c to 10 °c). in the superheated section the unnatural oscillations disappear as a result of the appropriate time step. dynamical behavior of a heat pump coaxial evaporator considering the phase border’s impact ... 257 fig. 6 refrigerant temperature changes in time and the pipe length is affected by water temperature jump of +6 °c (from 14 °c to 20 °c). in the superheated section unnatural oscillations are visible because of the wrong time step. fig. 7 refrigerant temperature changes in time and the pipe length is affected by water temperature jump of +6 °c (from 14 °c to 20 °c). in the superheated section unnatural oscillations disappear as a result of the appropriate time step. 6. conclusion in this section some remarks about the determination of the phase change point are given. without an accurate calculation of the location of this point no numerical convergence could be obtained in the newton method combined with the block-gauss elimination for linear systems. in our model the phase change point was determined by vapor quality x=1. the choice of the mathematical model was also influenced by our experiments with solving this problem; finally, as reported above, we have found it necessary to separate the evaporation region and 258 a. nyers, z. pek, j. nyers the superheated region and to write at the phase change point the equilibrium equation as well as the state equation of the superheated gas together with the other leading equations. using the recommended algorithm it is possible to determine the appropriate time step to ensure the convergence of solution. in the given case the convergence time step was 0.5s. as mentioned, the appropriate time step was obtained iteratively by using eq. (19). the advantage of the recommended method is that it is very simple and efficient. the disadvantage is that the appropriate time step cannot be calculated in an exact mathematical manner. rather, it can be defined only numerically in real time using the recommended iterative procedure. the diagrams in figs. 4-5 show the evaporator dynamic behavior for -4 °c unit jump of water temperature. as a response: the refrigerant input temperature decreases from 10.3 °c to 7.2 °c, the output vapor temperature barely varies whilst the evaporation length significantly reduces from 7.3m to 5.7m. the time duration of the transient process is 10 s. in the case of water temperature increase of +6 °c (fig 6-7), the refrigerant's input temperature increases from 10.5 °c to 12.6 °c, the vapor output temperature barely varies while the evaporation length reduces from 8.6 m to 7.5 m. references 1. macarthur, j.w., grald, e.w., 1989, unsteady compressible two-phase flow model for predicting cyclic heat 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series: mechanical engineering https://doi.org/10.22190/fume220903003a © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper experimental and numerical investigation on the performances of a small weapon barrel during its lifecycle walid boukera abaci1, nebojsa hristov1, igor radisavljevic2, lazar stojnic1, aleksa anicic3 1university of defense in belgrade, military academy, belgrade, serbia 2military technical institute, belgrade, serbia 3proof house for experimental testing, kragujevac, serbia abstract. the paper presents experimental and numerical investigations on the performances of a small caliber rifle barrel during its lifecycle. two 7.62 mm rifle barrels were used, the first was considered as a standard barrel, the second barrel was subjected to an accelerated life cycle test. measurements of the muzzle velocity, the rate of fire and the firearm accuracy and precision were carried out. the paper presents the correlation between the measured parameters and the total number of shots fired. after the durability tests, longitudinal cross sections were made by cutting the tested and the standard barrels. 3d scanning was employed to perform a comparison between the tested and the standard bore surfaces. ansys explicit dynamic analyses were performed based on the obtained surface scans. the numerical analyses results of the tested and the standard barrels showed good agreement with the experimental and the numerical internal ballistic model results. key words: accelerated lifecycle tests, muzzle velocity, rate of firing, accuracy, precision, bore damage, 3d scanning, explicit dynamic analysis 1. introduction the performance of weapon barrels decreases significantly during their usage. during the firing, the barrel is subjected to high thermo-mechanical stresses caused by the highly pressurized combustion gases and the interaction with the accelerated projectile. for the brief time that the firing process takes the barrel will be affected by the chemically reactive gases at elevated pressures and temperatures. inevitable damage will occur to the internal surfaces that progress with repetitive usage of the barrel. those damages are known as the received: september 03, 2022 / accepted: january 15, 2023 corresponding author: nebojsa hristov university of defence belgrade, military academy belgrade, veljka lukića kurjaka 33, 11042 belgrade e-mail: (nebojsahristov@gmail.com) 2 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic wear and erosion of the barrels. unwanted phenomena that lead to the degradation of the barrel performances. the exterior ballistic process highly depends on the initial directional and rotational muzzle velocities. internal surface damage will affect the projectile surface topography and the initial velocities which will affect the bullet’s aerodynamic parameters and induce a significant loss in the accuracy and precision of the rifle. much research has been established to study and predict the wear and erosion of the barrels experimentally and numerically [1-7]. jaramaz et al. [8] have developed a laboratory apparatus and a method for determining the erosion coefficient of gunpowder both experimentally and theoretically. abhilash et al. [9] and johnston [10] summarized in their reviews the various studies on erosion taking place in the weapon barrel and gave a list of various methods developed over the years to calculate and mitigate the barrel wear. new research has been conducted to study the bore damage of the barrel with the use of the new numerical advanced analysis tools to accurately predict its effect. shen et al. [11] studied the influences of bore damage on the bullet-barrel interaction process and the end of a 12.7 mm machine gun barrel's service life. based on the damage data obtained through barrel life tests they developed a novel finite element mesh generation method for the damaged barrel and a new transient coupled thermo-mechanical finite element (fe) model. based on the barrel's accelerated life test, and using numerical simulation technology and the ballistic performance change at each shooting stage, li et al. [12] reproduced the whole process of projectile shooting and established a relation between the angle of attack of the projectile and the wear of the barrel. they showed that the projectile muzzle spin rate decreases 57.5% than that of a barrel without wear. xiaolong et al. [13] also analyzed wear mechanism at different positions in the bore and established a three-dimensional barrel-wear finite element model using a parametric modeling method and concluded that the fundamental reason for interior ballistic performance degradation was wear on the initial position of the rifling. ding et al. [14] provided a new finite element meshing strategy for the worn barrel and a new parametric geometric modeling method for gun barrel using python code and abaqus software. by considering the gases propelling force and the changing friction stress effect, the user subroutine vuamp and vfriction were implemented to develop a transient coupled thermo-mechanical fe model to compute the plastic deformation of the rotating band and the performance of interior ballistics. in order to investigate the small caliber rifle performances during the course of its lifecycle, durability tests were conducted to accelerate the barrel lifespan, measures of the initial velocity, the rate of firing and the firing accuracy and precision were taken regularly. the internal surfaces of the used and standard barrels were obtained using 3d scanning. the obtained results were used to perform explicit dynamic analyses using ansys software. 2. experiments two 7.62 mm automatic rifle barrels were used. the first one is a new barrel and is considered as a standard barrel. the second one is a used barrel. it was used for 8000 rounds before performing a durability test. the durability test consisted of firing a large number of shots in different regimes using a standard nitrocellulose propellant. the experiments were performed with the standard atmospheric conditions. after every 1000 shots, a series of performance measurements were made: experimental and numerical investigation on the performances of a small weapon barrel... 3 the initial velocity of the v0 on a group of 10 shots (3 x 10 shots) the rate of firing on a group of 60 shots (2 x 30 shots) accuracy and precision tests on 3 targets in a group of 10 shots per target at a distance of 100 m. during the durability test, 13,860 rounds were fired (a total of 21,900 rounds were fired from the used rifle) of ammunition 7.62 × 39 mm. the velocity v0 was measured using a ballistic chronograph placed 2.5 m from the muzzle. the rate of fire was determined using a measuring system based on optical and sound sensors to register the cadence. the device is an integral part of the laboratory equipment for testing ballistic characteristics in the zastava arms test center. the system is equipped with sound and light sensors to be able to detect the sound produced by the shot, as well as the flash from the muzzle of the weapon when fired. the time between two firing was measured with an accuracy of 1 μs. the measurement results were collected using a specific acquisition system. the accuracy and precision of the barrels of the rifle are tested under favorable atmospheric conditions in a covered area of the shooting range protected from the wind and outside influences. before the start of the test, parts that could affect shooting accuracy were replaced and the weapon was cleaned and then inspected. precision and accuracy were tested by shooting 10 bullets of the same series and original packaging for each of the three targets. targets were placed at a distance of 100 m with sight. the shooting was performed by an excellent sitting down shooter using a tactical riflescope. the aiming point was the center of the black disk of the target that should be on the weapon horizon. after shooting, the target (fig. 1) was inspected and the coordinates of the scatter were collected. fig. 1 one of the three targets after shooting 10 shots at 17000 rounds 3. results of the experiments the durability test results are summarized in table 1. 4 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic table 1 durability test results n x 103 0 8 9 10 11 12 13 14 v0 [m/s] 720.21 690.83 692.02 691.18 687.61 692.58 688.40 681.84 rate of firing [shot/min] 620 622 627 627 635.5 629 631 635.5 n x 103 15 16 17 18 19 20 21 v0 [m/s] 685.59 675.38 677.49 673.64 667.01 669.89 664.81 rate of firing [shot/min] 638 633.5 635 641 640.5 641 642 figures 2 and 3 represent the initial muzzle velocity and the rate of firing as a function of the fired round number. fig. 2 muzzle velocity vs. number of rounds fig. 3 rate of fire vs. number of rounds the muzzle velocity experienced a decrease of 7.69% while the rate of fire experienced an increase of 3.55% between the results of the standard barrel and the results of the tested barrel after 21,800 rounds were fired. experimental and numerical investigation on the performances of a small weapon barrel... 5 the xy coordinates of the shots on the targets were collected and treated to obtain the coordinate of the center of shots, and the xy standard deviations of the shots as a function of the number of rounds (figs. 4 and 5). fig. 4 position of the center of shots vs. number of rounds fig. 5 standard deviations vs. number of rounds the circular error probable (cep) is the radius of a circle that has a 50% probability of containing the target. the cep of each of the measures was calculated approximately using eq. (1), [15], the obtained results were presented in fig. 6. 2 max( , ) (0.67 0.8 ) where if 0 0.5 if 0 max( , ) 0.5 . 19 (1 ) 5 / l x y l s x y l s l w cep w ww w                            (1) the accuracy of a rifle refers to how close the measurements are to the aiming point. the precision of the rifle refers to how close the measurements are to each other. the coordinate of the center of shots presented in fig. 4, showed that the accuracy of the tested barrel was independent of the number of the total round fired. however, the precision was 6 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic affected as it is shown in figs. 5 and 6. the standard deviation of the results and the cep increased with the total round fired. fig. 6 cep vs. number of rounds after the tests, longitudinal sections were made from the tested and the standard barrels using the wire-cut electrical discharge machining (edm) process. the edm process is a technique using sparks produced by electrical discharge to manufacture work-pieces in accurate dimensions and shape [16,17]. figures 7 and 8 show the longitudinal sections of the standard barrel and the used barrel, respectively. accurate views of the origin of rifling were obtained using an optical microscope as it can be seen in the figs. 7 and 8. fig. 7 longitudinal section of the standard barrel fig. 8 longitudinal section of the tested barrel 4. laser scanning of the interior surfaces for the purposes of measuring the interior surfaces and creating a 3d model, a 3d scanner hexagon rs5 was used, with a stabilized absolute arm platform 8525-7, shown experimental and numerical investigation on the performances of a small weapon barrel... 7 in fig. 9. with a precision of 0.048 mm, this system enables the reliable determination of 3d point clouds of the inner surface for the longitudinal cross-sections of the tested and the standard barrels. the technical characteristics of the laser scanner and the stabilization platform can be found in the hexagon brochure [18]. the rs5 laser scanner scans results of the used and the standard barrels were introduced to the polyworks software where preliminary preparations were made like setting the geometries approximately in the cartesian reference and deleting the majority of the unwanted scanned surfaces and points (fig. 10). the initial results were imported to matlab developed code as stl files, in order to extract the cartesian coordinates of the 3d scatter plot points. points that do not belong to the barrel bore surface were removed. the xyz coordinates of the center of two distant sections were determined using the average value of all the xyz specific section points. the rotation and translation matrix were calculated using the resulting vector between the two sections centers. using those rotation and translation matrixes the geometries were installed in the cartesian coordinate system. the average distance of the lands and grooves from the barrels axes were calculated following a cylindrical helix curve. the obtained results are presented in fig. 11. fig. 9 3d scanner hexagon rs5 with a stabilized absolute arm platform 8525-7 fig. 10 3d scans of the standard (left) and tested (right) barrels the origin of rifling was the most damaged area of the bore where the total land material was removed. this will increase the free travel distance of the projectile and delay the 8 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic proper engraving of the bullet jacket. increasing the free travel distance will decrease the total time of the firing process, which can explain the rise in the rate of fire. the thickness of the lands decreases unevenly with a maximum loss in the average land radius value of 0.17 mm at the axial position 76 mm from the bottom of the barrel. fig. 11 average radius of the internal surfaces (lands and grooves) of the used (ub) and the standard barrels (sb) as a function of the axial position 5. explicit dynamics analyses based on the 3d laser scanning data two cad models (fig. 12) for the standard and tasted barrels were created and used in an explicit dynamic analyses to simulate the interior ballistics process of a coupled projectile-barrel system. the explicit dynamic analysis is used to provide a firing process model considering its short-duration and high-pressure loadings. such a complex phenomenon requires advanced analyses tools to accurately predict its effect on the rifle and the bullet design. creating a high-quality mesh is one of the critical factors that must be considered to ensure simulation accuracy. a high-quality mesh means that there is an optimal balance between the computational cost and the level of fineness achieved. the meshing was made in a helicoidally way for both the barrels and the bullet to match the shape of deformations. fig. 12 cad models for the standard (left) and the tested (right) barrels experimental and numerical investigation on the performances of a small weapon barrel... 9 the three-dimensional refinement was made manually by retaining the element types and attempting to convert the elements in the regions where the interactions accrue between the barrel and the bullet into smaller elements of the same type as the original elements. this refinement effectively decreases the cell size of only the target area but does not increase the computational cost dramatically. in order to refine the mesh in the radial, axial and azimuthal directions special transitional mesh elements (fig. 13, left) were needed to increase the number of cells in the desired direction. more developed mesh elements (fig. 13, right) were needed to increase the number of cells in two directions simultaneously. the whole model was used for the analyses. fig. 13 transitional mesh elements in one (left) and two (right) directions fig. 14 represents the hex8 elements (8 nodes hexahedron) mesh of only a quarter of the projectile and the barrel to show the mesh refinement in all three (xyz) directions. fig. 14 helicoidal mesh of a quarter of the (left) projectile and the (right) barrel the number of nodes, elements and the type of elements are presented in the table 2. table 2 type and number of finite elements geometries barrels bullet number of nodes 517,932 159,435 number of elements 446,812 143,504 type of elements hex8 hex8 the projectile core and the barrels are made of the same material (steel), which is assumed to have a linear elastic behavior. the steel mechanical and thermal properties are presented in table 3. the projectile jacket material (brass) has a homogeneous, isotropic, elastic-plastic behavior with johnson-cook plastic deformation and failure models. the brass mechanical and thermal properties are summarized in table 4. the contact between the barrel and the projectile is a frictional contact with penalties. 10 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic table 3 steel mechanical and thermal properties [13-14] parameters e [gpa] poisson ratio density [kg/m3] yield stress [mpa] specific heat cp [j/kg k] thermal conductivity [w/m k] value 206 0.31 7850 933 483 30 table 4 brass jacket material mechanical and thermal parameters [11] elastic and thermal parameters e [gpa] poisson ratio density [kg/m3] specific heat [j/kg k] thermal conductivity [w/m k] value 115 0.31 8800 375 110 plastic parameters a [mpa] b [mpa] c n m tm [k] ε̇0 [s-1] value 206 505 0.01 0.42 1.68 1189 5x10-4 damage and fracture parameters d1 d2 d3 d4 d5 value 0.54 4.89 -3.03 0.014 1.12 the stresses in the barrel caused by the firing are the results of different loads. the main loads are the mechanical and thermal loads generated by the combustion of the propellant. for reasons of simplification in this study, all the loads are neglected except the stresses caused by the generated pressure of the propellant combustion and the interaction between the barrel and the bullet. fixed support boundary condition was applied on the rear surface of the barrel. moreover, the whole model was used, i.e. no symmetry conditions were applied. the loads introduced to the analyses are time-space dependent loads (fig. 15). they are applied as tabular data in a way that for each time step we introduced a pressure load applied on the interior walls of the barrel and the back of the projectile. the loads were calculated using the two-phase flow internal ballistic model presented in [19]. the twophase flow model considers both the solid (gunpowder grains) and the combustion gaseous component during the firing process and at a given time compute different pressure values along the barrel. fig. 15 time-space dependent loads on the barrel and the projectile experimental and numerical investigation on the performances of a small weapon barrel... 11 6. explicit dynamic analyses results discussion for a reasonable comparison between the analytical and the numerical analyses, both must have the same input parameters. for that reason, the bullet was suppressed in a preliminary simulation to neglect the effect of the bullet acceleration on the barrel. the total equivalent stresses obtained by the dynamic explicit analyses are compared with the analytical stress. the analytical equivalent stress is calculated by the von mises formula, 2 2 21 (( ) ( ) ( ) ) 2 e r t r z t z            (2) where r is radial stress, z is axial stress and t is tangential stress [20]. figure 16 represents the analytical and numerical equivalent total stresses as a function of the barrel radius (r) obtained in simulations with and without projectile for section 60 mm at 0.34 ms (fig. 16, left) and section 120 mm at 0.38 ms (fig. 16, right) from the bottom of the barrel. the equivalent stresses obtained by the simulations have the same trend as the analytical equivalent stresses for values of radii greater than 5 mm. the graphs of the simulations equivalent stresses show fluctuations for radii smaller than 5 mm, which is the region of the grooves and lands. those fluctuations could be explained by the creation of stress concentrations created in that region [21]. the equivalent stresses obtained by the simulation without projectile showed a good match with the analytical stresses, whereas the equivalent stresses obtained by the simulation with projectile were higher than the analytical stresses due to the presence of the projectile. the rotation and translation of the projectile generate other types of stresses on the barrel, such as the axial stress and torsional stress. fig. 16 analytical and numerical equivalent total stresses for section 60 mm at 0.34 ms (left) and 120 mm at 0.38 ms (right) from the bottom of the barrel the second and third comparisons were made between the position and the velocity of the projectile results calculated by the numerical model two-phase flow [19] and the results obtained by the explicit analysis of the standard barrel, as it is shown in fig. 17. the projectile position and velocity results obtained by the simulation showed good match with the results obtained by the internal ballistic model. 12 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic fig. 17 position (left) and velocity (right) of the projectile obtained by the numerical simulations and the internal ballistic model fig. 18 shows the position of the projectile at the end time where the projectile reaches the muzzle. fig. 18 position of the bullet at the end of the simulation figures 19 and 20 show a comparison between the equivalent total plastic strain obtained by the simulations and the deformations of the projectiles fired by the standard and the tasted barrels after the durability test. fig. 19 equivalent total plastic strain of the bullet fired by the standard barrel at the end of the simulation (left), experimental plastic deformations of the projectile (right) fig. 20 equivalent total plastic strain of the bullet fired by the used barrel at the end of the simulation (left), experimental plastic deformations of the projectile (right) experimental and numerical investigation on the performances of a small weapon barrel... 13 figures 19 and 20 show apparent similarity in the shape of the plastic deformation obtained by the experiments and the explicit dynamic analyses for both tested and standard barrels. at the beginning of the barrel lifespan, the engraving of the projectile outer surface is done at small accelerations. the uniform lands thickness is sufficient to allow the projectile to gain rotational velocity without further bullet-jacket deformation. the initial depth of the deformation will stabilize the smooth movement of the projectile on lands. however, with the continued use of the weapon, the interior surface will be deteriorated, and the thickness of the lands will decrease unevenly, the engraving of the projectile outer surface will be done far from the origin of refiling at significant accelerations. the bullet will no longer have a smooth movement on lands behavior and more azimuthal bullet-jacket deformation will occur, which will generate more vibration, more friction and material deformation energy loss, less rotational and directional velocities and a displacement from the initial spin axis. figure 21, left, shows the velocities of the projectiles as a function of time determined by the simulations. the projectiles velocities reached are 717.29 m/s and 669.95 m/s for the standard and tested barrel, respectively. fig. 21, right, shows the angular velocities of the projectiles as a function of time determined by the same simulations. the projectiles angular velocities reached are 171,824 rpm and 135,786 rpm for the standard and tested barrel, respectively. fig. 21 (left) projectile velocities and (right) angular velocities of the standard and tested barrels as a function of time from the simulations results, the projectiles initial axial directional velocities drops by 6.59% and 20.97% in the projectile initial angular velocities between the standard and tasted barrels. the initial angular velocity of the projectile is crucial in the projectile flight stability and affects the rifle precision. a reduction of 20% would affect the projectile flight stability and result in a sharp decline in the firing range and precision. table 5 shows the initial velocities comparison between the experimental and the simulation results for both the standard and used barrels. the percentage of the difference between the experimental and the simulation velocities were less than 1% for both the standard and used barrels. the satisfactory agreements between the presented results increase the credibility of this work. 14 w. boukera abaci, n. hristov, i. radisavljevic, l. stojnic, a. anicic table 5 initial velocities comparison standard barrel [m/s] used barrel [m/s] experimental results 720.21 664.82 simulation results 717.29 669.95 percentage of the difference experimental vs. simulation 0.41% 0.77% 7. conclusion the paper presented experimental and numerical investigations on the performances of a small caliber rifle barrel during its lifecycle. durability test experimentations were conducted on a 7.62 mm automatic rifle. appropriate measurements were made to track the rifle performances during the stages of its lifecycle. after the durability testes, the used and the standard barrels were cut longitudinally using the wire-cut edm process to enable the detection of the rifle’s internal surfaces damage using 3d scanning. based on the 3d scans, ansys explicit dynamic analyses were conducted to investigate the behavior of the tested and the standard barrels. special transitional mesh elements were used to be able to refine the regions where the interactions accrue between the barrel and the projectile. the following conclusions can be extracted from the experimental and simulation results:  the tests showed correlations between the performances of the rifle and the total number of shots fired. the muzzle velocity experienced a drop of 7.66% at the end of the tests while the rate of firing experienced an increase of 3.55%. the accuracy of the firing was independent of the total round fired number. according to the standard deviation of the shots and the cep tests the precision of the rifle decreases with the repetitive use of the weapon.  the optical microscope images showed the degradation of the bore internal surface in the region of the origin of rifling. moreover, according to the calculation of the average distance of the lands and grooves from the barrels axes the origin of rifling experienced the severest damage with total removal of the land material.  the numerical analyses showed satisfactory agreements with the experimental and the internal ballistic model results. the equivalent stresses obtained by the simulation without projectile showed good agreement with the analytical stresses. the projectile position and velocity results achieved by the simulation were similar to the internal ballistic model results. furthermore, visible similarities in the projectile surface plastic deformation were captured between the experiments and the numerical analyses. on the top of that, the differences between the experimental and the simulation velocities were less than 1% for both the standard and used barrels.  according to the simulation results, the projectile experienced a drop of 6.59% and 20.97% in the initial and rotational velocities respectively between the standard and tasted barrels. which also indicates the diminution of the rifle barrel performances. the explicit dynamic analyses were employed to provide a model of the short duration firing process for the standard and tested barrels. since it is too expensive to perform physical testing, success in modeling this process can help in the improvement of the rifle’s experimental and numerical investigation on the performances of a small weapon barrel... 15 design. a reliable explicit dynamic analysis can accurately predict complex responses, such as large material deformations and failure and interactions between the barrel and the projectile. based on the explicit dynamic analysis, the lifespan of the rifle barrel can be predicted more accurately. acknowledgement: the research was supported by test center of zastava arms kragujevac of the republic of serbia and hexagon manufacturing intelligence – hexagon metrology representative office serbia for experimental tests. references 1. ahmad, i., 1988, the problem of gun barrel erosion: an overview, gun propulsion technology, 109, pp. 311-355. 2. lawton, b., 2001, thermo-chemical erosion in gun barrels, wear, 251(1-12), pp. 827-838. 3. lawton, b., 2003, the influence of additives on the temperature, heat transfer, wear, fatigue life, and self ignition characteristics of a 155 mm gun, journal of pressure vessel technology, 125(3), pp. 315-320. 4. sopok, s., rickard, 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22-32. 15. nelson, w., 1988, use of circular error probability in target detection, report esd-tr88-109, massachusetts, usa. 16. akıncıoğlu, s., 2022, taguchi optimization of multiple performance characteristics in the electrical discharge machining of the tigr2, facta universitatis-series mechanical engineering, 20(2), pp. 237-253. 17. sidhu, a.s., 2021, surface texturing of non-toxic, biocompatible titanium alloys via electro-discharge, reports in mechanical engineering, 2(1), pp. 51-56. 18. www.hexagonmi.com/nl-nl/products/portable-measuring-arms/romer-absolute-arm-compact (last access: 08.08.2022) 19. boukera abaci, w., hristov, n. p., ziane, a. n., jerković, d. d., savić, s. r., 2021, analysis of thermal and gasdynamic characteristics of different types of propellant in small weapons, thermal science, 25(6 part a), pp. 4295-4306. 20. evci, c. isik, h., 2018, analysis of the effect of propellant temperature on interior ballistics problem, journal of thermal engineering, 4(4), pp. 2127-2136. 21. boukera abaci, w., kari, a.v., jerković, d.d., hristov, n.p., 2020, the influence of the internal ballistic pressure on the rifled barrel stress response, scientific technical review, 70(2), pp. 41-46. facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 31 42 https://doi.org/10.22190/fume190623005b © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  surface modification of ring-traveler of textile spinning machine for substantiality preetkanwal singh bains 1 , jasmaninder singh grewal 2 , sarabjeet singh sidhu 1 , sandeep kaur 3 , gurpreet singh 1 1department of mechanical engineering, bcet, gurdaspur, punjab, india 2gndec, gill road, ludhiana, punjab, india 3gndu (rc), gurdaspur, punjab, india abstract. in this report, a study of the wear mechanisms involved in the spinning ring and the traveler of textile industry are presented. these components, after surface processing with various coatings techniques, were analyzed on the test rig so as to analyze the wear mechanism. the objective was accomplished by comparing various plasma sprayed coatings on e52100 steel pins using a pin-on-disc machine. the surface morphology as well as mechanical properties of the deposited coatings, namely wc-cocr, al2o3+tio2 (alumina-titania) and cr3c2nicr, as well as uncoated e52100, were comparatively studied. this study elucidates towards improving the working life of the ring in a textile mill in the spinning operation. an x-ray diffractometer (xrd) and scanning electron microscope (sem) were employed to characterize the unworn and worn surfaces of the specimens. the study revealed that the wear rate of plasma sprayed thermal coatings enhanced with augmenting load. the plasma sprayed wc-co-cr, cr3c2nicr, al2o3+13tio2 coatings developed on workpiece pins exhibited a notable decrease in volume loss of the material as compared to uncoated e52100 substrate. wcco-cr coating turned out to be the best performer in terms of the lowest cumulative volume loss among all the variants of coatings. key words: pin-on-disc, thermal, al2o3, coatings, sem, traveler, xrd, ring 1. introduction the wear mechanisms amongst various parts of the spinning machine have been extensively studied and analyzed by myriad researchers over the past few years. for the efficient working of a textile mill, a ring and a traveler play a crucial role; the later impart twisting to yarn (thread), around the inner periphery of the ring, facilitating the winding received june 23, 2019 / accepted january 11, 2020 corresponding author: sarabjeet singh sidhu department of mechanical engineering, beant college of engineering & technology, gurdaspur – 143521, punjab, india e-mail: sarabjeetsidhu@yahoo.com 32 p.s. bains, j.s. grewal, s.s. sidhu, s. kaur, g. singh of the yarn on the bobbin. the centrifugal force between the two generates a high pressure during winding when the traveler glides around the ring along with the yarn (fig. 1). as a consequence, the inner surface of the ring deteriorates in the form of waviness depicting abnormal wear pattern. this problem, however, mushrooms in by the recurring high-temperature cycles in addition to the usual tribological contact between them. the waviness formation on the internal face of the ring interrupts the free sliding of the traveler along the ring, resulting in an irregular winding of the thread. the obstructed sliding motion is usually associated with abrasion and wears causing the uneven and rough internal surface of the ring. owing to abrasive wear, this causes the affected part to fail prematurely and demands regular replacements and reinstating of the rings. moreover, this causes a considerable breakage of the thread and hence results in reduction in production efficiency [1, 2]. in the light of this, the surface modification of this component was carried out in this study, to enhance the compatibility, durability and performance with the help of suitable surface coatings. fig. 1 new spinning ring (left) and damaged ring surface (right) ceramics possess infinite merits over metallic as well as polymeric materials and are known for high hardness and effective resistance to the thermal and corrosive environment [3, 4]. numerous oxide ceramics like alumina, silica, zirconia, titania, and chromium find widespread applications as potential materials for surface coatings and thermal barriers and for enhancing erosion, wear, abrasion, and corrosion resistance [5, 6]. plasma spraying has been the prominent thermal deposition method in this regard [79]. when spraying for coating, the molten particles collide to build a layer of coating with superior thermo-mechanical properties. the coating is generated high energy collision of particles to form solidified layer. al2o3 and cr2o3 coatings have been a point of interest for several industrial applications like turbines and pumps [10]. al2o3 is a ceramic material known to have exceptional resistance to wear along with exhibiting high surface waviness traveler ring surface modification of ring-traveler of textile spinning machine for substantiality 33 hardness at elevated temperatures [11, 12]. it has been witnessed from literature study that cr2o3 coating is a better candidate for a wear and erosion resistant coating in a gentle sliding condition, under the normal load and it may exhibit typical tribological behavior. rickerby and winstone [13] put forward that the surface coatings help to deal with the physical and chemical degradation of the surface. the fracture toughness of al2o3 coatings can be enhanced by the addition of tio2 particles clad in the form of layers [14]. thermal spraying is an effective and economical route to develop thick and uniform coatings for the surface modification of a part [15]. these spray coatings techniques are extensively used in automobile, textile, turbines, and aircraft industry [16]. the investigations revealed that the properties of the coatings primarily depend on the cohesion/bonding of particles which varies steeply with the temperature of sprayed particles [17]. although copious research work [18-21] associated with the wear resistant coatings developed by plasma spray and hvof techniques could be witnessed, much less work towards improving the wear resistance of distinct components of the textile industry has been traced. in cr3c2nicr coating, the nicr phase is responsible for the required corrosion resistance whereas the carbide ceramic phase provides the necessary abrasive wear resistance [22, 23]. this study, thus, aims at exploring the characteristics and wear behavior of plasma sprayed coated e52100 pins using wc-co-cr, al2o3+tio2 and cr3c2nicr ceramics. the wear trends of coating observed from these tests were used to envisage the working life of the actual rings in the textile mill. 2. materials and methods 2.1. preparation of substrate and coating deposition usually, the rings in the spinning machine are mainly subjected to friction and abrasive wear [24]; thus, to overcome such problems, wc-co-cr, al2o3+13tio2 and cr3c2nicr [25] opt as coating materials. the detail of the coating powders utilized in the present work is enlisted (table 1) as below. sulzer-metco f4 plasma gun for atmospheric plasma spraying is employed to deposit three coating variants. argon as carrier gas acts as a medium for both plasma-operation and coating powders. table 1 detail of the plasma sprayed powders powder preparation route powder size (µm) density (mg/mm3) shape of particles composition (wt. %) wc-co-cr sintering/agglomerated -45/ +10 5.8 spherical 86 wc, 10 co, 4 cr al2o3+13tio2 blending -45/ +11 3.43 angular 87 al2o3, 13 tio2 cr3c2nicr cladding -45/ +15 3.0 irregular 85 cr3c2, 15 nicr a high-carbon iron alloy, e52100, in the form of pins (50mm x 5mm) was used as a substrate material owing to its hardenability and resistance towards wear, making it suitable for industrial applications. al2o3+13tio2, wc-co-cr and cr3c2nicr powders were deposited with a thickness in the range of 100-150 µm on the end-faces of the e52100 substrate (fig. 2) material by the plasma spray. nine circular specimens (50mm of length, 5mm diameter) were grit blasted using alumina grits at 3 kg/cm2 pressure. the stand-off distance in shot blasting was kept between 150-200 mm. the average roughness of the surface to be coated was fixed as 6.8 μm (approx.) facilitating suitable adhesion of 34 p.s. bains, j.s. grewal, s.s. sidhu, s. kaur, g. singh the sprayed particles. the grit blasted specimens were cleaned with acetone in an ultrasonic cleaning unit followed by spraying. the chemical composition of the material used for testing (e52100) has been represented in table 2. fig. 2 pictorial view of (a) pins; (b) as-coated pin ends table 2 chemical composition of e52100 pins grade elements (%) e52100 cr c ni si cu p mn s 1.580 0.996 0.107 0.236 0.11 0.054 0.468 0.005 2.2. characterization of microstructure the microstructures and surface morphologies of the coating powder were examined using (jsm 6610lv, jeol) field emission scanning electron microscopy (fe-sem). in order to obtain coatings of uniform thickness, the same was noted precisely during the procedure of plasma spraying using a thin film thickness gauge (minitest-600b, precision ± 1µm) at six distinct points, considering the average thickness of the coating. to identify distinct phases, xrd analysis for coated specimens was carried out using the x’pert pro, panalytical advance diffractometer (netherland) with cukα radiation and nickel filter at 40 ma. the sem examination of the coated samples before the wear test was carried out showing the dense, even with few pores and proper deposition of coatings without unmelted particles, as shown in fig. 3. surface modification of ring-traveler of textile spinning machine for substantiality 35 fig. 3 morphology of as-coated specimens: (a) wc-co-cr; (b) al2o3 +13tio2; (c) cr3c2nicr 2.3. experimental procedure to examine the controlled wear, the comparative tests were carried out for uncoated and plasma sprayed cylindrical pins (e52100) by employing a pin-on-disc apparatus (wear and friction monitor, tr-201) (fig. 4) that incorporates a rotating disc and a sample holder loaded with dead weights. the weights (30n, 40n, 50n) were applied as a normal force on the sample for 30 minutes. the specimen pin was fixed in the holder at 80mm track diameter with the coated end facing the carbon steel (en-31) rotating disc of a hardness of 64 hrc. the parameters of wear test are presented in the table 3. the specimens were cleaned and degreased and weighed using a precision electronic balance (citizen, cy220). the loss of weight for each specimen was noted at a gap of 30 minutes. after each trial, the sample pin was allowed to cool at room temperature after removing from the test rig. the wear debris was removed with the help of brushing, the sample was re-weighed to determine the material loss and then fixed back in the holder in the same position. (a) (b) (c) 36 p.s. bains, j.s. grewal, s.s. sidhu, s. kaur, g. singh fig. 4 pin-on-disc wear apparatus table 3 parameters of the sliding wear test nomenclature value pin diameter (mm) 5.0 pin material e52100 bearing alloy steel disc material carbon steel (en 31) track diameter (mm) 80 velocity (m/s) 5.50 sliding distance (m) 5400 weight (n) 30, 40, 50 additionally, the volume loss represented in table 4 indicates the amount of wear encountered by each sample calculated according to below eq. (1), 3 3 weight loss (mg) cvl(mm )= densityof coating material (mg/mm ) (1) the density of the coating material was measured by the gravimetric method. moreover, the average porosity of as-sprayed coatings was less than 1%. the average thickness of selected coatings was recorded as 125 microns (approx) as measured using minitest-600b. fig. 6 shows the microstructure of the worn out surfaces of the coated specimens. typical x-ray diffractograms (fig. 7) for plasma sprayed coatings on e52100 specimens were carried out for phase identification with a rate of scanning ranging from 1º to 100º/min (2theta) with cu-kα radiation equals to 1.5418å. surface modification of ring-traveler of textile spinning machine for substantiality 37 table 4 experimentation design and results trials coatings load cvl (mm3) s/n ratio (db) r1 r2 1 wc-co-cr 30 0.059 0.061 24.4358 2 wc-co-cr 40 0.048 0.052 26.0137 3 wc-co-cr 50 0.090 0.101 20.3856 4 al2o3 +13tio2 30 0.084 0.084 21.5144 5 al2o3 +13tio2 40 0.083 0.086 21.4615 6 al2o3 +13tio2 50 0.099 0.111 19.5621 7 cr3c2nicr 30 0.300 0.299 10.4721 8 cr3c2nicr 40 0.665 0.665 3.5436 9 cr3c2nicr 50 0.528 0.531 5.5226 in order to draw valid conclusions, the prominent process factors, for instance, coating type and load (table 4) selected by screening trials were changed at three levels. the limited numbers of runs were achieved as the optimum conditions using the full factorial design. in this experiment, two different repetitions at random order were carried out to obtain the s/n ratio for more precise results. the s/n ratio is considered as a performance measure in terms of ratio of magnitude of the signal strength to the noise. 3. results and discussion the input variables chosen according to the control log of the standard full factorial method, and the recorded cumulative volume loss (cvl) values for wear of coatings along with their respective s/n ratio are arranged in table 4. it is noteworthy from fig. 5 that the coating is the most significant parameter affecting the volume loss of the coated samples; wherein, wc-co-cr coated pin (sample) exhibited very less wear damage. however, the severe abrasive wear was witnessed for cr3c2nicr variant of coating sample and was severe amid all the selected variants of coatings. moreover, the load applied during the pin-on-disc testing was observed as an insignificant factor affecting the wear of the coated surface (table 5). hence, the wear resistance of plasma sprayed coatings generated on e52100 substrate was recorded as wc-co-cr > al2o3 +13tio2 > cr3c2nicr in their decreasing order. fig. 5 main effects plot of s/n ratios for cvl 38 p.s. bains, j.s. grewal, s.s. sidhu, s. kaur, g. singh table 5 analysis of variance for cvl source df seq ss adj ss adj ms f-value p-value coatings 2 505.46 505.46 252.736 40.75 0.002* load (n) 2 19.99 19.99 9.996 1.61 0.307 residual error 4 24.81 24.81 6.203 total 8 550.27 *significant the post-wear characterization of sem micrographs of worn samples in fig. 6 shows that the surface deposition remained uniform, homogeneous entailing no visible cracks. fig. 6(a) illustrates that mostly spherical shaped particles were present with an elongated profile in few particulates resulting in denser coatings. the traces of carbide fragmentation followed by spalling were observed from fig. 6(a). some debris containing partially molten particles of the agglomerated and sintered alumina-titania had been noticed, as shown in fig. 6(b). also, sem fig. 6 sem of worn samples (a) wc-co-cr (b) al2o3 +13tio2 (c) cr3c2nicr (at 30n) (c) (b) (a) molten metal dense coating uneven pits spherical particles surface modification of ring-traveler of textile spinning machine for substantiality 39 micrographs revealed that oxidation might have occurred due to the presence of γ-al2o3, as disclosed in xrd analysis (fig. 7). however, in fig. 6(c) white layers revealing that stress concentration were more prominent in the cr3c2nicr variant of coating, exhibiting minimum resistance for abrasive wear amid all coatings. the wear mechanism has impacted the cr3c2nicr coating surface in form of uneven pits as a sign of brittle fracture accompanied by small and non-uniform plastically deformed region. a significant grain fracture and subsurface damage were witnessed in wc-co-cr coated specimen. the assessment of worn surface using x-ray diffraction measurements was carried out to analyze the surface modification of specimens. the xrd images depicted a notable quantity of wc as the major constituent phase which resulted in increased hardness of the coating, besides cr and co as a minor fraction in wc-co-cr coating. this ruled out any possibility of decarburization of wc as a result of cr addition during coating procedure as no other phase, as expected, was spotted, which could be attributed to the chromium addition [26, 27]. additionally, certain spectrum peaks showed an evident quantity of retained cobalt as a binder phase. x-ray diffractogram of al2o3+13tio2 coating specified the occurrence of the broad bulge between 45° and 65° in the concerned pattern that promoted the development of an amorphous phase. few peaks for rutile-tio2 (6% wt. fraction) could also be witnessed in x-ray analysis (fig. 7). the x-ray diffractogram in fig. 7 signified a high content of γ-al2o3 accompanied by almost equal amounts of tio2 along with a small amount of amorphous; al2o3 attributed to the rapid solidification of molten particles of alumina [28, 29]. the x-ray diffractogram for cr3c2nicr advocated a higher amount of chromium carbide as the major phase. nicr compound has been detected as a minor phase in the form of a small one. fig. 7 xrd spectra for coated e52100 samples after wear 40 p.s. bains, j.s. grewal, s.s. sidhu, s. kaur, g. singh three coated specimens with each coating were analyzed against wear on pin-on-disc apparatus with normal acting loads ranging between 30n, 40n and 50n, respectively (table 4) along with uncoated specimen of e52100 substrate. fig. 8 depicts the wear resistance of all the three variants of plasma sprayed coatings. furthermore, the rate of wear for cr3c2nicr coating augmented with normal load whereas wc-co-cr coating showed a negligible effect. the uncoated pins of e52100 exhibited cvl of 5.660, 7.245 and 6.904 mm3 at load of 30, 40 and 50 n, respectively. the wear resistance of coating with tungsten carbide powder is recorded to be higher amid all coatings. this is due to a relatively high amount of wear resistant fused carbide powder in this variant of coatings [30, 31]. this is also attributed to the finer carbide particles and a less carbon loss during abrasive wear. the percentage decrease in cvl for wc-co-cr coating has been recorded as 98.94% at 30 n, 99.3% at 40 n and 99.3% at 50 n as compared to the base metal. the reduction in the volume loss for al2o3+13tio2 coated pins was recorded as 98.58% at 30 n, 98.89% at 40 n and 99.3% at 50 n against e52100 material. this is related to lower hardness and lower cohesion due to high porosity in alumina-titania coatings [32]. finally, cr3c2nicr coating variant experienced 94.7% at 30 n, 90.75% at 40 n and 96.5% at 50 n as wear reduction, comparatively. this higher erosion is expected as the material removal occurred mainly by carbide particles fracture that involved crack initiation at a particlematrix interface which propagated to the surface of the coating. this phenomenon amplified with a higher porosity of the coating material being brittle in nature. better wear resistance of wc-co-cr coating over other counterparts may be due to the better fracture as well as adhesive strength of the co-cr matrix with the carbides. fig. 8 cumulative volume wear rate (mm3) for coatings and uncoated e52100 at 30n, 40n, and 50n surface modification of ring-traveler of textile spinning machine for substantiality 41 4. conclusions this study aimed at investigating the effect of various plasma sprayed coatings on wear 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cytocompatibility and tribological performance of co–cr implant, journal of materials research, 34(16), pp. 2837-2847. 31. nasiri-vatan, h., adabi, m., 2017, investigation of wear and corrosion resistance of nanocomposite coating formed on az31b mg alloy by plasma electrolytic oxidation, transactions of the imf, 95(6), pp. 308-315. 32. pinzon, a.v., urrego, k.j., gonzalez-hernandez, a., ortiz, m.r., galvis, f.v., 2018, corrosion protection of carbon steel by alumina-titania ceramic coatings used for industrial applications, ceramics international, 44(17), pp. 21765-21773. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 1, 2014, pp. 15 25 an approach to efficient fem simulations on graphics processing units using cuda udc 519.6+531 björn nutti 1 , dragan marinković 2,3 1 algoritmfabriken ab, stockholm, sweden 2 department of structural analysis, tu berlin, germany 3 faculty of mechanical engineering, university of niš, serbia abstract: the paper presents a highly efficient way of simulating the dynamic behavior of deformable objects by means of the finite element method (fem) with computations performed on graphics processing units (gpu). the presented implementation reduces bottlenecks related to memory accesses by grouping the necessary data per node pairs, in contrast to the classical way done per element. this strategy reduces the memory access patterns that are not suitable for the gpu memory architecture. furthermore, the presented implementation takes advantage of the underlying sparse-block-matrix structure, and it has been demonstrated how to avoid potential bottlenecks in the algorithm. to achieve plausible deformational behavior for large local rotations, the objects are modeled by means of a simplified co-rotational fem formulation. key words: co-rotational fem, graphics processing units, cuda, sparse blockmatrix, conjugate gradient solver, geometrical nonlinearity 1. introduction simulation of deformable objects plays an important role in different areas, ranging from engineering tasks in structural analysis, via virtual reality applications, such as surgery simulators, up to entertainment industry, e.g. games and animations, to name but a few. many of these areas impose two conflicting goals: 1) highly efficient computations of objects' deformational behavior, whereby 2) the geometrically nonlinear behavior due to large local rotations is to be recovered with sufficient accuracy. in the entertainment industry applications, it is often sufficient to utilize very simple mass-spring models [1] for this purpose, while fields such as virtual reality [2] or multibody system dynamics [3] demand more accurate methods for plausible results. with a received december 03, 2013 / accepted february 1, 2014  corresponding author: björn nutti algoritmfabriken ab, stockholm, sweden e-mail: bjorn@algoritmfabriken.se original scientific paper 16 b. nutti, d. marinković rapid pace of the hardware development, the finite element method (fem) has gained ground in the past decade. however, the application of nonlinear fem formulations results in simulations that are computationally more intensive thus putting a higher demand on computational resources. there are two complementary ways of approaching the problem: 1) application of simplified geometrically nonlinear fem formulations, and 2) shifting the computations to numerically more powerful hardware components. with the increasing computational power of graphics processing units (gpu), shifting the computational tasks to the gpu has become an attractive option. this approach also allows the cpu to deal with other tasks such as collision detection or haptics. the linear fem simulation of deformational structural behavior is rather efficient and stable but it also results in artifacts such as unrealistic growth in volume when the deformational behavior involves finite local rotations. a rigorous geometrically nonlinear fem formulation resolves this problem effectively, but in many cases rather inefficiently and accompanied by numerical stability issues. one of the possible ways of handling this problem was presented by mueller et al [4]. however, their method is based on rotations determined per element nodes and, as a consequence, the elastic forces are not guaranteed to sum up to zero (the authors refer to the unbalanced forces as "ghost forces"). in contrast to this approach, the improved method presented in [5] and [6] uses rotations on the element level. marinkovic et al. [7] discussed different aspects of this simplified corotational fem formulation and demonstrated its application in various fields. solving the fe system of equations is another key-element for highly efficient simulations. the conjugate gradient solver [8, 9] offers many advantages, as will be discussed in this paper. the advantages come particularly to the fore when the solver is used in combination with a general purpose gpu as a modified form of stream processor that provides a massive floating-point computational power (for the state-of-the-art gpus expressed in teraflops per second). this approach has already been a subject of interest of several authors in the recent years [10, 11, 12]. in this paper, an efficient simulation pipeline for deformable objects is presented. the fem models of deformable objects employ the linear tetrahedral element in combination with the above mentioned simplified co-rotational fem formulation that uses elementbased rotations. the implementation is realized with the nvidia's compute unified device architecture (cuda) paradigm suitable for a vast majority of modern gpus. to make the simulation pipeline feasible for simulating both soft and stiff objects at interactive frame rates, the implicit euler time integration scheme is employed. the bottlenecks in the simulation pipeline and the optimization of memory access patterns will be particularly addressed in order to achieve high gpu utilization efficiency. 2. the basics of the implemented co-rotational fem formulation the motivation behind the development of the simplified co-rotational fem formulation is to derive a geometrically nonlinear fem formulation that is stable and very efficient besides enabling sufficiently accurate simulation of the deformational behavior characterized by large local rotations. the idea arises from the principles of incorporating flexible bodies in multi-body-system (mbs) dynamics. in mbs dynamics, the overall motion of flexible objects is described as a superposition of large rigid-body motion and small an approach to efficient fem simulations on graphics processing units using cuda 17 deformation with respect to a body-fixed reference frame that performs the same rigidbody motion as the object. instead of assigning a single local reference frame to the object, a set of local reference frames can be assigned to different areas of the modeled object. in a rigorous co-rotational fem formulation, a local reference frame would be typically defined at each element integration point as those points are used in the evaluation of element matrices and vectors. however, in the present formulation, the idea is to account for the rigid-body rotation with a somewhat lower 'spatial resolution'. hence, an average rigid-body rotation is determined for each finite element. since co-rotational approaches decouple rigid-body motion from deformational motion, they allow the usage of engineering strain and stress measures in the formulation as well as decoupling of geometrical from material nonlinearities. the element-based rigid-body rotation is not the only simplification of this formulation compared to the rigorous geometrically nonlinear fem formulations. another important simplification lies in the fact that the behavior of finite elements remains purely linear with respect to the co-rotational reference frame. thus, the only geometrically nonlinear effect taken into account is the averaged rigid-body rotation of single finite elements. the very core of the formulation is given by the following equation that defines the vector of internal nodal forces on the element level: 1 ( )   inte e e e e 0ef r k r x x , (1) where re is the rotation matrix describing the element rigid-body rotation, ke is the linear element stiffness matrix, while x0e and xe are the initial and current element configurations (nodal coordinates), respectively. the first term in the parenthesis on the right-hand side of the equation yields the current element configuration rotated back to the original element orientation. thus, the entire term in the parenthesis yields the rotation-free nodal displacements. by multiplying them with the linear element stiffness matrix, one obtains the element internal forces with respect to the original element configuration, which are, finally, rotated to the current element configuration, i.e. through re. very simple algebra transforms eq. (1) into the following form: r uee r e0eeeeeeeinte fxkxkrxrkrf  1 , (2) where ke r is the rotated element stiffness matrix and fue r a contribution to the internal elastic forces, a part of which can be pre-computed for more efficient computation. it should be now more obvious where the efficiency of this formulation rests. in a presimulation step, the linear stiffness matrix of each element is computed. over the course of simulation, the information about the current and initial element configurations is used to extract the element rotation matrix describing the purely rotational part of the motion. the matrix is further used to obtain the rotated element stiffness matrix, re-assemble the current structural stiffness matrix and compute the internal elastic forces. 3. time integration for dynamics the properties of the presented co-rotational fem formulation come primarily to the fore in dynamics. the fem equations for dynamics can be given in the following form: 18 b. nutti, d. marinković intext ffcvma  , (3) where a and v are the vectors of nodal accelerations and velocities, m and c are the structural mass and damping matrices, while fext and fint are the vectors of nodal external and internal forces of the entire structure, respectively. the authors of the paper have used the lumped mass matrix. bearing in mind that the primary objective of the development is an interactive realtime simulation, the implicit euler time integration is chosen as a time-integration scheme suitable for both soft and stiff materials. this integration scheme uses the acceleration and velocity at the end of the time-step to update the current positions and velocities: t  δttttt avv , (4) t  δttttt vxx , (5) where δt is the time-increment and the right superscript denotes the moment in time at which the quantity is defined. in each time-step, v t+δt is obtained by solving the following system of linear algebraic equation: tttt bva  , (6) with 2δtδt t t t kcma  , (7) δt( )   t t t t rt t t u extb mv k x f f , (8) where kt t is the tangent stiffness matrix of the entire structure and fu rt is assembled from vectors fue r introduced in eq. (2) for an element, and both quantities are defined for time t. a detailed derivation of these equations can be found in [2]. for the implementation at hand, the damping is defined as rayleigh damping [13], thus: kmc  , (9) where α and β are the coefficients of the mass proportional damping and stiffness proportional damping, respectively. in this work, β=0 was adopted. 4. implementation considerations the cuda architecture is built around a scalable array of multithreaded streaming multiprocessors, designed to execute hundreds of threads simultaneously. during execution, threads are grouped into smaller units called warps. threads within a warp are scheduled to execute simultaneously. the warp scheduler swaps warps in and out during execution to maximize hardware utilization. resources, such as data registers, are shared among warps. hence the higher resource requirements a warp has, the fewer warps can be scheduled on each multiprocessor simultaneously, potentially decreasing the computational efficiency. an approach to efficient fem simulations on graphics processing units using cuda 19 in this paper, four categories of memory are considered: host memory, device memory, shared memory and textures. the host memory is accessible by processes running on the cpu. the device memory is the global memory accessible by the multiprocessor, while the shared memory is located on chip inside each multiprocessor unit and is, hence, much faster than the device memory. although it is possible to perform random accesses to the device memory, it is crucial to follow a set of rules to maximize the bandwidth utilization. the heuristic is to use linear access patterns (coalescing) within warps, both when reading and writing the memory. breaking this rule causes serialized memory accesses, which usually has a dramatic negative impact on the performance. once the memory is loaded into the onchip shared memory, data accesses are two orders of magnitude faster compared to the device memory. although there is a cache for the device memory, it is sometimes beneficial to use the texture memory, as it uses a cache separate from the other ones. a warp performing random reads and writes can benefit from the use of textures, as one cache can be used for reading while the other for writing. 5. assembling the structural stiffness matrix for each element, i, of the tetrahedral mesh, first the element stiffness and mass matrices are computed. within the framework of geometrically nonlinear structural analysis, the structural mass matrix remains constant regardless of the structural deformation or motion in general. on the other hand, the structural stiffness is defined by the tangent stiffness matrix that is configuration dependant and has to be updated upon each time increment. the tetrahedral element stiffness matrix, ker 1212 , has a form of 44 block matrices, knmr 33 , that define the relation between nodes n and m. according to the presented simplified co-rotational fem formulation, the global stiffness matrix is computed as a sum of the rotated element stiffness matrices:  i i r et kk , (10) and since the rotation matrix is orthogonal, the rotated element stiffness matrix is computed as: t eee r e rkrk  . (11) when the global stiffness matrix is assembled on a single-threaded machine, the computation is usually implemented as an element-wise summation, where block matrices of ke r are consecutively added to the corresponding positions in kt. an attempt to apply this approach on a multi-threaded machine would demand synchronization of all write accesses and serialization of all simultaneous accesses to a specific memory location. hence, such an implementation would perform poorly. assigning the task of computing a subset of knm-matrices to each of the kernels would resolve the problem of synchronization, as each position in kt is written once after the corresponding thread is finished with the summation of ke matrices. this approach would, however, break the rules for efficient coalesced memory accesses to the gpu's device memory, because a random access pattern is used when ke matrices are summed up. 20 b. nutti, d. marinković on the other hand, any off-diagonal block, knm, nm, of the element stiffness matrix is attributed to the element edge with nodes n and m, while diagonal blocks knn, are attributed to single element nodes. the same is valid for the global stiffness-matrix, in which off-diagonal and diagonal elements are denoted kij and kii, respectively. a possible approach to resolving the problem of memory accesses is to observe the mesh as a collection of edges and nodes rather than as a collection of elements. by gathering all kij block-matrices for each edge ij and kii for each node i in a pre-simulation step, the blocks in kt can be computed as a sum of piecewise linear memory throughout the simulation. this approach allows fast coalesced memory accesses when computing kt. the described pattern is visualized in fig 1. fig. 1 a) 'per-element' assemblage of the global stiffness matrix; b) 'per-edge and node' assemblage of the global stiffness matrix when assembling the diagonal part of the matrix, each block-matrix kii, at position i in k diag is computed as a sum of knn block-matrices belonging to the elements that share the node i. during the pre-simulation step, a two-dimensional array is created, where all blockmatrices contributing to kii are stored at row i. the maximum number of elements sharing the same node is determined in order to find the smallest block-width that will fulfill the requirements for coalesced memory reads. all rows are zero-padded to this width. over the course of simulation, the elements in k diag are computed with a block-parallel-reduction algorithm (see [14]), that is applied row-wise. within the framework of this research, it has been discovered that the use of thread-blocks of dimension [3block-width], with a grid dimension of [number-of-nodes1] is well suited for the computation. this layout puts an upper limit to the number of neighbors for a node, as there is an upper limit of the number of threads that is allowed in a thread-block. however, this should not introduce any practical limitation for conventional geometries and meshes. as previously discussed, the off-diagonal elements are attributed to the edges in the tetrahedral mesh. generally speaking, the edges are shared by fewer elements than it is the case with the nodes. if each thread-block had computed a single kij matrix, the benefits from coalesced memory accesses would have been poor. in order to improve this, each threadblock is given the task of computing multiple block-matrices, as shown in fig. 2. a) b) an approach to efficient fem simulations on graphics processing units using cuda 21 fig. 2 the strategy of efficient coalesced memory accesses – grouping and summing up the block-matrices related to the same node pair the matrices computed within a thread-block are all related to the same node pair. in the presented implementation, the thread-block size is set to a value that assures coalesced memory accesses. sets of sub-matrices are then stored in a memory-layout matching the thread-blocks. during the simulation, a thread-block reads a set of matrices into the shared memory, these matrices are then rotated, summed up and stored into the shared memory. when all computations are finished, the result is written back to the device memory. 6. improved equation assembly assembling the system of linear equations (eq. (6)) includes a range of arithmetic operations which are performed through a combination of kernel invocations. this section shows how to optimize the required amount of floating-points-operations, as well as the memory usage. to be more specific, the main issues addressed are:  when assembling a, each element in kt has to be multiplied by t 2 . this would not have had a noticeable impact if kt had been constant over the course of simulation. however, since kt depends on structural deformation (i.e. element rotations), this multiplication cannot be pre-computed. it is also not possible to perform this operation while assembling kt because kt is used again when assembling b.  the second issue is the summation that involves the sparse matrix kt and the diagonal matrix m. this requires attention because the block compressed row storage of the sparse matrix is not optimized for accessing diagonal elements. to address the first problem, the damping matrix in the form given in eq. (9) is introduced into eq. (6), which is further divided by t 2 . to improve the readability of the obtained equation, denotations 2 t a a   ~ , 2 t b b   ~ , 2 t m m   ~ and 2t t t1    are introduced, so that: t t t kma  ~~ t , (12) 1 ( ) δt t    t t t t t rt t u extb m(v x ) a x f f , (13) and the equation to solve reads: tttt bva ~~  . (14) 22 b. nutti, d. marinković both m ~ and t can be pre-computed. the only additional computational effort in eq. (14), compared to eq. (6), is vector tx t , which however requires only a low complexity linear operation. however, the presented approach permits computation of t a ~ as the first step. this matrix is further directly used to compute t b ~ . to address the second issue mentioned above, matrix t a ~ can be observed as a sum of an off-diagonal matrix and a diagonal matrix, i.e. t off t diag t aaa ~~~  . hence, the simple idea is to compute t offtk using one kernel, while computing t diagtkm ~ t by means of another kernel. the presented implementation optimizes the number of arithmetic operations as well as random accesses to device memory. however, it still requires uncoalesced memory accessed for rotation matrices. to reduce the amount of data that is read in this manner, it is possible to represent a rotation either as an axis-angle pair or as a quaternion. this strategy allows for storing and accessing rotations in single 128-bits memory transactions, in contrast to the approach based on matrices, which requires up to three times as much. the results presented in this paper are based on the matrix representation. the axis-angle and quaternion representations are beyond the scope of this paper. 7. solving the fe system of equations at each time-step, the fe system of equations (eq. (14)) is solved to obtain the nodal velocities and, thus, update the configuration. since the system matrix, ã t , is a positive definite matrix, the authors' choice is a preconditioned conjugate gradient solver, optimized for working on block compressed row storage sparse matrices. the basic idea of an iterative solver is that it starts out with an initial guess, i.e. initial solution vector, and goes through an iterative process to update the solution vector in each iteration. a pre-conditioner matrix is used to provide a faster convergence to the solution. a convergence criterion is used to determine whether the accuracy of the solution is acceptable or more iteration steps are needed to improve the accuracy. the preconditioned conjugate gradient method involves three matrix-vector products, three vector updates and four inner products per iteration. the number of operations is somewhat greater compared to the conjugate gradient method without preconditioning, but the "pre-conditioned version" is more effective provided the pre-conditioner is well chosen and reduces the system condition number. in this manner, the pre-conditioner improves the convergence rate of the method enough to make up for the additional cost caused by the conditioning. the solver also provides a very easy way of performing a trade-off between the solution accuracy and computational effort by limiting the number of performed iterations. furthermore, the efficiency of the iterative solver can be noticeably improved by a reasonable choice of starting vector of the iterative process. this is particularly interesting in dynamics with the system of equations solved for nodal velocities. namely, the nodal velocities typically do not change dramatically within a time-step. hence, a good choice would be to take the velocities from the previous time-step as a starting vector for the iterative solver in the next time-step. this improves the numerical efficiency of simulation as fewer iteration steps are needed to arrive at the solution. an approach to efficient fem simulations on graphics processing units using cuda 23 8. results to provide a glimpse of the results achievable by the presented co-rotational fem formulation, a liver model depicted in fig. 3 is used. the model implements the coupledmeshed technique [7] to couple the triangulated surface mesh (2598 vertices and 5192 faces) to the fem model (660 nodes and 2640 elements). this technique is adopted to enable modeling of rather complex geometries by means of computationally not too demanding fem models. the approach represents a promising solution for various types of virtual reality applications, such as virtual surgery in this specific case. a) b) c) fig. 3 liver model: a) surface vertices; b) triangulated surface; c) fe-mesh fig. 4 gives a few snapshots from an interactive simulation with the aforementioned liver model. although large, the deformations appear realistic and they do not exhibit artificial enlargement typical for linear analysis. fig. 4 large realistic liver model deformations during a dynamic interactive simulation to evaluate the computational speed-up that the proposed gpu implementation offers, the performance of a single-threaded dual-core intel core2duo t7200 processor running at 2 ghz is compared with the performance achieved by a geforce gtx 8800 graphics card with 128 stream processors and a core clock-speed of 575 mhz. a synthetic mesh generator that produces tetrahedron meshes with evenly distributed elements is used for generating fem models. the size of the generated fem models ranges from 2500 to 20000 elements. as for the boundary conditions, a chosen set of nodes is fixed, while the structure is exposed to the influence of gravity. the preconditioned conjugate gradient solver discussed above is set to run exactly 20 iterations in each time-step so that a direct comparison between the cpu and gpu performances can be done. the diagram in fig. 5, left, shows the number of frames per second for the considered models with the computations performed on the cpu and the gpu. one may notice that the single-threaded cpu implementation exhibits almost a linear dependence between the amount of data to be processed and the computational time. on the other hand, the gpu implementation demonstrates better utilization of available computational resources when a larger amount of data is to be processed. consequently, fig. 5, right, reveals greater simulation speed-up achieved by switching from the cpu to the gpu for larger models. 24 b. nutti, d. marinković one of the reasons for that is the fact that the warp scheduler can overlap memory transactions in a more efficient way when more active threads are available. fig. 5 simulations results using gpu and cpu implementations 9. conclusions the paper presents a very effective approach to performing real-time or nearly realtime interactive simulations with large, geometrically nonlinear deformations. the objective of real-time simulation is addressed in two complementary ways. the first consideration is related to the choice of an efficient fem formulation. the authors use the simplified co-rotational fem formulation that enables accounting for geometrically nonlinear effects to a large extent, whereby the efficiency and stability of numerical computation are kept over the course of simulation. this is achieved by extending the linear fem by the element-based rigid-body rotation and neglecting all other geometrically nonlinear effects. the resulting deformational behavior is rather realistic for deformations characterized by arbitrarily large rotations but relatively small strains. the second consideration is related to the choice of hardware components used to perform the computations. the general idea is simple and consists in switching the computation from the conventionally used cpu to modern gpus that offer ever greater computational power. however, typical fem computations involve geometries of irregular topology. this fact further implies random memory access patterns unsuitable for the modern gpu memory management. this paper presents an original solution as to how to increase the memory bandwidth usage by utilizing efficient memory access patterns. the presented solutions are very suitable for various virtual reality applications. however, as the computational efficiency strongly gains in importance in engineering applications, the development can be of large interest in this field as well. particularly the field of mbs dynamics with deformable bodies involved may benefit from the development. acknowledgement: this work is partially supported by the project iii41017 virtual human osteoarticular system and its application in preclinical and clinical practice, funded by the ministry of education and science of the republic of serbia. an approach to efficient fem simulations on graphics processing units using cuda 25 references 1. yang, y., xiao, r., he z., 2011, real-time deformations simulation of soft tissue by combining mass-spring model with pressure based method, proceedings of the 3rd ieee international conference on advanced computer control (icacc '11), harbin, china, pp. 506–510. 2. erleben, k., sporring, j., henrikssen k., dohlman, h., 2005, physics-based animation, charles river media, usa. 3. zehn, m., 2005, mbs and fem: a marriage-of-convenience or a love story?, benchmark int. magazine for eng. design&analysis, pp. 12-15. 4. mueller m., dorsey j., mcmillan l., jagnow r., cutler b., 2002, stable real-time deformations, proceedings of 2002 acm siggraph/eurographic symposium on computer animation, san antonio, usa, pp. 49-54. 5. hauth, m., strasser w., 2004, corotational simulation of deformable solids, journal of wscg, 12(1), pp. 137-144. 6. mueller, m., gross, m., 2004, interactive virtual materials, proceedings of graphics interface 2004, waterloo, canada, pp 239-246. 7. marinković, d., zehn, m., marinković z., 2012, finite element formulations for effective computations of geometrically nonlinear deformations, advances in engineering software, 50, pp. 3-11. 8. hestenes, m. r., stiefel, e., 1952, methods of conjugate gradients for solving linear systems, journal of research of the national bureau of standards, 49(6), pp. 409-436. 9. benzi, m. 2002, preconditioning techniques for large linear systems: a survey, journal of computational physics, 182(2), pp. 418-477. 10. cecka c., lew a. j., darve, e., 2011, assembly of finite element methods on graphics processors, international journal for numerical methods in engineering, 85(5), pp. 640-669. 11. mafi, r., sirouspour, s., 2013, gpu-based acceleration of computations in nonlinear finite element deformation analysis, international journal for numerical methods in biomedical engineering, doi: 10.1002/cnm.2607 12. georgescu s., chow, p., okuda, h., 2013, gpu acceleration for fem-based structural analysis, archives of computational methods in engineering 20(2), pp. 111-121 13. bathe, k. j., 1982, finite element procedures in engineering analysis, prentice-hall, inc., englewood cliffs, new jersey. 14. harris, m., optimizing parallel reduction in cuda, available at: http://developer.download.nvidia.com/compute/cuda/1.1beta/x86_website/projects/reduction/doc/reduction.pdf (access date: 14.02.2014.) pristup efikasnim mke simulacijama pomoću grafiĉkih karti primenom cuda-platforme rad predstavlja izuzetno efikasan pristup simulaciji dinamičkog ponašanja deformabilnih objekata primenom metode konačnih elemenata (mke) pri čemu se proračuni izvode primenom grafičkih karti (gk). predstavljeno rešenje smanjuje efekat „uskog grla” uzrokovanog memorijskim pristupima tako što grupiše podatke po parovma čvorova mke modela, nasuprot klasičnom pristupu kod koga se grupisanje vrši po elementima. time ova strategija redukuje memorijske pristupe neadekvatne sa stanovišta arhitekture memorije grafičkih karti. takođe, ovaj pristup koristi prednosti matrica zapisanih u kompaktnom obliku i pokazano je kako izbeći dalja potencijalna „uska grla” u algoritmu. da bi se deformaciono ponašanje opisalo na realističan način i pri velikim lokalnim rotacijama, iskoršćena je uprošćena korotaciona fem formulacija. ključne reči: korotaciona mke, grafička karta, cuda, kompaktna matrica, metoda konjugovanih gradijenata, geometrijska nelinearnost. http://developer.download.nvidia.com/compute/cuda/1.1http://developer.download.nvidia.com/compute/cuda/1.113162 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 773 785 https://doi.org/10.22190/fume241003045s © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dynamic stability analysis of a coupled moving bogie system vladimir stojanović1,2, lubomir dimitrov3, pancho tomov3, deli li2, vlastimir nikolić1 1univerity of niš, faculty of mechanical engineering, serbia 2lakehead university, ontario, canada 3technical university of sofia, bulgaria orcid ids: vladimir stojanović https://orcid.org/0000-0003-1011-4979 lubomir dimitrov https://orcid.org/0000-0001-8584-5958 pancho tomov https://orcid.org/0000-0003-1649-6947 deli li https://orcid.org/0009-0005-5375-402x vlastimir nikolić https://orcid.org/0009-0004-5752-5083 abstract. this study explores the dynamic behavior and vibration stability of a complex moving mechanical oscillator coupled with a three-part viscoelastically connected continuous beam-foundation system. elastic waves generated by the oscillator can destabilize the system, a scenario common in high-speed trains and vehicles. the paper presents an improved analytical approach, focusing on the effects of a variable primary stiffness suspension on vibration stability. using the d-decomposition method and the principle of the argument, instability intervals are determined for varying stiffness configurations, highlighting the benefits of nontraditional suspension systems in improving local stability. key words: instability, critical velocity, d-decomposition method, principle of argument 1. introduction the rapid expansion of road and railway transport has intensified the need for a detailed analysis of the dynamic effects of physical parameters on the stability of mechanical systems. vibration instability, often resulting from energy transfer through anomalous doppler waves [1, 2], is a particular concern when high-speed trains exceed the propagation speed of elastic waves within railway structures. this phenomenon has been extensively studied, particularly in simple mechanical models, leading to further investigations into more complex systems under moving loads. michaltsos [3] analyzed the linear dynamic response of a simply supported beam subjected to a moving load with constant magnitude and variable velocity. the study examines the effects of acceleration and deceleration on the beam's behavior under both a single-axle load and a two-axle vehicle model, with damping considered for the latter. subsequent studies, such as those by kim [4] and śniady [5], incorporated additional factors like rotary inertia and axial compressive forces into the received: october 03, 2024 / accepted november 06, 2024 corresponding author: vladimir stojanović faculty of mechanical engineering, university of niš, a. medvedeva 14, 18000 niš, serbia e-mail: vladimir.stojanovic@masfak.ni.ac.rs https://orcid.org/0000-0003-1011-4979 https://orcid.org/0000-0001-8584-5958 https://orcid.org/0000-0003-1649-6947 https://orcid.org/0009-0005-5375-402x https://orcid.org/0009-0004-5752-5083 774 v. stojanović, l. dimitrov, p. tomov, d. li, v. nikolić stability analyses, enriching the understanding of these dynamic systems. basu and rao [6] derived analytical solutions for the steady-state response of an infinite beam on a viscoelastic foundation under a constant-velocity moving load. similarly, froio et al. [7] explored the steady-state response of a uniform, infinite euler-bernoulli beam on a pasternak foundation subjected to a constant-velocity moving load. they obtained a universal closed-form solution using a fourier transform, capable of representing the response for any set of beam and foundation parameters. mazilu presented significant results in [8], addressing the interaction between a vehicle, simplified to a two-mass oscillator, and a slab track modeled as an infinite structure composed of elastically supported double euler-bernoulli beams. to carry out the time-domain analysis, a semi-analytical method was developed, leveraging the remarkable properties of the slab track's time-domain green's functions. meanwhile, karimi and ziaeirad [9] studied the nonlinear coupled vibration analysis of a beam with moving supports under the action of a moving mass. researchers such as verichev and metrikine [10, 11, 12] successfully applied the d-decomposition method to define stability regions for systems with moving masses along beams. similarly, dimitrovová and varandas [13] examined the transient dynamic response of a beam supported on a foundation with a sudden change in stiffness and subjected to a constant-velocity moving force, utilizing finite integral transformations. in a related study, dimitrovová [14] proposed a repetitive procedure for determining oscillation frequencies induced by a moving mass. additionally, erbas et al. [15] performed a three-dimensional analysis of near-resonant regimes of a point load moving uniformly along the surface of a coated elastic half-space. svedholm et al. [16] contributed valuable insights into the behavior of loads moving on beams of finite length, while lee and renshaw [17] introduced a new solution technique for solving the moving mass problem in nonconservative, linear, distributed parameter systems using complex eigenfunction expansions. all these works underscored the importance of advanced analytical methods, such as complex eigenfunction expansions and finite integral transformations, for understanding the interactions between continuous and discrete systems. these methods are critical for analyzing the dynamic challenges posed by high-speed trains, where the interaction between the moving oscillator (train) and the supporting structure (rail) produces complex responses. stojanović et al. [18] conducted an in-depth analysis of the vibration stability of a coupled bogie system moving uniformly along a flexibly supported infinite high-order shear deformable coupled beam system on a viscoelastic base. this paper presents a model of a complex coupled system with four supports, featuring varying suspension configurations. the stability of vibrations is analyzed by examining the interaction between an infinite beam and a complex mechanical oscillator moving uniformly. building upon the work of stojanović and petković [19], which explored the stochastic stability of complex finite beam and beam-foundation systems, this study extends the analysis to a high-order shear deformable model of an infinite beam system subjected to a moving oscillator. stability regions for the complexly coupled mechanical oscillator are determined for different stiffness moduli of the primary suspension, providing a comprehensive analysis of vibration stability in relation to varying suspension configurations. 2. modeling of the complex moving oscillator the model of the complex foundation system is illustrated in fig. 1. in this configuration, the rail is represented as a supported infinite beam, while the slab and base are modeled as dynamic stability analysis of a coupled moving bogie system 775 additional beams with free boundary conditions. these elements are interconnected by viscoelastic spring-damping components, which are crucial for capturing the real behavior of the track structure. these viscoelastic connections account for both stiffness and damping, significantly influencing the system's dynamic response to moving loads. to accurately capture the behavior of the multilayer beam system, the high-order shear deformable beam theory (also known as the reddy-bickford beam theory) is employed. this advanced theory extends beyond the classical beam theory by incorporating the effects of shear deformation and rotational inertia. the reddy-bickford theory allows for a more realistic and detailed representation of the coupled mechanical interactions between the beam layers in the system. the interaction between the infinite coupled beam-layer system is described by the following set of equations. ψ1+kf3(w1-w2)+νf3(ẇ1-ẇ2)=p(x,t), (1) φ1=0, (2) ψ2+kf2(w2-w3)-kf3(w1-w2)+νf2(ẇ2-ẇ3)-νf3(ẇ1-ẇ2)=0, (3) φ2=0, (4) ψ3+kf1w3-kf2(w2-w3)+νf1ẇ3-νf2(ẇ2-ẇ3)=0, (5) φ3=0, (6) where ψi=ci1ẅi-ci2(φi '+wi '')-ci3ẅi ''+ci4wi (4) +ci5φ̈ i '+ci6φ i ''', φi=di1φ̈ i -di2φ i ''+di3(φi +wi ')+di4ẅi '+di5wi '' ', i=1, 2, 3, and where cik and dij are constants obtained and documented in the paper by stojanović and petković [20] in the form for one beam (appropriate indices need to be added for each beam individually) and can be utilized for all others. e and g are the young’s modulus and the shear modulus of the beam material; ρ is the mass density of the beam material; kfi and νfi are the stiffness moduli and the viscosity shear moduli of the foundation per unit length. wi(x, t) are vertical deflections of the beams and φ i (x,t) are the angles of rotation of the cross-section of the beams. the following notation was used to write the expression ẅi= ∂ 2 wi ∂t2 , wi '= ∂wi ∂x , wi ''= ∂ 2 wi ∂x2 , φ̈ i = ∂ 2 wi ∂t2 , φ i '= ∂wi ∂x , φ i ''= ∂ 2 wi ∂x2 . right hand side of eq. (1) represents the reaction of the bogie system and has the following form p(x,t)=-δ(x-vt-d3) (m d 2 w2 01 dt2 +l̃214) -δ(x-vt-d2) (m d 2 w2 02 dt2 +l̃223) -δ(x-vt-d1) (m d 2 w1 01 dt2 +l̃112) -δ(x-vt) (m d 2 w1 02 dt2 +l̃121) . (7) w1 01(t), w1 02(t), w2 01(t), w2 02(t), are the vertical displacements of the masses (wheels); w1 0(t), w2 0(t) are the vertical displacements of the centers of mass of the bars that represent the bogies’ bodies, and w0 0(t) is the vertical displacement of the bar that represents the car’s 776 v. stojanović, l. dimitrov, p. tomov, d. li, v. nikolić body. θ0, θ1, and 𝜃2 are the angles of rotation of the bars around their centers of mass. m0, m1 and j0, j1 are the masses and moments of inertia of the bars; k0 and ϵ0 are the stiffness modulus and the damping factor of the primary suspension; k1 and ϵ1 are the stiffness modulus and the damping factor of the secondary suspension. the dirac deltafunction is designated with δ (∙). the factors ζ1, ζ2, ζ3, and ζ4 are multipliers for the stiffness modulus of the primary suspension. fig. 1 complexly coupled mechanical oscillator together with eqs. (1-3) the following set of eqs. (8-13) represent the full form of the dynamic equilibrium of complex moving oscillator m1 d 2 w2 0 dt2 -l̃214-l̃223+l̃2 =0, (8) (j1+m 1 r1 2) d 2 θ2 dt2 +l̃214 ζ4 ζ3+ζ4 dl-l̃223 ζ3 ζ3+ζ4 dl=0, (9) m1 d 2 w1 0 dt2 -l̃112-l̃121+l̃1=0, (10) (j1+m 1 r2 2) d 2 θ1 dt2 +l̃112 ζ2 ζ1+ζ2 dl-l̃121 ζ1 ζ1+ζ2 dl=0, (11) dynamic stability analysis of a coupled moving bogie system 777 m0 d 2 w0 0 dt2 -l̃1-l̃2=0, (12) j0 d 2 θ0 dt2 +(l̃2-l̃1) dl0 2 =0, (13) w2 01(t)=w(vt+d3,t), w2 02(t)=w(vt+d2,t), w1 01(t)=w(vt+d1,t), w1 02(t)=w(vt,t), lim |x-vt|→∞ wi(x,t)=0, lim |x-vt|→∞ φ i (x,t)=0, (14) where l̃ijp=ζpk 0 (wi 0j -wi 0+θilij(-1) j+1)+ϵ0 d dt (wi 0j -wi 0+θilij(-1) j+1), l̃1=k 1 (w1 0-w0 0-θ0 dl0 2 -r2θ1)+ϵ1 d dt (w1 0-w0 0-θ0 dl0 2 -r2θ1) , l̃2=k 1 (w2 0-w0 0+θ0 dl0 2 +r1θ2)+ϵ1 d dt (w2 0-w0 0+θ0 dl0 2 +r1θ2) , l11= ζ2 ζ1+ζ2 dl,l12= ζ1 ζ1+ζ2 dl,l21= ζ4 ζ3+ζ4 dl, l22= ζ3 ζ3+ζ4 dl, r1=l22dl 2 , r2=l11dl 2 . by using a moving reference system ξ=x-vt the previous equations can be rewritten. it is now possible to analyze the characteristic equation of vibrations by applying integral transforms to the modified system of equations wsi(ξ,s)=∫ wi(ξ,t) e-stdt ∞ 0 , φsi(ξ,s)=∫ φ i (ξ,t) e-stdt, ∞ 0 ŵk,si(k,s)=∫ wsi(ξ,s)e-ikξdξ ∞ -∞ , φ̂si(k,s)=∫ φsi(ξ,s)e-ikξdξ ∞ -∞ w1 01(s)=∫ w1 01(t)e-stdt ∞ 0 , w1 02(s)=∫ w1 02(t)e-stdt ∞ 0 , w2 01(s)=∫ w2 01(t)e-stdt ∞ 0 , w2 02(s)=∫ w2 02(t)e-stdt ∞ 0 , w0 0(s)=∫ w0 0(t)e-stdt, ∞ 0 w1 0(s)=∫ w1 0(t)e-stdt, ∞ 0 w2 0(s)=∫ w2 0(t)e-stdt, ∞ 0 θ0=θ 0 (s)=∫ θ0(t)e -stdt, ∞ 0 θ1=θ 1 (s)=∫ θ1(t)e -stdt, ∞ 0 θ2=θ 2 (s)=∫ θ2(t)e -stdt. ∞ 0 778 v. stojanović, l. dimitrov, p. tomov, d. li, v. nikolić by using the procedure given in stojanović et al. [18] we are presuming that the initial conditions of the problem eqs. (1-3) and eqs. (8-13) are insignificant (the initial conditions do not have an impact on the system stability because of the fact that the governing equations are linear) and with keeping in mind that: w1 02(s)= w(0,s), w1 01(s)= w(d1,s) ,w2 02(s)= w(d2,s) ,w2 01(s)= w(d3,s), the application of the transforms results in the system of algebraic equations after elimination φ̂si(k,s) in the present form f(k,s) ŵk,s(k,s)=e-id1k(-ζ2k0 w(d1,s) -ms2 w(d1,s) -sϵ0 w(d1,s) +ζ2k0w1 0(s)+sϵ0w1 0(s)ζ2 2 θ1k0dl ζ1+ζ2 ζ2θ1sϵ0dl ζ1+ζ2 ) +e-id2k(-ζ3k0 w(d2,s) -ms2 w(d2,s) -sϵ0 w(d2,s)+ζ3k0w2 0(s) +sϵ0w2 0(s)+ ζ3 2 θ2k0dl ζ3+ζ4 + ζ3θ2sϵ0dl ζ3+ζ4 ) +e-id3k(-ζ4k0w (d3,s)-ms2w(d3,s)-sϵ0w(d3,s) +ζ4k0w2 0(s)+sϵ0w2 0(s)ζ4 2 θ2k0dl ζ3+ζ4 ζ4θ2sϵ0dl ζ3+ζ4 ) -ms2w(0,s)-sϵ0w(0,s)+ζ1k0w1 0(s)+sϵ0w1 0(s)+ ζ1 2 θ1k0dl ζ1+ζ2 + ζ1θ1sϵ0dl ζ1+ζ2 , (15) m1s2w2 0(s)-ζ3k0 w(d2,s) -ζ4k0 w(d3,s) -sϵ0 w(d2,s) -sϵ0 w(d3,s) +ζ3k0w2 0(s)+ζ4k0w2 0(s)-k1w0 0(s)+k1w2 0(s)-sϵ1w0 0(s)+2sϵ0w2 0(s) +sϵ1w2 0(s)+θ2k1 ( ζ3dl ζ3+ζ4 dl 2 )+ ζ3 2 θ2k0dl ζ3+ζ4 ζ4 2 θ2k0dl ζ3+ζ4 + 1 2 θ0k1dl0+θ2sϵ1 ( ζ3dl ζ3+ζ4 dl 2 )+ ζ3θ2sϵ0dl ζ3+ζ4 ζ4θ2sϵ0dl ζ3+ζ4 + 1 2 θ0sϵ1dl0=0, (16) θ2j1s2+θ2m1s2 ( ζ3dl ζ3+ζ4 dl 2 ) 2 ζ3 2 k0dl w(d2,s) ζ3+ζ4 + ζ4 2 k0dl w(d3,s) ζ3+ζ4 ζ3sϵ0dlw(d2,s) ζ3+ζ4 + ζ4sϵ0dlw(d3,s) ζ3+ζ4 + ζ3 2 k0dlw2 0(s) ζ3+ζ4 ζ4 2 k0dlw2 0(s) ζ3+ζ4 + ζ3sϵ0dlw2 0(s) ζ3+ζ4 ζ4sϵ0dlw2 0 (s) ζ3+ζ4 + ζ3 3 θ2k0dl 2 (ζ3+ζ4) 2 + ζ4 3 θ2k0dl 2 (ζ3+ζ4) 2 + ζ3 2 θ2sϵ0dl 2 (ζ3+ζ4) 2 + ζ4 2 θ2sϵ0dl 2 (ζ3+ζ4) 2 =0, (17) m1s2w1 0(s)-ζ2k0 w(d1,s) -sϵ0 w(d1,s) -ζ1k0 w(0,s) -sϵ0 w(0,s) +ζ1k0w1 0(s)+ζ2k0w1 0(s)-k1w0 0(s)+k1w1 0(s)-sϵ1w0 0(s)+2sϵ0w1 0(s) dynamic stability analysis of a coupled moving bogie system 779 +sϵ1w1 0(s)-θ1k1 ( ζ2dl ζ1+ζ2 dl 2 )+ ζ1 2 θ1k0dl ζ1+ζ2 ζ2 2 θ1k0dl ζ1+ζ2 1 2 θ0k1dl0 -θ1sϵ1 ( ζ2dl ζ1+ζ2 dl 2 )+ ζ1θ1sϵ0dl ζ1+ζ2 ζ2θ1sϵ0dl ζ1+ζ2 1 2 θ0sϵ1dl0=0, (18) θ1j1s2ζ1 2 k0dlw(0,s) ζ1+ζ2 + ζ2 2 k0dlw(d1,s) ζ1+ζ2 ζ1sϵ0dlw(0,s) ζ1+ζ2 + ζ2sϵ0dlw(d1,s) ζ1+ζ2 + ζ1 2 k0dlw1 0(s) ζ1+ζ2 ζ2 2 k0dlw1 0(s) ζ1+ζ2 + ζ1sϵ0dlw1 0(s) ζ1+ζ2 ζ2sϵ0dlw1 0(s) ζ1+ζ2 + ζ1 3 θ1k0dl 2 (ζ1+ζ2) 2 + ζ2 3 θ1k0dl 2 (ζ1+ζ2) 2 +θ1m1s2 ( ζ2dl ζ1+ζ2 dl 2 ) 2 + ζ1 2 θ1sϵ0dl 2 (ζ1+ζ2) 2 + ζ2 2 θ1sϵ0dl 2 (ζ1+ζ2) 2 =0, (19) m0s2w0 0(s)+2k1w0 0(s)-k1w1 0(s)-k1w2 0(s)+2sϵ1w0 0(s)-sϵ1w1 0(s) -sϵ1w2 0(s)+θ1k1 ( ζ2dl ζ1+ζ2 dl 2 ) -θ2k1 ( ζ3dl ζ3+ζ4 dl 2 )+θ1sϵ1 ( ζ2dl ζ1+ζ2 dl 2 ) -θ2sϵ1 ( ζ3dl ζ3+ζ4 dl 2 )=0, (20) θ0j0s2+ 1 4 dl0(k1+sϵ1) [ 2(ζ1+ζ2)(ζ3+ζ4)(w2 0(s)-w1 0(s))+dl((ζ1+ζ2)(ζ3-ζ4)θ2-(ζ1-ζ2)(ζ3+ζ4)θ1) (ζ1+ζ2)(ζ3+ζ4) +2θ0dl0]=0. (21) the beam-foundation reaction is mathematically represented by the function f(k,s), which appears on the left-hand side of eq. (15). we obtain the following system of algebraic equations with respect to the laplace-displacements of the contact points ws(0,s), ws(d1,s), ws(d2,s), ws(d3,s) after using the inverse fourier transform over k to the system of eqs. (34-40), and eliminating w0 0, w 1 0 , w2 0, θ0, θ 1 , θ2 and subsequently setting ξ=0, ξ=d1, ξ=d2, ξ=d3 𝐆 ∙ws[(0, d1, d2, d3), s]=0 ⇔ [ g 11 ⋯ g 14 ⋮ ⋱ ⋮ g 41 ⋯ g 44 ] { ws(0,s) ws(d1,s) ws(d2,s) ws(d3,s)} ={ 0 0 0 0 }. (22) we get the system of integrals that can be seen in the eq. (22) as a result of the application of the inverse fourier transform. system of algebraic eqs. (15-21) possesses a non-trivial solution provided that the determinant of the matrix g satisfies the condition det g=0, (23) in the following sections the model stability is analyzed by investigating the eigenvalues of the characteristic eq. (23). 780 v. stojanović, l. dimitrov, p. tomov, d. li, v. nikolić 3. solution method and stability criteria having at least one root of the characteristic eq. (23) with a positive real part can lead to instability in the vibration of the complex moving oscillator. this equation is an integral equation corresponding to the laplace variables and the issue of finding its complex roots is crucial for the model under study. to simplify the solution, the d-composition method, presented in stojanović et al. [18], was used. in this approach, k0 is treated as a complex parameter, and the straight lines = iω is mapped onto the complex plane k0. the ddecomposition divides the complex plane k0 into regions with different numbers of roots for eq. (23), focusing on those with re(s)>0. by using the d-decomposition method we can determine a relative variation of the number of the unstable roots possessing the stiffness k0, however not the number itself. in order to find the number of unstable roots for any single value of stiffness k0 we are applying the argument principle, explained in stojanović et al. [18]. parametric analyses are performed for the super-critical case when the complex moving oscillator system’s velocity is higher than the certain critical velocity vcr, because in sub-critical case (v< vcr), there seems not to be any crossing point of the ddecomposition curves and the positive part of the real axes. this may indicate that in the sub-critical case the number of unstable roots (roots with a positive real part) of the charactieristic equation does not change with the stiffness of the supports of the complex moving oscillator k0. on the other hand, in the super-critical case, the number of unstable roots starts varying the moment when k0 passes the critical values of the stiffness. 4. results and discussion the d-decomposition curves were derived using the same model parameters as in stojanović et al. [18]. the referent case (referred to as case 1) represents the traditional primary suspension system, where the stiffness k0 is defined with factors ζ1=ζ2=ζ3=ζ4=1. variations in primary suspension stiffness for other cases are outlined in table 1. for different values of k0, the d-decomposition curves and the corresponding number of unstable roots were computed at supercritical velocity. the dynamic stability of the system was thoroughly analyzed, focusing on the instability that arises due to supercritical velocity. fig. 2 illustrates the d-decomposition curves for the referent case, while table 2 provides details on the number of unstable roots for different intervals of k0. table 1 varying non-dimensional parameter of the primary suspension varying factor ζ 1 ζ 2 ζ 3 ζ 4 case 1 1 1 1 1 case 2 8 1 1 1 case 3 1 8 1 8 case 4 1 8 1 4 dynamic stability analysis of a coupled moving bogie system 781 in this case a stable region (with no unstable roots) was identified within the stiffness range k0∈(3.067, 5.074)×107 n/m. it is important to note that for the selected supercritical velocity, other intervals of k0 exhibit varying numbers of unstable roots, from which the instability intervals were calculated. this reference case represents the classical coupled moving oscillator model. table 2 review of instability regions (v=1.12vcr) k0[107n/m] 0 2.573 3.067 5.074 5.076 +∞ case 1 4 2 0 2 4 fig. 2 d-decomposition curves (case 1) in case 2, where the first suspension has a higher stiffness (ζ1=8, while the other factors remaining ζ2=ζ3=ζ4=1), the stable region shifts to k0∈(2.8834, 5.0793)×107 n/m (see table 3). the factor ζ1 was selected after numerous numerical experiments showed no significant changes in dynamic stability within the range 1<ζ1<8. complete stable region is determined according to obtained d-decomposition curves depicted in fig.3. the complete stable region was determined based on the d-decomposition curves shown in fig. 3. these results demonstrate that adjusting the stiffness of the suspension can expand the stability range and shift it toward lower stiffness values. this comparison highlights the advantages of the proposed varying stiffness model in controlling stability regions. table 3 review of instability regions (v=1.12vcr) k0[107n/m] 0 0.4691 1.5851 2.8365 2.8834 5.0793 7.1094 +∞ case 2 4 2 4 2 0 2 4 782 v. stojanović, l. dimitrov, p. tomov, d. li, v. nikolić fig. 3 d-decomposition curves (case 2) in case 3, where the stiffness factors were chosen symmetrically (ζ1=ζ3=8 and ζ2=ζ4=1), the stable regions are distributed across two distinct intervals: k0∈(0.5077, 1.6516)×107n/m and k0∈(2.6090, 6.1575)×107 n/m. compared to case 1, where the stable region was limited k0∈(3.067, 5.074)×107 n/m, case 3 shows a significantly larger stability range, as depicted in fig. 4. finally, in case 4 the last suspension has reduced stiffness, with the ζ factors defined as ζ1=1, ζ2=8, ζ3=1, ζ4=4. the results for this case are presented in table 5 and illustrated in fig. 5. in this configuration, the stable regions are found in the intervals k0∈(0.9368, 1.9278)×107 n m⁄ and k0∈(2.3176, 5.7494)×107 n m⁄ . compared to the reference case 1, there is a clear expansion of the stability range. however, in comparison to case 3, the overall stable region is slightly smaller. nevertheless, the stability zones are shifted to lower values of the stiffness coefficient k0, indicating that stability can be achieved at reduced stiffness. these findings emphasize the potential benefits of varying the stiffness coefficients in the primary suspension system. the results suggest that further exploration and deeper investigation into this approach could yield additional insights and improvements in dynamic stability. table 4 review of instability regions (v=1.12vcr) k0[107n/m] 0 0.4394 0.5077 1.6516 2.6090 6.1575 6.2844 +∞ case 3 4 2 0 2 0 2 4 dynamic stability analysis of a coupled moving bogie system 783 fig. 4 d-decomposition curves (case 3) table 5 review of instability regions (v=1.12vcr) k0[107n/m] 0 0.4709 0.9368 1.9278 2.3176 5.7494 6.2270 +∞ case 3 4 2 0 2 0 2 4 fig. 5 d-decomposition curves (case 4) 784 v. stojanović, l. dimitrov, p. tomov, d. li, v. nikolić 5. conclusions the study presented a detailed analysis of the stability of a complex coupled mechanical system, specifically focusing on the effects of varying stiffness in the primary suspension system under moving loads. through the use of d-decomposition curves, the stability regions were identified for different configurations of suspension stiffness, revealing significant insights into the dynamics of the system. the results showed that varying the stiffness of the primary suspension significantly affects the stability of the system. in the referent case (case 1), a stable region was identified within the stiffness range k0∈(3.067, 5.074)×107 n/m. however, when the stiffness of the first suspension was increased in case 2, the stability region shifted slightly and expanded, demonstrating the potential benefits of tuning the system’s stiffness parameters. case 3, where the suspension was symmetrically adjusted, exhibited a significantly larger stability range compared to case 1, with stability occurring in two distinct regions. this indicates that symmetric variations in stiffness can lead to broader stability domains. and in case 4, where the stiffness of the last suspension was reduced, the stable regions were shifted to lower stiffness values, allowing for stability at reduced stiffness values. though the overall stable area was slightly smaller than in case 3, the findings suggest that varying suspension stiffness can optimize stability even at lower stiffness values. overall, this research highlights the importance of carefully adjusting suspension stiffness to enhance the dynamic stability of mechanical systems under moving loads. the results suggest that further exploration of suspension stiffness variations could offer valuable strategies for improving the stability and performance of complex mechanical systems in realworld applications, particularly in the context of transportation infrastructure. acknowledgement: we acknowledge the financial support from natural sciences and engineering research council of canada through discovery grants and from the government of canada’s new frontiers in research fund (nfrf) [ nfrfr-2021-00262 ] and research was also financially supported by the ministry of science, technological development and innovation of the republic of serbia (contract no. 451-03-47/2023-01/ 200109). references 1. fryba, l., 1999, vibration of solids and structures under moving loads, telford, london. 2. stojanović, v., kozić, p., petković, m.d., 2017, dynamic instability and critical velocity of a mass moving uniformly along a stabilized infinity beam, international journal of solids and structures, 108, pp. 164–174. 3. michaltsos, g.t., 2002, dynamic behavior of a single-span beam 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objects with stabilizers and unconventional couplings, journal of sound and vibration, 570, 118020. 19. stojanović, v., petković, m., 2013, moment lyapunov exponents and stochastic stability of a threedimensional system on elastic foundation using a perturbation approach, asme journal of applied mechanics, 80(5), 051009. 20. stojanović, v., petković, m.d., 2016, nonlinear dynamic analysis of damaged reddy–bickford beams supported on an elastic pasternak foundation, journal of sound and vibration, 385, pp. 239–266. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 1, 2015, pp. 21 25 1non-frictional damping in the contact of two fibers subject to small oscillations udc 539.3 mikhail popov 1,2 1 berlin university of technology, germany 2 national research tomsk state university, russia abstract structural damping is discussed for the contact of two fibers in a woven material. in the presence of both normal and tangential oscillations, structural (relaxation) damping takes place even with perfect sticking in the contact, where slip-related frictional damping disappears. for the case of an infinite coefficient of friction and small amplitudes a closed-form solution for energy lost during one oscillation cycle is obtained. key words: woven composites, structural damping, oscillating contacts 1. introduction the present paper is concerned with internal damping in woven materials. when fabrics are deformed, energy is dissipated in the contacts between fibers, and it is well known that at least part of this dissipation is due to friction in partial slip zones of the contact. exact solutions for frictional damping in the contact of spheres due to tangential oscillations go back to mindlin et. al. [1] and are also applicable to the contact of two crossed cylinders (such as the fibers in a woven material). damping in the presence of both normal and tangential oscillations, however, has never been described exhaustively and it remains a current research topic [2],[3]. recently it has been suggested [4] that the superposition of normal and tangential oscillations leads, in addition to slip-related frictional dissipation, to a new type of non-frictional damping that is caused by elastic relaxation due to variations in normal load and therefore contact area. it is found that in the absence of slip (an infinite coefficient of friction) and for small oscillation amplitudes, the energy dissipated during an oscillation cycle is described by received january 05, 2015 / accepted february 16, 2015 corresponding author: popov mikhail technische universität berlin, 10623 berlin, germany e-mail: mikhail.popov@tu-berlin.de original scientific paper 22 m. popov     2* 2 0* 20 0 2 8 sin 3 z n x z fg q u u e u     (1) where  0 xu and  0 zu are the tangential and normal oscillation amplitudes, 0 is the phase shift between the oscillations (the oscillation frequencies are identical), fn is the normal contact force, uz the indentation depth (relative approach of bodies). e * and g * are the reduced elastic and shear moduli and can be expressed through elastic modulus e and poisson's ratio v as follows, when both contact partners are made from the same linear, homogeneous, isotropic material. * 22(1 ) e e    (2) * (1 )(2 ) e g      (3) in the present work we specialize the above result for a system of two fibers crossed at right angles, representing a single mesh cell of a woven material. small oscillations that are applied to the end of one fiber produce oscillations in the contact, leading to structural damping (and frictional damping, if slip is permitted). all assumptions from [4] (linearly elastic materials without viscous effects, infinite coefficient of friction, small amplitudes) are used here as well, so as to isolate the contribution of structural damping to overall losses. 2. analysis we consider a very simple model representing a mesh cell of a woven material: two fibers with circular sections (with radius r) that are crossed at right angles, fig.1. the fibers lie in x, y plane, while the upward-facing axis is labeled z. three of the four fiber ends are rigidly embedded in the plane at z = 0, while one end is connected to a parallel guide that permits motion in x, y plane (horizontal and vertical). these boundary conditions are neither the only possible nor necessarily the most representative of real fabrics. the above model is chosen for its simplicity, while other possible configurations are left for future work. the movable end is pre-stressed by deflecting it downwards by wz,0, which is of the order of 2r in woven materials, due to symmetrical boundary conditions. through this initial displacement, contact between the fibers is established, and base loading f (0) is produced in the contact. in addition, the movable end of the fiber is forced to oscillate with amplitudes  xw ,  zw , a common frequency and phase shift 0. our general approach is as follows: firstly, the oscillation amplitudes of the movable fiber end are related to force oscillations, with certain amplitudes, in the contact. linear beam theory is used for this, while the influence of indentation depth uz is neglected (our general assumption is that wz,0 >> uz >> w). the force oscillations in the contact are then related to geometrical oscillations through the contact stiffness, which is itself determined by the contact configuration, and therefore wz,0. non-frictional damping in the contact of two fibers subject to small oscillations 23 x y z l l w0,x w0,z fig. 1 two crossed fibers are considered as elastic beams with circular cross-section consider a beam of length 2l that is stressed with a contact force fz in the middle and deflected by w0,z at one end. the deflection of the central point (at x = l) of the beam with these boundary conditions is known to be [5] 3 0,z , ( ) 24 2 z i z wf l i l e w    , (4) where i = r 4 /4 is the area moment of inertia. for the beam with two fixed ends, only the first component, due to the contact force in the middle, is present: 3 , ( 2 ) 4 z ii zw f l i l e  . (5) the difference between the two is equal to indentation depth uz in the contact: 3 0,z , , 12 ( ) ( ) 2 z i z iz i zu w l w l wf l ei     . (6) as noted above, we assume that the indentation depth is small compared to the deflection of the beam, and apply non-penetration condition, uz = 0, which leads to a linear relationship between contact force and deflection of the free end: 0, 3  z zf ei w l . (7) for tangential loading, the lower beam is stressed length-wise; its deformation therefore can be neglected. the equations in this case become 3 0, ,x 24 2 ( ) xx i wf l i l e w    , (8) , 0ii xw . (9) proceeding as above, we obtain 0,x 3 xf ei w l . (10) 24 m. popov the force is thus linearly proportional to the deflection of the end of the movable beam. if the latter is now oscillating according to ,0 sinz z zw w w t  and x 0sin( )xw w t    , then the amplitudes of the force oscillations in the contact are 3   z zw ei l f , (11) 3   xx w ei l f . (12) now, when the oscillation of contact forces is known, the corresponding components of relative displacement of contacting bodies, ux and uz can be found by dividing the force increments by contact stiffness kx in the tangential direction or kz in normal direction. the latter are known to be *2xk g a , (13) *2zk e a , (14) where a, the contact radius, is equal to zru in the contact of a sphere with a plane or the contact of two crossed cylinders [6]. the derivative of the normal contact stiffness with respect to indentation depth uz is given by 2 2 *      n z z z z f k r e u uu . (15) one last step is necessary to tie all equations together: indentation depth uz, which, for a spherical contact, is given by [6]   /3 * 0 2 3 4         z f u e r , (16) where , (0) 0 3zf w l ei   is the initial loading determined with eq. (7). substituting all factors into the relaxation-damping eq. (1) and simplifying, gives the following result: 2 4 / 32 / 3 5 5/ 3 2 02 1/ 3 ,0 2 (1 )(2 ) 4 sin 3 (1 ) z x z r e l w r q w w                           . (17) by introducing fiber aspect ratio l r  , the normalized initial displacement 0 /  zww r and grouping some of the factors under 2 / 3 5/ 3 2 1/ 3 2 (1 )(2 ) 4 3 (1 ) q             , (18) we can write the result more compactly as   25 4/3 2 0 0sin    x zq q w e ww . (19) non-frictional damping in the contact of two fibers subject to small oscillations 25 note that q varies only slightly for typical values of v. its numerical value is approximately 45 for 0.2  , 47 for 0.3  and 51 for 0.5  ). 3. discussion the obtained result for a system of two crossed fibers is similar to eq. (1) that describes relaxation damping when oscillations are applied to the contact directly. in particular, the proportionality to the square of the tangential oscillation amplitude, and the modulus of the normal oscillation amplitude is preserved, which is, of course, not surprising, since linearity is assumed in the derivation. more interesting is the inverse proportionality to the fifth power of the aspect ratio of the fibers, which means that the effect will be much more pronounced in densely woven fabrics than in sparse ones. the obtained result is only valid for an infinite coefficient of friction. an interesting avenue for future work would be to consider realistic coefficients of friction and to determine the relative importance of frictional and structural damping. also, although a physical interpretation of relaxation damping in perfect stick conditions is given in [4], the underlying mechanism in the presence of sliding is yet to be determined. other unexplored possibilities involve other boundary conditions for the mesh cell, embedding the cell in a viscous medium (which would extend the results to woven composites), as well as experimental verification. acknowledgements: the author would like to thank valentin l. popov for many useful discussions. this work is supported in part by tomsk state university academic d.i. mendeleev fund program. references 1. mindlin r.d., mason w.p., osmer j.f., deresiewicz h., 1952, effects of an oscillating tangential force on the contact surfaces of elastic spheres, pros. 1st us national congress of applied mechanics, asme, new york, pp. 203–208. 2. davies m., barber j.r., hills d.a., 2012, energy dissipation in a frictional incomplete contact with varying normal load, int. j. mech. sci., 55, pp. 13–21. 3. putignano c., ciavarella m., barber j.r., 2011, frictional energy dissipation in contact of nominally flat rough surfaces under harmonically varying loads, j. of the mechanics and physics of solids, 59, pp. 2442–2454. 4. popov m., popov v.l., pohrt r., 2014, relaxation damping in oscillating contacts, arxiv:1410.3238 [cond-mat.soft]. 5. landau l.d., lifshitz e.m., 1970, theory of elasticity, 2nd english edition, §2, pergamon press. 6. popov v.l., 2010, contact mechanics: physical principles and applications, springer-verlag. http://arxiv.org/abs/1410.3238 11095 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 711 720 https://doi.org/10.22190/fume230927014b © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper an integrated optimization of production and preventive maintenance scheduling in industry 4.0 samane babaeimorad1, parviz fattahi2, hamed fazlollahtabar3, mahmood shafiee4 1faculty of engineering, alzahra university, tehran, iran 2department of industrial engineering faculty of engineering, alzahra university, tehran, iran 3department of industrial engineering, school of engineering, damghan university, damghan, iran 4schools of engineering, university of kent, canterbury ct2 7nt, united kingdom orcid ids: samane babaeimorad https://orcid.org/0000-0001-5409-3343 parviz fattahi https://orcid.org/0000-0003-3039-2220 hamed fazlollahtabar https://orcid.org/0000-0003-3053-4399 mahmood shafiee https://orcid.org/0000-0002-9032-8354 abstract. preventive maintenance (pm) plays an important role in maximizing machine reliability, improving production efficiency and reducing repair costs. due to the importance of pm in manufacturing environments, it is necessary to develop an integrated model for scheduling production jobs and maintenance interventions on the machines. on the other side, with the advent of industry 4.0 and the transformation of factories into smart factories, the production and maintenance processes generate huge volume of data on realtime basis. despite the importance of the issue and the competitiveness of manufacturing companies in the world, past studies show that the integration of production scheduling and maintenance in industry 4.0 platform has not been paid much attention. therefore, in this paper, we propose an optimal parallel machine-scheduling problem with pm activities. a mathematical model is formulated to optimize the scheduling of production and maintenance operations. industry 4.0 conceptual model is also presented in this paper, in which the smart sensors are considered as the real enablers for industrial digitalization. the optimization problem is solved using gams software and branch and bound algorithm. the results of the model provide a suitable schedule for scheduling production and maintenance, and the comparison of solution methods shows that the branch-andbound algorithm achieves a suitable output in a shorter time. key words: production scheduling, preventive maintenance (pm), joint optimization, industry 4.0 received: september 27, 2023 / accepted march 15, 2024 corresponding author: hamed fazlollahtabar department of industrial engineering, school of engineering damghan university, university street, university square, damghan, iran, postal code: 36716-41167 e-mail: h.fazl@du.ac.ir https://orcid.org/0000-0001-5409-3343 712 s. babaeimorad, p. fattahi, h. fazlollahtabar, m. shafiee 1. introduction nowadays, manufacturing companies offer a variety of products to their customers at very competitive prices. time-based manufacturing systems can help companies shorten their production and delivery lead times, minimize changeover, reduce spare parts inventory levels and respond instantly to market changes. real-time decision-making can be considered as an important feature in time-based production and manufacturing systems. real-time describes various operations or processes that respond to inputs reliably within a relatively fast. in other words, it means that it uses logic and mathematics to process the data just a few seconds after it becomes available so that users can get valuable information when they need it. it can help manufacturing businesses identify anomalies and sudden changes in production environment as they occur (like when a spike in product demand or an unexpected breakdown in manufacturing unit occurs) in order to take effective action to control the negative impacts of such changes [1]. the equipment failure is known as one of the main reasons for reduced production efficiency in many manufacturing industries [2]. there are various examples of equipment failure in different industry sectors such as the thermoplastic industry (e.g, the molds in hydraulic presses), the semiconductor industry [3], cutting tools in drilling (e.g, machines, cooling systems), and sensors [4]. in some of these failure cases, the equipment or machinery would need to be repaired or replaced, which can be costly and/or time consuming. the preventive maintenance (pm) has been considered as one of the most effective strategies to avoid unexpected failures in production facilities and enhance the service life of machinery. despite the many benefits of pm in manufacturing systems, repair activities may sometimes interfere with production processes and result in a complete shutdown of the production line. furthermore, irregular pm actions may lead to safety hazards and huge losses to the company. therefore, the pm activities must be scheduled in advance and performed on a routine basis to minimize production interruptions and ensure quality compliance. in this paper, a new pm scheduling mathematical model is formulated for the integrated scheduling of parallel machine production and pm interventions such that the completion time as well as the expected cost are simultaneously minimized. an industry 4.0 conceptual model is also presented in this paper, in which the smart sensors are considered as the real enablers for industrial digitalization. in addition to periodic pm activities, corrective maintenance is also performed to fix machines when they experience accidental failures. in order to optimize the timeliness of production and maintenance, reliability is considered for each machine. the reliability function is to estimate the production efficiency and estimate the useful life of parts to prevent their failure. another innovation of this article is the simultaneous implementation of preventive and corrective maintenance in the production system with two parallel machines. corrective maintenance and repairs are used when parts need to be replaced after estimating their useful life. in addition to the above, the conceptual model of the production system has been presented in the context of industry 4.0. in today's competitive world, production data is very vital. to remain in the field of production, they must be analyzed. for this purpose, mastery of data science and its process is very significant. industry 4 is used to analyze data and predict production and maintenance schedules. the optimization problem is solved using gams software and branch and bound algorithm. the results of each two cases have been compared. the organization of this paper is as follows: section 2 reviews the related work on the integration of production, maintenance and industry 4.0. section 3 presents new pm scheduling an integrated optimization of production and preventive maintenance scheduling in industry 4.0 713 mathematical model. section 4 presents the numerical example, followed by the conclusion and future directions for research in section 5. 2. literature review there are often two challenges, which should be taken into account in a production system. the first challenge is associated with the production planning, aiming to determine optimal production lot size and evaluate the required production capacity; and the second challenge is associated with the scheduling and sequence of production operations, which aims to allocate the existing production capacity to the jobs and determine the sequence of production operations and their start time. the literature review in this paper is divided into two parts. the first part is related to scheduling and maintenance, and the next part is related to the performance of sensors in the manufacturing system and how to determine the time of maintenance. maintenance of production systems is basically of two types. in the first type, known as corrective maintenance (cm), there is no control over the condition of production facilities, meaning that the machines are repaired only after they are broken down. in the second type of maintenance, known as preventive maintenance (pm), there is a level of control over the condition of production facilities, meaning that the maintenance activities are performed to prevent the occurrence of failure. in this case, the scheduling of maintenance interventions plays a main role in production and operations management [5]. optimal maintenance planning can help reduce the frequency of breakdowns and production stoppages [2]. on the other hand, non-optimal maintenance schedules may lead to increase in machine downtime, loss of production, and operating costs. another type of integrated models under time-based pm activities is the classical single machine-scheduling problem with tool replacement. yazdani et al. [6] also studied the parallel machine-scheduling problem with tool change activities such that the makespan was minimized and the maximum makespan was kept below a certain level. also researchers introduced a multi-objective optimization model that had two machines operating in parallel. they examined the breakdown from two resources [7]. the first resource is the machine and the second resource is the moulds. pm activity was considered flexible. the purpose of presenting the model was to minimize the completion time of jobs and the unavailability of machines and moulds. none dominated sorting genetic algorithm (nsga-ii) was proposed to solves the model. gara-ali et al. [8] considered a general model for scheduling the production jobs on unrelated parallel-machines with pm interventions. the processing times were assumed to deteriorate according to their position in the production sequence and the goal of the maintenance was to help to restore good processing conditions. yoo and lee [9] determined a joint optimization of schedule for production jobs and pm activities on parallel machines such that the scheduling cost was minimized. zhang et al. [10] studied an integrated scheduling production jobs and pm interventions for disrupted parallel machines. they assumed that machines might be unavailable at some particular times but they can resume operations. the objective function was to minimize the expected sum of completion times for both the reusable and non-reusable cases. shen and zhu [11] studied a parallel-machine scheduling problem with pm in which the processing and maintenance times were assume to be uncertain variables. sin and chung [12] studied a single machine-scheduling problem with the application to energy-aware scheduling with 714 s. babaeimorad, p. fattahi, h. fazlollahtabar, m. shafiee pm operations under time-of-use (tou) pricing. the authors formulated the problem with a bi-objective mixed integer non-linear programming model and then solved it by metaheuristic algorithm. chen et al. [13] addressed the single machine-scheduling problem for a rotor production workshop and then solved the problem with a genetic algorithm (ga). the machine availability was evaluated by its reliability, which can be restored by a pm. branda et al. [14] studied a flow shop scheduling problem in which machines might be unavailable at some times during the production planning horizon due to random failures. to solve the problem, they proposed two novel meta-heuristic algorithms, including a standard ga and a harmony search method. bhattacharya and chakraborty [15] presented a model for using a past cnc face milling dataset and employed a random forest (rf) regressor to effectively predict the response values of the said process for given sets of input parameters. zhao and yuan [16] considered constraints as product delivery time and changing machine failure rate, and established a multi-objective optimization model aiming to minimize the processing cost and the product processing time. the model includes the changing machine failure rate into the integrated optimization of job-shop production scheduling and predictive maintenance, and enables the prediction of the machine state according to the processing time of the current job, laying the basis for the decision-making of the machine activity and the reasonable and effective production planning. today, the internet of things, machine learning, big data and sensors can be used to further monitor manufacturing technologies and systems, for example, to predict, reduce costs, improve production, predict maintenance and repair operations, and etc. there are various researches related to surveillance systems based on the internet of things, which contain positive and significant results and feedback. one of the applications of industry 4.0 in manufacturing is the use of sensors. smart sensors or monitoring devices enable processing of lots of data points [17]. cheung et al. [18] presented locations in manufacturing systems for monitoring by sensors. the basis of working with these sensors is that a set of sensor data is sent to a remote server. if an abnormal situation occurs, an alarm sounds and the process is executed with high precision. according to the above literature reviewed, the following research gaps are extracted: 1. lack of considering multiple objectives in decision making such as minimizing maintenance costs and the completion time; 2. reliability of determining time to maintenance operations were not considered deeply and neither integrated with economic maintenance decisions. the estimation of the useful life of the parts in the production system in order to increase the reliability of the machines has not been investigated; 3. by evolution of industry 4.0, past works did not integrate the corresponding technologies into real-time maintenance scheduling. to handle the aforementioned gaps, the contributions of this work are listed below: 1. providing a multi-objective model of reliability-based maintenance scheduling considering the industry 4.0; 2. developing an exact solution approach for the integrated maintenance scheduling within industry 4.0 paradigms; 3. using real sensor data to predict maintenance operations. https://www.sciencedirect.com/science/article/pii/s0360835220306598?casa_token=d79h3vf6rm4aaaaa:l3a6jsoy6useykyqlrdo_6v5ihrlshxwl2xbumrpx3pctiy9k3kro1bqaf3symhaciufmvxnwg#! an integrated optimization of production and preventive maintenance scheduling in industry 4.0 715 3. the proposed model in this section, in addition to explaining the industry 4.0-based information flow management framework, we propose our integrated production and maintenancescheduling model. industry 4.0 describes an environment in which flexible and intelligent processes utilize data obtained from sensors connected to all parts of a value chain simultaneously with the occurrence of an event to optimize business processes. these cyberphysical systems use the power of big data analytics to translate complete operational vision into unprecedented speed and efficiency in production processes. at present, these machines and equipment can be programmed to interact with each other, with their higher and lower operating systems, and to make intelligent decisions independent of human intervention being exactly the definition of a modern smart factory. in addition, a production system may fail due to various reasons, or the useful life of parts may end, in which case repair or replacement of parts is required. since the production line is completely stopped when performing maintenance and repair operations, it is very essential to determine the right time for maintenance and repair operations to prevent the production of defective products, machine breakdowns, and also the costs caused by stopping the production line. the analysis of the data received from the sensors of industrial systems is of particular importance and they are entered into the model as parameters and primary data so that their results can be used to produce a suitable schedule. in the following, the characteristics of the problem are explained. the problem involves i={1,2,…, n} independent job, which is in set n={j1,…, jn}. there are k={1,2,…, k} machines in parallel and independent in each period to process these jobs. in other words, the presented model can be used for k={1,2,…, k} that are placed in parallel. at zero time and before the start of processing, all jobs and machines are available. the problem is that each machine is able to perform a maximum of one job at a time, and each job must be processed on only one machine. preventive maintenance, mk={1,2,…, m} is provided periodically. moreover, after each maintenance operation, the machine becomes well in its new state. because the machines in question are considered safe, no net operations are performed at the start of the schedule. if the machines break down between the two intervals intended for preventive maintenance operations, corrective maintenance operations must be performed on them. in this system, both machines can be stopped at the same time for maintenance operations, and this depends on the schedule and breakdown of the machines. the objective function includes minimizing the time to complete the jobs and minimizing the average cost of maintenance in two separate sections. eqs. (1) and (2) represent the makespan and the cost of maintenance operations, respectively. eq. (3) represents a linear weighted combination of two above objective functions: 1 maxmin z c= (1) where cmax = max {cik} is completion time of the last job, tmk, start time of maintenance operations on the machine k; cik: completion time of the ith job on machine k, min 𝑧2 = ∑ 𝐶𝑝𝑚∗𝑅𝑘(𝑡𝑚𝑘)+𝐶𝑐𝑚∗𝐹𝑘(𝑡𝑚𝑘) 𝜇𝑝𝑚𝑘∗𝑅𝑘(𝑡𝑚𝑘)+𝜇𝑐𝑚𝑘∗𝐹𝑘(𝑡𝑚𝑘) ∫ 𝑅𝑘(𝑢)𝑑𝑢 𝑚 𝑚−1 𝐾 𝑘=1 (2) where cpm is cost of a pm action, r(t): r(t) = 1 − f(t) is machine reliability, f(t) is probability distribution function for time to failure of machines, cpm is cost of a cm action, 716 s. babaeimorad, p. fattahi, h. fazlollahtabar, m. shafiee µpmk is the average time for executing a pm action, µcmk is the average time for executing a cm action, * * 1 1 2 2 1 2* * 1 2 min ( ) ( ) z z z z z w w z z − − = + (3) where w1 and w2 are weights considered for the objective functions in order to use the lp metric method; the model constraints are presented in the following equations:    max 1,2,3,..., and 1,2,3,...,jkc c k k j n    (4)   1 0 1 and j 1,2,3,..., k n ijk k i x i j n = = =    (5) where xijk= 1 if job i precedes job j on machine k,   1 0 1 and i 1,2,3,..., k n ijk k j x i j n = =     (6)     0 , and i, 1,2,3,..., , 1,2,3,..., n hik ijk h x x i j i h j n k k =       (7)  0 0 1,2,3,...,kc k k=   (8)         (1 ) ( ( )* ( )) ( )* ( )* i 1,2,3,..., , , 1,2,3,..., , 1,2,3,..., , 1,2,3,..., k k k k k k m kk jk ijk ik jk m k pm m k m k cm m k m k ic m x c p t g t r t g t f t y n i j j n k k m m + −  + + + +          (9) where pik is processing time of the ith job on machine k, m is a large positive number, gpm(tmk) is probability density function associated with the duration of a pm action, gcm(tmk) is probability density function associated with the duration of a cm action, yimk, 1 if the mth maintenance activity is performed prior to the ith job on machine k,    1 i 1,2,3,..., , 1,2,3,..., k k m im k m y n k k    (10)     1 1 1,2,3,..., , 1,2,3,..., k n im k k i y k k m m = =    (11) where mk is number of maintenance activities on machine k,     1,2,3,..., , 1,2,3,...,mk mk kt l k k m m=    (12) where lmk is the latest maintenance start time for the mth maintenance activity on machine k,     1,1, ( ) ( ) ( ) ( )* ( ) 1,2,3,..., , 1,2,3,..., 1 m km k p m m k m k k k kl t g t r t l k k m m ++ = + +    − (13) 𝑝(𝜀 ≥ 𝐿𝑘) = 𝛽 (14) where β is a positive number between 0 and 1, ε is instant of failure in the machine, an integrated optimization of production and preventive maintenance scheduling in industry 4.0 717         and 0,1 and 0 1,2,3,..., , 1,2,3,..., , 1,2,3,..., kijk im k mk kx y t k k m m j n      (15) eq. (4) indicates the maximum time, when the processing of jobs is completed. eq. (5) states that each job must only be processed on one machine. eq. (6) ensures that the maximum number of successors of every job to be one. eq. (7) limits the number of successors of each job to a maximum of one on each machine. eq. (8) states that the zero job ends in zero time, in other words, no processing time is set for the zero job. eq. (9) shows how to calculate the processing time of jobs on machines. therefore, if job j is performed after job i to machine k, the processing time of job j will be greater than job i. eq. (10) indicates that after processing each job, a maximum of one maintenance operation can be performed on the machine. eq. (11) states that a maximum of one maintenance operation can be performed on each machine after each job and at any time interval. eqs. (12) and (13) explain how to calculate the latest maintenance time on machines. in eq. (14) to obtain the 𝐿𝑘 for maintenance on each machine, the maximum production efficiency of each machine must be considered, which is indicated here by β and is considered as a parameter. then, with efficiency and integration, the value of the upper limit of the integration interval indicates the latest start time of maintenance operations, which is given in eq. (14). eq. (15) indicates decision variables. 4. numerical example in this section, a numerical solution algorithm is provided to illustrate the proposed model. the following provided a numerical method for determining decision variables that minimizes completion time and total maintenance costs per unit time. since the proposed model is np-hard, solving it with gams software would be so times consuming. for solving in small scale, gams software has been used, and for large scale, branch and bound method has been used in matlab software. initially, after introducing the parameters, the upper limit of the integral (lk) shown in eq. (13) is calculated. then the latest maintenance time on each machine can be calculated. because by calculating the latest time of maintenance operations, it is possible to periodically determine the time of preventive maintenance. in the next step, the jobs are assigned to the machines and the time of preventive maintenance and repairs is determined. after processing each job, the system checks whether it is time for maintenance and corrective repairs. if it is time for maintenance, maintenance operations will be performed. after the maintenance operations are completed, job processing starts again and will be returned to eq. (13). if the condition fails, the allocation of jobs to the machines will continue. if all the jobs are assigned to the machines, the calculation of the objective function is performed and finally the output of the algorithm and the schedule of production, maintenance and repairs are reached. the branch and bound algorithm is used to schedule jobs and preventive maintenance. in this section, the input data for solving the model by the proposed algorithm is given. this model is solved by 10 problems with different processing times. in this section, the parameters are given. job execution times are generated using random data. however, for the duration of maintenance and repairs and their cumulative distribution, another article and previous studies have been used. in problem number 1, seven jobs j1, j2, j3, j4, j5, j6 and j7 are defined to run the model and obtain the solution. 718 s. babaeimorad, p. fattahi, h. fazlollahtabar, m. shafiee ▪ costs: cost of pm = 2$, cost of cm = 4$. ▪ the duration of pm task ~lognorm (mean µp=10, standard deviation δp=1.5). ▪ the duration of cm task ~lognorm (mean µc=20, standard deviation δc=2). ▪ the time to machine failure ~ weibull distribution (shape parameter=2, scale parameter =100), i.e., the average lifetime is μ=8.86-time units. ▪ β=90% ▪ number of machines =2. ▪ the number of jobs and their processing times is given in table 1. ▪ in this example, the jobs are assigned to machines and there are different sequences between the jobs of each machine. random numbers in accordance with previous articles have been used to calculate the proposed model. this model was run with 10 jobs each time by the proposed algorithm and in matlab software and 7 jobs by gams software. table 1 time required processing each job on both machines 10 9 8 7 6 5 4 3 2 1 p ro b le m n u m b er 12 5 12 6 11 7 10 1 processing times for the jobs on both machines(s) (pik) 5 17 18 12 7 10 6 2 20 19 8 19 7 20 6 3 13 6 10 6 5 8 11 4 16 18 8 5 14 12 9 18 20 18 5 5 17 12 14 7 6 12 8 11 6 6 14 6 14 15 15 17 20 16 10 5 7 10 16 15 20 17 15 15 15 12 13 8 8 19 14 15 19 12 5 9 10 19 9 14 20 15 5 5 17 9 7 13 11 10 the number of preventive maintenance operations was initially set equal to the number of jobs. in the worst case, after each job processing, a maintenance operation is performed on the machine and the required number is determined after the model is run. the results obtained from the proposed algorithm and gams software are given in table 2. schedule of jobs and anticipated is given in fig. 1. j3 cm j4 pm j1 j5 cm j6 j7 pm j2 machine 1 machine 2 12 13.92 14.88 20.88 21.47 33.47 35.4130.4129.9519.9512.9512.0811 jobs corrective maintenance corrective maintenance fig. 1 initial scheduling of the seven jobs on two parallel machines an integrated optimization of production and preventive maintenance scheduling in industry 4.0 719 table 2 the obtained results problem number objective function cmax gams cmax (branch and bound) solution time (branch and bound) solution time (gams) 1 17.623 35.41 35.41 121.18 524.12 2 19.713 39.23 39.23 99.89 335.92 3 26.273 52.35 52.35 101.22 529.32 4 15.723 31.25 31.25 133.37 658.18 5 33.995 66.53 143.269 6 37.043 72.58 187.191 7 47.478 93.45 135.594 8 48.373 95.24 258.427 9 33.328 65.15 227.686 10 40.448 79.39 167.517 in this section, the proposed model is solved using parameters. the output of the model is a schedule that, in addition to the production schedule, optimizes the maintenance and repair schedule in an integrated manner. this model is very serious to save time and money. according to the current conditions, many manufacturing plants still do not integrate the production schedule with the maintenance and repair schedule. each unit tries to optimize individually. in addition, due to today's competitive world, in order to optimize and remain in the field of competition, they must update themselves with the latest science. organizations, in addition to needing a comprehensive plan for production, also need to analyze data and use it in order to increase quality and competition. therefore, it is very necessary to provide a model that can optimize all these things and can use data science for data analysis. 5. conclusion in this paper, an optimization model was presented to schedule production operations as well as preventive maintenance interventions on multiple parallel machines in an integrated way. a conceptual model was presented in order to understand how industry 4.0 works. then the proposed model was solved by gams software and branch-and-bound algorithm. the output of the model was the integrated schedule of production and maintenance and repairs. this model is essential from the point of view that it optimizes both programs (scheduling and maintenance) simultaneously. in addition, the results show that for a high number of scales, the branch-and-bound algorithm provides better results in the shortest time. for future studies, online analytical processing (olap) tools can be used to examine changes in machines, as well as condition-based maintenance. furthermore, some other decision variables such as pricing, batch size, etc. can also be considered in our model. 720 s. babaeimorad, p. fattahi, h. fazlollahtabar, m. shafiee references 1. kazimieras zavadskas, e., turskis, z., stevic, z., mardani, a., 2020, modelling procedure for the selection of steel pipes supplier by applying fuzzy ahp method, operational research in engineering sciences: theory and applications, 3(2), pp. 39-53. 2. bajestani, m.a., banjevic, d., beck, j.c, 2014, integrated maintenance planning and production schedulingwith markova deteriorating machine conditions, international journal of production research, 52(24), pp. 7377–7400. 3. sloan, t. w., shanthikumar, j.g., 2000, combined production and maintenance scheduling for a multiple product, singlemachine production system. production and operations management, 9(4), pp. 379–399. 4. bajestani, m.a., 2014, 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availability constraints and preventive maintenance, international journal of production research, 59(9), pp. 2708–2721. 14. branda, a., castellano, d., guizzi, g., popolo, v., 2021, dataset of meta heuristics for the flow shop scheduling problem with maintenance activities integrated, data in brief, 36, 106985. 15. bhattacharya, s., chakraborty, s., 2023, prediction of responses in a cnc milling operation using random forest regressor. facta universitatis-series mechanical engineering, 21(4), pp. 685-700. 16. zhao, z. y., yuan, q. l., 2022, integrated multi-objective optimization of predictive maintenance and production scheduling: perspective from lead-time constraints, journal of intelligent management decision, 1(1), pp. 67-77. 17. friedemann, m., trapp, t.u., stoldt, j., langer, t., putz, m., 2016, a framework for information-driven manufacturing, procedia cirp, 57, pp. 38-43. 18. cheung, w.f., lin, t.h., lin, y.c, 2018, a real-time construction safety monitoring system for hazardous gas integrating wireless sensor network and building information modeling technologies. sensors, 18, 436. https://www.sciencedirect.com/science/article/pii/s2352340921002699 https://www.sciencedirect.com/science/article/pii/s2352340921002699 facta universitatis series: mechanical engineering vol. 18, no 1, 2020, pp. 57 68 https://doi.org/10.22190/fume171128002s © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper  numerical investigation of the influence of the doubly curved blade profiles on the reversible axial fan characteristics živan spasić, miloš jovanović, jasmina bogdanović-jovanović, saša milanović faculty of mechanical engineering, university of niš, serbia abstract. in reversible axial fans a change in the direction of the impeller rotation is accompanied with a change in the direction of the working fluid flow. to satisfy the flow reversibility, the impeller blades are usually designed with straight symmetrical profiles. the flow reversibility may also be achieved by using asymmetrical blade profiles in which, to satisfy the equality of the leading and trailing angle of the profiles, the mean line of the profile has to have a double curvature in the shape of the stretched letter 's'. the paper numerically investigates the influence of the doubly curved blade profiles on the reversible axial fan characteristics. numerical simulations are carried out on an axial fan only with the impeller, with the blades that have double-curved mean line profiles for different values of the angles at the profile ends. for numerical simulation the ansys cfx software package is used. results of the numerical simulation are shown in diagrams δp(q), (q) and p(q) at different angles of the profile ends. on the basis of the simulation and analysis of the characteristics, appropriate conclusions are proposed, along with the most advantageous profile of the blades. key words: reversible axial fan, curved profile, angle, characteristics, numerical simulations 1. introduction reversible axial fans are used to achieve a forced air-gas flow in the primary and reverse flow regime. the fans only have a single impeller whose reversibility of the flow is achieved by changing the direction of rotation [1, 2, 3]. to satisfy the reversibility of the flow, with the identical curve performance for both regimes, the impeller blades are received november 28, 2017 / accepted january 12, 2018 corresponding author: ţivan spasić faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš, serbia e-mail:zivans@masfak.ni.ac.rs 58 ţ. spasić, m. jovanović, j. bogdanović-jovanović, s. milanović usually designed with straight symmetrical profiles. efficiency of these fans is relatively low due to the large incidence angle flow on the profile of the blades [4, 5]. classical axial fans, designed for one direction of flow, can work as reversible fans as well. the characteristics of such fans in a reversible regime are worse due to unfavorable flow conditions. the fan characteristics are less different in the direct and reversible mode with a fan with a smaller curvature of the blade profile [2, 3]. the efficiency of the fan depends on the shapes of the blade profiles, the blade itself, and the ratio of the diameters of the impeller hub and the shroud [2, 6]. numerical flow simulations in fans can determine the best shape profiles for achieving better efficiency of the fans, which can serve for further experimental tests [7, 8, 9]. various commercial programs are available for numerical simulations. the k- model, employed in this study, is frequently used as the method for solving the turbulent flow in turbomachinery [9-11]. the reversible axial fan is designed with the blades having a double curved profile for increase pressure [1, 12]. this paper presents the original profile design with a slightly double curved mean line. the fan is designed based on the design of the fan blades with straight profiles. to form the profile with a doubly curved mean line, normally the thickness of the straight symmetric profile marked as pp2 is applied [1, 13]. the shape of the profile is determined after a series of numerical simulations of flow in these fans with different angles of the profile end curvature [1]. this paper presents the comparison of the characteristics of the fan with the straight profiles of the blade numerically obtained. we should be careful with the introduction of double curved profiles since any change in the curvature can lead to separation of the flow from the surface of the blade profile and thus to the degradation of the fans’ performance [14,15]. 2. the basic geometry of the reversible axial fan the basic geometry of the reversible axial fan is designed with the straight profiles of blades (fig. 1) [1, 13]. in order to achieve equality of fluid energy exchange along the radius of the impeller, the blades of the impeller are spatially curved. the fan impeller is designed under the principle of equal specific work of all elementary stages, that is, under the principle of the flow along axially symmetrical cylindrical surfaces [3, 6, 15]. to determine the shape of the fan blades, the profiles are designed in 13 elementary stages, approximately equally distributed along the height of the blade. the design was performed using the method of lift force for straight profiles [1, 13]. in a cascade with straight profiles the leading and trailing angle of blade profile (1l = 2l) equals the angle of inclination profile t in the cascade profile (fig. 1): 1 2l l t    (1) fig. 1 schematically shows a meridian section of the reversible axial fan, with a developed cylindrical section for the mean section of the impeller. the straight profiles are set to step t, the angle of inclination t profile that changes the height of the blade. the primary flow of the fluid is left-to-right flow. in fig. 1 the primary flow is indicated by speed c corresponding to the direction of circumferential speed u . with the change in the flow direction (circumferential velocity direction  u ), the flow becomes reversible (flow velocity  c ). numerical investigation of the influence of the doubly curved blade profiles on the reversible axial fan... 59 fig. 1 reversible axial fan the diameter of the fan impeller hub and other geometrical values are obtained according to the recommendations for optimal values of the dimensionless volume coefficient and the pressure coefficient. for the calculation parameters of flow, volumetric flow rate q = 3.61 m3/s, total pressure increase δptot = 180 pa, rotation speed n = 1405 rpm and assumed efficiency  = 0.65, the basic geometry of the fan impeller is obtained [1, 13]:  di = 300 mm diameter of the fan impeller hub,  de = 630 mm peripheral diameter of the fan impeller, and,  zk = 6 number of impeller blades. 3. profiles with a doubly curved mean line asymmetric profiles the flow reversibility at the reversible axial fan can be achieved by applying blades with asymmetrical profiles, which have a doubly curved profile shape mean line in the shape of a stretched 's' (fig. 2). in order to achieve the same characteristics for both directions of flow, the leading and trailing angles of the profile blades should be equal 1l =2l. they are different from the angle of inclination profile t in the cascade profile 1l =2l t depending on the angles of the curvature of the ends of profile l (fig. 4). for the purpose of creating a profile, the thickness of a straight symmetric profile pp2 is usually applied to the mean line [1, 13]. 3.1. mean line design in the profiles with a double curvature the design shape of the mean line of this profile is determined after a series of numerical simulations of flow in the straight profile cascades [12]. fig. 2 shows the design of the mean line of the profile with a double curvature: rays (a1 and a2) are drawn from the ends of the profile (point a1 and a2), under the chosen angle of curvature of profile mean line δβl. the rays with verticals, drawn along length l1 60 ţ. spasić, m. jovanović, j. bogdanović-jovanović, s. milanović (l1=0,2l) of the ends of the profile, make intersection points b1 and b2. the line joining points (b1 and b2) passes through the middle mean line of the straight profile, point o. curvature of the profile is obtained by connecting the lines ( 1 1a b and 1 2b b on the one hand, and 2 2a b and 2 1b b on the other) with radius r (r=l). the perpendicular distance between the rays (a1 and a2) drawn from the end points (point a1 and a2) is marked with e in fig. 2. relative size e ( /e e l ), in relation to the profile length, with the angle of profile curvature δβl, is expressed by the relation: sin le l    . fig. 2 design of the mean line of the doubly curved profile the profiles with a doubly curved mean line have equal angles of curvature of the profile ends, which along with the centerline of the profile (a1a2) amounts to l. 3.2. designing the curved profile the curved profile is designed by perpendicularly applying thickness δj of the straight symmetrical profile marked as pp2 [1, 13] along the doubly curved mean line lj for the intersection j (table 1, fig. 3). table 1 distribution of thickness along the profile mean line [1, 13] (lj/l ) 102 0.00 2.14 3.57 5.62 10.53 20.43 30.24 40.14 50.00 (δj/δmax) 102 0.00 43.33 50.00 58.33 73.33 86.67 95.00 96.67 100.0 (lj/l )  102 59.77 69.58 79.48 89.38 94.29 96.52 97.86 100.0 (δj/δmax) 102 96.67 95.00 86.67 73.33 58.33 50.00 43.33 0.00 fig. 3 geometry of the doubly curved profile l – the profile length maxthe maximum profile thickness, r – the radius of the curvature of the profile ends, r – the radius of the profile curvature,  – the angle of the curvature of the profile ends numerical investigation of the influence of the doubly curved blade profiles on the reversible axial fan... 61 if one observes a cascade with profiles that have a doubly curved mean line, the leading and trailing angles of blade profile (1l =2l) are greater or smaller than the angle of inclination of profile cascade t, for the angle of end curvature of profile mean line l (fig. 4): 1 2l l t l      (2) fig. 4 shows the position of the doubly curved profiles in the cascade profile with 1l =2l >t (fig. 4-a) and 1l =2l <t (fig. 4-b). a) b) fig. 4 cascades with doubly curved profiles: a) leading and trailing angles of the profile 1l =2l =t+l, (1l <t), b) leading and trailing angles of the profile 1l =2l =tl, (1l >t) 4. numerical simulation of flow in the reversible axial fan in order to determine the best profile shape, numerical simulations are carried out for the flow in an reversible axial fan with the blades which possess different angles of curvature of the profile mean line ends: l = 2.90 ( e = 0.05), l = 4.50 ( e = 0.078) and l = 60 ( e =0.1). 4.1. model of the reversible axial fan for numerical simulations the blades of the model fan for numerical simulations are formed on the basis of the fan designed with straight profiles for seven cylindrical sections of the impeller x (x=i-vii) (seven elementary stages) [1, 13], whose positions are defined by radii rx (table 2). the profile geometry of radii rx (table 2, fig. 5) is defined by profile length lx, 62 ţ. spasić, m. jovanović, j. bogdanović-jovanović, s. milanović thickness distribution δj along the mean line, as shown in fig. 3, the maximum thickness in the middle of the profile δmax,x, the profile leading and tail curvature radii r1x=r2x and the angle of profile inclination βtx. table 2 geometry of the profile cascade in cylindrical sections section: x rx [mm] tx [mm] lx [mm] tx [°] δmax,x [mm] (δmax/l)x [-] r1x= r2x [mm] i 150 157 144 53.7 12 0.083 2.4 ii 178 186 138 42.3 11 0.080 2.2 iii 205 215 133 36.4 10 0.075 2.0 iv 233 243 126 31.8 9 0.071 1.8 v 260 272 121 28.5 8 0.066 1.6 vi 288 301 114 26.0 7 0.061 1.4 vii 315 330 108 24.6 6 0.056 1.2 fig. 5 blade with the profile mean lines developed in a plane, l=29.1° the difference in the inclination angles of the profile mean line at the hub ((ti=ti =53.7°) and the shroud (te=tvii = 24.6°), the angular spatial blade curvature l (fig. 5), is l=t = ti. – te.= 53.7– 24.6 = 29.1° the blades are mounted on the hub under a specific blade angle (ti), which is defined in accordance with the profile inclination angle at the hub (l =ti). the angles of the profile ends for each cylindrical cross section (x) of the impeller are defined in relation to the inclination angle of the profile, i.e., in relation to the inclination angle of the mean line of the straight profiles (the profile pp2, table 2), as well as 1l =2l =t l. the mean lines of the blade profile developed in a plane with angles 1l =2l >t are shown in fig. 6-a, and the mean lines of the blade profile developed in a plane with angles 1l =2l <t are shown in fig. 6-b. numerical investigation of the influence of the doubly curved blade profiles on the reversible axial fan... 63 a) b) fig. 6 mean lines of the blade profile developed in a plane, a) profile with angles 1l =2l >t, b) profile with angles 1l =2l <t 4.2. numerical flow simulations in ansys cfx in order to investigate the influence of the doubly curved blade profiles on the performance of a reversible axial fan, numerical simulations of flow are carried out for the design of a fan with one impeller, with the blades that have doubly curved mean line profiles, for different flow through the fan q = (1100013500) m3/h, impeller speed n=1405 min-1 and air density  =1.2 kg/m3 [1]. numerical simulations of the fan flow can provide aerodynamic characteristics. one of the most popular software packages for turbomachinery flow simulation, ansys cfx, is used for the numerical simulation of fluid [9, 11, 16]. 4.2.1. model geometry formation the model geometry is formed by drawing a 3d model in a specialized part of software for the design of impellers of turbomachinery, ansys cfx-bladegen (fig. 7-a). flow space is defined by the borders of entrance and exit from the impeller hub, shroud and blades of the impeller. blade geometry is defined by several cylindrical sections (elementary stages). in this case, seven cylindrical sections are chosen for the application of the profile geometry (table 1 and table 2). profile geometry at each intersection is defined by a comprehensive profile angle r, profile inclination angle t in the radial direction or the profile inclination angle in the axial direction (a=900t), and a thickness profile along the mean line of the profiles with the profile leading and tail curvature radii. this program has the option to 'compare'; thus it is possible to make a comparison of the profile geometry of a particular section of the impeller blades with a different profile cross-section by overlapping on the same profile (fig. 7-b). this feature is particularly important when performing simulations to compare the performance of fans with various shapes of the profile. to assess the influence of the shape profile on the performance, it is 64 ţ. spasić, m. jovanović, j. bogdanović-jovanović, s. milanović necessary to set up profiles of different forms of the same angle of profile inclination t because the characteristics of the fan are influenced by the inclination angle of the profile in cascade t, in addition to effect of the profile shape. a) (-7.7497,151.5502) b) fig. 7 a) fan model for the numerical simulation, b) comparison of the geometry of straight and curved profiles 4.2.2. creating the mesh on the basis of the defined geometry model, a discretization mesh of the flow field is formed. this is one of the most difficult phases because the quality of the mesh depends on the accuracy of numerical simulation. on the other hand, one cannot ignore the constraints of the computing capacity, namely, those that limit the number of mesh nodes and the size of the mesh elements. when solving this problem a mesh of non-uniform flow field can be created, and it is significantly finer in those areas that are particularly important for research in the defined task. in turbomachinery, because of the symmetry of the impeller, only one of the blades can be considered along with a half of the space between the blades. this fact allows the formation of a finer mesh to shorten the calculation time of numerical simulations. there are a number of software versions for the creation of a mesh of the model, which in a sense makes it easier to prepare the model for numerical simulation. the mesh is formed in the software for creating a mesh for turbomachinery impellers turbogrid ansys, which is part of the ansys software package for simulating flow. in this program, the user first defines the input and output of the simulated domain. the input and output are at a distance of 100 mm in front of, or behind, the impeller axis perpendicular to the axis of rotation (fig. 8-a). the radial clearance between the blades of the impeller is 2.5 mm. the number of mesh elements of 1/6 of the fan impeller is about 1000000 for all simulations. the mesh is made up from the topology (h/f/c/l-grid) with a mesh around the profile (o-grid), the mesh setting is performed according to the criteria recommended for maximum and minimum values of the elements: the relation between the edges, volume ratio and angle elements. numerical investigation of the influence of the doubly curved blade profiles on the reversible axial fan... 65 4.2.3. defining the physical parameters of flow defining the initial and boundary conditions, fluid characteristics and other physical parameters is done in the pre-processor of ansys cfx-pre. this program defines the physics of the flow process. this determines whether the geometric model and its component parts are at rest or moving. rotating motion of the impeller is defined as the axis of rotation and rotational speed (n=1405 min-1). then, each of the interface geometry is assigned with the boundary conditions (input, output, solid surfaces-walls) and the initial value, the total pressure at the entrance (pitot=100 kpa) and the desired mass fan flow. when the model consists of several domains, it is necessary to define the places of their merger (the interface). due to the symmetry of the impeller, the simulation is performed only in the space surrounding one blade, so it is necessary to also define the periodic surfaces of the flow field (fig. 8-b). a) b) fig. 8 appearance of the simulated domain. the marks and values: di=300 mm – hub diameter, d’e=635 mm – shroud diameter, s – tip clearance, lg=120 mm – length of the hub, bi=116 mm – the width of the blade at di, be=45 mm – the width of the blade at de, l=200 mm – length of the simulated domain. this part of the program defines the type of fluid and its physical properties, the turbulence model (in this case, the k- model), and the criteria for numerical calculations (convergence residual 10-5, the maximum number of iterations, the level of resolution, etc.). 4.3. results of numerical simulation the results of numerical simulations are presented in diagrams p(q), (q) and p(q), which are given on the basis of the averaged values of simulation for the middle cylindrical section of the impeller, for different angles of the blade profile ends. all simulations are carried out for the angle of inclination of the impeller blades (mounting angle of blades) l=53.70, which is measured at the hub and is equal to the angle of inclination of the mean line of the profiles blade around the hub (l =ti). in order to assess the influence of the curvature of the blade profile on the performance of the fan, the diagrams also show the characteristics of the fan with the straight profile of the blades (profile pp2), also marked as l=00 in the diagrams. 66 ţ. spasić, m. jovanović, j. bogdanović-jovanović, s. milanović a) 1l =2l =t+l b) 1l =2l =t-l fig. 9 the influence of curvature on the total rise in pressure, ptot(q) a) 1l =2l =t+l b) 1l =2l =t-l fig. 10 the influence of curvature on the efficiency, (q) a) 1l =2l =t+l b) 1l =2l =t-l fig. 11 the influence of curvature on the power of the fan, p(q) figs. 9-a,b, 10-a,b, 11-a,b show, based on the performed simulation, the performance of the fan with doubly-curved profile of the blades (1l =2l =tl) and corresponding characteristics of the fan design with the straight profile of the blades (1l =2l =t, l=0), for comparison. numerical investigation of the influence of the doubly curved blade profiles on the reversible axial fan... 67 4.3.1. analysis of the results obtained for the curved profiles, 1l =2l >t (fig. 9-a, 10-a, 11-a) increasing the pressure in the fan with blades that have curved profile ends 1l =2l >t leads to approximately the same value for angles of curvature l =2.90 and l =40, which in relation to the fan with the straight profile is higher for about 2 % (fig. 9-a). for an angle of curvature l =60 this difference is the smallest. with the increase in the angle curvature profile the efficiency of the fan slightly falls. (fig. 9-b). the difference of efficiency was 0.5% for the highest flow rate (q=13500 m3/h), and 1% for the minimum flow, between the impeller with the straight profile of the blades and the curved profile, for l =60. approximately equal power is needed for all the fans with curved blade profiles, for all the angles of curvature, and it is larger than the power of the fan with straight profiles, an average of 1.3% for the entire range of the simulated flow (fig. 9-c). the optimal operating parameters (max) of the fan are achieved with the blades having the following profiles:  straight ( l =00 ): ptot=200 pa, q =12750 m3/h, p=0.80 kw, =0.868  curved, 1l =2l >t, l =2.90: ptot =194 pa, q =13000 m3/h, p=0.807 kw, =0.867  curved, 1l =2l >t, l =4.50: ptot =193 pa, q =13000 m3/h, p=0.807 kw, =0.865  curved, 1l =2l >t, l =60: ptot =192 pa, q =13000 m3/h, p=0.805 kw, =0.862 the optimal operating parameters of the fan with curved profiles 1l =2l >t are obtained at a higher flow rate in relation to the blades of the straight profile by about 2%. 4.3.2. analysis of the results obtained for the curved profiles, 1l =2l <t (fig. 9-b, 10-b, 11-b) with the decrease of the leading and trailing angle profile of the blades the fan pressure decreases compared to the straight profiles for the entire range of simulation. for the angle of curvature l=60 the reduction of the fan pressure is about 2.5%, for the angle of curvature l =4.50 the reduction is about 1.5%, and for the angle of curvature l =2.90 the reduction is approximately 1% (fig. 9-b). the efficiency throughout the range of the simulated flow rate slightly decreases with the increasing angle of curvature (fig. 10-b), for the angle of curvature l=60 the reduction of efficiency is about 0.7% for the calculated flow (q = 13000 m3/h). with the increase in the angles of curvature the fan power is reduced and it is less than the power of the fan blades with the straight profiles. for the angle of curvature l=60 the power reduction is approximately 2% in the whole range of the simulation (fig. 11-b). the optimal operating parameters (max) of the fan are achieved with the blades having the following profiles:  straight, 1l =2l =t ( l=0 ): ptot =200 pa, q =12750 m3/h, p=0.80 kw, =0.868  curved, 1l =2l <t , l=2.90: ptot =199 pa, q =12700 m3/h, p=0.815 kw, =0.866  curved, 1l =2l <t , l =4.50: ptot =197 pa, q =12700 m3/h, p=0.792 kw, =0.865  curved, 1l =2l <t, l =60: ptot =204 pa, q =12500 m3/h, p=0.823 kw, =0.862 68 ţ. spasić, m. jovanović, j. bogdanović-jovanović, s. milanović 5. conclusions on the basis of the conducted numerical simulations and the analysis of the obtained fan characteristics, it can be concluded that the profile curvature influences the performance of the axial reversible fan. the highest growth in the fan pressure is achieved with the blades that have the 1l =2l >t profile angles, with the angle of curvature l=2,90 for about 2% of the value of all the simulated flow. the efficiency in the optimal regime is slightly smaller, by about 0.1%. the smallest increment of pressure and the efficiency of the lowest level are achieved in the fan with the blades that have a curved profile with the 1l =2l <t angles. the optimal operating parameters of the fan with curved profiles 1l =2l >t (l=2,90) are obtained at a higher flow rate in relation to the blades of the straight profile by about 2%. the optimal operating parameters of the fan with curved profiles 1l =2l < t are achieved at a lower flow rate compared to the blades with straight profiles for about 0.4%. there is a risk involved and thus care should be taken when introducing a double curve in the profile of the blades. any change in the curve directly affects the separation of the flow from the blade profiles, which can lead to a drop in the characteristics of the fan. references 1. spasić, ţ., 2012, numerical and experimental investigation of the influence of the blade profile shape on the reversible axial fan characteristics (in serbian), phd thesis, faculty of mechanical engineering nis, university of nis, nis, 131 p. 2. brusilovskij, i.v., 1978, the aerodynamic schemes and characteristics of the axial fan cagi, (in russian), moscow, 197 p. 3. krsmanović, lj., gajić, a., 2000, turbomachinery-fans (in serbian), faculty of mechanical engineering belgrade, 289 p. 4. cebeci, t., platzer, m., chen, h., chang, k.-c., shao j.p., 2005, analysis of low-speed unsteady airfoil flows, springer, berlin, 171 p. 5. parker, d.e., simonson, m.r., 1982, transonic fan/compressor rotor design study, volume iii-final report, research laboratories, wright-patterson air force base, ohio. 6. еck, b., 1973, fans-design and operation of centrifugal, axial-flow and cross-flow fans, pergamont press, oxford, england, 592 p. 7. bogdanović-jovanović, j., milenković, d., stamenković, ţ., spasić, ţ, 2017, determination of averaged axisymmetric flow surfaces and meridian streamlines in the centrifugal pump using numerical simulation results, facta universitatis-series mechanical engineering, 15(3), pp. 479-493. 8. huang, c.h., gau, c.w., 2012, an optimal design for axial-flow fan blade: theoretical and experimental studies, journal of mechanical science and technology 26(2), pp. 427-436. 9. lin, s.c., shen, m.c., tso, h.r., yen, h.c., chen, y.c., 2017, numerical and experimental study on enhancing performance of the stand fan, applied sciences, 7(3), 267; doi:10.3390/app7030267. 10. bamberger, k., carolus, t., 2017, the development, application, and validation of a quick optimization method for the class of axial fans, asme journal of turbomachinery, 139, p. 111001. 11. ferziger, j. h., perić, m., 2002, computational methods for fluid dynamics, springer-verlag berlin heidelberg newyork, 423 p. 12. bogdanović, b., bogdanović-jovanović, j., spasić, ţ., 2009, reversible axial fan with blades created of slightly distorted panel profiles, facta univesitatis-series mechanical engineering, 7(1), pp 23-36. 13. spasić, ţ., milanović, s., šušteršič, v., nikolić, b., 2012, low-pressure reversible axial fan with straight profile blades and relatively high efficiency, thermal science, 16(2), pp. s593-s603. 14. ilikan, n,a., ayder e., 2015, influence of the sweep stacking on the performance of an axial fan, asme journal of turbomachinery, 137, p. 0610041. 15. wallis, a.r., 1983, axial flow fans and ducts, john wiey&sons, new yorker, 444 p. 16. еlhadi, e., kegi, w., 2002, simulation of vortex flows in axial flow fan using computational fluid dynamics, pakistan journal of information and technology, 1(3), pp 242-249. 12379 facta universitatis series: mechanical engineering vol. 23, no 1, 2025, pp. 161 170 https://doi.org/10.22190/fume240304036j © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper influencing factors of deformation and failure of porous coal under conventional loading xiaohua jin1, jiyu zheng2, shunheng hua1, xinru tong1 1school of energy & environment, zhongyuan university of technology, zhengzhou, china 2school of resource and safety engineering, henan university of engineering, zhengzhou, china orcid ids: xiaohua jin https://orcid.org/0009-0001-8619-7419 jiyu zheng https://orcid.org/0000-0001-7697-5400 shunheng hua https://orcid.org/0009-0002-7594-4818 xinru tong https://orcid.org/0009-0008-9176-2196 abstract. coal exploration requires draining the gas from the coal seam. for this purpose, drill holes in the coal seam are required. under the field conditions, the stress and strain states around a drill hole are not easy to determine and the same is valid for the material law defining their dependence. in this study, by arranging boreholes in coal samples, the effects of loading rate, borehole inclination, borehole diameter and other factors that influence the mechanical properties of porous coal are studied. the results show that the strength of porous coal increases with the increase of loading rate, which is consistent with the behavior of non-porous coal. the porous coal samples with certain inclination angles of the drill hole are easier to destroy compared to the porous coal samples with a horizontal drill hole. furthermore, the failure surface is consistent with the inclination angle, and a specimen with an upward inclination angle of the drill hole is easier to destroy compared to a specimen with a downward inclination angle. comparing the samples with the borehole diameters of 5 mm and 10 mm, the axial deformation and transverse deformation of the latter are 1.7 times and 1.63 times that of the former, respectively. regarding the failure mode, it is shown that the borehole plays a leading role in the process of fracture propagation. key words: conventional loading, porous coal, deformation and failure, influence factor 1. introduction coal exploration requires draining the gas from the coal seam. in the process of gas drainage, it is necessary to drill holes in the coal seam. however, those drill holes may collapse in the soft coal seam and this is one of the main factors affecting the gas drainage. received: march 04, 2024 / accepted july 22, 2024 corresponding author: xiaohua jin school of energy & environment, zhongyuan university of technology, zhengzhou 450007, china e-mail: 6561@zut.edu.cn https://orcid.org/0009-0001-8619-7419 https://orcid.org/0000-0001-7697-5400 https://orcid.org/0009-0002-7594-4818 https://orcid.org/0009-0008-9176-2196 162 x. jin, j. zheng, s. hua, x. tong it is caused by the local damage of the coal around the hole. this damage may be caused by continuous loading or creep. zeng et al. [1] studied the mechanical properties and cracking behavior of perforated specimens under uniaxial compression. the experimental results indicated that the stress field distribution before crack initiation varies with the shape of the preexisting hole, and differences primarily exist in the configuration of the pure tension triangle and the low force region. huang et al. [2] investigated the crack initiation, propagation and coalescence process, and acoustic emission (ae) characteristics using photographic and ae monitoring. three failure modes were identified: splitting failure, stepped path failure, and planar failure modes. ji et al. [3] also had a viewpoint that specimens with central holes have three different types of typical failure modes. duan et al. [4] studied the mechanical behavior of porous brittle materials by numerical simulation. they showed that the original initiation crack is not necessarily the main crack that finally generates the failure, as heterogeneity is an important factor of local crack propagation in brittle materials. creep loading is quite different from conventional loading. there is an obvious yield platform and greater deformation under creep loading [5]. under the influence of external stress, the coal around the borehole can easily become unstable, deformed and damaged, resulting in cracks and low gas drainage efficiency, which may lead to coalmine power disasters [6, 7]. feng et al. [8] carried out a numerical simulation study on the pre-holed coal specimens and found that there were "i" type cracks in the axial direction of the prefabricated holes. according to the typical tensile failure mode of porous coal samples, the crack propagation and the failure state of porous coal samples are more complex [9, 10]. haeri et al. [11] performed compression tests on pre-drilled brazilian disks and rectangular specimens. the breakage load is measured for the ring type disc specimens with an axial hole with varying diameters and the distribution of the induced lateral stress is obtained. li et al. [12] conducted uniaxial compression tests on defective coal samples with different inclination angles and found that the uniaxial compressive strength and elastic modulus increased with the increase of the inclination angle. the original holes and fissures have a great influence on the crack propagation of coal and rock. during the loading process, tensile cracks occur in the axial direction around the borehole and at the left and right ends of the far field. shear cracks occur in the transverse direction around the borehole, and the cracks in other parts are affected by tensile stress and shear stress [13-15]. in the case of a specimen with borehole, the area around the hole becomes a stress concentration area, which is, generally, symmetrically distributed along the hole. complex fractures are also related to coal structure and primary fractures [16, 17]. a hole pair dominates the direction of crack propagation and has a restraining effect on the original crack, which also has a certain impact on the crack propagation around the hole [18-22]. zheng and wang [23] studied creep loading of porous coal under different influencing factors, and analyzed the effects of borehole diameter, borehole depth, and borehole inclination angle on creep. through the conventional uniaxial loading test of porous coal, this paper studies the influencing factors of deformation and failure of porous coal, mainly focusing on three aspects: ▪ the influence of the loading rate on deformation and failure of porous coal, ▪ the influence of borehole inclination on the deformation and failure of porous coal, ▪ the influence of borehole diameter on the deformation and failure of porous coal. through the above tests, the influencing factors of deformation and failure of porous coal are mastered to provide guidance for drilling construction and hole protection. influencing factors of deformation and failure of porous coal under conventional loading 163 2. test scheme and process 2.1. test scheme the conventional loading test scheme of porous coal is shown in table 1. firstly, the conventional loading test of non-porous coal is carried out. the mechanical parameters of the coal are determined. furthermore, basic parameters are set for subsequent tests, and samples are provided to obtain the comparative test results for porous coal. the test parameters of the non-porous coal are: loading rate of 0.5 mpa/s, drilling depth of 25 mm and drilling diameter of 5 mm. loading tests of porous coal are carried out to investigate the following influencing factors: loading rate, borehole inclination and borehole diameter on the mechanical characteristics of porous coal. the loading rate is set to take three values: 0.1 mpa/s, 0.25 mpa/s and 0.5 mpa/s. the borehole inclination is set to be +30° and -30° compared with the horizontal borehole. the drilling diameter is set to take values: 2.5 mm, 5 mm and 10 mm. sample shape and borehole inclination angles are depicted in fig. 1. table 1 conventional loading test scheme of borehole coal test name no. drilling depth [mm] borehole inclination [°] loading rate [mpa/s] borehole diameter [mm] conventional non-porous c1 25 0 0.5 5 loading rate c2 25 0 0.1 5 c3 25 0 0.25 5 c4 25 0 0.5 5 borehole inclination c5 25 +30 0.25 5 c6 25 -30 0.25 5 borehole diameter c7 25 0 0.25 2.5 c8 25 0 0.25 5 c9 25 0 0.25 10 fig. 1 considered samples: a) sample shape, b) borehole inclination angles a) b) 164 x. jin, j. zheng, s. hua, x. tong 2.2. test process 2.2.1. preparatory work before turning on the power, the power switch on the back of the numerical control device is used to activate the preheating state. the readings of each sensor, which are displayed in the "self inspection" state, are observed via a computer to check that the system is operative. upon confirming it, the cooling water valve is opened and the hydraulic source started. thy system needs to run under low pressure for the first 5 minutes, after which it can operate under high pressure until the oil temperature is normal (pipeline heating), which indicates the end of the preparation stage. 2.2.2. conducting the experiment upon preparation, test pieces are installed according to different test types. the displacement sensor is adjusted to the appropriate position according to the reading value displayed in the "self-inspection" state. depending on how the control is performed, automatic and manual control modes are distinguished. the automatic mode can be selected for normal continuous loading, while the manual mode can be selected for creep loading tests. depending on what physical quantity is controlled, force and deformation modes are applicable. upon selection, the available waveform will be displayed. one has to select the loading parameters and enter the test piece parameters. at the beginning of the test, the test sample needs to be preloaded so that the indenter is in full contact with the test sample, i.e. the gap is eliminated. when the preloading is completed, the software indicates that the test can be started. 2.2.3. processing the test results rmt-301 test system provides users with conventional test result processing software. as long as the corresponding result column and the corresponding file name of the test result are entered, various test curves and data are obtained and can be printed out. additionally, the system can also provide unprocessed test data, which can be later processed by some other data processing software to represent the results in a suitable form. as already mentioned, the conventional loading test of non-porous coal is done first. the sample is continuously loaded at the loading rate of 0.5 mpa/s until the specimen gets damaged. fig. 2a shows the development of the axial displacement, d1, and the transverse displacement, d3. fig. 2 coal sample c1: a) force-displacement curve, b) failure mode 0 5 10 15 20 25 -3 -2 -1 0 1 2 displacement [mm] f o rc e [m p a] d 1d 3 a) b) influencing factors of deformation and failure of porous coal under conventional loading 165 the peak axial deformation of the non-porous specimen is 0.478 mm. the resulting peak strength of uniaxial compression is 20.7 mpa. obviously, the compressive strength of the specimen is relatively high, while the peak deformation of the specimen is small, and the specimen shows a brittle failure. in this case, the damage is mainly shear failure, with an obvious shear failure surface and single crack expansion, as can be seen in fig. 2b. 3. analysis of the test results 3.1. influence of loading rate on deformation and failure of porous coal the deformation and failure of coal samples experienced elastic stage, plastic yield stage and failure stage. in the elastic stage, the specimen has linear compression. after plastic deformation and yield, the stress peak is reached, and the specimen is then destroyed. the stress-strain curves and failure characteristics under different loading rates are shown in figs 3-5. when the loading rates are 0.1 mpa/s, 0.25 mpa/s and 0.5 mpa/s, the compressive strength is 14.58 mpa, 16.15 mpa and 17.54 mpa, respectively. obviously, the higher the loading rate, the greater the compressive strength. fig. 3 coal sample c2: a) force-displacement curve, b) failure mode fig. 4 coal sample c3: a) force-displacement curve, b) failure mode 0 5 10 15 20 -2 -1 0 1 displacement [mm] f o rc e [m p a] d 1d 3 0 5 10 15 20 -3 -2 -1 0 1 displacement [mm] f o rc e [m p a] d 3 d 1 a) b) a) b) 166 x. jin, j. zheng, s. hua, x. tong fig. 5 coal sample c4: a) force-displacement curve, b) failure mode regarding the damage, the fracture expansion form of porous coal is very different from that of non-porous coal. the fracture expansion of non-porous coal has an obvious shear plane, while the fracture expansion of porous coal has no obvious shear plane. the fracture expansion of porous coal is greatly affected by the borehole. due to the formation of stress concentration areas around the borehole during the loading process, the resulting stress around the borehole is greater, so the fracture expansion is centered on the borehole. the borehole controls the main fracture expansion direction, and the fracture expansion direction and the extend of fracture are more complex. 3.2. influence of borehole inclination on deformation and failure of coal with holes when the borehole inclination is +30°, the peak stress is 15.29 mpa and the peak deformation is 0.4744 mm. on the other hand, with the borehole inclination of -30°, the peak stress is 15.77 mpa and the peak deformation is 0.6847 mm, as shown in figs. 6a and 7a. compared with the horizontal boreholes, the elevation angle and depression angle of boreholes are 30°, and, in both cases, the coal strength is reduced. compared with + 30° drilling, the deformation of -30° drilling specimen is greater when it is damaged. fig. 6 coal sample c5 (borehole, inclination angle +30°): a) force-displacement curve, b) failure mode from the damage state, as shown in figs. 6b and 7b, the borehole inclination has a great impact on the damage of porous coal. when the borehole angle tilts upward by 30°, the upper part of the coal sample breaks seriously, and the fracture surface is consistent with 0 5 10 15 20 -1 -0.5 0 0.5 1 displacement [mm] f o rc e [m p a] d 3 d 1 0 4 8 12 16 -2 -1 0 1 displacement [mm] f o rc e [m p a] d 3 d 1 a) b) a) b) influencing factors of deformation and failure of porous coal under conventional loading 167 the direction of the borehole inclination. when the drilling angle is inclined downward 30°, the lower part of the coal sample is broken more severely, and the fracture surface is consistent with the direction of the drilling inclination. it can be noticed from this failure behavior that the failure surface is consistent with the dip direction regardless of the up dip angle or down dip angle. fig. 7 coal sample c6 (borehole, inclination angle -30°): a) force-displacement curve, b) failure mode 3.3. influence of borehole diameter on deformation and failure of porous coal with an increased borehole diameter, the lateral deformation is greater and the damage is more pronounced (figs. 8-10). changing the borehole diameter from 2.5 mm, via 5 mm to 10 mm, the peak stress is 18.93 mpa, 17.03 mpa and 10.55 mpa, respectively. the peak deformation is 0.6188 mm, 0.43635 mm and 0.7435 mm, respectively. the transverse deformation is 0.1007 mm, 0.0797 mm and 0.1303, mm respectively. it can be seen from the above test results that, with the increasing borehole diameter, the compressive strength decreases slightly, but the deformation is influenced by some factors, such as original cracks, impurities, etc., exhibiting certain discreteness. comparing the results for the borehole diameters of 5 mm and 10 mm, the axial deformation and transverse deformation of the latter are 1.7 times and 1.63 times that of the former, respectively. when the borehole diameter is small, the borehole has little effect on the coal sample structure, the compressive strength is still large, and the transverse deformation has little change compared with the loading of non-porous coal sample. when the borehole diameter is large, it has a great impact on the coal sample structure, the compressive strength decreases. when the drilling diameter is 10 mm, the compressive strength is 10.55 mpa, which represents a decrease of 44.3% compared to the compressive strength (18.93 mpa) obtained in the case with the drilling diameter of 2.5 mm. from the perspective of failure characteristics, when the diameter of the borehole is small, it has the influence of the borehole and also shows certain shear characteristics. the expansion of the fracture along the vicinity of the borehole is not obvious. when the borehole diameter is large, the main fracture expands centered on the borehole. 0 5 10 15 20 -1 -0.5 0 0.5 1 displacement [mm] f o rc e [m p a] d 3 d 1 a) b) 168 x. jin, j. zheng, s. hua, x. tong fig. 8 coal sample c7: a) force-displacement curve, b) failure mode fig. 9 coal sample c8: a) force-displacement curve, b) failure mode fig. 10 coal sample c9: a) force-displacement curve, b) failure mode 4. conclusion testing coal samples with boreholes of different parameters performed to determine the resulting stress and deformation are of practical guiding significance for better understanding of the dependence between the stress and deformation states around the coal seam boreholes. through the conventional loading test of porous coal, the effects of various 0 5 10 15 20 -1.5 -1 -0.5 0 0.5 1 1.5 displacement [mm] f o rc e [m p a] d 3 d 1 0 5 10 15 20 -4 -2 0 2 displacement [mm] f o rc e [m p a] d 3 d 1 0 2 4 6 8 10 12 -4 -3 -2 -1 0 1 2 displacement [mm] f o rc e [m p a] d 3 d 1 a) b) a) b) a) b) influencing factors of deformation and failure of porous coal under conventional loading 169 influencing factors on the mechanical properties of porous coal are studied, and the following conclusions are made: ▪ with the loading rates of 0.1 mpa/s, 0.25 mpa/s and 0.5 mpa/s, the resulting compressive strength is 14.58 mpa, 16.15 mpa and 17.54 mpa, respectively. hence, with the increasing loading rate, the strength of porous coal also increases, which is consistent with the effect of loading rate on non-porous coal. ▪ the test results of the uniaxial loading test of coal samples with drilling inclination of +30° and -30° show that the coal samples with holes at a certain inclination angle are easier to damage than the coal samples with horizontal holes. the failure surface is consistent with the inclination angle, and the specimens with the upward inclination angle are easier to damage compared to those with the downward inclination angle. ▪ the diameter of the drill hole has a great influence on the peak stress and peak deformation of coal. when the diameter of the drill hole is small, its influence on the structure of coal sample is small, the compressive strength is 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universitatis series: mechanical engineering vol. 21, no 1, 2023, pp. 151 173 https://doi.org/10.22190/fume200328038m © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper review of bone scaffold design concepts and design methods jelena milovanović, miloš stojković, milan trifunović, nikola vitković university of niš, faculty of mechanical engineering, serbia abstract. the paper brings out a review of existing, state-of-the-art approaches to designing the geometry of the scaffolds that are used for tissue engineering with a special emphasis on the macro scaffolds aimed for bone tissue recovery. similar concepts of different authors are organized into groups. the focus of the paper is on determining the existing concepts as well as their advantages and disadvantages. besides the review of scaffolds' geometry solutions, the analysis of the existing designs points to some serious misconceptions regarding the scaffold role within the (bone) tissue recovery. in the last section of the paper, the main requirements regarding geometry, that is, architecture and corresponding mechanical properties and permeability are reconsidered. key words: tissue engineering, bone tissue scaffolds, design concepts, design methods 1. introduction tissue engineering (te) is an interdisciplinary field that applies the principles of engineering and life sciences to the development of biological substitutes that restore, maintain, or improve tissue function or a whole organ [1]. the tissue is a biological formation built of numerous different but similar types of cells that are of the same origin. except for the cells, tissue is built of extra-cellular matrix (ecm), which is made of specific proteins and enzymes. the ecm has a role of spatial frame (honeycomb or armature) that provides primarily mechanical support to the cells as well as biochemical communication network among the tissue cells. in tissue engineering the term of tissue engineering scaffold (further in text te scaffold or just scaffold) is usually used to indicate the artificial ecm, that is, the ecm which is built artificially by the (humandeveloped) technology, which has or should have the same role as natural ecm: to provide for mechanical and biochemical support for the cells that should grow-out through the space of the scaffold, building a new piece of the tissue. received march 28, 2020 / accepted october 14, 2020 corresponding author: jelena milovanovic university of niš, faculty of mechanical engineering in niš, a. medvedeva 14, 18000 niš, serbia e-mail: jelena.milovanovic@masfak.ni.ac.rs j. milovanović, m. stojković, m. trifunović, n. vitković however, the te scaffold should not always be interpreted as an artificially made ecm. actually, the design of the scaffold architecture should not necessarily mimic the natural ecm. there are several cases [2, 3] of real application of scaffold, especially in the field of bone tissue engineering where the scaffold design is significantly different from the ecm design. these application cases helped to clarify and firm the term of te scaffold as a kind of structure that provides mechanical support (and biochemical connection network) to the tissue cells, and which should not be necessarily equivalent to the ecm. 1.1. classification of the bone te scaffolds architecture in terms of so-called architecture or conceptual design, the bone scaffold may be classified into two main different types: the first, porous, that is similar to the geometry of spongy bone tissue (fig. 1) unlike the second, which may be described as a lattice-like scaffold (fig. 2). in the first design concept, the main distinctive design characteristic is pore, i.e. void, its shape and size. the junction elements in this scaffold design concept are "in the function" of building the voids, that is, pores. usually, the junction elements are very complex shell-like (husk-like) shapes that fill the space in-between the pores. fig. 1 porous scaffold [4] the scaffold design concept that does not follow a spongy bone tissue as a sort of design template shifts the focus from the pore’s design towards the design of scaffold junction elements. the design parameters are related to the shape of the junction elements, i.e. struts, its cross-section profile and the guiding curves. often, this type of scaffolds resembles a three-dimensional lattice more than a porous structure. of course, the whole volume of space through which this kind of scaffold is being stretched is the void except for the struts, so the airiness and connection between the voids of this kind of scaffold is (or may be) even greater than in the porous scaffolds. however, the contact surface of the junction elements of this type of scaffold is far smaller than in the case of a porous scaffold. review of bone saffold design concepts and design methods 153 fig. 2 lattice-like scaffolds there is a third type of architecture of the bone scaffold, which resembles fabric [5]. it is usually made of layers placed one over another. layers are made of tiny fibers which are oriented randomly or according to some 2d pattern. considering the design of junction elements (fibers), however, these scaffolds more look like lattice-like scaffolds. in fact, this kind of scaffold design may be categorized as a specific sort of lattice-like scaffold. thus, the taxonomy of the bone tissue scaffolds regarding their architecture, i.e. design concept, may be proposed through a following tree (fig. 3): ▪ porous scaffold, (tissue-like), where the focus is on the design of pores o 3d pattern of pore units (pore-cells) (the generic units of pores are designed, and the pattern of their 3d disposition is parametrically controlled) ▪ where the design of generic units of pores and their 3d pattern are predefined, ▪ topologically optimized regarding − pores size (ratio of voids/junction elements volume) − voids connectivity − maximum or minimum of junction shell-like elements contact surface − required mechanical properties − multi-criteria ▪ lattice-like scaffold where the focus is on design of lattice struts o fabric-like scaffold (layers of fibers disposed in a different 2d patterns); this sub-type of the scaffold architecture is usually made by fdm additive manufacturing technology ▪ optimized regarding − ratio of voids/junction elements volume o lattice-like scaffold as a 3d pattern structure made of lattice struts units (the generic units of lattice are designed, and the pattern of their 3d disposition is parametrically controlled) ▪ where the design of generic units of lattice and the 3d pattern are predefined j. milovanović, m. stojković, m. trifunović, n. vitković ▪ topologically optimized regarding − required mechanical properties − ratio of voids/junction elements volume − multi-criteria o lattice-like scaffold as 3d non-patterned structure made of fully designed lattice struts ▪ topologically optimized regarding − required mechanical properties − ratio of voids/junction elements volume − multi-criteria fig. 3 bone te scaffold design concepts tree the most often scaffold design solutions in practice are a fabric-like scaffold or a simple porous scaffold, since they are the easiest to produce. for such type of scaffolds, the "fused deposition modeling" is the golden standard as the fabrication method. it is a cheap method which enables application of different materials including various biocompounds, bio-inks and gels. complex porous or lattice-like scaffolds are more difficult to produce by fdm due to the necessity for deposition of support structure along with deposition of the main materials. the post-processing can be very pernickety and timeconsuming. however, fabric-like and simple porous scaffolds are inapplicable for detailed topological optimization, especially regarding mechanical properties. this is an important shortcoming for the case of the scaffolds aimed for recovery of bone tissue that is usually (and considerably) affected by the mechanical loads. in addition, having on mind the limitations of the current manufacturing technologies (especially the amt) the authors of the paper consider that there is a need to propose the scaffold classification regarding scaffold size, too. the scaffolds, whose overall dimensions (height, width, length) are larger than 1 mm should be classified as macro scaffolds. the scaffolds, whose overall dimensions are less than 1 mm should be classified as micro te scaffold architecture design concepts porous scaffold (tissue-like) focus on design of pores lattice-like scaffold focus on design of struts 3d patterned lattice units fabric-like scaffold 3d pattern of pore units topological optimized unit design and pattern predefined unit design and pattern non-patterned lattice scaffold topological optimized unit design and pattern predefined unit design and pattern topological optimized ratio of voids/junction elements volume… optimized regarding ratio of voids/junction elements volume… review of bone saffold design concepts and design methods 155 scaffolds. the similar sub-classification may be done regarding the size of the scaffold design details such as junction elements dimensions or pore size. the scaffolds whose design details are smaller than 0.25 mm like diameter of a pore or a strut or thickness of a junction element should be classified as micro-detail scaffolds (e.g. pore volume is less than 0.01 mm3 and area of the junction element's cross-section is 0.196 mm2). so, regarding sizes, we propose the following taxonomy: ▪ macro scaffolds featured by o macro-details o micro-details ▪ micro scaffolds featured by o macro-details o micro-details according to the widely accepted requirements for the achievement of successful scaffolds for te application [6, 7], the scaffold should: 1) possess appropriately sized interconnecting voids (pores) to favor tissue integration, reinnervation and vascularization, 2) be made from material with controlled biodegradability or bio-resorbability so that tissue will eventually replace the scaffold, 3) have appropriate surface chemistry to favor cellular attachment, cell differentiation and proliferation, 4) possess adequate mechanical properties to match the intended site of implantation and handling, 5) not induce any adverse response, and 6) be easily fabricated into a variety of shapes and sizes. traditional conventional fabrication techniques [8] generate random architecture and provide minimal control over the internal architecture of the scaffolds, meaning that they are incapable to precisely and repeatably control the structure of the scaffold in terms of pore size, geometry interconnectivity and spatial distribution of pores. most scaffolds fabricated with these techniques suffer from a lack of mechanical strength and/or uniformity in pore disposition and size. on the other side, the advent of additive manufacturing technologies (amt or at) enabled production of complex three-dimensional structures of scaffolds of controlled internal architecture, that is, fabrication of scaffolds with precisely defined pore's shape and size as well as their spatial disposition. amt offers major advantages regarding fabrication of te scaffolds: customized design, computer-controlled fabrication, anisotropic scaffold structures, and application of various biomaterials. to fabricate a te scaffold by amt, it is first necessary to model the geometry of the scaffold in cad software. considering the importance of internal architecture of scaffolds and lack of adequate review in this field, the focus of this paper is to review the most important published designing approaches for scaffold internal architecture and corresponding design concepts. for the sake of terminological precision, it seems necessary to clarify the term "internal architecture" of the scaffold. actually, if one introduces this term, then it should be clarified what the term "external architecture" would refer to. the term internal architecture of the scaffold is usual, and it refers to the geometry of the scaffold junction elements that are located in the volume of the scaffold. however, it is possible to design and create specific junction elements of the scaffold that would be located in the so-called boundary surface (layer) of the scaffold. the boundary surface (layer) of the scaffold is j. milovanović, m. stojković, m. trifunović, n. vitković the imaginary surface that wraps the volume of the scaffold and usually imitates the shape of the boundary surface of the tissue region that should be replaced by the scaffold. thus, the term "external architecture" of the scaffold may refer to the geometry of these junction elements. 2. design concepts of scaffolds for bone tissue recovery as already stated hereinbefore, the design of te scaffolds usually attempts to mimic both the internal architecture of replaced tissue and the external shape (boundary surface, or “contour geometry”). the internal architecture is complex and consists of numerous pores (voids) interconnected by channels, which facilitate cell proliferation and nutrient flow, and consequently tissue regeneration [9]. there are many attempts to create scaffold design with controlled internal architecture by the application of different design concepts. the most characteristic approaches are presented in this (following) review. 2.1. unit cells-based design with this kind of design approach, the scaffold internal architecture is created by arranging junction elements as a sort of building blocks or so-called unit-cells in the space which is shrouded by the boundary (contour) surface of the bone region, substituted by the scaffold. the unit cells are designed in a cad application and have parametrically controlled geometry. the boundary surface of the scaffold is designed through the process of reverse modeling of the bone based on medical images data (mri, ct scans or x-ray images). this approach also allows creation of the heterogeneous scaffold internal architectures, by collocating the unit cells of various geometries (already predesigned and stored in the unit cells library) in the space occupied by scaffold. advances in computer technology and its use to aid tissue engineering have led to creation of a new field called computer-aided tissue engineering (cate). sun et al. [10] were among the first researchers to review advances in this field. cate encompasses the following three major applications in tissue engineering: 1) computer-aided tissue modeling, 2) computer-aided tissue informatics, and 3) computer-aided tissue scaffold design and manufacturing [10]. the same authors also discussed the application of cate to so-called biomimetic modeling and design of tissue scaffolds [11, 12]. considering that the biological tissue is inherently a heterogeneous structure regarding its porosity and mechanical features, to model te scaffold with such features it is required to apply, i.e. to embed the appropriate unit cells from the unit cell library (fig. 4), which meet required porosity, interconnectivity and mechanical properties. by collocating the unit cells, similar in size, but of different design (and, consequently, of different characteristics), in the space of tissue region that the scaffold should substitute, the designer can create the scaffold of the required characteristics (porosity, structural strength, elasticity, etc.). the authors identified this approach as the characterization of tissue structural heterogeneity through a homogenization technique. in situations when the “characterization” review of bone saffold design concepts and design methods 157 of bone tissue has to be defined per layers, the designer can build the scaffold by deposing layer by layer, where each layer consists of 2d array of one kind of unit-cell. fig. 4 samples of the designed scaffold unit cells [13] the geometry of the boundary surface of the scaffold model is usually formed by applying boolean operations of subtraction, where the raw block of scaffold is being pruned (trimmed) by the model of boundary surface of the bone. overall procedure of modeling and designing biomimetic bone scaffold is presented in fig. 5. fig. 5 biomimetic bone scaffold design procedure [13] j. milovanović, m. stojković, m. trifunović, n. vitković sun, starly and other authors [12, 13, 14], proposed the internal architecture design (iad) approach to overcome the issues encountered by the designers in cad software during recurrent emplacing of unit-cells featured by heterogeneous complex geometry in three dimensions. this issue becomes even more difficult to cope with when one should manufacture such intricate structures. keeping in mind that this kind of structures is possible to fabricate using amt, i.e. using the principle of solidification of material layer over layer, the iad approach offers to generate a kind of biomimetic designed tissue scaffolds through 2d (layered) interior pattern. this pattern is used to generate a processing tool path (fig. 6). fig. 6 methodology for internal architecture design [13, 14] authors designed cylindrically shaped bone scaffolds using the iad methodology and fabricated them using the theriform machine (fig. 7). fig. 7 cylindrical shaped and a bone scaffold designed using the iad [13] the scaffolds were fabricated of alumina. the main disadvantage of this design approach is the inability to represent (visualize) the final design of the scaffold as a whole, review of bone saffold design concepts and design methods 159 considering that this approach implies an implicit representation of unit cells geometry of different shapes. this inability gains the importance especially for the case of applying complex internal scaffold architecture. the same group of authors (gomez et al. [15, 16]) presented scaffold designing process that is based on applying unit geometric shapes in three scales (multi-scale: micro-, mesoand macroscale, fig. 8). at the micro-scale the geometry of unit building blocks (cells), their porosity grade, voids connectivity and mechanical properties are considered, that is, their geometric and mechanical congruency with the tissue these unit-cells should substitute. at the meso-scale, the scaffold design, featured by heterogeneous properties of the real bone tissue, begins to be considered as a whole. within the meso-scale, designing the heterogeneous scaffold involves the morphological, structural, and mechanical properties of the tissue, which are defined in the micro-scale, but also, the loading conditions for the tissue that are defined in macro-scale [16]. in the macro-scale, the design process is focused on scaffold boundary surfaces and implanting conditions, mechanical constraints and loads, and connections with neighboring tissue. thus, the meso-scale model may be perceived as the model which connects and integrates the data sets which come from the models built in microand in macro-scale. the changes that are being made in design of the model built in either microor macroscale are reflected in models design in two other scales, thereby integrating the scaffold design process between the scales. the data set inherent to each unit-cell, consisting of parameters relevant to its mechanical, biological and geometric properties as well as to its connectivity and manufacturability, makes an information chunk used for unit cell selection and assembling in a heterogeneous tissue scaffold. fig. 8 multi-scale modeling of a bone [16] j. milovanović, m. stojković, m. trifunović, n. vitković wettergreen et al. [16] recognized the lack of a generic geometric connection feature between unit-cells as one of the issues in the other approaches based on unit cells, which may result in emergence of critical stress lines and border fractures. to overcome this issue, the authors developed the library of unit-cells with generic interface in the form of a torus, capable to be merged seamlessly ensuring the required mechanical properties (like elasticity, stiffness, strength), porosity and perfusion of the scaffold. additionally, a series of structural analyses (using finite element method) has been conducted for different geometries of unit-cells to determine their stress and strain state under regular loads/constraints cases for a wide range of material and porosity grades. the approach was demonstrated on the example of computer-aided design of the scaffold that should substitute the human vertebra body tissue [18]. the scaffold is designed as a layered structure by arranging the unit cells of determined material in series or parallel, trying to provide the similar mechanical properties of the scaffold as the corresponding bone tissue material. chua et al. analyzed suitability of different polyhedral shapes for use as a scaffold unit cell [19, 20]. only open cellular (the cell is made just of cell edges and the cells connect through open faces) were accepted for porous scaffold constructions. total of 11 polyhedral shapes were selected and subsequently divided into two categories – cells that: 1) can fill space without leaving gaps, and 2) can fill space with leaving gaps. selected polyhedral shapes were modeled in cad software (creo, former pro/engineer) in a way that enables scaling to the appropriate pore size in accordance with the application of the scaffold). the geometry of the scaffold unit-cells is parametrically controlled. the scaffold model is generated by choosing an appropriate unit-cell from the library (depending on porosity grade, ratio between the unit-cell boundary area to its volume and strength requirements), sizing it and assembling automatically following the surface profile of the actual tissue/organ. to verify the concept and developed algorithm for automated scaffold assembling, scaled models of unit cells and scaffolds with different strut thickness were made from commercialized polyamide (pa) material using selective laser sintering (sls) technology (sinterstation 2500 machine). all the necessary actions are implemented through the system called casts (computer aided system for tissue scaffolds) [20]. the algorithm of casts is able to automatically generate a tissue-like (bio-mimetic) structure that is suitable for the specific application. to validate the system, a patient specific femur scaffold was generated and fabricated from duraform polyamide material via. automated scaffold design (asd) is another method for designing a 3d bone tissue scaffold introduced by mahmoud et al. [22]. asd covers segmentation, registration and 3d rendering visualization of the scaffold and defected bone. segmentation is performed on computed tomography (ct) images using k-means algorithm. registration is done in three stages and requires ct images of both legs of the patient (where only one is defected) (fig. 9). 3d visualization is obtained using the matlab function isosurface, implementing lorensen’s „marching cubes” algorithm. review of bone saffold design concepts and design methods 161 fig. 9 for the same slice: healthy bone image (left), defected bone image (middle), and difference between healthy and defected bone image (right) [22] after remodeling of external architecture of the scaffold, asd is applied to design the scaffold internal architecture, i.e. unit-cell geometry of appropriate pores size for the desired bone tissue (fig. 10). the final scaffold model is made by intersecting scaffold structure made of unit-cells with scaffold outer shape, that is, the boundary surface of the bone (fig. 11). fig. 10 differently designed internal architecture of the scaffold, i.e. differently designed unit-cells and their 3d arrangement [22] fig. 11 final scaffold model (left), and its positioning in the volume of the defective bone (right) [22] chantarapanich et al. evaluated library of 119 polyhedrons for modeling of so-called open-cellular and closed-cellular scaffolds [23]. each polyhedron was evaluated according to the criteria related to geometry, mechanical strength and manufacturability. the result of evaluation revealed that only four polyhedrons were suitable to be used for the creation of the closed-cellular scaffold, while six polyhedrons were suitable for opencellular scaffold creation. 2.2. image-based design image-based design approach is also focused on creating the biomimetic scaffold architecture featured by irregular or regular porous structure. it relies on radiographic images analysis, usually ct and µct. this approach was initially proposed by hollister et al. [24]. their design method (called image based engineering (ibe)) begins with creation of defect image (contour design of the implant) by inverting the contrast of the ct or magnetic resonance imaging (mri) image. scaffold internal architecture (3d array of structure units) is created by so-called image-based topology design method, which implies setting voxels within an image design cube to either “0” (void voxel) for no material or “1” (solid voxel) for material. the structure units may be created of entities which can be expressed by a geometric mathematical formula, such as cylinders j. milovanović, m. stojković, m. trifunović, n. vitković or spheres. the porous structure, that is, internal channels of these units, can have regular or random spatial disposition. random porous structure can be created by random setting voxels to 0 or 1. the image pore size is defined by the image resolution as mm/voxel. scaffold is created by combining defect image with architecture image. scaffolds made of epoxy using stereolithography (sla 250 machine) were created for orbital floor, and yucatan mini-pig temporomandibular joint condyle reconstruction. in vivo testing was conducted with scaffolds manufactured from hydroxyapatite (ha) (with different internal architectures) implanted in a yucatan mini-pig mandible. drawing on aforementioned work of hollister, taboas et al. developed methods for creating scaffolds that contain locally porous and globally porous internal architectures [25]. global porous architecture and scaffold exterior are created using ibe method [24]. local porous architecture is created using conventional techniques. scaffolds are produced by using indirect solid free form (sff) manufacturing technique developed by the authors. this technique is compatible with ibe method and combines the benefits of local pore manufacturing and direct sff fabrication. the main characteristic of indirect sff is that a mold is used to cast the final product. poly(l)lactide (pla) scaffolds were made with porogen leaching and emulsion-solvent diffusion casting of polymer into sff global pore molds. molds were created on a solidscape modelmaker ii 3d printer. porous discrete composites, including regions of pure sintered ceramic (ha), pure polymer (pla, polyglycolide (pga)), and combinations of the two in the same scaffold, were also fabricated. biomimetic pla scaffold, replicating human distal femoral trabecular bone structure, was produced with solvent casting. in accordance to the proposed taxonomy this kind of scaffold is a porous scaffold featured by random architecture. multi-scale voxel modeling approach presented by fung et al. [26] uses patient specific digital images as the basis for modeling the bone structure both at the macroscopic and microscopic levels. macroscopic geometry is acquired from low resolution digital images of the patient bone by traditional reverse engineering techniques. a high resolution image is used for microscopic geometry construction (fig. 12 (left)). randomness of the trabecular network was described by using correlation function, which can be thought of as the probability of finding randomly selected points that are both in the pore phase (fig. 12 (right)). fig. 12 scanning electron photomicrograph of transverse slab of vertebral trabecular bone (left). the sample image after thresholding (middle). 2-point correlation function of pore (r represents the distance of two randomly selected points) (right). [26] review of bone saffold design concepts and design methods 163 the basic idea of the authors was to reconstruct a target (micro)structure (starting from the initial regular structure) which would be statistically equivalent to the original (micro)structure (fig. 13). fig. 13 the initial regular cell structure (left). the final random cell structure (right). the comparison of correlation functions (down) [26] na-alginate sample, based on voxel model, was fabricated on an in-house built direct fabrication system (fig. 14). fig. 14 designed simple voxel model (left). fabricated na-alginate sample based on voxel model (right) [26] one of the possible approaches is direct reconstruction of 3d volumetric model from µct images, as presented by podshivalov et al [27]. in accordance to the proposed taxonomy this kind of scaffold may be categorized as a porous, featured by random architecture. they j. milovanović, m. stojković, m. trifunović, n. vitković presented micro-scale structure scaffolds made from a polymeric biocompatible material manufactured at different levels of resolution (24 µm and 48 µm) [27] voronoi tessellation method was used by gomez et al. [28] for the design of 3d trabecular bone-like structures. in accordance to the proposed taxonomy this kind of scaffold may be categorized as a lattice-like scaffold whose struts are topologically optimized. the core of the method is in disposition of the points within the volume-of-interest (voi) in accordance to the certain distribution, and subsequent creation of irregular polyhedral unit cells from these points. distribution of points is defined by using the µct images of the l3 human vertebra. images were split by voronoi tessellation method, which results in formation of voronoi cells. for every image, distribution of points in 2d is defined by creating center points of voronoi cells. 3d distribution of points is the result of summarizing successive 2d slides at a distance equal to the bone index “mean trabecular separation”. 3d voronoi cell structure is obtained by processing these points. polyhedral unit cells separated from each other at an equivalent distance to the bone trabecular thickness are created next. boolean operations give the final 3d porous interconnected structure. smoothing is being performed at the end. 2.3. implicit surfaces modeling implicit surfaces modeling is a highly flexible approach which allows complex scaffold internal architecture to be easily described using a single mathematical equation. application of triply periodic minimal surfaces (tpms) for the construction of scaffold internal architecture is a dominant approach among researchers nowadays [29, 30, 31, 32]. minimal surfaces may be characterized as surfaces of minimal surface area for given boundary conditions. tpms are minimal surfaces that are periodic in three independent directions, extending infinitely. the most important advantages of tpms in the field of scaffold internal architecture design are the following [29, 30, 31, 32]: 1) precise and easy controllability of internal pore architecture, 2) design process can be fully automated, and 3) a high surface area to volume (sa/v) ratio. tpms also appears in the natural and man-made worlds (silicates, bi-continuous composites, lyotropic colloids, detergent films, and lipid bilayers). starting from the tpms mesh surface (composed of simple trigonometric functions), through the offsetting procedure, yoo generated various types of thickened solids, suitable for representing scaffold internal architecture [29]. the scaffold architecture may be characterized as a porous, topologically optimized. he also presented a new method which uses tpms as the basic pore-making element and generates human bone scaffold models. this was the first attempt to use tpms for scaffold design. the same author proposed a new approach based on the multi-void tpms pore architectures [30]. review of bone saffold design concepts and design methods 165 the main advantage of multi-void tpms-based scaffolds is a dramatic increase of sa/v ratio compared to conventional tpms scaffolds. another contribution of yoo is a hierarchical porous scaffold design based on tpms [33]. this time the author used boolean operation of intersection to generate scaffold with controlled internal architecture. talus bone scaffold model was designed and fabricated using the sysopt eden 330 rp machine. scaffold was made of uv-curable polymer. like yoo, yang and zhou presented an effective method for multiple substructures combination [34]. the proposed method enables easy construction and direct fabrication of functional gradient porous scaffold (fgps). 2.4. specific approaches lal and sun [35] presented a computer modeling approach for constructing 3d microsphere-packed bone graft structure. basic microspheres packing model was created from scanning electron microscopy (sem) images of synthesized cylindrical bone grafts. two extreme cases of microspheres packing were examined: maximum packing density (minimum porosity and open-cell bone structure) and minimum packaging density (maximum porosity and closed-cell bone structure). for these cases, number of microspheres was determined. since bone is composed of open and closed-cell structures, the number of microspheres in synthetic bone graft (a combination of both packing cases) was calculated using a statistical approach. microsphere-packed 3d bone graft is formed by stacking randomly packed microsphere layers (randomly combined open/closed cell packing situations). parametric study on the impact of the microsphere’s diameter on pore size and number of packed microspheres was conducted. comparison between the cad model of bone graft showing bone ingrowth and histological image of in vitro bone ingrowth showed that the cad model resembles a histological image. this scaffold may be categorized as a pattern porous scaffold. lian et al. [36] discussed 3d concentric microstructure construction in artificial bone. 3d concentric architecture with gradient porosity is constructed by arranging 2d concentric structures which have the same mathematical model. 2d structures can have different structural patterns that are obtained by changing the model parameters. among the input parameters there are porosity, height and radius of the artificial bone. special software for design of concentric architecture (fiber structures) was developed. in accordance with the proposed taxonomy, this kind of scaffold is a fiber-liked lattice scaffold. these structures were incorporated into the calcium phosphate cement (cpc) matrix to form a fiber reinforced cpc composite artificial bone with controlled internal architecture and the desired porosity. it was also confirmed that these resorbable fibers incorporated in the artificial bone may provide short-term strength and can be degraded significantly faster than the ha leaving macro-pores suitable for bone ingrowth. cylindrical cpc-fiber scaffold with a height of 23 mm and a diameter of 10 mm was fabricated by indirect amt. ramin and harris [37, 38] developed a dedicated library of routines in order to interact with cad software and perform the automatic design of geometric elements representing scaffold internal architecture. they used multi-section solid as the basic element. developed routines were used for defining the pore shape and size, 3d path for each multi-section solid and designing the multi-section solids. this methodology allows rapid design and integration of a complex network of channels within scaffold, determined by the set of variable parameters that can be changed within the software, to match the desired characteristics j. milovanović, m. stojković, m. trifunović, n. vitković defined by internal architecture of tissue. five cubic scaffolds with interconnected pore channels that range from 200 to 800 μm in diameter were made using this methodology. this kind of scaffold is a porous scaffold with predefined 3d pattern of voids. cai and xi [39] introduced morphology-controllable modeling approach for constructing te bone scaffolds. the main advantage of this approach is the possibility to create scaffolds with various irregular pores by using finite element shape function. the pore shape is controlled by subdivided units. the volume (solid model) of the bone that should be substituted by the scaffold is being discretized into the 3d mesh of hexahedral elements (units). the vertices and edges of every hexahedral element (unit) are being used as vertices and edges of the control polyhedron of the surface subdivision sphere primitive. this primitive is introduced as a basic pore making unit, which can be mapped into various irregular pore units. the iso-parametric transformation was used for mapping the basic unit into an arbitrary unit (irregular pore). at the last step, the scaffold model is being created by boolean operation of subtraction. one bone scaffold model was fabricated by ink-jet printing. the scaffold made in this way obviously belongs to the porous kind of scaffolds with predefined void's geometry and predefined 3d pattern disposition. 2.4. lattice-like scaffold as 3d non-patterned structure made of fully designed lattice struts the so-called anatomically shaped lattice scaffold (аsls) that was developed by the research group from the university of niš [2, 40, 41,] is a kind of 3d lattice scaffold whose struts do not follow some 3d pattern. it is a design concept that aims to ensure high geometrical congruency to the particular anatomy, to provide maximal permeability as well as the simple and efficient fixation. the proposed lattice design concept is featured by two groups of struts (fig. 15). the enveloping struts are densely interlaced following the geometry of outer wrapping surface of the bone tissue. still the lattice of enveloping struts is designed sparse enough to enable easy penetration of vascular and nerve structures to the interior of the scaffold volume. the second group of struts (cross-linking struts) is stretched through the interior space of the scaffold volume, the space which should be taken by the spongy bone tissue. the cross-linking struts connect the struts in the enveloping lattice, providing the required strength and stiffness to the cage of the scaffold. low density of the inner structure is designed to assure profound vascularization and innervation of the bone graft. several scaffolds of this kind are applied for the in-vivo experiment, which is performed in order to explore their applicability for the real cases of missing large pieces of the bone. the scaffolds are designed for large trauma of proximal diaphysis of rabbit's tibia (fig. 16). they are made of ti-alloys by application of amt (in particular for the experiment, by using ebm and dmls). the newest research regarding this kind of scaffold is focused on making the whole implant assembly of bone graft and biodegradable scaffold at once by using bio-3dprinter. review of bone saffold design concepts and design methods 167 fig. 15 concept solution of asls developed for human tibia [2] fig. 16 scaffold designed and fabricated for large trauma of proximal diaphysis of rabbit's tibia (left), scaffold implantation into the defect area in a rabbit model (right) [42] 2.5. topology optimization in the last ten years, with the significant increase of the computer capabilities, and the emerging new algorithms for the so-called topological shape optimization [43, 44, 45, 46], 3d modeling of bone tissue scaffolds becomes an ideal field for applying topological optimization of scaffold architecture in order to provide the desired characteristics regarding the expected mechanical stresses and deformations. material distribution method has demonstrated its potential in a large number of case studies in this field [43]. in terms of scaffold design, one of main and unavoidable optimization goals should be minimal volume of the material of the scaffold, which leads to algorithm [47] to generate lattice like structures. at the same time, this goal ensures the lattice-like structure to be as much airy as possible. in case of this optimization goal, to algorithm should not consider the structures featured by “closed” pores, keeping in mind that the scaffold should enable maximal communication through its volume. j. milovanović, m. stojković, m. trifunović, n. vitković another optimization goal should be related to the minimization of strain in the structure. this optimization goal would direct to algorithm to generate a kind of anisotropic lattice-like structure optimal to bear the load typical for that bone region. however, for engaging this optimization goal into to algorithm, it is necessary to model the equivalent load case, which should approximately emulate the usual and real load cases [48]. in the field of scaffold design topology optimization techniques were used also by hollister [49, 50, 51, 52, 53], challis et al. [54] and many others. they considered stiffness and diffusive transport properties in their works. using topology optimization, hollister and co-workers also created an interbody fusion cage with porous architecture to help improve arthrodesis [55]. in this study a topology optimization algorithm is proposed as a technique to design scaffolds that meet specific requirements for mass transport and mechanical load bearing. 3. discussion almost all (except to) methods presented for modeling the geometry of bone tissue scaffolds are intended to mimic the complex geometry of the trabecular structure of spongy bone tissue. comparing to the spongy bone tissue, the geometry of the cortical bone structure is not in the focus of current research, probably for two reasons: the first relates to its density the structure of the cortical bone is too dense and it is not likely that the artificially created structure could allow required extent of communication of surrounding tissue to the bone interior, necessary for profound innervation and vascularization of proto-tissue within the volume of the scaffold; another reason is the limitations of the current additive manufacturing technologies to produce such a dense and, in the same time, geometrically complex structure. the resolution of solidification and/or deposition of materials that can be achieved by existing additive manufacturing technologies are insufficient for such fine details. 3.1. design of scaffold and its real implantation purpose regardless the type of bone tissue for which scaffold design methods are being developed spongy or cortical, it is important to emphasize, once again, that the vast majority of these design methods are aimed to mimic the geometry of the structure of natural bone tissue to a greater or lesser extent. however, the question is is it necessary at all to create a bone scaffold geometry which resembles the structures of natural bone tissue? such an approach could be justified if the goal is to produce a kind of tissue endoprosthesis that should completely and permanently replace the missing bone. in that case, the goal would be to model the geometry of both the outer, wrapping surface of the bone and the internal structure of the spongy bone, which matches as closely as possible the geometry of the natural tissue. it would not, however, be a scaffold, but rather an endoprosthesis with all the geometric details of the complete bone volume however, with bone scaffold implantation, the intent is substantially different from implantation of a bone endoprosthesis. first of all, the bone scaffold is aimed to reinforce the proto tissue in early stage of recovery, that is, to provide required temporary mechanical properties to the growing bone tissue. keeping that in mind, the scaffolds that are aimed for bone tissue recovery, but whose design imitates neither spongy bone tissue nor the ecm, are usually designed as a kind of three-dimensional lattice structure that resembles a cage. the cage holds the proto-tissue review of bone saffold design concepts and design methods 169 that should transform into true bone tissue with all its natural features including geometry during recovery. the early proto-tissue is a mixture of a crushed natural or artificial bone with the addition of fat tissue, blood plasma, progenitor cells and growth factors. due to its mushy consistence, that is, very low structural strength, the proto-tissue cannot be exposed to higher mechanical loads. on the other hand, the research and practice [56, 57] indicate the necessity of mechanical loads application to a portion of the traumatized bone as one of the main stimuli for the ossification process, first at the interface of the proto-tissue and surrounding healthy tissue, and later, at the depth of the proto-tissue itself. also, the scaffold geometry should enable smooth penetration of nutrients into the volume of prototissue without which it is not possible to expect the transformation of proto-tissue into genuine bone tissue. in fact, the scaffold geometry should not obstruct the proliferation of blood vessels and nerves into the proto-tissue volume. this scaffolding function indicates the maximum porosity or transparency of the lattice structure. scaffolds whose geometry mimics spongy bones, however, do not facilitate but hinder the sprouting of native tissue into the space of scaffolds. this function calls for maximum porosity or airiness of the lattice structure of the scaffold. another important feature of the scaffold geometry in the macro and micro scale is suitable adhesiveness of the scaffold strut surfaces that will help the proto-tissue particles to attach to the scaffold firmly. finally, the last, but no less important feature of the bone scaffold, which, however, is not directly related to its geometry is its biodegradability. since the proto tissue is expected to transform into a genuine bone tissue during the recovery process, growing up through the volume of the scaffold cage, it is necessary for the artificial structure of the scaffold to degrade and resorb over time, and the volume of degraded scaffold structures to be replaced with the real bone tissue. it is the most desirable scenario of recovery that would allow any artifactual structure, which could possibly be the source of infections and necrotic processes in the future, to disappear from the tissue. if the time-controlled biodegradability of the scaffolds could be achieved in near future, this would be indirectly related to the scaffold geometry. certain elements of the scaffold cage structure would degrade faster while the others would degrade slower, so the mechanical properties of the scaffold cage structure could change according to a predefined time plan. the initial load taken over by the scaffold at the beginning of recovery process could be transferred to the newly formed bone tissue during the recovery time gradually. having in mind the primary function of bone scaffold to reinforce the protobone tissue during the recovery, scaffold geometry (i.e. lattice structure of the scaffold) should match the required anisotropy of the mechanical properties. this is needed in order to ensure proper deformability according to the load conditions and bone characteristics specific for the particular patient. the scaffold models created as a three-dimensional pattern of shape units, i.e. unit cells (voids or struts) have small potential to adjust the anisotropy precisely, that is, to be personalized for the particular patient. in contrast, the algorithms of topological optimization coupled with modern cae software bring momentous advantage in designing of personalized lattice-like bone scaffolds that match the required anisotropy of mechanical properties. j. milovanović, m. stojković, m. trifunović, n. vitković 4. conclusion here are the concluding remarks regarding the presented methods for designing of scaffolds aimed for bone tissue recovery in four aspects: regarding geometry: the presented review of realized bone scaffolds concepts shows that there are many different approaches to the geometric modeling of bone scaffolds. also, the most of existing methods are focused on designing the scaffolds whose geometry mimic spongy bone tissue. the modern cad applications enable modeling of such shapes in an efficient manner, by recurrent laying of the three-dimensional shape units in the space, simultaneously controlling the size of pores (voids). even though numerous methods for bone scaffold designing are developed, it is important to notice that complex and multi-lateral requirements which a bone scaffold should meet are still not clearly defined and agreed. within the discussion section a thesis about an important misconception that seems to exist regarding the current scaffold design concepts is brought out: the determination to design the scaffold geometry congruently to the spongy bone tissue geometry is in contrast to the basic functions of scaffold. mechanics: a significant drawback of most of the design concepts of bone scaffolds, especially those that are unit-cell based (3d pattern), is their inability of adaptation to the required anisotropic mechanical properties. fabrication: considering the geometric complexity of the structures, additive manufacturing technologies seem as an optimal choice for the scaffold fabrication method (fdm, sls, dmls). however, in order to produce geometry details that can exist in the bone scaffolds as it is introduced in this paper, a significant improvement in hardware as well as the speed and resolution of rp machines are required. also, according to many authors, the future scaffold design concepts that should be personalized in terms of geometry, mechanics and time-controlling biodegradability, will require to be fabricated of multiple materials, which will also call for a significant improvement of fabrication process. very probably, we should expect to witness a new additive manufacturing technology which will be able to create the multi-material scaffold simultaneously infiltrated by personalized bio-material of bone graft. testing and application: regarding the experimental research of the bone scaffold design, it is worth mentioning that there were just a few research studies where the scaffolds of complex design (tmps, to) were applied in in-vivo experiments. mostly, the experimental research was done with simple three-dimensional pattern unit-cell design concepts of scaffolds in in-vitro experiments. as far as we have found out, there is lack of data on possible research cases of clinic application of biodegradable scaffolds for bone tissue regeneration. acknowledgements: this research was financially supported by the ministry of education, science and technological development of the republic of serbia. references 1. skalak, r., fox, c.f., 1988, tissue engineering, proceedings of a workshop held at granlibakken, lake tahoe, california, 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half-space caused facta universitatis series: mechanical engineering vol. 15, no 2, 2017, pp. 307 313 doi: 10.22190/fume170511014m © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper 1determination of important parameters for patent applications udc 911.2:556 dušan marković 1 , dalibor petković 2 , vlastimir nikolić 1 , miloš milovančević 1 , nebojša denić 3 1 university of niš, faculty of mechanical engineering 2 university of niš, pedagogical faculty in vranje 3 univerzitet of priština, faculty of science and mathematics, kosovska mitrovica abstract. this research study is an analysis of patent applications based on different input parameters. nine patent indicators for describing patent applications are retrieved from the world bank database. the method of anfis (adaptive neuro fuzzy inference system) is applied to selecting the most important parameters for patent applications. the inputs are: charges for the use of intellectual property for payments and receipts, research and development expenditure, trademark applications for residents and nonresidents, researchers in research and development (r&d), technicians in r&d and high-technology exports. as the anfis outputs, patent applications for nonresidents and residents are considered. the results show that the combination of research and development expenditure and technicians in r&d is the most influential combination of input parameters for patent applications. key words: anfis, patent applications, research, development 1. introduction since the development of new technologies is rapid and the knowledge of economics has increased, intangible assets are now more significant than before. patents have the main role in intangible assets. they are the main outcome of research and development and demonstrate the capability of innovation. companies are influenced by patent applications because of reputation and revenue. patents also present an essential issue in engineering management, technology management and finance management. received may 11, 2017 / accepted june 26, 2017 corresponding author: dalibor petković pedagogical faculty in vranje, partizanska 14, 17500 vranje, serbia e-mail: dalibortc@gmail.com 308 d. marković, d. petković, m. milovanĉević, n. denić several algorithms have been thus far presented for analyzing patent performance [1, 2, 3]. the investigation [4] proposed a patent portfolio-based approach for assessing potential research and development (r&d) partners that could consider the inter-partner resource fit in the assessment process. the study [5] investigated commercial application scenarios using patent analysis where five scenarios of application of patented commercial technology were obtained. an automatic patent quality analysis and classification system was developed in [6], where experimental results showed that the proposed system could capture the analysis effectively compared with the traditional manpower approach. the results from the article [7] showed that strengthening patent protection on horizontal r&d promotes vertical innovation (quality improvement) but hinders horizontal innovation (variety expansion). the investigation [8] showed that an increasing degree of competition enhances innovation and patent applications, which helps firms appropriating part of the benefits of their r&d investments. the article [9] showed that the later the timing of the patent, the higher the innovation performance, while under low uncertainty there was an early-mover advantage. in order to overcome high nonlinearity in patent applications models [10] in this study a soft computing method was applied. when structuring predictive models, it is crucial to include the most influential variables and discard the redundant and non-informative predictors. correct variable selection will result in increased model predictability and interpretability. in this paper, we use a soft computing approach to select the most influential variables for patent applications. the adaptive neuro-fuzzy inference technique (anfis) [11-15] is applied to the available data sets to select the predominant model variables. 2. materials and method table 1 shows nine input and two output parameters used in this research. the patent data is used from the world bank database. the first and second input parameters represent charges for the use of intellectual property for payments and recipients, respectively, between residents and nonresidents for the authorized use of proprietary rights (such as patents, trademarks, copyrights, industrial processes and designs including trade secrets, and franchises) and for the use, through licensing agreements, of produced originals or prototypes (such as copyrights on books and manuscripts, computer software, cinematographic works, and sound recordings) and related rights (such as those for live performances and television, cable, or satellite broadcast). the data for the first input parameter are in us dollars. the third input parameter represents expenditures for research and development for current and capital expenditures (both public and private) on creative work undertaken systematically to increase knowledge, including knowledge of humanity, culture, and society, and the use of knowledge for new applications. the fourth input parameter represents trademark applications filed for applications to register a trademark with a national or regional intellectual property (ip) office. a trademark is a distinctive sign which identifies certain goods or services as those produced or provided by a specific person or enterprise. a trademark provides protection to the owner of the mark by ensuring the exclusive right to use it to identify goods or services, or to authorize another to use it in return for payment. the period of protection varies, but a trademark can be renewed indefinitely beyond the time limit on payment of additional fees. it determination of the important parameters for patent applications 309 represents direct nonresident trademark applications for those filed by applicants from abroad directly at a given national ip office. the fifth parameter represents direct resident trademark applications for those filed by domestic applicants directly at a given national ip office. the sixth parameter represents the total trademark applications for those filed by domestic applicants directly at a given national ip office. the seventh input parameter represents researchers in research and development (r&d), which are professionals engaged in the conception or creation of new knowledge, products, processes, methods, or systems and in the management of the projects concerned. phd students engaged in r&d are included. the eighth input parameter represents technicians in r&d and equivalent staff, namely the people whose main tasks require technical knowledge and experience in engineering, physical and life sciences (technicians), or social sciences and humanities (equivalent staff). they participate in r&d by performing scientific and technical tasks involving the application of concepts and operational methods, normally under the supervision of researchers. the ninth input parameter represents hightechnology exports which are products with high r&d intensity, such as in aerospace, computers, pharmaceuticals, scientific instruments, and electrical machinery. the data are in us dollars. the first output parameter is patent applications of nonresidents for worldwide patent applications filed through the patent cooperation treaty procedure or with a national patent office for exclusive rights for an invention – a product or process that provides a new way of doing something or offers a new technical solution to a problem. a patent provides protection for the invention to the owner of the patent for a limited period, generally 20 years. the second output represents patent applications of residents for worldwide patent applications filed through the patent cooperation treaty procedure or with a national patent office for exclusive rights for an invention. table 1 input parameters for patent applications inputs names 1 charges for the use of intellectual property, payments (current us$) 2 charges for the use of intellectual property, receipts (current us$) 3 research and development expenditure (% of gdp) 4 trademark applications, direct nonresident 5 trademark applications, direct resident 6 trademark applications, total 7 researchers in r&d (per million people) 8 technicians in r&d (per million people) 9 high-technology exports (current us$) an anfis model will be established in this study to estimate the most important parameters for patent applications in serbia. the hybrid learning algorithms are applied to identify the parameters in the anfis architectures. a fuzzy inference system in the matlab software is employed in the whole process of the anfis training and evaluation. 310 d. marković, d. petković, m. milovanĉević, n. denić 3. results the input parameters with the lowest rmse values have the most relevance in regard to the outputs. in other words, the input parameter with the smallest rmse values has the most influence for the patent applications. the selection of the combinations of the two parameters is also performed in order to find the optimal combination of the two parameters that are of the greatest influence on the patent applications. since the results for some input combinations are redundant, tables 2-3 present only some of the selected input coronations (combinations, correlations?) of the two parameters. table 2 presents the input parameters influence on the prediction of patent applications in serbia for nonresidents. accordingly, table 3 presents the input parameters influence on the prediction of patent applications in serbia for residents. according to the results in table 2 one can conclude that the input parameter 7 (researchers in r&d (per million people)) is the most influential for patent applications in serbia for nonresidents. on the contrary, the rmse values for input parameter 1 (charges for the use of intellectual property, payments (current us$)) have the smallest influence for the prediction of patent applications in serbia for nonresidents. table 2 shows a part of two input combinations since the results for other combinations are redundant. according to the results, one can see that the input combination of parameter 3 and 8 (research and development expenditure (% of gdp) and technicians in r&d (per million people)) has the largest influence on patent applications in serbia for nonresidents. table 3 shows that the same parameter has the highest influence for patent applications in serbia for residents also. table 2 input parameters rmse values for the prediction of patent applications in serbia for nonresidents table 3 input parameters rmse values for the prediction of patent applications in serbia for residents one input two inputs one input two inputs 1 – rmse=78x1010 3, 4 rmse=0.1120 1 rmse=251x1010 3, 4 rmse=0.7572 2 rmse=51x107 3, 5 rmse=0.0213 2 rmse=19x108 3, 5 rmse=0.0805 3 rmse=8.7630 3, 6 rmse=0.6992 3 rmse=5.8842 3, 6 rmse=2.7461 4 rmse=3.6590 3, 7 rmse=0.0127 4 rmse=47.7747 3, 7 rmse=0.0466 5 rmse=1.3894 3, 8 rmse=0.0005 5 rmse=6.5772 3, 8 rmse=0.0018 6 rmse=8.6900 4, 5 rmse=0.5469 6 rmse=21.5403 4, 5 rmse=1.9533 7 rmse=0.0203 4, 6 rmse=0.7986 7 rmse=0.0179 4, 6 rmse=1.7458 8 rmse=22.9083 4, 7 rmse=0.2182 8 rmse=65.9266 4, 7 rmse=0.7052 9 rmse=21x108 4, 8 rmse=0.1335 9 rmse=45x1010 4, 8 rmse=0.4249 5, 6 rmse=0.4849 5, 6 rmse=1.7116 5, 7 rmse=0.0026 5, 7 rmse=0.0073 5, 8 rmse=0.0229 5, 8 rmse=0.0826 6, 7 rmse=0.3702 6, 7 rmse=1.2785 6, 8 rmse=0.8187 6, 8 rmse=3.3914 7, 8 rmse=0.0055 7, 8 rmse=0.0305 determination of the important parameters for patent applications 311 according to the results in table 4, the input parameter 5 (trademark applications, direct resident) is the most influential for the prediction of patent applications in croatia for nonresidents. on the contrary, the training error for input parameter 9 (high-technology exports (current us$)) has the smallest influence for the prediction of patent applications in croatia for nonresidents. according to the results in table 5, one can see the same input combination of parameters 3 and 8 (research and development expenditure (% of gdp) and technicians in r&d (per million people)) as in the case of serbia with the largest influence on the prediction of patent applications in croatia for nonresidents as well. table 5 presents the results for parameters selection for the prediction of patent applications in croatia for residents. thus, it is obvious, that the only difference between the results presented in table 4 and table 5 is a single input parameter influence. the most influential parameter for patent applications in croatia for residents is parameter 4 (trademark applications, direct nonresident). table 4 input parameters rmse values for the prediction of patent applications in croatia for nonresidents table 5 input parameters rmse values for the prediction of patent applications in croatia for residents one input two inputs one input two inputs 1 rmse=38x1010 3, 4 rmse=0.6246 1 rmse=21x1010 3, 4 rmse=0.4310 2 rmse=22x1010 3, 5 rmse=0.0693 2 rmse=18x1010 3, 5 rmse=0.0476 3 rmse=9.3460 3, 6 rmse=2.1536 3 rmse=2.1498 3, 6 rmse=0.8482 4 rmse=0.9596 3, 7 rmse=0.0469 4 rmse=0.3904 3, 7 rmse=0.0338 5 rmse=0.2246 3, 8 rmse=0.0070 5 rmse=0.6700 3, 8 rmse=0.0097 6 rmse=4.2449 3, 9 rmse=0.5223 6 rmse=2.0379 3, 9 rmse=1.0007 7 rmse=0.9240 4, 6 rmse=2.1184 7 rmse=0.6028 4, 5 rmse=0.7065 8 rmse=1.1260 4, 7 rmse=0.5094 8 rmse=1.2072 4, 6 rmse=0.2872 9 rmse=188x1010 4, 8 rmse=2.1156 9 rmse=1063x1010 4, 7 rmse=1.2742 4, 9 rmse=1.3059 4, 8 rmse=0.3427 5, 6 rmse=0.2419 4, 9 rmse=0.1697 5, 7 rmse=0.0218 5, 6 rmse=0.0292 5, 8 rmse=1.0865 5, 7 rmse=0.1646 5, 9 rmse=2.9935 5, 9 rmse=4.0226 6, 8 rmse=0.0488 6, 7 rmse=0.0627 table 6 shows the results for parameters influence on patent applications in hungary and the results are the same as for croatia and serbia, except for the single parameter influence on the prediction of patent applications. for nonresident patent applications in hungary the most influential parameter is the total trademark applications, while for resident patent applications in hungary the most influential parameter is the direct nonresident trademark applications, direct nonresident (table 7). 312 d. marković, d. petković, m. milovanĉević, n. denić table 6 input parameters rmse values for the prediction of patent applications in hungary for nonresidents table 7 input parameters rmse values for the prediction of patent applications in hungary for residents one input two inputs one input two inputs 1 rmse=44x1010 3, 4 rmse=1.2792 1 rmse=122x1010 3, 4 rmse=1.7950 2 rmse=218x1013 3, 5 rmse=0.2820 2 rmse=31x1014 3, 5 rmse=0.1559 3 rmse=94.9878 3, 6 rmse=6.4223 3 rmse=16.6644 3, 6 rmse=9.1210 4 rmse=9.5912 3, 7 rmse=0.0726 4 rmse=6.5445 3, 7 rmse=0.0916 5 rmse=128.6648 3, 8 rmse=0.0142 5 rmse=15.292 3, 8 rmse=0.0107 6 rmse=5.4916 3, 9 rmse=1.2275 6 rmse=13.0220 4, 5 rmse=2.2457 7 rmse=61.7852 4, 5 rmse=3.8079 7 rmse=17.9915 4, 6 rmse=6.4263 8 rmse=34.6907 4, 6 rmse=0.5635 8 rmse=19.2941 4, 7 rmse=0.5974 9 rmse=102x1014 4, 7 rmse=0.1682 9 rmse=175x1014 4, 8 rmse=0.2337 4, 8 rmse=1.1733 5, 6 rmse=2.2104 4, 9 rmse=0.6462 5, 7 rmse=0.1625 5, 6 rmse=0.5721 5, 8 rmse=0.0979 5, 7 rmse=6.1736 5, 9 rmse=8.5541 6, 7 rmse=4.6757 6, 7 rmse=6.6236 6, 8 rmse=0.1440 6, 8 rmse=0.2204 4. conclusion patents are intangible assets as well as the main outcome of research and development that demonstrate the capability of innovation. in this study the methodology is employed to determine the most influential parameters for patent applications. the adaptive neurofuzzy inference technique (anfis) is applied to the available data sets to select the 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assessment review, 65, pp.29-40. facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 633 656 https://doi.org/10.22190/fume201222024a © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load ahmed e. abouelregal1,2, hamid mohammad-sedighi3,4, seyed ali faghidian5, ali heidari shirazi3 1department of mathematics, college of science and arts, jouf university, al-qurayyat, saudi arabia 2department of mathematics, faculty of science, mansoura university, mansoura, egypt 3mechanical engineering department, faculty of engineering, shahid chamran university of ahvaz, ahvaz, iran 4drilling center of excellence and research center, shahid chamran university of ahvaz, ahvaz, iran 5department of mechanical engineering, science and research branch, islamic azad university, tehran, iran abstract. in this article, the influence of thermal conductivity on the dynamics of a rotating nanobeam is established in the context of nonlocal thermoelasticity theory. to this end, the governing equations are derived using generalized heat conduction including phase lags on the basis of the euler–bernoulli beam theory. the thermal conductivity of the proposed model linearly changes with temperature and the considered nanobeam is excited with a variable harmonic heat source and exposed to a time-dependent load with exponential decay. the analytic solutions for bending moment, deflection and temperature of rotating nonlocal nanobeams are achieved by means of the laplace transform procedure. a qualitative study is conducted to justify the soundness of the present analysis while the impact of nonlocal parameter and varying heat source are discussed in detail. it also shows the way in which the variations of physical properties due to temperature changes affect the static and dynamic behavior of rotating nanobeams. it is found that the physical fields strongly depend on the nonlocal parameter, the change of the thermal conductivity, rotation speed and the mechanical loads and, therefore, it is not possible to neglect their effects on the manufacturing process of precise/intelligent machines and devices. key words: nonlocal elasticity, rotating nanobeam, thermoelasticity, variable thermal conductivity, varying load received december 22, 2020 / accepted february 28, 2021 corresponding author: hamid mohammad-sedighi mechanical engineering department, faculty of engineering, shahidchamranuniversity of ahvaz, ahvaz, iran e-mails: h.msedighi@scu.ac.ir; hmsedighi@gmail.com mailto:h.msedighi@scu.ac.ir 634 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi 1. introduction thermomechanical effects resulting from the interaction of temperature and deformation fields are of particular significance in various modern designs such as high-speed aircrafts, jets, gas and steam turbines, nuclear reactors and missiles. both theoretical and practical concerns, including a rapid supply of thermal energy, have been gaining increasing attention. the conventional heat conduction theory, on the basis of the fourier’s law, takes an instant reaction to the temperature gradient and gives a distinctive parabolic equation for temperature change. to solve the infinite speed thermal propagation phenomena predicted by the classical thermoelasticity theory, modified generalized theories of thermoelasticity have been established. lord and shulman [1] suggested one of the modified generalized thermoelasticity theories containing a relaxation time by introducing a novel rule for thermal conductivity to modify the classic fourier law. the refined law includes the heat flux and its partial derivative with respect to time. among the models that have gained great popularity in recent years, the model presented by tzou [2-4] is of great interest to the researchers in the field. in this model, tzou introduced thermal heat conduction model with dual-phase lag (dpl) to include the effects of microscopic reactions in the rapid transit of heat transfer mechanism into a microscopic formula. in the constitutive relationship between the temperature gradient and heat flux, two different phase-lags have been introduced. many efforts were made to overcome the problem of unlimited speeds of heat waves predicted by the classical thermoelasticity theory [5-8]. at high operating temperatures, thermal conductivity becomes more significant in popular areas such as materials science, electronics and building insulation. temperature has a distinctive impact on thermal conductivity of metals and non-metal materials. thermal conductivity in physics is defined as the capacity of materials for heat transfer. this specifically occurs in the fourier's heat conduction theory. it was demonstrated through practical experiments that as temperature rises, thermal conductivity of pure metals decreases. in alloys, any rise in temperature results in increasing thermal conductivity, and generally, the variation of electrical conductivity is smaller than that for temperature [9]. the rapid development in micro-electromechanical systems (mems) led to more complex sensors and electronic devices. in order to satisfy the industrial requirements, new technologies have been exploited to design high-tech mems products. mems structures include mechanical components, e.g. small cantilevers, extensions and films that have been frequently filled with a variety of geometrical measurements and arrangements [10]. the influences of small-scale interactions between the neighboring material particles or constituents must be taken into account for these micro/nano-structures. this effect can be observed in miniaturized sub-micron systems like nano-actuators and nano-sensors, microscopes and micro/nano-electronic systems. it is important for designers and engineers to discover the accurate thermomechanical features of ultra-high sensitive devices in order to attain the true aim of providing for the measure of diversion from associated loads in order to forestall cracks, improve the execution and produce the suitable lifetime for mems devices [11]. from the discussion above, the principle of vibration is well-known and well-studied in dual beam systems. nevertheless, the scale-dependent vibration of beam systems makes few contributions. the structures of scale-dependent nanobeams are constructed from nano-materials with special properties because of their dimensions at the nanoscale. temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 635 nanoparticles, nanowires and nanotubes are common examples of the materials with attractive features in sub-micron scales [12]. nanomaterials are the base choices of technological products for the next century and have intensified the interest of the scientific community in advanced physics, chemistry, biomedicine and different branches of technological sciences. from mechanical point of view, due to their sub-micron dimensions, the impact of small-scale exhibits significant features resulting from the measurements of atomic fission. the strength of the mechanical piezomagnetic nanosized sensors was studied in post-buckling state by malikanet al. [13]. a powerful flexomagnetic characteristic is also considered in this research. furthermore, as a result of the flexomagnetic effects, the stability of the euler-bernoulli porous nanobeam was thoroughly investigated by malikan et al. [14]. the magnetization with strain gradients is related to the flexomagnetics. malikan et al. [15] created an euler-bernoulli beam model incorporating the combination of piezomagnetic-flexomagnetic properties and small-scale effects. flexoelectricity was also studied for a piezoelectric nanobeam by considering the internal viscoelasticity [16]. as mentioned earlier, the size effect has a great contribution in the static and dynamic behavior of micro and nanoscale structures and should not be neglected, that is, the classical continuum mechanics at nanoscale does not make any sense and must be modified accordingly. nanomaterials are used in complex nanoelectromechanical systems (nems) as well as nanocomposites due to their attractive properties. nanoscale devices are referred to the structures and systems constructed from nanosize materials. the smallscale impact in the nonlocal elasticity is defined by the assumption that stress vector is related to the entire strain field of the whole body (eringen, [17]). it is distinct from the classical theory of elasticity. nonlocal theory takes into account interatomic interactions over a long time and its findings depend on the dimensions of the body. the nonlocal theory includes the knowledge about the long-range forces between points along these lines and the internal length scale is essentially used as a material parameter to detect the small scale effects. any inconveniences of classical continuum theory can be easily avoided and then on local elasticity theory may appropriately describe the size-dependent phenomena. the use of the conventional theory in studying the behavior of nanostructures is inappropriate in many cases because the classical theories are not responsible for small-scale effects. recent nanotechnological considerations have led to the application of nonlocal elasticity in submicron scales and the literature indicates that the nonlocal theory of elasticity is typically used progressively to analyze the nanostructures in a consistent and fair manner. the fundamental contrast of the conventional local elasticity theory to those proposed by eringen called nonlocal one [17–19] substantially depends on the definition of the stress vector. the nonlocal theory of elasticity, however, has been applied in various fields including elastic wave dispersion, mechanical decomposition, etc. the eringen nonlocal elasticity theory (enet) has been applied in many theoretical investigations on the free vibrations of nanoparticles and nanobeams taking into account the assumptions of euler-bernoulli (ebt) and timoshenko beams (tbt) models [20-27]. another area of research which is emerging due to its current as well as future applications is therefore theoretical studies on the nanoscale behavior of rotating micro/nanobeams in different environment. rotating nanostructures include different fields like nano-turbines, nano gears and motors and miniaturized bearings. taking into account a great deal of studies on the sustainability of rotating nanoscale structures, this idea is expected to draw significant attention in the future; existing samples 636 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi include the analysis of molecular carbon nanotubes and gears, nano-motors and nano-shafts [28, 29]. the dynamics of rotating nanotubes and gears were investigated by srivastava [30] under a single imposed laser beam. the mathematical modeling for rotating nanomotor was carried out by lohrasebi and rafii-tabar [31]. nanoelectromechanical (nems) devices emerge as the next generation of advanced technology, which is capable of making a significant evolution in people's lives. considering that rotating motors and other devices are particularly of interest in advanced technology, nanobeams under rotation are analyzed in the present research to determine their vibrational characteristics. the analysis of vibrational behavior of rotating beam-like structures in the framework of classical theory, thanks to their significance in practical applications, has been an interesting and challenging question in recent decades. some of these fundamental studies are conducted on homogenous rotating beam structures [32–40]. research of different features of nanoscale beams is an attractive subject in nanotechnology as it encompasses the electronic and optical characteristics of nano-devices. the main objective of this article is to discover the vibrational behavior of rotating isotropic nanobeams in a mathematical framework using the eringen's nonlocal theory of elasticity. since the rotating devices such as rotary engines are of particular importance to such modern technologies, the vibration characteristics of rotating nanobeams are investigated in the current research work. the considered model is on the basis of the euler-bernoulli theory, generalized dual phase-lag heat conduction and includes the coriolis and spin-smoothing phenomena. for the first time ever, the features of the mentioned advanced nano-rotating mechanism are reported. this study would contribute to a thorough understanding of the dynamics of rotating nanobeams. different aspects of the studied model including thermomechanical vibration, point load and nonlocal effects are taken into account. moreover, numerical examples are presented to indicate the impact of several parameters on the deflection, temperature change and bending moment of rotating nanobeams. this study will recognize various requirements for design and production of environmentally sensitive resonator machines. 2. nonlocal model of thermoelasticity eringen [17] theoretically introduced a fundamental continuum mechanics in the context of nonlocal theory of elasticity to explore the structural issues at micro-scale dimensions. for nano-scale beams, the general governing equations of nonlocal constituent relations can be described as follows [18] [1 − (𝑒0𝑎)2∇2]𝜏𝑘𝑙 = 𝜎𝑘𝑙 , (1) where 𝑒0 denotes a correlation coefficient depending on the type of material, 𝑎 stands for the internal length scale, ∇2 refers to the laplacian operator,𝜏𝑘𝑙symbolizes the nonlocal stress tensor and 𝜎𝑘𝑙 represents the classic stress tensor. the classical elasticity equation is accessible if parameter 𝑎 takes zero value (𝑒0𝑎 = 0). the constitutive equation is expressed as 𝜎𝑖𝑗 = 2𝜇𝑒𝑖𝑗 + [𝜆𝑒𝑘𝑘 − 𝛾(𝑇 − 𝑇0)]𝛿𝑖𝑗 , (2) in which 𝜇 and 𝜆 represent the lame coefficients and 𝛿𝑖𝑗 refers to kronecker’s delta operator. moreover, 𝜃 = 𝑇 − 𝑇0 stands for a temperature change with respect to reference temperature temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 637 𝑇0, 𝑒 = div �⃗� is the volumetric strain and �⃗� denotes the displacement vector. lame’s constants and shear modulus are defined by: 𝜆 = 𝐸𝜈 (1+𝜈)(1−2𝜈) , 𝜇 = 𝐸 2(1+𝜈) . (3) the modified heat conduction equation that includes two phase lags is described as follows [2-4]: (1 + 𝜏𝜃 𝜕 𝜕𝑡 ) (𝐾𝜃,𝑖),𝑖 = (1 + 𝜏𝑞 𝜕 𝜕𝑡 + 1 2 𝜏𝑞 2 𝜕2 𝜕𝑡2) (𝜌𝐶𝐸 𝜕𝜃 𝜕𝑡 + 𝛾𝑇0 𝜕𝑒 𝜕𝑡 − 𝜌𝑄), (4) where 𝑄 represents the heat source, 𝛾 = 𝐸𝛼𝑇/(1 − 2𝜈), in which 𝛼𝑇 is the thermal expansion constant, 𝐸 denotes the module of elasticity, 𝐾 implies the thermal conductivity, 𝐶𝐸 refers to the specific heat and 𝜌 symbolizes the density of material. the theory of classical thermoelasticity, i.e. the lord and shulman model [1], and dual-phase-delay model [2-4] can be derived from eq. (3) for the selection of different phase lag parameters 𝜏𝜃 and 𝜏𝑞. • by putting 𝜏𝜃 = 𝜏𝑞 = 0 in eq. (3), we get the nonlocal theory of coupled thermoelasticity (cte). • the generalized theory of nonlocal thermoelasticity (ls theory) can be obtained by putting 𝜏𝜃 = 0 and 𝜏𝑞 = 𝜏0 (𝜏0refers to the relaxation time). • the nonlocal dual-phase-lag model (dpl) is accessible when 𝜏𝜃 falls within the interval [0, 𝜏𝑞]. it is worth mentioning that if one of the thermo-physical characteristics (𝐾,𝐶𝐸and 𝜌) depends on temperature, eq. (3) is converted to a nonlinear partial differential equation. 3. problem formulation fig. 1 displays the configuration of a thermoelastic rotating nanoscale beam. the considered nanobeam has thickness ℎ, width 𝑏 and length 𝐿. the 𝑥-axis lies in the axial direction of nanobeam and its width and thickness are in 𝑦 and 𝑧 directions, respectively. initially, it is supposed that the nanobeam is stress free and the temperature is tentatively at 𝑇0.the displacement components can be described as follows: 𝑢 = −𝑧 𝜕𝑤 𝜕𝑥 , 𝑣 = 0, 𝑤 = 𝑤(𝑥, 𝑡), (5) where 𝑤 denotes the transverse deflection. 638 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi fig. 1 schematic configuration of rotating nanobeam eq. (5) can be used to simplify the one dimension constitutive relationship (1) as 𝜎𝑥 − 𝜉 𝜕2𝜎𝑥 𝜕𝑥2 = −𝐸 (𝑧 𝜕2𝑤 𝜕𝑥2 + 𝛼𝑇𝜃), (6) in which 𝜎𝑥 indicates the nonlocal axial thermal stress, 𝛼𝑇 = 𝛼𝑡/(1 − 2𝜈) and 𝜉 = (𝑒0𝑎)2. bending moment m may be calculated by the following integration formula: 𝑀(𝑥, 𝑡) = ∫ 𝑧𝜎𝑥d𝑧 ℎ/2 −ℎ/2 . (7) thereby, the following partial differential equation is satisfied by the bending moment after substituting eq. (6) into eq. (7) 𝑀(𝑥, 𝑡) − 𝜉 𝜕2𝑀 𝜕𝑥2 = −𝐸𝐼 ( 𝜕2𝑤 𝜕𝑥2 + 𝛼𝑇𝑀𝑇), (8) where 𝐸𝐼 and 𝐼 = 𝑏ℎ3/12 represent the bending rigidity of nanobeam and cross-sectional moment of inertia, respectively. in eq. (8), 𝑀𝑇 indicates the thermal moment of nanobeam and can be written by: 𝑀𝑇 = 12 ℎ3 ∫ 𝜃(𝑥, 𝑧, 𝑡)𝑧d𝑧 ℎ/2 −ℎ/2 . (9) it is assumed that the nanoscale beam is distributed by a transversely varying force 𝑞(𝑥, 𝑡) and, therefore, the transversal equation of motion can be written as [37] 𝜕2𝑀 𝜕𝑥2 = −𝑞(𝑥, 𝑡) + 𝜌𝑏ℎ 𝜕2𝑤 𝜕𝑡2 , (10) it is supposed that the nanobeam rotates with angular velocity ω (about the axis which is parallel to 𝑧 axis) centered at a small distance 𝑟 from the left edge of the nanobeam. centrifugal tension force 𝑅(𝑥) is introduced as a result of rotation. in this case, the equation of motion (10) is given as [41] temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 639 𝜕2𝑀 𝜕𝑥2 + 𝜕 𝜕𝑥 (𝑅(𝑥) 𝜕𝑤 𝜕𝑥 ) + 𝑞(𝑥, 𝑡) = 𝜌𝑏ℎ 𝜕2𝑤 𝜕𝑡2 (11) centrifugal force 𝑅 vanishes when angular velocity ω is equal to zero and the nanobeam is in stationary state. the centrifugal stiffening effect can be modeled by including axial-type load 𝑅(𝑥) which can be described as [41, 42] 𝑅(𝑥) = ∫ 𝜌𝐴ω2(𝑟 + 𝑥)𝑑𝑥 𝐿 𝑥 (12) in which x represents the distance from the origin (see fig. 1). parameter 𝑟 refers to the distance from the rotation axis to the left edge of the nanobeam (a hub radius in fig. 1). after some computations, eq. (12) can be simplified as 𝑅(𝑥) = 𝜌𝐴ω2 2 [(𝐿 + 𝑟)2 − (𝐿 + 𝑥)2] = 𝜌𝐴ω2 2 [(𝑟 − 𝑥)(2𝐿 + 𝑟 + 𝑥)] (13) for calculating bending moment 𝑀(𝑥, 𝑡), eqs. (11) and (13) can be utilized 𝑀(𝑥, 𝑡) = 𝜉 (𝜌𝐴 𝜕2𝑤 𝜕𝑡2 − 𝑞(𝑥, 𝑡) − 𝜕 𝜕𝑥 (𝑅(𝑥) 𝜕𝑤 𝜕𝑥 )) − 𝐸𝐼 ( 𝜕2𝑤 𝜕𝑥2 + 𝛼𝑇𝑀𝑇) (14) if moment 𝑀 is omitted in eqs. (11) and (14), the governing equation can be expressed as [ 𝜕4 𝜕𝑥4 + 𝜌𝐴 𝐸𝐼 𝜕2 𝜕𝑡2 (1 − 𝜉 𝜕2 𝜕𝑥2)]𝑤 − 1 𝐸𝐼 (1 − 𝜉 𝜕2 𝜕𝑥2) [𝑞 + 𝜕 𝜕𝑥 (𝑅(𝑥) 𝜕𝑤 𝜕𝑥 )] + 𝛼𝑇 𝜕2𝑀𝑇 𝜕𝑥2 = 0 (15) in the absence of heat source (𝑄 = 0), the modified heat conduction relation (4) can be defined as (1 + 𝜏𝜃 𝜕 𝜕𝑡 ) ∇. (𝐾𝑥∇𝜃) = (1 + 𝜏𝑞 𝜕 𝜕𝑡 + 𝜏𝑞 2 2 𝜕2 𝜕𝑡2) 𝜕 𝜕𝑡 (𝜌𝐶𝐸𝜃 − 𝛾𝑇0𝑧 𝜕2𝑤 𝜕𝑥2) (16) 4. thermal properties of materials the identification of nonlinear problems will be carried out if the material properties are temperature-dependent which means specific heat 𝐶𝐸 and thermal conductivity 𝐾 depend upon the temperature distribution [28]. in the current work, it is assumed that other physical parameters are constant and not dependent on the temperature changes [43]. the material's thermal conductivity 𝐾 is supposed to be defined by a linear function of temperature change [44] as follows: 𝐾𝑥 = 𝐾𝑥(𝜃) = 𝐾0(1 + 𝐾1𝜃), (17) in which parameter 𝐾0 denotes the thermal conductivity at reference temperature t0 and 𝐾1 denotes a factor that characterizes thermal conductivity variability. by substituting eq. (17) into eq. (16), the partial nonlinear differential equation is obtained in the form 640 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi 𝐾0 (1 + 𝜏𝜃 𝜕 𝜕𝑡 )∇. ((1 + 𝐾1𝜃)∇𝜃) = (1 + 𝜏𝑞 𝜕 𝜕𝑡 + 𝜏𝑞 2 2 𝜕2 𝜕𝑡2) 𝜕 𝜕𝑡 (𝜌𝐶𝐸𝜃 − 𝛾𝑇0𝑧 𝜕2𝑤 𝜕𝑥2) (18) the previous equation can be converted to a linear equation by defining the mapping [43] 𝐾0𝜓 = ∫ 𝐾𝑥(𝜃)d𝜃 𝜃 0 , (19) after inserting eq. (17) into eq. (19) and computing the integration, we have [43] 𝜓 = 𝜃(1 + 1 2 𝐾1𝜃). (20) through differentiating relation eq. (20) with regard to distance variable and also with respect to time variable t, the following relationships are deduced ∇𝜓 = 𝐾𝑟(𝜃) 𝐾0 ∇𝜃, (21) 𝜕𝜓 𝜕𝑡 = 𝐾𝑟(𝜃) 𝐾0 𝜕𝜃 𝜕𝑡 . (22) by employing eqs. (21) and (22), therefore, the heat conduction equation (18) is simplified to (1 + 𝜏𝜃 𝜕 𝜕𝑡 ) ( 𝜕2 𝜕𝑥2 + 𝜕2 𝜕𝑧2)𝜓 = (1 + 𝜏𝑞 𝜕 𝜕𝑡 + 1 2 𝜏𝑞 2 𝜕2 𝜕𝑡2) 𝜕 𝜕𝑡 (𝜂𝜓 − 𝛾𝑇0 𝐾 𝑧 𝜕2𝑤 𝜕𝑥2). (23) in which 1/𝜂 = 𝐾/𝜌𝐶𝐸stands for thermal diffusivity. 5. sinusoidal solution in order to calculate the solution of the problem, we take the temperature change response as a sinusoidal solution {𝜓, 𝜃}(𝑥, 𝑧, 𝑡) = {ψ, θ}(𝑥, 𝑡) sin ( 𝜋 ℎ 𝑧). (24) substituting eq. (24) into eqs. (14), (15) and (23), which leads to [ 𝜕4 𝜕𝑥4 + 𝜌𝐴 𝐸𝐼 𝜕2 𝜕𝑡2 (1 − 𝜉 𝜕2 𝜕𝑥2)]𝑤 − 1 𝐸𝐼 (1 − 𝜉 𝜕2 𝜕𝑥2) 𝑞 + 24𝛼𝑇 𝜋2ℎ 𝜕2ψ 𝜕𝑥2 = 0, (25) 𝑀(𝑥, 𝑡) = 𝜉 (𝜌𝐴 𝜕2𝑤 𝜕𝑡2 − 𝑞) − 𝐸𝐼 ( 𝜕2𝑤 𝜕𝑥2 + 24𝑇0𝛼𝑇 𝜋2ℎ θ), (26) (1 + 𝜏𝜃 𝜕 𝜕𝑡 ) ( 𝜕2 𝜕𝑥2 − 𝜋2 ℎ2)ψ = (1 + 𝜏𝑞 𝜕 𝜕𝑡 + 1 2 𝜏𝑞 2 𝜕2 𝜕𝑡2) 𝜕 𝜕𝑡 (𝜂ψ − 𝛾𝑇0𝜋2ℎ 24𝐾 𝜕2𝑤 𝜕𝑥2). (27) some nondimensional variables are provided below for the sake of generality {𝑢′, 𝑥′, 𝐿′, 𝑤′, 𝑧′, ℎ′, 𝑏′} = 𝜂𝑐{𝑢, 𝑥, 𝐿, 𝑤, 𝑧, ℎ, 𝑏}, {𝑡′, 𝜏𝑞 ′ , 𝜏𝜃 ′ } = 𝜂𝑐2{𝑡, 𝜏𝑞 , 𝜏𝜃}, 𝜉′ = 𝜂2𝑐2𝜉, {θ′, ψ′} = 1 𝑇0 {θ,ψ}, 𝑀′ = 1 𝜂𝑐𝐸𝐼 𝑀, 𝑞′ = 𝐴 𝐸𝐼 𝑞, 𝑐2 = 𝐸 𝜌 . (28) after introducing dimensionless parameters in eqs. (25) to (27), one can obtain [ 𝜕4 𝜕𝑥4 + 12 ℎ2 𝜕2 𝜕𝑡2 (1 − 𝜉 𝜕2 𝜕𝑥2)]𝑤 − (1 − 𝜉 𝜕2 𝜕𝑥2) [𝑞 + 𝜕 𝜕𝑥 (𝑅(𝑥) 𝜕𝑤 𝜕𝑥 )] + 24𝑇0𝛼𝑇 𝜋2ℎ 𝜕2θ 𝜕𝑥2 = 0 (29) temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 641 (1 + 𝜏𝜃 𝜕 𝜕𝑡 ) ( 𝜕2 𝜕𝑥2 − 𝜋2 ℎ2)ψ = (1 + 𝜏𝑞 𝜕 𝜕𝑡 + 𝜏𝑞 2 2 𝜕2 𝜕𝑡2) 𝜕 𝜕𝑡 (ψ − 𝛾𝜋2ℎ 24𝐾𝜂 𝜕2𝑤 𝜕𝑥2) (30) 𝑀(𝑥, 𝑡) = 12𝜉 ℎ2 𝜕2𝑤 𝜕𝑡2 − 𝜉 [𝑞 + 𝜕 𝜕𝑥 (𝑅(𝑥) 𝜕𝑤 𝜕𝑥 )] − 𝜕2𝑤 𝜕𝑥2 − 24𝑇0𝛼𝑇 𝜋2ℎ θ (31) in eqs. (29)-(31), the primes are omitted for the sake of simplicity. a specific type of external transverse load is now taken into account. it is considered that the vertical load decays exponentially with time, acting towards the thickness of the beam [45] as follows 𝑞(𝑥, 𝑡) = −𝑞0(1 − 𝛿e−𝛽𝑡) (32) where 𝑞0 is the magnitude of the dimensionless point load and 𝛽 is the dimensionless decaying parameter of the external force, respectively (𝛿 = 0 can be used in the case of uniform distributed force). it is also assumed that the system works at constant rotating speed and centrifugal tension load 𝑅(𝑥) is considered to have its maximum value (at𝑥 = 0) which takes the following form [41] 𝑅𝑚𝑎𝑥 = ∫ 𝜌𝐴ω2(𝑟 + 𝑥)𝑑𝑥 = 1 2 𝜌𝐴ω2𝐿(2𝑟 + 𝐿) 𝐿 0 (33) thereby, the equation of motion (29) can be described as [ 𝜕4 𝜕𝑥4 + 12 ℎ2 𝜕2 𝜕𝑡2 (1 − 𝜉 𝜕2 𝜕𝑥2)]𝑤 − (1 − 𝜉 𝜕2 𝜕𝑥2) [𝑞 + 6𝐿ω2(2𝑟+𝐿) ℎ2 𝜕2𝑤 𝜕𝑥2 ] + 24𝑇0𝛼𝑇 𝜋2ℎ 𝜕2θ 𝜕𝑥2 = 0 (34) the bending moment in eq. (31) may also be defined as 𝑀(𝑥, 𝑡) = 12𝜉 ℎ2 𝜕2𝑤 𝜕𝑡2 − 𝜉 [𝑞 + 6𝐿ω2(2𝑟+𝐿) ℎ2 𝜕2𝑤 𝜕𝑥2] − 𝜕2𝑤 𝜕𝑥2 − 24𝑇0𝛼𝑇 𝜋2ℎ θ (35) the initial conditions are supposed to be ψ(𝑥, 0) = 𝜕ψ(𝑥,0) 𝜕𝑡 = 0, 𝑤(𝑥, 0) = 𝜕𝑤(𝑥,0) 𝜕𝑡 = 0. (36) the nanobeam subjects to the following boundary conditions: ▪ doubly-clamped ends with the following conditions 𝑤(𝑥, 𝑡)|𝑥=0,𝐿 = 0, 𝜕𝑤(𝑥,𝑡) 𝜕𝑥 | 𝑥=0,𝐿 = 0. (37) ▪ the nanobeam is harmonically heated as θ(0, 𝑡) = θ0 cos(𝜔𝑡) , 𝜔 > 0, (38) in which θ0 is its amplitude and 𝜔 denotes the thermal angular frequency. in the case of 𝜔 = 0, the above condition is utilized for thermal shock loading. using eq. (15) and eq. (20) yields ψ(0, 𝑡) = θ0 cos(𝜔𝑡) + 1 2 𝐾1[θ0 cos(𝜔𝑡)]2. (39) ▪ the surface 𝑥 = 𝐿 is assumed to be thermally isolated 𝜕ψ(𝐿,𝑡) 𝜕𝑥 = 0. (40) 642 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi 6. laplace transform strategy using laplace technique, eqs. (30), (34) and (35) are converted to the following equations [(1 + 6𝜉𝐿ω2(2𝑟+𝐿) ℎ2 ) d4 d𝑥4 − ( 12𝜉𝑠2 ℎ2 + 6𝐿ω2(2𝑟+𝐿) ℎ2 ) d2 d𝑥2 + 12𝑠2 ℎ2 ] �̅� = − 24𝑇0𝛼𝑇 𝜋2ℎ d2θ̅ d𝑥2 − �̅�(𝑠) (41) (1 + 𝜏𝜃𝑠) ( d2 d𝑥2 − 𝜋2 ℎ2) ψ̅ = 𝑠(1 + 𝜏𝑞s + 𝑠2𝜏𝑞 2/2) (ψ̅ − 𝛾𝜋2ℎ 24𝐾𝜂 d2�̅� d𝑥2) (42) �̅�(𝑥, 𝑠) = 12𝜉𝑠2 ℎ2 �̅� − (1 + 6𝜉𝐿ω2(2𝑟+𝐿) ℎ2 ) d2�̅� d𝑥2 − 24𝑇0𝛼𝑇 𝜋2ℎ θ̅ − 𝜉�̅�(𝑠) (43) where �̅�(𝑠) = 𝑞0 ( 1 𝑠 − 𝛿 𝛽+𝑠 ). the following differential equation can be obtained once function θ ̅is extracted from eqs. (41) and (42) [ d6 d𝑥6 − 𝐴 d4 d𝑥4 + 𝐵 d2 d𝑥2 − 𝐶] �̅� = 𝐴5�̅�(𝑠)/𝐴1 (44) where 𝐴 = 1 𝐴1 (𝐴5𝐴1 + 𝐴2 + 𝐴4𝐴6), 𝐵 = 1 𝐴1 (𝐴5𝐴2 + 𝐴3), 𝐶 = 𝐴5𝐴3 𝐴1 , 𝐴1 = (1 + 6𝜉𝐿ω2(2𝑟+𝐿) ℎ2 ) , 𝐴2 = ( 12𝜉𝑠2 ℎ2 + 6𝐿ω2(2𝑟+𝐿) ℎ2 ), 𝐴4 = 24𝑇0𝛼𝑇 𝜋2ℎ , 𝐴3 = 12𝑠2 ℎ2 , 𝐴5 = 𝜋2 ℎ2 + 𝑠(1+𝜏𝑞s+𝑠2𝜏𝑞 2/2) 1+𝜏𝜃𝑠 , 𝐴6 = 𝑠(1+𝜏𝑞s+𝑠2𝜏𝑞 2/2) 1+𝜏𝜃𝑠 ( 𝛾𝜋2ℎ 24𝐾𝜂 ) . (45) then the general solution for 𝑤 ̅̅ ̅can be achieved by solving the differential eq. (44) as follows �̅�(𝑥, 𝑠) = ∑ (𝐶𝑗e −𝑚𝑗𝑥 + 𝐶𝑗+3e 𝑚𝑗𝑥) − 𝐴5�̅�(𝑠) 𝐶𝐴1 3 𝑗=1 (46) from the given boundary conditions, undetermined parameters 𝐶𝑗, (𝑗 = 1,2. . ,6), can be calculated. parameters 𝑚1 2, 𝑚2 2 and 𝑚3 2 also satisfy the following equation 𝑚6 − 𝐴𝑚4 + 𝐵𝑚2 − 𝐶 = 0 (47) eq. (46) is incorporated into eq. (41) and yields ψ̅(𝑥, 𝑠) = − 1 𝐴4𝐴5 [𝐴1 d4�̅� d𝑥4 − (𝐴2 + 𝐴4𝐴6) d2�̅� d𝑥2 + 𝐴3�̅� + �̅�(𝑠)] (48) with the help of eq. (46), the solution of eq. (48) can be simplified as ψ̅(𝑥, 𝑠) = ∑ 𝐻𝑗(𝐶𝑗e −𝑚𝑗𝑥 + 𝐶𝑗+3e 𝑚𝑗𝑥)3 𝑗 =1 − 𝐻4, (49) where 𝐻𝑗 = − 1 𝐴4𝐴5 [𝐴1𝑚𝑗 4 − (𝐴2 + 𝐴4𝐴6)𝑚𝑗 2 + 𝐴3], 𝐻4 = �̅�(𝑠)(𝐴4−𝐶) 𝐶𝐴4𝐴5 (50) bending moment �̅� is determined from (43) using the solutions (46) and (49) �̅�(𝑥, 𝑠) = ∑ 𝐿𝑗(𝐶𝑗e −𝑚𝑗𝑥 + 𝐶𝑗+3e 𝑚𝑗𝑥)3 𝑗=1 + 𝐿4 (51) temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 643 where 𝐿𝑗 = −(𝐴1𝑚𝑗 2 + 𝐴4𝐻𝑗 − 𝐴0), 𝐿4 = �̅�(𝑠) (𝐴4𝐻4 − 𝐴0𝐴5 𝐶𝐴1 − 𝜉) , 𝐴0 = 12𝜉𝑠2 ℎ2 (52) axial displacement �̅� can be calculated by using eq. (46) �̅� = −𝑧 d�̅� d𝑥 = 𝑧∑ 𝑚𝑗(𝐶𝑗e −𝑚𝑗𝑥 − 𝐶𝑗+3e 𝑚𝑗𝑥)3 𝑗=1 . (53) boundary conditions (37)-(40) are reduced in the laplace transform field to �̅�(𝑥, 𝑠)|𝑥=0,𝐿 = 0, d2�̅�(𝑥,𝑠) d𝑥2 | 𝑥=0,𝐿 = 0, (54) ψ̅(𝑥, 𝑠)|𝑥=0 = θ0 [ 𝑠 𝑠2+𝜔2 + 𝐾1(𝑠2+2𝜔2) 2𝑠(𝑠2+4𝜔2) ] = �̅�(𝑠), (55) dθ̅ d𝑥 | 𝑥=𝐿 = 0. (56) by applying these conditions in eqs. (46) and (49), the following system of equations are obtained ∑ (𝐶𝑗 + 𝐶𝑗+3) = 𝐴5�̅�(𝑠) 𝐶𝐴1 3 𝑗=1 (57) ∑ (𝐶𝑗e −𝑚𝑗𝐿 + 𝐶𝑗+3e 𝑚𝑗𝐿) = 𝐴5�̅�(𝑠) 𝐶𝐴1 3 𝑗=1 (58) ∑ 𝑚𝑗 2(𝐶𝑗 + 𝐶𝑗+3) = 𝐴5�̅�(𝑠) 𝐶𝐴1 3 𝑗=1 (59) ∑ 𝑚𝑗 2(𝐶𝑗𝑒 −𝑚𝑗𝐿 + 𝐶𝑗+3𝑒 𝑚𝑗𝐿) = 𝐴5�̅�(𝑠) 𝐶𝐴1 3 𝑗=1 (60) ∑ 𝐻𝑗(𝐶𝑗 + 𝐶𝑗+3) 3 𝑗=1 = 𝐻4 + �̅�(𝑠), (61) ∑ 𝑚𝑗𝐻𝑗(𝐶𝑗𝑒 −𝑚𝑗𝐿 − 𝐶𝑗+3𝑒 𝑚𝑗𝐿)3 𝑗=1 = 0, (62) parameters 𝐶𝑗 , (𝑗 = 1,2. . ,6) are unknown and have to be determined and, thereby, the analytical solutions for the physical parameters in the laplace domain can be achieved. it is difficult to take the laplace inversion of the complicated transformed field variables expressions. within the next section, the results are evaluated numerically using the expansion technique of the fourier series. temperature �̅� can be obtained by solving eq. (20) after applying the laplace transform as �̅�(𝑥, 𝑠) = sin ( 𝜋 ℎ 𝑧) [ −1+√1+2𝐾1�̅� 𝐾1 ] (63) 7. laplace transform inversion with the aim of solutions in the physical domain, at last, we invert the transformation of laplace to the functions that govern the motion of the system. we now follow a numerical overlay strategy based on an extension of the fourier series [46]. any functions in 644 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi the laplace domain, i.e. �̅� (𝑥, 𝑠), is transferred to the time domain, i.e. 𝑔(𝑥, 𝑡), by using the following procedure: 𝑔(𝑥, 𝑡) = e𝑐𝑡 𝑡 {1 2 �̅�(𝑥, 𝑐) + re [∑ (−1)𝑛𝑁 𝑛=1 �̅� (𝑥, 𝑐 + i𝑛𝜋 𝑡 )]}, (64) where 𝑁 represents a large value denoting the number of truncated terms in the original fourier series and can be selected to satisfy e𝑐𝑡re {(−1)𝑛�̅� (𝑥, 𝑐 + i𝑛𝜋 𝑡 )} ≤ 휀1, (65) in which 휀1 is a small positive integer corresponds to achieve the desired accuracy. various numerical studies demonstrated that parameter 𝑐 can be determined as 𝑐𝑡 ≈ 4.7 for appropriate convergence [47]. 8. results and discussion throughout this section, some discussions and numerical results are provided to illustrate the general response of the problem. moreover, the numerical examples are presented to explore the impacts of the physical fields analyzed with three appropriate parameters. specific physical parameters are taken into consideration from the published works in the literature for computational purposes. in our analysis, the following properties are utilized for silicon-based material: 𝐸 = 169 gpa, 𝜌 = 2330 ( kg m3 ), 𝐶𝐸 = 713 ( j kg k ) , 𝛼𝑇 = 2.59 × 10−9 ( 1 k ) , 𝜈 = 0.22, 𝑇0 = 293 𝐾, 𝐾 = 156 ( w mk ) . we consider a nanobeam with dimensionless parameters as given in eq. (24). the following values are set in our calculations, i.e., 𝐿/ℎ = 10and 𝑏/ℎ = 0.5. the dimensionless length is taken as 𝐿 = 1 and it is assumed that 𝑡 = 0.1. as mentioned before, a linear function of temperature change 𝜃 is considered for the thermal conductivity of the material (see eq. (13)). figs. 2-5 show the variations of nondimensional displacement, bending moment and temperature gradient for different values of 𝐾1-parameter to illustrate the thermal properties of the nanobeam that vary in terms of temperature variable 𝜃. two different thermal conduction parameters will be considered. when the thermal conductivity depends on temperature, values 𝐾1 = −1 and 𝐾1 = −0.5 are taken into account. if the thermal conductivity remains constant, then 𝐾1 is equal to zero. it is assumed that the other parameters are fixed and take the values 𝜉 = 0.01, 𝜔 = 5, ω = 0.3, 𝜏𝑞 = 0.02 and 𝜏𝜃 = 0.01. from the illustrative results, it is observed that any change in parameter 𝐾1 has a remarkable influence on all studied fields that proves that the consideration of variable thermal conductivity is essential for this kind of analysis. it is also noted that the behavior of nanostructures are dependent on temperature changes and with the increase in the external temperatures, the results of small-scale nonlocal theories also increase. the variation of deflection 𝑤 versus distance x for different values of thermal conductivity parameter 𝐾1 is shown in fig. 2. as can be seen, increasing the values of parameter 𝐾1 results temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 645 in decreasing lateral vibration 𝑤. fig. 2 illustrates that the deflection distribution 𝑤 has zero values at both ends (i.e. disappears) and meets the limit conditions of the rotating nanobeam at 𝑥 = 0 and 𝑥 = 𝐿. temperature 𝜃 is also depicted in terms of parameter 𝐾1in fig. 3. temperature 𝜃 will decrease while distance x becomes larger to drive towards wave propagation, as shown in fig. 3. from theoretical experiments, it is found that the thermal conductivity of pure metals decreases as the temperature increases. however, the thermal conductivity drops dramatically as the temperatures exceed absolute zero [48]. fig. 3 indicates that any reduction in parameter 𝐾1 results in increasing temperature values 𝜃. this phenomenon is verified by the experimental results reported by abo‐dahab et al. [49]. fig. 2 deflection 𝑤 under the influence of variable thermal properties fig. 3 temperature 𝜃 under the influence of variable thermal properties fig. 4 demonstrates that the values of displacement decrease from 0 to 0.3 and consequently in the range 0.3 to 0.8 shift upward to the highest amplifications. in addition, the 646 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi values of 𝑢 vary linearly in the last interval 0.8 ≤ 𝑥 ≤ 1 of wave propagation. it should be noted that the displacement distribution is highly influenced by the variation of parameter 𝐾1. according to fig. 5, the effect of parameter 𝐾1is revealed as increasing the distribution of bending moment 𝑀 which implies that the impact of thermal conductivity variation cannot be disregarded [48]. the mechanical characteristics of the nanobeam emphasize that the wave propagates in the medium with finite speed [49]. fig. 4 displacement 𝑢 under the influence of variable thermal properties fig. 5 bending moment 𝑀 under the influence of variable thermal properties in the second case, the influence of angular rotation velocity ω on the dimensionless field quantities is illustrated. it is supposed that our findings are consistent with both nonlocal parameter 𝜉, the angular excitation of thermal load 𝜔, and phase lags 𝜏𝑞 and 𝜏𝜃. our calculations are performed using fixed values for system parameters as 𝜉 = 0.1, 𝜔 = 5, 𝜏𝑞 = 0.02 and 𝜏𝜃 = 0.01. temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 647 the variations of the fields studied for three assigned values of rotational speed (ω = 0, 0.1, 0.3) are shown in figs. 6-9. in the case of stationary nanobeam, the rotational speed is set to be zero (ω = 0), as a special case in our simulations. the variation of angular speed ω which cases the variation in deflection w, is depicted in fig. 6. as indicated, this parameter is found to have a substantial effect on the distribution of deflection 𝑤 by comparing the results of stationary beam with rotating one. when angular velocity ω becomes larger, deflection 𝑤 takes smaller values. the obtained results are consistent with those reported by ebrahimi et al. [50]. the variation in temperature 𝜃 with respect to distance 𝑥 is verified by assuming different values for rotational speed ω, as demonstrated in fig, 7. fig. 6 deflection 𝑤 vs angular velocity ω fig. 7 temperature 𝜃 vs angular velocity ω clearly, the temperature shifts upward as angular velocity ω increases which is also consistent with the observations of ebrahimi and shafie [50, 51]. 648 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi fig. 8 presents a survey of the impact of angular speed ω on the variation of displacement 𝑢. it is inferred that the curves that reflect the displacement field are influenced by the rotation. this figure shows that at certain ranges, by increasing the rotation speed, the distribution of displacement 𝑢 sharply decreases at first and then slightly increases. fig. 9 shows the influence of the variation of angular velocity ω on the distribution of bending moment 𝑀for rotating nanobeams. it is observed from the figure that the rotation of the nanobeam has a great effect on the values of bending moment and the amplitude of the moment becomes larger by increasing angular velocity ω. fig. 8 displacement 𝑢 vs angular velocity ω fig. 9 bending moment 𝑀 vs angular velocity ω one of the objectives of this study is to explore the mechanical features of certain nano-devices such as nano-turbine blades [52] in the presence of temperature field and angular velocity which may provide valuable insights for designers and engineers. from the previous observations, one can also deduce the major impact of rotation on the temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 649 distribution of physical properties of such devices. it should be emphasized that the findings are in line with the earlier results in refs. [53-55]. when the values of parameters (𝜔,𝐾1, ω, 𝜏𝑞 , 𝜏𝜃) are fixed, the effect of the nonlocal parameter 𝜉 on the non-dimensionalfield variables can be examined. through figs. 1013, along the axial direction, the thermoelastic behavior of rotating nanobeam is shown by plotting the variations of displacement u, bending moment m, deflection w and temperature 𝜃. the values of nonlocal parameter are considered to be 𝜉 = 0 (classical theory), 𝜉 = 0.01 and 𝜉 = 0.03 for comparison. the figures exhibit that the nonlocality of the nanobeam has distinct influences on all fields studied and illustrates the difference between classical thermoelasticity theories and nonlocal ones. for various three assigned values of nonlocal scaling parameter 𝜉, fig. 10 shows different curves for the deflection of non-rotating nanobeam. it can be seen that with an increase in the nonlocal parameter, the deflection becomes very small and the dispersed nature turns into a non-dispersed form. the temperature distributions are also shown for various nonlocal scaling parameter versus axial direction x, in fig. 11. it is concluded that the temperature convergence can be achieved by increasing the nonlocal scaling parameter. fig. 12 demonstrates that the displacement amplitude increases with the nonlocal parameter due to the inclusion of nonlocality in the thermoelastic model. the influence of the nonlocal parameter 𝜉 on the variation of bending moment is presented in fig. 13. one observes that as the nonlocal scale parameter increases, the bending moment tends to decrease substantially. it is worth mentioning that our findings and conclusions are in agreement with those reported in the literature [56]. the results typically confirm that one should expect significant diversity in the results by considering the nonlocality of nanobeam rather than classical one. the distinction between classical thermoelasticity models and nonlocal thermoelasticity theories in thermal fields is justified [57]. in the nanoscale systems and devices, the impact of this parameter should also be taken into account. fig. 10 deflection 𝑤 distribution vs nonlocal parameter 𝜉 650 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi fig. 11 temperature 𝜃 distribution vs nonlocal parameter 𝜉 fig. 12 displacement 𝑢 distribution vs nonlocal parameter 𝜉 fig. 13 bending moment 𝑀 distribution vs nonlocal parameter 𝜉 temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 651 as the last effort in this research, the variations of different fields variables in terms of the point load 𝑞0 are plotted in figs. 14-17. there are three different non-dimensional values for point load 𝑞0 which are taken into account for the sake of comparison. the plotted curves show the distinct effect of the point load on the results. in the case of uniform distributed load, it is assumed that 𝛿 = 0, and for the point load with exponential decay in time, it is considered that 𝛿 = 1. clearly, the difference between the outcomes becomes more significant with increasing amplitude of point load 𝑞0. figs. 14-17 indicate that there is a greater discrepancy between u and w quantities by increasing the point load compared to those for bending moment and temperature gradient. it is also noted from these figures that the absolute values of the field variables become larger when the exponential decay with respect to time is considered for point load 𝑞0 [58, 59]. fig. 14 deflection 𝑤 under the influence of point load 𝑞0 fig. 15 temperature 𝜃 under the influence of point load 𝑞0 652 a. e. abouelregal, h. m. sedighi, s. a. faghidian, a. h. shirazi fig. 16 displacement 𝑢 under the influence of point load 𝑞0 fig. 17 bending moment 𝑀 under the influence of point load 𝑞0 9. conclusions this research focused on the wave dispersion behavior of rotating nanobeam on the basis of the nonlocal elasticity theory in conjunction with generalized thermoelasticity with phase-lag. the systems of governing equations were derived in the context of the euler-bernoulli beam theory. the thermal conductivity of nanobeam and modulus of elasticity were considered to be temperature-dependent. for a uniform rotating nanobeam, the governing equations were extracted considering the variability of thermal conductivity and nonlocal scale effect. the nanobeam surface was assumed to be thermally loaded with uniformly heat source by considering the exponential decay with time. the effects of dynamic load, nonlocal parameter, rotating speed and thermal conductivity variations on the field variables were graphically described and examined. our findings showed the following important results: temperature-dependent physical characteristics of rotating nonlocal nanobeams subject to... 653 ▪ the variation of the thermal conductivity affects the wave propagation rate of all of field variables. physical fields strongly depend on the variation of the thermal conductivity. ▪ the dependency of the thermal conductivity on the temperature causes a significant influence on the mechanical and thermal interactions. ▪ the nonlocal parameter effects are considered to be significant on all fields studied. ▪ the presence of dynamic load has a considerable impact on the results of all physical quantities. ▪ there are significant differences between the results of point load with exponential decay and those obtained in the case of uniformly distributed one. the thermoelastic stresses and temperature gradient, on the other hand, are highly dependent on the angular frequency of the thermal loading. current research may be used in applications including micro and nano mechanical gears, micro-mirrors for light guiding purposes in image projection devises, micro-turbine blades, micro accelerometers (fig. 18), resonators, frequency filters and relay switches. fig. 18 (a) micro scale mechanical gear, (b)micro-mirror, (c) micro-turbine, (d) micro scale accelerometer acknowledgements: h.m. sedighi 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mechanics, germany abstract. in this study, we elaborate on the friction between a one-dimensional elastomer and a one-dimensional rigid randomly rough surface. special emphasis is laid on the energy dissipation in the elastomer. its subsequent temperature change is under inspection. the elastomer is modeled as a spring and a damper in parallel (kelvin model) in a one-dimensional substitute model according to the concept of the method of dimensionality reduction (mdr). the randomly rough surface is a self-affine onedimensional fractal whose hurst exponent h is varied in an extended range between -1 and 3. in full contact, the temperature shift is dominated by ratio between the typical power dissipated in the elastomer to the power that is led away. it is independent of the normal force and proportional to the sliding speed squared. the flash temperature behavior is discussed for different hurst exponents. key words: contact mechanics, elastomer friction, temperature dependence, method of dimensionality reduction 1. introduction the roughness of surfaces is considered to be the main source of friction since the seminal work of bowden and tabor [1]. greenwood and tabor [2] ascribe the frictional behavior of polymers to the deformation losses in the material. experimentally, grosch [3] found support for these thoughts by considering friction between rubber and hard specimens with controlled roughness. an insight into the role of rheology [4] and surface roughness [5, 6] in polymer friction was gained in the subsequent years. the concept of the coefficient of friction is used in most works in the field. hereby, amontons' laws are implicitly considered to hold: the frictional force is proportional to the normal force and hence the coefficient of friction is independent of the normal force [7, 8]. this is a widely spread law, which is a rather crude model of real frictional systems. it is well-known that both the static and the received january 11, 2015 / accepted fabruary 14, 2015 corresponding author: rainer heise institute for mechanics, technische universität berlin, straße des 17. juni 135, d10623 berlin, germany e-mail: rainer.heise@tu-berlin.de original scientific paper 48 r. heise dynamic coefficient of friction may be changed by a factor of four depending on geometrical and loading conditions of the tribological system under observation. schallamach [9] studied elastomer friction experimentally. deviations from amontons' laws were under investigation in recent studies [10, 11]. they can originate from macroscopic interfacial dynamics [12, 13, 14] or are related to the contact mechanics of rough interfaces. in this study, we consider the thermodynamic behavior of elastomers due to friction in situations where amontons' laws lose their validity. some basic understanding of the frictional response of a viscoelastic body is captured from a simple model: (i) the elastomer is modeled as a kelvin body, which is completely defined by a constant elastic modulus and a thermally activated viscosity, (ii) the undeformed surface is plain and exhibits no friction microscopically, (iii) the counter body is rigid and has a randomly rough, self-affine fractal surface, (iv) capillarity and adhesion are excluded, (v) the features are investigated in a one-dimensional substitute model. although these conditions seem to simplify the situation vastly we still obtain an interesting, non-trivial frictional and thermodynamic behavior of the elastomer. we cannot claim a direct one-to-one correspondence to a full-fledged three dimensional analysis but we see a broad applicability of our results to one-dimensional grounds if the rules put forward in the method of dimensionality reduction (mdr) [15-18] are taken into account. li et al. [19] provided a study of a one-dimensional rough surface in contact with an elastomer. dimaki and popov [20] investigated the temperature dependence of the coefficient of friction for elastomers for an indented cone. the aim of this note is to study the behavior of an elastomer in contact with a rough surface when temperature shifts in the contacts are admitted. the mdr maps three-dimensional frictional problems onto one-dimensional ones. thereby, two different steps have to be taken. firstly, the contact geometry of the surface has to be taken to a one-dimensional substitute model. secondly, physical quantities in the contact have to be calculated in the model. let us review a certain class of surfaces that we employ for our model building. 1.1 self-affine isotropic surfaces and fractal surfaces an arbitrary surface may be described by certain statistical key values. we want to mention here the root mean square (rms) roughness 2h h   and the rms gradient 2( )h h     , where  denotes the ensemble average over several realizations of the system. both quantities can be related to the power spectrum or autocorrelation function ( )c q of the surface under consideration. in a one-dimensional substitute model these moments of the autocorrelation function can be held constant from an original two-dimensional model 2 1 1 2 2 1 1 d   ( ) ( ) d     ( ). d d d d h qc q h q q c q              (1) fractal surfaces are self-affine surfaces. this means that the surface looks about the same as the resolution is increased or decreased. there is no natural length scale to be found in this kind of surfaces. hurst exponent h is another quantity to characterize fractal surfaces flash temperatures generated by friction of a viscoelastic body 49 2 ,fd h  (2) where df is the fractal dimension of the surface. under a rescaling of spatial coordinate x  x height coordinate h changes to  h h. the original hurst exponent ranges from 0 to1. low values of the hurst exponent correspond to high volatility: large values of the quantity under consideration have a high probability to be followed by low value. this means that a surface with low hurst exponent looks rather rugged. high values of the exponent point towards a low volatility, i.e. the values of the physical quantity have a high probability to be followed by another similar value. this corresponds to a smooth surface if height coordinates are under investigation. the range of the hurst exponent can be further extended. one-dimensional self-affine isotropic surface h is described by a power spectrum 2 1 3 1 0 ˆ ( ) , h d f i f f q c q c q q q q q              (3) with dimensionless strength 0ĉ and cut-off wave vectors qi and qf. the rms roughness and the rms slope of the surface are computed as 22 02 2 2 2 0 ˆ 1 , ˆ ( ) 1 . 1 h f f i h i f c q q h h q c q h h q                                 (4) eliminating strength 0ĉ , the gradient of the surface is given by 2 2 2 2 2 2 1 ( ) . 1 1 h i f f h f i q qh h q h h q q                     (5) dropping the requirement of the hurst exponent being in the range [0,1] we can identify three different behaviors of the slope. keeping in mind qf / qi >> 1 2 2 2 2 2 2 2 2 2 2 ( ) , 0 1 ( ) , 0 1 1 ( ) , 1. 1 f h i f f i h h q h h h qh h q h h h q h h q h h h                            (6) 50 r. heise for negative values of h, the short wavelengths dominate the slope. for values of h in the physically most interesting range, the slope is interpolating between the values for long respectively short wave vectors. finally, for high values of the hurst exponent the shortest wave number, i.e. the longest wavelength, dominates. for the generation of a randomly rough surface with the desired properties, we fall back on the fourier transform 1 1 1 d   ( ) exp[ ( )]. 2 d dh q b q i x      (7) the one-dimensional fourier coefficients of the surface are proportional to the square root of the power spectrum 1 1 2 .d db c l   (8) this choice ensures that the requirements on the roughness and slope are satisfied. note that b1d has to be symmetric about qf in order to get a real surface. the randomness of the different realizations of the surface is assured by uniformly distributed random phase . here, phase  has to be antisymmetric about qf to fulfill the reality condition. for a discrete realization, the maximal and minimal wave vectors depend on spatial step dx and system length l / / , 2 .f iq dx q l   (9) the one-dimensional surface is generated as the inverse discrete fourier transform (dft). as an example, the dft of one-dimensional height coordinate h is expressed as 1 1 1 2 ( ) exp ( 1)( ( ) 1) . n d d k i h j b kk j n            (10) in order to get a real surface, the symmetry conditions for a discrete realization translate into 1 1 1 1 ( ) ( 1) ( ) ( 1), 1, , / 2. d d d d b k b n k j n k k n           (11) we stress that there is a periodicity of l = ndx due to periodicity feature of the dft. 2. modeling after having described the generation of the one-dimensional substitute surface we now turn to the modeling of the contact itself. in the considered model, two one-dimensional substitute surfaces z and u are moved relative to each other with constant velocity v0. the surfaces are discretized and the individual sites are labeled by i = 1,…,n. in every site i there exists a spring damper combination. this combination is a kelvin body as shown in fig. 1. flash temperatures generated by friction of a viscoelastic body 51 fig. 1 kelvin element: spring with stiffness k = 2dxg/(1   in parallel with a damper dx it consists of a spring with stiffness k = 2 dx g/(1  , which is coupled in parallel with a damper with damping constantdx . the material of the original polymer is described by elastic modulus g and viscosity . in the one-dimensional model there exists no interaction between neighboring sites. applied external normal force fn is held constant during the frictional process. hence, it is a force controlled process. for simplicity, we employ periodic boundary conditions. one can imagine that the indenter is virtually infinitely many times repeated. due to the chosen boundary conditions the set-up has spatial periodicity l = dx n. another important ingredient in the model is the contact criterion. the surfaces are considered to be in contact if the coordinate of deformable surface u is larger than the one of rigid surface z: i iz  < u . (12) from the viscoelastic model mentioned earlier, one can easily find the equation of motion for the kelvin element at every site 0( )extf k u u u    for some reference coordinate u0. in terms of discretized variables, the force exerted at each site in contact is 1 ,i i i i u u f k u dt          (13) where   g is the relaxation time of the kelvin element (for incompressible media ). the spring experiences a force according to the deviation from the undisturbed soft surface u0i = 0 while the damper exerts a force that is proportional to the rate of change in the deformed surface. we define 0 0v v v kdx dt gdx      (14) as the ratio between the time scale of motion dt = dx/v0 and the time scale of the viscoelastic material (relaxation) k g. between the surfaces, a force acts according to viscoelastic model eq. (13) if it is in contact. the force is set to zero if surfaces are not in 52 r. heise contact. negative forces are not considered since this would correspond to adhesive effects which we exclude from our study. initially, the rigid surface described by coordinates zi is generated with a certain given rms roughness h, system length l, spacing dx and hurst exponent h. this fixes also the rms gradient 2 1 1 1/ (( ) / ) n i i i z n z z dx     and cut-off wave vectors qi and qf . note that the mean of z is zero by construction. the moving rigid surface is pressed into the deformable surface in such a way that the given normal force is sustained. at the same time the periodic boundary conditions are enforced. the soft deformed lower surface is now described by coordinate ui. the situation is viewed as a stationary system so that all transient features have disappeared. hence, time step dt for discretization step dx is rather a bookkeeping device. side step dx is fixed as well as time step dt throughout this study. in particular, this means that spatial derivatives are linked to time derivatives via 0 . d d v dt dx  (15) further on, we denote indices according to the scheme: i all sites, j sites in contact, k sites without contact. the one-dimensional chain can be seen in fig. 2. fig. 2 chain of discretized kelvin elements imagine we ride along with the indenter to the right. one point of the surface is transferred to a new position. this new position of relaxing surface ui is calculated according to eq. (13) without external force. in a continuous description, this leads to a simple ordinary differential equation 00 .u u u   (16) it is solved by ( ) (0)exp t u t u         (17) after some time t for an undisturbed surface at u0=0 for some constant u(0). in discretized coordinates, the solution relates one site to the coordinate one step to the right 1 exp .i i dt u u          (18) this solution corresponds to a free evolution and relaxation of the polymer. for the deformable surface at a single site, four distinct possible scenarios exist: flash temperatures generated by friction of a viscoelastic body 53 (i) the first possibility is a site that is already in contact and remains so. the old coordinate of the element is uj+1=zj+1. it evolves freely according to eq. (18). its new coordinate fulfills requirement uj≥ zj and hence stays in contact. the force at this site in contact is calculated according to eq. (13). the coordinate of the deformable surface is set to the rigid one, uj=zj. (ii) the second possible result of the evolution of a site, which already has been in contact, loses its contact uj ≤ zj. the force acting at this site is set to zero, fk = 0. (iii) another outcome of the evolution is that a former free site hits the rigid surface and thus gets into contact uj≥ zj. again force fj in this newly established contact site is calculated in accordance with eq. (13). finally, the coordinate is set to the rigid one, uj=zj. (iv) the last possibility of evolution is a free contact that stays free. its coordinate ku evolves according to the equation of motion eq. (13) with external force set to zero and reference surface u0k=0. eq. (18) gives the height of surface uk at this site. there is no force acting between the surfaces at this site, fk=0. a typical picture of the described situation can be found in fig. 3. fig. 3 typical picture of the surfaces: rigid rough surface (blue solid) and deformed elastomer (red dashed) summing local forces fi over all the sites yields the total force exerted between the surfaces. this force should equal given normal force fn. in order to achieve this requirement, the coordinate of the rigid body is adjusted by an overall shift of this surface by a quantity z 0 .i iz z z  (19) the adaptation to the new equilibrium is an iterative process since the contact region changes as well and defines it implicitly. 54 r. heise for the computation of frictional force fx the tangential force for all sites 1 i i x i i i i i z z f f z f dx       (20) is calculated. remember that the sites without contact sustain no force and hence the frictional force equals zero in non-contact sites. the coefficient of friction is defined as a ratio between the tangential and the total normal force . x x i n i f f f f     (21) the contact length is computed as the sum over the number of contact site times the spatial step . cont cont j l dx dxn  (22) another quantity of interest is the surface gradient in contact sites 2 11 , j j cont jcont z z z n dx           (23) which is a measure for the roughness that is actually experienced by the surfaces. 2.1. temperature dependence the temperature dependence of the simulated tribological contact is included in two ways. firstly, the viscoelastic properties of the material may be changed due to the temperature rise. a detailed analysis [21] reveals the following temperature dependence. suppose the elastomer can be thought of as an amorphous body consisting of molecules. they usually oscillate in a potential well formed by the surrounding molecules. but there is also the possibility of jumps whose energy is provided by thermal fluctuations within an activation volume. these rarely occurring jumps are the physical reason for a flow in the presence of shear stresses in fluids. without shear stress, a molecule can jump in any direction with a probability that is proportional to a boltzmann factor 0exp[ / ]bp u k t for some temperature t, activation potential u0, and boltzmann constant kb. in absence of a shear stress, there is no macroscopic motion in the fluid. when a shear stress s is imposed onto the system the situation changes in such a way that the potential barriers diminish in the direction of the shear stress by an amount sv0 for some activation volume. in the direction against the shear stress the potential increases by the same amount. summarizing the potentials for jumps along and opposite to the shear stress we get 0 0 0 0 , . a s o s u u v u u v       (24) flash temperatures generated by friction of a viscoelastic body 55 the activation volumes have the order of magnitude of an atomic volume a 3 where a is the atomic radius. the rate of shear deformation is proportional to the difference in flow of the molecular flow 0 0 0 0 0 0 1 exp exp exp sinh x s s o b b s b b dv u v u v c dz k t k t u v c k t k t                                       (25) for some (frequency) constant c1. from this expression, we see a deviation from the behavior of newtonian fluids. especially for small shear stress, i.e. sv0/(kbt)<<1, we retain only the leading term in the taylor expansion. inverting the shear rate expression 0 0 1 expx s s b b dv u v c dz k t k t              (26) we learn that there exists a prefactor t in addition to the thermal activation arrhenius term so that the viscosity reads 0 1 0 exp . b b uk t c v k t         (27) referring to a certain state ( the temperature dependence might be expressed as 0 0 0 0 0 0 0 0 00 0 0 0 1 1 exp 1 1 exp 1 1 , 1 b b b ut t k t t u ut t tt k t t k t t                                                        (28) where . lyashenko et al. [22] claim that a  rise in the temperature might cause a shift of one decade in the frequency of the viscoelastic spectrum. looking at a temperature  the activation coefficient amounts to 141 in units of kbt0. popov [21] says that an increase of the temperature by 30 k leads to a diminishing of the viscosity to a half. this corresponds to an activation energy of 8.67 in units of kbt0 at . we use the latter value in our calculations. 2.2. dissipated power the temperature change originates from the dissipated power in a viscoelastic contact. the energy loss in the viscous part of the element leads to a rise in temperature in the contact. simple considerations about the heat flow (power flow) in the contact and bulk give rise to a temperature field in the contact. in three dimensions, heat flow q in an isotropic continuum is proportional to the temperature gradient of temperature field t 56 r. heise , t q (29) according to the fourier law. proportionality constant  is the specific thermal conductivity. the rate of change of the temperature field is governed by the convection-diffusion equation · · t c c t r t          q v (30) for a medium of density  and specific heat capacity c. the temporal change in temperature is due to the divergence of the heat flow, a convective term due to the motion of the source, and the heat generation from a source r. including the fourier equation (29) the rate of change of the temperature for source free materials is then described by the heat exchange equation 2· · · . t c c t t c t t               q v v (31) for a stationary situation the time derivative vanishes. assuming a motion in x direction with velocity v0, i.e. v = (v0,0,0),we obtain an equation for the temperature field 2 0 , x v t t     (32) where   c) is the thermal conductance and 2 is the laplace operator. its solution consists of a factor that stems from the fundamental solution and a factor describing the motion 0 (| | ) 2 for ( , , ), 2 | | v xq t e x y z      x x x (33) where q is a point source at x = 0. this solution simplifies to the original fundamental solution of the homogeneous laplace equation if the exponent factor is close to unity, i.e. we get a constraint on the velocity 0 1 2 v dx   (34) for characteristic length dx. especially, at the surface of the half-space z = 0 the solution is 2 2fo r , 2 q t r x y r      (35) for point-like source q. if one considers a smeared-out source a distribution 0 2 | | , 1 q r a r a          q (36) leads to an isothermal plane at z = 0. this is the situation that occurs if two half-spaces are in thermal contact through a circular contact area of radius a in ideal contact. then the flash temperatures generated by friction of a viscoelastic body 57 relationship between the temperature difference between the half-spaces at infinite distance from the contact t and entire heat flow q through the passage is given by * 4 ,q a t  (37) where  is the harmonic mean of the specific thermal conductivities of the half-spaces 1/. this picture is in accordance with a normal pressure distribution loading an elastic half-space and leading to a constant strain. turning to a description in the one-dimensional substitute system we consider the surface as independent element in a winkler foundation. the thermal conductivity of a single such element is expressed as * . 2 dx  (38) through a passage of length 2a, the amount of * *2 4 a a a a q jdx tdx a t           (39) flows in accordance with the three-dimensional result eq. (37). the heat current through every element j is related to q and to the temperature difference via *2 . q j t dx      (40) the heat flow in single element of the winkler foundation is then given by *2 .p tdx  (41) reversing this relation we obtain for the temperature change in one-dimensional substitute model * . 2 p t dx    (42) for the kelvin model in a one-dimensional winkler foundation, the force exerted in one site in contact is given by eq. (13). the power dissipated in the contact amounts to ( ) .j j j j j j jp f u k u u u    (43) in non-contact sites, microscopically energy is dissipated through the relaxation of the kelvin element. the spring exerts a force on the damper during relaxation. the power dissipated in the element equals the difference of the mechanical energy stored in the spring 2 2 1 . 2 k k k u udw k p dt dt    (44) the temperature itself depends on how this heat is led away. for an element with thermal conductivity , 58 r. heise 2 2 2 2 1 0 1 2 ( ) , 2 . 4 j j j j j j j j j k k k k k duk t gu u u u du dx dt dt u u gv u uk t dx dt dx                        (45) keep in mind that the shift in temperature causes a change in the spectral behavior of the elastomer. this is seen in a changing viscosity and hence a shift in relaxation time i. for every site, viscosity i. is adapted to temperature ti in this contact. for points in contact, the change of viscosity leads to a different force experienced by the surface. in this way the entire contact configuration has to be adjusted to the new temperature. 3. results and discussion numerical simulations have been carried out for different ranges of the parameters involved. the viscosity is chosen to be   pa s, the modulus g   pa, the thermal conductivity w/m/k. the normal forces are varied logarithmically in a range from   n, to 20 n, the sliding velocity from 2   m/s to 2   m/s. the geometry is described by 5000 steps of with dx = 2m and a roughness h= 1m. the hurst exponent is extended to a range of [-1,3]. in fig. 4, the coefficient of friction as a function of both velocity and normal force is shown. the viscosity is adapted to the temperature in the contact sites in an iterative process. fig. 4 coefficient of friction normalized by the rms slope as a function of velocity and flash temperatures generated by friction of a viscoelastic body 59 normal force in logarithmic scale at h = 0.7 3.1. full contact if the surfaces are pressed with sufficiently high normal force against each other and the velocity is simultaneously rather low a full contact between the surfaces is established. in this situation, several quantities including the temperature can be estimated analytically. the mathematical condition of full contact is , , 1, .j ju z j n   (46) therefore, in all the expressions regarding the deformable surface it can be substituted by the unchanged rigid one. especially, the force in every site is given by 1 0 1( ) ( ) . 2 j j j j j j j j j z z v f k u u k z z dx                 (47) the averaging of the elastic force is introduced to avoid finite size effects and account for periodicity of the model. the sum of site forces sums up to the normal force 1 0 1( ) , 2 j j n j j j j j j z z v f f k z z knz dx                  (48) where 1/ j j n   . hence, the mean value of the rigid profile, i.e. the indentation, is . 4 n nf f z kn gl     (49) the parameter . 4 nfz f h glh n   (50) encodes the relation between indentation and roughness of the surface. we have introduced the dimensionless variable . 4 nf f gdxh  (51) for the frictional force, one encounters the following picture 1 1 10 1 2 1 2 2 0 0 02 ( ) 2 ( ) 4 j j j j j j x j j j j j j j j j j j j z z z z z zv f f z f k z z dx dx dx z z kv kn z lv z dx                                (52) 60 r. heise if one assumes that relaxation time j is approximately the same for all the sites. this is appropriate in the first estimate. the power in every site is found by looking at product of the force at this site and the rate of change of the surface 1 1 0 . j j j j j j j j j z z z z p f u f f v dt dx       (53) translating this to a temperature shift yields 0 . 2 j j j p t t t dx     (54) for the averaged temperature change, we get 1 10 1 0 22 10 2 22 10 2 2 0 0 0 0 0 0 1 1 2 1 ( ) 2 2 ( ) 2 ( )4 2 (2 1 1 exp 1 j j j j j j j j j j j j j j j j j j j j j jj b t t p n ndx z z z zv k z z v n dx dx z zkv ndx dx z zgdxv ndx g dx t zv u tn t k t t                                                               2 1 2 ) . j jz dx  (55) assuming that the temperature shift is about the same at every site jt t   and that this shift is small 0t t  we find an estimate for the temperature shift at full contact 0 00 2 2 00 0 1 1 2 = b t t ut k tv z       (56) from this expression we learn that there are several possible contributions. the first part of the sum in the denominator is the ratio between the typical power dissipated in the viscous material and the power led away. the second part gives a contribution from the thermal activation of the material. depending on the choice of the parameters the temperature shift can be tuned. in fig. 5, the proportionality of the temperature shift to the squared velocity is shown at low values of velocity. at these values the surfaces are in full contact. at higher velocities the number of contacts decreases until the number of contacts diminishes to single ones. in this region, it is more sensible to focus on the flash temperature in single contacts. flash temperatures generated by friction of a viscoelastic body 61 fig. 5 normalized shift of the temperature as a function of velocity for different hurst exponents. due to high normal force the elastomer is in full contact for low velocities. the slope is two. at higher velocities the contact becomes partial and the dependence is linear. this effect is occurs for lower hurst exponents at lower velocities 3.2. flash temperature in each element, the energy dissipation leads to rise in temperature as stated above. for partial contact of the surfaces, the temperature will shift differently at contacting and non-contacting sites. looking back at the motivation of this study we focus on the site of highest temperature. this temperature called flash temperature is the maximal temperature that we find over all sites. certainly, it depends on the actual realization of the surface. anyway, averaging over a large number of realizations (200 in our case) we find a rise of up to 5% of the temperature at a hurst exponent of h = 0.7 as shown in fig. 6. 62 r. heise fig. 6 normalized shift of the flash temperature as a function of normalized velocity v and normalized normal force f at h = 0.7. for small velocities and forces the flash temperature shift is small but is enhanced for higher values of these quantities as can be seen above the temperature shift is proportional to the power generated in each element. looking for its highest value one has to distinguish between contact and noncontact sites. in non-contact sites the power is proportional to the difference between the coordinates squared. obviously, this difference is largest when the detachment of the elastomer chain just occurred. the behavior of the free elastomer surface is governed by parameter dt/. in contact sites, the picture is different. here, the elastomer surface follows the rigid profile. hence it maps the surface topology of the rough surface into the dynamic behavior of the elastomer. from fig. 7 we see different behavior of the flash temperature depending on the contact configuration. for high normal force (lhs) the contacts saturate, i.e. there is full contact. the flash temperature shift increases with the normal load linearly. for lower loads, the relative contact number depends on the hurst exponent of the rigid surface. on the right hand side, the normal force is held constant. here the number of contacts is constant for low velocities. for higher speeds, more contacts are broken. for the flash temperature this means that, after having a linear dependence on the velocity, it saturates further on beginning for rough surfaces with a low hurst exponent. we propose the following power law dependencies of the flash temperature shift and the relative contacts 0 1 t t n n max cont t v f t n v f n        (57) with the empirical exponents flash temperatures generated by friction of a viscoelastic body 63 fig. 7 relative contacts (ncont / n, upper row) and normalized shift of the flash temperature (lower row) as a function of normalized normal force f (left, velocity constant 0.1v  ) and normalized velocity v (right, normal force constant 0.5f  ) for different values of the hurst exponent 1.6, 0 ( ) 1.69 0.64 , 0 1 1, 1 3 0.4, 0 ( ) 0.32 0.26 , 0 2 0.84, 2 3 1, 0 ( ) 0.87 0.84 , 0 1 0, 1 3 0.8, 0 ( ) 0.8 0.3 , 0 1 0.5, 1 3 t t n n h h h h h h h h h h h h h h h h h h h h                                                           (58) fig. 8 shows the numerical fits of the slope, i.e. the exponents of the normalized normal force and velocity in the ranges as indicated in fig.7. 64 r. heise fig. 8 numerical fits of the exponents of normalized normal force (left, velocity constant) and normalized velocity (right, force constant) for relative contacts (upper row) and normalized shift of the flash temperature (lower row) in the power law eq.(57) as a function of the hurst exponent the fit in the upper left corner resembles the result of pohrt and popov [23] for the purely elastic indentation. furthermore, as already seen in eq. (6), the most interesting range for the investigation of different hurst exponents is [0,1]. for hurst exponents up to 0, the flash temperature depends on the normal force with an exponent close to 0.4. in the intermediate range from 0 to 2, the exponent increases to 0.8. for constant forces (rhs), the flash temperature shift is depends on the velocity with an exponent of about 1.6 for low hurst exponents. the exponent then decreases to values about 1 for hurst exponents larger than 1. 4. conclusions and outlook we have found in the framework of the mdr a strong implication that we can study the temperature dependence in the frictional contact of an elastomer with a rough surface in a simple kelvin model. we have accomplished a shift in mean temperature as well in the flash temperature. the shift has been explored as a power law of the dimensionless normal force and velocity. the numerical fits of the exponents in the power laws have been carried out and the corresponding dependence on the hurst exponent proposed. further research can include more realistic elastomer models and lead to the temperature flash temperatures generated by friction of a viscoelastic body 65 dependence of the coefficient of friction itself. furthermore a reduction of the parameter space by dimensionless quantities is appropriate. acknowledgements: we are grateful to v. l. popov and s. kürschner for discussions. the author is financially supported by dfg project po 810/12-2. references 1. f. p. bowden, d. tabor, 1985, the friction and lubrication of solids, oxford university press, p. 452. 2. j. a. greenwood, d. tabor, 1958, the friction of hard sliders on lubricated rubber: the importance of deformation losses, proceedings of the physical society, 71, pp. 989 3. k. a. grosch, 1963, the relation between the friction and visco-elastic properties of rubber, proceedings of the royal society, 274, pp. 21-39 4. m. barquins, r. courtel, 1975, rubber friction and the rheology of viscoelastic contact, wear, 32, pp. 133-150. 5. m. klüppel, g. heinrich, 2000, rubber friction on self-affine road track, rubber chemistry and technology, 73, pp. 578-606. 6. b. n. j. persson, 2001, theory of rubber friction and contact mechanics, journal of chemical physics, 115, pp. 3840-3861. 7. b. lorenz, b. n. j. persson, g. fortunato, m. giustiniano, f. baldoni, 2013, rubber friction for tire tread compound on road surfaces, journal of physics: condensed matter, 25, pp. 095007. 8. v. l. popov, a. v. dimaki, 2011, using hierarchical memory to calculate friction force between fractal rough solid surface and elastomer with arbitrary linear rheological properties , technical physics letters, 37, pp. 8-11. 9. a. schallamach, 1952, the load dependence of rubber friction, proceedings of the physical society. section b, 65,pp. 657. 10. o. ben-david, j. fineberg, 2011, static friction coefficient is not a material constant, phys. rev. lett., 106, pp. 254301. 11. m. otsuki, h. matsukawa, 2013, systematic breakdown of amontons' law of friction for an elastic object locally obeying amontons' law, scientific reports, 3, pp. 1586. 12. s. m. rubinstein, g. cohen, j. fineberg, 2004, detachment fronts and the onset of dynamic friction, nature, 430, pp. 1005-1009. 13. o. ben-david, g. cohen, j. fineberg, 2010, the dynamics of the onset of frictional slip, science, 330, pp. 211-214. 14. d. s. amundsen, j. scheibert, k. thögersen, j. trömborg, a. malthe-sörenssen, 2012, 1d model of precursors to frictional stick-slip motion allowing for robust comparison with experiments, tribology letters, 45, pp. 357-369. 15. m. heß, 2011, about mapping of some three-dimensional contact problems to systems with a lower spatial dimensionality, cullier verlag, berlin. 16. m. heß, 2012, on the reduction method of dimensionality: the exact mapping of axisymmetric contact problems with and without adhesion, physical mesomechanics, 15, pp. 264-269 17. r. pohrt, v. l. popov, a. e. filippov, 2012, normal contact stiffness of elastic solids with fractal rough surfaces for oneand three-dimensional systems, phys. rev. e, 86, pp. 026710. 18. v. l. popov, m. heß, 2015, method of dimensionality reduction in contact and friction, springer, berlin heidelberg. 19. q. li, m. popov, a. dimaki, a. e. filippov, s. kürschner, v. l. popov, 2013, friction between a viscoelastic body and a rigid surface with random self-affine roughness, phys. rev. lett., 111, pp. 034301. 20. a.v. dimaki, v.l. popov; 2014, coefficient of friction between a rigid conical indenter and a model elastomer: influence of local frictional heating, phys. mech., 17 (5), p. 57-62. 21. v. l. popov, 2010, contact mechanics and friction, 1st ed., springer, berlin heidelberg, 362 p. 22. i. a. lyashenko, l. pastewka, b. n. j. persson, 2013, comment on “friction between a viscoelastic body and a rigid surface with random self-affine roughness”, phys. rev. lett., 111, 189401. 23. r. pohrt, v.l. popov, valentin l., 2012, normal contact stiffness of elastic solids with fractal rough surfaces, phys. rev. lett., 108, pp. 104301. 7427 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 145 156 https://doi.org/10.22190/fume210203037p © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper experimental research into marble cutting by abrasive water jet andrzej perec, aleksandra radomska-zalas, anna fajdek-bieda the jacob of paradies university, faculty of technology, gorzow wlkp., poland abstract. the article presents research on the erosion of the metamorphic rock marble by the abrasive water jet (awj). the fragmentation of abrasive grains during the erosion process is demonstrated. the effect of the cutting process's most important parameters as traverse speed, nozzle id, and abrasive mass flow rate, on the maximum cutting depth, is shown. to create a mathematical-statistic model of the erosion process, the methodology of the response surface (rsm) was used for modeling. the polynomial equation of the second degree is chosen for developing the regression model. studies have shown the optimal parameters of the process, to reach the highest depth of the cut. additionally, the erosion wear of a focusing tube under different process conditions is presented. key words: awj, abrasive waterjet machining, modeling, erosion wear, marble 1. introduction high-speed water jet machining is a fast-growing advanced manufacturing technology. the features of this technology are particularly environment friendly [1,2]. additionally, it successfully competes with traditional materials cutting methods. to a large extent, this is due to the wide possibilities of cutting various materials [3,4], including multi-layer materials with different properties [5,6] and precise cutting complex contour [7], or conducting them in uncommon conditions [8,9] (risk of detonation, conflagration, etc.) and low temperature of the process [10]. the materials treatment by high-speed awj is much further elaborate than the traditional cutting processes. a high-pressure pump (an intensifier or triplex pump) is a source of high pressure, which is, in the water nozzle, converted into a high-speed jet; next it grabs abrasive grains in the mixing chamber and accelerates them to a high speed. received february 03, 2021 / accepted april 14, 2021 corresponding author: andrzej perec ajp university in gorzow wlkp., teatralna 25, 66-400 gorzow wlkp. poland e-mail: aperec@ajp.edu.pl 146 a. perec, a. radomska-zalas, a. fajdek-bieda the admixture of abrasive grains to the water jet results in a dramatic growth of machining performance [11]. thanks to that, it is feasible to cut almost any material [12]. generally, the most utilized abrasive material is garnet [13]. other natural and synthetic abrasives, such as olivine [14], crushed glass [15], and aluminum oxide [16], may also be used. to achieve a trade-off between a long nozzle life and the big performance of the workpiece machining, the heedful selection of abrasive material is endorsed [17,18]. research on cutting rock materials was realized in different scientific centers. karakut et al. [19] presented research on the granite and aydin et al. [20] presented research on the marble cutting, in which it was noticed that increasing the feed reduces the slot width, while increasing the abrasive flow and stand-off distance increases the slot width. patel et al. [21] also conducted experimental investigations on the effect of abrasive water jet machining control parameters on the granite rock removal rate. khana et al. [22] published details on measuring the marble removal rate by awj. sitek et al. [23] introduced the effect of the traverse speed of the head and the stand-off distance on the quality of the processed surface and proposed the special variograms for the analysis of cut surface properties. however, the tests focused mainly on achieving the highest removal rate or the smallest roughness of the cut slot. arab and celestino [24] carried out research related to the cutting process of different rocks by awj and estimation of the impact of their properties on the awj erosion efficiency. the test effects demonstrate that the erosion phenomenon and thus the cutting performance depend on the rock kind and its microstructures. hloch et al. [25] demonstrated that the abrasive waterjet (awj) is a suitable technological method for sandstone cutting. the research was conducted on the surfaces that were created after machining at a pressure of 400 mpa with a focusing tube diameter of 1.02 mm. barton garnet 80 mesh was used as the abrasive material. oh et al. [26,27] also published research into abrasive waterjet cutting of granite and shale rocks, carried out under variables conditions of the water pressure (up to 314 mpa only), traverse speed, abrasive flow rate, cutting pass numbers, and stand-off distance. based on the above presented state of the art, it should be noted that cutting of rock materials, especially granite was the subject of research in various centers. this paper is focused on offering a model of the process of cutting another rock the marble. 2. materials and methods 2.1. cut material marble is a crystalline rock consisting mainly of calcium carbonate calcite grains. this type of rock was created by the transformation of limestone rocks. marble is a very valuable decorative stone and a construction material. it is widely utilized for carving, as an architectonic material, and in other different applications. it comes in a variety of colors: white, cream, red, up to shades of black. marble powder can be merged with cement or synthetic resins to perform restructured marble. the marble used for the test came from the nanutarra white marble quarry, nanutarra station, ashburton shire, western australia. this material is visually appealing, hard, and with high gloss. it is described by the following properties: experimental research into marble cutting by abrasive water jet 147 ▪ density: 2730 kg/dm3, ▪ compression strength: 45 47.5 mpa, ▪ mohs hardness: 7. 2.2. abrasive material in the research, garnet grains were used as abrasive. the garnets group of minerals contains closely related isomorphic minerals. garnet grains are isostructural, which means they have the identical crystal structure. this leads to similar form and properties of crystals. the most frequently used in the awj technology is almandine garnet. the chemical formula of this garnet is fe3al2(sio4)3. details of typical gma80 garnet particle distribution are shown in fig. 1. fig. 1 typical gma80 garnet grain size distribution a normal, almost symmetric, density function approximating the abrasive particle distribution is visible. the most important garnet properties are shown in table 1. table 1 properties of gma80 abrasive grains [28] crystal system cubic twinning none unit cell a = 11.53 å habit crystals usually dodecahedrons or trapezohedrons; also, in combination or with hexoctahedron; massive; granular cleavage 1; {110} parting sometimes distinct fracture conchoidal to uneven tenacity brittle color deep red to reddish-brown, sometimes with a violet or brown or brownish black hardness (mohs) 6.5-7.5 density 4.1-4.3 148 a. perec, a. radomska-zalas, a. fajdek-bieda the alluvial, almandine garnet comes from geraldton deposits in western australia, from the dune sands garnet-bearing. through a unique geological history of erosion and deposition, it contains the highest quality garnet [29]. 2.3. test procedure the experiments were done on the test rig with the i50 intensifier (kmt), and 2 axes cnc machine type ils55 by techni waterjet controlled by a pc system. the maximal pressure is 400 mpa at a flow rate of 5 dm3/min. to grab abrasive grains after they were shot out from the cutting head, a special collector was used [15]. the collector was customized to grab the abrasive grains and to preclude any extra grains disintegration. the underside pvc collector was shielded by a mild steel target to avert perforation. no wear marks were noticed on the safeguarding target after the termination of tests [2]. the caught abrasive grains are then dried. for the used abrasive grain size distribution tests, the retsch sieving system was used. the fragmented garnet left on the sieves was weighed on the laboratory digital scale. the materials were cut by pointing the jet at the material and moving it at a constant speed relative to the material. the cutting sample thickness was selected so that the undermost effective processing parameters do not result in a through-cutting. in this way, potential inaccurate measurements of cutting depth were eliminated. process parameters were chosen based on previous works involving authors of the present work [30,31], and the works of other investigators [22,32,33]. the abrasive concentration determines the ratio of the abrasive mass to the water mass in the awj. the mass of the abrasive is set on the feeder, while the mass of water in the jet arises from the flow rate for a given id of the water nozzle at a given pressure, considering discharge coefficient (cd). the maximum cutting depth was selected as the output parameter. this is a widely used parameter [31,37] that clearly defines the effectiveness of this process. measurements of cutting depth were made by a digital caliper altimeter. the experiment design was utilized to minimize the tests numbers and cut its time [34,35]. the tests were led with a full factorial design. the response surface methodology(rsm) with 36 tests [36] was used (table 2). rsm is a fusion of statistical and mathematical modeling methods. it can be utilized in multi-criteria optimization[37]. in addition, it also ensures a join amid process control parameters and the perceived responses. the polynomial equation for making the value of a regression model [38] follows: 𝑦 = 𝛽0 + ∑ 𝛽𝑖 𝑘 𝑖=1 𝑥𝑖 + ∑ 𝛽𝑖𝑖 𝑘 𝑖=1 𝑥𝑖 2 ±  (1) where y is dependent variable (response), xi is values of the i-th control parameter, k is number of control parameters, β0, βi, βii are the coefficients of regressions and ε is the error. experimental research into marble cutting by abrasive water jet 149 table 2 values of parameters used in experiments and results of cutting depth test no nozzle id [mm] abrasive concentration [%] traverse speed [mm/s] cutting depth [mm] 1 0.25 15 2 64.10 2 0.25 15 4 54.40 3 0.25 15 6 41.20 4 0.25 17.5 2 67.50 5 0.25 17.5 4 57.00 6 0.25 17.5 6 47.00 7 0.25 20 2 69.70 8 0.25 20 4 60.10 9 0.25 20 6 47.10 10 0.25 22.5 2 66.67 11 0.25 22.5 4 55.60 12 0.25 22.5 6 43.70 13 0.3 15 2 72.00 14 0.3 15 4 62.60 15 0.3 15 6 46.50 16 0.3 17.5 2 76.20 17 0.3 17.5 4 65.70 18 0.3 17.5 6 54.10 19 0.3 20 2 79.80 20 0.3 20 4 69.10 21 0.3 20 6 54.70 22 0.3 22.5 2 75.10 23 0.3 22.5 4 64.60 24 0.3 22.5 6 50.50 25 0.33 15 2 80.10 26 0.33 15 4 70.10 27 0.33 15 6 54.70 28 0.33 17.5 2 86.90 29 0.33 17.5 4 74.20 30 0.33 17.5 6 61.50 31 0.33 20 2 88.60 32 0.33 20 4 78.28 33 0.33 20 6 62.20 34 0.33 22.5 2 85.60 35 0.33 22.5 4 72.80 36 0.33 22.5 6 57.18 3. results and discussion 3.1. abrasive grain fragmentation gma80 abrasive grain fragmentation tests were performed for cutting heads with the water nozzle and focusing tube following sets: ▪ 0.25 mm/0.76 mm ▪ 0.33 mm/1.02 mm the fragmentation test results for a cutting head equipped with a 0.25 mm id water nozzle and a 0.76 mm id focusing tube are illustrated in fig. 2. 150 a. perec, a. radomska-zalas, a. fajdek-bieda fig. 2 the disintegration of the effect of the gma80 grain at water nozzle id 0.25 mm, focusing tube id 0.76 mm, pressure 390 mpa this distribution is similar to the bimodal one with negative asymmetry (skewness = 0.248), in which two clearly outlined observation focal points can be seen, for grains 180-150 m and with the particles below 53 m predominancies, which previously has not occurred. the number of 355-250 m particles reduced remarkably. abrasive grain distribution is very platykurtic, (kurtosis<0.67).overall, a considerable grain size reduction was observed. the abrasive concentration change had a very small impact on grain fragmentation [39]. fig. 3 shows the outcomes of the breakage of gma80 abrasive under 390 mpa pressure for a cutting head equipped with a 0.33 mm id water nozzle and a 1.02 mm id focusing tube. this distribution is also similar to the bimodal one, with similar focal points, for grains 180-150 m and for grains smaller than 53 m. the largest fraction (almost 20%) was smaller than 53 m. particle size also decreases significantly. in this case, abrasive concentration had almost no impact on grain fragmentation, either. fig. 3 the disintegration of the effect of the gma80 grain at water nozzle id 0.33 mm, focusing tube id 1.02 mm, and pressure 390 mpa fig. 4a shows exemplification particles of fresh (not used) abrasive. the abrasive grains are round and isometric. the grain size is not very diverse. fig. 4b presents the view of abrasive grains after escape the focusing tube. most grains have been fragmented. one can notice various experimental research into marble cutting by abrasive water jet 151 size grains, still, mainly isometric in the shape, although with sharp edges. between them, a small number of grains with bigger dimensions occur. a) b) fig. 4 gma80 abrasive grains a) fresh, b) after disintegration in the cutting head 3.2. cutting results the outcomes of studies on the impact of process control parameters (independent variables) on the cutting depth (dependent variable) are indicated in table 3. table 3 details of analysis of variance source df adj ss adj ms f-value p-value vif model 6 5499.48 916.58 278.06 0.000 linear 3 5305.94 1768.65 536.54 0.000 nozzle 1 1658.18 1658.18 503.03 0.000 1.02 concentration 1 63.64 63.64 19.31 0.000 1.00 traverse speed 1 3584.13 3584.13 1087.29 0.000 1.00 square 3 235.28 78.43 23.79 0.000 nozzle*nozzle 1 42.21 42.21 12.81 0.001 1.02 concentr. * concentr. 1 176.14 176.14 53.43 0.000 1.00 traverse speed*traverse speed 1 16.93 16.93 5.13 0.031 1.00 error 29 95.59 3.30 total 35 5595.08 s = 1.34181 r2 = 99.06% r2adj = 98.87% r2pred = 98.57% the method of analysis of variance (anova) for the 95% level of confidence ( = 0.05) was made. the model coefficient is statistically significant when it reaches p-value <0.05 [40]. to estimate multicollinearity, the variance inflation factor (vif) was calculated. it quantifies the intensity of multicollinearity. vif reveals how much the variance of the evaluated regression factor is inflated caused by multicollinearity in the model. when vif is 1.0, multicollinearity does not occur. for all tested factors, no multicollinearity was observed because vif ≤ 1.02. the regression standard error s = 1.34181 and all r2factors (r2, r2 adj, and r2 pred) are little differing and take on values over 98%. it confirms that the raw data satisfactory match to the line of regression. on the ground of research results, the final cutting depth model(2) was introduced: 152 a. perec, a. radomska-zalas, a. fajdek-bieda 𝐷𝑐 = 13.7 – 663𝑑𝑛 + 14.14𝐶𝑎– 3.055𝑆𝑡 + 1499𝑑𝑛 2 − 0.3655𝐶𝑎 2 − 0.378𝑆𝑡 2 (2) where dc is depth of cut [mm], dn is water nozzle id [mm], ca is abrasive content in the jet [%], and st is traverse speed [mm/s]. the scattering of the actual and predicted depth of cut values is shown in fig. 5. all points are localized near to a straight line; this confirms the formulated model is satisfactory. fig. 5 scattering plot for actual and predicted cutting depth the impact of traverse speed and abrasive concentration on the depth of cut is shown in fig. 6. the cutting depth is directly proportional to the diameter of the water nozzle. the highest value can be observed for a nozzle id of 0.33 mm. this is due to bringing the most energy to the cutting zone. fig. 6 effect of abrasive concentration and traverse speed on cutting depth with water nozzle diameter’s: a) 0.25 mm, b) 0.29 mm, c) 0.33 mm however, in the case of the impact of the abrasive concentration (fig. 7), which indirectly characterizes the mass flow rate of the abrasive, a clear extremum of the cutting depth for the abrasive concentration at the middle-value zone can be observed. the abrasive concentration value at this condition, calculated based on the model eq. (2) is 19.3%. exceeding this value results in a decrease in cutting depth. this is mainly due to the reduction in the speed of abrasive grains in the stream because the water energy is too low to keep the maximum speed of an increased number of abrasive grains in the jet. in addition, the interaction between abrasive grains in the jet is adversely affected on cutting depth. this was observed for the entire range of both tested nozzles and feed. experimental research into marble cutting by abrasive water jet 153 fig. 7 effect of traverse speed and water nozzle id on cutting depth with abrasive concentration: a) 15%, b) 18.75%, c) 22.5% the impact of the traverse speed on the cutting depth (fig. 8) is inversely proportional. the greatest cutting depths are achieved for the lowest feed rates. the maximum value of cutting depth was observed for a traverse speed of 2 mm/s. the nature of this relationship is primarily due to a greater number of abrasive grains affecting the workpiece in the cutting zone per unit of time. fig. 8 effect of water nozzle id and abrasive concentration on cutting depth with traverse speed: a) 2 mm/s, b) 4 mm/s, c) 6 mm/s 3.3. focusing tube wear erosion possibilities of awj have an impact similar to the one related to the cutting efficiency of the target material and the wear rate of the focusing tube [41, 42].the erosion wear rate was computed based on measuring the focusing tube mass loss at fixed time intervals. fig. 9 presents an exemplification view (after edm cutting along the axis) of the internal surface of the worn focusing tube. uneven wear of the interior surface was observed, created by the process of forming an abrasive waterjet. the pure water jet at high speed (over 700 m/s) comes into the focusing tube and reduces the practicable open inlet diameter for the abrasive particles. before abrasive particles come into the jet, they are bounced few times from the external water jet surface and the focusing tube inside the surface and cause its varied wear. 154 a. perec, a. radomska-zalas, a. fajdek-bieda fig. 9 exemplification view of the internal surface of the worn focusing tube fig. 10 shows the influence of focusing tube mass loss from abrasive flow for gma80 garnet. the average focusing tube wear factor (the slope of a line) reaches the value of 0.067 mg/s. fig. 10 effect of the running time on focusing tube mass loss for gma80 garnet 4. conclusion based on the conducted research related to the modeling of marble cutting, the following conclusions were obtained: ▪ abrasive concentration has no effect on abrasive particle fragmentation. ▪ each tested control factors of the awj have a significant influence on erosive abilities, measured in the form of cutting depth. ▪ in the entire tested range of control parameters, the biggest depth of cut was observed for the abrasive concentration = 19.3%. ▪ the cutting depth is directly proportional to the water nozzle id in the tested range. ▪ r-squared (the percentage of variation in the response that is explained by the model) over 99% shows the model fits very well to experimental data. ▪ adjusted r2 value = 98.87%, which is r2, adjusted for the number of predictors in the model relative to the number of tests, also confirms a very good model fit. ▪ predicted r2 over 98% shows a very good predicting of the model reaction for the new observations. ▪ for regression coefficients of the model was observed no multicollinearity. ▪ optimal settings of awj cutting parameters from the maximum cutting depth point of view for the examined area are as follows: nozzle id = 0.33 mm, abrasive experimental research into marble cutting by abrasive water jet 155 concentration = 19.3%, and feed speed = 2 mm/s. at the above parameters of cutting, the maximal depth of cut of more than 87.3 mm was attained. in further research, the machining model will be extended with additional control parameters, e.g., standoff distance and water pressure. references 1. kukiełka, k., 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[1]. gay and suong [2] stated that, sandwich structures were created when two thin facings or skins were bonded or welded together on a lighter core that kept the two skins apart. the core increased the panel's bending stiffness and resistance to buckling loads [3,4]. they also found that sandwich structures had fewer lateral deformations, more buckling resistance, and higher natural frequencies than other types of structures. douville and grognec [5] showed that sandwich structures were employed in a wide range of industrial applications due to their desirable combination of light weight and strong mechanical qualities. therefore, steel is commonly used for the skins [6,7]. hence, commonly used materials in the core are metallic foams and polymers [8–10]. rigid-foam core is commonly used and has moderate strength and stiffness [11]. moreover, one of the common cores used in sandwich panels is a honeycomb [12–14]. in general, sandwich structures are preferred over traditional materials due to their high corrosion resistance [15] and low thermal and acoustic conductivity [16,17]. the high thermal expansion coefficient of aluminum honeycomb sandwich constructions limits their use in thermal management systems [18]. the behavior and failure modes of sandwich structures in flexure have been observed in a variety of studies [17,19–21]. a sandwich structure can fail due to a variety of damage mechanisms, including tensile failure/skin compressive [22], local skin wrinkling [23,24], and core indentation failure [25,26]. the sensitivity of nap-core sandwich, a unique type of structural composite with different characteristics, was studied by ha et al. [27]. mccormack et al. [28] studied the failure of sandwich beams with metallic foam cores. ha et al. [29] studied the behaviors of nap-core sandwiches, with a particular emphasis on the effect of symmetry in nap cores. in thermal applications, heat conduction and heat flux in sandwich beams has been experimented. non-fourier heat conduction was investigated in a sandwich panel with a cracked foam core by fu et al. [30]. under a high-temperature environment, the vibration properties of a carbon nanotube-reinforced sandwich curved shell panel were explored by mehar and kumar panda [31]. sun et al. [32] investigated the convective cooling efficiency of a sandwich panel with a hierarchical corrugated core heated from face sheets and cooled actively through the core. onyibo and safaei [33] applied finite element analysis to honeycomb sandwich structures. su et al. [34] investigated the thermal insulation performance of structural insulated panels (sips) with glass fiber-reinforced polymer (gfrp) surfaces. moradi-dastjerdi and behdinan [35] presented a smart multifunctional sandwich plate with a central lightweight porous layer, two intermediate polymer/graphene nanocomposite layers, and two piezoceramic active faces. zhang et al. [36] proposed an energy harvester that used an arc-shaped piezoelectric sheet to scavenge rotational energy from rotating devices. zhao et al. [37] proposed a multifunctional carbon fiber honeycomb sandwich structure (mcfhs). safaei et al. [38] presented an analytical solution based on molecular mechanics model to estimate the elastic critical axial buckling strain of chiral multi-walled carbon nanotubes (mwcnts). using experimental, theoretical, and numerical simulation methodologies, chen et al. [39] investigated the heat transmission mechanism and thermal insulation performance of fabricated c/sic corrugated lcsp. safaei et al. [40] explored the thermoelastic responses of sandwich plates with porous polymeric core and proposed ultra-lightweight engineering structures of sandwich plates with one porous polymeric core and two carbon nanotube (cnt)/polymer nanocomposite outer layers. asmael et al. [41] reviewed the ultrasonic machining of carbon fiber–reinforced plastic composites. critical buckling bending analysis of sandwich structure under axial compression and thermal... 591 according to the law of vibration, everything (components and structures) is in constant motion, vibrating at a given frequency. safaei [42] investigated the damped vibrational behavior of a lightweight sandwich plate subjected to a periodic force over a short period of time. liu et al. [43] examined the nonlinear forced vibrations of functionally graded materials (fgms). hadji and avcar [44] investigated free vibration of a square sandwich plate with functionally graded (fg) porous face sheets and an isotropic homogenous core under different boundary conditions. li et al. [45] theoretically and experimentally studied the nonlinear vibrations of fiber-reinforced composite cylindrical shells (frccss) with bolted joint boundary conditions. liu et al. [46] developed a novel method for solving nonlinear forced vibrations of functionally graded (fg) piezoelectric shells in multiphysics fields. however, there have been many studies on nonlinear vibration behaviors of various materials [47-54]. the nonlinear forced vibration behaviors of third-order shear deformable nanobeams in the presence of both surface stress and surface inertia due to high surface to volume ratio were investigated by sahmani et al. [55]. yang et al. [56] carried out free vibration evaluations on fg graphene nanoplatelet-reinforced composite (fggplrc) arches in both in-plane and out-of-plane. katariya et al. [57] used higher-order shear deformation theory to construct a generic mathematical model for evaluating the bending and vibration responses of skew sandwich composite plates. enhanced donnell nonlinear shell theory and maxwell static electricity/magnetism equations were used by liu et al. [58] to construct a coupled nonlinear model for composite cylindrical shells to examine nonlinear forced vibrations in multi-physics fields. moradi-dastjerdi and behdinan [59] used an innovative and reliable approach to investigate the free vibration behaviors of multifunctional smart sandwich plates (mssps). buckling is one of the major failure causes of machine elements; therefore, buckling loads should be considered in designs. bažant and beghini [60] showed that sandwich shell failures caused by skin, core, and interface fracturing were often associated with instability and as a result, it was difficult to distinguish among delamination, fracturing, damage, buckling, face wrinkling and scaling. however, based on the type of loading, various buckling forms could occur, which can be global or local [61,62]. zhao [63] investigated new buckling characteristic equation for sandwich pipelines. alhijazi et al. [64] analyzed the elastic properties of luffa and palm natural fiber composites (nfcs) with epoxy and ecopoxy matrixes, taking fiber volume fractions into account. fan et al. [65] investigated the thermal post buckling characteristics of porous composite nanoplates. grygorowicz et al. [66] studied the elastic buckling of a three-layered beam with a metal foam core using analytical and numerical approaches. ansari et al. [67] studied the pull-in instability properties of hydrostatically and electrostatically actuated circular nanoplates, taking into account the influence of surface stress. fattahi and safaei, [68] studied the buckling properties of nanocomposite beams reinforced with single-walled carbon nanotubes (swcnts). liu et al. [69] used a size-dependent numerical solution methodology to investigate nonlinear buckling and post buckling of cylindrical micro sized shells made of checkerboard. nonlinear buckling of elastically supported functionally graded graphene platelet-reinforced composite (fg-gplrc) was analytically predicted by yang et al. [70]. under temperature conditions, yang et al. [71] performed an analytical investigation on the asymmetric static and dynamic buckling of a pinned-fixed fg-gplrc. safaei et al. [72] performed highprecision thermal and mechanical buckling analyses on ud porous sandwich plates with fg-cnt cluster-reinforced nanocomposite outer layers resting on elastic foundations. magnucki and magnucka-blandzi [73] developed an analytical model for sandwich 592 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu structures using a rectangular plate as an example. sahmani and aghdam [74] studied the size-dependent buckling and post buckling characteristics of microtubules embedded in cytoplasm under axial compressive load using the nonlocal strain gradient theory of elasticity, which included both nonlocality and strain gradient size dependency. this was done within the framework of a refined orthotropic shell theory with hyperbolic distribution of shear deformation. yunliang and junping [75] studied bifurcation point buckling and extreme point buckling of sandwich structures. sahoo et al. [76] investigated the finite element solutions of thermal buckling load values of graded sandwich curved shell structures using a higher-order kinematic model that included shear deformation effect. avilés and carlsson [77] developed an elastic foundation model for analysis of the local buckling behavior of foam core sandwich columns containing a through-width face/core debond. also, there have been numerous investigations on nonlinear buckling [78–80], sizedependent buckling, and bending behaviors [81–84]. jasion et al. [85] investigated the global and local buckling–wrinkling of the face sheets of sandwich beams and sandwich circular plates using computational and experimental methods. moreover, léotoing et al. [86] presented the first implementation of a revolutionary unified sandwich model considering closed-form solutions for both global and local buckling. in the presence of surface stress impact, fan et al. [87] predicted the shear buckling characteristics of skew nano-plates composed of a fgm. sahmani and aghdam [88] investigated the size-dependent buckling and post buckling responses of hybrid fg nanoshells integrated with surface-bonded piezoelectric nanolayers using a refined exponential shear deformation shell theory in conjunction with nonclassical eringen's nonlocal elasticity theory. under in-plane hydrostatic and uniaxial compression, elastic buckling of regular hexagonal thin sheets made of homogenous and isotropic materials with internal, translational and rotational elastic edge supports were studied by ghanati and safaei [89]. however, bending of a slender structural element characterizes its behavior under an external load applied perpendicularly to a longitudinal axis of element. furthermore, yan et al. [90] predicted the bending stiffness, peak load and initial failure load of sandwich structures. mehar and panda [91] numerically evaluated the deflection behaviors of carbon nanotube-reinforced composite plates using finite-element method and validated the accuracy of results using three-point experimental bending test data. barbaros et al. [92] carried out a review on fg porous nanocomposite materials considering manufacturing, application, and mechanical characteristics. liu et al. [93] investigated the impact responses of shear deformable sandwich cylindrical shells with two face layers and a fg porous core. aldakheel and miehe [94,95] investigated coupled thermo-mechanical strain gradient plasticity theory taking into account microstructure-based size effects. chen [96] used fem to investigate the collapse behaviors of corrugated cross section beams subjected to three-point bending. zhao et al. [97] reviewed the mechanical properties of graphene and graphene composites. wang et al. [98] proposed a novel multilevel modeling approach for calculating the young's modulus of polymers reinforced with graphene nanoplatelets. zhang et al. [99] studied the bending strength, stiffness, and energy absorption of a corrugated sandwich composite structure. few research works have been carried out on natural fiber composites [100,101]. mehar and panda [102] conducted deflection analysis to investigate the effect of mwcnt reinforcement on the rigidity of sandwich curved panels. sahmani and aghdam [103] estimated the young's modulus and poisson's ratio of nano porous biomaterials with refined truncated cube cells. the effect of adding calcium carbonate nanoparticles (nps) to cement plast on mechanical characteristics was experimentally explored by safaei et al. [104]. li et al. [105] investigated vibro-impact resistant performance of fiber reinforced composite (frc) critical buckling bending analysis of sandwich structure under axial compression and thermal... 593 plates with polyurea coating (pc) under four-edge elastic constraints. seong et al. [106] applied bending mechanism to determine the control parameters of core shear stress and found that, according to analytic calculations, the shear stress of core could be greatly decreased by increasing clearance. under the impact of mechanical loading and heat field, the frequency, deflection, and stress values of carbon nanotube-reinforced sandwich plate structures were numerically investigated by mehar et al. [107]. fattahi et al. [108] presented an experimental investigation on the effect of nanoparticle volume fraction on the elastic characteristics of a polymer-based nanocomposite and obtained comparable results to other existing theoretical models. moreover, sivaram [109] improved the strength and mechanical properties of aluminum materials by the inclusion of polypropylene foam sheets. hu and wang [110] created a thermal–mechanical analysis model to characterize thermal shock behaviors of auxetic honeycomb core ceramic sandwich structures (csss). daouas et al. [111] investigated thermal performance of walls under optimal conditions. wang et al. [112] derived an exact analytical solution for steady-state heat transfer in fg sandwich slabs under convective-radiative boundary conditions. geoffroy et al. [113] studied 3d printed sandwich materials filled with hydrogels at extremely low heat release rates. safaei et al. [114] used fem to examine the effect of static and harmonic loads on honeycomb sandwich beams. in this paper, the flexural strength of sandwich beams was improved, heat flux and temperature at which sandwich structures buckle were determined using finite element analysis (commercial software ansys). in a connected linear buckling analysis of sandwich beams, the load factor on the static load causing buckling was assessed, the critical buckling load and bending behavior of carbon fiber strut imbedded in sandwich core was determined. in general, sandwich beam heat transfer and thermal strain at corresponding temperature were investigated and compared with added silica aerogel material. 2. problem objective in this paper, the analysis of buckling caused by thermal expansion in a sandwich beam was presented for simply supported sandwich beams under axial compression and the critical buckling force and buckling mode shapes were investigated. furthermore, the bending deformation of cantilever sandwich beams with different imbedded carbon fiber materials was studied and results were compared with analytical calculations. in addition, the rate of heat conduction of the sandwich was analyzed to reduce the heat flux and temperature induced stresses and make it a favorable thermal insulator. 2.1. geometry and finite element model the geometry of sandwich beam solid model structure was used to investigate bending and critical bucking load, while carbon foam core was applied to analyze heat flux and thermal stresses of a given thickness using finite element analysis. the advantages of sandwich solid model are short computational time, good adhesion properties and more reliable results when compared to analytical solution. fig. 1(a) shows the solidworks commercial software design of 8 cylindrical bar struts of 1.5 mm diameter and 10 mm spacing. fig. 1(b) shows carbon fiber strut imbedded in the core in order to improve sandwich axial compression and fig. 1(c) depicts rectangle shape strut inserted in the core for absorbing bending stresses in sandwich beams. furthermore, fig. 1(d) depicts the sandwich with added aerogel layers. lastly, table 1 shows the dimensions of the used sandwich beam. 594 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu (a) (b) (c) (d) fig. 1 geometry of sandwich beam (a) cylindrical bar strut of 1.5 mm diameter and 10 mm spacing; (b) carbon fiber strut imbedded in core; (c) rectangle shape strut inserted in core (d) sandwich with added aerogel layers table 1 dimensions of sandwich beam name length x length y length z volume mm2 mass kg nodes element skin mm length 150 80 9 60000 0.471 12300 8880 1 2.2. loads and boundary conditions axial load case was considered in sandwich buckling while point load was applied in bending tests of sandwich. however, for the buckling of sandwich beam structures, simply supported (pinned at both ends) conditions were used which could only move laterally, and in bending tests, sandwich structures were treated as cantilever beams (fixed at one end and point load on the other end), as shown in fig. 2(a). in buckling, no displacement along z-direction, completely fixed on one side and moving on the other side along the direction of applied force, in this case x-direction, as shown in fig. 2(b). fig. 2(c) depicts thermal boundary conditions (high temperature applied on top skin and room temperature set at bottom skin). finally, fig. 2(d) shows thermal boundary conditions for the temperature that could buckle the sandwich beam. critical buckling bending analysis of sandwich structure under axial compression and thermal... 595 (a) (b) (c) (d) fig. 2 boundary conditions (a) cantilever for bending analysis; (b) simply supported, moving along horizontal direction; (c) steady-state temperature boundary condition (higher temperature applied on top skin); (d) thermal expansion of sandwich beam 2.3. material properties there has been extensive research on silica aerogels recently. as the lightest processed solid material on the planet, aerogels have the highest empty volume fraction in their structure. they are made by super critically drying silica gels, which was also the first inorganic aerogel to be developed by patil et al. [115]. in this analysis aerogel was used for the reduction of heat flux, thermal strain and expansion. sandwich beam materials consist of carbon fiber-polyimide for skins and carbon foam as core. ogasawara et al. [116] used the same structure to develop an effective heatresistant sandwich panel. table 2 summarizes the depicted material properties, carbon fiber-polyimide was used as face sheet and carbon foam was used as the core to analyze flexural strength in cantilever beam conditions and investigate strength along the axis (buckling). furthermore, the sandwich beam with the same material was subjected to thermal loads and the results with and without aerogel were evaluated. 596 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu table 2 material properties of sandwich beams material young’s modulus e (mpa) poisson’s ratio ν shear modulus g (mpa) density ρ (kg/m3) thermal conductivity (w/m°c) carbon fiber-polyimide 70000 0.36 25735 1410 2.16 carbon foam 123.79 0.33 46538 2267 0.11 silica aerogel 5 0.2 20833 105 0.016 3. fem analysis ansys software was applied to perform numerical analysis on sandwich beams with carbon foam core. a linear buckling model was used in finite element analyses. the conceptual buckling strength of an ideal elastic structure was calculated using linear buckling (also known as eigenvalue buckling) and nonlinear analyses to investigate buckling temperature of sandwich beam. however, in contrast to linear solutions, nonlinear solutions were also used in experiments to achieve the most precise results. thereby, this method gave closer buckling estimation. hence, critical buckling load (stress) and factors that optimize the sandwich structure were investigated and heat flow rate across sandwich layers was observed. all sandwich beam layers were modelled using solid model and was meshed using eight node solid elements solid186 (the total statistics of 8880 elements and 12300 nodes). the details of the sandwich structure were b = 80 mm, l = 150 mm, tf = 1 mm, tc = 4 mm, ef = 70000 mpa, p = 100 n, and ec = 5 mpa. the faces of sandwich beam were made of carbon fiber-polyimide and its core was made of carbon foam. tables 3, 4 and 5 show fem results and analytical calculations. table 3 bending of cantilever sandwich beam structure deformation (mm) fem 2.14 analytical 2.17 table 4 buckling of sandwich beam structure critical buckling load (n) fem 31097 analytical 3167.1 table 5 heat flow rate of sandwich beam structure heat flux (w/m2) fem 2090.85 analytical 2091.00 3.1. global buckling of sandwich column when studying sandwich systems, it is generally thought that the core only supports shear and that skins carry tensile and compressive loads under flexure [117]. the critical buckling bending analysis of sandwich structure under axial compression and thermal... 597 contributions of core and skin to flexural and shear stiffness were considered in the sandwich beams examined in this analysis. two modes of elastic buckling are possible: euler buckling mode with sandwich column bending and a core shear mode. the shear deformation of core was taken into account in a more precise approach [118,119]. fig. 3 shows the schematic model of buckling setup to find out plane displacement on sandwich beam. fig. 3 schematic model of buckling setup for case of a strut with built-in ends (which are constrained against rotation), the euler buckling load pe is: 𝑃𝐸 = 4𝜋2(𝐸𝐼)𝑒𝑞 𝑙2 (1) the equivalent flexural stiffness eieq of the sandwich beams was calculated by eq. (2). 𝐸𝐼𝑒𝑞 = 𝑏𝑡3 6 𝐸𝑓 + 𝑏𝑡𝑑2 2 𝐸𝑓 + 𝑏𝑐3 12 𝐸𝑐 (2) where ef and ec are the young's moduli of face sheet and core materials, respectively, b is the width of sandwich column, t is the thickness of face sheet, and c is the thickness of core. d ≡ t + c is the distance between the mid-planes of face sheets. the core shear buckling load ps was set by the shear stiffness of core. 𝑃𝑠 = (𝐴𝐺)𝑒𝑞 (3) where the corresponding shear rigidity of core (ag)eq is: (ag)𝑒𝑞≈bcg𝑐 (4) where ps ≈ ageq, a = (h + h)2/h, gc = ec/(1 + 2νc), gc is the shear modulus of the core and bc the cross-sectional area of the core. however, at transformation values of strut slenderness ratio, these buckling modes interacted to produce a combined collapse load (pcr) where: p𝑐𝑟=pe+p𝑠 (5) 598 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu 3.2. heat transfer equations local heat flux for one-dimensional steady-state heat transfer across a sandwich, according to the fourier heat transfer law, could be written as: 𝑞 = −𝑘𝛥𝑇 = −𝑘 𝑑𝑇 𝑑𝑥 (6) where q is local heat flux density (w/m2), k is material thermal conductivity (w/m*k), 𝛥𝑇 = 𝑑𝑇 𝒅𝒙 is temperature gradient (k/m). 3.3. thermal resistance thermal resistance is a thermal property and a measurement of how well a material resists a heat flow. 𝑄wall = −𝑘𝐴 𝑇2−𝑇1 𝐿 = − 𝑇2−𝑇1 𝑅thcond (7) rskin = 𝑡 𝐾𝑠𝐴 (8) rcore = 𝑡 𝐾𝑐𝐴 (9) 𝑅thcond = 2( rskin ) + 𝑅 core (10) where 𝛥𝑇 is the amount of heat transited through sandwich beam (t2 t1), rthcond represents thermal conductive resistance, l is length, t is thickness, a is area, ks is the thermal conductivity of skin and kc is the thermal conductivity of core. hence, in this analysis, temperature was applied on the bottom and top face sheets of the sandwich beam. fig. 4 depicts the schematic model of conductive heat distribution on sandwich beam and showed that high heat was transferred from the top skin, distributed through core and down to bottom skin. fig. 4 schematic model of conductive heat distribution on sandwich beam. critical buckling bending analysis of sandwich structure under axial compression and thermal... 599 4. results and observation 4.1. deformation of sandwich considering the deformation of sandwich beam shape due to applied forces and change in temperature, in order to investigate total deformation and the state of stresses. this analysis considered the buckling of sandwich beam due to applied forces and elevated temperature, bending of sandwich due to applied loads and heat flow rate across the sandwich beam. 4.2. buckling analysis in order to determine the failure mode in which relatively large deflections occurred on sandwich beam, buckling analysis was carried out to investigate what load and temperature level caused buckling and elastic instability on sandwich structures. moreover, sandwich deformation nature was investigated by determining the temperature that buckled the sandwich, according to the material properties used for sandwich beam. furthermore, simulation was carried out on two sandwich beams, one with carbon fiber strut in the core and the other without carbon fiber strut. fig. 5 depicts the buckling nature of the sandwich beam without carbon fiber strut. furthermore, the same procedure was applied to observe the buckling nature of the sandwich with imbedded strut as shown in fig. 6. in this case, temperature was not added in static structural analyzer and the results showed improvement in buckling load for the sandwich with carbon fiber circular strut, as shown in table 6. table 6 buckling load sandwich type buckling load (n) sandwich with circular strut imbedded in core 34413 sandwich without circular strut 31017 (a) (b) fig. 5 buckling of sandwich under axial compression (a) isometric view; (b) side view 600 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu (a) (b) (c) fig. 6 (a) cylindrical carbon fiber imbedded in the core (diameter of 1.5 mm); (b) buckling of sandwich under axial compression (isometric view); (c) buckling of sandwich under axial compression (side view) 4.3. buckling analysis with thermal expansion nonlinear buckling was simulated through a static structural analysis in ansys. temperature was raised and sandwich buckling was observed. to begin with, analytical settings in structural run were set and large deflection was turned on. in addition, boundary was assigned, x movements on both sides were constraint to confine the sandwich beam during thermal expansion in order to observe compressive stresses along x-direction. furthermore, sandwich vertex were prevented from moving along z-direction and finally the left and right sandwich beam edges were restricted from moving vertically. in general, sandwich beam could not expand thermally along x-direction, and expanded only along y and z directions due to the constraints; it was not free to translate and rotate globally. 4.3.1. load environment temperature of 70 °f was assigned to sandwich beam in static structural analysis and the thermal condition of sandwich beam was 100 °f. hence, in this analysis, there was a 30 °f temperature increase that caused thermal expansion. therefore, temperature that caused state of stresses on the sandwich beam were noted. hence, fig. 7(a) shows the first mode shape, the sandwich beam were quizzed due to thermal load and not allowed to move globally which caused it to buckle in the middle. however, as there was 30 °f difference which was the only load on sandwich beam and was multiplied by load multiplier of the first mode in eigenvalue buckling module and summed with 70°f environmental temperature assigned early which resulted to the buckling temperature of 140.47°f. the critical buckling bending analysis of sandwich structure under axial compression and thermal... 601 same analysis was repeated for sandwich structures with aerogel layers, in order to investigate effect of aerogel layers and the corresponding bucking temperature. fig. 7 shows the 3 modes of sandwich beam without silica aerogel. moreover, the first mode was considered the most. fig. 8 shows aerogel position in the sandwich beam and first 2 mode shapes of sandwich beam with silica aerogel material. table 7 summarizes the temperature causing buckling on both sandwich beams with and without aerogel layers. there was 6% increase in temperature that could buckle the sandwich beam when silica aerogel was used. (a) (b) (c) fig. 7 mode shapes of sandwich beam without silica aerogel (a) mode 1; (b) mode 2; and (c) mode 3 602 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu (a) (b) (c) fig. 8 mode shapes of sandwich beam with silica aerogel (a) sandwich beam with silica aerogel addition; (b) mode 1; (c) mode 2 table 7 buckling temperature sandwich type buckling temperature (°f) sandwich with silica aerogel 149.44 sandwich without silica aerogel 140.47 4.4. bending of sandwich to evaluate both the ductility and soundness of a material, sandwich structure was treated as cantilever beam with one end fixed and point load of 100 n at the other end. hence, deformation occurred and stress was observed and analysis was carried out on the three components of sandwich, top skin, core and bottom skin. therefore, it was observed that the maximum deformation and stress were on the top face sheet as depicted in fig. 9(a). also, fig. 9(b) shows deformation on sandwich core while fig. 9(c) depicts the bottom skin deformation where both the stress and deformation are at minimum. hence, with the same boundary and loading conditions, in order to improve the flexural stiffness, rectangular shaped carbon fiber was added to the core. moreover, fig. 10(a) and 10(b) depicts the detailed and overlay views of rectangular carbon fiber on sandwich core, respectively. fig. 10(c) shows the stresses absorbed by carbon fiber rectangular strut. in addition, fig. 11 illustrates the total deformation of the sandwich beam under different applied forces. as a results of that, the sandwich beam with imbedded rectangular shaped carbon fiber has less deformation when compared to sandwich beam without carbon fiber. hence, for sandwich beam deformation, fig. 12 shows the effect of core thickness on deformation. core thickness was increased by 0.2 mm which significantly reduced deformation. therefore, there was a good correlation between core thickness and deformation; as core thickness was increased, deformation was decreased. critical buckling bending analysis of sandwich structure under axial compression and thermal... 603 (a) (b) (c) fig. 9 deformation of sandwich (a) deformation at top skin; (b) deformation at core; (c) deformation at bottom skin (a) (b) (c) fig. 10 deformation of sandwich with reinforced carbon fiber (a) sandwich view 1; (b) sandwich view 2; (c) stresses acting on rectangular carbon fiber 604 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu fig. 11 applied force vs. total deformation of sandwich beam fig. 12 core thickness (mm) vs. total deformation (mm) 4.5. conduction thermal analysis of sandwich beam and stresses in order to investigate heat transfer, sandwich structure was made of more than one material, thermal loads and effective factors of the sandwich were observed. moreover, a change in temperature produced thermal strains which could cause sandwich deformation. the analysis was conducted under steady state thermal module in ansys and the solution was imported into static structural module for the observation of thermal strain and mechanical stress. temperature of 427 °c was applied to top skin and 22 °c to bottom skin. it was assumed that the sandwich beam was used as cover structure which emitted high temperature. the effects of temperature increase from bottom face sheet to top face sheet (t = 22°c to 427 °c) for the sandwich of carbon fiber-polyimide face skin and carbon foam core are shown in fig. 13. at high temperatures, thermal stress was increased from minimum to maximum causing expansion on the top face, as shown in fig. 14. hence, there critical buckling bending analysis of sandwich structure under axial compression and thermal... 605 was some deformation in bottom skin (contraction), as shown in fig. 15. similarly, silica aerogel with the thickness of 0.5 mm was added as shown in fig. 16. fig. 17 shows the corresponding thermal strain nature of the sandwich beam. as a result of considering the heat conduction of the sandwich, based on review [116], carbon fiber-polyimide and carbon foam materials were used. fig. 18(a) shows the deformation of aerogel layer on the hot side while fig. 18(b) depicts the thermal deformation of aerogel material only bonded between top skin and core. in addition, fig. 18(c) shows thermal strain on the bottom skin of the sandwich beam. it was found that thermal strain was minimal on the sandwich beam due to the addition of silica aerogel layers and the top skin did not deform so much as compared when silica aerogel was not added. in general, it was observed that there was significant decrease in heat flux when silica aerogel was added, as depicted in fig. 19. sandwich beam was subjected to high temperature ranging from 100 °c to 430 °c and the corresponding heat flux was recorded for sandwich beams with and without aerogel material. the results showed that the sandwich beam with silica aerogel had significant heat transfer resistance, as shown in fig. 19. similarly, simulation was repeated, in this case thermal deformation was considered and results showed that sandwich beam with aerogel layers had good thermal strain resistance, as depicted in fig. 20. however, thermal deformation was decreased by 17.6% when silica aerogel was added as layer. in this analysis, 0.5 mm aerogel layer improved thermal strain by about 17.6%. fig. 13 temperature distribution across the top and bottom skins, temperature increasing from t = 27 °c to 427 °c fig. 14 sandwich structure showing deformation on the top face at 427 °c (carbon foam core used) 606 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu fig. 15 sandwich structure showing deformation on the bottom face at 427 °c fig. 16 sandwich structure with silica aerogel layer (a) (b) (c) fig. 17 thermal strain on (a) top skin; (b) core; and (c) bottom skin (scale 0.5x auto) critical buckling bending analysis of sandwich structure under axial compression and thermal... 607 (a) (b) (c) fig. 18 sandwich with aerogel layer. (a) sandwich structure showing deformation on the top face at 427 °c (carbon foam core and aerogel layer used); (b) thermal strain on top aerogel layer; (c) thermal strain on bottom skin fig. 19 change in temperature vs. heat flux of sandwich 608 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu fig. 20 change in temperature vs. thermal deformation of sandwich beam 4.6. analysis and comparison the experimental approaches developed by goswami [120] were simulated and modified. goswami used aramid sheet for skin and aramid hexagonal honeycomb core, filled with silica aerogel. the setup was separated into two parts: heating system (part a) where the heater was installed on the top face sheet of the sandwich with thermocouple arrangements while only the thermocouple was installed on the bottom face sheet (part b). temperature increase stopped at 350 ºc because the manufacturer specified that the flash point of silica aerogel has to be 395 ºc. fig. 21 shows heat flux on honeycomb sandwich panel in three cases; 1) honeycomb filled with silica aerogel. 2) honeycomb with silica aerogel as auxiliary face sheet. 3) honeycomb without silica aerogel. the results showed that the honeycomb with silica aerogel as an auxiliary face sheet had lower heat flux, thereby providing good heat insulation in the honeycomb sandwich beam. in ansys workbench, a thermal analysis experiment was simulated and the results were validated. moreover, the honeycomb filled with silica aerogel as shown in fig. 22(a) has higher temperature difference (the heat will take longer to transfer from one side to the other) than the one without silica aerogel as depicted in fig. 22(b). furthermore, the methods used by goswami in the experiment was by no means the most effective approach towards providing heat insulation in the honeycomb sandwich as there was much heat flow in honeycomb cell walls, as shown in fig. 22(c). in general, silica aerogel was used as an auxiliary face sheet before the honeycomb hexagonal core as depicted in fig. 22(d). therefore, heat flow was minimal and evenly distributed through the honeycomb core, as shown in fig. 22(e). critical buckling bending analysis of sandwich structure under axial compression and thermal... 609 fig. 21 time vs. heat flux of sandwich panel (a) (b) (c) (d) (e) fig. 22 honeycomb sandwich panel: (a) honeycomb filled with silica aerogel; (b) honeycomb without silica aerogel; (c) heat flow in honeycomb cell walls on honeycomb filled with silica aerogel; (d) silica aerogel as auxiliary face sheet; (e) reduced heat flow on core cell walls (silica aerogel as auxiliary face sheet) 610 b. safaei, e. chukwueloka onyibo, d. hurdoganoglu 5. conclusion the findings of a detailed study and simulation were shown and plotted and important observations were recorded. thermal stresses and buckling temperature were improved by the addition of silica aerogel material. steady state condition of the sandwich beam was investigated in commercial software ansys and the temperature induced stress and the nature of thermal expansion-induced deformation were noted. hence, the parameters affecting thermal strain and heat flow were mechanical properties and the most significant one was thermal conductivity, as we considered the heat conduction of the sandwich beam (heat flux). in addition, experimental approach was simulated and modified. filling hexagonal honeycomb cells with silica aerogel was not the most effective way of heat reduction in the sandwich but it could be used as an auxiliary face sheet. moreover, carbon foam with lower thermal conductivity was further improved by the addition of silica aerogel layer which was the lightest and hence had the mass of 1.22 g and extreme heat flow resistance. as a result of flexural strength, the sandwich was imbedded with rectangular shaped and cylindrical bar carbon fiber strut in the core, thereby improving stiffness and buckling load. addition of carbon fiber strut was effective in 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there are different force transmission elements, which consist completely or partly of rubber. rubber parts or, generally, elastomers usually have amplitude-dependant and frequency-dependent force transmission properties. rheological models are used to describe these properties. a method for characterization of the amplitude and frequency dependence of such a rheological model is presented within this paper. within this method, the used rheological model can be reduced or expanded in order to illustrate various non-linear effects. an original result is given with the automated parameter identification. it is fully implemented in matlab. such identified rheological models are intended for subsequent implementation in a multi-body model. this allows a significant enhancement of the overall model quality. key words: experiment, rubber, hysteresis, nonlinear viscoelastic behavior, frequency, amplitude 1. introduction within the current studies, it is important to characterize nonlinear behavior of rubber parts or generally elastomers in view of the fact that they and their impact gain more and more in importance in modern simulation models. for this reason, nvh-researchers from various technical application fields deal with the description and characterization of rubber received may 05, 2014 / accepted july 10, 2014 corresponding author: christopher heine technical university berlin, germany; bsh bosch and siemens home appliances gmbh, wohlrabedamm15, 13629 berlin, germany e-mail: christopher.heine@bshg.com original scientific paper c. heine, m. plagemann 138 parts and their properties. puel et al. [5] have dealt with a linear viscoelastic rubber model with non-linear time dependence with the aim of integrating it into a multi one. a common method [2, 4, 6] for describing rubber parts implies the division of their properties into three parts, briefly described in the following. the first part deals with non-linear dependence of the force response on the displacement amplitude the amplitude response. the second part includes the force response dependence on the excitation frequency the frequency dependence. the third part comprises modeling the temperature dependence. in the present work, the temperature dependence is not taken into account. the two first types of dependence are generally shown in fig. 1. in order to describe these curves, rheological models are established. these models consist of different ways of coupling the basic elements of mechanics, e.g. spring, damper and friction elements. the amplitude dependence represents the static or quasi-static dependence of the force response on the excitation amplitude. these static hystereses are measured under quasistatic excitation with the excitation frequency ≤ 0.03 hz [6]. generally, elastomers behave increasingly stiffer for smaller amplitudes [6]. the hystereses for smaller amplitudes are applied to the reversal points of the larger angle amplitudes. with regard to the frequency dependence, the dynamic hysteresis becomes stiffer with increasing frequency. all graphs in this work were normalized and thus presented without units. fig. 1 representation of the amplitude dependence (left, internal test) and the frequency dependence (right, [7]) this splitting of the dependencies is also performed in the work of lou et al. [4] and jrad et al. [2]. in both works, rheological models are used to describe amplitude and frequency-dependence of the respective rubber parts. these tested rubber parts have shown only linear viscoelastic properties. the models are characterized on the basis of measurements. this paper is about rubber parts which are mounted inside of a washing appliance. the characteristics of these components will be characterized for use within a multi-body model. for this, the properties are split into amplitude and frequency dependence, which are individually detected by measurements, and are described below by means of rheological models. as a differentiation relative to the cited works, a special feature in this work is the occurrence of non-linear viscoelastic behavior, which is also characterized by means of advanced rheological models. this is an original feature of the conducted research. modeling of nonlinear viscoelastic material behavior 139 2. motivation the reason for this work is to optimize the multi-body predictive modeling performance. this has been undertaken by characterizing the force elements which are included in a model of a front loading washing machine. fig. 2 depicts a front loading washing machine. the individual parts are numbered. the washing machine can be divided in two assemblies. the first assembly is the oscillating system and the second is the housing. the front panel (fig. 2, no. 1; following part assignments refer to fig. 2, too), the side and back panel (no. 10) and the feet (no. 11) are parts of the housing. the drum (no.3), the drum support (no. 4), the pulley (no. 5), the tub (no. 7), the electrical drive (no. 12), the front counterweight (no. 6) and the top counterweight (no. 9) are parts of the oscillating system. the oscillating system is hooked with the aid of the force transmission elements in the housing. these force transmission elements are the springs (no. 8), the damper (no. 13) and the gasket (no. 2). fig. 2 construction of a washing machine in this work, models for characterizing the damper and their connections are to be found. inside the damper connections are small rubber parts to effect an elastic connection. these connections are to be measured and subsequently characterized. the measurements of these connections are divided with respect to the two above named properties of the elastomers (amplitudeand frequency dependence). both measurements are performed with an identical test setup. the using test setup is shown in fig. 3 as an example. on the right side of the fig. 2 the excitation (fig. 3, e) is carried out with varying amplitudes or different frequencies. using a displacement (fig. 3, ds) and a force sensor (fig. 3, fs), the response of the component to be tested is detected. the results of these measurements are shown in the next section. c. heine, m. plagemann 140 fig. 3 illustration of the test setup 3. rheological model for representing the amplitude dependence within the quasi-static measurements, two cases of the amplitude dependence occur. the first type of hysteresis is shown in fig. 4. all the given curves emanate from a series of characterization tests on a rubber specimen. angle m om en t fig. 4 hysteresis curves for different excitation amplitudes (type i) on the other hand, static hystereses are measured, where the hysteresis demonstrates a constriction or a decrease of force magnitudes in the width toward the zero crossing. these are shown exemplarily in fig. 5. modeling of nonlinear viscoelastic material behavior 141 angle m om en t fig. 5 constriction of the trapezoidal hysteresis curves for different amplitudes (type ii) as a basis for simulating the amplitude dependence, the so-called prandtlelement is used. this element consists of a serial arrangement of a spring and a friction element. the resultant force from this element is determined by using the following equation explained in beerens [1], where x represents the displacement, k is the stiffness of the spring, r represents the frictional force of the friction element and fpr describes the resulting force of the prandtl-element. 2 2 2 2 pr pr pr pr 1 ( )[1 ( ) ( ( ))(1 ( ))] 2 f kx t sign f r sign f x t sign f r= − − − + −& & & (1) the characteristic hysteresis of a prandtl-element is shown in fig. 6. -1 -0.5 0 0.5 1-1 -0.5 0 0.5 1 displacement fo rc e fig. 6 hysteresis of a prandtl-element c. heine, m. plagemann 142 in order to display measured hysteresis curves n prandtl-elements are connected in parallel. in addition to the n prandtl-elements a non-linear spring is connected in parallel. this results in the so-called masing model whose circuit diagram is shown in fig. 7. in the paper by beerens [1], the masing model is discussed extensively. fig. 7 masing model with n prandtl-elements and a non-linear spring for the description of the nonlinear spring force a modified third-order polynomial is used, in which the square shares are neglected. (2) xbaxff += 3 to make the number of parameter of the masing model manageable, two further simplifications are made. the first simplification indicates that the stiffnesses of the springs inside the prandtl-elements are identical. rrnrr kkkk ==== ...21 (3) the second simplification is concerned with the frictional forces of the friction elements of the prandtl-elements. these frictional forces are distributed with an exponential function. with αη−α= eri 1 1 − − = n iη with i = 1,…, n and [0,1]η∈ (4) as a result of both simplifications and the description of the non-linear spring via the said polynomial approach the used masing model is completely described by five parameters. these five parameters comprise parameters a and b for the description of the polynomial, k is the stiffness of the prandtl-elements, the number n of prandtlelements and parameter α to describe the distribution function of the frictional forces. the number n of prandtl-elements controls the smoothness of the modeled hysteresis and is set to seven within this work, identified by a short preliminary study. modeling of nonlinear viscoelastic material behavior 143 4. rheological model for representing the frequency dependence this section is to explain the frequency dependence. in order to display frequency dependence the so-called maxwell-element is used. this element consists of a serial connection of a spring and a damper. the resulting force of this element can be calculated by using the following equation. ( ) ( ) ( )max max cf t cx t f t d = −& & (5) in the present case, the frequency dependence is to be imaged from zero to 20 hz. pohl and wahle [6] have demonstrated that the efficiency of a maxwell-element is limited in the damping characteristics to a narrow frequency band of about two frequency decades. for this reason, it is sufficient for the description here to use two maxwellelements connecting in parallel to the masing model. the complete model is shown in fig. 8. a similar model has also been used in the work of pohl and wahle [6]. fig. 8 the complete model used 5. results of amplitude dependence for the calculation of the model parameters to be used in a matlab script has been programmed. with the help of this script, an optimal parameter set for the assigned model is computed completely automatically. within the measurement ten measurements with different excitation amplitude are carried out for each rubber part. in the first calculation step, the model parameters are individually determined for each amplitude level. the resulting data for various amplitudes are shown for four different amplitude levels in fig. 9. it is clearly seen that the used model can represent different amplitude levels very well. c. heine, m. plagemann 144 -0.05 0 0.05 -0.25 0 025 angle m om en t -0.25 0 0.25 -0,5 0 0,5 angle m om en t -0.5 0 0.5 -0.5 0 0.5 angle m om en t -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 angle m om en t measured replicated measured replicated measured replicated measured replicated fig. 9 simulated hysteresis for different amplitude levels within a second step, the calculated values of the model parameters have been modeled as a function of the excitation amplitude. the functions to be used to describe the amplitude dependence of the model parameters could be reduced to three functions. it is a polynomial of degree 4 and degree 1 as well as the exponential function. the four model parameters and their characteristic functions of the excitation amplitude are shown in fig. 10. amplitude va lu e a amplitude va lu e k prandtl amplitude va lu e 0.2 0.4 0.6 b amplitude va lu e calculated replicated calculated replicated calculated replicated calculated replicated α fig. 10 model parameters as a function of the amplitude what follows is a special case of the constricted hysteresis. the total force of the prandtl-elements must be reduced down to the zero crossing of the path excitation. to realize this, the resultant force of the seven prandtl-elements is multiplied with a scaling function which depends on the path excitation. pr (| tanh( ) |)( ) (1 )f px ksk x k + = + (6) modeling of nonlinear viscoelastic material behavior 145 the used scaling function enhances the complete model with two additional parameters. on the one hand parameter p which describes the slope of the hyperbolic tangent and on the other hand parameter k which describes the minimum scaling at the zero crossing. an example of the profile of this scaling function is shown in fig. 11. -1 -0.5 0 0.5 10 0.2 0.4 0.6 0.8 1 displacement sc al in g fa ct or fig. 11 example of the profile of the used scaling function this model with the described scaling function is implemented in matlab. an optimal set of parameters is calculated by using matlab for the model with implemented scaling function. fig. 12 shows two different amplitude levels. on the left side of the figure, the hysteresis can be reproduced without using the scaling function. however, on the right side, the scaling function is used. it is clearly seen that the use of the scaling function leads to a high precision representation. -0.5 0 0.5 -0.5 0 0.5 without scaling displacement fo rc e -0.5 0 0.5 -0.5 0 0.5 with scaling displacement fo rc e -1 0 1 -1 0-1 -0.5 0 0.5 1 displacement fo rc e -1 0 1 -1 0-1 -0.5 0 0.5 1 displacement fo rc e measured replicated measured replicated measured replicated measured replicated fig. 12 hysteresis with and without scaling function using the model with and without scaling function the amplitude dependence of the present elastomer could be clearly described and modeled. c. heine, m. plagemann 146 6. results of frequency dependence a matlab script has been programmed for the second step, which now calculates the parameters for the frequency dependence. the calculation of the maxwell parameters can be performed by determining the loss and storage stiffness as described in the work of lion [3]. it is not possible to use this method to calculate the maxwell parameters. the reason for this lies in the possibilities of measuring technology detection of the component properties. therefore, parameters c1, c2, d1 and d2 need to be calculated individually. what results from this matlab calculation is hysteresis for different excitation frequencies shown in fig. 13. -1 -0.5 0 0.5 1 -1 0 1 3.5 hz angle m om en t -1 -0.5 0 0.5 1 -1 0 1 8 hz angle m om en t -1 -0.5 0 0.5 1 -1 0 1 10.5 hz angle m om en t -1 -0.5 0 0.5 1 -1 0 1 16.5 hz angle m om en t measured replicated measured replicated measured replicated measured replicated fig. 13 measured (straight) and simulated (dotted) hysteresis for different excitation frequencies the calculation of the model parameters for describing the frequency dependence is carried out for a frequency range of two to 20hz with a step size of 0,5hz. from this calculation, the function of the model parameters of the excitation frequency could be calculated in a subsequent step. these dependencies could be described by linear functions and constant model parameters. the functions used to describe the frequency dependence of the model parameters are shown in fig. 14. 7. conclusion in this work, the material properties of elastomers in terms of amplitude and frequency dependence have been measured. then rheological mathematical models are identified to describe the dynamic behavior. in order to maximize the representation quality and to decrease modeling time a matlab script is programmed. this matlab script contain the possibility to expand or simplify the model to be calculated. by this fact various components could be described with the aid of this script. modeling of nonlinear viscoelastic material behavior 147 c1 frequency va lu e c2 frequency va lu e d1 frequency va lu e d2 frequency va lu e calculated replicated calculated replicated calculated replicated calculated replicated fig. 14 representation of the frequency dependence of the model parameters the simulated hysteresis describes the measured hysteresis in terms of amplitude and frequency dependence. the identified model describes the material behavior of the measured elastomer within the work area completely automatically. then the described models are successfully implemented in a multi-body simulation, leading to an improvement of the imaging quality of the multi-body simulation. references 1. beerens, c., 1994, zur modellierung nichtlinearer dämpfungsphänomene in der strukturmechanik, mitteiliungen aus dem institut fuer mechanik, ruhr-universität bochum 2. jrad, h., renaud, f., dion j.l., tawfiq, i., haddar, m., 2013, experimental characterization, modeling and parametric identification of hysteresis friction behavior of viscoelastic joints, international journal of applied mechanics, 5(2), pp. 1350018-1-21 3. lion, a., 2005, phenomenological modelling of strain.induced structural changes in filler-reinforced elastomers, kgk kautschuk gummi kunststoffe, 58(2), pp.157-162 4. lou, w., hu, x., wang, c., li, q., 2010, frequencyand strain-amplitude-dependent dynamical mechanical properties and hysteresis loss of cb-filled vulcanized natural rubber, international journal of mechanical sciences, 52(2), pp. 168-174 5. puel, g., bourgeteau, b., aubry, d., 2013, parameter identification of nonlinear time-dependent rubber bushings models towards their integration in multibody simulations of a vehicle chassis, mechanical systems and signal processing, 36(2), pp. 354-369. 6. pohl, r., wahle, m., 1999, entwicklung eines rechenmodells zur beschreibung von gummibauteilen bei statischer und dynamischer belastung (simulationsmodell für gummibauteile): schlußbericht zum bmbf-forschungsvorhaben, fachhochschule aachen 7. waltz, m., 2005, dynamisches verhalten von gummigefederten eisenbahnrädern, phd dissertation, fachhochschule aachen c. heine, m. plagemann 148 opšti pristup modeliranju nelinearnih histerezisnih efekata zavisnih od amplitude i frekvencije na osnovu eksperimenata sa gumenim delovima detaljni opis osobina gumenih delova dobija na značaju u sadašnjim modelima simulacije simulacija sa više tela. jedan primer primene je multi body simulacija kretanja mašine za pranje. unutar mašine za pranje javljaju se različiti elementi prenosa sile koji su potpuno ili delimično gumeni. gumeni delovi ili, uopšte, elastomeri, obično imaju svojstva prenosa sile koji su zavisni od amplitude i frekvencije. reološki modeli se koriste za opis ovih svojstava. jedna metoda karakterizacije zavisnosti amplitude i frekvencije takvog reološkog modela je predstavljena u ovom radu. po toj metodi, korišćeni reološki model se može svesti ili proširiti radi ilustracije raznih nelinearnih efekata. originalan rezultat je dat sa automatizovanom identifikacijom parametra. u potpunosti je primenjen u metlabu. takvi identifikovani reološki modeli namenjeni su potonjoj implementaciji u modelu sa više tela. to nam dopušta značajno pojačanje ukupnog kvaliteta modela.. ključne reči: eksperiment, guma, histereta, nelinearno visko-elastično ponašanje, frekvencija, amplituda a general approach to modeling nonlinear amplitude and frequency dependant hysteresis effects based on experiments with rubber parts christopher heine1,2, markus plagemann2 1. introduction 2. motivation 3. rheological model for representing the amplitude dependence 4. rheological model for representing the frequency dependence 5. results of amplitude dependence 6. results of frequency dependence 7. conclusion references facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 425 443 https://doi.org/10.22190/fume181204004s © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic emission kannivel saravanakumar 1 , balakrishnan sai lakshminarayanan 1 , vellayaraj arumugam 1 , carlo santulli 2 , ana pavlovic 3 , cristiano fragassa 3 1department of aerospace engineering, mit campus, chromepet, anna university, india 2school of architecture and design, università of camerino, ascoli piceno, italy 3department of industrial engineering, university of bologna, bologna, italy abstract. this paper aims at investigating the influence of the addition of milled glass fibers upon quasi-static indentation (qsi) properties of glass/epoxy composite laminates. the qsi behavior was experimentally studied by evaluating indentation force, residual dent depth, energy absorbed and size of the damaged area for different indentation depths. following the qsi tests, the filler-loaded glass/epoxy samples were subjected to three-point bending tests in order to measure residual flexural strength, and the results were compared with the baseline glass/epoxy samples. both tests were performed with online acoustic emission monitoring in order to observe damage progression and characterize different fracture mechanisms associated with loading. the results show that the filler-loaded laminates exhibit a substantial improvement in the peak force and contact stiffness, with a reduced permanent damage both in terms of depth and of area, in comparison with the baseline ones. it is found that the filler presence offers greater stiffness and higher energy dissipation through toughening mechanisms such as filler debonding/pullout and filler bridging/interlocking. key words: glass/epoxy, delamination, quasi-static indentation (qsi), residual flexural strength, acoustic emission received december 04, 2018 / accepted february 03, 2019 corresponding author: cristiano fragassa department of industrial engineering, alma mater studiorum university of bologna, viale risorgimento 2, 40136, bologna, italy e-mail: cristiano.fragassa@unibo.it 426 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. 1. introduction fiber-reinforced composites, made of glass, carbon or even natural fibers, have found their application in several industries such as aerospace, automobile, marine, wind turbines production, etc., due to their high specific stiffness/strength, chemical resistance and fatigue properties, which makes them a suitable alternative for metals [1-3]. on the other side, the composites are susceptible to delamination due to poor mechanical properties through their thickness [4]. this effect is evident, in particular, when adhesion between the fibers and the matrixes is not initially perfect or has deteriorated with use [5]. especially in the case of low velocity impacts [6] due to external objects, tool drop during service/maintenance, runway debris and other accidental events, etc., localized damage can result in drastic reduction in strength/stiffness and is likely to expand during service. it is noticed that the damage induced in low-velocity impact can be effectively simulated in quasi-static indentation tests regarding their advantage in providing longer times, hence enabling damage evolution monitoring [7, 8]. in practice, the quasi-static indentation tests supply information about contact behavior between the sample and the indenter as well as on the occurrence of sequential damage with varying indenter displacement/depth. abdallah and bouver [9] experimentally investigated the damage behavior and effects of permanent indentation on highly oriented composites plates. they observed that the peak force experienced during low velocity impact was higher than in the case of the quasi-static test. however, damage morphology and absorbed energy were equivalent for both tests. various works [10, 11] have established correlations between quasi-static indentation and dynamic falling weight impact tests. the damage initiation and propagation were investigated comprehensively by controlling peak force and deformation. the parameters, such as peak load or ultimate load, incident energy, absorbed energy, elastic energy and residual depth, were determined in order to quantify the local damage in the composite materials during quasi-static indentation tests [12, 13]. arabzadeh and zeinoiddini [14] studied quasi-static indentation response of flexibly supported pressurized pipes, suggesting a closed-form relationship between indentation force and dent depth by considering different boundary conditions, such as internal pressure soil stiffness and embedment between soil and pipe into their modeling. they observed that the influence of the surrounding soil was prominent when the fluid pressure inside the pipe is very low. sutherland and guedes soares [15] investigated the quasi-static indentation behavior of e-glass/polyester marine laminates observing that the chopped strand mat laminates exhibited better contact stiffness than the woven roving ones. they also report that the large global deflection in thinner samples leads to a larger contact area due to the wrapping of laminate around the indenter. in quasi-static punch shear tests on quasi-isotropic carbon/epoxy, a good correlation between load-displacement response and finite element modeling was reported, indicating that a slope change or a load drop indicates delamination initiation, which propagates through subsequent oscillations: further damage was produced by plug formation exit from the plate [16]. the progressive penetration mechanism of ultra-high molecular weight polyethylene reinforced cross-ply composite laminates was investigated by o’masta et al. [17]. they observed that sample penetration occurred by tensile ply rupture under the projectile, and higher penetration resistance and onset velocity occurred as the consequence of the sample being end-supported rather than back-supported. the effect of hybridization on quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 427 impact and post-impact performance of the composite laminates was investigated by suresh kumar et al. [18]. the low-velocity impact behavior was simulated by quasistatic indentation (qsi) tests on quasi-isotropic glass/epoxy, glass/basalt/epoxy (g/b/g, b/g/b) and glass/carbon/epoxy (g/c/g, c/g/c) laminates with acoustic emission monitoring. addition of basalt fiber and carbon fiber to glass fiber improved indentation damage resistance, while ae monitoring was reported to be a sensitive method to characterize damage evolution in the laminates. the conventional composite laminates fabricated with thermoset matrix, such as epoxy, suffer from low impact damage resistance, poor fiber/matrix interface bond strength, low fracture toughness, and poor transverse mechanical properties. their delamination resistance can be enhanced by incorporating micro/nano-sized fillers into the matrix. toughening mechanisms, such as cavitation, crack pinning, crack deflection, and crack bridging were observed to have improved interlaminar fracture toughness of the composites [19-21]. acoustic emission monitoring can be effectively used for monitoring/tracking the damage evolution and for identifying/characterizing failure modes in the fiberreinforced composites laminates during loading [22-24]. each acquired ae signal bears some relation with the damage mechanisms, in that it is associated with the specific amount of strain energy released during failure. it has been reported that the failure modes can be identified based on the signal-based approach utilizing ae waveforms, fast fourier transform (fft) and short time fast fourier transform (stfft), while the parametric-based approach uses ae parameters, such as counts, energy, rise time, rms, signal strength and duration, etc., [25, 26]. bussiba et al. [27] employed stfft analysis to discriminate different failure modes and studied sequential damage evolution in composite laminates. ramirez-jimenez et al. [28] discriminated damage mechanisms, such as matrix crack, debonding, delamination, and fiber breakage based on the peak frequency analysis. arumugam et al. [29] investigated the classification of failure modes for different ply orientation sequences, based on the frequency content of ae signals. this work focuses on investigating the effect of introducing a limited amount of milled glass fiber fillers upon the quasi static indentation behavior and residual performance of glass/epoxy laminates. the damage initiation, progression, and failure mechanism associated with quasi-static indentation and three-point bending test were also discussed with online ae monitoring. the glass/epoxy samples were subjected to quasi-static indentation test at different indentation depths (1 to 6 mm). the indentation test parameters in terms of peak force and absorbed energy, as well as residual (permanent) dent depth and damage area were evaluated, and the results were correlated with those from the baseline samples. the residual strength of the samples was also estimated in order to determine damage tolerance of the composite laminates. 2. experimental procedure 2.1. materials and fabrication of composite laminates the glass/epoxy composite laminates were fabricated by hand lay-up technique with a cross-ply stacking sequence of [0º/90º]4s configuration. unidirectional 220 g/m² glass fabric and ly556 epoxy resin with hy951 hardener were used as raw materials and taken in the ratio of 1:1 by weight for fabricating the laminates. milled glass fibers were 428 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. added to the epoxy resin (in a ratio of 5:100 by weight) through sonication and mechanical stirring to distribute them uniformly in the resin. the mixture was then degassed in order to remove entrapped air bubbles. the hardener was added to the mixture at a ratio of 1:10 by weight and further stirred to initiate the curing process. the mixture was then evenly distributed on the glass fabric with the aid of brush and roller to improve fiber impregnation. correspondingly, baseline glass/epoxy laminates without milled glass fibers were fabricated as above. the laminates, with dimensions of 500 x 500 mm, and nominal thickness of 4.5 (±0.25) mm, were allowed to cure at room temperature for 24 hours; then the samples were cut from them using abrasive water-jet cutting machine. also filler-loaded samples had the same objective thickness of 4.5 mm, and their weight was in excess with respect to the baseline ones by no more than around 2%, therefore basically included in the experimental error. 2.2. quasi-static indentation test quasi-static indentation tests were performed on the tinius olsen 100kn universal testing machine at crosshead speed of 1 mm/min. the test was carried out with the indentation fixture as per astm d6264/d6264m-17 standard [30], hence with a hemispherical indenter. the glass/epoxy samples with dimensions of 150 x 100 mm rested on the fixture and were clamped at both sides, as shown in fig. 1. later, the indentation test was performed directly on the center of the samples. the specimen was indented with a 12.7 mm diameter hemispherical end steel tup. load-displacement data were recorded via the digital data acquisition system from the universal testing machine. four specimens were tested in all the cases, and the average results were considered. qsi tests were carried out at a predetermined indentation depth of 1, 2, 3, 4, 5 and 6 mm, respectively. quasi-static indentation behavior and test parameters, such as indentation force, residual dent depth, energy absorbed, contact stiffness and damaged area at different indentation depths were evaluated. the evolution of damage and damage mechanism associated with indentation were monitored with online ae monitoring. fig. 1 quasi-static indentation test fixture quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 429 2.3. three-point bending test post-indentation flexural tests were performed on glass/epoxy samples, trimmed to a dimension of 150x50 mm using a diamond saw, with three-point bending fixture under displacement control regime. care was taken to ensure not to damage the indented zone during cutting, according to previous indications supplied by [31-33]. the tests were carried out at a constant cross-head speed of 1 mm/min. the span length was kept equal to 100 mm, and four repetitions were performed for each category of samples. the residual flexural strength was determined from the test, and the results were correlated with the non-indented baseline and filler-loaded samples. 2.4. acoustic emission monitoring of qsi and fai acoustic emission monitoring was employed during quasi-static indentation tests and flexural after indentation tests. an eight-channel ae system supplied by the physical acoustic corporation (pac) (princeton, nj, usa) with a sampling rate of 3mhz and a 40 db pre-amplification was used. a threshold of 45 db was fixed for filtering the ambient noise. two wideband (wd) sensors in a linear arrangement were employed for ae measurements, and these sensors were attached at a nominal distance of 100 mm along the sample length. high vacuum silicon grease was used as a coupling agent between the sensors and the sample surface. the wave velocity measurements and subsequent calibration of the sensors were performed by the typical pencil lead break test. the average wave velocity for both baseline and filler-loaded glass/epoxy samples were found to be 3120 m/s. the peak definition time (pdt), hit definition time (hdt) and hit lockout time (hlt) were set to be 30 µs, 150 µs, and 300 µs, respectively. 3. results and discussion 3.1. quasi-static indentation test quasi-static indentation tests facilitate investigating the contact behavior between the glass/epoxy laminates and the indenter during loading, as well as monitoring the occurrence of damage sequentially by varying indenter displacement/depth. usually, the onset of the damage during transverse loading of the composite laminates occurs by: (i) matrix cracking at the local indenter contact point; (ii) debonding between the fiber/matrix interfaces due to transverse matrix cracks; (iii) fiber buckling at the contact point on the compression side; (iv) delamination; and (v) fiber breakage on the tensile side, due to penetration/perforation [15]. the resulting dent force, energy absorbed (ea), residual dent and size of the damaged area were determined in terms of indentation depth. moreover, the residual strength of the laminates subjected to quasi-static indentation was evaluated to ensure integrity and damage tolerance of the laminates. 430 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. fig. 2 load-displacement curve for different indentation depth: (a) baseline samples (b) filler-loaded samples figure 2 shows a typical load-displacement curve for the quasi-static indentation test on both baseline and filler-loaded glass/epoxy samples, tested at different indentation depths. the curve profile was initially quasi-linear, which suggested a prevalently elastic behavior: this was followed by the onset of some permanent plastic deformation with increasing indentation displacement. the resistance offered by the samples was observed to increase with indentation depth. thus, the peak force increases consistently with indentation depth accompanied by a sequence of load drops associated with the occurrence of a significant damage, such as delamination and fiber failure. at a higher indentation depth, the onset of the process leading from fiber disruption to perforation through the appearance of back damage reduces the resistance of the samples through load drops to be ascribed to the indenter producing shear failure and local crushing of fibers during local bending [34]. these load drops are followed by the appearance of a plateau region in the curve, which can be attributed to frictional sliding between the indenter and the sample. in particular, it can be observed that beyond 4 mm no longer any major increase in the peak load is revealed: this was associated with the predominant fiber damage, leading to the appearance of back face damage. also, the results show that irrespective of indentation depth, the filler-loaded samples exhibited higher load carrying capacity (peak force) than the baseline samples. the slope of the load-displacement curve during quasi-elastic phase, preceding the first load drop, defines the initial contact stiffness of the glass/epoxy. further, as the indentation depth increases, non-linear behavior was observed associated with damage accumulation and progression through matrix cracking, while the onset of delamination at a higher delamination depth corresponds to stiffness degradation. more specifically, it was found that the filler-loaded samples exhibit initial contact stiffness of 2181 ± 75 n/mm and delaminated contact stiffness of 955 ± 72 n/mm, against 1902 ± 58 n/mm and 832 ± 75 n/mm for the baseline samples, respectively. in other words, it was observed that the filler-loaded samples exhibited approximately 15% improvement in initial and delaminated contact stiffness, compared to the baseline samples. quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 431 fig. 3 comparison of: (a) peak force (b) absorbed energy between baseline and fillerloaded samples for different indentation depths figures 3 (a) and (b) indicate peak force and absorbed energy variation with respect to indentation depth: both peak force and absorbed energy are slightly higher for the fillerloaded samples; though given the standard deviation, differences are minimal. it can also be observed that while the former shows an abrupt increase at a given indentation depth, particularly between 1 and 2 mm, the latter grows quasi-linearly with it. in general terms, this behavior depends on the fact that the peak force suddenly increases after the contact area overcomes the limit that is related to the dimension of the indenting tup, an evidence which is basically related to its hemispherical geometry [35]. however, at a higher indentation depth, beyond 4 mm, some failure of load-bearing fibers due to induced tensile stress resulted in some reduction in the peak load, which means that the peak loads for both the filler-loaded and the baseline samples were similar. in other words, the influence of the filler on absorbed energy was prominent at a lower indentation depth, while as the indentation depth increases, the contribution of the filler to energy dissipation deteriorates. this was attributed to the damage gradually extending from the matrix to the reinforcement through the intermediate occurrence of some debonding. what was expected was that the filler-loaded samples would exhibit a reduced damaged area and a smaller residual dent at respective indentation depths in comparison with the baseline ones. the relationship between the peak force and the absorbed energy with a residual (permanent) dent, i.e. the plastic deformation of the samples remaining after unloading for both the baseline and the filler-loaded samples is reported in fig. 4. in general, both the absorbed energy and the residual dent are increasing with the indentation force. the relation between the peak force and the residual dent indicates that the accumulation of damage is dependent on indentation displacement. the change in the slope was observed to be small/gradual at a lower indentation depth below 3 mm corresponding to matrix cracking and some delamination damage. as the indentation depth increases, the slope of the curve changes rapidly while the peak force remains almost unchanged. the occurrence of fiber breakage with matrix cracking and delamination contributes to an increase in the residual dent depth and with no raise in the peak force, suggesting at this point the transition of failure mechanism from gradual to severe, substantially indicating the occurrence of fiber failure, basically signifying that no resistance is offered by the broken fibers. it is observed that the filler-loaded samples exhibit a lower permanent depth 432 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. by an average of 25% than the baseline samples. the filler presence offers improved stiffness and also higher energy dissipation through plastic deformation. figure 4 (b) shows the relationship between the permanent dent depth and the absorbed energy for the baseline and the filler-loaded samples. it is well known that the energy absorbed by the samples is utilized for investigating damage development. thus, the responses of the absorbed energy and the residual dent are dependent and the results show that the trend of the absorbed energy and the residual dent was almost linear till an indentation depth below 4 mm. however, as the indentation depth increases further, the damage is close to saturation in the laminate, which causes marked non-linear behavior. irrespective of indentation depth, the residual dent depth induced on the filler-loaded samples was consistent and depended upon the absorbed energy, which, in its turn, was higher with the reduced residual dent comparing to the baseline samples. it was observed that the fillers presence enhanced the energy dissipation capacity of the laminates, which was attributed to the toughening mechanism, such as filler debonding/pullout, filler interlocking/ bridging of cracks, as will be seen in fig. 12. more indications were expected to come from the measurement of the extent of the damaged, hence delaminated, areas, by non-destructive backlight imaging of the damaged samples with imagej software for post-processing: these are reported in fig. 5 for various indentation depths. the delaminated area had irregular shape and perimeter, despite being centered on the point of indentation. the damage area at the back surface was observed to be greater comparing to the front surface, suggesting a larger disruption if the fiber layers with indenter progress in the laminate, significant for the reduction of flexural performance [36]. the damage area was found to be reduced in the filler-loaded samples for an average of 25% less than for the baseline samples. this is due to the higher rigidity/stiffness offered by the filler-loaded samples resulting in energy dissipation through toughening mechanism such as filler debonding/pullout, filler bridging/interlocking, as will be seen in sem images. this proves that the filler inclusion resulted in higher energy absorption with a reduced damage area, contributing to enhanced crashworthiness properties of the laminates. in particular, the effect of the filler introduction on the residual flexural strength after indentation allowed observing that the filler-loaded samples exhibited an average of 18% higher load carrying capacity than the baseline samples (fig. 6). in particular, three-point bending strength of non-indented laminates was 249  6.5 mpa for baseline ones and 284.5  8.3 mpa for filler-loaded ones. bending effects are the cause of shear forces leading to delamination propagation toward a free edge during the three-point bending test. this propagation expands from the former delamination damage developed during quasi-static indentation, followed by a newly generated damage. the damage evolution was observed to be gradual on the samples subjected to a low indentation depth showing matrix cracking, and debonding damage followed up by the ultimate failure by fiber breakage at compression or tension side of the samples. the residual flexural strength was observed to reduce gradually by an average of 15% for an indentation depth above 3 mm. in addition, the samples subjected to higher indentation exhibit penetration/ perforation damage leading to a large delamination area and intensive fiber breakage. quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 433 fig. 4 (a) peak force vs. residual dent (b) absorbed energy vs. residual dent: comparison between baseline and filler-loaded samples for different indentation depths fig. 5 damage area at different indentation depth for baseline and filler-loaded samples: (a) front surface (b) back surface fig. 6 residual flexural strength for baseline and filler-loaded samples at different indentation depths 434 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. 3.2. acoustic emission monitoring of quasi-static indentation test acoustic emission monitoring is widely employed for inspecting and identifying the sequence and the respective extent of damage mechanisms generated in fiber reinforced composites [37-39]. the microscopic failure events are detected during the tests by ae sensors as ae signals and the frequency analysis is employed to discriminate the failure modes in composite materials. each ae signal acquired during the tests belongs, therefore, to specific damage modes with a certain amount of strain energy released. the damage mechanisms such as matrix crack, debonding, delamination and fiber breakage were discriminated, based on the peak frequency – cumulative counts vs. time plot. the frequency analysis was performed on the glass/epoxy laminates subjected to quasi-static indentation and flexural after indentation tests. the load-displacement behavior of the glass/epoxy samples subjected to quasi-static indentation test with acoustic emission monitoring will be discussed in detail, as follows. in general, the ae events initiate after the occurrence of local plastic deformation in the samples. figures 7 and 8 show peak frequency & cumulative counts vs. time plot of corresponding load-displacement behavior for each indentation depth. the curve profile of the ae cumulative counts – time plot signifies the evolution of damage initiation and progression during loading, as previously discussed, e.g. in [24]. it was observed that the profile of the ae cumulative counts curve changes significantly with subsequent damage as loading progresses. initially, the damage starts with lower count rates, so that the ae cumulative counts curve is almost flat while the ae signals can be mostly attributed to matrix cracking. further loading intensifies the progression of matrix cracking within the ply at faster rates, promoting fiber/matrix debonding and fiber breakage, as will be shown in figs. 10 and 11. damage accumulation was indicated by a sudden and abrupt increase in the cumulative counts with a change in the slope associated with a major failure such as delamination. finally, a sharp increase in the cumulative counts with a steep slope corresponds to unstable crack growth, resulting in the ultimate failure of the laminates. also, the ae signals associated with different failure mechanisms were identified sequentially during the damage evolution from the peak frequency vs. time plots. it is suggested from previous literature that the peak frequency ranges in the gfrp correspond to different damage mechanisms: with some accuracy, these can be defined as 70-120 khz for matrix cracking, 120-190 khz for delamination, 190-260 khz for debonding and 260-320 khz for fiber failure. in particular, figs. 7 (a) and (b) show the results for a lower indentation depth of 1 mm. both in the baseline and the filler-loaded laminates, the damage initiates in the form of matrix cracking, while fiber/matrix debonding was observed only in the case of the fillerloaded samples; delamination was not yet evident at this indentation depth. this can suggest that the filler-loaded laminates may provide higher energy dissipation through an additional toughening mechanism such as filler/matrix debonding offered by the presence of milled glass fibers. in practical terms, at 1 mm indentation depth, no significant damage was visible in both the baseline and the filler-loaded samples, except minor local indentation at the contact of the indenter. this will be confirmed in fig. 9, with no delamination/debonding visible for 1 mm indentation depth in the baseline samples. figures 7(c) and (d) show the results for an indentation depth of 2 mm, where the damage onset load (or) delamination threshold load was indicated from the incipient point during quasi-static indentation. typically, the incipient point defines the initial change in the slope or a drop in the load where delamination occurs during testing. the baseline samples quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 435 show a change in the slope during 2050n and 1.3mm indentation depth, while the fillerloaded samples indicate a change in the slope only during 2550n and 1.6mm indentation depth. correspondingly to this load/indentation, damage accumulation was observed attributed to a sudden and abrupt increase in the ae cumulative counts with a change in the slope associated with a major failure, such as debonding/delamination, as observable from the peak frequency-time plot. these failure modes were nominal at 2 mm indentation depth and consequently exhibit a smaller damage area, as seen in fig. 9. the size of the damaged area, the intensity of damage and relevant damage mechanism were observed to increase with indentation depth. fig. 7 peak frequency & cumulative counts vs. time plot of corresponding loaddisplacement behavior for indentation depth. baseline samples: (a) 1mm (c) 2mm (e) 3mm & filler-loaded samples: (b) 1mm (d) 2mm (f) 3mm 436 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. the indentation response of the glass/epoxy samples subjected to 3 mm indentation depth can be seen in figs. 7(e) and (f). in this case, the change of the slope of the loaddisplacement curve indicates reduced contact stiffness of the material due to the initiation of damage mechanisms, such as matrix cracking, debonding and delamination, as can be seen in figs. 10 and 11. as previously discussed, the flat region of the curve of the ae cumulative counts indicates the damage initiation stage attributed to low-frequency failure mode matrix cracking, whereas the initiation of delamination in the baseline samples occurred at 2150n and 1.3mm, followed by a sudden load drop at 2850n and 1.98mm, corresponding to the fiber failure, as seen in figs. 10 and 11. in general, it is reported that the initial peak force increases consistently with indenter displacement, accompanied by a sequence of load drops associated with significant damages modes such as fiber failure and delamination. it can also be observed that the intensity of delamination signals observed in the baseline samples is higher than that of the filler-loaded samples which validates the correlation of damage area as seen in fig. 9. in contrast, the filler-loaded samples exhibit delamination threshold load of 2550n, corresponding to 1.7mm indentation depth, while a sudden load drop was observed at 3280n, corresponding to 2.57mm. in practice, it was observed that the damage onset load and the fiber breakage load exhibited by the filler-loaded samples was for 19% and 15% higher than for the baseline samples. this result shows that damage initiation and accumulation were delayed for the filler-loaded sample comparing to the baseline samples. in figs. 10 and 11, it can be observed that indentation depth of 3 mm results in compression fracture at the contact location of the indenter causes matrix shear cracking and fiber micro-buckling while delamination followed up by intra-laminar cracking and axial fiber splitting at the back side. in fig. 9, it appears that the filler-loaded samples exhibit a reduced damage area than the baseline samples, and, moreover, the filler-loaded samples have higher contact stiffness and load carrying capacity than the baseline samples. figures 8 (a) to (f) show the load-displacement behavior of the baseline and the fillerloaded samples indented at 4, 5 and 6mm. the same trend was observed with a flat profile of the ae cumulative counts indicating the damage initiation stage and a steep rise in the slope of the curve showing the damage accumulation. it was observed that the damage onset load in the baseline samples occurred at an average of 2100 n at 1.2mm followed by a sudden load drop at 2700n and 1.8mm corresponding to the fiber failure. in contrast, the filler-loaded samples exhibited delamination damage load and a load drop related to fiber damage at 2500n and 1.6mm, and 3100n and 2.1mm, respectively. it is significant to observe that the damage onset load and the fiber breakage load exhibited by the fillerloaded samples were for about 20% and 15% higher than the baseline samples, respectively. this is due to enhanced energy dissipation through toughening mechanism such as filler/matrix debonding, filler interlocking/bridging which contributes to higher load-bearing capacity. at a higher indentation depth, a fiber failure occurs during penetration/perforation of the samples which reduces the load resistance capability. beyond 4 mm indentation depth, all the samples show the same trend in peak load and absorbed energy, while no significant increase was observed due to a more intense fiber failure. the failure occurred through matrix shear cracking, and fiber micro-buckling at the local contact of the indenter, followed by delamination between the adjacent plies and intra-laminar cracking (or) axial fiber splitting through penetration/perforation of the indenter. it is clearly seen that the quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 437 delamination onset load was almost similar for all the samples while the improvement in the fiber breakage load reduces drastically with an increasing indentation depth due to the perforation/penetration of the samples. fig. 8 peak frequency & cumulative counts vs. time plot of corresponding loaddisplacement behavior for indentation depth. baseline samples: (a) 4mm (c) 5mm (e) 6mm & filler-loaded samples: (b) 4mm (d) 5mm (f) 6mm 438 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. fig. 9 images of projected indentation damaged area at different indentation depth figures 10 and 11 show optical microscope images of the fractured surfaces observed during the quasi-static indentation test in the baseline and the filler-loaded samples. at a lower indentation depth, matrix fracture damage occurs, which decreases the localized stiffness of the laminate, but does not result in any catastrophic failure. however, further increasing the indentation displacement can cause the matrix cracking to progress into delamination and fiber breakage, due to the tensile stress caused by the indenter contact. in general, the transverse matrix cracking develops into delamination between the adjacent plies, and finally, prevalent intra-laminar crack and in-plane fiber buckling/breakage cause penetration and perforation damage at a higher indentation depth (4 to 6 mm). figures 9, 10 and 11 show that the intensity of damage in the baseline samples was predominant comparing to the filler-loaded samples, especially at a higher indentation depth. in contrast, the damage exhibited by the filler-loaded samples was minor and delayed, due to the improved load carrying capacity and energy absorption characteristics offered by the presence of fillers. figures 12 (a) and (b) show scanning electron microscope images of the fractured surfaces of the baseline and the filler-loaded samples. the baseline samples quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 439 exhibit smooth and brittle fracture surface associated with no plastic deformation corresponding to low fracture toughness of epoxy matrix. however, the fracture surface of the filler-loaded samples indicates rough surface with intense scarps/river lines. the intense scarps/river lines indicate hindrance of crack growth. figure 12 (b) shows the fillers strongly bonded to the matrix and filler debonding/pullout, which can be attributed to the additional energy dissipation mechanisms. fig. 10 optical microscopic images of the fractured surfaces observed during quasistatic indentation test for baseline samples fig. 11 optical microscopic images of the fractured surfaces observed during quasi static indentation test for filler-loaded samples 440 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. fig. 12 sem micrograph of the fractured surfaces showing: (a) baseline samples (b) filler-loaded samples online ae monitoring was also employed during the three-point bending test, to monitor the evolution of damage and to characterize the damage modes associated with flexural loading. acoustic emission results of flexural after indentation and quasi-static indentation test showed mostly similar trends. therefore, the results of the normalized number of ae events for flexural after indentation test was illustrated to discuss the damage mechanisms. figure 13 shows the normalized number of ae events exhibited during flexural after indentation test associated with types of failure modes for baseline and filler-loaded samples. the damage mechanisms associated with flexural after indentation test were characterized from frequency ranges, as discussed above. it can be observed that the delamination signals intensity in the baseline samples are more intense and greater than the filler-loaded samples which validate the correlation between these data and those of the damage areas. it can be observed that the matrix cracking signals associated during the test were found to be concentrated and more intense since they start almost from the beginning of loading; this triggers the other failure modes in both the baseline and fillerloaded samples. the number of the ae events associated with the delamination signals for the baseline samples was found to be more intense comparing to the filler-loaded samples: this trend was observed to be prominent at a higher indentation depth (beyond 4 mm). it can be suggested that, at this point, quasi-static indentation on the glass/epoxy samples has caused sufficient delamination damage, which might progress during postindentation flexural testing. correspondingly, the number of the ae signals attributed to matrix cracking decreases, while those related to delamination increase for the samples indented beyond 4 mm. this evidences that during flexural loading, the damage initiates as matrix cracking in the non-indented samples and samples indented below 4 mm. in contrast, in the samples indented beyond 4 mm, the failure starts/develops from the former damage mode (usually delamination / debonding). consequently, the debonding and fiber breakage signals observed in all other cases were almost similar, and no significant changes in the intensity of ae signals were observed as seen in fig. 13(a). quasi-static indentation behavior of gfrp with milled glass fiber filler monitored by acoustic... 441 fig. 13 types of failure modes versus normalized number of ae events for flexural after indentation test: (a) baseline samples (b) filler-loaded samples similarly, fig. 13 (b) shows the normalized number of the ae events with corresponding types of failure modes exhibited by the filler-loaded samples during flexural after indentation test. it can be observed that the normalized number of the ae events corresponding to the matrix cracking was highly predominant for the filler-loaded samples in all the cases (virgin and all indented samples). in contrast, the number of the ae events corresponding to delamination signals was observed to be lower for the filler-loaded samples than the baseline samples, even for the samples subjected to a higher indentation depth. however, the corresponding debonding signals associated with the test in the fillerloaded samples were greater than the baseline samples. this is due to the filler presence in the interlaminar domain introducing toughening mechanism, such as filler debonding/ pullout resulting in more intense debonding ae events. subsequently, the number of the ae events related to the fiber breakage was almost similar in both the baseline and the filler-loaded samples for all the cases of flexural after indentation. 4. conclusions this present work investigates the influence of milled glass fiber filler on glass/epoxy samples subjected to quasi-static indentation and flexural after indentation tests with acoustic emission monitoring. the evolution of damage and characterization of damage mechanisms associated with loading were discussed, and the results were correlated with the baseline glass/epoxy samples. the results show that, irrespective of indentation depth, the filler-loaded samples exhibited a slightly higher peak force for about an average of 10% than the baseline samples. in all the cases beyond 4 mm, no significant difference in the peak load was observed, corresponding to the predominant fiber damage during penetration/perforation of samples. the initial contact stiffness as well as the delaminated one of the filler-loaded samples was 2181 ± 75 n/mm and 955 ± 72 n/mm, respectively, while the baseline samples showed the initial and the delaminated contact stiffness of 1902 ± 58 n/mm and 832 ± 75 n/mm, respectively. the filler-loaded samples showed 15% improvement in the initial and the delaminated contact stiffness in comparison with the baseline samples. 442 k. saravanakumar, b.s. lakshminarayanan, v.arumugam, et al. it was observed that the filler-loaded samples exhibited a reduced permanent depth and a damage area by an average of 25% than the baseline samples. the presence of the filler offers greater rigidity/stiffness and higher energy absorption through plastic deformation and toughening mechanism, such as filler debonding/pullout, filler bridging/interlocking. the residual flexural strength was observed to reduce gradually by an average of 15% for an indentation depth below 3 mm with a reduction of about 30% at a higher indentation depth in both the baseline and the filler-loaded samples, leading to a large delamination area and intensive fiber breakage causing penetration/perforation damage. this evidences that the beyond 3 mm, the load carrying capacity of the glass/epoxy samples decreases drastically due to the greater accumulated damage during indentation. besides, the filler-loaded samples exhibited an average of 18% higher residual strength than the baseline samples. damage mechanisms associated with quasi-static indentation and flexural after indentation test, as characterized from ae frequency data, were almost similar. however, in the former the presence of delamination and debonding was more evident for the highest indentation depths than in the latter samples. references 1. atas, c., icten, b.m., küçük, m., 2013, thickness effect on repeated impact response of woven fabric composite plates, compos part b, 49, pp. 80–85. 2. aktas, m., atas, c., icten, b.m., karakuzu, r. 2009, an experimental 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manufactured by vacuum infusion, compos part b, 69, pp. 507-515. 39. mohammadi, r., najafabadi, m.a., saeedifar, m., yousefi, j., minak, g., 2017, correlation of acoustic emission with finite element predicted damages in open-hole tensile laminated composites, compos part b, 108, pp. 427-435. experimental forced solar thin layer ginger drying facta universitatis series: mechanical engineering vol. 14, no 1, 2016, pp. 101 111 original scientific paper experimental forced solar thin layer ginger drying  udc 662.6 mahesh kumar mechanical engineering department, guru jambheshwar university of science & technology, india abstract. in this research paper, the convective and the evaporative heat transfer coefficients of ginger (zingiber officinale) drying in an indirect solar cabinet dryer under the induced forced convection mode is presented. experiments were conducted during the month of march 2015 under the climatic conditions of hisar, india (29°5’5”n latitude and 75°45’55”e longitude). the experimental data obtained for solar drying of a constant ginger mass of 150 g has been used to determine constants ‘c’ and ‘n’ in the nusselt number expression using linear regression analysis; consequently, the convective and the evaporative heat transfer coefficients have been evaluated. the average value of constants ‘c’ and ‘n’ were evaluated as 0.999 and 0.318, respectively. the average values of the convective and the evaporative heat transfer coefficients were found to be 3.95 w/m2 °c and 160.47 w/m2 °c, respectively, for the given mass samples of ginger. the average collector efficiency was observed to be 14.5%. the experimental error in terms of percentage uncertainty was found to be 20.87%. key words: ginger drying, indirect forced solar dryer, convective heat transfer coefficient, evaporative heat transfer coefficients, collector efficiency 1. introduction india is the largest producer of ginger (an herb obtained from zingiber officinale) in the world accounting for about one-third of the total world output followed by china, nigeria, indonesia, bangladesh and thailand [1]. ginger believed to have originated in south-east asia and is used worldwide as a spice for flavoring a multitude of food varieties and products. it is also used in medicine, particularly in traditional medicine of india. drying is received: august 14, 2015 / accepted: december 2, 2015 corresponding author: mahesh kumar guru jambheshwar university of science & technology, hisar 125001, haryana, india e-mail: mk_shandilya@yahoo.com 102 m. kumar essentially important for preservation of ginger for future use by removing enough moisture from it in order to avoid its decay and spoilage. the ginger dried in open sun is dependent on the availability of sunshine; it requires large space and a long drying time and is susceptible to contamination. solar drying is one of the most promising alternatives to the open sun drying which can be used in developing countries. the performance of the forced convection solar dryer was compared to the electrically operated tray type mechanical dryer for drying of groundnut, ginger, and garlic. it is found that cost benefit ratio for solar dryer was higher [2]. the dried ginger quality was reported better and the drying time was observed as less in a solar hybrid dryer than in the open sun drying (osd) [3]. the best quality dried ginger in the shortest drying time was obtained from the heat pump dehumidified dryer and the mixed mode solar dryer than the tray dryer [4]. the heat pump dehumidified ginger drying incorporated by the two stage-drying could reduce the drying time by 59.32% at 40 °c [5]. the ginger drying is found experimentally faster with the solar dryer than with the open air method [6]. a forced convective cabinet dryer was used for single layer ginger drying at different drying air temperatures at a fixed air velocity [7]. the photovoltaic powered indirect forced convection solar dryer was developed for drying ginger [8]. the quality of dried ginger was improved and the drying time was observed to be reduced by 66.7% against osd. the drying kinetics of ginger rhizomes under blanched and non-blanched conditions were studied by using the hybrid solar dryer and the mechanical tray dryer [9]. the page equation was found to be the best among the five thin layer drying models to predict the moisture content of the thin layered sliced ginger. the thin layer solar drying of ginger was studied with a different mass flow rate and the page model was found as the most appropriate for representing the ginger drying curve [10]. the solar drier with evacuated tube collectors was designed for drying ginger [11]. the drying kinetics of pre-treated and untreated ginger at selected temperatures and at a constant air velocity was investigated [12]. the drying time of pre-treated ginger was found shorter than the untreated one and the page model was reported to be best suited to describe the drying kinetics of ginger. the performance of an indirect forced convection solar dryer integrated with a thermal storage for drying ginger was studied [13]. the average efficiency of the dryer was reported to be 30%. drying characteristics of ginger rhizomes sliced to various lengths and the whole ginger were studied under sun drying, solar tunnel drying and cabinet tray drying at different temperatures [14]. the moisture content was observed to be reduced from 81.3 to 10% and slicing of ginger significantly reduced the drying time in all the drying methods. a mixed mode solar cabinet dryer was used to study the drying characteristics of ginger which was observed as better for ginger drying in aspects of quality, drying time, and power requirement as compared to osd [15]. in developing countries, the open sun drying is a popular and economical method used for drying and preservation of agricultural and food products. but it has a number of drawbacks which include degradation of product quality by wind-blowing, debris, rain, insects, and animals. to avoid these problems a solar dryer consisting of a box shaped housing unit having a transparent sunlight cover was developed. afterwards, many researchers have made several improvements in the solar drying technology utilizing natural convection circulation, forced convection circulation, and auxiliary source heating (i.e., electricity and fossil fuels) to achieve the desirable drying features. in the forced convection solar dryers, the air required for product drying is forced through the solar collector to dryer chamber by using a fan or a blower, whereas in the natural convection solar dryers, it is due to experimental forced solar thin layer ginger drying 103 natural action or buoyancy force action. the indirect solar dryer is the oldest type of solar dryers which consists of a separate solar collector with a transparent cover (glass sheet) on the top and a drying unit with an opaque cover on the top. these dryers are not standardized; they fulfill the special requirements for drying different products. in the present study, ginger samples of a constant mass (150 g) have been dried in an indirect forced convection solar dryer under the meteorological conditions of hisar (29°5’5”n latitude and 75°45’55”e longitude), india. for obtaining more accurate results, the experiment was repeated three times, namely, first sample drying, second sample drying, and third sample drying. the convective and the evaporative heat transfer coefficients of ginger dried under the forced convection mode have been evaluated which will be required for the design of an optimum dryer. the solar collector efficiency has also been evaluated. 2. material and methods 2.1. experimental setup the schematic view of the forced convection indirect solar dryer is shown in fig. 1. it mainly consists of a solar air collector for heating the air, a drying chamber containing wet product to dry, and a chimney fitted with an induced fan provided at the top of the drying chamber. drying unit chimney fresh air in abso rbing plate glas s stand weighing balance ginger samples on wire mesh tray pvc pipe stand hot air gt &et ground air out solar collector fan cit , cot , fig. 1 schematic view of indirect forced convection solar dryer the design details of the solar collector and the cabinet dryer are given in table 1. the solar air collector consists of a black coated galvanized iron sheet which heats up the air. the back part and side walls of the solar air collector have been insulated with glass wool. to achieve the desired glazing effects a transparent 8 mm thick glass sheet was fitted over the top of the collector. the drying chamber of size 0.4 m × 0.45 m × 0.5 m was thermally insulated with glass wool to minimize heat losses. the heated air in the collector was made to flow to 104 m. kumar the drying chamber through the glass wool insulated pvc pipe of 5 cm in diameter and of 40 cm in length. a rectangular shaped wire mesh tray of size 0.08 m × 0.20 m was used to accommodate the evenly placed ginger pieces of known mass in a single thin layer. the weight reduction of the ginger samples due to the evaporation of moisture by hot air was measured by using an electronic weighing balance (model tj – 6000, capacity 6 kg, least count of 0.1 g) placed inside the cabinet dryer. table 1 design details of solar air heater and cabinet dryer s. no. details of solar dryer dimensions/specifications 1. total area of flat plate collector 1.3 m × 1 m 2. aperture area of collector 1.26 m × 0.95 m 3. air inlet passage to the collector 0.5 m × 0.07 m 4. collector inclination (tilt angle) 30o 5. absorber plate gi sheet, 1.2 mm thick 6. bottom side of collector 1.2 cm thick wooden plate 7. glass thickness 8 mm 8. cabinet dryer 0.4 m × 0.45 m × 0.5 m 9. tray size 0.08 m × 0.20 m 10. fan (dc brushless, 12 vdc) 1.4 w, 2700 ± 300 rpm 11. solar panel (36 number of cells) 1.2 m × 0.5 m × 0.03 m 12. battery (12 v lead acid) 18 cm × 7.5 cm × 16.5 cm 2.2. sample preparation fresh ginger purchased from local market was washed, peeled, and cylindrically-shaped with 1.5 cm in diameter and 3 cm in length. the slices were kept at ambient for one and half hour to remove surface moisture. the samples were accommodated in a rectangular shaped wire mesh tray which was directly placed on the weighing balance. 2.3. experimental procedure and instrumentation experimental observations were recorded during day time (between 9 am to 6 pm) in the month of march 2015 at guru jambheshwar university of science and technology, hisar. experimental observations were recorded at every 1 hour time interval. the following steps were followed for the experimentation: (i) accurately measured fresh ginger mass samples of 150 g was accommodated on a rectangular shaped wire mesh tray (0.08 m × 0.20 m) placed on the weighing balance. (ii) a digital humidity meter (model ht 315) was used to measure temperature (te) and relative humidity ( or rh) just above the ginger samples. it was started 1 minute before recording the observations. (iii) dryer inlet temperature (to,c), product surface temperature (tg), collector inlet air temperature (ti,c), and absorbing plate average temperatures (ts,av) of the collector were measured by using calibrated thermocouples attached to the digital temperature meter (accuracy ± 0.1 °c). (iv) the air flow at the collector inlet air passage was measured by using a digital anemometer (model am-4201) having the least count of 0.1 m/s. experimental forced solar thin layer ginger drying 105 (v) the quantity of moisture content evaporated (mev) from the ginger samples during drying time interval was obtained by the difference in weight of the consecutive readings which was measured by a digital weighing balance (model tj – 6000, capacity 6 kg, least count of 0.1 g). (vi) the solar intensity data (i) have been collected hourly at ccshau, hisar. (vii) all the hourly data during the experiments were recorded from 9.00 am to 6.00 pm. (viii) for obtaining better accuracy in the results the experiments were repeated thrice. 2.4 thermal modeling the rate of heat utilized to evaporate moisture is given as [16-18]: . 0.016 [ ( ) ( )]e c g eq h p t p t  (1) convective heat transfer coefficient (hc) can be determined as [17], ( / ) (repr)n c vh k x c (2) . 0.016( / ) (repr) [ ( ) ( )]n e v g eq k x c p t p t  (3) where x is the characteristic dimension in m. 0.016( / ) (repr) [ ( ) ( )]n ev v g e tm k x c p t p t a t   (4) where at is the area of tray in m 2 and t is time in seconds. let 0.016( / )[ ( ) ( )]v g e tk x p t p t a t z    nev czm prre]/[  (5)  prrelnln]/ln[ nczmev  (6) eq. (6) is the form of a linear equation, cmxy  (7) where y = ln(mev/z), m = n, x = ln(re pr) and c = ln c. thus, c = e c . values of m and c in eq. (7) are obtained by using the simple linear regression formulae. evaporative heat transfer coefficients (he) were calculated as [18], 0.016 [{ ( ) ( )}/( )]e c g e g eh h p t p t t t   (8) the thermo-physical properties, namely, specific heat (cv), thermal conductivity (kv), density (v), viscosity (μv), and partial vapor pressure p(t) of humid air were evaluated for mean temperature ti = [(tg + te)/2] by using the following eqs. [16-18]: 3824 107581.610101.11434.02.999 iiiv tttc   (9) 40.0244 0.7673 10v ik t   (10) 106 m. kumar 353.44/( 273.15)v it   (11) iv t85 10620.410718.1   (12) ( ) exp[25.317 5144/( 273.15)]ip t t   (13) 2.5. solar air collector efficiency solar air collector efficiency (ηc) was calculated as the ratio of total amount of heat at solar flat plate collector outlet (qo) to the total amount of heat received by solar flat plate collector (qi) as [19]: qiqoc / (14) , , ,( )o o o c v o c i c vq v a t t c      (15) ci aiq  (16) 2.6. experimental error the experimental error was evaluated in terms of percent uncertainty (internal + external) for the mass of moisture evaporated. the following equations were used for internal uncertainty [18]: oni nu )( 22 2 2 1   (17) where σ is the standard deviation and no denotes the number of sets. % internal uncertainty = (ui / mean of the total observations) ×100 (18) for external uncertainty, the least counts of all the instruments used in measuring the observation data were considered. 3. results and discussion the experimental data collected for ginger drying under induced forced convection mode in an indirect solar dryer for three samples of constant mass (150 g) are given in tables 1 to 3. table 1 data of 1 st sample ginger drying in an indirect solar dryer (march 29, 2015) time (hour) tg (oc) te (oc)  (%) mev (g) ts,av (oc) ti,c (oc) to,c (oc) i (w/m2) 0 27.1 26.8 50.2 34.8 25.8 37.5 441 1 36.8 35.1 33.9 11.8 49.6 30.2 54.3 662 2 43.4 40.7 21.5 12.6 58.2 33.6 64.2 802 3 46.2 43.0 18.3 17.4 63.3 35.1 69.1 862 4 48.3 44.8 15.9 18.7 64.0 36.3 70.2 903 5 49.7 45.9 14.2 16.5 62.9 36.7 69.5 865 6 48.4 44.8 13.6 12.4 58.5 36.2 63.8 588 7 44.7 41.5 16.3 8.4 51.1 35.1 55.6 473 8 39.6 37.2 21.9 6.7 43.8 33.2 46.7 215 9 34.4 32.6 26.7 5.2 35.1 31.0 35.8 85 experimental forced solar thin layer ginger drying 107 table 2 data of 2 nd sample ginger drying in an indirect solar dryer (march 30, 2015) time (hour) tg (oc) te (oc)  (%) mev (g) ts,av (oc) ti,c (oc) to,c (oc) i (w/m2) 0 27.6 26.1 52.3 36.3 23.8 35.8 469 1 36.1 34.4 31.8 12.3 50.3 27.0 53.7 680 2 42.9 40.1 20.2 14.1 57.1 30.3 61.5 810 3 45.2 42.2 18.6 15.2 65.3 32.4 68.6 891 4 47.8 44.4 17.7 16.1 66.0 34.7 70.8 911 5 48.7 45.1 15.8 14.2 63.2 33.5 69.4 798 6 47.2 43.8 14.9 12.2 61.7 33.2 61.9 774 7 45.1 41.8 16.3 8.4 50.8 31.9 51.7 406 8 40.2 37.5 18.9 6.1 42.5 30.0 41.6 185 9 35.6 33.6 24.2 5.1 38.3 27.4 31.9 81 table 3 data of 3 rd sample ginger drying in an indirect solar dryer (march 31, 2015) time (hour) tg (oc) te (oc)  (%) mev (g) ts,av (oc) ti,c (oc) to,c (oc) i (w/m2) 0 27.9 26.5 55.3 37.1 24.9 37.9 521 1 36.5 34.8 33.0 11.8 49.5 28.4 54.1 637 2 43.2 40.3 21.0 13.2 57.5 31.6 63.7 817 3 45.7 42.6 18.0 16.1 66.5 33.9 72.8 911 4 48.1 44.6 17.7 16.6 66.6 35.3 72.9 912 5 49.2 45.4 15.0 14.7 64.3 34.7 71.3 825 6 47.8 44.2 14.0 11.9 60.5 34.9 61.8 636 7 44.9 41.6 17.0 8.2 49.5 32.4 50.5 343 8 39.9 37.3 21.0 5.7 41.3 31.5 42.4 173 9 35.0 33.1 26.0 4.5 37.4 28.3 32.7 79 the data given in tables 1 to 3 were used to determine the values of constants c and n in the nusselt number expression by simple linear regression method. the values of these coefficients were used in eq. (2) for evaluating the values of the convective heat transfer coefficients. the values of the convective heat transfer coefficients were used in eq. (8) for determining the values of the evaporative heat transfer coefficients. the efficiency of solar air collector has also been calculated. the evaluated values of the constants (c and n), convective heat transfer coefficients, evaporative heat transfer coefficients, and solar air collector efficiency are summarized in table 4. the product of reynolds number (re) and prandtl number (pr) has been observed less than equal to 10 5 , which indicates that the entire ginger drying for forced indirect solar drying mode falls within a laminar flow [16]. table 4 evaluated values of constants, convective and evaporative heat transfer coefficients, and collector efficiency c n hc (w/m2 °c) hc,av (w/m2 °c) he (w/m2 °c) he,av (w/m2 °c) c (%) c,av (%) first sample drying (29-03-2015) 0.999 0.322 4.00 4.06 4.04 145.47 -181.47 166.79 11.60 -20.58 14.16 second sample drying (30-03-2015) 0.999 0.318 3.94 3.98 3.96 141.73 -174.01 160.36 11.53 -20.63 14.41 third sample drying (31-03-2015) 0.999 0.315 3.82 3.87 3.85 137.2 -165.84 154.25 12.37 -20.80 14.93 108 m. kumar the average value of constant c was evaluated as 0.999 and the values of exponent n were found to vary from 0.315 to 0.322. the average value of exponent was found to be 0.318. the values of convective heat transfer coefficients for the dried samples of ginger were observed to vary from 3.82 to 4.06 w/m 2 °c. the average value of the convective heat transfer coefficients was found to be 3.95 w/m 2 °c. the variation in convective heat transfer coefficients with respect to drying time is illustrated in fig. 2. variation in the values of convective heat transfer coefficients for each day of drying is observed almost constant. different values of convective heat transfer coefficients have been obtained for different ginger drying samples. this variation could be due to change in the operating conditions with drying days. the convective heat transfer coefficients of six crops (green chillies, green peas, white gram, onion, potato, and cauliflower) were reported to vary from crop to crop with a range of 3.5 w/m 2 °c to 26 w/m 2 °c [20]. the evaluated values of the convective heat transfer coefficient of ginger are observed to lie within this range. 3 3.2 3.4 3.6 3.8 4 4.2 0 1 2 3 4 5 6 7 8 9 10 time (hours) h c ( w /m 2 o c ) first sample second sample third sample fig. 2 variation in convective heat transfer coefficients with time the values of the evaporative heat transfer coefficients for the dried samples of ginger were observed to vary from 137.2 to 181.47 w/m 2 °c. the average value of the evaporative heat transfer coefficients was found to be 160.47 w/m 2 °c. the variation in the evaporative heat transfer coefficients with respect to the drying time is illustrated in fig. 3. significant 100 120 140 160 180 200 0 1 2 3 4 5 6 7 8 9 10 time (hours) he ( w /m 2 o c ) first sample second sample third sample fig. 3 variation in the evaporative heat transfer coefficients with time experimental forced solar thin layer ginger drying 109 variations have been observed in the values of the evaporative heat transfer coefficients for each sample of ginger drying. the photographic views of the samples of ginger before and after drying are shown in fig. 4. before drying after drying fig. 4 ginger samples before and after drying for the given days of ginger drying the solar air collector efficiency was observed to vary from 11.53% to 20.80 %. the variation in the collector efficiency with respect to the drying time is shown in fig. 5. the average value of the collector efficiency was found to be 14.5%. 0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9 10 time (hours) (% ) e ff ic ie n c y first sample drying second sample drying third sample drying fig. 5 variation in solar air collector efficiency with time the experimental error in terms of percent uncertainty (internal + external) is evaluated as 20.87%. the different values of the convective and the evaporative heat transfer coefficients were found to be within the given range. the error bars (a graphical representation of the variability of data) is used to show the error or uncertainty in the measurement of the convective and the evaporative heat transfer coefficients. it gives a general idea of how precise a measurement is or conversely, how far from the reported value the true value might be. the variability of the convective and the evaporative heat transfer coefficients from their true value are illustrated by error bars (95% ci) in fig. 6. it is drawn with the help of the spss software (version 16.0). 110 m. kumar sample number sample number h c (w /m 2 o c ) h e (w /m 2 o c ) fig. 6 error bars 4. conclusions in this research, the values of the convective and the evaporative heat transfer coefficients for the forced convection indirect solar drying of ginger for a constant mass have been reported. the experimental data were analyzed by using the nusselt number expression with the help of simple linear regression analysis. the average values of constant c and exponent n in the nusselt number expression were found to be 0.999 and 0.318, respectively. the values of the convective and the evaporative heat transfer coefficients were observed to vary from 3.82 to 4.06 w/m 2 °c and 137.2 to 181.47 w/m 2 °c respectively. the average value of the convective and the evaporative heat transfer coefficients for ginger drying under the forced convection mode inside an indirect solar cabinet dryer was found to be 3.95 w/m 2 °c and 160.47 w/m 2 °c, respectively. the solar air collector efficiency was observed to vary from 11.53% to 20.80 % and its average value was evaluated as 14.5%. the experimental error in terms of percentage uncertainty (internal + external) was calculated to be 20.87%. this research work would be useful in designing a dryer for obtaining high quality dried ginger. it will also be helpful in obtaining an optimum moisture level of ginger for retaining its quality during storage period. acknowledgements: the paper is a part of the research done within the project under the scheme of minor research project from the budget head ‘daa, 18b (ii) research activities (r&d) of guru jambheshwar university of science & technology (gjus&t). the author would like to thank the gjus&t for financial support. references 1. dohrooc, n.p., kansal, ahluwalia, s. n., 2012, status of soft rot of ginger (zingiber officinale roscoe), department of vegetable science, dr y.s. parmar university of horticulture and forestry, nauni, solan (hp), india 2. jain, n. k., kothari, s., mathur, a. n., 2004, techno-economic evaluation of a forced convection solar dryer, journal of agricultural engineering, 41(3), pp. 6-12. 3. prasad, j., prasad, a., vijay, v., 2006, studies on the drying characteristics of zingiber officinale under open sun and solar biomass (hybrid) drying, international journal of green energy, 3, 79-89. experimental forced solar thin layer ginger drying 111 4. phoungchandang, s., 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journal of engineering science and technology, 8(2), pp. 191-207. 18. kumar, m., prakash, o., kasana, k. s., 2012, experimental investigation on natural convective heating of milk, journal of food process engineering, 35, pp. 715-726 . 19. pangavhane, d., sawhney, r., sarsavadia, p., 2002, design, development and performance testing of a new natural convection solar dryer, energy, 27, pp. 579-590. 20. anwar, s. i., tiwari, g. n., 2001, evaluation of convective heat transfer coefficient in crop drying under open sun drying, energy conversion and management, 42(5), pp. 627-637. 14015 facta universitatis series: mechanical engineering vol. 23, no 4, 2025, pp. 921 943 https://doi.org/10.22190/fume250830033c © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd review article cu-cu mechanical bonding for 3d integration of the next generation electronic chips: interfacial mechanisms, surface engineering, and emerging low-temperature strategies sooyong choi1, chih-ming chen2, byungil hwang3 1department of intelligent semiconductor engineering, chung-ang university, republic of korea 2department of chemical engineering, national chung hsing university, taiwan 3school of integrative engineering, chung-ang university, republic of korea orcid ids: sooyong choi https://orcid.org/0009-0004-1523-4122 chih-ming chen https://orcid.org/0000-0003-4481-1094 byungil hwang https://orcid.org/0000-0001-9270-9014 abstract. three-dimensional integrated circuit (3d ic) technology is essential for ultrahigh-density integration, and cu-cu direct mechanical bonding has emerged as a promising alternative due to the limitations of traditional soldering methods. this technology offers excellent power efficiency, high-density packaging, faster processing speeds, and improved heat dissipation. however, existing research has limitations in systematically analyzing the relationship between key parameters affecting cu-cu bonding quality such as bonding layer thickness, temperature, grain size, and surface roughness. a comprehensive review, particularly under the low-temperature conditions necessary for 3d ic processes, has been lacking. therefore, this review introduces the fundamental theory of cu-cu bonding and systematically discusses the effects of the parameters on bonding quality. by integrating these factors with a specific focus on lowtemperature cu-cu interfaces, this review aims to provide a practical and up-to-date guide on cu-cu bonding technologies. key words: cu, bonding, 3d packaging, low-temperature bonding 1. introduction technological advancements in computing and data storage systems have driven the demand for ultra-high-density integrated circuits. over the past few decades, reducing the transistor size has been the primary focus of research and development in the high-density integrated circuit (ic) field. the current state-of-the-art scaling technologies have achieved received: august 30, 2025 / accepted november 10, 2025 corresponding author: byungil hwang school of integrative engineering, chung-ang university, seoul 06974, republic of korea e-mail: bihwang@cau.ac.kr 922 s. choi, c. chen, b. hwang gate widths in the range of just a few nanometers [1-3]. however, further size reduction presents significant challenges owing to the quantum mechanical limitations of these materials [4]. consequently, the interest in three-dimensional (3d) ic technology has increased in recent years. 3d ics maximize the chip density by stacking multiple chips vertically on a limited substrate area, with high bandwidth memory (hbm) being a prime example [5-8]. in hbms, 16-high stacks have been formed to achieve dynamic random access memories (dram) with 32 gb capacity. traditionally, chip-to-chip connections have been created using external metal solder balls [9-18]. however, in 3d ics, where the gaps between the interconnections are in submicron scales, conventional soldering techniques become increasingly challenging. this has led to a growing interest in direct bonding technologies, where metal interconnects are bonded without the need for external solder balls [14-18]. among the various interconnect materials such as ag, au, and cu, cu-cu direct bonding stands out because of cu’s exceptional mechanical properties, superior electromigration resistance, high electrical conductivity, and excellent thermal conductivity. additionally, cu’s cost-effectiveness, compared with that of au and ag, makes it the preferred choice in industrial applications. consequently, cu-cu bonding has become the most widely adopted technology for packaging in 3d ics [19-24]. in cu-cu direct bonding, two flat cu surfaces are joined at high temperatures by applying external pressure. a bond is formed through the diffusion of atoms and grain growth between the surfaces. surface diffusion plays a critical role in facilitating cu-cu bonding as cu atoms migrate across the interface, forming a bonded structure. as the process continues, recovery and grain growth at both the interface and interior of cu further enhance the bond [25-27]. the quality of this bond is primarily determined by the bonding conditions such as time, temperature, pressure, and initial surface and microstructural characteristics of cu [28]. therefore, a thorough understanding of how these factors influence the bonding properties is essential to achieve high-quality cu-cu bonding. despite the increasing focus on developing reliable cu-cu bonding technologies, comprehensive reviews that address the critical factors affecting the bonding reliability are lacking. this review explores the fundamental mechanisms and theoretical background of cu-cu bonding under various conditions. it also highlights the state-of-the-art surface control technologies used to achieve high-quality cu-cu bonds at low temperatures, offering a future perspective on the role of cu-cu bonding in the next-generation electronic packaging. 2. factors affecting cu-cu mechanical bonding the grain size and surface roughness of cu are crucial factors for achieving high-quality bonding interfaces [28-38]. a reduction in the grain size decreases the likelihood of microcracks, thereby enhancing the shear strength. grain boundaries act as barriers to crack propagation; therefore, smaller grain sizes at the cu-cu bonding interface result in stronger bonds [39]. this relationship is described by the hall–petch equation, 50 0 . y,y kdσσ −+= (1) where σy is the yield strength related to the grain size, σy,0 is the starting yield stress for dislocation movement, k is a constant of strengthening coefficient, and d is the average grain size [40]. cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 923 kao et al. [41] confirmed the hall–petch relationship in electrically plated cu thin films. this directly supports the validity of the premise that “smaller grain size → higher yield strength/hardness of the material itself” in cu. grain size and mechanical response co-vary, and smaller grains tend to accompany higher yield stress. table 1 shows the electron backscatter diffraction (ebsd) mapping and its summary for various cu electroplated samples with different bis-(3-sulfopropyl) disulfide (sps) conditions. pc designates specimens electroplated with polyethylene glycol (peg) and chloride ions, whereas pcs refers to those containing sps as an accelerator in addition to the pc composition. pc, pcs0.2, pcs0.5, pcs1.0, and pcs2.0 represent samples with increasing sps concentration, from 0 ppm (pc) to 2.0 ppm (pcs2.0). the results indicate that pcs0.2 exhibits the smallest average grain size (0.29 μm), while higher sps specimens show larger grains (table 1, fig. 1). consistently, tensile data report the highest average yield stress for pcs0.2, with lower values for the coarser-grained specimens (table 2), in agreement with the hall–petch relation. these results suggest that grain refinement likely contributes to boundary strengthening and the observed increase in yield stress within this dataset. table 1 average grain sizes of various cu electroplated samples determined by the ebsd grain mappings in fig. 1 specimens pc pcs0.2 pcs0.5 pcs1.0 pcs2.0 grain size (μm) 0.55 0.29 1.97 3.79 3.77 fig. 1 ebsd grain mappings of the cu electroplated layers labelled as (a) pc, (b) pcs0.2, (c) pcs0.5, (d) pcs1.0, and (e) pcs2.0. the mappings are created by orientation imaging microscopy (oim) analysis v8 [41] 924 s. choi, c. chen, b. hwang table 2 yield stresses and elongations of various cu electroplated samples specimens pc pcs0.2 pcs0.5 pcs1.0 pcs2.0 yield stress (mpa) 351 ± 8 416 ± 8 290 ±12 261 ± 5 248± 2 elongation (%) 2.9 ± 0.2 4.3 ± 0.3 7.2 ± 0.9 9.0 ± 0.7 9.4 ± 0.4 somekawa et al. [42] studied the effect of grain size on the shear strength, bonding time, and bonding pressure by comparing two types of samples: a fine-grained sample with an 11 µm grain size and a coarse-grained sample with a 28 µm grain size. they found that, to achieve a comparable shear strength, the coarse-grained sample required thermal pressing for 3 h at 5 mpa, whereas the fine-grained sample required pressing for only 2 h at 3 mpa [42]. this indicated that finer grain sizes contributed to higher bond strengths under the same temperature, pressure, and time conditions. in general, in cu, there is a tendency for “fine grains → bonding promotion.” however, locally, there are cases where adhesion weakens at defect concentration points such as grain boundaries and triple points, so we must be cautious about the equation “many grain boundaries are equal to strong adhesion.” as shown in fig. 2, this is a regional property that varies significantly depending on location and microstructure, so generalization should be avoided [43]. for example, in fig. 2 (d), it can be observed that the adhesion strength at the junction area is significantly reduced. the surface roughness also significantly affects bond quality. to achieve high-quality bonding, it is critical to maximize the actual contact area between the two metal surfaces. gui et al. [44] demonstrated a strong correlation between the bonding energy and actual contact area. excessive roughness could promote void formation, which reduced the mechanical strength [45]. in the work by rebhan et al. [39], the bonding strength linearly increased as the roughness decreased from 8.0 to 1.5 nm. shigetou et al. [46] demonstrated that chemical mechanical polishing (cmp) polished cu films with a mean surface roughness of approximately 0.3 nm could be directly bonded at room temperature via the surface activated bonding (sab) method. under a background vacuum below ∼3×10⁻³ pa, these atomically smooth surfaces achieved full-area atomic-level bonding between cu grains with shear strengths exceeding 50 mpa. thus, optimizing the surface roughness is essential for maximizing the bonding strength of cu-cu bonding. as mentioned earlier, the bonding temperature, pressure, and time determine the degree of diffusion, which directly affects the interface quality of cu-cu joints. atoms in solid metals vibrate constantly, and their kinetic energy is proportional to the absolute temperature [26]. this energy distribution allows atoms to diffuse across lattice points when they exceed a specific energy threshold. the diffusion process can be described by fick’s first law [47]. from an atomic perspective, the heat energy accumulates as the temperature increases, causing the atoms to vibrate more intensely. this increased vibration allows the atoms to move more easily from their original positions, thereby facilitating greater diffusion. in other words, atomic vibration is proportional to the temperature; thus, higher temperatures result in faster diffusion and stronger bonds. therefore, increasing the bonding temperature enhances the diffusion of atoms and improves the bond strength between the two cu surfaces. jang et al. [48] studied the effects of bonding and annealing temperatures on cu-cu bonding. they measured the interfacial adhesion energy after bonding at 300, 350, and 400 °c for 1 h, without annealing. the interfacial adhesion energy increased with the bonding cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 925 temperature, showing a positive correlation. at 400 °c, a small error margin was obtained, indicating uniform bonding across the wafers. the annealing temperature also plays a crucial role in cu-cu bonding [28, 49, 50]. without annealing, voids are often present at the bonding interface. however, annealing at higher temperatures produces clean cu bonding layers. samples were annealed at 200, 250, and 300 °c for 1 h in an n₂ environment and the results of interfacial adhesion-energy measurements were compared. annealing at 200 °c had little effect; however, annealing at 250 and 300 °c increased the adhesion energy by 3.1 and 4.3 times, respectively [48]. these results show that, beyond a certain threshold, annealing significantly improves the adhesion energy at the cu bonding interface when conditions such as oxygen, pressure, and time are controlled [48]. lin et al. [51] investigated low-temperature cu-cu direct mechanical bonding using cmp highly (111)-oriented nanotwinned cu (nt-cu) films in the temperature range of 150– 200 °c, and quantitatively evaluated the interfacial void distribution by cross-sectional focused ion beam (fib) imaging. as shown in fig. 2 (a), for the bonding condition of 150 °c for 60 min, the columnar microstructure of the nt-cu films remains essentially intact, while a small number of tiny voids are still present along the bonding interface. in contrast, fig. 2 (c,d) demonstrates that, when the same cmp nt-cu films are bonded at 200 °c for 30 min, no discernible voids are observed at the cu-cu interface and a continuous, flat bonding interface is obtained. these observations indicate that, under identical surface roughness and bonding pressure, increasing the bonding temperature from 150 to 200 °c together with a sufficient bonding time effectively eliminates interfacial voids and improves the metallic continuity across the interface, which is expected to result in a significant enhancement of the practical bonding quality and interfacial adhesion/strength. furthermore, fig. 2 (b), corresponding to the condition of 200 °c for only 5 min, still exhibits a noticeable population of nanoscale voids at the bonding interface, corroborating that not only temperature but also the overall thermal budget (i.e., the temperature–time product) is a critical factor governing void healing by cu diffusion and creep during cucu bonding. fig. 2 cross-sectional fib image of the typical cmp nt-cu films bonded at (a) 150 ◦c for 60 min and (b) 200 ◦c for 5 min. (c,d) the typical cmp nt-cu films bonded at 200 ◦c for 30 min [51] 926 s. choi, c. chen, b. hwang li et al. [52] found a proportional relationship between the bonding pressure and shear strength in cu nanosolder paste bonding, where higher pressures generated stronger bonds at the same temperature. these studies demonstrated that both the bonding and annealing temperatures, as well as the bonding pressure, significantly influenced the quality and strength of the cu-cu bonding. voids at the metal-metal bond interface play a critical role in determining the bonding strength [53, 54]. even with the same overall void fraction, variations in the void shape and distribution can lead to different bonding outcomes [55]. therefore, understanding how the status of voids affects the strength of cu-cu bonding is essential. voids reduce the contact area between the metal interfaces, which in turn decreases the bonding strength and increases the resistance [56]. oh et al. [57] investigated the effects of temperature and pressure on the formation of local voids during cu-cu bonding. their results showed that increasing the pressure relative to the temperature significantly reduced the void size. specifically, the length and height of the voids decreased markedly when the pressure was increased to 30 mpa from 20 mpa. in contrast, increasing the temperature from 200 to 300 °c led to only minor reductions in the void size. void shapes are classified into three main categories: oblate, spherical, and prolate [58]. as the pressure decreased, the voids changed from oblate to spherical. these findings were consistent with that of other researchers who used both experimental and simulation methods [58, 59]. additionally, a simulation study revealed that under identical conditions, oblate voids grew faster than spherical voids, and spherical voids grew faster than prolate voids, during the bonding process. the study also found that the fracture initiation toughness was the highest when the void was prolate and lowest when it was oblate. in other words, the bonding was the most robust when the void was prolate and the weakest when it was oblate [58]. unfortunately, the voids formed at the bonding interface are usually oblate or spherical; therefore, more attention should be paid to the void formation and evolution during the cucu direct bonding process [58, 61-63]. void ripening was observed at the bonding interface subjected to thermal annealing at 200 °c for 30 to 120 min [63]. the absence of intergranular coarsening or interface breakup indicates that the microstructural change is dominated by void evolution rather than bulk recrystallization or grain growth. concomitantly, the void morphology evolves toward more circular shapes at longer times, with an apparent increase in the thickness dimension inferred from cross-sectional contrast. such circularization and coarsening of a subset of voids, together with the reduction in the count of smaller voids, is the characteristic fingerprint of curvature driven ostwald ripening at the bonding interface. the combined observations—decreasing void count with time, progressive circularization and coarsening of the surviving voids, and the lack of concurrent grain growth—substantiate the claim that void ripening occurs at the bonding interface during annealing at 200 °c for 30–120 min. a kinetic model was proposed to describe the relationship between the void radius and annealing time. a different void evolution phenomenon was observed under nanocrystalline cu bonding conditions [64]. the high thermal instability of nanocrystalline cu facilitated the grain growth and amalgamation at the bonding interface at low temperatures. however, the grain growth exerted a force on the voids, which split the initial large void into numerous nanovoids. these factors such as temperature and pressure can affect the formation of these voids, so care must be taken when joining cu-cu interface. cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 927 in summary, the reliability of direct cu-cu bonding is dictated by the coupled influence of four interdependent factors—grain size, surface roughness, thermo-mechanical processing parameters, and interfacial void behavior. refinement of the copper grain structure activates hall–petch strengthening, thereby elevating interfacial shear strength and suppressing microcrack propagation under identical thermal and pressure conditions. sub-nanometer surface roughness maximizes real contact area and therefore bonding energy, yet once a critical roughness threshold is exceeded, asperity-induced void formation offsets these benefits and degrades mechanical integrity. because the lattice diffusion coefficient follows an arrhenius relationship, concurrent application of moderately elevated temperatures (≈ 300–400 °c) and compressive stresses on the order of several tens of megapascals accelerates interfacial diffusion and markedly increases adhesion energy. nevertheless, voids persist as dominant reliability hazards; while elevated temperature enhances atomic mobility, empirical evidence indicates that increased external pressure more effectively collapses or heals voids, thereby restoring contact area and lowering electrical resistance. collectively, these observations establish a practical design framework—fine-grained microstructure, ultra smooth surfaces, judicious thermo-mechanical loading, and proactive void mitigation—through which high-integrity cucu interconnections can be consistently achieved. 3. low-temperature cu-cu mechanical bonding technologies 3.1. surface activated mechanical bonding sab combines two materials by leveraging the interatomic adhesion of atomically clean surfaces in an ultra-high vacuum (uhv) environment at 21-25 °c [60-64]. unlike the traditional bonding methods, sab achieves strong bonds without requiring hightemperature treatments, annealing, or toxic chemical cleaning processes. instead, a highly accelerated ar fast atom beam is employed to physically remove contaminants and oxides from the surface of the wafer. kim et al. [60] proposed a method that used an ar ion beam on a cu surface in a uhv environment to eliminate the oxide layer before performing sab. for successful thermo-compression bonding (tcb), the surface must be free of contaminants and have a flat oxide-free profile. cmp can be used to achieve a root mean square (rms) surface roughness of 1 nm or less; however, cu oxidizes quickly, forming an oxide layer over 1 nm thick within just 1 h of environmental exposure [60]. therefore, the removal of this oxide layer is critical for successful tcb. in sab, an ion or high-velocity atomic beam is used to activate the bonding surfaces in a uhv environment, facilitating the formation of chemical bonds between cu surfaces at room temperature [62]. the sab setup (as depicted in fig. 3) consists of various chambers designed for processing, analysis, heating, and alignment. after the surface activation in the process chamber, the two wafers are brought into contact and bonded in the bonding chamber. this enables wafer-level cu-cu bonding using an ar ion beam with sab technology. the surface activated cu-cu bonding process involves specialized treatments that remove or prevent the formation of an oxide layer on the cu surface. this results in a strong bond at room temperature without needing an annealing step to increase the bond strength. auger electron spectroscopy (aes) was used to assess the cleanliness of the cu surfaces prior to bonding. following the ar ion beam treatment, the transfer of the cu sample from the process chamber to the analysis chamber took approximately 15 min, during which 928 s. choi, c. chen, b. hwang time residual gases like o₂, h₂, and o could be generated. however, auger analysis confirmed that these gases had a minimal impact on the surface. fig. 3 schematic view of surface activated bonding machine the bond strength, a key measure of bonding quality, was assessed, and the tensile strength was found to exceed 6.47 mpa, with fractures occurring in the bulk or glue. this demonstrates that cu-cu bonding at room temperature can be successfully achieved using an ar ion beam during the sab process. shigetou et al. [65] demonstrated the successful bonding of 6 µm-pitch bumpless cu electrodes using the sab method; their technique also facilitated bonding of 20 µm-pitch au microbumps at room temperature [66]. 3.2. enhanced mechanical bonding strength by chemical pretreatment the formation of cuo is inevitable during cu-cu bonding [67-70], especially at high bonding temperatures. cu is highly susceptible to oxidation, with an oxide layer over 1 nm thick formed in less than 1 h of exposure [71]. although several methods have been proposed to remove this oxide layer [72-75], they are primarily limited to wafer-to-wafer bonding, highlighting the need for new techniques. consequently, various approaches have been explored to remove the oxide layer before bonding. exposure of the surface to various contaminants, as well as the oxide film, is not good for bonding quality as it causes voids. so chemical pretreatment of cu surfaces is employed prior to bonding or film growth to remove native oxides and organic contaminants and to smooth calendering induced roughness. for example, agents such as ammonium persulfate or acetic acid improve surface cleanliness and planarity, thereby promoting uniform nucleation, enlarging effective contact area, and reducing interfacial resistance and voiding. process conditions must avoid overetching that can introduce pits [76]. fig. 4 is a scanning electron microscopy (sem) image showing the differences before and after two chemical pretreatments (ap: ammonium persulfate, aa: acetic acid) on a single slide. the top row cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 929 (a–c) shows the surface of the pretreated cu foil itself, while the bottom row (d–f) shows the graphene surface grown on each cu foil. as shown in fig. 4 (a), untreated cu foil exhibits distinct stripe patterns and poor flatness due to marks from the commercial calendering process. (b) pretreatment with ap reduces the stripe patterns, significantly improving flatness, but in areas where the oxide layer is not sufficiently dense, the solution directly etches the cu itself, resulting in small particles/protrusions on the surface. (c) pretreatment with aa results in milder etching compared to ap, avoiding direct etching of the cu substrate, thereby achieving better flatness without small protrusions. in summary, surface flatness follows the order of aa treatment > ap treatment > untreated. (d) graphene grown on untreated cu shows numerous white spot like impurities, which are interpreted as particles formed by the melting and aggregation of cu surface oxides during the growth and cooling process. (e) graphene grown on ap pretreated cu shows a reduction in the number of white particles, confirming the oxide removal effect, but they are still scattered. (f) graphene grown on aa pretreated cu has a very smooth surface with extremely few impurities, making it the cleanest of the three in terms of surface cleanliness. in summary, fig. 4 illustrates the effect of pretreatment agents on cu surface flatness and oxide removal, and how these results directly impact graphene surface cleanliness, as shown intuitively by sem. aa produces the most uniform and clean cu/graphene surface, while ap is effective but leaves some overetching and residual particles. through this study, we can understand why chemical pretreatment is applied to metal surfaces. fig. 4 sem of (a) pristine copper foil without pre-treatment; (b) copper foil treated with ap solution; (c) copper foil treated with aa solution; (d) graphene grown on pristine copper foil without pre-treatment; (e) graphene grown on copper foil treated with ap solution; (f) graphene grown on copper foil treated with aa solution [76] jangam et al. [77] introduced a method that used formic acid to eliminate the oxide layer. fig. 5 depicts an integrated apparatus that both generates formic acid (hcooh) vapor and provides a treatment/bonding chamber for cu films. in this setup, nitrogen is 930 s. choi, c. chen, b. hwang passed through a reservoir of concentrated formic acid (98% hcooh) to create a dilute vapor stream, approximately 1% hcooh in 99% n₂, which is delivered to the chamber designed for surface treatment and subsequent bonding. the cu sample is mounted on a heated stage; prior to dosing, the chamber is evacuated to ~5 pa and the sample temperature is raised to 200 °c to establish a low oxygen environment and activate surface chemistry [78]. once the mixed hcooh/n₂ stream enters the chamber and impinges on the heated cu surface, gas phase hcooh adsorbs and dissociates to form surface formate and hydrogen species; near 200 °c the formate further decomposes to release co₂ and atomic hydrogen. the generated hydrogen reduces native copper oxides to metallic cu with evolution of co₂ and h₂o, thereby restoring a bondable surface. the volatile byproducts are continuously removed through the exhaust, and—using the same apparatus shown in fig. 5—the treated surfaces are subsequently brought into contact and bonded at 200 °c under load. this technique allowed for localized cleaning, completing the process in less than 10 s, thereby enhancing the overall efficiency. these reactions occurred between 100 and 150 °c, when formic acid reacted with the cuo layer to produce copper formate and vapor. above 200 °c, copper formate decomposed into co₂ and h₂ gas, leaving behind pure cu metal. after the oxide was removed, cu-cu bonding was performed at 240 °c for 5 s, resulting in excellent bonding with an average shear strength of over 150 mpa [78]. fig. 5 schematic of formic acid vapor generation and cu sample treatment/bonding machine. reproduced from [78] in addition to formic acid, other acids such as sulfuric [79], acetic [80], citric [81], hydrochloric [74, 75], and fluorine-containing plasma [65, 82] have been used to remove oxide layers. however, the use of acidic solutions is not always suitable for complementary metal oxide semiconductor (cmos) devices. in the cmos industry, acidic solutions are generally avoided because they may negatively affect the device performance and reliability. cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 931 3.3. cu-cu mechanical bonding by passivation 3.3.1. pd and sn passivation to enhance mechanical reliability of cu-cu bonding as mentioned earlier, acidic substances such as acetic acid can remove the cuo film, but they also etch the cu layer, making the removal of the oxide layer difficult to control [80, 83]. to address this issue, many researchers have focused not only on removing the oxide layer from preoxidized cu surfaces but also on preventing further oxidation [84-87]. the passivation method allows bonding at lower temperatures, eliminating the need for sputtering deposition and vacuum bonding, thereby offering a more efficient and costeffective solution. however, previous studies that used passivation for research purposes primarily relied on sputtered cu for bonding [84, 88]. sputtered films tend to have a lower roughness and higher uniformity than those formed by other deposition techniques. despite this, pd passivation stands out because of its stable electrical performance, reliability at low temperatures, and ability to resist corrosion and degradation. tsai et al. [89] used pd passivation to overcome the challenges of cu oxidation and facilitated cu-cu bonding at reduced temperatures. in their approach, a 50 nm ti layer and 200 nm cu seed layer were initially deposited onto a si wafer. a 2 µm photoresist layer was then coated and patterned on the seed layer, followed by electroplating to form the cu redistribution layer (rdl). after another lithography process to enhance the photoresist, a 10 µm photoresist layer was applied and patterned for cu pillar micro bumps, and 5 µm high cu pillars were electroplated. the wafer was then cleaned with dilute sulfuric acid to remove the surface oxides on the cu pillars, and a 10 nm pd passivation layer was deposited to prevent reoxidation. the bonding process was carried out at 1.91 mpa and 150 °c for 50 min, with an operating pressure of 7 × 10⁻³ torr and base pressure of 1 × 10⁻⁶ torr, without additional surface treatments. the structure with ti passivation exhibited a noncontact resistance of 7 × 10⁻⁴ ω·cm², while the structure with pd passivation showed significantly lower resistance at 2 × 10⁻⁷ ω·cm². reliability tests confirmed that the pd passivated material offered a high resistance to temperature and humidity degradation [89]. kung et al. [90] addressed cu surface oxidation and height nonuniformity by introducing a thin sn passivation layer atop electroplated cu and then employing lowtemperature, low-pressure thermo-compression to enable direct cu-cu bonding without cmp. sn’s low melting point (232 °c) and high deformability allow it to conformally bridge pad-to-pad topography, thereby accommodating surface roughness and pillar height mismatch while promoting intimate metallic contact. the process flow comprised sputtering a 50 nm ti adhesion layer and a 300 nm cu seed on si, electroplating ~5 µm cu, and depositing sn at controlled thicknesses of 1 µm, 800 nm, and 600 nm as the principal variable. after acetone and 50 vol% hcl cleans, wafers were bonded in ~10⁻² torr ambient with a 40 °c min⁻¹ ramp, 1 min dwell at the target temperature, and 1–2 mpa applied pressure, followed by n₂ purge cooling. two temperature windows were emphasized: 220 °c (solid sn, minimizing collateral reflow risks) and 250 °c (transient liquid assisted flow, maximizing gap fill and planarity compensation). fig. 6 (b) establishes the governing defect formation mechanism when sn is insufficient to fully mediate roughness during intermetallic growth. as sn is consumed, scalloped cu₆sn₅ and smoother cu₃sn emerge at the reaction front; if the residual sn thickness is too small, the opposing intermetallic compound (imc) surfaces first touch at asperities, concentrating load locally and leaving low pressure regions that preserve micron scale gaps. this “sn depletion → roughness amplification → stress concentration” pathway explains 932 s. choi, c. chen, b. hwang the observed transition from isolated microvoids to extended unbonded lines and large voids in thin sn regimes. experimentally, 800 nm sn at 220 °c even under 2 mpa produced large voids attributable to early rough imc contact, whereas 250 °c promoted more complete imc formation and improved integrity but remained thickness limited for gap accommodation. at 600 nm sn, residual micro gaps persisted at both temperatures despite raising pressure to 2 mpa, underscoring the need for a minimum sn budget to suppress roughness induced defects. fig. 6 schematic of (a) the cu-to-cu direct bonding using sn as the passivation layer and (b) imc formation and microstructure evolution of the cu-sn bonding when sn is not thick enough quantitatively, the most robust condition was 1 µm sn bonded at 250 °c and 1 mpa, yielding ~0.9% areal void fraction with ~13% unbonded length. under otherwise comparable settings, 1 µm sn at 220 °c (1 mpa) and 800 nm at 250 °c (2 mpa) both produced ~1.5% voiding with ~20–25% unbonded length, while the worst case—800 nm sn at 220 °c and 2 mpa—exhibited ~4.7% voiding and ~37.5% unbonded length due to the large void mechanism highlighted in fig. 6 (b). surface planarization strongly mitigated these fig. 6 (b) type defects: the rms (rq) of cu surface roughness dropped from ~22 nm (as plated) to ~1.3 nm (polish + cmp) and peak-to-valley height variation from ~0.96 µm to ~0.28 µm, driving, for example, the 800 nm/220 °c case from ~4.8% voiding and ~37.3% unbonded length down to ~0.3% and ~7.35%, respectively; 800 nm/250 °c then cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 933 met targets even at 1 mpa, and 600 nm/250 °c/2 mpa became acceptable. cross-sectional transmission electron microscopy (tem) further supports the microstructural soundness: at 1 µm/250 °c/1 mpa/1 min, a cu/cu₆sn₅/cu₃sn/cu₆sn₅/cu stack formed without interfacial flaws; at 800 nm/250 °c/2 mpa, the imcs converted fully to cu₃sn with clean cu-cu₃sn interfaces, consistent with literature reported high shear strengths for cu/cu₃sn/cu joints [90]. this study demonstrates that transient liquid assisted bonding with ≥1 µm sn achieves high-quality cu-cu interconnects at 250 °c and 1 mpa within 1 min, without cmp. a prudent caveat is that the optimization relies primarily on structural metrics (voiding, unbonded length, tem); for process window definition in product settings, complementary electrical contact resistance and long-term reliability under coupled thermo-humidity and thermal cycling loads should be quantified under the same conditions. in summary, the use of pd and sn as a passivation layer significantly reduced the contact resistance, resulting in superior electrical properties suitable for interconnect applications. these studies demonstrated that pd and sn passivation facilitates high-quality cu-cu bonding at low temperatures with stable electrical performance, making it ideal for low-thermal-budget applications in 3d integration. 3.3.2. au passivation and cr/au passivation in addition to pd passivation, we explored cu-cu bonding using au passivation. the key advantage of au passivation is that it enables bonding at a lower temperature than pd passivation, while improving the electrical properties and providing excellent bonding strength in tensile tests. moreover, au passivation can be applied directly at both the wafer and chip levels. however, one drawback is the formation of imcs during bonding, which exhibit poor electrical properties, low structural stability, and require large bump pitches, negatively affecting the bonding process. there are two methods for passivating ag or au: one uses sputtering and an e-beam [91], and the other uses a metal capping layer on top of rdl + low-temperature tcb [92, 93]. liu et al. [92] proposed using au as a passivation material. they conducted bonding at a low temperature of 150 °c for 30 min under a vacuum of 0.01 torr through tcb. during bonding, cu diffused through the au layer to the bonding interface, forming a strong bond. no imcs were observed, probably because of the difficulty in nucleation at low temperatures. the mechanical strength was evaluated via a pull test, which revealed a bonding strength of 0.94 mpa, indicating a high-quality bond. the electrical performance was also excellent, with a noncontact resistance of approximately 10⁻⁸ ω·cm², regardless of the contact size. liu et al. [93] discovered that a cr wetting layer between the rdl and au passivation layer reduced the bonding temperature to 70 °c. they found that bonding could be completed in 15 min at 100 °c for wafer-level applications and in 180 s at 70 °c for chiplevel applications. aes confirmed the diffusion of cu atoms from the cr wetting layer to the bonding surface. for comparison, atomic force microscopy (afm) was used to measure the rms surface roughness of various samples, including cu surfaces with 2 nm cr and 8 nm au layers, cu surfaces with 10 nm au layers, and cu rdls. the rms values were 3.2 nm for cu, 2.75 nm for cu/au, and 2.55 nm for cr/au. these findings demonstrated that the cr/au layer resulted in a more uniform surface roughness, thereby improving the bonding quality and minimizing voids. scanning acoustic tomography and sem further confirmed the superior bonding quality of the passivated layer, compared with that of the layers without passivation. 934 s. choi, c. chen, b. hwang a chip pull test was conducted to assess the mechanical strength, and revealed that the bonding strength at the wafer level exceeded the mechanical strength of the adhesive (11.89 mpa), as cracks appeared at the interface between the fixative adhesive and silicon substrate. the success of the chip-level bonding, as a function of temperature and time, is presented in table 3. table 3 bonding success as a function of temperature and time 180 s 60 s 30 s 15 s 200 °c success 180 °c success 150 °c success success success 120 °c success success success fail 100 °c success fail 70 °c success fail 60 °c fail despite the cr wetting layer, the noncontact resistance increased to approximately 10⁻⁷ ω·cm² at supply currents ranging from 0 to 100 ma. additional tests were conducted to assess the mechanical strength of chip-level bonding with au and cr/au passivation layers at 200 and 120 °c, as presented in table 4. table 4 bonding strength of au and cr/au passivation at 200 °c and 120 °c passivation parameter (temperature/time) force (mpa) au 200 °c/180 s 4.69 au 120 °c/180 s 1.19 cr/au 200 °c/180 s 5.11 cr/au 120 °c/180 s 1.75 as shown in table 4, the bonding strengths of chips with cr/au passivation are superior to those of the chips with au passivation at both 200 and 120 °c. this demonstrates that cr wetting and au passivation can improve the surface roughness, enhance the bonding quality, and ensure strong mechanical performance at both the chip and wafer levels. thus, we can conclude that au and cr/au passivation not only protects cu from oxidation prior to bonding at ultralow temperatures but also facilitates cu diffusion to the bonding surface. in particular, cr/au passivation enhances the surface roughness, further improving the bonding quality. lee et al. [91] reported a representative example of low-temperature cu-cu hybrid bonding enabled by noble-metal passivation nanolayers, specifically au and ag, deposited on cu pads for cu/sio₂ hybrid bonding applications. in their process, ~12 nm au and ~15 nm ag were formed by e-beam evaporation on 1 µm cu films, with the nanolayers acting simultaneously as oxidation barriers and diffusion mediators during bonding at 180 °c followed by annealing at 200 °c. fig. 7 (d) in their work schematically illustrates the heating test used to quantify diffusion between the passivation metal (au or ag) and cu. a ti adhesion layer is first sputtered on thermally oxidized si, followed by a 1 µm cu film, on top of which a nanometer-scale au or ag layer is evaporated. this stack is then subjected to isothermal heating at 25, 100, 150, and 200 °c for 1 h, and the evolution of the surface chemistry and depth profile of au/ag, cu, and o is analyzed by x-ray photoelectron cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 935 spectroscopy (xps) and tem. through this protocol, lee et al. [91] extract diffusion activation energies for both au↔cu and ag↔cu, showing that au diffuses more readily into cu than cu into au, whereas for ag–cu the opposite trend holds. these data position au and ag nanolayers as tunable passivation/diffusion layers that enable solid-state interdiffusion at ≤200 °c while maintaining cu surface protection during the pre-bond steps. fig. 7 (d) focuses specifically on the cu-cu bonding interface when an au nanolayer is employed as the passivation layer. sem and tem observations reveal a porous au nanolayer at the bonding interface, with the dark contrast region corresponding to cu together with oxygen that was introduced prior to bonding, and a bright narrow band associated with au along the interface. the accompanying energy dispersive spectroscopy (eds) line scan in fig. 7 (d) demonstrates a pronounced oxygen peak localized at the au/cu interface, indicating that oxygen contamination was already present at the au–cu interface before bonding and that the porous morphology of the au nanolayer allowed limited oxygen ingress without forming bulk intermetallics. nonetheless, after bonding and annealing, au is largely diffused into cu, and the interface is filled predominantly with cu and o residues rather than a distinct au layer. from the viewpoint of metal passivation engineering (e.g., au/ag/cr passivation schemes in cu hybrid bonding), fig. 7 shows that an ultrathin au passivation nanolayer can both suppress cu oxidation sufficiently for lowtemperature bonding and participate in controlled interdiffusion, while the porous nature and oxygen uptake at the interface must be carefully managed to optimize mechanical strength and interfacial cleanliness. fig. 7 (a) fe-sem and (b) tem analysis results of a bonded wafer using an ag passivation nanolayer and (c) eds scanning results and (d) schematic diagram of au and ag passivation on cu [91] 936 s. choi, c. chen, b. hwang 3.4. cu-cu mechanical bonding using cu (111) surface of nt-cu several studies have shown that cu diffusion on the (111) surface is the most favorable for cu-cu bonding [94-100]. the development of nt-cu through electrodeposition facilitates cu-cu bonding at temperatures below 250 °c by producing cu films dominated by (111)-oriented grains, which exhibit the highest surface diffusivity [101]. table 5 illustrates that the (111) surface has a diffusion coefficient three to four orders of magnitude higher than that of the other orientations. this higher diffusivity allows for lower bonding temperatures on the (111) surface. a (111) orientation for nt-cu can be achieved through electroplating or sputtering. for example, a 97% (111)-oriented cu film was fabricated by sequentially sputtering a 100 nm thick ti adhesion layer and 200 nm thick cu film onto an 8-inch si wafer using an oerlikon cluster line 300. however, the large-scale production of (111)-oriented cu remains challenging. to address this issue, the (111)-oriented single-crystal cu surface can be replaced with a highly (111)-oriented polycrystalline cu thin film. table 5 calculated cu surface diffusivity on (111), (100), and (110) planes at various temperatures ranging from 150 to 300 °c dsurf／temp. (cm2/s) (111) (100) (110) 300 °c 1.51×10-5 1.48×10-8 1.55×10-9 250 °c 1.22×10-5 4.74×10-9 3.56×10-10 200 °c 9.42×10-6 1.19×10-9 5.98×10-11 150 °c 6.85×10-6 2.15×10-10 6.61×10-12 for successful direct cu-cu bonding, at least one surface must be (111) oriented. the bonding proceeds more efficiently if both cu surfaces are aligned along the <111> axis [101]. researchers have bonded nt-cu pieces to randomly oriented cu pieces at different temperatures to study void formation. bonding at 150 °c for 1 h results in voids, but void formation is significantly reduced at 200 °c for 30 min or 250 °c for 10 min. ebsd was used to confirm the orientations of the bonded materials. the tensile strengths of five 2×2 cm² samples of 97% (111)-oriented cu films bonded at 200 °c for 1 h were tested. three of the five samples demonstrated tensile strengths above 4.3 mpa, while the remaining two showed fracture strengths of 2.7 and 1.3 mpa, confirming good tensile strength. shie et al. [102] reported that cu-cu bonding can be achieved at 300 °c in only 10 s by employing nt-cu with a <111> orientation. the nt-cu microstructure provides both excellent mechanical properties and strong resistance to electromigration. in their study, 30-µm-diameter nt-cu bumps were electrodeposited on 8-inch wafers, planarized by cmp to a surface roughness of rq ≈ 5 nm, and de-oxidized in a citric-acid solution for 30 s, followed by chip-to-chip bonding in n₂ at 300 °c under an applied pressure of approximately 90 mpa. under these conditions, the bonding interface after a 10 s process was nearly void-free; most of the (111)-oriented nt-cu grains were preserved, and only limited detwinning was observed near the passivation openings. single-joint resistance measurements using kelvin structures yielded low values of approximately 4.4–4.9 mω with only weak dependence on bonding time in the range of 10–60 s. these findings indicate that the high surface diffusivity of cu (111) at 300 °c plays a dominant role in enabling rapid bonding and that reliable cu-cu joints can be realized even when the (111)oriented surface fraction of the bumps is as low as about 40%. cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 937 lu et al. [103] fabricated approximately 6 μm-thick nt-cu films with a strong (111) texture by electroplating on a (111)-oriented cu seed layer followed by cmp, and then introduced an epoxy-induced surface modification to obtain an epoxy-treated nanotwinned cu (ent-cu) structure. as shown in fig. 8, ebsd and grain-size analyses indicate that the as-deposited nt-cu exhibits a columnar nanotwinned microstructure with a highly (111)oriented surface, whereas after repeated thermal cycling in contact with an epoxy the entcu develops a dual-layer structure in which the top ~1 μm is transformed into a finegrained, randomly oriented layer with an average grain size of about 0.2 μm, while the underlying nt-cu retains its strong (111) texture. using these films for cu-cu tcb in the temperature range of 200–300 °c, lu et al. [103] found that in the conventional nt-cu joints the bonding interface remains largely intact even at 300 °c, as evidenced in fig. 8 (a), where only a mildly zigzagged but still distinct boundary is observed with limited grain penetration across the interface. fig. 8 microstructures of initial cu films. (a) plane-view scanning ion microscopy (sim) image of nt-cu. (b) plane-view ebsd oim image of nt-cu. (c) distribution of grain sizes for the grains in (b). (d) plane-view sim image of ent-cu. (e) plane-view ebsd oim image of ent-cu. (f) distribution of grain sizes for the grains in (e) [103] in contrast, the ent-cu joints exhibit pronounced abnormal grain growth originating from the fine-grained interfacial layer; fig. 9 (b) demonstrates that large grains extend into both cu films and effectively consume the original cu-cu interface, leading to an almost interface-free microstructure after bonding at 300 °c. these observations imply that merely exposing a fully (111)-oriented nt-cu surface is insufficient for interface elimination, and that only when a highly defective, high grain-boundary-energy fine-grained surface layer is introduced on top of a (111)-textured nanotwinned underlayer, as characterized in fig. 938 s. choi, c. chen, b. hwang 9, can the interface-free cu-cu joint structure observed in fig. 9 be achieved at relatively low temperatures. the (111)-based cu-cu bonding route demonstrated by lu et al. [103] therefore offers the advantage of maintaining a reliable (111)-oriented nt-cu backbone for current conduction while eliminating the bonding interface without noble-metal interlayers, but it also suffers from the requirement of a still relatively high bonding temperature around 300 °c, increased process complexity due to epoxy thermal cycling and additional cmp, and partial loss of the (111) texture at the immediate bonding surface. fig. 9 cross-sectional sem images after bonding at 300 °c for 1 h: (a) nt-cu; (b) ent-cu [103] 4. conclusion this review explored the various factors influencing cu-cu mechanical bonding and provided a detailed analysis of the fundamental principles governing these effects. the key parameters affecting cu-cu bonding were identified as the thickness of the bonding layer, grain size, temperature, and surface roughness. increasing the cu thickness was found to enhance the bonding energy because of changes in the surface microstructure that depend on the thickness; these changes were examined in detail by introducing artificial voids and performing subsequent theoretical analyses. in addition, grain refinement was shown to suppress microcrack formation and propagation, thereby improving the bonding performance, while higher bonding and annealing temperatures promoted atomic diffusion and accelerated bond formation. with regard to surface roughness, the bonding quality improved as the roughness decreased and the true contact area increased, but beyond a critical condition void formation degraded the bonding strength. various methods for achieving low-temperature bonding, such as sab, passivation, and chemical pretreatment, have been widely studied to improve the cu-cu bonding quality, and the advantages, disadvantages, and characteristics of these techniques are summarized in table 6. at the same time, the high oxidation tendency of cu continues to impose severe constraints on the available thermal budget, giving rise to issues such as wafer deformation, increased integration complexity, and limited compatibility with front end device structures. future progress toward more reliable and manufacturable cu-cu joints will therefore depend on the combined optimization of microstructure design (including grain size, preferred orientation, and thickness), surface engineering (including roughness and oxide control), and thermo-mechanical loading paths, particularly under low-temperature and cu-cu mechanical bonding for 3d integration of the next generation electronic chips... 939 fine-pitch conditions. in addition, practical implementation in high-throughput 3d integration will require chemistries that are compatible with cmos processing, scalable cu architectures dominated by the (111) orientation or based on nanocrystalline cu, and robust inline metrology and defect screening schemes for interfacial voids and failures related to surface roughness. the continued development of such cu-cu bonding strategies under a low thermal budget is essential to fully exploit ultra-high-density interconnects in next-generation three-dimensional integration, where very high input and output density, power efficiency, and long-term reliability must be satisfied at the same time. table 6 comparison of cu-cu bonding methods bonding method bonding temperature advantages disadvantages reference sab 200 °c -250 °c high bonding quality without toxic chemical cleaning uhv environment low throughput complex process [60-64] chemical pretreatment 250 °c -300 °c low cost only wafer–wafer bonding low bonding quality [65, 74, 75, 77, 80-82] pd passivation 150 °c high bonding quality, high electrical performance and reliability limited application hard to control surface roughness [89] au, cr/au passivation 120 °c very low bonding temperature, high electrical performance high reliability imc formation [92, 93] (111) oriented nt-cu 150 °c -300 °c high cmos compatibility limitation of fine-pitch cu bump structure [101, 102] 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of medicine there is a great contribution of new techniques and technologies, which is reflected in an entire system of health care services. customized implants are both fully geometrically and topologically adjusted so as to meet the needs of individual patients, thus making each implant unique. their production requires joint efforts of a multidisciplinary team of different profile experts who combine their knowledge in the implant knowledge model. thus, we develop an expert system which should help or replace humans in the process of implant material selection. this paper gives an overview of the expert system concept for the given problem. its task is to carry out a selection of biomaterial (or class of material) for a customized implant. the model significantly improves the efficiency of preoperative planning in orthopaedics. key words: customized implant, biomaterial selection, expert system 1. introduction technological development influences all spheres of the society, especially the field of information technologies, economy, as well as the user’s needs. with constant innovation and invention, thousands of computer applications are being created every day, on various topics, available worldwide, whose functionality meets the customer needs and market demands. using the information integration capabilities of the computer integrated manufacturing system, which shortens product lead-time, improves its quality and reduces its cost, has led to the formation of multidisciplinary teams of different area experts [1]. moreover, the knowledge based technologies have provided the integration of different areas of knowledge into a single software environment. such systems are usually based on the application of received july 23, 2016 / accepted november 18, 2016 corresponding author: miloš ristić college of applied technical sciences niš, aleksandra medvedeva 20, 18000 niš, serbia e-mail: milos.ristic@vtsnis.edu.rs 134 m. ristić, m. manić, d. mišić, m. kosanović, m. mitković methodologies from the domain of artificial intelligence [2]. the most commonly used are expert systems, genetic algorithms and neural networks. their application in biomedicine is significant, both in the data monitoring systems, and in the advanced decision-making systems. in comparison to the personalization in industry, personalization in medicine has just recently begun to gain importance. personalized medicine derives from the belief that the same illnesses afflicting different patients cannot be treated in the same manner [3]. an implant is a medical device manufactured to replace a missing biological structure, support a damaged biological structure, or fix an existing biological structure [4 – 7]. implants must respond to the specific demands in patient treatment. as such, they are used in almost all the areas and fields of medicine. unlike standard orthopaedic implants, which have predetermined geometry and topology, customized implants are completely adjusted to match anatomy and morphology of the selected bone of the specific patient [8]. in this way they fully meet the needs of the patient, thus shortening a post-operative treatment period and significantly reducing adverse reactions to the acceptance of implants or possible pain. the patient-specific implant concept has been evidenced since 1996 as research of hip replacement implants for the sake of implant adaptation and customization [9]; then, since 1998, the first cases of patient-specific implants for the skull have been developed [10]. these kinds of implants are custom devices based on patient-specific requirements [11]. material selection is one of the most important steps in implant design and manufacturing. the selection and use of implant materials involve important prospective decisions [12]. each material has specific combinations and ranges of chemical, mechanical, electrical, thermal, and biologic performance characteristics. design requirements dictate material selection; however, once the material selection is made, they strongly affect the design process in both positive and negative ways [12]. the material used for implant manufacturing should, beside mechanical characteristics, be similar to the host bone with sufficient mechanical strength; it should have adequate porosity because it reduces mechanical properties such as compressive strength and resistance to corrosion [13]. the material should be reproducibly processable into a three-dimensional structure and it must tolerate sterilization according to the required international standards for clinical use [14]. moreover, the manufacturing costs of these materials should be reasonable and their implantation relatively simple, precise, and reproducible [15]. the selection of the most appropriate material, or combination of materials, is an important process in view of a large number of materials and their associated materials processes, necessitating the simultaneous consideration of many conflicting criteria [16]. the chart method, computer-aided materials selection and knowledge-based systems are common techniques in material screening. the material selection system developed by ashby [17] concentrates on the data modeling aspect of the problem by presenting the data in a chart format. cambridge engineering selector (ces) is a powerful selection and analysis tool that is based on the ashby’s materials selection methodology [18]. electre (elimination and choice expressing reality), topsis (technique for order of preference by similarity to ideal solution), and ahp (analytic hierarchy process) are utilized for material selection. fuzzy techniques have been employed either independently or with other techniques such as genetic algorithm, neural networks, kbs (knowledge-based system), and mcdm (multicriteria decision making) techniques [19]. implant material selection using expert system 135 dargie et al. [20] presented a computer-aided design system for suggesting candidate manufacturing process and material combinations. lai and wilson [21] suggested interactive computer program and artificial intelligence techniques to select candidate material and primary process combinations for a part, during the early stage of design. bamkin and piearcey [22] justified the development of a ‘design assistant’ program for the selection of materials according to knowledge-based system. sapuan et al. [23] demonstrated application of knowledge-based system in material selection of ceramic matrix composites for engine components. moreover, zha [24] described the work of selecting suitable manufacturing processes and materials in concurrent design according to a fuzzy knowledge based decision support method. in this paper, the knowledge based system for implant material selection by the production rules has been developed. in order to make reasoning closer to human nature, the system allows work with some uncertainties due to fuzzy logic. the implant material selection using expert system can be made by rankings properties such as strength, formability, corrosion resistance, biocompatibility and low implant price [19]. the application of quantitative decision-making methods for the purpose of biomaterial selection in orthopedic surgery is presented in the paper [25]. a decision support system based on the use of multi-criteria decision making (mcdm) methods, named mcdm solver, is developed in order to facilitate the selection process of biomedical materials selection and increase confidence and objectivity [26]. based on these research results, we propose an expert system. bearing in mind that the implants are complex geometric forms, the most commonly used method for their design is reverse engineering [27]. this paper presents an example of the expert system which is a decision support system used for the selection of materials, applied to the orthopaedic implants design. therefore, in the definition of the implant model, the implant knowledge is additionally inserted in the form of facts, which actually define a knowledge model about the implant. this knowledge, connected by appropriate relations to the rule databases for the material selection (or the material class selection), provides the prerequisites for the start of the customized implant material selection process. 2. expert system for implant material selection expert systems are meant to solve real complex problems by reasoning about knowledge which would normally require a specialized human expert (such as a doctor, e.g. orthopaedic surgeon). the typical structure of an expert system consists of: a knowledge base, an inference engine and an interface. since in the expert system the decision making process and the knowledge base are separated, parts of knowledge within the knowledge base can be easily supplemented or modified. the knowledge base contains rules, which describe the knowledge and work logic of a particular field expert. the task of the expert system presented in this paper is to recommend a suitable material to meet the requirements of a customized implant, and then to decide on the selection of the manufacturing technological process. this expert system is actually a rule-based application implemented by the jess rule engine [28]. 136 m. ristić, m. manić, d. mišić, m. kosanović, m. mitković 3. implant knowledge model for the needs of a missing bone part, a geometrically precise and anatomically conforming 3d model of a customized implant is designed. such model requires 3d bone model reconstruction, for which the implant is intended, most commonly on the basis of an incomplete bone image [29]. fig. 1 [30] presents the model of a tibia bone where the upper selvage is lacking. the upper selvage is designed to replace the missing part of a bone in the form of a volumetric bone implant. fig. 1 the model of proximal tibia and the missing customized bone implant [30] the presented model contains geometrical data which can be easily transferred from the model tree into the knowledge implant model, and, when necessary, be used in the work of a material selection system. in order for the expert system to begin its work, it is necessary for the implant model knowledge to be designed. the basic building block of every expert system is knowledge. knowledge in expert system consists of facts and heuristics. while heuristics is made of rules of judgment based on experience or intuition (tacit knowledge domain), the facts are widely distributed and publicly available information that are agreed upon at the expert level in subject areas (explicit knowledge domain). for a successful work of our expert system it is necessary to ensure an adequate knowledge transfer (fig. 2) from the field expert to the knowledge engineer, so that the engineer could insert accumulated knowledge in the knowledge base. fig. 2 knowledge transfer from an expert to an expert system knowledge base in order for a resulting database of expert knowledge to have its function, it needs to be connected, on one side, with the specific problem database (in our case it is the knowledge model about customized implant), and on the other, with reasoning mechanisms (which is a part of the expert shell). the following table gives a part of the knowledge base about customized implants. this knowledge base is adequately fulfilled by orthopaedist and engineers who have designed and manufactured the implant. since these parameters are implant material selection using expert system 137 essential, it is important to present a knowledge model about the customized implant with the facts, characteristics, as well as with the description of the facts or the definition of certain parameters values. table 1 query on the volume implant patient gender: male age: 56 cause: bone damage patient weight: 84 kg type of injury: disease cause of injury: cancer bone: tibia part of the bone: lateral proximal tibia should implant be inserted by internal or external fixation? internal fixation is implant permanent or temporary? permanent implant volume 10-15 cm3 in what way will the implant be fixed? with screws with how many screws and which type of screws? 2 or 3, depending on the patient age what is the connection of the implant and the adjacent tissue? towards bone (trabecular bone) should the implant have the same surface quality towards adjacent tissue or is it different in the area where it faces the bone, in the part where it is connected to cartilage/muscles…? cavities should be 500-900m soft tissues do not ingrown (lower roughness, polished surface) how much load should the implant endure during lifetime? high biocompatibility very high sterilizability very high the query shown in the table 1 was used for data acquisition from experts in this field. based on this knowledge, the rules were formed and written. for execution of the rules, we used the expert shell jess, a rule engine and scripting environment written entirely in the java language. the query can be sent to doctors, engineers and other experts so that they can give their suggestions and examples. the data collected this way can later be integrated into the system and also used for further development and improvement of the system. the system will, depending on the input values, show these suggestions as well as explanations why some material is selected. 138 m. ristić, m. manić, d. mišić, m. kosanović, m. mitković 4. biomaterial class knowledge base and example of decision–making process as there is no universal or optimal material, whose characteristics fit each implant model, it is necessary to choose from a large number of available biomaterials the one that, according to certain specific requirements, fully corresponds to the model. on the other hand, a wide range of materials ensures that the materials belonging to different classes of biomaterials will have certain properties. in order to decide upon the selection of a concrete material, it is often necessary to predict such a conflict resolution that will clearly define the procedure for determining priorities; thus, the process of material selection will be fully defined. the structure of the described expert system consists of 3 modules: a module for material class selection, a module for material type selection, and a module for customized implant manufacturing technology selection. the first module for biomaterial class selection is shown in fig. 3. based on the recognized class of materials, we can further narrow our search by selecting the specific material for implant manufacturing. in the module for customized implant manufacturing technology selection of the designed expert system, the manufacturing technology is determined according to available resources, restrictions and applicable technologies. fig. 3 model of an expert system module for material class selection in table 2 the rules for material class selection are given [31]. for defined parameters in the form of facts, there are three classes of biomaterials presented and their comparison is in a certain value range. after integrating the knowledge about the model, and the biomaterial classes and other necessary knowledge models, in the expert system, the user of such a proposed system, e.g. a doctor, can select material (or material class) for the customized implants. implant material selection using expert system 139 table 2 rule base on material classes (extract) [31] t en si le m o d u lu s y ie ld s tr en g th u lt im at e te n si le s tr en g th s tr ai n t o f ai lu re d u ct il it y t o u g h n es s r es is ta n ce t o i n v iv o a tt ac k l o ca l h o st r es p o n se ( b u lk ) m an u fa ct u ri n g l o ca ti o n metals m h h m m h l h o ceramics h / m l l m h l i/o polymers l l l h h l m m o explanations l – low; m – intermediate; h – high; o – out i – in i/o – in and out by inserting this knowledge in jess a code is in the following form: (defrule choose_m (and (or (not (feature_has_value (feature tm))) (feature_has_value (feature tm) (value m))) (or (not (feature_has_value (feature ys))) (feature_has_value (feature ys) (value h))) (or (not (feature_has_value (feature us))) (feature_has_value (feature us) (value h))) (or (not (feature_has_value (feature sf))) (feature_has_value (feature sf) (value m))) (or (not (feature_has_value (feature e))) (feature_has_value (feature e) (value h))) (or (not (feature_has_value (feature dt))) (feature_has_value (feature dt) (value m))) (or (not (feature_has_value (feature ut))) (feature_has_value (feature ut) (value h))) (or (not (feature_has_value (feature hrc))) (feature_has_value (feature hrc) (value m))) (or (not (feature_has_value (feature d))) (feature_has_value (feature d) (value h))) (or (not (feature_has_value (feature r))) (feature_has_value (feature r) (value l))) (or (not (feature_has_value (feature lhr))) (feature_has_value (feature lhr) (value h))) (or (not (feature_has_value (feature m))) (feature_has_value (feature m) (value h))) (or (not (feature_has_value (feature pp))) (feature_has_value (feature pp) (value p))) ) => (printout t "choose metal" crlf) (assert (materialclass (name metal))) ) as a result jess has, based on the criteria given by the user and the defined rule base, selected the biomaterial class [30]. in this scenario the suggested solution is the metallic biomaterial (fig. 5). 140 m. ristić, m. manić, d. mišić, m. kosanović, m. mitković fig. 5 material class recommended by jess [30] biomaterial class recommended by jess (fig. 5) is further presented to the user through the user interface. 5. implant material selection module for material class selection has shown a shortcoming as it cannot work with uncertain values. the proposed material class is, for the given criteria, better than the other classes, but there is no solution ranking capability which would indicate the extent to which the given solution is more acceptable. this is the reason why the material selection module was designed. this module also introduces the principles of a fuzzy expert system. when defining certain values, the experts use linguistic expressions more often than numerical values. thus, the statements become more general and imprecise, but are, as such, more understandable to the interlocutor. for example, for a doctor or an engineer who needs to describe the characteristics of the material it is much easier to quantify their values by using the linguistic expressions such as "the price of materials is low and tensile strength is extremely high," than to quantify his/her evaluation "the price is 3.7 € / kg, a tensile strength 1200 mpa." the fuzzy approach, in addition to relaxation, is characterized by softness, gradual transition from one to the other extreme, for example, from small, medium to large biocompatibility of materials. in the fuzzy logic, the statement is true to some degree. the fuzzy logic allows linguistic statements to be computer processed and, therefore, the technologies that use a fuzzy approach (fuzzy technologies), are considered human oriented. for each material, in addition to standard data such as name, group, chemical composition, status etc., the values of material characteristics, such as modulus of elasticity, ultimate elasticity, fatigue, tensile strength, density, biocompatibility, and other characteristics presented in appropriate units, are defined. in order to facilitate the insertion and updates, a module has been designed that inserts the values of the fuzzy variables from excel files. this enables people without any programming skills to change and add new materials in the excel file. new values and materials will be automatically read when the system restarts or runs again. each characteristic of the material has its defined minimum and maximum value as well as a fuzzy set of values that it uses for the proper linguistic value. in addition, each fuzzy value is defined by its membership function (triangle, rectangle, trapezoid or other) and the domain over which is defined. other functions such as sigmoid or gaussian curve can be used depending on the needs of the application. the example of fuzzy values is given in the fig. 6. implant material selection using expert system 141 fig. 6 the fuzzy values of material characteristics all the information about material and fuzzy values of the characteristics (features) are added to the jess file. rete.batch(clpfile); rete.reset(); rete.add(miv); rete.eval("(facts)"); rete.getglobalcontext().setvariable("lmsg", new value(lmsg)); for (material m:lmaterials) { rete.add(m); for (feature f:lfeatures) rete.add(mf.fv); afterwards we call the execution of the appropriate jess file which in this case is modul3.clp. rete.run(); jess attributes enable us to add and change rules in jess scripts without the need to change or compile the entire application. an example of a rule is: (defrule implant_volume_ex_low (materialinputvalues (implant_volume "ex_low" )) => (call ?lmsg add "consider the possibility of non-implementation of bone implant and consider the possibility of implementation of a scafold") this rule will be activated if the linguistic value of the implant volume is low. a message will be added in the list of messages informing the user that it is necessary to reconsider the implementation of bone implant and suggest the implementation of a scaffold. appropriate weight factor is assigned to each material characteristic. default value for each weight factor is assigned by the knowledge engineer and can be further modified by writing the jess rules for a specific case. the resulting score function for material quality then multiplies each of the obtained values for material features by weight factor and sums up these values. 142 m. ristić, m. manić, d. mišić, m. kosanović, m. mitković 1 n score i i i f f w    (1) by activating the rules, the system has performed material base search and then ranked the materials. an overview of the application results is presented in the fig. 7. using the resulting score function (equation 1) where fi is the truthfulness of fact for a specified characteristic, and wi is the weighting factor of that characteristic, a material candidate list is obtained, which is presented in a descending order starting form from the best solution (with the highest score function) downwards. by applying additional rules for displaying only materials in a certain range, only those materials that meet a desired range of values can be presented. fig. 7 recommended implant material by expert system presented results show that the optimal solution for customized implant material is ti6a14v alloy. second suggested solution is ti29nb13ta4.6zr alloy that has slightly better biomechanical characteristics, but, on the other hand, also a higher price which negatively influences the final score for the second material. 6. conclusion this paper presents the concept of the expert system applied to the decision making process for an appropriate selection of the material for customized implants. in order to choose a suitable material from a group of candidate materials, the system for material selection based on the expert system technology was developed. due to the rule based system and fuzzy logic a framework for fuzzy expert system for implant material selection development was created. the user interacts with the system through the user interface and defines certain parameters. thus he communicates with interface engine which, on one hand, reads the facts from a database or excel file into the working memory, and on the other, uses if-then rules that represent accumulated knowledge. by activating these rules and procedures, the expert system actually makes set of steps that ultimately provide a decision. at the moment the system considers 27 possible materials, and provides possibility for new material insertion by simply editing excel file. the rule base consists of 24 simple rules that illustrate possibilities of this system. new rules can be added, which makes this system adaptable. the system is designed as an open one for upgrading the knowledge base and the rule base; it gives a good basis for development of quality and applicable system for practical use. implant material selection using expert system 143 the presented system was successfully tested on a customized volumetric bone implant model. for the developed implant knowledge model, the expert system suggested a list of materials. this list of materials reflects the clinical practice experience data, and thus the expert system work results are verified. further development can secure the creation of software tool that can be used for educational purposes as the system can provide suggestions and explanations that normal human expert cannot. on the other hand, the system can be used to help, support, optimize and improve a complex decision making process for choosing customized implant material and manufacture technology. acknowledgements: the paper is a result of the project iii 41017, supported by the ministry of science and technological development of the republic of serbia. authors express their gratitude to sandia national laboratories from usa, for the license to use jess software for academic use, research agreement for jess, no. #15n08123. references 1. guangleng, x., yuyun, z., 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titanium alloy (ti6al4v) mehmet ali guvenc1, hasan huseyin bilgic2, selcuk mistikoglu1 1iskenderun technical university, department of mechanical engineering, hatay, turkey 2necmettin erbakan university, department of aeronautical engineering, konya, turkey abstract. in recent years, with the development of sensor technologies, communication platforms, cyber-physical systems, storage technologies, internet applications and controller infrastructures, the way has been opened to produce competitive products with high quality and low cost. in turning, which is one of the important processes of machining, chatter vibrations are among the biggest problems affecting product quality, productivity and cost. there are many techniques proposed to reduce chatter vibrations for which the exact cause cannot be determined. in this study, an active vibration control based on the sliding mode control (smc) has been implemented in order to reduce and eliminate chatter vibration, which is undesirable for the turning process. in this context, three-axis acceleration data were collected from the cutting tool during the turning of ti6al4v. finite impulse response (fir) filtering, fast fourier transform (fft) analysis and integral process were carried out in order to use the raw acceleration data collected over the system in control. the system is modeled mathematically and an active control block diagram is created. it is observed that chattering decreased significantly after the application of active vibration control. the surface quality formed by the amplitude of the graph obtained after active control has been compared and verified with the data obtained from the actual manufacturing result. key words: chatter, sliding mode controller, active vibration control, turning, ti6al4v 1. introduction in recent years, smart manufacturing applications have found a wide range of uses in machining [1–3]. increasing the quality in the machining process is possible by reducing and controlling the vibrations (chatter) that occur during production. chatter vibrations in received july 28, 2021 / accepted november 10, 2021 corresponding author: selcuk mistikoglu department of mechanical engineering, iskenderun technical university, 31200 hatay, turkey e-mail: selcuk.mistikoglu@iste.edu.tr 308 m. a. guvenc, h. h.bilgic, s.mistikoglu turning and milling, which are among the most widely used methods in machining, cause important problems and directly affect the cost and product quality [4–7]. chatter vibrations occurring during machining may cause early completion of the life of the cutting tool, increase in surface roughness of the workpiece and damage to machine parts. chattering is a classic problem that occurs frequently and reduces production efficiency. the operator often encounters it while its cause is not entirely clear. in a good quality turning process, it is desirable that the chip thickness should be standard at the beginning and end of production. however, chatter vibrations cause the thickness of the chip in turning to be variable. this negatively affects the roughness of the machined surface and causes noise during turning [8-10]. these chatter vibrations, which are constantly encountered in machining applications, should be avoided without sacrificing quality and efficiency [11]. for this reason, the identification of chatter vibration and chatter suppression has been a subject of interest to researchers working in the field of machining. to overcome these problems, various techniques have been developed by researchers to detect, reduce and control of chatter vibration. techniques suggested in the literature include determining of the effects of machining parameters on product quality, prediction of machining parameters for the best product quality, optimization of machining parameters, tracking/monitoring of the machining process, identification/detection of chatter vibrations, real-time control of chatter during machining, and also providing suitable numerical tools for modeling those effects [12-15]. passive control of the chatter vibrations during turning is an easy, applicable and efficient approach with correct element selection. the basis of the reduction of chatter vibration by passive control is the spring and damping element design suitable for the system frequency and amplitude [4]. however, the trend towards active control applications with different methods for a higher product quality has increased. in active vibration control applications in turning, there are stages of measuring the vibrations occurring during processing accurately, filtering/analyzing the measured data and creating an external impact force that can damp down the chatter vibration [16-18]. there are many studies in the literature on the determination, identification, prediction and control of chatter. alternative techniques to reduce chatter have been studied by j. munoa et al. they proposed progressive solutions to design and control approaches and examined the success of the existing methods used to improve the cutting process [4,16,19]. in the study in which passive control and active control applications and future expectations were discussed, it is argued that active vibration control promises hope for the future in reducing chattering [20]. the chatter vibration generated during turning has high frequency and low amplitude. therefore, sensors, data acquisition devices and actuators used in active vibration control must have high speed and precision. in a study conducted to measure the vibration values that occur on the cutting tool and are constantly increasing, jang and tarng used piezoelectric sensors and measured and compared the controlled and uncontrolled vibration frequencies occurring in the cutting tool. they observed that the controlled vibration frequency values are 90% less than the uncontrolled vibration frequency values [21]. magnetorheological (mr) and electro-rheological (er) fluids are among the systems commonly used in active vibration control applications. mei et al. performed mr-based active vibration control for the boring process and achieved great success [22]. in addition to these, piezoelectric patches and actuators have expanded the application area thanks to their high operating frequencies and precision [23]. harms et al. designed and implemented a piezoelectric-based cutting tool for active control on the lathe [24]. haifeng identification of chatter vibrations and active vibration control by using sliding mode controller... 309 et al. implemented a piezo-based active vibration control to suppress chatter vibrations that occur during the turning process [25]. as seen from the studies given above and in literature, there are many studies on the identification, examination and control of chatter vibrations occurring during turning. however, it has been seen that there is not sufficient study on the determination of chatter vibrations during the turning of the ti6al4v alloy and the active control of these vibrations. it has been observed that the active chatter suppression studies in turning are generally carried out with classical control applications. by this paper, it is aimed at identification of chatter vibrations and active control of chatter by using smc which is known robust, stable and optimal for ti6al4v. in this study, an active vibration control application working with a sliding mode based has been implemented in order to reduce the chatter, which adversely affects product quality, tool life and productivity in turning, during production. in this context, three-axis acceleration data were collected on the tool during machining, the collected data is filtered, and the double integral is taken for the displacement data, and then chatter analysis was performed using the fast fourier transform (fft) method. active force control is carried out with the help of the mathematical model created. 2. experimental study in the experimental studies, a quantum turning lathe and ti6al4v titanium alloy workpiece with dimensions of ø28 mm x 250 mm are used. experiments were carried out with the processing parameters determined for this alloy. the image of the experiment set is given in fig. 1. the constant parameters used in the experiments are given in table 1. table 1 the constant parameters used in experimental studies workpiece material ti6al4v cutting tool tungsten carbide depth of cut, a (mm) 1 cutting speed, v (m/min) 50 feed rate, vc(mm/rev) 0.3 spindle speed (rpm) 1200 ti6al4v (titanium grade 5) ti alloy used as a workpiece has a wide range of uses such as military equipment, navy, protection, aerospace, medical technology, ultrasonic thanks to its low specific gravity and high mechanical properties [26–28]. ti alloys are generally selected according to phase transformations and ti6al4v alloy is in the α/β group. this alloy is often seen in aircraft landing gear, gas turbines, and implant applications. it is possible to see that ti6al4v alloy is frequently produced by casting, forging and machining methods, as well as powder metallurgy [29,30]. the chemical composition and mechanical properties of ti6al4v alloy are given in table 2 and table 3, respectively. 310 m. a. guvenc, h. h.bilgic, s.mistikoglu fig. 1 experimental setup for vibration measurement table 2 the chemical composition of ti6al4v (%) al v c fe o n h ti 5,5-6,5 3,5-4,5 0,08 0,25 0,13 0,03 0,012 rest table 3 mechanical properties of ti6al4v mechanical properties density (g/cm3) 4,45 specific heat (j/(g x k)) 0,56 thermal conductivity (w/(m x k)) 7,1 elasticity modulus (kn/mm2) 114 thermal elongation (10-6/k) 8,9 tensile strength (mpa) 892 yield limit (mpa) 828 elongation (%) 10 2.1. theoretical background there are two types of vibration in machining: spontaneous and forced vibration. these vibrations cause deterioration of the surface quality. it also wears the cutting tool, shortening its life and damaging the components of the bench. forced vibration is a vibration that occurs as a result of the mechanical movements of the machine. the spontaneous vibration, on the other hand, occurs separately from the machine and external factors due to chip removal. this vibration is also called chatter vibration [12,31]. chatter vibrations, which are seen as spontaneous vibrations during turning, are caused by machining forces, chip cross-section and changes in tool geometry. due to the complex nature of the chatter vibrations, the cutting forces that occur during turning activate one of the structural modes of the cutting tool and the workpiece system, causing a wavy surface on data acquisition system 3 axis accelerometer workpiece identification of chatter vibrations and active vibration control by using sliding mode controller... 311 the workpiece. it is of great importance to recognize the system elements in order to identify and control the chatter vibrations. within the scope of this study, system modeling is carried out for the active vibration control application. the vibrations in the direction of the plunge axis with the highest vibration are taken into account and the mathematical model is created accordingly. the mechanism of chatter vibration is shown in fig. 2. in this mechanism, m, k, c are the tool parameters and refer to mass, spring constant and damping coefficient, respectively. fig. 2 system modeling in turning time dependent mathematical expression of sawdust is shown in eq. (1). in the equation, the displacement of the cutting point over time with concern to the position in the previous cycle is expressed as x(t) − x(t-), where hs(shown as h0(t) in fig. 2) is the thickness of the uncut chip. in the equation, it is defined as  = 60 ∕ 𝑁, where n represents the rotation speed of the spindle in rev/m. ( ) ( ) ( )sh t h x t x t = − + − (1) the equation expressing the system behavior during turning is presented in eq. (2). here, fc is the cutting force, b is the depth of cut, kf is the specific cutting coefficient. specific cutting coefficient varies depending on the physical and mechanical properties of the material. ( ) ( ) ( ) ( ) ( ( ) ( ))c f smx t cx t kx t f t k b h x t x t + + =  − + − (2) in active control applications, an external actuating force is required to keep the tool vibration at the desired level. as expressed in fig. 2, the acting force has been designed to affect the tool. in the case that the actuator force is added to the system, the mathematical model will be as in eq. (3). fact represents the impact force generated by the actuator. ( ) ( ) ( ) ( ( ) ( )) ( )f s actmx t cx t kx t k b h x t x t f t+ + = − + − + (3) the process parameters used to ensure real situations are [25,32,33]: mass (m) = 2.5 kg, stiffness (k) = 2x10 n/m, damping (c) = 1340 n.s/m and cutting coefficient (kf) = 2000 n/m2. eq.(4) is obtained if the transfer function between force and displacement for the cutting tool is defined as (s). with the addition of fact to the system for fig. 2, which is 312 m. a. guvenc, h. h.bilgic, s.mistikoglu adapted for the closed-loop feedback control strategy, the general transfer function of our system is obtained as in eq.(5) and the system transfer function in eq.(6). ( ) ( )( ( ) ( ))c actx s s f s f s= + (4) ( ) ( )( ( ( )( 1)) ( ))s f s actx s s k b h x s e f s −= + − + (5) 2 2 ( ) 1 ( ) ( ) x s c g s f s ms cs k s as b = = = + + + + (6) 2.2. data acquisition within the scope of this study, the acquisition of vibration data generated during surface turning of ti6v4al alloy workpiece is performed with dewesoft/sirius, which provides a flexible, modular, expandable and safe working environment (fig. 3a). the most important features of sirius are that it prevents errors frequently encountered during the measurement process, is simple to use and has user-friendly data processing software. a) b) fig. 3 a) dewesoft data acquisition device b) mems triaxle accelerometer mems type pcb/3713b1130g brand accelerometer has been used to collect 3-axis acceleration data on the machine tool (fig. 3b). the accelerometer has a measurement range of ±30 g and is produced with a titanium coating to prevent damage to interference data and external effects. the accelerometer has a measurement frequency of up to 1 khz. during turning, cutting force and tool vibration occur in three axes: rotation axis (radial), feed axis and plunge (depth) axis. reducing vibration in any direction increases turning reliability and product quality. however, in this study, the vibrations in the direction of the plunge axis with the highest vibration were taken into account and a mathematical model was created accordingly. when choosing the location of the sensor, a point where the vibration will occur the most and where active control can be applied is selected. for this reason, the acceleration sensor is placed on the cutting tool and in the area close to the cutting point. 2.3. data processing the collected raw data may be insufficient to reach the desired result or use in control applications. data processing is an important step in recalculating and filtering of signals, revealing signal statistics and ultimately achieving the desired result. identification of chatter vibrations and active vibration control by using sliding mode controller... 313 2.3.1. filtering of acceleration data it is of great importance to clean and analyze the signals in order to use the raw data collected from the system. filtering, one of the most basic applications of signal processing studies can be used for different purposes such as decreasing or increasing certain frequencies, decreasing or increasing the signal amplitude, cleaning and analyzing unwanted signals. filters can also be used to condition a time history signal by reducing unwanted frequency content. in active control applications, filtering processes are often used to eliminate a disturbance or model the response of a physical system. it is possible to see both iir (infinite impulse response) and fir (finite impulse response) applications in active control applications. the formulas for fir and iir are detailed in eqs. (7) and (8). 0 ( ) ( ) ( ) n k y n a k x n k = = − (7) 0 0 ( ) ( ) ( ) ( ) ( ) n p k j y n a k x n k b j y n j = = = − + −  (8) in these equations, x(n) is the input series in time, y(n) is the time series at the output, n is the total number of data points of the input signal in the time domain, with the series called a(k) filter a, n and p are the numbers of terms in the filters, also referred to as rank and number of links, respectively. for example, fir filter is implemented between k=0 and k=n. 2.3.2. integral of acceleration data it is often impossible to use the raw data collected from the system directly. especially, in order to use the acceleration data from the system in sensitive active control applications, the acceleration data must be converted into displacement data. thus, more accurate results can be obtained. the most important method used to convert acceleration data into position data is integrating [33,34]. the triaxial acceleration data collected during machining are arranged in the dewesoftx software interface and converted into displacement data to be input in active control applications. then, the displacement data were input to the control model from the same point as the point where data was collected under real conditions. 2.3.3. fast fourier transform (fft) the fourier transform, developed by the french mathematician jean-baptiste joseph fourier, argues that any function can be written as the sum of sine and cosine functions, and the amplitude of the sine and cosine waves can be determined using the integral. the basic equation used for the fourier transform is given in eq. (9). in this equation; sx(f) is the output of the fourier transform in the frequency domain, x(t), the function due to time and the frequency in radians 2πf [35]. 2( ) ( ) j ft xs f x t e dt + − − =  (9) fft can be used to determine the high-frequency values in any signal. these frequencies can cause unwanted noise or vibration. fft analysis is also widely used in the monitoring and 314 m. a. guvenc, h. h.bilgic, s.mistikoglu fault detection of the vibration state of machine elements in constant motion such as bearings and gears. fft analyses, which have an important place in the space, aviation and automotive industry, are performed by using multiples of two (512, 1024, 2048) samples in the time block [36]. the general equation for fft is given in detail in eqs. (10) and (11), and k row represents the frequency element, n number of samples, i square root (-1) value, x(i) sampled signal data, n represents the index of the sample waiting to be processed. 1 0 ,( ) 0,1,2, 1( ) n jn k n x ke nn  − − = =  − (10) 1 *(2 * * / ) 0 [ ]* n i k n n n xk x n e  − − = =  (11) within the scope of this study, in order to obtain information about the vibration that occurs during the turning process, fft analysis has been carried out and examined for the acceleration data collected. thus, information was obtained about what the operating frequency of the system was during processing, what the natural frequency range was and what the sampling number should be in data collection. 3. active vibration control 3.1. sliding mode control theory in recent years, sliding mode control (smc), which has been used in many areas with increasing acceleration and is among nonlinear control methods, is preferred especially for meeting high-speed control needs. among the biggest advantages of smc is the ability to change the dynamic behavior of the system, to reduce the degree of the system and to minimize disruptive effects. smc is preferred as an effective control approach against systems that cannot be modeled dynamically, variable parameters and distorting effects [37]. in the sliding mode control consisting of sliding and reaching stages, equivalent and switching rules must be realized, respectively [38]. so many functions such as tanh, sgn etc. are used to overcome the chatter problem, which is one of the biggest disadvantages of the sliding mode control [39, 40]. the transfer function given in eq.(6) is written as a differential equation, as in eq. (12). �̈�(t)=a�̇�+bx cu (12) tracking error e(t) to find control input u(t) in the sliding mode control can be defined as follows, ( ) ( ) ( )e t r t x t= − (13) in the eq. (13), e(t) is the tracking error, r(t) is the command signal, and x(t) is the measured signal. sliding surface function can be expressed as eq. (14). 1 ( ) ( ) n d s t e t dt  −   = +    (14) assuming n = 2, eq. (15) is obtained. identification of chatter vibrations and active vibration control by using sliding mode controller... 315 ( ) ( ) ( )s t e t e t= + (15) in this equation,  is a positive constant and if this equation is derivative, eq. (16) is obtained. ( ) ( ) ( )s t e t e t= + (16) substituting r-x for e in this equation, eq. (17) is obtained. s e r ax bx cu= + − − + (17) the control input u(t), which is the sum of equivalent control (ueq) and switching control (usw), is given in eq. (17). ( ) ( ) ( )eq swu t u t u t= + (18) due to the nature of the sliding mode controller, �̇� = 0 is expected. if eq. (18) is arranged for �̇� = 0, ueq can be expressed as eq. (19). ( ) 1 ( ) ( ) ( ) ( )equ e t r t ax t bx t c = − − + + (19) with ksw=0.5 and ω= 0.1, = switching function, usw can be expressed as in eq. (20). ( ) tanh( ( ) )sw swu t k s t=  (20) acceleration data received from the cutting tool was used as feedback information in the turning process. with the help of the acceleration data, the necessary reaction force for the actuator is calculated. the closed-loop used for control is shown in fig. 4. fig. 4 smc system block scheme 4. results and discussion within the scope of the experiments, chatter vibrations generated on the tool holder were collected and evaluated for three different cutting axes with the help of a triaxial pcb brand industrial accelerometer and dewesoft/sirius data collection device. the accelerometer is rigidly fixed on the tool holder without using any damping element. 316 m. a. guvenc, h. h.bilgic, s.mistikoglu for the active control application, a data set of 20 s was chosen from the data collected during turning. selected data is filtered, integrated and fft analysis is applied. the amplitude values of the collected data ranged between 110 m/s2 in the positive direction and -132 m/s2 in the negative direction, and the data collection rate was realized as 2000 (2000 samples/s) per second. in order to determine the operating frequencies of the system, the vibration data collected was first subjected to fft and critical frequency values were determined. after fft analysis, it is observed that 110 hz is the critical frequency value. this value is in the range of chatter vibration seen in the literature and confirms the study by munoa in 2016, where he also mentioned the range of chatter vibrations (20-200 hz) [1]. the fft analysis results are given in detail in fig. 5. fig. 5 fft analysis results in order to use the raw data collected from the system, it is of great importance to analyze the signals and to clean the unwanted signals. iir or fir filters are generally used in active control applications [2]. fir filtering method was used to clean the parasitic data contained in the acceleration data and to obtain a clean data set that can be used in active control. the triaxial acceleration data collected during turning are arranged in the dewesoftx software interface and converted into displacement data by taking its integral so that it can be input in active control applications. the 20-second data randomly selected from the raw data for the smc application is given in fig. 6. 11 m/s2, 108 hz fft acceleration data 0 100 200 300 400 500 f (h z) 0 .0 0 5 .0 0 1 0 .0 1 4 .0 0 t (s ) -1 3 2 .0 0 -5 0 .0 0 0 .0 0 5 0 .0 0 1 1 0 .0 0 03:20 03.25 03.30 03.40 a cc el er at io n (m /s 2 ) a cc el er at io n (m /s 2 ) identification of chatter vibrations and active vibration control by using sliding mode controller... 317 fig. 6 uncontrolled 20 seconds displacement data then, the displacement data were input to our smc model from the same point as the point where the data was collected under real conditions, and the control of the model during processing was provided for 20 seconds. the displacement graph obtained after active control is given in fig. 7. fig. 7 displacement data after active smc while the displacement data were between 300-500 µm before control, it was observed to be between 50-100 µm after the control. this significant difference will allow us to obtain a high-quality product during turning. according to the information obtained from previous studies, chatter vibrations occur with average displacement amplitude of 200-300 µm. in fig. 8, there is a visual of the surface roughness after normal turning and chatter vibration [1,3–5]. fig. 8 surface roughness after turning for ti6al4v workpiece chatter regionstandard region 318 m. a. guvenc, h. h.bilgic, s.mistikoglu in fig. 9, there is a visual where the data obtained as a result of the active control is compared with the chatter data received from the system. as can be deduced from the visual, there has been a serious improvement in the quality of the surface roughness due to the decrease in the vibration amplitudes that occur during turning as a result of the active smc of the cutting tool. fig. 9 data from controlled and uncontrolled situations it has been determined that the actuator generates a force of up to 25 n in order to decrease the amplitude during active control. as a result of the literature studies, it was seen that an actuator force of 18 n is required for active control of aa6025 aluminum alloy during turning, and 45 n for turning of c45 quality steel [6,7]. it is advocated that the force required by the actuator in active control applications varies depending on the mechanical properties of the workpiece, the properties of the cutting tool and the turning parameters. the information obtained from the literature confirms the results we have obtained [4145]. the actuator force graph is given in the fig. 10 for active control application. fig. 10 reaction force of the actuator identification of chatter vibrations and active vibration control by using sliding mode controller... 319 fft analysis of displacement data is given for uncontrolled and controlled situations, respectively, in figs. 11 and 12. according to the fft results, there has been a significant decrease in the critical frequency amplitude. this situation has revealed that the amplitude and frequency of the vibration that occurs after the control are too small to affect the product quality and system elements negatively. fig. 11 fft results for displacement data without active control fig. 12 fft results for displacement data after smc 5. conclusions in this study, it is aimed to define chatter vibrations that negatively affect product quality and productivity in turning and to reduce them with active control application. within the scope of the study, acceleration data in three axes were collected from the cutting tool during dry turning of titanium alloy ti6al4v quality material. the collected data is processed and then made available in active control applications. according to the results obtained from the study: ▪ it has been observed that the amplitude values of the acceleration data collected on the tool during the dry turning of ti6al4v alloy vary between 110 m/s2 in the positive direction and -132 m/s2 in the negative direction. 320 m. a. guvenc, h. h.bilgic, s.mistikoglu ▪ according to the fft result of the acceleration data, it is observed that the critical frequency values were 110 hz. ▪ it is observed that the displacement values before the control were in the range of 250350 µm. ▪ after the active smc application, it has occurred that the displacement data decreased to the range of 50-100 µm. ▪ it is concluded that the fir digital filtering method is an important step in active control applications. ▪ it is concluded that the chatter problem, which is one of the major problems of machining, can be solved with active vibration control applications, a correct data acquisition mechanism, controller, system recognition processes and correct actuator selection. ▪ it is concluded that the smc works with 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barcena, r., munoa, j., 2019, delayed feedback control for chatter suppression in turning machines, mechatronics, 63, pp. 102276. 44. zaeh, m.f., kleinwort, r., fagerer, p., altintas y., 2017, automatic tuning of active vibration control systems using inertial actuators, cirp annals manufacturing technology, 66, pp. 365–368. 45. guvenc, m. a., cakir, m., mistikoglu, s., 2019, experimental study on optimization of cutting parameters by using taguchi method for tool vibration and surface roughness in dry turning of aa6013, in 10th international symposium on intelligent manufacturing and service systems, pp. 1032-1040. 12145 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 631 656 https://doi.org/10.22190/fume230905046p © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd review article agro-waste natural fiber sample preparation techniques for bio-composites development: methodological insights resego phiri1, sanjay mavinkere rangappa1, suchart siengchin1, dragan marinkovic2,3 1natural composites research group lab, department of materials and production engineering, the sirindhorn international thai-german graduate school of engineering (tggs), king mongkut’s university of technology north bangkok (kmutnb), thailand 2university of niš, faculty of mechanical engineering, serbia 3tu berlin, department of structural analysis, berlin, germany abstract. in today's engineering industries, there is a growing focus on sustainable and eco-friendly products due to their recyclability, abundant availability and property variability. one key aspect of this sustainability effort is the development of eco-friendly materials, particularly bio composites derived from agricultural waste residues. the physical, mechanical and thermal properties as well as suitability of these fibers depend greatly on the methods used for extraction, processing, chemical modification and physical treatments. understanding these processes comprehensively is essential for obtaining desired natural fibers/fillers from agricultural waste for creating effective bio composites to meet specific application demands. this study aims to provide a comprehensive assessment of the various extraction and modification techniques employed for natural fibers. it offers an in-depth review of diverse extraction processes, ranging from the initial harvesting to the decortication stage. it has been revealed that the choice of extraction methodologies depends on climate, water resources, local traditions, and the desired fiber quality. additionally, the paper explores chemical and physical treatments, highlighting how each method influences the structure and properties of natural fibers. overall, this review offers practical insights into the steps taken to transform agro-waste biomass into desired natural fiber and in turn biocomposite material, while enhancing product quality and performance. key words: natural fibers, extraction processes, chemical treatments, physical/surface modifications, bio-composite development received: september 05, 2023 / accepted november 28, 2023 corresponding author: sanjay mavinkere rangappa natural composites research group lab, department of materials and production engineering, the sirindhorn international thai-german graduate school of engineering (tggs), king mongkut’s university of technology north bangkok (kmutnb), thailand e-mail: mavinkere.r.s@op.kmutnb.ac.th 632 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic 1. introduction in response to the escalating global apprehension and consciousness regarding environmental issues, numerous industrial sectors have progressively shifted their focus towards the adoption of sustainable and regenerative materials. natural fibers, being inherently renewable and readily accessible within the natural ecosystem, have emerged as a pivotal avenue in this pursuit. this inclination is largely attributed to their array of advantages, including but not limited to, cost-effectiveness, lightweight nature, inherent renewability, biodegradability and an array of distinctive high-specific properties [1–3]. with the rise of agriculture, deliberate cultivation of fiber-yielding plants became established. early civilizations explored plant fibers, particularly in textile weaving, long before perfecting practices like metalworking or pottery. various types of fibers held significant importance for their versatility. some were used for crafting ropes and yarns, essential for activities like hunting, fishing, netting, climbing, and transportation, while others found their place in the intricate art of textile weaving [4]. comprised mainly of lignocellulosic materials, natural fibers encompass a variety of resources, including wood, ramie, jute, hemp, kenaf, sisal, coir, flax, bamboo and various fruit fibers. effectively integrating natural fibers into composite materials is crucial for their practical application. substituting synthetic materials with natural fibers aligns with the principles of environmental stewardship, reflecting the contemporary shift towards sustainability. table 1 presents a comparison of attributes between natural and synthetic fibers. while natural fibers generally exhibit lower tensile strength and modulus compared to synthetic counterparts, they offer significant cost and ecological advantages. variations in fiber structure and growth conditions contribute to the diverse mechanical properties observed in natural fibers. through precise treatment procedures, these fibers can be tailored to serve as reinforcing fillers with various mechanical characteristics. moreover, careful application of specific treatments can even impart mechanical properties similar to those of synthetic fiber composites. table 1 assessment of key properties of major synthetic and natural fibers [5,6]. fibers density (g/cm3) tensile strength (mpa) elastic modulus (gpa) decomposition temperature (°c) cost (euros/kg) carbon 1.8-1.9 2090-5200 525 3647 33-166 kevlar (aramid) 1.4 2380-3100 124 496 16.7 synthetic zirconia 5.6 700 100 2497 41.7 e-glass 2.5 1625-3400 72 756 1.6-2 c-glass 2.5 2800 69 756 9.3-16 sisal 1.4 507-855 24 270 0.7 flax 1.4 250-1000 12-100 280 0.15-0.76 natural cotton 1.5-1.6 350 11 246 0.35 banana 1.3 791 30 320 0.7-0.9 jute 1.3-1.5 187-540 3-55 270 0.7 the environmental significance of natural fibers lies in their seamless integration with agricultural waste resources, promoting a holistic approach to waste management and valorization. this convergence taps into the untapped potential of discarded agricultural agro-waste natural fiber sample preparation techniques for bio-composites development... 633 by-products, granting them secondary utility and enhancing the sustainability of resulting composites. the global surge in waste production, fueled by rapid urbanization, population growth, and industrialization, has reached alarming levels. this surge encompasses various waste streams, including agricultural residues. table 2 provides an overview of the promising agricultural product resources and their respective countries of origin. these residues, once regarded as mere by-products, hold the significant potential to be transformed into valuable natural fibers. leveraging these residues addresses the dual challenge of waste reduction and resource optimization, offering a pathway to mitigate environmental impact. agricultural residues fibers encompass a diverse array of chemical substances, predominantly composed of cellulose, hemicellulose, lignin and smaller quantities of pectin [7]. the composition and proportions of these components can vary depending on the type of plant species, climatic conditions and the method of extraction and fiber preparation approaches employed. numerous comprehensive studies have evaluated the properties of agricultural wastebased composites [8–10]. these investigations include the isolation and characterization of agro-waste materials, involving analyses of their physical, mechanical and chemical attributes. however, a notable research gap exists in the exploration of agro-residues and their associated extraction and preparation strategies employed prior to their deployment into bio composites. this gap underscores the need for a more comprehensive understanding of the crucial pre-processing steps involved in optimizing the utilization of agricultural residues in biocomposite materials. achieving optimal performance in bio composites materials requires a comprehensive understanding of the interplay between fiber characteristics and processing techniques. by examining the techniques for fiber extraction, cleaning and modification, along with their impact on agro-waste fibers and their composite performance, this paper seeks to shed light on the intricate relationship between preparation methods and the final properties of the resulting composites. by understanding the fiber preparatory techniques, merits and obstacles, researchers, engineers and practitioners will be better prepared to fully exploit the capabilities of materials founded on natural fibers. table 2 production statistics of agricultural products that can be used as potential natural fiber resources [11] production (million tons) countries banana pineapple coconut sugarcane rice kenaf jute oil palm india 24.87 1.46 11.93 341.2 159.2 0.12 1.98 china 10.55 1 0.25 125.54 203.29 0.08 0.17 0.67 philippines 9.23 2.4 15.35 31.87 18.44 0.002 0.47 brazil 6.9 2.48 2.82 0.74 11.76 14.2 26.7 1.34 indonesia 6.19 1.78 18.3 33.7 71.28 4.35 0.007 120 thailand 1.65 2.65 1.01 100.1 38.79 1.3 0.06 12.8 vietnam 1.56 0.54 1.31 20.08 44.04 8.2 0.02 2. agro-residues as sources of natural fibers on a global scale, roughly 140 gigatons of biomass are discarded as waste annually, contributing significantly to environmental challenges such as the release of greenhouse gases and the deterioration of soil-water quality [12]. as the world's population continues 634 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic to grow and debates surrounding the allocation of resources for food and fuel intensify, immense pressure is placed on the agricultural sectors to ramp up their agro-food production. this heightened demand and supply of agricultural products have consequently led to a surge in the disposal of biomass residues. given these circumstances, the recycling and the secondary utilization of this waste have become a critical issue with notable social and economic implications. to address this, the strategic use of agricultural residue waste as natural fiber resources can be approached through three primary avenues: incorporating them as additives to improve compatibility among principal components, reinforcing them within plastic matrices to enhance structural integrity and utilizing them as cost-effective and renewable filler materials. agricultural residues can be categorized into two distinct groups: field residues, which are remnants left in agricultural fields after crop harvesting (examples include pineapple leaves, cornhusks, cotton seed pods, kapok seed pods and cotton stalks), and process residues, which are materials generated after processing crops into usable resources (examples include bagasse, pea peel, wheat straw and rice straw). table 3 provides an overview of some major agricultural residues harnessed for their natural fiber attributes [13]. table 3 overview of prominent agro-residues utilized for natural fiber resources. agro-residues synopsis fiber source utilization rice straw rice, a staple food for most of the world's population, contributes not only to sustenance but also to a significant agricultural residue known as rice straw. the leftover rice straw presents a valuable opportunity as a source of natural fibers with diverse applications. about 731 million metric tons of rice straw are generated globally each year. rice straw comprises the stalks and leaves of rice plants left behind after the grains are harvested. the primary fiber source within rice straw is the vascular bundles located within the stem. used for making paper, packaging materials and handicrafts. wheat straw this is a by-product resulting from the collection of wheat grains, consisting of segments between nodes, nodal points, foliage, husks, and central stem parts of wheat. around 600 million metric tons of wheat straw are generated globally annually. wheat straw encompasses the plant material that remains after wheat grains have been collected. main part is the vascular bundles found in the stem. primarily used in paper production and as packaging material. corn cob and husks these maize/corn cultivation residues are typically removed and discarded before consuming the corn. corn husks and cobs contribute to a substantial portion of agricultural waste, but specific statistics may vary by region. the corn cob is the innermost part of the corn plant, while the corn husks is the outer layer of green leaves that envelop the corn. used for crafts, textiles and cellulose for paper production. agro-waste natural fiber sample preparation techniques for bio-composites development... 635 sugarcane bagasse sugarcane bagasse is the aftermath of the sugarcane crushing process to extract the sugary juice. what is left behind is the fibrous material that comprises the stalks' outer layers, mainly consisting of the fibers and pulp that provided structural support to the plant. global sugarcane production yields about 1.9 billion metric tons of bagasse annually. fibrous residue left after sugarcane juice extraction. used for paper production, packaging and as a biomass source for energy generation. coconut husks this refers to the coarse and fibrous exterior layer that encases the coconut fruit. this protective covering acts as a natural shield for the inner seed of the coconut. the roughness and texture of the husk are due to the dense network of fibers that make up its composition. coconut husks contribute to a significant portion of waste from coconut processing, with specific quantities varying based on coconut production levels. outer fibrous shell of coconut. used for ropes, mats, brushes, geotextiles, and horticulture substrates. pineapple leaves pineapple leaves, an often-overlooked component of the fruit-bearing plant, hold remarkable value beyond their decorative appearance. these leaves, situated at the crown of the pineapple plant, are a source of natural fibers known as piña fibers. pineapple cultivation gives rise to a notable quantity of plant waste, about 250 metric tons per hectare. leaves of pineapple plants. used for delicate textiles like piña cloth, as well as crafts. fruits and vegetable waste fruits and vegetable waste refers to the discarded or unused parts of fruits and vegetables that are generated during the harvesting, processing, or consumption of these food items. estimates of global food waste, including fruits and vegetables, have suggested that roughly one-third of all food produced for human consumption is lost or wasted. peels, skins, stems and other parts not consumed as food. used as growing mediums for plants, helping improve soil structure, water retention, and aeration in horticultural settings. 3. natural fiber extraction processes natural fiber extraction serves as the fundamental and pivotal initial phase in the broader realm of fiber processing. this process encompasses a series of techniques aimed at isolating fibers from various plant sources [14]. achieving this separation requires the delicate task of disrupting the connections or bonds between the robust stem cores and the delicate fiber bundles surrounding them. the broader context of the fiber extraction process 636 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic extends beyond just straw; it encompasses processes of liberating fibers from diverse plants components. these methods play a crucial role in enhancing the quality, durability and usability of these fibers. in the context of agricultural waste, the residues undergo preparatory steps to make fiber extraction more efficient. depending on the type of residue, this might involve harvesting, cleaning, drying and removing any unwanted materials like dirt or impurities. figure 1 provides an overview of the key processes involved in the extraction of natural fibers for the development of bio composites. fig. 1 processes involved in the extraction of natural fiber from plant biomass to use as reinforcement phase in bio composites. 3.1. harvesting the process of harvesting natural fibers from various agricultural crop residues involves intricate systems that are often tailored to local conditions and practices. these systems are designed to effectively extract fibers from plants while considering factors such as the type of crop, the quality of the fibers needed and the prevailing agricultural practices in the region. as a result, harvesting methods can vary significantly depending on these factors. the development of harvesting systems is not a one-size-fits-all approach [15,16]. rather, it is a dynamic process that considers the unique characteristics of each crop, the resources available and the expertise of local farmers. this localized approach ensures that the methods used are efficient, sustainable and suited to the specific agricultural context. one of the primary distinctions in harvesting methods lies in the choice between manual and mechanized approaches as shown in figure 2. manual harvesting is prevalent in many regions, particularly where labour is readily available and cost-effective. skilled workers harvest fibers by hand, employing traditional techniques that have been refined over generations. in contrast, mechanized harvesting involves the of technology and innovation. this harvesting method have gained popularity in areas where labour costs are agro-waste natural fiber sample preparation techniques for bio-composites development... 637 higher or where the demand for efficiency is paramount. specialized machinery is used to streamline the process, reducing labour requirements, and increasing overall output. a) b) fig. 2 comparison of manual harvesting and machine harvesting techniques in the context of natural fiber extraction from agricultural crops; (a) manual harvesting, showcasing skilled workers meticulously handpicking fibers from the plants, (b) machine harvesting, where specialized machinery efficiently gathers fibers from the plants. moreover, within harvesting systems for natural fibers, two distinct methods often come into play: whole stem harvesting and cut stem harvesting. these methods are indicative of the intricate decision-making processes involved in optimizing the extraction of fibers from various agricultural produce. the whole stem harvesting approach involves gathering the entire plant, including the stalk, leaves and sometimes even the roots to extract fibers [17]. this method can result in minimal waste, as nearly all parts of the plant are used in some way. it can also simplify the processing steps, especially when the fibers are distributed along the entire length of the stem. in contrast, cut stem harvesting involves severing only the part of the plant that contains the desired fibers, often leaving behind leaves and other non-essential portions [18]. this method is frequently used for crops like sugarcane and bamboo. 3.2. fiber retting natural retting involves the controlled decomposition of non-fibrous materials that bind the fibers to the plant's stem. through retting, this plant structure matrix is deliberately broken down, effectively loosening the fibers and making them accessible for further processing. a comparison between retting methods is shown in table 4 [19–21]. the outcome of this transformative process has dual effect. first, the liberated fiber bundles, now freed from their binding matrix, emerge in a state that allows for their separation with greater ease. this sets the stage for subsequent processing steps, such as scutching and hackling, that refine the fibers and align them in a more uniform manner. secondly, the residual fibers that remain 638 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic after retting become enriched in cellulose content, a crucial factor contributing to the highstrength properties of the extracted fibers [22]. numerous extensive inquiries and studies have been dedicated to comprehending the intricate influence of retting processes on the characteristics of natural fibers and their respective composites. stawski et al. [23] examined the thermal and mechanical attributes of okra fibers acquired through water retting and dew retting methods. these fibers were subjected to different processing techniques for the removal of non-fibrous components. yu et al. [24] conducted a study involving the use of warm water retting as a pre-treatment method for banana stems. the research aimed to assess how varying durations of retting treatment impact both the physical and chemical properties of both the banana stems and the fibers derived from different parts of the plant. the findings revealed that longer retting periods were associated with a reduction in the presence of fiber impurities. however, this prolonged retting time also led to a decline in the strength and toughness of the resulting fibers. in their study, narkpiban & poonsawat [25] explored the impacts of hydrothermal pre-treatment on the characteristics of kenaf fibers subjected to water and enzyme retting. this involved treating the fibers in a high-pressure reactor at temperatures of 160, 180, and 200°c. the findings revealed that both water and enzymatic retting led to an increase in cellulose content within the fibers. the implications of retting processes are not confined solely to the fibers themselves. the interactions between retted fibers and the matrix materials used in bio composites are a matter of considerable interest. for instance, de prez et al. [26] conducted a study focusing on the mechanical properties of composites that were strengthened using fibers extracted after undergoing enzymatic treatments. the investigation also delved into factors such as fiber fineness, moisture absorption in the composites and the analysis of fracture surfaces. the findings indicated that all the enzymatic treatments exhibited encouraging results in terms of tensile strength and stiffness. moreover, rozyanty et al. [27] examined how different retting methods (water and mechanical retting) for kenaf bast fibers impact the tensile strength and water absorption properties of unsaturated polyester (upe) composites reinforced with these fibers. the study revealed that water-retted kenaf bast fibers possess surfaces characterized by cleanliness, smoothness, minimal fiber detachment and fewer gaps compared to the mechanically extracted samples. recent practices have witnessed the integration of advanced technologies and pretreatment methods into the retting process, exemplified by a study conducted by ruan et al. [28]. in their research, they employed radio frequency (rf) pre-treatment to effectively break down non-cellulosic components present in the material. this pre-treatment process was particularly advantageous as it prompted the rapid absorption of water and expansion of the polar constituents within the flax phloem. as a result, this action effectively loosened the connections between individual fibers and fiber bundles, creating a more accessible pathway for enzymes to penetrate the material. the outcomes of their investigation revealed that the implementation of rf pre-treatment had a positive impact on the overall retting process. the enhanced breakdown of non-cellulosic materials and the improved water absorption and swelling characteristics contributed to an increase in retting efficiency. the choice between the retting methodology depends on factors such as climate, available water resources, local traditions and the desired quality of the fibers. proper timing and monitoring are crucial in all methods to ensure that the fibers are neither underretted which can result in insufficient fiber separation nor over-retted leading to fiber damage. réquilé et al. [29] conducted a study that affirms how the desired fiber quality in agro-waste natural fiber sample preparation techniques for bio-composites development... 639 hemp can be ensured through the retting process by adjusting the duration of the retting process based on various weather conditions, such as the strength of solar radiation, the amount of rainfall and the surrounding temperatures. in situations where there is a shortage of rainfall or the air temperature is low, it becomes necessary to extend the retting period to allow for the attainment of an adequate level of retted biomass, which is crucial for achieving the desired fiber quality. retting not only facilitates fiber separation but also has environmental implications. improper disposal of retting water, which contains decomposed plant material, can lead to pollution of water bodies. therefore, responsible retting practices are essential to minimize environmental impact. table 4 the comparison between the natural fiber retting techniques. retting method process description merits limitations process duration water retting the process of water retting involves immersing plant stems in water and monitoring them at regular intervals. as the water seeps into the central part of the stem, it causes the inner cells to swell, leading to the rupture of the outermost layer. ▪ remarkable degree of consistency and excellence in their overall quality. ▪ resulting fibers are characterized by an even and uniform level of retting throughout their structure. ▪ associated with significant pollution stemming from anaerobic bacterial fermentation. ▪ produces foulsmelling odors. ▪ the process is resource intensive. 7-14 days dew retting plant stalks are evenly distributed across fields, allowing them to be exposed to ample sunlight, atmospheric air, and dew. this exposure facilitates the colonization of fungi, which in turn leads to the decomposition of cellular stem tissues and adhesives. ▪ minimal water requirement, making it a preferred option in regions facing water scarcity. ▪ relies on natural atmospheric conditions. ▪ reduced energy consumption and operational costs. ▪ it relies heavily on weather conditions, making it less predictable and potentially slower than other retting methods. ▪ it is highly susceptible to contamination with soil. 2-10 weeks enzymatic retting involves the application of specific enzymes to facilitate the breakdown of non-cellulosic components that bind the fibers to the plant material. enzymes facilitate the breakdown of gum and pectin materials present in the stem through a process known as hydrolysis. ▪ precise control over the retting process. ▪ reduced processing time compared to traditional retting methods. ▪ more potential to produce highquality fibers with specific characteristics ▪ high costs, as enzymes used can be expensive to acquire and apply. ▪ requires high level of expertise and training for the appropriate selection and implementation of enzymes. ▪ requires meticulous monitoring to achieve optimal conditions for enzyme activity. 1 -2 days 640 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic retting method process description merits limitations process duration chemical retting chemical retting is a fiber separation method that involves the use of chemical agents to break down the non-cellulosic components binding plant fibers to the stalks. these chemicals target and degrade the pectins, lignin and other adhesive substances, thus releasing the individual fibers. the plant material is typically immersed in the chemical solution for a specific period, allowing the agents to catalyze the breakdown of the binding materials. ▪ chemical agents enable manufacturers to finely regulate retting conditions, resulting in consistent and uniform fiber quality. ▪ it is generally faster compared to some natural retting methods, significantly reducing processing times and increasing overall efficiency. ▪ allows for greater flexibility in adapting to various fiber types, making it applicable to a wide range of plant materials. ▪ necessitate careful management and treatment to prevent contamination of water sources. ▪ the use of chemicals can lead to the loss of certain properties in the fibers, affecting their color, texture, and overall quality. ▪ the reliance on chemicals also introduces an additional cost factor, as procuring and applying the necessary agents can contribute to higher production expenses. 1-2 hours microbial retting microbial retting harnesses the action of microorganisms to break down the non-cellulosic components binding plant fibers to the stalks. the harvested plant material is exposed to a controlled environment conducive to microbial activity. the microorganisms, including bacteria and fungi, produce enzymes that target and degrade pectins, lignin and other adhesive substances present in the plant material thereby loosening the fibers. ▪ relies on natural processes involving microorganisms rather than chemical agents hence presents a more sustainable approach aligning well with ecological ideals. ▪ challenging to control consistency as the process relies heavily on factors such as temperature, humidity and the availability of suitable microorganisms. ▪ microbial retting also results in longer processing times. ▪ the process produces bad odors during the decomposition process. 10-30 days agro-waste natural fiber sample preparation techniques for bio-composites development... 641 3.3. decortication in situations where fibers are tightly bound to the plant stem, a process called decortication is employed. this process entails the mechanical removal of the non-fibrous outer layers of the plant material to expose the fibers. historically, hand decortication was the norm and involved pounding the leaves and scraping away the pulp with a knife, but this approach is labour-intensive and time-consuming [30]. nowadays, mechanical decorticators offer an efficient alternative. these devices utilize rotating wheels with blunt knives to crush and beat the leaves, leaving behind only the fibers. figure 3 illustrates how mechanical forces are applied to break the bonds between the fiber and its core. various types of decorticators, such as crushing rollers, ball mills, hammer mills and drop weights, have been used for fiber processing. for high-value applications, additional treatment is often required to reduce unwanted noncellulosic remains in the mechanically extracted fibers. however, mechanical decortication can disrupt fiber cell wall structures, leading to defects that may negatively impact mechanical properties and composite performance [31]. the extent of such defects depends on factors like previous retting treatments [32]. nevertheless, some research indicates that high-quality fibers can be obtained through mechanical decortication, especially in modern mills designed to maintain the integrity and length of the fibers by disentangling and aligning them [33]. the performance of a decorticator is influenced by several parameters, and understanding their effects can help improve current industrial decortication processes, even if there's a trade-off between fiber quantity and quality, depending on the chosen parameters and the condition of the natural fiber source. fig. 3 illustrations of mechanical decorticators in action; (a)automated decortication by hammering wherein drums with outward-extending hammers that repeatedly strike straw to break it down are employed and (b) automated decortication by roller milling wherein cylindrical rollers are used to break the biomass [36]. following the initial decortication process, a crucial step known as post-decortication is carried out to refine the fiber extraction. this step involves the separation and cleaning of detached fibers from a mixture containing fiber/core-bound components and fine particles. various methods are employed for this process. one common method is scutching, which involves further beating or mechanical action to separate the fibers from other components [34]. another approach involves sieving, utilizing differences in particle size to isolate the fibers. these techniques contribute significantly to the overall quality improvement of the fibers extracted after decortication, making them more suitable for a wide range of applications across 642 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic industries. following the extraction of fibers through any of the methods, it is crucial to initiate a comprehensive washing process to remove any residual impurities. subsequently, the fibers need to undergo a meticulous drying procedure. this is vital because the moisture content within the fibers can significantly impact their overall quality. notably, artificial drying allows for better control over drying conditions, ensuring that the fibers are dried uniformly and without the risk of overexposure to sunlight, which can lead to bleaching and degradation [35]. in instances where natural drying is chosen, it is important to ensure that the fibers are protected from direct sunlight, they should be dried under shade to prevent them from becoming discolored or weakened due to prolonged exposure to sunlight. 4. natural fiber chemical modifications one of the primary challenges encountered in utilizing natural fibers as reinforcements in polymer composites is the disparity between the hydrophilic nature of the fibers and the hydrophobic nature of the polymer matrix [37]. this incongruity can lead to poor interfacial adhesion and suboptimal composite properties. addressing this issue involves reducing the inherent hydrophilicity of natural fibers through chemical treatment and surface modification techniques. recent research endeavors have explored a range of chemical treatment processes targeted at enhancing the surface properties of natural cellulosic fibers. these treatments are aimed at refining the performance of these fibers across diverse applications. 4.1. mercerization this treatment procedure involves immersing the fibers in a solution containing strong alkaline substances like sodium hydroxide (naoh). the mechanism of alkali treatment of a typical cellulose based natural fiber is exemplified in figure 4. alkaline agents break down the hemicellulose and lignin components present in the fiber, leaving behind a more refined form of cellulose. this treatment enhances the adhesion between the fiber and the matrix in composite materials through a two-fold process: firstly, it diminishes the water-attracting characteristics of natural fibers and secondly, it clears away any impurities on the fiber's surface, resulting in a textured surface [38].the enhanced roughness texture of the natural fiber surface from this treatment facilitates a stronger connection between the fibers and the biopolymer matrix, achieved through mechanical interlocking. another benefit arises from the reduction in the fiber’s hydrophilic nature, which curtails the interaction between the fiber's hydroxyl groups and water molecules [39]. consequently, issues like fiber swelling, void formation and debonding in biopolymer composites are mitigated. this leads to the production of biopolymer composites exhibiting markedly advanced mechanical properties. fig. 4 illustration of a typical alkali treatment reaction of cellulose-based fibers with naoh agro-waste natural fiber sample preparation techniques for bio-composites development... 643 alkali treatment brings about a range of alterations in the characteristics of the natural fibers. the treatment's effects can encompass an elevation in the fiber's tensile strength, elongation at break and stiffness. the degree of these changes can be regulated by factors such as size and morphology of the fibers, soaking duration and concentration of the alkali solution [40]. the duration for which natural fibers are immersed in an alkali solution have been found to be a crucial factor for natural fiber performance. in a study conducted by setyayunita et al. [41], the structural and functional characteristics of natural fibers exhibited noticeable variations when exposed to varying soaking durations. in the experiment conducted by zin et al. [42] pineapple leaf fibers were immersed in alkali solution for durations of 1 and 3 hours. their findings suggested that the most favorable tensile strength was achieved after a 3-hour alkaline treatment. in essence, the duration of the treatment significantly impacts the inherent properties of the fibers, leading to observable distinctions in the resulting behavior and attributes of the composite materials they contribute to. the concentration of the alkaline treatment has also been identified as a significant factor that profoundly affects both the performance of natural fibers and the properties of the resulting composite materials. studies have consistently highlighted the substantial impact of the alkaline solution's concentration on the performance of natural fiber performance. among the notable contributions in concentration effects in mercerization of natural fibers is the work by boumaaza et al. [43], where the flexural properties of treated natural sisal, flax and jute fibers with respect to varying concentrations of naoh (1.5%, 2%, and 4%) were investigated. in the study conducted by cai et al. [44], abaca fibers were subjected to various concentrations (5%, 10%, and 15%) of aqueous naoh solution. the research highlighted that elevated alkaline concentrations led to reductions in both tensile strength and young’s moduli as a result of fiber breakage and complete removal of fiber binding materials. table 5 provides a synopsis of other investigations that have explored the alkali treatment of natural fibers for utilization in composites reinforced by natural fibers. table 5 examples of recent investigations related to alkaline treatments of natural fibers used in natural fiber reinforced composites. fiber matrix chemical treatment procedure outcome ref. abutilon indicum – epoxy (l12) initially, the stems were harvested and left to soak in water for approximately 21 days, facilitating the extraction of the fibers. following this soaking period, the fibers were cleaned and subjected to treatment with a 5% naoh solution. the alkali-treated fibers then underwent another round of washing using purified water, then dried at 80°c to eliminate any remaining moisture content. the application of alkali treatment to alter the surface of the fibers led to a notable enhancement in the mechanical properties of the composite samples. furthermore, the thermal stability of the composite samples treated with alkali exhibited an increase when compared to the untreated composites. [45] 644 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic fiber matrix chemical treatment procedure outcome ref. dichrostachys cinerea bark fibers (dcfs) epoxy the initial process involved immersing the untreated dcfs in a solution of 5% (w/v) naoh at room temperature. the immersion durations ranged from 30 to 120 minutes. subsequently, the treated fibers were cleaned thoroughly until the solution was neutralized. hydrochloric acid solution was added then final rinse with demineralized water was carried out before the fibers are dehydrated in an oven at 80°c for a duration of four hours. the alkali treatment exhibited a noteworthy impact on the crystalline index of the optimally treated dcfs due to the reduction of amorphous components. furthermore, the cellulose degradation temperature of the optimally treated dcf improved significantly. moreover, the mechanical characteristics of the chemically treated samples significantly outperformed those of the untreated raw samples. [46] pineapple leaves araldite epoxy resin initially, the pineapple leaves were cleaned and dried for 24 hours in an oven at 70°c. fibers were then cut into smaller lengths, approximately 2– 5 mm. following this, the fibers were subjected to sodium hydroxide solution of concentrations of 3%, 5%, 7% and 10% for an hour. thorough cleaning with deionized water was carried out to ensure the removal of any excess alkali chemicals that may have adhered to the surface of the fibers during the treatment. the alkaline treatment demonstrated a multifaceted enhancement in the fiber's attributes, influencing their strength, thermal behavior and various compositional characteristics. [47] salacca zalacca fiber (szf) high density polyethylene (hdpe) the treatment of szf involved immersing it in a 5% naoh solution, with varying immersion times of 1, 2, 3, 4, and 5 hours. the fiber was then washed with 1% acetic acid solution until a neutral ph value was achieved. subsequently, the szf was subjected to a rinse with distilled water, after which it was left to air dry at room temperature for approximately 3 days. a further dehydration process was undertaken using an oven set to a temperature of 35°c for 8 hours. applying alkali treatment to szf leads to improvements across several key parameters, including density, crystallinity index, tensile properties and the interfacial shear strength between the fibers and the matrix. [48] chestnut cupula fiber recycled highdensity polyethylene (r-hdpe) the cupula fibers were carefully dried in a conventional air oven until their moisture content was reduced to less than 1%. subsequently, the dried cupula fiber underwent treatment with a 5 wt% naoh solution for around 24 hours at room temperature. after the treatment, the fibers underwent multiple washes with distilled water to ensure the removal of excess alkali. the fibers were once again dried in an oven, this time at 60°c for a period of 24 hours. the composites consisting of alkali-treated cupula fibers exhibited superior mechanical properties in comparison to their untreated counterparts. this improvement can be attributed to heightened surface roughness and enhanced interface adhesion. [49] agro-waste natural fiber sample preparation techniques for bio-composites development... 645 fiber matrix chemical treatment procedure outcome ref. lagenaria siceraria fiber (lsf) – epoxy the fibers were washed and subsequently immersed in a solution containing naoh with a concentration of 4% (w/v) at room temperature. following this, the fibers were subjected to a rinsing process involving tap water. afterwards, the fibers were washed with a solution consisting of 1% (v/v) acetone and distilled water. fibers were then dried in an oven at 60°c for 24 hours. the compressive, flexural, impact, and tensile strengths of the composite made from 4% alkaline-treated lsf were found to surpass those of the untreated lsf composite. this improvement is attributed to the robust interlocking between the fibers and the matrix. [50] kenaf polyvinyl alcohol (pva) this treatment was carried out by immersing the fibers in a pre-prepared naoh solution with concentrations of 4, 6, 8, and 10 weight percentage (wt.%) for a duration of 4 hours, all at room temperature. the fibers were subjected to rinsing using deionized water until a neutral ph was achieved. treated fibers were then carefully dried in a drying oven set to a temperature of 60°c. the utilization of naoh treatment effectively removed impurities, resulting in a cleaner and more textured surface for the kenaf fibers. additionally, the treatment led to substantial improvements in the tensile properties of the fibers. [39] 4.2. silane treatment this procedure involves the interaction of the fiber with a silane coupling agent followed by fiber drying and curing stages. the application of silane treatment yields a range of potential modifications on natural fibers. these modifications can involve enhancing the hydrophobic nature of the fibers, along with altering their surface morphology and chemical characteristics. the process of applying silane treatment to natural fibers involves three distinct reactions: hydrolysis, self-condensation, absorption and chemical bonding as illustrated in figure 5 [51]. silane molecules undergo hydrolysis when exposed to water and ethanol, releasing alcohol and silanol groups. subsequently, the silanol groups initiate a selfcondensation process, which is deliberately moderated by maintaining an acidic ph environment. this control ensures the presence of smaller molecular-sized free silanol groups. upon introducing fibers to the silane solution, the hydroxyl groups found in the cellulose/hemicellulose of the fibers absorb the silanol groups through hydrogen bonding. this interaction leads to the formation of robust polysiloxanes (-si-o-si-) structures which ultimately establish connections with the matrix through chemical bonding [52]. the degree of modification in natural fibers can be controlled by adjusting key factors: the concentration and duration of the silane treatment, as well as the specific type of silane coupling agent used. this nuanced control has been extensively explored by numerous researchers and scientists, leading to valuable insights. for instance, youbi et al. [53] conducted a study evaluating the impact of silane treatment on surface modification of raphia vinifera fibers (rvf). in their research, fibers were subjected to various concentrations of silane solutions (1% and 5%) to gauge their effects. alshahrani & prakash [54] carried out investigations specifically focused on the effects of silane treatment on fibers and their subsequent impacts on composite properties. in their work, raw chopped fibers and peel646 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic dried particles were treated with amino silane using the acid hydrolysis method. in a separate study by sabarinathan et al. [55], a novel cellulosic fiber extracted from the peduncle portion of the fish tail palm tree underwent treatment with different concentrations (1%, 5%, and 9%) of silane solution. across these research works, a consistent trend emerged: silanetreated fibers exhibited superior mechanical properties when compared to untreated fibers. additionally, it was noted that higher concentrations of coupling agents and prolonged treatment durations led to more pronounced modifications in the properties of the fibers. in addition, certain investigations have delved into the impacts of various silane coupling agents. for instance, wang et al. [56] conducted a study focusing on the mechanical properties and thermal behavior of bamboo fibers reinforced polypropylene composites under the influence of different silane treatments. this study involved the use of three distinct silane coupling agents: γ-aminopropyltriethoxysilane, γ-methacryloxypropyltrimethoxysilane and 3glycidyloxypropyltriethoxysilane. the research found that the tensile strength and flexural strength of the silane-treated fiber composites displayed enhancements when compared to untreated fiber composites, this was attributed to an enhanced interfacial adhesion between the fibers and the matrix. this study underscored the importance of selecting the appropriate silane coupling agent to achieve desired improvements in the performance characteristics of composite materials. fig. 5 the general mechanism underlying the formation between silanes (e.g. 3glycidyloxypropyl trimethoxysilane shown here), and the surface of the fiber. 4.3. acetylation acetylation, a widely employed chemical treatment, serves as a versatile method for altering the characteristics of natural fibers. this treatment revolves around the interaction of cellulose with an acetylating agent, such as acetic anhydride or acetic acid, usually in the presence or absence of a catalyst as illustrated on figure 6. the fundamental concept agro-waste natural fiber sample preparation techniques for bio-composites development... 647 behind acetylation is the substitution of hydroxyl groups (oh) present in natural fibers with acetyl groups (ch3co), effectively rendering the fiber surface more hydrophobic [57]. the acetylation process can yield a spectrum of modifications to the properties of natural cellulosic fibers, encompassing heightened resistance to water, improved dimensional stability and increased thermal stability. furthermore, the treatment can induce alterations in the fiber's surface morphology and bring about changes in its chemical and physical attributes. the extent of modification achievable can be fine-tuned by adjusting the concentration and duration of the acetylation treatment. oladele et al. [58] conducted investigations which revealed that employing higher concentrations of acetylating agents and extending the treatment duration result in more pronounced transformations in the natural fiber's properties. beztout et al. [59] investigated how acetylation processes and cellulose content affect mechanical, thermal and rheological properties of pcl/cellulose composites. the treatment resulted in improved interfacial adhesion and dispersion of cellulose within the pvc matrix, leading to an enhanced surface morphology. furthermore, boustani et al. [60] conducted a comprehensive investigation into the impact of solventfree acetylation using acetic anhydride on fiber structures. in their study, they acetylated flax and wood pulp fibers at room temperature in a solvent-free system, employing acetic anhydride and sulfuric acid as a catalyst. the results revealed a direct relationship between the degree of fiber acetylation and the reaction duration. additionally, souza da rosa et al. [61] explored the feasibility of acetylation techniques applied to natural fibers derived from waste materials originating from pulp and paper mill operations. their study encompassed four distinct acetylation methodologies, broadening the understanding of the potential applications of acetylation across various fiber sources. fig. 6 acetylation mechanisms, (a) with acid catalyst and (b) without acid catalyst. 4.4. bleaching bleaching is a common chemical treatment employed to alter the characteristics of natural cellulosic fibers like cotton, bamboo and hemp. this treatment involves the use of various chemical agents, including hydrogen peroxide, chlorine dioxide and sodium hypochlorite [62]. the selection of a specific bleaching agent hinges on the unique requirements of the fiber in question and its intended application. through the bleaching process, natural fibers undergo several alterations, such as increased luminosity, enhanced capacity for dye absorption and 648 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic improved mechanical attributes. however, if not conducted properly, bleaching can also weaken the fibers. the extent of these modifications is influenced by factors like the type and concentration of the bleaching agent, the temperature, the ph conditions and the duration of treatment [63,64]. however, there are growing concerns regarding the ecological consequences of the chemicals utilized in conventional bleaching procedures. as a result, efforts are underway to develop more sustainable alternatives involving the use eco-friendly bleaching agents and enzymatic bleaching approaches. many investigative studies in this regard have been explored. for instance, pereira et al. [65] explored the influence of acetosolv and environmentally friendly bleaching treatment on orange bagasse fiber. their findings suggest that all the treatment variables employed in their investigation significantly contribute to the fiber's characteristics. the use of this eco-friendly bleaching approach holds promise for employing orange bagasse fibers as reinforcement in polymer matrix applications. in a study conducted by zhang et al. [66], an innovative and eco-friendly scouring and bleaching method for flax rove was investigated, utilizing supercritical carbon dioxide fluid instead of a traditional water-based medium. in comparison with the conventional water-based approach, this proposed supercritical bleaching process conserves substantial amounts of water and energy, establishing it as an eco-friendlier option for producing flax rove and other materials used in the textile industry. furthermore, sojka-ledakowicz et al. [67] introduced a bleaching method that employs the hydrogen peroxide in the gaseous phase. their results demonstrated that vaporized hydrogen peroxide-based bleaching is not only environmentally friendly but also operates at lower temperatures, saves energy and generates minimal waste compared to the conventional waterand energy-intensive bleaching process. regarding the use of enzymatic bleaching singh et al. [68], proposed the use of crude xylano-pectinolytic enzymes for bleaching ramie fibers intended for textile applications. their conclusion highlighted the environmental safety of enzymatic bleaching, which efficiently removed non-cellulosic impurities, rendering the fibers more suitable for textile use compared to chemical bleaching. similarly, hassan et al. [69] conducted bleaching experiments using protease, xylanase and alkaline protease enzymes to evaluate the effects of enzymatic bio-scouring on the physiochemical and mechanical properties of fiber fabrics. their enzymatic approach yielded results on par with the conventional chemical method. 4.5. grafting grafting treatment is a surface modification process employed to alter the properties of natural cellulosic fibers through the introduction of polymer chains onto the fiber surface, as illustrated in figure 7. it is achieved through the chemical bonding of monomers or polymers onto the fiber surface via covalent bonds, resulting in the creation of a polymer layer on the fiber surface [70]. various techniques, such as radiation-induced grafting, chemical grafting, and enzymatic grafting, can be used for the grafting process. the choice of technique is determined by the specific properties required for the fibers and the intended application. grafting can bring about diverse alterations in the properties of natural cellulosic fibers, including increased hydrophobicity, enhanced mechanical properties and improved thermal stability. the properties of the grafting layer can be finely tuned by adjusting the type and concentration of monomers or polymers utilized, as well as the reaction conditions. these variables have been thoroughly explored by researchers. in a study conducted by tataru et al. [71], the radiation grafting procedure of flax natural fibers was investigated. the grafting of epoxidized plant oils onto flax fabrics was found to facilitate the formation of an interphase when cured with an epoxy-amine thermoset matrix. agro-waste natural fiber sample preparation techniques for bio-composites development... 649 while the composite's tensile strength exhibited a significant improvement of 10-20% compared to untreated fabric composites, the study highlighted the potential of radiation grafting in enhancing composite properties. additionally, research by bao et al. [72] demonstrated the application of enzyme-initiated grafting for the modification of cellulose fibers. besides its excellent environmental friendliness, this method showcased the ability to graft chains with well-controlled structures and lengths onto cellulosic fiber surfaces. as a result, this approach offers an environmentally friendly and controllable means for grafting, which can be effectively employed in the production of various functional cellulosic materials. fig. 7 illustration of the grafting process [73] 5. surface/physical treatment physical treatments can also be used to modify the surface of natural cellulosic fibers to improve their compatibility with other materials or enhance their properties. physical treatments do not alter extensively the chemical composition of the fibers, and therefore the interface is generally enhanced via an increased mechanical bonding between the fiber and the matrix. one the most attractive feature of physical methods is the fact that they are ‘clean’ since they do not involve chemicals. some common physical treatment methods are outlined in table 6. table 6 physical treatment processes for natural fibers. surface treatment description benefits drawbacks sources plasma treatment plasma treatment is a process that enhances natural cellulosic fiber’s surface properties using low-pressure, nonthermal plasma. this plasma is created by applying an electric field to a low-pressure gas (e.g., oxygen, nitrogen, or argon), generating reactive species. these reactive species interact with the fiber surface, introducing functional groups, surface etching or crosslinking. ▪ effectively modifies the surface of natural polymers without altering their core properties. ▪ increases the wettability of the fiber-matrix interface, without causing pollution. ▪ the etching mechanism can lead to surface degradation and creates pits on the fiber surface. [74–76] ▪ ▪ 650 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic surface treatment description benefits drawbacks sources corona treatment corona treatment is an electrical process where high voltage is applied near the fiber surface, generating a corona discharge. this discharge produces reactive species like ozone, which introduce polar groups to enhance fiber surface energy and improve adhesion to polymer matrices. ▪ swiftly and efficiently enhances the bonding between natural fibers and diverse polymer matrices. ▪ enhances wettability by increasing surface polarity, promoting compatibility between hydrophilic fibers and hydrophobic matrices. ▪ extended treatment durations could lead to the development of a considerably uneven fiber surface. ▪ surface ablation and etching from the process diminishes fiber strength. [77–79] uv/ozone treatment uv/ozone treatment of natural fibers is a surface modification technique that involves exposing the fibers to ultraviolet (uv) light in the presence of ozone. this process induces photochemical reactions on the fiber's surface, leading to the formation of oxygencontaining functional groups. these newly introduced polar groups enhance the fiber's wettability and adhesion properties, making it more compatible with polymer matrices and other materials in various applications. ▪ enhanced fiber surface properties by introducing polar functional groups, resulting in improved wettability and enhanced adhesion to polymer matrices. ▪ highly versatile and can be applied to a wide range of natural fibers, making it adaptable to various industries. ▪ fibers can be subjected to uv degradation. ▪ the treatment's effectiveness may vary depending on the type of natural fiber. ▪ while the process is environmentally friendly, its energy consumption may be a concern in terms of overall sustainability. [80,81] thermal treatment thermal treatment of natural fibers is a surface modification process that involves subjecting the fibers to elevated temperatures in a controlled environment. during this treatment, the fibers are exposed to heat in an inert gas atmosphere. the high temperature induces chemical changes on the fiber's surface, which can include the removal of impurities, moisture and hemicellulose, resulting ▪ improved thermal stability of the fibers, this is beneficial where fibers may be exposed to elevated temperatures. ▪ enhanced flame resistance of natural fibers, reducing their flammability and making them less susceptible to catching fire or sustaining combustion. ▪ increased crystallinity of ▪ if not controlled precisely and subjected to long exposure, natural fibers mechanical properties become compromised. ▪ heat treatment of the fibers can also make them more brittle and reduce their flexibility, impacting their suitability for certain applications. [82–84] agro-waste natural fiber sample preparation techniques for bio-composites development... 651 in enhanced fiber purity and increased cellulose content. natural fibers, higher crystallinity often leads to improved mechanical properties. electron radiation electron radiation treatment of natural fibers is a surface modification technique that involves exposing the fibers to high-energy electrons. during this process, the electrons penetrate the fiber's surface, causing molecular changes and enhancing its surface properties. the energy from the electrons initiates chemical reactions on the fiber's surface, leading to the formation of new functional groups and increased wettability. ▪ this treatment is highly controllable, allowing for precise adjustments to achieve desired modifications. ▪ the exposure to irradiationinduced degradation will lower the peak thermal decomposition temperature of materials with high cellulose content. [85,86] dielectric barrier dielectric barrier treatment of natural fibers is a surface modification method that employs a high-frequency electrical discharge in a controlled gas environment to enhance the fiber’s surface properties. this process involves placing the fibers between two electrodes separated by a dielectric material, creating a barrier that prevents direct electrical discharge contact with the fibers. when high-voltage alternating current is applied, it ionizes the surrounding gas, forming a plasma. reactive species generated within the plasma chemically modify the fiber surface, introducing polar functional groups and increasing surface energy. ▪ reduces water absorption, enhances chemical resistance and fosters stronger interfacial bonding, ultimately increasing the overall performance and versatility of natural fibers in a wide range of applications. ▪ it requires specialized equipment and expertise to create and maintain the necessary electrical discharge conditions. ▪ the process may not be suitable for all types of natural fibers, and achieving uniform treatment across large fiber surfaces can be challenging. [87,88] 652 r. phiri, s. m. rangappa, s. siengchin, d. marinkovic 6. conclusions in response to the growing global concern for environmental issues, many industrial sectors have increasingly shifted their focus towards adopting sustainable and eco-friendly materials. the use of agro-waste residues as natural fibers for biocomposite development has emerged as a promising alternative, primarily because these resources are inherently renewable, readily available in the natural ecosystem and help address waste management challenges. despite the advantages these materials offer, they face challenges in commercialization due to issues such as poor compatibility with polymer matrices, limited wettability, high moisture absorption, and reduced durability. these limitations are closely tied to both the nature of the fibers themselves and the extraction and preparation processes involved in transforming them into final products. therefore, it is crucial to comprehend the impact of fiber extraction methods and fiber modification procedures to tailor fiber properties to specific application requirements. this review provided a comprehensive understanding of the preparation and processing methods of agro-waste residues into suitable natural fibers for composite development. it was highlighted that the selection of suitable harvesting, retting and decortication method plays a crucial role in the physical properties of the fibers. additionally, it was deliberated that the choice of retting method depends on factors such as climate, water resources, local traditions and desired fiber quality. furthermore, various chemical treatments approaches used for natural fibers offer a range of properties by chemically altering the chemical composition of the fiber thereby enhancing compatibility with matrices and modifying hydrophilicity. physical treatment approaches also play a vital role in modifying the surfaces of natural fibers without changing their chemical compositions. 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(eds.) green composites for automotive applications, pp. 3–41, woodhead publishing. 88. vander wielen, l.c., ragauskas, a.j., 2006, dielectric-barrier discharge initiated grafting to enhance fiber charge, chem eng commun. 193, pp. 683–688. 11687 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 199 216 https://doi.org/10.22190/fume230605035e © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper the up-grating rank approach to solve the forced fractal duffing oscillator by nonperturbative technique yusry o. el-dib1, nasser s. elgazery1, haifa a. alyousef2 1department of mathematics, faculty of education, ain shams university, roxy, cairo, egypt 2department of physics, college of science, princess nourah bint abdulrahman university, riyadh, saudi arabia orcid ids: yusry o. el-dib https://orcid.org/0000-0001-6381-5918 nasser s. elgazery https://orcid.org/0000-0003-4691-2526 haifa a. alyousef https://orcid.org/0000-0003-1727-0958 abstract. the current research studies a fractal duffing oscillator in the presence of periodic force. to find an analytic solution for this oscillator, the aspects explained in the following are considered. first, we obtain an alternative unforced fractal fourthorder equation and then convert it into a continuous space. therefore, the nonperturbative (np) approach is used to calculate the analytic solution for the alternate equation in the second-order form after reducing its rank. it is seen that the analytical and numerical solutions agree very well. the computations reveal that for every value of the fraction parameter, the approximation and numerical solutions are identical. the present study gives reliability in the technique of reducing the order of differential equations. furthermore, the required periodic solution is also obtained by galerkin’s technique. in contrast to the traditional technique, which works to transform the variable and is valid only in the absence of external forces, if there is an external force, it leads to significant mathematical difficulties. the current technique works on the operator, which is simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties. key words: forced fractal duffing oscillator, non-perturbative (np) technique, fractal space, galerkin’s approach received: june 05, 2023 / accepted october 07, 2023 corresponding author: nasser s. elgazery department of mathematics, faculty of education, ain shams university, el makrizy street, roxy, heliopolis: 11566, cairo, egypt e-mail: nasersaleh@edu.asu.edu.eg https://orcid.org/0000-0001-6381-5918 https://orcid.org/0000-0003-4691-2526 https://orcid.org/0000-0003-1727-0958 mailto:nasersaleh@edu.asu.edu.eg 200 y.o. el-dib, n.s. elgazery, h.a. alyousef 1. introduction fractals have been a fascinating characteristic of many natural occurrences since the seventeenth century, and they have a significant impact on a wide range of applications [1,2]. fractals have been widely applied in many fields, such as fluid mechanics, materials design, and petroleum engineering, as well as in other engineering problems to describe and express them clearly and effectively [3,4]. this also appears clearly in the application of the theory of fractals in the investigation within the branched networks to study thermal conductivity [5,6]. for the electrical and mechanical properties of composite materials, the fractal approach offers straightforward and effective tools [7]. moreover, the fractal rheological models provide a fast physical understanding of the rheological characteristics of the fluids [8]. due to its numerous large-scale applications for modeling a variety of phenomena that emerge in most domains, the idea of two-scale fractals theory has recently drawn significant attention from scientists [9]. where this theory enables each problem to be observed on two different scales, one of which is related to a continuous problem and the other to a discrete problem. the hidden qualities of a polymer procedure that cannot be exposed by other models can be determined using the fractal two-phase flow model [1012]. he et al. [13] used a novel fractal vibration architecture model using the two-scale fractals theory. anjum et al. [14,15] also investigate two different fractional models with two-scale for population expansion and tsunami waves, respectively. an interesting topic in mechanical engineering sciences and physics is nonlinear oscillations. one of the most well-liked nonlinear oscillators in engineering, physics, and biology was first introduced in 1918 by the german electrical engineer "georg duffing". since then, duffing oscillators have been widely applied in nature. using a traditional perturbation method, nayfeh and mook derived a periodic solution for the duffing oscillator, which simulates a nonlinear structural vibration. moreover, nayfeh used the multiple-scale methodology to produce a periodic analytical solution for a duffing oscillator in the presence of a damping coefficient [16]. the homotopy perturbation technique was later developed by renowned chinese scientist ji-huan he [17] and utilized in several engineering and physics investigations to determine analytical solutions for the duffing oscillator and other mechanical vibrations. on the other hand, recent years have seen an increase in interest in mechanical engineering applications for fractal oscillators, as a subfield of vibration theory. wang [18] used a novel frequency formulation to study a fractal nonlinear oscillator. the same author also found an analytical solution for a fractal nonlinear oscillator in [19] by employing the effective he's frequency formulation. further, wang and wei [20] have presented a newfangled, straightforward, and effective frequency formula to deal with fractal nonlinear oscillators. also, two approximate solutions for the fractal yao-cheng and duffing oscillators have been introduced in [21-23], built on the harmonic balance and antiquated chinese approaches. since the solution of systems related to the study of fractal oscillators plays an important and useful role in describing and explaining many phenomena that occur on irregular boundaries or in porous media. essentially, it is usually impossible to find exact solutions for these nonlinear oscillators. as a result, mathematicians have been developing different analytical methods to solve this type of nonlinear fractal problem, such as using extended approximation approaches. the most prominent of these methods are perturbation and multiplescale techniques [16]. sheng et al. [24] have presented a fractal model for a shale matrix the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 201 using a multi-scale porous structure. miao et al. [25] investigated another fractal model for a fractured porous medium. furthermore, other fractal models for describing the gas flow inside microfractures in layers of rocks in the depths of the oceans and underground have been mentioned in the literature [26-28]. on the other hand, yang et al. [29] used the local fractional differential transform approach to get the precise solutions in a fractal heat transfer. additionally, several efforts have been established to analyze these phenomena using the non-perturbative method. by using the non-perturbative method, el-dib and elgazery [30,31] established a practical method for solving fractal models with damping forces. on the other hand for more relevant works focusing on different methods and models in the literature, the semi-analytical technique is one of the methods that play an important role in finding solutions of physical problems [32,33]. furthermore, for some numerical solutions see sinc-bernoulli collocation procedure [34]. fractal oscillators have recently attracted more attention in mechanical engineering and physics. despite this, all previous studies depended on a classical technique that works to transform the variable. this traditional technique is limited only by the absence of external forces. here it should be noted that there were some timid attempts to deal with fractal oscillators under external forces, such as the attempt of elías-zúñiga et al. [35], where they faced many mathematical difficulties. regarding our new technique, it is a different and distinct technique from its predecessor, as it works on the operator. it is also simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties. the aforementioned factors serve as an inspiration for this investigation goal, which is to look into the fractal duffing oscillator in the presence of a periodic force via the nonperturbative technique. as opposed to earlier works, with direct handling of the operator, the present technique is simple to deal with forced fractal problems and easy to obtain analytic solutions. 2. problem statement indeed, ji-huan he’s publications [36,37] aiming to formulate the oscillation frequency have considered the conservative nonlinear oscillator: ( ) 0; (0) and (0) 0u f u u a u+ = = = (1) here, f(u) represents the nonlinear restoring force, u is the displacement, and a is a constant. in the mentioned publications, ji-huan he has used the function f (u) to established the oscillation frequency 2(), where  is the total frequency, in the form 2 1 2 ( ) ( ) lim u a df u du  →  = (2) this formula has been well worked for the cubic polynomial of the function f (u) provided that this function does not contain the velocity �̇� or ü and the combination of them. for these shortcomings, the superior and most effective formula was developed by el-dib [38], and it can be used to produce successive approximations to the solutions for non-linear 202 y.o. el-dib, n.s. elgazery, h.a. alyousef oscillations [39]. it is better to elongate this method to equations having damping forces such that ( ) ( , , ) 0u g u u f u u u u+ + = (3) if  is the total frequency to the above oscillator then it useful to write 2( ) ( ) 0u u u +  +  = (4) the following procedure is used to estimate () and 2(): the comparison of eqs. (3) and (4) leads to the following error: 2 2( , ) ( ( , , ) ) ( ) ( ( ) ) ( )e f u u u u t g u u t= − + −    (5) the mean square error is defined as ( ) 2 2 2 2 0 ( , ) ( ( , , ) ) ( ) ( ( ) ) ( ) ; 2 t e f u u u u t g u u t dt t= − + − =       (6) the minimum value requires that 2 2 2 2 2 2 2 0 (( ( , , ) ) ( ) 2( ( , , ) )( ( ) ) ) 0 ( ) ( ) tde d f u u u u t f u u u g u uu dt d d = − + − − =      (7) 2 2 2 2 0 (( ( ) ) ( ) 2( ( , , ) )( ( ) ) ) 0 tde d g u u t f u u u g u uu dt d d = − + − − =      (8) by suitable introducing trial solution in terms of the total frequency , the simplification leads to 2 2 2 0 0 ( ) ( ) ( , , ) 0 t t u t dt u t f u u u dt − =  (9) 2 2 0 0 ( ) ( ) ( ) 0 t t u t dt u t g u dt − =  (10) solving both eq. (9) and eq. (10) yields 2 2 0 2 0 ( ) ( , , ) ( ) ( ) t t u t f u u u dt u t dt   =   (11) 2 0 2 0 ( ) ( ) ( ) ( ) t t u t g u dt u t dt   =   (12) performing the aforementioned integrals gives the equivalent damping coefficient and the equivalent conservative frequency. the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 203 3. the fractal forced duffing equation ineffective modelling of the system is a major obstacle for a nonlinear oscillation system in the fractal space. particularly, the distribution of the periodic property's effect cannot be explained by differential equation models. in order to do this, the current work develops a fractal-differential model, and a forced fractal duffing oscillator is used as an example to show the fundamental characteristics of the fractal oscillator subjected to harmonic force. whereas a lot of research articles focused on a simple oscillation in fractal space [40-42], this work investigates a more generalized fractal vibration system that is modeled by the form: 2 3 02 cos ; 0 1 du u qu t dt    + + =    (13) here, , , q, , and  are the non-zero coefficients, fractal factor, vibration natural frequency, duffing coefficient, parametric force, and amplitude, respectively. assuming that the initial conditions are described as (0) (0) , 0 du u a dt = = (14) the fractal theory has grown in importance in both mathematics and mechanical engineering because it is useful in developing a governing equation in a fractal environment. there are numerous definitions for the fractional derivatives. most definitions often used ones include riemann-liouville, caputo, xiao-jun yang, and jumarie for further justification see [43-45]. the he's fractal derivative is defined as [7, 46, 47]: 0 0 00 ( ) ( ) (1 ) ( )t t t t u t u td u lim dt t t     − →   − =  + − (15) in this case, the fractal factor  runs along the t dimension. if we see a motion on a vast size, it might alter continuously, but if we observe it on a tiny scale, it might discontinue. as a result, the fractal theory is an effective mathematical instrument for conducting more thorough global research. moreover, this fractal derivative exhibits some of the following characteristics [48-50]: 0 1 2 , and → → → = = = du du du lim u lim u lim u dt dt dt     (16) here the over-dot refers to the time classical derivative. the fractal derivative given in eq. (15) has widely been used to deal with porous or hierarchical structures [46-50] with great success. if we see a motion on a vast size, it might alter continuously, but if we observe it on a tiny scale, it might discontinue. as a result, the fractal theory is an effective mathematical instrument for conducting more thorough global research. the technique of creating mathematical models and the geometric physical interpretation of fractal derivatives were discussed in [51]. defining the restoring force f (u) in the current problem as 2 3 0( )f u u qu= + (17) 204 y.o. el-dib, n.s. elgazery, h.a. alyousef accordingly, eq. (13) can be written as 2 ( ) cos du f u t dt  + =  (18) operating 𝑑 𝑑𝑡𝛼 on eq. (18) one time and 𝑑 𝑑𝑡2𝛼 another time gives 3 ( ) cos du d d f u t dt dt dt   + =  (19) 4 2 2 ( ) cos du df u d t dt dt dt   + =  (20) to estimate 𝑑 𝑑𝑡𝛼 cos ω𝑡 and 𝑑 𝑑𝑡2𝛼 cos ω𝑡 one may apply the following fractal derivative proposed by el-dib and elgazery [30,31]: 11 1 2 2 .. ( cos( ) sin( ) ) ..n n d s s d dt −= +     (21) therefore, one gets 11 1 2 2 cos cos( )cos sin( ) sin d t s t s t dt − =  −       (22) 2 2 2 2 2 cos ( )( ( ) )cos 2 ( ) ( ) sin d t b a t a b t dt  = −   −        (23) here, a() and b() are described by 11 1 2 2 ( ) cot( ) and ( ) sin( )a s b s −= =     (24) employing eqs. (22) and (23) into eqs. (19) and (20) by using eq. (18) yields 11 1 1 2 2 23 2 cos( ) ( ) cos( ) ( ) sin( )sin du du d s f u s f u s t dt dt dt −− + − = −            (25) ( )2 2 2 2 4 2 2 ( ) ( ) ( ) ( ) 2 ( ) ( )sin du df u du b a f u a b t dt dt dt          + − −  + = −      (26) removing sin t between eqs. (25) and (26) yields ( ) ( )2 2 21 24 3 2 2 2 cos ( ) ( ) ( ) 0 du du df du df s b a f u dt dt dt dt dt              − + + +  + + =        (27) this is the alternative form of the forced fractal eq. (18) in the form of a fractal fourthorder equation. this equation is subject to the following initial conditions: 1 22 3 (0 ) (0 ) (0 ) (0 ) , 0 , ( ), cos( ) du du du u a f a s dt dt dt = = = − = −             (28) the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 205 to convert eq. (27) and its corresponding initial conditions given in eq. (28) into continuous space, the transformation given in eq. (21) may be used. therefore, with some simplifications, one gets 4 3 2 2 2 2 2 2 1 ( ) 2 ( ) ( )( ) ( ) ( ) 0a d u a a d u b a a d u d f u b + − + + +  + +  + = (29) the operator d represents the derivative corresponding to the variable t. thus, eq. (29) is a nonlinear fourth-order equation free of the variable coefficients (4) (3) 2 2 2 2 2 2 3 2 2 2 4 2 2 3 2 2 2 2 02 2 ( ) 2 ( ( ) ) ( 6 3 ) 1 ( ( ) ) 0 q u au b a u a b a a u u u u u u b a b a b a u b + + +  + +  − +  + + + − + +  +  = (30) this is the alternative form of the forced fractal duffing eq. (13) in continuous space. also, the alternative initial conditions in the continuous space are performed in the form ( ) ( ) 2 2 3 2 1 (0) , (0) ( ), (0) ( ( )) ( ) , ( ) ( ) 0 ( ) 4 3 ( ) ( ) u a u aa u f a a a b a u a a f a b  = = − = − +    = − − −          (31) it is noteworthy, if the initial velocity is not zero such that u(0) =a and 𝑑𝑢(0) 𝑑𝑡𝛼 = 𝐵, the present approach has a good applicable so the application of the formula given in eq. (9) to the fractal initial condition has the form 11 1 2 2 (0) cos( ) (0) sin( ) (0) du s u s u dt −= +     (32) employing the non-zero initial conditions yields that �̇�(0) = 𝐵 𝑆𝛼−1𝑠𝑖𝑛( 𝝅𝛂 2 ) − 𝐴 𝑆 𝑐𝑜𝑡 ( 𝝅𝛂 2 ) and u(0)=a. therefore, as 𝐵 → 0 the converted initial conditions given in eq. (31) will arises. in the transformation given in eq. (21), the fractal parameter s could be estimated by comparing between the linear frequency given in eq. (30) with the linear one of the fractal eq. (27). this comparison shows that 3 2 2 2 2 4 4 20 0 2 2 ( ) (1 ) 0 ( 1) ( 1) b b b a a b b b b  −  − + + = − −   (33) here, b() ≠ 1 is considered, unless  = 1. employing the definition of a() given in eq. (24) in the above equation yields 2 1 2 2 2 2 4 2 2 2 2 2 42 0 0 0 tan ( ) ( ) ( ) ( ) 4(1 ) (1 ) 2 ( 1) s b b b b b b b b   = −  +  − + −  −   −     (34) 206 y.o. el-dib, n.s. elgazery, h.a. alyousef it is worthwhile to observe that the applied frequency  can be sought in terms of the fractal order . this can be read from eq. (33) to have 2 2 2 2 2 2 0 3 2 2 4 0 ( ( 1) ) ( ) (1 ) b b a b a b a b − −  = + −    (35) the trial solution for the nonlinear fourth-order eq. (30) may be suggested in the form 0 1 2 3 4( ) cos sin cos3 sin3u t k t k t k t k t   = + + + (36) here,  represents the oscillation frequency and the constant coefficients k’s can be estimated by using the initial conditions given in eq. (31). consequently, 2 2 2 1 2 2 2 2 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 4 2 3 1 ( ( ( ) 9 )), 8 (4 ( 3 ( ) 9 )), 8 1 ( ( ( ) )), 8 (4 ( 3 ( ) )) 24 k a a f a b b a k a a b f a b b k a a f a b b a k a a b f a b b  = + − +    = − + − +    = − + − +   = + − +               (37) since the original system is a second-order derivative, the fourth-order derivative in eq. (30) is artificial and should be reduced to the second-order one. in addition to reducing rank of eq. (29) to the second-order form, our goal is to construct a linearizing form for it. the linearizing method can be used to achieve this goal. the following process may be suggested [52]: rearranged eq. (30) in the form (4) (3) 2 2 2( , , ) 2 ( ( ) ) ( , ) 0g u u u u a b a a u f u u u+ + − + = (38) the above two functions g and f are defined as (4) (3) (4) (3) 2 2 2 2 4 2 2 3 2 2 2 2 2 2 2 02 2 1 3 ( , , ) 2 ( ) , 1 ( , ) ( ( ) ) ( 6 ) q g u u u u au b a u u u u b q f u u a b a b a u u b b   = + + +  +      = − + +  +  +  +   (39) apply caughey’s linearized approach [53] to eq. (38) with considered the suggested trial solution given in eq. (36) yields 2 2 2 0 1( ) 2 ( ( ) ) ( ) 0u a b a a u u+ +  − + =    (40) here, 0 () and 1 () are estimated using the following formula proposed by el-dib [38] the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 207 (4) (3) 2 0 0 0 0 0 0 2 0 0 ( , , ) ( ) 2 ( ) ; ( ) t t g u u u u t dt t u t dt = =       (41) 2 0 0 0 0 1 2 0 0 ( ) ( , ) ( ) ( ) t t u t f u u dt u t dt =     (42) the results of the above integrals are straightforward in their calculations and are available from the corresponding author. it should be noticed that eq. (40) can be expressed in the following manner: 22 ( ) ( ) 0u u u   + + = (43) here, the coefficients  () and 2() are constants functions of the frequency  given by 2 2 2 0 ( ( ) ) ( ) ( ) a b a a+  − =    (44) 2 1 0 ( ) ( ) ( )       = (45) eq. (43) is a linear second-order equation having a damping force. the damping force is characterized by the coefficient  (). the solution of eq. (43) has a damping behavior unless  is equal unity. more damping behavior is found as  having a greater decrease than unity [30,31,54]. the solution that covers this behavior is performed in the following form: ( ) 1 2 3 4( ) ( cos sin cos3 sin3 ) t u t e k t k t k t k t − = + + +       (46) the above solution is obtained due to applying the normal form technique. in this approach, the frequency  has the form 2 2 2( ) ( )    = − (47) 4. numerical estimation the numerical solution of the original forced fractal nonlinear oscillator given in eq. (13) at → 1 and analytical solution given in eq. (46) were compared and shown in fig. 1 for the system having a = 0 =1, q =  =0.01, and =10. the relative error is estimated in this calculation to be 0.1331 which reveals that the two solutions were in accord. further, the comparison of the analytical solution given in eq. (46) of the second-order eq. (43) with the numerical solution of the fourth-order eq. (30) has been displayed in fig. 2 for the case of  is close to unity. the system that is used in these calculations has a = 0 =1, q =  =0.01, and =10. the relative error is estimated in this calculation to be 0.006445 which 208 y.o. el-dib, n.s. elgazery, h.a. alyousef demonstrates that the analytical and numerical solutions are in good accord with one another. it is noted that the periodic solution is revealed in this calculation. this behavior is expected because → 1 the fractal parameter a() should tend to zero. consequently, the damping forces in both eq. (30) and eq. (43) will disappear. the damping behavior will appear due to the fractional order  decreasing away the unity as shown in fig. 3. in the graph of fig. 3 consequent values  are considered and collected together with the case of   1. the agreement between the numerical solution and the analytical solution is still observed even when the value of  decreases from one to zero. the comparison shows that the damping behavior increases with decreasing in . in a word, the computations revealed that for every value of the fraction parameter, the approximation and numerical solutions are identical. from the above figures, the present study gives reliability to the technique of reducing the order of differential equations. this makes us think that the alternative equation can be handled, discussed, and then the solutions can be used as a reliable indicator. fig. 1 comparison of the numerical solution of the original forced fractal nonlinear oscillator given in eq. (13) with the analytical solution given in eq. (46) at  is very close to unity for the system having a = 0 =1, q =  =0.01, and =10 fig. 2 comparison of the analytical solution given in eq. (46) with the numerical one of eq. (30) in the case of at → 1 for the system having a = 0 =1, q =  =0.01, and =10 the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 209 fig. 3 represents to the influence of the variation in  on the comparison of the analytical solution given in eq. (46) with the numerical one of eq. (30) for a = 0 =1, q =  = 0.01, and  = 10 in figs. 4-6 the examinations are done for the amplitude and the frequency of the applied periodic force. for this purpose, the fractional order is held at  = . the calculations are used in the same system as given in fig. 2, with alternative values of the parameters  and . it is noted that the relative error is 0.1189 when  =0.1, see fig. 4, but this error becomes 0.3281 as  =0.2 shown in fig. 5. therefore, when comparing fig. 4 with fig. 5, it becomes clear that small delta values make the solution more accurate than larger values. when  is held at  =0.1 with changes in  from 7 to 10 as observed in fig. 6, the relative error has increased for the case of large values of . hence, the comparison between figs. 4 and 6 means that the increase in  leads to improving the accuracy of the solution. it should be noted that one of the most important features of these calculations is that the fourth-order equation can be reduced to a second-order one via the non-perturbative approach. fig. 4 the relative error due to the comparison of the analytical solution given in eq. (46) with the numerical one of eq. (30) for a = 0 =1, q =0.01,  = 0.1,  = 0.9, and  = 7 210 y.o. el-dib, n.s. elgazery, h.a. alyousef fig. 5 the relative error due to the comparison of the analytical solution given in eq. (46) with the numerical one of eq. (30) for a = 0 =1, q =0.01,  = 0.2,  = 0.9, and  = 7 fig. 6 the relative error due to the comparison of the analytical solution given in eq. (46) with the numerical one of eq. (30) for a = 0 =1, q =0.01,  = 0.1,  = 0.9, and  = 10 5. periodic solution on the other side, the periodic solution of eq. (43) can be established, for a non-zero coefficient (). in this case, the frequency  will recall �̂� which is assumed to cover the periodic solution. to accomplish this periodic solution, galerkin’s technique may be applied to establish the frequency �̂�. to construct the residual function r(�̂�;t), one can insert the suggested solution given in eq. (36) into eq. (43), which gives 2 2 2 2 1 2 2 1 2 2 2 2 3 4 4 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ; ) (( ) 2 )cos (( ) 2 )sin ˆ ˆ ˆ ˆ ˆ ˆ(( 9 ) 6 )cos3 (( 9 ) 6 )sin3 r t k k t k k t k k t k k t = − + + − − + − + + − −                   (48) to estimate the total frequency �̂�, one may apply the following galerkin’s formula: ( ) 0 ˆ ˆ ˆ ˆ ˆ( ; ) cos sin cos3 sin 3 0 t r t t t t t dt    + + + = (49) the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 211 using the definitions given in eq. (37) and the residual given in eq. (48), the above integral leads to the following frequency equation: 4 3 2 4 3 2 0 ˆ ˆ ˆ 0c c c c  + + + = (50) here, the coefficients c’s are listed below: 2 4 2 4 2 2 2 2 2 3 0 4 2 2 2 2 2 4 2 2 3 2 2 0 0 4 2 2 2 2 2 0 0 ˆ9 ( 2 ( )) ˆ ˆ12( 2 ( ) ( )) ˆ ˆ(9 9 12 13 ) ( ) 6( ) ( ) 3 ˆ(3 3 4 ) ( ) c ab a c a a a q a aab ab c a a q a ab a a a q a aa b c a a q a aa b = −  = + − − − +   = + − − + + − − −   = + − −                      (51) given the real roots of the frequency in eq. (50), the periodic solution of eq. (43) can be obtained in the form 1 2 3 4 ˆ ˆ ˆ ˆ( ) cos sin cos3 sin3u t k t k t k t k t   = + + + (52) numerical illustration for the current state is shown in the figs. 7-12. the calculations are made for the periodic solution given in eq. (52) where the frequency �̂� is estimated from the frequency eq. (50). it is noted that the coefficients k’s are depend on the frequency �̂� as defined by eq. (37). the numerical system that is considered in these calculations is a = 0 =1, q =  =0.1, and =3 with the variation of the fractal order . in fig. 7, the smooth periodic curve is observed when the fractal order has the value  = 0.9. the periodic curve due to  = 0.8 that lies in fig. 8 appears similar to the case of the first successive solution in [39]. this behavior of the periodic solution has been affected by decreasing in the fractal order as shown in figs. 9-12. fig. 7 representation of the periodic solution given in eq. (52) for a system having a = 0 =1, q =  = 0.1,  = 0.9, and  = 3 212 y.o. el-dib, n.s. elgazery, h.a. alyousef fig. 8 representation of the periodic solution given in eq. (52) for a = 0 =1, q =  = 0.1,  = 0.9, and  = 3 fig. 9 representation of the periodic solution given in eq. (52) for a = 0 =1, q =  = 0.1,  = 0.7, and  = 3 fig. 10 representation of the periodic solution given in eq. (52) for a = 0 =1, q =  = 0.1,  = 0.5, and  = 3 the up-grating rank approach to solve a forced fractal duffing oscillator by the np technique 213 fig. 11 representation of the periodic solution given in eq. (52) for a = 0 =1, q =  = 0.1,  = 0.3, and  = 3 fig. 12 representation of the periodic solution given in eq. (52) for a = 0 =1, q =  = 0.1,  = 0.1, and  = 3 6. conclusions a fractal duffing oscillator with periodic force was investigated in the current work. ultimately, the present conclusion observations are summarized as follows: ▪ by using the rank upgrade technique, an alternative unforced fractal fourth-order equation was obtained and converted to its traditional derivative form in the continuous space. ▪ after reducing its rank, the non-perturbative approach was used to calculate the analytic solution for the alternate equation in the second-order form. ▪ it is seen that the analytical and numerical solutions agree quite well. ▪ the computations revealed that for every value of the fraction parameter , the approximation and numerical solutions are identical. ▪ 𝑑 𝑑𝑡𝛼 cos ω𝑡 and 𝑑 𝑑𝑡2𝛼 cos ω𝑡 was estimated, which is considered a new result that can be used in future works. 214 y.o. el-dib, n.s. elgazery, h.a. alyousef ▪ the rank upgrade/reducing technique was introduced as a reliable approach to overcome difficulties. ▪ the present study gives reliability in the technique of reducing the order of differential equations. ▪ one of the most important features of these calculations lies in the fact that a fourthorder equation can be reduced to a second-order one via the non-perturbative approach. ▪ galerkin's method was also used to find the required periodic solution. ▪ decreasing the fractal order has an impact on the behaviour of periodic solution. ▪ in contrast to the traditional technique, which worked to transform the variable and was valid only in the absence of external forces, if there was an external force, it leads to great mathematical difficulties. the current technique works on the operator, which is simple and effective when fractal oscillators with external forces are investigated, it is easy to obtain analytic solutions, and the technique doesn't lead to any mathematical difficulties. acknowledgement: the authors express their gratitude to princess 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weizhong wang4, ligang zhou5 1anhui electrical engineering professional technique college, hefei, anhui, china 2 school of economics and management, southeast university, nanjing, jiangsu, china 3 department of logistics, military academy, university of defence in belgrade, belgrade, serbia 4 school of economics and management, anhui normal university, wuhu, anhui, china 5 school of mathematical sciences, anhui university, hefei, anhui, china abstract. the power transformer is one of the most critical facilities in the power system, and its running status directly impacts the power system's security. it is essential to research the risk priority evaluation of the power transformer parts. failure mode and effects analysis (fmea) is a methodology for analyzing the potential failure modes (fms) within a system in various industrial devices. this study puts forward a hybrid fmea framework integrating novel hesitant fuzzy aggregation tools and critic (criteria importance through inter-criteria correlation) method. in this framework, the hesitant fuzzy sets (hfss) are used to depict the uncertainty in risk evaluation. then, an improved hfwa (hesitant fuzzy weighted averaging) operator is adopted to fuse risk evaluation for fmea experts. this aggregation manner can consider different lengths of hfss and the support degrees among the fmea experts. next, the novel hfwga (hesitant fuzzy weighted geometric averaging) operator with critic weights is developed to determine the risk priority of each fm. this method can satisfy the multiplicative characteristic of the rpn (risk priority number) method of the conventional fmea model and reflect the correlations between risk indicators. finally, a real example of the risk priority evaluation of power transformer parts is given to show the applicability and feasibility of the proposed hybrid fmea framework. comparison and sensitivity studies are also offered to verify the effectiveness of the improved risk assessment approach. key words: risk priority evaluation, power transformer parts, aggregation operator, critic method, hesitant fuzzy sets received february 23, 2022 / accepted march 20, 2022 corresponding authors: jing chen and qun wu anhui electrical engineering professional technique college, hefei, anhui, 230051, china, and school of economics and management, southeast university, nanjing, jiangsu, 211189, china e-mails: chenjingdz024@163.com (j. chen) and qunw_ahu@126.com (q. wu) 400 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou 1. introduction the power transformer is one of the most important and expensive facility in the power system [1]. it holds the balance in guaranteeing the secure operation of the power system. usually, power transformers in the operating process are often subject to various factors kinds of stresses as electrical, mechanical, chemical, and thermal stress from internal and external environments. if some accidents happen in the power transformer, it will destroy the normal running of the power systems, even thorough disruption of power systems. the power transformer is a complicated structure that involves several parts such as iron core, tap switch, cooling system, non-electric quantity protection, winding, bush, and body part [2]. the running failure risk of the power transformer is closely correlated with the risks of the power transformer parts. through decomposing the power transformer into several parts and achieving the quantitative risk assessment and priority evaluation of these parts, one can identify the weak link of the system and determine the risk factor which does highly affect the equipment reliability [3]. the risk priority evaluation of power transformer parts significantly enhances the pertinence on transformer repair and maintenance and prevents and reduces failure risk in electricity-management systems. consequently, the study on risk priority evaluation of power transformer parts has essential significance in both theory and practice [4]. the fmea (failure mode and effect analysis) method, known as one of the most effective risk analysis tools, has been widely applied in prioritizing the risk failure modes (fms) in electro-medic industry [5], supply chain system [6], and the chemical industry [7]. diverse disciplines and fields like evidential reasoning method [8], prospect theory [9], probabilistic graphic [10], and logistic regression [11] have also been performed to boost the development of the theoretical researches on fmea. as for the traditional fmea, experts employ a 1-10 numerical scale to offer the risk priority number (rpn) to measure the risk value and the overall risk prioritization. a large value implies a high-risk degree. for more details about the fmea implementation process, one can read [12, 13]. nevertheless, the rpn ignores uncertainty when experts present their reference opinions. owing to the uncertain and complex nature of risk, one cannot accurately estimate the risk priority evaluation in real life. plenty of uncertain tools have been set to overcome the deficiencies of rpn in classical fmea [14, 15]. the fuzzy technique is one of the most active and the broadest fields of application research in improving fmea [16-18]. the fuzzy technique is a flexible means of representing subjective uncertainty and fuzziness when experts declare their opinions. with the scholars increasingly focusing attention on fuzzy sets (fss) theory, there are various types of fss supporting the applicability and convenience of uncertain fmea for risk evaluation and prioritization. some famous fss such as the intuitionistic fuzzy sets (ifss) [19], z-numbers [20], and pythagorean fuzzy sets [21] were used in the fmea successfully, which improved the robustness of the risk analysis procedure. however, there often exist conditions that experts are hesitant to offer a precise risk value. then, the classical fss will be inadequate and incapable of describing such hesitant evaluation information. as an expansion of the traditional fss and ifss, the hesitant fuzzy sets (hfss) has several possible membership values, making it more flexible and comprehensive in representing fuzzy information [22]. the hfss also has the simple mathematical expression form and numerical computation [23]. it has already successfully been used in various decision-making fields like green sustainable airport design [24], supplier selection [25], and e-learning website evaluation [26]. this paper uses the hfss to appraise the fuzzy evaluation of risk indicators under each fm to better express the complicated vagueness and hesitancy in the risk assessment. risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 401 the risk assessment in the fmea framework could be thought of as a multi-criteria group decision-making (mcgdm) problem [27, 28]. under the mcgdm term for fmea, the fmea experts’ weights and the risk indicators’ weights play crucial roles in deciding the group risk evaluation and the final collective risk values of the fms. in the fmea team, the fmea experts might own different perspectives of the risk assessments as they have diverse professional backgrounds and specialized skills. concerning experts’ weights, because the risk analysis process needs to cooperate mutually, the principle of the support degrees among their opinions provides a new angle for fusing their risk evaluations. regarding the weights of risk indicators, existing objective weighting methods like the entropy method [9] and the maximizing deviation method [29] think mostly about the differences among the risk indicators. they do not take the possible correlations among risk indicators into account. hence the objective weighting method that can consider the correlations among risk indicators when operating the risk analysis is welcome. meanwhile, the subjective weights cannot be negligible when processing the risk evaluation. as a result, it could be imperative to incorporate both subjective and objective weights for fmea experts and risk indicators to achieve a comprehensive risk evaluation result. even though the classical and improved fmea framework for risk evaluation and prioritization have made fruitful achievements. as the current risk analysis environment gets more complex, fmea still holds some space for its self-exploitation in mathematical modeling. in summary, the following three research motivations will increase the feasibility and practicability of the fmea problems: ▪ the precise numerical risky evaluations are unavailable to the fmea experts. even though the fss have numerous domains to fmea problems, few studies paid attention to the hfss for fmea risk evaluation. it is meaningful to research the fmea disposing of uncertain information under the hfss environment. ▪ despite many successful application fields using the hfss, there is still a gap in the theoretical research results of the hfss, especially in dealing with hfss of different lengths. it is beneficial to develop a solution that can not only ensure the integrity of the hfss but also offer a computational model and some information measures for the hfss with different lengths. ▪ most of the existing fmea frameworks for risk evaluation support that mutual independence exists among the risk indicators. however, there are often some correlations among risk indicators in actual risk analysis. thus, it is necessary to integrate the objective weights for risk indicators with the fmea approach. this paper will develop a risk priority evaluation model using a hybrid fmea approach under the hfss environment, and the main contributions of our paper include: ▪ the hfss representation model is an efficient tool for expressing the hesitant fuzzy evaluation offered by the fmea experts. using the least common multiple rule to deal with hfss of different lengths and utilizing the novel hesitant fuzzy aggregation operators and distance measures for hesitant fuzzy modeling and calculation can enrich the theoretical research results of the hfss. ▪ consider the evaluation similarity between experts and the correlations among risk indicators. the novel hfwa operator with the support degree is developed to fuse the individual risk evaluation matrix. then the novel hfwga operator with critic weights is applied to decide the risk priorities of fms. the subjective and objective combined weights will help get a complete risk evaluation result. 402 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou ▪ to illustrate the validity of our proposed novel fmea methodology, a case study of risk priority evaluation of power transformer parts is provided. then the comparative analysis with some existing fmea models is implemented, which also vastly guarantees the persuasive power. the rest of our paper is: some literature is reviewed in section 2. section 3 are some basic theoretical concepts about the hfss, including the definition, new operational laws and aggregation operators, and the distance measure. section 4 constructs a hybrid fmea framework with hesitant fuzzy aggregation tools and combined weights for fmea experts and risk indicators. section 5 employs the proposed model for the risk prioritization of power transformer parts. in section 6, sensitivity and comparative analyses are performed to verify the validity of the proposed model. some conclusions are drawn in section 7. 2. literature review 2.1. fmea approach using fss theory fmea is one of the most powerful tools for analyzing system reliability and safety. research showed that the fmea plays a significant role in maintenance management strategies in risk analysis. with the relentless march of technological progress, the mathematic model, mcdm approach, and artificial intelligence are adopted more and more when solving fmea problems. the risk priority evaluation involves high complexity and various angles, and the fmea method is proven to be valid to assist experts in making a reasonable decision. in recent years, many scholars have focused on research in the fmea area using fuzzy techniques and achieved encouraging results [30, 31]. zhang et al. [32] developed a linguistic distribution opinion evolution-based social network consensus model to help fmea experts reach a consensus. using an extended copras method to solve the risk evaluation issues and determine the weights with the kemeny median and swara methods, shen and liu [33] proposed a novel risk assessment model in a free double hierarchy hesitant fuzzy linguistic term set environment. jin et al. [34] ranked fms using interval-valued q-rung orthoptic fuzzy sets (ivq-rofss). they developed a scientific risk evaluation model combining the ivq-rof-deviation maximization method with the ivq-rof-additive ratio assessment method. huang and xiao [35] developed a novel improved fmea framework for risk evaluation with a novel concept named the interval-valued intuitionistic fuzzy clouds. the risk analysis in fmea is uncertain, complex, and ambiguous. it would be an excellent choice for the experts to use the fss tools to represent their vagueness in the fmea execution. this way can preserve information integrity and improve the accuracy of quantitative analyzes. 2.2. decision-making evaluation model with hfss due to the subjective and objective complexity in real-world risk analysis, experts would have difficulties giving their perspectives precisely. by contrast, experts often feel more comfortable hesitantly giving their opinions. the hfss expressed by several possible membership values have proven solid in modeling the fuzzy evaluation. in recent years, many researchers have modelled the uncertain information using the hfss [36–40]. with the early screening of lung cancer is becoming exceedingly significant for the effective treatment of lung cancer, liao et al. [36] constructed a framework uses the double normalization-based multi-aggregation (dnma) and delphi methods with hfss to risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 403 solve the lung cancer screening problem. mishra et al. [37] established a hybrid framework containing the aras method and a novel divergence measure for hfss, to find the best antiviral therapy which can be reliable for covid-19 patients with mild symptoms. liao et al. [38] put forward a choquet integral-based hesitant fuzzy gained and lost dominance score method considering the interactions among criteria and the experts’ preference characteristics in mcgdm problems and then applied their model to the higher business education evaluation. considering that energy storage technologies (ests) enable coping with the intermittency of energy sources by storing excess energy for use when needed, colak and kaya [39] established an integrated mcdm model consolidating the delphi, ahp, and vikor methods with hfss to evaluate ests for turkey. as reviewed beforehand, it is regularly hard for fmea experts to give their appraisal utilizing exact value on a mathematical size of 1-10. compared with other fuzzy tools, the hfss is more successful in displaying the hesitant uncertainty in real applications. hence, hfss are applied to manage the ambiguous risk evaluations in this paper. 2.3. the objective critic weighting approach when computing the risk priority of fms, the final evaluation and ranking are highly dependent on the weights of the risk indicators. the weighting determination methods usually incorporate subjective and objective weighting strategies. the subjective weights mainly rely on the expert’s judgments, which are impacted by the experts’ professional knowledge structure and expertise and skills in the related areas. the objective weighting technique decides the weights with the given assessment data by addressing mathematical models. these objective weights are beneficial in cases where the subjective weights given by the experts are inconsistent so that one can achieve more objective results. the well-known objective critic weighting method was initiated by diakoulaki et al. [41] in 1995. it relies on the analysis of the assessment matrix for collecting all preference information included in the assessment criteria. the calculation of acquiring the objective weight contains the standard deviation of the criteria and the correlation between the criteria. critic method has been successfully utilized in accomplishing various decision-making scenarios [42–46]. lai and liao [42] focused on solving the blockchain platform evaluation issues by incorporating linguistic d numbers (ldns), dnma method, and critic method. in their model, the critic technique is combined with the ldn-based dnma approach to mirror the correlations between criteria in the blockchain platform assessment course. as for the sustainable supply chain risk management (sscrm), to characterize the principal risk factors and rank the business alternatives based on these factors, abdel-basset and mohamed [43] developed a joined pathogenic methodology dependent on the topsis-critic strategy to gauge the sscrm factors. peng and huang [44] presented a novel q-rof financial risk decision-making method based on cocoso with critic methods. they applied the combined weight method based on critic and the linear weighted comprehensive process to simultaneously consider subjective and objective information. wu et al. [45] offered a practical model for site determination of photovoltaic hydrogen production project based on the mcdm method. their model determines the subjective and objective weights by applying bwm and critic, respectively. then the collective weights are derived by using the game theory. the execution of fmea is a run of the mcgdm issue, and the weights of both fmea experts and risk indicators ought to be fundamental components in fmea. in a mcgdm, experts’ opinions will be supported by each other, and there are often correlations between 404 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou criteria in actual fmea implementation courses. thus, it is necessary to consider the similarity measure among fmea experts and the amount of information contained in the risk indicators to offer objective weights for real-world fmea problems. 3. methodology 3.1. hesitant fuzzy sets definition 1. [23] let x be a nonempty set. a hfss h on x is a function that maps each element of x to a subset of [0,1] such that { , ( ) | }h x h x x x=    (1) where h(x) is a set of possible values in [0,1], denoting the possible membership degrees of an element xx to the set h. xia and xu [23] called h(x)={γ1,…,γ#h} a hesitant fuzzy element (hfe), and let ω be the set of all hfes. γ1 (l=1,…,#h) and #h are the possible elements and the elements number of h, respectively. without loss of generality, the elements in h(x) are in increasing order, and #h is called the length of h. definition 2. [23] for a hfe h, # 1 1 ( ) # h ll s h h  = =  is called the score function of h. for two hfes h1 and h2, if s(h1)>s(h2), then h1 is superior to h2, denoted by h1≻h2; if s(h1)=s(h2), then h1 is indifferent to h2, denoted by h1~h2. 3.2. new operational laws and two novel aggregation operators for hfss one can find that different hfes may have different lengths, leading to difficulties in making operations on hfes. the least common multiple (lcm) rule is widely used in number theory and statistics. it gives valuable references and ideas in studying the multi-fuzzy sets. inspired by [47], the concept of r-hfe is: definition 3. let h={γ1,γ2,…,γ#h} be a hfe, and #h be its length, then hr is a bag (or multiset) with r#h elements in hr such that 1 1 1 2 2 2 # # #{ , ,..., , , ,..., ,..., , ,..., }.r h h h r times r times r times h         = (2) example 1. let h1={0.2,0.3} and h2={0.5,0.6,0.7} be two hfes. as the lengths of h1 and h2 are different, we use the proposed lcm principle to normalize the two hfes. the lcm of 2 and 3 is 6, then h 3 1 ={0.2,0.2,0.2,0.3,0.3,0.3} and h 2 2 ={0.5,0.5,0.6,0.6,0.7,0.7}. according to theorems 2-5 in [48], it has shown that some significant information measures of a hfe stay unaltered under the lcm rule, so next we propose some new operational laws and two novel aggregation operators for hfss based on the lcm rule definition 4. let h, h1, and h2 be three hfes, #h, #h1 and #h2 are their lengths, respectively, l=lcm(#h1,#h2), and λ be a real number. then risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 405 (1) addition: 1 1 2 2 1 1 2 21 2 1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 1 2 2 # # 1 1 2 2 # # /# /# /# /#/# /# { ,..., , ,..., ,..., ,..., } { ,..., , ,..., ,..., ,...,h h h h l h times l h times l h times l h timesl h times l h tim h h             =   1 2 1 2 ( ) ( ) ( ) ( ) 1,..., } . es l l l l l l        = = + −  (2) multiplication: 1 1 2 2 1 1 2 21 2 1 1 1 1 1 1 2 2 2 2 2 2 1 2 1 1 2 2 # # 1 1 2 2 # # /# /# /# /#/# /# { ,..., , ,..., ,..., ,..., } { ,..., , ,..., ,..., ,...,h h h h l h times l h times l h times l h timesl h times l h tim h h             =  1 2 ( ) ( ) 1,..., } { }. es l l l l    = =  (3) scalar-multiplication: {1 (1 ) }. h h      = − − (4) power operation: { }. h h     = property 1. let h, h1, and h2 be three hfes, λ, λ1, and λ2 be three real numbers, then (1) h1⊕h2=h2⊕h1; (2) h1⊗h2=h2⊗h1; (3) λ(h1⊕h2)=λh1⊕λh2; (4) (h1⊗h2)λ=h λ 1 ⊗ h λ 2 ; (5) λ1h λ2h=(λ1+λ2)h; (6) hλ1⊗ hλ2=hλ1+λ2. definition 5. let hj (j=1,2,…,n) be a collection of hfes, #hj (j=1,2,…,n) are the lengths of these hfles. l is the lcm of #hj (j=1,2,…,n), and w=(w1,w2,…,wn)t is the associated weighting vector satisfying wj ≥0 and ∑jwj=1. then a novel hesitant fuzzy weighted averaging operator is a mapping hfwa: ωn→ω such that: /# 1 2 ( ) ( ) 1 1,2,..., 1 ( , ,..., ) 1 (1 ) | .j j nn w l hj j w n j j l l j j l l j hfwa h h h w h h   = = =    =  = − −       (3) definition 6. let hj (j=1,2,…,n) be a collection of hfes, #hj (j=1,2,…,n) and l be defined before, a novel hesitant fuzzy geometric weighted averaging operator is a mapping hfgwa: ωn→ω such that: /# 1 2 ( ) ( ) 1 1,2,..., 1 ( , ,..., ) ( ) ( ) | .j j j nn w w l hj j w n j l l j j l l j hfwga h h h h h   = = =    =  =       (4) 3.3. novel distances between hfss definition 7. let h1 and h2 be two hfes, l be the lcm of #h1 and #h2, then the normalized generalized distance measure between h1 and h2 is defined as 1/ 1 2 1 2 ( ) ( ) 1 1 ( , ) , l l l l gd h h l      =   = −     (5) where γ 1 σ(l) and γ 2 σ(l) are the lth elements in h l/#h1 1 and h l/#h2 2 , respectively, and κ≠0 is the generalized distance parameter. 406 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou in particular, if κ=1 and κ=2, then eq. (5) becomes the hamming distance and the euclidean distance, respectively. similarly, the hausdorff distance can be also extended to hfss environment. the hausdorff distance between any two hfes h1 and h2 is: 1 2 1 2 ( ) ( )( , ) max , 1,2,..., .{| |}l l l haud h h l l  = − = (6) in addition, we can obtain some hybrid distance measures combining the above distance measures, such as the generalized hybrid distance between h1 and h2: 1/ 1 2 1 1 1 2 ( ) ( ) ( ) ( ) 1 1 1 ( , ) | | max | | . 2 { } l l l l l l l ghd h h l           =    = − + −       (7) property 2. the above distance measure d(h1,h2) between any two hfss h1 and h2 satisfies the following properties: p(1). reflexivity: d(h1,h2)=0 if and only if h1=h2; p(2). boundedness: 0≤d(h1,h2)≤1; p(3). symmetry: d(h1,h2)=d(h2,h1); p(4). triangle inequality: for any hfss h1, h2 and h3, d(h1,h3)≤d(h1,h2)+d(h2,h3). 4. proposed hybrid fmea framework for risk assessment under hfss environment 4.1. problem description the risk evaluation and prioritization in the fmea framework can be naturally view as a mcgdm problem, i.e., the m risk failure modes fmi (i=1,2,…,m) are alternatives, the n risk factors cj (j=1,2,…,n) are criteria, and there are g fmea experts ek (k=1,2,…,g) joining in the risk evaluation. the criteria and experts’ weighting vectors w=(w1,w2,…,wn)t and λ=(λ1,λ2,…,λg)t satisfying wj, λk ≥0, ∑jwj =1 and ∑kλk =1. let rk=(r k ij) m×n (k=1,2,…,g) be the risk evaluation matrix given by ek (see table 1), where r k ij implies the risk value for the fmi with respect to criterion cj offered by expert ek. table 1 risk priority evaluation in the fmea framework based on mcgdm method e1 e2 eg c1 c2 … cn c1 c2 … cn … c1 c2 … cn fm1 r 1 11 r 1 12 … r 1 1n r 2 11 r 2 12 … r 2 1n r g 11 r g 12 … r g 1n fm2 r 1 21 r 1 22 … r 1 2n r 2 21 r 2 22 … r 2 2n r g 21 r g 22 … r g 2n … … … … … … … … … … … … … … fmm r 1 m1 r 1 m2 … r 1 mn r 2 m1 r 2 m2 … r 2 mn r g m1 r g m2 … r g mn the hybrid fmea framework under the hfss environment with the mcgdm technique contains three major stages (also see the flowchart exhibited in fig. 1): 1. risk evaluation of each fm: the hfss are utilized to represent the risk evaluation. 2. aggregation of the individual risk evaluation: the new hfwa operator with combined weights is utilized to fuse the individual risk evaluation matrix. 3. risk prioritization of each fm: the novel hfwga operator with the critic weights is applied to determine the risk priorities of fms. risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 407 fig. 1 the flow chart of the proposed decision-making process 4.2. risk evaluation procedure step 2.1: determine all the potential fms first, invite the fmea experts to carry on the discussion on the risk analysis. according to the expertise and experience of the fmea experts, different experts can complement each other with their advantages to identify and determine all the potential fms. step 2.2: form the hesitant fuzzy risk evaluation matrix after identifying and selecting all the potential fms, each fmea expert is asked to offer their opinions on the risk levels towards the risk indicators o, s, d using several possible membership values concerning the linguistic term ‘very high’. then, all the hesitant fuzzy risk values of fmi on risk indicator cj offered by expert ek can be recorded as r k ij (i=1,…,m;j=1,…,n;k=1,…,g). finally, all the r k ij values form g decision matrix rk=(r k ij)m×n (k=1,2,…,g) corresponding to the fmea expert ek (k=1,2,…,g). 408 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou 4.3. risk evaluation information aggregation when it comes to aggregating the group risk evaluation information, in most existing fmea analyses, the weights of fmea experts are often subjectively given in advance. however, this will lead to the collective risk evaluation being more dependent on experts with larger weights. using objective weights for fmea experts, on the one hand, can eliminate the extreme aggregation mode and, on the other hand, may also achieve more objective aggregation results. based on the subjective trust relationship and the objective similarity measure, we propose a combined weighting method to determine the weights of fmea experts. then, we employ the newly developed hfwa operator to fuse the individual risk evaluation information under hfss environment. step 3.1: compute the support degree among the risk evaluation values as for the risk values r k ij representing fmi on risk indicator cj offered by fmea expert ek, if it is close to the risk evaluation r k’ ij (k’≠k) given by the other fmea experts, then it should have higher support degree. specifically, the support degree between r k ij and r k’ ij (k’≠k) can be formulated as: sup( , ) 1 ( , ).k k k k ij ij ij ijr r d r r   = − (8) obviously, when d(r k ij ,r k’ ij )=0, meaning that expert ek and expert ek’ have the same evaluation result concerning the fmi on risk indicator cj. they give the largest mutual support to each other. then, the support degree between r k ij and all the others experts is ( , ) 1 sup( ) sup( , ) 1 . 1 1 k k ij ij k k k k k ij ij ij k k d r r r r r g g    = = − − −   (9) step 3.2: calculate the weight of expert ek concerning the fmi on risk indicator cj based on the support degrees defined in eqs. (8) and (9), the weight of expert ek towards the fmi on risk indicator cj can be obtained directly by using each sup(r k ij) divide the sum of sup(r k ij) such that sup( , ) 1 ( , ) sup( ) . sup( ) sup( , ) ( 1) ( , ) k k k k k ij ij ij ij ijk k k k k ij k k k k k ij ij ij ij ij k k k k k k k r r g d r r r r r r k g d r r          − − = = = − −      (10) so an objective weight can be set for each fmea expert. assume the subjective weighting vector for the experts is λ=(λ1,λ2,…,λg)t. consequently, the combined weight for expert ek is (1 ) ,k k ij ij k   = + − (11) where α is a balancing parameter between the objective weight and subjective weight. step 3.3: form the group risk evaluation matrix applying the weights derived in step 3.2, the group risk evaluation matrix rc=(r c ij)m×n can be computed by using the new hfwa operator such that /#1 2 ( ) ( ) 1,2,..., 1 ( , ,..., ) 1 (1 ) | ( ) , k k ij ij ij ij ij ij ij g l rc g k k k ij ij ij ij l l ij l l k r hfwa r r r r      = =   = = − −      (12) where lij is the lcm of #r k ij (k=1,2,…,g). risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 409 4.4. risk priority calculation the conventional fmea method often sets the equal weight or the subjective known weights for the risk indicators. they cannot tackle the issue with risk indicators interaction and unknown weights. critic is one of the most effective mcdm methods that spotlights the objective criteria weight. this method determines the objective weight dependent on two elements of data, including the distinctions and correlations among various criteria all the while. the first is the contrast intensity using the standard deviation to illustrate each criterion separately. the second dimension is the conflict between criteria, the linear correlation coefficient between criteria measures the conflict. to solve the risky preference indicated by hfss, we investigate this strategy in the hfss environment as: step 4.1: decide the objective weights of the risk indicators: critic method as for group risk evaluation matrix rc=(r c ij)m×n derived in section 4.3, compute the score s c ij of each hesitant fuzzy risk evaluation value r c ij, then we form a score matrix sc=(s c ij) m×n as 11 12 1 21 22 2 1 2 . c c c n c c c c n c c c m m mn m n s s s s s s s s s s       =        (13) the standard deviation σj of the jth criterion, which is used to measure the contrast intensities of criteria and can be calculated by 1 1 ( ) with . m m c c c ij j ij ci i j j s s s s m m  = = − = =   (14) the correlation coefficient between the jth and lth criteria, which is used to measure the conflicts between criteria is calculated as 1 2 2 1 1 ( )( ) . ( ) ( ) m c c c c ij j il l i jl m m c c c c ij j il l i i s s s s s s s s  = = = − − = − −    (15) the quantity of information qj contained in the criterion cj is computed as 1 (1 ). n j j jl l q   = = − (16) then the objective weighting vector of the risk indicators can be derived as . jo j j j q w q =  (17) so an objective weight can be set for each risk indicator. assume the subjective weighting vector for the risk indicator is ws=(w s 1 ,w s 2 ,…,w s n )t. consequently, the combined weight for risk indicator cj is (1 ) ,o s j j jw w w = + − (18) where β is a balancing parameter between the objective and subjective weights. 410 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou step 4.2: calculate the global risk value of each fm combining the weights obtained in step 4.1 with the group risk matrix rc=(r c ij )m×n, applying the novel hfwga operator to fuse each risk indicator to calculate the global risk value of each fm such that: /# 1 2 ( ) ( ) 1,2,..., 1 ( , ,..., ) ( ) | ( ) , c j i ij i i i i n w l rc c c c j j c i w i i in l l ij l l j r hfwga r r r r   = =    = =       (19) where li is the lcm of #r c ij (j=1,2,…,n). step 4.3: determine the risk priorities of all the fms concerning the global risk value r c i , calculate their score values. then the risk priority ranking order of each fm can be determined according to their score values. that is, the bigger r c i is, the higher risk priority of fmi is. 5. illustrative example this section gives a case study on the risk priority evaluation of the power transformer parts to illustrate the applicability and effectiveness of the hybrid fmea framework for risk analysis. besides, the sensitivity analysis is conducted to expound the robustness of our model. furthermore, the comparative analysis is also implemented to verify the benefits of our hybrid fmea framework for risk evaluation and prioritization. 5.1. research object and problem description power transformers are critical equipment of substations, power plants, and electric management departments, the running conditions of which will affect the safety and stability of the system. the correct and in-time diagnosis of their faults and operation maintenance is crucial. based on the gross structure and constituent of a power transformer, the fmea experts unify all the potential fms as winding, iron core, bush, body part, non-electric quantity protection, tap switch, cooling. they are denoted by fmi (i=1,…,7). fig. 2 gives the statistical failure data of power transformers parts of one specific region in the past 10 years. then we employ the proposed hybrid fmea framework to evaluate and prioritize the identified fms of the power transformer parts. fig. 2 fault statistics of transformer parts risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 411 5.2. the decision-making steps 5.2.1 risk evaluation of each fm concerning the above selected fms, a fmea group including three experts ek (k=1,2,3) are asked to offer their opinions on the risk evaluations of fms using the hfss. in terms of multi-dimensional risk evaluation, each fm is evaluated under three risk indicators occurrence (o), severity (s), and detection (d). the hfss of the seven fms under the three risk indicators provided by all the fmea experts are put together in tables 2-4. table 2 the hesitant fuzzy evaluation provided by fmea expert e1 o s d fm1 {0.4,0.5} {0.8,0.9} {0.6,0.7} fm2 {0.3,0.4} {0.7,0.8} {0.7,0.8} fm3 {0.2,0.3} {0.7,0.8} {0.3,0.4} fm4 {0.1,0.2} {0.7,0.8,0.9} {0.4,0.5} fm5 {0.1} {0.6,0.7} {0.7,0.8} fm6 {0.1,0.2} {0.5,0.6} {0.4} fm7 {0.1} {0.3,0.4} {0.4,0.5} table 3 the hesitant fuzzy evaluation provided by fmea expert e2 o s d fm1 {0.3,0.4} {0.9,1} {0.7,0.8} fm2 {0.2} {0.8,0.9} {0.7,0.8} fm3 {0.1,0.2} {0.6,0.7} {0.4,0.5} fm4 {0.2,0.3} {0.7,0.8} {0.4,0.5,0.6} fm5 {0.1,0.2} {0.5,0.6,0.7} {0.7,0.8,0.9} fm6 {0.1} {0.6,0.7} {0.2,0.3,0.4} fm7 {0.1,0.2} {0.2,0.3,0.4} {0.2,0.3} table 4 the hesitant fuzzy evaluation provided by fmea expert e3 o s d fm1 {0.5} {0.9} {0.6,0.7,0.8} fm2 {0.1,0.2} {0.8,0.9,1} {0.6,0.7,0.8} fm3 {0.1} {0.6,0.7,0.8} {0.4,0.5,0.6} fm4 {0.2,0.3} {0.7,0.8,0.9} {0.3,0.4,0.5} fm5 {0.2} {0.5,0.6,0.7} {0.5,0.6,0.7} fm6 {0.2,0.3} {0.5,0.6} {0.3,0.4} fm7 {0.2,0.3} {0.3,0.4,0.5} {0.2,0.3,0.4} 5.2.2. risk evaluation information aggregation in this stage, we first need to derive the weights of the expert. on the one hand, the objective weights can be obtained based on the improved hesitant fuzzy hamming distance measure (κ=1 in eq. (5)) and the support degrees eqs. (8) and (9). specifically, the total support degrees of all the risk values constitute a support matrix, as shown in table 5. combining the support matrix and eq. (12), the objective weights of fmea experts can be determined, as shown in table 6. 412 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou table 5 the total support degrees of all the risk values e1 e2 e3 o s d o s d o s d fm1 0.925 0.925 0.925 0.875 0.925 0.925 0.900 0.950 0.950 fm2 0.825 0.875 0.975 0.900 0.925 0.975 0.875 0.900 0.950 fm3 0.875 0.925 0.875 0.925 0.925 0.925 0.900 0.950 0.900 fm4 0.900 0.975 0.950 0.950 0.950 0.925 0.950 0.975 0.925 fm5 0.925 0.950 0.900 0.950 0.975 0.875 0.925 0.975 0.825 fm6 0.925 0.950 0.925 0.900 0.900 0.925 0.875 0.950 0.950 fm7 0.900 0.950 0.825 0.925 0.925 0.875 0.875 0.925 0.900 table 6 the objective weights of the fmea experts e1 e2 e3 o s d o s d o s d fm1 0.343 0.330 0.330 0.324 0.330 0.330 0.333 0.339 0.339 fm2 0.317 0.324 0.336 0.346 0.343 0.336 0.337 0.333 0.328 fm3 0.324 0.330 0.324 0.343 0.330 0.343 0.333 0.339 0.333 fm4 0.321 0.336 0.339 0.339 0.328 0.330 0.339 0.336 0.330 fm5 0.330 0.328 0.346 0.339 0.336 0.337 0.330 0.336 0.317 fm6 0.343 0.339 0.330 0.333 0.321 0.330 0.324 0.339 0.339 fm7 0.333 0.339 0.317 0.343 0.330 0.337 0.324 0.330 0.346 assume the subjective weights for the experts are λ=(0.2,0.3,0.5)t. by using eq. (11), and let α=0.5, referring to the equal importance between the objective and subjective weights. consequently, table 7 provides the combined weight for expert ek. table 7 the combined weights of the fmea experts e1 e2 e3 o s d o s d o s d fm1 0.271 0.265 0.265 0.312 0.315 0.315 0.417 0.420 0.420 fm2 0.259 0.262 0.268 0.323 0.321 0.318 0.418 0.417 0.414 fm3 0.262 0.265 0.262 0.321 0.315 0.321 0.417 0.420 0.417 fm4 0.261 0.268 0.270 0.320 0.314 0.315 0.420 0.418 0.415 fm5 0.265 0.264 0.273 0.320 0.318 0.318 0.415 0.418 0.409 fm6 0.271 0.270 0.265 0.317 0.311 0.315 0.412 0.420 0.420 fm7 0.267 0.270 0.259 0.321 0.315 0.318 0.412 0.415 0.423 based on the combined fmea experts’ weights, one can apply the newly introduced hfwa aggregation operator eq. (12) to fuse the individual risk evaluation matrix into a group matrix rc=(r c ij)7×3. table 8 displays the collective risk evaluation information. table 8 the group risk evaluation matrix o s d fm1 {0.416,0.471} {0.880,1} {0.635,0.635,0.676,0.736,0.777,0.777} fm2 {0.188,0.257} {0.778,0.778,0.833,0.88,1,1} {0.662,0.662,0.7,0.764,0.8,0.8} fm3 {0.127,0.189} {0.629,0.629,0.672,0.731,0.773,0.773} {0.375,0.375,0.421,0.476,0.522,0.522} fm4 {0.175,0.275} {0.7,0.7,0.773,0.8,0.876,0.876} {0.36,0.36,0.434,0.461,0.534,0.534} fm5 {0.143,0.175} {0.529,0.529,0.6,0.629,0.7,0.7} {0.63,0.63,0.703,0.735,0.811,0.811} fm6 {0.143,0.214} {0.534,0.534} {0.299,0.299,0.328,0.370,0.4,0.4} fm7 {0.143,0.219} {0.267,0.267,0.343,0.37,0.444,0.444} {0.257,0.257,0.298,0.358,0.399,0.399} risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 413 5.2.3. calculate the risk priorities of fms regarding the group risk evaluation matrix table 8, we first utilize the critic weighting method to derive the objective weights. based on the score of hfss and eq. (14), the standard deviation of each risk indicator can be calculated as σ1=0.0932, σ2=0.1833, σ3=0.1666. using eq. (15), one can obtain the correlation coefficient matrix ρ=(ρjl)3×3 among the risk indicators. then, by using eqs. (16) and (17), the quantity of information contained in each risk indicator and the objective weights of the risk indicators can be derived as q1=0.9016, q2=0.6685, q3=0.8785 and w o 1 =0.2380, w o 2 =0.3472, w o 3 =0.4147. combining with the equal weights of the conventional fmea model and using eq. (18) with β=0.5 (implying the equal importance between the objective and subjective weights). then the combined weights for risk indicators are w1=0.2857, w2=0.3403, w3=0.3740. table 9 the global hesitant fuzzy risk evaluation of each fm global hesitant fuzzy risk evaluation score ranking fm1 {0.6289,0.6289,0.6439,0.7190,0.7338,0.7338} 0.681 1 fm2 {0.4882,0.4882,0.5103,0.5873,0.6242,0.6242} 0.554 2 fm3 {0.3286,0.3286,0.3507,0.4336,0.4461,0.4461} 0.387 5 fm4 {0.3675.0.3675.0.4073.0.4798.0.5229.0.5229} 0.445 3 fm5 {0.3886,0.3886,0.4227,0.4623,0.4974,0.4974} 0.443 4 fm6 {0.2948,0.2948,0.3051,0.3802,0.3914,0.3914} 0.343 6 fm7 {0.2210,0.2210,0.2534,0.3146,0.3483,0.3483} 0.284 7 finally, based on eq. (19), one can obtain the global hesitant fuzzy risk evaluation of each fm as shown in table 9. based on the score of global hesitant fuzzy risk value of each fm, then the risk priority ranking order are determined also shown in table 9. as a result, we can now rank the fms in light of the scores values, and the risk priority ranking of all the fms is fm1≻fm2≻fm4≻fm5≻fm3≻fm6≻fm7. apparently, fm1 winding owns the highest risk level, which deserves the greatest attention. 6. discussions 6.1. sensitivity analysis next, a sensitivity analysis is conducted to explore the effect of the related parameters in our model on the risk calculation and rankings. α and β indicate the importance degree of the objective weights on experts and risk indicators, respectively. larger α and β imply a larger preference towards the objective weights. here, α and β are from 0 to 1 to analyze the sensitivity. the fixed α and β are α=β=0.5. fig. 3 and fig. 4 indicate how different single α and β affects the final score values of the 7 fms, respectively. one can see from fig. 3 and fig. 4, changing a single parameter will affect the final score values of fms. the parameters α and β play certain roles in deciding the combined weights for the experts and the risk indicators and then affecting the final score values of fms. specifically, the scores of fms decrease monotonically with the increase of β, but the trends of the scores of fms are not the same with α increases. these are mainly because the parameter β directly affects the global weights of the risk indicators, while the parameter α only impacts the local weights of the experts. furthermore, one can draw the following conclusions based on fig. 3 and fig. 4. 414 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou fig. 3 variation of the score values of the fms with different α fig. 4 variation of the score values of the fms with different β ▪ the effect on the final score values of fms of α is less than that of β. the score values present a stable status as the single parameter α changes, while the score values decrease with an extensive variation range when a single parameter β changes. in essence, the objective weights for risk indicators have more influence on the final score values than the objective weights for experts. risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 415 ▪ changing either single α and β will make a different ranking for the fms, especially for fm4 and fm5. the ranking of fm5 increases with the increase of α, and the ranking of fm4 increases with the rise of β. although the other fms’ ranking stays the same, these dynamic changes cannot be negligible since they directly relate to the maintenance and inspection costs. ▪ the final score values affected by the two combined weighting methods with two parameters α and β can constitute a three-dimensional variation pattern. here are two two-dimensional images that present the changing trend of final score values to facilitate sensitivity analysis directly. according to the above calculations and analysis, it is apparent that the combined weights for risk assessment are flexible and easy to operate, and experts can select suitable parameter values to control the balance between the objective and subjective weights. due to the natural uncertainty and complexity in risk analysis, taking these factors into account is necessary and meaningful. 6.2. comparative analysis to verify the effectiveness of our hybrid fmea framework for risk assessment, this section we compare our method with some existing fmea models under the same case study. these comparative methods contain the classical rpn method [49], the hf-vikor method [50], the hf-topsis method [51], and the generalized todim method [52]. specifically, we first transform the hfss into precise numerical values in [0,10] based on their scores, and then we utilize the classical fmea method [49] to determine the rpn of each fm. next, according to the distance measure between the hfss [53] and the objective weights for risk indicators using the maximizing deviation method, we apply the hf-vikor [50], and hf-topsis [51] to derive the compromise solution and the closeness coefficient of each fm, respectively. finally, the generalized todim method [52] is extended into the hesitant fuzzy environment, and an extended generalized todim is developed for the risk prioritization. table 10 the comparisons of different methods classical rpn [49] hf-vikor [50] hf-topsis [51] generalized todim [52] rpn order si ri qi order d i d + i ccii order πi order fm1 299.78 1 0.139 0.074 0.106 1 0.078 0.497 0.864 1 1 1 fm2 134.35 2 0.354 0.221 0.287 2 0.163 0.417 0.719 2 0.803 2 fm3 47.19 5 0.645 0.272 0.459 5 0.328 0.240 0.423 5 0.346 5 fm4 80.48 3 0.551 0.241 0.396 4 0.283 0.291 0.507 4 0.526 3 fm5 71.73 4 0.521 0.255 0.388 3 0.269 0.304 0.530 3 0.477 4 fm6 38.00 6 0.736 0.301 0.518 6 0.394 0.171 0.302 6 0.179 6 fm7 22.27 7 0.856 0.320 0.588 7 0.480 0.100 0.172 7 0 7 the computed target values and the corresponding ranking orders of the fms’ risk prioritization gained by these methods are listed in table 10. from table 10, one can find some ponderable results. ▪ first, fm1 always has the highest level of risk priority, and fm7 always has the lowest risk priority in all the five risk evaluation methods. the most crucial and least severe fms are always the same under the five risk evaluation methods. this result coincides with the fault statistics of transformer parts concerning fm1 and fm7 shown in fig. 2. 416 b. zhou, j. chen, q. wu, d. pamučar, w. wang, l. zhou ▪ second, our risk priority ranking results of fms are consistent with the classical rpn method and the generalized todim method for fmea, which reveal a relatively high homogeneity between our proposed fmea approach and the other two methods. this also implies that our proposed hybrid fmea framework for risk assessment is valid to evaluate and rank the risk prioritization of fms. ▪ finally, regarding the hf-vikor and hf-topsis methods, fm5 has a higher risk priority than fm4, which is opposite with our approach and the other two methods. but on the whole, there are slight differences between the overall risk values of fm5 and fm4. the different decision-making mechanisms cause the ranking difference. more specifically, compared with the above four comparative models, our method has the following differences and advantages: ▪ first, the classical fmea [49] is based on the rpn method that employs precise values as the risk evaluation information and treats the risk indicators as equally important. compared with the classical fmea method, our proposed hybrid fmea model can successfully deal with uncertainty representation (hfss), uncertain information fusion (hesitant fuzzy aggregation techniques), and combined weights effects (combined weighting method). the proposed fmea framework for risk analysis is more comprehensive and objective. ▪ second, the hf-vikor [50], and hf-topsis [51] methods for fmea are based on the reference level models. the selection of the reference points has a strong influence on the final risk evaluation results. our fmea model directly fuses the risk evaluation, and it has a different decision-making mechanism with them. the maximizing deviation method is used to derive the objective weights for risk indicators. it is based on the contrast intensity of each risk indicator and does not consider the correlations between risk indicators. hence, our hybrid fmea model is more straightforward and produces more flexible weights, creating more objective and practical risk analysis results. ▪ third, the generalized todim method for fmea [52] is based on the dominant comparative model, and it can consider several parameters and embody diverse experts’ risk attitudes. nevertheless, certain limitations of this method will exist concerning the selection of related parameters in practice. our model produces consistent results with the generalized todim model. still, the parameters in our combined weights are more acceptable, and the decision-making mechanism also corresponds with the multiplicative property of the classical fmea. by providing a novel hybrid fmea framework integrating the hesitant fuzzy aggregation technique with the combined weights approach, the developed fmea model is forthright and flexible in handling risk assessments issues containing multiple experts and risk indicators. the combined weights help cover the group experts’ consensus and correlations between risk indicators. it is more reasonable and practicable than the fmea models that direct aggregate individual risk evaluation information and assume that the risk indicators are independent of each other in practical risk analysis. 7. conclusions, limitations, and future studies the fmea approach served as a practical risk analysis framework is widely adopted in risk assessment and management with remarkable results. with the increasing complexity of the engineering system, the uncertain tool is also being accepted by many experts when risk priority evaluation of power transformer parts based on hybrid risk fmea framework... 417 they express their risk evaluations. the hfss has been a very efficient representation model to describe the hesitant and fuzzy assessments. this paper develops a hybrid fmea framework for risk evaluation and prioritization with a novel hesitant fuzzy computational model and critic weighting method. our model efficaciously and comprehensively deals with information representation, information fusion, and risk evaluation. the novel hesitant fuzzy computational representation model can enrich the theoretical research results of the hfss. the combined weights can obtain a more accurate result as it considers both the support degrees among experts and the subjective power of fmea experts. the hfwga operator with critic weights can nicely satisfy the multiplicative property of the classical fmea model and consider the correlations between risk indicators. finally, the risk priority evaluation of power transformer parts with sensitivity and comparative analyses reveals that our model is efficient in providing risk analysis support. this paper has vital significance in theory and practical risk analysis. however, our model still has deficiencies, which point the way for future study. first, the risk indicators used to evaluate the fms might remain inadequate. in the fmea framework, only three risk indicators are considered. more aspects and risk indicators should help improve the risk assessment’s comprehensiveness. second, two common aggregation tools, including the weighted averaging and geometric weighted averaging operators, are utilized to fuse the hesitant fuzzy evaluation information. some more advanced aggregation techniques such as the choquet average operator [54] and geometric bonferroni mean [55] can also improve the interactive effect of the interaction relationships among risk factors. finally, although the critic weighting method can determine objective weights considering the differences and correlations among various risk indicators simultaneously, the inherent relation between the risk indicators is worthy of further investigation and exploitation. the research results offer a reference for the fmea and risk assessment. one can get a reasonable risk prioritization if the decision process is implemented correctly. the hesitant fuzzy aggregation technique is involved in these two stages to fuse the individual risk evaluation matrix and decide the total risk value. some mcdm methods like the vikor [56] and the generalized todim [57] can also be used in the second stage. the proposed fmea model uses hfss as the evaluation representations. some other uncertain or fuzzy techniques [58, 59] will also be feasible when expressing subjective uncertainties. finally, the proposed fmea model is applied to the risk prioritization 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2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper funnel flow of a navier-stokes-fluid with potential applications to micropolar media mariia fomicheva 1,2 , wolfgang h. müller 3 , elena n. vilchevskaya 1,2 , nikolay bessonov 1,2 1 institute for problems in mechanical engineering of the russian academy of sciences, st.-petersburg russia 2 peter the great saint-petersburg polytechnic university, st.-petersburg russia 3 institut für mechanik, kontinuumsmechanik und materialtheorie, technische universität berlin, germany abstract. in this paper foundations are laid for a future solution of a fully coupled flow problem for the micropolar medium undergoing structural change in a funnel-shaped crusher. initially the fundamental equations of micropolar media are revisited and the problem of structural changes of micropolar media moving in a crusher is explained. then a review of the current state-of-the-art is presented and a necessary extension of the problem is motivated. the need for using numerical methods of fluid mechanics is emphasized. as a prerequisite for the study of the fully coupled initial boundary value 2d-flow problem of a micropolar fluid the funnel flow of a navier-stokes fluid is investigated based on an implicit finite difference scheme using the thomas algorithm. numerical results for velocities, stresses, and for the pressure dependence of the funnel flow are presented. the correctness of the algorithm is checked by specializing to the case of a flow through a tunnel of constant cross-section under the influence of gravity, for which an analytical solution is available. key words: micropolar media, structural change, microinertia, viscous medium received april 01, 2019 / accepted june 14, 2019 corresponding author: wolfgang h. müller institut für mechanik, kontinuumsmechanik und materialtheorie, technische universität berlin, sekr. ms. 2, einsteinufer 5, 10587 berlin, germany e-mail: wolfgang.h.mueller@tu-berlin.de 256 m. fomicheva, w.h. müller, e.n. vilchevskaya, n. bessonov 1. micropolar media undergoing structural change 1.1. introductory remarks generalized continuum theories (gcts) have gained the attention of the materials science community because they allow modeling of materials with an inner structure. these are used in modern engineering constructions on the large as well as on the small scale, for example, in light-weight aerospace, automotive, microelectronic, and micromechanical designs. a particular type of gct describes micropolar media, and emphasizes the aspect of inner rotational degrees of freedom of a material (see [1] for a modern formulation). this theory is particularly suited for studies of soils, polycrystalline and composite matter, granular and powder-like materials, porous media and foams and, in particular, for materials that are ―somewhere in-between a solid or a fluid,‖ for example liquid crystals. the following should be noted. it is well known that the inertia tensor of a continuum particle, j, the so-called micro-inertia tensor, plays an important role in context with its rotational degree of freedom, specifically in combination with the angular velocity vector, ω, assigned to the continuum element. in eringen’s theory of micropolar media (see for example [2]) j is a conserved field quantity, unable of structural change and production, and not truly an independent variable such as the mass density, which characterizes the inertia of matter w.r.t. linear momentum and obeys its own kinetic equation (the mass balance), independently of the momentum balance. therefore, most recently, it has been emphasized in [3] that the inertia tensor should also be treated as a completely independent structural field variable. however, in contrast to the balance of mass, the micro-inertia tensor is not conserved. rather its governing equation contains a production term, χ, which within the framework of continuum theory must be considered as a constitutive quantity. in the following subsections it will be shown that it allows modeling additional features of materials, namely processes accompanied by a considerable structural change. the problem of a funnel flow will be used for demonstration, which can eventually will be investigated and later used for crushing of particles to smaller size. previously similar problems were considered in the articles [4-7]. 1.2. governing equations the motion of micropolar media is described by the following coupled system of differential equations:  balance of mass, δ 0 δt     v , (1)  balance of momentum, δ δt      v σ f , (2)  balance of spin, δ δt           ω j ω j ω μ σ m , (3) where ρ is the field of mass density, v and ω are the linear and angular velocity fields, σ is the nonsymmetric cauchy stress tensor, f is the specific body force, j is the specific funnel flow of a navier-stokes-fluid with potential applications to micropolar media 257 micro-inertia tensor, μ is the non-symmetric couple stress tensor, (a b)×=a×b is the gibbsian cross, and m are specific volume couples. we denote by δ( ) d( ) ( ) ( ) δ dt t       v w (4) the substantial derivative of a field quantity, d(·)/dt is the total derivative and w the mapping velocity of the observational point (see [8]). in the traditional micropolar theory, each material point or ―particle‖ of a micropolar continuum is phenomenologically equivalent to a rigid body. hence, its micro-inertia does not change intrinsically, see for example [2, 9-11]. even if a so-called micromorphic structure is considered, which in principle allows an intrinsic change of micro-inertia (following [2, 12, 13]), many papers use only the following additional equation for the conservation of inertia (e.g., see [14, 15]), which is an identity: δ δt     j ω j j ω . (5) note that the terms on the right hand side characterize the change of the inertia tensor, which is exclusively due to rigid body rotation. an extension to this approach was suggested in [16], where it was assumed that the inertia of polar particles may change as the continuum deforms. this idea was further elaborated in [3], where it was clearly stated that the tensor of inertia should be treated as an independent field. within that approach a fixed elementary volume v was treated as a micropolar (macro-) particle, as customarily done in spatial description. then its tensor of inertia is obtained by homogenization, namely by averaging the inertia tensors of microparticles within a representative volume. because of the movement of the medium, the elementary volume contains different micro-particles as time passes, and the inertia tensor of the volume will change due to the incoming or outgoing flux of inertia. however, internal structural transformations are also possible. these are due to combination or fragmentation of particles during mechanical crushing, to chemical reactions, or to changes of anisotropy of the material. these effects are explained in greater detail in [7, 17, 18]. in a nutshell, on the continuum scale all of this can be taken into account by adding a source term, χ, to the right-hand side of eq. (5), which now reads: δ δt      j ω j j ω χ . (6) on the continuum level this source term must be considered as a new constitutive quantity for which an additional constitutive equation has to be formulated. the form of the constitutive equation depends on the problem under consideration and can be a function of many physical quantities, such as temperature, pressure, flow rate, etc. we will now discuss an example pertinent to the intentions of this paper. 1.3. the crusher problem: an example for structural change in a micropolar medium a first non-trivial solution to the so-called crusher problem was presented in [7], where the situation depicted in fig. 1 (left inset) was analyzed: within an infinite onedimensional space, –∞ < x < +∞, a continuous flow of randomly oriented and randomly 258 m. fomicheva, w.h. müller, e.n. vilchevskaya, n. bessonov sized micro-particles is coming in from the left. on a continuum level this corresponds to a spherical tensor of microinertia of a fixed initial size. they keep moving to the right at a constant speed, v0, prescribed by a conveyor belt. in other words, the balance of linear momentum does not need to be considered. it is identically satisfied. on its way to the right the particles enter a region –δ ≤ 0 ≤ +δ, symmetrically arranged around the position x = 0, where they are continuously crushed to smaller and smaller sizes. on the continuum scale this means that the tensor of micro-inertia stays spherical but that its ―size‖ decreases. fig. 1 1d crusher problem and microinertia development for the production of the moment of inertia, χ = χ i the following relationship was postulated [7]: * f f 0 if ( ) [ ( , ) ] if ( , ) 0 if 0 if x x j x t j x x x t x x x                            , (7) where j and α are positive constants, which can intuitively be interpreted as being related to the minimum grain size the particles can be crushed to and to the inverse of the particle toughness, respectively. thus, because they are characteristics of the material and not of the crusher, they are constitutive properties. xf is the current location of the incoming shock front of the to-be-crushed particles. note that the front will eventually leave the crusher area. for this case eq. (6) can be solved in closed form using the method of characteristics. a typical result of the decreasing micro-inertia is shown in fig. 1, right. it should be emphasized that the predicted change in micro-inertia is an important result in itself. it is not necessarily connected to a concurrent solution for the angular velocity ω based on the balance of spin shown in eq. (3). in fact in the present and in the cited articles of the authors the spin balance was not touched at all. the presence of the linear velocity is sufficient to induce further change of j as we shall see now. in [5] the crusher model was extended in two ways: a transient two-dimensional flow of the couette type of a viscous medium of the navier-stokes type between two plates was considered, fig. 2. funnel flow of a navier-stokes-fluid with potential applications to micropolar media 259 fig. 2 crushing of a viscous material in this case the production term is also isotropic. it was assumed that it is given by the following expression: *tr ( )( ( ) ( ))j j h x h x l     iχ σ , (8) where h(x) is the heaviside step function. moreover, the ―pressure‖ term, tr σ, describes the conversion of the crusher action to a material response. in other words, it is related to the effectiveness of the crusher and to the transmission of its external forces into the material to be crushed. it is important to note that the problem decouples and that it was solved (numerically) in two steps, as follows. first, the velocity is determined numerically from the transient balance of momentum eq. (2) by using the navier-stokes law without bulk viscosity, dev( )p    i   σ v v , (9) where η is the shear viscosity coefficient, and p is the pressure. then, once the velocity profile is known, it can be used to determine the temporal development of the micro-inertia by a numerical solution of eq. (6). typical results in dimensionless form, 0/v v v , /z z h , 0 /t v t l , 0/j j j , are shown in fig. 3. a few comments are made in order to explain why a viscous constitutive equation is used in context with particle crushing. there are several reasons. first, crushing of (brittle) particles is often achieved in a slurry, which is viscous due to the added water. however, even if there is no water added, the particles will be in contact with each other giving rise to friction on the mesoscopic and to viscosity on the macroscopic scale. in fact, this is also acknowledged in the literature on crushing (see [4]). 260 m. fomicheva, w.h. müller, e.n. vilchevskaya, n. bessonov fig. 3 temporal development of the velocity and of the micro-inertia (horizontally at 2 / 3x  ) profiles consequentially, the next complication added to our crusher analysis should be the solution of a fully coupled problem, where the balances of momentum and of micro-inertia are solved concurrently. this type of problem arises, for example, during funnel flow of a viscous medium subjected to gravity (see fig. 4). it can only be solved numerically, unless the funnel angle degenerates to α = 90°. in this paper the fully coupled problem will not be treated. in the first step we will only concentrate on the numerical solution for the flow of a viscous medium of the navier-stokes type without micro-inertia through a funnel. the situation will be explained in more detail in the next section. 2. problem statement consider the (planar) situation shown in fig. 4. a continuous stream of bulk material flows from the top into the inlet orifice of a container of width 2l of height h1. the size of the particles entering the container region will be the same. it will also stay the same during the passage, because in this paper we consider the flow problem only and not its coupling to microinertia. therefore, as a boundary condition we assume that the infinite supply of particles enters the system with the same velocity pointing only in y-direction, 1 2 0( , ) yx y h h v   v e , (10) at all times. we assume that the material is pushed into the tunnel under a pressure p0. then the material finally enters a funnel region of height h2 where the width narrows down linearly to 2l0. in this configuration the angle between the walls of the funnel and of the horizon is given by α = arctan 1= 45°. gravity points in negative vertical direction, yg f e . (11) it attempts to accelerate the flow but, as we shall see, because of the viscous nature of the fluid the released potential energy will dissipate and the fluid is not accelerated as much as it could if there were no dissipation. as we shall explain later, it reaches a stationary state in the case of a long straight tunnel without a funnel, independently of falling funnel flow of a navier-stokes-fluid with potential applications to micropolar media 261 coordinate y. note that because of the planar nature of the presented problem there will never be a velocity component out-of-plane. hence initially the vertical component is constant, vy = v0, and the projection of the velocity on the x-axis vanishes, vx = 0. this will definitely change when the material enters the funnel region of height h2. here we will encounter both velocity components, ( , , ) ( , , )x x y yv x y t v x y t v e e . (12) we will now discuss the equations required to determine both velocity components. fig. 4 particle transport through the 2d-container the flow we consider is that of a viscous material of the navier-stokes type without bulk viscosity according to eq. (9). as it was mentioned above, the velocity cannot be regarded as constant and must be determined from the balance of linear momentum eq. (2). it is also assumed that the liquid under consideration is incompressible, satisfying the equation 0 v . (13) given eqs. (9) and (13), eqs. (2) take the form (ρ0 is the constant mass density and the velocity w of the observational point will be zero in the present case), 0 0 δ + + δ p t      v v f . (14) on the left and right borders of the channel, and in particular in the funnel region, no slip conditions are assumed, 0, 0y xv v  . (15) 262 m. fomicheva, w.h. müller, e.n. vilchevskaya, n. bessonov the following equations were used as boundary conditions for the pressures on the upper boundary of the system and at the end of the funnel, respectively (h = h1 + h2): 0( , , ) , ( , 0, ) 0p x y h t p p x y t    . (16) in general, eqs. (13)-(16) are evaluated numerically by using the explicit method of integration (see [19] for a detailed explanation of the numerical method that was used). however, for a special case it is possible to check the numerical results by means of an analytic steady-state solution. simply consider a straight tunnel of width 2l. in other words, put α = 90° in fig. 4. then consider stationary conditions and solve eqs. (13)-(15) with the following semi-inverse ansatz: ( , )y yv x yv e . (17) because of the incompressibility it turns out that this velocity profile must be independent of height, meaning of the coordinate y (far from the top inlet) and is given by: 2 2 0 1 ( ) ( ) 2 yv q g l x      , (18) where const. p q y     , (19) and 2l denotes the width of the tunnel. this solution shows that the medium is not accelerated such that the velocity in y-direction increases steadily. rather it reaches a stationary state because the released potential energy is dissipated. from a force-related viewpoint one might say that the gravitational forces acting on the bulk, the pressures acting on top and on the bottom, and the frictional forces are in static equilibrium. 3. results we first present the transition to the stationary state based on a numerical solution of eqs. (13)-(15). for this purpose the following (quasi-dimensionless) data was used (δ indicates grid and time spacings): 0 0 1 2 0 1, 4, 0.25, 0.01, 1.32, 1, 0.01, 3.36, 0.64, 0.36, 0.04, 0.04, 1. l h q v g h h l x y t                  (20) fig. 5 shows a comparison of stationary situations for a direct channel for a crosssection y = 1/2h. it can be seen that the analytical solution (18) and the numerical solution are in good agreement with each other. funnel flow of a navier-stokes-fluid with potential applications to micropolar media 263 fig. 5 distribution of the vertical velocities. comparison of analytical and numerical solutions (solid and dotted lines, respectively) the following figures show the distribution of the velocity projections on the x and y axes in various cross-sections at different points of time. note that the graphs are presented in dimensionless form. velocities shared initial velocity v0 on the upper border of the vessel (since the initial velocity was directed against the y axis, then v0 is a negative value). the axis y was divided by h2, and the axis x by l. also note that all calculations were performed for α = 45°. the three insets on the left of fig. 6 show the distribution of the horizontal velocity component vx as a function of height h as time passes, t1 < t2 < t3. the following can be said about the horizontal velocity components vx (x, y, t):  they are antisymmetric w.r.t. x = 0: vx (x, y, t) = -vx (-x, y, t). this is why only half of the distribution is shown.  they vanish at the wall as they should.  they become smaller with increasing height y, which makes sense because they result as a consequence of a narrowing cross-section.  they increase with time and a stationary state (if it exists) has not been reached at time t3 yet. the insets on the right of fig. 6 show the distribution of vertical velocity component vy as a function of height h. the following can be said about the vertical velocity components vy (x, y, t):  they are mirror-symmetric w.r.t. x = 0: vy (x, y, t) = vy (-x, y, t).  they vanish at the wall and at x = 0 as they should.  they become smaller with increasing height y, which we expect because there is more space for the fluid. also with increasing height y they look parabolic, which makes sense for a hagen-poiseuille type of flow.  they increase with time and a stationary state (if it exists) has not been reached at time t3 yet. 264 m. fomicheva, w.h. müller, e.n. vilchevskaya, n. bessonov fig. 6 distribution of the horizontal and of the vertical velocities (left and right insets, respectively) at increasing times at various vertical cross-sectional heights y funnel flow of a navier-stokes-fluid with potential applications to micropolar media 265 fig. 7 distribution of the horizontal and of the vertical velocities (left and right insets, respectively) at increasing times at various horizontal cross-sectional cuts x fig. 7 presents essentially the same information as the two previous sets of figures but from a different viewpoint. now both velocity components are depicted as a function of ―falling height‖ y for various cross-sectional cuts. for the horizontal component vx (x, y, t) the following can be said in addition to the previous statements: 266 m. fomicheva, w.h. müller, e.n. vilchevskaya, n. bessonov  for greater heights y the velocities increase if one moves away from the center towards x = l.  at smaller heights the behavior is more complex. in particular the curve at position x = l describing the situation at the end of the funnel, (y = 0) must go through zero, because of the boundary condition.  some curves are slightly jaggedly, which is because the points of integration were connected by linear lines, which does not depict the situation correctly. for the horizontal component vx (x, y, t) the following can be said in addition to the previous statements:  the velocities in the middle x = l are the greatest, as to be expected. fig. 8 shows the distribution of pressure isolines in the vessel at different points in time. in contrast to fig. 4 the container was rotated by 90 degrees for convenience of graphical representation. the direction of flow is indicated by an arrow. note that the greater the pressure, the brighter the isolines. these distributions will become important as a measure of the intensity of the crushing process in view of the equation for the production of microinertia shown in eq. (8). obviously there is now high potential for crushing or better microinertia production in the narrowing funnel. in contrast to that nothing will happen in the straight passageway. fig. 8 the distribution of the pressure isolines in the vessel at different points of time 4. conclusions and outlook in this paper the following was achieved:  the importance of studying micropolar media as an example of a generalized theory of continuum was emphasized, since they allow modeling materials with an internal structure besides taking into account the aspect of the internal rotational degrees of freedom of the material. this theory is suitable for studying many applied problems, for example, in soil mechanics.  an extension of the balance of microinertia was presented. it accounts for due intrinsic structural changes of microinertia and the need for the extension was physically motivated.  as a particular example of this kind of problem, the milling process in a crusher was used. funnel flow of a navier-stokes-fluid with potential applications to micropolar media 267  as a prerequisite for further study the flow of a navier-stokes fluid through a 2dfunnel was investigated numerically.  velocities and pressure distributions were obtained, discussed, and may now serve for future studies of the coupled problem of the funnel flow of a micropolar medium showing structural change due to milling. acknowledgements: support of this work through a stipend from tu berlin to m.f. is gratefully acknowledged. references 1. eremeyev, v.a., lebedev, l.p., altenbach, h., 2012, foundations of micropolar mechanics, springer science & business media, heidelberg, new york, dordrecht, london. 2. eringen, a.c., kafadar, c.b., 1976, polar field theories, in: continuum physics iv, academic press, london. 3. ivanova, e.a., vilchevskaya, e.n., 2016, micropolar continuum in spatial description, continuum mechanics and thermodynamics, 28(6), pp. 1759-1780. 4. bain, o., billingham, j., houston, p., lowndes, i., 2015, flows of granular material in two-dimensional channels, journal of engineering mathematics, 98(1), pp. 49-70. 5. fomicheva, m., vilchevskaya, e.n., müller, w., bessonov, n., 2019, milling matter in a crusher: modeling based on extended micropolar theory, continuum mechanics and thermodynamics (in print). 6. glane, s., rickert, w., müller, w.h., vilchevskaya, e., 2017, micropolar media with structural transformations: numerical treatment of a particle crusher, proceedings of xlv international summer school — conference apm 2017, pp. 197-211. 7. müller, w.h., vilchevskaya, e.n., weiss, w., 2017, a meso-mechanics approach to micropolar theory: a farewell to material description, physical mesomechanics, 20(3), pp. 13-24. 8. ivanova, e., vilchevskaya, e., müller, w.h., 2016, time derivatives in material and spatial description – what are the differences and why do they concern us?, in k. naumenko, m. aßmus (eds.), advanced methods of mechanics for materials and 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2277-2285. 17. müller, w.h., vilchevskaya, e.n., 2018, micropolar theory with production of rotational inertia: a rational mechanics approach, generalized models and non-classical approaches in complex materials, 1, pp. 195-229. springer, cham. 18. morozova, a.s., vilchevskaya, e.n., müller, w.h., bessonov, n.m., 2019, interrelation of heat propagation and angular velocity in micropolar media, dynamical processes in generalized continua and structures, pp. 413-425. springer, cham. 19. chorin, a.j., 1997, a numerical method for solving incompressible viscous flow problems, journal of computational physics, 135(2), pp. 118-125. 7514 facta universitatis series: mechanical engineering vol. 21, no 2, 2023, pp. 201 222 https://doi.org/10.22190/fume210403058s © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper parametric study of a cnc turning process using discriminant analysis baneswar sarker, shankar chakraborty department of production engineering, jadavpur university, india abstract. in the present day manufacturing scenario, computer numerical control (cnc) technology has evolved out as a cost effective process to perform repetitive, difficult and unsafe machining tasks while fulfilling the dynamic requirements of high dimensional accuracy and low surface finish. adoption of cnc technology would help an organization in achieving enhanced productivity, better product quality and higher flexibility. in this paper, an endeavor is put forward to apply discriminant analysis as a multivariate statistical tool to investigate the effects of speed, feed, depth of cut, nose radius and type of the machining environment of a cnc turning center on surface roughness, tool life, cutting force and power consumption. simultaneous discrimination analysis develops the corresponding discriminant function for each of the responses taking into account all the input parameters together. on the contrary, step-wise discriminant analysis develops the same functions while considering only those significant input parameters influencing the responses. higher values of hit ratio and cross-validation percentage prove the application of both the discriminant functions as effective prediction tools for achieving enhanced performance of the considered cnc turning operation. key words: cnc turning, discriminant analysis, process parameter, response, hit ratio, cross-validation 1. introduction in manufacturing and metalworking industries, turning is the most basic material removal process where a single-point wedge-shaped cutting tool is employed to remove material from the surface of a rotating cylindrical workpiece. the cutting tool is advanced linearly in a direction parallel to the axis of rotation of the workpiece [1]. turning is an extremely precise process that can attain a surface finish of 0.5-1 µm [2]. the turning center or lathe provides the power for turning the workpiece at a given rotational speed, and feeding the cutting tool at a specified rate and depth of cut, facilitating material received april 03, 2021 / accepted august 14, 2021 corresponding author: shankar chakraborty department of production engineering, jadavpur university, india. e-mail: s_chakraborty00@yahoo.co.in 202 b. sarker, s. chakroborty removal in the form of chips [3, 4]. in order to cope up with the present-day requirements of high productivity and low production cost with enhanced product quality, conventional multi-spindle lathes are now being gradually substituted by the high performance computer numerical control (cnc) machine tools due to their ease of setting, operation, repeatability and accuracy. in cnc machining technology, there is an automated control of machine tools through dedicated instructions stored in memory to machine complex workpieces to fulfill the requirements of higher dimensional accuracy and better surface finish under the occasional supervision of an operator. its various advantageous features, like program storage and editing facility, ability to store multi-part programs, tool offset and compensation, ability to send and receive data from a variety of sources etc. have made the cnc technology an almost indispensible tool in the present-day highly competitive manufacturing environment [5]. a schematic diagram illustrating the cnc turning process is shown in fig. 1. fig. 1 schematic representation of cnc turning process it has been observed that the machining performance of a cnc turning center with respect to material removal rate (mrr), surface roughness (sr), tool life (tl), cutting force (cf), power consumption (pc), tool wear, etc. is greatly affected by the settings of its different input parameters, like feed rate, spindle speed, depth of cut, type of the cutting fluid, machining environment, etc. [6]. researchers have already applied several approaches to identify the best settings of multiple input parameters of the cnc turning processes for attaining higher productivity with the desired quality level. occasionally, the manufacturer’s operating manuals are consulted or the expert operator’s knowledge is sought to determine the optimal parametric combination of a cnc turning process. unfortunately, these intuitive and conservative approaches do not always lead to the best machining performance of a cnc process under a given machining environment. thus, to determine the optimal operating levels of various input parameters during cnc turning operation on a given work material, it has become essential to examine the effects of those input parameters on the process outputs (responses). keeping this objective in mind, this paper aims at the application of discriminant analysis for a cnc turning process in order to develop the corresponding discriminant functions showing the influences of parametric study of a cnc turning process using discriminant analysis 203 the considered process parameters on the responses, as well as to single out the most significant parameter for each of the responses. in simultaneous estimation of discriminant analysis, the developed functions consist of all the input parameters of the cnc turning process, while in step-wise analysis, only the significant parameters are taken into account in the developed functions. the performance of both the estimation procedures is validated based on the values of hit ratio and cross-validation percentage. 2. survey of the literature considering feed rate, cutting speed and depth of cut as the input parameters during cnc turning operation of sae 8822 alloy steel, kanakaraja et al. [7] determined their best settings based on taguchi methodology. singh and sodhi [8] adopted response surface methodology (rsm) to determine the optimal settings of feed rate, depth of cut and cutting speed for attaining improved values of mrr and sr in cnc turning on aluminium-7020 alloy material. during hard turning operation of aisi 4340 steel on a cnc turret lathe, rashid et al. [9] investigated the influences of feed rate, spindle speed and depth of cut on sr values of the machined components using taguchi methodology. while taking into consideration depth of cut, spindle speed and feed rate as the parameters of a cnc turning process, rudrapati et al. [10] analyzed their effects on sr of the machined components. the said process was later optimized using teaching-learning-based optimization algorithm. park et al. [11] applied rsm technique for establishing the relationships between various machining parameters, i.e. cutting speed, feed rate, nose radius, edge radius, rake angle and relief angle, and cutting energy and energy efficiency. non-dominated sorting genetic algorithm-ii (nsga-ii) was adopted for multi-objective optimization and development of the pareto optimal solutions. the optimal parametric setting was finally determined using technique of order preference by similarity to the ideal solution (topsis). arunkumar et al. [12] applied taguchi methodology to establish the optimal intermixture of depth of cut, speed, feed rate and coolant type during cnc machining of lm6 aluminum alloy for having better sr values. applying rsm technique, nataraj and balasubramanian [13] established the optimal settings of cutting speed, depth of cut and feed rate for achieving better values of sr, intensity of vibration and work-tool interface temperature while machining hybrid metal matrix composites. gadekula et al. [14] employed taguchi methodology for optimization of a cnc turning process while treating feed rate, spindle speed and depth of cut as the input parameters, and mrr and sr as the responses. rathore et al. [15] studied the influences of feed rate, depth of cut, spindle speed and coolant type on sr properties of aa 6463 materials. the weights of the responses were determined using principal component analysis and the optimal parametric mix was identified based on grey relational analysis (gra) technique. sahoo et al. [16] applied weighted aggregate sum product assessment (waspas) method for parametric optimization of a cnc turning process for achieving minimum tool vibration and sr of 6063-t6 aluminum components. vijay kumar et al. [17] studied the effects of feed rate, depth of cut and spindle speed on sr and mrr during cnc turning on en 19 stainless steel material. based on taguchi’s l18 mixed orthogonal array experimental design plan, syed irfan et al. [18] optimized the settings of cutting speed, feed rate and depth of cut while performing cnc turning operation on en45 spring steel material. the mrr and sr were treated as the responses. while machining aluminium-2014 alloy, aswal et al. [19] considered cutting speed, depth of cut and feed rate as the input parameters of a cnc turning operation, and investigated their effects on sr. 204 b. sarker, s. chakroborty it has been revealed from the review of the existent literature that various multi-criteria decision-making tools, i.e. topsis, gra, waspas, etc. have already been employed by the past researchers for parametric optimization of cnc turning processes. taguchi methodology has become a popular technique among the research community for single objective optimization of cnc turning processes. the relationships between the cnc turning parameters and responses have also been investigated through the deployment of rsm technique. both rsm technique and discriminant analysis are explicit methods having clear, transparent and unambiguous underlying mathematical principles with similar computation time. however, there are some drawbacks of rsm technique. it attempts to fit data to a polynomial even though many systems cannot be well explained by second order polynomials. it becomes necessary to decrease the range of the independent variables, if the system cannot be explained by the regression equation computed through rsm technique. on the other hand, discriminant analysis develops a causal model which maximizes the group difference by computing weights associated with the independent variables. hence, it becomes an effective tool in evaluating the effect of each independent variable on the dependent variable based on its ability to separate the group differences. besides this, the range of the independent variable does not affect the solution accuracy. thus, it can be considered capable of effective parametric analysis of varied machining processes. 3. discriminant analysis discriminant analysis is a multivariate statistical technique used for categorizing a set of observations into predefined groups [20]. it can be considered as a profile analysis, where it evaluates differences between groups based on a set of independent variables. it establishes the link between the categorical (nominal or non-metric) dependent variables and metric independent variables. the discriminant function, computed from this analysis, has a linear relationship between two or more independent variables and can be expressed as below [21]: zqr= α +β1x1r + β2x2r+……….+ βnxnr (1) where zqr is the score of discriminant function q for object r, α is the intercept, xnr is the independent variable n for object r and βn is the discriminant coefficient for independent variable n. the discriminant analysis tests the hypothesis of equality of group means for each of the dependent variables. the group mean, also called group centroid, is the arithmetic mean of the discriminant scores for all the objects belonging to a single group. the group centroid denotes the most characteristic location of an object in a group, and the distance between the groups can be explained by comparing their centroids. it also enables prediction of the group where a certain element can be classified based on the closeness of its discriminant score to the group centroid. the discriminant function is said to be statistically significant if there is a substantial difference between the group centroids [21]. the statistical significance of the function is calculated by comparing the spread of the discriminant score for each group and therefore, by testing the intersection between the groups. a small intersection represents significant separation between the groups due to the discriminant function, while a large intersection denotes poor differentiating power of the function. multiple discriminant functions can be developed provided that the dependent variables comprise more than two groups. the number of functions computed equals to parametric study of a cnc turning process using discriminant analysis 205 (g – 1), where g is the number of groups, with different discriminant scores calculated by each function. in this paper, however, the analysis is conducted with each dependent variable consisting of two groups, where their relations with a combination of independent variables are established with the help of a single discriminant function. in this analysis, the responses of the considered cnc turning process are considered as the dependent variables, while the turning parameters are treated as the independent variables. the steps of discriminant analysis are illustrated through a flowchart in fig. 2. at first, the problem statement and purpose of the analysis are identified. the purpose of this fig. 2 flowchart showing steps of the discriminant analysis 206 b. sarker, s. chakroborty paper is to demonstrate the application of discriminant analysis to evaluating the effects of the considered cnc turning parameters on the responses while identifying the most significant parameter influencing each of the responses. the analysis framework is then formulated. determination of the independent and dependent variables takes place, followed by classification of the dependent variables into corresponding binary categories. if a dependent variable is metric, it needs to be transformed into non-metric data. checking of the sample size is also required in this step. pituch and stevens [22] advised that the ratio between the sample size and number of independent variables should be 20:1, with a minimum of 20 elements in the group containing the least number of objects. after this step, the corresponding assumptions of discriminant analysis, i.e. multivariate normality, multicollinearity and homogeneity of covariance matrices need to be validated. the independent variables can be tested for univariate normality while calculating their skewness and kurtosis values, which can be considered as adequate for validation of multivariate normality [21, 22]. multicollinearity indicates high inter-correlations between two or more independent variables. it poses problems in determination of the significance of an independent variable because the influences of the independent variables are confounded due to high correlations between them, making its absence as a mandatory requirement [22]. multicollinearity can be tested using variance inflation factor (vif) and tolerance values. the vif measures how much larger the variance would be for multicollinear data than the orthogonal data, where its most preferred value is 1 [23]. tolerance is the reciprocal of vif. homogeneity of covariance matrices or homoscedasticity specifies whether the covariance matrix for each group is equal to each other and is verified using the box’s m test, which considers equality of the within-class covariance matrices as the null hypothesis. thus, nonrejection of the null hypothesis is desired, which can be denoted by an insignificant result. the developed discriminant function can be interpreted by assessing the unstandardized and standardized coefficients of the independent variables and structure matrix. the contribution of an independent variable to the ability of the discriminant function to separate and classify objects into the related groups is determined by the absolute value of its standardized discriminant coefficient. as the independent variables are quantified in different scales, it is recommended to compare their relative contributions based on standardized coefficients. larger is the absolute value of the standardized coefficient, higher is the discriminating power of the independent variable. the influence of an independent variable on the discriminant function can also be explained using the corresponding structure matrix. the structure coefficients, also known as structure correlations, are the correlations between the independent variables and discriminant function. thus, structure coefficient can be treated as the factor loading of an independent variable on the discriminant function, allowing measurement of the relative closeness of the variable to the discriminant function. in step-wise discriminant analysis, structure correlations can be computed even for those variables not included in the model. the unstandardized coefficient, computed for each of the independent variables in the model, is utilized for formulating the discriminant function. the discriminant function yields the discriminant score for different values of the independent variables. these scores are instrumental in cross-validation and classification of the objects into the corresponding groups. the objects are classified into groups based on their discriminant scores and closeness to the group centroids. the cut-off scores, considered to determine the groups into which the related objects are classified, are computed using group centroids. the cut-off score (zc) between two groups is calculated using the following equations: parametric study of a cnc turning process using discriminant analysis 207 a) for unequal groups: zc= (nazb + nbza) / (na+nb) (2) where na and nb are the group sizes, and za and zb are the group centroids, respectively. b) for equal groups: zc= (za + zb) / 2 (3) in the validation stage, accuracy of the discriminant function in separating and classifying objects into the relevant groups is measured, based on two approaches, i.e. hit ratio and cross-validation. hit ratio is a measure of actual percentage of correct classification of objects by the developed discriminant function. along with the hit ratio calculation, cross-validation must be carried out to validate the results in order to apply the function for classification of the subsequent objects into appropriate groups. discriminant analysis aims at maximization of the separation between two groups based on sample-specific error [24]. since the errors may differ for different samples of objects, it becomes necessary to cross-validate the results, which provides the predictive accuracy of the function along with its suitability of application for a wider range of samples. in this paper, the leave-one-out procedure of cross-validation is applied [22], where one element from the sample is systematically excluded and the discriminant function is estimated based on the remaining elements in the sample. the excluded element is then classified into one of the two groups according to its discriminant score. this process repeats till every element in the sample is excluded and classified. higher values of hit ratio and cross-validation percentage are desired to validate the function’s suitability and potentiality as a multivariate prediction tool. discriminant analysis has certain similarities and dissimilarities with regression analysis and analysis of variance (anova). all these techniques have one dependent variable and one or more independent variables. however, anova and regression analysis are concerned with continuous dependent variables, while discriminant analysis has categorical dependent variables [25]. on the other hand, regression and discriminant analysis deal with continuous independent variables while anova has categorical independent variables. both regression and discriminant analysis can predict values (although of different data types) and study the influence of independent variables on dependent variables, while anova is used to ascertain the effects of independent variables on dependent variables. mathematically, discriminant analysis is similar to one-way multivariate anova (manova), with the difference being in the variable data types. in manova, like anova, the classification is on the basis of the categorical independent variables, while in discriminant analysis, the classification is on the basis of the values that the dependent variables obtain. 4. discriminant analysis for a cnc turning process gupta et al. [26] applied taguchi methodology along with fuzzy logic reasoning approach for multi-response optimization of a high speed cnc turning operation on aisi p20 tool steel material using tin coated tungsten carbide inserts. speed (s), feed (f), depth of cut (d), nose radius (nr) and environment (e) were selected as the input parameters (independent variables), and sr (in µm), tl (in min), cf (in n) and pc (in w) were the responses (dependent variables). taking three different operating levels for each of 208 b. sarker, s. chakroborty the turning parameters, gupta et al. [26] conducted 27 experimental runs and measured the corresponding responses values. the settings of the cnc turning parameters are provided in table 1 and the detailed experimental plan is shown in table 2. based on this dataset, both the simultaneous and step-wise estimation discriminant analyses are carried out to explore the influences of the considered cnc turning parameters on each of the responses. for this purpose, ibm spss statistics 25.0 software is employed. table 1 cnc turning parameters along with their levels [26] turning parameter symbol unit level 1 level 2 level 3 speed s m/min 120 160 200 feed f mm/rev 0.1 0.12 0.14 depth of cut d mm 0.2 0.35 0.5 nose radius nr mm 0.4 0.8 1.2 environment e dry wet cryogenic as all the response values for the said cnc turning operation are metric in nature, it is necessary to categorize them into two non-metric groups on the basis of their median values, as provided in table 2. the values of the responses which are higher than their corresponding medians are considered as high and are categorized into group 2. on the contrary, in group 1, values of the responses lower than the medians are classified as low. table 2 cnc turning parameters along with their levels [26] s f d nr e sr sr tl tl cf cf pc pc (m/min) (mm/rev) (mm) (mm) (µm) group (min) group (n) group (w) group 120 0.1 0.2 0.4 1 1.41 2 29 2 171.3 1 1066 1 120 0.1 0.35 0.8 5 0.71 2 34 2 147.5 1 1560 2 120 0.1 0.5 1.2 9 0.6 2 54.67 2 111.74 1 866 1 120 0.12 0.2 0.8 5 0.47 1 34.67 2 120.3 1 1493 2 120 0.12 0.35 1.2 9 0.19 1 51.66 2 180.6 2 987 1 120 0.12 0.5 0.4 1 1.18 2 27 1 236.2 2 1187 1 120 0.14 0.2 1.2 9 0.67 2 50 2 157.7 1 960 1 120 0.14 0.35 0.4 1 1.16 2 24.66 1 214.4 2 1134 1 120 0.14 0.5 0.8 5 0.92 2 28.33 2 286.9 2 1813 2 160 0.1 0.2 1.2 5 0.18 1 27.66 1 116.37 1 1586 2 160 0.1 0.35 0.4 9 0.45 1 47.66 2 133.33 1 1013 1 160 0.1 0.5 0.8 1 0.43 1 21.66 1 191.23 2 1240 1 160 0.12 0.2 0.4 9 0.58 1 45.66 2 125.4 1 893 1 160 0.12 0.35 0.8 1 0.72 2 20.33 1 149.43 1 1253 1 160 0.12 0.5 1.2 5 0.31 1 25.66 1 212.46 2 1773 2 160 0.14 0.2 0.8 1 0.66 2 20 1 162.93 1 1107 1 160 0.14 0.35 1.2 5 0.64 2 22.33 1 190.23 2 1533 2 160 0.14 0.5 0.4 9 0.75 2 41.33 2 177.76 2 1373 1 200 0.1 0.2 0.8 9 0.16 1 40 2 106.23 1 1053 1 200 0.1 0.35 1.2 1 0.23 1 15.67 1 208.5 2 1373 1 200 0.1 0.5 0.4 5 0.67 2 21.67 1 209.8 2 2094 2 200 0.12 0.2 1.2 1 0.4 1 14.67 1 200.2 2 1286 1 200 0.12 0.35 0.4 5 0.5 1 20.33 1 178.8 2 1866 2 200 0.12 0.5 0.8 9 0.18 1 37.66 2 168.7 1 1613 2 200 0.14 0.2 0.4 5 0.64 2 18 1 162 1 1573 2 200 0.14 0.35 0.8 9 0.31 1 34.33 2 162.5 1 1453 2 200 0.14 0.5 1.2 1 0.48 1 16.66 1 276.16 2 1667 2 median 0.58 27.66 171.3 1373 parametric study of a cnc turning process using discriminant analysis 209 it is worthwhile to mention here that among the responses, sr, cf and pc are smallerthe-better type of quality characteristics, and tl is the sole larger-the-better quality feature. since discriminant analysis cannot be performed with categorical independent variables, type of the cutting environment is converted into three distinct classes using a 1-9 point scale (where 9 = cryogenic environment, 5 = wet environment and 1 = dry environment). for carrying out a robust discriminant analysis, number of experimental runs plays an important role. pituch and stevens [22] suggested a ratio of 20:1 between the number of observations and the number of independent variables, with a minimum of 20 members in the smallest group. thus, 123 additional experimental runs are simulated to have a sample pool of 150 observations, which is in agreement with the guideline stated. all those independent and dependent variables are simulated in such a way that they must lie between their corresponding minimum and maximum values. table 3 exhibits the number of members in each group for discriminant analysis for the four responses. table 3 members in each group for discriminant analysis group number of members sr tl cf pc 1 77 76 79 85 2 73 74 71 65 now, the assumptions for normality, non-multicollinearity and homogeneity of covariance matrices need to be validated. for both the simultaneous and step-wise estimations of discriminant analysis, normality and multicollinearity tests would be the same, while the test for homogeneity of covariance matrices would be different. for normality test, the related skewness and kurtosis values are computed, and for multicollinearity test, tolerance and vif values are estimated. table 4 exhibits results of the normality and multicollinearity tests for the considered input (independent) variables. table 4 tests for normality and multicollinearity input normality test multicollinearity test variable skewness kurtosis tolerance vif s 0.111 -1.478 1 1 f 0.037 -1.493 1 1 d 0.000 -1.510 1 1 nr 0.000 -1.510 1 1 e 0.000 -1.510 1 1 according to pituch and stevens [22], when both skewness and kurtosis values for a distribution are between -2 and +2, it should be considered as normal. in this case, values of skewness lie in the range of 0 to 0.111, while the kurtosis values are between -1.510 and -1.478. as all the skewness and kurtosis values are within the prescribed range, it can be concluded that the considered input variables follow normal distribution. tolerance is a measure of variability in one independent variable that the other independent variables cannot explain. its value lies between 0 and 1, with lower values indicating presence of multicollinearity. the vif is the reciprocal of tolerance. the tolerance and vif values are 1 for all variables, indicating that the variables are orthogonal, without multicollinearity. 210 b. sarker, s. chakroborty 4.1. simultaneous estimation in this procedure, every single independent variable is involved in the analysis based on which the corresponding discriminant function is developed. however, before the analysis, the assumption of equality of covariance matrices needs to be tested using the box’s m value. the null hypothesis for this test is that the within-group covariance matrices are equal for the dependent variables. the box’s m values for sr, tl, cf and pc are computed as 76.789, 59.22, 60.568 and 74.954, respectively. the corresponding p-values are all less than 0.001, inferring that they are significant, thus rejecting the null hypothesis for the four dependent variables. however, the discriminant analysis may still be robust despite the violation of the above assumption of equality of covariance matrices as it has less importance during the analysis [27]. table 5 shows the assessment of model fit with the help of the wilks’ lambda value. the wilks’ lambda indicates the effectiveness of the discriminant function in differentiating objects into the related groups. the lower the value of wilks’ lambda is, the higher the discriminating power of the function is. smaller p-values (p < 0.05) also infer the same conclusion. in this case, all the four discriminant analyses exhibit low p-values, representing the functions’ ability to effectively distinguish objects between the groups. tables 6-8 collectively exhibit the influences of the independent variables (s, f, d, nr and e) on the responses (sr, tl, cf and pc) for the said cnc turning process. table 5 assessment of model fit for simultaneous estimation output variable eigenvalue canonical correlation wilks’ lambda chisquare df p-value sr 0.644 0.626 0.608 72.297 5 0.000 tl 2.17 0.827 0.315 167.862 5 0.000 cf 0.954 0.699 0.512 97.453 5 0.000 pc 0.17 0.382 0.854 22.904 5 0.000 table 6 group centroids for simultaneous estimation group group centroid sr tl cf pc 1 (low) 0.776 1.444 -0.92 -0.359 2 (high) -0.818 -1.483 1.023 0.469 table 7 standardized discriminant function and structure coefficients for simultaneous estimation sr tl cf pc input variable std. disc. func. coeff. str. coeff. std. disc. func. coeff. str. coeff. std. disc. func. coeff. str. coeff. std. disc. func. coeff. str. coeff. s 0.829 0.636 0.742 0.271 0.174 0.089 0.723 0.663 f -0.532 -0.35 0.196 0.062 0.4 0.214 0.478 0.419 d -0.198 -0.123 0.211 0.078 0.911 0.685 0.462 0.405 nr 0.504 0.338 0.235 0.056 0.18 0.084 0.311 0.281 e 0.369 0.249 -1.079 -0.701 -0.658 -0.395 0.23 0.2 parametric study of a cnc turning process using discriminant analysis 211 table 8 unstandardized discriminant function coefficients for simultaneous estimation input unstandardized discriminant function coefficient variable sr tl cf pc s 0.029 0.025 0.005 0.023 f -33.764 12.016 25.000 29.661 d -1.61 1.724 8.892 3.8 nr 1.589 0.717 0.55 0.952 e 0.115 -0.472 -0.214 0.07 constant -1.738 -4.116 -6.311 -9.609 4.1.1. discriminant analysis for sr table 6 shows that for sr response, group 2 with higher values of sr (> 0.58 µm) has a negative centroid, while group 1 having lower values of sr (≤ 0.58 µm) has a positive centroid. it indicates that the independent variables with negative standardized discriminant coefficients would influence the discriminant score of an observation towards the group with higher values of sr (group 2). similarly, the variables with positive coefficients would influence the discriminant score of an observation towards the group with lower sr values (group 1). table 7 shows that f and d have negative standardized discriminant function coefficients which would tend to decrease the discriminant score, moving it towards the centroid of group 2. as a result, when the values of f and d increase, sr also increases with deterioration of surface quality of the turned components. conversely, as s, nr and e have positive coefficients for sr in table 7, increase in their values would significantly reduce sr. the strength of influence of each independent variable on the discriminating power of the developed function is indicated by the absolute value of its coefficient, which in turn, can be employed to compare the level of its significance on the considered dependent variable. in this case, sr depends mostly on s, followed by f, although their nature of contribution is completely opposite. the structure coefficients, which show the correlations between the independent variables and discriminant function, are 0.636, -0.35, -0.123, 0.338 and 0.249 for s, f, d, nr and e, respectively. table 8 shows the unstandardized discriminant function coefficients, based on which the following discriminant function for sr is developed: zsr= -1.738 + 0.029×s – 33.764×f -1.61×d +1.589×nr + 0.115×e (4) the corresponding cut-off score is calculated as -0.042, which signifies that the observations with discriminant scores, estimated using eq. (4), less than -0.042 should be classified into group 2 (sr values more than 0.58 µm). similarly, the observations with discriminant scores of more than -0.042 should be categorized into group 1 (sr values less than 0.58 µm). 4.1.2. discriminant analysis for tl it can be observed from table 6 that group 2 with higher tl values has a negative centroid. on the other hand, the centroid of group 1 consisting of lower values of tl is positive. table 7 depicts that s, f, d and nr have positive coefficients, while e has negative coefficient. thus, it can be inferred that tl would decrease with increasing values of s, f, d and nr, while it would increase with increase in the scored value of e. the tl would mostly depend on e, followed by s. the correlations between the independent variables and discriminant function are 0.271, 0.062, 0.078, 0.056 and -0.701 for s, f, d, 212 b. sarker, s. chakroborty nr and e, respectively. now, based on the unstandardized discriminant function coefficients of table 8, the following discriminant function for tl is derived. ztl= -4.116 + 0.025×s + 12.016×f + 1.724×d + 0.717×nr 0.472×e (5) the cut-off score is equal to -0.039, which denotes that the observations whose discriminant scores, estimated using eq. (5), are less than -0.039, should be classified into group 2 with higher tl value (more than 27.66 min). on the other hand, observations with discriminant scores of more than the cut-off score would be in group 1 with lower tl values (less than 27.66 min). 4.1.3. discriminant analysis for cf table 6 shows that group 2 with higher cf values has a positive centroid and group 1 with lower cf values has a negative centroid. from table 7, it can be propounded that s, f, d and nr have positive coefficients, while the coefficient for e is negative. as a result, cf would increase with increasing values of s, f, d and nr, and increased score for e would result in decreased value of cf. the most important turning parameter influencing cf is d, followed by e, as noticed from the absolute values of their corresponding standardized discriminant coefficients. the structure coefficients, representing the correlations between the independent variables and discriminant function, are 0.089, 0.214, 0.685 and 0.084 and -0.395 for s, f, d, nr and e, respectively. now, based on table 8, the following discriminant function for cf is developed. zcf= -6.311 + 0.005×s + 25.0×f + 8.892×d + 0.550×nr 0.214×e (6) for cf response, the corresponding cut-off score is estimated to be 0.103. it symbolizes that the observations with discriminant scores higher than 0.103 would be assigned to group 2 with higher cf values (more than 171.3 n). in the similar direction, the observations having discriminant scores of less than the cut-off score would be allocated to group 1 (cf values less than 171.3 n). 4.1.4. discriminant analysis for pc from table 6, it can be noticed that the centroid for group 2 is positive, while its value is negative for group 1. thus, the independent variables whose standardized discriminant function coefficients are positive, would like to increase the discriminant scores of the observations moving them towards the centroid of group 2. in table 7, all the five independent variables have positive coefficients. hence, increasing values of s, f, d, nr and e are all responsible for higher pc during the cnc turning operation. it can also be revealed that s and f are the two most important turning parameters influencing pc. the correlations between the independent variables and discriminant function are estimated as 0.663, 0.419, 0.405, 0.281, 0.2 for s, f, d, nr and e, respectively. now, based on table 8, the following discriminant function for pc is established. zpc= -9.609 + 0.023×s + 29.661×f + 3.8×d + 0.952×nr + 0.070×e (7) for this response, the value of the cut-off score is calculated as 0.110. it indicates that the observations with discriminant scores of more than 0.110 would be classified into group 2 having higher pc (greater than 1373 w). similarly, observations with discriminant scores of less than 0.110 would be included in group 1 with lower pc values (less than 1373 w). parametric study of a cnc turning process using discriminant analysis 213 4.1.5. validation of the discriminant analysis results now, it is required to validate the results derived from the simultaneous estimationbased discriminant analysis in order to justify the corresponding prediction performance. it can be observed from table 3 that for sr response, among 150 original and simulated experimental runs, 77 have low sr values (less than 0.58 µm) and the remaining 73 observations have high sr values (more than 0.58 µm). in table 9, the discriminant function developed for sr can correctly identify 71 group 1 observations (out of 77) and 52 group 2 observations (out of 73). so, the percentages of correct classifications are 92.2% and 71.2%, respectively. hence, the hit ratio for the discriminant function for sr is 82% (123 out of 150), with a misclassification error of 18%. the prediction performance of this discriminant function is cross-validated based on leave-one-out approach, using ibm spss statistics 25.0 software. for sr, the percentages of correct classification for group table 9 classification results for simultaneous estimation method output variable type of validation count (%) group predicted group membership total 1 2 sr original count 1 71 6 77 2 21 52 73 % 1 92.2 7.8 100 2 28.8 71.2 100 crossvalidated count 1 65 12 77 2 21 52 73 % 1 84.4 15.6 100 2 28.8 71.2 100 tl original count 1 70 6 76 2 6 68 74 % 1 92.1 7.9 100 2 8.1 91.9 100 crossvalidated count 1 70 6 76 2 6 68 74 % 1 92.1 7.9 100 2 8.1 91.9 100 cf original count 1 68 11 79 2 16 55 71 % 1 86.1 13.9 100 2 22.5 77.5 100 crossvalidated count 1 57 22 79 2 16 55 71 % 1 72.2 27.8 100 2 22.5 77.5 100 pc original count 1 70 15 85 2 24 41 65 % 1 82.4 17.6 100 2 36.9 63.1 100 crossvalidated count 1 70 15 85 2 24 41 65 % 1 82.4 17.6 100 2 36.9 63.1 100 214 b. sarker, s. chakroborty 1 and group 2 objects based on cross-validation are 84.4% and 71.2%, respectively. hence, the hit ratio for cross-validation is 78% (117 out of 150). similarly, in case of tl, both the hit-ratio and cross-validation percentages are 92%. for cf response, the hit-ratio is 82%, while the cross-validation percentage is 74.7%. finally, for pc, 74% of the original and cross-validated grouped cases can be correctly classified. these higher values of hit-ratio prove that the discriminant functions developed on the basis of the simultaneous estimation method have the ability to correctly classify the responses into appropriate lower and higher groups. 4.2. step-wise estimation the step-wise estimation of discriminant analysis is appropriate when only the significant independent variables need to be included in the developed discriminant function. these independent variables are selected based on the wilks’ lambda value. the variables having smaller wilks’ lambda values and maximum ability to decrease the overall wilks’ lambda, are first chosen for inclusion in the model. before developing the model, it is assumed that the model does not have any independent variable. in each step, the variable whose ‘f to enter’ value is the largest and simultaneously higher than the entry criterion, is included in the model, while the ‘f to remove’ value is necessary to exclude any insignificant variable from further consideration. the ‘f to enter’ and ‘f to remove’ values, which would decide the entry and exit of the independent variables in the model, are estimated as 3.84 and 2.71, respectively, and are set as defaults in the software. these values relate to p-values of 0.05 and 0.10, respectively. this process is continued till all the significant variables, satisfying the entry criterion, are included in the model, while the insignificant variables are removed from the model. as mentioned earlier, before the start of this analysis, testing of the assumptions is mandatory. assumptions of normality and multicollinearity, as tested in table 4, also hold true for step-wise discriminant analysis. the box’s m test is conducted again to check whether the covariance matrices are homogenous or not. the values of the box’s m for sr, tl, cf and pc are determined as 55.583, 47.129, 6.697 and 21.78, respectively. the corresponding p-values for sr and tl are less than 0.001, while those for cf and pc are greater than 0.001 (0.365 and 0.002, respectively). hence, for sr and tl, the null hypothesis of equality of covariance matrices is rejected, while it cannot be rejected for cf and pc. even though for sr and tl, the assumption of equality of covariance matrices is violated, their discriminant analyses may still be considered robust. the model fit now needs to be validated applying the overall wilks’ lambda for the discriminant functions of all the four responses in order to check their ability to separate objects into separate groups. table 10 exhibits the eigenvalues and wilks’ lambda values for the dependent variables (sr, tl, cf and pc), testing the significance of the discriminant function for each of those variables. it can be noticed that all the p-values are less than 0.05, indicating satisfactory discriminating power of the developed functions. in tables 11-14, variables entered into the models and removed from the models during step-wise discriminant analysis for the four considered responses are provided. parametric study of a cnc turning process using discriminant analysis 215 table 10 assessment of model fit for step-wise estimation output variable eigenvalue canonical correlation wilks’ lambda chi-square df p-value sr 0.619 0.618 0.618 70.311 4 0.000 tl 2.088 0.822 0.324 164.63 4 0.000 cf 0.896 0.687 0.528 93.687 3 0.000 pc 0.144 0.355 0.874 19.753 3 0.000 table 11 variables included/not included in the model for sr variable included variable not included input variable tolerance f-value wilks’ lambda input variable tolerance min. tolerance f-value wilks’ lambda e 0.945 48.422 0.824 d 0.994 0.941 2.221 0.608 s 0.965 17.046 0.69 nr 0.97 15.11 0.682 d 0.983 8.096 0.652 table 12 variables included/not included in the model for tl variable included variable not included input variable tolerance f-value wilks’ lambda input variable tolerance min. tolerance f-value wilks’ lambda e 0.793 232.304 0.843 f 0.982 0.78 3.806 0.315 s 0.814 61.737 0.462 nr 0.969 5.361 0.336 d 0.983 4.369 0.334 table 13 variables included/not included in the model for cf variable included variable not included input variable tolerance f-value wilks’ lambda input variable tolerance min. tolerance f-value wilks’ lambda d 0.921 83.389 0.829 s 0.993 0.916 2.114 0.52 e 0.933 33.654 0.649 nr 0.989 0.917 2.277 0.519 f 0.967 11.539 0.569 table 14 variables included/not included in the model for pc variable included variable not included input variable tolerance f-value wilks’ lambda input variable tolerance min. tolerance f-value wilks’ lambda s 0.995 11.674 0.944 nr 0.998 0.994 2.177 0.861 f 0.997 4.891 0.903 e 0.998 0.995 1.229 0.867 d 0.997 4.536 0.901 216 b. sarker, s. chakroborty it can be revealed from table 11 that the independent variables included in step-wise estimation of the dependent variable sr are s, f, nr and e. the independent variables that significantly influence tl are e, s, nr and d. on the other hand, d, e and f are the significant variables for cf, while s, f and d maximally influence pc. from the computed f-values, s is the most significant independent variable for sr, followed by f. for tl, the most significant independent variable is e, followed by s. similarly, d has the maximum discriminating power on cf, followed by e. for response pc, s is identified as the most significant independent variable. conversely, d is singled out as the least significant contributor for sr, while f has no discriminating power on tl. in the similar direction, s and nr do not contribute significantly to cf, and for pc, the insignificant independent variables are nr and e. in discriminant analysis, an independent variable can significantly differentiate objects into the corresponding groups only when the difference between the means of the independent variables across the groups is significant. for insignificant independent variables, the difference between their means across the groups is not enough to create sufficient separation between those two groups. hence, for any variation in the values of insignificant variables, changes in the discriminant scores remain negligible with respect to their respective dependent variables. likewise the simultaneous estimation method of discriminant analysis, tables 15-17 also exhibit the effects of five independent variables of the cnc turning process on the dependent variables for step-wise estimation method. table 15 group centroids for step-wise estimation group group centroid sr tl cf pc 1 (low) 0.761 1.416 -0.891 -0.33 2 (high) -0.802 -1.455 0.992 0.432 table 16 standardized discriminant function and structure coefficients for step-wise estimation input variable sr tl cf pc std. disc. func. coeff. str. coeff. std. disc. func. coeff. str. coeff. std. disc. func. coeff. str. coeff. std. disc. func. coeff. str. coeff. s 0.833 0.649 0.737 0.277 -0.083 0.768 0.721 f -0.534 -0.357 -0.133 0.4 0.22 0.508 0.456 d 0.075 0.21 0.08 0.914 0.707 0.49 0.44 nr 0.504 0.344 0.233 0.057 -0.095 -0.041 e 0.375 0.254 -1.072 -0.715 -0.652 -0.407 -0.039 table 17 unstandardized discriminant function coefficients for step-wise estimation unstandardized discriminant function coefficient input variable sr tl cf pc s 0.029 0.024 0.024 f -33.915 25.012 31.516 d 1.713 8.919 4.025 nr 1.589 0.712 e 0.116 -0.469 -0.212 constant -2.312 -2.659 -5.051 -9.024 parametric study of a cnc turning process using discriminant analysis 217 table 15 shows that the centroid of group 2 with higher sr values is negative, while group 1 having lower sr values has a positive centroid. thus, it can be unveiled that s, nr and e with positive standardized discriminant coefficients have negative impacts on sr, while sr would increase with higher values of f. the structure coefficients which denote the correlations between the independent variables and discriminant function are estimated as 0.649, -0.357, 0.075, 0.344 and 0.254 for s, f, d, nr and e, respectively. based on the results of step-wise estimation, the values of the standardized coefficient and structure correlation show that s has the most discriminating power on sr, maximally influencing it. table 17 provides the unstandardized discriminant coefficients which lead to the subsequent development of the following discriminant function for sr: zsrs= -2.312 + 0.029×s 33.915×f + 1.589×nr + 0.116×e (8) the related cut-off score is estimated as -0.041. it denotes that the observations whose discriminant scores are higher than -0.041 would be classified into group 1 with lower sr values. similarly, the observations having discriminant scores of less than -0.041 would be assigned to group 2 with higher sr values. as in table 15, the centroid of group 2 is negative and that of group 1 is positive, the independent variables having positive standardized discriminant coefficients would cause the observations to move closer to group 1, thereby reducing tl. therefore, s, d and nr have negative influences on tl, while an increase in the score for e would increase tl. both the standardized coefficients and structure correlations establish that e has the maximum discriminating power on tl. the related discriminant function for tl is represented as below: ztls= -2.659 + 0.024×s + 1.713×d + 0.712×nr 0.469×e (9) the cut-off discriminant score for tl is -0.039. thus, the observations with discriminant scores of less than -0.039 would be assigned to group 2 with higher tl values. on the contrary, observations with discriminant scores higher than the corresponding cut-off score would be classified into group 1with lower tl values. from table 15, it can also be observed that as group 2 has a positive centroid value, the independent variables with positive standardized discriminant coefficients are expected to have positive impacts on cf. thus, with increasing values of f and d, cf would increase, while it would decrease with higher scores for e. both the standardized coefficients and structure correlations identify d as the most significant input variable for cf. the following equation shows the developed discriminant function for cf: zcfs= -5.051 + 25.012×f + 8.919×d 0.212×e (10) for this response, the cut-off score is calculated as 0.100. it denotes that the observations with discriminant scores higher than 0.100 would be added to group 2 with higher cf values. similarly, the observations with discriminant scores lower than the cutoff score would be included in group 1 with lower cf values. similarly for response pc, as the independent variables with positive standardized discriminant coefficients significantly influence it, increasing values of f and d would be responsible to increase pc. based on the standardized discriminant coefficients and structure correlations, it can be propounded that s has the maximum influence on pc. the related discriminant function is developed as given below: zpcs= -9.024 + 0.024×s + 31.516×f + 4.025×d (11) 218 b. sarker, s. chakroborty the cut-off score for these responses is calculated as 0.102, which signifies that the observations having discriminant scores of more than 0.102 would be assigned to group 2 with higher pc values. on the contrary, observations with discriminant scores of less than the cutoff score would be included in group 1 having lower pc values. the numbers of correctly classified items, indicated by hit ratio, along with the crossvalidation results for all the dependent variables are provided in table 18. for sr response, the hit ratio is 81.3% and the cross-validation percentage is 78%. the hit ratio and cross-validation percentage for tl are both 96%. for cf, both the hit ratio and crossvalidation percentage are 82%, while the hit ratio and cross-validation percentage for pc are both 74%. from these observations, it can be concluded that the developed step-wise discriminant functions for the responses have the ability to categorize the observations into the corresponding groups with minimum misclassification error. 5. results and discussion as mentioned earlier, the aim of this paper is to study the influences of different input parameters of a cnc turning process on its responses as well as to identify the most important parameter for each of the responses. it can be unveiled from both the simultaneous and step-wise estimation methods of discriminant analysis that speed is the most significant parameter for sr and pc. on the other hand, machining environment maximally influences tl and depth of cut is the most influential parameter for cf. the coefficients of these input parameters in the discriminant function for each of the responses, along with the structure correlations, indicate their comparative strengths of influence on the responses. in this analysis, it can be noticed that an increase in speed causes sr to decrease. the decrease in sr can be explained due to decrease in built-up-edge formation at higher temperature at the chip-tool interface at higher spindle speed [28]. an increase in feed rate leads to an increase in sr. as feed rate increases, wide and deep cracks are formed which are responsible for poor surface quality of the machined components [29]. an increase in feed rate also causes cf to increase due to the required plastic deformation and generation of excess heat in the machining area, thereby increasing tool wear and eventual deterioration of surface finish. although sr increases with increasing values of depth of cut, it is supposed to have negligible effect on sr. the slight variation in sr is due to tool chatter, occurring at higher values of depth of cut. better surface quality of the machined components can be achieved at higher nose radius. it can be attributed to lower strength of insert nose. at smaller nose radius of the tool, the contact length between insert tip of the tool and workpiece becomes narrower, thus reducing heat dissipation from the shear zone, causing higher stress and heat concentration at the zone, thereby increasing tool wear and sr [29]. it can also be observed that cryogenic machining environment improves sr because the machining zone temperature is effectively controlled at cryogenic environment, which simultaneously reduces adhesion between tool flank faces and chip, thus reducing tool wear and sr [30]. parametric study of a cnc turning process using discriminant analysis 219 table 18 classification results for step-wise discriminant analysis output variable type of validation count (%) group predicted group membership total 1 2 sr original count 1 65 12 77 2 16 57 73 % 1 84.4 15.6 100 2 21.9 78.1 100 crossvalidated count 1 65 12 77 2 21 52 73 % 1 84.4 15.6 100 2 28.8 71.2 100 tl original count 1 76 0 76 2 6 68 74 % 1 100 0 100 2 8.1 91.9 100 crossvalidated count 1 76 0 76 2 6 68 74 % 1 100 0 100 2 8.1 91.9 100 cf original count 1 68 11 79 2 16 55 71 % 1 86.1 13.9 100 2 22.5 77.5 100 crossvalidated count 1 68 11 79 2 16 55 71 % 1 86.1 13.9 100 2 22.5 77.5 100 pc original count 1 70 15 85 2 24 41 65 % 1 82.4 17.6 100 2 36.9 63.1 100 crossvalidated count 1 70 15 85 2 24 41 65 % 1 82.4 17.6 100 2 36.9 63.1 100 tool life can be defined as the time elapsed for the measured wear level of a tool to exceed an established critical value of wear. a standard measure of tl is the time to develop its maximum value of flank wear width [31]. increased values of speed and feed rate cause higher tool flank wear, thereby decreasing tl. an increase of tool flank wear can be attributed to the increase in the concentration of compressive stress at the tool rake face in the vicinity of the cutting edge. higher tool flank wear is also due to increase in temperature of the tool creating high cutting edge load or lowered tool hardness due to the phenomenon of thermal softening at the proximity of the cutting edge [29]. on the other hand, feed rate does not have any significant influence on tl. tool flank wear also increases with increasing values of nose radius due to increase in cf, which can be attributed to the increase in the thrust force component [32]. higher depth of cut implies that the contact length between the cutting edge and workpiece increases, causing deeper 220 b. sarker, s. chakroborty wear along the cutting edge, thereby decreasing tl [33]. machining environment has significant impact on tl. it can be noticed that tool wear is minimum at cryogenic machining environment. application of cryogenic environment improves wear resistance of the tool and decreases the temperature at the cutting zone, thereby reducing abrasion and adhesion. with increase in spindle speed of the cnc turning process, cf is found to increase, although insignificantly. it can be attributed to the material strengthening effect induced by the strain gradient [34]. similarly, nose radius positively influences cf. the increase in cf is due to increase of the thrust force component, along with a marginal increase in feed force and tangential force [35]. feed rate and depth of cut also have positive influences on cf. with increase in feed and depth of cut, cf increases because the sheared chip crosssection grows larger along with the deformed metal volume, which makes the workpiece material increasingly resistant to shearing, requiring more force to remove the chips [36]. application of cryogenic environment during cnc turning reduces cf, due to reduction in the coefficient of friction between the chip and the tool, and decrease in the chip contact length due to formation of smaller chips [37]. according to this discriminant analysis, all the five cnc turning parameters have positive influences on pc. higher power is required for higher cf, simultaneously caused by the increases in speed, feed, depth of cut and nose radius [38]. however, an increase in pc at cryogenic machining environment can be attributed to the increase in strength and hardness of the workpiece, when cooled by the cryogenic fluid [39]. this increase in strength and hardness of the workpiece may lead to an increase in energy consumption while removing material from the workpiece surface. however, it can be noted that both the nose radius and machining environment are insignificant parameters, while speed is the most significant parameter for pc. 6. conclusion this paper deals with the application of discriminant analysis in a cnc turning process to explore the influences of its five input parameters on four responses, and identify the most significant parameter for each of the considered responses. after validating the corresponding assumptions, like absence of multicollinearity and missing data, normality of the independent variables, etc., two sets of discriminant functions are developed. in simultaneous estimation method, all the independent variables are considered, while in step-wise estimation method, the insignificant independent variables are excluded while developing the respective models. based on the developed discriminant functions, it can be revealed that higher feeds are responsible for poor surface finish of the turned components, where better surface quality is achieved at higher values of speed and nose radius, and cryogenic machining environment. it is least affected by depth of cut. similarly, higher values of speed, depth of cut and nose radius are responsible for reduced tool life. it would increase at cryogenic machining environment and remain unaffected due to feed. cutting force would increase at higher values of feed and depth of cut. cryogenic machining environment would cause cutting force to decrease, and speed and nose radius have no significant roles on cutting force. finally, higher values of speed, feed and depth of cut are all responsible for more power consumption during cnc turning operation, while it remains unaffected due to changes in nose radius and machining environment. it can also be propounded that the reduced discriminant functions developed by step-wise estimation method has similar effectiveness as those formulated with the inclusion of all the independent variables. higher values of hit ratio and cross-validation percentage parametric study of a cnc turning process using discriminant analysis 221 conclude that both the functions are well capable of classifying objects into the corresponding binary groups. discriminant analysis has few limitations. it requires certain assumptions to be satisfied in order to provide satisfactory results. in discriminant analysis, with an increase in the number of independent variables, sample size must be increased as well. however, its advantages outweigh its limitations. discriminant analysis has several advantages as an effective prediction tool. the causal relationship between the independent and dependent variables can be envisaged based on the developed discriminant 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vol. 14, no 3, 2016, pp. i i editorial foreword to the thematic issue: tribology and contact mechanics in biological and medical applications applications tribology, contact mechanics and biomechanics belong to the group of extremely complex engineering disciplines and, not surprisingly, they have been a subject of interest for a long time now. the interest has particularly grown in the recent decades and the research in the field intensified dramatically. this may be partly attributed to the development of modern hardware tools which made computations and analyses that engineers dreamt of decades ago not only possible, but literally at their fingertips. equally important is the recognition that the research in those fields can only be successful if it involves a multidisciplinary approach. hence, it is of utmost importance that researchers should work together across those disciplines to tackle major challenges. this aspect was the major impetus for valentin l. popov (technische universität berlin) and sergey g. psakhie (russian academy of sciences) to bring the researchers together by organizing an international workshop entitled tribology and contact mechanics in biological and medical applications at the technische universität berlin. the present thematic issue contains a selection of papers which reflects the spectrum of topics addressed at the workshop. the subject of the workshop is the intersection of contact mechanics, tribology and medicine/biology. this area has been experiencing rapid development in recent years. current research objectives include the development of high-performance and low-wear materials in order to significantly increase the lifetime of medical prostheses such as artificial hips or knee joints and implants. tribological characteristics play a very important role here, so that their influence on the functional properties of biological joints and medical devices is of great interest. for example, ultrasound oscillations are used to produce or improve the function of biomedical instruments. the wear behavior, temperature development and damping properties can be controlled by a targeted use of gradient materials. another major area of research is the optimal adaptation and compatibility between artificial material and human tissue. a very important issue is the adhesive behavior of biological tissues. a further focus is on elastomers whose capabilities are tested for transferability to human tissue. due to the extreme interdisciplinary character of the field of bio-tribology, many, even fundamental, questions have not been answered yet. it is important in this area to look for complementary expertise and combine it. this is the main focus of the workshop. dragan marinković editor-in-chief editorial foreword to the thematic issue: tribology and contact mechanics in biological and medical applications applications facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 365 383 https://doi.org/10.22190/fume190605043d © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper optimization of multi-pass face milling parameters using metaheuristic algorithms sunny diyaley 1 , shankar chakraborty 2 1 department of mechanical engineering, sikkim manipal institute of technology, sikkim manipal university, majitar, sikkim, india 2 department of production engineering, jadavpur university, kolkata, india abstract. in this paper, six metaheuristic algorithms, in the form of artificial bee colony optimization, ant colony optimization, particle swarm optimization, differential evolution, firefly algorithm and teaching-learning-based optimization techniques are applied for parametric optimization of a multi-pass face milling process. using those algorithms, the optimal values of cutting speed, feed rate and depth of cut for both roughing and finishing operations are determined for having minimum total production time and total production cost. it is observed that the teaching-learning-based optimization algorithm outperforms the others with respect to accuracy and consistency of the derived solutions as well as computational speed. two statistical tests, i.e. paired t-test and wilcoxson signed rank test also confirm its superiority over the remaining algorithms. finally, these metaheuristics are employed for multi-objective optimization of the considered multi-pass milling process while concurrently minimizing both the objectives. key words: multi-pass milling, optimization, metaheuristic, objective, parameter 1. introduction in manufacturing industries, machining is considered as one of the most important processes to remove excess amount of material from a given workpiece so as to fulfill the end requirements of the customers. in machining processes, a cutting tool having material harder than the part/component being shaped is employed for material removal by shear deformation in the form of chips. during manufacturing, the workpiece may require different machining operations, like turning, milling, drilling, reaming, grinding, etc. in the milling process, the material is removed from the workpiece by a rotating multiple tooth cutter. with the rotation of the cutter, each tooth removes a small amount of the received june 05, 2019 / accepted november 10, 2019 corresponding author: s. chakraborty department of production engineering, jadavpur university, kolkata e-mail: s_chakraborty00@yahoo.co.in 366 s. diyaley, s. chakraborty material from the advancing workpiece during each spindle revolution. as the relative movement between the cutter and the workpiece can be in any direction, different workpiece surfaces, like flat, angular, curved or any combination of them can be machined in milling. in the face milling operation, the cutter, mounted on a spindle rotating perpendicular to the surface, provides a high material removal rate due to the cutting action of many teeth on its periphery and face. the milling operation is usually performed in a single pass or in multiple passes. the multi-pass milling is always preferable to a single pass operation for better machining economies and is utilized to shape workpieces that cannot be machined in a single pass. the multi-pass operation in the face milling often results in low power consumption, machining force, machine depreciation and chatter phenomena. the multipass milling, having complicated material removal mechanism, consists of multiple rough passes and one finish pass. major portion of the material is removed from the workpiece during multiple rough passes. the single finish pass is mainly responsible for surface smoothening. fig. 1 exhibits a typical multi-pass face milling operation. (a) roughing (b) finishing fig. 1 multi-pass face milling operation in the present day manufacturing environment, multi-pass milling processes are extensively used for production of low cost high quality parts/components. due to a high capital cost, it is always recommended to operate the multi-pass milling machines efficiently and economically. the machining performance of a multi-pass milling optimization of multi-pass face milling parameters using metaheuristic algorithm 367 operation is often characterized by the effects of its various input parameters, like cutting speed, feed rate, depth of cut, number of passes, etc. on different process outputs (responses), i.e. material removal rate, surface roughness, tool life, power consumption, amplitude of vibration, etc. the success of a multi-pass milling operation with respect to higher product quality, increased production rate, reduced machining cost and time depends on proper selection of its various input parameters [1, 2]. because of the complexity of this process, the determination of the optimal combination of its parameters has become a complex and challenging task [3]. it also depends on the workpiece shape, types of the cutter and work materials, technological requirements, capability of the milling machine, etc. it has been observed that in most of the cases, milling parameters are chosen from the machining database or handbook. in some occasions, the concerned machinists also participate in identifying the most tentative settings of various milling parameters based on their knowledge and expertise. but the milling parameters selected based on these resources are extremely conservative and far from their optimal values. various mathematical programming techniques, like dynamic programming, geometric programming, linear programming, etc. have also been applied in the past to determine the optimal settings of the machining parameters during a multi-pass face milling operation. these conventional optimization techniques usually lack robustness while solving complex optimization problems. thus, they are now being substituted by different metaheuristics which can efficiently explore the entire search space in order to find out the near optimal or global optimal solutions with a lower computational effort and time. therefore, these metaheuristic algorithms are popularly deployed by the researchers for deriving more robust and accurate solutions. thus, it is always advisable to implement the corresponding mathematical formulations and determine the best parametric setting of a multi-pass face milling operation based on the applications of robust optimization tools. in this paper, a multi-objective mathematical model for a multi-pass face milling operation is considered with the aim to simultaneously minimize the total production time and total production cost. the optimization problem is also subjected to various real time machining constraints with respect to different limiting values of cutting speed, feed rate, depth of cut, tool life, surface roughness, cutting force and machining power. it is subsequently solved using six most popular metaheuristic algorithms, i.e. artificial bee colony (abc) optimization, ant colony optimization (aco), particle swarm optimization (pso), differential evolution (de), firefly algorithm (fa) and teaching-learning-based optimization (tlbo) techniques. the optimization performance of the adopted metaheuristics is also validated with respect to accuracy and consistency of the derived solutions, and computational speed. 2. literature survey conceição antónio et al. [4] substituted depth of cut with a sequence of depths of cut and applied a genetic search approach for parametric optimization of a multi-pass milling process. zarei et al. [5] applied harmony search (hs) algorithm to optimize various multipass face milling parameters, i.e. number of passes, depth of cut in each pass, speed and feed in order to obtain minimum total production cost. the adopted optimization tool had high convergence speed and solution accuracy as compared to genetic algorithm (ga). rao and pawar [6] applied three non-traditional optimization techniques, i.e. abc, pso and simulated annealing (sa) for minimization of production time in a multi-pass milling 368 s. diyaley, s. chakraborty operation. while taking into account several practical machining constraints, an et al. [7] adopted ga technique to minimize the unit production cost for a multi-pass milling operation. yang et al. [8] presented the application of a fuzzy pso algorithm for optimization of various machining parameters in a multi-pass face milling process. the adopted technique would outperform other optimization tools with respect to minimum production cost under multiple technological constraints. yang et al. [9] adopted a fuzzy multi-objective pso technique for parametric optimization of a multi-pass face milling operation. it was observed that the proposed algorithm could generate efficient pareto optimal solutions with satisfactory convergence rate. using ga technique, an et al. [10] solved a multi-objective model so as to minimize unit production cost and total machining time, and maximize profit rate. for a face milling operation, various real time machining constraints were also considered while developing that model. nagarchi and patel [11] applied hs algorithm to determine the optimal values of depth of cut, speed and feed rate in a face milling operation with an aim to improve surface roughness of machined components at the minimum total production cost. its optimization performance was later validated with respect to ga technique. zhai et al. [12] proposed an improved tlbo algorithm, in the form of tlbo with dynamic assignment learning strategy (datlbo), for a multi-tool milling process in order to maximize profit rate under several machining constraints. its optimization performance was also compared with that of other techniques. mellal and williams [13] applied cuckoo optimization algorithm to minimize the total production time of a face milling operation and also validated its superior optimization performance. li et al. [14] presented the application of adaptive multi-objective pso algorithm to maximize energy efficiency and minimize production cost in a multi-pass face milling operation. khalilpourazari and khalilpourazary [15] employed a robust grey wolf optimizer for parametric optimization of a multi-pass milling operation, and also compared its performance against other optimization techniques. yang [16] proposed the combined application of chaotic operator and imperialist competitive algorithm to optimize various input parameters of a multi-pass face milling operation. the optimal values of speed, feed and depth of cut in each pass were determined for minimum total production cost. miodragović et al. [17] endeavored to modify the basic fa technique to optimize different machining parameters in multi-pass turning and multi-pass face milling operations subject to various practical operational constraints. it was concluded that the proposed algorithm had the ability to reach the global optimal solutions for complex optimization problems. it can be observed from the survey of the existing literature that optimization of multi-pass milling processes has already caught the attention of the past researchers, and different metaheuristic algorithms, mainly in the form of ga, sa, hs, pso, etc. have been applied for determination of the optimal values of cutting speed, feed rate, depth of cut and number of passes for the considered multi-pass milling processes. minimizations of total production time and of total production cost as well as maximization of profit rate have mainly been treated as the objectives. the past researchers have also endeavored to integrate the concept of the fuzzy set theory with some of the metaheuristic algorithms to solve the multi-pass milling process optimization problems. but, no research work has been conducted till date to study the comparative optimization performance of the most commonly adopted metaheuristic algorithms in this domain and to identify the best algorithm. thus, the objective of this paper is focused on the application of six metaheuristics, i.e. abc, aco, pso, de, fa and tlbo to solve a multipass milling process optimization problem, and compare their relative performance with respect to accuracy and consistency of the derived solutions, and computational speed. optimization of multi-pass face milling parameters using metaheuristic algorithm 369 3. mathematical model for multi-pass face milling operation a multi-pass milling operation usually consists of two operations, i.e. multiple roughing passes and a single finishing pass. for roughing operation, length of tool (cutter) path (lr) can be expressed as [17]: lr = l + ap+ e (1) where l is the length of the workpiece, ap is the approach distance and e is a small distance provided to avoid the chance of any possible damage. the value of e is generally considered as 2-5 mm. for a symmetrical milling operation, the approach distance can be given by: 22 222              wdd a p (2) where d is the diameter of the tool and w is the width of cut. if a single tool is employed to perform the entire machining operation, the total production time then consists of actual machining time (tm), machine idling time (ti) and tool changing time (tr). thus, total production time (ttotal) can be expressed as follows: ttotal = tm + ti + tr (3) the objective function can thus be stated as: f1(x) = min(ttotal) = min(tm + ti + tr) (4) the actual machining time for a single pass milling operation can be given by: vfz dl tm 1000   (5) where d is the tool diameter (in mm), l is the length of the workpiece (in mm), v is the cutting speed (in m/min), f is the feed rate per tooth (in mm/tooth) and z is the number of teeth. the actual machining time (in min) in a multi-pass milling operation consists of n roughing pass time (tmri) and one finishing pass time (tms). ms n i mrim ttt  1 (6) where n is the number of roughing passes. the actual roughing and finishing times for single pass are denoted as below: zfv dl t riri r mri 1000   (7) zfv dl t ss s ms 1000   (8) where vri and vs are the cutting speed in roughing and finishing operations, respectively, and fri and fs are the feed rate per tooth in roughing and finishing operations, respectively. 370 s. diyaley, s. chakraborty idling time (ti) consists of preparation time (tp) (in min/piece) and idle tool motion time (ti) (in min). thus, ti = tp + ti = tp + (h1ls + h2) + n(h1lr + h2) (9) where h1 (in min/mm) and h2 (in min) are the constants related to tool travel time and tool approach or depart time, and ls is the tool path length for finishing pass. as a single tool is utilized for the milling operation, it needs to be changed only after it is worn out. thus, the tool changing time is dependent on actual machining time and tool life. tool exchange time (tr) (in min) for n rough passes and one finish pass can be denoted as:            n i s ms tc ri mri tcr t t t t t tzt 1 (10) where l psy ri x riri q vv ri vvvv v zwfdv dkc t 1          (11) l psy s x ss q vv s vvvv v zwfdv dkc t 1          (12) where ttc is the tool changing time (in min/edge) for each cutting edge, tri and ts are the values of tool life for rough and finish passes, respectively, and cv, kv, xv, yv, sv, qv, pv and l are various constants and exponents related to tool and workpiece material. now, by substituting eqs. (6), (9) and (10) in eq. (3), the corresponding objective function for minimization of the total production time can be expressed as: ps n i ri tttt  1 (13) where )(1 21 hlh t t ztt r ri tc mriri          (14) )(1 21 hlh t t ztt s s tc mss          (15) where tri is the unit time for i th rough pass (in min/part) and ts is the unit time required for the finish pass (in min/part). the second objective considered here is to minimize total production cost (ctotal) which has four components, i.e. machining cost (cm), machine idling cost (ci), tool change cost (cr) and tool cost (ct). thus, the total production cost can be given as follows: ctotal = cm + ci + cr + ct (16) hence, the second objective function can be defined as: f2(x) = min(ctotal) = min(cm + ci + cr + ct) (17) optimization of multi-pass face milling parameters using metaheuristic algorithm 371 machining cost cm depends on actual machining time (tm) (in min), and unit labor and overhead cost (k0) (in usd/min). cm = k0tm (18) on the other hand, machine idling cost ci is: ci = k0ti (19) tool change cost cr can be given as below: cr = k0tr (20) tool cost ct can be expressed based on the following equation:            n i s ms t ri mri tt t t k t t kzc 1 (21) where kt is the cost of tool material (in usd/edge). now, by substituting eqs. (18)-(21) in eq. (16), the objective function for minimum production cost can be given as follows: ps n i ri tkccc 0 1   (22) where )( 210 0 0 hlhk t k z t tk zktc tr ri t ri tc mriri          (23) )( 210 0 0 hlhk t k z t tk zktc ts s t s tc mss          (24) where cri is the unit cost for i th rough pass (in usd/part), cs is the unit cost for finish pass (in usd/part), and ltr and lts are the cutting travel length for rough and finishing pass, respectively. the objective functions for minimum production time and minimum production cost are subjected to various constraints, like (a) parameter bounds, (b) cutting force constraint, (c) machine tool power constraint, (d) surface finish constraint, (e) tool life constraint and (f) number of rough passes constraint. these constraints are stated here-in-under. (a) parameter bounds: for i th rough pass, the range for cutting speed vri is vmin ≤ vri ≤ vmax (25) where vmin and vmax are the minimum and maximum cutting speeds, respectively. for i th rough pass, the range for feed rate fri is fmin ≤ fri ≤ fmax (26) where fmin and fmax are the minimum and maximum feed rate, respectively. 372 s. diyaley, s. chakraborty for i th rough pass, the range for depth of cut dri is dr min ≤ dri ≤ dr max (27) where dr min and dr max are the minimum and maximum depth of cut, respectively. there are also three similar constraints for cutting speed, feed rate and depth of cut for the finish pass. (b) cutting force constraints: 0)( max 11 1  f d dfzwkc xg u uuuu q x r y r ps uu (28) 0)( max 22 2  f d dfzwkc xg u uuuu q x r y r ps uu (29) 0)( max3  f d dfzwkc xg u uuuu q x s y s ps uu (30) where cu, ku, xu, yu, su, pu and qu are various constants and exponents. fr1, fr2 and fs are the feed rate for the first rough pass, second rough pass and finish pass, respectively, dr1, dr2 and ds are the depth of cut for the first rough pass, second rough pass and finish pass, respectively, and fmax is the maximum allowable cutting force. (c) available machine tool power constraints: 0)( max 111 4  p d vdfzwkc xg q r x r y r ps    (31) 0)( max 222 5  p d vdfzwkc xg q r x r y r ps    (32) 0)( max6  p d vdfzwkc xg q s x s y s ps    (33) where cλ, kλ, xλ, yλ, sλ, pλ and qλ are various constants and exponents, vr1, vr2 and vs are the cutting speed for the first rough pass, second rough pass and finish pass, respectively, and pmax is the maximum allowable machining power. (d) constraints for surface finish: 00321.0)( max 2 1 7  r e r r r f xg (34) 00321.0)( max 2 2 8  r e r r r f xg (35) 00321.0)( max 2 9  s e s r r f xg (36) where re is the nose radius of the tool edge, and rr max and rs max are the required surface roughness values for rough passes and finish pass, respectively. optimization of multi-pass face milling parameters using metaheuristic algorithm 373 (e) tool life constraint: 0)( 1 111 10           r l psy r x rr q vv t zwfdv dkc xg vvvv v (37) 0)( 1 222 11           r l psy r x rr q vv t zwfdv dkc xg vvvv v (38) 0)( 1 12           r l psy s x ss q vv t zwfdv dkc xg vvvv v (39) where tr is the tool replacement life (in min) as specified by the machinist. (f) constraint for number of rough passes: the total depth of cut (dt) for multi-pass milling operation is estimated as the sum of n number of rough passes depth of cut and one finish pass depth of cut.    n i srit ddd 1 (40) while optimizing different machining parameters for a multi-pass milling operation, the optimal number of passes needs to be determined. based on the maximum depth of cut allowed in the roughing operation and depth of cut for a finish pass, the smallest number of rough passes (nmin) can be obtained as:           max max min r st d dd gintn (41) where dt is the amount of total material to be removed, dr max is the maximum allowable depth of cut for rough passes, ds max is the maximum allowable depth of cut for finish pass and gint(·) is the greatest integer operator. based on the minimum depth of cut allowed in the roughing operation and depth of cut for a finish pass, the largest number of rough passes (nmax) can be derived as:           min min max r st d dd sintn (42) where dr min is the minimum allowable depth of cut for rough passes, ds min is the minimum allowable depth of cut for finish pass and sint(·) is the smallest integer operator. hence, the number of rough milling passes (n) required for machining the given part geometry can be estimated while considering a specific integer number between nmin and nmax. thus, the first objective function is formulated as:            n i psri ttttxf 1 1 min)min()( (43) 374 s. diyaley, s. chakraborty the second objective function is given by:            n i psri tkcccxf 1 02 min)min()( (44) the objective function for simultaneous optimization of production time and production cost (multi-objective optimization) is developed as below: min2 2 2 min1 1 13 )( )( )( )( )min()( xf xf w xf xf wzxf  (45) where f1(x)min and f2(x)min are the minimum values of production time and production cost, respectively achieved during single objective optimization, and w1 and w2 are the weights (relative importance) assigned to production time and production cost, respectively. 4. metaheuristic algorithms metaheuristic-based optimization is a competent method for solving hard optimization problems in order to provide acceptable solutions in a reasonable time. the suffix ‘meta’ in metaheuristic refers to upper level methodology, and the word ‘heuristic’ denotes a new approach to solving problems which helps in guiding design of underlying mathematical approaches to solve specific optimization problems [18]. these optimization methods utilize algorithms which are simple, flexible and well accepted for fast solving of large scale problems. they can be applied to any problem that can be formulated as function optimization problems and can be effectively hybridized with other optimization techniques. their major drawbacks are that most of them require fine-tuning of some algorithm specific parameters so as to obtain global optimal solutions; besides, they do not guarantee in repeatability of the optimal results with the same set of initial conditions [19]. however, their ability to solve large scale optimization problems easily and efficiently makes them a popular choice among the researchers for solving complex optimization problems. the two main components of metaheuristic algorithms are intensification and diversification [20]. intensification refers to focusing on the search process in a local region while exploiting the information that a current good solution has been found out in this region, while diversification signifies the process of producing distinct solutions so as to explore the search space on a global scale. a suitable balance needs to be maintained between these two components which is crucial to preserve the overall efficiency and performance of the algorithm; otherwise, it may cause the system to be trapped in local optima and increase difficulty in convergence [21]. these algorithms usually employ mechanisms inspired from social behavior, nature, physical laws, etc. to achieve near global solutions in the search space through efficient and comprehensive exploration. they can be classified into two broad categories based on the number of solutions used at the same time, i.e. single solutionand population-based metaheuristic algorithms. in single solution-based algorithms, only one single solution is considered at a time, whereas, population-based algorithms work with a population of solutions to create new solutions. greedy randomized adaptive search process (grasp), sa, tabu search (ts), etc. are some of the single solution-based algorithms. on the other hand, aco, evolutionary algorithms, like ga, de, etc. are few examples of population-based optimization of multi-pass face milling parameters using metaheuristic algorithm 375 metaheuristics. the metaheuristic algorithms are also categorized into four major groups based on their origin, e.g. bio-inspired, swarm intelligence-based, physics/chemistry-based and others [22]. ga and de are examples of the bio-inspired algorithms as they imitate the biological system of nature. swarm intelligence-based algorithms mimic the collective behavior of a group of animals, like a flock of bird or fish, and insects, like bees, ants, wasps and termites. the most popular swarm intelligence-based algorithm is pso technique which is also based on swarm behavior, like bird flocking and fish schooling. a cuckoo search algorithm is inspired by the process of breeding behavior of some species of cuckoo birds that lay eggs in the nests of host birds. the fa technique is another swarm intelligence-based algorithm that employs flashing behavior of swarming fireflies. the aco technique is based on the behavior of ants searching a path between the source of food and their colony. echolocation phenomenon of foraging bats is used by bat algorithm and foraging behavior of honey bees is employed in abc algorithm. thus, the phenomenon of ethology is employed in swarm intelligence-based algorithms [23]. physics and chemistry-based metaheuristic algorithms are developed based on certain physical or chemical phenomena as observed in the universe. the sa, gravitational search algorithm and curved spaced optimization are some of the examples of physics-based optimization techniques. there are also certain metaheuristic algorithms inspired by human behaviors, like tlbo, imperialistic competitive algorithm, ts, firework algorithm etc. in this paper, six metaheuristic algorithms, in the form of abc, aco, pso, fa, de and tlbo, are applied to solve the multi-objective mathematical model for a multi-pass face milling operation in order to determine the optimal values of cutting speed, feed rate and depth of cut for roughing and finishing operations. to achieve their best performance, the values of different algorithm-specific parameters are appropriately chosen, as provided in table 1. all these algorithms are coded in matlab 2013a, and run in 4.00 gb ram 2.9 ghz processor and 32-bit operating platform. these six algorithms are simple and easy to implement with high computational efficiency. they have already been extensively applied to solve many engineering optimization problems due to their exceptional ability to search near optimal or global optimal solutions. table 1 tuning parameters for six metaheuristic algorithms metaheuristic parameter abc algorithm number of iterations = 300, swarm size = 200, number of employed bees = 50% of the swarm size, number of onlooker bees = 50% of the swarm size, number of scouts per cycle = 1, number of cycles = 1000 and limit = 50. aco algorithm number of iterations = 300, sample size = 40, intensification factor = 0.5 and deviation distance ratio = 1. pso algorithm number of iterations = 300, population size = 200, inertia weight factor = 0.65, and acceleration coefficients = 1.65 and 1.75. fa number of iterations = 300, number of fireflies = 200, light absorption coefficient = 1, initial randomness = 0.9, randomness factor = 0.91 and randomness reduction = 0.75. de algorithm number of iterations = 300, population size = 200, lower bound of scaling factor = 0.5, upper bound of scaling factor = 0.8, crossover probability = 0.9. tlbo algorithm number of iterations = 300 and population size = 200. 376 s. diyaley, s. chakraborty 5. metaheuristics-based optimization of a multi-pass milling process as already mentioned, this paper deals with the application of six metaheuristic algorithms, i.e. abc, aco, pso, fa, de and tlbo techniques for parametric optimization of a multipass face milling operation. based on these algorithms, the optimal values of cutting speed, feed rate and depth of cut are determined with the aim to minimize total production time and total production cost separately. a multi-objective optimization model is also developed and subsequently solved using those algorithms while simultaneously minimizing both the objectives. thus, eqs. (43)-(45) are optimized subject to various practical machining constraints. the values of different machining data, constants and exponents, as used in the constraints, are provided in table 2. based on eqs. (41)-(42) and taking the value of total depth of cut (dt) as 8 mm, it is found that for the given part configuration, two rough passes are required, followed by one finish pass. table 2 values of various machining data [9] symbol and unit numerical data workpiece (grey cast iron 190 hb) length l (mm) 230 width w (mm) 90 face-milling cutter (cemented carbide) diameter d (mm) 160 number of teeth z 16 nose radius of the cutting edge re (mm) 1 milling cost and constraints overhead cost k0 (usd/min) 0.5 cost of tool material kt (usd/edge) 2.5 tool changing time ttc (min/edge) 1.5 preparation time tp (min/edge) 0.75 tool travel time h1 (min/mm) 0.00007 tool advance/depart time h2 (min) 0.3 depth of cut (roughing) [dr min, dr max] (mm) [2.0,4.0] depth of cut (finishing) [ds min, ds max] (mm) [0.5,2.0] feed rate [fmin, fmax] (mm/tooth) [0.1,0.6] cutting speed [vmin, vmax] (m/min) [50, 300] tool replacement life tr (min) 240 surface mean roughness rr max (µm) and rs max (µm) 25.0 and 2.5 force fmax (n) 8000 power pmax (kw) 8 constants and exponents in face milling equation constants/exponents tool life: cv = 445, kv = 1.0, xv= 0.15, yv = 0.35, pv = 0, qv = 0.2, sv = 0.2, l = 0.32 force: cu = 534.6, ku = 1, xu = 0.9, yu = 0.74, pu = 1, qu = 1.0, su = 1.0 power: cλ = 0.5346, kλ = 1, xλ = 0.9, yλ = 0.74, pλ= 0, qλ = 1, sλ = 1.0 optimization of multi-pass face milling parameters using metaheuristic algorithm 377 the results of solving the single objective optimization problems for minimum total production time and total production cost using the six adopted metaheuristic algorithms are exhibited in tables 3 and 4, respectively. it is interestingly observed that these algorithms successfully satisfy all the considered practical constraints, and tlbo algorithm supersedes the others with respect to both the minimum total production time and total production cost. for this algorithm, the minimum total production time is derived as 0.9805 min and the minimum total production cost is obtained as 1.44875 usd for the given part geometry. tables 3 and 4 also provide the optimal values of cutting speed, feed rate and depth of cut for both the roughing and finishing operations for all the considered algorithms. in fig. 2, the convergence diagrams for total production time and total production cost as the objective functions are shown. it can be clearly revealed from this figure that the tlbo algorithm reaches the minimum values for both these objectives quite quickly as compared to others. actually, this algorithm takes only 5-10 iterations to arrive at the minimal values. amongst the six algorithms, the optimization performance of fa technique is observed to be the worst. table 3 results for minimum total production time variable unit range/ limit abc aco pso fa de tlbo cutting parameters vr1 m/min 50-300 85.3967 85.2367 85 85.3349 85.3843 85.4 vr2 m/min 50-300 65.4576 65.4147 65.5 65.2577 65.4663 65.5 vs m/min 50-300 137.0874 137.0199 137 137.0848 137.0015 137.1 fr1 mm/tooth 0.1-0.6 0.6020 0.6020 0.6 0.6082 0.6001 0.6 fr2 mm/tooth 0.1-0.6 0.6001 0.5999 0.6 0.6013 0.6032 0.6 fs mm/tooth 0.1-0.6 0.2514 0.2552 0.2550 0.2560 0.255 0.252 dr1 mm 2-4 2.2731 2.2726 2.2811 2.2823 2.2957 2.27 dr2 mm 2-4 2.3173 2.3424 2.3 2.347 2.3359 2.3 ds mm 0.5-2 0.593 0.5928 0.5920 0.593 0.593 0.592 constraints g1(x) ≤ 0 -1079.9 -1081 -1105.12 -1001.56 -1034.05 -1105.12 g2(x) ≤ 0 -975.905 -908.391 -1023.17 -883.594 -897.329 -1023.17 g3(x) ≤ 0 -6908.06 -6906.09 -6908.06 -6903.22 -6906.4 -6908.06 g4(x) ≤0 -7963.07 -7963.14 -7963.37 -7962.67 -7962.83 -7963.2 g5(x) ≤ 0 -7971.26 -7971.01 -7971.44 -7970.97 -7970.94 -7971.44 g6(x) ≤0 -7850.31 -7850.11 -7850.29 -7849.65 -7850.17 -7850.4 g7(x) ≤ 0 -0.02355 -0.02355 -0.02352 -0.02355 -0.02355 -0.02355 g8(x) ≤ 0 -0.00427 -0.00428 -0.00432 -0.00441 -0.00485 -0.00432 g9(x) ≤ 0 -0.02474 -0.02474 -0.02474 -0.02474 -0.02474 -0.02474 g10(x) ≥ 0 55.503 57.191 61.04 52.239 55.21 56.655 g11(x) ≥ 0 434.4418 432.6057 435.5096 435.3286 427.8060 435.5096 g12(x) ≥ 0 83.6276 83.6108 83.5022 81.9771 83.9732 84.2407 production time min 0.9809 0.9811 0.9808 0.9826 0.9812 0.9805 378 s. diyaley, s. chakraborty table 4 results for minimum total production cost variable unit range /limit abc aco pso fa de tlbo cutting parameters vr1 m/min 50-300 85.3765 85.2473 85.3497 85.3545 85.3946 85 vr2 m/min 50-300 65.1546 65.016 65.2144 65.2265 65.2465 65.5 vs m/min 50-300 137.0030 137.0905 137.0129 137.0197 137.0746 137 fr1 mm/tooth 0.1-0.6 0.5998 0.5998 0.5941 0.612 0.6 0.6 fr2 mm/tooth 0.1-0.6 0.6 0.6003 0.6030 0.6005 0.6 0.6 fs mm/tooth 0.1-0.6 0.2553 0.2550 0.2553 0.2554 0.2554 0.2555 dr1 mm 2-4 2.2956 2.2922 2.296 2.285 2.2714 2.2712 dr2 mm 2-4 2.3191 2.3440 2.3591 2.3459 2.3004 2.3 ds mm 0.5-2 0.5928 0.5928 0.5929 0.5925 0.593 0.592 constraints g1(x) ≤0 -1043.77 -1038.46 -1038.38 -1053.87 -1105.12 -1105.12 g2(x) ≤0 -971.315 -900.531 -835.63 -893.6 -1022.07 -1023.17 g3(x) ≤0 -6906.73 -6906.73 -6905.61 -6905.96 -6906.4 -6906.4 g4(x) ≤0 -7962.88 -7962.91 -7962.86 -7962.95 -7963.2 -7963.27 g5(x) ≤0 -7971.38 -7971.15 -7970.8 -7971.03 -7971.54 -7971.44 g6(x) ≤0 -7850.22 -7850.12 -7850.08 -7850.09 -7850.09 -7850.4 g7(x) ≤0 -0.02356 -0.02355 -0.02356 -0.02356 -0.02356 -0.02356 g8(x) ≤0 -0.00386 -0.00392 -0.00385 -0.00405 -0.00432 -0.00432 g9(x) ≤0 -0.02474 -0.02474 -0.02474 -0.02474 -0.02474 -0.02474 g10(x) ≥0 55.8935 56.5830 55.8896 55.6956 56.7142 61.04004 g11(x) ≥0 444.4484 444.8605 433.6894 437.47 443.6846 435.5096 g12(x) ≥0 84.01337 83.3675 83.4983 83.4119 83.4336 84.2407 production cost usd 1.44892 1.44912 1.44916 1.44917 1.44882 1.44875 (a) (b) fig. 2 convergence diagrams for six metaheuristic algorithms on the other hand, the boxplots exhibiting the variability in the values of total production time and total production cost for the six algorithms are depicted in fig. 3. a higher length of the box represents more variability in the derived solutions. it can be optimization of multi-pass face milling parameters using metaheuristic algorithm 379 noticed from this figure that the tlbo algorithm provides more consistent solutions for both the objective functions with low variability. the de algorithm performs worst with quite high variability in the obtained solutions. thus, it can be concluded that the performance of the tlbo algorithm is superior to the remaining optimization techniques with respect to accuracy and consistency of the derived solutions as well as convergence speed. in order to validate the uniqueness of the tlbo algorithm, and identify the presence of any significant difference between the optimization performance of the considered metaheuristic algorithms, two statistical tests, i.e. paired t-test (parametric test) and the wilcoxson signed rank test (non-parametric test) are now performed with the following hypotheses:  h0 (null hypothesis): population means for two algorithms are equal (µ1 = µ2).  hα (alternate hypothesis): population means for two algorithms are not equal (µ1 ≠ µ2). (a) (b) fig. 3 boxplots for the considered metaheuristic algorithms in these hypotheses, the population mean signifies the average value of each of the objective functions calculated after 300 iterations for all the metaheuristics considered. the results of the paired t-test for total production time and total production cost are shown in tables 5 and 6, respectively. for both the objectives, as the absolute values of tstatistic for the paired comparisons between the tlbo and other algorithms are higher than the corresponding critical values of t at 5% level of significance, the null hypotheses for all the paired t-tests can thereby be rejected. similarly, tables 7 and 8, respectively, show the results of the wilcoxson signed rank test with the calculated p-values for all the pair-wise comparisons between the tlbo and remaining algorithms for both the objectives. the p-values are observed to be less than 0.05 at 5% level of significance for all the paired comparisons. the results of these statistical tests thus again prove the uniqueness and best optimization performance of the tlbo algorithm against the other metaheuristic algorithms. 380 s. diyaley, s. chakraborty table 5 results of paired t-test for total production time abc aco pso fa de -851.188 -1608.38 -155.843 -206.373 -869.86 table 6 results of paired t-test for total for production cost abc aco pso fa de -293.9 -1103.97 -1012.78 -12.83 -1383.58 table 7 results of the wilcoxson signed rank test for total production time abc aco pso fa de 0.0002 0.0001 0.0007 0.0008 0.0003 table 8 results of the wilcoxson signed rank test for total production cost abc aco pso fa de 0.0014 0.0018 0.0024 0.0028 0.0037 for multi-objective optimization of both the total production time and total production cost, the objective function of eq. (45) is now optimized using all the six metaheuristics. in this case, equal importance (weight) is provided to both the objectives. table 9 exhibits the results of this multi-objective optimization which again proves the excellent performance of the tlbo algorithm as compared to others. the minimum value of objective function (z) is observed to be 1.00007 with the total production time as 0.9552 min and total production cost as 1.4220 usd while satisfying all the given machining constraints. the optimal values of cutting speed, feed rate and depth of cut for roughing and finishing operations for the considered multi-pass face milling operation would thus simultaneously minimize both the total production time and total production cost. the corresponding convergence diagram and boxplots, as shown in fig. 4, also signify the best optimization performance of the tlbo algorithm. the results of paired t-test and wilcoxson signed rank test, as provided in tables 10 and 11, respectively, prove that the optimization performance of the tlbo algorithm is unique and best among all the considered metaheuristic algorithms. thus, it can be applied as an almost global optimization tool for parametric optimization of a multi-pass milling process. as already mentioned, this paper deals with comparing the optimization performance of six well known metaheuristic algorithms in order to determine minimum total production cost and total production time in a multi-pass face milling operation. each algorithm has its unique characteristics, and before its execution, its different parameters must be properly tuned; otherwise, it may lead to local optimal solution. these parameters are considered to be the vital elements in any algorithm. due to stochastic behavior of these algorithms, each algorithm generates different results in each run. therefore, in this paper, the best solution is determined after performing 30 independent runs for each algorithm. the best solution determined by each algorithm is recorded and subsequently compared. the process of performing independent runs for each algorithm thus reduces randomness in the derived results. optimization of multi-pass face milling parameters using metaheuristic algorithm 381 table 9 results of multi-objective optimization for multi-pass milling process variable unit range/ limit abc aco pso fa de tlbo cutting parameters vr1 m/min 50-300 85.38 85.2145 85 85.1619 85.3947 85.4 vr2 m/min 50-300 65.3296 65.4622 65.5 65.2529 65.4988 64 vs m/min 50-300 137.0614 137.0311 137.1 137.0862 137.0993 137 fr1 mm/tooth 0.1-0.6 0.5988 0.6 0.6 0.6 0.512 0.411 fr2 mm/tooth 0.1-0.6 0.6 0.6 0.6 0.6 0.6 0.6 fs mm/tooth 0.1-0.6 0.2557 0.2554 0.255 0.256 0.255 0.255 dr1 mm 2-4 2.2838 2.2903 2.27 2.2925 2.271 2.227 dr2 mm 2-4 2.3698 2.3773 2.3 2.35 2.38 2.3 ds mm 0.5-2 0.5926 0.5921 0.592 0.5929 0.5924 0.501 constraints g1(x) ≤0 -1069.12 -1049.65 -1105.12 -1143.64 -1868.67 -2877.73 g2(x) ≤0 -832.893 -812.481 -1023.17 -886.809 -805.135 -1023.17 g3(x) ≤0 -6904.84 -6906.62 -6908.06 -6903.39 -6907.39 -7060.35 g4(x) ≤0 -7963.02 -7962.98 -7963.37 -7962.97 -7967.28 -7972.66 g5(x) ≤0 -7970.74 7970.59 -7971.44 -7970.97 -7970.55 -7972.09 g6(x) ≤0 -7849.9 -7850.17 -7850.29 -7849.67 -7850.2 -7871.27 g7(x) ≤0 -0.02356 -0.02356 -0.02356 -0.02356 -0.02395 -0.02432 g8(x) ≤0 -0.00407 -0.02356 -0.00432 -0.00393 -0.00432 -0.0051 g9(x) ≤0 -0.02474 -0.02474 -0.02474 -0.00247 -0.02474 -0.02474 g10(x) ≥0 56.13855 57.43446 61.04004 57.8748 112.9195 212.7508 g11(x) ≥0 431.5535 426.3241 435.5096 435.3978 424.8074 486.2283 g12(x) ≥0 82.6644 83.5022 83.5022 81.9923 83.4049 110.6262 production time min 0.98090 0.98083 0.9808 0.9812 0.9677 0.9552 production cost usd 1.44892 1.44894 1.4487 1.4489 1.4349 1.4220 z 1.00033 1.00036 1.000241 1.00154 1.00019 1.00007 table 10 results of paired t-test for multi-objective optimization abc aco pso fa de -87.580 -78.704 -88.073 -618.248 -22.8169 table 11 results of wilcoxson signed rank test for multi-objective optimization abc aco pso fa de 0.0024 0.0021 0.0032 0.0031 0.0011 382 s. diyaley, s. chakraborty (a) (b) fig. 4 convergence diagram and boxplots for multi-objective optimization 6. conclusions in this paper, an endeavor is put forward for parametric optimization of a multi-pass face milling operation using six most popular metaheuristic algorithms. these algorithms are employed to determine the optimal values of cutting speed, feed rate and depth of cut for both roughing and finishing operations of the considered milling process while minimizing total production time and total production cost. based on the derived results, it can be revealed that the application of the tlbo algorithm estimates the minimum total production time and total production cost as 0.9805 min and 1.44875 usd, respectively, for the considered part geometry. as compared to the remaining algorithms, it converges quite quickly towards the almost global solution and the derived solutions are highly consistent with minimum variability. the applications of the paired t-test and the wilcoxson signed rank test also confirm its superiority over the other algorithms. the results of multi-objective optimization of the considered multi-pass milling process using the tlbo algorithm show that the values of total production time as 0.9552 min and total production cost as 1.4220 usd are obtained while simultaneously minimizing both the objectives. thus, it can be concluded that the tlbo algorithm can be employed as an effective tool for both single and multi-objective optimization of any of the machining processes. references 1. shunmugam, m.s., bhaskara reddy, s.v., narendran, t.t., 2000, selection of optimal conditions in multipass face-milling using a genetic algorithm, international journal of machine tools and manufacture, 40(3), pp. 401-414. 2. tolouei-rad, m., bidhendi, i.m., 1997, on the optimization of machining parameters for milling operations, international journal of machine 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metaheuristics in combinatorial optimization, annals of operations research, 140(1), pp. 189-213. 23. fister, jr.i., yang, x.s., fister, i., brest, j., fister, d., 2013, a brief review of nature-inspired algorithms for optimization, elektrotehniski vestnik, 80(3), pp.1-7. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 3, 2016, pp. 329 334 doi: 10.22190/fume1603329b original scientific paper numerical study of stress-strain localization in the titanium surface modified by an electron beam treatment udc 531.3 ruslan balokhonov 1,2 , varvara romanova 1 , alexey panin 1 , sergey martynov 1 , marina kazachenok 1 1 institute of strength physics and materials science, tomsk, russian federation 2 national research tomsk polytechnic university, tomsk, russian federation abstract. numerical simulation is performed to investigate the mesoscale stress-strain localization in a surface-modified commercial titanium alloy. the calculated crystalline microstructure corresponds to that observed in experiments and is accounted for in an explicit way as initial conditions of a dynamic boundary-value problem. the latter is stated in terms of plane strain developing in microstructure subjected to tension and is solved numerically by the finite-difference method. elastic-plastic constitutive models were built to describe the experimental mechanical response both of the substrate and of the modified layer. plastic strain localization is found to depend on the grain yield strength. key words: microstructure-based numerical simulation, plastic strain localization, titanium alloys, surface-modified materials 1. introduction stress-strain localization phenomena have been much studied both experimentally and theoretically (see, for instance, a review [1]) and may be due to different physical processes operating at different scale levels. at the microlevel, dislocations, slip bands, dislocation cells, fragmented structures, etc., are formed in single crystals and in grains of polycrystals. the macroscopic stress-strain localization may be due to the geometry of structure elements or specimens, or else, to loading conditions. in simulating localization effects observed in homogeneous specimens under loading, the form of the plastic received october 10, 2016 / accepted november 21, 2016 corresponding author: ruslan balokhonov institute of strength physics and materials science, russian academy of sciences, 2/4, pr. akademicheskii, 634055 tomsk, russia e-mail: rusy@ispms.tsc.ru 330 r. balokhonov, v. romanova, a. panin, s. martynov, m. kazachenok potential [2] generally plays a decisive role. the softening rate was a decisive factor for the onset of localization [3]. to describe the final fracture in the form of strain localization in a polycrystalline material, there was developed a crystal plasticity model with a hardening law accounting for slip and twin systems [4]. it was shown in [5, 6] that a large deformation gradient theory can provide a localized shear band thickness in the neck, no matter what the computational grid size is. a non-local constitutive formulation including a gradient internal length of poroplastic materials was given in [7] to derive analytical expressions of the critical hardening/softening modulus for localized failure. since the mid-1980s the mesoscopic stress-strain localization associated with different interfaces were studied numerically in many papers where an explicit account is taken of the material microstructure (see, e.g. [8-12]. in our earlier works we investigated stress-strain localization processes in polycrystalline materials [13, 14], metal-matrix composites [15, 16] and materials with coatings [17, 18]. it has been shown numerically that the curvature of the coating-substrate interface is responsible for the stress concentration that gives rise to plastic strain localization in metal substrates and causes fracture of ceramic coatings. the level of local stresses near an interface of certain curvature depends on the difference in the mechanical properties between the contacting materials at the interface. an explicit account of microstructure of coated titanium was not considered in the literature yet. the aim of the present work is to simulate numerically the evolution of the mesoscale stress concentration and plastic strain localization in surface modified commercial vt1-0 titanium samples subjected to tension, with the respective polycrystalline and submicrocrystalline structures of the titanium substrate and modified surface layer being taken into account. 2. formulation of the problem and experiment use was made of low-energy high-current electron beam (lehceb) treatment to provide a combination of melting and fast crystallization of a thin surface layer of vt1-0 titanium samples. the crystalline microstructure and modified surface layer thickness were found to depend on the energy density, with a curvilinear interface being formed between the substrate and the modified surface layer (fig. 1a) [19]. the experiments under consideration included the following stages. dumb-bell test pieces with a gage section of 2112 mm were cut out by wire electrical discharge machining. the test pieces were annealed in a vacuum chamber at a temperature of 750c for 1 h and polished mechanically. uniaxial tension of the titanium samples was performed in an instron 5582 tensile testing machine at room temperature and a loading rate of 0.3 mm/min. the stress-strain behavior is shown by dots in fig. 1b. the flat surfaces of the samples were exposed to three 50 µs pulses using a solo electron beam system (institute of high current electronics sb ras, tomsk, russia). the beam energy density was 18 j/cm 2 and the pulse repetition frequency was 0.3 hz. irradiation was performed in argon atmosphere with a residual pressure of 0.02 pa. a jem-2100 transmission electron microscope (tem) was employed to examine the microstructure of the samples before and after the electron beam processing. the hardness and elastic moduli of the titanium substrate and modified surface layer were measured with a nanotest nanotester equipped with berkovich pyramid. the calculations relied on an explicit account of the experimental microstructure (fig. 1c–d). the associated tension tests were simulated. the dynamic boundary-value problem http://www.multitran.ru/c/m.exe?t=6996605_1_2&s1=%e2%20%f1%e5%f0%e5%e4%e8%ed%e5%20%e2%ee%f1%fc%ec%e8%e4%e5%f1%ff%f2%fb%f5%20%e3%ee%e4%ee%e2%20%ef%f0%ee%f8%eb%ee%e3%ee%20%e2%e5%ea%e0 numerical study of stress-strain localization in titanium surface modified by an eelectron beam treatment 331 represented in the plane strain formulation was solved numerically by the finite-difference method [17]. elastic-plastic constitutive models were built to describe the experimental mechanical response both of the substrate and of the surface layer. the isotropic strain hardening function confines the equivalent stress: / 0( ) p p eq r eq s s         , (1) where p eq is the equivalent plastic strain, s and 0 are the ultimate stress and yield stress, and p r is a representative value of the equivalent plastic strain. to allow for the stochastic orientation of crystallites in the modified layer relative to the loading direction, the yield stress, elastic shear, and bulk moduli were varied in a random way from one crystallite to the other within the ranges presented in table 1. the average yield strength of the crystallites was calculated as the substrate yield strength multiplied by the ratio of surface layer-to-substrate hardness. the experimental ratio was found to be 1.25, whereas the ratio of surface layer-tosubstrate elastic moduli was 1.1. that is why different combinations of three values of the yield strength and shear () and bulk (k) moduli were used for the substrate grains (see table 1). the strain hardening function for grains characterized by average properties accounts for the experimental stress-strain behavior of the titanium substrate (fig. 1b). table 1 mechanical properties of grains and crystallites used in the calculations 0 [мpa] s [мpa] p r [ %] k [gpa]  [gpa] crystallites 113–338 388 0.13 73–102 52–74 grains 90, 180, 270 310 0.13 66, 79, 93 47, 57, 67 fig. 1 experimental (a) and simulated microstructures (b–c) of a commercial vt1-0 titanium substrate with a modified surface layer produced by means of lehceb surface irradiation and a plastic potential for a titanium sample (d) (d) grain 1 grain 2 grain 3 crystallites p eq [mpa] 0 2 4 6 8 200 250 300 experiment calculation p eq , % titanium modified surface layer titanium single grain crystallites x y (c) а) b) 332 r. balokhonov, v. romanova, a. panin, s. martynov, m. kazachenok the boundary conditions for the top and bottom surfaces correspond to a free surface and symmetry plane, respectively. the rightand left-hand surfaces simulate uniaxial elongation of the polycrystalline microstructures along the x-axis. 3. plastic strain localization at the mesoscale two sets of calculations were performed. one set deals with the elongation of a single titanium grain with a modified surface layer (fig. 1с). the yield strength of the substrate grain was 90, 180, and 270 mpa (table 1). the plastic strains were found to originate near the curvilinear interface between the substrate and the modified surface layer, giving rise to a system of slip bands that develop in conjugated directions and compensate for the local bending of the material (figs. 2a and 2b). in a qualitative sense, the same strain patterns are observed in the experiments (fig. 2c). the plastic strain localization is shown to be more pronounced in grains possessing higher resolved shear stress and stiffness than in those favorably oriented in the loading direction (figs. 2a and 2b). fig. 2 plastic strain relief in the tensile microstructure presented in fig. 1c with substrate yield strength of 270 (a) and 90 mpa (b) and an experimental tem image of the plastic strain localization in a titanium grain with a modified surface layer (c) fig. 3 equivalent stress patterns for the cases presented in fig. 2a and b a) c) 90 270 b) 90 270 a) b) numerical study of stress-strain localization in titanium surface modified by an eelectron beam treatment 333 note that the only difference between the cases presented in figs. 2a and 2b is the magnitude of the yield stress of the substrate grain, with all the other factors, among which are the microstructure and distribution of the mechanical properties of crystallites in the modified surface layer, being the same. despite this, the plastic strain patterns in the modified layer differ widely between the two cases at hand. for instance, there are crystallites and crystallite groups that experience large strains in one case or undergo no plastic flow at all in the other case (circles in figs. 2a and b), and vice versa (squares in figs. 2a and b). this is due to the difference in the plastic strain localization histories between the titanium substrate grains lying below the modified layer. for a titanium grain unfavorably oriented relative to the loading direction, the modified surface layer is exposed to low stresses (fig. 3a). plastic flow originates in the surface layer. on further loading the plastic strain localization pattern formed in the surface layer controls the plastic strain generation and localization in the substrate grain. other scenario is realized in the substrate grain possessing low yield stress, where the microcrystalline layer formed on the grain surface is hardened (fig. 3b). in this case, plastic flow originates in the grains lying near the interface between the substrate and the modified surface layer, and the plastic strain localization pattern in the grains depends on the degree of interfacial curvature. the other set of calculations is devoted to a simulation of deformation of the microstructure containing three grains in the substrate (fig. 4). the highest stress concentration and plastic strain localization are observed in a triple (hard-soft-hard grains) junction, and the plastic strain localization bands formed in hard grains are curved due to grain rotation caused by plastic straining of a soft grain lying in between (fig. 4a). fig. 4 plastic strain relief and stress patterns in the tensile microstructure given in fig. 1d b) а) c) 270 90 180 270 90 180 270 90 180 334 r. balokhonov, v. romanova, a. panin, s. martynov, m. kazachenok 4. conclusion stress-strain localization in surface modified titanium subjected to tension is investigated numerically. plastic strain patterns in a microcrystalline modified surface layer lying over titanium grains of different orientations relative to the loading direction are found to be dramatically different. plastic strain localization is most pronounced in titanium grains undergoing high resolved shear stresses, and the plastic strain localization bands assume a curvilinear form due to grain rotation. acknowledgements: the work is supported by russian science foundation (project no. 14-19-00766). references 1. antolovich, s.d., armstrong, r.w., 2014, plastic strain localization in metals: origins and consequences, progress in materials science, 59, pp. 1–160. 2. becker, r., needleman, a., 1986, effect of yield surface curvature on necking and failure in porous plastic solids, j. appl. mech., 53, pp. 491–499. 3. xue, l., 2010, localization conditions and diffused necking for damage plastic solids, engineering fracture mechanics, 77, pp. 1275–1297. 4. wang, y.y., sun, x., wang, y.d., hu, x.h., zbib, h.m., 2014, a mechanism-based model for deformation twinning in polycrystalline fcc steel, materials science & engineering a, 607, pp. 206–218. 5. borg, u., 2007, strain gradient crystal plasticity effects on flow localization. international journal of plasticity, 23, pp. 1400–1416 6. anand, l., aslan, o., 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elastoplastic behaviour of an artificial 3d-structure under dynamic loading, computational materials science, 28 (3-4), pp. 518–528. 16. balokhonov r.r., romanova v.a., schmauder s., 2006, computational analysis of deformation and fracture in a composite material on the mesoscale level, computational materials science, 37, pp. 110–118. 17. balokhonov, r.r., romanova, v.a., 2009, the effect of the irregular interface geometry in deformation and fracture of a steel substrate–boride coating composite, international journal of plasticity, 25 (11), pp. 2225-2248. 18. balokhonov r.r., zinovjev a.v., romanova v.a., bakeev r.a, zinovjeva o.s., 2016, numerical simulation of deformation and fracture in a material with a polysilazane-based coating, physical mesomechanics, 19 (4), pp. 430-440. 19. panin a.v., kazachenok m.s., kretova o.m., perevalova o.b., ivanov y.f., lider a.m., stepanova o.m., and kroening m.h., 2013, the effect of electron beam treatment on hydrogen sorption ability of commercially pure titanium, appl. surf. scie., 284, pp. 750–756. http://www.sciencedirect.com/science?_ob=articleurl&_udi=b7599-48cxgp5-fs&_user=10&_coverdate=10%2f31%2f1991&_alid=772263856&_rdoc=138&_fmt=high&_orig=search&_cdi=12949&_st=13&_docanchor=&view=f&_ct=257&_acct=c000050221&_version=1&_urlversion=0&_userid=10&md5=1b7a3f7d0e77483d04d2ae7767873d7c http://www.sciencedirect.com/science?_ob=articleurl&_udi=b7599-48cxgp5-fs&_user=10&_coverdate=10%2f31%2f1991&_alid=772263856&_rdoc=138&_fmt=high&_orig=search&_cdi=12949&_st=13&_docanchor=&view=f&_ct=257&_acct=c000050221&_version=1&_urlversion=0&_userid=10&md5=1b7a3f7d0e77483d04d2ae7767873d7c http://www.sciencedirect.com/science?_ob=articleurl&_udi=b6tw8-3ydg01n-7&_user=10&_coverdate=01%2f01%2f2000&_alid=794541343&_rdoc=9&_fmt=high&_orig=search&_cdi=5556&_sort=d&_docanchor=&view=c&_ct=10&_acct=c000050221&_version=1&_urlversion=0&_userid=10&md5=91e8121782a519eaaa223c342ccff127 http://www.scopus.com/record/display.url?eid=2-s2.0-68849095455&origin=resultslist&sort=cp-f&src=s&st1=balokhonov&sid=12304ef5325b7bb1ab9351666a3714b3.y7eslnddisn8ce7qwvy6w%3a20&sot=b&sdt=b&sl=42&s=author-name%28balokhonov%29+and+pubyear+%3e+2008&relpos=3&relpos=3&citecnt=5&searchterm=author-name%28balokhonov%29+and+pubyear+%26gt%3b+2008 http://www.scopus.com/record/display.url?eid=2-s2.0-68849095455&origin=resultslist&sort=cp-f&src=s&st1=balokhonov&sid=12304ef5325b7bb1ab9351666a3714b3.y7eslnddisn8ce7qwvy6w%3a20&sot=b&sdt=b&sl=42&s=author-name%28balokhonov%29+and+pubyear+%3e+2008&relpos=3&relpos=3&citecnt=5&searchterm=author-name%28balokhonov%29+and+pubyear+%26gt%3b+2008 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 1, 2018, pp. 77 86 https://doi.org/10.22190/fume180102012b © 2018 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper a numerical study of the microscale plastic strain localization in friction stir weld zones udc 539.4, 519.6 ruslan balokhonov 1 , varvara romanova 1 , ekaterina batukhtina 1 , maxim sergeev 2 , evgeniya emelianova 2 1 institute of strength physics and materials science, sb ras, tomsk, russia 2 tomsk state university, tomsk, russia abstract. a crystal plasticity approach was used to study the effects of grain shape and texture on the deformation behavior of friction stir weld (fsw) microregions. the explicit stress-strain analysis was performed for two representative grain structures with equiaxed and extended grains. grain orientations were assigned to simulate no texture or a weak or strong cubic texture. calculations have shown that the texture gave rise to earlier plastic strain localization on a larger scale. the highest stresses were found to develop in a non-textured specimen with equiaxed grains where the grain boundaries served as a barrier to dislocation motion. in both equiaxed and extended grain structures with a strong cubic texture no pronounced strain localization was seen on the grain scale but mesoscale shear bands appeared early in the deformation process. the calculations have shown that the microstructure-based simulation is a reasonable tool to study the deformation behavior of fsw materials, which is difficult to be predicted within macroscopic models alone. key words: friction stir welds, microstructure, texture, crystal plasticity simulation 1. introduction friction stir welding (fsw) [1] is an efficient way of joining metals with the use of severe plastic deformation and mechanical mixing of materials in the weld zone at temperatures below the melting point. the solid state of the joined metals is retained during fsw, which is an advantage of this technique over fusion welding. with the fsw received january 02, 2018 / accepted february 07, 2018 corresponding author: ruslan balokhonov institute of strength physics and materials science sb ras, 634055, tomsk, russia e-mail: rusy@ispms.tsc.ru 78 r. balokhonov, v. romanova, e. batukhtina, m. sergeev, e. emelianova process, it is possible to join materials otherwise difficult or impossible to be welded, specifically aluminum alloys. a peculiar feature of friction stir welds is a complex microstructure formed in different weld zones which drastically differs from that of the base metal [2-8]. three distinct regions are commonly observed in fsw joints where the microstructure and mechanical properties gradually vary with a distance from and a depth below the weld surface center line (fig. 1). the weld nugget (wn) zone is characterized by a fine or ultrafine equiaxed grain structure inherent in dynamic recrystallization [4, 5]. for different materials and fsw regimes, the grain orientations in the wn region were reported to vary from randomly distributed to those characteristic of a strong shear texture or a cubic texture [4-7]. a thermomechanically affected zone (tmaz) adjacent to the wn region demonstrates different microstructure patterns on the advancing and retreating sides of the rotated tool [2-8]. a well-defined elongated grain structure is formed on the retreating side. on the advancing side of the tmaz, columnar-shaped regions of stirringproduced fine grains alternate with the regions of grains inherent from the base metal. approaching the weld centerline, the grain columns tend to curve about the pin rotation trajectory. for many fsw materials, a two-component texture is shown to form in the tmaz regions. the heat affected zone (haz) has a microstructure similar to that of the base metal, with the mechanical properties being different from those of both base metal and tmaz due to a redistribution of precipitations [6, 9]. fig. 1 schematics of friction stir welding and typical microstructures observed in different fsw regions due to a peculiar microstructure, each fsw zone makes a specific contribution to the fsw material deformation and fracture behavior which is difficult to be predicted without consideration of the microstructural effects. although extensive numerical microstructurebased investigations of microand mesoscale deformation phenomena have been performed for many materials, we are aware of only few efforts made to analyze the microscale stress and strain evolution in fsw materials. in our recent work [9], the finite difference analysis of tensile loading was performed for a microstructure formed in the advancing side of the tmaz where fsw-affected grains acquired a quasi-rectangular shape and arranged in columns. this type of microstructure was referred to as an ordered microstructure, as distinct from the disordered polycrystalline a numerical study of the microscale plastic strain localization in friction stir weld zones 79 microstructure formed in the nugget and originally seen in the base material. the strength of the material was found to depend on the degree of ordering of the microstructure. a similar numerical analysis for a microstructure containing fine and coarse grains in the nugget and base materials and elongated grains in the tmaz was performed in [10]. maximum plastic strain localization and fracture sites were found to depend on the specimen strain and strength of the material. both sets of calculations [9, 10], however, were performed for rather idealized 2d microstructures, with the isotropic elastic-plastic models being used to describe the deformation response of individual grains. it is, therefore, a challenge to perform a more realistic 3d analysis of microscale stress and strain fields formed in different fsw microvolumes under loading with accounting for an anisotropic elastic-plastic material response on the grain scale. in this paper, we have investigated numerically the grain shape and crystallographic texture effects on the plastic strain localization in microstructures found in the selected fsw zones of an aluminum alloy. three-dimensional polycrystalline models were constructed to take into an explicit account the grain geometry and the elastic-plastic anisotropy of the facecentered cube (fcc) crystallites. 2. microstructure based simulation a large group of models implicitly accounting for the grain structure aims at describing a homogenized material response [11, 12]. another approach implies an explicit consideration of the grain structure and thus enables microscale stresses and strains to be estimated [13, 14]. going this way, we explicitly introduce three-dimensional microstructures found in different fsw zones in finite element calculations. the microstructure-based simulation procedure implies (i) the development of a micromechanical material model accounting for the grain geometry and constitutive behavior, (ii) the model implementation in a boundaryvalue problem complemented with initial and boundary conditions, and (iii) a numerical solution of the problem to study the evolution of stresses, strains, displacements, energy and other parameters of interest under loading. 2.1. microstructure model polycrystalline models with equiaxed and extended grains characteristic of the wn and tmaz, respectively, were generated on regular 3×10 6 element meshes using a method of step-by-step packing (ssp) [15, 16]. this method is based on a combination of analytical and simulation tools. first, a computational domain is discretized by a mesh, with coordinates being defined for nodal points. since the design of a microstructure is followed by simulation of its mechanical behavior, the discretization parameters are dictated by the numerical method to be further applied. in a general case, an arbitrarily-shaped computational domain can be discretized by a regular or irregular mesh. as input data, a number of grain seeds are distributed among the mesh elements, with each kind of seeds being associated with a certain analytical law of growth. the laws of seed distribution and growth are derived from experimental data to obtain a model microstructure with geometrical characteristics of the grains similar to real ones. at each further step in the ssp procedure, the volumes surrounding the seeds are incremented by preset values in accordance with the analytical laws of the grain growth. for each mesh element belonging to none of the grains it is checked 80 r. balokhonov, v. romanova, e. batukhtina, m. sergeev, e. emelianova whether coordinates of its central point fall within any of the incremented grain volumes. if so, the cell is considered to belong to this grain and excluded from further analysis. such a procedure is repeated until the grain structure covers the whole computational domain. to construct the equiaxed and elongated grain structures presented in fig. 2, 1600 and 450 grain seeds, respectively, were randomly distributed over a computational domain approximated by a 200×75×200 regular mesh with cubic elements [16]. in the former case (fig. 2a), all the grains grew at the same growth rate according to a spherical law, while the growth of extended grains (fig. 2b) obeyed an ellipsoidal law, with an aspect ratio of the ellipsoidal semi-axes being 1:3. a) b) c) fig. 2 polycrystalline models of 20075200 µm with equiaxed (a) and extended grains (b) and a local coordinate system associated with fcc crystallographic axes 2.2. constitutive description of grains it is common practice to describe the deformation behavior of a polycrystalline material in terms of the crystal plasticity theory where the polycrystal is treated as an aggregate of single crystals with different crystallographic orientations relative to a global coordinate system [11, 13, 14]. crystal plasticity models acquire a special significance for materials with a high degree of elastic-plastic anisotropy due to crystallographic texture or a limited number of slip systems [13]. plastic deformation inside the grains is assumed to occur by dislocation gliding in active slip systems, with the slip system being activated provided that the resolved shear stress becomes equal to a critical value. in a general case, the critical resolved shear stress is a function depending on the strain hardening, microstructure evolution, and etc. a large body of publications devoted to crystal plasticity simulations is available at present (see, e.g., overview [11]). particularly, a large group of crystal plasticity models were developed for aluminum alloys (e.g., [17]). the elastic response of grains is described by the generalized hooke's law for an anisotropic material written in a rate form. expressing the total strain rate tensor as the sum of the elastic and plastic parts, we get p ij e ijij    (1) and ( )p ij ijkl kl klc    , (2) where σij is the stress tensor and cijkl is the tensor of elastic moduli. a numerical study of the microscale plastic strain localization in friction stir weld zones 81 aluminum single crystals have an fcc crystal lattice characterized by a cubic symmetry. let us write the constitutive equations for an aluminum single crystal with respect to a local coordinate system xi associated with crystallographic axes ai as presented in fig. 2c. then, the constitutive equations take on the same form for all grains regardless of their orientations in the global coordinate system xi. due to the symmetry of the crystal lattice, the matrix of elastic moduli in eq. (2) contains 12 non-zero constants, with only three of them being independent. components of the plastic strain rate tensor are calculated in terms of the crystal plasticity theory as a summary slip on all active slip systems: ( )( ) ( )p ij i j j is m s m      (3) where si and mi are the slip direction and slip plane normal vectors in a slip system . the shear strain rate depends on the ratio between resolved shear stress () and threshold stress * () in this slip system: ( ) ( ) ( ) 0 *( / )       , ( ) ( ) ( ) i ij js m    . (4) here 0 is the reference slip rate taken to be the same for all slip systems and  is the strain rate sensitivity parameter. according to schmid’s law, a slip system is activated provided that the resolved shear stress () operative on the slip plane c is equal to or greater than threshold value * () . a large number of models were suggested in the literature to describe critical resolved shear stress * () with accounting for different strengthening mechanisms. the more effects are aimed at accounting for, the more fitting and physically-based parameters are required to be determined. moreover, the iterative methods employed in crystal plasticity calculations additionally increase computational costs. it is reasoned, therefore, to use simplified constitutive models with a reduced number of parameters where possible. in this paper, the following simple expression was used to describe an isotropic hardening of aluminum grains: ( ) ( ) ( ) * 0 b ak      , ( ) ( ) a t dt    , (5) where 0 () is the initial critical resolved shear stress of an aluminum single crystal, assumed to be the same for all slip systems. the second term in this equation defines the strain hardening: a () is the accumulated slip in the slip system  and k and b are the fitting constants. 2.3. finite element implementation the solution of a boundary-value problem with an explicit account of the microstructure requires for substantial computational resources. on the one hand, the considered microvolume must contain a sufficient number of structural elements (e.g. grains) to simulate microand mesoscale deformation processes as realistic as possible. on the other hand, structural elements and interface regions should be approximated in much detail to ensure acceptable accuracy of the solution. this necessitates the use of detailed meshes with a large number of elements. it is, therefore, important for the solution of micromechanical problems to minimize computational requirements without loss of information and solution accuracy. 82 r. balokhonov, v. romanova, e. batukhtina, m. sergeev, e. emelianova an approach that considerably reduces the computer memory, disk space, and computational time requirements suggests the solution of quasi-static problems in a dynamic formulation where equations of motion are solved instead of those of equilibrium in order to find corresponding displacement fields. this allows using explicit numerical methods which have significant advantages over implicit calculations from the viewpoint of computational capacity. the benefit of explicit methods becomes even more critical for solving nonlinear problems such as those concerned with the microstructure-based simulations. in this paper, the deformation behavior of model grain structures was calculated using the finite element software package abaqus/explicit. the equiaxed and extended grain models were imported in the abaqus/explicit by means of an input file containing an orphan fe mesh, element sets representing grains associated with different local orientations, and the material constitutive parameters appearing in eqs. (2)-(5). thus, the grains were associated with the same material but differed by orientations of the local coordinate axes relative to global coordinates, specified in terms of the euler angles. the material constitutive model, eqs. (2)−(5), written with respect to the local coordinates, has been imported into the abaqus/explicit by means of a vumat user subroutine, with the equations being solved by iterations. model parameters and loading conditions were chosen to provide close agreement of quasistatic and dynamic solutions. shear strain rate ( ) being in inverse proportion to the relaxation time was shown to mainly control the material strain rate sensitivity [18]. calculations for varied loading velocities have shown that the dynamic and static solutions are in a reasonable agreement provided that ( ) is approximately ten times as large as the total strain rate. the boundary conditions were formulated with respect to the global coordinate system xi (fig. 2a). the uniaxial tension was applied along the x1-axis in the case of the equiaxed grain structure and along the x1or x3-axis for the extended grain structure. to eliminate dynamic effects, the tension velocity in the both sets of calculations was gradually increased for a time necessary for the elastic wave to run over the computational domain more than 5 times and then kept constant. the top specimen surface was free of external forces and the bottom and lateral surfaces were treated as symmetry planes. 3. computational results 3.1. stress and strain fields in equiaxed grain structure for a polycrystal composed of equiaxed grains (fig. 2a), three sets of calculations were performed where the grain orientations were distributed at random or scattered about certain crystallographic direction within 5 and 22º to simulate a strong or weak {100}<100> cubic texture, respectively. in all calculations uniaxial tension was applied along the x1-axis. the equivalent stress and plastic strain fields calculated at the initial stage of plastic deformation in the textured and non-textured polycrystals are presented in fig. 3. the stress distributions in non-textured (fig. 3d) and weakly textured polycrystals (fig. 3e) exhibit significant microscopic non-uniformity on the grain scale. in both cases, the highest stress concentration is observed in the vicinity of the grain boundaries. the peak stresses depend on the degree of crystallographic misorientations of contacting grains. this is why a higher level of local stresses is observed in the non-textured material whose grains are characterized a numerical study of the microscale plastic strain localization in friction stir weld zones 83 by a wide range of crystallographic orientations. in the crystal with a strong cubic texture, (fig. 3f), equivalent stresses demonstrate quasi-uniform distributions. a few stress concentration zones are observed along the grain boundaries, with the stress magnitudes involved being merely twice as high as the average level. a) b) c) d) e) f) fig. 3 equivalent plastic strain (a–c) and stress fields (d–f) in non-textured (a and d), weakly textured (b and e), and strongly textured polycrystals (c and f) at an engineering strain of 1% the plastic strain distributions demonstrate two characteristic strain localization scales. smaller-scale strain localization zones are attributed to plastic shear strains developing along the grain boundaries, which gives rise to individual grain displacements and rotations. a distinct strain localization pattern such as this is seen in the non-textured sample (fig. 3a) and can hardly be found in the strongly textured material (fig. 3c). larger-scale strain localization is associated with the formation of shear bands running through the entire sample and involving the whole groups of grains in plastic deformation. the mesoscale bands seen on the top surface are oriented transversely to the tensile axis. on the lateral faces, the bands intersect at an angle of 45°. a well-pronounced mesoscale band pattern is known to be a peculiar feature of many aluminum alloys. 3.2. plastic strain localization in extended grain structure for a polycrystal consisting of extended grains (fig. 2b), a strong {100}<100> cubic texture was assigned wherein <100> crystallographic direction was chosen to be along the major grain axis. the deviations of the crystallographic axes of grains relative to this direction were specified within ± 5º. the polycrystal was subjected to uniaxial tension by two schemes: along and across major grain axis. 84 r. balokhonov, v. romanova, e. batukhtina, m. sergeev, e. emelianova a) b) c) d) e) f) fig. 4 equivalent plastic strains in the textured polycrystal composed of extended grains loaded along the x1(a–c) and x2-axes (d–f) at an engineering strain of 0.75% (a and d), 3.5% (b and e) and 7.5% (c and f) the plastic strain distributions seen in different tensile stages are presented in fig. 4. with both of the tensile schemes, the plastic strain distributions bear witness to the strong mesoscale plastic strain localization even on the early plastic flow stage. notably, the strain localization pattern is hardly affected by the grain boundaries. in other words, the latter are no obstacle to the plastic flow propagation. without giving illustrations to confirm the foregoing conclusion, it can be emphasized, however, that a stressed state analysis reinforces this statement. the equivalent stress fields exhibit a quasi-uniform pattern, and there are no distinct stress concentration zones even near the grain boundaries. the plastic deformation in the textured material develops at a much lower level of stresses than in a non-textured grain structure. the localized plastic deformation bands on the free surface are formed across the tensile direction irrespective of grain orientation, whereas on the lateral faces, the bands develop at 45 to the tensile direction. it should be noted that from the standpoint of the crystallographic texture, directions x1 and x3 appear to be equivalent. the major difference between the two cases of tension at hand is the relative orientation of the tensile and major grain axes. 4. conclusion numerical investigations of the plastic strain localization in the microstructures typical of different regions of aluminum friction stir welds. the grain shape and the texture effects were studied using the example of microstructures with equiaxed and extended grains assigned random orientations of crystallographic axes or a weak or strong cubic texture. a crystal plasticity model was applied to describe the elastic-plastic response of aluminum a numerical study of the microscale plastic strain localization in friction stir weld zones 85 grains. the boundary value problem in a dynamic formulation was solved by the explicit finite element method to investigate the deformation behavior of the model microstructures under uniaxial tension. calculations have shown that different fsw zones demonstrate distinct deformation response to loading due to peculiar combinations of the grain geometry and crystallographic texture. both grain shape and texture were found to affect the plastic strain localization patterns to a lesser or greater extent. in all cases, however, the presence of texture gave rise to earlier plastic strain localization on a larger scale. the highest stresses were found to develop in a non-textured specimen with equiaxed grains where the grain boundaries served as a barrier to dislocation motion. on the early deformation stage, the plastic strains in the equiaxed polycrystal mainly localized along grain boundaries, while the larger-scale strain localization in the form of shear bands involving whole groups of grains developed on a later deformation stage. in both equiaxed and extended grain structures with a strong cubic texture no pronounced strain localization was seen on the grain scale but mesoscale shear bands appeared early in the deformation process. regardless of the grain structure, the mesoscale shear bands run through the whole specimens across the tensile axes, that is, in agreement with experimental evidence for aluminum alloys [19]. in this paper we have considered only few of possible combinations of grain geometry and orientations peculiar for fsw materials with much more effects related to the fswproduced microstructures, textures and mechanical properties remaining beyond of our study. nevertheless, we can conclude that the microstructure-based simulation appears to be a reasonable tool to study the deformation behavior of fsw materials, which is difficult to be predicted within macroscopic models alone. another important problem disregarded in the paper is concerned with microcrack origination under the fsw processing. the damage problem is of critical significance for fsw applications although in some particular cases the presence of microflaws has negligible effect, e.g., on the adhesive and cohesive strength of internal interfaces [20]. acknowledgements: this work is supported by the russian foundation for basic research (grant no. 16-01-00469-a) references 1. nandan, r., debroy, t., bhadeshia, h.k.d.h., 2008, recent advances in 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n.t., sahoo, s.k., samajdar, i., dey, g.k., bhanumurthy, k., 2011, microstructure and microtextural studies of friction stir welded aluminium alloy 5052, mater. des., 32, pp. 1657–1666. 8. fonda, r.w., knipling, k.e., bingert, j.f., 2007, microstructural evolution ahead of the tool in aluminum friction stir welds, 58, pp. 343–348. ../../../users/varvara/appdata/users/varvara/appdata/local/e-library/aa7xxxx/s1005030210601181.htm#! ../../../users/varvara/appdata/users/varvara/appdata/local/e-library/aa7xxxx/s1005030210601181.htm#! ../../../users/varvara/appdata/users/varvara/appdata/local/e-library/aa7xxxx/s1005030210601181.htm#! ../../../users/varvara/appdata/users/varvara/appdata/local/e-library/aa7xxxx/s1005030210601181.htm#! 86 r. balokhonov, v. romanova, e. batukhtina, m. sergeev, e. emelianova 9. balokhonov, r.r., romanova, v.a., martynov, s.a., zinoviev, a.v., zinovieva, o.s., batukhtina, e.e., 2016, a computational study of the microstructural effect on the deformation and 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mesomechanical aspects of deformation-induced surface roughening in polycrystalline titanium, mater. sci. eng. a, 697, pp. 248-258. 19. wittridge, n., knutsen, r., 1999, a microtexture based analysis of the surface roughening behaviour of an aluminium alloy during tensile deformation, mater. sci. eng. a., 269, pp. 205–216. 20. popov, v.l., pohrt, r., li, q., 2017, strength of adhesive contacts: influence of contact geometry and material gradients, friction, 5, pp. 308–325. http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 http://www.sciencedirect.com/science/article/pii/s0927025615006527 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 3, 2014, pp. 251 260 1situation assessment through multi-modal sensing of dynamic environments to support cognitive robot control udc 681.5 atta badii, ali khan, rajkumar raval, hamid oudi, ricardo ayora, wasiq khan, amine jaidi, nagarajan viswanathan intelligent systems research laboratory, school of computer science and electronic engineering, university of reading, united kingdom abstract. awareness of emerging situations in a dynamic operational environment of a robotic assistive device is an essential capability of such a cognitive system, based on its effective and efficient assessment of the prevailing situation. this allows the system to interact with the environment in a sensible (semi)autonomous / pro-active manner without the need for frequent interventions from a supervisor. in this paper, we report a novel generic situation assessment architecture for robotic systems directly assisting humans as developed in the corbys project. this paper presents the overall architecture for situation assessment and its application in proof-of-concept demonstrators as developed and validated within the corbys project. these include a robotic human follower and a mobile gait rehabilitation robotic system. we present an overview of the structure and functionality of the situation assessment architecture for robotic systems with results and observations as collected from initial validation on the two corbys demonstrators. key words: situation assessment, cognitive control, dynamic environments, humanrobot interaction, human-robot co-working and mixed-initiative taking 1. introduction situation assessment is that capability of a system, which provides for an awareness of emerging situations. in a robotic environment, this typically includes updates on the relevant static and dynamic states of entities (objects, persons, spaces) with which the robot may or may not be interacting at the time [1-5]. this assessment allows the robotic system to interact with the environment in a sensible manner without the need for the intervention of a (human) supervisor. an innovative generic situation assessment architecture for robotic systems has received october 23, 2014 / accepted november 25, 2014 corresponding author: atta badii school of systems engineering, university of reading, whiteknights, jj thomson building, rg6 6ay, uk e-mail: atta.badii@reading.ac.uk original scientific paper 252 a. badii, a. khan, r. raval, h. oudi, r. ayora, w. khan, a. jaidi, n. viswanathan been developed as part of the european corbys project (ec fp7) [10]. this paper presents the overall architecture for this situation assessment capability and its application in proof-ofconcept demonstrators as developed and validated in the corbys project. these demonstrators include a gait rehabilitation robotic system and a robotic human follower (henceforth also referred to as demonstrator i and demonstrator ii respectively). the robotic human follower (corbys demonstrator ii) is an existing mobile robot that has been modified using the corbys system to autonomously follow a human coworker in exploratory settings. the mobile platform is equipped with sensors for perception of the environment including perception of the states and behaviour of humans in the ambient space. thus the corbys cognitive modules anticipate human behaviour in the environment and create appropriate inputs to the low-level controls so as to enable the robot to follow the human co-worker whilst avoiding obstacles. the mobile gait rehabilitation robotic system (demonstrator i) represents another very challenging application domain that has been developed as part of the corbys project. this consists of a mobile platform which hosts a powered robotic orthosis. the mobile platform facilitates mobility whereas the powered robotic orthosis assists a patient with their locomotion. the situation assessment architecture interprets the state and effort of the patient including the physical and psychological state. this interpretation is used to create appropriate commands for robot control adaptation. the corbys architecture enables this gait rehabilitation system to optimally support the rehabilitation requirements of the patient at different stages in a range of gait disorders. 2. situation assessment architecture the overall corbys system is comprised of four layers; the physical layer is at the bottom of the stack; this consists of the sensors and actuators that are plugged in to the corbys architecture, based on the specific application domain. the logical layers include the cognitive, executive and control layers. it is the cognitive layer where the situation assessment architecture resides. this architecture, the main focus of this paper, endows the robotic system with the cognitive capabilities by assessing the current states of the system and the environment (including humans) to inform decision making responsive to emerging situations by generating high-level inputs for the control system. 2.1. building situation awareness we define situation awareness as having knowledge about a particular situation. our architecture interprets data provided by the sensors and actuators coming from a robotic system to create such awareness. there are several factors a robot may need to be made aware of in an environment, therefore, we follow a suitably scoped but extensible approach in terms of cardinality of the states to be assessed using ontologies and formalise the definitions of context and situation in corbys. our definition of context for robotic applications is centred on the location of the robot in an environment and the entities (mainly objects but also co-workers) existing in such an environment and whose dynamic states would need to be continuously monitored and assessed. on the other hand, a situation is defined by events happening in a specific context. situation assessment through multi-modal sensing of dynamic environments to support cognitive robot 253 the context for a robot is the set of those predicates related to the spatio-temporally specific state vectors of the robot itself and the entities in its operational space relevant to the robot interactions in the recent past, present or immediate future. this includes all self-states such as space, position, place, location, resources, and the states of persons and objects with which the robot is interacting. the notion of context is a hierarchical (multilayered) abstraction space and its formulation and expression are strongly ontologically committed and have to be efficiently structured and de-limited to provide only the necessary and sufficient situation parametric values within a spatio-temporally logical analysis window so as to avoid computationally prohibitive complexity. we build a context by aggregating and interpreting the data coming from different sensors and actuators. therefore any information sensed from an environment becomes part of the context. a situation, on the other hand, is time and event critical and is highly dependent on the context. in other words, a situation relies on the current context and is defined by the co-occurrence of particular events happening over a temporal analysis window. in corbys, the gait rehabilitation robot uses the corbys cognitive architecture to control a powered orthosis to help patients with walking difficulties to overcome some of the challenges posed by their particular gait impairments. for this case, we analyse a patient’s gait to identify their specific locomotion difficulties as encountered in particular gait phases. fig. 1 the corbys situation assessment architecture 254 a. badii, a. khan, r. raval, h. oudi, r. ayora, w. khan, a. jaidi, n. viswanathan the results produced from the gait pattern analysis are then correlated with physiological data to track progress and state. the second corbys demonstrator is required to navigate in an environment and take deliberate actions responsive to its sensory input to detect obstacles and assert their presence to facilitate route planning and decision making. for demonstrator i, the location context remains the same given that the mobile robotic system with the powered orthosis will allow the patient to move about in a big hall space, however, the system and the human will not leave that space to enter a different location, e.g., an adjacent room. thus the focus in this demonstrator is on the representation of the human context that is interacting with the robot. this is based on the sensed gait motion, and the psycho-physiological states. given that each corbys demonstrator is intended for a different application domain; these would eventually come with different data requirements therefore we have provided for a flexible way to import new ontologies that can be domain-specific to extend the current general robot ontology. in this case although a context will reference something different in both demonstrators given the distinct application domains, the logical foundation, and the definitions remain re-usable throughout. the graphs which contain the built context and situations are then stored in a triple store which works as a long-term memory of experiences (lived experiences of situations created by the occurrence of events) and encounters (objects detected in an environment) by the robot. saving this data in memory, stored as a timestamped context cache, is crucial as it can be used later by the robot for offline processing or by the researcher for debugging and further research. the ontology works as a logical schema for the data that needs to be represented in a semantic graph, as well as provide a general definition in description-logic to support the dynamic binding and expression of contexts and situations as required by the demonstrators in this project. thus in corbys, having the data in such a semantic graph structure supports the process of coupling the inference information provided by the perception and comprehension modules with contexts and the identified situations. we start building contexts after identifying patterns of interest to give the robot an understanding of its environment to know which situations to expect or which ones are more likely to occur. the context built based on the logical definition in our ontology couples situations that occur over particular temporal analysis windows of interest. fig. 1 illustrates the final implemented version of the situation assessment architecture. the preceptors and state integrators process the raw device data to arrive at semantic knowledge upon which the context builder operates. this is used to arrive at the context and situations (that may exist at a given point in time) upon which to base the decision regarding environment impact. 2.2. framework architecture the modular situation assessment and decision sup-port system provides a hybrid architecture integrating machine learning and semantic technologies to enhance the robotic perception of low level (weak) data signals. it organises information as taxonomy of concepts conforming to knowledge as perceived and understood by humans as such remains open to integrating human-in-the-loop intervention if desired. therefore we make use of the decision support system to make inferences based on the information available for each situation and context. situation assessment through multi-modal sensing of dynamic environments to support cognitive robot 255 fig. 2 the situation assessment schema for the robotic follower in the corbys project the sensing modules or preceptors deal with processing sensory data signals. we identify patterns and information of interest using machine learning and data analysis to identify contexts. together with the state integrators, these build contexts from data acquired from sensors and select areas of interest based on the sought after patterns in data. the contexts are then persisted in memory and attached to situations where particular events of interest happen. the decision engine makes decisions based on multiple selected criteria given the situation at hand. the decision takes the form of an action that can be performed, e.g., stopping to avoid an obstacle, or adapting the orthotic actuation to better the (intended) movements of a patient in-session. although the two corbys demonstrators are different in their application, they share similarities in their data-driven modes of reasoning and the need to acquire ambient information to update the context and thus the situation assessment upon which to take the best action after careful analysis of the data and assessment of the situations at hand. the situation assessment schema with demonstrator ii as an example case can be seen in fig. 2. perception of the environment takes place based on the sensory data, followed by fusion in the comprehension phase, which allows for building of relations between perceived objects through fusion and inference to arrive at a richer context. this facilitates appropriate decision making which leads to an action execution. this is then fed back into the loop as it impacts the environment which is being sensed continuously. 256 a. badii, a. khan, r. raval, h. oudi, r. ayora, w. khan, a. jaidi, n. viswanathan 2.3. the human co-worker’s states pattern space one of the facets of the ambient pattern space as monitored by the system to decide what actions to take (e.g., adapting the current gait actuation level) is the psychophysiological state vector describing the person’s relevant conditions. this is built using heart rate, skin temperature, perspiration, posture and activity level. the applicability and usability of this facet span both corbys demonstrators. in demonstrator ii, the robotic follower benefits from an awareness of the interacting human’s physiological states in scenarios such as hazardous environment exploration. in demonstrator i, a brain computer interface is also employed, which processes electroencephalography (eeg) signals to detect intention of motion and attention to motion of the person undergoing gait therapy. this is particularly relevant in application domains requiring physical companion support responsive to (imminent) ambient conditions as is the case in gait rehabilitation. the architecture registers the time-stamped intended motion by the patient as detected through the bci module; such intended motion information is used to decide on the initial level for the patient’s supportive orthotic actuation. 2.4. static and dynamic activity recognition acceleration and rotation from an inertial measurement unit are used to detect static and dynamic activities including standing, walking, turning left and right. fig. 3 classification of static (standing) and dynamic (walking, turning) activities from inertial measurements (accelerometer and gyroscope): a) raw acceleration tri-axial signal (forward progression along the z-axis, all three axes are highly correlated), b) raw x-axis rotational signal from gyroscope, c) aratg feature extracted from gyroscope signal – turning activities can be distinguished from other activities (turning direction left for upward peaks and right for downward peaks, d) expectation-maximisation clustering results show linearly separable data space suited to a multi-class support vector machine (without kernel) situation assessment through multi-modal sensing of dynamic environments to support cognitive robot 257 activity classification is done using a multi-class support vector machine with features including the euclidean distance for walking and standing activities and the average rotation angles related to gravity direction [6] – also known as aratg – which simplifies the identification of left and right turns (see fig. 3). a classification accuracy of 88.73% with an average detection latency of 34µs is achieved in 10-fold cross validation. this facet also applies to both corbys demonstrators. the robotic follower becomes aware of the acceleration and orientation of the interacting human through this processing of the inertial measurements. in the robotic gait rehabilitation system, this information allows suitable adaptations in the gait trajectories depending on the activity (for instance, starting and stopping). 2.5. gait analysis (in corbys demonstrator i) it is the gait deviation analysis based on joint angles data that provides the information required for the gait rehabilitation system to make inferences regarding what supportive actuation needs to be applied at which point in the walk cycle to improve the person’s gait as best suited to the condition consistent with the clinical judgement of the gait therapist. fig. 4 hip joint angles (x-axis / sagittal movement) for one pathological gait cycle as extracted and segmented per phase by our gait analysis method. muscle activity in emg signal is also shown for the lower limb muscle groups: quadriceps, hamstring, tibialis and calf. top half of figure corresponds to the left leg while the bottom half depicts motion in the right leg. 258 a. badii, a. khan, r. raval, h. oudi, r. ayora, w. khan, a. jaidi, n. viswanathan a range of machine learning and classification techniques were evaluated with respect to the accuracy of their results in classifying the phases in a gait cycle based on the joint angles data. these included anns, naïve bayes, svms as well as unsupervised methods including expectation maximisation and k means using a joint angles dataset that comprised of gait trajectories for the hip, knee and ankle joints of both legs. trained classifiers were used to determine the optimal gait support that had to be provided to the patient as prescribed by the gait therapist based on clinical observation of the patients’ pathological gait. we also undertook phase classification (i.e. the determination of the walk cycle phase transition points) based on time domain processing as well as frequency domain spectral analysis of data which consisted of a number of statistical/time-domain features extracted from joint-angular motion data of the gait cycle. phase classification enabled the determination of the deviation in a patient’s gait from the optimal reference gait pattern at the sub-phase level. this approach has been tested with normal and pathological data demonstrating its suitability in classifying the phases of the gait regardless of type. gait analysis also entails muscle activation pattern detection and tracking against expected activation per previously reported studies [7]. the emg signal enables the detection of muscle activity orchestration per gait cycle which enables the determination of the expected muscle activation pattern (via a confidence measure) as exhibited in normal locomotion. the corbys system provides sensors for four muscles groups with electrodes placed on the vastus medialis (quadriceps), biceps femoris (hamstrings), tibialis anterior (anterior shank) and gastrocnemius (calf). in fig. 4, the trough (global minimum in this case) in sagittal movement (along the x-axis) of the right hip joint represents the start of the swing phase for the right leg. at this point, the left leg enters into the stance phase providing support for the right leg swing. conversely, leading up this trough in the right hip x-axis signal, the right leg is in stance phase supporting the swing of the left leg. the muscle activity analysis verifies that the muscle groups are activated consistent with gait motion; this is of interest to the therapist from the rehabilitation point of view. large-scale evaluation of gait analysis and muscle activity analysis are ongoing with a cohort of user groups as part of the corbys project evaluation activities. 2.6. decision making the decision engine determines the best suited action at each stage in each case based on reasoning over the situation assessment in given context. dempster-shafer method for evidence combination [8, 9] is used to combine information from several data sources to give a better overall picture of a system. this is done by combining beliefs given by several sensors or data points into a single set of beliefs. traditional probabilities are replaced with belief functions that provide a measure of certainty, for instance, regarding a classification label given to a particular instance. by combining several of these beliefs from different observers or sensors, we can be much more certain of the class label assigned. dempster shafer theory assigns a belief to each possible combination of states in a system. so a system with two states {a, b} will have values for {a}, {b], {a and b} and {}, the empty set. this is known as the power set of {a, b}. for a system with s unique states, there are 2 s belief assignments made by the system. if a combination of states is impossible, it is assigned a belief of 0. the empty set is also usually assigned a belief of 0. the dempster-shafer rule can be applied recursively without the combination situation assessment through multi-modal sensing of dynamic environments to support cognitive robot 259 order affecting the final results. thus to combine, for instance, a, b and c, we could first apply the rule to the masses of a and b, and then combine this result with the mass of c. we implemented the dempster-shafer theory of evidence combination to fuse the various states data inputs from the pre-processing modules to determine a current context and situation classification at any point in the gait cycle as is required for corbys demonstrator i. however, this implementation can be used to fuse any attributes or properties. 2.7. hardware-based reflexive behaviour adaptation an important capability of this architecture is the hard-ware-based reflexive module (rm) implemented with field programmable gate arrays (fpga). this module resides on the control layer of the architecture and monitors sensory data at high speeds to intervene at any point as needed to ensure safety protection. the situation awareness module provides updates to the reflexive behaviour adaptation module and thus to the fpga reflexive layer to enable real-time response to emerging safety situations. in demonstrator ii, the rm realises the real-time reflexive responses of the robotic follower by implementing low level reactive algorithms in the core module (e.g. obstacle detection). laser scanner data from an artificial rectangular area in front of the robot is processed by the rm module to detect obstacles in the path of the robotic follower and to ensure the safety of the robot by stopping immediately before hitting the obstacle. this is also applicable in other real-time safety protection contexts. for gait rehabilitation support (in demonstrator i), rm realises the real-time reflexive responses by monitoring sensors and actuators in real-time at high-speed in fpga. it detects (emerging) unsafe situations that could potentially harm the patient/demonstrator. upon detecting such a situation, the rm module cuts the power supply to the gait rehabilitator. the rm carries out detection of unsafe situations using complex event processing (cep). 3. conclusion in this paper, we have described the design and implementation of the corbys situation assessment architecture. this has included the various processing modules such as activity recognition, person states integration, gait analysis including sub-phase classification and deviation calculation, muscle activation analysis, overall context building and decision making regarding the assistive-remedial actuation that needs to be applied. all of the above modules have been implemented and tested individually with datasets made available over the course of the project. beyond the integrated conformance testing and performance evaluation of the modules of the situation assessment architecture, the larger scale evaluations are on-going. this includes decision engine performance evaluation with all semantic information available from the preprocessing modules determining the next best assistive-remedial actions for gait support. acknowledgement: this research was supported by the european commission as part of the corbys (cognitive control framework for robotic systems) project under contract fp7 ict-270219. 260 a. badii, a. khan, r. raval, h. oudi, r. ayora, w. khan, a. jaidi, n. viswanathan references 1. m. endsley, 2000, theoretical underpinnings of situation awareness: a critical review, situation awareness analysis and measurement, mahwah, nj. 2. blasch, e & s. plano, 2002, jdl level 5 fusion model: user refinement issues and applications in group tracking, spie vol. 4729, aerosense, pp. 270 – 279. 3. wahde, m., 2009, a general-purpose method for decision-making in autonomous robots. iea/aie, vol. 5579 of lecture notes in computer science. 4. ye, j., dobson, s., mckeever, s., 2011, situation identification techniques in pervasive computing, pervasive & mobile computing. 5. boytsov, a. & zaslavsky, a., 2011, from sensory data to situation awareness – enhanced context spaces theory approach. 9th ieee int. conf. dependable, autonomic and secure computing. 6. zhang, m. & sawchuk, a.a., 2011, a feature selection-based frame-work for human activity recognition using wearable multimodal sensors, proc. int. conf. body area networks, beijing. 7. vaughan, c.l., davis, b.l., o’connor, j.c., 1992, dynamics of human gait , 2nd ed., kiboho publishers south africa. 8. shafer, g., 1976, a mathematical theory of evidence, princeton university press. 9. dempster, a. p., 1967, upper and lower probabilities induced by a multivalued mapping, the annals of mathematical statistics 38 (2), pp. 325–339. 10. corbys project, 2014. corbys. [online] available at: http://www.corbys.eu (accessed 10. nov. 2014). procena situacije kroz multimodalnu detekciju dinamičkih okruženja za podršku kognitivnog upravljanja robotom svest o nastalim situacijama u dinamičkom operativnom okruženju robotskog sistema za podršku je ključna osobina takvog kognitivnog sistema bazirana na njegovoj efektivnoj i efikasnoj proceni preovlađujuće situacije. ovo omogućava sistemu da ostvari interakciju sa okolinom na razuman/semiautonoman/proaktivan način bez potrebe za čestom intervencijom od strane supervizora. u ovom radu je prikazana nova generalna arhitektura za procenu situacije za robotske sisteme koji direktno podržavaju ljude, kao što je sistem razvijen u okviru projekta corbys. ovaj rad predstavlja celokupnu arhitekturu za procenu situacije razvijenu u okviru corbys projekta kao i njenu primenu na “proof-of-concept” demonstratorima. razmatrani demonstratori su mobilni robot za praćenje ljudi i mobilni robotski sistem za rehabilitaciju hoda. rad predstavlja pregled strukture i funkcionalnosti arhitekture za procenu situacije kao i dobijene rezultate i sakupljene observacije tokom inicijalne validacije dva corbys demonstratora. ključne reči: procena situacije, kognitivno upravljanje, dinamičko okruženje, interakcija robota i čoveka, zajednički rad robota i čoveka i uzajamno preuzimanje inicijative plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 14, no 2, 2016, pp. 159 168 original scientific paper 1reduction of residual shear stress in the loaded contact using friction hysteresis udc 531.4 adrian kraft, roman pohrt berlin institute of technology, germany abstract. we investigate the tangential contact problem of a spherical indenter at constant normal force. when the indenter is subjected to tangential movement, frictional shear stresses arise at the interface and do not vanish when it is moved backwards. we study the evolution of shear stress when the indenter is moved back and forth at falling amplitude. the method of dimensionality reduction (mdr) is employed for obtaining the distribution of stick and slip zones as well as external forces and the final stress distribution. we find that the shear stress decreases. for the special case of linearly falling amplitude of the movement, we observe uniform peaks in the shear stress. the absolute value of the shear stress peaks is reduced best for a high number of back-andforth-movements with slowly decreasing amplitude. key words: coulomb friction, residual stress, contact mechanics, stick, slip 1. introduction loaded contacts exist in a broad variety of technical and natural situations. a basic frictional couple consists of two deformable bodies, which are pressed together and subjected to an additional tangential loading. when the tangential load is below the threshold of gross sliding, some micro-slip occurs nevertheless. in technical contact that experience cyclic loading, the relative movement of the bodies is often invisible to the naked eye. however, the friction involved can cause effects such as damping [3, 4] and fretting wear [8]. the same applies in a slightly more complicated fashion for biaxial loading [5]. consider a contact where the normal load is held constant, so that the bodies cannot separate. initially, the pure normal loading can take place without inducing shear stress in received february 24, 2016 / accepted july 10, 2016 corresponding author: roman pohrt technische universität berlin, straße des 17. juni 135, 10623 berlin, germany e-mail: roman.pohrt@tu-berlin.de 160 a. kraft, r. pohrt the contact interface. however, when some tangential load is applied, shear stress arises. upon unloading, the stress does not simply vanish due to the dual nature of the contact (stick zones vs. slip zones). tangential force and resulting tangential displacement together form a system with hysteresis, which can be described with a preisach formalism [10]. in 1928, prandtl developed a model for microtribology out of many micro-sliders, which also included hysteresis [12] (see [11] for english translation of the original paper). in his model, the exact state of the system at any moment of time depends on its prehistory and can be very complicated. prandtl poses the question of whether it is possible to restore the virgin state. with very simple arguments, he shows that if the contact partners start to oscillate with large amplitude and the amplitude then decreases slowly, then each slider finally comes to a neutral, non-stressed position and the system returns to the virgin state. he compares this result with demagnetization through slowly decreasing oscillating magnetic fields, first studied by e. madelung [6]. in this paper, we will apply this idea to the loaded contact with partial sliding. assume a typical hertzian contact. a spherical indenter with radius r is pressed into a plane. the sphere is approximated by a parabola and we assume that the plane is rigid. we keep normal force fn that we first applied constant throughout the following. this configuration is known as the hertzian contact with contact radius a and has been studied in great detail. normal stress distribution p as a function of radial coordinate r reads 1 2 2 max 2 2 / 31/ 3 * n max r p(r) p 1 a (6f ) e p r                . (1) here we used the reduced modulus of elasticity: 2 2 1 2 * 1 2 1 11 e e e      , (2) for two elastic bodies with moduli of elasticity e1 and e2 , shear moduli g1 and g2 and poisson's ratios 1 and 2, respectively. the equivalent reduced shear modulus is: 2 2 1 2 * 1 2 1 11 g g g      . (3) we now move the indenter horizontally along one spatial direction and assume that coulombs law of friction is valid in its simplest form τ(r) ≤ μp(r). here τ is the shear stress and μ is the coefficient of friction. this condition leads to the partitioning of the contact area into a stick area in the center and a slip area at the edges of it. for example, at the very edge of the contact surface, normal stress disappears and the slightest displacement of the indenter would lead to shear stress violating our friction condition if we assumed it would stick to the plane. the solution of this is that some parts of the contact area are being slipped over the plane while the others stick to it. at a certain point, the stick area completely disappears and the whole contact is slipping. tangential displacement ux (0) at this last point reads [2]: reduction of residual shear stress in the loaded contact using friction hysteresis 161 (0) *x m x * au e g   . (4) let us now drag the indenter back and forth along the same spatial direction on the rigid plane. we observe different shear stress distributions depending on the motion history. the shear stress created from the first movement will not disappear if we simply return to our starting point. in the following, we will analyze the shear stress under a back and forth movement with falling amplitude with the aim of reducing this stress. for the analysis of the system, the aleshins method of memory diagrams is well suited. because the contact stress distribution accumulates more and information of past reversal points, the hh-mode described in sec. 31 of [9] should be applied iteratively. using careful algebra, an explicit analytical solution might be achievable. however, we decided to use the method of dimensionality reduction for our analysis, which we describe in the next section. alternatively, one could employ a full boundary elements solution [7]. 2. mdr and discretized 1d-model we consider the three-dimensional contact of two elastic bodies. in the following, we presuppose axially symmetric profiles. let z=f (r) be the difference between the profiles of the bodies. according to the theorems of the method of dimensionality reduction (mdr), this contact can be exactly replaced by a contact with a one-dimensional linearly elastic foundation with independent springs. to reduce the initial three-dimensional contact to a one-dimensional one, two steps are required. first, we replace the elastic bodies by the one-dimensional linearly elastic foundation. normal stiffness δkz and tangential stiffness δkx of the springs are chosen according to: * z ek x   (5) and * xk xg   , (6) where x denotes the distance between two springs. second, we replace the three-dimensional profile z=f(r) with a one-dimensional profile in accordance to [2]: x 2 2 0 f (r) g(x) x dr x r     . (7) the reverse transformation is: r 2 2 0 2 g(x) f (r) dx r x   . (8) if now transformed profile g(x) is pressed with normal force fn and resulting indentation depth d into the elastic foundation, we obtain the displacement in the contact area: zu (x) d g(x)  . (9) 162 a. kraft, r. pohrt for non-adhesive contacts we can set: z a) 0 d (au g( : )    , (10) where a is the contact radius. we again denote the length a the ‘contact radius’ because following the theorems of mdr, all the lengths in the model are equal to the respective ones in a three-dimensional problem. the same applies to the contact area. the force of a spring at point x inside the contact area equals: z z z z(x) u (x) e*u (xf k x.)   (11) if we choose the spring separation distance to be infinitesimal, we get for the normal force: * a a a 0 * n zf : e u (x)dx 2e (d g(x))dx.      (12) after bringing the elastic bodies in normal contact with force fn, we apply tangential movement ux (0) to the indenter and assume that coulomb's law of friction is valid following: (r) p(r) for stick,   (13) (r) p(r) for slip.  (14) now every spring sticks to the indenter as long as δfx=δkx ux (0) < μ δfz. this results in the following conditions: (0) x x x x z z z z x (x) , if k u (x) u (x) (x) u u f k f u , k (x) in a state of slip.         (15) where ux(x) denotes here the horizontal displacement of a spring at point x. the sign of the second equation depends on the direction of the indenter motion. we denote the radius of the stick-area by c and find: * *) x (0 e (d g(g )).u c  , (16) for which the whole contact area is in slip state. similarly to the normal contact, the force of a spring at point x inside the contact area equals: * xx xx(x) u (x) g u (x)f k x    . (17) if we choose the spring separation distance to be infinitesimal, we get for the tangential force: * a x x a f g u (x)dx.    (18) reduction of residual shear stress in the loaded contact using friction hysteresis 163 the distribution of tangential stress in the original three-dimensional contact can be calculated according to [2]:       r 22 x * zr dx rx )x(ug )r( . (19) for our problem, we choose the classical contact between a sphere with radius r and a plane. the modified profile of z=f(r)=r 2 /2r is g(x)=x 2 /r. normal and tangential stiffness follow eqs. (5) and (6). with a given indentation depth d, we calculate contact radius a according to eq. (10). the length of the elastic foundation is set to 2a and consists of ns springs. distance δx between two adjacent springs is 2a/ns. the normal displacement is uz(x)=d-g(x)=d-x 2 /r. for the normal force, we get sn n k 1 zf 2 u (k)e x*    . with a given indenter displacement ux (0) , we first assume that every spring sticks. next, we check if tangential spring force is exceeding the maximum tangential force according to the first part in eq. (15) and where required, we adjust the displacement. for the tangential force, we get    sn 1k * xx xg)k(u2f . 3. reduction of residual shear stress magnitude we push a spherical indenter into an elastic plane and drag it horizontally. if we drag it not too far to one side, our contact area will be divided in a stick-area in the center and a slip-area on the outside. the stick-area will have the shape of a circle and the slip-area will encircle it in the shape of an annulus. by moving it further in the chosen direction, the stick-area gets smaller and smaller before it disappears completely and we reach macroscopic slipping. if we move the indenter in one direction without reaching macroscopic slipping and subsequently perform a small displacement in the opposite direction, we will observe that: the former stick-area is still in stick-state. more inner parts of the former slip-area are now also in stick-state. the outermost border of the former slip-area is in slip-state again. thus, we will observe different displacement areas: in the center, there will be a circle where the displacement equals the macroscopic displacement of the indenter, then there is an annulus in a stick-state, which was at some former time in a slip-state, and finally on the outermost border there will always be an annulus in slip-state. that is, by simply alternating the direction of the indenter several times, we can create a strongly fragmented displacement field. if now, for example, we drag the indenter until the whole contact is in slip-state, we could create a growing stick-area with a fragmented displacement field, by simply moving in a certain back-and-forth motion with falling amplitudes. as the displacement field determines the shear stress distribution, we can manipulate the stress by moving the indenter in a given way. in order to minimize the shear stress, the following procedure is discussed. 164 a. kraft, r. pohrt on a rigid plane base, we push a spherical indenter until an indentation depth d and related normal force fn are reached. we keep this configuration for the normal contact throughout the following procedure. the origin of coordinate ux (0) is placed on the rigid plane. subsequently we displace the indenter horizontally by uxmax (0) in ux (0) direction so that the stick area disappears completely according to eq. (4) and the indenter reaches macroscopic slipping. now we drag the indenter back and forth with linearly falling amplitude around the origin of ux (0) -coordinate. let nr be the total preselected amount of reversal points and k the current reversal point, then the analytic expression describing the position of the indenter relative to the rigid plane is:       (0) (k 1)0 x max x r r r u u (k) (n 1 k) 1 for k 1,2,..., n 1 n        . (20) for example, if we choose nr=1, the indenter moves to uxmax (0) and then back to 0. if we choose nr=2, the indenter goes to ux (0) =uxmax (0) , then to uxmax (0) /2 and back to 0. hence, the whole procedure is path-controlled by sequence ux (0) (k) . shear stress τ(r) and tangential force fx are only evaluated at each reversal point. note that eq. (20) only describes these points of the movement. the inertial forces of the indenter are not considered and the problem is seen as quasi-static. therefore, it does not matter in what exact fashion the indenter moves back and forth. instead of moving linearly, it might as well follow a sinusoidal movement, as depicted in fig. 1. finally, it should be pointed out that the movement always ends (for k=nr+1) at starting point ux (0) =0. fig. 1 normalized position of the indenter on the rigid plane versus the reversal points. uxmax (0) is the minimum displacement to reach macroscopic slipping. total amount of reversal points is 20. reduction of residual shear stress in the loaded contact using friction hysteresis 165 fig. 2 normalized tangential force versus reversal points. total amount nr of reversal points is 20. fn is the constant normal force as aforementioned, we have chosen to move the indenter path-controlled. from this, we obtained the tangential forces at each reversal point. fig. 2 shows the absolute value of the alternating forces. we would get exactly the same motion if we picked our reversal points according to this force sequence. for example the first reversal point (k=1) was defined so that we reach macroscopic slipping, that is: fx= μfn. fig. 3 shear stress in the contact area after 6 and 6.5 reversal points (a and b in [6]). a is the contact radius and pmax the maximum normal stress in the contact surface of a spherical indenter and a rigid plane. total amount nr of reversal points is 20. 166 a. kraft, r. pohrt at every state of the movement, we can calculate the shear stress distribution according to eq. (19). fig. 3 shows the stress distribution after six back and forth movements (continuous line; point a in fig. 1), thus exactly at a reversal point. we can see that inside the outer ring of the contact area (approximately 0.5a<r<a), the shear stress follows τ(r)=μp(r) from which we can conclude that the contact surface in this area is slipping over the plane. in the inside, the shear stress decreases and we have a stick area. with further back and forth movements at falling amplitude, the stick area will grow more and more towards the edge. the second curve shows the state if we move from the sixth reversal point back to our starting point (dashed line; point b in fig. 1). note that there is stick for approximately r<0.75a. in the outer ring, all points are sliding but the orientation has changed (negative shear stress in the figure). more generally, we can see that in the mdr space, every spring has a specific maximum tangential deflection. due to the higher normal deformation, this maximum deflection is greater for the springs in the middle of the contact. with falling amplitude of the movement, little by little those springs enter the stick state, so that they can follow the macroscopic movement of indenter ux (0) . at the outer edge, the springs can only slightly deform tangentially and are practically always in a slip-state, except shortly after switching orientation at each reversal point. fig. 4 shows the shear stress distribution after 20 back-and-forth displacements and return to the origin. as we can see, the residual stress alternates between positive and negative values. that is, the springs are pointing in both directions. it is interesting to see, that the different peaks take almost the same value. fig. 4 shear stress in the contact area after 20 reversal points. total amount nr of reversal points is 20 reduction of residual shear stress in the loaded contact using friction hysteresis 167 the ultimate purpose of the procedure is to minimize shear stress in the contact area. for this, we can evaluate the shear stress distribution that remains after the dragging as showed fig. 4. for the evaluation, we want to focus on the extreme values of the stress and calculate therefore the minimum, maximum and arithmetic mean of the absolute values of the shear stress at the end of the dragging. if we do this for different nr, we get fig. 5. as fig. 4 already illustrated, the peaks are quite evenly and that is why the minima and maxima of the absolute values are close together. as fig. 5 further shows, we can easily reduce the shear stress in the contact by moving a few times back and forth. if for example we created the maximum achievable shear stress by reaching macroscopic sliding, then it would be sufficient to invert the initial movement direction twice with falling amplitude according to eq. (20), to almost halve this stress (nr=2). fig. 5 minimum, maximum and arithmetic mean of the shear stress at the end of the dragging (k = nr+1) versus different reversal point amounts. 4. conclusion we have analyzed the hertz contact with the coulomb friction for the case that the spherical indenter is being moved with falling amplitudes tangentially to a plane. with the aid of the method of dimensionality reduction (mdr), we divided the contact area at every step into a stick and a slip area and subsequently determined the shear stress distribution. we find that after the movements, the remaining stress has comparable portion in both directions. with respect to the radial coordinate, it changes orientation in close succession. the peaks of this curve are equally large in good approximation. 168 a. kraft, r. pohrt the back-and-forth tangential movement proves to be a good way to reduce the residual shear stress inside a frictional contract. with only a few iterations, the maximum stress value can be reduced effectively. references 1. hertz, h., 1881, über die berührung fester elastischer körper, journal für die reine und angewandte mathematik , 92, pp. 156-171. 2. popov, v.l., heß, m., 2015, method of dimensionality reduction in contact mechanics and friction, springer. 3. mindlin, r., mason, w., osmer, j., deresiewicz, h., 1952, effects of an oscillating tangential force on the contact surfaces of elastic spheres, proc. 1st. us nat'l. congr. appl. mech. asme new york, pp. 203-208. 4. borri-brunetto, m., carpinteri, a., invernizzi, s. & paggi, m., 2005, micro-slip of rough surfaces under cyclic tangential loading, proceedings of the 4th contact mechanics international symposium, pp. 333-340. 5. ciavarella, m., 2013, frictional energy dissipation in hertzian contact under biaxial tangential harmonically varying loads, j strain analysis, 49(1), pp. 27-32. 6. e. madelung, 1905, über magnetisierung durch schnell verlaufende ströme und die wirkungsweise des rutherford-marconischen magnetdetektors, ann. phys. , 322(10), pp. 861-890. 7. pohrt, r., li, q., 2014, complete boundary element formulation for normal and tangential contact problems, physical mesomechanics, 17/4, pp. 334-340. 8. popov, v. l., 2014, analytic solution for the limiting shape of profiles due to fretting wear, scientific reports, 4, doi:10.1038/srep03749 9. aleshin, v., abeele, k. v. d., 2012, hertz–mindlin problem for arbitrary oblique 2d loading: general solution by memory diagrams, journal of the mechanics and physics of solids, 60/1, pp. 14-36. 10. aleshin, v., abeele, k. v. d., 2009, preisach analysis of the hertz–mindlin system, journal of the mechanics and physics of solids, 57, pp. 657-672. 11. popov, v., gray, j., 2012, prandtl-tomlinson model: history and applications in friction, plasticity, and nanotechnologies, z. angew. math. mech., 92/9, pp. 683-708. 12. prandtl, l., 1928, ein gedankenmodell zur kinetischen theorie der festen körper, z. angew. math. mech., 8, pp. 85-106. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 535 543 https://doi.org/10.22190/fume170504021z original scientific paper computerized radial artery pulse signal classification for lung cancer detection udc 004.8:616 zhichao zhang 1,2 , anton umek 2 , anton kos 2 1 shandong provincial key laboratory of network based intelligent computing, university of jinan, jinan, china 2 faculty of electrical engineering, university of ljubljana, slovenia abstract. pulse diagnosis, the main diagnosis method in traditional chinese medicine, is a non-invasive and convenient way to check the health status. doctors usually use three fingers to feel three positions; cun, guan, and chi of the wrist pulse, to diagnose the body’s healthy status. however, it takes many years to master the pulse diagnosis. this paper aims at finding the best position for acquiring wrist-pulse-signal for lung cancer diagnosis. in our paper, the wrist-pulse-signals of cun, guan, and chi are acquired by three optic fiber pressure sensors of the same type. twelve features are extracted from the signals of these three positions, respectively. eight classifiers are applied to detect the effectiveness of the signal acquired from each position by classifying the pulse signals of healthy individuals and lung cancer patients. the results achieved by the proposed features show that the signal acquired at cun is more effective for lung cancer diagnosis than the signals acquired at guan and chi. key words: radial artery pulse feature extraction, pulse signal classification, lung cancer detection 1. introduction lung cancer is one of the major public health problems worldwide [1]. pulse diagnosis is a safe and auxiliary way of detecting lung cancer in clinical medicine [2]. since pulse diagnosis heavily depends on physicians‟ experience, it takes a very long time and a lot of energy to master the technique of pulse diagnosis [3]. with the development of sensor technology, signal processing, data analysis techniques, and artificial intelligence, the pulse based diagnosis can be implemented by processing the radial artery pulse signal [4, 5]. to popularize the pulse diagnosis in clinical medicine, researchers have been trying to received may 04, 2017 / accepted october 31, 2017 corresponding author: anton umek faculty of electrical engineering, university of ljubljana, tržaška c. 25, 1000 ljubljana, slovenia e-mail: anton.umek@fe.uni-lj.si 536 z. zhang, a.umek, a. kos acquire and process the radial artery pulse signal using sensor technology and machine learning [6-7]. yong jun et al. [8] designed a sensor from the resonator to acquire the radial artery pulse signal based on the reflection coefficient; the sensor can be embedded in a wearable communication device. according to their experiment results, the acquired signal reflects the useful artery information and meets the requirements of the signal analysis for clinical purposes. zuo et al. [9] analyzed three types of radial artery sensors: pressure, photoelectric, and ultrasonic sensors according to the physical meanings, correlations of the acquired signal from these three sensors, and sensitivity factors. their conclusion shows: it is better to measure the transmural pressure to diagnose some special disease which can change the elastic property and thickness of the vessel wall. according to the pulse diagnosis theory [10], the lung cancer patients‟ vessel wall is different from that of healthy individuals; thus the signal of pulse pressure acquired from the optical sensor in our experiment has useful information for detecting lung cancer. machine learning is a popular and effective way of analyzing the radial artery pulse signal. khaire et al. [11] extracted the spectral features from the radial artery pulse signal and used the support vector machine to classify the individuals‟ pre-meal and post-meal signals. in rangaprakash‟s experiment [12], the pulse signal was taken from the volunteers before and after exercise. seven spatial features extracted from the pulse wave are used to classify the signals into these two groups using recursive cluster elimination based support vector machine. their work proved that the features obtained from the radial artery pulse signal can be used to detect the physiological state of the individuals. zhang and sun [13] utilize a convolutional neural network to classify twelve pulse waveforms; long pulse, feeble pulse, stringy pulse, thready pulse, deep pulse, rapid pulse, hesitant pulse, soft pulse, short pulse, slippery pulse, flood pulse, and faint pulse. half of the dataset with 200 samples was selected to train the classifier without feature extraction and the classification accuracy was found to be 93.49%. however, this is the basic traditional wrist pulse waveform classification which could not be used to disease diagnosis. chow et al. [14] defined the doppler parameters to be the disease sensitive features and applied the support vector machine to distinguish between acute appendicitis patients and healthy individuals. gong et al. [15] designed a wrist pulse sensing and analyzing system for recognition of cirrhosis patients with an accuracy of 87.09%. unlike the work they did, our research aims to identify lung cancer patients from healthy individuals. currently, in the radial-artery-signal processing research, only few works are directly related to the detection of some specific disease. to the best of our knowledge, no related work uses a radial artery pulse signal to identify lung cancer patients. in this work, we extract the features from the radial artery pulse signal for identification of lung cancer patients‟ and compare the effectiveness of the wrist-pulse-signals acquired from the cun, guan and chi positions as shown in fig. 1c. the rest of our paper is organized as follows: section ii briefly explains the data acquisition and pre-processing. the proposed features for lung cancer detection are listed in section iii. section iv presents and discusses the experiment result, and section v concludes the paper. computerized radial artery pulse signal classification for lung cancer detection 537 2. data acquisition and pre-processing the dataset is acquired from the shandong academy of chinese medicine. fig. 1a shows the pulse signal acquisition device. the signal sampling rate is 800 hz. the raw continuous signal was collected from the cun, guan, and chi wrist locations of 16 healthy individuals and 15 lung cancer patients. different levels of pressure are imposed on the radial artery by moving the robotic arm during the acquisition of the radial pulse signal. an example of a raw continuous signal with five pressure levels is shown in fig. 1b. the locations of “cun” and “guan” and “chi” of the wrist are shown in fig.1c. figure 2a shows the enlarged local signal at one pressure level. a) b) c) fig. 1 a) the pulse acquisition device; b) the acquired raw signal; c) the locations of cun, guan and chi in our previous work, we utilized the gaussian filter to de-noise the signal and designed an algorithm to remove the baseline wander and segment the continuous signal into single periods of the signal. fig. 2b shows the de-noised signal with the fitted baseline. the corresponding segmentation result is shown in fig. 2c. from fig. 2, it is evident that the baseline is removed and the single periods of the signal are segmented successfully using the developed algorithm. in addition, the average value of baseline is recorded to be used as one of the features in our experiment. after segmentation, we extracted features from the single periods and used classifiers to recognize the valid single periods of the signal. the examples of invalid and valid single wrist-pulse-signal periods are shown in fig. 3a to fig. 3d, respectively. finally, 8508 valid single periods of the radial-artery-signal are available for classification. 538 z. zhang, a.umek, a. kos a) b) c) fig. 2 a) an example of the enlarged raw signal; b) an example of the baseline fitting result; c) an example of the segmentation result computerized radial artery pulse signal classification for lung cancer detection 539 a) b) c) d) fig. 3 a) and b) examples of invalid periods; c) and d) examples of valid periods 3. feature extraction in our paper, we analyze single periods of the radial-artery-signal in the time domain. the features are extracted from the shapes of single periods. in fig. 4, it can be seen that the single period has a cardiac rapid ejection part and a heart slow-firing blood part (sfbp). the rapid ejection period lasts from the start of the period to the maximum. and sfbp lasts from the maximum to the end of the period. for more convenient features extraction, the sfbp is divided into 5 parts (a1, a2, a3, b1, b2) in the time domain, as shown in fig. 4. the parts of a1, a2, a3 account 1/8 respectively of sfbp, whereas b1 and b2 account 1/4 and 3/8 of sfbp, respectively. a) b) fig. 4 the single period radial-artery-signals: a) signal of the healthy individual, b) signal of the lung cancer patient 540 z. zhang, a.umek, a. kos the features are extracted from different parts of the period. the twelve features shown in table 1 are used to train the classifiers and identify the lung cancer patients‟ signals. table 1 definitions of features name definition pressure the depth of probe descent ave_single average baseline value of the single period of wrist pulse signal len_period length of the single period of wrist-pulse-signal pmax_index index of the sample of the maximum num_09 number of points that fit the condition (maximum-minimum >= 0.9) ave_a1 average value of a signal in b1 clip ave_a2 average value of a signal in b2 clip ave_a3 average value of a signal in b3 clip ave_b1 average value of a signal in c1 clip ave_b2 average value of a signal in c2 clip min_psub the distance between the peaks of heart slow-firing blood period location signal acquisition location (cun, guan, chi) the feature pressure represents different pressure levels imposed on the volunteers‟ radial artery. when changing the imposed pressure, the baseline of the signal changes as well. thus, we calculate the baseline‟s average value (ave_single) to be a feature as well. the length of the period (len_period) is given in seconds, and the average values of a1, a2, a3, b1, and b2 are extracted as features. a lung cancer patient‟s cardiac rapid ejection period is longer than that of a healthy individual as shown in fig.4. accordingly, we extract the index of the maximum amplitude (pmax_index). in addition, the lung cancer patients‟ heart slow-firing blood period is different in shape from that of healthy individuals. feature min_psub is the distance between the peaks in heart slow-firing blood period. since the radial-artery-signal is acquired from three probes on cun, guan, and chi, the acquisition location is also taken as a feature. 4. classification in this experiment, we have three different training sets. each training set consists of signals acquired from two healthy individuals and two lung cancer patients from one of the three positions, i.e. cun, guan, and chi. the rest of the individuals‟ data forms the three corresponding testing sets. at the extraction of the aforementioned features, the following classifiers are used to classify the periods of healthy individuals and lung cancer patients: linear svm (support vector machine), coarse gaussian svm, fine knn (k-nearest neighbors), cosine knn, subspace discriminant, subspace knn, simple tree and quadratic discriminant. each individual‟s classification results are counted respectively to calculate the lung cancer case rate. in this experiment, the one whose lung cancer periods are accounted for 50% or more is diagnosed to be a lung cancer patient. table 2 shows the diagnosis accuracy of lung cancer patients and healthy individuals using the abovementioned classifiers. each individual‟s diagnosis results coupled with the best classification accuracy are shown in table 3. computerized radial artery pulse signal classification for lung cancer detection 541 table 2 diagnosis accuracy of different classifiers classifier diagnosis accuracy cun guan chi linear svm 88.46% 80.00% 88.00% coarse gaussian svm 96.15% 88.00% 88.00% fine knn 96.15% 68.00% 88.00% cosine knn 96.15% 72.00% 88.00% subspace discriminant 88.46% 84.00% 76.00% subspace knn 65.38% 64.00% 88.00% simple tree 73.08% 64.00% 84.00% quadratic discriminant 73.08% 76.00% 84.00% table 3 diagnosis accuracy for individual subjects individual cun guan chi label 1 32.26% 82.61% 50.00% healthy 2 47.14% 96.50% 37.63% healthy 3 81.98% 66.67% 34.78% healthy 4 0.00% 7.06% 50.00% healthy 5 0.00% 4.52% 10.15% healthy 6 0.00% 42.78% 6.83% healthy 7 0.00% 17.56% 7.97% healthy 8 0.00% 17.56% 7.97% healthy 9 0.00% 45.51% 4.23% healthy 10 0.00% 38.64% 20.00% healthy 11 0.00% 34.78% 0.00% healthy 12 0.00% 0.00% 0.00% healthy 13 0.63% 0.00% 3.23% healthy 14 0.00% 0.00% 0.63% healthy 15 100.00% 100.00% insufficient lung cancer 16 100.00% insufficient 100.00% lung cancer 17 100.00% 100.00% 100.00% lung cancer 18 80.00% 100.00% 100.00% lung cancer 19 insufficient 100.00% 0.00% lung cancer 20 100.00% insufficient insufficient lung cancer 21 100.00% 100.00% 100.00% lung cancer 22 100.00% 100.00% 100.00% lung cancer 23 100.00% 100.00% 83.33% lung cancer 24 100.00% 99.50% 53.96% lung cancer 25 100.00% 100.00% 100.00% lung cancer 26 100.00% 100.00% 87.50% lung cancer 27 90.48% 92.03% 71.62% lung cancer note: the result of „insufficient‟ means the samples of the corresponding individual are not enough for classification 542 z. zhang, a.umek, a. kos 5. discussion as shown in table 2, the best result for the signal acquired from guan is 88.00%, and the corresponding classifier is coarse gaussian svm. while the best diagnosis accuracy of 88.00% for the pulse signal acquired from chi is achieved by the two svm models and the three knn models mentioned above. compared with the signal acquired from chi and guan, the signal acquired from cun shows the best result when using 5 classifiers (linear svm, coarse gaussian svm, fine knn, cosine knn, subspace discriminant), especially when using coarse gaussian svm, fine knn, and cosine knn, the diagnosis accuracy is 96.15%. in addition, we notice that coarse gaussian svm performs best for the signal acquired from all these three positions. thus, to show the performance of the coarse gaussian svm classifier in detail, we show each individual‟s diagnosis result in table 3 for coarse gaussian svm classifier. table 3 shows that two healthy individuals (individual 1 and 4) and a lung cancer patient (individual 19) are diagnosed incorrectly with the signal acquired from chi, whereas three healthy individuals (individual 1, individual 2, individual 3) are misdiagnosed to be lung cancer patients with the signals of guan. while, when using the signal acquired from guan and cun, all the lung cancer patients can be recognized correctly. there is only one healthy individual (individual 3) misdiagnosed with the signal acquired from cun. in summary, our proposed twelve features are effective to distinguish between the radial-artery-signal of healthy individuals and lung cancer patients, and compared with the signal acquired from chi and guan, the signal acquired from the position of cun gives the most accurate classification results, especially with the coarse gaussian svm classifier. in the current work, the radial-artery-signal related features are taken only from the time domain. in our future work, we plan to analyze the signal in the frequency domain by extracting its frequency domain features. meanwhile, we will try to get more lung cancer patients‟ and healthy individuals‟ radial-artery-signals to test the effectiveness of the pulse signal acquired from cun. moreover, we plan to design a more effective signal preprocessing procedure and build a more effective model to get more accurate classification results. 6. conclusion this paper is a study of the possibilities of lung cancer patient recognition using machine learning algorithms. the wrist pulse signal is acquired from healthy individuals and from lung cancer patients. the acquisition is performed simultaneously at the radial artery locations of cun, guan, and chi. in this work, we extract twelve features from the single period of signal and build eight classifiers to recognize the signal of lung cancer patients. the results show that the pulse signal acquired from cun performs the best compared with that from guan and chi. the extracted features are verified to be 96.15% accurate for the lung cancer diagnosis with the signal acquired from cun using the classifier of coarse gaussian svm. in our future work, we will improve the raw signal pre-processing algorithms, obtain more raw signals, and try to find even more effective features to increase the diagnosis accuracy of lung cancer detection with the emphasis on the pulse signal acquired from the position of cun. computerized radial artery pulse signal classification for lung cancer detection 543 acknowledgments: we appreciate the support given by the shandong academy of chinese medicine. this work was supported in part by the national natural science foundation of china under grant 61572231, in part by the shandong provincial key research & department project under grant 2017ggx10141, and in part by the slovenian research agency within the research program algorithms and optimization methods in telecommunications. the corresponding author is anton umek. references 1. siegel, r.l., miller, k.d., jemal, a., 2016, cancer statistics, 2016, ca: a cancer journal for clinicians, 66(1), pp. 7-30. 2. luo, c.h., su, c.j., huang, t.y., chung, c.y., 2016, non-invasive holistic health measurements using pulse diagnosis: i. validation by three-dimensional pulse mapping, european journal of integrative medicine, 8(6), pp. 921-925. 3. hsu, w.c., 2016, method and system for detecting signals of pulse 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recognition based on convolutional neural networks, proc. computer, consumer and control (is3c), 2016 international symposium, xian, ieee, pp. 366-369. 14. chow, w.h., wu, c.k., tsang, k.f., li, b.y.s., kwok, t.c., 2014, wrist pulse signal classification for inflammation of appendix, pancreas, and duodenum, proc. industrial electronics society, iecon 2014-40th annual conference, ieee, dallas, pp. 2479-2483. 15. gong, s., xu, b., sun, g., chen, m., wang, n., dong, c., wang, p., cui, j., 2012, accurate cirrhosis identification with wrist-pulse data for mobile healthcare, proc. second acm workshop on mobile systems, applications, and services for healthcare, acm, toronto, pp. 6. facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 601 612 https://doi.org/10.22190/fume210112025a © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper li-he’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system naveed anjum1,2,3, ji-huan he1,4,5, qura tul ain1,2, dan tian5 1national engineering laboratory for modern silk, college of textile and engineering, soochow university, suzhou, china 2school of mathematical sciences, soochow university, suzhou, china 3department of mathematics, government college university, faisalabad, pakistan 4school of mathematics and information science, henan polytechnic university, jiaozuo, china 5school of science, xi'an university of architecture and technology, xi’an, china abstract. this paper highlights li-he’s approach in which the enhanced perturbation method is linked with the parameter expansion technology in order to obtain frequency amplitude formulation of electrically actuated microbeams-based microelectromechanical system (mems). the governing equation is a second-order nonlinear ordinary differential equation. the obtained results are compared with the solution achieved numerically by the runge-kutta (rk) method that shows the effectiveness of this variation in the homotopy perturbation method (hpm). key words: microelectromechanical systems, enhanced perturbation method, parameter expansion method, nonlinear oscillator, amplitude-frequency relationship 1. introduction the last two decades have witnessed rapid advancement in nonlinear sciences arising in oscillation theory and other fields of physics [1-5]. several methods were developed to find periodic solutions of nonlinear oscillatory systems, for example, variation iteration method (vim) [6-7], homotopy perturbation method [8-9], hamiltonian approach (ha) [10], energy balance method (ebm) [11], spreading residue harmonic balance method received january 12, 2021 / accepted march 01, 2021 corresponding author: ji-huan he national engineering laboratory for modern silk, college of textile and engineering, soochow university, 199 ren-ai road, suzhou 215123, china. e-mail: hejihuan@suda.edu.cn 602 n. anjum, j.-h. he, q. t. ain, d. tian (srhbm) [12], iteration perturbation method (ipm) [13], and other methods [14-15]. hpm was proposed in the later 1990s [16-17] and now has been established into a mature phase for ordinary differential equations [18-19], partial differential equations [20-21], and differential equations of fractional order [22-24]. it is widely applied for nonlinear oscillation problems of conservative oscillators [19,25], attachment oscillator [26], fangzhu oscillator [27], micro systems’ oscillators [28-29], and fractional-order oscillators [30]. generally, a single iteration of this technique leads to a high accuracy of the solution. many researchers devoted their efforts and time to scrutinize the applicability of hpm for nonlinear problems and used it with the parameter expansion technology [31-32], the supporting terms [33-35], and the laplace transform [18,21]. recently, an adjustment in the perturbation method is proposed by filobello-nino [36], named the enhanced perturbation method. this method is highly accurate and provides better results because it deals with the problems with both small and large values of the perturbation parameters. li and he [37] adopted this modification to link the enhanced perturbation method with the parameter expansion technology [19,31], and highly accurate results can be achieved for nonlinear oscillators. ji et. al., [38] employed hybridization of li-he’s technique with ebm to find an approximate solution of the nonlinear problem of a tangent packing system. the microelectromechanical system (mems) refers to the high technology devices of small sizes; it has become a hot topic in both academic and industrial communities [3940]. the mems are intelligent structures and their systems are commonly micron or nanometer. microelectronic technology is the origin of these tiny devices used in vibrators, sensors, switches, and so on [41-42]. spring-base structures [43-45], nanotubes [15,46], and microbeams [11-12,14] can be considered as some of the potential and very applicable nano/microstructures in various sensing and actuating devices. these structures are modeled by generally using galerikin’s method and represented by nonlinear mathematical models. different types of forcing nonlinearities such as electrostatic force [11,43], electromagnetic force [41,44], and van der waals force [12] make the solution process extremely difficult. therefore, approximate solutions of these nonlinear models are important for predicting their dynamic behavior. recently fu et al. [11] studied electrically excited microbeams-based mems oscillator by employing the ebm [11]. the electrostatic force was used for actuation while the solution was depicted as an amplitude-frequency relationship. in this paper, we link the enhanced perturbation method with the parameter expansion technology [19,31] and propose an amplitude-frequency formula based on li-he’s approach [37] in order to find an approximate solution of the aforementioned model. the nonlinear frequency obtained from the proposed technology is compared with the frequency achieved numerically using the runge-kutta method (rk) for verification. we also match the results of li-he’s technique with those attained from ebm [11] to ensure the effectiveness of the suggested approach over ebm. 2. problem statement consider a doubly-clamped microbeam of length l, width b, thickness h and density ρ shown in fig. 1 with coordinate system oxyz. equation of motion as deflection of microbeam can be expressed with a partial differential equation as li-he’s modified homotopy perturbation method for doubly-clamped electrically actuated... 603 4 2 2 4 2 2 0 ( , ) 0 2 l w w es w w ei s n dx f x l xx x          + − + − =         (1) where w(x,) is the function of location x while time  represents the deflection of microbeam, e is the young’s modulus, i=bh3/12 and s=bh are moment of inertia about y-axis and area of cross section, respectively, ñ is the axial load between microbeam and its substrate and f(x, ) is the actuation force resulting from electrostatic excitation [42]. 2 2 2 1 1 ( , ) 2 ( ) ( ) b f x d w d w      = −  − +  (2) where ν denotes poisson ratio, εν is dielectric constant with usual value of 8.85 pfm−1 and d is the initial gap between the substrate and the beam. as the whole study is performed for a doubly-clamped microbeam, the boundary conditions will be (0, ) ( , ) (0, ) ( , ) 0, 0 l w w w w l x x      = = = =   (3) fig. 1 model of doubly clamped electrically actuated microbeam-based mems for simplicity, the variables of deflection of microbeam, location, and time in nondimensional form can be taken as w w d = , x l  = , t t  = (4) where 4hbl t ei  = (5) eq. (1) after substituting the nondimensional variables from eq. (4) is given by 14 2 2 2 4 2 2 2 2 0 1 1 0 4 (1 ) (1 ) w w w w d t w w               + − + − − =        − +      (6) where nondimensional parameters axial load n, aspect ratio α and parameter of electrostatic force v in eq. (4) are as follows 604 n. anjum, j.-h. he, q. t. ain, d. tian 2nl n ei = , 2 6 d h    =     , 4 2 3 3 24l v ed h   = (7) also the boundary conditions in nondimensional form can be expressed as (0, ) (1, ) (0, ) (1, ) 0, 0 w w w t w t       = = = =   (8) we apply method of separation of variables in order to find the solution of eq. (6) subject to boundary conditions of eq. (8). hence defection function w(η,t) can be written as the product of two functions. ( , ) ( ) ( )w t t   = (9) where χ(t) is the time function and ξ(η) is the trail function satisfying all the boundary conditions mentioned in eq. (8). in our study, we use trail function suggested as [11] 2 2( ) 16 (1 )   = − (10) substitute eq. (9) into eq. (6) and multiply the governing equation by φ(η)(1−w2)2 and then integrate over dimensionless domain in order to obtain 1 1 2 2 2 2 2 2 2 0 0 1 1 1 2 2 2 2 2 0 0 0 (1 ) (1 ) (1 ) 0 d d w n d d v d                      − + − −    − + − =           (11) where over dot ( )• represents differentiation with respect to time variable t and prime ( )• represents the partial differentiation with respect to coordinate variable η. eq. (11) can be rewritten as 2 4 3 5 7 0 1 2 3 4 5 6( ) 0c c c c c c c+ + + + + + =       (12) where coefficients c0, c1,…, c6 can be determined as follows 1 2 0 0 c d =  1 4 1 0 2c d = −  1 6 2 0 c d =  1 2 2 3 0 ( )c n v d = − −     li-he’s modified homotopy perturbation method for doubly-clamped electrically actuated... 605 1 1 3 3 2 4 0 0 2 2c n d d             = − + −      1 1 5 5 3 2 5 0 0 2c n d d              = − +      1 1 5 2 6 0 0 c d d        = −       eq. (12) is a second-order nonlinear ordinary differential equation and li-he’s approach will be employed for the solution under the following initial conditions (0) , (0) 0a = = (13) where a is the initial amplitude of the nonlinear oscillatory system. 3. basic idea of li-he’s approach to understand the idea of li-he’s approach, consider the linear oscillator 2 0x x + = (14) where x is the function of time t representing the general displacement and ω is the angular frequency of the oscillator. eq. (14) can be expressed in an operator form as 2 2( ) 0d x+ = (15) where d=d/dt is a differential operator. the enhanced perturbation method apply annihilator operator d2+ω2 to eq. (15); we have 2 2 2 2 2 4( )( ) 2 0d d x x x x + + = +  + = (16) this method can solve a wide class of non-linear problems. it is more effective in the case of nonlinear problems with the forced term but it can also apply to the problems without forced term [36]. after applying some suitable substitution, the eq. (16) is a higher-order equation and can be rewritten into linear ḻ and nonlinear ṉ operator form as: 0lx nx+ = (17) we can construct the homotopy equation for eq. (17) as 0( , ) (1 )[ ( ) ( )] [ ( ) ( )] 0, [0,1]h q q l l q l n q    = − − + + =  (18) where q is embedding parameter and ξ0 is initial solution of eq. (17). it is clear from eq. (8) 0( ,0) ( ) ( ) 0h l l  = − = (19) ( ,1) ( ) ( ) 0h l n  = + = (20) 606 n. anjum, j.-h. he, q. t. ain, d. tian hpm uses embedding parameter q as an expanding parameter [19], and basic assumption is that the solution of eq. (17) can be specified as a power series in q: 2 3 4 0 1 2 3 4q q q q     = + + + + + (21) setting q=1 results in the approximate analytic solution of eq. (17) 0 1 2 3 4 1 lim q x       → = = + + + + + (22) 4. solution of model problem to apply li-he’s technique discussed in the above section, eq. (12) can be expressed in the form 2 4 3 5 7 1 2 3 4 5 6(1 ) 0b b b b b b+ + + + + + =       (23) where 0 ( 1,2, ,6) j j c b j c = = . to reveal the solution process, consider eq. (23) which is hard to be resolved analytically specially when b1=−1, because the linear part has the form 0  − = which has no periodic solution. we express eq. (23) in an operator form as 2 3 2 4 6 1 2 3 4 5 6[ (1 ) ] 0d b b b b b b+ + + + + + =      (24) according to the enhanced perturbation method [36], we put on the annihilator operators d2+1 to eq. (14) 2 2 3 2 4 6 1 2 3 4 5 6( 1) (1 ) 0d d b b b b b b + + + + + + + =       (25) by applying the annihilator operators, a higher-order differential equation of eq. (24) can be written as 2 3 5 7 2 4 2 3 3 4 3 5 3 6 1 3 2 3 4 2 3 2 4 5 2 6 2 2 5 6 1 2 2 2 3 2 3 4 2 (6 3 ) (20 5 ) (42 7 ) (2 2 4 ) (12 4 8 ) 0 b b b b b b b b b b b b b b b b    − − − − − − + +        + + + + + +         + + + + + + =                                      (26) the linear part becomes now 2 3 0b  − = (27) which represents a linear oscillator. for eq. (26), the homotopy equation can be defined as 2 3 5 7 2 4 2 3 3 4 3 5 3 6 1 3 2 3 4 2 3 2 4 5 2 6 2 2 5 6 1 2 2 2 3 2 3 4 2 (6 3 ) (20 5 ) (42 7 ) (2 2 4 ) (12 4 8 ) 0 b p b b b b b b b b b b b b b b b    − + − − − − − +        + + + + + + +         + + + + + + =                                      (28) li-he’s modified homotopy perturbation method for doubly-clamped electrically actuated... 607 the solution and coefficient of the linear term can be expanding as 2 0 1 2p p   = + + + (29) 2 4 2 3 1 2b p p= +  +  + (30) where ω4 and ωi are constants and can be recognized by means of no secular term. substituting eq. (29) and eq. (30) into eq. (28) and continuing as that by the standard perturbation method, we have 4 0 0 0 00, (0) , (0) 0a    − = = = (31) 4 3 5 7 2 4 2 1 1 1 0 3 4 0 3 5 0 3 6 0 1 3 0 0 2 3 0 0 4 0 0 2 3 2 4 5 2 6 0 0 5 0 0 0 0 6 0 0 0 0 1 0 0 0 2 2 2 2 2 3 0 0 0 0 0 0 2 0 0 0 0 (6 3 ) (20 5 ) (42 7 ) (4 2 2 ) (12 4 b b b b b b b b b b b b b b b    − − − − − − − +       + + + + + + +       + + + +                                    2 3 4 0 0 0 0 0 08 ) 0  + + =      (32) by utilizing eq. (12), we can take the initial approximate solution as 0 cosa t =  (33) eq. (32) will get the form after employing the initial solution 4 3 3 5 5 7 7 1 1 1 3 4 3 5 3 6 3 2 3 5 2 5 3 2 3 2 3 1 3 2 3 4 5 2 3 5 2 5 7 2 5 7 2 7 5 6 3 4 1 cos [ cos ] [ cos ] [ cos ] [ cos ] [ cos ] [6 cos 9 cos ] [20 cos 25 cos ] [42 cos 49 cos ] [ 6 cos a t b b a t b b a t b b a t b b a t b b a t b a t a t b a t a t b a t a t b a  − −   −  −  −  +   +   +   −   +   −   +   −   + −     3 4 3 5 4 3 5 4 5 29 cos ] [ 20 cos 25 cos ] 0t a t b a t a t+   + −   +   = (34) after simple calculation, eq. (34) can be written 4 1 1 1 2 3 4cos cos3 cos5 cos7 0t t t t − +  +  +  +  = (35) where 3 5 7 3 2 5 23 4 3 5 3 6 54 1 1 7 2 3 2 5 2 3 4 5 46 1 2 1 2 3 5 35 53 4 8 64 4 8 35 3 5 3 5 64 4 8 4 8 b b b b b b bb a a a a a a b b b b b a a a a a  = −  − − − −  −  −  +  +  +  +   (36) 3 5 7 3 2 5 23 4 3 5 3 6 54 2 7 2 3 2 5 2 3 4 5 46 1 2 1 2 5 21 459 4 16 64 4 16 189 5 9 45 64 4 16 4 16 b b b b b b bb a a a a a b b b b b a a a a a   = − − − −  −  −  +  +  +  +  (37) 5 7 5 2 7 2 5 2 5 43 5 3 6 5 6 2 2 3 7 25 175 25 16 64 16 64 16 16 b b b b b b b b a a a a a a = − − −  −  +  +  (38) 7 7 23 6 6 4 49 64 64 b b b a a = − −  (39) 608 n. anjum, j.-h. he, q. t. ain, d. tian requirement of no secular term needs 3 5 7 3 2 5 23 4 3 5 3 6 54 1 7 2 3 2 5 2 3 4 5 46 1 2 1 2 3 5 35 53 4 8 64 4 8 35 3 5 3 5 0 64 4 8 4 8 b b b b b b bb a a a a a a b b b b b a a a a a  −  − − − −  −  −  +  +  +  +  = (40) if it is enough to obtain the first-order approximate solution, then from eq. (30), we yield 2 4 1  = − (41) solving ω from eqs.(40) and (41) we have 2 4 6 3 4 5 6 2 4 1 2 3 5 35 4 8 64 3 5 1 4 8 b b a b a b a b a b a + + +  = + + (42) and the corresponding approximate solution of eq. (12) is 2 4 6 3 4 5 6 2 4 0 1 2 64 48 40 35 ( ) cos 64 48 40 c c a c a c a t a t c c a c a   + + +  =  + +   (43) which is different from the solution gained by ebm [11]. 5. results and discussion the analytic nonlinear frequency and the approximate solution for the vibration of microbeam can be calculated from eqs. (42) and (43), respectively. a comparison between the frequencies obtained from li-he’s approach and those gained by rk is demonstrated in table 1 for different parameters. it displays the high correctness of the proposed solution as the maximum percentage error is not greater than 1%. this table and the graphs of figs. 2 and 3 specify that the analytic solution accomplished by li-he’s approach, eqs. (42) and (43), can estimate the dynamic vibrational activities of the microbeams acceptably. this approves the validity of the proposed solution. table 1 comparison of analytic frequency obtained by li-he and rk methods a n α v ωrk ωli-he error (%) 0.15 12 20 15 22.9224 22.8896 0.1431 0.15 18 50 15 29.8489 29.8222 0.0269 0.3 6 35 10 28.8737 28.8882 0.0502 0.3 24 10 20 16.6002 16.6422 0.2530 0.45 12 20 5 28.5210 28.4952 0.0905 0.45 24 10 10 26.4555 26.5042 0.1841 0.6 6 25 5 29.0216 29.0262 0.0159 0.6 18 40 5 33.8716 34.0674 0.5781 li-he’s modified homotopy perturbation method for doubly-clamped electrically actuated... 609 fig. 2 comparison of solutions and errors of li-he’s approach with ebm for the parameters a=0.3, n=10, α =24, v=10 fig. 3 comparison of solutions and errors of li-he’s approach with ebm for the parameters a=0.45, n=20, α =12, v=5 610 n. anjum, j.-h. he, q. t. ain, d. tian the top panels of figs. 2 and 3 compare the solutions obtained from li-he’s technique (red line) expressed in eq. (43), ebm (black line) depicted in ref. [28] with the solution achieved by rk method (blue line). this comparison validates that the findings from the proposed method and those attained by the rk method match remarkably well. we also show the variation of errors for the said system in the bottom panels of figs 2 and 3. errors of ebm (black stars with solid line) and errors of li-he’s approach (red squares with solid line) against time confirm the supremacy of the proposed technique over the ebm. 6. concluding remarks in this research study, we have applied hybridization of the enhanced perturbation method and the parameter expansion technology (collectively called li-he’s approach) to approximate the periodic behavior of electrically excited microbeams-based microelectromechanical system. the solution achieved from the proposed method has good agreement with the numerically ones gained by the runge-kutta technique. this method not only gives an alternative approximate solution to the oscillatory system by refining the order 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h.m., 2016, rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based nems actuators, international journal of non-linear mechanics, 87, pp. 97-108. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 16, no 2, 2018, pp. 273 283 https://doi.org/10.22190/fume171114026o © 2018 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper the study and the mechanism of nitrogen oxides’ formation in combustion of fossil fuels udc 691.141 bulbul ongar 1 , iliya k. iliev 2 , vlastimir nikolić 3 , aleksandar milašinović 4 1 almaty university of power engineering & telecommunications (aupet), faculty of heat energy, almaty, kazakhstan 2 university of ruse, department of thermotechnics, hydraulics and ecology, ruse, bulgaria 3 university of niš, faculty of mechanical engineering, serbia, niš 4 university of banja luka, department of vehicle and engine, b&h, banja luka abstract.the burning of all fossil fuels is accompanied by the production of large quantities of nitrogen oxides. nitrogen oxide from coal combustion is formed from the molecular nitrogen in the air and the nitrogen contained in the fuel. in accordance with the mechanism of formation of nitric oxide from fuel, it is desirable to increase the concentration of coal dust in the flame. the thermal regime of combustion accelerates the release of volatiles, with flames spreading out and the coke residue contributes to the chemical reduction of nox. in this work we consider the specific issues of the formation mechanism of nox fuel and ways to reduce their atmospheric emissions. presented are results from the calculation of the influence of the following on the level of nitric oxides during coal combustion: temperature, oxygen concentration and time of release of fuel nitrogen. it has been established that the influence of nitric oxide fuel on the total nitric oxide emissions is more noticeable at low temperatures of the combustion process. key words: coal gasification, concentration of nitrogen oxides, torch, recombination, burner received november 14, 2017 / accepted july 10, 2018 corresponding author: ongar bulbul almaty university of power engineering & telecommunications (aupet), faculty of heat energy, almaty 050060, kazakhstan e-mail: ongar_bulbul@mail.ru mailto:ongar_bulbul@mail.ru 274 b. ongar, i. k. iliev, v. nikolić, a. milašinović 1. introduction exhausted pollutants such as nox, sox, co, co2, and unburned hydrocarbon particles, have been considered as one of the major sources of air pollution in urban regions, which seriously affect human health, environment, and economic development [1]. the trend towards more stringent regulations on emissions has been an important driving force in the search for more environmentally friendly engines [2]. environmental pollution by toxic products from the combustion of organic fuels is one of the inevitable results of modern heat-power engineering. the main pollutants from that combustion are: coal fly ash, oxides of sulphur and nitrogen. when burning fuel oil those are the sulphur and nitrogen oxides, whilst when burning natural gas, the polluters are nitrogen oxides. fly ashes and oxides of sulphur are formed in quantities, determined by the ash and sulfur contents. a decrease in their contents in the products of combustion is reached by cleaning combustion gases or through preliminary removal from fuel (through enrichment – for decrease in the ash-content, removing of sulphur from fuel). in comparison to these pollutants, the level of formation of nitrogen oxides can be substantially regulated to a sufficient degree by means of flue/furnace gases. direct experimental data about the role of preliminary heat treatment of fuel on the formation level of fuel oxides of nitrogen are known. it is further known that at temperatures 7001200 k there are rather intensive recovery reactions of carbon dioxide and an oxide of nitrogen on carbon to carbon monoxide, and besides that the speed of the decomposition reaction of an oxide of nitrogen (at t > 1200 к the reaction is sharply inhibited) is two orders higher than the speed of decomposition of carbon dioxide [3]. research efforts continuously focus on modeling nox formation/destruction reactions in pulverized coal combustion [4-8]. nitric oxide (no) is the most abundant nox from coal combustion. the simulations often neglect fast no (significant only in strong fuelrich flames), while considering fuel no, typically accounting for 75-95% of the total no in coal combustors and thermal no, becoming important for the flame temperatures above 1600-1800 k. an in-house developed 3d differential mathematical model of furnace processes, including the fueland thermal-no formation/destruction reactions, validated against available measurements in the case-study boiler units [9-14], was used for the analysis. the comprehensive combustion code offers a balance between the sophistication of submodels describing individual processes and computational practicality. the no formation and depletion mechanisms were studied in dependence on twophase gas particle reactive turbulent flow and pulverized coal diffusion flame, accounting for extremely complex interactions between the influencing parameters. impact of various operating conditions in the case-study furnace on the nox emission and flame, like the fuel and the combustion air distribution over the burners and tiers, fuel-bound nitrogen content and grinding fineness of coal were investigated individually and in combination. complex numerical experiments of this kind can help to optimize flow, combustion and heat transfer and improve the furnace exploitation regarding emission and efficiency. the study and the mechanism of nitrogen oxides' formation in combustion of fossil fuels 275 2. decrease in emissions of nitrogen oxides by optimization of furnace aerodynamics complex suppression of formation of nitrogen oxides demands a combinatorial approach involving thermochemistry, the aerodynamics of burning, hydromechanics and heat mass exchange. problems of thermophysics and power systems cause some incomparable interest and have a high value in practice. the relevance of this problem and the growing attention to it are related to the increase of energy use efficiency and to the solution of environmental problems, as well as to the increase in the amount of polluting substances released into the atmosphere, but also to the operation of existing power stations, in particular with the creation of new combustion chambers. first of all, the power plants working with solid fuel are the main source of air pollution, water and the soil. this problem can be solved only on the basis of physical, mathematical and chemical modelling. in this regard numerical experiments become one of the most economical and convenient ways for the detailed analysis of the difficult physical and chemical phenomena occurring in the furnace camera. use of modern computers allows to solve these problems for specific power stations (thermal power plant, state district power station, etc.) and for any power fuel [15]. when burning any type of organic fuel, a large amount of the nitrogen oxides promoting pollution of the atmospheric air is formed. those include the following compounds: n2o, no, n2o3, no2, n2o4 and n2o5. in practice only the oxide no and the dioxide no2 of nitrogen matter from the point of view of ecology, their sum in terms of dioxide of nitrogen designate nox. as per the obtained data, the degree to which nitrogen oxides determine the toxicity of the flue gases when burning coal and fuel oil is 40-50%, rising to 90-95% when burning natural gas. furthermore, no accounts for about 95-99% of the nitrogen oxides nox formed in the boiler, whereas 4080% of the nitric oxide, contained in the flue gases is further oxidised to no2 in the atmosphere. nitrogen oxides when burning coal can be formed from molecular nitrogen in the air and nitrogen-containing components of the fuel. there are three types of mechanisms allowing for their formation: thermal, fuel and fast. during the combustion of natural gas thermal and fast nox, are formed, whereas during the combustion of heavy fuel oil or coal – thermal and fuel nox. formation of thermal nitrogen oxides occurs on the basis of the zeldovich mechanism. these reactions are characterized by a high activation energy and proceed at temperatures over 1800 к [3]. two balancing reactions offered by zeldovich are given below taking into account the chain mechanism of the formation no which is put forward by: kjnnono k k 197 1 3 2     (1) kjonoon k k 17 2 4 2      (2) where o – oxygen, %; n2 – molecular nitrogen, %; no concentration of an oxide of nitrogen, %; n – nitrogen, %. k1 and k2 – speed constants of the direct reaction in eqs. (1) and (2), k3 and k4 speed constants of the reverse reaction respectively. the differential equation for no formation due to excess concentration of oxygen is as follows: 276 b. ongar, i. k. iliev, v. nikolić, a. milašinović               2 43000 22 86000 2 11 3 64105 noenoe od dno rtrt  (3) where τ –the time, s; r – the universal gas constant, r=8.314 j/(molk); t – the absolute temperature, k. settlement calculations are provided in [16] during the analysis of the nox concentration in the flames of a swirl burner and a straight flow one. the amount of nitrogen oxides formed when using swirl burners depends on the intensity of the swirl of a stream and can be estimated by means of the following dependence: 0.8(1 )x xno no n   (4) where nox – the concentration in a swirl burner, g/nm 3 and n the amount of stream swirl in the given burner, g/nm 3 . it is known that no formation during the combustion process occurs at the end of the torch, in the stage of burning out of volatiles. according to the approximate mechanism of formation of no fuel, in the beginning fuel nitrogen passes in intermediate radical connections, after which it is partially oxidized to no. mass and spectroscopic analysis showed presence of active radicals of cn, hcn, nh, nh2 and oh at an initial stage of burning. furthermore, according to [17] there are the following reactions: conhohhcn  32 (5) 2223 2/1 hohnoonh  (6) it is established that the level of conversion (transition) of fuel nitrogen to nitrogen oxides significantly depends on the content of nitrogen in the initial fuel. at small contents of fuel nitrogen (less than one percent) it is almost entirely converted to nox. however, at a level of nitrogen content in the fuel 11.3%, merely 1625% is converted into nox. it is also accepted that the formation level of nitrogen oxides significantly depends on the concentration of oxygen in a torch’s zone of ignition and in this regard it depends on the excess of air in said torches. the maximum quantity of fuel nitrogen oxides is formed at excess air ratio α = 0.850.9. compared to thermal nitrogen oxides, the dependence of the formation level of nitrogen oxides on temperature has a more difficult character. in particular, in the temperature zone of 10001400 k an almost exponential dependence of formation level of the fuel takes place. at further temperature increase, (higher than 1400 k) this dependence gains a linear character [3]. fast fuel nitrogen oxides (typical for hydrocarbonic fuels) are formed in the root part of a torch. that reaction is characterized by a very high speed (considerably bigger than formation of thermal no). the study and the mechanism of nitrogen oxides' formation in combustion of fossil fuels 277 fenimore [18] made the assumption that the formation of fast nitrogen oxides is promoted by binding of molecules of nitrogen by the active radicals mentioned earlier, for example ch and c2 [19-21] binding reactions of n2 can look as follows: nhcnnch   2 (7) cnnc 22 2    (8) nhhcnch   22 (9) nohohn    (10) the existence of a large concentration of hcn near a zone of burning experimentally confirms the possibility of no formation according to the specified scheme. the share of fast no of the total formed nox is usually relatively small and makes up about 1015 %. in it is noted that the temperature of the most intensive formation of fast nitrogen oxides ranges between 12001600 к. the maximum level of formation of fast oxides is observed at excess air ratio α=0.650.8 which are values slightly less than those of excess of air of the fuel nitrogen oxides specified by our consideration. it is remarkable that at α<0.60.7 there is some decrease in the formation level of fast nitrogen oxides (fig. 1). more intensive formation of carbon monoxide and its partial restoration to molecular nitrogen can perhaps be the cause of this decrease. from fig. 1 we observe that the higher the temperature of the process, the faster the exit of atomic nitrogen and its recombination, the former of which happens in less than 0.04 s. fig. 1 a recombination of n in n2 depending on the temperature level of process at various values of concentration of oxygen 1 % (1), 2 % (2), 3 % (3), 4 % (4) and 5 % (5) of oxygen 278 b. ongar, i. k. iliev, v. nikolić, a. milašinović the recombination of atomic nitrogen in molecular nitrogen occurs as depicted in fig. 2, for a very short time, less than 0.04 s at a temperature t=8001600 к; at a content of oxygen o2 no more than 3 % in the combustion gases (recirculation to a torch root). fig. 2 a recombination of n in n2 depending on time at various o2: 1 % (1), 2 % (2), 3 % (3), 4 % (4), 5 % (5) and 6 % (6) thus, a decrease in the emissions of nitrogen oxides by influencing furnace aerodynamics and conditions of burning can be obtained by the following actions [22]:  burning of fuels at small excess of air or  recirculation of the products of combustion in the furnace camera (the greatest effect on decrease in an exit of nox is reached at recirculation introduction directly to the hot air before the torches). however, these methods have limitations, namely:  considerable complication of torch ignition at decrease in the oxygen content in the burning zone;  temperature increase of overheated steam in connection with a tightening of ignition and burning out of fuels;  some decrease in the completeness of the burning out of fuel (increase of heat losses due to unburnt or the emergence of unburnt chemicals). these actions are generally directed at the decrease in the maximum temperatures in a zone of active burning, and at the reduction of oxygen concentration, and furthermore at the creation of a recovery environment for the no formed. other possibilities of achieving a decrease in the formation level could be:  phasic (step) burning of organic fuels (reduction of oxygen concentration and temperature decrease in the ignition zone)  application furnace burner devices providing a realization of the noted conditions for the decrease in formation level of nitrogen oxides. it is specified that a decrease in the nox quantity of 4050 % when burning gas and of 25÷40 % when burning fuel oil has been successful. however, burning of fuels with small excess of air can lead to the formation of soot and a strong carcinogenic compound the study and the mechanism of nitrogen oxides' formation in combustion of fossil fuels 279 benzo[a]pyrene. the situation can be exacerbated considerably given an unsatisfactory state of the furnace burner devices and their imperfections. use of straight-flow burners (as a rule in tangential fire chambers) in combination with a certain aerodynamic scheme of burning allows for a phasic supply of an oxidizer in the horizontal direction and step-wise burning to be organised. in various schemes the question of introduction of gases of recirculation to the fuel ignition zone can be considered. thus, in such a scheme it is possible to implement all three mechanisms of influence on the level of formation of nitrogen oxides, which may contribute to the high effectiveness of straight-flow burners and tangential furnaces. 3. the determination of the level of formation of nitrogen oxides by varying the fuel concentration in the burner when complex burning of nitrogen oxide pollutants takes place, it is important to optimise the burning processes first and foremost by undertaking actions related to the burner’s mode of operation. according to the mechanism of formation of fuel nitrogen oxides, an increase of concentration of coal dust in the aero mix is necessary. the latter, respectively the thermal mode of burning, promotes an exit of volatiles and their ignition in the conditions of relative oxygen insufficiency. in this regard the application of supply of coal dust together with the overall concentration considerably exceeding the traditional ratio of air and coal dust can be one of the possible mechanisms to reach a decrease in the formation level of nitrogen oxides. such modes can analytically be predetermined and embodied in the designs of burner arrangements and it is easy for them be established during operation of regular installations, taking into account the concentration characteristics of the given coal dust. this idea was realized on the p-57-3m no. 8 eheps-1 package boiler under the possible conditions of operation of the burner arrangements [23]. tests were carried out at electric loading of the block with a power of 450 mw, a coefficient of excess of air behind a transitional zone 1.15, respectively behind an air heater 1.33, and temperature of feed water t=483 k in an operating mode of the package boiler without hdc. fuel consumption equalled 83.33 kg/s at the following parameters of the package boiler. qi r =15.56 mj/kg, a r =41%, w r =7%. the consumption of primary air in the three mills was on average 0.393 nm 3 /s, as for the others it was 0.16 nm 3 /s. the content of combustibles in slag was 7.5%, and in the flue gases 4.3 %. heat losses with exhausted flue gases, with mechanically unburnt fuel, and due to radiation along with the physical heat of slag equal respectively 4.09 %; 3.88 %; 0.29 %. gross hopper efficiency was 91.74 % and 92.03 % before and after tests respectively, and expenses on own needs increased by 0.12 %. sampling was carried out in the splitting of the flue at the level of the water economizer. the content of nitrogen oxides was determined by the device "eudiometer-1". the torch temperature was estimated by means of a pyrometer. expenses of air and other parameters were fixed on panel board devices. in processing thermos-aerodynamic and the chromatographic of data, representatives of kazniie and service tsnito egres-1 took part in the preparation of tests and the measurements of parameters. 280 b. ongar, i. k. iliev, v. nikolić, a. milašinović in normal operation of the boiler, the average value of the concentration of nitrogen oxides at the measuring point was nox=0.864 g/nm 3 with an average value of flame temperature at the outlet of the furnace of t=1618 k. as a result of the redistribution of air streams by way of the translation of primary air consumption to secondary air, a reduction of the content of nitrogen oxides in combustion products of 28 % on average is reached, i.e. decreased to 0.62 g/nm 3 , at an average value of temperature of the torch at the exit from the fire chamber of t=1598 k. thus, the results presented from the carried out test on the method of suppressing the formation of nitrogen oxides using measures related to the operational regime showed that a decrease in the formation level of nitrogen oxides is possible. regime actions represent part of the complex suppression of formation of nitrogen oxides, so we outlined a number of essential actions to be undertaken to aid in this aforementioned reduction on a global scale [23]. the method of preliminary gasification of fuel is one of the possible ways of global suppression of formation of nitrogen oxides. on the other hand, at the initial phase of ignition of the torch to a certain degree, is the process of coal gasification. in particular, it is known that at temperatures 700-1200 k quite intense redox reactions take place. thus, the speed of reaction of the formed nitrogen oxide decomposition is two orders higher than the speed of decomposition of carbon dioxide. the result is expected to reduce the concentration of nitrogen oxides in the combustion chamber of the ekibastuz condensing thermal power plant by means of fire-technical methods to a level of less than 0.2 g/nm 3 , i.e. by decreasing the concentration of nitrogen oxides this should take place 4-5 times. the nox emissions on level are reduced for the emulsion fuels compared to neat diesel: the higher the water content the smaller the nox emission. as the engine load is increasing, a slighter increase of nox emission, at engine speeds of 1100 and 2100 rpm, can be observed. at engine speed of 1700 rpm and engine load of 300 and 450 nm, the highest nox emission reduction was reached. the most notable reduction was of 67% for the emulsion of 10% water ratio and of 71% for the emulsion of 25% water ratio, at engine torque of 300 nm. this can be explained by higher carbon oxidation due to water droplet micro-explosion which provides a better mixing of fuel and in-cylinder air. this assumption is supported by the reduced levels of o2. 4. results and discussion according to fig. 2, the possible recombination of atomic nitrogen in molecular nitrogen occurs in a very short time, less than 0.04 seconds, in the wide range of change in the concentration of oxygen. the purpose of the experimental confirmation of this position is marked with a vertical tubular furnace upgraded as follows (fig. 3). tube 7 made from stainless steel with a diameter of 10 mm was installed in the furnace for air supply from below. preheated argon from electric furnace 6 was introduced in mixer 5 of the system with central cross streams. dust from ekibastuz coal was brought into the system through feeder 4, located above the mixer. the time of crossing of the air stream with the heated up by the argon in the mixer coal dust was regulated by accounting for the velocity of the the study and the mechanism of nitrogen oxides' formation in combustion of fossil fuels 281 gas-mix by the vertical displacement of tube 7 along the vertical combustion chamber. air in the form of transverse jets was supplied into the stream of gas-mix (fig. 3). fig. 3 schema of pilot unit: 1–furnace electric heater; 2–holes for thermocouples; 3–peephole; 4–burner pulverized feeder; 5mixing machine; 6–electric heating unit of a gas; 7–tube for air supply owing to the vertical supply of the stream during the experiment, the coal dust instantly mixes with the system of vertical argon streams, and consequentially the same momentary crossing of streams occurs between the gas mixture and the air. the concentration of oxygen in the system was regulated by the ratio between the air consumption and the argon gas. height of the furnace and the accepted flowmeters allowed setting time of shift of phases and temperature in the furnace with an accuracy of 20 %. experiments were made with dust of ekibastuz (kazakhstan) coal. the dust consumption in all experiments was kept constant at made 0.042 g/s. the excess air ratio was =1.2, and the temperature in the furnace changed from t=773 k to t=973 k. the oxygen concentration in the furnace was 11 %. it was determined by using the chromatograph lhm-8md. a determination of the concentration of nitrogen oxide was performed using, as noted, the device of the "eudiometer-1". the technique of carrying out gas analyses is described in [23]. a mathematical model of solidification works integrated with a genetic search algorithm and a knowledge base of operational parameters is presented. surekha et al. [24] presented multi-objective optimization of green sand mould system using evolutionary algorithms, such as genetic algorithm (ga) and particle swarm optimization (pso). dučić et al. [25] presented optimization of chemical composition in the manufacturing process of flotation balls based on intelligent soft sensing. the implementation of modern cad/cam software systems is frequent in the research projects of the process, as well as the combination of modern cad/cam software systems and methods of metaheuristic optimization. 282 b. ongar, i. k. iliev, v. nikolić, a. milašinović 5. conclusions at torch burning in furnace chambers of the boilers, at the initial stage of the burning process (during the release of the volatile components) the volatiles emit a part of the compound as water vapour and nitrogen-containing gases. the latter are oxidized with the formation of nitrogen oxides, the level of which depends on the content of oxygen and the temperatures in the fuel ignition zone. the fuel nitrogen which remained in coke at the secondary combustion is also partially converted to oxidized nitrogen, but its share in comparison with the emissions of nitrogen fuel oxides from the gas phase is insignificant. the maximum release of fuel nox is observed at the end side of the torch, at the stage of ignition and burning of volatiles, at temperatures of tmax=1200÷1800 k. the influence of fuel nitrogen oxides on the general emissions of nitrogen oxides is more significant at low temperatures of the burning process. acknowledgement: we express special gratitude to doctor of technical sciences, professor b.k. aliyarov, for his encouraging recommendations on research in the field of formation of nitrogen oxides in the combustion of partially gasified coal. the study was supported by the almaty university of energy and communications. references 1. fan, x., hu, w., yang, j., 2008, micro-emulsified diesel oil. petroleum science and technology, 26, pp. 2125-2136. 2. armas, o., ballesteros, r, martos, f.j., 2005, characterization of light diesel engine pollutant emissions using water – emulsified fuel, fuel, 84, pp. 1011-1018. 3. kotler, v.r., 1987, reduction of emissions of nitrogen oxides by boilers of thermal power plants during the burning of organic fuel. ser. 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sigal, i.y., 1989, ways of reducing nitrogen oxide emissions in thermal power plants, teploenergetika, 3, pp. 5-8. 17. kotler, v.r., 1987, nitrogen oxides in boiler flue gases, energoatomizdat, p.144. 18. fenimore, c.p., 1971, formation of nitric oxide in premixed hydrocarbon flames, thirteenth symposium on combustion – the combustion institute, pp. 374-384. 19. sigal, i.y., 1988, air protection when burning fuel, nedra, p. 312. 20. roslyakov, p.v., zakirov, i.a., 2001, non-stechiometric burning of natural gas and heavy fuel oil at thermal power plants, izdatel’stvo mei, p. 144. 21. sigal, i.y., 1983, development and tasks in the research on the development of nitrogen oxide formation conditons in furnace processes, teploenergetika, 9, pp. 5-108. 22. putilov, v.y., 2007, modern nature protection technologies in power industry, the information collection, mei publishing house, 388 p. 23. temirbaev, d.zh., 1992, the study of thermochemistry of oxides of nitrogen combustion of ekibastuz high concentration coal, final report on hdt №17/91. №gr 01910009373. – almaty aeu, 48 p. 24. surekha, b., kaushik, k.l., panduy, k.a., vundavilli, r.p., parappagoudar, b.m., 2012, multi-objective optimization of green sand mould system using evolutionary algorithms, the international journal of advanced manufacturing technology, 58(1), pp. 9-17. 25. dučić, n., ćojbašić, t., slavković, r., jordović, b., purenović, j., 2016, optimization of chemical composition in the manufacturing process of flotation balls based on intelligent soft sensing, hemijska industrija, 70(6), pp. 603-614. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 205 215 a data-mining based method for the gait pattern analysis  udc 004.8:617.3 marcelo rudek 1 , nicoli mariá silva 1 , jean-paul steinmetz 2 , andreas jahnen 3 1 pontifical catholic university of parana, department of production and system engineering, brazil 2 department of research and development, zitha senior centre on memory and mobility michel rodange, zitha, luxembourg 3 luxembourg institute of science and technology (list), luxembourg abstract. the paper presents a method developed for the gait classification based on the analysis of the trajectory of the pressure centres (cop) extracted from the contact points of the feet with the ground during walking. the data acquirement is performed ba means of a walkway with embedded tactile sensors. the proposed method includes capturing procedures, standardization of data, creation of an organized repository (data warehouse), and development of a process mining. a graphical analysis is applied to looking at the footprint signature patterns. the aim is to obtain a visual interpretation of the grouping by situating it into the normal walking patterns or deviations associated with an individual way of walking. the method consists of data classification automation which divides them into healthy and non-healthy subjects in order to assist in rehabilitation treatments for the people with related mobility problems. key words: gait analysis, pattern analysis, data mining, rehabilitation. 1. introduction actual monitoring technologies based on biometric signals have allowed important improvements in healthcare applications and assistive uses. the approaches that include biomechanics analysis and its functional signaling which focus on rehabilitation or early disease detection are being investigated in many ways as in [1-5, 19, 20]. new systems using cameras, forceplates, walkways and insole devices offer new perspectives to data received october 10, 2015 / accepted november 10, 2015 corresponding author: marcelo rudek pontifical catholic university of parana, pucpr/ production and system engineering graduate program – ppgeps, imaculada conceicao, 1155, 80215-030, curitiba, brazil e-mail: marcelo.rudek@pucpr.br original scientific paper 206 m. rudek, n. m. silva, j. p. steinmetz, a. jahnen analysis as in [6]. nowadays an increasing field is observed with the use of sensors to achieve informative data applied to the evaluation of movement of a person. the tactile sensors generate pressure distribution maps from the feet contact during walking [7]. this mapping might be used in the measurements of posture, equilibrium and motor control of the patients within the clinical evaluation of the walking process. measuring stability allows a better understanding of the people‟s necessities. in this context, the research demonstrates a special interest in the investigation of elderly people [8]. the goal is to provide support to the patients susceptible to the risk of falls due to the deterioration of their mobility caused by neurological disorders, diseases or surgery that affect their motor control [9]. for example, the systems can provide a data analysis during the patient treatment period in his physical rehabilitation to evaluate his self-control in the movements‟ progression. four essential features about the gait analysis are described in [9]. the first is how much and in what direction the given patient leans while standing still. the second feature is how much pressure the patient exerts under each foot and in what regions of the foot. the third clinical feature is how much the patient sways while attempting to stand still. the fourth clinical feature characterizes dynamic balance. in motion the whole set of features occurs at same time; that is why it is not possible to fully capture the essence of motion without tactile information [10]. there are several applications in which the medical professional needs information of healthy subjects that can be used as reference for comparison with the patient‟s behavior as in [6, 12, 13]. the pressure maps produce this relationship. one possible way of getting the visualization of the pressure map of walking is by the trajectory of the centre of pressure (cop) under each footprint in the walking map. in the standing position, as the method of [10, 11], it is used a sensor plate system to generate a graphical representation of the cop variation determined by averaging the pixel values on the location of the intensity-weighted pressure points of each foot. in that case, the cop changes according the equilibrium for the overall image, which could be considered an estimate of the person‟s centre of gravity (cog). as proved by [1, 8, 9, 10, 14 and 15], it is possible to estimate the balance information from cop. then, the cop analysis is very useful to understand the gait pattern of an individual for his rehabilitation evaluation. the information used in this research is from a tactile sensor mattress (walkway) [7]. this device generates a map containing the pressure variations of the contact points of foot during walking. this enables us to conduct the study by achieving the walking characteristics of each individual, i. e.; it provides a way to carry out balance measurement and cadence motor control analysis of each individual. the special interest is a gait analysis in elderly patients with a potential risk of falling; thus, it can act to prevent the risks and build support devices customized to the individual walking characteristics. in this wide context, the main objective of the research is to define a separation process using the automated data mining of the behavior pattern by classifying healthy or unhealthy people, or classifying different pathologies that interfere with the movement. 2. proposed method the functional structure of proposed method for the gait pattern classification is shown in fig. 1. basically, the process model has two main parts; a pool dedicated to the “data gathering and analysis” (i.e. clinical viewpoint) and the “data processing” contains four a data-mining based method for gait pattern analysis 207 inner process, as i) data normalization; ii) data warehouse; iii) data mining and iv) graphical representation. these steps are discussed in the following sections as well as the background concepts about data acquirement and their respective parameters. g ai t p at te rn c la ss if ic at io n d at a ga th er in g an d a n al ys is d at a p ro ce ss in g patient #1,...,n gathering data from walkway system ciclical exam sequence data normalization creating datawarehouse data mining procedure graphical pattern classification clinical analysis rules patient database specialized knownledge fig. 1 process modelling of the gait pattern classification 2.1. gait data background the gaitrite® [7] system is a complete tool for measuring the gait information to the indoor laboratorial approach. during the patient movement while crossing the walkway, the pressure points affected by the footfall generate two groups of measured values called spatial and temporal parameters. spatial parameters are the step length, stride length, base width, step width, stride width, toe in / toe out, leg length, length of foot, and width of foot. the temporal parameters are step time, stride time, stance time, swing time, ambulation time, velocity, cadence, single support, double support. fig.2 shows the spatial parameters used in our study and definitions are in [7]. fig. 2 parameters identification on footfall based on these generated parameters during the whole walking cycle, it is possible to compute the standard deviation and coefficient of variations for stride time, stride length, stance time and base of support in order to get complementary information about the gait 208 m. rudek, n. m. silva, j. p. steinmetz, a. jahnen (one cycle = stride). in addition, it is possible to record the footprints and respective pressure values stamped and also assess the respective cop values. as presented by [11], the risk of falls can be measured by correlation with (i) “stride-to-stride” variability in velocity, (ii) variability in stride width and (iii) double-support time. this information is implicit in the cop distribution of each foot. based on the spatial and temporal parameters, the doctor is able to evaluate the gait and provide the diagnosis related to a specific problem. some other additional information can be extracted from the cop‟s trajectory. fig 2 shows three bounded regions representing the foot position extracted from the walking along the indicated line of progression, and the inner points are the cops trajectory for each one. the positions labeled as „a‟, „b‟ and „c‟ are those of the heel of foot in its first point of contact. this is the pattern assumed by the gaitrite system as reference to stride length and step length calculation [7]. the stride width and step width might be measured by gaitrite and this relationship is also based on the heel position. toe–toe measures instead of heel–heel measures have been chosen in specific cases for calculating the gait parameters as in [16], because the measure of lengths of step and stride depends on how much pressure information exists in a specific region of the footprint. in our research we are adopting a centroid reference as indicated in fig. 2 based on the method of [14]. the cop values are exported by the gaitrite system, and it is the point of location of the vertical ground reaction force vector in each foot [14]. in motion, a set of the cop defines the trajectory of each foot while in the ground-contact during the time, based on its weighted average pressure values in the contact area. the cop position values change in various directions under the foot according to the individual sway. note that a different interpretation is given if the analysis is based on a sensor plate [17, 18] where the cop is a calculated median value between feet (cop = cog). the assumption is different in our study because the cops are individual set data from each individual footprint (cop ≠ cog). a graphical plot of its (x, y) coordinates gives a visual sense of foot movement during the footfall. to sum up, the cop is the median representation of all the affected sensors along the footfall during the foot sway and it gives a foot signature map that describes the footfall behavior [18, 21]. 2.2. data normalization and repository the gaitrite system exports a file containing the individual trajectories of the pressure centres of each foot of a given patient. each patient is submitted to five standardized testing protocols which are: 1) normal walking); 2) slow walking; 3) fast walking; 4) normal walking in conjunction with cognitive task (cognitive task is the patient speaking aloud the countdown by 2 in 2, starting with 50 or 30 depending on the condition of the patient); and 5) normal walking associated with motor activity (motor activity is the patient carrying a glass cup with 2/3 filled with water). each test combines two runs, i.e. going and coming back on the walkway. the generated file contains the gait data tabulated in ascii format as in fig 3. the file can be imported as an excel spreadsheet whose columns are: the parameter 'obj' , i.e.. the sequential number of each foot; the 'time' column records the time (in ms) in which each sensor has been activated; column 'l/r' is the respective foot in ground contact, where '0' represents the right foot, and value '1' to the left foot; columns 'x' and 'y' contain the coordinates of each centre of pressure (cop) forming part of the sensor activation trajectory line during the contact of each individual foot on mattress. a data-mining based method for gait pattern analysis 209 0.4 0.6 0.8 1 0.9 0.95 1 foot # 1 x y 0.8 0.9 1 0.8 0.9 1 foot # 2 x y 0.85 0.9 0.95 1 0.8 0.9 1 foot # 3 x y 0.9 0.95 1 0.9 0.95 1 foot # 4 x y 0.9 0.95 1 0.8 0.9 1 foot # 5 x y 0.9 0.95 1 0.9 0.95 1 foot # 6 x y 0.9 0.95 1 0.8 0.9 1 foot # 7 x y 0.85 0.9 0.95 1 0.9 0.95 1 foot # 8 x y 0.8 0.9 1 0.8 0.9 1 foot # 9 x y 0.7 0.8 0.9 1 0.95 1 foot # 10 x y fig. 3 file model to data exchange from gaitrite® and respective extracted footprints from the cop file we obtain all separate footprints graphs [23], as indicated by blue (right foot) and red (left foot) plots in fig. 3. each footprint represents the respective foot signature compound by their x and y coordinates normalized to [0-1] range. a repository database is organized by all the spreadsheets containing walking information for all patients. 210 m. rudek, n. m. silva, j. p. steinmetz, a. jahnen 2.3. data mining procedure as described by mirkovic in [25], data mining is a practical procedure based on clustering and it is widely applied to group selection. it represents roles to find patterns and also implements methods in order to describe the structural patterns in data repository. in our context, we apply this functionality in order to classify healthy or nonhealthy individuals by their cop footprints. the data is taken from a sample set with individual trajectories of the cops of each foot for a given patient. we expect that the output should give us a prediction about new samples. in our approach we have applied the rapidminer® studio software [24] to the data mining because it is an open source tool; it also has applicability for a wide range of problems. the steps of implementation are shown in fig 4. fig. 4 steps implementation of mining process as depicted in fig. 4, the mining process implementation has the following operations as defined in [24]: 2.3.1. retrieves each „retrieve‟ activity is an operator with reading function of an object from the data repository. thus, we can get data / metadata like subject and his particular set of parameters. for our process of the gait analysis we have two retrieves where one „retrieve‟ is the known data and the other is the testing data. the training set contains x and y coordinates of the footprints of all subjects, as plotted in fig. 5. 2.3.2. set roles a „role‟ operator is used to change the role for each of the attributes. the role of an attribute reflects its categorization and embeds some specific task. the special roles are: label, id, prediction, cluster, weight, and batch. for our proposal, we have two roles. a first operator set role is to assign the role identifier to the attribute "patient" just to its own identification as well as the “healthy” attribute is assigned as a label for it. the second role set is related to testing data and it carries the attribute "patient" only. (note that the healthy or non-healthy is a further output). a data-mining based method for gait pattern analysis 211 2.3.3. decision tree it is a graphical representation tool of the searching model composed by nodes and leafs [22]. the idea behind the decision tree is to create the classification model whose function is to predict the classification of a value. each inner node corresponds to one of the input attributes and each leaf node represents a value of the label attribute (the path from the root to the leaf). thus the decision tree implements the model. 2.3.4. apply model the model is first trained through a known sample, and the related information is learnt by the model. after this, the model (decision tree) can be applied to another set of values for prediction. this operation generates the processed output. 0 50 100 150 200 250 300 350 5 10 15 20 25 30 patient #2 cop [exam 1: normal walking] x (walking direction) y ( la te ra l w id th ) fig. 5 an example of footprints along walkway (from data repository) 3. results and discussion a testing condition is implemented in order to evaluate the proposed classification mechanism. the range of test includes the total of 22 healthy and 7 non-healthy patients through the amount of almost 200 footprints. fig. 6 shows an example of the cop distribution under one individual foot for both (a) healthy and (b) non-healthy cases. fig. 6 example of the cop distribution of one foot, for (a) healthy and (b) non-healthy individual x y x y a) b) 212 m. rudek, n. m. silva, j. p. steinmetz, a. jahnen in fig. 6, the points are, respectively, x and y cartesian coordinates of the cop trajectory. the normal walking protocol as experienced and the respective decision tree generated by the data mining is presented in fig. 7. fig.7 decision tree to pattern classification from footprint parameters the parameters of the searching model are, the contacting „time‟ of each foot (heeltoe) on ground and object „obj‟ (right or left foot) as the nodes in the decision tree. the leaves are „x‟ and „y‟ coordinates as a deviation in the cop trajectory in each foot on walkway (as normalized data examples previously presented in fig. 6). fig. 8, in (a) presents the comparative graphical representation about the gait pattern classification executed between the real data and the respective mining classification. a data-mining based method for gait pattern analysis 213 fig. 8 (a) real data from all patients; (b) classification for testing subject a) b) 214 m. rudek, n. m. silva, j. p. steinmetz, a. jahnen from graphs in fig. 8, in (a) presented are in red all the footprint plots for the healthy people set and the blue are footprints for a non-healthy group. the red or blue points are the trajectory of each patient along the walkway. the representation is based on a pair of coordinates of their respective cop trajectories. the graph in (b) shows the footprints classification result based on the decision tree. in this case all the parameters, i.e. „time‟, „object‟ and „x‟, „y‟ cop coordinates for each patient are classified according to the decision tree. in the same way the graph shows the footprints in red and blue for healthy and non-healthy subjects, respectively. thus it is possible to have a visual interpretation of the pattern classification. in graph (b) it possible to see that the resulting data for a testing subject are very similar to the classification ones. surely, we can generally understand how the classification works; however, the visual sense of the pattern is the only information possible to get from the graphs. the displacement measurement cannot be estimated and an additional mechanism should be implemented to improve this evaluation. 3. conclusions the text presents a proposal of the method developed to provide the classification of individuals in two possible conditions concerning their gait, so called healthy and nonhealthy subjects, through analysis of their gait patterns. the centre of pressure (cop) trajectory in each foot is calculated by the gaitrite system and the parameters (time, footprint position) can be used to define gait characteristics. a data mining approach is implemented as a computational tool for the support in the gait pattern classification. the main application of method is to assist the clinical diagnosis and respective treatment evaluation. a decision tree is modeled in order to provide the classification of objects according to the known training group. the example has shown that a selected patient is classified according its condition, i.e., his gait behavior is mapped inside the correct group. then, the tested features have contributed to the classification of individuals regarding their similarity, and as demonstrated, the proposal is a promising tool in the gait analysis context. for future works, we want to improve the classification (decision tree) for patient classification by their associate deceases adding machine learning strategies. 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waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 12, no 1, 2014, pp. 73 84 design of small bulb turbines with unequal specific work distribution of the runner's elementary stages  udc 621.22 jasmina bogdanović-jovanović 1 , božidar bogdanović 1 , ivan božić 2 1 university of niš, faculty of mechanical engineering 2 university of belgrade, faculty of mechanical engineering abstract. regarding the present state of knowledge in the field of the turbomachinery design, the method for designing small bulb turbines with unequal specific work distribution of the turbine runner's elementary stages near the hub is presented in the paper. the distribution function of specific work of all the elementary stages is obtained, according to which the averaged axisymmetric flow surfaces of the turbine runner have a negligibly small deviation from the cylindrical flow surfaces. the specific work of the near-the-hub elementary stages, in the given distribution function, can be reduced up to 60% of the required (design) specific work, still achieving nearly cylindrical flow surfaces. key words: bulb turbine, elementary stages, specific work, axisymmetric flow surfaces 1. introduction both the hydraulic turbines design and their operating performance analysis require the use of simpler methodologies in preliminary design phases, especially when the geometry of the turbine runner is not completely defined. in the field of the turbomachinery design, such attempts are continuously made thus providing us with different methodologies and procedures for the design process optimization [10, 11]. with the computer technology development, a significant breakthrough of numerical methods and numerical simulation of the fluid flow has been made, encouraging the numerical techniques incorporation into the design procedure and performance analysis of the turbomachinery [12, 13]. received november 15, 2013 / accepted january 19, 2014  corresponding author: jasmina bogdanovićjovanović university of niš, faculty of mechanical engineering, department of hydroenergetics, niš, serbia e-mail: bminja@masfak.ni.ac.rs original scientific paper 74 j. bogdanović-jovanović, b. bogdanović, i. božić a number of assumptions and simplifications, empirical equations as well as the designer's experience always accompany the designing process [1, 2, 5, 7, 9]. the basic assumption in the hydraulic turbine designing procedure is axisymmetric flow surfaces. an elementary stage is a flow space between two elementary immediate axisymmetric flow surfaces. the intersection of stay vane (sv), or turbine runner (tr), and axisymmetric flow surfaces, defines the vane or blade profiles in the turbine elementary stages. stay vanes and blades are formed by the stay vane profiles and the runner blade profiles in the turbine's elementary stages. the vane and blade profiles in the turbine elementary stages can be defined using suitable hydraulic calculations for a twodimensional fluid flow through the profile cascade of axisymmetric flow surfaces. in the cases when the flow surfaces are cylindrical, the theory of fluid flow through the straight plane profile cascade can be used in calculations [6]. in order to achieve cylindrical or almost cylindrical flow surfaces in the bulb turbine runners, these runners are usually designed to obtain an equal specific work of all the elementary stages. the turbine runners designed according to such a principle are made of blade profiles with a significantly larger inclination angle near the hub than near the shroud. in order to minimize the blades' spatial curvature and to reduce the axial length of the runner, it is eligible to design the runner with smaller specific work of the elementary stage nearer the hub than the blade periphery [3, 4]. in this way, the spatial curvature of the stay vanes is also reduced. the possibility to design such stay vanes and turbine runners is analyzed in the paper. only small bulb turbines, which can be used in small hydropower plants or as models of large water turbines (for the model testing purpose) are considered in this study. 2. basic formulas the scheme of a bulb turbine meridional cross-section, with traces of two elementary immediate axisymmetric flow surfaces, where one (below) is marked as sm, is presented in fig. 1. a flow space between two elementary immediate axisymmetric flow surfaces represents an elementary stage of the runner. to determine the shape of stay vanes and runner blades, it is sufficient (for small turbines) to define profiles in 7 to 12 elementary stages, approximately evenly distributed along the blade height. fig. 1 scheme of a bulb turbine meridional cross-section design of small bulb turbines with unequal specific work distribution of the runner's elementary stages 75 the runner is designed as a free-vortex flow (to obtain zero circumferential component of absolute flow velocity at outlet (cu2=0)); therefore, based on the euler's equation, the specific work of the turbine runner elementary stage is: 1 1( ) ( ) ( )k m m u my s r s c s   , za 2 ( ) 0u mc s  , (1) where: cu, cr, cz – circumferential, radial and axial component of absolute flow velocity [m/s] (absolute flow velocity o o oc= u r zu r zc c c  ), r1 – radius in cross-section 1 [m],  – angular velocity of turbine runner [s -1 ]. according to eq. (1), the required circumferential component of absolute flow velocity, in front of the inlet of the turbine elementary stage, can be defined by formula: 1 1 ( ) ( ) ( ) k m u m m y s c s r s   . (2) stay vanes produce circumferential components of absolute velocity in front of the runner inlet. neglecting the influence of viscous friction, the flow in area between sv and tr (vaneless area) is free-vortex flow: 1 1' 1 1( ) ( ) . ( ) ( ) ( ) ( )m u m m u m m u mr s c s const r s c s r s c s      , therefore, the circumferential components of absolute velocity of the stay vane elementary stages can be calculated as: 1 1' 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) m k m u m u m m m r s y s c s c s r s r s     . (3) streamline inclination angle (with respect to circumferential coordinate our  , where ou is the unit vector in the direction of circumferential velocity u , ou = uu  , = u r  ) in the outlet of stay vane area is calculated by the next formula: 1' 1 1' ( ) ( ) ( ) m m m u m c s s arctg c s   , where 2 2=m r zc c c . (4) to determine the inclination angles of vane profile chamber line, in the outlet of the stay vane elementary stages, l. l .1( ) ( ) ( )b m m b ms s s     , it should be assumed that the value of flow inclination angle in outlet of the stay vane is ( o .1 ( ) (2 5)b ms   ). in front of the inlet of the stay vane, there is a zero circumferential component of the absolute velocity (cu0 = 0), thus, the inclination angle of the camber line in the inlet of the turbine stay vane is 1.a(sm) = 90 o . design operating parameters of the turbine are: , q = q + , yt = yt + , where q + and yt + are the best efficiency (optimal) operating parameters (when the turbine operates with maximum efficiency,  = max =  + ), q [m 3 /s] is turbine volume flow rate and yt [j/kg] is turbine specific work ( t ty gh (g = 9.81 m/s 2 ), ht – net turbine head [m]). adopting the maximum value of hydraulic efficiency (h = h + ), design specific work of the turbine runner is yk = yk + = h + yt + . when designing a turbine using the model of equal specific work of turbine elementary stages (yk (sm) = const. = yk + ), and when flow surfaces in the runner are cylindrical (r1(sm) = r2(sm), cz1 = cz2 = const.), the theory of flow 76 j. bogdanović-jovanović, b. bogdanović, i. božić in the straight plane profile cascades is used. inclination angles of profile chord in elementary stages of the turbine runner (t = t (sm)) are defined by the lift force method, where, using the above mentioned theory, the well known formula is used [1, 2]: =2 u y wl t w    , (5) where: y – profile lift coefficient, l – length of the chord line, t – cascade spacing, wu = wu1  wu2 (wu = cu, for cu2 = 0), w = c u is relative velocity of water flow, o o ow= u r zu r zw w w  ( wu = cu  u, wr = cr , wz = cz) and w – value of averaged velocity in infinity ( 1 2w 0.5(w w )   ). for elementary stages of the runner there are: y = y (sm), t = t (sm), l = l (sm), wu = wu (sm) and w = w (sm). since coefficient y depends on inclination angle of the profile chord line, based on formula (5), for calculated valuey , the inclination angle of the profile chord line can be determined respectively. in the case where flow surfaces in the runner are not cylindrical (when flow components are not uniform along the radius), volume flow rate q (sm) passing through the flow space between the hub and flow surface sm, is calculated as follows: 1 2( ) ( ) 1 2( ) 2 ( ) 2 ( ) m m i i r s r s m z z r r q s c r rdr c r rdr     , (6) where, cz1(r) and cz2(r) are distribution functions of axial components of absolute flow velocities in the control cross-sections in front of and behind the runner. functions cz1(r) and cz2(r) should satisfy the following formula of volume flow through the runner: 1 22 ( ) 2 ( ) e e i i r r z z r r q c r rdr c r rdr     . (7) for given function yk (r), the specific work of turbine runner can be calculated using the formula: 2 2 1 1 ( ) 2 ( ) ( ) e i r k k k z a r y y r dq y r c r rdr q q              . (8) 3. conditions for obtaining cylindrical flow surfaces in the control crosssections in front of and behind the turbine runner the necessary condition is that the control cross-sections are placed in the space which is physically bounded by cylindrical surface of the hub (turbine hub radius ri = const.) and the shroud (turbine shroud radius re = const.), as shown in fig. 1. the equation of steady fluid flow of the inviscid fluid, disregarding gravity acceleration, can be written in the form: design of small bulb turbines with unequal specific work distribution of the runner's elementary stages 77   1 , tc rotc gradp    , (9) where p is static pressure [pa] and pt the total pressure ( 2 / 2tp p c  ) [pa]. for the fluid flow on cylindrical surfaces (cr = 0, o oc u zu zc c  ) in control cross-sections in front of and behind the turbine runner (where / 0   ), according to the component of vector eq. (9) in the direction of coordinate r, a differential equation is obtained: ( ) 1u u tz z c rc pc c r r r r         , (10) which defines the condition for cylindrical flow surfaces in control cross-sections 1-1 and 2-2. in control cross-section 2-2 (outlet of the runner) cu2 = 0 and pt = pt,2 = const., therefore, according to equation (10) it is cz = cz2 = const. defining the hydraulic efficiency of the runner elementary stage as: 1 2 1 2 . ( ) ( ) ( )= ( ) ( ) ( ) k m k m h k m t m t m t m y s y s s p s p s p s          . (11) and, assuming that hydraulic efficiency is the same for all elementary stages ( . .( ) = .h k m h ks const  ), the total pressure in control cross-section 1-1 (inlet of the runner) can be expressed as: 1 2 1 . ( ) ( ) k t t h k y r p r p     . (12) according to eq. (1): 1 1 1 ( )u krc y r   , i.e. 1 1 1 ( )u kc y r r  (13) where 1( ) ( )k k my r y s , due to 1 1( )mr r s . considering eqs. (12) and (13), eq. (10) for obtaining cylindrical surfaces in the control cross-sections 1-1 (for 1 1( )t tp r p , 1z zc c , 2u uc c ) yields: 11 1 12 2 . ( )1 1 ( ) kz z k h k y rc c y r r rr           . (14) where: 2 2 . ( )1 0k h k y r r     . for equal specific work of the runner elementary stages ( ( )k ky r const y   ), due to equation (14) it is obtained cz1 = const., and due to eqs. (6) and (7) cz1 = cz2 and r1(sm) = r2(sm), thus the assumption of the cylindrical flow surfaces in the runner is reasonable [8]. if / 0ky r   , according to equation (14), 1 / 0zc r   , thus, due to equation (5), for 2 zconst. czc   , can be resolved that 1 2( ) ( )m mr s r s and flow surfaces are approximately conical in the runner. 78 j. bogdanović-jovanović, b. bogdanović, i. božić further on, the relationship between the hydraulic efficiency of turbine runner ( . ( )h k ms ) and turbine hydraulic efficiency ( ( )h ms ) is obtained. denoting .t ip and .t iip as total pressures on inlet (i) and outlet (ii) of turbine, the hydraulic efficiency of the turbine elementary stage is defined by relation:   . . ( ) ( ) ( ) ( ) ( ) k m k m h m t m t i m t ii m y s y s s y s p s p s      . (15) since 1 1 2 2 1 1 2 2. .( ) ( ) i ii i iit i m t ii m t t t g g gp s p s p p p y y y                 , where 1igy  and 2 iigy  are specific losses of flow energy in the flow passages from i to 1 and the flow passages from 2 to ii, it is easy to predict that there is a relationship between ( )h ms and . ( )h k ms : 1 . 1 2 1 ( ) h h k i ii           , or . 1 1 2 1 ( ) h k h i ii           , (16) where, ( )h h ms  , . . ( )h k h k ms  , 1 1( )i i ms   and 2 2 ( )ii ii ms   are dimensionless losses of flow energy in the passages from i to 1 and the flow passages from 2 to ii (turbine diffuser), . 1 1 1 ( ) ( ) ( ) g i m i i m k m y s s y s       and .2 2 2 ( ) ( ) ( ) g ii m ii ii m k m y s s y s       . (16') since 1 2 .1 2 1 2( ) ( ) ( ) (1 ( )) ( )t m k m g m m k mp s y s y s s y s            , where 1 2 ( )ms   .1 2 ( ) / ( )g m k my s y s , based on eq. (11), it can be written: . 1 2 1 ( ) 1 ( ) h k m m s s     . (17) according to eq. (16), it can be written: 1 1 2 2 1 1 ( ) h i ii           , i.e. 1 1 2 2 1 1i ii h           . (17') dimensionless loss i1 contains flow energy losses of the stay vane. the designing assumption is that terms i1 and 12 are larger than dimensionless losses in the diffuser, 2ii. 4. suggestion for function ( )ky r in order to reduce specific work of elementary stages near the hub, and to achieve flow surfaces inside the runner that do not deviate much from the cylindrical surfaces, the distribution function of elementary stages specific work is suggested as follows: 2 0 .0 0 ( ) 2 , for and ( ) const., for k o i k k e y r a r b r b r r r r y r y r r r               (18) where is ( ) / 0ky r r   for 0r r . design of small bulb turbines with unequal specific work distribution of the runner's elementary stages 79 function yk (r) is defined according to radius r in the control cross-section 1-1 (in front of the runner). function graph yk (r), which is defined by equation (18) is presented in fig. 2. function yk (r) is defined according to accepted parameters ro and yk.i = yk (ri). to obtain flow surfaces in the turbine runner with only a small deviation of the cylindrical surfaces the following is recommended: 0 . 1 ( ) and (0.6 0.7) 2 i e k i kr r r y y     , where ky  is design specific work of the runner. the value of specific work yk.o is an unknown value and it can be defined using a requirement that specific work of runner (yk) obtained by eq. (8), is equal to the design specific work of runner (yk = yk + ). value .0ky is determined, as will be shown later, using an iterative procedure. the coefficients a and b in the first eq. (18) are calculated using formulae: 2.0 . .0 02 0 and a= ( ) k k i k i y y b y r b r r     , (19) obtained from the requirement .( )k i k iy r y , for r = ri and 0 .0( )k ky r y , for r=r0. assuming that the flow surfaces in the control cross-section in front of the runner (11) are cylindrical, based on eqs. (14) and (18), it is obtained: 2 1 02 1 1 0 0 , for and . ( ), for z i z z e c r r r r r r r c const c r r r r                     (20) where the coefficients are: 0 2 1 3 4 h b r b          , 2 1 4 h b b          , 2 0 2 2 4 a r b b    and 0 2 4r ab    (21) integral of the first eq. (20), for ri  r  ro, becomes: 2 2 2 1 1 1 1 1 ( ) ( ( )) ( ) ( ) ln 2 o z z o o o o r c r c r r r r r r r r                  . (22) in the previously shown function cz1(r), for ri  r  r0 , there is an unknown velocity value cz1(r0) (axial component of flow velocity cz1 = cz1(r0), for r0  r  re). this velocity can be defined according to the requirement that the flow rate calculated by eq. (7) is equal to the design turbine flow rate (q = q + ). since coefficients a and b, and also coefficients α, β, γ and δ, depend on unknown variable yk.0, it can be concluded that function cz1(r0) is determined using an iterative procedure. fig. 2 function yk (r) 80 j. bogdanović-jovanović, b. bogdanović, i. božić since cz1(r) = cz1(r0), for r0  r  re, and for ri  r  r0 z1c (r) is defined by eq. (22), eqs. (7) and (8), used for determination of values yk.0 and cz1(r0), can be derived into forms: 0 2 2 1 1 0 02 ( ) ( )( ) i r z z e r q c r rdr c r r r     (23) 0 2 2 1 .0 1 0 0 1 1 2 ( ) ( ) ( ( )( )) i r k k z k z e r y y r c r rdr y c r r r q q     . (24) 5. determination of .0ky and 1 0( )zc r values .0ky and cz1(r0) are determined by using an iterative procedure. perhaps, it is better to say that this is a double iterative procedure, since in each step of determining .0ky , cz1(r0) should be determined as well by another iterative procedure [3]. in order to reduce the number of iterative steps for determining .0ky , this value is obtained in the first iterative step, using a formula: (1) . .0 1 k k i k y k y y k      , (25) where k = k (r0, ri, re) is: 2 2 2 4 4 3 3 0 0 0 0 0 2 2 2 0 ( ) 0.5( ) 1.333 ( ) ( )( ) i i i e i i r r r r r r r r k r r r r         . (25') formula (25) is derived using the assumption that the axial components of flow velocities in the control cross-section in front of the runner change negligibly (for 1 const.z zc c  ). using value (1) .0ky , which was determined by formula (25), the procedure of determining yk.0 is completed in maximum of three iterative steps. the reason for this, as calculations showed, is that cz1(r) values change less than specific work yk (r), for ri  r  r0. in iterative steps of determining yk.0 correction of this value is performed, requiring that calculated specific work (yk) using formula (24) slightly differs from the design specific work of the turbine runner yk + ( / 0.005k k ky y y   ). in each iterative procedure of determining yk.0 the value cz1(r0) is also obtained by an iterative procedure. in the first iterative step it can be taken that z1 0c ( ) 1.2 zr c , where: 2 2/[ ( )]z e ic q r r  , for q = q + . (26) in iterative steps of determining cz1(r0) a correction of this value is performed, and according to formula (23), calculated flow rate q slightly differs from design flow rate q + ( / 0.005q q q   ). in the program for determining yk.0 and cz1(r0) [4], in the iterative procedure of solving the problem, cz1(r0) changes with the iterative step 0.005 zc  , and yk.0 changes with the design of small bulb turbines with unequal specific work distribution of the runner's elementary stages 81 step 0.005 ky   . the integrals in formulae (23) and (24) are calculated using a trapezoid rule, whereas in the integration area from ri to r0 calculation points are distributed for the iterative step δr = 0.0025m (2.5mm). besides printing the calculated values yk.0 and cz1(r0), program [4] is also developed to print values yk (r), cz1(r) and q(r), for r  [ri,re], for each iterative step δr = 0.0025m, where r – radius in control cross-section in front of the turbine runner and q(r) is the volume flow rate under the cylindrical flow surface of radius r. in the control cross-sections before (1-1) and behind (2-2) turbine runner, the flow surfaces are cylindrical, where z2c const. zc  . according to q(r) data, where r = r1(sm), the functional relationship of cylinder radius in the same axisymmetric flow surfaces can be obtained. regarding equation (6), for q(sm) = q(r), r2(sm) = r2(r) and z2c const. zc  it follows: 2 2 ( ) ( ) i z q r r r r c    , (27) where 1 = ( )mr r s and 2 2 ( )mr r s . for r2(r) < 1, as obtained for / 0ky r   , the flow surfaces in the turbine runner are approximately conical and r1(sm) = r2(sm). to determine the real flow surfaces' deviation from the cylindrical shape, the dimensionless function is 1/ 2 2( ) / ( )r r r r r . in cases where 1/ 2 ( ) 1.03r r  , it can be said that the real flow surface in the turbine runner negligibly vary from the cylindrical surface. the same calculation procedure has been used for determining values yk.0 and cz2(r0) and coefficients a, b, , ,  and  like in the computer program given in ref. [4]. even though the program is originally created for axial flow fans, the same code can be used for bulb turbine as well, replacing value h with 1/h and value cz2(r0) with cz1(r0). diagrams of yk (r), cz1(r) and 1/ 2 ( )r r are given in fig. 3, for yk.i = 0.7yk + and different values of r0 (r0 = 180, 210, 240, 280 and 310 mm) of one small bulb turbine, which is built in the small hydro power plant "grcki mlin" near prokuplje. the geometrical and design operating parameters of the turbine are: ri = 142 mm, re = 355 mm, q + = 1.4m 3 /s, yk + = 20.36 j/kg and  = 41.9 r/s (n = 400 min -1 ). for adopted η = 0.75 (ηm = 0,94, ηq = 0.96 and 1-2 = i-1 + 2-ii) it follows that h.k = 0.91. as can be concluded from fig. 3, for yk.i = 0.7yk + = 14.2 j/kg, 1/ 2 ( ) 1.03r r  is obtained (when flow surfaces in the runner can be considered cylindrical), for r0  210 mm. it can be shown easily that for values yk.i = 0.6yk + = 12.2 j/kg, 1/ 2 ( ) 1.03r r  is obtained for r0  190 mm. for the observed turbine it is ri / re = 0.40. the bulb turbine in the small hydro power plant "grcki mlin" on the river toplica, near prokuplje, is a small turbine (pt = 26 kw); therefore, the blades are shaped using panel-like profiles of the runner elementary stages. the runner consists of 4 blades, and due to the technical reasons the blade profile thickness is: i = 16 mm near the hub and e = 10 mm on the blade periphery. due to structural constraints, the profile lengths are li = 296 mm (ri = 142 mm) and le = 525 mm (re = 355mm). the requirement is to maintain geometric parameters ri = 142 mm and re = 355mm. 82 j. bogdanović-jovanović, b. bogdanović, i. božić the calculation based on the model of unequal specific work of the elementary stages, that are changed by the rule given in equation (18), for yk.i = 0.6yk + = 12.2 j/kg and r0  180 mm (when 1 0( ) 4.27 m/szc r  , yk.0 = 20.68 j/kg, a = 169.5, b = 5871,  = 47065,  = 104426,  = 7356,  = 408.2 and 1/2r (r) 1.023 ) the following is obtained: t.i = 31 o and t.e = 15.8 o . the inclination angle of the blade profile is t.i  t.e = 15.2 o , which is 6.2 o (29%) smaller than the blade profile designed based on the model of the equal specific work of the turbine elementary stages. the values of iterative steps and defined relative errors of specific work and volume flow rate can be selected in the program for determination of yk.0 and cz1(r0) [4]. however, the values applied in the above mentioned example of the bulb turbine give the results that are accurate enough for the technical practice. fig. 3 diagrams of ( )ky r , 1( )zc r and 1/ 2 ( )r r design of small bulb turbines with unequal specific work distribution of the runner's elementary stages 83 conclusion the distribution function of specific work in turbine elementary stages (yk(r)) are proposed in the paper. in addition, function 1/ 2 ( )r r is defined, according to which the flow surfaces deviation compared to the cylindrical shapes can be determined. using the proposed functional distribution of specific work of the turbine elementary stages, with the fulfilled condition that the flow surfaces negligibly deviate from the cylindrical surfaces, the designed bulb turbine blades are less twisted. this design method is applicable to stay vanes, obtaining less twisted stay vanes as well. acknowledgement: this paper is result of technological project no. tr33040, revitalization of existing and designing new micro and mini hydropower plants (from 100 to 1000 kw) in the territory of south and southeast serbia, supported by ministry of science and technological development of the republic of serbia. references 1. babić, m., stojković, s., 1990, basics of turbomachinery: operating principles and mathematical modeling, university of kragujevac, naučna knjiga, beograd, serbia (in serbian). 2. benišek, m., 1998, hydraulic turbines, faculty of mechanical engineering, belgrade (in serbian). 3. bogdanović, b., bogdanović-jovanović, j., spasić, ž., 2010, designing of low pressure axial flow fans with different specific work of elementary stages, the international conference proceedings: mechanical engineering in xxi century, pp. 99-102, niš. 4. bogdanović, b., bogdanović-jovanović, j., todorović, n., 2011, program for determination of unequal specific work distribution of elementary stages in the low-pressure axial flow fan design procedure, facta universitatis, series mechanical engineering, 9(2) pp.149-160. 5. obradović, n., 1974, turbocompressors, tehnička knjiga, beograd (in serbian). 6. bogdanović, b., bogdanović-jovanović, j., spasić, ž., milanović, s., 2009, reversible axial fan with blades created of slightly distorted panel profiles, , facta universitatis series mechanical engineering, 7(1) pp.23-36 7. ristić b., milenković, 1996, small hydropower plants, naučna knjiga, beograd (in serbian). 8. bogdanović-jovanović, j., bogdanović, b., milenković, d., 2012, determination of averaged axisymmetric flow surfaces according to results obtained by numerical simulation of flow in turbomachinery, thermal science, 16(2) pp. 647-662. 9. jovičić, n., babić, m., jovičić, g., gordić, d., 2005, performance prediction of hydraulic turbomachinery, facta universitatis, series mechanical engineering, 3(1) pp. 41-57. 10. albuquerque, r.b.f., manzanares-filho, n., oliveira, w., 2007, conceptual optimization of axial-flow hydraulic turbines with non-free vortex design, imeche 221, part a: j. power and energy, pp.713-725. 11. hofler, e., gale, j., bergant, a., 2011, hydraulic design and analysis of the saxo-type vertical axial turbine, transactions of the canadian society for mechanical engineering, 35(1), pp.119-143. 12. da cruz, a.g.b., luiz, a. mesquita, a.l.a., blanco, c.j.c, 2008, minimum pressure coefficient criterion applied in axial-flow hydraulic turbines, j.of the braz.soc. of mech. sci. & eng., 30(1), pp.30-38. 13. ferro, l.m.c., gato, l.m.c., falcão a.f.o., 2011, design of the rotor blades of a mini hydraulic bulbturbine, renewable energy 36, pp.2395-2403. 84 j. bogdanović-jovanović, b. bogdanović, i. božić projektovanje malih cevnih turbina sa različitim jediničnim radovima elementarnih stupnjeva obrtnog kola uzimajući u obzir dosadašnja saznanja iz oblasti projetovanja turbomašina, u ovom radu je prikazan metod projektovanja malih cevnih turbina sa različitim jediničnim radovima elementarnih stupnjeva turbinskog kola u okolini glavčine. data je funkcija raspodele jediničnih radova elementarnih stupnjeva turbinskog kola, pri kojoj osnosimetrične strujne površine u turbinskom kolu zanemarljivo malo odstupaju od cilindričnih strujnih površina. u datoj funkciji raspodele, jedinični rad elementarnog stupnja turbinskog kola može se, uz glavčinu, smanjiti i na 60% proračunskog jediničnog rada turbinskog kola, a da strujne površine u obtnom kolu budu priližno cilindrične. ključne reči: cevna turbina, elementarni stupanjevi, jedinični rad, osnosimetrične strujne površine. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 181 191 1non-linear mathematical models in the theory of experimental design: application in the manufacturing processes udc 621.7+519.2 anđela lazarević, dragoljub lazarević faculty of mechanical engineering, university of niš, serbia abstract. since real processes usually are not linear, different non-linear mathematical models are usually applied for studying their behavior: high order polynomials, power, exponential and other functions. in this paper, the application of multiple power functions is presented, with and without interactions between the influential factors, in modelling various technological problems. the modeling results show that the proper mathematical model is selected, one that guarantees high accuracy in the entire experimental space. furthermore, an important conclusion is reached, stating that interactions between the influential factors are not of importance in such mathematical models; thus they can be neglected. therefore, it is shown that it is more practical to use the basic mathematical model (without interactions) than the expanded mathematical model (with interactions) key words: experimental design, non-linear mathematical models, plasma cutting, face milling 1. introduction the theory of experimental design represents a qualitatively new approach to the theoretical-experimental analysis and optimization of complex processes/systems, with universal application and range of advantages in comparison to the concept and practice of the one-factor-at-a-time method [1-2]. the theory of experimental design addresses management of an experiment, i.e., its preparation and physical realization as well as processing and analysis of the experimental results according to the previously determined plan, which enables the variations of the influential factors simultaneously on various levels, in each of the following series of trials. the experimental design was proven as successful method in various fields, which is especially evident in complex research objects with a large number of influential factors. received june 4, 2015 / accepted july 20, 2015 corresponding author: anđela lazarević affiliation: faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, 18000 niš e-mail: andjela.lazarevic@gmail.com original scientific paper 182 a. lazarević, d. lazarević the basic characteristics and advantages of the theory of experimental design are as follows [1,5,6]: number of trials, successive experiment management, i.e., in stages (step by step), from simpler to more complex designs, a simple statistical (regression and dispersion) analysis of experimental results, a possibility of qualitative and quantitative assessment of effects of each of the influential factors and their interactions on the target function, easy optimization of a process/system which is the subject of research, on the basis of obtained empirical (regression) model of the target function that encompasses the complete experimental space, minimum time and material losses (expenses) for experiment realization, elimination of the subjective influence of the researcher, etc. [7]. the outline of recommended procedure in the theory of experimental design, as a statistical approach in designing and analysing an experiment, include [1-4]: recognition and statement of the problem, choice of factors, levels, and range, selection of the response variable, choice of experimental design, performing the experiment, statistical analysis of the data, and conclusions and recommendations. the selection of mathematical model (regression equation), which is used to establish a connection between influential factors and target function, depends on the research objective, the complexity of the phenomenon under study, the selected experimental design, as well as quantity and quality of available information. both theory and practice have shown that in most cases the best choice is a mathematical model in the form of polynomials (linear, quasi-linear, square, etc.). more complex mathematical models ensure higher accuracy in the prediction of researched system behaviour in the selected hyperspace. however, such models also imply a more complex experimental procedures, higher expenses, time-consuming analysis and interpretation of experimental results etc. on the other hand, the selection of the simplest linear mathematical model does not enable an analysis of the factor interactions’ effects; various examples undoubtedly show that the factor interactions’ effect on the target function can be more significant than that of separate factors. it should also be emphasized that it is the case of lower-order factor interactions, since the higher-order factor interactions can be neglected. namely, interactions of many factors, as a rule, do not influence the accuracy of the selected mathematical model. various experiments and practical knowledge point to the fact that the mathematical models used for the analysis and optimisation of complex processes are those in the form of multiple power, exponential, or some other function instead of polynomials. such mathematical models are easily transformed into linear functions (first order polynomials). therefore, the synergy of a complex non-linear mathematical model and simple processing and analysis of experimental data is established [8]. upon linearization of the aforementioned functions, a basic mathematical model without interactions is obtained. as it was mentioned, the influence of interactions should not be rejected a priori. the choice of appropriate criterion for building the mathematical model is not often obvious [9-10]. the accuracy of basic non-linear models without interactions and expanded non-linear mathematical models with interactions are comparatively analyzed on examples given in this paper. non-linear mathematical models in the theory of experimental design... 183 2. complex power functions as mathematical models numerous experiments have shown that a successful approximation can be made for modelling most diverse technological processes by using a complex power equation [1116] in the following form: ki p k p i ppp c xxxxxcf  321 321 (1) where xi (i=1,2,3…k) influential factors (natural coordinates), c, pi constants to be determined, k number of factors. by using logarithmic form for the eq. (1) it could be rewritten in a reduced linear form: kkii xbxbxbby  110 (2) where iiiic xxpbcbfy ln,,ln,ln 0  eq. (2) is represented in coded coordinates. the connection between natural and coded coordinates is established through the following transformation equations: ki xx xx x ii ii i ....3,2,1; lnln lnln 21 minmax max     (3) according to eq. (3), coded coordinates take integer values (xi=+1  0  -1). for the purposes of dispersion analysis it is necessary to repeat trials at certain points in the experimental hyperspace. the systems of trial repetition are as follows: 1. repetition for n0 times only in the central point of the experimental design (xi=0); 2. uniform repetition for n times in each vertex of an experimental hypercube (xi=1); 3. non-uniform repetition for nu times in certain vertices of an experimental hypercube, or possibly, only in one point. it should be emphasised that for such mathematical model the determination of basic levels of influential factors is conducted by applying the following relations: minmax 2 0 iii xxx  (4) after determining the coefficients of linear regression eq. (2), the unknown constants of target function eq. (1) are calculated using the following formulas: )ln(exp; )/ln( 2 max 10minmax i k i i k i i ii i i xpbc xx b p    (5) the mathematical model can be expanded by introducing interactions of influential factors, when eq. (2) turns into the quasi-linear form. by introducing transformation equations into this equation, one gets the expanded form of power function eq. (1) in the form: )ln...lnln...lnlnln ... 3 ln 2 ln 1 ln 123 ... 2 ln 1 ln 12 (exp...2 2 1 1 k x j x i x kijlm p l x j x i x mijl p xxxpxxp p x p xck      (6) 184 a. lazarević, d. lazarević 2.1 selected examples of mathematical models application example i: modelling of the kerf in plasma arc cutting the objective of the plasma cutting process is to concentrate a large amount of energy on a small surface of a workpiece which leads to intensive heating of the material surface. the source of energy is high temperature and high speed ionized gas. the gas is ionized by means of a direct current passing between the cathode (inside the nozzle) and the anode (workpiece). the plasma jet cuts the material by releasing the energy spent for the plasma gas ionization upon hitting the workpiece surface. the removal of the melted material from the cutting zone is done by the plasma jet kinetic energy. even though it is the case of a complex process, which is characterised by a large number of influential factors, the previous analysis has shown that this number can be reduced to three influential factors (input values, independent variables): cutting current (i), cutting speed (v) and material thickness (s) (table 1). as the target function (output value, dependent variable) one of the basic characteristics of the cutting quality is selected – kerf (w). influential factors were varied on two levels. table 1 cutting factors and their levels cutting factor symbol unit factor levels level 1 (low) level 2 (high) cutting current i a 45 80 material thickness s mm 4 6 cutting speed v m/min 0.9 1.3 for the purpose of this research, a number of experiments were done [17]. the entire experiment consists of 25 trials, while 8 measurement results are used for regression analysis (table 2). the main experimental matrix (used for all the examples in this paper) is represented by the full factorial design of type 2 3 . such a design enables the application of mathematical models including independent assessment of the main effects, as well as all influential factor interactions. straight-line cuts are made in the performed experiment by varying input process factors, according to table 1. test samples are made of stainless steel x10crnimn-1610-2 en designation (en10025). the experimental research of plasma cutting process is conducted on the cnc machine for plasma cutting, type hpm steel max 6.25. the given data on the kerf represent mean values from obtained 3 measurements. for the purposes of using eqs. (1-6), the following substitution of variables was introduced: fc=w; x1=i; x2=v; x3=s. the need to determine the unknown constant c and exponents p1; p2; p3 in this case arises from the basic eq. (1). the regression analysis is conducted, coefficients of the linear mathematical model are generated, and thus the regression equation in coded coordinates obtains the following form: 1 2 30.62338 0.03463 0.04263 0.008125y x x x    (7a) non-linear mathematical models in the theory of experimental design... 185 this equation is easily transformed, in the afore-described manner, into non-linear regression equation in natural coordinates: 04008.023186.012038.008073.1 sviw  (7b) the kerf increases with the increase in cutting current and material thickness since the exponents i and s are positive, whereas the exponent of cutting speed is negative, indicating decreasing tendency of kerf, with increasing cutting speed. eq. (7b) is suitable for application in engineering practice. by introducing interactions into the eq. (7a), the quasi-linear regression equation is obtained in the form: 3213231 21321 008125.0006125.0007375.0 001125.0008125.004263.003463.062338.0 xxxxxxx xxxxxy   (8a) in the natural coordinates, this regression equation has the form: )lnlnln7577.0lnln266585.3lnln185933.0ln ln225267.1(exp75.3 73409.050963.517675.0 1 svisvsiv isviw    (8b) for the purposes of verification of the generated mathematical models, the results from 17 trials are used, otherwise not used for the purposes of regression analysis, as shown in table 3. this experiment confirmation, which contains a great number of trails, should contribute to higher reliability of conclusions from the following analysis. table 2 main experiment i v s wexp wcal   1 80 (+1) 1.3 (+1) 6 (+1) 1.88 1.8776 1.8655 0.12770.7713 0.1277 0.7713 2 45 (-1) 1.3 (+1) 6 (+1) 1.75 1.7507 1.7406 0.0400 0.5371 0.0400 0.5371 3 80 (+1) 0.9 (-1) 6 (+1) 2.05 2.0483 2.0315 0.0829-0.9024 0.0829 0.9024 4 45 (-1) 0.9 (-1) 6 (+1) 1.86 8571.1 8955.1 0.15591.9086 0.1559 1.9086 5 80 (+1) 1.3 (+1) 4 (-1) 1.83 1.8276 1.8354 0.13110.2951 0.1311 0.2951 6 45 (-1) 1.3 (+1) 4 (-1) 1.70 1.6989 1.7126 0.06470.7412 0.0647 0.7412 7 80 (+1) 0.9 (-1) 4 (-1) 1.98 1.9779 1.9987 0.10610.9444 0.1061 0.9444 8 45 (-1) 0.9 (-1) 4 (-1) 1.91 1.9079 1,8650 0.1099-2.3560 0.1099-2.3560 note: results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model into table 2 and table 3, the calculation results obtained by applying eq. (7b) and eq. (8b) are also inserted, as well as corresponding errors. 186 a. lazarević, d. lazarević the following relations were used for the calculation of errors:  for the percentage error () and mean percentage error ( ), ny yycal       ;(%)100 exp exp (9a)  for absolute percentage error () and mean absolute percentage error ( δ ), ; n        (9b) where n the number of trials in the experiment. table 3 confirmation experiment i v s wexp wcal   80 1.025 6 2.02 1.9862 1.9711 1.6733-2.4208 1.6733 2.4208 80 1.075 6 1.96 1.9640 1.9495 0.2041 -0.5357 0.2041 0.5357 80 1.125 6 1.98 1.9430 1.9291 1.8687-2.5707 1.8687 2.5707 80 1.175 6 1.91 1.9231 1.9097 0.6859 -0.0157 0.6859 0.0157 80 1.225 6 1.93 1.9042 1.8913 1.3368-2.0052 1.3368 2.0052 80 1.275 6 1.92 1.8863 1.8739 1.7552-2.4010 7552.1 2.4010 80 1.0 4 1.83 1.9336 1.9505 5.6612 6.5847 5.6612 6.5847 80 1.2 4 1.82 1.8593 1.8698 2.1593 2.7363 2.1593 2.7363 45 1.0 6 1.81 1.8259 1.8498 0.8785 2.1989 0.8785 2.1989 45 1.1 6 1.84 1.7982 1.8094 2.2717-1.6630 2.2717 1.6630 45 0.95 4 1.80 1.8756 1.8418 4.2000 2.3222 4.2000 2.3222 45 1.0 4 1.80 1.8455 1.8200 2.5278 1.1111 2.5278 1.1111 45 1.05 4 1.72 1.8173 1.7995 5.6570 4.6221 5.6570 4.6221 45 1.1 4 1.75 1.7909 1.7802 2.3371 1.7257 2.3371 1.7257 45 1.15 4 1.67 1.7659 1.7619 5.7425 5.5030 5.7425 5.5030 45 1.20 4 1.63 1.7424 1.7446 6.8957 7.0307 6.8957 7.0307 45 1.25 4 1.62 1.7201 1.7282 6.1790 6.6790 6.1790 6.6790 note: results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model based on the data from table 2, one can conclude that the calculation errors are negligible, regardless whether the basic regression eq. (7b) or the expanded regression eq. (8b) is applied. as it could have been expected, calculation errors are larger in relation to confirmation experiment results (table 3). non-linear mathematical models in the theory of experimental design... 187 fig. 1 correlations between the experimental results and results predicted by the generated model for the kerf calculation: a.) for basic model, b) for expanded model the conformance assessment of the mathematical model prediction with the experimental results can also be conducted using the correlation coefficient (r). based on the fig. 1 it could be concluded that both mathematical models ensure high levels of correlation, which are very similar, although not identical. the perfect prediction implies that all points should lie on a straight line passing through the origin and inclined at 45º. from fig. 1 one can observe that there is a relatively small deviation of the line of regression (which represents the best linear approximation of the data) from the ideal line. table 4 adequacy assessment criteria of the mathematical models related for the plasma cutting process experiment basic mathematical model expanded mathematical model max δ r max δ r main experiment 2.36 1.06 0.911 0.16 0.1 0.925 confirm.experiment 7.03 3.07 6.9 3.06 taking three criteria (max,  , r) into consideration results in the fact that an expanded mathematical model cannot guarantee better prediction for randomly selected points within the chosen experimental space (table 4). example ii: modelling of the thermal stress in the face milling face milling process has periodical characteristics, since the number of teeth in contact with material change periodically. periodical heating and cooling of the teeth is very unfavourable, since it itself represents a thermal process. in this investigation three influential factors are chosen for modelling the face milling process: cutting speed (v), unit feed rate (fz) and depth of cut (a) (table 5). 188 a. lazarević, d. lazarević table 5 cutting factors and their levels cutting factor symbol unit factor levels level 1 (low) level 2 (high) cutting speed v m/s 2.32 3.67 unit feed rate fz mm/tooth 0.178 0.280 depth of cut a mm 1.00 2.25 thermo-electric current (u) is chosen as the target function. influential factors are varied on two levels. a sample of 16 trials is extracted from the literature [18]. one half of trials are used for conducting the regression analysis (table 6), while another half is used for the selected mathematical models verification (table 7). table 6 main experiment v fz a uexp ucal   3.67 (+1) 0.28 (+1) 2.25 (+1) 14.8 8000.14 8188.14 0000.0 1270.0 0000.0 1270.0 2.32 (-1) 0.28 (+1) 2.25 (+1) 13.4 3998.13 3733.13 0015.0 1993.0   0015.0 1993.0 3.67 (+1) 0.18 (-1) 2.25 (+1) 14.0 0000.14 0413.14 0000.0 2950.0 0000.0 2950.0 2.32 (-1) 0.18 (-1) 2.25 (+1) 12.7 6998.12 6716.12 0016.0 2236.0   0016.0 2236.0 3.67 (+1) 0.28 (+1) 1.00 (-1) 14.2 2000.14 1413.14 0000.0 4134.0 0000.0 4134.0 2.32 (-1) 0.28 (+1) 1.00 (-1) 12.7 7000.12 7618.12 0000.0 4866.0 0000.0 4866.0 3.67 (+1) 0.18 (-1) 1.00 (-1) 13.4 4001.13 3993.13 0007.0 0052.0 0007.0 0052.0 2.32 (-1) 0.18 (-1) 1.00 (-1) 12.1 1000.12 0922.12 0000.0 0645.0 0000.0 0645.0 note: results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model test samples are made of steel c 60 din designation. the experiment is done on the milling machine fs-gvk-3 (prvomajska). cutting head jal g-750 ( 125 mm) with hard material inserts span 12 03 er is used. the metering of the thermo-electric current (in mv) is done using thermocouples and heatmeter (digital multi-thermometer). the metering is done in a few seconds after the beginning of the cutting process, namely in the stationary phase of the process. non-linear mathematical models in the theory of experimental design... 189 it should be mentioned that in the confirmation experiment the first two trials are related to the central point of the experimental design. the other trails are related to the experimental points located on the coordinate axes of the experimental design, therefore, both models generate the same results. table 7 confirmation experiment v fz a uexp ucal   2.95 0.223 1.0 13.4 13.3863 13.3863 0.1022-0.1022 0.1022 0.1022 2.95 0.223 1.0 13.7 13.3863 13.3863 2.2898-2.2898 2.2898 2.2898 1.83 0.223 1.5 12.1 12.0805 12.0805 0.1612-0.1612 0.1612 0.1612 4.65 0.223 1.5 14.9 14.8332 14.8332 0.4483-0.4483 0.4483 0.4483 2.95 0.142 1.5 12.6 12.6839 12.6839 0.6659 0.6659 0.6659 0.6659 2.95 0.351 1.5 14.1 14.1276 14.1276 0.1957 0.1957 0.1957 0.1957 2.95 0.223 0.67 12.8 12.7742 12.7742 0.2016-0.2016 0.2016 0.2016 2.95 0.223 3.37 15.0 14.0276 14.0276 6.4827-6.4827 6.4827 6.4827 note: results in the numerator are related to the basic mathematical model, while in the denominator is related to the expanded mathematical model applying the same methodology as in the previous examples, the following regresion equations are generated:  for the mathematical model without interactions: 321 02340.002695.005132.059423.2 xxxy  (10a) 05771.011898.022380.029976.12 afvu z (10b)  for the mathematical model with interactions: 3213231 21321 000959.0000356.0002106.0 001439.002340.002695.005132.059423.2 xxxxxxx xxxxxy   (11a) )lnlnln06834.0lnln06930.0 lnln07980.0(exp1562.12 04102.011741.023298.0 afvaf avafvu zz z    (11b) from eqs. (10) and (11) it is obvious that the thermo-electric current increases with the increase of cutting speed, unit feed rate and depth of cut, since exponents v, fz and a are positive. the regression equation in the form of eq. (10b) is suitable for application in engineering practice. based on fig. 2 (which relates to the entire experiment) one can conclude that both mathematical models ensure high levels of correlations, which are almost identical. 190 a. lazarević, d. lazarević fig. 2 correlations between the experimental results and results predicted by the generated model for the face milling process: a) for basic model, b) for expanded model taking all criteria into consideration results in the fact that an expanded mathematical model cannot guarantee a better prediction for randomly selected points within the entire experimental hyperspace (table 8). table 8 adequacy assessment criteria of the mathematical models related to the face milling process experiment basic mathematical model expanded mathematical model max δ r max δ r main experiment 0.49 0.23 0.965 0.02 0.005 0.966 confirm. experiment 6.48 1.32 6.48 1.32 3. conclusions this paper presents the application of non-linear mathematical models for different technological process modelling. it is possible to draw the following conclusions from the analysis of described example: 1. the classical theory of experimental design shows that non-linear mathematical models in the form of higher-order polynomials ensure a higher accuracy than the linear ones. these models, especially models including factors’ interactions, make intricate the analysis and interpretation of modeling results. however, non-linear mathematical models in the form of complex power functions with interactions do not guarantee higher accuracy than the same models without interactions. this is the reason why the application of such mathematical models can be justified only in specific cases. 2. although non-linear mathematical models without factors’ interactions show lower accuracy in the points of experiment than expanded mathematical models, the mean error of both models within the entire experimental space is practically identical. non-linear mathematical models in the theory of experimental design... 191 3. it is shown that it is more practical and purposeful to use the basic mathematical model (without interactions) than the expanded mathematical model (with interactions). 4. the accuracy of selected mathematical models in this paper is significantly larger in experimental points than in other points of experimental hyperspace. 5. the analogous analysis should indicate whether the same conclusions can be reached for other mathematical models of this type. references 1. montgomery, d. c., 2001, design and analysis of experiments, john wiley & sons, inc., new york. 2. kim, w. k., kim, h. t., ryu, g. m., kim, g. h., 2013, study on high-speed cutting characteristics using design of experiments, international journal of precision engineering and manufacturing, 14(10), pp. 1869-1872. 3. dersjö, t., olsson, m., 2012, efficient design of experiments for structural optimization using significance screening, structural and multidisciplinary 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montgomery, d. c., myers, r. h., carter, w. h., vining, g. g., 2005, the hierarchy principle in designed industrial experiments, quality and reliability engineering international, 21, pp. 197-201. 10. heller, c., schmoeckel, d., 1988, umformen von aluminium-blechen bei erhöhten temperaturen, aluminium, 64(4), pp. 398-405. 11. choudhury, i. a., el-baradie, m. a., 1997, surface roughness prediction in the turning of high-strength steel by factorial design of experiment, journal of material processing technology, 67,pp. 55-61. 12. choudhury, i. a., el-baradie, m. a., 1998, tool-life prediction model by design of experiment for turning high strength steel (290 bhn), journal of material processing technology, 77, pp. 319-326. 13. shunmungam, m. s., bhaskara reddy, s. v., narendan, t. t., 2000, selection of optimal conditions in multi-pass face-milling using a genetic algorithm, international journal of machine tool & manufacture, 40, pp. 401-414. 14. dasgupta, r., thakur, r., govindrjan, b., 2002, regression analysis of factors affecting high stress abrasive wear behaviour, practical failure analysis, 2(2), pp. 65-68. 15. tosun, n., ozlar, l., 2002, a study of tool life in hot machining using artificial neural networks and regression analysis method, journal of materials processing technology, 124, pp. 99-104. 16. samanta, b., erevelles, w., omurtag, y., 2008, prediction of workpiece surface roughness using soft computing, proc, imeche, part b: j. engineering manufacture, 222, pp.1221-1232. 17. lazarević, a., 2010, modelling of the correlations between the parameters of the plasma arc cutting and heat balance analysis using the method of artificial intelligence, (in serbian) phd thesis, faculty of mechanical engineering, university of nis. 18. kovač, p., milikić, d., 2006, predicting equation of thermo-electric currents and cutting conditions generated in face milling, university of novi sad, faculty of technical sciences, novi sad. http://link.springer.com/journal/12541 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 427 437 https://doi.org/10.22190/fume160830010b © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper radial force impact on the friction coefficient and temperature of a self-lubricating plain bearing udc 669.141 nada bojić 1 , dragan milčić 1 , milan banić 1 , miroslav mijajlović 1 , ružica nikolić 2,3 1 university of nis, faculty of mechanical engineering nis, serbia 2 university of kragujevac, faculty of engineering kragujevac, serbia 3 university of žilina, research centre, slovakia abstract. self-lubricating bearings are available in spherical, plain, flanged journal, and rod end bearing configurations. they were originally developed to eliminate the need for re-lubrication, to provide lower torque and to solve application problems where the conventional metal-to-metal bearings would not perform satisfactorily, for instance, in the presence of high frequency vibrations. among the dominant tribological parameters of the self-lubricating bearing, two could be singled out: the coefficient of friction and temperature. to determine these parameters, an experimental method was applied in this paper. by using this method, the coefficient of friction and temperature were identified and their correlation was established. the aim of this research was to determine the effect of radial force on tribological parameters in order to predict the behavior of sliding bearings with graphite in real operating conditions. key words: radial force, coefficient of friction, temperature, experimental method 1. introduction self-lubricating sliding bearings possess a number of advantages and they enable significant savings both in maintenance and lubricants. when using bronze bushings with graphite inserts, the graphite forms a thin film on both contact surfaces, which is highly resistant to impacts; besides, it remains in its position even without rotation. those bushings are used for parts exposed to high loads and low speeds. the most influential received august 30, 2016 / accepted june 02, 2017 corresponding author: nada bojić university of nis, faculty of mechanical engineering, aleksandra medvedeva 14, 18000 nis, serbia e-mail: nalemfkg@gmail.com 428 n. bojić, d. milĉić, m. banić, m. mijajlović, r. nikolić tribological parameters of self-lubricating sliding bearings with graphite inserts are the material of the sliding pair, radial force, sliding speed, temperature, diameter of graphite inserts, their disposition within the bushing and the percentage coverage of graphite of the overall sliding surface, as well as radial clearance and surface roughness and hardness. as there are many influential parameters, the friction coefficient is usually determined experimentally. ďuriš and labašová [1] and labašová [2] experimentally obtained the value of the friction coefficient for the sliding pair aluminum steel. their tests were done on tribotestar m'89. the authors studied the influence of radial load and sliding speed on the value of the friction coefficient. by applying the aforementioned method, labašová [3, 4] examined the friction coefficient for bronze (cuzn25al6) with inserted lamellar graphite. the results showed that the friction coefficient decreases with an increasing radial load. although the given research investigated the friction of self-lubricating bearings with inserted lamellar graphite, the values of contact pressure are unknown as the diameter of the joint is not given. furthermore, the graphite coverage is also unknown as the authors only state that the coverage is between 20 and 30%. the given research was also performed for relatively low values of radial force – up to 600 n. ozcarac et al. [5] carried out wear bearing tests on the tribometer with three types of samples. those were bronzes based on tin and lead – rb-1, rb-7 rb-4. wear tests were conducted at loads of 10, 20 and 40 n and a sliding speed of 0.5 m/s with samples in the form of a ring. upon the completion of the tests, the weight of samples was measured and the value of the friction coefficient was calculated. optical and sem tests were also performed in order to characterize the wear of the above alloys. savaskan and bican [6] tested the friction and wear of the al-25zn-3cu-3 alloy si on the tribometer by varying the pressure and the sliding speed. they noticed that the friction coefficient of the alloy increases with the sliding speed, but decreases with the increasing pressure up to 1.5 mpa; above this pressure value the trend reverses and the friction coefficient increases. however, the temperature and the extent of wear of the alloy increased constantly with the increasing pressure and sliding speed. pawlak et al. [7] measured, by experimentally establishing the porosity of self-lubricating sliding bearings using the hexagonal boron nitride (bn-h) as an additive, the friction coefficient of the (h-gt + oil) substrate, at varying loads from 1.05 to 2.0 mpa and sliding speeds of 1.35 and 2.5 m/s. it was found that the addition of oil micro-particles reduces the friction coefficient by about half with respect to the self-lubricating bearings. omrani et al. [8] analyzed the tribological properties of al-16si-5ni-5graphite selflubricating metal matrix composite and compared its tribological properties with steel. they found out that in the case of limited lubrication the al-16si-5ni-5graphite had values of the friction coefficient lower than steel. furthermore, the friction coefficient of steel increased with the increase of the applied normal load, while for al-16si-5ni5graphite it decreased. the ideal material for sliding bearings would be a material that easily absorbs the abrasive particles and eliminates them if the contact is disturbed. the composite materials possess very good properties in this regard. suresh et al. [9] researched the composite materials under different loads and sliding speeds by the pin-on-the tests. larger wear was recorded with increasing load and sliding speeds. the values of the friction coefficient increased with the subsequent increase of load/sliding speed. it was noticed that the radial force impact on the friction coefficient and temperature… 429 graphite, filled by the g-e composite, exhibited a lower friction coefficient than the other two composites, regardless of variations in load/sliding speed. liu et al. [10] analyzed the effect of the percentage of graphite in the composite materials by using the umt 2mt tribometer. the given research concluded that the appropriate graphite content and hardness of the materials are the two most crucial factors to achieve the desired solid lubrication performance. the heating of a bearing is usually caused by speed, load and friction within the bearing. in addition to the above parameters, the cooling system failure and external heat sources also cause bearings to heat up. radila and zeskotekb [11] measured the temperature of bearing using thermocouples, and came to the conclusion that the greatest impact on temperature increase in a sliding bearing is exerted by an increased number of rpms of the shaft and the radial load. the objective of investigation by bonny et al. [12] was to establish the influence of the parameters – the radial forces and the oscillation speed – on the tribological characteristics of the wc-co cemented carbides during the sliding contact. prasad [13] recorded a significant increase in wear with increased load. from the above research, it can be concluded that the mutual relations of the load, the friction coefficient and the quantity of the generated heat are very complex. although the self-lubricating plain radial bearings with graphite inserts are very widely used, from the above analysis of the previous research one can conclude that there is very little data about their tribological performance in scientific publications. the manufacturers of such bearings claim that, as a rule, the total friction force increases with the load. however, this rule does not apply to the unit friction force and the friction coefficient since they are decreasing. there are two reasons why this happens: 1. the friction coefficient and the specific friction force depend on the real pressure, whose value changes only slightly with increasing load. the friction coefficient decreases with increasing load and thus reduces the tendency of metal to create heat on contact. 2. an increase in the actual contact area is slower than that in the loading. the increased load requires the necessary decrease of clearance because the carrying area of the bearing increases thus lowering the unit pressure. this paper presents the research on the influence of the radial force value on the friction coefficient and temperature of a self-lubricating plain radial bearing with graphite inserts. apart from the research by other authors, the effect of radial load was studied with two different values of diameter of graphite inserts, as well as for two values of graphite coverage – 20 and 30%. the values of the radial force were significantly increased in contrast to the above-discussed research of labašová [3, 4]. 2. friction and temperature of self-lubricating bearings the coefficient of friction is a complex tribological parameter, which has a stochastic character due to a number of influencing factors. therefore, the analytical procedure for the determination of the friction coefficient is rather complex. that is further complicated by the character of the contact between a self-lubricating bushing and a shaft, as well as the load of the self-lubricating bushing due to the radial force. namely, the self-lubricating bushing, through its active surfaces, has a variable contact with the shaft at variable speeds. therefore, the 430 n. bojić, d. milĉić, m. banić, m. mijajlović, r. nikolić application of an experimental method is a simpler procedure to determine the friction coefficient. the same applies for heat generation as it is a case-sensitive process depending on many parameters that can be estimated only experimentally [14]. to determine the friction coefficient one starts from the coulomb’s equation that relates the load and the friction coefficient in contact of two real bodies as a function of time:        ( ) ( ) ( ) ( ) ( ) ( ) n n f t f t t f t t f t (1) where (t) is the friction coefficient, f(t) is the friction force, fn(t) is the normal force and t is the time. any increase in friction leads to increased bearing temperature which eventually can cause the destruction of the bearing. the elevated temperature can be a result either of the heat generation or the conditions in which the contact is realized. the increase in temperature, as a result of the heat development caused by friction, is given by: t c p v     (2) where c is the total thermal resistance to the heat dissipation from the surface, p is the contact pressure and v is the sliding speed. the wear intensity depends on pressure (p) and sliding speed (v); thus one takes the value of their product (p∙v) of the selected material as a criterion for the design and calculation of the self-lubricating bearing. approximate values, recommended in the literature, may range from 0.8·10 -3 to 1.8·10 -3 °cms/n [15]. the friction moment is calculated as: ( ) ( )tr nm t f t l  (3) where l is the lever arm. in calculating the friction coefficient one starts from eq. (3). the friction force occurs when there is a relative movement of the shaft with respect to the self-lubricating bearing. it is considered that the friction force is a result of the load (the normal (radial) force) and the torque. the part of the torque that causes the rotation of the shaft, which overcomes the frictional resistance of the contact, is the friction moment, which is a function of time. the friction moment is calculated as the product of the normal force of the sleeve (obtained by the force sensors) and its lever arm. the friction coefficient is calculated by the condition of equality of the friction moment in contact of the bushing and the shaft and the torque by which the entire system acts on the support o2. 3. experiment experimental research of the self-lubricating bearings was conducted in order to determine the friction coefficient and temperature. figure 1 shows the configuration for measuring the variables necessary to determine the friction coefficient at the contact of the self-lubricating bearing and the shaft. the main rotational motion, provided by a working machine lathe (lt), was transferred to driving shaft (ds) by torque sensor (m1) and clutch (c1). self-lubricating bearing (sb) was mounted on the shaft sleeve. the radial force impact on the friction coefficient and temperature… 431 bearing was radially and axially fixed by the upper (us) and the lower (ls) part of the support. through the opening of upper support (us), by the lower addition of force sensor (as), the radial force, induced by tightening the screw, acted on the bearing, while the magnitude of the force was measured by force sensor (m2). the circumferential force, which represents the frictional force of the bearing, was measured by lever (l1), of length 150 mm, which was placed perpendicular to the axis of the bearing and rigidly mounted on upper support (us). sensor (s1) measured the intensity of this force. the temperature of the self-lubricating bearing was measured by thermocouples t(t) and t1(t), where thermocouple t(t) measured the temperature at the center of the bearing bushing, while thermocouple t1(t) measured the temperature at the end of bearing bushing (sb). thermal camera (t) recorded the temperature of the end surface of the bearing frontal bed. all the sensors, thermocouples were connected to computer unit (r), which performed the data acquisition. the images from the thermal camera were taken immediately when the lathe stopped, i.e. at the stopping of the shaft rotation. fig. 1 the scheme of the measurement configuration the lathe “potisje ada pa-c30” was used for testing the self-lubricating bearing, as well as the measuring system, which consisted of a measuring and amplifier device “national instruments cdaq-9178”, torque sensor “hottinger baldwin messtechnik t1” (100 nm), force sensors “hottinger baldwin messtechnik s7m” (500 n) and “hottinger baldwin messtechnik u9” (5 kn), two k-type thermocouples and thermal imaging camera flir e 50. the self-lubricating plain bearings with graphite, 50/40x40, were used in this experiment, manufactured by fasil a.d., whose beds were made of substrate high quality bronze cusn12 with inserts (lamellae) of graphite of diameter of 8 and 10 mm, which homogeneously adhered to the two surfaces occupying 20% or 30% of the casings. in total 4 bushings 432 n. bojić, d. milĉić, m. banić, m. mijajlović, r. nikolić were tested each with a unique combination of graphite inset diameter and graphite coverage. fig. 2 shows the technical drawing of the self-lubricating bearing with the diameter of inserts of 10 mm and graphite coverage of 20%. during the tests the ambient temperature value was 25 ° c and all the tests were performed with the rotation speed of 54 rpm which corresponded to sliding speed of 0,113 m/s. tolerances of all the bushings were checked before the experiment. the experimental testing was performed with two values of radial force – 1500 and 3000 n. the time of experiment was 1200s for every bushing and the measurements were performed five consecutive times on every individual bushing. the bushings were initially rotated for 10 min before the test with graphite polishing grease nlgi class 2. when this run was completed, the bushings were cleaned from the grease before the actual tests. fig. 2 the self-lubricating bearing ø50/40x40, with inserts (lamellae) of graphite of a diameter of 10 mm, coverage of 20% radial force impact on the friction coefficient and temperature… 433 4. results and discussion the following notation was used during the experiment: aa_bb_cc_dddd, where the first group of symbols (aa) designate the diameter of the graphite inserts, the second (bb) the percent of graphite coverage, the third (cc) sliding speed and the fourth (dddd) the value of the radial force. table 1 summarizes the results of the experiments for 4 tested samples at two values of radial force. the results presented in table 1 provided the data for the last test, as there were 5 tests per bushing in order to secure the repeatability of results. the results were monitored during all 5 tests in order to avoid possible experimental mistakes. the temperature value in table 1 was given at the end of the test, i.e after all 5 consecutive tests on an individual bushing with distinctive insert diameters and graphite coverages. it was observed during the experiments that, after 2 to 3 consecutive tests on an individual bushing, a thermal equilibrium was achieved, so there was no increase in the temperature of the bushing in the fourth and fifth tests. table 1 experimental results bushing real value of radial force, n friction coefficient temperature, °c 8_20_54_1500 1394.5 0.079 43 8_30_54_1500 1428.8 0.077 53 10_20_54_1500 1500.8 0.039 33.5 10_30_54_1500 1475.8 0.041 37 8_20_54_3000 2967 0.09 54.8 8_30_54_3000 3049.4 0.09 51.1 10_20_54_3000 2772.3 0.055 43.7 10_30_54_3000 2917.7 0.044 41.50 as the radial force is supplied by tightening of the screw there was a small variation in the obtained radial force from the desired values (1500 and 3000 n). furthermore, due to thermal expansion of the bushing the value of the radial force for each individual bushing was readjusted between the consecutive runs regardless of the fact that thermal equilibrium was obtained in the fifth run. figure 3 shows the value of radial force (a), friction coefficient (b), temperature measured by the thermocouple on the bushing frontal bed (c) and thermal image of the bushing frontal bed (d). the variation of the result for the friction coefficient in fig. 3b is a result of the lathe induced vibrations. however, as clearly shown in the figure, the average value of friction coefficient is µ = 0.039. the small oscillations of radial force value are also a consequence of the lathe induced vibration with average value of fr = 1500.8 n. from the figure it is also clear that there is a good agreement between the thermocouple and thermographic measurements. 434 n. bojić, d. milĉić, m. banić, m. mijajlović, r. nikolić a) b) c) d) fig. 3 results of experimental measurements for a bushing 10_20_54_1500: a) radial force (average value fr = 1500.8 n), b) friction coefficient (average value µ = 0.039), c) temperature measured by thermocouple (average value t = 33.49 °c), d) temperature measured by the thermal imaging camera of 33.5 °c at bushing frontal bed figure 4 shows the influence of radial force on the value of the friction coefficient for different diameters of graphite inserts and different graphite coverages. in contrast to the findings of the other authors and the data provided by the manufacturers of such bearings, as clearly shown in the figure, the coefficient of friction increases with an increase in radial load for values of contact pressure greater than approximately 0.95 mpa (radial load of 1500 n, bushing diameter x width 40 x 40 mm). the lowest values of the friction coefficient were obtained for 10 mm inserts and radial force value of 1500 n. obviously, the larger contact area of an individual graphite insert helps the better distribution of solid lubricant inside the bushing. it is also clear that the graphite coverage does not have a significant influence on a lower value of radial force (1500 n). the situation changes with the increase in the load. for instance, with 10 mm inserts and coverage of 20% there is a greatest increase in the friction coefficient value with an increase in radial load, while for 30% coverage there is almost no rise of the friction coefficient value. the increase of friction coefficient value is probably a consequence of plastic saturation of the contact. plastic saturation of a contact occurs when all (or almost all) asperities are in contact. in such circumstances the molecular component of the friction coefficient is independent of contour pressure, while the deformational component intensively increases with an increase radial force impact on the friction coefficient and temperature… 435 in contour pressure [16]. these findings should be further investigated in order to determine the exact reasons for the increase in the friction coefficient with radial load and mechanisms behind it. fig. 4 the influence of radial force on the friction coefficient for various graphite insert diameters and graphite coverage figure 5 shows the influence of radial force on the value of temperature of the selflubrication plain bearing with graphite inserts for different diameters of graphite inserts and different graphite coverages. fig. 5 the influence of radial force on the temperature for various graphite insert diameters and graphite coverages as in the case of the coefficient of friction, there is an increase in temperature with the increase in the load. the increase is more pronounced for the bushings with lower value 436 n. bojić, d. milĉić, m. banić, m. mijajlović, r. nikolić of graphite coverage. the results are expected as any increase in the friction coefficient leads to an increase in temperature. furthermore, the obtained temperatures are within the permissible limits as they are below 100 ºc, which is a limiting temperature for this type of bearings. as graphite has a near zero coefficient of thermal expansion, it will not expand with the increase in temperature as the bronze sleeve will. if the temperature is greater than the limiting temperature, the graphite inserts will become loose, which leads to lower lubrication and destruction of the bearing due to wear and scuffing. 5. conclusions this paper presents the experimental research into the influence of radial load on the coefficient of friction and temperature of self-lubricating plain radial bearings with graphite inserts. the effect of radial load was determined for two different values of graphite inserts, as well as for two values of graphite coverage. it is determined that the friction coefficient value increases with the increase in radial load probably due to plastic saturation of the contact at a nominal contact pressure greater than 1 mpa. mechanisms which lead to such behavior should be further investigated. the increase of diameter of graphite inserts from 8 to 10 mm leads to lower values of the coefficient of friction. the percent of graphite coverage does not influence the value of the friction coefficient significantly for the contact pressure of 1 mpa. with the increase in contact pressure, the influence of graphite coverage becomes more pronounced. the temperature of a self-lubricating plain radial bearing with graphite inserts in the given geometric configuration increases with the increase in radial load. such behavior is expected, as an increase in the friction coefficient is followed by an increase in temperature. as already noted, further research should be directed towards understanding the friction mechanisms above the contact pressure of 1 mpa, as well as the combined influence of radial load and sliding speed. 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http://www.sciencedirect.com.proxy.kobson.nb.rs:2048/science/article/pii/s0043164810001031?_alid=1824886703&_rdoc=165&_fmt=high&_origin=search&_docanchor=&_ct=32643&_zone=rslt_list_item&md5=2de4c5be4c2758f6b57a9711d2b296f8 facta universitatis series: mechanical engineering vol. 19, no 3, special issue, 2021, pp. 345 360 https://doi.org/10.22190/fume201125029f © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper green supplier selection based on the information system performance evaluation using the integrated best-worst method hamed fazlollahtabar1, navid kazemitash2 1department of industrial engineering, school of engineering, damghan university, damghan, iran 2department of management, faculty of economics and administrative sciences, university of mazandaran, babolsar, iran abstract. information systems (is) have become crucial for all the organizations to survive in contemporary technology-oriented environment. consequently, the number of companies and organizations which have invested widely in their is infrastructures to present better services and to produce higher value products is increasing. on the other hand, nowadays, because of the increase of governmental rules and serious requirements of more people in the case of environmental protection, it seems necessary for all the enterprises to follow these regulations if they want to survive in the global markets. however, what is at issue here is not just the companies’ agreement with the environmental laws; in addition, they should apply some strategies to decrease the negative environmental impacts of their products in some countries. thus, the aforementioned arguments are the reasons for the compulsory use of the green supplier selection (gss) in all firms. considering the mentioned contents, the purpose of this study is representation of the relation between iss and gss as two vital components of firms in a novel way which has not been done before. actually, it shows the iss' performance or effectiveness to select the green suppliers taking into account the different levels of importance of gss measures (including eight criteria and 31 sub-criteria), using a multicriteria decision-making method called best worst method (bwm) to identify the weights (importance) of gss measures and compute the gss performance of 10 iss in a company using the data gathered in a survey from iss' experts. key words: information systems, green supplier selection, best worst method, significance scoring received november 25, 2020 / accepted march 02, 2021 corresponding author: hamed fazlollahtabar department of industrial engineering, school of engineering, damghan university, damghan, iran e-mail: h.fazl@du.ac.ir 346 h. fazlollahtabar, n. kazemitash 1. introduction information systems (is) as an academic field, first of all, attracted research interest in the 1960s [1]. that was the time when the applied computer science emerged which in its turn aimed at the design and implementation of data processing applications. information systems (is) have become essential for all the organizations to survive in today’s technologyoriented environment. the number of companies and organizations which have invested widely in their is infrastructures to present better services and to produce better value products is increasing. this rise has led to the question of how much those systems add value to the business or to the organization compared to their investment. the role of information systems (is) in providing business a competitive edge has been the topic of so many discussions recently. the conclusion is that not the is solution but their utilization is what provides competitive advantages. thus, because of the aforementioned functions and importance of is, there are too many studies which show a big role the iss play in relation to the other fields such as health care and medicine [2, 3], transportation [4], energy [5], biology [6], education [7], environment [8, 9, 10], geography [11] and so many other disciplines. but one of the most important fields that the trace of iss has been seen is the selection of green suppliers. nowadays, because of the increase of governmental rules and serious requirements of more people in the case of environmental protection, it seems necessary for all the enterprises to follow these regulations if they want to survive in the global market. however, what is at issue here is not just the companies’ agreement with the environmental laws; in addition, they should apply some strategies to decrease the negative environmental impacts of their products in some countries. therefore, to gain sustainable development, the integration of environmental, economic and social performance turned into a complex challenge for them. because of the above reasons, the companies working on this matter buy their required materials and services from specific suppliers which can simultaneously fulfill their expectations like low-cost, high-quality, short lead-time and environmental criteria,. on the one hand, by considering the aforementioned information about the importance of iss, green supplier selection, and the direct impact of iss on the selection of green suppliers so that iss effect on numerous other factors that impress selecting process in modern organizations, and, on the other hand, because of the financial restriction in both fields, a good recognition of their relation could be helpful to the reduction of costs and their effectiveness. inasmuch as there are various applied iss in companies by different tasks, each of them could have individual influence on selecting process. hence, the purpose of this study is the evaluation of the impact of is on the green supplier selection and actually finding the level of effectiveness of each is on the green supplier selection process. therefore, this paper is going to represent a great framework to support its goals. at first, it examines other research projects, literatures and experts' opinion to gather the most important criteria and sub-criteria which have effects on the green supplier selection. then, through the best-worst method (bwm), the local and global weights of criteria and sub-criteria will be obtained by the experts' opinions. the next step is measuring the iss' performance in association with green supplier selection which is gained by the experts' opinions. ultimately, as a conclusion, companies could focus on the specific is or iss which play a more important role in the green supplier selection processes and reinforce them if necessary. green supplier selection based on information system performance evaluation using the integrated... 347 2. literature review because of the expansion of people awareness about the environmental changes and applying of the compulsory green rules which have been exerted by governments, approximately all companies should obey, to survive in these competitive markets. therefore, there are so many studies that deal with suppliers. for instance [12] represented a green supplier selection model based on the emission of co2 which is produced through the transportation and production processes by considering three models to maximize the total profits and the green factors, and minimize the co2 emissions. banaeian et al. [13] in their research have selected the green supplier using the fuzzy group decision-making methods. actually, they compared the result of three different techniquestopsis, vikor and gra methods in a fuzzy environment. kuo, et al. [14] developed a green supplier selection model in the electronic industry by using the new hybrid mcdm method including danp and vikor. environmental permits and reporting, pollution prevention and resource reduction, hazardous substances wastewater and solid waste, air emissions and product content restrictions were considered as environmental dimension, while company commitment, management accountability and responsibility, legal and customer requirements, risk assessment and risk management, improvement objectives, training, communication, worker feedback and participation, audits and assessments and documentation and records determined as management systems dimension. govindan et al., [15] reviewed all the research studies related to the green supplier from 1997 till 2011, in order to find the most common used approaches for evaluating and selecting the green supplier as well as the most common criteria which have been considered in this case and finally identifying the existent limitations. for instance, their study illustrated that the fuzzy single approach has been the most repetitive applied technique and the environmental management system has been the most selected criteria. jain et al. [16] presented an initiated decision-making process to evaluate the suppliers based on the green criteria in which there are a carbon market sensitive (cms) and a green decision-making approach based on data envelopment analysis (dea) called cms– gdea. their applied model in one case displayed that the “pay up” factor from carbon trading adds a new dimension to competition among suppliers and increases overall supply chain profitability; finally, they encourage companies to follow the green rules. gupta et al. [17] worked on the evaluation of supplier selection based on the green innovation abilities among small and medium companies. in their study, there are three different methodological phases including; selection of green innovative criteria through literature review and interviews with decision-makers, ranking of selection criteria using a novel best worst method (bwm) and ranking of suppliers with respect to selection criteria weights obtained in phase two using fuzzy topsis. in another research, galankashi et al. [18] prioritized the criteria which affect the green supplier selection through the fuzzy analytical network process. hamdan and cheaitou [19] by using the combination of three techniques including ahp, fuzzy topsis and multi-objective optimization approach, evaluated the supplier selection and the order allocation based on the green criteria that led into the flexible model. 348 h. fazlollahtabar, n. kazemitash the aim of [20] was to implement hybrid grey theory-marcos methods for decisionmaking regarding the selection of suppliers in the libyan iron and steel company (lisco). this hybrid model was divided into two phases: the first consists of determining the weights of the criteria that contribute to decision-making, which has been done using the grey theory, and the second phase consists of selecting the best supplier from among six suppliers, which has been completed using the marcos model. durmić et al. [21] performed supplier selection to achieve sustainability, taking into account all aspects: economic, social and environmental criteria. for this purpose, a combined fucom rough saw approach has been used. pamucar [22] provided a multi-criteria decision-making that combines interval grey numbers and normalized weighted geometric dombi-bonferroni mean operator to address the situations where attribute values take the form of interval grey numbers under uncertain information. there are too many research studies about gss and iss separately as two crucial parts of contemporary organizations, while, except for some limited studies in which is is considered as an effective factor for gss, there is no research that points to the relation between them. this is the exact reason why this paper works on. on the other hand, the second issue that is observed in the majority of the previous studies is using the complicated and time-consuming techniques like dematel, ahp, anp, danp, topsis and vikor to compute the needed requirements, although there are so many studies which have mentioned their weak points. and it is the exact reason that why this paper utilized a novel mcdm technique (bwm) that is simpler and more practical, functional and usable. 3. statement of the problem on the one hand, in this contemporary business world, there is not any enterprise which would be able to compete and even survive without having close relations with outer partners; that is exactly the point where the supplier chain management (scm) arises from and wants to optimize the information flow exchanges among all participating factors in the supplier chain. thus, the more effective supplier chain, the more competitive the advantages; so, because of the complex condition of today's business, all companies need to have a long-term relationship with their partners – this is the reason why all corporations should be aware and alert to identify and select supply resources. hence it can show the extreme importance of supplier selection. as mentioned, by considering the growth of worldwide awareness of environmental protection, green production has become an important issue for almost every manufacturer and will determine his long-term sustainability. thus, the green factors have changed the face of supplier selection. because of the growth of governmental rules and serious and increasing demands of the mass in the case of environmental protection, it seems crucial for all companies to follow these regulations if they want to remain in the market. however, the deal of companies with these environmental laws is not enough; they should apply some strategies and policies to reduce the negative environmental impacts of their products. consequently, to obtain sustainable development, the integration of environmental, economic and social performance has turned into a complex challenge for them. because of the above reasons, companies consider various criteria to assign their required materials to suppliers. green supplier selection based on information system performance evaluation using the integrated... 349 on the other hand, iss have become vital for all of companies to survive and remain in today’s technology-oriented market. the number of companies and firms which have invested widely in their is infrastructures to present better services and to produce better value products is increasing. nowadays in majority of the companies, there are some practical and basic information systems which have changed from competitive advantage to necessity and as it was expressed, because of the significance of the flow of information in the supplier chain, the information systems emerge as one of the most effective factors in the green supplier selection. but there are some barriers and even problems which have played hidden roles in selection of the best green supplier and never have been paid attention to: 1there is no localized green supplier selection model for the examined industry. 2although all the evidence shows the impact of iss on green supplier selection, there is not any study showing the impact of different types of iss on criteria and sub-criteria of the green supplier selection which leads to the evaluation of the iss' effectiveness regarding green supplier selection. 3 -there is not any research that indicates which is plays the most important role in connection with selecting the green supplier. as the aim of this paper is evaluating of each single is in effecting on the green suppliers’ selection and actually finding the level of effectiveness of each is on the green supplier selection process, it could cope with the aforementioned problems. at the first step, it represents a localized gss model including eight criteria and 31 sub-criteria of green supplier selection, based on the gss experts' opinions (first problem). then it illustrates the performance of every is in relation with green supplier selection process using the wbm (which computes the importance (weights) of every measure of gss model) and performance item-scores (which represents the effectiveness and performance of iss to select the green suppliers) of all existent iss in a company (second and third problem). after the presentation of the model, the procedures of problems solving are demonstrated as techniques, step by step. as shown above, there are three primitive operations in which 8 criteria and 31 subcriteria have been selected by 12 number of organization experts that have been extracted from the principal literature. then, the bwm as the mcdm technique consists of three sub-sections in which the local weights of criteria, the local weights of sub-criteria and finally the global weights of sub-criteria are computed, respectively. as the last step, by determining the iss' performances regarding meeting the green supplier selection criterion, the scores of the iss are calculated. ultimately, based on the computed final scores of iss, they are ranked. in this way, the determined goals of study are achieved, or indeed, the mentioned problems of the study are solved. 4. methods and materials the purpose of this study is to evaluate the performance of various iss of a company, in the green supplier selection process (gss). as this aim is met by a mcdm method to gain the global weights of the green supplier selection' sub-criteria, and another technique to rank the iss, based on their performances in connection with the gss, it looks necessary to show the steps of bwm as the mcdm method and item-scoring to rank the iss. the proposed conceptual model is depicted in fig. 1. 350 h. fazlollahtabar, n. kazemitash fig. 1 the conceptual model of green supplier selection's criteria and sub-criteria 4.1. best worst method as discussed above, because the green supplier selection is a multi-criteria concept, to measure it, we should use mcdm method. mcdm methods allow considering multiple criteria with different weights. there are several mcdm methods that have been applied in literature but in this study, a newly developed mcdm method called best worst method (bwm) is used [23, 24]. in comparison with similar existing mcdm methods, bwm needs less data as it does not require a full pairwise comparison matrix, and its results are more consistent due to its structured pairwise comparison system; that is the main reason why it is applied in this study. further, it is perceived by the decision-makers as simple and very close to the way they judge and reason while making decision. subsequently, the steps of the bwm are described briefly as follows: step 1 determine a set of decision criteria. in this study, the criteria are presented in two levels as criteria and sub-criteria. step 2 determine the best (b) (e.g., the most desirable, the most important) and the worst (w) (e.g., the least desirable, the least important) decision criteria based on the decision-maker(s)/expert(s) opinion. green supplier selection based on information system performance evaluation using the integrated... 351 step 3 determine the preference of the best decision criterion (b) over all the other decision criteria, using linguistic 9-point scale for pairwise comparison for best worst method (table 1). the result is a best-to-others (bo) vector as follows: a𝐵 = (𝑎𝐵1, 𝑎𝐵2 , … , 𝑎𝐵𝑛) whereas abj represents the preference of b over j and as expected abb =1. table 1 linguistic scale for pairwise comparison for best worst linguistic scale equally important equal to moderately more important moderately more important moderately to strongly more important strongly more important equivalent number 1 2 3 4 5 linguistic scale strongly to very strongly more important very strongly more important very strongly to extremely more important extremely more important equivalent number 6 7 8 9 step 4 determine the preference of all the decision criteria over the worst criterion (w), using linguistic 9-point scale for pairwise comparison for best worst method (table 1), which results in the others-to-worst (ow) vector as follows: aw = (𝑎1𝑊 , 𝑎2𝑊 , … , 𝑎𝑛𝑊)𝑇 whereas ajw represents the preference of j over w and, as expected, aww =1. step 5 find the optimal weights (w1 *, w2 *,…, wn *). the optimal weights should be determined so that the maximum absolute differences {|wb − 𝑎𝐵𝑗𝑊𝑗|, |𝑊𝑗 − 𝑎𝑗𝑊𝑊𝑤|}for all j is minimized, or equivalently; 𝑚𝑖𝑛 𝑚𝑎𝑥 {|wb − 𝑎𝐵𝑗𝑊𝑗|, |𝑊𝑗 − 𝑎𝑗𝑊𝑊𝑤|} subject to ∑ 𝑊𝑗 = 1𝑗 𝑊𝑗 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 (1) problem (2) is equal to the following linear problem: 𝑚𝑖𝑛 𝜉𝐿 subject to |wb − 𝑎𝐵𝑗𝑊𝑗| ≤ 𝜉𝐿 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 |𝑊𝑗 − 𝑎𝑗𝑊𝑊𝑤| ≤ 𝜉𝐿 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ∑ 𝑊𝑗 = 1𝑗 𝑊𝑗 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 j (2) 352 h. fazlollahtabar, n. kazemitash solving the above model (2), optimized weights (w1 *, w2 *,…, wn *) and the optimal objective function value ξl will be gained. for this model ξl can be directly considered as an indicator of the consistency of the comparisons (here we do not use consistency index, so that values close to zero show a high level of consistency of the pairwise comparisons provided by the decision-maker(s)/expert(s). for mcdm problems with more than one level of criteria such as this study, first of all, the weights for different levels should be obtained through the bwm steps, then, the weights of different levels have to be multiplied to determine the global weights [25]. 4.2. evaluation of iss by item-scoring using the bwm, the optimal weights of the criteria (w1 *, w2 *,…, wn *) are calculated. now the isi so that (i=1,…,m) with respect to its green supplier selection measurement j, so that (j=1,…,n) is obtained. therefore, xij using, for instance, a 7-point scale (very low to very high), to determine the overall green supplier selection's performance of is i. for the gssi the following formula: 𝐺𝑆𝑆𝑖 = ∑ 𝑊𝑗 ∗ 𝑥𝑖𝑗 𝑛 𝑗=1 , 𝑎𝑛𝑑 𝑖 = 1,2, … , 𝑚. (3) 5. case study the proposed information system effectiveness model is tested to evaluate and rank the use of iss in emdad-khodro company. saipa automotive group as one of the two biggest automotive companies of iran decided to found emdad-khodro firm as its subdivision company with the purpose of responding to relief requirements of their customers to complete the after-sale services network in 2003. to achieve the planned goals and be adaptable in the contemporary market compared with the rivals, it has implemented some management systems such as quality management systems based on iso9001, complaints management system based on iso10002, training management system based on iso10015 and risk management system based on iso31000 and information systems which are explained in detail in the following. according to the functions of the determined model, it is obviously necessary to specify using iss in the company and start the analysis. based on the record of company, there have existed windows-based iss for 15 years; although almost all the web-based systems which have been implemented since 2011, have replaced the windows-based iss as a major platform, there are some minor parts which still use windows-based systems. web-based systems work in the field of employees, customers and representations. finally, mobile iss have been applied since 2014 which actually cover the whole activities related to company. however, there are 10 active information systems such as transaction processing system (tps), electronic commerce (ec), customer relation management (crm), decision support system (dss), management information system (mis) and office automation system (oas), knowledge management (km), supply chain management (scm), enterprise resource planning (erp) and business intelligence (bi). but there is not any research of their performance and effectiveness in the case of gss; this is an opportunity to try to do that and that is the reason why it is selected as the case study. here, firstly, the conceptual framework of gss is presented which adopted from the literature to measure iss' performance as a multi-criteria decision-making problem, as green supplier selection based on information system performance evaluation using the integrated... 353 shown in fig. 1. in fact, it is a visualization of table 2, including eight perspectives (main criteria) to measure iss' performance (green design, service, green image, quality, environmental management, green product, delivery and cost), as well as the items (subcriteria) of each perspective (three sub-criteria to measure green design perspective, two sub-criteria to measure service perspective, two sub-criteria to measure green image perspective, three sub-criteria to measure quality perspective, six sub-criteria to measure environmental management perspective, seven sub-criteria to measure green product perspective, four sub-criteria to measure delivery perspective and ultimately four subcriteria to measure cost perspective). as mentioned before, to measure the iss' performance of a firm, it is necessary to have two sets of data: the optimal weight for the criteria (w1 *, w2 *,…, wn *) and the iss' score on various sub-criteria, xij. the optimal weights are obtained through the expert opinions, while the scores are computed based on the data from a survey among the 100 experts of iss. in the following sections, gaining weights is described firstly, and then the scores and, finally, the use of the equation (3) to calculate the overall performance of each is. 5.1. weights of green supplier selection measures to obtain the weights of the criteria and sub-criteria, the comparison data needed for bwm is gained by interviewing 20 experts in the field of green supplier selection, individually. next, the weights of criteria and their sub-criteria are determined using bwm. finally, the overall weights for the criteria and sub-criteria are computed by using the aggregation (based on a simple average). table 2 shows the aggregated weights of the eight main criteria and their items (sub-criteria) based on the inputs which are provided by the experts. the consistency ratios are all close to zero ranging from 0 to 0.17, which shows a high reliability of the results. as can be seen from table 2, column 2, green product (weight = 0.2468) criteria is the most important green supplier selection perspective, followed by green design perspective (weight = 0.1741), quality perspective (weight = 0.1330), delivery perspective (weight = 0.1219), service perspective (weight = 0.1008), environmental management perspective (weight = 0.0994), cost perspective (weight = 0.0894) and green image perspective (weight = 0.0346) which is by far the least important perspective of the green supplier selection based on the experts' opinion. the global weights of the sub-criteria (the multiplication of the weights of the subcriterion by the weights of the main criterion to which it belongs) are calculated in table 2, column 5. based on these results, design for reduction or elimination of hazardous materials as the third sub-criteria of the green design (weight = 0.1176) has the most weight which illustrates the most effectiveness role which a sub-criterion could play with respect to the green supplier selection, though the green product has the most amount of weight among the criteria. 354 h. fazlollahtabar, n. kazemitash table 2 relative importance (weights) of the criteria and sub-criteria criteria local weights sub-criteria local weights global weights of sub-criteria green design 0.1741 design for resource efficiency 0.0885 0.0154 design of products for reuse, recycle, and recovery of material 0.2360 0.0411 design for reduction or elimination of hazardous materials 0.6755 0.1176 service 0.1008 rate of processing order 0.2336 0.0235 service quality 0.7664 0.0773 green image 0.0346 ratio of green customers to total customers 0.8418 0.0291 green purchase trend of customers 0.1582 0.0055 quality 0.1330 quality-related certificates 0.6316 0.0840 capability of quality management 0.2535 0.0337 reject rate 0.1149 0.0153 environmental management 0.0994 environmental protection policies/plans 0.1472 0.0146 environment protection system certification 0.1106 0.011 eup 0.4448 0.0442 odc 0.0543 0.0054 rohs 0.1151 0.0114 wee 0.1280 0.0127 green product 0.2468 cost of component disposal 0.1376 0.034 green production 0.2922 0.0721 green certifications 0.1190 0.0294 green packaging 0.1351 0.0333 recycle 0.1272 0.0314 remanufacturing 0.0427 0.011 reuse 0.1463 0.0361 delivery 0.1219 order frequency 0.0866 0.011 order fulfillment rate 0.2520 0.0307 lead time 0.1810 0.0221 delivery efficiency 0.4804 0.0586 cost 0.0894 buying friendly materials 0.0837 0.0075 compliance with sectorial pricing 0.1422 0.0127 performance value/price 0.5277 0.0472 transportation cost 0.2463 0.0220 5.2. green supplier selection item-scores of iss as the first step, in a survey among the 50 iss' experts of the mentioned firm, their opinions about the iss performance and effectiveness with respect to the selection of green suppliers are provided, in which the respondents rated 10 most common iss level based on items from different gss determined sub-criteria on a nine-point likert type scale. and finally, the last operation of this step is that the experts' opinions for every single subcriterion are averaged (table 3). then, the aggregated gss performance of the various iss with respect to different perspectives (columns 2, 4, 6, 8, 10, 12, 14, 16), as well as the ranking of each is based on each perspective (columns 3, 5, 7, 9, 11, 13, 15, 17) are green supplier selection based on information system performance evaluation using the integrated... 355 computed (table 4). furthermore, the overall aggregated gss performance of each is based on items of all perspectives and overall ranking based on this aggregated number are shown in table 4, columns 18 and 19, respectively. assigning weights to different items (sub-criteria) and to different perspectives (main criteria) produces significant differences in the overall (and perspective-based) gss score of different iss. table 3 green supplier selection item-scores of 10 iss green design service green image quality g1 g2 g3 s1 s2 g1 g2 q1 q2 q3 tps 3.67 4.31 4.52 7.16 3.09 5.07 4.14 4.59 5.56 5.14 oas 3.54 3.78 4.63 7.86 3.96 5.26 4.39 4.48 6.21 5.49 mis 6.53 7.09 6.9 5.87 5.48 6.79 7.54 7.87 7.79 7.19 dss 5.92 6.73 7.08 6.16 4.97 6.66 6.92 6.67 6.86 6.97 ec 5.01 6.19 4.74 8.27 6.07 7.51 6.74 6.13 5.45 4.77 erp 8.69 4.99 7.21 7.12 5.13 5.12 6.59 6.19 7.37 4.16 scm 8.15 4.96 7.59 7.66 5.34 4.88 5.9 5.79 7.03 4.99 crm 5.79 8.14 6.68 7.7 7.93 7.99 8.43 7.28 7.84 6.18 km 4.91 5.61 5.63 6.02 5.18 5.64 6.47 6.47 5.72 5.45 bi 6.13 7.17 5.13 6.59 6.11 7.26 7.09 6.52 5.02 6.16 environmental management green product e1 e2 e3 e4 e5 e6 g1 g2 g3 g4 g5 g6 g7 tps 4.31 4.62 4.09 4.18 3.75 3.62 2.42 4.39 4.73 5.29 6.14 5.92 6.27 oas 4.57 5.26 3.68 4.3 4.12 3.79 3.59 4.79 5.02 5.4 6.52 6.3 6.55 mis 7.7 7.41 5.58 5.73 7.96 5.8 6.23 8.08 7.28 7.58 7.91 7.44 7.4 dss 6.23 6.52 4.78 6.14 6.47 6.26 5.89 7.53 6.46 6.79 6.57 6.81 6.78 ec 6.69 6.17 6.59 6.93 6.85 7.47 6.44 6.12 7.26 5.55 5.59 6.85 6.62 erp 7.85 7.55 5.18 5.63 7.69 7.15 7.13 7.71 7.93 8.32 8.21 8.06 7.79 scm 8.1 7.29 5.14 5.22 6.67 7.81 7.61 6.18 7.89 5.14 6.83 7.14 7.34 crm 5.94 6.26 4.49 5.06 4.53 5.34 5.7 5.18 6.23 6.67 7.43 6.99 6.76 km 5.63 5.2 5.01 5.29 5.37 5.76 6.09 7.03 5.35 5.81 6.86 7.16 7.51 bi 5.15 5.51 5.37 5.71 6.11 4.39 6.43 7.1 5.03 6.32 5.94 6.05 6.16 delivery cost d1 d2 d3 d4 c1 c2 c3 c4 tps 3.13 5.48 5.77 4.69 5.6 5.47 6.71 5.4 oas 3.47 6.23 6.12 5.66 6.07 5.9 6.8 5.81 mis 6.27 5.69 6.49 6.17 6.35 7.83 6.55 6.79 dss 6.14 4.18 5.83 5.67 5.71 6.79 6.97 5.54 ec 6.77 6.17 6.16 6.47 7.01 7.35 7.49 7.74 erp 6.52 7.38 7.59 7.53 8.14 5.15 6.73 5.18 scm 6.63 7.09 8.13 7.79 7.57 6.58 7.76 5.12 crm 7.19 5.57 5.88 7.63 5.04 7.12 6.37 4.65 km 5.12 4.25 5.27 6.92 5.44 6.03 6.34 4.3 bi 6.38 4.27 6.43 6.51 6.56 6.52 6.86 5.95 according to table 4, there are two different ways in which it is possible to evaluate and investigate the performance of iss to support the gss process it is based on. on the one hand, it is available to assess the performance of iss through their overall aggregations and rankings, so that the more overall aggregation, the better ranking. for instance, mis possesses the most overall aggregation (6.8800), so it is the first information system as the best one. 356 h. fazlollahtabar, n. kazemitash table 4 green supplier selection performance of 10 iss. g re en d es ig n r an k s er vi ce r an k g re en i m a g e r an k q u a li ty r an k e n vi ro n m en ta l m a n a g em en t r an k g re en p ro d u ct r an k d el iv er y r an k c o st r an k o v e ra ll r an k t p s 4 .3 9 5 2 9 4 .0 4 0 8 1 0 4 .9 2 2 9 1 0 4 .8 9 9 1 1 0 4 .0 8 6 6 9 4 .8 4 4 4 9 4 .9 4 9 5 1 0 6 .1 1 7 8 4 .7 4 6 1 0 o a s 4 .3 3 2 9 1 0 4 .8 7 1 9 5 .1 2 2 4 8 5 .0 3 4 6 9 4 .0 8 4 1 1 0 5 .2 7 7 2 1 0 5 .6 9 7 2 8 6 .3 6 6 6 5 .0 6 4 9 m is 6 .9 1 2 1 3 5 .5 7 1 1 6 6 .9 0 8 7 4 7 .7 7 1 6 1 6 .4 0 4 7 2 7 .5 1 5 1 2 6 .1 1 5 6 5 6 .7 7 4 2 6 .8 8 1 d s s 6 .8 9 4 7 4 5 .2 4 8 8 6 .7 0 1 1 5 6 .7 0 7 3 5 .6 4 3 7 5 6 .8 1 5 2 3 5 .3 6 4 2 9 6 .4 8 6 4 6 .3 2 1 5 e c 5 .1 0 6 9 8 6 .5 8 3 9 2 7 .3 8 8 2 2 5 .8 0 1 4 8 6 .7 1 9 3 1 6 .2 6 0 2 7 6 .3 6 7 3 4 7 .4 9 1 1 6 .1 9 3 6 e r p 6 .8 1 7 1 5 5 .5 9 4 9 5 5 .3 5 2 6 7 6 .2 5 5 9 4 6 .4 0 0 6 3 7 .8 2 9 8 1 7 .4 1 5 6 2 6 .2 4 1 7 6 .7 9 9 2 s c m 7 .0 1 8 9 1 5 .8 8 2 4 5 .0 4 1 4 9 6 .0 1 2 4 7 6 .3 3 5 7 4 6 .7 3 3 8 4 7 .5 7 4 7 1 6 .4 6 2 5 6 .5 7 6 4 c r m 6 .9 4 5 8 2 7 .8 7 6 3 1 8 .0 5 9 6 1 7 .2 9 5 6 2 5 .0 4 3 6 8 6 .1 7 3 8 6 .7 5 6 3 5 .9 4 1 9 6 .6 3 2 3 k m 5 .5 6 1 6 7 5 .3 7 6 2 7 5 .7 7 1 3 6 6 .1 6 2 7 5 5 .2 7 4 9 7 6 .5 9 0 8 5 5 .7 9 2 6 7 5 .7 1 8 1 0 5 .8 9 8 8 b i 5 .6 9 9 9 6 6 .2 2 2 1 3 7 .2 3 3 1 3 6 .0 9 8 4 6 5 .3 3 1 3 6 6 .3 2 6 9 6 5 .9 1 9 8 6 6 .5 6 2 3 6 .0 8 1 7 green supplier selection based on information system performance evaluation using the integrated... 357 it means that it has the most effectiveness and best performance in relation with gss. and after that, erp (6.7986), crm (6.6319), scm (6.5756), dss (6.3210), ec (6.1931), bi (6.0805), km (5.8977), oas (5.0642) and tps (4.7460) are placed in the following ranking, respectively. on the other hand, it is possible to investigate the iss based on their scores and rankings in every single part (the aggregation of every criterion). for example, mis performance as the best one among the 10 mentioned iss, is placed as the first one in the quality criteria, the second one in three criteria, including environmental management, green product and cost criteria, the third one in the green design criteria, the fourth one in the green image criteria and the sixth one in the service criteria. as this way evaluates the performance of iss in every gss criteria, it is the best one to compare two different iss which have close overall aggregations (not exactly the same). for example, there is a slight difference between the overall aggregation of mis and erp which are 6.8800 and 6.7986, respectively, thus in the eyes of someone, it could not clearly explain the superiority of mis over erp. therefore, they rely on the second way to describe the differences and performance of every one in comparison with others. in this case, erp's performance (rank or actually aggregated score) is better than mis in three criteria consisting of service, delivery and green product in which the erp has the best performance, while in the other criteria mis has better scores and rankings. based on the given information in table 4, crm as the third ranked is according to the overall aggregated has the best performance (score) in two perspectives including service and green image. in the same situation, scm as the fourth effective is, possesses two first ranks in green design and delivery criteria and finally as the sixth effective is in relation to gss, ec is placed as the first one in environmental management and cost criteria, even though it is mentioned as the most effective is, it is the best just in one perspective. 5.3. managerial implications our results possess critical managerial implications. firstly, positioning is an important participator in gss performance because it provides an acceptable basis for iss to compare their gss performance to that of other iss. secondly, regardless of positioning, having knowledge about the importance of different gss perspectives, and about the different items of each perspective, gss related managers can formulate more effective strategies to improve their gss performance based on their own purposes. the presented methodology in this study has been used to determine the weight and importance of different aspects of overall is performance. this gives managers a chance to have a good view of critical aspects of performance and allows them to focus more on the important aspects. this study has considered gss performance from eight perspectives which have been used in the literature. as such, gss related managers can enhance gss performance of their iss, based on their purposes. according to this study's findings, green product criteria of gss play the most important role in enhancing gss performance, which means that, if the green product aspects should be the purpose of a firm, focusing more on the green production measurements will improve the firm’s gss performance, as this measure is the most important of all green product measurements. furthermore, for other criteria, this study also determines the sub-criteria which are the most important and how the green supplier selection related managers can improve their firms’ gss performance based on different goals. 358 h. fazlollahtabar, n. kazemitash 6. conclusions although there are so many research studies of the gss and information systems separately, and there are some studies which refer to the is as one of the criteria or subcriteria to select the green supplier, there is not any study to evaluate and investigate the direct relation between these two vital elements of the disciplines that are related to firms and identify the performance of every is and its effectiveness with respect to gss, though iss have been turned into the necessity of all companies. actually, the advantages of this evaluation is that understanding the importance of different gss measures helps managers spend more time, money, energy and resources on the critical aspects on their objectives. the methodology proposed in this paper can be utilized in two general contexts: (1) as a systematic way to compare the gss performance of a set of iss. in this context, based on the results of similar evaluation and the determined purposes of the companies, the position of every iss and their superiorities could be found. plus, the results can also be used by other stakeholders, for instance allowing venture capitalists to identify the best investment opportunities; (ii) as a systematic way to specify the importance (weight) of different criteria and measurements for a single is. in this context, the results can be used by the firm in question to formulate effective gss strategies that are adjusted to its competitive strategy. 6.1. findings the results show the importance of taking into account the weights of different green supplier selection items, which allows each is to determine not only its overall aggregation position (quality), while at the same time providing accurate information of its position with regards to each criterion. moreover, iss can improve their green supplier selection performance based on the importance of each perspective. for example, if scm wants to maintain or improve this ranking, it should focus more on the ratio of green customers to total customers which has the highest importance among all items of green image perspective, in which scm is located in the ninth place (see table 2). moreover, in some situations, where the aim is not to compare the position of iss with each other or such a comparison is impossible to make, knowing the importance of each criterion and their subcriteria can help iss to improve their performance based on their main objectives. more precisely, if an is wants be prominent in green product as the most important criteria in gss process, it should focus on and invest in green production, since the given information in table 2 shows that the green production level is the most important item from a green product perspective. in addition, by changing their objectives, iss can change their strategy and invest more in specific perspective(s) in line with their new purposes. for instance, if an is has thus far concentrated more on the quality perspective, focusing more on green design aspects can help the is improve its gss by looking at design for reduction or elimination of hazardous materials since the results in table 2 show that it is the most important measurement within the green design perspective. therefore, regardless of knowing its position relative to other firms, based on the weight of the sub-criteria of different gss perspectives (table 2), an is can recognize which sub-criteria(s) can improve or change its green supplier selection performance from each perspective. as such, these results can help iss enhance their overall performances. green supplier selection based on information system performance evaluation using the integrated... 359 6.2 limitations the proposed methodology utilized to solve the defined problem, has no limitations and can be used for solving any mcdm problem. what makes this paper stand out from others is that it is the first time that one study evaluates the direct relation of iss and gss individually, and, on the other hand, it is the first time that a study measures the gss performance of iss in which the contributions of iss in related to gss are computed. but, like every other study, interviewing numerous experts in every part of methodology took the time out of standard range. plus, gathering the data through the questionnaire from 100 experts is too hard and time-consuming, and consequently, the calculations and operations of their opinions are so complicated. 6.3. future research direction in the future research, it could be possible to measure the same thing with different gss model including different criteria and sub-criteria. as another suggestion for future, it is possible to change the first or second part of this relation, for instance, the evaluation of online iss' performance in gss. references 1. langefors, b., 1966, theoretical analysis of information systems, sweden: studentlitteratur. 2. sirintrapun, s.j., artz, d.r., 2016, health information systems, clinics in laboratory medicine, 36(1), pp. 133-152. 3. sahay, s., nielsen, p., latifov, m., 2018, grand challenges of public health: how can health information systems support facing them?, health policy and technology, 7(1), pp. 81-87. 4. chen, j., li, m., jiang, r., hu, m.b., 2017, effects of the amount of feedback information on urban traffic with advanced traveler information system, physics letters a, 381(35), pp. 2934-2938. 5. sicilia, á.., madrazo, l., massetti, m., plazas, f.l., ortet, e., 2017, an energy information system for retrofitting smart urban areas, energy procedia, 136, pp. 85-90. 6. miller, w.b. 2017. biological information systems: evolution as cognition-based information management, progress in biophysics and molecular biology, 134, pp. 1-26. 7. köylüoğlu, a.s., duman, l., bedük, a., 2015, information systems in globalization process and their reflections in education, procedia-social and behavioral sciences, 191, pp. 1349-1354. 8. anjana, n.s., amarnath, a., nair, m.h., 2018, toxic hazards of ammonia release and population vulnerability assessment using geographical information system, journal of environmental management, 210, pp. 201-209. 9. durmić, e., 2019, evaluation of criteria for sustainable supplier selection using fucom method, operational research in engineering sciences: theory and applications, 2(1), pp. 91-107. 10. chatterjee, p., stević, ž., 2019, a two-phase fuzzy ahp-fuzzy topsis model for supplier evaluation in manufacturing environment, operational research in engineering sciences: theory and applications, 2(1), pp. 72-90. 11. wagner, k., 2017, geographic information systems and glacial environments, in past glacial environments (second edition), pp. 503-536. 12. yu, f., yang, y., chang, d., 2018, carbon footprint based green supplier selection under dynamic environment, journal of cleaner production, 170, pp. 880-889. 13. banaeian, n., mobli, h., fahimnia, b., nielsen, i.e., omid, m., 2018, green supplier selection using fuzzy group decision-making methods: a case study from the agri-food industry, computers & operations research, 89, pp. 337-347. 14. kuo, t.c., hsu, c.w., li, j.y., 2015, developing a green supplier selection model by using the danp with vikor. sustainability, 7(2), pp. 1661-1689. 15. govindan, k., rajendran, s., sarkis, j., murugesan, p., 2015, multi criteria decision-making approaches for green supplier evaluation and selection: a literature review, journal of cleaner production, 98, pp. 66-83. 360 h. fazlollahtabar, n. kazemitash 16. jain, v., kumar, s., kumar, a., chandra, c., 2016, an integrated buyer initiated decision-making process for green supplier selection, journal of manufacturing systems, 41, pp. 256-265. 17. gupta, h., barua, m. k., 2017, supplier selection among smes on the basis of their green innovation ability using bwm and fuzzy topsis, journal of cleaner production, 152, pp. 242-258. 18. galankashi, m.r., chegeni, a., soleimanynanadegany, a., memari, a., anjomshoae, a., helmi, s. a., dargi, a., 2015, prioritizing green supplier selection criteria using fuzzy analytical network process, procedia cirp, 26, pp. 689-694. 19. hamdan, s., cheaitou, a., 2017, supplier selection and order allocation with green criteria: an mcdm and multi-objective optimization approach, computers & operations research, 81, pp. 282-304. 20. badi, i., pamucar, d., 2020, supplier selection for steelmaking company by using combined grey-marcos methods, decision-making: applications in management and engineering, 3(2), pp. 37-48. 21. durmić, e., stević, ž.,, chatterjee, p., vasiljević, m., tomašević, m., 2020, sustainable supplier selection using combined fucom – rough saw model, reports in mechanical engineering, 1(1), pp. 34-43. 22. pamucar, d., 2020, normalized weighted geometric dombi bonferoni mean operator with interval grey numbers: application in multicriteria decision-making, reports in mechanical engineering, 1(1), pp. 44-52. 23. rezaei, j., 2015, best-worst multi-criteria decision-making method. omega, 53, pp. 49–57. 24. rezaei, j., 2016, best-worst multi-criteria decision-making method: some properties and a linear model, omega, 64, pp. 126–130. 25. salimi, n., rezaei, j., 2018, evaluating firms’ r&d performance using best worst method, evaluation and program planning, 66, pp. 147-155. facta universitatis series: mechanical engineering vol. 17, no 2, 2019, pp. 181 190 https://doi.org/10.22190/fume190330024a © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper1 from winkler’s foundation to popov’s foundation ivan argatov department of materials science and applied mathematics, faculty of technology and society, malmö university, malmö, sweden abstract. in recent years, the method of dimensionality reduction (mdr) has started to figure as a very convenient tool for dealing with a wide class of elastic contact problems. the mdr modeling framework introduces an equivalent punch profile and a one-dimensional winkler-type elastic foundation, called henceforth popov’s foundation. while the former mainly accounts for the geometry of contact configuration, the popov foundation inherits the main characteristics of both the contact interface (like friction and adhesion) and the contacting elastic bodies (e.g., anisotropy, viscoelasticity or inhomogeneity). the discussion is illustrated with an example of the kendall-type adhesive contact for an isotropic elastic half-space. key words: elastic contact, winkler foundation, method of dimensionality reduction, contact stiffness, adhesion strength 1. introduction contact phenomena [1,2] can be encountered in diverse applications ranging from engineering (wheel/rail contact [3], tribological systems [4], etc.) to medicine (e.g., contact of articular cartilage layers [5]). depending on the material’s deformation response to external loading, contact geometry configuration, and accompanying surface effects such as friction or adhesion, a particular contact problem can be very complicated for analytical treatment. while a number of numerical techniques have been developed in the last few decades [6,7], still analytical and semi-analytical models [8,9] of contact interactions are preferred over computer simulations, especially if qualitative understanding of the contact problem is required. for many years, the theory of local contact of elastic bodies, which was created by heinrich hertz [10], was one of the most difficult parts of the solid mechanics courses. in recent years, the method of dimensionality reduction (mdr) has been developed by received march 30, 2019 / accepted june 10, 2019 corresponding author: ivan argatov affiliation: malmö university, 21119 malmö, sweden e-mail: ivan.argatov@mau.se mailto:ivan.argatov@mau.se 182 i. argatov valentin l. popov and his collaborators [11] into a mature and effective semi-analytical framework for analysis of the hertzian type elastic contact. a great advantage of the mdr is that it reduces the contact problem to a much simpler one for a one-dimensional winkler foundation. however, the matter is not so simple, and a vulgar interpretation of the mdr methodology may lead to erroneous conclusions (see, e.g., the author’s discussion [12]). in the present paper, we consider the adhesion aspect of elastic contact, how it is introduced into the core mdr for axisymmetric jkr (johnson–kendall– roberts)-type contact [13], and how it could be generalized into the non-axisymmetric case. 2. winkler’s foundation in this section, we briefly outline the basics of the winkler foundation model in the light of the comparison with an elastic half-space model. 2.1. response equation of the winkler foundation consider an elastic foundation bounded by a flat surface, which is assumed to be smooth (that is, contact is frictionless) and non-sticky (non-adhesive). let an absolutely rigid body be pressed against the foundation surface by a normal force, . the external load is equilibrated by the contact pressures, ( ), distributed over a contact area. according to the hypothesis introduced by emil winkler (see, in particular, [14]), the contact pressures are determined solely by the local normal displacements, ( ), of the surface points which come into contact, i.e., 1 2 3 1 2( , ) ( , )p x x ku x x , (1) constant is called the coefficient of foundation. under the assumption that the contact is unilateral, so that the contact pressure density is not allowed to take negative values, the contact area will be determined by the displaced surface points. 2.2. contact stiffness and incremental contact stiffness in the case of a flat-ended rigid body, which touches the foundation surface over a certain domain, , generally speaking, we will have 3 1 2 1 2 2 1( , )u x x x x     , (2) where and are small angles of the rigid body’s rotation. in view of (1), the equilibrium equation implies that 1 1 2 2( )f k a s s     , (3) where is the area of , and are the first moments of area in the and directions, respectively, i.e., 1 2 1 2 2 1 d d . s x x x s x              from winkler's foundation to popov's foundation 183 let ( ) denote the center of mass of , that is 1 1 1 2 22 1 d d , c c x x x x xax                where the integration is carried out over the domain . then, in view of eqs. (1) and (2), eq. (3) can be rewritten as cf ka  , (4) where ( ) is the vertical displacement of the rigid body’s flat base at the mass center of . the quantity k ka (5) is called the contact stiffness. now, let a rigid body be bounded by a surface 3 1 2( , )x x x  , (6) such that ( ) and ( ) . (this is not a restrictive assumption.) if the rigid body is translationally displaced into the winkler foundation to some depth, , then the surface displacements are 3 1 2 1 2( , ) ( , )u x x x x  , (7) and contact domain will be determined by inequality ( ) . in this case, the contact force is given by 1 2 1 2( ( , )) d df k x x x x       , (8) where ( ) | | is the positive part function. the force-displacement relation (8) is nonlinear, but the incremental contact stiffness can be simply evaluated as follows (cf. eq. (5)): d d f ka   . (9) here, is the area of the domain . 2.3. comparison with the elasticity theory model let us compare the simple solution of the contact problem obtained in the framework of the winkler foundation model with the corresponding results for an isotropic elastic half-space (with young’s modulus and poisson’s ratio ). in particular, what we are interested in is the contact stiffness for a flat-ended indenter, which can be represented as *2k e  , (10) where ( ) is the reduced elastic modulus, and is the so-called harmonic capacity radius of the current contact area (see, e.g. [15]). 184 i. argatov the main difference between formulas (5) and (10) is that they predict different variations of the contact stiffness under a similarity scaling of the contact area, since and have dimensions of and , respectively, where is the dimension of length. explicit formulas for (or for the harmonic capacity) are known only in a limited number of cases (for instance, for an annular contact area [9]). also, some approximations for the -related characteristic (such as contact compliance or constriction resistance) can be found in the literature (see, e.g., [16,17]). in particular, using the cross-property connection established by sevostianov and kachanov [18], the following approximation can be written out for the harmonic capacity radius of a circular cluster of identical circular microcontacts: 1 h 1 1 ina r          . here, is the radius of identical microcontacts, is holm’s radius [19], whose value was estimated in [20,21]. finally, it is clear that for a circular contact area coincides with the contact radius. 3. popov’s foundation in this section we recast the underlying concept of the mdr into a simple form. for a complete and detailed review of the mdr, the reader is referred to [22–26]. 3.1. equivalent profile returning back to the contact problem for a rigid indenter, which is bounded by the surface (6), let us now assume that it is pressed against an elastic half-space. considering the indentation as a one-parametric process, that is the case for normal translational displacement of the indenter, both contact force and indenter displacement can be regarded as one-valued functions of harmonic capacity radius . in particular, let the latter function be designated as ( )g  (11) the mdr reduces the elastic contact problem to a much simpler contact problem for a one-dimensional winkler-type linearly deformable foundation, which will be called the popov foundation, and a rigid punch of equivalent profile. the latter is described by the equation (| |)z g x  , (12) where and are horizontal and vertical coordinates, and function ( ), , is defined according to eq. (11). note that, by definition, the equivalent punch is symmetric, and, therefore, the contact interval will be symmetric with respect to the -axis as well. from winkler's foundation to popov's foundation 185 3.2. stiffness coefficient of the popov foundation the intuitive simplicity of solving the mdr equivalent contact problem is explained not only by the simple character of the deformation response of the popov linearly deformable foundation, but by the exact correspondences (equalities) between the kinematic and force parameters, which are denoted by the same symbols and . while the first (kinematic) correspondence is facilitated by eqs. (11) and (12), the second (force) correspondence is achieved by tuning the popov foundation coefficient, . it is to note here that the mdr transformation rules for the exact mapping of any axisymmetric normal contact with and without adhesion have been first given by heß [27]. the incremental indentation stiffness for the popov foundation is equal to , since is the length of the one-dimensional contact interval. on the other hand, the incremental contact stiffness of an elastic half-space is given by eq. (10). by equating these two values, we readily arrive at the following relation [11]: * zk e . (13) let us emphasize [15] that, while in the one-dimensional equivalent contact problem is the half-width of the contact interval, in the original elastic contact problem has the meaning of the harmonic capacity radius of the current contact area, which corresponds to the same value of kinematic parameter . 4. adhesive strength of elastic contacts in this section, we touch on the adhesive aspect of unilateral frictionless contact. 4.1. adhesive strength of the winkler foundation consider first a flat-ended contact with a winkler foundation. to be more specific, we assume that a flat-ended indenter is pressed against a very thin compressible elastic layer bonded to an absolutely rigid substrate. it was shown by aleksandrov [28] that the normal deformation of such isotropic elastic layer coincides with that of a winkler foundation with the foundation constant ae k h  , (14) where is the layer thickness, and ( ) ( )( ) is the so-called aggregate elastic modulus. under certain conditions (when contour of contact area is smooth with a variable curvature radius much larger than the layer thickness), the jkr-type detachment criterion on the contact contour can be formulated as follows [29]: 2 , a c c e p p p h       , (15) where denotes the work of adhesion, and negative sign denotes tensile stresses. 186 i. argatov observe that eq. (15) was derived with the help of asymptotic modeling technique from the stress-intensity factor (sif) of the boundary layer (we refer to [30,31] for more details). now, let the flat-ended indenter is pulled from the layer surface with its base maintaining a horizontal position. the pull-off force, , thus, is given by c cf ap . (16) this means that the winkler foundation based model of adhesive contact, eqs. (14)–(16), predicts that the adhesive strength is proportional to the contact area. 4.2. adhesive strength of the kendall type contact in the case of an isotropic elastic half-space, the pull-off force of a circular cylindrical indenter of radius was evaluated by kendall as 3 *8cf a e    , (17) however, generalization of kendall’s formula (17) to the non-axisymmetric case is not trivial. following kendall [32], we consider an arbitrary flat-ended punch making perfect contact with an elastic half-space of reduced modulus . when a pull-off force is applied to the punch, its displacement is given by *2 f e   , where is the harmonic capacity radius of the contact area, and the elastic energy will be 2 * const. 4 e f u e    the surface energy is su a  , where is the area of contact. further, the potential energy of the load is 2 * const. 2 p f u e     thus, collecting the above formulas, we evaluate the total energy as follows: 2 * const. 4 t f u a e       (18) observe that and represent two different integral characteristics of the contact area. in the general case, the following inequality takes place [33]: a   . (19) from winkler's foundation to popov's foundation 187 so, differentiating eq. (18) with respect to , we readily get 2 * 24 tu f a ae           . (20) now, making use of the monotonicity property, which holds for both geometric quantities and , and assuming that for , we derive from (19) by differentiation that 1 2 aa      . (21) following the argumentation of kendall [29], we state that detachment is possible when 0tu a    . (22) thus, from eqs. (20)–(22), it follows that 2 * 28cf ae    . (23) or using the inequality (19) once again, we arrive at the following relation [34]: 2 * 38cf e    . (24) another estimate for the pull-off force can be derived from the approximate solution obtained by fabrikant [16] for an arbitrary flat-ended indenter 2 2 ( ) ( , ) 2 ( ) fa p r a a r      , (25) where the equation ( ) determines the boundary of the contact area in the polar coordinates ( ). formula (25) implies the following approximation for the maximum sif: 1max ( ) max ( ) 2 f k a a     . (26) recall that we assume that , i.e, the pull-off force is assumed to be negative. so, the substitution of (26) into the jkr detachment criterion (see [30,31]) * 1max ( ) 2k e   yields * 2 2 16 c e a f d     , (27) where is the diameter of the contact area. observe that formula (27) correctly predicts the onset of detachment for a flat-ended elliptical indenter. 188 i. argatov 4.3. on extension of the mdr to the kendall type adhesive contact in the case of a flat-ended cylindrical indenter, the shape function can be described as follows: 1 2 1 2 2 1 2 0, ( , ) , ( , ) , ( , ) \ . x x x x x x        here, is the contact area, which does not change during the normal translational indentation. let denote the harmonic capacity radius of  . then, the shape function of the equivalent one-dimensional punch will simply be   0, , , . x g x x        note that in the axisymmetric case, when equals the contact radius , the above definition coincides with that of [11]. this means that the pull-off force 12 c c df p  (28) must coincide with kendall’s result (17), when . according to the logic of the rule of heß for the adhesive contact between axiallysymmetric bodies (see [11], section 4.2), the critical value of pressure may depend on . the only question now is, what meaning should be attached to ? in view of the surprising difficulty of the evolution of the detachment process (see [35]), it is suggested to associate with critical force at which the detachment process starts (see [34] for details). observe that among the three upper estimates (23), (24), and (27), only the second one is a function of solely the harmonic capacity radius of the contact area. this fact limits the choice of approximations for to the following one: * 1 2c dp e    . (29) it is interesting and significant that the substitution of eq. (29) into eq. (28) yields kendall’s formula, eq. (17), in the axisymmetric case, when . 5. discussion and conclusion the model of winkler’s foundation is based on two concepts: linear deformation of spring elements according to hooke’s law and non-interaction between the spring elements. there are known many generalizations of the winkler model [36] and, in particular, its varied adaptations to the field of adhesion [37]. it is clear that popov’s foundation is a one-dimensional winkler foundation. but it is more than that. the mdr (and correspondingly the popov foundation) has been extended to cope with tangential [11] and torsional [38] contacts and to account for viscoelastic material’s constitutive relationship [39] as well as for material grading [40,41]. from winkler's foundation to popov's foundation 189 having been stemmed from a simple observation [22] that — as far as one is interested in the stiffness of the hertzian type contact — the three-dimensional contact problem can be reduced to a one-dimensional problem for a winkler foundation, the mdr has grown to become a comprehensive methodology for dealing with a wide class of elastic contacts. strictly speaking, a range of elastic contact problems solved by the mdr is characterized by a hierarchy of popov foundations, or to be more precise, each type of contact interaction, which is covered by the mdr, requires its own popov foundation. since distinct types of contact can differ by diverse surface effects (like friction and adhesion), one can consider a generic case with a combination of the effects and forms of loading. in the mdr framework, this can be done in a straightforward way, assuming superposition of the effects. however, a care should be taken in this regard since the superposition of physical effects is not always valid. to conclude, popov’s foundation, which serves as the basis for the mdr, represents a major advance in developing a unified approach to effective dealing with elastic contacts. references 1. johnson, k.l., 1985, contact mechanics, cambridge university press, cambridge. 2. popov, v.l., 2010, contact mechanics and friction: physical principles and applications, springer, heidelberg, new york. 3. kalker, j.j., 1990, three-dimensional elastic bodies in rolling contact, kluwer academic publications, dordrecht. 4. goryacheva, i.g., 1998, contact mechanics in tribology, kluwer academic publishers, dordrecht. 5. argatov, i., mishuris, g., 2015, contact mechanics of articular cartilage layers: asymptotic models, springer, cham. 6. wriggers, p., 2006, computational contact mechanics, springer, berlin. 7. yastrebov, v.a., 2013, numerical methods in contact mechanics, john wiley & sons, hoboken. 8. barber, j.r., 2018, contact mechanics, springer, cham. 9. argatov, i., mishuris, g., 2018, indentation testing of biological materials, cham, springer. 10. hertz, h., 1882, über die berührung fester elastischer körper, journal für die reine und angewandte mathematik, 92, pp. 156–171 (in german). 11. popov, v.l., heß, m., 2014, method of dimensionality reduction in contact mechanics and friction, springer, berlin. 12. argatov, i., 2016, a discussion of the method of dimensionality reduction, proceedings of the institution of mechanical engineers, pt. c: journal of mechanical engineering science 230(9), pp. 1424–1431. 13. johnson, k.l., kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids, proceedings of the royal society a, 324, pp. 301–313. 14. frýba, l., 1995, history of winkler foundation, vehicle system dynamics supplement 24, pp. 7–12. 15. argatov, i., hess, m., pohrt, r., popov, v.l., 2016, the extension of the method of dimensionality reduction to non‐ compact and non‐ axisymmetric contacts, journal of applied mathematics and mechanics (zamm), 96(10), pp. 1144–1155. 16. fabrikant, v.i., 1986, flat punch of arbitrary shape on an elastic half-space, international journal of engineering science 24, 1731–1740. 17. galin, l.a., 2008, contact problems: the legacy of l.a. galin., in: gladwell, g.m.l. 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2016, a simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, international journal of engineering science, 104, pp. 20–33. 41. argatov, i., hess, m., popov, v.l., 2018, the extension of the method of dimensionality reduction to layered elastic media, journal of applied mathematics and mechanics (zamm), 98(4), pp. 622–634. 7717 facta universitatis series: mechanical engineering vol. 21, no 2, 2023, pp. 323 338 https://doi.org/10.22190/fume210619072c © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fracture parameters evaluation for the cracked nonhomogeneous enamel based on the finite element method and virtual crack closure technique yan cai1,2, liming zhou1, ming li1 1school of mechanical and aerospace engineering, jilin university, china 2hospital of stomatology, jilin university, china abstract. to accurately solve the fracture parameters of enamel, we have established computational nonhomogeneous enamel models and constructed the fracture element of enamel dumb nodes, based on the enamel mineral concentration, nonhomogeneous mechanical properties, and virtual crack closure technique. through the commercial finite element software abaqus and the fracture element of the enamel dumb nodes, we have established the user subroutines umat and uel, which enabled solving of the energy release rates of the nonhomogeneous enamel structure with cracks. the stress intensity factors of central cracks, three-point bend and compact stretched enamels, and double-edge notched stretched enamels are determined. by comparing them with analytical solutions, we have proved that the fracture element of the enamel dumb nodes is highly accurate, simple, and convenient. in addition, the cracks can be other elements rather than singular or special elements; they show versatility and other advantages. the stress intensity factor of the dental enamel can be solved more realistically. thus, a new numerical method for prevention and treatment of dental diseases is provided. key words: nonhomogeneous enamel models, crack, virtual crack closure technique, finite element method, fracture element of enamel dumb nodes 1. introduction enamel is the most important component of the outermost tissue of human teeth [1-3], which serves as a barrier for protecting the vital pulp and the dentin. on a microstructural level, a sequence of crystal rods (or “prisms”) with diameters of 4–6 μm compose the enamel tissue. the crystal rods stretch from the dentin-enamel-junction (dej) to the occlusal surface and are arranged in a parallel manner [4]. the interface between the rods is received june 19, 2021 / accepted november 28, 2021 corresponding author: liming zhou, ming li school of mechanical and aerospace engineering, jilin university, changchun, jilin 130025, china e-mail: lmzhou@jlu.edu.cn (liming zhou), liming940411@gmail.com (ming li). mailto:lmzhou@jlu.edu.cn mailto:liming940411@gmail.com 324 y. cai, l. zhou, m. li a protein-rich district considered as the rod sheath. enamel is a composite material that has a unique and complex hierarchical structure. the unique properties of enamel distinguish it from other biological tissues like bone or dentin [5]. in the body, enamel is the most highly mineralized tissue, being composed of approximately 96% minerals (primarily apatite crystals containing ca2+ and p3-, small amounts of other phosphate crystals, and small amounts of other minerals), and only 4% organic matrix and water (3% water and 1% proteins) by weight [6]. teeth are well-designed to endure repeated chewing loads during a lifetime. to support oral function, the enamel should be equipped with adequate hardness, stiffness, and fracture resistance. although the toughness of the protective enamel coat is comparable to that of glass, this material is highly brittle [7]. in dental clinic, incomplete tooth or tooth fracture is a general pathological feature [8, 9]. considering the physiological characteristic of enamel and the circle force contact incurred during mastication, fatigue can lead to failure. fractures that run as ribbon cracks around the enamel shell walls may occur as a result of macro-contacts in the cusp regions with hard millimeter-scale food objects (e.g., nuts, seeds, and hard particulates) [10-12]. these cracks are distributed along a normal direction to the tooth surface orientation and remain entirely contained within the enamel. some cracks develop rapidly in clinic, and in severe cases, they can cause deterioration or even ultimate loss of tooth function. however, some fractured teeth can be used during life. therefore, considering how to effectively protect the tooth activity and reduce or even block the propagation of cracks, further analysis of the mechanical characteristics of the dental enamel is required. over the last decade, enamel has been modeled as a homogenous material [13, 14], but recently, it has been explored with variations in its mechanical characteristics. enamel hardness and elastic modulus decrease from the occlusal surface of enamel to the dej [15, 16]. the variations are probably due to chemistry, microstructure, and prism alignment. however, cuy et al. [17] suggested that enamel is significantly correlated with changes in average chemistry: the concentrations of hydroxyapatites p2o5 and cao are larger over the chewing surface and gradually decrease to the softer dej, but the concentrations of na2o and mgo change in the opposite direction. braly [18] and park [19] also reported similar variations of mechanical properties on the axial section of the enamel [20]. more recently, nanoindentation tests showed that the enamel elastic modulus and hardness decreased from ≥6 gpa and ≥120 gpa in the occlusal surface to less than 3 gpa and 70 gpa in the dej, respectively [21-23]. as reported, the crack growth resistance of enamel increases (rising r-curve) with crack extension from the outer to inner part; the law of variation of toughness is associated with the distance to dej [24]. due to the small size of cracks, it is difficult to build an in-vitro experimental model for early cracked teeth. thus, the finite element method (fem) becomes relatively practical for studies on cracked teeth [25-27]. recently, fem has been widely used to analyze the causes of cracked teeth, but the existing studies have focused on the stress distribution, biomechanical factors, crack expanding behavior, and fatigue life of cracked teeth [28-30]. moreover, due to methodological limitations, previous studies mostly regarded enamel as an isotropic material for mechanical analyses. using the analytical method, the fracture parameters of a simple geometric structure or a special load form of homogenous materials can be derived [31-33]. however, this method cannot be applied to complex systems or boundary conditions, as the calculation process becomes very expensive and time-consuming. nevertheless, these limitations can be overcome by using other numerical methods. fracture parameters evaluation for the cracked nonhomogeneous enamel based on... 325 nowadays, fem is extensively employed in fracture analysis of nonhomogeneous materials. the elastic modulus coefficient of nonhomogeneous materials is a smoothing function of specific coordinates, also associated with the establishment of element stiffness matrix. for two-dimensional (2d) problems, the stress intensity factors for modes (modes i and ii) can be comparatively evaluated through three nonhomogeneous material-tailored approaches: path-independent j-integral, virtual crack closure technique (vcct) [34-37], and displacement correlation. vcct was used for 2d cracked specimens and thereafter expanded to three-dimensional crack problems [38-40]. then, a method combining fem and vcct for the study of kinking cracks in a homogeneous material was introduced [41-43]. the energy release rates for cracks were computed by vcct [44], and the interface cracks were studied with the virtual crack extension method and vcct [45-47]. krueger [48] presented an overview of the vcct historical developments and discussed its different applications. leski [49] provided the general conditions for implementation of vcct in msc.patran. azimi et al. [50] probed into the special cracked lap shear specimen deboning using vcct. zeng et al. [51] extended vcct to investigate the fracture mechanics parameters and simulate crack propagation within the framework of cell-based smoothed fem. banks-sills and farkash [52] explored vcct to calculate stress intensity factors of two modes (modes i and ii) for an interface crack between two linear elastic, isotropic, and homogeneous materials. vcct was also extended to the investigation of the element-free galerkin method [53, 54]. based on graded fem, a 2d-vcct interface element was proposed for the dynamic fracture analysis issues [55]. despite the secondary development of the commercial finite element software (ansys or abaqus), the crack problem analysis is mainly focused on homogeneous materials, and rarely on nonhomogeneous materials [56-58]. in this study, we built nonhomogeneous enamel models based on the commercial finite element analysis (fea) software abaqus and vcct. a dummy node enamel fracture element was established, and the energy release rates of nonhomogeneous cracked enamel were solved. the method provided better results compared to the analytical method in addressing nonhomogeneous mechanical properties. 2. materials and methods 2.1. nonhomogeneous enamel model previous studies mostly considered the enamel as an isotropic material for mechanical analysis. however, the tooth mineral concentrations gradually change from tooth surfaces to the dej [17] (fig. 1). based on ref. [17], enamel is a nonhomogeneous material (from tooth surfaces to dej, the percentages of p2o5, cao, na2o, mgo and other components change in certain functions). for the model proposed, two materials were considered: material c, with a composition of 94% p2o5 + cao, 6% na2o + mgo and other components, and material m, with a composition of 89% p2o5 + cao, 11% na2o + mgo and other components. fig. 2 shows the gradient model of nonhomogeneous enamel materials. the enamels analyzed were composed of materials m and c, whose volume fractions continuously changed along certain dimensions of the model (fig. 2). the mori-tanaka models and the rule of mixture are common methods to evaluate the effective elastic properties of grade 326 y. cai, l. zhou, m. li composites. the effective moduli of the two materials were homogenized through these two methods. the effective property associated with volume fraction exponent follows a power law: ( ) m m c cp z p v pv= + , 1m cv v= − (1) (0.5 )n cv z h= + , ( 2 2, 0 )h z h n−      (2) where subscripts m and c in eq. (1) denote materials m and c, respectively; p denotes the poisson’s ratio or young’s modulus; z denotes the thickness coordinate; n is the power exponent factor. eq. (1) presents the volume fraction variation along non-dimensional thickness (z/h). volume fraction vc variation with different power exponent factors n is shown in fig. 3. fig. 1 map of young’s modulus throughout the enamel by nanoindentation material c material m 75% material c + 25% material m 50% material c + 50% material m 25% material c + 75% material m fig. 2 gradient model of nonhomogeneous enamel materials fracture parameters evaluation for the cracked nonhomogeneous enamel based on... 327 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 n=15.0 n=5.0 n=2.0 n=1.0 n=0.5 n=0.2 v c z/h n=0.0 fig. 3 variation of volume fraction vc function along the non-dimensional thickness with varying n 2.2. finite element method the elements were established by directly adopting the true parameter at the gauss integration points of each element [59]. the equilibrium equations are given as: e e e=k u f (3) where ue represents the nodal displacement vector and fe represents the load vector. the stiffness matrix of fem can be expresses as: e e e t e e e( ) ( ) d  = k b d x b (4) where e represents the problem domain of finite element e, be and de(x) represent the strain-displacement matrix and constitutive matrix variable, respectively. the strain-displacement is related to the gradients of the interpolating functions. in this study, the elasticity matrix de(x) = de(x, y) was assumed to be related with spatial coordinates and specified at each gaussian integration point. elastic modulus e and poisson’s ratio ν in elastic matrix de(x) can be established by interpolation for the nonhomogeneous materials [60]: 1 ε i i i e n e = = (5) 1 ε i i i v n v = = (6) where ni denotes the fem shape function corresponding to node i. ei and vi denote the material properties at node i of the element. ε is the node number of the element. by using eqs. (5) and (6), the actual variations of the material properties in an element can be approximated by polynomial forms. 328 y. cai, l. zhou, m. li 2.3. fracture element of enamel dumb nodes the enamel cracks commonly seen in the clinic are closely related to the acute bite force, temperature change, and congenital developmental defects of the enamel. the properties of the mechanical field near the crack tip in nonhomogeneous enamel are the same as in homogeneous materials [61]. however, the stress intensity factors and fracture toughness parameters follow gradient functions. for vcct, the strain energy was assumed as the same when releasing from a certain degree of crack extension or closing the crack of the same severity [54]. this reveals the similarity between crack extension from node b to c along one element and crack closure at node c (fig. 4), which is the case of only mode i loading. c c' ba a' δaδc x y b' c c' ba a' δaδc x y b' a b fig. 4 mesh configuration used for vcct element, which describes the growth process of the crack. crack closure at points (a) c and (b) b fig. 5 shows a crack inclination and crack tip with respect to the axes in the local coordinate system. θc c' b a a' crack element δa δc x y x y fig. 5 inclined crack in a plane with specific coordinates system. the crack inclination and crack tip in the local coordinate system are displayed the details of vcct elements can be found in refs. [45, 62]. in particular, each element has five nodes. nodes c and c' are located at the crack tip, node b is located ahead of the crack tip, and nodes a and a' are located behind the crack tip. the present element has two groups of nodes: a top group (nodes c, a, and b) and a bottom group (nodes c' and a'). a very stiff spring is placed between nodes c and c', and the crack-tip nodal forces related with node displacement are given as follows: fracture parameters evaluation for the cracked nonhomogeneous enamel based on... 329 ( )cx cx c cf k u u = − , ( )cy cy c cf k v v = − (7) where (uc, vc) and (uc’, vc’) represent the displacement of nodes c and c' at the specific coordinate system (x, y), respectively; kcx and kcy represent the xand y-direction stiffness, respectively. formerly, they are established with large values [45, 62], but when the crack growth is predicted, they reduce to zero. dummy nodes a, a', b are only employed to record the information on the displacement of the opening node behind the crack tip and the crack jump length before the tip, and do not contribute to the stiffness matrix. for nodes a and a', the displacement openings are calculated as: a au u u  = − , a av v v  = − (8) where (ua, va) and (ua’, va’) represent the displacement of nodes a and a' under the specific coordinate system (x, y), respectively. hence, the crack jump length is the distance between nodes c and b, whose relation is given as follows: 2 2( ) ( )b c b ca x x y y = − + − (9) where (xc, yc) and (xb, yb) are the coordinate values of nodes c and b in the global coordinate system, respectively. the strain energy release rates (gi and gii) under the specific coordinate system (x, y) attached to the crack tip was computed. thus, the fracture modes i and ii can be separated (fig. 5). the angle between x and x is given as: arccos arcsinb c b cx x y y a a  − −    = =         (10) by projecting the corresponding nodal forces and crack opening displacement into specific coordinates, we can obtain: cos sin sin cos cxcx cycy ff ff          =      −    (11) and ' ' ' ' cos sin sin cos aa aa aa aa u u v v       =      −         (12) considering 2d-vcct, we can express the approximate serrs as follows: i 2 cyf v g b a  =  , ii 2 cxf u g b a  =  , (13) where b represents the structure thickness. for the analysis of the dynamic loading fixed crack problem, we can describe the relationship between dynamic stress intensity factors ki (kii) and strain energy release rates gi (gii) as follows: 330 y. cai, l. zhou, m. li i i( )cyk sign f eg= (14) ii ii( )cxk sign f eg= (15) where e represents material matrix and has different expressions in plane stress and plane strain conditions (plane stress tip e e= , plane strain condition: 2(1 ( ) ) tip tip e e = − ); etip and μtip denote the crack tip elasticity modulus and crack tip poisson’s ratio, respectively [63]. 3. results and discussion 3.1. example 1 given the problem of central enamel cracks, we have selected finite-width central crack tension trials. the geometric model and the loading are shown in fig. 6, where l, w, and b represent length, width, and thickness of the model, respectively (l=2 mm, w=2 mm, b=0.1 mm). the crack length was 2a and the tension load is σ=30 mpa. the enamel is composed of materials m and c. the two materials were subjected to eqs. (1) and (2), where the power exponent factors were n=0, 0.5, 1.0, 5.0 and the elastic moduli were mm=90 gpa and mc=120 gpa. these two materials had the same poisson's ratio ν =0.3 (the properties of the enamel are the same in other examples). l σ σ w 2a fig. 6 geometric model of an enamel specimen containing a central crack to illustrate the convergence of this method, we have simulated a discrete model of 4 element meshes to the normalized dsifs ki (40×40 elements, 80×80 elements, 120×120 elements). the model was geometrically established with 2d cpe4 (2d standard plane strain elements) in abaqus®. we have employed no special means, such as the collapsed element technique at the crack tip or special singular elements. table 1 shows energy release rate gi from the computation model with power exponent factors n=0.0, 0.5, 1.0, and 5.0, four grids, and crack lengths a=0.2, 0.3, 0.4, 0.5, and 0.6 mm. by comparing with the analytical solutions, we validated the accuracy and effectiveness of the new method. fracture parameters evaluation for the cracked nonhomogeneous enamel based on... 331 fig. 7 shows energy release rate gi computed from the model with power exponent factors n=0.0, 0.5, 1.0, and 5.0, crack lengths a=0.3, 0.4, 0.5, and 0.6 mm, and cell division = 100×100. compared with the analytical solutions, the maximum error was only 3.2%, which indicates that the nonhomogeneous enamel calculation model and the calculation method proposed in this study is accurate and effective. table 1 energy release rate gi (n/mm) computed from the new model at power exponent factors of n=0.0, 0.5, 1.0, and 5.0 n method a (mm) 0.2 0.3 0.4 0.5 0.6 0.0 vcct 120120 0.00485 0.00792 0.01212 0.01856 0.02664 vcct 8080 0.00484 0.00791 0.01202 0.01863 0.02687 vcct 4040 0.00480 0.00790 0.01208 0.01838 0.02681 anal. solu. 0.00496 0.00807 0.01235 0.01896 0.02731 0.5 vcct 120120 0.00535 0.00854 0.01301 0.02006 0.02887 vcct 8080 0.00526 0.00852 0.01303 0.01999 0.02901 vcct 4040 0.00524 0.00851 0.01297 0.02016 0.02898 anal. solu. 0.00535 0.00871 0.01333 0.02046 0.02947 1.0 vcct 120120 0.00552 0.00901 0.01376 0.02116 0.03056 vcct 8080 0.00555 0.00903 0.01372 0.02113 0.03053 vcct 4040 0.00553 0.00904 0.01381 0.02130 0.03062 anal. solu. 0.00566 0.00922 0.01411 0.02167 0.03121 5.0 vcct 120120 0.00637 0.01062 0.01598 0.02435 0.03511 vcct 8080 0.00633 0.01043 0.01596 0.02433 0.03503 vcct 4040 0.00637 0.01039 0.01599 0.02448 0.03532 anal. solu. 0.00651 0.01065 0.01632 0.02504 0.03608 0.3 0.4 0.5 0.6 0.010 0.015 0.020 0.025 0.030 0.035 g i ( n /m m ) a (mm) anal. solu. n=0.0 anal. solu. n=0.5 anal. solu. n=1.0 anal. solu. n=5.0 vcct n=0.0 vcct n=0.5 vcct n=1.0 vcct n=5.0 fig. 7 energy release rate gi at crack lengths a=0.3, 0.4, 0.5, and 0.6 mm with different power exponent factors n=0.0, 0.5, 1.0, and 5.0 332 y. cai, l. zhou, m. li 3.2. example 2 given the three-point bending problems of enamel, we have conducted three-point bending trials. the geometric model with length w=4.4 mm, width h=1 mm, span s=4 mm, thickness b=0.25 mm is shown in fig. 8, where a denotes the length of the crack. the model is subject to a concentrated load p of 1.0 n. s p a w h fig. 8 enamel specimens for three-point bending trials fig. 9 shows energy release rate gi with power exponent factors n=0.0, 0.5, 1.0, and 5.0 and crack lengths a=0.45, 0.50, 0.55, and 0.6 mm. compared with the analytical solutions, the maximum error was only 3.9%, indicating that the nonhomogeneous enamel computation model and the method proposed are accurate and effective. 0.45 0.50 0.55 0.60 0.010 0.015 0.020 0.025 0.030 0.035 g i ( n /m m ) a (mm) anal. solu. n=0.0 anal. solu. n=0.5 anal. solu. n=1.0 anal. solu. n=5.0 vcct n=0.0 vcct n=0.5 vcct n=1.0 vcct n=5.0 fig. 9 energy release rate gi of three-point bending enamel specimens at crack lengths a=0.45, 0.50, 0.55, and 0.6 mm with different power exponent factors n=0.0, 0.5, 1.0, and 5.0 fracture parameters evaluation for the cracked nonhomogeneous enamel based on... 333 3.3. example 3 in the enamel compact tensile trials, the geometric model and loading are shown in fig. 10, where l, w, and b represent length, width, and thickness of the model, respectively (l=1 mm, w=1.25 mm, b=0.1 mm). the crack length is a and p=1.0 n is a concentrated load. l p a p w fig. 10 enamel specimen for compact tensile trials fig. 11 demonstrates energy release rate gi computed from power exponent factors n=0.0, 0.5, 1.0, and 5.0, and crack lengths a=0.3, 0.4, 0.5, and 0.6 mm. compared with the analytical solutions, the maximum error was only 3.3%, indicating that the nonhomogeneous enamel computation model and the method proposed are accurate and effective. 0.3 0.4 0.5 0.6 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 g i ( n /m m ) a (mm) anal. solu. n=0.0 anal. solu. n=0.5 anal. solu. n=1.0 anal. solu. n=5.0 vcct n=0.0 vcct n=0.5 vcct n=1.0 vcct n=5.0 fig. 11 energy release rate gi from compact tension enamel specimens at crack lengths a=0.3, 0.4, 0.5, and 0.6 mm with different power exponent factors n=0.0, 0.5, 1.0, and 5.0 334 y. cai, l. zhou, m. li 3.4. example 4 the geometric model and loading for double-edge-notched tension tests are shown in fig. 12, where l, w, and b represent length, width, and thickness of the model, respectively (l=2 mm, w=2 mm, b=0.1 mm). the crack length is a, and the load σ=40 mpa is uniformly distributed. l σ a a σ w fig. 12 enamel specimens for double-edge-notched tension tests fig. 13 shows energy release rate gi computed at power exponent factors n=0.0, 0.5, 1.0, and 5.0 and crack lengths a=0.2, 0.3, 0.4, and 0.5 mm. compared with the analytical solutions, the maximum error was only 4.1%, indicating that the nonhomogeneous enamel computation model and the method proposed are accurate and effective. 0.2 0.3 0.4 0.5 0.010 0.015 0.020 0.025 0.030 0.035 0.040 anal. solu. n=0.0 anal. solu. n=0.5 anal. solu. n=1.0 anal. solu. n=5.0 vcct n=0.0 vcct n=0.5 vcct n=1.0 vcct n=5.0 g i ( n /m m ) a (mm) fig. 13 energy release rate gi from compact tension enamel specimens at crack lengths a= 0.2, 0.3, 0.4, and 0.5 mm fracture parameters evaluation for the cracked nonhomogeneous enamel based on... 335 4. conclusions based on fem, the 2d-vcct enamel dumb element was constructed, which can be employed in the fracture analysis of enamel. we implemented the element into abaqus by the user element subroutine (umat and uel). in view of the numerical examples, the results were in agreement with the analytical solutions, which verified that this proposed interface element can directly apply fracture mechanics to solve fracture problems on abaqus. the proposed element has some advantages, such as the ability to extract fracture parameters without extra post-processing while the definition of body mesh is unconstrained. the proposed element can also be integrated with fem. thus, as the reliability of the proposed element was proved in various cases, it can be potentially employed in nonhomogeneous material fractures. in summary, the proposed method is simple, efficient, universal, and steady, and can be applied to the structure-level analysis of nonhomogeneous fracturing problems by resorting to the commercial fea codes. acknowledgments：the authors are grateful for the financial support to the national natural science foundation of china (grant no. 51975243), scientific research project of jilin provincial department of education (grant no. jjkh20190131kj) and graduate innovation fund of jilin university (grant no. 101832020cx099). references 1. schwartz, g.t., 2000, enamel thickness and the helicoidal wear plane in modern human mandibular molars, archives of oral biology, 45(5), pp. 401-409. 2. sajewicz, e., 2006, on evaluation of wear resistance of tooth enamel and dental materials, wear, 260(11-12), pp. 1256-1261. 3. baino, f., verne, e., 2017, production and characterization of glass-ceramic materials for potential use in dental applications: thermal and 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114, pp. 58-73. 8264 facta universitatis series: mechanical engineering vol. 23, no 2, 2025, pp. 211 225 https://doi.org/10.22190/fume220814041b © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature and sodium borohydride concentration manik barman, tapan kumar barman, prasanta sahoo department of mechanical engineering, jadavpur university, kolkata, india orcid ids: manik barman https://orcid.org/0009-0007-8345-6423 tapan kumar barman https://orcid.org/0000-0002-3435-809x prasanta sahoo https://orcid.org/0000-0002-1538-0646 abstract. previously electroless ni-b (enb) coatings were analyzed and optimized based on various coating parameters. however, variation of nano-indentation behaviour like nano-hardness, elastic modulus and scratch hardness variation with bath composition and heat treatment temperature has not been reported earlier. an attempt has been made to explore the same in the present study. enb coating layers are deposited on aisi 1040 steel specimen with varying concentration of sodium borohydride (nabh4) and heat-treated at 350°c, 450°c and 550°c to investigate the related effects. nanohardness and elastic modulus of as-coated specimens are found to improve with nabh4 concentration due to increased boron content and nodule size. both nano-hardness and elastic modulus are observed to improve further upon heat treatment because of incorporation of various boride phases leading to compact morphology and increased size of the nodules. scratch hardness value also increases with nabh4 concentration and it improves further upon heat treatment and reaches to its maximum at 450°c due to presence of compact and hard ni2b phase. compact homogeneous surface morphology enhances the friction and wear behaviour of the heat-treated coatings even though surface roughness deteriorates after heat treatment. key words: enb, heat-treatment, cof, wear, nano-hardness, micro-scratch 1. introduction enb alloy coatings are known to have improved tribological behaviour due to their cauliflower-like surface morphology which helps to improve frictional behaviour by reducing the real contact area [1, 2]. enb coatings also enhance the surface hardness of mild steel substrate [3, 4]. the coatings have also been investigated on the basis of different parameters like plating rate, thickness, surface morphology, surface roughness, surface received august 14, 2022 / accepted october 25, 2022 corresponding author: prasanta sahoo department of mechanical engineering, jadavpur university, kolkata 700032, india e-mail: psjume@gmail.com; prasanta.sahoo@jadavpuruniversity.in https://orcid.org/0009-0007-8345-6423 https://orcid.org/0000-0002-3435-809x https://orcid.org/0000-0002-1538-0646 212 m. barman, t. k. barman, p. sahoo hardness, tribological behaviour etc. [3, 4]. it was observed that the coating’s behaviour depend on coating composition which is a function of bath composition [3, 4] and bath temperature, ph etc. the amount of boron in the enb coated films is seen to depend on coating bath temperature [3,4] and nabh4 concentration [3, 5]. therefore, a small change in coating bath composition may modify the coating characteristics due to change in surface morphology, phase structure and coating composition. enb coatings exhibit crystalline structure at low boron content [3] while it turns to a combination of amorphous-crystalline structures with increase in boron content till medium level [3,5]. the same phase structure further transforms into x-ray amorphous with the increasing amount of boron [3]. therefore, the phase structure is a function of boron content. it was also reported that phase transformation of enb coatings starts at around 300°c [6-8]. the amorphous or amorphous-crystalline phase structure transforms into crystalline phase with the incorporation of ni, ni2b, ni3b due to heat treatment [6,7]. corrosion performance of the coatings with crystalline structure is seen to be better than amorphous structure [3]. hardness and wear resistance were seen to improve with the increase in boron content [3] possibly due to compact, homogeneous surface morphology and amorphous phase structure. hardness and wear resistance get improved further after heat treatment [9] with the incorporation of various hard phases like ni, ni2b, ni3b [6,10]. hardness of enb coatings having boron content up to 6% does not differ much while it gets improved upon rise in boron content beyond 6% [11]. moreover, hardness increases upon heat-treatment till 450°c. the same also increases with heat treatment duration. the transformation of hard ni2b, ni3b phases may be attributed for increase in hardness upon heat treatment [12]. binary alloy coatings are deposited with the addition of various hard particles like mo, w and investigated for evaluation of their characteristics [10,13,14]. incorporation of hard nano or micro sized particles into metal matrix may modify the coatings characteristics. different nickel based binary and ternary coatings are deposited with hard nano-particles like zro2, al2o3 to improve the surface hardness and elastic modulus through grain refinement [15]. hardness, corrosion performance and wear resistance of nano-composite coatings get improved due to addition of nano-particles like tio2, nd etc. into ni-b matrix [16]. therefore, it is apparent that enb coating characteristics depend on its surface morphology, phase structure and chemical composition. these parameters vary with the concentration of bath constituents. hence, the chemical content, surface morphology and phase structure may be altered by varying the bath composition. the operating conditions of the bath also influence the coating composition, plating rate, morphology and phase structure. enb coatings have been investigated to study the mechanical behaviour using nano-indentation technique at a fixed concentration of coating bath parameter. the surface morphology and phase structure also change with heat treatment which also impacts coating characteristics. the impact of reducing agent concentration and heat treatment temperature on nano-hardness, elastic modulus and scratch hardness has remained unexplored so far. thus, an attempt is made now to deposit enb coatings by varying the reducing agent (nabh4) concentration and then subjecting it to heat treatment at different temperatures to evaluate the related effect on nano-hardness, elastic modulus and scratch hardness. modification of surface morphology and phase structure with variation in bath composition and heat treatment temperature has been correlated with nano-hardness, elastic modulus and scratch hardness behaviour. tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature... 213 2. experimental details 2.1. coating deposition procedure enb coatings are deposited on aisi 1040 steel substrates. aisi 1040 steel contains 0.32-0.43 wt.% c, 0.10-0.25 wt.% si, 0.68-0.85 wt.% mn, 0.03 wt.% s, 0.06 wt.% p and the remainder is fe. square samples of size 15mm x 15mm x 2mm are used for the testing of mechanical properties. cylindrical samples of size ø6mm x 3cm long are employed for friction and wear tests. the specimens are smoothened and cleaned using soap water and rinsed in acetone to remove any oily substance. rust and oxide layers are removed by 50% hcl solution. finally, the smooth, defect-free specimens are emersed into chemical bath for coating deposition. steel surface is catalytically active to start the coating deposition process. however, the specimens are dipped into lukewarm pdcl2 to accelerate the deposition process further. a 200 ml coating solution bath is used for coating deposition. coating bath load is maintained within the range of 28.5 to 31.1 cm2/l. in the current study, no mechanical or ultrasonic agitation is used. alternatively, bath replenishment is chosen so that the coating bath stability and deposition rate is not hampered as well as the deposition of diffused layer does not take place due to the reduction of reactive species. the 200ml bath is replaced with another one after 2 hours as a measure of bath replenishment. therefore, a constant time duration of 4 hours is applied to deposit enb coatings on steel specimens. the coating bath contains nicl2 (20 g/l) and nabh4 (0.50, 0.80, and 1.10 g/l) as nickel and boron source in the solution, respectively. nabh4 concentrations of 0.50, 0.80 and 1.10 g/l are termed as low, medium and high concentrations in next sections. naoh (40 g/l) and c2h8n2 (59 g/l) are used as a buffer and complexing agent, respectively. pbno3 (0.0145 g/l) is utilized as a stabilizer. coating bath temperature is maintained at around 95±2ºc. on the other hand, bath ph is maintained at 12.5. after the deposition with varying nabh4 concentration is over, the specimens are heat-treated at 350°c (ht350), 450°c (ht450), 550°c (ht550) for 1 hour and cooled in the furnace itself. 2.2. coating characteristics surface morphology of enb coatings is analysed using sem (model: s3400n, make: hitachi, japan) accompanied by a secondary electron detector. elemental analysis of these coatings is done at an energy level of 10-15 kev with the help of edax. the setup is equipped with a super ultra-thin window for effective transmission of low energy x-rays. the edax system has been calibrated with si. the spectrum of the standard boron sample is compared with the as-deposited coated specimen spectrum to calculate the boron content [7, 17]. the xrd system uses cu-kα radiation with 2θ angle ranges from 20° to 90° and the scanning rate is kept constant at 0.02°/sec. 2.3. friction and wear measurement a pin-on-disc type multi-tribotester (model: tr-20le-chm-400, make: ducom, india) is employed for the analysis of tribological behaviour of the coated specimens under dry sliding conditions. the tribological tests are carried out as per astm standard g99-05 (reapproved 2010). the cylindrical specimens of ø6mm x 3cm in length are used for the tribo-tests. the rotating disc is made of en31 (58-68 hrc) having diameter of 11.5cm and thickness of 8mm. the cylindrical samples are pressed against the rotating disc with 50n of normal load. sliding velocity is maintained as 0.3925 m/s for a travel distance of 471m and a track diameter of 6 cm. friction force at the contact between coated surface and 214 m. barman, t. k. barman, p. sahoo rotating disc is recorded in the attached computer. mass of the specimen before and after each test is measured using a weighing balance having a precision of 0.01 mg. the difference between the two readings is the mass loss during tribo test. this mass loss is employed to calculate the specific wear rate. the following eq. (1) is employed to determine the specific wear rate (ws in kg/n.m) m ws s p =  (1) where, m is mass loss in kg, s is travel distance in m, and p is applied load in n. 2.4. mechanical properties nano-indentation tests are carried out using a nano-indentation test set up (nht-1, csm make) provided with a diamond made triangular base pyramid shape berkovich indenter. the maximum indentation depth is maintained as 500 nm [12]. the constant loading-unloading rate is 40 mn/min and idle time iss 2 seconds. the system considers oliver and pharr method to estimate hardness and elastic modulus [12]. the maximum indentation depth is kept below 1/10th of minimum coating thickness to avoid the substrate effect on test results [6]. the coated surfaces are polished with ultra-fine diamond paste to avoid roughness effect on indentation test results. the tests are conducted for 3 different samples obtained from same bath composition and at 10 locations along a line on each sample. the average of these data is considered here. scratch hardness is measured using a micro-scratch test set up (model: tr-101-ias, make: ducom, india) as per astm g171-03 standard. the test setup is provided with a diamond made rockwell c type indenter and the indenter tip radius is 200 µm. the tests are conducted against an applied load of 20n and the scratch length is kept as 5 mm. the scratch velocity is kept constant at 0.1 mm/sec. the scratch width is measured at 10 different locations and the average of those values is used to calculate the scratch hardness. the following eq. (2) is used to calculate the scratch hardness: 2 24.98p n hs x = gpa (2) where phs is scratch hardness, n is applied load in gm and x is scratch width in µm. 3. analysis of results 3.1. coating characterization the cross-cut thickness of enb samples is measured using the sem and the images are presented in fig. 1. the minimum coating thickness is measured to be 13μm at low nabh4 concentration [3]. the thickness values are measured to be 27.89μm and 30.69μm for medium and high nabh4 concentrations, respectively. this indicates an increasing trend of thickness with nabh4 concentration as deposition time is 4 hours for all the samples. higher concentration of nabh4 increases the reduction rate leading to increase in deposition rate leading to the increase in coating thickness. similar increasing trend has also been reported earlier [4]. the coating thickness of 9μm to 22μm is observed within the borohydride range of 0.50 to 1 g/l [4]. tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature... 215 fig. 1 sem image of cross-cut thickness of as-deposited enb coatings obtained at nabh4 concentration of, (a) 0.50 g/l, (b) 0.80 g/l and (c) 1.10 g/l boron contents in the coatings are found to be 3.5±0.40%, 6.60±0.50% and 8.70±0.40% for coatings obtained with low, medium and high nabh4 concentrations, respectively [3]. boron content is observed to increase with nabh4 concentration as reported earlier [7]. this agrees well with the current observation. the increase in nabh4 again enhances the reduction reaction leading to rise in boron content on coated layers. the coatings deposited with medium nabh4 concentration are found to possess 6.5%–7% boron which is in line with earlier observation [18]. surface morphology of the enb coatings has been displayed in fig. 2 to fig. 4. the sem images of as-deposited specimens displayed in figs. 2a, 3a and 4a show homogeneous distribution of nodules and uniform surface morphology throughout the working range. the coatings exhibit cauliflower or broccoli-like surface morphology in asdeposited conditions [1,2]. the nodules are observed to get separated and form broccolilike surface morphology at higher concentrations and an increase in nodule size is also displayed in fig. 4a. the broccoli-like surface morphology is usually formed with small granules [19]. the coatings displayed in figs. 2a, 2c, 3a and 3c are observed to maintain the cauliflower-like surface morphology even after heat treatment. the coating deposition starts at the metal surface and accumulates vertically. this vertical deposition leads to columnar growth which dominates over horizontal growth [14]. similar coloumnar growth may be seen in fig. 1. those columnar growths ultimately lead to cauliflower like surface morphology. the morphology displayed in figs. 2b, 2d, 3d, 4b, 4c and 4d shows the transformation of cauliflower to nodular structure due to the interaction effects or fusion 216 m. barman, t. k. barman, p. sahoo fig. 2 sem image of enb coated surface for 0.50 g/l nabh4 concentration and, (a) asdeposited, (b) ht350, (c) ht450, and (d) ht550 fig. 3 sem image of enb coated surface for 0.80 g/l nabh4 concentration and, (a) asdeposited, (b) ht350, (c) ht450, and (d) ht550 tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature... 217 of grains which resemble a bunch of grapes or blackberry [19]. the coating growth is observed to result a nodular surface morphology which bears a likeness to a bunch of grapes or blackberry due to the interaction effect with the rise in increasing nabh4 concentration and due to heat treatment [19]. fig. 3c shows the compact morphology with some prominent grain boundaries. the grains of the coatings get fused and become compact. but group-wise fusion makes some grain boundaries prominent. fig. 4 sem image of enb coated surface for 1.10 g/l nabh4 concentration and, (a) asdeposited, (b) ht350, (c) ht450, and (d) ht550 the x-ray diffraction (xrd) patterns of the enb coatings are displayed in fig. 5. it clearly shows that the coatings obtained with low nabh4 concentration exhibit a single sharp broad peak at about 53° accompanied by a hump in as-deposited condition [11] which is an indication of the co-existence of amorphous and nano-crystalline structure [1,10,11]. generally, coatings exhibit amorphous structure below 4.3% boron content [8,20]. a single hump only may be observed for coatings deposited with medium and high nabh4 concentrations in as-deposited condition [1,8,11]. the absence of a single broad crystallinity peak indicates the transformation of phase structure into amorphous structure due to a rise in boron content with nabh4 concentration [3,4]. 218 m. barman, t. k. barman, p. sahoo fig. 5 xrd patterns of enb coating in, (a) as-deposited, (b) deposited with 0.50 g/l nabh4 and heat-treated, (c) deposited with 0.80 g/l nabh4 and heat-treated, and (d) deposited with 1.10 g/l nabh4 and heat-treated crystallisation of enb coatings starts just below 300°c of heat-treatment temperature [6,7]. the ht350 enb coatings only exhibit crystalline peaks of ni and ni3b for low to high nabh4 concentrations [6]. precipitation of ni and ni2b phases can also be observed with the dissolution of ni3b phases at higher temperatures [6] which may be the reason for the presence of ni and ni3b crystalline phases only at 350ºc. the presence of crystalline phase ni2b may be observed with ni and ni3b at 450ºc [6,21]. similar results for coatings heat treated above 400ºc was reported in previous studies [4]. the presence of highintensity crystalline peaks in the xrd spectrums is the indication of phase transformation from amorphous to crystalline structure upon heat treatment [6,12]. study shows the coatings with 1-4.5% boron content do not crystalline completely [4,8]. this may be the reason for the gradual increase in sharp and high-intensity diffraction peaks with heat treatment temperature and nabh4 concentration [4]. but enb coatings do not contain ni2b phase at 550°c while the majority of the ni3b phases may be found besides ni phase. tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature... 219 3.2. mechanical properties the nano-indentation test results are displayed in table 1. enb coatings are observed to enhance the hardness of steel [3,4]. from table 1, it may be seen that the hardness and elastic modulus increase with nabh4 concentration in as-deposited condition [3]. the surface hardness of as-deposited coatings is seen to vary with boron content in coatings [11]. the boron content is observed to rise with nabh4 concentration [3,4]. hence, nanohardness also improves with nabh4 concentration, though it remains almost same till medium level. the hardness of the enb coatings till mid boron is seen to remain almost same [11]. in this study, the boron content is found to be around 6.6% till medium level which comes under mid boron content coatings. the current study agrees well with previous literature [11]. the nano-hardness is seen to improve after heat-treatment throughout the working range. the precipitation of various hard crystalline phases leads to this improvement in surface hardness. the presence of hard, compact crystalline phase ni2b leads to the highest hardness value of low nabh4 coatings at heat treatment temperature of 450°c [6,12] as the ni2b phase is known to be more compact phase than ni and ni3b [12]. the same decreases slightly upon further rise in temperature till 550°c due to absence of ni2b phase. the hardness of the medium and high nabh4 coatings improves with heat treatment temperature [12]. this improvement may be because of the addition of various crystalline phases which are reported earlier [4,8,22]. the coatings heat treated at 450°c contains ni2b phase [12] which is more compact leading to increase in surface hardness. it has been also reported earlier that the coatings with less than 4.5% boron content do not crystalline completely [2,8] which may be attributed to the lower hardness at low concentration and low heat treatment temperature. the hardness of the coatings obtained with medium and high nabh4 coatings increases with heat treatment temperature till 550°c possibly due to the hard crystalline phases as well as higher boron content. the elastic modulus is seen to increase with grain size [23]. this may be the cause of the rise in elastic modulus value of as-deposited enb coatings with nabh4 concentration. elastic modulus of enb coatings is found to be the functions of heat treatment temperature and duration [12]. the elastic modulus values presented in table 1 show an increasing trend with heat treatment temperature. it was also reported earlier that the grain size increased upon heat treatment temperature and duration [12]. the grain size of the coatings increases with heat treatment temperature which may be the possible reason for improvement in elastic modulus [12]. hence, the current study is in well accordance with the earlier reports [12]. the rise in elastic modulus is possibly because of addition of various crystalline phases which agrees well with previous studies as well [4,8,22]. the precipitation of various boride phases, as well as increased grain size, might have led to an improvement in the elastic modulus of the heat-treated coatings. the calculated scratch hardness value is displayed in table 1 which shows that it gets improved with nabh4 concentration in as-deposited condition. similar to nano-hardness, the rise in boron content in the coatings may be the reason for this rise in scratch hardness. the scratch hardness value reaches its maximum at 450ºc. the precipitation of the ni2b phase may be the reason for highest scratch hardness of the heat-treated coatings at 450ºc [12]. similar type of rise in nano-hardness may also be observed. the increase in nanohardness and scratch hardness from as-deposited to heat treated at 450ºc may be due to the transformation of various crystalline (ni, ni3b) phases [9]. the scratch hardness decreases upon further increase in heat treatment temperature at 550ºc possibly due to presence of 220 m. barman, t. k. barman, p. sahoo less compact phase ni3b phase instead of ni2b. moreover, the clustered surface morphology with several prominent grain boundaries may be another reason for the decrease in scratch hardness at 550ºc. the scratch hardness of the coatings in this current study is seen to be less relative to the as-deposited ni-b-w [14]. the incorporation of tungsten into ni-b matrix enhances the hardness due to solid solution strengthening of the coating matrix [14]. in this current study, the scratch hardness is found to improve with heat treatment temperature due to compact coating structure and precipitation of various crystalline phases but it is less than the as-deposited ni-b-w specimens [14]. the improvement in scratch hardness is correlated with the strain hardening of coatings [14]. nemane and chatterjee [14] have found that scratch resistance increases in multi-pass scratch test due to strain hardening. the current investigation is carried out with single pass scratch test leading to lower scratch hardness value of heat-treated enb coatings. table 1 tribo-mechanical properties parameters hardness (hv) vickers elastic modulus (e) in gpa scratch hardness (hsp) in gpa cof specific wear rate (ws) in x 10-8 kg/n.m ni-b coatings with 0.50 g/l nabh4 concentration as-deposited 388 82.70 0.80 0.46 23 ht 350°c 845 121 2.67 0.60 45.86 ht 450°c 1189 137 3.50 0.57 105.73 ht 550°c 1074 144 2.68 0.62 97.66 ni-b coatings with 0.80 g/l nabh4 concentration as-deposited 454.20 90.40 2.68 0.32 34 ht 350°c 992 131 3.71 0.77 33.97 ht 450°c 1156 178 4.43 0.72 19.10 ht 550°c 1208 195 3.45 0.78 11.08 ni-b coatings with 1.10 g/l nabh4 concentration as-deposited 867.90 121.30 3.50 0.16 80 ht 350°c 1030 144 4.90 0.65 16.14 ht 450°c 1244 195 6.35 0.70 17.83 ht 550°c 1388 218 4.00 0.67 39.92 3.3. tribological behaviour average cof values of enb coatings is also presented in table 1. cof value of the asdeposited coatings decreases with the rise in nabh4 concentrations. cauliflower like surface morphology is known to reduce friction between mating surfaces. this decrease in cof value with increased nabh4 concentration may be correlated with the cauliflower-like surface morphology and the increase in nodular size [3,4]. bigger nodules of cauliflower-like morphology increase the lubricating action and hence it reduces the cof value [3,4]. the coated surface becomes rough due to an increased reduction rate with a rise in nabh4 concentration [1,24]. but the cof value is found to decrease at high nabh4 concentration coatings [24]. the surface roughness asperity has a tendency to break into debris and fill the roughness valleys leading to smoothening of surfaces ultimately resulting in a reduction in average cof value [24]. the average cof value of all the heat-treated coatings is observed to increase relative to as-deposited conditions throughout the working range. the compact surface morphology leads to a slight reduction in cof value at 450°c. but the cluster aggregates forming a bunch tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature... 221 of grapes like surface morphology at 350ºc and 550ºc may have led to a high average cof value for coatings deposited with low nabh4 concentrations. cof value of the coatings obtained with medium nabh4 concentration gets increased upon heat treatment possibly because of rough surface due to the exposure of prominent grain boundaries. the cluster formation may also be another reason for increase in cof value. the cof value of heat treated enb coatings increases drastically at high level of nabh4 relative to as-deposited coatings. the surface roughness of enb coatings obtained with high nabh4 concentration is normally high which increases further after heat treatment leading to high cof value. the roughness may become also high because of the nodule formation and fused together ultimately transforming to blackberry like surface morphology and making the surface rough. this rough surface is expected to increase the cof value but crushing the asperity peaks and filling the roughness valleys made the surface smooth. the smooth surface reduces the cof value and remain almost same throughout the heat treatment temperature range. the specific wear rate values of the enb coatings are presented in table 1. it shows an increasing trend of wear rate for coatings obtained at low nabh4 concentration and heat treated till 450ºc. the same decreases slightly at 550ºc but remains higher than as-deposited one. the wear rate is found to decrease with the increase in surface hardness while it increases with rough surface [3]. coatings with low boron content leads to lower hardness but roughness increased upon heat treatment due to increased nodule size which may be the reason for this increase in specific wear rate. while the wear rate is observed to decrease with heat treatment temperature for medium and high nabh4 coatings. in contrary to the friction coefficient, wear rate is observed to rise with nabh4 concentration in the case of as-deposited enb coatings [11]. there is a chance of breaking the asperities of rough surfaces obtained from the high nabh4 concentration which led to an increment in wear rate [24]. the wear rate decreases for coatings obtained with medium and high nabh4 concentration with rise in temperature up to 550°c and that is possibly due to higher boron content as well as recrystallization of grains upon heat-treatment [8]. boron content is observed to increase with nabh4 concentration [3,4,11] which may improve the surface hardness of the enb coatings [3,4] and this may also be a possible reason for a decrease in specific wear rate. sem images of the worn-out zones of enb coatings have been displayed in figs. 6, 7 and 8. the figs. 6a, and 8a shows the presence of wear debris on wear track which confirms the adhesive type wear mechanism of the as-deposited coatings [17,24]. wear grooves may also be observed along the sliding direction which was caused by the repetitive nature of loading during the sliding [17,24]. fig. 6b and 7a show the delamination of the upper layer of the coatings with ploughing at certain areas. while heat treated coatings possess rough surface. those roughness asperities must have been crushed and ground severely. the wear debris is also found to be deposited along the sliding direction or attached to the sliding surface which acts as load-bearing areas leading to improvement in wear resistance with heat treatment temperature [8,11] and reduction in cof value sometimes [23]. overall, adhesive type of wear mechanism is observed to dominate for the borohydride reduced enb coatings over delamination of layer and abrasive type wear mechanism. a significant amount of mass loss occurs at the beginning of sliding due to grinding of roughness peaks which leads to increase in higher wear rate. but debris also found to attach with the sliding surface which separates the counter surface during tribological tests [11]. this also may lead to reduction in specific wear rate of the heat-treated coatings [8]. there are some spalling phenomenon or pitting may also be observed along the sliding direction because of the ground debris and adhesive type wear. therefore, the borohydride reduced enb coatings in as-deposited and heat-treated condition predominantly exhibit an adhesive type wear mechanism. 222 m. barman, t. k. barman, p. sahoo fig. 6 sem image of worn out enb coated surface for nabh4 concentration of 0.50 g/l in (a) as-deposited, (b) ht350, (c) ht450 and (d) ht550 fig. 7 sem image of worn out enb coated surface for nabh4 concentration of 0.80 g/l in (a) as-deposited, (b) ht350, (c) ht450 and (d) ht550 tribo-mechanical characterization of enb alloy coatings: effect of heat-treatment temperature... 223 fig. 8 sem image of worn out enb coated surface for nabh4 concentration of 1.10 g/l in (a) as-deposited, (b) ht350, (c) ht450 and (d) ht550 the electroless nickel-based coating deposition method is widely used for surface modification. the electroless deposition processes suffer some issues to dispose the toxic elements used in this process. some electroless coating bath uses pbno3 as stabilizer which is banned in many countries. accordingly, concept of cleaner production using chemical vapor deposition, physical vapor deposition etc. are being practiced. still, the electroless process cannot be overlooked because of their excellent tribological and mechanical behaviour. hence, the researchers are working on to develop lead free bath [25,26]. recently, several other stabilizers based on bismuth, tin are tried as replacement of lead based ones [27, 28]. thus incorporation of the concept of cleaner production in electroless coating processes by making the coating bath eco-friendly is attracting substantial attention of researchers over the globe [29]. 4. conclusions the outcome of the current investigation is summarized as given below: ▪ the amorphous or mixture of amorphous nano-crystalline structure of as-deposited enb coatings are transformed into crystalline phase structure throughout the working range. ▪ the nano-hardness value is also observed to improve with nabh4 concentration because of increased boron content. the same is also improved further with heat treatment temperature due to precipitation of hard crystalline phases like ni, ni2b, ni3b and compact surface morphology. 224 m. barman, t. k. barman, p. sahoo ▪ the elastic modulus is also increased with nabh4 concentration in as-deposited condition possibly due to the rise in nodular size. the nodules get enlarged further and form clustered morphology leading to further increase in elastic modulus upon heat treatment. 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by-nc-nd original scientific paper interactions and influences on coal miners' safety attention: an evaluation using improved dematel-ism de-yu xu1,2, hong-qing cui2,3, tian-yu wang1, jiang-wei fu1, guang-bo zhu4 1school of energy & environment engineering, zhongyuan university of technology, china 2school of safety science and engineering, henan polytechnic university, china 3central plains economic region coalbed (shale) gas innovation center, china 4faw-volkswagen co., ltd. qingdao branch, china orcid ids: de-yu xu https://orcid.org/0009-0005-0826-6692 hong-qing cui https://orcid.org/0000-0002-4282-2614 tian-yu wang https://orcid.org/0009-0009-4508-2859 jiang-wei fu https://orcid.org/0009-0003-4389-8121 guang-bo zhu https://orcid.org/0009-0007-4480-056x abstract. in coal mining, the myriad of factors influencing miners' attention to safety necessitates deeper exploration. particularly, discerning the significance and interplay of these factors offers crucial insights into the actual disparities in miners' safety attentiveness. yet, a limited number of comprehensive studies address this dimension. thus, an advanced decision making trial evaluation laboratory-interpretive structural model (dematel-ism) has been employed to probe the determinants impacting coal miners' safety focus and the mechanisms underpinning these interactions. the objective is to provide strategies that could diminish the occurrence of minor accidents. results revealed that there are 9 causative factors and 6 resultant factors shaping the coal miners' attention to safety. within the structural model of these factors, three layers and seven levels were identified. notably, the intricacy of relationships among these factors was found to be profound. emphasis is recommended on the management of these intricate deep-level causative factors boasting high driving power, and mid-level resultant factors characterized by both substantial driving force and dependence. key words: minor injury accidents, coal miners, safety attention, influencing determinants received: july 26, 2023 / accepted october 30, 2023 corresponding author: de-yu xu school of energy & environment engineering, zhongyuan university of technology, zhengzhou 451191, china e-mail: 6675@zut.edu.cn https://orcid.org/0009-0005-0826-6692 https://orcid.org/0000-0002-4282-2614 https://orcid.org/0009-0009-4508-2859 https://orcid.org/0009-0003-4389-8121 https://orcid.org/0009-0007-4480-056x 140 d.y. xu, h.q. cui, t.y. wang, j.w. fu, g.b. zhu 1. introduction in recent years, a steady improvement in coal mine safety has been observed in china, attributed largely to consistent investments and enhancements in safety technologies and equipment within coal mine enterprises. this enhancement has resulted in a notable decline in major accidents. yet, minor injury accidents, accounting for >90% of all accidents [1], persist as a prevalent concern. such accidents, apart from posing a significant threat to miner safety, also jeopardize the economic well-being and production of coal mines. mainly attributed to miners’ unsafe behaviors, minor injury accidents have been found to predominantly stem from a decline in coal miners' attention to safety [2-4]. emphasis on the factors that influence coal miners' attention to safety is underscored by its potential in mitigating these unsafe behaviors and consequently reducing minor injury accidents. several scholars have diligently studied these influences. it has been frequently postulated that a decrease in coal miners' attention to safety contributes to human error, and a strong correlation exists between safety attention and unsafe behaviors [5,6]. recent studies have highlighted relationships between individual characteristic factors and coal miners' safety attention [7-9]. mathematical methods have also been employed to elucidate the determinants of miners’ safety attention [10,11]. the prevailing research predominantly focuses on the influence mechanism of various factors on coal miners' safety attention [12-14]. it is well-acknowledged that diverse factors impact miners' safety attention [15-17]. a comprehensive understanding of the significance and interplay of these factors provides insights into actual disparities in safety attention among miners. nevertheless, a scarcity of in-depth studies on this crucial aspect limits the fulfillment of scientific requirements for coal mine safety management. the resulting challenge for practical management is the vast array of influencing factors and the evolving nature of safety information. these complexities make comprehensive control countermeasures elusive, potentially overlooking pivotal factors requiring attention in safety management. the combined usage of decision making trial evaluation laboratory (dematel) and interpretive structural modeling (ism) offers a framework to discern the importance and causal attributes of system factors, thereby shedding light on the intricate interactions between them [18-20]. moreover, the matrix impacts cross-reference multiplication applied to a classification (micmac) delves deeper, analyzing the driving force and dependencies among system factors [21, 22]. given their unique characteristics, an integration of the micmac method with dematel-ism has been proposed. such a methodological convergence can effectively rank each influencing factor, elucidate the multi-layered interpretation structure of the system, and through rigorous analysis of driving forces and dependencies, unveil the interplay mechanism among these factors. employing this improved approach can streamline the overwhelming workload in minor injury accident safety management and mitigate the potential oversight of key determinants. ultimately, the aim is to enhance safety management efficiency, providing mining companies with a solid foundation to address strategic challenges in minor injury accidents by bolstering miners’ safety attention. interactions and influences on coal miners' safety attention: an evaluation using improved... 141 2. improved dematel-ism method 2.1. value assignment of influencing factors the assignment of values to influencing factors predominantly relies on expert evaluations, fundamentally characterized by a systematic aggregation of expert opinions. subsequent processing of their responses culminates in the consolidation of results. a questionnaire-based approach was adopted for this expert evaluation. twenty experts, affiliated with three distinct research institutions in henan province and specializing in the study of coal miners’ safety behavior, were solicited for their insights. drawing from an array of studies and accident case analyses [7-11], these questionnaires distilled and highlighted 15 pivotal factors impacting miners' safety attention. these factors were compartmentalized into three distinct categories, as illustrated in table 1: individual factors, environmental factors, and management factors. leveraging the index system for these factors, experts were prompted to designate values to the interaction intensity among these 15 determinants, rooted in their professional acumen and prior experiences. to counteract potential individual biases in the evaluations, mean values were computed for each factor. table 1 the index system of influencing factors pertinent to coal miners' safety attention classification influencing factors individual factors safety cognition s1, safety consciousness s2, behavioral habit s3, safety self-control s4, risk perception s5, educational level s6, working years s7, work pressure s8, job burnout s9 environmental factors working environment s10, working strength s11, safety information stimulus s12 management factors safety supervision s13, safety education and training s14, safety atmosphere of the organization s15 the results from the value assignments were manifested as a matrix, denoted as the direct influence matrix a. this matrix, represented as a=(aij)n×n, is delineated in eq. (1). within this matrix, each constituent element aij signifies the potency of influence exerted by factor si on factor sj. the guiding metric for experts to designate these values is further elucidated in eq. (1). consequently, upon assimilating the experts' value designations, the direct influence matrix a was derived, as detailed in eq. (2). 0 has no influence on 1 has a weak influence on 2 has a moderate influence on 3 has a strong influence on i j i j ij i j i j s s s s a s s s s    =     (1) where, when i = j, aij = 0. 142 d.y. xu, h.q. cui, t.y. wang, j.w. fu, g.b. zhu 0 3 3 3 3 0 0 0 1 0 0 0 0 0 0 0 0 3 3 3 0 0 0 2 0 0 0 0 0 0 0 2 0 2 2 0 0 0 0 0 0 0 0 0 0 1 2 2 0 2 0 0 0 2 0 0 0 0 0 0 0 2 2 2 0 0 0 0 2 0 0 0 0 0 0 3 3 3 3 3 0 0 0 2 0 0 0 0 0 0 3 3 3 3 3 0 0 0 2 0 0 0 0 0 0 3 3 3 3 3 0 0 0 3 0 0 0 0 0 0 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 0 0 3 3 0 0 0 0 0 0 2 2 2 2 2 0 0 3 3 0 0 0 0 0 0 3 3 3 3 3 0 0 2 3 0 0 0 0 0 0 3 3 3 3 3 0 0 2 3 0 0 3 0 0 0 3 3 3 a = 3 3 0 0 1 1 0 0 0 0 0 0 3 3 3 3 2 0 0 2 2 0 0 3 3 3 0                                 (2) 2.2. enhanced dematel-ism analysis technique post the expert-driven value assignment of influential determinants, a pressing need emerges to determine both the centrality and cause degrees of each constituent factor using the refined dematel-ism analysis procedure. in initial phases, the method seeks to elucidate the significance and causative characteristics of each factor. successive stages demand the partitioning of system levels founded upon the centric and causative degrees of each variable. consequently, a multi-layered, recursive interpretative structure model is crafted to decipher intricate interactions between said factors. ultimately, the analytical procedure is harnessed to discern the drive and dependency values of each factor. an in-depth analysis of these values unravels the intricate interplay mechanisms among them. the ensuing subsections provide an exhaustive breakdown of the refined dematel-ism technique. 2.2.1. determining centrality and cause degrees the pivotal initiation step in the refined dematel-ism analysis method hinges on the calculations of the centrality and cause degrees. these degrees offer preliminary insights into the relative significance of each influencing factor. the methodical steps are delineated below: (1) system components are identified, denoted as s1-sn, wherein s1-sn pertain to the factors enumerated in table 1. (2) the direct influence matrix a is constructed. guided by expert insight and acumen, the degree of influence exerted by factor si on factor sj is assessed, resulting in the formation of the direct influence matrix a=(aij)n×n, as given in eq. (2). (3) the normalized direct influence matrix c, represented as c=(cij)n×n, is computed as follows: 1 max( ) ij ij n ij j a c a = =  (3) where, aij is an element of matrix a. interactions and influences on coal miners' safety attention: an evaluation using improved... 143 (4) the comprehensive influence matrix t is deduced, with the matrix presented as t=(tij)n×n: 1( )t c i c −= − (4) herein, c stands for the normalized direct influence matrix, and i symbolizes the unit matrix. the unit matrix i is characterized by diagonal elements equating to 1, with the remaining being 0. (5) for each factor, both the influential degree fi and affected degree ei are calculated. fi, representing the aggregate value of row i in t, mirrors the encompassing value of the influence factor si on its counterparts. conversely, ei, reflecting the summation of the value of column i in t, signifies the aggregate influence borne by factor si due to other factors. eqs. (5) and (6) represent fi and ei respectively: 1 , ( 1,2, , ) n i ij j f t i n = = = (5) 1 , ( 1,2, , ) n i ji j e t i n = = = (6) where, tij is an element of the matrix t. (6) subsequently, for each factor, the centrality degree mi and the cause degree ni are computed. mi, the cumulative value of both influential and affected degrees for each factor, pinpoints the prominence of factor si within the system. ni, the differential between the influential and affected degrees, sheds light on the causality dynamics among factors. eqs. (7) and (8) give mi and ni, respectively: i i im f e= + (7) i i in f e= − (8) where, a positive ni denotes si as a causative factor, while a negative ni classifies si as a resultant factor. 2.2.2. formulation of the multi-layer recursive interpretation structure model in the subsequent phase of the enhanced dematel-ism analysis method, the focus shifts to the conceptualization of a multi-layer recursive interpretation structure model. the processes entailed in this establishment are elucidated below. (1) the global influence matrix h is computed. this matrix, represented as h=(hij)n×n, is derived by incorporating the comprehensive influence matrix t with the unit matrix i. (2) the threshold value λ, essential for discerning the reachable matrix k, is derived from the comprehensive influence matrix t. following this determination, elements within the global influence matrix h undergo processing, leading to the acquisition of the reachable matrix k=(kij)n×n. the component kij is given in eq. (9): 1 0 ij ij ij h k h   =   ＜ (9) 144 d.y. xu, h.q. cui, t.y. wang, j.w. fu, g.b. zhu in this context, the threshold value λ is expressed as the sum of α and β, where α and β respectively denote the mean and standard deviations of the elements encapsulated in t. (3) stratification of the reachable matrix k is performed. from the matrix, both the reachable set r(si) and the antecedent set a(si) for factor si are extracted. here, r(si) represents {sj|sj∈s, kij=1} and a(si) denotes {si|si∈s, kji=1}. subsequently, the stratification aligns with eq. (10). the initial tier of factors is discerned, followed by the elimination of its corresponding row and column from k, aiding in the identification of the secondary tier of factors. this procedure is replicated iteratively until all system levels are demarcated. ( ) ( ) ( ),( 1,2, , )i i ir s a s r s i n= = (10) (4) depiction of the interconnected framework of influencing factors across diverse tiers is executed. this is premised on the prior stratification of the system, culminating in the illustration of the multi-layer recursive interpretation structure model for the system. 2.2.3. determination of driving force and dependence degrees in the progression of the advanced dematel-ism method, an essential step involves quantifying the driving force and dependence degrees of each factor. the subsequent delineation outlines the processes involved. the driving force degree, represented as pi, and the dependence degree, denoted as qj, for every factor are calculated. the quadrant classification diagram, a critical visual representation, is then constructed based on these values. derived from the reachable matrix k, the expressions for pi and qj are given by eqs. (11) and (12), respectively: 1 ( 1,2, , ) n i ij j p k i n = = = (11) 1 ( 1,2, , ) n j ij i q k j n = = = (12) within this context, kij is recognized as an element of matrix k. the promotional degree of factor si influencing other factors is signified by si, whereas the degree to which factor sj is influenced by other factors is represented by qj. in the quadrant diagram, the positions of factor si on the abscissa and ordinate are represented by pi and qj, respectively. the quadrant diagram, an essential analytical tool, is partitioned into four distinct quadrants. each quadrant epitomizes a unique cluster: the autonomous cluster (quadrant i), the independent cluster (quadrant ii), the connection cluster (quadrant iii), and the dependent cluster (quadrant iv). 3. analysis of results derived from the enhanced dematel-ism technique 3.1. centrality and cause degrees upon obtaining the direct influence matrix a, normalization was performed in accordance with eq. (3). subsequently, the comprehensive influence matrix t was deduced as per eq. (4). with reference to the comprehensive influence matrix t, various degrees – namely the influential degree fi, the affected degree ei, the centrality degree mi, and the cause degree ni interactions and influences on coal miners' safety attention: an evaluation using improved... 145 – associated with each influencing factor were computed, as detailed in eqs. (7) and (8). table 2 provides the calculated results concerning the centrality and cause degrees. further elucidation on the causal relationships among the factors shaping miners’ safety attention can be discerned from fig. 1. notably, these results reveal a distinction between 9 causative factors and 6 resultant factors. moreover, an elevated centrality degree is observed among the resultant factors. table 2 calculation results of centrality degree and cause degree factors fi ei mi ni sort by mi type s1 0.571 1.197 1.768 -0.627 6 resultant factor s2 0.484 1.743 2.227 -1.259 1 resultant factor s3 0.265 1.812 2.077 -1.546 4 resultant factor s4 0.402 1.812 2.214 -1.409 2 resultant factor s5 0.358 1.774 2.132 -1.417 3 resultant factor s6 0.758 0.000 0.758 0.758 12 causative factor s7 0.758 0.000 0.758 0.758 13 causative factor s8 0.806 0.482 1.288 0.323 8 causative factor s9 0.661 1.333 1.994 -0.672 5 resultant factor s10 0.740 0.000 0.740 0.740 14 causative factor s11 0.740 0.000 0.740 0.740 15 causative factor s12 0.909 0.222 1.131 0.687 10 causative factor s13 1.073 0.111 1.184 0.962 9 causative factor s14 0.761 0.111 0.872 0.650 11 causative factor s15 1.312 0.000 1.312 1.312 7 causative factor fig. 1 causal relationship of influencing factors 146 d.y. xu, h.q. cui, t.y. wang, j.w. fu, g.b. zhu 3.2. recursive structural interpretation of influencing factors the global influence matrix h was derived by integrating the unit matrix i with the comprehensive influence matrix t. with the introduction of the threshold value λ, elements within h were adjusted following eq. (9). consequently, the reachable matrix k, characterized in eq. (13), was formulated. the threshold λ was discerned from the summation of the average value α and the standard deviation β stemming from the elements in matrix t, yielding a computed threshold value of 0.107. 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 1 1 1 k = 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 1 1                                 (13) with the aid of the reachable matrix k, both the reachable set r(si) and the antecedent set a(si) pertaining to factor si were identified. factors of the primary level were extracted based on eq. (10). by eliminating corresponding rows and columns of level 1 factors within k, the level 2 factors were discerned. this methodological approach facilitated the classification of influencing factors into seven distinct levels, as cataloged in table 3. table 3 results of division of levels of the influencing factors level division element set level specification l1 s3, s4, s5 surface-level direct factor l2 s2 middle-level indirect factor l3 s1 middle-level indirect factor l4 s6, s7, s9, s14 middle-level indirect factor l5 s8, s12 deep-level fundamental factor l6 s10, s11, s13 deep-level fundamental factor l7 s15 deep-level root factor fig. 2 visually represents the multi-layer recursive interpretation structure delineating the factors influencing miners' attention to safety. an in-depth examination of the constructed model indicates a three-tiered structure comprising seven levels. furthermore, the presence of 12 intermediate and profound factors is evident, accounting for a significant 80% of the total influential factors. the intricate internal relationships and dynamic mechanisms linking these deep-level factors underscore the complexity inherent to this domain. interactions and influences on coal miners' safety attention: an evaluation using improved... 147 fig. 2 multi-layer recursive interpretation structure of influencing factors 3.3. driving force and dependence degrees of influencing factors drawing from the reachability matrix k, calculations for the driving force and dependence degrees of each influencing factor were meticulously executed, as delineated in eqs. (11) and (12). table 4 presents the computed values for both driving force and dependence degrees. the quadrant distribution map, showcased in fig. 3, plots the influencing factors based on their calculated degrees. table 4 driving force values and dependence values of influencing factors influencing factor driving force dependence s1 5 9 s2 4 10 s3 1 11 s4 1 11 s5 1 11 s6 6 1 s7 6 1 s8 7 3 s9 6 7 s10 3 1 s11 3 1 s12 7 3 s13 8 2 s14 6 2 s15 10 1 148 d.y. xu, h.q. cui, t.y. wang, j.w. fu, g.b. zhu fig. 3 classification of driving force and dependence of influencing factors the spatial distribution of influencing factors within the quadrant map conveys compelling insights. a striking 73% of the total influencing factors, amounting to 11 distinct factors, are situated within quadrants ii and iv. notably, these factors exhibit pronounced values in either driving force or dependence degrees. a significant observation concerns the positioning of safety cognition s1 and job burnout s9 within quadrant iii, suggesting a pronounced concurrent driving force and dependence for these two factors. further research could delve deeper into the underlying reasons for such classifications, offering a more nuanced understanding of the roles and interactions of these influencing factors within the mining safety landscape. this could pave the way for more targeted interventions and strategic policies to enhance miners' safety awareness and practices. 4. discussion 4.1. assessment of influencing factors’ importance degrees in the system depicted in table 2, the centrality degree serves as an indicator of the relative importance of the influencing factors. from the data, a specific hierarchical sequence regarding the factors' influence on miners' safety attention is discerned: s2 >s4 >s5 >s3 >s9 >s1 >s15 >s8 >s13 >s12 >s14 >s6=s7 >s10=s11. this hierarchy can be compartmentalized into three distinct levels. notably, factors such as safety consciousness s2, safety self-control s4, risk perception s5, behavioral habits s3, job burnout s9, and safety cognition s1 are classified under the first tier. as per fig. 1, the centrality degrees for these factors surpass 1.5, signifying their heightened importance within the system. these factors are thus inferred to exert a substantial impact on miners' attention to safety. the second echelon encompasses the safety atmosphere of the organization s15, work pressure s8, safety supervision s13, and safety information stimulation s12. these factors exhibit centrality degrees ranging between 1 and 1.5, denoting a medium level of systemic importance. interactions and influences on coal miners' safety attention: an evaluation using improved... 149 conversely, the education level s6, working years s7, working environment s10, and work intensity s11 are relegated to the third tier, characterized by their comparatively diminished importance. their centrality degrees, all falling below 1, suggest a relatively minor influence on coal miners' safety attentiveness. 4.2. examination of causative factors within the system presented in table 2, nine causative factors are identified. their hierarchical significance unfolds as follows: s15>s8>s13>s12>s14>s6>s7>s10>s11. it is posited that these causative elements play an instrumental role in shaping resultant factors and miners' emphasis on safety. however, an intricate scrutiny reveals that factors such as the education level s6, working years s7, and safety education and training s14 are situated within the mid-tier of the system as depicted in fig. 2. in table 2, a lower degree of significance is attributed to these elements. furthermore, as discerned from fig. 3, these factors are categorized within quadrant ii and are perceived as independent clusters with minimal dependency. a tepid response to the stimuli of profound root factors is observed. collectively, it is deduced that these components maintain a marginal and somewhat isolated impact on the overarching system. notably, while the working environment s10 and working intensity s11 are earmarked as profound factors in fig. 2, they are positioned within quadrant i and are associated with the autonomous cluster as per fig. 3. the driving forces underpinning these factors are delineated as being subpar, and their affiliations with mid-tier factors are not deemed robust. addressing these particular elements with timely, targeted interventions may proffer enhanced outcomes in bolstering coal miners' attention to safety. pivotal within the causative factors, the safety atmosphere of the organization s15 is discerned to reside at the l7 echelon. concurrently, working pressure s8, safety information stimulation s12, and safety supervision s13 are situated either at the l5 or l6 strata, as visualized in fig. 2. recognized as the system's profound layers, these factors—due to their pronounced importance are placed within quadrant ii in both figs. 1 and 3. being constituents of an independent cluster with a formidable driving force, their influence on both mid-tier and superficial elements is profound. thus, these are perceived as the linchpins modulating coal miners' dedication to safety. 4.3. exploration of resultant factors in table 1, six resultant factors are delineated, possessing a sequence of significance as follows: s2>s4>s5>s3>s9>s1. resultant from the cumulative impacts of other determinants, these factors are discerned to directly influence the coal miners' dedication to safety. an elevated level of importance is attributed to them within the system, as visualized in fig. 1, suggesting a critical role in shaping miners' safety attentiveness. upon examination, it was observed that factors such as behavioral habit s3, safety selfcontrol s4, and risk perception s5 reside within the l1 echelon in fig. 2, typifying the system's initial surface level. positioned in quadrant iv, these factors are identified as belonging to the dependent cluster in fig. 3. characterized by heightened dependency and attenuated driving forces, their manifestation is predominantly influenced by other variables, rendering them contingent on the modulation of external factors. conversely, safety cognition s1, safety consciousness s2, and job burnout s9 are mapped to the l2, l3, and l4 tiers, respectively, in fig. 2. these mid-tier factors within the ism serve as 150 d.y. xu, h.q. cui, t.y. wang, j.w. fu, g.b. zhu conduits, bridging antecedent and subsequent determinants. the driving forces intrinsic to these elements amplify the sway of superficial factors, while their dependencies bolster the imprint of profound determinants. however, an intriguing observation arises from the positioning of safety consciousness s2 within quadrant iv, aligning it with the dependent cluster. despite its inherent strong dependency, a subdued driving force has been noted, attenuating its potential reinforcement of surface elements. of particular note, safety cognition s1 and job burnout s9 are aligned with quadrant iii, categorizing them within the connection cluster as illustrated in fig. 3. their simultaneous possession of robust driving forces and dependencies suggests that perturbations within these factors could readily ripple through the system, influencing upper and lower echelons. such dynamism underscores the salience of these factors in determining coal miners' commitment to safety. the analysis infers that mid-level resultant factors, endowed with dominant driving forces and dependencies, significantly modulate coal miners’ focus on safety. 4.4. analysis of practical implications from influencing factors hierarchy is often employed as a critical technique in analyzing influencing factors and forms an integral foundation in safety management. within this hierarchy, surface-level factors have been determined to directly influence safety attentiveness. the direct and targeted intervention of these factors has been acknowledged to yield optimal results in the contemporary domain. deep-level factors, however, are perceived to be more pivotal, exerting their influence through mid and surface-level determinants. a vast array of deeplevel factors has been observed to influence other levels, rendering the nature of their influence intricate. for these deep-level determinants, comprehensive emphasis and preventive measures are generally proposed in current strategies. mid-level factors, traditionally, are not attributed significant influence over deep-level determinants but are discerned to primarily impact surface-level factors. the prevailing strategy recommends obstructing the connections between mid and surface levels. nonetheless, challenges are evident in these hierarchical solutions to influencing factors. an overwhelming number of factors demand attention, resulting in significant workload. given the dynamic nature of safety information, implementing these controls becomes challenging, thereby potentially overlooking certain mid-level factors over time. strikingly, analyses from improved methodologies suggest that specific mid-level factors play a pivotal role in modulating coal miners' safety attention, raising concerns over safety management, especially in minor injury incidents. it has been inferred that the nuanced interrelations among various factors cannot be aptly captured solely via hierarchical structures. instead, the implementation of refined methodologies is advocated to establish a hierarchical structure. within this, both driving forces and dependencies of influencing factors should be meticulously analyzed, unearthing the intricate interplay among them. such a comprehensive approach promises to yield insights of paramount practical relevance. upon scrutiny of the enhanced dematel-ism method and the intricate interplay among all influencing factors, two focal points emerge for mining entities. firstly, the deep-level causative factors manifesting pronounced driving forces demand attention. secondly, mid-level resultant factors characterized by robust driving forces and dependencies necessitate scrutiny. salient among these are safety atmosphere of the organization s15, work pressure s8, safety information stimulation s12, safety supervision s13, safety cognition s1, and job burnout s9. addressing these six pivotal determinants in isolation poses challenges, given their intertwined interactions and influences on coal miners' safety attention: an evaluation using improved... 151 nature and hierarchical interactions. yet, devising unified, systematic countermeasures tailored to their characteristics appears promising for mining entities, potentially alleviating concerns over minor injury incidents by enhancing miners' safety vigilance. the aforementioned six influential factors and their main associated personnel can be systematically approached in two dimensions. the initial dimension targets the triad of safety information stimulation s12, safety supervision s13, and safety atmosphere of the organization s15. integration of safety information stimulation s12 and safety supervision s13 into the overarching framework of organizational safety atmosphere s15 is recommended, elevating the significance of the former two within the latter. this amalgamation predominantly concerns the safety managers within mining firms, underscoring the pivotal role of establishing a scientifically rigorous safety organizational atmosphere to elevate the safety management caliber. the secondary dimension focuses on safety cognition s1, work pressure s8, and job burnout s9. primarily pertinent to coal miners, these factors, however, necessitate vigilant oversight from safety managers. proactive measures, both in terms of rewards and penalties, are suggested to foster miners' intrinsic motivation to hone their safety cognition and maintain optimal working conditions, thereby enhancing safety awareness. concurrently, vigilant monitoring is essential to promptly identify concerns related to miners' work pressure and burnout. decisive interventions should be swiftly deployed upon the identification of any detrimental behaviors jeopardizing miners' safety attention. the systematic management of these dimensions, centered on the highlighted six determinants, is postulated to hold profound practical and strategic implications in mitigating minor injury incidents within coal mining entities. 5. conclusions in the presented research, factors influencing coal miners' attention to safety were meticulously examined through an enhanced dematel-ism method. from this comprehensive analysis, critical determinants closely interlinked were extracted from the broader factor group. such extraction is postulated to diminish the substantial workload inherent in managing minor injury accidents and to counteract the potential oversight of pivotal factors. the findings elucidated that the coal miners' safety attention emanates from a composite interplay of factors, including their degree of importance, system level grade, driving force, and dependency characteristics. notably, two salient dimensions emerged as particularly significant for mining entities. the first dimension underscored that deep-level causative factors, characterized by an intense driving force, profoundly influence both middle and surface-level determinants, thus serving as pivotal elements modulating miners' safety focus. the second dimension accentuated the importance of mid-level resultant factors. endowed with a robust driving force and dependency, these factors are discerned to potentially sway deep and surfacelevel determinants, thereby critically affecting coal miners' safety attention. it is surmised that devising systematic response measures addressing the influencing factors from both dimensions holds potential for mining companies, offering strategic solutions to challenges posed by minor injury incidents and fostering enhanced safety attentiveness among miners. anticipated avenues 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kilibarda https://orcid.org/0000-0003-3794-7380 abstract. procurement logistics is one of the most important segments of the supply chain and one of the key factors of a company’s competitiveness. for that reason, many companies strive for constant optimization of this segment of the supply chain, both in terms of costs and in terms of time, reliability, etc. the aim of this paper is to develop a new approach based on dea-fucom-cocoso methods that aim to select efficient suppliers. the developed model was tested on the data of one trading company. the dea method was used in order to select only efficient ones from 29 observed in this paper. the fucom method was used to determine the weights of the 9 observed criteria used in the cocoso method for evaluation of 6 efficient suppliers. the results of the application of this method determined the final rank of suppliers, after which only the first 3 suppliers were considered. at the very end, a model for solving the problem of order allocation is defined in order to determine from which supplier it is necessary to order goods and in what quantity. by applying the defined model, the quantities that need to be ordered from certain suppliers in order to meet the demand on the market are obtained. based on the results, the developed approach showed the possibility of large application not only on the observed example but also on a larger problem. key words: procurement logistics, supplier selection, order allocation problem, fucom, cocoso 1. introduction procurement logistics is a segment of the supply chain that deals with the procurement of raw materials (products), inventory management, demand forecasting, selection of suppliers, tenders, etc. procurement starts a whole series of other activities that are realized in order to deliver a certain product to the end-user. also, procurement conditions all further activities and processes regardless of the type and size of the company. having in received: february 10, 2022 / accepted july 18, 2022 corresponding author: vukašin pajić university of belgrade, faculty of transport and traffic engineering v.pajic@sf.bg.ac.rs https://orcid.org/0000-0003-4918-4392 https://orcid.org/0000-0002-3711-2140 https://orcid.org/0000-0003-3794-7380 110 v. pajić, m. andrejić, m. kilibarda mind all these tasks, it can be concluded that efficient procurement logistics can achieve significant savings in the business. in order to achieve these savings, it is necessary to manage procurement processes, which is especially important if it is known that procurement costs can be up to 70% of the costs of the final product [1]. in addition, the share of procurement in the total turnover can range from 50-90% [2]. inventory management is just one of the places to achieve savings, where it is necessary to find the optimum between the quantity of goods in stock and the cost of purchasing new products. in addition to inventory, another place where savings can be made is when selecting suppliers. when selecting a supplier, it is necessary to take care to select a supplier who can follow the demand in the market, but also who has the necessary flexibility for sudden changes in demand. on that occasion, it is necessary to apply certain tools in order to facilitate the process of selecting a supplier. this problem has been recognized a long time ago in the literature where there are a large number of multi-criteria decision-making (mcdm) methods that can be applied to facilitate this problem. after selecting an adequate supplier, procurement logistics faces the next problem, which is the problem of order allocation, where it is now necessary to determine the quantities that need to be ordered from the supplier at the lowest possible cost. this problem is particularly complex given the many limitations that may arise on that occasion. like the previous one, this problem has also been recognized in the literature where there are a large number of papers proposing numerous models to facilitate solving this problem [3-7]. the aim of this paper is to define the procedure of procurement optimization by selecting efficient suppliers, as well as to define a model for solving the order allocation problem. to the best of the authors’ knowledge, there are no papers that simultaneously solve both problems in the way described in this paper (using mcdm methods and model for solving the problem of order allocation). this is exactly the gap that this paper deals with, which will enable managers to get a complete solution on the one hand, while on the other hand, it lays the foundation for an integrated solution for two groups of problems for future research in the literature. in order to achieve this, a hybrid data envelopment analysisfull consistency method-combined compromise solution (dea-fucom-cocoso) approach was applied in this paper. the dea method is a mathematical method using linear programming techniques to convert inputs to outputs with the purpose of evaluating the performance of comparable products. the aim is to determine relative efficiency which represents the ratio of the total weighted output to the total weighted input. the fucom method reduces the subjectivity of decision-makers, which leads to consistency in the weight values of the criteria. for this reason, this method was applied in this paper to determine the weights of the criteria. deviation from full consistency (dfc) was also obtained by applying this method. after determining the weights of the criteria, the cocoso method was applied in order to obtain the final rank of the suppliers. the cocoso method uses three aggregation strategies to generate measures of the overall utility of the alternatives. this method is characterized by an innovative structure based on the integration of compromise decisionmaking algorithms. in addition, the cocoso method is more reliable and stable than the available methods [8]. for this reason, this method has been applied in this paper in order to obtain the final rank of the suppliers. in addition, another reason for the application of this method is the fact that there are only a few papers in the field of logistics that applied this method. procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 111 the paper is organized as follows. after an introductory discussion, a description of the problem discussed in this paper was made as well as a review of the literature. the third chapter describes the formulations of the methods used in this paper, i.e., dea method, fucom, cocoso, and order allocation model. the next chapter presents the results and discussion of applying the described methodology on the example of a trading company. at the very end, concluding remarks are given as well as directions for future research. 2. problem description and literature review as previously mentioned, procurement logistics is one of the factors of a company’s competitiveness. this is due to the fact that procurement logistics is directly responsible for procuring the necessary raw materials or finished products in order to meet market demand. the inability to meet market demand in addition to a bad reputation has an impact on the creation of additional costs. for this reason, it is very important to effectively manage these processes. however, this is quite complicated given that a number of problems arise on this occasion. namely, it is first necessary to determine the need for procurement, then define the quantities as well as the type of product that needs to be procured. in practice, this is often done on the basis of forecasts that are determined on the basis of data from the previous period. it is then necessary to identify potential suppliers from whom products can be procured. once established, it is necessary to define the criteria in order to evaluate and rank them. these criteria are mainly financial and logistical. once the criteria are determined, the suppliers are evaluated according to them, after which the selection is made. after that, a tender is usually announced and negotiations are conducted. based on this procedure, the complexity of this problem faced by procurement logistics can be seen. however, even after selecting a supplier, it is necessary to constantly monitor and measure the results in order to achieve efficient business. monitoring of results is usually realized by applying certain key performance indicators (kpis) that are defined [9]. in order to increase business efficiency, as well as reduce costs, it is necessary to optimize procurement. this optimization is reflected either through an increase in the value of the defined kpis or through a decrease in the number of suppliers, i.e., cessation of work with inefficient suppliers. this problem has been recognized both in practice and in the literature where there is a large number of papers that relate only to the problem of supplier selection using various mcdm methods [10, 11]. mcdm methods are often used in solving various problems in logistics. these problems are most often related to the location of facilities (logistics centers, warehouses, etc.), the selection of suppliers, the selection of 3pl providers, etc. thus, ulutas et al. [12] in their paper applied the fuzzy swara and cocoso method to solve the problem of selecting the location of the logistics center. finally, the obtained results were compared with other mcdm methods. a similar problem was addressed by yazdani et al. [13] who developed a two-stage decision-making model consisting of the application of the dea method in the first phase, and the application of rough full consistency (r-fucom) and r-cocoso method. the dea method was applied to identify efficient and inefficient alternatives, while in the second phase the methods were applied to perform a ranking of efficient alternatives. principal component analysis (pca) was combined with the dea method in the paper [14] for measuring global logistics efficiency. in the paper [15] the dea method was used to measure transport efficiency and to identify the main factors that 112 v. pajić, m. andrejić, m. kilibarda affect transport efficiency. andrejić et al. [16] applied the dea method in their paper as well as the malmquist productivity index for measuring efficiency change in time for distribution centers. ayadi et al. [17] applied the fuzzy fucom method for determining the weights of criteria and sub-criteria, and then applied fuzzy multi-attribute ideal-real comparative analysis (f-mairca) and fuzzy preference ranking organization method for enrichment evaluation (f-promethee) for ranking the locations of the logistics platform. the problem of selecting a 3pl provider was addressed by wen et al. [18] who applied the cocoso method in a hesitant fuzzy linguistic environment to solve the multi-expert problem of selecting a 3pl provider. ecer and pamucar [5] applied the fuzzy best-worst method (fbwm) and fuzzy cocoso with the bonferroni method to select a sustainable supplier. mishra et al. [19] in their paper proposed a hesitant fuzzy cocoso framework based on discrimination measure for ranking sustainable 3pl reverse logistics provider. a review of the literature showed that in addition to mcdm methods, mathematical programming models based on genetic algorithm (ga), particle swarm optimization (pso), etc. are used in the selection and optimization. pan [4] proposed an optimization model of vendor selection based on fuzzy ga. the results of the application of the proposed model showed that it is possible to reduce the total procurement costs. the problem of supplier selection and order allocation was solved in the paper [3] in a closed-loop supply chain using monte carlo simulation and goal programming. rosyidi et al. [20] proposed a model for concurrent supplier selection model to minimize the purchasing cost and fuzzy quality loss considering process capability and assembled product specification. gheidar-kheljani et al. [21] solved the problem of supply chain optimization policy for supplier selection by applying a mathematical programming approach. in their model, the authors assumed that after ordering, the supplier divides the order into smaller lot sizes and delivers them over a period of time. masi et al. [22] developed a meta-model for choosing a supplier selection technique within an engineering, procurement, and construction (epc) company. choudhary and shankar [23] proposed an integer linear programming model to determine ordering times, lot-sizes, suppliers, and carriers to be selected at minimal costs. firouzi and jadidi [24] developed a multi-objective model for supplier selection and order allocation problem with fuzzy parameters in their paper. as the literature review found that there are not enough papers dealing with the described issues in the way described in this paper, the authors decided to apply dea-fucom-cocoso methods for supplier selection as well as the pure integer conic program (picone) model to solve the problem of order allocation. based on the above, it can be concluded that there is a significant number of papers dealing with the issue of supplier selection and the problem of order allocation, by applying various models. however, a review of the literature did not establish that there are papers dealing with the application of the mcdm methods for supplier selection and the model for solving order allocation problem. in order to select only efficient ones from the total number of suppliers, which will be later considered and evaluated using the fucom and cocoso methods, the dea method was applied in this paper. when implementing this method, it is necessary to define inputs and outputs. the procured quantities and purchase value were defined as inputs, while revenue, sales value, number of stores where the goods of that supplier are sold, write-off, costs of excessive stocks, and service level are defined as outputs. the procured quantity represents the total quantity of goods ordered from a particular expressed in tons (t). the purchase value represents the value (price) paid by the retailer to the supplier, expressed in the monetary unit (m.u). revenue represents the amount obtained when the value-added tax (vat) is deducted from the sales price, expressed in the m.u. sales value procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 113 is obtained by adding the retailer’s margin and the amount of vat to the purchase price also expressed in the m.u. the number of stores represents the total number of stores in which the goods of that supplier are sold (since the retailer has stores across the country). write-off represents all breaks, damages, expiration of the goods, etc., which is expressed in the m.u. excessive inventory costs occur due to poor inventory planning, which is expressed also in the m.u. the service level represents the accuracy of the supplier’s delivery (in terms of the completeness of the order and the timeliness of delivery), which is expressed in percentage (%). some of these indicators were also observed as criteria for evaluating suppliers in the fucom method. however, in addition to them, other logistical criteria have been defined. the reason for that lies in the fact that the observed company, when selecting a supplier, first contracts and defines financial indicators, and only after that the logistics ones. this is one of the main problems in practice, and in future models it is necessary to observe the financial and logistical criteria at the same time. for this reason, the fucom method considered the following criteria used to evaluate suppliers: ▪ quality ▪ price ▪ revenue ▪ excess inventory costs ▪ service level ▪ reliability ▪ flexibility ▪ write-off ▪ quality certification. in order to quantify the quality of the product, sales volume (t) in the past were observed (for a period of 3 months) and suppliers were evaluated on that basis. the price of the product is obtained by subtracting the purchase value from the sales value, expressed in the m.u. reliability was also obtained on the basis of historical data (how often the supplier complied with the agreed deadline, required quantity, and service level), expressed in %. flexibility represents the time needed and the ability to react to sudden changes in demand. in this paper, flexibility was determined based on the data from the retailer and represents the number of additional deliveries from the supplier needed in order to satisfy the demand. finally, the last criterion was the number of quality certifications that the supplier has, given that the quality of the considered company is an important parameter when selecting a supplier. 3. methodology in order to select the efficient suppliers a combination of dea-fucom-cocoso methods was used in this paper while to solve the order allocation problem, a model for order quantity allocation was used. the methodological steps of the application of these methods are presented below (fig. 1). 114 v. pajić, m. andrejić, m. kilibarda fig. 1 methodology 3.1. dea method for selecting efficient suppliers in order to single out only the efficient ones from the total number of suppliers, which will then be analyzed, the dea ccr output-oriented model was applied, which is presented below [25]: min ∑ 𝑣𝑟𝑖𝑥𝑖𝑗0𝑟 (1) s.t. ∑ 𝑢𝑟𝑦𝑟𝑗 − ∑ 𝑣𝑖𝑥𝑖𝑗𝑟 ≤ 0, ∀𝑗𝑟 (2) procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 115 ∑ 𝑢𝑖𝑦𝑖𝑗0 = 1𝑖 (3) 𝑢𝑟 , 𝑣𝑖 ≥ 0, ∀r, ∀i (4) where j represents the number of dmus (j=1, 2, …, n), m represents the number of inputs (xij = 1, 2, …, m) and s represents the number of outputs (yrj = 1, 2, …, s). yrj represents the amount of the rth output from dmuj; ur represents the weight given to the rth output; xij represents the amount of the ith input used by dmuj; vi represents the weight given to the ith input. 3.2. fucom method for determining criteria weights the procedure for determining the weight coefficients of criteria is comprised of three steps presented below [26, 27]: step 1 – all evaluation criteria are ranked according to the significance (from the most significant to the least significant). cj(1) > cj(2) > … > cj(k) (5) where k represents the rank of the observed criterion. step 2 – ranked criteria are compared in order to determine the comparative priority φk/(k+1), where k represents the rank of the criteria. ф = (φ1/2, φ2/3, …, φk/(k+1)) (6) where φk/(k+1) represents the significance that the criterion of the cj(k) rank has compared to the criterion of the cj(k+1) rank. step 3 – the final values of the weight coefficients of the evaluation criteria are determined. in order to determine these values two conditions must be met: 𝑤𝑘 𝑤𝑘+1 = φ𝑘/(𝑘+1) (7) 𝑤𝑘 𝑤𝑘+2 = φ𝑘/(𝑘+1) ⊗ φ(𝑘+1)/(𝑘+2) (8) after these two conditions are met, the final model for determining the final values of the weight coefficients of the evaluation criteria can be defined as: min χ (9) s.t. | 𝑤𝑗(𝑘) 𝑤𝑗(𝑘+1) − φ 𝑘 𝑘+1 | ≤ χ, ∀j (10) | 𝑤𝑗(𝑘) 𝑤𝑗(𝑘+2) − φ𝑘/(𝑘+1) ⊗ φ(𝑘+1)/(𝑘+2)| ≤ χ, ∀j (11) ∑ 𝑤𝑗 𝑛 𝑗=1 = 1, ∀j (12) 𝑤𝑗 ≥ 0, ∀j (13) after solving this model the final values of the evaluation criteria weights are determined (w1, w2, …, wn)t as well as the degree of dfc (χ). 116 v. pajić, m. andrejić, m. kilibarda 3.3. cocoso method for supplier ranking the procedure for the determination of final ranking by cocoso method includes the following steps [28]: step 1 – determine the initial decision-making matrix. step 2 – the normalization of criteria values is accomplished based on compromise normalization equation: 𝑟𝑖𝑗 = 𝑥𝑖𝑗−min 𝑖 𝑥𝑖𝑗 max 𝑖 𝑥𝑖𝑗−min 𝑖 𝑥𝑖𝑗 ; 𝑓𝑜𝑟 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛 (14) 𝑟𝑖𝑗 = max 𝑖 𝑥𝑖𝑗−𝑥𝑖𝑗 max 𝑖 𝑥𝑖𝑗−min 𝑖 𝑥𝑖𝑗 ; 𝑓𝑜𝑟 𝑐𝑜𝑠𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛 (15) step 3 – the total of the weighted comparability sequence and the whole of the power weight of comparability sequences for each alternative sum of the weighted comparability sequence and also an amount of the power weight of comparability sequences for each alternative as si and pi respectively are calculated: 𝑆𝑖 = ∑ (𝑤𝑗𝑟𝑖𝑗)𝑛 𝑗=1 (16) 𝑃𝑖 = ∑ (𝑟𝑖𝑗)𝑤𝑗𝑛 𝑗=1 (17) step 4 – relative weights of the alternatives using the following aggregation strategies are computed: 𝑘𝑖𝑎 = 𝑃𝑖+𝑆𝑖 ∑ (𝑃𝑖+𝑆𝑖)𝑚 𝑖=1 (18) 𝑘𝑖𝑏 = 𝑆𝑖 min 𝑖 𝑆𝑖 + 𝑃𝑖 min 𝑖 𝑃𝑖 (19) 𝑘𝑖𝑐 = 𝜆(𝑆𝑖)+(1−𝜆)(𝑃𝑖) (𝜆 max 𝑖 𝑆𝑖+(1−𝜆) max 𝑖 𝑃𝑖) ; 0 ≤ 𝜆 ≤ 1 (20) in eq. (20), λ is chosen by decision-makers and is usually λ = 0.5. step 5 – the final ranking of the alternatives is determined based on ki values (the higher the value the better): 𝑘𝑖 = (𝑘𝑖𝑎𝑘𝑖𝑏𝑘𝑖𝑐) 1 3 + 1 3 (𝑘𝑖𝑎 + 𝑘𝑖𝑏 + 𝑘𝑖𝑐) (21) 3.4. order allocation model in order to solve the problem of order allocation, the picone model was applied in this paper. the model presented is based on anna and fhiliantie [29] with the model being further complicated by the introduction of new constraints and the inclusion of transport costs in the objective function. the model notation which was used is shown below in table 1. procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 117 table 1 model notation indexes i (i = 1,2,…,m) index of period (month) j (j = 1,2,…, n) index of supplier sets m number of periods n number of suppliers parameters pij the price of raw material in the period i from supplier j (m.u./kg) tcij transportation cost in period i for supplier j (m.u./kg) ocij the ordering cost of goods during the period i from supplier j (m.u./order) hci the holding cost of goods in period i (m.u./kg) di demand of goods in period i (kg) scij supplier’s capacity in the period i from supplier j (kg) ssi safety stock in period i (kg) dtij delivery time in period i from supplier j (days) dtmax the maximum delivery time (days) dqij discount quantity (quantity for which discount is granted) in period i from supplier j (kg) daij discount amount granted if dq is achieved (%) dccap distribution center capacity (kg) decision variables xij the amount of goods ordered in the period i from supplier j (kg) yij binary variable (takes value 1 if order allocation will be carried out to a supplier and 0 otherwise) ii the quantity of goods stored in period i (kg) the model formulation (objective function and constraints) used in this paper is shown below. 𝑀𝑖𝑛 𝐶 = ∑ ∑ 𝑝𝑖𝑗𝑋𝑖𝑗 𝑛 𝑗=1 𝑚 𝑖=1 + ∑ ∑ 𝑂𝐶𝑖𝑗𝑌𝑖𝑗 𝑛 𝑗=1 𝑚 𝑖=1 + ∑ ∑ 𝑇𝐶𝑖𝑗𝑋𝑖𝑗 𝑛 𝑗=1 𝑚 𝑖=1 + ∑ 𝐻𝐶𝑖𝐼𝑖 𝑚 𝑖=1 (22) s.t. 𝑋𝑖𝑗 ≤ 𝑆𝐶𝑖𝑗 ∀𝑖, 𝑗 (23) 𝑋𝑖𝑗 ≤ 𝑀𝑆𝐶𝑖𝑗𝑌𝑖𝑗 ∀𝑖, 𝑗 (24) ∑ 𝑋𝑖𝑗 𝑛 𝑗=1 + 𝐼𝑖 ≥ 𝐷𝑖 ∀𝑖 (25) 𝐼𝑖 = 𝐼𝑖−1 + ∑ 𝑋𝑖𝑗 𝑛 𝑗=1 − 𝐷𝑖 ∀𝑖 (26) 𝐼1 = 𝐼0 + ∑ 𝑋1𝑗 𝑛 𝑗=1 − 𝐷1 (27) 𝐼𝑖 ≥ 𝑆𝑆𝑖 , ∀𝑖 (28) 𝑆𝑆𝑖 = 0,15 ∗ 𝐷𝑖 ∀𝑖 (29) 118 v. pajić, m. andrejić, m. kilibarda 𝐷𝑇𝑖𝑗𝑌𝑖𝑗 ≤ 𝐷𝑇𝑚𝑎𝑥 ∀𝑖, 𝑗 (30) 𝑋𝑖𝑗𝑌𝑖𝑗 ≥ 𝐷𝑄𝑖𝑗 ∀𝑖, 𝑗 (31) 𝑂𝐶𝑖𝑗 = 𝐷𝐴𝑖𝑗 ∗ 𝑂𝐶𝑖−1𝑗 ∀𝑖, 𝑗 (32) 𝑋𝑖𝑗 + 𝐼𝑖 + 𝑆𝑆𝑖 ≤ 𝐷𝐶𝑐𝑎𝑝 ∀𝑖, 𝑗 (33) 𝑌𝑖𝑗 ∈ (0,1), ∀𝑖, 𝑗 (34) 𝑋𝑖𝑗 , 𝐼𝑖 ≥ 0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, ∀𝑖, 𝑗 (35) 𝑂𝑖𝑗 ≥ 0, ∀𝑖, 𝑗 (36) in the developed model, the objective function has the task to minimize the total costs, i.e., costs of procurement, ordering, inventory, and transportation. constraint (23) refers to the capacity of a supplier which must be greater than the quantity ordered from that supplier. constraint (24) refers to order quantity allocation where that quantity must be less than the total capacity of the supplier for all periods. in order to meet the demand, the constraint (25) is set. the next two constraints (26) and (27) relate to inventory balance. since the lack of products can cause the inability to meet demand on the one hand, and on the other hand too much stock can generate excessive costs, it is necessary to find a balance in the quantity of goods in stock. when managing inventory, it is necessary to define a safety stock so that the company does not run out of products. constraints (28) and (29) refer to this issue. delivery time (which is the time that elapses from the moment of ordering to the moment of delivery) is also important for the ability to respond quickly to current demand. for this reason, a constraint (30) has been set. suppliers often grant discounts on certain quantities when ordering (thus stimulating customers to order from them). for this reason, the constraints (31) and (32) are included in the model developed in this paper. namely, if the ordered quantity is higher than the quantity for which the discount is granted, then the discount will be granted when ordering in the following period (i+1). constraint (33) refers to the capacity of the retailer’s distribution center which must not be exceeded. finally, constraints (34), (35), and (36) define the variables used in this paper. 4. results and discussion as already mentioned, the dea method was first used in this paper in order to single out only the efficient ones from the observed 29 suppliers. on that occasion, procured quantities and purchase value are defined as inputs, while revenue, sales value, number of stores where the goods of that supplier are sold, write-off, costs of excessive stocks and service level are defined as outputs. based on the data obtained from the analyzed company, the previously described output-oriented ccr model was applied. after solving the set model, it was concluded that only 6 suppliers are efficient out of the total number, which is shown in table 2. procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 119 table 2 results of the dea method dmu name objective value dmu name objective value supplier 1 0.830178741 supplier 16 1 supplier 2 0.993094408 supplier 17 0.912238298 supplier 3 0.894018835 supplier 18 0.762394285 supplier 4 0.653831891 supplier 19 0.935567367 supplier 5 0.952766145 supplier 20 0.839325219 supplier 6 0.850352713 supplier 21 0.727809542 supplier 7 1 supplier 22 0.653232181 supplier 8 1 supplier 23 0.852216565 supplier 9 1 supplier 24 0.731280219 supplier 10 0.716469484 supplier 25 0.85044203 supplier 11 0.938969404 supplier 26 0.697173369 supplier 12 1 supplier 27 1 supplier 13 0.973693036 supplier 28 0.716652258 supplier 14 0.856429161 supplier 29 0.985050685 supplier 15 0.812191691 since the application of the dea method separated efficient from inefficient suppliers, in the following parts of the paper only efficient suppliers were observed and analyzed. following the application of the dea method, the fucom method was applied to determine the weights of the criteria that were later used in the cocoso method for the final ranking of suppliers. criteria used in this paper for evaluation are product quality (c1), price (c2), revenue (c3), excess inventory costs (c4), service level (c5), reliability (c6), flexibility (c7), write-off (c8) and quality certification (c9). the first step in applying the fucom method is to rank the criteria from the most significant to the least significant, which is shown below. c1 > c2 > c3 > c4 > c5 > c6 > c7 > c8 > c9 after ranking the criteria by importance, the second step was conducted in order to determine the significance of the criteria (ωcj(k)). the comparison is made with respect to the first-ranked (c1) criterion. the presented values were obtained based on the assessment of 5 experts in the field of procurement of the analyzed company. the results of the comparison are shown in table 3. table 3 priorities of criteria criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 ωcj(k) 1 1.5 2.1 3.7 4.2 5 5 7 9 based on the value of ωcj(k) from table 3, it is possible to define a model in order to determine the weights of the criteria. the model used in this paper is presented below. min χ s.t. 120 v. pajić, m. andrejić, m. kilibarda | 𝑤1 𝑤2 − 1.5| ≤ 𝜒, | 𝑤2 𝑤3 − 1.4| ≤ 𝜒, | 𝑤3 𝑤4 − 1.76| ≤ 𝜒, | 𝑤4 𝑤5 − 1.13| ≤ 𝜒, | 𝑤5 𝑤6 − 1.19| ≤ 𝜒, | 𝑤6 𝑤7 − 1| ≤ 𝜒, | 𝑤7 𝑤8 − 1.4| ≤ 𝜒, | 𝑤8 𝑤9 − 1.28| ≤ 𝜒, | 𝑤1 𝑤3 − 2.1| ≤ 𝜒, | 𝑤2 𝑤4 − 2.46| ≤ 𝜒, | 𝑤3 𝑤5 − 1.99| ≤ 𝜒, | 𝑤4 𝑤6 − 1.34| ≤ 𝜒, | 𝑤5 𝑤7 − 1.19| ≤ 𝜒, | 𝑤6 𝑤8 − 1.4| ≤ 𝜒, | 𝑤7 𝑤9 − 1.79| ≤ 𝜒, ∑ 𝑤𝑗 9 𝑗=1 = 1, 𝑤𝑗 ≥ 0, ∀𝑗 lingo software was used to solve the presented model. by solving the model, the values of the criteria weights were obtained (0.3015, 0.2010, 0.1438, 0.0817, 0.0724, 0.0610, 0.0610, 0.0436, 0.0341)t as well as the dfc of the results χ = 0.00. after determining the weights of the criteria, the cocoso method was applied in order to obtain the final rank of the suppliers. the first step in applying this method involves defining an initial decision matrix, which is shown in table 4. table 4 initial decision-making matrix criterion c1 c2 c3 c4 c5 c6 c7 c8 c9 optimal value m ax m in m ax m in m ax m ax m ax m in m ax s1 6.207 5.119.942 12.441.184 13.278 99 98.8 651 379.872 1 s2 4.141 2.656.669 9.070.183 121.953 96 98.3 1521 468.742 3 s3 9 1.333 4.454 0 67 99.6 25 176 4 s4 9.561 4.437.933 7.802.354 45.416 45 97.9 671 631.287 2 s5 21.116 4.576.211 11.061.054 118.626 62 99.2 1518 326.144 1 s6 148 29.337 116.760 37.059 100 98.5 32 61.294 3 after determining the initial matrix, the second step was performed, i.e., normalization according to the type of criteria. based on table 4 it can be concluded that in this paper, 6 criteria that are max type and 3 criteria that are min type were observed. normalization was performed using eqs. (14) and (15). the results of normalization are shown in table 5. table 5 normalized matrix criterion c1 c2 c3 c4 c5 c6 c7 c8 c9 weight 0.3015 0.2010 0.1438 0.0817 0.0724 0.0610 0.0610 0.0436 0.0341 s1 0.29 0.00 1.00 0.89 0.98 0.53 0.42 0.40 0.00 s2 0.20 0.48 0.73 0.00 0.93 0.24 1.00 0.26 0.67 s3 0.00 1.00 0.00 1.00 0.40 1.00 0.00 1.00 1.00 s4 0.45 0.13 0.63 0.63 0.00 0.00 0.43 0.00 0.33 s5 1.00 0.11 0.89 0.03 0.31 0.76 1.00 0.48 0.00 s6 0.01 0.99 0.01 0.70 1.00 0.35 0.00 0.90 0.67 procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 121 the next step involves determining the values of si and pi for each alternative (supplier) using eqs. (16) and (17). the results of this step are shown in tables 6 and 7. table 6 weighted comparability sequence and si criterion c1 c2 c3 c4 c5 c6 c7 c8 c9 si s1 0.09 0.00 0.14 0.07 0.07 0.03 0.03 0.02 0.00 0.45 s2 0.06 0.10 0.10 0.00 0.07 0.01 0.06 0.01 0.02 0.44 s3 0.00 0.20 0.00 0.08 0.03 0.06 0.00 0.04 0.03 0.45 s4 0.14 0.03 0.09 0.05 0.00 0.00 0.03 0.00 0.01 0.34 s5 0.30 0.02 0.13 0.00 0.02 0.05 0.06 0.02 0.00 0.60 s6 0.00 0.20 0.00 0.06 0.07 0.02 0.00 0.04 0.02 0.42 table 7 exponentially weighted comparability sequence and pi criterion c1 c2 c3 c4 c5 c6 c7 c8 c9 pi s1 0.69 0.00 1.00 0.99 1.00 0.96 0.95 0.96 0.00 6.55 s2 0.61 0.86 0.96 0.00 0.99 0.92 1.00 0.94 0.99 7.27 s3 0.00 1.00 0.00 1.00 0.94 1.00 0.00 1.00 1.00 5.94 s4 0.79 0.67 0.94 0.96 0.00 0.00 0.95 0.00 0.96 5.27 s5 1.00 0.64 0.98 0.75 0.92 0.98 1.00 0.97 0.00 7.24 s6 0.22 1.00 0.51 0.97 1.00 0.94 0.72 1.00 0.99 7.34 finally, the last step in the application of the cocoso method involves determining three aggregate strategies as well as the value of ki in order to determine the final rank of the suppliers. these coefficients were determined using eqs. (18)-(21). the obtained results are shown in table 8. table 8 final aggregation and cocoso ranking of the alternatives supplier kia kib kic ki rank s1 0.166 2.563 0.882 1.924 4 s2 0.182 2.658 0.970 2.048 2 s3 0.151 2.443 0.804 1.799 5 s4 0.133 2.000 0.706 1.518 6 s5 0.185 3.137 0.987 2.268 1 s6 0.183 2.610 0.976 2.033 3 based on the results of table 8, it can be concluded that suppliers s5, s2, and s6 stood out as the first three most favorable solutions. these three suppliers are then further analyzed in the second part of the paper, which deals with solving the problem of allocating quantities during ordering. to solve this problem, the previously described model in chapter 3 was applied. the input data of the model used in this paper are shown in table 9. the model was used to allocate orders for 3 suppliers in the next 3 months. based on the data of the analyzed company, it was concluded that in the previous month the stock level was 5.000 kg (i0). also, based on the forecasts made in the company, the demand for the observed goods in the next period (for the next 3 months) is defined d1 = 15.000 kg, d2 = 17.500 kg, and d3 = 19.000 kg. in order to calculate the cost of ordering (ocij) it is necessary to define the cost of ordering in the previous period, which is defined 122 v. pajić, m. andrejić, m. kilibarda on the basis of data from the company and whose amount can be seen in table 9 (m0). the maximum delivery time (dtmax) is the time prescribed by the contracts (which are signed annually) and is 4 days for all suppliers. the last data required for the model is the capacity of the distribution center which in this case is 50.000 kg (dccap). it should be noted here that the displayed capacity refers only to the analyzed goods, and not to the capacity of the entire dc, which is significantly higher. based on these data, a picone model was formed, which was solved using lingo software. the output results of the model are shown in table 10. table 9 input data of the model parameter period (month) supplier 1 supplier 2 supplier 3 pij m1 430 450 440 m2 430 450 440 m3 430 450 440 tcij m1 0.58 0.56 0.60 m2 0.58 0.56 0.60 m3 0.58 0.56 0.60 ocij m0 1450 1425 1500 hcij m1 25 25 25 m2 25 25 25 m3 25 25 25 scij m1 9000 11000 8500 m2 9000 11000 8500 m3 9000 11000 8500 dtij m1 2 3 2 m2 2 3 2 m3 2 3 2 dqij m1 7000 8500 6000 m2 7200 8500 6100 m3 7400 8800 6500 daij m1 10 9 10 m2 12 12 11 m3 13 15 14 table 10 output results of the model objective function value variable value c 30.304.800 o13 1.350 variable value o21 1.148,4 x11 7.000 o22 1.141,14 x12 8.500 o23 1.201,5 x13 6.000 o31 999,108 x21 7.200 o32 969,969 x22 8.500 o33 1033,29 x23 6.100 i1 11.500 x31 7.400 i2 15.800 x32 8.800 i3 19.500 x33 6.500 ss1 2.250 o11 1.305 ss2 2.625 o12 1.296,75 ss3 2.850 procurement optimization by selecting efficient suppliers using dea-fucom-cocoso approach... 123 based on the results from table 10, it can be concluded that the total costs for the observed period are 30.304.800 m.u. in addition, it can be seen that all suppliers are engaged in all months. in the first month, it is necessary to order 7.000 kg of goods from supplier 1, 8.500 kg of goods from the second and 6.000 kg from the third. since the quantities shown are equal to the quantities necessary to grant a discount, based on the values of the variables o11, o12 and o13, it can be seen that the discount has been calculated and that the ordering costs amount to 1.305 m.u. for supplier 1, 1.296,75 m.u. for supplier 2 and 1.350 m.u. for supplier 3. the values of the quantity of goods in stock (11.500 kg) and safety stocks (2.250 kg) can also be seen for the first month. for the second month, according to the results of table 10, it is necessary to order 7.200 kg of goods from supplier 1, 8.500 kg from supplier 2 and 6.100 kg of goods from supplier 3. the costs of ordering are 1.148,4 m.u.; 1.141,14 m.u.; 1.201,5 m.u. respectively. at the end of the second month, it can be seen that there are 15.800 kg of goods left in stock, and 2.625 kg of safety stocks. in the third month, it is necessary to order 7.400 kg of goods from supplier 1, 8.800 kg of goods from supplier 2 and 6.500 kg of goods from supplier 3, where the ordering costs are 999,108 m.u.; 969,969 m.u. and 1033,29 m.u. respectively. based on these data, it can be seen that the ordering costs are the lowest for this month. at the end of the third month 19.500 kg of goods are left in stock and 2.850 kg of safety stocks. the results of the implementation showed that the proposed approach can provide significant decision support. also, the analyzed data in this paper refers to only 1 of the over 30 categories that exist within the analyzed company. this approach contributes because it observes all aspects of the real problem that has been solved. in addition, the proposed approach is fully applicable in practice and suitable for real-time problem solving (decision-support tool). the selection of financial and logistical criteria in the selection of suppliers as well as the costs of procurement, ordering, inventory and transport with certain constraints fully covered all real cases. one of the more significant constraint implemented in the model is the one related to the discount that is granted in case of ordering a certain quantity of products. the advantage of the proposed model is reflected in the applicability for other categories of goods, but also other types and sizes of companies, product types, etc. with minimal changes. the developed model represents a new integrated (hybrid approach) that solves two problems at once and which can be a goods basis for future research. 5. conclusion the aim of this paper was to present the complexities and problems faced by procurement logistics. the selection of suppliers, as well as the order allocation problem stand out as one of the most significant problems of procurement logistics that are recognized both in practice and in the literature. the problem of supplier selection is a frequent topic of papers in the literature where there are numerous mcdm methods used in solving this problem. on the other hand, the order allocation problem is often solved by applying numerous models and algorithms. however, a review of the literature found that there are no papers that combine these two approaches to solve the problems of supplier selection and order allocation. for this reason, a combination of dea-fucom-cocoso methods for the selection of efficient suppliers was applied in this paper, after which a model for solving the order allocation problem was defined. the dea method was applied 124 v. pajić, m. andrejić, m. kilibarda in this paper in order to single out efficient from inefficient suppliers. in the example of a trading company, 6 efficient suppliers were identified, out of 29, which were then further analyzed in the paper. according to the data from the company, and also on the basis of a review of the literature, 9 criteria for evaluating suppliers have been defined. as the weights of these criteria are not the same, the fucom method for determining weights was applied. after solving the fucom model, the weights of the criteria were obtained which were then applied in the cocoso method to obtain the final ranking of suppliers, since the goal was to determine the 3 best suppliers (according to the observed criteria) to then determine the quantities to be ordered from them, for the next 3 months. this problem is solved in the second part of the paper where a model is defined that aims to minimize the costs of procurement, ordering, inventory, and transport with certain constraints. after solving the developed model, the results showed that it is necessary to engage all 3 suppliers each month in order to meet the demand in the market. the limitation of this research is reflected in the application of a described methodology on a relatively small example, i.e., one product segment. the results of the analyzed example showed that the proposed approach can provide significant decision support, not only in the observed example but also beyond. namely, the developed approach can be applied to other problems, and not only to those related to procurement, with certain changes in the model. the application of the developed approach to the problems that occur in other segments of the supply chain stands out as one of the future directions of research. in addition, the application of algorithms, such as pso and ga, to optimize the amount of procurement also stands out as one of the directions of future research. acknowledgement: this paper was supported by the ministry of education, science and technological development of the republic of serbia, through the project tr 36006. references 1. bottani, e., centobelli, p., murino, t., shekarian, e., 2018, a qfd-anp method for supplier selection with benefits, opportunities, costs and risks considerations, international journal of information 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programming, international journal of asro, 9(1), pp. 98-105. 10643 facta universitatis series: mechanical engineering vol. 22, no 2, 2024, pp. 165 186 https://doi.org/10.22190/fume220523045n © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper maximization of wear rates through effective configuration of standoff distance and hydraulic parameters in ultrasonic pulsating waterjet akash nag1, amit rai dixit2, jana petrů1, petra váňová3, kateřina konečná3, sergej hloch1,4 1technical university of ostrava, faculty of mechanical engineering, czech republic 2indian institute of technology (ism) – dhanbad, india 3technical university of ostrava, faculty of materials science and technology, czech republic 4institute of geonics of the czech academy of sciences, ostrava-poruba, czech republic orcid ids: akash nag https://orcid.org/0000-0003-0400-7739 amit rai dixit https://orcid.org/0000-0001-6135-8098 jana petrů https://orcid.org/0000-0002-2378-5678 petra váňová https://orcid.org/0000-0002-0607-1278 kateřina konečná https://orcid.org/0000-0002-9633-0164 sergej hloch https://orcid.org/0000-0003-4066-0620 abstract. a pulsating waterjet is a technological modification of a conventional waterjet that utilizes ultrasonic vibrations to generate a modulated jet, resulting in repetitive fatigue loading of the material. the erosion efficiency of the ultrasonic pulsating waterjet is majorly determined by the hydraulic factors and its interaction with standoff distance. however, the dependency of the wear rates on different hydraulic factors and formulation of an implicit prediction model for determining effective standoff distance is still not present to date. therefore, in this study, the combined dependency of the supply pressure (20-40 mpa), nozzle diameter (0.3-1.0 mm), and standoff distance (1-121 mm) on wear rates of aw-6060 aluminum alloy are studied. statistical analysis is used to determine the statistically significant factors and formulate regression equations to determine output responses within the experimental domain. the surface topography and sub-surface microhardness of the eroded grooves were studied. the results show that both the disintegration depth and the material removal increase with an increase in the nozzle diameter and supply pressure. however, the dependency of the output responses on nozzle diameter is statistically more evident than supply pressure and two-way interactions. cross-sectional images of the grooves showed typical hydrodynamic erosion characteristics in erosion cavities, subsurface voids, and material upheaving. the results of microhardness analysis showed an approximately 15-20% increase in hardness values compared to the untreated samples. key words: pulsating waterjet, disintegration depth, volume removal, anova, microhardness received: may 23, 2022 / accepted november 30, 2022 corresponding author: amit rai dixit indian institute of technology (ism) – dhanbad, 826001, india e-mail: amitraidixit@iitism.ac.in https://orcid.org/0000-0003-0400-7739 https://orcid.org/0000-0001-6135-8098 https://orcid.org/0000-0002-2378-5678 https://orcid.org/0000-0002-0607-1278 https://orcid.org/0000-0002-9633-0164 https://orcid.org/0000-0003-4066-0620 166 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch 1. introduction waterjet technology and its technological modifications have historically been put into practice depending on the current level of knowledge and technical possibilities [1]. the aim of waterjet technology is to take advantage of its associated benefits, such as the cold and selective disintegration method, based on the mechanical properties of the material [2,3]. however, the broader use of this technology is limited due to a lack of high-efficiency abrasives [4], the operational issues of high-pressure pumps, and related maintenance of the components making it still not competitive with conventionally existing methods [5]. therefore, extensive research is being carried out in this sector to significantly improve the performance of waterjet technologies. it has been observed that a maximum erosion rate is achieved during the initial phase of the jet impact with the material surface [6]. this is attributed to the fact that during the initial phase, impact pressure given by eq. (1), generated by the jet is induced into the material, followed by stagnation pressure given by eq. (2), after a specific period. the magnitude of impact pressure is multiple times higher than the stagnation [7]. however, this impact pressure acts for an infinitesimal period. therefore, various ways to transform the continuous jet into a series of drops are being investigated extensively. 𝑝𝑖 = 𝑝𝑙𝑣𝑙𝑐𝑙 1+ 𝑝𝑙𝑐𝑙 𝑝𝑠𝑐𝑠 (1) 𝑝𝑠 = 1 2 𝜌𝑙𝑣𝑙 2 (2) a pulsating waterjet induces periodic loading into the material due to accelerated changes in impact and stagnation pressures decreasing the time interval for the mechanical relaxation of the material. these alternating pressures induce stress waves inside the material with the speed of sound [8]. the reflected wave also interacts and affects the induced wave and makes a local hotspot of constructive and destructive interference of stress waves. these hotspots help in propagating cracks on the surface or inside the material and cause intense fatigue failure, resulting in material erosion [9]. at the point of impact of the jet with the material, jet velocity instantly reduces to zero, leading to conversion of the kinetic energy into deformation energy, which further leads to the formation of the water shock or water hammer phenomenon. the pressure induced in the material is known as impact pressure [10]. this impact pressure is induced for a short period, after which the stagnation pressure starts to act on the material [11]. however, neither the impact pressure nor the stagnation pressure depends on the size of the droplet. only the duration for which the impact pressure is applied depends upon the size of the droplet. several methods for achieving a discontinuity from a continuous waterjet have been investigated and are still being researched [12]. the natural breakup of the jet is observed in cases where the water stream exiting from the nozzle is acted upon by cohesive and disruptive forces leading to perturbations and oscillations in the jet. these oscillations are intensified under favorable flow conditions and break the continuous flow into drops. surface tension was considered the main factor for the jet's natural breaking up, which restricted it to small modulation amplitudes and was observed for low-speed jets [13]. therefore, the inertial method is used for high-speed jets with a high reynolds number, modulation amplitude, and frequency, leading to bunching and forced breakup of the jet into a series of water droplets. in a forced breakup jet, the instability of the jet is due to the maximization of wear rates through effective configuration of standoff distance and hydraulic... 167 finite magnitude of the initiating amplitude. the jet exits the nozzle as a continuous jet with unsteady cyclical modulated velocity, resulting in slow and fast bunches of the jet flowing together. eventually, the faster section overtakes the slower section in each cycle, resulting in the separation of the continuous stream into discrete bunches of water elements. the drop formation obtained from the inertial method using axial velocity modulation was analyzed and concluded that it differed from rayleigh's theory. another apparatus to generate flow modulation used a rotary disc upstream of the nozzle [14], which causes resistance in the flow and induces a periodical variation in the discharge rate. however, its bulky design and quick-wearing rotating components were not suitable for practical purposes. another method for modulating jet generation is self-resonating nozzles (srns) [15] or helmholtz resonators. in this method, the flow modulation is due to the sudden change in the geometry of the nozzle or the resonating chamber [16]. various materials have been efficiently eroded or disintegrated using this approach to modulation. however, the system's performance is determined by the viscosity and temperature of the liquid, and modulation of the jet depends significantly on the nozzle design accuracy, limiting its utilization over a wider application area. another method for modulating the jet is to induce ultrasonic vibration into the water stream using an ultrasonic generator. apart from the ultrasonic generator, this assembly consists of an ultrasonic sonotrode in the form of a conical needle present near the nozzle exit and an electromechanical piezoelectric-based ultrasonic transducer [17]. the ultrasonic needle vibrates axially with the desired amplitude, restricting the course of water flow from the maximum to minimum value periodically. this results in the generation of oscillations downstream in the direction of the jet, focusing the energy at one point. however, with lower supply pressure (p = 10 mpa) and oscillation frequency (f = 15 khz), the ultrasonic needle submerged in the liquid cavitates and reduces its efficiency. additionally, at very high pressure (p > 100 mpa), the sonotrode oscillations are insignificant in generating discrete bunching of the waterjet, reducing its wear rates. recently, another assembly based on the induction of ultrasonic vibration into a continuous jet for generating a modulated jet was developed at the institute of geonics v.v.i. of cas, czech republic [18]. this setup consists of an ultrasonic cylindrical sonotrode fitted with a piezoelectric transducer that oscillates inside a high-pressure acoustic chamber lc [mm] whose length can be varied as desired. the ultrasonic sonotrode interacts with the incoming pressurized water p [mpa] inside this chamber and induces pressure fluctuations [19]. these fluctuations in pressure amplify while flowing through the chamber due to their converging shape [20]. the length of the acoustic chamber lc is varied to account for the liquid compressibility and tuning of the acoustic system. this effectively transfers the ultrasonic energy from the generator to the nozzle exit d [mm], where the pressure pulsations transform into velocity fluctuations [21]. these velocity fluctuations generate cyclic axial jet velocity vw [m/s] at the exit generating clusters of varying velocity water elements in the air. after traveling a specific distance, known as the breakup length, the modulated jet splits into discrete clusters of water elements. the breakup length of the modulated jet depends significantly upon the mean velocity of the jet, velocity modulation ratio, and modulation frequency. these discrete bunches of water elements impact the material surface periodically and accelerate the wear [22]. the performance of the pulsating water jet (pwj) depends on various technological parameters and their interactions, which can be categorized into ultrasonic factors, erosion factors, hydraulic factors, and target material properties. the response to variation of these parameters can be observed via the efficiency of the pwj in terms of disintegration depth, 168 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch volume removal, micro-hardness, and surface integrity. several researchers have conducted various experiments to understand the influence of these input parameters on the material erosion capability of the pulsating waterjet. however, it was observed that wear rates of pwj majorly depended on the hydraulic factor and standoff distance. hydraulic parameters comprise of supply pressure, hydraulic power, nozzle diameter, nozzle shape, and liquid density. supply pressure determines the flow rate and hydraulic energy of the pwj. an increase in the supply pressure generates a high-speed jet having higher kinetic energy. this increased kinetic energy leads to more significant erosion effects [23]. supply pressure also determines the standoff distance range corresponding to visible disintegration. an increase in supply pressure shifts the entire erosion range to a higher standoff distance [24]. with the increase in supply pressure, the breakup length of the pwj leads to the formation of a bunching phenomenon at a higher standoff distance [25]. the type of nozzle used in pwj depends upon its application. a circular nozzle focuses the jet into a specific material area compared to the flat nozzle, which guides the jet over a wider area. however, it was concluded by srivastava m. et. al. [26] that enhancement of mechanical properties is better when using circular nozzle than when using a flat nozzle due to the concentration of hydraulic energy during peening. nozzle dimension determines the flow rate of the jet at constant pressure. with an increase in the nozzle diameter, the modulated jet's flow rate and hydraulic energy increase, increasing the disintegration depths [27]. the density of the liquid used plays an important role. a comparative study performed by hloch s. et al. [28] showed that deeper and higher volume removal was calculated using 0.9% physiological saline compared to water during the disintegration of bone cement. this study also confirmed that pwj with saline as a working fluid could extract bone cement. standoff distance also plays a significant role in determining the erosion mechanism. standoff distance divides the whole erosion region into five segments: incubation, acceleration, culmination, depletion, and termination for a constant flow rate. all these stages are formed depending upon the morphology of the water clusters formed after the nozzle exit. with the increase in the standoff distance, the disintegration depth increases up to a specific limit and decreases with a further increase in the standoff distance [29]. to summarize, state of the art, it was observed that the output response of the pwj in the form of erosion rate majorly depends on the supply pressure, nozzle diameter, standoff distance, acoustic chamber, and excitation frequency. it can be implicitly defined by eq. (3). 𝐸 = 𝑓(𝑝, 𝑑, 𝑧, 𝑙𝑐 , 𝑓) (3) moreover, among these variables, lc and z are dependent variables and depend on p, d, and f, which are independent. the nature of the dependency of lc on p, d, and f values have been discussed in a previous study [30]. however, the dependency of z on p and d values is still missing in the literature. additionally, there is no implicit prediction model to predict the effective range of z for given p and d values. moreover, in the previous studies, a single hydraulic parameter, i.e., either p or d, was used to predict the optimal standoff distance and its limits corresponding to material erosion. however, it could not estimate the z value when both p and d varied simultaneously, i.e., no relationship of z in terms of hydraulic power was given. maximization of wear rates through effective configuration of standoff distance and hydraulic... 169 2. problem statement therefore, in this study, the simultaneous effect of nozzle diameter (d = 0.3 to 1.0 mm), supply pressure (p = 20, 30 and 40 mpa), and standoff distance (z = 1 to 121 mm) on disintegration depth (h), volume removal, and the microhardness of aluminum alloy were studied. statistical methods were used to determine input factors' dependency and interactions on output responses within the selected experimental domain. analytical equations were formulated to predict the effective standoff distance range and its optimal level for disintegration using supply pressure, nozzle diameter, and hydraulic power as variables. the microhardness and microstructural analysis of the disintegrated grooves was also carried out to understand the interaction of pwj with the material. 3. materials and methods the aluminum alloy aw-6060 was used to determine the optimum erosion capability of pwj based on the variation of nozzle diameters, supply pressure, excitation frequency, and standoff distance. aluminum alloy was selected as the workpiece material due to the advantage of being able to observe the eroded grooves generated by the pwj, which were easily attributed to its ductile nature. the chemical composition and mechanical properties of the aluminium alloy aw-6060 used for the experiments are shown in table.1. the sample used for the experiments was in plates of length 500 mm, width 100 mm, and thickness 5 mm. the microstructure of the sample used is shown in fig. 1. this micrograph shows the crosssectional plane of the sample perpendicular to the direction of the pwj trajectory. the effect of rolling during the fabrication of the sample resulting in deformed grains, can be observed on both sides of the sample as compared to the middle section of the sample. table 1 chemical and mechanical properties of the aluminium alloy. chemical composition al si mg fe mn ti zn cr cu 98.85 0.44 0.44 0.2 0.03 0.015 0.01 0.004 0.0071 mechanical composition rm [mpa] rp0.2 [mpa] a [%] hardness [hb] 218 185 12.56 79 the sample plates were attached to the catcher tank using jigs and fixtures to avoid movement during the tests. a hammelmann pump supplied the pressurized water with a maximum operating pressure of 160 mpa at a flow rate of 67 dm3/min. in the current study, water supply pressure was kept at p = 20, 30, and 40 mpa for each nozzle diameter. nozzle diameter variation was carried out using hammelmann nozzle inserts. the diameter variation used ranged from d = 0.3 mm to 1.0 mm. in this study we used a 20 khz prototype of the pwj available at the institute of geonics of cas for experimental investigation. this prototype uses a 20 khz ultrasonic generator to excite the piezoelectric ceramics attached to the sonotrode, enabling them to vibrate at their spatial frequency. this vibration-induced pressurized water travels through an acoustic chamber to reach the nozzle exit. the acoustic chamber is designed to modulate or tune the system to the resonant frequency (f = 19 – 21 khz), which is approximately ± 5% of the fundamental 170 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch frequency (20 khz), depending upon the physical conditions of the flow. this frequency modulation can be achieved by changing the length of the acoustic chamber. a detailed procedure for selecting the optimal acoustic chamber length for various combinations of supply pressure and diameter can be found in previous work [30]. therefore, all the experimental runs were carried out in the present study with optimal acoustic chamber length for all flow combinations. fig. 1 microstructure of the whole cross-section of the aluminum alloy aw-6060 sample the standoff distance of the nozzle head from the sample was varied using a robotic arm which was also used to determine the trajectory and traverse speed (v = 5 mm/s) of the jet over the sample. a step trajectory with minimum step height z = 1 mm was selected for every experimental run with an increment of 2 mm in the vertical direction and 15 mm in the horizontal direction between consecutive steps. the maximum step height, z = 121 mm, was determined when no visible disintegration trace was observed using p = 40 mpa and d = 1 mm, which was carried out as a pilot study. a schematic representation of the experimental setup is shown in fig. 2, and the experimental runs were carried out according to the parameters shown in table 2. fig. 2 schematic representation of the experimental setup maximization of wear rates through effective configuration of standoff distance and hydraulic... 171 table 2 experimental conditions run frequency [khz] nozzle diameter [mm] supply pressure [mpa] acoustic chamber length [mm] flow rate [dm3/min] hydraulic power [w] standoff distance [mm] 1 20 0.3 20 16 0.46 152.83 1 – 121 2 30 18 0.56 280.77 3 40 19 0.65 432.28 4 0.4 20 17 0.82 271.71 5 30 19 1.00 499.15 6 40 20 1.15 768.50 7 0.5 20 18 1.27 424.54 8 30 19 1.56 779.93 9 40 20 1.80 1200.78 10 0.6 20 20 1.83 611.34 11 30 20 2.25 1123.10 12 40 21 2.59 1729.12 13 0.7 20 21 2.50 832.10 14 30 21 3.06 1528.66 15 40 22 3.53 2353.53 16 0.8 20 21 3.26 1086.82 17 30 22 3.99 1996.62 18 40 22 4.61 3074.00 19 0.9 20 21 4.13 1375.51 20 30 22 5.05 2526.97 21 40 22 5.84 3890.53 22 1.0 20 21 5.09 1698.16 23 30 22 6.24 3119.72 24 40 22 7.20 4803.12 *discharge coefficient cd = 0.9 for hammelmann nozzle insert. all 24 erosion grooves generated on the aluminum sample by combining the input parameters mentioned in table 1 were scanned using a non-contact-type profilometer. the scanned data were analyzed using spip software to extract the groove information such as disintegration depth and volume removal for each groove generated by the pwj. specific samples produced by the pwj using the extreme level of technological parameters, i.e., p = 20 and 30 mpa with d = 0.3 and 1.0 mm, were selected for detailed investigations. the cross-sectional topography of the generated grooves was studied using an olympus ix70 light microscope. the cross-section of the sample's surface was polished and etched to observe the microstructural orientation of the material before and after the pwj treatment. the morphology of the disintegrated groove was studied using a jeol 6490lv scanning electron microscope (sem) operating in secondary electron mode (se). the overview micrographs were captured at a magnification of 13x, and the magnified images showing erosion characteristics were captured at 100x magnification. a microhardness study was also carried out on selected samples to observe the variation in the subsurface strengthening with different technological settings. the measurements were performed on a durascan g5 microhardness tester following the čsn en iso 6507-1 standard for metallic samples. indents were made with a load of 0.1 n, which was applied for 10 seconds on the crosssectional plane of the sample perpendicular to the erosion direction. the indents were formed below the disintegrated groove at an equidistance of 60 µm between two consecutive indents up to a 2 mm depth from the starting point. 172 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch 4. results and discussion 4.1. effect on disintegration depth fig. 3 shows the variation in the disintegration depth with increasing standoff distance, supply pressure, and nozzle diameter. each graph corresponds to a specific nozzle diameter kept constant, and the influence of supply pressure and standoff distance is observed. it is observed that the trend of the disintegration depth remains the same with increasing standoff distance for every combination of nozzle diameter and supply pressure. the erosion starts at a certain standoff distance (zvll) and reaches a maximum value for a certain standoff distance (zopt). a further increase in the standoff distance decreases the erosion capability, reduces the disintegration depth, and finally shows no visible erosion after a specific limit (zvul). these lower and upper vertical limits of the standoff distance corresponding to measurable erosion range largely depend on the nozzle diameter and the supply pressure of the pwj. the pulsating jet transforms into discrete bunches of water elements after covering a distance known as the breakup length [31]. this breakup length for a modulated jet depends significantly on the mean velocity of the jet, velocity modulation, and modulation frequency. however, it does not include the effect of the initial diameter of the jet. additionally, the magnitudes of the induced impact and stagnation pressure in the material are independent of the nozzle diameter. however, the time duration for which impact pressure acts over the material surface depends upon the nozzle diameter before starting lateral jetting. therefore, the erosive power increases either by increasing the nozzle diameter or the supply pressure when the nozzle diameter is constant. with the increase in the standoff distance, the pulsating jet that exits in continuous form starts to modulate and transform into discrete bunches of water elements. the phase from the exit until the breaking up of the jet shows no erosion or minimum erosion and is termed an incubation phase. internal microstructural changes can occur in this phase, but no visible external erosion is observed on the material surface. when the standoff distance reaches a specific distance, where the distinct cluster formations start to generate, visible disintegration is observed. these repetitive impacts of water elements lead to the induction of impact pressure causing material disintegration. with the increase in the standoff distance, the disintegration depth also increases up to a specific limit. this phase is known as the acceleration phase. the phase at which the deepest disintegration is observed for each particular experimental condition corresponding to a certain standoff distance is the culmination phase. with a greater increase in the standoff distance, the aerodynamic effect starts affecting the modulation of the jet, and it starts flaring out, losing its effectiveness. therefore, the disintegration depth values obtained decrease and are known as the depletion phase. with a further increase in the standoff distance of more than a specific limit, no erosion is observed due to the conversion of the pwj into mono-dispersed droplets. this phase is known as the termination phase. all five phases for fixed technological parameters (p, d, v, lc, and f) depend solely on the position of the sample material in relation to the nozzle exit. therefore, proper setting up of standoff distance is required for the optimal utilization of the technology for material disintegration. maximization of wear rates through effective configuration of standoff distance and hydraulic... 173 fig. 3 influence of nozzle diameter (d = 0.3 to 1 mm), supply pressure (p = 20, 30, and 40 mpa), and standoff distance (z = 1 to 121 mm) on disintegration depth (h) 174 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch hydraulic power considers the effects of both pressure and nozzle diameter and, therefore, serves as a suitable parameter for predicting the erosion range in the standoff distance. fig. 4 shows the variation in the standoff distance range corresponding to measured erosion with an increase in the hydraulic power for each combination of supply pressure and nozzle diameter. it can be observed that with the rise in hydraulic power, the lower vertical standoff distance (zvll) marking the erosion initiation shifts to higher standoff distance values. additionally, the erosion trace upper vertical limit (zvul) increases, increasing the hydraulic power leading to the generation of deeper grooves for a larger range of vertical limits (z). the zvll, zopt, and zvul of the standoff distance for a specific hydraulic power can be calculated using eqs. (4), (5) and (6), respectively. the coefficients of determination, r-sq (adj), are 85.6%, 89.6%, and 95.2% for the lower, optimal, and upper limits of standoff distance, respectively. therefore, the prediction of the range of z for which detectable erosion can be observed can be easily determined for any specific hydraulic power value corresponding to the combination of p and d values within the current experimental domain. 𝑧𝑣𝑙𝑙 = 6.335 + 0.01274 ∗ 𝑃ℎ − 0.000001 ∗ 𝑃ℎ 2 (4) 𝑧𝑜𝑝𝑡 = 5.367 + 0.01843 ∗ 𝑃ℎ − 0.000002 ∗ 𝑃ℎ 2 (5) 𝑧𝑣𝑢𝑙 = 6.068 + 0.03584 ∗ 𝑃ℎ − 0.000003 ∗ 𝑃ℎ 2 (6) fig. 4 predicted and actual vertical lower, optimal and upper limit of standoff distance corresponding to visible erosion range for each combination of d = 0.3 to 1.0 mm and p = 20, 30, and 40 mpa. 4.2. effect on volume removal the volume of material removed during the interaction of pwj with the aluminum surface for all combinations of the technological parameters is shown in fig. 5. p o te n ti a l co re maximization of wear rates through effective configuration of standoff distance and hydraulic... 175 fig. 5 influence of nozzle diameter (d = 0.3 to 1 mm), supply pressure (p = 20, 30 and 40 mpa), and standoff distance (z = 1 to 121 mm) on material removal. 176 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch the disintegration length for each standoff distance was kept constant at 15 mm. therefore, the volume of material removed depends significantly on the width and the depth of the eroded grooves. however, the trend of material removal is similar to the disintegration depth (fig. 3) obtained for entire experimental runs. it can be observed that with the increase in both the supply pressure (p = 20 to 40 mpa) and nozzle diameter (d = 0.3 to 1.0 mm), material volume removal increases. the range of maximum volume removal varies from 0.48 to 20.17 mm3 for p = 20 mpa, d = 0.3 mm to p = 40 mpa, and d = 1.0 mm, respectively. this increase in material volume removal is attributed to an increase in the flow rate from 0.46 to 7.20 dm3/min and hydraulic power from 152 to 4803 w for p = 20 mpa, d = 0.3 mm to p = 40 mpa, and d = 1.0 mm, respectively. 4.3. statistical analysis to observe the dependency of the disintegration depth achieved on the technological parameters, i.e., supply pressure and nozzle diameter, a full factorial design with three levels for supply pressure (p = 20, 30, and 40 mpa) along with eight levels for nozzle diameter (d = 0.3 to 1.0 mm) was used. maximum values of both the responses (disintegration depth and material removal) at each combination of the experimental run were used for the analysis. an analysis of variance (anova) [32,33] (table 3) and a pareto chart were used to identify the parameters or interaction of statistically significant (p < 0.05) parameters in determining the output responses, i.e., the disintegration depth and material volume removal. table 3 analysis of variance for disintegration depth and material removal disintegration depth source df contribution f-value p-value vif model 3 85.15% 38.22 0 linear 2 78.00% 52.52 0 p 1 31.56% 42.49 0 1 d 1 46.45% 62.54 0 1 two-way interactions 1 7.14% 9.62 0.006 p*d 1 7.14% 9.62 0.006 1 error 20 14.85% total 23 100.00% modal summary: s = 128.483, r-sq = 85.15%, r-sq(adj) = 82.92%, r-sq(pred) = 77.53% disintegration depth = 239 7.32 * p 399 * d + 43.5 * p * d material removal source df contribution f-value p-value vif model 3 95.86% 154.47 0 linear 2 91.60% 221.40 0 p 1 20.87% 100.88 0 1 d 1 70.73% 341.92 0 1 two-way interactions 1 4.26% 20.61 0 p*d 1 4.26% 20.61 0 1 error 20 4.14% total 23 100.00% modal summary: s = 1.19972, r-sq = 95.86%, r-sq(adj) = 95.24%, r-sq(pred) = 94.29% material removal = -2.36 0.0850 * p + 1.94 * d + 0.594 * p * d maximization of wear rates through effective configuration of standoff distance and hydraulic... 177 the results show that the models for depth and volume are both statistically significant. for the disintegration depth model, linear and two-way interaction models contribute 78.00% and 7.14%. nozzle diameter (46.45%) is the most statistically significant parameter for determining disintegration depth, followed by supply pressure (31.56%) and the interaction of both supply pressure and nozzle diameter (7.14%) for the selected experimental range domain. the anova results also show that the presented statistical model can accurately predict 77.53% of the disintegration depth values within the experimental domain. for material removal, the linear model (91.60%) contributes significantly compared to the two-way interaction model (4.26%). additionally, for material removal, nozzle diameter (70.73%) is the most influential parameter as compared to supply pressure (20.87%) and the interaction of supply pressure and nozzle diameter (4.26%). the predictability of the material removal values is nearly 94.29% with the presented statistical model within the selected experimental domain. therefore, it can be concluded that variation in the nozzle diameter affects the responses, i.e., the depth and the material volume, significantly, followed by supply pressure within the selected experimental domain. pareto charts of effects for both disintegration depth and material volume were plotted to determine the magnitude and importance of the effects (fig. 6). the bar corresponding to the effect that crosses the reference line (2.086 for depth and 2.09 for material removal) was considered to be statistically significant for the given output response. the pareto chart agrees with the anova results, i.e., for both the responses, nozzle diameter is the most influential parameter followed by supply pressure and then the interaction of both the nozzle diameter and the supply pressure. however, for material removal, the nozzle diameter is a significantly effective parameter for determining its value compared to any other parameters determining the disintegration depth. fig. 6 pareto chart of effects for a) disintegration depth and b) material removal fig. 7 shows the maximum disintegration depth response surface and material removal with supply pressure and nozzle diameter variation. these surface plots depict how the response trends change with a simultaneous change in input parameters. for a lower nozzle diameter (d = 0.3 mm), with an increase in supply pressure from p = 20 to 40 mpa, the differences in the maximum disintegration depth and material volume removal are observed to be 114 µm and 2.4175 mm3, respectively, as compared to 840 µm and 10.1528 mm3 for a larger nozzle diameter, d = 1.0 mm. this significant variation in response values, even keeping the same pressure range, is due to the variation in the flow rate and the hydraulic power for d = 0.3 mm (0.19 dm3 and 279.45 w) compared to d = 1.0 mm (2.11 a) b) 178 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch dm3 and 3104.96 w). therefore, the surface plot provides a better selection of input parameters according to the desired output response. it is recommended to use a larger nozzle diameter with lower pressure values for applications where deeper disintegration and higher material removal are required. however, for applications with a precise and limited area for extracting or disintegrating material, a smaller nozzle diameter with higher supply pressure can be used efficiently to achieve the desired results. fig. 7 response surface plot of disintegration depth and material removal with nozzle diameter and supply pressure. statistical analysis can also determine the zvll, zopt, and zvul for any combination of p and d within the selected experimental domain. this aids in the prediction of the effective range of z, where detectable erosion can be observed. the predictability values of zvll, zopt, and zvul with the regression equations mentioned in eqs. (7), (8) and (9) are calculated to be r-sq (pred) = 86.66%, 89.40%, and 96.07%, respectively. 𝑧𝑣𝑙𝑙 = −19.11 + 0.625 ∗ 𝑝 + 32.86 ∗ 𝑑 (7) 𝑧𝑜𝑝𝑡 = −29.95 + 0.85 ∗ 𝑝 + 47.62 ∗ 𝑑 (8) 𝑧𝑣𝑢𝑙 = −11.1 − 0.06 ∗ 𝑝 − 0.7 ∗ 𝑑 + 3.226 ∗ 𝑝 ∗ 𝑑 (9) 4.4. surface topography fig. 8 represents the cross-sectional images of the disintegration groove generated at optimal standoff distance for extreme levels of both supply pressure and nozzle diameter selected in this study, i.e., p = 20 and 40 mpa and d = 0.3 mm and 1.0 mm. fig 8a represents the erosion at the minimum flow rate, i.e., q = 0.43 dm3/min and 152.83 w hydraulic energy. a disintegration trace with a mean groove depth of h = 134 µm is obtained. isolated voids formed due to micro-channeling in the sub-surface region of the impacted area are observed (fig. 8b). these micro-channels originate from the groove cavity and propagate in different directions throughout the vicinity of the eroded groove [34]. a broader and deeper groove (h = 248 µm) is obtained with an increase in supply pressure from p = 20 to 40 mpa, keeping nozzle diameter the same, as d = 0.3 mm (fig. 8c). an asymmetric erosion groove is formed, which can be attributed to the localized material properties during the interaction of the pwj with the material surface. material upheaving is observed at the boundary of the disintegrated groove (fig. 8d). this material upheaving is significantly due maximization of wear rates through effective configuration of standoff distance and hydraulic... 179 to the radial outflow of the impacted water stream known as lateral jetting. the shear stress generated pushes the material towards the erosion periphery resulting in curved material flow. fig. 8 cross-sectional view of deepest disintegration trace generated at a) d = 0.3 mm, p = 20 mpa, b) magnified view of impacted region of fig. a, c) d = 0.3 mm, p = 40 mpa, d) magnified view of impacted region of fig. c, e) d = 1.0 mm, p = 20 mpa and f) d = 1.0 mm, p = 40 mpa. with the increase in the nozzle diameter, from d = 0.3 mm to 1.0 mm, the disintegration groove volume increases. an erosion crater, isolated subsurface voids, and material upheaving are also shown in fig. 8e. it can be observed that groove width significantly depends on the nozzle diameter. however, the depth of the groove is determined by the 180 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch supply pressure. from a volume removal point of view, selecting a larger nozzle diameter is more beneficial even with lower pressure levels. statistical analysis resulted in the same observation. however, if a specific groove width or disintegration of the intricate area is required, a smaller nozzle diameter with more considerable supply pressure can be used. with a further increase in the supply pressure p = 40 mpa, a deeper erosion groove than p = 20 mpa is obtained. the disintegration trace is heavily eroded with an asymmetric groove profile (fig. 8f). eroded material fragments dislodged due to the jet impact, and lateral jetting, can be observed near the eroded groove wall. isolated holes formed by the microjet streams penetrating the material and localized material interaction can be observed in the subsurface region around the disintegrated groove. these craters or voids can also be present in the parent material and act as stress concentration sites that promote or accelerate material failure under hydraulic impact [35]. 4.5. surface morphology fig. 9 shows the micrographs of the disintegrated grooves of the selected sample produced using extreme parameter levels (p = 20 and 40 mpa, d = 0.3 and 1.0 mm). the material damage and erosion are largely caused by the induction of compressive stress during the pwj impact, followed by shear stress due to the lateral flow of the jet. when they exceed the ultimate tensile limit of the aluminum alloy (215 mpa), the combined induced stresses result in material failure and removal. fig. 9a shows the groove generated using d = 0.3 mm and p = 20 mpa at z = 5 mm. at these technological conditions, the average width and depth of the groove were measured as w = 940 µm and h = 164 µm, respectively. the compressed area due to the pwj impact followed by water flowing away from the pwj footprint center was observed in fig. 9b. cavity rims and entrances are generated due to the jet's repetitive impact, causing the aluminum alloy to undergo material damage. with the increase in the pressure, p = 40 mpa, a larger erosion groove is obtained, having an approximate width of w = 1400 µm and a mean depth of h = 248 µm (fig. 9c). pwj under these technological conditions possesses higher hydraulic energy (432 w), producing greater erosion damage. dimple formation due to the interaction of pwj with ductile material can be seen near the pwj centerline. moreover, sites acting as an opening for the water flow to penetrate deeper into the materials are also observed in fig. 9d. fig. 9e shows the disintegration trace produced by pwj with d = 1.0 mm and p = 20 mpa at z = 33 mm. the hydraulic energy and flow rate of the pwj at this parameter setting are p = 1698 w and q = 5.09 dm3/min. this results in higher impact pressure on the surface inducing larger compressive stress in the material. a wider disintegration groove, w = 3100 µm, with a mean depth, h = 379 µm, is obtained. micro-cavities of varying dimensions are randomly generated through the disintegrating surface of the material (fig. 9f). fig. 9g represents the micrograph of the disintegrated area produced by pwj with d = 1.0 mm and p = 40 mpa at z = 125 mm. average disintegration width, w = 4700 µm, and depth, h = 1211 µm, were obtained during this experimental run. the image also shows the material upheaval caused by the lateral flow of the pwj, generating an approximate boundary of the erosion groove. the bottom of the groove is heavily articulated with several cavity openings caused by material deformation. from a fractographic point of view, the image shows the area with different modes of material erosion. a section of the compressed material surface formed along the pwj footprint centerline can be observed in fig. 9. maximization of wear rates through effective configuration of standoff distance and hydraulic... 181 fig. 9 sem images of disintegrated samples generated by pwj at different technological conditions a) d = 0.3 mm, p = 20 mpa, z = 5 mm, c) d = 0.3 mm, p = 40 mpa, z = 17 mm, e) d = 1.0 mm, p = 20 mpa, z = 33 mm and g) d = 1.0 mm, p = 40 mpa, z = 125 mm. the magnified images of the marked areas (b, d, f, and h) are also shown. 182 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch 4.6. microhardness microhardness analysis was performed to determine the change in the sub-surface hardness near the periphery of the eroded groove. the measurements were carried out with a load of 0.1 n starting below the disintegrated groove on the cross-sectional plane until a depth of 2 mm. the distance between two consecutive indents was kept at 60 µm. the sites for microhardness tests for all four samples are shown in fig. 10. fig. 10 microhardness measurement sites on different samples eroded with different parameter conditions: a) d = 0.3 mm, p = 20 mpa, z = 5 mm, b) d = 0.3 mm, p = 40 mpa, z = 17 mm, c) d = 1.0 mm, p = 20 mpa, z = 33 mm, and d) d = 1.0 mm, p = 40 mpa, z = 125 mm. fig. 11 shows the variation in microhardness data with increasing depth from the bottom of the groove on the cross-sectional plane. it can be observed that an increase of 15 – 20 % is recorded for the eroded samples throughout the depth as compared to original samples in their received form. for pwj, the impact pressure induced on the surface leads to plastic deformation in the form of increased dislocation density. this enhancement in dislocation density is attributed to grain refinement and the formation of slip lines near the impacted surface. the cyclical impact induces compressive stresses that propagate along the material cross-section leading to strengthening of the sub-surface in form of enhanced microhardness. however, the influence of the levels of the technological parameters (supply pressure, nozzle diameter, and standoff distance) on the subsurface strengthening determined by microhardness values is not visible. the subsurface hardening obtained at maximization of wear rates through effective configuration of standoff distance and hydraulic... 183 d = 0.3 mm and p = 20 mpa is almost similar to that obtained at d = 1.0 mm and p = 40 mpa. however, the disintegration depth or material removal varies significantly for these technological levels. this non-significant difference in microhardness values can be associated with the small range ( ̴ 25 mpa) (tensile strength rm = 215 mpa and yield point rp0.2 = 190 mpa) of the plastic zone for the aw-6060 aluminum alloy used in the current study. these properties of aw-6060 aluminum alloy create a small range for generating plastic deformation without affecting the tensile strength and lead to the material removal process at any technological level. therefore, the surfaces exposed to pwj are damaged and form disintegrated grooves at given technological levels. the variation in the microhardness of the unaffected material throughout the depth is also attributed to the distribution of the grain structure and its orientation. the microstructural distribution near the sample surfaces is randomly oriented and deformed in size compared to the grains present in the center of the sample due to the rolling action during the fabrication of the samples. fig. 11 microhardness trend along the cross-sectional plane below the disintegrated groove formed under different technological conditions. 5. conclusions in this study we investigated the utilization of the water hammer effect produced by an ultrasonic pulsating waterjet for disintegration of materials. the appropriate setting of technological parameters is not only necessary to maximize the disintegration but also by merely changing the distribution of impacts, pwj can be used for surface roughening and increasing fatigue life through peening. in this study, the nozzle diameter, supply pressure, 184 a. nag, a.r. dixit, j. petrů, p. váňová, k. konečná, s. hloch and standoff distance were selected to observe their effect on disintegration depth, volume removal, microhardness, and the surface morphology of the disintegrated material. statistical analysis was used to determine the statistically significant input parameters for the output responses and to formulate statistical regression equations to predict effective range and optimal standoff distance. the major findings of the study are summarized as follows: 1) a simultaneous increase in supply pressure (p = 20, 30, and 40 mpa) and nozzle diameter (d = 0.3 – 1.0 mm) increases the disintegration depth and material removal values. additionally, with a specific combination of p and d, the disintegration depth and material removal increase with the standoff distance up to a certain limit and then decrease with a further increase in the standoff distance. 2) sharp increase in erosion compared to gradual decrease due to air drag was observed while increasing the z values (z = 1 to 121 mm) (fig.5). this corresponds to smaller difference between the optimal standoff distance (zopt) and vertical lower limit of standoff distance (zvll) as compared to vertical upper limit of standoff distance (zvul) and zopt for all experimental conditions. 3) the hydraulic power of pwj was observed as a suitable indicator to identify the effective limits of standoff distance. eqns. 6, 7, and 8 showed that lower, optimal, and higher limits of the standoff distance could be easily predicted using hydraulic power as a variable. 4) analysis of variance results showed that the dependence of disintegration depth and material removal is higher for nozzle diameter (46.45% and 70.73%) as compared to supply pressure (31.56% and 20.87%) and two-way interactions (7.14% and 4.26%). the regression equations formulated allow the prediction of the values of the output responses within the selected experimental domain. 5) statistical regression equations were formulated to predict the effective range and optimal standoff distance using nozzle diameter and supply pressure values. 6) the cross-sectional topography of the selected samples shows the extent of the disintegration grooves inside the material and the upheaved material section formed at the periphery of the groove due to the lateral jetting. 7) sem images captured the overall disintegration traces and detailed erosion characteristics such as the formation of crater openings and compressed and ductile erosion regions for the selected samples. 8) a microhardness study showed an increase of 15 – 20% in microhardness values after impact with the pwj compared to the original state. however, no significant variation in the values was noticed due to a change in the parameter levels. this is attributed to the slight difference ( ̴ 25 mpa) between the yield and ultimate tensile values of the aw-6060 aluminum alloy that was used. the outcomes of this study would help to quickly tune the technology to resonance while also using it for practical purposes. a tuning methodology is proposed according to the implicit function for the determination of the optimal and effective range of standoff distance, which simplifies further research. acknowledgement: this work was supported by financial support by the czech science foundation, project gacr 19-00408s. maximization of wear rates through effective configuration of standoff distance and hydraulic... 185 references 1. perec, a., radomska-zalas, a., fajdek-bieda, a., 2022, experimental research into marble cutting by abrasive water jet, facta universitatis-series mechanical engineering, 20(1), pp. 145–156. 2. natarajan, y., murugesan, p.k., mohan, m., khan, s.a.l.a., 2020, 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ductile material caused by multiple droplet impingement at subsonic speeds, wear, 490–491, 204176. 35. poloprudský, j., chlupová, a., šulák, i., kruml, t., hloch, s., 2021, surface and subsurface analysis of stainless steel and titanium alloys exposed to ultrasonic pulsating water jet, materials, 14(18), 5212. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 385 396 https://doi.org/10.22190/fume190327034d © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper new class of digital malmquist-type orthogonal filters based on the generalized inner product; application to the modeling dpcm system nikola danković, dragan antić, saša nikolić, marko milojković, staniša perić university of niš, faculty of electronic engineering, department of control systems, serbia abstract. a new class of cascade digital orthogonal filters of the malmquist type based on bilinear transformation for mapping poles to zeroes and vice versa is presented in this paper. in a way, it is a generalization of the majority of the classical orthogonal filters and some newly designed filters as well. these filters are orthogonal with respect to the generalized inner product which is actually a generalization of the classical inner product. outputs of these filters are obtained by using polynomials orthogonal with respect to the new inner product. the main quality of these filters is that they are parametric adaptive. the filter with six sections is practically realized in the laboratory for modeling, simulation and control systems. performances of the designed filter are proved on modeling and identification of the system for differential pulse code modulation. real response and response from the proposed filter are compared with regard to the chosen criteria function. also, a comparative analysis of the proposed filter with some existing filters is performed. key words: digital orthogonal filters, malmquist functions, müntz polynomials, bilinear transformation, inner product, differential pulse code modulation 1. introduction the majority of the already existing classes of orthogonal rational functions are obtained by using one of the two simple transformations. the first one is a linear transformation s→as+b which was used for obtaining classical orthogonal functions as legendre, müntzlegendre, jacobi, and some other classes of orthogonal functions and appropriate filters as almost orthogonal [1–3], quasi-orthogonal [4, 5], and some generalized classes of received march 27, 2019 / accepted july 25, 2019 corresponding author: nikola danković university of niš, faculty of electronic engineering, department of control systems, aleksandra medvedeva 14, 18000 niš, republic of serbia e-mail: nikola.dankovic@elfak.ni.ac.rs 386 n. danković, d. antić, s. nikolić, m. milojković, s. perić orthogonal functions and filters [6]. the second transformation is a reciprocal transformation of poles to zeroes s→1/s (malmquist, laguerre functions) [7, 8]. in [9] and [10] the relation between generalized malmquist functions and new müntz polynomials is given. in [11] this relation is generalized using bilinear transformation for mapping poles to zeroes and vice versa. a new class of müntz polynomials, orthogonal with respect to the inner product obtained by this transformation of poles to zeroes, is derived. the filters based on the generalized malmquist orthogonal functions are designed in [12]. these filters are orthogonal with respect to the new inner product which is derived from symmetric reciprocal transformation s→b/(cs) which is a generalization of the reciprocal transformation given in [7, 8]. two types of these filters are designed in [12], namely, analogue and digital ones. further generalization of these orthogonal functions is based on the above mentioned symmetric bilinear transformation s→(as+b)/(cs-a) [11]. this transformation includes the above mentioned linear and reciprocal transformations. in [13], analogue orthogonal filters based on these functions are designed. since the first pulse-code modulation transmission of digitally quantized speech, in world war ii, digital signal processing (dsp) began to proliferate to all areas of human life. classical digital systems are known to possess poor parameters under finite-precision arithmetic, like frequency response sensitivity to changes in the structural parameters, noise, inner oscillations, and limit cycles [14]. these effects have led to development of wave filters [15] and orthogonal filters. rapid development of the digital orthogonal filters started in early 1980's [16, 17]. the most common approaches to the orthogonal filter synthesis are transfer function decomposition [16, 18] and the state space approach [19, 20]. various realizations of the digital orthogonal filters developed in the last several decades. for example, in [18], the development of the pipelined structure of digital orthogonal filters started. it continued in the 1990's till present days [14]. when the two most common approaches are compared, state-space realization has advantages in the case of mimo filters; it provides a better insight into their structure. however, in this paper the authors use transfer function realization of the proposed filters that is suitable for our purpose of modeling real systems. it was noted that iir digital filter can be applied to modern dsp applications (e.g. mobile communication), to multichannel prediction, etc. because of all this, the main goal of this paper is design and later practical realization of a new class of digital orthogonal filters. in this paper a bilinear transformation for mapping poles to zeroes and vice versa in the case of digital system is performed: z→(az+b)/(cz-a). in this way, the filters, orthogonal with respect to a more generalized inner product, are obtained both in z-domain and discretetime domain. the class of discrete orthogonal polynomials derived from [9, 10] will be used for determining outputs of the proposed filters. the proof of this orthogonality is given in the appendix. in this way, the designed orthogonal digital filter is a generalization of the majority of classical orthogonal filters (legendre, müntz-legendre, laguerre, jacobi, malmquist) as well as the most recently designed filter based on the reciprocal transformation [12]. the generalized malmquist filters designed in this paper are parametric adaptive. the proposed digital orthogonal filter is practically realized in the laboratory for modeling, simulation and control systems [21] and it will be applied to modeling of the well-known digital system for signal transmission, differential pulse code modulation (dpcm) system [22–24]. performances of the new filter are verified by comparing them with some other classes of digital orthogonal filters. new class of digital malmquist-type orthogonal filters based on generalized inner product... 387 the rest of the paper is organized as follows. in section 2, new digital orthogonal filters based on bilinear transformation are developed; first theoretically, later by simulation; finally, they are practically realized. the proposed filter is used to modeling a linear part of the dpcm system in section 3. comparison between the practically realized filter and some other filters is performed in section 4. in section 5, the authors give conclusions and discuss a possibility for further development in this area. finally, at the end, the appendix includes the proof of orthogonality of these filters in discrete-time domain, and also relations for the inner products and norms. 2. digital malmquist-type orthogonal filters: mathematical background, design and practical realization the generalized malmquist filters based on bilinear transformation in discrete-time domain can be derived from corresponding generalized malmquist filters in continuoustime domain [13] with the same procedure given in [12]. namely, operator s (operator of differentiation) is substituted by operator z (operator of prediction) to design corresponding digital filters. the transfer function of new digital filters has the following form: * *1 10 ( ) , , . n k k n k k k k k z a bz w z r z z c a                   (1) cascade scheme based on (1) is given in fig. 1, where k is the number of samples, and t is a sample period. the authors assume the sample period is one second because it is not important for further analysis, i.e. k≡kt (discrete time). fig. 1 block diagram of a digital orthogonal filter based on bilinear transformation a sequence of functions on the outputs of cascades of the proposed digital filter (1) obtained mathematically corresponds to responses by simulation and from practically realized orthogonal digital filter. these outputs for the specific case a=0, b=1, c=1 (reciprocal transformation) are already given (α0=1/2, α1=1/3, α2=1/4, α3=1/5) in [12] illustratively. these filters are orthogonal in complex z-plane: * 2 , 1 ( ( ) ( )) ( ) ( ) 2 n m n m n n mw z w z w z w z ds n i      , (2) where 1 1 10 ( ) k n k n k k a b z c az w z z z               , ,n m represents kronecker symbol, and contour г surrounds all the poles of wn(s). 388 n. danković, d. antić, s. nikolić, m. milojković, s. perić the proof of orthogonality is similar as in the case of continuous-time systems [11, 13]. if in transfer function wn(z) bilinear transformation is performed, the authors obtain wn * (z) whose poles are equal to zeroes of wn(z) and zeroes of wn * (z) are equal to poles of wn(z). thereby, all poles of wn * (z) are outside contour г, and zeroes of wn * (z) are inside contour г. if m≠n due to symmetry of the bilinear transformation, all the poles of the integrand (2) that lie inside contour г are annulled with appropriate zeros of wm * (s), so the contour integral (2) is equal to zero. in the case of m=n, there exists one first-order pole inside contour г. after applying the cauchy theorem, the following expression is obtained: (wn(s), wm(s))=nn 2 ≠0. finally, all the expressions stated above imply (2). practical realization of the generalized malmquist digital orthogonal filter is shown in fig. 2. fig. 2 practical realization of the generalized digital orthogonal filter based on bilinear transformation – printed circuit board outputs from the filter in z-domain φl(z)=u(z)wl(z), l=0, 1, 2, ..., n are orthogonal [12, 13, 25]: 2 ,( ( ) ( ))n m n n mz z n    . (3) outputs from this digital filter in time domain are obtained using inverse z-transformation: 1 11 ( ) { ( )} ( ) 2 k l l lk ζ z z z dz i          , (4) where contour γ surrounds all the poles of wl(z). this contour can be obtained by mapping sto z-domain: γ={c|z| 2 -2arez-b=0}[25]. a new inner product for the filter outputs is: 2* * * * , , 1 ( ( ), ( )) ( ( ), ( )) ( ) ( )n m n m n m n m n n m k j k k k k k k n              . (5) new class of digital malmquist-type orthogonal filters based on generalized inner product... 389 the relation for jn,m and nn 2 is given in the appendix. otherwise, this inner product is generated in the practically realized filter thanks to its structure. these filters are adjustable. it means that their parameters can be changed: numerical values of poles and parameters of bilinear transformation. for example, for c=0 well-known classical orthogonal filters based on linear transformation are obtained, and for a=0 orthogonal filters based on the reciprocal transformation (malmquist type). finally, in the case of a≠0, b≠0, c≠0 the most generalized orthogonal digital filter can be obtained. this digital cascade orthogonal filter will be practically applied to modeling a prediction filter in a well-known digital system in telecommunications, the dpcm system [22]. because of the cascade structure of the realized filter, i.e. a possibility to append more sections, the authors suppose that the filter will be suitable for modeling dpcm systems which can theoretically have an arbitrary high order predictor. a cascade structure is already verified in the case of appropriate class of analogue generalized malmquist filters [4]. 3. application to modeling dpcm system a new digital cascade orthogonal filter based on bilinear transformation will be applied to modeling and signal identification of a linear part of the dpcm system [22]. the dpcm is a well-known and commonly used technique for signal transmission in telecommunications. this system has a wide usage in different areas, starting from speech and image coding to the latest medical research [23]. an estimate, i.e. a prediction of the present value of the input signal is based on the knowledge of its earlier values [24]. that is why one of the most important parts of every dpcm and adpcm (adaptive differential pulse code modulation) is a predictor (a linear part of the system). for modeling of the prediction filter the authors use an adjustable model based on the proposed generalized digital filter (fig. 1). a block diagram of the digital orthogonal adjustable model based on bilinear transformation is shown in fig. 3. in this case the authors use a filter with six sections and real poles αk * =(aαk+b)/(cαk-a), k=0, 1, ..., n. fig. 3 block diagram of an adjustable model with the proposed orthogonal digital filter based on bilinear transformation it can be noticed from fig. 3 that the orthogonal model output is: 0 ( ) ( ) n m k k k y k b k   , (6) where k is the number of samples, n=5 in our case. 390 n. danković, d. antić, s. nikolić, m. milojković, s. perić the desired model of the linear part of dpcm system (exactly, the prediction filter in the encoder) is obtained by adjusting the following parameters: αk (k=0,1,…,5), summation coefficients bk (k=0,1,…,5), and parameters of bilinear transformation a, b, and c. in the case of modeling a particular unknown system, the parameters of the model should be adjusted in such a way that the model (fig. 3) corresponds as closely as possible to the unknown system. the process of modeling is performed in the well-known manner by introducing the same input to the system itself and to its adjustable model based on the new cascade orthogonal digital filter [12]. this input signal is shown in fig. 4. fig. 4 the input of dpcm linear part and the adjustable model the next step is measuring the outputs from system ys(t) and filter ym(t) and calculating the mean squared error (criteria function): 2 0 1 ( ( ) ( )) n s m k j y k y k n    . (7) optimal values of unknown parameters which lead to minimization of mean squared error can be obtained by using genetic algorithm with j is fitness function. the specific genetic algorithm used in experiments has the following parameters: initial population of 1000 individuals, a number of generations of 300, a stochastic uniform selection, a reproduction with 12 elite individuals, and gaussian mutation with shrinking. the used structure of chromosome was with eight parameters coded by real numbers: a, b, c, b0, b1, b2, b3, b4, and b5 (eq. 1). poles a0, a1, a2, a3, a4, a5 are also adjustable, and in these experiments their values are fixed. a series of experiments can be performed with other values of poles until the obtained model fill requirements in advance. the main goal of the experiment was to obtain the best model of the unknown system in regard to the criteria function, i.e. mean squared error. the original signal (output from the dpcm linear part) and the signal from the adjustable model based on the orthogonal digital filter of the generalized malmquist type are given in fig. 5. new class of digital malmquist-type orthogonal filters based on generalized inner product... 391 fig. 5 outputs from the dpcm linear part and the adjustable model the authors performed experiments with six sections. in the case of the digital orthogonal filter based on reciprocal transformation the filter with six sections is verified as better related to the criteria function than one with four or five sections [12]. obtained optimal values for parameters of adjustable orthogonal model are presented in table 1. mean squared error is: jmin=5.7216∙10 -3 . table 1 values for parameters of the adjustable orthogonal model parameters numerical values b0 0.56713 b1 0.59111 b2 0.32604 b3 0.61674 b4 -0.21408 b5 0.25053 α0 0.91286 α1 0.87552 α2 0.77119 α3 0.82101 α4 -0.15647 α5 0.79444 a 0.28327 b 0.09449 c 0.73071 in fig. 5 one representative sample is zoomed for illustrative purposes because of very small differences between sample values. measured outputs in discrete-time periods (sample periods) of the prediction filter and the adjustable model based on the new digital orthogonal filter are given in table 2. 392 n. danković, d. antić, s. nikolić, m. milojković, s. perić table 2 obtained outputs from the dpcm prediction filter and the adjustable model based on the digital orthogonal filter k output from the dpcm prediction filter output from the adjustable orthogonal model 0 0.82500 0.82500 1 2.09125 2.09215 2 3.68806 3.68806 3 5.71962 5.70912 4 7.18302 7.18302 5 9.53761 9.51500 6 10.89502 10.86011 7 12.05639 12.03209 8 14.99215 14.97022 9 16.44441 16.39001 10 17.93314 17.91510 11 19.39597 19.36826 12 19.98442 19.97384 13 20.04930 20.03347 14 19.59736 19.57832 15 20.33942 20.31451 from fig. 5 and table 2 a high level of matching can be noticed between signals from the dpcm linear part and the proposed orthogonal digital filter. finally, the model of the prediction filter in the dpcm encoder is formed as: *5 *1 1 0 0 ( ) , 0, k i m k k i i z w z b z              (8) where αk * =(aαk+b)/(cαk-a), and appropriate values of parameters are given in table 1. the proposed filter model (8) using numerical values in table 1 can be written in the following form:   5 4 3 2 6 5 4 3 2 0.527 0.661 0.893 1.131 1.432 1 0.619 0.077 0.125 0.592 0.958 1.732 1.214 1 m z z z z z w z z z z z z z                 . (9) the relation (9) for the transfer function of the system is more suitable for control system theory analysis, while the relation (8) is more suitable for system modeling. 4. comparative analysis between the proposed filter and some other classes of digital orthogonal filters in order to verify the quality of the model based on the new filter with six sections, a comparison with the models based on the some already existing digital orthogonal filters (generalized legendre and generalized malmquist filters) is performed. the criteria function is mean squared error again and the number of sections is six. the transfer functions of all the filters used in experiments are shown in table 3 (αi-1 is assumed to have constant value equal to zero). new class of digital malmquist-type orthogonal filters based on generalized inner product... 393 table 3 transfer functions of digital orthogonal filters used in experiments orthogonal filter type transfer function legendre (müntz-legendre) 5 1 0 0 ( ) ( ) ( ) k i k k i i z w z b z             generalized malmquist 5 1 0 0 ( ) k i k k i i b z w z b z           filter based on bilinear transformation 1 5 1 0 0 ( ) i k i k k i i a b z c a w z b z               the outputs of the models based on these filters are calculated as 5 0 ( ) ( )m l l l y t b t   . table 4 values for parameters of the adjustable orthogonal models based on new and existing types of digital orthogonal filters criteria function value and orthogonal model i 0 1 2 3 4 5 j=16.370∙10-3 orthogonal model with legendre filter (λ=0.67) αi 0.1677 0.3162 0.4777 0.6164 0.7011 0.8168 bi 0.7821 1.6234 0.9102 0.8648 0.5519 0.3271 j=13.022∙10-3 orthogonal model with generalized malmquist filter (b=0.92) αi 0.2681 0.1155 -0.1733 0.3124 0.2178 0.1665 bi 3.6410 -0.9131 2.1616 1.9872 0.8764 0.3558 j=5.721∙10-3 orthogonal model with a new filter based on bilinear transformation (a=0.2834, b=0.094, c=0.731) αi 0.9129 0.8755 0.7712 0.8210 -0.1565 0.7944 bi 0.5671 0.5911 0.3260 0.6167 -0.2141 0.2505 the structure of chromosome that is used in experiments is with 6 standard parameters coded by real numbers: α0, α1 ,..., α5, and one additional; for legendre filter λ and for generalized malmquist filter b. in table 4 the obtained parameters for proposed and existing filters are given. from table 4 it can be seen that the mean squared error is much bigger for the generalized malmquist [12] and the legendre digital orthogonal filters than for the filter presented in this paper. the excellent matching between the system output and the output of the adjustable model based on the proposed filter has shown the quality and need for new filters described in this paper. of course, the model of the linear part of dpcm system can be derived by using other methods, but in this paper the authors used new orthogonal filter to verify its performances. 394 n. danković, d. antić, s. nikolić, m. milojković, s. perić 5. conclusion and future work in this paper the authors gave mathematical background, simulation and practical realization of new digital orthogonal filters based on bilinear transformation of poles to zeroes and vice versa. it is an extension of generalizations of traditional orthogonal filters starting with filters based on reciprocal transformation. all good performances of already existing filters are included in this class of filters. the great quality of this filter is parametric adaptivity, i.e., possibility of adjusting values of poles and parameters of bilinear transformation. also, it is demonstrated that by setting specific values for parameters of bilinear transformation, most of the classes of realized filters can be obtained. in the case when c=0, these filters degenerate into classical orthogonal filters (legendre, laguerre, jacobi), and when is a=0, they degenerate into classical malmquist and generalized malmquist filters. in the future work, the authors could try to derive appropriate classes of orthogonal filters with complex poles which in some practical cases could be even better. further generalization could also be in the usage of more general symmetric transformations than the bilinear one. in this way, the study of generalized class of orthogonal analogue and digital filters will be concluded. appendix polynomials obtained by using bilinear transformation of poles to zeroes are orthogonal on the contour:  2 2 re 0c z a z b     . (a1) contour (a1) is a circle with radius 2 a b r c c        and center in ,0 a c       . using transformation z=rz * +a/c, i.e., z * =(za/c)/r, the circle is mapped into the unit circle with the center at the origin. the model of these polynomials is shown in fig. 1. outputs φm(z) for the unit input are: 0 10 0 1 ( ) ,..., ( ) , 1,2,..., i m i m i i a b z c az z z m n z z z                  . (a2) by development of eq. (a2) in partial fraction it is obtained: , 0 ( ) m m j m j j a z z      , (a3) where : , lim( ) ( ) i m j j m z a z z       , 1 0 , 0 ( ) m i i i m j m j i i i j a b z c a a                   . new class of digital malmquist-type orthogonal filters based on generalized inner product... 395 by using inverse z-transformation of φm(z) outputs in time-domain are obtained: ( 1) , 0 ( ) m k m m j j j k a      . (a4) by applying linear transformation onto linear systems, orthogonality is held. when linear transformation z=rz * +a/c is used, the region of orthogonality of filters based on bilinear transformation is mapped into the unit circle with the center at the origin, i.e. these polynomials are mapped into classical discrete-time orthogonal polynomials where orthogonality in the classical sense is valid: 1 * * *( ) ( ) ( ) ( )n m n mz z dz z z dz         , (a5) where: , ,* * * *0 0 ( ) , ( ) n m n j m j n m j j j j a a z z a a rz rz c c              . by using inverse z-transformation it is obtained: * ( 1) * ( 1) , , 0 0 ( ) , ( ) n m k k n n j j m m j j j j k a k a             . (a6) the norm for outputs of filters obtained by bilinear transformation is: 2 * * * * 1 ( ( ), ( )) ( ( ), ( )) ( ) ( )n n n n n n n k n k k k k k k            . (a7) therefore, a new inner product for outputs of the filter is obtained: 2* * * * , , 1 ( ( ), ( )) ( ( ), ( )) ( ) ( )n m n m n m n m n n m k j k k k k k k n              . 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comparison between the classical approach and the monte carlo method, information technology and control, 46(2), pp. 28-38. 25. danković, n., 2018, development of a new class of orthogonal filters with applications in modelling, analysis and synthesis of differential pulse code modulation system, phd thesis, university of niš, faculty of electronic engineering, serbia. facta universitatis series: mechanical engineering vol. 17, no 3, 2019, pp. 333 344 https://doi.org/10.22190/fume190426038p © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper edge detection parameter optimization based on the genetic algorithm for rail track detection milan pavlović 1 , vlastimir nikolić 2 , miloš simonović 2 , vladimir mitrović 3 , ivan ćirić 2 1college of applied technical sciences niš, serbia 2faculty of mechanical engineering, university of niš, serbia 3forstehd.o.o, belgrade, serbia abstract. one of the most important parameters in an edge detection process is setting up the proper threshold value. however, that parameter can be different for almost each image, especially for infrared (ir) images. traditional edge detectors cannot set it adaptively, so they are not very robust. this paper presents optimization of the edge detection parameter, i.e. threshold values for the canny edge detector, based on the genetic algorithm for rail track detection with respect to minimal value of detection error. first, determination of the optimal high threshold value is performed, and the low threshold value is calculated based on the well-known method. however, detection results were not satisfactory so that, further on, the determination of optimal low and high threshold values is done. efficiency of the developed method is tested on set of ir images, captured under night-time conditions. the results showed that quality detection is better and the detection error is smaller in the case of determination of both threshold values of the canny edge detector. key words: edge, canny, threshold, optimal, genetic algorithm 1. introduction image processing is a widely used method in machine vision systems for performing of certain operations on an image in order to extract useful information. edge detection is a part of image processing that can be used for reducing of the amount of data to be processed with the final goal to provide useful structural information about the boundaries of the object. the main goal of the edge detection is to locate and identify sharp discontinuities from an image. these discontinuities are effects of a significant local change in image intensity. an edge represents a point or a set of points that create a received april 26, 2019 / accepted june 30, 2019 corresponding author: milan pavlović college of applied technical sciences niš, aleskandra medvedeva 20, 18000 niš, serbia e-mail: milanpavl@gmail.com 334 m. pavlović, v. nikolić, m. simonović, v. mitrović, i. ćirić curve that follows a path of that change, and essentially it defines boundaries between two distinctly different regions. existence of the edge can be caused by variations in reflectance, illumination, color, shade, texture, orientation and depth of scene surfaces. however, image intensity is often proportional to scene radiance, so the edges are represented in the image by changes in the intensity function [1-4]. there are many uses of the edge detection for different purposes, such as vehicle and transport applications [4-8], human applications [9-11], medical applications [12-15], etc. however, quality of the edge detection is dependent on lighting conditions, the presence of objects of similar intensities, density of the edges in the scene, and noise [1], as well as the edge detector used in a detection process. many edge detectors have been proposed, tested, and compared, such laplacian of gaussian (log), prewitt, roberts, sobel and canny. however, the canny edge detector has shown the best results in the edge detection with good noise immunity and detection of the true edge points with minimum error [1, 3, 4, 16, 17]. one of the key factors for great accuracy of the canny edge detector is setting up the optimal threshold values. determination of that parameter can be long and difficult for different images, so its adaptability enables robustness and a wide range of use. in [18], an improved method for setting up the thresholds according to the gray-scale histogram is proposed. this method gave good results but it may cause some fake edges. in [19], the otsu algorithm is used in order to get a value of high threshold, while a value of low threshold is obtained by multiplying the high threshold value by a coefficient less than one, specifically 0.5. moreover, this method has proved to be effective for edge extraction, the adopted two threshold values are two global values, which are obtained based on the whole image. on the other hand, in [20] calculation of the low threshold value is based on a probability model; the otsu method is used for determination of the high threshold value, while the adaptive particle swarm optimization (apso) is employed instead of the traditional gradient descent method in order to get the optimal solutions of the both the otsu algorithm and the probability model algorithm. in [21], otsu and canny operators that choose thresholds adaptively by using a new adaptive grey mapping function combined with the shape identification algorithm in order to segment the area of the target leaf are presented. combination of the maximum entropy method with the otsu method for determination of the high and low thresholds of the canny algorithm is shown in [22]. the results showed that the proposed algorithm has better performance for the images which have complex distributions of grey level histogram. in order to overcome the difficulty of threshold selecting in the canny algorithm, in [23] is presented the method based on the otsu algorithm and mathematical morphology. this method chooses the threshold adaptively and simultaneously; it applies the improved canny operator and the image morphology separately to the image edge detection, and then performs image fusion of the two results using the wavelet fusion technology to obtain the final edge-image. the proposed algorithm in [24] utilizes the local threshold values to detect particles from the images along with the selection of how many sub-images to use and automatically segment the whole image into the required number. after that, the calculation of the local threshold values of each sub-image is done with the otsu algorithm for high threshold value (th), while the low threshold value is calculated as 0.4th. the algorithm which can adaptively determine the two thresholds based on the gradient histogram and the minimum interclass variance, is presented in [25]. in order to detect retinal blood vessels, the detector as a local dynamic hysteresis thresholding value generator based on the canny edge detector, is presented in [26]. the method based on applying of different values of sigma and thresholding in different parts of the image edge detection parameter optimization based on genetic algorithm for rail track detection 335 instead of processing the entire image with a single value of sigma and thresholding, is presented in [27]. after dividing the image, the mean pixel value of each sub-image is calculated and, depending upon these values, each sub-image will be processed by a gaussian filter with different sigma and thresholding values. in the proposed method [28], given a set of candidates for hysteresis thresholds, the basic idea was to combine gradient information with information obtained when the linking process is applied to all candidates, and to obtain the hysteresis thresholds from the previous fused information. however, the use of type-2 fuzzy sets to handle uncertainties that automatically select the threshold values, is presented in [29]. in [30], unsupervised determination of threshold values for the canny edge detector, based on the bi-dimensional maximum conditional entropy, is presented. on the other hand, determination of adjustable high and low threshold values of the canny edge detector for the gradient magnitude image is shown in [31]. the parameters are determined based on maximum cross-entropy between inter-classes and bayesian judgment theory, without any manual operation. genetic algorithm (ga) is a widely used method for different applications in the fields of gaming, real time systems, job shop scheduling, etc. [32, 33]. in the field of image processing, ga is used for image enhancement and segmentation [34, 35], different types of detection from images, e.g. geometric shape [36], medical [37], sar images [38], etc., as an optimization tool, in order to increase accuracy, quality and speed of detection, as well as for feature selection. in order to achieve desired color image enhancement, a genetic algorithm approach is used in [39]. the fitness function is formed and utilized for determination of the optimal set of generalized value. experimental results showed that the enhanced color images by the genetic algorithm approach are better than those obtained by any of the three existing approaches for comparison. also, for solving problem with image contrast enhancement, the method based on genetic algorithm is presented in [40], where a simple and novel chromosome representation together with corresponding operators, is used. based on experimental results, the proposed method gave good results in natural looking images, especially when the dynamic range of input image is high. however, genetic algorithm is used for other purposes in image processing, such as segmentation and feature selection. for developing of machine vision-based raisin detection technology for various lighting conditions, supervised color image segmentation using a permutation-coded genetic algorithm is implemented [41]. this segmentation identifies regions in hue–saturation–intensity (hsi) color space (gahsi) for desired and undesired raisin detection in various lighting conditions. in [42], the method based on genetic algorithm for evolving adaptive procedures for the contour-based segmentation of anatomical structures in 3d medical data sets is presented. the role of genetic algorithm was to evolve detector of 2d contours of an anatomical structure in order to obtain full segmentation of the structure. for purpose of selection of a set of features to discriminate the targets from the natural clutter false alarms in sar images, a genetic algorithm is used in [43]. this algorithm is driven by a new fitness function based on minimum description length principle (mdlp), and results showed that the proposed genetic algorithm selected a good subset of features to discriminate the targets from clutters effectively. utilization of a genetic algorithm for image feature selection, as a part of classifier, in the task of differentiating regions of interest on mammograms as either mass or normal tissue, is presented in [44]. compared to stepwise feature selection, ga-based feature selection gave better results. this leads to indication that genetic algorithm can provide many possibilities for linear or nonlinear classifiers. 336 m. pavlović, v. nikolić, m. simonović, v. mitrović, i. ćirić in this paper, the genetic algorithm application for optimization threshold values for the canny edge detection is presented. first, calculation of the rail track detection error with additional condition is performed. for minimization of error, genetic algorithm is used, in order to determine optimal high threshold value. however, detection results were not satisfactory, so determination of low and high threshold values is done in order to get more accurate detection. the developed method is tested on set of ir images captured under night-time conditions with the aim to detect edges of rail tracks. 2. theoretical background 2.1. canny edge detection the canny edge detector includes a list of criteria for successful edge detection: good detection, good localization and a single response. good detection means that edge detection should be with a low error rate, i.e. minimum number of false edges; good localization means that the points found by the detector should be as close as possible to the center of the true edge, while a single response provides that the detector must give only one response to a single edge and where possible, image noise should not create false edges [16, 45, 46]. the canny edge detector uses a multi-stage algorithm with four steps: image smoothing, calculation of value and direction of gradients, non-maxima suppression and checking and connecting edges [1, 20, 22, 47]. 2.1.1. image smoothing smoothing of the image is done by gaussian smoothing filter in order to remove the noise. this filter is linearly separable, and it can be divided into two parts. convolution with this filter can be done in order to smooth image according to the row and column respectively. the mathematical expression of gaussian smoothing filter is [22]: 2 2 2 2 1 ( , ) exp 2 2 x y g x y          (1) where σ is the standard deviation of gaussian smoothing filter, and it controls the degree of smooth. in the case that value of σ is small, it will be good localization and lower snr (signal-to-noise ratio), but if the value of σ is big, location accuracy will be lower and less noise. after applying gaussian smooth filter, the image will be [22]: ( , ) ( , ) ( , )i x y g x y f x y  (2) where f(x,y) is original image, and i(x,y) is image after filtering. 2.1.2. calculation of value and direction of gradients this step gives two results, the gradient in the x direction and the gradient in the y direction and it shows changes in intensity on image that indicates the presence of edges. for calculation of horizontal direction derivative px and vertical direction derivative py, the algorithm adopts first order limited difference of 22 neighboring area. the mathematical expression of px and py is, as follows [22]: edge detection parameter optimization based on genetic algorithm for rail track detection 337 ( 1, ) ( , ) ( 1, 1) ( , 1) ( , ) 2 x i i j i i j i i j i i j p i j         (3) ( , 1) ( , ) ( 1, 1) ( 1, ) ( , ) 2 y i i j i i j i i j i i j p i j         (4) the value of gradient m(i, j) can be determined as [22]: 2 2( , ) ( , ) ( , )x ym i j p i j p i j  (5) the direction of gradient is [22]: ( , ) ( , ) arctan ( , ) y x p i j i j p i j   (6) 2.1.3. calculation of value and direction of gradients gradient image cannot ensure edges of an image, and criteria where one accurate response to the edge should be satisfied. the algorithm compares the value of the gradient of current pixel and pixels in neighboring, in either the positive or the negative direction perpendicular to the gradient. if the value of current pixel is not greater than both, it suppresses it; otherwise, it preserved it. 2.1.4. checking and connecting edges after applying of non-maxima suppression, false edges caused by noise and color variation should be reduced as much as possible. in this step, filtering out of edge pixels with weak value of the gradient and preserve edge pixels with a high value of the gradient. in this step, two thresholds, low threshold tl and high threshold th, are determined and set by experience. if the value of the gradient of pixel (i, j) is bigger than th then this point is set as an edge pixel, and edge map t1(i, j) is got. if the value of the gradient of pixel (i, j) is smaller than tl, then this point is never set as an edge pixel. if the value of the gradient of pixel (i, j) is bigger than tl and smaller than th, edge map t2(i, j) is got. if pixel from edge map t2(i, j) is found at location in 8 neighborhood of an edge pixel, then it will be connected to that pixel. 2.2. genetic algorithm genetic algorithm (ga) is inspired by the process of natural selection, i.e. evolution, often used as an optimization tool, although the range of problems to which ga have been applied is quite wide. an implementation of a ga starts with a randomly generated population of chromosomes and it represents an iterative process, with the population in each iteration called a generation. in each generation, the fitness of every chromosome is evaluated, where fitness is value of objective function in solving of the optimization problem. fit chromosomes are selected and its genome is modified in order to form a new generation. a formed new generation is used in the next iteration of the algorithm. the algorithm can terminate when the maximum number of generations are produced or satisfactory fitness level is reached for population [48]. in fact, the genetic algorithm has following steps [34, 49]: 338 m. pavlović, v. nikolić, m. simonović, v. mitrović, i. ćirić 1. starting with a randomly generated population of n chromosomes, where n is the size of population, l is length of chromosome x. 2. calculation of fitness ƒ(x) of each chromosome x in the population. 3. repeating of the following steps until n offspring has been created: a. selecting of a pair of parent chromosomes from the current population, where the probability of selection being an increasing function of fitness. process of selection is done "with replacement", so the same chromosome can be selected more than once to become a parent. b. with probability pc (the "crossover probability" or "crossover rate"), crossing over the pair at a randomly chosen point (chosen with uniform probability) to form two offspring. if no crossover takes place, form two offspring that are exact copies of their respective parents. c. mutation of the two offspring at each locus with probability pm (the mutation probability or mutation rate), and placing of the resulting chromosomes in the new population. in case that n is odd, one new population member can be discarded at random. 4. replace the current population with the new formed population. 5. go to step 2. each iteration of this process is called a generation, and entire set of generations is called a run. at the end of a run, there are often one or more highly fit chromosomes in the population. 3. determination of optimal threshold value determination of the threshold, as the canny edge detection parameter, is important in order to get high quality and useful edges on the image. however, that parameter can be different for each image, so the traditional canny edge detector cannot set threshold adaptively, which makes the algorithm robustness weak. manual determination of optimal threshold is difficult, especially for infrared (ir) images because of their usually low quality, caused by performance of infrared camera. in addition, great effects on its quality have conditions for capturing, for example, in low-light and night-time conditions. for determination of the optimal threshold value minimization of error in rail track detection process is required, as one of the most important steps. the detection error is defined as follow ratio wrong detected pixels (wd) and right detected pixels (rd): d d w error r  (7) however, in order to prevent the occurrence where both parameters have low values, and thus the error is low, additional condition is the number of total detected pixels. in the case of a very low value of total detected pixels, the value of error is low, but the obtained image is not useful. the minimal value of total detected pixels depends on image, as well as quality of the detected edges. further on, genetic algorithm is used in order to find the minimum of the fitness function, i.e. for which value of thresholds, the error will have minimal value. in the first case, an optimal value of high threshold was determined, while in the second case, optimal values of low and high thresholds were determined. edge detection parameter optimization based on genetic algorithm for rail track detection 339 4. results and discussion in order to determine the optimal high threshold value of for the canny edge detector through minimization of error, genetic algorithm is used. the fitness function is obtained by fitting data for the dependence of the error on the threshold, as fourth-order polynomial (eq. 8). the goal of genetic algorithm was to find the minimum of the fitness function, i.e. the threshold value for which the error will have minimal value. the general parameters of genetic algorithm are shown in table 1. best and mean fitness with 25 iterations performed are shown in fig. 1, and average distance between individuals is shown in fig. 2. the algorithm needs about 3 seconds to converge after 25 iterations. 3 4 3 3 3 2 3 30.7714 10 1.1422 10 0.3584 10 0.1071 10 0.0480 10y x x x x            (8) table 1 parameters of genetic algorithm for determination of high threshold value parameter value population size 20 generations 50 function tolerance 1e-6 stall generations 20 fig. 1 best and mean fitness – high threshold value fig. 2 average distance between individuals – high threshold value 340 m. pavlović, v. nikolić, m. simonović, v. mitrović, i. ćirić the developed method based on genetic algorithm is tested on a set of different ir images, captured under night-time conditions. one of the tested scenarios is shown in fig. 3 (left). in this case, the task was to perform edge detection only for rail tracks. determined optimal value of high threshold is 0.4867, and the value of low threshold is calculated based on the otsu method [19]. minimal error, defined in the above manner, is 58.5697. after applying low and high threshold values, the edge detection for rail tracks is successfully done (shown with purple color in fig. 3 (right)). fig. 3 one of the tested scenarios (left), detected rail tracks with determination of high threshold value (right) however, quality of the rail track edge detection was not satisfactory. further on, genetic algorithm is used for determination of low and high threshold values of the canny edge detector. the fitness function is obtained by fitting data for the dependence of the error on the threshold, as five-order polynomial with two variables (eq. 9), x is low, y is high threshold value. in this case, the goal of genetic algorithm was to find the minimum of the fitness function, i.e. the values of low and high threshold for which the error will have minimal value. the general parameters of genetic algorithm for this case were the same as in the case with only high threshold (table 1). best and mean fitness with 22 iterations performed are shown in fig. 4, and average distance between individuals is shown in fig. 5. the algorithm needs about 40 seconds to converge after 22 iterations. 2 2 3 2 2 3 4 3 2 2 3 4 5 4 3 2 2 3 4 5 46.42 645.3 505.6 4286 1167 3038 8943 4028 5186 4607 6052 9582 1447 0.0001114 128.4 596.5 6004 448 480.1 6713 2193 z x y x x y y x x y x y y x x y x y x y y x x y x y x y x y y                                                    (9) edge detection parameter optimization based on genetic algorithm for rail track detection 341 fig. 4 best and mean fitness low and high threshold values fig. 5 average distance between individuals low and high threshold values determined optimal threshold values are 0.000142814916122408, as low threshold value, and 0.413898286851236, as high threshold value. after applying determined values, the edge detection for rail tracks is done (shown with purple color in fig. 6 (right)), with minimal error of 50.4768. compared to the case with determination of only a high threshold value, the detection error is smaller and quality of detection is better and useful for further image processing. although algorithm in the second case needs more time for convergence, it also depends on used hardware. variable quality of ir images and great variety of image intensity limit the edge detector, so not all edges of rail tracks are detected, but results are satisfactory. fig. 6 one of the tested scenarios (left), detected rail tracks with determination of low and high threshold values (right) 342 m. pavlović, v. nikolić, m. simonović, v. mitrović, i. ćirić 5. conclusions edge represents a significant local change in image intensity. in the edge detection process, important information about certain regions on the image through its boundaries, is provided. existence of edge can be caused by variations of different parameters, for example, illumination, color, shade, etc. on the other hand, that variation can be result of the existence of some object. however, one of the influential factors for successful edge detection is the determination of the optimal value of threshold, as a parameter of edge detector. that parameter can be different for each image, which further slows and complicates edge detection process. in addition, manual determination and setting parameter make edge detectors robustness weak. in this paper, optimization of threshold values for the canny edge detector is presented. the rail track the detection error with additional condition is defined. the optimal value of high threshold is determined using of genetic algorithm, based on the minimal value of error. however, quality of the rail track edge detection was not satisfactory, so genetic algorithm is used for determination of low and high threshold optimal values of the canny edge detector. testing of the developed method is done on set of ir images, captured under night-time conditions. results showed that, in the case of determination of optimal values of both thresholds, the detection error is smaller, quality of detection is better, and it can be used for further image processing. however, ir images have great variety of image intensity and quality that can additionally complicate edge detection. 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and technology, babol, iran 2department of industrial engineering, school of engineering, damghan university, damghan, iran abstract. the existence of a comprehensive and complete model, along with accurate and reliable data, helps to evaluate the performance of the supply chain. given the different layers and various performances in designing the supply chain, a method that can analyze and evaluate such network structure is required. moreover, data and conditions’ uncertainty highlight the need for a method that can also include uncertainty in evaluation. in this paper, designing a multi-period network is carried out with rough data to embed in various layers and levels of supply chain. the supply chain performance evaluation is performed using rough network data envelopment analysis. rough network data envelopment analysis (rndea) is a proper method since it analyzes all the current factors in evaluation; besides, it provides efficiency scores for inefficient decisionmaking units and boundary forecasting for these units on an efficient border. the study’s outcomes reveal the efficiency of different factors in the designed network. on the other hand, unlike common data envelopment analysis that indicates the maximum of a factor efficiency, the efficiency priority is calculated in the proposed rough network model, and divisional efficiency also is determined in each step. key words: network data envelopment analysis, rough numbers, multi-period, performance evaluation 1. introduction performance evaluation is essential because it acquaints us regarding the amounts of activities’ favorability and quality, creates a positive sign in societies, and improves the received: november 15, 2023 / accepted may 20, 2024 corresponding author: hamed fazlollahtabar department of industrial engineering, school of engineering damghan university, university street, university square, damghan, iran, postal code: 36716-41167 e-mail: h.fazl@du.ac.ir 2 s. ebadi, a. tajdin, i. mahdavi, h. fazollahtabar services’ quality, particularly in dynamic environments. recognizing various attitudes regarding performance evaluation indicate that evaluation discipline must be proportional to the growth and development of organizations and can respond to their various and numerous dimensions. on the other hand, if control and evaluation in a system are not carried out, communication with the internal and external environment of the organization is not established leading to make it impossible to performing the required reforms for the growth, development, and improvement of the organization’s activities. in two recent decades, there has been growing interest in evaluating the local governments’ performance worldwide [1]. implementation and performing a performance evaluation system is one factor that ensures organizations’ growth and development. according to global competition, organizations continuously improve their quality processes to stay in competition. companies concluded that competition enhancement supports them in detecting performance enhancement and provides cheaper products than efficient production competitors [2]. the competitiveness of organizations depends on their performance levels. for such cases, it is crucial to have a measurement and evaluation system that provides sets of indicators and reliable information for reflecting their goals and performance evaluation. in preparing and implementing organizational policies, performance evaluation is a key section. as a principle, each organization must evaluate its performance capacities as far as possible. several factors influence the countries’ growth and development. based on the performed studies in this regard, efficiency lead to the enhancement of economic development speed. hence, enhancing organizations’ performance and efficiency is an inevitable necessity for survival in today’s global market. the mentioned issue is not exclusive to a particular sector or industry and includes all economic sectors within a limited period [3]. the performance measurement system is critical in business management since it provides the required information for decision-making proceedings. therefore, measuring the proper cases at the appropriate time is critical in the supply chain. nevertheless, companies often fail in maximizing the limpid strategies benefits due to their failure to develop required performance measurement indicators for inefficiencies evaluation and efficiency improvement [4]. 2. literature review the customers’ requirements have become more differentiated and diverse, along with the intensification of competitiveness in the market more than ever. in this regard, companies seeking the main market competition must convert the unit operation into a status in which suppliers, customers, and partners can cooperate in various aspects. accordingly, the necessity of supply chain management is gradually enhanced. in these market spaces, competition between companies converts to competition among supply chains gradually [5]. nowadays, competition among supply chains has replaced competition among individual companies. hence, assessing the supply chain efficiency is one of the critical and fundamental issues since the management of each process is impossible without its efficiency assay. one of the most interesting fields in data envelopment analysis (dea) is the analysis of the efficiency of the network and multi-stage systems [6]. the supply chain performance issues are evaluated using dea. in common dea models (to be used for evaluating the rough multiperiod network data envelopment analysis for evaluation of supply chain… 3 efficiency of decision-maker units), there are several inputs and outputs. ignoring the intermediate products and link activities is one of the problems of these models [7]. there is extensive literature for assessing and evaluating the performance or efficiency of various systems using dea. a summary of performed researches for evaluating the supply chain performance is provided in table 1. table 1 summary of evaluation methods for the supply chain performance year important variables research methods title reference 2012 fuzzy number, non radial model, performance evaluation, intermediate production ndea evaluating performance supply chain by a new nonradial network dea model with fuzzy data [8] 2014 economic performance, environmental performance, dual-role factors, undesirable outputs, fuzzy set ndea a novel network data envelopment analysis model for evaluating green supply chain management [9] 2014 financial factors, performance indicators, intellectual capital, responsiveness, cooperation factor analysis a network data envelopment analysis model for supply chain performance evaluation: real case of iranian pharmaceutical industry [10] 2018 performance, humanitarian disasters ndea developing a new chance constrained ndea model to measure the performance of humanitarian supply chains [11] 2018 performance, stochastic data ndea developing a new chance constrained ndea model to measure performance of sustainable supply chains [12] 2019 performance, process, consulting, educational services ndea multi-period network data envelopment analysis to measure efficiency of a real business [13] 2020 overall weights, efficiency, intuitionistic fuzzy, ranking intuitionistic fuzzy anp -ndea efficiency measurement for hierarchical network systems using network dea and intuitionistic fuzzy anp [14] there are several applications of dea in evaluating the efficiency of training units [15]. dea is a strong performance evaluation method when the decision-making units (dmus) are homogeneous [16]. this method can solve problems with different inputs and outputs 4 s. ebadi, a. tajdin, i. mahdavi, h. fazollahtabar without pre-defined weights; hence, it is used more than other performance evaluation methods [17]. this method has been used in various fields such as transportation, supply chain, production [18], training, insurance [19], financial affairs [20], and energy systems [21, 22]. estimating the system’s general efficiency based on initial inputs and final outputs and bypassing the role of the internal structure of production processes are problems of the traditional dea method. some production processes produce intermediate products that are used as input for the next step of the final output production [23]. chen et al. [24] performed a study and recommended a novel fuzzy bwm-dea approach for selecting the smart product service module. providing the applicable model for evaluating the performance of supply chain networks was the purpose in [25]. in this work, the rough set theory has been used to construct a model for evaluating the performance of the supply chain in uncertain conditions. this model creates a novel point of view for utilizing a performance evaluation model to support managerial decisionmaking in a dynamic environment and uncertain conditions. 3. problem definition and modelling in dea, various methods are used to assess and evaluate efficiency and performance [26]. in multi-stage supply chains, outputs of a process stage are used as inputs of subsequent stages and have intermediate status. therefore, the structure of the network data envelopment analysis (ndea) model is evolved. several researchers extensively applied the series-dea model [2]. the considered supply chain system is in series. in most situations that dmus have a two-step structure, the outputs of the first step are used as input for the next step. the existing dea multi-step models in literature can be classified into two categories: closed system models and open system models [27]. in closed system dea models, the intermediate outputs have remained without change from one step to another step. contrary, in open dea system models, intermediate outputs of a step are partial inputs of the next step [28]. various models of dea have been extended in recent years to use in distinct fields. selection of models or appropriate models is one of the crucial steps before evaluating the studied units. thereby, evaluating the efficiency of these types of network structures is critical. the system’s general efficiency is decomposed into two steps in the structure of a two-step network. kao reformed the dea standard model considering the multi-step series relationship in general processes [29]. in the two-step structure problem, j𝜖j={1, …, n} is the index set of the n dmus; j0𝜖j denotes the evaluated dmu; xij is the vector of stage -1 external inputs i used by dmu j; zpj is the vector of intermediate measures p for dmu j; yrj is the vector of stage-2 final outputs r produced by dmu j; v=(v1, …, vm) is the vector of weights of the stage-1 external inputs; w=(w1, …, wq) is the vector of weights for the intermediate measures; u=(u1, …, us) is the vector of weights of the stage-2 0utputs. suppose a case where each dmu converts the x external inputs to y final outputs through z intermediate indicators with a two-step process. nothing except first step inputs are entered in these initial settings, and nothing except second step outputs are not exited from the system. e1(vrs) j is the efficiency of the first stage for dmu j. commonly, the efficiency of dmuj in the first step is defined as follows: rough multiperiod network data envelopment analysis for evaluation of supply chain… 5 𝐸𝑗 1(𝑣𝑟𝑠) = ∑ 𝑤𝑝 𝑧𝑝𝑗 𝑞 𝑝=1 ∑ 𝑣𝑖𝑥𝑖𝑗 𝑚 𝑖=1 (1) the efficiency of dmu j in the second step is defined as follows: 𝐸𝑗 2(𝑣𝑟𝑠) = ∑ 𝑢𝑟𝑦𝑟𝑗 𝑠 𝑟=1 ∑ 𝑤𝑝𝑧𝑝𝑗 𝑞 𝑝=1 (2) the overall efficiency of dmu j is defined as follows: 𝐸𝑗 (𝑣𝑟𝑠) = ∑ 𝑢𝑟𝑦𝑟𝑗 𝑠 𝑟=1 ∑ 𝑣𝑖𝑥𝑖𝑗 𝑚 𝑖=1 (3) after simplification, we have model i as follows: 𝑚𝑎𝑥 𝐸𝑘 (𝑣𝑟𝑠) = ∑ 𝑢𝑟𝑦𝑟𝑘 𝑠 𝑟=1 (4) s.t. ∑ 𝑣𝑖𝑥𝑖𝑘 = 1𝑚 𝑖=1 (5) ∑ 𝑤𝑝𝑧𝑝𝑗 − ∑ 𝑣𝑖𝑥𝑖𝑗 ≤ 0, 𝑗 = 1,… , 𝑛𝑚 𝑖=1 𝑞 𝑝=1 (6) ∑ 𝑢𝑟𝑦𝑟𝑗 − ∑ 𝑤𝑝𝑧𝑝𝑗 ≤ 0, 𝑗 = 1,… , 𝑛 𝑞 𝑝=1 𝑠 𝑟=1 (7) 𝑣𝑖 ≥ 0, 𝑖 = 1,… ,𝑚 (8) 𝑤𝑝 ≥ 0, 𝑝 = 1,… , 𝑞 (9) 𝑢𝑟 ≥ 0, 𝑟 = 1,… , 𝑠 (10) since the designed model of the problem is examined in real conditions in different periods, designing the evaluation model as a multi-period is required due to assessing the evaluation output in different periods, and the requirement of revision or improvement is planned for each period exclusively [30]. model ii: periods are added to examine the mathematical model of ndea as multiperiod in model i; therefore, the proposed model having period index t is given below: 𝑚𝑎𝑥 𝐸𝑘 (𝑣𝑟𝑠) = ∑ 𝑢𝑟𝑡𝑦𝑟𝑘𝑡 𝑠 𝑟=1 (11) s.t. ∑ 𝑣𝑖𝑡𝑥𝑖𝑘𝑡 = 1𝑚 𝑖=1 (12) ∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 − ∑ 𝑣𝑖𝑡𝑥𝑖𝑗𝑡 ≤ 0, 𝑗 = 1,… , 𝑛𝑚 𝑖=1 𝑞 𝑝=1 (13) ∑ 𝑢𝑟𝑡𝑦𝑟𝑗𝑡 − ∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 ≤ 0, 𝑗 = 1,… , 𝑛 𝑞 𝑝=1 𝑠 𝑟=1 (14) 𝑣𝑖 ≥ 0, 𝑖 = 1,… ,𝑚 (15) 𝑤𝑝 ≥ 0, 𝑝 = 1,… , 𝑞 (16) 𝑢𝑟 ≥ 0, 𝑟 = 1,… , 𝑠 (17) 6 s. ebadi, a. tajdin, i. mahdavi, h. fazollahtabar 3.1 rough network data envelopment analysis in recent decades, the performance evaluation with rough network data envelopment analysis is attracted scholars worldwide. the outputs and inputs in rough envelopment analysis are interval numbers obtained based on the up and down approximate of the rough set theory. the rough network data envelopment analysis (rndea) process has been used in this research due to subjective and ambiguous skill learners’ judgments and opinions regarding obtained feedback from the training process. there are studies that made use of uncertainty in evaluation processes in supply chain and production scheduling [31]. further, possibility theory using correlation study was applied in mechanical specification analysis [32]. the following steps of the rndea process are presented. according to the rough data envelopment analysis model, the envelopment analysis model with rough multiperiod network data is defined as follows: 𝑀𝑎𝑥 𝑒̃ 𝑘 = ∑ 𝑢𝑟𝑡⌈𝑦𝑟𝑘𝑡 𝐿 , 𝑦𝑟𝑘𝑡 𝑈 ⌋𝑠 𝑟=1 (18) s.t. ∑ 𝑣𝑖𝑡⌈𝑥𝑖𝑘𝑡 𝐿 , 𝑥𝑖𝑘𝑡 𝑈 ⌋𝑚 𝑖=1 = 1 (19) ∑ 𝑤𝑝𝑡⌈𝑧𝑝𝑗𝑡 𝐿 , 𝑧𝑑𝑗 𝑈 ⌋ 𝑞 𝑝=1 ∑ 𝑣𝑖𝑡⌈𝑥𝑖𝑗𝑡 𝐿 , 𝑥𝑖𝑗𝑡 𝑈 ⌋ ≤ 0 , 𝑗 = 1,2, … , 𝑛𝑚 𝑖=1 (20) ∑ 𝑢𝑟𝑡⌈𝑦𝑟𝑗𝑡 𝐿 , 𝑦𝑟𝑗 𝑈 ⌋𝑠 𝑟=1 − ∑ 𝑤𝑝𝑡⌈𝑧𝑝𝑗𝑡 𝐿 , 𝑞 𝑝=1 𝑧𝑝𝑗𝑡 𝑈 ⌋ ≤ 0 , 𝑗 = 1,2, … , 𝑛 (21) 𝑢𝑟𝑡 , 𝑤𝑝𝑡 , 𝑣𝑖𝑡 ≥ 𝜀 > 0 (22) here, yl rjt and yu rjt are the lower and upper limits of network outputs, respectively, obtained from the calculation of the rough number. moreover, zl pjt and zu pjt are the lower and upper limits of the input of the second sub-system. xl ijt and xu ijt are lower and upper limits of inputs of the first sub-system in the network series. urt, wpt, and vit are the weights of the network’s inputs and outputs. in this model, the p index indicated the p-th decision-making unit. furthermore, all weights are considered greater than a very small positive number (𝜀). the method of modeling in pessimistic state is as follows: 𝑀𝑎𝑥 𝑒̃𝑝 𝐿 = ∑ 𝑢𝑟𝑡𝑦𝑟𝑘𝑡 𝐿 𝑟 (23) s.t. ∑ 𝑣𝑖𝑡𝑥𝑖𝑘𝑡 𝑈 𝑖 = 1 (24) ∑ 𝑢𝑟𝑡𝑦𝑟𝑗𝑡 𝐿 𝑟 −∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝑈 𝑝 ≤ 0 (25) ∑ 𝑢𝑟𝑡𝑦𝑟𝑗𝑡 𝑈 −∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝐿 𝑝 ≤ 0, 𝑗 = 1,2, … , 𝑛, 𝑗 ≠ 𝑝𝑟 (26) ∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝐿 − ∑ 𝑣𝑖𝑡𝑥𝑖𝑗𝑡 𝑈 𝑖𝑝 ≤ 0 (27) ∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝑈 𝑝 − ∑ 𝑣𝑖𝑡𝑥𝑖𝑗𝑡 𝐿 𝑖 ≤ 0, 𝑗 = 1,2, … , 𝑛, 𝑗 ≠ 𝑝 (28) 𝑢𝑟 , 𝑤𝑑 , 𝑣𝑖 ≥ 𝜀 > 0 (29) the method of modeling in optimistic state is as follows: 𝑀𝑎𝑥 𝑒̃𝑝 𝑈 = ∑ 𝑢𝑟𝑡𝑦𝑟𝑘𝑡 𝑈 𝑟 (30) rough multiperiod network data envelopment analysis for evaluation of supply chain… 7 s.t. ∑ 𝑣𝑖𝑡𝑥𝑖𝑘𝑡 𝐿 𝑖 = 1 (31) ∑ 𝑢𝑟𝑡𝑦𝑟𝑗𝑡 𝑈 𝑟 −∑ 𝑤𝑝𝑡𝑍𝑧𝑝𝑗𝑡 𝐿 𝑝 ≤ 0 (32) ∑ 𝑢𝑟𝑡𝑦𝑟𝑗𝑡 𝐿 −∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝑈 𝑝 ≤ 0, 𝑗 = 1,2, … , 𝑛, 𝑗 ≠ 𝑝𝑟 (33) ∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝑈 − ∑ 𝑣𝑖𝑡𝑥𝑖𝑗𝑡 𝐿 𝑖𝑝 ≤ 0 (34) ∑ 𝑤𝑝𝑡𝑧𝑝𝑗𝑡 𝐿 𝑝 − ∑ 𝑣𝑖𝑡𝑥𝑖𝑗𝑡 𝑈 𝑖 ≤ 0, 𝑗 = 1,2, … , 𝑛, 𝑗 ≠ 𝑝 (35) 𝑢𝑟 , 𝑤𝑑 , 𝑣𝑖 ≥ 𝜀 > 0 (36) 4. numerical implementation the employment rate is critical among the policymakers in various countries, given the population growth, the unskillfulness of many people (even academic graduates), and eventually the unemployment of mass ranges of society. by reviewing the researches conducted in the field of designing education supply chains, it was observed that the supply chain for higher education and post-graduate education had been established, but so far there was no supply chain for skill training in the literature. it is due to the more or less encountering of most countries (from industrial societies to third world developing countries) with unemployment problems in different ways. skill training can be considered the main priority of the educational system as an essential tool for cultivating and enhancing the efficiency and productivity of human forces to reform and solve unemployment and economic development. there are two steps in the supply chain of skill training. generally, it can be mentioned that the outputs of a process step are considered the inputs of the next step, and it has intermediate status. therefore, the structure of the present study is a two-step network. only three main levels are considered to evaluate this chain as follows: first level (suppliers), second level (training process), and third level (consumers). the components of the first level: higher education graduates, educational and educational aids equipment, supplies equipment, training workshop, training course, the cost of center overhead, training support force, training content, training portal, welfare; the components of the second level: consultation and guidance of career, training scheduling, designing of training model, training forms, and theoretical and operational evaluation; the components of the third level): results of employment, final products. this research was carried out for five popular standards in the industry cluster and skill training field. for this purpose, five influential provinces in iran with higher amount of population, including alborz, east azerbaijan, isfahan, razavi khorasan, and tehran are investigated. initially, the collected data in five provinces were converted to rough data using related input data to factors’ performance in different layers in the first period. the summary is given in tables 2, 3, and 4. 8 s. ebadi, a. tajdin, i. mahdavi, h. fazollahtabar table 2 rough output related to the first layer dmu x1 x2 x3 x4 l u l u l u l u motoring 40.02 42.56 147.16 152 334.06 343.85 31.97 35.43 electricity 22.5 25.5 246.14 251.25 278.75 286.25 22.04 25.95 turning 16.43 18.97 156.14 165.93 237.5 252.5 16.02 18.56 welding 26.70 29.12 123.75 131.25 157.16 162 24.62 28.02 cartography 8.75 10.25 56.77 60.81 31.70 35.29 10.22 12.18 dmu x5 x6 x7 x8 l u l u l u l u motoring 34.5 37.5 133.75 138.64 207.5 222.5 3.75 4.72 electricity 22.04 25.95 89.56 93.02 128.62 135.20 3.75 4.72 turning 18.29 20.70 59.85 63.56 133.75 141.25 5.27 6.25 welding 33.38 28.56 93.27 97.97 137.72 140.64 3.75 5.25 cartography 9.29 11.70 41.43 43.97 64.41 67.75 4.75 6.25 dmu x9 x10 l u l u motoring 31.62 36.70 31.62 36.70 electricity 21.29 23.70 21.97 25.43 turning 16.16 18.83 17.29 19.70 welding 21.89 26.85 23.52 27.89 cartography 11.29 13.70 10.08333 13.08 uncertainty of data and conditions determines the need for a method that can apply uncertainty in evaluation. as a result, the design of a multi-period network is done with raw data to embed suitable factors in different layers and levels. evaluation of supply chain performance is done through the rough network data envelopment analysis method. rough network data envelopment analysis (rndea) is a suitable method because it analyzes all factors in the evaluation. considering that employment is one of the important and discussed issues of policy makers in different countries of the world and most of the countries are more or less facing the issue of unemployment. on the other hand, due to the fact that there is a deep and inferable link between education and employment, and education is considered an important factor in economic growth and employment, addressing the issue of education, especially skill training, which has not been addressed so far, is an inevitable necessity. this is why one of the important priorities in educational investments is the development and expansion of skill-based education. rough multiperiod network data envelopment analysis for evaluation of supply chain… 9 table 3 rough output related to the second layer dmu z1 z2 z3 l u l u l u motoring 979.58 994.58 4.58 5.41 986.77 1001.14 electricity 338 342.83 5.27 6.25 338.31 345.43 turning 258.5 264.16 5.12 5.87 262.85 267.68 welding 455.43 474.14 9.70 13.29 465.85 475.39 cartography 172.18 184.89 3.75 5.25 179.06 188.85 dmu z4 z5 l u l u motoring 33 37.83 986.14 995.93 electricity 21.29 23.70 334.04 341.79 turning 14.56 18.02 257.64 266.93 welding 24.70 28.29 468.45 475.70 cartography 11.29 13.70 184.06 193.85 table 4 rough output related to the third layer dmu y1 y2 y3 y4 l u l u l u l u motoring 83 87.83 104.18 108.22 64.06 73.85 168.60 175.97 electricity 27.43 32.14 40.12 44.87 22.72 27.68 53.60 60.97 turning 22.66 13.83 25.60 31.81 17.60 21.64 42.04 47.12 welding 37.18 43.29 53.37 57.62 33.77 37.81 83 87.83 cartography 22.39 27.02 29.97 33.43 17.12 20.70 42.45 47.54 by providing informal skill training along with high formal training, a very effective step can be taken in the development of human resource employment. therefore, due to increasing the efficiency and effectiveness of training in skill-oriented organizations, checking the progress towards the set educational goals, correct formulation of educational policies, identifying the components that need more attention and obtaining information in order to make the right decisions, it is necessary to enforce the training performance evaluation. having a chain of training for this type of training can help a lot in training a capable and trained workforce ready to work in the society. designing a performance evaluation model for skill training that leads to sustainable political, economic, social, cultural and environmental development is an undeniable necessity. considering the special and important role of skill training in increasing people's skills and employment, designing and evaluating the supply chain of these trainings is considered very important. the need to evaluate the performance of the education supply chain is important because it increases 10 s. ebadi, a. tajdin, i. mahdavi, h. fazollahtabar the efficiency and effectiveness of education. the findings of the research show the efficiency of various factors in the stages of the designed network. in addition, it provides efficiency scores for inefficient decision-making units and boundary prediction for these units in an efficient boundary. then, performance evaluation is carried out using input data and the proposed rndea model. the outputs of each step of efficiency evaluation in the rough form in the first period are indicated in table 5. table 5 output for period 1 rank rough dmu no 5 [0.285,0.571] motoring 1 1 [0.639,1] electricity 2 3 [0.450,0.767] turning 3 4 [0.346,0.648] welding 4 2 [0.633,0.855] cartography 5 notably, the mentioned process is repeated; performance evaluation is performed in different periods, and an improvement solution is provided. the results of this study show the effectiveness of different factors in the stages of network design. by designing the performance evaluation model of the skill training supply chain network, we tried to identify the best performance for the supply chain. 5. conclusion based on the outputs table, the best efficiency is related to the electrical field due to having the maximum efficiency in the first, second, and third layers. besides, the lowest efficiency is related to motoring. thereby: electricity > cartography > turning > welding > motoring; it can be described that these results were obtained given the supply chain components of skill training that are used in this research and declared in the numerical example section. moreover, the following reasons can be pointed out for the low efficiency of debated fields: lack of updated proper workshop equipment based on the requirement of the labor market, no existence of sufficient consumer goods due to the budget deficit, not the proportion of knowledge and experience of instructors with skill artisan and updated changes of technology, lack of insurance of employed skill learners in the repair shop, also skill learners that established their considered business, and lack of providing accurate information in employment tracking to the staff of career monitoring. another point is that most skill courses in electrical, building cartography, order-oriented turning and welding, and proportion to work market requirements. in contrast, the most presented periods in the motoring field are supply-oriented. thereby, most of the skilled learners participate in four required fields in 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https://www.sciencedirect.com/science/article/abs/pii/s0377221722002235#! https://www.sciencedirect.com/science/article/abs/pii/s0377221722002235#! https://www.sciencedirect.com/journal/european-journal-of-operational-research facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 39 45 https://doi.org/10.22190/fume190112007p © 2019 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper generalized archard law of wear based on rabinowicz criterion of wear particle formation valentin popov technische universität berlin, berlin, germany institute of strength physics and materials science, russian academy of sciences abstract. according to the archard law of adhesive wear, the wear volume is proportional to the normal force, the sliding distance, and inversely proportional to the hardness of the softer of contact partners. this law does not contain any properties characterizing “adhesion” of materials, e.g. the work of separation, either inside of the material or at the interface. the criterion for formation of wear particles, first formulated by rabinowicz in 1958, on the contrary, is based on the interplay of elastic energy and work of adhesion and contains as governing parameters the modulus of elasticity, hardness and the work of separation. following recent advances in understanding and simulation of wear, we discuss the ways how both laws could be melted together to a “generalized” archard-rabinowicz law of wear. key words: adhesive wear, archard, rabinowicz, particle formation, roughness 1. introduction in 1953, archard published an article with the modest title “contact and rubbing of flat surfaces” [1], in which he analyzed the contact of rough surfaces and on this basis, among other things, formulated the law of adhesive wear. this law is very simple and states that the volume v of worn material is proportional to the normal force, f, the sliding distance, s, and inversely proportional to the hardness of the material, 0: adh 0 fs v k  , (1) the constant kadh is the so-called adhesive wear coefficient. received january 12, 2018 / accepted march 01, 2018 corresponding author: valentin l. popov technische universität berlin, sekr. c8-4, straße des 17. juni 135, 10623 berlin, germany e-mail: v.popov@tu-berlin.de 40 v.l. popov as the law of amonton-coulomb for the force of friction, this is of course only a very rough empirical estimation. it is used primarily because of its simplicity and due to the absence of a generally accepted alternative. there exists a widespread opinion that the higher the hardness, the smaller is the wear, since the hardness is in the denominator of the archard law. this would be true if it were not for the coefficient of adhesive wear, which stands in the archard law as a multiplier. in reality, this coefficient can take values that differ by at least seven orders of magnitude, which deprives the law of any predictive power. indeed, kragelsky [2] formulated the exact opposite condition of low wear the principle of a positive gradient of hardness, which states that the surface layers should be softer than the deeper ones, otherwise catastrophic wear will occur. in the problem of wear, not only the total worn volume is of interest, but also the distribution of wear particles by size. wear debris formation and transportation is, after all, the physical mechanism of all types of “mesoscopic” wear (contrary to chemical, atom-by-atom wear). archard's law says nothing about the particle size. the first researcher who investigated this topic was ernest rabinowicz. in 1958, he wrote a five page-article in wear, in which he put forward the hypothesis of a mechanism determining the size of wear particles [3, 4]. he considered two micro-heterogeneities which collide and form a bridge, as suggested by bowden and tabor [5]. due to the plastic properties of the material, the maximum stress that can be achieved in this case is on the order of magnitude of the material hardness, 0. elastic energy stored in the material is proportional to the third degree of the size d of the contact: (0 2 /2g)d 3 , where g is shear modulus of the medium. this energy can relax by dislodging wear particles, but only if the stored elastic energy is larger than the energy needed to create new free surfaces, ~ wd 2 , where w is the specific work of separation: 2 3 20 2 d wd g   . (2) from this energetic criterion, it follows that only particles with a size larger than a certain critical size can detach spontaneously: 2 0 2c w d d g    . (3) the details of how plastic deformation and detachment occur during the wear process were not clear at that time. only 58 years later, in 2016, aghababaei, warner and molinari published an article in nature communications [6], which returns to the idea of rabinowicz, but at the level of physical mechanisms. aghababaei and colleagues did a very simple thing: they implemented rabinowicz' thought experiment in a mesomolecular model by generating two bodies with micro asperities. they forced the asperities to collide and carefully observed the results of their simulations. the medium generated by aghababaei and co-authors did not correspond to any real material, but it contained all the parameters that enter into the rabinowicz' equation: elastic modulus, specific work of separation and hardness. the behavior of asperities at impact did depend on the size of asperities. if the initial size of asperities was small enough, the main process was plastic deformation and gradual smoothing of the roughness. the process proceeded in a completely different way when two large asperities collided. in this case, from the very generalized archard law of wear based on rabinowicz criterion of wear particle formation 41 beginning, the process of cracking and formation of wear particles dominated. the breakthrough work by aghababaei, warner and molinari created a solid concept basis for model-based simulation of wear. the work of aghababaei, warner and molinari confirmed the rabinowicz criterion and lead to appearance of a new paradigm as represented by the recent works [7-9]. in particular, in [8], a generalization of the rabinowicz' criterion to heterogeneous media (e.g. layered materials) and in [9] an "asperity free" generalization of the rabinowicz' criterion have been suggested. this latter generalization is a principal step forward as the asperity size is a very poorly defined quantity for multiscale surface topographies. the approach suggested in [9] allowed the simulation of the development of the surface topography during the wear process. the simulation procedure included numerical determination of the contact regions having enough elastic energy for dislodging a particle of corresponding size with its subsequent "disappearance" from the system. the detachment of a particle changes the topography so that the critical conditions are created at another position. this process leads to an overall change in surface topography. after running the system for a long time, one can look at the distribution of particle sizes or calculate the total amount of wear. simulations carried out in [9] revealed a number of regimes. depending on the normal pressure and initial topography, the authors observed different regimes of wear from settling type to continuous and catastrophic wear. in the region of "mild wear", which is the main interest in the present study, a power-law dependency of the worn volume on the normal load was found which corresponds to experimental findings. indeed, the non-linear dependencies of the worn volume on the normal force in the case of large force interval have been reported as early as in 1970s [10]. an extensive experimental analysis of polymer materials can be found in [11] and a review for very broad material classes in [12]. it is important to note that the deviations from the archard law of wear (proportionality of wear volume to the normal force) may be the key to solving the riddle of the huge variation in the coefficient of adhesive wear. indeed, is it not paradoxical that archard's equation, which describes adhesive wear, does not contain any parameter characterizing adhesion? moreover, from considerations of dimension, it follows that the coefficient of wear cannot depend on the specific work of adhesion, since the available parameters do not allow building a dimensionless combination. the situation would change qualitatively if wear would be nonlinear with force. then the specific work of separation could be included in the equation of wear in a natural way. in the present paper, we analyze the possible power-law dependencies on system parameters. we consider stationary wear and two main modes of wear particle transport corresponding to open and closed tribological systems for both elastic and elastoplastic media. 2. power law wear equations we distinguish between the initial stage of wear, when surfaces with given surface topography are brought into contact and then forced in relative tangential movement and the state of stationary wear after the running-in stage. here, we focus on the stage of stationary wear. the current surface topography determines the contact configuration and 42 v.l. popov the detailed stress distribution at any time moment and thus determines the position and the size of wear particles (e.g. according to rules sketched in [9]). however, the wear process changes the surface topography, so that the properties of the surface topography cannot be considered as given but are also a product of the tribological process. following this line of argumentation, rabinowicz [13] came to the conclusion that the stationary roughness of contacting surfaces is on the same order of magnitude as the critical size (3) of junctions. as a matter of fact, it is not just the “roughness” but the “surface topography” which is determined by the characteristic length, eq. (3). in the paper [14] of the present author, it was argued that formation of the near-surface properties can be considered as a key problem and a great current challenge of tribology. at the present, no recognized concepts exist of how "friction machine" leads to formation of the stationary topography, or more generally, to formation of the third body. we only assume at this point that such stationary state does exist and that all its properties including rms roughness, wavelength cutoffs and others do not represent independent properties but are all functions of material properties. wear is not only the problem of cracking and detachment of wear debris but also of their transport out of the wear zone. in the present paper, we consider two cases: 1. the case of “immediate disappearance” of wear particles. this is the case considered in the majority of theories of wear. physically, it corresponds to the case of an open system in which the wear region regularly is cleaned from debris or wear particles have opportunity to leave the frictional zone. in this case, it is reasonable to assume that the wear is occurring homogeneously in the whole contact area, so that the wear intensity depends on the pressure and the wear volume is proportional to the square l 2 of the size l of the contact region. 2. the case of continuous transport of wear debris towards the boundary of the contact. this is typical condition for a closed tribological system. this diffusion like process can be accompanied be reintegration of the particles into the surface of by transfer of material from one partner to the other. under assumption of the existence of some characteristic size of wear particles, it was shown in [15] that the wear intensity (defined as the volume of the worn material per unit sliding length) at the given force is inversely proportional to the square of the size of the contact region. correspondingly, at a given pressure it does not depend on the size of the contact. (it is proportional to l 2 due to homogeneity of the process and to l -2 due to “diffusion like” transport of wear debris, so that both factors cancel each other). 2.1. immediate disappearance of debris (elastoplastic materials) our main intention is to modify the archard wear equation in such a way that the critical rabinowicz' length (3) comes into play. as explained above, it is necessary that the dependency on the normal force becomes non-linear; we thus assume a power-law dependency with some exponent . we search for a dependency in the form of a product of powers of all relevant governing parameters: 2 0 cv kp d l s   , (4) where p is the apparent pressure in the contact region, 0 is the hardness, dc the rabinowicz' critical length, l the linear size of the system and s the sliding distance. generalized archard law of wear based on rabinowicz criterion of wear particle formation 43 the main assumptions behind this form are the following:  we consider the stationary wear, therefore the wear volume is proportional to the sliding distance, s.  the wear process occurs homogeneously in the whole contact area. therefore, the wear intensity is proportional to l 2 and is determined locally by the pressure, p = f/l 2 .  all geometrical parameters of the contacting surfaces and thus the stationary wear process are determined solely by the rabinowicz' critical length, dc. from dimensional analysis, it follows that  = 0,  = - which leads to the following modified form of archard's wear law (1): 2 2(1 ) 0 0 p f v k l s k l s                   . (5) we see that the assumption of a stationary homogeneous wear intensity is compatible with a non-linear dependency of the wear intensity on the normal force, however, only at the expense of an additional dependency of the wear intensity on the system size. however, the dependence on adhesive properties of material is still absent. 2.2. continuous transport of debris (elastoplastic materials) according to the comments above, we search in this case for the wear law in the form 0 cv p d s   . (6) from the dimensional analysis, it follows: + = 0,  = 2, or explicitly 2 2 2 4 2 2 02 4 0 0 f w v k g s kf l g w s l                . (7) if  = 1, so that the classical archard's law (proportionality to the normal force) is valid, it follows for the adhesive wear coefficient 2 2 adh 2 0 cdgw k k ll             . (8) according to this equation, the wear coefficient is inversely proportional to the size of the system (as suggested in [15]) and is determined by the square of the ratio of the rabinowicz' critical length to the system size. 2.3. immediate disappearance of debris (elastic materials) as illustrated in paper [9], the application of the ideas behind the rabinowicz' criterion is not restricted to materials which can be deformed plastically. if the material is purely elastic and the tangential stresses which appear in the contact are characterized by the coefficient of friction, then the conditions for detaching of wear particles can still be 44 v.l. popov fulfilled. in this case however, the hardness is not available as a governing parameter. similarly to the section 2.1., we assume the following:  we consider the stationary wear, therefore the wear volume is proportional to the sliding distance, s.  the wear process occurs homogeneously in the whole contact area. therefore, the wear intensity is proportional to l 2 and is determined locally by the pressure, p = f/l 2 . 2v p g w l s   . (9) from the dimensional analysis, it follows: + = 0,  = 0. thus, the wear intensity again does not depend on the work of separation: 2 2f v k l s g         . (10) in the simplest case of linear proportionality to the normal force, we will get fs v k g  . (11) let us also discuss what happens if we renounce the claim that the wear intensity is proportional to the area, l 2 , but still assume stationary wear (and thus direct proportionality to s). the general form of the power-law dependency is under these assumptions v f g l w s    . (12) dimensional analysis gives: 2 2     , 2     (13) and 2 2 2v kf l g w s         (14) which finally brings in play also the specific work of separation (compare with [11]). 3. discussion in the present paper we analyzed power-law wear equations under conditions of stationary wear. we find that under the additional assumption of homogeneity of wear in the contact plane, the work of separation does not enter the wear equation. only deviation from this bound (for example due to transport of wear particles) makes the dependency of wear intensity on the specific work of separation possible. with other words, the specific work of separation can only enter the wear equation provided the wear process is characterized by some characteristic length. this can be the characteristic rabinowicz' length or other structural parameter. it is extremely interesting to check the found dependencies both experimentally and using direct numerical simulations as suggested in [9]. generalized archard law of wear based on rabinowicz criterion of wear particle formation 45 acknowledgement: this work has been conducted under partial financial support from the german ministry for research and education bmbf, grant no. 13nke011a and by the russian science foundation (project 17-11-01232). references 1. archard, j. f., 1953, contact and rubbing of flat surfaces, journal of applied physics, 24, pp. 981-988. 2. kragelski, i.v., 1965, friction and wear, butter worth. 3. rabinowicz, e., 1958, the effect of size on the looseness of wear fragments, wear, 2, pp. 4–8. 4. popova, e., popov, v.l., kim, d.-e., 2018, 60 years of rabinowicz’ criterion for adhesive wear, friction, 6(3), pp.341–348. 5. bowden, f.p., tabor, d., 2001, the friction and lubrication of solids, clarendon press. 6. aghababaei, r., warner, d.h., molinari, j.-f., 2016, critical length scale controls adhesive wear mechanisms. nature communications. 7, 11816.rabinowicz, e., 1995, friction and wear of 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popov, v.l., 2018, is tribology approaching its golden age? grand challenges in engineering education and tribological research, frontiers in mechanical engineering, 4, 16. 15. popov, v.l., smolin, i.yu., gervé, a., kehrwald, b., 2000, simulation of wear in combustion engines, computational materials science, 19, pp. 285-291. 12165 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 591 614 https://doi.org/10.22190/fume231004045e © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper finite element analysis for dynamic response of viscoelastic sandwiched structures integrated with aluminum sheets yasser hamed elmoghazy1,2, babak safaei1,2,3, saeid sahmani4 1nanotechnology and multifunctional structures research center (nmsrc), eastern mediterranean university, famagusta, turkey 2department of mechanical engineering, eastern mediterranean university, famagusta, turkey 3department of mechanical engineering science, university of johannesburg, south africa 4school of science and technology, the university of georgia, tbilisi, georgia abstract. passive vibration attenuation of modern mechanical structures is one of the most essential technologies applied to the arsenal of modern mechanical structures. in this work, dynamics analysis is performed on viscoelastic (ve) sandwich beam and plate by using finite element method. the proposed structure is composed of a ve core and aluminum face sheets as substrate layers on both sides of the structure. small-strain ve material is modeled based on complex constant moduli model and numerical method is used to develop the finite element (fe) shear model based on first-order shear deformation theory (fsdt) and hamilton principle. in modal analysis, model-effective mass analysis is performed to investigate its dominant mode shape and sweet spot at resonance using c3d20 and cps8 elements. the former is indicated as a 3-d element with 20 nodes while the latter is indicated as a 2-d plane strain element with eight nodes. in harmonic analysis, resonance frequency is obtained based on mode superposition method to evaluate the steady-state response of ve sandwich beam via maximum deformation. thereafter, the results are compared against analytical solutions from the literature. moreover, a parametric study shows that the natural frequency of the beam did not change with the increase in core thickness. however, normalized loss factor of ve sandwich beam 𝜂𝑛 𝜂𝑣⁄ is directly proportional to ve damping factor 𝜂𝑣 and the thickness of ve core layer. based on frequency response function, the results of resonance frequencies for ve beam are in the range of modal natural frequency. key words: viscoelastic, sandwich structure, dynamics analysis, complex constant moduli, normalized loss factor received: october 04, 2023 / accepted november 26, 2023 corresponding author: babak safaei department of mechanical engineering, eastern mediterranean university, famagusta, north cyprus via mersin 10, turkey department of mechanical engineering science, university of johannesburg, gauteng 2006, south africa e-mail: babak.safaei@emu.edu.tr 592 y. elmoghazy, b. safaei, s. sahmani 1. introduction the key parameters to obtain an optimum design of modern mechanical structures with ideal performance are lightweight and noise control which directly lead to the wide applications of thin-walled structures. these structures can be represented as thin beams and thin plates that are extensively used in many engineering applications such as aerospace [1], automobile industry [2], and civil engineering [3]. since thin-walled structures are "light" and "thin”, it is more likely to augment the induced vibrations under external excitations, which compromises the stability of such structures. as a result, the vibration of thin-walled structures must be controlled. one of the approaches to the problem is to use active piezoelectric elements which may offer an optimal damping performance of thin-walled structures since it includes the ability of sensing and actuation control [4]. applying a damping layer to thin-walled structures to control their vibrations is a widely utilized strategy in engineering practice [5]. among them, viscoelastic (ve) materials, which have good damping qualities, are most widely applied in structural vibration as a passive damping treatment [6, 7]. a typical passive damping treatment structure consists of two elastic layers and a ve core sandwiched between constraining layer and base beam. the ve core allows vibration energy to dissipate through a cyclic deformation, while the main advantage of constraining layer is to increase periodic shear deformation and stress resistance of the core, hence directly leading to an increase in damping energy within ve core [8]. in engineering practice, sandwich structures have complex geometric shapes and boundary conditions thus, such problems cannot be effectively solved by the conventional analytical process, accordingly finite element (fe) approach is commonly used which may facilitate the computational complexity of such problem [9–12]. arvin et al. [13] developed four nodes ve sandwich beam element based on higher-order shear deformation theory to explore the steady-state and transient responses of a ve composite sandwich beam. wang and inman [14] developed a 2d fe model to study the dynamic performance of a hybrid composite sandwich beam by using ve golla–hughes model (ghm). lin and rao [15] proposed fe model for vibration performance of ve multiple-layered sandwich beam by incorporating ve biot model into fe model. reddy and panda [16] developed fe model to explore the nonlinear frequency response for harmonically excited ve sandwich beam based on harmonic balance method considering different ve constitutive models. jahan et al. [17] performed fe modal study to examine the effectiveness and influential parameters of a passive rail damper system using lab-scaled model. araújo et al. [18] proposed fe model to explore the dynamic response of active-passive hybrid sandwich plate using mixed layerwise approach. although fe models have been widely used in the dynamic modeling of complex structures, some fe models are difficult to implement, thus some researchers have tried to ensure accurate modeling based on searching for ways to improve computational efficiency, such that, some models between the analytical method and the fe method, called semi-analytical semi-numerical method. lewandowski and baum [19] developed mixed model to anticipate the free vibration response of ve sandwich beam using virtual work principle and ve zener model. daya and potier-ferry [20] established a shell fe model using homotopy algorithm and asymptotic method to solve the nonlinear eigenvalue problem of ve sandwich structures. koutsawa et al. [21] investigated the non-linear behavior of a laminated glass beam by applying asymptotic numerical technique to establish a diamond toolbox. the performance of the diamond toolbox was demonstrated numerically and experimentally. also, lampoh et al. [22] performed sensitivity studies on the non-linear mechanical behavior of ve laminated glass beam. 593 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets recently, small-size composite structures with ve material have been explored by many researchers. for instance, loghman et al. [23] used finite difference method coupled with galerkin bases to examine the size-dependent vibration behavior of a ve micro-beam employing modified couple stress theory (mcst). li et al. [24] analytically and experimentally investigated the nonlinear vibration response of multiple layered ve composite sandwich plate and the equation of motion derived based on von karman approach. their analytical model was developed based on strain energy principle and validated against experimental nonlinear vibration characteristics. rahmani et al. [25] utilized a comprehensive mathematicalmechanical model based on higher-order, nonlocal eringen’s theory, and mcst theory to explore the wave propagation of rotating ve nanobeams including the influence of many parameters on wave propagation frequencies, such as nonlocal parameter and thermal gradient. yuan et al. [26] studied the dynamic stability of fgm composite microcells integrated with a nonlinear ve foundation under size-dependent gradient. jalaei et al. [27] proposed the transient response of small-size visco-porous fg structure exposed to transverse loads and magnetic field based on eringen’s nonlocal strain gradient theory. youzera et al. [28] proposed an applicable analytical model for the forced vibration analysis and nonlinear associated loss factor of ve-fgm composite beam by using zig-zag theory coupled with hsdt. the governing sixth-order differential equilibrium equation of motion is derived using virtual work principle. karmi et al. [29] established a numerical approach based on modal stability procedure (msp) to explore the vibroacoustic behavior of sandwich beam integrated with frequency and temperature-dependent ve core. hereby, research methodology for dynamic analysis of ve sandwich structures is proposed by using finite element method. in the pre-processing step, the integral fe shear beam element is developed based on first-order shear deformation theory (fsdt) since the ve core layer demonstrates considerable shear deformation behavior under different loading conditions. the beam’s equation of motion is derived based on hamilton principle. the numerical results are compared against analytical models from previous literature. further, the steady-state frequency response is obtained for optimizing the design and ensuring the structural integrity of such system. this work aims to achieve an appropriate design for both ve beam and plate with integrated aluminum sheets by investigating their damping characteristics, including the effects of ve loss factor 𝜂𝑣 and thickness of the ve core layer. 2. shear model kinematics relationships the geometric configurations and deformation of typical ve sandwich beams are demonstrated in fig. 1. here. ℎ1, ℎ2 and h are the thickness of upper layer, lower layer and the total thickness of the ve sandwich beam, respectively, and l is the span of ve sandwich beam. the shear strain of ve layer 𝛾𝑥𝑧 and rotation of ve sandwich beam 𝜃 could be expressed as [30]: 𝛾𝑥𝑧 = 𝜃 + 𝜑𝑥 (1) where 𝜑𝑥 is the shear angle of ve core since the two elastic layers are treated as eulerbernoulli beams neglecting shear deformation. 594 y. elmoghazy, b. safaei, s. sahmani to simplify fe modeling and formulation the following basic assumptions are proposed: 1. shear deformation is negligible for elastic layers. 2. the vibrations of the ve sandwich beam is dissipated through periodic shear deformation of the ve layer. 3. all layers of ve sandwich beam have the same transverse displacement 𝑤. 4. all layer interaction is constrained. 5. ve material is considered as small strain linear ve. fig. 1 passive damping treatment illustration for typical ve sandwich beam with a deformed shear configuration axial and transverse displacements 𝑢(𝑖)(𝑥, 𝑧, 𝑡), 𝑤(𝑖)(𝑥, 𝑧, 𝑡) respectively, at any point within the elastic layers could be expressed as [31]: 𝑢(𝑖)(𝑥, 𝑧, 𝑡) = 𝑢𝑖(𝑥, 𝑡) − 𝑧𝑖 𝜕𝑤𝑖(𝑥,𝑡) 𝜕𝑥 𝑤(𝑖)(𝑥, 𝑧, 𝑡) = 𝑤𝑖(𝑥, 𝑡) } 𝑖 = 𝑐, 𝑏 (2) where 𝑢𝑖(𝑥, 𝑡) and 𝑤𝑖 (𝑥, 𝑡) are the axial and transverse displacements, at the centroid point of elastic layers, respectively. the relative kinematics relations of ve core shear strain 𝛾𝑥𝑧 and axial displacement 𝑢𝑣 can be derived using local coordinate system and fsdt [32] as: 𝑢𝑣 = 1 2 (𝑢𝑐 + 𝑢𝑏) + 1 4 (ℎ1−ℎ2) 𝜕𝑤 𝜕𝑥 (3) 𝛾𝑥𝑧 = 1 𝐻−(ℎ1+ℎ2) [(𝑢𝑐 − 𝑢𝑏) + ( ℎ1+ℎ2 2 + (𝐻 − (ℎ1+ℎ2)) 𝜕𝑤 𝜕𝑥 ] (4) 595 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets 3. finite element modelling 3.1. degrees of freedom and shape functions of proposed beam model the integral shear beam element with two nodes i and j is shown in fig. 2. each node had four dof with lateral displacement 𝑤 along z-direction, rotation 𝜃 about x-axis, and axial displacements of elastic layers 𝑢𝑐 and 𝑢𝑏. the total set of element nodal displacements {𝑑𝑒} are given as: {𝑑𝑒} = { 𝑑𝑖 𝑑𝑗 } = {𝑤𝑖 𝜃𝑖 𝑢𝑏𝑖 𝑢𝑐𝑖 𝑤𝑗 𝜃𝑗 𝑢𝑏𝑗 𝑢𝑐𝑗}𝑇 (5) θi θj wi wj uci ubi ucj ubj i j zc zb u1 u2 fig. 2 integral shear beam element the transverse displacement field 𝑤(𝑥), rotation field 𝜃(𝑥) = 𝜕𝑤 𝜕𝑥 , and axial displacement field of elastic layers 𝑢𝑏(𝑥) and 𝑢𝑐(𝑥) are expressed as a cubic polynomial function: 𝑤(𝑥) = 𝑐0 + 𝑐1𝑠 + 𝑐2𝑠 2 + 𝑐3𝑠 3 (6) 𝜃(𝑥) = 𝑐1 + 2𝑐2𝑠 + 3𝑐3𝑠 2 (7) 𝑢𝑏(𝑥) = 𝑐4 + 𝑐5𝑠, 𝑢𝑐(𝑥) = 𝑐6 + 𝑐7𝑠 (8) where 𝑠 = 𝑥/𝑙𝑒 represents the normalized coordinate with 𝑙𝑒 being equivalent element length. from eqs. (6), (7), and (8) the element displacement field within the element could be expressed in matrix form as: { 𝑤 𝜃 𝑢𝑏 𝑢𝑐 } = [𝑇]{𝑐} = [ 1 𝑠 𝑠2 𝑠3 0 0 0 0 0 1 𝑙𝑒 2𝑠 𝑙𝑒 3𝑠2 𝑙𝑒 0 0 0 0 0 0 0 0 1 𝑠 0 0 0 0 0 0 0 0 1 𝑠] { 𝑐0 𝑐1 𝑐3 ⋮ 𝑐7} (9) where 𝑠 = 𝑥/𝑙𝑒 represents the normalized coordinate with 𝑙𝑒 being equivalent element length. {𝑐} and [𝑇] are the coefficient matrix and polynomial matrix, respectively which can be represented by element nodal displacements {𝑑𝑒} explicitly in matrix form as [32]: {𝑑} = [𝑇]{𝑐} = [𝑇][𝑄−1]{𝑑𝑒} = [𝑁]{𝑑𝑒} (10) where n is the shape function matrix corresponding to the four nodal dof of each node in the form of 4×8 matrix and 𝑄 is 8×8 matrix that depends on the applied boundary condition. using eq. (10) and applying the c-f boundary condition, the 𝑄 matrix is obtained in a matrix form as: 596 y. elmoghazy, b. safaei, s. sahmani 𝑄 = [ 1 1 1 1 0 0 0 0 0 1 𝑙𝑒 2 𝑙𝑒 3 𝑙𝑒 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 𝑙𝑒 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0] (11) the shape function is obtained after the decoupling-inversion operation using 𝑄−1 as: [𝑁𝑥 𝑁𝑥 ′ 𝑁1 𝑁2] 𝑇 = 𝑇𝑄−1 (12) the generalized shape function matrix is obtained as a column matrix [13]: 𝑁𝑥 = [ 1 − 3𝑠2 + 2𝑠3 𝑥 − 2𝑥2 𝑙𝑒 + 𝑥3 𝑙𝑒 2 0 0 3𝑠2 − 2𝑠3 𝑥3 𝑙𝑒 2 − 𝑥2 𝑙𝑒 0 0 ] 𝑇 , 𝑁𝑥 ′ = [ 6( 𝑥2 𝑙𝑒 3 − 𝑥 𝑙𝑒 2) 1 − 4𝑠 + 3𝑠2 0 0 −6( 𝑥2 𝑙𝑒 3 − 𝑥 𝑙𝑒 2) −2𝑠 + 3𝑠2 0 0 ] 𝑇 , 𝑁1 = [ 0 0 1 − 𝑠 0 0 0 𝑠 0 ] 𝑇 , 𝑁2 = [ 0 0 0 1 − 𝑠 0 0 0 𝑠 ] 𝑇 the displacements filed within the element could be then determined by the nodal displacement of element through shape functions interpolation: 𝑤 = 𝑁𝑥𝑑 𝑒, 𝜃 = 𝑁𝑥 ′𝑑𝑒 , 𝑢𝑏 = 𝑁1𝑑 𝑒, 𝑢𝑐 = 𝑁2𝑑 𝑒, 𝑢𝑣 = 𝑁3𝑑 𝑒, 𝛾 = 𝑁4𝑑 𝑒 where 𝑁3 and 𝑁4 were derived based on the relative kinematics relations 𝛾𝑥𝑧 and 𝑢𝑣 in eqs. (3) and (4) as follows: 𝑁3 = 1 2 [(𝑁1 +𝑁2) + ( ℎ1+ℎ3 2 )𝑁𝑥 ′] (13) 𝑁4 = 1 𝐻−(ℎ1+ℎ2) [(𝑁2 − 𝑁1) + ( ℎ1+ℎ3 2 + (𝐻 − (ℎ1+ℎ2))𝑁𝑥 ′] (14) 3.2. element stiffness matrix the principle of strain energy can be applied to obtain element stiffness matrix of ve sandwich beam. in general, the elastic strain energy is expressed as: 𝑈 = ∫ 𝜎2 2𝐸 𝑉 𝑑𝑉 (15) 3.2.1. strain energy due to axial loading based on eq. (15), the strain energy due to axial loading of elastic layer 𝑈𝑎 is: 𝑈𝑎 = 𝐸𝐴 2 ∫ 휀2𝑑𝑥 (16) 597 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets the definition of axial strain was defined as the rate change of the displacement: 휀 = 𝜕𝑢 𝜕𝑥 (17) substituting eq. (17) into eq. (16) yields [32]: 𝑈𝑎 = 1 2 𝐸𝐴∫ ( 𝜕𝑢 𝜕𝑥 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 𝑑𝑒𝑇𝐾𝑒𝑖 𝑒 𝑑𝑒 (18) 𝑈𝑎𝑐 = 1 2 𝐸𝑐𝐴𝑐 ∫ ( 𝜕𝑢𝑐 𝜕𝑥 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 𝑑𝑒 𝑇 𝐾𝑒𝑐 𝑒 𝑑𝑒 (19) 𝑈𝑎𝑏 = 1 2 𝐸𝑏𝐴𝑏 ∫ ( 𝜕𝑢𝑏 𝜕𝑥 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 𝑑𝑒 𝑇 𝐾𝑒𝑏 𝑒 𝑑𝑒 (20) the element stiffness matrix of base and constraining layers, corresponding to axial strain energy 𝐾𝑎𝑏 𝑒 and 𝐾𝑎𝑐 𝑒 , respectively can be expressed as: 𝐾𝑎𝑏 𝑒 = 𝐸𝑏𝐴𝑏 ∫ [ 𝜕𝑁1 𝜕𝑥 ] 𝑙𝑒 0 𝑇 [ 𝜕𝑁1 𝜕𝑥 ] 𝑑𝑥 (21) 𝐾𝑎𝑐 𝑒 = 𝐸𝑐𝐴𝑐 ∫ [ 𝜕𝑁2 𝜕𝑥 ] 𝑙𝑒 0 𝑇 [ 𝜕𝑁2 𝜕𝑥 ] 𝑑𝑥 (22) 3.2.2. strain energy due to bending loading the elastic strain energy due to flexural bending in the beam is given by: 𝑈𝑏 = ∫ 𝑀2(𝑥) 𝑦2 2𝐸𝐼2 𝑉 𝑑𝑉 (23) the flexural bending of the element is given by: 𝑀(𝑥) 𝐸𝐼 = 𝜕2𝑤 𝜕𝑥2 (24) substituting eq. 24 into eq. 23, the bending strain energy of elastic layers is given as: 𝑈𝑏 = 𝐸 2 ∬( 𝜕2𝑤 𝜕𝑥2 ) 2 𝑦2𝑑𝐴 𝑑𝑥 (25) 𝑈𝑏 = 1 2 𝐸𝐼 ∫ ( 𝜕2𝑤 𝜕𝑥2 ) 2 𝑑𝑥 = 1 2 𝑑𝑒𝑇𝐾𝑏 𝑒𝑑𝑒 (26) 𝑈𝑏𝑐 = 1 2 𝐸𝑐𝐼𝑐 ∫ (( 𝜕2𝑤 𝜕𝑥2 ) 2 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 𝑑𝑒𝑇𝐾𝑏𝑐 𝑒 𝑑𝑒 (27) 𝑈𝑏𝑏 = 1 2 𝐸𝑏𝐼𝑏 ∫ (( 𝜕2𝑤 𝜕𝑥2 ) 2 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 𝑑𝑒𝑇𝐾𝑏𝑏 𝑒 𝑑𝑒 (28) the element stiffness matrix of constraining and base layers, corresponding to bending strain energy 𝐾𝑏𝑐 𝑒 and 𝐾𝑏𝑏 𝑒 , respectively can then be expressed as: 𝐾𝑏𝑏 𝑒 = 𝐸𝑏𝐼𝑏 ∫ [ 𝜕𝑁𝑥 𝜕𝑥 ] 𝑙𝑒 0 𝑇 [ 𝜕𝑁𝑥 𝜕𝑥 ] 𝑑𝑥 (29) 𝐾𝑏𝑐 𝑒 = 𝐸𝑐𝐼𝑐 ∫ [ 𝜕𝑁2 𝜕𝑥 ] 𝑙𝑒 0 𝑇 [ 𝜕𝑁2 𝜕𝑥 ] 𝑑𝑥 (30) 598 y. elmoghazy, b. safaei, s. sahmani 3.2.3. strain energy due to shear loading strain energy due to shear loading can be derived as 𝑈𝑠ℎ𝑒𝑎𝑟 = ∫ 𝜏2 2𝐺 𝑑𝑣 = 𝐺𝛾2 2 ∫𝑑𝑣 (31) 𝑈𝑠ℎ𝑒𝑎𝑟 = 𝐺∗𝛾2 2 ∬𝑑𝐴 𝑑𝑥 (32) 𝑈𝑠ℎ𝑒𝑎𝑟 = 1 2 𝐺∗𝐴𝑣 ∫ 𝛾2 𝑙𝑒 0 𝑑𝑥 = 1 2 𝑑𝑒𝑇𝐾𝑠 𝑒𝑑𝑒 (33) 𝐾𝑠 𝑒 is the shear element stiffness matrix associated with shear strain energy such that: 𝐾𝑠 𝑒 = 𝐺∗𝐴𝑣 ∫ 𝑁4 𝑇𝑙𝑒 0 𝑁4𝑑𝑥 (34) the total elastic element stiffness matrix 𝐾𝑒 𝑒 can be expressed as: 𝐾𝑒 𝑒 = 𝐾𝑒𝑏 𝑒 + 𝐾𝑒𝑐 𝑒 + 𝐾𝑏 𝑏 𝑒 + 𝐾𝑏 𝑐 𝑒 (35) the element stiffness matrix then could be expressed as: 𝐾𝑒⏟ ⌊8×8⌋ = [ [𝐾𝑒𝑏 𝑒 ] [𝐾𝑏𝑏 𝑒 ] [𝐾𝑏𝑐 𝑒 ] [𝐾𝑒𝑐 𝑒 ] ] + 𝐾𝑠 𝑒⏟ ⌊8×8⌋ (36) where 𝐾𝑒 is the total element stiffness matrix obtained as the summation of elastic element stiffness matrix 𝐾𝑒 𝑒 and shear stiffness element matrix 𝐾𝑠 𝑒 of ve sandwich beam. 3.3. element mass matrix to obtain the element mass matrix, the principle of kinetic energy is applied. by considering the axial and rotational loading associated with kinetic energy, the total kinetic energy 𝑇 of ve sandwich beam is evaluated as: 𝑇 = 1 2 (𝜌𝑐𝐴𝑐 + 𝜌𝑣𝐴𝑣 + 𝜌𝑏𝐴𝑏) ∫ �̇� 2 𝑑𝑥 + 1 2 (𝜌𝑐𝐴𝑐 + 𝜌𝑣𝐴𝑣 + 𝜌𝑏𝐴𝑏) ∫ �̇� 2 𝑑𝑥 (37) the kinetic energy corresponding to the axial and bending loading of each layer is: 𝑇𝑎𝑐 = 1 2 𝜌𝑐𝐴𝑐 ∫ ( 𝜕𝑢𝑐 𝜕𝑡 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 �̇�𝑒 𝑇 𝑀𝑒𝑐 𝑒 �̇�𝑒 (38) 𝑇𝑏𝑐 = 1 2 𝜌𝑐𝐴𝑐 ∫ ( 𝜕𝑤 𝜕𝑡 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 �̇�𝑒 𝑇 𝑀𝑏𝑐 𝑒 �̇�𝑒 (39) 𝑇𝑎𝑏 = 1 2 𝜌𝑏𝐴𝑏 ∫ ( 𝜕𝑢𝑏 𝜕𝑡 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 �̇�𝑒 𝑇 𝑀𝑒𝑏 𝑒 �̇�𝑒 (40) 𝑇𝑏𝑏 = 1 2 𝜌𝑏𝐴𝑏 ∫ ( 𝜕𝑤 𝜕𝑡 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 �̇�𝑒 𝑇 𝑀𝑏𝑏 𝑒 �̇�𝑒 (41) 𝑇𝑎𝑣 = 1 2 𝜌𝑣𝐴𝑣 ∫ ( 𝜕𝑢𝑣 𝜕𝑡 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 �̇�𝑒 𝑇 𝑀𝑒𝑐 𝑒 �̇�𝑒 (42) 𝑇𝑏𝑣 = 1 2 𝜌𝑣𝐴𝑣 ∫ ( 𝜕𝑤 𝜕𝑡 ) 𝑙𝑒 0 2 𝑑𝑥 = 1 2 �̇�𝑒 𝑇 𝑀𝑏𝑣 𝑒 �̇�𝑒 (43) where 𝑇𝑎𝑐, 𝑇𝑎𝑏and 𝑇𝑎𝑣 are kinetic energies of the constraining, base, and core layers that correspond to axial loading, respectively. 𝑇𝑏𝑐, 𝑇𝑏𝑏 and 𝑇𝑏𝑣 are kinetic energies of constraining, base, and core layers corresponding to bending loading, respectively. 599 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets the element mass matrices associated with elastic layers and ve core layer corresponding to axial and bending loading 𝑀𝑎𝑐 𝑒 , 𝑀𝑏𝑐 𝑒 , 𝑀𝑎𝑏 𝑒 , 𝑀𝑏𝑏 𝑒 , 𝑀𝑎𝑣 𝑒 and 𝑀𝑏𝑣 𝑒 are expressed using the element shape functions as: 𝑀𝑎𝑐 𝑒 = 𝜌𝑐𝐴𝑐 ∫ 𝑁2 𝑇𝑙𝑒 0 𝑁2𝑑𝑥,𝑀𝑏𝑐 𝑒 = 𝜌𝑐𝐴𝑐 ∫ 𝑁𝑥 𝑇𝑁𝑥 𝑙𝑒 0 𝑑𝑥 (44) 𝑀𝑎𝑏 𝑒 = 𝜌𝑏𝐴𝑏 ∫ 𝑁1 𝑇𝑁1 𝑙𝑒 0 𝑑𝑥,𝑀𝑏𝑏 𝑒 = 𝜌𝑏𝐴𝑏 ∫ 𝑁𝑥 𝑇𝑁𝑥 𝑙𝑒 0 𝑑𝑥 (45) 𝑀𝑎𝑣 𝑒 = 𝜌𝑣𝐴𝑣 ∫ 𝑁3 𝑇𝑁3 𝑙𝑒 0 𝑑𝑥,𝑀𝑏𝑣 𝑒 = 𝜌𝑣𝐴𝑣 ∫ 𝑁𝑥 𝑇𝑙𝑒 0 𝑁𝑥𝑑𝑥 (46) the element mass matrix 𝑀𝑒 is given as: 𝑀𝑒⏟ ⌊8×8⌋ = 𝑀𝑎𝑐 𝑒 + 𝑀𝑏𝑐 𝑒 + 𝑀𝑎𝑏 𝑒 + 𝑀𝑏𝑏 𝑒 +𝑀a𝑣 𝑒 +𝑀𝑏𝑣 𝑒 (47) 3.4. finite element equation of motion after assembling the global stiffness and mass matrix of the structure, the hamilton variation principle is used to derive the dynamic equation of the ve sandwich beam: ∫ 𝛿(𝑇 − 𝑈) 𝑡2 𝑡1 𝑑𝑡 + ∫ 𝛿𝑤 𝑡2 𝑡1 𝑑𝑡 = 0 (48) after applying eq. (48), the ve sandwich beam equation of motion is expressed as: 𝑀𝑒�̈�𝑒 + 𝐾𝑒 𝑒𝑑𝑒 + 𝐾𝑣 𝑒𝑑𝑒 = 𝐹𝑒 (49) 3.5. linear vibration eigenvalue problem (constant complex modulus) the complex constant elastic modulus of ve material e is expressed as: 𝐸∗ = 𝐸′ + 𝑗𝐸′′ (50) where 𝐸′ is the real part, also called storage modulus, and 𝐸′′ is the imaginary part which is called energy dissipation modulus. 𝜂𝑣 = 𝐸′′ 𝐸′ is ve loss factor, which is the ratio of energy dissipation modulus to storage energy modulus. the loss factor of ve material physically represents the ratio of the energy consumed from ve material to total energy. many researchers have used constant shear complex modulus to obtain a simple dynamic equation by applying a simple harmonic excitation force [33] and eq. (49) could be expressed as: 𝑀𝑒�̈�𝑒 + 𝑑𝑒𝐾𝑒 = 0 (51) where 𝐾𝑒, 𝑀𝑒 are the total stiffness matrix and total mass matrix of the ve sandwich beam, respectively. then, the eigenvalue problem is expressed as (𝐾𝑒 − 𝜔∗2𝑀𝑒)𝑋 = 0 (52) where 𝜔∗ is the complex frequency, and x is the complex eigenmodes of ve sandwich beam. natural frequency 𝜔𝑛 and model loss factor 𝜂𝑛 are calculated as follows: 𝜔𝑛 = √𝑅𝑒𝜔 ∗ (53) 600 y. elmoghazy, b. safaei, s. sahmani 4. numerical results and validation 4.1. normal mode analysis (free vibration) normal modal analysis is a prerequisite step in modeling the dynamic behavior of ve sandwich structures. it is a semi-final step that gives theoretical values for eigenvalues and eigenvectors that have no physical meaning unless normal model analysis is paired with harmonic or time response analysis. the mechanical and geometrical properties of elastic and core layers of ve sandwich beam are summarized in table 1. table 1 materials mechanical properties and geometrical specifications of ve sandwich beam [33] aluminum layers ve core layer young modulus (mpa) 6.9× 104 1.794×(1+0.3i) poisson’s ration 0.3 0.3 loss factor 0.1,0.6,1,1.5 density (kg/m3) 2766 968.1 length (mm) 177.8 177.8 width (mm) 12.7 12.7 thickness (mm) 1.52 0.127 4.1.1. case study #1: c-f ve sandwich beam to study the linear vibration of ve sandwich structures, a cantilever beam was discretized into 30 elements. in this case, ve material is independent from young’s modulus e. the mechanical and geometrical properties of elastic and core layers of ve sandwich beam are summarized in table 1. a matlab code was developed to solve the eigenvalues problem. in addition, 2d and 3d finite element models were built using fe solver software abaqus to show the efficiency of the numerical approach. for mesh modeling, 2d and 3d fe models were built using abaqus software. the eight-node continuum quadratic brick element cps8 is used to discretize the 2d model, while the 20-node c3d20 was used to discretize the 3d model. the sizing control technique is used with a maximum mesh global size of 0.5 mm and maximum deviation factor of 0.1. table 1 gives the specifications of cps8 and c3d20. based on euler-bernoulli beam theory, the natural frequencies of a base beam without ve core and constraining layer are calculated as: 𝜔𝑛 = 𝐴𝑛 2𝜋𝐿2 √ 𝐸𝐼 𝜌𝐴 (54) where 𝐸 is the modulus of elasticity, 𝐼 is the cross-sectional moment of inertia of the base beam, 𝜌 is material density, and 𝐴 is the cross-sectional area of the beam. according to eq. (54), the first fundamental natural frequencies of the base beam without ve core and constraining layer are calculated as 38.83hz and 247.1hz. 601 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets it is worth pointing out that, with the addition of a ve layer and a constrained layer, the stiffness and mass are increased accordingly. from eq. (54), an increase in structure stiffness leads to an increase in the natural frequency of the structure, while an increase in the additional mass leads to a decrease in the natural frequency. according to table 1, the density of ve core is about half of elastic layer's density, however, the stiffness of the ve core is almost negligible compared to stiffness of the elastic layers. accordingly, the effect of the additional mass on the natural frequency is much greater than the effect the additional stiffness. for the sake of comparison, the first six natural frequencies and normalized loss factor of ve sandwich beam were calculated using three different elements based on shear energy consumption assumption when the loss factor 𝜂𝑣= [0.1, 0.6, 1.5]. table 2 mesh specifications of cps8 and c3d20 elements implemented in abaqus element description schematic ndof/node cps8 ▪ continuum biquadratic. ▪ 8-nodes. ▪ plane stress 2 translations c3d20 ▪ continuum quadratic brick. ▪ 20-nodes. 3 translations table 3 shows that fe shear models, real eigenmodes (rm) and asymptotic numerical method (anm) all had good accuracy in calculating the first six natural frequency of ve sandwich beam under c-f boundary condition against the analytical solution [34]. table 4 shows normalized loss factor of c-f ve sandwich beam with different 𝜂𝑣. in comparison, the accuracy of fe shear model was the highest compared with the analytical method [34], the error range was 0-0.15% followed by c3d20 element with an error range of 0-5.918% and an average error of 0.2%; cps8 element was the third with an error range of 0-6.565% and an average error of 0.858% and rm method had the lowest accuracy, with the lowest error value of 0%, the highest error value of 8.892% and an average error of 0.97%. 602 y. elmoghazy, b. safaei, s. sahmani although rm approach had a slightly higher average error than anm method, rm method could not be deemed accurate. because rm approach only considers the real part of natural frequency, which contradicts reality. although the prediction results of natural frequency by rm method were close to analytical solutions when the loss factor 𝜂𝑣 was small but when the loss factor 𝜂𝑣 was increased to 1.5, the prediction error of natural frequency reached 8.892%. on the other hand, the prediction error of anm for natural frequency was very consistent and did not fluctuate much, thus the calculated loss factor of anm for natural frequency was generally accurate. as a result, the prediction accuracies of the five numerical approaches for the natural frequency of c-f ve sandwich beam in this work were ranked as follows: integral shear element> c3d20 > cps8 > anm> rm. table 3 linear frequencies of the c-f ve sandwich beam with different 𝜼𝒗 𝜼𝒗 analytical [34] cps8 fe model rm [35] c3d20 anm [36] 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 0.1 64.1 64.1 64.1 64.1 64.5 64.5 296.4 296.5 296.3 296.6 298.6 298.9 743.7 743.6 743.4 744.3 750.1 746.5 1393.9 1392.6 1393.4 1395.2 1408.3 1407.7 2261.1 2256.7 2260.2 2263.4 2288.4 2286.2 3343.6 3332.8 3342.3 3347.3 3388.7 3385.7 0.6 65.5 64.3 65.4 64.1 64.7 65.9 298.9 297.4 298.8 296.6 299.5 303.1 745.5 744.5 745.2 744.3 751 752.3 1394.9 1393.1 1394.3 1395.2 1409 1412.7 2261.7 2257 2260.8 2263.4 2289 2290.6 3344 3333 3342.6 3374.3 3389 3389.5 1 67.4 64.7 67.4 64.1 65.1 67.8 302.8 299.1 302.7 296.6 301 309.1 748.6 746.1 748.4 744.3 753 761.1 1396.6 1394 1396 1395.2 1410 1420.6 2262.8 2257.6 2261.9 2263.4 2289 2297.9 3345 3333.3 3343.4 3374.3 3389 3395.9 1.5 69.8 65.5 69.9 64.1 65.9 67.8 308.8 302.3 308.7 296.6 304 309.1 754 749 754.2 744.3 756 761.1 1399.7 1395.8 1399.1 1395.2 1412 1420.6 2265 2258.7 2264.1 2263.4 2290 2297.9 3346 3333.9 3344.7 3374.3 3390 3395.9 average error (%) ≈ 0.858 0.03 0.94 0.2 0.97 603 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets table 4 shows that the accuracy of the fe shear element models implemented in this study is very high in predicting model loss factor for the first six modes of ve sandwich beam structure. with an error range of 0% to 11.45% and an average error of 2.38%, anm [36] was the most accurate one. in addition, c3d20 element had an error range of 0-14.7%, with an average of 4.19%, which placed it second. rm [35], which had a minimum error of 0% and a maximum error of 8.892%, had the lowest accuracy. the results showed a significant improvement in the overall damping performance of the c-f ve sandwich beam as the loss factor 𝜂𝑣was increased. when low frequencies were close to loading frequencies, they could generate huge amplitudes, causing what is known as resonance. table 4 normalized loss factor of c-f ve sandwich beam with different 𝜼𝒗 𝜼𝒗 analytical [34] cps8 rm [35] c3d20 anm [36] 𝜂𝑛 𝜂𝑣⁄ 𝜂𝑛 𝜂𝑣⁄ 𝜂𝑛 𝜂𝑣⁄ 𝜂𝑛 𝜂𝑣⁄ 𝜂𝑛 𝜂𝑣⁄ 0.1 0.281 0.283 0.283 0.282 0.281 0.242 0.242 0.243 0.241 0.242 0.154 0.154 0.154 0.153 0.154 0.088 0.089 0.089 0.088 0.089 0.057 0.057 0.057 0.056 0.057 0.039 0.039 0.039 0.038 0.039 0.6 0.246 0.281 0.283 0.280 0.247 0.232 0.241 0.243 0.239 0.224 0.152 0.154 0.154 0.152 0.150 0.088 0.089 0.089 0.088 0.088 0.057 0.057 0.057 0.056 0.057 0.039 0.039 0.039 0.038 0.039 1 0.202 0.277 0.283 0.276 0.204 0.217 0.238 0.243 0.236 0.201 0.150 0.153 0.154 0.152 0.142 0.088 0.089 0.089 0.088 0.086 0.057 0.057 0.057 0.056 0.057 0.038 0.039 0.039 0.038 0.037 1.5 0.153 0.271 0.283 0.270 0.155 0.197 0.232 0.243 0.231 0.176 0.146 0.152 0.154 0.150 0.131 0.087 0.088 0.089 0.087 0.083 0.056 0.057 0.057 0.056 0.056 0.038 0.039 0.039 0.038 0.039 average error (%) ≈ 5.42 6.02 4.19 2.38 fig. 3 shows relative changes in normalized deflection amplitudes for the first three eigenmodes of c-f ve sandwich beam with a different loss factor 𝜂𝑣= [0.1, 0.6, 1.5]. it could be noticed that both real and imaginary parts had different amplitudes but the same mode shapes. for high loss factor 𝜂𝑣, the imaginary part of the complex eigenmodes was significant which affected the damping performance of the whole structure. 604 y. elmoghazy, b. safaei, s. sahmani fig. 3 the complex normalized eigenmodes of c-f ve sandwich beam with 𝜼𝒗= [0.1, 0.6, 1.5] (a) mode 1; (b) mode 2; (c) mode 3 4.1.2. case study #2: s-s ve sandwich beam for the same mechanical and geometrical properties shown in table 1, the linear vibration characterizations of s-s supported boundary ve sandwich beam are considered. in this study, based on rao analytical model [37], the prediction accuracy of the natural frequency and loss factor of the ve sandwich beam structure are compared. it is seen from tables 5 and 6 that all models, had high accuracy in calculating the first six natural frequencies of ve sandwich beam in comparison with analytical solution [37]. for s-s boundary condition, cps8 had 0.59% average relative error and c3d20 had the lowest accuracy of 3.6%. on the other hand, from table 6 it was noticed that both models had relatively high accuracy in predicting the first six loss factors of s-s ve sandwich beam compared with analytical solution [37]. as a result, it means that the prediction accuracy of the numerical approaches for the natural frequency of s-s ve sandwich beam in this work is ranked as follows: cps8> c3d20. the dynamic characteristics of ve sandwich beam were studied, and the natural frequency of ve sandwich beam system was larger under s-s boundary conditions. the fundamental loss factor of ve sandwich beams with c-f boundary conditions was higher than that for ve sandwich beams with s-s boundary (c) (a) (b) 605 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets conditions. for high values of loss factor of ve material, the imaginary part of the stiffness matrix was considered which implied the existence of not only storage modulus but, also loss modulus. as can be noticed from fig. 4, the first normalized fundamental complex vibration modes of s-s ve sandwich beam with different damping factor of ve core 𝜂𝑣= [0.1,0.6,1.5] using the 2d plan stress element cps8 implemented in abaqus. the results show that the relative amplitude difference between real and imaginary modes was increased as the loss factor increased so that the imaginary part of the complex fundamental vibration modes was significant and could not be neglected. the flowchart of the employed methodology for dynamics analysis of sandwich beam is shown in fig. 5. in the pre-processing stage, material model and tie constraints between the layers are defined. also, the process involves assigning appropriate element models and defining c-c and s-s boundary conditions. in post-processing, the free vibration response results are interpreted based on tracking model effective mass. in addition, the parametric study stage involves investigation of ve parameters' effects on free vibration response of sandwich beam. in the last stage, steady state forced vibration response with different values of 𝜂𝑣 is obtained by utilizing mode superposition method that implies the representation of each eigenmode as an independent pattern of motion. table 5 linear frequencies of s-s ve sandwich beam with different 𝜼𝒗 𝜼𝒗 analytical [37] cps8 element c3d20 element 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 𝜔𝑛(𝐻𝑧) 0.1 148.51 148.44 148.55 488.47 488.19 489.797 1034.69 1033.5 1040.57 1795.13 1791.56 1408.3 2771.49 2763 2288.4 3964.28 3946.95 3388.7 0.6 150.71 149.23 149.34 489.75 489.01 490.61 1035.38 1034.01 1041.08 1795.54 1791.89 1811.05 2771.76 2763.23 2802.92 3964.47 3947.11 4016.7 1 154.42 150.62 150.73 492.06 490.48 492.08 1036.63 1034.95 1042.02 1796.3 1792.5 1811.65 2772.27 2763.64 2803.33 3964.83 3947.41 4016.99 1.5 160.72 153.16 153.27 496.49 493.29 494.89 1039.07 1036.78 1043.84 1797.78 1793.68 1812.82 2773.25 2764.45 2804.12 3965.52 3947.99 4017.55 average error (%) ≈ 0.59 3.60 606 y. elmoghazy, b. safaei, s. sahmani table 6 normalized loss factor of s-s ve sandwich beam with different 𝜼𝒗 𝜼𝒗 analytical [37] cps8 element c3d20 element 𝜂𝑛 𝜂𝑣⁄ 𝜂𝑛 𝜂𝑣⁄ 𝜂𝑛 𝜂𝑣⁄ 0.1 0.107 0.1069 0.1063 0.065 0.0651 0.0644 0.043 0.0433 0.0426 0.031 0.0307 0.03 0.3328 0.3468 0.3465 0.1943 0.195 0.1945 0.6 0.1068 0.1068 0.1062 0.0652 0.0651 0.0644 0.0433 0.0433 0.0426 0.0308 0.0307 0.03 0.305 0.3405 0.3402 0.192 0.1938 0.1933 1 0.106 0.1066 0.106 0.065 0.0651 0.0644 0.043 0.0433 0.0426 0.031 0.0307 0.03 0.2626 0.3292 0.27 0.1871 0.1916 0.231 1.5 0.1059 0.1063 0.15 0.065 0.065 0.087 0.0433 0.0432 0.056 0.0308 0.0307 0.038 0.0410 0.0432 0.0510 3346 3333.9 3390 average error (%) ≈ 1.8 6.28 4.1.3. case study #3: four-sided s-s sandwich plate the material and geometrical specifications of ssss ve plate structure are shown in table 7. the natural frequencies and loss factors of the structure are calculated using the developed fe model implemented in abaqus and compared with the results obtained by johnson and kienholz [38], who used modal strain energy method (mse) implemented in nastran commercial software to calculate the natural frequency and loss factor of ve sandwich structures. the ve sandwich plate in this study is discretized into 2200×120 and 1050×255 meshed areas using c3d20 element along with the length and width directions during the calculation. in adition, the mesh specification of c3d20 element elements implemented in abaqus is shown in table 2. 607 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets fig. 4 the complex normalized eigenmodes of s-s ve sandwich beam with 𝜂𝑣= [0.1, 0.6, 1.5] (a) mode 1; (b) mode 2; (c) mode 3 table 7 mechanical and geometrical specifications of ve ssss ve sandwich plate [38] properties elastic layers ve core layer young modulus (pa) 6.89× 1010 1.794×106 poisson’s ration 0.3 0.49 density (kg / m3) loss factor 2740 999 0.5 length (mm) 348 348 width (mm) 304.8 304.8 thickness (mm) 0.762 0.254 the mode shapes of the ve sandwich plate associated with the first fundamentals natural frequency obtained using c3d20 element are shown in fig. 6. the sweet spot could be visualized which showed the maximum deformation location points. (c) (a) (b) 608 y. elmoghazy, b. safaei, s. sahmani fig. 5 flow chart of the proposed dynamics analysis methodology pre-processing meshing test no yes no normal mode analysis problem definition -mechanical and geometrical specifications kinematics formulation -fsdt degree of freedom -interpolation shape function fem mechanical assembly -𝐾𝑒 -𝑀𝑒 discretization (ke −ω∗2me)x = 0 integral beam element fe model -abaqus/explicit boundary conditions model meshing contact -tie constrains cps8/c3d20 element complex frequency 𝝎∗ material loss factor 𝜼𝒗 ve core thickness model frequency 𝝎𝒏 model loss factor 𝜼𝒏 𝝎𝒏 = √𝑹𝒆𝝎∗ harmonic load excitations frf steady-state response parametric study 609 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets fig. 6 fundamental eigenmodes shape of viscoelastic sandwich ssss plate: (a) first mode, (b) second mode, (c) third mode, (d) forth mode tables 8 and 9 show the calculated results of natural frequency and normalized loss factor, respectively. it is clear from tables 8 and 9 that c3d20 element had good accuracy in predicting the damping characteristics of ve sandwich ssss plate against analytical solution with an average error in predicting the fundamental natural frequency of less than 0.4% and an average error of less than 6.5% in predicting the damping loss factor 𝜂𝑛 𝜂𝑣⁄ of ve sandwich plate. in addition, it can be noticed from tables 8 and 9 that, while increasing the mesh size the error slightly decreases but this increases the time computation of the whole fe process. thus, there exists a trade-off between mesh refinement and computational efficiency in the fe analysis. table 8 linear complex natural frequencies of ve sandwich plate obtained for different mesh sizes of c3d20 element mode mse method [38] c3d20 2200×120 c3d20 1050×255 natural frequency 𝜔𝑛 (hz) natural frequency 𝜔𝑛 (hz) error (%) natural frequency 𝜔𝑛 (hz) error (%) 1 57.4 57.3 0.174 57.2 0.348 2 113.2 113.5 0.265 113.3 0.088 3 129.3 129.5 0.155 129.1 0.155 4 179.3 177.7 0.892 177.1 1.227 average 0.372 0.454 table 9 normalized loss factor of ve sandwich plate obtained for different mesh sizes of c3d20 element mode mse method [38] c3d20 2200×120 c3d20 1050×255 loss factor 𝜂𝑛 𝜂𝑣⁄ loss factor 𝜂𝑛 𝜂𝑣⁄ error (%) loss factor 𝜂𝑛 𝜂𝑣⁄ error (%) 1 0.352 0.366 3.98 0.366 3.963 2 0.376 0.394 4.79 0.396 5.342 3 0.376 0.390 3.72 0.391 4.038 4 0.306 0.346 13.07 0.348 13.835 average 6.39 6.794 a) first mode b) second mode c) third mode d) forth mode 610 y. elmoghazy, b. safaei, s. sahmani 4.2. parametric studies it is required to examine the effects of various parameters on the vibration characteristics of ve sandwich beam. the thickness of the core and ve core loss factor are varied which affects natural frequency and mode shape. 4.2.1 effect of ve material loss factor on the dynamic characteristics of ve sandwich beam to study the influence of ve loss factor on the dynamic characteristics of natural frequency, ve loss factor is varied in the range of 0.1 to 1.5 and the change trends of natural frequency and normalized loss factor corresponding to the fundamental modes of c-f ve sandwich beam are illustrated in fig. 7. the fundamental natural frequencies of ve sandwich beams only increased minimally when the loss factor was increased from 0.1 to 1.5, indicating that 𝜂𝑣 have little effect on the natural frequencies of ve sandwich beams. constant complex module 𝐸′(1 + 𝑗𝜂𝑣), showed that the complex elastic modulus 𝐸∗ of ve material is directly proportional to the loss factor. in addition, the stiffness of ve layer and the stiffness of the entire ve sandwich beam increased. since the mass of ve sandwich beam did not vary, the fundamental frequency of the beam was increased. however, as shown in table 1, the elastic modulus of ve layer was four times lower than the elastic modulus of elastic layers, thus the increase in stiffness induced by the loss factor of ve layer could only make ve sandwich c-f beam stiffer. under the condition of constant mass, the natural frequencies of ve sandwich c-f beam are only increased slightly with a small increase in stiffness. in addition, when the loss factor of ve material 𝜂𝑣 is increased, the normalized loss factor 𝜂𝑛 𝜂𝑣⁄ of ve sandwich beam increased linearly, implying that the loss factor of ve material 𝜂𝑣 had a significant impact on the loss factor of ve sandwich beam and energy dissipation capacity of ve material is characterized by its loss factor. fig. 7 the effect of 𝜂𝑣 on the dynamic characteristics of c-f ve sandwich beam 611 dynamics analysis of viscoelastic sandwiched structures integrated with aluminum sheets 4.2.2. effect of ve core layer thickness on the dynamic characteristics of ve sandwich beam core thickness is varied from 0.5 to 6.5 mm to study its influence on the dynamic characteristics 𝜔𝒏 and normalized loss factor 𝜂𝑛 𝜂𝑣⁄ . indeed, the natural frequency of the system slightly decreased as the thickness of ve layer increased, as seen in fig. 8. increase in the stiffness of ve sandwich beam system generated by increasing the thickness of the ve layer was much smaller than the increase in the whole c-f ve sandwich beam stiffness since the elastic modulus of ve layer was four times lower than elastic modulus of elastic layers and the mass of the core was about 50% of elastic layers mass. the changing of the normalized loss factor 𝜂𝑛 𝜂𝑣⁄ for c-f beam was more significant than the changing trend of natural frequency, which was related to the damping nature of ve material. fig. 8 the effect of core thickness on the dynamic characteristics of c-f ve sandwich beam 4.3. frequency response analysis harmonic analysis is required to ensure that the system will successfully overcome resonance and harmful effects and mode superposition method is used based on linear iterations of mode shapes to obtain the frequency response function (frf). the frequency response curves were then generated to capture the steady-state vibration response of ve sandwich beam across specific frequency range. 4.3.1. steady-state dynamics of ve sandwich beam/plate to excite the first three fundamental mode shapes, a harmonic concentrated force was applied at point p = (59.27, 6.3) mm located along beam length, with magnetite of 𝐹0 = 100 n. the system linear frequency response (frf) is shown in fig. 9 with different loss factor 𝜂𝑣= [0.3, 0.6, 1]. it could be noticed from fig. 9 that the effect of the loss factor was significant, as the system had minimum amplitude response when loss factor 𝜂𝑣= 1. the maximum response of the system was within the range of the first fundamental natural frequency that had the highest value of effective mass along the direction of applied load. 612 y. elmoghazy, b. safaei, s. sahmani in addition, fig. 9 shows that the system tends to vibrate with maximum amplitude at natural frequency values. this occurs since the natural frequency of the structure matches the excitation frequency, leading to a resonance condition that can potentially cause failure effects in the system. fig. 9 linear frequency response curve of fundamental complex modes for c-f ve sandwich beam with 𝜂𝑣= [0.3, 0.6, 0.1] 5. conclusions in the present work, dynamic analysis was examined to evaluate the damping characteristics of ve sandwich structures, including natural frequency and normalized loss factor 𝜂𝑛 𝜂𝑣⁄ based on longitudinal shear deformation assumption. the obtained numerical results were compared against analytical model to validate the proposed fe models. the numerical results showed that all numerical fe models are applicable for approximating the damping characteristics of ve sandwich beam. the integral shear element model was developed using fe formulation had the lowest error range among other models. in addition, the linear complex natural frequency of ve sandwich plate were obtained for different mesh sizes of c3d20 element. the element shows high accuracy in predicting damping 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713 726 https://doi.org/10.22190/fume201211051b © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper numerical investigation of the influence of the link positioning in the coronary stent inside the normal artery: a comparative study of two commercial stent designs chandrakantha bekal1, satish shenoy1, ranjan shetty2 1department of aeronautical & automobile engineering, manipal institute of technology, manipal academy of higher education (mahe) manipal, india 2manipal hospitals, bangalore, india abstract. this paper investigates the performance of two commercial stent designs inside the normal artery for induced von mises stress and radial displacement pattern. investigation focuses on identifying the key design feature of the stent structure responsible for varied stress and displacement pattern. two commercial stent designs, supraflex (stent s) and yukon choice (stent t),are modeled using micro ct images and mimics® while idealized models are used for investigation. ansys workbench is used to numerically expand the stent inside an idealized normal artery with inflation pressure. the stent and the artery are modeled using elastic-plastic and hyperelastic material models, respectively. the results suggest crucial influence of the link positioning in inducing an area of higher von mises stress and stress gradient. the locations of a higher stress gradient are those in line with unbound stent crowns. also, higher and uniform arterial displacement can be observed in the locations in line with the bound crown. results also suggested a considerable difference in arterial distortion induced by two designs, causes for which can also be attributed to the differences in the link placement. the study suggests that the link connections play a crucial role in setting up stress field/radial displacement. suitable modification of the link positioning can reduce the higher stress gradient and arterial distortion, which probably can reduce arterial injury. key words: coronary stents, finite element analysis, arterial stress, artery radial displacement, arterial distortion received december 11, 2020 / accepted may 28, 2021 corresponding author: satish shenoy b department of aeronautical & automobile engineering, manipal institute of technology, manipal academy of higher education (mahe) manipal, india e-mail: satish.shenoy@manipal.edu  c. bekal, s. shenoy, r. shetty 1. introduction it has been reported that cardiovascular disease (cvd) is a major cause of mortality (about 30%) world over [1-2]. stenosis of the coronary artery is a major health issue involving accumulation of fatty deposits thereby narrowing the lumen area leading to myocardial infraction (mi). stenosis is conveniently treated using coronary stents [3]. percutaneous coronary intervention (pci) has revolutionalized the treatment of stenosis and there has been tremendous increase in the number of stents being used worldwide [1], [4]. mechanical interaction between the stent and the artery plays a crucial role in setting up a non-physiological stress field. it is reported that higher arterial stress can lead to greater intimal thickening [5]. smooth muscle cell proliferation and stent migration immediately post stenting can lead to narrowing of vessel, the condition known as instent restenosis [6]. clinicians are encountered with a variety of stent designs to choose from with a variety of geometrical configurations [7]. measuring immediate stress field post stenting is experimentally nonviable. data about comparative advantages/disadvantages of different stent designs are seldom available since assessment of long-term efficacy of implanted stent needs a long clinical trial period. hence a promising alternative for predicting the performance of the coronary stent is numerical investigation which has been successfully utilized to investigate different generic as well as realistic stent designs [8-10], also supported by clinical implications of design parameters [11-13]. azaouzi et al [14], investigated the effect of ‘link’ on bending and torsional capabilities without considering the artery interaction and suggested that bending and torsion are greatly affected by the link shape and placement. bukala et al [15] investigated for expansion of the stent inside the stenotic artery and discussed the mechanical interaction between stent/artery surface highlighting stress and strain pattern. chua et al [16] investigated interaction between the stent and the balloon during expansion and discussed stress distribution on the stent surface. a paper by bedoya et al [17] investigated the effect of the stent design parameters such as curvature radius, axial strut spacing and amplitude on biomechanical responses. martin et al [18] investigated the influence of the balloon folding configuration on arterial stress and suggested that the balloon configuration significantly affects the stress on the stent and the artery. though researchers investigated stress and displacement pattern, the rationale for a particular stress field and displacement pattern as well as identification of the key design feature responsible for the same are rarely reported. in this study, we tried to identify the key components of the stent design crucial for inducing a stress field and displacement pattern. our study investigates the comparative performance of two different stent designs using commercial finite element analysis (fea) software ansys workbench (ansys inc.). this paper involves realistic modeling of stent models in order to identify the geometrical arrangements of stent struts and links, and a numerical analysis of the expansion of the idealized stent model inside the idealized normal artery. the main outcome of the investigation is identification of the key design feature, which is responsible for differences in induced arterial von mises stress (vms) and differences in the arterial displacement pattern. the investigation findings are expected to serve as a qualitative tool to underscore the importance of the key design feature in the construction of the stent design and its clinical implications. numerical investigation of influence of link positioning in coronary stents inside normal artery...  2. methods and materials 2.1. geometry 2.1.1. stent idealized geometry of stents, supraflextm (sahajanand medical technologies pvt. ltd, india), hereafter referred as stent s, and yukon choice pc (translumia therapeutics, india), hereafter referred as stent t, are modeled using 3d image reconstructed using micro computed tomography (xradia versa xrm500, iisc, bangalore) and materialize mimics® innovation suite. the main dimensions of the stent models in crimped form are tabulated (table 1). table 1 crimped stent dimensions (diameters are actual values as measured on the models) stent inner radius (mm) outer radius (mm) length (mm) strut thickness (mm) stent s 0.36 0.46 4 0.1 stent t 0.46 0.56 4 0.1 reconstructed models from mimics® and idealized stent models (created by wrapping stent profile) (simplified as a rectangle section) are as shown in figs. 1 and 2, respectively. fig. 1 3-dimensional realistic geometry from mimics, stent s (a) and stent t (b) fig. 2 idealized geometry of stent s (a) and stent t (b)  c. bekal, s. shenoy, r. shetty 2.1.2. artery a healthy artery is modeled as a straight cylinder with a representative dimension of 1mm inner radius and 0.4mm thickness [19]; the artery length is taken to accommodate an unstented region into its value (1mm each on each side). the stent models are centrally placed as concentric with the artery model as shown in fig. 3. fig. 3 concentrically placed stent/artery meshed model (mesh size: artery-0.2mm, stent0.05mm) 2.2. material medical grade stainless steel (aisl316l) is extensively used for manufacturing a balloon expandable stent. bilinear elastic-plastic material model is assumed for this class of material with properties listed in table 2 [20-21]. table 2 bilinear elastic-plastic material constants used for stent modeling stent material young’s modulus yield strength poison’s ratio tangent modulus density aisl316l 196gpa 205mpa 0.3 692mpa 7850kg/m3 the artery is modeled as hyperelastic with 5 parameter mooney-rivlin model with constants as tabulated in table 3 [22]. numerical investigation of influence of link positioning in coronary stents inside normal artery...  table 3 5-parameter mooney-rivlin hyperelastic material constants material model constants values(kpa) artery 3rd order 5 parameter mooney rivlin c10 18.9 c01 2.75 c20 85.72 c11 590.43 c02 0 2.3. simulation conditions the crimped stent model is expanded in the first load step (subsequent removal of pressure in the second load step to simulate final condition, achieving final diameter of stent in the range of 1.6mm to 1.7mm) inside the idealized normal artery by 2mpa inflation pressure directly applied to the internal surface of the stent in radially outward direction [15-16]. rigid body motion of the stent is prevented by restricting few nodes of approximately central section of the central stent cell in longitudinal and tangential direction allowing only radial movement. a frictional contact with coefficient 0.2 is defined between the stent and the artery surface [23]. the artery ends are tethered in all directions (longitudinal, tangential and radial) to prevent a rigid body movement. a large deformation option is activated to perform a nonlinear analysis. augmented lagrangian contact formulation with penetration tolerance of 0.1 and normal stiffness 0.1 is used (these values specify the maximum penetration and stiffness allowed when contacts are encountered during simulation). intraluminal pressure caused by pressurized blood and in-vivo longitudinal prestretch are neglected in this study. in post processing, percentage of luminal surface subjected to critical von mises stress (vms) level is identified. these critical stress level (class i>545kpa and class iii>475kpa) are adapted from ref. [17]. apart from this, a plot of vms gradient (ratio of difference between vms between two consecutive nodal points and the difference in distance between those two points) on the arterial inside surface along arbitrary paths are plotted to identify the locations of a high stress gradient [24]. variations of radial displacement of the arterial surface through the same paths are plotted. the plots of radial displacement of the arterial inside surface at different sections, particularly to highlight the displacements near the link locations, identify distortion of the arterial inside surface post expansion. arterial distortion is presented as difference between maximum to minimum radial displacement of the artery surface across left, central and right stent sections. 2.4. mesh density test mesh density test result (table 4) suggested a large variation in the peak vms on the artery surface but the maximum radial displacement is unaffected by element size. we choose radial displacement as criterion for a mesh density test since the peak vms is observed only at few nodal locations, whereas the maximum radial displacement on the artery surface prevailed over larger nodal locations. for the chosen criterion, though found to be independent of the mesh size, we chose a mesh size of 0.05mm for stent and 0.2mm for artery, respectively, with a linear tetrahedron element. fig. 4 shows two of such meshes used for this study.  c. bekal, s. shenoy, r. shetty table 4 summary of mesh sensitivity test on artery expansion artery mesh size (mm) stent mesh size (mm) peak vms artery (mpa) peak vms artery (elemental mean) (mpa) peak vms stent (mpa) peak vms stent (elemental mean) (mpa) artery displacement (max) (mm) stent inside displacement (max/min) (mm) number of nodes/elements 0.4 0.1 0.20 0.27 263.6 254.1 0.26 0.80/0.70 83852/15519 0.2 0.1 0.39 0.50 263.8 252.9 0.27 0.81/0.70 101784/25494 0.1 0.1 0.65 0.94 263.7 252.9 0.26 0.81/0.70 173631/66601 0.08 0.05 0.88 1.35 291.5 291.1 0.26 0.80/0.70 285783/130363 a) b) fig. 4 different meshes used for mesh density study artery-0.1mm, stent 0.1mm (a); artery 0.08mm, stent -0.05mm (b) fig. 5 plots the radial displacement of the luminal surface for a default mesh and a mesh of the chosen element size. a) b) fig. 5 radial displacement of artery inside surface default mesh (a); and refined mesh (b) numerical investigation of influence of link positioning in coronary stents inside normal artery...  3. results and discussion the following section discusses the results of investigations in the form of plots, tables and graphs. fig. 6 vms distribution on artery luminal surface (class i-0.545mpa to 0.658mpa and class iii0.475mpa to 0.545mpa) for stent s at 2mpa inflation pressure (legend unit, mpa) fig. 7 vms distribution on artery luminal surface (class i-0.545mpa to 1.65mpa and class iii0.475mpa to 0.545mpa) for stent t at 2mpa inflation pressure (legend unit, mpa) table 5 percentage of luminal surface of arterial tissues subjected to class iii critical stress level stent percentage of luminal surface above 475kpa at 2mpa inflation pressure stent s 0.27 stent t 9.1 we have observed vms concentration predominantly on the internal surface of artery segments, with a diminishing stress across section, suggesting a large percentage volume of the artery remaining much below non-physiological stress (above class iii). figs. 6 and 7 show contour plots of vms distribution on the inside surface of artery for stent s and stent  c. bekal, s. shenoy, r. shetty t, respectively. it suggests that the artery segment in line with the stent crown is the area more susceptible for restenosis (class i stress level). stent s induced considerably a smaller area of critical stress region (table 5). this can be attributed to a lower stent/artery contact area (3.77mm2) compared to that of stent t (5.5mm2). for stent s, the critical stress region is observed along the stent-artery contact region while majority of the stented surface remained much below the critical stress (between 0mpa-0.475mpa). for stent t, approximately the entire stented region is subjected to critical stress state (between 0.475mpa-1.65mpa). a) b) fig. 8 distribution of class i and class iii stress level. stent crown unbound by link (thick circle) and stent crown bound by link (thin circle) for stent s(a), stent t (b) a closer look at the location of critical stress level (fig. 8) shows that higher vms areas (class i) are essentially located near the stent crowns unbound by connecting links, while near the bound crown a more distributed stress pattern is observed. this can be attributed to the fact that the stent crown unbound by links is more likely to open up and penetrate into the artery during expansion which is also evident from fig. 9 showing prolapse of arterial tissues through the stent struts at these locations. fig. 9 artery and stent surface after expansion, indicating larger tissue prolapse through stent struts at location without link (a) and no prolapse at location with link (b) (legend unit, mm) a plot of vms gradient on the inside surface of the artery along artery length (figs. 10 and 11) suggests the existence of certain high stress gradient points. plot essentially shows the vms gradient calculated along two random longitudinal paths (path 1 and path 2) on the internal surface of the artery. incidentally, path 1 passes through the location where it encounters the stent crown unbound by link. interestingly, the locations of high vms gradient happened to be on path 1 promptly suggesting consequence of tissue numerical investigation of influence of link positioning in coronary stents inside normal artery...  prolapse as discussed previously. stent t induced a comparatively higher stress gradient (5mpa/mm) in comparison with stent s (3mpa/mm). in stent t, the links are connected at the center of the stent struts unlike in stent s where the links are connected to the stent crown, which would have resulted in a greater number of high gradient points for stent t. high stress and stress gradient stimulate the regrowth of arterial tissue through the stent strut. this phenomenon is called re-stenosis which makes revascularization inevitable. fig. 10 vms gradient on artery inside surface along artery length for stent s at 2mpa inflation pressure fig. 11 vms gradient on artery inside surface along artery length for stent t at 2mpa inflation pressure the following plots (figs. 12 and 13) show radial displacement of the arterial inside surface at three circular sections at left, right and central region stented portion of the artery. incidentally, the left and right sections happened to pass through the linked region while the  c. bekal, s. shenoy, r. shetty central section passes through a region devoid of links. plots indicates more uniform (circular) displacement for the central section compared to the left and right sections. at the left and right regions, maximum displacement is observed at the locations of links (2 diametrically opposite locations for stent s and 3 locations at 1200 apart for stent t). this finding suggests the influence of the links in achieving higher radial displacements. fig. 12 radial displacement of arterial inside surface at left, center and right sections for stent s at 2mpa inflation pressure (legend unit, mm) fig. 13 radial displacement of arterial inside surface at left, center and right sections for stent t at 2mpa inflation pressure (legend unit, mm) for stent s, the difference between maximum and minimum radial displacement is found to be 0.11mm for both the left and right sections while in the central section the difference is found to be 0. 05mm. for stent t the difference is found to be 0.15mm at left ,0.13mm at right and 0.05mm at central section, thus rightfully suggesting the necessity of a closed cell stent design (more links) to achieve circular expansion [25]. numerical investigation of influence of link positioning in coronary stents inside normal artery...  fig. 14 variation radial displacement of artery inside surface along artery length for stent s at 2mpa inflation pressure fig. 15 variation radial displacement of artery inside surface along artery length for stent t at 2mpa inflation pressure non-uniformity of radial displacement is observed along longitudinal direction as well for both stent designs. the plot (figs. 14 and 15) shows the variation of radial displacement of the arterial inside surface along longitudinal direction of the artery. radial displacement is plotted through two longitudinal paths, path 1 and path 2, so that these paths pass through the locations of the stent crown. interestingly, path 2 encountered a pair of stent crown bound by the links. the plots suggest that the radial displacements increase gradually from the ends of the artery through the stented region. the maximum displacement value is located near the stent crown for both path 1 (0.26 mm) and path 2 (0.29 mm) for stent s. larger maximum radial displacement for path 2 propose the influence of the link. the links  c. bekal, s. shenoy, r. shetty not only achieved larger maximum displacement at this location but also contributed to higher displacement in the vicinity of the stent crown. similar trend is observed for stent t. for stent t the maximum radial displacements observed are 0.29 mm for path 1 and 0.34 mm for path 2. fig. 16 below takes a closer look on displacement of the luminal surface near the stent crown with and without link for stent s. it is observed that the stent crown unbound by links is bound to misalign when expanded and may result in twisting of the artery surface damaging endothelium surface. fig. 16 radial displacement of arterial inside surface near stent crown without link (a); and with link (b) for stent s (legend unit, mm) 4. conclusion, limitations and scope for future study this study investigates the performance of two commercial stent designs inside the normal artery particularly highlighting an induced stress field, an arterial displacement pattern and the factor affecting it. the result suggests that the link contributes crucially to maintaining a uniform stress field in the stented region. the presence of the link reduced the vms gradient and maintained uniform arterial displacements as well. uniformity of expansion is greatly affected near the stent crown unbound by link. the link also contributed to lesser tissue prolapse and lesser misalignment. the results suggest that stent designs with bound stent crowns can induce diffused stress pattern and uniform arterial displacement. it can also result in minimal arterial distortions thereby minimizing detrimental effect arising from non-native biomechanical environment on tissues. 3-dimensional idealized geometry is used in this investigation with simplifications (round edges of the strut are avoided). using realistic models directly for investigation can simulate the effect of surface irregularities caused by drug coating and crimping process. idealized artery geometry used here can certainly help in differentiating performance of different designs but realistic stenosed artery obtained from imaging techniques such as intra vascular ultra sound ivus [26] can improve the investigation results in terms of applicability of stent design for specific lesion type. the boundary conditions used for avoiding rigid body motions of artery and stents in this study are simplified. it is assumed that few nodes of central stent cells move only in radial direction. in actual case, all the points on stent surface move in longitudinal and tangential directions. hence, the boundary conditions that are more realistic can improve the validity of numerical investigation and translation of numerical inferences into clinical applications will be accomplished. though we reported the values of stress and stress gradients in this study, these values are to be numerical investigation of influence of link positioning in coronary stents inside normal artery...  considered with almost care as mesh density study suggests considerable variation in maximum von mises stress for different mesh sizes. nevertheless, the pattern of stress and stress gradient can be certainly relied upon. acknowledgements: this work is supported by ta pai phd scholarship program (mu/dreg/ muscholar/phd/2014), from manipal academy of higher education (mahe). references  mathers, c.d., boerma, t., ma fat, d., 2009, global and regional causes of death, br. med. bull., 92(1), pp. 7-32.  mozaffarian, d., 2016, aha statistical update heart disease and stroke statistics — 2016 update a report from the american heart association, circulation, 133(4), pp. e38-e360.  whittaker, d.r., fillinger, m.f., 2006, the engineering of endovascular stent technology: a review, vasc. endovascular surg., 40(2), pp. 85-94.  ramakrishnan, s., mishra, s., chakraborty, r., sarat chandra, k., mardikar, h.m., 2013, 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huber, k., minar, e., 2008, does carotid stent cell design matter?, stroke, 39(3), pp. 905-909.  buccheri, d., piraino, d., andolina, g., cortese, b., 2016, understanding and managing in-stent restenosis : a review of clinical data, from pathogenesis to treatment, j. thorac. dis., 8(3), pp. 1150-1162. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 241 252 https://doi.org/10.22190/fume201228011b © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper computational microstructure-based analysis of residual stress evolution in metal-matrix composite materials during thermomechanical loading ruslan balokhonov1, varvara romanova1, eugen schwab2, aleksandr zemlianov1,3, eugene evtushenko1 1institute of strength physics and materials science, sb, ras, russia 2doppelmayr seilbahnen gmbh, austria 3national research tomsk state university, russia abstract. a technique for computer simulation of three-dimensional structures of materials with reinforcing particles of complex irregular shapes observed in the experiments is proposed, which assumes scale invariance of the natural mechanical fragmentation. two-phase structures of metal-matrix composites and coatings of different spatial scales are created, with the particles randomly distributed over the matrix and coating computational domains. using the titanium carbide reinforcing particle embedded into the aluminum as an example, plastic strain localization and residual stress formation along the matrix-particle interface are numerically investigated during cooling followed by compression or tension of the composite. a detailed analysis is performed to evaluate the residual stress concentration in local regions of bulk tension formed under all-round and uniaxial compression of the composite due to the concave and convex interfacial asperities. key words: structure of materials, computer simulation, metal-matrix composites, residual stresses, strain localization 1. introduction wear is an extremely complex process during which material surface layers are subjected to a combination of different types of loading: tension, compression and shear. non-homogeneous degradation of the material during wear is mainly caused by the geometrical stress concentration due to surface asperities [1-3] or microstructural received december 28, 2020 / accepted february 08, 2021 corresponding author: ruslan balokhonov institute of strength physics and materials science sb ras 634055, tomsk, pr. akademicheskii, 2/4 e-mail: rusy@ispms.tsc.ru 242 r.balokhonov, v.romanova, e. schwab, a. zemlianov, e. evtushenko inhomogeneity [4], which is clearly expressed, for instance, in composite materials and coatings. the formation of composite coatings on metallic parts, from one hand, improves the strength and wear and corrosion resistance of their surface layers due to reinforcing particles. from the other hand, complex hierarchical structure of the composites, containing components of intricate irregular shapes with differing physical-mechanical properties: ceramic particles, metal matrices and substrates, transition zones between the matrix and particles, polycrystalline grains, etc., results in the stress concentration and plastic strain localization. even minor changes in the structure of a composite can result in a change of the deformation and fracture mechanisms at different scale levels and appreciably affect its strength. the composite’s structural inhomogeneity gives rise to generation of residual stresses (rss) during manufacture and coating deposition [5-7], which influence the material strength and durability in its future service. solutions of these problems can be directly associated with investigations of the stress concentration and strain localization mechanisms under thermal-mechanical loading of composites by the microstructurebased numerical simulation. an advantage of a numerical analysis over an experiment consists in a possibility of examining the stressed state evolution in the bulk of the composites and during their deformation. the ceramic particles used in numerical simulations of metal-matrix composites and coatings, which explicitly include structure inhomogeneity into consideration, are mostly characterized either by an ideal round shape and the problem is solved in a two-dimensional formulation [8-11], or by a spherical shape [11-13] in a three-dimensional analysis. such important issues as, for instance, the influence of the relative positions of ceramic particles, their dimensions and volume fractions on the composite’s fracture character are reported in the current literature [8, 9]. the discrete element method using a thermal expansion model taking into account damage effects and interfacial debonding was applied in [10] to simulate the thermal induced cracking due to thermal expansion mismatch between the constituent materials. the mechanisms of particle fracture are included into consideration in [11] as the processes of pore formation and coalescence. the influence of adhesion at the particle – matrix interface is reported in [12, 13]. in particular, tension of a composite with a single particle is simulated in [12], and tension-compression of a composite is numerically investigated in [13], taking into account the residual stresses developing in the composite after its quenching into water. a comparison of the stress and plastic strain distribution is performed including adhesion and without adhesion at the interfaces taken into consideration. an elastoplastic model based on a unit cell consisting of 8 subcells representing elastic particle and plastic matrix materials is developed in [14] to compute elastic moduli and residual stresses at both the constituent and composite levels. there are much fewer studies taking into account nonideal shapes of the reinforcing particles. 2d model metal-matrix composite with an interface zone between polyhedral ceramic particles and the metallic binder is studied in [15]. in [16], the effect of the composite's two-dimensional structure with polyhedral particles on the local characteristics of temperature distributions under thermal loading is discussed. meaningful simulation of the composite’s strain and damage in a three-dimensional structure with a few particles of regular polyhedral shapes is carried out in [17], without including any residual stresses, while in [18] effective elastic moduli and the coefficient of thermal expansion for the aluminum matrix reinforced by the particles of equal icosahedral shapes and sizes are predicted. plastic strain localization and fracture across multiple spatial scales in metal-matrix composite and coated materials are investigated by solving two[19-21] and three-dimensional [22] problems without taking into computational microstructure-based analysis of residual stress evolution in metal-matrix composite...243 consideration cooling-induced rss, while in [21] the 3d numerical analysis of the residual stress formation under cooling is performed. the purpose of this work is to develop a technique for designing three-dimensional structures of metal-matrix composites and coatings including complex irregular shapes of reinforcing particles into consideration, which is observed in the experiments. furthermore, using a computer simulation of cooling followed by compression or tension as an example, the aim is to investigate in detail the residual stress concentrations and their influence on the strength of the composites fabricated by ex-situ technologies, where the pre-synthesized particles are embedded into the matrix during a secondary process such as selective laser melting, stir casting, solid state sintering during hot pressing, etc. 2. microstructure generation technique for composite and coated materials the initial step in the microstructure-base analysis is the development of procedures for an explicit inclusion of their three-dimensional structure into the model. in this study we present a straightforward method for constructing numerical models of metal-matrix composites. the reinforcing carbide particles have irregular angular shapes qualitatively similar to rock fragments (fig. 1). therefore, in developing geometrical models of composite materials and coatings having complicated microstructure it is proposed to use macroscopic naturally broken rock stones. this approach allows assuming a scale invariance of the natural mechanisms of spalling during mechanical grinding. fig. 1 examples of the boron (a), titanium (b) and tungsten carbide particles (c) and chip stones (d) for this procedure, a few hundred chip stones were arbitrarily selected, which measured from 80 to 100 mm in their maximum section. in order to transform the stone surface into a 3d digital analog 3dsom pro software was used, which is readily available and sets minimum requirements to the hardware tools. the following equipment was used in obtaining the initial data: illuminated rotary stage, photo-studio for filming the specimen surfaces, additional light sources, and camera on a tripod. all specimens were coated with thin layers of white paint in order to speed up the process of automated determination of their contours. after performing the required adjustment, about a hundred frames were obtained from every evenly rotating stone specimen in equal time intervals (fig. 2a). 244 r.balokhonov, v.romanova, e. schwab, a. zemlianov, e. evtushenko fig. 2 photo of a stone (а), numerical model of the stone-particle (b) and the particle embedded into a cubic aluminum microvolume (c), different side views of another particle (d-f) and models of the composite (g) and coated materials (h) then using 3dsom pro software 3-d models of these specimens were constructed. their photographic images were imported as the initial data of a new project. the specimen surfaces were recognized in the manual and semiautomatic modes and separated from the image background and shadows, thus masks were created from the objects under study. following this, the process of designing a three-dimensional model continued in an automatic mode via the recognition of the camera and specimen positions in space. for every specimen, a 3-d model was designed, wherein the number of surfaces was varied from 1000 to 10000 depending on the surface geometry complexity, so that no fewer than 50 facets corresponded to a structure irregularity. examples of the stone models are presented in figs. 2b, 2d-f. using an stp file format, the resulting threedimensional surfaces of stones-particles were imported into abaqus and by a random spread of csys orientations and coordinates of the stone centers three-dimensional material structures with particles of irregular shapes of different scales (figs. 2c, 2g and 2h) were designed. computational microstructure-based analysis of residual stress evolution in metal-matrix composite...245 3. special aspects of residual stress formation in composites during cooling followed by mechanical loading consider the approach of multilevel numerical simulation with an explicit inclusion of the material structure by the example of evolution of residual stresses (rss) generating in metal-matrix composites and coatings during their manufacture and subsequent use. a special attention would be given to the formation of local regions subjected to tensile stresses. it is in these regions that local defects and cracks can nucleate under highintensity thermo-mechanical loading, resulting in an early loss of macroscopic strength of the composites and parts manufactured from them. the formation of tensile regions under external compressive loading is a controversial issue requiring detailed studies. in this work in order to distinguish between the individual mechanisms of stress concentration under different types of thermo-mechanical loading three series of numerical experiments on deforming aluminum–titanium carbide composite were successively performed: 1. uniaxial compression and tension of a specimen without its preliminary cooling (not including rss into consideration). the results and conclusions made in this part are valid for the specimens subjected to, for instance, stress-relieving treatment. 2. cooling from the melt to room temperature, which is often the case during manufacture of the composites under the conditions close to those of all-round compression. 3. cooling followed by uniaxial compression or tension of the composite. the focus was on the structure presented in fig. 2с, with the volume fraction of the particle material being 1 %. it should be noted that in this case all the results obtained are valid for the composites and coatings with a small volume fraction of particles in the absence of their mutual influence. three-dimensional static problems were solved numerically by the finite-element method using abacus. in the simulation of thermal loading, all faces of the representative volume element (rve) presented in fig. 2с are free from loading. we consider a steady-state process, wherein the temperature in all local regions of the rve, including the matrix and particle materials, in every loading step is taken to be equal and is linearly decreased from the melting to room temperature – from 660 to 24 degrees celsius. unlike a purely mechanical problem, during thermal loading the duhamel – neumann’s relations are solved, for which an additional contribution into the cubic part of the stress tensor is included into consideration – in calculating the cubic strain a term responsible for thermal expansion is added 0 2 ( ) 2 3 ( ) 3 ij ij kk ij ijk k t t      = − + − − the results of simulations are aggregated in fig. 3; they allow making the following conclusions. firstly, it has been determined that local regions undergoing bulk tensile stress appear under compression of composites. these regions are formed as early as in the elastic stage of composite deformation in the areas of concavities of the interface between the less stiff matrix and stiffer particle, with the maximum pressure values observed in the concavities whose position is perpendicular to the direction of compression (fig. 3a, c-type regions). these are the most vulnerable regions of possible nucleation of defects and cracks under high-intensity loading. secondly, comparing the calculations for the elastic matrix-elastic particle and plastic matrix-elastic particle composites, a conclusion can be drawn that the volume fraction of tensile regions during uniaxial compression is larger in the case of plastic matrix than it is if the matrix is 246 r.balokhonov, v.romanova, e. schwab, a. zemlianov, e. evtushenko assumed to be elastic (cf. figs. 3а and 3b). generalizing, this means that, other conditions being equal, single cracks are formed inside the particles in the ceramic composites, while in metal-matrix composites multiple cracking of the particles is possible. fig. 3 pressure in the elastic tic particle under different loading of the al-tic composite shown in fig. 2c: compression for the cases of elastic (a) and plastic matrices (b), cooling from 660°с to room temperature (c), tension (d) and cooling followed by compression (e) and tension of the composite (f) a pressure distribution pattern in a ceramic particle during cooling is shown in fig. 3с: the composite is compressed by about 1.6 %, i.e., to the same degree to which uniaxial compression was simulated (fig. 3b). it has been determined here that unlike uniaxial compression, in which case different regions of the particle are subjected to both negative and positive pressures, during cooling of the structure no regions of bulk tension are formed in the particle at all, they are rather concentrated solely in the matrix (which will be shown in what follows). this conclusion drawn from the results of numerical simulations is entirely consistent with the experimental data [3]. moreover, the average level of pressure is higher under all-round compression (cooling) than during uniaxial compression. the respective pressure and equivalent plastic strain distributions in the aluminum matrix are presented in fig. 4, showing only the matrix material without the particle. computational microstructure-based analysis of residual stress evolution in metal-matrix composite...247 fig. 4 pressure and corresponding equivalent plastic strain patterns in the aluminum matrix under different conditions: cooling (a) and cooling followed by compression (b) or tension of the composite (c) 248 r.balokhonov, v.romanova, e. schwab, a. zemlianov, e. evtushenko for the case of cooling it is shown that the area of the particle influence on the stressed state in the matrix is on the order of an average particle diameter (fig. 4a). figure 6a is also a good illustration of the above conclusion that under the conditions of all-round compression a local bulk tension region (red color) is formed in the matrix only. it has to be noted that the equivalent stress in this region is minimal, and aluminum is not plastically deformed here (fig. 4a). in the cases of a combined thermomechanical loading, the external tensile or compressive loads would be added to already existing stresses in the material caused by cooling, thus forming the patterns differing from each other (cf. figs. 3e and 3f). the main difference consists in the fact that the stresses in the c-type region are considerably higher in the tensile case than under compression. there is a more unexpected and exciting conclusion resulting from a comparison of the states formed during compression and tension for the cases where residual stresses are and are not taken into account. it is clear that the patterns in the c-type region under compression are very similar (cf. figs. 3b and 3e), while under tension they are quite different (cf. figs. 3d and 3f). therefore, a preliminary qualitative conclusion is as follows: residual stresses have little effect on the stress concentration in the c-type region during compression and strongly affect it under tension. this conclusion is quite natural for the tensile case, though least important. it is natural because the compressive stresses, generated in the c-type region during cooling (fig. 3c), are enhanced by the compressive loads appearing in this region during tension (fig. 3d). it is least important, because these stresses are compressive, so the particle would not be fractured despite the high level of the equivalent stress in these regions. hence, the first part of this interim conclusion becomes a preliminary general conclusion – residual stresses have little effect on the mechanical properties of the composites. at a first glance, this conclusion is odd. seemingly, the compressive loads, developing in the particle during cooling (fig. 3 c), would counteract the tensile stresses generated in the ctype region in the course of subsequent uniaxial compression (fig. 3b) and reduce the hazardous stress concentration in these areas. then there is a question: why does not the concentration finally decrease (cf. figs. 3b and 3e)? an answer to this question required a more detailed analysis of the evolution of the maximum stress concentration formed in the c-type region of the particle. the results of this analysis are aggregated in fig. 5. the evolution of stress concentration during tension without preliminary cooling (without rss) or during cooling followed by tension (with rss) is depicted in red curves; compression or cooling followed by compression are shown by black curves. zero strain corresponds to the state of the melt before cooling. there is all-round compression to about -1.6 %, resulting from cooling, where the red and black curves coincide. then from the state at -1.6 % the structure is additionally loaded by compression or tension. for the sake of convenience, the dependences for tension or compression without rss also start at -1.6 %. additionally the stress concentration evolution in the matrix material near the c-type region is presented by circles for the cases of cooling followed by tension or compression of the composite. it is clear that the stresses under tension are on the whole higher than under compression (compare the black and red curves in fig. 5); note that the stress values in the entire range of composite elongations are by 1.5 – 4 factors higher for the case where rss are taken into account than where they are not (compare red triangles and squares in fig. 5). for the above-described reasons, the plots under tension are less appealing from the practical perspective, since region c is under the conditions of bulk compression – computational microstructure-based analysis of residual stress evolution in metal-matrix composite...249 pressure is positive in the entire range of strains (fig. 5a, red curves). from the fundamental viewpoint, the following facts are interesting. in the initial stages of tensile loading after cooling, the stresses increase, then drop, and after that they start gradually increasing again. the sharp drop of the stresses results from plastic flow in the aluminum matrix, during which the stressed state is rearranged from all-round compression to uniaxial tension. as soon as this adjustment to the uniaxial tensile conditions is completed, the local compressive stresses in the c-type region of the particle start monotonically increasing again. it is seen in fig. 5 that the stress decreases in the aluminum matrix near the c-type region, which was caused by the rearrangement, then on further tension its decrease terminates, the stress remains constant, with its value being considerably lower than the yield point (fig. 5b, red circles). this suggests that the matrix material in the c-type region after adaptation to the uniaxial tension conditions remains in the elastic stage of the deformation; it is in the ‘silent’ zone not suffering from any additional loads. fig. 5 maximum pressure (a) and equivalent stress (b) in the c-type region of the particle shown in fig. 3 and in the corresponding near-interface region of the matrix 250 r.balokhonov, v.romanova, e. schwab, a. zemlianov, e. evtushenko during cooling followed by external uniaxial compression, there is a completely different situation. during cooling the local compressive pressure increases in the c-type region (fig. 5a, black-red triangles). then after additional loading by uniaxial compression, the conditions of bulk tension start forming in this region. the pressure that increased during cooling begins sharply decreasing, while remaining to be positive, i.e. compressive (fig. 5a, black triangles). at a certain point of compression equal to -2.1 %, the pressure becomes negative and region c transfers into a state of bulk tension. starting from this time point the equivalent stress stops decreasing and starts increasing (fig. 5b, black triangles). this point corresponds to the minimum equivalent stress, about 250 mpa, which is by a factor of three lower than that under the conditions of compression without including residual stresses into account (cf. black triangles and squares in fig. 5b). comparing the black curves with and without rss being taken into consideration (cf. black triangles and squares in fig. 5 a), it is evident that the rate of pressure decrease in the initial stages of structure compression after cooling (particle, with rss in fig. 5a) is considerably higher than if the structure was compressed from the initial state without preliminary cooling (particle, without rss in fig. 5a). upon further deformation, the rates of pressure drop equalize and these curves converge, demonstrating an equal level of stress concentration in the c-type local regions of bulk tension. as this goes on the aluminum matrix material undergoes plastic flow, maintaining the equivalent stress at a level of the current yield stress (fig. 5b, black circles). the residual stresses therefore play a positive role in the initial composite loading stages (up to 3-4 % compression), essentially reducing the level of hazardous stresses in the local regions of tension (indicated by a dashed circle in fig. 5b) and do not strongly affect the stressed state at larger strain degrees. 4. conclusions an example of using the microstructure-based numerical simulation technique for investigating the formation and evolution of residual stresses in metal-ceramic composites has been discussed. a numerical model has been proposed for constructing three-dimensional structures of materials with particles of irregular shapes. using the proposed technique, structures of composites and coatings of different scales have been designed. the details of strain localization and stress concentration in the interface regions have been investigated numerically during cooling from the melt to room temperature and subsequent uniaxial loading of the aluminum – titanium carbide composite. the principal conclusions drawn from the numerical simulations are the following: 1. in the composites subjected to compression local regions of bulk tension are formed near the interfaces. in the cases of uniaxial compression such regions are observed both in the matrix and in the particles, while under all-round compression – in the matrix only. the volume fraction of tensile regions in the particles is larger and the concentration of stresses in them is lower during plastic flow in the matrix than in the case where the matrix deforms elastically. to sum up, fracture in metal-matrix composites occurs later and can be characterized by multiple cracking of the particles, while in ceramic composites it takes place earlier and is associated with propagation of isolated cracks. stress concentration is by a factor of 1.5 times higher under all-round than under uniaxial compression. computational microstructure-based analysis of residual stress evolution in 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evtushenko, e.p., romanova, v.a., schwab, e.a., bakeev, r.a., emelyanova, e.s., zinovyeva, o.s., zinovyev, a.v., sergeev, m.v., 2020, formation of bulk tensile regions in metal matrix composites and coatings under uniaxial and multiaxial compression, physical mesomechanics, 23(2), pp. 135-146. 22. balokhonov, r., romanova, v., kulkov, a., 2020, microstructure-based analysis of deformation and fracture in metal-matrix composite materials, engineering failure analysis, 110, 104412. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 187 198 https://doi.org/10.22190/fume210102006r © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper mesoscale deformation-induced surface phenomena in loaded polycrystals varvara romanova, ruslan balokhonov, olga zinovieva institute of strength physics and materials science, tomsk, russia abstract. the paper reviews the results of numerical analyses for the micro-and mesoscale deformation-induced surface phenomena in three-dimensional polycrystals with the explicit account for the grain structure. the role of the free surface and grain boundaries in the appearance of the grain-scale stress concentrations and plastic strain nucleation is illustrated on the examples of aluminum polycrystals. special attention is paid to the discussion of mesoscale deformation-induced surface roughening under uniaxial tension. key words: polycrystals, free surface, grain boundaries, deformation-induced roughness, microstructure-based calculations, mesomechanics 1. introduction the mechanical and physical properties of material surface layers are among the key factors determining the service life of machine parts and components. this is especially evident for tribological systems where the load is directly applied to the material surface through the contact region to cause wear of the surface layers [1-4]. while it is not so evident in the absence of external forces, the free surface still plays a vital role in the deformation behavior of materials. many surface phenomena observed in real materials take their origin from the microstructural effects and thus can hardly be described within a classical phenomenologically-based approach of continuum mechanics. for example, stresses and strains in a homogeneous specimen in the absence of geometrical nonlinearities (notches, necks, surface roughness, etc.) are distributed uniformly over the specimen sections so that cracks have even chances to nucleate near the free surface or in the bulk. however, experimental studies (e.g., [5-8]) have reported that cracks under tensile or cyclic loadings primarily nucleated on the surface or within a thin subsurface layer. in order to describe this effect, we should go down from the macro to lower scales where the material is inhomogeneous. received january 02, 2021 / accepted january 27, 2021 corresponding author: varvara romanova institute of strength physics and materials science sb ras, akademicheskii prospect 2/4, 634055 tomsk, russia e-mail: varvara@ispms.tsc.ru 188 v. romanova, r. balokhonov, o. zinovieva another prominent example of the microstructure-related surface phenomena is deformation-induced roughening, inevitably accompanying plastic deformation in polycrystalline metals and alloys. the specimen free surface becomes rough even in the conditions of uniaxial or biaxial tension where no external forces act perpendicular to the specimen surface to cause any out-of-plane surface displacements [8-11]. again this behavior is failed to be predicted by macroscopic continuum mechanics but should be described in terms of micromechanics to take into account non-linear deformation modes attributed to the microstructural inhomogeneity. this paper reviews the results of numerical analyses for the free surface effects on the stress-strain evolution in quasistatically loaded polycrystals where the 3d grain structure is taken into account explicitly. special attention is given to the examination of surface phenomena at the mesoscale where individual grains are united into clusters to form rough patterns on the free surface. 2. microstructure-based simulation let us briefly discuss the computational approach in use while its detailed description is given elsewhere (see, e.g., [12-14]). the numerical simulation involves the finitedifference [12] or finite-element solutions [13-15] to the dynamic boundary-value problem for grain conglomerates. in the numerical implementation, mesh elements are united into sets to represent different grains. the elements belonging to a certain grain have the same physical and mechanical characteristics and contiguously occupy a closed region to fit the grain shape. neighboring grains interact with each other through a common boundary going through nodal points shared by the boundary elements. the adjacent grains are thought to be in permanent contact during the entire deformation process (i.e., cracks are supposed not to appear). in the numerical examples presented in this paper the grain structures were constructed with the step-by-step packing method providing fast generation of 3d microstructures on a hexahedral mesh [16]. a series of calculations was performed for polycrystals containing different numbers of equiaxed grains approximated by hexahedral meshes with different resolutions. the boundary-value problem was solved by the finite-difference method using inhouse software package dlhm-3d. two basic approaches involving either phenomenological [12, 13] or crystal plasticity models [17, 18] are commonly used to describe the grain constitutive behavior. in the former case, the grains are associated with a set of the stress-strain curves scattered about an average curve fitting experimental data for the polycrystalline material. in such a simplified way, the orientation dependence of grain ability to resist deformation is taken into account. crystal plasticity models, in their turn, rely on the physical mechanisms of plastic deformation related to a crystalline structure. along with the grain-scale inhomogeneity, the models are capable of describing an anisotropic mechanical response at the intragrain scale. the crystal plasticity formulation is particularly important for polycrystalline materials with a limited number of slip systems (e.g., hexagonal metals) or with a pronounced crystallographic texture. however, in the case of non-textured polycrystals with a cubic crystal lattice, the two kinds of models provide similar results, and the use of simplified models is quite feasible. in what follows, we analyze the results calculated for nontextured aluminum polycrystals in terms of the phenomenological description. mesoscale deformation-induced surface phenomena in loaded polycrystals 189 the polycrystalline grains in real materials are distinguished by their crystallographic orientations and, consequently, by their elastic and plastic properties. however, the difference between the elastic moduli of grains is frequently ignored since the elastic strain in metals is moderate compared to possible plastic strain. at the same time, this can be an important factor governing the grain-scale stress concentration and thus affecting the nucleation of plastic shear strains. in our calculations, the difference in the elastic characteristics of grains is accounted for by a random scatter of the elastic moduli about the mean within 10%. the elastic-to-plastic transition of grains is calculated in terms of the von mises yield criterion where the yield stress values are determined with allowance for the strain hardening and grain-boundary strengthening ( )0 i i p y eqi k f d   = + + , (1) where 0 i is the initial yield stress of the ith grain. in the numerical implementation the 0 i values for different grains were scattered within 20% about 10 mpa. the second term of the right-hand sum is the hall-petch relation, where 875 mpa mk =  is the coefficient of grain-boundary strengthening and id is the ith grain diameter calculated as a diameter of a sphere of the same volume. the third term of the sum is the strain hardening function constructed to fit the experimental data for an al6061-t3 ( ) (1 exp( / ))p p eq eqf a b= − −  . (2) here p eq is the equivalent plastic strain accumulated in a local region (mesh element); a = 65 mpa and b = 0.048 are the fitting constants. in order to simulate uniaxial tension, two opposite surfaces of the model specimens were specified to undergo axial displacements. the top surface was thought to be free of external forces, and the bottom surface was a symmetry plane. the lateral sides in the calculations were set to be either free surfaces or symmetry planes. 3. grain-scale stress-strain evolution in a subsurface layer the interfaces between materials with different elastic-plastic properties are known to be the sources of stress concentrations under loading [17, 19-22]. the difference in the material characteristics, the interface curvature and the loading conditions are among the governing factors responsible for the stress values in the vicinity of interfaces. noticeably, the difference in the elastic moduli of individual grains results in stress concentration near the grain boundaries already at the elastic stage of loading. even a small increase in local stresses in the near-boundary regions in comparison to the average stress level favors plastic strain nucleation here. the larger the difference in the mechanical characteristics of contacting grains, the higher the stress concentration near the boundary between them. the most favorable conditions for the nucleation of plastic strains are realized in the vicinity of triple junctions of grains markedly different in their elastic modules. the specimen surface can be treated as an interface between materials with essentially different mechanical properties (a metal-air interface). thus, in combination with geometrical 190 v. romanova, r. balokhonov, o. zinovieva irregularities, it can be the most powerful source of stress concentration. this agrees with the experimental and theoretical studies [5-11, 17] showing that the specimen surface plays a key role in the appearance of stress concentrations and plastic strain nucleation. this conclusion is supported by the calculation results for the polycrystalline conglomerate consisting of 280 grains (fig. 1b). the model size is 480×375×480 µm. the equivalent stresses in local regions of the surface layer become higher than those developing in the bulk grains from the very beginning of tension. as an illustration, the stress and plastic strain patterns in the surface layer and in a section located at a depth of 250 µm below the surface are plotted in fig. 2a and b. a scatter of local stresses and strains about their mean values is more pronounced in the surface layer than in the bulk of the material. this is the reason for first plastic deformation to nucleate on the surface as it is observed experimentally [5-8]. -50 0 50 100 0 1 2 3  11  22  33  12  13  23  ij , mpa f re q u e n c y , % a b fig. 1 frequency distributions of the stress tensor components (a) calculated for the polycrystalline model (b) under uniaxial tension along the 1x -axis let us analyze the frequency distributions of different stress and strain tensor components plotted in fig. 1a. it is not surprising that the stress and plastic strain tensor components along the tensile axis make a major contribution to the stress-strain state. the average values of other stress and strain tensor components have to be zero, which corresponds to macroscopic conditions of uniaxial tension. however, in local regions, particularly near the grain boundaries, all components of the stress and strain tensors take on nonzero values. what is more, they exhibit a symmetrical deviation of positive and negative values about the zero level. in other words, tensile regions alternate with compressive ones. the reason for the scatter of the stress and strain values about the zero mean is as follows. macroscopically, the stress components acting perpendicular to the tensile axis must approach zero because external forces do not act on the specimen in these directions. the local tensile and compressive regions compensate each other to provide overall zero stress. thus the macroscopic condition of mechanical equilibrium is satisfied. the same conclusion is valid for the strain tensor components whose average value is zero. a maximum scatter of local values of the stresses and strains about the mean is observed in the subsurface layer (fig. 2c). the scatter exhibits a nonlinear decrease as the depth below the surface increases, and levels out at a certain depth. mesoscale deformation-induced surface phenomena in loaded polycrystals 191 0 100 200 300 300 450 600 f re e s u rf a c e distance below free surface, m mean maximum minimum -60 -30 0 30 -60 -30 0 30  e q m p a  3 3 m p a  2 2 m p a 300 450 600  1 1 , m p a a b c fig. 2 equivalent stress and equivalent plastic strain distributions in the surface layer of polycrystal (a) and at a depth of 250 µm below the surface (b) and the maximum, minimum and mean values of stress tensor components at different depth below the free surface at 2% strain 4. “stick-slip” mode of plastic deformation a detailed study of plastic strain nucleation near grain boundaries is conducted for the model containing 10 grains approximated by a regular 100×100×100 mesh with a step of 1 µm (fig. 3a). the mechanical properties of grains in fig. 3 correspond to the color scale as follows: the darker the grain, the lower the elastic moduli and the yield stresses. in macroscopic terms, the model under consideration is not representative but it allows investigating the nucleation and development of plastic shear strains at the stage of microplastic deformation with reasonable time and space resolutions. the nucleation and propagation of microplastic strains are described with allowance for a “stick-slip” mode of plastic deformation, basing on the experimental observations for dislocation behavior. it has been shown in [23] that dislocations initially presenting in some steels and aluminum alloys are immobile. the stress necessary to start off the movement of existing dislocations or to nucleate new defects is assumed to be higher than the stress required to keeping propagation of the dislocations already involved in motion. a similar “stick-slip” mode of plastic deformation is typical for, e.g., lüders band propagation [24-26] or plc effect [23] where an extra stress is necessary to detach dislocations from interstitial atoms. an analogy can be drawn with friction [1], where the static friction force must be overcome by certain applied forces to disrupt the surface bonds between contacting bodies, whereupon their subsequent relative motion may occur at a lower force. in the numerical implementation, such a “stick-slip” dislocation behavior was simulated through a two-limit yield criterion [26], according to which two critical values are introduced in the von mises yield criterion. the higher stress limit, h, is specified for an elastically deformed 192 v. romanova, r. balokhonov, o. zinovieva material and the lower value, l, for a material involved in plastic deformation. mathematically, 0 in eq. (1) is set equal to h for the regions where 0p eq = and 0 = l if 0p eq  . as soon as a local region of the material becomes plastic, the yield stress drops down to the lower limit value and subsequent plastic deformation proceeds at the lower stress. at the elastic loading stage, the grains exhibit nonuniform stress distributions due to the difference in the elastic moduli of contacting grains. the maximum stress develops near the triple junctions of grains with greatly different elastic properties (fig. 3c). these regions are sources of plastic strain nucleation in the bulk of the polycrystalline material. one more stress concentration source and hence a plastic strain source is larger-scale geometric features — corner points and edges of the polycrystalline model. a b c fig. 3 polycrystalline model (a) and stress concentration in the vicinity of triple junction of grains (c) in the internal section (b) figure 4 illustrates the evolution of plastic flow in a bulk section and on the surface of the polycrystalline aggregate. first, plastic strains appear in isolated regions mainly near the parts of grain boundaries where higher stresses take place. the plastic strains at this loading stage have but a little effect on the deviation of the macroscopic stress-strain curve from the linear behavior, and thus this deformation stage can be classified as microplastic. on further loading, plastic strains in the form of localized shear bands begin to propagate from the near-boundary sources into the bulk of grains. simultaneously, new grain boundary regions become involved in plastic deformation. the analysis of shear band propagation with a high time resolution reveals a vortex motion in the elastic material ahead of the plastic front, which is well seen in the velocity field in fig. 4b at a strain of 0.35%. this motion is necessary to accommodate the elastoplastic deformation of the neighboring regions. the existence of the vortex patterns has been predicted by molecular dynamics simulations in [27, 28] and got experimental support in [5, 6]. at this deformation stage, the stress concentration and plastic strain patterns are still affected by the grain geometry, whereas the stress-strain curve approaches the macroscopic yield point. finally, some of the plastic fronts propagating from the grain boundaries form a through-the-thickness network on a larger scale. subsequent plastic deformation mainly localizes in these regions, resulting in crystal separation into several parts shifting to each other on further loading. new regions of plastic strain nucleation are unlikely to appear at this deformation stage and the finer bands stop developing either. correspondingly, the microplastic deformation stage gives way to the macroplasticity where the stress-strain curve goes over to the horizontal portion. mesoscale deformation-induced surface phenomena in loaded polycrystals 193 a b fig. 4 plastic strain patterns and velocity vector fields on the surface (a) and in an internal section (b) of a polycrystal at the microplastic deformation stage 5. mesoscale deformation-induced surface roughening the phenomenon of deformation-induced surface roughening acquires a special significance in tribological applications. being one of the essential factors influencing friction and adhesion (see, e.g., [1, 2, 29, 30]), this phenomenon can be treated as a positive or negative effect depending on the requirements on operative conditions. along with other methods to control adhesion and friction behavior (see, e.g., [30, 31]), the idea of controllable roughening sounds attractive which requires a deep understanding of the roughening mechanisms at different scales. the deformation-induced roughening is especially pronounced in polycrystalline metals where the surface out-of-plane displacements develop throughout all length scales from micro to macro [8-11]. a special role belongs to the mesoscale roughening, where the surface undulations occasionally take a form of ridges, ropes, wrinkles, checkered distributions of hills and hollows, etc. the width of the surface folds for different materials is reported to vary from several tens to several hundreds of micrometers while the height evolves from a few to several tens of microns in the course of deformation. of fundamental importance is the fact that not individual grains but their conglomerates are involved in cooperative motion to form the surface folds. thus, the mechanisms of the mesoscale surface roughening can hardly be described either in terms of dislocation theory or within macroscopic continuum mechanics alone. in this contribution, the evolution of mesoscale surface morphology is examined numerically in terms of mesomechanics. 194 v. romanova, r. balokhonov, o. zinovieva 0 2 4 6 8 10 0 200 400 600 =8.1% =1.1% =0.1% =0.01% s tr e s s , m p a tensile strain, % a b fig. 5 polycrystalline model (a) and its stress-strain curve (b) fig. 6 surface roughness in the polycrystalline model at a tensile strain of 0.01(a), 0.1(b), 1.1(c), and 8.1% (d) mesomechanics [32] treats inhomogeneous materials with interfaces of different nature and configurations. the presence of a microstructure gives rise to specific nonhomogeneous deformation modes at larger scales. we suggest that the term “mesoscale” does not imply a certain length-scale of investigation in contrast to, e.g., terms of “micromechanics” or “nanomechanics”, but relates to the specific phenomena produced by collective low-scale structural effects. for example, the mesomechanical approach is successfully applied in geodynamical problems [33], where representative structure elements are enormous tectonic plates. contrastingly, a nanomaterial where a characteristic length scale is about several nanometers can be considered within the mesomechanical approach as well [34]. in polycrystalline metals, the mesoscale phenomena are attributed to the cooperative effects of grain conglomerates. mesoscale deformation-induced surface phenomena in loaded polycrystals 195 fig. 7 equivalent plastic strain (b) and roughness patterns (c, d) on the free surface of the polycrystalline specimen (a) at a tensile strain of 0.4%; the boundary conditions at the lateral sides simulate the symmetry planes (b, d) or free surface (c) figure 6 illustrates roughness formation at the initial stage of tension on the top free surface of the polycrystalline model measuring 1000×500×500 µm presented in fig. 5a. corresponding deformation states are depicted on the stress-strain curve in fig. 5b. the material free surface is getting rough already at the elastic deformation stage (fig. 6a), although these kinds of surface undulations are difficult to be identified experimentally. one can reveal two distinct roughening patterns – the finer is attributed to the relative shift of individual grains, and the larger scale folds can be classified as mesoscale roughness. in the latter case, groups of grains located close to the specimen surface are set in the cooperative motion perpendicular to the free surface to form extended ridges and valleys. the mesoscale folds going through the entire surface of the specimen at a certain angle to the tensile axis exhibit a spatial periodicity. the transverse size of the mesoscale surface ridges and valleys exceeds the average grain size by severalfold, which agrees with experiments. obviously, the microstructure plays a primary role in the formation of the mesoscale roughening with the subsurface layer of grains making a major contribution to this process. the grain influence on the roughness pattern diminishes nonlinearly with the depth below the free surface. calculations where the thickness of the computational domain was varied revealed that grains lying within two average grain diameters below the free surface were a major contributor to the formation of mesoscale surface folds. for a larger thickness of the polycrystalline models, the deformation relief formed on the free surface remained nearly unchanged. one more factor which may affect the free surface pattern is the method of load application. in two sets of calculations for the polycrystalline model of 1500×500× 1500 µm shown in fig. 7a the lateral surfaces were set to be symmetry planes (fig. 7d) or 196 v. romanova, r. balokhonov, o. zinovieva free of a load (fig. 7c). in the former case, the top free surface shows a tendency to form longitudinal folds perpendicular to the load direction (fig. 7d), whereas in the latter the surface folds are formed at an angle of 45 degrees to the axis of tension (fig. 7c). 6. conclusion the mesoscale deformation-induced surface phenomena have been numerically analyzed on the examples of three-dimensional polycrystals under uniaxial tension. the equiaxial grain structures typical for aluminum alloys were implemented in the calculations. the three-dimensional formulation of the boundary-value problem enabled direct simulating the free surface effects in the polycrystals under deformation. in accordance with experimental evidence, the calculation results showed plastic strain nucleation primarily near the grain boundaries which were the sources of stress concentrations at the grain scale. the highest stress level is observed near the triple junctions of grains with strongly different mechanical characteristics. the stress and strain tensor components acting along the axis of tension made a major contribution to the deformation response both at the microand macroscales. while the rest stress and strain components were zero at the macroscale, they demonstrated positive and negative values deviating about the zero level at the grain scale with the highest quantities taking place near grain boundaries. the local tensile and compressive regions compensated each other to satisfy the macroscopic condition of mechanical equilibrium. the specimen surface was found to be the most powerful source of stress concentrations and thus played a key role in the nucleation of plastic strains. a scatter of local stresses and strains about their mean values was considerably more pronounced in the subsurface layer than at a distance below the free surface. this was the reason for plastic deformation to originate on the specimen surface which is consistent with experimental observations. finally, the appearance of mesoscale surface roughening at the initial stage of plastic deformation was analyzed. it was shown that originally flat free surface began to experience mesoscale morphological changes at the outset of the loading process. the calculation results suggested that the polycrystalline microstructure in the subsurface layer was responsible for the formation of the mesoscale surface folds. the width of the resulting folds exceeded the characteristic size of polycrystalline grains and interfacial asperities. the height of the surface folds increased in the course of deformation, but the 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pp. 376–383. 12430 facta universitatis series: mechanical engineering vol. 22, no 4, 2024, pp. 759 772 https://doi.org/10.22190/fume240118018y © 2024 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas storage hai-yan yang1, yu wang1, sen-lin liu1, qing tu2, gang chen1 1school of mechanical engineering, xihua university, sichuan, china 2chengdu technological university, school of intelligent manufacturing, chengdu, sichuan, china orcid ids: hai-yan yang https://orcid.org/0000-0003-4363-505x yu wang https://orcid.org/0000-0001-5147-7931 sen-lin liu https://orcid.org/0009-0007-7626-8731 qing tu https://orcid.org/0000-0003-3332-2412 gang chen https://orcid.org/0009-0005-4919-7830 abstract. the sonar detection of salt cavern gas storage (scgs) has a low accuracy due to sound wave attenuation. to solve the problem, this paper analyzes the attenuation features of sound waves in scgs, based on the mechanical wave equation and urick sound wave attenuation theory, as well as expression of sound velocity in dilute solution and the general empirical formula of solution density. specifically, three attenuation forms of sound waves in dilute solutions with different concentrations were studied, and a theoretical model for the total attenuation of sound waves in scgs was established. the model was adopted to explore the effect of solution temperature and concentration on sound wave attenuation. the results yield some interesting findings. it is shown that the temperature has a small overall effect on the attenuation of sound waves. also, the total attenuation coefficient of sound waves in suspensions with different concentrations increases with the frequency of the sound waves. furthermore, when the frequency of the sound waves remains unchanged, the total attenuation coefficient increases with the concentration of the suspension. finally, when the solution concentration is less than 10%, the total attenuation coefficient of the sound waves depends on scattering attenuation, and the sound wave attenuation is not greatly affected by the viscous attenuation and thermal attenuation. four salt solutions were tested to verify the correctness of the theoretical research. the experimental results show that, in the scgs environment, the salt solutions are ranked in a descending order of the influence over sonar detection accuracy starting from potassium chloride solution, via magnesium sulfates solution and calcium chloride solution to sodium chloride solution. the research provides a strong theoretical guidance for sonar detection of scgs, and a powerful engineering reference for sonar detection in brine. key words: salt cavern gas storage, sonar detection, acoustic attenuation, brine, wave frequencies received: january 18, 2024 / accepted april 11, 2024 corresponding author: yu wang school of mechanical engineering, xihua university, 9999 hongguang avenue, pidu district, chengdu city, sichuan province 611730, china. e-mail: xhwangyu@yeah.net https://orcid.org/0000-0003-4363-505x https://orcid.org/0000-0001-5147-7931 https://orcid.org/0009-0007-7626-8731 https://orcid.org/0000-0003-3332-2412 https://orcid.org/0009-0005-4919-7830 760 h.-y. yang, y. wang, s.-l. liu, q. tu, g. chen 1. introduction natural gas, as a clean and efficient low-carbon fossil energy, has experienced explosive growth in consumption. it is of particular importance to ensure the supply of natural gas through peak regulation. an underground gas storage offers an effective means of natural gas peak regulation. such an underground gas storage is mainly deployed in halite and other underground rock layers, which feature extremely low porosity and permeability, as well as stable mechanical properties [1]. halite is internationally recognized as the best place for energy storage [2]. however, salt cavern gas storage (scgs) often has a complex internal environment, and lies deep underground. the cavity may creep and collapse, exerting a great impact on the shape of the dissolved cavity of the underground gas storage. to ensure the security of scgs, it is extremely important to track the shape of the salt cavern in real time, and regularly explore and evaluate the shape of the dissolved cavity. currently, ultrasonic detection is the most popular way to detect the shape of scgs. but the detection accuracy would be significantly affected by the water and various salt media inside scgs. the main effect is the attenuation of ultrasonic waves. hence, the key and difficulty of ultrasonic detection are to clarify the attenuation features of ultrasonic waves in the gas storage media. ultrasonic wave attenuation in solutions can be divided into three forms: viscous attenuation, thermal attenuation, and scattering attenuation. harker et al. [3] discussed the scattering theory of compression waves in viscous fluids, and deduced the variation of ultrasonic phase velocity and attenuation with the changes of fluid viscosity, particle, elastic modulus, and concentration. prek [4] proposed a method to determine the acoustic wave characteristics in an elastoplastic fluid-filled tube using the frequency domain analysis and applied it to calculate the acoustic wave number from a transfer function between three pressure measuring devices, which showed that the acoustic wave number was complex and related to the wave speed and decay rate. ross [5] studied the propagation of ultrasonic waves in multi-layer media, composite materials and viscoelastic materials, and created the three-media viscous attenuation model for ultrasonic waves in multi-layer media. queiros et al. [6] explored the velocity and attenuation of sound waves in anisotropic materials. after solving the cross-correlation of the transmitted and received signals, they performed sinusoidal fitting on the two signals, measured the sound velocity with high precision, and constructed a comprehensive finite-element model for two kinds of defects. hutchinson and bevis [7] found that, as the heating temperature of low carbon steel rises, the enhancement of ferrite crystal anisotropy and the subsequent phase transformation at high temperature will lead to an attenuation peak of sound wave, which is consistent with rayleigh zone attenuation theoretical model. through experiments, hutchins et al. [8] weakened the sound signal attenuation induced by scattering and viscoelastic effects, using piezoelectric composite transducers and pulse compression processing. li et al. [9] investigated the attenuation of ultrasound in composites and proposed a time-domain analysis method for the propagation of ultrasound waves based on the finite element analysis of a bilayer fiber/matrix. the results show that the interfacial energy dissipation of the acoustic wave is the main factor affecting the ultrasonic attenuation characteristics. based on the epstein-carhart-allegrahawley (ecah) scattering model, tsuji et al. [10] constructed a technique called multiple echo reflection ultrasonic spectroscopy, which makes the viscoelastic ecah model applicable when the particle size and wavelength are comparable. yu et al. [11] proposed a method attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas... 761 to measure the concentration of hydrate-water dispersion particles based on multifrequency ultrasonic attenuation. the ultrasonic scattering attenuation model that accounts for multiple scattering was developed to predict the attenuation of liquid-solid dispersions. zhao et al. [12] developed a new predictive model for the ultrasonic attenuation of spherically mixed particles in gaseous media using the monte carlo method. the numerical simulation results for the system with a single particle type are in good agreement with the results of various standard models, such as classical ecah model and mcc model. through investigation and analysis, it can be seen that the propagation process of sound waves in the salt cavern environment is complex, involving regional differences. therefore, it is an urgent problem to master the attenuation features of sound waves in scgs media. starting with the attenuation form of ultrasonic waves in complex environment, this paper analyzes the main attenuation modes of ultrasonic waves in scgs environment. according to the wave equation and the specific environment of gas storage, the authors established the theoretical relationship of ultrasonic attenuation in scgs, carried out orthogonal analysis of the three attenuation forms to grasp the change features of ultrasonic attenuation with scgs medium and ultrasonic excitation frequency. in addition, the theoretical results were verified through experiments, providing a strong theoretical guidance for the sonar detection of scgs. 2. wave equation ultrasound, as a mechanical wave, is essentially a mechanical vibration propagating in a continuous medium. the wave equation for a non-ideal medium can be expressed as [13]: 𝜕2𝑝 𝜕𝑥2 = 1 𝑐2 𝜕2𝑝 𝜕𝑡2 (1) where, 𝑐 = ( 𝑘𝑠 𝜌 ) ( 1 2 ) , 𝜌 is the density of the medium, 𝑘𝑠=−𝑣 𝑑𝑃 𝑑𝑉 and it is the elastic modulus of the insulator, v is the volume, 𝑥 is the propagation distance, t is the propagation time, p is the sound pressure and 𝜉 is the displacement. hence, 𝑝 = 𝑘𝑠 ∗ 𝑑𝑉 𝑉 , and 𝑑𝑉 𝑉 = 𝑑𝜉 𝑑𝑥 . since the propagating medium in the salt cavern is a solid-liquid mixture, the adjacent particles in the propagating medium do not move at the same velocity. according to newton's law of internal friction in fluid mechanics, the viscous force per unit area of a one-dimensional (1d) planar wave with the dynamic viscosity of η can be expressed as: 𝑇′ = 𝜂 𝜕𝑣 𝜕𝑥 (2) therefore, the acoustic equation can be written as: 𝜌 ∂2𝜉 ∂𝑡2 = 𝐾𝑠 ∂2𝜉 ∂𝑥2 + 𝜂 ∂3𝜉 ∂𝑥2 ∂𝑡 (3) here, 𝐾𝑠 is the adiabatic volume elasticity coefficient of the fluid. for simple harmonic waves, the general solution of differential eq. (3) is usually represented by a combination of complex numbers. let  be the sound absorption coefficient under viscous fluid. then, the wave equation of sound waves under non-ideal conditions can be obtained as: 𝜉 = 𝜉0𝑒−𝛼𝜂𝑥𝑒𝑗𝜔(𝑡− 𝑥 𝑐 ) (4) 762 h.-y. yang, y. wang, s.-l. liu, q. tu, g. chen the medium inside the underground cavity is very complex. in such an environment, the attenuation mechanism of sound waves is often the combined effect of multiple attenuation mechanisms. as shown in fig. 1, during the actual propagation, the sound wave would experience viscous attenuation due to the internal friction of the medium, scattering attenuation due to the undissolved medium, and thermal attenuation due to the heat conduction [14-17]. this study intends to adopt the sound velocity and sound attenuation coefficient derived in the urick model [18], and explore the case of dilute solutions within 10%, without considering insoluble particles. fig. 1 schematic diagram of acoustic attenuation in complex environment 3. attenuation feature analysis 3.1. viscous attenuation features based on urick model [18], the sound velocity in suspensions can be expressed as: 𝑐 = (𝑃eff 𝛽𝑒𝑓𝑓) −1/2 (5) 𝑃eff = 𝜌(1 − 𝜑) + 𝑝′𝜑 (6) 𝛽𝑒𝑓𝑓 = 𝛽𝑐(1 − 𝜑) + 𝛽𝑐 ′𝜑 (7) where, peff is the effective density, 𝛽eff is the compression coefficient, φ is the liquid concentration, 𝑝′ is the particle density, 𝛽c ′ is the isothermal compression coefficient of solid particles and 𝛽c is the liquid isothermal compression coefficient. eqs. (5-7) show that the velocity of the sound wave decreases with the rise of concentration. also, with a higher the particle density, the sound velocity is slower and the velocity decreases faster. according to yang et al. [19], this paper adopts the empirical relationship between the density of the solution, the mass content of the substance, and the temperature: 𝜌 = 1006.0 + 737.7𝐶 − 0.311𝑇 − 1.993 × 10−3𝑇2 (8) where, c is the solute mass fraction and t is the absolute temperature. the formula was analyzed to calculate the change of solution density when the temperature is in the range attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas... 763 between 0°c and 100°c and the mass fraction of solute falls between 0% and 10%. the calculation results are shown in fig. 2. it can be seen from fig. 2 that the density of the solution changes with the temperature and the mass fraction of the solute. the higher the temperature, the smaller the density of the solution. but the density decrement is quite small. the temperature rise of 120°c only leads to 6.5% decline in solution density. by contrast, the solute mass fraction has a large impact on solution density: the solution becomes 18% denser, as the mass fraction of the solute grows by 25%. under the joint influence of temperature and solute mass fraction, the solute mass fraction has a greater impact on the solution density than the temperature. in the subsequent analysis, the solution density and sound wave attenuation are investigated, in view of the solute mass fraction and solution temperature. the liquid concentration normally grows exponentially with its viscosity. this paper fits the relationship between the two factors as a quadratic equation: 𝜂 = 10𝜑2 (9) fig. 2 the relationship between solution density, temperature and solute mass fraction the viscous attenuation coefficient of the sound wave derived from the navier-stokes (n-s) equation can be expressed as: 𝛼𝜂 = 2𝜋2𝑓2𝜂 𝜌𝑐3 (10) where, f is the frequency of the sound wave, c is the speed of sound, η is the viscosity of the medium and ρ is the density of the medium. it is assumed that the mixed medium is the sodium chloride solution, the largest component of scgs brine. substituting eqs. (5) and (9) into eq. (10), we have the viscous attenuation coefficient of sound wave in the mixed solution. next, the emission frequency of sound wave was set to be in the range 100 khz-1 mhz, the solution concentration to 2%, 4%, 6%, and 8%, and the solution temperature to 0-50°c. the viscous attenuation coefficients calculated for these parameters are given on the diagram in fig. 3. as shown in fig. 3, when the frequency of the sound wave is in the range of 100 khz1 mhz, the viscous attenuation coefficient of the sound wave increases with the concentration of the mixed solution. when the concentration of the mixed solution remains constant, the coefficient increases with the frequency of the sound wave. 764 h.-y. yang, y. wang, s.-l. liu, q. tu, g. chen fig. 3 variation of the solution viscosity attenuation coefficient 3.2. scattering attenuation features the scattering attenuation of the sound wave stems from the uneven acoustic impedance of the material. in general, the scattering attenuation of the sound wave is calculated by the rayleigh scattering theory. the scattering attenuation coefficient of the sound wave can be calculated by: 𝛼𝑠 = 8𝑟3𝑓4𝜋4 3𝑐4 (11) where, r is the diameter of the insoluble particles, c is the speed of sound, f is the frequency of the sound wave. suppose the insoluble particles in the mixed medium are silica with a density of 2.2 g/cm3, and the medium is water with a density of 1 g/cm3. substituting eqs. (5) and (9) into eq. (11), a formula for the scattering attenuation coefficient of the sound wave in the mixed solution is obtained. to display the scattering attenuation of the sound wave in the mixed media more intuitively, the parameter sensitivity analysis of sound wave scattering attenuation was carried out in matlab. again, the emission frequency of the sound wave was set to 100 khz-1 mhz, and the solution concentration to 2%, 4%, 6%, and 8%. the calculation results are shown in fig. 4. as shown in fig. 4, the changing solution concentration exerts a slight impact on the scattering attenuation of the sound wave. the higher the wave frequency, the greater the scattering attenuation coefficient. attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas... 765 fig. 4 the relationship between acoustic scattering attenuation and frequency 3.3. thermal attenuation features the thermal attenuation of sound waves is usually related to the thermal conductivity, constant volume specific heat, and constant pressure specific heat. the sound wave thermal attenuation coefficient can be calculated by: 𝛼𝑡 = 2𝜋2𝑓2𝑥 𝜌𝑐3 ( 1 𝐶𝑉 − 1 𝐶𝑃 ) (12) where x is the thermal conductivity, 𝐶𝑣 and 𝐶𝑝 are the constant volume and constant pressure specific heat capacities, respectively, f is the frequency of the sound wave and c is the velocity of the sound wave. to display the thermal attenuation of the sound wave more intuitively, the solution concentration was set to 2%-8% substituting eqs. (5) and (9) into eq. (11), one obtains the formula for the thermal attenuation coefficient of the sound wave in the mixed solution. the parameter sensitivity analysis of sound wave thermal attenuation was carried out in matlab. the emission frequency of the sound wave was set to 100 khz-1 mhz, the thermal conductivity to 0.65 w/m∙℃, the constant pressure specific heat to 𝐶𝑝=10.392 kj/kg°c, and the constant heat ratio k to 1.32. then, 𝐶𝑣 = 𝐶𝑝/k = 7.873. the results are shown in fig. 5. as shown in fig. 5, the thermal attenuation coefficient of the sound wave increases with the sound wave frequency. the larger the thermal attenuation coefficient, the slower the sound propagation. when the frequency remains constant, the solution is denser, the thermal attenuation coefficient is smaller and the sound propagation is faster. 766 h.-y. yang, y. wang, s.-l. liu, q. tu, g. chen fig. 5 variation of thermal attenuation coefficient of acoustic with frequency 3.4. total attenuation analysis from the above analysis on three kinds of attenuations during sound wave propagation, it is clear that the attenuation mechanism is very complex when the sound wave passes through a mixed medium. two or multiple attenuation forms may occur simultaneously, and act together on how the sound wave attenuates. according to the above theoretical calculation, scattering attenuation has the largest effect, while viscous attenuation and thermal attenuation have relatively small effects. when the wavelength of the propagating sound wave is much larger than the diameter of the particles in the medium, the attenuation of the ultrasonic wave in the mixed system can be directly added. the ultrasonic attenuation coefficient is expressed by the following formula [20]: 𝛼 = 𝛼𝜂 + 𝛼𝑠 + 𝛼𝑡 = 2𝜋2𝑓2𝜂 𝜌𝑐3 + 8𝑟3𝑓4𝜋4 3𝑐4 + 2𝜋2𝑓2𝑥 𝜌𝑐3 ( 1 𝐶𝑉 − 1 𝐶𝑃 ) (13) where 𝛼𝜂 is the viscous attenuation coefficient, 𝛼s is the scattering attenuation coefficient, 𝛼𝑡is the thermal attenuation coefficient, while, 𝑐, 𝑃eff and 𝛽𝑒𝑓𝑓 are given in eqs. (5-7), respectively. substituting the solution concentrations of 2%, 4%, 6%, and 8% into eq. (13), we obtained the change of the total attenuation of the sound wave in the suspension with different concentrations. the emission frequency of the sound wave was set to 100 khz1 mhz, the thermal conductivity to 0.65 w/m∙℃, the constant pressure specific heat to 𝐶𝑝 = 10.392 kj/kg ∙℃, and the constant heat ratio k to 1.32. then, 𝐶𝑣 = 𝐶𝑝/k = 7.873. the results are shown in fig. 6. attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas... 767 fig. 6 relationship between total attenuation coefficient of acoustic and frequency it can be seen that the total attenuation coefficient of the sound wave in suspensions with different concentrations increases the frequency of the sound wave. when the frequency of the sound wave remains constant, the total attenuation coefficient of the sound wave increases with the suspension concentration. when the solution concentration is below 10%, the total attenuation coefficient of the sound wave depends on scattering attenuation, while viscous attenuation and thermal attenuation has little influence on the sound wave attenuation. therefore, the scattering attenuation is regarded as the main attenuation form when the concentration of the suspension is less than 10%. in this case, the viscous and thermal attenuations are approximately negligible. 4. experimental analysis to simulate the sound wave propagation in the complex environment of scgs, the authors built an experimental platform and prepared different types of salt solutions (fig. 7). on the platform, an ultrasonic apparatus drives the ultrasonic transducer to transmit and receive sound waves. the sound wave data were collected by external equipment and the experimental data were processed. the vessel and foam plates were kept warm by external heaters to keep the test temperature close to the set level, in order to reveal the sound wave attenuation. a) b) fig. 7 experimental platform (left) and different types of salts (right) 768 h.-y. yang, y. wang, s.-l. liu, q. tu, g. chen the ultrasonic attenuation coefficient was measured by calculating the relationship between amplitude and the propagation distance of two ultrasonic waves: 𝛼 = − ln(𝐸/𝐸0) 𝐿 (14) where, 𝐸0 is the amplitude of the first sound wave, e is the amplitude of the second sound wave, l is the distance between the ultrasonic transducer and the reflecting wall and it is a fixed value. the two amplitudes can also be determined by experiments. the above formula helps to solve the attenuation coefficient of the sound wave. in the scgs, the cavern is created through dissolution. thus, the liquid medium of the cavern often covers various soluble salts, which are dissolved and mixed. this complicates the salt cavern medium. according to the existing surveys, the composition analysis of some salt mine media shows that the content of the same component varies significantly in salt mines at different places. besides, different salt mines differ in the main components. in general, the primary content is sodium chloride, plus some salt solutions like potassium chloride, magnesium sulfate and calcium chloride. in our experiments, the sound attenuation features are examined with the four salt solutions as the objects. first, the height of pure water was set to 160 mm, the distances between the ultrasonic transducers of different frequencies and the bottom surface of vessels to 145 mm (0.5 mhz), 117 mm (1 mhz), 127 mm (2 mhz), 127 mm (3.1 mhz), 129 mm (4 mhz), and 114 mm (5 mhz), in turn. then, 131 g, 267 g, 409 g, and 557 g of sodium chloride were added to the solution, in turn. the resulting sodium chloride solutions were of the concentration of 2%, 4%, 6%, and 8%, respectively. after the sodium chloride was fully dissolved, the temperature of the solution was controlled to 20°c. for the ultrasonic transducer, the excitation voltage was set to 150 v, the pulse width to 80 ns, the sampling frequency to 2mhz, and the repetitive pulse frequency (rpf) to 2 khz. the excitation frequency of the transducer was set to 0.5 mhz, 1 mhz, 2 mhz, 3.1 mhz, 4 mhz, and 5 mhz, in turn. the operating frequency of the ultrasonic transducer was obtained as 0.5 mhz, 1 mhz, 2 mhz, 3.1 mhz, 4 mhz, and 5 mhz, in turn. once the solution temperature was stabilized, the mean of multiple measurements was taken. the results are listed in table 1. table 1 sonic data in different concentrations of sodium chloride solution medium concentration peak 0.5 mhz 1 mhz 2mhz 3.1 mhz 4 mhz 5 mhz voltage [v] voltage [v] voltage [v] voltage [v] voltage [v] voltage [v] 2% 1 57.35 59.31 28.63 34.02 51.2 94.35 2 19.64 15.8 5.23 5.1 6.87 14.21 4% 1 57.88 58.44 29.41 36.12 51.35 93.78 2 19.21 16.52 5.48 5.1 5.98 12.45 6% 1 57.65 60.02 81.07 63.14 50.98 95.03 2 15.74 10.12 9.13 6.45 4.57 9.67 8% 1 58.33 59.46 80.87 85.47 87.45 95.12 2 12.57 9.15 7.88 7.54 6.72 8.54 it can be seen from table 1 that the sound wave attenuation coefficient increased with the medium concentration. when the medium temperature remains constant, the coefficient increases with the sound wave frequency. therefore, it is necessary to experimentally investigate the attenuation features of sound waves under the effect of changing temperature. attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas... 769 the solution was heated up to 50°c. during the heating process, the experimental data were recorded every for each temperature increment of 3°c. the operating frequency of the ultrasonic transducer was set to 0.5 mhz, 1 mhz, 2 mhz, 3.1 mhz, 4 mhz, and 5 mhz, in turn. the other settings remained the same as in earlier experiments. once the solution temperature was stabilized, the mean of multiple measurements was taken. figure 8 presents the results on sodium chloride solutions of the concentrations of 2%, 4%, 6%, and 8%, respectively. it can be learned that, when the concentration of sodium chloride solution remained constant, the sound wave attenuation coefficient is positively correlated with the solution temperature. the higher the temperature, the higher the sound wave attenuation coefficient, and the more serious the attenuation. however, the temperature had little effect on the attenuation of the sound wave in the sodium chloride solution. for every growth of 1°c in temperature, the attenuation coefficient of the sound wave increased by about 0.17 v/m. giving overall consideration of concentration and temperature variations, the attenuation coefficient of the sound wave was the smallest in the sodium chloride solution, when the sound wave frequency was 0.5 and 1 mhz. fig. 8 variation of acoustic attenuation coefficient in sodium chloride solution at different temperatures and concentrations without changing the above experimental conditions, sodium chloride was replaced with potassium chloride, magnesium sulfate and calcium chloride in turn for the subsequent experiments. figure 9 reports the variation of sound wave attenuation coefficients in solutions of different concentrations. 770 h.-y. yang, y. wang, s.-l. liu, q. tu, g. chen fig. 9 variation of acoustic attenuation coefficient in different concentrations solution it can be seen from fig. 9 that, when the medium temperature remains constant, the influence of medium concentration on sound wave attenuation features has a consistent trend with that of the sodium chloride solution. the sound wave attenuation coefficients are smallest at the wave frequencies of 0.5 mhz and 1 mhz, except for the case of sodium chloride solution. furthermore, the experimental data were extracted with solution concentrations of 2% and 6% (fig. 10). fig. 10 comparison of acoustic attenuation coefficients in different concentration solutions as shown in fig. 10, when the temperature remains the same, the different types of salt solutions of the same concentration can be ranked in descending order starting from potassium chloride solution, via magnesium sulfate solution and calcium chloride solution, to sodium chloride solution. hence, the sound wave was attenuated most significantly in potassium chloride solution. in addition, the urick model yields relatively accurate results when the sound wave frequency is high, indicating that the theoretical model has certain limitations in the calculation of sound wave attenuation in salt caverns. further experiments were conducted on the influence of temperature of sound wave attenuation features. figure 11 shows the results based on the experimental data of 8% magnesium sulfate solution. the temperature influence over sound wave attenuation features in different solutions is the same as that in the case of sodium chloride solution. for different salt solutions, when the solution temperature rises by 1℃, the attenuation coefficient of the sound wave increases by about 0.83 v/m, 0.2 v/m, and 0.12 v/m in potassium chloride solution, magnesium sulfate solution, and calcium chloride solution, respectively. overall, attenuation characteristics of ultrasonic waves from sonar logging tools in salt cavern gas... 771 temperature has the greatest influence on the attenuation of the sound wave in the potassium chloride solution, while the temperature influence on the other three solutions is similar. fig. 11 variation of sound attenuation coefficient in magnesium sulfate solution 5. conclusions this paper mainly theoretically analyzes the attenuation features of sound waves in salt cavities. based on the urick sound wave attenuation theory, the expression of sound velocity in dilute solution and the general empirical formula of solution density were added to study three attenuation forms of sound waves in dilute solutions with different concentrations. the following conclusions were drawn through theoretical and experimental analyses: ▪ when the solution concentration is less than 10%, the total attenuation coefficient of the sound waves depends on scattering attenuation, and the sound wave attenuation is not greatly affected by viscous attenuation and thermal attenuation. ▪ when the medium temperature remains the same, the sound wave attenuation coefficient increases with medium concentration, and sound wave frequency. ▪ the higher the temperature, the greater the sound wave attenuation. temperature has the greatest effect on sound wave attenuation in potassium chloride solution. overall, temperature has a small overall effect on the attenuation of sound waves. ▪ among the four salt solutions, the most severe sound wave attenuation was observed in potassium chloride solution. the salt solutions can be ranked in descending order of sound wave attenuation coefficient as potassium chloride solution>magnesium sulfates solution > calcium chloride solution > sodium chloride solution. ▪ without considering the distance requirement of sound wave detection, the sound wave attenuation obtained through theoretical and experimental analyses both reach a weak state at the sound wave frequencies 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(ed.), proceedings of the 2015 international conference on testing and measurement: techniques and applications (tmta 2015), 16-17 january 2015, phuket island, thailand. 14. fa, l., li, l., gong, h., chen, w., jiang, j., you, g., liang, j., zhang, y., zhao, m., 2022, investigation of the physical mechanism of acoustic attenuation in viscous isotropic solids, micromachines, 13(9), 1526. 15. shin, y., 2019, signal attenuation simulation of acoustic telemetry in directional drilling, journal of mechanical science and technology, 33, pp. 5189–5197. 16. hu, j., fu, l.y., wei, w., zhang, y., 2018, stress-associated intrinsic and scattering attenuation from laboratory ultrasonic measurements on shales, pure and applied geophysics, 175, pp. 929–962. 17. grigorieva, n.s., legusha, f.f., safronov, k.s., 2023, scattering of a plane sound wave by a spherical interface of two media with sound absorption in the acoustic boundary layer, acoustical. physics, 69, pp. 325-329. 18. han, q.b., xu, s., 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ćojbašić3 1university of kragujevac, faculty of technical sciences čačak, serbia 2research and development institute lola l.t.d., belgrade, serbia 3university of niš, faculty of mechanical engineering, niš, serbia 4deprtament of structural analysis, tu berlin, germany abstract. production of three-dimensional parts in 3d printing process gains growing importance in various fields, such as: aviation and car industry, architecture, medicine, dentistry, etc. mechanical performance is an important users’ requirement for manufacturers of 3d printed parts. furthermore, printed part highly depends on process parameters, position and orientation of the printed part, and performances of the 3d printer which prints the part. in this paper, based on experimental results, an artificial neural network has been used for modeling the dependence of process parameters and object orientation during printing, on the one side, and tensile strength as very important mechanical performance, on the other side. after establishing abovementioned dependence, the developed neural network has been used as a fitness function for the genetic algorithm while the genetic algorithm has been created for the optimization process. the result of optimization process was a set of optimal process parameters and part orientation giving the maximum tensile strength. the results have shown acceptable potential of the developed methodology for optimizing the 3d printing process as a complex engineering problem. key words: 3d printing, optimization, tensile strength, neural network, genetic algorithm 1. introduction 3d printing or additive manufacturing is a modern manufacturing process of making a three-dimensional part from a digital 3d model. it allows making the part from several different materials that have different mechanical and physical properties, in a unique received: april 29, 2022 / accepted september 12, 2022 corresponding author: nedeljko dučić university of kragujevac, faculty of technical sciences čačak, sv. save street, 32000 čačak, serbia e-mail: nedeljko.ducic@ftn.kg.ac.rs 2 i. milićević, m. popović, n. dučić, v. vujičić, p. stepanić, et al. process, usually, layer by layer. additive manufacturing technologies have been initially used only for prototyping in order to accelerate part development and minimize development costs. however, with the improvement of technology and materials, a development level allowing the production of final, functional parts or tools has been reached. in addition to prototyping, 3d printers offer a great potential for the production of various applications in the field of industrial design, aviation and car industry, architecture, medicine, dentistry, etc. fused deposition modeling (fdm) is one of the most important 3d printing methods which deliver parts made of thermoplastic filaments. the layers are stacked on top of each other in order to create a final 3d part. fdm 3d printing process uses different types of plastic materials and composite materials made of plastics and metal particles, ceramics, various carbon fibers, etc. due to its ease of use and low cost, this method has become very popular and is widely used for prototyping as well as for manufacturing functional industrial products. however, the default settings of printing process parameters, set by a manufacturer in some cases, do not guarantee the quality (dimensional errors and unsatisfactory mechanical characteristics of the part) of the printed part because of a significant number of process parameeters to be taken into consideration. for the parts made in this way to be completely functional, it is necessary to meet the requirements on the mechanical characteristics that depend on choice of basic material, technological process parameters as well as post-processing procedures. fdm 3d printing process consists of a fairly large number of parameters that can affect the final appearance and mechanical characteristics of the part. these parameters can be classified into: process parameters, fdm printer parameters, 3d model orientation mode, material type, 3d model and environment parameters [1], [2]. analysis of the process parameters influence on the quality of printed part and its mechanical properties in the fdm printing process has been the subject of numerous scientific research papers [3-6]. the most analyzed have been influences of the following parameters: layer thickness, type of internal filling, percentage of internal filling, distance between rasters, raster width, raster angle, material density, part orientation by x, y and z axis, number of contour walls, printer head velocity, nozzle temperature in the printer head, etc. [7-12]. research that includes testing the influence of all abovementioned parameters and their combinations requires a large number of samples, so the literature usually states the impact of only certain ones [13-18]. furthermore, due to the complexity of process parameters and an unclear relationship between process parameters and part performances, the influence of each parameter on 3d printed parts has not been universally defined yet. therefore, it is necessary to consider the method of choosing optimization parameters of process that will improve the quality of 3d printing [19-23]. in previous scientific research papers the mechanical properties of fdm printed samples have been most frequently tested by tensing [4,8,10,17,25-29]. tensile strength testing provides the most complete image of material’s mechanical properties. test samples (test pipes) have been defined by relevant standards. during testing, forces in the direction of the axis are acting on test pipe, tending to extend and break it. tensile testing machines, also known as tension test machines, have been used for this type of testing. after selecting the appropriate influential process parameters, it is possible to improve quality and mechanical properties of 3d printed part by their optimization. optimization of process parameters is a research challenge that has been addressed by many researchers [7,3032]. the research field of this challenge is wide because a significant number of process improving the mechanical characteristics of the 3d printing objects using hybrid machine learning... 3 parameters and their combinations lie in a design of the experiment while a range of materials to be processed is wide. it is also important to note that every 3d printer has its own specifics. the present paper deals with the influence that process parameters and test specimen orientation in 3d printer's workspace have on the mechanical characteristics, in particular case – on the tensile strength. this paper represents in a way an improvement over research studies that have been investigating only the influence of certain process parameters without taking into account the orientation of the test specimen in the 3d printer workspace. the present paper is divided into four chapters. after introduction, the second chapter describes the experimental part related to 3d printing and mechanical testing. the second chapter describes the process parameters and test specimen orientations used in the design of the experiment and their influence on tensile strength. then follows the mechanical testing of tensile strength. in the third chapter of the paper, a hybrid intelligent neuro-genetic system for the selection of process parameters and test specimen orientation that provides the best mechanical characteristics of the printed part has been developed. conclusions on the application of such a hybrid intelligent system for optimization the process parameters in order to maximize the tensile strength of the part made by 3d printing, are given in the fourth chapter. 2. experimental setup 2.1 3d printing process parameters and conditions for the purposes of the experiment, at first a 3d model of a standard test specimen, type 1ba (for tensile tests, according to iso 527-2:2012, annex a) has been created, from which an stl file have been obtained (fig. 1). the stl file has been imported into creatware software, in which the g-code has been generated. each g-code consisted of 3 test specimens that were positioned on a base at an angle 0, 45° and 90° to the y-axis (fig. 2), in order to determine how the direction of printing affects the tensile strength of the printed model. along with three different angles, the experiment has been repeated with different parameters of print speed, printer head temperature and print density. speed has varied from 40 to 80 mm/s, temperature from 200 °c to 220 °c and print density from 5 to 25%. thus 81 samples are obtained. the samples have been sorted and marked. a creatbot dx2 3d printer, that uses fdm printing technology, has been used for making the samples. pla plastics in the form of a wire (3 mm diameter) have been used as a material. all input data, i.e. process parameters, are displayed in the diagram at fig. 3. fig. 1 test specimen: a) stl model; b) a sample made on a 3d printer 4 i. milićević, m. popović, n. dučić, v. vujičić, p. stepanić, et al. fig. 2 generation of the g-code in the creatware software of the creatbot dx2 3d printer fig. 3 input data – process parameters improving the mechanical characteristics of the 3d printing objects using hybrid machine learning... 5 2.2 tensile strength testing procedure for determination of tensile strength of printed test pipes have been used tensile load tests in accordance with requirements defined by iso 527-1:2019 (plastics – determination of tensile properties – general principles) and iso 527-2:2012 (plastics – determination of tensile properties – test conditions for molding and extrusion plastics). a machine type skaz 2000 has been used for testing, equipped with 1st class forcemeasuring cell up to 20 kn in accordance with iso 7500-1:2018 and 1st class extensometer in accordance with iso 9513:2012. the appropriate software generated a dependence diagram of tensile force on deformation, the value of tensile strength as well as the relative elongation when tearing the test specimen for each test sample. tensile strength values for all tested samples are given in the diagram shown in fig. 4. fig. 4 output data tensile strength 3. neuro-genetic optimization of process parameters 3.1 tensile strength prediction model a neural network is a massively parallelized distributed processor with a natural ability to store experiential knowledge and ensure its use. through the training process, synaptic weights are systematically changed in an artificial neural network in order to achieve the desired network performance. the basic computing power of neural networks is massive parallelism, the ability to train and generalize. in order for the system modeling process through training to be as high quality as possible, it is necessary to cover the widest possible range of operating modes of the dynamic system. a neural network architecture consists of an input layer, one or more hidden layers, and an output layer. the most commonly used algorithm for training multilayer neural networks with a supervisor is error propagation backwards and is only applicable to feedforward networks. the basic idea is that the input signal propagates forward, from the input to the 6 i. milićević, m. popović, n. dučić, v. vujičić, p. stepanić, et al. output layer while the error signal propagates backwards, from the output to the input layer, changing the weights of the network in that order. the structure of the i-th artificial neuron with r inputs 1 2, ,..., rx x x multiplied by appropriate weighting coefficients ,1 ,2 ,, ,...,i i i r   and added bias can be represented in the following from [33]: 1 ,1 2 ,2 ,...i i i r i r ia x x x b       (1) where ai represents argument of transfer function (called activation function). the i-th neuron produces output data yi    ,1 r i i j i r ij y f a f x b     (2) this output then represents the input for the neurons of the next layer or represents an element of the output vector of the artificial neural network. in this paper, the levenbergmarquardt backpropagation algorithm has been used to train neural networks. this algorithm is a standard method for minimizing the mean square error (mse) criterion and is characterized by fast convergence and robustness [34]. the mse is calculated as follows:        22 1 1 1 1q q k k mse e k t k y k q q      (3) where q denotes number of experiments, e(k) denotes error, t(k) denotes target values, while y(k) are predicted values. updating the weights of network with the levenbergmarquardt algorithm [35] is presented by the equation:   1 1 t k k k k k kw w j j i j e      (4) which is based on the approximation of hessian matrix h. th jj i  (5) where i is identity unit matrix,  is a learning parameter and j is the jacobian matrix (the levenberg-marquardt algorithm requires the computation of the jacobian j matrix at each iteration step and the inversion of jtj square matrix, the dimension of which is n×n). to predict the tensile strength, the feedforward backpropagation neural network described above has been developed, whose training algorithm is the levenberg–marquardt algorithm. a set of data (81 samples) was used for the development of neural networks, which has been divided into three groups: training 80% (65 samples), validation 10% (8 samples) and testing 10% (8 samples). several different neural network architectures have been developed (with different number of hidden layers and neurons in them), and network performance has been evaluated based on mean squared error. the best performance in the prediction of tensile strength was shown by a network with one hidden layer and 12 neurons in the hidden layer (fig. 5). the neurons in input and hidden layers of neural networks had the sigmoid transfer function, while the neuron of the output layer has the linear transfer function. the process of creating a neural network ends in 15 epochs when the values of the mean square error are in the phase of training, validation and testing: 0.34, 5.17 and 9.25, respectively (fig. 6). improving the mechanical characteristics of the 3d printing objects using hybrid machine learning... 7 fig. 5 neural network architecture 4-12-1 fig. 6 mean square error during the epochs 8 i. milićević, m. popović, n. dučić, v. vujičić, p. stepanić, et al. as another criterion for confirming the validity of the created neural network, the regression criterion has been used, which compares the outputs from the neural network and the target outputs (tensile strength values obtained by experimental testing) in the training, validation, training and total in all phase. a graphical presentation of the regression criterion in the training, validation and testing phase as well as the total for all the above phases, is given in fig. 7. fig. 7 regression criterion as an assessment of neural network performance the listed performances of the neural network ann 4-12-1 are given in table 1. table 1 performance of the neural network 4-12-1 mean squared error (mse) regression training 0.34 0.996 validation 5.17 0.965 testing 9.25 0.931 the created neural network ann 4-12-1 is used below as a fitness function of the genetic algorithm for optimization of the process parameters with the aim of maximizing tensile strength. improving the mechanical characteristics of the 3d printing objects using hybrid machine learning... 9 3.2 optimization of 3d printing process parameters genetic algorithms (ga) are a search technique that uses heuristic optimization methods to find an exact or approximate solution simulating the mechanism of natural evolution. the algorithm uses four genetic operators: selection, reproduction, crossover and mutation. the second basic component of the genetic algorithm is the fitness function, which is a quantitative measure of quality, i.e., correctness of the proposed solution. the strength of the algorithm lies in its ability to determine the position of the global optimum in a space with multiple local extremes in the so-called multidimensional space. unlike most deterministic algorithms, the characteristic of genetic algorithms is that they begin the search for the optimal solution from a series of possible solutions that represent the initial population of the genetic algorithm. it is common to generate the initial population (initialization) by randomly selecting solutions from the domain. this includes defining individuals and associating them with possible solutions. genetic algorithms use individuals that describe a set of objects. each individual is assigned a fitness score that evaluates the quality of a given individual. the genetic algorithm must provide a way to continuously improve, from generation to generation, the absolute adaptation of each individual in the population as well as the mean adaptation of the entire population. this is achieved by successive application of genetic operators of selection, crossing and mutation, which provides better solutions to a given specific problem (fig. 8). fig. 8 flowchart of genetic algorithm the main disadvantage of ga is that the speed of convergence close to the global optimum can be quite slow. if the population has become sufficiently similar, but an 10 i. milićević, m. popović, n. dučić, v. vujičić, p. stepanić, et al. optimal solution has not been reached, the mean fitness of all individuals in the population is large, and the differences between the best individual and other individuals in the population are small. therefore, there is an insufficient gradient in the fitness function that would help the genetic algorithm to reach the optimal value. to overcome the above-mentioned anomalies simple ga and the successful execution of the genetic algorithm used various techniques adapted to the nature of the problem to solve. these are: various types of coding, customized more complex genetic operators, several types of fitness functions, generation replacement policy, fixed or adaptive parameter change during ga execution. methods of calculating the fitness function can also greatly affect ga performance. more detailed descriptions and analysis of the operation principles of genetic algorithms are given by references [36-39]. fig. 9 ga/ann-based optimization algorithm [40] the principle of hybrid neuro-genetic combination for optimization of 3d printing process parameters in order to maximize tensile strength is shown in fig. 9. the selection of the initial population is generated randomly. the fitness function of the genetic improving the mechanical characteristics of the 3d printing objects using hybrid machine learning... 11 algorithm is the neural network (ann 4-12-1), described in the previous section of this paper. the fitness of each member of the initial population is calculated using the ann model and is followed by selection, crossing and mutation as parts of the reproduction process of a new generation. the process is repeated until one of the multiple requirements for termination of ga is satisfied. the population range is 30 individuals, with 100 generations at maximum. stochastic universal sampling algorithm is used in the selection. uniform crossover served as the crossing operator. the coefficient of the crossover fraction is 0.8. crossover fraction defines a portion of the new population derived from a crossing (non-elite individuals), its value being between 0 and 1. not more than two elite individuals are to be transferred to the next generation. the optimization ends in 53 iterations (fig. 10), and gives us a maximized value of tensile strength of 46.47mpa, and the values of the parameters are angle = 0.005 (°), speed = 69.36 (mm/min), temperature 218 (°c), density 24 (%). fig. 10 optimization process during iterations the optimization results indicate that by increasing the velocity by 10 and slightly shifting the temperature and density, ga has found the maximum function generated by the neural network ann 4-12-1. 4. conclusion the present study describes a joint influence of process parameters and part orientation on the tensile strength as a very important mechanical characteristic of the part made bay 3d printing process. in order to establish a connection between inputs (process parameters and part orientation) and outputs (tensile strength), and based on 12 i. milićević, m. popović, n. dučić, v. vujičić, p. stepanić, et al. experimental results, a neural network has been developed that showed a high degree of efficiency in establishing correlations between inputs and outputs. the best performance (mse=0.931 in testing phase) has been shown by a neural network with 12 neurons in a hidden layer and it was determined as a fitness function for the optimization process resulting in a set of process parameters (speed = 69.36 (mm/min), temperature 218 (°c), density 24 (%)) and part orientation (angle = 0.005 (°)) for which the tensile strength has maximum value. the genetic algorithm has been used as an optimization technique and it showed great potential in optimizing the process parameters of 3d printing and the orientation of the part being created. the very special importance of the present paper is that it considers a joint influence of process parameters and orientation of the part to be printed on the tensile strength. it is also of a great importance for practical workers who set 3d printer settings and prepare process, and who need to know the influence of process parameters on mechanical characteristics of the part. the application of the presented methodology eliminates the expensive principles of "trial and error" to achieve high product quality. thus, the study promotes the methodology of obtaining a set of process parameters and the orientation of the part for which the tensile strength is the maximum value. in practice, maximum mechanical characteristics are often not required, so some other values of process parameters can produce an acceptable printed part. the application of the neural network model developed in this study for any set of input parameters (within defined limits) gives a stable and relatively accurate prediction of tensile strength. this practically means that the developed neural network can be used separately as a highprecision model of 3d printing process in the cases when maximum tensile strength is not sought, which gives the presented study additional significance. future research will be directed in three research directions. the first one is to increase the precision of the process model through a larger number of experiments. the second one will include a wider class of materials used in 3d printing technology, while the third one will include the application of the presented methodology to other processes from the family of rapid prototyping technology. 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silk, college of textile and clothing engineering, soochow university, suzhou, china 2school of mathematics and information science, henan polytechnic university, jiaozuo, china 3school of science, xi’an university of architecture and technology, xi’an, china 4school of mathematics, china university of mining and technology, xuzhou, jiangsu, china 5department of mathematics, faculty of science, king abdulaziz university, jeddah, saudi arabia abstract. a nonlinear vibration system, over a span of convincing periodic motion, might break out abruptly a catastrophic instability, but the lack of a theoretical tool has obscured the prediction of the outbreak. this paper deploys the amplitude-frequency formulation for nonlinear oscillators to reveal the critically important mechanism of the pseudo-periodic motion, and finds the quadratic nonlinear force contributes to the pull-down phenomenon in each cycle of the periodic motion, when the force reaches a threshold value, the pull-down instability occurs. a criterion for prediction of the pull-down instability is proposed and verified numerically. key words: micro-electromechanical system, eardrum oscillator, pull-in instability, asymmetrical oscillation, vibration attenuation, energy harvesting 1. introduction the nonlinear oscillators with even nonlinearities, especially the quadratic nonlinear oscillators [1-5], became a hot topic recently in both mathematics and engineering, because scientists and engineers cannot exactly differentiate between the periodic motion and the fake periodic motion with ease. scientists have now found that a convincing periodic received: january 14, 2023 / accepted february 08, 2023 corresponding author: ji-huan he and chun-hui he affiliation, national engineering laboratory for modern silk, college of textile and clothing engineering, soochow university, suzhou, p. r. china e-mails: hejihuan@suda.edu.cn (j.h. he), mathew_he@yahoo.com (c.h. he) mailto:hejihuan@suda.edu.cn mailto:mathew_he@yahoo.com 192 j.h. he, q. yang, c.h. he, a.a. alsolami motion might be a fake one, which is called as the pseudo-periodic motion [6,7], after some cycles of seeming periodic motion, its motion property will be suddenly changed, and the instability occurs, and scientists are often unable to spot the pseudo-periodic motion. though mathematicians can reveal easily the bifurcation property of such a nonlinear oscillator [8], the pseudo-periodic motion of the quadratic nonlinear oscillator has never been reported, and the critically important mechanism of the pseudo-periodic motion needs to be revealed, so that its instability can be completely avoided. we consider an important nonlinear oscillator which involves even nonlinearities, the toda oscillator [9,10,11], which plays an important role in physics: 2 2 ( 1) 0+ − =ud u k e dt (1) where k is constant. eq. (1) can be written equivalently in the following form 2 2 3 4 2 0 2 6 24 + + + + + = d u k k k u u u u dt (2) the toda oscillator behaves periodically when k is small, however the amplitude might lead to either infinity or zero when the absolute value of k reaches a threshold value. as another example, we consider a microelectromechanical system, which can be modelled by the following differential equation [12]: 2 2 0 1 + − = − d u k u udt (3) where u is dimensionless displacement and k is a voltage-related parameter. eq. (3) can be written as 2 2 3 4 2 (1 ) 0+ − + + + + = d u u k u u u dt (4) the even nonlinearities in eq. (4) are the main contributor to the pull-in instability, which should be avoided in the practical applications. the pull-in instability has been widely studied in the micro electromechanical systems (mems) [13-17], when the voltage is larger than its threshold value, the pull-in instability occurs, otherwise the systems move periodically. in this paper we focus on a similar phenomenon in quadratic nonlinear oscillators [18,19], and we call it as the pull-down instability. 2. the quadratic nonlinear oscillator we consider the following quadratic nonlinear oscillator 2 2 2 0+ + = d u u u dt  (5) with the following initial conditions pull-down instability of the quadratic nonlinear oscillators 193 (0) , (0) 0= =u a u (6) where u is the displacement, β is a positive constant, a is the initial amplitude. eq. (5) is called as the eardrum oscillator [19]. for small values of β, a periodic motion is predicted; however, when it increases to a threshold value, the kinetic energy will be gradually consumed, and finally the periodic motion is forbidden, this phenomenon is called as the pull-down instability. fig. 1 shows the dynamical properties of eq. (5) when a=0.1. when β<<1, a periodic solution is predicted as shown in fig. 1(b). when β is larger than a threshold value, the pull-down instability occurs as shown in fig. 1(d). for β near to the threshold value, the system will have a periodic motion first, and after some cycles, the pull-down instability happens suddenly. this phenomenon is called as the pseudo-periodic motion [6,7] as shown in fig. 1(c). (a) (b) (c) (d) fig. 1 periodic motion and pull-down instability of the quadratic nonlinear oscillator when a=0.1. (a) phase diagram; (b) periodic motion; (c) pseudo-periodic motion; and (d) pull-down instability a periodic motion can be generally expressed by a sinusoidal function u(t) = bsin(ωt+θ), where b and θ are determined by the initial conditions. the displacement of the harmonic motion varies from b to negative b periodically. 194 j.h. he, q. yang, c.h. he, a.a. alsolami in fig. 1, we find that during the periodic motion (fig. 1(b)) and the pseudo-periodic motion (fig. 1(c)), the quadratic force pulls down the system in each cycle. the amplitude changes from 0.1 to negative 0.11 in fig. 1(b) and negative 0.2 in fig. 1(c), respectively. this unsymmetrical oscillation phenomenon is interesting and amazing because it cannot be seen in any a conservative nonlinear oscillator [20-22], and it can be concluded that the quadratic nonlinear oscillator cannot be always used for vibration attenuation and an energy harvesting system [23,24]. the unsymmetrical amplitude in each cycle certainly frustrates scientists and engineers, because even the van der pol oscillator [25] leads to a sinusoidal motion when time tends to infinity. when β increases to a threshold value, the pull-down instability occurs, and a lack of a theoretical model to predict the time and the condition of the pull-down outbreak has greatly reduced the operation reliability, and hindered its applications. 3. periodic motion with asymmetric amplitudes this section introduces the frequency formulation for nonlinear oscillators [26] and extends to the nonlinear oscillators with even nonlinearities. considering a nonlinear equation in the form 2 2 ( ) 0+ = d u u dt  , (0) , (0) 0= =x a x (7) where λ is a nonlinear function of u, generally it can be expressed as 3 2 1 1 3 2 1( ) + += + + + n nu a u a u a u (8) for a nonlinear oscillator, it requires ( ) 0 u u  (9) the frequency formulation is [26] 3 2 2 ( ) = =u a u u   (10) the frequency formulation is simple but it gives a relatively high accuracy of the frequency-amplitude relationship, and it becomes a universal tool to various vibration systems [27-29]. for λ(u)=a1u+a3u3, eq. (7) is the duffing equation, and eq. (10) leads to the following result 2 2 1 3 3 = 4 +a a a (11) eq. (11) is same as that obtained by the homotopy perturbation method [30]. using the frequency formulation of eq. (10), for eqs. (5) and (6), we have pull-down instability of the quadratic nonlinear oscillators 195 3 = 1 2 + a  (12) table 1 shows eq. (12) has good accuracy of the approximate period, t=2π/ω, for βa<<1, and its accuracy deteriorates greatly when the value of βa becomes large. the errors arise in the quadratic force, f=βu2. when the quadratic force is small, eq. (5) can be solved by the perturbation method [19], but in practical applications, we do not know the criterion for the small value. table 1 comparison the approximate periods with the exact one a β exact period approximate period eq.(12) error approximate period eq.(20) error (%) 0.1 1 6.266 6.0276 3.8% 6.300 0.55% 0.3 1 6.627 5.5979 15.5% 6.450 2.66% 0.5 0.3 6.308 5.9109 6.2% 6.323 0.24% 0.7 0.3 6.399 5.7795 9.6% 6.363 0.56% eq. (10) is valid for nonlinear oscillators with only odd nonlinear terms, our problem has a quadratic nonlinear term, so the frequency formulation has to be modified. we re-write eq. (5) in the form 2 2 (1 ) 0, 0+ − =  d u u u u dt  (13) 2 2 (1 ) 0, 0+ + =  d u u u u dt  (14) eqs. (13) and (14) give, respectively, the following frequency formulations: 2 1 3 =1 2 − a  (15) 2 2 3 =1 2 + a  (16) the solution can be approximately expressed as 2 1 cos , 0 ( ) ( )cos , 0 a t u u t a a t u    =  +   (17) where δa in the increment of the amplitude when u<0. the periods of eqs. (13) and (14) are, respectively, given as 1 1 2 2 = 3 1 2 = − t a     (18) 196 j.h. he, q. yang, c.h. he, a.a. alsolami 2 2 2 2 = 3 1+ 2 =t a     (19) so the period of eq. (5) can be calculated as 1 2 = 2 3 3 1 1+ 2 2 + = + − t t t a a     (20) eq. (20) shows much better accuracy for βa<0.5, see table 1. when βa=1.1547, the period by eq. (20) becomes infinite large. 4. pull-down instability when βa tends to the threshold value, the pull-down instability occurs. the motion of eq. (5) can be decomposed into two parts as given in eqs. (13) and (14). it is obvious that eq. (13) has a tendance of a pull-down motion, and eq. (14) gives an opposite motion. we consider an extreme case when u1=-2a for eq. (13), and u2=a for eq. (14), the combined displacement is u1+u2=-2a+a=-a, so eq. (5) is still periodic. however if u1<-2a, we have |u1+u2|>a, under such case, the motion is pulled down, and its periodic motion is forbidden. we consider the extreme condition of u1=-2a, eq. (13) can be written approximately as 2 2 (1 2 ) 0+ − = d u a u dt  (21) this predicts 2 1 =1 2− a  (22) the pull-down instability occurs when 2 1 0  , that is 0.5a (23) so βa=0.5 is the threshold value. we give another approach to determination of the threshold value. eq. (5) can be written in the form 2 2 31 1 1 ( ) 2 2 3 + + = du u u h dt  (24) where h is the hamilton constant. by the initial conditions given in eq. (6), eq. (24) becomes 2 2 2 3 31 1 1 ( ) ( ) ( ) 2 2 3 = − + − du a u a u dt  (25) when u = -2a, the kinetic energy tends to zero, so the pull-down instability occurs when 2 2 2 3 31 1 1 ( ) ( 2 ) ( ( 2 ) ) ( ( 2 ) ) 0 2 2 3 = − = − − + − −  du u a a a a a dt  (26) pull-down instability of the quadratic nonlinear oscillators 197 eq. (26) results in βa>0.5 as shown in eq. (23). when βa<0.5, a periodic motion or a pseudo-periodic motion is predicted. when βa>0.5, the pull-down instability occurs. to verify this criterion, we consider various values of a and β, and the results are illustrated in figs. 2-4. when the value of βa is less than the threshold value, 0.5, but it is near to the value, the pseudo-periodic motion is observed, for example, βa=0.495 in fig. 2(c); while it is far from the value, for example, βa=0.3 in fig. 2(c), a periodic motion is seen. so far, we have no criterion for exactly predicting the value of βa for a pseudo-periodic motion, where an unsymmetrical oscillation deteriorates gradually in each cycle, and when u reaches negative 2a, the convincing periodic motion stops suddenly. (a) (b) (b) (d) fig. 2 the criterion of the pull-down instability when a=0.3. (a) phase diagram; (b) periodic motion βa=0.3; (c) pseudo-periodic motion βa=0.495 and (d) pull-down instability βa=0.51 198 j.h. he, q. yang, c.h. he, a.a. alsolami (a) (b) (b) (d) fig. 3 the criterion of the pull-down instability when a=0.5. (a) phase diagram; (b) periodic motion βa=0.15; (c) pseudo-periodic motion βa=0.495 and (d) pull-down instability βa=0.55 in figs. 2-4, we can see that when βa is small, a periodic motion is predicted. with gradually increasing βa, a pseudo-periodic motion might occur. finally, when βa is larger than 0.5, the pull-in instability is observed. 5. conclusion this paper proposes an extremely important concept of the pull-down breakout for nonlinear oscillators with even nonlinearities, and it will become a powerful research tool in nonlinear vibration theory. a sophisticated modification of the frequency formulation is suggested, and a criterion is built for judging the pull-down instability, the unsymmetrical amplitude motion, and the pseudo-periodic motion. the amplitude increment, δa, in eq. (17) will be discussed in a forthcoming paper, and when the absolute value of δa is larger than a, the pull-down instability occurs abruptly. hence, the amplitude change is the best parameter to predict the pseudo-periodic motion and the time when the pull-down instability occurs. pull-down instability of the quadratic nonlinear oscillators 199 (a) (b) (c) (d) fig. 4 the criterion of the pull-down instability when a=0.7: (a) phase diagram; (b) periodic motion βa=0.21; (c) pseudo-periodic motion βa=0.49 and (d) pull-down instability βa=0.56 references 1. beléndez, a., pascual, c., beléndez, t., hernández, a., 2009, solution for an anti-symmetric quadratic nonlinear oscillator by a modified he’s homotopy perturbation method, nonlinear analysis: real world applications, 10(1), pp. 416-427. 2. yeasmin, i.a., rahman, m., alam, m., 2021, the modified lindstedt–poincare method for solving quadratic nonlinear oscillators, journal of low frequency noise, vibration and active control, 40(3), pp. 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belendez, a., hernandez, a., belendez, t., fernandez, e., et al., 2007, application of he's homotopy perturbation method to the duffing-harmonic oscillator, international journal of nonlinear sciences and numerical simulation, 8(1), pp. 79-88. https://www.webofscience.com/wos/author/record/33361634 https://www.webofscience.com/wos/author/record/14768325 7904 facta universitatis series: mechanical engineering vol. 21, no 4, 2023, pp. 685 700 https://doi.org/10.22190/fume210728071b © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper prediction of responses in a cnc milling operation using random forest regressor shibaprasad bhattacharya, shankar chakraborty department of production engineering, jadavpur university, kolkata, west bengal, india abstract. in the present-day manufacturing environment, the modeling of a machining process with the help of statistical and machine learning techniques in order to understand the material removal mechanism and study the influences of the input parameters on the responses has become essential for cost optimization and effective resource utilization. in this paper, using a past cnc face milling dataset with 27 experimental observations, a random forest (rf) regressor is employed to effectively predict the response values of the said process for given sets of input parameters. the considered milling dataset consists of four input parameters, i.e. cutting speed, feed rate, depth of cut and width of cut, and three responses, i.e. material removal rate, surface roughness and active energy consumption. the rf regressor is an ensemble learning method where multiple decision trees are combined together to provide better prediction results with minimum variance and overfitting of data. its prediction performance is validated using five statistical metrics, i.e. mean absolute percentage error, root mean squared percentage error, root mean squared logarithmic error, correlation coefficient and root relative squared error. it is observed that the rf regressor can be deployed as an effective prediction tool with minimum feature selection for any of the machining processes. key words: cnc milling, random forest regressor, prediction, decision tree 1. introduction in the manufacturing domain, machining is the process of removing unwanted material from a given workpiece to provide the desired shape geometry while fulfilling the requirements of better surface quality and close dimensional tolerance. in the milling process, the material is removed from the workpiece with the help of an advancing multipleteeth cutter. as the milling cutter enters the workpiece, its cutting edges repeatedly cut into and exit from the materials, removing material from the workpiece with each pass due to received july 28, 2021 / accepted december 02, 2021 corresponding author: shankar chakraborty department of production engineering, jadavpur university, kolkata, west bengal, india. e-mail: s_chakraborty00@yahoo.co.in 686 s. bhattacharya, s. chakraborty shear deformation. the milling operation generally consists of four indispensable components, i.e. milling machine, workpiece, fixture and a suitable cutter. from small individual components to large heavy-duty products, the paradigm of milling covers a wide variety of operations. based on the motion of the rotary cutter, milling operations can largely be divided into two categories, i.e. face milling and peripheral milling. in the face milling, the rotary cutter is placed perpendicular to the workpiece while generating a surface normal to the axis of rotation [1]. on the other hand, in the peripheral milling, the cutter is placed parallel to the workpiece so that its sides always come into contact with the top of the workpiece. a large variety of materials, like aluminum, brass, magnesium, nickel, steel, zinc etc. can be machined in conventional milling machines; but when high precision and close dimensional tolerance are required, computer numerical control (cnc) milling machines may be employed. as the dynamic nature of the face milling operation requires close control, investigation of the material removal mechanism, modeling of the interrelationship between the milling parameters and responses, prediction of the responses and optimization of the process have been found to be of utmost importance [2]. like all other machining operations, the process outputs (responses) of a face milling operation, like material removal rate (mrr), average surface roughness (ra), directional cutting force (fc), active energy consumption (aec) etc. are also observed to be influenced by its various input parameters, such as spindle speed (s), cutting speed (n), depth of cut (ap), width of cut (ae), feed rate (f), cutting power, etc. these process outputs usually determine the quality of the end products in order to satisfy the consumers’ requirements. for this reason, it has become essential for the designer/process engineer to have a close control and better understanding of various milling parameters along with their interactions with the responses. based on the available experimental dataset, these interrelationships between the milling parameters and responses can be effectively modeled with the help of various statistical and machine learning techniques [3]. the developed models would also act as the prediction tools to envisage the tentative values of the considered responses for the given sets of different milling parameters. the main advantage of machine learning techniques lies with their ability to solve complex problems while reducing the complicacy of the dataset and making the models more interpretable [4]. with the help of these techniques, predictive monitoring of the process outputs has become easier, while integrating customers’ demands and taking care of other external factors affecting the process under consideration [5]. they also provide a broader scope for continuous improvement while automating the related decision-making tasks by efficiently manipulating the huge volume of available dataset. there are mainly two types of machine learning techniques, i.e. supervised machine learning and unsupervised machine learning. in supervised learning technique, the learning algorithm is usually trained on the basis of labeled data, and when the training data are not labeled, it is called unsupervised learning technique. classification and regression are the two popular examples of supervised learning technique, while unsupervised learning technique primarily encompasses clustering and association. in general terms, regression deals with quantitative anticipation of the responses, whereas, prediction of a qualitative response is termed as classification. in real time manufacturing environment, supervised learning algorithms are usually preferred due to availability of huge experimental datasets which would finally help in quantitative prediction of different responses based on the given sets of various machining parameters. in the domain of milling operation, the past researchers have already applied various statistical and machine learning techniques, mainly in the form of linear regression, k-nearest neighbors (knn) prediction of responses in a cnc milling operation using random forest regressor 687 regression, support vector regression (svr), artificial neural network (ann), adaptive neurofuzzy inference system (anfis) etc. for predictive modeling of the considered processes. although all of those techniques have provided satisfactory results, they also have their own limitations which often hinder their widespread applications as effective prediction tools. the main disadvantages of different statistical and machine learning techniques are summarized in table 1. table 1 disadvantageous features of some popular statistical and machine learning techniques method disadvantages linear regression it assumes normal distribution of the input variables, and presence of a linear relationship between the dependent and independent variables. in reality, these assumptions are often not valid. it is also quite sensitive to the presence of outliers. knn regression it is highly sensitive to the scale of the data. it also does not perform well for large datasets and widely varying dimensional data. ridge regression it includes bias in the model output. selection of hyperparameters may also affect its accuracy. lasso regression its prediction performance largely depends on the variability of the data under consideration. the selected features may result in higher bias. svr in this technique, selection of the appropriate kernel influences its prediction performance. it is also not at all suitable for large datasets and suffers from poor interpretability. ann its performance greatly depends on the system configuration and volume of the training data. being a black box type approach, it has poor interpretability. selection of the right activation function along with the number of hidden layers and number of nodes per layer affects its prediction accuracy. anfis it is highly sensitive to the number and type of the membership functions selected. cross-validation error would largely differ from the actual error for a smaller dataset. decision tree regressor being a highly unstable technique, a small change in the dataset may cause a significant change in the developed tree structure. it cannot be employed for continuous numerical variables. from table 1, it can be clearly noticed that all the considered statistical and machine learning techniques have some deficiencies, especially with respect to either flexibility or interpretability. thus, a trade-off has become essential between prediction accuracy and model interpretability. many of the machine learning algorithms are unstable, showing high variance resulting in poor prediction for the test datasets. using ensemble learning, these unstable and weak learners can be combined together to bring stability in the prediction process. the random forest (rf) regressor is based on ensemble learning, and can effectively bridge the gap between prediction accuracy and model interpretability. it is an aggregation of decision trees having more stability and capability to deal with continuous numerical variables. the rf is an example of the bagging method, which is an amalgamation of tree predictors that operates by constituting a profusion of decision 688 s. bhattacharya, s. chakraborty trees, making it less prone to bias. despite having a wide range of flexibility, it has not been specifically applied to predict responses for any of the machining processes. this paper lays down a framework to model a cnc face milling process using the rf regressor. unlike linear regression, it does not assume the presence of any existent relationship between the dependent and independent variables, and also does not need the dataset to be normally distributed. its prediction accuracy would suppose to increase with large datasets, unlike svr or knn regressor. its application does not require any supersophisticated hardware configuration (like ann); neither is there any need to choose any membership function (like anfis). there is also no requirement to scale the training and testing data before its application. all these advantageous features of the rf regressor make it a suitable machine learning technique having good prediction accuracy without a convoluted feature selection process. in this paper, an endeavor is thus put forward to explore the application potentiality of the rf regressor to predict values of mrr, ra and aec based on 27 experimental observations with n, f, ap and ae as the input cnc face milling parameters. this paper is organized as follows: section 2 presents a brief literature survey on the applications of different statistical and machine learning techniques in face milling operation. in section 3, the experimental details are presented, while in section 4, the theoretical and application framework for the rf regressor is laid down. section 5 introduces different statistical metrics along with the prediction results. conclusions are drawn in section 6. 2. literature survey it has already been mentioned that different statistical and machine learning techniques have been deployed by the past researchers as effective prediction tools for milling operation. table 2 provides a comprehensive review of the past literature mainly focusing on different milling parameters, responses and prediction tools considered for milling operations. it has become quite clear that although several forms of regression analysis, ann, anfis, svr, etc. have been adopted by the past researchers, the literature seriously lacks the application of the rf regressor as an effective prediction tool in the machining domain. to the best of the authors’ knowledge, there is no application of the rf technique as a predictive and regressor model in cnc milling operation. in order to validate the performance of the previously adopted prediction tools, a limited number of statistical metrics has been considered by the past researchers. in this paper, five statistical metrics in the form of mean absolute percentage error (mape), root mean squared percentage error (rmspe), root mean squared log error (rmsle), correlation coefficient (r) and root relative squared error (rrse) are taken into account to evaluate the performance of the rf regressor as an effective prediction tool during cnc face milling operation. prediction of responses in a cnc milling operation using random forest regressor 689 table 2 list of milling parameters, responses and prediction tools considered by the past researchers author(s) milling parameters response(s) prediction model(s) lo [6] s, f, ap ra anfis radhakrihnan and nandan [7] s, f, ap fc regression analysis, ann ozcelik and bayramoglu [8] s, ap, f, step over ra regression analysis lela et al. [9] n, f, ap ra regression analysis, svr, bayesian neural network rashid et al. [10] s, f, ap ra regression analysis dave and raval [11] n, f, ap fx, fy (cutting force along x and y directions) regression analysis, ann sharkawy [12] s, f, ap ra anfis, radial basis function network, genetically evolved fuzzy inference system durakbaşa et al. [13] n, ap, f ra regression analysis zhang et al. [14] s, f, ap ra gaussian process regression rubeo and schmitz [15] s, feed per tooth, radial immersion fc regression analysis bandapalli et al. [16] s, f, ap ra anfis yeganefar et al. [17] s, f, axial and radial depth of cut ra svr, ann, regression analysis lin et al. [18] s, f, ap ra regression analysis, ann this paper n, f, ap, ae mrr, ra, aec rf regressor table 3 milling parameters and their operating levels [19] parameter symbol unit level 1 level 2 level 3 cutting speed n rev/min 1200 1700 2200 feed rate f mm/min 220 270 320 depth of cut ap mm 0.3 0.4 0.5 width of cut ae mm 5 10 15 690 s. bhattacharya, s. chakraborty 3. experimental data using a cnc machine tool (carver 400m_rt) with a spindle power of 5.6 kw and a maximum rotational speed of 6000 rpm, khan et al. [19] performed face milling operations on aisi-1045 steel material. a three-fluted carbide cutting tool with 24 mm diameter was deployed for the milling operations. during the milling operation, four input parameters, i.e. n, f, ap and ae were considered and their settings were varied at three different operating levels. those input milling parameters and their varying operating levels are shown in table 3. based on l27 orthogonal array, 27 experiments were conducted while treating mrr (mm3/min), ra (µm) and aec (kj) as the process outputs/responses. the experimental plan and values of the measured responses are exhibited in table 4. during the application of the rf regressor as a prediction tool for this cnc face milling operation, among the 27 experimental runs, 21 trials are randomly selected for the training purpose and the remaining six trials are considered for testing of the developed model. table 4 experimental dataset [19] sl. no. n f ap ae mrr ra aec purpose 1 1200 220 0.3 5 330 3.30 535.802 training 2 1200 220 0.4 10 880 2.95 184.929 training 3 1200 220 0.5 15 1650 1.41 88.519 training 4 1200 270 0.3 5 405 3.83 426.109 training 5 1200 270 0.4 10 1080 3.87 146.050 testing 6 1200 270 0.5 15 2025 1.68 69.823 training 7 1200 320 0.3 5 480 3.97 361.832 training 8 1200 320 0.4 10 1280 3.53 122.976 testing 9 1200 320 0.5 15 2400 2.29 53.988 training 10 1700 220 0.3 10 660 1.81 337.042 training 11 1700 220 0.4 15 1320 1.13 142.727 testing 12 1700 220 0.5 5 550 3.47 299.031 training 13 1700 270 0.3 10 810 2.85 269.604 training 14 1700 270 0.4 15 1620 1.41 113.648 training 15 1700 270 0.5 5 675 3.91 238.476 training 16 1700 320 0.3 10 960 2.55 213.559 testing 17 1700 320 0.4 15 1920 1.39 92.551 training 18 1700 320 0.5 5 800 4.12 193.109 training 19 2200 220 0.3 15 990 1.76 244.303 training 20 2200 220 0.4 5 440 3.33 425.797 testing 21 2200 220 0.5 10 1100 2.36 165.620 training 22 2200 270 0.3 15 1215 1.17 193.939 training 23 2200 270 0.4 5 540 3.72 338.579 training 24 2200 270 0.5 10 1350 2.58 131.343 testing 25 2200 320 0.3 15 1440 1.41 160.886 training 26 2200 320 0.4 5 640 3.86 286.850 training 27 2200 320 0.5 10 1600 2.76 108.147 training prediction of responses in a cnc milling operation using random forest regressor 691 4. application of rf as a prediction tool during any machining operation, depending on the experimental plan employed, a large volume of useful dataset is usually generated. to better understand the machining operation and study the influences of the input parameters on the responses, a suitable model needs to be developed so as to extract valuable information from the experimental dataset. a subfield of artificial intelligence which mainly focuses on various ways of training the machines for having a better understanding of a problem/system, is called machine learning. as it can be interpreted, the goal of a machine learning algorithm is to better generalize an existing problem while providing the desired solutions. to achieve the desired outputs, a designer needs to train different learners. often, due to presence of noise in the training data, the designed learners turn out to be occasionally weak. ensemble learning is a machine learning archetype [20] where multiple learners are combined together to predict the response values. two of the most commonly employed ensemble learning approaches are bagging and boosting [21]. bagging or bootstrap aggregation is a parallel ensemble method, whereas boosting is considered as a sequential ensemble method. ensemble learning models perform best for machine learning techniques that are generally unstable, like decision trees, anns etc. [22]. the main reason behind using unstable learners for ensemble learning is that they can produce different generalization patterns which help in minimizing variability to some extent [23]. the rf is an example of the bagging method [24], which is an amalgamation of tree predictors that operates by constituting a profusion of decision trees. it can be effectively employed for both classification and regression. the basic function of rf can be understood using the schematic diagram, as depicted in fig. 1. fig. 1 schematic diagram of a random forest 692 s. bhattacharya, s. chakraborty from this diagram, it can be observed that, based on the training dataset, several decision trees are created which are assumed to be uncorrelated. each of the decision trees is developed based on subsets of variables and samples from the training data. again, each of the subsets of variables is considered with replacement. in the rf regressor, the final prediction is performed after averaging the outputs of all the developed decision trees. while employing the rf regressor as an effective prediction tool, there are few parameters to be tuned by the concerned designer, mainly based on intuition [25]. in table 5, some important parameters of the rf regressor are provided, where p is the number of input variables. table 5 parameters of a rf regressor parameter default value number of decision trees 500 number of variables per split √p maximum number of terminal nodes unrestricted resampling scheme with replacement while employing the rf regressor as a prediction tool for the cnc face milling operation, the number of decision trees generated plays a significant role. a smaller number of decision trees leads to underfitting of data, whereas a large number of decision trees are responsible for data overfitting. when each decision tree is framed, there is a scope of feature selection where the designer can choose all the input variables under consideration or set them accordingly. the maximum number of terminal nodes, as the name suggests, is the upper limit of number of nodes that each tree can have. now, when a subset of training data is adopted to model the rf, the designer may wish to set features in such a way that if a subset is once used, it would not be used again. in this case, the resampling scheme needs to be considered without replacement. among various parameters employed for modeling a rf regressor, number of decision trees and number of variables selected per split mostly affect the prediction accuracy. the default value for number of variables per split is the squared root of the number of input variables, but for datasets with a smaller number of input variables (preferably less than 13), number of variables per split is generally set equal to the number of input variables [25]. on the other hand, the optimal number of decision trees to be framed is identified after simulating the model for up to 500 decision trees and then selecting the number which would yield the lowest value of mean squared error (mse). the variations of mse value with changing number of decision trees for mrr, ra and aec are portrayed in figs. 2-4, respectively. from these figures, the optimal number of decision trees for each rf is selected having the lowest mse value for each of the responses under consideration. the optimal numbers of decision trees to be developed for mrr, ra and aec are provided in table 6. table 6 optimal number of decision trees for each response response number of decision trees mrr 367 ra 25 aec 195 prediction of responses in a cnc milling operation using random forest regressor 693 fig. 2 number of decision trees against mse for mrr fig. 3 number of decision trees against mse for ra 694 s. bhattacharya, s. chakraborty fig. 4 number of decision trees against mse for aec in this paper, the entire modeling based on the past experimental data for the cnc face milling operation is performed with the help of random forest package available in the statistical programming software r [26], which is an open-sourced, robust and easy to apprehend language [19]. based on the training dataset, the developed rf regressor generates a large number of decision trees (as mentioned in table 6) for each of the responses under consideration which are finally aggregated to predict the corresponding response values. some typical examples of the framed decision tress for mrr, ra and aec are respectively provided in figs 5-7. in fig. 5, for mrr, the rf regressor first treats width of cut (ae) as the predictor variable in the root node. now, depending on its value, two branches emerge from the root node. when its value is observed to be greater than 5 mm, feed rate (f) is considered as the next predictor variable. the rf regressor predicts the mrr value as 1628.8 mm3/min for feed rate greater than 220 mm/min. on the other hand, when the corresponding feed rate is less than or equal to 220 mm/min, it envisages the value of mrr as 1056 mm3/min. in the experimental dataset, there are eight observations satisfying the condition of width of cut greater than 5 mm and feed rate greater than 220 mm/min. similarly, five observations fulfill the condition of width of cut greater than 5 mm and feed rate less than or equal to 220 mm/min. in this decision tree, when the value of width of cut is less than or equal to 5 mm, spindle speed (n) is considered as the succeeding predictor variable. prediction of responses in a cnc milling operation using random forest regressor 695 fig. 5 a sample decision tree for mrr fig. 6 a sample decision tree for ra 696 s. bhattacharya, s. chakraborty fig. 7 a sample decision tree for aec when the corresponding spindle speed is noticed to be greater than 1700 rev/min, it leads to a terminal node with the predicted mrr value as 590 mm3/min. but, for spindle speed less than or equal to 1700 rev/min, depth of cut (ap) is adopted to generate two more child nodes. the rf regressor predicts the mrr value as 675 mm3/min when the depth of cut is found to be more than 0.4 mm, and for depth of cut less than or equal to 0.4 mm, the predicted value of mrr is 405 mm3/min. the decision tress for ra and aec, in figs. 6 and 7, can also be similarly explained. 5. prediction performance of the rf regressor in this paper, the prediction performance of the proposed rf regressor is validated using five statistical metrics, i.e. mape, rmspe, rmsle, r and rrse. the mathematical formulations of all these measures are presented as below: mape: 100 1 1  −  = n i i ii a pa n (1) rmspe: 100 1 1 2         −  = n i i ii a pa n (2)  = +−+ n i ii ap n 1 2))1log()1(log( 1 (3) r:   = = −− −− n i n i i ppaa ppaa 1 22 1 )()( ))(( (4) prediction of responses in a cnc milling operation using random forest regressor 697 rrse:   = = − − n i i n i ii aa ap 1 2 1 2 )( )( (5) where ai and pi are the actual and predicted response values, a and p are the means of all the actual and predicted response values, and n is the number of observations in the test dataset. the mape measures the absolute percentage error between the actual and predicted response values. its main problem is that it introduces a heavy penalty when the actual value is close to 0. on the contrary, rmspe provides an estimation of the standard deviation of the residuals. but its value is significantly affected by the presence of outliers in the dataset. this problem can be avoided to some extent by the application of rmsle along with rmspe. the degree of association between the actual and predicted response values is computed using r value. finally, rrse calculates the total squared error and normalizes it while dividing by the total squared error of the simple predictor. while taking the square root of the relative squared error, the error is reduced to the same dimension as the response being predicted. among all these measures, a higher value is always preferable for r, while for the remaining measures, lower values would indicate better prediction performance of the rf regressor [27]. in table 7, the predicted values of all the responses for the considered testing dataset are provided. on the other hand, table 8 shows the computed values of the five statistical metrics used to evaluate the prediction performance of the developed rf regressor. table 7 predicted response values using the rf regressor sl. no. response mrr ra aec actual predicted actual predicted actual predicted 1 1080 1026.07 3.87 2.66 146.050 182.17 2 1280 1173.09 3.53 2.67 122.976 174.08 3 1320 1546 1.13 1.53 142.727 132.51 4 960 1093.37 2.55 2.62 213.559 253.95 5 440 626.87 3.33 3.46 425.797 320.95 6 1350 1125.97 2.58 2.57 131.343 155.14 table 8 values of the statistical measures for the responses statistical measure mrr ra aec mape 17.23 16.34 22.51 rmspe 21.00 21.78 24.77 rmsle 0.19 0.16 0.23 r 0.85 0.78 0.92 rrse 0.53 0.7 0.5 698 s. bhattacharya, s. chakraborty from table 8, it can be noticed that although ra has the lowest r value, its corresponding mape score is the best. for ra, the model has overestimated and underestimated the true ra value more frequently than the other responses. but, even after this, the relative deviation is the lowest. it can be noticed from both these tables that rf regressor performs satisfactorily while foreseeing the values of all the three responses of the cnc face milling operation under consideration. from the results of the test dataset, it can be noticed that the model has not overfitted the data. otherwise, the outcome of the test dataset would have been far worse even after getting a good result from the training dataset. non-parametric machine learning techniques do not assume anything about the dataset, whereas, in parametric techniques, some assumptions are made with respect to the underlying distribution of the dataset as well as the relationship between the dependent and independent variables. thus, modeling of a machining operation and prediction of the corresponding responses using a non-parametric machine learning technique with a small dataset is quite challenging, but the proposed rf regressor yields satisfactory results even with a small experimental dataset for cnc face milling operation. while predicting the corresponding response values, it also employs minimum number of milling parameters as the predictor variables in the decision trees. 6. conclusions over the years, application of different machine learning techniques in the manufacturing domain has increased exponentially. it has now become a challenging task to choose an appropriate machine learning technique to depict the relationship between the dependent and independent variables of any machining process. in this paper, an attempt is put forward to employ the rf regressor as an effective prediction model based on a small experimental dataset of cnc face milling operation. it has several advantageous features as compared to other statistical and machine learning models. its main advantage is that it does not consider the inherent distribution of the input data or existent relationship between the dependent and independent variables. number of optimal decision trees and number of input variables per split are enough to develop this prediction tool. its robustness makes it suitable for generalizing different machining-related applications. machining conditions with binary or more than two categorical input variables can be accommodated in this tool without much effort. but, it has also some limitations. it fails in the cases when the data is outside the ‘scope’ of the model. suppose that there is a training space where each input parameter has a particular range. if the test data has some values totally outside the range, the model would fail. this problem would not occur for linear regression. the rf regressor would also perform poorly in the case of sparse data where certain expected values do not exist at all. as a future scope, the prediction performance of the rf regressor can be explored using large datasets, although it may lead to overfitting of data. its application potentiality can also be validated based on experimental datasets from other metal removal processes, like cnc turning, cnc end milling, and especially non-traditional 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harbour road, west bengal, india 2military academy, university of defence in belgrade, belgrade, serbia 3university of belgrade, faculty of organizational sciences, belgrade, serbia 4college of engineering, yuan ze university, taoyuan city, taiwan 5department of computer science and mathematics, lebanese american university, lebanon 6faculty of mechanical engineering, university of niš, niš, serbia 7tu berlin, department of structural analysis, berlin, germany abstract. researchers work hard to embrace technological changes and redefine the quality management as quality 4.0 (q 4.0). in this context, the purpose of the current work is twofold. first, it aims to compare the preparedness of the small and medium enterprises (smes) for sustaining in q4. second, it intends to propose a novel hybrid spherical fuzzy based multi-criteria group decision-making (magdm) framework with einstein aggregation (ea). a real-life case study on six smes is carried out with the help of three experts. for aggregating the individual responses (using spherical fuzzy numbers or sfns), ea is used. then two very recent models such as simple ranking process (srp) and symmetry point of criterion (spc) are extended using sfn to rank the smes. finally, the validation tests and sensitivity analysis are carried out. it is noted that the application of analytical tools, knowledge management and use of technology under the support and mentorship of visionary leadership are the key criteria for building up the capability to embrace q 4.0. interestingly, it is noted that medium scale firms are better prepared than small-scale enterprises. this work is apparently a first of its kind that focuses on smes for assessing their quality management practices in industry 4.0 era. key words: quality 4.0, spherical fuzzy sets (sfs), einstein aggregation, simple ranking process (srp), symmetry point of criterion (spc) received: august 31, 2023 / accepted october 18, 2023 corresponding author: darko božanić military academy, university of defence in belgrade, veljka lukica kurjaka 33, 11040, belgrade, serbia e-mail: darko.bozanic@va.mod.gov.rs 454 s. biswas, d. božanić, d. pamučar, d. marinković 1. introduction quality has been one of the critical factors for the success, survival and growth of any organization irrespective of its scale, products/services, structure and processes. over the years with the industrial revolutions powered by technological advancement, the fundamental concepts and practices of the quality management have undergone a notable change. presently, the world is experiencing the phenomenal impact of the fourth industrial revolution (aka industry 4.0) on the socio-economic, cultural and environmental wellbeing. industry 4.0 (i 4.0) is “the current trend of automation and data exchange in manufacturing technologies, including cyber-physical systems, the internet of things, cloud computing and cognitive computing and creating the smart factory” that is characterized by seamless connection and coordination among people, system and technology [1]. the enhanced operational efficiency, creation of new business models, services, and goods, i 4.0 promises to change industrial production. in contrast to traditional forecasting, it allows for real-time production planning and dynamic optimization. nowadays, affordable and intelligent sensors and wireless networks can capture the product lifecycle data [2-5]. analysis of data does not require supercomputers anymore [6], and economical cloud and fog computing make data easily approachable [7], for instance, cloud computing services provided by amazon and alibaba [8]. iiot platforms are used for data transmission, storage, and analysis [9]. i 4.0 has prompted society 5.0 (soc. 5.0) to place the human being at the center of innovation, technological revolution, and industrial automation. soc. 5.0, also known as the super intelligent society, aims to leverage sophisticated technology from i 4.0 for the benefit of humanity, promoting connectivity between people and systems in cyberspace with artificial intelligence-assisted results optimization. soc.5.0 intends to utilize technological developments for improving the quality of living of society by solving socioeconomic issues and acts as a driving force in redefining human life. in a nutshell, soc. 5.0 is a juxtaposition of cyberspace and physical world [10] with the objective of meeting sustainable development goals (sdg). the bridge between i 4.0 and soc. 5.0 is by means of internet of things (iot). case study [11] revealing the enhanced efficacy in the manufacturing industry by reducing the downtime costs have been seen with the use of industrialiot combined with predictive analytics. i 4.0 focuses on enhancing manufacturing efficiency through high-tech techniques [12]. in certain circumstances, the implementation of i 4.0 shows that linkages between persons, systems, and objects can result in an efficient, full, dynamic, and real-time network [13]. soc. 5.0 is a companion to i 4.0 and promises to transform society for the better welfare of humanity [14]. this civilization serves as a vehicle for social growth, with the goal of having a significant influence on society at all levels, including quality of life and sustainability [15]. evolution of quality 4.0 or q 4.0 [16] alongside the dependency of traditional quality tools unlike pareto charts and control charts within organizations have been found to emerge. people, process and technology are the three driving forces for q 4.0. a natural evolution from total quality management (tqm) to q 4.0, not replacing the traditional quality methods; activities inside organizations emerged rapidly and smartly, due to the confluence of big data and artificial intelligence (ai), united with traditional quality methods. as per kpmg reports, india’s budget announcement of an inr50,000 crore fund for the national research foundation (nrf) aims to boost the overall research environment in the country, resulting in high-quality innovation. the operational ability of q 4.0 will a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 455 encompass a self-diagnostic and self-healing systems with faster delivery for an optimized operating model by providing transformational results in quality [17]. the quality of any product and services is one of the most crucial factor, it gives the company an edge on competitive advantage in the global market. the progression of i 4.0 has provided a catalytic effect to the development in the field of quality management. it is an information driven, real-time, innovation focused, analytical and integrated ecosystem that drives the quality management in the organizations [18-19]. in this regard, we may define q 4.0 as the digital technology and high-end computation enabled process management to take agile measures for ensuring customer value by providing products and services [20]. q 4.0 helps to take almost real-time decisions on the product quality using iot and ai/ml enabled systems and subsequently to discriminate the information to the stakeholders [21-22]. in effect, q 4.0 provides the organizations the ability to cope up disruptions and uncertainties in a dynamic environment and support the transformation and growth of the organizations [23-27]. however, the concept of q 4.0 has not been explored comprehensively yet both from conceptual and application perspectives [28-30]. the existing literature shows some recent contributions in exploring various factors that decide the success in embracing q 4.0. this includes: top management commitment, leadership, ownership to work, organizational culture and transparent communication, capability to critical analysis and problem solving, innovative mind-set, training and development, ability to capture and handle big data, awareness and skill set for conceiving, understanding and applying digital technologies and analytical models for data analysis and drawing actionable insights, organization of wide integration of activities, collaboration of the stakeholders, fund support, etc. [27, 28, 31-39]. future of q 4.0 lies in imbibing the essence of quality of work life, ensuring psychological well-being, boosting up morale and character development, empowerment, sharing of knowledge and embracing spirituality coupled with advancement in technology for mankind. the coming era of society 5.0 calls for upgradation of q 4.0 to next level, i.e., q 5.0 [40]. in other words, q 5.0 shall compose sociology, science, technology, and religion (ssrt) seamlessly for the development and progress of the humanity and civilization. all these concepts converge to a single point, i.e., inclusive development wherein small and medium-sized enterprises (sme) are essential part of the movement. therefore, the need of smes to embrace q 4.0 emerges as a critical issue. the present work focuses on the assessing the present state of the small and medium enterprises (smes) to harness q 4.0. smes play a significant role in gearing india's economic growth in any manufacturing company. they act as effective support systems to large-scale manufacturing, thereby fostering the industries' growth. many prominent business organizations rely on smes as they are the suppliers of the products, and hence smes play a vital role in the value creation besides the supply chain management. this sector is currently termed as "engine growth of india” [41]. as of date, approximately 42.50 million registered and unregistered smes are operational in india. the indian economy experts have said that the sme sector has a huge potential and can contribute to achieving the target of the nation's manufacturing policy, which is assumed to contribute 25% of india's gdp in 2022. in india, smes generate employment at a cheaper cost of capital, and it also helps balance the imbalances amongst various regions [42]. i 4.0 or smart manufacturing has many offerings regarding opportunities for the sme sector, which can help them enhance their competitiveness. since smes do not operate on a very large scale, it becomes comparatively easy for them to implement and experiment with digital transformation 456 s. biswas, d. božanić, d. pamučar, d. marinković faster than the bigger organizations. in recent times, smes have been witnessing significant transformations due to new-age technologies driven mainly by ai/ml and iot. the advanced technologies nowadays are helping smes to automate and generate intelligence from their day-to-day activities. the adaptation of new-gen technology has become essential for the growth and survival of smes in the new normal [43]. the analysis of the scientific literature also says that there has been a shift in production management towards intelligent manufacturing, which is based on data analytics techniques. sme's needs to embrace i 4.0 at a much faster pace as tech-based solutions have immense potential to enrich and fetch greater revenues for the sme's and help in the continuity of the business during pandemic disruption. the effectiveness in adapting quality management system for attaining improved performance in smes can be found established in iso 9001:2015. the laid down provision establishes the generic requirements that are applicable to any organization, regardless of its type or size, or the products and services it provides in a two-fold way i.e., to comply by the regulations and meet consistency and enhance customer satisfaction [44]. effective adaptation of the philosophy of tqm and right intent to implement the soft (e.g., leadership and commitment of the top management, customer focus, organization wide quality culture, training and development of the employees) and hard practices (for instance, process analysis and continuous improvement, adopting advanced technology, implementation of standards and specifications, information analysis) significantly help the organizations to improve the sustainability performance [22]. implementation of tqm is one of the major sources of competitive advantage for the smes [45]. in this paper we present a case study on six smes. these smes mostly deal with engineering products for supplying to the large organizations from the public sector like indian railways and private sectors like automobile and industrial products. the smes are located in the eastern part of india. we aim to investigate their capabilities or readiness in sustaining in the era of i 4.0 and soc. 5.0. we derive a list of critical factors that decide the success or failure in i 4.0 and soc. 5.0 for the smes. a group of three experts participated in the investigation process. the experts have substantial expertise in quality engineering, lean manufacturing, process analysis, strategy formulation and technology management with work experiences of 18 years, 21 years and more than 25 years respectively. the experts visited the plants of the smes along with a support team, observed the process, discussed with the management and staff and collected evidences. based on their opinions the alternatives were compared on the attributes that manifest the critical success factors as identified through literature review and experts’ views. for analysis purpose, an integrated magdm framework using spc (used to derive the criteria weights) and srp (applied for ranking the smes). to deal with the subjective bias associated with the opinions of the experts, sfs is used. the original framework of spc and srp are extended by using sfns. this paper offers two contributions: managerial contributions: the present paper provides a comparison of readiness of smes based on critical success factors of total quality management in i 4.0 while heading towards soc. 5.0. the growth of the economy of a country largely depends on the contributions of the smes. the competitiveness of the smes is significantly dependent on quality. furthermore, as the large organizations have been gearing to adapt new technologies, it is quite imperative for the smes to rise on the occasion and usher in implementing q 4.0. a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 457 based on the findings within our search we contend that this kind of study in indian context adds value due to scantiness of literature vis-à-vis our problem. technical contributions: the ongoing work proposes a new hybrid sfn based decision support system using an integrated spc-srp model applying ea. the srp method is free from the effect of selection of appropriate normalization scheme, which acts as a potential cause of improper outcome and instability of the solution. further, srp model is able to consider individual criterion impact and works considerably under complexities [46]. to complement the benefits of the srp method, the present work uses another newly developed model such as spc [47] for deriving criteria weights. spc method considers the symmetry point (a middle point of the interval decided by the upper and lower values) for each criterion. the method balances the effect of the criteria and is free from normalization. further, it considers absolute deviation and is very easy to implement. the concept of sfs has evolved gradually as an advanced extension of fuzzy sets (fs) [48], intuitionistic fuzzy sets (ifs) [49], type 2 ifs [50], interval valued ifs [51], neutrosophic fuzzy sets (nfs) [52], picture fuzzy sets (pfs) [53-54], pythagorean fuzzy sets (pyfs) [55], and q rung orthopair fuzzy sets (qrofs) [56]. sfs uses three dimensional geometrical properties with degree of positive (μ), neutral (η) and negative memberships (ϑ) [57]. the relative advantages of sfs [57-59] over the others are given below: ▪ sfs considers μ, η and ϑ unlike fs, ifs, pyfs and qrofs, ▪ greater flexibility in selection of membership values such that sum of square of such values lie between 0 and 1 and that way it solves the issue with pfs, ▪ enables the researchers to better work with vagueness and uncertainty for a granular analysis, ▪ three-dimensional space is comparatively lesser complex than qrofs and fermatean fuzzy sets (ffs) [60]. the existing literature shows a very limited number of applications of spc method [6163] and almost no application of srp model. also, prior to the present work no other studies have attempted to combine these two models that do not require any consideration for the type of normalization. furthermore, the present work provides a novel sfs based extension of srp and spc methods using ea as an aggregate operator (ao). ea provides flexibility to the researchers and enables a granular analysis under uncertainty. hence, the framework proposed in this paper extends substantial technical benefits in solving complex real-life problems. 2. preliminaries in this section, some of the preliminary definitions and operations related to sfs are briefly described. for detailed descriptions of the concepts and operations and their proofs the readers may refer to the past contributions [57-59, 64-67]. sfs has been extensively used by the researchers in building decision-making models and their applications in several real-life problems. some of the recent applications are mentioned in table 1. 458 s. biswas, d. božanić, d. pamučar, d. marinković table 1 examples of applications of sfs application area references medical diagnosis [68-69] 3d printer selection [70] location selection for waste disposal [71] insurance policy selection [72] supplier selection [73] process analysis [74-75] planning for drone based city logistics services [66] energy management [76] formulation of advertisement strategy [77] agricultural management [78] personnel selection for sales profession [79] definition 1. a spherical fuzzy set (sfs) is defined as  , ( ( ), ( ), ( ))= s x x x x x u   (1) where u is the universe of discourse, μ, η and ϑ: u→ [0, 1]. the relationship is given below: 2 2 20 1 + +    (2) definition 2. basic operations. let, a spherical fuzzy number (sfn) is represented as s = {μ, η, ϑ} and s1 and s2 are the two sfns. some of the basic operations are defined as: addition: 1 1 2 2 2 2 2 2 2 2 2 22 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2( ) , , ((1 ) (1 ) )    = + − − + − −    s s            (3) multiplication: 1 1 2 2 2 2 2 2 2 2 2 22 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2, ( ) , ((1 ) (1 ) )    = + − − + − −    s s             (4) multiplication by a scalar; w > 0 1 1 2 2 2 22 2(1 (1 ) ) , , ((1 ) (1 ) )   = − − − − − −    w w w wws      (5) power of sfn let w > 0 denotes the power of s wherein w is a scalar quantity. then we have the following definition 1 1 2 2 2 22 2, (1 (1 ) ) , ((1 ) (1 ) )   = − − − − − −    w w w w ws      (6) a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 459 compliment of s  , ,=cs    (7) definition 3. weighted aggregation operators (ao). let w = {w1, w2 …wn} be the weights of the sfns s1, s2….sn, n is a finite number and sum of weights = 1. then the basic weighted averages are defined as follows: spherical fuzzy weighted arithmetic average (sfwaa) 1 2 3 1 1 2 2 2 22 2 1 1 1 1 ( , , ,...... ) (1 (1 ) ) , , ( (1 ) (1 ) ) = = = = =   − − − − − −       i i i i w n n n n n w w w w i i i i i i i i i sfwaa s s s s      (8) spherical fuzzy weighted geometric average (sfwga) 1 2 3 1 1 2 2 2 22 2 1 1 1 1 ( , , ,...... ) ,(1 (1 ) ) , ( (1 ) (1 ) ) = = = = =   − − − − − −       i i i i w n n n n n w w w w i i i i i i i i i sfwga s s s s      (9) definition 4. einstein t-norm and s-norm. the t-norm and s-norm are defined as: 2 21 (1 ).(1 ) = + − − e dl t d l (10) 2 2 2 2 ( , ) 1 + = +  e d l s d l d l (11) for details about the properties of einstein t-norm and s-norm and related spherical fuzzy einstein operators, one may refer to the work of [80]. definition 5. spherical fuzzy einstein weighted average (sfewa) is defined [80] as: 2 2 1 1 2 2 1 1 1 1 2 3 2 2 1 1 1 2 2 1 1 (1 ) (1 ) , (1 ) (1 ) 2 ( ) ( , , ,...... ) , (2 ) ( ) 2 ( ) (2 ) ( ) = = = = = = = = = =   + − −       + + −         =    − +             − +             k k k k k k k k k k n n w w k k k k n n w w k k k k n w k k n n n w w k k k k n w k k n n w w k k k k sfewa s s s s            (12) 460 s. biswas, d. božanić, d. pamučar, d. marinković definition 6. score and accuracy function. the score and the accuracy functions are defined [59, 64] as: 1 (2 ) 3  = + − −   (13) = −  (14) the certainty function is found as: =  (15) rules for comparison of the sfns if 1 2   s s then 1 2s s else if 1 2   s s then 1 2s s if 1 2  = s s then if 1 2  s s then 1 2s s otherwise, if 1 2  s s then 1 2s s if 1 2  = s s and 1 2  =s s then if 1 2 s s then 1 2s s definition 7. defuzzification the defuzzified value of a sfn is obtained as: 1 2 2 2(100 [(3 ) ( ) ] ) 2 2  =  − − −     (16) 3. proposed framework using sf-spc-srp with ea the present section illustrates the procedural steps of the proposed framework briefly. step 1. identification of the criteria for comparing the smes included in the study. it is seen from the past studies that there is a scantiness of work concentrating comprehensively on exploring the critical issues pertaining to adaptation of q 4.0 and subsequently, soc. 5.0. further, it is observed that apparently no prior work exists that attempted to compare the readiness of the smes to embrace q 4.0. however, from the allied studies [22, 27-28, 31-39, 44] it can be made out that for a successful transition to q 4.0 and eventually, to soc. 5.0, the smes need to be mentored by the visionary leadership, supported with resources in a congenial work environment, trained to use advanced technologies and analytical tools and develop a knowledge base. in line with the crux of the past studies, the present work decided to include the criteria listed in table 2 for assessing the preparedness of the smes. a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 461 table 2 description of the criteria s/l criteria description effect c1 organizational structure lean and flat structure (+) c2 leadership support governance, commitment, resource support, adaptable, clarity in the vision and mission, approachable, fast decision making (+) c3 organizational culture focus on process, problem solving and team work attitude, open to ideas and suggestions, innovativeness (+) c4 level of automation use of technology in process (+) c5 information sharing assimilation and dissipation of information, access to information, open communication, transparency (+) c6 data management and use of analytical tools data capturing, storage, database management and analysis of data using scientific methods (+) c7 collaboration collaboration with the suppliers and customers, co-makers, collaboration and coordination with the stakeholders (+) c8 knowledge management continuous learning and use in action, training and upskilling, manpower planning, retention, r&d (+) c9 business risk fund and resource shortage, vulnerability to disruptions, absence of risk management plans, security and privacy (-) step 2. selection of the linguistic scale for rating of the criteria and compared units (i.e., alternatives). the linguistic scales and corresponding sfns used for finding the comparative weights of the criteria and rating of the alternatives are given in table 3. step 3. selection of the expert panel members. in this study, a group of three experts (e1, e2 and e3) took part in evaluating the alternatives. the experts have substantial work experience of 15, 22 and 28 years respectively and expertise in quality management, lean manufacturing, process management and analysis, technology implementation, manufacturing and managing large-scale operations. table 3 linguistic scale and sfns used for rating linguistic term code μ η ϑ very high/ very good vh/ vg 0.9 0.1 0.1 high/ good h/ g 0.7 0.3 0.3 medium m 0.5 0.5 0.5 low/ poor l/ p 0.3 0.3 0.7 very low/ very poor vl/ vp 0.1 0.1 0.9 step 4. rating of the criteria and alternative units. let i = 1, 2…m is the number of alternatives (i.e., the number of sample units under comparison) ai being the ith alternative; cj denotes the jth criterion (j = 1,2….n); k=1,2…r is the number of experts/decision maker who took part in the rating/ evaluation. the sfn corresponding to the rating of the performance of ai with respect to cj as opined by the kth decision maker in linguistic term is given by: { , , }r r r r ij ij ij ij   = 462 s. biswas, d. božanić, d. pamučar, d. marinković step 5. aggregation of individual ratings and formulation of the decision matrix. using ea as the ao (see expression (12)), the aggregation is done as follows: 2 2 1 1 2 2 1 1 1 2 3 1 2 2 1 1 1 2 (1 ( ) ) (1 ( ) ) , (1 ( ) ) (1 ( ) ) 2 ( ) { , , } ( , , ,.... ) , (2 ( ) ) (( ) ) 2 ( ) (2 ( ) ) k k k k k k k k r r w wk k ij ij k k r r w wk k ij ij k k r wk ij r k ij ij ij ij ij ij ij ij n n w wk k ij ij k k r wk ij k wk ij sfewa                  = = = = = = = = + − − + + − = = = − + −         2 1 1 (( ) )k k r r wk ij k k  = =                                   +      (17) similarly, the performance values of all the alternatives with respect to all criteria are obtained and the decision matrix is formulated as: 11 12 1 21 22 2 1 2 ... ... ... ... ( ) ... ... ... ... ... ... ... ... ... ... ... ... n n ij m n m m mn m n d                    = =         (18) it is to be noted that all the elements of the decision matrix are sfns, i.e.: 11 11 11 11 12 12 12 12{ , , }, { , , }, { , , }mn mn mn mn           = = = step 6. derive the score matrix. using the definition given by eq. (13), the scores of the elements of the decision matrix are obtained as 1 ( ) (2 ) 3  = = + − −ij ij ij ij ijscore     (19) accordingly, the score matrix is given as: 11 12 1 21 22 2 1 2 ... ... ... ... ( ) ... ... ... ... ... ... ... ... ... ... ... ... n n ij m n m m mn m n               =  =          (20) a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 463 now the weights of the criteria are calculated using the conventional steps of the spc method [47]. step 7. calculate the symmetry points of the performance values with respect to each criterion. the symmetry point for each criterion is the middle value of the interval and is calculated as: ( ) ( ) ; 1,2... ; [1, ] 2  +  = =   ij ij j min max sp i m j n (21) step 8. construct the absolute distance matrix. the elements of the absolute distance matrix are given by: 11 12 1 21 22 2 1 2 ....... ....... ( ) ..... ..... ....... ..... ....... n n ij m n m m mn m n                   = =       (22) 11 11 1 12 12 2, ,.......,=  − =  − =  −mn mn nsp sp sp   step 9. construct the matrix of the moduli of the symmetry. the modulus of the ith alternative with respect to the jth criterion is defined as the averaged absolute distance from cj. the values (rij) are calculated as: 1= =   m in i mn mn mr  (23) accordingly, the matrix r is formulated. step 10. derive the modulus of the symmetry criterion. the modulus of the symmetry criterion is obtained by averaging the elements of the corresponding column of r matrix. accordingly, a row vector is formulated whose elements are given by: 1 2 1 1 1 1 1( ) , ,...., ; 1,2... m m m i i in i i i j n r r r q q j n m m m = = =       = = =          (24) 464 s. biswas, d. božanić, d. pamučar, d. marinković step 11. derive the weights of the criteria. the weight of the jth criterion is obtained as: 1 1 ; 1 = = = =  n j j jn j j j q q   (25) using the weights as derived at step 11, the ranking of the alternatives are done in the subsequent steps based on srp [46]. step 12. formulation of the ranking matrix. using the score matrix (obtained at step 6) the alternatives are ranked with respect to each criterion. the alternative with highest ℑij value will be ranked first if the desirable effect is maximization whereas the alternative with lowest ℑij value will be ranked first if the desirable effect is minimization. accordingly, the ranking matrix is formulated as φ whose each element is the rank of the ith alternative with respect to the jth criterion, oi(j). step 13. formulate the weighted ranking matrix. the elements of the weighted ranking matrix is given by: ( ) ( )=i j j i jv o (26) step 14. calculate the total ranking score for each alternative. the total ranking score for each alternative is calculated as: ( ) 1= = n i i j j rs v (27) step 15. find out the priority score of the alternatives. the priority score for each alternative is calculated as: = −i ip m rs (28) decision rule: the higher is the value of the priority score, preferred is the alternative over the others. 4. case study in this paper we present a case study on six smes (which we call as alternatives in our study), located in the eastern part of india. we aim to investigate their capabilities or readiness in sustaining in the era of i 4.0 and soc. 5.0. our analysis is grounded on the theoretical perspectives of q 4.0. we derive a list of critical factors that decide the success or failure in i 4.0 and soc. 5.0 for the smes. a group of three experts have participated in the investigation process. the experts visited the plants along with a support team, observed the process, discussed with the management and staff and collected evidences. based on their opinions the alternatives have been compared on the attributes that manifest a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 465 the critical success factors as identified through literature review and experts’ views. for analysis purpose, we use a new integrated sf-spc-srp framework in this paper. the research framework for this paper is illustrated by fig. 1. fig. 1 flow chart of the research table 4 provides the short description of the sample units. the sample units have been visited, diagnosed and rated with respect to the criteria as given in the table 2 by the experts. the rating and concerned data analysis is presented in the next section. 466 s. biswas, d. božanić, d. pamučar, d. marinković table 4 sme organizations under study unit scale of business product/service annual turnover (rs. crore) a1 small fabrication, forging, and heat treatment machine items 28 a2 small machine tools and inspection instruments, and spare parts 20 a3 medium welding items 200 a4 small battery charger, transformer, l.t. control panel 30 a5 medium conveyor components, heavy structural fabrication items 15 a6 small pvc pipe and related products 12 5. findings in this section, we present the brief summary of the data analysis. table 5 provides the rating given by the experts for evaluating the smes under study. the corresponding sfn values can be obtained from the table 3. putting the corresponding sfns we apply the ao such as sfewa (see eq. (17)) to aggregate the opinions of the experts (in our case the number of experts, r = 3). all experts are given equal priority (i.e., 1/3). table 6 provides the sfn based decision matrix, obtained by aggregating the ratings of the three experts on the six smes under consideration. table 5 evaluation of smes by the experts expert criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 alternatives (+) (+) (+) (+) (+) (+) (+) (+) (-) expert 1 a1 vg vg m g g m m vg m a2 g vg m g m p p m m a3 vg vg g m g g vp g p a4 m g p vg vp m m p vp a5 g m vg m g m g m m a6 p g m g p vp m m vp expert 2 a1 g vg m g g g g vg p a2 g m m m m p m g m a3 m g g p p vg p m vp a4 g m vp vg vp m vp p vp a5 m vp vg m g p g m m a6 p m m g vp vp m m vp expert 3 a1 vg vg g vg m g m g m a2 vg g g m g vp p m p a3 g g m p m g vp m m a4 p m p g vp m p vp p a5 g p g m g p m p m a6 vp m m g p vp m m vp a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 467 1 3 1 3 1 3 1 1 3 3 1 3 1 3 1 2 3 14 14 14 14 14 14 14 13 3 2 23 14 14 1 1 13 2 2 3 14 14 1 1 3 14 1 3 3 2 2 14 14 1 1 3 14 1 3 2 14 1 { , , } ( , , ) (1 ( ) ) (1 ( ) ) , (1 ( ) ) (1 ( ) ) 2 ( ) , (2 ( ) ) (( ) ) 2 ( ) (2 ( ) ) (( k k k k r k k k k k k k k k k k k k k sfewa                = = = = = = = = = = = + − − + + − = − + − +          1 3 3 2 14 1 (0.791,0.209,0.209) ) )k k  =                     =                      table 6 sfn based decision matrix criteria c1 c2 c3 alternatives (+) (+) (+) a1 0.85 0.145 0.145 0.900 0.100 0.100 0.579 0.424 0.424 a2 0.791 0.209 0.209 0.751 0.250 0.250 0.579 0.424 0.424 a3 0.751 0.250 0.250 0.791 0.209 0.209 0.645 0.357 0.357 a4 0.535 0.357 0.479 0.579 0.424 0.424 0.252 0.209 0.766 a5 0.645 0.357 0.357 0.344 0.250 0.693 0.855 0.145 0.145 a6 0.252 0.209 0.766 0.579 0.424 0.424 0.500 0.500 0.500 criteria c4 c5 c6 alternatives (+) (+) (+) a1 0.791 0.209 0.209 0.645 0.357 0.357 0.645 0.357 0.357 a2 0.579 0.424 0.424 0.579 0.424 0.424 0.252 0.209 0.766 a3 0.380 0.357 0.629 0.535 0.357 0.479 0.791 0.209 0.209 a4 0.855 0.145 0.145 0.100 0.100 0.900 0.500 0.500 0.500 a5 0.500 0.500 0.500 0.700 0.300 0.300 0.380 0.357 0.629 a6 0.700 0.300 0.300 0.252 0.209 0.766 0.100 0.100 0.900 criteria c7 c8 c9 alternatives (+) (+) (-) a1 0.579 0.424 0.424 0.855 0.145 0.145 0.445 0.424 0.562 a2 0.380 0.357 0.629 0.579 0.424 0.424 0.445 0.424 0.562 a3 0.192 0.145 0.833 0.579 0.424 0.424 0.344 0.250 0.693 a4 0.344 0.250 0.693 0.252 0.209 0.766 0.192 0.145 0.833 a5 0.645 0.357 0.357 0.445 0.424 0.562 0.500 0.500 0.500 a6 0.500 0.500 0.500 0.500 0.500 0.500 0.100 0.100 0.900 the next step is to derive the score values of all elements of the decision matrix for constructing the score matrix (see table 7). 468 s. biswas, d. božanić, d. pamučar, d. marinković table 7 score matrix criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 alternatives (+) (+) (+) (+) (+) (+) (+) (+) (-) a1 0.855 0.900 0.577 0.791 0.643 0.643 0.577 0.855 0.486 a2 0.791 0.750 0.577 0.577 0.577 0.426 0.464 0.577 0.486 a3 0.750 0.791 0.643 0.464 0.566 0.791 0.405 0.577 0.467 a4 0.566 0.577 0.426 0.855 0.367 0.500 0.467 0.426 0.405 a5 0.643 0.467 0.855 0.500 0.700 0.464 0.643 0.486 0.500 a6 0.426 0.577 0.500 0.700 0.426 0.367 0.500 0.500 0.367 using the definition given by eq. (19), the score of the element of the decision matrix are obtained as: 14 14 14 14 14 1 1 ( ) (2 ) (2 0.791 0.209 0.209) 0.791 3 3  = = + − − = + − − =score     now the weights of the criteria are calculated using the conventional steps of the spc method. first, the symmetry points of the performance values with respect to each criterion are calculated. for example, the symmetry point for the 4th criterion is calculated as: 4 4 4 ( ) ( ) 0.855 0.464 ( , 1,2...6) 0.660 22  +  + = = = =i imin max sp where i similarly, the other values are calculated as: 1 2 3 5 6 7 8 9 0.640; 0.683; 0.640; 0.533; 0.579; 0.524; 0.640; 0.433 = = = = = = = = sp sp sp sp sp sp sp sp the next step is to construct the absolute distance matrix as given in table 8. table 8 absolute distance matrix alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 a1 0.2148 0.2166 0.0631 0.1311 0.1100 0.0646 0.0532 0.2148 0.0528 a2 0.1505 0.0669 0.0631 0.0826 0.0438 0.1532 0.0595 0.0631 0.0528 a3 0.1100 0.1074 0.0031 0.1953 0.0326 0.2121 0.1194 0.0631 0.0334 a4 0.0743 0.1062 0.2148 0.1953 0.1667 0.0787 0.0572 0.2148 0.0288 a5 0.0031 0.2166 0.2148 0.1597 0.1667 0.1143 0.1194 0.1541 0.0667 a6 0.2148 0.1062 0.1403 0.0403 0.1078 0.2121 0.0239 0.1403 0.0667 11 11 1 14 14 4 63 63 3 0.855 0.640 0.215 0.791 0.660 0.131 0.500 0.640 0.140 ... =  − = − = =  − = − = =  − = − = sp sp sp    in the subsequent step, the matrix of the moduli of the symmetry is constructed (see table 9). for example: a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 469 6 6 6 6 1 2 1 9 1 1 1 1 11 12 61 69 11 12 61 69 6 6 6 60.1496, 0.1519, 0.3005, 0.1369 = = = = = = = = = = = =        i i i i i i i i r r r r     table 9 matrix of moduli of symmetry alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 a1 0.1496 0.1519 0.2019 0.1695 0.1626 0.2163 0.1250 0.1657 0.1033 a2 0.1617 0.1822 0.2019 0.2323 0.1812 0.3270 0.1553 0.2455 0.1033 a3 0.1705 0.1728 0.1811 0.2886 0.1848 0.1760 0.1783 0.2455 0.1075 a4 0.2260 0.2368 0.2738 0.1568 0.2853 0.2783 0.1545 0.3330 0.1241 a5 0.1988 0.2928 0.1363 0.2681 0.1494 0.2997 0.1121 0.2915 0.1004 a6 0.3005 0.2368 0.2331 0.1915 0.2458 0.3795 0.1442 0.2834 0.1369 now we proceed to derive the modulus of the symmetry criterion (qj) using eq. (23) and subsequently, the criteria weights (wj) are calculated using eq. (24). the results are summarized in table 10. table 10 moduli of the symmetry criteria and weights c1 c2 c3 c4 c5 c6 c7 c8 c9 sum q 0.2012 0.2122 0.2047 0.2178 0.2015 0.2795 0.1449 0.2608 0.1126 1.835 wj 0.1096 0.1156 0.1115 0.1187 0.1098 0.1523 0.0790 0.1421 0.0613 1 1 1 1 1 0.1496 0.1617 0.1705 0.2260 0.1988 0.3005 0.2012 66 0.2012 0.2012 0.1096 0.2012 0.2112 ..... 0.1126 1.835 m i i r q w = + + + + + = = = = = = + + +  it is seen that enhanced use of analytical tools (c6), retention and management of knowledge (c8) and automation (c4) under the guidance and support of the top management (c2) are given top priorities by the expert team. the alternatives are now ranked in the subsequent steps based on srp method. using the score matrix (see table 9) the alternatives are ranked with respect to each criterion. accordingly, the ranking matrix is formulated (see table 11). table 11 ranking matrix criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 alternatives (+) (+) (+) (+) (+) (+) (+) (+) (-) a1 1 1 3 2 2 2 2 1 4 a2 2 3 4 4 3 5 5 2 5 a3 3 2 2 6 4 1 6 3 3 a4 5 4 6 1 6 3 4 6 2 a5 4 6 1 5 1 4 1 5 6 a6 6 5 5 3 5 6 3 4 1 470 s. biswas, d. božanić, d. pamučar, d. marinković now the weighted ranking matrix is obtained by using the criteria weights (given in table 10). table 12 provides the weighted ranking matrix. table 12 weighted ranking matrix criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 alternatives (+) (+) (+) (+) (+) (+) (+) (+) (-) a1 0.1096 0.1156 0.3346 0.2374 0.2196 0.3046 0.1579 0.1421 0.2454 a2 0.2193 0.3469 0.4462 0.4747 0.3295 0.7615 0.3948 0.2842 0.3067 a3 0.3289 0.2313 0.2231 0.7121 0.4393 0.1523 0.4737 0.4263 0.1840 a4 0.5481 0.4626 0.6692 0.1187 0.6589 0.4569 0.3158 0.8526 0.1227 a5 0.4385 0.6938 0.1115 0.5934 0.1098 0.6092 0.0790 0.7105 0.3681 a6 0.6578 0.5782 0.5577 0.3560 0.5491 0.9138 0.2369 0.5684 0.0613 finally, the alternatives are ranked using the steps (see steps 14 and 15) defined by eqs. (26) and (27). table 13 provides the ranking of the alternatives. table 13 ranking of the alternatives alternatives rsi pi rank a1 1.8668 4.1332 1 a2 3.5637 2.4363 3 a3 3.1710 2.8290 2 a4 4.2055 1.7945 5 a5 3.7138 2.2862 4 a6 4.4792 1.5208 6 it is noted that the medium sized firm a1 has an edge over the others in terms of their readiness to embrace q 4.0. however, the result also suggests that it is not necessary that micro firms are worst as we see a3 holds the 2nd position. the order of performance is as follows: a1 > a3> a2 > a5> a4 > a6. 5.1. comparison with other mcdm models it is very important to examine the validity and stability of the results obtained by using a composite mcdm framework like what we have done in our paper [81-82]. mcdm results in many instances get affected significantly by external changes in given conditions relating to criteria and alternative sets and their values, any algorithmic changes and so on [83-87]. the existing literature shows evidences of comparison approach for validation of mcdm results [88-92]. in this regard, we compare the result obtained by using the framework used in the present work with other models such as eamr [93], copras [94] and cradis [95] based on the score matrix. table 14 provides the result of spearman’s rank correlation test among the ranking results of various mcdm models. table 14 result of spearman’s ranking test (comparison of mcdm models) model copras eamr cradis srp 0.829* 0.771* 0.943* *correlation is sig at 0.05 level (2-tails) a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 471 therefore, it is contended that the framework used in this work provides a comparable and consistent result with respect to other models. 5.2. sensitivity analysis it is quite imperative to investigate about the stability aspect of the result. accordingly, we perform the sensitivity analysis. there are several ways for carrying out sensitivity analysis such as exchange of criteria weights, vary the weights proportionately among others etc. [96-100]. in this work firstly the weights of the criterion c6 (having highest value) has been decreased by 10% at each step while increasing the weights of the other criteria have been increased proportionately. then the weight of the criterion c9 (having the lowest value) have been increased by 10% while decreasing the weights of the other criteria proportionately. in this way, beside the initial case 15 experimental cases (exp. 1 to exp. 15) have been instituted and the alternatives have been ranked. table 15 shows the criteria weights under various experimental cases and fig. 2 depicts the result of the sensitivity analysis. it is seen from the fig. 2 that except for some cases (exp. 9 to exp. 12), a6 and a4 could hold their initial positions. the alternatives a2 and a3 have interchanged their position only for exp. 11. the best alternative (a1) remains at the top under all conditions and a5 also remains at its initial position. hence, the result of the sensitivity analysis indicates that our framework provides a reasonable stable outcome. table 15 experimental cases and criteria weights (sensitivity analysis) criteria weights cases c1 c2 c3 c4 c5 c6 c7 c8 c9 sum initial 0.1096 0.1156 0.1115 0.1187 0.1098 0.1523 0.0790 0.1421 0.0613 1.00 exp. 1 0.1115 0.1175 0.1134 0.1206 0.1117 0.1371 0.0809 0.1440 0.0632 1.00 exp. 2 0.1132 0.1193 0.1152 0.1223 0.1134 0.1234 0.0826 0.1457 0.0650 1.00 exp. 3 0.1148 0.1208 0.1167 0.1238 0.1150 0.1110 0.0841 0.1473 0.0665 1.00 exp. 4 0.1162 0.1222 0.1181 0.1252 0.1164 0.0999 0.0855 0.1487 0.0679 1.00 exp. 5 0.1174 0.1234 0.1193 0.1265 0.1176 0.0899 0.0868 0.1499 0.0691 1.00 exp. 6 0.1185 0.1246 0.1205 0.1276 0.1187 0.0809 0.0879 0.1510 0.0703 1.00 exp. 7 0.1196 0.1256 0.1215 0.1286 0.1197 0.0728 0.0889 0.1520 0.0713 1.00 exp. 8 0.1205 0.1265 0.1224 0.1295 0.1207 0.0656 0.0898 0.1529 0.0722 1.00 exp. 9 0.1213 0.1273 0.1232 0.1303 0.1215 0.0590 0.0906 0.1538 0.0730 1.00 exp. 10 0.1220 0.1280 0.1239 0.1311 0.1222 0.0531 0.0914 0.1545 0.0737 1.00 exp. 11 0.1089 0.1149 0.1108 0.1179 0.1091 0.1515 0.0782 0.1413 0.0675 1.00 exp. 12 0.1080 0.1140 0.1099 0.1171 0.1082 0.1507 0.0773 0.1405 0.0742 1.00 exp. 13 0.1071 0.1131 0.1090 0.1161 0.1073 0.1498 0.0764 0.1396 0.0816 1.00 exp. 14 0.1061 0.1121 0.1080 0.1151 0.1063 0.1487 0.0754 0.1385 0.0898 1.00 exp. 15 0.1049 0.1110 0.1069 0.1140 0.1051 0.1476 0.0743 0.1374 0.0988 1.00 472 s. biswas, d. božanić, d. pamučar, d. marinković fig. 2 result of sensitivity analysis 5.3. rank reversal test (rrt) rank reversal (rr) is a phenomenon wherein the initial ranking order obtained by using a mcdm model gets influenced significantly as a consequence of changes in the given setup such as changes in the alternative and criteria set [101]. in this paper rrt is performed. given the initial order a1 > a3 > a2 > a5 > a4 > a6 at each stage one alternative was removed and ranking of the remaining alternatives was done. table 16 exhibits the result of the rrt. the result indicates that our sfn-spc-srp framework with ea does not suffer from rr phenomenon. table 16 result of rrt ranking positions cases 1 2 3 4 5 6 initial a1 a3 a2 a5 a4 a6 removal of a6 a1 a3 a2 a5 a4 removal of a4, a6 a1 a3 a2 a5 removal of a4, a5, a6 a1 a3 a2 removal of a2, a4, a5, a6 a1 a3 removal of all except a1 a1 therefore, overall we contend that the result obtained in the present work is considerably reliable, stable and free from rr phenomenon. 6. discussion the present study reveals some interesting observations. it is seen that enhanced use of analytical tools (c6), retention and management of knowledge (c8) and automation (c4) under the guidance and support of the top management (c2) are given top priorities by the a spherical fuzzy based decision making framework with einstein aggregation for comparing ... 473 expert team. it is observed that c6 and c8 are quite nearby to each other based on their calculated weight while c4 and c2 are very close to each other. the preferential order supports the general observations made in the past studies [102-103]. top management commitment and support is necessary to instill quality culture and implement the practices in the organizations with proper strategic intent. the present study finds that understanding and implementation of analytical tools and techniques with extensive applications of automation in redefining the processes and creating a knowledge base are more important. i 4.0 demands a lot of use of automation and data analytics for ensuring the effectiveness and efficiency of the processes towards achieving excellence in quality [104-105]. however, most of the smes are driven by the owners’ decisions. hence, top management support is very important. the closeness of the weights of c6, c8, c4 and c2 suggests that these four criteria are not distantly apart from each other. now, looking at the final ranking of the alternatives, it is observed that a small-scale unit (a1) is better prepared as compared with the medium enterprises. this result is also fascinating. there is a general perception that because of a higher turnover and stronger financial position, a medium scale business is better prepared to invest and adopt new technologies. however, our work reminds that if there is an intent and effective teamwork under the able guidance of the top management, a small-scale unit can adopt technology driven quality practices. technically, it is seen that the framework used in this work is free from the selection of normalization schemes. that way it provides a lot of relaxation to the analysts. further, the outcome is found stable and comparative with no effect of rank reversal. the use of ea as an ao extends the benefit of forming a collective group decision making. however, the spc model consists of slightly more number of computational steps. srp also dependent on the initial score matrix. therefore, changes in ao may have some impact on the outcome of srp model. 7. conclusions the present age is characterized by cyber-physical space as we call i 4.0, which is progressing toward technology-society space soc. 5.0. at this juncture, it is important to focus on holistic development in industrial space. smes act as a backbone for any industrial revolution and is instrumental in inclusive growth of society and nation. keeping pace with technological revolution, traditional quality management has evolved to a stage known as q 4.0. therefore, it is essential that smes embrace q 4.0. in this paper, we shed light on this aspect. an attempt has been made to present a primary level analysis of the readiness of the smes vis-à-vis implementation of q 4.0. for this purpose, we have presented a new sfn based integrated mcdm framework of spc-srp method with ea as the ao. with a real-life field study on six smes in india, we have demonstrated the comparative analysis. the criteria are selected in tune with the conceptual background and theoretical foundation of i 4.0 and soc. 5.0 as found in the literature. we have used nine criteria such as leadership support, organizational culture, organizational structure, data management and use of analytical tools, information sharing, knowledge management, level of automation, collaboration and business risk to compare four small and two medium sized enterprises. we notice that the experts have given more emphasis on use of analytical tools (c6), retention and management of knowledge (c8) and automation (c4) under the guidance and support of the top management (c2) while evaluating the smes. 474 s. biswas, d. božanić, d. pamučar, d. marinković further analysis has revealed that a small size enterprise (a1) outperformed the medium scale organizations. our study provides a guideline to the analyst and strategic decisionmakers. to our understanding, the present paper is a unique contribution. there are some further possibilities for future studies. for example, the framework can be applied to a large number of smes located at various locations and belong to different types of industries. furthermore, an empirical scale development research can be done to investigate the impact of the criteria (considered in this paper) on the firm performance. an interesting future study may attempt to apply the same model on large-scale corporates. there is an opportunity to apply our model to solve various other real-life problems. our model provides some advantages, such as flexibility and granularity in analysis, reasonably accurate and stable solution. however, with an increasing number of criteria having at par importance level may impose a limitation to the method used. therefore, one may attempt to conduct a simulation-based experiment to check the efficiency of our framework. also, a future study may be planned to see the effect of changing the ao. references 1. masood, t., sonntag, p., 2020, industry 4.0: adoption challenges and benefits for smes, 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(eds), advances in industrial and production engineering. flame 2022. lecture notes in mechanical engineering. springer, singapore, pp. 103-113. https://doi.org/10.1007/s10479-022-04856-4 https://doi.org/10.1007/s10479-022-04856-4 plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 2, 2015, pp. 109 121 1software testing of the rail vehicle dynamic characteristics udc 629.4 sanjin troha 1 , miloš milovančević 2 , alireza kuchak 3 1 faculty for technical engineering, university in rijeka, croatia 2 faculty for mechanical engineering, university in niš, serbia 3 department of structural analysis, tu berlin, germany abstract. the modern construction concept and the determination of the machine system characteristics anticipate cad design. creating model that will be tested using fem and other methods for determining stress-strain is a very important part of rail vehicle construction. applicative software package consists of linear and non–linear methods for the prediction of railway vehicle behavior and various methods of analysis have been assembled into a single coherent package in order to allow real problems in railway vehicle dynamics to be solved. key words: software, virtual testing, rail vehicle 1. introduction the dynamic behavior of railway vehicles has been a research topic for more than a century. the dynamic fluctuations that occur as a result of the wheel rails contacts were described for the first time by george stephenson in 1821. since then a lot has been done concerning the control system of the railway vehicle vibration, and, in particular, the dynamic behavior of vehicle in a curve. thus, at the sixties when computers became more accessible and mathematical formulas more accurate, quality results in determining the railway vehicle dynamics were obtained. further advance in computer technology has led to the rapid development of numerical techniques for problem solving of vehicle dynamics [1, 2]. as part of the british association for research railways (now the aea technology rail) a large number of programs were developed in last twenty five years in order to solve the movement dynamics. this development includes the creation of linear models used in the received june 2, 2015 / accepted july 18, 2015 corresponding author: sanjin troha faculty of engineering, university of rijeka, vukovarska 58, 51000 rijeka, croatia e-mail: sanjin.troha@riteh.hr original scientific paper 110 s. troha, m. milovanĉević, a. kuchak linear analysis and programs based on the linear theory of movement. the linear program for vehicle dynamics in curve track is time progressed and has basic understanding of nonlinear force on the wheel-rail contact. further development led to the non-linear program analysis of transitional states that are researching the characteristics of the vehicle using discrete inputs and features rail road on the basis of known vertical and lateral displacement. it has been concluded that the linear program has some limitations related to the theoretical assumptions about wheel-rail contact, but, if recognized, these linear programs are extremely useful [3, 4]. 2. development of linear analysis of rail vehicle 2.1. basic software characteristics the equations of the rolling process are generally nonlinear. the first studies that examined the dynamics of vehicles used the linearization techniques to obtain a set of linear equations of motion that were then solved leading to understanding the vehicle dynamic behavior. these early works of the british institute for the development of railways resulted in the cei award 1975. they still represent a large portion of the linear analysis techniques used in the software. this involved the use of linearized models of contact-point rail (which is generated by the static procedure booton) and linearization forces that occur during a slow rolling process (creep). also back then there were some simplifications related to the symmetry of the vehicle. it was noted that the coordinates that describe the movement of vehicles can be divided into groups of vertical and lateral. vertical coordinates relate to the movement of vehicles together with those that cause displacement in the vertical and longitudinal plane. the group gives the lateral movement of the coordinate administration compared to the previously stated levels. to simplify small movements of vertical and lateral groups the coordinate-equation can be divided; a large number of the first models were actually only vertical or lateral models. today, exclusively three-dimensional models are used and the only disadvantage of the linear software is that it is limited to small displacements. linear methods allow for an analysis of the dynamic behavior of the vehicle on the right tracks provided that linearization of the suspension elements is done. in essence, the basic problem related to the linear model is that the mathematical model is fully adequate but it is very difficult to assign adequate parameters of the model. giving the correct parameters is the main factor affecting the accuracy of predicting the dynamic behavior of the vehicle. the linear techniques limitations exist only in the data selected for analysis. it also automatically performs suspension elements linearization. linear solving of the wheelrail contact has its limitations but if recognized the program is still very useful because it allows for the variation of the taper point parameters and understanding of the dynamics of movement [5, 6, 7]. 3. development of nonlinear analysis of rail vehicle experiences with the linear theory of the dynamics of movement have led to the studies on the behavior of the vehicle in a curve and the vehicle's behavior when there are discrete inputs; it has led to the use of nonlinear equations of motion. the quasi-static theory of software testing of rail vehicle dynamic characteristics 111 movement has been developed in 1978. this is essentially a nonlinear theory that deals with the problems of the geometry of the wheel-rail contact and the forces that occur in slow moving vehicle (creep, creep forces), the kalker's method. two different methods are used for calculating and solving equations and both give similar results. the first method uses a partial linearization of the contact wheel-rail information so that it can carry out their interpolation; quasi-static force equilibrium solution is obtained by rapid convergence. the second method uses step-by-step integration which includes creating time range with the vehicle in a curve; such convergence is also a quasi-static solution. the integration technique is slower but it involves attenuation of the integration that arises as a result of lateral and longitudinal accelerations. the advantage of this second method is that it can be extended to predict the behavior of the vehicle passing through the switch, that is, generally speaking, for any kind of prediction of nonlinear behavior. the developed theoretical method that allows for an analysis of the friction braking torque, has, in addition to the creep forces arising from kelkerovog work, variations of the ellipse contact surface size in order to predict the behavior of the axle assembly and derailing the rails. nonlinear programming method is tested by the movement simulation of a two-axle rail vehicle-trolley at a certain line. the simulation results are lined with experimental results on a given line. the simulation results comparing to the results of the previous measurements are located in the range of up to 20% deviation. it is experimentally confirmed that the movement simulation can be performed in order to predict the vehicle dynamic behavior on a straight track [8, 9, 10, 11]. 4. program analysis of transient states of rail vehicle program analysis of transient states is used to calculate the vehicle’s dynamic response to external effect of railroad geometric imperfections and external forces in general. the calculation covers full nonlinear model vehicles and oscillations that occur as a result of movement or wheel-rail contact. the program analysis of transient state in the software package is presented through a complex and highly demanding user interface. a dynamic vehicle’s response to outside influences is conditioned by a number of different parameters of the vehicle, wheel-rail or rails contact and geometry, geometry point. the transient states analysis represents the most complex form of analysis in addition to the contents of the calculation inputs and double wheel-rail contact. the process of analyzing the rail vehicle movement begins with defining vehicles (geometry and physical properties) in terms of defining and determining the file from which the software that reads the data on the vehicle is taken, as well as the appointment of the analysis. the file is used to define geometry and physical characteristics of vehicles (trolleys). since this is a dynamic test, the trolleys in a file bearing the code record on trolleys enter the following data:  trolley is a vehicle with two axles (2 axles)  the weight of the axle assembly 1313.63 kg  the total maximum weight of loaded trolley 15373 kg  stiffness spring primary suspension is mn/m 1.1  modulus 80 kn/mm 2  the distance between the axles 3.9 m furthermore, it is necessary to define the conditions of dynamic testing of vehicles, in terms of defining precisely the desired speed of the vehicle (in our case, 50 km/h and 60 km/h). it is 112 s. troha, m. milovanĉević, a. kuchak also possible to define the range of speeds at which we test. since the standard uic 518 defines the length of test track sections, it must also be defined in the software on which the dynamic response of the vehicle is determined. furthermore, it is necessary to define the time steps for the calculation and the recommended values are 1-5 milliseconds. quality of rail road greatly affects the quality of the movement of rail vehicles. software package that is used to simulate the movement of the trolley calculates the impact of rail road defined through the file that carries information about the track. since this is a rail road which is in the third category and whose tracks are observed, according to the type of deformation (stability, warping and camber). some data on the state of rail are captured using measuring equipment given in table 1. the data presented in table 1 shows data discrepancies rail geometry of constructive listed values (found rates). we can see that the camber on the part of the railroad from 4716 m – 4718 m railway line is -15 mm compared to constructively inventoried value, while in the act of 4979 m – 4982 m railway line is -17 mm. table 1 rail way track geometry parameters type of deformation stationing length measures from to (m) overshoot 4.716 4.718 2 -15 mm stability-d 4.808 4.811 3 -23 mm stability-d 4.815 4.817 2 25 mm stability-l 4.815 4.818 3 25 mm twisting 3.5m 4.979 4.981 2 -19 mm overshoot 4.979 4.982 3 -17 mm the complexity of nonlinear dynamic model demonstrates the need to define the parameters required for the calculation of the relative speed of the axle assembly. in this sense, it is necessary to define the profile of wheelrail contact, geometry axle assembly, coefficients of friction at the contact point between the wheel surface and rails as well as the coefficient of friction between the side rails and wreaths bandage given that there is a possibility of contact in two points. profile point trolley is defined by uic regulations as well as for all other rail vehicles so that the profile data point contained in the fail s1002-sw is determined on the basis of standard uic 510-2. profile rail is also regulated by the uic regulations so that the data file on the profile rail is fixed. defining the geometry of the axle assembly is based on adding the value of half the distance between the wheels and the gap between the wheel and rail. the wheel-rail contact is fully determined; it is necessary to define the coefficients of friction at the contact point between the tread and the rail for the wheels, as well as the coefficients of friction between the wheel and the side rails. in the case of tested trolley, all these coefficients are 0.3. since the contact nonlinearity is based on the complex conditions determined by geometry of vehicle and track, it is necessary to define the parameters of the tracks. in the context of defining the track characteristics it is possible to enter values for the lateral and vertical damping and rigidity. after complete definition of all the previously mentioned parameters it is necessary to define the parameters that should be output from the software testing of rail vehicle dynamic characteristics 113 analysis. considering that there are experimental data on the vertical acceleration at the axle bearing housing assembly in the center of mass, the defined output channels carry supple on the vertical accelerations on the bearing housing and the center of gravity of mass trolleys. 5. results of dynamic analyses 5.1. testing of the vertical acceleration vertical acceleration significantly affects the stability and tranquility of rail vehicle. the standard uic 518 fully defined the conditions of measurement, analysis and limit values of vertical acceleration. in addition to the general conditions, there are those related to the testing of vehicles which limit the speed or when the measurement is done at speeds below 120 km/h as well as axle loads must not exceed the limit value of 2q0 < 225 kn, provided that the test speed: v=1.1vdoz with a minimum vdoz+10[km/h] of tolerance: ± 5km/h. the conditions related to the state and the selection of tracks for testing are also listed as follows: the test is performed on the track that is used for regular excavation, on the condition that rail must be dry. the measurement is carried out at speeds of less than 120 km/h. the limit values of vertical acceleration on the body -for vehicles towing vehicles (with single suspension (zs)lim = 4 m/s 2 . the virtual measuring of vertical acceleration in a box above the front axle assembly trolley and above the point on the bearing housing shaft has been done. the measured values are analyzed in the time and frequency domain. maximum value of lateral acceleration is (ys)lim = 4 m/s 2 . software package has the ability to identify and graphically display lateral acceleration, as well as the relationship between lateral and vertical forces y/q and other dynamic parameters that determine the safety and tranquility of vehicles. the values of vertical acceleration under different conditions are given in the time and frequency domain. the vertical acceleration, at the bearing of axle, is presented in fig. 1. fig. 1 vertical acceleration in time domain 114 s. troha, m. milovanĉević, a. kuchak the values of the maximum amplitude of acceleration are in the range of 0-60 s. the time range of 0-60 s corresponds with the scope of the distance from 0-1000 m presented in fig. 2. in the period of 80-230 s the movement stabilization occurs and the maximum amplitude of acceleration is in the range of from -10 to 10 m/s 2 . then, in the period of 230270 s amplitude appears in the range of from -20 to 20 m/s 2 , and then the vehicle is stable. fig. 2 vertical acceleration with respect to distance traveled 0-1000m fig. 3 vertical acceleration in frequency domain displaying data in frequency domain s(f) depending on frequency f is presented in fig.3. frequency functions of the signal have the unit of acceleration ((mm/s 2 ) 2 )/hz. if amplitude a mm/s 2 of a signal is g(f) =s(f)∆f the acceleration can be calculated software testing of rail vehicle dynamic characteristics 115 2 ( )a g f as stated in report erri b 153/rp 21 [12]. length of fft analysis in post processing is 1024 points, which gives 512 sampling points in the frequency domain ∆f=1/(1024∆t), time step is set by software while the maximum frequency can be fmax=1/(2∆t). the result of measuring the vertical acceleration over the trolleys bearing housing axle assembly at a speed indicating extreme values of vertical acceleration -29.21 and 18.64 m/s 2 , fig. 4. these values greatly exceed the limit, according to uic 518, of 4 m/s 2 or are not relevant to the comparison given that it is not a suspended part. fig. 4 vertical accelerations at speed of 60 km/h fig. 5 vertical acceleration in frequency domain at speed of 60 km/h 116 s. troha, m. milovanĉević, a. kuchak the result of measuring the vertical acceleration of trolleys at the measuring point shaft bearing housing assembly at a speed indicating extreme values of vertical acceleration -47.25 m/s 2 and 27.28 m/s 2 . value of vertical acceleration at the axle bearing housing assembly reaching a value of 47.25 m/s 2 indicates that there are very large dynamic changes "dynamic impact" on the bearing. such data are expected considering that it is a simulation of movement of not suspended wheel-rail contact. such high values, peaks of maximum amplitude of vertical acceleration are a consequence of information entered on the state of the rails. it is a track that, according to uic regulations falls into category 3. the vertical acceleration presented in the frequency domain (fig. 5) is the pillar of the development of software diagnostics of railway vehicles. substantial research in the field of application of fft analysis in the diagnosis of the working state condition of rail vehicles are made, and are intended to determine the significant parameters that define the image of the fft of response. based on previously presented fft diagram it can be concluded that the maximum amplitude of the acceleration is in the range of 15-25 hz. also, with the increase of speed, the amplitude range is shifted towards higher frequencies. 5.2. the results of measurements of vertical acceleration on the floor of the trolley acceleration values under different conditions are given in the time and frequency domain. the vertical acceleration at the measuring point on the trolleys floor is presented in fig.6. thus, in the range from 0-100 s, there are peaks of acceleration that exceed 1 m/s 2 . then, in the range of 100-250 s stable movement of the vehicle occurs, and dynamic instability again. in fig. 7 range 0-80 s is not present in the time domain, but in the domain of the transit distance 0-1000 m. fig. 6 vertical acceleration in time domain (50 km/h) software testing of rail vehicle dynamic characteristics 117 fig. 7 vertical acceleration with respect to distance traveled 0-1000 m fig. 8 vertical acceleration in the frequency domain (50 km/h) the result of measuring the vertical acceleration of trolleys at the floor measuring point at the speed of 50km/h indicates the extreme values of vertical acceleration of 2.54m/s 2 and 2.946 m/s 2 , these values are less than the maximum allowed vertical acceleration defined by the uic 518, fig. 8. the result of measuring the vertical acceleration of trolleys at the floor measuring at the speed of 60km/h indicates the maximum value of the vertical acceleration -2814 m/s 2 and 3122 m/s 2 , fig. 9. the maximum values of vertical acceleration are less than the maximum permissible value according to uic 518. this indicates the following: provided that a software machine that simulates the dynamics is adequate, trolleys can safely run 118 s. troha, m. milovanĉević, a. kuchak on the railway track class 3 at speed of 60 km/h, without the risk from the aspect of allowed vertical acceleration. the fact that there is a difference in the diagrams of vertical acceleration at speeds of 50 and 60 km/h indicates that there is no linear correlation of speed and dynamic response of the vehicle. the diagrams of the frequency spectrum of the vertical acceleration indicate that the maximal acceleration amplitude of suspended part of the trolleys, are in the range of from 0-5hz. fig. 9 vertical acceleration in time domain (60 km/h) fig. 10 vertical acceleration in frequency domain (60 km/h) software testing of rail vehicle dynamic characteristics 119 6. conclusions results of the simulation of trolleys speeds 50 to 60 km/h clearly indicate the following: the vertical acceleration at the axle bearing housing assembly is significantly greater than that on the trolleys floor. this is due to the fact that the bearing housing has no damper for axle assembly oscillations, while on the floor the oscillations are damped; damping is in proportion to the suspension (springs and friction dampers). the vertical accelerations on the trolleys floor and on the bearing housings are greater with increasing speed; this is a consequence of the fact that increase in speed increases the kinetic and potential energy of the vehicle and thus amplitude of oscillation. in comparative display in the frequency domain differences in frequencies are observed that are accompanied by maximum amplitude of vertical acceleration. fig. 11 presents a comparative view of vertical acceleration on the bearing housing in the frequency domain at 50 and 60 km/h. from the diagram, it is evident that there is certain amplitude schedule dependence in the frequency range of 0-25hz. it is also easily noticeable that the maximum amplitude of vertical acceleration at a speed of 60km/h occurs in the range of oscillation frequency of 20-25hz, and the maximum amplitude of vertical acceleration at a speed of 50km/h occur in the range of oscillation frequency of 15-20hz. fig. 12 shows a comparative view of vertical acceleration on the trolleys floor in the frequency domain of 50 and 60 km/h. from the diagram, it is evident that there is certain amplitude dependence schedule in the frequency range of 0-15hz. it is also easy noticeable that the amplitude of vertical acceleration at speeds of 50 km/h and 60 km/h occurring in the range of the oscillation frequency of 0-5hz. fig. 11 frequency spectrum of vertical acceleration of bogie frame elements of suspension (springs and dampers) have a role to absorb fluctuations that occur as a result of geometric wheel-rail imperfections. the importance of the suspension 120 s. troha, m. milovanĉević, a. kuchak elements is presented on comparative diagram in fig. 13 with the vertical amplitude of vibration in the frequency domain 0-25hz at the trolleys floor measuring points and bearing housing at a speed of 60km/h. fig. 12 frequency spectrum of vertical acceleration at center of mass fig. 13 compiled frequency spectrum of bogie frame and center of the mass of vertical acceleration software testing of rail vehicle dynamic characteristics 121 references 1. uškalov v. f., reznikov l. m., 1985, statistička dinamika šinskih kola, akademija nauka, kijev. 2. pfeiffer f., 2008, mechanical system dynamics,springer-verlag, berlin heidelberg. 3. ahmed a., 2001, computational dynamics, john wiley & sons, inc., 605 third avenue, new york, ny. 4. coutinho m., 2001, dynamic simulation of multibody systems, birkäuser. 5. schielen w. (ed.), 1990, multibody systems handbook, springer-verlag, berlin. 6. roberson r.e., schwertassek r., 1988, dynamics of multibody systems, springer-verlag, berlin. 7. park j.h., koh h.i.k., kim n.p., 2011, parametric study of lateral stability for railway vehicle, journal of mechanical science and technology, 25(7), pp. 1657-1666. 8. polach o., 2006, on non-linear methods of bogie stability assessment using computer simulations, proceedings of the institution of mechanical engineers, part f, journal of rail and rapid transit, 220, pp. 13-27. 9. true h., 1999, on the theory of nonlinear dynamics and its application in vehicle systems dynamics , vehicle system dynamics, 31(5-6), pp. 393-421. 10. polach o., 2007, einfluss der nichtlinearitäten des systems fahrzeug-gleis auf die stabilität, vdi-berichte nr.2022: nichtlineare schwingungen – reibung und kontaktmechanik, tagung kassel, pp. 35-46. 11. simić g., bernhard r., 2013, improving railway boarding accessibility, facta universitatis, series: mechanical engineering, 11(1), pp. 103-112. 12. erri b 153/rp 21, 1993, application of iso standard 2631 to railway vehicles, comfort index nmv – comparison with the iso/sncf comfort note and with the wz. utrecht. facta universitatis series: mechanical engineering vol. 19, no 4, 2021, pp. 767 779 https://doi.org/10.22190/fume191124002k © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper* bergström-boyce vs. hyperelastic rubber models in structural analysis of tires nikola korunović1, milan banić1, milan trifunović1, ana pavlović2 1university of niš, faculty of mechanical engineering in niš, serbia 2university of bologna, department of industrial engineering, bologna, italy abstract. new viscoelastic/viscoplastic material models, such as the bergström-boyce (bb) model, are known to bring advantages in finite element analysis (fea) of rubberbased components. to test if the same is true in fea of tires, a study was performed in which the hyperelastic yeoh model, as well as the bb and dynamic bergström-boyce (dbb) models were used to characterize the tread of an existing 205/65 r16 tire. curve fitting results for all material models as well as the results of the footprint and steady state rolling tire analyses are presented in the paper. there are notable differences between the obtained results at higher strain rates, when viscoelastic material behavior is dominant. key words: tire, viscoleastic/viscoplastic, material models, structural analysis, curve fitting, finite element analysis 1. introduction the accuracy of the results of the tire analysis based on fem, as well as the results of the analysis of any product made partly or wholly of rubber, largely depends on the material model used to describe the mechanical behavior of its structural components [1, 2]. those components can be either entirely made of rubber or rubber-based composites. if a rubberbased composite is modeled using rebar elements, its rubber matrix is modeled using the same material models that are used for modeling of rubber components [3]. in numerous research studies, the mechanical behavior of the tire was modeled as time independent, nonlinear elastic [4]. thereby, hyperelastic material models [5] were used, which neglect the viscoelastic characteristics of the rubber that are reflected in phenomena such as hysteresis, relaxation or creep. however, these characteristics may be very pronounced in various types of rubber with fillers that are normally used in tire design. thus, it may be expected that the use of more advanced material models, which account for time-dependent received november 24, 2019 / accepted january 20, 2020 corresponding author: nikola korunović university of niš, faculty of mechanical engineering in niš e-mail: nikola.korunovic@masfak.ni.ac.rs 768 n. korunović, m. banić, m. trifunović, a. pavlović rubber behavior, may lead to more accurate modeling of mechanical behavior of rubber, especially when phenomena like rolling resistance or thermal dissipation are modeled. we performed an extensive curve-fitting study, involving a number of hyperelastic and viscoelastic material models applied to different sets of experimental data on filled rubbers. due to a large volume of data, its complete results will be published separately, and only the most relevant subset is presented here. material models considered in curve fitting were: ▪ yeoh (y) [6, 7] ▪ bergstöm-boyce (bb) [8, 9] ▪ dynamic bergstöm-boyce (dbb) [10] the first two models are amongst the most popular ones used in tire design, while bb and dbb represent new viscoelastic/viscoplastic material models that are expected to have a higher accuracy of prediction than hyperelastic ones. prediction accuracy of the bergstöm-boyce model was assessed in several studies [11-14]. ghoreishy et al. [15] studied the effect of the use of different carbon black and their blends on the time-dependent mechanical behavior of a typical rubber compounds used in the tread of passenger car tire as well as the accuracy of prediction of the bergström-boyce material model, where the equilibrium response was modeled by the yeoh model. they have performed tests according to the astm: d412-c and iso7743 and compared the test results with simulation results. only at higher strains there was a slight deviation of predicted behavior from the experimentally determined one. they explained this deviation as a consequence of debonding phenomenon between polymer-filler and filler–filler that occur at higher strains which change the structure of the material. since the bergström–boyce model does not take the interaction between polymer networks and fillers into consideration, the accuracy of the model decreased. in another study by ghoreishy et al. [16] it was shown that bergstömboyce was quite capable of predicting even the mechanical behavior of the tire tread compounds reinforced by silica and carbon black at both low and high strain values. in this paper, the accent is put on a study in which the tire tread was modeled using the three mentioned rubber models while tire behavior during vertical loading, acceleration and braking was simulated by means of fea [17, 18]. 2. curve fitting experimental data was obtained by uniaxial compression testing of the rubber mixture tg-615, manufactured by “tigar technical rubber” serbia, which has mechanical characteristic similar to the ones of tire tread. it is characterized by shore hardness value of 60 and its base is a pure natural rubber. the mixture is reinforced by moderate amount of carbon black particles. testing was performed through loading-unloading cycle of uniaxial compression at strain rates -0.078, -0.28 and stress relaxation. curve fitting was performed using mcalibration software. the obtained material parameters were transferred to fea software abaqus for further analysis, via specialized software polyumod. mechanical behavior was predicted for a full load-unload cycle. the comparison of experimental data and behavior of rubber specimen predicted by yeoh model is shown in fig. 1. due to the mentioned limitations of hyperelastic models, unloading behavior could not be predicted. the average r2 value for all tests was 0.972. bergström-boyce vs. hyperelastic rubber models in structural analysis of tires 769 the yeoh model fit tends towards the unloading curve at low strains and towards the loading curve for higher strains. fig. 1 comparisons of experimental data and behavior predicted by yeoh model the comparison of experimental data and behavior of rubber specimen predicted by the bergstöm-boyce model is shown in fig. 2. average r2 value for all tests was 0.895. the prediction was unrealistic at strains greater than 0.4. the predicted strain rate dependence and hysteresis are less pronounced than in the experimental data set, while the stress relaxation prediction is in agreement with the experimental data set. despite the differences between the predicted and the experimental behavior, the obtained model parameters can be used for further simulations with confidence, especially in the lower strain region important in tire analysis (20-40%), as the average error is slightly above 10%. the comparison of experimental data and behavior of rubber specimen predicted by the dynamic bergstöm-boyce model is shown in fig. 3. average r2 value for all tests was 0.997, which indicates almost a perfect fit of predicted and experimental data sets. moreover, strain rate dependence, stress relaxation and value of dissipated energy (hysteresis) are also predicted with a very high accuracy and the dynamic bergstöm-boyce model achieves almost a perfect fit to test data. 770 n. korunović, m. banić, m. trifunović, a. pavlović fig. 2 experimental data vs prediction by the bergstöm-boyce model 3. trial simulations in abaqus using viscoelastic/viscoplastic rubber material models the curve-fitting study described in previous chapter had shown clear advantages of viscoelastic/viscoplastic material models in fea of rubber-based components. thus, it was expected that these would also bring the advantage in tire fea. a comprehensive study was performed in which several rubber models were used to perform the footprint and steady state rolling analyses of an existing 205/65 r16 tire. all the analyses in the study were conducted using simplified tread tire fem model [17, 19] with rather coarse mesh, as a very large number of analyses had to be run. all rubber materials except tire tread were defined as hyperelastic, yeoh. tread material was defined by curve fitting in mcalibration to tg-615 mixture experimental data, using four selected material models and analysis options shown in table 1. the first subset of analyses was based on viscoelastic fea procedures in which the surface was moved 30 mm towards the tire axis and back to starting position, with analysis times varying from 0.001 to 10000 seconds. when analysis time was set to 0.001 sec, the tire was moving relative to the surface at the velocity of 60 / 0.001 = 60000 mm/sec, i.e. 216 km/h. analysis time of 0.01 sec corresponded to tire velocity of 21.6km/h, 0.1 sec to 2.16 km/h, etc. those speeds were expected to cover the range of velocities that may be expected in tire exploitation. to capture the full range of viscoelastic rubber behavior, smaller velocities were also considered. finally, to approximate the strain rate in tire during bergström-boyce vs. hyperelastic rubber models in structural analysis of tires 771 steady state rolling analyses, the velocities of 50 and 80 km/h were also taken into account, with rather coarse mesh, as a very large number of analyses had to be run. all rubber materials except tire tread were defined as hyperelastic, yeoh. tread material was defined by curve fitting in mcalibration to tg-615 mixture experimental data, using four selected material models and analysis options shown in table 1. fig. 3 comparisons of experimental data and behavior predicted by dynamic bb model table 1 material models used in viscoelastic/viscoplatic tire material modeling study no. material model viscoelastic material behavior definition material model parameters abaqus analysis procedure 1. yeoh *viscoelastic, prony series (3 terms) mcalibration built-in 2. bergstrom boyce *hysteresis mcalibration built-in 3. bergstrom boyce user mcalibration user subroutine, polyumod library 4. dynamic bergstrom boyce user mcalibration user subroutine, polyumod library the next set of figures (fig. 4 7) brings the comparison of footprint results obtained using the yeoh material model with prony series (y+p), the bergstrom-boyce (bb) model (built-in or user) and the dynamic bergstrom-boyce (dbb) model for several values of tire axle velocity (relative and normal to the ground). on one hand, it may be 772 n. korunović, m. banić, m. trifunović, a. pavlović seen that the difference between predictions by various models gets smaller with decrease of speed. on the other, the bb model predicts the largest difference in sensitivity of loaddeflection curves to strain rate, as well as the largest hysteresis. the y+p model curve is constantly above the dbb curve except for very small velocities, while the bb curve is the highest of all for large velocities and the lowest for lowest velocities. the yeoh model constantly predicts very small hysteresis, while the bb model predicts very large hysteresis for velocities between 13889 and 600 mm/s. all models converge at very small velocities, with hysteresis becoming minor or negligible. 0 5 10 15 20 25 30 0 1000 2000 3000 4000 5000 6000 footpint analysis 13889 mm/s (50 km/h) yeoh + prony built-in bergstrom-boyce built-in or user dynamic bb user f z uz fig. 4 loading-unloading curve obtained by footprint analysis, at tire axle velocity of 13889 mm/s (50 km/h) relative to the surface, loading-unloading time 0.00432 s 0 5 10 15 20 25 30 0 1000 2000 3000 4000 5000 6000 footpint analysis 2777.78 mm/s (10 km/h) yeoh + prony built-in bergstrom-boyce built-in or user dynamic bb user f z uz fig. 5 loading-unloading curve obtained by footprint analysis, at tire axle velocity of 2777.78 mm/s (10 km/h) relative to the surface, loading-unloading time 0.0216 s bergström-boyce vs. hyperelastic rubber models in structural analysis of tires 773 0 5 10 15 20 25 30 0 1000 2000 3000 4000 5000 6000 footpint analysis 6 mm/s (0.0216 km/h) yeoh + prony built-in bergstrom-boyce built-in or user dynamic bb user f z uz fig. 6 loading-unloading curve obtained by footprint analysis, at tire axle velocity of 6 mm/s (0.0216 km/h), loading-unloading time 10 s -4 0 4 8 12 16 20 24 28 32 -600 0 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 footpint analysis 0.6 mm/s (0.00216 km/h) yeoh + prony built-in bergstrom-boyce built-in or user dynamic bb user f z uz fig. 7 loading-unloading curve obtained by footprint analysis, at tire axle velocity of 0.6 mm/s (0.00216 km/h), loading-unloading time 100 s a series of figures showing contact stress at the footprint at deflection of 30 mm, obtained using different material models and different strain rates, is presented next (fig. 8 fig. 10). the differences in predicted stress at high strain rates are very obvious. the bb model predicts the largest differences and the contact pressure it predicts at high strain rates is very uneven. on the other hand, the dynamic bb was quite insensitive to strain rate change. behavior of the y+p model was something in between those two. 774 n. korunović, m. banić, m. trifunović, a. pavlović yeoh + prony dynamic bb bb built-in bb user fig. 8 contact stress at the footprint, at deflection of 30mm, and tire axle velocity of 6000 mm/s (21.6 km/h), corresponding to loading-unloading time of 0.01 s yeoh + prony dynamic bb bb built-in bb user fig. 9 contact stress at the footprint, at deflection of 30mm, and tire axle velocity of 600 mm/s (2.16 km/h), corresponding to loading-unloading time of 0.1 s bergström-boyce vs. hyperelastic rubber models in structural analysis of tires 775 yeoh + prony dynamic bb bb built-in bb user fig. 10 contact stress at the footprint, at deflection of 30mm, and tire axle velocity of 6 mm/s (0.0216 km/h), corresponding to loading-unloading time of 10 s at this moment, there are no experimental results related to contact pressures of modeled tire at different speeds that can be compared with numerically obtained data. experimentally obtained contact pressure distribution at the footprint, for a similar, statically loaded, 175/70 r13 tire is shown in fig. 11. although there exists a significant difference in local contact pressures between experimental and numerical results, which is a consequence of the use of the simplified model, global predictions are generally similar, as largest values of contact pressure are located close to tire shoulders. in order to make a relevant comparison of the results, experimental data should be obtained for the 205/65 r16 tire moving towards the axle or rolling at different rolling speeds and compared to numerical results obtained using detailed tread tire models. fig. 11 contact pressure at the footprint of a 175/70 r13 tire obtained using xsensor tire sensors although it is stated in the abaqus documentation that hysteresis part of the built-in bergstrom-boyce material model is not active in steady state analysis, results obtained by 776 n. korunović, m. banić, m. trifunović, a. pavlović this procedure and various material models are shown in next section. inactivity of hysteresis in ssr means that only hyperelastic part of material model is active, which is a notable limitation concerning tire analysis. it should be examined if this is the case with user subroutine-based bb model and dbb model. fig. 12 and fig. 13 show the dependence of longitudinal force and moment on the axle obtained by steady state rolling analysis. the responses obtained using different models are notably diverse, even in cases when only the hyperelastic part is active. 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 ssr analysis, mi = 0.6, v = 10 km/h yeoh + prony bb built-in bb user dynamic bb l o n g it u d in a l fo rc e f x [ n ] angular velocity (rad/s) fig. 12 longitudinal force obtained by steady state rolling analysis at 10km/h using different material models 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 -1000000 -500000 0 500000 1000000 ssr analysis, mi = 0.6, v = 10 km/h yeoh + prony bb built-in bb user dynamic bb m o m e n t o n t ir e a xi s [ n ] angular velocity (rad/s) fig. 13 moment on tire axis obtained by steady state rolling analysis at 10km/h using different material models bergström-boyce vs. hyperelastic rubber models in structural analysis of tires 777 finally, fig. 14 and 15 bring comparison of footprint stress at various moments of braking-to-acceleration analysis, obtained using different material models. yeoh + prony dynamic bb bb built-in bb user fig. 14 contact stress at the footprint at full braking, 50 km/h, obtained using different models yeoh + prony dynamic bb bb built-in bb user fig. 15 contact stress at the footprint at 40% of braking-to-acceleration step, 50 km/h, obtained using different models 778 n. korunović, m. banić, m. trifunović, a. pavlović 4. conclusion structural analyses of a vertically loaded and rolling tire were performed, in which the selected viscoelastic/viscoplastic and hyperelastic rubber material models were used to model the tread. the results of the analyses were presented and compared. significant differences in the results obtained by various models were demonstrated, especially at higher strain rates when the contribution of viscoelastic part of material models was the greatest. at lower strain rates, all material models except the yeoh + prony yielded similar results. at higher strain rates, the results obtained using the yeoh + prony and the dbb models were similar, while the results obtained using the bb model were notably different. the results of the current study are considered as giving a general insight into behavior of viscoelastic/viscoplastic models in fea of tires. further studies, in which all rubber materials included in tire structure would be accurately described using viscoelastic/viscoplastic material models, should give a clearer picture of the possible advantages/disadvantages of viscoelastic/viscoplastic models. numerical results ought to be compared with experimental ones, obtained at various tire velocities, to establish the reliable conclusions concerning the use of new material models. references 1. charlton, d., yang, j., teh, k., 1994, a review of methods to characterize rubber elastic behavior for use in finite element analysis, rubber chemistry and technology, 67(3), pp. 481-503. 2. boyce, m.c., arruda, e.m., 2000, constitutive models of rubber elasticity: a review, rubber chemistry and technology, 73(3), pp. 504-523. 3. korunović, n., fragassa, c., marinković, d., vitković, n., trajanović, m., 2019, performance evaluation of cord material models applied to structural analysis of tires, composite structures, 224, pp. 111006. 4. ghoreishy, m.h.r., 2008, a state of the art review of the finite element modelling of rolling tyres, iranian polymer journal, 18(8), pp. 571-597. 5. marckmann, g., verron, e., 2006, comparison of hyperelastic models for rubber-like materials, rubber chemistry and technology, 79(5), pp. 835-858. 6. yeoh, o.h., 1990, characterization of elastic properties of carbon-black-filled rubber vulcanizates, rubber chemistry and technology, 63(5), pp. 792-805. 7. yeoh, o.h., fleming, p., 1997, a new attempt to reconcile the statistical and phenomenological theories of rubber elasticity, journal of 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14. banić, m.s., stamenković, d.s., miltenović, v.d., milošević, m.s., miltenović, a.v., djekić, p.s., rackov, m.j., 2012, prediction of heat generation in rubber or rubber-metal springs, thermal science, 16, pp. 527-539. 15. ghoreishy, m.h.r., firouzbakht, m., naderi, g., 2014, parameter determination and experimental verification of bergström–boyce hysteresis model for rubber compounds reinforced by carbon black blends, materials & design, 53(0), pp. 457-465. bergström-boyce vs. hyperelastic rubber models in structural analysis of tires 779 16. ghoreishy, m., alimardani, m., mehrabian, r., gangali, s., 2013, modeling the hyperviscoelastic behavior of a tire tread compound reinforced by silica and carbon black, journal of applied polymer science, 128(3), pp. 1725-1731. 17. korunović, n., trajanović, m., stojković, m., 2008, finite element model for steady-state rolling tire analysis, journal of serbian society for computational mechanics, 2(1), pp. 63-79. 18. kabe, k., koishi, m., 2000, tire cornering simulation using finite element analysis, journal of applied polymer science, 78(8), pp. 1566-1572. 19. korunović, n., trajanović, m., stojković, m., vitković, n., trifunović, m., milovanović, j., 2012, detailed vs. simplified tread tire model for steady-state rolling analysis, strojarstvo: časopis za teoriju i praksu u strojarstvu, 54(2), pp. 153-160. facta universitatis series: mechanical engineering vol. 19, no 2, 2021, pp. 271 284 https://doi.org/10.22190/fume201212003z © 2021 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper fractal approach to mechanical and electrical properties of graphene/sic composites yu-ting zuo1,2, hong-jun liu1,2 1school of materials science and engineering, lanzhou university of technology, china 2state key laboratory of advanced processing and recycling of non-ferrous metals, lanzhou university of technology, china abstract. graphene and carbon nanotubes have a steiner minimum tree structure, which endows them with extremely good mechanical and electronic properties. a modified hall-petch effect is proposed to reveal the enhanced mechanical strength of the sic/graphene composites, and a fractal approach to its mechanical analysis is given. fractal laws for the electrical conductivity of graphene, carbon nanotubes and graphene/sic composites are suggested using the two-scale fractal theory. the steiner structure is considered as a cascade of a fractal pattern. the theoretical results show that the two-scale fractal dimensions and the graphene concentration play an important role in enhancing the mechanical and electrical properties of graphene/sic composites. this paper sheds a bright light on a new era of the graphene-based materials. key words: steiner minimum tree structure, 3d printing, graphene-based composites, two-scale fractal dimension 1. introduction graphene is a two-dimensional nanomaterial with a steiner minimum tree structure [1,2] characterized by the most stable structure and network optimization, see fig. 1. the steiner structure is also seen in natural phenomena, for example, multiple bubbles’ interaction leads to such a structure and it is used for nanofiber fabrication by the bubble electrospinning [3,4,5]. fig. 2 is a photograph of dry cracked soil, and it can be seen that the cracks meet the requirements of the steiner minimum tree structure. the trusswork with the steiner minimum tree structure has the most stable and good mechanical property. likewise, graphene also has very stable chemical and mechanical properties. the network optimization characteristics of graphene give it excellent thermal and electrical conductivity properties; as a result high electronic and thermal conductivities are predicted. received december 12, 2020 / accepted january 05, 2020 corresponding author: yu-ting zuo lanzhou university of technology, lanzhou 730050, china e-mail: zuoyuting2020@126.com 272 y.t. zuo, h.j. liu graphene/sic composites are widely used in various fields, from ceramics to building materials, and the three-dimensional printing technology is now available for advanced fabrication of graphene/sic composites [6-9]. a) b) fig. 1 (a) steiner minimum tree structure and (b) steiner minimum tree structure in graphene fig. 2 cracked land with minimal tree structure graphene is the hardest material found so far, and has an excellent mechanical property as well; it is one of the materials with the highest known strength. at the same time, it has good toughness and can be bent with ease. the specific surface area of graphene is up to 2600m2/g. in addition, graphene has good electrical conductivity; at room temperature, its electron mobility can be as high as 20000cm2/(v·s), which is more than 10 times that of silicon and more than twice that of indium antimonide (insb). table 1 compares the mechanical, thermodynamic and electrical properties of graphene with other materials. due to its unique properties, graphene is widely used to enhance the mechanical and electronic properties of various functional composites. fractal approach to mechanical and electrical properties of graphene/sic composites 273 table 1 properties of graphene, cnt, nano-sized steel and polymers [10] materials tensile strength thermal conductivity at room temperature (w/mk) electronic conductivity(s/m) graphene 130±10gpa (4.84±0.44)×103 to (5.30±0.48)×103 7200 cnt 60~150gpa 3500 3000~4000 nano-sized steel 1769mpa 5~6 1.35×106 hdpe plastic 18~20mpa 0.46~0.52 insulator natural rubber 20~30 0.13~0.142 insulator kevlar fiber 3620mpa 0.04 insulator 2. mechanical property a smaller grain size results in a stronger material; this can be explained by the well-known hall-petch effect [11,12], which is widely used in material design, that is, adding nano or micro particles can improve the mechanical properties of materials. in 1951, hall found the relationship between crystal particle size and yield strength. in 1953, petch discovered that crystal size also had a similar relationship with brittle fracture. the hall -petch effect was described in a mathematical expression [11,12]: 0 k d   = + (1) here, σ is the elastic modulus or strength, σ0 is the parent property of the material, k is the material constant, d is the average diameter of the particle, β is the geometric constant, and it is related to the volume percentage of the additive. the hall-petch effect cannot describe the effects of the size distribution surface morphology of particles on the mechanical property; additionally, when the particles tend to be small enough, an inverse hall-petch effect occurs [11], so a modified hall-petch effect is much needed to take into account the particles’ surface morphology. for graphene-reinforced composites, see fig.3, considering the two-dimensional characteristics of graphene, d, can be obtained by using the equivalent diameter: 1/36 ( ) a d   = (2) where a is the average area of graphene and δ is the thickness of graphene. we obtain the following modified hall-petch effect formula 0 /3 k a    = + (3) where k’ is the material constant. 274 y.t. zuo, h.j. liu fig. 3 microstructure of graphene composites the mechanical properties of composites can be enhanced by adding graphene, which can be qualitatively described by the hall-petch effect. for isotropic elastic materials, the stress formula is a f = (4) here σ is the stress, f is the external force applied, and a is the area of the cross section of the object. fig. 4(a) is a force analysis diagram of a control body containing graphene. the control body is fractured under external forces, and the fracture surface is unsmooth as shown in fig. 4(b). a) b) fig. 4 force analysis of the composite control body containing graphene the control body in fig. 4(a) can be divided into two parts, sic matrix and graphene. before breaking, the matrix needs to overcome the friction resistance between the matrix and the graphene, which is very high due to an extremely large specific surface area of the graphene. in addition, the fracture surface contains a lot of graphene, which is not smooth. to sum up, the force balance equation can be written f a f= + (5) fractal approach to mechanical and electrical properties of graphene/sic composites 275 where f is the external force, f is the frictional resistance between the matrix and the graphene, and a is the area of the non-smooth section, which can be expressed as: 0 +a a a=  (6) where a0 is the smooth cross section area, a is proportional to the specific surface parameters of graphene. the real stress of the matrix of the composite material is 0 = + f f a a  −  (7) the frictional resistance between the matrix and the graphene is proportional to the surface area of graphene; if the percentage of the volume of graphene is φ, we have 2/3f s   (8) 2/3a s    (9) where φ is the volume concentration. eq. (7) becomes 2/32/3 0 2/3 2/3 = = + 1+ af a a b b      −− (10) where a and b are constants, 0 = /f a , /a a a= and /b b a= . fig. 5 shows the effect of graphene content on the stress of the composites under the same force. fig. 5 the effect of the graphene content on the stress of the composites under the same force ( =100a b= , 0 =1000) to obtain a formula similar to the hall-petch effect, we make a simple mathematical approximation to formula (10). let 276 y.t. zuo, h.j. liu 2/3x = (11) eq. (10) becomes ( )= + f ax x a bx  − (12) differentiating eq. (12) with respect to x, we have 2 ( + ) ( ) ( )= ( + ) a a bx b f ax x a bx  − − −  (13) setting x=0, we have (0)= f a  (14) 2 (0)= aa bf a  − −  (15) its first order taylor series approximation is 0 2 ( ) (0) (0) = aa bf x x x a     + = + + (16) that is 2/3 0 k  = + (17) so we obtain the following formulation similar to the hall-petch effect 0 2/3 = 1 k k   − + − (18) where 2 aa bf k a + = . table 2 shows the effect of the graphene content on the fracture strength of pva/ graphene composites [10]. table 2 influence of the graphene content on the fracture strength of pva/ graphene composites [10] φ (%) 0 0.3 0.5 0.7 σ0(mpa) 49.7 68.2 72.2 87.8 according to the data in table 2, we can approximately determine that k=13.2165, so that 0 2/3 13.2165 1 1 3 65= .21   − + − (19) fig. 6 shows the stress-strain behavior for pva/graphene nanocomposites with different go weight loadings. with only 0.7 wt.% go, the tensile strength of the nanocomposite increased by 76% from 49.9 to 87.6 mpa [10]. fractal approach to mechanical and electrical properties of graphene/sic composites 277 fig. 6 stress-strain behavior for pva/graphene nanocomposites with different go weight loadings. with only 0.7 wt.% go, the tensile strength of the nanocomposite increased by 76% from 49.9 to 87.6 mpa [10] 3. electronic property according to ohm’s formulation, the resistance of a metal rod can be expressed as kl r s = (20) where r, l, and s represent, respectively, the resistance, the length and the area of the conductor, k is a constant. however, for a carbon nanotube, the resistance can be expressed as [13] r l (21) where α is the fractal dimensions of the carbon nanotube, and it can be calculated as α=2.52 [13]. the fractal theory has become a useful tool to various engineering problems, for examples, non-smooth fibers [14], the fractal current law [15], the fractal pressure drop through a cigarette filter[16], fractal-like multiple jets in electrospinning process [17], the fractional fokker-planck equation [18], the fractional kundu–mukherjee–naskar equation [19], the fractal telegraph equation[20], and fractal integral equations and fractal oscillators [21,22], the fractal boussinesq equation [23], the fractal microgravity [24],the fractal bratu-type equation [25], the fractal diffusion [26], the fractal two-phase flow [27], the fractal boundary problems [28], and the fractal convection-diffusion law [29]. in this paper we will apply the fractal theory to the study of the electronic properties of carbon nanotubes [30,31], graphene and graphene-based composites. the fractal dimension can be calculated as ln ln m n  = (22) 278 y.t. zuo, h.j. liu where m is the new number of the measured units under a reduced size of 1/n. for example, the fractal dimensions for the koch curve is ln4/ln3, while the cantor set is ln2/ln3, and the sierpinski triangle is ln3/ln2. in ref. [13], the graphene or the carbon nanotube is considered as a graphene fractal as illustrated in fig. 7. the fractal dimension of the graphene fractal is ln 5 =1.16096 ln 4  = (23) eq. (23) describes exactly a fractal pattern as illustrated in fig. 7, the initial iteration is a graphene unit, and the iteration process can continue to infinity. the value of the fractal dimensions is an important factor to describe the self-similarity of a geometric patterner. however, for a practical application, a discontinuous geometric patterner can be considered as some cascade of a self-similar fractal one. for example, we have two adjacent cascades of the sierpinski triangle, see fig. 8. their porosity factors are quite different, and we need the two-scale fractal dimensions to measure the difference [32-35]. the two-scale fractal theory has been widely used as an effective mathematics tool to analyze various discontinuous problems, for examples, fractional camassa-holm equation [36], biomechanism of silkworm cocoon[37], snow’s thermal insulation [38], fractal calculus for analysis of wool fiber [39] and polar bear hairs [40,41]. fig. 7 the graphene fractal fig. 8 two adjacent levels of the sierpinski triangle with fractal dimensions of ln3/ln2. the two-scale fractal dimension is defined as [32-35] 0 0 l l  =  (24) where α and α0 are dimensions for the two scales, respectively, one is large and the other is smaller; l and l0 are, respectively, the measured units (e.g. length, area or volume) for the two scales. for the two adjacent levels of the sierpinski triangle given in fig. 8, the two-scale fractal dimensions are 3/2 and 9/8, respectively. fractal approach to mechanical and electrical properties of graphene/sic composites 279 consider a unit of the graphene as illustrated in fig. 9; we have l/l0=5/4 and α0=2, and 5 2 =2.5 4  =  (25) according to eq. (21), the resistance of a carbon nanotube can be expressed as 2.5r kl= (26) using the experimental data given in ref. [42], we can identify the value of k in eq. (26), and obtain the following relationship 2.594.5r l= (27) this is quite close to r=95l2.52, which was given in ref. [13]. comparison of our theoretical result given by eq. (27) with the experimental data from ref. [42] is depicted in fig.10. fig. 9 a unit of the two-dimensional graphene fig. 10 resistance (kω) versus length (μm) for swnts. the dots are experimental data [42], the continuous line is the theoretical prediction according to the two-scale fractal theory graphene/sic composites are widely used in engineering, especially advanced materials science and smart ceramics, and their electrical property is the focus for various applications. 280 y.t. zuo, h.j. liu consider a rod of graphene/sic composite as illustrated in fig. 11. graphene-based composite materials have attracted much attention due to their extremely good thermal and electronic properties [10,42]. fig. 11 graphene/sic composite electrical conductivity of continuous non-metal materials scales with its length with a negative power -c l (28) we already know that the addition of graphene can greatly enhance the electrical conductivity of the composite. the equivalent length of graphene and the equivalent length of sic material can be expressed as 1/2 graphene graphene v l         (29) 1/3( )sic sicl v (30) where lgraphene is the equivalent length of graphene, δ is the thickness of the graphene, lsic is the equivalent length of sic material, vgraphene and vsic are, respectively, the total volumes of graphene and sic in the composite. we assume that the graphene concentration in the composite is φ, which implies 0graphenev v = (31) 0 (1 )sicv v= − (32) where v0 is the total volume of the composite. after a simple operation, we have the following scaling laws 1/2 graphenel  (33) 1/3(1 )sicl  − (34) fractal approach to mechanical and electrical properties of graphene/sic composites 281 the electrical conductivity of graphene and the resistance of sic are, respectively, as follows 1 /2 graphenec  (35) 2 /3(1 )sicc  − (36) where α1 and α2 are two-scale dimensions for graphene and sic material, respectively. according to the analysis given in the above section, we have α1=2.5, which implies 1.25 graphenec  (37) the total resistance of the graphene/sic composite can be expressed as 2 /31.25 1 2 (1 )graphene sicr r r k k  −= + = + − (38) where k1 and k2 are constants in this study. electrical conductivity of graphene/sic composites can be expressed as 2 /31.25 1 2 1 (1 ) c k k  − = + − (39) where k1 and k2 are constants in this study. in most cases, k1>>k2, and eq. (39) can be approximated as 1.25 0c c k= + (40) in ref. [42] three powder compositions were prepared with increasing gnps contents, specifically 5, 10 and 20 vol.% were used to measure the electrical conductivity of the graphene/sic composite. the results are listed in table 3. table 3 electrical conductivity of the graphene/sic composite with different gnps contents [42] φ (%) 0 0.05 0.10 0.20 c(s/m) 17 305 933 2306 porosity(%) 0.5 0.6 0.65 0.70 α2 1.5 1.8 1.95 2.1 according to the above experimental data, we know c0=17 s/m, and constant k involved in eq. (40) can be approximately determined; finally, the following equation is obtained 1.2517 15195c = + (41) 282 y.t. zuo, h.j. liu fig. 12 shows the comparison between our theoretical prediction and the experimental data, and a good agreement is obtained. fig. 12 the electrical conductivity of the graphene/sic composite rod 4. discussion and conclusion many experiments have proved that an increase of go concentration will enhance the sic/graphene composite’s mechanical property; however, similarly to the inverse hall-petch effect, when the concentration of go increases to a threshold value, an inverse effect can be predicted. recently le et al. suggested a theoretical method for graphene-reinforced composites, showing a bright light on the optimal design of sic/ graphene composites for advanced applications [43]. this paper suggests, for the first time, some new concepts for graphene/sic composites, e.g., the graphene fractal and the two-scale porosity. a fractal modification of ohm’s resistance formulation is suggested, and our theoretical prediction sees a good agreement with the experiment results given in ref. 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2568-2573. 43. le, n.y., nguyen, t.p., vu, h.n., nguyen, t.t., vu, m.d., 2020, an analytical approach of nonlinear thermo-mechanical buckling of functionally graded graphene-reinforced composite laminated cylindrical shells under compressive axial load surrounded by elastic foundation, journal of applied and computational mechanics, 6(2), pp. 357-372. 13348 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume241230020b © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a top-down multiscale analysis for deriving local material properties of additively manufactured al-si alloy ruslan balokhonov1, aleksandr zemlianov1, artur shugurov1,3, diana gatiyatullina1,2, ivan ivashov1,3, vasilii balokhonov1,2, varvara romanova1 1institute of strength physics and materials science, sb ras, russia 2national research tomsk state university, russia 3national research tomsk polytechnic university, russia abstract. a top-down approach is proposed for the numerical-experimental determination of local material properties of an additively manufactured alsi12 alloy possessing clearly expressed hierarchical structure. a thin-walled product was fabricated by wire electron beam additive technology. the alloy multiscale structure is studied experimentally by optical, scanning and transmission electron microscopy. the compression and nanoindentation mechanical tests are carried out. based on the experimental data, the finite element models of a layered structure at the macrolevel, dendritic and composite cellular structures at the mesolevel, and a composite structure comprising an aluminum matrix and silicon particles at the microlevel are created. the proposed top-down analysis assumes sequential macro-meso-micro structure-based numerical simulations to derive the mechanical properties of aluminum in dendrites at the microlevel and aluminum in the eutectic at the submicron level. the stress concentration and the plastic strain localization in dendritic, cellular and composite structures are analyzed. it was found at the mesoscale that the eutectic material experiences more shear stresses than the aluminum dendrites, with the highest stresses being observed in between the closely located dendrites. the volumetrically tensile and pure shear regions, as well as the regions of low elastic strains, are found after 30% compression. key words: multiscale analysis, additive manufacturing, aluminum-silicon alloys, microstructure-based numerical simulation, plastic deformation received: december 23, 2024 / accepted may 30, 2025 corresponding author: ruslan balokhonov institute of strength physics and materials science sb ras 634055, tomsk, pr. akademicheskii, 2/4, russia e-mail: rusy@ispms.ru 2 r. balokhonov, a. zemlianov, a. shugurov, et al. 1. introduction the design of multiscale models adequately describing deformation and fracture of newly developed advanced materials has been a continuing challenge for mechanics and materials science. it is primarily necessary to develop and study such models as applied to composite materials demonstrating pronounced structural inhomogeneity. it is the complex hierarchical structure of composites which makes their behavior hardly predictable in the frame of single-scale approaches. the stresses may concentrate at the interfaces between the structure components and the plastic flow can localize in their vicinity, which would affect the macroscopic material response and result in a premature loss of the strength. a number of present-day studies in the field of multiscale modeling have been thoroughly reviewed [1]. the authors discuss the analytical, semi-analytical and numerical methods for modeling composites and analyzing their effective properties. there are two major approaches in the multiscale numerical modeling: bottom-up and top-down. the former assumes the solution of a series of direct problems on structure deformation at different scale levels (e.g. micro-, meso-, and macroscales), starting from the lowest level. the effective material properties are determined at every level via homogenization and then used at higher levels as the model parameters. note that this approach assumes the material properties at the lowest level to be known in advance: they are frequently selected from the data of macroscopic experiments or estimated from indirect experiments. however it is the derivation of the local material properties, presumably differing at different scales, which is the most acute problem of multiscale approaches. it has been noted [1] that the finer the scale, the more difficult it is to experimentally verify the material constants in the model and to measure the properties of microstructure inhomogeneities. there are some recent examples of an implementation of the bottom-up approach [2, 3]. in a two-level modeling scenario [2], the authors introduced the strain rate and the matrix and fiber properties at the microlevel aiming at determining the fiber characteristics at the mesolevel, where the dynamic properties of braided composites are predicted. the purpose of a multiscale approach [3] is to identify the effect of porosity on mechanical properties and damage mechanisms of multidirectional carbon-fiber-reinforced polymer (cfrp) composites. the data on the direct finite-element modeling of cfrp composites with varying pore volume fractions and pore sizes are used for a stochastic multiscale prediction of mechanical response of the laminates. the principal task of a top-down approach is the solution of an inverse problem of determining the local material properties at the lower scales using the data of macroscopic and other special experiments. the bottom-up and top-down approaches can be combined to provide an iterative determination and refinement of the local material characteristics [1, 4], while multi-criteria decision-making framework can be used in composite material selection by comprehensive sensitivity analysis methods [5]. additively manufactured composites represent a class of advanced materials for which the issues a multiscale analysis are most important. this is due to the fact that a natural structural inhomogeneity of the composites, manufactured using conventional processes and associated with the availability of heterogeneous materials (particles, fibers, and matrix) in them, is added with the morphological and crystallographic anisotropy related to the manufacturing method – 3d printing. due to multiple thermal cycling and high temperature gradients in the additive manufacturing, a complex, hierarchically organized structure is formed, which contains melt pools and interlayers between the deposited layers, as well as a specific structure within the melt pools and at the boundaries in between. there a top-down multiscale analysis for deriving local material properties of additively… 3 are two most versatile additive-manufacture processes: powder bed fusion (pbf) and directed energy deposition (ded). in the former case, the components are manufactured by a preliminary deposition of thin powder layers on a platform and their compaction with a roller, followed by the layer-by-layer laser (l) or electron beam (e) fusion [6]. in the epbf, the powder is preheated and the particles fuse in vacuum [7, 8]. the l-pfd process, also referred to as selective laser melting (slm), uses a high-power laser to melt the powders in an inert atmosphere without any pre-heating. similar to 3d-printing polymeric materials [9], the metallic materials used in the ded process are either powders or wires fed through a nozzle, which are melted by a laser, an electric arc, or an electron beam. the ded method is perfect for repairing high-cost components with a negligible material loss [10]. the resulting material structure and texture heavily depend on the additive manufacturing process and its modes. for instance, the particle size or wire diameter in ded are relatively large compared to those in pbf, which results in a larger melt pool or layer and interlayer thickness. by giving preference to e-pbf instead of slm, one can increase the alloy hardness and ductility due to structure refinement and second-phase precipitation prevailingly at the grain boundaries [11]. a repeated high-energy electronbeam irradiation of the material surface can improve the strength of a sample fabricated by the wire-arc additive manufacturing (waam) technology [12]. the spatial grain-size distribution, inhomogeneous in three dimensions, might strongly depend on the e-ded strategy, and an additional annealing treatment can increase the texture strength and structure homogenization [13]. the resulting e-ded alloy structure can be affected by the thickness and material properties of the substrate [14] and the final sample shape: whether it is a thin-walled product or a bulk item [15]. furthermore, the beam power and the wire feed and displacement rates can significantly influence the final product shape and quality [16]. aluminum alloys and composites thereof have been extensively used in the aerospace industry due to their high energy efficiency [17] and are, therefore, frequently applied in additive manufacturing. the latest advances in this field have been overviewed, for instance, in [18], where the authors provided a detailed description of the structure formation and mechanical properties of aluminum alloys, al-si among them, and aluminum-matrix composites manufactured using a variety of am processes. in the wirefeed additive manufacturing, the energy of a laser is frequently used. in a different study [19], a hot al-si6 wire was deposited on an al6061-t6 substrate. using the method of digital image correlation (dic), the authors demonstrated that in the samples in tension the deformation localizes in the near-boundary regions of the melt pool, which results in a characteristic serration of the stress-strain curve. this is due to the fact that primary aluminum dendrites in the regions adjacent to the melt pool boundaries are coarser than those in the material bulk. the waam technology is currently becoming increasingly popular. the size effect of ceramic powder particles, added between the layers of al-si6 alloy, on the fusion process and material microstructure was studied elsewhere [20]. the authors demonstrated that a coating waam-manufactured from nanosized powder allows significantly decreasing the arc fluctuation and provides its correct deflection, which improves the molten pool fluidity and the forming quality. an interlayer coating can favor a grain size decrease and a transition from columnar crystals to fine equiaxed dendrites, which improves the rupture strength of al-si6 alloy both along and across the printing direction. a combined laser-arc technology for manufacturing al-si12 alloy was proposed in [21]. it is shown that high-frequency circular vibrations of a laser beam improve the 4 r. balokhonov, a. zemlianov, a. shugurov, et al. forming precision, decrease the structure element size, and increase the yield strength and relative elongation of the alloy. the use of an electron-beam energy impact in wireand powder-feed additive manufacturing of alloys and composites, al-si among them, has been described in a recent review [22]. the authors analyzed the features of defect structure formation in dependence of manufacturing modes, wire and powder states, scanning strategy, and conditions of additional ultrasonic processing, laser treatment or shot peening. the wirefeed electron-beam additive manufacturing (webam) demonstrates clear advantages over the laser additive process, since it occurs under the vacuum conditions, which allows preventing oxidation of aluminum alloys, and the energy density and the heat penetration depth are much higher than those of the laser technology [23, 24]. earlier we performed a multiscale analysis of deformation and fracture in an aluminum matrix – carbide particle composite material [25] and used a bottom-up approach in a numerical study of deformation of a webam eutectic al-si12 alloy [26–28]. we demonstrated that this alloy is a multiscale composite material. at the submicron scale, its eutectic network contains disperse silicon particles measuring hundreds of nanometers, which are surrounded by an aluminum matrix. the peculiarities of formation of residual stresses during cooling of this composite and its subsequent mechanical loading were investigated using the literature data on the properties of aluminum and silicon [26]. at a higher scale level of tens of microns we revealed the formation of a dendritic structure. a two-level approach is illustrated elsewhere [27], wherein the effective eutectic network properties were determined via homogenization of the solutions to the problem of deformation of a silicon particles–aluminum matrix composite at the submicron level. when solving a series of direct problems, we studied the influence of mechanical properties of the aluminum matrix on the composite structure strength. more recently [28], the twolevel approach was applied in the investigation of the effect of an interlayer on the deformation and fracture of dendritic structures. in so doing, the properties of aluminum at the submicron and micron levels were determined using indirect experimental data on the grain-size dependence of its yield stress and strength. the purpose of the present study is to apply a top-down approach to determining the plastic properties of aluminum at the micro and submicron levels, relying on the data of nanoindentation macroscopic experiments. 2. additive manufacturing and microstructure analysis of al-si12 alloy the samples were printed using a wire-feed electron-beam additive manufacturing technology in a laboratory setup designed at the institute of strength physics and materials science sb ras (tomsk, russia). its operating principle is the following: an electron gun generates an electron beam to form a stable melt pool. to this aim, a larger beam sweep diameter is selected than that of the wire. therefore, part of the beam heats and melts the upper substrate layer and the other part heats and melts the wire. the resulting liquid metal via the jet or drop transfer is fed into the liquid metal pool formed in the substrate. in order to form directed layers, use is made of a three-coordinate, water-cooled table. during printing, the beam is not shifted, only the table is displaced at the controlled velocity and direction. the electron beam parameters (current and accelerating voltage) can be regulated during printing, which, together with the regulation of the table motion and wire-feed, provides a flexible control over the printing modes. the manufacturing process is carried out in vacuum. a top-down multiscale analysis for deriving local material properties of additively… 5 in this study we printed a thin-walled sample with a length of 8 cm, a single layer thickness of 5 mm and a height of 3 cm [27]. a eutectic alsi12 alloy esab оk autrod 4047 1.2 mm wire was deposited layer-by-layer onto a аlmg5 alloy substrate, using an optimal deposition mode with an exponentially decreased energy input to compensate for the fast cooling of the first layers from the substrate. this allowed forming the samples with minimal defects of porosity and discontinuities and prevented spreading of the top layers [27]. fig. 1a shows the sample section. we examined the ebam product structure using metallography and scanning and transmission electron microscopy. the metallographic studies were performed according to a standard procedure. the samples measuring 10х10 mm were cut from the middle part of the product, ground on an abrasive paper of 200 to 2000 grits, and polished using a diamond paste and a suspension. the surfaces of the metallographic sections were examined in an olympus lext ols4100 microscope. the sample microstructure was studied in a tescan vega ii lmu scanning electron microscope (sem) and in a leo evo-50 microscope equipped with a system for the energy dispersive microanalysis (eds). thin foils for the tem studies in a jem-2100 transmission electron microscope (tokyo boeki ltd., japan) were prepared by mechanical grinding on abrasive paper and ion thinning in an em-09100is ion slicer. fig. 1 multiscale nature of webam alsi12 alloy. optical images of the thin-walled printed sample cross section (a), inlayer dendritic structure (b) and interlayer cellular structure (c). bright-field tem image of the eutectic al-si composite (d) and corresponding eds maps showing distribution of al (e) and si (f) 6 r. balokhonov, a. zemlianov, a. shugurov, et al. the microstructural investigations have demonstrated that the additively manufactured alsi12 alloy has a complex multiscale structure and represents a multiscale additive composite. as compared to conventionally manufactured alsi12 alloys there are the following specific features unique to the webam alloys. at the macrolevel, this thinwalled sample represents a layered product in which the fused layers alternate with interlayers – direct sites of fusion in the heat affected zone (haz) (fig. 1a). the material structure in the printed layer and at the fusion interface is different (cf. fig. 1b and c). in the former case, we observe coarse aluminum dendrites of a few tens of microns, which are surrounded by a eutectic network. the eutectic network, in its turn, was revealed by the tem examination as the volumetric silicon particles of hundreds of nanometers having different shapes, which are surrounded by the aluminum matrix (fig. 1d–f). the interlayer structure is absolutely different. between the layers in the zone affected by the electron beam there is a 30–50 µm thick interlayer formed from the fused aluminum containing agglomerated silicon particles (fig. 1c). these particles have more complicated shapes and an order of magnitude larger size than the submicron particles in the eutectics; they are distributed over the volume as a cellular structure. 3. multiscale numerical modeling problem formulation 3.1 model structures of different spatial scales using the experimental images given in fig. 1, four geometrical models were designed at every spatial scale level. a model of the layered structure of a macrosample, wherein the printed layers alternate with thin interlayers at the layer fusion boundary, is presented in fig. 2а. the dimensions of the model macrosample correspond to those of the experimental sample subjected to uniaxial compression in the scanning direction. according to the experimental data (fig. 1а), the average printed layer thickness is on the order of 400 µm, with an interlayer of an average thickness of 50 µm observed between the layers (fig. 1с). thus, the volume fraction of the printed layer and the interlayer is about 89% (fig. 2b) and 11% (fig. 2c), respectively. we included a fourth part of the sample into our calculations; the symmetry conditions were set on the right and back surfaces. to sum up, the geometrical model of this macrosample contains 6 layers and 6 interlayers. the number of finite elements in the layered structure is about 2 million. the model structures at the mesoand microlevels (fig. 2d–f) were designed based on the following considerations. we take that the structures at the mesolevel are representative. therefore, the total volume fractions of aluminum and silicon have to be 86% and 14%, respectively, i.e. equal to their percentages in the initial feed wire for printing. in the experiment, the volume fraction of aluminum dendrites after additive printing is found to be 42 %, and that of the eutectics – 58 %. the same ratio was used at the mesolevel when designing the model dendritic structure whose morphology (fig. 2d) corresponds to the experimentally observed morphology (fig. 1b). the size of the designed structure is on the order of 100×120 µm, and it is digitized by a regular rectangular mesh of 500×600 elements. since at the microlevel the eutectic material contains the entire 14 % of silicon, the volume fraction of silicon in the model composite eutectic material would be 24% (fig. 2f). the numerical model of the al-si composite structure has the size of about 2.5×2.5 µm; the computational domain contains 230400 finite elements. the size of a top-down multiscale analysis for deriving local material properties of additively… 7 the cellular structure in the interlayer at the mesolevel (fig. 2e) is 100×30 µm and the computational domain size is 720×250 elements. the volume fraction of silicon particles is 14 %; when the next layer is printed, the material in the heat affected zone is remelted, and silicon particles contained in the eutectics agglomerate to form coarser particles surrounded by the aluminum inherited from the eutectics and the dendrites. fig. 2 layered macrosample cut from thin wall (a–c), dendritic (d) and cellular structures at mesolevel (e) and al-si composite at microlevel (f) 3.2 formulation of the mechanical problem the dynamic boundary-value problems on loading the layered structures and composites are solved in a three-dimensional formulation in the abaqus/explicit solver. the printed layer and interlayer material in the layered structure at the macrolevel, the aluminum dendrites and the eutectic network in the dendritic structure and the remelted dendrites in the cellular structure of the interlayer at the mesolevel, and the aluminum matrix in the al-si composite are taken to be elastoplastic. their mechanical response is described using the hooke’s law: 2 ( / 3)p ij ij ij kk ij ij kk ij ijp s k            (1) 8 r. balokhonov, a. zemlianov, a. shugurov, et al. where σij and sij are the stress tensor and stress deviator components, p is the pressure, εij and p ij are the total and plastic strain components, δij is the kronecker delta, k and µ are the volumetric and shear moduli, the upper dot designates a time derivative. the plastic flow is described using the plastic flow law p ij ijs  associated with the yielding condition given by ( ) 0p eq eq    , where λ is the scalar factor that is identically zero in the elastic region. the function of isotropic strain hardening of the materials represents a combination of the exponential and linear laws   0( )exp( / )p p p eq s s eq r          (2)  p p eq eqa b    , (3) where σs and σ0 are the ultimate stress and yield point, p r characterizes the strain hardening degree, a and b are the linear hardening constants, p eq and σeq are the stress and accumulated plastic strain intensities. the response of silicon particle in the cellular structure of the interlayer at the mesolevel and in the eutectic al-si composite at the microlevel is elastic and is described by eq. (1), where 0p ij  . the uniaxial compression in the scanning direction х is modeled by the kinematic boundary conditions for all the calculations performed in this study (fig. 2). particle velocities on the left and right surfaces were gradually increased to avoid dynamic effects. the top surface is free of loads, while the symmetry conditions are set on the bottom surface. here, the elastic moduli at different spatial scales were determined from the mixture model. the eutectic modulus at the mesolevel and that of the layer at the macrolevel were calculated via the volume fractions of aluminum and silicon particles at the microlevel and via those of dendrites and eutectics at the mesolevel, respectively. the interlayer moduli at the macrolevel were calculated using the volume fraction of silicon particles and aluminum in the cellular structure at the mesolevel. for instance, for the modulus of volumetric compression we have (1 )eut al sik k f k f   , (1 )layer al eutk k k    , (1 )interlayer al sik k k    , (4) where ƒ, φ and η are the volume fractions of aluminum in the eutectics, aluminum dendrites in the layer, and aluminum in the interlayer, respectively. 4. deriving local material properties by top-down analysis in a microstructure-based numerical simulation, it is critically important to determine the local characteristics of the structure component materials at different spatial scales – their values could differ from the experimentally determined macroscopic properties by a few times due to the scaling factor. it is the main aim of the present study to determine the mechanical properties of materials in an additively manufactured al-si12 alloy. it is necessary to derive the properties of aluminum in dendrites and eutectics. in the course of a comparatively fast crystallization after its additive melting, aluminum formed in the a top-down multiscale analysis for deriving local material properties of additively… 9 eutectic network between the submicron silicon particles can have a nanodefective structure. its plasticity characteristics would be different from those in larger dendrites, where the dislocation mean free path is an order of magnitude larger. unfortunately, it is impossible to determine the plastic properties of these materials necessary for the calculations, specifically the dependence of stresses on strains. firstly, this is due to the fact that it is impossible to cut the samples of tens of microns, not to mention microns. secondly, even given this possibility, there are no such testing machines to perform mechanical tensile or compressive tests of the samples as small as these. on the other hand, it is possible to carry out mechanical indentation testing, which would provide some indirect information on the local hardness and elasticity [29]. however, these data can be obtained from a limited scale – in our case at the mesolevel of the dendritic structure. it is impossible to do it at the microlevel, since in order to achieve a local area of indentation of tens of nanometers one needs a load of fractions of a millinewton. these indentations, though technically not improbable, lie beyond the testing machine resolution and are not correct. therefore, at the scales of microns and tens of microns the material properties are difficult or impossible to determine. in view of this fact, the use of combined numericalexperimental approaches is required, aimed at identifying the properties that would be further used in the calculations for the sake of identifying the features of deformation and fracture of additively manufactured alloys having a complex multiscale structure. the top-down numerical-experimental approach proposed in this work consists in a derivation of the local properties of aluminum in the dendrites and, further, its properties in the eutectics. in so doing, we consistently use the macroscopic experiments on compression of the samples of an additively manufactured al-si12 alloy (fig. 3а) and the nanoindetation data (fig. 3b). the samples for mechanical tests were cut from the material along the scanning direction. the compressive tests were performed in a testsystems machine (testsystems, russia). the loading rate was 1mm/min. fig. 3 experimental response of macroscopic ebam alsi12 sample in compression along scanning direction х shown in figs. 1 and 2 and its approximation by exponential and linear laws (a) and indentation points made by forces of 400, 100 and 10 mn (b) 10 r. balokhonov, a. zemlianov, a. shugurov, et al. 4.1 nanoindentation in this work, we propose to estimate the inlayer, interlay, eutectic and aluminum material properties by the method of indentation. the hardness of the ebam alsi12 samples prepared for metallography was performed in a nanotest machine (micro materials ltd., wrexham, uk) using a berkovich diamond indenter under the applied load control mode. the maximum load was varied from 2 to 400 mn. at the loads of 200 and 400 mn, fifteen indents were made in every case, and at 2 and 10 mn – 100 indentations separated by a distance of 100 µm (fig. 3b), which allowed ruling out the effect of plastic deformation on the measured characteristics. the loading and unloading durations were 20 s each. in order to correct the temperature drift after the 90% unloading, the samples were held at a constant load of 60 s. by analyzing the indentation curves using the oliver-pharr method we determined the sample hardness h and the reduced young’s modulus e*. the indentations at a hold-down load of 400 mn demonstrate some of the integral characteristics of the dendritic material structure within the layer (fig. 4а) and the cellular structure with the silicon particles surrounded by aluminum in the haz (fig. 4b), since they cover a comparatively large area for averaging. out of 15 indentations (10 are shown in fig. 3b), only one fell within the interlayer (fig. 4b); the images of the other 14 indentations in different sites of the layers are qualitatively similar to that given in fig. 4а. it is from these 14 measurements that we determined the average material nanohardness – 0.71 gpa. inside the interlayer, the material hardness is found to be 0.59 gpa. we can, therefore, take the layer-to-interlayer ratio of the mechanical properties of the materials to be 1.2. the curves of the load against the indentation area for all of the 15 indentations are qualitatively similar to those given in fig. 4с, indicating that all measurements are correct. fig. 4 indenter imprints in printed layer (a) and in interlayer (b) and nanoindentation curves at a load of 400 mn (c) the lower the indentation force, the stronger the influence of inhomogeneities on the measured results, therefore the less smoother and monotonic could be the indentation curves. on the other hand, in order to derive the mechanical properties of the micron structure elements, we need the smaller indentation imprints, which are formed at weaker loading forces. therefore, for the sake of estimation of mechanical properties of aluminum in the dendrites and the eutectic material, we used the data measured at the least possible maximum force for this alloy of 2 mn, at which a sufficiently large percentage of correct measurement is available (fig. 5). to begin with, from 100 measurements we removed 18 incorrect indentations identified by nonmonotonic curves (allowed monotonicity is shown a top-down multiscale analysis for deriving local material properties of additively… 11 in fig. 5c). then two sets were compiled from the remaining measurements – the indentation traces falling within the center of either the dendrite (fig. 5a) or the eutectics (fig. 5b). the data in these two sets were averaged, and the hardness values of the eutectic and aluminum materials were determined, and so was their ratio – 1.7. fig. 5 indenter imprints traces in dendrite (a) and eutectic network (b), indentation curves at a load of 2 mn (c) 4.2 multiscale top-down numerical simulation a top-down numerical analysis assumes a solution of boundary-value problems on deformation of structures at different spatial scales, starting from the macrolevel. at every spatial scale, an inverse problem is solved in order to select the constants of eqs. (2) and (3) for the structure component materials. at the macrolevel, we solved the problem of compression of a layered structure shown in fig. 2a. the aim of the calculations was to select the plastic properties of the layer and interlayer materials to make sure that the calculated integral curve under the conditions of the layered structure compression would coincide with the experimental stress-strain curve given in fig. 3а. in the initial iteration, the model parameters were determined from the mixture model using the results of nanoindentation at 400 mn (fig. 4). it has been taken that the ratio of the layer and interlayer materials characterizes the difference in the flow stresses, σs and σ0 in the exponential law and in the linear equation constants a and b. based on the tilt of the indentation curves (fig. 4c), we take that parameter p r is the same for the layer and interlayer materials. since hlayer / hinterlayer =1.2, at the macrolevel we obtain a system of equations: 0 0 0 (1 )macro layer interlayer       ; 0 0/ 1.2layer interlayer   , (1 )macro layer interlayer s s s       ; / 1.2layer interlayer s s   , (5) (1 )macro layer interlayera a a    ; / 1.2layer interlayera a  , (1 )macro layer interlayerb b b    ; / 1.2layer interlayerb b  . denoting all 4 constants in eq. (5) by the same symbol for the sake of convenience, we have 12 r. balokhonov, a. zemlianov, a. shugurov, et al. (1 )macro layer interlayer       ; / 1.2layer interlayer   , (6) where ξlayer and ξinterlayer are the derived properties of the layer and interlayer materials and ξmacro are the known constants in the experimental flow curve approximation for the macrosample in compression (fig. 3а). since the volume fraction of the layer material is also known 0.89  , from eq. (6) we uniquely obtain the model layer and interlayer parameters that are further used as the first iteration in modeling the layered structure compression. this solution of the boundary-value problem at the macrolevel is a test solution, since in the case of a uniaxial compression along direction х, in which case the kinematic loading is effected along the layers and interlayers and the interlayer boundaries are presented by straight lines (fig. 2a), the stress and strain distributions within the layers have to be homogeneous without any localization. therefore, the integral calculation curve for this structure has to completely coincide with that obtained in accordance with the mixture model. this was the case in the calculations: the constants obtained using eq. (6) did not require any iterative refinement; they provided a complete coincidence of the calculated and experimental curves (table 1). the respective curves are shown in fig. 6а. here and further in the figures, the stress and strain in the calculated curves of structure deformation represents the stress and plastic strain intensities averaged over the computational domain. table 1 plastic properties of the layer and interlayer materials determined from a numerical calculation of deformation of an alsi12 layered sample at the macrolevel σs, mpa σ0, mpa p rε , % a, мpа b, мpа layer interlayer 266 219 114 94 4,2 4,2 210 170 430 360 now let us turn to a lower scale level. the task in this stage is to determine plastic properties of the eutectic material and aluminum in the dendrites. we have to select the model parameters so as to let the calculated integral curve in the case of a dendritic structure compression at the mesolevel (fig. 2d) coincide with the earlier obtained response of the layer material at the macrolevel, described by the derived constants ξlayer. at this level, we use the data of nanoindentation at 2 mn (fig. 5), which characterize the difference in the mechanical properties of aluminum in the dendrites and the eutectic material by comparing their hardness values. therefore, at the mesolevel eq. (6) acquires the following form: (1 )layer al eutectic meso       ; / 1.7eutectic al meso   , (7) where al meso and ξeutectic are the derived properties of aluminum in the dendrites and the eutectic material. the volume fraction of aluminum dendrites is known χ = 0.42, so using eq. (7) we uniquely determine the model parameters as the first iteration. the aluminum dendrites inside the layer at the mesolevel have complicated shapes, there are stress concentrations and strain localization near the curvilinear interfaces between the eutectics and the dendrites. therefore, the constants al meso and ξeutectic determined from the mixture model fail to provide an agreement between the calculated integral flow curve of the dendritic structure and the stress-strain curve of the layer derived at the macrolevel. in this case, it is necessary to derive these parameters via their successive variation when solving the direct a top-down multiscale analysis for deriving local material properties of additively… 13 boundary-value problems on the dendritic structure compression. the resulting parameters are listed in table 2, and the respective curves obtained at the mesolevel – in fig. 6b. table 2 plastic properties of alsi eutectics and al dendrites determined from numerically calculated dendritic structure deformation at mesolevel σs, mpa σ0, mpa p rε , % a, мpа b, мpа alsi eutectic al in dendrites 337 198 143 84 4,4 4,4 270 160 500 290 to sum up, in this step of the top-down numerical-experimental analysis we have determined the plastic material properties at the mesolevel, specifically we have derived the stress-strain curves of the eutectics and dendrites, which are impossible to obtain experimentally. in addition, at this level of consideration it is also possible to estimate the local mechanical properties using nanoindentation, which was successfully done. at the microlevel, a still lower spatial scale, it is hardly possible to experimentally estimate the local material properties by nanoindentation. the results that allow deriving the material properties at the submicron level through numerical simulations are therefore ever more valuable. similarly, if we follow the above-described procedure, an etalon curve for deriving the properties of aluminum surrounding silicon particles measuring hundreds of nanometers (fig. 1d–f) is the flow curve for the eutectics material obtained at the mesolevel and determined by the constants ξeutectic (table 2). no additional conditions based on the indentation data are required at the microlevel, since silicon particles are not subjected to plastic deformation, and the elastic moduli are known and are taken to be independent of the spatial scale in the range up to 100 nm. in view of these considerations, at the microlevel we solved an inverse problem of selection of the model parameters for aluminum in eutectics al micro via performing a series of numerical experiments on compression of a structure given in fig. 2f. the derived mechanical properties of aluminum at the submicron level are presented in table 3. the flow curve calculated for the al-si composite and the derived aluminum properties are given in fig. 6c. table 3 plastic properties of aluminum in eutectics, determined from numerical calculation of deformation of al-si composite at microlevel material σs, mpa σ0, mpa p rε , % a, мpа b, мpа al in eutectic 315 133 8,5 260 180 it was a priori assumed that at the mesolevel aluminum in dendrites inside the layer and in the cellular structure in the interlayer has the same defect structure, i.e. it is the same material with the same mechanical properties. therefore, one could derive them without modeling the dendritic structure deformation, as it was mentioned above (path 1), but determine the constants, al meso and ξeutectic, via solving the problem on deformation of a cellular structure shown in fig. 2e to achieve a consistency between the calculated integral curve and the interlayer material properties derived at the macrolevel and characterized by its constants ξinterlayer (path 2). let us verify this assumption by calculating the cellular structure deformation using the aluminum properties already determined at the mesolevel 14 r. balokhonov, a. zemlianov, a. shugurov, et al. via path 1 al meso , which are highlighted in fig. 6b by a cyan curve with stars. the respective calculation results are presented in fig. 6d. it is seen that the calculated flow curve (orange with triangles) perfectly coincides with the mechanical interlayer properties, determined at the macrolevel under compression of the structure to 8 %, and starts deflecting upwards at larger strains. this implies that the properties of aluminum in the interlayer and in the dendrites under intensive straining might differ. the reason for the differences could stem from the differences in the aluminum properties in the cell walls and inside the cells, since in the former the dislocation mean free path between the closely located silicon particles is small, which provides a better strain hardening effect. fig. 6 stress-strain curves of structure elements of additively manufactured al-si12 alloy at macro(a), meso(b, d) and microscales (c) figs. 7 and 8 illustrate an inhomogeneous stress-strain state at the mesolevel. it is seen that a complex stressed state is formed in the dendritic structure (fig. 7a). in the eutectics, the level of stress intensity is higher than that in the dendrites. both the eutectics and the dendrites are prevailingly found in the state of volumetric compression (fig. 7b). in the vicinity of the upper free surface above two elongated dendrites, there are two local volumetric tension regions formed due to bending of the eutectic framework. it is evident from fig. 7b that the regions of approximately the same highly compressive pressure are a top-down multiscale analysis for deriving local material properties of additively… 15 formed both in minor-, average-size dendrites, and eutectic network between the dendrites located on top of each other. it is noteworthy that the stress intensity in the dendrites is much lower than that in the network. this suggests that the eutectic network undergoes larger shear loading that do the dendrites. moreover, the pressure between the adjacent intergrown dendrites is low, and in some regions is close to zero (blue and green areas in fig. 7b), considering that the stress intensity is at its maximum (fig. 7a). this indicates that the eutectic network material is in the state close to pure shear. the stresses in the eutectics and dendrites concentrate due to the plastic flow localization; therefore, the distributions in fig. 7a and c are qualitatively similar. it is worth noting that even at a 32 % compression there are regions in the eutectic network, wherein the plastic deformation is on the order of 2–5% (blue areas in fig. 7с). the stress intensity in these regions is low (fig. 7а), while the pressure is quite high (fig. 7b), implying that they undergo merely pure volumetric compression. fig. 7 compression of dendritic structure shown in fig. 2d to 32 %. stress intensity (a) and pressure patterns (b). accumulated plastic strain patterns presented separately for the eutectic network and dendrites (c). undeformed shape a different pattern is observed in the case of the cellular structure deformation in the interlayer (fig. 8). a distinguishing feature is the fact that, considering the overall compression, both the aluminum matrix and the silicon particles contain regions of volumetric tension (red areas in fig. 8a). in the matrix, these regions are found near the particle convexities in the direction perpendicular to that of compression, while in the 16 r. balokhonov, a. zemlianov, a. shugurov, et al. particles near the matrix–particle interface concavities along the direction of compression. the values of both the negative pressure (fig. 8a) and the stress intensity are comparable and large, implying that these are the regions of a high-intensity volumetric-shear straining. between the regions of volumetric tension and compression, i.e. where the pressure changes its sign (green area sin fig. 8a), the stress intensity is, though not the highest, but quite high, measuring hundreds of megapascals. therefore, the conditions of moderate pure shear are realized there. the maximum compressive and tensile stresses develop in the intricately-shaped particles with curvilinear interfaces with the matrix (fig. 8a). the highest plastic strain localization is observed in the cell walls between the closely spaced silicon particles, while inside most cells the localization is not so pronounced (fig. 8с). due to a complex morphological anisotropy of this structure, however, the stresses might concentrate in the centers of individual cells and there might develop a stress concentration resulting in a developed plastic flow (dashed circle in fig. 8c). in the matrix of the cellular structure there are also weakly deformed regions (blue areas in fig. 8c). these regions are more numerous, and the plastic strain intensity in them is an order lower than that in the dendritic structure (cf. figs. 7с and 8с). in fact, these are the regions of nearly elastic deformation that might be found in the conditions of shear or volumetric compression, or their combination (cf. fig. 8а–с). the elastic regions are located both near the concavities fig. 8 compression of interlayer structure shown in fig. 2e to 32%. pressure distributions presented separately for the aluminum matrix and silicon particles (a), stress intensity (b) and accumulated plastic strain patterns in the aluminum matrix (c). deformed shape between the elongated particles and the matrix, and between the closely spaced silicon particles. at the microlevel, the same qualitative behavior patterns are observed in the a top-down multiscale analysis for deriving local material properties of additively… 17 composite material of the eutectics as in the cell walls, since it represents the same aluminum matrix–silicon particles composite material. from the quantitative point, the difference consists in the following: firstly, the average stresses at the microlevel are higher than at the mesolevel, since aluminum in the eutectics possesses higher mechanical properties than aluminum in the dendrites. secondly, above-described peculiarities of formation of the volumetric compression and tension regions, as well as the weakly deformed regions, are not so strongly pronounced, since the submicron silicon particles have less complicated shapes than those of large silicon particles in the interlayer. 5. conclusions using an alsi12 alloy fabricated by a wire-feed electron-beam additive manufacturing, a top-down approach to a numerical-experimental derivation of local mechanical properties of hierarchically organized composite materials has been proposed. the structure of this webam alsi12 alloy has been examined by the methods of optical, scanning and transmission electron microscopy at different spatial scale levels. the experiments on compression and nanoindentation of the webam alsi12 alloy samples have been performed. using the experimentally obtained images, a number of finite-element models were generated of the layered structure at the macrolevel, of the dendritic and composite cellular structure at the mesolevel, and of the aluminum matrix–silicon particles composite structure – at the microlevel. these models were integrated into the abaqus/explicit solver and the respective boundary-value problems were solved. the principle of the proposed top-down analysis consists in a successive series of numerical calculations of compression of the structures at different spatial scale, starting from the macro scale. an inverse problem of derivation of the mechanical properties of aluminum in the dendrites of about tens of microns and aluminum in the eutectic material at the submicron scale was solved. to begin with, we solved the problem of the layered structure compression. the macroscopic flow curve and the data of large indentation loads were used to derive the material properties inside the printed layer and in the interlayer. these properties and the indentation data at small loads were then used at the mesolevel to derive the properties of aluminum dendrites and eutectic network in modeling the dendritic structure compression. finally, using the obtained eutectic properties, we solved the problems on compression of the particle-reinforced al-si composite at the microlevel and derived the plastic properties of aluminum surrounding silicon particles measuring hundreds of nanometers. an analysis of the inhomogeneous stressed state of the structures of different spatial scales under the conditions of their compression has prompted the following conclusions: – in the case of compression of the dendritic structure inside the layer, the eutectic network is more prone to shear stresses than the dendrites, and the regions between the neighboring dendrites experience a state close to pure shear; – in the case of 30 % deformation, there are weakly deformed regions in the eutectic network, which undergo a state close to a purely volumetric compression, implying that the plastic deformation is small; it is found to be about 3 %; – in the case of compression of the cellular structure observed in the interlayer, both in the aluminum matrix and in the silicon particles there are regions of high-intensity volumetric tension and shear and the regions of pure shear. the largest compressive and 18 r. balokhonov, a. zemlianov, a. shugurov, et al. tensile stresses develop in large particles of complicated shapes having curvilinear interfaces with the matrix; – maximum plastic strain localization is found in the cell walls between the closely located silicon particles, and inside most of the cells it is less clearly pronounced; – there are a larger number of weakly deformed regions in the cellular structure than in the dendritic structure; minimum plastic strain is found to be 0.2 %. acknowledgement: the work was supported by the russian science foundation (grant no. 23-1100222, https://rscf.ru/en/project/23-11-00222/). basic 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2centre for nonlinear systems, chennai institute of technology, chennai, tamilnadu, india 3faculty of engineering, dogus university, istanbul, türkiye 4department of industrial engineering & management, yuan ze university, taoyuan city, taiwan 5department of applied mathematical science, college of science and technology, korea university, sejong, republic of korea 6department of industrial and management engineering, institute of digital anti-aging health care, inje university, gyeongsangnam-do, republic of korea orcid ids: michael sandra https://orcid.org/0009-0008-1913-1998 samayan narayanamoorthy https://orcid.org/0000-0002-3782-4666 krishnan suvitha https://orcid.org/0000-0003-1408-1393 dragan pamucar https://orcid.org/0000-0001-8522-1942 daekook kang https://orcid.org/0000-0002-7861-1544 abstract. traditional geothermal detection methods, such as extensive ground-based surveys and drillings, are often costly, time-consuming, and environmentally intrusive. to address these challenges, this study presents a novel hybrid fuzzy multi-criteria decision-making model to evaluate and prioritize non-invasive, cost-effective remote sensing (rs) techniques. this model uses t-spherical dual-hesitant fuzzy set to manage the inherent ambiguities in the evaluation of multiple criteria. the logarithmic percentage change-driven objective weighting technique assigns the relative importance of criteria, and the multiple triangle scenarios-ii methodology helps in comprehensive evaluation and ranking. by incorporating expert judgment and addressing inherent uncertainties, this model provides a systematic framework for optimizing rs technique selection. findings indicate that thermal infrared imaging, with a significance score of 0.7187, holds transformative potential for geothermal energy development. sensitivity and comparative analyses further confirm the robustness of this approach. this research offers a valuable resource for energy developers and policymakers aiming to leverage rs technologies for efficient geothermal resource management and development. key words: mcdm, t-spherical dual-hesitant fuzzy set, multiple triangles scenarios-ii, objective weighting, geothermal detection received: november 16, 2024 / accepted april 08, 2025 corresponding authors: samayan narayanamoorthy, dragan pamucar, daekook kang department of mathematics, bharathiar university, coimbatore, india e-mail: snmphd@buc.edu.in, dpamucar@gmail.com, dkkang@inje.ac.kr https://orcid.org/0009-0008-1913-1998 https://orcid.org/0000-0002-3782-4666 https://orcid.org/0000-0003-1408-1393 https://orcid.org/0000-0001-8522-1942 https://orcid.org/0000-0002-7861-1544 mailto:snmphd@buc.edu.in 534 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang 1. introduction the global population is projected to reach 9.7 billion by 2050 [1], presenting significant challenges in meeting the escalating energy demands driven by increased urbanization, industrialization, and technological advancements. the depletion of conventional energy sources such as coal, petroleum, and natural gas not only threatens to exacerbate global warming but also poses severe environmental risks [2]. according to chhandama et al., [3], carbon dioxide (co2) emissions are expected to reach 40 million kilograms by 2030, potentially contributing to a rise in global temperatures exceeding 20c. this abrupt upsurge in temperature could lead to the extinction of up to 1 million species and place hundreds of millions of humans at risk. furthermore, it is predicted that around 50,000 tw of electrical energy will be required by 2050 [4], underscoring the urgent need for sustainable energy solutions. immediate and sustainable interventions are essential to replenish these supplies and mitigate the associated environmental impacts to avert the impending energy crisis. renewable energy (re) offers a sustainable alternative to finite resources, meeting growing energy demands while reducing environmental impact. among re sources, the transition to geothermal energy (ge) is particularly crucial due to its stability and reliability. unlike solar and wind, ge provides a consistent energy supply, unaffected by weather variability, making it a crucial solution for seamless grid integration and long-term decarbonization. despite being underutilized, ge’s potential to deliver dependable, lowemission power highlights its importance in advancing the energy transition and ensuring sustainable energy security. ge originates from the vast reservoir of thermal energy generated by the radioactive decay of minerals and the primordial heat from the earth's formation. as a non-variable and renewable resource, ge can be used for baseload power generation, reducing overdependence on fossil fuels and hydropower plants [5]. compared to other energy sources, the earth has an essentially infinite supply of this energy stored within its core. this thermal energy is particularly abundant around the pacific ocean, including the northern hemisphere, where active volcanic regions contribute to significant geothermal resources. effective exploitation of these resources requires thorough exploration. research into ge exploration has heavily relied on standard methods such as geophysical [6], geospatial [7], and electromagnetic [8] techniques. however, unanticipated geological intricacies and reservoir characteristics that were not accounted for during the prediction process could present operational obstacles or safety hazards during extraction, exacerbating the effects of erroneous predictions. therefore, ensuring precise and reliable projections of geothermal reserves is critical for increasing energy production, efficiency, enhancing safety, and ensuring the long-term sustainability of this significant re resource. remote sensing (rs) has the potential to accurately identify locations showing geothermal anomalies [9]. prospective geothermal sites can be found in remote or challenging terrains due to rs, which provides an inexpensive and non-invasive way to explore large, often inaccessible areas. there are distinct types of rs techniques, such as aster and tir, applicable in various circumstances. by utilizing satellite or aerial sensors, rs can detect subtle temperature anomalies and surface manifestations associated with geothermal activity. evaluating and selecting rs techniques involves multiple, often conflicting criteria, necessitating a multicriteria model for efficient use of ge. multi-criteria decision making (mcdm) can help identify the most beneficial alternative by balancing these conflicting factors [10]. optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 535 the limited availability of non-renewable resources presents a significant challenge in balancing future energy demand and production. this underscores the urgent need for more sustainable energy solutions. however, there is a notable gap in the existing literature regarding the optimal selection of rs techniques for geothermal reservoir exploration. current methods for evaluating rs techniques often fail to fully capture the complexities of expert preferences, especially across large and diverse geographical areas. additionally, mcdm models have limitations, including an inability to effectively convey expert assessments in natural language, and they tend to be time-consuming and inefficient. as a result, these models struggle to provide a decision order that reflects the real-world, in-depth process of human decision making (dm), hindering their effectiveness in selecting the most appropriate rs techniques for geothermal exploration. the motivation for this study is to identify the optimal rs technique for geothermal reservoir exploration. to achieve this, a novel hybrid mcdm paradigm is introduced, applying a fuzzy approach to assess various rs techniques. however, choosing the ideal solution in mcdm is challenging due to inherent uncertainties, such as incomplete or ambiguous information, dynamic external factors, and the subjective judgments of decisionmakers. by addressing these challenges, this study aims to improve decision-making processes, facilitating the efficient and accurate exploration of geothermal resources. this is crucial for advancing ge as a practical and reliable renewable energy source. the novelty of this study lies in the development of a comprehensive decision-making framework that integrates the t-spherical dual hesitant fuzzy (t-sdhf) set to handle high uncertainty and hesitation, employs the logarithmic percentage change-driven objective weighting (lopcow) method for precise criterion weighting, and utilizes the multiple triangles scenarios-ii (mutriss-ii) technique for accurate alternative ranking. this is the first study to combine lopcow and mutriss-ii for evaluating and selecting remote sensing techniques in geothermal reservoir exploration, offering a balanced and reliable assessment. the proposed approach is rigorously validated through robustness, sensitivity, and comparison analyses, ensuring its effectiveness in addressing complex decisionmaking challenges under imprecision and ambiguity. 2. literature review in the multifaceted and constantly changing world of today, decision-makers face a slew of issues that necessitate an organized and well-informed process [11]. a systematic framework for handling decision issues combining numerous objectives, various criteria, and dynamic preferences is provided by mcdm techniques [12]. traditional methods such as dematel [13], vikor [14], topsis [15], promethee [16], and electre [17] laid the groundwork by structuring decision problems, organizing options, and establishing preference relationships. however, as decision-making scenarios grew more intricate, newer techniques like waspas [18], comet [19] and fradar [20] emerged. these advanced approaches better handle competing goals and incorporate subjective assessments from multiple decision-makers, providing a balanced and flexible framework that enhances decision quality and inclusiveness. some of the subjective weighting techniques include ahp [21], swara [22] while objective methods include entropy [23] and merec [24]. besides these mcdm models, ecer and pamucar [25] introduced a novel objective weighting technique named lopcow. its 536 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang benefits include removing gaps in data because of the size, producing more realistic weightings, and taking into account positive as well as negative data when weighting. tadic et al., [26] used modified fuzzy topsis and fuzzy copras methods for evaluation and ranking of electric vehicles. nila et al., [27] employed triangular fuzzy lopcow-fucom technique for the evaluation and selection of third-party logistics service. ulutas et al., [28] used grey numbers based lopcow framework for the evaluation of third-party logistic providers for automobile production firms. biswas and joshi [29] compared the post-listing performance of ipos in the indian stock market (ism) using lopcow, highlighting that market performance is not solely driven by fundamental efficiency and equity ownership has little impact. the study suggested that other factors contribute to ipo performance beyond these traditional metrics. every mcdm technique that has been devised so far has encountered some restrictions such as the subjective nature of dm, reliance on data quality, and the challenge of model complexity [30]. to address these challenges, zakeri et al., [31] presented a novel mcdm approach, mutriss-ii that could compute the areas filled by options in n-dimensional space. the material selection challenges were addressed using this mutriss approach. making decisions frequently requires navigating subjectivity and ambiguity. fuzzy-based mcdm techniques have been introduced to address unpredictability and inaccuracy in dm systems [32]. multiple fuzzy sets (fs) have been proposed so far in the literature, including intuitionistic fs, interval-valued fs, neutrosophic fs, picture fs [33], bipolar fs, and lineardiophantine fs. however, among all of these fs, an innovative fs, spherical fs, introduce by kutlu and kahraman, has piqued the interest of academics due to the benefits it offers [34]. bonab et al., [35] utilized spherical fs and choquet integral to evaluate autonomous cars for the logistics sector. nguyen et al., [36] assessed the wire and cable industry's governance, social, and environmental performance using the waspas and spherical fuzzy dea-ahp approaches. gamal et al. [37] developed an ecologically sound computational technique for evaluating the optimal energy storage systems by integrating ahp-macont in a spherical fuzzy environment. spherical linear diophantine fs and its accompanying aggregated geometric and arithmetic operators were developed by riaz et al. [38] in a study, and they are employed in many real-world applications, such as network systems, voting, digital image processing and so on. further kakati et al. [39] introduced rectified complex t-sf set employing the dombichoquet integral operator to diagnose diabetic retinopathy through fundus images. later, alamoodi et al., [40] integrated 2-tuple linguistic t-sf set and entropy-fdosm for the effective appraisal of electric bus. conventional models typically require experts to provide single values for membership parameters, which can be restrictive and less expressive, particularly in situations with competing criteria or uncertain evaluations. to address these issues, the t-sdhf set combines the t-spherical fuzzy (t-sf) and dual hesitant fuzzy (dhf) sets, incorporating positive, negative, and neutral membership functions. this integration allows t-sdhf sets to capture both degrees of membership and non-membership simultaneously, giving experts a more flexible and realistic way to convey hesitation and preferences. by doing so, the t-sdhf set improves the robustness of dm processes, providing a structured framework that can more accurately reflect expert input and enhance the reliability of decision outcomes, even in intricate and high-dimensional dm scenarios. mcdm techniques have proven effective in diverse fields such as business, engineering, healthcare, and energy, addressing complex decision-making challenges [41]. their adaptability optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 537 and versatility make them suitable for various decision-making scenarios. mostafaeipour et al., [42] used the fuzzy-delphi-ahp methodology to investigate the difficulties in ge extraction in india. using the swara-aras technique, puppala et al., [43] investigated the location selection for geothermal projects in afghanistan in 2022. ghose et al., [44] then used triangular fuzzy topsis technique to evaluate varied re in india. gudala et al., [45] used horizontal wells to analyse the puga geothermal reservoir. a triplet of horizontal wells was evaluated and improved for co2 plume ge harvesting by nematollahi et al., [46]. in their study, ngethe et al., [47] examined the selection of ge resources for direct use in kenya. in the northeastern region of anatolia, zorlu and dede [48] assessed the possible geoheritage in glacial and periglacial deposits. 3. preliminaries 3.1. dual hesitant fuzzy set a dual hesitant fuzzy (dhf) set defined on the universal set ℧ is represented by,  , ( ), ( ) |x h x g x x =  (1) where, h(x) and g(x) are two sets of some values in [0,1] denoting the possible grades of membership and non-membership of the member x ϵ ℧ to the set ρ respectively, satisfying the condition 0 , 1,0 1   + +   +  (2) where ζ ϵ h(x), η ϵ g(x), ζ+ ϵ h+(x) = uζ ϵ h(x) max {ζ}, η+ ϵ g+(x) = uη ϵ g(x) max {η} for all x ϵ ℧. for ease, the pair (h(x), g(x)) is termed as dhf element denoted by ξ = (h,g), satisfying the condition, ζ ϵ h, η ϵ g, ζ+ ϵ h+ = uζ ϵ h max {ζ}, η+ ϵ g+ = uη ϵ g max {η}, 0 ≤ ζ , η ≤ 1, 0 ≤ ζ+ + η+ ≤ 1. 3.2. t-spherical fuzzy set a t-spherical fuzzy (t-sf) set on ℧ is stated as,  , ( ), ( ), ( )t t tt x x x x x  =  (3) where, αt (x) : ℧ → [0,1], βt (x) : ℧ → [0,1] and γt (x) : ℧ → [0,1] signifies the positive grade of membership (pgm), abstain grade of membership (agm) and negative grade of membership (ngm) to t respectively, fulfilling the condition, ( ) ( ) ( )0 ( ) ( ) ( ) 1 n n n t t tx x x   + +  (4) for some z+ n with the triplet (α, β, γ) known as t-sf elements. 538 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang 3.3. t-spherical dual hesitant fuzzy set a t-spherical dual hesitant fuzzy (t-sdhf) set on ℧ is defined by,  , ( ), ( ), ( )x x x x x     =  (5) where ϕϖ (x), µϖ (x), ψϖ (x) are three sets of some possible different values between [0,1] signifying pgm, agm and ngm of the member x ϵ ℧ to the set ϖ respectively, with the condition, ( ) ( ) ( )0 max ( ) min ( ) min ( ) 1 n n n x x x     + +  (6) here ϕϖ (x) = max {ξϕ}, µϖ (x) = max {ξµ} and ψϖ (x) = max {ξψ}, in which ξϕ, ξµ and ξψ are dhf elements for some z+ n. for convenience, the triplet ϕϖ (x), µϖ (x), ψϖ (x) is termed as t-spherical dual hesitant number (t-sdhfn) denoted by ρ = (ϕ, µ, ψ). the refusal grade of membership is defined as, ( )  ( )  ( )   1 1 max ( ) min ( ) min ( ) n n n n x x x      = − + +    (7) 3.4. score and accuracy function the score function s(ρ) and accuracy function p(ρ) of t-sdhfn ρ are defined by, ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) 2 n n n n n n i i i in h n g n h n g s                  = = = =       + − − −            =     (8) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) 2 n n n n n n i i i in h n g n h n g p                  = = = =       + − + −            =     (9) where n(hϕ), n(gϕ), n(hψ), and n(gψ) represent the number of elements contained respectively in ϕ and ψ for some z+ n. the value of the s(ρ) ϵ [0,1]. consider ρ1 and ρ2 be the two t-sdhfns. let s(ρ1) and s(ρ2) be the score functions with p(ρ1) and p(ρ2) as the accuracy functions of ρ1 and ρ2 respectively. then if s(ρ1) ˃ s(ρ2) then ρ1 ˃ ρ2 if s(ρ1) = s(ρ2) then either p(ρ1) ˃ p(ρ2) then ρ1 ˃ ρ2 or p(ρ1) = p(ρ2) then ρ1 = ρ2 4. proposed methodology in this study, lopcow method is applied to calculate the criterion weights and mutriss-ii method is applied to rank the alternatives. the graphical representation of this framework is given in fig. 1. theoretical explanations of these methods are presented below. optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 539 fig. 1 proposed mcdm framework 4.1. logarithmic percentage change-driven objective weighting method step 1: create the initial decision matrix, ij m n d   =   (10) here m and n represent the number of alternatives and criteria of the complex problem respectively. each performance value ℘ij determined by the relevant experts are based on the t-sdhf set provided in eq. (5). then the t-sdhf decision matrix is defuzzified employing the score function provided in eq. (8). step 2: the normalized decision matrix is determined by employing linear max-min normalization technique using the following equations, min max min max max min for beneficial criteria, for cost criteria ij j i i ij j ij i i r   −   − =   −  − (11) step 3: the pv for every criterion is determined by taking the natural log of the mean square value and expressing it as a percentage of the standard deviation. this stage aids in reducing the weights' unequal distribution. as a result, pv is determined as, 2 1ln 100 m ij i  =       =         (12) 540 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang where the standard deviation and number of alternatives are denoted by σ and m respectively. step 4: the relative significance of each criterion is determined using the equation given below, 1 ij n ij j=  =  (13) 4.2. multiple triangles scenarios-ii method the algorithm for the ith alternative in the proposed mutriss-ii method is given as follows: step 1: the normalization of the matrix is done using the equation provided below, 1 1 for beneficial criteria max min for cost criteria ij ij j n ij ij j n ij r        =     (14) step 2: construct the following equation by placing each ℘j of the ith alternative in descending order, max min max max 1 min n min : { , , , , }ij mn ij mn mn mn mn mn− −   → → =     (15) step 3: the subsequent equation is employed to compute the angles of each ith alternative, ( ) 1 1 1 1 1( ) ( ) 90, 1, , 1 ij ij ij ijj w w w w j n     − − −   −   −=  = − (16) 1ij ij w w    − step 4: calculate the overall score of the alternatives by calculating the areas that the alternatives occupy using the formula below, 1 1 sin 0.5 ij ij n i j j av w w     − = = (17) in line with avi, the alternatives are arranged in descending order. 5. case study in this section, we have demonstrated the proposed novel hybrid mcdm approach through the selection of the most beneficial remote sensing technique for the exploration of geothermal reservoirs in india. india, the third-largest global power consumer after the us and china, has an annual demand of 1.54 trillion kwh, with over 45% met by fossil fuels, 26% by petroleum, and the rest from biomass and re sources [49]. the country’s large population drives increasing energy needs. research by the indian institute of science reveals 86 gw of installed re optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 541 capacity, including 34 gw from solar and 37.5 gw from wind power [49]. while solar and wind provide significant returns, their output can be inconsistent. a case study is conducted for the discovery of promising and untapped geothermal reservoirs in the indian region. to make sure that the dm process is strategic, and in line with the objective of identifying the most promising geothermal reservoir, an expert with insights in the pertinent field is selected to advise and validate the various remote sensing choices. prior to extraction of the energy, a number of essential factors are taken into account to make an informed choice, the description of which is shown in fig. 2. fig. 2 description of the criteria the following gives a brief description about the remote sensing alternatives. light detection and ranging (lidar) (o1) the lidar remote sensing technique operates by producing laser pulses from an aerial or terrestrial platform and measuring the time it takes for the pulses to return after striking the earth's surface. lidar sensors generate precise three-dimensional point clouds, allowing for thorough mapping of the earth's topography and surface features. by providing realistic terrain models and recognizing structural patterns, lidar aids in mapping fractures, fault lines, and other subsurface features that may indicate the presence of geothermal reservoirs. thermal infrared (tir) (o2) this technique detects geothermal spots by gathering and analyzing thermal radiation released by the earth's surface. the approach is based on sensors that capture infrared wavelengths linked with temperature fluctuations. the presence of underlying heat in geothermal locations causes various thermal signatures on the earth's surface. tir sensors detect these temperature variations, allowing for the exact identification of prospective geothermal areas. tir sensing offers beneficial insights into the thermal properties of the landscape by measuring the heat emitted from the surface, allowing for the recognition and mapping of regions with temperatures that are elevated, which indicate underlying geothermal activity. 542 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang radio detection and ranging (radar) (o3) – this technique functions by radiating microwave pulses at the earth's surface and capturing the signals that bounce back. radar is very valuable because it can penetrate clouds and function in all-weather situations. radar sensors can detect small surface deformations and topographical changes caused by subsurface geothermal activity. these modifications may include variations in ground elevation or surface roughness. the radar data can reveal these underlying structures, providing insights into feasible geothermal reservoirs. advanced spaceborne thermal emission and reflection radiometer (aster) (o4) – aster operates using multispectral and thermal infrared capabilities. in the context of geothermal detection, aster's thermal infrared bands (8-12 micrometers) are very relevant. these bands allow for the measurement of surface temperatures, which allows for the detection of thermal anomalies associated with probable geothermal locations. aster helps to identify and characterize subsurface heat sources by evaluating temperature changes and surface features. aster imagery's high degree of spatial accuracy makes it easier to identify geological structures and features essential to geothermal exploration. visible and near-infrared to shortwave infrared (vnir, 350 to 1300 nm-swir, 1300 to 2500 nm) (o5) – this remote sensing technology captures electromagnetic radiation in certain spectral bands extending from the visible to the shortwave infrared regions. surface temperatures, vegetation and minerals all have distinct spectral signatures in these bands. vnir bands are sensitive to differences in vegetation health and land cover, whereas swir bands are sensitive to temperature-related characteristics. 5.1. determination of criteria weights the developed fusion fuzzy mcdm framework is employed to determine the weights of each criterion and probable alternative. an expert in the pertinent field evaluates each of the alternatives oi, i = 1, ..., 5, for the circumstance in hand in accordance with each of the attributes sj, j = 1, ..., 6, and offers their assessment of performance in the form of a t-sdhfn. the steps adapted from the weighting and ranking technique is as follows: step 1: the expert evaluates each alternative's performance using the t-sdhfn specified in eq. (5). table 1 represents the initial t-sdhf matrix. table 2 presents the defuzzified score matrix using the expression given in eq. (5). an illustration of the score function of the first element (that is, ℘11) is shown below, 3 3 1 1 1 1 1 (0.52 0.26 0.15) (0.24 0.3) (0.35 0.5) (0.22 0.15) 3 2 2 2 0.4931 2      + + + − + − + − +          = step 2: the defuzzified matrix is normalized using the eq. (11) for beneficial and cost criteria respectively. 11 0.4931 0.3862 0.8665 0.5096 0.3862 r − = = − optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 543 table 1 the t-sdhf decision matrix s1 s2 s3 o1 (<{0.52,0.26,0.15},{0.24,0.30}>, <{0.52,0.36,0.66},{0.12,0.09}>, <{0.35,0.50},{0.22,0.15}>) (<{0.39,0.29},{0.13}>, <{0.42},{0.14}>, <{0.26,0.31,0.45},{0.25,0.14,0.45}>) (<{0.64,0.58,0.47},{0.15,0.21}>, <{0.15},{0.11}>, <{0.33,0.35},{0.14}>) o2 (<{0.45,0.67},{0.28}>, <{0.55,0.05},{0.31,0.43}>, <{0.25},{0.11}>) (<{0.32},{0.19}>, <{0.15,0.25},{0.12,0.15}>, <{0.45,0.57,0.59},{0.17,0.35}>) (<{0.34,0.67},{0.15,0.12,0.29}>, <{0.15,0.35},{0.19}>, <{0.25,0.35,0.45},{0.30,0.40,0.30} >) o3 (<{0.52,0.56},{0.33}>, <{0.62,0.53},{0.10,0.05}>, <{0.55},{0.36}>) (<{0.43,0.37},{0.32}>, <{0.25,0.55},{0.13}>, <{0.47},{0.47,0.49}>) (<{0.35},{0.11}>, <{0.47,0.15},{0.15}>, <{0.51,0.64},{0.15}>) o4 (<{0.39,0.41},{0.31}>, <{0.35,0.55},{0.21,0.34}>, <{0.25,0.34,0.48},{0.05,0.09}>) (<{0.25},{0.12}>, <{0.15,0.35},{0.05,0.10}>, <{0.59,0.32},{0.36}>) (<{0.15,0.39,0.25},{0.12,0.27}>, <{0.61,0.21},{0.05}>, <{0.42,0.34},{0.19}>) o5 (<{0.47,0.49},{0.26}>, <{0.69,0.15},{0.08}>, <{0.74},{0.20,0.05,0.11}>) (<{0.35},{0.15}>, <{0.15,0.05},{0.60,0.40}>, <{0.40},{0.57}>) (<{0.63,0.34},{0.11,0.09,0.13}>, <{0.21,0.15},{0.14}>, <{0.35,0.45},{0.06,0.21}>) s4 s5 s6 (<{0.68,0.55},{0.11}>, <{0.25,0.35},{0.17}>, <{0.37,0.15},{0.19,0.11}>) (<{0.72,0.57},{0.16}>, <{0.15},{0.04}>, <{0.24,0.39},{0.14}>) (<{0.69,0.75},{0.11,0.15}>, <{0.55},{0.14}>, <{0.51,0.15},{0.06,0.14}>) (<{0.91,0.51},{0.04}>, <{0.34},{0.11}>, <{0.52,0.49},{0.24}>) (<{0.65,0.54,0.61},{0.14,0.11}>, <{0.46},{0.22}>, <{0.62,0.49},{0.15,0.09}>) (<{0.45,0.49,0.51},{0.21,0.14}>, <{0.45,0.35},{0.09,0.12}>, <{0.54},{0.30}>) (<{0.59,0.43},{0.17}>, <{0.39,0.47},{0.11}>, <{0.43,0.41},{0.17,0.15,0.11}>) (<{0.41,0.65},{0.26}>, <{0.45,0.55},{0.14,0.16}>, <{0.54},{0.08,0.31}>) (<{0.61},{0.11}>, <{0.45,0.55},{0.04,0.16}>, <{0.65},{0.11,0.15}>) (<{0.69,0.66,0.42},{0.11,0.21}>, <{0.55,0.65},{0.16,0.17}>, <{0.41,0.33,0.27},{0.14}>) (<{0.61,0.59},{0.17}>, <{0.65},{0.20}>, <{0.61},{0.12}>) (<{0.65,0.74},{0.18}>, <{0.65,0.55},{0.17,0.03}>, <{0.45,0.55,0.19},{0.13}>) (<{0.79,0.57},{0.59}>, <{0.35,0.55},{0.33,0.01}>, <{0.15,0.25},{0.21}>) (<{0.68,0.54},{0.31}>, <{0.35},{0.21}>, <{0.25,0.15},{0.10,0.16}>) (<{0.65,0.75},{0.14,0.15}>, <{0.45},{0.11}>, <{0.15,0.45},{0.22,0.27}>) table 2 the defuzzified t-sdhf decision matrix s1 s2 s3 s4 s5 s6 o1 0.4931 0.5045 0.5086 0.4995 0.5544 0.5966 o2 0.5096 0.4905 0.5161 0.6411 0.5124 0.5077 o3 0.5012 0.5003 0.4685 0.5091 0.4893 0.4922 o4 0.4886 0.5007 0.4967 0.5360 0.4809 0.5588 o5 0.3862 0.5065 0.5171 0.5744 0.5133 0.5854 step 3: the pv of each criterion is computed using the eq. (12) and is provided in table 3. the pv of first criterion is given below, 1 0.8134 ln 100 79.4613 0.3674    =  =    544 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang step 4: table 3 shows the relative significance of each criterion which is calculated using eq. (13). 1 79.4613 0.2476 320.8675  = = table 3 standard deviation, percentage value and significance of criteria criteria σ pv relative weight s1 0.3674 79.4613 0.2476 s2 0.3452 38.9030 0.1212 s3 0.3705 73.4468 0.2289 s4 0.3643 35.5419 0.1108 s5 0.3476 41.7180 0.1300 s6 0.3975 51.7965 0.1614 5.2. identifying the rank of alternatives step 1: a t-sdhf decision matrix is constructed in the form of eq. (10) (refer step 1 of lopcow method). the defuzzified matrix is normalized using the eq. (14) for beneficial and cost criteria respectively. for instance, 11 0.4931 0.9677 0.5096 r = = step 2: each ℘j of ith alternative is arranged in their descending order using eq. (15). table 4 shows the example of alternative-1. that is, {s6, s5, s3, s2, s1, s4}. step 3: the angles of each triangle is computed using the eq. (16). table 4 shows the angles of each triangle for alternative-1. ( )0.1614 (1/13) 0.1557 90 17.3678j =    = step 4: table 4 shows overall areas occupied by alternative-1 using the eq. (17). the angle and area occupied by rest of the alternatives are computed in the same way as alternative-1. table 5 shows the overall score and ranking of each alternative. for instance, ( )1 1 1 sin(17.3978) 0.5 0.0681 0.2131 0.0562 0.1961 0.6831av =    + + + + = table 4 area occupied by alternative 1 criteria alternative-1 weight angle θj radian area s6 1.0000 0.1614 s5 1.0000 0.1300 1.2416 17.3978 0.3038 0.1496 s3 0.9837 0.2289 0.5680 7.9591 0.1390 0.0681 s2 0.9722 0.1212 1.8879 26.4549 0.4619 0.2131 s1 0.9677 0.2476 0.4896 6.8603 0.1198 0.0562 s4 0.7792 0.1108 2.2357 31.3279 0.5470 0.1961 0.6831 optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 545 table 5 rank and area occupied by each alternative alternative avi rank o1 0.6831 3 o2 0.7187 1 o3 0.6252 5 o4 0.6532 4 o5 0.6914 2 6. results and discussion in this study, the optimal rs technique for the maximal energy detection of the geothermal reservoir is explored through novel hybrid fuzzy mcdm under a t-sdhf environment. the spherical framework of the t-sdhf set enabled a smoother transition between varying levels of uncertainty. by incorporating the flexibility of dhf set, which accommodated multiple membership and non-membership degrees, the t-sdhf set accurately depicted complex and multidimensional data, by allowing choice-makers to capture intricate interconnections within a decision context. in scenarios where standard fuzzy sets or hf sets, which address only membership hesitation, fall short, the t-sdhf set provides a more comprehensive solution. tsdhf set-based techniques additionally enhanced dm resilience by providing a systematic framework for dealing with ambiguities and vagueness, hence increasing the dependability and stability of decision outputs. the case study in this research involved six criteria and five alternatives. the significance of each criterion was computed using the t-sdhf lopcow method, and the ranking of the alternatives was done using the mutriss-ii method. the lopcow approach leverages objective information to generate the criteria weights. the criterion weights have a relatively even distribution. furthermore, this technique proves to efficiently handle an enormous number of parameters and alternatives. contrarily, mutriss-ii gets beyond the shortcomings of the existing mcdm approach, which include inconsistent ranking, identifying several possibilities as preferred alternatives, and failing to consider the input of experts during the dm process. the versatility of the suggested hybrid technique, as well as its ability to give precise information, contribute to its usefulness in assisting geothermal exploration dm. furthermore, the proposed approach aims to provide accurate solutions using robust but simple algorithmic procedures. from the results of the t-sdhf lopcow method, it is found that spatial resolution (s1) obtained the highest weightage of 0.2476, followed by spectral bands (s3) with a value of 0.2289, and thirdly temporal resolution (s6) with 0.1614. the spatial resolution determined the level of clarity in the image. it accurately detected tiny features such as temperature anomalies and surface manifestations, which were critical for identifying probable geothermal sites. on the other hand, even though the area with coverage (s4) of the distinct remote sensing techniques were significant, this criterion obtained the least value of 0.1108. tir (o2) obtained 0.7187 and constituted the leading remote sensing technique for identifying geothermal reserves. tir has the ability to detect small temperature variations, which is critical for efficient and targeted geothermal exploration and resource assessment. tir sensing provides unique insights into the thermal features of the landscape by measuring heat released from the surface, allowing for the identification and mapping of 546 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang areas with elevated temperatures that indicate underlying geothermal activity. the tir remote sensing for detecting potential geothermal sites is shown in fig. 3. different triangles covered by tir are displayed in fig. 4. the second favored technique was vnir-swir (o5), which achieved a value of 0.6914. these sensors are very useful for studying the earth's surface features. the technique uses sensitivity to detect minor changes in surface composition and temperature that indicate geothermal activity. vnir-swir remote sensing identifies and maps probable geothermal energy locations by analyzing reflectance patterns and thermal anomalies. fig. 3 tir detecting heat anomalies even though lidar (o1) is extremely accurate for topographic mapping, it is limited by its reliance on direct line-of-sight. this means that elements obscured by dense foliage or structures may not be fully recorded. furthermore, lidar data collecting and processing can be expensive and resource-intensive, providing obstacles for projects with little funding. this puts lidar in third place for geothermal location detection. aster has a limited revisit frequency, which means that revisit times might be relatively long, ranging from weeks to months depending on the location. this infrequent visitation complicates the monitoring of dynamic geothermal phenomena that may change rapidly over shorter timescales. whereas radar's (o3) poor ability to penetrate dense foliage limits its effectiveness in heavily forested areas. furthermore, radar often has lower spatial resolution than optical sensors such as aster (o4). this places aster fourth, with radar being the least recommended alternative. despite tir's high ranking in geothermal resource exploration, challenges persist due to the ill-posed nature of surface temperature data, which is constrained by limited spectral information and assumptions about air conditions or emissivity. to address these limitations, it is crucial to integrate multiple remote sensing methods such as optical, infrared, and radar and employ high-spectral tir imaging for comprehensive data fusion [50]. optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 547 fig. 4 different triangles covered by tir this approach enhances the accuracy and consistency of geothermal resource exploration by mitigating issues like uneven surface temperatures and atmospheric interference. combining data from various sensors and increasing observation frequencies can provide more precise insights and capture complex geothermal resource characteristics. this study benefits stakeholders and the government by highlighting the risks of inaccurate geothermal reserve estimation. overestimating reserves may lead to excessive costs and environmental impact, while underestimating them could cause premature operation shutdowns and financial losses. erroneous forecasts can also hinder reservoir management, threatening long-term sustainability. 6.1. comparative analysis of different ranking techniques every mcdm technique has a unique way for carrying out dm examination. in this section, the outcomes of the proposed hybrid mcdm methodology are compared to those of existing mcdm techniques. we compared the rankings of our proposed technique to distance-based (topsis) [15], score-additive (copras) [51], trace-based (mcrat) [52], perimeter similarity (raps) [53], outranking (promethee-ii) [16], and aggregated sum product (waspas) [18] methods. from table 6, it is seen that o2 consistently ranks as the highest performing alternative in most methods. for instance, in the topsis method, o2 has the highest score, compared to o1, o3, o4, and o5, indicating that it is the most optimal choice based on relative closeness to the ideal solution. similarly, in copras, o2 achieves a perfect score of 1, outperforming the other alternatives. in contrast, the mcrat method shows a smaller gap between alternatives, with o1 (0.1780) and o2 (0.1787) having nearly identical values, but o2 still slightly edges ahead. for the raps method, o2 remains the best performer, while o1 is worst. in the promethee-ii method, o2 also outperforms all other alternatives with a positive value of 0.038, while o3, o4, and o5 have negative values, indicating poorer 548 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang relative performance. lastly, in the waspas method, o2 marginally outperforms o1, but the difference between the alternatives is minimal overall. table 6 comparison of the proposed model with the existing models topsis copras mcrat raps promethee-ii waspas o1 0.6956 0.9932 0.1780 0.9692 0.0314 0.4802 o2 0.7239 1.0000 0.1787 0.9753 0.0380 0.4829 o3 0.5480 0.9366 0.1688 0.9244 -0.0370 0.4529 o4 0.6355 0.9647 0.1730 0.9454 -0.0033 0.4664 o5 0.3710 0.9438 0.1664 0.9117 -0.0291 0.4560 fig. 5 comparison of the proposed technique with existing models the variation in rankings across these methods highlights the sensitivity of the results to the chosen decision-making approach, emphasizing the need to select the method that best aligns with the decision context and priorities. the results reveal that the integrated mcdm framework produces more flexible solutions than the individual techniques. however, in contrast to the aforementioned mcdm methodologies, the suggested strategy is compatible for our application. fig. 5 shows a grouped bar plot illustrating the ranks acquired using various mcdm approaches. to go deeper into these rankings, spearman's rank correlation coefficient is employed. fig. 6 shows the results of the correlation coefficient. optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 549 fig. 6 spearman’s rank correlation coefficient 6.2. sensitivity analysis in this section, we evaluated the level of sensitivity of our suggested system. the coherence of the findings obtained with the proposed approach is assessed by varying the significance level of each criterion. to evaluate the reliability of the gathered results, we examine two cases. in the first case, the beneficial criteria, high desired value, is set to one, while the cost criteria, least desired value, is assigned to zero. that is, s1, s3, s4, s5, s6, which are considered as the beneficial criteria are given the value of 0.2 and the cost criterion, namely, (s2) is asset to zero. in the second case, both the beneficial and non-beneficial criteria are set to be equal. here, all the criteria are assigned equal values of 0.167. table 7 and fig. 7 illustrate the impact of adjusting significant parameters on the ranking of alternatives in two different cases, highlighting the responsiveness of the model to changes in criterion weighting. in case-i, the alternatives are ranked as follows: o4 is the top choice followed by o5, o3, o1, and o2 in the last position. in case-ii, however, the rankings change considerably: o4 remains the highest ranked, but o2 rises to the second position, followed closely by o3, o5, and o1 drops to the lowest rank. these shifts demonstrate that even slight variations in the weighting of criteria can lead to a reordering of alternatives, emphasizing the model’s sensitivity to the values assigned to different factors. this investigation underlines the importance of proper weight adjustment to ensure that the chosen alternative aligns with the desired priorities in each scenario. 550 m. sandra, s. narayanamoorthy, k. suvitha, d. pamucar, d. kang table 7 results of the sensitive analysis alternative case i case ii avi rank avi rank o1 0.0170 4 0.02146 5 o2 0.0164 5 0.02151 2 o3 0.0181 3 0.02149 3 o4 0.0233 1 0.02152 1 o5 0.0200 2 0.02148 4 fig. 7 radar representation of the outcome of sensitive analysis further, spearman rank correlation coefficient was conducted in the study. it showed that the proposed rank showed a negative correlation of -0.5 with case-i, indicating a moderate inverse relationship between the two. similarly, the proposed rank and case-ii exhibited a negative correlation of -0.1, suggesting a very weak inverse association. in contrast, the correlation between case-i and case-ii was positive, with a value of 0.3, indicating a weak positive relationship between the two cases. these results provide insights into how the different cases and the proposed rank interact and highlight varying degrees of association among them. 7. conclusion as the entire world grapples with the challenges of a burgeoning population, the transition to re emerges as a critical strategy for ensuring a robust and sustainable future. the need to investigate different sources of re is critical for producing energy in situations where the conventional high energy return renewable resources become inconsistent and unreliable. in this study, geothermal reserves were identified using the most promising rs approach through a unique hybrid fuzzy mcdm technique. one of the distinguishing features of the ge is its reliability and consistency. geothermal power generation, unlike optimizing non-invasive remote sensing for geothermal exploration with t-spherical dual hesitant... 551 other re sources such as solar or wind, is not weather-dependent. it delivers continuous and baseload power, making it an important and consistent contributor to the global energy mix. rs technologies allow for the detailed mapping of surface temperatures, geological formations, and vegetation stress, which are indicators of geothermal activity. this minimizes the need for extensive ground-based surveys and drilling, reducing environmental disruption and lowering exploration costs. additionally, continuous monitoring through rs supports the efficient management of geothermal fields, ensuring the long-term viability and minimal ecological impact of geothermal energy projects. the t-sdhf-lopcow-mutriss-ii model is used to assess the difficulty of selecting an appropriate remote sensing technique for detecting probable geothermal reserves in india. understanding the characteristics of geothermal spots prior to extraction is critical for drilling activities to reduce the risk of resource depletion or reservoir damage, thereby contributing to the overall sustainability, efficiency, and responsible management of ge resources. therefore, five rs techniques were investigated under several critical criteria in a newly introduced t-sdhf environment. the combination of the t-sf set with the dhf set has proven to be a promising technique for dealing with ambiguity as well as vagueness in the dm process. the lopcow approach assessed the relative importance of each crucial criterion. the lopcow results revealed that spatial resolution and area with coverage were the most and least critical parameters for precisely locating the site for maximum energy extraction. mutriss-ii method uses triangles to express criteria-based performance values and prioritizes remote sensing approaches based on the area of the shapes formed by these triangles. the angles between the triangles are dynamically calculated in mutriss-ii. this technique revealed that the tir approach improved the ability to identify minor temperature differences in a concise manner, allowing for more efficient exploration and usage of geothermal resources. this approach also offered useful information on the thermal properties of the ground. according to the sensitivity analysis, the suggested method is sensitive to the weights of the attributes, and the comparative analysis confirmed that our established structure is capable of providing a reliable remote sensing technique for potential geothermal reserves. this study, like many other scientific investigations, has some limitations. the data focused solely on prospective rs techniques for exploring geothermal resources in the indian region, and only one dm expert was involved, limiting the breadth of informed dm. additionally, the novel mutriss-ii approach, which utilized analytic geometry to determine areas filled by alternatives in n-dimensional space, struggled with handling unknown variables in the application. while tir is the top strategy for exploring geothermal resources, it might be challenging to obtain data just from this method. surface temperature estimation is challenging due to limited spectral information and assumptions about air conditions or emissivity, resulting in inaccurate results. to overcome this constraint, varied rs data must be integrated. future research could broaden the scope by incorporating data from countries across various regions of the world and enhancing the analysis process by including group knowledge. interdisciplinary research is crucial for exploring and evaluating geothermal resources. collaboration among geology, geophysics, geochemistry, geography, and hydrology enhances resource detection accuracy and provides a comprehensive assessment of potential, providing reliable foundations for energy development. furthermore, the newly developed set 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turkiye case, international journal of disaster risk reduction, 111, 104744. 52. stević, ž., ersoy, n., başar, e.e., baydaş, m., 2024, addressing the global logistics performance index rankings with methodological insights and an innovative decision support framework, applied sciences, 14(22), 10334. 53. saleh, n., hassan, m.a., salaheldin, a.m., 2024, skin cancer classification based on an optimized convolutional neural network and multicriteria decision-making, scientific reports, 14(1), 17323. facta universitatis series: mechanical engineering vol. 18, no 3, 2020, pp. 419 437 https://doi.org/10.22190/fume200602034p © 2020 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method – fucom-f dragan pamucar 1 , fatih ecer 2 1 department of logistics, university of defence in belgrade, military academy, belgrade, serbia 2 department of business administrative, faculty of economics and administrative sciences, afyon kocatepe university, afyonkarahisar, turkey abstract. values, opinions, perceptions, and experiences are the forces that drive almost each and every kind of decision-making. evaluation criteria are considered as sources of information used to compare alternatives and, as a result, make selection easier. seeing their direct effect on the solution, weighting methods that most accurately determine criteria weights are needed. unfortunately, the crisp values are insufficient to model real life problems due to the lack of complete information and the vagueness arising from linguistic assessments of decision-makers. therefore, this paper proposes a novel subjective weighting method called the fuzzy full consistency method (fucom-f) for determining weights as accurately as possible under fuzziness. the most prominent feature of the proposed method is obtaining the most accurate weight values with very few pairwise comparisons. consequently, thanks to this model, consistency and reliability of the results increase while the processing time and effort decrease. moreover, an illustrative example related to the green supplier evaluation problem is performed. finally, the robustness and effectiveness of the proposed fuzzy model is demonstrated by comparing it with fuzzy best-worst method (f-bwm) and fuzzy ahp (f-ahp) models. key words: fuzzy full consistency method (fucom-f), weighting of criteria, mcdm, subjective weighting, fuzzy bwm, fuzzy ahp received june 02, 2020 / accepted august 18, 2020 corresponding author: fatih ecer department of business administrative, faculty of economics and administrative sciences, afyon kocatepe university, ans campus, 03030 afyonkarahisar, turkey e-mail: fatihecer@gmail.com 420 d. pamucar, f. ecer 1. introduction multi-criteria decision-making (mcdm), which is a very important component of the decision-making theory, is usually divided into two classes with regard to the solution area of the problem, as continuous and discrete. in order to address continuous problems, multi-objective decision-making (modm) methods are adopted. however, discrete problems are solved by using multi-attribute decision-making (madm) methods. nonetheless, mcdm is widely used to describe the discrete mcdm i.e. madm in existing literature [1]. the mcdm methods aim at selecting the best alternative among those available. one of the two main components of the mcdm methods is represented by the weights of the criteria that define the process under consideration. the weights of the criteria express the importance and the effects of the criteria on the evaluation results. criterion weights can be determined subjectively or objectively. moreover, opinions, thoughts, and experiences of experts play a crucial role whilst criterion weights are subjectively determined [2]. as mentioned above, determining the weights of the criteria is one of the key problems that arise in multi-criteria analysis models. the problem of selecting an appropriate method for defining the weight coefficients of criteria is a very important step in the models of mcdm. the impartial determination of weight coefficients and the transformation of stated expert preferences into weight coefficients are basic requirements that are posed before the subjective group of models. if we bear in mind that the weight coefficients significantly influence the outcome of a decision-making process, it is clear that particular attention has to be paid to the models for determining the weights of criteria. numerous authors [3,4,5] agree that the values of the criteria weights are significantly conditioned by the methods of their determination. in addition, there is no agreement on the best method of determining criteria weights. however, in the literature there is an agreement that the weights calculated by certain methods are significantly more accurate than those obtained by expert evaluations. in their study, zavadskas et al. [6] found that the analytic hierarchy process (ahp) is the most commonly used model for determining the weight coefficients of criteria and/or the evaluation of alternatives. one of the benefits and reasons as to why the authors opt for the application of the ahp model is due to the ability to validate results by determining the degree of the model consistency. however, according to some psychological research [7], in the ahp method it is very difficult to perform completely consistent pairwise comparisons over nine criteria since this requires a large number of comparisons (n(n-1)/2). a model that has managed to overcome some of the above-mentioned ahp model constraints is the best-worst method (bwm) [1]. one of the greatest advantages of the bwm is a significantly lower number of pairwise comparisons compared to the ahp, only 2n-3. a smaller number of pairwise comparisons of criteria have a direct impact on higher consistency of the model, i.e. greater reliability of the results. additionally, the application of the bwm is not limited to comparing up to nine criteria as it requires a lower number of comparisons. by forming best-to-others (bo) and others-to-worst (ow) vectors, the data that are more consistent are obtained through the ahp model with a lower number of pairwise comparisons, at the same time. however, one of the problems with the bwm is determination of the optimum values of weight coefficients in the case of major deviations in the degree of consistency. in such situations, rezaei [1] proposes prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 421 the determination of interval values and use of the mean value of intervals as the final value of the weight coefficient. however, there is no guarantee that the central part of the interval will represent the optimal weight coefficients values. the optimum value may be closer to the left or right limit of the interval. in the cases of greater inconsistency of results, the optimum values of weight coefficients are not even covered by the defined interval weight values [8]. one of the newer models for determining the criteria weights, based on the principles of the pairwise comparisons of criteria and the validation of results throughout a deviation from maximum consistency, is the full consistency method (fucom) [9]. the fucom is a model that, to some extent, eliminates the stated deficiencies of the bwm and ahp models. as shown in fig. 1, the advantages that are determinative for the application of the fucom include a small number of pairwise comparisons of criteria (only n-1 comparison), the ability to validate the results by defining the deviations from the maximum consistency (dmc) of comparisons, and appreciation of transitivity during the pairwise comparison of criteria. as with other subjective models for determining the weights of criteria, in the fucom model there is a subjective influence of decisionmakers on the final values of the weights of criteria. this particularly refers to the first and second step of the fucom in which decision-makers rank the criteria according to their personal preferences and perform pairwise comparisons of the criteria ranked. however, unlike other subjective models, the fucom has shown minor deviations in the obtained values of the weights of criteria from the optimum values [9]. moreover, the methodological procedure of the fucom eliminates the problem of redundancy of pairwise comparisons of criteria, which is present in some subjective models for determining the weights of criteria [10, 11, 12]. in addition to being a new model, there are a number of studies in which the benefits of the fucom are exploited. for example, pamucar et al. [8] demonstrated the application of the fucom-mairca multi-criteria model for evaluating the railway crossings. 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 ahp bwm fucom n u m b e r o f c ri te ri a number of pairwise comparisons fig. 1 number of pairwise comparison of different weighting methods 422 d. pamucar, f. ecer badi and abdulshahed [13] showed the application of the fucom to evaluation of the line in air traffic. noureddine and ristic [14] used a hybrid fucom-mabac model for evaluating transport routes for dangerous goods in road traffic. in addition to these studies, the fucom was applied to supply chain management [15,16]. in the real world, it often happens that, due to partial knowledge of attributes or the lack of information regarding the problem, decision-makers prefer to evaluate attributes using linguistic variables instead of crisp values. in such situations, information on the attributes obtained from decision-makers may be unclear, imprecise or incomplete. the fuzzy set theory introduced by [17] is one of the tools successfully used to present such inaccuracies in a mathematical form. since the creation of fuzzy sets, mcdm problems with imprecise information have been successfully modeled using the theory of fuzzy sets. according to the best knowledge of the authors, the application of the fucom in the fuzzy environment has not yet been shown and this paper is targeted at filling this gap in the literature. thus, it is one of the motives for creating this extension of the fucom's work in the fuzzy environment. therefore, the aims of this paper are as follows.  to improve the methodology for defining the weight coefficients of criteria by developing a fucom-f.  to determine the weights of criteria using the fucom throughout a detailed algorithm in the fuzzy environment.  to bridge the gap that exists in the methodology for determining the weight coefficients of criteria throughout a new model in treating uncertainty, which is based on fuzzy numbers. to achieve this, the rest of the paper was organized as follows. the proposed fucom-f model was introduced in detail in the next section. in the third section of the paper, an illustrative example was conducted to reveal the steps of the proposed method. at the same time, the results obtained from comparisons of three models i.e. fucom-f, fbwm, and fahp were discussed. finally, conclusions are presented in section 4. 2. fuzzy full consistency mcdm method in this section, the fucom-f has been discussed in detail after giving information on triangular fuzzy numbers (tfns) which are the expressions of linguistic variables as fuzzy numbers. 2.1. triangular fuzzy numbers the fuzzy set theory assigns membership degrees to linguistic variables and considers them as probability distribution. to achieve this, it utilizes fuzzy numbers. although there are various shapes of fuzzy numbers like trapezoidal, triangular or gaussian, the tfn is the most preferred by researchers in literature [18]. the outlines of fuzzy sets and tfns have been given below briefly [18,19,20]. definition 1: a fuzzy number is a special fuzzy set {( , ( )), }ff x x x  , where x takes its values on the real line, : x     and ( )f x is a membership function in the closed interval [0,1]. prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 423 definition 2: a tfn expresses the relative strength of each pair of elements in the same hierarchy, and can be denoted as ( , , ),t l m u where .l m u  parameters l, m, u indicate the lower bound value, the center, and the upper bound value in a fuzzy event, respectively. triangular type membership function of t fuzzy number can be described as in eq. (1) 0 , ( ) / ( ) , ( ) ( ) / ( ) , 0 , t x l x l m l l x m x u x u m m x u x u                   (1) consider two tfns 1 1 1 1( , , )t l m u and 2 2 2 2( , , ).t l m u the following describes the basic operations of two fuzzy numbers, t1 and t2, respectively: 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )l m u l m u l l m m u u     (2) 1 1 1 2 2 2 1 2 1 2 1 2( , , ) ( , , ) ( , , )l m u l m u l l m m u u  (3) 1 1 1 2 2 2 1 2 1 2 1 2( , , ) / ( , , ) ( / , / , / )l m u l m u l u m m u l for 0, 0, 0i i il m u   (4) 1 1 1 1 ( , , ) , ,i i i i i i l m u u m l         for 0, 0, 0i i il m u   (5) definition 3: the graded mean integration representation (gmir) let ( , , )j j j ja l m u be a tfn and gmir ( )jr a of ia can be calculated as: 4 ( ) 6 j j j j l m u r a    (6) 2.2. fuzzy fucom (fucom-f) assume that in a mcdm problem, there are n evaluation criteria that are denoted as wj, j = 1,2, ...,n, and their weight coefficients need to be determined. subjective models for determining weights based on pairwise comparison of criteria require decision-makers to determine the degree of impact of criterion i on criterion j. the degree of influence criterion i has on criterion j is presented as the value of comparison (aij). since the obtained values of comparison aij are not based on accurate measurements, but on subjective estimates, it is expected that existing uncertainties will be presented with fuzzy numbers. in the application of fuzzy numbers in the mcdm models, linguistic scales are most frequently used. thus, throughout this paper, a fuzzy linguistic scale [20], described by triangular fuzzy numbers, is used to present expert preferences in the fucom-f (table 1). because it is a multi-criteria model, it should be emphasized that the model 424 d. pamucar, f. ecer presented can be also used to determine the weight coefficients of alternatives, and, therefore, the final rank and the selection of the optimum one from the set of alternatives observed. table 1 fuzzy linguistic scale [20] linguistic terms membership function equally important (ei) (1,1,1) weakly important (wi) (2/3,1,3/2) fairly important (fi) (3/2,2,5/2) very important (vi) (5/2,3,7/2) absolutely important (ai) (7/2,4,9/2) based on the main settings of the fucom [9], the extension of the traditional model in a fuzzy environment has been carried out. accordingly, the fucom-f algorithm has been presented in detail in four steps. step 1 determine the decision criteria. an initial step in multi-criteria models for evaluating alternatives is defining a set of evaluation criteria. as defined in the beginning of this chapter, supposing that there are n (j=1,2,...,n) evaluation criteria that are represented by a set 1 2{ , ,..., }nc c c c . step 2 rank the decision criteria. experts determine the rank of criteria in accordance with their preferences regarding the significance of the criteria. the first rank is assigned to a criterion that is expected to have the highest weight coefficient and so on, towards the criterion of the least significance. the last place is held by the criterion for which we expect to have the lowest value of the weight coefficient. thus, the criteria ranked according to the expected impact on decision-making in a mcdm model is obtained. (1) (2) ( )...j j j kc c c   (7) where k represents the rank of the criterion observed. if two or more criteria have the same ranking, the equality sign is placed between the criteria instead of ">". step 3 comparisons of the criteria using tfns. the criteria are compared to each other using fuzzy linguistic expressions from a defined scale (table 1). the comparison is made with respect to the first-ranked (most significant) criterion. thus, we obtain the fuzzy criterion significance ( ( )j kc ) for all the criteria that are ranked in step 2. since the first-ranked criterion is compared with itself (its significance is (1)jc ei  ), n1 comparison of the remaining criteria must be performed. based on the defined significance of criteria, fuzzy comparative significance /( 1)k k  is determined by applying eq. (8). ( 1) ( 1) ( 1) ( 1) ( ) ( ) ( ) ( ) /( 1) ( , , ) ( , , ) j k j k j k j k j k j k j k j k l m u c c c c k k l m u c c c c                (8) prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 425 thus, a fuzzy vector of comparative significance of the evaluation criteria is obtained using eq. (9). 1/ 2 2/3 /( 1)( , ,..., )k k     (9) where /( 1)k k  represents the significance that the criterion of cj(k) rank has in relation to the criterion of cj(k+1) rank. step 4 calculate the optimal fuzzy weights. in the fourth step, the final values of the fuzzy weight coefficients of criteria 1 2( , ,..., )t nw w w are calculated. the final values of weight coefficients should satisfy two conditions: condition 1 the ratio of weight coefficients of the observed criteria (cj(k) and cj(k+1)) should be equal to their comparative significance (k/(k+1)) defined in step 2, i.e. that it fulfills the condition: /( 1) 1 k k k k w w     (10) condition 2 in addition to the condition defined by expression (9), the final values of weight coefficients should satisfy transitivity, i.e. that /( 1) ( 1)/( 2) /( 2) k k k k k k       , i.e. that 1 1 2 2 k k k k k k w w w w w w       . thus, another condition that needs to be satisfied by the final values of weight coefficients is obtained: /( 1) ( 1)/( 2) 2 k k k k k k w w        (11) minimum dmc, i.e.  = 0, is satisfied only if the transitivity among weight coefficients is completely satisfied. then, it can be said that /( 1) 1 0 k k k k w w      and /( 1) ( 1)/( 2) 2 0 k k k k k k w w         . for such obtained values of weight coefficients, dmc is  = 0. in order to satisfy these conditions, it is necessary to determine the values of the weight coefficients of criteria 1 2( , ,..., )t nw w w that satisfy the condition that /( 1) 1 k k k k w w      and /( 1) ( 1)/( 2) 2 k k k k k k w w          with the minimization of value . based on the settings defined, the final nonlinear model for determining the optimal fuzzy values of the weight coefficients of the evaluation criteria can be set 1 2( , ,..., )t nw w w . 426 d. pamucar, f. ecer /( 1) 1 /( 1) ( 1)/( 2) 2 1 min . . , , 1, , , 0, 1,2,..., k k k k k k k k k k n j j l m u j j j l j s t w j w w j w w j w w w w j j n                                            (12) where ( , , )l m u j j j jw w w w and /( 1) /( 1) /( 1)/( 1) ( , , )l m u k k k k k kk k       . in order to achieve the highest consistency, it is necessary to satisfy the condition that /( 1) 1 0 k k k k w w      and /( 1) ( 1)/( 2) 2 0 k k k k k k w w         . thereby, the model given by eq. (12) can be transformed into a fuzzy linear model, eq. (13). the optimal fuzzy values of weight coefficients are obtained 1 2( , ,..., )t nw w w , if it is solved. 1 /( 1) 2 / ( 1) ( 1)/ ( 2) 1 min . . , , 1, , , 0, 1, 2,..., k k k k k k k k k k n j j l m u j j j l j s t w w j w w j w j w w w w j j n                                         (13) where ( , , )l m u j j j jw w w w and /( 1) /( 1) /( 1)/( 1) ( , , )l m u k k k k k kk k       . 3. illustrative example this section of the paper presents the application of the fucom-f using an example of determining the weight coefficients of the criteria for evaluating green suppliers. with its application in the example shown, the model verification and comparison of results with other models from the literature, i.e. fuzzy bwm (fbwm) [20] and fuzzy ahp (fahp) [21] models, have been performed. prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 427 based on a literature analysis, 15 representative criteria have been identified for the evaluation of green suppliers, table 2. the criteria are grouped within three dimensions: economic (c1), environmental (c2), and social (c3). as can be seen in table 2, the criteria are arranged at two hierarchical levels. table 2 dimensions and their factors for evaluating green suppliers dimension criteria code economic (c1) cost/price c11 quality c12 delivery c13 technology c14 flexibility c15 financial capability c16 environmental (c2) pollution production c21 eco-design c22 environmental management system c23 green image c24 environmental training c25 social (c3) social responsibility c31 commitment to health and safety of employees c32 ethical issues c33 the interests and rights of employee c34 the first level includes economic, environmental, and social dimensions. the second level is presented by the groups of criteria within c1, c2, and c3. the aim of the fucom-f application is to determine the global values of weight coefficients of the second-level criteria. the solution of this problem using the fucom-f is performed by defining four models: model 1 – determining the local values of weight coefficients of c1, c2 and c3, model 2 – determining the local values of weight coefficients within the c1, model 3 – determining the local values of weight coefficients within the c2 and, model 4 – determining the local values of weight coefficients within the c3. by multiplying the local values of the weight coefficients of dimensions with corresponding local values of the criteria (within the observed dimension), the global optimal values of the weights of the criteria are obtained. a detailed overview of models 1-4 is presented in the next sub-section. 3.1. determining the fuzzy weights 3.1.1. model 1 – weight coefficients of c1, c2, and c3 dimensions after defining the first-level criteria, in the second step their ranking was performed. dimensions were ranked as follows: environmental (c1) > economic (c2) > social (c3). in the next step (step 3), based on the preferences of decision-makers, the linguistic variables of the comparative significance of the criteria ranked were determined (table 3). table 3 linguistic evaluations of main dimensions dimensions c1 c2 c3 linguistic variables ei wi fi 428 d. pamucar, f. ecer by applying the fuzzy linguistic scale, linguistic variables were transformed into tfns, shown in table 4. table 4 tfn transformations of evaluations dimensions c1 c2 c3 tfn (1,1,1) (2/3,1,3/2) (3/2,2,5/2) applying expression (8), the comparative significance of the criteria has been defined as follows.  1 21/ 2 2/3,1,3/2 2/3,1,3/2( ) (1,1,1)=c cc c    2 32/ 3 3/2,2,5/2 2 )2( 0/3,1,3) ( )=(1.00,2. 0,3/ .73c cc c    by calculating the comparative significance of the criteria, the vector of comparative significance  (0.67,1.00,1.50),(1.00,2.00,3.73)  was defined. in the following section (step 4), the constraints of the model (12) were defined based on the vector of comparative significance. by applying expression (10), we have defined the first group of constraints:  1 2 2/3,1,3/2/c cw w  and 2 3/ (1.00,2.00,3.73)c cw w  . based on expression (11), a constraint that arises from the conditions of relation transitivity 1 3/c cw w    (1.00,2.00,3.730.67,1.00, ) (0.67,2.001.50 ,5.60)  was defined. based on the constraints defined, a model (13) for determining the optimal values of the weight coefficients of dimensions was formed. by solving the model, the optimum local values of the weight coefficients:       0.261,0.3891,0.5831 0.3881,0.3881,0.3881 0.1038,0.1945,0.38, , 91 t jw  and  = 0.001 were obtained (fig. 2). the lingo 17.0 software has been used to solve the model presented. c1 c2 c3 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 w ig h t criteria fig. 2 fuzzy criteria weights for model 1 prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 429 3.1.2. model 2 – weight coefficients within the c1 dimension a similar methodology was also applied to solving model 2. after defining the criteria within the c1 dimension, the ranking of criteria was performed: c11> c13> c15> c16> c12> c14. in the next step, the comparison of the criteria was performed (table 5). table 5 linguistic evaluations of economic factors factors c11 c13 c15 c16 c12 c14 linguistic variables ei wi fi vi ai ai then, the comparative significance of the criteria was defined as follows. 11/ 13 (2/3,1,3/2)/(1,1,1) (2/3,1,3/2)c c   , 13/ 15 ( / / )/(2/3,1,3/2)=(1.0,2.0,3.73)3 2,2,5 2c c  , 15/ 16 (5/ / )/( / / )=(1.0,1.5,2.33)2,3,7 2 3 2,2,5 2c c  , 16/ 12 (7/ / )/(5/2,3,7/2)=(1.0,1.33,1.8)2,4,9 2c c  , 12/ 14 2,4,9 2 2 )2(7/ / )/(7/ / )= 8,4 ,,9 (0.7 1.0,1.29c c  based on the comparative significance of the criteria, a vector of comparative significance was defined as 0.67,1,1.5 1,2,3.73 1,1.5,2.33 1,1.33,1.8(( ),( ),( ),( ), 0.78,1,1.( ))29  and by applying expression (10), the first group of constraints of the fuzzy linear model was defined as 1 2 0.67,/ 1( , . )1 5c cw w  , 3 5 1.0,2.0/ ( ,3 3).7c cw w  , 5 6 1.0,1.5/ ( ,2 3).3c cw w  , 6 2 1.0,1.3/ ( 3, 8)1.c cw w  and 2 4 0.78,1.0/ , 2 ).( 1 9c cw w  . by applying eq. (11), the second group of constraints was defined as 1 5 0.67,2./ ( 0, 6)5.c cw w  , 3 6 1.0,3.0/ ( ,8 1).7c cw w  , 5 2 1.0,2./ ( 0, .2)4c cw w  , and 6 4 0.78,1.33/ , 3 )2. 1(c cw w  . the optimum values of the criteria were obtained by solving the fuzzy linear model presented below. by solving the above model with lingo 17.0, the weight coefficients of the criteria within the c1 dimension with a deviation from the maximum consistency 0.05  were obtained. 430 d. pamucar, f. ecer             0.1525,0.3100,0.3125 0.0479,0.0939,0.0992 0.1726,0.2970,0.2975 0.0747,0.1200,0.1204 0.0654,0.1475,0.1475 0.0403,0.1170,0.11 , , , ( 1 0 ) , 9 , jw c                       . 3.1.3. model 3 – weight coefficients within the c2 dimension within the second dimension (c2), five criteria have been identified which, based on expert preferences, were ranked as follows: c22> c21> c24> c25> c23. based on the preferences of decision-makers, the linguistic values of the comparative significance of the criteria ranked have been determined (table 6). table 6 linguistic evaluations of environmental factors factors c22 c21 c24 c25 c23 linguistic variables ei wi wi fi ai in the next step, based on expert comparisons of the criteria (table 6), using eq. (8), the comparative significance of the criteria 22/ 21 0.67,1.0,( 1.5)c c  , 21/ 24 0.45,1.0,2 2 ). 4(c c  , 24/ 25 1.67,3.0,( 5.2)c c  , 25/ 23 1.0,1.3,( 1.8)c c  , and a vector of comparative significance  0.67,1.0,1.5 0.45,1.0,2.24 1.67,3.0,5( ),( ),( ),(.2 1.0,1.3, )1.8  were defined. from vector , applying eq. (10), the first group of the constraints of the fuzzy linear model was defined as 22 21 0.67,1./ ( 0, 5)1.c cw w  , 21 24 0.45,1.0 2/ )4( , .2c cw w  , 24 25 1.67,3./ ( 0, 2)5.c cw w  , 25 23 1.0,1./ ( 3, .8)1c cw w  , while by applying eq. (11), the second group of constraints was defined as 22 24 0.3,1.0/ ( ,3 6).3c cw w  , 21 25 0.74,3.0 1/ )7( , 1.c cw w  , 24 23 1.67,4./ ( 0, 4)9.c cw w  . the optimum values of the criteria within the c2 dimension were obtained by solving the following fuzzy linear model. prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 431 2 1 2 1 2 1 2 1 2 1 2 1 1 4 1 4 1 4 1 4 1 4 1 4 4 min . . ( 0.67) ; ( 0.67) ; ( ) ; ( ) ; ( 1.5) ; ( 1.5) ; ( 0.45) ; ( 0.45) ; ( ) ; ( ) ; ( 2.24) ; ( 2.24) ; ( l u l u m m m m u l u l l u l u m m m m u l u l l s t w w w w w w w w w w w w w w w w w w w w w w w w w                                                    4 5 4 5 4 5 4 5 5 3 5 3 5 3 5 3 5 3 5 3 2 4 5 4 5 ( 3) ; ( 3) ; ( 5.2) ; ( 5.2) ; ( ) ; ( ) ; ( 1.33) ; ( 1.33) ; ( 1.8) ; ( 1.8) ; ( 0.3) 1.67) ; ( 1.67) ; m m m m u l u l l u l u m m m m u l u l l u u l u w w w w w w w w w w w w w w w w w w w w w w w w w                                                        2 4 1 5 1 5 1 5 1 5 1 5 1 5 4 3 4 3 2 4 2 4 2 4 ( 3.36) ; ( 0.74) ; ( 0.74) ; ( 3) ; ( 3) ; ( 11.7) ; ( 11.7) ; ( 1.67) ; ( 1.67) ; ; ( 0.3) ; ( ) ; ( 3.36) ; u l l u l u m m m m u l u l l u l u l u m m u l w w w w w w w w w w w w w w w w w w w w w w w w                                                       4 3 4 3 4 3 4 3 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 1 1 1 2 2 2 3 3 3 4 4 ( 4) ; ( 4) ; ( 9.4) ; ( 9.4) ; ( 4 ) / 6 ( 4 ) / 6 ( 4 ) / 6 ( 4 ) / 6 ( 4 ) / 6 1; ; ; ; m m m m u l u l l m u l m u l m u l m u l m u l m u l m u l m u l m w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w                                              4 5 5 5 1 2 3 4 5 ; ; , , , , 0. u l m u l l l l l w w w w w w w w w                                      if the above model is solved, the weight coefficients of the criteria within the c2 dimension with a deviation from the maximum consistency 0.07  are obtained. 0.1589,0.3314,0.3332 0.1517,0.3100,0.3131 0.0353,0.0828,0.1096 0.1154,0.2567,0.2567 0.0349,0.1105,0.11 ( ), ( ), ( 2) ( ), ( ) 38 , ( ) jw c                 3.1.4. model 4 – weight coefficient within the c3 dimension in model 4 within the third dimension (c3), four criteria have been identified. the criteria were ranked on the basis of expert preferences c34> c31> c32> c33 and the comparison of the criteria was made using tfns. the linguistic variables of the comparative significance of the criteria ranked are shown in table 7. 432 d. pamucar, f. ecer table 7 linguistic evaluations of social factors factors c34 c31 c32 c33 linguistic variables ei wi fi vi therefore, based on expert comparisons of the criteria, the comparative significance of the criteria 34/ 31 0.67,1.0,( 1.5)c c  , 31/ 32 1.0,2.0,3( .73)c c  , 32/ 33 1.0,1.5,( 2.3)c c  , and a vector of comparative significance  0.67,1.0,1.5 1.0,2( ),( ),.0,3.7 1.0,1.5,( )2.3  were defined. from vector  , by applying eqs. (10) and (11), two groups of constraints were defined: first group: 34 31 0.67,1./ ( 0, 5)1.c cw w  , 31 32 1.0,2.0/ ( ,3 3).7c cw w  , 32 33 1.0,1./ ( 5, .3)2c cw w  and second group: 34 32 0.67,2./ ( 0, 6)5.c cw w  and 31 33 1.0,3./ ( 0, .7)8c cw w  . the optimum values of the criteria within the c3 dimension were obtained by solving the following fuzzy linear model. 1 24 1 1 24 1 2 34 1 4 1 4 1 4 1 1 2 1 2 1 2 1 2 min . . ( 3.7) ;( 0.67) ; ( 3.7) ;( 0.67) ; ( ) ;( ) ; ( ) ; ( 1.5) ; ( 1.5) ; ( ) ; ( ) ; ( 2) ; ( 2) ; u ll u u ll u l um m m m u l u l l u l u m m m m s t w ww w w ww w w ww w w w w w w w w w w w w w w w                                                 4 2 4 2 4 2 4 22 3 1 32 3 1 32 3 2 3 2 3 4 2 4 2 ( 2) ; ( 2) ; ( 5.6) ; ( 5.6) ;( ) ; ( ) ;( 1.5) ; ( )( 1.5) ; ( 2.3) ; ( 2.3) ; ( 0.67) ; ( 0.67) ; m m m m u l u ll u l um m l um m u l u l l u l u w w w w w w w ww w w ww w w ww w w w w w w w w w                                                   1 3 1 3 1 3 1 3 1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4 1 2 3 ; ( 3) ; ( 3) ; ( 8.7) ; ( 8.7) ; ( 4 ) / 6 ( 4 ) / 6 ( 4 ) / 6 ( 4 ) / 6 1; ; ; ; ; , , , m m m m u l u l l m u l m u l m u l m u l m u l m u l m u l m u l l l w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w                                            4 0.lw                            prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 433 if the above model is solved, the weight coefficients of the criteria within the c3 dimension with a deviation from the maximum consistency 0.06  are obtained. 0.2155,0.3698,0.3714 0.0816,0.1842,0.1842 0.0503,0.1461,0.1486 0.1904,0.3900,0.390 ( ), ( ), ( 3) ( ), ( )2 jw c              as previously mentioned, for the evaluation of green suppliers in a multi-criteria model, the weight coefficients of the second-level criteria were used. since the criteria were divided into two hierarchical levels, the values of the criteria by the hierarchical levels represent local fuzzy values. the global values of the weights of the criteria were defined by multiplying the weight coefficients of the first hierarchical level with the groups of criteria of the second hierarchical level. the final global fuzzy values of the weights of the criteria are presented in fig. 3. c11 c12 c13 c14 c15 c16 c21 c22 c23 c24 c25 c31 c32 c33 c34 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 w e ig h ts criteria fig. 3 final fuzzy values of the criteria the validation of the results was performed by comparing the results obtained with the application of the fbwm and fahp models. for the research shown in this paper, all data were collected in the form of fuzzy matrices of pairwise comparisons. the fuzzy matrices of pairwise comparisons were formed for the criteria of both hierarchical levels. the data were the basis for the application of all the models considered: the fahp, fbwm, and fucom-f. all three models were based on the principles of pairwise comparison of criteria and relations based on transitivity. since all three models have similar basic mathematical foundations, it is possible to perform a comparison of results as well as testing based on the same data set. thus, in this study, in order to compare the results of all 434 d. pamucar, f. ecer three models, the data collected for the application of the fahp model were used. for the pairwise comparison of all models, the same scale shown in table 1 was used. in order to form a fbwm mathematical model, fuzzy bo (best-to-others) and ow (others-to-worst) vectors must be formed. the fuzzy bo and ow vectors were formed on the basis of the comparisons made for the best/worst criterion in fahp matrices. similarly, the data from the ahp comparison matrices were used to form a mathematical model of the fucom-f. the comparisons were made for the most significant criterion and all criteria were ranked according to the data. consequently, in fig. 4 the final (global) values of the weight coefficients of the criteria applying the fahp, fbwm, and fucom-f are shown. in fig. 4, it can be observed that the values of the weight coefficients of the criteria by the models considered are approximately the same if the central values of the interval of fuzzy numbers are taken into consideration. the greatest varieties of the fuzzy values of the weight coefficients of the criteria have been obtained with the fahp model. for the majority of criteria, the weight values obtained using the fucom-f fit into the upper and lower limits of the fuzzy interval of the ahp model. it is similar to the fbwm. the values of weight coefficients using the fbwm also follow the ahp model intervals, except for the c11 and c13 criteria where the interval limits are shifted relative to the fahp and the fucom-f. however, if we consider the maximum value belonging to fuzzy weight coefficient c13, it is established that it is approximately the same as with the fahp and the fucom-f. thus, it can be concluded that, with respect to minor deviations, approximately the same values of weight coefficients have been attained from all of the models. in the following section, the comparison of the results was performed based on the deviations of the models considered from the maximum consistency. c11 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 f u z z y w e ig h ts legend: fuzzy fucom fuzzy bwm fuzzy ahp c11 c11 c12 c12 c12 c13 c13 c13 c14 c14 c14c15 c15 c15c16 c16 c16c21 c21 c21c22 c22 c22 c23 c23 c23c24 c24 c24c25 c25 c25c31 c31 c31 c32 c32 c32 c33 c33 c33 c34 c34 c34 criteria fig. 4 final values of the weights calculated by fahp, fbwm, and fucom-f prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 435 3.2. checking consistency the deviations were determined by hierarchical levels and the mean values of deviations by levels were identified. as with the fucom-f, one model was formed for the first level, while four models were formed for the second level. by means of the application of the fbwm, fahp, and fucom-f, the following deviations from the maximum values of the degree of consistency with the models were obtained: (1) fbwm: cr(model 1)=0.059, cr(model 2)=0.049, cr(model 3)=0.078 and cr(model 4)=0.074; (2) fahp: cr(model 1)=0.071, cr(model 2)=0.084, cr(model 3)=0.081 and cr(model 4)=0.080; (3) fucom-f: dfc(model 1)=0.001, dfc(model 2)=0.053, dfc(model 3)=0.074 and dfc(model 4)=0.066. through analyzing the obtained deviations from the maximum consistency (dmc) for all three models, it can be perceived that the minimum consistency conditions defined by the fahp model (crmin = 0.10) have been satisfied. the greatest deviations were obtained with the fahp model, which was expected since it requires a greater number of comparisons of criteria (n(n1)/2) compared to the fbwm (2n3), and the fucom-f (n1). in addition, it has been perceived that the dmc values with the fucom-f and fbwm are approximately equal, with only slight differences. yet, the fucom-f shows superior consistency with respect to the fbwm in three out of four models. however, it is necessary to emphasize that the deviations between these two models are minimal; so, greater dominance of the fucom-f over the fbwm cannot be discussed. additionally, it is necessary to take into account the fact that there is a major difference in the number of the comparisons of criteria, especially regarding the relationship with the fahp method. as a result, it can be anticipated that in certain cases, different results of the same problem, resolved by different methods, may be obtained. in the examples provided and comparisons with the fbwm and fahp methods, there was no such case, but such a possibility should not be excluded from consideration. 4. conclusions when selecting the most suitable alternative for mcdm problems, different importance levels of the criteria are taken into consideration. in order to determine the importance levels with respect to the opinions of experts, a number of weighting methods such as the saw, ahp/anp, swara, bwm, and fucom have been used in existing literature. the fuzzy set theory can be used to overcome problems containing ambiguity and vagueness. if such weighting methods are integrated with the fuzzy set theory, which best expresses the human thought and reasoning structure, more reliable results can be obtained. in this study, therefore, fuzzy sets were combined with the fucom method and the fuzzy fucom (fucom-f) has been proposed. moreover, pairwise comparisons for criteria were conducted using linguistic variables instead of crisp values in the decisionmaking process. one of the most important advantages of the proposed model is the provision of similar results as with the fbwm and the fahp models by means of conducting solely n1 pairwise comparisons. thereby, the influence of the inconsistency of expert preferences regarding the final values of the weights of criteria is reduced. due to the minimum number of expert comparisons required, the fucom-f is considered to be the best way to determine the criteria weights. furthermore, it is a simple mathematical apparatus that 436 d. pamucar, f. ecer provides credible weight coefficients which contribute to rational judgment in decisionmaking [22]. as a result, the fucom-f is an effective decision-tool that aids decisionmakers in dealing with their own subjectivity when prioritizing criteria. in the present paper, additionally, the robustness and objectivity of the proposed model has been demonstrated by comparing it with the fbwm and the fahp models. the impressive consistency of the results obtained has been presented as well. furthermore, it has been shown that the model is adjustable and suitable for application to various measuring scales in order to express expert preferences. for future research, the proposed model could be applied in all areas of science, engineering, and social sciences. when combined with other ranking methods (topsis, aras, edas, codas, mairca, copras, etc.), it could be utilized reliably in deciding on the best alternative for mcdm problems. references 1. rezaei, j., 2015, best-worst 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vasiljevic, m., gavranovic, m. a., 2018, new methodology for improving service quality measurement: delphi-fucom-servqual model, symmetry, 10, pp. 757. prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method... 437 17. zadeh, l.a., 1975, the concept of a linguistic variable and its application to approximate reasoning part i, information sciences, 8(3), pp. 199–249. 18. ecer, f., 2014, a hybrid banking websites quality evaluation model using ahp and copras-g: a turkey case, technological and economic development of economy, 20(4), pp. 758-782. 19. ecer, f., 2018, an integrated fuzzy ahp and aras model to evaluate mobile banking services, technological and economic development of economy, 24(2), pp. 670-695. 20. guo, s., zhao, h., 2017, fuzzy best-worst multi-criteria decision-making method and its applications, knowledge-based systems, 121, pp. 23–31. 21. wang, b., songa, j., ren, j., lib, k., duana, h., wanga, x., 2019, selecting sustainable energy conversion technologies for agricultural residues: a fuzzy ahp-vikor based prioritization from life cycle perspective, resources, conservation & recycling, 142, pp. 78-87. 22. fazlollahtabar, h., smailbasic, a., stevic, z., 2019, fucom method in group decision-making: selection of forklift in a warehouse, decision making: applications in management and engineering, 2(1), pp. 49-65. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 13, no 3, 2015, pp. 259 268 reverse engineering of the mitkovic type internal fixator for lateral tibial plateau  udc 617:621.7 nikola vitković 1 , milan m. mitković 2 , milorad b. mitković 2 , nikola korunović 1 , dalibor stevanović 1 , marko veselinović 1 1 faculty of mechanical engineering, university of niš, serbia 2 faculty of medicine, university of niš, serbia abstract. in orthopaedic surgery it is very important to use proper fixation techniques in the treatment of various medical conditions, i.e. bone fractures or other traumas. if an internal fixation method, such as plating, is required, it is possible to use dynamic compression plates (dcp) or locking compression plates (lcp) and their variants. for dcp implants it is important to match the patient's bone shape with the most possible accuracy, so that the most frequent implant bending is applied in the surgery. for lcp implants it is not so important to match the patient’s bone shape, but additional locking screw holes are required. to improve the geometrical accuracy and anatomical correctness of the shape of dcp and to improve the lcp geometric definition, new geometrical modelling methods for the mitkovic type internal fixator for lateral tibia plateau are developed and presented in this research. the presented results are quite promising; it can be concluded that these methods can be applied to the creation of geometrical models of internal fixator customized for the given patient or optimized for a group of patients with required geometrical accuracy and morphological correctness. key words: cad, orthopaedic, fixator, parametric models, maf, mitkovic 1. introduction orthopaedic surgery deals with the treatment of various medical conditions related to the human musculoskeletal system, such as bone fractures, various diseases like osteoporosis or hereditary ones. for the treatment of bone fractures the orthopaedic surgeons use methods of external and internal fixation. external fixation is performed by the use of elements that are positioned outside the human body, above the skin. internal received june 29, 2015 / accepted august 10, 2015 corresponding author: nikola vitković faculty of mechanical engineering, university of niš, aleksandra medvedeva 14, niš, serbia e-mail:vitko@masfak.ni.ac.rs, nvitko@gmail.com original scientific paper 260 n. vitković, m. m. mitković, m.b. mitković, n. korunović, d. stevanović, m. veselinović fixation implies a surgical insertion of the implant in the human body, under the skin. geometrically precise and anatomically correct 3d geometrical models of human bones are of great importance in the internal fixation implants modelling. with such models it is possible to adequately prepare surgical intervention, create geometrical models of the customized fixation elements, manufacture fixation elements, etc. in the sense of geometrical modelling, the quality of the internal fixation can be improved by the creation of more accurate geometrical models of both human bones and internal fixators. for the creation of the geometrical models of the human bones two general approaches are used. the first one is based on the generation of 3d geometrical models of bones belonging to a particular patient, on the basis of complete geometrical data gathered by the use of medical imaging methods. such images may be obtained by the volumetric scanning methods (computer tomography ct or magnetic resonance imaging mri) or by multiple 2d scanning (xray, ultrasound) [1-4]. this approach involves the generation of 3d geometrical models using software packages which are an integral part of the medical scanner (e.g. vitrea) [5], or by subsequent processing of medical images in medically oriented cad programmes (e.g. materialise mimics) [6], or by application of various algorithms for the 3d model creation based on 2d contouring [7, 8]. one of the main disadvantages of this approach is the inability to generate the whole bone models in the cases when the medical images are incomplete or of poor quality due to osteoporosis, arthritis, or a comminuted bone [9, 10]. a complete bone model can be created, but with big approximation of its shape, which considerably reduces the quality of the model and its further application [9, 11]. the second approach to the generation of 3d geometrical models of bones or bone segments is based on the use of pre-defined predictive models of human bones [9-12]. with predictive models, the geometrical entities have been described by mathematical functions, the arguments of which are morphometric parameters or some other legible ones which can be read from medical images of a particular patient. the morphometric parameters are measurable dimensions, which can be acquired from medical images, for example ct or x-ray. through such an approach, it is possible to generate a 3d geometrical model which best suits the physical model of the patient's bone, even in the cases when only partial data about bone shape and geometry are available [9-12]. the potential shortcomings of such an approach can be either an insufficient number of parameters or inadequately selected parameters [9, 10] or inadequately chosen prediction surfaces [11]. the common fixation technique for the internal bone fracture fixation presumes application of compression plates [13]. this group of implants can be separated on dynamic compression plates (dcp) [14] and locking compression plates (lcp)[15], or some type of their combination. dcp implants require a direct contact between the bone outer tissue (periosteum) and the implant. the pressure on the periosteum can lead to devitalisation of the underlying bone and it can result in plate breaking. lcp implants include screws which are locked into the plate so that they provide more stability to the assembly of the bone and the implant. there is no need for any direct contact of the bone outer surface and the implant, but it is better if the implant geometry and shape follow the bone geometry [13-16]. according to the references related to the biomechanical properties of unique plates (lcp) vs. standard plates (dcp) [17], it is concluded that the unique plates provide better fixation and they can withstand more load. in addition to the type of fixation, quality of fracture reduction, soft-tissue handling, injury characteristics reverse engineering of mitkovic type internal fixator for lateral tibial plateau 261 and the patient’s general health status significantly influence the results. there is no evidence that the use of lcp will overcome these other factors [18]. the research in this paper is focused on the application of the newly developed method of anatomical features (maf) [6, 9, 10] for the creation of geometrically accurate and morphologically/anatomically correct models of the mitkovic type internal fixator for lateral tibial plateau (mif-ltp). three geometrical models of mif-ltp for human tibia bone (tibia is one of bones of lower leg under the knee) are created and they are:  solid model of the mif-ltp. this model is a classical cad model which can be used for the manufacturing of the fixator by the application of classical manufacturing or by the application of the additive manufacturing processes.  optimal parametric model (opm) based on the median bone geometry. this is a solid model whose dimensions can be changed according to the geometry of the specific bone. the median bone geometry is defined for the input set of bone samples, which defines the group of people for which the fixator is applicable. this means that the fixator can be used for the patient whose bone geometric characteristics fall within a group.  parametric points model (ppm) of the proximal part of the mif-ltp which is based on the parametric surface model of the human bone developed by the application of maf. the contact surface between the fixator and the bone is defined by the points whose values of the coordinates are defined as parametric functions [9, 10]. the geometry of all three fixator models can be adjusted in such a way that they can be used as dcp or lcp fixators. the shape is already precontoured (of course more for the ppm than for the opm) and the holes for screws can be adjusted to achieve screw locking. it is also important to mention that both the parametric models can be transformed into solid ones by applying the generally known cad techniques, e.g. closed surface technical feature. the main objective of this research is to create procedures for the creation of geometrical models of the mif-tpl which will enable orthopaedic surgeons and engineers to quickly create (manufacture) adequate physical models of the fixator for the given patient in different clinical situations. this paper consists of four main sections. the first section contains short introduction to the maf method. next, mif-ltp is described in short. in the third part of the paper, the method for the creation of the opm of mif-ltp is presented. the next section demonstrates the method for the creation of the ppm of the mif-ltp. finally, the conclusion is given including some references for the future work. 2. method of anatomical features (maf) method of anatomical features (maf) is developed in order to create patient specific geometrical models (polygonal, surface, and volume) and parametric (predictive) models of the human bones. the inputs for the maf are scans of bones created by the use of medical imaging methods. these scans can be volumetric (e.g. ct) or 2d scans (e.g. xray). by the application of techniques and procedures created by maf it is possible to create complete bone models even in the cases when input scans provide incomplete data 262 n. vitković, m. m. mitković, m.b. mitković, n. korunović, d. stevanović, m. veselinović about bone geometry, topology and anatomy, as presented in fig. 1. for these purposes two important components of maf are developed. fig. 1 the components of maf and its application the first component includes reverse engineering procedures which are based on the referential geometrical entities (rges) of the human bones. the rges represent basic geometrical elements (points, planes, axis, etc.) and they are constructed with respect to the morphology and anatomy of the specific bone. they are described in detail in the literature [9, 10]. with these procedures it is possible to create various geometrical models of the human bones customized to the specific patient. the second component of the maf is a parametric model which enables creation of 3d geometrical models of the human bones even in the cases when the geometrical data about specific bone is incomplete (e.g. bone fractures or diseases, only single 2d images is available, etc.). the parametric model consists of parametric functions with arguments defined as morphometric parameters. the morphometric parameters are in most cases dimensions which can be easily acquired (measured) from medical images and they are defined individually for an explicit bone (e.g. radius of femoral head is a unique morphometric reverse engineering of mitkovic type internal fixator for lateral tibial plateau 263 dimension for the femur). the geometrical models of a particular bone are created by the application of acquired values in parametric functions. more detailed description of the maf and its various applications are presented in the literature [9, 10]. 3. mitkovic type internal fixator for lateral tibial plateau mif-ltp this paper describes a method for creating a geometric model of the mif-ltp. this fixator consists of two parts: distal and proximal. the distal part of the fixator is constructed as a cylinder so that the contact surface with the bone surface is minimized. the proximal part of the fixator is constructed as a plate whose shape is already precontoured to match that of the bone in the best possible way. the fixator can be manufactured in different sizes so that it can be applied to tibia bones of different sizes and shapes. the main characteristics (advantages) of the mif-ltp are:  it can be used in the treatment of different tibial fracture types: epiphyseal, metaphyseal and upper diaphyseal fractures of tibia [18],  minimal contact between implant and bone provides for better vascularisation of the tissue,  axial dynamization is possible,  rotational stability of the fracture is achieved,  application is minimally invasive,  hospitalization of the patient is shorter, and,  postoperative recovery process takes lesser time. 3.1. optimal parametric model (opm) of mif-ltp this paper describes a method for the creation of opm of the mif-ltp. based on the fact that the bones are not shaped as typical geometrical forms and that two identical bones do not exist in reality, there is a requirement to define optimal geometry of the fixator, which can be applied on more than just one tibia. the mif-ltp consists of two parts, distal (cylinder) and proximal (plate), whose geometrical models are made separately and then connected in one whole model. the first step in creating a geometrical model of mif-ltp is to create the fixator’s outer contour projection, perpendicular to tibial shaft tangential plane, on the tibial surface, fig. 2. the mif-ltp contour geometry is created on the basis of existing (real) mif-ltp geometry and topology. it is measured on the physical sample of the fixator. a projection curve is used to create the proximal and distal curve guidelines and limitations for the design of optimal mif-ltp geometrical model. the process of creating proximal part guiding curves consists of two individual steps: creating the outer (lateral) and inner (medial) circles with adequate radius values on the tibial model surface. the proximal part of guiding curves is created by cutting and connecting the created circles, fig. 3a,b. the proximal fixator part volume model is created by dragging the rectangle profile along the contour curves, fig. 3c. the distal fixator part volume model is formed by extrusion of adequate profiles to the merging point with the proximal model. the complete fixator solid model for the specific bone (patient) is created by merging proximal and distal part models. 264 n. vitković, m. m. mitković, m.b. mitković, n. korunović, d. stevanović, m. veselinović fig. 2 mif-ltp contour projection fig. 3 a, b) constructed circles, c) mif-ltp volume model 3.2. use case and results in order to obtain optimal values of guiding curves radiuses (and thus fixing the model so that it can be applied to a number of different patients) measurements on ten different tibias (patients age 20 to 55, male and female) are performed and mean values are obtained and presented in table 1. the values given in table 1 are used to create a guiding curve with mean radiuses values, which is used to create a fixator volume model with optimal geometry opm, which is presented in fig. 4. table 1 specific guiding curves radiuses mean values for ten different tibia bones tibia radius r1 radius r2 radius r3 radius r4 mean value 15.5969 26.1637 83.0417 17.9543 fig. 4 opm of mif-ltp defined as solid geometrical model reverse engineering of mitkovic type internal fixator for lateral tibial plateau 265 3.2. definition of the parametric points model (ppm) of the proximal part of the mif-ltp ppm defines a contact surface between the inner surface of proximal mif-ltp and the outer surface of the human proximal tibia, as a parametric model. in order to construct such surface, the input (imported) filtered polygonal model of the human tibia bone is used. the already mentioned projected boundary curve of the mif-ltp is used as the limiting curve for the definition of the points which form the contact surface, fig. 5a. the points are defined as anatomical and supporting [9]. the anatomical points are presented in fig. 5b as crosses, and they closely define the geometry of anatomical surfaces (dents, crests, etc.). the supporting points are added in order to create surface of a greater geometric precision and anatomical correctness, and they are presented in fig. 5b as circles. these points are used for the creation of the preliminary surface model of the contact surface by the use of automatic surface patch (technical feature) in catia. this surface model is presented in fig. 5c. fig. 5 definition of mif-ltp contact surface with initial parametric points: a) projected curve on bone polygonal model; b) anatomical and supporting points; c) surface created over the constructed points; d) surface of the input bone model; e) deviation analysis between input and created surface models a) b) c) d) e) 266 n. vitković, m. m. mitković, m.b. mitković, n. korunović, d. stevanović, m. veselinović 3.3. geometrical accuracy of the created model to test the accuracy of the created surface, a deviation analysis is performed between the preliminary contact surface and the input surface of the tibia bone. the input surface model of the tibia bone in the section of interest is created by the use of the same technical feature (automatic patch) on the polygonal model of the bone, and presented in fig. 5d. the limiting curve for the input surface section is a projected boundary curve of the preliminary surface. this means that the surface model of the proximal tibial part is divided by the limiting curve; as a result of the operation, the input contact surface of the tibia is created and presented in fig. 5d. the deviation analysis is presented in fig. 5e. it can be seen that maximum deviation is 1.213 mm (close to 0%, this means only one, or just few points), and minimum value is 0.286 mm (about 0.9%). the deviations for most points are around 1 mm or below (about 70%). these values are acceptable; yet the model is improved by the application of more supporting points in adequate areas as presented in fig. 6a (points shown as squares). after the addition of the points the procedure fig. 6 definition of mif-ltp contact surface with additional parametric points: a) additional points (square); b) new surface model with added points; c) deviation analysis between input and newly created surface models a) b) c) reverse engineering of mitkovic type internal fixator for lateral tibial plateau 267 is performed again and the deviation analysis is done for the improved contact surface. the results are presented in fig. 6c. it can be seen the maximum deviation for the improved surface is below 1 mm, precisely 0.89 mm. the deviations around 0.5 mm are about 70% of all the deviations which is more than enough as confirmed by experienced medical workers included in this research. these points are accepted as the parametric points for the definition of the parametric contact surface of the mif-ltp. of course, additional testing with more tibial samples will be performed in the future research in order to improve the quality of contact surface of the mif-ltp proximal part. the solid model of the proximal part of the mif-ltp is created by adding thickness to the created contact surface. by applying classical technical features of trimming and merging with already created solid model of distal part of the mif-ltp, the solid model of the whole fixator is created. 4. conclusion the methods presented in this paper enable the creation of surface, solid and parametric geometrical models of the mitkovic type internal fixator for lateral proximal tibia (mif-ltp). they enable the creation of the optimal parametric model (opm) and parametric points model (ppm) of the mif-ltp. both geometrical models can be used for dynamic compression plates (dcp) and locking compression plates (lcp) with adequate geometrical accuracy and anatomical correctness. opm is good choice when there is a requirement to quickly create an implant’s geometrical model because its geometry can be adjusted to the patient's bone by simply changing the values of defined radiuses. in this case, the implant inner surface will not follow the bone outer surface exactly, and the locking screws will have to be implemented in the model in order to achieve lcp fixation stability. ppm is adequate choice when there is enough time for the fine adjustment of the fixator geometry. in such cases ppm can be modified to achieve both lcp and dcp fixation by modifying the position of the anatomical and structural points. these models can be used for the manufacturing of customized implants, for the creation of preliminary models for the fea analysis, for the preoperative planning in orthopaedics, and for other applications in medicine and engineering. the authors are aware that the current number of bone samples and parameters is not sufficient for the final claim about general application of the created fixator models. this would require an increasing number of input bone samples as well as to include more parameters; this would hopefully be done in the future for both opm and ppm of the mif-ltp. currently, the geometry of only one mode of the mitkovic type internal fixator is analyzed. in the future work more fixation implants will be designed and parameterized. in that way database of geometrical models of internal fixators can be made and used when it is necessary. acknowledgements: the paper is part of the project iii41017 virtual human osteoarticular system and its application in preclinical and clinical practice, sponsored by republic of serbia for the period of 2011-2015. 268 n. vitković, m. m. 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rouhi2, mahsa nazeri mehrabani4, erfan babaee tirkolaee5,6,7 1department of software engineering, istinye university, istanbul, türkiye 2department of software engineering, azarbaijan shahid madani university, tabriz, iran 3faculty of science and engineering, southern cross university, gold coast, australia 4department of electrical and computer engineering, university of tabriz, tabriz, iran 5department of industrial engineering, istinye university, istanbul, türkiye 6department of industrial engineering and management, yuan ze university, taiwan 7department of industrial and mechanical eng., lebanese american university, byblos, lebanon abstract. achieving timely access to data objects is a major challenge in big distributed systems like the internet of things (iot) platforms. therefore, minimizing the data read and write operation time in distributed systems has elevated to a higher priority for system designers and mechanical engineers. replication and the appropriate placement of the replicas on the most accessible data servers is a problem of np-complete optimization. the key objectives of the current study are minimizing the data access time, reducing the quantity of replicas, and improving the data availability. the current paper employs the olympiad optimization algorithm (ooa) as a novel population-based and discrete heuristic algorithm to solve the replica placement problem which is also applicable to other fields such as mechanical and computer engineering design problems. this discrete algorithm was inspired by the learning process of student groups who are preparing for the olympiad exams. the proposed algorithm, which is divide-andconquer-based with local and global search strategies, was used in solving the replica placement problem in a standard simulated distributed system. the 'european union database' (eudata) was employed to evaluate the proposed algorithm, which contains 28 nodes as servers and a network architecture in the format of a complete graph. it was revealed that the proposed technique reduces data access time by 39% with around six replicas, which is vastly superior to the earlier methods. moreover, the standard deviation of the results of the algorithm's different executions is approximately 0.0062, which is lower than the other techniques' standard deviation within the same experiments. key words: olympiad optimization algorithm, distributed systems, replica placement, data access time, stability, availability received: july 07, 2023 / accepted october 05, 2023 corresponding author: erfan babaee tirkolaee department of industrial engineering, istinye university, 34396 istanbul, türkiye e-mail: erfan.babaee@istinye.edu.tr 502 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. 1. introduction quick access to data items would be critical when designing big, distributed systems like cloud computing and the internet of things (iot). data retrieval takes a longer time in distributed systems. as a result, system designers now place a high priority on lowering the operation time of data read and write in distributed systems. this issue can be addressed by storing several copies of the data across different servers, allowing for faster data access from distant places. replication in this context refers to generating identical duplicates of the data on other servers. the placement of copies, which could be done either statically or dynamically, is often important to the performance of distributed systems. decreasing access time to data objects in distributed systems is a challenging issue. data objects and their replicas are distributed across numerous servers geographically isolated in a distributed system such as iot. the primary objective of data replication in such systems is to improve the system's dependability and efficiency. the number and location of replicas stored on various servers are critical parameters in the problem of replica placement. the research problem includes determining the optimal quantity of replicas, searching for optimal servers, storing data in the servers, and finally reducing the overall cost of the system for data processing. determining the data replicas' optimal locations is an np-complete problem. to address the problems of placement and replacement of data replicas within distributed systems, the current study proposes a novel discrete optimizer algorithm. the new algorithm is inspired by the process of group teaching and learning of students of a class for an olympiad exam, which was named the olympiad optimization algorithm (ooa). in ooa, the population of individuals includes and simulates the behavior of a set of students who attempt to get the best learning from their teammates. there is a competition between the individuals (students) to receive the best level of learning and preparation for the olympiad exam. ooa solves an optimization problem based on the divide and conquer approach, including local and global search strategies. in each iteration, the population is divided into groups of students, and the students are engaged in teaching and learning activities. in ooa, no teacher individuals are involved in the learning process, and the population evolution is performed based on swarm intelligence and group-based learning. the relevant solution spaces for the optimal solution are searched by each subgroup. solutions describe the quantity of replicas for each data object and their locations on the data servers; the fitness of a solution indicates the data access time provided by the solution. furthermore, ooa can be utilized in addressing discrete optimization problems. the primary achievements of this study can be outlined as follows: i. olympia optimization algorithm (ooa) is applied to address the problem as a novel discrete optimization algorithm. this method draws inspiration from the collaborative learning dynamics of students preparing for an olympiad exam, employing a swarmbased approach. throughout each iteration, the algorithm simulates the teaching and learning processes among the population members (akin to students), driving the evolution of the population. this proposed algorithm adopts a divide-and-conquer strategy, incorporating both local and global search techniques. the individuals within the population are partitioned into subgroups, each employing a specific imitation mechanism to explore distinct regions within the solution space. during the global search phase, students emulate the behaviors of the best-performing students in the entire population. ii. ooa can be used to address the majority of graph-based optimization problems. data replication in distributed systems using olympiad optimization... 503 iii. ooa is adopted in solving the issue of replica placement in distributed systems as a routing problem in a graph. iv. a minimum data access time with a lower quantity of produced replicas is provided. v. the development of a standard case study platform to evaluate the replica placement algorithms is the other contribution of this study. in the case study, the european union database (eudata) was employed to evaluate the suggested algorithm; this database contains 28 nodes as servers and a network architecture in the format of a complete graph. the experiments involved running the case study using varying quantities of data items. the remainder of the current paper is structured as follows. fundamental definitions associated with the problem of replica placement as well as previous works are reviewed in section 2. section 3 states the proposed ooa and its adoption to the replica placement problem. the case study and developed simulation platform were explained in the first part of section 4. the results attained by the ooa are evaluated and compared with the results of other methods in the second part of section 4, followed by broad conclusions in section 5. 2. literature survey 2.1 background 'replicas' are extra copies of data objects stored on various servers to provide quicker remote data access, and 'replica replacement' is the process of generating perfect clones of the same data on various servers. generally, the performance of distributed systems like iots depends on the placement of copies (it can be accomplished either statically or dynamically). in recent years, various strategies have been developed addressing the problem of replica placement, each with its unique structure, advantages, and disadvantages. algorithms like greedy, tree-based, hot spot, and hot region are this field's primary replica placement techniques. these algorithms consider the majority of client load while randomly distributing m copies of data items across n servers. the data is distributed using the normal distribution approach in this manner. this process must be repeated several times to achieve the desired result, with the best result being chosen. one of the main benefits of this method is its ease of implementation; however, due to the algorithm's randomness, obtaining the most and the best-desired results is difficult [1]. the following is a review of some of the pertinent algorithms applied to the problems of replica placement. 2.2. greedy and tree-based algorithms after comparing all n servers, the best server, as well as a replica version, is selected by this algorithm. the lowest-cost service provider is chosen for delivering the replica version after evaluating the costs of each customer's access to it. a similar method is applied to the remaining versions in the next cycles. when addressing data access, it is assumed that the most recent replica is used by each client. the main advantage of this algorithm is its quick execution, whereas the main drawback is that determining the appropriate data distribution strategy could be challenging [1]. this is implemented through dynamic programming. although this method was initially used in positioning the web proxy, it could also be employed in placing the replicas in distributed systems. network connectivity in distributed systems will likely take the form of a tree. at the 504 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. uppermost level of this technique's smaller sub-trees, each one owns a communication interface with the web server. a request is sent by each client to the corresponding interface, which then forwards it to the webserver to connect with that [2]. 2.3. hot spot-based and workload-aware algorithms the other technique is storing multiple data replicas close to clients with heavy workloads. accordingly, data is split among service providers according to the volume of work produced by their clients; to m service providers that are surrounded by many workloads, m replicas are distributed. this algorithm estimates the volume of work traffic by using the radius of the neighborhood around each service provider [1]. szymaniak et al. [3] and ng and zhang [4] introduced a common area-based placement approach. this algorithm's main objective is to accelerate placement operations. through the application of this algorithm, the network is divided into multiple regions based on node delays; each region is known as a cell. the cells are then sorted by a radix-type technique, and k cells are selected as placement agent servers. the algorithm uses the gnp approach to split the network and map it to euclidean space. the typical node delay is determined by the gnp approach, before converting the space of the network to euclidean coordinates. this algorithm computes the density of nodes for each coordinate within every network area and subsequently selects the areas with the node with the highest density. this method investigates the process of automated scaling as well as dynamic replica placement. to reduce the scaling replicas' and other resources' overall cost, the automatic scaling method based on service overhead is first recommended. before resource assignment, a hybrid load forecasting technique is utilized for workload prediction. the overall cost is calculated based on the workload. when the constraints are recognized, the optimization problem could be considered a problem of linear programming. then using the tabu algorithm, the optimum scaling strategy is chosen. the introduced data placement technique reduces the normal processing time of dynamic replica placement. this replica placement technique offers a more evenly distributed storage capacity than the relevant methods [5]. 2.4. user experience-based algorithms this approach offers an adaptable strategy for replica placement within an edge computing environment, with the objective of evaluating the relationships among user access patterns, the quantity of replicas, and their specific placements. in this study, the replica placement approach was created to identify the ideal locations for replicas. this would improve the performance of data access and cloud storage. this method employs the dynamic replica creation algorithm (drc-gm). drc-gm addresses the irregular nature of user access by utilizing the data block as the data unit. to meet the data availability requirement, drcgm dynamically adjusts the quantity of data copies, taking into account the connection between data access frequency and the number of replicas. the findings indicate that the drc-gm algorithm can greatly improve system performance in an edge computing environment. this improvement includes enhanced prediction accuracy, faster access response times, increased utilization of network and storage space, and improved data availability [6]. this method employs a dynamic replica allocation strategy to enhance user experience. the replica consistency preservation method ensures that the data is correct and consistent. the recommended replica management strategy may greatly decrease the rented nodes' total cost (server) as the duration lengthens. in comparison with previous data replication in distributed systems using olympiad optimization... 505 strategies, the suggested method can cut overall costs by up to 32.27% and 53.65%, respectively. the proposed replica allocation technique has the potential to significantly minimize storage overhead as well as the delay of data transmission [7]. 2.5. heuristic-based replica placement algorithms this approach employs genetic algorithm (ga) to identify the optimal locations for the generated replicas. the algorithm takes two key inputs: the euclidean coordinates representing the network and the required number of servers for placement. the algorithm unfolds in two main steps. first, it calculates the euclidean coordinates of the network. second, it assesses the density of each region, including the count of nodes within the target region. subsequently, it chooses the center for each region to serve as a pivotal point or as the cluster's core after each region is aggregated. in the following phase, the density of each cluster is determined through the compatibility function. based on the density magnitude, two clusters are consistently selected from the pool of good clusters, and the intersection operation is performed. the prior coordinates are then swapped out for the new coordinates in the third stage, which involves a random leap on the new coordinates. the approach is repeated until many excellent clusters are found as placement servers [8]. this strategy increases availability by enhancing the number of existing clones. usually, the two objectives of increasing the distance between comparable exchanges while decreasing the distance between distinct exchanges must be accomplished. by measuring the bandwidth of the shortest link between two sites, dijkstra's approach can be used to determine their distance. while the plan is carried out, the network is fixed. the websites are colorized after the list of every file on the network is collected. a collection of locations is created that consists of an original file (data item) and optionally multiple copies (replicas) of the original file. subsequently, the file that has the greatest distance from the nearest one is selected and substituted [9]. table 1 demonstrates a list of relevant methods with their advantages and disadvantages. the replica placement algorithm can be investigated in static and dynamic models. the dynamic nature of the requests and resources has not been considered in the proposed method. hierarchical and greedy-based algorithms [14, 15, 16] can be used to replicate the data objects in the dynamic distributed systems. in order to alleviate the replication time, the data servers can be clustered by different clustering and machine learning algorithms [17, 18, 19]. furthermore, the data servers can be grouped (partitioned) based on their features before being selected by the replication algorithms [20, 21, 22]. different methods have been proposed to manage the big data objects over the distributed systems’ servers [23, 24, 25]. detecting anomalies in multivariate time series data is of paramount importance for ensuring the overall performance and reliability of cloud-based distributed systems. given the intricate and rapidly changing nature of these systems, anomaly detection poses a significant challenge. to address these issues, different deep learning-based methods were proposed [26, 27]. the iot as a large distributed system with numerous diverse physical devices needs a dynamic services coordination method; the coordinating method can integrate the heterogeneous devices and data storages into the context aware iot infrastructure. different studies have proposed the situation aware iot services coordination approach [28]. different clustering and classification methods that have been developed in the previous studies [29, 30, 31] can be used to cluster the data objects and data storages in the distributed systems. furthermore, different metaheuristic algorithms with swarm intelligence features can be used to find out the best location of the data replicas [32, 33]. 506 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. table 1 related methods and their pros and cons method advantage disadvantage flexible replica placement [6] quick response time, high data availability data-file types and node types are not considered experience-based replica placement [7] high performance in big distributed systems and low financial and storage costs collaborative resource management is not considered replica placement using ga [8] reduced run time and optimum placement in large distributed systems, it shows a low performance consistency-based replica placement [9] low response time and higher data availability it is only suitable for fixed networks during the execution of the method pso and fuzzy-based replica placement algorithm [10] n/a the data write transaction performance is low replica placement with service and content delivery networks [11] high stability there is a probability of being trapped in a local optimum correlated data-replicas placement [12] low response time the performance for the data writes transaction is low priority-based replica management [13] low average response time and capability of fault tolerance it is only suitable for static systems 3. proposed method data retrieval could take a very long time in a distributed system. therefore, decreasing the required time for data objects to read and write in distributed systems is particularly difficult. as discussed earlier, a common approach to address this problem is to store numerous clones of the data on multiple servers (replicas) for easier access from a particular distance. a novel discrete heuristic swarm-based algorithm, ooa, has been proposed in the current study to address the problem of replica placement. compared to various other methods, the application of ooa can reduce data access time in static contexts. 3.1. olympiad algorithm 3.1.1. algorithm structure the proposed ooa employs a swarm-based imitation technique as both its local and global search methods [34]. this heuristic method operates with a population and groupbased approach, solving optimization problems through a divide-and-conquer framework. in this approach, each member of the population within ooa mimics the behavior of a student in a classroom setting, particularly those students preparing for an olympiad exam. the iterative teaching and learning processes among the population members (kind to students) drive the evolution of the population. this novel algorithm follows a divide-andconquer strategy, integrating both local and global search strategies. individuals within the population are grouped into subgroups, each employing a specific imitation approach to explore distinct regions within the overall solution space. fig. 1 provides an overview of ooa's general workflow, illustrating its applicability to solving optimization problems. competition exists among the individuals (students) as they learn from one another. each student maintains a memory of their learning rate, representing their position in the learning data replication in distributed systems using olympiad optimization... 507 process. in the ooa, each student is represented as a numeric array, and the student population consists of various solutions. fig. 1 workflow of the proposed ooa [34] 508 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. as depicted in fig. 1, the initial phase of ooa, following the sorting process, entails dividing the student population into n teams, where each team consists of m students. the foremost team is recognized as the global best team, while the final team assumes the role of the global worst team. within each team, comprised of students, exploration of their respective local solution space takes place. the primary student in the initial team is designated as the global best student, while in each successive team, the first student serves as the local best student. 3.1.2. learning procedure within ooa, students within teams aim to acquire knowledge from the best student in either the adjacent team (referred to as the local best student) or the top-performing global best team. the learning operator, a fundamental component of ooa, facilitates both local and global search operations. in essence, learning serves as the primary mechanism for ooa to seek out the optimal solution, with the aim of enhancing the population's overall knowledge or fitness. the introduction of the learning operator is intended to bolster the population's knowledge. individuals (students) are sorted based on their knowledge, as demonstrated in fig. 2, illustrating ooa's endeavor to enhance the population's knowledge through this operator. the learning operator encompasses four key steps. in the initial step, students within each team seek to learn from the students in their adjacent team. following the initial phase of the learning operator, knowledge from the first team is propagated to other teams in a manner reminiscent of the bubble sort algorithm. fig. 3 visually illustrates this first step of the proposed learning operator, wherein a team's knowledge is transferred to the neighboring team. this knowledge transfer process is carried out sequentially, mirroring the operation of the bubble sort algorithm, as knowledge gradually moves from one team to its adjacent counterpart. fig. 2 first step of learning algorithms as portrayed in fig. 3, all students within a learning team absorb knowledge from their corresponding peers in the adjacent team located on the left side. if there is no noticeable data replication in distributed systems using olympiad optimization... 509 improvement in the knowledge levels of the less proficient students in the weak team, the second stage of the learning process is initiated. during this second improvement stage, the best student from the learning team (belonging to the right-side team) undergoes a mutation process to introduce diversity. the mutation operator conducts a localized search on the best student from the weaker team. in situations where no substantial improvement is observed in this second step, the third step is put into action. in the third step, all students in the learning team (the weaker team) acquire knowledge from their counterparts in the global best team (the first team). fig. 3 learner team's students learn from the adjacent team's students in situations where the students from the global-best team are unable to effectively instruct the students in the worst team, the mutation operator is invoked on the global best student. mutating the global best student introduces a slight degree of diversity, which can help prevent the algorithm from converging to a local optimum. ultimately, students (referred to as search agents) from various teams are amalgamated to generate a new population. the learning operator is executed iteratively on this student population, as part of the ongoing optimization process. this iterative learning process allows for the continual refinement of the population's knowledge and, consequently, enhances the algorithm's ability to find optimal solutions to the given problem. 3.2. adapting the olympiad algorithm to replica placement problem 3.2.1. mathematical specification of the problem in distributed systems, replicating data objects on multiple servers is necessary to improve data access time. the problem at hand involves server number s(n) as well as data item number (𝑘), where k is within the range 1 < 𝑘 ≤ 𝐾 and 𝑛 is within the range 1 < 𝑛 ≤ 𝑁. the corresponding capacities of service provider n and data item k are denoted by 𝐶(𝑛) and 𝑉(𝑘), respectively. the communication cost between service providers 𝑆(𝑛) and 𝑆(𝑚) is represented by integer 𝑙(𝑛𝑚), and the data transport cost between servers 𝑆(𝑛) and 𝑆(𝑚) is equivalent to the communication cost. it is assumed that 𝑙(𝑛𝑚) is equal to 𝑙(𝑚𝑜𝑛) in this context. additionally, the variables read(𝑛, 𝑘) and write(𝑛, 𝑘) indicate the number of read and write requests for the object (𝑘) from servers s(n). each data object 510 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. has a primary server, 𝑃(𝑘), which possesses the initial version of the object. it is important to mention that object (𝑘) cannot be directly transferred to a different server in its original state. the servers where the object (𝑘) is replicated are recorded in a distinct list referred to as 𝑅𝑆(𝑘) for each primary server. when server 𝑆(𝑛) intends to perform a read operation on data object k, it should select the nearest server that holds either the original or replica versions of the object. during the writing process, the associated server sends a data update request (𝑘) to the main server. in this scenario, the primary server 𝑃(𝑘) broadcasts a message to all the servers that host object (𝑘), and every server updates the corresponding data item accordingly. the primary goal of the problem of replica placement is distributing the replicas among servers to minimize the total read and write operations cost and enhance the data availability. the proposed approach's initial step involves transforming replica placement into the well-known common traveling salesperson problem (tsp). to illustrate this, the performancesensitive genetic algorithm (psgwa) was employed to assess various methods to replicate 2 data items across 6 distinct servers in a scenario where there exist 2 data objects and 6 servers. each data item is assigned a unique id number (kn denotes the unique id of the nth data object). there are four distinct modes of replicating 2 data items (k1 and k2) on a given server (s). fig. 4(a) presents the configurations of replicating 2 data items across the 6 data servers. various modes are as follows: ▪ 1st mode: neither k1 nor k2 is cloned on server s. ▪ 2nd mode: only k1 is cloned on server s, but not k2. ▪ 3rd mode: only k2 is cloned on server s, but not k1. ▪ 4th mode: both k1 and k2 are cloned on server s. the columns of the generated graph represent the different modes of the servers, while each row corresponds to a specific server. each solution path acts as a representation of the desired locations for the data item's replica. each path's fitness value in the graph is determined based on the objective function. like the traveling salesperson problem, the optimal path is the one that has the shortest distance according to the fitness function. fig. 4(b) illustrates the process of replica placement to replicate 2 data objects across 6 servers. it presents a placement model that indicates the specified path of the 2 data-object copies on the 6 servers ([2,3,1,1,3,4]). this path can be interpreted as follows: ▪ server (1) exclusively stores the k1 data item's replica. ▪ server (2) exclusively stores the k2 data item's replica. ▪ server (3) possesses no replicas. ▪ server (4) possesses no replicas. ▪ server (5) exclusively stores the replica of the k2 data item. ▪ server (6) stores both k1 and k2 data items' replicas. ultimately, the problem of replica placement was successfully mapped into the traveling salesperson problem. fig. 5 showcases the ultimate representation of the graph of replication as depicted in fig. 4(b). the same problem has been addressed using a different technique [26, 27]. data replication in distributed systems using olympiad optimization... 511 mode 1s1 mode 2 mode 3 mode 4 s2 s6 s3 s4 s5 begining of the path end of the path mode 1 mode 2 mode 3 mode 4 begining of the path end of the path fig. 4 mapping the replica placement problem's search space to the tsp problem's search space: (a) two data object's potential replication model over six 2 data items in 6 distinct servers and (b) a placement path example when replicating distinct data servers. fig. 5 replica placement graph 3.2.2. individual structure each student is represented by a numeric array in the replica placement issue. fig. 6 displays a student's structure in the replica placement issue. each cell in a student array contains information about the server mode for that instance. the server index could be calculated using the student array’s index. each student has a replication path within the population. the path in fig. 4(b) is the same as the one in the student array depicted in fig. 6. fig. 6 structure of a student in ooa is used in solving the problem of replica placement in the case of six data servers 512 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. each replication array's fitness is assessed by applying the fitness cost function. when the replication array is sorted according to fitness, the population is categorized into n teams. the term "student knowledge" in this study refers to a student's fitness; that fitness (replication array) reflects the proximity to the ideal solution. fitness is proportional to the time it takes to get the data. the required cost and time to access data in the server will be changed by any replica placement. the best student specifies the location of a data replica with the shortest access time. each student array in the initial population is randomly initialized. if one of the edges (or nodes) in the graph of the problem of replica placement fails, the algorithm could still quickly determine the best new condition. as a result, if a node is removed, ooa can quickly determine the optimal (shortest) path because it is no longer required to start from the beginning. 3.2.3. fitness function the total cost of operations associated with the data item k is calculated using eq. (1), where 𝐴𝑐𝑐𝑒𝑠𝑠𝑅(𝑘) indicates all read operations for data item k from all servers that have received read requests for item k. eq. (2) calculates the value of 𝐴𝑐𝑐𝑒𝑠𝑠𝑅(𝑘). in eq. (2), 𝑁𝑆(𝑛𝑘) denotes the nearest or the least cost server that possesses a clone of object 𝑘. furthermore, in eq. (3), all write or update operations on data item k from all servers involved in the update request are referred to as 𝐴𝑐𝑐𝑒𝑠𝑠𝑊(𝑘). eq. (3) calculates the value of 𝐴𝑐𝑐𝑒𝑠𝑠𝑊(𝑘), and eq. (4) is used to calculate the total data operation cost (toc) of the whole system for all data items. the aim is to reduce the toc parameters, and ooa was used to find the best data replica placement with a minimum amount of toc. 𝑇𝑂𝐶(𝑘) = 𝐴𝑐𝑐𝑒𝑠𝑠𝑅(𝑘) + 𝐴𝑐𝑐𝑒𝑠𝑠𝑊(𝑘) (1) 𝐴𝑐𝑐𝑒𝑠𝑠𝑅 = 𝑉(𝑘) × (∑ 𝑟𝑒𝑎𝑑(𝑛, 𝑘)𝑁 𝑛=1 × 𝑙(𝑛 × 𝑁𝑆(𝑛, 𝑘))) (2) 𝐴𝑐𝑐𝑒𝑠𝑠𝑤 = 𝑉(𝑘) × ∑ [𝑤𝑟𝑖𝑡𝑒(𝑛, 𝑘) × [𝑙(𝑛, 𝑝(𝑘)) + ∑ 𝑙(𝑝(𝑘), 𝑗)∀𝑗∈𝑅𝑆(𝑘),𝑗≠𝑛 ]]𝑁 𝑛=1 (3) 𝑇𝑂𝐶 = ∑ 𝑇𝑂𝐶(𝑘)𝐾 𝑘=1 (4) 4. case study and results 4.1. case study and simulation platform the 'european union standard database' (eudata) was employed, as a case study, to examine the suggested algorithm; this case study database contains 28 nodes as servers and a network architecture in the format of a complete graph. the vertices of the graph 𝐺(𝑉, 𝐸) depict the nations of the ‘european union’ (|𝑉| = 28). furthermore, 𝐸 represents the network communication equipment's cost which is equal to the edges of the graph. to evaluate the introduced method's performance in the eudata case study, a simulation platform was developed in matlab. in the developed platform, the other techniques, such as genetic algorithm (ga), ant colony optimization (aco), particle swarm optimization (pso), shuffle frog leaping algorithm (sfla), and grey wolf optimization (gwo), were implemented along with the ooa. the implemented algorithms were executed with different settings in the developed platform. the simulations were carried out for all algorithms on the same hardware with microsoft windows using matlab. table 2 demonstrates the calibration parameters. data replication in distributed systems using olympiad optimization... 513 table 2 calibration parameters of various replication methods adjusted experimentally algorithm parameters value ga quantity of chromosomes 150 chromosome length 28 × k crossover rate 0.7 mutation rate 0.05 sfla quantity of frogs 96 size of memeplex 12 quantity of memeplexes 8 fla iterations (beta) 7 quantity of iterations 100 gwa quantity of wolves 50 𝑎 [0, 2] 𝐶 variable 𝐴 variable 𝑟1, 𝑟2 random values in [0, 1] pso quantity of particles 30 inertia weight 0.8 ratio of inertia weight damping 0.99 particle.c1 and particle.c2 1.8 quantity of iterations 100 ooa quantity of students 40 quantity of teams 4 size of teams 10 learning rate random values in [0.2, 0.8] imitation count 1 quantity of iterations 100 the following are the examination criteria applied in the current study: ▪ total operation cost (toc), which is the data access operations' cost (read and write) ▪ the quantity of generated replica items by the replica placement algorithm. ▪ the algorithm's convergence speed. ▪ algorithm's reliability in addressing the problem of the replica placement. ▪ algorithm's stability during various executions for the same data item. the experiments involved running simulations using varying quantities of data items, ranging from one to five. the connectivity costs between 28 data servers were uploaded as a matrix and uploaded in a free access repository [58]. the details of 28 data servers employed in the experiment are demonstrated in table 3 and geographically distributed over the 28 data servers situated all around europe. in the simulations, servers’ number' 5', '17', '9', '22', and '11' hold significant importance as they are considered key servers for storing original data items. 514 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. table 3 details of data servers server number server name country name server capacity in gbs 1 fi finland 10021 2 se sweden 21451 3 ee estonia 21110 4 lv lithonia 30000 5 lt lithuania 32002 6 dk denmark 34555 7 pl poland 46111 8 cz czech 48121 9 sk slovakia 42121 10 hu hungary 10001 11 at austria 10021 12 ro romania 21451 13 it italy 21110 14 sl slovenia 30000 15 bg bulgaria 32002 16 gr greece 34555 17 cy cyprus 46111 18 mt malta 48121 19 pt portugal 42121 20 es spain 10001 21 fr france 34555 22 de deutschland 46111 23 lu luxembourg 48121 24 be belgium 42121 25 nl netherlands 10001 26 gb great britain 34555 27 ie ireland 46111 28 hr croatia 48121 table 4 lists the five original data items employed in the simulations. all simulations incorporated various quantities of data pieces, ranging from 1 to 5 (where 1 < k < 5). each data access request in the simulations involved accessing five data items, with each number representing the corresponding server. table 5 lists the data-access requests employed in the simulations, encompassing both read and write operations and request settings. table 4 volume/location of original data items employed by simulation data object k1 k2 k3 k4 k5 volume (in gbs) 0.1 0.15 0.16 0.18 0.2 primary hosted server 5 (lithuania) 17 (cyprus) 9 (slovakia) 22 (deutschland) 11 (austria) data replication in distributed systems using olympiad optimization... 515 table 5 read/write requests employed by simulation (adapted from [28]) request number read request (array 28×5) write request (array 28×5) 1 5, 3, 3, 0, 1 1, 1, 0, 2, 1 2 0, 4, 1, 1, 1 0, 5, 2, 0, 2 3 1, 2, 7, 6, 0 1, 1, 1, 0, 0 4 4, 3, 8, 0, 6 0, 1, 0, 1, 0 5 0, 6, 11, 2, 1 0, 2, 0, 1, 0 6 9, 0, 4, 0, 9 0, 0, 0, 0, 0 7 3, 5, 2 1, 10 0, 0, 0, 0, 0 8 1, 4, 2, 0, 1 0, 0, 1, 0, 0 9 3, 5, 0, 8, 0 0, 0, 0, 1, 0 10 2, 3, 2, 4, 5 0, 1, 1, 0, 1 11 2, 1, 1, 8, 0 0, 0, 0, 1, 0 12 2, 0, 7, 4, 9 0, 1, 2, 0, 0 13 1, 1, 2, 4, 0 0, 0, 0, 0, 0 14 1, 1, 4, 1, 0 1, 2, 1, 0, 0 15 0 ,4, 3, 0, 7 0, 1, 0, 2, 0 16 1, 1, 2, 2, 1 0, 1, 0, 1, 0 17 3, 6, 5, 4, 5 0, 0, 0, 0, 2 18 6, 0, 4, 0, 6 0, 1, 0, 1, 0 19 0, 4, 2, 3, 0 0, 0, 1, 0, 4 20 5, 3, 4, 0, 3 0, 1, 2, 4, 0 21 2, 10, 2, 6, 0 0, 0, 2, 0, 0 22 5, 4, 4, 0, 2 1, 0, 0, 1, 4 23 3, 0, 2, 3, 1 0, 0, 1, 0, 0 24 2, 7, 9, 0, 5 0, 3, 0, 0, 3 25 1, 4, 3, 3, 5 0, 0, 0, 2, 0 26 3, 6, 2, 10, 5 1, 0, 0, 1, 1 27 6, 0, 4, 0, 6 0, 0, 3, 0, 0 28 0, 4, 4, 3, 0 0, 2, 0, 0, 2 4.2. results when it comes to the challenge of replica placement in distributed systems, k data items are required to be cloned across n servers. in such a situation, s(n) indicates the nth instance of the server, whereas item(k) represents data object k. to identify the most suitable replica locations, a series of experiments were conducted using five different optimization algorithms: ga, aco, sfla, gwo, pso, and ooa. to identify the optimal location for the clones, five sets of experiments were undertaken using ga, aco, sfla, gwa, pso, and ooa algorithms. in the primary set of the test, the data items were replicated on twenty-eight servers. toc and the quantity of the generated data item clones were utilized for results comparison. every technique produces a different number of data replicates over different servers. consequently, there is variation in the time of accessing data and the number of replicas for each algorithm. the first experiment's findings are depicted in fig. 7. twenty-eight servers were used in the initial stage of this experiment to hold copies of data items created by each algorithm. the overall data access time was computed by conducting 56 read/write operations on the replicas. various algorithms may employ a varying quantity and distribution of servers for storing data items. therefore, the quantity and placement of copies impact total access time. 516 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. an experiment with different numbers of data items was conducted similarly. the proposed technique is to identify the optimal replica’s locations over the data servers. fig. 7 displays the optimal toc obtained from the ga, aco, sfla, gwa, pso, and ooa algorithms. if the value of k is set to 5, the total operation counts for the 56 operations of accessing data are as follows for ooa, pso, gwa, sfla, aco, and ga algorithms: 0.673900 seconds, 0.802200 seconds, 0.816300 seconds, 0.824100 seconds, 1.057800 seconds, and 0.930660 seconds, respectively. according to the results, ooa outperforms the other methods in data success time. when four data objects are replicated, similar results are achieved; the total replica data access time generated by ooa, pso, gwa, sfla, aco, and ga is 0.638000, 0.760350, 0.791470, 0.768980, 0.945750, and 0.928160 seconds, respectively. ooa manages replicas with a lower toc compared to the previous techniques. fig. 7 shows that, in terms of the toc criterion, pso and gwa perform similarly in the replica placement problem. ooa takes advantage of both algorithms' strengths and outperforms gwa and pso. in all benchmarks, ooa has a lower toc in comparison with the previous techniques. fig. 8 demonstrates how different methods reduce data-access time. in terms of reduction of toc, ooa outperforms the prior approaches in all benchmarks, and all algorithms perform equally in the modest benchmarks (k=1). the toc of various methods varies significantly across all benchmarks. the toc criteria of the proposed technique demonstrate an almost linear increase as the quantity of data objects grows. when k exceeds five, a statistical interpolation approach is employed to estimate an unknown toc value. in the context of large datasets, interpolation is an approach employed to estimate the replica-placement algorithms' performance, which involves utilizing additional known values within the same sequence as the unknown value to estimate its value. if a consistent pattern can be observed among the collected data points, it is possible to predict the success of the replica placement techniques. fig. 7 greatest time of data access achieved by various methods for various quantities of data items 0 0.2 0.4 0.6 0.8 1 1.2 k1 k2 k3 k4 k5 avg to c ( se co n d s) ooa pso gwo sfla aco ga data replication in distributed systems using olympiad optimization... 517 fig. 8 replica-placement algorithms achieve varying levels of reduction in data-access time percentages for various quantities of data items fig. 9 exhibits the trendlines produced by multiple methods for the toc, which show the methods' behavior for various data items. the slope of the trendlines represents the new algorithm's performance. figs. 9 and 10 depict the linear behavior of ooa and the slope of the trendlines for the different methods, respectively. the ooa trendline slope is around 0.045. ooa and pso show the lowest and highest amount of trendline slope, respectively. pso's toc value exhibits a more pronounced increase as the amount of data increases, making it suitable for handling a large quantity of data items in distributed systems like iot and cloud systems. another factor for evaluation is the quantity of generated clones (replicas) by replica placement methods. the higher the storage cost, the greater the number of created clones. fig. 9 trendlines representing the obtained values of toc through various algorithms -10% 0% 10% 20% 30% 40% 50% 60% k1 k2 k3 k4 k5 avg to c r e d u ct io n ooa pso gwo sfla aco ga 518 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. fig. 10 trendline slopes are generated for different algorithms fig. 11 shows the replica counts generated by various methods for varying quantities of data objects. when k=1, ga, aco, sfla, gwa, pso, and ooa generated five, six, five, five, and four copies, respectively. the number of created copies in the huge benchmark (𝑘 = 5) is 24, 30, 8, 12, 9, and 7, correspondingly. the ooa produces fewer replicas than the other algorithms in most benchmarks. overall, the proposed algorithm can save more time with a lower number of replications. ooa requires less storage compared to other methods in addressing replica placement. the significant reduction in time achieved with a limited number of replications indicates that ooa is more efficient compared to the other methods. as depicted in figs. 9 and 10, ooa exhibits linear growth in the quantity of replicas but with a smaller slope compared to previous techniques. the convergence criterion is the heuristic algorithms' other performance requirement. in order to examine the convergence of the algorithms, an additional series of experiments was conducted. fig. 11 replica-placement algorithms generate varying number of replicas for various numbers of data items in optimal scenarios 0.0459 0.27 0.0522 0.0518 0.113 0.087 0 0.05 0.1 0.15 0.2 0.25 0.3 ooa pso gwo sfla aco ga toc trend slop 0 5 10 15 20 25 30 35 k1 k2 k3 k4 k5 n u m b e r o f r e p lic as ooa pso gwo sfla aco ga data replication in distributed systems using olympiad optimization... 519 notably, the proposed methodology demonstrates significantly faster convergence compared to other methods. the ooa algorithm achieves the optimal server selection for replicas of a single data object within five iterations. in a specific test (k1), 4 replicas were distributed across the available servers, which resulted in a toc reduction of 51%. the results vividly exhibit the superiority of the ooa algorithm over pso, gwa, sfla, aco, and ga concerning the reduction of toc and speed of convergence. the 2nd experiment evaluated the performance of various methods in replicating and storing 2 data items; the convergence of these methods when confronted with the task of replicating and storing two data items in the appropriate servers. the findings indicate the ooa's superior performance in terms of convergence and toc. in this benchmark experiment, the ooa algorithm can find the optimal replica site before the 10th iteration. with only four copies, the proposed approach reduced toc by 45%. the new method's toc value is lower than the toc values produced by the other techniques in this experiment. in this benchmark scenario, ooa achieved a 41% reduction in toc by utilizing six replicas across the 28 data servers. the experimental results demonstrate the superior performance of ooa compared to other algorithms regarding the reduction of toc and speed of convergence. the speed of convergence and obtained toc by different algorithms in the fourth benchmark (k=4) have been shown in fig. 12. as shown in the results, ooa selects the best servers for replicating 4 data items before the 20th iteration in this benchmark scenario (k=4). when there are four data items, ooa reduces toc by 36% with seven replicates. pso and sfla perform similarly in this benchmark (23% toc reduction). similar to other algorithms, gwo achieves a 20% reduction in data access time. additionally, it requires a lower number of data servers compared to other techniques. in the remaining studies, various algorithms' performance was assessed using five data items. fig. 13 depicts the methods' performance when there are five data items for replication. ooa decreases toc by 32% with only seven data replicates, as demonstrated in fig. 13. pso, gwa, and sfla all perform similarly in this benchmark. ooa demonstrates a faster convergence rate in comparison with the other algorithms. gwo reduces toc by 18% by producing twelve clones. following the suggested algorithm, pso requires fewer data servers to manage five data items compared to ga, gwa, aco, and sfla. fig. 12 convergence of various algorithms towards optimal solutions for k=4 (four data objects) 520 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. fig. 13 convergence of various algorithms towards optimal solutions for k=5 (five data items) the average value of toc reduction by various approaches for five benchmarks is demonstrated in fig. 14. on average, the reduction of toc achieved by ooa, pso, gwa, sfla, aco, and ga algorithms is 39%, 25%, 28%, 30%, 18%, and 21%, respectively. the proposed ooa algorithm achieves an average toc reduction of 39%. furthermore, the average quantity of copies generated by ooa, pso, gwa, sfla, aco, and ga algorithms is 6.25, 7.2, 8.2, 12.5, 19.75, and 14.25, respectively. overall, the suggested strategy maintains data with fewer replicas and reduces data access time the most. in terms of outcomes, ooa greatly surpasses the other algorithms about toc reduction, quicker convergence, and fewer created clones (lower storage spaces). the findings were analyzed regarding deviation value to determine all mentioned methods' stability. ten iterations of each method were reviewed to assess the range of probable outcome changes. fig. 14 toc average reduction data replication in distributed systems using olympiad optimization... 521 table 6 indicates the toc values obtained by six algorithms when k=5 during 10 executions. according to the findings, the average value of toc obtained from ten executions of ooa is about 0.6917; whereas this figure for ga, aco, sfla, gwa, and pso are 1.0266, 1.1723, 0.8421, 0.8237 and 0.8022 respectively. indeed, the ooa has more performance compared to the other algorithms in terms of the toc reduction criteria. furthermore, the minimum value of toc provided by ooa is lower than the other algorithms. the values in the final row of table 6 illustrate the standard deviation (stdv) among the produced toc in 10 executions. the least stdv indicates the ooa's higher stability in this benchmark. the increased stability observed in the results obtained by ooa reflects a higher level of reliability. gwo and sfla have approximately similar performance in this benchmark (k=5). table 6 toc values obtained by various algorithms undergo a range of changes during ten executions when k=5 #runs ga aco sfla gwa pso ooa 1 1.0240 1.0578 0.8625 0.8163 0.7888 0.6981 2 0.9306 1.1785 0.8315 0.8166 0.7977 0.6911 3 1.1277 1.1698 0.8625 0.8315 0.7965 0.6987 4 0.9552 1.2245 0.8241 0.8214 0.8139 0.6787 5 1.1698 1.1554 0.8241 0.8237 0.8116 0.6921 6 1.0578 1.2258 0.8315 0.8312 0.7972 0.6962 7 0.9306 1.1412 0.8287 0.8241 0.7965 0.6917 8 0.9564 1.1941 0.8625 0.8166 0.7843 0.6961 9 1.1412 1.1412 0.8312 0.8287 0.7999 0.6897 10 0.9731 1.2353 0.8625 0.8267 0.835 0.6850 min 0.9306 1.0578 0.8241 0.8163 0.7843 0.6787 max 1.1698 1.2353 0.8625 0.8315 0.835 0.6987 avg 1.0266 1.1723 0.8421 0.8237 0.8024 0.6917 stdv 0.0919 0.0531 0.0177 0.0058 0.0147 0.0062 regarding the results obtained by six algorithms when k=4 during ten executions, the average value of toc obtained by ooa is about 0.6528; whereas this figure for ga, aco, sfla, gwa, and pso are 0.9581, 1.0019, 0.7770, 0.8018 and 0.7778 respectively. on average, the ooa has a lower toc than the other algorithm. like the other benchmarks, ooa provides lower data access time with a limited quantity of produced replicas. the other important criterion is the reliability of the results. in this study, the lower amount of deviation among the obtained results during 10 executions indicates the reliability of the results. as shown in table 6, gwa has the lowest stdv and highest reliability. after gwa, ooa is more stable (reliable) than the other algorithms. the smaller the stdv, the more stable the ooa in this benchmark. the greater the stability, the greater the dependability of the ooa results. analyzing results from 10 executions while 𝑘 = 5, the average quantity of replicas produced by ooa is 7, which is smaller compared to the other techniques. the average quantity of replicas generated by ga, aco, sfla, gwa, and pso algorithms is 25.3, 34.4, 12, 12.4, and 9.97, respectively. the average quantity of replicas produced for five data items by gwa and pso is 12.4 and 9.9 in the simulated distribution systems. indeed, ooa shows the lowest time of data access with around seven replicas for five data items in the simulated distributed system. ooa is superior to the 522 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. existing methods when it comes to the quantity of produced replicas. the value of stdv among the results achieved by ga, aco, sfla, gwa, pso, and ooa are 1.6363, 2.4129, 2.3094, 0.5163, 1.3703, and 1.1547, respectively. in this benchmark, gwa has the highest performance from the stability perspective. after gwa, ooa has the lowest stdv and, consequently, the highest stability of the results. based on the results of ten executions on the four data object, the average quantity of copies created by ooa is 6.6, which is lower than the other methods. ga, aco, sfla, gwa, and pso generated an average of 21.2, 25.1, 12.9, 10.6, and 9.1 copies, respectively. in the simulated distribution systems, the average quantity of copies generated by gwo and pso for four data items is 10.6 and 9.1. in the simulated distributed system, ooa delivers the shortest data access time with around 6.6 replicates for four data items. in terms of the quantity of created clones, the ooa outperforms the other methods. stdv values generated by ga, aco, sfla, gwa, pso, and ooa are 3.7947, 2.4698, 2.5582, 0.5163, 0.316228, and 1.3498, respectively. gwa has the highest performance in this benchmark in terms of stability. following gwa, ooa has the lowest stdv and hence the best results stability. to confirm and trust the results, four distinct settings were used in similar investigations. in every experiment, the initial servers of the initial data items are different. moreover, depending on the layout, data item sizes fluctuate. each replica-placement method has been thoroughly tested in a variety of settings (configurations). table 7 shows the final experiment parameters. on each arrangement, 4 various data items with various quantities were stored in chosen servers. each data object's initial server is specified in the settings. the findings reveal the ooa's performance in determining the optimal replica placement in different configurations. in the conducted testing with varied settings, ooa has better performance with regard to convergence speed and toc reduction. regarding the results, sfla, gwa, pso, and ooa converged respectively to 0.7377, 0.7069, 0.6987, and 0.7029 after 100 iterations. the results of this experiment indicate that the ooa is superior to the other algorithms. the 2nd experiment was conducted with the 2nd configuration. the provided toc by sfla, gwa, pso, and ooa are 0.8167, 0.7899, 0.7977, and 0.6981, respectively. the ooa is superior to the other algorithms from the toc and convergence points of view. in this experiment, the sfla, pso, and gwa all performed similarly. the third experiment was conducted in different configurations. the obtained toc by sfla, gwa, pso, and ooa are 0.7274, 0.7045, 0.8357, and 0.6787. in this experiment, similar to the previous experiments, the ooa is superior to the other algorithms from the toc and convergence points of view. finally, the fourth experiment was conducted in different configurations. the provided toc by sfla, gwa, pso, and ooa are 0.7763, 0.7776, 0.7843, and 0.6850. in most of the benchmarks, ooa finds the optimum solution before the other algorithms. in most trials, the sfla and gwa perform similarly, whereas ga and aco perform worse than the other algorithms. overall, the ooa outperforms both the sfla, gwa, and pso. table 7 four workload configs to validate various algorithms' performance config. (workload) number primary servers data objects' volume in gigabytes 1 [15, 7, 3, 17, 27] [0.10, 0.15, 0.16, 0.10, 0.20] 2 [1, 28, 11, 5, 3] [0.70, 0.15, 0.86, 0.90, 0.70] 3 [19, 2, 17, 15, 6] [0.30, 0.25, 0.10, 0.70, 0.25] 4 [7, 1, 18, 25, 9] [0.90, 0.15, 0.30, 0.50, 0.60] data replication in distributed systems using olympiad optimization... 523 fig. 15 illustrates the toc achieved by different replica placement algorithms across four distinct configurations. the proposed ooa algorithm outperforms other methods in performance and efficiency. sfla, gwa, and pso algorithms exhibit similar performance in most configurations and benchmarks. additionally, the quantity of replicas generated by ooa is smaller compared to other methods. this results in a reduced total data access time in distributed systems by minimizing the quantity of data replicas located on multiple servers. among the algorithms evaluated, pso, gwa, and sfla exhibit better performance and efficiency compared to aco and ga. on average, pso generates 7.2 replicas, gwa generates 8.2 replicas, and ooa generates approximately six replicas. on the other hand, the sfla algorithm generates an average of approximately twelve replicas. fig. 15 replica-placement algorithms’ performance within different configurations the average reduction in total toc obtained by the ooa, pso, gwa, and sfla algorithms is 39%, 25%, 28%, and 30%, respectively, with sfla exhibiting a higher reduction compared to pso and gwa. among the replica placement methods used in the current study, the aco algorithm demonstrates the least performance and efficiency. ooa, on the other hand, maintains consistent performance regardless of the specific characteristics of the data objects. it exhibits reliable and predictable behavior, as observed in fig. 15, with reduced fluctuations indicating its reliability. the linear trend with a slope close to 0 degrees further illustrates the effectiveness of the proposed ooa algorithm in the given scenario. 4.3. discussion regarding the toc reduction criterion, ooa exhibits superior performance compared to the others. figs. 8 and 9 illustrate the relatively poor performance of aco when dealing with a larger quantity of data items, as it incurs an overhead of performance in some of the benchmarks. based on the simulation results, ooa offers the lowest data access time and achieves a higher toc reduction compared to other algorithms. figs. 10 and 11 show the linear behavior of ooa and the slopes of the trend lines for different techniques. the slope of the ooa trendline is approximately 0.045. ooa and pso indicate the lowest and highest trend line slopes, respectively. pso is suitable for processing large amounts of data in distributed systems such as iot and cloud systems, as the toc value increases significantly 524 b. arasteh, a. bouyer, r. ghanbarzadeh, et al. as the amount of data increases. the ooa’s toc value is lower than the toc values produced by the other techniques in this experiment. the results demonstrate the superior performance of ooa compared to other algorithms regarding reduction of toc and speed of convergence. on average, the toc reductions achieved by ooa, pso, gwa, sfla, aco, and ga algorithms are 39%, 25%, 28%, 30%, 18%, and 21%, respectively. notably, the proposed ooa algorithm achieves an average toc reduction of 39%. furthermore, the results show that the average number of copies generated by the ooa, pso, gwa, sfla, aco, and ga algorithms are 6.25, 7.2, 8.2, 12.5, 19.75, and 14, respectively. overall, the proposed strategy manages data with fewer replicas and provides the fastest data access times. ooa far outperforms other algorithms in terms of reduced toc, faster convergence, and fewer clones created. the stability of the results is one of the main criteria that should be considered in metaheuristic algorithms. the findings were analyzed regarding deviation value to determine all mentioned methods' stability. ten iterations of each method were reviewed to assess the range of probable outcome changes. regarding the findings, the average value of toc obtained from ten executions of ooa is about 0.6917; whereas this figure for ga, aco, sfla, gwa, and pso are 1.0266, 1.1723, 0.8421, 0.8237 and 0.8022 respectively. indeed, the ooa has more performance compared to the other algorithms in terms of the toc reduction criteria. the lowest stdv among the obtained results indicates high ooa stability in this benchmark. on average, ooa has a lower toc than other algorithms. like other benchmarks, ooa provides faster data access times while limiting the number of replicas generated. another important criterion is the reliability (stability) of the results. in this study, the small variability between the results obtained during the 10 runs indicates the reliability of the results. according to the results, gwa has the lowest stdv and the highest reliability. according to gwa, ooa is more stable (reliable) than other algorithms. the higher the stability, the more reliable the ooa results. the average total toc reductions achieved by the ooa, pso, gwa, and sfla algorithms were 39%, 25%, 28%, and 30%, respectively, with sfla showing higher reductions compared to pso and gwa. among the replica placement methods used in current research, the aco algorithm has the lowest performance and efficiency. ooa, on the other hand, guarantees consistent performance regardless of the specific properties of the data object. as shown in fig. 15, it exhibits reliable and predictable behavior, with small variations demonstrating its reliability. a linear trend with a slope close to 0 degrees further clarifies the effectiveness of his proposed ooa algorithm in certain scenarios. to find the optimal values of the ooa parameters, different experiments have been performed with different parameter values. workload 3 in table 7 was used in this series of experiments. the experiments have been performed with different values. table 8 shows the performance of ooa with different values of configuration parameters. the best performance was obtained when the learning rate was about 0.5 or 0.6. the small team size leads to better performance. table 8 calibrating the parameters of the ooa algorithm in workload 3 in table 7 ooa calibration parameters’ values performance num. of teams size of teams learning rate toc reduction 4 10 0.4 29% 6 7 0.2 30% 8 5 0.6 32% 10 4 0.8 31% data replication in distributed systems using olympiad optimization... 525 5. conclusion rapid data access must be considered while designing big dispersed systems. the task of replicating data in distributed systems like iot is a problem that falls under the category of npcomplete problems. the ooa approach, as a novel discrete heuristic algorithm, is introduced as a population-based algorithm to address replica management in order to reduce the cost of data operations. in this study, the eudata was used as a case study. the case study included 28 data servers that were modelled by a complete graph. regarding the case study, a simulation platform was developed using matlab. different replica placement algorithms were implemented in the simulated platform. this method was evaluated in a simulated distributed system with 28 data servers using the matlab platform. ooa outperforms the previous methods when it comes to data access time, quantity of replicas generated, result stability, and dependability and decreases data-access time by around 39% on average; this value for ga, pso, gwa, aco, and sfla, is 21%, 25%, 28%, 18%, and 30%, respectively. in order to save allocation resources, the next issue is to limit the number of replicas generated for data items. ga, pso, gwa, sfla, aco, and ooa produced an average of 14, 7.2, 8.2, 12.50, 20, and 6 data object replicas, respectively. ooa obtains the least data access time while using the least storage areas. furthermore, the suggested technique converges quicker than ga, pso, sfla, and aco. ooa yielded lower sds compared to the other algorithms in this example. in repeated executions, ooa is more stable in comparison with the other algorithms and can provide similar results again and over again. therefore, ooa shows more reliability and efficiency in addressing the problem of replica placement. in this analysis, the overall stability of the system is assumed, although only a combination of read and write transactions is considered, while other data transactions are not accounted for. the study does not analyze the execution time of the algorithm to reach the optimal response in an unstable state. future efforts could focus on parallelizing this method. additionally, the dynamic nature of the current method is expected to be a topic for future research, as the migration of initial and replica data in a dynamic setting presents unique challenges. various heuristic/metaheuristics [35-50] and machine learning methods have been developed and utilized in 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voxel-mars: a method for early detection of alzheimer’s disease by classification of structural brain mri, annals of operations research, 258, pp. 31-57. 57. graczyk-kucharska, m., olszewski, r., golinski, m., spychala, m., szafranski, m., weber, g. w., miadowicz, m. ,2022, human resources optimization with mars and ann: innovation geolocation model for generation z, journal of industrial and management optimization, 18(6), pp. 4093-4110. 58. https://drive.google.com/drive/folders/1o50_l9hgmwaa1imrnz5o06n46xc2o80d?usp=sharing (last access: 05.07.2023). https://doi.org/10.1007/s42235-023-00356-8 10575 facta universitatis series: mechanical engineering vol. 20, no 3, 2022, pp. 479 501 https://doi.org/10.22190/fume220307022m © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper assessment and performance analysis of machine learning techniques for gas sensing e-nose systems lubna mahmood1, zied bahroun2, mehdi ghommem3, hussam alshraideh2 1engineering systems management graduate program, american university of sharjah, sharjah, united arab emirates 2department of industrial engineering, american university of sharjah, sharjah, united arab emirates 3department of mechanical engineering, american university of sharjah, sharjah, united arab emirates abstract. e-noses that combine machine learning and gas sensor arrays (gsas) are widely used for the detection and identification of various gases. gsas produce signals that provide vital information about the exposed gases for the machine learning algorithms, rendering them indispensable within the smart-gas sensing arena. in this work, we present a detailed assessment of several machine learning techniques employed for the detection of gases and estimation of their concentrations. the modeling and predictive analysis conducted in this paper are based on knn, ann, decision trees, random forests, svm and other ensembling-based techniques. predictive models are implemented and tested on three different mox gas sensor-based experimental datasets as reported in the literature. the assessment includes a delineated analysis of the different models’ performance followed by a detailed comparison against results found in the literature. it highlights factors that play a pivotal role in machine learning for gas sensing and sheds light on the predictive capability of different machine learning approaches applied on experimental gsa datasets. key words: gas sensor arrays, e-nose, machine learning, feature extraction, feature selection, classification, regression received: march 07, 2022 / accepted may 10, 2022 corresponding author: mehdi ghommem department of mechanical engineering, american university of sharjah, p.o. box 26666, sharjah, united arab emirates. e-mail: mghommem@aus.edu 480 l. mahmood, z. bahroun, m. ghommem, h. alshraideh 1. introduction gas sensing technology which now encompasses the e-nose systems has witnessed tremendous growth in the past few decades. since the advent of e-noses in the late 20th century, gas sensing technology has experienced an accelerating trend in the development of different types of gsas that can be employed in e-noses [1-3]. these developments come in light of different technologies that have been deployed for manufacturing gas sensors. sensors based on metal-oxide (mox) semiconductors [4,5], carbon-nano materials [6,7] and polymer composites [8,9] operate on electrochemical principles, whereas, acoustic [10] and infrared [11] sensors focus on non-electrochemical operating principles. as a consequence, e-noses are currently used in a plethora of applications that require detection of the gas type and in many cases, the estimation of the gas concentrations. given the correlation of radon with the occurrence of lung cancer, blanco-novoa et al. proposed an iot system for estimating radon concentrations that could further activate mitigation devices to reduce its concentration in indoor environments [12]. besides air quality control, e-noses have been employed for detecting the freshness levels of different kinds of meat based on their odor level, thereby, showing promising results for food quality control [13]. wireless sensor networks are also used to detect and keep ammonia gas levels in check to minimize the environmental hazard they might pose within the livestock environment [14]. furthermore, the ability of the e-noses to offer a relatively inexpensive and fast mechanism for the detection of various chemical compounds has made them increasingly viable for medical diagnosis, for example, monitoring diabetes based on exhaled acetone levels [15]. machine learning within the e-nose forms a vital component of the smart gas-detection platform. the sensing materials of the gsas develop a sensory response, based on the type of sensor, to the exposed chemical compounds which is then exploited by machine learning algorithms for their identification [16]. despite their supreme functionality, gsas are often riddled with challenges that impact their selectivity, sensitivity, stability and their ability to render a viable sensor response [17]. while many of these challenges are influenced by the sensors’ external environment and have been tackled by implementing different sensor fabrication technologies, some challenges such as sensor drift [18,19] remain widely adverse. the two primary types of drift include the ‘first-order’ drift which occurs due to the chemical interaction between the sensor surface and the exposed gas. this could result in aging and an irreversible poisoning of the sensor surface due to prolonged gas exposure. ‘second-order’ drift, on the other hand, is attributed to external factors such as fluctuations in the environment or aberrant changes in the experimental conditions. to overcome the challenges of a single gas sensor, e-noses combine gsas with machine learning to mimic the human olfactory system in detecting/identifying various chemical compounds. machine learning for e-nose systems generally addresses two primary tasks, classification in which the type of chemical compound is identified and regression in which chemical compound concentration is estimated [20]. based on the sensory response to a particular chemical compound, which represents the sensor signal, machine learning techniques attempt to extract relevant characteristics from the sensor signals using feature extraction and feature selection [21]. this approach of feature extraction was implemented by vergara et al. to classify six different gases, i.e., ethylene, ethanol, ammonia, acetaldehyde, acetone and toluene, keeping in mind the challenge of sensor drift [19]. an extensive dataset that rendered the gas type and concentration obtained from 16 mox sensors, was used to implement support vector machine (svm) ensemble classifiers to improve gas detection and counteract assessment and performance analysis of machine learning techniques for gas sensing e-nose... 481 sensor drift simultaneously. akin to the work presented in [19], adhikari and saha proposed the use of artificial neural network (ann) as a deep learning network [22,23] and k-nearest neighbors (knn) ensembles to correctly identify the six gases in the face of drift [24]. the dataset consisting of six gas classes, posed a multi-class classification task that was tackled using ensembles combining several machine learning algorithms to render the final classification result. in addition to being used for classification, the dataset was employed to perform regression to predict the gas concentrations. rehman and bermak proposed a machine learning algorithm that incorporated feature selection within the regression model [25]. through feature selection, the predictive model was able to identify relevant characteristics from the sensor responses to help estimate the concentrations for each of the six gases. using predictive models, feature extraction, and feature selection for classification and regression using different gsa datasets, previous studies have worked towards improving the performance of the e-nose system. consequently, significant advancements in different machine learning techniques continue to help tackle the challenges faced by gsas while ameliorating their performance for different applications. this paper first presents the general computational framework used for smart gas sensing and discusses several predictive techniques used in an e-nose system. the framework is applied to assess the performance of different machine learning techniques using three different experimental datasets that were previously obtained from mox gas sensors. the assessment entails a detailed analysis of their implementation on the selected datasets and a comparison of their results against those obtained from previously-published studies. finally, we summarize the main findings of the present work and conclude with some remarks. 2. machine learning-based approach for gas sensing smart gas sensing using e-noses for classification and regression involves three vital steps as shown in fig. 1. multiple sensors in the gsa generate a unique sensory response in the form of a time-series signal for each gas they are exposed to, depending on the type of sensor being used and its operating principle. in several cases, the sensors undergo a physical or chemical change that results in a measurable sensor output. for instance, the sensor output could be measured as the change in electrical resistance or conductance of a sensitive material when exposed to a particular gas. as a result, the sensor response is typically referred to as a unique fingerprint that is specific to a certain gas and a particular concentration. once the signal is acquired, the next pivotal step of signal pre-processing helps in the useful and meaningful transformation of the original raw data into representative features through feature extraction. this results in the conversion of the sensor response into a set of n features that capture vital information from the sensor response. in many cases, the transformation can render a high-dimensional dataset due to the large number of features that can be extracted. this might cause complexities while implementing machine learning models, thereby, resulting in the poor performance of the overall e-nose system [26,27]. this can be encountered using feature selection techniques that help retain important features subset from the original feature set to improve model performance [20,21,28,29]. consequently, the generated dataset can be divided into training and testing datasets. the training dataset is used for model building, whereas, the testing dataset is used to test for the model performance. classification-based models attempt to identify and segregate the different gases using class labels, for example, segregating the quality of wine 482 l. mahmood, z. bahroun, m. ghommem, h. alshraideh using wine odors as conducted in [30], whereas, regression-based models aim to establish a relationship between the features and the gas concentration as a numeric variable [31]. fig. 1 block diagram representation of the steps involved in the e-nose operation given the primary focus of this work on sensor data obtained from mox sensors, fig. 2 shows a typical mox sensor response on exposure to a gas. the response depicts the change in conductance over time during the gas exposure with three different phases; phase i (initial phase), phase ii (exposure phase), and phase iii (regeneration phase) [32]. it is quite evident that the raw time-series data from the sensor contains numerous data points. as a consequence, feature extraction and feature selection that form a subset of dimensionality reduction techniques, focus on extracting/selecting representative features from the sensor data [33]. the extracted features could represent information such as the slope of the signal in each phase, the rise time of the signal, the peak value of the signal, the trough of the signal and the overall average value of the signal. as a result, dimensionality reduction techniques are utilized to provide characteristic features that summarize a major portion of the information contained in the original raw data. fig. 2 typical response of mox gas sensor on exposure to ethylene. adapted from [32] assessment and performance analysis of machine learning techniques for gas sensing e-nose... 483 3. assessment of machine learning techniques the adopted e-nose machine learning-based framework is applied to three mox gas sensor datasets to assess various machine learning models’ performance. whenever possible, a comparison of the adopted framework results with previously-published work is presented. each dataset was obtained through different experimental protocols for detecting different gases and their concentrations [18,19,25,34,37]. the experimental protocol, assessment and the results for each dataset are discussed in the following sections. 3.1. dataset 1experimental dataset of six gases with varying concentrations this dataset contains measurements collected over three years for the detection of six analytes, namely, ammonia, acetaldehyde, acetone, toluene, ethanol and ethylene using 16 commercial mox sensors in a gsa [19,25,34]. the complete experimental setup, fully controlled to provide high reliability and measurement reproducibility, consisted of four different types of mox sensors (four of each type), thereby, resulting in 16 mox sensors. each gas and its respective concentration was exposed to the gsas in no particular order to yield a complete dataset of 13,910 sensor signals, split into ten different batches. further details regarding the experimental procedure, various gas concentrations tested and different batches can be found in the work presented by vergara et al. [19]. 3.1.1. methodology following the signal acquisition, fonollosa et al. resorted to extracting two types of features from the sensor response [19,34]. for r[k], that represents the sensor resistance at time step k, the steady-state feature ∆r represents the difference of the maximum resistance change and the baseline resistance value, and its normalized value ‖∆r‖ is depicted as the ratio of this difference with respect to the baseline resistance value. the transient features represent the sensors’ dynamic response by accounting for the rising/decaying potion of the signal using the exponential moving average (ema) for different smoothing parameter values. the complete feature dataset consists of 128 features (8 main features x 16 sensors) and can be found in [25]. prior to the implementation of machine learning models, the features for each sensor were labeled and numbered accordingly. 3.1.2. part 1: machine learning – regression the main outline of the regression performed on this dataset followed the approach undertaken by rehman and bermak [25] for a fair comparison of the implemented models with the models presented in [25] and is shown in fig. 3. for this analysis, we used data from batch 1 to train the machine learning algorithm and develop the model, whereas, batches 2-5 were used to test the model for the final prediction results. the final results were quantified by the mean absolute percentage error (mape) metric for numerical predictions. this process was repeated for each of the six gases to estimate their concentrations separately. during the cross-validation phase, the model hyperparameters were tuned through parameter optimization to avoid the occurrence of any model overfitting or underfitting. 484 l. mahmood, z. bahroun, m. ghommem, h. alshraideh fig. 3 methodology outline for regression part 1 the complete list of algorithms applied in the previous study [25] and those applied in this work are shown in table 1. the knn algorithm was initially applied to the training dataset using k as 5 and the distance metric as euclidean distance. to optimize the model performance, these hyperparameters were tuned during the training phase. prior to training the knn algorithm, the training dataset was normalized to minimize the impact of the scale ranges for different features. for both the decision trees (dt) and the random forests (rf) models, regression was performed using the least squares regression splitting criterion. regression results of the techniques used in this work for all the six analytes is shown in table 2. table 1 list of machine learning models used in previous work and present work machine learning techniques previous work [25] present work levenberg marquardt (lmnn) bagging using random forests: bagging (rf) scaled conjugate gradient (scgnn) knn bayesian regularization (brnn) vote using gradient boosted trees and rf: vote (gbt,rf) average ensemble (avgensemble) rf(r)* support vector machines (svm) gbt gaussian processes (gp) decision trees (dt) random forests (rf) vote using knn, rf and dt: vote (knn,rf,dt) heuristic random forests (hrf) *since rf is implemented in the previous and present work, the rf of this research is referred to as rf(r). as shown in table 2, in the case of ethanol, when testing batch 2, knn (mape of 35.35%) achieved the best performance in comparison to all the models reported in the previous study [25]. despite the lack of high concentrations present in the training dataset, the present implementation of the models enabled to obtain relatively lower errors. this can be supported by the performance of dt (mape of 45.42%), vote using knn,rf,dt (mape of 51.05%) and rf(r) (mape of 55.02%), that had mapes lower than the hrf model [25]. for testing batches 2-5, knn provided the least mape of 39.23%, that was comparable with the mape of hrf. other models that delivered a decent performance include the vote using knn,rf,dt (mape of 52.26%) and rf(r) (mape of 59.18%). for batch 2 of ethylene, knn (mape of 15.66%) demonstrated the best performance amongst assessment and performance analysis of machine learning techniques for gas sensing e-nose... 485 the implemented models and those presented in the previous study [25]. in addition to knn, bagging using rf (mape of 21.82%) and rf(r) (mape of 23.47%) also delivered good performances. in case of batches 2-5, the lowest achievable mape was 58.22% from bagging using rf, wherein, the high mapes can be attributed to the sensor drift experienced over the course of the experiment. nevertheless, bagging using rf (mape of 58.22%), knn (mape of 64.97%), dt (mape of 71.44%) and rf(r) (mape of 63.03%) performed better than scgnn and svm [25]. table 2 mapes for six gases using different machine learning models present work gas test batch machine learning model/mape(%) bagging (rf) knn vote (gbt,rf) rf(r) gbt dt vote (knn,rf,dt) 1 batch 2 61.94 35.35 87.39 55.02 124.39 45.42 51.05 batches 2-5 60.35 39.23 72.08 59.18 89.74 72.35 52.26 2 batch 2 21.82 15.66 61.29 23.47 102.55 33.16 32.98 batches 2-5 58.22 64.97 73.83 63.03 88.34 71.44 89.55 3 batch 2 35.06 11.46 59.44 37.93 81.47 18.79 19.93 batches 2-5 54.23 38.22 81.15 56.15 106.31 39.79 41.69 4 batch 2 22.71 55.05 25.99 20.47 39.74 38.73 37.65 batches 2-5 22.14 59.77 26.55 19.8 41.49 49.07 41.81 5 batch 2 30.74 25.39 40.26 26.12 57.46 85.76 59.64 batches 2-5 33.17 32.54 53.75 32.09 77.3 86.25 58.54 6 batch 2 38.2 28.28 41.74 40.02 43.46 52.22 59.00 batches 2-6 38.6 64.02 50.66 41.63 59.68 84.83 65.98 gas 1 – ethanol, gas 2 – ethylene, gas 3 – ammonia, gas 4 – acetone, gas 5 – acetaldehyde, gas 6 – toluene when it comes to ammonia, the lowest mape of 11.46% was achieved by knn for batch 2 as seen in table 2, which is lower than the error achieved by all the models presented in the previous study [25]. this was closely followed by dt and vote using knn,rf,dt with mapes of 18.79% and 19.93% respectively. as a consequence, knn, dt and vote using knn,rf,dt collectively outperformed lmnn, scgnn, svm, gp and rf [25]. on the other hand, for testing batches 2-5, the lowest mapes were achieved by knn (mape of 38.22%) and dt (mape of 39.79%). hyperparameter tuning of knn (setting k=7), helped knn achieve the lowest mape in comparison to other models tested in this work, and those in the previous study [25]. for acetone, rf(r) provided the lowest mape of 20.47% amongst all the implemented models. although the best performance was attributed to svm (mape of 10.16%) from the previous study [25], the performance of rf(r) was comparable to that of hrf (mape of 20.44%) that had the second-best performance. the training dataset had low frequencies of high concentrations, whereas, the testing dataset had much higher frequencies of high concentrations. despite this discrepancy, rf(r) rendered the lowest mape of 19.8% for testing batches 2-5 in this work. besides rf(r), bagging using rf (mape of 22.14%) and vote using gbt,rf (mape of 26.55%) also delivered a good performance. for testing batch 2 of acetaldehyde, knn demonstrated the best performance with a staggering low mape of 25.39%, followed by rf(r) (mape of 26.12%) and bagging using rf (mape of 30.74%) (see table 2). on the other hand, for testing batches 2-5, the best performance was observed for rf(r) with the lowest mape of 32.09%, closely 486 l. mahmood, z. bahroun, m. ghommem, h. alshraideh followed by knn (mape of 32.54%) and bagging using rf (mape of 33.17%). for toluene, batches 2-6 were tested as opposed to batches 2-5 since no measurements were recorded for toluene in batches 3, 4 and 5. based on the regression results, the best performing model for testing batch 2 was knn (mape of 28.28%) with a performance comparable to hrf (mape of 21.48%) and gp (mape of 26.58%) from the previous study [25]. other models that performed equivalently well included bagging using rf (mape of 38.2%) and rf(r) (mape of 40.02%) which had mapes lower than the models used in the previous study [25]. for batches 2-6, the best results were attained using bagging using rf (mape of 38.6%), vote using gbt,rf (mape of 50.66%) and rf(r) (mape of 41.63%), which collectively had errors lower than all the models except hrf from the previous study [25]. going a step further, feature selection techniques of sequential forward selection (sfs) and sequential backward elimination (sbe), and the dimensionality reduction technique of principle component analysis (pca) were applied to investigate any potential change in the regression performance. pca for the given dataset resulted in 128 pcs as the dataset contained 128 features. to minimize dimensionality while retaining maximum data variance, the limit for data variance was set to 95%, where the number of independent pcs that yielded 94.3% variance in the data was found to be 4. a summary of the best performing models for each gas and the associated mapes is shown in table 3. feature selection or pca helped to minimize the redundancy in the feature datasets and thereby, improve the performance of the implemented models. moreover, models that did not perform well earlier were able to gain an advantage to their benefit. in case of sfs, 82.61% of the features selected represented the transient response of the sensors, whereas, quite a few pertained to the steady-state response of the sensors. similarly, in case of sbe, the transient response of the sensors formed 74.76% of the remaining features. though more importance is attributed to the transient features, our work implies the importance of both steady-state and transient features for estimating the gas concentrations. table 3 summary of best performing models and associated mapes present work gas 1 – ethanol, gas 2 – ethylene, gas 3 – ammonia, gas 4 – acetone, gas 5 – acetaldehyde, gas 6 – toluene gas test batch machine learning model technique mape (%) 1 batch 2 rf(r) sfs 29.75 batches 2-5 dt pca 36.59 2 batch 2 vote (knn,rf,dt) sbe 15.31 batches 2-5 rf(r) sfs 47.69 3 batch 2 knn sbe 8.19 batches 2-5 bagging (rf) sfs 11.38 4 batch 2 vote (gbt,rf) pca 13.45 batches 2-5 bagging (rf) sbe 29.59 5 batch 2 rf(r) sbe 12.94 batches 2-5 knn pca 10.39 6 batch 2 vote (knn,rf,dt) pca 39.9 batches 2-6 bagging (rf) sfs 38.48 assessment and performance analysis of machine learning techniques for gas sensing e-nose... 487 3.1.3. part 2: machine learning – regression so far, we focused on estimating concentrations of the six gases for batch 2, batches 25 (ethanol, ethylene, ammonia, acetone, acetaldehyde) and batches 2-6 (toluene). for the other batches of data, a different approach was followed, where, the machine learning algorithms were trained on batches 1 to (m-1) and tested on the mth batch. to clarify further, for ethanol, ethylene, ammonia, acetone and acetaldehyde, m ∈ {6,7,8,9,10}; whereas, for toluene m ∈ {7,8,9,10}. since batch 6 was already tested for toluene in the previous analysis, batch 6 for toluene was not included in this section. one of the main reasons for following this approach for batches 6-10 was sensor drift. the sensor data was collected over the course of 36 months, with batch 6 measurements beginning in month 17. training on batch 1 where measurements were taken in months 1 and 2, and testing on batches with measurements taken much later in time could result in inaccurate prediction results due to changes in the sensor response over time. as a consequence, all the previous batches prior to the testing batch were included in the model training to estimate the gas concentrations. results of the best performing regression models in the present work are shown in table 4. akin to the previous analysis, the knn table 4 mapes for gases on testing batches 6-10 present work gas training batches test batch machine learning model mape (%) 1 batches 1-5 batch 6 knn 22.17 batches 1-6 batch 7 rf 22.22 batches 1-7 batch 8 rf 23.73 batches 1-8 batch 9 dt 38.14 batches 1-9 batch 10 dt 54.02 2 batches 1-5 batch 6 dt 20.1 batches 1-6 batch 7 rf 54.01 batches 1-7 batch 8 rf 29.57 batches 1-8 batch 9 rf 29.46 batches 1-9 batch 10 rf 85.45 3 batches 1-5 batch 6 dt 14.58 batches 1-6 batch 7 knn 5.98 batches 1-7 batch 8 knn 58.9 batches 1-8 batch 9 bagging (rf) 8.89 batches 1-9 batch 10 bagging (rf) 52.21 4 batches 1-5 batch 6 rf 4.01 batches 1-6 batch 7 rf 37.26 batches 1-7 batch 8 rf 7.59 batches 1-8 batch 9 bagging (rf) 17.96 batches 1-9 batch 10 dt 44.87 5 batches 1-5 batch 6 dt 19.87 batches 1-6 batch 7 rf 18.55 batches 1-7 batch 8 dt 30.82 batches 1-8 batch 9 rf 6.35 batches 1-9 batch 10 dt 59.61 6 batches 1-6 batch 7 bagging (rf) 46.72 batches 1-7 batch 8 bagging (rf) 45.62 batches 1-8 batch 9 dt 40.9 batches 1-9 batch 10 bagging (rf) 24.6 gas 1 – ethanol, gas 2 – ethylene, gas 3 – ammonia, gas 4 – acetone, gas 5 – acetaldehyde, gas 6 – toluene 488 l. mahmood, z. bahroun, m. ghommem, h. alshraideh model was initially used with k as 5 using the euclidean distance metric. the mapes are a result of this selection whereas, in some cases, the hyperparameters were tuned accordingly. as seen in table 4, some models rendered remarkably low mapes such as knn (mape of 22.17%) for testing batch 6 of ethanol, dt (mape of 20.1%) for testing batch 6 of ethylene, knn (mape of 5.98%) for testing batch 7 of ammonia, bagging using rf (mape of 8.89%) for testing batch 9 of ammonia, rf (mape of 4.01%) for testing batch 6 of acetaldehyde, rf (mape of 6.35%) for testing batch 9 of acetone and bagging using rf (mape of 24.6%) for testing batch 10 of toluene. the low mapes in majority of the cases except for when testing batch 10, indicated that using several batches of data for model training helped in addressing sensor drift. using more recent data samples resulted in good predictions, with rf repeatedly delivering the best mapes, thereby, fortifying the ability of ensemble methods to deliver good prediction results. furthermore, sfs, sbe and pca using 4 pcs were applied along with the machine learning models to improve their performance. for some models, the mapes remained the same despite the use of feature selection or dimensionality reduction. this was seen from testing batch 10 of ethanol using dt and sfs (mape of 54.02%), batch 7 of ammonia using knn and sbe (mape of 5.98%), batch 10 of acetone using dt and sbe (mape of 59.61%). alternatively, in cases such as testing batch 6 of ethanol using knn, sfs reduced the mape drastically from 22.17% to 8.15% and for testing batch 10 of ethylene using rf, sbe reduced the mape from 85.45% to 80.03%. testing batch 8 of acetaldehyde with rf and pca reduced the mape from 7.59% to 6.24%. consequently, the use of either wrapper-based feature selection techniques or pca helped to ameliorate the performance of the implemented models. 3.1.4. machine learning – classification in addition to regression, classification was performed to predict the gas type using two different approaches: ▪ approach 1: training the algorithm on batch m and testing the model on the remaining consecutive batches, where m  {1,2,3,4,5,6,7,8,9}. ▪ approach 2: training the algorithm on batch (m-1) and testing the model on the mth consecutive batch, where m  {2,3,4,5,6,7,8,9,10}. the two approaches implemented are similar to those implemented in their respective previously published counterparts in [19,24]. this was done to enable a fair comparison of the results obtained. we implemented machine learning models such as rf, ann, knn, svm, dt, bagging using rf and vote using gbt,rf for both aforementioned approaches. for ann, a back propagation multi-layer perceptron (bp-mlp) model with 2 hidden layers and 15 nodes was used. moreover, to get appropriate results, the training datasets were normalized to lie between -1 and 1 for svm, ann and knn. using 10-fold, leave-one-out cross-validation, the classification accuracies for the machine learning models using approach 1 are shown in table 5. from table 5, it is observed that training on batch 1 and testing on all consecutive batches results in the highest average classification accuracy of 60.75% using svm. a similar observation can be made for batch 2 (accuracy of 60.31%) and batch 3 (accuracy of 63.59%), which rendered accuracies in the same range but using knn. for knn, the value of k was initially set to 5, however, on tuning the model hyperparameters, setting the value of k equal to 7 yielded the best results for classification. irrespective of the technique that rendered the assessment and performance analysis of machine learning techniques for gas sensing e-nose... 489 highest accuracy, it was quite evident that the classification accuracy decreased on going from training batch 1 to batch 9 due to sensor drift. despite the occurrence of drift, the classification accuracy of 60.75% was higher than the accuracies rendered by ann (accuracy of 42.43%) and knn (accuracy of 45.77%) as reported in [24]. our results were also compared to those presented by vergara et al. [19], who used an svm-based machine learning model for classification. on comparing accuracies, it was observed that the implemented svm classifier (accuracy of 60.75%) performed better than the svm classifier in [19]. besides svm, knn also provided a higher accuracy 60.31% when trained on batch 2 and tested on batches 3-10 in comparison to svm (accuracy of 58.35%) in [19]. a similar claim can be made for training batch 4 (accuracy of 44.46% with bagging using rf), batch 5 (47.23% with svm) and batch 8 (56.44% with rf). as a result, the classifiers in our research were successful in outperforming the svm classifier in [19] in 50% of the cases. using the second approach, i.e., training classifiers using only the previous batch and testing on the next immediate consecutive batch, we set the classifiers to their initial parameters to perform classification. results of approach 2 for different models are shown in table 6. it is quite clear that the accuracy for each test batch reaches its highest value when the model is trained with a previous batch. moreover, classifiers such as vote (gbt,rf), rf, svm, bagging (rf) and rf rendered quite high accuracies for 6 out of the table 5 average classification accuracies for training with different batches – approach 1 training batch test batches average classification accuracy (%) ann knn svm rf dt bagging (rf) vote (gbt,rf) batch 1 batches 2-10 53.19 58.72 60.75 49.25 50.8 44.11 45.32 batch 2 batches 3-10 56.47 60.31 50.97 54.07 49.99 56.36 49.46 batch 3 batches 4-10 54.78 63.59 60.74 58.98 45.5 57.87 51.6 batch 4 batches 5-10 35.03 36.98 40.77 43.88 37.25 44.46 43.65 batch 5 batches 6-10 32.19 32.07 47.23 27.06 19.28 25.1 25.76 batch 6 batches 7-10 48.82 57.53 54.13 54.13 48.7 53.49 53.61 batch 7 batches 8-10 47.82 52.66 64.54 58.77 56.48 57.93 62.4 batch 8 batches 9-10 39.06 36.42 47.76 56.44 47.84 52.44 50.79 batch 9 batch 10 17.22 15.36 15.25 13.83 16.83 14.19 11.83 table 6 classification accuracies when trained on a previous batch and tested only on the subsequent batch – approach 2 training batch test batch classification accuracy (%) ann knn svm rf bagging (rf) vote (gbt,rf) dt batch 1 batch 2 45.42 70.58 53.62 77.25 74.36 77.33 76.29 batch 2 batch 3 85.75 91.55 79.07 91.42 94.96 88.91 77.8 batch 3 batch 4 61.49 70.19 67.7 83.85 81.99 78.88 80.12 batch 4 batch 5 58.88 50.25 54.82 89.34 89.85 93.4 88.32 batch 5 batch 6 57.3 37.48 67.96 39.74 38.57 37.96 37 batch 6 batch 7 69.28 74.4 78.19 74.31 74.62 71.52 65.15 batch 7 batch 8 89.12 70.07 92.86 86.73 83.33 80.27 66.67 batch 8 batch 9 54.04 48.51 72.98 92.98 91.91 86.38 73.62 batch 9 batch 10 17.22 15.36 15.25 13.83 14.19 11.83 16.83 490 l. mahmood, z. bahroun, m. ghommem, h. alshraideh 9 cases, which reflects their superior performance when compared to the svm classifier, svm-based ensembles and the pca models implemented in [19]. thus, approach 2 indicates that the most recent data samples provide a better prediction for the subsequent data samples, thereby, helping in reducing the impact of drift. as the test batch number increases, impact of drift also increases which further reduces the classification accuracy. using feature selection or pca using 4 pcs, the average classification accuracy obtained for the machine learning models, either improved slightly or remained the same. many classification accuracies especially when testing batches 9-10 and batch 10 remained the same, indicating that their optimal performance had already been achieved prior to feature selection or pca. just like for regression, more transient features were selected in sfs (70% transient features), and a higher proportion of transient features remained after implementing sbe (76% transient features). a similar observation was made by using filter-based feature selection to determine feature importance using weights. using gini index, the weights of the top 30 features are shown in fig. 4. the weights represented in fig. 4 depict the decreasing order of importance for the features, where 27% of the features represent steady-state features and 73% represent transient features. with more importance given to the transient features, these features were capable of capturing more information from the sensor signals to yield good prediction results. fig. 4 feature weights for top 30 features in the dataset 1 the impact of different number of features on the classification accuracy is shown in fig. 5, which shows the average classification accuracies for the best performing models for training batch 1 and testing batches 2-10. rf and bagging using rf reach their peak classification accuracies at the top 40 features, whereas, svm uses the top 35 features to attain the highest classification accuracy. on the other hand. knn, uses the top 50 features to reach its highest accuracy. despite the different number of features used, it was evident that once the peak classification accuracy is attained by each model, the accuracy begins to stabilize and maintain a nearly steady value, with a minimum threshold of 30 features. with at least top 30 highest weighted features, the classification accuracy increased, following which it slowly waned off to provide a steadier value. although, this filter-based feature selection technique used a different approach to identify the most relevant features, the results remained consistent with those obtained earlier using sfs and sbe. assessment and performance analysis of machine learning techniques for gas sensing e-nose... 491 fig. 5 average classification accuracy using different number of features – dataset 1 3.2. dataset 2 – experimental dataset of four gases with varying concentrations the following dataset was used for the detection of four gases: ethylene (ey), ethanol (ea), carbon monoxide (co) and methane (me). the experiment involved testing for 10 different concentrations of each analyte over a course of 22 days, resulting in a total of 40 different concentrations and 640 measurement samples. the experiment used eight mox sensors in the gsas referred to as chemical detection platforms (boards). varying responses for each gas and its concentration were obtained using five identical boards, with any variability eliminated by maintaining the same configuration in all five boards. further details pertaining to the experimental procedure can be found in the work of fonollosa et al. [18]. the signals acquired from the eight sensors in a given board across different days, as reported in [18], are displayed in fig. 6. all the sensors follow a similar trend during the initial phase (i) for 50s, exposure phase (e) for 100s and the recovery phase (r) for 450s. the exposure phase can be easily identified by the dip in the sensor response when being exposed to the target gas and it is quite evident that the sensors render slightly varied responses on different days. this can be seen in fig. 6, wherein, the stable response of each fig. 6 sensor signals acquired for detecting 62.5 ppm of ethylene on board 1, on day 4 and day 21 492 l. mahmood, z. bahroun, m. ghommem, h. alshraideh sensor on board 1 varies from day 4 to day 21 for ethylene. despite testing the same board over different days, the variability in the sensor responses can be attributed to drift, which could be a consequence of external factors. 3.2.1. methodology as an initial step, signal pre-processing using weighted moving average was performed to eliminate noise levels from the signals. weighted moving average is the most common filter used for signal noise processing [35,36], which despite its simplicity, renders a highattenuation ability, by reducing noise while retaining the shape of the signal. given the sensor responses from the five boards, two main window lengths of 5 and 10 were used for noise removal, where fig. 7 shows the impact of using weighted moving average with a window length of 5 on sensors 1 and 3 from board 1, for the detection of 12.5ppm of ethanol. fig. 7 implementation of weighted moving average on sensor signals from sensor 1 and 3 on board 2 and day 1 for 12.5 ppm of ethanol following the pre-processing, relevant signal characteristics were extracted from 𝑟[𝑘 + ∆𝑥] that represents the sensor resistance, with 𝑘 as the discrete time-step, where 𝑘 ∈ [0,600] and ∆𝑥 is used to indicate any incremental change in the time-step. the signal characteristics extracted from 𝑟[𝑘 + ∆𝑥] in this work are discussed below: drop value as the signal transitions from the initial to the exposure phase is given as, 𝒅𝒓𝒐𝒑 = 𝑎𝑣𝑔 𝑘=0,1,…50 𝑟[𝑘 + ∆𝑥] − 𝑎𝑣𝑔 𝑘=51,52,…150 𝑟[𝑘 + ∆𝑥] (1) whereas, the transition rate is given by the slope, 𝒔𝒍𝒐𝒑𝒆 = 𝑟[𝑘1+∆𝑥1]−𝑟[𝑘2+∆𝑥2] ∆𝑥2−∆𝑥1 , 𝑘1 = 50, 𝑘2 = 51 (2) average exposure value attained when the target gas is exposed to the sensor is expressed as, 𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒆𝒙𝒑𝒐𝒔𝒖𝒓𝒆 = 𝑎𝑣𝑔 𝑘=51,52,…150 𝑟[𝑘 + ∆𝑥] (3) assessment and performance analysis of machine learning techniques for gas sensing e-nose... 493 fall time (denoted by ft) from 10% to 90% of the signal while transitioning from the initial to exposure phase is given as, 𝑟[𝑘 + ∆𝑥]10% = 𝑎𝑣𝑔 𝑘=0,1,…50 𝑟[𝑘 + ∆𝑥] − − [0.1 × ( 𝑎𝑣𝑔 𝑘=0,1,…50 𝑟[𝑘 + ∆𝑥] − 𝑎𝑣𝑔 𝑘=51,52,…150 𝑟[𝑘 + ∆𝑥])] (4) 𝑟[𝑘 + ∆𝑥]90% = 𝑎𝑣𝑔 𝑘=0,1,…50 𝑟[𝑘 + ∆𝑥] − − [0.9 × ( 𝑎𝑣𝑔 𝑘=0,1,…50 𝑟[𝑘 + ∆𝑥] − 𝑎𝑣𝑔 𝑘=51,52,…150 𝑟[𝑘 + ∆𝑥])] (5) 𝒇𝒕 = 𝑡𝑖𝑚𝑒|(𝑟[𝑘 + ∆𝑥] = 𝑟[𝑘 + ∆𝑥]90%) − 𝑡𝑖𝑚𝑒|(𝑟[𝑘 + ∆𝑥] = 𝑟[𝑘 + ∆𝑥]10%) (6) rise time (denoted by rt) from 10% to 90% of the signal while transitioning from the exposure to recovery phase is expressed as, 𝑟[𝑘 + ∆𝑥]10% = 𝑎𝑣𝑔 𝑘=151,152,…600 𝑟[𝑘 + ∆𝑥] − [0.1 × ( 𝑎𝑣𝑔 𝑘=151,152,…600 𝑟[𝑘 + ∆𝑥] − 𝑎𝑣𝑔 𝑘=51,52,…150 𝑟[𝑘 + ∆𝑥])] (7) 𝑟[𝑘 + ∆𝑥]90% = 𝑎𝑣𝑔 𝑘=151,152,…600 𝑟[𝑘 + ∆𝑥] − [0.9 × ( 𝑎𝑣𝑔 𝑘=151,152,…600 𝑟[𝑘 + ∆𝑥] − 𝑎𝑣𝑔 𝑘=51,52,…150 𝑟[𝑘 + ∆𝑥])] (8) 𝒓𝒕 = 𝑡𝑖𝑚𝑒|(𝑟[𝑘 + ∆𝑥] = 𝑟[𝑘 + ∆𝑥]10%) − 𝑡𝑖𝑚𝑒|(𝑟[𝑘 + ∆𝑥] = 𝑟[𝑘 + ∆𝑥]90%) (9) fig. 8 illustrates the five different features in eqs. (1)-(3), (6) and (9) extracted from sensor 4 of board 1 for the detection of 12.5ppm of ethanol. the five features ensure that fig. 8 visual depiction of the five features extracted from sensor signal of sensor 4 on board 1 and day 1 for 12.5 ppm of ethanol 494 l. mahmood, z. bahroun, m. ghommem, h. alshraideh they cover all the three phases of the signal and are able to extract relevant information from the raw signal. the pre-processing and feature extraction was achieved for eight sensors on all five boards, for all the four gases, thereby, resulting in a total of 40 features (5 main features x 8 sensors) to be used for classification and regression. the summary of the complete feature dataset is shown in table 7. table 7 summary of the complete feature set dataset 2 sensor no. feature name/feature number drop value average exposure value slope value in the ie fall time from 10% to 90% in the ie rise time from 10% to 90% in the er 1 f1 f2 f3 f4 f5 2 f6 f7 f8 f9 f10 3 f11 f12 f13 f14 f15 4 f16 f17 f18 f19 f20 5 f21 f22 f23 f24 f25 6 f26 f27 f28 f29 f30 7 f31 f32 f33 f34 f35 8 f36 f37 f38 f39 f40 3.2.2. machine learning classification the complete feature dataset of 40 features was subjected to machine learning for classification. 480 measurements (75% of the dataset) from boards 1, 2 and 3 were used to train the machine learning models, whereas, 160 measurements (25% of the dataset) from boards 4 and 5 were used for model testing. the training data was subjected to 10-fold, leave-one-out cross-validation to minimize the possibility of any model underfitting or overfitting. classification models such as the knn, svm, dt and rf were implemented and the corresponding results are shown in fig. 9. using k as 5 with the manhattan distance metric, the classification accuracy was found to be 96.25%. it was observed that changing the value of k beyond 5 and lower than 3 reduced the classification accuracy, as a result, the tested k values were limited to 5,4, and 3. for both knn and svm, the features were normalized to attain the same scale range following which svm rendered an accuracy of 99.38%. besides svm, the rf model using 100 decision trees, a tree depth of 10 and the information gain splitting criterion rendered a good performance with 99.38% accuracy. fig. 9 classification accuracies of machine learning models – dataset 2 assessment and performance analysis of machine learning techniques for gas sensing e-nose... 495 for knn, sbe improved the performance of the model from 96.25% to 98.75% using 38 features. on the other hand, using pca with 15 pcs along with dt enhanced the performance of the model from 77.5% to 90%. amongst the top 20 features shown in fig. 10, drop value features accounted for 35% and slope value features accounted for 30%. this clearly shows that although drop value features accounted for a higher percentage, ample importance was given to the features from the other phases of the signal response as well. fig. 10 feature weights for top 20 features in the dataset dataset 2 on implementing the machine learning models with different number of features based on their gini index weights, for all the four models, the accuracy steadily increased until the top 20 highest weighted features were used, following which it maintained a steady value (see fig. 11). this shows that the top 20 features are capable of providing a steadily increasing accuracy and attaining a more stable value requires the use of at least 20 features. fig. 11 classification accuracy using different number of features – dataset 2 496 l. mahmood, z. bahroun, m. ghommem, h. alshraideh 3.3. dataset 3 – experimental dataset of co with varying concentrations this dataset was collected through the detection of different carbon monoxide (co) levels using fourteen mox sensors under fluctuating humidity conditions [37]. the experiment employed seven different mox sensors (2 sensors of each type), resulting in a total of 14 mox sensors for detecting 10 different concentrations. each concentration was tested 10 times, yielding a complete dataset of 1,300 measurement samples. more details regarding the experimental procedure can be found in the work presented by burgués et al. [37]. each sample of co was measured for 900s, during which the sensor heater voltage levels were fluctuated to generate high and low sensor temperatures in cycles of 25s. as a result, the duration of each measurement being 900s, with one cycle of 25s, resulted in approximately 36 complete cycles during each measurement taken by burgués et al. [37], as shown in fig. 12. fig. 12 sensor response from sensor 1 and 2 to 20ppm of co 3.3.1. methodology based on the sensor responses, the three main features were extracted from 𝑟[𝑘 + ∆𝑥] that represents the sensor resistance, with 𝑘 as the discrete time-step, where 𝑘 ∈ [0,900] and ∆𝑥 is used to indicate any incremental change in the time-step as shown below: the maximum value of the signal amongst all the 36 cycles for a given gas is given as, maximum peak = abs. max k=0,1,…900 (r[k + ∆x]) (10) and the time-step at which the maximum peak occurs is defined as, timemax. peak = k + ∆x (11) drop value as the signal transitions from the peak value to the lowest value in that cycle is given as, drop = maximum peaklocal min k=0,1,…900 (r[k + ∆x]) (12) here, 𝑘 < 𝑡𝑖𝑚𝑒𝑚𝑎𝑥. 𝑝𝑒𝑎𝑘. on the other hand, the transition rate is determined through the slope as follows: assessment and performance analysis of machine learning techniques for gas sensing e-nose... 497 minimum value = local min k=0,1,…900 (r[k + ∆x]) (13) here, 𝑘 > 𝑡𝑖𝑚𝑒𝑚𝑎𝑥. 𝑝𝑒𝑎𝑘 and the time-step at which the minimum value occurs is defined as, timemin. value = k + ∆x (14) consequently, 𝒔𝒍𝒐𝒑𝒆 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑒𝑎𝑘 – 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑡𝑖𝑚𝑒𝑚𝑎𝑥.𝑝𝑒𝑎𝑘 – 𝑡𝑖𝑚𝑒𝑚𝑖𝑛.𝑣𝑎𝑙𝑢𝑒 (15) fig. 13 illustrates the three different features extracted as shown in eqs. (10), (12) and (15) from sensor 1 for the detection of 20ppm of co. this was done for the responses from all 14 sensors for each of the co concentrations, thereby, resulting in 42 features (3 main features x 14 sensors) to be used for regression. the feature and the feature number associated with each sensor is shown in table 8. fig. 13 visual depiction of the three features extracted from sensor signal of sensor 1 for 20ppm of co table 8 summary of the complete feature set dataset 3 sensor no. feature name/feature number sensor no. feature name/feature number maximum peak value drop value slope value maximum peak value drop value slope value 1 f1 f2 f3 8 f22 f23 f24 2 f4 f5 f6 9 f25 f26 f27 3 f7 f8 f9 10 f28 f29 f30 4 f10 f11 f12 11 f31 f32 f33 5 f13 f14 f15 12 f34 f35 f36 6 f16 f17 f18 13 f37 f38 f39 7 f19 f20 f21 14 f40 f41 f42 498 l. mahmood, z. bahroun, m. ghommem, h. alshraideh 3.3.2. machine learning – regression the complete dataset of 42 features with 1,300 measurement samples was used for regression. the dataset was split using the 70:30 ratio, where 70% of the original dataset was reserved for training and the remaining 30% was used for testing the machine learning models. both, classical models such as dt, knn, svm, ann and ensemble machine learning models such as rf, voting and bagging were used to perform the regression, wherein, the performance of the models was compared using the symmetric mean absolute percentage error (smape) to account for 0ppm concentrations present in the dataset. results of regression are shown in fig. 14, where the smapes for all the models hovered around 60%. however, the lowest smape was attained by rf (smape of 58.82%), followed by bagging using rf (smape of 60.28%). hyperparameter tuning for rf resulted in an optimal value of 48 trees, whereas, the hyperparameters for ann were tuned to 2 hidden layers with 2 neurons each. for knn, the optimal value of k was found to be 7 with manhattan distance. the dataset contains 10 repetitions for each concentration of co with varying levels of humidity for each repetition. as a result, the high smapes can be attributed to the lack of repeatability in the sensor responses due to fluctuating humidity levels. when determining the weights of the features using correlation, each type of feature, i.e., the drop value, the slope value and the peak value accounted for an equal portion of the top 21 features (33.3% each), which highlights their equal importance for regression. furthermore, to alleviate the impact of high smapes, sfs and sbe were implemented to reflect on any performance improvement that could be achieved. the results of using sfs and sbe indicated that feature selection proved to be successful in reducing the smape for all the models. for some models such as bagging using rf, the smape reduced significantly from 60.28% to 55.73% and rf, where the smape reduced to 52.71% from 58.82%. however, for some models such as svm, the smape remained almost the same irrespective of feature selection. in any case, the best performing models were found to be the rf, bagging using rf, knn and ann models. using sfs, the three types of features in the dataset were equally present in most of the cases, whereas, for sbe, an equal number of all the three features were dropped while training the models. fig. 14 regression performance of machine learning models – dataset 3 assessment and performance analysis of machine learning techniques for gas sensing e-nose... 499 using pca, 19 pcs were used along with the models to achieve a 95% variance, where models of rf (smape of 51.82%), knn (smape of 56.81%) and bagging using rf (smape of 54.12%) provided the lowest smapes. in other cases, the error remained the same or slightly decreased. this shows that for rf, knn and bagging using rf, a linear combination of the features through pca could be used to obtain a better performance, whereas, the remaining models relied on the original feature dataset to estimate the concentrations of co. on further observation, our results indicate the ability of the implemented models to be applied to regression tasks. they also highlight the importance of feature selection and pca in reducing computational time to improve the model performance. consequently, other factors that might have played a crucial role could be the sensor data itself. the inability of sensors to reproduce the same sensor response owing to external factors such as humidity can easily compromise the performance of the machine learning model for either classification, regression or both. 4. conclusion e-noses are used profusely for smart gas sensing applications such as medical diagnosis, environmental monitoring and food quality control. their ability to identify different gases and estimate their concentrations makes them extremely viable for gas sensing. in this work, we provided a detailed assessment of different machine learning techniques used in e-noses for the purpose of classification and regression using three different experimental gsa datasets. the present analysis discusses concepts of signal pre-processing, feature extraction, feature selection followed by classification and regression using machine-learning techniques. feature selection using wrapper-based techniques of sfs and sbs, and filter-based techniques accounting for gini index and correlation to determine feature importance is studied in detail to investigate the predictive capability of the developed models. in addition, the machine learning techniques employed were not limited to classical models such as dt, knn and svm but also incorporated ensemble techniques of voting, bagging and rf to offer a more comprehensive discussion. the performance of the models was assessed using classification accuracies and regression errors, followed by a comparison of the results obtained from models presented in previously published studies. in most cases, classical models such as knn, dt and svm showed great capability for classification problems as demonstrated by the obtained high level of accuracies. ensemble techniques such as bagging (rf) and rf outperformed their classical counterparts presented in this work and published in previous works. the pivotal role of feature extraction was demonstrated by the good performance of models. moreover, the models implemented in this work were able to offer more accurate prediction than most of the reported results obtained from the same datasets. in terms of dimensionality reduction, pca demonstrated improvement in the performance of the machine learning algorithms. however, linear discriminant analysis, a technique widely used for feature extraction and classification, also helps minimize data dimensionality and can be used with gsa datasets. in addition, regression results revealed the impact of several factors that come into play when applying machine-learning techniques. these factors include the size of the training datasets, where it is crucial to pay heed to the size of the dataset, so as to avoid model underfitting or overfitting to reduce their impact on the models’ performance. moreover, a large number of features, non-representative features and poor quality of data for training the models can be a massive challenge for both classification and regression. 500 l. mahmood, z. bahroun, m. ghommem, h. alshraideh despite the lack of publicly available gsa datasets, this work was able to identify three mox sensor-based gsa datasets to provide a delineated assessment of different machine learning models for classification and regression. diverse machine learning algorithms and techniques with different working principles were selected to perform classification and regression to provide a holistic discussion of machine learning in gas sensing. in all the three case studies, feature selection and dimensionality reduction were able to improve the predictive capability of machine learning models. in many cases, 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design and tuning of the lyapunov based nonlinear position control of electrohydraulic servo systems mitja kastrevc, nenad gubeljak, edvard detiček university of maribor, faculty of mechanical engineering, slovenia abstract. a complete study of the development of a nonlinear backstepping controller for an electrohydraulic servo system is shown in this paper. the authors present an optimized nonlinear mathematical model used as fundamental for computer simulation. a proposed nonlinear controller is suitable for research of behavior of the complete system in control. special attention is paid to the selection of tuning parameters. using the experience of earlier studies of the state-space controller where the additional feedback signals such as velocity and acceleration signal increase the frequency and damping factor of the system, the results were proved by computer simulation. the results show that by appropriate selection of tuning parameters the system can achieve the best reference signal tracking performance with a small tracking error. the proposed approach seems to be adequate not only for step reference signals but also for ramp and sinusoidal reference signals. however, the parameters of the backstepping controller can be optimized manually to achieve the best results required. key words: nonlinear control, nonlinear modeling, lyapunov methods, electrohydraulic servo system, computer simulation 1. introduction electrohydraulic servo systems (ehs-systems) are used in several fields of industry and represent an important part of automatization. the systems like robots, computer numerical control machines (cnc), press brakes, and computer-controlled testing machines use the advantage of hydraulic power systems. additionally to this, the positioning systems have some specific use spatially in testing machines, where the dynamic and accuracy are very important. smooth response characteristics with the received: june 07, 2022 / accepted july 28, 2022 corresponding author: mitja kastrevc university of maribor, faculty of mechanical engineering, smetanova 17, maribor slovenia e-mail: mitja.kastrevc@um.si 2 m. kastrevc, n. gubeljak, e. detiček capacity of generating high force and positioning accuracy are important for use in the area of testing machines ideal for material or testing of constructions. this capability in positioning exerts a significant impact on modern equipment for position control applications [1,2]. different authors present research proof of the usage of a different approach to improving the positioning of electro-hydraulic servo (ehs) systems [3,4]. the development of a suitable controller, which could cover most demands is very significant. it is well known that the dynamics of ehs systems are highly nonlinear [5-8]. the system model is normally stiff with fast dynamics for the hydraulics and relatively slow dynamics for the mechanical parts. models usually contain strong nonlinear elements such as the flow in orifices, friction, valve overlap, and input saturation. the oil characteristics depend on parameters such as temperature and air content [9-11]. therefore, the control strategy must be based on a nonlinear type of controller such as the nonlinear integrator backstepping approach. emphasis is essentially on the effect of the tuning parameters and on how it influences the dynamic behavior of the system. the backstepping control ensures the global asymptotic stability, the tuning parameters of the controller, nonetheless do greatly affect desired dynamic behavior [4]. a commonly used approach in the design of feedback systems, use the method of feedback linearization to eliminate nonlinearities. feedback linearization employs changes of coordinates to transform a given nonlinear system into an equivalent linear one. a major advantage of the feedback linearization approach is related to the cancelations of the system's nonlinear dynamics that are introduced in the design process [9-11]. some nonlinearities have positive effects on system stability; therefore, their cancellation can achieve the opposite effect, which can lead to unstable operation of the system. as a solution to this problem the integrator backstepping approach is an appropriate solution. the fundamental concept of the backstepping method is introduced by krstic et al. in their book [12]. the approach focusing on the stabilization problem in stochastic nonlinear systems is developed in the extension of this book. the backstepping control method is also presented in [13-15] where this technique is explained in detail for regulating and tracking problems. numerous applications in the industry were used to obtain successful control of electric machines, wind turbines, based power production, robotic production systems, and flight trajectory control. a backstepping controller is one of the recursive designed controllers. it is designed by step back toward the control input starting with the scalar equation which is separated from it by the largest number of integrators. because of this recursive structure, the design process of this controller starts at the known stable system and back out new controllers that progressively stabilize each other subsystem. the process terminates when the final external control is reached [9-11]. without uncertainties, backstepping can be used to force a nonlinear system to behave like a linear system in a new set of coordinates. backstepping can avoid cancellations of useful nonlinearities and pursue the objectives of stabilization and tracking reference input. for tracking problem backstepping always use the error between the actual and desired input to start the design process. design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 3 this method of control also provides a solution for optimal control problems [16], estimating parameters and adaptive control design as well as the development of robust nonlinear controllers. the observer-based backstepping technique has been designed for force control of the electrohydraulic actuator system [17] and an active suspension control system [18]. to obtain the best value for its tuning parameters the particle swarm optimization tool was proposed [19]. it is proved that the technique improves the transient stability and damping presented in the system. the performance of the backstepping controller depends on its gains or controller parameters. with the use of a simplified mathematical model, the value of control parameters k1, k2, and k3 are tuned manually to prevent the control signal from chattering. 2. mathematical model of the electro-hydraulic servo system a typical electrohydraulic positioning servo system consists of a hydraulic power supply (fig. 1a, pos. 1) e.g. constant pressure hydraulic pump with a relief valve and accumulator (fig. 1a, pos. 1), a flow control servo valve (sv) (fig. 1a, pos. 2), a hydraulic cylinder (fig. 1a, pos. 3), moving mass (fig. 1a, pos. 4), a position sensor and an electronic control unit. fig. 1 represents electrohydraulic servo drive (a-schematic representation, b-photo of the testing rig with added mass in moving trolley). because designing positioning servo systems are complex spatially for additional boundary conditions, computer simulation of the behavior of such systems is necessary. the analysis of the system dynamics is realized through mathematical modeling and computer simulation of dynamic behavior [20]. the mathematical model is based on physical laws that express the dynamic behavior of the ehs system through differential equations. however, this task is troublesome due to the multidisciplinary nature of the electrohydraulic system that requires electric, magnetic, mechanical, and hydraulic knowledge. nonlinearities of systems used in ehs drives consist of nonlinearities, which contribute to the dynamical response of the complete system. a) b) fig. 1 a) ehs positioning system b) electrohydraulic servo system with load 4 m. kastrevc, n. gubeljak, e. detiček the detailed description of different components used in the ehs system, the authors described in the articles [9-11]; therefore, for giving some explanation only the important facts will be presented. 2.1 servo valve dynamics a detailed description of the dynamic response of servo valve (sv) can be presented with the use of basic equations based on physics laws (detail description given in [9, 10, 21]. the dynamic behavior of sv involves a large number of parameters. to obtain an optimal solution, the inspection of step responses and frequency diagrams suggests an approximation of a second-order model of the form [22].   2 21 sgnv v v v hs v v vv x x x f x k u         (1) where: v – valve natural frequency, v damping coefficient, xv–valve spool displacement, fhs – valve hysteresis and response sensitivity, kv – valve flow gain, u – valve input signal. most producers of sv give information on the most needed parameters needed to obtain constants in eq. (1). additionally, the limitation of the highest spool speed is recommended for the simulation process. 2.2 hydraulic actuator dynamics the hydraulic actuator (cylinder) represents the next important part of the ehs system as the converter of hydraulic power into mechanical power. for calculation, the behavior of pressure in both chambers is needed. eqs. (2) and (3) give the chamber pressure dynamics by [21], as follows: aeff a a p p li lea a p q a x q q v        (2) beff b b p p li leb b p q a x q q v        (3) in eqs. (2) and (3) the effective bulk module is also variable, which depends on the pressure. in literature, there is also given the dependence on temperature, which can be also included in the calculation. the empirical equation for calculation of the effective bulk modulus presented by lee [21] for hydraulic cylinders is expressed as: 1 max 2 3 max log a aeff p a a a p            (4) 1 max 2 3 max log b beff p a a a p            (5) design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 5 with parameters a1=0.5, a2=90, a3=3, max=1800 mpa, pmax=28 mpa. the cylinder chambers volumes are given by  0 0a a p pv v a x x       (6)  0 0b b p pv v a x x        (7) where: x0 initial piston position, xp actual piston position, va0,vb0 the initial chamber volumes. these volumes consist of an efficient part (the volume required to fill only the chambers) and an inefficient part (volume of the pipelines between the valve and actuator) for side a and b, respectively. aa= ap is a piston surface area and the ab=ap rode side area where  =ab/aa is a ratio of cylinder bore area and the annulus effective area at the rod side [9, 21]. furthermore, in eqs. (2) and (3), qli and qle denote the internal leakage flow and the external leakage flow, respectively. leakage from one cylinder chamber to another, known as internal leakage flow, by an assumption of laminar flow can be calculated:  li l b aq k p p  (8) where: kl the internal leakage flow coefficient. the external leakage can be neglected. the dynamic subsystem of the piston and the moving mass is described by:    p f p a b p extm x f x p p a f       (9) where: mt the total mass, ff friction force, fext external load force. total mass m consists of the piston mass and the mass of hydraulic fluid in the cylinder chambers and the pipelines and the mass of the load. however, the mass of fluid can usually be neglected with the piston mass. in eq. (9) an important part is given to friction force. different authors give various solutions for representing the influence of friction. as a compromise for calculation presented equation is used:            sgn sgn p s x c f p v p c p s p vf p co p p sof x f x f x f x x f x x f e            (10) with: vf the parameter for viscous friction, fco static friction force, fso – stribeck friction force, cs – stribeck velocity. this friction force depends on the velocity sign and is restricted to viscous friction and columb friction with the stribeck effect. moreover, the friction force must be considered to obtain acceptable tracking accuracy. the natural frequency of components and the whole system is important data in the dynamical analysis. the natural frequency for the overall system equals the natural frequency of the hydro-mechanical part; therefore, the electro-hydraulic control system's natural frequency will be: h h c m   (11) 6 m. kastrevc, n. gubeljak, e. detiček to calculate the natural frequency of the hydraulic cylinder, the hydraulic cylinder stiffness must be found in advance. the total stiffness of the differential cylinder is defined by the following equation [21]: 2 2 a a b b h a b a a c v v      (12) in the case of a synchronous cylinder (double rod of equal area), aa is equal ab. for performance calculation, only the minimum stiffness will be considered since it has the worst effect on the system dynamics. 2.3 verification of mathematical model for simulation of electrohydraulic servo system (ehs), the matlab-simulink software is used. the parameters used in the simulation are given in tab. 1. external force fext can be used in the case where the system has additionally load opposite moving direction (in our case is no additional external force present). table 1 main parameters of the ehs system parameter value parameter value ap 6.4×10-4 m2 m 200 kg vf 70 (n s)/m vt 131.85×10-6 m3 fco 19.62 n  1.5×109 pa fext 0 n ps 21 mpa kl 3×10-13 m5/ns  850 kg/m3 cd 0.63 kv 5.53×10-7 m2/v the validation of the dynamical model of the ehs system has been proved by earlier experimental testing, where the ramp reference was used (fig. 2). the error difference between experimental and simulated responses is shown in fig. 3, where the difference between simulated and experimental is also shown. the mathematical model perfectly emulates the real system dynamics; therefore, the results of computer simulation of the system by using a nonlinear controller are relevant. design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 7 fig. 2 comparison of nonlinear mathematical model and the experimental response of the ehs system with ramp response fig. 3 the error of simulated and experimental response to reference 3. controller design the backstepping controller design in the case of nonlinear dynamical systems is a successful control strategy. nonlinear systems can be divided into subsystems, whose numbers depend on dynamical model order. because of the recursive structure, the designer can start the design process at the known-stable system and design,” back out”, new controllers that progressively stabilize each outer subsystem. the process is finished when the final external control is reached [9, 11]. 8 m. kastrevc, n. gubeljak, e. detiček the mathematical description of a nonlinear system in a new set of coordinates allows us to use the backstepping approach to force it to behave like a linear one [12]. the ability to avoid the cancelation of useful nonlinearities enables a backstepping approach to pursue the objectives of stabilization and tracking rather than that of the linearization method. therefore, the use of a backstepping controller is possible for solving tracking and regulation problems. for tracking problems, backstepping always uses the error between the actual and desired input to start the design process. 3.1 model simplification due to the complexity of the ehs system, the simplification of the model is preferable. because of the construction of the system the dynamics of components must be taken into consideration. if we compare the dynamical properties of sv and other parts of the system, we can see that the dynamic of the servo valve with control electronics is much higher than the dynamic of the cylinder and load. for a hydraulic system with ap=6.4×10-4 m2, vt=131.85×10-6 m3 and m=200 kg, the minimum natural frequency is: 2 min 2 215.866 34.35 p h t a rad hz m v s        (13) natural frequency can be read out, from the manufacturer's datasheet for sv moog 769 [20] we can read out fv=325hz. because of a very high natural frequency of the servo valve in comparison to hydraulic cylinder v >> hmin ), taking into account also a dynamic effect of the manifold which decreases the hydraulic natural frequency, the dynamic of sv can be neglected. the servo valve can be described only by the static relationship between the spool position and the valve current input [9, 22]. moreover, the combined assembly of the servo valve and the electronic amplifier can be described by the equation: v vx k u  (14) where: kv (m/v) combined sv and electronic amplifier gain. simplified system equations take the following form:       1 sgn sgn . sgn4 4 4 p s p p x p c ext p l vf p co p p so s l l d v p p l l t t t x x a f v p x f x x f e m m m p u p p c k u a x k p v v v                                    (15) where: xp actual piston position, vp piston velocity, pl load pressure. design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 9 3.2 nonlinear controller design the system state equations can be represented in “strict feedback form”:     1 2 2 1 3 2 2 4 3 5 3 6 2 7 3 sgn sgns x x x a x a x a x a p u x u a x a x            (16) where: x1 = xp, x2 = vp , x3 = pl, with: 1 2 3 4 5 6 7 4 4 4 , , , , , , p vf co ext d v p l t tt a f f a a a a a c k a a a k m m m m v vv                  the designing process of a backstepping controller is the usage of linear terms which corresponds to the lipschitz inequality given by:    f x f y l x y   (17) to satisfy the condition in eq. (17) the substitute of the “sgn” function with the continuously differentiable function “tanh = th” (hyperbolic tangent) is recommended. 1 2x x (18a)  2 1 3 2 2 3 2 4x a x a x a th x a     (18b)  3 5 3 6 2 7 3sgnsx a p u x u a x a x      (18c) where  free coefficient (selected value of is =2). the control task is to stabilize the plant and track the given reference signal asymptotically. the detailed derivations of finding backstepping control input are covered in the three steps [9, 11]. step 1 the first step is to find the first error variable and its derivate as shown in eq. (19) and (20). 1 1z x r  (19) 1 1z x r  (20) where: r(t) the reference input , x1 – position variable. defining virtual control 1 follows from in consideration of eq. (18a). the difference between actual and virtual control can be expressed with: 2 2 1 2 2 1z x x z      (21) now we can define a candidate control of lyapunov functional for this equation: 2 1 1 1 2 v z  (22a) 10 m. kastrevc, n. gubeljak, e. detiček 2 1 1 2 v z    (22b) 2 1 1 1 2 v z    (22c) authors rozali [3] and kadisi [4] proposed an additional constant  for fine-tuning needs shown in eqs. (22b) and (22c). this enables more flexibility in the tuning process, but also cause additional estimation for values in the tuning process. with the conventional method (without added fine-tuned parameter) the results can be achieved successfully in comparison to the classical linear control. the derivative of v yields:      1 1 2 1 1v x r z x r r       (23) now we can select a virtual control for the first-order system: 1 1 1 1 1 1r k z k x k r r       (24) where k1>0. hence, v can be rewritten as: 2 1 1 1 1 2v k z z z   (25) step 2 we use a similar method to achieve define a second virtual control as given in eqs. (26), (27), (28), (29) and finally (30). 3 3 2 3 3 2z x x z      (26) 2 2 1z x   (27) 2 2 1 2 1 2 v v z   (28) 2 2 1 2 2v v z z   (29)   2 2 2 3 2 4 1 1 2 2 1 1 a x a th x a z k z a           (30) where k2>0, therefore: 2 2 2 1 1 2 2 1 2 3v k z k z a z z        (31) step 3 this step is the final step to become the final control for the system: 3 3 2z x   (32) 3 2 3 1 2 v v z   (33) design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 11 3 2 3 3v v z z   (34) 1 2 6 2 7 3 2 3 3 5 3 1 1 s u a z a x a x k z a p x           (35) where k3>0. then we get: 2 2 2 3 1 1 2 2 3 3v k z k z k z       (36) where k1, k2, k3>0. note that eq. (28) is the lyapunov function of the system defined by eqs. (18a) to (18c) and that the control law is given by eqs. (24), (30), and (35) render its derivative negative semidefinite [9, 11] 4. selection of the design parameters before we start with the selection of the parameters of the backstepping controller the desired dynamic behavior of the system should be discussed first. different authors proposed different approaches to selecting the proper controller and parameters for it [23, 24]. references of servo-hydraulic drives are mostly: step, ramp, and sinusoidal signals. by use of conventional controllers such as pid controllers, the control parameters should be selected according to the dynamic characteristics of the hydraulic drive. in the case of positioning the system contains an integral dynamic behavior. the tuning method of the pid controller is based on the fact that i part of the controller should be avoided due to stability reasons. also, the d part of the controller should be avoided due to signal noise and the low damping factor of the hydraulic system. in some special cases, it is also possible to use a lag-lead controller. generally, only the p part of the pid controller is appropriate for position control. in most industrial cases the selection of the gain factor of the p controller should follow the demand of the aperiodic step response without any overshoot. to achieve these, only small gain factors of the p controller can be applied. similarly, the same restrictions about the dynamic response of the ehs system are taken into account in the case of the lyapunov-based backstepping controller. to choose the design parameters of the above-developed lyapunov-based position tracking controller we should inspect the closed-loop equations again. in our case, we have two virtual control equations 1 (24), 2 (30), with final control law (35) which contains three tuning parameters k1, k2, k3. k1, k2, k3 were determined by an experimental computer simulation method. in the beginning, we choose a step reference signal with a magnitude of 1mm. this value corresponds to velocity error by the ramp reference signal. starting point where a selection of equal values of all three parameters k1 = k2 = k3.the result is shown in fig.4. 12 m. kastrevc, n. gubeljak, e. detiček fig. 4 step response of ehs system by the backstepping controller in fig. 4 p-cont. means p controller with amplification k is equal to kp. the notification bs-1, bs-2 and bs-3 represents backstepping controllers with different parameters k1 , k2 and k3. aiming at achieving a fast response we try with a higher value of k2, while k1 and k3 remain the same as in the previous experiment (fig. 5). in the next step, we try with different values of k3 (fig. 6). higher values of k2 increase dynamic response (fig. 5). higher values of k3 increase the damping factor (fig. 6). fig. 5 step response of ehs system by different values of k2 design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 13 fig. 6 step response of ehs system by different values of k3 improvement of the previous result can be reached by increasing both k2 and k3, see fig. 7. fig. 7 step response of ehs system with different values of k2 and k3 finally, we realize the ramp reference response with previous values of k2 and k3. the results are shown in figs. 8 and 9. the differences between time responses by the p– controller and back-stepping controller are brightened in the detail. 14 m. kastrevc, n. gubeljak, e. detiček fig. 8 ramp response of ehs system with different values of k2 and k3 fig. 9 detail by approaching the final value of a ramp response of the ehs system with different values of k2 and k3 fig. 10 shows the response of the system on a sinusoidal reference signal. in this case, we use the parameters as were used in previous cases. we can see the influence of nonlinear control in comparison to the p controller. design and tuning of lyapunov based nonlinear position control of electrohydraulic servo systems 15 fig. 10 sinus reference response of ehs as shown in figs. 9 and 10, the behavior of the system controlled with the backstepping controller is significantly better than from linear p controller. we can see that the chosen parameters enabled better dynamical behavior of the system. the parameters with equal values of k (k1 =k2 =k3) give a better dynamical response than the p controller. however, the comparison of both shows significant improvement of system dynamic behavior in the case of backstepping controllers. 5. conclusion electrohydraulic servo actuators will still be the right choice for many application domains, for many years. the actuators play a vital role in maneuvering industrial processes and manufacturing lines. especially important is successfully closed-loop position, velocity, and force control of such drives. in the last years the clear goal of many researchers and engineers was where to find an improved control strategy. to acquire the highest performance of electro-hydraulic actuators a suitable controller has to be designed. this paper deals with the development of a lyapunov-based nonlinear controller based on the integrator backstepping approach. described procedures are focused on the selection of tuning parameters of such a controller. the proposed approach shows that the selection of appropriate parameters for the backstepping controller can be satisfactorily realized by the computer simulation method. the results of computer simulation are represented and conclusions are explained. the results of computer simulation and the 16 m. kastrevc, n. gubeljak, e. detiček proposed method for select parameters can be used in different reference signals. the proposed method allows setting the best starting parameters for use in the real system. the fact, that computer simulation enables to predict critical operations conditions, allows developers to avoid damage to the system. finally, by using these experiences, the electro-hydraulic servo system's performances can be essentially improved and the best position reference tracking performance with small steady-state error can be achieved. our future research will include the real-time and practical realization of the abovedeveloped nonlinear controller. acknowledgements: the authors are grateful for the financial support to the slovenia research agency arrs, for the research program p2-0137 numerical and experimental analysis of nonlinear mechanical systems. references 1. khalil, h.k., 2009, interactive analysis of closed-loop electro-hydraulic control systems, 13th international conference on aerospace scienties & aviation technology, asat-13-hc-01. 2. kovari, a., 2015, effect of leakage in electrohydraulic servo systems based on complex nonlinear mathematical model and experimental results, acta polytechnica hungarica, 12(3), pp. 129-146. 3. rozali, s.md., rahmat, m.f., rashid husain, a., 2012, backstepping design for position tracking control of nonlinear system, ieee international conference on control system, 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181-193. 20. wright, h., alleyne, a., liu, r., 1997, on the stability and performance of two-state hydraulic servovalves, proceedings of the asme dynamic systems and control division, 63, pp. 215-222. 21. jelali, m., kroll, a., 2003, hydraulic servo-systems:modelling, identification and control, springer – verlag, london, berlin, heidelberg. 22. moog. servovalves with integrated electronics d769 series. rapport technique, moog inc. 23. mavrinac, m., lorencin, i., car, z., šercer, m., 2022, genetic algorithm-based parametrization of a pi controller for dc motor control, technical gazette, 16(1), pp.16-22. 24. pingale, r., shinde, s.n., 2021, fuzzy logic approach for routing in internet of things network, technical gazette, 15(1), pp. 18-24. 6053 facta universitatis series: mechanical engineering vol. 20, no 1, 2022, pp. 37 52 https://doi.org/10.22190/fume200310039h © 2022 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds riad harouz1, abdelaziz lakehal1, khaled khelil2, olivier dedry3, neda hashemi3, said boudebane4 1department of mechanical engineering, mohamed cherif messaadia university, algeria 2department of electrical engineering, mohamed cherif messaadia university, algeria 3metallic and materials science unit (mms), university of liege, belgium 4department of metallurgy, badji mokhtar annaba university, algeria abstract. the paper examines the friction and wear behavior of four different wc/ticco cermets, where three of them are composed of 5%, 10% and 15% tic additions, and a wc-co grade without tic, taken as a reference material for comparison purpose. the principal aim is to improve wear resistance to high sliding speeds (hot rolling) of the wc-co material as a reference by adding previously-listed percentage of tic. the samples (cermets) were prepared according to the powder metallurgy procedure, which includes the preparation of the powder mixture, its compression shaping and liquid phase sintering. sintering was carried out at 1460 ° c, for 14 hours, in a reducing medium (h2). the tic materials are added in order to boost hardness of the wc-15co cermet and, consequently, its resistance to wear under thermomechanical conditions. the experiments are conducted using a pin-on-disc tribometer in contact with al2o3 alumina ball at two sliding speeds of 0.5m/s and 0.75m/s, at a high temperature of 450°c, and a 20 n load. it has been noticed that some recorded friction coefficients are unstable and exhibit many peaks during almost the whole friction test period. the obtained results from the sem microscope show that the wear behavior of the new proposed material is improved, where it has been shown that, at the sliding speed of 0.75m/s, the greater the tic percentage is, the lower the average friction coefficient will be. also, for the speed of 0.5 m/s, the average friction coefficient is relatively more stable with the tic percentage increase. moreover, the obtained experimental results show an average wear rate decrease, with respect to reference grades (na), that amounts to nearly 36% and 41% at the two sliding speeds p1 (0.5m/s) and p2 (0.75m/s), respectively. key words: friction, wear, sliding speed, cermets, wc/tic-co received march 10, 2020 / accepted october 19, 2020 corresponding author: abdelazizlakehal department of mechanical engineering,mohamedchérifmessaadia university, p.o. box 1553 souk-ahras, algeria e-mail: lakehal21@yahoo.fr 38 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane 1. introduction titanium carbide (tic) based cermets, used in many industrial applications, have received considerable attention due to several interesting properties at a high temperature, including a good tribological behavior, a better strength to weight ratio, and a higher wear resistance [1-3]. also, the additions of tic in the mixture of co wc powders are justified by a better hot stability of titanium carbide compared to tungsten carbide. on the other hand, tungsten may produce volatile oxides which then accelerate the degradation of the surface of the material subjected to high temperature wear. the manufacture of cermets (wc-co) could be accomplished by various techniques such as liquid phase sintering and conventional molding processes including injection molding. yang et al. [4] give reviews of the additive manufacturing (am) process, which appears to be an alternative to the traditional manufacturing processes able to produce parts with very complicated geometry. however, in this work we have adopted the classical technique based on liquid phase sintering because of its relative simplicity and good control of its parameters. even though wc-co cermets, without any addition, are widely employed as a cutting tool for high speed machining, hard metals and hard high-alloy steels, they exhibit poor oxidation resistance and corrosion [5, 6]. also, it has been shown that these materials present surface degradation under high thermomechanical stresses and a high sliding speed [7]. several authors have reported that adding tic to tungsten carbide-cobalt (wc–co) results in cermets with better performance just as they significantly enhance wear resistance of the wc/tic cermets. duman, et al. [8] investigated these performances in thermo-mechanical and high sliding speed applications. yang at al. [9] examined the impact of tic addition on the oxidation behavior of the wc-co cermet, following the appearance of complex oxides (w-ti-o) which are more stable and more adherent than wo3 oxide, volatile at medium temperature. furthermore, these wc/tic grades show interesting properties, with respect to conventional wc-co cemented carbides, including high hardness, good wear and oxidation resistance, high thermal and chemical stability, and a relatively low friction coefficient at a high temperature [10-12]. these materials are used in many industrial devices, involving high temperature dry contact components, such as hot-rolling dies, and aerospace equipment [13]. furthermore, many authors have recently reported a considerable improvement of wear resistance of the wc-co cermets, through the incorporation of tic carbide particles [14-16]. the aim of the work is to examine the influence of the tic addition on the cermet wear behavior at a high temperature for two different sliding speeds. in this study, four grades, prepared by the procedure of powder metallurgy (wc, co, tic), are employed. the four shades of cermets were prepared with different ratios of wc/tic-co contents (5%, 10%, and 15%) of tic including a reference grade without tic (wc-co). the tribological behaviors of these cermets were studied by severe friction tests for dry rotary sliding. hence, this paper investigates the wear behavior of these wc-cotic grades at a high temperature and high sliding speed, and compared to the reference wc–co cermet. the experimental tests have been carried out under a load of 20 n at a temperature of 450°c using two sliding speeds of 0.50 m/s and 0.75 m/s. these parameters have been selected depending on the test machine (high temperature tribometer) characteristics and the use conditions of the wc-co-tic material. the chosen temperature represents an average value of the working temperature of the considered material. for the two sliding speeds, our choice is due to the fact that the underlined material employs an average speed of about 0.5-0.75 m/s in hot rolling applications. the load of 20n is an average machine value. the dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds 39 obtained results have shown that adding tic carbide particles has a positive effect on the wear behavior of the underlying cermets for the two considered speeds. moreover, it has been noticed that addition of tic stabilizes, somewhat, the friction coefficient even at a high speed. also, we have observed the formation of an oxide film made of tungsten and cobalt for the two considered speeds of 0.50 m/s and 0.75 m/s, respectively. also, lemboub et al. [17] show by microstructure analysis the formation of a mixed carbide (w, ti) c of « core-rim » type morphology. 2. experimentation and materials 2.1. cermet preparation the mean size of the particles and powder density of wc, co and tic, manufactured by norinco group china, are given in table 1. table 1 the mean density and particle size of various powders wc, co and tic various powders wc co tic particle size(µm) 1.1 2 4 average density of the powder (g/cm3) 4.5 <0.75 4 the employed cermets, made of various compositions, reported in table 2, are prepared using a classical procedure of powder metallurgy. the cobalt composition has been chosen to be 15% co. this percentage allows better heat removal when the cermet is heated during operation and guarantees a compromise between hardness and toughness [18,19]. table 2 weight percentage of the samples grades composition in wt.% wc % %co % tic na (reference shade) 85 15 nb 80 15 5 nc 75 15 10 nd 70 15 15 in order to guarantee homogeneity of the microstructure of each shade, a grinding of the powder prepared in mass percentage of the shades is poured into a ball mill (fig. 1). this operation makes it possible to obtain homogeneity of the different mixtures of the powder of the samples, prepared in a humid medium formed of c2h5oh alcohol during an average grinding of 24 hours. fig. 1 powder ball mill 40 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane the mixture of powders is dried in an oscillating dryer (oven) intended for drying the powders of the composition (wc-co-tic). the average drying time of 30 to 40 min causes a vaporization of the alcohol c2h5oh; then the dried powder is mixed with an adhesive to ensure good agglomeration of the three different powders of the composition. finally, a forced sieving of the agglomerated powder allows the separation of other elements (large grain sizes) and obtaining grains of the same size. 2.1.1. the microstructure of grades before wear tests the microstructure of grades before wear tests, depicted in fig. 2, shows a sem micrograph of the wc-co-5tic sinter which clearly distinguishes different constituents and their distribution in the structure. the compression of the dried and sieved powder is carried out by a hydraulic press. the pressing values are related to the density (compaction) and the geometry of the specimen. the geometry of the specimen (15mm thick and 35mm in diameter) is calculated while considering the shrinkage of the specimen after cooling. finally, an average pressing of 15 mpa is applied, followed by a steaming operation for an average duration of 3 to 4 hours in an oven in order to vaporize the glue of the test pieces. fig. 2 microstructure sem image of grade (nb) 5% tic before wear tests the sample sintering cycle (cermets) is a standard cycle with three stages: heating to 600 ° c followed by a rise to 1460 ° c, and cooling for an overall average time of 14 hours in the oven. the sintering operation consists of heating the test pieces to such a temperature that the powder grains weld together. the sintering cycle is carried out in a tubular chamber in the furnace blown by an argon gas in a non-oxidizing atmosphere (ar + h2). 2.1.2. characterization of samples the hardness and the density tests, carried out using induno olona-va-italy device and archimedes method, respectively, have shown that the obtained samples comply with iso standards as reported in table 3. the microstructure of the wear track is analyzed by employing a philips-branded feg esem xl 30 microscope. dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds 41 table 3 the mean density and particle size of various powders wc, co and tic grades density(g/cm3) hardness (hrc) na (reference shade) 13.86 69 nb 13.48 71.6 nc 11.74 74.4 nd 10.7 77.7 2.2. xrd phase analysis fig. 3 reports the xrd diffraction spectrum obtained on the wc-5%tic-co cermet. the specifications of the used xrd machine are: (operation angle range: 15-90°, step angle: 0.02°, kcuanticathod: 0.154056 nm, power 3kw, max tube current 60 ma). note that the identification of the phases was carried out through the use of xpert high score plus and confirmed on the basis of the jcpds international center for diffraction data 1996 files. besides wc, which is the main phase of cermet, the figure shows peaks of the complex carbide co3w3c, formed from the interaction between cobalt and wc during sintering in the liquid phase. the spectrum also highlights the simple tic carbide, represented by very distinct peaks. the superposition of the wc and tic peaks, corresponds to the mixed carbide (w, ti) c, resulting from the solubility between the tungsten (wc) and titanium (tic) monocarbons. fig. 3 xrd patterns of grade (nb) 5% tic before wear tests 3. tests of friction wear tests were performed at 450°c for two sliding speeds and without lubricant on a csm high temperature pin-on-disc tribometer (fig. 4).the test process involving the al2o3 ball and the two cermets (wc/tic-co) and the reference (wc-co) samples have undergone friction and rotary sliding tests at a high temperature using a tribometer (csm 42 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane instruments switzerland) (fig. 4)[20]. the four considered cermets have experienced tests, at two speeds (table 4), using a 6 mm diameter alumina ball with hardness (hv) and density values of 1800 and 3.2, respectively. for each grade, three friction tests, corresponding to three radii, are carried out for two parameter sets p1 and p2 (force, temperature and sliding speed) (see fig. 4). thus, the estimated average value of the coefficient of friction is recorded. fig. 4 friction test process between the al2o3 ball and the cermets table 4 wear test employed parameters thermo-mechanical parameters of wear tests parameters applied load (n) sliding speed (m/s) contact temperature (°c) ambient temperature contact (°c) distance traveled (m) time of test p 1 20 0.5 450 20 5000 2h 46min p 2 20 0.75 450 20 5000 1h 53min a wear track of 5 mm radius and centered on the test cermet disc (fig. 5) is obtained due to the ball rubbing against the sample surfaces. fig. 5 friction wear track of the samples after tests the software tribometer module (version 4.4.q), with a sampling time of one second, is used to regularly record the force and coefficient of friction, humidity, the penetration depth of the ball, the grades and furnace temperatures (fig. 6). dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds 43 fig. 6 schematic illustration of the pin-on-disc tribometer two sliding speeds of 0.50m/s and 0.75m/s at 450°c are employed to test our wc/tic-co cermets usually employed in hot rolling tools. the test durations, for the 5000 m travelled distance, are 2 h 46 min and 1h 53 min for the two considered speeds of 0.50m/s and 0.75m/s, respectively. the choice of 450°c is due to the fact that this temperature could be considered as an average operating temperature of the cermet, intended for the preparation of hot rolling mill rollers. the selected distance will allow a wear normal regime operation of the materials and, therefore, obtain a significant wear volume. 4. wear rate measuring procedure and friction track characterization the track wear rates are measured by a profilometer (3d optical microscope contour gt-i brukernano surfaces division tucson, az usa). the wear track image, illustrated in (fig. 5), is analyzed using the vision software. the analysis, using an optical profiler (fig. 7), gives a geometrical profile of the sample track whose volume is characterized by three parameters namely, depth (in μm), width and perimeter (in µm). fig. 7 depth profile (μm) vs. wear track width (mm) and four images were taken at 90° from each other (fig. 8), where a mean profile is obtained and the corresponding worn surface is calculated for each one. then, an average worn surface is computed and multiplied by the track perimeter in order to obtain wear profile worn volume v [16,21,22]. zhu, et al. [23] estimated the rate of wear (k) from the worn volume for a covered distance of 5000 m and using a load of 20 n. however, for a load p in (n) and a traveled distance d in (m), the wear rate is given as: 44 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane v k p d =  (1) fig. 8 the four optical profilometer measured sections spaced apart by 90° 5. results and discussion 5.1. friction and wear behavior of the cermets long-term friction tests of 5000 m covered distance for the wc-co-tic and wc-co cermets, with severe tribological parameters, were conducted. for two parameters p1 (0.5 m/s) and p2 (0.75 m/s) (table 4) used to test the considered cermets, no significant difference concerning the friction coefficients was observed between the two speeds. the long time period friction tests, for such hard materials, were chosen in order to surpass the running-in regime (severe wear) and achieve the mild wear regime. employing parameters p1 and p2, friction coefficient (cof) against the travelled distance for four considered samples nb 5% tic, nc 10% tic, nd 15% tic and reference na with no tic addition, are given in fig. 9. fig. 9 variation of the friction wear coefficient with respect to reference grade (na) for two sliding speeds: a) p1 (0.5m/s) and b) p2 (0.75m/s) for parameter p1(0.5m/s) in (fig.9a)) and after a short sliding distance of about 400 m, the cof for grades (nb), (nc) and (nd) exhibit a relatively slow variation with a mean value of µ=0.58 which is smaller than that of reference grade (na). grade (nc) shows a better stability of the friction coefficient, with an average value of μ = 0.56 along the whole test. reference grade (na) is practically unstable with an increasing cof showing small ripples from the point 1500 m up to the end of the friction test. the instability observed for na (the basic composition, i.e. without the addition of tic) on the coefficient of friction curve could be explained by brittleness of the angular-shaped wc particles where the dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds 45 constraints are concentrated. these are the cause of cracking of the rupture of the surface layers of the cermet during sliding. the debris thus formed constitutes a third body which intensifies and regenerates friction. in (fig. 9.b) for parameter p2 (0.75m/s), the cof shows a stable variation with relatively large average values of about μ=0.4 for na, μ=0.9 for nb. the friction behavior of sample nb containing 5% tic could be attributed to structural heterogeneity due to poor distribution of tic particles during the homogenization operation of the powder mixture. the cof of the nc grade drops sharply after 3000 m of the test which shows heterogeneity within this region, then rises and stabilizes, until the end of the test, around an average cof value of approximately μ = 0.6. the nd grade indicates stability with an average value of μ = 0.57, showing micro peaks along the whole test probably due to heterogeneity of the microstructure. hence, for this large sliding speed of 0.75 m/s under the considered test conditions, it seems that the addition of tic has a negative influence on the friction coefficient. the impact of the tic addition on the cermet tribological behavior, at a high temperature, could be explained by the graph of (fig.9) which exhibits a stabilization level during the surface degradation process. on the other hand, the curve of the reference cermet, without addition of tic, which presents, in a first stage, lower values of the cof, increases monotonically with the increase in the sliding distance. this will lead to much higher values compared to those recorded on the samples containing 5% tic. this could be explained by the stabilization of the layer due to the formation of complex oxides. 5.2 wear microstructure analysis the microstructure analysis is represented by the sem image as shown in fig. 10. the microstructure of the wear trace of the wc-co-5% tic material after sliding against an al2o3 alumina ball with two sliding speeds p1 (0.5m/s) and p2 (0.75m/s) shows wear mechanisms which produce two types of wear represented by three zones as shown in figs. 10b) and d): ▪ zone a: weak cracked oxidized zone, abrasion wear. ▪ zone b: strongly oxidized and weakened zone, detachment of oxides, formation of cavities by decohesion. ▪ zone c: weakly oxidized zone, homogeneous wear by abrasion. in figs. 10 a) and c) are shown the traces of the wear track for the two speeds which indicates the influence of the increase in speed fig. 10c) an abrasive wear layer, which are micro grains on the track. also for p1 (0.5m/s) sliding velocity shows also that the observed wear track is essentially created by debris and tiny oxidized zones. the process of sliding velocity increasing p2 (0.75m/s) illustrate that the generated track is caused by larger oxidized debris of the abrasive wear. an oxide layer is generated which facilitates the development of wear resistance proven by the observed stability variation and the obtained friction coefficient value. moreover, in both figs. 10b) and d) few hard metal micro cracks, in various directions, are noticed at the interface. 46 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane fig. 10 sem micrograph of the grade (nb) wear track for: a) and b) sliding speed p1 (0.5m/s), c) and d) sliding speed p2 (0.75m/s) fig.11 illustrates the same sem/eds analysis at points (a), (b) and (c) at sliding speed p1(0.5m/s) of grade (nb) wc-co-5%tic showing spectra of different peaks containing elements (c, o, al, ti, co, w) whose composition percentages are given in table 5. the figure shows homogeneity of small debris of tungsten oxide whose existence is confirmed by the elements of the ball (o, al) (table 5) found in the wear track. hence, this analysis may prove that the presence of tic in the grade (nb) has delayed the oxidation, improving, therefore, the wear and oxidation resistance. the microstructure analysis using sem/ eds at points (a), (b) (c) and (d) with sliding speed p2(0.75m/s) of grade (nb) wc-co-5%tic (fig.12) illustrates the formation of an oxidized layer (tribofilm) composed, mainly, of an oxide rich in cobalt at points (a) and (c), and an oxide rich in tungsten at points (b) and (d) as reported in table 6. furthermore, the analysis reveals that the formed oxide is adherent, compact and spread over the whole wear track (oxide film formation) which boosts the wear resistance at high temperature. besides, the eds spectra of the wear tracks shows transfer of a small amount of the material (al) from the alumina ball to the track indicating the occurrence of a wear process. the structure of the wc-5% tic-co cermet, obtained by liquid phase sintering, is shown in fig. 11. it consists of a set of wc particles welded by the complex compound co3w3c. dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds 47 other wc particles bind to tic, through a dissolution-precipitation mechanism in liquid phase, to form the mixed carbide (w, ti) c which acquires a "core-rim" morphology. fig. 11 sem/eds micrographs of the grade (nb) cermet wear track at sliding speed p1 (0.5m/s) 48 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane fig. 12 sem/eds micrographs of the grade (nb) cermet wear track at sliding speed p2 (0.75m/s) dry sliding friction and wear of the wc/tic-co in contact with al2o3 for two sliding speeds 49 table 5 the spectra (a, b, c) elements on different points for the sliding speed p1(0.50m/s) tracks elements c o al ti co w total spectrum (a) 7.024561 0.68357 0.112209 0.064895 2.608032 89.50673 100% spectrum (b) 6.597754 11.20896 1.057664 3.137928 19.13305 58.86465 100% spectrum (c) 5.511438 10.03151 0.691847 3.510701 12.28522 67.96928 100% table 6 the spectra (a, b, c) elements on different points for the sliding speed p2(0.75m/s) tracks elements c o al ti co w total spectrum (a) 4.010348 5.024438 1.187481 0 81.87428 7.903455 100% spectrum (b) 3.343464 20.52027 1.055945 0.350977 31.37472 43.35462 100% spectrum (c) 4.194662 22.07891 3.678074 0.131206 65.34973 4.567423 100% spectrum (d) 4.003289 14.01891 0.652586 0.105596 13.12244 68.09717 100% 5.3. wear rate comparison of the wear rate for the three considered grade cermets, with respect to reference grade (wc-co), in terms of tic addition and using two sliding speeds p1(0.5m/s) and p2(0.75m/s), is illustrated in fig. 13. it is clear that adding tic results in a considerable decrease in the wear rate as reported in table 7. also, for the three cases, without tic, 5% tic and 10% tic, the average difference of the wear rate between the two speeds is about 55%, whereas for 15% tic addition, the wear rate difference has decreased to nearly 39%. in other words, it can be concluded that increasing the tic percentage may reduce the speed influence on the wear rate. fig. 13 wear rate of different grades cermets according to reference grade (na) for two sliding speeds p1(0.5m/s) and p2(0.75m/s) 50 r. harouz, a. lakehal, k. khelil, o. dedry, n. hashemi, s. boudebane table 7 wear rate decrease in % with respect to reference grades (na) at two sliding speeds p1(0.5m/s) and p2(0.75m/s) for the travelled distance of 5000 m tic percentage (%) siding speed (m/s) 5% 10% 15% 0.50 14.3% 43.9% 51.2% 0.75 13.3% 44.3% 63.9% the improvement in the wear resistance of the wc-co cermet, observed during the addition of tic, could be explained as follows: titanium carbide modifies the morphology of the wc-particles, which passes from an angular or faceted shape to a rounded shape. according to several previous works [24-26], the absence of sharp angles on the rounded particle eliminates the concentration of stresses and, therefore, the risks of failure by propagation of brittle cracks in the material. this leads to a significant enhancement in resistance. on the other hand, the appearance of a “core-rim” type structure following a phenomenon of dissolution of fine particles of carbides and reprecipitation of mixed carbide during sintering in the liquid phase could have a positive effect on the wear resistance of the cermet. the hardness of the cermet will then increase with the rate of the tic additions. 6. conclusion in this work a comparative analysis of the friction and wear rate for three cermets nb, nc and nd with tic addition ratios of 5%, 10% and 15%, respectively, has been carried out. the comparison has been conducted with respect to a reference sample na, with no tic addition, for dry rotational sliding friction tests against al2o3 ball, at two sliding speeds p1(0.5m/s) and p2(0.75m/s) and a fixed temperature t=450°c. in addition, using sem and eds micrographs, the nb grade microstructure has been examined. from the obtained experimental results, some conclusions are drawn: ▪ for the sliding speed of 0.5 m/s, the friction coefficient of grade (nc) has revealed a better stability with respect to (nb) and (nd) cermets which, in turn, have shown an acceptable stability of the cof coefficient. reference grade (na) is practically unstable with an increasing cof with small ripples. ▪ for the second speed of 0.75m/s, the cof has exhibited a quite stable variation for the four considered cermets, with a relatively large average value due, probably, to tic addition. ▪ the addition of tic has resulted in a significant decrease of the wear rate. also, adding tic has clearly reduced the influence of the speed on the wear rate. ▪ through sem/eds images of grade (nb) wear tracks, an oxidation layer made, mostly, of tungsten, cobalt and oxygen, has been noticed. the formed layer (tribofilm), is compact, adherent and spread over the whole wear track improving, therefore, the wear resistance at the considered temperature of t=450°c. moreover, these images show the formation of mixed carbide (w, ti) c with a "core-rim" morphology type. 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2014, cemented carbides for mining, construction and wear parts, in comprehensive hard materials, elsevier science and technology, editor: vinod k. sarin, pp. 425-251. 11849 facta universitatis series: mechanical engineering vol. 21, no 3, special issue, 2023, pp. 433 451 https://doi.org/10.22190/fume230521023r © 2023 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper multi-criteria selection of standards for system analyst activities in organizations miomir rakić1, miodrag žižović2, boža miljković3, angelina njeguš4, mališa r. žižović5, igor đorđević6 1informatika inc., belgrade, serbia 2economics institute, belgrade, serbia 3faculty of education sombor, university of novi sad, sombor, serbia 4faculty of informatics and computing, singidunum university, belgrade, serbia 5faculty of technical sciences čačak, university of kragujevac, čačak, serbia 6faculty of computer science, megatrend university, belgrade, serbia abstract. the paper defines the methodology for the selection of standards and tools that system analysts use for the purposes of analyzing the computer infrastructure of organizations. the goal of system analyst analysis is primarily to increase: the speed of processing information, the efficiency of data processing and data exchange. a large number of standards and tools complicates an adequate choice, and on the other hand system analysts are not the only ones who influence the choice. in the analysis of the choice of standards and tools, in the second chapter an algorithm for determining the weight coefficients for the criteria selected by the company and the ranking of the standards for selection is presented through an example. in order to choose the best standard, the choice was made between four standards described in the third chapter. the chapter defines the criteria that cover it activities for the selection of standards according to selected criteria and defined weight coefficients in the observed company. finally, on the assumption that the company together with the analyst chose “k” criteria, where the methodology (lbwa) for the selection of weight coefficients for the criteria was proposed, a new model for calculating the ranking of standards was presented, i.e. the choice of the best. the conclusion of the paper is that the presented procedure for choosing standards is not complicated, that it is very successful, and that in this way the proposals were more easily accepted and implemented in the fastest way. key words: standard, recording processes, analyzing processes, lbwa method, multi-criteria analysis, bpmn, idef0, idef3, dfd received: may 21, 2023 / accepted august 05, 2023 corresponding author: boža miljković faculty of education sombor, university of novi sad, podgorička 4, 25000 sombor, serbia e-mail: boza.miljkovic@pef.uns.ac.rs 434 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević 1. introduction system analysts are it professionals who work on analyzing the needs of an organization’s computer infrastructure. their primary goal is to increase information processing speed and efficiency of data processing, and to utilize existing computer resources (hardware, software, and personnel) in the most effective way to achieve better economic results in the organization's operations. their goal is also to assess the organization's position in the market in the same field, and whether organizational changes and actions are needed to improve business processes and procedures in the organization in the near future, in order for the organization to be among the leaders in the field and to meet the assigned tasks. similarly, system analysts work on long-term research, identifying and studying problems. they propose solutions chosen from multiple possibilities in coordination with all participants in the organization's activities who may be interested in these solutions or who may be affected by them, with the aim that everyone works and collaborates to make the proposals more easily accepted and later implemented in the fastest way possible. system analysts can be experts who work within an organization. in this case, they hold a high position in the organization hierarchy (usually with few superiors). alternatively, they can be employed by another organization and perform these tasks upon the request of a given organization. in the first case, it can be said that system analysts continuously work on organizational changes aimed at achieving better business results for the organization, while in the other case, they are engaged by the organization's management to study its position in the relevant field and to work on organizational changes as needed. this paper does not discuss organizational changes or types of business organization. we note that this job is almost always difficult, requiring immense energy and knowledge in the areas of information technology, organization, and the technology that people in the organization are involved in, as well as psychology, as it is necessary to ensure that individuals within the organization accept potential changes as a change that will also benefit them, as well as the organization itself. likewise, sociological aspects of these tasks must be considered, as they are often related to relationships among interest groups within the organization, as well as relationships within the broader community, as organizations are almost always an important factor for the life of the local community, or if we look more broadly, state communities. therefore, organizational changes that system analysts design are mostly changes related to technology and the use of information technology, and they are independent of the type of organizational structure of the organization, i.e. they must fit into every type of organization. simply put, the principle is used that what is technologically better should be applied (similar to james miller's idea of "living systems", good new theories do not eliminate previous good theories but incorporate them into the new theory this theory by j.g. miller is highly applicable for explaining the work of an organization as a whole and of each of its individual parts). when starting a job, system analysts first need to record the organization's operations (focusing on what is relevant to their work), and for that, they need to use certain tools and standards. today, numerous tools and standards are available for this purpose. however, all these tools and standards have certain advantages and disadvantages because they were created mostly for specific purposes, so the right choice is often crucial for successful work. in practice, the choice is primarily made through recommendations from colleagues who multi-criteria selection of standards for system analyst activities in organizations 435 have used certain standards and tools in specific situations, and system analysts themselves typically choose the tools and standards they are accustomed to in their practice or those they have previously used. if the situation arises where something else must be used, usually it is done on the recommendation of colleagues. in existing literature [1], we find that the number of these standards and tools is continuously increasing, but there are no defined rules on how to make a choice between these standards and tools based on certain criteria and precisely defined indicators related to those criteria. this can be explained by the fact that system analysts are not the only ones making this choice. managers of organizations have a crucial role in the selection process, as they will be the ones who will primarily use these standards and tools for their daily work. analysts certainly have great influence with their experience and knowledge, but they are not the deciding factor. however, we must note that analysts as experts have a significant impact on the selection of standards and tools, as they define the criteria that these standards and tools individually fulfill and evaluate how well the proposed criteria meet specific standards. in this paper, experts have defined possible criteria (there are 28 of them), and ratings from 0 to 1 have been given for each criterion. the assumption is always regardless of the nature of the criterion that it is a maximizing criterion, that is, 1 is the best rating and 0 is the worst rating. however, in the observed case, it is acceptable to choose a criterion with a rating of 0, and it is not discarded for that reason. therefore, minimizing criteria have been translated into maximizing criteria. in the specific analysis, it was found that in each particular case, the choice of criteria is made in coordination between the management of the organization and system analysts who deal with organizational changes, or it organizational changes in the organization. therefore, each organization will have its own criteria for choosing standards and tools [2]. however, since the standards and tools for this work are not developed for a specific organization, it can be noted that they are developed for use in various organizations that will choose the standards and tools that have criteria they need. the paper deals with criteria that exist in various standards. these criteria are of a general nature and are differently fulfilled in different standards the ratings of the fulfillment of a certain criterion in a standard are given by experts, i.e. experienced system analysts, for each standard and each criterion individually. clearly, the organizations themselves will rank their selected criteria according to importance for their work, and this will be the basis for choosing standards. this ranking will be done by determining the weight coefficients of the criteria by the organizations themselves. the method of analysis and synthesis examples according to the experiences of experts has been used. in this paper, we limit ourselves to proposed criteria, and solutions. according to the experiences of experts, these criteria often have emerged in different cases of organizations and institutions that they have observed and worked in as system analysts. concrete numerical data to process analysis was used as an example to open questions about the advantages and disadvantages of chosen criteria and method. the disadvantages of this method are the limit of chosen criteria and the limit of concrete numerical data of weight coefficients for example. however, at the end of the work, the case of adding new 436 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević criteria for the system analysis of a particular organization was analyzed. adding new criteria is done by simply introducing new criteria into the existing model as shown. the major problem in mca (multi-criteria analysis) models is determining the importance of criteria and ist weight coefficients. generally, authors tend to divide models for determining the weight of criteria into subjective and objective ones [3]. the subjective approach to determining weight criteria is based on information obtained from decision makers or experts, as is the case with the swara (step‐wise weight assessment ratio analysis) method [4], bwm (best worst method) [5] or fucom method (full consistency method) [6], etc. in the case of an objective approach for determining the weighting factors, the opinions of the decision-makers are ignored, and decisions are made based on the information obtained in the decision-making matrix. such a method is e.g. fanma method [7]. also, methods that are often used to determine the measure of preference of one criterion over another are methods based on pairwise comparisons [3], and they are the ahp method, bwm, and dematel method [8]. nevertheless, the method most often used for determining the mutual influence of criteria and for determining the diagram of relational relationships is the dematel method (except for the widely used bmw method due to the comparison of fewer pairs) [9]. each of these methods, in addition to their qualities, functionality, simplicity, and applicability, also has disadvantages. one of the methods that is often used, which relatively well overcomes the shortcomings of the mentioned methods, is based on the level based weight assessment model (lbwa), whose algorithm requires a small number of comparisons in pairs of criteria and whose mathematical algorithm is rational and logical [10]. through research conducted on various examples [11, 12, 13], and by comparing the values of weighted coefficients obtained by applying the lbwa model with the values of weighted coefficients in other studies, almost identical weight values were obtained. thus, validating the lbwa model. the lbwa model has a number of advantages that favor it for application, especially because it is a model that is flexible to additional corrections of the values of the weighted coefficients. in addition, this model is suitable for modeling problems in an uncertain environment, because even in that case it enables the processing of expert preferences [10]. therefore, the subject of this paper is to determine the weight coefficients for the criteria that are selected by the organization and then rank the standards accordingly. before defining the criteria and rules for selecting standards and tools, the next chapter provides a description of the four most commonly used standards that will be the subject of selection in this paper. in the third chapter, criteria (28 in total) covering it activities in almost all observed organizations are defined. in addition, expert assessments from this field are given on the fulfillment of these criteria in the proposed selection standards. finally, in the chapter of standard selection on the assumption that the company chose the methodology lbwa (level based weight assessment) for the selection of weight coefficients for the criteria was proposed, a new model for calculating the ranking of standards was presented, i.e. the choice of the best. the conclusion of the paper is that the presented procedure for choosing standards is not complicated, that it is very successful, and that in this way the proposals were more easily accepted and implemented in the fastest way. multi-criteria selection of standards for system analyst activities in organizations 437 2. selection standard alternatives (a1, a2, a3 i a4) are presented through the most widespread standards currently applied for recording and subsequent process analysis: ▪ dfd (data flow diagram) (a1) ▪ idef0 (icam definition for function modeling, where icam is an acronym for integrated computer aided manufacturing) (a2) ▪ idef3 (integrated definition for process description capture method) (a3), i ▪ bpmn (business process model and notation) (a4). each of these standards has its own capabilities, specifics, advantages and disadvantages, and they have certain limitations in relation to the type of process being recorded. 2.1. data flow diagram data flow diagram (dfd) or data flow chart (dfc) is used in process modeling, and it focuses on the flow of data between processes while analyzing data storage for maximum availability and reduced search time. this is a set of processes that are executed in parallel and represents a set of parallel processes and connections of data flow and storage between them. it is graphically described through processes, data flow, data storage, and external entities. a process is a set of operations that transform input data into output data and define the process with input data flow based on a clearly specified processing logic, using data from the input flow or data storage, with the result of processing being output data flow and/or updated data in data storage. it is indicated by a numerical label and the name of the process. data flow is a path (graphically indicated by an arrow) through which groups of objects (documents, forms, templates, etc.) pass, showing the elements between which the data flow occurs. data storage (file, folder, database, etc.) is used for storing data and is defined as data flow at rest, connected only to system processes via data flow. it allows for the accumulation of data flow content and is connected to processes based on the principle of input/output to data storage. external objects used in the standard are expected to establish connections with objects outside the context of the observed system and represent sinks and/or sources of data flow. data flow enables a view of all activities within the process but is limited to a so-called "paged" display where the process is divided into pages related to the display format. this way, the functionality of a global overview of the process is significantly hindered, especially when complex processes are involved. likewise, it fully covers the recognition of participants in the process but less so organizational structure. fig. 1 shows a part of the diagram that contains process for liquid assets loan approval. 438 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević fig. 1 example of a part of diagram for liquid assets loan approval according to dfd standard the dfd standard has been the only standard for decades for defining processes within a quality system and is supported by numerous (or almost all) software tools. in addition to the mentioned limitation related to the "paged" format, another limitation occurs when there are multiple transitions from one diagram to several separate diagrams that represent parts of the process, where the global overview of the entire process is greatly hindered.the standard does not support conditional branching and merging of data flow, and does not provide insight into the duration of activities or the starting/ending time and conditional points of activities. as this standard has been in use for the longest time, numerous software tools allow for transferring recognized elements into other standards. 2.2. icam definition for function modeling this standard (idef0) allows us to have a detailed overview of all activities by using a depth-based view, where at each deeper level we have more detail regarding activities and the flow between them. the standard enables the display of the same process depending on the depth used for observation, with the tracking of certain constraints related to grouping arrows (flow). multi-criteria selection of standards for system analyst activities in organizations 439 the disadvantage of this standard is that there is no comprehensive, unified, detailed overview of the entire model, while at the same time, it is an advantage because the designer can focus on a specific detail within the process. when it comes to the criterion of software support, this standard has been covered by numerous high-quality tools for its application since its publication, and this remains true today. the standard defines certain limitations that apply to the number of allowed activities per diagram (3-6, max 8), the obligation to align input/output arrows on hierarchical diagrams (also 3-6, max 8), division of arrows according to the icom (inputs, controls, outputs, mechanisms) rule, etc. the tool has also enabled the creation of entities at a global level, their structure through attributes, as well as crud (create, read, update, delete) irun (insert, retrieve, update, nullified) rules for their use, but this does not belong to the characteristics of the standard and is not shown in this paper. fig. 2 presents one of the diagrams of the cost calculation process in production. fig. 2 example of diagram of cost calculation according to idef0 standard given the criteria necessary for multi-criteria analysis [14, 15], the standard fully recognizes the elements related to objects (input, output, and internal), enables the identification of elements that control and manage the process, the structure of the process and description of process activities, participants in the process, and is easy to use. however, the standard performs poorly or does not recognize the sequence, branching, and objects that affect activities. also, a major drawback is that it does not recognize the ability to define the duration of each activity and the moment of its start and end. 440 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević in the case of complex processes that have a deep hierarchical structure, it is difficult to move from level to level and understand their connectivity and activity integrity across levels. the second constraint refers to the "stacking-positioning" of activities on the diagram, where the most important activity is positioned in the top left corner. the key constraint that has exceptional significance in interpreting the flow of the process and the impact of arrows on activities is the reversed position of the arrow entering the activity (icom rule), which is specific for this standard. the standard defines the possibility of transferring entities identified in the idef3 standard to the idef1x standard (integration definition for information modeling), but this possibility is not elaborated idef3. 2.3. integrated definition for process description capture method the idef3 standard [16] describes the behavior of a system. this model includes certain constraints of the system, including resources as well as time relationships. the basic building blocks of idef3 diagram descriptions are: activities, connections, references, decision points and decompositions. fig. 3 presents an example of one of the diagrams of equipment maintenance flow. fig. 3 first level of equipment maintenance flow according to idef3 standard during the analysis, uobs (units of behavior) and their mutual links that determine the dynamics of the process were identified. through decision points, the sequence and simultaneity of activities, as well as the parallelism at the beginning and end of certain https://en.wikipedia.org/wiki/idef https://en.wikipedia.org/wiki/idef https://en.wikipedia.org/wiki/idef https://en.wikipedia.org/wiki/idef multi-criteria selection of standards for system analyst activities in organizations 441 uobs, were defined. this allows for the determination of the shortest, longest, and optimal routes for performing certain segments of uobs, as well as the entire observed process. the standard enables the process and activity-structural system analysis to decompose into the atomic level of indivisible uobs. the rules on the number of activities and arrows are the same as those in the idef0 standard, but unlike it, the option of determining the duration and cost of each activity is defined within the standard itself, but not the start and end times of activities. when it comes to the criterion of software tool prevalence, this standard has not been recognized to a sufficient extent by software manufacturers, so it has not had the same level of prevalence as, for example, idef0. an important characteristic of this standard, which sets it apart from others, was its application for defining simultaneity and sequencing in the execution of uobs. based on these parameters, as well as the duration of each uob, critical and optimal paths were defined and direct process optimization was enabled. within the standard, "data" objects are not provided, which on the other hand significantly limits the possibility of process analysis, especially in determining all elements that participate in the process, as well as the mutual influences of objects and uobs. 2.4. business process model and notation this standard (bpmn) [17] is the newest of these four and enables structural system analysis to be performed, as well as activities, flows, objects, and decision points to be recognized. in this standard, the order of activity execution is respected, so the first activity in the sequence is always on the left side and goes to the right. arrows, which indicate flow, have inputs/outputs on the left and right sides and only serve to emphasize the process realization flow. the specificity of this standard is the richness of graphical symbols [18] related to decision-making, as well as symbols indicating the way a particular activity is started depending on the initiator. this standard, like the previous two, has the ability to display structural system analysis, or a more detailed representation of a certain complex activity, which is decomposed [19]. according to the standard, this decomposition takes place on the same diagram, although some tools, such as powerdesigner, allow for the generation of a separate diagram. the standard includes objects (dataobject and datastore) that provide a generalized description of the structure of an object involved in the process, but the standard does not define the possibility of recognizing the impact of that object on an activity and vice versa, nor does it define the structure of that object. the standard [20] also does not define or recognize the possibility of exchanging a datastore object with a data model. fig. 4 presents one example of bpmn model of data migration. considering the trend in information technology, where there has been a lot of emphasis on graphic design lately, and the fact that this standard has visually appealing symbols, both in shape and possible colors, software manufacturers have produced numerous tools that use this standard as primary or one of the standards for process modeling, both in terms of quantity and quality [21]. 442 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević fig. 4 an example of an external communications office activity according to bpmn standard multi-criteria selection of standards for system analyst activities in organizations 443 3. selection criteria and their fulfillment in standards based on the review of relevant literature [22, 15], and consultations with experts in this field (people with years of experience working with case (computer aided software engineering) tools for process recording, analysis, and improvement in various organizations with various technologies) [23, 24], we propose the following criteria with expert ratings: "how well the criteria are fulfilled by each standard". industry criteria, are presented in table 1 and those are: 1. recognition and description of objects that have emerged in the field and that enter the observed process, their structures and interactions with the process and activities within the process. these objects cannot be changed within the process, 2. description of internal objects that are formed during the process and define the impact of objects on activities and vice versa, the impact of activities on objects, as well as their structure, 3. description of objects that are the result of the process and exit the process, and affect the execution of other processes in the environment, 4. recognition of all activities within the process and defining their functions, 5. recognition of the start and end of the activity's duration, 6. recognition of participants (owners, performers, and other stakeholders) in the process, 7. recognition of the sequence of events, as well as parallel and simultaneous execution of activities within the observed process, 8. recognition of branching and merging, a criterion that allows showing that certain activities can occur in parallel or sequentially, 9. the ability to structure (decompose) the process, which allows complex activities to be broken down into sub-activities, 10. recognition of process control possibilities, which ensures the identification of control functions within the process, 11. recognition and structuring of data storage, which enables the decomposition of data storage into basic attributes, type, and unambiguousness, 12. recognition of the structure of an object, which enables the object to be decomposed into basic attributes, type, and unambiguousness, 13. recognition of organizational structure, which enables the definition of the structure, interconnection, and hierarchy of an organization, 14. recognition of relationships between systems, which enables precise and unambiguous definition of the possible connections between different processes within one system, as well as the connection of systems at a higher level. 15. creating custom properties allows for the creation of new properties within a standard that are not otherwise provided by the standard, through the use of some existing ones, 16. recognition of activity costs, which enables the valuation or estimation of the cost of conducting an activity, 17. recognition of the frequency of activity occurrence enables the recognition and valuation of the number of times an activity is repeated during the process, 18. conditioning the start of an activity enables the recognition of triggers that influence the initiation of the activity itself, 19. conditioning the completion of an activity enables the recognition of conditions that allow the activity to be completed. 444 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević table 1 industry criteria criteria with expert ratings criteria dfd idef0 idef3 bpmn c1 1.0 1.0 1.0 1.0 c2 1.0 1.0 1.0 1.0 c3 1.0 0.7 0.8 0.9 c4 0.8 0.8 0.8 0.9 c5 0.8 0.8 0.9 0.9 c6 0.8 0.8 0.2 0.4 c7 1.0 0.4 0.7 1.0 c8 0.0 0.1 0.8 0.2 c9 0.1 0.7 0.7 0.9 c10 0.5 0.5 0.4 0.7 c11 1.0 0.0 0.0 1.0 c12 0.6 0.0 0.0 0.5 c13 0.0 0.0 0.5 0.0 c14 0.3 0.8 0.6 0.7 c15 0.0 0.5 0.5 0.0 c16 0.0 0.2 0.7 0.0 c17 0.0 0.0 0.3 0.0 c18 0.0 0.0 0.7 0.7 c19 0.0 0.0 0.7 0.7 general criteria, are presented in table 2 and those are: 20. the prevalence of software tools and their quality and degree of support for the standard, 21. simplicity, a criterion that indicates an easy, intuitive, and graphically acceptable standard to users, 22. upgradeability is a criterion that depends on options available in the standard itself, 23. universality of standard application to different types of processes, 24. application in military projects, which have their own specificity, 25. application in quality systems, which have their own specificity, 26. application in complex projects, 27. required designer experience defines how much experience a designer needs to be able to apply the standard, 28. multiplication of projects gives value to the possibility of multiplying the standard by transferring recognized elements to another standard. table 2 general criteria criteria dfd idef0 idef3 bpmn c20 0.9 0.3 0.6 0.9 c21 0.9 0.9 0.3 0.8 c22 1.0 0.2 0.3 0.7 c23 0.3 1.0 0.5 0.5 c24 1.0 0.6 0.4 0.5 c25 0.9 0.8 0.7 0.5 c26 0.5 0.8 0.6 0.9 c27 0.8 0.8 0.6 0.7 c28 0.4 0.8 0.6 0.9 multi-criteria selection of standards for system analyst activities in organizations 445 according to the experiences of experts, these criteria have emerged in cases of organizations and institutions that they have observed and worked in as system analysts. in the case of a need for new criteria in a particular organization, they could be introduced into the model in the same way as those listed [25, 26]. in this paper, we limit ourselves to the criteria listed. 4. standard selection 4.1. criteria and weight coefficient selection one of the newer methods for determining weight coefficients of the criteria is lbwa method. žižović and pamučar [10] were the first authors who demonstrated the method. characteristics of this method are that it can be used in both individual and group decisionmaking, it is a relatively simple mathematical calculation and rational mathematical algorithm, where the simplicity of the method does not depend on the number of criteria [10]. the first thing we do while using lbwa method is defining criteria. if k is the number of criteria, then we have a set of criteria s = {c1, c2, ... , ck}. after that, we start using the lbwa method according to the steps by žižović and pamučar [10]. the choice of criteria is made within the organization that is undergoing organizational change [27, 28, 29]. which of the listed 28 criteria the organization will take into account depends on various factors. primarily, it is a question of what needs to be changed, what is the technology of work, what are the work processes in the organization, what are the suggestions of analysts, what are the desires and ambitions of management, what are the requirements of the organization's owner, etc. in this paper, we assume that the organization, together with the analyst, has chosen k criteria (this number is generally less than or equal to 28). therefore, let's assume that the criteria have been selected and arranged in order of importance for the decision maker, i.e., top management of the organization: cx1, cx2, . . . cxk (1) as we assumed that these k criteria were selected from the listed 28, it means that we can represent the problem of choosing the most optimal standard as a multi-criteria model of the following form, as presented in table 3. table 3 multi-criteria model criteria cx1 cx2 . . . cxk s t a n d a r d s a1 p11 p12 p1k a2 p21 p22 p2k . . . at pt1 pt2 ptk in our case t = 4, i.e. a1 ≡ dfd, a2 ≡ idef0, a3 ≡ idef3, a4 ≡ bpmn, and p1r, p2r, p3r, p4r (r = 1, ... , k) are the corresponding values provided in the preceding part. 446 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević in order to set the procedure for this problem, it is necessary to determine the weight coefficients of the criteria cx1, ... cxk, i.e. to determine numbers wx1, ... , wxk, ((wxi > 0) and ∑ wxi = 1)k i=1 . the procedure will be set using lbwa method provided in [10]. as the criteria cx1, ... cxk are ordered by strength, it follows that: wx1 ≥ wx2 ≥ . . . ≥ wxk > 0 following the lbwa method, the criteria are to be grouped as follows: ▪ group 1 are criterion cx1 and criteria that are of equal strength or between that and half of cx1 ▪ group 2 are criteria that are exactly half of cx1 and between that and one third of cx1, ▪ group 3 are criteria that are exactly one third of cx1 and between that and one quarter of cx1, ▪ ... group x grouping ends with a group in which we place the weakest criterion cxk. then, we select number r0 ∈ r, such that it is higher than the number of criteria in any of the previous groups. furthermore, let an optional group m be made out of criteria: 𝐶𝑥𝑝, . . . 𝐶𝑥𝑞 (2) which are such that criterion cxj (𝐼𝑝 ≤ 𝐼𝑗 ≤ 𝐼𝑞) is assigned number f(cxj) = r0 mr0+ixj (3) where 0 ≤ ixj < r0, and with ro being the selected fixed positive number and where: 0 ≤ ixp ≤ . . . ≤ ixj ≤ . . . ≤ ixq < ro (4) for cx1 it is understood that 𝐼 𝑥1 = 0, 𝑓(cx1) = r0 r0 = 1 ) then the criterion cx1 is assigned number wx1, wx1 = 1 1+f(cx2)+⋯+f(cxk) (5) and criterion cxk is assigned number wxk = f(cxi) ∗ wx1 (i = 2, 3, ..., k) (6) nb 1: it is obvious that the selection ixj in the requested range is reached as wxj = 1 𝑚 w1 for ixj = 0 for 0 < ixj < ro we get 1 𝑚+1 < wxj < 1 𝑚 . for 0 < ixj < ixj + 1 < ro we get 1 𝑚+1 < wxj + 1 < < wxj < 1 𝑚 if wxj is to be in the first half of the interval ( 1 𝑚+1 < 1 𝑚 ), ixj must be selected so that ixj ∈ ( 𝑟o 2𝑚+1 , ro) (7) multi-criteria selection of standards for system analyst activities in organizations 447 if wxj is to be in the second half of the interval ( 1 𝑚+1 < 1 𝑚 ), ixj must be selected so that ixj ∈ (0 , mro 2m+1 ) (8) in a similar way, the interval ( 1 𝑚+1 , 1 𝑚 ) can be set to four equal parts by selecting points from the interval ( 0, ro) that will be reflected in the appropriate points, ( 0, mro 4m+3 , 2mro 2m+1 , 3mro 4m+3 , ro) (9) and selecting appropriate points for ixj depending on where we want wxj to be located. clearly, this is possible for any for any desired division of the interval ( 1 𝑚+1 , 1 𝑚 ). 4.2. ranking alternatives (standards) the next step in the standard selection process is the matrix transformation table 3 by multiplying each column in the matrix table 3 with the weight coefficient matching the criterion in the column, as presented in table 4. so we form the new matrix: table 4 matrix of decision making criteria cx1 cx2 . . . cxk s t a n d a r d s a1 wx1*p11 wx2*p12 wxk*p1k a2 wx1*p21 wx2*p22 wxk*p2k . . . at wx1*pt1 wx2*pt2 wxk*ptk an exception to the previous one is the case when t = 4, as presented in the example in table 5. table 5 matrix of decision making example criteria cx1 cx2 . . . cxk s t a n d a r d s a1 wx1*p11 wx2*p12 wxk*p1k a2 wx1*p21 wx2*p22 wxk*p2k a3 wx1*p31 wx2*p32 wxk*p3k a4 wx1*p41 wx2*p42 wxk*p4k therefore, in this matrix we add up all the elements for each type and we get 448 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević v(ai) = ∑ wxj pij k j=1 (10) it is obvious that 0 ≤ w(ai) ≤ 1 (because ∑ wxik j=1 = 1 i 0 ≤ pij ≤ 1 for each j = 1, ...k i i=1, ..., s. additionally we define that ai is better than aj if and only if v(ai) > v(aj). in case that v(ai) = v(aj) the better alternative is the one with higher value according to the first criterion. if both values are equal, then the better alternative is the one with higher value according to the second criterion and so on until the last criterion. if v(ai) = v(aj) and all values are equal, then the selection is random. this selection method is called simple additive weighting [30]. naturally, the best ranked standard is selected. example 1: let us assume that system analysts and managers of an organization have selected the criteria: c1, c2, c4, c5, c6, c9, c10, c14, c15, c26 i c27 from the criteria list and ordered them in strength: c1→ c2→ c4→ c5→ c14→ c15→ c26→ c27→ c6→ c9→ c10 and then determined that group 1 contains the criteria c1, c2, c4, c5, while the remaining criteria are in group 2, where c14, c15, c26 are exactly half of the strongest, c6 and c27 so that they are equal in the first quarter to c1, while c9 and c10 are equal in the second quarter to c1. this is some examples how to selecting the best standard! solution: we select r0 = 10. from the problem, it follows that the first four are in group 1 i1 = 0, i2 = 0, i3 = 0, i4 = 0 => f(c1) = f(c2) = f(c3) = f(c4) = 1 the rest are in group 2, such that: i14 = 0, i15 = 0, i26 = 0 => f(c14) = f(c15) = f(c26) = 10 2∗10 = 0.5 interval [0, 10] is divided by points 20 11 , 20 5 , 60 9 and we arrive at (0, 20 11 , 20 5 , 60 9 , 10) according to the problem, we select i27 i i6 from the first quarter 0, 20 11 , i27 = i6 = 1, while we select i9 i i10 from the second quarter i0 = i10 = 2 and we get 20 11 , 20 5 . f(c27) = f(c6) = 10 21 = 0.476 as well as f(c9) = f(c10) = 10 22 = 0.455 so that w1 = 1 1+𝑓(𝐶2)+ 𝑓(𝐶4)+ 𝑓(𝐶5)+ 𝑓(𝐶14)+ 𝑓(𝐶15)+ 𝑓(𝐶16)+ 𝑓(𝐶27)+ 𝑓(𝐶6)+ 𝑓(𝐶9)+ 𝑓(𝐶10) = 1 7.362 w1 = 0.136 i.e.:w2 = 0.136, w4 = 0.136, w5 = 0.136, w14 = 0.068, w15 = 0.068, w26 = 0.068, w27 = 0.065, w6 = 0.065, w9 = 0.062 i w10 = 0.062. as the starting matrix of the problem is provided in a table 6, it follows: multi-criteria selection of standards for system analyst activities in organizations 449 table 6 starting matrix of problem c1 c2 c4 c5 c14 c15 c26 c27 c6 c9 c10 a1 1.0 1.0 0.8 0.8 0.3 0.0 0.5 0.8 0.8 0.1 0.5 a2 1.0 1.0 0.8 0.8 0.8 0.5 0.8 0.8 0.7 0.7 0.5 a3 1.0 1.0 0.8 0.9 0.6 0.5 0.6 0.6 0.2 0.7 0.4 a4 1.0 1.0 0.9 0.9 0.7 0.0 0.9 0.7 0.4 0.9 0.7 by multiplying elements of each column with the previously calculated matching weight coefficients, we get the complete matrix of the problem, table 7: table 7 matrix of decision making c1 c2 c4 c5 c14 c15 c26 c27 c6 c9 c10 a1 0.136 0.136 0.109 0.109 0.020 0.000 0.034 0.052 0.052 0.006 0.031 a2 0.136 0.136 0.109 0.109 0.054 0.034 0.054 0.052 0.052 0.043 0.031 a3 0.136 0.136 0.109 0.122 0.041 0.034 0.041 0.039 0.013 0.043 0.025 a4 0.136 0.136 0.122 0.122 0.048 0.000 0.061 0.046 0.026 0.056 0.043 by adding up lines, we get: v(a1) = 0.685 v(a2) = 0.810 v(a3) = 0.739 v(a4) = 0.796 therefore, the best standard is a2, i.e. idef0, followed by bpmn, idef3 and finally dfd. 5. conclusion management of business processes in any business segment or development is crucial for achieving maximum efficiency and attaining strategic objectives [31,32]. standards for modeling business processes are always a current topic, and there has always been an effort towards their improvement. standards define the documentation, analysis, and reengineering of existing processes. they serve as a foundation for generating program code to support business operations or the development of new systems. additionally, they are used as a basis for blockchain algorithms in creating smart contracts and serve as a foundation for implementing robotic process automation (rpa) and other related technologies. the system analyst plays a crucial role in organizing the modeling of business processes, selecting and implementing standards, as well as choosing analytical methods for system design to solve business problems and improve operations [33]. the selection of criteria depends on various factors such as work technology, organizational workflows, 450 m. rakić, m. žižović, b. miljković, a. njeguš, m. r. žižović, i. đorđević organizational changes, organization goals, management experience, the architecture of the developing system, and more. these factors directly influence the choice of standards. in the paper, 28 key criteria covering it activities were mapped. experts, system analysts for selecting standards for business process modeling, provided assessments of the representation of these criteria in the proposed standards (dfd, idef0, idef3, and bpmn). based on their needs and capabilities, the company proposed the use of a multi-criteria model for determining weight coefficients for the selected criteria, known as the level-based weight assessment (lbwa) model. after selecting the criteria and determining the weight coefficients, the alternatives, in this case, the standards, were ranked using the simple weighted sum method. the paper presents four different scenarios for ranking the standards. according to the first scenario, the best standard is idef0, followed closely by bpmn. in the second scenario, bpmn is determined to be the best standard. in the third scenario, idef3 is identified as the best standard. finally, in the fourth scenario, the dfd standard is ranked as the best. in cases where two standards have the same significance, the final decision regarding the choice will be left to the 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kozina, g., 2021, influence of digital technologies and its technological dynamics on company management, tehnički vjesnik, 28(4), pp. 1262-1267. 33. narang, m., kumar, a., dhawan, r., 2023, a fuzzy extension of merec method using parabolic measure and its applications, journal of decision analytics and intelligent computing, 3(1), pp. 33–46. 34. khan, m. r., ullah, k., khan, q., 2023, multi-attribute decision-making using archimedean aggregation operator in t-spherical fuzzy environment, reports in mechanical engineering, 4(1), pp. 18–38. 11718 facta universitatis series: mechanical engineering vol. 23, no 1, 2025, pp. 49 63 https://doi.org/10.22190/fume230405019a © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper effect of isothermal and isochronal aging on the microstructure and precipitate evolution in beta-quenched n36 zirconium alloy ali w. aldeen1,2, dina y. mahdi2, chen zhongwei1, imad a. disher3, barhm mohamad4 1state key laboratory of solidification processing, northwestern polytechnical university, shaanxi, china 2department of materials engineering, college of engineering, university of kufa, najaf, iraq 3department of ceramics and building materials, college of materials engineering, university of babylon, babil, iraq 4department of petroleum technology, koya technical institute, erbil polytechnic university, erbil, iraq orcid ids: ali w. aldeen https://orcid.org/0000-0002-7983-5435 dina y. mahdi https://orcid.org/0009-0002-7985-9303 chen zhongwei https://orcid.org/0000-0003-4303-6049 imad a. disher https://orcid.org/0000-0001-8876-8214 barhm mohamad https://orcid.org/0000-0001-8107-6127 abstract. in this study, the effect of isothermal and isochronal aging is reported to investigate the precipitate evolution and recrystallization of n36 zirconium alloy after β-quenching. two groups of samples were cut from the as-received sheet of n36 zirconium alloy and subjected to solution treatment and subsequent aging at 580, 640, and 700 °c for 40 and 600 min, respectively. optical microscopy (om), scanning electron microscopy (sem), transmission electron microscopy (tem), energy dispersive spectroscopy (eds), and electron backscattering diffraction (ebsd) were utilized to characterize the microstructure and second-phase particle (spps) evolution. results show that the implemented quenching after solution treatment produces fine interlaced α-plates structure conserved inside prior β grain boundaries with 12 variant directions that follow burger misorientation characteristics. after aging for a short time, initial α-plates conserve their shape and become softer, and spps spread along their boundaries. recrystallizations are finished for specimens aged at a higher temperature or for a longer time. the recrystallized structure exhibits non-uniform grains and a random spps distribution. despite the differences in morphology, some recrystallization grains retain the orientation feature from the initial α-plates. hardness declines as temperature and time rise, and no hardness peak is seen. roughness and wettability rise with increasing ageing temperatures. key words: n36 zirconium alloy, aging, second phase particles, recrystallization received: april 05, 2023 / accepted june 24, 2023 corresponding author: zhongwei chen state key laboratory of solidification processing, northwestern polytechnical university, xi'an, 710072, shaanxi, china e-mail: chzw@nwpu.edu.cn https://orcid.org/0000-0002-7983-5435 https://orcid.org/0009-0002-7985-9303 https://orcid.org/0000-0003-4303-6049 https://orcid.org/0000-0001-8876-8214 https://orcid.org/0000-0001-8107-6127 50 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad 1. introduction the corrosion resistance and mechanical properties of zirconium alloys depend on the features of their microstructure, which include those of the parent phase as well as those of the second phase particles. the presence of the second phase particles may not only promote the nucleation of recrystallized grains during the annealing process but also hinder the migration of grain boundaries during recrystallization and grain growth. therefore, it is of great scientific and engineering significance to study second phase precipitation behavior along with the development of the microstructure of zirconium alloys [1, 2]. it is well known that the chemical composition plays a vital role in the development of the microstructure of an alloy and determines its response to heat treatment during manufacturing processes. the optimal compositions for novel zirconium alloys have thus been the subject of numerous research efforts [1–5]. at present, in the process of deepening burnup and prolonging the refueling cycle life, a large number of nb-containing alloys are used as fuel element cladding. among these new alloys, n36 was independently developed in china from the zr-sn and zr-nb series. to obtain a zirconium alloy with specific properties and performance that meet the requirements of cladding applications, a detailed study of the microstructure is indispensable. many efforts have been made in the past few years to study the microstructure of spps; these include the phase identification of spps in pre-deformed and annealed alloys [6–8], the chemical composition, and the crystal determination of spps [9-12]. studies have also looked at how microstructure evolution alters mechanical properties [13-16]. however, studying the effect of spps distribution on the recrystallization process during the heat treatment is still under consideration due to the lack of research coverage on this topic. the production process of n36 alloy sheet is passed through: ingot β-phase quenching, αphase hot rolling, several intermediate annealing’s, 30%-50% cold working after each annealing is finished, and final annealing to proceeds recrystallization [17]. heating into the β phase and then cooling rapidly to room temperature makes the microstructure of zirconium alloy very complex, and it is expected that the martensitic structure will be formed during very rapid cooling. the matrix resulted in super saturation with alloying elements [18,19]. therefore, the processes of recrystallization and precipitation will be more complicated during the further thermal treatment that follows β-quenching. chai et al. [20] studied the effect of aging temperatures in the range 600-700°c in different periods of time on zr-0.85sn-0.4nb-0.4fe0.1cr-0.05cu alloy. they found a linear distribution of spps on the boundary of α-plates after a short time and low temperature of aging, while as temperature or time increases, heterogeneity of both recrystallization and the random distribution of spps occurs. on the other hand, gopalan et al. [21] performed positron annihilation measurements to study the aging response of βquenched zircaloy-2. by detecting positron lifetime and the s-parameter, they have tried to correlate those defect-sensitive signals to various microstructural features. jayakumar et al. [22] used 2-100 mhz ultrasonic attenuation and velocity measurements to test the hard intermetallic in β-quenched zr-2 during aging. velocity measurements were found to be more dependable and repeatable for characterizing zircaloy-2 microstructure than attenuation measurements. in the current study, the effect of aging following β-quenching on the recrystallization and precipitation behavior of n36 alloy with nb as the dominant element was mainly investigated. this is of crucial importance in order to understand the relationship between the microstructure and the aging heat treatment, provide practical guidance for the evolution of spps distribution, and find the relation between grain orientation prior to and effect of isothermal and isochronal aging on the microstructure and spps evolution... 51 following recrystallization, the mechanism through which precipitation and recrystallization are interacted, and their effect on hardness, roughness, and wettability. 2. experimental method fully recrystallized sheet material of n36 zirconium alloy provided by the china nuclear power institute with a chemical composition of zr-0.85sn-1nb-0.3fe-0.1o (wt.%) is used in current work. the original sheet was cut into a group of samples measuring 10 mmx6 mmx1 mm. in order to prevent oxidation during heating, the samples were sealed in a vacuum glass tube with 4.7x10-3 pa. firstly, all samples were subjected to β solution treatment at 1050°c for a holding time of 35 min; after that, the samples were quenched in cold water immediately after crushing the vacuum tube and denoted as wq. then, the samples were aged at 580, 640, and 700°c for 40 min and 600 min, respectively. the samples with a 40 min holding time are denoted as 580a, 640a, and 700a, while the samples with a 600 min holding time are denoted as 580b, 640b, and 700b, as shown in fig. 1. fig. 1 schematic representation of heat treatment the microstructures of the samples in both groups were characterized using an optical microscope (om) and the sem secondary electron (se) image. for om and sem observation, the samples were gradually polished using sic papers with a final grit number of 5000, and after that, they underwent chemical etching using a solution of h2o, hno3, and hf as etching agents prepared in a ratio of 45:45:10. fei tecnai g2 f20 supertwin transmission electron microscopy with an accelerating voltage of 200 kv was used to obtain some results. an energy-dispersive x-ray spectroscopy (eds) equipped with tem to get the chemical identification of some particles. conventional twin-jet polishing was 52 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad used as a surface preparation technique for the tem imaging with a mixed solution of 90% ethanol and 10% perchloric acid below -35°c. an electron backscatter diffraction (ebsd) system is also equipped with a sem system to characterize the microstructure of some aging samples. the electrochemical polishing parameters used for ebsd sample preparation were 20 v voltages, 0.4-0.6 a current, and a polishing time of 10-40 s. in order to ensure the validity of the statistical results, at least 300 second phase particles were counted for each aged sample. a vickers indentation machine (dhv-1000 hardness tester model) was used to measure hardness in each state. the average result of ten indentations on each surface of the sample was adopted, and the loading force of the hardness tester was 9.8 n with a loading time of 15 s. an atomic force microscope (afm, ntegra from nt-mdt), an optical contact angle meter, and interfacial tension were used to evaluate surface properties. 3. experimental results 3.1. β-quenched microstructure figure 2a shows an optical micrograph image of n36 alloy after solution treatment and quenching in cold water. the high temperature β phase converted to a fine interlaced αplates structure with different crystalline orientations that were surrounded by prior β-grain fig. 2 β-quenching microstructure: (a) optical micrograph (b) bf-tem image (c) ebsdipf image (d) mad histogram effect of isothermal and isochronal aging on the microstructure and spps evolution... 53 boundaries, indicating the growth trends of the α-plates. such a fine plate with a thickness of about 1µm results from a high cooling rate [19]. figure 2b represents a bright field (bf) tem image of β-quenched sample. three evidences can be drawn, the first of which is the absence of precipitate, which indicates the super saturation of alloying elements inside the matrix; the second is the irregularity of the grain boundaries; and lastly, the presence of large dislocations, which indicates that the energy is high, which will encourage the precipitate to nucleate and grains to recrystallize to reach the state of stability during further thermal processes. figure 2c is an inverse pole figure (ipf) ebsd image showing the crystallographic orientation of the α-plates inside a large β-grain boundary, with each color indicating that α-plates have different orientations. combined with figure 2a, it can be found that α-plates clusters in the same direction have the same color and, therefore, the same crystallographic orientations. using the misorientation angle distribution mad in fig. 2d, it can be obtained that a large number of α-plates are gathered around 60°, and at the same time, there is a relatively small amount aggregated around 10° and 90°. the existence of such a special distribution is due to burger's relation, where a single β grain can give 12 α variants directions [23]. 3.2. recrystallization and spps evolution during aging figure 3 shows a group of images for samples aged at 580-700°c after β-quenching. according to samples aged for a short time (group a), it seems that the time is very short to induce the recrystallization process even at high temperatures of 700°c, and the α-plates structure kept well. as time prolongs in group b, a change in the microstructure becomes apparent compared with the initial β-quenching state. some ancient interlaced α-plates still kept in the 580b sample beside the new recrystallization microstructure, while samples 640b and 700b showed an almost striking change of structure towards recrystallization. fig. 3 optical micrograph of microstructure development for samples aged at different times and temperatures 54 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad figures 4a and 4b represent bf-tem images of 640a and 640b aged samples, respectively. fig. 4a shows that the α-plates structure is still conserved with regular plate boundaries compared with that in β-quenching, see fig. 2b, with less dislocation area, which indicates that with a short aging time, α-plates are oriented towards dynamic recovery. also, there is some small precipitate scattered along the plate’s boundaries, which proves that aging has indeed exerted a significant influence on the β-quenching microstructure. on the other hand, in fig. 4b striking changes in morphologies occur compared with the β-quenching microstructure. in this figure, the interlaced α-plate is totally replaced by an irregular grain shape with spps distributed randomly both interand intra-granularly. another phenomenon that should be noticed is that the precipitate on the grain boundaries is larger than that inside the grains, following the ostwald ripening mechanism [19]. a high magnification bf-image in fig. 4c (integrated with fig. 4b) confirmed a hexagonal closed-packed structure with ellipsoidal morphology of spp as the almost nature of the precipitate. fig. 4 bf-tem comparison images of microstructure for (a) 640a sample (b) 640b sample (c) spp chemical identification the spps distribution results are shown in fig. 5. for samples aged in group a, spps are distributed linearly, noting that, as the temperature increases, the particle size increases a little, as does the transverse distance between lines. for group b, spps are distributed randomly, and as the temperature increases, the particles size apparently increases. the general shape of the spps is semi-spherical in the early stages of precipitation, indicating the formation of gp zones, i.e, the precipitates are fully coherent with the parent phase. as the temperature or time increases, their shape becomes polyhedral, indicating the loss of coherency with the parent phase [24], and tends to coarsen, as shown in the histogram of fig. 6. effect of isothermal and isochronal aging on the microstructure and spps evolution... 55 fig. 5 spps morphology and distribution after aging of samples a and b fig. 6 average size of spps in different aging samples fig. 7a represents the ipf-ebsd image and the related misorientation information shown in fig. 7b after aging sample 580b. according to the result of the microstructure, two regions can be observed separately: still conserved -plates obtained by β-quenching that need more driving force to overcome the high stress in those regions, and recrystallized structure in some grains inside huge prior β-grain boundaries. the misorientation angle in fig. 7b is almost identical to the features of misorientation distribution inherited from βquenching, where the misorientation information is distributed in the same way as on parent plates. finally, some small grains recrystallized inside massively recrystallized grains. 580a 580b 640a 640b 700a 700b 0 20 40 60 80 100 a v e ra g e s iz e o f s p p s , n m sample 56 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad fig. 7 ipf-ebsd image of (a) 580b aged sample (b) corresponding mad fig. 8 shows the ipf-ebsd and misorientation distribution histogram after aging sample 640b. from the figure, one can see that striking changes in morphologies occur compared with the β-quenching microstructure. all β-quenching microstructures totally disappear and are replaced by recrystallization characteristics as clearly visible, where the small recrystallized grains are almost sited near the β-boundary while the large grains grow inside it. another phenomenon one should notice is that the recrystallized grains are inhibited from penetrating the β-grain boundary. one can conclude from figs. 7(a) and 8(a) that, as the aging temperature increases, the driving force continues to grow protrusion small grain boundaries and convert them to small recrystallized grains while the previous recrystallized grains continue in their growth. also, the size of the recrystallized grains well reflected the effects of precipitates on their size. in addition, despite the stark change in morphologies, the mad data from ebsd shows that the recrystallized structure still carries over some characteristics from the β-quenching structure, as illustrated in fig. 8(b). however, it appears that the maximum density has been significantly reduced as a result of the spread of other misorientations following recrystallization. fig. 8 640b aging sample (a) ipf-ebsd image (b) mad histogram 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (b) f re q u e n c y ( % ) misorientation angle (o) 640°c effect of isothermal and isochronal aging on the microstructure and spps evolution... 57 3.3. micro-hardness, afm and wettability testing to better reveal the effect of microstructure and precipitate evolution on surface characterizations of n36 zirconium alloy during the aging process, vickers micro-hardness, afm, and wettability tests were performed. the degree of specimen hardness can serve as a reflection of the simultaneous processes of precipitation and recrystallization during ageing. the results of average vickers microhardness are shown in fig. 9 for both β-quenching and aged samples. it can be seen that the hardness graph lines show a decrease in values from β-quenching to increasing time at any fixed aging temperature. even at low temperatures, there is no hardness peak. therefore, the hardness values decrease as the recrystallization proceeds in the samples. note that sample 700b shows some inconsistency in its value despite the progress of its recrystallization process. another incidence that should be noted is that the hardness values of group a samples decreased even though they conserved the plate structure at a lower aging time compared to the hardness of β-quenching sample. fig. 9 vickers micro-hardness results of β-quenching and aging samples due to the afm's ability to detect height variations on the sample in the range of nanometers, this method is excellent for showing the impact of spps changes in surface topography. fig. 10 shows 2d and 3d afm results of surface characterization in the 10x10 µm2 scan area of β-quenching and 580b and 700b aging samples after polishing, cleaning, and drying the surface to reduce the influence of surface defects during the heating and quenching processes. the β-quenching surface is essentially featureless, though some mounts or pits with an average roughness (ra) below 2 nm are still present, as shown in fig. 10 (a, b). in comparison, the ra of aging surface samples confirms a slight increase in roughness due to the precipitation of particles during the aging process. the surface roughness increased from 2 nm in β-quenching sample to 8.7 and 13.4 nm in the 580b and 700b aging samples, as shown in fig. 10 (c, d) and (e, f). a small protrusion, located within a black circle, could be seen on the surface of the 700b-aged sample in fig. 10 (e, f). 0 100 200 300 400 500 600 200 220 240 260 h a rd n e s s , h v aging time, min 580°c 640°c 700°c 58 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad fig. 10 2d and 3d afm micrograph images: (a-b) β -quenching sample; (c-d) 580b; (e-f) 700b figure 11 shows the contact angle results of a microliter droplet on the β-quenching, 580b, and 700b surfaces. it can be seen clearly that the shape of ultrapure water droplets on all surfaces did not change much after aging compared with that of β-quenching, all droplets show a hydrophilic property; and the contact angle decreased from 70.26° in βquenching substrate to 63.25° in 580b substrate and 57.16° in 700b substrate. therefore, the surface's wettability increased after aging. effect of isothermal and isochronal aging on the microstructure and spps evolution... 59 fig. 11 contact angle test of three surface: (a) β-quenching substrate; (b) 580b substrate; (c) 700b substrate 4. discussion 4.1. β-quenching microstructure the appearance of such a fine interlaced α-plates structure in figs. 2a and 2c is essentially due to rapid quenching immediately after solution treatment. previous studies [23] confirmed that the rapid cooling from the β-solution treatment region for zirconium alloys containing β-stabilizer elements larger than α-stabilizer could induce such a fine martensitic structure. in addition to the large number of defects that are introduced during the rapid cooling process, including vacancies, dislocations, and interfaces, the alloying elements are also saturated in matrix. the misorientation distribution of the interlaced αplates obeys strictly burger relationship shear transformation during transformation from β to α. burger suggested that 12 α variants could arise from prior β parent grain based on the symmetry of both phases. all burger distributions appeared during the results; these values precisely take place only at 10°, 59°-64°, and 90° [23]. the weaker frequency distribution of α-plate at 10° and 90° could be attributed to difficult coverage over a large area by the ebsd test, specifically with a complex α-plate structure. some small grains deviate from the burger distribution values. this could be due to an error during testing. 4.2. precipitate and grain growth relationship during aging the rapid cooling rate produces a high driving force stored inside the β-quenching structure. this driving force will release and have the priority of contributing to the recrystallization of the alloy matrix during the subsequent aging to reduce the surface energy. during aging, the alloying elements in their supersaturated state in the matrix will also be polarized and eventually precipitate as second-phase particles to reduce the energy of the system. couet et al. [25] studied the transformation kinetics of zr-4 alloy during aging after water quenching by tep. the results showed that the full precipitation of the second phase always occurs earlier than the recrystallization in the single α-phase region. additionally, a study by garner et al. [26] showed that the second phase of the zr-4 alloy was fully separated after aging at 610°c for only 1 min. when the aging temperature of 580°c was used, the precipitation of the second phase particles also preceded the recrystallization process. in our previous study [19], the growth kinetics of spps calculated after aging of the current alloy, and the results show agreement with the fact that the precipitation of the second phase particles preceded the recrystallization process during aging. all previous studies give a good explanation of why spps are distributed linearly at α-plate grain boundary during aging for a short time. 60 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad therefore, after β-quenching a large amount of α-plates boundaries is produced, and the prior β-grain boundaries are still kept well. the second phase will have priority for nucleation at such sites due to the smoothing of the nucleation path. so, the linear distribution of spps on α-plat boundaries begins to be more apparent for short-period aging. after prolonged aging time, the recrystallization will proceed and make the spps distributed interand intragranularly. however, big particles will coarsen at the grain boundary under the conditions of the ostwald repining mechanism [19]. with extended aging time, the recrystallization process will expand into nearby regions with larger dislocation densities after they have formed. after aging the 580b sample, two sites were observed to be well divided into those that still conserved the initial α-plates and the preferred recrystallization zone starting from the prior β-grain boundary. while, regarding the 640b and 700b samples, α-plate structure disappeared and changed to fully recrystallized grains. luan et al. [18] found that 2 and 4 hours are not enough to induce a fully recrystallized state during the aging of zr-4 alloy at 650 °c; instead, it takes about 72 hours to obtain a fully recrystallized state. in the current investigation, 10 h is enough time to induce a fully recrystallized structure at 640°c, or it may be less. chai et al. [20] found 90 min to be enough time to complete recrystallization at 650°c in zr–0.85sn–0.4nb– 0.4fe–0.1cr–0.05cu alloy and gave a good explanation. although water quenching is still a common factor in the starting state, the variation in cooling rate may still exist due to the difference in sample size in the working condition. according to the irregular shape of recrystallized grains, it is known that the high cooling rates induce the formation of dense dislocation, as seen in fig. 2b, which will assist α-plates boundaries in migration during the recrystallization process. strain-induced boundary migration (sibm) with dislocation density is suggested as a responsible mechanism for inducing the recrystallization process and the production of massive grains by rumball and coleman. [9]. during recrystallization, the early precipitated particles have a big effect on the recrystallization process [27]. when the α-plates boundary moves to recrystallize, it meets the precipitated particles and is affected by zener pinning resistance (pz) [28]. if the driving force of recrystallization is greater than pz, the grain boundary can continue to advance around the second phase particles. if at a particular location the driving force is less than pz, the grain boundary will be pinned in its original direction and continue to grow in the direction of smaller resistance. therefore, movement of recrystallized grain boundaries in association with many defect dispersions in the matrix towards the distribution of second phase particles leads to interaction, eventually forming a highly irregular shape of the grain. on the other hand, when the boundary of α-plates moves towards the prior β-zr grain boundary during recrystallization, it collides with a large number of spps stationed there, since these places are the preferred sites for the growth of such particles due to the smooth route of nucleation. so pz was much larger than the recrystallization driving force, and eventually the recrystallized grains still took the original β-zr grain boundary as part of the grain boundary. thus, these particles will be responsible for obstructing the boundary of α-plates and preventing them from penetrating the prior β grain boundary during recrystallization [29, 30]. regarding the relatively good agreement between the inherited misorientation angle distribution after recrystallization and that in β-quenching state according to the concept of burger [23], chai et al. [20] found that the misorientation distribution inherited from the two states of β-quenching and recrystallization is also almost the same in both, with a relatively weak frequency in the recrystallization state. therefore, the growth of the plate structure during recrystallization is almost in the same preferred direction. effect of isothermal and isochronal aging on the microstructure and spps evolution... 61 4.3. effect of solution treatment and aging on hardness, roughness and wettability in agreement with previous studies [20, 31], the hardness values decreased during the aging process compared with those in the β-quenching state, which has a maximum value of 245 hv. when samples are aged for a short time, the hardness values decrease, although the samples still conserve the α-plate structure and induce spps nucleation. the decrease in hardness caused by the recovery process and changes in texture. during extended aging periods, the decrease in hardness becomes more pronounced. a slight increase in the hardness value of the 700b sample could be due to the coarsening of spps at this stage. generally, the main purpose of the alloying elements in zirconium alloy is still highlighted as improving corrosion resistance rather than the mechanical properties of the alloy [10,32]. the surface topography of the β-quenching, 580b, and 700b samples confirmed the presence of a slight increase in the surface roughness despite the use of the vacuum tube glass protection during heating [33]. the difference in the surface topography is mainly related to the absence of spps in the β-quenching sample and their presence in the aging samples [34]. the increase in the size of the protrusions is probably due to the increase in the size of the oxidized spps compared to the oxidized surface due to the migration of the alloying elements to the free surface and then oxidation [35]. hence, the increasing ra values between 580b and 700b aged samples are attributed to the increasing size of the spps underneath the surface and their concentration and oxidation in a specific place. regarding contact angle results, the surface wettability increased with increasing aging temperature, in agreement with increasing surface roughness. therefore, it’s possible that the spps size contributed to the surface roughness and played the same role in increasing surface wettability, where increasing particle size leads to an increase in surface wettability. also, it is stated that fine spps improve corrosion resistance compared to coarse spps [2, 36]. based on what was discussed, coarsening spps leads to increased surface wettability. 5. conclusion due to the significance of thermal treatment during the manufacturing process, the effect of isothermal and isochronal aging following β-quenching on the microstructure and spps evolution of n36 zirconium alloy is investigated. some important conclusions are as follows: 1. after β-quenching, the results show a structure of interlaced α-plates that follows burger orientation distribution, and the supersaturating of the alloying elements in the matrix are the main characteristics of this stage. 2. for a short aging period, α-plates structure is well conserved in all temperature ranges (580°c–700°c), and the spps distributed linearly in the boundary of α-plate structure. with increasing aging temperatures, the size of the particles increases. 3. for longer aging time, the 580b sample show two regions; still kept α-plate and recrystallized grains, while in 640b and 700b, α-plates disappeared and totally changed to irregular recrystallized grain. some grains that have undergone recrystallization have successfully inherited the misorientation distribution characteristic of α plates. in addition, the large size of spps sited on the prior β-grain boundary prohibited α-plate from penetrating during recrystallization. 62 a.w. aldeen, d.y. mahdi, z. chen, i.a. disher , b. mohamad 4. the hardness values decreased from their maximum 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by environmental sem and analysis by afm, fib, and tem, j. nucl. mater.. 404, pp. 97-108. 35. abolhassani, s., dadras, m., leboeuf, m., gavillet, d., 2003, in situ study of the oxidation of zircaloy-4 by esem, j. nucl. mater., 321, pp. 70-77. 36. park, j.-y., choi, b.-k., yoo, s.j., jeong, y.h., corrosion behavior and oxide properties of zr–1.1 wt% nb–0.05 wt% cu alloy, j. nucl. mater., 359, pp. 59-68. plane thermoelastic waves in infinite half-space caused facta universitatis series: mechanical engineering vol. 15, no 3, 2017, pp. 545 563 https://doi.org/10.22190/fume170104004f © 2017 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper *thermal buckling analysis of functionally graded circular plate resting on the pasternak elastic foundation via the differential transform method udc 624.04 fatemeh farhatnia 1 , mahsa ghanbari-mobarakeh 1 , saeid rasouli-jazi 1 , soheil oveissi 2 1 department of mechanical engineering, khomeinishahr branch, islamic azad university, khomeinishahr, iran 2 department of mechanical engineering, najafabad branch, islamic azad university, najafabad, iran abstract. in this paper, we propose a thermal buckling analysis of a functionally graded (fg) circular plate exhibiting polar orthotropic characteristics and resting on the pasternak elastic foundation. the plate is assumed to be exposed to two kinds of thermal loads, namely, uniform temperature rise and linear temperature rise through thickness. the fg properties are assumed to vary continuously in the direction of thickness according to the simple power law model in terms of the volume fraction of two constituents. the governing equilibrium equations in buckling are based on the von-karman nonlinearity. to obtain the critical buckling temperature, we exploit a semi-numerical technique called differential transform method (dtm). this method provides fast accurate results and has a short computational calculation compared with the taylor expansion method. furthermore, some numerical examples are provided to consider the influence of various parameters such as volume fraction index, thicknessto-radius ratio, elastic foundation stiffness, modulus ratio of orthotropic materials and influence of boundary conditions. in order to predict the critical buckling temperature, it is observed that the critical temperature can be easily adjusted by appropriate variation of elastic foundation parameters and gradient index of fg material. finally, the numerical results are compared with those available in the literature to confirm the accuracy and reliability of the dtm to determine the critical buckling temperature. key words: thermal buckling, orthotropic plate, functionally graded materials (fgm), pasternak elastic foundation, differential transform method (dtm) received january 4, 2017 / accepted april 22, 2017 corresponding author: fatemeh farhatnia affiliation: department of mechanical engineering, khomeinishahr branch, islamic azad university, khomeinishahr, iran e-mail: farhatnia@iaukhsh.ac.ir mailto:farhatnia@iaukhsh.ac.ir 546 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi 1. introduction one of the most important undesired phenomena in a mechanical structure as observed in plates is thermal buckling. the response of the plate to buckling depends on its mechanical properties. functionally graded materials (fgm) are made of ceramic and metal constituents, including high mechanical strength, good machinery ability, and high thermal resistance. fgms are a good choice to be employed as the material constituents in a plate exposed to high thermal gradients. they are less likely to delaminate at high temperatures due to the continuity of their physical and mechanical properties. in addition, the orthotropic properties of fgms have received great attention because they increase the tolerance of plates to various types of loading. numerous studies have been conducted on thermal plate buckling. in what follows, we mention the studies related to the subject of this paper. dewey and costello [1] proposed an analytical and experimental method to investigate the thermal buckling of flat plates where the modulus of elasticity changes due to the thermal gradient. najafizadeh and eslami [2] discussed the thermo-mechanical response of plates based on the first-order shear deformation theory. they derived nonlinear and linear governing equations from the energy method and by using the calculus of variations. in another work, najafizadeh and eslami [3] discussed the thermoelastic buckling of the circular orthotropic composite plates under various kinds of thermal loading. they obtained the governing equations based on the love-kirchhoff hypothesis and the sanders’ nonlinear strain-displacement relation. li et al. [4] examined the nonlinear vibration and thermal buckling of orthotropic annular plates with a centric rigid mass. they employed the hamilton’s principle to derive the governing equations based on the vonkarman nonlinearity. najafizadeh and heydari [5] assessed the thermal buckling of circular plates in functionally graded materials under uniform radial compression subject to various types of thermal loads. they established the governing equations in buckling, using variational method and solved them via bessel functions. prakash and ganapathy [6] applied the finite element method to analyze the vibrations and thermal buckling of circular fgm plates. zhao et al. [7] studied the thermal and mechanical buckling behavior of plates with arbitrary geometry, including plates containing square and circular holes in the center. zenkour and sobhy [8] investigated the thermal buckling of fg sandwich plates, using sinusoidal shear deformation plate theory. the thermal loads are assumed to have uniform, linear, and non-linear distribution through thickness. jalali et al. [9] assessed the thermal buckling of circular fgm plates with varying thickness, using the pseudo-spectral method (psm). evaluating the reaction of plates resting on elastic foundations and subject to different types of loads has a great scientific importance, particularly in modern engineering structures. kiani and eslami [10] studied the exact solution of thermal buckling in annular fgm plates resting on the pasternak elastic foundation. they examined the effects of geometrical parameters, power-law index, and coefficients of the elastic foundation on the critical buckling temperature. jabbari et al. [11] studied the buckling of a solid porous circular plate subjected to thermal loading. they derived the governing equations based on the sanders’ nonlinear strain-displacement relation and determined the pre-buckling temperatures and the critical buckling temperatures. yaghoobi and fereidoon [12] proposed the thermal and mechanical buckling of functionally graded (fg) plates resting on elastic foundation, using nth-order shear deformation theory. they obtained the governing equations via exploiting the minimum total potential energy method. mansouri and shariat [13] predicted the thermal buckling response of heterogeneous orthotropic plates based on thermal buckling analysis of functionally graded circular plate ... 547 the high-order theories. they employed a new version of the dqm (differential quadrature method) to solve the governing differential equations. mirzaei and kiani [14] investigated the thermally induced bifurcation buckling of rectangular composite plates reinforced with single-walled carbon nanotubes. they revealed that in most cases, the fg-x pattern of cnts is the most influential case since it results in higher critical buckling temperatures. yu et al. [15] evaluated the new numerical results of thermal buckling of functionally graded plates (fgps) with internal defects (for example, crack or cutout), using an effective numerical method. they employed the new formulation of the first-order shear deformation plate theory associated with extended isogeometric analysis (xiga) and level sets. moreover, they investigated the influences of various aspect ratios, including gradient index, crack length, plate thickness, cutout size, and boundary conditions on the critical buckling temperature rise (cbtr). tung [16] studied the nonlinear bending and post-buckling behavior of functionally graded sandwich plates resting on elastic foundations and subject to uniform external pressure, thermal loading, and uniaxial compression in the thermal environment. sun et al. [17] numerically investigated the thermomechanical buckling and post-buckling of a functionally graded material (fgm) timoshenko beam resting on a two-parameter non-linear elastic foundation and subject to only a temperature rise, using the shooting method. in the present research, we accomplish the thermal buckling of an orthotropic fg circular plate. to that end, the differential transformation method (dtm) is applied to solve the governing equation of thermal buckling. the literature review indicates that the present work is the first attempt to exploit dtm to evaluate the critical buckling temperature. a number of studies have been carried out by this method [18-24]. the results are presented in four categories based on linear and uniform thermal load and simply supported and clamped edge conditions in the proceeding sections. the effects of parameters such as volume fraction index, stiffness of the pasternak elastic foundation, thickness-to-radius ratio, and modulus ratio of orthotropic material to critical temperature are also investigated. 2. mathematical formulation of constitutive equations consider a functionally graded circular solid supported on the pasternak elastic foundation (fig. 1). a polar coordinate system (r, θ, z) is employed to label the material points of the plate in radial, circumferential, and thickness directions. fig. 1 schematic of problem based on the sanders’ kinematic relations and the von-karman nonlinear assumption, the strain-displacement relationships are written in polar coordinates system as follows: 548 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi ,       ,        2rr rr rr r r rzk zk zk                (1) where rr ,  , and r are mid-surface strains and krr, kθθ, and krθ are the curvatures defined as follows: 2 2 2 1 ,   2 rr rr u w w k r r r               2 2 2 2 2 1 1 1 1 ,   2 v u w w w k r r r rr r                       (2) 2 2 1 v 1 1 1  ,     r r u v w w w w k r r r r r r rr                                where u, v, and w represent the middle-plane displacements in the polar coordinates. stress-strain relationships in polar coordinates are expressed as follows: 11 12 21 22 66 ( ) ( ) ( ( ) ( ) ( ) ( )) ( ) rr rr t rr t rr t rr t r r a a a a a                             (3) in which, 11 21 12 22 66 0 ( ) ( ) ( )( ) , ( ) , , , 1 1 1 1 , , ( ) , ( ) , ( ) r rr r r r r r r r r r r r r r t rr r t e e ee a a a a a g e e z z zz t t t t z z t                                                        (4) where εt is the thermal strain, αr and αθ are the thermal expansion coefficients in radial and circumferential directions, respectively. moreover, t and t0 are the current and reference temperatures, respectively. the readers interested in the thermo-elastic stressstrain relations in a symmetrical case of deformation can find more details in the study conducted by kiani et al. [10]. upon substituting eq. (4) into eq. (3), the constitutive relations for orthotropic fg plate can be re-written as follows: 11 12 11 12 21 22 21 22 66 ( ) ( ) rr rr r rr r r r a a a a t a a a a t a                               (5) in addition, grθ is shear modulus, and er and eθ are young’s modulus of the plate in radial and circumferential directions, respectively. based on the power-law model in polar coordinates, the material properties are as follows:   1 ( ) ,       ( )        2 n r r r r r rcm m cm c m z e e e e e e h z      (6)   1 ( ) ,   2 ( ) n r r r r r rcm m cm c m z z h           (7) thermal buckling analysis of functionally graded circular plate ... 549   1 ( ) ,     )   2 (  n cm m cm c m z h z               (8)   1 ( ) ,         2 ( ) n r r r r r rcm m cm c m z g g g g g g h z          (9) where h, n and subscripts m and c represent the thickness, fg power index, the metal and ceramic properties, respectively. by assuming the polar orthotropic characteristics of plate, young’s modulus in the circumferential coordinate is defined as follows: 1   (( ) ( )  ( )  ) 2 n r cm m z e e e e e h z z z             (10) where μ is the orthotropic modulus ratio. 3. governing equations in thermal buckling stress resultants and stress couples are obtained as follows: 2 2 ( , , )  ( , , )  h rr r rr r h n n n dz         (11) 2 2 ( , , )   ( , , )   h rr r rr r h m m m z dz         (12) substituting eq. (5) into eqs. (11) and (12) yields the stress resultants and stress couples as follows: 1 1 2 2 1rr rr rrn a b k a b k c t        (13) 2 2 3 3 2rr rrn a b k a b k c t         (14) 1 1 2 2 3rr rr rrm b d k b d k c t        (15) 2 2 3 3 4rr rrm b d k b d k c t         (16) where,   1 2 3 2 1 2 3 11 12 22 2 1 2 3 2 2 2 1 3 11 12 2 4 21 22 2 2 1 ( , ) (1, )( )  , ( , ) (1, )( ) h h h h r r h h a a a b b b z a a a dz d d d z c c z a a dz c c z a a dz                                   550 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi when the plate is subjected to the mechanical loading, the total energy is given by: v u  (17) where ω and u are the potential energy of the external loading and the strain energy, respectively. ω is the summation of the potential energy of mechanical loading and two parameters elastic foundation reaction. the elastic foundation is exerted on the lower surface of the plate, as shown in fig. 1. in this research, the mechanical loading is absent. u is the summation of thermal strain energy, membrane strain energy, bending strain energy, coupled bending-membrane strain energy, and elastic foundation strain energy: 2 /2 0 0 /2 ( )   1 2 r h h rr rr rr rr r u t rdzdrt d                           (18) by integrating through thickness, u can be expressed as follows: 2 0 0 2 2 2 2 1     (         2 ) 2   1 r rr rr rr rr r r r r w g g u n k m n k m n rdrd k m w w k w k k r r                                     (19) where kw and kg denote the stiffness of pasternak elastic foundation (tension and shear foundation parameters, respectively). by substituting eqs. (13-16) into eq. (19), setting the resultant expression into the expression of the total potential energy function, eq. (17), and employing euler equations [5], the governing equilibrium equations in the buckling of the plate resting on the pasternak foundation based on the von-karman nonlinearity are obtained as follows [10]: , , , , , , , , , , ,2 2 , , , ,2 2 1 0 1 2 0  2 1 1 2 2 1 1 1 ( ( ) 1 , 0) r r r r r r r r rr r r r r r r r rr r w r rg r n n n n r r n n n r r m m m m m m n w r r r r r n w w k w k w w w r r r r                                     (20) the governing equations of equilibrium can be expressed in terms of displacement components by assuming the symmetric state of buckling which gives the variation with respect to the circumferential direction set to zero, substituting eqs. (13-16) into eqs. (20), and utilizing eq. (2):   22 2 12 2 2 2 3 2 2 3 2 2 11 1 2(1 ) 1 0 r r t t r ku u u w w w e r r r r rr r rk n nw w w e r r rr r r                                               (21) thermal buckling analysis of functionally graded circular plate ... 551 2 3 2 3 2 2 22 3 2 2 3 3 2 2 4 3 2 3 , ,4 3 2 2 3 , , (2 )1 2 (1 ) 2 1 1 0 r r t t r rr r w g r rr ku u u u w w w w w e r r r r rr r r r r r rk w w w w e n w n w r r rr r r r r k w k w w r                                                                    (22) where, 2 1 2 3 3 1 1 , , 1 2 2 2 1 1 1 12 3 4 4 2 cm m cm m cm e e h e e h e n n n e e h e n n n                             3.1. types of thermal loading in this study, two cases of temperature rise are considered: uniform and linear temperature differences. when the plate is not thick enough, the assumption of linear distribution is reasonable [10]. but when the temperature on the top and bottom surfaces of the plate is constant and no source of heat is available in it, the temperature variation can be defined as a linear function of thickness coordinate; for instance, the temperature distribution in aircraft window, walls of the building, or furnace. by assuming that the reference temperature of plate is t0 and the displacement in the radial direction is prevented due to restraint on the edge of the plate, n t r and n t θ as the representatives of stress resulting from the thermal gradients can be determined in two categories due to the temperature rise across thickness as follows:  uniform thermal loading: 1 2 (1 ) 1 2 1(1 ) t cm m m cm cm cmr r m m r e e e n c t h e t n n                   (23) 2 2 ( ) 1 2 1(1 ) t cm m m cm cm cmr m m r e e e n c t h e t n n                     (24)  u linear thermal loading: 1 ( ) ( )    2 m c m z t z t t t h          (25) 02 2 (1 ) ( ) 1 2 1(1 ) (1 )           2 2 2 2 (1 ) t cm m m cm cm cmr r m m m r m m cm m m cm cm cmr r e e e n h t t e n n e e e e h t n n                                 (26) 552 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi 02 2 ( ) ( ) 1 2 1(1 ) ( )          2 2 2 2(1 ) t cm m m cm cm cmr m m m r m m cm m m cm cm cmr r e e e n h t t e n n e e e eh t n n                                    (27) where ∆t=t(z)-t0. by eliminating the radial displacement components of u and after performing some mathematical manipulations, the equations of the equilibrium can be summarized in one equation in terms of out-plane displacement component of w as follows: 4 3 2 2 4 3 2 2 3 2 2 1 )( ) ( 0t t e r g g w w w w w w w d n k n k k w r r r rr r r r r r                          (28) where, 2 2 2 1 3 1( ) / e (1 )e re e ed    . based on the adjacent equilibrium criteria, the state of stable equilibrium may be designated by w0; in addition, it is w0+w1 in the neighborhood of stability state when w1 can be represented to any small increment of displacement. similar to out-plane displacement, the stress resultants are divided into two terms representing the stable equilibrium and the neighboring state [10]. upon substituting w0+w1 and stress resultants in two terms into the governing equation (28) and performing some mathematical manipulations, the stability equation is obtained as follows: 4 3 2 2 1 1 1 1 1 04 3 2 2 3 2 1 0 1 2 ( ) 1 ( ) 0 t e r g t g w w w w w w d n k r rr r r r r r w n k k w r r                         (29) where n t r0 and n t θ0 are the pre-buckling thermal loading, and the following relation can be established: n t r0= n t θ0= -nt . the following dimensionless quantities are defined to deal with the problem under consideration in the dimensionless forms: 22 42 1φ , , ,  , , tt gt t wr r g w e e e e k rn r k rw n rr n n k k h r d d d d        4 3 2 2 3 2 3 4 3 2 2 2 3 φ φ φ φ 2 ( ) φ ( ) φ 0 t r g t g w d d d d d n k dd d d d d n k k d                        (30) 4. solving the governing equation by dtm the differential transform method (dtm) is a numerical method based on the taylor series expansion that proposes the solution in the form of polynomials [24]. this method is a fast convergent method in comparison with the taylor series in order to solve the differential equations. the advantage of this method is its low computational manipulation and its applicability to handle linear and non-linear ordinary and partial differential equations. by thermal buckling analysis of functionally graded circular plate ... 553 exploiting the dtm, the differential equations are reduced to the recurrence relations and convert the boundary conditions into a set of algebraic equations. the differential transform of the k-th derivative of function f(r) is defined as follows: 0 1 ( ) [ ]    ! k k r r d f r f k k dr         (31) where f(r) and f[k] are the original function and transform function, respectively. the inverse differential transform of f[k] is defined as follows [5]: 0 0 ( ) [ ]( )k k f r f k r r     (32) in the domain of r, original function f(r) is considered to be analytical, and r=r0 represents any point in r. f(r) is represented by power series whose center is located at r0. from eqs. (31) and (32), it can be concluded that: 0 0 0 ( ) ( ) ( )      ! k k k k r r r r d f r f r k dr            (33) table 1 demonstrates the fundamental mathematical properties of the differential transform method (dtm): table 1 fundamental theorems of dtm [22] original function transformed function ( ) ( ) ( )f r y r z r  [ ] [ ] [ ]f k y k z k  ( ) ( )  f r y r  [ ] [ ]f k y k  ( ) ( ). ( )f r y r z r 1 0 1 1[ ]   [ ]   [ ]   k k f k y zk k k    ( ( )) ( ) ( ) m m d y r f r dr  ! [ ] [ ] ! ( )m k f k y k k m      n f r r 0         [ ] ( )   1           if k n f k if k n k n      hence, function ф[ξ] is obtained as follows: 0 1 2 0 φ[ ] [ ] [0] [1] [2]   n k k k               (34) to solve eq. (35), we use the differential transform relationships of the k-th derivative of the function of non-dimensional out-plane displacement ф; moreover, dtm theorems are listed in table 1: 554 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi 1 4 3 1 1 1 1 1 14 0 φ ( 3)( 4)( 3)( 2)( 1) [ 4] k k d k k k k k k k k k k k d                  1 3 2 1 1 1 1 13 0 φ ( 2)( 3)( 2)( 1) [ 3] k k d k k k k k k k k k d                1 2 1 1 1 12 0 φ ( 1)( 2)( 1) [ 2] k k d k k k k k k k d              φ ( 1) [ 1] d k k d      (35) 1 2 3 1 1 1 12 0 φ ( 3)( 2)( 1) [ 2]     k k d k k k k k k k d              1 2 1 1 1 0 φ ( 2)( 1) [ 1] k k d k k k k k d            1 3 1 1 0 φ ( 3) [ ] k k k k k       substituting the aforementioned relationships into the governing equation in thermal buckling eq. (28) yields: 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 2 ( 3)( 4)( 3)( 2)( 1) [ 4] 2 ( 2)( 3)( 2)( 1) [ 3] ( 1)( 2)( 1) [ 2] ( 1) [ 1] ( 3)( 2)( 1) [ 2] k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k a k k k k k k k a                                                             1 1 1 1 1 1 1 0 0 ( 2)( 1) [ 1] ( 3)  [ ] 0 k k w k k k k k k k k k k                (36) by utilizing the appropriate theorems of the dt method (see table 1 and the simplified form of eq. 36), we have the following recurrence relation: 1 2 2 2 ( 2 )( 1) [ 1] [ 3] [ 1] ,               3  ( )( 1 ) ) ( wa k a k k k k k k k k                 (37) where a1=n t r+kg and a2=n t θ+kg. the following boundary conditions are imposed as clamped and simply supported boundary conditions on the edges of the plate. however, by applying differential transformation method to boundary conditions, we can obtain: thermal buckling analysis of functionally graded circular plate ... 555  non-dimensional clamped edge: 1 1 0 ,    0 d d        (38)  differential transform of clamped edge condition with dtm: 0 0 [ ] 0, [ ] 0  n n k k k k k       (39)  non-dimensional simply supported boundary condition: 2 21 1 0,  0e d d d r dd                 (40)  differential transform of simply supported edge condition with dtm: 0 0 [ ] 0,   ( 1 ) [ ] 0  n n k k k k k k          (41) in this study, the symmetrical thermal buckling behavior of a plate is considered, and the regularity condition is imposed besides the boundary conditions. the non-dimensional form of regularity condition and its differential transform are as follows:   0 0, 1 0   d d        (42) upon substituting k=3,5,7,… into the recurrence eq. (37), we get: 2 1 2 1 2 [1] φ[3] 0 3 12 (6 3 ) [3] [1]   (20 5 ) [5] [3]  φ[5] 0,  φ[7] 0 ,   15 240 35 1260 w w a a a k a a k                       (43) it can be concluded that for odd values of k in eq. (34), φ[k] equals zero [23-24]. therefore, by using recurrence eq. (37), we can find that φ[k] can be determined in terms of φ[0] and φ[2]. by using recurrence relation (37) for k=2, 4, 6,…, we can obtain the following equations: 1 2 1 2 1 2 2( ) [2] [0] [4]       8 72 (12 4 ) [4] [2] (30 6 ) [6] [4] [6]  , [8] ,   24 600 48 2352 w w w a a k a a k a a k                         (44) hence, all the φ[k] with even values of k in eq. (34) can be expressed in terms of φ[2], φ[0]. 556 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi in order to determine the critical buckling temperature, recurrence relation (42) and imposed boundary conditions of eq. (39) are simultaneously employed for clamped edges. therefore, a set of two homogenous equations are established in terms of φ[0] and φ[2], as follows: 11 12 21 22 [0] [2] 0 [0] [2] 0 h h h h          (45) where h11, h12, h21, and h22 are the coefficients of polynomials of n-th order. 1 2 11 1 2 1 2 1 2 12 1 2 21 1 2 1 2 1 2 22 (3 ) 1 8 72 (6 150)(8 72) (3 )( ) 1 4 36 24 600 (6 150)(4 36) 6(3 ) 2 18 (6 150)(2 36) (12 4 )( ) 2 9 4 100 (4 100)(4 36) w w w w w w k a a k h ka a a a a a h k a a k h ka a a a a a h                                                       (46) for a non-trivial solution of eq. (45), the determinant of coefficients must vanish, leading to the eigenvalue problem. hence, we have: 11 12 21 22        0         h h h h  (47) in a similar manner, we can obtain the solution for the simply supported plate by utilizing the recurrence relation (37) and the imposed boundary conditions of eq. (41). 5. numerical results in this section, some numerical results are presented for the thermal buckling of orthotropic fg circular plates of two categories of uniform and linear temperature rise under clamped and simply supported boundary conditions. due to the high volume of the calculations required, the system of algebraic equations presented in the preceding section are implemented in a computer code in matlab software, and the numerical results are presented in a tabulated form. for the numerical results, an fg plate composed of aluminum (as metal) and alumina (as ceramic) is considered. young’s modulus of aluminum (al) and alumina are em=70 gpa and ec=380 gpa, respectively. the thermal expansion coefficients are αm=23×10 -6 k -1 and αc=7.4×10 -6 k -1 for metal and ceramic constituents, respectively. poisson’s ratio remains constant at ν=0.3. it is assumed that the material properties are assumed to be temperature-independent. to examine the convergence rate of dt method, we obtain the results for a clamped isotropic homogeneous circular plate with the thickness-toradius ratio of h/r=0.01. we observe that the number of terms (k=20) is sufficient to get precise values of the critical temperature. this trend remains constant in other numerical results as presented in the following section. thermal buckling analysis of functionally graded circular plate ... 557 5.1. uniform loading clamped boundary condition table 2 shows the variation of the critical buckling temperature with respect to fg power-law index, thickness-to-radius ratio, and orthotropic ratio. as it is observed, the critical temperature increases as the value of μ rises. according to table 2, for a specific h/r ratio, the buckling critical temperature is reduced by increasing the value of n; however, increasing the h/r ratio increases the values of the critical temperature. table 2 variation of the critical temperature ( o k) of the clamped plate versus fg power-law index, orthotropic ratio, and thickness-to-radius ratio in case of uniform thermal loading n μ h/r 0 0.5 1 2 5 10 0.50 0.010 0.150 0.020 0.030 0.040 7.070 15.911 28.280 63.641 113.132 4.081 9.104 16.081 36.153 64.214 3.163 7.390 12.851 28.934 52.560 2.914 6.552 11.652 26.214 46.603 2.151 6.060 11.141 26.042 46.142 2.002 5.950 10.860 25.851 45.823 0.70 0.010 0.150 0.020 0.030 0.040 13.780 31.020 55.131 124.061 220.552 7.813 17.570 31.232 70.280 124.944 6.422 14.414 25.611 55.412 98.503 5.684 12.780 22.714 51.091 90.831 5.164 11.181 21.421 52.732 93.721 5.010 10.551 21.090 50.261 92.361 0.9 0.010 0.150 0.020 0.030 0.040 21.381 48.110 85.541 192.462 342.152 12.112 27.260 48.460 109.033 193.842 9.5404 21.480 38.881 89.412 152.872 8.812 19.823 35.234 79.270 140.920 9.150 20.444 36.341 81.841 145.382 9.040 20.121 33.370 80.124 142.534 1 0.010 0.150 0.020 0.030 0.040 24.950 56.150 99.824 224.581 399.270 14.142 31.813 56.552 127.240 226.214 11.594 25.083 46.370 104.343 185.480 10.280 23.132 41.113 92.504 164.451 10.001 21.863 38.411 89.432 161.652 8.901 20.520 38.604 88.104 160.412 1.5 0.010 0.150 0.020 0.030 0.040 51.633 114.280 203.141 456.602 807.412 29.262 64.744 115.081 258.660 457.411 23.480 53.091 94.421 213.192 360.590 21.260 47.120 83.662 188.150 332.542 21.941 48.562 80.311 194.121 343.150 22.561 49.933 78.760 199.480 352.761 2 0.010 0.150 0.020 0.030 0.040 87.150 200.051 349.374 794.632 1408.221 49.371 113.361 197.942 450.160 797.630 40.634 89.381 162.401 355.991 651.631 35.890 82.393 143.881 327.270 579.880 33.151 78.050 141.450 322.644 558.262 31.134 77.412 140.661 320.180 555.180 558 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi table 3 exhibits the critical buckling temperature with respect to elastic foundation coefficient, orthotropic ratio, and fg power-law index. as elastic foundation coefficients increase by increasing the orthotropic ratio, the critical temperature increases. on the other hand, when the fg power-law index increases, the critical temperature decreases. this shows that the pure ceramic plate, compared to the metal-ceramic plate, is more stable at the elevated working temperature. table 3 variation of the critical temperature ( o k) of the clamped plate versus fg power-law index, orthotropic ratio, and elastic foundation on the critical temperature in case of uniform thermal loading, h/r=0.020 kw, kg μ n (0,0) (100,0) (200,0) (500,0) (100,10) (200,20) 0 55.131 63.121 67.341 81.470 77.470 96.911 0.50 31.232 35.690 38.150 46.151 43.880 54.880 0.7 1 25.611 29.341 31.280 37.851 35.903 44.841 2 22.714 26.030 27.741 33.581 31.960 39.801 5 21.422 23.822 24.633 31.610 30.841 37.250 0 99.824 114.293 121.780 147.531 140.050 175.360 0.5 56.552 64.520 69.270 83.812 79.512 99.5281 1 1 46.370 53.141 56.343 68.301 65.381 81.334 2 41.113 47.282 50.480 61.012 57.722 73.982 5 38.411 44.690 47.742 59.553 53.801 71.552 0 203.142 232.570 248.150 300.222 285.450 357.153 0.5 115.081 131.764 140.580 170.244 161.711 202.300 1.5 1 94.421 108.111 115.342 139.544 132.680 165.823 2 83.662 95.781 102.233 123.644 117.583 147.221 5 80.313 92.812 100.433 120.810 115.283 145.342 5.2. linear loading clamped boundary conditions table (4) represents the effect of increasing the thickness-to-radius ratio on increasing the critical temperature in case of linear thermal loading condition. this table clearly shows that the trend of variation of the critical temperature is confirmed in table 2 as well. the comparison of the two tables indicates that the rate of increase is higher for linear thermal loading as compared to the uniform one. nevertheless, for small h/r and non-zero quantities of n, this trend is reversed. thermal buckling analysis of functionally graded circular plate ... 559 table 4 variation of the critical buckling temperature ( o k) for the clamped plate in case of linear thermal loading, (kw, kg)=0 μ h/r n 0 0.5 1 2 5 0.50 0.010 0.150 0.020 0.030 0.040 8.480 26.531 50.740 67.592 220.670 2.652 12.780 26.521 37.052 122.801 0.843 8.673 19.272 27.860 93.292 0.351 6.640 15.472 22.873 77.060 0.305 6.062 14.763 20.160 73.893 0.70 0.010 0.150 0.020 0.030 0.040 15.173 43.640 93.142 234.551 432.540 4.750 21.021 48.682 128.581 240.714 1.511 14.272 35.372 96.700 182.851 0.523 10.920 28.403 79.370 151.231 0.451 10.284 27.932 77.370 150.681 0.9 0.010 0.150 0.020 0.030 0.040 22.772 76.230 151.082 364.921 664.300 7.123 36.710 78.962 200.153 369.6821 2.274 24.922 57.371 150.444 280.832 0.650 19.110 46.122 123.480 231.901 0.563 18.713 44.920 122.212 231.493 1 0.010 0.150 0.020 0.030 0.040 29.912 92.291 179.632 429.171 778.532 8.273 43.632 93.131 234.563 432.552 2.980 30.171 68.214 176.932 329.124 0.731 23.101 54.772 145.231 271.880 0.660 21.860 53.790 143.053 264.810 1.5 0.010 0.150 0.020 0.030 0.040 62.510 197.501 355.670 931.304 1572.631 19.562 95.113 185.882 510.520 875.182 15.032 71.954 151.512 390.711 671.040 10.080 57.064 118.450 322.131 555.614 9.840 55.860 115.042 315.793 551.260 2 0.010 0.150 0.020 0.030 0.040 101.694 325.971 617.394 1511.632 2266.591 31.813 156.981 322.672 828.650 1261.371 30.290 124.653 251.290 639.903 971.882 22.394 100.291 205.614 528.782 805.652 21.350 98.140 205.380 524.433 803.570 table (5) shows the effect of the presence of elastic foundation on the critical buckling temperature for linear loading condition. as it is observed, the critical temperature decreases when the elastic foundation coefficient gets more values. in order to validate the present solutions, we compare them with the results obtained from the study conducted by ghiasian et al. [25]. table (6) shows the critical temperature for the clamped plate without elastic foundation. the values are em=201 gpa, αm=12.33×10 -6 k -1 , ec=350 gpa, and αc=5.87×10 -6 k -1 , and the poisson’s ratio is ν=0.3. 560 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi table 5 effect of elastic foundation on the critical buckling temperature for linear temperature rise, h/r=0.020 (kw, kg) μ n (0,0) (100,0) (200,0) (500,0) (100,10) (200,20) 0 93.140 107.621 114.160 136.523 142.211 163.120 0.50 48.681 56.560 60.163 71.362 74.610 54.442 1 35.371 41.212 43.363 51.851 54.661 61.681 0.7 2 28.401 33.280 35.222 41.634 43.362 49.742 5 27.931 33.111 35.022 41.411 43.331 48.040 0 179.632 207.291 219.152 264.062 274.664 312.511 0.5 93.131 105.611 111.761 134.290 141.562 158.322 1 68.210 79.390 83.900 100.812 104.361 118.692 1 2 54.772 64.361 66.274 80.2380 85.331 117.322 5 54.390 65.830 65.182 79.012 83.030 98.190 0 355.670 416.954 439.520 521.322 545.123 622.433 0.5 185.880 217.913 229.722 272.253 284.620 325.342 1 151.510 177.622 187.244 222.363 234.163 265.253 1.5 2 118.450 138.863 146.370 173.632 180.853 207.283 5 117.841 140.721 145.342 172.884 180.052 206.220 table 6 comparison of the results of the present study with those of ref. [25] h r p.e. * 0.040 p.e. * 0.030 p.e. * 0.020 p.e. * 0.010 n 0.857 199.644 0.884 112.291 0.893 49.910 1.11 12.731 present 201.369 113.293 50.360 12.591 0 ref. [25] 0.015 141.149 0.264 83.142 0.146 37.001 2.88 9.532 present 141.170 83.362 37.055 9.265 0.5 ref. [25] 0.038 133.358 0.105 74.921 0.654 33.120 2.41 8.536 present 133.307 75.000 33.338 8.335 1 ref. [25] 0.676 122.314 0.768 68.752 0.792 30.554 0.792 7.761 present 123.147 69.284 30.798 7.700 2 ref. [25] 0.253 114.452 0.356 64.326 0.732 28.486 2.718 7.370 present 114.742 64.556 28.696 7.175 5 ref. [25] *the percentage of error thermal buckling analysis of functionally graded circular plate ... 561 5.3. effect of boundary conditions on the critical buckling temperature in circular fg plates, due to the stretching-bending coupling exposed to uniform thermal loading, the asymmetric material distribution induces the pre-buckling thermal moments. therefore, the bifurcation buckling may not occur, and the buckling critical temperature is not available [26]. however, the plates with clamped edges can tolerate the bending moments and remain in an un-deformed configuration. in this study, we consider that the variation of the critical buckling temperature is tabulated for a homogeneous orthotropic circular plate under simply supported boundary condition, as shown in table 7. table 7 variation of the critical buckling temperature for a homogeneous orthotropic circular plate under simply supported boundary condition and linear and uniform temperature rises μ h r 0.50 0.70 0.90 uniform linear uniform linear uniform linear 0.010 2.06 4.12 3.72 7.45 5.37 10.74 0.150 4.63 9.26 8.37 16.74 12.08 24.17 0.020 8.23 16.46 14.88 29.76 21.48 42.96 0.030 18.52 37.04 33.49 66.97 48.34 96.67 0.040 32.92 65.84 59.53 119.04 85.93 171.86 h r 1.0 1.50 2.00 uniform linear uniform linear uniform linear 0.010 6.03 12.07 10.31 20.76 15.80 34.28 0.150 13.58 27.15 23.29 46.24 39.64 79.55 0.020 24.13 48.27 41.46 73.41 70.46 126.31 0.030 54.30 108.60 93.10 187.88 159.56 318.12 0.040 96.54 193.08 165.95 333.14 253.41 569.59 6. conclusions in this paper, we have analyzed the thermal buckling of a circular fgm plate resting on the pasternak elastic foundation and subjected to uniform and linear thermal loading by employing the differential transform method (dtm) to obtain the solutions. some remarkable conclusions obtained in this study are as follows:  by utilizing the differential transform method (dtm), the governing differential equation in the thermal buckling can be transformed to algebraic equations in the sub-domains. dtm is capable of deriving the analytical solution to determine the critical buckling temperature for fg orthotropic plate resting on two-parameter (pasternak) foundations.  comparing the results with those existing in the literature indicates that dtm is a fast convergent, precise, and cost-efficient tool to analyze the thermal buckling behavior of functionally graded plates. 562 f. farhatnia, m. ghanbari-mobarakeh, s.r. jazi, s. oveissi  by increasing volume fraction n, the flexural rigidity is reduced as the plate becomes more metal-rich. consequently, the critical buckling temperature is reduced. this leads to the conclusion that for a ceramic-rich plate, the value of the critical temperature is the maximum.  the present study has investigated the effect of the presence of elastic foundation, as a controlling parameter, on the critical temperature for two boundary conditions, namely clamped and simply supported edges. the numerical results 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universitatis series: mechanical engineering vol. 23, no 3, 2025, pp. 479 509 https://doi.org/10.22190/fume240801007v © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper a new framework for risk assessment of road transportation of hazardous substances sahand vahabzadeh1, sina s. haghshenas2, saeid j. ghoushchi1, giuseppe guido2, vladimir simic3,4,5, dragan marinkovic6,7 1faculty of industrial engineering, urmia university of technology, urmia, iran 2department of civil engineering, university of calabria, via bucci, rende, italy 3university of belgrade, faculty of transport and traffic engineering, belgrade, serbia 4yuan ze university, college of engineering, department of industrial engineering and management, taoyuan city, taiwan 5department of computer science and engineering, college of informatics, korea university, seoul, republic of korea 6tu berlin, department of structural analysis, berlin, germany 7mechanical science institute, vilnius gediminas technical university, lithuania orcid ids: sahand vahabzadeh https://orcid.org/0009-0007-5326-4957 sina s. haghshenas https://orcid.org/0000-0003-2859-3920 saeid j. ghoushchi https://orcid.org/0000-0003-3665-9010 giuseppe guido https://orcid.org/0000-0001-7570-4123 vladimir simic https://orcid.org/0000-0001-5709-3744 dragan marinkovic https://orcid.org/0000-0002-3583-9434 abstract. various elements including environmental, public safety, economic, and security concerns define the evaluation of threats connected with the movement of dangerous materials. it is imperative to undertake a comprehensive investigation to address the multifaceted problem of risk assessment concerning the road transportation of dangerous materials. the primary aim of this study is to present a framework for threat assessment in this domain. to achieve this objective, an integrated approach involving stepwise weight assessment ratio analysis (swara) with the multi-attributive border approximation area comparison (mabac) approach under the z-number theory is introduced. preliminary investigations and expert opinions are taken into consideration, and 17 risks are identified for developing the failure mode and effect analysis (fmea) technique for rural roads located in cosenza, a region in southern italy. a qualified analysis is conducted between the outcomes of the fmea approach implemented by zswara-mabac and those of the conventional fmea method. this investigation is undertaken to achieve sustainable mobility goals by evaluating the risks and enhancing the safety of the transportation of dangerous substances via roadways. therefore, it is essential to re-evaluate the laws and measures required to mitigate hazards on the regional road network of southern italy. received: august 1, 2024 / accepted february 5, 2025 corresponding author: vladimir simic university of belgrade, faculty of transport and traffic engineering, vojvode stepe, belgrade, serbia e-mail: vsima@sf.bg.ac.rs 480 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. key words: risk assessment, hazardous substances, sustainable mobility, swara, z-number theory 1. introduction road safety guarantees are essential for preventing crashes and protecting lives [1-3]. this is especially critical when carrying dangerous substances, as any occurrence could have serious consequences. the transportation of dangerous materials is a complex issue in road transportation that requires careful consideration from various perspectives [4-5]. hazardous substances are utilized in numerous industries as raw materials for industrial production, such as petrochemicals used in plastic industries or various chemical derivatives utilized in the health and cosmetic industries [6]. consequently, the transportation of dangerous materials is considered a critical matter for various industries to operate efficiently from an economic standpoint [7]. however, with the growth of various industries and their need for abundant raw materials for increased production, the escalation in the volume of transportation of hazardous materials is indisputable [8]. thus, particularly in developed and developing countries, concerns about the safety of individuals who are directly and indirectly involved in the transportation of these materials on roads arise [9]. in recent years, there has been an increased emphasis on the transportation of dangerous substances, with a specific emphasis on road transportation [10]. the transportation of dangerous substances through densely populated and confined spaces has experienced a rise, prompting concerns about the likelihood of mishaps and their ramifications. moreover, the ecological consequences of transporting dangerous substances have emerged as a significant issue. the possibility of spills or leaks occurring during transportation has the potential to result in lasting harm to both the environment and public health [11-12]. mcdm models are crucial in risk assessment due to their ability to handle complex decision-making scenarios involving multiple conflicting criteria [13]. the use of integrated fuzzy mcdm methods allows for the consideration of subjective assessments, thereby refining risk rankings and prioritization [14]. the systematic approach of mcdm models leads to more informed and effective risk management strategies, ultimately contributing to better project outcomes and safety measures [15]. mcdm frameworks facilitate the aggregation of diverse expert insights, which incorporates multiple decisionmakers to identify and prioritize alternatives [16]. the extended z-number based mcdm methods enhance this capability by incorporating uncertainty and reliability into the decision-making process, which is particularly important in risk assessment. these methods provide a more nuanced approach to evaluating risks by considering both the fuzziness of information and its reliability, thus offering a comprehensive framework for decision-making under uncertainty. z-numbers, introduced by zadeh, are used to manage the fuzziness and reliability of information, which is crucial in risk assessment where data is often uncertain or incomplete [17]. the use of z-numbers in mcdm allows for a more accurate representation of uncertainty, as demonstrated in the evaluation of driving behavior risks, where dual perspectives and credibility values are calculated to assess risk scores [18]. the integration of z-numbers in mcdm enhances the decision-making process by providing additional certainty and guidance, as seen in the valuation of uncertain information, which leads to more confident and comprehensive judgments [19]. a likelihood-based a new framework for risk assessment of road transportation of hazardous substances 481 approach to ranking z-numbers further refines decision-making by comparing the randomness and reliability of information, thus supporting more informed decisions in complex scenarios [13]. given the multifaceted nature of the issue, it is imperative to undertake a thorough investigation into the associated risks [20]. to this end, a comprehensive understanding and evaluation of the processes governing the transportation of hazardous materials are essential for effective risk prevention and mitigation [21]. the transportation of such materials by road involves a multitude of variables, necessitating the incorporation of various data inputs, including information on the type of materials being transported, geometric road data, vehicle data, and other pertinent details, in the risk assessment process [22]. many strategies have been developed to assess the possible risks connected to the movement of hazardous products [23-25]. multi-criteria decision-making (mcdm) approaches and failure modes and effects analysis (fmea) are widely adopted techniques for assessing risk [26-29]. despite the extensive efforts in this domain, a number of challenges remain unsolved. specifically, most of the existing methods in risk assessment cannot handle the uncertainty and reliability of information simultaneously. classical fuzzy approaches consider only uncertainty and do not take into account the reliability of data, which is an important factor in decision-making. this has resulted in incomplete evaluations and unstable outcomes in many cases. in that respect, this paper fills these gaps by developing a new framework that incorporates both stepwise weight assessment ratio analysis (swara) and multi-attributive border approximation area comparison (mabac) methods into the fuzzy environment. the key novelty is using the theory of z-number in handling uncertainty and reliability of information together, which is an important feature of the approach for assessing risks in hazardous materials transportation because data are often incomplete or fuzzy under real-life conditions. subsequently, a team of experienced technicians identified 17 potential risks associated with the transportation of dangerous materials. to evaluate these risks and compare the outcomes of the proposed approach, a real-life case study was conducted alongside two other approaches. it is noteworthy that the novelty of this research lies in the utilization of the z-number theory to address the uncertainties associated with these factors. this approach considers the reliability of the factors, as well as their fuzziness, in the evaluation process. the main contributions, their impacts, and their insights to this study are as follows: ▪ this paper represents a new framework that combines swara and mabac methods under a fuzzy environment. the integration of these methods into one would provide a systematic and structured approach to the evaluation and prioritization of risks, hence enhancing overall accuracy and consistency in risk assessments concerning the complex domain of hazardous materials transportation. ▪ this paper applies the theory of z-number, put forward by zadeh, which accounts simultaneously for the fuzziness and the reliability of information. the presented dual consideration of uncertainty and reliability is an important limit of the methodologies at present available that increases the potential applicability of the proposed framework under real-world operative conditions characterized by incomplete, imprecise, or uncertain data. ▪ the proposed framework eliminates the limitations of classical fuzzy methods, which take into consideration only uncertainty without respect to data reliability. overcoming 482 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. such weaknesses, the framework gives more full and reliable risk assessments with smaller odds on unstable outcomes and incomplete analyses. ▪ the framework was then validated by a real case study in which 17 potential risks associated with hazardous materials transportation were identified and evaluated. the practical application of the framework proves its effectiveness and adaptability to realworld scenarios, hence providing evidence of relevance for industry professionals and decision-makers. ▪ the performance of the proposed framework was compared to two different existing approaches in order to emphasize its strengths and advantages. this comparative analysis reinforces not only the reliability and robustness of the proposed model but also its superiority in handling the complexities related to risk assessment under uncertainty. the subsequent sections of this investigation are structured in an ensuing manner: section 2 presents a succinct compendium of the literature review. the methodologies, comprising the fuzzy sets concept, z-number theory, the z-swara method, and the zmabac method are expounded upon in section 3. section 4 explores the suggested approach. section 5 explicates the distinguishing features of a case study. section 6 examines the use of three strategies and discusses the ranking of the most important risks in the road traffic of hazardous materials. 2. literature review the transportation of dangerous materials has been the subject of valuable research [3035]. deng et al. [36] proposed a generic approach for preventing dangerous chemical accidents, which utilizes k-means clustering analysis of incident data to address the associated challenges. to demonstrate the effectiveness of their approach, they developed a database of dangerous chemical incidents and employed a k-means clustering algorithm to categorize them. the results obtained revealed that the suggested approach significantly enhances accident categorization and enables the identification of the most suitable order of crucial objectives to prioritize, the prerequisites for accident prevention, and the development of preventive measures. on the other hand, hong et al. [37] utilized the association rules mining (arm) method to detect the contributory crash-risk effects of hazardous material (hazmat) vehicle-involved accidents on expressways. they conducted a case study and analyzed accident data from the crash database of the korea expressway corporation between 2008 and 2017. according to their findings, the use of arm as a data mining methodology established correlations between crashes involving hazardous material vehicles and crucial factors that increase the risk of such incidents. moreover, the implementation of arm held promise for producing easily understandable results and valuable insights to improve the safety of expressways. extensive research has been conducted on the parameters that influence accidents during the transportation of dangerous substances, the severity of injuries, and the risks that contribute to such accidents [38-42]. büyüközkan et al. [43] introduced a comprehensive and structured mcdm approach based on the analytical hierarchy process (ahp) and vikor in an intuitionistic fuzzy (if) environment to evaluate the selection process of hazardous waste transportation. to overcome bias, uncertainties, and partiality in decision-making processes, they adopted a group decision-making approach using if. the efficacy of this proposed framework was validated by applying it to a real-world case study in turkey. ma a new framework for risk assessment of road transportation of hazardous substances 483 et al. [44] analyzed the statistical distribution features of several factors, including hazardous materials, transport crashes, driver attributes, vehicle attributes, environmental factors, and road conditions. to handle unobserved variability among data, they proposed an ordered logit regression model. the findings of their model estimation revealed a significant correlation between the harshness of accidents involving dangerous material transportation and 10 elements, including violations, risky driving behaviors, and vehicle faults. noguchi et al. [45] developed an innovative methodology for examining accident scenarios relating to hazmat transportation. their proposed technique utilized network theory as a means to represent the intricate crash process. the methodology employed in their research involved the selection of accident scenarios from accident processes through the amalgamation of the hazmat transportation crash network and transportation-related environmental features based on crash statistics. the research applied the methodology to conduct a case study on the transportation of liquefied petroleum gas (lpg) via roadways in japan. the findings suggested that the methodology presented for the investigation of crash scenarios had the potential to expedite the process of risk assessment in the transport of hazardous materials while also facilitating the transition from risk assessment to risk management. weng et al. [46] built a quantitative risk assessment (qra) model to evaluate the risk of hazmat transportation accidents. the qra model combined the frequency and consequences of all potential accident scenarios. an event tree, which consisted of six intermediate events, was utilized to identify the potential accident scenarios. to prove the efficacy of the proposed qra model in evaluating the danger of hazardous material transportation accidents, a case study was carried out using relevant hazmat transportation data from shanghai. the outcomes showed that the proposed model had reliable performance. ren and yang [47] analyzed the risk factors of hazmat road transportation accidents with a bayesian network model, hrt-bn, developed upon tree augmented naive bayes (tan), reflecting dependencies such as those of accident types and rescue times. the results point out that human factors are the main cause of slight accidents with short rescue times, while the interaction of multiple factors results in more serious and varied accidents. other factors, such as seasonal and regional factors, come into play such as longer rescue times during summer. their analysis concluded with targeted recommendations to reduce risks. kanj et al. [48] proposed a novel method to enhance transportation safety related to hazardous goods with the help of real-time information. they have worked out a hybrid methodology called fuzzy ahp-topsis (technique for order preference by similarity to ideal solution) for risk analysis and optimized route selection on three criteria, namely: cost, duration, and risk. the approach proposed aimed at finding the safest route by assigning calculated weights to these criteria through pairwise comparisons. they examined both static and dynamic environments where the static decisions remained fixed while dynamic decisions changed with the flow during transport. their outcome indicated an enhancement of safety and reduced risks on hazardous material transportation. hsu et al. [49] proposed the integrated quality function deployment and multicriteria decisionmaking (qfd-mcdm) framework for selecting the key drivers to i5.0 in order to enhance supply chain resilience while concurrently handling hazardous material transportation risk. some of the proposed measures that will contribute to sustainability and stability of firms, as proposed by them, include safe and inclusive working environments, customized products and services, improved flexibility of production, enhanced control redundancy, and full utilization of real-time data analysis. their findings showed how the integration of i5.0 and scr can have a synergistic effect in mitigating hmtr. the findings also provided useful insights and implications for enterprises from different industries. 484 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. 3. methodology in this section, we will outline the concepts of fuzzy set theory and z-number theory, followed by a discussion of the z-swara and z-mabac methodologies. additionally, we will elucidate the advantages of employing the swara and mabac methods in comparison to other weighting and ranking approaches, respectively. the swara method involves a hierarchical ranking of criteria, prioritizing the most significant criteria over those of lesser importance. the participation of experts in assessing the weight of these criteria is essential to this process. swara allows decision-makers to establish priorities aligned with predefined policies, particularly in contexts where these priorities reflect known situations. it effectively assesses the relative significance of multiple criteria concurrently. its versatility makes swara suitable for various decision-making scenarios, and its use among researchers has notably increased in recent years. in terms of contextual consideration, comprehensive evaluation of multiple criteria, and broad applicability, the swara method presents several advantages over other methods such as the analytic hierarchy process (ahp), the analytic network process (anp), and the best worst method (bwm) [16]. the mabac method offers several advantages in mcdm over traditional methods. it is particularly effective in handling complex decision-making scenarios by incorporating various innovative approaches, such as time-series analysis, fuzzy logic, and probabilistic linguistic terms. these enhancements allow mabac to provide more accurate and reliable decision-making outcomes. the mabac method can incorporate time-series data, allowing it to handle dynamic decision-making scenarios where criteria and preferences may change over time. this is achieved through time weights, which enhance the reliability of the decisionmaking process by accounting for temporal variations. it allows for bidirectional adjustments, enabling decision-makers to refine and adjust criteria weights and preferences dynamically, which is not typically possible with traditional static methods [50]. moreover, integrating fuzzy numbers and dual probabilistic linguistic term sets allows mabac to effectively manage uncertainty and fuzziness in decision-making. this capability is crucial for scenarios where decision criteria are not precisely defined or are subject to interpretation. the method's ability to incorporate fuzzy logic and probabilistic assessments provides a more nuanced and comprehensive evaluation of alternatives, enhancing decision accuracy [51]. additionally, mabac's approach of ranking alternatives based on their distances from an approximate boundary region limits unconditional compensation among attribute values, leading to more balanced and fair evaluations [50]. the method has been successfully applied in various practical scenarios, such as determining winners in competitions and selecting sustainable suppliers, demonstrating its versatility and effectiveness in real-world applications [52]. the mabac method offers distinct advantages in terms of robustness, flexibility, and ease of implementation, making it a compelling choice for multi-criteria decision-making. while methods like topsis, vikor, and others have their strengths, mabac’s focus on border approximation and comprehensive evaluation positions it effectively for scenarios where nuanced decision-making is critical. 3.1. fuzzy set theory the idea of fuzziness presents a membership function to manage many language variables [53−55]. within the realm of individuals' internal cognition, inference, and perception, a certain a new framework for risk assessment of road transportation of hazardous substances 485 extent of ambiguity exists. fuzzy sets handle sources of imprecision and uncertainty that are non-statistical and ambiguous in nature. the fundamental descriptions of fuzzy number sets used in this work are clarified in the next part. definition 1: a set a that is fuzzy, and is defined on the reference set x, can be expressed through the use of eq. (1): 𝐴 = {(𝑥, 𝜇𝐴(𝑥))|𝑥 ∈ 𝑋} (1) the membership function denoted as 𝜇𝐴(𝑥): 𝑋 → [0,1] in eq. (1) characterizes the fuzzy set a. the value of membership 𝜇𝐴(𝑥) signifies the extent to which 𝑥 ∈ 𝑋 pertains to a. the membership degree is the extent that an element, 𝑥 ∈ 𝑅, belongs to the fuzzy set 𝐴 ̃. this parameter conveys the level of assent or conviction in x being a member of the fuzzy set 𝐴 ̃, or the degree of conformity of x with the intended concept of the set 𝐴 ̃. definition 2: a triple (l, m, u) can be employed to denote a triangular fuzzy number 𝐴 ̃, with its membership function being articulated in eq. (2). 𝜇𝐴(𝑥) = { 0 𝑥 ≤ 𝑙 𝑥−𝑙 𝑚−𝑙 𝑙 ≤ 𝑥 ≤ 𝑚 𝑢−𝑥 𝑢−𝑚 𝑚 ≤ 𝑥 ≤ 𝑢 0 𝑥 ≥ 𝑢 (2) definition 3: let 𝐴 ̃= (𝑙1, 𝑚1, 𝑢1) and 𝐵 ̃= (𝑙2, 𝑚2, 𝑢2) represent two triangular fuzzy numbers, and let λ be a positive constant. in this scenario, mathematical actions on these fuzzy numbers are executed based on eqs. (3−7): �̃� ⊕ �̃� = (𝑙1 + 𝑙2, 𝑚1 +𝑚2, 𝑢1 + 𝑢2) (3) �̃�⨂�̃� = (𝑙1. 𝑙2, 𝑚1. 𝑚2, 𝑢1. 𝑢2), (4) �̃� − �̃� = (𝑙1 − 𝑢2, 𝑚1 −𝑚2, 𝑢1 − 𝑙2), (5) 𝐴 �̃� = ( 𝑙1 𝑢2 , 𝑚1 𝑚2 , 𝑢1 𝑙2 ), (6) 𝜆�̃� = 𝜆(𝑙1, 𝑚1, 𝑢1) = (𝜆𝑙1, 𝜆𝑚1, 𝜆𝑢1), (7) definition 4: let 𝐴 ̃ = (𝑙1, 𝑚1, 𝑢1) and 𝐵 ̃ = (𝑙2, 𝑚2, 𝑢2) denote two positive triangular fuzzy numbers. the measure of separation among 𝐴 ̃ and 𝐵 ̃ is ascertained by means of eq. (8): 𝑑(�̃�, �̃�) = √ ((𝑙1−𝑙2) 2+(𝑚1−𝑚2) 2+(𝑢1−𝑢2) 2) 3 (8) definition 5: a triple (l, m, u) is utilized to define a triangular fuzzy number. the change of said number to a crisp number is achieved through the utilization of eq. (9): 𝐴 = 𝑙+4×𝑚+𝑢 6 (9) 486 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. 3.2. z-number theory the z-number theory was anticipated as a thorough representation of uncertainty theory, which addresses the calculation of non-reliable numbers [56]. unlike fuzzy theory, the z-number theory considers the concept of reliability. it can be merged with mcdm approaches, such as vikor [57−58]. the z-number represented as a custom pair z = (a, b), is a type of fuzzy data that is utilized to characterize the value of the random variable x. it is important to note that in the aforementioned statement, a is a form of fuzzy constraint that is applied to the value of the random variable x, while b is the partial dependability of probability criteria connected with a. eq. (10) presents the triad (x, a, b) as a comprehensive constraint on x, and designates it as z-valuation, akin to an assignation statement: prob(𝑋 is 𝐴) is 𝐵 (10) this particular constraint is acknowledged as a probability constraint that connotes a function of probability distribution. additionally, it can be given in the following manner: 𝑅(𝑋): 𝑋 𝑖𝑠 → 𝑝𝑜𝑠𝑠(𝑋 = 𝑢) = 𝜇𝐴(𝑢) (11) in the aforementioned equation, the symbol 𝜇𝐴 represents the membership function of a, while u denotes a generic value of x. the function 𝜇𝐴 can be interpreted as a limitation on r(x), signifying that it encompasses 𝜇𝐴(𝑢), denoting the extent to which u can be satisfied. consequently, x presents a stochastic variable with a probability distribution which acts as a potential restriction on x. the potential constraints and probability density function are illustrated in eqs. (12) and (13), respectively: 𝑅(𝑋): 𝑋 𝑖𝑠 𝑝 (12) 𝑅(𝑋): 𝑋 𝑖𝑠 𝑝 → 𝑝𝑟𝑜𝑏(𝑢 ≤ 𝑋 ≤ 𝑢 + 𝑑𝑢) = 𝑝(𝑢)𝑑𝑢 (13) in eq. (13), the term du denotes the constituent of u's derivatives. to effectuate the conversion of a z-number into a fuzzy number, one must execute the ensuing mathematical operations: if z=(a, b) represents a z-number (where a has been identified as the verbal variable illustrated in table 1, while b has been identified as the verbal variable accessible in table 2 and 𝐴 = {(𝑋, 𝜇𝐴)|𝑋 ∈ [0,1]} and 𝐵 = {(𝑋, 𝜇�̃�)|𝑋 ∈ [0,1]} denote fuzzy triangular numbers, then the conversion of fuzzy number reliability to a certain number can be accomplished in the following manner: 𝛼 = ∫𝑥𝜇�̃�(𝑥)𝑑𝑥 ∫𝜇�̃�(𝑥)𝑑𝑥 (14) and the second part is added to the first part as follows: 𝑍𝛼 = {(𝑋, 𝜇𝐴𝛼)|𝜇𝐴𝛼(𝑥) = 𝛼𝜇𝐴𝛼 , 𝑋 ∈ [0,1]} (15) in the aforementioned eqs. (14) and (15), 𝛼 denotes the reliability weight, 𝜇𝐵(𝑥) is an indicator of the degree of dependence of 𝑥 ∈ 𝑋 in b, and 𝜇𝐴𝛼(𝑥) is an indicator of the degree of dependence of 𝑥 ∈ 𝑋 in 𝐴𝛼. a new framework for risk assessment of road transportation of hazardous substances 487 table 1 risk prioritization in fmea with extended swara linguistics terms membership functions equally-significant (es) (1,1,1) moderately-less-significant (mos) (2/3,1, 3/2) less-significant (ls) (2/5,1/2,2/3) very-less-significant (vls) (5/2,3,7/2) much-less-significant (mus) (2/7,1/3,2/5) table 2 conversion guidelines of linguistics variables of reliability linguistic variables very weak (vw) weak (w) medium (m) strong (s) very strong (vs) tfns (0,0,0.3) (0.1,0.3,0.5) (0.3,0.5,0.7) (0.5,0.7,0.9) (0.7,1.0,1.0) 3.3. z-swara method achieving best results in challenging situations depends on evaluating many approaches of decision-making. these techniques enable methodically evaluating and ranking many elements engaged in the decision-making process [59-61]. the technique known as swara is a method for making decisions by determining values for weight that play a pivotal role in the process [62-63]. keršuliene et al. [64] developed this approach, and it has a major advantage in that it allows one to evaluate the opinions of professionals on criteria of relevance all the while figuring their relative weights. in the context of a fuzzy environment, the fuzzy swara technique serves as an adapted decision-making approach that is leveraged to calculate the weights of criteria and sub-criteria. the fuzzy swara methodology operates in a way that is like to that of the swara method. however, it has been extended to accommodate uncertainties in decision-making or a dearth of knowledge, hence the term "fuzzy" [65−66]. the weights of the criteria in the fuzzy swara method are established according to expert judgments, underscoring the indispensable role of researchers in the process. to heighten the level of confidence in the resultant outcomes, the fuzzy swara method has been extended to encompass the z-swara technique, which takes into account a reliability factor. the z-swara methodology involves the ensuing steps: step 1: academics in the field of expertise commonly sort criteria in a descending manner based on their level of significance, where the utmost crucial factors are assigned greater precedence than their less noteworthy counterparts. this approach is attributed to their vast knowledge and proficiency. step 2: during the preliminary assessment stage, proficient individuals should allocate linguistic qualities to the comparative significance of criterion j concerning the preceding j−1 criteria. subsequently, the specialists employ table 1 to ascertain the worth of the primary component (�̃�𝑗). table 2 is utilized for the computation of the dependability component (�̃�𝑗). the outcome is a z-value allocated to every individual state. step 3: to obtain a precise numerical value for the second aspect, which is reliability, the utilization of eq. (14) is deemed necessary in order to transform the z-number that was obtained during step 2 into a triangular fuzzy number (tfn) as elaborated in eq. (12). upon completion of the equation, the weight is subsequently integrated into the original component as stipulated in eq. (13). 488 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. to determine the relative significance of linguistic variables on the j-th criteria, the znumber transforms by adjusting the appropriate tfn values, as presented in tables 1−2, respectively. furthermore, table 3 provides additional z-number to tfn conversions for a comprehensive understanding of the phenomenon. table 3 guidelines are provided for transforming z-number linguistics variables into fuzzy numbers linguistics terms membership functions linguistics terms membership functions (es, vw) (1,1,1) (es, w) (1,1,1) (es, m) (1,1,1) (es, s) (1,1,1) (es, vs) (1,1,1) (mos, vw) (0.212,0.316,0.474) (mos, w) (0.367,0.548,0.822) (mos, m) (0.474,0.707,1.061) (mos, s) (0.561,0.837,1.255) (mos, vs) (0.636,0.949,1.423) (ls, vw) (0.126,0.158,0.212) (ls, w) (0.219,0.274,0.367) (ls, m) (0.283,0.354,0.474) (ls, s) (0.355,0.418,0.561) (ls, vs) (.379,0.474,.0636) (vls, vw) (0.092,0.104,0.126) (vls, w) (0.159,0.181,0.219) (vls, m) (0.205,0.233,0.283) (vls, s) (0.243,0.276,0.335) (vls, vs) (0.275,0.313,0.379) (mus, vw) (0.069,0.079,0.092) (mus, w) (0.120,0.137,0.159) (mus, m) (0.155,0.177,0.205) (mus, s) (0.184,0.209,0.243) (mus, vs) (0.209,0.237,0.275) step 4: to determine the coefficient �̃�𝑗, refer to eq. (16): 𝐾𝑗 = { 1 𝑗 = 1 𝑆𝑗 + 1 𝑗 > 1 , 𝑗 = 1,2, … , 𝑛. (16) step 5: the weight coefficient �̃�𝑗, which is indistinct, is established in the following manner, considering the outcomes obtained from step 4: �̃�𝑗 = �̃�𝑗−1 𝑧𝑗 , 𝑗 = 1,2, … , 𝑛, (17) where �̃�𝑗 is tfn and �̃�𝑗= (1, 1, 1). step 6: lastly, the scheming of the relative weights of the j-th evaluation criteria is performed, which carefully considers all n evaluation criteria. this is achieved by applying eq. (18): �̃�𝑗 = �̃�𝑗 ∑ 𝑞𝑗 𝑛 𝑗=1 , 𝑗 = 1,2, … , 𝑛, (18) where �̃�𝑗 is a tfn. 3.4. z-mabac method the mabac method [67] is a contemporary approach to decision-making that was formulated in 2015. ever since its inception, the mabac method has been extensively utilized and adapted to tackle a plethora of problems within the realm of mcdm [68-69]. the mabac method's foundational framework involves the assessment of the criterion function's distance for each alternative observed from the boundary approximation domain. the ensuing section expounds on the six-step procedure for the application of the mabac method. a new framework for risk assessment of road transportation of hazardous substances 489 the z-mabac technique is a solution to address the issue of decision-making within a fuzzy environment. following the acquisition of attribute weights, the mabac method is utilized to compute the value of the standard function for each alternative, with the distance of said function from the margin approximation area subsequently defined. once the distance of the standard function from the margin approximation area is determined, the alternatives are ranked, and the optimal choice is selected. step 1: involves the construction of an initial decision matrix z utilizing the application of z-numbers. in the formation of the primary decision matrix (z), a crucial step involves the assessment of m alternatives based on n criteria. these alternatives are appropriately represented as vectors, denoted by 𝐴𝑖 = (𝑋𝑖1, 𝑋𝑖2, … , 𝑋𝑖𝑛), where 𝑋𝑖𝑗 represents the value of the j-th alternative by the i-th criterion (𝑖 = 1,2, … ,𝑚; 𝑗 = 1,2, … 𝑛): 𝐶1 𝐶2 … 𝐶𝑛 𝑍 = 𝐴1 𝐴2 … 𝐴𝑚 [ 𝑥11 𝑥12 … 𝑥1𝑛 𝑥21 𝑥22 … 𝑥2𝑛 … … … … 𝑥𝑚1 𝑥𝑚2 … 𝑥𝑚𝑛 ] , (19) step 2: involves the conversion of z-numbers into fuzzy numbers, with the intent of facilitating informed decision-making. the initial decision matrix (z) contains inherent linguistic values (table 4) that undergo a conversion process into fuzzy numbers, resulting in the matrix 𝑍. table 4 language-based variables for ranking the failure modes linguistic variables very weak (vw) weak (w) moderately weak (mw) medium (m) moderately strong (ms) strong (s) very strong (vs) tfns (0,0,1) (0,1,3) (1,3,5) (3,5,7) (5,7,9) (7,9,10) (9,10,10 ) in order to adequately address the language-based elements present in z-numbers, while the second element b is turned into a triangle fuzzy number, the initial element a becomes a trapezoidal fuzzy number (table 5). a z-number is a mathematical entity comprising two unique constituents. within the current framework, the 𝑍 matrix is first solved employing the centroid approach of the second segment, subsequently amalgamating into an assemblage of trapezoidal fuzzy numbers, designated as 𝑍𝛼: 𝐶1 𝐶2 … 𝐶𝑛 𝑍𝛼 = 𝐴1 𝐴2 … 𝐴𝑚 [ 𝑍𝛼11 𝑍𝛼12 … 𝑍𝛼1𝑛 𝑍𝛼21 𝑍𝛼22 … 𝑍𝛼2𝑛 … … … … 𝑍𝛼𝑚1 𝑍𝛼𝑚2 … 𝑍𝛼𝑚𝑛] , (20) 490 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. table 5 conversion guidelines for z-number linguistics based variables to fuzzy numbers linguistics terms membership functions linguistics terms membership function (vw, vw) (0,0,0.32) (vw, w) (0,0,0.55) (vw, m) (0,0,0.71) (vw, s) (0,0,0.84) (vw, vs) (0,0,0.95) (w, vw) (0,0.32,0.95) (w, w) (0,0.55,1.64) (w, m) (0,0.71,2.12) (w, s) (0,0.84,2.51) (w, vs) (0,0.95,2.85) (mw, vw) (0.32,0.95,1.58) (mw, w) (0.55,1.64,2.74) (mw, m) (0.71,2.12,3.54) (mw, s) (0.84,2.51,4.18) (mw, vs) (0.95,2.85,4.74) (m, vw) (0.95,1.58,2.21) (m, w) (1.64,2.74,3.83) (m, m) (2.12,3.54,4.95) (m, s) (2.51,4.28,5.86) (m, vs) (2.85,4.74,6.64) (mh, vl) (1.58,2.21,2.85) (ms, w) (2.74,3.84,4.93) (ms, m) (3.54,4.95,6.36) (ms, s) (4.18,5.86,7.53) (mh, vh) (4.74,6.64,8.54) (s, vw) (2.21,2.85,3.16) (s, w) (3.84,4.93,5.48) (s, m) (4.95,6.36,7.07) (s, s) (5.86,7.53,8.37) (s, vs) (6.64,8.54,9.49) (vs, vw) (2.85,3.16,3.16) (vs, w) (4.93,5.48,5,48) (vs, m) (6.36,7.07,7.07) (vs, s) (7.53,8.37,8.37) (vs, vs) (8.54,9.49,9.49) step 3: the process of normalizing the aggregated fuzzy matrix yields the matrix 𝑍ˊ, which conforms to a standard measurement range as an alternative to the conventional [0,1]. for every benefit criterion j in the decision matrix 𝑍𝛼, the trapezoidal fuzzy number element 𝑍𝑖𝑗 𝛼= (𝑍𝑖𝑗1, 𝑍𝑖𝑗2, 𝑍𝑖𝑗3) is subjected to normalization by separating the maximum value in 𝑍𝑖𝑗3: 𝑍𝑖𝑗 ˊ =( 𝑍𝑖𝑗1 𝑚𝑎𝑥𝑖(𝑍𝑖𝑗3) , 𝑍𝑖𝑗2 𝑚𝑎𝑥𝑖(𝑍𝑖𝑗3) , 𝑍𝑖𝑗3 𝑚𝑎𝑥𝑖(𝑍𝑖𝑗3) ) , 𝑗 ∈ benefit-related criteria (21) for each j-th cost criterion in the decision matrix 𝑍𝛼 , standardization is carried out by separating the minimum value of 𝑍𝑖𝑗1 for each component 𝑍𝑖𝑗 𝛼= (𝑍𝑖𝑗1, 𝑍𝑖𝑗2, 𝑍𝑖𝑗3) and subsequently taking the inverse in order: 𝑍𝑖𝑗 ˊ =( 𝑚𝑖𝑛𝑖(𝑍𝑖𝑗1) 𝑍𝑖𝑗3 , 𝑚𝑖𝑛𝑖(𝑍𝑖𝑗1) 𝑍𝑖𝑗2 , 𝑚𝑖𝑛𝑖(𝑍𝑖𝑗1) 𝑍𝑖𝑗1 ) , 𝑗 ∈ cost-related criteria (22) step 4: the computation of the constituent element of a more intricate matrix denoted as (v) is performed through the utilization of eq. (24): 𝑣𝑖𝑗 = 𝑤𝑗(�̃� + 𝑍𝑖𝑗 ˊ )𝐶,̃ 𝑗 = 1,2, … , 𝑛, 𝑖 = 1,2, … ,𝑚, (23) 𝑉 = [ 𝑣11 𝑣12 … 𝑣1𝑛 𝑣21 𝑣22 … 𝑣2𝑛 … … … … 𝑣𝑚1 𝑣𝑚2 … 𝑣𝑚𝑛 ] = [ 𝑤1. �̃� + 𝑍11 ˊ 𝑤2. �̃� + 𝑍11 ˊ … 𝑤𝑛. �̃� + 𝑍1𝑛 ˊ 𝑤1. �̃� + 𝑍21 ˊ 𝑤2. �̃� + 𝑍22 ˊ … 𝑤𝑛. �̃� + 𝑍2𝑛 ˊ … … … … 𝑤1. �̃� + 𝑍𝑚1 ˊ 𝑤2. �̃� + 𝑍𝑚2 ˊ … 𝑤𝑛 . �̃� + 𝑍𝑚𝑛 ˊ ] (24) a new framework for risk assessment of road transportation of hazardous substances 491 step 5: the computation of the matrix of bordering approximative fields (g) is undertaken herein. the bordering approximative field is established as follows: 𝑔𝑖 = (∏ 𝑣𝑖𝑗 𝑚 𝑗=1 ) 1 𝑚⁄ , 𝑖 = 1,2, … ,𝑚. (25) the components of the weighted matrix (v) are denoted by 𝑣𝑖𝑗 , while m signifies the total number of alternatives. upon computation of the value 𝑔𝑖, a matrix of approximative fields is constructed per criteria g (as expressed in eq. (26)) in the format of n, which denotes the total number of criteria utilized in selecting the alternatives: �̃� = (𝑔1, 𝑔2, … , 𝑔𝑛). (26) step 6: an alternative methodology for ascertaining the boundary approximative area (q) involves the computation of the distance matrix element: 𝑄 = [ 𝑞11 𝑞12 … 𝑞1𝑛 𝑞21 𝑞22 … 𝑞2𝑛 … … … … 𝑞𝑚1 𝑞𝑚2 … 𝑞𝑚𝑛 ]. (27) the determination of the distance of every alternative from the boundary approximative area (𝑞𝑖𝑗) is achieved through the subtraction of the values of the bordering approximative areas (g) from the corresponding elements in the heavier matrix (v): 𝑄 = 𝑉 − 𝐺 = [ 𝑣11 𝑣12 … 𝑣1𝑛 𝑣21 𝑣22 … 𝑣2𝑛 … … … … 𝑣𝑚1 𝑣𝑚2 … 𝑣𝑚𝑛 ] − [𝑔1 𝑔2 … 𝑔𝑛], (28) 𝑄 = [ 𝑣11 − 𝑔1 𝑣12 − 𝑔2 … 𝑣1𝑛 − 𝑔𝑛 𝑣21 − 𝑔1 𝑣22 − 𝑔2 … 𝑣2𝑛 − 𝑔𝑛 … … … … 𝑣𝑚1 − 𝑔1 𝑣𝑚2 − 𝑔2 … 𝑣𝑚𝑛 − 𝑔𝑛 ] = [ 𝑞11 𝑞12 … 𝑞1𝑛 𝑞21 𝑞22 … 𝑞2𝑛 … … … … 𝑞𝑚1 𝑞𝑚2 … 𝑞𝑚𝑛 ], whereas 𝑔𝑖 denotes the contiguous approximative regions for criterion 𝐶𝑖, and 𝑣𝑖𝑗 denotes the constituents of the denser matrix (v). alternative 𝐴𝑖 may pertain to a contiguous approximative region (g), an upper contiguous approximative region (𝐺+), or a lower contiguous approximative region (𝐺−). for each i, 𝐴𝑖 is a member of the set {𝐺 ∨ 𝐺+ ∨ 𝐺−}. in order to identify the region where the ideal alternative 𝐴+ is located, the upper approximative area (𝐺+) must be determined. similarly, the lower approximative area (𝐺−) must be identified to represent the space where the anti-ideal alternative (𝐴−) is located. this is described in fig. 1 [57]. the determination of the affiliation of alternative 𝐴𝑖 with respect to the approximative area 𝐺, 𝐺+ or 𝐺−is conducted through the utilization of eq. (29): 𝐴𝑖 ∈ { 𝐺+ 𝑖𝑓 𝑞𝑖𝑗 > 𝑔𝑖 𝐺 𝑖𝑓 𝑞𝑖𝑗 = 𝑔𝑖 𝐺− 𝑖𝑓 𝑞𝑖𝑗 < 𝑔𝑖 , 𝑖 = 1,2, … ,𝑚. (29) for a specific set, the optimal solution is chosen only if it is a member of the upper approximating field (𝐺+) for as many criteria as possible. 492 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. fig. 1 styles and sizes for equations step 7: the prioritizing of the alternatives is determined through a computing of the values of the criteria functions, as per eq. (30). this equation involves the summation of distances between the alternatives and the surrounding approximative fields, indicated as (𝑞𝑖). to arrive at the final values of the criterion functions, one must add the elements of the q matrix along the rows in accordance with eq. (30). this process ultimately facilitates the prioritizing of the alternatives. 𝑆𝑖 = ∑ 𝑞𝑖𝑗 𝑛 𝑗=1 , 𝑗 = 1,2, … , 𝑛, 𝑖 = 1,2, … ,𝑚. (30) 4. proposed approach in this particular section, the proposed methodology for prioritizing risks is introduced by utilizing fmea, fuzzy-swara-mabac, and z-mabac approaches. the suggested method is delineated in three stages. the preliminary stage of this recommended approach entails the identification of risks within the ambit of risk evaluation by the fmea team, while the values of the three factors are assigned based on table 6. furthermore, in this phase, the pertinent team determines the dependability of each identified risk. table 6 traditional ratings for sod factors rating severity (s) occurrence (o) detection (d) 10 hazardous-without-warning very-high: ʺfailure is almost inevitableʺ ʺabsolute uncertaintyʺ 9 hazardous-with-warning 8 very-high high: ʺrepeated failuresʺ high: ʺrepeated failuresʺ 7 high 6 moderate moderate: ʺoccasional failuresʺ moderate: ʺoccasional failuresʺ 5 low 4 very-low 3 minor low: ʺrelatively few failuresʺ low: ʺrelatively few failuresʺ 2 very-minor 1 none remote: ʺfailure is unlikelyʺ remote: ʺfailure is unlikelyʺ a new framework for risk assessment of road transportation of hazardous substances 493 in the subsequent phase, the employment of the fuzzy swara approach is implemented to account for varying levels of significance pertaining to the triple criteria. upon identification of the most favorable and unfavorable criteria by the proficient team, pairwise comparisons of the criteria are carried out utilizing linguistic variables. the linguistic variables proffered by the experts are subsequently subjected to conversion into fuzzy numbers utilizing table 1. subsequently, the execution of the fuzzy swara mathematical model is performed on the aforementioned values, yielding the optimal weights of the triple factors. in the next phase of the study, the z-mabac method is utilized to rank the identified risks. this methodology takes into account the differing levels of significance of the triple criteria, according to the outputs of the initial and secondary phases. this method combines the fuzzy values with the dependability of each criterion for the three elements of each risk, in contrast to the conventional mabac technique. as demonstrated in this study, this more nuanced approach yields superior results. in the suggested approach, table 5 transforms the values into fuzzy numbers after the development of the decision matrix, which consists of both fuzzy and z-numbers. thereafter, various models are executed within a fuzzy environment. the outcome from the application of these models yields an indistinguishable result from that of the primary ranking of the identified risks in the preliminary phase. the definitive ranking of the risks is established based on the dominance ranking theory, with the triple approaches being juxtaposed against one another. fig. 2 shows the application procedure of the proposed approach. fig. 2 an overview of the proposed methodology 494 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. 5. case study the research region falls in the province of cosenza's municipality of rende (fig. 3). [70]. the territory of the city of rende has an extension of 55 sq km and is located north of cosenza between the municipalities of montalto uffugo, castiglione cosentino, castrolibero, cosenza, san vincenzo la costa, san fili, marano principato, rose [71]. fig. 3 location of the municipality of rende (cs) rende has a population of about 36,500 inhabitants. within the modern city, which developed downstream from the historic center, numerous commercial activities have sprung up in recent decades and public services and equipment of general interest, both public and private, are concentrated. to the north-east of the municipal area, there is also an industrial zone, which represents a strong attraction pole for commercial vehicles. this zone, localized in contrada lecco, is part of the city's industrial development area, and therefore characterized by a high volume of vehicles which, on a daily basis, transport freights with origin or destination in the urban area of the city of rende [72-73]. the main road axes are oriented along the south-north direction, also influenced by the morphology of the territory. however, significant vehicular flows also occur on the transversal routes along the east-west axis (particularly on state road 107), which cross the city, connecting locations on the ionian coast with those on the tyrrhenian coast, and vice versa. within the municipal territory of rende, there is an interchange of the a2 mediterranean motorway (cosenza nord-rende), which provides a connection to state road 107 silana-crotonese. in the urban area, provincial road 241 (formerly state road 19/19bis) plays an important role, serving as the structural axis of rende's settlement development along the crati plain. the a2 motorway and sp241 simultaneously function as collectors for supramunicipal traffic, particularly for the montalto uffugo-cosenza connection, and as primary roads connecting cosenza and rende. some roads serve as transversal links but face significant performance issues (low capacity values compared to high demand peaks), leading to instances of congestion at times, especially on routes connecting to the university campus, which accommodates a new framework for risk assessment of road transportation of hazardous substances 495 thousands of daily trips. a strong polarization of people's mobility between the territories of cosenza and rende is evident. the estimation of vehicular traffic volumes affecting the area was derived from on-site surveys and an analysis of data from the "15th general census of population and housing regarding commuter movements for study or work purposes" (istat). the analysis reveals that the area is affected by a total of 10,320 internal trips (with origins and destinations within the municipal territory), 21,497 inbound trips (with origins outside the municipal territory and destinations within it), and 6,050 outbound trips (with origins within the municipal territory and destinations outside it). the total number of road accidents occurred in the last 5 years (2020-2024) in the study area is equal to 482, involving 735 injured and 5 dead. fig. 4 shows the heavy vehicles impact on traffic flow in the whole study area. the percentage of heavy vehicles on the total volume of vehicles observed in the road network varies from 3% (secondary roads) to 30% (access/exit roads to the industrial area). fig. 4 an overview of traffic flows on the road network in the case study 496 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. among the goods transported in the case study area, a share between 5% and 8% is related to hazardous substances, which are mainly toxic waste, electronic equipment with release of polluting substances, exhausted vegetable oils. the transportation of these types of substances in the area is due to the presence of a waste storage and treatment plant. these values lead to an analysis of the effects that the transport of dangerous substances can generate on the environment and human health. 6. results this specific section focuses on the outcomes of the suggested strategy used to assess possible risks resulting from the transporting of hazardous substances on highways. in accordance with the initial phase of this approach, the team responsible for fmea endeavors to detect and pinpoint all plausible risks that may surface during the process. besides this, the team also defines triple factor values for every identified risk, which are shown in table a.1 in the appendix. it is worth noting that identifying the 17 risks related to the carriage of hazardous substances was not a linear process. to begin with, we conducted a critical review of related literature to identify widely reported risks in the transportation of hazardous materials. complementary to this were expert consultation sessions with persons with substantial experience in logistics, transportation safety, and hazardous materials handling. their input was important to ascertain that the identified risks were not only of practical relevance but also representative of realistic situations. we analyzed historical data on past incidents and accidents involving dangerous goods to further support and consolidate this selection process. the following risks were fully identified through a multi-step process, as shown in table a.1. these risks could be used as a basis for the next steps in assessing and ranking risks. the z-number theory is applied aptly to accommodate the uncertainty imbued inherently in these factors. in this way, dependability and ambiguity of these factors are remarkably acknowledged. table a.2 illustrates the z-number values of the triple factors of each risk based on input from the fmea team. following the use of the swara approach in the second phase of the study's methodology, the evaluation of risks described in table 7 is carried out for each decisionmaker by determining the values of coefficient k and the weights q and w derived using eqs. (16–18). table 7 the weights of the threat factors by swara team number risk factor comparative importance coefficient recalculated weight weight s 1 1 1 1.000 1.000 1.000 0.428 0.444 0.466 tm-1 o 0.285 0.333 0.4 1.285 1.333 1.4 0.714 0.750 0.778 0.306 0.333 0.362 d 0.4 0.5 0.666 1.4 1.5 1.666 0.429 0.500 0.556 0.183 0.222 0.259 s 1 1 1 1.000 1.000 1.000 0.414 0.425 0.440 tm-2 d 0.285 0.333 0.4 1.285 1.333 1.4 0.714 0.750 0.778 0.295 0.319 0.342 o 0.222 0.25 0.285 1.222 1.25 1.285 0.556 0.600 0.636 0.230 0.255 0.280 o 1 1 1 1.000 1.000 1.000 0.434 0.454 0.483 tm-3 s 0.4 0.5 0.666 1.4 1.5 1.666 0.600 0.667 0.714 0.260 0.303 0.345 d 0.222 0.25 0.285 1.222 1.25 1.285 0.467 0.533 0.584 0.202 0.242 0.282 a new framework for risk assessment of road transportation of hazardous substances 497 the transformation of linguistic variables into triangular fuzzy numbers is accomplished during this stage. upon successful conversion, the coefficient 𝑘𝑗 from eq. (16), the fuzzy weight 𝑞𝑗 from eq. (17), and the final weight of the factors in the form of fuzzy numbers 𝑤𝑗 from eq. (18) can be ascertained. table 8 includs the final fuzzy weight of the main criteria for each decision-maker (dm). table 8 ultimate swara weights of the threat factors risk factors ultimate weights s 0.3680 0.3910 0.4176 o 0.3238 0.3477 0.3757 d 0.2275 0.2612 0.2949 following the application of the swara and z-number approaches in the third phase of the study methodology, the rate of the coefficient k and the weights of q and w were calculated for every decision-maker assessing the offered risks in table 9. throughout this particular procedure, the linguistic variables undergo a transformation into triangular fuzzy numbers by virtue of the equations that have been explicitly outlined in tables 1−2. once fuzzy numbers have been generated from the linguistic variables, the coefficient, fuzzy weight, and ultimate weight of the components in the structure of fuzzy numbers are all determined. table 10 shows each decision-maker's final fuzzy weight for every important criteria. table 9 the weights of the risk factors by z-swara team number risk factor comparative importance coefficient recalculated weight weight s 1 1 1 1.000 1.000 1.000 0.407 0.419 0.438 tm-1 o 0.205 0.233 0.283 1.205 1.233 1.283 0.799 0.811 0.830 0.317 0.340 0.364 d 0.335 0.418 0.561 1.335 1.418 1.561 0.499 0.572 0.622 0.203 0.240 0.272 s 1 1 1 1.000 1.000 1.000 0.379 0.385 0.395 tm-2 d 0.159 0.181 0.219 1.159 1.181 1.219 0.820 0.847 0.863 0.311 0.326 0.341 o 0.12 0.137 0.159 1.12 1.137 1.159 0.708 0.745 0.770 0.268 0.287 0.304 o 1 1 1 1.000 1.000 1.000 0.419 0.436 0.463 tm-3 s 0.335 0.418 0.561 1.335 1.418 1.561 0.641 0.705 0.749 0.268 0.308 0.347 d 0.184 0.209 0.243 1.184 1.209 1.243 0.515 0.583 0.633 0.216 0.254 0.293 table 10 final z-swara weights of the threat factors risk factors final weights s 0.3522 0.3712 0.3939 o 0.3355 0.3549 0.3776 d 0.2439 0.2739 0.3025 upon the completion of normalizing the fuzzy assessment matrix, which is represented in table a.3, the ensuing step involves deriving the weighted normalized matrix zswara. this is accomplished by assimilating the weights of the distinct threat factors, as elaborated in table a.4. finally, after the completion of the normalization process for the final weights, the risks are subjected to a ranking procedure utilizing the fuzzy swara-mabac, z-macab, 498 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. and fmea techniques. this particular section of the study implements three distinct approaches in accordance with table a.3−a.4, whereby the outcomes are subsequently presented, while also considering the general dependability of the risks and the inherent uncertainty related with the sod variables. upon conclusion of this analysis, the various options are meticulously compared and contrasted by means of their respective rankings, which are presented in table 11. based on the findings presented in table 11, it can be determined that the fmea approach identified risk modes r14, r7, and r8 as the top three risks, with rpn scores of 280.77, 186.666, and 144.444, respectively. these risks are deemed critical and necessitate the implementation of corrective or preventive measures. conversely, risk mode r1, with an rpn score of 37.333, is ranked last and is currently not in need of corrective actions due to financial constraints. additionally, the fuzzy swara-mabac approach ranked risk modes r12, r13, and r7 as the top three risks, with scores of s12=0.1313, s13=0.1140, and s7=0.1040, respectively. risk mode r2, with a score of s2=−0.1656, was ranked last using this approach. furthermore, it is worth noting that z-mabac ranked risk modes r16, r12, and r9 as the top three risks, with scores of s16=0.1350, s12=0.1333, and s9=0.1196, respectively. lastly, risk mode r2 has been ranked seventeenth with a score of s2=−0.2022 by the z-mabac approach. table 11 comparison of risk prioritization based on three approaches z-mabac fuzzy swara-mabac conventional fmea risk rank si rank si rank rpn 16 -0.124 16 -0.1258 16 37.333 r1 17 -0.202 17 -0.1656 8 95.333 r2 7 0.0917 6 0.0654 12 69 r3 11 -0.400 10 0.0212 14 46.666 r4 15 -0.072 15 -0.0758 13 65 r5 8 0.0556 9 0.0253 4 125 r6 5 0.1029 3 0.1040 2 186.666 r7 9 0.0472 8 0.0415 3 144.444 r8 3 0.1196 7 0.0593 4 125 r9 14 -0.059 14 -0.0475 9 83.333 r10 10 -0.013 11 0.0047 6 120.888 r11 2 0.1333 1 0.1313 10 82.962 r12 6 0.1023 2 0.1140 5 124.444 r13 12 -0.048 13 -0.0368 1 280.777 r14 13 -0.048 12 -0.0255 11 80 r15 1 0.1350 4 0.0862 7 100 r16 4 0.1101 5 0.0703 15 41.481 r17 in fig. 5, we present the outcomes of the prioritization of risks based on three different approaches. furthermore, we executed the assessment and prioritization of risks. the results of the analysis highlight the distinctions in prioritization among the three proposed approaches. in fact, the fuzzy approach has triumphantly overcome the inadequacies of the traditional fmea methodology and provided an absolute priority by considering different weights for each of the 17 risks, utilizing diverse weights for prioritization, and applying fuzzy theory to leverage uncertainty. this approach brings the subject closer to the realworld and engages expert opinions more than ever to obtain accurate results. a new framework for risk assessment of road transportation of hazardous substances 499 fig. 5 a comparison between the prioritization of risks based on three different approaches in contrast, the approach proposed in this study extends the notion of data reliability for each criterion by employing the z-number theory, in addition to covering the benefits of z-mabac and fuzzy swara-mabac techniques. the opinions of experts are incorporated with greater accuracy in calculations, and consequently, our responses will be more dependable. apart from obtaining outcomes that are close to the real-world, the proposed approach is also simple in its solution method. 6.1. validation of proposed approach this section presents two validation studies designed to evaluate the reliability and validity of the rankings derived from the proposed z-swara-mabac approach. the first study involved a comparative analysis with two established methods, z-waspas and zmarcos, to assess the consistency and effectiveness of z-swara-mabac in prioritizing risks associated with hazardous material transportation. the second study employed pearson's correlation analysis to further validate the results produced by the mcdm methods. 6.1.1. comparative analysis the objective of this section is to provide empirical evidence regarding the reliability, accuracy, and efficacy of the proposed methodology. to achieve this, a thorough comparative analysis has been conducted, contrasting the results obtained from the z-swara-mabac technique with those derived from established methods, specifically the extended z-number with weighted aggregated sum product assessment (z-waspas) and the extended znumber with measurement of alternatives and ranking according to compromise solution (zmarcos). the results derived from the application of diverse methodologies underscore the efficacy of the z-waspas and z-marcos approaches, as illustrated in table 12 and figure 6. a detailed examination reveals that criteria a16, a12, and a9 consistently ranked first, second, and third, respectively, across all employed methods, highlighting the robustness of these findings. the comprehensive analysis of risk prioritization using the waspas and marcos methodologies reveals significant parallels in the identified risks. notably, the top four risks consistently rank the same across all three approaches, indicating a strong 500 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. consensus. while minor variations do exist in the prioritization of other risks, these generally do not exceed two ranks, which is considered acceptable given the low magnitude of the discrepancies. therefore, it can be concluded that the proposed method is both valid and reliable, producing results that closely align with those from the comparative methodologies. table 12 risks rankings based on three various ranking methodologies risk z-mabac z-marcos z-waspas score rank score rank score rank a1 -0.124 16 0.4562 15 0.591 17 a2 -0.202 17 0.4348 17 0.596 16 a3 0.0917 7 0.5382 9 0.675 8 a4 -0.4 11 0.5114 12 0.646 10 a5 -0.072 15 0.4901 13 0.604 15 a6 0.0556 8 0.5763 8 0.677 7 a7 0.1029 5 0.6157 6 0.709 5 a8 0.0472 9 0.5882 7 0.649 9 a9 0.1196 3 0.7136 2 0.718 3 a10 -0.059 14 0.4384 16 0.606 14 a11 -0.013 10 0.5183 11 0.646 11 a12 0.1333 2 0.7092 3 0.725 2 a13 0.1023 6 0.6577 5 0.686 6 a14 -0.048 12 0.5289 10 0.628 12 a15 -0.048 13 0.4564 14 0.626 13 a16 0.135 1 0.7356 1 0.736 1 a17 0.1101 4 0.6955 4 0.718 4 the consistent identification of risks across the mabac, waspas, and marcos methods reinforces the robustness and validity of the proposed approach. this alignment enhances confidence in the accuracy of the prioritization process and affirms the reliability of the findings. by employing multiple techniques that yield consistent results, this study underscores its credibility in risk assessment. such agreement among diverse methodologies bolsters the conclusion that the proposed approach is both valid and dependable for effective risk evaluation. fig. 6 chart of risks rankings based on three various ranking methodologies a new framework for risk assessment of road transportation of hazardous substances 501 6.1.2. pearson's correlation the pearson correlation coefficient is one of the most important statistical tools that describes the strength and direction of the linear relationship between two continuous variables. in the mcdm framework, pearson's correlation bears great relevance and importance both as a useful tool in analysis and interpretation and as a support method in decision-making for choosing the best alternative according to the preset criteria. usually, the decision-makers have to select the best alternative among various options on the basis of multiple criteria. pearson's correlation coefficient plays a very important role not only in the selection but also in the comparison and validation of different ranking methods applied in the mcdm frameworks. decision-makers may employ different mcdm methods (e.g., mabac, waspas, and marcos). pearson’s correlation can be used to compare the rankings produced by these methods by analyzing the correlations among the scores derived from different methods. this comparative analysis can highlight inconsistencies or affirm the robustness of the rankings. pearson's correlation is an efficient tool that can be used to verify the relationship between ranking and scoring functions. in this way, it is possible to determine if the rankings created by different methods of scoring are either consistent or correlated, which gives a better understanding of the reliability of the decision-making process. the use of pearson's correlation in assessing the aforementioned relationship in mcdm will provide the ability for decision-makers to assess the stability and dependability of the outcomes. consistent rankings from the various scoring methods confirm the validity of the decisionmaking process, while inconsistencies in the rankings may signal a need to revisit criteria weighting or the scoring methodologies themselves. the decision-maker will be able to identify which methods yield similar results and which may lead to divergent rankings by calculating the correlation coefficients between the rankings produced by different methods. a high value of the correlation coefficient between two methods will indicate that these methods produce consistent rankings, while a low coefficient will point to significant differences. this evaluation is important for establishing the robustness of the alternatives identified. the decision-maker could be less uncertain about the reliability of the selections when several techniques are suggesting similar top choices. pearson's correlation coefficient formula is presented below. 𝑟 = 1 𝑛 ∑ (𝑥𝑖−�̅�)∗(𝑦𝑖−�̅�) 𝑛 𝑖=1 √( 1 𝑛 ∑ (𝑥𝑖−�̅�) 2𝑛 𝑖=1 )∗( 1 𝑛 ∑ (𝑦𝑖−�̅�) 2)𝑛 𝑖=1 (31) the results (figure 7) from the pearson’s correlation analysis between three different ranking methods— mabac, marcos, and waspas— provide valuable insights into the accuracy and appropriateness of these methods for evaluating risks in transporting hazardous materials. fig. 7 pearson's correlations between mabac, marcos, and waspas 502 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. the pearson's correlations between mabac and the other two methods (marcos and waspas) indicate a strong degree of alignment in the rankings produced. specifically: ▪ a correlation of 0.9926 between mabac and waspas suggests an extremely close resemblance in the rankings generated by these two methods. this high correlation indicates that both methods assess alternatives similarly in terms of their effectiveness in addressing the criteria for evaluating risks. ▪ a correlation of 0.9657 between mabac and marcos also reflects a robust agreement, indicating that both methods yield comparable rankings for the alternatives considered. the consistency between mabac and the other ranking methods suggests that the mabac method is generating stable and reliable rankings for the evaluation of risks associated with transporting hazardous materials. a strong correlation implies that the method is not only accurate but also resilient to variations in input data across different methods. this reliability is important for risk assessment because the decision-makers should rely on a valid and believable result. both marcos and waspas also show strong correlations amongst themselves, from the pearson correlation coefficient of 0.9608. that means the three methods operate on kindred principles to evaluate risks, hence building a case towards the use of any of them. however, since mabac shows the highest correlation with waspas (0.9926), it positions mabac as a particularly accurate and effective method within this group for this specific application. based on the analysis of pearson's correlation coefficients, we validate that the mabac method provides accurate ranking results for evaluating risks in transporting hazardous materials. the high correlation coefficients among mabac with the other ranking methods, marcos and waspas, denote a high level of agreement in results from risk evaluations and show that mabac correctly captured the inherent relationships among risk factors. further, in tune with other methods and having the robustness in providing consistent rankings, the method of mabac is recommended to be most appropriate in this study. it allows drawing accurate results that enhance decision-making capabilities in the realm of hazardous material transportation and, finally, to safer and more effective risk management. 7. discussion the current study reveals some convergences and divergences in risk prioritization by applying different methodologies in the assessment of risk in hazardous material transportation. fmea, based on quantitative data and historical trends, gives operational risks like unfitness of personnel on appropriate training in safety matters and poor maintenance of vehicles. on the other hand, the fuzzy swara-mabac approach that integrated expert judgment listed organizational risks related to management inefficiency, such as r12 and r13, in the highest rank order for priority. furthermore, the z-swara-mabac method has identified vehicle compatibility with hazardous materials (r16), drawing on an important perspective rooted in regulatory considerations and contextual factors. the differences among these approaches are due to their methodological frameworks. fmea focuses strongly on measurable factors, often disregarding such qualitative elements as organizational culture. on the other hand, the fuzzy swara-mabac method captures those qualitative dimensions of risk. integration of z-number theory in the z-swara-mabac approach introduces sophisticated modeling of uncertainty that allows the risks to be reevaluated in a new framework for risk assessment of road transportation of hazardous substances 503 light of contextual and regulatory considerations. for example, certain risks-like r2-that have been rated low in conventional methods start assuming great significance when viewed through the prism of compliance and contextual relevance. these results highlight how much the success of risk assessment depends on methodology. while fmea represents one systematized data-driven approach, with roots in historical insights, the fuzzy swara-mabac and z-swara-mabac methods make use of expert opinions and uncertainty modeling in order to uncover risks that otherwise may be overlooked. each method adds something different, which indicates that a comprehensive assessment framework is vital when taking into consideration the complex nature of hazardous material transportation risks. the study also carries practical implications for stakeholders. transportation planners can use these insights to enhance infrastructure and optimize routing strategies, particularly in high-risk areas. safety regulators can revise the priority listing of standards and training that must be made mandatory. logistics operators can take up risk mitigation measures based on their particular needs, such as in-vehicle monitoring and emergency response plans. emergency responders can focus resources on high-hazard areas and improve multiagency coordination. combining findings from all three approaches gives the decision-maker a comprehensive view of the priorities of risk and thus the safest, most efficient ways of transporting hazardous materials. 8. policy implications to mitigate the critical risks associated with hazardous material transportation, policymakers must adopt a multi-dimensional strategy informed by the study’s findings. the integration of diverse methodologies provides a robust framework for risk assessment, enabling targeted interventions to enhance safety and operational efficiency. some of the most important of these measures are as follows: ▪ strengthening training and safety protocols: the drivers and personnel who come into contact with the hazardous materials are to undergo extensive training courses on safe handling, emergency response, and compliance with local and international regulations. this would be further developed through regular updates and practical drills in order to avoid human error. ▪ implementing rigorous vehicle maintenance standards: vehicle safety requires planned inspections, proper maintenance records, and real-time monitoring. it will identify the majority of imminent risks long before they can cause an accident. additional utilization of telematics and predictive analytics will achieve even more efficiencies and reliability for maintenance. ▪ enhancing regulatory compliance: policymakers should establish strict audits and incentives for adhering to safety standards. working in collaboration with transportation companies to develop specific benchmarks with regard to regulatory compliance can provide a culture of accountability and continuous improvement. ▪ improving risk communication and public awareness: safety risks associated with hazardous material transportation should be made transparent. communities will be informed through public awareness campaigns; the involvement of local authorities and emergency services will increase their risk management and preparedness. public confidence will also be aided by regular reporting on safety practices and incidents. 504 s. vahabzadeh, s. s. haghshenas, s. j. ghoushchi, g. guido, et al. ▪ developing robust contingency plans: contingency planning will minimize the effects of any probable accident. to this end, comprehensive emergency response plansdeveloping evacuation plans and resource allocation, among others-are to be developed with frequent simulation tests. coordinated collaboration between transportation operators and emergency services ensures that responses are effective. ▪ leveraging advanced technologies: safety may be significantly enhanced by technological investment. it preserves real-time monitoring systems, automatic reporting tools, and predictive analytics that detect developing threats and guarantee compliance. algorithms for route planning that include risk evaluations may further reduce exposure to high-risk zones. these strategies will, therefore, allow stakeholders to mitigate most of the critical risks identified in this study while engendering a safety and accountability culture in the industry. these policies will continuously need to be evaluated and adapted to respond to evolving challenges in hazardous material transportation, assuring long-term sustainability and public safety. 9. conclusions the transportation of hazardous materials by road is a very complex issue and demands rigorous analysis from different perspectives. this work proposed an integrated approach that used the swara and mabac methods under z-number theory for the assessment and prioritization of risks. by applying expert judgment, critical risks were identified and developed in a tailored fmea approach with regard to rural roads in cosenza, southern italy. results underlined clearly the differences between the three proposed approaches in prioritizing risks, being more performative: the fuzzy approach treated the uncertainty and overcame the limitations set by traditional fmea. by using distinct weights for all criteria, this provided a strong priority ranking that is really useful for decision making under uncertainty. the study significantly provides new insights to enhance methodologies in risk evaluation for transportation of hazardous material. its results provide practical insights for possible future applications on different roadway systems or risk assessment scenarios. the present study predominantly relates to the conditions of rural roads in cosenza, italy, and for this reason, the regional effects it may present will not be directly applicable in other areas or networks of urban roads. the narrowness of the scope limits generalization since the road infrastructure, traffic patterns, and regulatory environment can vary a lot across different regions. it is further recommended that the proposed framework be put to test in heterogeneous environments, including urban as well as other areas with different road conditions, to ensure generalization and adaptability. conducting comparative studies across different geographical and regulatory contexts can provide deeper insights into the framework’s robustness. involving more stakeholders in expert opinions, such as local authorities, transport companies, and affected communities, will be attempted in future work in order to decrease the subjectivism of reliance on expert opinions. such a mechanism would 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names severity (s) occurrence (o) detection (d) tm-1 tm-2 tm-3 tm-1 tm-2 tm-3 tm-1 tm-2 tm-3 r1 unsuitable condition of the roads 6 7 5 6 4 4 1 2 1 r2 unsuitable meteorological conditions 4 3 4 7 5 6 5 4 4 r3 improper packaging 8 8 7 4 2 3 3 2 4 r4 improper loading and unloading 5 7 6 3 3 4 3 1 3 r5 lack of protective equipment 5 4 6 3 2 4 4 5 4 r6 depreciation of vehicles 9 9 7 4 6 5 2 4 3 r7 failure to monitor and maintain vehicles 7 7 6 4 5 3 8 6 7 r8 failure to comply with the special requirements for transporting hazardous materials by vehicle 6 6 8 5 3 5 6 4 5 r9 lack of awareness of the driver 9 8 8 7 5 3 2 3 4 r10 failure to comply with traffic regulations 4 5 6 2 5 3 4 6 5 r11 failure to comply with the requirements related to the transportation of hazardous substances by the driver 6 4 7 4 5 3 5 5 6 r12 over transport 5 8 7 2 3 2 4 5 7 r13 poor management of facilities 4 7 5 3 5 2 8 6 7 r14 inadequate safety training 8 6 5 7 6 6 7 6 8 r15 illegal operation 5 5 6 2 4 3 6 4 5 r16 vehicle does not match hazmat 8 9 8 4 6 2 3 2 4 r17 natural disaster 3 2 3 1 2 2 9 10 9 table a.2 the decision matrix that utilizes z-numbers symbols severity (s) occurrence (o) detection (d) tm1 tm2 tm3 tm1 tm2 tm3 tm1 tm2 tm3 r1 (m,w) (ms,m) (m,m) (m,w) (mw,m) (m,m) (vw,vw) (w,w) (vw,w) r2 (m,w) (mw,w) (m,s) (s,m) (ms,s) (m,vs) (ms,w) (m,m) (mw,s) r3 (s,s) (vs,s) (m,s) (mw,vw) (w,w) (mw,m) (w,w) (w,w) (mw,m) r4 (ms,m) (s,s) (ms,w) (w,vw) (mw,w) (m,m) (w,vw) (vw,w) (mw,m) r5 (mw,w) (w,w) (ms,m) (mw,vw) (w,m) (mw,m) (w,vw) (mw,w) (mw,m) r6 (vs,s) (h,vs) (s,vs) (mw,w) (ms,m) (m,m) (w,w) (mw,w) (mw,m) r7 (s,m) (ms,s) (m,s) (m,w) (m,m) (mw,s) (s,m) (ms,s) (ms,vs) r8 (ms,s) (ms,m) (s,vs) (m,m) (w,m) (ms,s) (m,w) (mw,m) (m,s) r9 (vs,vs) (s,s) (s,vs) (ms,w) (mw,w) (w,m) (w,w) (w,m) (mw,m) r10 (mw,m) (mw,w) (m,w) (w,w) (mw,w) (mw,m) (mw,w) (m,m) (mw,m) r11 (m,vw) (mw,m) (ms,s) (mw,w) (mw,w) (w,m) (mw,w) (mw,m) (m,m) r12 (mw,w) (s,m) (ms,s) (w,vw) (w,vw) (w,vw) (mw,m) (m,m) (ms,s) r13 (m,w) (s,m) (m,s) (w,vw) (m,w) (w,m) (s,m) (m,m) (m,s) r14 (s,m) (m,m) (mw,s) (ms,m) (m,m) (m,s) (m,m) (m,m) (s,s) r15 (mw,vw) (mw,m) (m,m) (w,m) (mw,m) (w,m) (m,w) (mw,m) (mw,s) r16 (s,h) (vs,vs) (s,vs) (mw,w) (m,m) (w,m) (w,w) (w,w) (mw,m) r17 (w,vw) (w,vw) (w,w) (vw,vw) (w,w) (w,w) (vs,s) (vs,vs) (vs,vs) a new framework for risk assessment of road transportation of hazardous substances 509 table a.3 the matrix of normalized fuzzy evaluations symbol s o d r1 0.5029 0.6126 0.7377 0.4109 0.5216 0.6502 0.2275 0.2700 0.3440 r2 0.4538 0.5604 0.6820 0.3238 0.4146 0.5346 0.2958 0.3919 0.5013 r3 0.6010 0.7038 0.7934 0.4856 0.6018 0.7225 0.2351 0.3048 0.4030 r4 0.5765 0.6908 0.8073 0.4607 0.5751 0.6936 0.2351 0.2961 0.3833 r5 0.4415 0.5344 0.6542 0.4856 0.6018 0.7225 0.2427 0.3222 0.4226 r6 0.6501 0.7559 0.8352 0.3860 0.4949 0.6213 0.2427 0.3222 0.4226 r7 0.5519 0.6647 0.7795 0.4109 0.5216 0.6502 0.3565 0.4616 0.5701 r8 0.5765 0.6908 0.8073 0.4109 0.5216 0.6358 0.2806 0.3745 0.4816 r9 0.6501 0.7559 0.8352 0.4358 0.5483 0.6647 0.2351 0.3048 0.4030 r10 0.4293 0.5344 0.6542 0.4856 0.6018 0.7225 0.2654 0.3571 0.4620 r11 0.4783 0.5865 0.7099 0.4856 0.6018 0.7225 0.2654 0.3571 0.4620 r12 0.5274 0.6386 0.7516 0.5354 0.6553 0.7514 0.2958 0.3919 0.5013 r13 0.5274 0.6386 0.7516 0.4856 0.6018 0.7080 0.3261 0.4267 0.5308 r14 0.5029 0.6126 0.7238 0.3611 0.4681 0.5924 0.3261 0.4267 0.5308 r15 0.4293 0.5344 0.6542 0.5105 0.6286 0.7369 0.2654 0.3571 0.4620 r16 0.6501 0.7559 0.8352 0.4607 0.5751 0.6936 0.2351 0.3048 0.4030 r17 0.3680 0.4301 0.5428 0.5604 0.6687 0.7514 0.4323 0.5225 0.5897 table a.4 weighted normalized purpose approach matrix symbol s o d r1 0.4462 0.5237 0.6120 0.4764 0.5694 0.6757 0.2439 0.2794 0.3303 r2 0.4148 0.4953 0.5845 0.3355 0.4264 0.5422 0.2947 0.3729 0.4580 r3 0.5569 0.6451 0.7194 0.5642 0.6493 0.7368 0.2502 0.3061 0.3779 r4 0.5085 0.5334 0.6021 0.5345 0.6179 0.7077 0.2502 0.2983 0.3582 r5 0.4049 0.4681 0.5486 0.5566 0.6466 0.7368 0.2551 0.3147 0.3825 r6 0.6202 0.7167 0.7879 0.4364 0.5325 0.6419 0.2551 0.3170 0.3901 r7 0.4938 0.5828 0.6718 0.4663 0.5629 0.6734 0.3675 0.4627 0.5584 r8 0.5371 0.6339 0.7307 0.4589 0.5560 0.6545 0.2872 0.3654 0.4488 r9 0.6231 0.7182 0.7879 0.5164 0.6063 0.6966 0.2502 0.3077 0.3832 r10 0.3895 0.4595 0.5395 0.5459 0.6377 0.7327 0.2740 0.3470 0.4267 r11 0.4274 0.5010 0.5852 0.5510 0.6431 0.7356 0.2740 0.3470 0.4267 r12 0.4768 0.5594 0.6437 0.6260 0.6937 0.7551 0.3064 0.3901 0.4797 r13 0.4694 0.5528 0.6353 0.5620 0.6468 0.7260 0.3293 0.4159 0.5003 r14 0.4540 0.5397 0.6273 0.3998 0.4964 0.6099 0.3339 0.4202 0.5046 r15 0.3928 0.4609 0.5390 0.5481 0.6506 0.7425 0.2723 0.3477 0.4302 r16 0.6476 0.7311 0.7879 0.5160 0.6113 0.7077 0.2502 0.3061 0.3779 r17 0.3522 0.3874 0.4449 0.6141 0.6914 0.7551 0.4633 0.5478 0.6050 13862 facta universitatis series: mechanical engineering vol. 23, no 3, 2025, pp. 579 604 https://doi.org/10.22190/fume250610029y © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper development of a fuzzy-based decision support system for sustainable tractor selection in green ports galip cihan yalçın ostim technical university, türkiye orcid id: galip cihan yalçın https://orcid.org/0000-0001-9348-0709 abstract. tractors play a critical role in the operational processes of green ports. the primary objective of this study is to develop a decision support system (dss) for the selection of tractors suitable for green port operations. in this context, a hybrid multicriteria decision-making (mcdm) approach based on fuzzy logic—namely the ffhamacher-cimas-lodeci-radar (fermatean fuzzy–hamacher-criteria importance assessment-logarithmic decomposition of criteria importanceranking based on the distances and range) hybrid method is proposed. this hybrid model enables the simultaneous integration of both quantitative and qualitative criteria into the decisionmaking process. expert weight vectors are determined using fermatean fuzzy sets, while the overall criteria weight vector is constructed through a combination of subjective (ffhamacher-cimas) and objective (ff-hamacher-lodeci) weighting techniques. the performance ranking of tractor alternatives is obtained using the radar method. the proposed methodology was applied to a tractor selection problem for a green port in türkiye. the decision model was established based on the evaluations of ten experts, involving eight criteria (two quantitative and six qualitative) and five alternative tractors. according to the results of the case study, towing capacity emerged as the most influential criterion. among the alternatives, the mafi t 230e tractor demonstrated the highest performance. the robustness of the proposed hybrid method was supported through three sensitivity analysis scenarios. additionally, comparative analyses revealed a high level of consistency in the results, confirming the reliability of the method. based on the findings, practical implications and recommendations were provided to support decision-making processes in green port operations. key words: tractor selection, fermatean fuzzy sets, ff-hamacher-cimas, ffhamacher-lodeci, radar received: june 10, 2025 / accepted august 05, 2025 corresponding author: galip cihan yalçın ostim technical university, 06374 ankara, türkiye. e-mail: galipcihan.yalcin@ostimteknik.edu.tr https://orcid.org/0000-0001-9348-0709 mailto:galipcihan.yalcin@ostimteknik.edu.tr 580 g. c. yalçin 1. introduction the increasing volume of global trade has placed ports at the center of not only economic growth but also environmental sustainability. the increasing global trade volume also leads to increased greenhouse gas (ghg) and particulate matter (pm) emissions from logistics equipment [1]. today, ports are not only cargo handling areas; they have also become one of the important sources of ghg emissions with their energy-intensive, fossil fuel-based equipment [2]. increasing efficiency and reducing environmental impacts in port operations have become both political and economic imperative. sustainability goals in port operations necessitate the transformation of not only ships but also in-port equipment such as cranes, carrier vehicles and terminal tractors [3]. in this context, terminal tractors play a critical role in the transportation of containers within the terminal and account for a significant portion of port-related emissions [4]. in large commercial ports such as the san pedro bay port complex in california, terminal tractors contribute to 28% and 33% of total oxides of nitrogen (noₓ) and pm emissions, respectively [5]. as in the case of ambarlı port, diesel-fueled terminal tractors produce the highest carbon emissions among in-port equipment [6]. this high contribution rate has placed the conversion of terminal tractors at the center of ports’ carbon emission reduction strategies. regulatory policies and technological advances developed in recent years encourage the replacement of diesel engines with low or zero-emission alternative power systems [7-9]. traditional diesel engine tractors attract attention with their high fuel consumption and emission production due to the characteristics of in-port operations such as low speed, frequent stop/start and high idle rate [1]. real field tests have shown that diesel terminal tractors emit 2 to 3 times more noₓ and pm2.5 than their emission certification levels [10]. the evaluation of alternative energy systems has added a new dimension to the selection of terminal tractors. electric and hydrogen fuel cell tractors promise zero emissions and are also promising in terms of energy efficiency. studies conducted in recent years reveal that electric versions of terminal tractors offer advantages in terms of both fuel consumption and operating costs [11]. comparative total cost of ownership analyses show that hydrogen fuel cell tractors will be cost-competitive with their diesel counterparts [1]. thanks to adaptive energy management systems designed specifically for variable load profiles, these vehicles provide efficient energy consumption and operational flexibility [12]. the technological transition process affects not only environmental but also operational decisions. studies conducted in port terminals have shown that optimal planning and task assignment strategies of terminal tractors can significantly reduce loading/unloading times and empty trip rates [13]. in addition, evaluation of various transfer scenarios with different tractor models (e.g. semi-trailer or full trailer) once again emphasizes the importance of flexibility and engineering standards in equipment selection [14]. the primary motivation of this study is to develop a decision support system (dss) for the selection of tractors used in green ports. the main objective is to address the tractor selection problem through a multi-criteria decision-making (mcdm) approach by proposing a hybrid method as a dss. to handle complex and sensitive computations, the use of fermatean fuzzy (ff) sets [15], which are based on fuzzy logic, is proposed. ff sets not only offer enhanced capability in managing high levels of uncertainty but also allow for more flexible expression of expert judgments. development of a fuzzy-based decision support system for sustainable tractor selection... 581 moreover, to strengthen the operations and aggregation processes of ff sets, hamacher t-norm and t-conorm-based ff sets have been utilized as bonferroni aggregation [16]. specifically, the ffhwa (fermatean fuzzy hamacher weighted average) aggregation operator [17] was employed for aggregating expert evaluations. the proposed hybrid dss is the ff-hamacher-cimas-lodeci-radar (ffhamacher-criteria importance assessment-logarithmic decomposition of criteria importance) method. in this framework, ff-hamacher-cimas is adopted as the subjective criteria weighting method, where the cimas technique [18] is adapted to the ff-hamacher environment for the first time. for objective criteria weighing, the ff-hamacher-lodeci method was developed by extending the lodeci method [19] using ff-hamacher sets. both weighting methods incorporate ff-hamacher sets for the first time in literature. the ff-hamacher-cimas method was chosen due to its ability to provide consistencybased weighting calculations, whereas the ff-hamacher-lodeci method offers precise weight computations based on logarithmic decomposition. for ranking the tractor alternatives, the radar method [20] was utilized. the proposed hybrid model enables the simultaneous evaluation of both quantitative and qualitative criteria, and the ranking process is performed using the radar approach [21]. the applicability, robustness, and consistency of the proposed hybrid method were validated through a real-world case study conducted at a green port in türkiye. this study presents a novel dss for selecting the most suitable tractor for green port operations by integrating fuzzy logic and mcdm techniques. a new hybrid methodology (ff-hamacher-cimas-lodeci-radar) is proposed, combining subjective and objective weighting methods with an advanced ranking model. subjective weights are derived using ff-hamacher-cimas to reflect expert judgments, while objective weights are calculated through ff-hamacher-lodeci based on decision matrix data. the model evaluates both quantitative (towing capacity, turning radius) and qualitative (energy efficiency, emission level, maintenance cost, total cost of ownership, ease of use, port infrastructure suitability) criteria simultaneously. applied to a real-world case in türkiye involving five tractor alternatives and ten experts, the mafi t 230e emerged as the top choice, with towing capacity as the most critical criterion. the methodology’s robustness was confirmed through sensitivity analysis and comparison with 15 established mcdm methods, showing high consistency. this research contributes both methodologically and practically to sustainable decision-making in green port logistics and beyond. this paper is structured into six sections. section 2 presents the literature review. section 3 outlines the methodology adopted in the study. section 4 details the case study conducted to demonstrate the applicability of the proposed approach. section 5 discusses the results and their practical implications. finally, section 6 provides the conclusions drawn from the study. 2. literature review 2.1. green ports: a paradigm for sustainable port management green ports represent a strategic approach aimed at mitigating environmental impacts while enhancing energy efficiency and sustainable management practices within port operations. traditional port activities often contribute significantly to environmental externalities, including elevated ghg emissions, and air and water pollution. consequently, 582 g. c. yalçin there is an imperative for ports to adopt sustainable practices to address these challenges effectively. the operational components of ports, such as terminal equipment, ships, and land transportation systems, are primary sources of carbon emissions and pollutants, thereby impacting air quality. various global ports have undertaken concerted efforts to implement emission reduction measures, with outcomes influenced by factors such as population density, business models, and specialization in container transportation [22]. effective collaboration between public authorities and port administrations is pivotal in implementing policy instruments aimed at reducing ghg emissions. research underscores the role of environmental incentive systems, automation, and digitalization in facilitating the transition towards green port operations, particularly highlighted in asian ports [23]. renewable energy technologies, such as solar energy, wind turbines, fuel cells, and ocean energy systems, emerge as promising solutions to diminish fossil fuel dependency in ports. studies confirm the technical and economic viability of these technologies in significantly reducing carbon emissions and enhancing sustainability [24]. modeling energy consumption and enhancing the efficiency of port facilities are critical components of operational strategies. it has been demonstrated that optimizing port operations can lead to substantial reductions in energy consumption, with potential savings estimated at up to 34% [25]. integrating green port initiatives with logistics efficiency further underscores the potential for reducing carbon emissions and operational costs, exemplified by initiatives such as transitioning hazardous material transportation vehicles to electric systems [26]. the advancement of digital technologies plays a transformative role in shaping green ports. innovations such as internet of things (iot), artificial intelligence (ai), remote monitoring systems, and autonomous technologies are instrumental in optimizing energy consumption, emission control, and logistics management within port environments [27]. in conclusion, the evolution towards green ports necessitates a holistic approach encompassing policy interventions, technological advancements, and collaborative governance to achieve sustainable development goals. by integrating these strategies, ports worldwide can effectively mitigate environmental impacts while fostering economic growth and operational resilience in a rapidly evolving global context. 2.2. green ports: a paradigm for sustainable port management the vision of green ports fundamentally seeks to minimize the environmental impacts associated with port operations, enhance energy efficiency, and reduce carbon footprints. within this framework, cargo handling equipment (che) represents one of the most significant sources of port-related emissions. key machinery, including terminal tractors, rubber-tired gantry (rtg) cranes, straddle carriers, forklifts, and reach stackers, are indispensable for maintaining port operational efficiency, yet they concurrently contribute substantially to greenhouse gas emissions and overall energy consumption. terminal tractors have been identified as primary contributors to emissions within port environments. empirical data from chinese ports reveal that although terminal tractors comprise just over one-third of all equipment in operation, they account for nearly half of the total fuel consumption among che [28]. a parallel trend has been observed in turkish ports, where terminal tractors constitute many carbon emissions within che fleets [6]. development of a fuzzy-based decision support system for sustainable tractor selection... 583 these findings highlight the critical need to reconsider the technological design and environmental impact of such equipment. the energy consumption profile of che is influenced by multiple factors, including fuel type, engine specifications, and operational duty cycles. notably, equipment with larger engine capacities (though fewer in number) may disproportionately contribute to emissions due to prolonged operational hours or intensive workload demands [28]. this recognition has catalyzed accelerated efforts toward the electrification of port equipment, particularly in regions with stringent policy frameworks or explicit carbon reduction targets. the transition to electric or hybrid che alternatives presents several challenges. research conducted in california ports identifies key barriers such as high upfront investment costs, limitations in grid capacity, and the need for workforce adaptation [29]. nonetheless, with strategic long-term planning, electrification initiatives have demonstrated considerable potential not only to reduce emissions but also to modernize port infrastructure in a manner that promotes inclusivity and resilience [30]. beyond technological substitution, optimizing the deployment and scheduling of che is critical for reducing emissions and improving operational efficiency. coordinated scheduling among terminal tractors, cranes, and yard vehicles can significantly reduce idle times, lower energy consumption, and enhance overall system productivity. emerging optimization frameworks have demonstrated effectiveness in integrating various types of equipment within unified planning routines, thereby reducing empty trips and operational delays [31-32]. the selection of cargo handling equipment is a strategic element within sustainable port management. mcdm methodologies, such as the ahp, promethee, and topsis, are extensively employed to evaluate alternative equipment based on a comprehensive set of criteria. these criteria typically encompass environmental factors (e.g., co₂, nox, particulate matter emissions), economic considerations (investment and operational costs), technical specifications (power output, efficiency), and operational characteristics (flexibility, cycle time) [33]. in summary, the progression toward green ports necessitates a dual focus on mitigating the environmental footprint of cargo handling equipment and enhancing their operational efficiency. key strategies include the electrification of high-emission equipment (particularly terminal tractors) the integration of these assets within optimized scheduling models, and the adoption of life-cycle management approaches. collectively, these measures constitute essential components for achieving long-term sustainability and environmental stewardship in port operations. 3. methodology 3.1. hamacher t-norm and t-conorm based aggregation operator definition 1. the ff set (ℱ̃) defined as ℱ̃ = {〈𝑓, 𝑥ℱ̃(𝑓), 𝑦ℱ̃(𝑓) | 𝑓 ∈ 𝐹〉} is a fuzzy logic-based set using functions defined by the universe 𝐹 and element 𝑓. the ff numbers defined in this set are defined as follows: 𝑥ℱ̃(𝑓): the degree of membership, 𝑦ℱ̃(𝑓): the degree of non-membership, and (𝑧ℱ̃(𝑓)): the indeterminacy degree. the ff numbers fulfill 584 g. c. yalçin the following conditions: 0 ≤ (𝑥ℱ̃(𝑓)) 3 + (𝑦ℱ̃(𝑓)) 3 ≤ 1 and 𝑧ℱ̃(𝑓) = √1 − (𝑥ℱ̃(𝑓)) 3 − (𝑦ℱ̃(𝑓)) 33 [15]. definition 2. the accuracy function is calculated when the score functions used to convert ff sets into numerical values are equal. consider two ff sets as ℱ̃1 = {〈𝑓, 𝑥ℱ̃1(𝑓), 𝑦ℱ̃1(𝑓) | 𝑓 ∈ 𝐹〉} and ℱ̃2 = {〈𝑓, 𝑥ℱ̃2(𝑓), 𝑦ℱ̃2(𝑓) | 𝑓 ∈ 𝐹〉}. score function (𝑆𝑐(ℱ̃1)) calculation is shown in eq. (1) and accuracy function (𝐴𝑐(ℱ̃1)) calculation is shown in eq. (2) [15]: 𝑆𝑐(ℱ̃1) = 1 2 (1 + (𝑥ℱ̃1(𝑓)) 3 − (𝑦ℱ̃1(𝑓)) 3 ) (1) 𝐴𝑐(ℱ̃1) = (𝑥ℱ̃1(𝑓)) 3 + (𝑦ℱ̃1(𝑓)) 3 (2) definition 3. the hamacher t-norm and t-conorm are computed by applying eq. (3) and eq. (4) respectively [17]: 𝑇 − 𝑛𝑜𝑟𝑚 (𝑎, 𝑏) = 𝑎𝑏 𝜗+(1−𝜗)(𝑎+𝑏−𝑎𝑏) (3) 𝑇 − 𝑐𝑜𝑛𝑜𝑟𝑚 (𝑎, 𝑏) = 𝑎+𝑏−𝑎𝑏−(1−𝜗)𝑎𝑏 1−(1−𝜗)𝑎𝑏 (4) herein, 𝑎, 𝑏 ∈ [0,1] and 𝜗 > 0. definition 4. for a group of ff sets defined as ℱ̃𝑟 = {〈𝑓, 𝑥ℱ̃𝑟(𝑓), 𝑦ℱ̃𝑟(𝑓) | 𝑓 ∈ 𝐹〉}, the ffhwa aggregation operator is computed by applying eq. (5) [17]: 𝐹𝐹𝐻𝑊𝐴 =⊕𝑟=1 ɍ 𝜏𝑟ℱ̃𝑟 = ( √ ∏ (1+(𝜗−1)(𝑥ℱ̃𝑟 (𝑓)) 3 ) 𝜏𝑟 ɍ 𝑟=1 −∏ (1−(𝑥ℱ̃𝑟 (𝑓)) 3 ) 𝜏𝑟 ɍ 𝑟=1 ∏ (1+(𝜗−1)(𝑥ℱ̃𝑟 (𝑓)) 3 ) 𝜏𝑟 ɍ 𝑟=1 +(𝜗−1)∏ (1−(𝑥ℱ̃𝑟 (𝑓)) 3 ) 𝜏𝑟 ɍ 𝑟=1 3 , √𝜗 3 ∏ (𝑦ℱ̃𝑟 (𝑓)) 𝜏𝑟 ɍ 𝑟=1 √∏ (1+(𝜗−1)(1−(𝑦ℱ̃𝑟 (𝑓)) 3 )) 𝜏𝑟 ɍ 𝑟=1 +(𝜗−1)∏ (𝑦ℱ̃𝑟 (𝑓)) 3𝜏𝑟 ɍ 𝑟=1 3 ) (5) 3.2. the novel ff-hamacher-cimaslodeci-radar hybrid method in this study, the elements of the decision model are defined as experts (𝛦 = {𝜖1, 𝜖2, … , 𝜖ӗ, … , 𝜖ӗ} (ӗ = 1,2, … , ӗ)), quantitative criteria (𝑁 = {𝜂1, 𝜂2, … , 𝜂ň, … , 𝜂ň} (ň = 1,2, … , ň)), qualitative criteria (𝐿 = {ȴ1, ȴ2, … , ȴĺ, … , ȴĺ} (ĺ = 1,2, … , ĺ)), overall criteria (𝑇 = {ȶ1, ȶ2, … , ȶţ, … , ȶţ} (ţ = 1,2, … , ţ); (ň + ĺ = ţ)), and alternatives (𝛢 = {𝑎1, 𝑎2, … , 𝑎ą, … , 𝑎ą} (ą = 1,2, … , ą)). the hybrid method consists of four stages: stage 1: subjective criteria weights are determined based on the experts' assessment of all criteria. expert weights are also determined. stage 2: objective criterion weights are determined by experts evaluating only the alternatives according to qualitative criteria and are combined with other quantitative criteria. stage 3: the criteria obtained because of subjective and objective criteria weighing development of a fuzzy-based decision support system for sustainable tractor selection... 585 methods are combined. stage 4: based on the final criteria weights, terminal tractor for sustainable port management performance levels are calculated and ranked. the diagram of the ff-hamacher-cimas-lodeci-radar hybrid method is presented in fig. 1. fig. 1 the diagram of the ff-hamacher-cimas-lodeci-radar hybrid method 586 g. c. yalçin the application steps of the ff-hamacher-cimas-lodeci-radar hybrid method are summarized as follows: stage 1: subjective weighting of criteria using the ff-cimas method [18]: step 1-1: it is important to calculate expert weights according to the levels of expertise, which play an important role in decision-making processes, and to integrate these weights into the decision-making process. in this step, the expertise levels presented in table 1 are determined and the expertise level matrix (�̃� = [�̃�ӗ]ӗ) is created accordingly. table 1 linguistic expressions for expertise levels [34] linguistic expressions ffns very-poor (vp) (0.21, 0.70) poor (p) (0.36, 0.41) medium (m) (0.42, 0.52) good (g) (0.73, 0.10) very-good (vg) (0.82, 0.50) step 1-2: the linguistic expressions are transformed into ffns and after obtaining the score function with eq. (6), experts’ weights are calculated with eq. (7): �̃�ӗ = 𝑆𝑐(�̃�ӗ) = 1 2 (1 + (𝑥�̃�ӗ(𝑓)) 3 − (𝑦�̃�ӗ(𝑓)) 3 ) ; 𝑆𝑐(�̃�ӗ) ∈ [0,1] (6) 𝛹ӗ = �̃�ӗ ∑ �̃�ӗ ӗ ӗ=1 ; (ӗ = 1,2, … , ӗ ) (7) step 1-3: to derive the criteria evaluation matrix, experts (𝜖ӗ) evaluate each overall criterion (ȶţ) using the linguistic expressions as presented in table 2. then, these linguistic expressions are transformed into ff numbers, and the criteria evaluation matrix (�̃� = [�̃�ţӗ]ţӗ ) is created. table 2 linguistic expressions for evaluating criteria/alternatives [34] linguistic expressions ffns exceptionally low (exl) (0.30, 0.50) extremely low (el) (0.35, 0.43) very low (vl) (0.36, 0.56) low (l) (0.40, 0.73) below average (ba) (0.42, 0.30) average (a) (0.47, 0.21) above average (aa) (0.50, 0.62) high (h) (0.55, 0.38) very high (vh) (0.60, 0.18) extremely high (eh) (0.72, 0.50) exceptionally high (exh) (0.83, 0.42) step 1-4: to derive the weighted criteria evaluation matrix (�̃� = [𝛨ţӗ]ţӗ ), the expertise level matrix and the criteria evaluation matrix are aggregated via ff-hamacher product operation (eq. (8)). development of a fuzzy-based decision support system for sustainable tractor selection... 587 𝛨ţӗ = �̃�ӗ⨂�̃�ţӗ = { ( ( 𝑥�̃�ӗ (𝑓)𝑥�̃�ţӗ (𝑓) √𝜗+(1−𝜗)((𝑥�̃�ӗ (𝑓)) 3 +(𝑥�̃�ţӗ (𝑓)) 3 −((𝑥�̃�ӗ (𝑓)) 3 (𝑥�̃�ţӗ (𝑓)) 3 )) 3 ) , √ (𝑦�̃�ӗ (𝑓)) 3 +(𝑦�̃�ţӗ (𝑓)) 3 −((𝑦�̃�ӗ (𝑓)) 3 (𝑦�̃�ţӗ (𝑓)) 3 )−(1−𝜗)((𝑦�̃�ӗ (𝑓)) 3 (𝑦�̃�ţӗ (𝑓)) 3 ) 1−(1−𝜗)((𝑦�̃�ӗ (𝑓)) 3 (𝑦�̃�ţӗ (𝑓)) 3 ) 3 ) | 𝑓 ∈ 𝐹 } (8) step 1-5: the score function (𝑆𝑐(�̃�ţӗ)) used to convert the weighted criteria evaluation matrix into crisp values (eq. (9)) yields the crisp weighted criteria evaluation matrix (𝛨 = [𝛨ţӗ]ţӗ ). 𝛨ţӗ = 𝑆𝑐(�̃�ţӗ) = 1 2 (1 + (𝑥�̃�ţӗ(𝑓)) 3 − (𝑦�̃�ţӗ(𝑓)) 3 ) ; (𝑆𝑐(𝛨ţӗ) ∈ [0,1]) (9) step 1-6: the normalized criteria evaluation matrix (𝛬 = [𝛬ţӗ]ţӗ ) is computed applying eq. (10). 𝛬ţӗ = 𝛨ţӗ ∑ 𝛨ţӗ ţ ţ=1 ; (ţ = 1,… , ţ; ӗ = 1,2, … , ӗ ) (10) step 1-7: eq. (11) is used to determine the weighted criteria evaluation matrix(𝛲 = [𝛲ţӗ]ţӗ ). 𝛲ţӗ = (𝛬ţӗ ∗ 𝛹ӗ); (ţ = 1,… , ţ; ӗ = 1,2, … , ӗ ) (11) step 1-8: the maximum value matrix (𝛲ţ 𝑚𝑎𝑥 = [𝛲ţ 𝑚𝑎𝑥] ţ ) is calculated by eq. (12) and the minimum value matrix (𝛲ţ 𝑚𝑖𝑛 = [𝛲ţ 𝑚𝑖𝑛] ţ ) is calculated by eq. (13). 𝛲ţ 𝑚𝑎𝑥 = 𝑚𝑎𝑥 1≤ţ≤ţ 𝛲ţ ; (ţ = 1,… , ţ ) (12) 𝛲ţ 𝑚𝑖𝑛 = 𝑚𝑖𝑛 1≤ţ≤ţ 𝛲ţ ; (ţ = 1,… , ţ ) (13) step 1-9: the matrix representing the differences between minimum and maximum values (𝛭 = [𝛭ţ]ţ ) is computed applying eq. (14). 𝛭ţ = (𝛲ţ 𝑚𝑎𝑥 − 𝛲ţ 𝑚𝑖𝑛); (ţ = 1,… , ţ) (14) step 1-10: the criteria weight matrix (ɰ = [ɰţ]ţ ) is computed applying eq. (15). 588 g. c. yalçin ɰţ = 𝛭ţ ∑ 𝛭ţ ţ ţ=1 ; (ţ = 1,… , ţ) (15) herein, ɰţ = (ɰ1, ɰ2, … ,ɰţ), ɰţ ∈ [0,1], and ∑ ɰţ ţ ţ=1 = 1. if the (𝑅𝐼)values fall below 0.1, the criteria’s weights are consistent, and the next step is taken. stage 2: objective weighting of criteria using the ff-lodeci method [18] step 2-1: each alternative (𝑎ą) is evaluated by each expert (ϵӗ) against attribute criteria only (ȴĺ) using the linguistic expressions defined in table 2. following this evaluation, the lvs are transformed into ff sets as detailed in table 2. thus, an initial decision matrix (�̃�(𝜖ӗ) = [�̃�(𝜖ӗ)ąĺ]ą𝑥ĺ ) where �̃�(𝜖ӗ)ąĺ = (𝑥�̃�(𝜖ӗ)ąĺ (𝑓), 𝑦 �̃�(𝜖ӗ)ąĺ (𝑓)) (ą = 1,2, … , ą; ĺ = 1,2, … , ĺ; ӗ = 1,2, … , ӗ) is created. step 2-2: the evaluations made by each expert are combined using the ffhwa aggregation operator shown in eq. (16). thus, the aggregated decision matrix (𝕃 = [𝕃ąĺ]ą𝑥ĺ ) is composed. 𝕃 = �̃�ӗ⨂�̃�ţӗ = { ( ( 𝑥�̃�ӗ (𝑓)𝑥�̃�ţӗ (𝑓) √𝜗+(1−𝜗)((𝑥�̃�ӗ (𝑓)) 3 +(𝑥�̃�ţӗ (𝑓)) 3 −((𝑥�̃�ӗ (𝑓)) 3 (𝑥�̃�ţӗ (𝑓)) 3 )) 3 ) , √ (𝑦�̃�ӗ (𝑓)) 3 +(𝑦�̃�ţӗ (𝑓)) 3 −((𝑦�̃�ӗ (𝑓)) 3 (𝑦�̃�ţӗ (𝑓)) 3 )−(1−𝜗)((𝑦�̃�ӗ (𝑓)) 3 (𝑦�̃�ţӗ (𝑓)) 3 ) 1−(1−𝜗)((𝑦�̃�ӗ (𝑓)) 3 (𝑦�̃�ţӗ (𝑓)) 3 ) 3 ) | 𝑓 ∈ 𝐹 } (16) herein, 𝜗 > 0. step 2-3: to convert the ff fuzzy sets into crisp values, the score functions (𝑆𝑐(𝕃ąĺ)) is computed using eq. (17). thus, the crisp aggregated decision matrix (𝑆𝑐(𝕃ąĺ) = [𝕃ąĺ]ą𝑥ĺ ) is derived. 𝑆𝑐(𝕃ąĺ) = 1 2 (1 + (𝑥�̃�ąĺ (𝑓)) 3 − (𝑦�̃�ąĺ (𝑓)) 3 ) ; (𝑆𝑐(𝐶ąţ) ∈ [0,1]) (17) step 2-4: in order to create the initial decision matrix including the overall criteria (𝐶 = [𝐶ąţ]ą𝑥ţ ), it is necessary to include the initial decision matrix for both the qualitative criteria (𝕃 = [𝕃ąĺ]ą𝑥ĺ ) and the quantitative criteria (ℕ = [ℕąň]ą𝑥ň ). step 2-5: normalization is accomplished using eq. (18) for the cost and benefit criteria. thus, a normalized decision matrix (𝜅 = [𝜅ąţ]ą𝑥ţ ) is obtained. development of a fuzzy-based decision support system for sustainable tractor selection... 589 𝜅ąţ = { 𝐶ąţ 𝐶ţ 𝑚𝑎𝑥 𝑓𝑜𝑟 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 𝐶ţ 𝑚𝑖𝑛 𝐶ąţ 𝑓𝑜𝑟 𝑐𝑜𝑠𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 } ; (ą = 1,… , ą; ţ = 1,… , ţ) (18) step 2-6: the decomposition is accomplished using eq. (19). thus, the decomposed decision matrix (𝐷 = [𝐷ąţ]ą𝑥ţ ) is obtained. 𝐷ąţ = 𝑚𝑎𝑥{|𝜅ąţ − 𝜅𝜂ţ|} ; (ą. = 1,… , ą; ţ = 1,… , ţ; 𝜂 = 1,2, … ň), (𝜂 ≠ ą) (19) step 2-7: logarithmic decomposition is accomplished using eq. (20). thus, a logarithmic decomposed decision matrix (𝛶 = [𝛶ţ]ţ ) is obtained. 𝛶ţ = 𝑙𝑛 (1 + ∑ 𝐷ąţ ą ą=1 ą ) ; (ą = 1,… , ą; ţ = 1,… , ţ) (20) step 2-8: using eq. (21), the matrix of the weights (ϣ = [ϣţ]ţ ) of the criteria is obtained. ϣţ = 𝛶ţ ∑ 𝛶ţ ţ ţ=1 (21) stage 3: final weighs of the criteria based on ff-cimas and ff-lodeci methods. step 3-1: the criteria weights obtained by the subjective criteria weighting method ff cimas method and the objective criteria weighting method fflodeci method are combined with eq. (22) to obtain the final criteria weight matrix (𝓌 = [𝓌ţ]ţ ). 𝓌ţ = 𝜆ɰţ + (1 − 𝜆)ϣţ; (ţ = 1,… , ţ) (22) where 𝜆 is the parameter expressing the degree of importance of the subjective criterion weights and 𝜆 ∈ [0,1]. stage 4: ranking the alternatives based on ff-radar method [20-21] step 4-1: the initial decision matrix (𝐶 = [𝐶ąţ]ą𝑥ţ ) is created as in step 2-4. step 4-2: using eq. (23), the maximum proportion matrix is obtained separately for the benefit and cost criteria. 𝑅𝐴𝐷𝐴𝑅 = { 𝛼ąţ = 𝑚𝑎𝑥 ą 𝐶ąţ 𝐶ąţ 𝑚𝑎𝑥 ą 𝐶ąţ 𝐶ąţ + 𝐶ąţ 𝑚𝑖𝑛 ą 𝐶ąţ , 𝑓𝑜𝑟 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 𝛼ąţ = 𝐶ąţ 𝑚𝑖𝑛 ą 𝐶ąţ 𝑚𝑎𝑥 ą 𝐶ąţ 𝐶ąţ + 𝐶ąţ 𝑚𝑖𝑛 ą 𝐶ąţ , 𝑓𝑜𝑟 𝑐𝑜𝑠𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 (23) step 4-3: using eq. (24), the minimum proportion matrix is obtained separately for the benefit and cost criteria. 590 g. c. yalçin 𝑅𝐴𝐷𝐴𝑅 = { 𝛽ąţ = 𝐶ąţ 𝑚𝑖𝑛 ą 𝐶ąţ 𝑚𝑎𝑥 ą 𝐶ąţ 𝐶ąţ + 𝐶ąţ 𝑚𝑖𝑛 ą 𝐶ąţ , 𝑓𝑜𝑟 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 𝛽ąţ = 𝑚𝑎𝑥 ą 𝐶ąţ 𝐶ąţ 𝑚𝑎𝑥 ą 𝐶ąţ 𝐶ąţ + 𝐶ąţ 𝑚𝑖𝑛 ą 𝐶ąţ , 𝑓𝑜𝑟 𝑐𝑜𝑠𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 (24) step 4-4: the values obtained in step 4-2 and step 4-3 are used in eq. (25) to obtain the empty range matrix. 𝜃ąţ = |𝛼ąţ − 𝛽ąţ| (25) step 4-5: the relative relationship matrix is obtained using eq. (26). 𝛪ąţ = 𝛼ąţ 𝛽ąţ+𝜃ąţ (26) step 4-6: the relative relationship matrix and the criteria weights are multiplied as in eq. (27) to obtain the weighted relative relationship matrix. 𝛱ąţ = 𝛪ąţ ∗ 𝓌ţ (27) step 4-7: eq. (28) is used to calculate the aggregated ranking index of the alternatives. the alternative with the highest value among the obtained values becomes the best alternative. 𝜓ą = 𝑚𝑖𝑛 (∑ 𝛱ąţ ţ ţ=1 ) ∑ 𝛱ąţ ţ ţ=1 (28) 4. case study this case study was conducted to support the selection of a tractor for a green port in türkiye. a panel consisting of experts specializing in green port operations was established to guide the evaluation process. a total of eight criteria (two quantitative and six qualitative) were identified to assess the alternatives. additionally, five alternative tractor models suitable for green port operations were selected. this section presents detailed information regarding the experts’ levels of expertise, descriptions of the evaluation criteria, and the specifications of the alternative tractors considered in the analysis. 4.1. decision model 4.1.1. experts a panel of ten experts working in green port operations in türkiye was formed for this study. these experts possess varying levels of experience and domain-specific expertise. the composition of the expert panel based on their expertise levels is presented in table 3. through face-to-face interviews conducted with the expert group, the tractor alternatives were evaluated in accordance with the established criteria. this collaborative assessment development of a fuzzy-based decision support system for sustainable tractor selection... 591 ensured that both technical knowledge and practical experience were incorporated into the decision-making process. table 3 the expert group for assessing the green maritime transport performance criteria experts expertise level professions 𝜖1 medium m specialists in green port operations-1 𝜖2 good g specialists in green port operations-2 𝜖3 good g specialists in green port operations-3 𝜖4 very good vg specialists in green port operations-4 𝜖5 very good vg specialists in green port operations-5 𝜖6 medium m specialists in green port operations-6 𝜖7 very good vg specialists in green port operations-7 𝜖8 very good vg specialists in green port operations-8 𝜖9 good g specialists in green port operations-9 𝜖10 very good vg specialists in green port operations-10 4.1.2. criteria definition in this study the definition of the criteria for the terminal tractor selection for green ports is based on several technical, economic, environmental and operational factors. the criteria are individually explained as follows: towing capacity (ton) (𝜂1 − ȶ1): it is referred to as the maximum horizontal pulling force that can be safely and effectively exerted by the tractor to pull handling equipment or connected trailers. in full trailer models commonly used in ports, the load is completely supported by the trailer itself, and the terminal tractor merely provides the pulling force necessary to counteract friction as well as rolling resistance [14]. towing capacity is a crucial performance indicator for ensuring that tractors can maneuver efficiently in tight spaces, maintain safe speeds under load, and endure the demanding, continuous operation cycles typical of terminal environments [35]. additionally, having an adequate towing capacity is essential to maintaining smooth traffic flow and achieving high operational efficiency within the terminal [36]. turning radius (m) (𝜂2 − ȶ2): it is an important measure of performance that defines the smallest circular path a tractor-trailer combination can negotiate and has a direct impact on maneuverability within the limited areas typical of ports and terminals. according to ma et al. terminal tractor design prefers narrow working spaces, frequent steering, and complex road surfaces, all needing a smaller turning radius for effective operation [14]. reducing the turn radius allows terminal tractors to better fit into operational requirements where available space is narrow, such as trailer alignment, yard stacking, or vessel loading operations. energy efficiency (ȴ1 − ȶ3): it is a term that defines the capability to save energy consumption while maintaining the optimum working performance. as brzeziński et al. pointed out, energy efficiency at terminals is crucial since handling vehicles like tractors can significantly affect the terminal daily energy requirement [37]. in addition, martínezmoya et al. demonstrate that yard tractors generate substantial amounts of co₂ emissions and energy consumption in container terminals, underlining the need for more energyefficient models and operational practices [38]. in this context, transitioning from diesel592 g. c. yalçin powered to electric or hybrid has been identified as a key strategy to enhance energy efficiency while supporting environmental goals. emission level (ȴ2 − ȶ4): it refers to the quantity and mixture of pollutants released into the environment during operation, playing a critical role in contributing to air quality around port terminals. terminal tractors have been identified as a major port-associated emission source, particularly co₂ and noₓ, often accounting for over 25% of overall emissions at major terminals [4]. emission rates vary depending on the fuel type: diesel tractors tend to have higher noₓ, and pm emissions compared to liquefied natural gas (lng) or renewable natural gas-powered tractors [5]. in addition, electrification of terminal tractors has been proposed as a feasible alternative to lower life-cycle emissions significantly, especially in regions aiming for net-zero port operations. improvement in emission control technologies and cleaner energy sources are critical pathways to achieving sustainable and low-emission port operations [38]. maintenance cost (ȴ3 − ȶ5): it refers to the total expenses incurred to ensure the continuous operational reliability, safety, and longevity of the terminal tractors, covering activities such as regular inspections, repairs, part replacements, and preventive maintenance. maintenance cost is a significant part of total lifecycle expenses, especially when transitioning to electric terminal tractors, where although the upfront investment is high, the simplified mechanical systems can substantially reduce ongoing maintenance needs. traditional terminal tractors driven by diesel engines, while more affordable to purchase initially, become more expensive in terms of maintenance costs because of such variables as engine deterioration, transmission overhaul, and greater component failure under port use [39]. additionally, operational challenges such as tire damage, particularly in harsh terminal environments, contribute notably to the maintenance costs, necessitating systematic inspection and failure analysis programs. total cost of ownership (tco) (ȴ4 − ȶ6): it considers all the direct and indirect expenses of purchasing, operating, maintaining, energy consumption, and final disposal of the vehicle for its life cycle. terminal tractor tco analysis points out that while electric or fuel cell-based models cost more to buy than their diesel equivalents, they can reach the same cost after a few years due to the lower maintenance and fuel expenses [1]. according to olivari et al., terminal tractor electrification is more economically viable when taking into consideration possible future advancements in battery technology and forecasted decreases in energy prices [11]. ease of use (ȴ5 − ȶ7): it refers to the degree to which operators can efficiently, comfortably, and safely operate the control systems and interfaces of the vehicle during various cargo handling activities. in port environments, where frequent maneuvering and coupling/decoupling of trailers is required, an ergonomic and user-friendly tractor design is crucial in order to prevent operator fatigue and reduce operational errors [14]. recent studies on dashboard and control interface design for electric terminal tractors emphasize that user-centered design approaches, such as employing familiar graphical elements and minimizing unnecessary complexity, significantly improve driver comfort and operational smoothness during the transition from diesel to electric models. port infrastructure suitability (ȴ6 − ȶ8): it refers to the compatibility between the vehicle's operational requirements with the port environment's design and technical requirements. as ma et al. have pointed out, terminal tractors must be designed for narrow working sites, complex road surfaces, frequent breaking, and long continuous working hours, so their design is highly coupled with the physical characteristics and operation development of a fuzzy-based decision support system for sustainable tractor selection... 593 flows of the terminal [14]. moreover, yang observes that the adoption of green and automated technologies in terminals requires a high degree of infrastructural adjustment, particularly about energy supply networks and layout planning optimization [40]. infrastructure readiness also affects how well electric, or hybrid tractor adoption can be achieved, since charging infrastructure, turning radii, and trailer handling areas must be adapted to tractor design specifications. 4.1.3. alternatives definition in the specific case study addressing the issue of the terminal tractor selection for green ports, the potential options are explained below: kalmar ottawa (a1): these terminal tractors have a high usage rate in terminal operations worldwide. the kalmar ottawa t2 model stands out with its compact chassis design, optimized field of view and fast maneuverability. the modular structure of the vehicle provides easy maintenance and low total cost of ownership [41]. terberg yt (a2): these series terminal tractors are durable and flexible tractors widely preferred in european and asian markets. models such as yt193 and yt220 offer high efficiency in narrow port areas thanks to their strong chassis structure and optimized turning circle. the vehicles are equipped with an adjustable fifth wheel structure that adapts to different trailer heights [42]. tico pro-spotter (a3): it has a high market share especially in domestic terminal and storage areas in the united states. the pro-spotter series is known for its spacious cabin design and durable powertrain systems that focus on operator comfort. the use of volvo penta electric powertrains in the latest generation models has increased the energy efficiency of the vehicles and reduced maintenance costs [43]. capacity tj series (a4): they are durable vehicles designed specifically for heavy-duty transportation and high-density terminal operations. models such as the tj5000 and tj6500 offer a wide range of usage flexibility with both off-road and dot (highway) compliant versions. the new generation models developed by capacity trucks feature high-lift capacity hydraulic fifth wheel systems and optimized chassis weight distribution [44]. the mafi t 230e (a5): it is specially designed for sustainable port operations as a fully electric terminal tractor. it maximizes energy efficiency thanks to its high-capacity lithiumion batteries and regenerative braking system. mafi has significantly improved operator ergonomics in the t 230e model by offering low chassis height and increased visibility [45]. 4.1.4. evaluating tractor using the ff-hamacher-cimas-lodeci-radar hybrid model the tractor selection process based on expert evaluations was conducted by sequentially applying the steps of the ff-hamacher-cimas-lodeci-radar hybrid method. the expert proficiency levels are presented in table 4, while the experts’ evaluation vectors for the criteria are provided in table 5. the expert-based evaluation matrix of the tractor alternatives according to the criteria is shown in table 6. as a result of the application, the criteria weight vector is presented in table 7, and the tractor ranking vector is given in table 8. according to the results, towing capacity was identified as the most significant criterion, and the mafi t 230e was determined to be the highestperforming tractor alternative. 594 g. c. yalçin table 4 the significant levels of the experts expert experience ff numbers score functions 𝛹ӗ 𝑥�̃�(𝑓) 𝑦�̃�(𝑓) 𝜖1 medium m 0.42 0.52 0.47 0.0709 𝜖2 good g 0.73 0.10 0.69 0.1054 𝜖3 good g 0.73 0.10 0.69 0.1054 𝜖4 very good vg 0.82 0.50 0.71 0.1084 𝜖5 very good vg 0.82 0.50 0.71 0.1084 𝜖6 medium m 0.42 0.52 0.47 0.0709 𝜖7 very good vg 0.82 0.50 0.71 0.1084 𝜖8 very good vg 0.82 0.50 0.71 0.1084 𝜖9 good g 0.73 0.10 0.69 0.1054 𝜖10 very good vg 0.82 0.50 0.71 0.1084 table 5 the criterion assessment matrix with lvs experts ȶ1 − 𝜂1 ȶ2 − 𝜂2 ȶ3 − ȴ1 ȶ4 − ȴ2 ȶ5 − ȴ3 ȶ6 − ȴ4 ȶ7 − ȴ5 ȶ8 − ȴ6 𝜖1 h vh vh h a h exh vh 𝜖2 vh vh eh aa aa h eh vh 𝜖3 h h h aa h h eh h 𝜖4 aa vh h h h aa vh h 𝜖5 h h vh h a h exh h 𝜖6 vh h vh a a h vh vh 𝜖7 h vh h h a aa vh h 𝜖8 h h vh a h h eh vh 𝜖9 aa vh vh aa aa aa exh h 𝜖10 aa vh h a aa aa eh aa table 6 the initial decision matrix with linguistic expressions experts alternatives ȶ3 − ȴ1 ȶ4 − ȴ2 ȶ5 − ȴ3 ȶ6 − ȴ4 ȶ7 − ȴ5 ȶ8 − ȴ6 𝜖1 𝑎1 vh eh aa vh aa h 𝑎2 eh eh vh h aa h 𝑎3 h eh h eh h aa 𝑎4 vh eh a a ba a 𝑎5 eh exh h vh h h 𝜖2 𝑎1 vh vh h eh aa h 𝑎2 eh vh vh vh aa eh 𝑎3 h vh vh vh h a 𝑎4 vh vh ba a aa aa 𝑎5 eh eh aa vh h vh 𝜖3 𝑎1 vh eh aa h h h 𝑎2 eh eh h h vh h 𝑎3 h eh h eh h aa 𝑎4 h eh a a ba aa 𝑎5 vh exh h h vh vh 𝜖4 𝑎1 eh eh vh aa h eh 𝑎2 eh eh h h h vh 𝑎3 vh eh h vh aa aa 𝑎4 h eh a aa aa h 𝑎5 eh exh h vh h vh development of a fuzzy-based decision support system for sustainable tractor selection... 595 𝜖5 𝑎1 eh eh vh aa vh eh 𝑎2 eh eh h h vh vh 𝑎3 vh eh h vh aa aa 𝑎4 h eh aa h h h 𝑎5 eh exh vh h vh h 𝜖6 𝑎1 eh h vh h vh vh 𝑎2 eh h aa h vh vh 𝑎3 h h h vh aa aa 𝑎4 h h aa h h h 𝑎5 eh vh aa vh aa h 𝜖7 𝑎1 eh eh vh vh vh vh 𝑎2 vh eh h vh h vh 𝑎3 h eh vh vh h h 𝑎4 aa eh aa h h h 𝑎5 vh exh vh h aa h 𝜖8 𝑎1 eh vh vh vh eh vh 𝑎2 eh vh vh h h h 𝑎3 vh vh vh vh h h 𝑎4 h vh h h h vh 𝑎5 eh eh a vh h vh 𝜖9 𝑎1 h vh h eh vh eh 𝑎2 h vh h vh aa vh 𝑎3 aa vh h eh aa vh 𝑎4 a vh aa vh aa eh 𝑎5 h eh vh aa a eh 𝜖10 𝑎1 eh eh h vh a aa 𝑎2 eh eh a eh a vh 𝑎3 vh eh vh exh aa a 𝑎4 eh eh h h ba a 𝑎5 eh exh h eh a aa table 7 the final criteria weights ȶ1 − η1 ȶ2 − η2 ȶ3 − ȴ1 ȶ4 − ȴ2 ȶ5 − ȴ3 ȶ6 − ȴ4 ȶ7 − ȴ5 ȶ8 − ȴ6 𝓌ţ 0.1549 0.0857 0.1070 0.1156 0.1297 0.1433 0.1196 0.1443 rank 1 8 7 6 4 3 5 2 table 8 the aggregated ranking index matrix kalmar ottawa (𝑎𝟏) terberg yt (𝑎𝟐) tico pro-spotter (𝑎𝟑) capacity tj series (𝑎𝟒) the mafi t 230e (𝑎𝟓) 𝜓ą 0.9600 0.9940 0.8920 0.9390 1.0000 rank 3rd 2nd 5th 4th 1st 5. results in this study, a novel hybrid decision-making methodology (ff-hamacher-cimaslodeci-radar) was proposed and implemented to identify the most suitable tractor among various cargo handling equipment alternatives for green port operations. the methodology was applied to a real-life case involving a port currently operating in türkiye. 596 g. c. yalçin the decision-making model involved ten expert decision-makers, eight evaluation criteria, and five different tractor alternatives. as a result of the application of the hybrid method, three primary outcomes were obtained: determination of criterion importance levels: through the integration of the ffhamacher-cimas-lodeci methodology, the weights of the evaluation criteria were calculated by combining both subjective (expert opinions) and objective (data-driven) perspectives. the weights of the criteria as follows: “towing capacity (ton) (ȶ1) (𝓌1 = 0.1549) > port infrastructure suitability (ȶ8) (𝓌8 = 0.1443) > total cost of ownership (tco) (ȶ6) (𝓌6 = 0.1433) > maintenance cost (ȶ5) (𝓌5 = 0.1297) > ease of use (ȶ7)(𝓌7 = 0.1196) > emission level (ȶ4) (𝓌4 = 0.1156) > energy efficiency (ȶ3) (𝓌3 = 0.1070) > turning radius (m) (ȶ2) (𝓌2 = 0.0857)” according to the final importance ranking of criteria, towing capacity (ton) was identified as the most influential factor in the decision-making process, while turning radius (m) was ranked as the least important. this outcome can be attributed to the specific characteristics of the port under consideration; namely, the port area offers sufficient spatial allowance, thus diminishing the criticality of turning radius in operational scenarios. conversely, towing capacity directly reflects the operational power and efficiency of the tractors, indicating its fundamental role in sustainable cargo handling processes. ranking of tractor alternatives: using the radar method, tractor alternatives were evaluated and ranked according to their performance across the determined criteria. the ranking of the tractors as follows: “the mafi t 230e (𝑎5) (𝜓5 = 1.000) > terberg yt (𝑎2) (𝜓5 = 0.994) > kalmar ottawa (𝑎1) (𝜓1 = 0.960) > capacity tj series (𝑎4) (𝜓4 = 0.939) > tico pro-spotter (𝑎3) (𝜓5 = 0.892)” the final ranking revealed that the mafi t 230e tractor emerged as the most suitable alternative for green port implementation. this model was followed by other alternatives in descending order of performance, consistent with the aggregated performance scores derived from the decision model. validation of methodological applicability: the successful application of the ffhamacher-cimas-lodeci-radar hybrid methodology demonstrated its capability to effectively support multi-criteria decision-making in the selection of eco-efficient cargo handling equipment. the method enabled a transparent, structured, and rigorous evaluation process by integrating fuzzy logic with subjective and objective weight derivation techniques, as well as a robust ranking algorithm. in conclusion, the findings of this study underscore the practical relevance and decisionmaking robustness of the proposed hybrid methodology. by guiding the selection of the most appropriate tractor aligned with green port objectives, the method provides a replicable framework for other ports seeking to enhance their sustainability performance through informed equipment procurement strategies. 5.1. sensitivity analysis for robustness to validate the reliability and robustness of the tractor selection results obtained through the ff-hamacher-cimas-lodeci-radar hybrid methodology for green port operations, a series of sensitivity analysis scenarios were designed. the primary objective of these scenarios was to observe the stability of the results under varying conditions and to assess the resilience of the proposed method against changes in methodological parameters. within this scope, three distinct sensitivity scenarios were developed and analyzed. development of a fuzzy-based decision support system for sustainable tractor selection... 597 scenario 1 impact of varying weighting techniques: this scenario was designed to examine how changes in the relative contribution of the ff-hamacher-cimas and ffhamacher-lodeci weighting techniques would influence the final ranking of tractor alternatives. a parameter denoted as 𝜆 was introduced to represent the contribution coefficient of the ff-hamacher-cimas method in the overall weighting process. by incrementally adjusting the 𝜆 parameter from 0 to 1, the model simulated varying levels of influence from the respective methods. the findings, illustrated in fig. 2 (tractor performance scores) and fig. 3 (ranking orders), reveal that when the 𝜆 coefficient approaches zero (indicating minimal contribution from the cimas method) the highest-ranking tractor shifts to terberg yt. conversely, as the influence of cimas increases (𝜆 → 1), the mafi t 230e regains its position as the top performer. the analysis concludes that an 𝜆 value of 0.5 yields the most balanced outcome, suggesting that equal contributions from both weighting methods enhance the objectivity and consistency of the decision-making process. scenario 2 influence of qualitative vs. quantitative criteria: the second scenario was developed to test the performance of the hybrid method when only qualitative or quantitative criteria were used independently in the decision model. this aimed to evaluate the hybrid methodology's capability to integrate both types of data simultaneously, a key advantage in complex multi-criteria decision environments. according to the results, depicted in fig. 4 (performance scores) and fig. 5 (alternative rankings), if only qualitative criteria were considered, kalmar ottawa would have been selected as the best alternative. conversely, if only quantitative criteria were employed, terberg yt would have emerged as the top choice. however, when both types of criteria were integrated using the proposed hybrid method, the mafi t 230e was consistently identified as the most suitable tractor. these findings emphasize the methodological limitation of relying solely on one data type and demonstrate how the hybrid approach effectively mitigates potential decision-making biases, ensuring a more holistic and accurate evaluation. fig. 2 results according to first sensitivity scenario 598 g. c. yalçin fig. 3 ranks according to first sensitivity scenario fig. 4 results according to the second sensitivity scenario fig. 5 ranks according to second sensitivity scenario development of a fuzzy-based decision support system for sustainable tractor selection... 599 scenario 3 evaluation of tractor suitability under alternative conditions: the third sensitivity scenario aimed to verify the consistency of the selection of the mafi t 230e as the top-performing tractor by testing its dominance across various sub-scenarios. in each sub-scenario, the lowest-performing tractor (based on the original ranking) was systematically removed from the model, and the evaluation process was repeated to identify the new top performer. as shown in fig. 6 and summarized in table 9, the mafi t 230e remained the top-ranked tractor across all sub-scenarios, thereby confirming its robust performance and suitability for green port operations under varying configurations. this consistency reinforces the accuracy and stability of the proposed hybrid methodology in identifying optimal alternatives. fig. 6 results according to third sensitivity scenario table 9 the ranks according to third sensitivity scenario sas-3 scenarios ranking best performance result 𝑎𝟓 > 𝑎𝟐 > 𝑎𝟏 > 𝑎𝟒 > 𝑎𝟑 the mafi t 230e sas-3a: removed 𝑎𝟑 𝑎𝟓 > 𝑎𝟐 > 𝑎𝟏 > 𝑎𝟒 the mafi t 230e sas-3b: removed 𝑎𝟒 𝑎𝟓 > 𝑎𝟐 > 𝑎𝟏 the mafi t 230e sas-3c: removed 𝑎𝟏 𝑎𝟓 > 𝑎𝟐 the mafi t 230e in conclusion, the outcomes of all three sensitivity scenarios confirm the robustness, reliability, and adaptability of the ff-hamacher-cimas-lodeci-radar hybrid methodology. the consistent identification of the same optimal tractor under different analytical conditions validates the method as an effective decision-support tool for equipment selection in sustainable port management. therefore, the proposed hybrid methodology can be confidently considered as an ideal approach for supporting strategic decision-making in the context of green port development. 600 g. c. yalçin 5.2. comparative analysis for consistency to evaluate the effectiveness and reliability of the proposed ff-hamacher-cimaslodeci-radar hybrid method for tractor selection in green port operations, a comparative analysis was conducted using alternative ranking methodologies widely recognized in the literature. specifically, fifteen alternative multi-criteria decision-making (mcdm) methods were applied to the same dataset to derive tractor performance scores and corresponding rankings. these methods include radar ii [46], rawec [47,48], coraso [49], alwas [50], aroman [51], ratgos [52], mabac [53-55], marcos [56], ram [57], saw, waspas [58], arlon [59], opara [60], wedba [61], and cocoso [62-64]. the ranking outcomes derived from these methods are presented in fig. 7. upon examination of the results, it was found that the top-performing tractor (the mafi t 230e) and the lowest-performing tractor (tico pro-spotter) remained consistent across all fifteen methods, thereby confirming the robustness of these alternatives within varying methodological contexts. however, minor deviations were observed in the middle ranks. specifically, four of the methods (mabac, marcos, wedba, and cocoso) produced a slight change in the order between the third-ranked tractor (kalmar ottawa) and the fourth-ranked tractor (capacity tj series) when compared to the proposed hybrid method. this variation highlights the sensitivity of certain methods to slight differences in performance criteria weighting and aggregation procedures. furthermore, a correlation analysis was conducted to examine the statistical relationship between the performance scores of tractors as determined by the proposed hybrid method and each of the alternative methods. the correlation coefficients are presented in table 10. the results reveal that the ff-hamacher-cimas-lodeciradar hybrid method demonstrates a very high level of correlation with all alternative methods, indicating strong consistency and alignment in decision outcomes. in summary, the findings from this comparative analysis support the validity and reliability of the proposed hybrid methodology. its ability to produce consistent rankings across a wide range of established mcdm methods reinforces its suitability as a robust decision-support tool for selecting environmentally suitable tractors in green port operations. fig. 7 ranks according to comparative analysis development of a fuzzy-based decision support system for sustainable tractor selection... 601 table 10 the correlation analysis results methods t h e re su lt s r a d a r i i r a w e c c o r a s o a l w a s a r o m a n r a t g o s m a b a c m a r c o s r a m s a w w a s p a s a r l o n o p a r a w e d b a c o c o s o the results 1.00 0.92 0.98 0.98 0.95 1.00 0.99 0.98 0.98 0.99 0.98 0.98 0.99 0.97 0.94 0.97 radar ii 1.00 0.87 0.87 0.99 0.95 0.89 0.89 0.87 0.90 0.87 0.87 0.94 0.87 0.80 0.88 rawec 1.00 1.00 0.89 0.97 1.00 1.00 1.00 1.00 1.00 1.00 0.98 1.00 0.99 1.00 coraso 1.00 0.89 0.97 1.00 1.00 1.00 1.00 1.00 1.00 0.98 1.00 0.99 1.00 alwas 1.00 0.98 0.91 0.91 0.89 0.92 0.89 0.90 0.95 0.89 0.82 0.89 aroman 1.00 0.98 0.97 0.97 0.98 0.97 0.97 0.99 0.96 0.92 0.96 ratgos 1.00 1.00 1.00 1.00 1.00 1.00 0.99 1.00 0.98 1.00 mabac 1.00 1.00 1.00 1.00 1.00 0.99 1.00 0.98 1.00 marcos 1.00 1.00 1.00 1.00 0.98 1.00 0.99 1.00 ram 1.00 1.00 1.00 0.99 1.00 0.98 0.99 saw 1.00 1.00 0.98 1.00 0.99 1.00 waspas 1.00 0.99 1.00 0.99 1.00 arlon 1.00 0.98 0.95 0.99 opara 1.00 0.99 1.00 wedba 1.00 0.99 cocoso 1.00 6. conclusion this study developed a decision support system aimed at selecting the most suitable tractor for green port operations. the proposed system integrates the ff-hamacher-cimas-lodeciradar hybrid method, which effectively combines subjective and objective criteria weighting through fuzzy hamacher operations and utilizes the radar technique for ranking alternatives. the novelty of this approach lies in the simultaneous inclusion of both qualitative and quantitative criteria within a unified decision-making framework, enabling a comprehensive evaluation of alternative tractors tailored to the specific requirements of green ports. the applicability of the hybrid method was demonstrated through a case study involving a green port in türkiye, where a panel of ten experts evaluated five tractor types across eight criteria, including both technical and environmental factors. the findings identified the mafi t 230e as the optimal tractor choice, with towing capacity emerging as the most significant criterion and turning radius as the least influential, justified by the expansive operational area of the port. sensitivity analyses further confirmed the robustness and reliability of the model, while comparative assessments with fifteen alternative methods showed strong consistency and high correlation, validating the effectiveness of the proposed system. overall, this research contributes a novel, reliable, and practical decision support tool for green port management, facilitating informed and balanced equipment selection that aligns with sustainability goals. the results offer 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journal of economic sciences, 14(1), 123-146. 66. więckowski, j., sałabun, w., 2025, comparative sensitivity analysis in composite material selection: evaluating oat and comsam methods in multi-criteria decision-making. spectrum of mechanical engineering and operational research, 2(1), pp. 1-12. facta universitatis series: mechanical engineering vol. 17, no 1, 2019, pp. 95 102 https://doi.org/10.22190/fume190118011t © 2019 by university of niš, serbia | creative commons licence: cc by-nc-nd original scientific paper  on adhesive theories in multilayered interfaces, with particular regard to "surface force apparatus" geometry michele tricarico 1 , antonio papangelo 2 , andrei constantinescu 3 , michele ciavarella 1 1 politecnico di bari, department of mechanics, mathematics and management, italy 2 hamburg university of technology, department of mechanical engineering, germany 3 laboratoire de mécanique des solides, ecole polytechnique, palaiseau cedex, france abstract. adhesion is a key factor in many tribological processes, especially wear. we generalize a recent formulation for the indentation of a multilayered material using an efficient integral transform method, to the case of adhesion, using a simple energetic transformation in the jkr regime. then, we specialize the study for the geometry of the surface force apparatus, which consists of two thin layers on a substrate, where the intermediate layer is softer than the other two. we find the pull-off force under "force control" (i.e. for "soft" loading systems), as well as under "displacement control" (i.e. for "rigid" systems), as a function of the geometrical thicknesses and material properties ratios, and the method is fully implemented in a fast mathematica code, available to the public (see appendix). key words: jkr theory, surface force apparatus, adhesion 1. introduction adhesion forces are more and more of interest in many areas of engineering, both as the basic building block to the theories of frictional interaction, or wear, particularly at micro and nano scales, i.e. at the level of asperities. wear in particular does not occur when asperities deform plastically but when they adhere to the countersurface so strongly that they in fact detach a particle of material. received january 18, 2019 / accepted march 08, 2019 corresponding author: michele ciavarella department of mechanics, mathematics and management, politecnico di bari, viale japigia, italy e-mail: mciava@poliba.it 96 m. tricarico, a. papangelo, a. constantinescu, m. ciavarella in a recent paper [1], the role of adhesion in contact mechanics has been reviewed starting in particular from the fundamental contribution of johnson, kendall and roberts (jkr) [2], who generalized hertz' theory to include van der waals forces described as infinitely short range forces, so that a "contact area" can still be defined, which includes both compressive and tensile stresses. jkr theory has been confirmed in a number of investigations: in principle it should hold only for soft materials and large sphere radius, but in practice for a single, smooth asperity, it holds approximately even for dimensions appropriate to the atomic force microscope (afm), or the surface force apparatus (sfa). the latter is a scientific instrument which measures the interaction force of two surfaces as they are brought together and retracted using multiple beam interferometry to monitor surface separation, directly measure the contact area and observe any surface deformations occurring in the contact zone. developed by tabor and winterton [3] and israelachvili and tabor [4], it comprises thin sheets of molecularly smooth mica, or similar material, glued to the cylindrical glass lenses of equal radii, which are then pressed into elastic contact with their axes at right angles. the sfa is frequently used in conjunction with the jkr theory to extract the surface energy of the contacting sheets. errors may arise since the jkr theory accounts for the contact of homogeneous, isotropic and elastic cylinders (equivalent to the contact of a sphere with a flat surface). sridhar et al. [5] extended the jkr theory to the layered structure of the sfa. the main condition for applying the jkr is perhaps that surfaces should be very smooth, and this is why the surface layer of the sfa is an extremely flat, optically transparent, mica layer (in turn backed with an ultra-thin silver layer to reflect light), and further material or molecules of interest are then coated or adsorbed onto the mica layer. the mica layer is mounted on a glass cylinder and two such cylinders are put in contact in cross-perpendicular configuration, which is equivalent to the contact of a sphere with a plane (see fig.1). sfa is extremely sensitive as it uses piezoelectrics to position with a force accuracy in forces at the 10 -8 n level, and optical interferometry to measure distances to within 0.1 nanometer, and is similar in some respects to the atomic force microscope (afm), except that it is a surfacesurface apparatus rather than a tip-surface one, and it can measure much longer-range forces, although as a technique it is probably more laborious. fig. 1 the geometry of the sfa apparatus contact. in each of the crossing cylinders, a mica layer is covering a glass cylinder (a bilayer is in this configuration adsorbed) on adhesive theories in multilayered interfaces... 97 in the sfa, many authors probably use the jkr original formulation for homogeneous halfspaces to extract the surface work of adhesion of the contacting sheets. sridhar et al. [5] already proposed a hybrid finite element method-analytical technique to obtain an extension of the jkr solution. their work plots the force as a function of the contact area, and hence reveals only the maximum force under force control (when the loading system can be considered "soft"), which they find only weakly dependent on moduli or thicknesses ratios. as an example, we can consider the case of functionally graded materials with power law elastic modulus, where e0 is the characteristic modulus at length z0 0 0 ( ) k z e z e z        with 1 1k   (1) this case can be solved analytically [6], to obtain that the pull-off force remains independent on the elastic modulus 3 2 c k p r     (2) [for example under force control] where ω is the work of adhesion of the interface. hence, pc is very weakly dependent on power exponent k and for example for a case k = 0.5 the pull-off changes from the jkr value 3 2 cp r   by only +14%. however, a combination of elastic moduli which is not monotonically varying leads to more interesting results. for example, stan and adams [7] use a mathematical solution similar to what we shall use here, and show the results for three samples having layer structure (from top to bottom) indicated in tab. 1. the samples differ from each other through the elastic modulus of the second layer where sample #3 is very soft. each layer was 2 nm thick, the tip was considered rigid and of radius 20 nm, and w = 0.1j/m 2 for all the samples. sample #1, #2 give almost identical results, while sample #3 gives a very different force-indentation curve, with less pull-off under force control, but much more pull-off under displacement control, perhaps by +30-40%, than the other two. table 1 examples of 3 layers structure on a substrate of much larger modulus (100 gpa, poisson’s ratio 0.25), in stan and adams [7] e[gpa] #1 #2 #3 top 15 15 15 mid 50 25 5 bott 10 10 10 mcguiggan et al.[8] used the fem technique of [5] to find more extensive results for the sfa including some experiments, and found that, realistically, the pull-off force can vary between ... 2cp r r      (3) 98 m. tricarico, a. papangelo, a. constantinescu, m. ciavarella i.e. a variation with respect to the jkr value of −33%... + 33%. material properties and layer thicknesses in [8] are reported in tab. 2, and can be considered typical for sfa. between the glass substrate (silica) and the mica surface, which are almost of the same material properties, there is an intermediate layer with more than one order of magnitude softer modulus, which is typically an epoxy glue. table 2 examples of 3 layers typical of sfa with their properties according to [8] layer thickness [µm] e [gpa] ν mica 5.5 62 0.21 epoxy 25 3.4 0.5 silica substrate 72 0.25 however, notice once again that all the results are typically plotted in terms of contact area vs. load, which does not permit us to distinguish between force or displacement control. a realistic setup, of course, is somewhere in between load control and displacement control, the latter being the limit when the stiffness of the system tends to very high values. in this paper, we shall extend the method published by constantinescu et al. [9] to adhesive configurations and study some implications for sfa apparatuses jkr adhesive curves. 2. from non-adhesive to adhesive solution the original adhesionless method [9] writes the harmonic papkovich-neuber displacement potentials as the hankel transform of four unknown arbitrary functions 1 2 3 4( ), ( ), ( ), ( )i i i ia a a a    for each layer and two other functions 5( )a  and 6 ( )a  for the substrate. in order to get the solution of the problem, the determination of 4n+2 unknown functions is needed. then we write the boundary conditions of continuity of displacements and traction among the layers, and finally the contact problem at the surface. the algebraic computation of displacements and stresses is done symbolically in mathematica. the system has a solution in terms of 1 1 ( )a  , which depends on an unknown pressure distribution (function h()). this function is the solution of the fredholm equation of the second kind 1 0 1 ( ) ( , ) ( ) ( )h m y h y dy f      , for 0 1  (4) where f depends only on the imposed indentation depth and on the indenter shape, and τ is the normalized radial coordinate in the contact area. for kernel m(y,τ), infinite integrals are given in constantinescu et al. [9] which unfortunately contain highly oscillatory integrand if a/h1 (the contact area over thickness of the first layer) is high. to improve the range of a/h1 < 100 with sufficient accuracy, we modified the original code in the supplementary material of constantinescu et al. [9], by splitting the integration intervals of the infinite integrals (which are obviously already truncated in practice to where the integrand is significantly nonzero) in 10 parts, where on each of them gauss-legendre quadrature with 25 points is done. on adhesive theories in multilayered interfaces... 99 2.1. transformation into adhesive solution the original jkr theory is derived and it determined the elastic strain energy u for the sphere problem by following a two-step scenario as follows. we first load the contact in compression to load p1 up to a contact area a1. we then hold the contact area constant as to make the system linear, and then reduce the load to p2. hence we can write 1 2 1 1 2( ) p p p              (5) assuming the final value of displacement δ2 [displacement control], contact areas a1 and δ1 are obtained by minimizing the total potential energy 1u a   (6) therefore, we obtain 1 0 a    and hence 1 1 1 1 u u a a           (7) and hence 2 1 1 2 1 2 1 1 2 a p            (8) as obtained in [10], which permits us to derive a general relation between the adhesive solution and that without adhesion, which is exact in axisymmetric problems as is the present one. a similar derivation was later also suggested by popov [11], and more restricted cases also treated in [12,13]. we use non-dimensional parameters according to [1]: 2ˆˆ ˆ; ; p a p a r r r          (9) where 1/3 * 1e r          (10) with * 1 1 2 11 e e    being the reduced elastic modulus of the first layer. we obtain that pull-off under displacement control is ˆ 5 / 6ap   and under force control it is ˆ 1.5ap   for the original jkr spherical case. 100 m. tricarico, a. papangelo, a. constantinescu, m. ciavarella 3. some examples let us gives a few examples of interest for sfa apparatuses. in particular we consider a rigid sphere of radius r = 4h1 indenting a layered flat surface. we take, therefore, as in fig.1, 2 layers on a substrate having the same young's modulus of to the top layer, with e1 = e3 = 70 gpa, ν1 = 0.21, ν2 = 0.5, ν3 = 0.25 and we fix e1/e2 = 20 and h = 1 nm, varying the thickness of intermediate layer h2/h1 = {1, 5, 10, 20, 50}. the solution is calculated over a list of adhesiveless indentation depth values. we use two discretizations: in the interval δ1 = [0.001, 2] nm we use a step of 0.02 nm, while for δ1 = [2, 50] we use a step of 0.5 nm. the finer discretization for small values of δ1 aims at obtaining a good estimation of the pull-off in displacement control. it should be noted that it is not possible to compute the solution at point δ1 = 0, because of numerical limitation in the code. as shown in fig.2, the curves of load-indentation displacement or contact radius vs. displacement vary considerably their shape, and in particular the pull-off load vary both if under displacement control (the load at the lowest displacement, ˆ ap ) or under forcecontrol ˆ bp (the absolute minimum of the load). fig. 2 curves of load vs. displacement (a) and contact radius vs. load (b), for a case "sfa-like" (2 layers on a substrate having properties identical to the top layer), with e1 = 70 gpa; e1/e2 = 20; but varying the thickness of the intermediate layer h2/h1 = {1, 5, 10, 20, 50} fig. 3 pull-off load under displacement-control ˆ ap (a) or load control ˆ bp (b) for a case "sfa-like" (2 layers on a substrate having properties identical to the top layer), with e1 = 70 gpa; e1/e2 = 20; but varying the thickness of the intermediate layer h2/h1 a) b) a) b) on adhesive theories in multilayered interfaces... 101 in particular, fig. 3 shows only the pull-off loads and it is clear that the variation is particularly relevant under displacement control, which is a case not so documented in the literature, whereas under force control the variation is relatively small, as already wellknown. we now move to consider a fixed ratio of thicknesses, namely h2/h1 = 5, which is typical for sfa, and vary modulus ratio e1/e2 = {1, 5, 10, 20, 50}, see figs. 4 and 5. the variation of pull-off forces is shown to depend strongly on the modulus ratio initially and then asymptotically reach some limit values. fig. 4 curves of load vs. displacement (a) and contact radius vs. load (b), for a case "sfa-like" with h2/h1 = 5, and vary the modulus ratio e1/e2 = {1, 5, 10, 20, 50} fig. 5 pull-off load under displacement-control ˆ ap (a) or load control ˆ bp (b) for a case "sfa-like" with h2/h1 = 5 and vary the modulus ratio e1/e2 = {1, 5, 10, 20, 50} 4. conclusions we have obtained a general method for solving multilayered problems indentation with jkr adhesion and we have made some considerations for the sfa type of geometry. it is shown that the thickness ratio and modulus ratio of the second to first layers (assuming the substrate is almost identical to the top layer) can vary in a relatively modest way the pull-off under force control, and in a more pronounced way the pull-off under displacement control. the curves vary their shape considerably. the model provides a fast evaluation of the entire curve. a) b) a) b) 102 m. tricarico, a. papangelo, a. constantinescu, m. ciavarella appendix: mathematica code the code used for this work, implemented in the symbolic software mathematica, is available as supplementary resource with the paper [13] at the following link: http://dx.doi.org/10.1016/j.ijsolstr.2013.04.017 acknowledgements: antonio papangelo is thankful to the dfg (german research foundation) for funding the projects ho 3852/11-1 and pa 3303/1-1. references 1. ciavarella, m., joe, j., papangelo, a., barber, j.r., 2019, the role of adhesion in contact mechanics, journal of the royal society interface, 16(151), 20180738. 2. johnson, k.l, kendall, k., roberts, a.d., 1971, surface energy and the contact of elastic solids, proceedings of the royal society a, 324(1558), pp. 301—313. 3. tabor, d., winterton, r.h.s, 1969, the direct measurement of normal and retarded 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13326 facta universitatis series: mechanical engineering https://doi.org/10.22190/fume241129002e © 2025 by university of niš, serbia | creative commons license: cc by-nc-nd original scientific paper application of concrete canvas for enhancing railway substructure performance under static and dynamic loads balázs eller1,2, szabolcs fischer1 1széchenyi istván university, department of transport infrastructure and water resources engineering, hungary 2university of pécs, department of civil engineering, hungary abstract. this study evaluates the potential of concrete canvas (cc), a geosynthetic cementitious composite mat (gccm), in improving railway substructure performance under static and dynamic loads. while cc is widely used for drainage and slope protection, its behavior under railway-specific dynamic forces remains insufficiently studied. addressing this gap, full-scale laboratory tests, including advanced gom atos 3d scanning and four-point bending evaluations, were conducted to assess deformation, settlement, and shear resistance. results showed that cc integration reduced settlement by 44.9% compared to unreinforced tracks after one million load cycles, while inner shear resistance can be improved by up to 52-57%. cc's quasi-rigid nature allowed it to withstand over 2.3 million dynamic load cycles – equivalent to 25.875 million tons of mixed traffic – without significant cracking or material failure. the material maintained structural cohesion, with microfractures contributing to its enhanced durability and resistance to deformation. bending tests revealed that fiber reinforcement in the case of longitudinal weaving direction increased flexural strength by 55.1% after 7 days and 81.7% after 28 days of curing, further emphasizing cc's robustness. these results highlight cc's ability to address issues such as weak subgrades and poor drainage, making it a promising solution for localized track repairs and challenging operational environments. future research should explore long-term field performance, extreme environmental conditions, and advanced modeling techniques to optimize cc applications in railway infrastructure. key words: railway, ballasted track, deterioration, concrete canvas, cc, geosynthetic cementitious composite mat, gccm, dynamical effect, gom atos received: november 29, 2024 / accepted january 12, 2025 corresponding author: szabolcs fischer department of transport infrastructure and water resources engineering, faculty of architecture, civil engineering and transport sciences, széchenyi istván university, h-9026 győr, egyetem tér 1., hungary e-mail: fischersz@sze.hu 2 b. eller, s. fischer 1. introduction railway tracks are one of the most significant forms of fixed-rail transportation within the domain of land-based travel. railroads enable the transport of substantial quantities of goods and passengers at relatively high speeds, with low energy consumption and minimal environmental impact (this claim applies predominantly to railways powered by electric traction). for freight transport, longer distances (exceeding 1,000 km) are typically relevant, whereas for passenger transport, shorter distances (generally under 1,000 km) are more significant. railway operations rely on the collaboration and contributions of various disciplines, functioning continuously, 24/7. this typically involves a broad range of engineering tasks and their coordination, such as structural designs, load-bearing capacity, and stability (primarily the field of civil engineering [1-7]); the structural configuration and reliable operation of vehicles [8,9]; logistics and traffic management tasks [10-12]; high-voltage traction systems and battery-hybrid systems [13,14]; signaling and safety devices; telecommunications systems; and more. railway track structures, whether utilizing traditional crushed stone ballast or ballastless designs, consist of two primary components: the superstructure and the substructure. this article focuses primarily on thin structural elements used at the interface between the ballast and the substructure, exploring their applicability through laboratory testing. both static and dynamic analyses are considered in this context. the appropriate drainage and support from the subgrade are essential at the railway structures. railway track maintenance organizations need solutions that can perform several functions simultaneously and economically. in this case, it means that the operator highly recommended striving for the usage of technologies that can be easily installed and require fewer human and mechanical resources while the solution fulfills the requirements of the renewal. this makes it easy to postpone more serious and expensive interventions, which are necessary but delayed because of the possible cost implications. this deterioration mostly comes from inappropriate drainage, weak subsoil or mudding effect, too. the authors investigate these problems with the experiences of hungarian state railways (máv) and the raaberbahn's hungarian division (roeee, i.e., győr-sopronebenfurth railways), but most of the problems can be projected onto the railways of other countries as well. following this idea, the authors investigate the application of gccm (geosynthetic composite cementitious mat) layers in railway circumstances. these materials primarily serve drainage or slope protection purposes; however, the authors see much greater potential in them. among the gccm layers, the most promising material is the concrete canvas (cc) [15], which was a pioneer in these solutions, was examined. thanks to the fast early cement curing effect, in addition to good drainage and layer separation, much less installing time and track closure (or, in other words, derail – but not derailment) are necessary. the authors of the current paper (where "authors" is written later in the paper, as it relates to this given meaning) previously made several tests with gccm material to check the applicability from railway aspects [16,17]. in addition, numerous case studies can be found on the official website of concrete canvas [15]. extreme rainfall and hot weather are no problem. the material is protected against frost damage to a certain extent under the railway ballast, but it is also used for channel lining where there is no frost protection. thanks to the internal fiber reinforcement and the application of concrete canvas for enhancing railway substructure performance… 3 double-sided geotextile covering, the material's size change is less; thus, it is much more resistant to frost [15]. in the case of the material suits trench lining, the service life can be extended further with crushed stone covering. eller executed research on the behavior of the material under static forces in his previous studies [16-18]. during the research, it was confirmed that the shear resistance showed a more significant effect compared to most types of geogrids [19], which means more stability in the railway ballast. for these types of materials, drainage function is one of the primary solution options; therefore, that solution was not examined separately. the results proved the applicability, but from a railway point of view, one of the most important parts of these experiments is summarized in this paper. that means the load under a railway track is exceptional. in addition to the dead load, the elements of the track structure suffer and are also subjected to significant dynamic forces. considering that cc in bond state forms a rigid layer structure, the question arises regarding the behavior of the material under dynamic forces. it is important to note that the solution is not cost-effective and time-consuming in the case of new construction of longer lengths. on the other hand, the authors believe that the investigated solution is an appropriate possibility for short renewable track sections, where the problem comes from the weak subgrade and the inappropriate drainage. when the full-scale sample was constructed, the properties of the applied tensar triax geogrid were considered according to al-sumaiday et al. [20]. as the scope of the cc material is primarily related to the railway topic, it is first necessary to examine the main substructure improvements used in the present. indraratna et al. [21] showed that the substructure is responsible for the failures of the ballast, which is approx. 58%. the mixing of the materials or the weak subgrade properties causes big problems. roshan et al. [22] published a review article on railway track degradation, failures, and solutions. the mentioned ground improvement is also discussed by liu et al. [23], where they discuss the subgrade engineering aspect from the high-speed railway aspect, and by le kouby et al. [24], where their approach is generalized in the professional field of railways. both articles contain ground improvement technologies like geosynthetic-reinforced pile-supported embankments, pile-raft structures and pile-plate structures, or deep mixing. laboratory studies and field tests were presented by fortunato et al. [25] about soil-binder columns. the full-scale laboratory measurements presented in this article have inspired the authors' large-scale studies. the traditional reinforcement system, i.e., stabilizations, was collected by solihu [26]. this review article includes both the advantages and disadvantages of this technology. in the topic discussed, the number of publications is smaller; what is more, the application of cc in the railway environment is not dealt with by anyone other than the authors. liu et al. [27] reviewed the design and application of concrete canvas reinforced with spacer fabric. the design of the internal fiber structure is also touched upon and is an integral part of this publication, as one of the aims of the paper is to demonstrate the effectiveness of warp-knitted spacer fabric, weft-knitted spacer fabric or woven spacer fabric in improving specific mechanical properties. this solution is not new in industrial applications and has been studied by zhao et al. [28], while the bending and compression test aspects are discussed considering the ordinary mechanical behavior [29,30]. ko et al. [31] investigated that regardless of whether or not the blowing agent was added, the bending (flexural) strength tended to increase as the filling factor of the concrete increased, and it was confirmed that the concrete filling rate should be more than 100% to fully exhibit 4 b. eller, s. fischer the physical properties of gccm. in terms of the characteristics of gccms, this may be true for different types of cc materials in general. the use of concrete canvas or gccms has many possibilities beyond drainage tasks. as envisioned in its rail application, it can also be used to perform other tasks. one of them is emergency engineering, which is part of the cc material portfolio [32]. among the many possibilities, tents, underwater pipeline protection, and mine wall reinforcement are among the examples to be mentioned. on the other hand, unique structures were investigated by mohammadsalehi et al. [33], which were obtained by deforming already solidified materials and joining them as sandwich structures. joining in a lattice structure increased the flexural strength. circular concrete columns were covered by a cc layer and tested under axial compression [34,35]. the effect of the reinforcement was significant; compression strength increased by 70-90%. eller et al. [18] previously performed deflection and other measurements, the application of which will also be part of this article. however, it is necessary to mention [36], which has the same topic in parts. it provides suitable literature for comparing the results. the astm international released several guidelines to help the engineering applications, from which the relevant pieces were taken. in 2018, astm international field preparation, layout, installation, and hydration standard of geosynthetic concrete composite mat [37] was published, while in 2019, another guideline was published on the topic of the three-point bending test of gccms [38]. the third relevant regulation on the subject was issued in 2021, which was the measurement of the compression strength of the gccms [39]. all three requirements were taken into account during the measurements; however, some tests had minor differences due to technical limitations. despite the increasing use of geosynthetic materials in railway infrastructure, there is still much to learn about the performance and long-term behavior of cc as a reinforcement solution for railway substructures. while cc has proven effective in applications such as drainage, slope stabilization, and even military construction, its potential under the dynamic loads typical of railway traffic has not been thoroughly explored. most research to date has focused on static load conditions, offering limited insight into how cc responds to the repeated stresses of moving trains or how well it maintains its structural integrity over time. additionally, many studies have been confined to small-scale laboratory tests, leaving a lack of real-world data to guide practical applications. comparisons between cc and more traditional reinforcement methods, such as geogrids, are also scarce, making it difficult for engineers to evaluate their suitability for specific challenges like weak subgrades or poor drainage. this study seeks to fill these gaps by investigating cc's performance under both static and dynamic loading conditions, using full-scale testing and advanced measurement techniques. by addressing these unanswered questions, the research aims to provide a clearer picture of cc's capabilities and expand its role as a reliable and innovative solution in railway engineering. the structure of the study is as follows: section 2 deals with materials and methods, section 3 is about the results and discussion, and section 4 contains the derived conclusions as so-called "thesis statements". application of concrete canvas for enhancing railway substructure performance… 5 2. materials and methods 2.1. materials 2.1.1. real-size (full-scale) tests in all cases, the same type of stone ballast was applied in current tests. the ballast crushed stone was andesite, with 31.5/50 mm grain sizes (i.e., the nominal minimum grain size was 31.5 mm; hence, the maximum is 50 mm) according to the msz en standard [40]. the sample was given by colas északkő ltd., from the quarry of szob, hungary. it can be said that all ten investigations on this topic were made from the same type of stone particles for the correct future comparisons. in the current article, the cc was purchased from concrete canvas ltd. the dynamical tests were made in 2022 when concrete canvas ltd. modified the composition of the cc. therefore, the first test was made with type cc13 (where 13 means the thickness of the material in the 'mm' unit); on the other hand, the new tests were made with the newer type cct3. this is because the manufacturer has modified the structure and reduced the maximum thickness to 11 mm. so, the distributed types nowadays are the cct1, cct2 and cct3 types, where the thicknesses are 5, 7 and 11 mm. these materials are relatively thin compared to the whole layer structure of a railway track. because of that, these can mainly be compared to geopolymers. there are other notable types of cc, like the cch or ccx, but thanks to their characteristics, these types are weaker in terms of railway loads. the protection layer (like in every other measurement) was formed by z0/22 (0/22 mm) crushed stone. this material was ordered from a local mine near győr (hungary). the particle size distribution (psd) was measured for adequate future comparisons because the significant standard deviation was experienced at the previous measurements. according to psd, the z0/22 complies with the normal standard from [41]. the measured psds are shown in fig. 1. fig. 1 psd from the full-scale samples' z0/22 protection layer (the applied sieves: 0.032, 0.063, 0.125, 0.25, 0.50, 1, 2, 4, 8, 16, 32, 63 mm) permanent displacements and changes were measured using the digital image correlation (dic) method. the dic is a widely used non-contact surface deformation 0 10 20 30 40 50 60 70 80 90 100 0.03 0.13 0.51 2.05 8.19 32.77 p er ce n t p a ss in g [ % ] approx. particle/sieve diameter [mm] borderline of distribution (lower limit) borderline of distribution (upper limit) normal distribution stronger sample 6 b. eller, s. fischer measurement technology; the environmental requirements, while the measurement is simple, the measurement accuracy is high, and it is a widely applicable method [42-47]. today, dic measurement is used in many fields of industry and science, including biomechanics, architecture and materials science, but it is not used in railway construction and investigations. this is due to the large number of particles in the railway ballast and the difficulty of accessing them. nevertheless, it is an excellent solution for representing displacement and deformation in laboratory conditions. in this study, the measurements made by gom atos 3d were combined with gom tritop measurements to increase accuracy. the method was detailed by szalai et al. [47] and szalai et al. [48]. in the full-scale sample, an l4 marked (type) maba hungaria ltd. type sleeper was used. its height is 23 cm, while on the longitudinal aspect, a slice was cut out from the original sleeper. thanks to this, the length of the sleeper was 39 cm (its centerline is the symmetrical axis of the rail foot), and it follows that the contact area was 29×39 cm2. 2.1.2. 4-point bending and compression tests eller et al. [18] published the main properties by measurement and evaluation. these results were for specimens that were bonded at rest, without load. the fe (finite element) modeling of the previous bending and compression tests has shown that the 3d fiber reinforcement and fabric covering in the material significantly increases the resistance to the different forces. the aim was to investigate whether this reinforcement could be quantified or not. the cement powder material was not available separately, so the powder was collected from the excess materials that came from previous measurements. finally, the specimens required for the tests were prepared from the collected cement powder. the specimens prepared were 4×4×16 cm3 and 7×7×21 cm3 brick bodies. on the other hand, another basis for comparison is the examination of the material taken from the full-scale test. in this case, the specimen is given; it has a preloaded, distorted, uneven surface. thanks to this, the used cc material was mentioned in section 2.1.1. 2.2. methods 2.2.1. real-size (full-scale) tests like the previous investigations on this topic, the tests performed in the laboratory of széchenyi istván university, győr. this article contains different types of measurements, so these will be presented in two parts. the common parts are the following. for the dynamical investigation test, a full-scale sample was built under laboratory circumstances. the first try of this test was published by szalai et al. [49]. initially, the frame of the sample was made from two layers of wooden sheets/boards, while the filler material was simple sandy gravel with low e2=16.33 mpa bearing capacity. the measurement was made according to msz 2509-3:1989 [50]. after the initial test, the frame was built up from steel u profiles, which ensured 0.25 m thick of 0/22 mm crushed stone protection layer. the thicker protection layer was significant because the sample was built on concrete ground in the laboratory. with this formation, the bearing capacity could be decreased to ensure real circumstances. the other important part was to decrease the rigidity due to the base layer. to avoid the concrete slab's rigidity, under ballast mats (ubms) were laid on the base ground. its thickness is ~3 cm (the average density is 800 kg/m3, the average hardness on the upper application of concrete canvas for enhancing railway substructure performance… 7 surface is 44 shore a, hence on the bottom surface is 46 shore a; the vertical static bedding modulus is 0.12 n/mm3, while the vertical dynamic bedding modulus is 0.2 n/mm3 at 5 hz and 0.242 n/mm3 at 10 hz). on the ubms, one layer of geotextile was used as a separation layer. in the 0.25 m thick 0/22 mm protection layer for better bearing capacity, one layer of triax tx130s geogrid, the effect was studied by fischer [19]. it was built in the middle of the layer. the protection layer was compacted at 0.12 m and 0.25 m using a 68 kg heavy l-2/c plate vibrator. the size of the plate vibrator is 500×500 mm2, with a nominal vibration rate of about 3000/min. on the ready layer structure, bearing capacity was measured on the upper plane by static plate load test. the reached bearing capacity was e2=34.65 mpa, which was a satisfying result from the aspect of the railway loads. this value satisfies the circumstances and conditions for which the authors would apply this solution. under the ballast layer, one more geotextile layer was laid to prevent the mixing of the different layers. on the other hand, knowing the contact area of the used sleeper part, the area under the full-scale angle load distribution has been painted out on the white geotextile surface. editing this "core area" was important because the cc sample had been cut to cover this surface. on the other hand, the edge of the ballast was also signed with orange (color) to ensure an adequate track structure profile. the blue crosses signed the possible dynamic load-bearing capacity measurement points to get appropriate feedback on the homogeneity of the additional layer. it is seen in fig. 2. there were 4 measurements with different number measurement cycles. 2 tests were applied with cc and 2 tests without it. these measurements were made in 2022; thanks to this, the applied cc specimen was cc13. the authors decided that between the cc13 and the new type cct3, the difference is mainly the layer thickness; the experienced behavior will not be modified significantly. that means the results can be compared to the previous measurements. fig. 2 set-up of the protection layer, border signs of the materials the hydration is a critical moment. according to the authors' experience, under laboratory conditions, water easily flowed laterally, so the water had to be collected on the cc surface, while deformation must be minimized. on the other hand, if a lot of water flows out, it is absorbed in the 0/22 mm layer, which can distort the measurements due to 8 b. eller, s. fischer high moisture. the manufacturer's recommendation for hydration for this cc type is 7.5 liter/m2 of water. the watering was made two times, first directly on the surface of the cc and then on the 10 cm thick ballast layer after the compaction. the amount of water was not measured precisely every time; the addition was approximately as prescribed. it should be noted that cc cannot be overhydrated, so the principle of more rather than less applies. based on later results, hydration was adequate. the first gom atos (discussed below, in this section) measurement had to be made on the cc specimen (fig. 3): the flatness value. the flatness of cc after loading was analyzed [16,17]. after these experiences, the flatness values could be comparable. fig. 3 flatness measurement before hydration on the protection layer, the standard nominal ballast thickness was used according to 225 kn axle loads, which resulted in 0.32 m crushed stone with 31.5/50 mm particle size distribution. 0.32 m is also ideal because the structural height of the measuring instrument limited the further increase in ballast bed thickness. the compaction of the railway ballast material was executed in 3 series/phases on each 10 cm height. on the top of the lower ballast layer, a 23 cm height and 39 cm length reinforced concrete sleeper was placed. after that, the ballast was filled until the level of the top of the sleeper. according to the standards, an adequate ballast shoulder was created and compacted. the final cross-sectional design is shown in fig. 4. the best way to measure movements and displacements (i.e., deformation) is to use contactless measurement, one of the modern forms of testing. this 3d technology allows the non-contact measurement of displacements and deformations of objects subjected to mechanical or environmental loads, which is difficult to achieve at a practical level in railway (civil) engineering but can provide excellent results at the laboratory level. the railway ballast and movements, as well as deformations, were measured using the gom atos triple scan, a high-precision industrial optical 3d scanner [51-53]. it measures the application of concrete canvas for enhancing railway substructure performance… 9 entire surface of the part; the two-axis sensor captures a dense point cloud or polygon mesh that accurately describes the geometry of the object, indicating invisible defects. the projector unit projects a pattern of stripes (gray-code-based) of different widths and light intensities onto the object to be immersed, whereby the ccd detects the phase-shifted sinusoidal light intensity variation reflected from the object. these patterns are captured by two cameras [51-53]. the instrument operates on the heterodyne light interference principle, utilizing multiple phase shifts to achieve maximum sub-pixel accuracy. using the transformation equations of optical imaging, the computer calculates the 3d coordinates of each camera pixel with high precision in just a few seconds. fig. 4 the set-up of the full-scale test multiple independent measurements from different viewpoints are necessary to digitize an object's entire surface. since both the cameras and the projector are calibrated, 3d surface points can be triangulated from any two of the three distinct views provided by the three units. the projection unit employs low-frequency blue light, which effectively reduces the impact of interfering ambient light [51-53]. the measuring instrument has an accuracy of 17 µm or 0.02 mm. for the measurements, 3d-printed plastic particles (i.e., "eikosi-tessera-eder") are used with point stickers as reference points. these were placed between the variously spaced ballast particles, excluding on the slope or horizontal surface. in addition, point stickers were also glued to the frame of the loading instrument for appropriate measurements. these measurement elements are shown in fig. 5. with these, the ready full-scale sample; the initial state was measured by gom atos and tritop. digitization requires system calibration, which involves scanning the caliper plate from various distances and angles. following the measurement, the scanned models are compared with the cad model of the original piece using atos professional 2017 software [51-53]. 10 b. eller, s. fischer fig. 5 the reference points in reality (left) and at visualization (right) – green dots, elements if a cross-section is to be made using the gom measurement, meshing would be required due to the highly variable and irregular surface of the crushed stone bedding. this would require a significant amount of data, which would require an extensive it infrastructure. for this reason, the measurement strategy was changed, feature points were selected, and their displacement was taken into account. the gom tritop measurement was suitable for this measurement. this measurement was better than a simple geodetic measurement because the specific points are always the same, marked with a marker so that their deformation can be better detected. this method of recording data also allows the displacement of the entire full-scale sample to be investigated since the number of points marked is simply a matter of choice. after the measurement, these can be stored as data and can be easily recalled and compared with other measurements for future analysis. such a measurement is performed on about 28 to 35 million measurement points, so as mentioned above, an adequate it background and performance are essential. after the digitalization, the cyclical dynamical loading was started immediately. this move was significant because the ballast layer needed loading to be pushed into the cc layer. it was a necessary movement to earn the lower stone-cemented state. the full-scale sample was constructed to meet the conditions for an axle load of 225 kn. the l4 sleeper piece lower plane was 0.29 m wide and 0.39 m long, so the loading area was 0.1131 m2. to determine the dynamical loading of the device, the calculation method was used from [54]. according to the planned axle load, with these parameters and method, the required dynamic load was fd,max=70.83 kn (see fig. 4). there were 2 measurements with the cc layer in the structure. gom measurements were made at the following states: (i) 100,000 cycles, (ii) 200,000 cycles, (iii) 500,000 cycles, (iv) 1.0 million cycles, (v) 2.0 million cycles (only at the first test, due to technical due to technical reasons), (vi) 2.3 million cycles (only at the first test, due of technical due to technical reasons). after the tests, the sleeper and the ballast were removed. because of the loading, the particles were pushed into the cc and cemented. the phenomenon was the same as after statical loads. the stone particles had to be removed by force to measure the flatness of the end state. the flatness was measured by gom atos according to the previous investigations. application of concrete canvas for enhancing railway substructure performance… 11 2.2.2. 4-point bending and compression tests as described in section 2.1.2, the original deflection measurements are compared with unreinforced and preloaded specimens. the considered professional literature and tests were described previously [55-57], like the puncture test, 3-point bending test, tensile test, etc. from that, a four-point bending test, compression strength test and puncture test were made. with the exception of the puncture test, the measurements were repeated with specimens without fiber reinforcement. the other option is the bending test of cc material with a deformed surface subjected to dynamic loading. in this case, the bending test was planned after the flatness test. one of the possibilities was the four-point bending measurements after the flatness examination. the test and the devices were the same as in eller et al. [18]. 350×200 mm2 specimens were cut from the 1.1×1.2 m2 area specimen. the movement of the edges was measured in the middle. it is important to note that each specimen was cut from the 45° nominal loading area only. it is discussed in section 3.1. the mechanical strength and the volume stability were described by han et al. [58,59]. as it is known, cc material has two weaving (woven) directions, which show different results in the tests. for this reason, the 12 specimens are divided into 6 pieces per woven direction. in addition to the bending test, it was important to prove that the material did not break under dynamic loading. the second comparison option is the specimen without 3d fiber reinforcement. a standard specimen shall be used to test the strength properties of concrete, which may be cubic, cylindrical, or slab-shaped. according to this, the authors created 4×4×16 cm3 and 7×7×21 cm3 brick bodies (so-called beams) from the aforementioned cement powder. the tested specimen is seen in fig. 6. the specimen size was chosen according to the available measuring instruments and the amount of material. first, the smaller specimens were tested in a 3-point bending test; then, a compression test was performed on the intact side. the 3-point bending measurements were discarded due to measurement device errors, but the cubic sections taken from the specimens were usable. so, the compression test corresponded to the compression test of a standard 4×4×4 cm3 cube. the 7×7×21 cm3 specimens were measured by 4-point bending tests. according to the previous methods, the specimens were tested after 1, 7, and 28 days of curing. fig. 6 the set-up of the four-point bending test and the broken cc specimen 12 b. eller, s. fischer compressive strength tests were made to calculate the material's young's modulus (e [mpa]), published by eller et al. [18]. the specimens were 300×300 mm2 size tables from cc13. the material was loaded slowly until 200 kn. thanks to the wide and thin specimen, 200 kn was an upper limit due to the wide and thin material, as a spectacular breakage would hardly ever have occurred in this size dimension. this new compressive strength (destructive) test was carried out according to msz en 206 [60]. during the test, the surface of the pressure plate was larger than the test specimen. the load was applied in a central and uniformly increasing manner. the load was applied until the specimen was destroyed. the purpose of these methods is to determine to what extent the 3d fiber reinforcement strengthens the material and to what extent the embedded stone particles reduce it. it has to be mentioned that no safety factor was considered in the publishing of strength values. 3. results and discussion 3.1. real-size full-scale test as it was mentioned, two measurements were used of cc in the railway structure. first, the number of loaded cycles stopped at 2.3 million, so the simulated mixed mass rolled over is 25.875 million tons. the second test stopped at 800,000 loaded cycles, which means 9.0 million tons. an additional number of cycles was not possible due to instrument failure. in order to compare the adequacy of the samples, a reference measurement was executed. this measurement was taken up to 10 million load cycles, which means 112.5 million through-rolled tons. as described in the methods section, gom measurements were performed after different loading intervals. this data recording method can be used to test the displacement measurement of the entire full-scale sample, about 28-35 million measurement points. in this evaluation, the average heights of the points at the four corners of the sleeper were taken to determine the settlement, and the settlement curve was determined on this basis. the advantage of this measurement is that additional points can be evaluated later on the basis of the test; on the other hand, 3d images and cross-sections can be created, and the point cloud is available. the recording method can be seen in fig. 7. in addition to scanning, a number of highresolution images need to be taken in order to process the data correctly. the measured settlement in each condition can be represented and compared to the previous or initial measurements. in fig. 8, the full-scale sample's settlement is marked by yellow arrows. the arrows act as vectors, which also show the direction of movement. as mentioned, other fixed points could have been chosen, but the authors focused exclusively on the settling of the railway sleeper. using the settlement data from the measurement, the settlement of the sample is presented in fig. 9. the measured values come from the gom and the continuous control measurements. unfortunately, due to technical reasons, the measurement of the full-scale sample with cc ended at 2.3 million load cycles, so someone could only compare it with the sample 10 million load-cycle sample. on the graph, according to the measured settlements, a power function was applied. this fits appropriately to the measured settlements after different amounts of loads. it can be concluded that at 1 million load application of concrete canvas for enhancing railway substructure performance… 13 cycles, the use of the cc specimen more than halved the settlement of the reference track structure. no further measurements were made for technical reasons. the results can be compared to the ones reported by fischer [5], where the investigation was related to field tests and measurements. according to [5], tracks behave like logarithmic settlements; however, the current tests showed power functions. this is not a clear and problematic discrepancy, but it does require further investigation, in particular to identify the causes and factors that have modified the curves in the studies related to section 2.1.1, where both the unreinforced and cc-reinforced designs showed power function settlement trends. eller et al. [17] proved that the cc layer can increase the inner shear resistance by 52-57% if the cc layer gets the necessary preloading or normal loading. in these previous tests, there were two options. at first, the layer was loaded by 100 kn for 7 days, while the second series was loaded by 100 kn for only 30 minutes, both statically. as it was emphasized above, the considered samples were loaded immediately after the building. after removing the ballast layer, the surface of the cc layer was the same as under the static loading tests. the penetration effect of the ballast particles showed the same state, which means the immediate loading on the structure helps to achieve the same results. from this, the same inner shear resistance is estimated. fig. 7 scanning, in addition to surround imaging fig. 8 the state after a given load, representing the rate and vector direction of settling 14 b. eller, s. fischer fig. 9 comparison of the settlements of the full-scale samples, with and without cc layer in the track structure the surface was measured by gom atos to compare the penetration between the static and the dynamic loading cases. before the 3d scanning, all the stone particles had to be removed from the layer structure. in some cases, the particles could be moved only by force. after scanning the surface, it was seen that the flatness value was nearly the same. these results are summarized in table 1. it is seen that the 7 days of static preloading had a more significant effect on the cc, while the other two options showed similar results. from this, it can be assumed that the dead load alone provides enough load for the ballast particles to penetrate into the cc layer. under the sleeper, the deflection was larger a bit, but not significantly (fig. 10). on the other hand, the unevenness and the penetrations are clearly visible, which is presented in fig. 10. due to the compressive force of the particles, the cement powder separated from the cc layer and surrounded the ballast particles. the bonding happened with this deformation. as was written above, the main question at this solution was the cc's behavior under dynamic loads. at the laboratory, after the sample was removed, it was seen that the cc layer was not cracked/broken. to be able to prove this scientifically, an additional measurement had to be used. according to section 2.1.2, the cc was cut into pieces (fig. 11.), and the pieces were measured in a four-point-bending test. fig. 11 shows the planned cut lines and the cross-sections covered by the 4-point bending measurement (red bar), the number of specimens and the direction of weaving of the specimens. the green edge represents the border of the nominal 45° load distribution. in the figures and tables given below, the following abbreviations are used: av stands for ‘average’, sd for ‘standard deviation’, rsd for ‘relative standard deviation’. s = 0.9419t0.2273 r² = 0.9336 s = 0.1833t0.2838 r² = 0.97 0 5 10 15 20 25 30 35 40 0 5,000,000 10,000,000 s et tl em en t, s [m m ] axle load, t [tons] without cc layer with cc layer power (without cc layer) power (with cc layer) application of concrete canvas for enhancing railway substructure performance… 15 table 1 comparison of the flatness values, using previous results from [15,17] specimens hydration av flatness [-] sd flatness [-] rsd flatness [%] before hydration 0 day 12.68 cc after seven days of permanent 100 kn load 7 days 28.63 5.799 20.30 cc after 30 min. of permanent 100 kn load 7 days 21.08 1.598 7.60 cc in the full-scale sample after immediate dynamic loads 28+ days 20.99 2.660 12.70 fig. 10 scanned cc layer from the full-scale sample after dynamical loads (left), as well as the surface deformation of the cut-out specimen (right) as a result of the absorbed forces, the specimen resists and breaks, so the authors look for the magnitude of the resistance up to the moment of the first crack. so, if all graphs have the same slope up (i.e., tangent) to a certain point, the material is considered rigid, not broken. if the tangent is much lower than the other specimens, the specimen is considered broken. the measured bending at the first crack can be seen in table 2. in figs. 12 and 13, it is seen that the steepness of the force resistance is nearly the same in every specimen until a certain amount of loading. after the first cracks had occurred, the loading continued, and then after 9-10 mm bending, the loading was removed. without loading, the deflection of the specimens "went back". the different directions in the woven fabric showed different final conditions, too. in the longitudinal direction went back to 3.0 mm (±0.5 mm), while in the transversal direction went back to 4.0 mm (±0.3 mm). in the latter case, the standard deviation was more minor. in summary, the pattern of the graphs has three stages. the first is until the first crack; hence, the third is after removing the loads. these are nearly the same in all specimens. however, in the middle (second) section variable, the slope of the increase in bending is similar. the difference between the flexural strengths was not significant. 16 b. eller, s. fischer the specimens after the test are seen in fig. 14. even after the tests, no serious fracture was visible after 9-10 mm of bending; the structure remained intact, so a certain degree of flexibility was ensured. from a railway point of view, this demonstrates that the formation of cracks in slight bending does occur when tearing in the textile or pvc layers. this means that drainage will not be a problem in the event of a breakage under dynamic loads. fig. 11 plan of the line (direction) of cuts on the dynamically loaded specimen table 2 four-point-bending test results based on force-displacement graphs longitudinal woven transversal woven av load at first crack [kn] 0.260 0.190 sd load at first crack [kn] 0.003 0.026 av displacement at first crack [mm] 1.990 1.690 sd displacement at first crack [mm] 0.470 0.650 application of concrete canvas for enhancing railway substructure performance… 17 fig. 12 results of four-point bending tests with different weavings – longitudinal woven fig. 13 results of four-point bending tests with different weavings – transversal woven fig. 14 specimens nr. 1-6 after the four-point bending test 0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36 0.41 0.46 0 2 4 6 8 10 v er ti ca l o a d , f v [k n ] vertical deflection at the center of the sample, ycent. [mm] 2 3 4 5 6 0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36 0 2 4 6 8 10 v er ti ca l lo a d , f v [k n ] vertical deflection at the center of the sample, ycent. [mm] 7 8 9 10 11 12 18 b. eller, s. fischer 3.2. 4-point-bending and compression tests as mentioned above, one of the aims is to show the difference in deflection between reinforced and unreinforced cc material. the authors assumed, based on measurement and modeling experience, that the mechanical resistance of materials with 3d fiber reinforcement is much higher than that of unreinforced specimens. thus, as described in section 2, the specimens were prepared, and measurements were performed. measurements of the material taken from the full-scale sample were also taken to complete the measurements. naturally, a significant variance was expected due to the variable deformation/deposition caused by the bedding particles, but it can be said that throughout the research, the variance was always significant when testing the cc material. the results of the measurements are summarized in table 3. the original deflection studies were presented by eller et al. [18] and are not included in table 3. in addition to summarizing the measurements of the cc material at 1, 7 and 28 days of curing age, the measurements of the material taken from the full-scale sample were also examined at curing ages older than 28 days. since the total curing time is 28 days, no distinction was made between specimens "older" than 28 days and specimens 28 days old. the results shown in table 3 indicate that 3d fiber reinforcement has a significant effect on the bending resistance, but not in all weaving directions. previous measurements show that there is a significant difference between longitudinal and transversal weaving. this difference is also shown in the same way for the specimen taken from under the stone bedding, but starting from a lower value, so the numerical difference is also smaller. it is also seen that the standard deviation increases as the flexural strength increases. as expected, the standard deviation of the cc material with a deformed surface was more significant than that of the other specimens. table 3 comparing unreinforced and reinforced cc's flexural strengths (fl. str.) with [18] and related deflections until the first crack during 4-point bending tests (long. stands for ‘longitudinal’, transv. for ‘transversal’) cc without 3d fiber results in 1 day (left) and 7 days (right) av fl. str. [mpa] 3.57 not relevant av fl. str. [mpa] 3.61 not relevant sd fl. str. [mpa] 0.39 sd fl. str. [mpa] 0.57 diff. from [18] 10.84% diff. from [18] 15.86% av ycent. [mm] 2.25 av ycent. [mm] 1.43 cc without 3d fiber results in 28 days av fl. str. [mpa] 4.31 not relevant sd fl. str. [mpa] 0.76 diff. from [18] 17.53% av ycent. [mm] 1.92 cc from the full-scale sample after 2 mgt (see fig. 9) in longitudinal and transversal woven directions – they can be compared to 28-day-old specimens av fl. str. [mpa] 3.52 long. av fl. str. [mpa] 2.80 transv. sd fl. str. [mpa] 0.27 sd fl. str. [mpa] 0.60 diff. from [18] 7.62 diff. from [18] 21.64 av ycent. [mm] 2.07 av ycent. [mm] 1.82 application of concrete canvas for enhancing railway substructure performance… 19 in addition, it should be mentioned that the structural width of the cc is 1.1 m in the transversal direction and 73-200 m in the longitudinal direction, depending on the structural thickness. in fig. 15, the comparison of the flexural strengths of cc specimens and cc beams can be seen. the blue and orange columns mean the same specimens but in different weaving directions, while the grey columns represent the 4×4×16 cm3 cc beams without the 3d fiber matrix. fig. 15 comparison of the flexural strengths of cc specimens in both weaving direction and cc beams, using the results from [18] as can be seen, the standard deviation is considerable for the cc specimens but much smaller for the cc beams produced. this is probably due to the fact that in the former case, the hydration was not even, the waviness of the base material collecting water that cannot be absorbed immediately. when mixing the prepared beams, the material was evenly watered, while the bonding in the template was also done under equal conditions. as has been demonstrated so far, the weaving direction of the material has a significant influence on the flexural strength. the longitudinal weaving of cc provides a significant strength gain compared to the unreinforced condition, but this is not true for the transversely woven cc material. the flexural strength was close to the same or lower values compared to the unreinforced one. an exact summary of the reinforcement values is given in table 4. table 4 compares the measured values for unreinforced and reinforced cc materials. where the reinforced material did not show a higher value, it was marked as "ineffective". overall, for longitudinal weaving, fiber reinforcement increased flexural strength by 55.1% after only 7 days of curing (from 3.61 to 5.60 mpa) and rose to 81.7% after 28 days of curing (from 4.31 to 7.83 mpa). despite this, transversal weaving did not increase 3.01 5.60 7.83 2.15 3.87 2.38 3.57 3.61 4.31 0 1 2 3 4 5 6 7 8 9 10 1 7 28 s tr es s,  [m p a ] curing time [days] cc with longitudinal woven cc with transversal woven cc beams 20 b. eller, s. fischer proportionally. the reason for this can only be speculated. on the one hand, non-uniform hydration could have caused the discrepancy, and on the other hand, it is more likely, as was already apparent from the original measurements, that the transversal weaving is much weaker. as mentioned, the transversal direction length is 1.1 m, while the longitudinal direction can be 73-200 m (depending on the type of material). therefore, based on the authors' tests, it can be said that the flexural strength without fiber reinforcement is almost the same as the reinforced material for transversal weavings. table 4 the effect of the 3d reinforcement of the cct3's flexural strength curing time cc nonreinforced cc with 3d fibre – longitudinal woven cc with 3d fibre – transversal woven av fl. str. [mpa] av fl. str. [mpa] increase [%] av fl. str. [mpa] increase [%] 1 day 3.57 3.01 not affect 2.15 not affect 7 days 3.61 5.60 55.1% 3.87 7.2% 28 days 4.31 7.83 81.7% 2.38 not affect in fig. 16, the flexural strengths of cc specimens with flat surfaces and the cc specimens with deformed surfaces are compared. it was found that the results did not change significantly for the weaker transversal weaving, whereas the decrease was 4.31 mpa for the longitudinal weaving. the numerical comparison can be seen in table 5. it is seen that the magnitude of the decrease is significant; however, as described in section 3.1, the material still had sufficient cohesion after fracture, i.e., even in the case of possible cracking or fracture, drainage is not affected. based on the experiences from the full-scale measurement, the decreased value is, therefore, tolerable in practical terms. fig. 16 comparing flexural strength of the flat surface and the deformed surface ccs 7.83 3.52 2.38 2.80 0 2 4 6 8 10 12 cc with normal curing cc under loading s tr es s,  [m p a ] samples after 28 days of curing longitudinal woven transversal woven application of concrete canvas for enhancing railway substructure performance… 21 table 5 comparison of the flexural strength of plain and deformed cc for both longitudinal and transversal weaving (bw. stand for ‘between’) cc specimens with a flat surface cc specimens from the fullscale test longitudinal woven transversal woven longitudinal woven transversal woven curing time: 28 days av strength [mpa] 7.83 2.38 3.52 2.80 difference bw. long. wov. specimens [mpa] –4.31 difference bw. long. wov. specimens [%] decreasing by 44.9% difference bw. tr. wov. specimens [mpa] +0.42 difference bw. tr. wov. specimens [%] increasing by 17.7% the subsequent measurement was the compression test of 4×4×4 cm3 cc concrete blocks. this type of test was conducted to compare the results of the cc plates initially tested at 30×30×1.1 cm3 dimension. that measurement was made to determine an adequate, usable young's modulus. the process of analysis used the "least squares" method. a straight line was calculated that best fit the data, and then an array that described the line was returned [18]. the standard deviation was significant after the measurements. to ensure a proper comparison, the same evaluation was performed for the new specimens. it should be mentioned that the compression of the original specimens was affected by both the textile and the pvc layer and was not the case for the unreinforced cubes. in table 6, the relevant results of compression strength measurements are seen at 1-7-28 days of curing. it is seen that young's modulus ratio (ymr) of the cc cube at 1 day is almost 10, while the 7 and 28-day specimens are almost 20. the difference is enormous, but it does not affect your experience of the product. under 225 kn railway axle load, thanks to the load distribution, the distributed load on the subgrade surface is ~10 n/mm2. from the results, it is seen that the material is capable of supporting the loads from the railway traffic and the track after any days of curing. it is also seen that the compression strength at the cc cubes reached ~50% of the final strength, although this was measured with a relatively large standard deviation. this was below the value that characterizes the fast cure behavior of the material, which is 80% of the strength after one day of curing. the results and the standard deviations are graphically presented in fig. 17. the young's modulus of cc specimens and cc cubes can be seen at 1-7-28 days of curing. by comparing the results of the cc material and the cc cube, it can be seen that the standard deviation is much smaller for the cc cube than for the cc specimen. the reason for this is the same as in the other study; the answer is hydration. 22 b. eller, s. fischer table 6 comparison of the flexural strength of plain and deformed cc for both long. and trans. woven with using of [18] – comp. means compression, ymr means young's modulus ratio 300×300×13 mm2 cc specimens 40×40×40 mm3 cubes without 3d fiber reinforcement curing 1 day 7 days 28 days 1 day 7 days 28 days av e [mpa] 480.70 515.50 523.60 4,548.12 10,161.42 10,737.76 sd e [mpa] 180.72 128.56 102.18 422.26 1,068.59 1,549.56 av ymr [-] 9.46 19.71 20.51 av comp. [mm] 1.08 0.81 0.97 0.91 0.63 0.65 sd comp. [mm] 0.36 0.15 0.06 0.20 0.17 0.11 av fv,max [kn] 199.20 not broken 192.83 not broken 188.91 not broken 80.18 broken 112.51 broken 133.09 broken sd fv,max [kn] 1.19 11.32 19.44 3.84 8.56 5.70 avmax [mpa] 2.21 not broken 2.14 not broken 2.14 not broken 41.78 70.32 83.18 sdmax [mpa] 0.01 0.13 0.22 20.61 5.35 3.57 fig. 17 comparing compression strength of the 300×300×13 mm3 cc specimen and the 40×40×40 mm3 cc cubes using the results from [18] 4,548.1 10,161.4 10,737.8 480.7 515.5 523.6 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 1 day 7 days 28 days y o u n g 's m o d u lu s, e [m p a ] curing time 40 mm × 40 mm × 40 mm specimens 300 mm × 300 mm × 13 mm specimens application of concrete canvas for enhancing railway substructure performance… 23 after evaluating the results, it was concluded that the evaluated young's modulus of the 3d fiber reinforcement is significantly weaker due to the dense weaving. the fiber reinforcement strengthens the cc material for some measurements and also weakens it for some measurements. all this is a scientific approach; the practical use of this fact does not change because its relatively thin design and its flexible laying on the ground improves its functionality. for these reasons, although the measurements give the reader a good picture, the use of the results in modeling did not yield the same results when comparing the modeling of the different tests. it can be seen that the 3d fiber reinforcement and the double-sided overlay only allow the material to be tested as a complex composite structure and that the reinforcement has different effects on the resistances encountered in each test. the fiber reinforcement apparently weakens the material so much that the middle of the layer cannot be taken as one material in modeling. in summary, if someone wants to model with the cc material, one would need to define 4 material relationships to get valid values. for these reasons, data from cc tests without reinforcement, although showing the extent of reinforcement, are still insufficient for fe modeling. 4. conclusions the main question of this paper was whether cc can be applied under real-scale dynamic railway loads. according to the results, this study highlights the significant advantages of using cc to renew local failures and improve the lifetime of the railway track. the authors examined the deformation of the surface with a flatness test and found that the surface is continuous, and no fracture is visible. after that, the material was cut, which was used for a deflection test. with the deflection test, it was found that, due to the fiber structure, the cc layer is more of a semi-rigid material than a rigid one. although the first crack or fracture occurred at a deflection of 1-3 mm – after loading up to a deflection of 9-10 mm – the material returns to a deflection of 3-4 mm in a resting state. during this time, micro-fractures occurred in the structure, but the surface remained "intact". its quasi-rigid behavior under cyclic loading halved settlement compared to unreinforced tracks after one million load cycles. moreover, cc proved durable, maintaining its structural integrity and cohesion under prolonged dynamic loading with no signs of significant cracking or failure. key performance metrics, such as flatness and shear resistance, underscored its robustness, while its drainage properties and enhanced shear resistance (achieved through preloading) reduced risks associated with weak subgrades and poor water management. it was also seen that the 3d reinforcement in the cc significantly increases the flexural strength in the longitudinal weaving direction, while the effect in the transversal direction is not relevant. fiber reinforcement, particularly with longitudinal weaving, significantly enhances the flexural strength of cement composite materials, with a 55.1% increase after 7 days and an 81.7% increase after 28 days of curing. these improvements highlight the effectiveness of fiber reinforcement in enhancing the mechanical properties of cc materials over time. the dense distribution of the 3d matrix structure led to a significant property reduction in the cc, causing the material's elastic modulus to decrease by a factor of 10-20 compared 24 b. eller, s. fischer to the non-reinforced version. however, other tests indicate that, with adequate plane support, the material is capable of withstanding railway loads without substantial damage. based on these variable results, it is clear why previous models could not be adequately validated. the four materials of the cc layer should be modeled as composite layers, but due to their varying design and hydration, deriving valid values for other types of modeling is challenging. basically, the technology is less economical in the case of new construction due to the high cost of materials. in the authors' opinion, the application of the material is suitable for repairing local defects, or it can be attached to a screening machine and "pulled" under the crushed stone ballast, similar to the geotextile and/or geogrids in rolls. based on these experiences, different options and layer structures should be determined in the future. this study identified some limitations. variations in material thickness and weaving influenced performance, pointing to the need for standardized manufacturing processes. additionally, the controlled hydration conditions used in the laboratory might not fully replicate real-world field scenarios, potentially affecting the material's performance in practice. the research also focused mainly on static and cyclic loads, leaving other factors, such as extreme temperatures and chemical exposure, unexplored. future research should emphasize field testing and long-term monitoring to evaluate cc's performance in diverse environmental and operational settings. advanced finite element modeling could provide deeper insights into its behavior under complex loading scenarios, such as high-speed rail traffic and impact forces. exploring alternative material compositions, including enhanced fiber reinforcements or hybrid composites, may further improve its flexibility and load distribution properties. in conclusion, this study establishes cc as an auspicious material for reinforcing railway substructures with local failures. with further innovation, field validation, and development of its material properties, cc has the potential to expand its applications in railway infrastructure. acknowledgement: this paper received technical support from the research team "sze-rail." special gratitude is extended to dr. szabolcs szalai, brigitta fruzsina szívós, hanna csótár, and vivien nemes, all affiliated with széchenyi istván university (siu), for their invaluable assistance in performing the gom measurements and evaluations. acknowledgments are also due to dr. majid movahedi rad, dr. attila németh, dr. dániel harrach, gusztáv baranyai, and erika huschekjuhász, also from széchenyi istván university (siu), for their significant contributions to the execution of laboratory tests. the authors appreciate the technical and moral support provided by so de go consulting ltd., led by ceo györgy bozi, as well as concrete canvas ltd. for their assistance in the procurement, delivery, and logistics of the cc samples from hungary and the uk. the related investigations were conducted in the building materials and structural testing laboratory (siu, faculty of architecture, civil engineering and transport sciences) and the 3d surface digitization and length measurement technology laboratory (siu, audi hungaria faculty of automotive engineering). the english language proofreading of the article was carried out by bettina gerencsér. this research was conducted as part of the siu foundation's project titled 'sustainable railways – investigation of the energy efficiency of electric rail vehicles and their infrastructure'. application of concrete 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